JOURNAL OF COMPUTATIONAL ANALYSIS AND APPLICATIONS VOLUME 22, 2017

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Volume 22, Number 1 ISSN:1521-1398 PRINT,1572-9206 ONLINE

January 2017

Journal of Computational Analysis and Applications EUDOXUS PRESS,LLC

Journal of Computational Analysis and Applications ISSNno.’s:1521-1398 PRINT,1572-9206 ONLINE SCOPE OF THE JOURNAL An international publication of Eudoxus Press, LLC (fourteen times annually) Editor in Chief: George Anastassiou Department of Mathematical Sciences, University of Memphis, Memphis, TN 38152-3240, U.S.A [email protected] http://www.msci.memphis.edu/~ganastss/jocaaa The main purpose of "J.Computational Analysis and Applications" is to publish high quality research articles from all subareas of Computational Mathematical Analysis and its many potential applications and connections to other areas of Mathematical Sciences. Any paper whose approach and proofs are computational,using methods from Mathematical Analysis in the broadest sense is suitable and welcome for consideration in our journal, except from Applied Numerical Analysis articles. Also plain word articles without formulas and proofs are excluded. The list of possibly connected mathematical areas with this publication includes, but is not restricted to: Applied Analysis, Applied Functional Analysis, Approximation Theory, Asymptotic Analysis, Difference Equations, Differential Equations, Partial Differential Equations, Fourier Analysis, Fractals, Fuzzy Sets, Harmonic Analysis, Inequalities, Integral Equations, Measure Theory, Moment Theory, Neural Networks, Numerical Functional Analysis, Potential Theory, Probability Theory, Real and Complex Analysis, Signal Analysis, Special Functions, Splines, Stochastic Analysis, Stochastic Processes, Summability, Tomography, Wavelets, any combination of the above, e.t.c. "J.Computational Analysis and Applications" is a peer-reviewed Journal. See the instructions for preparation and submission of articles to JoCAAA. Assistant to the Editor: Dr.Razvan Mezei,Lenoir-Rhyne University,Hickory,NC

28601, USA.

Journal of Computational Analysis and Applications(JoCAAA) is published by EUDOXUS PRESS,LLC,1424 Beaver Trail Drive,Cordova,TN38016,USA,[email protected] http://www.eudoxuspress.com. Annual Subscription Prices:For USA and Canada,Institutional:Print $700, Electronic OPEN ACCESS. Individual:Print $350. For any other part of the world add $130 more(handling and postages) to the above prices for Print. No credit card payments. Copyright©2017 by Eudoxus Press,LLC,all rights reserved.JoCAAA is printed in USA. JoCAAA is reviewed and abstracted by AMS Mathematical Reviews,MATHSCI,and Zentralblaat MATH. It is strictly prohibited the reproduction and transmission of any part of JoCAAA and in any form and by any means without the written permission of the publisher.It is only allowed to educators to Xerox articles for educational purposes.The publisher assumes no responsibility for the content of published papers.

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Editorial Board Associate Editors of Journal of Computational Analysis and Applications Francesco Altomare Dipartimento di Matematica Universita' di Bari Via E.Orabona, 4 70125 Bari, ITALY Tel+39-080-5442690 office +39-080-3944046 home +39-080-5963612 Fax [email protected] Approximation Theory, Functional Analysis, Semigroups and Partial Differential Equations, Positive Operators.

Fractional Differential Equations Nonlinear Analysis, Fractional Dynamics Carlo Bardaro Dipartimento di Matematica e Informatica Universita di Perugia Via Vanvitelli 1 06123 Perugia, ITALY TEL+390755853822 +390755855034 FAX+390755855024 E-mail [email protected] Web site: http://www.unipg.it/~bardaro/ Functional Analysis and Approximation Theory, Signal Analysis, Measure Theory, Real Analysis.

Ravi P. Agarwal Department of Mathematics Texas A&M University - Kingsville 700 University Blvd. Kingsville, TX 78363-8202 tel: 361-593-2600 [email protected] Differential Equations, Difference Equations, Inequalities

Martin Bohner Department of Mathematics and Statistics, Missouri S&T Rolla, MO 65409-0020, USA [email protected] web.mst.edu/~bohner Difference equations, differential equations, dynamic equations on time scale, applications in economics, finance, biology.

George A. Anastassiou Department of Mathematical Sciences The University of Memphis Memphis, TN 38152,U.S.A Tel.901-678-3144 e-mail: [email protected] Approximation Theory, Real Analysis, Wavelets, Neural Networks, Probability, Inequalities.

Jerry L. Bona Department of Mathematics The University of Illinois at Chicago 851 S. Morgan St. CS 249 Chicago, IL 60601 e-mail:[email protected] Partial Differential Equations, Fluid Dynamics

J. Marshall Ash Department of Mathematics De Paul University 2219 North Kenmore Ave. Chicago, IL 60614-3504 773-325-4216 e-mail: [email protected] Real and Harmonic Analysis

Luis A. Caffarelli Department of Mathematics The University of Texas at Austin Austin, Texas 78712-1082 512-471-3160 e-mail: [email protected] Partial Differential Equations George Cybenko Thayer School of Engineering

Dumitru Baleanu Department of Mathematics and Computer Sciences, Cankaya University, Faculty of Art and Sciences, 06530 Balgat, Ankara, Turkey, [email protected]

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Dartmouth College 8000 Cummings Hall, Hanover, NH 03755-8000 603-646-3843 (X 3546 Secr.) e-mail:[email protected] Approximation Theory and Neural Networks

Partial Differential Equations, Semigroups of Operators H. H. Gonska Department of Mathematics University of Duisburg Duisburg, D-47048 Germany 011-49-203-379-3542 e-mail: [email protected] Approximation Theory, Computer Aided Geometric Design

Sever S. Dragomir School of Computer Science and Mathematics, Victoria University, PO Box 14428, Melbourne City, MC 8001, AUSTRALIA Tel. +61 3 9688 4437 Fax +61 3 9688 4050 [email protected] Inequalities, Functional Analysis, Numerical Analysis, Approximations, Information Theory, Stochastics.

John R. Graef Department of Mathematics University of Tennessee at Chattanooga Chattanooga, TN 37304 USA [email protected] Ordinary and functional differential equations, difference equations, impulsive systems, differential inclusions, dynamic equations on time scales, control theory and their applications

Oktay Duman TOBB University of Economics and Technology, Department of Mathematics, TR06530, Ankara, Turkey, [email protected] Classical Approximation Theory, Summability Theory, Statistical Convergence and its Applications

Weimin Han Department of Mathematics University of Iowa Iowa City, IA 52242-1419 319-335-0770 e-mail: [email protected] Numerical analysis, Finite element method, Numerical PDE, Variational inequalities, Computational mechanics

Saber N. Elaydi Department Of Mathematics Trinity University 715 Stadium Dr. San Antonio, TX 78212-7200 210-736-8246 e-mail: [email protected] Ordinary Differential Equations, Difference Equations

Tian-Xiao He Department of Mathematics and Computer Science P.O. Box 2900, Illinois Wesleyan University Bloomington, IL 61702-2900, USA Tel (309)556-3089 Fax (309)556-3864 [email protected] Approximations, Wavelet, Integration Theory, Numerical Analysis, Analytic Combinatorics

Christodoulos A. Floudas Department of Chemical Engineering Princeton University Princeton,NJ 08544-5263 609-258-4595(x4619 assistant) e-mail: [email protected] Optimization Theory&Applications, Global Optimization

Margareta Heilmann Faculty of Mathematics and Natural Sciences, University of Wuppertal Gaußstraße 20 D-42119 Wuppertal, Germany, [email protected]

J .A. Goldstein Department of Mathematical Sciences The University of Memphis Memphis, TN 38152 901-678-3130 [email protected]

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Hrushikesh N. Mhaskar Department Of Mathematics California State University Los Angeles, CA 90032 626-914-7002 e-mail: [email protected] Orthogonal Polynomials, Approximation Theory, Splines, Wavelets, Neural Networks

Approximation Theory (Positive Linear Operators) Xing-Biao Hu Institute of Computational Mathematics AMSS, Chinese Academy of Sciences Beijing, 100190, CHINA [email protected] Computational Mathematics

Ram N. Mohapatra Department of Mathematics University of Central Florida Orlando, FL 32816-1364 tel.407-823-5080 [email protected] Real and Complex Analysis, Approximation Th., Fourier Analysis, Fuzzy Sets and Systems

Jong Kyu Kim Department of Mathematics Kyungnam University Masan Kyungnam,631-701,Korea Tel 82-(55)-249-2211 Fax 82-(55)-243-8609 [email protected] Nonlinear Functional Analysis, Variational Inequalities, Nonlinear Ergodic Theory, ODE, PDE, Functional Equations.

Gaston M. N'Guerekata Department of Mathematics Morgan State University Baltimore, MD 21251, USA tel: 1-443-885-4373 Fax 1-443-885-8216 Gaston.N'[email protected] [email protected] Nonlinear Evolution Equations, Abstract Harmonic Analysis, Fractional Differential Equations, Almost Periodicity & Almost Automorphy

Robert Kozma Department of Mathematical Sciences The University of Memphis Memphis, TN 38152, USA [email protected] Neural Networks, Reproducing Kernel Hilbert Spaces, Neural Percolation Theory Mustafa Kulenovic Department of Mathematics University of Rhode Island Kingston, RI 02881,USA [email protected] Differential and Difference Equations

M.Zuhair Nashed Department Of Mathematics University of Central Florida PO Box 161364 Orlando, FL 32816-1364 e-mail: [email protected] Inverse and Ill-Posed problems, Numerical Functional Analysis, Integral Equations, Optimization, Signal Analysis

Irena Lasiecka Department of Mathematical Sciences University of Memphis Memphis, TN 38152 PDE, Control Theory, Functional Analysis, [email protected]

Mubenga N. Nkashama Department OF Mathematics University of Alabama at Birmingham Birmingham, AL 35294-1170 205-934-2154 e-mail: [email protected] Ordinary Differential Equations, Partial Differential Equations

Burkhard Lenze Fachbereich Informatik Fachhochschule Dortmund University of Applied Sciences Postfach 105018 D-44047 Dortmund, Germany e-mail: [email protected] Real Networks, Fourier Analysis, Approximation Theory

Vassilis Papanicolaou Department of Mathematics

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National Technical University of Athens Zografou campus, 157 80 Athens, Greece tel:: +30(210) 772 1722 Fax +30(210) 772 1775 [email protected] Partial Differential Equations, Probability

Approximation Theory, Banach spaces, Classical Analysis T. E. Simos Department of Computer Science and Technology Faculty of Sciences and Technology University of Peloponnese GR-221 00 Tripolis, Greece Postal Address: 26 Menelaou St. Anfithea - Paleon Faliron GR-175 64 Athens, Greece [email protected] Numerical Analysis

Choonkil Park Department of Mathematics Hanyang University Seoul 133-791 S. Korea, [email protected] Functional Equations

H. M. Srivastava Department of Mathematics and Statistics University of Victoria Victoria, British Columbia V8W 3R4 Canada tel.250-472-5313; office,250-4776960 home, fax 250-721-8962 [email protected] Real and Complex Analysis, Fractional Calculus and Appl., Integral Equations and Transforms, Higher Transcendental Functions and Appl.,q-Series and q-Polynomials, Analytic Number Th.

Svetlozar (Zari) Rachev, Professor of Finance, College of Business, and Director of Quantitative Finance Program, Department of Applied Mathematics & Statistics Stonybrook University 312 Harriman Hall, Stony Brook, NY 11794-3775 tel: +1-631-632-1998, [email protected] Alexander G. Ramm Mathematics Department Kansas State University Manhattan, KS 66506-2602 e-mail: [email protected] Inverse and Ill-posed Problems, Scattering Theory, Operator Theory, Theoretical Numerical Analysis, Wave Propagation, Signal Processing and Tomography

I. P. Stavroulakis Department of Mathematics University of Ioannina 451-10 Ioannina, Greece [email protected] Differential Equations Phone +3-065-109-8283 Manfred Tasche Department of Mathematics University of Rostock D-18051 Rostock, Germany [email protected] Numerical Fourier Analysis, Fourier Analysis, Harmonic Analysis, Signal Analysis, Spectral Methods, Wavelets, Splines, Approximation Theory

Tomasz Rychlik Polish Academy of Sciences Instytut Matematyczny PAN 00-956 Warszawa, skr. poczt. 21 ul. Śniadeckich 8 Poland [email protected] Mathematical Statistics, Probabilistic Inequalities Boris Shekhtman Department of Mathematics University of South Florida Tampa, FL 33620, USA Tel 813-974-9710 [email protected]

Roberto Triggiani Department of Mathematical Sciences University of Memphis Memphis, TN 38152 PDE, Control Theory, Functional

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Analysis, [email protected]

USA. Tel 334-844-6557 office 678-642-8703 home Fax 334-844-6555 [email protected] Approximation Theory, Chebychev Systems, Wavelet Theory

Juan J. Trujillo University of La Laguna Departamento de Analisis Matematico C/Astr.Fco.Sanchez s/n 38271. LaLaguna. Tenerife. SPAIN Tel/Fax 34-922-318209 [email protected] Fractional: Differential EquationsOperators-Fourier Transforms, Special functions, Approximations, and Applications

Ahmed I. Zayed Department of Mathematical Sciences DePaul University 2320 N. Kenmore Ave. Chicago, IL 60614-3250 773-325-7808 e-mail: [email protected] Shannon sampling theory, Harmonic analysis and wavelets, Special functions and orthogonal polynomials, Integral transforms

Ram Verma International Publications 1200 Dallas Drive #824 Denton, TX 76205, USA [email protected] Applied Nonlinear Analysis, Numerical Analysis, Variational Inequalities, Optimization Theory, Computational Mathematics, Operator Theory

Ding-Xuan Zhou Department Of Mathematics City University of Hong Kong 83 Tat Chee Avenue Kowloon, Hong Kong 852-2788 9708,Fax:852-2788 8561 e-mail: [email protected] Approximation Theory, Spline functions, Wavelets

Xiang Ming Yu Department of Mathematical Sciences Southwest Missouri State University Springfield, MO 65804-0094 417-836-5931 [email protected] Classical Approximation Theory, Wavelets

Xin-long Zhou Fachbereich Mathematik, Fachgebiet Informatik Gerhard-Mercator-Universitat Duisburg Lotharstr.65, D-47048 Duisburg, Germany e-mail:[email protected] Fourier Analysis, Computer-Aided Geometric Design, Computational Complexity, Multivariate Approximation Theory, Approximation and Interpolation Theory

Lotfi A. Zadeh Professor in the Graduate School and Director, Computer Initiative, Soft Computing (BISC) Computer Science Division University of California at Berkeley Berkeley, CA 94720 Office: 510-642-4959 Sec: 510-642-8271 Home: 510-526-2569 FAX: 510-642-1712 [email protected] Fuzzyness, Artificial Intelligence, Natural language processing, Fuzzy logic Richard A. Zalik Department of Mathematics Auburn University Auburn University, AL 36849-5310

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J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 22, NO.1, 2017, COPYRIGHT 2017 EUDOXUS PRESS, LLC

SOME PERTURBED VERSIONS OF THE GENERALIZED TRAPEZOID INEQUALITY FOR FUNCTIONS OF BOUNDED VARIATION WENJUN LIU AND JAEKEUN PARK

Abstract. In this paper, we establish some perturbed versions of the generalized Trapezoid inequality for functions of bounded variation in terms of the cumulative variation function.

1. Introduction In the past few years, many authors have considered various generalizations of some kinds of integral inequalities, which give explicit error bounds for some known and some new quadrature formulae. For example, in [6], Dragomir established the following generalized trapezoidal inequality for functions of bounded variation: Theorem 1.1. Let f : [a, b] → R be a function of bounded variation. Then [ ] b ∨ 1 ∫ b (x − a)f (a) + (b − x)f (b) 1 x − a+b 2 f (t)dt − + (1.1) (f ), ≤ b − a a b−a 2 b−a a ∨b where x ∈ [a, b] and a (f ) denotes the total variation of f on the interval [a, b]. The constant 12 cannot be replaced by a smaller one. The best inequality one can derive from (1.1) is the trapezoid inequality b 1 ∫ b f (a) + f (b) 1 ∨ (1.2) (f ). f (t)dt − ≤ b − a a 2 a 2 Here the constant

1 2

is also best possible.

For a function of bounded variation v : [a, b] → C, the Cumulative Variation Function (CVF) V : [a, b] → [0, ∞) is defined by t ∨ V (t) := (v), a

the total variation of v on the interval [a, t] with t ∈ [a, b]. Recently, Dragomir [7] considered the refinement of (1.1) in terms of the cumulative variation function. Theorem 1.2. Let f : [a, b] → C be a function of bounded variation on [a, b]. Then [∫ ( t ) ) ] ∫ b (∨ b 1 ∫ b x ∨ (x − a)f (a) + (b − x)f (b) 1 (1.3) f (t)dt − (f ) dt + (f ) dt ≤ b − a a b−a a b−a x a t [ ] x b ∨ ∨ 1 ≤ (x − a) (f ) + (b − x) (f ) b−a a x [ ] b a+b  1 x − 2 ∨    (f ), +   2 b−a a ≤ [ x ] b b ∨  ∨ ∨  1 1   (f ) + (f ) − (f ) ,   2 2 a x a for any x ∈ [a, b]. 2010 Mathematics Subject Classification. 26D15, 26A45, 26A16, 26A48. Key words and phrases. Generalized Trapezoid inequality, Cumulative variation, Function of bounded variation, Lipschitzian function, Monotonic function.

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WENJUN LIU et al 11-18

J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 22, NO.1, 2017, COPYRIGHT 2017 EUDOXUS PRESS, LLC

W. J. LIU AND J. K. PARK

In order to extend the classical Ostrowski’s inequality for differentiable functions with bounded derivatives to the larger class of functions of bounded variation, Dragomir obtained the following result in [13]: Theorem 1.3. Let f : [a, b] → R be a function of bounded variation on [a, b]. Then, for all x ∈ [a, b], we have the following inequality [ ] b ∫ b ∨ 1 x − a+b 1 2 f (t)dt ≤ + (1.4) (f ). f (x) − b−a a 2 b−a a The constant (1.5)

1 2

is the best possible. The best inequality one can obtain from (1.4) is the midpoint inequality ( ) ∫ b b 1∨ 1 a+b − f (t)dt ≤ (f ), f 2 a 2 b−a a

for which the constant

1 2

is also sharp.

Recently, Dragomir [8] considered the refinement of (1.4) in terms of the cumulative variation function. Theorem 1.4. Let f : [a, b] → C be a function of bounded variation on [a, b]. Then [∫ ( t ) ) ] ∫ b ∫ b (∨ b x ∨ 1 1 (1.6) f (t)dt ≤ (f ) dt + (f ) dt f (x) − b−a a b−a a x a t [ ] x b ∨ ∨ 1 ≤ (x − a) (f ) + (b − x) (f ) b−a a x [ ] b a+b  1 x − 2 ∨    + (f ),   2 b−a a ≤ [ x ] b b ∨  ∨ ∨  1 1   (f ) + (f ) − (f ) ,   2 2 a

a

x

for any x ∈ [a, b]. Very recently, Dragomir [9] obtained the following perturbed Ostrowski type inequality for functions of bounded variation, in which he denoted ℓ : [a, b] → [a, b] the identity function: Theorem 1.5. Let f : [a, b] → C be a function of bounded variation on [a, b], and x ∈ [a, b]. Then for any λ1 (x) and λ2 (x) complex numbers, we have ∫ b [ ] 1 1 (1.7) (b − x)2 λ2 (x) − (x − a)2 λ1 (x) − f (t)dt f (x) + 2(b − a) b−a a [∫ ( x ) ( ) ] ∫ b ∨ t x ∨ 1 (f − λ1 (x)ℓ) dt + (f − λ2 (x)ℓ) dt ≤ b−a a x t x [ ] x b ∨ ∨ 1 ≤ (x − a) (f − λ1 (x)ℓ) + (b − x) (f − λ2 (x)ℓ) b−a a x  {x } b ∨ ∨     max (f − λ1 (x)ℓ), (f − λ2 (x)ℓ) ,   a x ≤ [ ] ( ) x b  a+b ∨ ∨  x − 1  2  (f − λ1 (x)ℓ) + (f − λ2 (x)ℓ) ,   2 + b − a a x where

d ∨

(g) denotes the total variation of g on the interval [c, d].

c

For related results, see [1]-[5], [11]-[12], [14]-[32]. Motivated by the above works, the purpose of this paper is to establish some perturbed versions of the generalized trapezoid inequality (1.3) for functions of bounded variation in terms of the cumulative variation function.

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WENJUN LIU et al 11-18

J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 22, NO.1, 2017, COPYRIGHT 2017 EUDOXUS PRESS, LLC

SOME PERTURBED VERSIONS OF THE GENERALIZED TRAPEZOID INEQUALITY

2. Inequalities for functions of bounded variation As in [7], it is known that the CVF is monotonic nondecreasing on [a, b] and is continuous at a point c ∈ [a, b] if and only if the generating function v is continuous at that point. If v is Lipschitzian with the constant L > 0, i.e., |v(t) − v(s)| ≤ L|t − s| for any t, s ∈ [a, b], then V is also Lipschitzian with the same constant. The following lemma is of interest in itself as well, see also [10]. Lemma 2.1. Let f, u : [a, b] → C. If f is continuous on [a, b] and u is of bounded variation on [a, b], then ∫ ∫ ( t ) b b b ∨ ∨ (2.1) f (t)du(t) ≤ |f (t)|d (u) ≤ max |f (t)| (u). a t∈[a,b] a a a We have the following result: Theorem 2.1. Let f : [a, b] → C be a function of bounded variation on [a, b] and x ∈ [a, b]. Then for any λ(x) complex number, we have the inequalities ) ( 1 ∫ b (x − a)f (a) + (b − x)f (b) a + b (2.2) f (t)dt − − λ(x) x − b − a a b−a 2 [∫ ( t ) ( ) ] ∫ b ∨ b x ∨ 1 ≤ (f − λ(x)ℓ) dt + (f − λ(x)ℓ) dt b−a a x a t [ ] x b ∨ ∨ 1 ≤ (x − a) (f − λ(x)ℓ) + (b − x) (f − λ(x)ℓ) b−a a x [ ] b a+b  1 x − 2 ∨    + (f − λ(x)ℓ)   2 b−a a ≤ x b b  ∨  1 ∨ 1 ∨  (f − λ(x)ℓ) + (f − λ(x)ℓ) − (f − λ(x)ℓ) ,  2 2 a a x where

d ∨

(g) denotes the total variation of g on the interval [c, d] and ℓ : [a, b] → [a, b] is the identity

c

function. Proof. We shall start with the identity obtained in [6] ∫ b ∫ b (2.3) f (t)dt − [(x − a)f (a) + (b − x)f (b)] = (x − t)df (t), a

a

in which the integrals in the right hand side are taken in the Riemann-Stieltjes sense. If we replace f (t) with f (t) − λ(x)t in (2.3), then we can get the following equation: ) ∫ b ( ∫ b a+b = (x − t)d [f (t) − λ(x)t] . (2.4) f (t)dt − [(x − a)f (a) + (b − x)f (b)] − λ(x)(b − a) x − 2 a a Taking the modulus in (2.4) and using the property (2.1), we have ( ) 1 ∫ b [(x − a)f (a) + (b − x)f (b)] a + b (2.5) f (t)dt − − λ(x) x − b − a a b−a 2 ∫ 1 b ≤ (x − t)d [f (t) − λ(x)t] b−a a ( t ) ∫ b ∨ 1 (f − λ(x)ℓ) ≤ |x − t|d b−a a a

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WENJUN LIU et al 11-18

J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 22, NO.1, 2017, COPYRIGHT 2017 EUDOXUS PRESS, LLC

W. J. LIU AND J. K. PARK

1 = b−a

[∫

(

x

(x − t)d a

t ∨

) (f − λ(x)ℓ)

∫

b

(t − x)d

+

( t ∨

x

a

)] (f − λ(x)ℓ)

.

a

Integrating by parts in the Riemann-Stieltjes integral we have x ( t ) ) ∫ x ∫ x (∨ t t ∨ ∨ (2.6) (x − t)d (f − λ(x)ℓ) = (x − t) (f − λ(x)ℓ) + (f − λ(x)ℓ) dt a a a a a t=a ) ∫ x (∨ t = (f − λ(x)ℓ) dt a

and ∫

b

(t − x)d

(2.7) x

( t ∨

a

) (f − λ(x)ℓ)

= (t − x)

t ∨ a

a

b ∨

=(b − x) ∫

b

=

(

b (f − λ(x)ℓ)

x

b

(

−

t=x

∫

b

(f − λ(x)ℓ) −

a b ∨

∫

x

)

(

t ∨

x t ∨

) (f − λ(x)ℓ) dt

a

)

(f − λ(x)ℓ) dt

a

(f − λ(x)ℓ) dt.

t

Using (2.5)-(2.7), we deduce the first inequality in (2.2). Since t x ∨ ∨ (f − λ(x)ℓ) ≤ (f − λ(x)ℓ) for t ∈ [a, x] a

a

and b ∨

(f − λ(x)ℓ) ≤

t

then

∫

x a

and

∫

b x

b ∨

(f − λ(x)ℓ) for t ∈ [x, b],

x

(

t ∨

) (f − λ(x)ℓ) dt ≤ (x − a)

a

(b ∨

x ∨

(f − λ(x)ℓ)

a

) (f − λ(x)ℓ) dt ≤ (b − x)

t

b ∨

(f − λ(x)ℓ),

x

which prove the second inequality in (2.2). With the max properties we have (x − a)

x ∨

(f − λ(x)ℓ) + (b − x)

a

b ∨

(f − λ(x)ℓ)

x

 b ∨    max {x − a, b − x} (f − λ(x)ℓ)    a {x } ≤ b ∨ ∨     max (f − λ(x)ℓ), (f − λ(x)ℓ) (b − a)   a

x

[ ] ∨  b   1 (b − a) + x − a + b  (f − λ(x)ℓ)   2 2 a ≤ [ ] b x b ∨  ∨ ∨  1 1   (f − λ(x)ℓ) + (f − λ(x)ℓ) − (f − λ(x)ℓ) (b − a),   2 2 a a x 

which completes the proof.

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SOME PERTURBED VERSIONS OF THE GENERALIZED TRAPEZOID INEQUALITY

The following trapezoid type inequality holds: Corollary 2.1. Let f : [a, b] → C be a function of bounded variation on [a, b]. Then for any λ ∈ C, we have the inequalities [∫ a+b ( t ) ) ] ∫ b (∨ b 1 ∫ b ∨ 2 f (a) + f (b) 1 f (t)dt − (f − λℓ) dt + (2.8) (f − λℓ) dt ≤ a+b b−a a b − a a 2 a t 2 1∨ (f − λℓ), 2 a b

≤

which is equivalent to [∫ a+b ( t ) ) ] ∫ b (∨ b 1 ∫ b ∨ 2 f (a) + f (b) 1 (2.9) f (t)dt − inf (f − λℓ) dt + (f − λℓ) dt ≤ a+b b − a a b − a λ∈C a 2 a t 2 [b ] ∨ 1 ≤ inf (f − λℓ) . 2 λ∈C a 3. Inequalities for Lipschitzian functions We can state the following result: Theorem 3.1. Let f : [a, b] → C be a function of bounded variation on [a, b] and x ∈ (a, b). If λ(x) is a complex number and there exists the positive number L(x) such that f − λ(x)ℓ is Lipschitzian with the constant L(x) on the interval [a, b], then ( ) 1 ∫ b a + b (x − a)f (a) + (b − x)f (b) − λ(x) x − (3.1) f (t)dt − b − a a b−a 2 [( ] )2 a+b (b − a)2 L(x) x− + . ≤ b−a 2 4 Proof. It’s known that, if g : [c, d] → C is Riemann integrable and u : [c, d] → C is Lipschitzian with the ∫d constant L > 0, then the Riemann-Stieltjes integral c g(t)du(t) exists and ∫ ∫ d d g(t)du(t) ≤ L (3.2) |g(t)|dt. c c Taking the modulus in (2.4) and using the property (3.2) we have ( ) 1 ∫ b (x − a)f (a) + (b − x)f (b) a + b (3.3) f (t)dt − − λ(x) x − b − a a b−a 2 ∫ 1 b ≤ (x − t)d [f (t) − λ(x)t] b−a a [∫ ] ∫ x b L(x) (x − t)dt + (t − x)dt ≤ b−a a x [( ] )2 L(x) a+b (b − a)2 = x− + , b−a 2 4 

which proves the result.

Corollary 3.1. Let f : [a, b] → C be a function of bounded variation on [a, b]. If λ is a complex number and there exists the positive number L such that f − λℓ is Lipschitzian with the constant L on the interval [a, b], then 1 ∫ b f (a) + f (b) 1 (3.4) f (t)dt − ≤ L(b − a). b − a a 4 2

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4. Inequalities for Monotonic functions Now, the case of monotonic integrators is as follows: Theorem 4.1. Let f : [a, b] → C be a function of bounded variation on [a, b] and x ∈ (a, b). If λ(x) is a real number such that f − λ(x)ℓ is monotonic nondecreasing on the interval [a, b], then ( ) 1 ∫ b (x − a)f (a) + (b − x)f (b) a + b f (t)dt − − λ(x) x − (4.1) b − a a b−a 2 [ ] ∫ b 1 1 2 2 (b − x)f (b) − (x − a)f (a) − λ(x)[(b − x) + (x − a) ] − sgn(t − x)f (t)dt ≤ b−a 2 a 1 {(x − a)[f (x) − f (a) − λ(x)(x − a)] + (b − x)[f (b) − f (x) − λ(x)(b − x)]} b−a Proof. It’s known that, if g : [c, d] → C is continuous and u : [c, d] → C is monotonic nondecreasing, then ∫d the Riemann-Stieltjes integral c g(t)du(t) exists and ∫ ∫ d d g(t)du(t) ≤ (4.2) |g(t)|du(t). c c ≤

Taking the modulus in (2.4) and using the property (4.2) we have ( ) 1 ∫ b [(x − a)f (a) + (b − x)f (b)] a + b f (t)dt − − λ(x) x − (4.3) b − a a b−a 2 ∫ 1 b ≤ (x − t)d [f (t) − λ(x)t] b−a a [∫ ] ∫ b x 1 (x − t)d[f (t) − λ(x)t] + (t − x)d[f (t) − λ(x)t] . ≤ b−a a x Integrating by parts in the Riemann-Stieltjes integral we have ∫ x (x − t)d[f (t) − λ(x)t] a x ∫ x =(x − t)[f (t) − λ(x)t] + [f (t) − λ(x)t]dt t=a

∫

a

x2 − a2 f (t)dt − λ(x) 2 a ∫ x x2 − a2 = − (x − a)f (a) + λ(x)a(x − a) + f (t)dt − λ(x) 2 a ∫ x 1 = − (x − a)f (a) − λ(x)(x − a)2 + f (t)dt 2 a x

= − (x − a)[f (a) − λ(x)a] +

and

∫

b

(t − x)d[f (t) − λ(x)t] x

b =(t − x)[f (t) − λ(x)t]

∫

b

− ∫

[f (t) − λ(x)t]dt x

t=x

b

=(b − x)[f (b) − λ(x)b] −

f (t)dt + λ(x) x

b2 − x2 2

∫ b 1 =(b − x)f (b) − λ(x)(b − x)2 − f (t)dt. 2 x If we add these equalities, we get ∫ x ∫ b (x − t)d[f (t) − λ(x)t] + (t − x)d[f (t) − λ(x)t] a

x

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SOME PERTURBED VERSIONS OF THE GENERALIZED TRAPEZOID INEQUALITY

1 =(b − x)f (b) − (x − a)f (a) − λ(x)[(b − x)2 + (x − a)2 ] − 2

∫

b

sgn(t − x)f (t)dt a

and by (4.3) we get the first inequality in (4.1). Now, since f − λ(x)ℓ is monotonic nondecreasing on the interval [a, b], then ∫ x (x − t)d[f (t) − λ(x)t] a

≤(x − a)[f (x) − λ(x)x − f (a) + λ(x)a] =(x − a)[f (x) − f (a) − λ(x)(x − a)] and

∫

b

(t − x)d[f (t) − λ(x)t] x

≤(b − x)[f (b) − λ(x)b − f (x) + λ(x)x] =(b − x)[f (b) − f (x) − λ(x)(b − x)], 

which completes the proof.

Corollary 4.1. Let f : [a, b] → C be a function of bounded variation on [a, b]. If λ is a real number such that f − λℓ is monotonic nondecreasing on the interval [a, b], then 1 ∫ b f (a) + f (b) 1 (4.4) f (t)dt − ≤ [f (b) − f (a) − λ(b − a)]. b − a a 2 2 5. Conclusions Some explicit error bounds for known or new quadrature formulae are given recently through various generalizations of some kinds of integral inequalities. In this paper, by using the ideas of Dragomir in [9], we establish some perturbed versions of the generalized trapezoid inequality for functions of bounded variation in terms of the cumulative variation function. These results can be regarded as further generalizations of [6], in which the generalized trapezoidal inequality for functions of bounded variation are established. Acknowledgments. This work was partly supported by the National Natural Science Foundation of China (Grant No. 11301277), the Natural Science Foundation of Jiangsu Province (Grant No. BK20151523), the Six Talent Peaks Project in Jiangsu Province (Grant No. 2015-XCL-020) and the Qing Lan Project of Jiangsu Province. References

∫ [1] M. W. Alomari, A companion of Ostrowski’s inequality for the Riemann-Stieltjes integral ab f (t)du(t), where f is of bounded variation and u is of r-H-H¨ older type and applications, Appl. Math. Comput. 219 (2013), no. 1, 4792–4799. [2] N. S. Barnett and S. S. Dragomir, A perturbed trapezoid inequality in terms of the third derivative and applications, in Inequality theory and applications. Vol. 5, 1–11, Nova Sci. Publ., New York,2007. [3] N. S. Barnett and S. S. Dragomir, Perturbed version of a general trapezoid inequality, in Inequality theory and applications. Vol. 3, 1–12, Nova Sci. Publ., Hauppauge, NY,2003. [4] N. S. Barnett and S. S. Dragomir, A perturbed trapezoid inequality in terms of the fourth derivative, Korean J. Comput. Appl. Math. 9 (2002), no. 1, 45–60. [5] N. S. Barnett, S. S. Dragomir and I. Gomm, A companion for the Ostrowski and the generalized trapezoid inequalities, Math. Comput. Modelling 50 (2009), no. 1-2, 179–187. [6] P. Cerone, S. S. Dragomir and C. E. M. Pearce, A generalized trapezoid inequality for functions of bounded variation, Turkish J. Math. 24 (2000), no. 2, 147–163. [7] S. S. Dragomir, Refinements of the generalized trapezoid inequality in terms of the cumulative variation and applications, RGMIA Research Report Collection, 16 (2013), Article 30, 15 pp. [8] S. S. Dragomir, Refinements of the Ostrowski inequality in terms of the cumulative variation and applications, Analysis, 34 (2014), no. 2, 223–240. [9] S. S. Dragomir, Some perturbed Ostrowski type inequalities for functions of bounded variation, RGMIA Research Report Collection, 16 (2013), Article 93, 14 pp. [10] S. S. Dragomir, Refinements of the generalised trapezoid and Ostrowski inequalities for functions of bounded variation, Arch. Math. (Basel) 91 (2008), no. 5, 450–460. [11] S. S. Dragomir, On the trapezoid quadrature formula and applications, Kragujevac J. Math. 23 (2001), 25–36.

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[12] S. S. Dragomir, Some inequalities of midpoint and trapezoid type for the Riemann-Stieltjes integral, Nonlinear Anal. 47 (2001), no. 4, 2333–2340. [13] S. S. Dragomir, The Ostrowski integral inequality for mappings of bounded variation, Bull. Austral. Math. Soc. 60 (1999), no. 3, 495–508. [14] S. S. Dragomir and A. Mcandrew, On trapezoid inequality via a Gr¨ uss type result and applications, Tamkang J. Math. 31 (2000), no. 3, 193–201. [15] G. Helmberg, Introduction to spectral theory in Hilbert space, North-Holland Series in Applied Mathematics and Mechanics, Vol. 6, North-Holland, Amsterdam, 1969. [16] V. N. Huy and Q. -A. Ngˆ o, A new way to think about Ostrowski-like type inequalities, Comput. Math. Appl. 59 (2010), no. 9, 3045–3052. [17] A. I. Kechriniotis and N. D. Assimakis, Generalizations of the trapezoid inequalities based on a new mean value theorem for the remainder in Taylor’s formula, JIPAM. J. Inequal. Pure Appl. Math. 7 (2006), no. 3, Article 90, 13 pp. (electronic). [18] W. J. Liu, Some Ostrowski type inequalities via Riemann-Liouville fractional integrals for h-convex functions, J. Comput. Anal. Appl. 16 (2014), no. 5, 998–1004. [19] W. J. Liu, Some Simpson type inequalities for h-convex and (α, m)-convex functions, J. Comput. Anal. Appl. 16 (2014), no. 5, 1005–1012. [20] W. J. Liu and X. Y. Gao, Approximating the finite Hilbert transform via a companion of Ostrowski’s inequality for function of bounded variation and applications, Appl. Math. Comput. 247 (2014), 373–385. [21] W. J. Liu, Y. Jiang and A. Tuna, A unified generalization of some quadrature rules and error bounds, Appl. Math. Comput. 219 (2013), no. 9, 4765–4774. [22] W. J. Liu, Q. A. Ngo and W. Chen, On new Ostrowski type inequalities for double integrals on time scales, Dynam. Systems Appl. 19 (2010), no. 1, 189–198. [23] W. J. Liu, W. S. Wen and J. Park, A refinement of the difference between two integral means in terms of the cumulative variation and applications, J. Math. Inequal. 10 (2016), no. 1, 147–157. [24] W. J. Liu, W. S. Wen and J. Park, Hermite-Hadamard type inequalities for MT-convex functions via classical integrals or fractional integrals, J. Nonlinear Sci. Appl. 9 (2016), no. 3, 766–777. [25] Z. Liu, Some inequalities of perturbed trapezoid type, JIPAM. J. Inequal. Pure Appl. Math. 7 (2006), no. 2, Article 47, 9 pp. (electronic). [26] P. R. Mercer, Hadamard’s inequality and trapezoid rules for the Riemann-Stieltjes integral, J. Math. Anal. Appl. 344 (2008), no. 2, 921–926. [27] M. Z. Sarikaya and N. Aktan, On the generalization of some integral inequalities and their applications, Math. Comput. Modelling 54 (2011), no. 9-10, 2175–2182. [28] K.-L. Tseng, G.-S. Yang and S. S. Dragomir, Generalizations of weighted trapezoidal inequality for mappings of bounded variation and their applications, Math. Comput. Modelling 40 (2004), no. 1-2, 77–84. [29] K.-L. Tseng, G.-S. Yang and S. S. Dragomir, Generalizations of a weighted trapezoidal inequality for monotonic functions and applications, ANZIAM J. 48 (2007), no. 4, 553–566. [30] N. Ujevi´ c, Error inequalities for a generalized trapezoid rule, Appl. Math. Lett. 19 (2006), no. 1, 32–37. [31] Z. Wang and S. Vong, On some Ostrowski-like type inequalities involving n knots, Appl. Math. Lett. 26 (2013), no. 2, 296–300. [32] Q. Wu and S. Yang, A note to Ujevi´ c’s generalization of Ostrowski’s inequality, Appl. Math. Lett. 18 (2005), no. 6, 657–665. (W. J. Liu) College of Mathematics and Statistics, Nanjing University of Information Science and Technology, Nanjing 210044, China E-mail address: [email protected] (J. K. Park) Department of Mathematics, Hanseo University, Chungnam-do, Seosan-si 356-706, Republic of Korea E-mail address: [email protected]

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J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 22, NO.1, 2017, COPYRIGHT 2017 EUDOXUS PRESS, LLC

A COMPANION OF OSTROWSKI LIKE INEQUALITY AND APPLICATIONS TO COMPOSITE QUADRATURE RULES WENJUN LIU AND JAEKEUN PARK

Abstract. A companion of Ostrowski like inequality for mappings whose second derivatives belong to L∞ spaces is established. Applications to composite quadrature rules are also given.

1. Introduction In 1938, Ostrowski established the following interesting integral inequality (see [24]) for differentiable mappings with bounded derivatives: Theorem 1.1. Let f : [a, b] → R be a differentiable mapping on (a, b) whose derivative is bounded on (a, b) and denote ∥f ′ ∥∞ = sup |f ′ (t)| < ∞. Then for all x ∈ [a, b] we have t∈(a,b)

[ ] ∫ b 2 1 (x − a+b 1 2 ) f (t)dt ≤ + (b − a)∥f ′ ∥∞ . f (x) − b−a a 4 (b − a)2

(1.1) The constant

1 4

is sharp in the sense that it can not be replaced by a smaller one.

This inequality has attracted considerable interest over the years, and many authors proved generalizations, modifications and applications of it. For example, the early work of Milovanovi´c and Peˇcari´c [21, 23] extended this inequality for differentiable mappings with bounded derivatives, to functions f that are n times differentiable with |f (n) | ≤ M and gave an application to quadrature. In [8], motivated by [12], Dragomir proved some companions of Ostrowski’s inequality, as follows: Theorem 1.2. Let f : [a, b] → R be an absolutely continuous function on [a, b]. Then the following inequalities ∫ b f (x) + f (a + b − x) 1 − f (t)dt 2 b−a a  [ ( 3a+b )2 ] x− 4 1   (b − a)∥f ′ ∥∞ , f ′ ∈ L∞ [a, b],  8 +2 b−a    [(  )q+1 ( a+b )q+1 ]1/q  −x 21/q x−a + 2b−a (b − a)1/q ∥f ′ ∥p , (1.2) ≤ b−a (q+1)1/q    p > 1, p1 + 1q = 1 and f ′ ∈ Lp [a, b],   [  3a+b  1 x− 4 ] ′  f ′ ∈ L1 [a, b] 4 + b−a ∥f ∥1 , hold for all x ∈ [a, a+b 2 ]. Recently, Alomari [1] introduced a companion of Dragomir’s generalization of Ostrowsk’s inequality for absolutely continuous mappings whose first derivatives are belong to L∞ ([a, b]). Theorem 1.3. Let f : [a, b] → R be an absolutely continuous mappings on (a, b) whose derivative is bounded on [a, b]. Then the inequality [ ] ∫ b f (x) + f (a + b − x) f (a) + f (b) 1 +λ − f (t)dt (1 − λ) 2 2 b−a a 2010 Mathematics Subject Classification. 26D15, 41A55, 41A80. Key words and phrases. Ostrowski like inequality; twice differentiable mapping; L∞ spaces; composite quadrature rule.

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W. J. LIU AND J. K. PARK

 (1.3)

(

1 ≤  (2λ2 + (1 − λ)2 ) + 2 8

x−

(3−λ)a+(1+λ)b 4

(b − a)2

)2   ′  (b − a)∥f ∥∞

a+b holds for all λ ∈ [0, 1] and x ∈ [a + λ b−a 2 , 2 ].

In (1.3), choose λ = 12 , one can get [ ∫ b 1 f (x) + f (a + b − x) f (a) + f (b) ] 1 + − f (t)dt 2 2 2 b−a a [ ] 2 (x − 5a+3b 3 8 ) (1.4) ≤ +2 (b − a)∥f ′ ∥∞ . 32 (b − a)2 And if choose x = (1.5)

a+b 2 ,

then one has [ ( ) ] ∫ b 1 1 a+b f (a) + f (b) 1 + − f (t)dt ≤ (b − a)∥f ′ ∥∞ . f 2 8 2 2 b−a a

It’s shown in [1] that the constant 18 is the best possible. In related work, Dragomir and Sofo [10] developed the following Ostrowski like integral inequality for twice differentiable mapping. Theorem 1.4. Let f : [a, b] → R be a mapping whose first derivative is absolutely continuous on [a, b] and assume that the second derivative f ′′ ∈ L∞ ([a, b]). Then we have the inequality [ ] ( ) ∫ b 1 f (a) + f (b) 1 a+b 1 ′ − x− f (x) − f (t)dt f (x) + 2 2 2 2 b−a a ] [ 3 1 |x − a+b 1 2 | (1.6) + (b − a)2 ∥f ′′ ∥∞ , ≤ 48 3 (b − a)3 for all x ∈ [a, b]. In (1.6), the authors pointed out that the midpoint x = a+b 2 gives the best estimator, i.e., [ ( ) ] ∫ b 1 a+b f (a) + f (b) 1 1 (1.7) + − f (t)dt ≤ (b − a)2 ∥f ′′ ∥∞ . f 2 48 2 2 b−a a 1 in inequality (1.7) is sharp. In fact, we can choose f (t) = (t − a)2 in (1.7) to prove that the constant 48 For other related results, the reader may be refer to [2, 3, 4, 5, 7, 9, 11, 13, 14, 15, 16, 17, 18, 19, 20, 22, 25, 26, 27, 28, 29, 30] and the references therein. Motivated by previous works [1, 6, 8, 10], we investigate in this paper a companion of the above mentioned Ostrowski like integral inequality for twice differentiable mappings. Our result gives a smaller estimator than (1.7) (see (2.9) below). Some applications to composite quadrature rules are also given.

2. A companion of Ostrowski like inequality The following companion of Ostrowski like inequality holds: Theorem 2.1. Let f : [a, b] → R be a mapping whose first derivative is absolutely continuous on [a, b] and assume that the second derivative f ′′ ∈ L∞ ([a, b]). Then we have the inequality [ ] 1 f (x) + f (a + b − x) f (a) + f (b) + 2 2 2 ( ) ′ ∫ b ′ a + b f (x) − f (a + b − x) 1 1 x− − f (t)dt − 2 2 2 b−a a [ ] 2 3 1 ( a+3b 1 ( a+b 4 − x)(x − a) 2 − x) (2.1) ≤ + (b − a)2 ∥f ′′ ∥∞ 3 3 3 (b − a) 3 (b − a) 1 for all x ∈ [a, a+b 2 ]. The first constant 3 in the right hand side of (2.1) is sharp in the sense that it can not be replaced by a smaller one provided that x ̸= a+3b and x ̸= a. 4

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A COMPANION OF OSTROWSKI LIKE INEQUALITY

Proof. Define the kernel K(t) : [a, b] → R by  t ∈ [a, x],  t − a, K(t) := (2.2) t − a+b , t ∈ (x, a + b − x], 2  t − b, t ∈ (a + b − x, b], for all x ∈ [a, a+b 2 ]. Integrating by parts, we obtain (see [8]) ∫ b ∫ b 1 g(x) + g(a + b − x) 1 (2.3) K(t)g ′ (t)dt = − g(t)dt. b−a a 2 b−a a ′ Now choose in (2.3), g(x) = (x − a+b 2 )f (x), to get [ ( ) ] ∫ b 1 a+b K(t) f ′ (t) + t − f ′′ (t) dt b−a a 2 ( ) ) ∫ b( 1 a+b 1 a+b ′ ′ (2.4) = x− [f (x) − f (a + b − x)] − t− f ′ (t)dt. 2 2 b−a a 2

Integrating by parts, we have ) ∫ b( ∫ b 1 a+b f (a) + f (b) 1 (2.5) t− f ′ (t)dt = − f (t)dt. b−a a 2 2 b−a a Also upon using (2.3), we get [ ( ) ] ∫ b 1 a+b K(t) f ′ (t) + t − f ′′ (t) dt b−a a 2 ( ) ∫ b ∫ b 1 1 a+b ′ = K(t)f (t)dt + K(t) t − f ′′ (t)dt b−a a b−a a 2 ( ) ∫ b ∫ b f (x) + f (a + b − x) 1 1 a+b = (2.6) − f (t)dt + K(t) t − f ′′ (t)dt. 2 b−a a b−a a 2 It follows from (2.4)–(2.6) that

(2.7)

( ) ∫ b a+b 1 K(t) t − f ′′ (t)dt 2(b − a) a 2 [ ] ∫ b 1 1 f (x) + f (a + b − x) f (a) + f (b) = f (t)dt − + b−a a 2 2 2 ( ) 1 a + b f ′ (x) − f ′ (a + b − x) + x− . 2 2 2

Now using (2.7) we obtain [ ] 1 f (x) + f (a + b − x) f (a) + f (b) + 2 2 2 ( ) ′ ∫ b ′ 1 a + b f (x) − f (a + b − x) 1 − x− − f (t)dt 2 2 2 b−a a ∫ b ′′ ∥f ∥∞ a + b (2.8) ≤ dt. |K(t)| t − 2(b − a) a 2 Since x ∈ [a, a+b 2 ], we have ∫ b a + b I := |K(t)| t − dt 2 a )2 ∫ x ∫ a+b−x ( ∫ b a + b a+b a + b = (t − a) t − dt + t− dt + (b − t) t − dt 2 2 2 a x a+b−x ( ) )2 ( ) ∫ x ∫ a+b−x ( ∫ b a+b a+b a+b = (t − a) − t dt + t− dt + (b − t) t − dt 2 2 2 a x a+b−x ( )3 (a + 3b − 4x)(x − a)2 2 a+b (a + 3b − 4x)(x − a)2 = + −x + 12 3 2 12

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=

(a + 3b − 4x)(x − a)2 2 + 6 3

(

)3 a+b −x , 2

and referring to (2.8), we obtain the result (2.1). The sharpness of the constant 13 can be proved in a special case for x = (1.7)).

a+b 2

(see the line behind 

Remark 1. If we take x = a+b 2 in (2.1), we recapture the sharp inequality (1.7). If we take x = a in (2.1), we obtain the perturbed trapezoid type inequality ∫ b (b − a)2 f (a) + f (b) b − a 1 ′ ′ − [f (b) − f (a)] − f (t)dt ≤ ∥f ′′ ∥∞ , 2 8 b−a a 24 which has a smaller estimator than the sharp trapezoid inequality ∫ b f (a) + f (b) (b − a)2 1 − f (t)dt ≤ ∥f ′′ ∥∞ 2 b−a a 8 stated in [11, Proposition 2.7]. Remark 2. Consider

( F (x) =

) ( )3 a + 3b a+b − x (x − a)2 + −x 4 2

3a+b for x ∈ [a, a+b 2 ]. It’s easy to know that F (x) obtains its minimal value at x = 4 . Therefore, in (2.1), 3a+b the point x = 4 gives the best estimator, i.e., [ ] ∫ b 1 f ( 3a+b ) + f ( a+3b ) f (a) + f (b) ′ 3a+b ′ a+3b f ( ) − f ( ) b − a 1 4 4 4 4 + + − f (t)dt 2 2 2 8 2 b−a a

(2.9)

≤

1 (b − a)2 ∥f ′′ ∥∞ , 64

the right hand side of which is smaller than that of (1.7). 3. Application to Composite Quadrature Rules In [10], the authors utilized inequality (1.6) to give estimates of composite quadrature rules which was pointed out have a markedly smaller error than that which may be obtained by the classical results. In this section, we apply our previous inequality (2.1) to give us estimates of new composite quadrature rules which have a further smaller error. Theorem 3.1. Let In : a = x0 < x1 < · · · < xn−1 < xn = b be a partition of the interval [a, b], i+1 hi = xi+1 − xi , ν(h) := max{hi : i = 1, · · ·, n}, ξi ∈ [xi , xi +x ], and 2 S(f, In , ξ) =

then

n−1 1∑ [f (xi ) + f (ξi ) + f (xi + xi+1 − ξi ) + f (xi+1 )] hi 4 i=0 ( ) n−1 1∑ xi + xi+1 hi ξi − [f ′ (ξi ) − f ′ (xi + xi+1 − ξi )] , − 4 i=0 2

∫

b

f (x)dx = S(f, In , ξ) + R(f, In , ξ) a

and the remainder R(f, In , ξ) satisfies the estimate [n−1 ( ) )3 ] n−1 ∑ xi + 3xi+1 ∑ ( xi + xi+1 1 ′′ 2 |R(f, In , ξ)| ≤ ∥f ∥∞ (3.1) − ξi (xi − ξi ) + − ξi . 3 4 2 i=0 i=0

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A COMPANION OF OSTROWSKI LIKE INEQUALITY

Proof. Inequality (2.1) can be written as ∫ b 1 f (t)dt − [f (a) + f (x) + f (a + b − x) + f (b)] (b − a) a 4 ( ) b−a a+b ′ ′ + x− [f (x) − f (a + b − x)] 4 2 [( ) ( )3 ] 1 a + 3b a+b 2 ≤ (3.2) − x (x − a) + −x ∥f ′′ ∥∞ . 3 4 2 i+1 Applying (3.2) on ξi ∈ [xi , xi +x ], we have 2 ∫ xi+1 1 f (t)dt − [f (xi ) + f (ξi ) + f (xi + xi+1 − ξi ) + f (xi+1 )] hi 4 xi ( ) hi xi + xi+1 + ξi − [f ′ (ξi ) − f ′ (xi + xi+1 − ξi )] 4 2 [( ) ( )3 ] 1 xi + 3xi+1 xi + xi+1 2 ≤ − ξi (xi − ξi ) + − ξi ∥f ′′ ∥∞ . 3 4 2

Now summing over i from 0 to n − 1 and utilizing the triangle inequality, we have ∫ n−1 ∫ n−1 b ∑ xi+1 1∑ f (t)dt − S(f, In , ξ) = [f (xi ) + f (ξi ) + f (xi + xi+1 − ξi ) + f (xi+1 )] hi f (t)dt − a 4 i=0 i=0 xi ( ) n−1 xi + xi+1 1∑ ′ ′ hi ξi − [f (ξi ) − f (xi + xi+1 − ξi )] + 4 i=0 2 [ ( ) ( )3 ] n−1 ∑ xi + 3xi+1 xi + xi+1 1 ′′ 2 − ξi (xi − ξi ) + − ξi ≤ ∥f ∥∞ 3 4 2 i=0 

and therefore (3.1) holds. i+1 Corollary 3.1. If we choose ξi = 3xi +x , then we have 4 ( ) ( ) ] n−1 [ 1∑ 3xi + xi+1 xi + 3xi+1 S(f, In ) = f (xi ) + f +f + f (xi+1 ) hi 4 i=0 4 4 [ ( ) ( )] n−1 1 ∑ ′ 3xi + xi+1 xi + 3xi+1 ′ + f −f h2i 16 i=0 4 4

and ∑ 1 ′′ ∥f ∥∞ h3i . 64 i=0 n−1

(3.3)

|R(f, In )| ≤

Remark 3. It is obvious that inequality (3.3) is better than [10, inequality (3.1)] due to a smaller error. Acknowledgments. This work was partly supported by the National Natural Science Foundation of China (Grant No. 11301277), the Natural Science Foundation of Jiangsu Province (Grant No. BK20151523), the Six Talent Peaks Project in Jiangsu Province (Grant No. 2015-XCL-020) and the Qing Lan Project of Jiangsu Province. The authors would like to thank Professor J. Duoandikoetxea and Professor G. V. Milovanovi´c for bringing reference [11] and references [21, 22, 23] to their attention, respectively. References [1] M. W. Alomari, A companion of Dragomir’s generalization of Ostrowski’s inequality and applications in numerical integration, RGMIA Res. Rep. Coll., 14 (2011) article 50. [2] M. W. Alomari, A companion of Ostrowski’s inequality with applications, Transylv. J. Math. Mech. 3 (2011), no. 1, 9–14. [3] M. W. Alomari, A companion of Ostrowski’s inequality for mappings whose first derivatives are bounded and applications in numerical integration, RGMIA Res. Rep. Coll., 14 (2011) article 57.

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[4] M. W. Alomari, A generalization of companion inequality of Ostrowski’s type for mappings whose first derivatives are bounded and applications in numerical integration, RGMIA Res. Rep. Coll., 14 (2011) article 58. [5] N. S. Barnett, S. S. Dragomir and I. Gomm, A companion for the Ostrowski and the generalised trapezoid inequalities, Math. Comput. Modelling 50 (2009), no. 1-2, 179–187. [6] S. S. Dragomir, A companions of Ostrowski’s inequality for functions of bounded variation and applications, RGMIA Res. Rep. Coll., 5 (2002), Supp., article 28. [7] S. S. Dragomir, Ostrowski type inequalities for functions defined on linear spaces and applications for semi-inner products, J. Concr. Appl. Math. 3 (2005), no. 1, 91–103. [8] S. S. Dragomir, Some companions of Ostrowski’s inequality for absolutely continuous functions and applications, Bull. Korean Math. Soc. 42 (2005), no. 2, 213–230. [9] S. S. Dragomir, Ostrowski’s type inequalities for continuous functions of selfadjoint operators on Hilbert spaces: a survey of recent results, Ann. Funct. Anal. 2 (2011), no. 1, 139–205. [10] S. S. Dragomir and A. Sofo, An integral inequality for twice differentiable mappings and applications, Tamkang J. Math. 31 (2000), no. 4, 257–266. [11] J. Duoandikoetxea, A unified approach to several inequalities involving functions and derivatives, Czechoslovak Math. J. 51(126) (2001), no. 2, 363–376. [12] A. Guessab and G. Schmeisser, Sharp integral inequalities of the Hermite-Hadamard type, J. Approx. Theory 115 (2002), no. 2, 260–288. [13] A. R. Hayotov, G. V. Milovanovi´ c and K. M. Shadimetov, On an optimal quadrature formula in the sense of Sard, Numer. Algorithms 57 (2011), no. 4, 487–510. [14] Vu Nhat Huy and Q. -A. Ngˆ o, New bounds for the Ostrowski-like type inequalities, Bull. Korean Math. Soc. 48 (2011), no. 1, 95–104. [15] W. J. Liu, Some Ostrowski type inequalities via Riemann-Liouville fractional integrals for h-convex functions, J. Comput. Anal. Appl. 16 (2014), no. 5, 998–1004. [16] W. J. Liu and X. Y. Gao, Approximating the finite Hilbert transform via a companion of Ostrowski’s inequality for function of bounded variation and applications, Appl. Math. Comput. 247 (2014), 373–385. [17] W. J. Liu, Q. A. Ngo and W. Chen, On new Ostrowski type inequalities for double integrals on time scales, Dynam. Systems Appl. 19 (2010), no. 1, 189–198. [18] W. J. Liu, Y. Jiang and A. Tuna, A unified generalization of some quadrature rules and error bounds, Appl. Math. Comput. 219 (2013), no. 9, 4765–4774. [19] W. J. Liu, W. S. Wen and J. Park, A refinement of the difference between two integral means in terms of the cumulative variation and applications, J. Math. Inequal. 10 (2016), no. 1, 147–157. [20] W. J. Liu, W. S. Wen and J. Park, Hermite-Hadamard type inequalities for MT-convex functions via classical integrals or fractional integrals, J. Nonlinear Sci. Appl. 9 (2016), no. 3, 766–777. [21] G. V. Milovanovi´ c, On some integral inequalities, Univ. Beograd. Publ. Elektrotehn. Fak. Ser. Mat. Fiz. No. 498-541 (1975), 119–124. [22] G. V. Milovanovi´ c, Generalized quadrature formulae for analytic functions, Appl. Math. Comput. 218 (2012), no. 17, 8537–8551. [23] G. V. Milovanovi´ c and J. E. Peˇ cari´ c, On generalization of the inequality of A. Ostrowski and some related applications, Univ. Beograd. Publ. Elektrotehn. Fak. Ser. Mat. Fiz. No. 544-576 (1976), 155–158. [24] D. S. Mitrinovi´ c, J. E. Peˇ cari´ c and A. M. Fink, Classical and new inequalities in analysis, Mathematics and its Applications (East European Series), 61, Kluwer Acad. Publ., Dordrecht, 1993. [25] M. Z. Sarikaya, On the Ostrowski type integral inequality, Acta Math. Univ. Comenian. (N.S.) 79 (2010), no. 1, 129–134. ˇ [26] M. Z. Sarikaya, New weighted Ostrowski and Cebyˇ sev type inequalities on time scales, Comput. Math. Appl. 60 (2010), no. 5, 1510–1514. [27] E. Set and M. Z. Sarıkaya, On the generalization of Ostrowski and Gr¨ uss type discrete inequalities, Comput. Math. Appl. 62 (2011), no. 1, 455–461. [28] K.-L. Tseng, S.-R. Hwang and S. S. Dragomir, Generalizations of weighted Ostrowski type inequalities for mappings of bounded variation and their applications, Comput. Math. Appl. 55 (2008), no. 8, 1785–1793. [29] K.-L. Tseng, S.-R. Hwang, G.-S. Yang and Y.-M. Chou, Weighted Ostrowski integral inequality for mappings of bounded variation, Taiwanese J. Math. 15 (2011), no. 2, 573–585. [30] S. W. Vong, A note on some Ostrowski-like type inequalities, Comput. Math. Appl. 62 (2011), no. 1, 532–535. (W. J. Liu) College of Mathematics and Statistics, Nanjing University of Information Science and Technology, Nanjing 210044, China E-mail address: [email protected] (J. K. Park) Department of Mathematics, Hanseo University, Chungnam-do, Seosan-si 356-706, Republic of Korea E-mail address: [email protected]

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A modified shift-splitting preconditioner for saddle point problems ∗ Li-Tao Zhang† Department of Mathematics and Physics, Zhengzhou University of Aeronautics, Zhengzhou, Henan, 450015, P. R. China

Abstract Recently, Cao, Du and Niu [Shift-splitting preconditioners for saddle point problems, Journal of Computational and Applied Mathematics, 272 (2014) 239-250] introduced a shiftsplitting preconditioner for saddle point problems. In this paper, we establish a modified shift-splitting preconditioner for solving the large sparse augmented systems of linear equations. Furthermore, the preconditioner is based on a modified shift-splitting of the saddle point matrix, resulting in an unconditional convergent fixed-point iteration, which is a generalization of shift-splitting preconditioners. Finally, numerical examples show the spectrum of the new preconditioned matrix for the different parameters. Key words: Saddle point problem; Shift-splitting; Krylov subspace methods; Convergence; Preconditioner. MSC : 65F10; 65F15; 65F50

1

Introduction

For solving the large sparse augmented systems of linear equations ( )( ) ( ) f A BT x Au = = ≡ b, g y −B 0

(1)

∗

This research of this author is supported by NSFC Tianyuan Mathematics Youth Fund (11226337), NSFC(11501525,11471098,61203179,61202098,61170309,91130024,61272544, 61472462 and 11171039), Science Technology Innovation Talents in Universities of Henan Province(16HASTIT040), Aeronautical Science Foundation of China (2013ZD55006), Project of Youth Backbone Teachers of Colleges and Universities in Henan Province(2013GGJS-142,2015GGJS-179), ZZIA Innovation team fund (2014TD02), Major project of development foundation of science and technology of CAEP (2012A0202008), Defense Industrial Technology Development Program, China Postdoctoral Science Foundation (2014M552001), Basic and Advanced Technological Research Project of Henan Province (152300410126), Henan Province Postdoctoral Science Foundation (2013031), Natural Science Foundation of Zhengzhou City (141PQYJS560). † E-mail: [email protected].

1

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where A ∈ R is a symmetric and positive definite matrix and B ∈ R is a matrix of n m full row rank and m < n, x, f ∈ R , y, g ∈ R . It appears in many different applications of scientific computing, such as constrained optimization [32], the finite element method for solving the Navier-Stokes equation [24, 25, 26], and constrained least squares problems and generalized least squares problems [1, 29, 35, 36]. There have been several recent papers [2-24,25-29,30,31,33,37-40] for solving the augmented system (1). Santos et al. [29] studied preconditioned iterative methods for solving the singular augmented system with A = I. Yuan et al. [35, 36] proposed several variants of SOR method and preconditioned conjugate gradient methods for solving general augmented system (1) arising from generalized least squares problems where A can be symmetric and positive semidefinite and B can be rank deficient. The SOR-like method requires less arithmetic work per iteration step than other methods but it requires choosing an optimal iteration parameter in order to achieve a comparable rate of convergence. Golub et al. [27] presented SOR-like algorithms for solving system (1). Darvishi et al. [23] studied SSOR method for solving the augmented systems. Bai et al. [2, 3, 22, 40] presented GSOR method, parameterized Uzawa (PU) and the inexact parameterized Uzawa (PIU) methods for solving systems (1). Zhang and Lu [37] showed the generalized symmetric SOR method for augmented systems. Peng and Li [28] studied unsymmetric block overrelaxation-type methods for saddle point. Bai and Golub [4, 5, 6, 7, 11, 31] presented splitting iteration methods such as Hermitian and skew-Hermitian splitting (HSS) iteration scheme and its preconditioned variants, Krylov subspace methods such as preconditioned conjugate gradient (PCG), preconditioned MINRES (PMINRES) and restrictively preconditioned conjugate gradient (RPCG) iteration schemes, and preconditioning techniques related to Krylov subspace methods such as HSS, block-diagonal, block-triangular and constraint preconditioners and so on. Bai and Wang’s 2009 LAA paper [31] and Chen and Jiang’s 2008 AMC paper [22] studied some general approaches about the relaxed splitting iteration methods. Wu, Huang and Zhao [33] presented modified SSOR (MSSOR) method for augmented systems. Recently, Cao, Du and Niu [19] introduced a shift-splitting preconditioner and a local shift-splitting preconditioner for saddle point problems (1). Moreover, the authors studied some properties of the local shift-splitting preconditioned matrix and numerical experiments of a model Stokes problem are presented to show the effectiveness of the proposed preconditioners. For large, sparse or structure matrices, iterative methods are an attractive option. In particular, Krylov subspace methods apply techniques that involve orthogonal projections onto subspaces of the form { K(A, b) ≡ span b, Ab, A2 b, ..., An−1 b, ...}. The conjugate gradient method (CG), minimum residual method (MINRES) and generalized minimal residual method (GMRES) are common Krylov subspace methods. The CG method is used for symmetric, positive definite matrices, MINRES for symmetric and possibly indefinite matrices and GMRES for unsymmetric matrices [30]. In this paper, based on shift-splitting preconditioners presented by Cao, Du and Niu [19], we establish a modified shift-splitting preconditioner for saddle point problems. Furthermore, the preconditioner is based on a modified shift-splitting of the saddle point matrix, resulting in an unconditional convergent fixed-point iteration, which is a generalization of shift-splitting preconditioners. Finally, numerical examples show the effectiveness of the proposed preconditioners. However, the relaxed parameters of the modified shift-splitting methods are not optimal and only lie in the convergence region of the method. 2

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2

Modified shift-splitting preconditioner

Recently, for the coefficient matrix of the augmented system (1), Cao, Du and Niu [19] presented a shift-splitting precnditioner 1 PSS = (αI + A), 2 where α is a positive constant and I is an identity matrix. This shift-splitting precoditioner PSS is constructed by the shift-splitting of the matrix A 1 1 A = PSS − QSS = (αI + A) − (αI − A), 2 2 which naturally leads to the shift-splitting scheme uk+1 = (αI + A)−1 (αI − A)uk + 2(αI + A)−1 b, k = 0, 1, 2, ... Based on shift-splitting preconditioners presented by Cao, Du and Niu [19], we establish a modified shift-splitting preconditione, which is as follows ( ) ( ) 1 1 1 αI1 + A B T 1 αI1 − A −B T A = (Ω + A) − (Ω − A) = − , (2) −B B βI2 βI2 2 2 2 2 ( ) αI1 0 where α ≥ 0, β > 0 is a constant, Ω = and I1 ∈ Rn×n , I2 ∈ Rm×m are the 0 βI2 identity matrix. By this special splitting, the following shift-splitting iteration method can be defined for the saddle point problems (1). The modified shift-splitting iteration method(MSS): Given an initial vector u0 , for k = 0, 1, 2, ..., until {uk } converges, compute ( ( ) ( ) ) 1 αI1 + A B T 1 αI1 − A −B T f k k+1 , (3) u + u = −B B βI g βI 2 2 2 2 where α ≥ 0, β > 0 is a constant and I1 ∈ Rn×n , I2 ∈ Rm×m are the identity matrix. Remark 2.1. When the relaxed parameters α = β, the modified shift-splitting iteration method (MSS) reduces to the shift-splitting iteration method (SS); When the relaxed parameters α = 0, the modified shift-splitting iteration method (MSS) reduces to the local shift-splitting iteration method (LSS). So, MSS iteration method is the generalization of SS iteration method and LSS iteration method. Furthermore, when doing numerical examples, we may choose appropriate parameters to improve the convergence speed. Obviously, the modified shift-splitting iteration method can naturally induce a splitting preconditioner for the Krylov subspace method. The splitting preconditioner based on iterative scheme (3) is as follows ( ) 1 αI1 + A B T 1 . (4) PM SS = (Ω + A) = −B βI2 2 2

3

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On iterative scheme (3), at each step of applying the modified shift-splitting preconditioner PM SS within a Krylov subspace method, we need to solve a linear system with PM SS as the coefficient matrix, which is as follows: ) ( 1 αI1 + A B T z=r −B βI2 2 for a given vector r at each step. Moreover, the preconditioner PM SS can do the following matrix factorization )( )( ) ( 1 I1 β1 B T αI1 + A + β1 B T B 0 I1 0 PM SS = . (5) − β1 B I2 I2 0 βI2 2 0 Let r = [r1T , r2T ] and z = [z1T , z2T ], where r1 , z1 ∈ Rn and r2 , z2 ∈ Rm . So we can obtain ( ) ( )( )−1 ( )( ) I1 0 αI1 + A + β1 B T B 0 I1 − β1 B T r1 z1 . =2 1 B I r2 z2 0 βI2 0 I2 2 β

(6)

Hence, the algorithmic on the modified shift-splitting iteration method (MSS) is as follows: Algorithm 2.1: For a given vector r = [r1T , r2T ], we can compute the vector z = [z1T , z2T ] by (6) from the following steps: (a) t1 = r1 − β1 B T r2 ; (b) solve (αI1 + A + β1 B T B)z1 = 2t1 ; (c) z2 = β1 (Bz1 + 2r2 ). Remark 2.2. From Algorithm 2.1 in this paper and Algorithm 2.1 in [19], we can see that steps (a) ∼ (c) are different because of using different parameter β. In the second step of Algorithm 2.1, we need to solve sub-linear system with the coefficient matrix αI1 + A + β1 B T B. Since the matrix αI1 + A + β1 B T B is symmetric positive definite, we may employ the CG or preconditioned CG method to solve step (b) in Algorithm 2.1.

3

Convergence of MSS method

Now, we will analyze the unconditional convergence property of the corresponding iterative method for saddle point problems. At first, similar to the proving process in [19], we can obtain the following Lemmas. Lemma 3.1. Let A be a symmetric positive definite matrix, and B have full row rank. If λ is an eigenvalue of TM SS , then λ ̸= ±1, where TM SS is the iteration matrix of the modified shift-splitting iteration, which is as follows ) )−1 ( ( αI1 − A −B T αI1 + A B T . (7) TM SS = B βI2 −B βI2 Lemma 3.2. Assume that A is symmetric positive definite, B has full row rank. Let λ be an eigenvalue of TM SS and [x∗ , y ∗ ] be the corresponding eigenvector with x ∈ Cn and y ∈ Cm . Moreover, if y = 0, then |λ| < 1. 4

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Lemma 3.3. [34] Consider the quadratic equation x2 − bx + c = 0, where b and c are real numbers. Both roots of the equation are less than one in modulus if and only if |c| < 1 and |b| < 1 + c. Theorem 3.4. Let A ∈ Rn×n be a symmetric and positive definite matrix, B ∈ Rm×m have full row rank and let α ≥ 0, β > 0 be constant numbers. Let ρ(TM SS ) be the spectral radius of the modified shift-splitting iteration matrix. Then it holds that ρ(TM SS ) < 1, ∀α ≥ 0, β > 0, i.e., the modified shift-splitting iteration converges to the unique solution of the saddle point problems (1). ( ) x Proof. Let λ be an eigenvalue of ρ(TM SS ) and be the corresponding eigenvector. y Then we have ( )( ) ( )( ) αI1 − A −B T x αI1 + A B T x =λ , (8) B βI2 y −B βI2 y Expanding out (8) we obtain { (λ − 1)ξx + (λ + 1)Ax + (λ + 1)B T y = 0, (λ + 1)Bx + (1 − λ)βy.

(9)

By Lemma 3.1, we know that λ ̸= 1. So, we may get from the first equation of (9) that y=

λ+1 Bx. β(λ − 1)

(10)

Substituting (10) into the first equation of (9) yields α(λ − 1)x + (λ + 1)Ax +

(λ + 1)2 T B Bx = 0. β(λ − 1)

(11)

∗

By Lemma 3.2, we know that x ̸= 0. Multiplying xx∗ x on both sides of the equation (11), we have x∗ Ax x∗ B T Bx αβ(λ − 1)2 + β(λ2 − 1) ∗ + (λ + 1)2 = 0. (12) xx x∗ x Let x∗ Ax x∗ B T Bx a = ∗ > 0, b = ≥ 0. xx x∗ x Then, from (12) we know that λ satisfies the following real quadratic equation λ2 +

2b − 2αβ αβ − βa + b λ+ . αβ + βa + b αβ + βa + b

By Lemma 3.3, |λ| < 1 if and only if αβ − βa + b αβ + βa + b < 1

(13)

(14)

5

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and

2b − 2αβ αβ − βa + b αβ + βa + b < 1 + αβ + βa + b .

(15)

Obviously, the equations (14) and (15) hold for any α ≥ 0, β > 0. Hence, we have ρ(TM SS ) < 1, ∀α ≥ 0, β > 0. Remark 3.1. Obviously, from Theorem 3.4, we know that the modified shift-splitting iteration method is converent unconditionally. Remark 3.2. In actual operation, when using the Krylov subspace method like GMRES or CG method, we may choose PM SS as the preconditioner to accelerate the convergence. Actually, the left-preconditioned linear system based on the preconditioner PM SS is as follows −1 −1 (I − TM SS )u = PM SS Au = PM SS b.

4

Numerical examples

In this section, to further assess the effectiveness of the modified shift-splitting preconditioned −1 matrix PM SS A combined with Krylov subspace methods, we present a sample of numerical examples which are based on a two-dimensional time-harmonic Maxwell equations in mixed form in a square domain (−1 ≤ x ≤ 1, −1 ≤ y ≤ 1). For the simplicity, we take the generic source: f = 1 and a finite element subdivision such as Figure 1 based on uniform grids of √ √ √ 2 2 2 triangle elements. Three mesh sizes are considered: h = 8 , 12 , 18 . The solutions of the preconditioned systems in each iteration are computed exactly. Information on the sparsity of relevant matrices on the different meshes is given in Table 1, where nz(A) denote the nonzero elements of matrix A.

Figure 1: A uniform mesh with h =

√ 2 4

Since the modified shift-splitting preconditioners have two parameters, in numerical experiments we will test different values. Numerical experiments show the spectrum of the −1 new preconditioned matrix PM SS A for the different parameters. −1 In Figures 2, 3 and 4 we display the eigenvalues of the preconditioned matrix PM SS A in √ 2 the case of h = 8 for different parameters. In Figures 5, 6 and 7 we display the eigenvalues √ −1 2 of the preconditioned matrix PM SS A in the case of h = 12 for different parameters. In 6

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1

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−1 Figure 2: The eigenvalue distribution for the modified shift-splitting preconditioned matrix PM SS A

when α = 0.01, β = 1(the first), α = 0.01, β = 0.1(the second) and α = 0.01, β = 0.01(the third), √ 2 respectively. Here, h = 8 .

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−1 Figure 3: The eigenvalue distribution for the modified shift-splitting preconditioned matrix PM SS A

when α = 0.1, β = 1(the first), α = 0.1, β = 0.1(the second) and α = 0.1, β = 0.01(the third), √ 2 respectively. Here, h = 8 .

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−1 Figure 4: The eigenvalue distribution for the modified shift-splitting preconditioned matrix PM SS A

when α = 1, β = √ 1(the first), α = 1, β = 0.1(the second) and α = 1, β = 0.01(the third), respectively. Here, h = 82 .

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−1 Figure 5: The eigenvalue distribution for the modified shift-splitting preconditioned matrix PM SS A

when α = 0.01, β = 1(the first), α = 0.01, β = 0.1(the second) and α = 0.01, β = 0.01(the third), √ 2 respectively. Here, h = 12 .

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−1 Figure 6: The eigenvalue distribution for the modified shift-splitting preconditioned matrix PM SS A

when α = 0.1, β = 1(the first), α = 0.1, β = 0.1(the second) and α = 0.1, β = 0.01(the third), √ 2 respectively. Here, h = 12 .

0.8 0.6

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−1 Figure 7: The eigenvalue distribution for the modified shift-splitting preconditioned matrix PM SS A

when α = 1, β = √ 1(the first), α = 1, β = 0.1(the second) and α = 1, β = 0.01(the third), respectively. Here, h = 122 .

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−1 Figure 8: The eigenvalue distribution for the modified shift-splitting preconditioned matrix PM SS A

when α = 0.01, β = 1(the first), α = 0.01, β = 0.1(the second) and α = 0.01, β = 0.01(the third), √ 2 respectively. Here, h = 18 .

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−1 Figure 9: The eigenvalue distribution for the modified shift-splitting preconditioned matrix PM SS A

when α = 0.1, β = 1(the first), α = 0.1, β = 0.1(the second) and α = 0.1, β = 0.01(the third), √ 2 respectively. Here, h = 18 .

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Figure 10: The eigenvalue distribution for the modified shift-splitting preconditioned matrix

−1 PM first), α = 1, β = 0.1(the second) and α = 1, β = 0.01(the SS A when α = 1, β = 1(the √ 2 third), respectively. Here, h = 18 .

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Table 1: datasheet for different grids Grid m n nz(A) nz(B) nz(W ) order of A 8×8 176 49 820 462 217 225 16 × 16 736 225 3556 2190 1065 961 32 × 32 3008 961 14788 9486 4681 3969 64 × 64 12160 3969 60292 39438 19593 16129 Table 2: Iteration counts and relative residual about the modified shift-splitting precondi−1 tioned matrix PM SS A when choosing different parameters, where the number of iterations and relative residual of unpreconditioned BICGSTAB and GMRES are − and −, 171(1) and √ 2 −7 7.4545 × 10 , respectively. Here, h = 8 denotes the size of the corresponding grid. α 0.01 0.01 0.01 0.1 0.1 0.1 1 1 1

β 1 0.1 0.01 1 0.1 0.01 1 0.1 0.01

ItBICGST AB(P −1 A) M SS 6 3 2 21.5 6.5 5 82.5 31 13

ResBICGST AB(P −1 A) M SS 7.6716 × 10−7 5.4416 × 10−7 8.7718 × 10−7 5.4960 × 10−7 6.2392 × 10−7 3.8958 × 10−7 4.2920 × 10−7 6.0454 × 10−7 6.3508 × 10−7

ItGM RES(P −1 A) M SS 10(1) 6(1) 5(1) 24(1) 12(1) 8(1) 65(1) 33(1) 20(1)

ResGM RES(P −1 A) M SS 7.4779 × 10−7 7.4225 × 10−7 1.8299 × 10−7 9.6647 × 10−7 9.3667 × 10−7 7.3712 × 10−7 6.5701 × 10−7 8.5683 × 10−7 5.1740 × 10−7

−1 Figures 8, 9 √and 10 we display the eigenvalues of the preconditioned matrix PM SS A in the 2 case of h = 18 for different parameters. Figures 2 ∼ 10 show that the distribution of eigenvalues of the preconditioned matrix confirms our above theoretical analysis. In Tables 2 ∼ 4 −1 we show iteration counts and relative residual about preconditioned matrices PM SS A when choosing different parameters and applying to BICGSTAB and GMRES Krylov subspace iterative methods on three meshes, where ItBICGST AB(P −1 A) and ResBICGST AB(P −1 A) are M SS M SS −1 the iteration numbers and relative residual of the preconditioned matrices PM SS A when applying to BICGSTAB Krylov subspace iterative methods, respectively. ItGM RES(P −1 A) M SS and ResGM RES(P −1 A) are the iteration numbers and relative residual of the preconditioned M SS −1 matrices PM A when applying to GMRES Krylov subspace iterative methods, respectively. SS

Remark 4.1. From the above figures and tables, we know that the smaller the parameter β is, the gather the eigenvalues are and the fewer the iteration counts are. −1 Remark 4.2. From Tables 2, 3 and 4, it is very easy to see that the preconditioner PM SS A will improve the convergence of BICGSTAB and GMRES iteration efficiently when they are applied to the preconditioned BICGSTAB and GMRES to solove the Stokes equation and two-dimensional time-harmonic Maxwell equations by choosing different parameters.

5

Conclusions

In this paper, we establish the modified shift-splitting preconditioner for solving the large sparse augmented systems of linear equations. Furthermore, the preconditioner is based on a modified shift-splitting of the saddle point matrix, resulting in an unconditional conver10

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Table 3: Iteration counts and relative residual about the modified shift-splitting precondi−1 tioned matrix PM SS A when choosing different parameters, where the number of iterations and relative residual of unpreconditioned BICGSTAB and GMRES are − and −, 362(1) and √ 2 −7 9.4148 × 10 , respectively. Here, h = 12 denotes the size of the corresponding grid. α 0.01 0.01 0.01 0.1 0.1 0.1 1 1 1

β 1 0.1 0.01 1 0.1 0.01 1 0.1 0.01

ItBICGST AB(P −1 A) M SS 14.5 5.5 3 63.5 13.5 7.5 216.5 88 27.5

ResBICGST AB(P −1 A) M SS 4.1689 × 10−7 9.0310 × 10−7 5.2030 × 10−7 5.2347 × 10−7 6.1091 × 10−7 4.5380 × 10−7 4.7653 × 10−7 9.6032 × 10−7 1.1257 × 10−7

ItGM RES(P −1 A) M SS 19(1) 9(1) 6(1) 50(1) 23(1) 12(1) 123(1) 65(1) 34(1)

ResGM RES(P −1 A) M SS 5.2459 × 10−7 7.4043 × 10−7 9.3857 × 10−7 6.7889 × 10−7 4.9215 × 10−7 8.6233 × 10−7 8.0138 × 10−7 7.5718 × 10−7 8.5489 × 10−7

Table 4: Iteration counts and relative residual about the modified shift-splitting precondi−1 tioned matrix PM SS A when choosing different parameters, where the number of iterations and relative residual of unpreconditioned BICGSTAB and GMRES are 742 and 8.0810×10−7 , √ 1− and −, respectively. Here, h = 182 denotes the size of the corresponding grid. α 0.01 0.01 0.01 0.1 0.1 0.1 1 1 1

β 1 0.1 0.01 1 0.1 0.01 1 0.1 0.01

ItBICGST AB(P −1 A) M SS 58 7.5 4 2644.5 34.5 13 8517.5 116 93

ResBICGST AB(P −1 A) M SS 6.7835 × 10−7 7.7089 × 10−7 6.1349 × 10−7 4.2297 × 10−7 8.1807 × 10−7 9.4646 × 10−7 9.3710 × 10−7 7.8164 × 10−7 6.9354 × 10−7

ItGM RES(P −1 A) M SS 34(1) 16(1) 9(1) 94(1) 43(1) 21(1) 229(1) 132(1) 66(1)

ResGM RES(P −1 A) M SS 8.5510 × 10−7 3.1469 × 10−7 2.6837 × 10−7 9.9981 × 10−7 7.0956 × 10−7 5.0204 × 10−7 9.1052 × 10−7 9.2308 × 10−7 8.8886 × 10−7

gent fixed-point iteration, which is a generalization of shift-splitting preconditioners. Fi−1 nally, numerical examples show the preconditioner PM SS A will improve the convergence of BICGSTAB and GMRES iteration efficiently when they are applied to the preconditioned BICGSTAB and GMRES to solove the Stokes equation and two-dimensional time-harmonic Maxwell equations by choosing different parameters.

References [1] M. Arioli, I. S. Duff and P. P. M. de Rijk, On the augmented system approach to sparse least-squares problems, Numer. Math., 1989, 55:667-684. [2] Z.-Z. Bai, B. N. Parlett and Z. Q. Wang, On generalized successive overrelaxation methods for augmented linear systems, Numer. Math., 2005, 102:1-38. [3] Z.-Z. Bai, Z.-Q. Wang, On parameterized inexact Uzawa methods for generalized saddle point problems, Linear Algebra Appl., 2008, 428:2900-2932. [4] Z.-Z. Bai, X. Yang, On HSS-based iteration methods for weakly nonlinear systems, Appl. Numer. Math., 2009, 59:2923-2936.

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[5] Z.-Z. Bai, G. H. Golub, K. N. Michael, On inexact hermitian and skew-Hermitian splitting methods for non-Hermitian positive definite linear systems, Linear Algebra Appl., 2008, 428:413-440. [6] Z.-Z. Bai, Several splittings for non-Hermitian linear systems, Science in China, Series A: Math., 2008, 51:1339-1348. [7] Z.-Z. Bai, G. H. Golub, L.-Z. Lu, J.-F. Yin, Block-Triangular and skew-Hermitian splitting methods for positive definite linear systems, SIAM J. Sci. Comput., 2005, 26:844-863. [8] Z.-Z. Bai, G. H. Golub, M. K.Ng, Hermitian and skew-Hermitian splitting methods for nonHermitian positive definite linear systems, SIAM J. Matrix. Anal. A., 2003, 24:603-626. [9] Z.-Z. Bai, G. H. Golub, M. K. Ng, On successive-overrelaxation acceleration of the Hermitian and skew-Hermitian splitting iteration. Available online at http://www.sccm.stanford.edu/wrap/pub-tech.html. [10] Z.-Z. Bai, G. H. Golub and C.-K. Li, Optimal parameter in Hermitian and skew-Hermitian splitting method for certain twoby- two block matrices, SIAM J. Sci. Comput., 2006, 28:583603. [11] Z.-Z. Bai, G. H. Golub and M. K. Ng, Hermitian and skew-Hermitian splitting methods for non-Hermitian positive definite linear systems, SIAM J. Matrix Anal. Appl., 2003, 24:603-626. [12] Z.-Z. Bai, G. H. Golub and M. K. Ng, On successive-overrelaxation acceleration of the Hermitian and skew-Hermitian splitting iterations, Numer. Linear Algebra Appl., 2007, 14:319-335. [13] Z.-Z. Bai, G. H. Golub and J.-Y. Pan, Preconditioned Hermitian and skew-Hermitian splitting methods for non-Hermitian positive semidefinite linear systems, Numer. Math., 2004, 98:1-32. [14] Z.-Z. Bai and M. K. Ng, On inexact preconditioners for nonsymmetric matrices, SIAM J. Sci. Comput., 2005, 26:1710-1724. [15] Z.-Z. Bai, M. K. Ng and Z.-Q. Wang, Constraint preconditioners for symmetric indefinite matrices, SIAM J. Matrix Anal. Appl., 2009, 31:410-433. [16] Z.-Z. Bai, Optimal parameters in the HSS-like methods for saddle-point problems, Numer. Linear Algebra Appl., 2009, 16:447-479. [17] Z.-Z. Bai, J.-F. Yin and Y.-F. Su, A shift-splitting preconditioner for non-Hermitian positive definite matrices, J. Comput. Math., 2006, 24:539-552. [18] Z.-Z. Bai, G. H. Golub, J.-Y. Pan, Preconditioned Hermitian and skew-Hermitian splitting methods for non-Hermitian positive semidefinite linear systems. Technical Report SCCM-0212, Scientific Computing and Computational Mathematics Program, Department of Computer Science, Stanford University, Stanford, CA, 2002. [19] Y. Cao, J. Du, Q. Niu, Shift-splitting preconditioners for saddle point problems, J. Comput. Appl. Math., 2014, 272: 239-250 [20] Y. Cao, L.-Q. Yao and M.-Q. Jiang, A modified dimensional split preconditioner for generalized saddle point problems, J. Comput. Appl. Math., 2013, 250:70-82. [21] Y. Cao, L.-Q. Yao, M.-Q. Jiang and Q. Niu, A relaxed HSS preconditioner for saddle point problems from meshfree discretization, J. Comput. Math., 2013, 31:398-421.

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[22] F. Chen, Y.-L. Jiang, A generalization of the inexact parameterized Uzawa methods for saddle point problems, Appl. Math. Comput, 2008, 206:765-771. [23] M. T. Darvishi and P. Hessari, Symmetric SOR method for augmented systems, Appl. Math. Comput, 2006, 183:409-415. [24] H. Elman and D. Silvester, Fast nonsymmetric iterations and preconditioning for Navier-Stokes equations, SIAM J. Sci. Comput., 1996, 17:33-46. [25] H. Elman and G. H. Golub, Inexact and preconditioned Uzawa algorithms for saddle point problems, SIAM J. Numer. Anal., 1994, 31:1645-1661. [26] B. Fischer, A. Ramage, D. J. Silvester and A. J. Wathen, Minimum residual methods for augmented systems, BIT, 1998, 38:527-543. [27] G. H. Golub, X. Wu and J.-Y. Yuan, SOR-like methods for augmented systems, BIT, 2001, 55:71-85. [28] X.-F. Peng and W. Li, On unsymmetric block overrelaxation-type methods for saddle point, Appl. Math. Comput, 2008, 203(2):660-671. [29] C. H. Santos, B. P. B. Silva and J.-Y. Yuan, Block SOR methods for rank deficient least squares problems, J. Comput. Appl. Math., 1998, 100:1-9. [30] H. A. Van der Vorst, Iterative Krylov Methods for Large Linear Systems, Cambridge Monographs on Applied and Computational Mathematics, Cambridge University Press, Cambridge, UK, 2003. [31] L. Wang, Z.-Z. Bai, Convergence conditions for splitting iteration methods for non-Hermitian linear systems, Linear Algebra Appl., 2008, 428:453-468. [32] S. Wright, Stability of augmented system factorizations in interior-point methods, SIAM J. Matrix Anal. Appl., 1997, 18:191-222. [33] S.-L. Wu, T.-Z. Huang and X.-L. Zhao, A modified SSOR iterative method for augmented systems, J. Comput. Appl. Math., 2009, 228(1):424-433. [34] D. M. Young, Iteratin Solution for Large Systems, Academic Press, New York, 1971. [35] J.-Y. Yuan, Numerical methods for generalized least squares problems, J. Comput. Appl. Math., 1996, 66:571-584. [36] J.-Y. Yuan and A. N. Iusem, Preconditioned conjugate gradient method for generalized least squares problems, J. Comput. Appl. Math., 1996, 71:287-297. [37] G.-F. Zhang and Q.-H. Lu, On generalized symmetric SOR method for augmented systems, J. Comput. Appl. Math., 2008, 1(15):51-58. [38] L.-T. Zhang, A new preconditioner for generalized saddle matrices with highly singular(1,1) blocks, Int. J. Comput. Math., 2014, 91(9):2091-2101. [39] L.-T. Zhang, T.-Z. Huang, S.-H. Cheng, Y.-P. Wang, Convergence of a generalized MSSOR method for augmented systems, J. Comput. Appl. Math., 2012, 236:1841-1850. [40] B. Zheng, Z.-Z. Bai, X. Yang, On semi-convergence of parameterized Uzawa methods for singular saddle point problems, Linear Algebra Appl., 2009, 431:808-817.

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CLOSED-RANGE GENERALIZED COMPOSITION OPERATORS BETWEEN BLOCH-TYPE SPACES CUI WANG DEPARTMENT OF MATHEMATICS, TIANJIN UNIVERSITY, TIANJIN 300072 P.R. CHINA. [email protected] ZE-HUA ZHOU∗ DEPARTMENT OF MATHEMATICS, TIANJIN UNIVERSITY, TIANJIN 300072, P.R. CHINA. [email protected]; [email protected]

Abstract. Let ϕ denote a nonconstant analytic self-map of the open unit disk D, g be an analytic function on D. In this paper, we characterize the necessary or sufficient conditions for generalized composition operators Z z g Cϕ f (z) = f 0 (ϕ(ξ))g(ξ)dξ, 0

on the Bloch-type spaces to have a closed range. Moreover, if g ∈ H ∞ , according to relationship between α and β, we show several conclusions.

1. Introduction Let H(D) be the class of all holomorphic functions on D, where D is the open unit disk in the complex plane C. Denote by H ∞ = H ∞ (D) the space of all bounded holomorphic functions on D with the supremum norm kf k∞ = supz∈D |f (z)|. For 0 < α < ∞, a holomorphic function f is said to be in the Bloch-type space B α or α−Bloch space, if kf kα = sup(1 − |z|2 )α |f 0 (z)| < ∞. z∈D

The little Bloch-type space B0α , consists of all f ∈ B α , such that lim (1 − |z|2 )α |f 0 (z)| = 0.

|z|→1

It is well-known that both B α and B0α are Banach spaces under the norm kf kBα = |f (0)| + sup(1 − |z|2 )α |f 0 (z)|. z∈D

Moreover, B0α is the closure of polynomials in B α . When 0 < α < 1, B α is the analytic Lipschitz space Lip1−α , which consists of all f ∈ H(D) satisfying |f (z) − f (w)| ≤ C|z − w|1−α , for some constant C > 0 and all z, w ∈ D. When α = 1, B α becomes the classical Bloch ∞ space B. When α > 1, B α is equivalent to the weighted Banach space Hα−1 . Let Hα∞ be the weighted Banach space of holomorphic functions f on D satisfying sup(1 − |z|2 )α |f (z)| < ∞. z∈D

We refer the readers to the book [13] by K. Zhu, which is an excellent resource for the development of the theory of function spaces. ∗ Corresponding author. This work was supported in part by the National Natural Science Foundation of China (Grant Nos. 11371276; 11301373; 11401426). 2010 Mathematics Subject Classification. Primary: 47B38; Secondary: 46E15, 26A24, 30H30,47B33 Key words and phrases.closed-range, bounded below, generalized composition operator, Bloch-type space.

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Wang and Zhou: Closed-range generalized composition operators We say that a subset H of D is called a sampling set for the Bloch-type space B α , if there is k > 0 such that sup{(1 − |z|2 )α |f 0 (z)|, z ∈ D} ≤ k sup{(1 − |z|2 )α |f 0 (z)|, z ∈ H}. The pseudo-hyperbolic metric is given by a−z , z, a ∈ D. 1−a ¯z σa (z) is the automorphism of D which changes 0 and a. It is well-known that the pseudo1−|a|2 hyperbolic metric is M¨ obius-invariant. Moreover, we have that σa0 (z) = (1−az) 2. A subset G of D is an r-net for some r ∈ (0, 1), if for every w ∈ D, there exists a z ∈ G such that ρ(z, w) < r. If we define ρ(z, E) = inf{ρ(z, w) : w ∈ E} for a set E ⊂ D, then a relatively closed subset E of D is an r-net if and only if ρ(z, E) ≤ r. For every analytic self-map ϕ of D and g ∈ H(D), the generalized composition operator Cϕg is defined by Z z Cϕg f (z) = f 0 (ϕ(ξ))g(ξ)dξ, z ∈ D, ρ(z, a) = |σa (z)|, where σa (z) =

0

which was firstly introduced by Li and Stevi´c [9]. For further references and details about the generalized composition operator, we refer the readers to [10, 11] and their references. S. Li and S. Stevi´c [9] gave the boundedness and compactness of Cϕg : B α → B β , which will play a central roll in our paper, so we use the notation τα,β (z) to state the results. For α > 0 and β > 0, let (1 − |z|2 )β |g(z)| τα,β (z) = , z ∈ D. (1 − |ϕ(z)|2 )α Theorem A. Let α, β > 0, g ∈ H(D) and ϕ be an analytic self-map of D. Then Cϕg : B α → B β is bounded if and only if sup τα,β (z) < ∞. z∈D

Theorem B. Let α, β > 0, g ∈ H(D) and ϕ be an analytic self-map of D. Then Cϕg : B α → B β is compact if and only if Cϕg : B α → B β is bounded and lim

|ϕ(z)|→1

τα,β (z) = 0.

The composition operator is defined by Cϕ (f )(z) = f (ϕ(z)) on the spaces of analytic functions on D. In 2000, Gathage, Yan and Zheng [7] characterized closed-range composition operators on Bloch spaces firstly. Chen [5] not only added a sufficient condition for [7], but also studied a sufficient and necessary condition of the boundedness from below for Cϕ on the Bloch space of the unit ball. Then Gathatage, Zheng and Zorboska [8] introduced the notion of sampling sets for the bloch space and gave a necessary and sufficient condition for Cϕ on the Bloch space to have closed-range. This result has been extended by Chen and Gauthire [4] to α−Bloch spaces with α ≥ 1. Soon after Zorboska [14] added new and general results on the closed-range determination of Cϕ on Bloch-type spaces. There are also many articles on various other holomorphic function spaces. G. R. Chac´on [3] provided a geometric characterization for those composition operators having closed-range on Dirichlet-type spaces. Recently, necessary and sufficient conditions for a closed-range composition operator on Besov spaces and more generally on Besov type spaces were given by M. Tjani [12]. Akeroyd and Fulmer [1, 2] characterized the closed range composition operators on weighted Bergman spaces. In this paper, we give some results to determine when the generalized composition operator Cϕg has closed-range. To some extent, our results generalize some existing results. For example, the results obtained in this paper also hold for the classical composition operator Cϕ : B α → B β , which we get by choosing g = ϕ0 , so some results of [14] can be got easily by this paper. In section 2, we show several necessary and sufficient conditions for the generalized composition operator Cϕg between Bloch-type spaces to have closed-range; apart from

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Wang and Zhou: Closed-range generalized composition operators these, we use a set to describe when Cϕg : B α /C → B β is bounded below. In section 3, if g ∈ H ∞ , according to relationship between α and β, we show several conclusions. In order to state our main results conveniently, from now on we note Ωε,α,β = {z ∈ D, τα,β (z) ≥ ε} and Gε,α,β = ϕ(Ωε,α,β ). Throughout the remainder of this paper, C will denote a positive constant, the exact value of which will vary from one appearance to the next. The notations A  B, A  B, A  B mean that there exist different positive constants C such that B/C ≤ A ≤ CB, A ≤ CB, CB ≤ A. 2. Sampling sets and r-net A bounded generalized composition operator Cϕg : B α → B β is said to be bounded below, if there exists a constant k > 0 such that kCϕg f kBβ ≥ kkf kBα . Meanwhile, we know that Cϕg maps any constant function to 0 function, so it is only useful to consider spaces of analytic functions modulo the constants. It follows that we can replace the norm kf kBα with the seminorm kf kα in the definition of boundedness below. Therefore, in this paper, we just show some results on X/C, which means that a Banach space X of analytic functions on D modulo the constants. Lemma 1. Let X be Banach spaces of analytic functions. If ϕ is a nonconstant analytic self-map of D, then Cϕg is one-to-one on X/C. Proof. If Cϕg f1 = Cϕg f2 , we obtain f10 (ϕ(z))g(z) = f20 (ϕ(z))g(z). Excluding the isolated points where g vanishes, since f1 and f2 are analytic, ϕ is a nonconstant analytic self-map of D, the open mapping theorem for analytic functions ensures that f10 (z) = f20 (z) for every  z ∈ D, and hence Cϕg is one-to-one on X/C. A basic operator theory result asserts that a one-to-one operator has a closed range if and only if it is bounded below. Therefore, Lemma 1 implies the following theorem. The detailed proof is similar to Proposition 3.30 of [6], and so we omit it. Theorem 1. Let 0 < α, β < ∞, ϕ be a nonconstant analytic self-map of D. Then Cϕg : B α /C → B β has a closed range if and only if it is bounded below from B α /C to B β . This is equivalent to the condition that there exists M > 0 such that kCϕg f kβ ≥ M kf kα , ∀f ∈ B α /C. Remark 1. Since ϕ is an open map, a generalized composition operator Cϕg never has a finite rank. However, the closed subspaces of the range of a compact operator are only the finite dimensional ones, so a compact generalized composition operator can never have a closed range. Theorem 2. Let 0 < α, β < ∞, ϕ be a nonconstant analytic self-map of D. Suppose that Cϕg : B α → B β is bounded. Then Cϕg : B α /C → B β has a closed range if and only if there exists ε > 0 such that the set Gε,α,β is a sampling set on B α /C. Proof. Suppose that there exists ε > 0 such that the set Gε,α,β is a sampling set on B α /C. In this case, we can find a constant k > 0 such that kf kα

≤ k sup{(1 − |ϕ(z)|2 )α |f 0 (ϕ(z))|, z ∈ Ωε,α,β } ≤ k sup{

(1 − |ϕ(z)|2 )α (1 − |z|2 )β |f 0 (ϕ(z))g(z)|, z ∈ Ωε,α,β } (1 − |z|2 )β |g(z)|

= k sup{

1 (1 − |z|2 )β |f 0 (ϕ(z))g(z)|, z ∈ Ωε,α,β } τα,β (z)

≤ ≤

k sup{(1 − |z|2 )β |f 0 (ϕ(z))g(z)|, z ∈ D} ε k g kC f kβ ε ϕ

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Wang and Zhou: Closed-range generalized composition operators and because Cϕg : B α → B β is bounded, it is bounded below. By Theorem 1, we obtain that Cϕg : B α /C → B β has a closed range. Conversely, assume that Cϕg : B α /C → B β has a closed range. Then there exists k > 0, such that for ∀f ∈ B α /C, supz∈D (1 − |z|2 )β |f 0 (ϕ(z))g(z)| ≥ kkf kα . Without loss of generality, we suppose that kf kα = 1. Thus, by the definition of supremum, we can choose ω ∈ D, such that (1 − |ω|2 )β |f 0 (ϕ(ω))g(ω)| ≥ k/2, that is to say, (1 − |ω|2 )β |f 0 (ϕ(ω))g(ω)| =

(1 − |ω|2 )β |g(ω)| (1 − |ϕ(ω)|2 )α |f 0 (ϕ(ω))| (1 − |ϕ(ω)|2 )α

= τα,β (w)(1 − |ϕ(ω)|2 )α |f 0 (ϕ(ω))| k . (2.1) ≥ 2 Since (1 − |ϕ(ω)|2 )α |f 0 (ϕ(ω))| ≤ 1, τα,β (w) ≥ k/2. If ε = k2 , then Ωε,α,β contains the point ω, and so ϕ(ω) ∈ Gε,α,β . On the other hand, Cϕg : B α → B β is bounded, Theorem A implies that there exists a constant M > 0, such that τα,β (w) ≤ M. Combining the above inequality with (1), we conclude that M (1 − |ϕ(ω)|2 )α |f 0 (ϕ(ω))| ≥ τα,β (w)(1 − |ϕ(ω)|2 )α |f 0 (ϕ(ω))| ≥

k . 2

Thus (1 − |ϕ(ω)|2 )α |f 0 (ϕ(ω))| ≥

k . 2M

Since ϕ(ω) ∈ Gε,α,β , sup{(1 − |z|2 )α |f 0 (z)|, z ∈ Gε,α,β } ≥ (1 − |ϕ(ω)|2 )α |f 0 (ϕ(ω))| ≥

k . 2M

Hence Gε,α,β is a sampling set on B α /C.



Theorem 3. Let 0 < α, β < ∞, and ϕ be a nonconstant analytic self-map of D. Suppose that Cϕg : B α → Bβ is bounded. If Cϕg : B α /C → B β has a closed range, then there exist c > 0 and 0 < r < 1, such that Gc,α,β is an r-net for D. Proof. We assume that Cϕg is bounded and has a closed-range. By Theorem A, there exists K > 0 such that sup τα,β (z) = K for z ∈ D. Meanwhile, there exists M > 0 such that kCϕg f kβ ≥ M kf kα for all f ∈ B α /C. Let ω ∈ D and consider the function ϕω (z) with ϕω (0) = 0 and ϕ0ω (z) = (σω0 (z))α , where ω−z σω (z) = 1−ωz . We have that ϕω (z) ∈ B α /C and kϕω kα

=

sup(1 − |z|2 )α |ϕ0ω (z)|

=

sup(1 − |σω0 (z)|2 )α

=

1.

z∈D

z∈D

In the above equation we use the fact that 1 − |σω (z)|2 =

(1 − |ω|2 )(1 − |z|2 ) = |σω0 (z)|(1 − |z|2 ). |1 − ωz|2

Thus, kCϕg ϕω kβ

=

sup(1 − |z|2 )β |ϕ0ω (ϕ(z))g(z)| z∈D

=

sup z∈D

=

(1 − |z|2 )β |g(z)| (1 − |ϕ(z)|2 )α |σω0 (ϕ(z))|α (1 − |ϕ(z)|2 )α

sup τα,β (z)(1 − |σω (ϕ(z))|2 )α . z∈D

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Wang and Zhou: Closed-range generalized composition operators We shall frequently get that K ≥ sup τα,β (z)(1 − |σω (ϕ(z))|2 )α ≥ M (1 − |σω (ϕ(z))|2 )α ≥ M, z∈D

which reveals that there exists z0 ∈ D such that τα,β (z0 ) ≥ M/2, (1 − |σω (ϕ(z0 ))|2 )α ≥ M/2K. p Thus let ε = M/2, r = 1 − (M/2K)1/α , we have for all ω ∈ D, there exists z0 ∈ Ωε,α,β such that ρ(ω, ϕ(z0 )) < r, and so Gc,α,β is an r-net for D.  Theorem 4. Let 0 < α, β < ∞, and ϕ be a nonconstant analytic self-map of D. Suppose that Cϕg : B α → B β is bounded. If there exist ε > 0 and 0 < r < 1, such that Gε,α,β contains the annulus A = {z : r < |z| < 1}, then Cϕg : B α /C → B β has a closed range. Proof. Suppose that Cϕg : B α /C → B β is not bounded below. Then there exists a sequence of functions {fn } with kfn kα = 1 and kCϕg fn kβ → 0. It follows that for ∀ε > 0, there exists Nε when n > Nε , we have kCϕg fn kβ < ε. Then sup (1 − |ω|2 )α |fn0 (ω)| =

ω∈Gε,α,β

=

sup (1 − |ϕ(z)|2 )α |fn0 (ϕ(z))|

ω∈Ωε,α,β

sup ω∈Ωε,α,β

=

sup ω∈Ωε,α,β

≤ =
Nε . Because Gε,α,β contains the annulus A = {z : r < |z| < 1}, there exists r0 < r such that |zn | ≤ r0 ≤ 1 and zn → z0 with |z0 | < r0 . Since kfn kα = 1, by Montel’s theorem, there exists a subsequence fnk → f uniformly on every compact subsets of D, where f ∈ B α /C. Cauchy’s estimate gives that fn0 k → f 0 uniformly on every compact subsets of D. By (2), supω∈Gε,α,β (1 − |ω|2 )α |fn0 (ω)| → 0 as n → ∞. On the other hand, Gε,α,β contains an infinite compact subset of D, we get that f 0 ≡ 0. This contradicts the fact that |(1 − |z0 |2 )α fn0 (z0 )| ≥ 1/2. Hence, Cϕg : B α /C → B β has a closed range.  3. the case of g ∈ H ∞ In this section we will give a special case g ∈ H ∞ . Combine α and β, we get several results. Theorem 5. Let ϕ be a nonconstant analytic self-map of D, ϕ(0) = 0, g ∈ H ∞ and Cϕg : B α → B β is bounded. (i) If 0 < α < β < ∞ then Cϕg : B α /C → B β can not have a closed range. (ii) If α > β > 0 and β < 1 then Cϕg : B α /C → B β can not have a closed range. Proof. (i) Since g ∈ H ∞ , there exists a constant k > 0, such that |g(z)| ≤ k, for every z ∈ D. For ϕ(0) = 0, by Schwarz-Pick Theorem in [6], we know 1 − |z|2 ≤ 1, z ∈ D. 1 − |ϕ(z)|2

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Wang and Zhou: Closed-range generalized composition operators So we have τα,β (z)

(1 − |z|2 )β |g(z)| (1 − |ϕ(z)|2 )α k(1 − |z|2 )β (1 − |ϕ(z)|2 )α k(1 − |z|2 )α (1 − |z|2 )β−α (1 − |ϕ(z)|2 )α

= ≤ =

≤ k(1 − |ϕ(z)|2 )β−α . Since 0 < α < β < ∞, as |ϕ(z)| → 1, τα,β (z) converges to 0. By Theorem B, Cϕg : B α → B β is compact. Hence Cϕg : B α /C → B β can not have a closed range. (ii) Replacing φ by ϕ, φ0 by g in the proof of (i) of Theorem 3.6 in [14], we can get this result easily, so we omit the details here.  Remark 2. (i) Let ϕ be a nonconstant analytic self-map of D, ϕ(0) = 0, g ∈ H ∞ . If α = β, then (1 − |z|2 )β |g(z)| (1 − |ϕ(z)|2 )α k(1 − |z|2 )β ≤ (1 − |ϕ(z)|2 )α ≤ k.

τα,β (z)

=

By Theorem A, we obtain Cϕg : B α → B β is bounded. While apart from this, we can not get whether Cϕg : B α /C → B β has a closed range or not. (ii) Under the conditions of Theorem 5, if α > β ≥ 1, whether Cϕg : B α /C → B β has a
closed range or not is uncertain. We just give an example (ii) of Example 1 showing that this operator sometimes do not have a closed range. While, we fail to give the concrete proof that this operator do not have a closed range always or an example to show this operator has a closed range sometimes. So this can be an open problem. Example 1. Let ϕ(z) = z, g(z) = 1. (i) If α = β = 2, then τα,β (z) =

(1 − |z|2 )2 |g(z)| =1 (1 − |ϕ(z)|2 )2

and so Ωε,α,β = D for every 0 < ε < 1. In addition, ϕ(z) = z is a one-to-one analytic map of the disk onto itself, therefore, Gε,α,β = ϕ(Ωε,α,β ) = D. Then Gε,α,β is a sampling set on B α /C, and by Theorem 2, Cϕg : B α /C → B β has a closed range. (ii) If α = 3, β = 2, then τα,β (z)

=

(1 − |z|2 )β (1 − |ϕ(z)|2 )α

=

(1 − |z|2 )β−α → ∞

as ϕ(z) → 1. By Theorem A, Cϕg : B α → B β is not bounded. Hence Cϕg : B α /C → B β can not have a closed range. Example 2. Let g(z) = z + 1, ϕ(z) = τα,β (z)

z−1 2 .

= ≤ =

43

If α = β, then

(1 − |z|2 )α |g(z)| (1 − |ϕ(z)|2 )α 4(1 − |z|2 )α |z + 1| (1 − |z|)α (3 + |z|)α 4(1 + |z|)α |z + 1| →0 (3 + |z|)α

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Wang and Zhou: Closed-range generalized composition operators as z → −1. By Theorem B, Cϕg : B α → B β is compact. Hence Cϕg : B α /C → B β can not have a closed range. References [1] J.R. Akeroyd and S.R. Fulmer, Closed-range composition operators on weighted Bergman spaces, Integr. Equ. Oper. Theory, 72 (2012), 103-114. [2] J.R. Akeroyd and S.R. Fulmer, Erratum to: Closed-range composition operators on weighted Bergman spaces, Integr. Equ. Oper. Theory, 76 (2013), 145-149. [3] G.R. Chac´ on, Closed-range composition operators on Dirichlet-type spaces, Complex Anal. Oper. Theory, 7 (2013), 909-926. [4] H. Chen and P. Gauthier, Boundedness from below of composition operators on α-Bloch spaces, Canad. Math. Bull. 51 (2008), 195-204. [5] H. Chen, Boundedness from below of composition operators on the Bloch spaces, Sci. China Ser. 46 (2003), 838-846. [6] C.C. Cowen and B.D. MacCluer, Composition operators on spaces of analytic functions, CRC Press, Studies in Ad-vanced Mathematics, CRC, Boca Raton, FL, 1995. [7] P. Ghatage, J. Yan and D. Zheng, Composition operators with closed range on the Bloch space. Proc. Amer. Math. Soc. 129 (2000), 2039-2044. [8] P. Ghatage, D. Zheng and N. Sampling sets and closed-Range composition operators on the Bloch space, Proc. Amer. Math. Soc. 133 (2004), 1371-1377. [9] S. Li and S. Stevi´ c, Generalized composition operators on Zygmund spaces and Bloch type spaces, J. Math. Anal. Appl. 338 (2008), 1282-1295. [10] S. Stevi´ c and A. K. Sharma, Generalized composition operators on weighted Hardy spaces, Appl. Math. Comput. 218 (2012), 8347-8352. [11] S. Stevi´ c, Generalized composition operators between mixed-norm and some weighted spaces, Numer. Funct. Anal. Optim. 29 (2008), 959-978. [12] M. Tjani, Closed range composition operators on Besov type spaces, Complex Anal. Oper. Theory, 8 (2014), 189-212. [13] K.H. Zhu, Operator Theory in Function Spces, Marcel Dekker, New York, 1990. [14] N. Zorboska, Isometric and closed-range composition operators between Bloch-type spaces, Int. J. Math. Math. Sci. (2011), Article ID 132541, 15 pages, doi:10.1155/2011/132541.

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Approximate ternary Jordan bi-derivations on Banach Lie triple systems Madjid Eshaghi Gordji1 , Vahid Keshavarz1 , Choonkil Park2 and Jung Rye Lee3∗ 1

Department of Mathematics, Semnan University, P. O. Box 35195-363, Semnan, Iran 2

Research Institute for Naturan Sciences, Hanyang University, Seoul 133-791, Korea 3

Department of Mathematics, Daejin University, Kyeonggi 487-711, Korea

E-mail: [email protected], [email protected], [email protected], [email protected] Abstract. We prove the Hyers-Ulam stability of ternary Jordan bi-derivations on Banach Lie triple systems associated to the Cauchy functional equation.

1. Introduction and preliminaries We say that a functional equation (Q) is stable if any function g satisfying the equation (Q) approximately is near to true solution of (Q). Ternary algebraic operations were considered in the 19th century by several mathematicians and physicists. Cayley [8] introduced the notion of cubic matrix which in turn was generalized by Kapranov, Gelfand and Zelevinskii [6]. As an application in physics, the quark model inspired a particular brand of ternary algebraic systems. The so-called Nambu mechanics which has been proposed by Nambu [11], is based on such structures. There are also some applications, although still hypothetical, in the fractional quantum Hall effect, the non-standard statistics (the anyons), supersymmetric theories, Yang-Baxter equation, etc, (cf. [15, 27]). The comments on physical applications of ternary structures can be found in [1, 5, 10, 14, 17, 23, 24, 29]. A normed (Banach) Lie triple system is a normed (Banach) space (A, k · k) with a trilinear mapping (x, y, z) 7→ [x, y, z] from A × A × A to A satisfying the following axioms: [x, y, z]

=

− [y, x, z] ,

[x, y, z]

=

− [y, z, x] − [z, x, y] ,

[u, v, [x, y, z]]

=

[[u, v, x] , y, z] + [x, [u, v, y] , z] + [x, y, [u, v, z]] ,

k [x, y, z] k

≤

kxkkykkzk

for all u, v, x, y, z ∈ A (see [12, 16]). Definition 1.1. Let A be a normed Lie triple system with involution ∗. A C-bilinear mapping D : A × A → A is called a ternary Jordan bi-derivation if it satisfies D([x, x, x], w)

=

[D(x, w), x, x] + [x, D(x, w∗ ), x] + [x, x, D(x, w)],

D(x, [w, w, w])

=

[D(x, w), w, w] + [w, D(x∗ , w), w] + [w, w, D(x, w)]

for all x, w ∈ A. 0

2010 Mathematics Subject Classification. Primary 39B52; 39B82; 46B99; 17A40. Keywords: Hyers-Ulam stability; bi-additive mapping; Lie triple system; ternary Jordan bi-derivation. 0∗ Corresponding author. 0

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Approximate ternary Jordan bi-derivations The stability problem of functional equations originated from a question of Ulam [28] concerning the stability of group homomorphisms. Hyers [13] gave a first affirmative partial answer to the question of Ulam for Banach spaces. Hyers’ Theorem was generalized by Aoki [3] for additive mappings and by Th.M. Rassias [21] for linear mappings by considering an unbounded Cauchy difference. J.M. Rassias [20] followed the innovative approach of the Th.M. Rassias theorem in which he replaced the factor kxkp + kykp by kxkp kykp for p, q ∈ R with p + q 6= 1. The stability problems of various functional equations have been extensively investigated by a number of authors (see [2, 7, 9, 10, 18, 19, 22, 23, 24, 25, 26, 30, 31]). 2. Hyers-Ulam stability of ternary Jordan bi-derivations on Banach Lie triple systems Throughout this section, assume that A is a normed Lie triple system. For a given mapping f : A × A → A, we define Dλ,µ f (x, y, z, w) = f (λx + λy, µz + µw) + f (λx + λy, µz − µw) +f (λx − λy, µz + µw) + f (λx − λy, µz − µw) − 4λµf (x, z) for all x, y, z, w ∈ A and all λ, µ ∈ T1 := {ν ∈ C : |ν| = 1}. From now on, assume that f (0, z) = f (x, 0) = 0 for all x, z ∈ A. We need the following lemma to obtain the main results. Lemma 2.1. ([4]) Let f : A × A → B be a mapping satisfying Dλ,µ f (x, y, z, w) = 0 for all x, y, z, w ∈ A and all λ, µ ∈T1 . Then the mapping f : A × A → A is C-bilinear. Lemma 2.2. Let f : A × A → A be a bi-additive mapping. Then the following assertions are equivalent: f ([a, a, a], [w, w, w]) = [f (a, w), a, a] + [a, f (a, w∗ ), a] + [a, a, f (a, w)], (2.1)

f ([a, a, a], [w, w, w]) = [f (a, w), a, a] + [a, f (a∗ , w), a] + [a, a, f (a, w)] for all a, w ∈ A, and f ([a, b, c] + [b, c, a] + [c, a, b], [w, w, w]) = [f (a, w), b, c] + [a, f (b, w∗ ), c] + [a, b, f (c, w)] + [f (b, w), c, a] + [b, f (c, w∗ ), a] + [b, c, f (a, w)] + [f (c, w), a, b] + [c, f (a, w∗ ), b] + [c, a, f (b, w)], f ([a, a, a], [b, c, w] + [c, w, b] + [w, b, c]) = [f (a, b), c, w] + [b, f (a∗ , c), w] + [b, c, f (a, w)]

(2.2)

+ [f (a, c), w, b] + [c, f (a∗ , w), b] + [c, w, f (a, b)] + [f (a, w), b, c] + [w, f (a∗ , b), c] + [w, b, f (a, w)] for all a, b, c, w ∈ A. Proof. Replacing a by a + b + c in the first equation of (2.1), we have f ([a + b + c, a + b + c, a + b + c], [w, w, w]) = [f (a + b + c, w), a + b + c, a + b + c] + [a + b + c, f (a + b + c, w∗ ), a + b + c] + [a + b + c, a + b + c, f (a + b + c, w)]. Then we have f ([a + b + c, a + b + c, a + b + c], [w, w, w]) = f ([a, a, a], [w, w, w]) + f ([a, b, a], [w, w, w]) + f ([a, c, a], [w, w, w]) + f ([b, a, a], [w, w, w]) + f ([b, b, a], [w, w, w]) + f ([b, c, a], [w, w, w]) + f ([c, a, a], [w, w, w]) + f ([c, b, a], [w, w, w]) + f ([c, c, a], [w, w, w]) + f ([a, a, b], [w, w, w]) + f ([a, b, b], [w, w, w]) + f ([a, c, b], [w, w, w]) + f ([b, a, b], [w, w, w]) + f ([b, b, b], [w, w, w]) + f ([b, c, b], [w, w, w])

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M. Eshaghi Gordji, V. Keshavarz, C. Park, J. R. Lee + f ([c, a, b], [w, w, w]) + f ([c, b, b], [w, w, w]) + f ([c, c, b], [w, w, w]) + f ([a, a, c], [w, w, w]) + f ([a, b, c], [w, w, w]) + f ([a, c, c], [w, w, w]) + f ([b, a, c], [w, w, w]) + f ([b, b, c], [w, w, w]) + f ([b, c, c], [w, w, w]) + f ([c, a, c], [w, w, w]) + f ([c, b, c], [w, w, w]) + f ([c, c, c], [w, w, w]) = [f (a, w), a, a] + [a, f (a, w∗ ), a] + [a, a, f (a, w)] + [f (a, w), b, a] + [a, f (b, w∗ ), a] + [a, b, f (a, w)] + [f (a, w), c, a] + [a, f (c, w∗ ), a] + [a, c, f (a, w)] + [f (b, w), a, a] + [b, f (a, w∗ ), a] + [b, a, f (a, w)] + [f (b, w), b, a] + [b, f (b, w∗ ), a] + [b, b, f (a, w)] + [f (b, w), c, a] + [b, f (c, w∗ ), a] + [b, c, f (a, w)] + [f (c, w), a, a] + [c, f (a, w∗ ), a] + [c, a, f (a, w)] + [f (c, w), b, a] + [c, f (b, w∗ ), a] + [c, b, f (a, w)] + [f (c, w), c, a] + [c, f (c, w∗ ), a] + [c, c, f (a, w)] + [f (a, w), a, b] + [a, f (a, w∗ ), b] + [a, a, f (b, w)] + [f (a, w), b, b] + [a, f (b, w∗ ), b] + [a, b, f (b, w)] + [f (a, w), c, b] + [a, f (c, w∗ ), b] + [a, c, f (b, w)] + [f (b, w), a, b] + [b, f (a, w∗ ), b] + [b, a, f (b, w)] + [f (b, w), b, b] + [b, f (b, w∗ ), b] + [b, b, f (b, w)] + [f (b, w), c, b] + [b, f (c, w∗ ), b] + [b, c, f (b, w)] + [f (c), a, b] + [c, f (a∗ ), b] + [c, a, f (b)] + [f (c), b, b] + [c, f (b∗ ), b] + [c, b, f (b)] + [f (c, w), c, b] + [c, f (c, w∗ ), b] + [c, c, f (b, w)] + [f (a, w), a, c] + [a, f (a, w∗ ), c] + [a, a, f (c, w)] + [f (a, w), b, c] + [a, f (b, w∗ ), c] + [a, b, f (c, w)] + [f (a, w), c, c] + [a, f (c, w∗ ), c] + [a, c, f (c, w)] + [f (b, w), a, c] + [b, f (a, w∗ ), c] + [b, a, f (c, w)] + [f (b, w), b, c] + [b, f (b, w∗ ), c] + [b, b, f (c, w)] + [f (b, w), c, c] + [b, f (c, w∗ ), c] + [b, c, f (c, w)] + [f (c, w), a, c] + [c, f (a, w∗ ), c] + [c, a, f (c, w)] + [f (c, w), b, c] + [c, f (b, w∗ ), c] + [c, b, f (c, w)] + [f (c, w), c, c] + [c, f (c, w∗ ), c] + [c, c, f (c, w)] for all a, b, c, w ∈ A. On the other hand, for the right side of equation, we have [f (a + b + c, w), a + b + c, a + b + c] + [a + b + c, f (a + b + c, w∗ ), a + b + c] + [a + b + c, a + b + c, f (a + b + c, w)] = [f (a, w), a, a] + [f (a, w), a, b] + [f (a, w), a, c] + [f (a, w), b, a] + [f (a, w), b, b] + [f (a, w), b, c] + [f (a, w), c, a] + [f (a, w), c, b] + [f (a, w), c, c] + [f (b, w), a, a] + [f (b, w), a, b] + [f (b, w), a, c] + [f (b, w), b, a] + [f (b, w), b, b] + [f (b, w), b, c] + [f (b, w), c, a] + [f (b, w), c, b] + [f (b, w), c, c] + [f (c, w), a, a] + [f (c, w), a, b] + [f (c, w), a, c] + [f (c, w), b, a] + [f (c, w), b, b] + [f (c, w), b, c] + [f (c, w), c, a] + [f (c, w), c, b] + [f (c, w), c, c] + [a, f (a, w∗ ), a] + [a, f (a, w∗ ), b] + [a, f (a, w∗ ), c] + [b, f (a, w∗ ), a] + [b, f (a, w∗ ), b] + [b, f (a, w∗ ), c] + [c, f (a, w∗ ), a] + [c, f (a, w∗ ), b] + [c, f (a, w∗ ), c] + [a, f (b, w∗ ), a] + [a, f (b, w∗ ), b] + [a, f (b, w∗ ), c] + [b, f (b, w∗ ), a] + [b, f (b, w∗ ), b] + [b, f (b, w∗ ), c] + [c, f (b, w∗ ), a] + [c, f (b, w∗ ), b] + [c, f (b, w∗ ), c] + [a, f (c, w∗ ), a] + [a, f (c, w∗ ), b] + [a, f (c, w∗ ), c] + [b, f (c, w∗ ), a] + [b, f (c, w∗ ), b] + [b, f (c, w∗ ), c] + [c, f (c, w∗ ), a] + [c, f (c, w∗ ), b] + [c, f (c, w∗ ), c] + [a, a, f (a, w)] + [a, b, f (a, w)] + [a, c, f (a, w)] + [b, a, f (a, w)] + [b, b, f (a, w)] + [b, c, f (, w)] + [c, a, f (a, w)] + [c, b, f (a, w)] + [c, c, f (a, w)] + [a, a, f (b, w)] + [a, b, f (b, w)] + [a, c, f (b, w)] + [b, a, f (b, w)] + [b, b, f (b, w)] + [b, c, f (b, w)] + [c, a, f (b, w)] + [c, b, f (b, w)] + [c, c, f (b, w)] + [a, a, f (c, w)] + [a, b, f (c, w)] + [a, c, f (c, w)] + [b, a, f (c, w)] + [b, b, f (c, w)] + [b, c, f (c, w)] + [c, a, f (c, w)] + [c, b, f (c, w)] + [c, c, f (c, w)] for all a, b, c, w ∈ A. It follows that f ([a, b, c] + [b, c, a] + [c, a, b], [w, w, w]) = [f (a, w), b, c] + [a, f (b, w∗ ), c] + [a, b, f (c, w)] + [f (b, w), c, a] + [b, f (c, w∗ ), a] + [b, c, f (a, w)] + [f (c, w), a, b] + [c, f (a, w∗ ), b] + [c, a, f (b, w)]

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Approximate ternary Jordan bi-derivations for all a, b, c, w ∈ A. Hence (2.2) holds. Similarly, we can show that f ([a, a, a], [b, c, w] + [c, w, b] + [w, b, c]) = [f (a, b), c, w] + [b, f (a∗ , c), w] + [b, c, f (a, w)] + [f (a, c), w, b] + [c, f (a∗ , w), b] + [c, w, f (a, b)] + [f (a, w), b, c] + [w, f (a∗ , b), c] + [w, b, f (a, w)] for all a, b, c, w ∈ A. For the converse, replacing b and c by a in the first equation of (2.2), we have f ([a, a, a] + [a, a, a] + [a, a, a], [w, w, w]) = [f (a, w), a, a] + [a, f (a, w∗ ), a] + [a, a, f (a, w)] + [f (a, w), a, a] + [a, f (a, w∗ ), a] + [a, a, f (a, w)] + [f (a, w), a, a] + [a, f (a, w∗ ), a] + [a, a, f (a, w)], and so   f ([a, a, a], [w, w, w]) + ([a, a, a], [w, w, w]) + ([a, a, a], [w, w, w]) = 3([f (a, w), a, a] + [a, f (a, w∗ ), a] + [a, a, f (a, w)]). Thus   f 3([a, a, a], [w, w, w]) = 3([f (a, w), a, a] + [a, f (a, w∗ ), a] + [a, a, f (a, w)]) and so f ([a, a, a], [w, w, w]) = [f (a, w), a, a] + [a, f (a, w∗ ), a] + [a, a, f (a, w)] for all a, w ∈ A. Similarly, we can show that f ([a, a, a], [w, w, w]) = [f (a, w), a, a] + [a, f (a∗ , w), a] + [a, a, f (a, w)] for all a, w ∈ A. This completes the proof.



Now we prove the Hyers-Ulam stability of ternary Jordan bi-derivations on Banach Lie triple systems. Theorem 2.3. Let p and θ be positive real numbers with p < 2, and let f : A × A → A be a mapping such that kDλ,µ f (x, y, z, w)k ≤ θ(kxkp + kykp + kzkp + kwkp ),   kf ([x, y, z] + [y, z, x] + [z, x, y]), w − [f (x, w), y, z] + [x, f (y, w∗ ), z] − [x, y, f (z, w)] − [f (y, w), z, x] − [y, f (z, w∗ ), x] − [y, z, f (x, w)] − [f (z, w), x, y] − [z, f (x, w∗ ), y] − [z, x, f (y, w)]k   + kf x, ([y, z, w] + [z, w, y] + [w, y, z]) − [f (x, y), z, w] − [y, f (x∗ , z), w] − [y, z, f (x∗ , w)] − [f (x, z), w, y]

(2.3)

(2.4)

− [z, f (x∗ , w), y] − [z, w, f (x, y)] − [f (x, w), y, z] − [w, f (x∗ , y), z] − [w, y, f (x, z)]k ≤ θ(kxkp + kykp + kzkp + kwkp ) for all λ, µ ∈ T1 and all x, y, z, w ∈ A. Then there exists a unique ternary Jordan bi-derivations D : A × A → A such that kf (x, y) − D(x, y)kB ≤

2θ (kxkp + kykp ) 4 − 2p

(2.5)

for all x, y ∈ A. Proof. By the same reasoning as in the proof of [4, Theorem 2.3], there exists a unique C-bilinear mapping D : A × A → A satisfying (2.5). The C-bilinear mapping D : A × A → A is given by D(x, y) := lim

n→∞

1 f (2n x, 2n y), 4n

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M. Eshaghi Gordji, V. Keshavarz, C. Park, J. R. Lee for all x, y ∈ A. It is easy to show that D(x, y) = lim

n→∞

1 1 f (8n x, 2n y) = lim f (2n x, 8n y) n→∞ 16n 16n

for all x, y ∈ A, since f is bi-additive. It follows from (2.4) that   kD ([x, y, z] + [y, z, x] + [z, x, y]), w) − [D(x, w), y, z] − [x, D(y, w∗ ), z] − [x, y, D(z, w)] − [D(y, w), z, x] −[y, D(z, w∗ ), x] − [y, z, D(x, w)] − [D(z, w), x, y] − [z, D(x, w∗ ), y] − [z, x, D(y, w)]k   + kD x, ([y, z, w] + [z, w, y] + [w, y, z]) − [D(x, y), z, w] − [y, D(x∗ , z), w] − [y, z, D(x∗ , w)] − [D(x, z), w, y] − [z, f (x∗ , w), y] − [z, w, f (x, y)] − [f (x, w), y, z] − [w, f (x∗ , y), z] − [w, y, f (x, z)]k   1  1 1 = lim k n f 23n [x, y, z] + 23n [y, z, x] + 23n [z, x, y], 2n w − [ n f (2n x, 2n w), y, z] − [x, n f (2n y, 2n w∗ ), z] n→∞ 16 4 4 1 1 1 1 − [x, y, n f (2n z, 2n w)] − [ n f (2n y, 2n w), z, x] − [y, n f (2n z, 2n w∗ ), x] − [y, z, n f (2n x, 2n w)] 4 4 4 4 1 1 1 − [ n f (2n z, 2n w), x, y] − [z, n f (2n x, 2n w∗ ), y] − [z, x, n f (2n y, 2n w)]k 4 4 4  1   1 1 + k n f 2n x, 23n [y, z, w] + 23n [z, w, y] + 23n [z, w, y] − [ n f (2n x, 2n y), z, w] + [y, n f (2n x∗ , 2n z), w] 16 4 4 1 1 1 1 − [y, z, n f (2n x, 2n w)] − [ n f (2n x, 2n z), w, y] − [z, n f (2n x∗ , 2n w), y] − [z, w, n f (2n x, 2n y)] 4 4 4 4 1 1 1 − [ n f (2n x, 2n w), y, z] − [w, n f (2n x∗ , 2n y), z] − [w, y, n f (2n x, 2n z)]k 4 4 4 2np θ(kxkp + kykp + kzkp + kwkp ) = 0 ≤ lim n→∞ 16n for all x, y, z, w ∈ A. So   kD ([x, y, z] + [y, z, x] + [z, x, y]), w) − [D(x, w), y, z] − [x, D(y, w∗ ), z] − [x, y, D(z, w)] − [D(y, w), z, x] − [y, D(z, w∗ ), x] − [y, z, D(x, w)] − [D(z, w), x, y] − [z, D(x, w∗ ), y] − [z, x, D(y, w)]k and   + kD x, ([y, z, w] + [z, w, y] + [w, y, z]) − [D(x, y), z, w] − [y, D(x∗ , z), w] − [y, z, D(x∗ , w)] − [D(x, z), w, y] − [z, f (x∗ , w), y] − [z, w, f (x, y)] − [f (x, w), y, z] − [w, f (x∗ , y), z] − [w, y, f (x, z)]k for all x, y, z, w ∈ A. By Lemma 2.2, the mapping D is a unique ternary Jordan bi-derivation satisfying (2.5).



For the case p > 4, one can obtain a similar result. Theorem 2.4. Let p and θ be positive real numbers with p > 4, and let f : A × A → A be a mapping satisfying (2.3) and (2.4). Then there exists a unique ternary Jordan bi-derivation D : A × A → A such that kf (x, y) − D(x, y)k ≤

6θ (kxkp + kykp ) 2p − 4

for all x, y ∈ A. Proof. The proof is similar to the proof of Theorem 2.3.



Theorem 2.5. Let p and θ be positive real numbers with p < 21 , and let f : A × A → A be a mapping such that kDλ,µ f (x, y, z, w)k ≤ θ · kxkp · kykp · kzkp · kwkp ,

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Approximate ternary Jordan bi-derivations  kf ([x, y, z] + [y, z, x] + [z, x, y]), w − [f (x, w), y, z] + [x, f (y, w∗ ), z] − [x, y, f (z, w)] − [f (y, w), z, x] 

− [y, f (z, w∗ ), x] − [y, z, f (x, w)] − [f (z, w), x, y] − [z, f (x, w∗ ), y] − [z, x, f (y, w)]k   + kf x, ([y, z, w] + [z, w, y] + [w, y, z]) − [f (x, y), z, w] − [y, f (x∗ , z), w] − [y, z, f (x∗ , w)] − [f (x, z), w, y] − [z, f (x∗ , w), y] − [z, w, f (x, y)] − [f (x, w), y, z] − [w, f (x∗ , y), z] − [w, y, f (x, z)]k ≤ θ · kxkpA · kykpA · kzkpA · kwkpA for all λ, µ ∈ T1 and all x, y, z, w ∈ A. Then there exists a unique ternary Jordan bi-derivations D : A × A → A such that 2θ kxk2p kyk2p 4 − 24p

kf (x, y) − D(x, y)k ≤

(2.6)

for all x, y ∈ A. Proof. By the same reasoning as in the proof of [4, Theorem 2.6], there exists a unique C-bilinear mapping D : A × A → A satisfying (2.6). The C-bilinear mapping D : A × A → A is given by D(x, y) := lim

n→∞

1 f (2n x, 2n y), 4n

for all x, y ∈ A. The rest of the proof is similar to the proof of Theorem 2.3.



References [1] V. Abramov, R. Kerner, B. Le Roy, Hypersymmetry: A Z3 graded generalization of supersymmetry, J. Math. Phys. 38 (1997), 1650–1669. [2] M. Adam, On the stability of some quadratic functional equation, J. Nonlinear Sci. Appl. 4 (2011), 50–59. [3] T. Aoki, On the stability of the linear transformation in Banach spaces, J. Math. Soc. Japan 2 (1950), 64–66. [4] J. Bae, W. Park, Approximate bi-homomorphisms and bi-derivations in C ∗ -ternary algebras, Bull. Korean Math. Soc. 47 (2010), 195–209. [5] F. Bagarello, G. Morchio, Dynamics of mean-field spin models from basic results in abstract differential equations, J. Stat. Phys. 66 (1992), 849–866. [6] M. Bavand Savadkouhi, M. Eshaghi Gordji, J. M. Rassias, N. Ghobadipour, Approximate ternary Jordan derivations on Banach ternary algebras, J. Math. Phys. 50, Art. ID 042303 (2009). [7] L. C˘ adariu, L. G˘ avruta, P. G˘ avruta, On the stability of an affine functional equation, J. Nonlinear Sci. Appl. 6 (2013), 60–67. [8] A. Cayley, On the 34concomitants of the ternary cubic. Amer. J. Math. 4 (1881), 1–15. [9] A. Chahbi, N. Bounader, On the generalized stability of d’Alembert functional equation, J. Nonlinear Sci. Appl. 6 (2013), 198–204. [10] Y. Cho, C. Park, M. Eshaghi Gordji, Approximate additive and quadratic mappings in 2-Banach spaces and related topics, Int. J. Nonlinear Anal. Appl. 3 (2012), No. 1, 75–81. [11] Y. L. Daletskii, L. A. Takhtajan, Leibniz and Lie algebra structures for Nambu algebra, Lett. Math. Phys. 39 (1997), 127–141. [12] T. Hopkins, Nilpotent ideals in Lie and anti-Lie triple systems, J. Algebra 178 (1995), 480–492. [13] D.H. Hyers, On the stability of the linear functional equation, Proc. Natl. Acad. Sci. U.S.A. 27 (1941), 222–224. [14] M. Kapranov, I. M. Gelfand and A. Zelevinskii, Discriminants, Resultants and Multidimensional Determinants, Birkh¨ auser, Berlin, 1994. [15] R. Kerner, The cubic chessboard: Geometry and physics, Class. Quantum Grav. 14 (1997), A203, 1997. [16] W.G. Lister, A structure theory of Lie triple systems, Trans. Amer. Math. Soc. 72 (1952), 217–242. [17] J. Nambu, Generalized Hamiltonian dynamics, Physical Review D (3) 7 (1973), 2405–2412. [18] C. Park, K. Ghasemi, S. G. Ghaleh, S. Jang, Approximate n-Jordan ∗-homomorphisms in C ∗ -algebras, J. Comput. Anal. Appl. 15 (2013), 365-368. 50

Eshaghi Gordji et al 45-51

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M. Eshaghi Gordji, V. Keshavarz, C. Park, J. R. Lee [19] C. Park, A. Najati, S. Jang, Fixed points and fuzzy stability of an additive-quadratic functional equation, J. Comput. Anal. Appl. 15 (2013), 452–462. [20] J. M. Rassias, On approximation of approximately linear mappings by linear mappings, J. Funct. Anal. 46 (1982), 126–130. [21] Th.M. Rassias, On the stability of the linear mapping in Banach spaces, Proc. Amer. Math. Soc. 72 (1978), 297–300. [22] S. Schin, D. Ki, J. Chang, M. Kim, Random stability of quadratic functional equations: a fixed point approach, J. Nonlinear Sci. Appl. 4 (2011), 37–49. [23] S. Shagholi, M. Bavand Savadkouhi, M. Eshaghi Gordji, Nearly ternary cubic homomorphism in ternary Fr´echet algebras, J. Comput. Anal. Appl. 13 (2011), 1106–1114. [24] S. Shagholi, M. Eshaghi Gordji, M. Bavand Savadkouhi, Stability of ternary quadratic derivation on ternary Banach algebras, J. Comput. Anal. Appl. 13 (2011), 1097–1105. [25] D. Shin, C. Park, Sh. Farhadabadi, On the superstability of ternary Jordan C ∗ -homomorphisms, J. Comput. Anal. Appl. 16 (2014), 964–973. [26] D. Shin, C. Park, Sh. Farhadabadi, Stability and superstability of J ∗ -homomorphisms and J ∗ -derivations for a generalized Cauchy-Jensen equation, J. Comput. Anal. Appl. 17 (2014), 125–134. [27] L. A. Takhtajan, On foundation of the generalized Nambu mechanics, Commun. Math. Phys. 160 (1994), 295–315. [28] S.M. Ulam, Problems in Modern Mathematics, Chapter VI, Science ed., Wiley, New York, 1940. [29] L. Vainerman, R. Kerner, On special classes of n-algebras, J. Math. Phys. 37 (1996), 2553–2565. [30] C. Zaharia, On the probabilistic stability of the monomial functional equation, J. Nonlinear Sci. Appl. 6 (2013), 51–59. [31] S. Zolfaghari, Approximation of mixed type functional equations in p-Banach spaces, J. Nonlinear Sci. Appl. 3 (2010), 110–122.

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SOME GENERALIZED DIFFERENCE SEQUENCE SPACES OF IDEAL CONVERGENCE AND ORLICZ FUNCTIONS KULDIP RAJ1 , AZIMHAN ABZHAPBAROV2 AND ASHIRBAYEV KHASSYMKHAN3

Abstract. In this paper we shall introduce some generalized difference sequence spaces by using Musielak-Orlicz function, ideal convergence and an infinite matrix defined on n-normed spaces. We shall study these spaces for some linear toplogical structures and algebraic properties. We also prove some inclusion relations between these spaces

1. Introduction and Preliminaries The notion of statistical convergence was introduced by Fast [5] and Schoenberg [31] independently. Over the years and under different names, statistical convergence has been discussed in the theory of Fourier analysis, ergodic theory and number theory. Later on, it was further investigated from the sequence space point of view and linked with summability theory by Fridy [6], Connor [1], Salat [29], Isik [14], Sava¸s [30], Malkowsky and Sava¸s [19], Kolk [16], Tripathy and Sen [32] and many others. In recent years, generalizations of statistical convergence have appeared in the study of strong integral summability and the structure of ideals of bounded continuous functions on locally compact spaces. Statistical convergence and its generalizations are also connected with subsets of the Stone-Cech compactification of natural numbers. Moreover, statistical convergence is closely related to the concept of convergence in probability. The notion depends on the density of subsets of the set N of natural numbers. A subset E of N is said to have the natural density δ(E) if the following limit exists: n 1X δ(E) = lim χE (k), where χE is the characteristic function of E. It is clear that n→∞ n k=1 any finite subset of N has zero natural density and δ(E c ) = 1 − δ(E). The notion of ideal convergence was first introduced by P.Kostyrko et.al [13] as a generalization of statistical convergence which was further studied in topological spaces by Das, Kostyrko, Wilczynski and Malik (see [2]). More applications of ideals can be seen in ([2], [3]). We continue in this direction and introduce I-convergence of generalized sequences in more general setting. A family I ⊂ 2Y of subsets of a non empty set Y is said to be an ideal in Y if (1) φ ∈ I; 2000 Mathematics Subject Classification. 40A05, 40B50, 46A19, 46A45. Key words and phrases. Orlicz function,Musielak-Orlicz function, statistical convergence, ideal convergence, solid, infinite matrix, n-normed space. 1

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2

(2) A, B ∈ I imply A ∪ B ∈ I; (3) A ∈ I, B ⊂ A imply B ∈ I, while an admissible ideal I of Y further satisfies {x} ∈ I for each x ∈ Y (see [11]). Given I ⊂ 2N be a non trivial ideal in N. A sequence n (xn )n∈N in X is said o to be Iconvergent to x ∈ X, if for each  > 0 the set A() = n ∈ N : ||xn − x|| ≥  belongs to I (see [10]). The notion of difference sequence spaces was introduced by Kızmaz [15], who studied the difference sequence spaces l∞ (∆), c(∆) and c0 (∆). The notion was further generalized by Et and C ¸ olak [4] by introducing the spaces l∞ (∆n ), c(∆n ) and c0 (∆n ). Let w be the space of all complex or real sequences x = (xk ) and let m, n be non-negative integers, then for Z = l∞ , c, c0 we have sequence spaces m Z(∆m n ) = {x = (xk ) ∈ w : (∆n xk ) ∈ Z}, m−1 m xk+1 ) and ∆0n xk = xk for all k ∈ N, which is xk − ∆m−1 where ∆m n n x = (∆n xk ) = (∆n equivalent to the following binomial representation ! m X m m v ∆ n xk = (−1) xk+nv . v v=0

Taking n = 1, we get the spaces which were studied by Et and C ¸ olak [4]. Taking m = n = 1, we get the spaces which were introduced and studied by Kızmaz [15]. The concept of 2-normed spaces was initially developed by G¨ahler [7] in the mid of 1960’s, while that of n-normed spaces one can see in Misiak[19]. Since then, many others have studied this concept and obtained various results, see Gunawan ([8], [9]) and Gunawan and Mashadi [10] and many others. Let n ∈ N and X be a linear space over the field K, where K is field of real or complex numbers of dimension d, where d ≥ n ≥ 2. A real valued function ||·, · · · , ·|| on X n satisfying the following four conditions: (1) (2) (3) (4)

||x1 , x2 , · · · , xn || = 0 if and only if x1 , x2 , · · · , xn are linearly dependent in X; ||x1 , x2 , · · · , xn || is invariant under permutation; ||αx1 , x2 , · · · , xn || = |α| ||x1 , x2 , · · · , xn || for any α ∈ K, and ||x + x0 , x2 , · · · , xn || ≤ ||x, x2 , · · · , xn || + ||x0 , x2 , · · · , xn ||

is called a n-norm on X, and the pair (X, ||·, · · · , ·||) is called a n-normed space over the field K. For example, we may take X = Rn being equipped with the n-norm ||x1 , x2 , · · · , xn ||E = the volume of the n-dimensional parallelopiped spanned by the vectors x1 , x2 , · · · , xn which may be given explicitly by the formula ||x1 , x2 , · · · , xn ||E = | det(xij )|, where xi = (xi1 , xi2 , · · · , xin ) ∈ Rn for each i = 1, 2, · · · , n, where script E denotes Euclidean space. Let (X, ||·, · · · , ·||) be an n-normed space of dimension d ≥ n ≥ 2 and

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{a1 , a2 , · · · , an } be linearly independent set in X. Then the following function ||·, · · · , ·||∞ on X n−1 defined by ||x1 , x2 , · · · , xn−1 ||∞ = max{||x1 , x2 , · · · , xn−1 , ai || : i = 1, 2, · · · , n} defines an (n − 1)-norm on X with respect to {a1 , a2 , · · · , an }. A sequence (xk ) in a n-normed space (X, ||·, · · · , ·||) is said to converge to some L ∈ X if lim ||xk − L, z1 , · · · , zn−1 || = 0 for every z1 , · · · , zn−1 ∈ X.

k→∞

A sequence (xk ) in a n-normed space (X, ||·, · · · , ·||) is said to be Cauchy if lim ||xk − xi , z1 , · · · , zn−1 || = 0 for every z1 , · · · , zn−1 ∈ X.

k,i→∞

If every Cauchy sequence in X converges to some L ∈ X, then X is said to be complete with respect to the n-norm. Any complete n-normed space is said to be n-Banach space. An Orlicz function M : [0, ∞) → [0, ∞) is a continuous, non-decreasing and convex function such that M (0) = 0, M (x) > 0 for x > 0 and M (x) −→ ∞ as x −→ ∞. Lindenstrauss and Tzafriri [17] used the idea of Orlicz function to define the following sequence space, ∞ n  |x |  o X k `M = (xk ) ∈ w : M < ∞, for some ρ > 0 ρ k=1

which is called as an Orlicz sequence space. Also `M is a Banach space with the norm ∞ o n  |x |  X k ≤1 . ||(xk )|| = inf ρ > 0 : M ρ k=1

Also, it was shown in [17] that every Orlicz sequence space `M contains a subspace isomorphic to `p (p ≥ 1). An Orlicz function M satisfies ∆2 −condition if and only if for any constant L > 1 there exists a constant K(L) such that M (Lu) ≤ K(L)M (u) for all values of u ≥ 0. An Orlicz function M can always be represented in the following integral form Z x M (x) = η(t)dt 0

where η is known as the kernel of M , is right differentiable for t ≥ 0, η(0) = 0, η(t) > 0, η is non-decreasing and η(t) → ∞ as t → ∞. A sequence M = (Mk ) of Orlicz functions is called a Musielak-Orlicz function see ([18], [25]). A sequence N = (Nk ) is defined by Nk (v) = sup{|v|u − Mk (u) : u ≥ 0}, k = 1, 2, · · · is called the complementary function of a Musielak-Orlicz function M. For a given Musielak-Orlicz function M, the Musielak-Orlicz sequence space tM and its subspace hM are defined as follows n o tM = x ∈ w : IM (cx) < ∞ for some c > 0 ,

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n o hM = x ∈ w : IM (cx) < ∞ for all c > 0 , where IM is a convex modular defined by IM (x) =

∞ X

Mk (xk ), x = (xk ) ∈ tM .

k=1

We consider tM equipped with the Luxemburg norm o n x ≤1 ||x|| = inf k > 0 : IM k or equipped with the Orlicz norm ||x||0 = inf

n1 k

 o 1 + IM (kx) : k > 0 .

A Musielak-Orlicz function (Mk ) is said to satisfy ∆2 -condition if there exist constants 1 1 a, K > 0 and a sequence c = (ck )∞ k=1 ∈ `+ (the positive cone of ` ) such that the inequality Mk (2u) ≤ KMk (u) + ck holds for all k ∈ N and u ∈ R+ whenever Mk (u) ≤ a. Let X be a linear metric space. A function p : X → R is called paranorm, if (1) (2) (3) (4)

p(x) ≥ 0 for all x ∈ X, p(−x) = p(x) for all x ∈ X, p(x + y) ≤ p(x) + p(y) for all x, y ∈ X, if (λn ) is a sequence of scalars with λn → λ as n → ∞ and (xn ) is a sequence of vectors with p(xn − x) → 0 as n → ∞, then p(λn xn − λx) → 0 as n → ∞.

A paranorm p for which p(x) = 0 implies x = 0 is called total paranorm and the pair (X, p) is called a total paranormed space. It is well known that the metric of any linear metric space is given by some total paranorm (see [33], Theorem 10.4.2, pp. 183). For more details about sequence spaces (see [21], [22], [23], [24], [26], [27], [28]) and reference therein. A sequence space E is said to be solid(or normal) if (xk ) ∈ E implies (αk xk ) ∈ E for all sequences of scalars (αk ) with |αk | ≤ 1 and for all k ∈ N. Let I be an admissible ideal of N, let p = (pk ) be a bounded sequence of positive real numbers for all k ∈ N and A = (ank ) be an infinite matrix. Let M = (Mk ) be a MusielakOrlicz function, u = (uk ) be a sequence of strictly positive real numbers and (X, ||., ..., .||) be a n-normed space. Further w(n − x) denotes the space of all X-valued sequences. For every z1 , z2 , ..., zn−1 ∈ X, for each  > 0 and for some ρ > 0 we define the following sequence spaces: n   n W I A, ∆m n , M, u, p, ||., ..., .|| = x = (xk ) ∈ w(n − x) : for given  > 0, n ∈ N : ∞ X k=1

ipk o o h  u ∆m x − L k n k ank Mk || , z1 , z2 , ..., zn−1 || ≥  ∈ I, for L ∈ X and n ∈ N , ρ

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n   n , M, u, p, ||., ..., .|| = x = (x ) ∈ w(n − x) : for given  > 0, n∈N: W0I A, ∆m k n ∞ X k=1

h  u ∆m x ipk o o k n k ank Mk || , z1 , z2 , ..., zn−1 || ≥ ∈I ρ

and n   n I W∞ A, ∆m n , M, u, p, ||., ..., .|| = x = (xk ) ∈ w(n − x) : ∃ k > 0, n ∈ N : ∞ X k=1

ipk o o h  u ∆m x k n k , z1 , z2 , ..., zn−1 || ank Mk || ≥K ∈I . ρ

Some special cases of the above defined sequence spaces are arises: If m = n = 0, then we obtain the spaces as follows n   n W I A, M, u, p, ||., ..., .|| = x = (xk ) ∈ w(n − x) : for given  > 0, n ∈ N : ∞ X k=1

ipk o o h  u x −L k k , z1 , z2 , ..., zn−1 || ≥  ∈ I, for L ∈ X and n ∈ N , ank Mk || ρ

n  n  W0I A, M, u, p, ||., ..., .|| = x = (xk ) ∈ w(n − x) : for given  > 0, n ∈ N : ∞ X k=1

ipk o o h  u x k k , z1 , z2 , ..., zn−1 || ≥ ∈I ank Mk || ρ

and n   n I W∞ A, M, u, p, ||., ..., .|| = x = (xk ) ∈ w(n − x) : ∃ k > 0, n ∈ N : ∞ X k=1

ipk o o h  u x k k ank Mk || , z1 , z2 , ..., zn−1 || ≥K ∈I . ρ

If m = n = 1, then the above spaces are as follows n   n I W A, ∆, M, u, p, ||., ..., .|| = x = (xk ) ∈ w(n − x) : for given  > 0, n ∈ N : ∞ X k=1

h  u ∆x − L ipk o o k k ank Mk || , z1 , z2 , ..., zn−1 || ≥  ∈ I, for L ∈ X and n ∈ N , ρ

n   n W0I A, ∆, M, u, p, ||., ..., .|| = x = (xk ) ∈ w(n − x) : for given  > 0, n ∈ N : ∞ X k=1

h  u ∆x ipk o o k k ank Mk || , z1 , z2 , ..., zn−1 || ≥ ∈I ρ

and n   n I W∞ A, ∆, M, u, p, ||., ..., .|| = x = (xk ) ∈ w(n − x) : ∃ k > 0, n ∈ N : ∞ X k=1

h  u ∆x ipk o o k k ank Mk || , z1 , z2 , ..., zn−1 || ≥K ∈I . ρ

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If M(x) = x for all x ∈ [0, ∞), then we have n   n I m W A, ∆n , u, p, ||., ..., .|| = x = (xk ) ∈ w(n − x) : for given  > 0, n ∈ N : ∞ X k=1

 pk  u ∆m x − L o o k n k , z1 , z2 , ..., zn−1 || ank || ≥  ∈ I, for L ∈ X and n ∈ N , ρ

n   n W0I A, ∆m n , u, p, ||., ..., .|| = x = (xk ) ∈ w(n − x) : for given  > 0, n ∈ N : ∞ X k=1

 pk o o  u ∆m x k n k , z1 , z2 , ..., zn−1 || ≥ ∈I ank || ρ

and n   n I W∞ A, ∆m n , u, p, ||., ..., .|| = x = (xk ) ∈ w(n − x) : ∃ k > 0, n ∈ N : ∞ X k=1

 pk o o  u ∆m x k n k ank || , z1 , z2 , ..., zn−1 || ≥K ∈I . ρ

If p = (pk ) = 1 for all k, then n the above spaces are as follows n   W I A, ∆m , M, u, ||., ..., .|| = x = (xk ) ∈ w(n − x) : for given  > 0, n ∈ N : n ∞ X k=1

 u ∆m x − L  o o k n k ank Mk || , z1 , z2 , ..., zn−1 || ≥  ∈ I, for L ∈ X and n ∈ N , ρ

n  n  W0I A, ∆m n , M, u, ||., ..., .|| = x = (xk ) ∈ w(n − x) : for given  > 0, n ∈ N : ∞ X k=1

 u ∆m x  o o k n k ank Mk || , z1 , z2 , ..., zn−1 || ≥  ∈ I ρ

and n  n  I W∞ A, ∆m n , M, u, ||., ..., .|| = x = (xk ) ∈ w(n − x) : ∃ k > 0, n ∈ N : ∞ X k=1

 o o  u ∆m x k n k ank Mk || , z1 , z2 , ..., zn−1 || ≥ K ∈ I . ρ

If A = (C, 1), the Ces` aro matrix, then the above spaces are as follows n  m  n I W ∆n , M, u, p, ||., ..., .|| = x = (xk ) ∈ w(n − x) : for given  > 0, n ∈ N : ∞ h X k=1

 u ∆m x − L ipk o o k n k Mk || , z1 , z2 , ..., zn−1 || ≥  ∈ I, for L ∈ X and n ∈ N , ρ

n   n W0I ∆m n , M, u, p, ||., ..., .|| = x = (xk ) ∈ w(n − x) : for given  > 0, n ∈ N : ∞ h  u ∆m x ipk o o X k n k Mk || , z1 , z2 , ..., zn−1 || ≥ ∈I ρ

k=1

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and n  m  n I ∆n , M, u, p, ||., ..., .|| = x = (xk ) ∈ w(n − x) : ∃ k > 0, n ∈ N : W∞ ∞ h X k=1

 u ∆m x ipk o o k n k Mk || , z1 , z2 , ..., zn−1 || ≥K ∈I . ρ

If we take A = (ank ) is a de La Valee Poussin mean i.e. ( 1 if k ∈ In = [n − λn + 1, n] λn , ank = 0, otherwise where (λn ) is a non-decreasing sequence of positive numbers tending to ∞ and λn+1 ≤   λn +1, λ1 = 1, then the above sequence spaces are denoted by W I λ, ∆m n , M, u, p, ||., ..., .|| ,     I m W0I λ, ∆m n , M, u, p, ||., ..., .|| and W∞ λ, ∆n , M, u, p, ||., ..., .|| . By a lacunary sequence θ = (kr ); r = 0, 1, 2, ... where k0 = 0, we shall mean an increasing sequence of non-negative integers with kr −kr−1 → ∞ as r → ∞. The intervals determined by θ will be denoted by Ir = (kr−1 , kr ] and hr = kr − kr−1 . We finally arrived, let ( 1 if kr−1 < k < kr hr , ank = 0, otherwise.    I m Then the above classes of sequences are denoted by W I θ, ∆m n , M, p, ||., ..., .|| , W0 θ, ∆n , M,    I θ, ∆m p, ||., ..., .|| and W∞ n , M, p, ||., ..., .|| . The following inequality will be used throughout the paper. If 0 ≤ pk ≤ sup pk = G, D = max(1, 2G−1 ) then (1.1)

|ak + bk |pk ≤ D{|ak |pk + |bk |pk }

for all k and ak , bk ∈ C. Also |a|pk ≤ max(1, |a|G ) for all a ∈ C. The main aim of this paper is to introduce some generalized difference sequence spaces defined by ideal convergence, a Musielak-Orlicz function M = (Mk ) and an infinite matrix A = (ank ). I have also make an effort to study some inclusion relations and their topological properties.

2. Main Results Theorem 2.1 Let M = (Mk ) be a Musielak-Orlicz function, p = (pk ) be a bounded sequence of positive real numbers and u = (uk ) be a sequence of strictly positive real numbers.      I m I m Then W I A, ∆m n , M, u, p, ||., ..., .|| , W0 A, ∆n , M, u, p, ||., ..., .|| and W∞ A, ∆n , M,  u, p, ||., ..., .|| are linear spaces over the field of complex numbers C.   Proof. We shall prove the result for the space W0I A, ∆m n , M, u, p, ||., ..., .|| . Let x = (xk )

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  and y = (yk ) be two elements of W0I A, ∆m n , M, u, p, ||., ..., .|| . Then there exists ρ1 > 0 and ρ2 > 0 and for z1 , z2 , ..., zn−1 ∈ X such that ∞ n h  u ∆m x ipk X o k n k A 2 = n ∈ N : ∈I ank Mk || , z1 , z2 , ..., zn−1 || ≥ ρ1 2 k=1

and ∞ n h  u ∆m y ipk X o k n k B 2 = n ∈ N : ank Mk || ∈ I. , z1 , z2 , ..., zn−1 || ≥ ρ2 2 k=1

Let α, β ∈ C. Since ||., ..., .|| is a n-norm, ∆m n is linear and the contributing of M = (Mk ), the following inequality holds: ∞ h  u ∆m (αx + βy ) ipk X k n k k ank Mk || , z1 , z2 , ..., zn−1 || |α|ρ1 + |β|ρ2 k=1

≤ D

+ D

∞ X k=1 ∞ X

ank

h

ipk  u ∆m x |α| k n k Mk || , z1 , z2 , ..., zn−1 || |α|ρ1 + |β|ρ2 ρ1

ank

h

ipk  u ∆m y |β| k n k Mk || , z1 , z2 , ..., zn−1 || |α|ρ1 + |β|ρ2 ρ2

k=1 ∞ X

≤ DK

+ DK

k=1 ∞ X k=1

ipk h  u ∆m x k n k , z1 , z2 , ..., zn−1 || ank Mk || ρ1 h  u ∆m y ipk k n k ank Mk || , z1 , z2 , ..., zn−1 || ρ2

n o |β| , where K = max 1, |α|ρ1|α| +|β|ρ2 |α|ρ1 +|β|ρ2 . From the above relation , we get ∞ ipk n h  u ∆m (αx + βy ) o X k n k k , z1 , z2 , ..., zn−1 || n∈N: ank Mk || ≥ |α|ρ1 + |β|ρ2 k=1

⊆

∞ n h  u ∆m x ipk X o k n k n ∈ N : DK ank Mk || , z1 , z2 , ..., zn−1 || ≥ ρ1 2

∪

∞ n h  u ∆m y ipk X o k n k n ∈ N : DK ank Mk || , z1 , z2 , ..., zn−1 || ≥ . ρ2 2

k=1

k=1

Since both the sets on the R.H.S of above relation are belongs to I, so the set on the L.H.S of the inclusion relation belongs to I. Similarly we can prove other cases. This completes the proof of the theorem. Theorem 2.2 Let M0 = (Mk0 ) and M00 = (Mk00 ) be two Musielak-orlicz functions. Then we      0 I m 00 I m 0 have W0I A, ∆m n , M , u, p, ||., ..., .|| ∩ W0 A, ∆n , M , u, p, ||., ..., .|| ⊆ W0 A, ∆n , M +  M00 , u, p, ||., ..., .|| .     0 I m 00 Proof. Let x = (xk ) ∈ W0I A, ∆m n , M , u, p, ||., ..., .|| ∩ W0 A, ∆n , M , u, p, ||., ..., .|| .

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Then we get the result by the following inequality: ∞ ipk h  u ∆m x X k n k ank (Mk0 + Mk00 ) || , z1 , z2 , ..., zn−1 || ρ k=1

≤ D

+ D

∞ X k=1 ∞ X k=1

ipk h  u ∆m x k n k , z1 , z2 , ..., zn−1 || ank Mk0 || ρ ipk h  u ∆m x k n k , z1 , z2 , ..., zn−1 || ank Mk00 || . ρ

Hence ∞ ipk o n h  u ∆m x X k n k , z1 , z2 , ..., zn−1 || ≥ n∈N: ank (Mk0 + Mk00 ) || ρ k=1

⊆

∞ ipk n h  u ∆m x X o k n k , z1 , z2 , ..., zn−1 || ≥ ank Mk0 || n∈N:D ρ 2

∪

∞ n h  u ∆m x ipk X o k n k n∈N:D ank Mk00 || , z1 , z2 , ..., zn−1 || ≥ ρ 2

k=1

k=1

Since both the sets on the R.H.S of above relation are belongs to I, so the set on the L.H.S of the inclusion relation belongs to I. This completes the proof of the theorem.     Theorem 2.3 The inclusions Z ∆m−1 , M, u, p, ||., ..., .|| ⊆ Z A, ∆m n , M, u, p, ||., ..., .||  nm−1    are strict for m ≥ 1. In general Z ∆n , M, u, p, ||., ..., .|| ⊆ Z A, ∆m n , M, u, p, ||., ..., .|| , I I I for m = 0, 1, 2, ... where Z = W , W0 , W∞ .   Proof. We give the proof for W0I A, ∆m−1 , M, u, p, ||., ..., .|| only. The others can be n   proved by similar argument. Let x = (xk ) be any element in the space W0I A, ∆m−1 , M, u, p, ||., ..., .|| . n Let  > 0 be given. Then there exists ρ > 0 such that the set

∞ ipk o n h  u ∆m−1 x X k n k , z1 , z2 , ..., zn−1 || ≥  ∈ I. n∈N: ank Mk || ρ k=1

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Since M = (Mk ) is non-decreasing and convex for every k, it follows that ∞ h  u ∆m x ipk X k n k ank Mk || , z1 , z2 , ..., zn−1 || 2ρ

k=1

=

∞ X

ipk h  u ∆m−1 x m−1 xk k n k+1 − uk ∆n , z1 , z2 , ..., zn−1 || ank Mk || 2ρ

k=1 ∞ X

≤ D

+ D

k=1 ∞ X

ank ank

k=1 ∞ X

≤ DH

k=1 ∞ X

 u ∆m−1 x ipk k n k+1 Mk || , z1 , z2 , ..., zn−1 || 2 ρ

h1

 u ∆m−1 x ipk k n k Mk || , z1 , z2 , ..., zn−1 || 2 ρ

h1

ipk h  u ∆m−1 x k n k+1 , z1 , z2 , ..., zn−1 || ank Mk || ρ

ipk h  u ∆m−1 x k n k , z1 , z2 , ..., zn−1 || , ank Mk || ρ k=1 o n where H = max 1, ( 12 )G . Thus we have ∞ n h  u ∆m x ipk o X k n k n∈N: ank Mk || , z1 , z2 , ..., zn−1 || ≥ 2ρ + DH

k=1

⊆

∞ ipk n h  u ∆m−1 x X o k+1 k n , z1 , z2 , ..., zn−1 || ≥ ank Mk || n∈N: ρ 2

∪

∞ n h  u ∆m−1 x ipk X o k n k n∈N: ank Mk || , z1 , z2 , ..., zn−1 || ≥ ρ 2

k=1

k=1

Since both the sets in right hand side of the above relation belongs to I, therefore we get the set ∞ n ipk o h  u ∆m x X k n k n∈N: ank Mk || , z1 , z2 , ..., zn−1 || ≥  ∈ I. ρ k=1

This inclusion is strict follows from the following example. Example. Let Mk (x) = x, for all k ∈ N, uk = pk = 1 for all k ∈ N and A = (C, 1), the Cesaro matrix. Now consider a sequence x = (xk ) = (k s ). Then for n = 1, x = (xk ) be m−1    I longs to W0I ∆m , M, u, p, ||., ..., .|| , n , M, u, p, ||., ..., .|| but does not belongs to W0 ∆n m−1 because ∆m xk = (−1)m−1 (m − 1)!. n xk = 0 and ∆n Theorem 2.4 For any two sequences p = (pk ) and q = (qk ) of positive real numbers and for any two n-norms ||., ..., .||1 and ||., ..., .||2 on X, we have the following     m I I I Z A, ∆m n , M, u, p, ||., ..., .||1 ∩Z A, ∆n , M, u, q, ||., ..., .||2 6= φ where Z = W , W0 and W∞ . Proof. Since the zero element belongs to both the classes of sequences, so the intersection is non-empty.

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SOME GENERALIZED DIFFERENCE SEQUENCE SPACES

11

    m I Theorem 2.5 The sequence spaces W0I A, ∆m n , M, u, p, ||., ..., .|| and W∞ A, ∆n , M, u, p, ||., ..., .|| are normal as well as monotone.   Proof. We shall prove the theorem for W0I A, ∆m n , M, u, p, ||., ..., .|| . Let x = (xk ) ∈   W0I A, ∆m n , M, u, p, ||., ..., .|| and α = (αk ) be a sequence of scalars such that |αk | ≤ 1 for all k ∈ N. Then for given  > 0, we have ∞ n h  u ∆m (α x ) ipk o X k n k k n∈N: ank Mk || , z1 , z2 , ..., zn−1 || ≥ ρ k=1

∞ ipk n o h  u ∆m (x ) X k n k , z1 , z2 , ..., zn−1 || n∈N: ≥  ∈ I. ank Mk || ρ k=1     I m Hence αk xk ∈ W0I A, ∆m n , M, u, p, ||., ..., .|| . Thus the space W0 A, ∆n , M, u, p, ||., ..., .||   is normal. Therefore W0I A, ∆m n , M, u, p, ||., ..., .|| is monotone also (see [12]). Similarly we can prove the theorem for other case. This completes the proof of the theorem.

⊆

References [1] J. S. Connor, The statistical and strong p-Ces` aro convergence of sequences, Analysis (Munich) 8(1988), 47-63. [2] P. Das, P. Kostyrko, W. Wilczynski and P. Malik, I and I* convergence of double sequences, Math. Slovaca, 58(2008), 605-620. [3] P. Das and P.Malik, On the statistical and I-variation of double sequences, Real Anal. Exchange, 33(2007-2008), 351-364. [4] M. Et and R. C ¸ olak, On generalized difference sequence spaces, Soochow J. Math. 21(1995), 377-386. [5] H. Fast, Sur la convergence statistique, Colloq. Math. 2(1951), 241-244. [6] J. A. Fridy, On the statistical convergence, Analysis 5(1985), 301-303. [7] S. G¨ ahler, Linear 2-normietre Rume, Math. Nachr., 28(1965), 1-43. [8] H. Gunawan, On n-inner product, n-norms, and the Cauchy-Schwartz inequality, Sci. Math. Jpn., 5(2001), 47-54. [9] H. Gunawan, The space of p-summable sequence and its natural n-norm, Bull. Aust. Math. Soc., 64(2001), 137-147. [10] H. Gunawan and M. Mashadi, On n-normed spaces, Int. J. Math. Math. Sci., 27(2001), 631-639. [11] M. Gurdal, and S. Pehlivan, Statistical convergence in 2-normed spaces, Southeast Asian Bull. Math., 33(2009), 257-264. [12] P. K. Kamthan and M. Gupta, Sequence spaces and series, Marcel Dekkar, New York (1981). [13] P. Kostyrko, T. Salat and W. Wilczynski, I-Convergence, Real Anal. Exchange, 26(2000), 669-686. [14] M. Isik, On statistical convergence of generalized difference sequence spaces, Soochow J. Math. 30(2004), 197-205. [15] H. Kızmaz, On certain sequence spaces, Canad. Math-Bull., 24(1981), 169-176. [16] E. Kolk, The statistical convergence in Banach spaces, Acta. Comment. Univ. Tartu, 928(1991), 41-52. [17] J. Lindenstrauss and L. Tzafriri, On Orlicz sequence spaces, Israel J. Math., 10(1971), 345-355. [18] L. Maligranda, Orlicz spaces and interpolation, Seminars in Mathematics 5, Polish Academy of Science, (1989).

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12

KULDIP RAJ1 , AZIMHAN ABZHAPBAROV2 AND ASHIRBAYEV KHASSYMKHAN3

[19] E. Malkowsky and E. Sava¸s, Some λ-sequence spaces defined by a modulus, Arch. Math., 36(2000), 219-228. [20] A. Misiak, n-inner product spaces, Math. Nachr., 140(1989), 299-319. [21] M. Mursaleen, On statistical convergence in random 2-normed spaces, Acta sci. Math. (szeged), 76(2010), 101-109. [22] M. Mursaleen and A. Alotaibi, On I-convergence in random 2-normed spaces, Math. Slovaca, 61(2011), 933-940. [23] M. Mursaleen and S. A. Mohiuddine, Statistical convergence of double sequences in intuitionistic fuzzy normed spaces, Chaos Solitons Fractals, 41(2009), 2414-2421. [24] M. Mursaleen, S. A. Mohiuddine and O. H. H. Edely, On the ideal convergence of double sequences in intuitionistic fuzzy normed spaces, Comp. Math. Appl., 59(2010), 603-611. [25] J. Musielak, Orlicz spaces and modular spaces, Lecture Notes in Mathematics, 1034(1983). [26] K. Raj, A. K. Sharma and S. K. Sharma, A Sequence space defined by Musielak-Orlicz functions, Int. J. Pure Appl. Math., 67(2011), 475-484. [27] K. Raj, S. K. Sharma and A. K. Sharma, Some difference sequence spaces in n-normed spaces defined by Musielak-Orlicz function, Armen. J. Math., 3(2010), 127-141. [28] K. Raj and S. K. Sharma, Some sequence spaces in 2-normed spaces defined by Musielak-Orlicz function, Acta Univ. Sapientiae Math., 3(2011), 97-109. [29] T. Salat, On statictical convergent sequences of real numbers, Math. Slovaca, 30(1980), 139-150. [30] E. Sava¸s, Strong almost convergence and almost λ-statistical convergence, Hokkaido Math. J., 29(2000), 531-566. [31] I. J. Schoenberg, The integrability of certain functions and related summability methods, Amer. Math. Monthly, 66(1959), 361-375. [32] B. C. Tripathy and M. Sen, Vector valued paranormed bounded and null sequence spaces associated with multiplier sequences, Soochow J. Math., 29(2003), 379-391. [33] A. Wilansky, Summability through Functional Analysis, North- Holland Math. Stud.,(1984). 1 School

of Mathematics Shri Mata Vaishno Devi University, Katra-182320, J & K, India.

2,3 Science-Pedagogical

Faculty, M. Auezov South Kazakhstan State University, Tauke Khan

Avenue 5, Shymkent 160012, Kazakhstan E-mail address: [email protected] E-mail address: azeke [email protected] E-mail address: [email protected]

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A general stability theorem for a class of functional equations including quadratic-additive functional equations Yang-Hi Lee and Soon-Mo Jung Department of Mathematics Education, Gongju National University of Education, Gongju 314-711, Republic of Korea E-mail: [email protected] Mathematics Section, College of Science and Technology, Hongik University, 339-701 Sejong, Republic of Korea E-mail: [email protected]

Abstract. We prove a general stability theorem of an n-dimensional quadratic-additive type functional equation Df (x1 , x2 , . . . , xn ) =

m ∑

( ) ci f ai1 x1 + ai2 x2 + · · · + ain xn = 0

i=1

by using the direct method. AMS Subject Classification: 39B82, 39B52 Key Words: generalized Hyers-Ulam stability; functional equation; n-dimensional quadraticadditive type functional equation; quadratic-additive mapping; direct method.

1

Introduction

Let G1 and G2 be abelian groups. For any mapping f : G1 → G2 , let us define Af (x, y) := f (x + y) − f (x) − f (y), Qf (x, y) := f (x + y) + f (x − y) − 2f (x) − 2f (y) for all x, y ∈ G1 . A mapping f : G1 → G2 is called an additive mapping (or a quadratic mapping) if f satisfies the functional equation Af (x, y) = 0 (or Qf (x, y) = 0) for all x, y ∈ G1 . We notice that the mappings g, h : R → R given by g(x) = ax and h(x) = ax2 are solutions of Ag(x, y) = 0 and Qh(x, y) = 0, respectively. A mapping f : G1 → G2 is called a quadratic-additive mapping if and only if f is represented by the sum of an additive mapping and a quadratic mapping. A functional equation is called a quadratic-additive type functional equation if and only if each of its solutions is a quadratic-additive mapping (see [9]). For example, 1 64

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2 the mapping f (x) = ax2 + bx is a solution of the quadratic-additive type functional equation. In the study of stability problems of quadratic-additive type functional equations, we have followed out a routine and monotonous procedure for proving the stability of the quadratic-additive type functional equations under various conditions. We can find in the books [2, 3, 7, 8] a lot of references concerning the Hyers-Ulam stability of functional equations (see also [1, 4, 5, 6, 14, 15]). Throughout this paper, let V and W be real vector spaces, let X and Y be a real normed space resp. a real Banach space, and let N0 denote the set of all nonnegative integers. In this paper, we prove a general stability theorem that can be easily applied to the (generalized) Hyers-Ulam stability of a large class of functional equations of the form Df (x1 , x2 , . . . , xn ) = 0, which includes quadratic-additive type functional equations. In practice, given a mapping f : V → W , Df : V n → W is defined by Df (x1 , x2 , . . . , xn ) :=

m ∑

( ) ci f ai1 x1 + ai2 x2 + · · · + ain xn

(1.1)

i=1

for all x1 , x2 , . . . , xn ∈ V , where m is a positive integer and ci , aij are real constants. Indeed, this stability theorem can save us much trouble of proving the stability of relevant solutions repeatedly appearing in the stability problems for various functional equations (see [11, 12, 13]).

2

Preliminaries

Let V and W be real vector spaces and let X and Y be a real normed space resp. a real Banach space. For a given mapping f : V → W , we use the following abbreviations fo (x) :=

f (x) − f (−x) 2

and fe (x) :=

f (x) + f (−x) 2

for all x ∈ V . We now introduce a lemma from the paper [10, Corollary 2]. Lemma 2.1 Let k > 1 be a real constant, let ϕ : V \{0} → [0, ∞) be a function satisfying either ∞ ∑ 1 ϕ(k i x) < ∞ Φ(x) := ki

(2.1)

i=0

for all x ∈ V \{0} or Φ(x) :=

∞ ∑ i=0

(

x k ϕ i k 2i

) 1 be a real number, let ϕ, ψ : V \{0} → [0, ∞) be functions satisfying each of the following conditions ( ) ∞ ∞ ∑ ∑ 1 x i k ψ i < ∞, ϕ(k i x) < ∞, k k 2i i=0

i=0

˜ Φ(x) :=

∞ ∑ i=0

(

x kϕ i k i

) ˜ < ∞, Ψ(x) :=

∞ ∑ 1 ψ(k i x) < ∞ k 2i i=0

for all x ∈ V \{0}, and let f : V → Y be an arbitrarily given mapping. If there exists a mapping F : V → Y satisfying the inequality ˜ ˜ ∥f (x) − F (x)∥ ≤ Φ(x) + Ψ(x)

(2.5)

for all x ∈ V \{0} and the conditions in (2.4) for all x ∈ V , then F is a unique mapping satisfying (2.4) and (2.5).

3

Main results

In this section, let a be a real constant with a ̸∈ {−1, 0, 1}. Lemma 2.1 plays an important role in the proofs of the following two main theorems. Theorem 3.1 Let n be a fixed integer greater than 1, let µ : V \{0} → [0, ∞) be a function satisfying the condition  ∞ ∑ µ(ai x)    < ∞ when |a| < 1,    i=0 a2i (3.1) ∞  i x) ∑  µ(a   < ∞ when |a| > 1   |a|i i=0

for all x ∈ V \{0}, and let φ : (V \{0})n → [0, ∞) be a function satisfying the condition  ∞ ∑ φ(ai x1 , ai x2 , . . . , ai xn )    < ∞ when |a| < 1,   a2i  i=0 (3.2) ∞  i x , ai x , . . . , ai x ) ∑  φ(a  1 2 n  < ∞ when |a| > 1   |a|i i=0

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4 for all x1 , x2 , . . . , xn ∈ V \{0}. If a mapping f : V → Y satisfies f (0) = 0,

2 2

f (ax) − a + a f (x) − a − a f (−x) ≤ µ(x)

2 2

(3.3)

for all x ∈ V \{0}, and ∥Df (x1 , x2 , . . . , xn )∥ ≤ φ(x1 , x2 , . . . , xn )

(3.4)

for all x1 , x2 , . . . , xn ∈ V \{0}, then there exists a unique mapping F : V → Y such that DF (x1 , x2 , . . . , xn ) = 0

(3.5)

for all x1 , x2 , . . . , xn ∈ V \{0}, Fe (ax) = a2 Fe (x) for all x ∈ V , and ∥f (x) − F (x)∥ ≤

and

Fo (ax) = aFo (x)

∞ ( ∑ µ(ai x) + µ(−ai x) i=0

2a2i+2

(3.6)

µ(ai x) + µ(−ai x) + 2|a|i+1

) (3.7)

for all x ∈ V \{0}. Proof. First, we define A := {f : V → Y | f (0) = 0} and a mapping Jm : A → A by f (am x) + f (−am x) f (am x) − f (−am x) + 2a2m 2am for x ∈ V and m ∈ N0 . It follows from (3.3) that Jm f (x) :=

∥Jm f (x) − Jm+l f (x)∥ ≤ =

m+l−1 ∑

∥Ji f (x) − Ji+1 f (x)∥

i=m m+l−1 ∑ i=m

f (ai x) + f (−ai x) f (ai x) − f (−ai x)

+

2a2i 2ai

f (ai+1 x) + f (−ai+1 x) f (ai+1 x) − f (−ai+1 x)

− −

2a2i+2 2ai+1

( ) m+l−1 2+a 2−a ∑ a a 1 i i i

− (3.8) =

2ai+1 f (a · a x) − 2 f (a x) − 2 f (−a x) i=m ( ) 1 a2 + a a2 − a i i i + i+1 f (−a · a x) − f (−a x) − f (a x) 2a 2 2 ( ) 1 a2 + a a2 − a i i i − 2i+2 f (a · a x) − f (a x) − f (−a x) 2a 2 2 ( )

a2 + a a2 − a 1 − 2i+2 f (−a · ai x) − f (−ai x) − f (ai x)

2a 2 2 m+l−1 ∑ ( µ(ai x) + µ(−ai x) µ(ai x) + µ(−ai x) ) ≤ + 2a2i+2 2|a|i+1 i=m

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5 for all x ∈ V \{0}. In view of (3.1) and (3.8), the sequence {Jm f (x)} is a Cauchy sequence for all x ∈ V \{0}. Since Y is complete and f (0) = 0, the sequence {Jm f (x)} converges for all x ∈ V . Hence, we can define a mapping F : V → Y by ( ) f (am x) + f (−am x) f (am x) − f (−am x) F (x) := lim Jm f (x) = lim + m→∞ m→∞ 2a2m 2am for all x ∈ V . We easily obtain from the definition of F and (3.4) that F (ax) + F (−ax) 2 f (am+1 x) + f (−am+1 x) = lim m→∞ 2a2m f (am+1 x) + f (−am+1 x) = a2 lim m→∞ 2a2m+2 2 = a Fe (x), F (ax) − F (−ax) Fo (ax) = 2 f (am+1 x) − f (−am+1 x) = lim m→∞ 2am m+1 f (a x) − f (−am+1 x) = a lim m→∞ 2am+1 = aFo (x) Fe (ax) =

for all x ∈ V , and by (1.1) and (3.2), we get ∥DF (x1 , x2 , . . . , xn )∥ ) ) ( (

Df am x1 , am x2 , . . . , am xn + Df − am x1 , −am x2 , . . . , −am xn

= lim m→∞ 2a2m ) ) ( ( m Df a x1 , am x2 , . . . , am xn − Df − am x1 , −am x2 , . . . , −am xn

+

2am ) ) ( ( ( m φ a x1 , am x2 , . . . , am xn + φ − am x1 , −am x2 , . . . , −am xn ≤ lim m→∞ 2a2m ) ( )) ( m φ a x1 , am x2 , . . . , am xn + φ − am x1 , −am x2 , . . . , −am xn + 2|a|m =0 for all x1 , x2 , . . . , xn ∈ V \{0}, i.e., DF (x1 , x2 , . . . , xn ) = 0 for all x1 , x2 , . . . , xn ∈ V \{0}. Moreover, if we put m = 0 and let l → ∞ in (3.8), then we obtain the inequality (3.7). Notice that the equalities ( ) x Fe (x) 2 Fe (|a|x) = |a| Fe (x), Fe = , |a| |a|2 ( ) x Fo (x) = Fo (|a|x) = |a|Fo (x), Fo |a| |a|

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6 are true in view of (3.6). When |a| > 1, in view of Lemma 2.1, there exists a unique mapping F : V → Y satisfying the equalities in (3.6) and the inequality (3.7), since the inequality ∥f (x) − F (x)∥ ≤ ≤ ≤

∞ ( ∑ µ(ai x) + µ(−ai x) i=0 ∞ ∑ i=0 ∞ ∑ i=0

2a2i+2

µ(ai x) + µ(−ai x) + 2|a|i+1

)

ϕ(|a|i x) |a|i ϕ(k i x) ki

holds for all x ∈ V \{0}, where we set k := |a| and ϕ(x) := µ(x)+µ(−x) . + µ(x)+µ(−x) 2|a| 2a2 When |a| < 1, in view of Lemma 2.1, there exists a unique mapping F : V → Y satisfying the equalities in (3.6) and the inequality (3.7), since the inequality ∥f (x) − F (x)∥ ≤

∞ ( ∑ µ(ai x) + µ(−ai x)

2a2i+2

i=0 ∞ ∑

µ(ai x) + µ(−ai x) + 2|a|i+1

)

ϕ(|a|i x) |a|2i i=0 ( ) ∞ ∑ x 2i = k ϕ i k

≤

i=0

holds for all x ∈ V \{0}, where k :=

1 |a|

and ϕ(x) :=

µ(x)+µ(−x) 2a2

+

µ(x)+µ(−x) . 2|a|



The proof of the following theorem runs analogously to that of the previous theorem. Theorem 3.2 Let n be a fixed integer greater than 1, let µ : V \{0} → [0, ∞) be a function satisfying the condition  ∞ ( ) ∑  x i   |a| µ < ∞ when |a| < 1,  i  a  i=0 (3.9) ( ) ∞  ∑  x   a2i µ i < ∞ when |a| > 1   a i=0

for all x ∈ V \{0}, and let φ : (V \{0})n → [0, ∞) be a function satisfying the condition  ∞ ( ) ∑  x1 x2 xn i   |a| φ i , i , . . . , i < ∞ when |a| < 1,   a a a  i=0 (3.10) ( ) ∞  ∑  x x x  1 2 n  a2i φ i , i , . . . , i < ∞ when |a| > 1   a a a i=0

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7 for all x1 , x2 , . . . , xn ∈ V \{0}. If a mapping f : V → Y satisfies f (0) = 0, (3.3) for all x ∈ V \{0}, and (3.4) for all x1 , x2 , . . . , xn ∈ V \{0}, then there exists a unique mapping F : V → Y satisfying (3.5) for all x1 , x2 , . . . , xn ∈ V \{0} and the conditions in (3.6) for all x ∈ V , and such that ( ( ) ( )) ∞ ∑ a2i + |a|i x −x ∥f (x) − F (x)∥ ≤ µ i+1 + µ i+1 2 a a

(3.11)

i=0

for all x ∈ V \{0}. Proof. First, we define A := {f : V → Y | f (0) = 0} and a mapping Jm : A → A by ( ( ) ( )) ( ( ) ( )) x −x am x −x a2m f m +f + f m −f Jm f (x) := 2 a am 2 a am for all x ∈ V and m ∈ N0 . It follows from (3.3) that ∥Jm f (x) − Jm+l f (x)∥ ≤

m+l−1 ∑

∥Ji f (x) − Ji+1 f (x)∥ i=m

( )) m+l−1 ∑ a2i ( ( x ) −x

( ( ) ( )) x ai −x f i +f f i −f = + a ai 2 a ai i=m ( ( ( ( ) ( )) ) ( ))

x x a2i+2 −x ai+1 −x

f i+1 + f i+1 f i+1 − f i+1 − −

2 a a 2 a a

m+l−1 ∑ a2i ( ( x ) a2 + a ( x ) a2 − a ( −x ))

(3.12) =

2 f a ai+1 − 2 f ai+1 − 2 f ai+1 i=m ( ( ) ( ) ( )) a2i −x a2 + a −x a2 − a x + f a i+1 − f i+1 − f i+1 2 a 2 a 2 a ( ( ) ( ) ( ) i 2 2 a x a +a x a −a −x + f a i+1 − f i+1 − f i+1 2 a 2 a 2 a ( ( ) ( ) ( ))

ai −x a2 + a −x a2 − a x

f a i+1 − f i+1 − f i+1 −

2 a 2 a 2 a ( )) ( ( ) ( ))] m+l−1 ∑ [ a2i ( ( x ) −x |a|i x −x ≤ µ i+1 + µ i+1 + µ i+1 + µ i+1 2 a a 2 a a

2

i=m

for all x ∈ V \{0}. On account of (3.9) and (3.12), the sequence {Jm f (x)} is a Cauchy sequence for all x ∈ V \{0}. Since Y is complete and f (0) = 0, the sequence {Jm f (x)} converges for all x ∈ V . Hence, we can define a mapping F : V → Y by [ 2m ( ( ) ( )) ( ( ) ( ))] a x −x am x −x F (x) := lim f m +f + f m −f m→∞ 2 a am 2 a am for all x ∈ V . Moreover, if we put m = 0 and let l → ∞ in (3.12), we obtain the inequality (3.11).

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8 In view of the definition of F and (3.4), we get the equalities in (3.6) for all x ∈ V and ∥DF (x1 , x2 , . . . , xn )∥

2m ( ( ) ( ))

a x1 x2 xn −x1 −x2 −xn = lim Df , , . . . , + Df , , . . . , m→∞ 2 am am am am am am ( ( ) ( )) m

a x1 x2 xn −x1 −x2 −xn

+ Df m , m , . . . , m − Df , m ,..., m

m 2 a a a a a a [ 2m ( ( ) ( )) a x1 x2 −x1 −x2 xn −xn ≤ lim φ m, m,..., m + φ , m ,..., m m m→∞ 2 a a a a a a ( ( ) ( ))] |a|m x1 x2 xn −x1 −x2 −xn + φ m, m,..., m + φ , ,..., m 2 a a a am am a =0 for all x1 , x2 , . . . , xn ∈ V \{0}, i.e., DF (x1 , x2 , . . . , xn ) V \{0}. We notice that the equalities ( ) x 2 = Fe (|a|x) = |a| Fe (x), Fe |a| ( ) x = Fo (|a|x) = |a|Fo (x), Fo |a|

= 0 for all x1 , x2 , . . . , xn ∈ Fe (x) , |a|2 Fo (x) |a|

hold in view of (3.6). When |a| > 1, according to Lemma 2.1, there exists a unique mapping F : V → Y satisfying the equalities in (3.6) and the inequality (3.11), since the inequality ( ( ) ( )) ) ( ))] ∞ [ 2i ( ( ∑ a x x −x |a|i −x ∥f (x) − F (x)∥ ≤ µ i+1 + µ i+1 µ i+1 + µ i+1 + 2 a a 2 a a i=0 ) ( ∞ ∑ x ≤ |a|2i ϕ |a|i i=0 ( ) ∞ ∑ x 2i = k ϕ i k i=0

( ) ( ) holds for all x ∈ V \{0}, where k := |a| and ϕ(x) := µ xa + µ −x a . When |a| < 1, according to Lemma 2.1, there exists a unique mapping F : V → Y satisfying the equalities in (3.6) and the inequality (3.11), since the inequality ) ( )) ( ( ) ( ))] ∞ [ 2i ( ( ∑ a x −x |a|i x −x ∥f (x) − F (x)∥ ≤ + µ i+1 + µ i+1 µ i+1 + µ i+1 2 a a 2 a a i=0 ( ) ∞ ∑ x ≤ |a|i ϕ |a|i i=0

≤

∞ ∑ ϕ(k i x) i=0

ki

holds for all x ∈ V \{0}, where k :=

1 |a|

and ϕ(x) := µ

71

(x) a

( ) + µ −x a .



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9 Lemma 2.2 is necessary for the proof of the following main theorem. Theorem 3.3 Let n be a fixed integer greater than 1, let µ : V \{0} → [0, ∞) be a function satisfying the condition  ∞ ( ) ∞ ∑ ∑ µ(ai x)  x i   < ∞ and |a| µ i < ∞ when |a| > 1,  2i  a a  i=0 i=0 (3.13) ( ) ∞ ∞  ∑ ∑ µ(ai x)  x   < ∞ and a2i µ i < ∞ when |a| < 1   |a|i a i=0

i=0

for all x ∈ V \{0}, and let φ : (V \{0})n → [0, ∞) be a function satisfying the conditions  ∞ ( ) ∞ ∑ φ(ai x1 , ai x2 , . . . , ai xn ) ∑  x1 x2 xn  i  < ∞ and |a| φ i , i , . . . , i < ∞   a2i a a a   i=0 i=0      when |a| > 1, ( ) ∞ ∞  ∑ ∑  φ(ai x1 , ai x2 , . . . , ai xn ) x1 x2 xn  2i  < ∞ and a φ i , i ,..., i < ∞    |a|i a a a  i=0 i=0     when |a| < 1 (3.14) for all x1 , x2 , . . . , xn ∈ V \{0}. If a mapping f : V → Y satisfies f (0) = 0 and the inequality (3.3) for all x ∈ V \{0} and (3.4) for all x1 , x2 , . . . , xn ∈ V \{0}, then there exists a unique mapping F : V → Y satisfying the equality (3.5) for all x1 , x2 , . . . , xn ∈ V \{0}, the equalities in (3.6) for all x ∈ V , and  ∞ [ ( ))] ∑ µ(ai x) + µ(−ai x) |a|i ( ( x )  −x   + µ i+1 + µ i+1   2a2i+2 2 a a   i=0       when |a| > 1, ∥f (x) − F (x)∥ ≤ ( ( ) ( )) ] ∞ [   x −x µ(ai x) + µ(−ai x)  ∑ a2i  µ i+1 + µ i+1 +    a a 2|a|i+1  i=0 2      when |a| < 1 (3.15) for all x ∈ V \{0}. Proof. We will divide the proof of this theorem into two cases, one is for |a| > 1 and the other is for |a| < 1. Case 1. Assume that |a| > 1. We define a set A := {f : V → Y | f (0) = 0} and a mapping Jm : A → A by ( ( ) ( )) f (am x) + f (−am x) am x −x Jm f (x) := + f m −f 2a2m 2 a am

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10 for all x ∈ V and m ∈ N0 . It follows from (3.3) that ∥Jm f (x) − Jm+l f (x)∥ ≤

m+l−1 ∑

∥Ji f (x) − Ji+1 f (x)∥

i=m m+l−1 ∑

( ( ) ( ))

f (ai x) + f (−ai x) ai x −x

+ f i −f =

2a2i 2 a ai i=m ( ( ) ( ))

f (ai+1 x) + f (−ai+1 x) ai+1 x −x

− − f − f

2a2i+2 2 ai+1 ai+1

( ) m+l−1 ∑ a2 + a a2 − a i i

− 1 = f (a · a x) − f (a x) − f (−ai x) (3.16)

2a2i+2 2 2 i=m ( ) 1 a2 + a a2 − a i i i − 2i+2 f (−a · a x) − f (−a x) − f (a x) 2a 2 2 ( ( ( ( ) ) ) x −x ai x a2 + a a2 − a + f i+1 − f i+1 f a i+1 − 2 a 2 a 2 a ( ( ( ( ) ) )) i 2 2

a −x −x x a +a a −a

− f a i+1 − f i+1 − f i+1

2 a 2 a 2 a ( ))] m+l−1 ∑ [ µ(ai x) + µ(−ai x) |a|i ( ( x ) −x µ i+1 + µ i+1 + ≤ 2a2i+2 2 a a i=m

for all x ∈ V \{0}. In view of (3.13) and (3.16), the sequence {Jm f (x)} is a Cauchy sequence for all x ∈ V \{0}. Since Y is complete and f (0) = 0, the sequence {Jm f (x)} converges for all x ∈ V . Hence, we can define a mapping F : V → Y by ( ( ) ( ))] [ x −x f (am x) + f (−am x) am f m −f + F (x) := lim m→∞ 2a2m 2 a am for all x ∈ V . Moreover, if we put m = 0 and let l → ∞ in (3.16), we obtain the first inequality of (3.15). Using the definition of F , (3.4), and (3.14), we get the equalities in (3.6) for all x ∈ V and ∥DF (x1 , x2 , . . . , xn )∥ ( ) ( )

Df am x1 , am x2 , . . . , am xn + Df − am x1 , −am x2 , . . . , −am xn = lim m→∞ 2a2m ( ( ) ( )) m

a x1 x2 xn −x1 −x2 −xn

+ Df m , m , . . . , m − Df , m ,..., m

m 2 a a a a a a ) ( ) [ ( m φ a x1 , am x2 , . . . , am xn + φ − am x1 , −am x2 , . . . , −am xn ≤ lim m→∞ 2a2m ( ( ) ( ))] |a|m x1 x2 xn −x1 −x2 −xn + φ m, m,..., m + φ , ,..., m 2 a a a am am a =0

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11 for all x1 , x2 , . . . , xn ∈ V \{0}, i.e., DF (x1 , x2 , . . . , xn ) = 0 for all x1 , x2 , . . . , xn ∈ V \{0}. We notice that the equalities Fe (|a|x) = |a|2 Fe (x)

Fo (|a|x) = |a|Fo (x)

and

are true in view of (3.6). Using Lemma 2.2, we conclude that there exists a unique mapping F : V → Y satisfying the equalities in (3.6) and the first inequality in (3.15), since the inequality ∥f (x) − F (x)∥ ≤ ≤

∞ [ ∑ µ(ai x) + µ(−ai x) i=0 ∞ ( ∑ i=0

2a2i+2 ( x ψ(k i x) i + k ϕ k 2i ki

( ( ) ( ))] |a|i x −x µ i+1 + µ i+1 2 a a )) +

( )

µ

x a

+µ 2

(

−x a

)

holds for all x ∈ V \{0}, where k := |a|, ϕ(x) := , and ψ(x) := µ(x)+µ(−x) . 2a2 Case 2. We now consider the case of |a| < 1 and define a mapping Jm : A → A by ( ( ) ( )) a2m x −x f (am x) − f (−am x) Jm f (x) := f m +f + 2 a am 2am for all x ∈ V and n ∈ N0 . It follows from (3.3) that ∥Jm f (x) − Jm+l f (x)∥ ≤

m+l−1 ∑

∥Ji f (x) − Ji+1 f (x)∥ i=m

( )) m+l−1 ∑ a2i ( ( x ) −x

f (ai x) − f (−ai x) ai ai 2ai i=m

( ( ) ( )) a2i+2 x −x f (ai+1 x) − f (−ai+1 x)

f i+1 + f i+1 − −

2 a a 2ai+1

m+l−1 ∑ a2i ( ( x ) a2 + a ( x ) a2 − a ( −x ))

(3.17) =

2 f a ai+1 − 2 f ai+1 − 2 f ai+1 i=m ( ( ) ( ) ( )) a2i −x a2 + a −x a2 − a x + f a i+1 − f i+1 − f i+1 2 a 2 a 2 a ( ) 2 2 1 a +a a −a − i+1 f (a · ai x) − f (ai x) − f (−ai x) 2a 2 2 ( ) 2

1 a +a a2 − a i i i + i+1 f (−a · a x) − f (−a x) − f (a x)

2a 2 2 ( )) ] m+l−1 ∑ [ a2i ( ( x ) −x µ(ai x) + µ(−ai x) ≤ µ i+1 + µ i+1 + 2 a a 2|a|i+1 =

2

f

+f

+

i=m

for all x ∈ V \{0}.

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12 On account of (3.13) and (3.17), the sequence {Jm f (x)} is a Cauchy sequence for all x ∈ V \{0}. Since Y is complete and f (0) = 0, the sequence {Jm f (x)} converges for all x ∈ V . Hence, we can define a mapping F : V → Y by [ 2m ( ( ) ( )) ] a x −x f (am x) − f (−am x) F (x) := lim f m +f + m→∞ 2 a am 2am for all x ∈ V . Moreover, if we put m = 0 and let l → ∞ in (3.17), we obtain the second inequality in (3.15). By the definition of F , (3.4), and (3.14), we get the equalities in (3.6) for all x ∈ V and ∥DF (x1 , x2 , . . . , xn )∥

2m ( ( ) ( ))

a x x −x −x x −x 1 2 n 1 2 n Df m , m , . . . , m + Df , ,..., m = lim m→∞ 2 a a a am am a ( m ) ( ) m m m m Df a x1 , a x2 , . . . , a xn − Df − a x1 , −a x2 , . . . , −am xn

+

2am [ 2m ( ( ) ( )) a x1 x2 xn −xn −x1 −x2 ≤ lim φ m, m,..., m + φ , m ,..., m m m→∞ 2 a a a a a a )] ) ( ( m m m m m φ a x1 , a x2 , . . . , a xn + φ − a x1 , −a x2 , . . . , −am xn + 2|a|m =0 for all x1 , x2 , . . . , xn ∈ V \{0}, i.e., DF (x1 , x2 , . . . , xn ) = 0 for all x1 , x2 , . . . , xn ∈ V \{0}. We remark that the equalities ( ) ( ) x Fe (x) Fo (x) x Fe = = and Fo 2 |a| |a| |a| |a| hold by considering (3.6). Using Lemma 2.2, we conclude that there exists a unique mapping F : V → Y satisfying the equalities in (3.6) and the second inequality in (3.15), since the inequality ) ( )) ] ∞ [ 2i ( ( ∑ x a −x µ(ai x) + µ(−ai x) ∥f (x) − F (x)∥ ≤ µ i+1 + µ i+1 + 2 a a 2|a|i+1 i=0 ( ( ) ( ))] ∞ [ ∑ x −x µ(k i+1 x) + µ(−k i+1 x) k i+1 = + µ i +µ 2k 2i 2 k ki i=0 ( )) ∞ ( ∑ ψ(k i x) x i ≤ +k ϕ i 2i k k i=0

holds for all x ∈ V \{0}, where k :=

1 |a| ,

µ(kx)+µ(−kx) . 2

ϕ(x) :=

k 2

(

) µ(x) + µ(−x) , and ψ(x) := 

In the following corollary, we investigate the Hyers-Ulam-Rassias stability version of Theorems 3.1, 3.2, and 3.3.

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13 Corollary 3.4 Let X and Y be a real normed space and a real Banach space, respectively. Let p, θ, ξ be real constants such that p ̸∈ {1, 2}, a ̸∈ {−1, 0, 1}, ξ > 0, and θ > 0. If a mapping f : X → Y satisfies f (0) = 0 and

2 2

f (ax) − a + a f (x) − a − a f (−x) ≤ ξ∥x∥p (3.18)

2 2 for all x ∈ X\{0}, as well as if f satisfies the inequality ( ) ∥Df (x1 , x2 , . . . , xn )∥ ≤ θ ∥x1 ∥p + · · · + ∥xn ∥p

(3.19)

for all x1 , x2 , . . . , xn ∈ X\{0}, then there exists a unique mapping F : X → Y satisfying (3.5) for all x1 , x2 , . . . , xn ∈ X\{0}, and the equalities in (3.6) for all x ∈ X, as well as ∥f (x) − F (x)∥ ≤

ξ∥x∥p ξ∥x∥p + |a2 − |a|p | ||a| − |a|p |

(3.20)

for all x ∈ X\{0}. ( ) Proof. If we put φ(x1 , x2 , . . . , xn ) := θ ∥x1 ∥p + · · · + ∥xn ∥p for all x1 , x2 , . . . , xn ∈ X\{0}, then φ satisfies (3.2) when either |a| > 1 and p < 1 or |a| < 1 and p > 2, and φ satisfies (3.10) when either |a| > 1 and p > 2 or |a| < 1 and p < 1. Moreover, φ satisfies (3.14) when 1 < p < 2. Therefore, by Theorems 3.1, 3.2, and 3.3, there exists a unique mapping F : X → Y such that (3.5) holds for all x1 , x2 , . . . , xn ∈ X\{0}, and (3.6) holds for all x ∈ X, and such that (3.20) holds for all x ∈ X\{0}. 

4

Quadratic-additive type functional equations

In this section, let a be a rational constant such that a ̸∈ {−1, 0, 1}. Assume that the functional equation Df (x1 , x2 , . . . , xn ) = 0 is a quadratic-additive type functional equation. Then F : V → Y is a solution of the functional equation Df (x1 , x2 , . . . , xn ) = 0 if and only if F : V → Y is a quadratic-additive mapping. If F : V → Y is a quadratic-additive mapping, then Fe (x) and Fo (x) are a quadratic mapping and an additive mapping, respectively. Hence, Fe (ax) = a2 Fe (x) and Fo (ax) = aFo (x) for all x ∈ V , i.e., F satisfies the conditions in (3.6). Therefore, the following theorems are direct consequences of Theorems 3.1, 3.2, and 3.3.

Theorem 4.1 Let n be a fixed integer greater than 1, let µ : V → [0, ∞) be a function satisfying the condition (3.1) for all x ∈ V , and let φ : V n → [0, ∞) be a function satisfying the condition (3.2) for all x1 , x2 , . . . , xn ∈ V . If a mapping f : V → Y satisfies f (0) = 0, (3.3) for all x ∈ V , and (3.4) for all x1 , x2 , . . . , xn ∈ V , then there exists a unique quadratic-additive mapping F : V → Y such that (3.7) holds for all x ∈ V .

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14 Theorem 4.2 Let n be a fixed integer greater than 1, let µ : V → [0, ∞) be a function satisfying the condition (3.9) for all x ∈ V , and let φ : V n → [0, ∞) be a function satisfying the condition (3.10) for all x1 , x2 , . . . , xn ∈ V . If a mapping f : V → Y satisfies f (0) = 0, (3.3) for all x ∈ V , and (3.4) for all x1 , x2 , . . . , xn ∈ V , then there exists a unique quadratic-additive mapping F : V → Y such that (3.11) holds for all x ∈ V .

Theorem 4.3 Let n be a fixed integer greater than 1, let µ : V → [0, ∞) be a function satisfying the condition (3.13) for all x ∈ V , and let φ : V n → [0, ∞) be a function satisfying the condition (3.14) for all x1 , x2 , . . . , xn ∈ V . If a mapping f : V → Y satisfies f (0) = 0, (3.3) for all x ∈ V , and (3.4) for all x1 , x2 , . . . , xn ∈ V , then there exists a unique quadratic-additive mapping F : V → Y satisfying the inequality (3.15) for all x ∈ V .

Corollary 4.4 Let X and Y be a real normed space and a real Banach space, respectively. Let p, θ, ξ be real constants such that p ̸∈ {1, 2}, a ̸∈ {−1, 0, 1}, p > 0, ξ > 0, and θ > 0. If a mapping f : X → Y satisfies (3.18) for all x ∈ X and the inequality (3.19) for all x1 , x2 , . . . , xn ∈ X, then there exists a unique quadratic-additive mapping F : X → Y such that (3.20) holds for all x ∈ X.

Corollary 4.5 Let X and Y be a real normed space and a real Banach space, respectively. Let θ and ξ be real constants such that a ̸∈ {−1, 0, 1}, ξ > 0, and θ > 0. If a mapping f : X → Y satisfies f (0) = 0, and

2−a 2+a

a a

f (ax) − f (x) − f (−x)

≤ξ

2 2 for all x ∈ X, as well as if f satisfies the inequality ∥Df (x1 , x2 , . . . , xn )∥ ≤ θ for all x1 , x2 , . . . , xn ∈ X, then there exists a unique quadratic-additive mapping F : X → Y such that ∥f (x) − F (x)∥ ≤

ξ∥x∥p ξ∥x∥p + |a2 − 1| ||a| − 1|

for all x ∈ X.

Acknowledgment. This research was supported by Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education (No. 2013R1A1A2005557).

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15

References [1] N. Brillou¨et-Belluot, J. Brzd¸ek and K. Ciepli´ nski, On some recent developments in Ulam’s type stability, Abstr. Appl. Anal. 2012 (2012), Article ID 716936, 41 pages. [2] Y.-J. Cho, Th. M. Rassias and R. Saadati, Stability of Functional Equations in Random Normed Spaces, Springer Optimization and Its Applications Vol. 86, Springer, New York, 2013. [3] S. Czerwik, Functional Equations and Inequalities in Several Variables, World Scientific, Hackensacks, New Jersey, 2002. [4] P. G˘avrut¸a, A generalization of the Hyers-Ulam-Rassias stability of approximately additive mappings, J. Math. Anal. Appl. 184 (1994), 431–436. [5] P. G˘avrut¸a, On a problem of G. Isac and Th. M. Rassias concerning the stability of mappings, J. Math. Anal. Appl. 261 (2001), 543–553. [6] D. H. Hyers, On the stability of the linear functional equation, Proc. Natl. Acad. Sci. USA 27 (1941), 222–224. [7] D. H. Hyers, G. Isac and Th. M. Rassias, Stability of Functional Equations in Several Variables, Birkh¨auser, Boston, 1998. [8] S.-M. Jung, Hyers-Ulam-Rassias Stability of Functional Equations in Nonlinear Analysis, Springer Optimization and Its Applications Vol. 48, Springer, New York, 2011. [9] Y.-H. Lee, On the quadratic additive type functional equations, Int. J. Math. Anal. (Ruse) 7 (2013), 1935–1948. [10] Y.-H. Lee and S.-M. Jung, A general uniqueness theorem concerning the stability of additive and quadratic functional equations, J. Funct. Spaces 2015 (2015), Article ID 643969, 8 pages. [11] Y.-H. Lee and S.-M. Jung, On the stability of a mixed type functional equation, Kyungpook Math. J. 55 (2015), 91–101. [12] Y.-H. Lee and S.-M. Jung, Generalized Hyers-Ulam stability of a 3-dimensional quadratic-additive type functional equation, Int. J. Math. Anal. (Ruse) 11 (2015), 527–540. [13] Y.-H. Lee, S.-M. Jung and M. Th. Rassias, On an n-dimensional mixed type additive and quadratic functional equation, Appl. Math. Comput. 228 (2014), 13–16. [14] Th. M. Rassias, On the stability of the linear mapping in Banach spaces, Proc. Amer. Math. Soc. 72 (1978), 297–300. [15] S. M. Ulam, A Collection of Mathematical Problems, Interscience Publ., New York, 1960.

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A Dynamic Programming Approach to Subsistence Consumption Constraints on Optimal Consumption and Portfolio Ho-Seok Lee∗

Yong Hyun Shin†

We investigate an optimal consumption and portfolio selection problem of an infinitely-lived economic agent with a constant relative risk aversion (CRRA) utility function who faces subsistence consumption constraints. We provide the closed form solutions for the optimal consumption and investment policies by using the dynamic programming method and compare the solutions with those obtained by the martingale method. We show that they coincide with each other. Comparison of optimal policies with and without subsistence consumption constraints shows that the constraints have effect on the optimal consumption and portfolio policies even when the constraints do not bind. ∗

Department of Mathematics, Kwangwoon University, 20, Kwangwoon-ro, Nowon-gu,

Seoul 01897, Republic of Korea, e-mail: [email protected] † Department of Mathematics, Sookmyung Women’s University, Seoul 04310, Republic of Korea, Corresponding author. e-mail: [email protected], Tel.: +82-2-20777682, FAX: +82-2-2077-7323

1

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Keywords : Consumption, portfolio selection, subsistence consumption constraints, dynamic programming method, CRRA utility.

1

Introduction

Following the seminal contributions of Merton [6, 7] on continuous-time optimal consumption and portfolio selection problems, there have been a number of research works on the optimization problems under various economic constraints. One of the most interesting topics is optimal consumption and portfolio selection with subsistence consumption constraints (see [1, 4, 5, 8, 10, 11, 12]). Subsistence consumption constraints mean that there exists a positive minimum consumption level (that can be a constant or a deterministic/stochastic process) such that the agent can live with. We consider the optimal consumption and investment problem with subsistence consumption constraints and a constant relative risk aversion (CRRA) utility function. We derive the optimal solutions in closed form by using the dynamic programming approach based on Karatzas et al. [2]. We also compare the solutions with those of Shin et al. [11] by using the martingale duality approach for the same optimization problem. We show that they agree with each other. Besides the methodological contribution through the dynamic programming method, we quantitatively compare our results to those of the agent without subsistence consumption constraints. The comparison shows that the existence of the subsistence consumption constraints affects the optimal consumption and portfolio policies even when the constraints do not bind.

2

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The prospect that the subsistence consumption constraints become binding later compels the agent to consume less and to invest in the risky asset more conservatively. The rest of this paper is organized as follows. The financial market is introduced in Section 2. In Section 3 the optimal consumption and investment problem is considered with subsistence consumption constraints. Section 4 demonstrates the impact of the subsistence consumption constraints on the optimal policies. Section 5 summarizes the paper.

2

The Economy

In a financial market, we assume that an economic agent has investment opportunities given by a riskless asset with a constant rate of return r > 0 and one risky asset St which follows a geometric Brownian motion with a constant mean rate of return µ and a constant volatility σ, dSt /St = µdt + σdBt , where Bt is a standard Brownian motion on a probability space (Ω, F, P) and {Ft }t≥0 is the P-augmentation of the filtration generated by the standard Brownian motion {Bt }t≥0 . A portfolio process π := {πt }t≥0 meaning amounts of money invested in the risky asset at time t is a measurable process adapted to {Ft }t≥0 and satisfies t

Z

πs2 ds < ∞, for all t ≥ 0 a.s.

(1)

0

A consumption process c := {ct }t≥0 is a measurable nonnegative process adapted to {Ft }t≥0 and satisfies Z t cs ds < ∞, for all t ≥ 0 a.s. 0

3

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Then, with a given initial endowment X0 = x > 0, the agent’s wealth process Xt at time t evolves according to dXt = [rXt + πt (µ − r) − ct ] dt + πt σdBt .

3

(2)

The Optimization Problem

Now we investigate the agent’s optimization problem with subsistence consumption constraints. Given a positive subsistence level of consumption R > 0, the agent’s problem is to maximize the total expected discounted utility from consumption with the constraint ct ≥ R, for all t ≥ 0.

(3)

In this paper, we assume that the utility function u(·) is of the CRRA type u(c) :=

c1−γ , γ > 0 (γ 6= 1), 1−γ

where γ is the agent’s coefficient of relative risk aversion. A pair (c, π) of the optimal consumption/investment processes is called admissible at initial capital x > 0, if the wealth process Xt in (2) is strictly positive and it satisfies the constraint (3). Let A(x) denote the set of all admissible consumption/investment pair at x > 0. Then, the agent’s optimization problem is given by V (x) :=

max

(c,π )∈A(x)

J(x; c, π),

(4)

where "Z

∞

J(x; c, π) := E 0

# 1−γ c e−ρt t dt , 1−γ

(5)

4

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subject to the budget constraint (2) and the subsistence consumption constraint (3). Here ρ > 0 is the subjective discount factor. In addition, we should impose a lower bound on initial wealth x as follows: x>

R r

such that a pair (c, π) corresponding to the wealth dynamics (2) should be admissible (see Lemma 3.1 of Gong and Li [1]). By the dynamic programming principle, the value function V (x) in the optimization problem (4) satisfies the following Bellman equation   1 2 2 00 c1−γ 0 max {rx + π(µ − r) − c} V (x) + σ π V (x) − ρV (x) + = 0. c≥R,π 2 1−γ (6) We assume that the wealth process Xt satisfies a transversality condition lim e−ρt V (Xt ) = 0,

t→∞

(7)

if V (·) is the solution to the Bellman equation (6). The first order conditions (FOCs) of the Bellman equation (6) for the optimal consumption/portfolio (c∗ , π ∗ ) imply
− 1 c∗ = (V 0 (x) γ and π∗ = −

µ − r V 0 (x) . σ 2 V 00 (x)

(8)

The subsistence consumption constraint (3) forces us to impose a threshold wealth level x e > 0 such that   R, c∗ =  (V 0 (x))− γ1 ,

for R/r < x < x e,

(9)

for x ≥ x e. 5

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Substituting the FOCs (8) and (9) into the equation (6) yields 1 (V 0 (x))2 R1−γ (rx − R)V 0 (x) − θ2 00 − ρV (x) + = 0, for R/r < x < x e (10) 2 V (x) 1−γ and 1 (V 0 (x))2 γ − 1−γ e, (11) rxV 0 (x) − θ2 00 − ρV (x) + V 0 (x) γ = 0, for x ≥ x 2 V (x) 1−γ where θ := (µ − r)/σ is the market price of risk. Moreover, we define a Merton constant K such that K := r +

ρ−r γ−1 2 + θ γ 2γ 2

(12)

and assume that K > 0 to guarantee the well-definedness of the optimization problem (4). Lemma 3.1. The value function V(x) in (4) is strictly concave and strictly increasing for x > R/r. Proof. The proof follows a similar line to that of Proposition 2.1 in Zariphopoulou [14]. Remark 3.1. For later use, we define two quadratic algebraic equations as follows:   1 2 f (m) := rm − ρ + r + θ m + ρ = 0 2

(13)

  1 2 2 1 2 g(n) := θ n + ρ − r + θ n − r = 0. 2 2

(14)

2

and

f (m) = 0 has two real roots m1 and m2 satisfying m1 > 1 > m2 > 0 and g(n) = 0 has two real roots n1 and n2 satisfying n1 > 0 and n2 < −1. Also

6

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we have the following relationships between roots of two quadratic equations (13) and (14): n1 =

1 1 , n2 = . m1 − 1 m2 − 1

(15)

Theorem 3.1. Assume that a strictly increasing and strictly concave function v(·) such that v(·) ∈ C 2 (R/r, ∞) solves the Bellman equation (6) for x > R/r. Then v(x) ≥ J(x; c, π) for all admissible pair (c, π). If (c∗t , πt∗ ) is the maximizer of the Bellman equation (6), then we derive v(x) = V (x) =

max J(x; c, π) = J(x; c∗ , π ∗ ). (c,π )∈A(x)

Proof. Let us define a function U (·, ·) as follows: U (t, Xt ) := e−ρt v(Xt ).

(16)

The Itˆo’s formula implies dU (t, Xt ) = e

−ρt



 1 2 2 00 {rXt + πt (µ − r) − ct } v (Xt ) + σ πt v (Xt ) − ρv(Xt ) dt + e−ρt σπt v 0 (Xt )dBt 2

≤ −e−ρt

0

c1−γ t dt + e−ρt σπt v 0 (Xt )dBt 1−γ

(17)

for any admissible pair (ct , πt ) of consumption/portfolio processes. For any t ≥ 0, we obtain Z v(X0 ) ≥

t

e 0

−ρs

c1−γ s ds + e−ρt v(Xt ) − 1−γ

Z

t

e−ρs σπs v 0 (Xs )dBs .

(18)

0

From (1), the second integral of the right-hand side of (18) is a bounded local martingale and hence a martingale, so we have "Z # 1−γ t c s v(x) ≥ E e−ρs ds + e−ρt v(Xt ) . 1−γ 0

(19)

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Letting t ↑ ∞ and using the monotone convergence theorem, the Lebesgue dominated convergence theorem and the transversality condition in (7), we derive "Z

∞

v(x) ≥ E 0

# 1−γ c s ds = J(x; c, π). e−ρs 1−γ

(20)

If (ct , πt ) is the maximizer of the Bellman equation (6), the inequality in (20) becomes the equality and consequently we obtain v(x) = V (x). Theorem 3.2. The value function V (x) of the optimization problem (4) is given by 
 C2 x − R m2 + R1−γ , r ρ(1−γ) V (x) =  r− 21 θ2 n1 D ξ −γ(n1 +1) + where

 D1 =

K(1−γ) ,

1

ρ

m2 −1 γ

+1

for R/r < x < x e, ξ 1−γ



1 K

−

(m2 − 1)n1 − 1

1 r

(21)

for x ≥ x e,

Rγn1 +1 , x e = D1 R−γn1 +

R K

(22)

and 1 C2 = m2

  R 1−m2 −γ x e− R . r

For x ≥ x e, ξ is determined from the following algebraic equation x = D1 ξ −γn1 +

ξ . K

Proof. For R/r < x < x e, trying a homogeneous solution of the form x −


R m r

to the equation (10), then we obtain the algebraic equation f (m) = 0 in (13). Thus we can find the homogeneous solution Ve (x) to the equation (10) as follows:     R m1 R m2 e V (x) = C1 x − + C2 x − , r r 8

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for some constants C1 and C2 . The particular solution

R1−γ ρ(1−γ)

to the equation

(10) can be easily derived. Thus V (x) is given by     R m1 R m2 R1−γ R1−γ e V (x) = V (x) + = C1 x − + C2 x − + . ρ(1 − γ) r r ρ(1 − γ) If C1 = 0 and C2 > 0, then V (x) is a concave function. Thus in order to guarantee the existence of the well-defined value function V (x) we set C1 = 0 and we will prove that C2 > 0 in Proposition 3.1 later. Therefore V (x) is given by   R m2 R1−γ V (x) = C2 x − + . r ρ(1 − γ)

(23)

For x ≥ x e, we set the optimal consumption c = C(x) and X(·) = C −1 (·), that is, X(c) = X(C(x)) = x. Then, from the FOCs (9), we obtain V 0 (x) = C(x)−γ , V 00 (x) = −γ

C(x)−γ−1 . X 0 (c)

(24)

Plugging the conditions (24) into the equation (11), we have rc−γ X(c) +

1 2 1−γ 0 γ 1−γ θ c X (c) − ρV (X(c)) + c = 0. 2γ 1−γ

Taking the derivative of (25) with respect to c implies   1 2 2 00 1−γ 2 θ c X (c) + r − ρ + θ cX 0 (c) − rγX(c) + γc = 0. 2γ 2γ

(25)

(26)

Trying a homogeneous solution of the form c−γn to the equation (26), then we obtain the algebraic equation g(n) = 0. Thus the homogeneous solution e X(c) is given by e X(c) = D1 c−γn1 + D2 c−γn2 , for some constants D1 and D2 . The particular solution

c K

to the equation

(26) can be easily derived. Thus X(c) is given by e X(c) = X(c) +

c c = D1 c−γn1 + D2 c−γn2 + . K K 9

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Now we should discard the rapidly growing term by setting D2 = 0. Therefore X(c) is given by X(c) = D1 c−γn1 +

c . K

(27)

We will prove that X 0 (c) > 0 in Proposition 3.1 later. Thus, from (24), we obtain V 00 (x) = −γ

C(x)−γ−1 0 and x e > R/r. Also C2 > 0 as promised before. Proof. From (29) and (34) we have    m2 −1 1 1 + 1 γ K − r 1 x e= + R (m2 − 1)n1 − 1 K   (m2 − 1) γ1 + n1 K1 − 1r = R. (m2 − 1)n1 − 1 Thus x e is a linear function of R and is an increasing function with respect to R since (m2 − 1)



1 γ

+ n1



1 K

(m2 − 1)n1 − 1

−

1 r

> 0,

because of m2 − 1 < 0. Now we use the Merton constant K in (12) and the quadratic equation (14) to obtain the inequality 2 n1 (ρ − r) + n1 γ−1 γn1 γn1 1 γn1 K − rγn1 − r 2γ θ − r − − = = r K K rK
rK ρ − r + 12 θ2 n1 − n2γ1 θ2 − r − 12 θ2 n21 − n2γ1 θ2 = = < 0. rK rK

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Thus we have 

0

X (R) = −γn1 D1 R =

−γn1 −1

m2 −1 γ



1 1 K − r 1 1 + = −γn1 + K (m2 − 1)n1 − 1 K

+1

γn1 r

1 1 − γn K − K > 0. (m2 − 1)n1 − 1

From the fact c > R, we have  γn1 +1 R 1> c

(36)

1 1 > K K

and

 γn1 +1 R . c

(37)

Thus we have  0

X (c) = −γn1  > −γn1

m2 −1 γ

+1



1 K

−

1 r

(m2 − 1)n1 − 1  m2 −1 1 1 + 1 γ K − r

(m2 − 1)n1 − 1  γn1 +1 R X 0 (R) = c

 γn1 +1 R + c  γn1 +1 R + c

1 K 1 K

 γn1 +1 R c

> 0, where the first inequality is obtained from (37) and the second inequality is obtained from (36). Consequently, from (33), we see that x e > R/r and C2 > 0 from (35). Remark 3.2. For R/r < x < x e, V 00 (x) has a lower bound. From Proposition 3.1 and (24), V 00 (x) has a lower bound for x ˜ ≤ x. From Lemma 3.1, V 0 (x) is bounded away from zero. Hence, π ∗ in (8) is bounded away from zero and the Bellman equation (6) is uniformly elliptic. Therefore the solution in Theorem 3.2 is the unique solution to the Bellman equation (6) by Krylov [3]. Vila and Zariphopoulou [13] provided an alternative proof by a similar argument. 12

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Now we will describe the related results of Shin et al. [11] in the following remark. They also pay their attention to the optimal consumption and portfolio selection problem with a subsistence consumption constraint, but they use the martingale method with Lagrangian duality to derive their solutions. Remark 3.3. With the notations in this paper, the value function V S (x) and the threshold wealth level x eS based on Section 4 of Shin et al. [11] are given as follows:     p2  1  R R p −1 p −1   d2 r −x 2 + x − R
r −x 2 + d2 p2 r d2 p2 S V (x) =  1−γ   c1 (λ∗ )p1 + γ (λ∗ )− γ + (λ∗ ) x, K(1−γ)

R1−γ ρ(1−γ) ,

for R/r < x < x eS , for x ≥ x eS (38)

and x eS = −c1 p1 R−γ(p1 −1) + where c1 =

1 K

γp2 1−γ

 +1 +

p2 −1 r

−

p2 ρ(1−γ)

p1 − p2

and d2 =



R , K

1 K



γp1 1−γ

 +1 +

p1 −1 r

−

p1 − p2

p1 ρ(1−γ)

R1−γ+γp1

(39)

R1−γ+γp2 .

(40)

p1 > 1 and p2 < 0 are two real roots of the following quadratic algebraic equation   1 2 2 1 2 h(p) := θ p + ρ − r − θ p − ρ = 0, 2 2

(41)

and λ∗ is determined by the following algebraic equation x = −c1 p1 (λ∗ )p1 −1 +

1 ∗ − γ1 (λ ) . K

(42)

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Lemma 3.2. 1

m2 C2 = (−d2 p2 ) 1−p2

and

D1 = −c1 p1 .

(43)

Proof. From (29), (34) and (15), we have   m2 −1 1 1 + 1 γ K − r R R −γn1 x e = D1 R + = R+ K (m2 − 1)n1 − 1 K   n2 1 1 + n 2 K − r γ R = R+ . n1 − n2 K It can be easily shown that p1 = n1 + 1, p2 = n2 + 1.

(44)

Thus we obtain  x e=

1 γ

 + p2 − 1 K1 − p1 − p2

p2 −1 r

R+

R . K

From (35), we have   R 1−m2 −γ m2 C2 = x e− R r !1−m2 R1+γ(p2 −1) = x eR − r   1−m2   1−p1 1 1   γ + p1 − 1 K + r  R1+γ(p2 −1) =    p1 − p2 γ(p2 −1)

= (−d2 p2 )1−m2 , where the last equality is obtained from the following relationships between roots and coefficients of the quadratic equation h(p) = 0 in (41) p1 + p2 =

θ2 − 2ρ + 2r 2ρ , p1 p2 = − 2 2 θ θ

(45)

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and (40). Therefore we obtain 1

m2 C2 = (−d2 p2 ) 1−p2 . From (34), we have  D1 =

=

m2 −1 γ

+1



1 K

−

1 r

(m2 − 1)n1 − 1   1 1 + p − 1 2 γ K − p1 − p2

Rγn1 +1 p2 −1 r

R1+γ(p1 −1)

(46)

= −c1 p1 , where the last equality is also obtained from the relationships (45) and (39). Corollary 3.1. D1 in (22) is positive. Proof. For p2 < x < p1 , we define a decreasing function F (x) as follows: h(x) 1 = − θ2 (x − p1 ) > 0. x − p2 2   γ−1 1 1 < Since 0 < F (1) < F , we have  and γ−1 γ F (1) F γ F (x) := −



1 + p2 − 1 γ



1 p2 − 1 − >0 K r

(see also Shim and Shin [9]). From (46), we have D1 > 0. Proposition 3.2. The value function V (x) and the threshold wealth level x e in our optimization problem coincide with V S (x) and x eS of Shin et al. [11], respectively.

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Proof. From (43) and (44), we can easily show that x e=x eS . For R/r < x < x e, we have d2

R r

−x d2 p2

!

p2 p2 −1

  R + x− r

R r

−x d2 p2

!

1 p2 −1

  p2 p2 −1 1 R p2 − 1 = (−d2 p2 ) 1−p2 x − p2 r p   2 R p2 −1 p2 − 1 = m2 C2 x − p2 r  m2 R = C2 x − , r

where the second equality is obtained from (43) and the third equality is obtained from (15) and (44). This equality means V (x) = V S (x) for R/r < x 0. If we let R → 0 to the consumption and portfolio pair (c∗ , π ∗ ) M in Theorem 3.3, we also arrive at (cM t , πt ). Due to the subsistence con-

sumption constraints, it is natural to consider the myopic strategies defined by cmyopic := max{R, cM t }. t But the myopic strategies are not optimal and the existence of the subsistence consumption constraints affect the consumption and portfolio policies even at the wealth level where the subsistence consumption constraints do not bind. This is because it is possible that the constraints will become binding later. The following proposition demonstrates quantitatively the impact of the subsistence consumption constraints on the consumption and portfolio policies when the constraints are not binding. ∗ M Proposition 4.1. For Xt ≥ x ˜, c∗t < cM t and πt < πt .

Proof. From (47) and (48), the optimal wealth process is given by 1 Xt = D1 c∗−γn + t

Since D1 > 0 and X(c) := D1 c−γn1 +

c∗t cM = t . K K

c is an increasing function from K

Proposition 3.1, we obtain c∗t < cM t = KXt .

(50)

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Also we derive πt∗

θ = σγ

  c∗t θ c∗t θ ∗−γn1 −γn1 D1 ct + < < Xt = πtM , K σγ K σγ

where the first inequality follows from D1 > 0 and the second one from (50).

5

Concluding Remarks

In this paper we study an optimal consumption and investment problem with subsistence consumption constraints. We use the dynamic programming method to derive the closed form solutions with a CRRA utility function. We also compare our solutions with those of Shin et al. [11] derived by the martingale approach. We show that they coincide with each other. In addition, we point out that the optimal consumption and portfolio policies may alter even when the constraints do not bind. This is attributed to the prospect that the subsistence consumption constraints become binding later. In this case, the agent consume less and invest in the risky asset more conservatively.

References [1] N. Gong and T. Li, Role of index bonds in an optimal dynamic asset allocation model with real subsistence consumption, Appl. Math. Comput., 174, 710–731 (2006).

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[2] I. Karatzas, J.P. Lehoczky, S.P. Sethi, and S.E. Shreve, Explicit solution of a general consumption/investment problem, Math. Oper. Res., 11, 261–294 (1986). [3] N.V. Krylov, Nonlinear elliptic and parabolic equations of the second order, Reidel, Dordrecht, 1987. [4] P. Lakner and L.M. Nygren, Portfolio optimization with downside constraints, Math. Finance, 16, 283–299 (2006). [5] B.H. Lim, Y.H. Shin, and U.J. Choi, Optimal investment, consumption and retirement choice problem with disutility and subsistence consumption constraints, J. Math. Anal. Appl., 345, 109–122 (2008). [6] R.C. Merton, Lifetime portfolio selection under uncertainty:

The

continuous-time case, Rev. Econom. Stat., 51, 247–257 (1969). [7] R.C. Merton, Optimum consumption and portfolio rules in a continuous-time model, J. Econom. Theory, 3, 373–413 (1971). [8] G. Shim and Y.H. Shin, Portfolio selection with subsistence consumption constraints and CARA utility, Math. Probl. Eng., 2014, Article ID 153793, 6 pages (2014). [9] G. Shim and Y.H. Shin, An optimal job, consumption/leisure, and investment policy, Oper. Res. Lett., 42, 145–149 (2014). [10] Y.H. Shin and B.H. Lim, Comparison of optimal portfolios with and without subsistence consumption constraints, Nonlinear Anal. TMA, 74, 50–58 (2011). 20

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[11] Y.H. Shin, B.H. Lim, and U.J. Choi, Optimal consumption and portfolio selection problem with downside consumption constraints, Appl. Math. Comput., 188, 1801–1811 (2007). [12] H. Yuan and Y. Hu, Optimal consumption and portfolio policies with the consumption habit constraints and the terminal wealth downside constraints, Insurance Math. Econom., 45, 405–409 (2009). [13] J.-L. Vila and T. Zariphopoulou, Optimal consumption and portfolio choice with borrowing constraints, J. Econom. Theory, 77, 402–431 (1997). [14] T. Zariphopoulou, Consumptioninvestment models with constraints, SIAM J. Control Optim., 32, 59–85 (1994).

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THE STABILITY OF CUBIC FUNCTIONAL EQUATION WITH INVOLUTION IN NON-ARCHIMEDEAN SPACES CHANG IL KIM AND CHANG HYEOB SHIN*

Abstract. In this paper, using fixed point method, we prove the Hyers-Ulam stability of the following functional equation f (2x + y) + f (2x + σ(y)) − 2f (x + y) − 2f (x + σ(y)) − 12f (x) = 0 with involution.

1. Introduction and Preliminaries In 1940, Ulam [18] proposed the following problem concerning the stability of group homomorphism: Let G1 be a group and let G2 a meric group with the metric d(·, ·). Given  > 0, does there exist a δ > 0 such that if a mapping h : G1 −→ G2 satisfies the inequality d(h(xy), h(x)h(y)) < δ for all x, y ∈ G1 , then there exists a homomorphism H : G1 −→ G2 with d(h(x), H(x)) <  for all x ∈ G1 ? Hyers [7] solved the Ulam’s problem for the case of approximately additive functions in Banach spaces. Since then, the stability of several functional equations have been extensively investigated by several mathematicians [2, 3, 5, 8, 9, 13, 14, 15, 16]. Jun and Kim [11] introduced the following functional equation (1.1)

f (2x + y) + f (2x − y) = 2f (x + y) + 2f (x − y) + 12f (x)

and they established the general solution and generalized Hyers-Ulam-Rassias stability problem for this functional equation. It is easy to see that the function f (x) = cx3 is a solution of the functional equation (1.1). Thus, it is natural that (1.1) is called a cubic functional equation and every solution of the cubic functional equation is said to be a cubic function. Let X and Y be real vector spaces. For an additive mapping σ : X −→ X with σ(σ(x)) = x for all x ∈ X, then σ is called an involution of X [1, 17]. Stetkær [17] introduced the following quadratic functional equation with involution (1.2)

f (x + y) + f (x + σ(y)) = 2f (x) + 2f (σ(y))

and solved the general solution, Belaid et al. [1] established generalized Hyers-Ulam stability in Banach space for this functional equation. Jung and Lee [12] investigated the Hyers-Ulam-Rassias stability of (1.2) in a complete β-normed space, using fixed point method. For a given involution σ : X −→ X, the functional equation (1.3)

f (2x + y) + f (2x + σ(y)) = 2f (x + y) + 2f (x + σ(y)) + 12f (x)

for all x, y ∈ X is called the cubic functional equation with involution and a solution of (1.3) is called a cubic mapping with involution. In this paper, using fixed point method, we prove the generalized Hyers-Ulam stability of the following functional equation (1.4)

f (2x + y) + f (2x + σ(y)) − 2f (x + y) − 2f (x + σ(y)) − 12f (x) = 0.

2010 Mathematics Subject Classification. 39B82, 39B52. Key words and phrases. cubic functional equation, involution, fixed point method, non-Archimedean space. *Corresponding Author. 1

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A valuation is a function | · | from a field K into [0, ∞) such that for any r, s ∈ K, the following conditions hold: (i) |r| = 0 if and only if r = 0, (ii) |rs| = |r||s|, and (iii) |r + s| ≤ |r| + |s|. A field K is called a valued field if K carries a valuation. The usual absolute values of R and C are examples of valuations. If the triangle inequality is replaced by |r + s| ≤ max{|r|, |s|} for all r, s ∈ K, then the valuation | · | is called a non-Archimedean valuation and the field with a non-Archimedean valuation is called non-Archimedean field. If | · | is a non-Archimedean valuation on K, then clearly, |1| = | − 1| and |n| ≤ 1 for all n ∈ N. Definition 1.1. Let X be a vector space over a scalar field K with a non-Archimedean nontrivial valuation | · |. A function k · k : X −→ R is called a non-Archimedean norm if satisfies the following conditions: (a) kxk = 0 if and only if x = 0, (b) krxk = |r|kxk, and (c) the strong triangle inequality (ultrametric) holds, that is, ||x + y|| ≤ max{kxk, kyk} for all x, y ∈ X and all r ∈ K. If k · k is a non-Archimedean norm, then (X, k · k) is called a non-Archimedean normed space. Let (X, k · k) be a non-Archimedean normed space. Let {xn } be a sequence in X. Then {xn } is said to be convergent if there exists x ∈ X such that limn−→∞ kxn − xk = 0. In that case, x is called the limit of the sequence {xn }, and one denotes it by limn−→∞ xn = x. A sequence {xn } is said to be a Cauchy sequence if limn−→∞ kxn+p − xn k = 0 for all p ∈ N. Since kxn − xm k ≤ max{kxj+1 − xj k | m ≤ j ≤ n − 1} (n > m), a sequence {xn } is Cauchy in (X, k · k) if and only if {xn+1 − xn } converges to zero in (X, k · k). By a complete non-Archimedean space we mean one in which every Cauchy sequence is convergent. In 1897, Hensel [6] discovered the p-adic numbers as a number theoretical analogue of power series in complex analysis. Fix a prime number p. For any nonzero rational number x, there exists a unique integer nx ∈ Z such that x = ab pnx , where a and b are integers not divisible by p. Then |x|p := p−nx defines a non-Archimedean norm on Q. The completion of Q with respect to the metric d(x, y) = |x − y|p is denoted P by Qp , which is called the p-adic number field. In ∞ k fact, Qp is the set of all formal series x = k≥nx ak p , where |ak | ≤ p − 1 are integers. The and multiplication between any two elements of Qp are defined naturally. The norm addition P∞ ak pk = p−nx is a non-Archimedean norm on Qp and it makes Qp a locally compact field. k≥nx p Let (X, d) be a generalized metric space. An operator T : X −→ X satisfies a Lipschitz condition with Lipschitz constant L if there exists a constant L ≥ 0 such that d(T x, T y) ≤ Ld(x, y) for all x, y ∈ X. If the Lipschitz constant L is less than 1, then the operator T is called a strictly contractive operator. Note that the distinction between the generalized metric and the usual metric is that the range of the former is permitted to include the infinity. Theorem 1.2. [4] Let (X, d) be a complete generalized metric space and let J : X −→ X be a strictly contractive mapping with some Lipschitz constant L with 0 < L < 1. Then for each given element x ∈ X, either d(J n x, J n+1 x) = ∞ for all nonnegative integers n or there exists a positive integer n0 such that (1) d(J n x, J n+1 x) < ∞ for all n ≥ n0 ; (2) the sequence {J n x} converges to a fixed point x∗ of J ; (3) x∗ is the unique fixed point of J in the set Y = {y ∈ X | d(J n0 x, y) < ∞} and 1 (4) d(y, y ∗ ) ≤ d(y, Jy) for all y ∈ Y . 1−L In 1996, Issac and Rassias [10] were the first to provide applications of stability theory of functional equations for the proof of new fixed point theorem with applications. By using fixed point methods, the stability problems of several functional equations have been extensively investigated by a number of authors.

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Throughout this paper, we assume that X is a non-Archimedean normed space and Y is a complete non-Archimedean normed space. 2. The generalized Hyers-Ulam stability for (1.4) Using the fixed point methods, we will prove the generalized Hyers-Ulam stability of the cubic functional equation (1.4) with involution σ in non-Archimedean normed spaces. For a given mapping f : X −→ Y , we define the difference operator Df : X 2 −→ Y by Df (x, y) = f (2x + y) + f (2x + σ(y)) − 2f (x + y) − 2f (x + σ(y)) − 12f (x) for all x, y ∈ X. Theorem 2.1. Assume that φ : X 2 −→ [0, ∞) is a mapping and there exists a real number L with 0 < L < 1 such that (2.1)

φ(2x, 2y) ≤ |8|Lφ(x, y), φ(x + σ(x), y + σ(y)) ≤ |8|Lφ(x, y)

for all x, y ∈ X. Let f : X −→ Y be a mapping such that f (0) = 0 and kDf (x, y)k ≤ φ(x, y)

(2.2)

for all x, y ∈ X. Then there exists a unique cubic mapping C : X −→ Y with involution such that kf (x) − C(x)k ≤

(2.3)

1 Φ(x) |2|4 (1 − L)

for all x ∈ X, where Φ(x) = max{φ(x, 0), φ(0, x)}. Proof. Consider the set S = {g | g : X −→ Y } and the generalized metric d in S defined by d(g, h) = inf {c ∈ [0, ∞)| kg(x) − h(x)k ≤ c Φ(x) for all x ∈ X}. Then (S, d) is a complete metric space(See [12]). Define a mapping J : S −→ S by Jg(x) =

1 {g(2x) + g(x + σ(x))} 8

for all x ∈ X and all g ∈ S. Let g, h ∈ S and d(g, h) ≤ c for some non-negative real number c. Then by (2.1), we have 1 kg(2x) + g(x + σ(x)) − h(2x) − h(x + σ(x))k |8| 1 ≤ max{kg(2x) − h(2x)k, kg(x + σ(x)) − h(x + σ(x))k} |8| ≤ cLΦ(x)

kJg(x) − Jh(x)k =

for all x ∈ X. Hence we have d(Jg, Jh) ≤ Ld(g, h) for any g, h ∈ S and so J is a strictly contractive mapping. Next, we claim that d(Jf, f ) < ∞. Putting y = 0 in (2.2), we get (2.4)

kf (2x) − 8f (x)k ≤

1 φ(x, 0) |2|

for all x ∈ X and putting x = 0 in (2.2), we get (2.5)

kf (y) + f (σ(y))k ≤ φ(0, y)

for all y ∈ X and putting y = x + σ(x) in (2.5), we get (2.6)

kf (x + σ(x))k ≤

1 φ(0, x + σ(x)) |2|

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for all x ∈ X. By (2.4) and (2.6), we have

1

kJf (x) − f (x)k =

f (2x) − 8f (x) + f (x + σ(x)) |8| n o 1 ≤ max kf (2x) − 8f (x)k, kf (x + σ(x))k |8| 1 ≤ 4 Φ(x) |2| for all x ∈ X. Hence d(Jf, f ) ≤

(2.7)

1 < ∞. |2|4

By Theorem 1.2, there exists a mapping C : X −→ Y which is a fixed point of J such that d(J n f, C) → 0 as n → ∞. By induction, we can easily show that 
o 1 n (J n f )(x) = 3n f (2n x) + (2n − 1)f 2n−1 x + σ(x) 2 for all x ∈ X and n ∈ N. Since d(J n f, C) → 0 as n → ∞, there exists a sequence {cn } in R such that cn → 0 as n → ∞ and d(J n f, C) ≤ cn for every n ∈ N. Hence, it follows from the definition of d that k(J n f )(x) − C(x)k ≤ cn Φ(x) for all x ∈ X. Thus for each fixed x ∈ X, we have lim ||(J n f )(x) − C(x)|| = 0

n−→∞

and so 
o 1 n n f (2 x) + (2n − 1)f 2n−1 x + σ(x) . 3n n−→∞ 2

(2.8)

C(x) = lim

It follows from (2.2) and (2.8) that kC(2x + y) + C(2x + σ(y)) − 2C(x + y) − 2C(x + σ(y)) − 12C(x)k 1 max{φ(2n x, 2n y), |2n − 1|φ(2n−1 (x + σ(x)), 2n−1 (y + σ(y)))} |8|n ≤ lim Ln max{φ(x, y), |2n − 1|φ(x, y)} = lim Ln φ(x, y) = 0 ≤ lim

n−→∞ n−→∞

n−→∞

for all x, y ∈ X, because |2n − 1| ≤ 1 for all n ∈ N. Hence C satisfies (1.4), C is a cubic mapping with involution. By (4) in Theorem 1.2 and (2.4), f satisfies (2.3). Assume that C1 : X −→ Y is another solution of (1.4) satisfying (2.3). We know that C1 is a fixed point of J. Due to (3) in Theorem 1.2, we get C = C1 . This proves the uniqueness of C.  Theorem 2.2. Assume that φ : X 2 −→ [0, ∞) is a mapping and there exists a real number L with 0 < L < 1 such that L (2.9) φ(x, y) ≤ φ(2x, 2y), φ(x + σ(x), y + σ(y)) ≤ φ(2x, 2y) |8| for all x, y ∈ X. Let f : X −→ Y be a mapping satisfying (2.2) and f (0) = 0. Then there exists a unique cubic mapping C : X −→ Y with involution such that (2.10)

kf (x) − C(x)k ≤

L Φ(x) |2|4 (1 − L)

for all x ∈ X, where Φ(x) = max{φ(x, 0), φ(0, x)}.

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Proof. Consider the set S = {g | g : X −→ Y } and the generalized metric d in S defined by d(g, h) = inf {c ∈ [0, ∞)| kg(x) − h(x)k ≤ c Φ(x) for all x ∈ X}. Then (S, d) is a complete metric space. Define a mapping J : S −→ S by n x  x + σ(x) o −g Jg(x) = 8 g 2 4 for all x ∈ X and all g ∈ S. Let g, h ∈ S and d(g, h) ≤ c for some non-negative real number c. Then by (2.9), we have

x  x + σ(x)  x  x + σ(x) 

−g −h +h kJg(x) − Jh(x)k = |8| g

2 4 2 4 n  x   x   x + σ(x)   x + σ(x)  o



≤ |8|max g −h −h

, g

2 2 4 4 ≤ cLΦ(x) for all x ∈ X. Hence d(Jg, Jh) ≤ Ld(g, h) for any g, h ∈ S and so J is a strictly contractive mapping. Next, we claim that d(Jf, f ) < ∞. By (2.4), (2.5) and (2.6), we have

x

 x + σ(x)  L

kJf (x) − f (x)k = 8f − 8f − f (x) ≤ 4 Φ(x) 2 4 |2| for all x ∈ X and hence L < ∞. |2|4 By Theorem 1.2, there exists a mapping C : X −→ Y which is a fixed point of J such that d(J n f, C) → 0 as n → ∞. By induction, we can easily show that n x  x + σ(x) o (J n f )(x) = 23n f n − f 2 2n+1 d(Jf, f ) ≤

for each n ∈ N. Since d(J n f, C) → 0 as n → ∞, there exists a sequence {cn } in R such that cn → 0 as n → ∞ and d(J n f, C) ≤ cn for every n ∈ N. Hence, it follows from the definition of d that k(J n f )(x) − C(x)k ≤ cn Φ(x) for all x ∈ X. Thus for each fixed x ∈ X, we have lim k(J n f )(x) − C(x)k = 0

n−→∞

and n x  x + σ(x) o C(x) = 23n f n − f . 2 2n+1 Analogously to the proof of Theorem 2.2, we can show that C is a unique cubic mapping with involution satisfying (2.10)  We can use Theorem 2.1 and Theorem 2.2 to get a classical result in the framework of nonArchimedean normed spaces. Taking φ(x, y) = θ(kxkp + kykp ) or φ(x, y) = θ(kxkp kykp + kxk2p + kyk2p ), we have the following examples. Example 2.3. Let θ ≥ 0 and p be a positive real number with p 6= 3. Let f : X −→ Y be a mapping satisfying (2.11)

kDf (x, y)k ≤ θ(kxkp + kykp )

for all x, y ∈ X. Suppose that kx + σ(x)k ≤ |2|kxk for all x ∈ X. Then there exists a unique mapping C : X −→ Y with involution such that the inequality

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θ||x||p , if p > 3, 3 − |2|p ) ||f (x) − C(x)|| ≤ |2|(|2| p θ||x||   , if 0 < p < 3  |2|(|2|p − |2|3 )    

holds for all x ∈ X. Proof. Let φ(x, y) |8||2|p−3 φ(x, y) for |8||2|p−3 φ(x, y) for Suppose that L

= θ(kxkp + kykp ) for all x, y ∈ X and L = |2|p−3 . Then φ(2x, 2y) = all x, y ∈ X. Since kx + σ(x)k ≤ |2|kxk for all x ∈ X, φ(x + σ(x), y + σ(y)) ≤ all x, y ∈ X. Hence if p > 3, then we have the results of Theorem 2.1. 3−p = |2|3−p . Then φ(x, y) = |2||8| φ(2x, 2y) for all x, y ∈ X and φ(x + σ(x), y + 3−p

σ(y)) ≤ |2|p φ(x, y) = |2||8| φ(x, y) for all x, y ∈ X. Hence if 0 < p < 3, then we have the results of Theorem 2.2. Thus the proof is complete.  Example 2.4. Let θ ≥ 0 and p be a positive real number with p 6= mapping satisfying (2.12)

3 2.

Let f : X −→ Y be a

kDf (x, y)k ≤ θ(kxkp kykp + kxk2p + kyk2p )

for all x, y ∈ X. Suppose that kx + σ(x)k ≤ |2|kxk for all x ∈ X. Then there exists a unique mapping C : X −→ Y with involution such that C is a solution of the functional equation (1.4) and the inequality  θ||x||p   , if p > 32 ,  3 − |2|2p ) |2|(|2| ||f (x) − C(x)|| ≤ θ||x||p   , if 0 < p < 32  |2|(|2|2p − |2|3 ) holds for all x ∈ X. Using Theorem 2.1 and Theorem 2.2, we obtain the following corollary concerning the stability of (1.4). Corollary 2.5. Let αi : [0, ∞) −→ [0, ∞) (i = 1, 2, 3) be increasing mappings satisfying (i) 0 < αi (|2|) < 1 and αi (0) = 0, (ii) αi (|2|t) ≤ αi (|2|)αi (t) for all t ≥ 0. Let f : X −→ Y be a mapping such that for some δ ≥ 0 (2.13)

kDf (x, y)k ≤ δ[α1 (kxk)α1 (kyk) + α2 (kxk) + α3 (kyk)]

for all x, y ∈ X. Suppose that kx + σ(x)k ≤ |2|kxk for all x ∈ X. Then there exists a unique cubic mapping C : X −→ Y with involution such that  1  e Φ(x), if 0 < M < |2|3 ,  3 |2|(|2| − M ) ||f (x) − C(x)|| ≤ 1  e  Φ(x), if N > |2|3 |2|(N − |2|3 ) holds for all x ∈ X, where M = max{(α1 (|2|))2 , α2 (|2|), α3 (|2|)}, N = min{(α1 (|2|))2 , α2 (|2|), α3 (|2|)} e and Φ(x) = δ max{α2 (kxk), α3 (kxk)}. As examle of Corollary 2.5, we can take α1 (t) = α2 (t) = α3 (t) = tp for all t ≥ 0. Then we have the following example. Example 2.6. Let δ ≥ 0 and p be a positive real number with p 6= mapping satisfying (2.14)

3 2.

Let f : X −→ Y be a

kDf (x, y)k ≤ δ(kxkp kykp + kxkp + kykp )

and kx + σ(x)k ≤ |2|kxk for all x, y ∈ X. Then there exists a unique mapping C : X −→ Y with involution such that the inequality

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δ||x||p , if p > 3, 3 p ||f (x) − C(x)|| ≤ |2|(|2| − p|2| ) δ||x||   , if 0 < p <  |2|(|2|2p − |2|3 )    

3 2

holds for all x ∈ X. Acknowledgements The first author was supported by the research fund of Dankook University in 2015. References [1] B. Boukhalene, E. Elqorachi, and Th. M. Rassias, On the generalized Hyers-Ulam stability of the quadratic functional equation with a general involution, Nonlinear Funct. Anal. Appl. 12. no 2 (2007), 247-262. [2] , On the Hyers-Ulam stability of approximately pexider mappings, Math. Ineqal. Appl. 11 (2008), 805818. [3] S. Czerwik, Functional equations and Inequalities in several variables, World Scientific, New Jersey, London, 2002. [4] J. B. Diaz, Beatriz Margolis A fixed point theorem of the alternative, for contractions on a generalized complete metric space Bull. Amer. Math. Soc. 74 (1968), 305-309. [5] G. L. Forti, Hyers-Ulam stability of functional equations in several variables, Aequationes Math. 50 (1995), 143-190. ¨ ber eine neue Begr¨ [6] K. Hensel, U undung der Theorie algebraischen Zahlen, Jahresber. Deutsch. Math. Verein. 6 (1897), 83-88. [7] D. H. Hyers, On the stability of linear functional equation, Proc. Natl. Acad. Sci. USA 27 (1941), 222-224. [8] D. H. Hyers, G. Isac, and T. M. Rassias, Stability of functional equations in several variables, Birkh¨ auser, Boston, 1998. [9] D. H. Hyers, T. M. Rassias, Approximate homomorphisms, Aequationes Math. 44 (1992), 125-153. [10] G. Isac, Th. M. Rassias, Stability of ψ-additive mappings: Applications to nonlinear analysis, Internat. J. Math. & Math. Sci. 19 (1996), 219-228. [11] K.-W. Jun and H.-M. Kim The generalized Hyers-Ulam-Rassias stability of a cubic functional equation, J. Math. Anal. Appl. 274 (2002), 867-878. [12] S. M. Jung, Z. H. Lee, A fixed point approach to the stability of quadratic functional equation with involution, Fixed Point Theory Appl. 2008. [13] S. M. Jung, On the Hyers-Ulam stability of the functional equation that have the quadratic property, J. Math. Anal. Appl. 222 (1998), 126-137. [14] M. S. Moslehian, T. M. Rassias, Stability of functional equations in non-Archimedean spaces, Applicable Analysis and Discrete Mathematics, 1 (2007), 325-334. [15] M. S. Moslehian, GH. Sadeghi, Stability of two types of cubic functional equations in non-Archimedean spaces, Real Analysis Exchange 33(2) (2007/2008), 375-384. [16] F. Skof, Approssimazione di funzioni δ-quadratic su dominio restretto, Atti. Accad. Sci. Torino Cl. Sci. Fis. Mat. Natur. 118 (1984), 58-70. [17] H. Stetkær, Functional equations on abelian groups with involution, Aequationes Math. 54 (1997), 144-172. [18] S. M. Ulam, A collection of mathematical problems, Interscience Publ., New York, 1960. Department of Mathematics Education, Dankook University, Yongin 448-701, Republic of Korea E-mail address: [email protected] Department of Mathematics, Soongsil University, Seoul 156-743, Republic of Korea E-mail address: [email protected]

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VALUE SHARING RESULTS FOR MEROMORPHIC FUNCTIONS WITH THEIR Q-SHIFTS XIAOGUANG QI, JIA DOU AND LIANZHONG YANG Abstract. This research is a continuation of a recent paper [16, 17]. Shared value problems related to a meromorphic function f (z) and its q-shift f (qz + c) are studied. Moreover, we also consider uniqueness problems on meromorphic functions f (z) share sets with f (qz + c).

1. Introduction We assume that the reader is familiar with the elementary Nevanlinna Theory, see, e.g. [8, 18]. Meromorphic functions are always non-constant, unless otherwise specified. As usual, by S(r, f ) we denote any quantity satisfying S(r, f ) = o(T (r, f )) for all r outside of a possible exceptional set of finite linear measure. In particular, we denote by S1 (r, f ) any quality satisfying S1 (r, f ) = o(T (r, f )) for all r on a set of logarithmic density 1. For a meromorphic function f and a set S of complex numbers, we define the S set E(S, f ) = a∈S {z|f (z) − a = 0}, where a zero of f − a with multiplicity m counts m times in E(S, f ). As a special case, when S = {a} contains only one element a, if E(a, f ) = E(a, g), then we say f (z) and g(z) share a CM ; if E(a, f ) = E(a, g), then we say f (z) and g(z) share a IM , see [18]. The classical results due to Nevanlinna [14] in the uniqueness theory of meromorphic functions are the five-point, resp. four-point, theorems: Theorem A. If two meromorphic functions f (z) and g(z) share five distinct values a1 , a2 , a3 , a4 , a5 ∈ C ∪ {∞} IM, then f (z) ≡ g(z). Theorem B. If two meromorphic functions f (z) and g(z) share four distinct values a1 , a2 , a3 , a4 ∈ C ∪ {∞} CM, then f (z) ≡ g(z) or f (z) ≡ T ◦ g(z), where T is a M¨ obius transformation. It is well-known that 4 CM can not be improved to 4 IM, see [6]. Further, Gundersen [7, Theorem 1] has improved the assumption 4 CM to 2 CM+2 IM, while 1 CM+3 IM is still an open problem. Heittokangas et al. [9, 10] considered the uniqueness of a finite order meromorphic function sharing values with its shift. They proved the following theorem: Theorem C. Let f (z) be a meromorphic function of finite order, let c ∈ C, and let a1 , a2 , a3 ∈ S(f )∪{∞} be three distinct periodic functions with period 2010 Mathematics Subject Classification. 30D35, 39A05. Key words and phrases. Q-shift; Meromorphic functions; Value sharing, Nevanlinna theory. 1

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c. If f (z) and f (z + c) share a1 , a2 CM and a3 IM, then f (z) = f (z + c) for all z ∈ C. Here, denote by S(f ) the family of all meromorphic functions a(z) that satisfy T (r, a) = o(T (r, f )), for r → ∞ outside a possible exceptional set of finite logarithmic measure. Some improvements of Theorem C can be found in [1, 11, 12, 15]. A natural question is: what is the uniqueness result in the case when f (z) shares values with f (qz + c) for a zero-order meromorphic function f (z). Corresponding to this question, we got the following result in [16]: Theorem D. Let f (z) be a zero-order meromorphic function, and q ∈ C \ {0}, c ∈ C, and let a1 , a2 , a3 ∈ C ∪ {∞} be three distinct values. If f (z) and f (qz + c) share a1 , a2 CM and a3 IM, then f (z) = f (qz + c) and |q| = 1. Theorem E. Let f (z) be a zero-order entire function, q ∈ C \ {0}, c ∈ C, and let a1 , a2 ∈ C be two distinct values. If f (z) and f (qz + c) share a1 and a2 IM, then f (z) = f (qz + c) and |q| = 1. It seems natural to ask whether the assumption ”constants ai ” can be replaced by ”small functions ai ” in Theorem E. We will give a positive answer in this paper. The reminder of this paper is organized as follows: Firstly, Section 2 contains some auxiliary results. We consider the value sharing problem for f (z) and f (qz + c) in Section 3. Section 4 is devoted to proving some uniqueness results for meromorphic functions f (z) share sets with f (qz + c). 2. Some Lemmas Lemma 2.1. [13, Theorem 2.1] Let f (z) be a zero-order meromorphic function, and q ∈ C \ {0}, c ∈ C. Then ¶ µ f (qz + c) = S1 (r, f ). m r, f (z) Lemma 2.2. [16, Theorem 3.2] Let f (z) be a zero-order meromorphic function, and q ∈ C \ {0}, c ∈ C. Then ¶ µ f (z) m r, = S1 (r, f ) (2.1) f (qz + c) and T (r, f (qz + c)) = T (r, f (z)) + S1 (r, f ).

(2.2)

Lemma 2.3. [13, Theorem 2.4] Let f (z) be a zero-order meromorphic solution of f (z)n P (z, f ) = Q(z, f ), where P (z, f ) and Q(z, f ) are q-shift difference polynomials in f (z). If the degree of Q(z, f ) as a polynomial in f (z) and its q-shifts is at most n, then m(r, P (z, f )) = S1 (r, f ).

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3. Improvement of Theorem E Next we show that ”constants ai ” in Theorems E can be replaced by ”small functions ai ”. Theorem 3.1. Let f (z) be a zero-order entire function, q ∈ C \ {0}, c ∈ C, and let a1 , a2 ∈ S(f ). If f (z) and f (qz + c) share a1 and a2 IM, then f (z) = f (qz + c) and |q| = 1. Remarks. (1). Theorem E and Theorem 3.1 seem to be so similar. However, our proof is different to the one in Theorem E. (2). We tried to improve Theorem D, unfortunately, we cannot get any improvement in this paper. Proof of Theorem 3.1. From the fact that a non-constant meromorphic function of zero-order can have at most one Picard exceptional value (see, e. 1 1 g., [3, p. 114]), it can be concluded that N (r, f −a ) 6= 0 and N (r, f −a ) 6= 0. 1 2 Define H1 (z)(f (z) − f (qz + c)) H(z) = , (3.1) (f (z) − a(z))(f (z) − b(z)) where H1 (z) = (f (z) − a(z))(f 0 (z) − b0 (z)) − (f 0 (z) − a0 (z))(f (z) − b(z)). And G(z) =

G1 (z)(f (z) − f (qz + c)) , (f (qz + c) − a(z))(f (qz + c) − b(z))

(3.2)

where G1 (z) = (f (qz+c)−a(z))(f 0 (qz+c)−b0 (z))−(f 0 (qz+c)−a0 (z))(f (qz+c)−b(z)). Equation (3.1) can be rewritten as µ 0 ¶ f (z) − b0 (z) f 0 (z) − a0 (z) H(z) = − (f (z) − f (qz + c)) f (z) − b(z) f (z) − a(z) µ ¶ H1 (z)(f (z) − a(z) + a(z)) f (qz + c) = 1− . (f (z) − a(z))(f (z) − b(z)) f (z)

(3.3)

Note H1 (z) = (f (z) − a(z))(f 0 (z) − b0 (z)) − (f 0 (z) − a0 (z))(f (z) − b(z)) = (f (z) − b(z))(a0 (z) − b0 (z)) − (f 0 (z) − b0 (z))(a(z) − b(z)), hence equation (3.3) can be expressed as µ ¶µ ¶ f (qz + c) H1 (z) H1 (z) H(z) = 1 − + a(z) f (z) f (z) − b(z) (f (z) − a(z))(f (z) − b(z)) µ ¶µ 0 f (qz + c) (f (z) − b(z))(a (z) − b0 (z)) − (f 0 (z) − b0 (z))(a(z) − b(z)) = 1− f (z) f (z) − b(z) ¶ 0 0 0 (f (z) − a(z))(f (z) − b (z)) − (f (z) − a0 (z))(f (z) − b(z)) + a(z) . (f (z) − a(z))(f (z) − b(z)) (3.4)

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By the assumption f (z) and f (qz + c) share a(z), b(z) IM and equation (3.3), we get N (r, H(z)) ≤ N (r, a(z)) + N (r, b(z)) = S(r, f ).

(3.5)

From equation (3.4), Lemma 2.1 and the lemma of logarithmic derivative, we know m(r, H(z)) = S1 (r, f ). Hence, T (r, H(z)) = S1 (r, f ). (3.6) Similarly as above, we know µ 0 ¶ f (qz + c) − b0 (z) f 0 (qz + c) − a0 (z) G(z) = − (f (z) − f (qz + c)). (3.7) f (qz + c) − b(z) f (qz + c) − a(z) Using a similar way, we obtain that T (r, G(z)) = S1 (r, f ).

(3.8)

Denote U (z) = mH(z) − nG(z). (3.9) Next, suppose on the contrary that f (z) 6= f (qz + c), and head for a contradiction. Case 1. Assume that there exists two integers m, n such that U (z) = 0. Then from (3.3) and (3.7), we deduce that ¶ µ 0 ¶ µ 0 f (qz + c) − b0 (z) f 0 (qz + c) − a0 (z) f (z) − b0 (z) f 0 (z) − a0 (z) − =n − , m f (z) − b(z) f (z) − a(z) f (qz + c) − b(z) f (qz + c) − a(z) which implies that ¶ µ µ ¶ f (qz + c) − b(z) n f (z) − b(z) m =A , f (z) − a(z) f (qz + c) − a(z) where A is a non-zero constant. If m 6= n, then we get from above equality and (2.2) that mT (r, f (z)) = nT (r, f (qz + c)) + S1 (r, f ) = nT (r, f (z)) + S1 (r, f ), which is a contradiction. If m = n, then we get f (z) − b(z) f (qz + c) − b(z) =B , f (z) − a(z) f (qz + c) − a(z) where B satisfies B m = A.

(3.10)

If B = 1, then we obtain f (z) = f (qz +c), which contradicts the assumption f (z) 6= f (qz + c). It remains to consider the case that B 6= 1. The equation (3.10) gives f (z)((B−1)f (qz+c)+a(z)−Bb(z)) = (Ba(z)−b(z))f (qz+c)+(1−B)a(z)b(z). Apply Lemma 2.3 to the above equation, resulting in m(r, ((B − 1)f (qz + c) + a(z) − Bb(z))) = S1 (r, f ). Consequently, T (r, f (qz + c)) = T (r, f ) + S1 (r, f ) = S1 (r, f ), which is impossible.

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Case 2. There does not exist two positive integers m, n such that U (z) = 0. In what follows, we denote Sf ∼g(n,m) (a) for the set of those points z ∈ C such that z is an a-point of f with multiplicity n and an a-point of g with multiplicity m such that a(z) 6= ∞, b(z) 6= ∞, a(z) − b(z) 6= 0. Let 1 1 N(n,m) (r, f −a ) and N (n,m) (r, f −a ) denote the counting function and reduced counting function of f (z) with respect to the set Sf ∼g(n,m) (a), respectively. Take z0 such that z0 ∈ Sf (z)∼f (qz+c)(n,m) (a(z)), we have mn 6= 0, since a(z) is not a Picard exceptional value of f (z) as we discuss above. Combining (3.3), (3.7) with (3.9), by calculating carefully, it follows that U (z0 ) = 0. From (3.6), (3.8) and (3.9), we have µ ¶ µ ¶ µ ¶ 1 1 1 N (n,m) r, ≤ N r, = N r, = S1 (r, f ). f (z) − a(z) U (z) mH(z) − nG(z) Using the same reason, we get µ ¶ µ ¶ µ ¶ 1 1 1 N (n,m) r, ≤ N r, = N r, = S1 (r, f ). f (z) − b(z) U (z) nH(z) − mG(z) Consequently, µ N (n,m) r,

1 f (z) − a(z)

¶

µ + N (n,m)

1 r, f (z) − b(z)

¶ = S1 (r, f ).

(3.11)

Combining (2.2) with (3.11), it follows that ³ ´ ³ ´ 1 1 T (r, f (z)) ≤ N r, + N r, + S1 (r, f ) f (z) − a(z) f (z) − b(z) ¶ Xµ 1 1 N (n,m) (r, = ) + N (n,m) (r, ) + S1 (r, f ) f (z) − a(z) f (z) − b(z) n,n ¶ X µ 1 1 = N (n,n) (r, ) + N (n,m) (r, ) + S1 (r, f ) f (z) − a(z) f (z) − b(z) m+n≥5 µ 1 X 1 1 ≤ N(n,m) (r, ) + N(n,m) (r, ) 5 f (z) − a(z) f (z) − b(z) m+n≥5 ¶ 1 1 + N(n,m) (r, ) + N(n,m) (r, ) + S1 (r, f ) f (qz + c) − a(z) f (qz + c) − b(z) 2 2 ≤ T (r, f ) + T (r, f (qz + c)) + S1 (r, f ) 5 5 4 = T (r, f ) + S1 (r, f ), 5 which is a contradiction. Therefore, we get f (z) = f (qz + c). The rest of proof consists of the conclusion that |q| = 1. The proof is similar as [10, Theorem 1.5]. In fact, we have given the proof in [16]. The proof is stated explicitly for the convenience of the reader. If f (z) is transcendental and suppose first |q| < 1. It can be assumed that there exists one point z0 such that f (z0 ) = a1 and that z0 is not a fixed point of qz + c. By the sharing assumptions of Theorem 3.1, we get f (qz0 + c) = a1 as well. By calculation, we know f (q n z0 + c(1 + · · · + q n−1 )) = a1 for all n ∈ N. Letting c n → ∞, it is concluded that a1 -points of f accumulate to z = 1−q , which

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is a contradiction. If |q| > 1, then set g(z) = f (qz + c). Assume that g has at least one a1 point, say at z0 . From the sharing assumptions, we get g( q1n z − c( 1q + · · · + q1n )) = a1 for all all n ∈ N. Using the same way above, c , which is a contradiction. Hence we get a1 -point of g accumulate to z = 1−q |q| = 1. If f is a rational function, then set f (z) = Pm i i=1 ai (qz+c) Pn . j j=1 bi (qz+c)

Pm i i=1 ai z Pn j b z i j=1

and f (qz + c) =

By simply calculations, it follow that |q| = 1. This com-

pletes the proof of Theorem 3.1. 4. Sharing sets results Gross [4, Question 6] asked the following question: Question. Can one find (even one set) finite sets Sj (j = 1, 2) such that any two entire functions f (z) and g(z) satisfying E(Sj , f ) = E(Sj , g) (j = 1, 2) must be identical? Since then, many results have been obtained for this and related topics (see [2, 19, 20, 21]). Here, we just recall the following two results only. Theorem F [5]. Let S1 = {1, −1}, S2 = {0}. If f (z) and g(z) are entire functions of finite order such that E(Sj , f ) = E(Sj , g) for j = 1, 2, then f (z) = ±g(z) or f (z)g(z) = 1. Theorem G [22]. Let S1 = {1, ω, . . . ω n−1 } and S2 = {∞}, where ω = cos(2π/n) + i sin(2π/n) and n ≥ 6 be a positive integer. Suppose that f (z) and g(z) are meromorphic functions such that E(Sj , f ) = E(Sj , g) for j = 1, 2, then f (z) = tg(z) or f (z)g(z) = t, where tn = 1. It is natural to ask what will happen if g(z) is replaced by q-shift of f (z) in Theorems F and G. In the following, we answer this problem, and get shared sets results for f (z) and its q-shift f (qz + c). Theorem 4.1. Let S1 , S2 be given as in Theorem G, and let f (z) be a zero-order meromorphic function satisfying E(Sj , f (z)) = E(Sj , f (qz + c)) for j = 1, 2, c ∈ C and q ∈ C \ {0}. If n ≥ 4, then f (z) = tf (qz + c), tn = 1 and |q| = 1. By the same reasoning as in the proof of Theorem 4.1, we obtain the following result. We omit the proof here. Corollary 4.2. Theorem 4.1 still holds if f is a zero-order entire function and n ≥ 3. In the following, we give a partial answer as to what may happen if n = 2 in Corollary 4.2, which can be seen an analogue for q-shift of Theorem F. Theorem 4.3. Suppose f (z) is a zero-order entire function and q ∈ C\{0}, c ∈ C. If f (z) and f (qz + c) share the set {a(z), −a(z)} CM , where a(z) is a non-vanishing small function of f (z), then one of the following statements hold: (1). C 2 f (z) = f (q 2 z + qc + c), where C is a constant such that C 2 6= 1; (2). f (z) = ±f (qz + c), and |q| = 1.

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Corollary 4.4. Suppose a is a non-zero constant in Theorem 4.3, then we get f (z) = ±f (qz + c), where |q| = 1. Corollary 4.5. Under the assumptions of Theorem 4.3, if f (z) and f (qz+c) share sets {a(z), −a(z)}, {0} CM , then f (z) = ±f (qz + c), where |q| = 1. Proof of Theorem 4.1. By the sharing assumption, we get f (z)n and f (qz + c)n share 1 and ∞ CM . This implies, f (qz + c)n − 1 = γ, f (z)n − 1

(4.1)

where γ is a non-zero constant. This gives f (qz + c)n = γ(f (z)n − 1 + Denote G(z) =

1 ). γ

(4.2)

f (z)n . 1 − γ1

Suppose γ 6≡ 1, then by the second main theorem and Lemma 2.2 to G(z), it follows that µ ¶ µ ¶ 1 1 + N (r, G) + N r, + S(r, G) nT (r, f ) + S(r, f ) = T (r, G) ≤ N r, G G−1 ! Ã ¶ µ 1 1 + S(r, f ) + N (r, f ) + N r, ≤ N r, f f (z)n − 1 + γ1 µ µ ¶ ¶ 1 1 ≤ N r, + N (r, f ) + N r, + S(r, f ) f f (qz + c) ≤ 2T (r, f ) + T (r, f (qz + c)) + S(r, f ) ≤ 3T (r, f ) + S1 (r, f ). This together with the assumption n ≥ 4 results in a contradiction. Hence, γ ≡ 1, that is, f (z)n = f (qz + c)n . This yields f (z) = tf (qz + c) for a constant t with tn = 1. Let F (z) = f (z)n and F (qz + c) = f (qz + c)n , then we get F (z) = F (qz + c). Similarly as Theorem 3.1, we have |q| = 1. The conclusion follows. Proof of Theorem 4.3. It follows by the assumptions that (f (qz + c) − a(z))(f (qz + c) + a(z)) = C 2 (f (z) − a(z))(f (z) + a(z)), (4.3) where C is a non-zero constant. Case 1. Suppose first that C 2 6≡ 1. Denote h1 (z) = f (z) −

1 f (qz + c), C

h2 (z) = f (z) +

1 f (qz + c). C

Then 1 f (z) = (h1 (z) + h2 (z)), 2 Moreover, we have

f (qz + c) =

h1 (z)h2 (z) = (1 −

113

C (h2 (z) − h1 (z)). 2

1 2 )a (z). C2

(4.4)

(4.5)

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From above equation, we get µ ¶ 1 = S(r, f ), N r, h1

µ N

1 r, h2

¶ = S(r, f ).

(4.6)

By definitions of h1 (z) and h2 (z), Lemma 2.2 yields T (r, hi ) ≤ 2T (r, f ) + S1 (r, f ), which means S1 (r, hi ) = o(T (r, f )) for all r on a set of logarithmic density 1, i = 1, 2. Denote h1 (qz + c) h2 (qz + c) α(z) = , β(z) = . h1 (z) h2 (z) From (4.6) and Lemma 2.1, we obtain that µ ¶ 1 = S1 (r, f ), T (r, α) = m(r, α) + N r, h1 µ ¶ (4.7) 1 = S1 (r, f ). T (r, β) = m(r, β) + N r, h2 From definitions of h1 (z), h2 (z) and equation (4.4), we conclude that Ch2 (z) − Ch1 (z) = h1 (qz + c) + h2 (qz + c). Dividing above equation with h1 (z)h2 (z), we obtain (α + C)h1 (z) = (C − β)h2 (z).

(4.8)

By combining (4.5) and (4.8), it follows that 1 2 )a (z) = 0. (4.9) C2 From (4.7) and (4.9), we get α = −C and β = C. Otherwise, we know T (r, h1 ) = S1 (r, f ), which means T (r, f ) = S1 (r, f ) from (4.4) and (4.5), a contradiction. Hence, we have (α + C)h21 (z) − (C − β)(1 −

h1 (qz + c) = −Ch1 (z),

h2 (qz + c) = Ch2 (z),

from definitions of α(z) and β(z), that is ½ −C(f (z) − C1 f (qz)) = f (qz) − C1 f (q(qz + c) + c), C(f (z) + C1 f (qz)) = f (qz) + C1 f (q(qz + c) + c). The above equations give C 2 f (z) = f (q 2 z + qc + c). Case 2. C 2 ≡ 1. The equation (4.3) implies that f (z) = ±f (qz + c). Using a similar way as Theorem 3.1, we get |q| = 1 in Case 2. Proof of Corollary 4.4. Similarly as Theorem 4.3, we obtain equations (4.4) and (4.5) hold as well. Equation (4.5) and the assumption that a is non-zero constant give ¶ µ ¶ µ 1 1 = 0, N r, = 0. (4.10) N r, h1 h2 Combining (4.10) with the definitions of h1 (z) and h2 (z), we conclude that h1 (z) and h2 (z) are constants. From (4.4), we get f (z) is a constant, which contradicts the assumption. Hence, only Case 2 of Theorem 4.3 holds, the conclusion follows.

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Proof of Corollary 4.5. It suffices to prove the case C 2 f (z) = f (q 2 z + qc + c) in Theorem 4.3 cannot hold. Suppose that f (z0 ) = 0, then by the sharing assumption and (4.4), it follows that h1 (z0 ) + h2 (z0 ) = 0,

h1 (qz0 + c) + h2 (qz0 + c) = 0.

Hence, h1 (qz0 + c) h2 (z0 ) = 1. h1 (z0 ) h2 (qz0 + c) From the proof of Theorem 4.3, we know α=

h1 (qz0 + c) = −C, h1 (z0 )

β=

h2 (qz0 + c) = C, h2 (z0 )

which means that h1 (qz0 + c) h2 (z0 ) = −1. h1 (z0 ) h2 (qz0 + c) which is impossible. This contradiction is only avoided when 0 is the Picard exceptional value of f (z) and f (qz + c). Since f (z) is a zero-order entire function, we conclude that f (z) must be a constant, which contradicts the assumption. Hence, f (z) = ±f (qz + c), where |q| = 1. Acknowledgements The authors thank the referee for his/her valuable suggestions to improve the present paper. This work was supported by the National Natural Science Foundation of China (No. 11301220 and No. 11371225) and the Tianyuan Fund for Mathematics (No. 11226094), the NSF of Shandong Province, China (No. ZR2012AQ020) and the Fund of Doctoral Program Research of University of Jinan (XBS1211). References [1] Z. X. Chen and H. X. Yi, On sharing values of meromorphic functions and their differences, Results Math. 63 (2013), 557-565. [2] G. Frank and M. Reinders, A unique range set for meromorphic functions with 11 elements, Complex Var. Theory Appl. 37 (1998), 185-193. [3] A. A. Goldberg and I. V. Ostrovskii, Value Distribution of Meromorphic Functions, Transl. Math. Monogr., vol. 236, American Mathematical Society, Providence, RI, 2008, translated from the 1970 Russian original by Mikhail Ostrovskii, with an appendix by Alexandre Eremenko and James K. Langley. [4] F. Gross, Factorization of meromorphic functions and some open problems. In: Complex analysis (Proc. Conf., Univ. Kentucky, Lexington, Ky., 1976), Lecture Notes in Math., Vol. 599, Springer, Berlin, 1977, pp. 51–67. [5] F. Gross and C. F. Osgood, Entire functions with common preimages. In: Factorization Theory of Meromorphic Functions, Marcel Dekker, 1982, pp. 19–24. [6] G. G. Gundersen, Meromorphic functions that share three or four values, J. London Math. Soc. 20 (1979), 457–466. [7] G. G. Gundersen, Meromorphic functions that share four values, Trans. Amer. Math. Soc. 277 (1983), 545–567. [8] W. Hayman, Meromorphic Functions, Clarendon Press, Oxford, 1964. [9] J. Heittokangas, R. Korhonen, I. Laine, J. Rieppo and J. L. Zhang, Value sharing results for shifts of meromorphic functions, and sufficient conditions for periodicity, J. Math. Anal. Appl. 355 (2009), 352–363.

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[10] J. Heittokangas, R. Korhonen, I. Laine and J. Rieppo, Uniqueness of meromorphic functions sharing values with their shifts, Complex Var. Elliptic Equ. 56 (2011), 8192. [11] S. Li and Z. S. Gao, Entire functions sharing one or two fiite values CM with their shifts or difference operators, Arch Math. 97 (2011), 475-483. [12] X. M. Li, Meromorphic functions sharing four values with their difference operators or shift, submitted. [13] K. Liu and X. G. Qi, Meromorphic solutions of q-shift difference equations, Ann. Polon. Math. 101 (2011), 215-225. [14] R. Nevanlinna, Le th´eor`eme de Picard-Borel et la th´eorie des fonctions m´eromorphes, Gauthiers-Villars, Paris, 1929. [15] X. G. Qi, Value distribution and uniqueness of difference polynomials and entire solutions of difference equations, Ann. Polon. Math. 102 (2011), 129-142. [16] X. G. Qi, L. Z. Yang and Y. Liu, Nevanlinna theory for the f (qz + c) and its applications, Acta. Math. Sci. 33 (2013), 819-828. [17] X. G. Qi, L. Z. Yang, Sharing sets of q-difference of meromorphic functions, Math. Slovak. 64 (2014), 51-60. [18] C. C. Yang and H. X. Yi, Uniqueness Theory of Meromorphic Functions, Kluwer Academic Publishers, 2003. [19] H. X. Yi, A question of Gross and the uniqueness of entire functions, Nagoya Math. J. 138 (1995), 169-177. [20] H. X. Yi, Unicity theorems for meromorphic or entire functions II, Bull. Austral. Math. Soc. 52 (1995), 215-224. [21] H. X. Yi, Unicity theorems for meromorphic or entire functions III, Bull. Austral. Math. Soc. 53 (1996), 71-82. [22] H. X. Yi, and L. Z. Yang, Meromorphic functions that share two sets, Kodai Math. J. 20 (1997) 127-134. Xiaoguang Qi University of Jinan, School of Mathematics, Jinan, Shandong, 250022, P. R. China E-mail address: [email protected] or [email protected] Jia Dou School of Mathematics, Shandong Normal University, Jinan, Shandong, 250358, P. R. China E-mail address: [email protected] Lianzhong Yang Shandong University, School of Mathematics, Jinan, Shandong, 250100, P. R. China E-mail address: [email protected]

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RANDOM NORMED SPACE AND MIXED TYPE AQ–FUNCTIONAL EQUATION ICK-SOON CHANG* AND YANG-HI LEE Abstract. We investigate the stability problems for the following functional equation f (x + ay) + f (x − ay) − 2f (x) +

a − a2 a + a2 f (y) − f (−y) − f (ay) = 0 2 2

in random normed spaces.

1. Introduction and Preliminaries We first demonstrate the usual terminology, notations and conventions of the theory of random normed spaces [7, 8]. The space of all probability distribution functions is denoted by ∆+ : = {F : R ∪ {−∞, ∞} → [0, 1] F is left-continuous and nondecreasing on R, where F (0) = 0 and F (+∞) = 1}. And let D+ := {F ∈ ∆+ | l− F (+∞) = 1}, where l− f (x) denotes the left limit of the function f at the point x. The space ∆+ is partially ordered by the usual pointwise ordering of functions, i.e., F ≤ G if and only if F (t) ≤ G(t) for all t ∈ R. The maximal element for ∆+ in this order is the distribution function ε0 : R ∪ {0} → [0, ∞) given by { 0, if t ≤ 0, ε0 (t) = 1, if t > 0. Definition 1.1. ([7]) A mapping τ : [0, 1] × [0, 1] → [0, 1] is called a continuous triangular norm (briefly, a continuous t-norm) if τ satisfies the following conditions : (TN1) τ is commutative and associative ; (TN2) τ is continuous ; (TN3) τ (a, 1) = a for all a ∈ [0, 1] ; (TN4) τ (a, b) ≤ τ (c, d) whenever a ≤ c and b ≤ d for all a, b, c, d ∈ [0, 1]. Typical examples of continuous t-norms are τP (a, b) = ab, τM (a, b) = min(a, b) and τL (a, b) = max(a + b − 1, 0). Definition 1.2. ([8]) A random normed space (briefly, RN-space) is a triple (X, µ, τ ), where X is a vector space, τ is a continuous t-norm and µ is a mapping from X into D+ such that the following conditions hold : *Corresponding author 2010 Mathematics Subject Classification : 39B52, 39B82, 46S10. Keywords and phrases : Random normed space; AC-functional equation. The first author was supported by Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education (No. 2013R1A1A2A10004419). 1

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(RN1) µx (t) = ε0 (t) for all t > 0 if and only if x = 0, (RN2) µαx (t) = µx (t/|α|) for all x ∈ X, α ̸= 0 and all t ≥ 0, (RN3) µx+y (t + s) ≥ τ (µx (t), µy (s)) for all x, y ∈ X and all t, s ≥ 0. t If (X, ∥ · ∥) is a normed space, we can define a mapping µ : X → D+ by µx (t) = t+∥x∥ for all x ∈ X and all t > 0. Then (X, µ, τM ) is a random normed space, which is called the induced random normed space.

Definition 1.3. Let (X, µ, τ ) be an RN -space. (A1 ) A sequence {xn } in X is said to be convergent to a point x ∈ X if for every t > 0 and ε > 0, there exists a positive integer N such that µxn −x (t) > 1 − ε whenever n ≥ N. (A2 ) A sequence {xn } in X is called a Cauchy sequence if for every t > 0 and ε > 0, there exists a positive integer N such that µxn −xm (t) > 1−ε whenever n ≥ m ≥ N . (A3 ) An RN-space (X, µ, τ ) is said to be complete if and only if every Cauchy sequence in X is convergent to a point in X. Theorem 1.4. ([7]) If (X, µ, τ ) is an RN-space and {xn } is a sequence such that xn → x, then limn→∞ µxn (t) = µx (t). The concept of stability for a functional equation arises when we replace the functional equation by an inequality which acts as a perturbation of the equation. The stability problem for functional equations originated from questions of Ulam [9] concerning the stability of group homomorphisms. Hyers [2] had answered affirmatively the question of Ulam for Banach spaces. A generalized version of the theorem of Hyers for additive mappings was given by Aoki [1] and for linear mappings was presented by Rassias [6]. Since then, many interesting results of the stability of various functional equation have been extensively investigated. Now we take into account the following mixed type additive-quadratic functional equation (briefly, AQ-functional equation) a − a2 a + a2 f (y) − f (−y) − f (ay) = 0. (1.1) 2 2 Here we promise that each solution of equation (1.1) is said to be an additive-quadratic mapping. Quite recently, the stability of functional equation (1.1) in the case when a = 1 was investigated in [3, 4, 5]. The main aim of this work is to establish the stability for the functional equation (1.1) in random normed spaces. f (x + ay) + f (x − ay) − 2f (x) +

2. Main results Let E1 and E2 be vector spaces. For convenience, we use the following abbreviations for a given mapping f : E1 → E2 , Af (x, y) := f (x + y) − f (x) − f (y), Qf (x, y) := f (x + y) + f (x − y) − 2f (x) − 2f (y), a − a2 a + a2 f (y) − f (−y) − f (ay) 2 2 is a rational number.

Df (x, y) := f (x + ay) + f (x − ay) − 2f (x) + for all x, y ∈ E1 , where a >

1 2

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3

A solution of Af = 0 is said to be an additive mapping and a solution of Qf = 0 is called a quadratic mapping. If a mapping f is represented by sum of additive mapping and quadratic mapping, we say that f is an additive-quadratic mapping. Lemma 2.1. A mapping f : E1 → E2 satisfies the functional equation Df (x, y) = 0 for all x, y ∈ E1 if and only if there exist a quadratic mapping g : E1 → E2 and an additive mapping h : E1 → E2 such that f (x) = g(x) + h(x) for all x ∈ E1 . Proof. (Necessity) We decompose f into the even part and the odd part by considering g(x) =

f (x) + f (−x) f (x) − f (−x) , h(x) = 2 2

for all x ∈ E1 . It is note that f (0) =

−Df (0,0) a2 +1

= 0. The following functional equalities

Qg(x, y) =Dg(x, y/a) − Dg(0, y/a) = 0, (x + y x + y ) ( x − y) ( x + y) (x + y x − y ) , + Dh , + Dh 0, − Dh 0, Ah(x, y) = − Dh 2 2a 2 2a 2a 2a =0 give that g is a quadratic mapping and h is an additive mapping. (Sufficiency) Assume that there exist a quadratic mapping g : E1 → E2 and an additive mapping h : E1 → E2 such that f (x) = g(x) + h(x) for all x ∈ E1 . Then we see that Df (x, y) = Dg(x, y) + Dh(x, y) = Qg(x, ay) + g(ay) − a2 g(y) − Ah(x + ay, x − ay) + Ah(x, x) + ah(y) − h(ay) =0 for all x, y ∈ E1 . Therefore we arrive at the desired conclusion.



In the following theorem, we establish the stability of the functional equation (1.1) in random normed spaces. Theorem 2.2. Let (Y, µ, τM ) and (Z, µ′ , τM ) be a complete RN-space and an RN-space, respectively. Suppose that V is a vector space and f : V → Y is a mapping with f (0) = 0 for which there exists a mapping φ : V 2 → Z such that µDf (x,y) (t) ≥ µ′φ(x,y) (t)

(2.1)

for all x, y ∈ V and all t > 0. If a mapping φ satisfies one of the following conditions : (i) µ′αφ(x,y) (t) ≤ µ′φ(2ax,2ay) (t) for some 0 < α < 2a, (ii) µ′φ(2ax,2ay) (t) ≤ µ′αφ(x,y) (t) ≤ µ′φ((2a)2 x,(2a)2 y) (t) for some 2a < α < (2a)2 , (iii) µ′φ((2a)2 x,(2a)2 y) (t) ≤ µ′αφ(x,y) (t) for some (2a)2 < α for all x, y ∈ V and all t > 0, then there exists a unique F : V → Y such that   α)t′ )} if  supt′ 0. Let c > 0 and ε > 0 be given. Since limt→∞ µ′z (t) = 1 for all z ∈ Z, there is some t0 > 0 such that M (x, t0 ) ≥ 1 − ε. Fix some t > t0 . Since α < 2a, we ∑ 4αj t know that the series ∞ j=0 (2a)j+1 converges. It guarantees that there exists some n0 ≥ 0 ∑ 4αj t such that n+m−1 < c for all n ≥ n0 and all m > 0. Together with (RN3) and j=n (2a)j+1 (2.4), this implies that µJn f (x)−Jn+m f (x) (c) ≥ µJn f (x)−Jn+m f (x)

( n+m−1 ∑ 4αj t ) (2a)j+1 j=n

≥ M (x, t) ≥ M (x, t0 ) ≥ 1 − ε

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for all x ∈ V. Hence {Jn f (x)} is a Cauchy sequence in the complete RN-space (Y, µ, τM ) and so we can define a mapping F : X → Y by F (x) := limn→∞ Jn f (x). Moreover, if we put m = 0 in (2.4), we have   t µf (x)−Jn f (x) (t) ≥ M x, ∑n−1 4αj t  (2.5) j=0 (2a)j+1

for all x ∈ V. Next we are in the position to show that F is an additive-quadratic mapping. In view of (RN3), we figure out the relation { ( t ) ( t ) ( t ) µDF (x,y) (t) ≥ τM µ(F −Jn f )(x+ay) , µ(F −Jn f )(x−ay) , µ2(Jn −F f )(x) , 12 12 12 ( t ) ( t ) ( t ) µ a−a2 (F −J f )(y) , µ− a+a2 (F −J f )(−y) , µ−(F −Jn f )(ay) , (2.6) n n 12 12 12 2 2 ( t )} µDJn f (x,y) 2 for all x, y ∈ V and all n ∈ N. The first six terms on the right hand side of the previous inequality tend to 1 as n → ∞ by the definition of F. Also we consider that (t) { (t) (t) µDJn f (x,y) ≥ τM µ Df ((2a)n x,(2a)n y) , µ Df (−(2a)n x,−(2a)n y) , 2 8 8 2·(2a)2n 2·(2a)2n (t) ( t )} , µ Df (−(2a)n x,−(2a)n y) µ Df ((2a)n x,(2a)n y) 8 8 2·(2a)n 2·(2a)n { (t) (t) ≥ τM µ φ((2a)n x,(2a)n y) , µ φ(−(2a)n x,−(2a)n y) , 8 8 2·(2a)2n 2·(2a)2n (t) ( t )} µ φ((2a)n x,(2a)n y) , µ φ(−(2a)n x,−(2a)n y) 8 8 2·(2a)n 2·(2a)n { ( (2a)2n t ) ( (2a)2n t ) ≥ τM µφ(x,y) , µφ(−x,−y) , 4αn 4αn ( (2a)n t )} ( (2a)n t ) µφ(x,y) , µ , φ(−x,−y) 4αn 4αn which tends to 1 as n → ∞ by (RN3). It follows from (2.6) that µDF (x,y) (t) = 1 for all x, y ∈ V and all t > 0. By (RN1), this means that DF (x, y) = 0 for all x, y ∈ V. We now approximate the difference between f and F. Fix x ∈ V, t > 0 and choose t′ < t. For arbitrary ε > 0, by F (x) := limn→∞ Jn f (x), there is a n ∈ N such that µF (x)−Jn f (x) (t − t′ ) ≥ 1 − ε. It follows by (2.5) that µF (x)−f (x) (t) ≥ τM {µF (x)−Jn f (x) (t − t′ ), µJn f (x)−f (x) (t′ )} { ( )} t′ ≥ τM 1 − ε, M x, ∑n−1 4αj t j=0 (2a)j+1

{ ( (2a − α)t′ )} ≥ τM 1 − ε, M x, . 4

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Because ε > 0 is arbitrary, we find that µF (x)−f (x) (t) ≥ M (x, (2a − α)t′ ) for all x ∈ V and t′ < t. The first inequality in (2.2) follows from the previous inequality. In order to prove the uniqueness of F, we assume that F ′ is another additive-quadratic mapping from V to Y satisfying the first inequality in (2.2) with F ′ (0) = f (0). Note that if F ′ is an additive-quadratic mapping, then we have by (2.3) ′

n−1 ∑

′

(Jj F ′ (x) − Jj+1 F ′ (x)) = 0

F (x) − Jn F (x) =

j=0

for all x ∈ V and all n ∈ N. With the help of (RN3) and the first inequality in (2.2), this result yields that for all x ∈ V and all n ∈ N, µF ′ (x)−Jn f (x) (t) = µJn F ′ (x)−Jn f (x) (t) (t) { (t) (t) ≥ τM µ (F ′ −f )((2a)n x) , µ (F ′ −f )(−(2a)n x) , µ (F ′ −f )((2a)n x) , 4 4 4 2·(2a)n 2·(2a)2n 2·(2a)2n ( t )} µ (F ′ −f )(−(2a)n x) 4 2·(2a)n { { ( ( 2a )n (2a − α)t′ ) { ( ( 4a2 )n (2a − α)t′ )} ≥ τM sup M x, , sup M x, . α 4 α 4 t′ 0. Therefore the Cauchy sequence {Jn f (x)} has the limit F (x) := limn→∞ Jn f (x) for all x ∈ V and   t µf (x)−Jn f (x) (t) ≥ M x, ∑n−1 ( (2.9) ( α )j 4 ( (2a) )j )  4 + 2 2 j=0 (2a) (2a) α α for all x ∈ V. Now, to prove that DF (x, y) = 0 for all x, y ∈ V, we consider (2.6) in case 1. By virtue of (RN3) and (2.1), we see that (t) (t) (t) { µDJn f (x,y) ≥ τM µ Df ((2a)n x,(2a)n y) , µ Df (−(2a)n x,−(2a)n y) , 2 8 8 2·(2a)2n 2·(2a)2n ( ) ( )} ) t , µ (2a)n ( ) t µ (2a)n ( x y −y −x Df (2a)n , (2a)n − 2 Df (2a)n , (2a)n 8 8 2 { ( (2a)2n t ) ( (2a)2n t ) ≥ τM µφ(x,y) , µφ(−x,−y) , 4αn 4αn ( αn t )} ( αn t ) µφ(x,y) , µφ(−x,−y) 4(2a)n 4(2a)n for all x, y ∈ V and all t > 0, which tends to 1 as n → ∞. It implies that all the terms of (2.6) are equal to 1 as n → ∞ and then we know that F is an additive-quadratic mapping. Employing the same argument as in the proof of case 1, the second inequality in (2.2) follows from (2.9).

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Finally, it remains to prove the uniqueness of F. Let us assume that F ′ : V → Y is another additive-quadratic mapping satisfying (2.2). Note that if F ′ is an additivequadratic mapping then by (2.7) ′

′

F (x) − Jn F (x) =

n−1 ∑

(

) Jj F ′ (x) − Jj+1 F ′ (x) = 0

j=0

for all x ∈ V and all n ∈ N. This relation with (RN3) and (2.2) imply that µF ′ (x)−Jn f (x) (t) = µJn F ′ (x)−Jn f (x) (t) { (t) (t) , µ (F ′ −f )(−(2a)n x) , ≥ τM µ (F ′ −f )((2a)n x) 4 4 2·(2a)2n 2·(2a)2n (t) ( t )} µ (2a)n ′ ( x ) , µ (2a)n ′ ( −x ) (F −f ) (2a)n (F −f ) (2a)n 4 4 2 2 { { ( ((2a)2 − α)(2a − α)t′ ( α )n ) , ≥ τM sup M x, 2((2a)2 − 2a) 2a t′ n ≥ 0, then we get the inequality µJn f (x)−Jn+m f (x)

( n+m−1 ∑ (( (2a)2 )j 4t )) j=n

≥

µ∑n+m−1 j=n

α

α

( n+m−1 ∑ (( (2a)2 )j 4t )) (Jj f (x)−Jj+1 f (x))

α

j=n

α

{ (( (2a)2 )j 4t )} ≥ τM n+m−1 µJj f (x)−Jj+1 f (x) j=n α α { { ) ( j j ( ) (2a) ((2a) + 1)t , ≥ τM n+m−1 τ µ j ((2a)j +1) M (2a) x x j=n Df , 2αj+1 2 2(2a)j (2a)j+1 ( 3(2a)j ((2a)j + 1)t ) µ −3(2a)j ((2a)j +1) ( −x ) , Df 0, 2αj+1 2 (2a)j+1 ( ) j j ) (2a) ((2a) − 1)t µ (2a)j ((2a)j −1) ( −x Df , −x 2αj+1 2 2(2a)j (2a)j+1 )}} ( j j ) 3(2a) ((2a) − 1)t µ 3(2a)j ((2a)j −1) ( x Df 0, 2αj+1 2 (2a)j+1 ≥ M (x, t) for all x ∈ V and all t > 0. And so we can define a mapping F : V → Y by F (x) := limn→∞ Jn f (x) for all x ∈ V and    µf (x)−Jn f (x) (t) ≥ M x, ∑

n−1 j=0

(

t (2a)2 α

 )j 

(2.11)

4 α

for all x ∈ V. Note that for all x, y ∈ V and all t > 0, (t) (t) { (t) ) ( ) µDJn f (x,y) ≥ τM µ (2a)2n ( x , µ , 2n (2a) y −y −x Df (2a)n , (2a) Df (2a) n n , (2a)n 2 8 8 2 2 (t) ( t )} ) ) ( n µ (2a)n ( x , µ (2a) y −y −x Df (2a)n , (2a) − 2 Df (2a) n n , (2a)n 8 8 2 { ( αn t ) ( αn t ) ≥ τM µφ(x,y) , µφ(−x,−y) , 4(2a)2n 4(2a)2n ( αn t )} ( αn t ) µφ(x,y) , µ , φ(−x,−y) 4(2a)n 4(2a)n which tends to 1 as n → ∞. Therefore we can show that F is an additive-quadratic mapping by using the similar fashion after (2.6). By the same reasoning as in the proof of case 1, the relation (2.2) yields the third inequality in (2.11). To complete the proof of the theorem, we are enough to show the uniqueness of F. Suppose that F ′ : V → Y is another mapping satisfying the third inequality in (2.2). If g is an additive-quadratic mapping, then, by (2.9), we have g(x) = Jn g(x) for all x ∈ V

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and all n ∈ N. Observe that µF (x)−F ′ (x) (t) = µJn F (x)−Jn F ′ (x) (t) (t) ( t )} { ≥ τM µJn F (x)−Jn f (x) , µJn f (x)−Jn F ′ (x) 2 2 { (t) (t) ( ) ( ) ≥ τM µ (2a)2n , µ , 2n (2a) x x (F −f ) (2a) (f −F ′ ) (2a) n n 8 8 2 2 ( ) ( ) ( ) t , µ (2a)2n ( ) t , µ (2a)2n −x −x (F −f ) (2a) (f −F ′ ) (2a) n n 8 8 2 2 (t) (t) ( ) ( ) µ (2a)n , µ (2a)n , x x (F −f ) (2a) (f −F ′ ) (2a) n n 8 8 2 2 ( ) ( )} ( ) t , µ (2a)n ( ) t µ (2a)n −x −x (F −f ) (2a)n (f −F ′ ) (2a)n 8 8 2 2 { ( (α − n2 )t′ ( α )n )}} { { ( (α − n2 )t′ ( α )m ) , sup M x, ≥ τM sup M x, 4 n 4 (2a)2 t′ 2 (2a)p −4a2 for all x ∈ X. Acknowledgement. The authors would like to thank the referees for giving useful suggestions and for the improvement of this manuscript. The first author was supported by Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education (No. 2013R1A1A2A10004419). References [1] T. Aoki, On the stability of the linear transformation in Banach spaces, J. Math. Soc. Japan 2 (1950), 64–66. [2] D.H. Hyers, On the stability of the linear functional equation, Proc. Natl. Acad. Sci. USA 27 (1941), 222–224. [3] G.-H. Kim and Y.-H. Lee, Stability of the Cauchy additive and quadratic type functional equation in non-archimedean normed spaces, Far East J. Math. Sci. 76 (2013), 147–157. [4] H.-M. Kim and Y.-H. Lee, Stability of fuctional equation and inequality in fuzzy normed spaces, J. Chungcheong Math. Soc. 26 (2013), 707–721. [5] Y.-H. Lee, Hyers-Ulam-Rassias stability of a quadratic-additive type functional equation on a restricted domain, Int. J. Math. Analysis 7 (2013), 2745–2752. [6] Th.M. Rassias, On the stability of the linear mapping in Banach spaces, Proc. Amer. Math. Soc. 72 (1978), 297–300. [7] B. Schweizer and A. Sklar, Probabilistic metric spaces, Elsevier, North Holand, New York, (1983). ˇ [8] A.N. Serstnev, On the motion of a random normed space, Dokl. Akad. Nauk SSSR 149 (1963), 280–283. [9] S.M. Ulam, A Collection of Mathematical Problems, Interscience, New York, 1960. Ick-Soon Chang*, Department of Mathematics, Chungnam National University, 79 Daehangno, Yuseong-gu, Daejeon 305-764, Republic of Korea E-mail address: [email protected] Yang-Hi Lee, Department of Mathematics Education,, Gongju National University of Education, Gongju 314-711, Republic of Korea E-mail address: [email protected]

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Blow-up of solutions for a vibrating riser equation with dissipative term Junping Zhao College of Science, Xi’an University of Architecture & Technology, Xi’an 710055, China. e-mail: [email protected] Abstract: In this paper we consider a vibrating riser equation with dissipative term and the homogeneous Dirichlet boundary condition. By developing the method in [9] and [16], we establish a blow-up result for certain solutions with non-positive initial energy as well as positive initial energy. Estimates of the lifespan of solutions are also given. Keywords: Blow-up of solution, quasilinear riser problem, positive initial energy AMS Subject Classification (2000): 35L70, 35L15

1

Introduction and main result

In this paper we consider the problem  q   utt + put + 2quxxxx − 2[(ax + b)ux ]x + (u3x )xxx   3    3 2 −[(ax + b)ux ]x − q(uxx ux )x = f (u), (x, t) ∈ [0, 1] × (0, T ),    u(0, t) = u(1, t) = uxx (0, t) = uxx (1, t) = 0, t ∈ (0, T ),     u(x, 0) = u0 (x), ut (x, 0) = u1 (x), x ∈ [0, 1],

(1.1)

where a, b, p, q are nonnegative constant, f (u) is a C(R) function satisfying some conditions to be special later. Problem (1.1) models the behavior of a riser vibrating due to effects of waves and current [14]. In 1997, Bayrack and Can [1] studied problem (1.1) and proved that, under suitable conditions on f and the initial data, all solutions of (1.1) blow up in finite time in the L2 space. To establish their result, the authors used the standard concavity method due to [7]. Gmira and Guedda [4] extended the result of [1] to the multi-dimensional version of the problem (1.1) by using the modified concavity method introduced in [6]. More recently, Hao et al. [5] discussed (1.1) and showed that, under suitable conditions, the solution blows up in finite time with a negative initial energy while exists globally with a nonnegative initial energy for the case p = 0. Precisely, the following blow-up result was established. Theorem 1 Let u(x, t) be a classical solution of the system (1.1). Assume that there exists a positive constant A such that the function f (s) satisfies ∫ s sf (s) ≥ (4 + A) f (υ)dυ for s ∈ R,

(1.2)

0

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and the initial values satisfy ∫ 1 1 q 2 2 E(0) = ∥u1 ∥2 + q∥u0xx ∥2 + (ax + b)u20x dx + ∥u0x u0xx ∥22 2 2 0 ∫ ∫ 1 ∫ u0 1 1 + (ax + b)u40x dx − f (υ)dυdx < 0 4 0 0 0 ∫

and

(1.3)

1

u0 u1 dx > 0.

(1.4)

0

Then the solution u(x, t) of the system (1.1) blows up in a finite time. In the present paper, we shall improve the results of [5] and derive the blow-up properties of solutions of problem (1.1) with non-positive initial energy as well as positive initial energy by developing the method in [9] and [16] (see Remark 2). Estimates of the lifespan of solutions will also be given. For the convenience of our computation, we set p = q = 1 and f (s) = |s|r−1 s. Then the condition (1.2) holds when r > 4. We define the energy function for the solution u of (1.1) by 1 E(t) = ∥ut ∥22 + ∥uxx ∥22 + 2

∫ 0

1

1 1 (ax + b)u2x dx + ∥ux uxx ∥22 + 2 4

∫

1

0

1 (ax + b)u4x dx − ∥u∥rr . (1.5) r

Then E ′ (t) = − ∥ut ∥22 ≤ 0, for t ≥ 0, ∫

and

t

E(t) = E(0) −

(1.6)

∥uτ (τ )∥22 dτ, t ≥ 0.

(1.7)

0

We also set

( α1 =

2 Br

)

1 r−2

,

E1 =

r−2 2 α1 = r

(

1 1 − 2 r

) B r α1r .

(1.8)

where B is the optimal constant of the embedding inequality ∥u∥r ≤ B∥uxx ∥2 , u ∈ H 2 ([0, 1]) ∩ H01 ([0, 1]), for 2 < r < +∞, that is B −1 =

inf

u∈H 2 ([0,1])∩H01 ([0,1]),u̸=0

(1.9)

∥uxx ∥2 . ∥u∥r

We introduce the functionals ∫

∫ t∫

1

1

2

a(t) =

u dx + 0

0

u2 dxdt, t ≥ 0

(1.10)

0

and G(t) = [a(t) + (T1 − t)∥u0 ∥22 ]−δ , t ∈ [0, T1 ], where δ =

r−2 4

(1.11)

and T1 > 0 is a certain constant to be specified later.

Our main result reads as follows.

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Theorem 2 Let u(x, t) be a classical solution of the system (1.1). Assume that r > 4 and either one of the following four conditions is satisfied: 1. E(0) < 0, 2. E(0) = 0 and

∫1 0

u0 u1 dx > 0,

3. 0 < E(0) < E1 and ∥u0xx ∥2 > α1 , { } ∫ √ 2 [( )∫ ] ( 01 u0 u1 dx) 1 r−2 r+2 2 4. E1 ≤ E(0) < min 2r 1 + r+2 0 u0 u1 dx − 2∥u0 ∥2 , 2(1+T )∥u ∥2 . 1

0 2

Then the solution u of the problem (1.1) blows up in a finite time T ∗ in the sense of (2.25). Moreover, the upper bounds for T ∗ can be estimated according to the sign of E(0): For the case 1, T ∗ ≤ t0 − Furthermore, if G(t0 ) < min{1,

√

α −β },

G(t0 ) . G′ (t0 )

then √

1 T ≤ t0 + √ ln √ −β ∗

α −β

α −β

− G(t0 )

.

For the case 2, T∗ ≤ −

G(0) 2(T1 − t + 1)∥u0 ∥22 G(0) = or T ∗ ≤ √ . ∫1 G′ (0) α (r − 2) 0 u0 u1 dx

For the case 3, T ∗ ≤ t0 − √ Furthermore, if G(t0 ) < min{1,

α′ −β ′ },

G(t0 ) . G′ (t0 )

then √

1 T ∗ ≤ t0 + √ ′ ln √ −β

α′ −β ′

α′ −β ′

− G(t0 )

.

For the case 4, δc T ∗ ≤ 2(3δ+1)/2δ √ {1 − [1 + cG(0)]−1/2δ }. α where c = (α/β)2+1/δ . Here α, β, α′ and β ′ are given in (2.23), (2.24), (2.27) and (2.28), respectively. And t0 = t∗ is given by (2.12) for the case 1 and t0 = t∗1 is given by (2.13) for the case 3. Remark 1 Compared with Theorem 1, we have no the restriction

∫1 0

u0 u1 dx > 0 in Theorem 2

when E(0) < 0. Remark 2 E1 defined in (1.8) is exactly the potential well depth obtained by Payne and Sattinger (see [13]). In [16], a global nonexistence theorem for abstract evolution equations with

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nonlinear damping terms was proved by combining the arguments in [3] and [8], where positive initial energy less than E1 was demanded while we allow here a larger positive initial energy (see the case 4). In this work, we divide the case E(0) > 0 into two cases: the case 3 and 4. Unlike [9], we discuss cautiously the case 3 by combining the method of [16] (see Lemma 7). We also note that the case 4 is allowed here since the damping term involved in problem (1.1) is linear. There are many related works on the existence and non-existence of global solutions to the hyperbolic equations with dissipative terms and damping terms, please see [2, 11, 12, 15] and the references therein.

2

Blow-up of the solutions

In this section, we shall prove Theorem 2. We start with a series of Lemmas. Lemma 3 Suppose u(x, t) is a classical solution of the system (1.1). Assume that E(0) < E1 and ∥u0xx ∥2 > α1 . Then there exists a positive constant α2 > α1 , such that ∥uxx (·, t)∥2 ≥ α2 , ∀ t ≥ 0,

(2.1)

∥u(·, t)∥r ≥ Bα2 , ∀ t ≥ 0.

(2.2)

and

Proof. The idea follows from [16] where different type of equations were discussed. We first note that, by (1.5) and (1.9), 1 1 1 E(t) ≥ ∥uxx ∥22 − ∥u∥rr ≥ ∥uxx ∥22 − B r ∥uxx ∥r2 = α2 − B r αr := g(α), r r r

(2.3)

where α = ∥uxx ∥2 . It is easy to verify that g is increasing for 0 < α < α1 , decreasing for α > α1 ; g(α) → −∞ as α → +∞ and g(α1 ) = E1 , where α1 is given in (1.8). Since E(0) < E1 , there exists α2 > α1 such that g(α2 ) = E(0). Let α0 = ∥u0xx ∥2 , then by (2.3) we have g(α0 ) ≤ E(0) = g(α2 ), which implies that α0 ≥ α2 . To establish (2.1), we suppose by contradiction that ∥uxx (t0 )∥2 < α2 for some t0 > 0. By the continuity of ∥uxx (·, t)∥2 we can choose t0 such that ∥uxx (t0 )∥2 > α1 . It follows from (2.3) that E(t0 ) ≥ g(∥uxx (t0 )∥2 ) > g(α2 ) = E(0). This is impossible since E(t) ≤ E(0) for all t ≥ 0. Hence (2.1) is established. To prove (2.2), we exploit (1.5) to see that 1 ∥uxx ∥22 ≤ E(0) + ∥u∥rr . r Consequently, 1 1 ∥u∥rr ≥ ∥uxx ∥22 − E(0) ≥ α22 − E(0) ≥ α22 − g(α2 ) = B r α2r . r r

(2.4)

Therefore (2.2) is concluded.

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Lemma 4

[9]

Let δ > 0 and B(t) ∈ C 2 (0, ∞) be a nonnegative function satisfying B ′′ (t) − 4(δ + 1)B ′ (t) + 4(δ + 1)B(t) ≥ 0.

(2.5)

If B ′ (0) > r2 B(0) + k0 , (2.6) √ then B ′ (t) > k0 for t > 0, where r2 = 2(δ + 1) − 2 (δ + 1)δ is the smallest root of the equation r2 − 4(δ + 1)r + 4(δ + 1) = 0. Lemma 5

[9]

If G(t) is a non-increasing function on [t0 , +∞), t0 ≥ 0 and satisfies the differ-

ential inequality G′ (t)2 ≥ a + bG(t)2+ δ , for t ≥ 0, 1

(2.7)

where a > 0, δ > 0 and b ∈ R, then there exists a finite time T ∗ such that lim G(t) = 0

t→T ∗−

and the upper bound of T ∗ is estimated respectively by the following cases: √ a (i) If b < 0 and G(t0 ) < min{1, −b }, then √ 1 T ≤ t0 + √ ln √ −b ∗

a −b

− G(t0 )

a −b

.

(ii) If b = 0, then G(t0 ) T ∗ ≤ t0 + √ . a (iii) If b > 0, then δc T ∗ ≤ t0 + 2(3δ+1)/2δ √ {1 − [1 + cG(t0 )]−1/2δ }, a where c = (a/b)2+1/δ . Lemma 6 Assume that r > 4, a(t) is defined by (1.10) and let u be a solution of (1.1), then we have a′′ (t) − 4(δ + 1)∥ut ∥22 ≥ Q1 (t), ∫

where Q1 (t) = (−4 − 8δ)E(0) + 2r

t

(2.8)

∥uτ ∥22 dτ + 2(r − 2)∥uxx ∥22 .

0

Proof.

By the definition of a(t), we have a′ (t) = 2

∫

∫

1

0

1

u2 dx,

uut dx +

(2.9)

0

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and by (1.1) and the divergence theorem we get ∫ 1 ∫ 1 ∫ 1 ′′ 2 a (t) = 2 ut dx + 2 uutt dx + 2 uut dxdx 0 0 0 ∫ 1 ∫ 1 ( =2 u2t dx + 2 u |u|r−2 u + 2[(ax + b)ux ]x + [(ax + b)u3x ]x 0 0 ) 1 3 2 +(uxx ux )x − 2uxxxx − (ux )xxx dx 3 ∫ 1 ∫ 1 2 = 2∥ut ∥22 + 2∥u∥rr − 4 (ax + b)u2x dx − 2 (ax + b)u4x dx − 4∥ux uxx ∥22 − 4∥uxx ∥(2.10) 2. 0

0

Using (1.5) and (1.7) we get

= =

≥

≥

a′′ (t) − 4(δ + 1)∥ut ∥22 1 a′′ (t) − 2∥ut ∥22 − (8δ + 4)∥ut ∥22 2 ∫ 1 ∫ 1 (ax + b)u4x dx − 4∥ux uxx ∥22 − 4∥uxx ∥22 (ax + b)u2x dx − 2 2∥u∥rr − 4 0 ( 0 ∫ 1 ∫ t 1 2 2 (ax + b)u2x dx − ∥ux uxx ∥22 ∥uτ ∥2 dτ − ∥uxx ∥2 − −2r E(0) − 2 0 0 ) ∫ 1 1 1 − (ax + b)u4x dx + ∥u∥rr 4 0 r ∫ t ∫ 1 2 2 (−4 − 8δ)E(0) + 2r ∥uτ ∥2 dτ + 2(r − 2)∥uxx ∥2 + 2(r − 2) (ax + b)u2x dx 0 0 ∫ 1 1 + (r − 4) (ax + b)u4x dx + (r − 4)∥ux uxx ∥22 2 0 ∫ t ∥uτ ∥22 dτ + 2(r − 2)∥uxx ∥22 (2.11) (−4 − 8δ)E(0) + 2r 0

since r > 4. Lemma 7 Assume that r > 4 and that either one of the following is satisfied: 1. E(0) < 0, 2. E(0) = 0 and

∫1 0

u0 u1 dx > 0,

3. 0 < E(0) < E1 and √∥2 >)α∫1 , [(∥u0xx ] 1 r+2 2 . 4. E1 ≤ E(0) < 2r 1 + r−2 u u dx − 2∥u ∥ 0 1 0 2 r+2 0 Then a′ (t) > ∥u0 ∥22 for t > t0 , where t0 = t∗ is given by (2.12) for the case 1, t0 = 0 for the cases 2 and 4, and t0 = t∗1 is given by (2.14) for the case 3. Proof.

We consider different cases on the sign of the initial energy E(0).

1. If E(0) < 0, then from (2.8), we have a′ (t) ≥ a′ (0) − 4(1 + 2δ)E(0)t, t ≥ 0. Thus a′ (t) > ∥u0 ∥22 for t > t∗ , where { ∫1 } { ′ } 2 u u dx a (0) − ∥u ∥ 0 1 0 2 0 t∗ = max , 0 = max , 0 . 4(1 + 2δ)E(0) 2(1 + 2δ)E(0)

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(2.12)

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2. If E(0) = 0, then a′′ (t) ≥ 0 for t ≥ 0. Furthermore, if a′ (0) > ∥u0 ∥22 (i.e., then

a′ (t)

>

∥u0 ∥22 ,

∫1 0

u0 u1 dx > 0),

t ≥ 0.

3. If 0 < E(0) < E1 , then using Lemma 3 and (1.8) we see that Q1 (t) ≥ −(4 + 8δ)E(0) + 2(r − 2)α22 > (4 + 8δ)(−E(0) + E1 ) := C1 > 0, t > 0.

(2.13)

Thus, from (2.8), we have a′′ (t) ≥ Q1 (t) > C1 > 0, t > 0. Hence a′ (t) > ∥u0 ∥22 for t > t∗1 , where t∗1 = max

{

{ } ∫1 } −2 0 u0 u1 dx ∥u0 ∥22 − a′ (0) , 0 = max , 0 . C1 C1

4. If E(0) ≥ E1 , we first note ∫ 1 ∫ 2 u dx − 0

∫ t∫

1

u20 dx

1

=2

0

(2.14)

uut dxdt. 0

(2.15)

0

By the H¨older inequality and Young inequality, we have ∫ t ∫ t ∫ 1 ∫ 1 ∥uτ ∥22 dτ. ∥u∥22 dt + u20 dx + u2 dx ≤ 0

0

0

0

By the H¨older inequality, Young inequality again, and (2.15), it follows from (2.9) that ∫ 1 ∫ 1 ∫ t a′ (t) ≤ a(t) + u20 dx + u2t dx + ∥uτ ∥22 dτ. (2.16) 0

0

0

In view of (2.8) and (2.16), we obtain a′′ (t) − 4(δ + 1)a′ (t) + 4(δ + 1)a(t) + K1 ( ) ∫ t ′′ 2 2 2 ≥ a (t) + 4(δ + 1) −∥u0 ∥2 − ∥ut ∥2 − ∥uτ ∥2 dτ + K1 ∫ ≥ (−4 − 8δ)E(0) + 2r ∫ ≥ 4δ

0 t

∫

∥uτ ∥22 dτ + 2(r − 2)∥uxx ∥22 − 4(δ + 1)∥u0 ∥22 − 4(δ + 1)

0 t

t

∥uτ ∥22 dτ + K1

0

∥uτ ∥22 dτ + 2(r − 2)∥uxx ∥22 ≥ 0,

0

where K1 = (4 + 8δ)E(0) + 4(δ + 1)∥u0 ∥22 . Let b(t) = a(t) +

K1 , t > 0. 4(1 + δ)

Then b(t) satisfies (2.5). By (2.6), we see that if ( ) K1 a′ (0) > r2 a(0) + + ∥u0 ∥22 , 4(1 + δ)

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(2.17)

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i.e., r+2 E(0) < 2r

[(

√ 1+

r−2 r+2

)∫

1

] u0 u1 dx −

2∥u0 ∥22

,

0

then a′ (t) > ∥u0 ∥22 , t > 0. The proof is completed. Hereafter, we will find an estimate for the life span of a(t) and prove Theorem 2. Proof of Theorem 2.

By the definition of G(t), we have G′ (t) = −δG(t)1+1/δ (a′ (t) − ∥u0 ∥22 ) G′′ (t) = −δG1+2/δ (t)V (t),

(2.18)

V (t) = a′′ (t)[a(t) + (T1 − t)∥u0 ∥22 ] − (1 + δ)(a′ (t) − ∥u0 ∥22 )2 .

(2.19)

where

For simplicity of calculation, we denote ∫ t 2 ∥u∥22 dt, P = ∥u∥2 , Q =

∫ R=

∥ut ∥22 ,

t

S=

∥uτ ∥22 dτ.

0

0

From (2.9), (2.15) and the H¨older inequality, we get (√ √ ) ∫ a (t) ≤ 2 P R + QS + ′

1

u20 dx.

(2.20)

0

For the case 1 and 2, it follows from (2.8) that a′′ (t) ≥ (−4 − 8δ)E(0) + 4(1 + δ)(R + S).

(2.21)

Applying (2.20) and (2.21), it yields V (t) ≥ [(−4 − 8δ)E(0) + 4(1 + δ)(R + S)][a(t) + (T1 − t)∥u0 ∥22 ] − 4(1 + δ)

(√ √ )2 P R + QS .

Applying (1.11) and (1.10), it follows V (t) ≥ (−4 − 8δ)E(0)G−1/δ (t) + 4(1 + δ)(R + S)(T1 − t)∥u0 ∥22 [ ] (√ √ )2 +4(1 + δ) (R + S)(P + Q) − P R + QS ≥ (−4 − 8δ)E(0)G−1/δ (t). In view of (2.18) we have G′′ (t) ≤ δ(4 + 8δ)E(0)G1+1/δ (t), t ≥ t0 .

(2.22)

Note that by Lemma 7, G′ (t) < 0 for t > t0 . Multiplying (2.22) by G′ (t) and integrating it from t0 to t, we obtain G′ (t)2 ≥ α + βG2+1/δ (t), for t ≥ t0 ,

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where

[ ] α = δ 2 G(t0 )2+2/δ (a′ (t0 ) − ∥u0 ∥22 )2 − 8E(0)G−1/δ (t0 ) > 0

(2.23)

β = 8δ 2 E(0).

(2.24)

and Then by Lemma 5, there exists a finite time T ∗ such that limt↗T ∗− G(t) = 0. Therefore (∫

)

1

2

lim

t↗T ∗−

∫ t∫

1

2

u dx +

u dxdt

0

0

= ∞.

(2.25)

0

For the case 3: 0 < E(0) < E1 , it follows from (2.8) and (2.13) that a′′ (t) ≥ (−4 − 8δ)E(0) + 2(r − 2)∥uxx ∥22 + 4(1 + δ)(R + S) > C1 + 4(1 + δ)(R + S).(2.26) Then using the same arguments as in (1), we have G′′ (t) ≤ −δC1 G1+1/δ (t), G′ (t)2 ≥ α′ + β ′ G2+1/δ (t), [

where ′

2

α =δ G

2+2/δ

′

(t0 ) (a (t0 ) −

∥u0 ∥22 )2

t ≥ t0 ,

] 2C1 −1/δ + G (t0 ) > 0 1 + 2δ

(2.27)

and

2C1 δ 2 . 1 + 2δ Then by Lemma 5, there exists a finite time T ∗ such that (2.25) holds. β′ = −

(2.28)

For the case 4: E(0) ≥ E1 , applying the same discussion as in the case 1, we may get the equalities (2.23) and (2.24) under the condition

E(0)
0,

(1.1)

with boundary condition u = △u = 0, x ∈ ∂Ω, t > 0,

(1.2)

and initial condition u(x, 0) = u0 (x),

x ∈ Ω.

(1.3)

¯ Here Ω ⊂ RN is a bounded domain with smooth boundary, p(x) is a function defined on Ω and k > 0 is the viscosity coefficient. The term k ∂∆u ∂t in (1.1) is interpreted as due to viscous relaxation effects, or viscosity. Equation (1.1) arises as a regularization of the pseudo-parabolic equation ∂u ∂∆u −k = ∆u, ∂t ∂t

(1.4)

which arises in various physical phenomena. (1.4) can be assumed as a model for diffusion of fluids in fractured porous media [1, 5, 6], or as a model for heat conduction involving a thermodynamic temperature θ = u − k∆u and a conductive temperature u [4, 13]. In [2], Bernis investigates a class of higher order parabolic with degeneracy depending on both the unknown functions and its derivatives, the fourth order case of which is the equation ∂ (|u|q−1 sgnu) + D2 (|D2 u|p−1 sgnD2 u) = f ∂t

(1.5)

where p > 1, q > 1 are constants. Some existence result of energy solutions was proved by energy method (see also [12, 17]).

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Motivated by (1.4) and (1.5), we study the problem (1.1)-(1.3) in this paper. Under some assumptions on the initial value, we will establish the existence, uniqueness and asymptotic behavior of weak solutions by the time-discrete method as used in [10, 11]. Equation (1.1) is something like the p-Laplacian equation, but many methods which are useful for the p-Laplacian equation are no longer valid for this equation. Because of the degeneracy, problem (1.1)-(1.3) does not admit classical solutions in general. So, we study weak solutions in the sense of following Definition A function u is said to be a weak solution of (1.1)-(1.3), if the following conditions are satisfied: 1. u ∈ L∞ (0, T ; W0 (Ω))∩C(0, T ; H 1 (Ω)), is the conjugate space of W 2,p(x) (Ω). 2,p(x)

∂u ∂t

∈ L∞ (0, T ; (W 2,p(x) )′ (Ω)), where (W 2,p(x) )′ (Ω)

2. For any φ ∈ C0∞ (QT ) and QT = Ω × (0, T ), the following integral equality holds ∫∫ ∫∫ ∫∫ ∂∇φ ∂φ ∇u dx dt − |△u|p(x)−2 △u△φdx dt = 0 . u dx dt + k ∂t ∂t QT QT QT 3. u(x, 0) = u0 (x). We need some theories on spaces W m,p(x) which we call generalized Lebesgue-Sobolev spaces. We refer the reader to [8] (see also [7, 9]) for some basic properties of spaces W m,p(x) which will be used later. For simplicity we set k = 1 in this paper. This paper is arranged as following. We first discuss the existence of weak solutions in Section 2. Our method for investigating the existence of weak solutions is based on the time discrete method to construct an approximate solutions. By means of the uniform estimates on solutions of the time difference equations, we prove the existence of weak solutions of the problem (1.1)-(1.3). We also prove the uniqueness and asymptotic behavior in Section 3 and Section 4 subsequently.

2

EXISTENCE OF WEAK SOLUTIONS

In this section, we are going to prove the existence of weak solutions. 2,p(x)

Theorem 1 If u0 ∈ W0

¯ p(x) satisfies for some constant L (Ω), p(x) ∈ C(Ω),

¯ −|p(x) − p(y)| ln |x − y| ≤ L, for any x, y ∈ Ω and p = min p(x) > 2. Then the problem (1.1)-(1.3) has at least one solution. ¯ Ω

We use the a discrete method for constructing an approximate solution. First, divide the T interval (0, T ) in N equal segments and set h = N . Then consider the problem 1 1 (uk+1 − uk ) − (∆uk+1 − ∆uk ) + △(|△uk+1 |p(x)−2 △uk+1 ) = 0, h h uk+1 |∂Ω = △uk+1 |∂Ω = 0, k = 0, 1, . . . , N − 1,

(2.1) (2.2)

where u0 is the initial value.

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Lemma 2 For a fixed k, if uk ∈ H01 (Ω), problem (2.1)-(2.2) admits a weak solution uk+1 ∈ 2,p(x) W0 (Ω), such that for any φ ∈ C0∞ (Ω), have ∫ ∫ ∫ 1 1 (uk+1 − uk )φdx + (∇uk+1 − ∇uk )∇φdx + |△uk+1 |p(x)−2 △uk+1 △φdx = 0. (2.3) h Ω h Ω Ω 2,p(x)

Proof. Let us consider the following functionals on the space W0 (Ω) ∫ ∫ ∫ 1 1 1 F1 [u] = |△u|p(x) dx, F2 [u] = |u|2 dx, F3 [u] = |∇u|2 dx, p(x) 2 2 Ω Ω Ω ∫ 1 1 H[u] = F1 [u] + F2 [u] + F3 [u] − f udx, h h Ω where f ∈ H −1 (Ω) is a known function. Using Young’s inequality, there exist constants C1 > 0, such that ∫ ∫ ∫ ∫ 1 1 1 p(x) 2 2 H[u] = |△u| dx + |u| dx + |∇u| dx − f u dx 2h Ω 2h Ω Ω p(x) Ω ∫ 1 ≥ |△u|p(x) dx − C1 ∥f ∥−1 . p+ Ω We need to check that H[u] satisfies the coercive condition. For this purpose, we notice that by u|∂Ω = △u|∂Ω = 0 and using the Lp theory for elliptic equation ([4]), ∥u∥W 2,p(x) ≤ C|△u|p(x) . Therefore, we have H[u] → ∞, as ∥u∥W 2,p(x) → +∞. ∫ Since the norm is lower semi-continuous and Ω f udx is a continuous functional, H[u] is 2,p(x)

weakly lower semi-continuous on W0 (Ω) and satisfying the coercive condition. From [3] we 2,p(x) conclude that there exists u∗ ∈ W0 (Ω), such that H[u∗ ] = inf H[u], and u∗ is the weak solutions of the Euler equation corresponding to H[u], 1 1 u − ∆u + △(|△u|p−2 △u) = f. h h Taking f = (uk −∆uk )/h, we obtain a weak solutions uk+1 of (2.1)–(2.2). The proof is complete. Now, we construct an approximate solution uh of the problem (1.1)-(1.3) by defining kh < t ≤ (k + 1)h, k = 0, 1, . . . , N − 1,

uh (x, t) = uk (x),

uh (x, 0) = u0 (x). The desired solution of the problem (1.1)-(1.3) will be obtained as the limit of some subsequence of {uh }. To this purpose, we need some uniform estimates on uh . Lemma 3 The weak solutions uk of (2.1)–(2.2) satisfy N ∫ ∑ h |△uk |p(x) dx ≤ C, k=1

∫

|△uh (x, t)|p(x) dx ≤ C,

sup 0 0, define jh (s) = h j( h ) and ∫

t−t1 −2h

jh (s)ds.

ηh (t) = t−t2 +2h

Clearly ηh (t) ∈ C0∞ (t1 , t2 ), limh→0+ ηh (t) = 1, for all t ∈ (t1 , t2 ). In the definition of weak solutions, choose φ = φk (x, t)ηh (t), we have ∫ ∫

t2

∫

t1

∫

∫

t2

uφk jh (t − t1 − 2h)dx dt − Ω

∫

∇u∇φk jh (t − t1 − 2h)dx dt −

+ t1

t2

Ω

∫

t2

∫

+

∫

t1

t2

uφk jh (t − t2 + 2h)dx dt Ω

∫

∇u∇φk jh (t − t2 + 2h) dx dt

t1

Ω

∫

t2

∫ ∇u∇φkt ηh dx dt

uφkt ηh dx dt + t1

Ω

∫

t2

∫

143

Ω

|△u|p(x)−2 △u△φk ηh dx dt = 0.

+ t1

t1

Ω

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Observe that ∫ t2 ∫ ∫ uφk jh (t − t1 − 2h)dx dt − (uφk )|t=t1 dx t1 Ω Ω ∫ t1 +3h ∫ ∫ t1 +3h ∫ uφk jh (t − t1 − 2h)dx dt − (uφk )|t=t2 jh (t − t1 − 2h)dx dt = t1 +h Ω t1 +h Ω ∫ (uφk )|t − (uφk )|t dx, ≤ sup 1 t1 +h 0, we have K[u(τ + h)] − K[u(τ )] ≥ ⟨u(τ + h) − u(τ ), u(x, τ ) − △u(x, τ )⟩. By δK[v] δv = v − △v, for any fixed t1 , t2 ∈ [0, T ], t1 < t2 , integrating the above inequality with respect to τ over (t1 , t2 ), we have ∫

t2 +h

t2

∫ K[u(τ )]dτ −

t1 +h

∫ K[u(τ )]dτ ≥

t1

t2

⟨u(τ + h) − u(τ ), u − △u⟩dτ.

t1

Multiplying the both side of the above inequality by 1/h, and letting h → 0, we obtain ⟩ ∫ t2 ⟨ ∂u K[u(t2 )] − K[u(t1 )] ≥ , u − △u dτ. ∂t t1

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Similarly, we have K[u(τ )] − K[u(τ − h)] ≤ ⟨u(τ ) − u(τ − h), u − △u⟩. ∫

Thus

t2

K[u(t2 )] − K[u(t1 )] ≤

⟨

⟩ ∂u , u − △u dτ, ∂t

⟨

⟩ ∂u , u − △u dτ. ∂t

t1

∫

and hence

t2

K[u(t2 )] − K[u(t1 )] =

t1

Taking t1 = 0, t2 = t, we get from the definition of solutions that ⟩ ∫ t⟨ ∂u ∂∆u K[u(t)] − K[u(0)] = − , u(τ ) dτ. ∂t ∂t 0 ∫ t⟨ ⟩ =− △(|△u|p(x)−2 △u), u(τ ) dτ ∫0∫ =− |△u|p(x) dx dτ. Qt

Theorem 8 Let u be the weak solution of the problem (1.1)-(1.3), p > 2. Then ∫ ∫ 2 C3 2 . |∇u(x, t)| dx + , Ci > 0 (i = 1, 2, 3), α = |u(x, t)|2 dx ≤ α (C1 t + C2 ) p −2 Ω Ω Proof. By (4.2), we have f ′ (t) = −

∫ |△u|p(x) dx ≤ 0. Ω

By u ∈

2,p(x) W0 (Ω),

we see that

∫

∫ |∇u(x, t)| dx +

Ω

(∫

∫ |u(x, t)| dx ≤ C

2

|△u| dx ≤ C

2

Ω

|△u|

2

Ω

p(x)

)2/p dx ,

Ω

that is f (t) ≤ C|f ′ (t)|2/p . Again by f ′ (t) ≤ 0, we have f ′ (t) ≤ −Cf (t)p /2 , and hence we complete the proof.

ACKNOWLEDGMENTS This work was supported by the TianYuan Special Funds of the National Natural Science Foundation of China (Grant No. 11526161).

References [1] G. I. Barwnblatt, Iv. P. Zheltov, and I. N. Kochina, Basic concepts in the theory of seepage of homogeneous liquids in fissured rocks, J. Appl. Math. Mech., 24(1960), 1286-1303. [2] F. Bernis, Qualitative properties for some nonlinear higher order degenerate parabolic equations, Houston J. Math., 14(3)(1988), 319-352.

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[3] Kungching Chang, Oritical point theory and its applications, Shanghai Sci. Tech. Press, Shanghai, 1986. [4] P. J. Chen and M. E. Gurtin, On a theory of heat conduction involving two temperatures, Z. Angew. Math. Phys., 19(1968), 614-627. [5] B. D. Coleman, R. J. Duffin and V. J. Mizel, Instability, uniqueness and non-existence theorems for the equations, ut = uxx − uxtx on a strip, Arch. Rat. Mech. Anal., 19(1965), 100-116. [6] E. DiBenedtto and M. Pierre, On the maximum principle for pseudoparabolic equations, Indiana Univ. Math. J., 30(6)(1981), 821-854. [7] X. Fan, J. Shen and D. Zhao, Sobolev embedding theorems for spaces W k,p(x) (Ω), J. Math. Anal. Appl., 262(2001), 749-760. [8] X. Fan and D. Zhao, On the spaces Lp(x) (Ω) and W m,p(x) (Ω), J. Math. Anal. Appl., 263(2001), 424-446. [9] O. Kovacik and J. Rakosnik, On spaces Lp(x) and W k,p(x) , Czechoslovak Math. J., 41:116:4 (1991), 592-618. [10] C. Liu, Weak solutions for a viscous p-Laplacian equation, Electronic Journal of Differential Equations, Vol. 2003(2003), No. 63, pp. 1-11. [11] C. Liu and T. Li, A fourth order degenerate parabolic equation with p(x)-growth conditions, Soochow J. Math., 33 (4)(2007), 813-828. [12] W. Liu and K. Chen, Existence and general decay for nondissipative hyperbolic differential inclusions with acoustic/memory boundary conditions, Math. Nachr., 289 (2-3) (2016), 300-320. [13] W. Liu, K. Chen and J. Yu, Extinction and asymptotic behavior of solutions for the omega-heat equation on graphs with source and interior absorption, J. Math. Anal. Appl., 435 (1) (2016), 112-132. [14] V. R. G. Rao and T. W. Ting, Solutions of pseudo-heat equation in whole space, Arch. Rat. Mech. Anal., 49(1972), 57-78. [15] R. E. Showalter and T. W. Ting, Pseudo-parabolic partial differential equations, SIAM J. Math. Anal., 1(1970), 1-26. [16] T. W. Ting, Parabolic and pseudoparabolic partial differential equations, J. Math. Soc. Japan, 21(1969), 440-453. [17] J. Yin, On the classical solutions of degenerate quasilinear parabolic equations of the fourth order, J. Partial Differential Equations, 2(2)(1989), 39-52.

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Generalizations on some meromorphic function spaces in the unit disc A. El-Sayed Ahmed and M. Al Bogami Sohag University Faculty of Science, Department of Mathematics, 82524 Sohag, Egypt Current Address: Taif University, Faculty of Science, Mathematics Department Box 888 El-Hawiyah, El-Taif 5700, Saudi Arabia

e-mail: [email protected]

Abstract In this paper, we define a general spherical derivative. Making use of this general derivative, we introduce some new classes of meromorphic functions in the unit disk. Also, we introduce some new classes of meromorphic functions which are defined by means of a general chordal distance.

1

Introduction

Let ∆ be the unit disk in the complex plane C, and let dA(z) be the Euclidean area element on ∆. Let H(∆) (resp. M (∆)) denote the class of functions that are analytic (resp.meromorphic) in ∆. The Green’s function in a−z is the M o¨bius transformation ∆ with singularity at a ∈ ∆ is given by g(z, a) = log |ϕa1(z)| , where ϕa (z) = 1−az of ∆. For 0 < r < 1, let ∆(a, r) = {z ∈ ∆ : |ϕa (z)| < r} be the pseudohyperbolic disk with center a ∈ ∆ and radius r. For 0 < p < ∞, the spaces Qp and Mp are defined by (see [1]): Z Z Qp = {f ∈ H(∆) : sup |f 0 (z)|2 (g(z, a))p dA(z) < ∞}, a∈∆

∆

Z Z |f 0 (z)|2 (1 − |ϕa (z)|)p dA(z) < ∞}.

Mp = {f ∈ H(∆) : sup

a∈∆

∆

The Bloch space B (cf. [1] and [16]), is the space of all analytic functions belonging to H(∆), for which B = {f ∈ H(∆) : kf kB = sup (1 − |z|2 )|f 0 (z)| < ∞}. z∈∆

When we study meromorphic functions in ∆, it is natural to replace |f 0 (z)| in these expressions by the spherical # derivative f # (z) = |f 0 (z)|/(1 + |f (z)|2 ) and obtain the classes Q# p , Mp and N , the class of normal function in ∆, respectively (see, for example, Aulaskari, Xiao and Zhao [4] and Wulan [19]). 2010 AMS: Primary 46 E 15, Secondary 30D45 . Key words and phrases: meromorphic functions, QK,ω spaces, chordal distance.

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2 The meromorphic counterpart of BMOA is the set UBC of meromorphic functions of uniformly bounded char# acteristic introduced by Yamashita [21]. It turns out that we have Qp = Mp ([3]), Q# p $ Mp ([5] and [19]). Now, let K : [0, ∞) → [0, ∞) be a right-continuous and nondecreasing function, then the spaces QK and Q# K are defined as follows (see [10, 20]): Definition 1.1 f ∈ H(∆) belongs to the space QK if Z Z kf k2K = kf k2QK = sup |f 0 (z)|2 K(g(z, a)) dA(z) < ∞.

(1)

Definition 1.2 f ∈ M (∆) belongs to the class Q# K if Z Z sup (f # (z))2 K(g(z, a)) dA(z) < ∞.

(2)

a∈∆

a∈∆

∆

∆

# Remark 1.1 It should be remarked that the space Q# K is not a linear space. It is clear that QK and QK are M o¨bius invariant. −2t p Remark 1.2 For 0 < p < ∞, K(t) = tp gives the space Qp and the class Q# ) , p . Choosing K(t) = (1 − e # we obtain Mp and Mp .

Remark 1.3 Choosing K(t) = 1, we get the Dirichlet space D and the spherical Dirichlet class D# . For a fixed r, 0 < r < 1, we choose   1, t ≥ log(1/r), K0 (t) =  0, 0 < t < log(1/r). Then, we obtain

Z Z

Z Z |f 0 (z)|2 K0 (g(z, a)) dA(z) =

|f 0 (z)|2 dA(z)

∆

and

∆(a,r)

Z Z

Z Z #

2

(f # (z))2 dA(z).

(f (z)) K0 (g(z, a)) dA(z) = ∆

∆(a,r)

# , where B# is the class of spherical Bloch functions We conclude that QK0 = B (cf. Axler [6]) and Q# K0 = B (cf. Section 3 in [10] ). It is easy to see that N ⊂ B # (cf. Lappan [14] and the discussion after Definition 2.1 in Wulan [19]). Now, let us introduce the following notation general spherical derivative

fn# (z) =

|f (n) (z)| ; n ∈ N. 1 + |f (z)|n+1

This general derivative gives a plethora of new results on the meromorphic function spaces. Note that if n = 1, we obtain the usual spherical derivative as defined above. α let ω : (0, 1] → (0, ∞) be a nondecreasing function. Let Nn,ω be the class of all normal functions in ∆. We recall that a function f meromorphic in ∆ is said to be ω−normal if and only if (1 − |z|2 )α fn# (z) < ∞. ω(1 − |ϕ (z)|) a z∈∆ sup

Now, we define some general meromorphic classes as follows:

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3 Definition 1.3 Let K : [0, ∞) → [0, ∞) be a nondecreasing function. For n ∈ N, a function f meromorphic in ∆ is said to belong to the class Q# K,n,ω if Z K(g(z, a)) sup (fn# (z))2 dA(z) < ∞. (3) ω(1 − |ϕa (z)|) a∈∆ ∆ Definition 1.4 A function f meromorphic in ∆ is said to be a general spherical Bloch function, denoted by # f ∈ Bn,ω , if there exists an r, 0 < r < 1, such that Z (fn# (z))2 sup dA(z) < ∞. (4) a∈∆ ∆ ω(1 − |ϕa (z)|) # It is easy to see that a normal function is a spherical Bloch function, that is, Nn,ω ⊂ Bn,ω , but the converse is not true . For more information of some related meromorphic function spaces, we refer to [1, 2, 7, 8, 9, 10, 11, 18] and others. For a nondecreasing function K : [0, ∞) → [0, ∞), we say that the space QK is trivial if QK contains only constant functions. Whether our space QK is trivial or not depends on the integral Z 1/e Z ∞ K(log(1/ρ))ρ dρ = K(t)e−2t dt. (5) 0

1

The notation A . B means that there exists a positive constant C such that A ≤ CB. The symbol & is understood in a similar fashion.

2

General meromorphic classes

It is necessary to know for which functions K the classes Q# K,n will be trivial. Here, the square of the general spherical derivative (fn# (z))2 is not necessarily subharmonic, where fn# (z) = Theorem 2.1 If the integral

Z

r 0

is divergent, then the space Z Z ∆

(fn# (z))2

Q# K,n,ω

|f (n) (z)| 1+|f (z)|n+1 ; n

∈ N.

K(log(1/R)) R dR ω(1 − R)

contains only constant functions.

K(g(z, a)) dA(z) ω(1 − |ϕa (z)|)

Z Z

K(g(z, a)) dA(z) ω(1 − |ϕa (z)|) ∆(a,r) µ ¶2 Z Z |f (n) (z)| K(g(z, a)) = dA(z) n+1 1 + |f (z)| ω(1 − |ϕa (z)|) ∆(a,r) µ ¶2 Z Z |f (n) ϕa (z)| K(log(1/|z|)) = |ϕ0a (z)|2 dA(z) n+1 1 + |f (ϕa (z))| ω(1 − |z|) |ϕa (z)| log(1/r)}, Z Z (fn# (z))2 dA(z) ∆(a,r) 2

ω (1 − r) ≤ K(log(1/r))

Z Z ∆(a,r0 )

(fn# (z))2

K(g(z, a)) ω 2 (1 − r) dA(z) ≤ λ 0 and set K1 (r) = inf(K(r), K(1)). # (i) If K is bounded, then Q# K,n,ω = QK1 ,n,ω . # (ii) If K is unbounded, then Q# K,n,ω = Nn,ω ∩ QK1 ,n,ω . Proof: (i) If K is bounded , we have K1 (r) ≤ K(r) ≤

K(∞) K1 (r) K(1)

# and it is clear that Q# K,n,ω = QK1 ,n,ω . # # (ii) By Proposition 2.1, we have Q# K,n,ω ⊂ Nn,ω ∩ QK1 ,n,ω . Now assume that f ∈ Nn,ω ∩ QK1 ,n,ω . We note that K(g(z, a)) = K1 (g(z, a)) in ∆/∆(a, 1/e). (In this domain, we have g(z, a) ≤ 1 ) . To compare the two # suprema in the integrals defining Q# K,n,ω and QK1 ,n,ω , it suffices to deal with integrals over ∆(a, 1/e). Using our assumption that f ∈ Nn,ω , we see that Z Z Z Z K(g(z, a)) 2 # 2 dA(z) ≤ kf kNn,ω (1 − |z|2 )−2 K(g(z, a)) dA(z) (fn (z)) 2 ω (1 − |ϕa (z)|) ∆(a,1/e) ∆(a,1/e) Z Z 1 = kf k2Nn,ω (1 − |z1 |2 )−2 K(log ) dA(z1 ) r ∆(0,1/e) 0 . # # (iv) If Dn,ω = Q# K,n,ω = QK,n,ω,0 , then K(0) = 0 . Proof: # To prove (i), we assume that K(0) > 0 and note that Dn# ⊂ Bn,ω,o = Nn,ω,0 ⊂ Nn,ω . If K is bounded, it is # # # clear that QK,n,ω = Dn,ω . If K is unbounded, we use Theorem 2.3 and the fact that Q# K1 ,n,ω = Dn,ω (we use # # # the notation of Theorem 2.3) to obtain that Q# K,n,ω = Nn,ω ∩ QK1 ,n,ω = Nn,ω ∩ Dn,ω = Dn,ω the proof of (i) is completely established. The proof of (ii) uses the same argument as the proof of Theorem 2.7 in [10] with some simple modifications # # ⊂ Bω,0 = Nn,ω,0 . except that we again use the fact that Dn,ω # To prove (iii), we remark that assumptions imply that Dn,ω * Q# K,n,ω,0 and use (ii). # # If the assumptions of (iv) hold, we have Dn,ω ⊂ QK,n,ω,0 and the conclusion follows from (ii). # Corollary 2.2 Dn,ω ⊂ Q# p,n,ω,0 for all p ,0 < p < ∞.

3

General chordal distance

In this section, we introduce and study some certain new scales of meromorphic functions in the unit disk and solve some problems connected with a general Chordal distance in these scales of spaces. b = C ∪ {∞} is The chordal distance between the points z and w in the extended complex plane C  |z−w|n  if z, w 6= ∞; n ∈ N. 1 1   (1+|z|2 ) n+1 (1+|w|2 ) n+1 χn (z, w) =  1   if w = ∞. 1 (1+|z|2 ) n+1

Remark 3.1 If, we put n = 1 in the general chordal distance, we obtain the usual chordal distance see [2]. The meromorphic Bergman class MαP is defined as the set of those f ∈ M (∆) for which Z (1 − |z|2 )α kf kpMα,ω = χn (f (z), 0)p dA(z) < ∞. p ω(1 − |z|) ∆ Now, we give the following result: Theorem 3.1 Let 1 ≤ p < ∞ , and −1 < α < ∞ and let f ∈ M (∆). Suppose that ¢α Z 1 ¡ 1 − |w| dt t < ∞. ¡ |w| ¢ t3 |w| ω 1 − t Then there exists a positive constant C, depending only on p and α, such that Z Z (1 − |z|2 )α (1 − |z|2 )p+α dA(z) . χn (f (z), f (0))p dA(z) ≤ C (fn# (z))p ω(1 − |z|) ω(1 − |z|) |z| ∆ ∆

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9 Proof: First let p = 1 and let 0 < t < 1. Since Z

1

χn (f (z), f (0)) ≤ 0

Fubini’s theorem and integration by parts yield Z (1 − |z|2 )α dA(z) . χn (f (z), f (0)) ω(1 − |z|) ∆

fn# (tz)|z| dt,

Z Z

1

(1 − |z|2 )α dA(z) ω(1 − |z|) ∆ 0 ¢α ¡ Z 1Z 1 − |w| dt t dA(w) = fn# (w)|w| ¡ |w| ¢ t3 ω 1− t 0 D(0,t) ¢α Z Z 1 ¡ 1 − |w| dt # t dA(w) = fn (w)|w| ¡ |w| ¢ t3 ∆ |w| ω 1 − t Z . (fn# (w))|w|dA(w), fn# (tz) dt|z|

∆

which is the desired asymptotic inequality for p = 1. If p > 1, choose q > ((p − 1)/p) such that α − pq + p > 0. By H¨ older’s inequality, we obtain Z 1 Z 1 |z| dt # χn (f (z), f (0)) ≤ fn (tz)|z| dt = fn# (1 − t|z|)q (1 − t|z|)q 0 0 Z 1 Z 1 pq ¢1/p ¡ ¡ ¢(p−1)/p |z|(p−1)/p dt (1 − t|z|) dt ≤ fn# (tz)p p −p ω (1 − t|z|) 0 ω p−1 (1 − t|z|)(1 − t|z|)pq/(p−1) 0 Z 1 ¡ ¢1/p . fn# (tz)p (1 − t|z|)pq dt|z|(1 − |z|)p−1−pq 0

from which Fubinis theorem yields Z (1 − |z|2 )α dA(z) . χn (f (z), f (0))p ω(1 − t|z|) ∆ = = .

Z Z

1

¡

¢p (1 − |z|)α+p−1−pq fn# (tz) (1 − t|z|)pq dt|z| dA(z) ω(1 − t|z|) ∆ 0 ¢α−pq+p−1 ¡ Z 1Z ¡ # ¢p 1 − |w| dt pq t dA(w) fn (w) (1 − |w|) |w| |w| t3 ω(1 − t ) 0 D(0,t) ¢α+p−1 Z Z 1 ¡ ¡ # ¢p 1 − |w| dt t dA(w) fn (w) |w| 3 |w| ω(1 − t ) t ∆ |w| Z fn# (w)p |w|dA(w). ∆

Theorem 3.2 Let 1 ≤ p < ∞ and −1 < α < ∞, and let f ∈ M (∆). Suppose that Z dA(w) < C |ϕ0w (z)|α+2 ω(1 − |ϕw (z)|)|ϕw (z)|(1 − |w|2 )2 ∆ where C is a positive constant. Then, Z Z Z dA(w) (χn (f (z), f (w))p (1 − |ϕw (z)|2 )α dA(w) ≤ λ |ϕ0w (z)|α+2 . 2 2 4 |1 − wz| ω(1 − ω(1 − |ϕ (z)|) |ϕ (z)|)|ϕ w (z)|(1 − |w| ) w w ∆ ∆ Proof: By the change of variable z = ϕw (u), Theorem 3.1 and Fubini’s theorem, Z Z (χn (f (z), f (w)))p (1 − |ϕw (z)|2 )α I(f ) = dA(z) dA(w) |1 − wz|4 ω(1 − |ϕw (z)|) ∆

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10 Z Z

(1 − |u|2 )α dA(w) dA(u) ω(1 − (1 − |w|2 )2 |u|) ∆ Z Z 2 p+α dA(u) dA(w) p (1 − |u| ) . ((f ◦ ϕw )# n (u)) ω(1 − |u| (1 − |w|2 )2 |u|) ∆ Z Z (1 − |u|2 )α dA(u) dA(w) = (fn# (ϕw (u)))p (1 − |ϕw (u)|2 )p ω(1 − |u|) |u| (1 − |w|2 )2 Z Z ∆ dA(w) |ϕ0w (z)|α+2 dA(z). = (fn# (z))p (1 − |z|2 )p+α 2 2 ω(1 − |ϕ (z)|)|ϕ w w (z)|(1 − |w| ) ∆ ∆ (χn ((f ◦ ϕw )(u), (f ◦ ϕw )(0)))p

=

But since Z ∆

|ϕ0w (z)|α+2

dA(w) < C. ω(1 − |ϕw (z)|)|ϕw (z)|(1 − |w|2 )2

Then,

Z I(f ) ≤ λ ∆

(fn# (z))p (1 − |z|2 )p+α dA(z).

Remark 3.2 In Theorem 3.2, if we put n = 1, we obtain theorem 1.2 in [2]. Corollary 3.1 Let 2 < p < ∞ and f ∈ M (∆). Then there exists a positive constant C, depending only on p, such that Z Z ∆

(p/2)−2 p ¢¡ (1 − |w|2 )(p/2)−2 ¢ χn (f (z) − f (w)) ¡ (1 − |z|2 ) dA(z) dA(w) ≤ Ckf kp # . Bp,n |1 − wz| ω(1 − |z|) ω(1 − |w|)

An application of Theorem 3.1 with α = 0 to the function .(f oϕw )(rz) yields Z

Z p

χn (f (z), f (w)) dA(z) . ∆(w,r)

∆(w,r)

(fn# (z))

p

p ¡ (1

− |z|2 ) ¢ dA(z) , ω(1 − |z|) |ϕw (z)|

(12)

where ∆(w, r) = {z : |ϕw (z)| < r} is the pseudohyperbolic disc of (pseudohyperbolic) center w ∈ ∆ and radius r ∈ (0, 1), and the constant of comparison depends only on r . This fact can be used to prove Theorem 3.3. # The class Mn,ω (p, q, s) consists of those f ∈ M (∆) for which Z kf kp

# Mn,ω (p,q,s)

= sup a∈∆

∆

(fn# (z))

p ¡ (1

q

s

− |z|2 ) ¢¡ (1 − |ϕa (z)|2 ) ¢ dA(z) < ∞. ω(1 − |z|) ω(1 − |ϕa (z)|)

For the next result, let |D(z, r)| denote the Euclidean area of D(z, r), so by [[12], p. 3], we have that |D(z, r)| = πr

(1 − |a|2 )2 (1 − |a|2 r2 )2

(13)

Theorem 3.3 Let 1 ≤ p < ∞, −2 < q < ∞, 0 ≤ s < ∞ and o < r < 1. Let α, β, γ, δ ∈ R such that α + β = q − p, and γ + δ = s, and let f ∈ M (∆). Then Z

¡

sup a∈∆

.

∆

1 |D(z, r)|

Z χn (f (z), f (w))p D(z,r)

¡ (1 − |z|2 )α ¢¡ (1 − |w|2 )β ¢ ω(1 − |z|) ω(1 − |w|)

¡ (1 − |ϕa (z)|2 )γ ¢¡ (1 − |ϕa (w)|2 )δ ¢ ¢ dA(w) dA(z) ≤ kf kp # . Mn,ω (p,q,s) ω(1 − |ϕa (z)|) ω(1 − |ϕa (w)|)

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11 Proof: Routine calculations and (15) show that for w ∈ D(z, r) and a ∈ ∆, 1 − |z|2 ' 1 − |w|2 ' 1 − |wz|2 ' |D(z, r)|1/2 ,

(14)

1 − |ϕa (z)|2 ' 1 − |ϕa (w)|2 ,

(15)

and

where the constants of comparison depend only on r . By (16), (17) and (14), Z Z ¡ ¡ ¢p ¡ (1 − |z|2 )α ¢¡ (1 − |w|2 )β ¢ 1 I = sup χn (f (z), f (w)) ω(1 − |z|) ω(1 − |w|) a∈∆ ∆ |D(z, r)| D(z,r) . .

¢ ¡ (1 − |ϕa (z)|2 )γ ¢¡ (1 − |ϕa (w)|2 )δ ¢ dA(w) dA(z) ω(1 − |ϕa (z)|) ω(1 − |ϕa (w)|) Z Z s 2 p¢ ¡ ¡ dA(w) ¢¡ (1 − |z|2 )q−p−2 ¢ ¡ (1 − |ϕa (z)|2 ) ¢ # p (1 − |w| ) . sup (fn (w)) dA(z) ω(1 − |w|) |ϕz (w)| ω(1 − |z|) ω(1 − |ϕa (z)|) a∈∆ ∆ D(z,r)

from which (16), (17) and Fubini’s theorem yield Z I

.

a∈∆

=

∆

Z

sup a∈∆

'

Z ¡

sup

∆

Z

sup a∈∆

∆

D(z,r)

Z ¡

(fn# (w))p

¡ (1 − |w|2 )q−2 ¢¡ (1 − |ϕa (w)|2 )s ¢ dA(w) ¢ dA(z) ω(1 − |w|) ω(1 − |ϕa (w)|) |ϕz (w)| s

D(z,r)

¡ (1 − |w|2 )q−2 ¢¡ (1 − |ϕa (w)|2 ) ¢ dA(z) ¢ # (fn (w))p dA(w) |ϕz (w)| ω(1 − |w|) ω(1 − |ϕa (w)|)

(fn# (w))p

¡ (1 − |w|2 )q ¢¡ (1 − |ϕa (w)|2 )s ¢ dA(w). ω(1 − |w|) ω(1 − |ϕa (w)|)

The class N of normal functions consists of those f ∈ M (∆) for which the family {f oϕ}, where ϕ is a M o¨bius transformation of ∆, is normal in ∆ in the sense of Montel. It is known that f ∈ M (∆) is all normal if and only if (1 − |z|2 ) kf kNn ,ω = sup fn# (z) < ∞. ω(1 − |z|) z∈∆ The following result establishes a sufficient condition for the general normal meromorphic functions to belong # to Mn,ω (p, q, s). Theorem 3.4 Let 1 ≤ p < ∞, −2 < q < ∞, 0 ≤ s < ∞ and 0 < r < 1,and let f ∈ Nn,ω . Let α, β, γ, δ ∈ R such that α + β = q − p, and γ + δ = s. Then Z Z ¡ ¡ (1 − |w|2 )α/p ¢¡ (1 − |z|2 )β/p ¢ 1 P kf kM # (p,q,s) . sup χn (f (z), f (w)) n,ω ω(1 − |w|) ω(1 − |z|) a∈∆ ∆ |D(z, r)| D(z,r) ¡ (1 − |ϕa (w)|2 )γ/p ¢¡ (1 − |ϕa (z)|2 )δ/p ¢ ¢p . dA(w) dA(z). ω(1 − |ϕa (w)|) ω(1 − |ϕa (z)|) b and define Proof: Let z, w ∈ C,   Fn (z, w) =



w−z 1+wz 1 z

if w ∈ C. if w = ∞.

b Denote the pseudohyA direct calculation shows that |Fn (z, w)|2 = χ2n (z, w)/(1 − χ2n (z, w)) for all z, w ∈ C. perbolic distance between the points z and w in ∆ by ρ(z, w) = |ϕz (w)|. By the uniform (ρ, χ)-continuity of f,

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12 there is an r1 ∈ (0, 1) such that χn (f (z), f (w)) < C, for ρ(z, w) < r1 [13], where C is a positive constant. Then, it follows that χn (f (z), f (w)) < Cχn (f (z), f (w)) |Fn (f (z), f (w))| = p 1 − χ2n (f (z), f (w))

(16)

for ρ(z, w) < r1 . Since f ∈ M (∆), there is an r2 ∈ (0, 1) such that the function gz (w) = Fn ((f oϕz )(w), f (z))is analytic in D(0, r2 ) = {w : ρ(0, w) = |w| < r2 } for all z ∈ ∆, and hence its Maclaurin series is of the form ∞ P ak (z)wk in D(0, r2 ).Therefore k=1

fn# (z)(1

2

− |z| ) = ≤

Z ¯ 2 ¯¯ wgz (w) dA(w)¯ |a1 | = 4 r D(0,r) Z ¯ ¯ 2 ¯Fn ((f oϕz )(w), f (z))¯ dA(w) r3 D(0,r)

for any r ∈ (0, r2 ). Now let r < min{r1 , r2 }. Then, we obtain that Z ¡ (1 − |z|2 )p ¢¡ (1 − |z|)q−p ¢¡ (1 − |ϕa (z)|2 )s ¢ dA(z) I(f ) = (fn# (z) ω(1 − |z|) ω(1 − |z|) ω(1 − |ϕa (z)|) ∆ Z Z q−p ¢¡ ¯ ¯ ¡2 ¢ ¡ (1 − |ϕa (z)|2 )s ¢ ¯Fn ((f oϕz )(w), f (z))¯ dA(u) p (1 − |z|) ≤ dA(z) 3 ω(1 − |z|) ω(1 − |ϕa (z)|) D(0,r) ∆ r Z Z q−p ¢¡ ¯ ¯ ¡2 ¢ ¡ (1 − |ϕa (z)|2 )s ¢ ¯Fn ((f (u), f (z))¯|ϕ0z (u)|2 dA(u) p (1 − |z|) = dA(z) 3 ω(1 − |z|) ω(1 − |ϕa (z)|) D(z,r) ∆ r Z Z ¡C ¢p ¡ (1 − |z|)q−p ¢¡ (1 − |ϕa (z)|2 )s ¢ ≤ dA(z). χn (f (u), f (z))|ϕ0z (u)|2 dA(u) 3 ω(1 − |z|) ω(1 − |ϕa (z)|) D(z,r) ∆ r

(17)

(18)

from which the assertion for r < min{r1 , r2 } follows by (16) and (17). If r ≥ min{r1 , r2 } , choose c > 1 such that r∗ = r/c < min{r1 , r2 }. Then, we easily obtain the assertion for r∗ . To obtain the assertion for r , it remains to make the set of integration larger by replacing D(z, r∗ ) by D(z, r) and note that there is a constant C, depending only on c, such that |D(z, r∗ )| ≥ C|D(z, r)| for all z ∈ ∆.

References [1] R. Aulaskari and P. Lappan, Criteria for an analytic function to be Bloch and a harmonic or meromorphic function to be normal , in: Complex Analysis and Its Applications (Hong Kong, 1993), Pitman Research Notes in Mathematics, 305 (Longman Scientific & Technical, Harlow, 1994), pp. 136–146. ¨ A, ¨ New characterizations of meromorphic Besov, Qp and related [2] R. Aulaskari, S. Makhmutov and J. RAtti classes, Bull. Aust. Math. Soc. 79(2009), 49-58. [3] R. Aulaskari, D. Stegenga and J. Xiao, Some subclasses of BMOA and their characterization in terms of Carleson measures , Rocky Mountain J. Math, 26(1996), 485–506. [4] R. Aulaskari, J. Xiao and R. Zhao, On subspaces and subsets of BMOA and UBC, Analysis, 15(1995), 101–121. [5] R. Aulaskari, H. Wulan and R. Zhao, Carleson measure and some classes of meromorphic functions , Proc. Amer. Math. Soc, 128(2000), 2329-2335. [6] S. Axler, The Bergman space, the Bloch space, and the commutators of multiplication operators, Duke Math. J , 53 (1986), 315-332. [7] C. Chuang, Normal families of meromorphic functions, Singapore: World Scientific. xi, (1993).

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13 [8] A. El-Sayed Ahmed, Lacunary series in some weighted meromorphic function spaces, Math. Aeterna 3(9)(2013), 787-798. [9] A. El-Sayed Ahmed and M. A. Bakhit, Sequences of some meromorphic function spaces, Bull. Belg. Math. Soc. - Simon Stevin 16(3)(2009), 395-408. [10] M. Ess´en and H. Wulan, On analytic and meromorphic functions and spaces of QK type, Illinois J. Math. 46(2002), 1233–1258. [11] C. Fefferman, Characterizations of bounded mean oscillation, Bull. Amer. Math. Soc. 77 (1971), 587-588. [12] J. B. Garnett, Bounded Analytic Functions, Pure and Applied Mathematics, Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], New York-London, (1981). [13] P. Lappan, Some results on harmonic normal functions, Math. Z. 90(1965), 15-159. [14] P. Lappan, A non-normal locally uniformly univalent function, Bull. London Math. Soc. 5(1973), 491–495. [15] P. Lappan and J. Xiao, Q# α -bounded composition maps on normal classes, Note Math. 20 (2000/01), 65–72. [16] C. Pommerenke, Boundary behaviour of conformal maps , Springer-Verlag, Berlin, 1992. [17] R. A. Rashwan, A. El-Sayed Ahmed and A. Kamal, Some characterizations of weighted holomorphic Bloch space, European Journal of Pure and applied Mathematics, 2(2)(2009), 250–267. [18] R. A. Rashwan, A. El-Sayed Ahmed and A. Kamal, Integral characterizations of weighted Bloch space and QK,ω (p, q) spaces, Mathematica, Tome 51 (74)(2009), 63–76. [19] H. Wulan, On some classes of meromorphic functions, Ann. Acad. Sci. Fenn. Ser.A Math. Diss. 116(1998),1– 57. [20] H. Wulan and P. Wu, Characterizations of QT spaces, J. Math. Anal. Appl. 2549(2)(2001), 484–497. [21] S. Yamashita, Functions of uniformly bounded characteristic,Ann. Acad. Sci.Fenn. Ser. A Math. 7(1982), 349–367.

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Maximum Norm Superconvergence of the Trilinear Block Finite Element Jinghong Liu∗and Yinsuo Jia† In this article we discuss a pointwise superconvergence post-processing technique for the gradient of the trilinear block finite element for the Poisson equation with homogeneous Dirichlet boundary conditions over a fully uniform mesh of the three-dimensional domain Ω. First, the supercloseness of the gradients between the piecewise trilinear finite element solution uh and the trilinear interpolant Πu is given. Secondly, we analyze a superconvergence post-processing scheme for the gradient of the finite element solution by using the Z-Z recovery technique, which shows that the recovered gradient of uh is superconvergent to the gradient of the true solution u in the pointwise sense of the L∞ -norm. Finally, a numerical example is given.

1

Introduction

Superconvergence of the gradient for the finite element approximation is a phenomenon whereby the convergent order of the derivatives of the finite element solutions exceeds the optimal global rate. Up to now, superconvergence is still an active research topic; see, for example, Babu˘ska and Strouboulis [1], Chen [2], Chen and Huang [3], Lin and Yan [4], Wahlbin [5] and Zhu and Lin [6] for overviews of this field. Nevertheless, how to obtain the superconvergent numerical solution is an issue to researchers. In general, it needs to use post-processing techniques to get recovered gradients with high order accuracy from the finite element solution. Usual post-processing techniques include interpolation technique, projection technique, average technique, extrapolation technique, superconvergence patch recovery (SPR) technique introduced by Zienkiewicz and Zhu [7–9] and polynomial patch recovery (PPR) technique raised by Zhang and Naga [10]. In previous works, for the linear tetrahedral element, Chen and Wang [11] obtained the recovered gradient with O(h2 ) order accuracy in the average sense of the L2 -norm by using the SPR technique. Using the L2 -projection technique, in the average sense of the L2 -norm, Chen [12] got the recovered gradient 1 with O(h1+min(σ, 2 ) ) order accuracy. Goodsell [13] derived by using the average ∗ Department of Fundamental Courses, Ningbo Institute of Technology, Zhejiang University, Ningbo 315100, China, email: [email protected] † School of Mathematics and Computer Science, Shangrao Normal University, Shangrao 334001, China, email: [email protected]

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technique the pointwise superconvergence estimate of the recovered gradient with O(h2−ε ) order accuracy. Brandts and Kˇr´ıˇzek [14] obtained by using the interpolation technique the recovered gradient with O(h2 ) order accuracy in the average sense of the L2 -norm. Zhang [15, 16] gave the theoretical analysis for the SPR technique for the one-dimensional two points boundary value problem and two-dimensional Laplacian equations, which proved two orders higher than the optimal convergence rate of the finite element solution at the internal nodal points over uniform meshes. Zhang and Victory [17] presented the theoretical justification for superconvergence of the SPR technique for a general secondorder elliptic equation over the quadrilateral meshes. Zhang and Zhu [18, 19] also analyzed the SPR technique in details as well as its applications to a posteriori error estimation. In this article, we consider a SPR recovery scheme by using the Z-Z technique, by which the pointwise superconvergence recovered gradient from the trilinear finite element approximation can be obtained. We shall use the letter C to denote a generic constant which may not be the same in each occurrence and also use the standard notations for the Sobolev spaces and their norms.

2

Maximum Norm Supercloseness

Suppose Ω ⊂ R3 is a rectangular block with boundary, ∂Ω, consisting of faces parallel to the x-, y-, and z-axes. Moreover, Ω is partitioned into a uniform ¯ = ∪ ¯. We rectangulation T h with mesh size h ∈ (0, 1) such that Ω e∈T h e consider the following Poisson equation with homogeneous Dirichlet boundary value conditions { −∆u = f, in Ω (2.1) u = 0, on ∂Ω. The corresponding weak form is a(u , v) = (f , v), ∀ v ∈ H01 (Ω),

(2.2)

∫

where a(u , v) ≡ (∇u , ∇v) =

∇u · ∇v dxdydz. Ω

We introduce a trilinear polynomial space Q1 , namely ∑ aijk xi y j z k , q ∈ Q1 , q(x, y, z) = (i,j,k)∈I

where the indexing set I is as follows: I = {(i, j, k)|0 ≤ i, j, k ≤ 1}. Denote the trilinear finite element space by } { ∩ ¯ S0h (Ω) = v ∈ C(Ω) H01 (Ω) : v|e ∈ Q1 (e), ∀ e ∈ T h .

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Thus the finite element method is to find uh ∈ S0h (Ω) such that a(uh , v) = (f , v), ∀ v ∈ S0h (Ω). Obviously, there is the following Galerkin orthogonality relation a(u − uh , v) = 0, ∀ v ∈ S0h (Ω).

(2.4)

Let the element e = (xe − he , xe + he ) × (ye − ke , ye + ke ) × (ze − de , ze + de ) ≡ I1 × I2 × I3 , ∞ ∞ ¯ ˜ and let {lj (x)}∞ j=0 , {lj (y)}j=0 , {lj (z)}j=0 be the normalized orthogonal Legen2 2 dre polynomial systems on L (I1 ), L (I2 ), and L2 (I3 ), respectively. It is easy to see that {li (x)˜lj (y)¯lk (z)}∞ i,j,k=0 is the normalized orthogonal polynomial system 2 on L (e). Set ∫ x ω0 (x) = ω ˜ 0 (y) = ω ¯ 0 (z) = 1, ωj+1 (x) = lj (ξ) dξ, xe −he

∫ ω ˜ j+1 (y) =

∫

y ye −ke

˜lj (ξ) dξ, ω ¯ j+1 (z) =

z

ze −de

¯lj (ξ) dξ, j ≥ 0.

Define the trilinear interpolation operator of projection type by Πe : H 3 (e) → Q1 (e) such that ∑ βijk ωi (x)˜ ωj (y)¯ ωk (z). (2.5) Πe u(x, y, z) = (i,j,k)∈I

∫ where β000 = u(xe −he , ye −ke , ze −de ), βi00 = I1 ∂x u(x, ye −ke , ze −de )li−1 (x) dx, ∫ ∫ β0j0 = I2 ∂y u(xe −he , y, ze −de )˜lj−1 (y) dy, β00k = I3 ∂z u(xe −he , ye −ke , z)¯lk−1 (z) dz, ∫ ∫ βij0 = I1 ×I2 ∂x ∂y u(x, y, ze − de )li−1 (x)˜lj−1 (y) dxdy, β0jk = I2 ×I3 ∂y ∂z u(xe − ∫ he , y, z)˜lj−1 (y)¯lk−1 (z) dydz, βi0k = I1 ×I3 ∂x ∂z u(x, ye −ke , z)li−1 (x)¯lk−1 (z) dxdz, ∫ βijk = e ∂x ∂y ∂z u li−1 (x)˜lj−1 (y)¯lk−1 (z) dxdydz, i, j, k ≥ 1. In addition, we define (Πu)|e = Πe u. Thus we have the global interpolation operator of projection type Π: H 3 (Ω) → S0h (Ω). In [20], we obtained the following supercloseness estimate Lemma 2.1. Let∩{T h } be a regular family of rectangular partitions of Ω, and u ∈ W 3, ∞ (Ω) H01 (Ω). For uh and Πu, the trilinear block finite element approximation and the corresponding interpolant of projection type to u, respectively. Then we have the following supercloseness estimate 4

|uh − Πu|1, ∞, Ω ≤ Ch2 |ln h| 3 ∥u∥3, ∞, Ω .

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3

Maximum Norm Superconvergence

SPR is a gradient recovery method introduced by Zienkiewicz and Zhu. This method is now widely used in engineering practices for its robustness in a posterior error estimation and its efficiency in computer implementation. For v ∈ S0h (Ω), we denote by Rx the SPR-recovery operator (or Z-Z recovery operator) with respect to the x-derivative, and begin by defining the point values of Rx v at the element nodes. After the recovered derivative values at all nodes are obtained, we construct a piecewise trilinear interpolant by using these values to obtain a global recovered derivative, namely SPR-recovery derivative Rx v. Obviously Rx v ∈ S0h (Ω). Similarly, we can define by Ry and Rz the recovered derivatives with respect to the y-derivative and the z-derivative, respectively. Consequently, we get a recovered gradient operator Rh = (Rx , Ry , Rz ). In the following, we mainly discuss the recovery operator Rx and its superconvergence properties. The superconvergence properties of Ry and Rz can be similarly derived. Let us first assume N is an interior node of the partition T h , and denote by ω the element patch around N containing eight elements (see Fig.1).

c S2

c S3 c S1

c S4 tN c S7

c S6

cS8

c S5

FIG. 1. Element Patch Containing Eight Elements Under the local coordinate system centered N , we let Sj be the barycenter of an element ej ⊂ ω, j = 1, 2, · · · , 8. SPR uses the discrete least-squares fitting to seek linear function p ∈ P1 (ω), such that |∥p − ∂x v|∥ = min |∥g − ∂x v|∥,

(3.1)

g∈P1 (ω)

∑8 1 where |∥w|∥ = ( j=1 |w(Sj )|2 ) 2 . Obviously, for w ∈ P1 (ω), we have |∥w|∥ = 0 ⇐⇒ w = 0

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. It is easy to verify that the problem (3.1) is equivalent to the following problem 8 ∑ [p(Sj ) − ∂x v(Sj )]g(Sj ) = 0, ∀g ∈ P1 (ω).

(3.2)

j=1

Then we define Rx v(N ) = p(0, 0, 0). If N is a node on the boundary, ∂Ω, of Ω, we can calculate Rx v(N ) by the linear extrapolation from the values of Rx v already obtained at two neighboring interior nodes, N1 and N2 , namely Rx v(N ) = 2Rx v(N1 ) − Rx v(N2 ).

(3.3)

Lemma 3.1. Let ω be the element patch around an interior node N , Sj the barycenter of the element ej ⊂ ω, j = 1. · · · .8, and Π the trilinear interpolation operator of projection type. For every u ∈ P2 (ω), we have ∂x (u − Πu)(Sj ) = 0.

(3.4)

Proof. Obviously, Sj is a Gauss point of the element ej ⊂ ω. From the definition of the operator Π,   1 ∑ 1 ∑ ∞ 1 ∑ ∞ ∑ ∞ ∞ ∑ ∞ ∑ ∞ ∑ ∑ ∑  βijk ωi (x)˜ u − Πu =  + + ωj (y)¯ ωk (z). i=0 j=0 k=2

i=0 j=2 k=0

i=2 j=0 k=0

By the representation of the coefficient βijk and the orthogonality of the Legendre polynomial system, we obtain for u ∈ P2 (ω), ∂x (u − Πu)(Sj ) = 0, which is the desired result (3.4). Lemma 3.2. Let ω be the element patch around an interior node N and Π the trilinear interpolation operator of projection type. For every u ∈ P2 (ω), we have ∂x u − Rx Πu = 0 in ω. (3.5) Proof. From (3.4) and the definition (3.1) of the recovery operator Rx , we have for u ∈ P2 (ω), Rx u = Rx Πu. (3.6) Since u ∈ P2 (ω), thus ∂x u ∈ P1 (ω). So we obtain Rx u = ∂x u.

(3.7)

Combining (3.6) and (3.7) yields the desired result (3.5). Lemma 3.3. For Πu ∈ S0h (Ω) the trilinear interpolant of projection type to u, the solution of (2.2), and Rx the x-derivative recovered operator by SPR, we have the superconvergent estimate |∂x u − Rx Πu|0, ∞, Ω ≤ Ch2 ∥u∥3, ∞, Ω .

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Proof. By the triangle inequality, the norms equivalence of the finitedimensional space, and the inverse property, we have |∂x(u − Rx Πu|0, ∞, Ω = |∂x u)− Rx Πu| ( 0, ∞, e ≤ |∂x u|0, ∞, e +) |Rx Πu|0, ∞, e

≤ C |∂x u|0, ∞, e + |∥Rx Πu|∥ ≤ C |∂x u|0, ∞, e + |∥∂x Πu|∥ ( ) ( ) ≤ C |∂x u|0, ∞, ω + |∂x Πu|0, ∞, ω ≤ C |∂x u|0, ∞, ω + h−1 |u|0, ∞, ω ,

(3.9) where ω is an element patch containing the element e. Let uI ∈ P2 (ω) be a quadratic interpolant to u. From (3.5) and (3.9), we obtain by using the interpolation error estimate, |∂x u − Rx Πu|0, ∞, Ω

= |∂x((u − uI ) − Rx Π(u − uI )|0, ∞, e ) ≤ C |∂x (u − uI )|0, ∞, ω + h−1 |u − uI |0, ∞, ω , ≤ Ch2 ∥u∥3, ∞, Ω .

This proves the statement. As for the y-derivative recovery operator Ry and the z-derivative recovery operator Rz , we have the following results similar to (3.8). |∂y u − Ry Πu|0, ∞, Ω ≤ Ch2 ∥u∥3, ∞, Ω .

(3.10)

|∂z u − Rz Πu|0, ∞, Ω ≤ Ch2 ∥u∥3, ∞, Ω .

(3.11)

Set Rh = (Rx , Ry , Rz ). Combining (3.8), (3.10) and (3.11) yields |∇u − Rh Πu|0, ∞, Ω ≤ Ch2 ∥u∥3, ∞, Ω .

(3.12)

In the following, we give the main result of this article. Theorem 3.1. For uh ∈ S0h (Ω) the trilinear block finite element approximation to u, the solution of (2.2), and Rh the gradient recovered operator by SPR, we have the superconvergent estimate 4

|∇u − Rh uh |0, ∞, Ω ≤ Ch2 | ln h| 3 ∥u∥3, ∞, Ω . Proof. Using the triangle inequality and the norms equivalence of the finitedimensional space, we have |∇u − Rh uh |0, ∞, Ω

≤ = ≤ ≤ ≤

|Rh (uh − Πu)|0, ∞, Ω + |∇u − Rh Πu|0, ∞, Ω |R( h (uh − Πu)|0, ∞, e + |∇u − Rh Πu|0, ∞, Ω ) C |∥Rh (uh − Πu)|∥ + |∇u − Rh Πu|0, ∞, Ω ( ) C |∥∇(uh − Πu)|∥ + |∇u − Rh Πu|0, ∞, Ω ( ) C |uh − Πu|1, ∞, Ω + |∇u − Rh Πu|0, ∞, Ω .

(3.13)

Combining (2.6), (3.12) and (3.13) yields 4

|∇u − Rh uh |0, ∞, Ω ≤ Ch2 | ln h| 3 ∥u∥3, ∞, Ω . This proves the statement.

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4

A Numerical Example

Example 1. Consider the following Poisson’s equation: { −∆u = f in Ω = [0, 1] × [0, 1] × [0, 1], u=0 on ∂Ω, where f

=

(−ex (ey − (e − 1)y − 1) − ey (ex − (e − 1)x − 1) +π 2 (ex − (e − 1)x − 1)(ey − (e − 1)y − 1)) sin(πz).

The exact solution is u = (ex − (e − 1)x − 1)(ey − (e − 1)y − 1) sin(πz). Let uh be the trilinear block finite element approximation to the exact solution u and N0 = (0.5, 0.5, 0.5). We solve Example 1 and obtain the following numerical results: Table 4.1 Results of the derivatives post-processing at the interior vertex N0

|∂x u(N0 ) − Rx uh (N0 )| 1.8364e-003 4.0003e-004 9.6873e-005

h 0.25 0.125 0.0625

Acknowledgments This work is supported by the National Natural Science Foundation of China (Grant 11161039). References [1] I. Babu˘ska and T. Strouboulis,The finite element method and its reliability, Numerical Mathematics and Scientific Computation, Oxford Science Publications, 2001. [2] C. M. Chen, Construction theory of superconvergence of finite elements, Hunan Science and Technology Press, Changsha, China, 2001 (in Chinese). [3] C. M. Chen and Y. Q. Huang, High accuracy theory of finite element methods, Hunan Science and Technology Press, Changsha, China, 1995 (in Chinese). [4] Q. Lin and N. N. Yan,Construction and analysis of high efficient finite elements, Hebei University Press, Baoding, China, 1996 (in Chinese). [5] L. B. Wahlbin,Superconvergence in Galerkin finite element methods, Springer Verlag, Berlin, 1995.

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[6] Q. D. Zhu and Q. Lin,Superconvergence theory of the finite element methods, Hunan Science and Technology Press, Changsha, China, 1989 (in Chinese). [7] O. C. Zienkiewicz and J. Z. Zhu, A simple estimator and adaptive procedure for practical engineering analysis, International Journal for Numerical Methods in Engineering, vol. 24, pp. 337–357, 1987. [8] O. C. Zienkiewicz and J. Z. Zhu, The superconvergence patch recovery and a posteriori error estimates. Part 1: The recovery techniques, International Journal for Numerical Methods in Engineering, vol. 33, pp. 1331–1364, 1992. [9] O. C. Zienkiewicz and J. Z. Zhu, The superconvergence patch recovery and a posteriori error estimates. Part 2: Error estimates and adaptivity, International Journal for Numerical Methods in Engineering, vol. 33, pp. 1365–1382, 1992. [10] Z. M. Zhang and A. Naga, A new finite element gradient recovery method: Superconvergence property, SIAM Journal on Scientific Computing, vol. 26, pp. 1192–1213, 2005. [11] J. Chen and D. S. Wang, Three-dimensional finite element superconvergent gradient recovery on Par6 patterns, Numerical Mathematics: Theory, Methods and Applications, vol. 3, no. 2, pp. 178–194, 2010. [12] L. Chen, Superconvergence of tetrahedral linear finite elements, International Journal of Numerical Analysis and Modeling, vol. 3, no. 3, pp. 273–282, 2006. [13] G. Goodsell, Pointwise superconvergence of the gradient for the linear tetrahedral element, Numerical Methods for Partial Differential Equations, vol. 10, pp. 651–666, 1994. [14] J. H. Brandts and M. Kˇr´ıˇzek, Gradient superconvergence on uniform simplicial partitions of polytopes, IMA Journal of Numerical Analysis, vol. 23, pp. 489–505, 2003. [15] Z. M. Zhang, Ultraconvergence of the patch recovery technique, Mathematics of Computation, Vol. 65, pp. 1431–1437, 1996. [16] Z. M. Zhang, Ultraconvergence of the patch recovery technique II, Mathematics of Computation, Vol. 69, pp. 141–158, 2000. [17] Z. M. Zhang and H. D. Victory Jr., Mathematical analysis of ZienkiewiczZhu’s derivative patch recovery technique, Numerical Methods for Partial Differential Equations, vol. 12, pp. 507–524, 1996.

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[18] Z. M. Zhang and J. Z. Zhu, Analysis of the superconvergence patch recovery techniques and a posteriori error estimator in the finite element method (I), Computer Methods in Applied Mechanics and Engineering, vol. 123, pp. 173–187, 1995. [19] Z. M. Zhang and J. Z. Zhu, Analysis of the superconvergence patch recovery techniques and a posteriori error estimator in the finite element method (II), Computer Methods in Applied Mechanics and Engineering, vol. 163, pp. 159–170, 1998. [20] J. H. Liu and Q. D. Zhu, Maximum-norm superapproximation of the gradient for the trilinear block finite element, Numerical Methods for Partial Differential Equations, vol. 23, pp. 1501–1508, 2007.

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HYERS-ULAM STABILITY OF AN ADDITIVE FUNCTIONAL INEQUALITY MING FANG AND DONGHE PEI∗ Abstract. In this paper, we prove that the generalized Hyers-Ulam stability of the additive functional inequality kf (2x + y + 2z) + f (2x + 3y + 3z) + f (4x + 4y + 3z)k ≤ k8f (x + y + z)k in β-homogeneous F -spaces.

1. Introduction and preliminaries The stability problem of functional equations originated from a question of Ulam [32] concerning the stability of group homomorphisms. Hyers [11] gave a first affirmative partial answer to the question of Ulam for Banach spaces. Hyers’ Theorem was generalized by Aoki [1] for additive mappings and by Th.M. Rassias [22] for linear mappings by considering an unbounded Cauchy difference. The paper of Rassias [22] has provided a lot of influence in the development of what we call generalized Hyers-Ulam stability of functional equations. A generalization of the Th.M. Rassias theorem was obtained by G˘avruta [9] by replacing the unbounded Cauchy difference by a general control function in the spirit of Th.M. Rassias’ approach. The stability problems for several functional equations or inequations have been extensively investigated by a number of authors and there are many interesting results concerning this problem (see [2]–[8],[10], [12]–[15], [21]–[24],[25]-[30],[34]). We recall a fundamental result in fixed point theory. Let X be a set. A function d : X × X → [0, ∞] is called a generalized metric on X if d satisfies (1) d(x,y)=0 if and only if x=y; (2) d(x,y)=d(y,x) for all x, y ∈ X; (3) d(x, z) ≤ d(x, y) + d(y, z) for all x, y, z ∈ X. Theorem 1.1 (see[6],[7]). Let (X, d) be a complete generalized metric space and let J : X → X be a strictly contractive mapping with Lipschitz constant L < 1. Then for 2010 Mathematics Subject Classification. Primary 39B62, 39B52, 46B25. Key words and phrases. additive functional equation; Hyers-Ulam stability; fixed point; βhomogeneous F -spaces. ∗ Corresponding author:[email protected] (D.Pei). 1

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each given element x ∈ X, either d(J n x, J n+1 x) = ∞

(1.1)

for all nonnegative integers n or there exists a positive integer n0 such that (1) (2) (3) (4)

d(J n x, J n+1 x) < ∞, for all n ≥ n0 ; the sequence {J n x} converges to a fixed point y ∗ of J; y ∗ is the unique fixed point of J in the set Y = {y ∈ X|d(J n0 x, y) < ∞}; 1 d(y, Jy) for all y ∈ Y . d(y, y ∗ ) ≤ 1−L

By the using fixed point method, the stability problems of several functional inequations have been extensively investigated by a number of authors(see[5][6][14][17]-[18]). We recall some basic facts concerning β-homogeneous F -spaces. Definition 1.2. Let X be a linear space. A nonnegative valued function k · k is an F -norm if it satisfies the following conditions: (FN1 ) kxk = 0 if and only if x = 0; (FN2 ) kλxk = kxk for all x ∈ X and all λ with |λ| = 1; (FN3 ) kx + yk ≤ kxk + kyk for all x, y ∈ X; (FN4 ) kλn xk → 0 provided λn → 0; (FN5 ) kλxn k → 0 provided kxn k → 0. Then (X, k · k) is called an F ∗ -space. An F -space is a complete F ∗ -space. A F -norm is called β-homogeneous (β > 0) if ktxk = |t|β kxk for all x ∈ X and all t ∈ R (see [31]). 2. HYers-Ulam Stability In β-homogeneous F -spaces From now on , Let X be a normed linear space and Y a β-homogeneous F -spaces. This paper,we prove that the generalized Hyers-Ulam stability of the additive functional inequality kf (2x + y + 2z) + f (2x + 3y + 3z) + f (4x + 4y + 3z)k ≤ k8f (x + y + z)k in β-homogeneous F -spaces. Lemma 2.1. Let f : X → Y be a mapping with f (0) = 0. Then it is additive if and only if it satisfies kf (2x + y + 2z) + f (2x + 3y + 3z) + f (4x + 4y + 3z)k ≤ k8f (x + y + z)k

(2.1)

for all x, y, z ∈ X . Proof. If f is additive, then clearly kf (2x + y + 2z) + f (2x + 3y + 3z) + f (4x + 4y + 3z)k = k8f (x + y + z)k

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HYERS-ULAM STABILITY OF AN ADDITIVE FUNCTIONAL INEQUALITY

3

for all x, y, z ∈ X . Assume that f satisfies (2.1). Suppose that f (0) = 0. putting z = 0 and replacing y by −x in (2.1), we get kf (x) + f (−x)k ≤ k8f (0)k = 8β kf (0)k = 0 and so f (−x) = −f (x) for all x ∈ X . Replacing y by −x − z in (2.1), we have kf (−y) + f (−x) + f (x + y)k ≤ 0 for all x, y ∈ X . We obtain f (x + y) = f (x) + f (y) for all x, y ∈ X .



Theorem 2.2. Let f : X → Y be a mapping with f (0) = 0. If there is a function ϕ : X 3 → [0, ∞) such that kf (2x + y + 2z) + f (2x + 3y + 3z) + f (4x + 4y + 3z)k ≤ k8f (x + y + z)k + ϕ(x, y, z)

(2.2)

and ∞ X
1 j j j ϕ(x, e y, z) := ϕ (−2) x, (−2) y, (−2) z l,



1
1 l m

(2.5)

(−2)l f (−2) x − (−2)m f ((−2) x) ≤

m−1 X i=l


1 i i i ϕ −(−2) x, −(−2) x, (−2) 2x 2βi

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n

f ((−2)k x) (−2)k

o

for all x ∈ X . It follows from (2.5) that the sequence is a Cauchy sequence n o k x) for all x ∈ X . Since Y is an F -space, the sequence f ((−2) converges. So one may (−2)k define the mapping A : X → Y by   f ((−2)k x) A(x) := lim , ∀x ∈ X . k→∞ (−2)k Taking m = 0 and letting l tend to ∞ in (2.5), we have the inequality (2.4). It follows from (2.2) that kA(2x + y + 2z) + A(2x + 3y + 3z) + A(4x + 4y + 3z)k 1 f ((−2)k (2x + y + 2z)) + f ((−2)k (2x + 3y + 3z)) = lim k→∞ (−2)kβ

+f ((−2)k (4x + 4y + 3z)) (2.6) 1

8f ((−2)k (x + y + z)) + lim 1 ϕ((−2)k x, (−2)k y, (−2)k z) ≤ lim k→∞ (−2)kβ k→∞ (−2)kβ ≤ k8A(x + y + z)k for all x, y, z ∈ X . One see that A satisfies the inequality (2.1) and so it is additive by Lemma (2.1). Now, we show that the uniqueness of A. Let T : X → Y be another additive mapping satisfying (2.4). Then one has

1

1 k k

kA(x) − T (x)k = A (−2) x − T (−2) x

(−2)k (−2)k

1 ≤ kβ A (−2)k x − f (−2)k x

2


+ T (−2)k x − f (−2)k x
1 ≤ 2 kβ ϕ e −(−2)k x, −(−2)k x, (−2)k 2x 2 which tends to zero as k → ∞ for all x ∈ X. So we can conclude that A(x) = T (x) for all x ∈ X.  Theorem 2.3. Let f : X → Y be a mapping with f (0) = 0. If there is a function ϕ : X 3 → [0, ∞) satisfying (2.2) such that   ∞ X x y z βj ϕ(x, e y, z) := 2 ϕ , , 0 [14, p. 67–70]. In this paper, we assume that the system (1)-(2) has a unique Carath´eodory solution for all (u, x0 ) ∈ L∞ (R+ , Rn ) × Rm . Consider the time scale change sγ (t) = t/γ, ∀γ > 0, ∀t ≥ 0. When the input u ◦ sγ is used instead of u, system (1)-(2) becomes
x˙ γ (t) = f xγ (t), u ◦ sγ (t) , t ≥ 0, (3) xγ (0) = x0 , (4) which can be written for all γ > 0 as Zt σγ (t) = x0 + γ


f σγ (τ ) , u (τ ) dτ, ∀t ∈ [0, ωγ ),

(5)

0

where σγ = xγ ◦ s1/γ and [0, ωγ ) is the maximal interval for the existence of solutions σγ . We seek necessary conditions and also sufficient conditions for the uniform convergence of the sequence of functions σγ .

3

Necessary Conditions

Our aim in this section is to derive necessary conditions for the uniform covergence of the sequence of functions σγ . 2

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Lemma 3.1. Assume that the maximal solution of system (1)-(2) is defined on R+ for all (u, x0 ) ∈ L∞ (R+ , Rn ) × Rm . Suppose that there exists a function h : R+ × R+ → R+ such that
|x (t)| ≤ h |x0 | , kuk∞ , ∀t ≥ 0, (6) for each initial state x0 ∈ Rn and each input u ∈ L∞ (R+ , Rn ). Assume that there exists a function qu ∈ L∞ (R+ , R
m )∩C 0 (R+ , Rm ) such that limγ→∞ kσ
γ − qu k∞ = 0. Then, we have f x0 , u(0) = 0, qu (0) = x0 , and f qu (t) , u (t) = 0, ∀t ≥ 0. Proof. From the fact that kuk∞ = ku ◦ sγ k , ∀γ > 0 and Inequality (6) it comes that kxγ k∞ ≤ h (|x0 | , kuk∞ ) = a. ∀γ > 0, Thus, we get from the continuity of σγ that |σγ (t)| ≤ a, ∀t ≥ 0, ∀γ > 0.

(7)

Inequality (7) along with the continuity of function f and the boundedness of the input u imply that there exists a constant r > 0 independent of γ, such that |f (σγ (τ ) , u (τ ))| ≤ r, ∀τ ≥ 0, ∀γ > 0. This means that we can apply the Dominated Lebesgue Theorem in Equation (5) and get Z lim

γ→∞

t

Z

t

f (qu (τ ) , u (τ )) dτ, ∀t ≥ 0,

f (σγ (τ ) , u (τ )) dτ = 0

(8)

0

where the continuity of f and the fact that limγ→∞ kσγ − qu k∞ = 0 are used. By Equation (7) we have kσγ − x0 k∞ /γ → 0 as γ → ∞. Thus, we obtain from (5) and (8) that Z t f (qu (τ ) , u (τ )) dτ = 0, ∀t ≥ 0, 0


which gives f qu (t) , u (t) = 0 for almost all t ≥ 0. From the continuity of functions f , qu , and u it comes that
f qu (t) , u (t) = 0, for all t ≥ 0. (9) Since σγ (0) = x0 , ∀γ > 0 it comes that qu (0) = x0 .

(10)

Finally, taking t = 0 in (9) and using (10) provides the necessary condition
f x0 , u(0) = 0, (11) which completes the proof. Remark 1. Once chosen an input u, the term u(0) is given so that any initial condition x0 for which we have limγ→∞ kσγ − qu k∞ = 0 should satisfy (11).

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4

Sufficient Conditions

In this section, we derive sufficient conditions to ensure that the sequence of functions σγ converges uniformly as γ → ∞. Definition 4.1. [7] A continuous function β : R+ → R+ is said to belong to class K∞ if it is strictly increasing, satisfies β (0) = 0, and limt→∞ β (t) = ∞. Lemma 4.1. [11] Consider a function z : [0, ω) ⊂ R+ → R+ , where ω may be infinite. Assume the following (i) The function z is absolutely continuous on each compact subset of [0, ω). (ii) There exist z1 , z2 ≥ such that z1 , z (0) < z2 and z˙ (t) ≤ 0 for almost all t ∈ [0, ω) that satisfy z1 < z (t) < z2 .
Then, z(t) ≤ max z (0) , z1 , ∀t ∈ [0, ω). Corollary 4.1. Consider a function z : [0, ω) ⊂ R+ → R+ , where ω may be infinite. Assume the following (i) The function z is absolutely continuous on each compact subset of [0, ω). (ii) There exist a class K∞
function β : R+ → R+ and z1 , z2 , z3 ≥ 0 such that max β −1 (z3 ) , z1 , z (0) < z2 , and z˙ (t) ≤ −β (z (t))+z3 for almost all t ∈ [0, ω) that satisfy z1 < z (t) < z2 .
Then, z (t) ≤ max z (0) , z1 , β −1 (z3 ) , ∀t ∈ [0, ω).
Proof. We have z˙ (t) ≤ 0 for almost all t ∈ [0, ω) that satisfy max β −1 (z3 ) , z1 < z (t) < z2 , and hence the result follows directly from Lemma 4.1. Lemma 4.2. Assume that there exists qu ∈ W 1,∞ (R+ , Rn ) such that
f qu (t) , u (t) = 0, ∀t ≥ 0, qu (0) = x0 .

(12) (13)

Define yγ : R+ → Rm as yγ (t) = σγ (t) − qu (t) = xγ (γt) − qu (t) , ∀γ > 0,

(14)

for all t ∈ [0, ωγ ). Suppose that we can find a continuously differentiable function V : Rm → R+ that satisfies the following: (i) V is positive definite, that is V (0) = 0 and V (α) > 0, ∀ 0 6= α ∈ Rm . (ii) V is proper, that is V (α) → ∞ as |α| → 0. (iii) There exist δ > 0 and β ∈ K∞ satisfying: 

  dV (α) · f yγ (t) + qu (t) , u (t) ≤ −β |yγ (t)| , dα α=yγ (t)   for all t ∈ [0, ω ) and ∀γ > 0 that satisfy |y (t)| < δ. γ γ

(15)

Then, • ωγ = +∞, ∀γ > 0. Furthermore, there exist E, γ ∗ > 0 such that kxγ k∞ ≤ E, ∀γ > γ ∗ , for any solution xγ of the system (3)-(4). 4

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• limγ→∞ kσγ − qu k∞ = 0. Proof. Since V is positive definite and proper, there exists β1 , β2 ∈ K∞ such that (see [7, p. 145]) β1 (|α|) ≤ V (α) ≤ β2 (|α|) , ∀α ∈ Rm .

(16)

From (5), we get for almost all t ∈ [0, ωγ ), ∀γ > 0 that
y˙ γ (t) = γ f yγ (t) + qu (t), u(t) − q˙u (t), (17) yγ (0) = 0. (18)
For any γ > 0, define Vγ : R+ → R+ as Vγ (t) = V yγ (t) , ∀t ∈ [0, ωγ ). Note that the function Vγ is absolutely continuous on each compact subset of [0, ωγ ) as a composition of a continuously differentiable function V and an absolutely continuous function yγ . Then, we get for almost all t ∈ [0, ωγ ) and all γ > 0 that h i
dV (α) dV (α) ˙ · y ˙ (t) = · γf y (t)+q (t) , u (t) − q ˙ (t) . Vγ (t) = γ γ u u dα dα α=yγ (t)

α=yγ (t)

(19)
Let Ω = 0, β1 (δ) . By (16) we have for any γ > 0, and for almost all t ∈ [0, ωγ ) that Vγ (t) ∈ Ω ⇒ |yγ (t)| < δ. (20) We conclude from (15), (19), and (20) that dV (α) ˙ , for almost all t ∈ [0, ωγ ),∀γ > 0 that satisfy Vγ (t) ∈ Ω. Vγ (t) ≤ −γ β (|yγ (t)|)+kq˙u k∞ dα α=yγ (t) (α) Thus, we deduce from the continuity of dVdα , the boundedness of q˙u , and (20) there exists some b > 0 independent of γ such that

V˙ γ (t) ≤ −γ β (|yγ (t)|)+b, for almost all t ∈ [0, ωγ ),∀γ > 0 that satisfy Vγ (t) ∈ Ω. Hence, (16) implies
V˙ γ (t) ≤ −γ β◦β2−1 Vγ (t) +b, for almost all t ∈ [0, ωγ ), ∀γ > 0 that satisfy Vγ (t) ∈ Ω. Thus, Corollary 4.1 and the fact that Vγ (0) = 0, ∀γ > 0, imply that Vγ (t) ≤   b −1 b β2 ◦ β γ , ∀γ > γ0 , ∀t ∈ [0, ωγ ) where γ0 = β◦β2−1 ◦β1 (δ) . Therefore, (16) implies that   b −1 |yγ (t)| ≤ β1 ◦ β2 ◦ β , ∀γ > γ0 , ∀t ∈ [0, ωγ ). (21) γ Thus, ωγ = +∞, ∀γ > γ1 for some γ1 > 0, and limγ→∞ kyγ k∞ = 0, which is equivalent to limγ→∞ kσγ − qu k∞ = 0. On the other hand, (21) and the fact that σγ = yγ + qu imply that there exists some E, γ ∗ > 0 such that kσγ k∞ ≤ E, ∀γ > γ ∗ , and hence kxγ k∞ ≤ E, ∀γ > γ ∗ . Lemma 4.3. Consider the nonlinear system [13] x˙ = f (x, u) = Ax + Φ (x) + R (u) , x (0) = x0 , y = Dx,

(22) (23) (24)

5

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where x0 ∈ Rm , A is an m × m Hurwitz matrix2 , D is an m × m matrix, input u ∈ L∞ (R+ , Rn ), state x, output y take values in Rm , function R ∈ C 0 (Rn , Rm ), and a locally Lipshitz function Φ ∈ C 0 (Rm , Rm ). Assume the following: (i) The exists qu ∈ W 1,∞ (R+ , Rm ) such that qu (0) = x0 and

Aqu (t) + Φ qu (t) + R u (t) = 0, ∀t ≥ 0. (ii) There exist c1 > 0, c2 > 0, ξ > 0 and r > 2 such that 

 α· Φ α+qu (t) −Φ qu (t) ≤ c1 |α|2 +c2 |α|r , for almost all t ≥ 0, ∀α ∈ Rm that satisfy |α| < ξ. (iii) One has c1 < 2 λ1max , where λmax is the largest eigenvalue for the m × m positive-definite symmetric matrix P that satisfies3 P A + AT P = −Im×m .

(25)

Let xγ , yγ be respectively the state and the output of (22)-(24) when we use the input u ◦ sγ instead of u. Then, • All solutions of (22)-(24) are bounded. Furthermore, there exist E, γ ∗ > 0 such that kxγ k∞ ≤ E, ∀γ > γ ∗ , for any solution xγ of the system (3)-(4). • limγ→∞ kFγ − Dqu k∞ = 0, where Fγ : R+ → Rm is defined as Fγ (t) = yγ (γt) , ∀t ≥ 0, ∀γ > 0. Proof. Since Φ is locally Lipschitz, the right-hand side of (22) is locally Lipschitz relative to x and hence the system (22) has a unique solution. The function qu satisfies (12)-(13) in Lemma 4.2 because of (i). Consider the continuously differentiable quadratic Lyapunov function candidate V : Rm → R such that V (α) = αT P α, ∀α ∈ Rm . Since P is symmetric, we have ∀α ∈ Rm that 2

2

λmin |α| ≤ V (α) = αT P α ≤ λmax |α| , where λmin is the smallest eigenvalue of the matrix P . Thus V is positive definite and proper. Since P is symmetric we have dV (α) m (26) dα = 2 |P α| ≤ 2λmax |α| , ∀α ∈ R . We have by (25) that
dV (α) 2 · Aα = 2P α · Aα = αT P A + AT P α = − |α| , ∀α ∈ Rm . dα

(27)

From Condition (i) we get for all γ > 0 that h i dV (α) dV (α) · f (yγ + qu , u) = · Ayγ + Aqu + Φ (yγ + qu ) + R (u) dα α= yγ dα α=yγ h i dV (α) = · Ayγ + Φ (yγ + qu ) − Φ (qu ) . dα α=yγ (28) 2 that 3 the

is each eigenvalue of A has a strictly negative real part. existence of the matrix P in (25) is guaranteed because A is Hurwitz [7, p.136].

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where yγ is defined in (14). We get from (28), (27), (26) and Condition (ii) that
dV (α) 2 r ·f yγ (t) + qu (t) , u (t) ≤ (−1 + 2c1 λmax ) |yγ (t)| + 2c2 λmax |yγ (t)| , dα α=yγ (t) ∀γ > 0 for almost all t ∈ [0, ωγ ) that satisfy |yγ (t)| < ξ, (29) where [0, ωγ ) is the maximal interval of existence of σγ and yγ . This leads to
1 − 2c1 λmax dV (α) 2 · f yγ (t) + qu (t) , u (t) ≤ − |yγ (t)| , dα α=yγ (t) 2 ! r r−2 1 − 2c1 λmax ∀γ > 0, for almost all t ∈ [0, ωγ ) that satisfy |yγ (t)| < min ,ξ . 4c2 λmax (30)  q

 1 λmax Thus, (15) in is satisfied with β (v) = 1−2c12 λmax v 2 , ∀v ≥ 0 and δ = min r−2 1−2c 4c2 λmax , ξ . Hence all conditions of Lemma 4.2 are satisfied so that the solution of (22) is bounded. Morover, there exist E, γ ∗ > 0 such that kxγ k∞ ≤ E, ∀γ > γ ∗ . Futhermore, we have limγ→∞ kσγ − qu k∞ = 0. Thus, we deduce from (24) that limγ→∞ kFγ − Dqu k∞ = 0. Example. Consider the system x˙ = −x + x3 − u, x (0) = 0.

(31) (32)

where state x takes values in R and input u ∈ W 1,∞ (R+ , R) is defined as u(t) = 0.1 sin (t), ∀t ≥ 0. The system (31)-(32) has the form (22)-(24), with x = y, m = n = 1, A = −1, Φ (α) = α3 , R(α) = −α, ∀α ∈ R, and D = 1. Observe that P in (25) equals 1/2 which mean that λmin = λmax = 1/2. We have u (0) = 0 and u is bounded with u(·) ∈ [umin , umax ] = [−0.1, 0.1] . (33) h i h i 2 2 √ Define the function χ : − √13 , √13 → − 3√ , as χ (v) = −v + v 3 , ∀v ∈ 3 3 3 i h − √13 , √13 . The function χ is strictly decreasing, bijective and its inverse function is continuous.h Hence, there exists a function qu ∈ C 0 (R+ , R) ∩ L∞ (R+ , R) i such that qu (·) ∈ − √13 , √13 , qu (0) = 0 and
χ qu (t) = −qu (t) + qu3 (t) = u (t) , ∀t ≥ 0.

(34)

It can be checked using (33) that kqu k∞ < 0.11 (see Figure (1b)). Thus
qu (·) 6= √13 . This fact and (34) implies that the function q˙u = u/ ˙ 1 − 3qu2 is bounded so that qu ∈ W 1,∞ (R+ , R). Hence Condition (i) of Lemma 4.3 is satisfied. On the other hand, we have for all α ∈ R that α (Φ (α + qu ) − Φ (qu )) = 3qu2 α2 + 3qu α3 + α4 .

(35)

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Since kqu k∞ < 0.11, one has 3qu2 ∞ < 0.0363 = c1 . Hence it follows from (35) that for any ξ > 0 we have 

 α Φ α + qu (t) − Φ qu (t) ≤ c1 α2 + (3 kqu k∞ + ξ) α3 ∀α ∈ Rm that satisfy |α| < ξ, for almost all t ≥ 0. (36)

0.2

0.2

0.1

0.1

0

qu(t)

σγ(t) = Fγ(t)

Thus, Condition (ii) in Lemma 4.3 is satisfied with c2 = 3 kqu k∞ + ξ. Moreover, we have c1 < 1 = 2 λ1max which implies that Condition (ii) in Lemma 4.3 is also satisfied. Therefore, the solution of (31)-(32) is bounded, that there exist E, γ ∗ > 0 such that kxγ k∞ ≤ E, ∀γ > γ ∗ , and that limγ→∞ kσγ − qu k∞ = limγ→∞ kFγ − qu k∞ = 0 (observe that σγ (·) = Fγ (·) because x (·) = y (·)). This is illustrated in Figure 1a.

γ=1 γ=3 γ=6 q (t)

−0.1

0 −0.1

u

−0.2 0

1

2

3

4 t

5

6

7

8

(a) Fγ (t) versus t for system (31).

−0.2 −0.1

−0.05

0 u(t)

0.05

0.1

(b) qu (t) versus u(t) for system (31).

Figure 1: Simulations.

5

Conclusion

In [5] a rule for deciding whether a process may or may not be a hysteresis is proposed for causal operators such that a constant input leads to a constant output. That rule involves checking whether the so-called consistency and strong consistency properties hold. In this paper we derived necessary conditions and sufficient ones for the uniform convergence of the shifted solutions σγ : t → xγ (γt) of the system x˙ = f (x, u ◦ sγ ). This uniform convergence is related to consistency. Does this mean that the concept of consistency can be extended to study operators for which the property that a constant input leads to a constant output, that property does not hold? This paper explores this issue for systems of the form x˙ = f (x, u), however, no clear cut answer may be drawn for the obtained results. Indeed, the necessary conditions alone cannot guarantee whether the uniform convergence of σγ when γ → ∞ happens or not. The sufficient conditions do imply that convergence but do not guarantee that the hysteresis loop of the operator is not trivial. In the example, we have seen that qu is a function of u so that the hysteresis loop is a curve and we cannot acertain from this whether system (31) is a hysteresis or not. This is a future research line.

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References [1] M. Brokate, and J. Sprekels, Hysteresis and phase transitions, SpringerVerlag, New York, 1996. [2] R. Dong, Y. Tan, H. Chen, and Y. Xie, “A neural networks based model for rate-dependent hysteresis for piezoceramic actuators”, Sensors and Actuators A: Physical, vol. 143, no. 2, pp. 370-376, 2008. [3] C. Enachescu, R. Tanasa1, A.Stancu1, G. Chastanet, J.-F. Ltard, J. Linares, and F. Varret, “Rate-dependent light-induced thermal hysteresis of [Fe(PM-BiA)2(NCS)2] spin transition complex”, Journal of Applied Physics, vol. 99, 08J504, 2006. [4] J. Fuzi, and A. Ivanyi, “Features of two rate-dependent hysteresis models”, Physica B: Condensed Matter, vol. 306, no. 1-4, pp. 137-142, 2001. [5] F. Ikhouane, “Characterization of hysteresis processes”, Mathematics of Control, Signals, and Systems, vol. 25, no. 3, pp. 294-310, 2013. [6] F. Ikhouane, and J. Rodellar, Systems with hysteresis: analysis, identification and control using the Bouc-Wen model, Wiley, Chichester, UK, 2007. [7] H. K. Khalil, Nonlinear systems, third edition, Prentice Hall, Upper Saddle River, New Jersey, 2002, ISBN 0130673897. [8] M. A. Krasnosel’skii, and A. V. Pokrovskii, Systems with hysteresis, Springer-Verlag, Berlin Heidelberg, 1989. [9] I. Mayergoyz, Mathematical models of hysteresis, Elsevier Series in Electromagnetism, New-York, 2003. [10] J.W. Macki, P. Nistri, and P. Zecca, “Mathematical models for hysteresis”, SIAM Review, vol. 35, no. 1, pp. 94-123, 1993. [11] M. F. M. Naser, and F. Ikhouane (2013). Consistency of the Duhem model with hysteresis, Math. Problems in Eng., 2013, Article ID 586130, 1-16. [12] M. F.M. Naser, and F. Ikhouane, “Hysteresis loop of the LuGre model”, Automatica, vol. 59, pp. 48-53, 2015. [13] J. Oh, B. Drincic and D.S. Bernstein, “Nonlinear feedback models of hysteresis”, IEEE Control Systems, vol. 29, no. 1, pp. 100–119, 2009. [14] K. Schmitt, and R. Thompson, Nonlinear analysis and differential equations: an introduction, lecture notes, University of Utah, Department of Mathematics, 1998. [15] A. Visintin, Differential models of hysteresis, Springer-Verlag, Berlin, Heidelberg, 1994.

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TABLE OF CONTENTS, JOURNAL OF COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 22, NO. 1, 2017

Some Perturbed Versions of the Generalized Trapezoid Inequality for Functions of Bounded Variation, Wenjun Liu and Jaekeun Park,…………………………………………………11 A Companion of Ostrowski Like Inequality and Applications to Composite Quadrature Rules, Wenjun Liu and Jaekeun Park,……………………………………………………………..19 A Modified Shift-Splitting Preconditioner for Saddle Point Problems, Li-Tao Zhang,……25 Closed-Range Generalized Composition Operators Between Bloch-Type Spaces, Cui Wang, and Ze-Hua Zhou,………………………………………………………………………………38 Approximate Ternary Jordan Bi-Derivations on Banach Lie Triple Systems, Madjid Eshaghi Gordji, Vahid Keshavarz, Choonkil Park, and Jung Rye Lee,………………………………45 Some Generalized Difference Sequence Spaces of Ideal Convergence and Orlicz Functions, Kuldip Raj, Azimhan Abzhapbarov, and Ashirbayev Khassymkhan,………………………52 A General Stability Theorem for a Class of Functional Equations Including Quadratic-Additive Functional Equations, Yang-Hi Lee and Soon-Mo Jung,……………………………………64 A Dynamic Programming Approach to Subsistence Consumption Constraints on Optimal Consumption and Portfolio, Ho-Seok Lee and Yong Hyun Shin,……………………………79 The Stability of Cubic Functional Equation with Involution in Non-Archimedean Spaces, Chang Il Kim and Chang Hyeob Shin,………………………………………………………………100 Value Sharing Results for Meromorphic Functions with Their q-Shifts, Xiaoguang Qi, Jia Dou, and Lianzhong Yang,…………………………………………………………………………107 Random Normed Space and Mixed Type AQ-Functional Equation, Ick-Soon Chang, and YangHi Lee,…………………………………………………………………………………………117 Blow-up of Solutions for a Vibrating Riser Equation with Dissipative Term, Junping Zhao,..128 Existence, Uniqueness and Asymptotic Behavior of Solutions for a Fourth-Order Degenerate Pseudo-Parabolic Equation with p(x)-Growth Conditions, Junping Zhao,……………………138 Generalizations on Some Meromorphic Function Spaces in the Unit Disc, A. El-Sayed Ahmed and M. Al Bogami,…………………………………………………………………………….148

TABLE OF CONTENTS, JOURNAL OF COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 22, NO. 1, 2017 (continued)

Maximum Norm Superconvergence of the Trilinear Block Finite Element, Jinghong Liu, and Yinsuo Jia,………………………………………………………………………………………161 Hyers-Ulam Stability of an Additive Functional Inequality, Ming Fang and Donghe Pei,……170 Characterization of a Class of Differential Equations, Mohammad Fuad Mohammad Naser, Omar M. Bdair, and Fayçal Ikhouane,………………………………………………………....179

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J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 22, NO.2, 2017, COPYRIGHT 2017 EUDOXUS PRESS, LLC

Fuzzy analytical hierarchy process based on canonical representation on fuzzy numbers Yong Denga,b,c,∗ a

School of Electronics and Information, Northwestern Polytechnical University, University, Xi’an, 710072, China b Big Data Decision Institute, Jinan University, Tianhe, Guangzhou, 510632, China c Department of Civil Engineering and Environment Science, School of Engineering, Vanderbilt University, Nashville, TN, 37235, USA

Abstract Fuzzy analytical hierarchy process(FAHP) is widely used in multi-criteria decision making (MCDM) under uncertain environments. Many works have been proposed. However, the existing methods are complex and time consuming. What’s more, the conflict management in AHP is still an open issue. To solve these issues, a novel and simple FAHP method is proposed based on the canonical representation of multiplication operation on fuzzy numbers in this paper. We adopt the main idea of classical AHP, that is the weight of each criterion can be determined by its relative ratio. The relative ratio can be easily determined in the proposed method. In addition, the average method is adopted to handle conflicts in AHP. An example on supplier selection is used to illustrate the efficiency of our proposed method. Keywords: Analytical Hierarchical Process, fuzzy numbers, fuzzy AHP, canonical representation of fuzzy numbers, supplier selection. 1. Introduction Analytical Hierarchy Process(AHP) is a powerful tool for handling both qualitative and quantitative multi-criteria factors in decision-making problems, developed by Saaty [1] in the 1970s. This method has been extensively ∗

Corresponding author: Yong Deng, Department 801, School of Electronics and Information, Northwestern Polytechnical University, Xian, Shaanxi, 710072, China. Email: [email protected]; [email protected] Preprint submitted to JOCAAA

February 3, 2016

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studied and refined since then. It provides a comprehensive and rational framework for structuring a decision problem, for representing and quantifying its elements, for relating those elements to overall goals, and for evaluating alternative solutions. With this method, a complicated problem can be converted to an ordered hierarchical structure. AHP method has been widely applied to multi-criteria decision making situations[2], such as: web sites selection[3], tools’ evaluation[4], e-business [5], drugs selection[6], group decision [7, 8]and so on[9, 10, 11, 12]. Multi-Criteria analysis problems require the decision maker to make qualitative assessments regarding the performance of the decision alternatives with respect to each independent criterion and the relative importance of each independent criterion with respect to the overall objective of the problem [13, 14]. As a result, uncertain subjective data are present which make the decision making process complex. Many math tools are developed. For example, evidence theory is heavily studied since it can fuse different data which make it widely used in multi-criteria decision making [15, 16, 17]. Due to the flexibility to handle linguistic information [18], the fuzzy sets theory is also widely used in many uncertain decision makings [19, 20, 21, 22, 23]. As a result, the classical AHP is extended to fuzzy AHP (FAHP) [24] and is applied to many MCDM applications under uncertain environment,such as environmental assessment and management[25, 26, 27], supplier management[28], group decision making [29], fuzzy MCDM[30], fuzzy MADM [31], and so on [32]. Two key issues should be solved in the application of fuzzy AHP. One issue is that how to determine the weight of each criterion when the elements of comparison matrix are fuzzy numbers. Unlike the classical AHP, the eigenvector of fuzzy comparison matrix cannot be obtained directly. Hence, some other steps are inevitable to get the final weights in most existing fuzzy AHP methods[24, 33],which makes the FAHP more completed to some degrees. The other key problem when applying the AHP is to avoid rank reversal[34]. Due to the different preference and subjective and objective factors in decision making, evidence connected from different sources are often conflicting[35, 36, 37, 38]. How to deal with conflict and dependence in AHP is still an open issue [39, 40, 41, 42]. In classical AHP, a well known coefficient, called as Consistency index(CI), is used to measure the conflicting degree in decision making. In some application systems, the AHP model should be adjusted when the CI is higher than a certain threshold value. The problem still exists in fuzzy AHP. Many methods have been proposed to handle this 2

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problem[43, 44]. In order to construct decision matrices of pairwise comparisons based on additive transitivity, Herrera-Viedma et al. propose consistent fuzzy preference relations[45]. In [43], the distance function between two linguistic preference relations is defined, then a new CI is defined based on the distance function. In [44, 46], a method is proposed to construct fuzzy linguistic preference relations, called as fuzzy LinPreRa method. However, it should be pointed out that is difficult to give a corresponding CI in fuzzy AHP. To handel these two issues mentioned above, we propose a novel and simple FAHP in this paper. On the one hand, we use the canonical representation of multiplication operation on fuzzy numbers, presented in [47], to obtain the weigh of each criterion in a straight and easy manner. On the other hand, we suggest to use average method to deal with conflicts in AHP decision making. The numerical example on supplier selection shows the efficiency of our proposed method. The paper is organized as follows. Section 2 begins with a brief introduction to the basic theory used in the proposed method,including AHP, fuzzy set theory and genetic algorithm. A typical fuzzy AHP is also introduced in this section. The proposed methodology is detailed in section 3. In section 4, our proposed method is applied to supplier selection. Section 5 concludes the paper. 2. Preliminaries 2.1. Analytical Hierarchy Process[1] The first step of AHP is to establish a hierarchical structure of the problem. Then, in each hierarchical level, use a nominal scale to construct pairwise comparison judgement matrix. Definition 2.1. Assuming (E1 , · · · , Ei , · · · , En ) are n decision elements, the pairwise comparison judgement matrix is denoted as Mn×n = [mij ], which satisfies: mij =

1 mji

(1)

where each element mij represents the judgment concerning the relative importance of decision element Ei over Ej . With the matrix constructed, the third step is to calculate the eigenvector of the matrix. 3

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Definition 2.2. Eigenvector of n×n pairwise comparison judgement matrix can be denoted as: w ⃗ = (w1 , · · · , wi , · · · , wn )T , which is calculated as follows: λmax ≥ n

Aw ⃗ = λmax w, ⃗

(2)

where λmax is the maximum eigenvalue in the eigenvector w ⃗ of matrix Mn×n . Before we transform the eigenvector into the weights of elements, the consistency of the matrix should be checked. Definition 2.3. Consistency index(CI)[1] is used to measure the inconsistency within each pairwise comparison judgement matrix, which is formulated as follows: CI =

λmax − n n−1

(3)

Accordingly, the consistency ratio(CR) can be calculated by using the following equation: CI RI

CR =

(4)

where RI is the random consistency index. The value of RI is related to the dimension of the matrix, which is listed in Table 1. Table 1: The value of RI(random consistency index)

dimension RI

1 2 0

0

3

4

5

6

7

8

9

10

0.52

0.89

1.12

1.26

1.36

1.41

1.46

1.49

If the result of CR is less than 0.1, the consistency of the pairwise comparison matrix M is acceptable. Moreover, the eigenvector of pairwise comparison judgement matrix can be normalized as final weights of decision elements. Otherwise, the consistency is not passed and the elements in the matrix should be revised.

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2.2. Fuzzy sets In 1965, the notion of fuzzy sets was firstly introduced by Zadeh[18], providing a natural way of dealing with problems in which the source of imprecision is the absence of sharply defined criteria of class membership[48]. A brief introduction of Fuzzy sets are given as follows. Definition 2.4. A fuzzy set A is defined on a universe X may be given as: A = {⟨x, µA (x)⟩ |x ∈ X } where µA : X → [0, 1] is the membership function A. The membership value µA (x) describes the degree of belongingness of x ∈ X in A. For a finite { set A = {x1 , . . . , xi , . . . , xn }, the fuzzy } set (A, m) is often µ (x ) µ (x ) µ (x ) A 1 / ,..., A i / ,..., A n/ denoted by x1 xi xn . In real application, the domain experts may give their opinions by fuzzy numbers. For example, in a new product price estimation, one expert may give his opinion as: the lowest price is 2 dollars, the most possibility price of the product may be 3 dollars, the highest price of this product will not be in excess of 4 dollars. Hence, we can use a triangular fuzzy number (2,3,4) to represent the expert’s opinion. The triangular fuzzy numbers can be defined as follows. Definition 2.5. A triangular fuzzy number A˜ can be defined by a triplet (a, b, c) , where the membership can be determined as follows A triangular fuzzy number A˜ = (a, b, c) can be shown in Fig.(1).  0, x < a    x−a , a 6 x 6 b b−a µA˜ (x) = c−x ,b 6 x 6 c    c−b 0, x > c

(5)

In Fig2. N 1, N 3, N 5, N 7 and N 9 are used to represent the pairwise comparison of decision variables from Equal to Absolutely preferred, and TFNs N 2, N 4, N 6 and N 8 represent the middle preference values between them.

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P A ( x) 1

0

a

b

X

c

Figure 1: A triangular fuzzy number.

Equal

Moderate

N1

N2

N3

N4

1

2

3

4

Very Fairly Absolute Strong Strong N9 N5 N6 N7 N8

μN(x)

1.0

0.5

0.0 5

6

7

8

9

Figure 2: Nine fuzzy numbers

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2.3. Canonical representation operation on fuzzy numbers In this section, the canonical representation of operation on triangular fuzzy numbers which are based on the graded mean integration representation method [47], is used to obtain the weight of each criterion in a simple manner. The canonical representation operation on fuzzy numbers is applied to many decision makings [49, 50]. Definition 2.6. Given a triangular fuzzy number A˜ = (a1 , a2 , a3 ), the graded mean integration representation of triangular fuzzy number A˜ is defined as: ˜ = 1 (a1 + 4 × a2 + a3 ) P (A) (6) 6 ˜ = (b1 , b2 , b3 ) be two triangular fuzzy numbers. Let A˜ = (a1 , a2 , a3 ) and B By applying Eq.(6), the graded mean integration representation of triangular ˜ can be obtained, respectively, as follows: fuzzy numbers A˜ and B ˜ = 1 (a1 + 4 × a2 + a3 ) P (A) 6 ˜ = 1 (b1 + 4 × b2 + b3 ) P (B) 6 The representation of the addition operation ⊕ on triangular fuzzy numbers ˜ can be defined as : A˜ and B ˜ = P (A) ˜ + P (B) ˜ = 1 (a1 + 4 × a2 + a3 ) + 1 (b1 + 4 × b2 + b3 ) (7) P (A˜ ⊕ B) 6 6 The canonical representation of the multiplication operation on triangular ˜ is defined as : fuzzy numbers A˜ and B ˜ = P (A) ˜ × P (B) ˜ = 1 (a1 + 4 × a2 + a3 ) × 1 (b1 + 4 × b2 + b3 ) (8) P (A˜ ⊗ B) 6 6 2.4. FAHP In this section, we briefly introduce a typical FAHP method . For detailed information, please refer [51, 52]. In the first step,triangular fuzzy numbers are used for pair-wise comparisons. Then, by using extent analysis method the synthetic extent value Si of the pair-wise comparison is introduced and by applying the principle of the comparison of fuzzy numbers, the weight vectors with respect to each element under a certain criterion is calculated. The details of the methodology are presented in the following steps: 7

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Let X = {x1 , x2 , . . . , xn } be an object set,and U = {u1 , u2 , . . . , um } be a goal set.According to the method of Changs extent analysis, each object is taken and an extent analysis for each goal, gi , is performed.Therefore,m extent analysis values for each object can be obtained, with the following signs: Mgi1 , Mgi2 , . . . , Mgim , i = 1, 2, . . . , n, where all the Mgij (j = 1, 2, . . . , m) are TFN’s. Step 1: The value of fuzzy synthetic extent with respect to the ith object is defined as { n m }−1 m ∑ ∑∑ j j Si = Mgi ⊗ Mgi (9) j=1

In order to obtain

m ∑

i=1 j=1

Mgij , perform the fuzzy addition operation of m extent

j=1

analysis values for a particular matrix such that ( m ) m m m ∑ ∑ ∑ ∑ j Mgi = lj , mj , uj j=1

{ To obtain

m n ∑ ∑

j=1

j=1

(10)

j=1

}−1 Mgij

, perform the fuzzy addition operation of Mgij (j =

i=1 j=1

1, 2, . . . , m) values such that n ∑ m ∑ i=1 j=1

( Mgij =

n ∑

li ,

i=1

n ∑

mi ,

i=1

n ∑

) ui

(11)

i=1

and then compute the inverse of the vector. Step 2: The degree of possibility of M2 = (l2 , m2 , u2 ) ≥ M1 = (l1 , m1 , u1 ) is expressed as: V (M2 ≥ M1 ) = hgt(M1 ≥ M2 )  if m2 ≥ m1  1, (l1 −u2 ) otherwise =  ((m2 −u2 )−(m1 −l1 )) 0, if l1 ≥ u2

(12)

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To compare M1 and M2 both V (M2 ≥ M1 ) and V (M1 ≥ M2 ) are required. Step 3: The degree of possibility for a convex fuzzy number to be greater than k convex fuzzy numbers Mi (i = 1, 2, . . . , k) can be defined as: V (M ≥ M1 , M2 , . . . , Mk ) = V [(M ≥ M1 ) and (M ≥ M2 ) and ... and (M ≥ Mk )] = min V (M ≥ Mi ), i = 1, 2, . . . , k

(13)

Let d′ (Ai ) = min V (Si ≥ Sk ), for k = 1, 2, . . . , n; k ̸= i. Then the weight vector is given by: W ′ = (d′ (A1 ), d′ (A2 ), . . . , d′ (An ))T

(14)

Step 4: The weight vector obtained in step 3 is normalized to get the normalized weights. 3. The proposed methodology One of the most key issue in fuzzy AHP is how to determine the weights given the fuzzy pairwise comparison judgement matrix. For example, given the linguistic data in Table 2, how can we get the weight of each criterion? In the following of this section, we solve the problem step by step. Table 2: The Fuzzy evaluation of criteria with respect to the overall objective

C1

C2

C3

C4

C5

C1 (1,1,1) (3/2,2,5/2) (3/2,2,5/2) (5/2,3,7/2) C2 (2/5,1/2,2/3) (1,1,1) (3/2,2,5/2) (5/2,3,7/2) C3 (2/5,1/2,2/3) (2/5,1/2,2/3) (1,1,1) (3/2,2,5/2) C4 (2/7,1/3,2/5) (2/7,1/3,2/5) (2/5,1/2,2/3) (1,1,1) C5 (2/7,1/3,2/5) (2/7,1/3,2/5) (2/5,1/2,2/3) (2/5,1/2,2/3)

(5/2,3,7/2) 0.3283 (5/2,3,7/2) 0.2839 (3/2,2,5/2) 0.1798 (3/2,2,5/2) 0.1262 (1,1,1) 0.0818

9

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3.1. Transformation of fuzzy comparison matrix Let’s consider the element in the comparison matrix classical AHP. The rating in the matrix means the relative importance of the criterion. For example, suppose only two criterion in a comparison matrix, listed as follows. Example 3.1. The comparison matrix is given as follows C1 C2

[ C1 C2 ] 1 3 1/3 1

From the matrix, the element C12 = 3 means that weight of the second criterion C2 is three times of that of the first criterion C1 . In addition, the eigenvector of comparison matrix can be easily obtained as follows: [ ] [ ] w1 0.75 = w2 0.25 Two important points should be noticed: First point, the sum of the eigenvector of comparison matrix should be ONE. For example,w1 + w2 = 0.75 + 0.25 = 1. Second point, the ratio of the weight should be coincide with the corresponding element in comparison matrix. In Example 3.1, we can get w1 /w2 = 0.75/0.25 = 3 = C12 . This idea of AHP can be easily adopted in fuzzy AHP. For example, in the Table 2, the element C12 = (3/2, 2, 5/2). According to the above analysis, we understand that the weight of the second criterion C2 is (3/2,2,5/2) times of that of the first criterion C1 (Notice: for the sake of simplicity, we suppose that (3/2, 2, 5/2) is not a linguistic variable N2 shown in Fig.2, but a simple fuzzy number to model the fuzzy variable ”ABOUT 2”). The only difference between this case with Example 3.1 is that one is a crisp number 3 while the other is a fuzzy number (3/2, 2, 5/2). How to represent the weight of the second criterion C2 is (3/2,2,5/2) times of that of the first criterion C1 in the canonical representation of multiplication operation on fuzzy numbers? According to the Eq.(8), we obtain the follow result. ˜ P (A˜ ⊗ B) = (1, 1, 1) ⊗ (3/2, 2, 5/2) = 61 (1 + 4 × 1 + 1) × 16 (3/2 + 4 × 2 + 5/2) =1×2 =2

(15)

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The Eq.(15) means that the the weight of the second criterion C2 is (3/2,2,5/2) times of that of the first criterion C1 could also be stated as ”the weight of the second criterion C2 is 2 times of that of the first criterion C1 under the canonical representation of multiplication operation on fuzzy numbers”. The other element of the canonical representation of multiplication operation on fuzzy numbers can also be determined and shown in Table3. Table 3: Evaluation of criteria with respect to the overall objective based on canonical representation of multiplication operation

C1

C2

C3

C4

C1 1 2 2 3 C2 46/90 1 2 3 C3 46/90 46/90 1 2 C4 212/630 212/630 46/90 1 C5 212/630 212/630 46/90 46/90

C5 3 3 2 2 1

We call the matrix in Table3 the comparison matrix with canonical representation of multiplication operation (CMCRMO) Let’s us see the first row of Table 3. If we suppose that the relative weight of the first criterion is 1, then we get that: 1)both the the relative weight of the second and the third criterion is 2; 2)both the the relative weight of the fourth and the fifth criterion is 3. Then, a straight way to obtain the corresponding weight is with the simple normalization of these relative weights. The result can be shown as follows. wC1 1 =

1 1+2+2+3+3

=

1 11

wC2 1 =

2 1+2+2+3+3

=

2 11

wC3 1 =

2 1+2+2+3+3

=

2 11

wC4 1 =

3 1+2+2+3+3

=

3 11

wC5 1 =

3 1+2+2+3+3

=

3 11

(16)

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In Eq.(16), the subscript C1 means that the weight is obtained according to the criterion C1 . The weight in Eq.(16) is not the final weight of each criterion since that there exists conflict in this situation, also called rank reversal[34, 43, 44, 46, 45]. This problem will be handled in the following part. 3.2. Conflict management with average method It should also be mentioned that the second point, namely ”the ratio of the weight should be coincide with the corresponding element in comparison matrix ” can be satisfied on some ideal situations. However, the preference order will not be always keep coincided in the whole AHP process. In real application, the comparison matrix given by experts may not strictly obey the preference order as shown in Example3.2. Example 3.2. The comparison matrix is given as follows C1 C2 C3 C4

C1 C2 C3 1 3 5  1/3 1 1/3   1/5 3 1 1/7 1/3 1/2

C4  7 3   2  1

From the first row of above comparison matrix, we can see that the importance ranking corresponding to C1 is C1 < C2 < C3 < C4 . However, from the second row of above comparison matrix, we can see that the importance ranking corresponding to C2 is C1 < C3 < C2 < C4 The consistency index index defined in Definition2.3 show the conflict in preference. In classical AHP, the CI is used to determine how consistence of the comparison matrix. If the value of CI is higher than a threshold, then some adjustments to deal with rank reversal should be made. Though many methods have been proposed on this filed, it is still an open issue. In decision making with fuzzy AHP, it is also inevitable. For example, see the first line of the Table 3, we get the following preference ranking order. 12

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Table 4: The Fuzzy evaluation of criteria with respect to the overall objective

Preference ranking order C1 C2 C3 C4 C5

C1 < C2 = C3 < C4 = C5 C1 < C2 < C3 < C4 = C5 C1 = C2 < C3 < C4 = C5 C1 = C2 < C3 < C4 < C5 C1 = C2 < C3 = C4 < C5

As can be seen from Table4, for C1 , the weight of C2 is equal to C3 . However, for C2 ,C3 ,C4 and C5 , the weight of C2 is less than C3 . There are many other conflicts in the ranking order. In this paper, we use average to decrease the conflict in the preference order. We average the weights of all criteria to get the final weight of each criterion. That is, if we get the the comparison matrix with canonical representation of multiplication operation (CMCRMO) shown in Table3, we can obtain the final weight of each criterion with the normalization of average weight of each criterion. Example 3.3. Suppose we get the comparison matrix with canonical representation of multiplication operation (CMCRMO) shown in Table3, we can get the average weight of the five criteria, respectively as follows wCav1 =

1 5

wCav2 =

1 5

wCav3 =

i

5 ∑ i

5 ∑ 1 5

wCav4 =

1 5

wCav5

1 5

=

5 ∑

i 5 ∑ i

5 ∑ i

wCCR = 51 (3 + 3 + 2 + 2 + 1) 5 wCCR = 4

1 5

wCCR = 3

1 5

wCCR = 2

1 5

wCCR = 1

1 5

( ( ( (

3+3+2+1+ 2+2+1+ 2+1+ 1+

46 90

46 90

+

46 90

+

46 90

)

46 90

+

212 630

+

46 90

+

212 630

) 212 630

+

(17) )

212 630

)

wCavi

Here, means the average weight of the ith’s criterion, the superscript av denotes average. wCCR means the canonical representation of multiplicai tion operation of the ith’s criterion, the superscript CR denotes canonical 13

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representation. The final weight of the ith’s criterion, wCf i , can be obtained with the normalization of average weight of each criterion wCavi and listed as follows wCf 1 =

av wC

1

av +w av +w av +w av +w av wC C C C C 1

2

3

4

=

5

11 1698 + 2636 + 3794 + 5992 +11 630 630 630 630

= 0.3283

wCf 2 =

av wC 2 av av av +w av +w av wC +wC +wC C4 C5 1 2 3

=

5992 630 1698 2636 + 630 + 3794 + 5992 +11 630 630 630

= 0.2839

wCf 3 =

av wC 3 av +w av +w av av av wC +wC +wC C5 C4 3 2 1

=

3794 630 1698 2636 + 630 + 3794 + 5992 +11 630 630 630

= 0.1798

=

2636 630 1698 2636 3794 + + + 5992 +11 630 630 630 630

= 0.1262

=

1898 630 1698 2636 + 630 + 3794 + 5992 +11 630 630 630

= 0.0818

wCf 4 = wCf 5 =

av wC

4

av +w av +w av +w av +w av wC C C C C 1

2

3

4

5

av wC 5 av av av +w av +w av wC +wC +wC C4 C5 1 2 3

(18)

Note that to detail our proposed method in a easily understood way, we suppose that the fuzzy number C12 = (3/2, 2, 5/2) means that the the weight of the second criterion C2 is (3/2, 2, 5/2)times of that of the first criterion C1 . However, according to the Figure2, the case is verse, where C12 = (3/2, 2, 5/2) means that the the weight of the second criterion C1 is (3/2, 2, 5/2)times of that of the first criterion C2 . As a result, if we use the linguistic variables shown in Figure2, the final weight of each criterion are shown in the right row in Table2. 3.3. The proposed fuzzy AHP algorithm Here we detail the proposed fuzzy AHP algorithm to determine weight vector under uncertain environment step by step. Step1: Construct the analytical hierarchy structure by domain experts. In this step, the experts will determine the objective of decision making, the relative criteria. In addition, the rating of the comparison matrix, modelled by fuzzy numbers can be given by experts through linguistic variables(for example, shown in Figue2), listed in Table2. Step2: For each criterion, using the canonical representation of multiplication operation on fuzzy numbers to obtain the comparison matrix with 14

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canonical representation of multiplication operation (CMCRMO), shown in Table3. Step3: Determine the average weight of the ith’s criterion wCavi , respectively by Eq.19. wCavi

N 1 ∑ CR = w N i Ci

(19)

where wCCR means the canonical representation weight of multiplication i operation of the ith’s criterion, the superscript CR denotes canonical representation. In this average process, the conflict in preference is handled to achieve a consensus preference. Step4: Determine the final weight of the ith’s criterion, wCf i , with the normalization of average weight of the ith’s criterion wCavi , respectively by Eq.20. wCf i

wCavi = N ∑ av wCi

(20)

i

4. Numerical Example Decision making is widely used in supplier management and selection [51, 53, 54, 55, 56, 57, 58]. In this section, a numerical example originated from [51] is presented to illustrate the procedure of the proposed model. Owing to the large number of factors affecting the supplier selection decision, an orderly sequence of steps should be required to tackle it. The problem taken here has four level of hierarchy, and the different decision criteria, attributes and the decision alternatives, will be further discussed. The main objective here is the selection of best global supplier for a manufacturing firm. Application of common criteria to all suppliers makes objective comparisons possible. The criteria which are considered here in selection of the global supplier are: (C1)Overall cost of the product (C2)Quality of the product (C3)Service performance of supplier (C4)Supplier profile (C5)Risk factor 15

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The AHP model of supplier selection can be constructed as shown in Fig 3

Global supplier selection

Level 1:Overall Objective(O’

Level 2:Criteria(c’

Cost(C1’

Quality(C2’

Service performance(C’

Supplier s profile(C’

Risk factor(C’

Level 3:Attributes(A’ A2

A1

A3

A4

A5

A8

A9

A12

A13

A16

A17

A6

A7

A10

A11

A14

A15

A18

A19

Level 4:Decision Alternatives(S’ Supplier1(S1’

Supplier2(S2’

Supplier3(S3’

Figure 3: Hierarchy for the global supplier selection.

As been seen from Fig 3., the overall cost of the product (C1) has three factors (attributes): (A1) Product price , (A2) Freight cost (A3) Tariff and custom duties . The quality of the product (C2) has four factors: (A4) Rejection rate of the product , (A5) Increased lead time , (A6) Quality assessment (A7)Remedy for quality problems. The service performance (C3) has four attributes: (A8) Delivery schedule , 16

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(A9)Technological and R&D support , (A10)Response to changes (A11) Ease of communication . The suppliers profile (C4) has four attributes: (A12)Financial status , (A13)Customer base , (A14)Performance history (A15)Production facility and capacity. The Risk factor (C5) has four attributes: (A16)Geographical location , (A17)Political stability , (A18) Economy (A19) Terrorism. Refer [51] for more detailed information about the attributes mentioned above. After the construction of the decision hierarchy of supplier selection, the fuzzy evaluation matrix of the criteria is constructed by the pairwise comparison of the different criterion relevant to the overall objective using triangular fuzzy numbers, which is shown in Table 2. The fuzzy evaluation of criteria with respect to the overall objective can be listed in Table 2. The final weights of each criteria can be determined by the GA method. The detailed calculation process is given in Section 3. The results are listed in right side of Table 2. In a similar way, the The fuzzy evaluation of the attributes with respect to criterion C1 to C6 can be given by domain experts and there corresponding results based on GA are listed in Table 5 to Table 9, respectively. For the criterion C1, the summary combination of priority weights can be listed in Table 10. Also, the others summary combination of priority weights of C2 to C5 are shown in Table 11 to Table 14. The Fuzzy evaluation of criteria with respect to the overall objective can be shown in 15. As can be seen from Table 15 and Figure 4, the best supplier is S1, which is the same to the works in [51] using the commonly used fuzzy AHP method mentioned in Section 2.4. 5. Conclusions In this paper, a novel and simple fuzzy AHP is proposed to handle MCDM. In our new method, the weight of each criterion can be determined by 17

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Table 5: The fuzzy evaluation of the attributes with respect to criterion C1

C1

A1

A2

A1 (1, 1, 1) A2 (2/5, 1/2, 2/3) A3 (2/5, 1/2, 2/3)

A3

WC1

(3/2, 2, 5/2) (3/2, 2, 5/2) 0.4747 (1, 1, 1) (3/2, 2, 5/2) 0.3333 (2/5, 1/2, 2/3) (1, 1, 1) 0.1920

Table 6: The fuzzy evaluation of the attributes with respect to criterion C2

C2

A4

A5

A6

A7

WC2

A4 A5 A6 A7

(1, 1, 1) (2/5, 1/2, 2/3) (2/3, 1, 3/2) (2/7, 1/3, 2/5)

(3/2, 2, 5/2) (2/3, 1, 3/2) (5/2, 3, 7/2) 0.3703 (1, 1, 1) (2/3, 1, 3/2) (3/2, 2, 5/2) 0.2391 (2/3, 1, 3/2) (1, 1, 1) (3/2, 2, 5/2) 0.2663 (2/5, 1/2, 2/3) (2/5, 1/2, 2/3) (1, 1, 1) 0.1243

Table 7: The fuzzy evaluation of the attributes with respect to criterion C3

C3

A8

A9

A10

A11

WC3

A8 (1, 1, 1) (3/2, 2, 5/2) (5/2, 3, 7/2) (7/2, 4, 9/2) 0.4264 A9 (2/5, 1/2, 2/3) (1, 1, 1) (5/2, 3, 7/2) (5/2, 3, 7/2) 0.3274 A10 (2/7, 1/3, 2/5) (2/7, 1/3, 2/5) (1, 1, 1) (3/2, 2, 5/2) 0.1566 A11 (2/9, 1/4, 2/7) (2/7, 1/3, 2/5) (2/5, 1/2, 2/3) (1, 1, 1) 0.0895

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Table 8: The fuzzy evaluation of the attributes with respect to criterion C4

C4

A12

A13

A14

A15

WC4

A12 A13 A14 A15

(1, 1, 1) (2/5, 1/2, 2/3) (2/5, 1/2, 2/3) (2/9, 1/4, 2/7)

(3/2, 2, 5/2) (1, 1, 1) (2/7, 1/3, 2/5) (2/5, 1/2, 2/3)

(3/2, 2, 5/2) (7/2, 4, 9/2) 0.4880 (2/5, 1/2, 2/3) (3/2, 2, 5/2) 0.2030 (1, 1, 1) (3/2, 2, 5/2) 0.1942 (2/5, 1/2, 2/3) (1, 1, 1) 0.1148

Table 9: The fuzzy evaluation of the attributes with respect to criterion C5

C5

A16

A17

A18

A19

WC5

A16 A17 A18 A19

(1, 1, 1) (2/3, 1, 3/2) (2/3, 1, 3/2) (2/5, 1/2, 2/3)

(2/3,1,3/2) (2/3, 1, 3/2) (2/3, 1, 3/2) 0.2331 (1, 1, 1) (3/2, 2, 5/2) (3/2, 2, 5/2) 0.3438 (2/5, 1/2, 2/3) (1, 1, 1) (3/2, 2, 5/2) 0.2741 (2/5, 1/2, 2/3) (2/5, 1/2, 2/3) (1, 1, 1) 0.1489

Table 10: Summary combination of priority weights: attributes of criterion C1

Weight

A1 A2 0.4747 0.3333

Alternatives S1 0.71 S2 0.13 S3 0.16

0.44 0.36 0.20

A3 0.1920

Alternative priority weight

0.69 0.08 0.23

0.6217 0.1920 0.1862

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Table 11: Summary combination of priority weights: attributes of criterion C2

Weight

A4 A5 A6 0.3703 0.2391 0.2663

Alternatives S1 0.51 S2 0.23 S3 0.26

0.51 0.23 0.26

0.69 0.08 0.23

A7 0.1243

Alternative priority weight

0.87 0.00 0.13

0.6027 0.1615 0.2359

Table 12: Summary combination of priority weights: attributes of criterion C3

Weight

A8 A9 A10 0.4264 0.3274 0.1566

Alternatives S1 0.27 S2 0.18 S3 0.55

0.69 0.08 0.23

0.05 0.64 0.31

A11 0.0895

Alternative priority weight

0.49 0.32 0.19

0.3927 0.2318 0.3754

Table 13: Summary combination of priority weights: attributes of criterion C4

Weight

A11 A12 A13 A14 Alternative priority 0.4880 0.2030 0.1942 0.1148 weight

Alternatives S1 0.83 S2 0.17 S3 0.00

0.45 0.45 0.10

0.69 0.08 0.23

0.33 0.33 0.34

0.6683 0.2277 0.1040

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Table 14: Summary combination of priority weights: attributes of criterion C5

Weight

A16 A17 A18 A19 Alternative priority 0.2331 0.3438 0.2741 0.1489 weight

Alternatives S1 0.72 S2 0.00 S3 0.28

0.49 0.32 0.19

0.83 0.17 0.00

0.27 0.18 0.55

0.6040 0.1834 0.2125

Table 15: Summary combination of priority weights: main criteria of the overall objective

Weight

C1 0.3542

Alternatives S1 0.6217 S2 0.1920 S3 0.1862

C2 C3 0.2696 0.1692

C4 0.1147

C5 0.0923

0.6027 0.3927 0.6683 0.6040 0.1615 0.2318 0.2277 0.1834 0.2359 0.3754 0.1040 0.2125

Alternative priority weight 0.5815 0.1938 0.2246

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7KLVSDSHU



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Figure 4: Comparison of proposed mthod with the previous work [21].

the the canonical representation of multiplication operation on fuzzy numbers. Instead of obtaining the eigenvector of the fuzzy comparison matrix, we get the weight simply by the ratio of each criterion. In addition, we get the final weight of each criterion by average method, which can deal with conflicts in an efficient manner. The proposed method is applied to supplier management under linguistic environment. The results show the efficiency of the proposed method. The method can be easily used in other fuzzy decision making problems. disclosure The author declares that there is no conflict of interests regarding the publication of this paper. Acknowledgment The work is partially supported by National High Technology Research and Development Program of China (863 Program) (Grant No. 2013AA013801), 22

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National Natural Science Foundation of China (Grant Nos. 61174022, 61573290, 61503237), China State Key Laboratory of Virtual Reality Technology and Systems, Beihang University (Grant No.BUAA-VR-14KF-02). References [1] T. L. Saaty, The analytic hierarchy process: Planning, priority setting, resources allocation, McGraw-Hill International Book Co. (New York and London), London: McGraw-Hill, 1980. [2] O. S. Vaidya, S. Kumar, Analytic hierarchy process: An overview of applications, European Journal of Operational Reaserch 169 (2006) 1– 29. [3] E. Ngai, Selection of web sites for online advertising using the AHP tools, Information & Management 40 (2003) 233–242. [4] E. Ngai, E. Chan, Evaluation of knowledge management tools using AHP, Expert Systems with Applications 29 (2005) 889–899. [5] Y. Lee, K. A. Kozar, Investigating the effect of website quality on ebusiness success: An analytic hierarchy process AHP approach, Decision Support Systems 42 (2006) 1383–1401. [6] L. A. Vidal, E. Sahin, N. Martelli, M. Berhoune, B. Bonan, Applying AHP to select drugs to be produced by anticipation in a chemotherapy compounding unit, Expert Systems with Application 37 (2010) 1528– 1534. [7] Y. C. Dong, G. Q. Zhang, W. C. H. et al, Consensus models for AHP group decision making under row geometric mean prioritization method, Decision Support Systems 49 (2010) 281–289. [8] X. Su, S. Mahadevan, P. Xu, Y. Deng, Dependence assessment in Human Reliability Analysis using evidence theory and AHP, Risk Analysis 35 (2015) 1296–1316. [9] M. K. Chen, S. C. Wang, The critical factors of success for information service industry in developing international market: Using analytic hierarchy process (AHP) approach, Expert Systems with Application 37 (2010) 694–704. 23

223

Yong Deng 201-228

J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 22, NO.2, 2017, COPYRIGHT 2017 EUDOXUS PRESS, LLC

[10] X. Deng, Y. Hu, Y. Deng, S. Mahadevan, Supplier selection using AHP methodology extended by D numbers, Expert Systems with Applications 41 (2014) 156–167. [11] A. Frikha, H. Moalla, Analytic hierarchy process for multi-sensor data fusion based on belief function theory, European Journal of Operational Research 241 (2015) 133–147. [12] G. Fan, D. Zhong, F. Yan, P. Yue, A hybrid fuzzy evaluation method for curtain grouting efficiency assessment based on an AHP method extended by D numbers, Expert Systems with Applications 44 (2016) 289–303. [13] Y. Deng, Y. Liu, D. Zhou, An improved genetic algorithm with initial population strategy for symmetric TSP, Mathematical Problems in Engineering 2015 (2015) 212794. [14] W. B. Du, Y. Gao, C. Liu, Z. Zheng, Z. Wang, Adequate is better: Particle swarm optimization with limited-information, Applied Mathematics & Computation 268 (2015) 832–838. [15] X. Deng, Y. Hu, Y. Deng, S. Mahadevan, Environmental impact assessment based on D numbers, Expert Systems with Applications 41 (2014) 635–643. [16] X. Deng, X. Lu, F. T. Chan, R. Sadiq, S. Mahadevan, Y. Deng, D-CFPR: D numbers extended consistent fuzzy preference relations, Knowledge-Based Systems 73 (2015) 61–68. [17] Z.-g. Liu, Q. Pan, J. Dezert, A belief classification rule for imprecise data, Applied intelligence 40 (2014) 214–228. [18] L. A. Zadeh, Fuzzy sets, Information and Control 8 (1965) 338–353. [19] E. K. Zavadskas, J. Antuchevicience, S. H. R. Hajiagha, The intervalvalued intuitionistic fuzzy multimoora method for group decision making in engineering, Mathematical Problems in Engineering 2015 (2015) 560690. [20] W. Jiang, Y. Yang, Y. Luo, X. Qin, Determining basic probability assignment based on the improved similarity measures of generalized 24

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J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 22, NO.2, 2017, COPYRIGHT 2017 EUDOXUS PRESS, LLC

fuzzy numbers, International Journal of Computers Communications & Control 10 (2015) 333–347. [21] A. Mardani, A. Jusoh, E. K. Zavadskas, Fuzzy multiple criteria decisionmaking techniques and applications - two decades review from 1994 to 2014, Expert Systems with Applications 42 (2015) 4126–4148. [22] W. Jiang, Y. Luo, X. Qin, J. Zhan, An improved method to rank generalized fuzzy numbers with different left heights and right heights, Journal of Intelligent & Fuzzy Systems 28 (2015) 2343–2355. [23] Y. Deng, A threat assessment model under uncertain environment, Mathematical Problems in Engineering 2015 (2015) 878024. [24] P. M. J. Laarhoven, W. Pedrycz, A fuzzy extension of Satty’s priority theory, Fuzzy Sets and Systems 11 (1983) 229–241. [25] M. Yang, F. I. Khan, R. Sadiq, Prioritization of environmental issues in offshore oil and gas operations: A hybrid approach using fuzzy inference system and fuzzy analytic hierarchy process, Process Safety and Environmental Protection 89 (2011) 22–34. [26] N. Rikhtegar, N. Mansouri, A. A. Oroumieh, A. Yazdani-Chamzini, E. K. Zavadskas, S. Kildien, Environmental impact assessment based on group decision-making methods in mining projects, Economic Research 27 (2014) 378–392. [27] B. Srdjevic, Y. D. P. Medeiros, Fuzzy ahp assessment of water management plans, Water Resource Management 22 (2008) 877–894. [28] F. T. S. Chan, N. Kumar, M. K. Tiwari, Global supplier selection: a fuzzy-AHP approach, International Journal of Production Research 46 (2008) 3825–3857. [29] C. S. Yu, A GP-AHP method for solving group decision-making fuzzy AHP problems, Computers & Operations Reserach 29 (2002) 1969–2001. [30] S. H. Tsaur, T. Y. Chang, C. H. Yen, The evaluation of airline service quality by fuzzy MCDM, Tourism Managenment 23 (2002) 107–115.

25

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Yong Deng 201-228

J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 22, NO.2, 2017, COPYRIGHT 2017 EUDOXUS PRESS, LLC

[31] C. Kahraman, U. Cebeci, D. Ruan, Multi-attribute comparison of catering service companies using fuzzy AHP: The case of turkey, International Journal of Production Economics 87 (2004) 171–184. [32] L. Mikhailov, Fuzzy analytical approach to partnership selection in formation of virtual enterprises, OMEGA-International Journal of Managenment Sciences 30 (2002) 393–401. [33] X. Zhang, Y. Deng, F. T. Chan, S. Mahadevan, A fuzzy extended analytic network process-based approach for global supplier selection, Applied Intelligence 43 (2015) 760–772. [34] Y. M. Wang, T. M. S. Elhag, An approach to avoiding rank reversal in AHP, Decision Support Systems 42 (2006) 1474–1480. [35] Y. Deng, W. K. Shi, Z. Z. Zhu, Q. Liu, Combining belief functions based on distance of evidence, Decision support systems 38 (2004) 489–493. [36] Y. Yang, D. Han, A new distance-based total uncertainty measure in the theory of belief functions, Knowledge-Based Systems 94 (2016) 114 – 123. [37] Y. Deng, S. Mahadevan, D. Zhou, Vulnerability assessment of physical protection systems: a bio-inspired approach, International Journal of Unconventional Computing 11 (2015) 227–243. [38] Y. Deng, Generalized evidence theory, Applied Intelligence 43 (2015) 530–543. [39] K. Yuan, F. Xiao, L. Fei, B. Kang, Y. Deng, Modeling sensor reliability in fault diagnosis based on evidence theory, Sensors 16 (2016) 113,DOI:10.3390/s16010113. [40] X. Su, S. Mahadevan, P. Xu, Y. Deng, Handling of dependence in dempstercshafer theory, International Journal of Intelligent Systems 30 (2015) 441–467. [41] W. Jiang, B. Wei, C. Xie, D. Zhou, An evidential sensor fusion method in fault diagnosis, Advances in Mechanical Engineering (2016) DOI:10.1177/1687814016632990.

26

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J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 22, NO.2, 2017, COPYRIGHT 2017 EUDOXUS PRESS, LLC

[42] X. Su, S. Mahadevan, W. Han, Y. Deng, Combining dependent bodies of evidence, Applied Intelligence (2015) DOI 10.1007/s10489–015–0723–5. [43] Y. C. Dong, Y. F. Xu, H. Y. Li, On consistency measures of linguistic preference relations, European Journal of Operational Research 189 (2008) 430–444. [44] T. C. Wang, Y. H. Chen, Incomplete fuzzy linguistic preference relations under uncertain environments, Information Fusion 11 (2010) 201–207. [45] E.Herrera-Viedma, F. Herrera, F. Chiclana, M. Luque, Some issues on consistency of fuzzy preference relations, European Journal of Operational Research 154 (2004) 98–109. [46] T. C. Wang, Y. H. Chen, Applying fuzzy linguistic preference relations to the improvement of consistency of fuzzy ahp, Information Sciences 178 (2008) 3755–3765. [47] C. C. Chou, The canonical representation of multiplication operation on triangular fuzzy numbers, Computers and Mathematics with Applications 45 (2003) 1601–1610. [48] A. Kaufmann, G. G. Gupta, Introduction to Fuzzy Arithmetic: Theory and Applications, Van Nostrand-Reinhold, New York, 1991. [49] Y. Deng, Q. Liu, A TOPSIS-based centroid-index ranking method of fuzzy numbers and its application in decision-making, Cybernetics and Systems 36 (2005) 581–595. [50] Y. Deng, Plant location selection based on fuzzy TOPSIS, International Journal of Advanced Manufacturing Technology 28 (2006) 839–844. [51] F. T. S. Chan, N. Kumar, Global supplier development considering risk factors using fuzzy extended AHP-based approach, OMEGAInternational Journal of Managenment Sciences 35 (2007) 417–431. [52] D. Y. Chang, Applications of the extent analysis method on fuzzy AHP, European Journal of Operational Research 95 (1996) 649–655. [53] H. K. Chan, F. T. S. Chan, A review of coordination studies in the context of supply chain dynamics, International Journal of Product Reaserch 48 (2010) 2793–2819. 27

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J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 22, NO.2, 2017, COPYRIGHT 2017 EUDOXUS PRESS, LLC

[54] F. T. S. Chan, V. Kumar, Performance optimization of a leagility inspired supply chain model: a CFGTSA algorithm based approach, International Journal of Product Reaserch 47 (2009) 777–799. [55] Y. Deng, F. T. S. Chan, A new fuzzy dempster MCDM method and its application in supplier selection, Expert Systems with Applications 38 (2011) 777–799. [56] Y. Deng, F. T. S. Chan, Y. Wu, D. Wang, A new linguistic MCDM method based on multiple-criterion data fusion, Expert Systems with Applications 38 (2011) 9854–9861. [57] W. P. Wang, A fuzzy linguistic computing approach to supplier evaluation, Applied Mathematic Modelling 34 (2010) 3130–3141. [58] R. H. Lin, An integrated FANP-MOLP for supplier evaluation and order allocation, Applied Mathematic Modelling 33 (2009) 2730–2736.

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A QUADRATURE RULE FOR THE FINITE HILBERT TRANSFORM VIA SIMPSON TYPE INEQUALITIES AND APPLICATIONS SHUNFENG WANG, NA LU AND XINGYUE GAO

Abstract. In this paper, a quadrature rule on an equidistant partition of the interval [a, b] for the finite Hilbert Transform of different classes of absolutely continuous functions via Simpson type inequalities is given, which may have the better error bounds than those obtained via trapezoid type inequalities. Some numerical experiments for different divisions of the interval [a, b] are also presented.

1. Introduction The finite Hilbert transform plays an important role in scientific and engineering computing. Denote by (T f )(a, b, ·) the finite Hilbert transform of the function f : [a, b] → R, i.e., we recall it [∫ ∫ b ∫ b ] t−ε 1 f (τ ) 1 f (τ ) (1.1) dτ := lim dτ, + (T f )(a, b; t) = P V ε→0 π π a τ −t a t+ε τ − t where P V has the usual meaning of the Cauchy principle value. There are some important approaches for evaluating finite Hilbert transforms, such as the Gaussian, Chebyshev, TANH, Iri-Moriguti-Takasawa, and double exponential quadrature methods. And for classical results on the finite Hilbert transform, see [4, 5, 6, 9, 11, 12, 13, 17]. In [5], by the use of trapezoid type rules taken on an equidistant partition of the interval [a, b], Dragomir et al. proved the following inequalities for the finite Hilbert transform of different classes of absolutely continuous functions. Theorem 1.1. Let f : [a, b] → R be a differentiable function such that its derivative f ′ is absolutely continuous on [a, b]. If ] n−1 [ f ′ (t)(b − a) + f (b) − f (a) b − a ∑ t−a b−t Tn (f ; t) = (1.2) + f; t − · i, t + ·i , 2πn πn i=1 n n then we have the estimate ( ) f (t) b−t (1.3) ln − Tn (f ; t) (T f )(a, b; t) − π t−a [ ]  ( ) 2  (b − a)2 a+b 1   + t − ∥f ′′ ∥[a,b],∞ , if f ′′ ∈ L∞ [a, b];   4πn 4 2     [ ]  q 1 1 1+ q1 1+ q1 (t − a) + (b − t) ∥f ′′ ∥[a,b],p , if f ′′ ∈ Lp [a, b], p > 1, + = 1; ≤ 1 1+ q p q  2πn(q + 1)   [ ]      1 1 (b − a) + t − a + b ∥f ′′ ∥  [a,b],1 ,  2πn 2 2  1 2 ′′   if f ′′ ∈ L∞ [a, b];  8πn (b − a) ∥f ∥[a,b],∞ ,     q 1 1 1+ q1 ∥f ′′ ∥[a,b],p , if f ′′ ∈ Lp [a, b], p > 1, + = 1; ≤ 1 (b − a) 1+ p q  2πn(q + 1) q       1 (b − a)∥f ′′ ∥[a,b],1 , 2πn for all t ∈ (a, b), where [f ; c, d] denotes the divided difference [f ; c, d] :=

f (c)−f (d) . c−d

Key words and phrases. Finite Hilbert transform, Simpson type inequalities.

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S. F. WANG, N. LU AND X. Y. GAO

Theorem 1.2. Let f : [a, b] → R be a twice differentiable function such that the second derivative f ′′ is absolutely continuous on [a, b]. Then ( ) b−t f (t) ln − Tn (f ; t) (1.4) (T f )(a, b; t) − π t−a [ ]  ( )2 (b − a)2 a+b 1    + t − (b − a)∥f ′′′ ∥[a,b],∞ , if f ′′′ ∈ L∞ [a, b];  2π  12n 12 2     1  ] q[B(q + 1, q + 1)] q [ 1 1 2+ q1 2+ q1 ≤ (t − a) + (b − t) ∥f ′′′ ∥[a,b],p , if f ′′′ ∈ Lp [a, b], p > 1, + = 1; 2π  2(2q + 1)n p q   [ ]  ( )  2   (b − a)2 a+b 1   + t− ∥f ′′′ ∥[a,b],1 ,  2 8πn 4 2  (b − a)3 ′′′   if f ′′′ ∈ L∞ [a, b];  36πn2 ∥f ∥[a,b],∞ ,    1  2+ q1 q 1 1 ≤ q[B(q + 1, q + 1)] (b − a) ∥f ′′′ ∥[a,b],p , if f ′′′ ∈ Lp [a, b], p > 1, + = 1; 2  2π(2q + 1)n p q      1  (b − a)2 ∥f ′′′ ∥[a,b],1 , 16πn2 for all t ∈ (a, b), where Tn (f ; t) is defined by (1.2). An extensive literature such as [1, 2, 3, 7, 8, 10, 14, 15, 16, 18, 19, 20, 21, 22] deal with Simpson type inequalities. In this paper, motivated by [5], by the use of Simpson type inequalities taken on an equidistant partition of the interval [a, b], a quadrature formula for the Finite Hilbert transform of different classes of absolutely continuous functions is obtained. Estimates for some error bounds and some numerical examples for the obtained approximation will also be presented. 2. THE RESULTS Lemma 2.1. Let u : [a, b] → R be an absolutely continuous function on [a, b]. Then one has the inequalities: ∫ ( ) b u(a) + 4u a+b + u(b) 2 u(s)ds − (b − a) (2.1) a 6  2 5(b − a)   ∥u′ ∥[a,b],∞ if u′ ∈ L∞ [a, b];   36    ) q1 1 (  1 1 2(b − a)1+ q 6q+1 + 3q+1 1 1 ≤ ∥u′ ∥[a,b],p , if u′ ∈ Lp [a, b], p > 1, + = 1; 1  p q q  (q + 1)       b − a ∥u′ ∥ [a,b],1 . 3 A simple proof of this fact can be done by using the identity [ ∫ a+b ( ( ) ) ∫ b 2 + u(b) u(a) + 4u a+b 5a + b 2 (2.2) u(s)ds − (b − a) = − s− u′ (s)ds 6 6 a a ] ) ∫ b ( a + 5b ′ u (s)ds , + s− a+b 6 2 and we omit the details. The following lemma holds. Lemma 2.2. Let u : [a, b] → R be an absolutely continuous (a, b), t ̸= τ and n ∈ N, n ≥ 1, we have the inequality: [ ( ) ( n−1 1 ∫ τ ( 1 ∑ τ −t u(s)ds − u t+i· + 4u t + i + τ − t t 6n i=0 n

230

function on [a, b]. Then for any t, τ ∈ 1) τ − t · 2 n

)

( ) ] τ −t + u t + (i + 1) · n

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J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 22, NO.2, 2017, COPYRIGHT 2017 EUDOXUS PRESS, LLC

A QUADRATURE RULE FOR THE FINITE HILBERT TRANSFORM

 5|τ − t| ′   ∥u ∥[t,τ ],∞ if u′ ∈ L∞ [a, b];   36n    ) q1 1 (  1 1 + 3q+1 2|τ − t| q 6q+1 1 1 ≤ ∥u′ ∥[t,τ ],p , if u′ ∈ Lp [a, b], p > 1, + = 1; 1  p q  (q + 1) q n       1 ∥u′ ∥ [t,τ ],1 , 3n where ∫ ∥u′ ∥[t,τ ],∞ :=ess sup |u′ (s)|, and ∥u′ ∥[t,τ ],p := s∈|t,τ |

t

τ

p1 |u′ (s)p |ds , p ≥ 1.

Proof. Consider the equidistant division of [t, τ ] (if t < τ ) given by En : xi = t + i ·

τ −t , i =0, n. n

If we apply the inequality (2.1) on the interval [xi , xi+1 ], we may write that: ) ( ( ) ) ( ) ( ∫ 1 τ −t τ −t τ −t xi+1 + 4u t + i + · + u t + (i + 1) · u t + i · n 2 n n τ − t · u(s)ds − xi 6 n  5(τ − t)2 ′   ∥u ∥[xi ,xi+1 ],∞ , if u′ ∈ L∞ [a, b];  2  36n    ) q1 1 (  1 1 2|τ − t|1+ q 6q+1 + 3q+1 1 1 ≤ ∥u′ ∥[xi ,xi+1 ],p , if u′ ∈ Lp [a, b], p > 1, + = 1; 1 1 1+  p q  n q (q + 1) q       |τ − t| ∥u′ ∥ [xi ,xi+1 ],1 , 3n from which we get [ ( ) ( ) ( ) ] 1 ∫ xi+1 ( 1 τ −t 1) τ − t τ −t u(s)ds − u t+i· + 4u t + i + · + u t + (i + 1) · τ − t xi 6n n 2 n n  5|τ − t| ′   ∥u ∥[xi ,xi+1 ],∞ , if u′ ∈ L∞ [a, b];  2  36n    ) q1 1 (  1 1 2|τ − t| q 6q+1 + 3q+1 1 1 ≤ ∥u′ ∥[xi ,xi+1 ],p , if u′ ∈ Lp [a, b], p > 1, + = 1; 1 1  1+ p q  (q + 1) q n q      1 ′  ∥u ∥[xi ,xi+1 ],1 . 3n Summing over i from 0 to n − 1 and using the generalised triangle inequality, we may write [ ( ) ( ) ( ) ] n−1 1 ∫ τ ( 1 ∑ τ −t 1) τ − t τ −t u(s)ds − u t+i· + 4u t + i + · + u t + (i + 1) · τ − t t 6n i=0 n 2 n n  n−1 ∑    5|τ − t|  ∥u′ ∥[xi ,xi+1 ],∞ , if u′ ∈ L∞ [a, b];  2  36n  i=0      2|τ − t| q1 ( 1 + 1 ) q1 n−1 ∑ 1 1 6q+1 3q+1 ≤ ∥u′ ∥[xi ,xi+1 ],p , if u′ ∈ Lp [a, b], p > 1, + = 1; 1 1 1+ q  p q q  (q + 1) n  i=0    n−1   1 ∑ ′   ∥u ∥[xi ,xi+1 ],1 .   3n i=0

However, n−1 ∑

∥u′ ∥[xi ,xi+1 ],∞ ≤ n∥u′ ∥[t,τ ],∞ ,

i=0

231

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J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 22, NO.2, 2017, COPYRIGHT 2017 EUDOXUS PRESS, LLC

S. F. WANG, N. LU AND X. Y. GAO n−1 ∑

′

∥u ∥[xi ,xi+1 ],p =

i=0

n−1 ∑ ∫ xi+1 i=0



xi

and n−1 ∑

∥u′ ∥[xi ,xi+1 ],1

i=0

[ (n−1 ∫ p1 ∑ 1 |u (s)| ds ≤ n q ′

xi+1

p

i=0

xi

p1 )p ] p1 1 = n q ∥u′ ∥[t,τ ],p , |u (s)| ds ′

p

n−1 ∫ ∫ ∑ xi+1 τ ′ ′ ≤ |u (s)|ds = |u (s)|ds = ∥u′ ∥[t,τ ],1 , xi t i=0



and the lemma is proved.

The following theorem in approximating the Hilbert transform of a differentiable function whose derivative is absolutely continuous holds. Theorem 2.1. Let f : [a, b] → R be a differentiable function such that its derivative f ′ is absolutely continuous on [a, b]. If ] n−1 [ t−a b−t f ′ (t)(b − a) + f (b) − f (a) b − a ∑ (2.3) + f; t − · i, t + ·i Tn (f ; t) = 6πn 3πn i=1 n n [ ] ) ) ( ( n−1 2(b − a) ∑ t−a b−t 1 1 + f; t − ,t + , · i+ · i+ 3πn i=0 n 2 n 2 then we have the estimate ( ) f (t) b−t ln − Tn (f ; t) (2.4) (T f )(a, b; t) − π t−a [ ]  ( ) 2  (b − a)2 a+b 5   + t− ∥f ′′ ∥[a,b],∞ , if f ′′ ∈ L∞ [a, b];   36πn 4 2      ) q1 [  ( 1 1 ] 2q 6q+1 + 3q+1 1 1 1 1 ≤ (t − a)1+ q + (b − t)1+ q ∥f ′′ ∥[a,b],p , if f ′′ ∈ Lp [a, b], p > 1, + = 1; 1  p q  π(q + 1) q   [ ]    1 1  a + b ′′    3πn 2 (b − a) + t − 2 ∥f ∥[a,b],1 ,  5   (b − a)2 ∥f ′′ ∥[a,b],∞ , if f ′′ ∈ L∞ [a, b];   72πn    ) q1  ( 1 1 2q 6q+1 + 3q+1 1 1 1 ≤ (b − a)1+ q ∥f ′′ ∥[a,b],p , if f ′′ ∈ Lp [a, b], p > 1, + = 1; 1  p q  π(q + 1) q     1   (b − a)∥f ′′ ∥[a,b],1 , 3πn for all t ∈ (a, b). Proof. Applying Lemma 2.2 for the function f ′ , we may write that [ ) ) n−1 n−1 f (τ ) − f (t) ( ∑ ( ∑ ( 1 τ −t 1) τ − t ′ ′ ′ − f (t) + (2.5) f t+i· +4 f t+ i+ · τ −t 6n n 2 n i=1 i=0 ] ) n−2 ∑ ( τ −t + f ′ (τ ) + f ′ t + (i + 1) · n i=0    5|τ − t| ∥f ′′ ∥[t,τ ],∞ , if f ′′ ∈ L∞ [a, b];   36n    1 1  1 1 2|τ − t| q ( 6q+1 + 3q+1 ) q ′′ 1 1 ≤ ∥f ∥[t,τ ],p , if f ′′ ∈ Lp [a, b], p > 1, + = 1; 1  p q q  (q + 1) n       1 ∥f ′′ ∥[t,τ ],1 . 3n

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A QUADRATURE RULE FOR THE FINITE HILBERT TRANSFORM

However, ( ) n−2 ) ∑ ( τ −t τ −t = f ′ t + (i + 1) · f′ t + i · n n i=0 i=1

n−1 ∑

and then by (2.5), we may write: [ ( ) ( ) ] n−1 n−1 f (τ ) − f (t) ( f ′ (t) + f ′ (τ ) 1 ∑ ′ τ −t 2 ∑ ′ 1) τ − t − + + · (2.6) f t+i· f t+ i+ τ −t 6n 3n i=1 n 3n i=0 2 n  5|τ − t| ′′   ∥f ∥[t,τ ],∞ ,   36n    ) q1 1 (  1 1 + 3q+1 2|τ − t| q 6q+1 ≤ ∥f ′′ ∥[t,τ ],p , 1  qn  (q + 1)       1 ∥f ′′ ∥ [t,τ ],1 , 3n for any t, τ ∈ [a, b], t ̸= τ . Consequently, we have ) ( ∫ b ∫ b[ ′ n−1 1 1 1 ∑ ′ τ −t f (τ ) − f (t) f (t) + f ′ (τ ) (2.7) dτ − P V + f t+i· PV π τ −t π 6n 3n i=1 n a a ] ( ) n−1 ( 2 ∑ ′ 1) τ − t + f t+ i+ · dτ 3n i=0 2 n  ∫ b  5   P V |τ − t|∥f ′′ ∥[t,τ ],∞ dτ,   36πn  a    ) q1 ∫ b  ( 1 1 2 6q+1 + 3q+1 1 ≤ P V |τ − t| q ∥f ′′ ∥[t,τ ],p dτ, 1   a (q + 1) q πn    ∫ b   1   PV ∥f ′′ ∥[t,τ ],1 dτ.  3πn a Since

( ( ) )] n−1 n−1 ( τ −t 1) τ − t f ′ (t) + f ′ (τ ) 1 ∑ ′ 2 ∑ ′ f t+i· f t+ i+ dτ PV + + · 6n 3n i=1 n 3n i=0 2 n a (∫ ( ) ∫ b )( ′ n−1 t−ε 1 ∑ ′ τ −t f (t) + f ′ (τ ) = lim+ + f t+i· + 6n 3n i=1 n ε→0 a t+ε ) ( ) n−1 ( 2 ∑ ′ 1) τ − t + f t+ i+ · dτ 3n i=0 2 n [ (∫ ) )] ∫ b )( ( n−1 t−ε f ′ (t)(b − a) + f (b) − f (a) 1 ∑ τ −t ′ = + lim + f t+i· dτ 6n 3n i=1 ε→0+ n a t+ε [ (∫ ) )] ∫ b )( ( n−1 t−ε ( 2 ∑ 1) τ − t ′ + · lim + f t+ i+ dτ 3n i=0 ε→0+ 2 n a t+ε [ [ ]] ( ) ( n−1 ) 1 ∑ τ −t f ′ (t)(b − a) + f (b) − f (a) n t−ε b + ·f t+i· + = lim 6n 3n i=1 ε→0+ i n a t+ε [ [ ]] ( ) ( n−1 ) ( 2 ∑ 2n 1) τ − t t−ε b + lim+ ·f t+ i+ · + 3n i=0 ε→0 2i + 1 2 n a t+ε [ ( ) ( )] n−1 f ′ (t)(b − a) + f (b) − f (a) 1 ∑n b−t a−t = + f t+i· −f t+i· 6n 3n i=1 i n n ∫

b

[

233

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S. F. WANG, N. LU AND X. Y. GAO

[ ( ) ( )] n−1 ( ( 1) b − t 1) a − t 2 ∑ 2n f t+ i+ · −f t+ i+ · + 3n i=0 2i + 1 2 n 2 n ] n−1 [ f ′ (t)(b − a) + f (b) − f (a) b − a ∑ a−t b−t = + ,t + i · f; t + i · 6n 3n i=1 n n ] [ n−1 ( ( 2(b − a) ∑ 1) a − t 1) b − t + · ,t + i + · , f; t + i + 3n 2 n 2 n i=0 and

∫

b

PV

∫

|τ − t|∥f ′′ ∥[t,τ ],∞ dτ ≤∥f ′′ ∥[a,b],∞ P V

a

∫

b

PV

b

|τ − t|dτ = ∥f ′′ ∥[a,b],∞

a

|τ − t| q ∥f ′′ ∥[t,τ ],p dτ ≤∥f ′′ ∥[a,b],p P V 1

a

∫ PV

b

∥f ′′ ∥[t,τ ],1 dτ = P V

a

∫

[∫ a

b

1

|τ − t| q dτ = a t

∥f ′′ ∥[τ,t],1 dτ +

[

] ( 1 a + b) (b − a)2 + t − , 4 2

] q∥f ′′ ∥[a,b],p [ 1 1 (t − a)1+ q + (b − t)1+ q , q+1 ] ∫ b

∥f ′′ ∥[t,τ ],1 dτ

t

] a + b 1 (b − a) + t − ∥f ′′ ∥[a,b],1 , ≤∥f ∥[a,t],1 (t − a) + ∥f ∥[t,b],1 (b − t) ≤ 2 2 ′′

[

′′

then, by (2.7) we get ] ∫ b n−1 [ 1 t−a f ′ (t)(b − a) + f (b) − f (a) b − a ∑ b−t f (τ ) − f (t) f; t − (2.8) P V dτ − − · i, t + ·i π τ −t 6πn 3πn i=1 n n a ] n−1 [ 2(b − a) ∑ t−a ( 1) b−t ( 1 ) − f; t − · i+ ,t + · i+ 3πn i=0 n 2 n 2 [ ]  ( 5∥f ′′ ∥[a,b],∞ 1 a + b)  2  (b − a) + t − , if f ′′ ∈ L∞ [a, b];   36πn 4 2    (  ) q1 ′′  1 1 ]  2q q+1 + 3q+1 ∥f ∥[a,b],p [ 1 1 1+ q1 1+ q1 6 + (b − t) , if f ′′ ∈ Lp [a, b], p > 1, + = 1; (t − a) ≤ 1 1+ q p q  π(q + 1) n   [ ]   ′′    ∥f ∥[a,b],1 1 (b − a) + t − a + b .   3πn 2 2 On the other hand, as for the function f0 : (a, b) → R, f0 (t) = 1, we have ( ) 1 b−a (T, f0 )(a, b; t) = ln , t ∈ (a, b), π t−a then obviously

∫ b ∫ b ∫ b 1 f (τ ) − f (t) + f (t) 1 f (τ ) − f (t) f (t) dτ PV dτ = P V dτ + PV , π τ −t π τ −t π a a a τ −t from which we get the equality: ( ) ∫ b f (t) b−t 1 f (τ ) − f (t) (2.9) (T f )(a, b; t) − ln dτ. = PV π t−a π τ −t a (T f )(a, b; t) =



Finally, using (2.8) and (2.9), we deduce (2.4).

Before we proceed with another estimate of the remainder in approximating the Hilbert Transform for functions whose second derivatives are absolutely continuous, we need the following lemma. Lemma 2.3. Let u : [a, b] → R be a function such that its derivative is absolutely continuous on [a, b]. Then one has the inequalities: ∫ ( ) b u(a) + 4u a+b + u(b) 2 (2.10) u(s)ds − (b − a) a 6

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A QUADRATURE RULE FOR THE FINITE HILBERT TRANSFORM

 (b − a)3 ′′    ∥u ∥[a,b],∞ , if u′′ ∈ L∞ [a, b];   81   1  (b − a)2+ q 1 1 ≤ Λ∥u′′ ∥[a,b],p , if u′′ ∈ Lp [a, b], p > 1, + = 1;  2 p q     2    (b − a) ∥u′′ ∥[a,b],1 , 24 where (2.11)

[( ∫

(

1 3

q

s

Λ= 0

(∫

2 3

+ 1 2

)q )q ) q1 ∫ 21 ( 1 1 q − s ds + s s− ds 1 3 3 3 (

(1 − s)

q

)q )q ) q1 ] ( ∫ 1 2 2 q − s ds + ds . (1 − s) s − 2 3 3 3

A simple proof of the fact can be done by the use of the following identity: ( ) ( ) ∫ b ∫ a+b 2 u(a) + 4u a+b + u(b) 1 2a + b 2 (2.12) u(s)ds − (b − a) = − (s − a) s − u′′ (s)ds 6 2 a 3 a ( ) ∫ 1 b a + 2b − (s − b) s − u′′ (s)ds, 2 a+b 3 2 and we omit the details. The following lemma also holds. Lemma 2.4. Let u : [a, b] → R be a differentiable function such that u′ : [a, b] → R is absolutely continuous on [a, b]. Then for any t, τ ∈ (a, b), t ̸= τ and n ∈ N, n ≥ 1, we have the inequality: [ ( ) ( ) ( ) ] n−1 1 ∫ τ ( 1 ∑ τ −t 1) τ − t τ −t u(s)ds − u t+i· + 4u t + i + · + u t + (i + 1) · τ − t t 6n i=0 n 2 n n  |τ − t|2 ′′   ∥u ∥[t,τ ],∞ , if u′′ ∈ L∞ [a, b];   81n2    1+ q1 1 1 ≤ |τ − t| Λ∥u′′ ∥[t,τ ],p , if u′′ ∈ Lp [a, b], p > 1, + = 1; 2  2n p q      |τ − t|  ∥u′′ ∥[t,τ ],1 . 24n2 Proof. Consider the equidistant division of [t, τ ] (if t < τ ) En : xi = t + i ·

τ −t , i =0, n. n

If we apply the inequality (2.10), we may state that ( ) ( ( ) ) ( ) ∫ τ −t 1 τ −t τ −t xi+1 u t + i · + 4u t + i + · + u t + (i + 1) · n 2 n n τ − t u(s)ds − · xi 6 n  |τ − t|3 ′′   ∥u ∥[xi ,xi+1 ],∞ , if u′′ ∈ L∞ [a, b];    81n3   1  |τ − t|2+ q 1 1 ≤ Λ∥u′′ ∥[xi ,xi+1 ],p , if u′′ ∈ Lp [a, b], p > 1, + = 1; 1 2+  p q q  2n    2    |τ − t| ∥u′′ ∥[xi ,xi+1 ],1 . 24n2 Dividing by |τ − t| > 0 and using a similar argument to the one in Lemma 2.2, we conclude that the desired inequality holds.  The following theorem in approximating the Hilbert transform of a twice differentiable function whose second derivative f ′′ is absolutely continuous also holds.

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J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 22, NO.2, 2017, COPYRIGHT 2017 EUDOXUS PRESS, LLC

S. F. WANG, N. LU AND X. Y. GAO

Theorem 2.2. Let f : [a, b] → R be a twice differentiable function such that the second derivative f ′′ is absolutely continuous on [a, b]. Then ( ) f (t) b−t ln − Tn (f ; t) (2.13) (T f )(a, b; t) − π t−a [ ]  ( ) 2 (b − a)2 a+b 1    + t− (b − a)∥f ′′′ ∥[a,b],∞ , if f ′′′ ∈ L∞ [a, b];  2  81n π 12 2     1 1  1 1 q[(t − a)2+ q + (b − t)2+ q ]Λ ′′′ ≤ ∥f ∥[a,b],p , if f ′′′ ∈ Lp [a, b], p > 1, + = 1; 2π  2(2q + 1)n p q   [ ]  ( )  2   1 (b − a)2 a+b   + t− ∥f ′′′ ∥[a,b],1 ,  2 24n π 4 2  3   (b − a) ∥f ′′′ ∥[a,b],∞ , if f ′′′ ∈ L∞ [a, b];   2  243n π   1  q(b − a)2+ q Λ ′′′ 1 1 ≤ ∥f ∥[a,b],p , if f ′′′ ∈ Lp [a, b], p > 1, + = 1; 2  2π(2q + 1)n p q     2    (b − a) ∥f ′′′ ∥[a,b],1 , 48n2 π for all t ∈ (a, b), where Tn (f ; t) is defined by (2.3) and Λ is defined by (2.11). Proof. Applying Lemma 2.4 for the function f ′ , we may write that (see also Theorem 2.1) [ ( ( ) ) ] n−1 n−1 f (τ ) − f (t) ( τ −t 1) τ − t f ′ (t) + f ′ (τ ) 1 ∑ ′ 2 ∑ ′ f t+i· f t+ i+ − + + · (2.14) τ −t 6n 3n i=1 n 3n i=0 2 n  |τ − t|2 ′′′   ∥f ∥[t,τ ],∞ , if f ′′′ ∈ L∞ [a, b];  2  81n    1+ q1 1 1 ≤ |τ − t| Λ∥f ′′′ ∥[t,τ ],p , if f ′′′ ∈ Lp [a, b], p > 1, + = 1; 2  2n p q      |τ − t|  ∥f ′′′ ∥[t,τ ],1 , 24n2 for any t, τ ∈ [a, b], t ̸= τ . Consequently, we may write: ( ) ∫ b ∫ b[ ′ n−1 1 f (τ ) − f (t) 1 f (t) + f ′ (τ ) 1 ∑ ′ τ −t (2.15) dτ − P V + f t+i· PV π τ −t π 6n 3n i=1 n a a ] ) ( n−1 ( 2 ∑ ′ 1) τ − t + f t+ i+ · dτ 3n i=0 2 n  ∫ b  1   PV |τ − t|2 ∥f ′′′ ∥[t,τ ],∞ dτ,  2π  81n  a   ∫ b  1 Λ ≤ P V |τ − t|1+ q ∥f ′′′ ∥[t,τ ],p dτ, 2  2n π  a   ∫ b   1   PV |τ − t|∥f ′′′ ∥[t,τ ],1 dτ.  24n2 π a since

∫

b

PV

|τ − t|2 ∥f ′′′ ∥[t,τ ],∞ dτ ≤ ∥f ′′′ ∥[a,b],∞ P V

a

[

′′′

=∥f ∥[a,b],∞ ∫ PV

b

∫

b

|τ − t|2 dτ

[ ] ( )2 ] (t − a)3 + (b − t)3 (b − a)2 a+b ′′′ (b − a), = ∥f ∥[a,b],∞ + t− 3 12 2 a

|τ − t|1+ q ∥f ′′′ ∥[t,τ ],p dτ ≤ ∥f ′′′ ∥[a,b],p P V 1

a

∫

b

1

|τ − t|1+ q dτ a

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J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 22, NO.2, 2017, COPYRIGHT 2017 EUDOXUS PRESS, LLC

A QUADRATURE RULE FOR THE FINITE HILBERT TRANSFORM

] q∥f ′′′ ∥[a,b],p [ (b − t)2+ q + (t − a)2+ q 2+ q1 2+ q1 (b − t) = + (t − a) 2q + 1 2 + 1q 1

1

=∥f ′′′ ∥[a,b],p and

∫

b

PV a

[

( )2 ] (b − a)2 a+b |τ − t|∥f ∥[t,τ ],1 dτ ≤ + t− ∥f ′′′ ∥[a,b],1 . 4 2 ′′′



Then by (2.15), we deduce the first part of (2.13). 3. NUMERICAL EXPERIMENTS For a function f : [a, b] → R, we may consider the quadrature formula ( ) f (t) b−t En (f ; a, b, t) := ln + Tn (f ; t), t ∈ [a, b]. π t−a

As shown in the above section, En (f ; a, b, t) provides an approximation for the Finite Hilbert Transform (T f )(a, b; t) and the error estimate fulfils the bounds described in (2.4) and (2.15). If we consider the function f : [1, 2] → R, f (x) = exp(x), the exact value of the Hilbert transform is exp(t)Ei(2 − t) − exp(t)Ei(1 − t) , t ∈ [1, 2]. π and the plot of this function is embodied in Figure 1. (T f )(a, b; t) =

−12

Figure 1 10

0

Figure 2

x 10

−0.2 5 −0.4 0

−0.6

−0.8

−5

−1 −10 −1.2

−15

1

1.2

1.4

1.6

1.8

−1.4

2

1

1.2

1.4

t

1.6

1.8

2

t

If we implement the quadrature formula provided by En (f ; a, b, t) using Matlab and chose the value of n = 100, the error Er (f ; a, b, t) := (T f )(a, b; t) − En (f ; a, b, t) has the variation described in Figure 2. −14

0

Figure 3

x 10

Figure 4

1.5

−1 −2

1

−3

0.5

−4

0

−5

−0.5

−6

−1

−7 −8

−1.5

1

1.2

1.4

1.6

1.8

2

1

t

1.2

1.4

1.6

1.8

2

t

For n = 200, the plot of Er (f ; a, b, t) is embodied in the following Figure 3, from which we can see that the precision of the error gets higher when n gets bigger. Now, if we consider another function f : [1, 2] → R, f (x) = sin(x), then the exact value of Hilbert transform is −Si (−2 + t) cos(t) + Ci (2 − t) sin(t) + Si (t − 1) cos(t) − sin(t)Ci (t − 1) (T f )(a, b; t) = , t ∈ [1, 2] π having the plot embodied in Figure 4. If we choose the value of n = 50, then the error Er (f ; a, b, t) := (T f )(a, b; t) − En (f ; a, b, t) for the function f (x) = sin x, x ∈ [a, b] has the variation described in Figure 5. Moreover, for n = 100, the behaviour of Er (f ; a, b, t) is plotted in Figure 6.

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J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 22, NO.2, 2017, COPYRIGHT 2017 EUDOXUS PRESS, LLC

S. F. WANG, N. LU AND X. Y. GAO −14

−15

Figure 5

x 10

Figure 6

x 10

5 4 2

0

0 −2

−5

−4 −10

−6 −8

−15

−10 −12

−20 −14 −16 1

1.2

1.4

1.6

1.8

2

1

t

1.2

1.4

1.6

1.8

2

t

Remark 1. When n = 100, for function f (x) = exp(x), the precision of the error is 10−06 in [5], while the precision obtained here is 10−12 . When n = 200, we also have the higher precision. For function f (x) = sin(x), it’s the same situation. Therefore, our results may have the better error bounds. Acknowledgments This work was supported by the Natural Science Foundation of Jiangsu Province (BY2014007-04). References [1] M. Alomari and S. Hussain, Two inequalities of Simpson type for quasi-convex functions and applications, Appl. Math. E-Notes 11 (2011), 110–117. [2] M. Bencze and C. Zhao, About Simpson-type and Hermite-type inequalities, Creat. Math. Inform. 17 (2008), 8–13. ˇ [3] V. Culjak, J. Peˇ cari´ c and L. E. Persson, A note on Simpson type numerical integration, Soochow J. Math. 29 (2003), no. 2, 191–200. [4] N. M. Dragomir, S. S. Dragomir and P. Farrell, Approximating the finite Hilbert transform via trapezoid type inequalities, Comput. Math. Appl. 43 (2002), no. 10-11, 1359–1369. [5] N. M. Dragomia, S. S. Dragomir, P. M. Farrell and G. W. Baxter, A quadrature rule for the finite Hilbert transform via trapezoid type inequalities, J. Appl. Math. Comput. 13 (2003), no. 1-2, 67–84. [6] S. S. Dragomir, Approximating the finite Hilbert transform via an Ostrowski type inequality for functions of bounded variation, JIPAM. J. Inequal. Pure Appl. Math. 3 (2002), no. 4, Article 51, 19 pages. [7] S. S. Dragomir, On Simpson’s quadrature formula for mappings of bounded variation and applications, Tamkang J. Math. 30 (1999), no. 1, 53–58. [8] A. Ghizzetti and A. Ossicini, Quadrature formulae, Academic Press, New York, (1970), 192 pages. [9] W. J. Liu, Some Ostrowski type inequalities via Riemann-Liouville fractional integrals for h-convex functions, J. Comput. Anal. Appl. 16 (2014), no. 5, 998–1004. [10] W. J. Liu, Some Simpson type inequalities for h-convex and (α, m)-convex functions, J. Comput. Anal. Appl. 16 (2014), no. 5, 1005–1012. [11] W. J. Liu and X. Y. Gao, Approximating the finite Hilbert transform via a companion of Ostrowski’s inequality for function of bounded variation and applications, Appl. Math. Comput. 247 (2014), 373–385. [12] W. J. Liu and N. Lu, Approximating the finite Hilbert Transform via Simpson type inequalities and applications, Politehnica University of Bucharest Scientific Bulletin-Series A-Applied Mathematics and Physics, 77 (2015), no. 3, 107-122. [13] W. J. Liu, Q. A. Ngo and W. Chen, On new Ostrowski type inequalities for double integrals on time scales, Dynam. Systems Appl. 19 (2010), no. 1, 189–198. [14] W. J. Liu, W. S. Wen and J. Park, A refinement of the difference between two integral means in terms of the cumulative variation and applications, J. Math. Inequal. 10 (2016), no. 1, 147–157. [15] Z. Liu, An inequality of Simpson type, Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci. 461 (2005), no. 2059, 2155–2158. [16] Z. Liu, More on inequalities of Simpson type, Acta Math. Acad. Paedagog. Nyh´ azi. (N.S.) 23 (2007), no. 1, 15–22 . [17] S. Okada and D. Elliott, H¨ older continuous functions and the finite Hilbert transform, Math. Nachr. 169 (1994), 219–233. [18] J. Peˇ cari´ c and S. Varoˇsanec, A note on Simpson’s inequality for functions of bounded variation, Tamkang J. Math. 31 (2000), no. 3, 239–242. [19] Y. Shi and Z. Liu, Some sharp Simpson type inequalities and applications, Appl. Math. E-Notes 9 (2009), 205–215. [20] N. Ujevi´ c, Double integral inequalities of Simpson type and applications, J. Appl. Math. Comput. 14 (2004), no. 1-2, 213–223. [21] A. Vukeli´ c, Estimations of the error for general Simpson type formulae via pre-Gr¨ uss inequality, J. Math. Inequal. 3 (2009), no. 4, 559–566. [22] G.-S. Yang and H.-F. Chu, A note on Simpson’s inequality for function of bounded variation, Tamsui Oxf. J. Math. Sci. 16 (2000), no. 2, 229–240. (S. F. Wang, N. Lu and X. Y. Gao) College of Mathematics and Statistics, Nanjing University of Information Science and Technology, Nanjing 210044, China E-mail address: [email protected]

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J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 22, NO.2, 2017, COPYRIGHT 2017 EUDOXUS PRESS, LLC

A QUADRATURE FORMULA IN APPROXIMATING THE FINITE HILBERT TRANSFORM VIA PERTURBED TRAPEZOID TYPE INEQUALITIES SHUNFENG WANG, XINGYUE GAO AND NA LU

Abstract. In this paper, we obtain the error estimation of a quadrature formula in approximating the finite Hilbert transform on an equidistant partition of the interval [a, b]. Some numerical examples for the obtained approximation are also presented.

1. Introduction In the recent year, many authors tried to consider error inequalities for some known and some new quadrature rules. For example, the well-known trapezoid and midpoint quadrature rules were considered (see [1], [4], [6], [9], [11], [12], [14], [15], [18], [19] and [20]). In [5], the authors proved the following theorem: Theorem 1.1. Let f : [a, b] → R be a mapping such the derivative f (n−1) (n ≥ 1) is absolutely continuous on [a, b]. Then ∫ ∫ b n−1 [ ] ∑ 1 b 1 (1.1) (x − a)k+1 f (k) (a) + (−1)k (b − x)k+1 f (k) (b) + (x − t)n f (n) (t)dt f (t)dt = (k + 1)! n! a a k=0

for all x ∈ [a, b]. Specially, we can obtain the following identity from (1.1) with x = a+b 2 : )n ∫ b n−1 ] (−1)n ∫ b ( ∑ (b − a)k+1 [ a+b (k) k (k) (1.2) f (a) + (−1) f (b) + t − f (n) (t)dt. f (t)dt = k+1 (k + 1)! 2 n! 2 a a k=0

In (1.2), for n = 1, we obtain the trapezoid rule ) ∫ b ∫ b( f (b) + f (a) a+b (1.3) f (t)dt = (b − a) + − t f ′ (t)dt. 2 2 a a The finite Hilbert transform of the function f : (a, b) → R is defined as [∫ ∫ b ∫ b ] t−ε 1 f (τ ) f (τ ) T (f )(a, b; t) = P V dτ = lim + dτ ε→0 π τ − t π(τ − t) a a t+ε where P V has the usual meaning of the Cauthy principle value (see [3]). In [7], the authors used the inequality (1.3) to approximate the finite Hilbert transform and obtain the following theorem: Theorem 1.2. Let f : [a, b] → R be such that f ′ : [a, b] → R is absolutely continuous on [a, b]. Then we have the bounds ( ) T (f )(a, b; t) − f (t) ln b − t − 1 [f (b) − f (a) + f ′ (t)(b − a)] π t−a 2π  [ ( )]2  ∥f ′′ ∥∞ (b − a)2 b+a   + t − , f ′′ ∈ L∞ [a, b];    4π 4 2    [ ] q ∥f ′′ ∥p 1 1 1+ q1 1+ q1 ≤ (1.4) (t − a) + (b − t) , p > 1, + = 1 and f ′′ ∈ Lp [a, b]; 1  1+ q p q  2π(q + 1)     ′′  ∥f ∥1   (b − a), f ′′ ∈ L1 [a, b], 2π for all t ∈ (a, b), where ∥ · ∥p are the usual Lebesgue norms in Lp [a, b] (1 ≤ p ≤ ∞). Key words and phrases. perturbed trapezoid type inequality; numerical integration; finite Hilbert transform.

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J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 22, NO.2, 2017, COPYRIGHT 2017 EUDOXUS PRESS, LLC

S. F. WANG, X. Y. GAO AND N. LU

In [8], by the use of trapezoid type rules taken on an equidistant partition of the interval [a, b], Dragomir et al. proved the following inequalities for the finite Hilbert transform of different classes of absolutely continuous functions. Theorem 1.3. Let f : [a, b] → R be a differentiable function such that its derivative f ′ is absolutely continuous on [a, b]. If ] n−1 [ b−t t−a f ′ (t)(b − a) + f (b) − f (a) b − a ∑ (1.5) + · i, t + ·i , f; t − Tn (f ; t) = 2πn πn i=1 n n then we have the estimate ( ) f (t) b−t ln − Tn (f ; t) (1.6) (T f )(a, b; t) − π t−a [ ]  ( ) 2  (b − a)2 a+b 1   + t− ∥f ′′ ∥[a,b],∞ , if f ′′ ∈ L∞ [a, b];   4πn 4 2     [ ]  q 1 1 1+ q1 1+ q1 (t − a) + (b − t) ∥f ′′ ∥[a,b],p , if f ′′ ∈ Lp [a, b], p > 1, + = 1; ≤ 1 1+ p q  2πn(q + 1) q   [ ]     1 1 a + b ′′    2πn 2 (b − a) + t − 2 ∥f ∥[a,b],1 ,  1   (b − a)2 ∥f ′′ ∥[a,b],∞ , if f ′′ ∈ L∞ [a, b];   8πn    q 1 1 1+ q1 ∥f ′′ ∥[a,b],p , if f ′′ ∈ Lp [a, b], p > 1, + = 1; ≤ 1 (b − a) 1+ p q q  2πn(q + 1)       1 (b − a)∥f ′′ ∥[a,b],1 , 2πn for all t ∈ (a, b), where [f ; c, d] denotes the divided difference [f ; c, d] :=

f (c)−f (d) . c−d

If we put n = 2 in (1.2), we can get the perturbed trapezoid rule )2 ∫ b ∫ ( (b − a)2 ′ 1 b f (b) + f (a) a+b (1.7) (b − a) − [f (b) − f ′ (a)] + f ′′ (t)dt, f (t)dt = t− 2 8 2 2 a a Recently, Liu and Pan [16] proved the following inequalities for the finite Hilbert transform of different classes of absolutely continuous functions via the above rule (1.7) (see also [13] for other related results). Theorem 1.4. Let f : [a, b] → R be such that f ′′ : [a, b] → R is absolutely continuous on [a, b]. Then we have the bounds ( ) f (t) b−t f ′ (t) 5 (1.8) T (f )(a, b; t) − π ln t − a − 2π (b − a) − 8π [f (b) − f (a)] f ′′ (t) f ′ (a) f ′ (b) (b − t) + (a − b)(a + b − 2t) − (t − a) + 8π 16π 8π  [ ] ( ) 2  b+a (b − a)3 ∥f ′′′ ∥∞   (b − a) t − + , f ′′′ ∈ L∞ [a, b];    24π 2 12     [ ]  q ∥f ′′′ ∥p 1 1 2+ q1 2+ q1 (t − a) + (b − t) , p > 1, + = 1 and f ′′′ ∈ Lp [a, b]; ≤ 1 1+ q p q  8π(2q + 1)    [( ]  ) 2   ∥f ′′′ ∥1 a+b (b − a)2   t − + , f ′′′ ∈ L1 [a, b];   8π 2 4 for all t ∈ (a, b), where ∥ · ∥p (1 ≤ p ≤ ∞) are the usual Lebesgue norms in Lp [a, b]. In this paper, inspired by [8], we shall derive a quadrature formula in approximating the finite Hilbert transform of different classes of absolutely continuous functions. Some numerical examples for the obtained approximation will be presented in Section 3.

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J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 22, NO.2, 2017, COPYRIGHT 2017 EUDOXUS PRESS, LLC

A QUADRATURE FORMULA IN APPROXIMATING THE FINITE HILBERT TRANSFORM

2. A quadrature formula for equidistant divisions Lemma 2.1. Let u : [a, b] → R be an absolutely continuous function on [a, b]. Then one has the inequalities: ∫ b (b − a)2 ′ u(a) + u(b) ′ (b − a) + [u (b) − u (a)] (2.1) u(s)ds − a 2 8  3 (b − a)   ∥u′′ ∥[a,b],∞ if u′′ ∈ L∞ [a, b];   24     (b − a)2+ q1 1 1 ′′ ≤ if u′′ ∈ Lp [a, b], p > 1, + = 1; 1 ∥u ∥[a,b],p ,  p q 8(2q + 1) q     2    (b − a) ∥u′′ ∥ [a,b],1 . 8 A simple proof of this fact can be done by using the identity ) ∫ ( ∫ b (b − a)2 ′ 1 b a+b u(a) + u(b) ′ (2.2) (b − a) + [u (b) − u (a)] = s− u′′ (s)ds u(s)ds − 2 8 2 a 2 a and we omit the details. The following lemma holds. Lemma 2.2. Let u : [a, b] → R be an absolutely continuous function on [a, b]. Then for any t, τ ∈ (a, b), t ̸= τ and n ∈ N, n ≥ 1, we have the inequality: [ ( ) ( )] n−1 1 ∫ τ 1 ∑ τ −t τ −t (2.3) u(s)ds − u t+i· + u t + (i + 1) · τ − t t 2n i=0 n n [ ( ] ) ( ) n−1 τ −t τ −t τ −t ∑ ′ ′ u t + (i + 1) · − u t + i · + 8n2 i=0 n n  |τ − t|2 ′′   ∥u ∥[t,τ ],∞ if u′′ ∈ L∞ [a, b];  2  24n    1  |τ − t|1+ q 1 1 ′′ ≤ if u′′ ∈ Lp [a, b], p > 1, + = 1; 1 ∥u ∥[t,τ ],p ,  2 p q q  8n (2q + 1)       |τ − t| ∥u′′ ∥[t,τ ],1 , 8n2 where ∫ τ p1 ′′ ′′ ′′ ′′ p ∥u ∥[t,τ ],∞ := ess sup |u (s)|, and ∥u ∥[t,τ ],p := |u (s) |ds , p ≥ 1. s∈|t,τ |

t

Proof. Consider the equidistant division of [t, τ ] (if t < τ ) given by τ −t En : xi = t + i · , i =0, n. n If we apply the inequality (2.1) on the interval [xi , xi+1 ], we may write that: ( ) ( ) ∫ τ −t τ −t xi+1 u t + i · + u t + (i + 1) · n n τ −t u(s)ds − · xi 2 n [ ( ] ) ( ) (τ − t)2 ′ τ −t τ −t ′ + u t + (i + 1) · − u t + i · 8n2 n n  3 |τ − t|   ∥u′′ ∥[xi ,xi+1 ],∞ , if u′′ ∈ L∞ [a, b];   24n3    1  1 1 |τ − t|2+ q ′′ ≤ if u′′ ∈ Lp [a, b], p > 1, + = 1; 1 ∥u ∥[xi ,xi+1 ],p , 2+ q1  p q 8n (2q + 1) q     2    |τ − t| ∥u′′ ∥ [xi ,xi+1 ],1 , 8n2

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S. F. WANG, X. Y. GAO AND N. LU

from which we get

[ ( ] 1 ∫ xi+1 ( 1 τ − t) τ − t) u(s)ds − u t+i· + u t + (i + 1) · τ − t xi 2n n n [ ( ( ) ) ] τ −t τ −t (τ − t) ′ + u t + (i + 1) · − u′ t + i · 8n2 n n  |τ − t|2 ′′   ∥u ∥[xi ,xi+1 ],∞ , if u′′ ∈ L∞ [a, b];  3  24n    1  |τ − t|1+ q 1 1 ′′ ≤ if u′′ ∈ Lp [a, b], p > 1, + = 1; 1 ∥u ∥[xi ,xi+1 ],p , 1 2+  p q  8n q (2q + 1) q       |τ − t| ∥u′′ ∥[x ,x ],1 , i i+1 8n2 Summing over i from 0 to n − 1 and using the generalised triangle inequality, we may write ] n−1 [ 1 ∫ τ ( 1 ∑ ( τ − t) τ − t) u(s)ds − + u t + (i + 1) · u t+i· τ − t t 2n i=0 n n ] [ ) ) ( ( n−1 (τ − t) ∑ ′ τ −t τ −t ′ − u + u t + i · t + (i + 1) · 8n2 i=0 n n [ ( ) ( )] ∫ n−1 xi+1 ∑ 1 τ −t τ −t 1 u t+i· + u t + (i + 1) · ≤ u(s)ds − τ − t xi 2n n n i=0 [ ( ] ) ( ) (τ − t) ′ τ −t τ −t ′ + u t + (i + 1) · − u t + i · 8n2 n n  n−1  |τ − t|2 ∑ ′′   ∥u ∥[xi ,xi+1 ],∞ , if u′′ ∈ L∞ [a, b];  3  24n   i=0    1 n−1  ∑ |τ − t|1+ q 1 1 ≤ ∥u′′ ∥[xi ,xi+1 ],p , if u′′ ∈ Lp [a, b], p > 1, + = 1; 1 1 2+ q  p q q (2q + 1) i=0   8n    n−1   |τ − t| ∑ ′′   ∥u ∥[xi ,xi+1 ],1 ,  8n2 i=0 However, n−1 ∑

∥u′′ ∥[xi ,xi+1 ],∞ ≤n∥u′′ ∥[t,τ ],∞ ,

i=0 n−1 ∑

∥u′′ ∥[xi ,xi+1 ],p =

n−1 ∑ ∫ xi+1

i=0

i=0



xi

and n−1 ∑

′′

∥u ∥[xi ,xi+1 ],1

i=0

[ (n−1 ∫ p1 ∑ 1 |u′′ (s)|p ds ≤ n q i=0

xi+1 xi

p1 )p ] p1 1 |u′′ (s)|p ds = n q ∥u′′ ∥[t,τ ],p ,

∫ ∫ xi+1 n−1 τ ∑ ′′ |u (s)|ds = |u′′ (s)|ds = ∥u′′ ∥[t,τ ],1 , ≤ xi t i=0



and the lemma is proved.

The following theorem in approximating the Hilbert transform of a differentiable function whose second derivative is absolutely continuous holds. Theorem 2.1. Let f : [a, b] → R be a differentiable function such that its derivative f ′ is absolutely continuous on [a, b]. If ] n−1 [ f ′ (t)(b − a) + f (b) − f (a) b − a ∑ t−a b−t Tn (f ; t) = (2.4) + f; t − · i, t + ·i 2πn πn i=1 n n

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[ ] 1 1 ′′ ′ ′ f (t)(a − b)(a + b − 2t) − f (b) + f (a) + f (b)(b − t) − f (a)(a − t) , − 2 8n 2 then we have the estimate ( ) b−t f (t) ln − Tn (f ; t) (2.5) (T f )(a, b; t) − π t−a [ ]  ( )2  a + b 1 4   (b − a) t − + (b − a)3 ∥f ′′′ ∥[a,b],∞ , if f ′′′ ∈ L∞ [a, b];   24πn2 2 3     [ ]  1 1 q 2+ q1 2+ q1 (t − a) + (b − t) ∥f ′′′ ∥[a,b],p , if f ′′′ ∈ Lp [a, b], p > 1, + = 1; ≤ 1 1+ p q  8πn2 (2q + 1) q   [ ]  ( )    1 1 a+b 2   ∥f ′′′ ∥[a,b],1 ,  8πn2 4 (b − a) + t − 2  1   (b − a)3 ∥f ′′′ ∥[a,b],∞ , if f ′′′ ∈ L∞ [a, b];  2  18πn    ( )2+ q1  b−a q 1 1 ≤ ∥f ′′′ ∥[a,b],p , if f ′′′ ∈ Lp [a, b], p > 1, + = 1; 1+ q1  2 2 p q  4πn (2q + 1)     1   (b − a)2 ∥f ′′′ ∥[a,b],1 , 32πn2 for all t ∈ (a, b). Proof. Applying Lemma 2.2 for the function f ′ , we may write that [ ] ) n−2 ) n−1 f (τ ) − f (t) ∑ ( ∑ ( 1 τ −t τ −t ′ ′ ′ ′ (2.6) − f (t) + f t+i· + f t + (i + 1) · + f (τ ) τ −t 2n n n i=1 i=0 [ ] ) n−1 ) n−2 ∑ ( ∑ ( τ −t τ −t τ − t ′′ ′′ ′′ ′′ f (τ ) + f t + (i + 1) · − f t + i · − f (t) + 8n2 n n i=0 i=1  2 |τ − t|   ∥f ′′′ ∥[t,τ ],∞ if f ′′′ ∈ L∞ [a, b];   24n2    1  |τ − t|1+ q 1 1 ′′′ ≤ ∥[t,τ ],p , if f ′′′ ∈ Lp [a, b], p > 1, + = 1; 1 ∥f  2 p q  8n (2q + 1) q       |τ − t| ∥f ′′′ ∥[t,τ ],1 , 8n2 However, ) n−2 ) n−1 ) n−2 ) n−1 ∑ ( ∑ ( ∑ ( ∑ ( τ −t τ −t τ −t τ −t ′ ′ ′′ ′′ f t+i· = f t + (i + 1) · , f t+i· = f t + (i + 1) · n n n n i=1 i=0 i=1 i=0 and then by (2.6), we may write: [ [ ] ( )] n−1 f (τ ) − f (t) f ′ (t) + f ′ (τ ) 1∑ ′ τ −t τ − t ′′ ′′ (2.7) − + f t+i· + f (τ ) − f (t) τ −t 2n n i=1 n 8n2  2 |τ − t|   ∥f ′′′ ∥[t,τ ],∞   24n2    1  |τ − t|1+ q ′′′ ≤ ∥[t,τ ],p , 1 ∥f  2 (2q + 1) q  8n       |τ − t| ∥f ′′′ ∥[t,τ ],1 , 8n2 for any t, τ ∈ [a, b], t ̸= τ . Consequently, we have ( )] ∫ b ∫ b[ ′ n−1 1 f (τ ) − f (t) 1 f (t) + f ′ (τ ) 1∑ ′ τ −t (2.8) dτ − P V + f t+i· dτ PV π τ −t π 2n n i=1 n a a

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∫ b τ − t ′′ 1 ′′ [f (τ ) − f (t)]dτ + PV 2 π 8n a  ∫ b 1    PV |τ − t|2 ∥f ′′′ ∥[t,τ ],∞ dτ,  2  24πn  a   ∫ b  1 1 |τ − t|1+ q ∥f ′′′ ∥[t,τ ],p dτ, ≤ 1 PV 2  q a 8πn (2q + 1)    ∫ b    1   PV |τ − t|∥f ′′′ ∥[t,τ ],1 dτ, 8πn2 a Since

[

( )] n−1 f ′ (t) + f ′ (τ ) 1∑ ′ τ −t PV + dτ f t+i· 2n n i=1 n a ] n−1 [ f ′ (t)(b − a) + f (b) − f (a) b − a ∑ a−t b−t = + ,t + i · , f; t + i · n n i=1 n n ∫

∫ PV a

b

b

[ ] 1 1 ′′ τ − t ′′ ′′ ′ ′ [f (τ ) − f (t)]dτ = f (t)(a − b)(a + b − 2t) − f (b) + f (a) + f (b)(b − t) − f (a)(a − t) , 8n2 8n2 2

and

∫

b

′′′

∫

′′′

b

|τ − t| ∥f ∥[t,τ ],∞ dτ ≤ ∥f ∥[a,b],∞ P V |τ − t|2 dτ a a ] [ )2 ( (b − t)3 − (a − t)3 4 a+b ′′′ ′′′ 3 =∥f ∥[a,b],∞ = ∥f ∥[a,b],∞ (b − a) t − + (b − a) , 3 2 3 2

PV

∫

b

1+ q1

′′′

∫

′′′

b

1

∥f ∥[t,τ ],p dτ ≤ ∥f ∥[a,b],p P V |τ − t| |τ − t|1+ q dτ a a ] [ 2+ q1 2+ q1 [ ] ′′′ q∥f ∥ + (b − t) (t − a) [a,b],p 2+ q1 2+ q1 =∥f ′′′ ∥[a,b],p = (t − a) + (b − t) , 2q + 1 2 + 1q PV

and

∫

b

PV

[ |τ − t|∥f ′′′ ∥[t,τ ],1 dτ ≤ ∥f ′′′ ∥[t,τ ],1

a

( )2 ] a + b 1 (b − a)2 + t − , 4 2

then, by (2.8) we get ] ∫ b n−1 [ 1 f (τ ) − f (t) f ′ (t)(b − a) + f (b) − f (a) b − a ∑ t−a b−t (2.9) P V dτ − − f; t − · i, t + ·i π τ −t 2πn πn i=1 n n a ] [ 1 1 ′′ f (t)(a − b)(a + b − 2t) − f (b) + f (a) + f ′ (b)(b − t) − f ′ (a)(a − t) + 2 8n 2 [ ]  ′′′ ( )2 ∥f ∥[a,b],∞ a+b 4    (b − a) t − + (b − a)3 , if f ′′′ ∈ L∞ [a, b];  2  24πn 2 3     ]  q∥f ′′′ ∥[a,b],p [ 1 1 2+ q1 2+ q1 (t − a) + (b − t) , if f ′′′ ∈ Lp [a, b], p > 1, + = 1; ≤ 1 1+ p q  8πn2 (2q + 1) q   [ ]  ( )  2  ∥f ′′′ ∥  a+b [a,b],1 1   (b − a)2 + t − ,  8πn2 4 2 On the other hand, as for the function f0 : (a, b) → R, f0 (t) = 1, we have ( ) 1 b−a (T, f0 )(a, b; t) = ln , t ∈ (a, b), π t−a

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A QUADRATURE FORMULA IN APPROXIMATING THE FINITE HILBERT TRANSFORM

then obviously (T f )(a, b; t) =

1 PV π

∫

b a

1 f (τ ) − f (t) + f (t) dτ = P V τ −t π

from which we get (T f )(a, b; t) −

(2.10)

f (t) ln π

(

b−t t−a

) =

∫

b a

1 PV π

f (t) f (τ ) − f (t) dτ + PV τ −t π

∫

b a

∫ a

b

dτ , τ −t

f (τ ) − f (t) dτ. τ −t 

Finally, using (2.9) and (2.10), we deduce (2.5). 3. Some numerical examples For a function f : [a, b] → R, we may consider the quadrature formula ( ) f (t) b−t En (f ; a, b, t) := ln + Tn (f ; t), t ∈ [a, b]. π t−a

As shown in the above section, En (f ; a, b, t) provides an approximation for the Finite Hilbert Transform (T f )(a, b; t) and the error estimate fulfils the bounds described in (2.5). If we consider the function f : [1, 2] → R, f (x) = exp(x), the exact value of the Hilbert transform is exp(t)Ei(2 − t) − exp(t)Ei(1 − t) , t ∈ [1, 2]. π and the plot of this function is embodied in Figure 1. (T f )(a, b; t) =

−6

Figure 1 10

4.5

Figure 2

x 10

4 5

3.5 3

0

2.5 −5

2 1.5

−10

1 −15

1

1.2

1.4

1.6

1.8

0.5

2

1

1.2

1.4

1.6

1.8

2

t

If we implement the quadrature formula provided by En (f ; a, b, t) using Matlab and chose the value of n = 200, the error Er (f ; a, b, t) := (T f )(a, b; t) − En (f ; a, b, t) has the variation described in Figure 2. −7

1.8

Figure 3

x 10

Figure 4

1.5

1.6 1.4

1

1.2

0.5

1

0

0.8

−0.5

0.6

−1

0.4

−1.5

0.2

1

1.2

1.4

1.6

1.8

1

2

1.2

1.4

1.6

1.8

2

t

For n = 1000, the plot of Er (f ; a, b, t) is embodied in Figure 3, from which we can see that the precision of the error gets higher when n gets bigger. Now, if we consider another function f : [1, 2] → R, f (x) = sin(x), then −Si (−2 + t) cos(t) + Ci (2 − t) sin(t) + Si (t − 1) cos(t) − sin(t)Ci (t − 1) , t ∈ [1, 2] π having the plot embodied in Figure 4. If we choose the value of n = 200, then the error Er (f ; a, b, t) := (T f )(a, b; t) − En (f ; a, b, t) for the function f (x) = sin x, x ∈ [a, b] has the variation described in Figure 5. Moreover, for n = 1000, the behaviour of Er (f ; a, b, t) is plotted in Figure 6. (T f )(a, b; t) =

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S. F. WANG, X. Y. GAO AND N. LU −8

−9

Figure 5

x 10

2

Figure 6

x 10

4 1

2 0

0

−2

−1

−4 −2 −6 −3

−8 −10

−4

−12 1

1.2

1.4

1.6

1.8

2

1

t

1.2

1.4

1.6

1.8

2

t

Acknowledgments This work was supported by the Natural Science Foundation of Jiangsu Province (BY2014007-04). References [1] [2] [3] [4] [5] [6] [7] [8] [9]

[10] [11] [12] [13] [14]

[15] [16] [17] [18] [19] [20] [21]

G. A. Anastassiou, Ostrowski type inequalities, Proc. Amer. Math. Soc. 123 (1995), no. 12, 3775–3781. N. S. Barnett and S. S. Dragomir, On the perturbed trapezoid formula, Tamkang J. Math. 33 (2002), no. 2, 119–128. P. L. Butzer and R. J. Nessel, Fourier Analysis and Approximation Theory, Birkhaiiser Verlag, Basel, 1 (1977). D. Cruz-Uribe and C. J. Neugebauer, Sharp error bounds for the trapezoidal rule and Simpson’s rule, JIPAM. J. Inequal. Pure Appl. Math. 3 (2002), no. 4, Article 49, 22 pp. P. Cerone and S. S. Dragomir, Trapezoidal Type Rules from an Inequalities Point of View. RGMIA research report collection, 2 (1999), no. 6. P. Cerone and S. S. Dragomir, Trapezoidal-type rules from an inequalities point of view, in Handbook of analyticcomputational methods in applied mathematics, 65–134, Chapman & Hall/CRC, Boca Raton, FL. N. M. Dragomir, S. S. Dragomir and P. Farrell, Approximating the finite Hilbert transform via trapezoid type inequalities, Comput. Math. Appl. 43 (2002), no. 10-11, 1359–1369. N. M. Dragomia, S. S. Dragomir, P. M. Farrell and G. W. Baxter, A quadrature rule for the finite Hilbert transform via trapezoid type inequalities, J. Appl. Math. Comput. 13 (2003), no. 1-2, 67–84. S. S. Dragomir, P. Cerone and J. Roumeliotis, A new generalization of Ostrowski’s integral inequality for mappings whose derivatives are bounded and applications in numerical integration and for special means, Appl. Math. Lett. 13 (2000), no. 1, 19–25. S. S. Dragomir, P. Cerone and A. Sofo, Some remarks on the trapezoid rule in numerical integration, Indian J. Pure Appl. Math. 31 (2000), no. 5, 475–494. W. J. Liu, Some Ostrowski type inequalities via Riemann-Liouville fractional integrals for h-convex functions, J. Comput. Anal. Appl. 16 (2014), no. 5, 998–1004. W. J. Liu, Some Simpson type inequalities for h-convex and (α, m)-convex functions, J. Comput. Anal. Appl. 16 (2014), no. 5, 1005–1012. W. J. Liu and X. Y. Gao, Approximating the finite Hilbert transform via a companion of Ostrowski’s inequality for function of bounded variation and applications, Appl. Math. Comput. 247 (2014), 373–385. W. J. Liu and N. Lu, Approximating the finite Hilbert Transform via Simpson type inequalities and applications, Politehnica University of Bucharest Scientific Bulletin-Series A-Applied Mathematics and Physics, 77 (2015), no. 3, 107-122. W. J. Liu, Q. A. Ngo and W. Chen, On new Ostrowski type inequalities for double integrals on time scales, Dynam. Systems Appl. 19 (2010), no. 1, 189–198. W. J. Liu and Y. X. Pan, Approximating the finite Hilbert Transform via perturbed trapezoid type inequalities, submitted. W. J. Liu, W. S. Wen and J. Park, A refinement of the difference between two integral means in terms of the cumulative variation and applications, J. Math. Inequal. 10 (2016), no. 1, 147–157. C. E. M. Pearce et al., Generalizations of some inequalities of Ostrowski-Gr¨ uss type, Math. Inequal. Appl. 3 (2000), no. 1, 25–34. E. Set and M. Z. Sarıkaya, On the generalization of Ostrowski and Gr¨ uss type discrete inequalities, Comput. Math. Appl. 62 (2011), no. 1, 455–461. K.-L. Tseng, S.-R. Hwang and S. S. Dragomir, Generalizations of weighted Ostrowski type inequalities for mappings of bounded variation and their applications, Comput. Math. Appl. 55 (2008), no. 8, 1785–1793. K.-L. Tseng, S.-R. Hwang, G.-S. Yang and Y.-M. Chou, Weighted Ostrowski integral inequality for mappings of bounded variation, Taiwanese J. Math. 15 (2011), no. 2, 573–585.

(S. F. Wang, X. Y. Gao and N. Lu) College of Mathematics and Statistics, Nanjing University of Information Science and Technology, Nanjing 210044, China E-mail address: [email protected]

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Pointwise Superconvergence of the Displacement of the Six-Dimensional Finite Element Yinsuo Jia∗and Jinghong Liu†

In this article we first introduce definitions of the regularized Green’s function, the discrete Green’s function, the discrete δ function, and the L2 -projection operator in six dimensions. Then the W 2,1 -seminorm estimates for the regularized Green’s function and the discrete Green’s function are derived. Finally, pointwise superconvergence of the displacement of the six-dimensional finite element is obtained.

1

Introduction

There have been many studies concerned with superconvergence of the finite element method for partial differential equations. Books and survey papers have been published. For the literature, we refer to [1–17] and references therein. It is well known that estimates for the Green’s function play very important roles in the study of the superconvergence (especially, pointwise superconvergence) of the finite element method (see [4, 5, 8, 12, 13, 14, 17]). For one- and twodimensional elliptic problems, one have obtained many optimal estimates for the Green’s function (see [17]). Recently, for three-dimensional elliptic problems, the 2 W 2,1 -seminorm optimal estimate with order O(| ln h| 3 ) for the discrete Green’s function was derived (see [12]). In this article, we will discuss estimate for the the discrete Green’s function based on the 6D Poisson equation. we shall use the symbol C to denote a generic constant, which is independent from the discretization parameter h and which may not be the same in each occurrence and also use the standard notations for the Sobolev spaces and their norms. We consider the following Poisson equation: Lu ≡ −∆u = f in Ω,

u = 0 on ∂Ω,

(1.1)

where Ω ⊂ R6 is a bounded polytopic domain. The weak formulation of (1.1) ∗ School of Mathematics and Computer Science, Shangrao Normal University, Shangrao 334001, China, email: [email protected] † School of Mathematics and Computer Science, Shangrao Normal University, Shangrao 334001, China, email: [email protected]

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JIA, LIU: SUPERCONVERGENCE OF THE SIX-DIMENSIONAL FEM

reads,

{

Find u ∈ H01 (Ω) satisfying a(u , v) = (f , v) for all v ∈ H01 (Ω). ∫

where a(u , v) ≡

∇u · ∇v dX, Ω

∫

and (f , v) ≡

f v dX. Ω

¯ Denote by S h (Ω) a continuous Let {T h } be a regular family of partitions of Ω. piecewise m(m ≥ 2)-degree (or tensor-product m-degree) ∩ polynomials space regarding this kind of partitions and let S0h (Ω) = S h (Ω) H01 (Ω). Discretizing the above weak formulation using S0h (Ω) as approximating space means, { Find uh ∈ S0h (Ω) satisfying a(uh , v) = (f , v) for all v ∈ S0h (Ω). Thus, the following Galerkin orthogonality relation holds. a(u − uh , v) = 0 ∀ v ∈ S0h (Ω).

(1.2)

h h For every Z ∈ Ω, we define the discrete ∩ 1 δ function δZ ∈ S0 (Ω), the reg∗ 2 ularized Green’s function GZ ∈ H (Ω) H0 (Ω), the discrete Green’s function GhZ ∈ S0h (Ω) and the L2 -projection Ph u ∈ S0h (Ω) such that (see [17]) h (v, δZ ) = v(Z) ∀ v ∈ S0h (Ω),

(1.3)

h a(G∗Z , v) = (δZ , v) ∀ v ∈ H01 (Ω),

(1.4)

a(G∗Z

(1.5)

−

v) = 0 ∀ v ∈ S0h (Ω), v) = 0 ∀ v ∈ S0h (Ω),

GhZ ,

(u − Ph u,

In this article, we will bound the terms |G∗Z |2, 1 h e∈T h GZ 2, 1, e .

∑

2

(1.6) h h h h and GZ 2, 1 . Here GZ 2, 1 =

Estimates for the Regularized Green’s Function

We first introduce the weight function defined by ( ) ¯ ¯ 2 + θ2 −3 ∀ X ∈ Ω, ϕ ≡ ϕ(X) = |X − X|

(2.1)

¯ ∈Ω ¯ is a fixed point, θ = γh, and γ ∈ [6, +∞) is a suitable real number. where X For every α ∈ R, we give the following notations: (∫ ) 12 m ∑ ∑ 2 2 2 2 n 2 β n α n D v , |∇ v| α = |∇ v| = ϕ |∇ v| dX , ∥v∥ = |∇n v|ϕα , α ϕ m, ϕ |β|=n

Ω

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JIA, LIU: SUPERCONVERGENCE OF THE SIX-DIMENSIONAL FEM

where β = (β1 , β2 , β3 , β4 , β5 , β6 ), |β| = β1 + β2 + β3 + β4 + β5 + β6 , and βi ≥ 0, i = 1, · · · , 6 are integers. In particular, for the case of m = 0, we write (∫ ∥v∥ϕα =

ϕ |v| dX α

2

) 12 .

Ω

We assume there exists a real number q0 (1 < q0 ≤ ∞) such that ∩ ∥v∥2, q ≤ C(q)∥Lv∥0, q ∀ v ∈ W 2, q (Ω) W01, q (Ω), 1 < q < q0 ,

(2.2)

which is the so-called a priori estimate (see [17]). As in the two-dimensional case (see [17]), we can obtain the following Lemma 2.1. Lemma 2.1. For ϕ the weight function defined by (2.1), we have the following estimates: n |∇n ϕα | ≤ C(α, n)ϕα+ 6 , α ∈ R, n = 1, 2, (2.3) ∫ ∫

ϕ dX ≤ C(k)| ln θ|, θ ≤ k < 1,

(2.4)

ϕα dX ≤ C(α − 1)−1 θ−6(α−1) ∀ α > 1.

(2.5)

Ω

Ω h For the L2 -projection operator Ph and the discrete δ function δZ , similar to the arguments in the two-dimensional setting (see [17]), we have the following results (2.6)–(2.8). Lemma 2.2. For Ph w the L2 -projection of w, we have the following stability estimate: ∥Ph w∥0, q ≤ C∥w∥0, q , 1 ≤ q ≤ ∞, (2.6)

∥Ph w∥1, q ≤ C∥w∥1, q , 6 < q ≤ ∞. Lemma 2.3. For lowing estimate:

h δZ

(2.7)

the discrete δ function defined by (1.3), we have the folh (X)| ≤ Ch−6 e−Ch |δZ

−1

|X−Z|

,

(2.8)

¯ and C is independent of h, X, and Z. where X, Z ∈ Ω, h In additon, we have the following weighted-norm estimate for δZ . h Lemma 2.4. For δZ the discrete δ function defined by (1.3) and ϕ defined by (2.1), we have the following estimate:

h

δZ −1 ≤ C. (2.9) ϕ Proof. From (2.1) and (2.8), ∫

h 2 ( )3 −1

δZ −1 ≤ C |X − Z|2 + θ2 h−12 e−Ch |X−Z| dX ϕ ∫Ω∞ ( 2 )3 −1 ≤ C r + θ2 h−12 e−Ch r r5 dr. 0

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Set h−1 r = t, then

h 2

δZ −1 ≤ C ϕ

∫

∞

(2 )3 t + γ 2 e−Ct t5 dt ≤ C,

0

which is the result (2.9). Lemma 2.5. For G∗Z the regularized Green’s function defined by (1.4) and ϕ defined by (2.1), we have the following weighted-norm estimate: ∥G∗Z ∥

5

1

ϕ− 3

≤ C |ln h| 6 .

(2.10)

Proof. From (1.3), (1.4), (1.6), (2.2), (2.6), the inverse estimate, the Sobolev Embedding Theorem (see [18]), and the Poincar´e inequality, we have ( ) 1 2 h G∗Z , ϕ− 3 G∗Z = a(G∗Z , w) = (δZ , w) = Ph w(Z) ≤ |Ph w|0, ∞ ∥G∗Z ∥ − 31 = ϕ

≤ Ch− q |Ph w|0, q ≤ Ch− q |w|0, q ≤ Ch− q q 6 ∥w∥1, 6 6 6 5 5 1

≤ Ch− q q 6 ∥w∥2, 3 ≤ Ch− q q 6 ϕ− 3 G∗Z 0, 3 6 5 1 ≤ Ch− q q 6 ϕ− 3 G∗Z , 6

6

6

5

1

∩

− 31

W01, 6 (Ω)

G∗Z .

(2.11) Set q = | ln h| in (2.11), and

where w ∈ W (Ω) and Lw = ϕ by the Young inequality, we get 2 1 1 5 5 2 ∥G∗Z ∥ − 31 ≤ C |ln h| 6 ϕ− 3 G∗Z ≤ C(ε) |ln h| 3 + ε ϕ− 3 G∗Z 2, 3

ϕ

1

(2.12)

1

In addition, from (1.4) and (2.3), − 13 ∗ 2 ϕ GZ 1

) ( 1 1 ≤ Ca ϕ− 3 G∗Z , ϕ− 3 G∗Z ( ) 2 1 ≤ C a(G∗Z , ϕ− 3 G∗Z ) + C (G∗Z )2 , |∇(ϕ− 3 )|2 2 2 ≤ C a(G∗Z , ϕ− 3 G∗Z ) + C ∥G∗Z ∥ − 13 ϕ

h 2 2 ≤ C˜ δ −1 + C˜ ∥G∗ ∥ − 1 . Z ϕ

Z ϕ

(2.13)

3

Combining (2.9), (2.12), (2.13), and choosing ε such that εC˜ = 21 , we immediately obtain the result (2.10). Theorem 2.1. For G∗Z the regularized Green’s function defined by (1.4), we have the following W 2, 1 -seminorm estimate: 4

|G∗Z |2, 1 ≤ C |ln h| 3 . Proof. Obviously, 2 |G∗Z |2, 1

∫ ≤ Ω

2 ϕ dX · ∇2 G∗Z ϕ−1 .

250

(2.14)

(2.15)

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Furthermore,

≤

≤ ≤ ≤ ≤ ≤ ≤

∫ ∫ (

2 ∗ 2 )2 −1 2 ∗ 2 − 12 2 ∗

∇ GZ −1 = ϕ ∇ G dX = ϕ ∇ G dX Z Z ϕ Ω Ω (∫ ( ∫ ) 2 − 21 ∗ 2 2 − 21 ∗ 2 C ∇ ϕ GZ dX + ∇ ϕ GZ dX Ω Ω ) ∫ − 12 2 2 + ∇ϕ |∇G∗Z | dx Ω ) ( ( )

2 − 12 ∗ 2 ∗ 2 ∗ 2 C ∇ ϕ GZ + ∥GZ ∥ − 31 + |GZ | − 32 1, ϕ ϕ 0 ) ( ( ) 2

∗ 2 ∗ 2 − 12 ∗ C L ϕ GZ + ∥GZ ∥ − 13 + |GZ | − 32 1, ϕ ϕ 0 ) ( 2 2 2 C ∥LG∗Z ∥ϕ−1 + ∥G∗Z ∥ − 13 + |G∗Z | − 32 1, ϕ ϕ ( )

h 2 2 2 ∗ − ∗ C δZ ϕ−1 + C a GZ , ϕ 3 GZ + C ∥G∗Z ∥ − 13 ϕ ( )

h 2 h − 32 ∗ ∗ 2

C δZ ϕ−1 + C δZ , ϕ GZ + C ∥GZ ∥ − 13 ϕ

h 2 2 ∗ C δZ ϕ−1 + C ∥GZ ∥ − 13 , ϕ

combined with (2.9) and (2.10), we have

2 ∗ 2 5

∇ GZ −1 ≤ C |ln h| 3 . ϕ

(2.16)

By (2.4), (2.15), and (2.16), we immediately obtain the result (2.14).

3

Estimates for the Discrete Green’s Function

The definition (1.5) shows that GhZ is a finite element approximation to G∗Z . In this section, we give the W 2, 1 -seminorm estimate for GhZ . Lemma 3.1. For G∗Z and GhZ , the regularized Green’s function and the discrete Green’s function, respectively, we have the following estimate: ∗ 4 GZ − GhZ ≤ Ch |ln h| 3 . 1, 1

(3.1)

Proof. Obviously, ∗ GZ − GhZ 2 ≤ 1, 1

∫ Ω

2 ϕ dX · G∗Z − GhZ 1, ϕ−1 .

(3.2)

Similar to the proof of (2.43) in [13], and using (2.16), we have ∗ G − Gh 2 −1 Z Z 1, ϕ

2

2 ≤ Ch2 ∇2 G∗Z ϕ−1 + Cˆ G∗Z − GhZ ϕ− 23

2 5 ≤ Ch2 |ln h| 3 + Cˆ G∗ − Gh − 2 . Z

251

Z ϕ

(3.3)

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In addition

∗

G − Gh 2 − 2 = (ϕ− 23 (G∗ − Gh ), G∗ − Gh ) = a(w, G∗ − Gh ) Z Z ϕ 3 Z Z Z Z ∗ Z h 2 Z 2 ∗ h = a(w − Πw, G − G ) ≤ ε G − G . −1 + C(ε) |w − Πw| Z

Z

Z

Z 1, ϕ

1, ϕ

(3.4) 2 where Lw = ϕ− 3 (G∗Z − GhZ ) and Π is an interpolation operator. Using the weighted interpolation error estimate in (3.4) (similar to pp.110 Lemma 4 in [17]) yields

∗



GZ − GhZ 2 − 2 ≤ ε G∗Z − GhZ 2 −1 + C(ε)h2 ∇2 w 2 . ϕ 3 ϕ 1, ϕ

(3.5)

Further, from the a priori estimate (2.2), (2.5), and the Sobolev Embedding Theorem [19], 2 2 ∇ w

ϕ

2

2

2

2 ≤ ∥ϕ∥0, 3 ∇2 w 0, 6 ≤ Cθ−2 ∥w∥2, 6 ≤ Cθ−2 ϕ− 3 (G∗Z − GhZ ) 2

2

( ) 2 0, 6 2

2

≤ Cθ−2 ϕ− 3 (G∗Z − GhZ ) ≤ Cθ−2 L ϕ− 3 (G∗Z − GhZ ) 2 )0 (

2

∗

∗ h 2 h 2 2 −2 − 23 h

≤ Cθ

ϕ δZ + GZ − GZ 1, ϕ−1 + GZ − GZ ϕ− 3 . 0

(3.6) Similar to the proof of (2.9), we can obtain

2 2

−3 h

ϕ δZ ≤ Ch2 .

(3.7)

0

Combining (3.5)–(3.7) yields

∗

G − G h 2 − 2 Z Z ϕ 3

≤

2 ( ) ε + C(ε)γ −2 G∗Z − GhZ 1, ϕ−1

2 +C(ε)γ −2 G∗ − Gh − 2 + C(ε)γ −2 h2 . Z

Z ϕ

(3.8)

3

Choosing suitable ε and γ ∈ [6, +∞) such that 0 < (2ε + 1)Cˆ < 1 as well as C(ε)γ −2 = 21 . From (3.8),

∗

GZ − GhZ 2 − 2 ≤ (2ε + 1) G∗Z − GhZ 2 −1 + h2 . 1, ϕ ϕ 3

(3.9)

From (3.3) and (3.9), ∗ 5 GZ − GhZ 2 −1 ≤ Ch2 |ln h| 3 . 1, ϕ

(3.10)

The result (3.1) immediately follows the results (2.4), (3.2), and (3.10). Theorem 3.1. For GhZ the discrete Green’s function, we have the following estimate: h h GZ ≤ C| ln h| 34 . (3.11) 2, 1

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Proof. By the triangle inequality, the interpolation error estimate, and the inverse property, we have h h h G ≤ G∗Z − GhZ 2, 1 + |G∗Z |2, 1 Z 2, 1 h h ≤ |G∗Z |2, 1 + |G∗Z − ΠG∗Z |2, 1 + ΠG∗Z − GhZ 2, 1 ≤ C |G∗Z |2, 1 + Ch−1 ΠG∗Z − GhZ 1, 1 (3.12) ≤ C |G∗Z |2, 1 + Ch−1 |G∗Z − ΠG∗Z |1, 1 +Ch−1 G∗Z − GhZ 1, 1 ≤ C |G∗Z |2, 1 + Ch−1 G∗Z − GhZ 1, 1 . Combining (2.14), (3.1), and (3.12) yields the result (3.11).

4

Superconvergence of the Displacement of the Finite Element

In this section, we give an application of the estimate for the discrete Green’s function in finite element superconvergence. Let Πu and uh be the interpolant and the finite element approximation to u, the solution of (1.1), respectively. Similar to the proof of [13], we can obtain the following lemma. m-degree finite element space. Lemma 4.1. Let S0h (Ω) be the tensor-product ∩ Suppose v ∈ S0h (Ω) and u ∈ W m+2, ∞ (Ω) H01 (Ω). Then we have the following weak estimate of the second type: |a(u − Πu, v)| ≤ Chm+2 ∥u∥m+2, ∞ |v|h2, 1 , m ≥ 2, (4.1) ∑ where |v|h2, 1 = e∈T h |v|2, 1, e . Finally, we give the following superconvergent estimate. h ¯ Theorem 4.1. ∩ 1Let {T } be a regular family of partitions of Ω and u ∈ m+2, ∞ W (Ω) H0 (Ω). For uh and Πu, the tensor-product m-degree finite element approximation and the corresponding interpolant to u, respectively. Then we have the following superconvergent estimates: 4

|uh − Πu|0, ∞, Ω ≤ Chm+2 |ln h| 3 ∥u∥m+2, ∞ , m ≥ 2.

(4.2)

Proof. For every Z ∈ Ω, applying the definition of GhZ and the Galerkin orthogonality relation (1.2), we derive (uh − Πu)(Z) = a(uh − Πu , GhZ ) = a(u − Πu , GhZ ).

(4.3)

From (3.11), (4.1), and (4.3), we immediately obtain the result (4.2). Acknowledgments This work was supported by the National Natural Science Foundation of China Grant 11161039. References

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1. J. H. Brandts and M. Kˇr´ıˇzek, Gradient superconvergence on uniform simplicial partitions of polytopes, IMA J. Numer. Anal. 23 (2003), 489–505. 2. J. H. Brandts and M. Kˇr´ıˇzek, Superconvergence of tetrahedral quadratic finite elements, J. Comput. Math. 23 (2005), 27–36. 3. C. M. Chen, Optimal points of stresses for the linear tetrahedral element (in Chinese), Nat. Sci. J. Xiangtan Univ. 3 (1980), 16–24. 4. C. M. Chen, Construction theory of superconvergence of finite elements (in Chinese), Hunan Science and Technology Press, Changsha, China, 2001. 5. C. M. Chen and Y. Q. Huang, High accuracy theory of finite element methods (in Chinese), Hunan Science and Technology Press, Changsha, China, 1995. 6. L. Chen, Superconvergence of tetrahedral linear finite elements, Internat J. Numer. Anal. Model. 3 (2006), 273–282. 7. G. Goodsell, Gradient superconvergence for piecewise linear tetrahedral finite elements, Technical Report RAL-90-031, Science and Engineering Research Council, Rutherford Appleton Laboratory, 1990. 8. G. Goodsell, Pointwise superconvergence of the gradient for the linear tetrahedral element, Numer. Meth. Part. Differ. Equ. 10 (1994), 651–666. 9. M. Kˇr´ıˇzek and P. Neittaanm¨ aki, Finite element approximation of variational problems and applications, Longman, Harlow, 1990. 10. Q. Lin and N. N. Yan, Construction and analysis of high efficient finite elements (in Chinese), Hebei University Press, Baoding, China, 1996. 11. R. C. Lin and Z. M. Zhang, Natural superconvergent points in 3D finite elements, SIAM J. Numer. Anal. 46 (2008), 1281–1297. 12. J. H. Liu, B. Jia, and Q. D. Zhu, An estimate for the three-dimensional discrete Green’s function and applications, J. Math. Anal. Appl. 370 (2010), 350-363. 13. J. H. Liu and Q. D. Zhu, Pointwise supercloseness of tensor-product block finite elements, Numer. Meth. Part. Differ. Equ. 25 (2009), 990-1008. 14. J. H. Liu and Q. D. Zhu, The W 1,1 -seminorm estimate for the four-dimensional discrete derivative Green’s function, J. Comp. Anal. Appl. 14 (2012), 165-172. 15. A. H. Schatz, I. H. Sloan, and L. B. Wahlbin, Superconvergence in finite element methods and meshes that are locally symmetric with respect to a point, SIAM J. Numer. Anal. 33 (1996), 505–521. 16. Z. M. Zhang and R. C. Lin, Locating natural superconvergent points of finite element methods in 3D, Internat J. Numer. Anal. Model. 2 (2005), 19–30. 17. Q. D. Zhu and Q. Lin, Superconvergence theory of the finite element methods (in Chinese), Hunan Science and Technology Press, Changsha, China, 1989. 18. R. A. Adams, Sobolev Spaces, Academic Press, New York, 1975.

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Estimates for Discrete Derivative Green’s Function for Elliptic Equations in Dimensions Seven and Up Jinghong Liu∗and Yinsuo Jia† This article will discuss estimates for discrete derivative Green’s function for elliptic equations in dimensions seven and up. First, the definitions of some terms are given. Then the estimates for the regularized derivative Green’s function are derived. Finally, using the triangular inequality, we obtain the estimates for discrete derivative Green’s function. The results of the article play important roles in the research of superconvergence of finite element methods.

1

Introduction

It is well known that estimates for the Green’s function play very important roles in the study of the superconvergence (especially, pointwise superconvergence) of the finite element method (see [1–8]). For one- and two-dimensional elliptic problems, one have obtained many optimal estimates for the Green’s function (see [8]). Recently, for dimensions three to five, we have obtained some optimal estimates for the discrete Green’s function (see [4–7]). At present, we also consider the six-dimensional discrete Green’s function and its estimates, and some results have been submitted to some Journals. In this article, we will discuss estimates for the discrete derivative Green’s function in dimensions seven and up. we shall use the symbol C to denote a generic constant, which is independent from the discretization parameter h and which may not be the same in each occurrence and also use the standard notations for the Sobolev spaces and their norms. We consider the following Poisson equation: Lu ≡ −∆u = f in Ω,

u = 0 on ∂Ω,

(1.1)

where Ω ⊂ Rd (d ≥ 7) is a bounded polytopic domain. The weak formulation of ∗ Department of Fundamental Courses, Ningbo Institute of Technology, Zhejiang University, Ningbo 315100, China, email: [email protected] † School of Mathematics and Computer Science, Shangrao Normal University, Shangrao 334001, China, email: [email protected]

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(1.1) reads,

{

Find u ∈ H01 (Ω) satisfying a(u , v) = (f , v) for all v ∈ H01 (Ω). ∫

where a(u , v) ≡

∇u · ∇v dX, Ω

∫

and (f , v) ≡

f v dX. Ω

¯ Denote by S h (Ω) a continLet {T h } be a regular family of partitions of Ω. uous finite S0h (Ω) = ∩ 1elements space regarding this kind of partitions and let h h S (Ω) H0 (Ω). Discretizing the above weak formulation using S0 (Ω) as approximating space means, { Find uh ∈ S0h (Ω) satisfying a(uh , v) = (f , v) for all v ∈ S0h (Ω). h For every Z ∈ Ω, we define the discrete derivative δ function ∂Z,ℓ δZ ∈ S0h (Ω) 2 h and the L -projection Ph u ∈ S0 (Ω) such that h (v, ∂Z,ℓ δZ ) = ∂ℓ v(Z) ∀ v ∈ S0h (Ω).

(1.2)

(u − Ph u, v) = 0 ∀ v ∈ S0h (Ω).

(1.3)

Here, for any direction ℓ ∈ R , |ℓ| = 1, and ∂ℓ v(Z) stand for the following onesided directional derivatives, respectively. d

h ∂Z,ℓ δZ =

h ∂Z,ℓ δZ

h h δZ+∆Z − δZ v(Z + ∆Z) − v(Z) , ∂ℓ v(Z) = lim , ∆Z = |∆Z|ℓ. |∆Z| |∆Z| |∆Z|→0 |∆Z|→0

lim

Let ∂Z,ℓ G∗Z ∈ H 2 (Ω)∩H01 (Ω) be the solution of the elliptic problem −∆∂Z,ℓ G∗Z = h ∂Z,ℓ δZ . We may call ∂Z,ℓ G∗Z the regularized derivative Green’s function. Further, let the discrete derivative Green’s function ∂Z,ℓ GhZ ∈ S0h (Ω) be the finite element approximation to ∂Z,ℓ G∗Z . Thus, a(∂Z,ℓ G∗Z − ∂Z,ℓ GhZ , v) = 0 ∀ v ∈ S0h (Ω). In this article, we will bound the terms |∂Z,ℓ G∗Z |1, 1 and ∂Z,ℓ GhZ 1, 1 .

2

(1.4)

Regularized Derivative Green’s Function and Its Estimates

We first introduce the weight function defined by ( ) d ¯ ¯ 2 + θ2 − 2 ∀ X ∈ Ω, ϕ ≡ ϕ(X) = |X − X|

256

(2.1)

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¯ ∈Ω ¯ is a fixed point, θ = γh, and γ ∈ [d, +∞) is a suitable real number. where X For every α ∈ R, we give the following notations: (∫ )2 m ∑ ∑ 2 2 β 2 n α n 2 |∇ v| = D v , |∇ v|ϕα = ϕ |∇ v| dX , ∥v∥m, ϕα = |∇n v|ϕα , 1

n

2

Ω

|β|=n

n=0

where β = (β1 , β2 , · · · , βd ), |β| = β1 + β2 + · · · + βd , and βi ≥ 0, i = 1, · · · , d are integers. In particular, for the case of m = 0, we write (∫ ∥v∥ϕα =

ϕ |v| dX α

2

) 12 .

Ω

We assume there exists a real number q0 (1 < q0 ≤ ∞) such that ∥v∥2, q ≤ C(q)∥Lv∥0, q ∀ v ∈ W 2, q (Ω) ∩ W01, q (Ω), 1 < q < q0 ,

(2.2)

which is the so-called a priori estimate (see [8]). As in the two-dimensional case (see [8]), we can obtain the following Lemma 2.1. Lemma 2.1. For ϕ the weight function defined by (2.1), we have the following estimates: n |∇n ϕα | ≤ C(α, n)ϕα+ d , α ∈ R, n = 1, 2, (2.3) ∫ ϕ dX ≤ C(k)| ln θ|, θ ≤ k < 1,

(2.4)

ϕα dX ≤ C(α − 1)−1 θ−d(α−1) ∀ α > 1.

(2.5)

Ω

∫ Ω

∫

ϕα dX ≤ C(1 − α)−1 ∀ 0 < α < 1.

(2.6)

Ω

In addition, we also have the following Lemmas. Lemma 2.2. For Ph w the L2 -projection of w, we have the following stability estimate: ∥Ph w∥0, q ≤ C∥w∥0, q , 1 ≤ q ≤ +∞. (2.7) h Lemma 2.3. For ∂Z,ℓ δZ the discrete derivative δ function defined by (1.2), we have the following estimate: h |∂Z,ℓ δZ (X)| ≤ Ch−d−1 e−Ch

−1

|X−Z|

,

(2.8)

¯ and C is independent of h, X, and Z. where X, Z ∈ Ω, h As for ∂Z,ℓ δZ , we have the following important estimate. h Lemma 2.4. For ∂Z,ℓ δZ the discrete derivative δ function defined by (1.2) and ϕ defined by (2.1), when α > 0, we have the following estimate:

h

∂Z,ℓ δZ

ϕ−α

≤ Ch

257

d(α−1)−2 2

.

(2.9)

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Proof. From (2.1) and (2.8), ∫

( ) dα −1 h 2

∂Z,ℓ δZ ≤ C |X − Z|2 + θ2 2 h−2d−2 e−Ch |X−Z| dX ϕ−α ∫Ω∞ ( 2 ) dα −1 ≤ C r + θ2 2 h−2d−2 e−Ch r rd−1 dr. 0

Set h−1 r = t, then

h 2

∂Z,ℓ δZ ≤ Chd(α−1)−2 ϕ−α

which is the result (2.9). Lemma 2.5. Suppose q0 > 2 and 0 < ε < 1. For ∂Z,ℓ G∗Z the regularized derivative Green’s function defined by (1.4) and ϕ defined by (2.1), we have the following weighted-norm estimate: ∥∂Z,ℓ G∗Z ∥ϕ1−ε ≤ Ch1−d+ 2 . εd

Proof. Set r =

1+ε ′ 1−ε , r

=

(2.10)

thus 1r + r1′ = 1. From (2.5), ∫ 2 = ϕ1−ε |∂Z,ℓ G∗Z | dX

1+ε 2ε ,

∥∂Z,ℓ G∗Z ∥ϕ1−ε 2

Ω

(∫ ≤

ϕ1+ε dX

) 1−ε 1+ε

∥∂Z,ℓ G∗Z ∥0, 1+ε 2

ε

Ω

( ) 1−ε 2 ≤ C ε−1 θ−dε 1+ε ∥∂Z,ℓ G∗Z ∥0, 1+ε . ε

Further, 1+ε

∥∂Z,ℓ G∗Z ∥0,ε1+ε ε

(

) 1 ∂Z,ℓ G∗Z , |∂Z,ℓ G∗Z | ε sgn∂Z,ℓ G∗Z ( ) h = a(∂Z,ℓ G∗Z , w) = ∂Z,ℓ δZ , w = ∂ℓ Ph w(Z)

=

≤

|Ph w|1, ∞ ≤ Ch− q −1 ∥Ph w∥0, q d

≤ Ch− q −1 ∥w∥0, q , d

where q ≥ 1, and w ∈ H01 (Ω) satisfies ( ) 1 a(v, w) = v, |∂Z,ℓ G∗Z | ε sgn∂Z,ℓ G∗Z ∀ v ∈ H01 (Ω). d(1+ε) Taking q = d−2(1+ε) > 1 and p1 = 1q + d2 , we have p = 1 + ε < 2. By the a priori estimate (2.2) and the Sobolev Embedding Theorem (see [9]), we get 1

ε ∥w∥0, q ≤ C ∥w∥2, p ≤ C ∥∂Z,ℓ G∗Z ∥0, 1+ε . ε

Thus

∥∂Z,ℓ G∗Z ∥0, 1+ε ≤ Ch− 2

2d q −2

2d

= Ch2− 1+ε ,

ε

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which results in

∥∂Z,ℓ G∗Z ∥ϕ1−ε ≤ Ch2−2d+εd . 2

The proof of the result (2.10) is completed. Lemma 2.6. For ∂Z,ℓ G∗Z the regularized derivative Green’s function defined by h (1.4) and ∂Z,ℓ δZ the discrete derivative δ function defined by (1.2), we have the following weighted-norm estimate:

2 h 2 ∗ 2 (2.11) ∥∇(∂Z,ℓ G∗Z )∥ϕ−α ≤ C ∂Z,ℓ δZ −α− 2 + C ∥∂Z,ℓ GZ ∥ −α+ 2 ∀α ∈ R. d ϕ d ϕ

Proof. Obviously, ∥∇(∂Z,ℓ G∗Z )∥ϕ−α ≤ a(∂Z,ℓ G∗Z , ϕ−α ∂Z,ℓ G∗Z ) + C ∥∂Z,ℓ G∗Z ∥ 2

2 −α+ 2 d ϕ

.

(2.12)

Further, a(∂Z,ℓ G∗Z , ϕ−α ∂Z,ℓ G∗Z )

h = (∂Z,ℓ δZ , ϕ−α ∂Z,ℓ G∗Z ) h ≤ ∥∂Z,ℓ δZ ∥ −α− d2 ∥∂Z,ℓ G∗Z ∥ ϕ 1 h 2 2 2 (∥∂Z,ℓ δZ ∥ϕ−α− d

≤

−α+ 2 d

ϕ

+ ∥∂Z,ℓ G∗Z ∥2 −α+ 2 ). ϕ

(2.13)

d

Combining (2.12) and (2.13) immediately yields the result (2.11). Lemma 2.7. Suppose − d2 < α < d2 and q0 > 2. For ∂Z,ℓ G∗Z the regularized derivative Green’s function defined by (1.4), we have the following weightednorm estimate: d(α−1) ∥∇(∂Z,ℓ G∗Z )∥ϕ−α ≤ Ch 2 (2.14) Proof. From (2.9),

h

∂Z,ℓ δZ

−α− 2 d

ϕ

≤ Ch

d(α−1) 2

≤ Ch

d(α−1) 2

.

(2.15)

.

(2.16)

From (2.10), ∥∂Z,ℓ G∗Z ∥

−α+ 2 d ϕ

Combining (2.11), (2.15) and (2.16) immediately yields the result (2.14). Theorem 2.1. Suppose q0 > 2 and d ≥ 7. For ∂Z,ℓ G∗Z the regularized derivative Green’s function defined by (1.4), we have the following estimate: |∂Z,ℓ G∗Z |1,1 ≤ Ch

2−d 2

(2.17)

Proof. Obviously, |∂Z,ℓ G∗Z |1, 1 ≤ When 0 < α
2 and 0 < α < min{ d4 , 1 − q20 + d2 }, then we have

d(α−1)

∂Z,ℓ G∗Z − ∂Z,ℓ GhZ −α ≤ Ch 2 . 1, ϕ

(3.2)

Proof. From (3.1),

∂Z,ℓ G∗Z − ∂Z,ℓ GhZ 2 −α 1, ϕ

2

2 2 ∗ 2 ≤ Ch ∇ (∂Z,ℓ GZ ) ϕ−α + C ∂Z,ℓ G∗Z − ∂Z,ℓ GhZ ϕ−α+ d2 ( )

2

2 ≤ Cˆ h2 ∇2 (∂Z,ℓ G∗Z ) ϕ−α + ∂Z,ℓ G∗Z − ∂Z,ℓ GhZ ϕ−α+ d2 . Similar to the Lemma 6 in [8, Chapter 3], we obtain



∂Z,ℓ G∗Z − ∂Z,ℓ GhZ 2 −α+ 2 ≤ 2 ∂Z,ℓ G∗Z − ∂Z,ℓ GhZ 2 −α . d ϕ 1, ϕ ˆ 3C Then we have



∂Z,ℓ G∗Z − ∂Z,ℓ GhZ 2 −α ≤ Ch2 ∇2 (∂Z,ℓ G∗Z ) 2 −α . ϕ 1, ϕ Similar to the arguments of the result (2.14), when 0 < α < can get

2

d(α−1)−2

∇ (∂Z,ℓ G∗Z ) −α ≤ Ch 2 . ϕ

4 d

(3.3)

and q0 > 2, we (3.4)

Combining (3.3) and (3.4) immediately yields the result (3.2). 2d Lemma 3.3. Suppose q0 > d−2 . For ∂Z,ℓ G∗Z and ∂Z,ℓ GhZ , the regularized derivative Green’s function and the discrete derivative Green’s function, respectively, we have the following estimate: ∂Z,ℓ G∗Z − ∂Z,ℓ GhZ ≤ Ch 4−d 2 . (3.5) 1, 1 Proof. Obviously, ∂Z,ℓ G∗Z − ∂Z,ℓ GhZ 2 ≤ 1, 1

∫ Ω

2 ϕα dX · ∂Z,ℓ G∗Z − ∂Z,ℓ GhZ 1, ϕ−α .

260

(3.6)

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LIU, JIA: ESTIMATES FOR DISCRETE DERIVATIVE GREEN’S FUNCTION

When d ≥ 7 and 0 < α < min{ d4 , 1 − q20 + d2 }, from (2.6), (3.2) and (3.6), ∂Z,ℓ G∗Z − ∂Z,ℓ GhZ 2 ≤ C(1 − α)−1 hd(α−1) . 1, 1 2d , we have d4 < 1 − q20 + d2 . Thus, Since q0 > d−2 ∂Z,ℓ G∗Z − ∂Z,ℓ GhZ 2 ≤ C inf (1 − α)−1 hd(α−1) = Ch4−d , 1, 1 4 0 d−2 and d ≥ 7. For ∂Z,ℓ GhZ the discrete derivative Green’s function defined by (1.4), we have the following estimate: ∂Z,ℓ GhZ ≤ Ch 2−d 2 . (3.7) 1, 1 Proof. By the triangular inequality, ∂Z,ℓ GhZ ≤ |∂Z,ℓ G∗Z | + ∂Z,ℓ G∗Z − ∂Z,ℓ GhZ . 1, 1 1, 1 1, 1

(3.8)

From (2.17), (3.5) and (3.8), we immediately obtain the result (3.7). Acknowledgments This work was supported by the National Natural Science Foundation of China Grant 11161039, and the Zhejiang Provincial Natural Science Foundation Grant LY13A010007. References 1. C. M. Chen, Construction theory of superconvergence of finite elements (in Chinese), Hunan Science and Technology Press, Changsha, China, 2001. 2. C. M. Chen and Y. Q. Huang, High accuracy theory of finite element methods (in Chinese), Hunan Science and Technology Press, Changsha, China, 1995. 3. G. Goodsell, Pointwise superconvergence of the gradient for the linear tetrahedral element, Numer. Meth. Part. Differ. Equ. 10 (1994), 651–666. 4. J. H. Liu, B. Jia, and Q. D. Zhu, An estimate for the three-dimensional discrete Green’s function and applications, J. Math. Anal. Appl. 370 (2010), 350-363. 5. J. H. Liu and Q. D. Zhu, Pointwise supercloseness of tensor-product block finite elements, Numer. Meth. Part. Differ. Equ. 25 (2009), 990-1008. 6. J. H. Liu and Q. D. Zhu, The W 1,1 -seminorm estimate for the four-dimensional discrete derivative Green’s function, J. Comp. Anal. Appl. 14 (2012), 165-172. 7. J. H. Liu and Y. S. Jia, Five-dimensional discrete Green’s function and its estimates, J. Comp. Anal. Appl. 18 (2015), 620-627. 8. Q. D. Zhu and Q. Lin, Superconvergence theory of the finite element methods (in Chinese), Hunan Science and Technology Press, Changsha, China, 1989. 9. R. A. Adams, Sobolev Spaces, Academic Press, New York, 1975.

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Existence of Solutions to a Coupled System of Higher-order Nonlinear Fractional Differential Equations with Anti-periodic Boundary Conditions Huina Zhanga,b , Wenjie Gaob,∗ a College

of Science, China University of Petroleum (East China), Qingdao, 266555, P.R.China b Institute of Mathematics, Jilin University, Changchun, 130012, P.R.China

Abstract In this paper, the authors study a coupled system of nonlinear fractional differential equations of order α, β ∈ (4, 5), the differential operator is taken in the Caputo sense. By using the Schauder fixed point theorem and the contraction mapping principle, the existence and uniqueness of solutions to the system with anti-periodic boundary conditions are obtained. Two examples are given to demonstrate the feasibility of the results. Keywords: Coupled system; Fractional differential equations; Anti-periodic boundary conditions; existence; uniqueness.

1. Introduction Recently, fractional differential equations have proved to be valuable tools in various fields of science and engineering. Indeed, we can find numerous applications in control, porous media, fluid flows, chemical physics and many other branches of science, see[1–3]. As a result, there are many papers dealing with the existence and uniqueness of solutions to nonlinear fractional differential equations, see[4–10]. Anti-periodic boundary value problems arise in the mathematical modeling of a variety of physical process, many authors have paid much attention in such problems, for examples and details of anti-periodic boundary conditions, the interested readers may refer to [11–17]. On the other hand, the coupled systems of nonlinear fractional differential equations have been a subject of intensive studies [17–21]. It should be noted that in [18–21], the study objects are coupled systems, but not the case of Caputo fractional derivatives. In [11–16], the authors only studied the existence of solutions for anti-periodic boundary value problems of fractional differential equation but not the coupled system. Motivated by [17], we consider a coupled system of nonlinear fractional differential equations in the sense of Caputo with a nonlinear term containing the derivatives of unknown functions. In this paper, we study the existence and uniqueness of solutions to the following coupled system of ∗ Corresponding

author: +86 431 85166425 Email addresses: [email protected] (Huina Zhang ), [email protected] (Wenjie Gao)

Preprint submitted to Journal of Computational Analysis and Applications

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nonlinear fractional differential equations  c α  D x(t) + f (t, y(t),c Dp y(t)) = 0, t ∈ [0, T ],     c Dβ y(t) + g(t, x(t),c Dq x(t)) = 0, t ∈ [0, T ],  x(k) (0) = −x(k) (T ), k = 0, 1, 2, 3, 4,     (k) y (0) = −y (k) (T ), k = 0, 1, 2, 3, 4,

(1.1)

where 4 < α, β < 5, α − q ≥ 1, β − p ≥ 1, c Dα denotes the Caputo fractional derivative of order α, f, g : [0, T ] × R × R → R are given continuous functions. This paper is organized as follows. In Section 2, we recall some basic definitions and preliminary results. In Section 3, we prove the existence of solutions to (1.1) by means of the Schauder fixed point theorem. Then, we obtain the uniqueness of solutions to the system by the contraction mapping principle. At the end, two examples are given to illustrate the applicability of our results. 2. Background Materials For the convenience of the readers, we present here the necessary definitions and lemmas [2], which are used throughout this paper. Definition 2.1. The Riemann-Liouville fractional integral of order α > 0 of a function y : (0, ∞) → R is given by I α y(t) =

1 Γ(α)

∫

t

(t − s)α−1 y(s)ds, 0

provided the right hand side is pointwise defined on (0, ∞). Definition 2.2. The Caputo fractional derivative of order α > 0 of a continuous function y : (0, ∞) → R is given by c

Dα y(t) =

1 Γ(n − α)

∫

t 0

y (n) (s) ds, (t − s)α−n+1

where n = [α] + 1, [α] denotes the integer part of number α, provided that the right side is pointwise defined on (0, ∞). Lemma 2.3. For any y ∈ C[0, T ], the unique solution of the boundary value problem { c q D x(t) = y(t), t ∈ [0, T ], 4 < q ≤ 5, x(k) (0) = −x(k) (T ), k = 0, 1, 2, 3, 4 can be written as

∫

(2.1)

T

G(t, s)y(s)ds,

x(t) = 0

where G(t, s) is the Green’s function given by  −s)q−2 −s)q−3 2(t−s)q−1 −(T −s)q−1  + (T −2t)(T + t(T −t)(T  2Γ(q) 4Γ(q−1) 4Γ(q−2)     + (6T t2 −4t3 −T 3 )(T −s)q−4 + (2T t3 −tT 3 −t4 )(T −s)q−5 , 0 < s < t < T, 48Γ(q−3) 48Γ(q−4) G(t, s) = (T −s)q−1 (T −2t)(T −s)q−2 t(T −t)(T −s)q−3  + + −  2Γ(q) 4Γ(q−1) 4Γ(q−2)     + (6T t2 −4t3 −T 3 )(T −s)q−4 + (2T t3 −tT 3 −t4 )(T −s)q−5 , 0 < t < s < T. 48Γ(q−3) 48Γ(q−4) 2

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Let

 (t−s)α−1 − 12 (T −s)α−1 −s)α−2 −s)α−3  + (T −2t)(T + t(T −t)(T   Γ(α) 4Γ(α−1) 4Γ(α−2)    + (6T t2 −4t3 −T 3 )(T −s)α−4 + (2T t3 −tT 3 −t4 )(T −s)α−5 , 0 < s < t < T, 48Γ(α−3) 48Γ(α−4) G1 (t, s) = α−2 −s)α−3  − 1 (T − s)α−1 + (T −2t)(T −s) + t(T −t)(T  2Γ(α) 4Γ(α−1) 4Γ(α−2)     + (6T t2 −4t3 −T 3 )(T −s)α−4 + (2T t3 −tT 3 −t4 )(T −s)α−5 , 0 < t < s < T. 48Γ(α−3)

48Γ(α−4)

48Γ(β−3)

48Γ(β−4)

 (t−s)β−1 − 12 (T −s)β−1 −s)β−2 −s)β−3  + (T −2t)(T + t(T −t)(T   Γ(β) 4Γ(β−1) 4Γ(β−2)    + (6T t2 −4t3 −T 3 )(T −s)β−4 + (2T t3 −tT 3 −t4 )(T −s)β−5 , 0 < s < t < T, 48Γ(β−3) 48Γ(β−4) G2 (t, s) = (T −2t)(T −s)β−2 −s)β−3 1 β−1  − (T − s) + + t(T −t)(T   2Γ(β) 4Γ(β−1) 4Γ(β−2)    + (6T t2 −4t3 −T 3 )(T −s)β−4 + (2T t3 −tT 3 −t4 )(T −s)β−5 , 0 < t < s < T. We call (G1 , G2 ) Green’s function for Problem (1.1). Define the space { } C = x(t) : x(t) ∈ C 4 [0, T ], x(k) (0) = −x(k) (T ), k = 0, 1, 2, 3, 4 , and { } X = x(t) : x(t) ∈ C and (c Dq x)(t) ∈ C[0, T ] endowed with the norm ∥x∥X = max max | x(i) (t) | + max | (c Dq x)(t) |, 0≤i≤4 t∈[0,T ]

t∈[0,T ]

where i ∈ N. Lemma 2.4. (X, ∥ · ∥X ) is a Banach space. P roof. Apparently X is a subspace of C 4 [0, T ], so we only need to prove that X is closed. Let xn (t) be a sequence converging to some x(t) in (X, ∥ · ∥X ), then it is clear that xn (t) is a converging sequence in the space C 4 [0, T ] and hence x ∈ C. Furthermore, the uniform convergence of (c Dq xn )(t) to (c Dq x)(t) implies that (c Dq x)(t) ∈ C[0, T ] and therefore x(t) ∈ X. The proof is complete. Similarly, we can define the Banach space { } Y = y(t) : y(t) ∈ C and (c Dp )y(t) ∈ C[0, T ] endowed with the norm ∥y∥Y = max max | y (i) (t) | + max | (c Dp y)(t) |, 0≤i≤4 t∈[0,T ]

t∈[0,T ]

where i ∈ N. For (x, y) ∈ (X, Y ), let ∥(x, y)∥X×Y = max{∥x∥X , ∥y∥Y }. Then clearly (X × Y, ∥ · ∥X×Y ) is a Banach space. 3

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Consider the following coupled system of integral equations: { ∫T x(t) = 0 G1 (t, s)f (s, y(s),c Dp y(t))ds, ∫T y(t) = 0 G2 (t, s)g(s, x(s),c Dq x(t))ds.

(2.2)

The following lemma states that Problem (1.1) is equivalent to Problem (2.2) and therefore the study of a system of differential equations is turn into the study of a system of integral equations. Lemma 2.5. Assume that f, g : [0, T ] × R × R → R are continuous functions. Then (x, y) ∈ (X, Y ) is a solution of (1.1) if and only if (x, y) ∈ (X, Y ) is a solution of system (2.2). P roof. The proof is immediate from the discussion above, we omit the details here. Let F : X × Y → X × Y be an operator defined as F (x, y)(t) = (F1 y(t), F2 x(t)), where ∫

∫

T

T

G1 (t, s)f (s, y(s),c Dp y(t))ds, F2 x(t) =

F1 y(t) =

G2 (t, s)g(s, x(s),c Dq x(t))ds.

0

0

It is obvious that a fixed-point of the operator F is a solution of Problem (1.1). Now we present the main results of this paper. 3. Main Results In this section, we will discuss the existence and uniqueness of solutions to Problem (1.1). Lemma 3.1.

[17]

The Green’s functions G1 (t, s), G2 (t, s) satisfy the following estimates: ∫T 0

∫T 0

| G1 (t, s) | ds ≤

3 Tα Γ(α+1) ( 2

+

5α4 −14α3 +55α2 +146α ) 768

= U1 , t ∈ [0, T ],

+55β +146β T | G2 (t, s) | ds ≤ Γ(β+1) ( 32 + 5β −14β 768 ) = U2 , t ∈ [0, T ], ∫ T ∂G1 (t,s) 3 2 α−1 +14α−12 ) = U3 , t ∈ [0, T ], ds ≤ TΓ(α) ( 32 + α −3α 48 ∂t 0 ∫ T ∂G2 (t,s) 3 2 β−1 +14β−12 ) = U4 , t ∈ [0, T ]. ds ≤ TΓ(β) ( 23 + β −3β 48 ∂t 0 β

4

3

2

(3.1) (3.2) (3.3) (3.4)

Theorem 3.2. Assume that one of the following conditions is satisfied: (H1 ) there exist positive constants A, B and constants bi , ci > 0, 0 < ρi , θi < 1 for i = 1, 2 such that | f (t, x, y) |≤ A + b1 | x |ρ1 +b2 | y |ρ2 , | g(t, x, y) |≤ B + c1 | x |θ1 +c2 | y |θ2 ; (H2 ) there exist constants li , ki > 0, 0 < γi , φi < 1 for i = 1, 2 such that | f (t, x, y) |≤ l1 | x |γ1 +l2 | y |γ2 , | g(t, x, y) |≤ k1 | x |φ1 +k2 | y |φ2 ; (H3 ) there exist constants di , σi > 0, δi , εi > 1 for i = 1, 2 such that | f (t, x, y) |≤ d1 | x |δ1 +d2 | y |δ2 , | g(t, x, y) |≤ σ1 | x |ε1 +σ2 | y |ε2 ; then Problem (1.1) has a solution. 4

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Before proving Theorem 3.2, we define a ball B in the Banach space X × Y as B = {(x(t), y(t))|(x(t), y(t)) ∈ X × Y, ∥(x, y)∥X×Y ≤ R, t ∈ [0, T ]}, where { } 1 1 1 1 R ≥ max 3U Aλ1 , (3U b1 λ1 ) 1−ρ1 , (3U b2 λ1 ) 1−ρ2 , 3KBλ2 , (3Kc1 λ2 ) 1−θ1 , (3Kc2 λ2 ) 1−θ2 . U = max{U1 , U3 , U5 , U6 , U7 }, where U5 =

T α−2 3 Γ(α−1) ( 2

2

+α

−α−2 ), 16

U6 =

T α−3 (α+3) 4Γ(α−2) ,

3T α−4 2Γ(α−3) , K is defined Γ([q]−q+2)+T [q]−q+1 , λ2 = Γ([q]−q+2)

U7 =

by the expression of U by replacing the corresponding α with β in each case, λ1 = Γ([p]−p+2)+T [p]−p+1 . Γ([p]−p+2)

P roof . Part 1: Let (H1 ) be valid. Step 1 : F : B → B. ∫ 0

T

∫ T ∂G21 (t, s) (T − s)α−3 (T − 2t)(T − s)α−4 t(T − t)(T − s)α−5 (t − s)α−3 ds − + + ≤ ds ∂t2 Γ(α − 2) 2Γ(α − 2) 4Γ(α − 3) 4Γ(α − 4) 0 T α−2 T α−2 T α−2 T α−2 + + + ≤ Γ(α − 1) 2Γ(α − 1) 4Γ(α − 2) 16Γ(α − 3) T α−2 3 α2 − α − 2 = ( + ) = U5 , Γ(α − 1) 2 16 ∫

T 0

∫ T ∂G31 (t, s) (T − s)α−4 (T − 2t)(T − s)α−5 (t − s)α−4 ds − + ≤ ds 3 ∂t Γ(α − 3) 2Γ(α − 3) 4Γ(α − 4) 0 T α−3 T α−3 T α−3 ≤ + + Γ(α − 2) 2Γ(α − 2) 4Γ(α − 3) T α−3 (α + 3) = = U6 , 4Γ(α − 2) ∫ 0

T

∫ T (T − s)α−5 ∂G41 (t, s) (t − s)α−5 ds − ≤ ds ∂t4 Γ(α − 4) 2Γ(α − 4) 0 T α−4 T α−4 ≤ + Γ(α − 3) 2Γ(α − 3) 3T α−4 = = U7 . 2Γ(α − 3)

Let U = max{U1 , U3 , U5 , U6 , U7 }, when k = 0, 1, 2, 3, 4, we have ∫

T

∂Gk1 (t, s) f (s, y(s),c Dp y(t))ds | ∂tk

| (F1 y)(k) (t) | =| ∫

0 T

|

≤ 0

∂Gk1 (t, s) | (A + b1 Rρ1 + b2 Rρ2 )ds ∂tk

≤ U (A + b1 Rρ1 + b2 Rρ2 ) = M.

5

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On the other hand, we can get ∫

1 | D F1 y(t)| = Γ([q] + 1 − q) c

t

(t − s)[q]−q |(F1 y)([q]+1) (s)|ds

q

∫

M ≤ Γ([q] + 1 − q) ≤

0 t

(t − s)[q]−q ds 0

M T [q]−q+1 . Γ([q] + 2 − q)

As a result ∥F1 y∥X = max max | (F1 y)(i) (t) | + max | (c Dq F1 y)(t) | 0≤i≤4 t∈[0,T ]

≤M+

t∈[0,T ]

[q]−q+1

MT = M λ1 Γ([q] + 2 − q)

= U (A + b1 Rρ1 + b2 Rρ2 )λ1 ≤

R R R + + = R. 3 3 3

Similarly ∥F2 x∥Y = max max | (F2 x)(i) (t) | + max | (c Dp F2 x)(t) | 0≤i≤4 t∈[0,T ]

t∈[0,T ]

≤ K(B + c1 R

θ1

+ c2 R )λ2 ≤ R. θ2

Hence, we conclude that ∥F (x, y)∥X×Y = max{∥F1 y∥X , ∥F2 x∥Y } ≤ R, in consequence, F : B → B. Step 2: F is continuous. This follows easily from the continuity of f, g, x(t), y(t) and G1 (t, s), G2 (t, s). Step 3: F (B) is relatively compact. Let us set M1 = max{| f (t, y(t),c Dp y(t)) |: t ∈ [0, T ], ∥y∥Y ≤ R, ∥c Dp y∥ ≤ R}, N1 = max{| g(t, x(t),c Dq x) |: t ∈ [0, T ], ∥x∥X ≤ R, ∥c Dq x∥ ≤ R}. ∫ | (F1 y) (t) | = ′

∂G1 (t, s) f (s, y(s),c Dp y(s))ds ∂t 0 ∫ T ∂G1 (t, s) ≤ M1 ds ≤ M1 U3 . ∂t 0 T

Hence, for t1 , t2 ∈ [0, T ], we have ∫ | (F1 y)(t2 ) − (F1 y)(t1 ) |≤

t2

| (F1 y)′ (s) | ds ≤ M1 U3 | t2 − t1 | .

t1

Similarly, we can get ∫ | (F2 x)(t2 ) − (F2 x)(t1 ) |≤

t2

| (F2 x)′ (s) | ds ≤ N1 U4 | t2 − t1 | .

t1

By the Arzel` a-Ascoli theorem, we can obtain that F (B) is an equicontinuous set, the operator F : B → B is completely continuous. Thus, Problem (1.1) has one solution by the Schauder fixed-point theorem. 6

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Part 2: Let (H2 ) be valid. In this part, let { } 1 1 1 1 R ≥ max (2U l1 λ1 ) 1−γ1 , (2U l2 λ1 ) 1−γ2 , (2Kk1 λ2 ) 1−φ1 , (2Kk2 λ2 ) 1−φ2 . We can also get the result by repeating arguments similar to part 1. Part 3: Let (H3 ) be valid. In this part, let { } 1 1 1 1 0 ≤ R ≤ min (2U d1 λ1 )− δ1 −1 , (2U d2 λ1 )− δ2 −1 , (2Kσ1 λ2 )− ε1 −1 , (2Kσ2 λ2 )− ε2 −1 . We can also get the result by repeating arguments similar to part 1. Here we omit it. This completes the proof. Example 3.1. Consider the system  2/5 c 17/4  D x(t) + sin t + (y(t))2/3 + (c D5/2 y(t)) = 0, 0 < t < 1,    4/7  c 9/2 1/2 1/3 c 11/4 D y(t) + t + (x(t)) + ( D x(t)) = 0, 0 < t < 1, (k) (k)  x (0) = −x (1), k = 0, 1, 2, 3, 4,     (k) y (0) = −y (k) (1), k = 0, 1, 2, 3, 4.

(3.5)

The system satisfies (H1 ) and hence Theorem 3.2 implies the existence of the solution to system (3.5). Theorem 3.3. Let f and g satisfy the following growth conditions : (H1 ) there exist four positive constants L1 , L2 , H1 , H2 such that | f (t, x1 , y1 ) − f (t, x2 , y2 ) |≤ L1 |x1 − x2 | + L2 |y1 − y2 |, | g(t, x1 , y1 ) − g(t, x2 , y2 ) |≤ H1 |x1 − x2 | + H2 |y1 − y2 |, t ∈ [0, T ], xi , yi ∈ R, i = 1, 2. (H2 ) max{L1 , L2 }U1 = Q1 < 1, max{H1 , H2 }U2 = Q2 < 1. Then Problem (1.1) has a unique solution. P roof. Let (x1 , y1 ), (x2 , y2 ) ∈ X × Y , then ∫ | (F1 y1 − F1 y2 )(t) | = ∫ ≤

∫

T

T

G1 (t, s)f (s, y1 (s),c Dp y1 (s))ds −

G1 (t, s)f (s, y2 (s),c Dp y2 (s))ds

0

0 T

| G1 (t, s) || f (s, y1 (s), D y1 (s)) − f (s, y2 (s),c Dp y2 (s)) | ds c

p

0

( ) ≤ U1 L1 |y1 (s) − y2 (s)| + L2 |c Dp y1 (s) −c Dp y2 (s)| ≤ max{L1 , L2 }U1 ∥y1 − y2 ∥Y . Analogously, ∫

T

| (F2 x1 − F2 x2 )(t) | ≤

| G2 (t, s) || g(s, x1 (s),c Dq x1 (s)) − g(s, x2 (s),c Dq x2 (s)) | ds 0

( ) ≤ U2 H1 |x1 (s) − x2 (s)| + H2 |c Dq x1 (s) −c Dq x2 (s)| ≤ max{H1 , H2 }U2 ∥x1 − x2 ∥X . 7

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Thus, || F (x1 , y1 ) − F (x2 , y2 ) ||X×Y =|| (F1 y1 − F1 y2 , F2 x1 − F2 x2 ) ||X×Y = max{|| F1 y1 − F1 y2 ||X , || F2 x1 − F2 x2 ||Y } ≤ max{Q1 || y1 − y2 ||Y , Q2 || x1 − x2 ||X } ≤ max{Q1 , Q2 } max{|| y1 − y2 ||Y , || x1 − x2 ||X } = max{Q1 , Q2 } || (x1 , y1 ) − (x2 , y2 ) ||X×Y . Hence, we conclude that Problem (1.1) has a unique solution by (H2 ) and the contraction mapping principle, this ends the proof. Example 3.2. Consider the system  c 5/2 y(t) c 17/4  D x(t) + L1 sin y(t) + L2 1+D  c D 5/2 y(t) = 0, 0 < t < 1,    c 11/4  c 9/2 x(t) D y(t) + H1 arctan x(t) + H2 1+D c D 11/4 x(t) = 0, 0 < t < 1,   x(k) (0) = −x(k) (1), k = 0, 1, 2, 3, 4,     y (k) (0) = −y (k) (1), k = 0, 1, 2, 3, 4. c

(3.6)

5/2

y(t) c q Where T = 1, f (t, y(t),c Dp y(t)) = L1 sin y(t) + L2 1+D c D 5/2 y(t) , g(t, x(t), D x(t)) = H1 arctan x(t) + c

11/4

x(t) H2 1+D c D 11/4 x(t) , α =

17 4 ,

β = 92 , p = 5/2, q = 11/4 and L1 , L2 , H1 , H2 > 0.

Noting that | (sin y)′ |=| cos y |≤ 1, | (arctan x)′ |=

v ′ 1 1 ≤ 1, ( ) = ≤ 1, 2 1+x 1+v (1 + v)2

we have | f (t, y1 (t),c Dp y1 (t)) − f (t, y2 (t),c Dp y2 (t)) | c D5/2 y (t) c 5/2 D y2 (t) 1 − ≤ L1 | sin y1 (t) − sin y2 (t) | +L2 1 +c D5/2 y1 (t) 1 +c D5/2 y2 (t) ≤ L1 | y1 (t) − y2 (t) | +L2 |c D5/2 y1 (t) −c D5/2 y2 (t) | ≤ max{L1 , L2 } || y1 − y2 ||Y ,

| g(t, x1 (t),c Dq x1 (t)) − g(t, x2 (t),c Dq x2 (t)) | c D11/4 x (t) c 11/4 D x2 (t) 1 ≤ H1 | arctan x1 (t) − arctan x2 (t) | +H2 − c 11/4 c 11/4 1+ D x1 (t) 1 + D x2 (t) ≤ H1 | x1 (t) − x2 (t) | +H2 |c D11/4 x1 (t) −c D11/4 x2 (t) | ≤ max{H1 , H2 } || x1 − x2 ||X , U1 =

Tα 3 5α4 − 14α3 + 55α2 + 146α ( + ) ≈ 0.1229, Γ(α + 1) 2 768

U2 =

Tβ 3 5β 4 − 14β 3 + 55β 2 + 146β ( + ) ≈ 0.0920. Γ(β + 1) 2 768

as long as we let max{L1 , L2 }
0, and for each ε > 0,     1 1 1 δ(r, ε) = inf 1 − d x ⊕ y, a : d(x, a) ≤ r, d(y, a) ≤ r, d(x, y) ≥ rε > 0. r 2 2 From now onward we assume that X is a hyperbolic metric space and if (X, d) is uniformly convex, then for every s ≥ 0, ε > 0, there exists η(s, ε) > 0 depending on s and ε such that δ(r, ε) > η(s, ε) > 0 for any r > s.

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Remark 2.3. If (X, d) is uniformly convex, then we have the following: (1) δ(r, 0) = 0 and δ(r, ε) is an increasing function of ε for every fixed r. (2) For r1 ≤ r2 , the following holds:    r2 r1 1 − δ r2 , ε 1− ≤ δ(r1 , ε). r1 r2 (3) If (X, d) is uniformly convex, then (X, d) is strictly convex, that is, whenever   1 1 d(x, a) = d(y, a) = d x ⊕ y, a , 2 2 for any x, y, a ∈ X, then we must have x = y. Recall that a hyperbolic metric space X is said to have property (R) [10] if any nonincreasing sequence of nonempty, convex, bounded, and closed sets has a nonempty intersection. The following theorem was proved by Khamsi and Khan [10]. Theorem 2.4. ([10]) Assume that (X, d) is complete and uniformly convex. Let C be nonempty, convex, and closed. Then for any x ∈ X, there exists a unique best approximant of x in C, that is, a unique x0 ∈ C such that d(x, x0) = d(x, C). Note that any complete and uniformly convex metric space has the property (R) (see [10]). We need the following results for our main results. Lemma 2.5. ([10] Lemma 2.2) Let (X, d) be uniformly convex. Assume r ≥ 0 such that  1 lim sup d(xn, a) ≤ r, lim sup d(yn , a) ≤ r and lim d a, xn ⊕ n→∞ 2 n→∞ n→∞

that there exists 1 yn 2



= r.

Then limn→∞ d(xn , yn ) = 0. The following metric version of the parallelogram identity, also known as the inequality of Bruhat and Tits, has been established in [10]. Theorem 2.6. ([10]) Let (X, d) be uniformly convex. Fix a ∈ X. For each r > 0 and for each ε > 0, denote    1 2 1 1 1 Ψ(r, ε) = inf d (a, x) + d2 (a, y) − d2 a, x ⊕ y , 2 2 2 2 where the infimum is taken over all x, y ∈ X such that d(a, x) ≤ r, d(a, y) ≤ r and d(x, y) ≥ rε. Then Ψ(r, ε) > 0 for any r > 0 and each ε > 0. Moreover, for a fixed r > 0, we have

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D. R. Sahu, S. Dashputre and S. M. Kang (i) Ψ(r, 0) = 0; (ii) Ψ(r, ε) is non-decreasing function of ε;

(iii) if limn→∞ Ψ(r, tn ) = 0, then limn→∞ tn = 0. The notion of p-uniform convexity was studied extensively by Xu [28], its nonlinear version for p = 2 has been introduced by Khamsi and Khan [10] using the above function Ψ as follows. Definition 2.7. ([10]) We say that (X, d) is 2-uniformly convex if   Ψ(r, ε) CX = inf : r > 0, ε > 0 > 0. r 2 ε2 From the definition of CX , we obtain the following inequality:   1 1 1 1 d2 a, x ⊕ y + CX d2 (x, y) ≤ d2 (a, x) + d2 (a, y) 2 2 2 2 for any a ∈ X and x, y ∈ X. Example 2.8. Let (X, d) be a metric space. A geodesic from x to y in X is a mapping c from a closed interval [0, l] ⊂ R to X such that c(0) = x, c(l) = y, and d(c(t), c(t0)) = |t−t0 | for all t, t0 ∈ [0, l]. In particular, c is an isometry and d(x, y) = l. The image α of c is called a geodesic (or metric) segment joining x and y. The space (X, d) is said to be a geodesic space if every two points of X are joined by a geodesic, and X is said to be uniquely geodesic if there is exactly one geodesic joining x and y for each x, y ∈ X, which will be denoted by [x, y], and called the segment joining x to y. A geodesic triangle ∆(x1 , x2 , x3 ) in a geodesic metric space (X, d) consists of three points x1 , x2 , x3 in X (the vertices of ∆) and a geodesic segment between each pair of vertices (the edges of ∆). A comparison triangle for the geodesic triangle ∆(x1 , x2, x3 ) in (X, d) is a triangle ¯ 1 , x2 , x3 ) := ∆(¯ ∆(x x1 , x ¯2, x ¯3 , ) in R2 such that dR2 (¯ xi , x ¯j ) = d(xi , xj ) for i, j ∈ {1, 2, 3} such triangle exists (see [4]). A geodesic space is said to be a CAT(0) space if all geodesic triangles of appropriate size satisfy the following comparison axiom. ¯ ⊂ R2 be a comparison triangle for ∆. Let ∆ be a geodesic triangle in X and let ∆ Then ∆ is said to satisfy the CAT(0) inequality if d(x, y) ≤ d(¯ x, y¯). ¯ for all x, y ∈ ∆ and all comparison points x ¯, y¯ ∈ ∆. Complete CAT(0) spaces are often called Hadamard spaces (see [16]). If x, y1 , y2 are points of a CAT(0) space and if y0 is the midpoint of the segment [y1 , y2 ], which will be 2 denoted by y1 ⊕y 2 , then the CAT(0) inequality implies   1 1 1 1 1 2 d x, y1 ⊕ y2 ≤ d2 (x, y1 ) + d2 (x, y2 ) − d2 (y1 , y2 ). (CN ) 2 2 2 2 4

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This inequality is the (CN) inequality of Bruhat and Tits [4]. As for the Hilbert space, the (CN) inequality implies the CAT(0) spaces are uniformly convex with r ε2 δ(r, ε) = 1 − 1 − . 4 The (CN) inequality also implies that Ψ(r, ε) =

r 2 ε2 . 4

Thus, a CAT(0) space is 2-uniformly convex with CX = 41 . We need the following more general inequality for convergence of Mann iterations. Theorem 2.9. ([7]) Let (X, d) be 2-uniformly convex. Then, for any α ∈ (0, 1), there exists CX > 0 such that d2 (a, αx ⊕ (1 − α)y) + CX min{α2 , (1 − α)2 }d2 (x, y) ≤ αd2 (a, x) + (1 − α)d2 (a, y) for any a, x, y ∈ X. Recall that Φ : X → R+ is called a type if there exists {xn } in X such that Φ(x) = lim sup d(x, xn). n→∞

Theorem 2.10. ([10, Theorem 2.4]) Assume that (X, d) is a complete and uniformly convex. Let C be a nonempty closed bounded and convex subset of X. Let Φ be a type defined on C. Then any minimizing sequence of Φ is convergent. Its limit is independent of the minimizing sequence. In fact, if X is 2-uniformly convex, and Φ is a type defined on a nonempty closed convex bounded subset C of X, then there exists a unique x0 ∈ C such that Φ2 (x0 ) + 2CX d2 (x0 , x) ≤ Φ2 (x)

(2.1)

for any x ∈ C. In this inequality, one may find an analogy with Opial property used in the study of the fixed point property in Banach and metric spaces.

2.2

Pointwise Lipschitzian type mappings and fixed points

First, we extend some wider classes of nonlinear mappings studied by Sahu et al. [24] in a metric space setting. Definition 2.11. ([24]) Let C be a nonempty subset of a metric space (X, d). A mapping T : C → C is said to be (1) pointwise nearly Lipschitzian with sequence {(αn (·), an)} if, there exists a sequence {an } in [0, ∞) with an → 0 and for each n ∈ N, there exists a function αn (.) : C → (0, ∞) such that d(T nx, T n y) ≤ αn (x)(d(x, y) + an )

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for all x,y ∈ C; (2) pointwise nearly uniformly α(·)-Lipschitzian with sequence {an } if, there exists a sequence {an } in [0, ∞) with an → 0 and there exists a function α(·) : C → (0, ∞) such that d(T n x, T n y) ≤ α(x)(d(x, y) + an ) for all x,y ∈ C; (3) asymptotic pointwise nearly Lipschitzian with sequence {(αn (·), an)} if, there exists a sequence {an } in [0, ∞) with an → 0 and for each n ∈ N, there exists a function αn (.) : C → (0, ∞) and with αn → α : C → (0, ∞) pointwise such that d(T nx, T n y) ≤ αn (x)(d(x, y) + an ) for all x,y ∈ C. We say that, an asymptotic pointwise nearly Lipschitzian mapping is (1) pointwise nearly asymptotic nonexpansive if αn (x) ≥ 1 for all n ∈ N and αn (x) → 1 pointwise, (2) pointwise asymptotically nonexpansive [18] an = 0 and αn (x) ≥ 1 for all n ∈ N and αn (x) → 1 pointwise. (3) a asymptotic pointwise nearly contraction if αn → α pointwise and α(x) ≤ k < 1 for all x ∈ C. A point x ∈ C is called a fixed point of T if T (x) = x. The fixed point set of T is denoted by F ix(T ).

3

Existence theorem

First, we prove the existence of fixed point for a pointwise nearly asymptotically nonexpansive mapping in a 2-uniformly convex metric space. Theorem 3.1. Let C be nonempty closed convex and bounded subset of a complete hyperbolic 2-uniformly convex metric space (X, d). Let T : C → C be a continuous pointwise nearly asymptotically nonexpansive mapping. Then T has a fixed point in C. Moreover, the set of fixed points is closed and convex. Proof. Fix x ∈ C. Define the function Φ(y) = lim supn→∞ d(T n (x), y) on C. By (2.1), there exists a unique ω ∈ C such that Φ2 (ω) + 2CX d2 (ω, y) ≤ Φ2 (y) for all y ∈ C. In particular, we have Φ2 (ω) + 2CX d2 (ω, T n(ω)) ≤ Φ2 (T n (ω))

(3.1)

for all n ≥ 1. Observe that Φ(T n (ω)) = lim sup d(T m (x), T n(ω)) m→∞

≤ lim sup d(T n (T m−n (x)), T n(ω)) m→∞   ≤ lim sup αn (ω)(d(T m−n(x), ω) + an ) m→∞

≤ αn (ω)(Φ(ω) + an )

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for all n ≥ 1. Hence, from (3.1) and (3.2), we have Φ2 (ω) + 2CX d2 (ω, T n(ω)) ≤ Φ2 (T n (ω)) ≤ αn (ω)(Φ(ω) + an )


2

= α2n (ω)[Φ2(ω) + a2n + 2Φ(ω)an ] for all n ≥ 1. By the definition of T, αn (ω) → 1 pointwise and an → 0 as n → ∞. Thus, limn→∞ d(ω, T n(ω)) = 0, i.e., T n (ω) → ω as n → ∞. By the continuity of T , we have   T (ω) = T lim T n (ω) = lim T n+1 (ω) = ω. n→∞

n→∞

Closedness of F ix(T ): Let {xn } be a sequence in F ix(T ) such that limn→∞ xn = x for some x ∈ C. Now it remains to show that x ∈ F ix(T ). Note that d(T n (xn ), T n (x)) ≤ αn (x)(d(xn, x) + an ), which implies that lim d(T n (x), xn) = 0.

n→∞

Since d(x, T n(x)) ≤ d(x, xn) + d(xn , T n(x), we have, limn→∞ d(x, T n(x)) = 0. By continuity of T , we have T x = x. Convexity of F ix(T ): Let x, y ∈ F ix(T ). We only need to prove that z = F ix(T ). Without loss of generality, we assume that x 6= y. Note that

x⊕y 2

∈

d(x, T n(z)) = d(T n(x), T n(z)) ≤ αn (x)(d(x, z) + an )     x⊕y ≤ αn (x) d x, + an 2   1 = αn (x) d(x, y) + an 2 for all n ≥ 1. Similarly, we have d(y, T n(z)) ≤ αn (y)



1 d(x, y) + an 2



for all n ≥ 1. By triangular inequality, we have d(x, y) ≤ d(x, T n(z)) + d(T n(z), y)     1 1 ≤ αn (x) d(x, y) + an + αn (y) d(x, y) + an 2 2   1 = (αn (x) + αn (y)) d(x, y) + an , 2 it follows that lim d(x, T n(z)) = lim d(T n (z), y) = d(x, y).

n→∞

n→∞

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D. R. Sahu, S. Dashputre and S. M. Kang Note   1 1 z ⊕ T n (z) ≤ d(x, z) + d(x, T n(z)) d x, 2 2 2   1 1 1 ≤ d(x, z) + αn (x) d(x, z) + an , 2 2 2   1 + αn (x) αn (x)an = d(x, z) + . 2 2 Similarly, we have   z ⊕ T n (z) 1 1 d y, ≤ d(y, z) + d(y, T n(z)) 2 2 2   1 1 1 ≤ d(y, z) + αn (y) d(y, z) + an 2 2 2   1 + αn (y) αn (y)an = d(y, z) + . 2 2

Thus,     z ⊕ T n (z) z ⊕ T n (z) d(x, y) ≤ d x, +d ,y 2 2 1 + αn (x) d(x, y) 1 + αn (y) d(x, y) αn (x) + αn (y) ≤ + + an . 2 2 2 2 2 Taking limit as n → ∞ both the sides, Hence, we have     z ⊕ T n (z) z ⊕ T n (z) d(x, y) lim d x, = lim d y, = . n→∞ n→∞ 2 2 2 Using Lemma 2.5, we conclude that limn→∞ d(z, T n(z)) = 0. Therefore, we must have T (z) = z, i.e., x⊕y 2 ∈ F ix(T ). This completes the proof. Remark 3.2. Theorem 3.1 is a natural generalization of Proposition 3.4 and Theorem 3.8 of Sahu et al. [24] in the framework of a hyperbolic 2-uniformly convex metric space. Theorem 3.1 extends the results of Dehaish et al. [7, Theorem 3.1], Goebel and Kirk [8, Theorem 1], and Kirk and Xu [18, Theorem 3.4] to the class of pointwise nearly Lipschitzian mappings which essentially wider than the class of mappings appearing in [7], [8] and [18].

4

Convergence of Mann iteration process

Lemma 4.1. Let C be nonempty, closed, convex, and bounded subset of a complete hyperbolic 2-uniformly convex metric space (X, d). Let T : C → C be a pointwise nearly asymptotically nonexpansive with sequence {(αn (·), an)} such that T is uniformly continP∞ P∞ uous. Assume that n=1 an < ∞ and n=1 (αn (p) − 1) < ∞ for all p ∈ F ix(T ). Let {tn } ⊂ [0, 1] be bounded away from 0 and 1, i.e., there exist two real numbers a, b such

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that 0 < a ≤ tn ≤ b < 1. The modified Mann iteration process is defined by (1.4). Then we have the following: (a) limn→∞ d(xn , p) exists for all p ∈ F ix(T ). (b) limn→∞ d(xn , T n(xn )) = 0 and limn→∞ d(xn, T m (xn )) = 0 for all m ≥ 1, provided that L = supn∈N supx∈C αn (x) < ∞. Proof. (a) Let δ(C) = supx,y∈C d(x, y) be the diameter of C. Let ω ∈ F ix(T ). Set δn := d(xn, ω), βn = αn (ω) and γn = an L. From (1.4), we have δn+1 = d(xn+1 , ω) = d(tn T n (xn ) ⊕ (1 − tn )xn , ω) ≤ tn d(T n(xn ), ω) + (1 − tn )d(xn , ω) ≤ tn d(T n(xn ), T n(ω)) + (1 − tn )d(xn , ω) ≤ tn αn (ω)(d(xn, ω) + an ) + (1 − tn )d(xn, ω) ≤ tn αn (ω)d(xn, ω) + an tn αn (ω) + (1 − tn )d(xn , ω) ≤ αn (ω)d(xn, ω) + αn (ω)an ≤ βn δn + γn P∞ P for all n ≥ 1. Noticing that ∞ n=1 γn < ∞.for all n ≥ 1. Therefore, n=1 (βn −1) < ∞ and from [1, Lemma 6.1.5], we conclude that limn→∞ d(xn , ω) exists. (b) First, we prove that limn→∞ d(xn, T n (xn )) = 0. By Theorem 3.1, T has a fixed point ω ∈ C. Lemma 4.1 implies that limn→∞ d(xn , ω) exists. Set r = limn→∞ d(xn , ω). Without loss of generality, we may assume r > 0. Note lim sup d(T n(xn ), ω) = lim sup d(T n (xn ), T n(ω)) n→∞

n→∞

≤ lim sup(αn (ω)(d(xn, ω) + an )) = r. n→∞

On the other hand, from (1.4), we have d(xn+1 , ω) ≤ tn d(T n(xn ), ω) + (1 − tn )d(xn , ω) for all n ≥ 1. Let U be a non-trivial ultrafilter over N. Then limU tn = t ∈ [a, b]. Hence r = lim d(xn+1 , ω) ≤ t lim d(T n(xn ), ω) + (1 − t)r. U

U

Since t 6= 0, we get limU d(T n(xn ), ω) ≥ r. Hence r ≤ lim inf d(T n (xn ), ω) ≤ lim sup d(T n (xn ), ω) ≤ r. n→∞

n→∞

So limn→∞ d(T n(xn , ω) = r. Since X is 2-uniformly convex, Theorem 2.9 implies CX min{t2n , (1 − tn )2 }d2 (xn , T n(xn )) ≤ tn d2 (xn , ω) + (1 − tn )d2 (T n (xn ), ω) −d2 (xn+1 , ω), where CX > 0 depends only on X. Since CX min{t2n , (1 − tn )2 } ≥ min{a2 , (1 − b)2 } > 0,

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and

  2 2 n 2 lim tn d (xn , ω) + (1 − tn )d (T (xn ), ω) − d (xn+1 , ω) = 0,

n→∞

we get lim d(xn , T n (xn )) = 0,

n→∞

which finish the prove of our claim. Next, we prove that limn→∞ d(xn , T m(xn ) = 0, for all m ≥ 1. The uniform continuity of T implies that lim d(T xn , T n+1 (xn )) = 0. n→∞

From (1.4), we have d(xn+1 , xn ) ≤ d(xn , T n (xn )) → 0 as n → ∞. Note that d(xn , T (xn)) ≤ d(xn , xn+1 ) + d(xn+1 , T n+1 (xn+1 )) + d(T n+1 (xn+1 ), T n+1 (xn )) +d(T n+1 (xn ), T (xn)) ≤ d(xn , xn+1 ) + d(xn+1 , T n+1 (xn+1 )) + αn+1 (xn )(d(xn+1 , xn ) +an+1 ) + d(T n+1 (xn ), T (xn)) ≤ d(xn , xn+1 ) + d(xn+1 , T n+1 (xn+1 )) + L(d(xn+1 , xn) + an+1 ) +d(T n+1 (xn ), T (xn)) for all n ≥ 1. Hence, we get limn→∞ d(xn T (xn )) = 0. Again, from the uniform continuity of T, we have lim d(T (xn ), T 2(xn )) = 0, n→∞

it follows that d(xn , T 2(xn ) ≤ d(xn , T (xn)) + d(T (xn), T 2 (xn )) → 0 as n → ∞. Inductively, we have lim d(xn , T m(xn ) = 0

n→∞

for all m ≥ 1. This completes the proof. We now establish main result of this section. Theorem 4.2. Let C be nonempty, closed, convex, and bounded subset of a complete hyperbolic 2-uniformly convex metric space (X, d). Let T : C → C be a pointwise nearly asymptotically nonexpansive with sequence {(αn (·), an)} such that T is uniformly continuP P∞ ous and supn∈N supx∈C αn (x) < ∞. Assume that ∞ n=1 an < ∞ and n=1 (αn (p)−1) < ∞ for all p ∈ F ix(T ). Let {tn } ⊂ [0, 1] be bounded away from 0 and 1, i.e., there exist two real numbers a, b such that 0 < a ≤ tn ≤ b < 1. The modified Mann iteration process is defined by (1.4). Consider the type Φ(x) = lim supn→∞ d(xn , x) on C. If ω is the minimum point of Φ, that is, Φ(ω) = inf{Φ(x) : x ∈ C}, then T (ω) = ω.

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Iteration Process for Pointwise Lipschitzian Type Mappings

13

Proof. Suppose that ω is the minimum point of Φ. For any m, n ≥ 1, we have   1 ω ⊕ T m (ω) 1 2 + CX d2 (ω, T m(ω)) ≤ d2 (xn , ω) + d2 (xn , T m (ω)). d xn , 2 2 2 Letting limit as n → ∞, we get   m 1 1 2 ω ⊕ T (ω) Φ + CX d2 (ω, T m(ω)) ≤ Φ2 (ω) + Φ2 (T m (ω)) 2 2 2

(4.1)

for any m ≥ 1. Using Lemma 4.1, we get Φ(T m(ω)) = lim sup d(xn , T m(ω)) n→∞   m m m ≤ lim sup d(xn , T (xn ) + d(T (xn ), T (ω)) , n→∞

≤ lim sup d(T m (xn ), T m(ω)) n→∞

≤ lim sup(αm (ω)(d(xn, ω) + am )) n→∞

= αm (ω)(Φ(ω) + am ) for any m ≥ 1. Since ω is the minimum point of Φ,we have   ω ⊕ T m (ω) Φ(ω) ≤ Φ 2 for any m ≥ 1. From (4.1), we have  ω ⊕ T m (ω) Φ (ω) + CX d (ω, T (ω)) ≤ Φ + CX d2 (ω, T m(ω)) 2 1 1 ≤ Φ2 (ω) + Φ2 (T m (ω)) 2 2 1 2 1 ≤ Φ (ω) + [αm (ω)(Φ(ω) + am )]2 2 2 2

2

m

2



for m ≥ 1. Taking limit superior as m → ∞, we get Φ2 (ω) + CX lim sup d2 (ω, T m(ω)) ≤ Φ2 (ω). m→∞

This implies that limm→∞ d(ω, T m(ω)) = 0. Therefore, T (ω) = ω, i.e., ω ∈ F ix(T ). This completes the proof. Remark 4.3. Theorem 4.2 extends the result of Dehaish et al. [7, Theorem 4.1] to pointwise nearly Lipschitzian mapping which essentially wider than the mapping appearing in [7].

Acknowledgment The author Samir Dashputre is grateful to Department of Mathematics and DST-CIMS, BHU, Varanasi for supporting to necessary facility for preparing successfully the manuscript.

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D. R. Sahu, S. Dashputre and S. M. Kang

References [1] R. P. Agarwal, D. O’Regan, and D. R. Sahu, Fixed point theory for Lipschitzian-type mappings with applications, Topological Fixed Point Theory and Its Applications, Vol. 6, Springer, New York, 2009. [2] Ya. I. Alber, C. E. Chidume and H. Zegeye, Approximating fixed points of total asymptotically nonexpansive type mappings, Fixed Point Theory Appl. 2006 (2006), Atricle id 10673, 20 pages. [3] L. P. Belluce and W.A. Kirk, Fixed point theorems for certain classes of nonexpansive mappings, Proc. Amer. Math. Soc. 20 (1969), 141–146. ´ [4] F. Bruhat and J. Tits, Groupes r´eductifs sur un corps local, Inst. Hautes Etudes Sci Pub. Math. 41 (1972), 5-251. [5] R. E. Bruck, T. Kuczumow and S. Reich, Convergence iterates of asymptotically nonexpansive mappings in Banach spaces with uniform Opial property, Colloq. Math. 65 (1993), 169–179. [6] H. Busemann, Spaces with non-positive curvature, Acta. Math. 80 (1948), 259–310. [7] B. A. Ibn Dehaish, M. A. Khamsi and A. R. Khan, Mann iteration process for asymptotic poitwise nonexpansive mappings in metric spaces, J. Math. Anal. Appl. 397 (2013), 861–868. [8] K. Goebel and W. A. Kirk, A fixed point theorem for asymptotically nonexpansive mappings, Proc. Amer. Math. Soc. 35 (1972), 171–174. [9] K. Goebel and S. Reich, Uniform Convexity, Hyperbolic Geometry, and Nonexpansive Mappings, Monographs and Textbooks in Pure and Applied Mathematics, Vol. 83. Marcel Dekker, Inc., New York, 1984. [10] M. A. Khamsi and A. R. Khan, Inequalities in metric spaces with applications, Nonlinear Anal. 74 (2011), 4306–4045. [11] A. R. Khan, M.A. Khamsi and H, Fukhar-ud-Din, Strong convergence of a general iteration scheme in CAT(0) spaces, Nonlinear Anal. 74 (2011), 783–791. [12] W. A. Kirk, Mappings of generalized contractive type, J. Math. Anal. Appl. 32 (1970), 567–572. [13] W. A. Kirk, Fixed point theorems for non-Lipschitzian mappings of asymptotically nonexpansive type, Israel J. Math. 17 (1974), 339–346. [14] W. A. Kirk, Fixed Point Theory for Nonexpansive Mappings I and II, Lecture Notes in Mathematics, Vol. 886, Springer, Berline, 1981, pp. 485-505. [15] W. A. Kirk, Fixed points of asymptotic contractions, J. Math. Anal. Appl. 227 (2003), 645–650.

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15

[16] W. A. Kirk, A fixed point theorem in CAT(0) spaces and R-tree, Fixed Point Theory Appl. 4 (2004), 309–316. [17] W. A. Kirk, Asymptotic pointwise contractions, Plenary Lecture, in: The 8th International Conference on Fixed Point Theory and Its Applications, Chiang Mai University, Thailand, July 16-22, 2007. [18] W. A. Kirk and H. K. Xu, Asymptotic pointwise contractions, Nonlinear Anal. 69 (2008), 4706–4712. [19] L. Leu¸stean, A quadratic rate of asymptotic regularity for CAT(0) spaces, J. Math. Anal. Appl. 325 (2007), 386–399. [20] K. Menger, Untersuchunge u ¨ber allgemeine Metrik, Math. Ann 100 (1928), 75–163. [21] S. Reich and I. Shafrir, Nonexpansive iterations in hyperbolic spaces, Nonlinear Anal. 15 (1990), 537–558. [22] D. R. Sahu, Fixed points of demicontiunous neraly Lipschitzian mappings in Banach spaces. Comment Math. Univ. Carolin. 46 (2005), 653–666. [23] D. R. Sahu and I. Beg, Weak and strong convergence of fixed points of nearly asymptotically nonexpansive mappings, Int. J. Mod. Math. 3 (2008), 135–151. [24] D. R. Sahu, A. Petru¸sel and Y. C. Yao, On fixed points of pointwise lipschitzian type mappings, Fixed Poit Theory 14 (2013), 171–184. [25] D. R. Sahu, N. C. Wong and J. C. Yao, A generalized hybrid steepest-descent method for variational inequalities in Banach spaces. Fixed Point Theory Appl. 2011, Article ID 754702, 28 pages. [26] D. R. Sahu, N. C. Wong and J. C. Yao, A unified hybrid iterative method for solving variational inequalities involving generalized pseudo-contractive mappings. SIAM J. Control Optim. 50 (2012), 2335–2354. [27] J. Schu, Iterative construction of fixed points of asymptotically nonexpansive mappings. J. Math. Anal. Appl. 158 (1991), 407–413. [28] H. K. Xu, Inequalities in Banach spaces with applications, Nonlinear Anal. 16 (1991), 1127–1138.

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Regularity of the American Option Value Function in Jump-Diffusion Model

Sultan Hussain1

∗

and Nasir Rehman2

1

Department of Mathematics, COMSATS Institute of Information Technology, Abbottabad Pakistan. 2 Department of Mathematics and Statistics, Allama Iqbal Open University H-8, Islamabad Pakistan.

Abstract This work is devoted to the regularity properties of the American options value function, when there are brusque variations in prices. We assume that there are finite number of jumps in each finite time interval and the asset price jumps in the proportions which are independent and identically distributed. These properties can be used to investigate the optimal hedging strategies, optimal exercise boundaries etc. for the options in jump-diffusion process. Keywords: American Option, Jump-Diffusion Model, Poisson Process, Lipschitz Continuity, Weak Derivatives.

1

Introduction

The pricing of options and the corporate liabilities have been developed significantly after the classical paper by Black and Scholes (1973). Although several techniques for the calculation of the value of the European option have been proposed in closed-form, the American options are still open for further research and consideration, causing an extensive literature on numerical methods. Recently, in Israel and Rincon (2008), the American options problem using inequality variational systems, and numerical methods based on finite elements and finite difference ∗

Corresponding author (Sultan Hussain) Tel.: +92 (0)333 9485988. E-mail address: [email protected]

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techniques is solved properly. Indeed, as it has been proposed also in Jaillet et. al. (1990), the problem to find the value of a put American option can be equivalent to getting the solution of a system of variational inequalities provided that this formulation respects some necessary hypotheses, see also Isreal and Rincon (2008). Jaillet, Lamberon and Lapeyre (1990) rely on the link between the optimal stopping and variational inequality in order to exploit the theory of American options. Pham (1997) investigated the regularity of the value function of the put American option in jump-diffusion process using the properties of the optimal exercise boundary. For more detailed discussion on the value function of the American options we refer the readers to the papers by Chiarella and Kang (2011), ElKaroui, et. al. (1998), ellot and Kopp (1990), Hussain and Shashiashvili (2010), Hussain and Rehman (2012), and books Glowinski, et. al. (1981), Karatzas and Shereve (1998), Lamberton and Lapeyre (1997), Shreve (2004) etc. We assume the interest rate and volatility are Lipschitz functions of time, payoff is arbitrary bounded from below convex function, and use purely probabilistic approach to obtained rigorous estimates for the first and second order derivatives of the value function of the put American options in order to use these results in our next work to construct uniform approximations for the discrete time hedging strategies as well as for the investigation of the optimal exercise boundary of the put American options. In Section 2, we set the basic notation and we formulate our model. Thus, we consider a financial market with two assets, i.e. the value of a money market account and the share of a stock whose price jumps proportionally at some times τj following the Poisson process, similarly as in Pham (1997). Note that in order to deal with this problem, following also the existing literature, we recall that the American options value function can be considered as the value of a function of an equivalent optimal stopping time problem. Thus, some preliminary results are presented here. Finally, in Section 3, the regularity properties of a put American option are derived solving a system of variational inequalities.

2

Notation - Preliminary Results

Let (Ω, F, P) be a probability space on which we define a standard Wiener process W = (Wt )0≤t≤T , a Poisson process N = (Nt )0≤t≤T with intensity λ and a sequence (Uj )j≥1 of independent, identically distributed random variables taking values in (−1, ∞). Assume that the time horizon T < ∞ is finite and the σ-algebras generated respectively by (Wt )0≤t≤T , (Nt )0≤t≤T , and (Uj )j≥1 are independent. Denote by (Ft )0≤t≤T the Pcompletion of the natural filtration of (Wt ), (Nt ) and (Uj )Ij≤Nt , j ≥ 1, 0 ≤ t ≤ T . On a filtered probability space (Ω, F, Ft , P)0≤t≤T consider a financial market with two assets mt , 0 ≤ t ≤ T , the price of a unit of a money market account at time t, and St , 0 ≤ t ≤ T, the value at time t of the share of a stock whose price jumps in the proportions U1 , U2 , ..., at some times τ1 , τ2 , . . ., see also Pham (1997). We assume that the τj ’s correspond to the jump times of a Poisson process. The evolution of the assets mt and St obeys the following ordinary and stochastic differential equations respectively, dmt = r(t)mt dt, m0 = 1, 0 ≤ t ≤ T,

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

  Nt X dSt = St− b(t)dt + σ(t)dWt + d  Uj  . j=1

We assume that (b(t), Ft )0≤t≤T is a certain, progressively measurable process; the deterministic time-varying interest rate r(t) and the volatility σ(t) are continuously differentiable functions of time, and the following requirements are satisfied: 0 ≤ r(t) ≤ r¯, 0 < σ ≤ σ(t) ≤ σ ¯ , |b(t)| ≤ r¯, |r(t) − r(s)| + |σ(t) − σ(s)| ≤ K|t − s|,

(1)

where s, t ∈ [0, T ] and r¯, σ, σ ¯ and K are some positive constants. From the above stochastic differential equation, the dynamics of St can be described by:   Z t    Z t Nt Y σ 2 (u)   b(u) − St = S0 (1 + Uj ) exp du + σ(u)dWu . 2 0 0 j=1 It is known, see for instance Lamberton and Lapeyre (1997), that the discounted stock Rt price S˜t = e− 0 r(u)du St is a martingale if and only if Z t Z t b(u)du = r(u)du − λtE(U1 ). (2) 0

0

In this brief paper, we investigate the regularity properties of the American option value function with a nonnegative, non-increasing convex payoff function g(x), x ≥ 0. We assume that g(0) = g(0+). Of course, a typical example of this family of functions is the put American option with payoffs g(x) = (L − x)+ where L is the exercise price. In the next paragraphs of this section, we present some necessary and preliminary results for the better understanding, and evaluation of our main outputs. First, it is necessary to recall that the American option value function v(t, x), x ≥ 0, 0 ≤ t ≤ T, can be considered as the value function of a relevant optimal stopping problem (see, for instance Karatzas and Shereve (1998), Section 2.5). In particular   Z τ   v(t, x) = sup E exp − r(v)dv g(Sτ (t, x)) , x ≥ 0, 0 ≤ t ≤ T, (3) τ ∈Tt,T

t

where Tt,T denotes the set of all stopping times τ such that t ≤ τ ≤ T , and the stochastic process Su (t, x), t ≤ u ≤ T satisfies the same stochastic differential equation as above, i.e.    Nu X dSu (t, x) = Su− (t, x) b(u)du + σ(u)dWu + d  Uj  , t ≤ u ≤ T, (4) j=1+Nt

with the initial condition St (t, x) = x, x ≥ 0. The unique solution (Su (t, x), Fu )t≤u≤T of (4) is given by the exponential   Z u    Z u Nu Y σ 2 (u)   Su (t, x) = x (1 + Uj ) exp b(u) − du + σ(u)dWu . 2 t t j=1+Nt

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Condition (2) leads to " Su (t, x) = exp ln x +

Ru t

r(u) − λE(U1 ) −

σ 2 (u) 2



du +

Ru t

σ(u)dWu +

#

N Pu

ln(1 + Uj ) .

j=Nt +1

Now, we can introduce the new stochastic process (Xu (t, x), Fu )t≤u≤T Z

u

Xu (t, y) = y + t



σ 2 (v) r(u) − λE(U1 ) − 2



Z

u

dv +

σ(v)dWv + t

Nu X

ln(1 + Uj ),

j=Nt +1

t ≤ u ≤ T, −∞ < y < ∞, Uj ∈ (−1, ∞), j = 1, 2, . . .. Remark 2.1. Profoundly, Su (t, x) = exp [Xu (t, ln x)] , t ≤ u ≤ T x > 0,

(5)

and for an arbitrary stopping time τ , t ≤ τ ≤ T , we obtain g(Sτ (t, x)) = ψ(Xτ (t, ln x)), where ψ(y) = g(ey ), −∞ < y < ∞ is the new payoff function. Now, it is clear that the corresponding optimal stopping time problem is derived straightforwardly by just substituting (5) into (3), having now   Z τ   u(t, y) = sup E exp − r(v)dv ψ(Xτ (t, y)) , (6) τ ∈Tt,T

t

with 0 ≤ t ≤ T and −∞ < y < ∞, then we obtain v(t, x) = u(t, ln x), x > 0, 0 ≤ t ≤ T. In what follows, the next known result, from Hussain and Shashiashvili (2010), is needed Lemma 2.2. Let g(x), x ≥ 0 be a nonnegative, non-increasing convex function. Then the new payoff function defined by ψ(y) = g(ey ), −∞ < y < ∞ is Lipschitz continuous, that is, |ψ(y2 ) − ψ(y1 )| ≤ g(0)|y2 − y1 |, y1 , y2 ∈ R. Thus, using the scaling property of the Brownian motion we can express the value function u(t, y) of the optimal stopping time problem (6), see Jaillet, et. al. (1990), as follows "  R    R t+τ (T −t)  2 t+τ (T −t) u(t, y) = sup E exp − t r(v)dv ψ y + t r(v) − λE(U1 ) − σ 2(v) dv τ ∈T0,1

Z +

τ

√ T − t σ(t + v(T − t))dWv +

0

Nt+τ (T −t)

X

# ln(1 + Uj ) ,

(7)

j=1

where T0,1 denotes the set of all stopping times τ with respect to the filtration (Fu )0≤u≤1 taking values in [0, 1]. Finally, we conclude the preliminary results of this section by proving the following theorem.

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Theorem 2.3. The value function u(t, y), 0 ≤ t ≤ T, −∞ < y < ∞ of the optimal stopping problem (6) is Lipschitz continuous in the argument y and locally Lipschitz continuous in t i.e. |u(t, y) − u(t, z)| ≤ g(0) |y − z|, y, z ∈ R, 0 ≤ t ≤ T, (8) |u(t, y) − u(s, y)| ≤ √

A |t − s|, T −t

(9)

where A is some nonnegative constant depending on parameters r¯, σ ¯ , g(0), λ, E(U1 ), K and T . Proof. Fixing any τ in Tt,T and y, z ∈ R, and using Lemma 2.2, we take  Z τ   Z τ  E exp − r(v)dv ψ(Xτ (t, y)) − E exp − r(v)dv ψ(Xτ (t, z)) t

t

≤ E|Xτ (t, y) − Xτ (t, z)| ≤ g(0) |y − z|. Benefiting ourselves by the well-known property that the difference between supremums is less or equal than the supremum of difference leads to the result (8). To show the second part of the theorem, i.e. (9), we shall use the expression (7) for the value function u(t, y). Take any τ ∈ T0,1 we can write   Z t+τ (T −t)  Z τ √ σ 2 (v) − Rtt+τ (T −t) r(v)dv ψ y+ r(v) − λEU1 − dv + T − t σ(t + v(T − t))dWv Ee 2 t 0 Nt+τ (T −t)

+

  Z R s+τ (T −s) r(v)dv ln(1 + Uj ) − Ee− s ψ y+

X

 r(v) − λEU1 −

s

j=1

√ + T −s

s+τ (T −s)

Ns+τ (T −s)

τ

Z

σ(s + v(T − s))dWv + 0

X j=1

R t+τ (T −t)  Z R s+τ (T −s) r(v)dv r(v)dv ≤ E e− t − e− s ψ y+

t

Z

Nt+τ (T −t)

τ

σ(t + v(T − t))dWv + 0

X

 dv

 ln(1 + Uj )

"

√ + T −t

σ 2 (v) 2

t+τ (T −t)

 r(v) − λEU1 −

σ 2 (v) 2

 dv

 R s+τ (T −s) r(v)dv ln(1 + Uj ) + e− s ×

j=1

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  Z t+τ (T −t)  Z τ √ σ 2 (v) r(v) − λEU1 − σ(t + v(T − t))dWv dv + T − t ψ y + 2 t 0 Nt+τ (T −t)

+

X

 ln(1 + Uj )

j=1 s+τ (T −s)

 Z τ √ σ 2 (v) −ψ y + r(v) − λEU1 − σ(s + v(T − s))dWv dv + T − s 2 s 0 Ns+τ (T −s)  # X + ln(1 + Uj ) j=1 "  Z t+τ (T −t)  R t+τ (T −t) R σ 2 (v) − t − ss+τ (T −s) r(v)dv r(v)dv r(v) − λEU1 − −e ≤ g(0)E e dv + 2 t  Z s+τ (T −s)  σ 2 (v) − r(v) − λEU1 − dv 2 Zs τ   √ √ + T − tσ(t + v(T − t)) − T − sσ(s + v(T − s)) dWv 0 # Nt+τ (T −t) X (10) ln(1 + Uj ) . + 

Z



j=Ns+τ (T −s) +1

Let us denote R(u) = can write

Ru 0

r(v)dv, 0 ≤ u ≤ T, and using the mean value theorem, we

Z Z s+τ (T −s) t+τ (T −t) R t+τ (T −t) R − t r(v)dv − ss+τ (T −s) r(v)dv −e r(v)dv − r(v)dv e ≤ t s ≤ |(R(t + τ (T − t)) − R(t)) − (R(s + τ (T − s)) − R(s))| ≤ 2 r¯|t − s|. (11) Similarly, we use the same arguments and obtain Z   Z s+τ (T −s)  t+τ (T −t)  σ 2 (v) σ 2 (v) r(v) − λE(U1 ) − dv − r(v) − λE(U1 ) − dv t 2 2 s   σ ¯2 |t − s|. ≤ 2 r¯ + λE|U1 | + 2 (12) Moreover, we fix τ, 0 ≤ τ ≤ 1, and 0 ≤ s ≤ t < T, using the requirement (1), on σ(t) we

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write 2  √ T − t σ(t + v(T − t)) − T − s σ(s + v(T − s)) dWv 0 Z τ 2 √ √ ≤ E T − t σ(t + v(T − t)) − T − s σ(s + v(T − s)) dv

Z E

τ

√

0 1

Z ≤

√

T − t σ(t + v(T − t)) −

2 √ T − s σ(s + v(T − s)) dv

0 1

Z

2

(T − t) (σ(t + v(T − t)) − σ(s + v(T − s))) dv

≤ 2 0

Z +2

1

√ 2 √ T − t − T − s σ 2 (t + v(T − t))dv.

0

From here we obtain 2 Z τ   √ √ T − t σ(t + v(T − t)) − T − s σ(s + v(T − s)) dWv E 0

≤

2K 2 T 2 + σ ¯2 (t − s)2 . T −t

(13)

Since (Uj )j≥1 be a sequence of independent, identically distributed, integrable random variables, therefore we can find Nt+τ Nt+τ (T −t) (T −t) X X E ln(1 + Uj ) = E |ln(1 + Uj )| j=Ns+τ (T −s) +1 j=Ns+τ (T −s) +1 Nt+τ (T −t) −Ns+τ (T −s)

X

= E

|ln(1 + Uj )|

j=1 N(t−s)(1−τ )

= E

X

|ln(1 + Uj )| .

j=1

Since Nt is an increasing function of time and τ ≤ 1 so we can write Nt+τ N(t−s) (T −t) X X E ln(1 + Uj ) ≤ E |ln(1 + Uj )| j=Ns+τ (T −s) +1 j=1 = E (Nt−s ) E| ln(1 + U1 )| = λ E |ln(1 + U1 )| (t − s),

(14)

Substituting (11)-(14) in (10) and using the fact that the difference between supremums is less or equal than difference supremum of the difference, we complete the proof. In the next section, the main results of the paper are presented.

3

Variational Inequalities

In this section, the variational inequalities of the value function are developed in order to Rt investigate the regularity results of the value function (3). Let S˜t = e− 0 r(u)du St is the

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discounted stock price, then the discounted price function v˜(t, x) = e−

Rt 0

r(u)du

Rt

v(t, xe

0

r(u)du

), 0 ≤ t ≤ T, x > 0

(15)

of the option at time t is C 2 on [0, T ) × R+ (see, Laberton and Lapeyre (1997)) and between the jump times, satisfies Z t Z t ∂˜ v ∂˜ v v˜(t, S˜t ) = v(0, S0 ) + (u, S˜u )du + (u, S˜u )S˜u (−λE(U1 )du + σ(u)dWu ) 0 ∂u 0 ∂x Z Nt   X 1 t ∂ 2 v˜ ˜u )σ 2 (u)S˜2 du + ˜τ ) − v˜(τj , S˜τ − ) . (u, S v ˜ (τ , S (16) + j u j j 2 0 ∂x2 j=1 The function v˜(t, x) is Lipschitz of order 1 with respect to x and with Sτj− = Sτj (1 + Uj ), j = 1, 2, . . .. The process Z tZ  Nt    X v˜(τj , S˜τj ) − v˜(τj , S˜τj − ) − λ Mt = v˜(u, S˜u (1 + z)) − v˜(u, S˜u ) dν(z)du (17) 0

j=1

is a square integrable martingale, where ν(z) is the law of the process U . Combining (16) and (17) we obtain that Z t 1 ∂˜ v ∂ 2 v˜ ∂˜ v (u, S˜u ) − λEU1 S˜u (u, S˜u ) + σ 2 (u)S˜u2 2 (u, S˜u ) v˜(t, S˜t ) − ∂u ∂x 2 ∂x 0  Z   −λ v˜(u, S˜u (1 + z)) − v˜(u, S˜u ) dν(z) du is a martingale, see Israel and Rincon (2008), and therefore ∂˜ v ∂˜ v (u, S˜u ) − λEU1 S˜u (u, S˜u ) ∂u ∂x Z   1 2 ∂ 2 v˜ ˜ 2 ˜ + σ (u)Su 2 (u, Su ) − λ v˜(u, S˜u (1 + z)) − v˜(u, S˜u ) dν(z) ≤ 0 2 ∂x

(18)

a.e. in [0, T ) × R. From Pham (1997), we know that if the payoff function is convex and non-increasing then the price function of the put American contingent claim is a convex function of the stock. Therefore, we can write ∂ 2 v(t, x) ≥0 (19) ∂x2 a.e. in [0, T ) × R. Theorem 3.1. The mapping ς(t, x) = x v(t, x) is Lipschitz continuous in x and locally Lipschitz continuous in the argument of t, i.e. |ς(t, x) − ς(t, y)| ≤ 2 g(0)|x − y|, 0 ≤ t ≤ T, 0 < x ≤ y < ∞, C x |ς(t, x) − ς(s, x)| ≤ √ |t − s|, 0 ≤ s ≤ t < T, x > 0, T −t where the constant C is the function of r¯, σ ¯ , g(0), λ, E(U1 ), K and T .

(20) (21)

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Proof. Consider that v(t, x) = u(t, ln x), x > 0, 0 ≤ t ≤ T , we can write |ς(t, x) − ς(t, y)| = |x u(t, ln x) − y u(t, ln y)| ≤ |x u(t, ln x) − x u(t, ln y)| + |x u(t, ln y) − y u(t, ln y)|. Using the bound (8) and the mean value theorem we arrive to (20). The expression (21) derives using the same arguments as previously, and the bound (9). 2

v(t,x) of the value function Proposition 3.2. The second order weak partial derivative ∂ ∂x 2 (3) satisfies with respect to x the local Holder estimate 2 D 2 ∂ v(t, x) x ≤√ , x > 0, 0 ≤ t < T, ∂x2 T −t

where  Dis a nonnegative constant depends on the parameters r, σ, σ, g(0), λ, E|U1 |, |U1 | E 1+U , K, T . 1 Proof. Using the expression (15), and from (18) and (19), we obtain the system of inequalities  R R 2 ∂v(t,x) x2 2 − 0t r(u)du ∂v(t,x) −2 0t r(u)du ∂ v(t,x)  x) + − λxEU e + σ (t)e  −r(t)v(t, 1 ∂t ∂x 2 ∂x2 R −λ (v(t, x(1 + z)) − v(t, x)) dν(z) ≤ 0 a.e. in [0, T ) × R,   ∂ 2 v(t,x) ≥ 0, x > 0. ∂x2 (22) Also since v(t, x) = u(t, ln x), x > 0, 0 ≤ t ≤ T , we have ( ∂v(t,x) x) x) = ∂u(t,ln , ∂v(t,x) = x1 ∂u(t,ln , ∂t ∂t ∂x ∂y (23) 2 2 ∂ v(t,x) 1 ∂ u(t,ln x) 1 ∂u(t,ln x) = x2 − x2 , 0 ≤ t < T, x > 0. ∂x2 ∂y 2 ∂y Substituting the latter relations and using the results of the Theorem 2.3 in the system of inequalities (22), we have 2 ∂ v(t, x) ∂x2 Rt Z   ∂v(t, x) 2 e2 0 r(v)dv + λxE|U1 | ∂v(t, x) + λ (v(t, x(1 + z)) − v(t, x)) dν(z) ≤ r(t)v(t, x) + 2 2 x σ ∂t ∂x    2 e2rT A |U1 | ≤ rg(0) + √ + λg(0)E|U1 | + λg(0)E x2 σ 2 1 + U1 T −t D √ ≤ . x2 T − t Thus, the required result is derived. Before, we proceed with the main result of this section, we need to state and prove the following result.

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Lemma 3.3. For the function γ(t, y) = y ∂v(t,y) ∂y , 0 ≤ t1 ≤ t2 < T, y > 0, of the value function (3) we have the following bound  Z y+h Z y+h 1 |γ(t2 , y) − γ(t1 , y)| ≤ |γ(t2 , y) − γ(t2 , z)|dz + |γ(t1 , y) − γ(t1 , z)|dz h y y (y + h)|v(t2 , y + h) − v(t1 , y + h)| + y|v(t2 , y) − v(t1 y)|  Z y+h + |v(t2 , z) − v(t1 , z)|dz ,

+

y

where h > 0. Proof. We can express the difference γ(t2 , y) − γ(t1 , y) = γ(t2 , y) − γ(t2 , z) + γ(t2 , z) − γ(t1 , z) + γ(t1 , z) − γ(t1 , y), for any positive real number z. Integrating both sides with respect to z over the interval [y, y + h], we obtain  Z y+h Z y+h 1 γ(t2 , y) − γ(t1 , y) = (γ(t2 , y) − γ(t2 , z))dz + (γ(t2 , z) − γ(t1 , z))dz h y y  Z y+h + (γ(t1 , z) − γ(t1 , y))dz . (24) y

Simplifying the second integral, we have  Z y+h Z y+h  ∂v(t2 , z) v(t1 , z) − dz (γ(t2 , z) − γ(t1 , z))dz = z ∂z ∂z y y (y + h)(v(t2 , y + h) − v(t1 , y + h)) − y(v(t2 , y) − v(t1 , y)) Z y+h − (v(t2 , z) − v(t1 , z))dz. =

y

Combining the latter expression with (24), the proof is complete. In the next, a very interesting result for the value of a put American option is derived. satisfies with respect to time argument Theorem 3.4. The mapping γ(t, x) = x ∂v(t,x) ∂x local H¨ older estimate with exponent 21 , i.e., 1 G+x H |γ(t, x) − γ(s, x)| ≤ √ |t − s| 2 , 0 ≤ s ≤ t < T, x > 0, T −t

(25)

where ¯ , σ, g(0), λ, E(U1 ),  G and H are positive constants depend on the parameters r¯, σ |U1 | E 1+U1 , K and T . Proof. From the continuity of ∂v(t,x) and the relations (23), using Proposition 3.2 we can ∂x write ∂v(t, x) ∂v(t, y) D |γ(t, x)−γ(t, y)| = x −y ≤√ |x−y|, 0 ≤ t < T, 0 < x ≤ y < ∞. ∂x ∂y T −t (26)

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Application of the bounds (21) and (26) in Lemma 3.3 gives  Z y+h Z y+h 1 D D √ √ |γ(t2 , y) − γ(t1 , y)| ≤ (z − y)dz + (z − y)dz h y T − t2 T − t1 y  Z y+h C C(y + h) C y √ + √ |t2 − t1 | + √ |t2 − t1 | + |t2 − t1 |dz T − t2 T − t2 T − t2 y   1 2D 2C y C h C h √ = h2 + √ |t2 − t1 | + √ |t2 − t1 | + √ |t2 − t1 | . h T − t2 T − t2 T − t2 T − t2 1

Let us choose h = C ∗ |t2 − t1 | 2 from the latter estimate we get    1 1 2C y 2 + 2 C|t − t | |t − t | |γ(t2 , y) − γ(t1 , y)| ≤ √ 2 D C∗ + 2 1 2 1 C∗ T − t2   1 2 2C y + 2 T C |t2 − t1 | 2 , ≤ √ 2 D C∗ + C∗ T − t2 q the minimum of which is attained at the point C ∗ = CDy . From here, the required result is derived. Acknowledgements: The first author is very grateful to RCMM and School of Physical Sciences grant scheme for the financial support to carry out this research project. Moreover, the first author would like to thank the Institute for Financial and Actuarial Mathematics, Department of Mathematical Sciences, University of Liverpool for the accommodation provided during the Summer 2012. The main results of the paper have been presented in the Actuarial and Financial Mathematics: Theory and Practice Workshop, 7th June 2012, Liverpool, UK.

References [1] F. Black and M. Scholes, The Pricing of Options and Corporate Liabilities, Journal of Political Economy, 81(3), 637-654 (1973). [2] C. Chiarella and B. Kang, The Evaluation of American Compound Option Prices Under Stochastic Volatility and Stochastic Interest Rates, Journal of Computational Finance, 17(1), 71-92 (2011). [3] El-Karoui, et al., Robustness of the Black and Scholes Formula, Mathematical Finance, 8(2), 93-126 (1998). [4] R. Elliott and P. Kopp, Option Pricing and Hedge Portfolios for Poisson Process, Stochstic Analysis and Applications, 9, 429-444 (1990). [5] R. Glowinsky, et al. , Numerical Analysis of Variational Inequalities. New York, Amsterdam: North-Holland Publishing Company, 1981. [6] S. Hussain and M. Shashiashvili, Discrete Time Hedging of the American Option, Mathematical Finance, 20(4), 647-670 (2010). [7] S. Hussain and N. Rehman, Estimate for the Discrete Time Hedging Error of the American Option on a Dividend-Paying Stock, Mathematical Inequalities and Applications, 15(1), 137-163 (2012).

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[8] V.P. Israel and M.A. Rincon,Variational Inequalities Applied to Option Market Problem, Applied Mathematics and Computation, 201, 384-397 (2008). [9] P. Jaillet, et al., Variational Inequalities and the Pricing of American Options, Acta Applicandae Mathematicae, 21(3), 263-289 (1990). [10] I. Karatzas and S.E. Shreve,Methods of Mathematical Finance, Springer, New York, 1998. [11] D. Lamberton and B. Lapeyre, Stochastic Calculus Applied to Finance, Champan and Hall, UK, 1997. [12] H. Pham,Optimal Stopping, Free Boundary, and American Option in a JumpDiffusion Model, Applied Mathematics and Optimization, 35, 145-164 (1997). [13] S.E. Shreve, Stochastic Calculus for Finance Vol. II, Springer, New York, 2004.

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On a summation boundary value problem for a second-order difference equations with resonance Saowaluk Chasreechai and Thanin Sitthiwirattham

1

Nonlinear Dynamic Analysis Research Center, Department of Mathematics, Faculty of Applied Science, King Mongkut’s University of Technology North Bangkok, Thailand E-mail: [email protected], [email protected]

Abstract In this paper, we obtain a sufficient condition for the existence of the solution for a second-order difference equation with summation boundary value problem at resonance, by using some properties of the Green’s function, the Schaefer’s fixed point theorem and intermediate value theorem. Finally, we present an example to show the importance of these result.

Keywords: boundary value problem; resonance; fixed point theorem; existence. (2010) Mathematics Subject Classifications: 39A05; 39A12.

1

Introduction

The study of the existence of solutions of boundary value problems for linear secondorder ordinary differential and difference equations was initiated by Ilin and Moiseev [1]. Then Gupta [2] studied three-point boundary value problems for nonlinear second-order ordinary differential equations. Since then, nonlinear second-order threepoint boundary value problems have also been studied by many authors, one may see [3-6] and references therein. Also, there are a lot of papers dealing with the resonant case for multi-point boundary value problems, see [7-11]. In [8], J.Liu, S.Wang and J.Zhang studied the existence of multiple solutions for boundary value problems of second-order difference equations with resonance: ∆2 u(t − 1) = g(t, u), u(0) = 0,

t ∈ {1, 2, ..., T },

u(T + 1) = 0.

(1.1) (1.2)

Using Morse theory, critical point theory, minimax methods and bifurcation theory. 1

Corresponding author

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S. Chasreechai and T. Sitthiwirattham

In this paper, we study the existence of solutions of a second-order difference equation with summation boundary value problem at resonance ∆2 u(t − 1) + f (t, u(t)) = 0, t ∈ {1, 2, ..., T }, η ∑ u(0) = 0, u(T + 1) = α u(s),

(1.3) (1.4)

s=1

where

2(T + 1) = 1, T ≥ 3, η ∈ {1, 2, ..., T − 1} and f is continuous function. αη(η + 1)

In this paper, we are interested in the existence of the solution for problem (1.3)2(T +1) (1.4) under the condition αη(η+1) = 1, which is a resonant case. Using some properties of the Green’s function G(t, s), intermediate value theorems and Schaefer’s fixed point theorem, we establish a sufficient condition for the existence of positive solutions of 2(T +1) problem αη(η+1) = 1. Let N be a nonnegative integer, Ni,j = {k ∈ N| i ≤ k ≤ j} and Np = N0,p . Throughout this paper, we suppose the following conditions hold: (H) f (t, u) ∈ C(NT +1 × R, R) and there exist two positive continuous functions p(t), q(t) ∈ C(NT +1 , R+ ) such that |f (t, tu)| ≤ p(t) + q(t)|u|m , t ∈ NT +1 , where 0 ≤ m ≤ 1. Furthermore,

(1.5)

lim f (t, tu) = ∞, for any t ∈ N1,T .

u→±∞

To accomplish this, we denote C(NT +1 , R),the Banach space of all function u with the norm defined by ∥u∥ = max{u(t) | t ∈ NT +1 }. The proof of the main result is based upon an application of the following theorem. Theorem 1.1. ([12]). Let X be a Banach space and T : X → X be a continuous and compact mapping. If the set {x ∈ X : x = λT (x), for some λ ∈ (0, 1)} is bounded, then T has a fixed point. The plan of the paper is follows. In Section 2, we recall some lemmas. In Section 3, we prove our main result. Illustrate example is presented in Section 4.

2

Preliminaries

We now state and prove several lemmas before stating our main results.

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On a summation boundary value problem for a second-order difference equations... 3 Lemma 2.1. The problem (1.3)-(1.4) is equivalent to the following u(t) =

T ∑

G(t, s)f (s, u(s)) +

s=1

u(T + 1) t, T +1

(2.1)

where

  αt(T       1 G(t, s) = αt(T (T + 1)(α − 1)    αt(T    αt(T

+ 1 − s) − 21 αt(η − s)(η − s + 1) − (T + 1)(α − 1)(t − s), s ∈ N1,t−1 ∩ N1,η−1 + 1 − s) − 12 αt(η − s)(η − s + 1), s ∈ Nt,η−1 + 1 − s) − (T + 1)(α − 1)(t − s), s ∈ Nη,t−1 + 1 − s), s ∈ Nt,T ∩ Nη,T (2.2)

Proof. Assume that u(t) is a solution of problem (1.3)-(1.4), then it satisfies the following equation: u(t) = C1 + C2 t −

t−1 ∑

(t − s)f (s, u(s)),

s=1

where C1 , C2 are constants. By the boundary value condition (1.3), we obtain C1 = 0. So, u(t) = C2 t −

t−1 ∑

(t − s)f (s, u(s)).

(2.3)

s=1

From (2.3), η ∑

∑∑ η(η + 1) ly(s) u(s) = C2 − 2 s=1 s=1 l=1 η−1 η−s

η(η + 1) 1∑ = C2 − (η − s)(η − s + 1)y(s). 2 2 s=1 η−1

From the second boundary condition, we have (2T + 2 − αη(η + 1))C2 = 2

T ∑

(T + 1 − s)f (s, u(s)) + α

s=1

η−1 ∑

(η − s)(η − s + 1)f (s, u(s)).

s=1

(2.4) Since

2(T +1) αη(η+1)

= 1, then (2.4) is solvable if and only if

α∑ (η − s)(η − s + 1)f (s, u(s)). (T + 1 − s)f (s, u(s)) = 2 s=1 s=1

T ∑

η−1

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Note that u(T + 1) −

η ∑ s=1

u(s) = (T + 1)C2 −

T ∑

(T + 1 − s)f (s, u(s))

s=1

1∑ η(η + 1) C2 + − (η − s)(η + 1 − s)f (s, u(s)), 2 2 s=1 η−1

and then [ η T ∑ ∑ 2 C2 = u(T + 1) − u(s) + (T + 1 − s)f (s, u(s)) 2T + 2 − η(η + 1) s=1 s=1 ] η−1 1∑ − (η − s)(η + 1 − s)f (s, u(s)) 2 s=1 [ η T ∑ ∑ α = u(T + 1) − u(s) + (T + 1 − s)f (s, u(s)) (T + 1)(α − 1) s=1 s=1 ] η−1 1∑ − (η − s)(η + 1 − s)f (s, u(s)) . 2 s=1 2(T + 1) ∑ We now use that u(T + 1) = u(s) to get η(η + 1) s=1 η

] [ η ∑ α u(T + 1) u(s) = u(T + 1) − , (T + 1)(α − 1) T + 1 s=1 and ] [∑ η−1 T α 1∑ C2 = (T + 1 − s)f (s, u(s)) − (η − s)(η + 1 − s)f (s, u(s)) (T + 1)(α − 1) s=1 2 s=1 +

u(T + 1) . T +1

Hence the solution of (1.3)-(1.4) is given, implicity as [∑ ] η−1 T αt 1∑ u(t) = (T + 1 − s)f (s, u(s)) − (η − s)(η + 1 − s)f (s, u(s)) (T + 1)(α − 1) s=1 2 s=1 t−1 ∑ u(T + 1) − (t − s)f (s, u(s)) + t. T +1 s=1

(2.5)

According to (2.5) it is easy to show that (2.1) holds. Therefore, problem (1.3)(1.4) is equivalent to the equation (2.1) with the function G(t, s) defined in (2.2). The proof is completed. 

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On a summation boundary value problem for a second-order difference equations... 5 Lemma 2.2. For any (t, s) ∈ NT +1 × NT +1 , G(t, s) is continuous, and G(t, s) > 0 for any (t, s) ∈ N1,T × N1,T . Proof. The continuity of G(t, s) for any (t, s) ∈ NT +1 × NT +1 , is obvious. Let 1 g1 (t, s) = αt(T + 1 − s) − αt(η − s)(η − s + 1) − (T + 1)(α − 1)(t − s), 2 where s ∈ N1,t−1 ∩ N1,η−1 . Here we only need to prove that g1 (t, s) > 0 for s ∈ N1,t−1 ∩ N1,η−1 , the rest of the proof is similar. So, from the definition of g1 (t, s), η ∈ N1,T −1 and the resonant 2(T +1) = 1, we have condition αη(η+1) 1 g1 (t, s) = αt(T + 1 − s) − αt(η − s)(η − s + 1) − (T + 1)(α − 1)(t − s) 2 1 = (T + 1)(t − s) + αs(T + 1 − t) − αt(η − s)(η − s + 1) 2 α > (T + 1)(t − s) − [tη(η + 1) − 2s(T + 1 − t)] 2 α > (T + 1)(t − s) − 2 T +1 > (T + 1)(t − s) − η(η + 1) > (T + 1)(t − s − 1) ≥ 0, for s ∈ N1,t−1 ∩ N1,η−1 . Since t > s and η(η + 1) ≥ 2(T + 1 − t) where T ≥ 3. The proof is completed.  Let

1 G∗ (t, s) = G(t, s). t

(2.6)

Then

  α(T       1 ∗ G (t, s) = α(T (T + 1)(α − 1)   α(T    α(T

+ 1 − s) − 21 α(η − s)(η − s + 1) − 1t (T + 1)(α − 1)(t − s), + 1 − s) − 12 α(η − s)(η − s + 1), + 1 − s) − 1t (T + 1)(α − 1)(t − s), + 1 − s),

s ∈ N1,t−1 ∩ N1,η−1 s ∈ Nt,η−1 s ∈ Nη,t−1 s ∈ Nt,T ∩ Nη,T . (2.7)

Thus, problem (1.3)-(1.4) is equivalent to the following equation: u(t) =

T ∑

tG∗ (t, s)f (s, u(s)) +

s=1

302

u(T + 1) t. T +1

(2.8)

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By a simple computation, the new Green’s function G∗ (t, s) has the following properties. Lemma 2.3. For any (t, s) ∈ NT +1 × NT +1 , G∗ (t, s) is continuous, and G∗ (t, s) > 0 for any (t, s) ∈ N1,T × N1,T . Furthermore, lim G∗ (t, s) := G∗ (0, s) t→0

1 = (T + 1)(α − 1)

{ α(T + 1 − s) − 12 α(η − s)(η − s + 1), α(T + 1 − s),

s ∈ N1,η−1 s ∈ Nη,T . (2.9)

Lemma 2.4. For any s ∈ N1,T , G∗ (t, s) is nonincreasing with respect to t ∈ NT +1 , and ∗ ∗ for any s ∈ NT +1 , △t G△t(t,s) < 0, and △t G△t(t,s) = 0 for t ∈ Ns . That is, G∗ (T + 1, s) ≤ G∗ (t, s) ≤ G∗ (s, s) where G∗ (t, s) ≤ G∗ (s, s) 1 = (T + 1)(α − 1)

{ α(T + 1 − s) − 12 α(η − s)(η − s + 1), s ∈ N1,η−1 α(T + 1 − s), s ∈ Nη,T (2.10)

∗

∗

G (t, s) ≥ G (T + 1, s)

{ (T + 1)(T + 1 − s) − 12 α(η − s)(η − s + 1), s ∈ N1,η−1 1 = (T + 1)(α − 1) (T + 1)(T + 1 − s), s ∈ Nη,T . (2.11)

Let u(t) = tw(t).

(2.12)

Then u(T + 1) = (T + 1)w(T + 1), and equation (2.8) gives w(t) =

T ∑

G∗ (t, s)f (s, sw(s)) + w(T + 1).

(2.13)

s=1

Now we have y(t) = w(t) − w(T + 1).

(2.14)

Then y(T + 1) = w(T + 1) − w(T + 1) = 0, and equation (2.13) gives 1 ∑ ∗ y(t) = G (t, s)f (s, s(y(s) + w(T + 1))). T + 1 s=1 T

(2.15)

We replace w(T + 1) by any real number λ, then (2.15) can be rewritten as 1 ∑ ∗ G (t, s)f (s, s(y(s) + λ)). y(t) = T + 1 s=1 T

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(2.16)

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On a summation boundary value problem for a second-order difference equations... 7 The following result is based on the Schaefer’s fixed point theorem. We define an operator T on the set Ω = C(NT +1 ) as follows: 1 ∑ ∗ T y(t) = G (t, s)f (s, s(y(s) + λ)). T + 1 s=1 T

(2.17)

∑ Lemma 2.5. Assume that f ∈ C(NT +1 × R, R), Ts=1 G∗ (t, s)q(s) < T + 1 and (1.5) holds. Then the equation (2.16) has at least one solution for any real number λ. Proof. We divide the proof into four steps. Step I. T maps bounded sets into bounded sets in Ω. Let us prove that for any R > 0, there exists a positive constant L such that for each y ∈ BR = {y ∈ C(NT +1 × R) : ∥y∥ ≤ R}, we have ∥(T y)(t)∥ ≤ L. Indeed, for any y ∈ BR , we obtain T 1 ∑ | (T y)(t)| = G∗ (t, s)f (s, s(y(s) + λ)) T + 1 s=1 ≤

1 ∑ ∗ 1 ∑ ∗ G (t, s)p(s) + G (t, s)|q(s) + λ|m T + 1 s=1 T + 1 s=1

≤

1 ∑ ∗ 1 ∑ ∗ G (t, s)p(s) + G (t, s)q(s)(∥y(s)∥ + ∥λ∥)m T + 1 s=1 T + 1 s=1

≤

T T 1 ∑ ∗ (R + ∥λ∥)m ∑ ∗ G (s, s)p(s) + G (s, s)q(s) T + 1 s=1 T +1 s=1

T

T

T

T

:= L.

(2.18)

Step II. Continuity of T . Let ϵ > 0, there exists δ > 0 such that for all t ∈ NT +1 and for all x, y ∈ BR with |(t, t(x(t) + λ) − (t, t(y(t) + λ)| < δ, we have f (t, t(x(t) + λ) − f (t, t(y(t) + λ) < ϵ. Then, we obtain T 1 ∑ ∗ | (T x)(t) − (T y)(t)| ≤ G (t, s)[f (s, s(x(s) + λ)) − f (s, s(y(s) + λ))] T + 1 s=1 T ϵ ∑ ∗ ≤ G (t, s) = ϵ. T +1 s=1

This means that T is continuous in Ω. Step III. T (BR ) is equicontinuous with BR defined as in Step II. bounded, then there exists M > 0 such that |f | ≤ M .

304

Since BR is

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8

S. Chasreechai and T. Sitthiwirattham For any ε > 0, there exists δ > 0 such that for t1 , t2 ∈ NT +1 |G∗ (t2 , s) − G∗ (t1 , s)| ≤

ϵ . M

Then we have

T 1 ∑ | (T y)(t2 ) − (T y)(t1 )| ≤ |G∗ (t2 , s) − G∗ (t1 , s)||f (s, s(y(s) + λ)) T + 1 s=1 M ∑ ∗ ≤ |G (t2 , s) − G∗ (t1 , s)| T + 1 s=1 ϵ = M· ≤ ϵ. M T

This means that the set T (BR ) is an equicontinuous set. As a consequence of Steps I to III together with the Arzela’-Ascoli theorem, we get that T is completely continuous in Ω. Step IV. A priori bounds. We show that the set E = {y ∈ C(NT +1 , R) / y = µT y for some µ ∈ (0, 1)} is bounded. By Lemma 2.1, assume that there exist y ∈ ∂BR with ∥y(t)∥ = R and µ ∈ (0, 1) such that y = µT y. It follows that T µ ∑ ∗ | y(t)| = G (t, s)f (s, s(y(s) + λ)) T +1 ≤ < ≤

µ T +1 1 T +1 1 T +1

s=1 T ∑

G∗ (s, s) |f (s, s(y(s) + λ))|

s=1

[ T ∑ s=1 T ∑

G∗ (s, s)p(s) +

T ∑

] G∗ (s, s)q(s) (∥y(s)∥ + ∥λ∥)m

s=1

T (R + ∥λ∥)m ∑ ∗ G (s, s)p(s) + G (s, s)q(s) T +1 s=1 s=1 ∗

:= L.

(2.19)

This shows that the set E is bounded. By the Schaefer’s fixed point theorem, we conclude that T has a fixed point which is a solution of problem (1.1). 

3

Main Results

In this section, we prove our result by using Lemmas 2.5-2.7 and the intermediate value theorem.

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On a summation boundary value problem for a second-order difference equations... 9 ∑ Theorem 3.1. Assume that (H1) holds. If Ts=1 G∗ (s, s)q(s) < 1, then the problem (1.3)-(1.4) has at least one solution, where { α(T + 1 − s) − 12 α(η − s)(η − s + 1), s ∈ N1,η−1 1 G∗ (s, s) = (T + 1)(α − 1) α(T + 1 − s), s ∈ Nη,T Proof. Since (2.19) is continuously dependent on the parameter λ. So, we should only investigate λ such that y(T + 1) = 0 in order that u(T + 1) = λ. Equation (2.16) is rewrite as 1 ∑ ∗ yλ (t) = G (t, s)f (s, s(yλ (s) + λ)), t ∈ NT +1 . T + 1 s=1 T

(3.1)

where λ is any given real number. Equation(3.1) show that there exists λ such that 1 ∑ ∗ L(λ) := yλ (T + 1) = G (T + 1, s)f (s, s(yλ (s) + λ)) T + 1 s=1 T

(3.2)

and we can observe that, yλ (T + 1) is continuously dependent on the parameter λ. To prove that there exists λ∗ such that yλ∗ (T + 1) = 0, we must to show that lim L(λ) = ∞ and lim L(λ) = −∞.

λ→∞

λ→−∞

Firstly, we prove that lim L(λ) = ∞ by supposing that lim L(λ) < ∞ as aconλ→∞

λ→∞

tradiction. Therefore there exists a sequence {λn } with lim L(λ) = ∞ such that n→∞

lim L(λn ) < ∞. This implies that the sequence {L(λn )} is bounded. Since the

λn →∞

function f (t, ty) is continuous with respect to t ∈ NT +1 and y ∈ R, we have f (t, t(yλn (t) + λn )) ≥ 0 , t ∈ NT +1

(3.3)

where λn is large enough, Assuminh that (3.3) is true and using (3.1), we have yλ ≥ 0 , t ∈ NT +1 .

(3.4)

lim f (t, t(yλn (t) + λn )) = ∞ , t ∈ NT +1 .

(3.5)

lim f (t, tu) = ∞ , t ∈ NT +1 .

(3.6)

Therefore, λn →∞

From (H), we get λ→∞

From (3.2),(3.5) and (3.6), we have lim yλn (T + 1) =

λn →∞

lim

λn →∞

T ∑

G∗ (T + 1, s)f (s, s(yλn (s) + λn ))

(3.7)

s=1

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10

S. Chasreechai and T. Sitthiwirattham 3 (T −1) 4

≥

lim

λn →∞

∑

G∗ (T + 1, s)f (s, s(yλn (s) + λn ))

s= 41 (T −1)

= ∞,

(3.8)

we find that this result contradicts our assumption. We define Sn = {t ∈ NT +1 | f (t, t(yλn (t) + λn )) < 0}. where λn is large. Note that Sn is not empty. Secondly, we divide the set Sn into set Sen and set Sbn as follows: Sen = {t ∈ Sn | yλn + λn > 0} and Sbn = {t ∈ Sn | yλn + λn ≤ 0} where Sen ∩ Sbn = ∅, Sen ∪ Sbn = Sn . So, we have from (H) that Sbn is not empty. In addition, we find from (H) that the function f (t, tu) is bounded below by a constant for t ∈ NT +1 and λ ∈ [0, ∞). Thus, there exists a constant M (< 0) which is independent of t and λn , such that f (t, t(yλn (t) + λn )) ≥ M , t ∈ Sen ,

(3.9)

Let h(λn ) = mint∈Sn yλn (t) and using the definitions of Sen and set Sbn , we have h(λn ) = min yλn (t) = −∥yλn (t)∥Sbn . bn t∈S

It follows that h(λn ) → −∞ as λn → ∞ since if h(λn ) is bounded below by a constant as λn → ∞, then (3.7) holds. Therefore, we can choose largeλn1 such that { } ∑ ∑ M Ts=1 G∗ (s, s) − Ts=1 G∗ (s, s)p(s) 1 h(λn ) < max −1, (3.10) ∑ T +1 1 − Ts=1 G∗ (s, s)q(s) for n > n1 . Employing (H), (3.1), (3.8), (3.9), the definitions of Sen , and set Sbn , for any λn > λn1 , we have 1 ∑ ∗ G (s, s)f (s, s(yλn (s) + λn )) T + 1 s∈S n 1 ∑ ∗ ≥ G (s, s)f (s, s(yλn (s) + λn )) T +1

yλn (t) ≥

en s∈S

+

1 ∑ ∗ G (s, s) (−p(s) − q(s)|yλn (s) + λn |m ) T +1 bn s∈S

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On a summation boundary value problem for a second-order difference equations...11  ≥

1  M T +1

∑ en s∈S

G∗ (s, s) −

∑

G∗ (s, s)p(s) −

bn s∈S

∑

 G∗ (s, s)q(s)∥yλn (s) + λn ∥m  .

bn s∈S

It follows that

[ ] T T T ∑ ∑ ∑ 1 yλn (t) ≥ G∗ (s, s)p(s) − G∗ (s, s)q(s)∥yλn (s) + λn ∥m M G∗ (s, s) − Sn , T +1 s=1 s=1 s=1 [ ] T T T ∑ ∑ ∑ 1 ≥ G∗ (s, s)p(s) − G∗ (s, s)q(s)h(λn ) , t ∈ Sn , M G∗ (s, s) − T +1 s=1 s=1 s=1 which implies that

[ ∑ ] ∑ M Ts=1 G∗ (s, s) − Ts=1 G∗ (s, s)p(s) 1 h(λn ) ≥ . ∑ T +1 1 − Ts=1 G∗ (s, s)q(s)

This result contradicts (3.9). Thus, the proof that lim L(λ) = ∞ is done. using a λ→∞

similar method, we can prove that lim L(λ) = −∞. λ→−∞

Notice that L(λ) is continuous with respect to λ ∈ (−∞, ∞). From the intermediate value theorem, there exists λ∗ ∈ (−∞, ∞) such that L(λ∗ ) = 0 , that is, y(T + 1) = yλ∗ (T + 1) = 0, which satisfies the second boundary value condition of (1.2). The proof is completed. 

4

Example

In this section, we give an example to illustrate our result. Example Consider the BVP 1 t ∈ N1,4 , ∆2 u(t − 1) + t2 + u(t) = 0, 2 2 5∑ u(0) = 0, u(5) = u(s). 6 s=1

(4.1) (4.2)

Set α = 65 , η = 2, T = 4, f (t, u) = t2 + 12 u(t). So we have αη(η + 1) = 1 and f (t, tu) = t2 + 2t u(t). 2(T + 1) Now we take q(t) = 5t . It is easy to check that 4 4 ∑ 1 ∑ 4 lim f (t, tu) = ±∞ and G∗ (s, s)q(s) ≤ (5 − s)s = < 1. u→±∞ 25 s=1 5 s=1 Thus the conditions of Theorem 3.1 are satisfied. Therefore problem (4.1)-(4.2) has at least a nontrivial solution.  Acknowledgements. This research was funded by King Mongkut’s University of Technology North Bangkok. Contract no. KMUTNB-GEN-57-18.

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S. Chasreechai and T. Sitthiwirattham

References [1] V. A. Ilin and E. I. Moiseev, Nonlocal boundary-value problem of the first kind for a Sturm-Liouville operator in its differential and finite difference aspects, J. Differ. Equations. 23(1987), 803-810. [2] C. P. Gupta, Solvability of a three-point nonlinear boundary value problem for a second order ordinary differential equations, J. Math. Anal. Appl. 168:2(1992), 540-551. [3] X. Lin, W. Lin, Three positive solutions of a secound order difference Equations with Three-Point Boundary Value Problem, J.Appl. Math. Comut. 31(2009), 279-288. [4] G. Zhang, R. Medina, Three-point boundary value problems for difference equations, Comp. Math. Appl. 48(2004), 1791-1799. [5] J. Henderson, H.B. Thompson, Existence of multiple solutions for second order discrete boundary value problems, Comput. Math. Appl. 43 (2002), 1239-1248. [6] T. Sitthiwirattham, J. Tariboon, Positive solutions to a generalized second order difference equation with summation boundary value problem. J. Appl. Math. (2012), Article ID 569313, 15 pages. [7] R. Ma, Multiplicity results for an m-point boundary value problem at resonance, Indian J. Math. 47:1(2005), 15-31. [8] X. L. Han, Positive solutions of a nonlinear three-point boundary value problem at resonance, J. Math. Anal. Appl. 336(2007), 556-568. [9] J.Liu, S.Wang, J.Zhang, Multiple solutions for boundary value problems of second-order difference equations with resonance, J. Math. Anal. Appl. 374 (2011), 187196. [10] H.Liu, Z.Ouyang, Existence of the Solutions for Second Order Three-point Integral Boundary Value Problems at Resonance. Bound. Value. Probl. (2013), 2013:197. [11] J.J.Nieto, Existence of a solution for a three-point boundary value problem for a second-order differential equation at resonance. Bound. Value. Probl. (2013), 2013:130. [12] H. Schaefer, U¨ ber die Methode der a priori-Schranken. Mathematische Annalen. 129(1955), 415416.

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Fuzzy quadratic mean operators and their use in group decision making Jin Han Park, Seung Mi Yu Department of Applied Mathematics, Pukyong National University, Pusan 608-737, South Korea [email protected] (J.H. Park), [email protected] (S.M. Yu) Young Chel Kwun∗ Department of Mathematics, Dong-A University, Pusan 604-714, South Korea [email protected]

Abstract Quadratic mean in statistics is a statistical measure defined as the square root of the mean of the squares of a sample. In this paper, we investigate the situations in which the input data are expressed in fuzzy values and develop some fuzzy quadratic mean operators, such as fuzzy weighted quadratic mean operator, fuzzy ordered weighted quadratic mean operator, and fuzzy hybrid quadratic mean operator. Especially, all these operators can reduce to aggregate interval or real numbers. Then based on the developed operators, we present an approach to group decision making and illustrate it with a practical example.

1

Introduction

Information aggregation is an essential process of gathering relevant information from multiple sources by using a proper aggregation technique. Many techniques, such as the weighted average operator [5], the weighted geometric mean operator [1], harmonic mean operator [2], weighted harmonic mean (WHM) operator [2], ordered weighted average (OWA) operator [17], ordered weighted geometric operator [3, 13], weighted OWA operator [8], induced OWA operator [21], induced ordered weighted geometric operator [15], uncertain OWA operator [14], hybrid aggregation operator [10] and so on, have been developed to aggregate data information. However, yet most of existing aggregation operators do not take into account the information about the relationship between the values being fused. Yager [18] introduced a tool to provide more versatility in the ∗ Corresponding

author: [email protected]

1

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information aggregation process, i.e., developed a power-average (PA) operator and a power OWA (POWA) operator. In some situations, however, these two operators are unsuitable to deal with the arguments taking the forms of multiplicative variables because of lack of knowledge, or data, and decision makers’ limited expertise related to the problem domain. Based on this tool, Xu and Yager [16] developed additional new geometric aggregation operators, including the power-geometric (PG) operator, weighted PG operator and power-ordered weighted geometric (POWG) operator, whose weighting vectors depend upon the input arguments and allow values being aggregated to support and reinforce each other. Quadratic mean in statistics is a statistical measure defined as the square root of the mean of the squares of a sample, which is a conservative average to be used to provide for aggregation lying between the max and min operators. Consider that, in the existing literature, the quadratic mean is generally considered as a fusion technique of numerical data, in the real-life situations, the input data sometimes cannot be obtained exactly, but fuzzy data can be given. Therefore, how to aggregate fuzzy data by using the quadratic mean? is an interesting research topic and is worth paying attention to. In this paper, we develop some fuzzy quadratic mean (FQM) operators. To do so, the remainder of this paper is arranged in the following sections. Section 2 reviews some basic aggregation operators. Section 3 develops some FQM operators, such as fuzzy weighted quadratic mean (FWQM) operator, fuzzy ordered weighted quadratic mean (FOWQM) operator, fuzzy hybrid quadratic mean (FHQM) operator, and so on. Section 4 presents an approach to multiple attribute group decision making based on the developed operators. Section 5 illustrates the presented approach with a practical example. Section 6 ends the paper with some concluding remarks.

2

Basic aggregation operators

We review some basic aggregation techniques and some of their fundamental characteristics. Definition 2.1 [5] Let WAA : Rn → R, if WAA(a1 , a2 , . . . , an ) =

n ∑

w j aj ,

(1)

j=1

where R is the set of real numbers, aj (j = 1, 2, . . . , n) is a collection of positive real numbers, and w = (w1 ,∑ w2 , . . . , wn )T is the weight vector of aj n (j = 1, 2, . . . , n), with wj ≥ 0 and j=1 wj = 1, then WAA is called the weighted arithmetic averaging (WAA) operator. Especially, if wi = 1, wj = 0, j ̸= i, then WAA(a1 , a2 . . . , an ) = ai ; if w = ( n1 , n1 , . . . , n1 )T , then the WAA

2

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operator is reduced to the arithmetic averaging (AA) operator, i.e., 1∑ aj . n j=1 n

AA(a1 , a2 , . . . , an ) =

(2)

Definition 2.2 [2] Let WQM : (R+ )n → R+ , if  WQM(a1 , a2 , . . . , an ) = 

n ∑

 12 wj a2j  ,

(3)

j=1

where R+ is the set of all positive real numbers, aj (j = 1, 2, . . . , n) is a collection of positive real numbers, and w = (w1∑ , w2 , . . . , wn )T is the weight vector of aj n (j = 1, 2, . . . , n), with wj ≥ 0 and j=1 wj = 1, then WQM is called the weighted quadratic mean (WQM) operator. Especially, if wi = 1, wj = 0, j ̸= i, then WQM(a1 , a2 . . . , an ) = ai ; if w = ( n1 , n1 , . . . , n1 )T , then the WQM operator is reduced to the quadratic mean (QM) operator, i.e., ( ∑n j=1

QM(a1 , a2 , . . . , an ) =

a2j

) 12

n

.

(4)

The WAA and WQM operators first weight all the given data, and then aggregate all these weighted data into a collective one. Yager [17] introduced and studied the OWA operator that weights the ordered positions of the data instead of weighting the data themselves. Definition 2.3 [17] An OWA operator of dimension n is a mapping OWA : Rn → that has an associated vector ω = (ω1 , ω2 , . . . , ωn )T such that ωj ≥ 0 ∑R n and j=1 ωj = 1. Furthermore, OWA(a1 , a2 , . . . , an ) =

n ∑

wj bj ,

(5)

j=1

where bj is the jth largest of ai (i = 1, 2, . . . , n). Especially, if wi = 1, wj = 0, j ̸= i, then bn ≤ OWA(a1 , a2 , . . . , an ) = bi ≤ b1 ; if w = ( n1 , n1 , . . . , n1 )T , then 1∑ 1∑ bj = aj = AA(a1 , a2 , . . . , an ). n j=1 n j=1 n

OWA(a1 , a2 , . . . , an ) =

3

n

(6)

Fuzzy quadratic mean operators

The above aggregation techniques can only deal with the situation that the arguments are represented by the exact numerical values, but are invalid if the aggregation information is given in other forms, such as triangular fuzzy number [9], which is a widely used tool to deal with uncertainty and fuzzyness, described as follows: 3

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Definition 3.1 [9] A triangular fuzzy [aL , aM , aU ]. The membership function  0,     x−aL , aM −aL µaˆ (x) = x−aU    aM −aU ,  0,

number a ˆ can be defined by a triplet µaˆ (x) is defined as: x < aL ; aL ≤ x ≤ aM ; aM ≤ x ≤ aU ; x > aU ,

where aU ≥ aM ≥ aL ≥ 0, aL and aU stand for the lower and upper values of a ˆ, respectively, and aM stands for the modal value [9]. Especially, if and two of aL , aM and aU are equal,then a ˆ is reduced to an interval number; if all aL , aM U and a are equal, then a ˆ is reduced to a real number. For convenience, we let Ω be the set of all triangular fuzzy numbers. Let a ˆ = [aL , aM , aU ] and ˆb = [bL , bM , bU ] be two triangular fuzzy numbers, then some operational laws defined as follows [9]: (1) a ˆ + ˆb = [aL , aM , aU ] + [bL , bM , bU ] = [aL + bL , aM + bM , aU + bU ]; (2) λˆ a = λ[aL , aM , aU ] = [λaL , λaM , λaU ]; (3) a ˆ × ˆb = [aL , aM , aU ] × [bL , bM , bU ] = [aL bL , aM bM , aU bU ] (4) a1ˆ = [aL ,a1M ,aU ] = [ a1U , a1M , a1L ]. In order to compare two triangular fuzzy numbers, Xu [12] provided the following definition: Definition 3.2 [12] Let a ˆ = [aL , aM , aU ] and ˆb = [bL , bM , bU ] be two triangular fuzzy numbers, then the degree of possibility of a ˆ ≥ ˆb is defined as follows: { ( ) } bM − aL ˆ p(ˆ a ≥ b) = δ max 1 − max ,0 ,0 aM − aL + bM − bL { ( ) } bU − aM +(1 − δ) max 1 − max , 0 , 0 , δ ∈ [0, 1] (7) aU − aM + bU − bM which satisfies the following properties: 0 ≤ p(ˆ a ≥ ˆb) ≤ 1, p(ˆ a≥a ˆ) = 0.5, p(ˆ a ≥ ˆb) + p(ˆb ≥ a ˆ) = 1.

(8)

Here, δ reflects the decision maker’s risk-bearing attitude. If δ > 0.5, then the decision maker is risk lover; If δ = 0.5, then the decision maker is neutral to risk; If δ < 0.5, then the decision maker is risk avertor. In the following, we shall give a simple procedure for ranking of the triangular fuzzy numbers a ˆi (i = 1, 2, . . . , n). First, by using Eq. (7), we compare each a ˆi with all the a ˆj (j = 1, 2, . . . , n), for simplicity, let pij = p(ˆ ai ≥ a ˆj ), then we develop a possibility matrix [14] as   p11 p12 . . . p1n  p21 p22 . . . p2n  , P = (9) ..   . pn1 pn2 . . . pnn 4

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where pij ≥ 0, pij + pji = 1, pii = 12 , i, j = 1, 2, . . . , n. ∑n Summing all elements in each line of matrix P, we have pi = j=1 pij , i = 1, 2, . . . , n. Then, in accordance with the values of pi (i = 1, 2, . . . , n), we rank the a ˆi (i = 1, 2, . . . , n) in descending order. Now, based on operational laws, we extend the WQM operator (3) to fuzzy environment: M U Definition 3.3 Let a ˆj = [aL j , aj , aj ] (j = 1, 2, . . . , n) be a collection of triangular fuzzy numbers, and let FWQM : Ωn → Ω, if

  12 n ∑ FWQM(ˆ a1 , a ˆ2 , . . . , a ˆn ) =  wj a ˆ2j  ,

(10)

j=1 T where w = (w∑ ˆj (j = 1, 2, . . . , n), with 1 , w2 , . . . , wn ) be the weight vector of a n wj ≥ 0 and j=1 wj = 1, then FWQM is called a fuzzy weighted quadratic mean (FWQM) operator.

Especially, if wi = 1, wj = 0, j ̸= i, then FWQM(ˆ a1 , a ˆ2 , . . . , a ˆn ) = a ˆi ; if w = ( n1 , n1 , . . . , n1 )T , then the FWQM operator is reduced to the fuzzy quadratic mean (FQM) operator: ( ∑n j=1

FQM(ˆ a1 , a ˆ2 , . . . , a ˆn ) =

a ˆ2j

) 12

n

.

(11)

By the operational laws and Eq. (10), we have  FWQM(ˆ a1 , a ˆ2 , . . . , a ˆn ) = 

n ∑

 21



wj a ˆ2j  = 

j=1

n ∑

 12 M U 2 wj [aL j , a j , aj ]

j=1

  21   21   12  n n n ∑ ∑ ∑  2 2 2  =  wj (aL , wj (aM , wj (aU  (12) j) j ) j ) j=1

j=1

j=1

and then by Eq. (12), we have

FQM(ˆ a1 , a ˆ2 , . . . , a ˆn ) ( ∑ ) 12 ( ∑n ) 21 ( ∑n ) 12  n L 2 M 2 U 2 (a ) (a ) (a ) j=1 j j=1 j j=1 j . = , , n n n

(13)

M U Especially, if the triangular fuzzy numbers a ˆj = [aL j , aj , aj ] (j = 1, 2, . . . , n) L U are reduced to the interval numbers a ˜j = [aj , aj ] (j = 1, 2, . . . , n), then the

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FWQM operator is reduced to the uncertain weighted quadratic mean(UWQM) operator:   21 n ∑ wj a ˜2j  UWQM(˜ a1 , a ˜2 , . . . , a ˜n ) =  j=1

  12   12  n n ∑ ∑  2  2 , wj (aU =  wj (aL  . (14) j ) j) j=1

j=1

If w = ( n1 , n1 , . . . , n1 )T , then the UWQM operator is reduced to the uncertain quadratic mean(UQM) operator: )1 ( ∑n a2j ) 2 j=1 (˜ UQM(˜ a1 , a ˜2 , . . . , a ˜n ) = n ( ∑ ) 12 ( ∑n ) 21  n 2 2 (aL (aU j) j ) j=1 j=1  . (15) = , n n U If aL j = aj = aj , for all j = 1, 2, . . . , n, then Eqs. (14) and (15) are, respectively, reduced to the WQM operator (3) and the QM operator (4).

Example 3.4 Given a collection of triangular fuzzy numbers: a ˆ1 = [2, 3, 4], a ˆ2 = [1, 2, 4], a ˆ3 = [2, 4, 6], a ˆ4 = [1, 3, 5], let w = (0.3, 0.1, 0.2, 0.4)T be the weight vector of a ˆi (i = 1, 2, 3, 4), then by Eq. (12), we have   21   12   12  n n n ∑ ∑  ∑ 2 2 2  FWQM(ˆ a1 , a ˆ2 , a ˆ3 , a ˆ 4 ) =  wj (aL , wj (aM , wj (aU  j) j ) j ) j=1

j=1

j=1

= [1.5811, 3.1464, 4.8580]. Based on the OWA and FQM operators and Definition 3.2, we define a fuzzy ordered weighted quadratic mean (FOWQM) operator as below: M U Definition 3.5 Let a ˆj = [aL j , aj , aj ] (j = 1, 2, . . . , n) be a collection of triangular fuzzy numbers. A FOWQM operator of dimension n is a mapping FOWQM : Ωn →∑Ω, that has an associated vector ω = (ω1 , ω2 , . . . , ωn )T such n that ωj ≥ 0 and j=1 ωj = 1. Furthermore,

 FOWQM(ˆ a1 , a ˆ2 , . . . , a ˆn ) = 

n ∑

 12 ωj a ˆ2σ(j) 

j=1

  12   21   12  n n n ∑ ∑ ∑  2 2 2  =  ωj (aL , wj (aM , wj (aU  , (16) σ(j) ) σ(j) ) σ(j) ) j=1

j=1

j=1

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[ ] M U where aσ(j) = aL σ(j) , aσ(j) , aσ(j) (j = 1, 2, . . . , n), and (σ(1), σ(2), . . . , σ(n)) is a permutation of (1, 2, . . . , n) such that a ˆσ(j−1) ≥ a ˆσ(j) for all j. However, if there is a tie between a ˆi and a ˆj by their average (ˆ ai + a ˆj )/2 in process of aggregation. If k items are tied, then we replace these by k replicas of their average. The weighting vector w = (w1 , w2 , . . . , wn )T can be determined by using some weight determining methods like the normal distribution based method, see Refs [11, 20] for more details. Similarly to the OWA operator, the FOWQM operator has the following properties: M U Theorem 3.6 Let a ˆj = [aL j , aj , aj ] (j = 1, 2, . . . , n) be a collection of triangular fuzzy numbers, the following are valid: (1) Idempotency: If all a ˆj (j = 1, 2, . . . , n) are equal, i.e., a ˆj = a ˆ, for all i, then

FOWQM(ˆ a1 , a ˆ2 , . . . , a ˆn ) = a ˆ. U M ˆ+ = (2) Boundedness: Let a ˆ− = [minj (aL j ), minj (aj ), minj (aj )] and a L M U [maxj (aj ), maxj (aj ), maxj (aj )], then

a ˆ− ≤ FOWQM(ˆ a1 , a ˆ2 , . . . , a ˆn ) ≤ a ˆ+ . M∗ U∗ ∗ (3) Monotonicity: Let a ˆ∗j = [aL j , aj , aj ] (j = 1, 2, . . . , n) be a collection U∗ M∗ L M ∗ and aU of triangular fuzzy numbers, then if aL j ≤ aj for all j ≤ aj , aj ≤ aj j, then

FOWQM(ˆ a1 , a ˆ2 , . . . , a ˆn ) ≤ FOWQM(ˆ a∗1 , a ˆ∗2 , . . . , a ˆ∗n ). ′

′

′

M U (4) Commutativity: Let a ˆ′j = [aL j , aj , aj ] (j = 1, 2, . . . , n) be a collection of triangular fuzzy numbers, then

FOWQM(ˆ a1 , a ˆ2 , . . . , a ˆn ) = FOWQM(ˆ a′1 , a ˆ′2 , . . . , a ˆ′n ), ˆ′n ) is any permutation of (ˆ a1 , a ˆ2 , . . . , a ˆn ). ˆ′2 , . . . , a where (ˆ a′1 , a Especially, if w = ( n1 , n1 , . . . , n1 )T , then the FOWQM operator is reduced M U to the FQM operator; if the triangular fuzzy numbers a ˆj = [aL j , aj , aj ] (j = L U 1, 2, . . . , n) are reduced to the interval numbers a ˜j = [aj , aj ] (j = 1, 2, . . . , n), then the FOWQM operator is reduced to the uncertain ordered weighted quadratic mean (UOWQM) operator:   21 n ∑ UOWQM(˜ a1 , a ˜2 , . . . , a ˜n ) =  ωj a ˜2σ(j)  j=1

  12   12  n n ∑ ∑  2 2  =  ωj (aL , ωj (aU  ,(17) σ(j) ) σ(j) ) j=1

j=1

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U where a ˜σ (j) = [aL σ(j) , aσ(j) ], (σ(1), σ(2), . . . , σ(n)) is a permutation of (1, 2, . . . , n) such that a ˜σ(j−1) ≥ a ˜σ(j) for all j. If there is a tie between a ˜i and a ˜j , then we replace each of a ˜i and a ˜j by their average (˜ ai + a ˜j )/2 in process of aggregation. If k items are tied, then we replace these by k replicas of their average. U If aL i = ai = ai , for all i = 1, 2, . . . , n, then the UOWQM operator is reduced to the ordered weighted quadratic mean (OWQM)operator:

 OWQM(a1 , a2 , . . . , an ) = 

n ∑

 12 ωj b2j  ,

(18)

j=1

where bj is the jth largest of aj (j = 1, 2, . . . , n). The OWQM operator (18) has some special cases: (1) If ω = (1, 0, . . . , 0)T , then OWQM(a1 , a2 , . . . , an ) = max{ai } = b1 .

(19)

(2) If ω = (0, 0, . . . , 1)T , then OWQM(a1 , a2 , . . . , an ) = min{ai } = bn .

(20)

(3) If ωj = 1, wi = 0, i ̸= j, then bn ≤ OWQM(a1 , a2 , . . . , an ) = bj ≤ b1 .

(21)

(4) If ω = ( n1 , n1 , . . . , n1 )T , then ( ∑n

2 j=1 bj

OWQM(a1 , a2 , . . . , an ) =

n

) 12

( ∑n =

= QM(a1 , a2 , . . . , an ).

j=1

a2j

) 12

n (22)

Clearly, the fundamental characteristic of the FWQM operator is that it considers the importance of each given triangular fuzzy number, whereas the fundamental characteristic of the FOWQM operator is the reordering step, and it weights all the ordered positions of the triangular fuzzy numbers instead of weighing the given triangular fuzzy numbers themselves. By combining the advantages of the FWQM and FOWQM operators, in the following, we develop a fuzzy hybrid quadratic mean (FHQM) operator that weights both the given triangular fuzzy numbers and their ordered positions. M U Definition 3.7 Let a ˆj = [aL j , aj , aj ] (j = 1, 2, . . . , n) be a collection of triangular fuzzy numbers. A FHQM operator of dimension n is a mapping FHQM : Ωn → Ω, which has an associated vector ω = (ω1 , ω2 , . . . , ωn )T with ωj ≥ 0 and ∑ n j=1 ωj = 1, such that

 12  n ∑ 2 FHQM(ˆ a1 , a ˆ2 , . . . , a ˆn ) =  ωj a ˆ˙ σ(j)  j=1

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  21   12   21  n n n ∑ ∑ ∑  2 2 2  =  , , ωj (a˙ L ωj (a˙ M ωj (a˙ U  , (23) σ(j) ) σ(j) ) σ(j) ) j=1

j=1

j=1

[ ] M U where a ˆ˙ σ(j) = a˙ L , a ˙ , a ˙ σ(j) σ(j) σ(j) is the jth largest of the weighted triangular T fuzzy numbers a ˆ˙ j (a ˆ˙ j = nwj a ˆj , j = 1, 2, . . . , n), w = 1 , w2 , . . . , wn ) is the ∑(w n weight vector of a ˆj (j = 1, 2, . . . , n) with wj ≥ 0 and j=1 wj = 1, and n is the balancing coefficient. Especially, if w = ( n1 , n1 , . . . , n1 )T , then a ˆ˙ j = a ˆj , j = 1, 2, . . . , n, in this case, the FHQM operator is reduced to the FOWQM operator; if ω = ( n1 , n1 , . . . , n1 )T , then   21 n ∑ 2 ˆn ) =  wj a ˆ˙ σ(j)  FHQM(ˆ a1 , a ˆ2 , . . . , a j=1

  12   21   21  n n n ∑ ∑ ∑  2 2 2  =  , , nwj2 (aL nwj2 (aM nwj2 (aU  (24) σ(j) ) σ(j) ) σ(j) ) j=1

j=1

j=1

which we call the generalized fuzzy weighted quadratic mean (GFWQM) operator. M U Moreover, if the triangular fuzzy numbers a ˆj = [aL j , aj , aj ] (j = 1, 2, . . . , n) U are reduced to the interval numbers a ˜j = [aL j , aj ] (j = 1, 2, . . . , n), then the FHQM operator is reduced to the uncertain hybrid quadratic mean (UHQM) operator:  12  n ∑ 2 UHQM(˜ a1 , a ˜2 , . . . , a ˜n ) =  ωj a ˜˙ σ(j)  j=1

  21   21  n n ∑ ∑  2 2  =  , nwj2 (aL nwj2 (aU  , (25) σ(j) ) σ(j) ) j=1

j=1

where a ˜˙ σ(j) is the jth largest of the weighted interval numbers a ˜˙ j (a ˜˙ j = nwj a ˜j , j = T 1, 2, . . . , n), w = (w , w , . . . , w ) is the weight vector of a ˜ (j = 1, 2, . . . , n) 1 2 n j ∑n with wj ≥ 0 and j=1 wj = 1, and n is the balancing coefficient. Especially, if w = ( n1 , n1 , . . . , n1 )T , then a ˜˙ j = a ˜j , j = 1, 2, . . . , n, in this case, the UHQM operator is reduced to the UOWQM operator. U If aL i = ai = ai , for all i = 1, 2, . . . , n, then the UHQM operator is reduced to the hybrid quadratic mean (HQM) operator:   21 n ∑ (26) HQM(a1 , a2 , . . . , an ) =  ωj a˙ 2σ(j)  , j=1

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where a˙ σ(j) is the jth largest of the weighted interval numbers a˙ j (a˙ j = nwj a ˜j , j = T 1, 2, . . . , n), w = (w , w , . . . , w ) is the weight vector of a (j = 1, 2, . . . , n) n j ∑1n 2 with wj ≥ 0 and j=1 wj = 1, and n is the balancing coefficient. Especially, if w = ( n1 , n1 , . . . , n1 )T , then a˙ j = aj , j = 1, 2, . . . , n, in this case, the HQM operator is reduced to the OWQM operator. Example 3.8 Given a collection of triangular fuzzy numbers: a ˆ1 = [2, 4, 5], a ˆ2 = [1, 3, 4], a ˆ3 = [2, 3, 5], a ˆ4 = [3, 4, 5], and a ˆ5 = [2, 5, 8], and w = (0.20, 0.25, 0.15, 0.25, 0.15)T be the weight vector of a ˆj (j = 1, 2, 3, 4, 5). Then we get the weighted triangular fuzzy numbers: a ˜˙ 1 = [2, 4, 5], a ˜˙ 2 = [1.25, 3.75, 5], a ˜˙ 3 = [1.5, 2.25, 3.75], a ˜˙ 4 = [3.75, 5, 6.25], a ˜˙ 5 = [1.5, 3.75, 6]. By using Eq. (9) (without loss of generality, set δ following matrix:  0.5000 0.5833 0.9545 0.0385 0  0.4167 0.5000 0.8462  P =  0.0455 0.1538 0.5000 0  0.9615 1 1 0.5000 0.5136 0.5846 0.8750 0.1429

= 0.5), we construct the  0.4864 0.4154   0.1250  .  0.8571 0.5000

Summing all elements in each line of matrix P , we have p1 = 2.5628, p2 = 2.1782, p3 = 0.8243, p4 = 4.3187, p5 = 2.6160 and then we rank the triangular fuzzy number a ˆi (i = 1, 2, 3, 4, 5) in descending order in accordance with the values of pi (i = 1, 2, 3, 4, 5): a ˆ˙ σ(1) = a ˆ˙ 4 , a ˆ˙ σ(2) = a ˆ˙ 5 , a ˆ˙ σ(3) = a ˆ˙ 1 , a ˆ˙ σ(4) = a ˆ˙ 2 , a ˆ˙ σ(5) = a ˆ˙ 3 . Suppose that ω = (0.1117, 0.2365, 0.3036, 0.3265, 0.1117)T is the weighting vector of the FHQM operator (derived by the normal distribution based method [11]), then by Eq. (23), we get   21 n ∑ 2 FHQM(ˆ a1 , a ˆ2 , a ˆ3 , a ˆ4 , a ˆ5 ) =  ωj a ˆ˙ σ(j)  j=1

  12   21   21  n n n ∑ ∑  ∑ 2 2 2  =  ωj (a˙ L , ωj (a˙ M , ωj (a˙ U  σ(j) ) σ(j) ) σ(j) ) j=1

j=1

j=1

= [2.0196, 4.0166, 5.4955].

4

Approaches to multiple attribute group decision making with triangular fuzzy information

For a group decision making with triangular fuzzy information, let X={x1 , x2 , . . . , xn } be a discrete set of n alternatives, and G = {G1 , G2 , . . . , Gm } be the set of 10

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T m ∑mattributes, whose weight vector is w = (w1 , w2 , . . . , wm ) with wi ≥ 0 and makers, whose i=1 wi = 1, and let D = {d1 , d2 , . . . , ds } be the set of decision ∑s weight vector is v = (v1 , v2 , . . . , vs )T , where vk ≥ 0 and k=1 v k = 1. Suppose [ ] (k)

(k)

L(k)

M (k)

U (k)

that A(k) = (ˆ aij )m×n is the decision matrix, where a ˆij = aij , aij , aij is an attribute value, which takes the form of triangular fuzzy number, of the alternative xj ∈ X with respect to the attribute Gi ∈ G. Then, we utilize the FWQM and FHQM operators to propose an approach to multiple attribute group decision making with triangular fuzzy information, which involves the following steps: (k)

Step 1. Normalize each attribute value a ˆij in the matrix A(k) into a corre[ ] (k) (k) sponding element in the matrix R(k) = (ˆ rij )m×n (ˆ rij = rij L(k) , rij M (k) , rij U (k) ) using the following formulas: [ ] (k) a ˆij aij L(k) aij M (k) aij U (k) (k) rˆij = ∑n = ∑n , ∑n , ∑n , (k) U (k) M (k) L(k) a ˆ j=1 aij j=1 aij j=1 aij j=1

ij

for benefit attribute Gi , (k)

rˆij = ∑n [

(27)

(k) 1/ˆ aij

(k) aij ) j=1 (1/ˆ

] 1/aij U (k) 1/aij M (k) 1/aij L(k) = ∑n , ∑n , ∑n , L(k) ) M (k) ) U (k) ) j=1 (1/aij j=1 (1/aij j=1 (1/aij for cost attribute Gi ,

(28)

where i = 1, 2, . . . , m, j = 1, 2, . . . , n, k = 1, 2, . . . , s. Step 2. Utilize the FWQM operator: (m ) 12 ∑ (k) (k) (k) (k) (k) 2 rˆj = FWQM(ˆ r1j , rˆ2j , . . . , rˆmj ) = wi (ˆ rij ) i=1

( ) 12 ( m ) 12  ) 12 ( m m ∑ ∑ ∑ M (k) U (k) L(k) wi (ˆ rij )2 wi (ˆ rij )2 (29) = wi (ˆ rij )2 , , i=1

i=1

i=1

to aggregate all the elements in the jth column of R(k) and get the overall (k) attribute value rˆj of the alternative xj corresponding to the decision maker dk . Step 3. Utilize the FHQM operator: ( s ) 21 ∑ (σ(k)) 2 (1) (2) (s) rˆj = FHQM(ˆ r , rˆ , . . . , rˆ ) = ωk (rˆ˙j ) j

j

j

k=1

( ) 21 ( s ) 21 ( s ) 21  s ∑ ∑ ∑ L(σ(k)) 2 M (σ(k)) 2 U (σ(k)) 2  (30) = ωk (r˙j ) , ωk (r˙j , ωk (r˙j ) ) k=1

k=1

k=1

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(k)

to aggregate the overall attribute values rˆj (k = 1, 2, . . . , s) corresponding to the decision maker dk (k = 1, 2, . . . , s) and get the collective overall attribute (σ(k)) L(σ(k)) M (σ(k)) U (σ(k)) value rˆj , where rˆ˙j = [r˙ , r˙ , r˙ ] is the kth largest of the j

j

j

(k) (k) (k) weighted data rˆ˙j (rˆ˙j = svk rˆj , k = 1, 2, . . . , s), ω = (ω1 , ω2 , . . . , ωs )T is ∑s the weighting vector of the FHQM operator, with ωk ≥ 0 and k=1 ωk = 1. Step 4. Compare each rˆj with all rˆi (i = 1, 2, . . . , n) by using Eq. (9), and let pij = p(ˆ ri ≥ rˆj ), and then construct a possibility matrix P = (pij )n×n , where pij ≥ 0, pij + pji = 1, pii = 0.5, i, j ∑ = 1, 2, . . . , n. Summing all elements n in each line of matrix P , we have pi = j=1 pij , i = 1, 2, . . . , n, and then reorder rˆj (j = 1, 2, . . . , n) in descending order in accordance with the values of pj (j = 1, 2, . . . , n). Step 5. Rank all the alternatives xj (j = 1, 2, . . . , n) by the ranking of rˆj (j = 1, 2, . . . , n), and then select the most desirable one. Step 6. End.

5

Illustrative example

In this section, we use a multiple attribute group decision making problem of determining what kind of air-conditioning systems should be installed in a library(adopted from [6, 7, 12, 22]) to illustrate the proposed approach. A city is planning to build a municipal library. One of the problems facing the city development commissioner is to determine what kind of air-conditioning systems should be installed in the library. The contractor offers five feasible alternatives, which might be adapted to the physical structure of the library. The alternatives xj (j = 1, 2, 3, 4, 5) are to be evaluated using triangular fuzzy numbers by the three decision makers dk (k = 1, 2, 3) (whose weight vector is v = (0.4, 0.3, 0.3)T ) under three major impacts: economic, functional, and operational. Two monetary attributes and six nonmonetary attributes (that is, G1 : owning cost ($/ft2 ), G2 : operating cost ($/ft2 ), G3 : performance (∗ ), G4 : noise level (Db), G5 : maintainability (∗ ), G6 : reliability (%), G7 : flex-

Table 1: Triangular fuzzy number decision matrix A(1)

G1 G2 G3 G4 G5 G6 G7 G8

x1 [3.5, 4.0, 4.7] [5.5, 6.0, 6.5] [0.7, 0.8, 0.9] [35, 40, 45] [0.4, 0.45, 0.5] [95, 98, 100] [0.3, 0.35, 0.5] [0.7, 0.74, 0.8]

x2 [1.7, 2.0, 2.3] [4.8, 5.1, 5.5] [0.5, 0.56, 0.6] [70, 73, 75] [0.4, 0.44, 0.6] [70, 73, 75] [0.7, 0.75, 0.8] [0.5, 0.53, 0.6]

x3 [3.5, 3.8, 4.2] [4.5, 5.2, 5.5] [0.5, 0.6, 0.7] [65, 68, 70] [0.7, 0.76, 0.8] [80, 83, 90] [0.8, 0.9, 1.0] [0.6, .68, 0.7]

x4 [3.5, 3.8, 4.5] [4.5, 4.7, 5.0] [0.7, 0.85, 0.9] [40, 42, 45] [0.9, 0.97, 1.0] [90, 93, 95] [0.6, 0.75, 0.8] [0.7, 0.8, 0.9]

x5 [3.3, 3.8, 4.0] [5.5, 5.7, 6.0] [0.6, 0.7, 0.8] [50, 55, 60] [0.5, 0.54, 0.6] [85, 90, 95] [0.4, 0.5, 0.6] [0.8, .85, 0.9]

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ibility (∗ ), G8 : safety (∗ ), where ∗ unit is from 0 − 1 scale, three attributes G1 , G2 , and G4 are cost attributes, and the other five attributes are benefit attributes, suppose that the weight vector of the attributes Gi (i = 1, 2, . . . , 8) is w = (0.05, 0.08, 0.14, 0.12, 0.18, 0.21, 0.05, 0.17)T ) emerged from three impacts is Tables 1-3. Table 2: Triangular fuzzy number decision matrix A(2)

G1 G2 G3 G4 G5 G6 G7 G8

x1 [4.0, 4.3, 4.5] [6.0, 6.3, 6.5] [0.7, 0.8, 0.9] [37, 38, 39] [0.4, 0.5, 0.6] [92, 93, 95] [0.4, 0.45, 0.5] [0.6, 0.7, 0.8]

x2 [2.1, 2.2, 2.4] [5.0, 5.1, 5.2] [0.4, 0.5, 0.6] [70, 73, 75] [0.5, 0.55, 0.6] [70, 75, 80] [0.8, 0.85, 0.9] [0.6, 0.65, 0.7]

x3 [5.0, 5.1, 5.2] [4.5, 4.7, 5.0] [0.5, .55, 0.6] [65, 66, 67] [0.8, 0.85, 0.9] [83, 84, 85] [0.7, 0.73, 0.8] [0.5, 0.6, 0.7]

x4 [4.3, 4.4, 4.5] [5.0, 5.1, 5.3] [0.7, 0.75, 0.8] [40, 42, 45] [0.8, 0.95, 1.0] [90, 91, 92] [0.7, 0.85, 0.9] [0.7, 0.76, 0.8]

x5 [3.0, 3.3, 3.5] [7.0, 7.5, 8.0] [0.7, 0.8, 0.9] [50, 52, 55] [0.4, 0.44, 0.5] [90, 93, 95] [0.4, 0.45, 0.5] [0.7, 0.8, 0.9]

Table 3: Triangular fuzzy number decision matrix A(3)

G1 G2 G3 G4 G5 G6 G7 G8

x1 [4.3, 4.4, 4.6] [6.4, 6.7, 7.0] [0.8, 0.85, 0.9] [36, 38, 40] [0.4, 0.46, 0.5] [93, 94, 95] [0.4, 0.5, 0.6] [0.7, 0.78, 0.8]

x2 [2.2, 2.4, 2.5] [5.0, 5.2, 5.5] [0.5, 0.6, 0.7] [72, 73, 75] [0.4, 0.45, 0.6] [77, 78, 80] [0.8, 0.9, 1.0] [0.5, 0.55, 0.6]

x3 [4.5, 4.8, 5.0] [4.7, 4.8, 4.9] [0.6, 0.7, 0.8] [67, 68, 70] [0.8, 0.95, 1.0] [85, 87, 90] [0.8, 0.86, 0.9] [0.6, 0.68, 0.7]

x4 [4.7, 4.9, 5.0] [5.5, 5.7, 6.0] [0.7, 0.8, 0.9] [45, 48, 50] [0.8, 0.85, 0.9] [90, 94, 95] [0.6, 0.7, 0.8] [0.8, 0.85, 0.9]

x5 [3.1, 3.2, 3.4] [6.0, 6.5, 7.0] [0.7, 0.75, 0.8] [55, 57, 60] [0.5, 0.55, 0.6] [90, 96, 100] [0.5, 0.57, 0.6] [0.8, 0.85, 0.9]

To select the best air-conditioning system, we first utilize the approach based on the FWQM and FHQM operators, the main steps are as follows: (k) Step 1. By using Eqs. (27) and (28), we normalize each attribute value a ˆij in the matrices A(k) (k = 1, 2, 3) into the corresponding element in the matrices R(k) = (ˆ rij )8×5 (k = 1, 2, 3) (Tables 4-6): Step 2. Utilize the FWQM operator (29) to aggregate all elements in the (k) jth column R(K) and get the overall attribute value rˆj : (1)

(1)

(1)

(1)

rˆ1 = [0.1736, 0.2029, 0.2436] , rˆ2 = [0.1473, 0.1751, 0.2167] , rˆ3 = [0.1689, 0.1985, 0.2354] , rˆ4 = [0.2043, 0.2422, 0.2759] , (1)

rˆ5 = [0.1687, 0.1991, 0.2370] , (2)

(2)

rˆ1 = [0.1770, 0.2044, 0.2417] , rˆ2 = [0.1622, 0.1878, 0.2191] , 13

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Table 4: Normalized triangular fuzzy number decision matrix R(1)

G1 G2 G3 G4 G5 G6 G7 G8

x1 [0.12, 0.16, 0.21] [0.15, 0.18, 0.21] [0.18, 0.23, 0.30] [0.22, 0.26, 0.32] [0.11, 0.14, 0.17] [0.21, 0.22, 0.24] [0.08, 0.11, 0.18] [0.18, 0.21, 0.24]

x2 [0.25, 0.32, 0.43] [0.18, 0.21, 0.24] [0.13, 0.16, 0.20] [0.13, 0.14, 0.16] [0.11, 0.14, 0.21] [0.15, 0.17, 0.18] [0.19, 0.23, 0.29] [0.13, 0.15, 0.18]

x3 [0.14, 0.17, 0.21] [0.18, 0.20, 0.25] [0.13, 0.17, 0.23] [0.14, 0.15, 0.17] [0.20, 0.24, 0.28] [0.18, 0.19, 0.21] [0.22, 0.28, 0.36] [0.15, 0.19, 0.21]

x4 [0.13, 0.17, 0.21] [0.20, 0.23, 0.25] [0.18, 0.24, 0.30] [0.22, 0.25, 0.28] [0.26, 0.31, 0.34] [0.20, 0.21, 0.23] [0.16, 0.23, 0.29] [0.18, 0.22, 0.27]

x5 [0.14, 0.17, 0.22] [0.16, 0.19, 0.21] [0.15, 0.20, 0.27] [0.16, 0.19, 0.23] [0.14, 0.17, 0.21] [0.19, 0.21, 0.23] [0.11, 0.15, 0.21] [0.21, 0.24, 0.27]

Table 5: Normalized triangular fuzzy number decision matrix R(2)

G1 G2 G3 G4 G5 G6 G7 G8

x1 [0.15, 0.16, 0.19] [0.17, 0.18, 0.19] [0.18, 0.24, 0.30] [0.25, 0.27, 0.29] [0.11, 0.15, 0.21] [0.21, 0.21, 0.22] [0.11, 0.14, 0.17] [0.15, 0.20, 0.26]

x2 [0.28, 0.32, 0.36] [0.21, 0.22, 0.23] [0.11, 0.15, 0.20] [0.13, 0.14, 0.15] [0.14, 0.17, 0.21] [0.16, 0.17, 0.19] [0.22, 0.26, 0.30] [0.15, 0.19, 0.23]

x3 [0.13, 0.14, 0.15] [0.21, 0.24, 0.26] [0.13, 0.16, 0.20] [0.15, 0.15, 0.16] [0.22, 0.26, 0.31] [0.19, 0.19, 0.20] [0.19, 0.22, 0.27] [0.13, 0.17, 0.23]

x4 [0.15, 0.16, 0.17] [0.20, 0.22, 0.23] [0.18, 0.22, 0.27] [0.22, 0.24, 0.27] [0.22, 0.29, 0.34] [0.20, 0.21, 0.22] [0.19, 0.26, 0.30] [0.18, 0.22, 0.26]

x5 [0.19, 0.21, 0.25] [0.13, 0.15, 0.17] [0.18, 0.24, 0.30] [0.18, 0.20, 0.21] [0.11, 0.13, 0.17] [0.20, 0.21, 0.22] [0.19, 0.14, 0.17] [0.18, 0.23, 0.29]

Table 6: Normalized triangular fuzzy number decision matrix R(3)

G1 G2 G3 G4 G5 G6 G7 G8

x1 [0.15, 0.17, 0.18] [0.16, 0.17, 0.19] [0.20, 0.23, 0.27] [0.26, 0.28, 0.31] [0.11, 0.14, 0.17] [0.21, 0.22, 0.24] [0.08, 0.11, 0.18] [0.18, 0.21, 0.24]

x2 [0.28, 0.30, 0.35] [0.20, 0.22, 0.24] [0.12, 0.16, 0.21] [0.14, 0.15, 0.16] [0.11, 0.14, 0.21] [0.15, 0.17, 0.18] [0.19, 0.23, 0.29] [0.13, 0.15, 0.18]

x3 [0.14, 0.15, 0.17] [0.22, 0.24, 0.25] [0.15, 0.19, 0.24] [0.15, 0.16, 0.17] [0.20, 0.24, 0.28] [0.18, 0.19, 0.21] [0.22, 0.28, 0.36] [0.15, 0.19, 0.21]

(2)

x4 [0.14, 0.15, 0.16] [0.18, 0.20, 0.22] [0.17, 0.22, 0.27] [0.21, 0.22, 0.25] [0.26, 0.31, 0.34] [0.20, 0.21, 0.23] [0.16, 0.23, 0.29] [0.18, 0.22, 0.27]

x5 [0.20, 0.23, 0.25] [0.16, 0.17, 0.20] [0.17, 0.20, 0.24] [0.17, 0.19, 0.20] [0.14, 0.17, 0.21] [0.19, 0.21, 0.23] [0.11, 0.15, 0.21] [0.21, 0.24, 0.27]

(2)

rˆ3 = [0.1744, 0.1974, 0.2314] , rˆ4 = [0.1977, 0.2342, 0.2676] , (2)

rˆ5 = [0.1717, 0.1979, 0.2333] , (3)

(3)

(3)

(3)

rˆ1 = [0.0714, 0.0795, 0.0892] , rˆ2 = [0.0573, 0.0638, 0.0772] , rˆ3 = [0.0699, 0.0831, 0.0959] , rˆ4 = [0.0782, 0.0879, 0.1004] , (3)

rˆ5 = [0.0704, 0.0781, 0.0890] . Step 3. Utilize the FHQM operator (30) (suppose that its weight vector is w = (0.243, 0514, 0.243)T determined by using the normal distribution based (k) method [11], let σ = 0.5) to aggregate the overall attribute value rˆj (k = 1, 2, 3) corresponding to the decision maker dk (k = 1, 2, 3), and get the collective overall 14

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attribute value rˆj : rˆ1 = [0.1568, 0.1818, 0.2160] , rˆ2 = [0.1385, 0.1619, 0.1939] , rˆ3 = [0.1536, 0.1771, 0.2086] , rˆ4 = [0.1791, 0.2119, 0.2417] , rˆ5 = [0.1523, 0.1771, 0.2095] . Step 4. Compare each rˆj with all rˆi (i = 1, 2, 3, 4, 5) by using Eq. (9) (without loss of generality, set δ = 0.5), and let pij = p(ˆ ri ≥ rˆj ), and then construct a possibility matrix:   0.5 0.8558 0.5869 0.0553 0.5882 0.5 0.2209 0 0.2301   0.1442   P =  0.4131 0.7791 0.5 0 0.5031  .   0.9447 1 1 0.5 1 0.4118 0.7699 0.4969 0 0.5 Summing all elements in each line of matrix P , we have p1 = 2.5861, p2 = 1.0952, p3 = 2.1953, p4 = 4.4447, p5 = 2.1786 and then we reorder rˆj (j = 1, 2, 3, 4, 5) in descending order in accordance with the values of pj (j = 1, 2, 3, 4, 5): rˆ4 > rˆ1 > rˆ3 > rˆ5 > rˆ2 . Step 5. Rank all the alternatives xj (j = 1, 2, 3, 4, 5) by the ranking of rˆj (j = 1, 2, 3, 4, 5): x4 ≻ x1 ≻ x3 ≻ x5 ≻ x2 and thus the most desirable alternative is x4 . Table 7: Comparison of the proposed approach with other approaches Xu’s approach [12]

Park et al.’s approach [7]

Proposed approach

FWHM operator FHHM operator x4 ≻ x5 ≻ x3 ≻ x1 ≻ x2

FWCHM operator FHCHM operator x4 ≻ x1 ≻ x3 ≻ x5 ≻ x2

FWQM operator FHQM operator x4 ≻ x1 ≻ x3 ≻ x5 ≻ x2

Solution method Aggregation stage Exploitation stage Ranking of alternatives

From the above analysis, the results obtained by the proposed approach are slightly different to the ones obtained Xu’s [12] approach but the same with Park et al. [7] approach (see Table 7). It perfectly depends on how we look at things, and not on how they are themselves. Therefore, depending on aggregation operators used, the results may lead to different decisions. However, the best alternative is x4 . 15

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6

Conclusions

In this paper, we have extended the traditional quadratic mean to fuzzy environments and introduced the FWQM operator. Based on the FWQM operator and Yager’s OWA operator [17], we have developed the FOWQM operator and the FHQM operator. It has been shown that both the FOWQM and FWQM operators are the special cases of the FHQM operator. It has also been pointed out that if all the input fuzzy data are reduced to the interval or numerical data, then the FHQM operator is reduced to the UHQM operator and the HQM operator, respectively. In these situations, the WQM operator and the OWQM operator are the two special cases of the HQM operator; the UWQM operator and the UOWQM operator are the two special cases of the UHQM operator. Then, based on the FWQM and FHQM operators, we present an approach to multiple attribute group decision making with triangular fuzzy information and illustrate it with a practical example.

Acknowledgement This study was supported by research funds from Dong-A University.

References [1] J. Acz´el and T.L. Saaty, Procedures for synthesizing ratio judgements, J. Math. Psychol. 27 (1983), 93-102. [2] P.S. Bullen, D.S. Mitrinovi and P.M. Vasi, Means and Their Inequalities, Dordrecht, The Netherlands: Reidel, 1988. [3] F. Chiclana, F. Herrera and E. Herrera-Viedma, Integrating multiplicative preference relations in a multipurpose decision-making model based on fuzzy preference relations, Fuzzy Sets Syst. 122 (2001), 277-291. [4] G. Faccinetti, R.G. Ricci and S. Muzzioli, Note on ranking fuzzy triangular numbers, Int. J. Intell. Syst. 13 (1998), 613-622. [5] J.C. Harsanyi, Cardinal welfare, individualistic ethics, and interpersonal comparisons of utility, J. Polit. Econ. 63 (1955), 309-321. [6] J.H. Park and E.J. Park, Generalized fuzzy Bonferroni harmonic mean operators and their applications in group decision making, J. Appl. Math., 2013 (2013), Article ID 604029, 14 pages. [7] J.H. Park, S.M. Yu and Y.C. Kwun, An approach based on fuzzy contraharmonic mean operators to group decision making, J. Comput. Anal. Appl. 17 (2014), 389-406. [8] V. Torra, The weighted OWA operators, Int. J. Intell. Syst. 12 (1997), 153-166. 16

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[9] P.J.M. Van Laarhoven and W. Pedrycz, A fuzzy extension of Saaty’s priority theory, Fuzzy Sets Syst. 11 (1983), 199-227. [10] Z.S. Xu, Uncertain Multiple Attribute Decision Making: Methods and Applications, Beijing, China: Tsinghua Univ. Press, 2004. [11] Z.S. Xu, An overview of methods for determining OWA weights, Int. J. Intell. Syst. 20 (2005) 843-865. [12] Z.S. Xu, Fuzzy harmonic mean operators, Int. J. Intell. Syst. 24 (2009) 152-172. [13] Z.S. Xu and Q.L. Da, The ordered weighted geometric averaging operators, Int. J. Intell. Syst. 17 (2002), 709-716. [14] Z.S. Xu and Q.L. Da, The uncertain OWA operators, Int. J. Intell. Syst. 17 (2002), 569-575. [15] Z.S. Xu and Q.L. Da, An overview of operators aggregating information, Int. J. Intell. Syst. 18 (2003), 953-969. [16] Z.S. Xu and R.R. Yager, Power-geometric operators and their use in group decision making, IEEE Trans. Fuzzy Syst. 18 (2010), 94-105. [17] R.R. Yager, On ordered weighted averaging aggregation operators in multi-criteria decision making, IEEE Trans. Syst. Man Cybern. 18 (1988), 183-190. [18] R.R. Yager, The power average operator, IEEE Trans. Syst. Man Cybern. A. Syst. Humans 31 (2001), 724-731. [19] R.R. Yager, Generalized OWA aggregation operator, Fuzzy Optim. Decision Making 3 (2004), 93-107. [20] R.R. Yager, Centered OWA operators, Soft Computing 11 (2007), 631639. [21] R.R. Yager and D.P. Filev, Induced ordered weighted averaging operators, IEEE Trans. Syst. Man Cybern. 29 (1999), 141-150. [22] K. Yoon, The propagation of errors in multiple-attribute decision analysis: A practical approach, J. Oper. Res. Soc. 40 (1989) 681-686.

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Sensitivity Analysis for General Nonlinear Nonconvex Set-Valued Variational Inequalities in Banach Spaces Jong Kyu Kim Department of Mathematics Education, Kyungnam University, Changwon, Gyeongnam 51767, Korea e-mail: [email protected] Abstract. In this paper, we show that the parametric general nonlinear nonconvex set-valued variational inequality is equivalent to the parametric general Wiener-Hopf equations. We used the equivalence formulation to study the sensitivity analysis for general nonlinear nonconvex set-valued variational inequalities without assuming the differentiability of the given data. Keywords: Sensitivity analysis, general nonlinear nonconvex variational set-valued inequalities, fixed point, general Wiener-Hopf equations, relaxed ϕ-accretive mapping, locally Lipschitz continuous mappings, uniformly r-prox regular sets, uniformly smooth Banach spaces. 2010 AMS Subject Classification: 49J40, 47H06.

1

Introduction

Variational inequality theory has become a very effective and powerful tools for studying a wide range of problems arising in pure and applied sciences which include the work on differential equations, mechanics, control problems in elasticity, general equilibrium problems in economics and transportation, obstacle, moving, and free boundary problems (see [1,3,5,8-10]). Sensitivity analysis for the solutions of variational inequalities with single-valued mappings has been studied by many authors by quite different techniques. By using the projection methods, Anastassiou et al. [2], Agarwal et al. [4], Dafermos [6], Faraj and Salahuddin [7], Kim et al. [11], Kyparisis [12], Khan and Salahuddin [13], Liu [14], Lee and Salahuddin [15], Noor and Noor [16], Qiu and Magnanti [18], Salahuddin [19,20], Yen and Lee [23], and Verma [24] studied the sensitivity analysis for the solutions of some variational inequalities with single-valued mappings in finite dimensional spaces, Hilbert spaces and Banach spaces. Noor and Noor [16] introduced and considered a new class of variational inequalities on the uniformly prox regular sets which are called the general nonlinear nonconvex variational inequalities. We note that the uniformly prox regular sets are nonconvex and include the convex sets as a special cases (see [5,17]). In this paper, we developed the general framework of sensitivity analysis for the general nonlinear nonconvex set-valued variational inequalities. For this, we established the equivalence between the parametric general nonlinear nonconvex set-valued variational inequalities and parametric general Wiener-Hopf equations by using the 0 This

work was supported by the Kyungnam University Research Fund, 2015.

1

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projection techniques (see [11,21]). This fixed point formulation is obtained by a suitable and approximate rearrangement of the parametric general Wiener-Hopf equations. We would like to point out that the Wiener-Hopf equations technique is quite general unified flexible and provides us with new approach to study the sensitivity analysis of general nonlinear nonconvex set-valued variational inequalities and related optimization problems. We used this equivalence to develop the sensitivity analysis for general nonlinear nonconvex set-valued variational inequalities without assuming the differentiability of the given data.

2

Preliminaries

Let X be a real Banach space with dual space X ∗ , h·, ·i be the dual pairing between X and X ∗ , and CB(X) denotes the family of all nonempty closed bounded subsets ∗ of X. The generalized duality mapping Jq : X → 2X is defined by Jq (u) = {f ∗ ∈ X ∗ : hu, f ∗ i = kukq , kf ∗ k = kukq−1 }, ∀u ∈ X, where q > 1 is a constant. In particular J2 is a usual normalized duality mapping. It is known that in general Jq (u) = kukq−2 J2 (u) for all u 6= 0 and Jq is single-valued if X ∗ is strictly convex. In the sequel, we always assume that X is a real Banach space such that Jq is a single-valued. If X is a Hilbert space then Jq becomes the identity mapping on . The modulus of smoothness of X is the function ρX : [0, ∞) → [0, ∞) is defined by   1 (ku + vk + ku − vk) − 1 : kuk ≤ 1, kvk ≤ t . ρX (t) = sup 2 A Banach space X is called uniformly smooth if lim

t→0

ρX (t) = 0. t

X is called q-uniformly smooth if there exists a constant c > 0 such that ρX (t) < ctq , q > 1. It is well known that the Hilbert spaces, Lp ( or lp ) spaces, 1 < p < ∞ and the Sobolev spaces W m,p , 1 < p < ∞ are all q-uniformly smooth. Note that Jq is single-valued if X is uniformly smooth. Concerned with the characteristic inequalities in q-uniformly smooth Banach spaces. Xu [22] proved the following results. Lemma 2.1. [22] The real Banach space X is q-uniformly smooth if and only if there exists a constant cq > 0 such that for all u, v ∈ X, ku + vkq ≤ kukq + qhv, Jq (u)i + cq kvkq . Let K be a nonempty closed subsets of X and we denote dK (·) or d(·, K) the usual distance function to the subset K, that is, dK (u) = inf ku − vk. v∈K

The set of all projections of u onto K is given by  PK (u) = v ∈ K : dK (u) = ku − vk . 2

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Definition 2.2. The proximal normal cone of K at a point u ∈ X is given by P NK (u) = {ζ ∈ X : u ∈ PK (u + αζ) for some α > 0}.

P Lemma 2.3. [5] Let K be a nonempty closed subset of X. Then ζ ∈ NK (u) if and only if there exists a constant α = α(ζ, u) > 0 such that

hζ, v − ui ≤ αkv − uk2 , ∀v ∈ K.

Lemma 2.4. [5] Let K be a nonempty closed and convex subset in X. Then ζ ∈ P NK (u) if and only if hζ, v − ui ≤ 0, ∀v ∈ K.

Definition 2.5. Let f : X → R be a locally Lipschitz continuous mapping with constant τ near a given point u ∈ X, i.e., for some  > 0, | f (v) − f (w) |≤ τ kv − wk, ∀ v, w ∈ B(u; ), where B(u; ) denotes the open ball of radius r > 0 and centered at u. The generalized directional derivative of f at u in the direction z, denoted by f o (u; z) is defined as follows: f (v + tz) − f (v) , f o (u; z) = lim sup t v→ut↓0 where v is a vector in X and t is a positive scalar. Definition 2.6. The tangent cone TK (u) to K at a point u ∈ K is defined as follows: TK (u) = {v ∈ X : doK (u; v) = 0}. The normal cone of K at u by polarity with TK (u) is defined as follows: NK (u) = {ζ : hζ, vi ≤ 0, ∀v ∈ TK (u)}. C The Clarke normal cone NK (u) is given by

 P C (u) , NK (u) = co NK where co(S) is the closure of the convex hull of S. P C It is clear that NK (u) ⊆ NK (u). The converse is not true in general. Note that P is always closed and convex, where as NK (u) is always convex but may not be closed (see [5,17]). C NK (u)

Definition 2.7. [17] For any r ∈ (0, +∞], a subset Kr of X is said to be normalized uniformly r-prox regular (or uniformly r-prox regular) if every nonzero proximal norP mal to Kr can be realized by an r-ball, that is, for all u ∈ Kr and all 0 6= ζ ∈ NK (u) r with kζk = 1, 1 hζ, v − ui ≤ kv − uk2 , ∀ v ∈ Kr . 2r

3

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Proposition 2.8. [17] Let r > 0 and Kr be a nonempty closed and uniformly r-prox regular subset of X. Set  U(r) = u ∈ X : 0 ≤ dKr (u) < r . Then we have the following statements: (i) For all u ∈ U(r), we have PKr (u) 6= ∅; (ii) For all r0 ∈ (0, r), PKr is Lipschitz continuous with constant δ =

r r−r 0

on U(r0 );

(iii) The proximal normal cone is closed as a set-valued mapping. ∗

Assume that T : X → 2X is a set-valued mapping and h : X → X is a nonlinear single-valued mapping. For any constant ρ > 0, we consider the problem of finding u ∈ X, x ∈ T (u) such that h(u) ∈ Kr and hρx + h(u) − u, v − h(u)i +

1 kv − h(u)k2 ≥ 0, ∀v ∈ Kr . 2r

(2.1)

The equation (2.1) is called a general nonlinear nonconvex set-valued variational inequality. Now we consider the problem of solving general Wiener-Hopf equations. To be more precise, let QKr = I − h−1 PKr where PKr is the projection operator, h−1 is the inverse of nonlinear operator h and I is an identity operator. For given nonlinear operators, z, u ∈ X, x ∈ T (u) such that T PKr z + ρ−1 QKr z = 0

(2.2)

is a called general Wiener-Hopf equation. Lemma 2.9. [17] u ∈ X, x ∈ T (u), h(u) ∈ Kr is a solution of (2.1) if and only if u ∈ X, x ∈ T (u), h(u) ∈ Kr satisfies the relation h(u) = PKr [u − ρx],

(2.3)

where PKr is the projection of X onto the uniformly r-prox regular set Kr . Lemma 2.9 implies that the general nonlinear nonconvex set-valued variational inequality (2.1) is equivalent to the fixed point problem (2.3). Now, we consider the parametric version of equations (2.1) and (2.2). To formulate the problem, let Γ be an open subset of X in which parameter λ takes values. Let xλ (u) ∈ Tλ (u) be a given operator defined on X × Γ and takes values in X × X.From now, we denote xλ (u) ∈ Tλ (u) unless otherwise specified. The parametric general nonlinear nonconvex set-valued variational inequality problem is to find (u, λ) ∈ X × Γ, xλ (u) ∈ Tλ (u) such that hρxλ (u) + hλ (u) − u, v − hλ (u)i ≥ 0, ∀v ∈ Kr .

(2.4)

We also assume that for some λ ∈ B, problem (2.4) has a unique solution u. Related to parametric general nonlinear nonconvex set-valued variational inequality problem (2.4), we consider the parametric general Wiener-Hopf equation. We consider the problem of finding (z, λ) ∈ X × Γ, xλ (u) ∈ Tλ (u) such that Tλ PKr z + ρ−1 QKr z = 0,

(2.5)

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where ρ > 0 is a constant and QKr z is defined on the set (z, λ) with λ ∈ Γ and takes values in B. Equation (2.5) is called a parametric general Wiener-Hopf equation. Lemma 2.10. Let X be a real Banach space. Than the following two statements are equivalent: (i) An element u ∈ X, xλ (u) ∈ Tλ (u) is a solution of (2.4), (ii) The mapping Fλ (u) = u − hλ (u) + PKr [u − ρxλ (u)] has a fixed point. One can established the equivalence relation between inequality (2.4) and equation (2.5) by using the projection techniques. Lemma 2.11. Parametric general nonlinear nonconvex set-valued variational inequality (2.4) has a solution (u, λ) ∈ X × Γ, xλ (u) ∈ Tλ (u) if and only if parametric general Wiener-Hopf equation (2.5) has a solution (z, λ) ∈ X × Γ, xλ (u) ∈ Tλ (u), where hλ (u) = PKr z (2.6) and z = u − ρxλ (u).

(2.7)

From Lemma 2.11, we know that Parametric general nonlinear nonconvex setvalued variational inequality (2.4) and parametric general Wiener-Hopf equation (2.5) are equivalent. We used these equivalence to study the sensitivity analysis of general nonlinear nonconvex set-valued variational inequalities. We assume that for some λ ∈ Γ, problem (2.5) has a solution z and B is a closure of a ball in X centered at z. We want to investigate those condition under which for each λ in a neighbourhood of λ, then (2.5) has a unique solution z(λ) near z and the function z(λ) is (Lipschitz) continuous and differentiable. ∗

Definition 2.12. Let T : X × Γ → 2X be a set-valued mapping. Then the operator Tλ (·) is said to be locally relaxed ϕ-accretive if there exists a constant ϕ > 0 such that hxλ (u) − xλ (v), jq (u − v)i ≥ −ϕku − vkq , ∀u, v ∈ X, λ ∈ Γ, and locally D-Lipschitz continuous if there exists a constant β > 0 such that kxλ (u) − xλ (v)k ≤ D(Tλ (u), Tλ (v)) ≤ βku − vk, S where D : 2 × 2 → (−∞, ∞) {+∞} is the Hausdorff metric i.e.,   ∗ D(A, B) = sup inf ku − vk, sup inf ku − vk , ∀A, B ∈ 2X . X∗

X∗

u∈A v∈B

u∈B v∈A

Definition 2.13. A single-valued mapping h : X × Ω → X is said to be locally Lipschitz continuous if there exists a constant γ > 0 such that khλ (u) − hλ (v)k ≤ γku − vk, ∀u, v ∈ X, and locally strongly accretive if there exists a constant ξ > 0 such that hhλ (u) − hλ (v), jq (u − v)i ≥ ξku − vkq , ∀u, v ∈ X, λ ∈ Γ.

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3

Main Results

In this section, we derive the main results of this paper. We consider the case when the solutions of the parametric general Wiener-Hopf equation (2.5) lies in the interior of B. We consider the map: for all (z, λ) ∈ X × Γ, xλ (u) ∈ Tλ (u), Fλ (z) = PKr z − ρxλ (u) = u − ρxλ (u),

(3.1)

hλ (u) = PKr z.

(3.2)

where We have to show that the map Fλ (z) has a fixed point, which is a solution of parametric general Wiener-Hopf equation (2.5). First of all we prove the map Fλ (z) defined by (3.1) is contractive with respect to z uniformly in λ ∈ Γ. Lemma 3.1. Let PKr be a locally Lipschitz continuous operator with constant δ = r r−r 0 . Let h : X × Γ → X be a locally Lipschitz continuous with constant γ > 0 and locally strongly accretive mapping with respect to the constant ξ > 0. Let T : ∗ Γ × X → 2X be a locally D-Lipschitz continuous with respect to the constant β > 0 and locally relaxed ϕ-accretive mapping with respect to the constant ϕ > 0. Then for all z1 , z2 ∈ X and λ ∈ Γ, we have kFλ (z1 ) − Fλ (z2 )k ≤ θkz1 − z2 k, where θ=

δ

p q

p 1 + qρϕ + cq ρq β q , κ = q 1 − qξ + cq γ q 1−κ

for p q

ρq cq β q + ρqϕ + 1
0 and locally strongly accretive mapping with constant ξ > 0, we have ku − v − (hλ (u) − hλ (v))kq

ku − vkq − qhhλ (u) − hλ (v), jq (u − v)i +cq khλ (u) − hλ (v)kq ≤ ku − vkq − qξku − vkq + cq γ q ku − vkq ≤ (1 − qξ + cq γ q )ku − vkq .

≤

It implies that ku − v − (hλ (u) − hλ (v))k ≤

p q 1 − qξ + cq γ q ku − vk.

(3.11)

From (3.10) and (3.11), we have ku − vk ≤ κku − vk + δkz1 − z2 k, where κ =

p q

1 − qξ + cq γ q . From which we have ku − vk ≤

δ kz1 − z2 k. 1−κ

(3.12)

Combining (3.9),(3.12) and (3.3), we have p q

kFλ (z1 ) − Fλ (z2 )k

1 + qρϕ + cq ρq β q kz1 − z2 k (3.13) 1−κ (1 − α)kz1 − z2 k + αθkz1 − z2 k.

≤ (1 − α)kz1 − z2 k + αδ =

It follows from (3.4) that θ < 1. Hence the mapping Fλ (z) defined by (3.1) is contractive and has a fixed point z(λ) which is the solution of parametric general Wiener-Hopf equation (2.5). Remark 3.2. From Lemma 3.1, we see that the map Fλ (z) defined by (2.4) has a unique fixed point z(λ), that is, z(λ) = Fλ (z). Also by assumptions, the function z for λ = λ is a solutions of parametric general Wiener-Hopf equation (2.5). Again by Lemma 3.1, we know that z for λ = λ is a fixed point of Fλ (z) and it is also a fixed point of Fλ (z). Consequently, we conclude that z(λ) = z = Fλ (z(λ)). Using Lemma 3.1, we can prove the continuity of the solution z(λ) of parametric general Wiener-Hopf equation (2.5). However for the sake of completeness and to convey the idea of the technique involved, we give the proof. Lemma 3.3. Assume that the operator Tλ (·) is locally D-Lipschitz continuous with respect to the parameter λ and hλ (·) is a locally Lipschitz continuous mapping. If Tλ (·) is a locally Lipschitz continuous mapping and the mapps λ → PKr λ z, λ → hλ (u), λ → Tλ (u) are continuous (or Lipschitz continuous), then the function z(λ) satisfying (3.3) is (Lipschitz) continuous at λ = λ. 7

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Proof. For all λ ∈ Γ invoking Lemma 3.1 and the triangle inequality, we have kz(λ) − z(λ)k

≤

kFλ (z(λ)) − Fλ (z(λ))k + kFλ (z(λ)) − Fλ (z(λ))k

(3.14)

≤ θkz(λ) − z(λ)k + kFλ (z(λ)) − Fλ (z(λ))k. From (3.1) and the fact that the operator Tλ (·) is locally D-Lipschitz continuous with respect to the parameter λ, we have kFλ (z(λ)) − Fλ (z(λ))k = ku(λ) − u(λ) − ρ(Tλ (u(λ)) − Tλ (u(λ)))k

(3.15)

≤ ρβkλ − λk. Combining (3.14) and (3.15), we obtain kz(λ) − z(λ)k ≤

ρβ kλ − λk, ∀λ, λ ∈ Γ. 1−θ

This completes the proof. Now, we are in a position to state and prove the main result of this paper. Theorem 3.4. Let u be a solution of parametric general nonlinear nonconvex setvalued variational inequality (2.4) and z be a solution of parametric general WienerHopf equation (2.5) for λ = λ. Let hλ (u) be a locally strongly accretive and locally Lipschitz continuous mapping. Let Tλ (u) be a locally D-Lipschitz continuous and locally relaxed ϕ-accretive mapping with respect to ϕ > 0 for all u ∈ B. If the mapps λ → PKr , λ → hλ (u), λ → Tλ (u) are Lipschitz (continuous) at λ = λ, then there exists a neighbourhood M of Γ of λ such that for λ ∈ M, parametric general Wiener-Hopf equation (2.5) has a unique solution z(λ) in the interior of B, z(λ) = z and z(λ) is (Lipschitz) continuous at λ = λ. Poof. The proof follows from Lemma 3.1, 3.3 and Remark 3.2.

References [1] J.P. Aubin and I. Ekeland, Applied Nonlinear Analysis, Wiley, New York, 1984. [2] G.A. Anastassiou, Salahuddin and M.K. Ahmad, Sensitivity analysis for generalized set valued variational inclusions, J. Concrete Appl. Math., 11(2013), 292-302. [3] C. Baiocchia and A. Capelo, Variational and Quasi Variational Inequalities, Wiley London, 1984. [4] R.P. Agarwal, Y.J. Cho and N.J. Huang, Sensitivity analysis for strongly nonlinear quasi variational inclusions, Appl. Math. Lett., 13(2000), 19-24. [5] F.H. Clarke, Y.S. Ledyaw, R.J. Stern and P.R. Wolenski, Nonsmooth analysis and control theory, Springer-Verlag, New York, 1998. [6] S. Dafermos, Sensitivity analysis in variational inequalities, Math. Oper. Res., 13(1988), 421-434. [7] A. Farajzadeh and Salahuddin, Sensitivity analysis for nonlinear set-valued variational equations in Banach framework, Journal of Function Spaces and Appl., Volume 2013, Article ID: 258543(2013), 6-pages. 8

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[8] J.K. Kim, D.S. Kim, A new system of generalized nonlinear mixed variational inequalities in Hilbert spaces, J. Convex Anal. 11(2004), 235243. [9] J.K. Kim, Y.P. Fang and N.J. Huang, An existence result for a system of inclusion problems with applications, Appl. Math. Lett., 21(2008), 1209-1214. [10] J.K. Kim and T.M. Tuyen, Regularization proximal point algorithm for finding a common fixed point of a finite family of nonexpansive mappings in Banach spaces, Fixed Point Theory Appl. 2011, 52(2011), 10-pages. [11] J.K. Kim, H.Y. Lan and Y.J. Cho, Solution sensitivity of generalized nonlinear parametric (A, η, m)-proximal operator system of equations in Hilbert spaces, Jour. Inequalities and Appl., 2014:362 (2014), 17-pages. [12] J. Kyparisis, Sensitivity analysis frame work for variational inequalities, Math. Prog., 38(1987), 203-213. [13] M.F. Khan and Salahuddin, Sensitivity analysis for completely generalized nonlinear variational inclusions, East Asian Math. Jour., 25(2009), 45-53. [14] J. Liu, Sensitivity analysis in nonlinear programs and variational inequalities via continuous selection, SIAM J. Control Optim., 33(1995), 1040-1068. [15] B.S. Lee and Salahuddin, Sensitivity analysis for generalized nonlinear quasivariational inclusions, Nonlinear Anal. Forum, 8(2004), 223-232. [16] M.A. Noor and K.I. Noor, Sensitivity analysis of general nonconvex variational inequalities, Jour. Inequalities and Appl., 2013:302 (2013), [17] R.A. Poliquin, R.T. Rockafellar and L. Thibault, Local differentiability of distance functions, Trans. Amer. Math. Soc., 352(2000), 5231-5249. [18] Y. Qiu and T.L. Magnanti, Sensitivity analysis for variational inequalities defined on polyhedral sets, Math. Oper Res., 14(1989), 410-432. [19] Salahuddin, Some aspects of variational inequalities, Ph. D. Thesis, AMU, Aligarh, India, 2000. [20] Salahuddin, Parametric generalized set valued variational inclusions and resolvent equations, J. Math. Anal. Appl., 298(2004), 146-156. [21] P. Shi, Equivalence of Wiener-Hopf equations with variational inequalities, Proc. Amer. Math. Soc., 111(1991), 339-346. [22] H.K. Hu, Inequalities in Banach spaces with applications, Nonlinear Anal. TMA., 16(12)(1991), 1127-1138. [23] N.D. Yen and G.M. Lee, Solution sensitivity of a class of variational inequalities, J. Math. Anal. Appl., 215(1997), 46-55. [24] R.U. Verma, Sensitivity analysis for generalized strongly monotone variational inclusions based on (A, η)-resolvent operator technique, Appl. Math. Lett., 19(2006), 1409-1413.

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Common Fixed Point Theorems for Non-compatible Self-mappings in b-Fuzzy Metric Spaces Jong Kyu Kim1 , Shaban Sedghi2 , Nabi Shobe3 and Hassan Sadati4 1 Department

of Mathematics Education, Kyungnam University, Changwon, Gyeongnam 51767, Korea e-mail: [email protected]

2,4 Department

of Mathematics, Qaemshahr Branch, Islamic Azad University, Qaemshahr, Iran e-mail: sedghi [email protected] e-mail: sadati [email protected]

3 Department

of Mathematics, Babol BranchIs, Islamic Azad University, Babol, Iran e-mail: nabi [email protected]

Abstract. By using R-weak commutativity of type (Ag ) and non-compatible conditions of selfmapping pairs in a b-fuzzy metric space, without the conditions for the completeness of space and the continuity of mappings, we establish some new common fixed point theorems for two self-mappings. An example is provided to support our new result. Keywords: b-fuzzy metric space, common fixed point theorem, R-weakly commuting mappings of type (Ag ), non-compatible mapping pairs. 2010 AMS Subject Classification: 47H10, 54H25.

1

Introduction and Preliminaries

The concept of fuzzy sets was introduced initially by Zadeh [17] in 1965. Since then, to use this concept in topology and analysis, many authors have expansively developed the theory of fuzzy sets and applications. George and Veeramani [5], Kramosil and Michalek [7] have introduced the concept of fuzzy topological spaces induced by fuzzy metric which have very important applications in quantum particle physics, particularly in connections with both string and E-infinity theory which were given and studied by El Naschie [1-4]. Many authors [6,9,10,13-15] have proved fixed point theorems in fuzzy (probabilistic) metric spaces. Definition 1.1. A binary operation ∗ : [0, 1] × [0, 1] → [0, 1] is a continuous t-norm if it satisfies the following conditions: (1) ∗ is associative and commutative, (2) ∗ is continuous, (3) a ∗ 1 = a, for all a ∈ [0, 1], 0 Corresponding

author: Jong Kyu Kim([email protected])

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(4) a ∗ b ≤ c ∗ d, whenever a ≤ c and b ≤ d, for each a, b, c, d ∈ [0, 1]. Two typical examples of a continuous t-norm are a ∗ b = ab and a ∗ b = min(a, b). Definition 1.2. [11] A 3-tuple (X, M, ∗) is called a fuzzy metric space if X is an arbitrary (non-empty) set, ∗ is a continuous t-norm and M is a fuzzy set on X 2 × (0, ∞), satisfying the following conditions for each x, y, z ∈ X and t, s > 0, (1) M (x, y, t) > 0, (2) M (x, y, t) = 1 if and only if x = y, (3) M (x, y, t) = M (y, x, t), (4) M (x, y, t) ∗ M (y, z, s) ≤ M (x, z, t + s), (5) M (x, y, .) : (0, ∞) → [0, 1] is continuous. Definition 1.3. [11] A 3-tuple (X, M, ∗) is called a b-fuzzy metric space for b ≥ 1 if X is an arbitrary nonempty set, ∗ is a continuous t-norm and M is a fuzzy set on X 2 × (0, ∞), satisfying the following conditions for each x, y, z ∈ X and t, s > 0, (1) M (x, y, t) > 0, (2) M (x, y, t) = 1 if and only if x = y, (3) M (x, y, t) = M (y, x, t), (4) M (x, y, bt ) ∗ M (y, z, sb ) ≤ M (x, z, t + s), (5) M (x, y, .) : (0, ∞) → [0, 1] is continuous. It should be noted that, the class of b-fuzzy metric spaces is effectively larger than that of fuzzy metric spaces, since a b-fuzzy metric is a fuzzy metric when b = 1. We present an example shows that a b-fuzzy metric on X need not be a fuzzy metric on X. −|x−y|p

Example 1.4. Let M (x, y, t) = e t , where p > 1 is a real number. We show that M is a b-fuzzy metric with b = 2p−1 . In fact, obviously conditions (1),(2),(3) and (5) of definition 1.3 are satisfied. Let f (x) = xp (x > 0). Then we know that it is a convex function, for 1 < p < ∞. So, we have  p a+c 1 ≤ (ap + cp ) , 2 2 p

it implies that (a + c) ≤ 2p−1 (ap + cp ). Therefore, we have |x − y|p t+s

|x − z|p |z − y|p + 2p−1 t+s t+s p |x − z| |z − y|p ≤ 2p−1 + 2p−1 t s |x − z|p |z − y|p = + . t/2p−1 s/2p−1 ≤ 2p−1

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Thus, for each x, y, z ∈ X we obtain M (x, y, t + s)

= e

−|x−y|p t+s

≥ M (x, z,

t 2p−1

) ∗ M (z, y,

s 2p−1

),

where a ∗ c = ac for all a, c ∈ [0, 1]. So condition (4) of definition 1.3 is hold and M is a b-fuzzy metric. It should be noted that in preceding example, for p = 2 it is easy to see that (X, M, ∗) is not a fuzzy metric space. −d(x,y)

t , where d is a b-metric Example 1.5. Let M (x, y, t) = e t or M (x, y, t) = t+d(x,y) on X and a ∗ c = ac for all a, c ∈ [0, 1]. Then it is easy to show that M is a b-fuzzy metric. In fact, obviously conditions (1),(2),(3) and (5) of definition 1.3 are satisfied. Since d is a b-metric, for each x, y, z ∈ X we have   d(x, y) ≤ b d(x, z) + d(z, y) .

Therefore, we obtain M (x, y, t + s)

= e

−d(x,y) t+s d(x,z)+d(z,y)

t+s ≥ e−b    d(x,z) d(z,y) −b t+s −b t+s = e e    −d(x,z) −d(z,y) t/b s/b ≥ e e

t s = M (x, z, ) ∗ M (z, y, ). b b So condition (4) of definition 1.3 is hold and M is a b-fuzzy metric. Similarly, we can t is also a b-fuzzy metric. show that M (x, y, t) = t+d(x,y) Next, we need the following definitions and propositions in b-metric spaces for our main theorems. Definition 1.6. Let f : R → R be a function. Then f is called b-nondecreasing, if x > by implies that f (x) ≥ f (y) for each x, y ∈ R. Lemma 1.7. [11] Let (X, M, ∗) be a b-fuzzy metric space. Then M (x, y, t) is bnondecreasing with respect to t, for all x, y in X. Also, M (x, y, bn t) ≥ M (x, y, t), ∀n ∈ N. Let (X, M, ∗) be a b-fuzzy metric space. For t > 0, the open ball B(x, r, t) with center x ∈ X and radius 0 < r < 1 is defined by  B(x, r, t) = y ∈ X : M (x, y, t) > 1 − r . We recall the notions of convergence and completeness in a b−fuzzy metric space. Let (X, M, ∗) be a b-fuzzy metric space. Let τ be the set of all A ⊂ X with x ∈ A if and only if there exists t > 0 and 0 < r < 1 such that B(x, r, t) ⊂ A. Then 3

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τ is a topology on X (induced by the b-fuzzy metric M ). A sequence {xn } in X converges to x if and only if M (xn , x, t) → 1 as n → ∞, for each t > 0. It is called a Cauchy sequence if for each 0 < ε < 1 and t > 0, there exists n0 ∈ N such that M (xn , xm , t) > 1 − ε, for each n, m ≥ n0 . The b-fuzzy metric space (X, M, ∗) is said to be complete if every Cauchy sequence is convergent. A subset A of X is said to be F-bounded if there exists t > 0 and 0 < r < 1 such that M (x, y, t) > 1 − r, for all x, y ∈ A. Lemma 1.8. [11] Let (X, M, ∗) be a b-fuzzy metric space. Then the following assertions hold: (i) If sequence {xn } ⊂ X converges to x, then x is unique, (ii) The convergent sequence {xn } ⊂ X is Cauchy. We have the following propositions in a b-fuzzy metric space. Proposition 1.9. [11] Let (X, M, ∗) be a b-fuzzy metric space and suppose that {xn } and {yn } are convergent to x, y respectively. Then we have M (x, y,

t ) ≤ lim supn→∞ M (xn , yn , t) ≤ M (x, y, b2 t) b2

M (x, y,

t ) ≤ lim inf n→∞ M (xn , yn , t) ≤ M (x, y, b2 t). b2

and

Proposition 1.10. [12] Let (X, M, ∗) be a b-fuzzy metric space and suppose that {xn } is convergent to x. Then, for all y ∈ X we have t M (x, y, ) ≤ lim supn→∞ M (xn , y, t) ≤ M (x, y, bt) b and

t M (x, y, ) ≤ lim inf n→∞ M (xn , y, t) ≤ M (x, y, bt). b

Lemma 1.11. A b-fuzzy metric is not continuous in general. Throughout, in this paper we assume that limt→∞ M (x, y, t) = 1. Lemma 1.12. Let (X, M, ∗) be a b-fuzzy metric space and suppose that M (x, y, kt) ≥ M (x, y, t), for all x, y ∈ X, 0 < k < 1 and t > 0. Then x = y. Proof. Since, M (x, y, kt) ≥ M (x, y, t), it follows that t t M (x, y, t) ≥ M (x, y, ) ≥ · · · ≥ M (x, y, n ). k k Hence, we can get M (x, y, t) ≥ limn→∞ M (x, y, ktn ) = 1, therefore, x = y. In 2010, Vats et al. [16] introduced the concept of weakly compatible. Also, in 2010, Manro et al. [8] introduced the concepts of weakly commuting, R-weakly commuting mappings, and R-weakly commuting mappings of type (P ), (Af ), and (Ag ) in a G-metric space. We will introduce these concepts in a b-fuzzy metric space.

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Definition 1.13. The self-mappings f and g of a b-fuzzy metric space (X, M, ∗) are said to be compatible if lim M (f gxn , gf xn , t) = 1 n→∞

and lim M (gf xn , f gxn , t) = 1,

n→∞

whenever {xn } is a sequence in X such that limn→∞ f xn = limn→∞ gxn = z, for some z ∈ X. Definition 1.14. A pair of self-mappings (f, g) of a b-fuzzy metric space (X, M, ∗) is said to be (1) weakly commuting if M (f gx, gf x, t) ≥ M (f x, gx, t), for all x ∈ X. (2) R-weakly commuting if there exists some positive real number R such that M (f gx, gf x, t) ≥ M (f x, gx, Rt ), ∀x ∈ X. Remark 1.15. If R ≤ 1, then R-weakly commuting mappings are weakly commuting. Definition 1.16. A pair of self-mappings (f, g) of a b-fuzzy metric space (X, M, ∗) are said to be (1) R-weakly commuting mappings of type (Af ) if there exists some positive real number R such that M (f gx, ggx, t) ≥ M (f x, gx, Rt ), for all x ∈ X. (2) R-weakly commuting mappings of type (Ag ) if there exists some positive real number R such that M (gf x, f f x, t) ≥ M (gx, f x, Rt ), for all x ∈ X. (3) R-weakly commuting mappings of type (P ) if there exists some positive real number R such that M (f f x, ggx, t) ≥ M (f x, gx, Rt ), for all x ∈ X. Remark 1.17. The self-mapping f of a b-fuzzy metric space (X, M, ∗) is said to be b-continuous at x ∈ X if and only if it is b-sequentially continuous at x, that is, whenever {xn } is b-convergent to x, {f (xn )} is b-convergent to f (x). −|x−y|2

Example 1.18. Let M (x, y, t) = e t , f x = 1 and  1, x ∈ Q, gx = −1, otherwise, for each x, y ∈ R, where a ∗ c = ac. Then it is easy to see that a pair of self-mappings (f, g) of a b-fuzzy metric space is weakly commuting, R-weakly commuting, and Rweakly commuting of type (P ), (Af ), and (Ag ).

2

The Main Results

Now we are in a position to introduce the main results of this paper. Theorem 2.1. Let (X, M, ∗) be a b-fuzzy metric space and (f, g) be a pair of noncompatible self-mappings with f X ⊆ gX (f X denotes the closure of f X). Assume

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that the following condition is satisfied: M (f x, f y, kt) ≥ min{M (gx, gy, b2 t), M (f x, gx, b2 t), M (f y, gy, b2 t)},

(2.1)

for all x, y ∈ X and 0 < k < 1. If (f, g) is a pair of R-weakly commuting mappings of type (Ag ), then f and g have a unique common fixed point (say z) and both f and g are not b-continuous at z. Proof. Since f and g are non-compatible mappings, there exists a sequence {xn } ⊂ X, such that lim f xn = lim gxn = z, z ∈ X, n→∞

n→∞

but either limn→∞ M (f gxn , gf xn , t) or limn→∞ M (gf xn , f gxn , t) does not exist or exists and is different from 1. Since z ∈ f X ⊂ gX, there must exist a u ∈ X satisfying z = gu. We can assert that f u = gu. If not, from condition (2.1) and Propsition 1.10, we obtain

≥ ≥ ≥ =

M (f u, gu, bkt) lim supn→∞ M (f u, f xn , kt)  lim supn→∞ min M (gu, gxn , b2 t), M (f u, gxn , b2 t), M (f xn , gu, b2 t)  min M (gu, gu, bt), M (f u, gu, bt), M (f u, gu, bt) M (f u, gu, bt),

that is, M (f u, gu, kt) ≥ M (f u, gu, t). Hence, by Lemma 1.12, we get f u = gu. Since (f, g) is a pair of R-weakly commuting mappings of type (Ag ), we have M (gf u, f f u, t) ≥ M (gu, f u,

t ) = 1. R

It means that f f u = gf u. Next, we prove f f u = f u. From condition (2.1), f u = gu and f f u = gf u, we have M (f u, f f u, kt) ≥ min{M (gu, gf u, b2 t), M (f u, gf u, b2 t), M (gu, f f u, b2 t)} = M (f u, f f u, b2 t) ≥ M (f u, f f u, t). From Lemma 1.12, we have f u = f f u, which implies that f u = f f u = gf u, and so z = f u is a common fixed point of f and g. Next we prove that the common fixed point z is unique. Actually, suppose that w is also a common fixed point of f and g. Then using the condition (2.1), we have M (z, w, kt)

= ≥ = ≥

M (f z, f w, kt) min{M (gz, gw, b2 t), M (f z, gw, b2 t), M (f w, gz, b2 t)} M (z, w, b2 t) M (z, w, t),

which implies that z = w, so that uniqueness is proved. Now, we prove that f and g are not b-continuous at z. In fact, if f is b-continuous at z, then for the b-convergent sequence {xn } to x, we have lim f f xn = f z = z and

n→∞

lim f gxn = f z = z.

n→∞

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Since f and g are R-weakly commuting mappings of type (Ag ), we get M (gf xn , f f xn , t) ≥ M gxn , f xn ,

t
. R

Hence, by Proposition 1.9, we have M ( lim gf xn , z, b2 t) ≥ lim supn→∞ M (gf xn , f f xn , t) n→∞

≥ lim supn→∞ M gxn , f xn , ≥ M z, z,

t
R

t
= 1, Rb2

it follows that limn→∞ gf xn = z. Hence, we can get lim M (f gxn , gf xn , t) ≥ M (z, z,

n→∞

t ) = 1. b2

Therefore, we have lim M (f gxn , gf xn , t) = 1.

n→∞

This contradicts with f and g being non-compatible. So f is not b-continuous at z. If g is b-continuous at z, then for the b-convergent sequence {xn } to x, we have lim gf xn = gz = z and lim ggxn = gz = z.

n→∞

n→∞

Since f and g are R-weakly commuting mappings of type (Ag ), we get M (gf xn , f f xn , t) ≥ M gxn , f xn ,

t
. R

Hence, we have M (z, lim f f xn , b2 t) ≥ lim supn→∞ M (gf xn , f f xn , t) n→∞

≥ lim supn→∞ M gxn , f xn , ≥ M z, z,

t
R

t
= 1, Rb2

it implies that lim f f xn = z = f z.

n→∞

This contradicts with f being not b-continuous at z, which implies that g is not b-continuous at z. This completes the proof. For the case b = 1 in Theorem 2.1, we have the following corollary. Corollary 2.2. Let (X, M, ∗) be a fuzzy metric space and (f, g) be a pair of noncompatible selfmappings with f X ⊆ gX. Assume that the following condition is satisfied:  M (f x, f y, kt) ≥ min M (gx, gy, t), M (f x, gx, t), M (f y, gy, t) , (2.2) for all x, y ∈ X and 0 < k < 1. If (f, g) is a pair of R-weakly commuting mappings of type (Ag ), then f and g have a unique common fixed point (say z) and both f and g are not b-continuous at z. 7

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3

Example

Next, we give an example to support for the main Theorem 2.1. Example 3.1. Let X = [2, 20] and a ∗ c = ac, for all a, c ∈ [0, 1] and let M be a fuzzy set on X × X × (0, +∞) defined as follows: M (x, y, t) = e

−|x−y| t

,

for all t ∈ R+ . Then (X, M, ∗) is a fuzzy metric space. We define mappings f and g on X by  2, x = 2 or x ∈ (5, 20], fx = 6, x ∈ (2, 5] and   2, 18, gx =  x+1 3

,

x = 2, x ∈ (2, 5], x ∈ (5, 20].

Clearly, from the above definitions, we know that f (X) ⊆ g(X), and (f, g) is a pair of non-compatible self-mappings. To see that f and g are non-compatible, consider a sequence {xn } = {5 + n1 }. Then we have f xn → 2, gxn → 2, f gxn → 6 and gf xn → 2. Thus 4 lim M (gf xn , f gxn , t) = e− t 6= 1. n→∞

On the other hand, there exists R = 1 such that  (2−2) −   e 7t , ( −2) 3 M (gf x, f f x, t) = e− t ,   − (2−2) e t , and

x = 2, x ∈ (2, 5], x ∈ (5, 20]

 (2−2) − t  , x = 2,   e (18−6) e− t , x ∈ (2, 5], M (f x, gx, t) = x+1    − ( 3 t −2) e , x ∈ (5, 20],

for all x ∈ X. Hence, it is easy to see that in every cases, we have M (gf x, f f x, t) ≥ M (gx, f x, t). That is, (f, g) is a pair of R-weakly commuting mappings of type (Ag ). Now we prove that the mappings f and g satisfy the condition (2.1) of Theorem 2.1 with k = 12 . To do this, we consider the following cases: Case (1) If x, y ∈ {2} ∪ (5, 20], then we have M (f x, f y, kt)

= M (2, 2, kt) = 1  ≥ min M (gx, gy, t), M (f x, gx, t), M (f y, gy, t) ,

and hence (2.1) is obviously satisfied. 8

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Case (2) If x, y ∈ (2, 5], then we have M (f x, f y, kt)

= M (6, 6, kt) = 1  ≥ min M (gx, gy, t), M (f x, gx, t), M (f y, gy, t) ,

and hence (2.1) is obviously satisfied. Case (3) If x ∈ {2} ∪ (5, 20] and y ∈ (2, 5], then we have 4

M (f x, f y, kt) = M (2, 6, kt) = e− kt and ( M (gx, gy, t) =

e− e−

|2−18| t |

,

x+1 −18| 3 t

x = 2, ,

.

x ∈ (5, 20].

Thus we obtain  M (f x, f y, t) ≥ min M (gx, gy, t), M (f x, gx, t), M (f y, gy, t) , for all x, y in X. Thus all the conditions of Theorem 2.1 are satisfied and 2 is a uniquecommon fixed point of f and g. Acknowledgments: This work was supported by the Basic Science Research Program through the National Research Foundation(NRF) Grant funded by Ministry of Education of the republic of Korea(2014046293).

References [1] M.S. El Naschie, On the uncertainty of Cantorian geometry and two-slit experiment. Chaos, Solitons and Fractals,9(1998), 17-29. [2] M.S. El Naschie, A review of E-infinity theory and the mass spectrum of high energy particle physics. Chaos, Solitons and Fractals, 19(2004), 20-36. [3] M.S. El Naschie, On a fuzzy Kahler-like Manifold which is consistent with twoslit experiment. Int. J. of Nonlinear Science and Numerical Simulation, 6(2005), 5-8. [4] M.S.El Naschie, The idealized quantum two-slit gedanken experiment revisitedCriticism and reinterpretation. Chaos, Solitons and Fractals, 27(2006), 9-13. [5] A. George and P. Veeramani, On some result in fuzzy metric space. Fuzzy Sets Syst, 64(1994), 5-9. [6] V. Gregori and A. Sapena, On fixed-point theorem in fuzzy metric spaces. Fuzzy Sets and Sys, 125( 2002), 45–52. [7] Kramosil I, Michalek J. Fuzzy metric and statistical metric spaces. Kybernetica, 11( 1975), 26–34. [8] S. Manro, S.S. Bhatia and S. Kumar, Expansion mapping theorems in G-metric spaces. Int. J. Contemp. Math. Sci., 5(51)(2010), 2529-2535. [9] D. Mihet¸, A Banach contraction theorem in fuzzy metric spaces. Fuzzy Sets Sys., 144(2004), 1-9. 9

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[10] B. Schweizer, H. Sherwood and R.M. Tardiff, Contractions on PM-space examples and counterexamples. Stochastica, 1(1988), 5-17. [11] S. Sedghi and N. Shobe, Common fixed point Theorem in b-fuzzy metric space. Nonlinear Functional Analysis and Applications Vol. 17(3)(2012), 349-359. [12] S. Sedghi and N. Shobe, Common fixed point Theorem for R-Weakly Commuting Maps in b-fuzzy Metric Space. Nonlinear Functional Analysis and Applications Vol. 19(2)(2014), 285-295. [13] S. Sedghi, N. Shobe and M.A. Selahshoor, A common fixed point theorem for Four mappings in two complete fuzzy metric spaces, Advances in Fuzzy Mathematics, Vol. 1, 1 (2006), 1-8. [14] S. Sedghi, D. Turkoglu and N. Shobe, Generalization common fixed point theorem in complete fuzzy metric spaces, Journal of Computational Analysis and Applictions Vol. 9(3)(2007), 337-348. [15] Y. Tanaka, Y. Mizn and T. Kado, Chaotic dynamics in Friedmann equation. Chaos, Solitons and Fractals, 24(2005), 7-22. [16] R.K. Vats, S. Kumar and V. Sihag, Some common fixed point theorem for compatible mappings of type (A) in complete G-metric space. Adv.Fuzzy Math., Vol. 6(1)(2011), 27–38. [17] L.A. Zadeh, Fuzzy sets. Inform and Control, 8( 1965), 38-53.

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On hesitant fuzzy filters in BE-algebras Young Bae Jun1 and Sun Shin Ahn2,∗ 1

Department of Mathematics Education, Gyeongsang National University, Jinju 52828, Korea 1

Department of Mathematics Education, Dongguk University, Seoul 04620, Korea

Abstract. The notions of hesitant fuzzy subalgebras and hesitant fuzzy filters are introduced and related properties are investigated. Relations between a hesitant fuzzy subalgebras and a hesitant fuzzy filters are discussed. The problem of classifying hesitant fuzzy filters by their γ-inclusive filter will be solved. Given a special set, we provide conditions for this set to be a hesitant fuzzy filter.

1. Introduction In 2007, Kim and Kim [4] introduced the notion of a BE-algebra, and investigated several properties. In [1], Ahn and So introduced the notion of ideals in BE-algebras. They gave several descriptions of ideals in BE-algebras. Song et al. [7] considered the fuzzification of ideals in BEalgebras. They introduced the notion of fuzzy ideals in BE-algebras, and investigated related properties. They gave characterizations of a fuzzy ideal in BE-algebras. The notions of Atanassov’s intuitionistic fuzzy sets, type 2 fuzzy sets and fuzzy multisets etc. are a generalization of fuzzy sets. As another generalization of fuzzy sets, Torra [8] introduced the notion of hesitant fuzzy sets which are a very useful to express peoples hesitancy in daily life. The hesitant fuzzy set is a very useful tool to deal with uncertainty, which can be accurately and perfectly described in terms of the opinions of decision makers. Also, hesitant fuzzy set theory is used in decision making problem etc. (see [6, 10, 11, 12, 13, 14]), and is applied to residuated lattices and M T L-algebras (see [3, 5]). In this paper, we introduce the notions of hesitant fuzzy subalgebras and hesitant fuzzy filters of BE-algebras, and investigate their relations and properties. We consider characterizations of hesitant fuzzy fuzzy subalgebras and hesitant fuzzy filters of BE-algebras. Given a special set, we provide conditions for this set to be a hesitant fuzzy filter. Given a special set, we provide conditions for this set to be a hesitant fuzzy filter.

0

2010 Mathematics Subject Classification: 08A72, 06F35. Keywords: (Transitive, self distributive) BE-algebra, Filter, Hesitant fuzzy subalgebra(filter). The corresponding author. Tel.: +82 2 2260 3410, Fax: +82 2 2266 3409 (S. S. Ahn). 0 E-mail: [email protected] (Y. B. Jun); [email protected] (S. S. Ahn). 0

∗

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2. Preliminaries Let K(τ ) be the class of all algebras of type τ = (2, 0). By a BE-algebra we mean a system (X; ∗, 1) ∈ K(τ ) in which the following axioms hold (see [4]): (∀x ∈ X) (x ∗ x = 1),

(2.1)

(∀x ∈ X) (x ∗ 1 = 1),

(2.2)

(∀x ∈ X) (1 ∗ x = x),

(2.3)

(∀x, y, z ∈ X) (x ∗ (y ∗ z) = y ∗ (x ∗ z)). (exchange)

(2.4)

A relation “≤” on a BE-algebra X is defined by (∀x, y ∈ X) (x ≤ y ⇐⇒ x ∗ y = 1).

(2.5)

A BE-algebra (X; ∗, 1) is said to be transitive (see [1]) if it satisfies: (∀x, y, z ∈ X) (y ∗ z ≤ (x ∗ y) ∗ (x ∗ z)).

(2.6)

A BE-algebra (X; ∗, 1) is said to be self distributive (see [4]) if it satisfies: (∀x, y, z ∈ X) (x ∗ (y ∗ z) = (x ∗ y) ∗ (x ∗ z)).

(2.7)

Every self distributive BE-algebra (X; ∗, 1) satisfies the following properties: (∀x, y, z ∈ X) (x ≤ y ⇒ z ∗ x ≤ z ∗ y and y ∗ z ≤ x ∗ z) ,

(2.8)

(∀x, y ∈ X) (x ∗ (x ∗ y) = x ∗ y) ,

(2.9)

(∀x, y, z ∈ X) (x ∗ y ≤ (z ∗ x) ∗ (z ∗ y)) .

(2.10)

Note that every self distributive BE-algebra is transitive, but the converse is not true in general (see [1]). Definition 2.1. ([4]) Let (X; ∗, 1) be a BE-algebra and let F be a non-empty subset of X. Then F is a filter of X if (F1) 1 ∈ F ; (F2) (∀x, y ∈ X)(x ∗ y, x ∈ F ⇒ y ∈ F ). 3. Hesitant fuzzy filters Definition 3.1. ([8]) Let E be a reference set. A hesitant fuzzy set on E is defined in terms of a function that when applied to E returns a subset of [0, 1], which can be viewed as the following mathematical representation: HE := {(e, hE (e))|e ∈ E} where hE : E → P([0, 1]).

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Definition 3.2. Given a non-empty subset A of X, a hesitant fuzzy set HX := {(x, hX (x))|x ∈ X} on satisfying the following condition: hX (x) = ∅ for all x ∈ /A

(3.1)

is called a hesitant fuzzy set related to A (briefly, A-hesitant fuzzy set) on X, and is represented by HA := {(x, hA (x)) | x ∈ X}, where hA is a mapping from X to P([0, 1]) with hA (x) = ∅ for all x ∈ / A. Definition 3.3. Given a non-empty subset (subalgebra as much as possible) A of X, let HA := {(x, hA (x)) | x ∈ X} be an A-hesitant fuzzy set on X. Then HA := {(x, hA (x)) | x ∈ X} is called a hesitant fuzzy subalgebra of X related to A (briefly, A-hesitant fuzzy subalgebra of X) if it satisfies the following condition: (∀x, y ∈ A) (hA (x ∗ y) ⊇ hA (x) ∩ hA (y)) .

(3.2)

An A-hesitant fuzzy subalgebra of X with A = X is called a hesitant fuzzy subalgebra of X. Example 3.4. Let X = {0, 1, a, b, c} be a BE-algebra with the following Cayley table: ∗ 1 a b c

1 1 1 1 1

a a 1 1 1

b b a 1 a

c c a a 1

For a subalgebra A = {1, a, b} of X, let HA := {(x, hA (x)) | x ∈ X} be an A-hesitant fuzzy set on X defined by { } HA = (1, [0, 1]), (a, (0, 12 ]), (b, ( 14 , 43 ]), (c, ∅) Then HA is an A-hesitant fuzzy subalgebra of X. Definition 3.5. Given a non-empty subset (subalgebra as much as possible) A of X, let HA := {(x, hA (x)) | x ∈ X} be an A-hesitant fuzzy set on X. Then HA := {(x, hA (x)) | x ∈ X} is called a hesitant fuzzy filter of X related to A (briefly, A-hesitant fuzzy filter of X) if it satisfies the following condition: (∀x ∈ A) (hA (x) ⊆ hA (1)) ,

(3.3)

(∀x, y ∈ A) (hA (x ∗ y) ∩ hA (x) ⊆ hA (y)) .

(3.4)

An A-hesitant fuzzy filter of X with A = X is called a hesitant fuzzy filter of X.

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Example 3.6. (1) Consider a BE-algebra X = {1, a, b, c} as in Example 3.4. Let HX := {(x, hX (x)) | x ∈ X} be a hesitant fuzzy set on X defined by { } HX = (1, [0, 1]), (a, (0, 81 )), (b, ( 14 , 34 ])), (c, (0, 41 )) Then HX is a hesitant fuzzy subalgebra of X, but not a hesitant fuzzy filter of X since hA (b ∗ a) ∩ hA (b) = hA (1) ∩ hA (b) = [0, 1] ∩ ( 41 , 34 ] ⊈ hA (a) = (0, 18 ). (2) Let X = {0, 1, a, b, c} be a BE-algebra with the following Cayley table: ∗ 1 a b c

1 1 1 1 1

a a 1 1 a

b b a 1 a

c c a a 1

Let HX := {(x, hX (x))|x ∈ X} be a hesitant fuzzy set defined by { } HX = (1, [0, 1]), (a, (0, 41 )), (b, (0, 14 )), (c, (0, 12 )) It is routine to verify that HX := {(x, hX (x))|x ∈ X} is a hesitant fuzzy filter of X. Proposition 3.7. Let HA := {(x, hA (x))|x ∈ X} be an A-hesitant fuzzy filter of X where A is a subalgebra of X. Then the following assertions are valid. (i) (∀x, y ∈ A)(x ≤ y ⇒ hA (x) ⊆ hA (y)), (ii) (∀x, y, z ∈ A)(hA (x ∗ (y ∗ z)) ∩ hA (y) ⊆ hA (x ∗ z)), (iii) (∀a, x ∈ A)(hA (a) ⊆ hA ((a ∗ x) ∗ x). Proof. Let x, y ∈ A be such that x ≤ y. Then x ∗ y = 1. It follows from (3.3) and (3.4) that hA (x) = hA (1) ∩ hA (x) = hA (x ∗ y) ∩ hA (x) ⊆ hA (y). (ii) Using (3.4) and (2.4), we have hA (x ∗ z) ⊇ hA (y ∗ (x ∗ z)) ∩ hA (y) = hA (x ∗ (y ∗ z)) ∩ hA (y) for all x, y, z ∈ A. (iii) Take y := (a ∗ x) ∗ x and x := a in (3.4). Then we have hA ((a ∗ x) ∗ x)) ⊇ hA (a ∗ ((a ∗ x) ∗ x)) ∩ hA (a) = hA ((a ∗ x) ∗ (a ∗ x)) ∩ hA (a) = hA (1) ∩ hA (a) = hA (a) □

by using (2.4), (2.1) and (3.3).

Corollary 3.8. Every hesitant fuzzy filter HX := {(x, hX (x))|x ∈ X} of X satisfies the following properties: (i) (∀x, y ∈ X)(x ≤ y ⇒ hX (x) ⊆ hX (y)), (ii) (∀x, y, z ∈ X)(hX (x ∗ (y ∗ z)) ∩ hX (y) ⊆ hX (x ∗ z)),

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(iii) (∀a, x ∈ X)(hA (a) ⊆ hA ((a ∗ x) ∗ x). We provide conditions for a hesitant fuzzy set to be a hesitant filter. Theorem 3.9. Let A be a subalgebra of a BE-algebra X. Every A-hesitant fuzzy set satisfies (3.3) and Proposition 3.7(ii). Then it is an A-hesitant fuzzy filter of X. Proof. Taking x := 1 in Proposition 3.7(ii) and using (2.3), we obtain hA (z) = hA (1 ∗ z) ⊇ hA (1 ∗ (y ∗ z)) ∩ hA (y) = hA (y ∗ z) ∩ hA (y) for all y, z ∈ A. Hence HA = {(x, hA (x))|x ∈ X} is an A-hesitant fuzzy filter of X. □ Corollary 3.10. Let HA := {(x, hA (x))|x ∈ X} be an A-hesitant fuzzy set for a subalgebra A of X. Then hA is an A-hesitant fuzzy filter of X if and only if it satisfies (3.3) and Proposition 3.7(ii). Theorem 3.11. An hesitant fuzzy set HA of X, where A is a subalgebra of X, is an A-hesitant fuzzy filter of X if and only if it satisfies the following conditions: (i) (∀x, y ∈ A)(hA (y ∗ x) ⊇ hA (x)), (ii) (∀x, a, b ∈ A)(hA ((a ∗ (b ∗ x)) ∗ x) ⊇ hA (a) ∩ hA (b)). Proof. Assume that HA := {(x, hA (x))|x ∈ X} is an A-hesitant fuzzy filter of X. Using (3.3), (3.4), (2.4),(2.1) and (2.2), we get hA (y∗x) ⊇ hA (x∗(y∗x))∩hA (x) = hA (1)∩hA (x) = hA (x) for all x, y ∈ A. It follows from Proposition 3.7 that hA ((a∗(b∗x))∗x) ⊇ hA ((a∗(b∗x))∗(b∗x))∩hA (b) ⊇ hA (a) ∩ ha (b) for all x, a, b ∈ X. Conversely, let HA (X) = {(x, hA (x))|x ∈ A} be an A-hesitant fuzzy set of X satisfying conditions (i) and (ii). If we take y := x in (i), then hA (1) = hA (x ∗ x) ⊇ hA (x) for all x ∈ A. Using (ii), we obtain hA (y) = hA (1 ∗ y) = hA (((x ∗ y) ∗ (x ∗ y)) ∗ y) ⊇ hA (x ∗ y) ∩ hA (x) for all x, y ∈ A. Hence HA is an A-hesitant fuzzy filter of X. □ Proposition 3.12. Let HX := {(x, hX (x)|x ∈ X} be a hesitant fuzzy set on X. Then HX is a hesitant fuzzy filter of X if and only if (∀x, y, z ∈ X)(z ≤ x ∗ y ⇒ hX (y) ⊇ hX (x) ∩ hX (z)).

(3.5)

Proof. Assume that HX is a hesitant fuzzy filter of X. Let x, y, z ∈ X be such that z ≤ x ∗ y. By Proposition 3.7 and Definition 3.5, we have hX (y) ⊇ hX (x ∗ y) ∩ hX (x) ⊇ hX (z) ∩ hX (x). Conversely, suppose that HX satisfies (3.5). By (2.2), we have x ≤ x ∗ 1 = 1. Using (3.5), we have hX (1) ⊇ hX (x) for all x ∈ X. It follows from (2.1) and (2.4) that x ≤ (x ∗ y) ∗ y for all x, y ∈ X. Using (3.5), we have hX (y) ⊇ hX (x ∗ y) ∩ hX (x). Therefore HX is a hesitant fuzzy filter of X. □

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As a generalization of Proposition 3.12, we have the following results. Theorem 3.13. Let HX := {(x, hX (x)|x ∈ X} be a hesitant fuzzy filter of X. Then n ∏ i=1

for all x, w1 , · · · , wn ∈ X, where

wi ∗ x = 1 ⇒ hX (x) ⊇ ∩ni=1 hX (wi ) ∏n i=1

(3.6)

wi ∗ x = wn ∗ (wn−1 ∗ (· · · w1 ∗ x) · · · )).

Proof. The proof is by induction on n. Let HX be a hesitant fuzzy filter of X. By Proposition 3.7(i) and (3.6), we know that the condition (3.6) is true for n = 1, 2. Assume that HX satisfies ∏ the condition (3.6) for n = k, i.e., ki=1 wi ∗ x = 1 ⇒ ∩ki=1 hX (wi ) for all x, w1 , · · · , wk ∈ X. ∏ Suppose that k+1 i=1 wi ∗ x = 1 for all x, w1 , · · · , wk , wk+1 ∈ X. Then hX (w1 ∗ x) ⊇ ∩k+1 i=2 hX (wi ). Since HX is a hesitant fuzzy filter of X, it follows form (3.4) that hX (x) ⊇ hX (w1 ∗ x) ∩ hX (w1 ) ⊇ (∩k+1 i=2 hX (wi )) ∩ hX (w1 ) = ∩k+1 i=1 hX (wi ). □

This completes the proof.

Theorem 3.14. Let HX = {(x, hX (x))|x ∈ X} be a hesitant fuzzy set of a BE-algebra satisfying (3.6). Then HX is a hesitant fuzzy filter of X. Proof. Let x, y, z ∈ X be such that z ≤ x ∗ y. Then z ∗ (x ∗ y) = 1 and so hX (y) ⊇ hX (x) ∩ hX (z) by (3.6). Using Proposition 3.12, HX is a hesitant fuzzy filter of X. □ Theorem 3.15. A hesitant fuzzy set HX := {(x, hX (x)|x ∈ X} of a BE-algebra X is a hesitant fuzzy filter of X if and only if the set HX (γ) := {x ∈ X|hX (x) ⊇ γ} is a filter of X for all γ ∈ P([0, 1]) whenever it is nonempty. Proof. Suppose that HX is a hesitant fuzzy filter of X. Let x, y ∈ X and γ ∈ P([0, 1]) be such that x ∗ y ∈ HX (γ) and x ∈ HX (γ). Then hX (x ∗ y) ⊇ γ and hX (x) ⊇ γ. It follows from (3.3) and (3.4) that hX (1) ⊇ hX (y) ⊇ hX (x ∗ y) ∩ hX (x) ⊇ γ. Hence 1 ∈ HX (γ) and y ∈ HX (γ), and therefore HX (γ) is a filter of X. Conversely, assume that HX (γ) is a filter of X for all γ ∈ P([0, 1]) with HX (γ) ̸= ∅. For any x ∈ X, let hX (x) = γ. Then x ∈ HX (γ). Since HX (γ) is a filter of X, we have 1 ∈ hX (γ) and so hX (x) = γ ⊆ hX (1). For any x, y ∈ X, let hX (x ∗ y) = γx∗y and hX (x) = γx . Take x ∗ y ∈ HX (γ) and x ∈ HX (γ) which imply that y ∈ HX (γ). Hence hX (y) ⊇ γ = γx∗y ∩ γx = hX (x ∗ y) ∩ hX (x). Thus HX is a hesitant fuzzy filter of X. □

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The filter HX (γ) in Theorem 3.15 is called the hesitant γ-inclusive set of HX := {(x, hX (x))|x ∈ X}. We make a new hesitant fuzzy filter from old one. Theorem 3.16. Let HX := {(x, hX (x))|x ∈ X} be a hesitant fuzzy set on a BE-algebra X. ∗ Define a hesitant fuzzy set HX on X by { hX (x) if x ∈ HX (γ) ∗ hX : X → P([0, 1]), x 7→ δ otherwise where γ is any subset of [0, 1] and δ is a subset of [0, 1] satisfying δ ⊊ ∩x∈H / X (γ) hX (x). If HX is a ∗ hesitant fuzzy filter of X, then so is HX . Proof. Assume that HX is a hesitant fuzzy filter of X. Then HX (γ) is a filter of X for all γ ∈ P([0, 1]) by Theorem 3.15. Hence 1 ∈ HX (γ) and so h∗X (1) = hX (1) ⊇ hX (x) ⊇ h∗X (x) for all x ∈ X. Let x, y ∈ X. If x ∗ y ∈ HX (γ) and x ∈ HX (γ), then y ∈ HX (γ). Hence h∗X (y) = hX (y) ⊇ hX (x ∗ y) ∩ hX (x) = h∗X (x ∗ y) ∩ h∗X (x). If x ∗ y ∈ / HX (γ) or x ∈ / HX (γ), then ∗ ∗ ∗ ∗ ∗ ∗ hX (x ∗ y) = δ or hX (x) = δ. Thus hX (y) ⊇ δ = hX (x ∗ y) ∩ hX (x). Therefore HX is a hesitant fuzzy filter of X. □ a,b = {(x, hX (x))|x ∈ X} where For two elements a and B of X, consider a hesitant fuzzy set HX { γ1 if a ∗ (b ∗ x) = 1 a,b hX : X → P([0, 1]), x 7→ γ2 otherwise

where γ1 and γ2 are subsets of [0, 1] with γ2 ⊊ γ1 . In the following example, we know that there a,b exist a, b ∈ X such that HX is not a hesitant fuzzy filter of X. Example 3.17. Let X = {0, 1, a, b, c} be a BE-algebra with the following Cayley table: ∗ 1 a b c

1 1 1 1 1

a a 1 1 a

b b a 1 b

c c c c 1

Let HX := {(x, hX (x))|x ∈ X} be a hesitant fuzzy set defined by { } HX = (1, [0, 1]), (a, (0, 14 )), (b, ( 41 , 34 )), (c, ( 68 , 78 )) 1,a 1,a 1,a 1 Then HX is not a hesitant fuzzy filter of X since h1,a X (a ∗ b) ∩ hX (a) = [0, 1] ⊈ hX (b) = (0, 4 ). a,b Now we provide a condition for the hesitant fuzzy set HX to be a hesitant fuzzy filter of X for all a, b ∈ X.

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a,b Theorem 3.18. If X is a self distributive BE-algebra, then the hesitant fuzzy set HX is a hesitant fuzzy filter of X for all a, b ∈ X. a,b Proof. Let a, b ∈ X. Obviously, ha,b X (1) ⊇ hX (x) for all x ∈ X. Let x, y ∈ X be such a,b that a ∗ (b ∗ (x ∗ y)) ̸= 1 or a ∗ (b ∗ x) ̸= 1. Then ha,b X (x ∗ y) = γ2 or hX (x) = γ2 . Hence a,b a,b ha,b X (x ∗ y) ∩ hX (x) = γ2 ⊆ hX (y). Assume that a ∗ (b ∗ (x ∗ y)) = 1 and a ∗ (b ∗ x) = 1. Then

1 = a ∗ (b ∗ (x ∗ y)) = a ∗ ((b ∗ x) ∗ (b ∗ y)) = (a ∗ (b ∗ x)) ∗ (a ∗ (b ∗ y)) = 1 ∗ (a ∗ (b ∗ y)) = a ∗ (b ∗ y) a,b a,b a,b and so ha,b X (x ∗ y) ∩ hX (x) = γ1 = hX (y). Therefore HX is a hesitant fuzzy filter of X for all a, b ∈ X. □

Theorem 3.19. Every filter of a BE-algebra can be represented as γ-inclusive set of a hesitant fuzzy filter. Proof. Let F be a filter of a BE-algebra X. For a subset γ of [0, 1], define a hesitant set HX by { γ if x ∈ F hX : X → P([0, 1]), x 7→ ∅ if x ∈ /F Obviously, F = HX (γ). We now prove that HX is a hesitant fuzzy filter of X. Since 1 ∈ HX (γ), we have HX (1) = γ ⊇ hX (x) for all x ∈ X. Let x, y ∈ X. If x ∗ y, x ∈ F , then y ∈ F since F is a filter of X. Hence hX (x ∗ y) = hX (x) = hX (y) = γ and so hX (x ∗ y) ∩ hX (x) ⊆ hX (y). If x ∗ y ∈ F and x ∈ / F , then hX (x ∗ y) = γ and hX (x) = ∅ which imply that hX (x ∗ y) ∩ hX (x) = γ ∩ ∅ = ∅ ⊆ hX (y). Similarly, if x ∗ y ∈ / F and x ∈ F , then hX (x ∗ y) ∩ hX (x) ⊆ hX (y). Obviously, if x ∗ y ∈ / F and x ∈ / F , then hX (x ∗ y) ∩ hX (x) ⊆ hX (y). Therefore HX is a hesitant fuzzy filter of X. □ Let HX = {(x, h(x))|x ∈ X} be a hesitant fuzzy set on X. For any a, b ∈ X and k ∈ N, consider the set hX [ak ; b] := {x ∈ X|hX (ak ∗ (b ∗ x)) = hX (1)} where hX (a ∗ (a ∗ (· · · ∗ (a ∗ (a ∗ x)) · · · ))) in which a appears k-times. Note that 1, a, b ∈ HX [ak ; b] for all a, b ∈ X and k ∈ N. Proposition 3.20. Let HX := {(x, hx (x))|x ∈ X} be a hesitant fuzzy set on X satisfying (3.3) and hX (x ∗ y) = hX (x) ∪ hX (y) for all x, y ∈ X. For any a, b ∈ X and k ∈ N, if x ∈ hX [ak ; b], then y ∗ x ∈ hX [ak ; b] for all y ∈ X.

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Proof. Assume that x ∈ hX [ak ; b]. Then hX (ak ∗ (b ∗ x)) = hX (1) and so hX (ak ∗ (b ∗ (y ∗ x))) = hX (ak ∗ (y ∗ b ∗ x)) = hX (y ∗ (ak ∗ (b ∗ x))) = hX (y) ∪ hX (ak ∗ (b ∗ x)) = hX (y) ∪ hX (1) = hX (1) for all y ∈ X by (2.4). Hence y ∗ x ∈ hX [ak ; b] for all y ∈ X.

□

Proposition 3.21. Let HX := {(x, hX ))|x ∈ X} be a hesitant fuzzy set on a BE-algebra X. If an element a ∈ X satisfies a ∗ x = 1 for all x ∈ X, then hX [ak ; b] = X = [bk ; a] for all b ∈ X and k ∈ N. Proof. For any x ∈ X, we have hX (ak ∗ (b ∗ x)) =hX (ak−1 ∗ (a ∗ (b ∗ x))) =hX (ak−1 ∗ (b ∗ (a ∗ x))) =hX (ak−1 ∗ (b ∗ 1)) =hX (1), and so x ∈ hX [ak ; b]. Similarly, x ∈ hX [bk ; a].

□

Proposition 3.22. Let X be a self distributive BE-algebra and let HX := {(x, hX (x))|x ∈ X} be a order reversing hesitant fuzzy set of X with the property (3.3). If b ≤ c in X, then hX [ak ; c] ⊆ hX [ak ; b] for all a ∈ X and k ∈ N. Proof. Let a, b, c ∈ X be such that b ≤ c. For any k ∈ N, if x ∈ hX [ak ; c], then hX (1) = hX (ak ∗ (c ∗ x)) = hX (c ∗ (ak ∗ x)) ⊆ hX (b ∗ (ak ∗ x)) = hX (ak ∗ (b ∗ x)) by (2.4) and (2.8). Hence hX (ak ∗ (b ∗ x)) = hX (1). Thus x ∈ hX [ak ; b], which completes the proof. □ The following example shows that there exists a hesitant fuzzy set HX of X, a, b ∈ X and k ∈ N such that hX [ak ; b] is not a filter of X.

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Example 3.23. Let X = {0, 1, a, b, c} be a BE-algebra with the following Cayley table: ∗ 1 a b c

1 1 1 1 1

a a 1 1 a

b b a 1 a

c c a a 1

Let HX := {(x, hX (x))|x ∈ X} be a hesitant fuzzy set defined by } { HX = (1, [0, 1]), (a, ( 14 , 43 )), (b, ( 34 , 12 )), (c, ( 68 , 87 )) Then hX [c; b] = {x ∈ X|hX (c ∗ (b ∗ x)) = hX (1)} = {1, a, b} is not a filter, since a ∗ c = a ∈ hX [c; b] and c ∈ / hX [c; b]. Theorem 3.24. HX := {(x, hX (x))|x ∈ X} be a hesitant fuzzy set on a self distributive BEalgebra X in which hX is injective. Then hX [ak ; b] is a filter of X for all a, b ∈ X and k ∈ N. Proof. Assume that X is a self distributive BE-algebra and hX is injective. Obviously, 1 ∈ hX [ak ; b]. Let a, b, x, y ∈ X and k ∈ N be such that x ∗ y ∈ hX [ak ; b] and x ∈ hX [ak ; b]. Then hX (ak ∗ (b ∗ x)) = hX (1) which implies that ak ∗ (b ∗ x) = 1, since hX is injective. Using (2.7), we have hX (1) = hX (ak ∗ (b ∗ (x ∗ y))) = hX (ak−1 ∗ (a ∗ (b ∗ (x ∗ y)))) = hX (ak−1 ∗ (a ∗ ((b ∗ x) ∗ (b ∗ y)))) = ··· = hX ((ak ∗ (b ∗ x)) ∗ (ak ∗ (b ∗ y))) = hX (1 ∗ (ak ∗ (b ∗ y)) = hX (ak ∗ (b ∗ y)) which imply that y ∈ hX [ak ; b]. Therefore hX [ak ; b] is a filter of X for all a, b ∈ X and k ∈ N. □ Theorem 3.25. HX := {(x, hX (x))|x ∈ X} be a hesitant fuzzy set of a self distributive BEalgebra X satisfying the condition (3.3) and hX (x ∗ y) = hX (x) ∩ hX (y), for all x, y ∈ X. Then hX [ak ; b] is a filter of X for all a, b ∈ X and k ∈ N. Proof. Let a, b ∈ X and k ∈ N. Obviously, 1 ∈ hX [ak ; b]. Let x, y ∈ X be such that x∗y ∈ hX [ak ; b] and x ∈ hX [ak ; b]. Then hX (ak ∗ (b ∗ (x ∗ y))) = hX (1) and hX (ak ∗ (b ∗ x)) = hX (1), which implies

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from the hypothesis that hX (1) = hX (ak ∗ (b ∗ (x ∗ y))) = hX (ak−1 ∗ (a ∗ (b ∗ (x ∗ y)))) = hX (ak−1 ∗ (a ∗ ((b ∗ x) ∗ (b ∗ y)))) = ··· = hX ((ak ∗ (b ∗ x)) ∗ (ak ∗ (b ∗ y))) = hX (ak ∗ (b ∗ x)) ∩ hX (ak ∗ (b ∗ y)) = hX (1) ∩ hX (ak ∗ (b ∗ y)) = hX (ak ∗ (b ∗ y)). Hence y ∈ hX [ak ; b] and therefore hX [ak ; b] is a filter of X for all a, b ∈ X and k ∈ N.

□

Proposition 3.26. HX := {(x, hX (x))|x ∈ X} be a hesitant fuzzy set of a BE-algebra X in which hX is injective. If F is a filter of X, then the following holds. (∀a, b ∈ F )(∀k ∈ N)(hX [ak ; b] ⊆ F ).

(3.7)

Proof. Assume that F is a filter of X and let a, b ∈ F and k ∈ N. If x ∈ hX [ak ; b], then hX (a ∗ (ak−1 ∗ (b ∗ x))) = hX (ak ∗ (b ∗ x)) = hX (1) and so a ∗ (ak−1 ∗ (b ∗ x)) = 1 ∈ F since hX is injective. Since F is a filter of X, it follows from (F2) that ak−1 ∗ (b ∗ x) ∈ F . Continuing this process, we obtain b ∗ x ∈ F and so x ∈ F . Therefore hX [ak ; b] ⊆ F for all a, b ∈ F and k ∈ N. □ Theorem 2.27. HX := {(x, hX (x))|x ∈ X} be a hesitant fuzzy set of a BE-algebra X. For any subset F of X, if the condition (3.7) holds, then F is a filter of X. Proof. Suppose that the condition (3.7) holds. Obviously, 1 ∈ hX [ak ; b] ⊆ F . Let x, y ∈ X be such that x ∗ y ∈ F and x ∈ F . Then hX (xk ∗ ((x ∗ y) ∗ y)) = hX (xk−1 ∗ (x ∗ ((x ∗ y) ∗ y))) = hX (xk−1 ∗ ((x ∗ y) ∗ (x ∗ y))) = hX (xk−1 ∗ 1) = hX (1) and hence y ∈ hX [ak ; b] ⊆ F , where b = x ∗ y. Therefore F is a filter of X.

□

Theorem 3.28. HX := {(x, hX (x))|x ∈ X} be a hesitant fuzzy set of a BE-algebra X. If F is a filter of X, then (∀k ∈ N)(F = ∪{hX [ak ; b]|a, b ∈ F }).

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Proof. Let F is a filter of X. By Proposition 3.26, the inclusion ∪{hX [ak ; b]|a, b ∈ F } ⊆ F holds. Let x ∈ F . Since x ∈ hX [1k ; x] for all k ∈ N, it follows that F ⊆ ∪ {hX [1k ; x]|x ∈ F } ⊆ ∪ {hX [ak ; b]|a, b ∈ F }. □

This completes the proof. Theorem 3.29. If HX := {(x, hX (x))|x ∈ X} is a hesitant filter of X, then the set Ha := {x ∈ X|hX (a) ⊆ hX (x)} is a filter of X for all a ∈ X.

Proof. Let x, y ∈ X be such that x ∗ y ∈ Ha and x ∈ Ha . Then hX (a) ⊆ hX (x ∗ y) and hX (a) ⊆ hX (y). By (3.3) and (3.4), we have hX (a) ⊆ hX (x ∗ y) ∩ hX (x) ⊆ HX (y) ⊆ hX (1) and so 1 ∈ Ha and y ∈ Ha . Therefore Ha is a filter of X. □ Theorem 3.30. Let a ∈ X and HX := {(x, hX (x))|x ∈ X} be a hesitant fuzzy set on X. Then the following properties are valid: (i) if Ha is a filter of X, then HX := {(x, hX (x))|x ∈ X} satisfies: (∀x, y ∈ X)(hX (a) ⊆ hX (x ∗ y) ∩ hX (x) ⇒ hX (a) ⊆ hX (y)).

(3.8)

(ii) if HX := {(x, hX (x))|x ∈ X} satisfies the condition (3.3) and (3.8), then Ha is a filter of X. Proof. (i) Assume that Ha is a filter of X and let x, y ∈ X be such that hX (a) ⊆ HX (x∗y)∩HX (x). Then x ∗ y ∈ Ha and y ∈ Ha . Since Ha is a filter of X, we obtain x ∈ Ha . Therefore hX (a) ⊆ hX (y). (ii) Let HX := {(x, hX (x))|x ∈ X} be a hesitant fuzzy set on X in which the conditions (3.3) and (3.8) hold. Then 1 ∈ Ha . Let x, y ∈ X be such that x ∗ y ∈ Ha and x ∈ Ha . Then hX (a) ⊆ hX (x ∗ y) and hX (a) ⊆ hX (x). Hence HX (a) ⊆ hX (x ∗ y) ∩ hX (x). Using (3.8), we have hX (a) ⊆ hX (y), i.e., y ∈ Ha . Thus Ha is a filter of X. □ References [1] S. S. Ahn and K. S. So, On ideals and upper sets in BE-algerbas, Sci. Math. Jpn. 68 (2008), 279–285 . [2] Y. B. Jun and S. S. Ahn, Hesitant fuzzy soft theory applied to BCK/BCI-algebras, The Scientific World Journal (submitted). [3] Y. B. Jun and S. Z. Song, Hesitant fuzzy set theory applied to filters in M T L-algebras, The Scientific World Journal (to appear). [4] H. S. Kim and Y. H. Kim, On BE-algerbas, Sci. Math. Jpn. 66 (2007), no. 1, 113–116. [5] G. Muhiuddin and Y. B. Jun, Hesitant fuzzy filters and hesitant fuzzy G-filters in residuated lattices, J. Appl. Math. (submitted).

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[6] Rosa M. Rodriguez, Luis Martinez and Francisco Herrera, Hesitant fuzzy linguistic term sets for decision making, IEEE Trans. Fuzzy Syst. 20(1) (2012), 109–119. [7] S. Z. Song, Y. B. Jun and K. J. Lee, Fuzzy ideals in BE-algebras, Bull. Malays. Math. Sci. Soc. 33 (2010), 147-153. [8] V. Torra, Hesitant fuzzy sets, Int. J. Intell. Syst. 25 (2010), 529–539. [9] V. Torra and Y. Narukawa, On hesitant fuzzy sets and decision, in: The 18th IEEE International Conference on Fuzzy Systems, Jeju Island, Korea, 2009, 1378-1382. [10] F. Q. Wang, X. Li and X. H. Chen, Hesitant fuzzy soft set and its applications in multicriteria decision making, J. Appl. Math. Volume 2014, Article ID 643785, 10 pages. [11] G. Wei, Hesitant fuzzy prioritized operators and their application to multiple attribute decision making, Knowledge-Based Systems 31 (2012), 176–182. [12] M. Xia and Z. S. Xu, Hesitant fuzzy information aggregation in decision making, Internat. J. Approx. Reason. 52(3) (2011), 395–407. [13] Z. S. Xu and M. Xia, Distance and similarity measures for hesitant fuzzy sets, Inform. Sci. 181(11) (2011), 2128–2138. [14] Z. S. Xu and M. Xia, On distance and correlation measures of hesitant fuzzy information, Int. J. Intell. Syst. 26(5) (2011), 410–425.

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A fixed point approach to the stability of nonic functional equation in non-Archimedean spaces∗ Tian-Zhou Xu1,†, Yali Ding1 , John Michael Rassias2 (1. School of Mathematics and Statistics, Beijing Institute of Technology, Beijing 100081, P. R. China) (2. Pedagogical Department E.E., Section of Mathematics and Informatics, National and Capodistrian University of Athens, 4, Agamemnonos Str., Aghia Paraskevi, Athens 15342, Greece)

Abstract In this paper, a new nonic functional equation is introduced. The solution of this functional equation can also be determined in certain type of groups using two important results due to Sz´ekelyhidi. Using the fixed point theorems due to Brzd¸ek and Ciepli´ nski, we give some Ulam–Hyers stability results for the nonic functional equation in non-Archimedean spaces. Keywords Ulam–Hyers stability; nonic functional equation; non-Archimedean space; fixed point method. Mathematics Subject Classification(2010) 39B82; 39B52; 46H25.

1

Introduction and preliminaries

In this paper R and N denote the sets of reals and positive integers, respectively. Moreover, R+ := [0, ∞) and N0 := N ∪ {0}. A valuation is a function | · | from a field K into R+ such that 0 is the unique element having the 0 valuation, |rs| = |r| · |s| and the triangle inequality holds, i.e., |r + s| ≤ |r| + |s|,

∀r, s ∈ K.

A field K is called a valued field if K carries a valuation. The usual absolute values of R and C are examples of valuations. Let us consider a valuation which satisfies a stronger condition than the triangle inequality. Let K be a field. A non-Archimedean valuation on K is a function | · | : K → R such that (1) |r| ≥ 0 and equality holds if and only if r = 0. (2) |rs| = |r||s|,

r, s ∈ K.

(3) |r + s| ≤ max{|r|, |s|},

r, s ∈ K.

Any field endowed with a non-Archimedean valuation is said to be a non-Archimedean field. In any such field we have |1| = | − 1| = 1 and |n × 1| ≤ 1 for all n ∈ N, where 1 is the neutral element of the semigroup (K, ·), 1 × 1 = 1 and (n + 1) × 1 = (n × 1) + 1 for n ∈ N. Let X be a linear space over a field K with a non-Archimedean valuation | · |. A function ∥ · ∥ : X → R+ is a non-Archimedean norm if it satisfies the following conditions: ∗ The

first author was supported by the National Natural Science Foundation of China (Grant No. 11171022). author. E-mail addresses: [email protected] (T.Z. Xu), [email protected] (Y. Ding), [email protected] (J.M. Rassias). † Corresponding

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(1) ∥x∥ = 0 if and only if x = 0; (2) ∥rx∥ = |r|∥x∥ for all r ∈ K and x ∈ X; (3) ∥x + y∥ ≤ max{∥x∥, ∥y∥} for all x, y ∈ X. If ∥ · ∥ : X → R+ is a non-Archimedean norm in X, then the pair (X, ∥ · ∥) is called a non-Archimedean normed space. Then (X, ∥ · ∥) is called a non-Archimedean normed space. Let X be a non-Archimedean normed space. Let {xn } be a sequence in X. Then {xn } is said to be convergent if there exists x ∈ X such that lim ∥xn − x∥ = 0. In that case, x is called the limit of the sequence {xn } and n→∞

we denote it by lim xn = x. A sequence {xn } in X is said to be a Cauchy sequence if lim ∥xn+p − xn ∥ = 0 for n→∞ n→∞ all p = 1, 2, . . .. Due to the fact that ∥xn − xm ∥ ≤ max{∥xj+1 − xj ∥ : m ≤ j ≤ n − 1} (n > m) a sequence {xn } is Cauchy if and only if {xn+1 − xn } converges to zero in a non-Archimedean normed space. The most important examples of non-Archimedean spaces are p-adic numbers. The p-adic numbers have gained the interest of physicists because of their connections with some problems coming from quantum physics, p-adic strings and superstrings (see [15]). In this paper, we first introduce the following new nonic functional equation f (x + 5y) − 9f (x + 4y) + 36f (x + 3y) − 84f (x + 2y) + 126f (x + y) − 126f (x)+ 84f (x − y) − 36f (x − 2y) + 9f (x − 3y) − f (x − 4y) = 9!f (y).

(1)

It is easy to see that the function f (x) = ax9 is a solution of the functional equation (1). Every solution of the functional equation (1) is said to be a nonic mapping. The study of stability problems for functional equations is related to a question of Ulam [20] concerning the stability of group homomorphisms and affirmatively answered for Banach spaces by Hyers [10]. The result of Hyers was generalized by Aoki [2] for approximate additive mappings and by Rassias [17] for approximate linear mappings by allowing the Cauchy difference operator CDf (x, y) = f (x + y) − [f (x) + f (y)] to be controlled by ϵ(∥x∥p + ∥y∥p ). In 1994, a further generalization was obtained by G˘avrut¸a [7], who replaced ϵ(∥x∥p + ∥y∥p ) by a general control function φ(x, y). We refer the reader to (see for instance [1, 3–6, 8, 11–14, 16, 18, 21, 22]) and references therein for more information on Ulam’s problem during the last seventy years. From now on S denotes a nonempty set and X stands for a complete non-Archimedean normed space. Given S S S a set Z ̸= ∅ and functions φ : S → S and F : S × Z → Z, we define an operator LF φ : Z → Z (Z denotes the family of all functions mapping a set S into a set Z) by

LF φ (α)(t) := F (t, α(φ(t))),

α ∈ Z S , t ∈ S.

Moreover, if Λ : S × R+ → R+ , then we write Λt := Λ(t, ·), t ∈ S. For explicitly later use, we recall the following results by Brzd¸ek and Ciepli´ nski [4]. Theorem 1 Let Λ : S × R+ → R+ , f : S → X, T : X S → X S , φ : S → S, ε : S → R+ and ∥T (α)(t) − T (β)(t)∥ ≤ Λ(t, ∥α(φ(t)) − β(φ(t))∥),

α, β ∈ X S , t ∈ S.

(2)

n Assume also that Λt is nondecreasing for every t ∈ S, lim (LΛ φ ) (ε)(t) = 0(t ∈ S) holds and n→∞

∥T (f )(t) − f (t)∥ ≤ ε(t),

t ∈ S.

(3)

Then for each t ∈ S the limit lim T n (f )(t) =: A(t)

(4)

n→∞

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exists and the function A ∈ X S is the unique fixed point of T with n ∥f (t) − A(t)∥ ≤ sup (LΛ φ ) (ε)(t) =: h(t),

t ∈ S.

(5)

n∈N0

Corollary 1 Let F : S × X → X, φ : S → S, Λ : S × R+ → R+ , f : S → X, ε : S → R+ and ∥F (t, x) − F (t, y)∥ ≤ Λ(t, ∥x − y∥),

t ∈ S, x, y ∈ X.

(6)

n Assume also that, for every t ∈ S, Λt is nondecreasing, lim (LΛ φ ) (ε)(t) = 0(t ∈ S) holds and n→∞

∥f (t) − F (t, f (φ(t)))∥ ≤ ε(t),

t ∈ S.

(7)

Then for each t ∈ S the limit n lim (LF φ ) (f )(t) =: A(t)

(8)

n→∞

exists and the function A ∈ X S is the unique solution of the functional equation A(t) = F (t, A(φ(t)))

(9)

such that (5) holds. We end this section with two corollaries, which are immediate consequences of Corollary 1. Corollary 2 Let a : S → K\{0}, φ : S → S, f : S → X, δ : S → R+ , ∥f (φ(t)) − a(t)f (t)∥ ≤ δ(t), and

δ(φn (t)) ∏n = 0, i n→∞ | i=0 a(φ (t))| lim

t∈S

(10)

t ∈ S.

(11)

Then there exists a unique solution A ∈ X S of the functional equation A(φ(t)) = a(t)A(t) such that

(12)

δ(φn (t)) ∥f (t) − A(t)∥ ≤ sup ∏n , i n∈N0 | i=0 a(φ (t))|

t ∈ S.

(13)

Corollary 3 Let b : S → K, ψ : S → S, f : S → X, ε : S → R+ , ∥f (t) − b(t)f (ψ(t))∥ ≤ ε(t), and

t∈S

n ∏ i lim b(ψ (t)) ε(ψ n+1 (t)) = 0, n→∞

(14)

t ∈ S.

(15)

i=0

Then there exists a unique solution B ∈ X S of the functional equation B(t) = b(t)B(ψ(t)) such that

n } ∏ i n+1 ∥f (t) − B(t)∥ ≤ max ε(t), sup b(ψ (t)) ε(ψ (t)) , n∈N0

(16)

{

t ∈ S.

i=0

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(17)

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2

Solution of the nonic functional equation on commutative groups In this section, we solve the functional equation (1) on commutative groups with some additional requirements.

A group S is said to be divisible if for every element b ∈ S and every n ∈ N, there exists an element a ∈ S such that na = b. If this element a is unique, then S is said to be uniquely divisible. In a uniquely divisible group, this unique element a is denoted by nb . That the equation na = b has a solution is equivalent to saying that the multiplication by n is surjective. Similarly, that the equation na = b has a unique solution is equivalent to saying that the multiplication by n is bijective. The following two important results due to Sz´ekelyhidi (see [19] for the details). Theorem 2 Let G be a commutative semigroup with identity, S a commutative group and n a nonnegative integer. Let the multiplication by n! be bijective in S. The function f : G → S is a solution of Fr´echet functional equation △x1 ,...,xn+1 f (x0 ) = 0 (18) for all x0 , x1 , . . . , xn+1 ∈ G if and only if f is a polynomial of degree at most n, i.e., f is given by f (x) = An (x) + · · · + A1 (x) + A0 (x),

x ∈ G,

(19)

where A0 (x) = A0 is an arbitrary element of S and An (x) is the diagonal of an n-additive symmetric function An : Gn → S. Theorem 3 Let G and S be commutative groups, n a nonnegative integer, φi , ψi additive functions from G into G and φi (G) ⊆ ψi (G)(i = 1, 2, . . . , n + 1). If the functions f, fi : G → S(i = 1, 2, . . . , n + 1) satisfy f (x) +

n+1 ∑

fi (φi (x) + ψi (y)) = 0,

(20)

i=1

then f satisfies Fr´echet functional equation △x1 ,...,xn+1 f (x0 ) = 0. Using the results, we have the following theorem. Theorem 4 Let S be a commutative group and V be a linear space. Then the function f : S → V satisfies the functional equation (1) for all x, y ∈ S, if and only if f is of the form f (x) = A9 (x),

x ∈ S,

where A9 (x) is the diagonal of the 9-additive symmetric map A9 : S 9 → V . Proof.

Assume that f satisfies the functional equation (1). We can rewrite the functional equation (1) in the

form

2 1 2 1 f (x + 5y) + f (x + 4y) − f (x + 3y) + f (x + 2y) − f (x + y) 126 14 7 3 2 2 1 1 − f (x − y) + f (x − 2y) − f (x − 3y) + f (x − 4y) + 2880f (y) = 0. 3 7 14 126 Thus by Theorems 2 and 3, f is of the form f (x) −

f (x) =

9 ∑

Ai (x),

x ∈ S,

(21)

(22)

i=0

where A0 (x) = A0 is an arbitrary element of V , and Ai (x) is the diagonal of the i-additive symmetric map Ai : S i → V for i = 1, 2, . . . , 9. Replacing x = 0, y = 0 in (1), one finds f (0) = 0. Hence A0 (x) = A0 = 0. 362

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Replacing x = 0, y = x and x = x, y = −x in (1) and adding the two resulting equations, we get f (−x) = −f (x) for all x ∈ S. So the function f is odd. Thus we have A8 (x) = A6 (x) = A4 (x) = A2 (x) = 0 for all x ∈ S. It follows that f (x) = A9 (x) + A7 (x) + A5 (x) + A3 (x) + A1 (x). Replacing (x, y) with (0, 2x) in (1), one obtains f (10x) − 8f (8x) + 27f (6x) − 48f (4x) − (9! − 42)f (2x) = 0.

(23)

Replacing (x, y) with (5x, x), one gets f (10x) − 9f (9x) + 36f (8x) − 84f (7x) + 126f (6x) − 126f (5x) + 84f (4x) −36f (3x) + 9f (2x) − (9! + 1)f (x) = 0.

(24)

Subtracting equations (23) and (24), we find 9f (9x) − 44f (8x) + 84f (7x) − 99f (6x) + 126f (5x) − 132f (4x) +36f (3x) − (9! − 33)f (2x) + (9! + 1)f (x) = 0.

(25)

Replacing (x, y) with (4x, x), and multiplying the resulting equation by 9, one obtains 9f (9x) − 81f (8x) + 324f (7x) − 756f (6x) + 1134f (5x) − 1134f (4x) +756f (3x) − 324f (2x) − 9(9! − 9)f (x) = 0.

(26)

Subtracting equations (25) and (26), we get 37f (8x) − 240f (7x) + 657f (6x) − 1008f (5x) + 1002f (4x) − 720f (3x) −(9! − 357)f (2x) + (10! − 80)f (x) = 0.

(27)

Replacing (x, y) with (3x, x), and multiplying the resulting equation by 37, one finds 37f (8x) − 333f (7x) + 1332f (6x) − 3108f (5x) + 4662f (4x) − 4662f (3x) +3108f (2x) − 37(9! + 35)f (x) = 0.

(28)

Subtracting equations (27) and (28), we arrive at 93f (7x) − 675f (8x) + 2100f (5x) − 3660f (4x) + 3942f (3x) − (9! + 2751)f (2x) +(47 · 9! + 1215)f (x) = 0.

(29)

Replacing (x, y) with (2x, x), and multiplying the resulting equation by 93, one finds 93f (7x) − 837f (6x) + 3348f (5x) − 7812f (4x) + 11718f (3x) − 11625f (2x) −93(9! − 75)f (x) = 0. Subtracting equations (29) and (30) and then dividing by 2, we arrive at 1 81f (6x) − 624f (5x) + 2076f (4x) − 3888f (3x) − (9! − 8874)f (2x) 2 +(70 · 9! − 2880)f (x) = 0.

(30)

(31)

Replacing (x, y) with (x, x), and multiplying the resulting equation by 81, one finds 81f (6x) − 729f (5x) + 2916f (4x) − 6723f (3x) + 9477f (2x) − 81(9! + 90)f (x) = 0.

(32)

Subtracting equations (31) and (32), we arrive at 1 105f (5x) − 840f (4x) + 2835f (3x) − (9! + 10080)f (2x) + (151 · 9! + 4410)f (x) = 0. 2 Replacing (x, y) with (0, x), and multiplying the resulting equation by 105, one finds 105(5x) − 840f (4x) + 2835f (3x) − 5040f (2x) − 105(9! − 42)f (x) = 0.

(33)

(34)

Subtracting equations (33) and (34), we arrive at f (2x) = 29 f (x).

(35)

By (35) and An (rx) = rn An (x) whenever x ∈ S and r ∈ Q, we obtain 29 (A9 (x)+A7 (x)+A5 (x)+A3 (x)+A1 (x)) = 29 A9 (x) + 27 A7 (x) + 25 A5 (x) + 23 A3 (x) + 2A1 (x). It follows that A7 (x) = A5 (x) = A3 (x) = A1 (x) = 0 for all x ∈ S. Hence f (x) = A9 (x). The converse is easily verified. 

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3

Stability results

Throughout this section, we assume that S is a commutative group and X is a complete non-Archimedean normed space. For a given mapping f : S → X, we define the difference operators Df (x, y) := f (x + 5y) − 9f (x + 4y) + 36f (x + 3y) − 84f (x + 2y) + 126f (x + y) −126f (x) + 84f (x − y) − 36f (x − 2y) + 9f (x − 3y) − f (x − 4y) − 9!f (y) for all x, y ∈ S. Theorem 5 Let φ : S 2 → R+ be a function such that lim |2|−9n φ (2n x, 2n y) = 0,

x, y ∈ S.

n→∞

(36)

Assume also that f : S → X be a mapping such that ∥Df (x, y)∥ ≤ φ(x, y),

x, y ∈ S.

(37)

Then there exists a unique nonic mapping T : S → X such that ∥f (x) − T (x)∥ ≤ sup |2|−9(n+1) δ(2n x),

x ∈ S,

(38)

n∈N0

where δ(x)

Proof.

:=

{ 1 |210| |210| |15| max |210|φ(0, x), φ(0, 3x), φ(3x, −3x), φ(2x, −2x), |9!| |8!| |8!| |6!| |2940| |210| |210| |35| φ(x, −x), φ(0, 0), φ(0, 4x), φ(4x, −4x), |162|φ(x, x), |6!| |8!| |9!| |9!| |18| φ(3x, −3x), |93|φ(2x, x), |37|φ(3x, x), |9|φ(4x, x), φ(5x, x), φ(0, 2x), |8!| } 1 1 1 1 φ(0, 8x), φ(8x, −8x), φ(0, 6x), φ(6x, −6x) . |9!| |9!| |8!| |8!|

Replacing x = y = 0 in (37), we get ∥f (0)∥ ≤

1 φ(0, 0). |9!|

(39)

Replacing x and y by 0 and x in (37), respectively, we get ∥f (5x) − 9f (4x) + 36f (3x) − 84f (2x) + 126f (x) − 126f (0) + 84f (−x) −36f (−2x) + 9f (−3x) − f (−4x) − 9!f (x)∥ ≤ φ(0, x)

(40)

for all x ∈ S. Replacing x and y by x and −x in (37), respectively, we have ∥f (−4x) − 9f (−3x) + 36f (−2x) − 84f (−x) + 126f (0) − 126f (x) + 84f (2x) −36f (3x) + 9f (4x) − f (5x) − 9!f (−x)∥ ≤ φ(−x, x)

(41)

for all x ∈ S. By (40) and (41), we obtain ∥f (x) + f (−x)∥ ≤

1 max{φ(0, x), φ(x, −x)} |9!|

(42)

for all x ∈ S. Replacing x and y by 0 and 2x in (37), respectively, and using (39) and (42), we find ∥f (10x) − 8f (8x) { + 27f (6x) − 48f (4x) − (9! − 42)f (2x)∥ 1 1 1 1 ≤ max φ(0, 2x), φ(0, 8x), φ(8x, −8x), φ(0, 6x), φ(6x, −6x), |9!| |9!| |8!| |8!| } |4| |4| |84| φ(0, 4x), φ(4x, −4x), φ(2x, −2x) |8!| |8!| |9!| 364

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J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 22, NO.2, 2017, COPYRIGHT 2017 EUDOXUS PRESS, LLC

for all x ∈ S. Replacing x and y by 5x and x in (37), respectively, we get ∥f (10x) − 9f (9x) + 36f (8x) − 84f (7x) + 126f (6x) − 126f (5x) + 84f (4x) −36f (3x) + 9f (2x) − (9! + 1)f (x)∥ ≤ φ(5x, x)

(44)

for all x ∈ S. By (43) and (44), we obtain ∥9f (9x) − 44f (8x) + 84f (7x) − 99f (6x) + 126f (5x) − 132f (4x) +36f { (3x) − (9! − 33)f (2x) + (9! + 1)f (x)∥ 1 1 1 ≤ max φ(5x, x), φ(0, 2x), φ(0, 8x), φ(8x, −8x), φ(0, 6x), |9!| |9!| |8!| } 1 |4| |4| |84| φ(6x, −6x), φ(0, 4x), φ(4x, −4x), φ(2x, −2x) |8!| |8!| |8!| |9!|

(45)

for all x ∈ S. Replacing x and y by 4x and x in (37), respectively, and using (39) we have ∥f (9x) − 9f (8x) + 36f (7x) − 84f (6x) + 126f (5x) − 126f { (4x) +84f (3x) − 36f (2x) − (9! − 9)f (x)∥ ≤ max φ(4x, x),

} 1 φ(0, 0) |9!|

(46)

for all x ∈ S. By (45) and (46), we get ∥37f (8x) − 240f (7x) + 657f (6x) − 1008f (5x) + 1002f (4x) −720f (3x) − (9! − 357)f (2x) + (10! − 80)f (x)∥ { 1 1 1 ≤ max |9|φ(4x, x), φ(0, 0), φ(5x, x), φ(0, 2x), φ(0, 8x), φ(8x, −8x), |8!| |9!| |9!| } 1 1 |4| |4| |84| φ(0, 6x), φ(6x, −6x), φ(0, 4x), φ(4x, −4x), φ(2x, −2x) |8!| |8!| |8!| |8!| |9!|

(47)

for all x ∈ S. Replacing x and y by 3x and x in (37), respectively, then using (39) and (42), we have ∥f (8x) − 9f (7x) + 36f (6x) − 84f (5x){+ 126f (4x) − 126f (3x) + 84f (2x) } 1 1 1 −(9! + 35)f (x)∥ ≤ max φ(3x, x), φ(0, 0), φ(0, x), φ(x, −x) |8!| |9!| |9!|

(48)

for all x ∈ S. By (47) and (48), we get ∥93f (7x) − 675f (6x) + 2100f (5x) − 3660f (4x) +3942f (3x) − (9! + 2751)f (2x) + (47 · 9! + 1215)f (x)∥ { |37| |37| |37| ≤ max |37|φ(3x, x), φ(0, 0), φ(0, x), φ(x, −x), |9|φ(4x, x), |8!| |9!| |9!| 1 1 1 φ(5x, x), φ(0, 2x), φ(0, 8x), φ(8x, −8x), φ(0, 6x), |9!| |9!| |8!| } 1 |4| |4| |84| φ(6x, −6x), φ(0, 4x), φ(4x, −4x), φ(2x, −2x) |8!| |8!| |8!| |9!|

(49)

for all x ∈ S. Replacing x and y by 2x and x in (37), respectively, then using (39) and (42), we have ∥f (7x) − 9f (6x) { + 36f (5x) − 84f (4x) + 126f (3x) − 125f (2x) − (9! − 75)f (x)∥ } 1 1 1 1 |4| ≤ max φ(2x, x), φ(0, 2x), φ(2x, −2x), φ(0, x), φ(x, −x), φ(0, 0) |9!| |9!| |8!| |8!| |8!|

(50)

for all x ∈ S. By (49) and (50), we get 1 ∥81f (6x) − 624f (5x) + 2076f (4x) − 3888f (3x) − (9! − 8874)f (2x) 2 +(70{· 9! − 2880)f (x)∥ 1 |93| |93| |93| |372| ≤ max |93|φ(2x, x), φ(2x, −2x), φ(0, x), φ(x, −x), φ(0, 0), |2| |9!| |8!| |8!| |8!| 1 1 |37|φ(3x, x), |9|φ(4x, x), φ(5x, x), φ(0, 2x), φ(0, 8x), φ(8x, −8x), |9!| |9!| } 1 1 |4| |4| φ(0, 6x), φ(6x, −6x), φ(0, 4x), φ(4x, −4x) |8!| |8!| |8!| |8!| 365

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for all x ∈ S. Replacing x and y by x and x in (37), respectively, then using (39) and (42), we have ∥f (6x) − 9f (5x) { + 36f (4x) − 83f (3x) + 117f (2x) − (9! + 90)f (x)∥ |4| 1 |84| |4| ≤ max φ(x, x), φ(0, 0), φ(0, x), φ(x, −x), φ(0, 2x), |9!| |8!| |8!| }|8!| 1 1 1 φ(2x, −2x), φ(0, 3x), φ(3x, −3x) |8!| |9!| |9!|

(52)

for all x ∈ S. By (51) and (52), we get 9! ∥105f (5x) − 840f (4x) + 2835f (3x) − ( + 5040)f (2x) + (151 · 9! + 4410)f (x)∥ 2 { |756| |324| |324| |81| ≤ max |81|φ(x, x), φ(0, 0), φ(0, x), φ(x, −x), φ(2x, −2x), |8!| |8!| |8!| |8!| |9| |37| |9| |9| |93| φ(0, 3x), φ(3x, −3x), φ(2x, x), φ(3x, x), φ(4x, x), |8!| |8!| |2| |2| |2| 1 1 1 1 φ(5x, x), φ(0, 2x), φ(0, 8x), φ(8x, −8x), |2| |2| |2 · 9!| |2 · 9!| } 1 |2| |2| 1 φ(0, 6x), φ(6x, −6x), φ(0, 4x), φ(4x, −4x) |2 · 8!| |2 · 8!| |8!| |8!|

(53)

for all x ∈ S. Replacing x and y by 0 and x in (37), respectively, then using (39) and (42), we have ∥f (5x) − 8f (4x) { + 27f (3x) − 48f (2x) − (9! − 42)f (x)∥ 1 1 |4| |4| ≤ max φ(0, x), φ(0, 3x), φ(3x, −3x), φ(0, 2x), φ(2x, −2x), |8!| |8!| |8!| |8!| } |14| 1 1 |84| φ(x, −x), φ(0, 0), φ(0, 4x), φ(4x, −4x) |9!| |8!| |9!| |9!| for all x ∈ S. By (53) and (54), we get ∥f (2x) − 2 f (x)∥ ≤ 9

=:

{ 1 |210| |210| max |210|φ(0, x), φ(0, 3x), φ(3x, −3x), |9!| |8!| |8!| |15| |35| |2940| |210| φ(2x, −2x), φ(x, −x), φ(0, 0), φ(0, 4x), |6!| |6!| |8!| |9!| |18| |210| φ(4x, −4x), |162|φ(x, x), φ(3x, −3x), |93|φ(2x, x), |9!| |8!| 1 |37|φ(3x, x), |9|φ(4x, x), φ(5x, x), φ(0, 2x), φ(0, 8x), |9!| } 1 1 1 φ(8x, −8x), φ(0, 6x), φ(6x, −6x) |9!| |8!| |8!| δ(x), x ∈ S.

(54)

(55)

By Corollary 2, there exists a unique mapping T : S → X such that T (2x) = 29 T (x) and (38) holds. By (8) in Corollary 1, n −9n T (x) := lim (LF f (2n x), φ ) (f )(x) = lim 2 n→∞

n→∞

x ∈ S.

(56)

It remains to show that T is a nonic map. By (37), we have ∥Df (2n x, 2n y)/29n ∥ ≤ |2|−9n φ(2n x, 2n y)

(57)

for all x, y ∈ S and n ∈ N. So, by (36), (54) and (55), ∥DT (x, y)∥ = 0 for all x, y ∈ S. Thus the mapping T : S → X is nonic.



Similar to Theorem 5, one can prove the following result. Theorem 6 Assume that the multiplication by 2n (n ∈ N) be bijective in S. Let φ : S 2 → R+ be a function such that (x y ) x, y ∈ S. (58) lim |2|9n φ n , n = 0, n→∞ 2 2 366

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Assume also that f : S → X be a mapping such that ∥Df (x, y)∥ ≤ φ(x, y),

x, y ∈ S.

Then there exists a unique nonic mapping T : S → X such that { ( ) ( x )} x 9(n+1) ∥f (x) − T (x)∥ ≤ max δ , sup |2| δ n+2 , 2 n∈N0 2

(59)

x ∈ S,

(60)

where δ(x) is defined as in Theorem 5. Proof.

From (55), we have

( x ) (x)

, x ∈ S. (61)

f (x) − 29 f

≤δ 2 2 By Corollary 3, there exists a unique mapping T : S → X such that T (x) = 29 T ( x2 ) and (60) holds. By (8) in Corollary 1, ( ) t F n 9n T (x) := lim (Lφ ) (f )(t) = lim 2 f , x ∈ S. (62) n→∞ n→∞ 2n 

The rest of the proof is similar to the proof of Theorem 5.

Corollary 4 Let S be a non-Archimedean normed space and X be a complete non-Archimedean normed space with |2| < 1. Let ϵ, λ be positive numbers with λ ̸= 9, and f : S → X be a mapping satisfying ∥Df (x, y)∥ ≤ ϵ(∥x∥λ + ∥y∥λ ), Then there exists a unique nonic mapping T : S → X such that  2ϵ∥x∥λ   ,  |9!|2 · |2|9 ∥f (x) − T (x)∥ ≤ 2ϵ∥x∥λ    , |9!|2 · |2|λ Proof.

x, y ∈ S.

λ > 9, x ∈ S; λ < 9, x ∈ S.

Let φ : S 2 → R+ be defined by φ(x, y) = ϵ(∥x∥λ + ∥y∥λ ) for all x, y ∈ S. Then the corollary is followed 

from Theorems 5 and 6. Similar to Corollary 4, one can obtain the following corollary.

Corollary 5 Let S be a non-Archimedean normed space and X be a complete non-Archimedean normed space with |2| < 1. Let ϵ, λ, µ be positive numbers with λ + µ ̸= 9, and f : S → X be a mapping satisfying ∥Df (x, y)∥ ≤ ϵ∥x∥λ · ∥y∥µ , Then there exists a unique nonic mapping T : S → X such that  ϵ∥x∥λ+µ   ,  |9!|2 · |2|9 ∥f (x) − T (x)∥ ≤ λ+µ ϵ∥x∥    , |9!|2 · |2|λ+µ

x, y ∈ S.

λ + µ > 9, x ∈ S; λ + µ < 9, x ∈ S.

References [1] R. P. Agarwal, B. Xu, W. Zhang, Stability of functional equations in single variable, J. Math. Anal. Appl., 288(2003), 852–869. 367

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[2] T. Aoki, On the stability of the linear transformation in Banach spaces, J. Math. Soc. Japan, 2(1950), 64–66. [3] J. Brzd¸ek, K. Ciepli´ nski, A fixed point approach to the stability of functional equations in non-Archimedean metric spaces, Nonlinear Analysis, 74(2011), 6861–6867. [4] J. Brzd¸ek, K. Ciepli´ nski, A fixed point theorem and the Hyers–Ulam stability in non-Archimedean spaces, J. Math. Anal. Appl., 400(2013), 68–75. [5] L. C˘adariu, V. Radu, On the stability of the Cauchy functional equation: A fixed point approach, Grazer Math. Ber, 346(2004), 43–52. [6] L. C˘adariu, V. Radu, Fixed point methods for the generalized stability of functional equations in a single variable, Fixed Point Theory Appl., 2008(2008), Art ID 749392, 1–15. [7] P. G˘avrut¸a, A generalization of the Hyers-Ulam-Rassias stability of approximately additive mappings, J. Math. Anal. Appl., 184(1994), 431–436. [8] M. E. Gordji, H. Khodaei, A. Najati, Fixed points and quartic functional equations in β-Banach modules, Results Math., 62(2012), 137–155. [9] M. Hossz´ u, On the Fr´echet’s functional equation, Bul. Isnt. Politech. Iasi, 10(1964), 27–28. [10] D. H. Hyers, On the stability of the linear functional equation, Proc. Nat. Acad. Sci. USA, 27(1941), 222–224. [11] D. H. Hyers, G. Isac, Th. M. Rassias, Stability of Functional Equations in Several Variables, Birkhauser, Boston, Basel, Berlin, 1998. [12] S. -M. Jung, Hyers–Ulam–Rassias Stability of Functional Equations in Nonlinear Analysis, Springer, New York, 2011. [13] P. I. Kannappan, Functional Equations and Inequalities with Applications, Springer, New York, 2009. [14] H. A. Kenary, S. Y. Jang, C. Park, A fixed point approach to the Hyers–Ulam stability of a functional equation in various normed spaces, Fixed Point Theory Appl., 2011(2011), Article ID 67, 1–14. [15] A. Khrennikov, Non-Archimedean Analysis: Quantum Paradoxes, Dynamical Systems and Biological Models, Kluwer Academic Publishers, Dordrecht, 1997. [16] Z. Moszner, On the stability of functional equations, Aequationes Math., 77(2009), 33–88. [17] Th. M. Rassias, On the stability of the linear mapping in Banach spaces, Proc. Am. Math. Soc., 72(1978), 297–300. [18] R. Saadati, S. M. Vaezpour, C. Park, The stability of the cubic functional equation in various spaces, Math. Commun., 16(2011), 131–145. [19] L. Sz´ekelyhidi, Convolution Type Functional Equations on Topological Abelian Groups, World Scientific, Singapore, 1991. [20] S. M. Ulam, A Collection of the Mathematical Problems, Interscience, New York, 1960. [21] T. Z. Xu, On fuzzy approximately cubic type mapping in fuzzy Banach spaces. Information Sciences, 278(2014), 56–66. [22] T. Z. Xu, C. Wang, Th. M. Rassias, On the stability of multi-additive mappings in non-Archimedean normed spaces, Journal of Computational Analysis and Applications, 18(2015), 1102–1110.

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Global Attractivity and Periodicity Behavior of a Recursive Sequence E. M. Elsayed1,2 , and Abdul Khaliq1 Mathematics Department, Faculty of Science, King Abdulaziz University, P. O. Box 80203, Jeddah 21589, Saudi Arabia. 2 Department of Mathematics, Faculty of Science, Mansoura University, Mansoura 35516, Egypt. 1

E-mails: [email protected], [email protected] ABSTRACT Our aim in this paper is to study the global stability character and the periodic nature of the solutions of the difference equation bxn−k + cxn−s , n = 0, 1, ..., xn+1 = axn−l + d + exn−t where the initial conditions x−r , x−r+1 , x−r+2 , ..., x0 are arbitrary positive real numbers, r = max{l, k, s, t} is nonnegative integer and a, b, c, d, e are positive constants. Keywords: stability, periodic solutions, global attractor, difference equations. Mathematics Subject Classification: 39A10 –––––––––––––––––

1. INTRODUCTION Our goal in this paper is to investigate the global stability character and the periodicity of the solutions of the difference equation bxn−k + cxn−s , n = 0, 1, ..., (1) xn+1 = axn−l + d + exn−t where the initial conditions x−r , x−r+1 , x−r+2 , ..., x0 are arbitrary positive real numbers, r = max{l, k, s, t} is nonnegative integer and a, b, c, d, e are positive constants. Recently there has been a lot of interest in studying the global attractivity, the boundedness character and the periodicity nature of nonlinear difference equations see for example [1-20]. The study of the nonlinear rational difference equations is interesting and attractive to many researchers working in this field It is quite challenging and rewarding, many real life phenomena are modelling using these equations. Examples from economy, biology,etc. may be obtained in [3,7,11,12] The study of some properties of these equations via the global attractivity, the boundedness and the periodicity of these equations is of great interest. For examples in the articles [11,12,15]. Recently, many researchers have investigated the behavior of the solution of difference equations for example: In [1] Ahmed investigated the behavior of the solutions of the difference equation xn−2k+1 . xn+1 = Q ±1 ± ki=1 xn−2i+1 Elabbasy et al. [8] studied the boundedness, global stability, periodicity character and gave the solution of some special cases of the difference equation αxn−k . xn+1 = Qk β + γ i=0 xn−i

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Elabbasy et al. [9] investigated the global stability, periodicity character and gave the solution of some special cases of the difference equation dxn−l xn−k . xn+1 = a + cxn−s − b Yalçınkaya [32] has studied the following difference equation xn+1 = α +

xn−m . xkn

For some related work see [21—35]. Here, we recall some basic definitions and some theorems that we need in the sequel. Let I be some interval of real numbers and let F : I k+1 → I, be a continuously differentiable function. Then for every set of initial conditions x−k , x−k+1 , ...,x0 ∈ I, the difference equation (2) xn+1 = F (xn , xn−1 , ..., xn−k ), n = 0, 1, ..., has a unique solution {xn }∞ n=−k .

A point x ∈ I is called an equilibrium point of Eq.(2) if x = f (x, x, ..., x).

That is, xn = x for n ≥ 0, is a solution of Eq.(2), or equivalently, x is a fixed point of f . Definition 1.1. (Periodicity)

A sequence {xn }∞ n=−k is said to be periodic with period p if xn+p = xn for all n ≥ −k.

Definition 1.2. (Stability)

(i) The equilibrium point x of Eq.(2) is locally stable if for every x−k , x−k+1 , ..., x−1 ,x0 ∈ I with

> 0, there exists δ > 0 such that for all

|x−k − x| + |x−k+1 − x| + ... + |x0 − x| < δ, we have |xn − x|
0, such that for all x−k , x−k+1 , ..., x−1 , x0 ∈ I with |x−k − x| + |x−k+1 − x| + ... + |x0 − x| < γ, we have lim xn = x.

n→∞

(iii) The equilibrium point x of Eq.(2) is global attractor if for all x−k , x−k+1 , ..., x−1 , x0 ∈ I, we have lim xn = x.

n→∞

(iv) The equilibrium point x of Eq.(2) is globally asymptotically stable if x is locally stable, and x is also a global attractor of Eq.(2). (v) The equilibrium point x of Eq.(2) is unstable if x is not locally stable.

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The linearized equation of Eq.(2) about the equilibrium x is the linear difference equation k P

yn+1 =

i=0

∂F (x,x,...,x) yn−i . ∂xn−i

(3)

Theorem A [26] Assume that p, q ∈ R and k ∈ {0, 1, 2, ...}. Then |p| + |q| < 1, is a sufficient condition for the asymptotic stability of the difference equation xn+1 + pxn + qxn−k = 0, n = 0, 1, ... . Remark 1. Theorem A can be easily extended to a general linear equations of the form xn+k + p1 xn+k−1 + ... + pk xn = 0, n = 0, 1, ...,

(4)

where p1 , p2 , ..., pk ∈ R and k ∈ {1, 2, ...}. Then Eq.(4) is asymptotically stable provided that k P

i=1

|pi | < 1.

Consider the following equation xn+1 = g(xn , xn−1 , ...xn−K )

n = 0, 1, 2...

(5)

The following theorem will be useful for the proof of our results in this paper. Theorem B [27]: Let [α, β] be an interval of real numbers and assume that g : [α, β]k+1 → [α, β], is a continuous function satisfying the following properties : (a) g(x1 , x2 , ..., xk+1 ) is non-increasing in one component (for example xσ ) for each xr (r 6= σ) in [α, β], and is non-increasing in the remaining components for each xσ ∈ [α, β]; (b) If (m, M ) ∈ [α, β] × [α, β] is a solution of the system

M = g(m, m, ..., m, M, m, ...m, m) and m = g(M, M, ..., M, m, M, ..., M, M ), then m = M. Then Eq.(5) has a unique equilibrium x ∈ [α, β] and every solution of Eq.(5) converges to x.

2. LOCAL STABILITY OF THE EQUILIBRIUM POINT OF EQ.(1) In this section we study the local stability character of the solutions of Eq.(1). The equilibrium points of Eq.(1) are given by the relation bx + cx . x = ax + cd + dex If a 6= 1, then the equilibrium points of Eq.(1) is given by x=0

and

x=

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b + c + d(a − 1) . e(1 − a)

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Let f : (0, ∞)4 −→ (0, ∞) be a function defined by f (u0 , u1 , u2 , u3 ) = au0 + Therefore at x =

bu1 + cu2 . d + eu3

b+c+d(a−1) e(1−a)

∂f (x, x, x, x) ∂u0 ∂f (x, x, x, x) ∂u2

∂f (x, x, x, x) b(a − 1) = −c1 , =− ∂u1 (b + c) c(a − 1) ∂f (x, x, x, x) = −c2 , = − = (a−1)(b+c−d+ad) = −c3 (b+c) (b + c) ∂u3 = a = −c0 ,

Then we see that at x = 0 ∂f (x, x, x, x) ∂u0 ∂f (x, x, x, x) ∂u1

∂f (x, x, x, x) b = = −c1 ∂u1 d c ∂f (x, x, x, x) = −c2 , = 0 = −c3 d ∂u1

= a = −c0 , =

Then the linearized equation of Eq.(1) about x is yn+1 + c0 yn−l + c1 yn−k + c2 yn−s + c3 yn−t = 0. Theorem 2.1. Assume that 1
4ad. Therefore Inequality (7) holds. Second suppose that Inequality (7) is true. We will show that Eq.(1) has a prime period two solution. Assume that √ √ −eAB + ξ ξ , = p = −e(a+1)(b+c+d+ad)+ 2ae2 (a+1) 2ae2 A

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and q=

√ −eAB− ξ 2ae2 A , where

A = (a + 1), B = (a + b + c + d + ad)

where ξ = e2 (a + 1)2 (b + c + d + ad)2 − 4ade2 (a + 1)(b + c + d + ad).

We see from Inequality (7) that

e2 (a + 1)2 (b + c + d + ad)2 − 4ade2 (a + 1)(b + c + d + ad) > 0, then after dividing by e2 (a + 1)(b + c + d + ad) we see that ⇒

(a + 1)(b + c + d + ad) > 4ad,

Therefore p and q are distinct real numbers. Set x−l x−s

= p, x−l+1 = q, , x−k = q, x−k+1 = p, = p, x−s+1 = q, x−t = p, x−t+1 = q, and x0 = q.

We wish to show that x1 = x−1 = p and

x2 = x0 = q.

It follows from Eq.(1.1) that x1 = ax−l +

(b + c)p bx−k + cx−s bp + cp = ap + = ap + d + ex−t d + ep d + ep ³ √ ´ ξ (b + c) −eAB+ 2ae2 A ³ = ap + √ ´ . ξ d + e −eAB+ 2 2ae A

Multiplying the denominator and numerator of the right side by 2ae2 A gives x1 = ap +

√ (b+c)(−eAB+ ξ) √ , 2 2 2ae Ad−e AB+e ξ

√ Multiplying the denominator and numerator of the right side by {2ae2 Ad − e2 AB − e ξ} x1

= ap +

√ √ (b+c)(−eAB+ ξ)(2ae2 Ad−e2 AB−e ξ) √ √ , (2ae2 Ad−e2 AB+e ξ)(2ae2 Ad−e2 AB−e ξ)

= ap +

√ (b+c)[−2ade3 A2 B+e3 A2 B 2 −e(e2 A2 B 2 −4ade2 AB)+2ade2 A ξ] , √ 2 (2ade2 A−e2 AB)2 −(e ξ )

= ap +

√ (b+c)[2ade3 AB(2−A)+2ade2 A ξ] e4 A2 (4a2 d2 +B 2 −4adB)−e4 A2 B 2 +4ade4 AB ,

Replacing A = (a + 1) and B = (b + c + d + ad) in denominator of above equation gives x1

= ap + = ap + = ap −

√ (b+c)[2ade3 AB(1−a)+2ade2 A ξ ] 4a2 d2 e4 (a+1)2 −4ade4 (a+1)2 (b+c+d+ad)+4ade4 (1+a)(b+c+d+ad) √ (b+c)[2ade3 AB(1−a)+2ade2 A ξ ] 2 2 4 2 4 2 4a d e (a+1) −4ade (a+1) (b+c+d+ad)+4ade4 (1+a)(b+c+d+ad) √ (b+c)[2ade3 AB(1−a)+2ade2 A ξ ] 4a2 de2 (a+1)(b+c)

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Dividing numerator and denominator by(b + c) we get x1 = ap −

√ 2ade3 AB(1−a)+2ade2 A ξ 4a2 de2 (a+1)

Now inserting the value of p we get √ √ −eAB + ξ eB(1 − a) + ξ x1 = − 2e2 A 2ae2 ³ √ 1 −eAB+ ξ eB(1−a)+√ξ ´ = − = A a 2e2 √ −eaB(a + 1) − eB + eBa2 − ξ = 2ae2 (a + 1)

= ap −

eB(1 − a) + 2ae2

√

ξ

√ √ −eAB+a ξ−eB(1−a)(1+a)−(a+1) ξ 2ae2 (a+1)

putting the value of B = (b + c + d + ad) we get x1 =

√ −e(b+c+d+ad)(a+1)− ξ 2 2ae (a+1)

=q

Similarly as before one can easily show that x2 = p. Then it follows by induction that x2n = p

and

x2n+1 = q

for all

n ≥ −1.

Thus Eq.(1) has the positive prime period two solution ...,p,q,p,q,..., where p and q are the distinct roots of the quadratic equation (13) and the proof is complete. The following Theorems can be proved similarly. Theorem 3.2. Eq.(1) has a prime period two solutions if and only if e2 (a + 1)2 (d + ad + b + c)2 − 4e2 (ad + b + c)(a + 1)(d + ad + b + c) > 0, t − odd , l, k, s − even. Theorem 3.3. Eq.(1) has a prime period two solutions if and only if e2 (a + 1)2 (d + ad − b − c)2 − 4e2 ad(a + 1)(d + ad − b − c) > 0 , l − even , s, k, t − odd. Theorem 3.4. Eq.(1) has a prime period two solutions if and only if e2 (d − ad + b + c)2 (a − 1)2 − 4e2 (a − 1)2 (b + c)(d − ad + b + c) > 0, l, t − odd, s, k − even. Theorem 3.5. Eq.(1) has a prime period two solutions if and only if e2 (a + 1)2 (b + c − d − ad)2 + 4ae2 (a + 1)(b + c − d − ad)(d − b − c) > 0, l, t − even, s, k − odd.

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4. GLOBAL ATTRACTIVITY OF THE EQUILIBRIUM POINT OF EQ.(1) In this section we investigate the global attractivity character of solutions of Eq.(1). Theorem 4.1. The equilibrium point x of Eq.(1) is global attractor. Proof: Let p, q are a real numbers and assume that f : [p, q]4 −→ [p, q] be a function defined by f (u0 , u1 , u2 , u3 ) = au0 +

bu1 + cu2 . d + eu3

We can easily see that the function f (u0 , u1 , u2 , u3 ) increasing in u0 , u1 , u2 and decreasing in u3 . Suppose that (m, M ) is a solution of the system m = f (m, m, m, M )

and

M = f (M, M, M, m).

Then from Eq.(1), we see that m = am +

(b + c)m , d + eM

That is 1−a=

b+c , d + eM

M = aM +

1−a=

(b + c)M , d + em

b+c , d + em

or, b+c b+c = , d + eM d + em then d + eM = d + em. Thus M = m. It follows by the Theorem B that x is a global attractor of Eq.(1) and then the proof is complete.

5.

NUMERICAL EXAMPLES

For confirming the results of this paper, we consider numerical examples which represent different types of solutions to Eq. (1). Example 1. We assume l = 3, k = 2, s = 3, t = 2 x−3 = 7, x−2 = 2, x−1 = 1, x0 = 9, a = 0.1, b = 0.2, c = 0.9, d = 0.6 e = 0.3. See Fig. 1. Example 2. See Fig. 2, since l = 4, k = 3, x−4 = 12, x−3 = 7, x−2 = 9, x−1 = 10, x0 = 5, a = 0.9, b = 2, c = 7, d = 3. 2.5

7 x 10 plot of x(n+1)= a.X(n−l)+((b.X(n−k)+c.X(n−s))/((d+e.X(n−t))))

plot of x(n+1)= a.X(n−l)+((b.X(n−k)+c.X(n−s))/((d+e.X(n−t)))) 20

2

15

x(n)

x(n)

1.5

1 10 0.5

0

0

200

400

600

800

1000

n

5

0

200

400

600

800

1000

n

Figure 1. Figure 2. Example 3. We consider l = 3, k = 2, x−3 = 12, x−2 = 7, x−1 = 9, x0 = 10, a = 0.3, b = 1.5, c = 11, d = 8. See Fig. 3.

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Example 4. See Fig. 4, since l = 3, k = 4, x−4 = 12, x−3 = 7, x−2 = 9, x−1 = 10, x0 = 5, a = 0.6, b = 2, c = 7, d = 4. plot of x(n+1)= a.X(n−l)+((b.X(n−k)+c.X(n−s))/((d+e.X(n−t)))) 12

10

10

8

8

x(n)

x(n)

plot of x(n+1)= a.X(n−l)+((b.X(n−k)+c.X(n−s))/((d+e.X(n−t)))) 12

6

6

4

4

2

2

0

0

200

400

600

800

1000

n

0

0

200

400

600

800

1000

n

Figure 3.

Figure 4.

Acknowledgements This article was funded by the Deanship of Scientific Research (DSR), King Abdulaziz University, Jeddah. The authors, therefore, acknowledge with thanks DSR technical and financial support.

References [1] A.M. Ahmed, and A.M. Youssef, A solution form of a class of higher-order rational difference equations, J. Egyp. Math. Soc., 21 (2013), 248—253. [2] M. Aloqeili, Global stability of a rational symmetric difference equation, Appl. Math. Comp., 215 (2009) 950-953. [3] M. Aprahamian, D. Souroujon, and S. Tersian, Decreasing and fast solutions for a second-order difference equation related to Fisher-Kolmogorov’s equation, J. Math. Anal. Appl. 363 (2010), 97-110. xn + αxn−1 , Appl. [4] H. Chen and H. Wang, Global attractivity of the difference equation xn+1 = β + xn Math. Comp., 181 (2006) 1431—1438. axn−1 , Appl. Math. [5] C. Cinar, On the positive solutions of the difference equation xn+1 = 1 + bxn xn−1 Comp., 156 (2004) 587-590. [6] S. E. Das and M.Bayram, On a System of Rational Difference Equations, World Applied Sciences Journal 10(11) (2010), 1306-1312. [7] Q. Din, and E. M. Elsayed, Stability analysis of a discrete ecological model, Computational Ecology and Software 4 (2) (2014), 89—103. αxn−k , [8] E. M. Elabbasy, H. El-Metwally and E. M. Elsayed, On the difference equations xn+1 = Qk β + γ i=0 xn−i J. Conc. Appl. Math., 5(2) (2007), 101-113.

[9] E. M. Elabbasy, H. El-Metwally and E. M. Elsayed, Qualitative behavior of higher order difference equation, Soochow Journal of Mathematics, 33 (4) (2007), 861-873.

[10] E. M. Elabbasy, H. El-Metwally and E. M. Elsayed, On the Difference Equation xn+1 = a0 xn +a1 xn−1 +...+ak xn−k b0 xn +b1 xn−1 +...+bk xn−k , Mathematica Bohemica, 133 (2) (2008), 133-147.

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[11] H. El-Metwally, Global behavior of an economic model, Chaos, Solitons and Fractals, 33 (2007), 994—1005. [12] H. El-Metwally and M. M. El-Afifi, On the behavior of some extension forms of some population models, Chaos, Solitons and Fractals, 36 (2008), 104—114. [13] H. El-Metwally and E. M. Elsayed, Form of solutions and periodicity for systems of difference equations, Journal of Computational Analysis and Applications, 15(5) (2013), 852-857. [14] E. M. Elsayed, Behavior and expression of the solutions of some rational difference equations, Journal of Computational Analysis and Applications, 15 (1) (2013), 73-81. [15] E. M. Elsayed, Solution for systems of difference equations of rational form of order two, Computational and Applied Mathematics, 33 (3) (2014), 751-765. [16] E. M. Elsayed and H. El-Metwally, Global Behavior and Periodicity of Some Difference Equations, Journal of Computational Analysis and Applications, 19 (2) (2015), 298-309. [17] E. M. Elsayed, Dynamics of a Recursive Sequence of Higher Order, Communications on Applied Nonlinear Analysis, 16 (2) (2009), 37—50. [18] E. M. Elsayed, Qualitative behavior of difference equation of order three, Acta Scientiarum Mathematicarum (Szeged), 75 (1-2), 113—129. [19] E. M. Elsayed, Qualitative behavior of difference equation of order two, Mathematical and Computer Modelling, 50 (2009), 1130—1141. [20] R. Karatas, C. Cinar and D. Simsek, On positive solutions of the difference equation xn+1 = xn−5 , Int. J. Contemp. Math. Sci., Vol. 1, 2006, no. 10, 495-500. 1 + xn−2 xn−5 [21] V. L. Kocic and G. Ladas, Global Behavior of Nonlinear Difference Equations of Higher Order with Applications, Kluwer Academic Publishers, Dordrecht, 1993. [22] M. R. S. Kulenovic and G. Ladas, Dynamics of Second Order Rational Difference Equations with Open Problems and Conjectures, Chapman & Hall / CRC Press, 2001. [23] A. S. Kurbanli, On the Behavior of Solutions of the System of Rational Difference Equations, World Applied Sciences Journal, 10 (11) (2010), 1344-1350. [24] R. Memarbashi, Sufficient conditions for the exponential stability of nonautonomous difference equations, Appl. Math. Letter 21 (2008) 232—235. [25] A. Neyrameh, H. Neyrameh, M. Ebrahimi and A. Roozi, Analytic solution diffusivity equation in rational form, World Applied Sciences Journal, 10 (7) (2010), 764-768. yn with A < 0, Appl. Math. [26] M. Saleh and M. Aloqeili, On the difference equation yn+1 = A + yn−k Comp., 176(1) (2006), 359—363. xn [27] M. Saleh and M. Aloqeili, On the difference equation xn+1 = A + , Appl. Math. Comp., xn−k 171 (2005), 862-869. xn−1 [28] T. Sun and H. Xi, On convergence of the solutions of the difference equation xn+1 = 1 + , xn J. Math. Anal. Appl. 325 (2007) 1491—1494. [29] C. Wang, S. Wang, L. LI, Q. Shi, Asymptotic behavior of equilibrium point for a class of nonlinear difference equation, Advances in Difference Equations, Volume 2009, Article ID 214309. 8pages. [30] C. Wang, S. Wang, Z. Wang, H. Gong, R. Wang, Asymptotic stability for a class of nonlinear difference equation, Discrete Dynamics in Natural and Society, Volume 2010, Article ID 791610, 10pages. [31] I. Yalçınkaya, On the global asymptotic stability of a second-order system of difference equations, Discrete Dynamics in Nature and Society, Vol. 2008, Article ID 860152, 12 pages, doi: 10.1155/2008/ 860152.

378

E. M. Elsayed et al 369-379

J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 22, NO.2, 2017, COPYRIGHT 2017 EUDOXUS PRESS, LLC

[32] I. Yalçınkaya, On the difference equation xn+1 = α + xn−m , Discrete Dynamics in Nature and xk n Society, Vol. 2008, Article ID 805460, 8 pages, doi: 10.1155/2008/ 805460. [33] E. M. E. Zayed and M. A. El-Moneam, On the rational recursive sequence xn+1 = Communications on Applied Nonlinear Analysis, 12 (4) (2005), 15—28. [34] E. M. E. Zayed and M. A. El-Moneam, On the rational recursive sequence xn+1 = Comm. Appl. Nonlinear Analysis, 12 (2005), 15-28.

α+βxn +γxn−1 A+Bxn +Cxn−1 ,

αxn +βxn−1 +γxn−2 +δxn−3 Axn +Bxn−1 +Cxn−2 +Dxn−3 ;

[35] B. G. Zhang, C. J. Tian and P. J. Wong, Global attractivity of difference equations with variable delay, Dyn. Contin. Discrete Impuls. Syst., 6(3) (1999), 307—317.

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TABLE OF CONTENTS, JOURNAL OF COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 22, NO. 2, 2017

Fuzzy Analytical Hierarchy Process Based On Canonical Representation on Fuzzy Numbers, Yong Deng,…………………………………………………………………………………201 A Quadrature Rule for the Finite Hilbert Transform via Simpson Type Inequalities and Applications, Shunfeng Wang, Na Lu, and Xingyue Gao,…………………………………229 A Quadrature Formula in Approximating the Finite Hilbert Transform via Perturbed Trapezoid Type Inequalities, Shunfeng Wang, Xingyue Gao, and Na Lu,……………………………239 Pointwise Superconvergence of the Displacement of the Six-Dimensional Finite Element, Yinsuo Jia, and Jinghong Liu,………………………………………………………………247 Estimates for Discrete Derivative Green's Function for Elliptic Equations in Dimensions Seven and Up, Jinghong Liu, and Yinsuo Jia,………………………………………………………255 Existence of Solutions to a Coupled System of Higher-order Nonlinear Fractional Differential Equations with Anti-periodic Boundary Conditions, Huina Zhang, and Wenjie Gao,………262 Iteration Process for Pointwise Lipschitzian Type Mappings in Hyperbolic 2-uniformly Convex Metric Spaces, D. R. Sahu, Samir Dashputre, and Shin Min Kang,…………………………271 Regularity of the American Option Value Function in Jump-Diffusion Model, Sultan Hussain, and Nasir Rehman,…………………………………………………………………………..286 On a Summation Boundary Value Problem for a Second-Order Difference Equations with Resonance, Saowaluk Chasreechai and Thanin Sitthiwirattham,……………………………298 Fuzzy Quadratic Mean Operators and Their Use In Group Decision Making, Jin Han Park, Seung Mi Yu, and Young Chel Kwun,………………………………………………………………310 Sensitivity Analysis for General Nonlinear Nonconvex Set-Valued Variational Inequalities in Banach Spaces, Jong Kyu Kim,………………………………………………………………327 Common Fixed Point Theorems for Non-compatible Self-mappings in b-Fuzzy Metric Spaces, Jong Kyu Kim, Shaban Sedghi, Nabi Shobe, and Hassan Sadati,……………………………336 On Hesitant Fuzzy Filters in BE-Algebras, Young Bae Jun, and Sun Shin Ahn,…………….346

TABLE OF CONTENTS, JOURNAL OF COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 22, NO. 2, 2017 (continued) A Fixed Point Approach to the Stability of Nonic Functional Equation In Non-Archimedean Spaces, Tian-Zhou Xu, Yali Ding, and John Michael Rassias,………………………………359 Global Attractivity and Periodicity Behavior of a Recursive Sequence, E. M. Elsayed, and Abdul Khaliq,…………………………………………………………………………………369

Volume 22, Number 3 ISSN:1521-1398 PRINT,1572-9206 ONLINE

March 2017

Journal of Computational Analysis and Applications EUDOXUS PRESS,LLC

Journal of Computational Analysis and Applications ISSNno.’s:1521-1398 PRINT,1572-9206 ONLINE SCOPE OF THE JOURNAL An international publication of Eudoxus Press, LLC (fourteen times annually) Editor in Chief: George Anastassiou Department of Mathematical Sciences, University of Memphis, Memphis, TN 38152-3240, U.S.A [email protected] http://www.msci.memphis.edu/~ganastss/jocaaa The main purpose of "J.Computational Analysis and Applications" is to publish high quality research articles from all subareas of Computational Mathematical Analysis and its many potential applications and connections to other areas of Mathematical Sciences. Any paper whose approach and proofs are computational,using methods from Mathematical Analysis in the broadest sense is suitable and welcome for consideration in our journal, except from Applied Numerical Analysis articles. Also plain word articles without formulas and proofs are excluded. The list of possibly connected mathematical areas with this publication includes, but is not restricted to: Applied Analysis, Applied Functional Analysis, Approximation Theory, Asymptotic Analysis, Difference Equations, Differential Equations, Partial Differential Equations, Fourier Analysis, Fractals, Fuzzy Sets, Harmonic Analysis, Inequalities, Integral Equations, Measure Theory, Moment Theory, Neural Networks, Numerical Functional Analysis, Potential Theory, Probability Theory, Real and Complex Analysis, Signal Analysis, Special Functions, Splines, Stochastic Analysis, Stochastic Processes, Summability, Tomography, Wavelets, any combination of the above, e.t.c. "J.Computational Analysis and Applications" is a peer-reviewed Journal. See the instructions for preparation and submission of articles to JoCAAA. Assistant to the Editor: Dr.Razvan Mezei,Lenoir-Rhyne University,Hickory,NC

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Editorial Board Associate Editors of Journal of Computational Analysis and Applications Francesco Altomare Dipartimento di Matematica Universita' di Bari Via E.Orabona, 4 70125 Bari, ITALY Tel+39-080-5442690 office +39-080-3944046 home +39-080-5963612 Fax [email protected] Approximation Theory, Functional Analysis, Semigroups and Partial Differential Equations, Positive Operators.

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J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 22, NO.3, 2017, COPYRIGHT 2017 EUDOXUS PRESS, LLC

PERIODIC ORBITS OF SINGULAR RADIALLY SYMMETRIC SYSTEMS SHENGJUN LI

1,2 ,

WULAN LI

3

AND

YIPING FU

1

Abstract. We study the existence of periodic orbits of planar radially symmetric systems with a singularity. These orbits have periods which are large integer multiples of the period of the forcing, and rotate exactly once around the origin in their period time. The proof is based on the use of topological degree theory and a fixed point theorem in cones.

1. Introduction In the paper [11], Fonda and J.Ure˜ na have studied the periodic, subharmonic and quasi-periodic orbits for the radially symmetric system x (1.1) x ¨ + f (t, |x|) = 0, x ∈ R2 \ {0}, |x| where f ∈ C((R/T Z) × (0, ∞), R) may be singular at the origin. As mentioned in [10], many phenomena of the nature obey to laws of (1.1), such as the Newtonian equation for the motion of a particle subjected to the gravitational attraction of a sun which lies at the origin. Setting ρ(t) = |x(t)|, they proved the following result: Theorem 1.1 Suppose that f (t, ρ) > 0 for t ∈ [0, T ], ρ > 0 and satisfies the following conditions: (A1 ) lim f (t, ρ)/ρ = 0, for a.e. t ∈ R. ρ→∞

(A2 ) There exists some function h ∈ L1loc (R) and some number r0 > 0 such that |f (t, ρ)| ≤ h(t)ρ, on R × [r0 , +∞]. Then, there exists a connect set C of T -radially periodic solutions of (1.1) which goes from zero to infinity. We look for solutions x(t) ∈ R2 which never attain the singularity, in the sense that x(t) 6= 0,

for every t ∈ R.

Using the same idea in [8], we may write the solutions of (1.1) in polar coordinates x(t) = ρ(t)(cos ϕ(t), sin ϕ(t)). 2000 Mathematics Subject Classification. Primary 34C25. Key words and phrases. Periodic orbits, singular radially symmetric systems, topological degree theory, fixed point theorem in cone. 1

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2

SHENGJUN LI,

WULAN LI

AND

YIPING FU

Then we have the collisionless orbits if ρ(t) > 0 for every t. Moreover, equation (1.1) is equivalent to the following system ( 2 ρ¨ + f (t, ρ) − µρ3 = 0, (1.2) ρ2 ϕ˙ = µ, where µ is the angular momentum of x(t). Recall that µ is constant in time along any solution. If x is a T -radially periodic, then ρ must be T -periodic. We will prove the existence of a T -periodic solution ρ of the first equation in (1.2). We thus consider the boundary value problem ( 2 ρ¨ + f (t, ρ) = µρ3 , (1.3) ρ(0) = ρ(T ), ρ(0) ˙ = ρ(T ˙ ). Let µ = 0, (1.3) can be written the singular T -periodic problem (1.4)

ρ¨ + f (t, ρ) = 0.

The question about the existence of non-collision periodic orbits for scalar equations and dynamical systems with singularities has attracted much attention of many researchers over many years. See[5, 7, 12, 13, 15, 24]. Usually, the proof is based on variational approach [1, 2, 6, 16, 22], the method of upper and lower solutions [3, 21], some fixed point theorems [19, 26, 27, 28, 29] or the topological degree theory [17, 18, 23, 30]. In particular, several existence results for the following scalar differential equation (1.5)

x ¨ + a(t)x = f (t, x)

has been established in [23, 25, 27]. Note that (1.5) is a nonlinear perturbation of Hill equation x ¨ + a(t)x = 0. Moreover, it has been found that a particular case of (1.5), the Ermakov–Pinney equation, plays an important role in studying the Lyapunov stability of periodic solutions of Lagrangian equations [20]. Our main motivation is to obtain by the above papers [9, 17, 27], by the use of topological degree theory and a well-known fixed point theorem in cones, we prove the existence of large-amplitude periodic orbits whose minimal period is an integer multiple of T , and rotate exactly once around the origin in their period time. The rest of this paper is organized as follows. In Section 2, some preliminaries are given. In Section 3, , we give the main results. 2. Preliminaries We say that the scale linear equation (2.1)

x ¨ + a(t)x = 0

is nonresonant if its unique T -periodic solution is the trivial one. When (2.1) is nonresonant, as a consequence of Fredholm’s alternative, the nonhomogeneous equation x ¨ + a(t)x = h(t)

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admits a unique T -periodic solution which can be written as Z T G(t, s)h(s)ds, x(t) = 0

where G(t, s) is the Green’s function of (2.1), associated with periodic boundary conditions (2.2)

x(0) = x(T ),

x(0) ˙ = x(T ˙ ).

Throughout this paper, we always assume that the following standing hypothesis is satisfied: (H) a(t) is a continuous T -function and the Green’s function of (2.1) is positive for all (t, s) ∈ [0, T ] × [0, T ]. In other words, the strict anti-maximum principle holds for (2.1)-(2.2). It is prove in [25] that if a(t) satisfies a  0 and λ1 (a) > 0, then condition (H) is satisfied; here the notation a  0 means that a(t) ≥ 0 for all t ∈ [0, T ] and a(t) > 0 for t in a subset of positive measure, λ1 (a) denotes the first anti-periodic eigenvalue of x00 + (λ + a(t))x = 0 subject to the anti-periodic boundary conditions x(0) + x(T ) = 0,

x(0) ˙ + x(T ˙ ) = 0.

Now we make condition (H) clear. When a(t) ≡ k 2 , condition (H) is equivalent to saying that 0 < k 2 ≤ λ1 = (π/T )2 , where λ1 is the first eigenvalue of the homogeneous equation x00 + k 2 x = 0 with Dirichlet boundary conditions x(0) = x(T ) = 0. For a non-constant function a(t), there is an Lp -criterion proved in [25]. To describe these, we use k · kq to denote the usual Lq -norm over (0, T ) for any given exponent q ∈ [1, ∞]. The conjugate exponent of q is denoted by p : p1 + 1q = 1. Let M(q) denote the best Sobolev constant in the following inequality Ckuk2q ≤ ku0 k22

for all u ∈ H01 (0, T ).

The explicit formula for M(q) is ( 1−2/q   2π

M(q) =

qT 1+2/q 4 T,

2 q+2

Γ(1/q) Γ(1/2+1/q)

2

,

for 1 ≤ q < ∞, for q = ∞,

where Γ(·) is the Gamma function of Euler. Let us define (2.3) A = {a ∈ Lp [0, T ] : a  0, kakp < M(2q) for some 1 ≤ p ≤ +∞} Lemma 2.1[25] Assume that a(t) ∈ A, then (2.1) satisfies the standing hypothesis (H), i.e, G(t, s) > 0 for all (t, s) ∈ [0, T ] × [0, T ]. Remark 2.2 If p = 1, condition kakp < M(2q) can be weakened to kak1 ≤ M(∞) = 4 by the celebrated stability criterion of Lyapunov. In case p = ∞, condition kakp < M(2q) reads as kak∞ < M(2) = π 2 , which is a well known criterion for the anti-maximum principle used in related literature. In this case,kakp < M(2q) can be weakened to a(t) ≺ π 2 . Under hypothesis (H), we always denote m (2.4) M = max G(t, s), m = min G(t, s), σ= . 0≤s,t≤T 0≤s,t≤T M Thus M > m > 0 and 0 < σ < 1.

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SHENGJUN LI,

WULAN LI

AND

YIPING FU

In order to prove our results, we need two preliminary results. The first one is a well-known fixed point theorem in cones, which can be found in [14]. Theorem 2.3 Let X be a Banach space and K a cone in X. Assume Ω1 , Ω2 are ¯ 2 \Ω2 . Let open subsets of X with 0 ∈ Ω1 , Ω ¯ 2 \Ω1 ) → K T : K ∩ (Ω be a continuous and completely continuous operator such that (i) kT xk ≤ kxk for x ∈ K ∩ ∂Ω1 ; (ii) There exist ψ ∈ K {0} such that x 6= T x + λψ for x ∈ K ∩ ∂Ω2 and λ > 0. ¯ 2 \ Ω1 ). The same conclusion remains valid if (i) Then T has a fixed point in K ∩ (Ω holds on K ∩ ∂Ω2 and (ii) holds K ∩ ∂Ω1 . In applications below, we take X = C[0, T ] with the supremum norm k · k and define K = {x ∈ X : x(t) ≥ 0 for all t ∈ [0, T ] and min ≥ σkxk}. 0≤t≤T

where σ is as in (2.4). One can readily verify that K is a cone in X. Define an operator T : X → X by Z T (T x)(t) = G(t, s)F (s, x(s))ds 0

for x ∈ X and t ∈ [0, T ], where F : [0, T ] × R → [0, ∞) is continuous and G(t, s) is the Green’s function of (2.1). Lemma 2.4 T is well defined and maps X into K. Moreover, T is continuous and completely continuous. Proof It is easy to see that T is continuous and completely continuous since F is a continuous function. Thus, we only need to show that T (X) ⊂ K. Let x ∈ X, then we have Z T min (T x)(t) = min G(t, s)F (s, x(s))ds 0≤x≤T

0≤x≤T

0

T

Z ≥m

F (s, x(s))ds 0

Z

T

= σM

F (s, x(s))ds 0

Z ≥ σ max

0≤x≤T

T

G(t, s)F (s, x(s))ds 0

= σkT xk. This implies that T (X) ⊂ K and the proof is completed. To state the second preliminary result, we recall some notation and terminology from [9]. Let X be a Banach space of functions, such that C 1 ([0, T ]) ⊆ X ⊆ C([0, T ]), with continuous immersions, and set X∗ = {ρ ∈ X : min ρ > 0}. Define the following two operators: D(L) = {ρ ∈ W 2,1 (0, T ) : ρ(0) = ρ(T ), ρ(0) ˙ = ρ(T ˙ )}, (2.5)

L : D(L) ⊂ X → L1 (0, T ),

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Lρ = ρ¨,

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and N : X∗ → L1 (0, T ),

(N ρ)(t) = −f (t, ρ(t))

Taking σ ∈ R not belonging to the spectrum of L, (1.5) can be translated to the fixed problem ρ = (L − σI)−1 (N − σI)ρ. We will say that a set Ω ⊆ X is uniformly positively bounded below if there is a constant δ > 0 such that min ρ ≥ δ for every ρ ∈ Ω. we need the following theorem, which has been proved in [9]. Theorem 2.5 Let Ω be an open bounded subset of X, uniformly positively bounded below. Assume that there is no solution of (1.5), on the boundary ∂Ω, and that deg(I − (L − σI)−1 (N − σI), Ω, 0) 6= 0. Then, there exists a k1 ≥ 1 such that, for every integer k ≥ k1 , systems (1.1) has a periodic solution xk (t) with minimal period kT , which makes exactly one revolution around the origin in the period time kT . The function |xk (t)| is T periodic and, when restricted to [0, T ], it belongs to Ω. Moreover, if µk denotes the angular momentum associated to xk (t), then lim µk = 0.

k→∞

3. Main Results In this section, we state and prove the main results. First we recall that A denotes the set defined by (2.3). Theorem 3.1 Suppose that there exist a(t) ∈ A and 0 < r < R such that (H1 ) −a(t)ρ ≤ f (t, ρ) ≤ σ/r − 1/σr, ∀ρ ∈ [σr, r], (H2 ) f (t, ρ) ≥ 0, ∀ρ ∈ [σR, R]. Then, equation(1.4) has a T -periodic solution, and there exists a k1 ≥ 1 such that, for every integer k ≥ k1 , system (1.1) has a periodic solution with minimal period kT , which makes exactly one revolution around the origin in the period time kT. Moreover, exist constant C > 0 (independent of µ and k) such that 1 < |xk (t)| < C, for every t ∈ R and every k ≥ k1 , C and, if µk denotes the angular momentum associated to xk (t) then lim µk = 0.

k→∞

Now we begin by showing that Theorem 3.1, and use topological degree theory. To this end, we deform (1.4) to a simpler singular autonomous equation ρ¨ +

1 1 ρ − = 0. 2 r ρ

where r is as in Theorem 3.1. In order to apply Theorem 2.5, we consider the µ = 0 and study for τ ∈ [0, 1], the following homotopy equation (3.1)

ρ¨ + f (t, ρ; τ ) = 0,

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associated to periodic boundary conditions ρ(0) = ρ(T ),

ρ(0) ˙ = ρ(T ˙ ),

where f (t, ρ; τ ) = τ f (t, ρ) + (1 − τ )(

ρ 1 − ). r2 ρ

Note that f (t, ρ; τ ) satisfies the conditions: (H01 ) f (t, ρ; τ ) + a(t)ρ ≥ 0, ∀ρ ∈ [σr, R], (H02 ) f (t, ρ; τ ) ≤ 0, ∀ρ ∈ [σr, r] and f (t, ρ; τ ) ≥ 0, ∀ρ ∈ [σR, R]. uniformly with respect to τ ∈ [0, 1]. We need to find a priori estimates for the possible positive T -periodic solutions of (3.1). The important point to be proved is the following. Proposition 3.2 Suppose that there exist a ∈ A and 0 < r < R such that f (t, ρ; τ ) satisfies (H01 ) and (H02 ). Then, equation (3.1) has at least one T -periodic solution. Proof The existence is established using Theorem 2.3. To do so, let us write equation (3.1) as ρ¨ + a(t)ρ = f (t, ρ; τ ) + a(t)ρ. Define the open sets Ω1 = {ρ ∈ X : kρk < r,

Ω2 = {ρ ∈ X : kρk < R}.

Let K be a cone defined by (2.5) and define an operator on K by Z T (Φρ)(t) = G(t, s) [f (s, ρ(s); τ ) + a(t)ρ] ds. 0

¯ R \Ωr ) → C[0, T ] is continuous and completely continuous since Clearly, Φ : K ∩ (Ω f : [0, T ] × [σr, R] × [0, 1] is continuous. Also we have Φ(K) ⊂ K. By the first inequality of condition (H01 ), we have f (t, ρ; τ ) + a(t)ρ ≥ a(t)ρ, ∀ρ ∈ [σr, r]. Let ψ ≡ 1, so ψ ∈ K. Now we prove that (3.2)

ρ 6= Φρ + λρ, ∀ρ ∈ K ∩ ∂Ωr and λ > 0.

Suppose not, that is, suppose there exist ρ0 ∈ K ∩ ∂Ωr and λ0 > 0 such that ρ0 = Φρ0 + λ0 ψ. Now since ρ0 ∈ K ∩ ∂Ωr , then ρ0 (t) ≥ σkρ0 k = σr. Let µ = min ρ0 (t). t∈[0,T ]

Then we have ρ0 (t) = (Φρ0 )(t) + λ0 Z T = G(t, s)[f (s, ρ0 (s); τ ) + a(s)ρ0 (s)]ds + λ0 0 T

Z ≥

G(t, s)a(s)ρ0 (s)ds + λ0 0

Z ≥µ

T

G(t, s)a(s)ds + λ0 = µ + λ0 , 0

RT note 0 G(t, s)a(s)ds = 1. This implies µ ≥ µ + λ0 , a contradiction. Therefore, (3.2) holds. On the other hand, by the second inequality of condition (H02 ), we have f (t, ρ; τ ) + a(t)ρ ≤ a(t)ρ,

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Now we prove that kΦxk ≤ kxk,

(3.3)

x ∈ K ∩ ∂ΩR .

In fact, for any ρ ∈ K ∩ ∂ΩR , we have Z T (Φx)(t) = G(t, s) [f (s, ρ0 (s); τ ) + a(s)ρ(s)] ds 0 T

Z ≤

G(t, s)a(s)ρ(s)ds 0 T

Z ≤

G(t, s)a(s)ds · max ρ(t) = kρk. t∈[0,T ]

0

Therefore, kΦρk ≤ kρk, that is, (3.3) holds. It follows from Theorem 3.3, (3.2) and (3.3) that Φ has a fixed point ρ ∈ K ∩ ¯ R \Ωr ), the proof is finished.  (Ω Proof of Theorem 3.1 Now, from Proposition 3.2, this fixed point is a positive solution of (3.1) satisfying r ≤ kxk ≤ R. Notice the boundary condition ρ(0) ˙ = ρ(T ˙ ). Integrate (3.1) from 0 to T , we get Z T Z T ρ¨(t)dt = − f (t, ρ(t); τ )dt = 0. 0

0 +

Thus kf (t, ρ(t); τ )k1 = 2kf (t, ρ(t); τ )k1 . Since ρ(0) = ρ(T ), there exists t1 ∈ [0, T ] such that ρ(t ˙ 1 ) = 0. Therefore Z t kρk ˙ = max |ρ(t)| ˙ = max ρ¨(s)ds 0≤t≤T 0≤t≤T t1 Z T Z T + f (s, ρ(s); τ ) ds ≤ |f (s, ρ(s); τ )| ds = 2 0

0 T

Z ≤2

|a(s)ρ(s)| ds 0

≤ 2Rkak1 := H. where f + (t, ρ(t); τ ) = max{f (t, ρ(t); τ ), 0}. Define the linear operator L as in (2.5) and the Nemytzkii operator Nτ : X∗ → L1 (0, T ), (Nτ ρ)(t) = −f (t, ρ(t); τ ), (3.1) also can be translated to the fixed problem ρ = (L − σI)−1 (Nτ − σI)ρ,

(3.4)

since L − σI is invertible. Take C = max{1/r, R, H} and let the open bounded in X be Ω = {ρ ∈ X :

1 < ρ(t) < C C

and |ρ(t)| ˙ 0. By Lemma the result of Capietto, Mawhin and Zanolin [4], the Leray-Schauder degree of I −L−1 N (µ, ·) is equal to the Brouwer degree of F, i.e., 1 dL (I − L−1 N (µ, ·, ), Ω, 0) = dB (F, ( , C) × (−C, C)) = 1. C By Theorem 2.1, the proof of Theorem 3.1 is thus completed. It is a direct consequence of Theorem 3.1 taking r and R small and big enough, respectively. We obtain Corollary 3.3 Assume that the following two conditions hold: (H3 ) lim+ f (t, ρ)/ρ = −∞, uniformly for t ∈ [0, T ], ρ→0

(H4 )

lim f (t, ρ)/ρ = +∞, uniformly for t ∈ [0, T ]

ρ→+∞

Then problem (1.1) has the same conclusion of Theorem 3.1. Acknowledgment This work is supported by the National Natural Science Foundation of China (Grant No.11461016), Hainan Natural Science Foundation (Grant No.113001,No.20151002), Scientific Research Development Fund of Wenzhou Medical University(No:QTJ11014) and Zhejiang Provincial Department of Education Research Project(No:Y201328047). References 1. A. Ambrosetti, V. Coti Zelati, Periodic solutions of singular Lagrangian systems, Birkh¨ auser Boston, Boston, MA, 1993. 2. A. Bahri, P. H. Rabinowitz, A minimax method for a class of Hamiltonian systems with singular potentials, J. Funct. Anal. 82 (1989), 412-428. 3. D. Bonheure, C. De Coster, Forced singular oscillators and the method of lower and upper solutions, Topol. Methods Nonlinear Anal. 22 (2003), 297-317. 4. A. Capietto, J. Mawhin, F. Zanolin, Continuation theorems for periodic perturbations of autonomous systems, Trans. Amer. Math. Soc. 329 (1992), 41-72. 5. Z. Cheng, J. Ren, Periodic and subharmonic solutions for Duffing equation with a singularity, Discrete Contin. Dyn. Syst. 32 (2012), 1557C1574. 6. V. Coti Zelati, Periodic solutions for a class of planar, singular dynamical systems, J. Math. Pures Appl. 68 (1989), 109-119. 7. M. A. del Pino, R. F. Man´ asevich, Infinitely many T -periodic solutions for a problem arising in nonlinear elasticity, J. Differential Equations, 103 (1993), 260-277. 8. A. Fonda, R. Toader, Periodic orbits of radially symmetric Keplerian-like systems: A topological degree approach, J. Differential Equations, 244 (2008), 3235-3264. 9. A. Fonda, R. Toader, Periodic solutions of radially symmetric perturbations of Newtonian systems, Proc. Amer. Math. Soc. 140 (2012), 1331-1341.

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SHENGJUN LI et al 393-401

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PERIODIC ORBITS OF SINGULAR RADIALLY SYMMETRIC SYSTEMS

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10. A. Fonda, R. Toader, F. Zanolin, Periodic solutions of singular radially symmetric systems with superlinear growth, Ann. Mat. Pura Appl. 191 (2012), 181-204. 11. A. Fonda, A. J. Ure˜ na, Periodic, subharmonic, and quasi-periodic oscillations under the action of a central force, Discrete Contin. Dyn. Syst. 29 (2011), 169-192. 12. P. Habets, L. Sanchez, Periodic solution of some Li´ enard equations with singularities, Proc. Amer. Math. Soc. 109 (1990), 1135-1144. 13. R. Hakl, P. J. Torres, On periodic solutions of second-order differential equations with attractive-repulsive singularities, J. Differential Equations, 248 (2010), 111-126. 14. M. A. Krasnosel’skii, Positive Solutions of Operator Equations, Noordhoff, Groningen, 1964. 15. A. C. Lazer, S. Solimini, On periodic solutions of nonlinear differential equations with singularities, Proc. Amer. Math. Soc. 99 (1987), 109-114. 16. J. Li, S. Li, Z. Zhang, Periodic solutions for a singular damped differential equation, Bound. Value Probl. 2015, 2015(5). 17. S. Li, F. Liao, J. Sun, Periodic solutions of radially symmetric systems with a singularity, Bound. Value Probl. 2015, 2013(110). 18. S. Li, F. Liao, W. Xing, Periodic solutions of Li´ enard differential equations with singularity, Electron. J. Differential Equation, 151 (2015), 1-12. 19. S. Li, F. Liao, H. Zhu, Periodic solutions of second order non-autonomous differential systems, Fixed Point Theory, 15 (2014), 487-494. 20. R. Ortega, Periodic solution of a Newtonian equation: stability by the third approximation, J. Differential Equations, 128 (1996), 491-518. 21. I. Rachunkov´ a, M. Tvrd´ y, I. Vrko˘ c, Existence of nonnegative and nonpositive solutions for second order periodic boundary value problems, J. Differential Equations, 176 (2001), 445-469. 22. M. Ramos, S. Terracini, Noncollision periodic solutions to some singular dynamical systems with very weak forces, J. Differential Equations, 118 (1995), 121-152. 23. J. Ren, Z. Cheng, S. Siegmund, Positive periodic solution for Brillouin electron beam focusing system, Discrete Contin. Dyn. Syst. Ser. B. 16(2011), 385-392. 24. S. Solimini, On forced dynamical systems with a singularity of repulsive type, Nonlinear Anal. 14 (1990), 489-500. 25. P. J. Torres, Existence of one-signed periodic solutions of some second-order differential equations via a Krasnoselskii fixed point theorem, J. Differential Equations, 190 (2003), 643-662. 26. H. Wang, On the number of positive solutions of nonlinear systems, J. Math. Anal. Appl. 281 (2003) 287-306. 27. H. Wang, Positive periodic solutions of singular systems with a parameter, J. Differential Equations, 249 (2010), 2986-3002. 28. F. Wang, Y. Cui, On the existence of solutions for singular boundary value problem of thirdorder differential equations, Math. Slovaca, 60 (2010), 485-494. 29. F. Wang, F. Zhang, Y. Ya, Existence of positive solutions of Neumann boundary value problem via a convex functional compression-expansion fixed point theorem, Fixed point theory, 11(2010), 395-400. 30. P. Yan, M. Zhang, Higher order nonresonance for differential equations with singularities, Math. Methods Appl. Sci. 26 (2003), 1067-1074. 1 College of Information Sciences and Technology, Hainan University, Haikou, 570228, China 2 School of Matheematical Sciences and Computing Technology, Central South University, Changsha, 410083, China 3 College of Information Science and Computer Engineering, Wenzhou Medical University, Wenzhou, 325035, China E-mail address: [email protected] (S. Li) E-mail address: [email protected] (W. Li) E-mail address: [email protected] (Y. Fu)

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Approximate ternary Jordan ring homomorphisms in ternary Banach algebras M. Eshaghi Gordji1 , Vahid Keshavarz2 , Jung Rye Lee3 , Dong Yun Shin4∗ and Choonkil Park5 1,2

Department of Mathematics, Semnan University, P. O. Box 35195-363, Semnan, Iran 3

Department of Mathematics, Daejin University, Kyeonggi 487-711, Korea 4

5

Department of Mathematics, University of Seoul, Seoul 130-743, Korea

Department of Mathematics, Hanyang Universityl, Seoul 133-791, Korea

e-mail: [email protected], [email protected], [email protected], [email protected], [email protected] Abstract. Let A and B be real ternary Banach algebras. An additive mapping = : (A, [ ]A ) −→ (B, [ ]B ) is called a ternary Jordan homomorphism if =([x, x, x]A ) = [=(x), =(x), =(x)]B for all x ∈ A. In this paper, we investigate the stability and superstability of ternary Jordan ring homomorphisms in ternary Banach algebras by using the fixed point method.

1. Introduction Ternary algebraic operations were considered in the 19th century by several mathematicians. Cayley [5] who introduced the notion of cubic matrix which in turn was generalized by Kapranov, Gelfand and Zelevinskii in 1990 [13]. As an application in physics, the quark model inspired a particular brand of ternary algebraic systems. There are also some applications, although still hypothetical, in the fractional quantum Hall effect, the nonstandard statistics (the anyons), supersymmetric theories, Yang-Baxter equation, etc, (cf. [1, 26]). The comments on physical applications of ternary structures can be found in [2, 8, 9, 21, 22, 23, 26]. Let A and B be ternary Banach algebras. An additive mapping = : (A, [ ]A ) → (B, [ ]B ) is called a ternary ring homomorphism if =([x, y, z]A ) = [=(x), =(y), =(z)]B for all x, y, z ∈ A. An additive mapping = : (A, [ ]A ) → (B, [ ]B ) is called a ternary Jordan ring homomorphism if =([x, x, x]A ) = [=(x), =(x), =(x)]B for all x ∈ A. We say that a functional equation (Q) is stable if any function g satisfying the equation (Q) approximately is near to true solution of (Q). Also, we say that a functional equation is superstable if every approximately solution is an exact solution of it. 0

2010 Mathematics Subject Classification. Primary 39B52; 39B82; 46B99; 17A40. Keywords: stability, superstability, ternary Jordan ring homomorphism; ternary Banach algebra. 0∗ Corresponding uuthor: [email protected] (Dong Yun Shin). 0

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Approximate ternary Jordan ring homomorphisms The study of stability problems originated from a famous talk given by Ulam [25] in 1940: “Under what condition does there exist a homomorphism near an approximate homomorphism?” In 1941, Hyers [12] answered affirmatively the question of Ulam for additive mappings between Banach spaces. A generalized version of the theorem of Hyers for approximately additive maps was given by Rassias [20] in 1978. For more details about various results concerning such problems the reader is referred to [3, 6, 7, 10, 11, 15, 16, 17, 18, 19, 24]. We need the following fixed point theorem. Theorem 1.1. [14] Suppose that (Ω, d) is a complete generalized metric space and T : Ω → Ω is a strictly contractive mapping with the Lipschitz constant L. Then, for any x ∈ Ω, either d(T n x, T n+1 x) = ∞,

∀n ≥ 0,

or there exists a positive integer n0 such that (1) d(T n x, T n+1 x) < ∞ for all n ≥ n0 ; (2) the sequence {T n x} is convergent to a fixed point y ∗ of T ; (3) y ∗ is the unique fixed point of T in Λ = {y ∈ Ω : d(T n0 x, y) < ∞}; (4) d(y, y ∗ ) ≤

1 1−L d(y, T y)

for all y ∈ Λ.

In this paper, we prove the stability and superstability of ternary Jordan ring homomorphisms in ternary Banach algebras by using the fixed point method. 2. Stability of ternary Jordan ring homomorphisms In this section, we establish the stability ternary Jordan ring homomorphisms in ternary Banach algebras. Throughout this section, assume that A and B are ternary Banach algebras. Lemma 2.1. [9] Let f : A → B be an additive mapping. Then the following assertions are equivalent f ([a, a, a]) = [f (a), f (a), f (a)]

(2.1)

f ([a, b, c] + [b, c, a] + [c, a, b]) = [f (a), f (b), f (c)] + [f (b), f (c), f (a)] + [f (c), f (a), f (b)]

(2.2)

for all a ∈ A, and

for all a, b, c ∈ A. Theorem 2.2. Let f : A → B be a mapping for which there exist functions ϕ : A × A → [0, ∞) and ψ : A × A × A → [0, ∞) such that kf (x + y) − f (x) − f (y)k ≤ ϕ(x, y),

(2.3)

kf ([x, y, z] + [y, z, x] + [z, x, y]) − [f (x), f (y), f (z)] − [f (y), f (z), f (x)] − [f (z), f (x), f (y)]k ≤ ψ(x, y, z) for all x, y, z ∈ A. If there exists a constant 0 < L < 1 such that x y L ϕ , ≤ ϕ(x, y), 2 2 2

403

(2.4)

(2.5)

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M. Eshaghi Gordji, V. Keshavarz, J. Lee, D. Shin, C. Park x y z  L ≤ 3 ψ(x, y, z) , , 2 2 2 2 for all x, y, z ∈ A, then there exists a unique ternary Jordan ring homomorphism = : A → B ψ

kf (x) − =(x)k ≤

L ϕ(x, x) 2(1 − L)

(2.6)

(2.7)

for all x ∈ A. Proof. It follows from (2.5) and (2.6) that x y  (2.8) lim 2n ϕ n , n = 0, n→∞ 2 2 x y z  lim 23n ψ n , n , n = 0 (2.9) n→∞ 2 2 2 for all x, y, z ∈ A. By (2.5), limn→∞ 2n ϕ(0, 0) = 0 and so ϕ(0, 0) = 0. Letting x = y = 0 in (2.3), we get f (0) ≤ ϕ(0, 0) = 0 and so f (0) = 0. Let Ω = {g : A → B, g(0) = 0}. We introduce a generalized metric on Ω as follows: d(g, h) = dϕ (g, h) = inf{K ∈ (0, ∞) : kg(x) − h(x)k ≤ Kϕ(x, x), ∀x ∈ A}

.

It is easy to show that (Ω, d) is a generalized complete metric space. Now, we consider the mapping T : Ω → Ω defined by T g(x) = 2g( x2 ) for all x ∈ A and g ∈ Ω. Note that, for all g, h ∈ Ω and x ∈ A, d(g, h) < K ⇒ kg(x) − h(x)k ≤ Kϕ(x, x) x x x x ⇒ k2g( ) − 2h( )k ≤ 2 K ϕ( , ) 2 2 2 2 x x ⇒ k2g( ) − 2h( )k ≤ L K ϕ(x, x) 2 2 ⇒ d(T g, T h) ≤ L K. Hence we show that d(T g, T h) ≤ L d(g, h) for all g, h ∈ Ω, that is, T is a strictly contractive mapping of Ω with the Lipschitz constant L. Putting y = x in (2.3), we get kf (2x) − 2f (x)k ≤ ϕ(x, x) for all x ∈ A. So

x x L  x 

, ≤ ϕ(x, x)

f (x) − 2f

≤ϕ 2 2 2 2 for all x ∈ A, that is, d(f, T f ) ≤ L2 < ∞. Let us denote =(x) = lim 2n f n→∞

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Approximate ternary Jordan ring homomorphisms for all x ∈ A since limn→∞ d(T n f, =) = 0. By the result in [4] , = is a ternary Jordan mapping and so it follows from the definition of =, (2.4) and (2.9) that k=([x, y, z] + [y, z, x] + [z, x, y]) − [=(x), =(y), =(z)] − [=(y), =(z), =(x)] − [=(z), =(x), =(y)]k

 x y z z y x z x y 

= lim f 23n [ n , n , n ] + 23n [ n , n , n ] + 23n [ n , n , n ] n→∞ 2  x  2 y2 2  z  2 2  2y   z2 2  x  z  x  y 

3n 3n − 2 [f n , f n , f n ] − 2 [f n , f n , f n ] − 23n [f n , f n , f n ] 2  2 2 2 2 2 2 2 2 x y z  ≤ lim 23n ψ n , n , n = 0 n→∞ 2 2 2 and so =([x, y, z] + [y, z, x] + [z, x, y]) = [=(x), =(y), =(z)] + [=(y), =(z), =(x)] + [=(z), =(x), =(y)] for all x ∈ A. According to Theorem 1.1, since F is the unique fixed point of T in the set Λ = {g ∈ Ω : d(f, g) < ∞}, F is the unique mapping such that kf (x) − F(x)k ≤ K ϕ(x, x) for all x ∈ A and K > 0. Again, using Theorem 1.1, we have d(f, F) ≤

1 L d(f, T f ) ≤ 1−L 2(1 − L)

and so kf (x) − F(x)k ≤

L ϕ(x, x) 2(1 − L)

This completes the proof.



Corollary 2.3. Let θ, p be nonnegative real numbers with r, p > 1 and

r−3p 2

≥ 1. Suppose that f : A → B is a

mapping such that kf (x + y) − f (x) − f (y)k ≤ θ(kxkr + kykr ), kf ([x, y, z] + [y, z, x] + [z, x, y]) − [f (x), f (y), f (z)] − [f (y), f (z), f (x)] − [f (z), f (x), f (y)]k ≤ θ(kxkp .kykp .kzkp ) for all x, y, z ∈ A. Then there exists a unique ternary Jordan ring homomorphism F : A → B satisfying kf (x) − F(x)k ≤

θ kxkr (2r − 2)

for all x ∈ A. Proof. The proof follows from Theorem 2.2 by taking ϕ(x, y) := θ(kxkr + kykr ),

ψ(x, y, z) := θ(kxkp .kykp .kzkp )

for all x, y ∈ A. Then we can choose L = 21−r and so the desired conclusion follows.

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M. Eshaghi Gordji, V. Keshavarz, J. Lee, D. Shin, C. Park Remark 2.4. Let f : A → B be a mapping with f (0) = 0 such that there exist functions ϕ : A × A → [0, ∞) and ψ : A × A × A → [0, ∞) satisfying (2.3) and (2.4). Let 0 < L < 1 be a constant such that ϕ(2x, 2y) ≤ 2Lϕ(x, y),

ψ(2x, 2y, 2z) ≤ 23 Lψ(x, y, z)

for all x, y, z ∈ A. By the similar method as in the proof of Theorem 2.2, one can show that there exists a unique ternary Jordan ring homomorphism F : A → X satisfying kf (x) − F(x)k ≤

1 ϕ(x, x) 2(1 − L)

for all x ∈ A. For the case ϕ(x, y) := δ + θ(kxkr + kykr ),

ψ(x, y, z) := δ + θ(kxkp · kykp · kzkp )

(where θ, δ are nonnegative real numbers and r > 0, p < 1 and

r−3p 2

≥ 1), there exists a unique ternary Jordan

ring homomorphism = : A → X satisfying kf (x) − F(x)k ≤

δ θ + kxkr r (2 − 2 ) (2 − 2r )

for all x ∈ A.

3. Superstability of ternary Jordan ring homomorphisms In this section, we formulate and prove the superstability of ternary Jordan ring homomorphisms. Theorem 3.1. Suppose that there exist function ψ : A × A × A → [0, ∞) and a constant 0 < L < 1 such that x y z  L ψ , , ≤ 3 ψ(x, y, z) 2 2 2 2 for all x, y, z ∈ A. Moreover, if f : A → B is an additive mapping such that kf ([x, y, z] + [y, z, x] + [z, x, y]) − [f (x), f (y), f (z)] − [f (y), f (z), f (x)] − [f (z), f (x), f (y)]k ≤ ψ(x, y, z) for all x, y, z ∈ A, then f is a ternary Jordan ring homomorphism.

Proof. The proof of this theorem is omitted as similar to the proof of Theorem 2.2.



Corollary 3.2. Let θ, r, s be nonnegative real numbers with r > 1 and s > 3. If f : A → B is an additive mapping such that kf ([x, y, z] + [y, z, x] + [z, x, y]) − [f (x), f (y), f (z)] − [f (y), f (z), f (x)] − [f (z), f (x), f (y)]k ≤ θ(kxks + kyks + kzks ) for all x, y, z ∈ A, then f is a ternary Jordan ring homomorphism.

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Approximate ternary Jordan ring homomorphisms Remark 3.3. Let θ, r be nonnegative real numbers with r < 1. Suppose that there exists a function ψ : A × A × A → [0, ∞) and a constant 0 < L < 1 such that ψ(2x, 2y, 2z) ≤ 23 Lψ(x, y, z) for all x, y, z ∈ A. If f : A → B is an additive mapping such that kf ([x, y, z] + [y, z, x] + [z, x, y]) − [f (x), f (y), f (z)] − [f (y), f (z), f (x)] − [f (z), f (x), f (y)]k ≤ ψ(x, y, z) for all x, y, z ∈ A, then f is a ternary Jordan ring homomorphism.

References [1] V. Abramov, R. Kerner, and B. Le Roy, Hypersymmetry: a Z3 graded generalization of supersymmetry, J. Math. Phys. 38 (1997), 1650–1669. [2] F. Bagarello and G. Morchio, Dynamics of mean-field spin models from basic results in abstract differential equations, J. Stat. Phys. 66 (1992), 849–866. [3] L. C˘adariu, L. G˘ avruta, P. G˘ avruta, On the stability of an affine functional equation, J. Nonlinear Sci. Appl. 6 (2013), 60–67. [4] L. Cˇadariu, V. Radu, On the stability of the Cauchy functional equation: a fixed point approach, Grazer Math. Ber. 346 (2004), 43–52 [5] A. Cayley, On the 34concomitants of the ternary cubic, Amer. J. Math. 4 (1881), 1–15. [6] A. Chahbi, N. Bounader, On the generalized stability of d’Alembert functional equation, J. Nonlinear Sci. Appl. 6 (2013), 198–204. [7] Y. Cho, C. Park, M. Eshaghi Gordji, Approximate additive and quadratic mappings in 2-Banach spaces and related topics, Int. J. Nonlinear Anal. Appl. 3 (2012) No. 1, 75–81. [8] Y. L. Daletskii, L. A. Takhtajan, Leibniz and Lie algebra structures for Nambu algebra, Lett. Math. Phys. 39 (1997), Art. ID 127. [9] M. Eshaghi Gordji, V. Keshavarz, Ternary Jordan homomorphisms between unital ternary C ∗ -algebras (preprint). [10] P. G˘aavruta, L. G˘ aavruta, A new method for the generalized Hyers-Ulam-Rassias stability, Int. J. Nonlinear Anal. Appl. 1 (2010), No. 2, 11–18. [11] N. Ghobadipour, C. Park, Cubic-quartic functional equations in fuzzy normed spaces, Int. J. Nonlinear Anal. Appl. 1 (2010), 12–21. [12] D. H. Hyers, On the stability of the linear functional equation, Proc. Natl. Acad. Sci. U. S. A. 27 (1941), 222–224. [13] M. Kapranov, I. M. Gelfand, A. Zelevinskii, Discriminants, Resultants and Multidimensional Determinants, Birkh¨auser, Berlin, 1994. [14] B. Margolis, J.B. Diaz, A fixed point theorem of the alternative for contractions on the generalized complete metric space, Bull. Amer. Math. Soc. 126 (1968), 305–309. [15] C. Park, K. Ghasemi, S. G. Ghaleh, S. Jang, Approximate n-Jordan ∗-homomorphisms in C ∗ -algebras, J. Comput. Anal. Appl. 15 (2013), 365–368. [16] C. Park, S. Kim, J. Lee, D. Shin, Quadratic ρ-functional inequalities in β-homogeneous normed spaces, Int. J. Nonlinear Anal. Appl. 6 (2015), No. 2, 21–26.

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M. Eshaghi Gordji, V. Keshavarz, J. Lee, D. Shin, C. Park [17] C. Park, A. Najati, Generalized additive functional inequalities in Banach algebras, Int. J. Nonlinear Anal. Appl. 1 (2010), No. 2, 54–62. [18] C. Park, A. Najati, S. Jang, Fixed points and fuzzy stability of an additive-quadratic functional equation, J. Comput. Anal. Appl. 15 (2013), 452–462. [19] C. Park, Th. M. Rassias, Isomorphisms in unital C ∗ -algebras, Int. J. Nonlinear Anal. Appl. 1 (2010), No. 2, 1–10. [20] Th. M. Rassias, On the stability of the linear mapping in Banach spaces, Proc. Amer.Math. Soc., 72 (1978), 297–300. [21] S. Shagholi, M. Bavand Savadkouhi, M. Eshaghi Gordji, Nearly ternary cubic homomorphism in ternary Fr´echet algebras, J. Comput. Anal. Appl. 13 (2011), 1106–1114. [22] S. Shagholi, M. Eshaghi Gordji, M. Bavand Savadkouhi, Stability of ternary quadratic derivation on ternary Banach algebras, J. Comput. Anal. Appl. 13 (2011), 1097–1105. [23] D. Shin, C. Park, Sh. Farhadabadi, On the superstability of ternary Jordan C ∗ -homomorphisms, J. Comput. Anal. Appl. 16 (2014), 964–973. [24] D. Shin, C. Park, Sh. Farhadabadi, Stability and superstability of J ∗ -homomorphisms and J ∗ -derivations for a generalized Cauchy-Jensen equation, J. Comput. Anal. Appl. 17 (2014), 125–134. [25] S. M. Ulam, Problems in Modern Mathematics, Chapter VI, Science ed. Wiley, New York, 1940. [26] L. Vainerman, R. Kerner, On special classes of n-algebras, J. Math. Phys. 37 (1996), 25–53.

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Approximate controllability of fractional impulsive stochastic functional differential inclusions with infinite delay and fractional sectorial operators Zuomao Yan∗ and Xiumei Jia January 30, 2016 Abstract: In this paper, the approximate controllability of fractional impulsive stochastic functional differential inclusions with infinite delay and fractional sectorial operators is considered. By using the stochastic analysis, the fractional sectorial operators and a fixed-point theorem for multi-valued maps, a new set of necessary and sufficient conditions are formulated which guarantees the approximate controllability of the fractional impulsive stochastic system. The results are obtained under the assumption that the associated linear system is approximately controllable. Finally, an example is also given to illustrate the obtained theory. 2000 MR Subject Classification: 34A37; 60H15; 26A33; 93B05; 93E03 Keywords: Approximate controllability; Fractional impulsive stochastic functional differential inclusions; Infinite delay; Fractional sectorial operators; Fixed point theorem

1

Introduction

The notion of controllability has played a central role throughout the history of modern control theory. Moreover, approximate controllable systems are more prevalent and very often approximate controllability is completely adequate in applications; see [1-3]. Therefore, various approximate controllability problems for different kinds of dynamical systems have been investigated in many publications; see [4,5] and references therein. The fractional differential equations has received a great deal of attention, and they play an important role in many applied fields, including viscoelasticity, electrochemistry, control, porous media, electromagnetic and so on. In recent years, several papers have studied the approximate controllability of semilinear fractional differential systems without delay and infinite delay (see [6-9]). As a result of its widespread use, the approximate controllability of stochastic systems have received extensive attention. More recently, there are very few contributions regarding the approximate 1

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controllability of fractional stochastic control system. For example, Sakthivel et al. [10], Kerboua et al. [11], Muthukumar and Rajivganthi [12], Farahi and Guendouzi [13]. Impulsive partial functional differential equations or inclusions have become an active area of investigation due to their applications in fields such as mechanics, electrical engineering, medicine biology (see [14]). Recently, the approximate controllability for some fractional impulsive semilinear differential systems have been studied in several papers. For example, Liu and Bin [15] studied the approximate controllability for a class of Riemann-Liouville fractional impulsive differential inclusions. Balasubramaniam et al. [16] derived sufficient conditions for the approximate controllability of impulsive fractional integro-differential systems with nonlocal conditions in Hilbert space. Chalishajar et al. [17] discussed the approximate controllability of abstract impulsive fractional neutral evolution equations with infinite delay in Banach spaces. However, besides impulse effects and delays, stochastic effects likewise exist in real systems. For semilinear impulsive stochastic control systems in Hilbert spaces, there are several papers devoted to the approximate controllability (see [18,19]). Zang and Li [20] obtained the approximate controllability of fractional impulsive neutral stochastic differential equations with nonlocal conditions by using Krasnoselskii-Schaefer-type fixed point theorem. Motivated by the researches mentioned previously, in this paper we consider the approximate controllability of a class of fractional impulsive stochastic functional differential inclusions with infinite delay in Hilbert spaces of the form c

dw(t) , dt α ∈ (0, 1), t ∈ J = [0, b], t 6= tk ,

Dα N (xt ) ∈ AN (xt ) + Bu(t) + F (t, xt )

x0 = ϕ ∈ B, ∆x(tk ) = Ik (xtk ),

k = 1, . . . , m,

(1)

(2) (3)

where the state x(·) takes values in a separable real Hilbert space H with inner product h·, ·iH and norm k · kH . Here c Dα is the Caputo fractional derivative of the order α ∈ (0, 1) with the lower limit zero, A is a fractional sectorial operator defined on (H, k · kH ). Let K be another separable Hilbert space with inner product h·, ·iK and norm k · kK . Suppose {w(t) : t ≥ 0} is a given Kvalued Wiener process with a covariance operator Q > 0 defined on a complete probability space (Ω, F, P ) equipped with a normal filtration {Ft }t≥0 , which is generated by the Wiener process w. The control function u ∈ LpF (J, U ), a Hilbert space of admissible control functions, p ≥ 2 be an integer, and B is a bounded linear operator from a Banach space U to H. The time history xt : (−∞, 0] → H given by xt (θ) = x(t+θ) belongs to some abstract phase space B defined axiomatically; F, G, Ik (k = 1, . . . , m), N (ψ) = ψ(0) − G(t, ψ), ψ ∈ B, are given functions to be specified later. Moreover, let 0 < t1 < · · · < tm < b, − − are prefixed points and the symbol ∆x(tk ) = x(t+ k ) − x(tk ), where x(tk ) and + x(tk ) represent the right and left limits of x(t) at t = tk , respectively. The 2

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initial data {ϕ(t) : −∞ < t ≤ 0} is an F0 -adapted, B-valued random variable independent of the Wiener process w with finite second moment. To the best of our knowledge, the approximate controllability of fractional impulsive stochastic functional differential inclusions with infinite delay and fractional sectorial operators and the form (1)-(3) is an untreated topic in the literature. To close the gap in this paper, we study this interesting problem. Sufficient conditions for the approximate controllability results are derived by a fixed-point theorem for multi-valued maps combined with the stochastic analysis and the fractional sectorial operators. The known results appeared in [15-20] are generalized to the fractional impulsive stochastic systems settings and the case of infinite delay. The rest of this paper is organized as follows. In Section 2, we introduce some notations and necessary preliminaries. In Section 3, we give our main results. In Section 4, an example is given to illustrate our results.

2

Preliminaries

Let (Ω, F, P ) be a complete probability space with probability measure P on Ω and a normal filtration {Ft }t≥0 . Let H, K be two real separable Hilbert spaces and we denote by h·, ·iH , h·, ·iK their inner products and by k · kH , k · kK their vector norms, respectively. L(H, K) be the space of bounded linear operators mapping K into H equipped with the usual norm k · kH and L(H) denotes the Banach space of bounded linear operators from H to H. Let {w(t) : t ≥ 0} denote an K-valued Wiener process defined on the probability space (Ω, F, P ) with covariance operator Q. We assume that there exists a complete orthonormal ∞ system {en }∞ n=1 in K, a bounded sequence of nonnegative real numbers {λn }n=1 such that Qen = λn en , n = 1, 2, . . . , and a sequence βn of independent Brownian motions such that hw(t), ei =

∞ p X

λn hen , eiβn (t), e ∈ K, t ∈ J,

n=1

and Ft = Ftw , where Ftw is the σ-algebra generated by {w(s) : 0 ≤ s ≤ t}. Let L02 = L2 (K0 , H) be the space of all Hilbert-Schmidt operators from K0 to H with the norm k ψ k2L0 = Tr((ψQ1/2 )(ψQ1/2 )∗ ) for any ψ ∈ L02 . Clearly for 2

any bounded operators ψ ∈ L(K, H) this norm reduces to k ψ k2L0 = Tr(ψQψ ∗ ). 2 Let Lp (Fb , H) be the Banach space of all Fb -measurable pth power integrable random variables with values in the Hilbert space H. Let C([0, b]; Lp (F, H)) be the Banach space of continuous maps from [0, b] into Lp (F, H) satisfying the condition supt∈J E k x(t) kpH < ∞. We use the notations P(H) is the family of all subsets of H. Let us introduce the following notations: Pcl (H) = {x ∈ P(H) : x is closed},

Pbd (H) = {x ∈ P(H) : x is bounded},

Pcv (H) = {x ∈ P(H) : x is convex},

Pcp (H) = {x ∈ P(H) : x is compact}. 3

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Consider Hd : P(H) × P(H) → R+ ∪ {∞} given by ½ ¾ e B) e = max sup d(˜ e sup d(A, e a Hd (A, a, B), ˜) , ˜ e e a ˜∈A b∈B ˜ e ˜b) = inf e = inf ˜ d(˜ where d(A, a, ˜b), d(˜ a, B) e d(˜ e a, b). Then, (Pbd,cl (H), Hd ) is a ˜∈A b∈B a metric space and (Pcl (H), Hd ) is a generalized metric space. In what follows, we briefly introduce some facts on multi-valued analysis. For more details, one can see [21,22]. A multi-valued map Φ : J → Pcl (H) is said to be measurable if for each x ∈ H, the function Y : J → R+ defined by Y (t) = d(x, Φ(t)) = inf{d(x, z) : z ∈ Φ(t)} is measurable. Φ has a fixed point if there is x ∈ H such that x ∈ Φ(x). The set of fixed points of the multi-valued operator Φ will be denoted by FixΦ. Definition 2.1. A multi-valued operator Φ : H → Pcl (H) is called: (a) γ-Lipschitz if there exists γ > 0 such that Hd (Φ(x), Φ(y)) ≤ γd(x, y), x, y ∈ H. (b) a contraction if it is γ-Lipschitz with γ < 1. In this paper, we assume that the phase space (B, k · kB ) is a seminormed linear space of F0 -measurable functions mapping (−∞, 0] into H, and satisfying the following fundamental axioms due to Hale and Kato (see e.g., in [23]). (A) If x : (−∞, σ + b] → H, b > 0, is such that x|[σ,σ+b] ∈ C([σ, σ + b], H) and xσ ∈ B, then for every t ∈ [σ, σ + b] the following conditions hold: (i) xt is in B; ˜ k xt kB ; (ii) k x(t) kH ≤ H (iii) k xt kB ≤ K(t − σ) sup{k x(s) kH : σ ≤ s ≤ t} + M (t − σ) k xσ kB , ˜ ≥ 0 is a constant; K, M : [0, ∞) → [1, ∞), K is continuous where H ˜ K, M are independent of x(·). and M is locally bounded, and H, (B) For the function x(·) in (A), the function t → xt is continuous from [σ, σ+b] into B. (C) The space B is complete. The next result is a consequence of the phase space axioms. Lemma 2.1. Let x : (−∞, b] → H be an Ft -adapted measurable process such that the F0 -adapted process x0 = ϕ(t) ∈ L02 (Ω, B) and x|[0,b] ∈ PC([0, b], H), then k xs kB ≤ Mb E k ϕ kB +Kb sup E k x(s) kH , 0≤s≤b

where Kb = sup{K(t) : 0 ≤ t ≤ b}, Mb = sup{M (t) : 0 ≤ t ≤ b}. 4

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We introduce the space PC formed by all Ft -adapted measurable, H-valued stochastic processes {x(t) : t ∈ [0, b]} such that x is continuous at t 6= tk , + x(tk ) = x(t− k ) and x(tk ) exists for all k = 1, ..., m. In this paper, we always assume that PC is endowed with the norm 1

k x kPC = ( sup E k x(t) kpH ) p . 0≤t≤b

Then (PC, k · kPC ) is a Banach space. Definition 2.2 ([24]). The fractional integral of order γ with the lower limit zero for a function h ∈ L1 (J, H) is defined as Itγ h(t)

Z

1 = Γ(γ)

0

t

h(s) ds, (t − s)1−γ

t > 0, γ > 0

provided the right side is point-wise defined on [0, ∞), where Γ(·) is the gamma function. Definition 2.3 ([24]). The Riemann-Liouville derivative of order γ with the lower limit zero for a function h ∈ L1 (J, H) can be written as Dtγ h(t)

dn 1 = Γ(n − γ) dtn

Z 0

t

h(s) ds, (t − s)γ+1−n

t > 0, n − 1 < γ < n.

Definition 2.4 ([24]). The Caputo derivative of order γ for a function h ∈ L1 (J, H) can be written as Dtγ h(t) = Dtγ (h(t) − h(0)),

t > 0, 0 < γ < 1.

Next, we are ready to recall some facts of fractional Cauchy problem. c

Dtα x(t) = Ax(t),

t ≥ 0,

(4)

x0 = ϕ ∈ B,

(5)

where A is linear closed and D(A) is dense. Definition 2.5 ([25]). A family {Sα (t) : t ≥ 0} ⊂ L(H) is called a solution operator for (4)-(5) if the following conditions are satisfied: (a) Sα (t) is strongly continuous for t ≥ 0 and Sα (0) = I. (b) Sα (t)D(A) ⊂ D(A) and ASα (t)ϕ = Sα (t)Aϕ for all ϕ ∈ D(A), t ≥ 0. (c) Sα (t)ϕ is a solution of (4)-(5) for all ϕ ∈ D(A), t ≥ 0. Definition 2.6 ([24]). An operator A is said to be belong to eα (M, ω) if the solution operator Sα (·) of (4)-(5) satisfies k Sα (t) kL(H) ≤ M eωt ,

t≥0

for some constants M ≥ 1 and ω ≥ 0. 5

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Definition 2.7 ([24]). A solution operator Sα (·) of (4)-(5) is called analytic if it admits an analytic extension to a sector Σθ0 = {λ ∈ C − {0} : arg λ < θ0 } for some θ0 ∈ (0, π2 ]. An analytic solution operator is said to be of analyticity type (θ0 , ω0 ) if for each θ < θ0 and ω > ω0 there is an M = M (θ, ω) such that k Sα (t) kL(H) ≤ M eωRet , Set

eα (ω) =

[

t ∈ Σθ .

{eα (M, ω) : M ≥ 1}, eα :=

[

{eα (ω) : ω ≥ 0},

and Aα (θ0 , ω0 ) = {A ∈ eα : A generates an analytic solution operator Sα of type (θ0 , ω0 )}. Remark 2.3 ([25, Theorem 2.14]). Let α ∈ (0, 2). A linear closed densely defined operator A belongs to Aα (θ0 , ω0 ) if and only if λα ∈ ρ(A) for each λ ∈ Σθ0 + π2 (ω0 ) = {C − {0} : | arg(λ − ω0 )| < θ0 + π2 } and for any ω > ω0 , θ < θ0 there is a constant C = C(θ, ω) such that C |λ − ω|

k λα−1 R(λα , A) kL(H) ≤

for λ ∈ Σθ0 + π2 . According to the proof of Theorem 2.14 in [25], if A ∈ Aα (θ0 , ω0 ) for some θ0 ∈ (0, π) and ω0 ∈ R, the solution operator for the Eq. (4)-(5) is given by Z 1 Sα (t) = eλt λα−1 R(λα , A)dλ 2πi Γ for a suitable path Γ. Next, a mild solution of the Cauchy problem c

Dtα x(t) = Ax(t) + f (t),

t ∈ J,

x0 = ϕ ∈ B, can be defined by Z

t

x(t) = Sα (t)ϕ +

Tα (t − s)f (s)ds, 0

where Tα (t) =

1 2πi

Z eλt R(λα , A)dλ Γ

for a suitable path Γ and f : J → H is continuous. Lemma 2.2 ([25]). If A ∈ Aα (θ0 , ω0 ) then k Sα (t) kL(H) ≤ M eωt ,

k Tα (t) kL(H) ≤ Ceωt (1 + tα−1 )

for every t > 0, ω > ω0 . So putting ˜ S := sup k Sα (t) kL(H) , M

˜ T := sup Ceωt (1 + tα−1 ), M

0≤t≤b

0≤t≤b

6

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we get

˜S, k Sα (t) kL(H) ≤ M

˜T . k Tα (t) kL(H) ≤ tα−1 M

Based on the above consideration, we introduce the definition of mild solution for (1)-(3). Definition 2.8. Let A ∈ Aα (θ0 , ω0 ) with θ0 ∈ (0, π2 ] and ω0 ∈ R. An Ft adapted stochastic process x : (−∞, b] → H is called a mild solution of the system (1)-(3) if x0 = ϕ ∈ B satisfying x0 ∈ L02 (Ω, H), x|[0,b] ∈ PC, and  Sα (t)[ϕ(0) − G(0, ϕ)] + G(t, xt )   Rt   + 0 Tα (t − s)Bu(s)ds   R  t   + 0 Tα (t − s)f (s)dw(s),     Sα (t)[ϕ(0) − G(0, ϕ)] + G(t, xt )     +Sα (t − t1 )I1 (xt1 )   R   + t Sα (t − s)Bu(s)ds R0t x(t) = + 0 Sα (t − s)f (s)dw(s),    ..    .     Sα (t)[ϕ(0)  Pm − G(0, ϕ)] + G(t, xt )    + S (t − tk )Ik (xtk )   R t k=1 α     + R0t Sα (t − s)Bu(s)ds  + 0 Sα (t − s)f (s)dw(s),

t ∈ [0, t1 ],

t ∈ (t1 , t2 ],

t ∈ (tm , b],

where f ∈ SF,x = {f ∈ Lp (J, L02 ) : f (t) ∈ F (t, xt ) a.e. t ∈ J}. Let x(t; ϕ, u) denotes state value of the system (1)-(3) at time t corresponding to the control u ∈ LpF (J, U ). In particular, the state of system (1)-(3) at t = b, x(b; ϕ, u) is called the terminal state with control u and the initial value ϕ. Introduce the set B(b; ϕ, u) = {x(b; ϕ, u), u(·) ∈ LpF (J, U )} is called the reachable set of the system (1)-(3), where LpF (J, U ) is the closed subspace of LpF (J ×Ω, U ), consisting of all Ft -adapted, U -valued stochastic processes. Definition 2.9. The system (1)-(3) is said to be approximately controllable on the interval J if B(b; ϕ, u) = Lp (Fb , H), where B(b; ϕ, u) is the closure of the reachable set. It is convenient at this point to define operators Z Γbτ

b

= τ

Sα (b − s)BB ∗ Sα∗ (b − s)ds, Z

Γb0

= 0

b

0 ≤ τ < b,

Sα (b − s)BB ∗ Sα∗ (b − s)ds,

R(a, Γbτ ) = (aI + Γbτ )−1 , R(a, Γb0 ) = (aI + Γb0 )−1 for a > 0, where B ∗ denotes the adjoint of B and Sα∗ (t) is the adjoint of Sα (t). It is straightforward that the operator Γbτ is a linear bounded operator. Lemma 2.4 ([3]). For any x ˜b ∈ Lp (Fb , H) there exists φ˜ ∈ LpF (Ω; L2 (0, b; L02 )) Rb ˜ such that x ˜b = E x ˜b + 0 φ(s)dw(s). 7

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Now for                                   a ux (t) =                                 

any a > 0 and x ˜b ∈ Lp (Fb , H) we define the control function · Rb ˜ S ∗ Tα∗ (b − t)(aI + Γb0 )−1 E x ˜b + 0 φ(s)dw(s) ¸ −Sα (b)[ϕ(0) − G(0, ϕ)] − G(b, xb ) Rb −B ∗ Tα∗ (b − t) 0 (aI +· Γbs )−1 Sα (b − s)f (s)dw(s), t ∈ [0, t1 ], Rb ˜ S ∗ Tα∗ (b − t)(aI + Γb0 )−1 E x ˜b + 0 φ(s)dw(s) ¸ −Sα (b)[ϕ(0) − G(0, ϕ)] − G(b, xb ) −B ∗ Tα∗ (b − t)(aI + Γbs )−1 Sα (t − t1 )I1 (xt1 ) Rb −B ∗ Tα∗ (b − t) 0 (aI + Γbs )−1 Sα (b − s)f (s)dw(s),

t ∈ (t1 , t2 ],

· Rb ˜ S − t)(aI + Ex ˜b + 0 φ(s)dw(s) ¸ −Sα (b)[ϕ(0) − G(0, ϕ)] − G(b, xb ) Pm −B ∗ Tα∗ (b − t) k=1 (aI + Γbs )−1 Sα (t − tk )Ik (xtk ) Rb −B ∗ Tα∗ (b − t) 0 (aI + Γbs )−1 Sα (b − s)f (s)dw(s),

t ∈ (tm , b],

.. .

∗

Tα∗ (b

Γb0 )−1

where f ∈ SF,x = {f ∈ Lp (J, L02 ) : f (t) ∈ F (t, xt ) a.e. t ∈ J}. Lemma 2.5 ([26]). For any p ≥ 1 and for arbitrary L02 -valued predictable process φ(·) such that wZ w sup E w w

s∈[0,t]

0

s

w2p µZ t ¶p w 2p 1/p p w (E k φ(s) kL0 ) ds , t ∈ [0, ∞). φ(v)dw(v)w ≤ (p(2p−1)) 0

H

2

In the rest of this paper, we denote by M1 =k B kH , Cp = (p(p − 1)/2)p/2 . Our main results are based on the following lemma. Lemma 2.6 ([27]). Let (H, d) be a complete metric space. If Φ : H → Pcl (H) is a contraction, then Fix Φ 6= ∅.

3

Main results

In this section we shall present and prove our main results. Let us list the following hypotheses. (H1) The function G : J × B → H is continuous, and there exists a positive constant LG such that E k G(t, ψ1 ) − G(t, ψ2 ) kpH ≤ LG k ψ1 − ψ2 kpB for t ∈ J, ψ1 , ψ2 ∈ B.

8

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(H2) The function F : J × B → Pcp (L02 ) is a multifunction such that (·, φ) → F (t, φ) is measurable for each φ ∈ B. 1

(H3) There exists a function l(t) ∈ L q (J, R+ ), q ∈ (0, α) such that EHdp (F (t, φ1 ), F (t, φ2 )) ≤ l(t) k φ1 − φ2 kpB for t ∈ J, φ1 , φ2 ∈ B, and dp (0, F (t, 0)) ≤ l(t) for a.e. t ∈ J. (H4) The functions Ik : B → H are continuous and there exist constants ck such that E k Ik (ψ1 ) − Ik (ψ2 ) kpH ≤ ck k ψ1 − ψ2 kpB for ψ1 , ψ2 ∈ B, k = 1, . . . , m. (H5) For each 0 ≤ t < b, the operator aR(a, Γbτ ) → 0 in the strong operator topology as a → 0+ i.e., the linear differential Cauchy problem corresponding to system (1)-(3) is approximately controllable on J. Theorem 3.1. Let A ∈ Aα (θ0 , ω0 ) with θ0 ∈ (0, π2 ] and ω0 ∈ R. If the assumptions (H1)-(H4) are satisfied, then the system (1)-(3) has at least one mild solution on J, provided that µ ¶1−q · m X 1−q p ˜ ˜p c + C M 4p−1 Kbp LG + mp−1 M i p T S p(1 − α) + 1 − q i=1 ¸· ¸ 2p 2p 1 bp(2α−1) p(α−1/2)−q p−1 ˜ ×b k l k q1 1 + 3 MT M1 ap 2p(α−1)+1 < 1. + L (J,R )

Proof. We introduce the space Bb of all functions x : (−∞, b] → H such that x0 ∈ B and the the restriction x|[0,b] ∈ PC. Letk · kb be a seminorm in Bb defined by 1 k x kb =k x0 kB +( sup k x(s) kpH ) p , x ∈ Bb . 0≤s≤b

We consider the multi-valued map Φ : Bb → P(Bb ) by Φx the set of ρ ∈ Bb such that  Sα (t)[ϕ(0) − G(0, ϕ)] + G(t, xt )     + R t T (t − s)Bua (s)ds + R t T (t − s)f (s)dw(s), t ∈ [0, t ],  1  x 0 α 0 α    S (t)[ϕ(0) − G(0, ϕ)] + G(t, x ) α t    +S (t − t )I (x )  1 1 t1   Rαt Rt + S (t − s)Buax (s)ds + 0 Sα (t − s)f (s)dw(s), t ∈ (t1 , t2 ], α ρ(t) = 0    ...      Sα (t)[ϕ(0)  Pm − G(0, ϕ)] + G(t, xt )    + k=1 Sα (t − tk )Ik (xtk )    + R t S (t − s)Bua (s)ds + R t S (t − s)f (s)dw(s), t ∈ (t , b], m x 0 α 0 α 9

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where f ∈ SF,x = {f ∈ Lp (J, L02 ) : f (t) ∈ F (t, xt ) a.e. t ∈ J}. For ϕ ∈ B, we define ϕ˜ by ½ ϕ(t), −∞ < t ≤ 0, ϕ(t) ˜ = Sα (t)ϕ(0), 0 ≤ t ≤ b, then ϕ˜ ∈ Bb . Set x(t) = y(t) + ϕ(t), ˜ −∞ < t ≤ b. It is clear to see that x satisfies Definition 2.8 if and only if y satisfies y0 = 0 and  Sα (t)[ϕ(0) − G(0, ϕ)] + G(t, yt + ϕ˜t )     + R t T (t − s)Bua (s)ds + R t T (t − s)f (s)dw(s),  t ∈ [0, t1 ],  y 0 α 0 α    S (t)[ϕ(0) − G(0, ϕ)] + G(t, y + ϕ ˜ ) α t t    +S (t − t )I (y + ϕ˜ )  1 1 t1 t1   Rαt Rt a + S (t − s)Bu (s)ds + 0 Sα (t − s)f (s)dw(s), t ∈ (t1 , t2 ], α y y(t) = 0  .    ..     Sα (t)[ϕ(0) − G(0, ϕ)] + G(t, yt + ϕ˜t )  P    + m S (t − tk )Ik (ytk + ϕ˜tk )   R t k=1 α Rt  + 0 Sα (t − s)Buay (s)ds + 0 Sα (t − s)f (s)dw(s), t ∈ (tm , b], where

uay (t) =

·  Rb  b −1 ∗ ∗ ˜  Ex ˜b + 0 φ(s)dw(s) S Tα (b − t)(aI + Γ0 )    ¸      −Sα (b)[ϕ(0) − G(0, ϕ)] − G(b, xb )     Rb    −B ∗ Tα∗ (b − t) 0 (aI +· Γbs )−1 Tα (b − s)f (s)dw(s),    Rb   b −1 ∗ ∗ ˜  (b − t)(aI + Γ ) E x ˜ + S T φ(s)dw(s) b  α 0 0   ¸      −S (b)[ϕ(0) − G(0, ϕ)] − G(b, x ) α b                                   

.. .

−B ∗ Tα∗ (b − t)(aI + Γbs )−1 Sα (t − t1 )I1 (xt1 ) Rb −B ∗ Tα∗ (b − t) 0 (aI + Γbs )−1 Tα (b − s)f (s)dw(s),

t ∈ [0, t1 ],

t ∈ (t1 , t2 ],

· Rb ˜ ˜b + 0 φ(s)dw(s) S ∗ Tα∗ (b − t)(aI + Γb0 )−1 E x ¸ −Sα (b)[ϕ(0) − G(0, ϕ)] − G(b, xb ) Pm −B ∗ Tα∗ (b − t) k=1 (aI + Γbs )−1 Sα (t − tk )Ik (xtk ) Rb −B ∗ Tα∗ (b − t) 0 (aI + Γbs )−1 Tα (b − s)f (s)dw(s), t ∈ (tm , b],

and f ∈ SF,y = {f ∈ Lp (J, L02 ) : f (t) ∈ F (t, ys + ϕ˜s ) a.e. t ∈ J}. Let Bb0 = {y ∈ Bb : y0 = 0 ∈ B}. For any y ∈ Bb0 , 1

1

k y kb =k y0 kB +( sup k y(s) kpH ) p = ( sup k y(s) kpH ) p , 0≤s≤b

0≤s≤b

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¯ : B 0 → P(B 0 ) thus (Bb0 , k · kb ) is a Banach space. Define the multi-valued map Φ b b 0 ¯ the set of ρ¯ ∈ B such that ρ¯(t) = 0, t ∈ [−∞, 0] and by Φy b  −Sα (t)G(0, ϕ) + G(t, yt + ϕ˜t )   Rt Rt    + 0 Tα (t − s)Buay (s)ds + 0 Tα (t − s)f (s)dw(s), t ∈ [0, t1 ],     −Sα (t)ϕ(0) + G(t, yt + ϕ˜t )       +SRαt (t − t1 )I1 (yt1 + ϕ˜t1 ) R t + 0 Sα (t − s)Buay (s)ds + 0 Tα (t − s)f (s)dw(s), t ∈ (t1 , t2 ], ρ¯(t) =  .  ..       −Sα P (t)ϕ(0) + G(t, yt + ϕ˜t )   m   S (t − tk )Ik (ytk + ϕ˜tk ) +   R t k=1 α Rt  + 0 Sα (t − s)Buay (s)ds + 0 Tα (t − s)f (s)dw(s), t ∈ (tm , b], where f ∈ SF,y . Obviously, the operator Φ has a fixed point if and only if ¯ has a fixed point, to prove which we shall employ Lemma 2.6. For operator Φ better readability, we break the proof into a sequence of steps. ¯ Step 1. We show that (Φy)(t) ∈ Pcl (Bb0 ). (n) ∗ ¯ Indeed, let y (t) → y (t), (¯ ρn )n≥0 ∈ (Φy)(t) such that ρ¯n (t) → ρ¯∗ (t) in 0 0 Bb . Then ρ¯∗ (t) ∈ Bb and there exists fn ∈ SF,y(n) such that, for each t ∈ [0, t1 ], (n)

ρ¯n (t) = −Sα (t)G(0, ϕ) + G(t, yt + ϕ˜t ) Z t Z t a + Tα (t − s)Buy(n) (s)ds + Tα (t − s)fn (s)dw(s), 0

0

where · Z uay(n) (t) = B ∗ Tα∗ (b − t)(aI + Γb0 )−1 E x ˜b +

b

˜ φ(s)dw(s) ¸ (n) −Sα (b)[ϕ(0) − G(0, ϕ)] − G(b, yb + ϕ˜b ) 0

Z −B ∗ Tα∗ (b − t)

b

0

(aI + Γbs )−1 Tα (b − s)fn (s)dw(s).

Using the fact that F has compact values and (H3) holds, we may pass to a subsequence if necessary to obtain that fn converges to f∗ in Lp ([0, t1 ], L02 ), hence, f∗ ∈ SF,y∗ . Then, for each t ∈ [0, t1 ], ρ¯n (t) → ρ¯∗ (t) = −Sα (t)G(0, ϕ) + G(t, yt∗ + ϕ˜t ) Z t Z t Tα (t − s)fn (s)dw(s), + Tα (t − s)Buay∗ (s)ds + 0

0

where uay∗ (t)

=S

∗

Tα∗ (b

− t)(aI +

Γb0 )−1

· Z Ex ˜b +

b

˜ φ(s)dw(s)

0

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¸ −Sα (b)[ϕ(0) − G(0, ϕ)] − G(b, yb∗ + ϕ˜b ) Z −B

∗

Tα∗ (b

b

− t) 0

(aI + Γbs )−1 Sα (b − s)f∗ (s)dw(s).

Similarly, for each t ∈ (tk , tk+1 ], k = 1, . . . , m, we have (n) G(t, yt

ρ¯n (t) = −Sα (t)G(0, ϕ) + Z

t

+ 0

+ ϕ˜t ) +

(n)

Sα (t − ti )Ii (yti + ϕ˜ti )

i=1

Z Tα (t − s)Buay(n) (s)ds +

k X

t

Tα (t − s)fn (s)dw(s), 0

where · Z uay(n) (t) = S ∗ Tα∗ (b − t)(aI + Γb0 )−1 E x ˜b +

b

˜ φ(s)dw(s) ¸ (n) −Sα (b)[ϕ(0) − G(0, ϕ)] − G(b, yb + ϕ˜b ) 0

−B ∗ Tα∗ (b − t)(aI + Γbs )−1

(n)

Sα (t − ti )Ii (yti + ϕ˜ti )

i=1

Z −B ∗ Tα∗ (b − t)

k X

b

0

(aI + Γbs )−1 Tα (b − s)fn (s)dw(s).

Using the fact that F has compact values and (H3) holds, we may pass to a subsequence if necessary to obtain that fn converges to f∗ in Lp ([tk , tk+1 ], L02 ), hence, f∗ ∈ SF,y∗ . Then, for each t ∈ [tk , tk+1 ], k = 1, . . . , m, ρ¯n (t) → ρ¯∗ (t) = −Sα (t)G(0, ϕ) + G(t, yt∗ + ϕ˜t ) +

k X i=1 t

Sα (t − ti )Ii (yt∗i + ϕ˜ti )

Z +

0

Z Tα (t − s)Buay∗ (s)ds +

t

Tα (t − s)fn (s)dw(s), 0

where uay∗ (t)

· Z Ex ˜b +

b

˜ φ(s)dw(s) ¸ ∗ −Sα (b)[ϕ(0) − G(0, ϕ)] − G(b, yb + ϕ˜b )

=S

∗

Tα∗ (b

− t)(aI +

Γb0 )−1

−B ∗ Tα∗ (b − t)(aI + Γbs )−1 Z −B ∗ Tα∗ (b − t)

0

0

k X

Sα (t − ti )Ii (yt∗i + ϕ˜ti )

i=1 b

(aI + Γbs )−1 Tα (b − s)f∗ (s)dw(s). 12

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¯ ¯ Therefore, ρ¯∗ (t) ∈ (Φy)(t) and (Φy)(t) ∈ Pcl (Bb0 ). ¯ Step 2. We show that (Φy)(t) is a contractive multi-valued map for each y(t) ∈ Bb0 . ¯ Let t ∈ [0, t1 ] and y(t), yˆ(t) ∈ Bb0 and let ρ¯(t) ∈ (Φy)(t). Then there exists f ∈ SF,y such that ρ¯(t) = −Sα (t)G(0, ϕ) + G(t, yt + ϕ˜t ) Z t Z t + Tα (t − s)Buay (s)ds + Tα (t − s)f (s)dw(s). 0

0

From (H3), there exists v(t) ∈ F (t, yˆt + ϕ˜t ) such that E k f (t) − v(t) kpL0 ≤ l(t) k yt − yˆt kpB . 2

P(L02 ),

Consider Λ : [0, t1 ] →

given by

Λ(t) = {v(t) ∈ H : E k f (t) − v(t) kpL0 ≤ l(t) k yt − yˆt kpB }. 2

Since the multi-valued operator W (t) = Λ(t) ∩ F (t, yˆt + ϕ˜t ) is measurable (see [28], Proposition III.4), there exists a function fˆ(t), which is a measurable selection for W. So, fˆ(t) ∈ F (t, yˆt + ϕ˜t ) and E k f (t) − fˆ(t) kpL0 ≤ l(t) k yt − yˆt kpB . 2

For each t ∈ [0, t1 ], we define ρˆ(t) = −Sα (t)G(0, ϕ) + G(t, yˆt + ϕ˜t ) Z t Z t a + Tα (t − s)Buyˆ(s)ds + Tα (t − s)fˆ(s)dw(s). 0

0

Then, for each t ∈ [0, t1 ], we have E k ρ¯(t) − ρˆ(t) kpH ≤ 3p−1 E k G(t, yt + ϕ˜t ) − G(t, yˆt + ϕ˜t ) kpH wZ t wp w w a a p−1 w Tα (t − s)B[uy (s) − uyˆ(s)]dsw +3 E w w 0 wZ t wpH w w Tα (t − s)[f (s) − fˆ(s)]dw(s)w +3p−1 E w w w 0

H

≤ 3p−1 LG k yt − yˆt kpB Z t p−1 ˜ p p−1 (t − s)p(α−1) E k B[uay (s) − uayˆ(s)] kpH ds +3 MT t1 0

¸2/p ¸p/2 ·Z t · p(α−1) ˆ(s) kp 0 ˜p ds (t − s) E k f (s) − f +3p−1 Cp M T L 2

0

≤3

p−1

L k yt −

yˆt kpB 13

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· Z t 2p 2p 1 p−1 p(α−1) ˜ [(t − s)(b − s)] L k yt − yˆt kpB +6 MT M1 p t1 a 0 ¸ Z b p/2−1 ˜ p p p(α−1) +Cp t1 MT (b − τ ) l(τ ) k yτ − yˆτ kB dτ ds p−1

0

p/2−1

+3p−1 Cp t1

Z

˜p M T

≤ 3p−1 Kbp LG k y − yˆ

t

0 p kb

(t − s)p(α−1) l(s) k ys − yˆs kpB ds

· Z t p(α−1) ˜ 2p M 2p 1 tp−1 [(t − s)(b − s)] LG +6p−1 Kbp M 1 T ap 1 0 µZ b ¶1−q ¸ p(α−1) p/2−1 ˜ p 1−q +Cp t1 (b − τ ) MT dτ k l k q1 ds k y − yˆ kpb + 0

p/2−1

+3p−1 Kbp Cp t1 ×klk

1

L q (J,R+ )

µZ

˜p M T

L (J,R )

t

(t − s)

¶1−q

p(α−1) 1−q

ds

0

k y − yˆ kpb

· 1 2p 2p 1 p−1 2p(α−1)+1 ˜ b ≤3 LG LG + 2 MT M1 p t1 a 2p(α − 1) + 1 µ ¶1−q ¸ 1−q p/2−1 ˜ p MT +Cp t1 bp(α−1)+1−q k l k q1 L (J,R+ ) p(1 − α) + 1 − q ¶1−q µ 1−q p/2−1 ˜ p +3p−1 Kbp Cp t1 MT p(1 − α) + 1 − q ¶ p(α−1)+1−q k y − yˆ kpb . ×t1 k l k q1 + µ

p−1

Kbp

p−1

L (J,R )

Similarly, for each t ∈ (tk , tk+1 ], k = 1, . . . , m. Let y(t), yˆ(t) ∈ Bb0 and let ρ¯(t) ∈ ¯ (Φy)(t). Then there exists f ∈ SF,y such that ρ¯(t) = −Sα (t)G(0, ϕ) + G(t, yt + ϕ˜t ) + Z +

Z

t

Tα (t − 0

s)Buay (s)ds

+

k X

Sα (t − ti )Ii (yti + ϕ˜ti )

i=1 t

Tα (t − s)f (s)dw(s). 0

From (H3), there exists v(t) ∈ F (t, yˆt + ϕ˜t ) such that E k f (t) − v(t) kpL0 ≤ l(t) k yt − yˆt kpB . 2

Consider Λ : (tk , tk+1 ] → P(L02 ), given by Λ(t) = {v(t) ∈ H : E k f (t) − v(t) kpL0 ≤ l(t) k yt − yˆt kpB }. 2

Since the multi-valued operator W (t) = Λ(t) ∩ F (t, yˆt + ϕ˜t ) is measurable (see [28], Proposition III.4), there exists a function fˆ(t), which is a measurable se14

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lection for W. So, fˆ(t) ∈ F (t, yˆt + ϕ˜t ) and E k f (t) − fˆ(t) kpL0 ≤ l(t) k yt − yˆt kpB . 2

For each t ∈ (tk , tk+1 ], k = 1, . . . , m, we define ρˆ(t) = −Sα (t)G(0, ϕ) + G(t, yˆt + ϕ˜t ) + Z

t

+ 0

Sα (t − ti )Ii (ˆ yti + ϕ˜ti )

i=1

Z Tα (t − s)Buayˆ(s)ds +

k X

t

Tα (t − s)fˆ(s)dw(s).

0

Then, for each t ∈ (tk , tk+1 ], k = 1, . . . , m, we have E k ρ¯(t) − ρˆ(t) kpH ≤ 4p−1 E k G(t, yt + ϕ˜t ) − G(t, yˆt + ϕ˜t ) kpH w k wp wX w w +4p−1 E w S (t − t )[I (y + ϕ ˜ ) − I (ˆ y + ϕ ˜ )] α i i ti ti i ti ti w w

H

i=1

wp wZ t w w a a w T (t − s)B[u (s) − u (s)]ds +4p−1 E w α y yˆ w w 0 wZ t wpH w w +4p−1 E w Tα (t − s)[f (s) − fˆ(s)]dw(s)w w w 0

H

≤ 4p−1 LG k yt − yˆt kpB ˜p +4p−1 k p−1 M S

k X

E k Ii (yti + ϕ˜ti ) − Ii (ˆ yti + ϕ˜ti ) kpH

i=1 p−1

+3

Z

˜ p (tk+1 − tk )p−1 M T

˜p +4p−1 Cp M T

t

0

(t − s)p(α−1) E k B[uay (s) − uayˆ(s)] kpH ds

·Z t · ¸2/p ¸p/2 (t − s)p(α−1) E k f (s) − fˆ(s) kpL0 ds 2

0

k X

˜p ≤ 4p−1 LG k yt − yˆt kpB +4p−1 k p−1 M S ˜ 2p M 2p +12p−1 M 1 T

1 (tk+1 − tk )p−1 ap

Z

ci k yti − yˆti kpB

i=1 t

[(t − s)(b − s)]p(α−1)

0

· k X ˜p × LG k yt − yˆt kpB +k p−1 M ci k yti − yˆti kpB S Z ˜p +Cp (tk+1 − tk )p/2−1 M T p−1

+4

p/2−1

Cp (tk+1 − tk )

0

i=1

b

¸ (b − τ )p(α−1) l(τ ) k yτ − yˆτ kpB dτ ds Z

˜p M T

0

t

(t − s)p(α−1) l(s) k ys − yˆs kpB ds

15

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˜p ≤ 4p−1 Kbp LG k y − yˆ kpb +4p−1 Kbp k p−1 M S ˜ 2p M 2p +12p−1 Kbp M 1 T

1 (tk+1 − tk )p−1 ap

Z

k X

ci k y − yˆ kpb

i=1 t

[(t − s)(b − s)]p(α−1)

0

· k X ˜p ˜p × LG + k p−1 M ci + Cp (tk+1 − tk )p/2−1 M S T i=1

¶1−q µZ b p(α−1) klk (b − τ ) 1−q dτ ×

1 Lq

0 p−1

+4

Kbp Cp (tk+1

×klk

1

L q (J,R+ )

≤4

Kbp

− tk )

˜p M T

t

(t − s)

p(α−1) 1−q

¶1−q ds

0

k X

˜ 2p M 2p 1 (tk+1 − tk )p−1 ci + 3p−1 M 1 T ap i=1 · k X 2p(α−1)+1 p−1 ˜ p b LG + k MS ci

˜p LG + k p−1 M S

1 2p(α − 1) + 1

×

p/2−1

k y − yˆ kpb

µ p−1

(J,R+ )

µZ

¸ ds k y − yˆ kpb

µ ˜p +Cp (tk+1 − tk )p/2−1 M T ×bp(α−1)+1−q k l k

i=1 ¶1−q

1−q p(1 − α) + 1 − q ¸

1

L q (J,R+ )

¶1−q 1−q p(1 − α) + 1 − q ¶ k l k q1 k y − yˆ kpb . + µ

˜p +Kbp Cp (tk+1 − tk )p/2−1 M T ×(tk+1 − tk )p(α−1)+1−q

L (J,R )

Thus, for all t ∈ [0, b], we have ˜ k y − yˆ kp , k ρ¯ − ρˆ kpb ≤ L b and

¯ Φˆ ¯ y) ≤ L ˜ k y − yˆ kp , Hdp (Φy, b

where · µ m X p p−1 p p−1 ˜ p ˜ ˜ L = 4 Kb LG + m MS ci + Cp MT i=1

p(α−1/2)−q

×b

1−q p(1 − α) + 1 − q

¶1−q

¸· ¸ bp(2α−1) 2p 1 p−1 ˜ 2p k l k q1 1 + 3 MT M 1 p < 1. L (J,R+ ) a 2p(α − 1) + 1

¯ is a contraction on B 0 . In view of Lemma 2.6, we conclude that Φ ¯ has Hence, Φ b ∗ 0 ∗ at least one fixed point y ∈ Bb . Let x(t) = y (t) + ϕ(t), ˜ t ∈ (−∞, b]. Then, x 16

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is a fixed point of the operator Φ, which implies that x is a mild solution of the problem (1)-(3) and the proof of Theorem 3.1 is complete. Theorem 3.2. Assume that assumptions of Theorem 3.1 and (H5) are satisfied and {Tα (t) : t ≥ 0} is compact. Moreover, if F is uniformly-bounded, then the system (1)-(3) is approximately controllable on J. Proof. Let xa (·) be a fixed point of Φ in Bb . By Theorem 3.1, any fixed point of Φ is a mild solution of the system (1)-(3). This means that there is xa ∈ Φ(xa ), that is, there is f ∈ SF,xa such that  Sα (t)[ϕ(0) − G(0, ϕ)] + G(t, xat )     + R t T (t − s)Buaa (s)ds + R t T (t − s)f (s)dw(s), t ∈ [0, t ],  1  x 0 α 0 α   S (t)[ϕ(0) a a  − G(0, ϕ)] + G(t, x ) + S (t − t )I (x ) α α 1 1  t t 1  Rt Rt   + 0 Sα (t − s)Buaxa (s)ds + 0 Sα (t − s)f (s)dw(s), t ∈ (t1 , t2 ], a x (t) = ..   .     Sα (t)[ϕ(0) xat )  Pm − G(0, ϕ)] + G(t,  a   + k=1 Sα (t − tk )Ik (xt )    + R t S (t − s)Bua (s)dsk + R t S (t − s)f (s)dw(s), t ∈ (t , b], m xa 0 α 0 α where

·  Rb  b −1 ∗ ∗ ˜  (b − t)(aI + Γ ) Ex ˜b + 0 φ(s)dw(s) S T  α 0   ¸      −S (b)[ϕ(0) − G(0, ϕ)] − G(b, x )  α b    Rb    −B ∗ Tα∗ (b − t) 0 (aI +· Γbs )−1 Sα (b − s)f (s)dw(s),    Rb   ∗ ∗ b −1 ˜  ) E x ˜ + φ(s)dw(s) S T (b − t)(aI + Γ b  α 0 0   ¸      −S (b)ϕ(0) − G(0, ϕ)] − G(b, x ) α b   a −1 b ∗ ∗ ux (t) = −B Tα (b − t)(aI + Γs ) Sα (t − t1 )I1 (xt1 )  Rb    −B ∗ Tα∗ (b − t) 0 (aI + Γbs )−1 Sα (b − s)f (s)dw(s),    ..    .  ·   Rb   S ∗ T ∗ (b − t)(aI + Γb )−1 E x ˜  ˜b + 0 φ(s)dw(s)  α 0   ¸      −Sα (b)ϕ(0) − G(0, ϕ)] − G(b, xb )    Pm    −B ∗ Tα∗ (b − t) k=1 (aI + Γbs )−1 Sα (t − tk )Ik (xtk )   Rb  −B ∗ Tα∗ (b − t) 0 (aI + Γbs )−1 Sα (b − s)f (s)dw(s),

t ∈ [0, t1 ],

t ∈ (t1 , t2 ],

t ∈ (tm , b].

By using the stochastic Fubini theorem, it is easy to see that xa (b) = Sα (t)[ϕ(0) − G(0, ϕ)] + G(b, xab ) + Z + 0

b

Z Sα (b − s)Buaxa (s)ds +

m X

Sα (b − tk )Ik (xatk )

k=1 b

Sα (t − s)f (s)dw(s) 0

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=x ˜b − a(aI +

Γb0 )−1

· Z Ex ˜b +

b

˜ φ(s)dw(s) − Sα (b)[ϕ(0) − G(0, ϕ)]

0

¸ m X −G(b, x ¯ab ) − a (aI + Γbs )−1 Sα (b − tk )Ik (xtk ) Z

k=1 b

−a 0

(aI + Γbs )−1 Sα (b − s)f (s)dw(s).

By the assumption that the sequences {f (s)} is uniformly bounded on J. Thus there is a subsequence, still denoted by {f (s)} that converge weakly to say f ∗∗ (s) in L02 . Now, the compactness of Tα (t), t > 0 which implies that Tα (b − s)[f (s) − f ∗∗ (s)] → 0. Also, by (H5), for all t ∈ J, a(aI + Γbs )−1 → 0 strongly as a → 0+ and k a(aI +Γbs )−1 k≤ 1. Thus, for t ∈ [0, b], by the Lebesque dominated convergence theorem it follows that E k xa (b) − x ˜b kpH ¯ab )] kpH ≤ 5p−1 E k a(aI + Γb0 )−1 [E x ˜b − Sα (b)[ϕ(0) − G(0, ϕ, 0)] − G(b, x wX wp w m w b −1 w a(aI + Γ ) S (b − t )I (x +5p−1 E w ) α k k tk w s w H

k=1

µZ +5p−1 E

b

0

µZ +5p−1 E

b

0

µZ +5p−1 E

0

→0

b

˜ k2 ds k a(aI + Γb0 )−1 φ(s) H

¶p/2 ¶p/2

k a(aI + Γbs )−1 Tα (b − s)[f (s) − f ∗∗ (s)] k2H ds ¶p/2 k a(aI + Γbs )−1 Tα (b − s)f ∗∗ (s) k2H ds

as a → 0+ .

So xa (b) → x ˜b holds, which shows that the system (1)-(3) is approximately controllable and the proof is complete.

4

Application

Consider the fractional impulsive partial stochastic neutral functional differential inclusions in the following form ∂2 N (zt )(x) + u ˜(t, x) ∂x2 Z t ˜b1 (t, s − t, x, z(s, x))ds w(t) , + dt −∞ 0 ≤ t ≤ b, 0 ≤ x ≤ π,

Dtα N (zt )(x) ∈

z(t, 0) = z(t, π) = 0,

0 ≤ t ≤ b,

(6)

(7)

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z(τ, x) = ϕ(τ, x), τ ≤ 0, 0 ≤ x ≤ π, Z tk 4z(tk , x) = ηk (s − tk )z(s, x)ds, k = 1, 2, . . . , m,

(8) (9)

−∞

where Dtα is a Caputo fractional partial derivative of order 0 < α < 1, and u ˜(·) is a real function of bounded variation on [0, b]. w(t) denotes a standard cylindrical Wiener process in H defined on a stochastic space (Ω, F, P ). In this system, Z t

N (zt )(x) = z(t, x) −

b1 (s − t)z(s, x)ds. −∞

Let H = L2 ([0, π]) with the norm k · k and define the operators A : D(A) ⊆ H → H by Aω = ω 00 with the domain D(A) := {ω ∈ H : ω, ω 0 are absolutely continuous, ω 00 ∈ H, ω(0) = ω(π) = 0}. It is well known that A is the infinitesimal generator of an analytic semigroup (T (t))t≥0 in H. Furthermore, A has a discrete spectrum with eigenvalues of 2 the form −nq , n ∈ N and corresponding normalized eigenfunctions are given

2 π sin(nz). In addition {xn : n ∈ N} is an orthonormal basis for P∞ 2 H, T (t)y = n=1 e−n t (y, xn )xn for all y ∈ H, and every t > 0. From these expressions it follows that (T (t))t≥0 is a uniformly bounded compact semigroup, −1

by xn (z) =

so that R(λ, A) = (λ − A) is a compact operator for all λ ∈ ρ(A) i.e. A ∈ Aα (θ0 , ω 0 ). ˜ : (−∞, −r] → R be a nonnegative measurable Let r ≥ 0, 1 ≤ p < ∞ and let h function which satisfies the conditions (h-5), (h-6) in the terminology of Hino ˜ is locally integrable and there is a nonet al. [29]. Briefly, this means that h ˜ + τ ) ≤ γ(ξ)h(τ ˜ ) negative, locally bounded function γ on (−∞, 0] such that h(ξ for all ξ ≤ 0 and θ ∈ (−∞, −r) \ Nξ , where Nξ ⊆ (−∞, −r) is a set whose ˜ H) the set consists of all Lebesgue measure zero. We denote by PC r × Lp (h, classes of functions ϕ : (−∞, 0] → H such that ϕ|[−r,0] ∈ PC([−r, 0], H), ϕ(·) ˜ k ϕ kp is Lebesgue integrable on is Lebesgue measurable on (−∞, −r), and h (−∞, −r). The seminorm is given by µ Z −r ¶1/p ˜ ) k ϕ kp dτ k ϕ kB = sup k ϕ(τ ) k + h(τ . −r≤τ ≤0

−∞

˜ H) satisfies axioms (A)-(C). Moreover, when r = 0 The space B = PC r × Lp (h, R ˜ )dτ )1/2 , ˜ = 1, M (t) = γ(−t)1/2 and K(t) = 1+( 0 h(τ and p = 2, we can take H −t for t ≥ 0 (see [29, Theorem 1.3.8] for details). Additionally, we will assume that R 2 ˜ 1 = ( 0 (b1 (s)) ds) 12 < ∞, (i) The function b1 : R → R, is continuous, and L ˜ −∞ h(s) (ii) The function ˜b1 : R4 → R, is continuous and there exist continuous functions aj : R → R, j = 1, 2, such that |˜b1 (t, s, x, y)| ≤ a1 (t)a2 (s)|y|,

(t, s, x, y) ∈ R4 ,

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and |˜b1 (t, s, x, y1 ) − ˜b1 (t, s, x, y2 )| ≤ a1 (t)a2 (s)|y1 − y2 |, (t, s, x, y1 ), (t, s, x, y2 ) ∈ R4 ˆ1 = ( with L

R0 −∞

1 (a2 (s))2 ds) 2 ˜ h(s)

< ∞.

(iii) The functions ηk : R → R, k = 1, 2, . . . , m, are continuous, and Lk = R0 2 1 k (s)) ( −∞ (ηh(s) ds) 2 < ∞ for every k = 1, 2, . . . , m, ˜ ˜ H) with ϕ(θ)(x) = ϕ(θ, x), (θ, x) ∈ (−∞, 0] × B. Take ϕ ∈ B = PC 0 × L2 (h, Let G : [0, b] × B → H, F : [0, b] × B → P(H) be the operators defined by N (ψ)(x) = ψ(0, x) − G(t, ψ)(x), Z

0

G(t, ψ)(x) = Z

b1 (s)ψ(s, x)ds, −∞ 0

F (t, ψ)(x) =

˜b1 (t, s, x, ψ(s, x))ds.

−∞

Also defining the maps Ik and B by Z 0 Ik (ψ)(x) = ηk (s)ψ(s, x)ds, (Bu)(t)(x) = u ˜(t, x). −∞

Using these definitions, we can represent the system (6)-(9) in the abstract form (1)-(3). Moreover, for any t ∈ [0, b], ψ, ψ1 ∈ B, we have that E k G(t, ψ) − G(t, ψ1 ) kp ≤ LG k ψ − ψ1 kpB , E k F (t, ψ) − F (t, ψ1 ) kp ≤ LF k ψ − ψ1 kpB , E k Ik (ψ) − Ik (ψ1 ) kp ≤ (Lk )p k ψ − ψ1 kpB , k = 1, 2, . . . , m, and F is bounded linear ˜ 1 )p , LF = (k a1 k∞ L ˆ 1 )p . operators with E k F kp ≤ LF , where LG = (L L(B,H)

Further, we can impose some suitable conditions on the above-defined functions to verify the assumptions on Theorem 3.2. Hence by Theorems 3.2, the system (6)-(9) is approximately controllable on [0, b]. Acknowledgments The first author’s work was supported by NNSF of China (11461019), the President Fund of Scientific Research Innovation and Application of Hexi University (xz2013-10, XZ2014-22), the Scientific Research Project of Universities of Gansu Province (2014A-110).

References [1] R. Triggiani, A note on the lack of exact controllability for mild solutions in Banach spaces. SIAM J. Control Optim. 15 (1977), 407-411.

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[2] N.I. Mahmudov, A. Denker, On controllability of linear stochastic systems, Internat. J. Control 73 (2000), 144-151. [3] J.P. Dauer, N.I. Mahmudov, Controllability of stochastic semilinear functional differential equations in Hilbert spaces, J. Math. Anal. Appl. 290 (2004), 373-394. [4] N. I. Mahmudov, Approximate controllability of semilinear deterministic and stochastic evolution equations in abstract spaces, SIAM J. Control Optim. 42 (2003), 1604-1622. [5] P. Balasubramaniam, J.Y. Park, P. Muthukumar, Approximate controllability of neutral stochastic functional differential systems with infinite delay, Stoch. Anal. Appl. 28 (2010), 389-400. [6] R. Sakthivel, Y. Ren, N.I. Mahmudov, On the approximate controllability of semilinear fractional differential systems, Comput. Math. Appl. 62 (2011), 1451-1459. [7] S. Kumar, N. Sukavanam, Approximate controllability of fractional order semilinear systems with bounded delay, J. Differential Equations 252 (2012), 6163-6174. [8] Z. Yan, Approximate controllability of partial neutral functional differential systems of fractional order with state-dependent delay, Internat. J. Control 85 (2012), 1051-1062. [9] A. Debbouche, D.F.M. Torres, Approximate controllability of fractional nonlocal delay semilinear systems in Hilbert spaces, Internat. J. Control 86 (2013), 1577-1585. [10] R. Sakthivel, R. Ganesh, S. Suganya, Approximate controllability of fractional neutral stochastic system with infinite delay, Rep. Math. Phys. 70 (2012), 291-311. [11] M. Kerboua, A. Debbouche, D. Baleanu, Approximate controllability of sobolev type nonlocal fractional stochastic dynamic systems in Hilbert spaces, Abstr. Appl. Anal. 2013 (2013), Article ID 262191, 1-10. [12] P. Muthukumar, C. Rajivganthi, Approximate controllability of fractional order neutral stochastic integro-differential system with nonlocal conditions and infinite delay, Taiwanese J. Math. 17 (2013), 1693-1713. [13] S. Farahi, T. Guendouzi, Approximate controllability of fractional neutral stochastic evolution equations with nonlocal conditions, Results. Math. 65 (2014), 501-521. [14] M. Benchohra, J. Henderson, S.K. Ntouyas, Impulsive Differential Equations and Inclusions, Hindawi Publishing Corporation, New York, 2006.

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[15] Z. Liu, M. Bin, Approximate controllability for impulsive riemann-liouville fractional differential inclusions, Abstr. Appl. Anal. 2013 (2013), Article ID 639492, 1-17. [16] P. Balasubramaniam, V. Vembarasan, T. Senthilkumar, Approximate controllability of impulsive fractional integro-differential systems with nonlocal conditions in Hilbert space, Numer. Funct. Anal. Optim. 35 (2014), 177197. [17] D.N. Chalishajar, K. Malar, K. Karthikeyan, Approximate controllability of abstract impulsive fractional neutral evolution equations with infinite delay in Banach spaces, Electron. J. Differential Equations 275 (2013), 121. [18] R. Subalakshmi, K. Balachandran, Approximate controllability of nonlinear stochastic impulsive intergrodifferential systems in Hilbert spaces, Chaos Solitons Fractals 42 (2009), 2035-2046. [19] L. Shen, J. Sun, Approximate controllability of stochastic impulsive functional systems with infinite delay, Automatica 48 (2012), 2705-2709. [20] Y. Zang, J. Li, Approximate controllability of fractional impulsive neutral stochastic differential equations with nonlocal conditions, Bound. Value Probl. 193 (2013), 1-14. [21] K. Deimling, Multi-Valued Differential Equations, De Gruyter, Berlin, 1992. [22] S. Hu, N. Papageorgiou, Handbook of Multivalued Analysis, Kluwer Academic Publishers, Dordrecht, Boston, 1997. [23] J.K. Hale, J. Kato, Phase spaces for retarded equations with infinite delay, Funkcial. Ekvac. 21 (1978), 11-41. [24] A.A. Kilbas, H.M. Srivastava, J.J. Trujillo, Theory and applications of fractional differential equations, North Holland Mathematics Studies, vol. 204, Elsevier Science Publishers BV, Amsterdam, 2006. [25] E. Bajlekova, Fractional evolution equations in Banach spaces (Ph.D. thesis), Eindhoven University of Technology, 2001. [26] G. Da Prato, J. Zabczyk, Stochastic Equations in Infinite Dimensions, Cambridge University Press, Cambridge, 1992. [27] H. Covitz, Jr.S.B. Nadler, Multivalued contraction mappings in generalized metric spaces, Israel J. Math. 8 (1970), 5-11. [28] C. Castaing, M. Valadier; Convex Analysis and Measurable Multifunctions, Lecture Notes in Mathematics, vol. 580, Springer-Verlag, Berlin, Heidelberg, New York, 1977. 22

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[29] Y. Hino, S. Murakami, T. Naito, Functional-Differential Equations with Infinite Delay, in: Lecture Notes in Mathematics, vol. 1473, Springer-Verlag, Berlin, 1991. Zuomao Yan and Xiumei Jia Department of Mathematics, Hexi University, Zhangye, Gansu 734000, P.R. China ∗ Corresponding author. E-mail address: [email protected]

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HYERS-ULAM STABILITY OF GENERAL ADDITIVE MAPPINGS IN C ∗ -ALGEBRA GANG LU, GUOXIAN CAI, YUANFENG JIN∗ , AND CHOONKIL PARK Abstract. In this paper, we prove that the generalized Hyers-Ulam stability of homomorphisms in C ∗ -algebras and Lie C ∗ -algebras and also of derivations on C ∗ -algebras and Lie C ∗ -algebras for an 4-variable additive functional equation

1. Introduction and preliminaries The stability problem of functional equations originated from a question of Ulam [28] concerning the stability of group homomorphisms. Hyers [11] gave a first affirmative partial answer to the question of Ulam for Banach spaces. Hyers’ Theorem was generalized by Aoki [1] for additive mappings and by Th.M. Rassias [19] for linear mappings by considering an unbounded Cauchy difference. The paper of Rassias has provided a lot of influence in the development of what we call generalized Hyers-Ulam stability of functional equations. A generalization of the Th.M. Rassias theorem was obtained by G˘avruta [9] by replacing the unbounded Cauchy difference by a general control function in the spirit of Th.M. Rassias’ approach. The stability problems for several functional equations or inequations have been extensively investigated by a number of authors and there are many interesting results concerning this problem (see [2]–[8],[10], [12]–[14], [18]–[21],[22]-[27],[29]). We recall a fundamental result in fixed point theory. Let X be a set. A function d : X × X → [0, ∞] is called a generalized metric on X if d satisfies (1) d(x,y)=0 if and only if x=y; (2) d(x,y)=d(y,x) for all x, y ∈ X; (3) d(x, z) ≤ d(x, y) + d(y, z) for all x, y, z ∈ X. Theorem 1.1 (see[6],[7]). Let (X, d) be a complete generalized metric space and let J : X → X be a strictly contractive mapping with Lipschitz constant L < 1. Then for 2010 Mathematics Subject Classification. Primary 39B62, 39B52, 46B25. Key words and phrases. additive functional equation; Hyers-Ulam stability; fixed point; derivation on C ∗ -algebras and Lie C ∗ -algebras. ∗ Corresponding author: [email protected](Y. Jin). 1

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each given element x ∈ X, either d(J n x, J n+1 x) = ∞

(1.1)

for all nonnegative integers n or there exists a positive integer n0 such that (1) (2) (3) (4)

d(J n x, J n+1 x) < ∞, fir all n ≥ n0 ; the sequence {J n x} converges to a fixed point y ∗ of J; y ∗ is the unique fixed point of J in the set Y = {y ∈ X|d(J N0 x, y) < ∞}; 1 d(y, Jy) for all y ∈ Y . d(y, y ∗ ) ≤ 1−L

By the using fixed point method, the stability problems of several functional equations have been extensively investigated by a number of authors(see[5][6][16][17]). This paper, using the fixed point method, we prove the generalized Hyers-Ulam stability of homomorphisms in C ∗ -algebras and of derivations on C ∗ -algebras for the general Jensen-type functional equation. And, we prove that the generalized Hyers-Ulam stability of homomorphisms in Lie C ∗ -algebras and of derivations on Lie C ∗ -algebras for the following additive functional equation: f (dx1 + ax2 + bx3 + cx4 ) + f (ax1 + dx2 + cx3 + bx4 ) + f (bx1 + cx2 + dx3 + ax4 ) + f (cx1 + bx2 + ax3 + dx4 )

(1.2)

= (d + a + b + c)f (x1 + x2 + x3 + x4 ) Here a, b, c and d are real numbers with a + b + c + d 6= 0. Throughout the paper, assume that k is a + b + c + d. 2. Stability of Homomorphisms and Derivations in C ∗ -Algebras Throughout this section, assume that X is a C ∗ -algebras with norm k · kX and that Y is a C ∗ -algebra with norm k · kY . For a given mapping f : X → Y , we define Fµ f (x1 , x2 , x3 , x4 ) := µf (dx1 + ax2 + bx3 + cx4 ) + µf (ax1 + dx2 + cx3 + bx4 ) + µf (bx1 + cx2 + dx3 + ax4 ) + µf (cx1 + bx2 + ax3 + dx4 )

(2.1)

− (d + a + b + c)f (µ(x1 + x2 + x3 + x4 )) for all µ ∈ T1 := {v ∈ C : |v| = 1} and x1 , · · · , xm ∈ X. Note that a C-linear mapping H : X → Y is called a homomorphism in C ∗ -algebras if H satisfies H(xy) = H(x)H(y) and H(x∗ ) = H(x)∗ for all x, y ∈ X. Now we prove the Hyers-Ulam stability of homomorphisms in C ∗ -algebras for the functional equation Fµ f (x, y) = 0.

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Theorem 2.1. Let a, b, c and d be fixed nonzero real numbers. Let f : X → Y be an mapping for which there exists a function ϕ : X 4 → [0, ∞), such that kFµ f (x1 , x2 , x3 , x4 )kY ≤ ϕ(x1 , x2 , x3 , x4 ),

(2.2)

kf (xy) − f (x)f (y)kY ≤ ϕ(x, x, y, y),

(2.3)

kf (x∗ ) − f (x)∗ kY ≤ ϕ(x, x, x, x)

(2.4)

for all µ ∈ T1 := {v ∈ C : |v| = 1} and all x1 , x2 , x3 , x4 , x, y ∈ X. If there exists an
4 4 4 4 0 < L < 1 such that ϕ(x1 , x2 , x3 , x4 ) ≤ |k| Lϕ x , x , x , x for all x1 , x2 , x3 , x4 ∈ 1 2 3 4 4 k k k k ∗ X, d, a, b, c, α4 ∈ R with 4 < |k|, then there exists a unique C -algebra homomorphism H : X → Y such that x x x x 4 ϕ , , , (2.5) kf (x) − H(x)kY ≤ |k|(1 − L) 4 4 4 4 for all x ∈ X.
4 4 4 4 Lϕ x , x , x , x Proof. It follows that ϕ(x1 , x2 , x3 , x4 ) ≤ |k| that 1 2 3 4 4 k k k k   (k)j (k)j (k)j (k)j 4j lim ϕ x , x , x , x4 = 0 (2.6) 1 2 3 j→∞ |k|j 4j 4j 4j 4j for all x, y ∈ X. Consider the set A := {g : X → Y }

(2.7)

and introduce the generalized metric on A: d(g, h) = inf{C ∈ R+ : kg(x) − h(x)kY ≤ Cϕ

x x x x , , , , ∀x ∈ X}. 4 4 4 4

It is easy to show that (A, d) is complete. Now we consider the linear mapping J : A → A such that   4 k Jg(x) := g x |k| 4 for all x ∈ X. By Theorem 3.1 of [6] d(Jg, Jh) ≤ Ld(g, h)

(2.8)

(2.9)

(2.10)

for all g, h ∈ A. Letting µ = 1 and x1 = x2 = x3 = x4 = x in (2.2), we get

 

4

k 1 x x x x 4 x x x x

f x − f (x) ≤ ϕ( , , , ) ≤ ϕ( , , , )

k 4 |k| 4 4 4 4 |k| 4 4 4 4 for all x ∈ X. Hence d(f, Jf ) ≤

4 . |k|

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By Theorem 1.1, there exists a mapping H : X → Y such that (1) H is a fixed point of J, that is,   4 k H x = H(x) |k| 4 for all x ∈ X. The mapping H is a unique fixed point of J in the set B = {g ∈ A : d(f, g) < ∞}.

(2.11)

(2.12)

This implies that H is a unique mapping satisfying (2.24) such that there exists C ∈ (0, ∞) satisfying x x x x kH(x) − f (x)kY ≤ Cϕ , , , (2.13) 4 4 4 4 for all x ∈ X. (2) d(J n f, H) → 0 as n → ∞. This implies the inequality  n  (k) x 4n f lim = H(x) (2.14) n n→∞ |k| 4n for all x ∈ X. 1 d(f, Jf ), which implies the ineqality (3) d(f, H) ≤ 1−L d(f, H) ≤

4 . |k|(1 − L)

(2.15)

This implies that the inequality (2.5) holds. Next, we show that H(x) is additive map. kH(dx1 + ax2 + bx3 + cx4 ) + H(ax1 + dx2 + cx3 + bx4 ) +H(bx1 + cx2 + dx3 + ax4 ) +H(cx1 + bx2 + ax3 + dx4 ) − (d + a + b + c)H(x1 + x2 + x3 + x4 )k

l  l 

4 (k) = lim f (dx1 + ax2 + bx3 + cx4 ) l→∞ |k|l 4l  l  4l (k) + lf (ax1 + dx2 + cx3 + bx4 ) |k| 4l  l  (k) 4l + lf (bx1 + cx2 + dx3 + ax4 ) |k| 4l  l  4l (k) + lf (cx1 + bx2 + ax3 + dx4 ) |k| 4l  l 

4l (k) −(d + a + b + c) l f (x1 + x2 + x3 + x4 )

l |k| 4   4l (k)l (k)l (k)l (k)l ϕ x1 , l x2 , l x3 , l x4 = 0 ≤ lim l→∞ |k|l 4l 4 4 4

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Therefore, the mapping H : X → Y is Cauchy additive. By a similar method with above, we may get µH(x) = H(µx) for all µ ∈ T1 and all x ∈ X. Thus one can show that the mapping H : X → Y is C-linear. It follows from (2.3)that kH(xy) − H(x)H(y)kY 2n  2n   n   n  4 (k) xy (k) x (k) y

f − f f = lim

n→∞ k 42n 4n 4n n  2n   n   n  4 (k) x (k) y (k) xy

≤ lim f − f f

n→∞ k 42n 4n 4n n  n  4 (k) x (k)n x (k)n y (k)n y , n , n , n =0 ≤ lim ϕ n→∞ k 4n 4 4 4 for all x, y ∈ X. So H(xy) = H(x)H(y)

(2.16)

for all x, y ∈ X. It follows from (2.4) that n  n ∗   n ∗ 4 (k) x (k) x



kH(x ) − H(x) kY = lim f − f

n→∞ k 4n 4n Y n  n  n n n 4 (k) x (k) x (k) x (k) x , n , n , n =0 ≤ lim ϕ n→∞ k 4n 4 4 4 ∗

∗

for all x ∈ X. So H(x∗ ) = H(x)∗ for all x ∈ X. Thus H : X → Y is C ∗ -algebra homomorphism satisfying (2.5), as desired.  Theorem 2.2. Let a, b, c and d be fixed nonzero real numbers. Let f : X → Y be a mapping for which there exists a function ϕ : X 4 → [0, ∞) satisfying (2.2),(2.3) and
4 (2.4). If there exists an L < 1 such ϕ(x1 , x2 , x3 , x4 ) ≤ |k| Lϕ k4 x1 , k4 x2 , k4 x3 , k4 x4 for all x1 , x2 , x3 , x4 ∈ X with |k| < 4, then there exists a unique C ∗ -algebra homomorphism H : X → Y such that x x x x 1 kf (x) − H(x)k ≤ ϕ , , , 4 − 4L k k k k for all x ∈ X.

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Proof. Consider the set A := {g : X → Y } and introduce the generalized metric on A: x x x x , , , , ∀x ∈ X} k k k k We consider the linear mapping J : A → A such that   |k| 4 Jg(x) := g 4 k for all x ∈ X. It follow from (2.2) that

   

f (x) − k 4 x ≤ 1 ϕ x , x , x , x

4 k 4 k k k k d(g, h) = inf{C ∈ R+ kg(x) − h(x)kY ≤ Cϕ

(2.17)

(2.18)

(2.19)

for all x ∈ X. Hence d(f, Jf ) ≤ 41 . The rest of the proof is similar to the proof of Theorem 2.1.



Recall that a C-linear mapping δ : X → Y is called a derivation on X satisfies δ(xy) = δ(x)y + xδ(y) for all x, y ∈ X. Theorem 2.3. Let a, b, c, d be the fixed real numbers. Let f : X → Y be an mapping for which there exists a function ϕ : X 4 → [0, ∞), such that kFµ f (x1 , x2 , x3 , x4 )kY ≤ ϕ(x1 , x2 , x3 , x4 ),

(2.20)

kf (xy) − f (x)y − xf (y)kY ≤ ϕ(x, x, y, y),

(2.21)

for all µ ∈ T1 := {v ∈ C : |v| = 1} and all x1 , x2 , x3 , x4 , x, y ∈ X. If there exists an
4 4 4 4 L < 1 such that ϕ(x1 , x2 , x3 , x4 ) ≤ |k| Lϕ x , x , x , x for all x1 , x2 , x3 , x4 ∈ X 1 2 3 4 4 k k k k with |k| > 4, then there exists a unique derivation δ : X → X such that x x x x 1 kf (x) − δ(x)kY ≤ ϕ , , , (2.22) |k|(1 − L) 4 4 4 4 for all x ∈ X.
Proof. It follows from ϕ(x1 , x2 , x3 , x4 ) ≤ |k| Lϕ k4 x1 , k4 x2 , k4 x3 , k4 x4 that 4 j  j  j  j  j ! 4 k k k k lim ϕ x1 , x2 , x3 , x4 = 0 j→∞ |k| 4 4 4 4 for all x1 , x2 , x3 , x4 ∈ X. Consider the set A := {g : X → X} and introduce the generalized metric on A: d(g, h) = inf{C ∈ R+ : kg(x) − h(x)k ≤ Cϕ

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It is easy to show that (A, d) is complete. Now we consider the linear mapping J : A → A such that   4 k Jg(x) := g x k 4 for all x ∈ X. By Theorem 3.1 of [6] d(Jg, Jh) ≤ Ld(g, h)

(2.23)

for all g, h ∈ A. Letting µ = 1 and x1 = x2 = x3 = x4 = x in (2.2), we get

 

4

f k x − f (x) ≤ 1 ϕ( x , x , x , x )

|k| 4 4 4 4

k 4 for all x ∈ X. 1 Hence d(f, Jf ) ≤ |k| . By Theorem 1.1, there exists a mapping δ : X → Y such that (1) δ is a fixed point of J, that is,   k 4 δ x = δ(x) k 4

(2.24)

for all x ∈ X. The mapping δ is a unique fixed point of J in the set B = {g ∈ A : d(f, g) < ∞}.

(2.25)

This implies that δ is a unique mapping satisfying (2.24) such that there exists C ∈ (0, ∞) satisfying x x x x kδ(x) − f (x)kY ≤ Cϕ , , , (2.26) 4 4 4 4 for all x ∈ X. (2) d(J n f, δ) → 0 as n → ∞. This implies the inequality  n  4n (k) x lim f = δ(x) (2.27) n n→∞ |k| 4n for all x ∈ X. 1 (3) d(f, δ) ≤ 1−L d(f, Jf ), which implies the ineqality d(f, H) ≤

1 . |k|(1 − L)

(2.28)

This implies that the inequality (2.22) holds. The rest of the proof is similar to the proof of Theorem 2.1.

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Theorem 2.4. Let a, b, c, d be the fixed real numbers. Let f : X → Y be an mapping for which there exists a function ϕ : X 4 → [0, ∞), such that kFµ f (x1 , x2 , x3 , x4 )kY ≤ ϕ(x1 , x2 , x3 , x4 ),

(2.29)

kf (xy) − f (x)y − xf (y)kY ≤ ϕ(x, x, y, y),

(2.30)

for all µ ∈ T1 := {v ∈ C : |v| = 1} and all x1 , x2 , x3 , x4 , x, y ∈ X. If there exists an
4 Lϕ k4 x1 , k4 x2 , k4 x3 , k4 x4 for all x1 , x2 , x3 , x4 ∈ X L < 1 such that ϕ(x1 , x2 , x3 , x4 ) ≤ |k| with |k| < 4, then there exists a unique derivation δ : X → X such that x x x x 1 ϕ , , , (2.31) kf (x) − δ(x)kY ≤ 4(1 − L) k k k k for all x ∈ X. Proof. The proof is similar to the proof of 2.3.



3. Stability of homomorphisms in Lie C ∗ -algebras A C ∗ -algebra C, endowed with the Lie product xy − yx [x, y] := 2 ∗ on C, is called a Lie C -algebras(see[5],[15]). Definition 3.1. Let X and Y be Lie C ∗ -algebras. A C-linear mapping H : X → Y is called a Lie C ∗ -algebras homomorphism if H([x, y]) = [H(x), H(y)] for all x, y ∈ X. Throughout this section, assume that X is a Lie C ∗ -algebras with a norm k · kX and B is a Lie C ∗ -algebras with a norm k · kY . Now, we prove the generalized Hyers-Ulam stability of homomorphisms in Lie C ∗ algebras for the functional equation Dµ f (xx1 , x2 , x3 , x4 ) = 0. Theorem 3.2. Let a, b, c, d be the fixed real numbers. Let f : X → Y be an mapping for which there exists a function ϕ : X 4 → [0, ∞), such that kFµ f (x1 , x2 , x3 , x4 )kY ≤ ϕ(x1 , x2 , x3 , x4 ),

(3.1)

kf ([x, y]) − [f (x), f (y)]kY ≤ ϕ(x, x, y, y),

(3.2)

for all µ ∈ T1 := {v ∈ C : |v| = 1} and all x1 , x2 , x3 , x4 , x, y ∈ X. If there exists an
4 4 4 4 0 < L < 1 such that ϕ(x1 , x2 , x3 , x4 ) ≤ |k| Lϕ x , x , x , x for all x1 , x2 , x3 , x4 ∈ 1 2 3 4 4 k k k k X with |k| > 4, then there exists a unique derivation H : X → X such that x x x x 1 ϕ , , , (3.3) kf (x) − H(x)kY ≤ |k|(1 − L) 4 4 4 4 for all x ∈ X.

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Proof. By the same method as in the proof of Theorem 2.1, we can get the mapping H : X → Y given by  n  4n k H(x) = lim f x n n→∞ |k| 4n for all x ∈ X. Thus it follows from 3.2that

 2n    n   n 

42n k k k

f

kH([x, y]) − [H(x), H(y)]kY = lim [x, y] − f x , f x

n→∞ |k|2n 42n 4n 4n Y   kn kn 42n ϕ x, y = 0 ≤ lim n→∞ |k|2n 4n 4n for all x, y ∈ X, and so H([x, y]) = [H(x), H(y)] for all x, y ∈ X. Therefore, H : X → Y is a Lie C ∗ -algebras homomorphism satisfying 3.3. This completes the proof.  Theorem 3.3. Let a, b, c, d be the fixed real numbers. Let f : X → Y be an mapping for which there exists a function ϕ : X 4 → [0, ∞), such that kFµ f (x1 , x2 , x3 , x4 )kY ≤ ϕ(x1 , x2 , x3 , x4 ),

(3.4)

kf ([x, y]) − [f (x), f (y)]kY ≤ ϕ(x, x, y, y),

(3.5)

for all µ ∈ T1 := {v ∈ C : |v| = 1} and all x1 , x2 , x3 , x4 , x, y ∈ X. If there exists an
4 0 < L < 1 such that ϕ(x1 , x2 , x3 , x4 ) ≤ |k| Lϕ k4 x1 , k4 x2 , k4 x3 , k4 x4 for all x1 , x2 , x3 , x4 ∈ X with |k| < 4, then there exists a unique derivation H : X → X such that x x x x 1 kf (x) − H(x)kY ≤ ϕ , , , (3.6) 4(1 − L) k k k k for all x ∈ X. Proof. By the same method as in the proof of Theorem 2.1, we can get the mapping H : X → Y given by  n  4 |k|n H(x) = lim n f x n→∞ 4 kn for all x ∈ X. Thus it follows from 3.5that

 2n    n   n 

|k|2n 4 4 4

kH([x, y]) − [H(x), H(y)]kY = lim 2n f [x, y] − f x , f x

n→∞ 4 k 2n kn kn Y  n  2n n |k| 4 4 ≤ lim 2n ϕ x, n y = 0 n n→∞ 4 k k for all x, y ∈ X, and so H([x, y]) = [H(x), H(y)] for all x, y ∈ X. Therefore, H : X → Y is a Lie C ∗ -algebras homomorphism satisfying 3.6. This completes the proof. 

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4. Stability of derivations in Lie C ∗ -algebras Definition 4.1. Let X be a Lie C ∗ -algebra. A C-linear mapping δ : X → X is called a Lie derivation if δ([x, y]) = [δ(x), y] + [x, δ(y)] for all x, y ∈ X. Throughout this section, assume that X is a Lie C ∗ -algebra with a norm k · kX . Finally, we prove the generalized Hyers-Ulam stability of derivations on Lie C ∗ -algebras for the functional equation Dµ f (x1 , x2 , x3 , x4 ) = 0. Theorem 4.2. Let a, b, c, d be the fixed real numbers. Let f : X → Y be an mapping for which there exists a function ϕ : X 4 → [0, ∞), such that kFµ f (x1 , x2 , x3 , x4 )kY ≤ ϕ(x1 , x2 , x3 , x4 ),

(4.1)

kf ([x, y]) − [f (x), y] − [x, f (y)]kY ≤ ϕ(x, x, y, y),

(4.2)

for all µ ∈ T1 := {v ∈ C : |v| = 1} and all x1 , x2 , x3 , x4 , x, y ∈ X. If there exists an
4 4 4 4 Lϕ x , x , x , x for all x1 , x2 , x3 , x4 ∈ 0 < L < 1 such that ϕ(x1 , x2 , x3 , x4 ) ≤ |k| 1 2 3 4 4 k k k k X with |k| > 4, then there exists a unique derivation δ : X → X such that x x x x 1 ϕ , , , (4.3) kf (x) − δ(x)kY ≤ |k|(1 − L) 4 4 4 4 for all x ∈ X. Proof. By the same method as in the proof of Theorem 2.1, we can get the mapping δ : X → Y given by  n  k 4n f x δ(x) = lim n n→∞ |k| 4n for all x ∈ X. Thus it follows from 4.2that kδ([x, y]) − [δ(x), y] − [x, δ(y)]kY

 2n    n  n   n  n 

42n k k k k k

f = lim [x, y] − f x , y − x, f y

n→∞ |k|2n 42n 4n 4n 4n 4n Y   42n kn kn ≤ lim ϕ x, y = 0 n→∞ |k|2n 4n 4n for all x, y ∈ X, and so δ([x, y]) = [δ(x), y] + [x, δ(y)] for all x, y ∈ X. Therefore, δ : X → X is a Lie C ∗ -algebras homomorphism satisfying 4.3. This completes the proof.  Theorem 4.3. Let a, b, c, d be the fixed real numbers. Let f : X → Y be an mapping for which there exists a function ϕ : X 4 → [0, ∞), such that kFµ f (x1 , x2 , x3 , x4 )kY ≤ ϕ(x1 , x2 , x3 , x4 ),

(4.4)

kf ([x, y]) − [f (x), y] − [x, f (y)]kY ≤ ϕ(x, x, y, y),

(4.5)

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for all µ ∈ T1 := {v ∈ C : |v| = 1} and all x1 , x2 , x3 , x4 , x, y ∈ X. If there exists an
4 Lϕ k4 x1 , k4 x2 , k4 x3 , k4 x4 for all x1 , x2 , x3 , x4 ∈ 0 < L < 1 such that ϕ(x1 , x2 , x3 , x4 ) ≤ |k| X with |k| < 4, then there exists a unique derivation δ : X → X such that x x x x 1 kf (x) − δ(x)kY ≤ ϕ , , , (4.6) 4(1 − L) k k k k for all x ∈ X. Proof. By the same method as in the proof of Theorem 2.1, we can get the mapping δ : X → Y given by  n  |k|n 4 δ(x) = lim n f x n→∞ 4 kn for all x ∈ X. Thus it follows from 4.5 that kδ([x, y]) − [H(x), y] − [x, H(y)]kY

 2n    n  n   n  n 

4 4 4 4 4 |k|2n

[x, y] − f x , y − x, f x = lim 2n

n→∞ 4 k 2n kn kn kn kn Y  n  2n n |k| 4 4 ≤ lim 2n ϕ x, n y = 0 n n→∞ 4 k k for all x, y ∈ X, and so δ([x, y]) = [δ(x), y] + [x, δ(y)] for all x, y ∈ X. Therefore, δ : X → X is a Lie C ∗ -algebras homomorphism satisfying 4.6. This completes the proof.  Acknowledgments G. Lu was supported by Doctoral Science Foundation of ShenYang University of Technology and the Scientific Research Foundation for the Returned Overseas Chinese Scholars, State Education Ministry(No.2014-62009). Y.jin was supported by National Natural Science Foundation of ChinaThe study of high-precision algorithm for high dimensional delay partial differential equations.2014-2017 (11361066) References [1] T. Aoki, On the stability of the linear transformation in Banach spaces, J. Math. Soc. Japan 2 (1950), 64–66. [2] J. Aczel and J. Dhombres, Functional Equations in Several Variables, Cambridge Univ. Press, Cambridge, 1989. [3] P.W. Cholewa, Remarks on the stability of functional equations, Aequationes Math. 27 (1984), 76–86. [4] Y. Cho, C. Park and R. Saadati, Functional inequalities in non-Archimedean Banach spaces, Appl. Math. Lett. 23 (2010), 1238-1242. [5] Y.Cho, R.Saadati and Y.Yang,Approximation of Homomorphisms and Derivations on Lie C ∗ -algebras via Fixed Point Method,Journal of Inequalities and Applications,2013:415,http://www.journalo?nequalitiesandapplications.com/content/2013/1/415.

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[6] L.C˘ adariu and V.Radu, Fixed Points and the Stability of Jensen’s Functional equation, Journal of Inequalities in Pure and Applied Mathematics,vol.4, no.1, article4,7papers,2003. [7] J.B.Diaz and B.Margolis, A Fixed Point Theorem of the Alternative, for Contractions on a Generalized Complete Metric Space, Buletin of American Mathematical Sociaty, vol.44,pp.305-309,1968. [8] A. Ebadian, N. Ghobadipour, Th. M. Rassias, and M. Eshaghi Gordji, Functional inequalities associated with Cauchy additive functional equation in non-Archimedean spaces, Discrete Dyn. Nat. Soc. 2011 (2011), Article ID 929824. [9] P. Gˇavruta, A generalization of the Hyers-Ulam-Rassias stability of approximately additive mappings, J. Math. Anal. Appl. 184 (1994), 431–436. [10] I. S. Chang, M. Eshaghi Gordji, H. Khodaei, and H. M. Kim, Nearly quartic mappings in βhomogeneous F-spaces, Results Math. 63 (2013) 529-541. [11] D.H. Hyers, On the stability of the linear functional equation, Proc. Natl. Acad. Sci. USA 27 (1941), 222–224. [12] D.H. Hyers, G. Isac and Th.M. Rassias, Stability of Functional Equations in Several Variables, Birkh¨auser, Basel, 1998. [13] G. Isac and Th.M. Rassias, On the Hyers-Ulam stability of ψ-additive mappings, J. Approx. Theory 72 (1993), 131–137. [14] G. Lu and C. Park, Hyers-Ulam Stability of Additive Set-valued Functional Equations, Appl. Math. Lett. 24 (2011), 1312-1316. [15] G. Lu and C.Park,Hyers-Ulam Stability of General Jensen-Type Mappings in Banach Algebras,Result in Mathematics, 66(2014), 385-404. [16] C.Park, Fixed Point and Hyers-Ulam-Rassias Stability of Cauchy-Jensen Functional Equations in Banach Algebras, Fixed Point Theory and Applications, vol.2007, Article ID 50175, 15pages,2007. [17] C.Park and J.M.Rassias, Stability of the Jensen-Type Functional Equation In C ∗ -Algebras: A fixed point Approach, Abstract and Applied Analysis, vol.2009, Article ID 360432, 17pages, 2009. [18] Th.M. Rassias (Ed.), Functional Equations and Inequalities, Kluwer Academic, Dordrecht, 2000. [19] Th.M. Rassias, On the stability of the linear mapping in Banach spaces, Proc. Amer. Math. Soc. 72 (1978), 297–300. [20] Th.M. Rassias, On the stability of functional equations in Banach spaces, J. Math. Anal. Appl. 251 (2000), 264–284. [21] Th.M. Rassias, On the stability of functional equations and a problem of Ulam, Acta Math. Appl. 62 (2000), 23–130. [22] J.M. Rassias, Onapproximation of approximately linear mappings by linear mappings,J.Funct.Anal.46(1982)126-130. [23] J.M. Rassias, Onapproximation of approximately linear mappings by linear mappings,Bull.Sci.Math.108(1984)445-446. [24] J.M. Rassias, Solution of a problem of Ulam,J.Approx.Theory57(1989)268-273. [25] J.M. Rassias, Complete solution of the multi-dimentional problem of Ulam,Discuss.Mathem.14(1994)101-107. [26] J.M. Rassias, On the Ulam stibility of Jensen and Jensen type mappings on restricted domains,J.Math.Anal.Appl.281(2003)516-524. [27] J.M. Rassias, Refined Hyers-Ulam Approximation of Approximately Jensen Type Mappings, Bull.Sci.Math.131(2007)89-98. [28] S.M. Ulam, Problems in Modern Mathematics, Chapter VI, Science ed., Wiley, New York, 1940. [29] T.Z. Xu, J.M. Rassias and W.X. Xu, A fixed point approach to the stability of a general mixed additive-cubic functional equation in quasi fuzzy normed spaces, Internat. J. Phys. Sci. 6 (2011), 313–324.

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Gang Lu 1.Department of Mathematics, School of Science, ShenYang University of Technology, Shenyang 110870, P.R. China 2.Department of Mathematics, Zhejiang University, Hangzhou 310027,Peoples Republic of China E-mail address: [email protected] Guoxian Cai Department of Mathematics,Ajou University, Suwon 443-749, Korea E-mail address: [email protected] Yuanfeng Jin Department of Mathematics, Yanbian University, Yanji 133001, P.R. China P.R. China E-mail address: [email protected] Choonkil Park Department of Mathematics, Research Institute for Natural Sciences, Hanyang University, Seoul 133-791, Korea E-mail address: [email protected]

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A Higher Order Multi-step Iterative Method for Computing the Numerical Solution of Systems of Nonlinear Equations Associated with Nonlinear PDEs and ODEs Malik Zaka Ullaha,b,∗, S. Serra-Capizzanob , Fayyaz Ahmadc , Arshad Mahmooda , Eman S. Al-Aidarousa a Department of Mathematics, King Abdulaziz University, Jeddah 21589, Saudi Arabia Dipartimento di Scienza e Alta Tecnologia, universita dell’Insubria, Via Valleggio 11, 22100 Como, Italy Dept. de F´ısica i Enginyeria Nuclear, universitat Polit`ecnica de Catalunya, Comte d’Urgell 187, 08036 Barcelona, Spain b

c

Abstract The main focus of research in the current article is to address the construction of an efficient higher order multi-step iterative methods to solve systems of nonlinear equations associated with nonlinear partial differential equations (PDEs) and ordinary differential equations (ODEs). The construction includes second order Frechet derivatives. The proposed multi-step iterative method uses two Jacobian evaluations at different points and requires only one inversion (in the sense of LU-factorization) of Jacobian. The enhancement of convergence-order (CO) is hidden in the formation of matrix polynomial. The cost of matrix vector multiplication is expensive computationally. We developed a matrix polynomial of degree two for base method and degree one to perform multi-steps so we need just one matrix vector multiplication to perform each further step. The base method has convergence order four and each additional step enhance the CO by three. The general formula for CO is 3s − 2 for s ≥ 2 and 2 for s = 1 where s is the step number. The number of function evaluations including Jacobian are s + 2 and number of matrix vectors multiplications are s. For s-step iterative method we solve s upper and lower triangular systems when right hand side is a vector and 1 pair of triangular systems when right hand side is a matrix. It is shown that the computational cost is almost same for Jacobian and second order Frechet derivative associated with systems of nonlinear equations due to PDEs and ODEs. The accuracy and validity of proposed multi-step iterative method is checked with different PDEs and ODEs. Keywords: Multi-step, Iterative methods, Systems of nonlinear equations, Nonlinear partial differential equations, Nonlinear ordinary differential equations

1. Introduction A valuable discussion can be found about Frechet derivatives in [1]. We will show that why higher order Frechet derivatives are avoided in the construction of iterative methods for general systems of nonlinear equations and why there are suitable with for a particular class of systems of nonlinear equations associated with ODEs and PDEs. To make things simpler, consider a system of three nonlinear equations F(y) = [ f1 (y), f2 (y), f3 (y)]T = 0,

(1)

∗

Corresponding author Email addresses: [email protected] (Malik Zaka Ullah ), [email protected] (S. Serra-Capizzano ), [email protected] (Fayyaz Ahmad), [email protected] (Arshad Mahmood), [email protected] (Eman S. Al-Aidarous)

February 4, 2016

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where y = [y1 , y2 , y3 ]T . The first order Frechet derivative (Jacobian) of (1) is ∂f  1  ∂y  1  ∂ f2 F0 (y) =   ∂y1  ∂ f3 ∂y1

∂ f1 ∂y2 ∂ f2 ∂y2 ∂ f3 ∂y2

∂ f1    ∂y3   f11 ∂ f2    =  f ∂y3   21 f31 ∂ f3   ∂y3

 f13   f23   f33

f12 f22 f32

(2)

Next we proceed for the calculation of second-order Frechet derivative. Suppose h = [h1 , h2 , h3 ]T is a constant vector.   h1 f11 + h2 f12 + h3 f13     F0 (y)h = h1 f21 + h2 f22 + h3 f23  ,   h1 f31 + h2 f32 + h3 f33      f111 f122 f133  h21   f121         2  00 2 F (y)h =  f211 f222 f233  h2  + 2  f212    2   f311 f322 f333 h3 f312

(3) f113 f213 f313

  f123  h1 h2     f223  h1 h3  .    f323 h2 h3

(4)

Clearly the computational cost for second-order Frechet derivative is high in the case of general systems of nonlinear equations. Many systems of nonlinear equations associated with PDEs and ODEs can be written as    F(y) = L(y) + f (y) + w = 0,   F(y) = Ay + f (y) + w = 0,

(5)

where A is the discrete approximation to linear differential operator L(·) and f (·) is the nonlinear function. If we write down the second-order Frechet derivative of (5) by using (4) we get  00  f (y1 )   0  00 2 F (y)h =  0  .  ..  0

0 00 f (y2 ) 0 0

0 ··· 0 ··· f 00 (y3 ) · · · 0

···

  2  h1    2   h2    2   h3    .    .    .    00 f (yn ) h2n 0 0 0

(6)

For the further analysis , we introduce some notation. If a = [a1 , a2 , , · · · , an ]T and b = [b1 , b2 , , · · · , bn ]T are vectors then the diagonal matrix of a vector and point-wise product we define as  a1   0 diag(aa) =  .  ..  0

0 a2

0 ··· 0 ···

0

0 ···

 0   0   , a b = diag(aa) b = [a b , a b , · · · , a b ]T . 1 1 2 2 n n    an

(7)

For the motivation of readers we list some famous nonlinear ODEs and PDEs and their first- and second-order derivatives in scalar and vectorial forms (Frechet derivatives). Let D x and Dt are the discrete approximations of differential operators in spatial and temporal dimensions and u is the function of spatial variables and in some cases temporal variable is also taken. We also introduce a function h which is independent from u and It , I x are identity matrices of the size number of nodes in temporal and spatial dimensions respectively.

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1.1. Bratu problem The Bratu problem is discussed in [2] and it is stated as   f (u) = u00 + λeu = 0, u(0) = u(1) = 0,      d f (u)    h = h00 + λeu h,    du     d2 f (u) 2  u 2     du2 h = λe h ,     F(u) = D2x u + λeu = 0 ,       F0 h = D2x h + λeu h,   
   F0 = D2x + λ diag eu ,     F00 h2 = λeu h2 .

(8)

The closed form solution of Bratu problem can be written as !   cosh((x − 0.5)(0.5θ))    u(x) = −2log ,  cosh(0.25θ)   √   θ = 2λcosh 0.25θ
.

(9)

p The critical value of λ satisfies 4 = 4λc sinh(0.25θc ). The Bratu problem has two solution, unique solution and no solution if λ < λc , λ = λc and λ > λc respectively. The critical value λc = 3.51383071912516. 1.2. Frank-Kamenetzkii problem The Frank-Kamenetzkii problem [3] is written as  1    u0 (0) = u(1) = 0, u00 + u0 + λeu = 0,    x    1    F(u) = D2x u + D x u + λeu = 0 ,    x    0 1 2  F h = D h + D x h + λeu h,  x   x !   
1  0 2   F = D x + diag D x + λ diag eu ,    x     F00 h2 = λeu h2 .

(10)

The Frank-Kamenetzkii problem has no solution (λ > 2), (λ = 2) and two solution (λ < 2). The closed form solution of (10) is given as !  p    2(2 − λ) , c = log 2(4 − λ) ± 4  1     √ !     4 − λ) ± 2 2(2 − λ)    ,  c2 = log  2λ2 ! (11)  c   16e 1    , u(x) = log    (2λ + ec1 x2 )2 !      16ec1    . u(x) = log (1 + 2λec2 x2 )2 1.3. Lane-Emden equation The Lane-Emden equation is classical equation [4] which is introduced in 1870 by Lane and later Emden (1907) studied it. Lane-Emden equation deals with mass density distribution inside a spherical star when it is in hydrostatic

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equilibrium. The lane-Emden equation for index n = 5 can be written as  2    u00 + u0 + u5 = 0, u(0) = 1, u0 (0) = 0,    x    1    F(u) = D2x u + D x u + u5 ,    x    0 1 2  F h = D x h + D x h + 5u4 h,    x !   
1  0 2   F = D x + diag D x + 5 diag u4 ,    x     F00 h2 = 20 u3 h2 .

(12)

The closed form solution of (12) can be written as u(x) = 1 +

x2 3

!− 21

.

(13)

1.4. Klien-Gordan equation Klien-Gordan equation is discussed and solved in [5].    utt − c2 u xx + f (u) = p, −∞ < x < ∞, t > 0     2 2 2   F(u) = (Dt − c D x )u + f (u) − p ,     0 F h = (D2t − c2 D2x )h + f 0 (u) h ,       F0 = D2t − c2 D2x + diag( f 0 (u)),      F00 h2 = f 00 (u) h2 , where f (u) is the odd function of u and initial conditions are     u(x, 0) = g1 (x),    ut (x, 0) = g2 (x).

(14)

(15)

We have calculated the second-order Frechet derivatives of four different nonlinear ODEs and PDEs. Clearly the computational cost of second-order Frechet derivatives are not higher than first-order Frechet derivatives or Jacobians. So we insist that the second-order Frechet derivatives for particular class of ODEs and PDEs are not expensive as they are in the case of general systems of nonlinear equations. The main source of information about iterative methods is the manuscript written by J. F. Traub [6] in 1964. Recently many researchers have contributed in the area of iterative method for systems of nonlinear equations [7–16]. The major part of work is devoted for the construction iterative methods for the single variable nonlinear equations[17]. According to Traub’s conjecture if we use n function evaluations, then the maximum CO is 2n in the case of single variable nonlinear equation but for multi-variable case we do not have such claim. In the case of systems of nonlinear equations the multi-steps iterative methods are interesting because with minimum computational cost we are aimed to construct higher-order convergence iterative methods. For the better understanding we can divide multi-steps iterative methods in two parts one is called base method and second part is called multi-steps. In the base method we construct an iterative method in way that it provides maximum enhancement in the convergence-order with minimum computational cost when we perform multi-steps. Malik et. al.

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[18] proposed the following multi-step iterative method (MZ1 ) :   Number of steps       CO        Function evaluations       Inverses     MZ1 =  Matrix vector multiplications      Number of solutions of systems        of linear equations       when right hand side is matrix     when right hand side is vector

 0     φ1 = F(x) F (x)φ        =m≥2   2      y1 = x − φ 1     = 2m   3        1 0   Base-Method →  =m+1  W = 3F (y1 ) − F0 (x)       2       =2    WT = 3F0 (y1 ) + F0 (x)         1 =1      φ1 y2 = x − Tφ    4        for s = 1, m − 2              φ s+1 = F(y s+1 ), Wφ    =1  (m − 2)-steps →       y s+2 = y s+1 − φ s+1 ,     =m−1        end

In [19] F. Soleymani and co-researchers constructed an other multi-step iterative method (FS):   0    φ1 = F(x) F (x)φ        Number of steps =m≥2       2          y1 = x − φ 1    CO = 2m       3          1 0   0 Base-Method →   Function evaluations = m + 1     W = 3F (y ) − F (x)   1        2      Inverses =2          WT = 3F0 (y1 ) + F0 (x)       FS =  Matrix vector multiplications = 2m − 3       φ1 y = x − Tφ      2   Number of solutions of systems         for s = 1, m − 2            of linear equations        φ s+1 = F(y s+1 ), F0 (x)φ        (m − 2)-steps →    when right hand side is matrix =1         y s+2 = y s+1 − T2φ s+1 ,     when right hand side is vector    =m−1      end H. Montazeri et. al. [20] developed the more efficient multi-step iterative methods (HM):   Number of steps       CO        Function evaluations       Inverses     HM =  Matrix vector multiplications      Number of solutions of systems        of linear equations       when right hand side is matrix     when right hand side is vector

  0   φ1 = F(x) F (x)φ       =m≥2    2      y1 = x − φ 1   = 2m      3   Base-Method →    F0 (x)T = F0 (y1 )   =m+1      !       23 9 2   =1     I − 3T + T φ1  y2 = x −  8 8 =m        for s = 1, m − 2              φ s+1 = F(y s+1 ), F0 (x)φ     !     (m − 2)-steps →  3 5      =1 I − T φ s+1 , y = y −   s+2 s+1     2 2      =m−1      end

2. The proposed new multi-step iterative method We proposed a new multi-step iterative method (MZ2 ):

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   Number of steps       CO       Function evaluations        Inverses       Matrix vector     MZ2 =  multiplications      Number of solutions        of systems of linear       equations when       right hand side is matrix      right hand side is vector

  0   φ1 = F(x) F (x)φ    =m≥2    !       4   00 0  = 3m − 2    φ2 = F x − φ 1 φ 21 F (x)φ       9!     =m+2       3     Base-Method →  y1 = x − φ 1 + φ 2  =1     2         0 0     F (x)T = F (y )   1   ! !   =m    7 7 2 3       y = x − I − 6T + T φ + φ  1 2   2   2 2 2         for s = 1, m − 2              φ s+2 = F(y s+1 ), F0 (x)φ       (m − 2)-steps →     =1   y = y s+1 − 2I − T φ s+2 ,   s+2          end  =m

We claim that the convergence-order of our proposed multi-step iterative method is    m = 1, 2 CO =   3m − 2 m ≥ 2,

(16)

where m is the number of steps of MZ2 . The computational costs of MZ1 and FS are high because both methods use two inversions of matrices. The multi-step iterative method HM use only one inversion of Jacobian and hence is a good candidate for the performance comparison. For further discussion we will not consider MZ1 and FS methods. We presented comparison between MZ2 and HM in Table1 and 2. The Table 1 tells us if the number of function evaluations and number of solutions of system of linear equations are equal then the performance of MZ2 in terms of convergenceorder is better than HM when number of step of MZ2 are grater or equal to four. When the convergence-orders of both iterative methods are equal then we can see from Table 2 that the computation effort of HM is always more than that of MZ2 for m ≥ 2. The performance index to measure the efficiency of an iterative method to solve systems of nonlinear equation is defined as 1

ρ = CO f lops .

(17)

In Table 3 we provided the computational cost of different operation and Table 4 shows the performance index as defined in (20) for a particular case when HM and MZ2 have the same convergence-order. Clearly the performance index of MZ2 is better than that of HM.

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Table 1: Comparison between multi-steps iterative method MZ2 and HM if number of function evaluations and solutions of system of linear equations are equal.

Number of steps Convergence-order Function evaluations Solution of system of linear equations when right hand side is vector Solution of system of linear equations when right hand side is matrix Matrix vector multiplications

MZ2 (m ≥ 2)

HM (m ≥ 2)

MZ2 (m = 2)

HM (m = 3)

MZ2 (m = m1 )

HM (m = m1 + 1)

Difference MZ2 − HM

m 3m − 2 m+2

m 2m m+1

2 4 4

3 6 4

m1 3m1 − 2 m1 + 2

m1 + 1 2(m1 + 1) m1 + 2

m

m−1

2

2

m1

m1

0

1

1

1

1

1

1

0

m

m

2

3

m1

m1 + 1

1 m1 − 4 0

−1

Table 2: Comparison between multi-steps iterative method MZ2 and HM if convergence-orders are equal.

Number of steps Convergence-order Function evaluations Solution of system of linear equations when right hand side is vector Solution of system of linear equations when right hand side is matrix Matrix vector multiplications

MZ2 (m ≥ 1)

HM (m ≥ 1)

Difference HM − MZ2

2m 6m − 2 2m + 2

3m − 1 6m − 2 3m

m−1 0 m−2

2m

3m

m

1

1

0

2m

3m − 1

m−1

7

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Table 3: Computational cost of different operations (the computational cost of a division is three times to multiplication).

LU decomposition Multiplications Divisions Total cost n(n − 1)(2n − 1) n(n − 1) n(n − 1)(2n − 1) n(n − 1) +3 6 2 6 2 Two triangular systems (if right hand side is a vector) Multiplications Divisions Total cost n(n − 1) n n(n − 1) + 3n Two triangular systems (if right hand side is a matrix) Multiplications Divisions Total cost n2 (n − 1) n2 n2 (n − 1) + 3n2 Matrix vector multiplication n2

Table 4: Comparison of performance index between multi-steps iterative methods MZ2 and HM.

Iterative methods Number of steps Rate of convergence Number of functional evaluations The classical efficiency index Number of Lu factorizations Cost of Lu factorizations Cost of linear systems Matrix vector multiplications Flops-like efficiency index

HM 5 10

MZ2 4 10

6n

6n

21/(6n)

21/(6n)

1

1

n(n − 1)(2n − 1) n(n − 1) +3 6 2 4(n(n − 1) + 3n) + n2 (n − 1) + 3n2 5n2

n(n − 1)(2n − 1) n(n − 1) +3 6 2 4(n(n − 1) + 3n) + n2 (n − 1) + 3n2 4n2

101/

4n3 3




+12n2 + 38 3 n

101/

4n3 3




+11n2 + 38 3 n

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3. Convergence Analysis In this section, we will prove that the local convergence-order of MZ2 is seven for m = 3 and later we will establish a proof for the convergence-order of multi-step iterative scheme MZ2 , by using mathematical induction. Theorem 3.1. Let F : Γ ⊆ Rn → Rn be sufficiently Frechet differentiable on an open convex neighborhood Γ of x∗ ∈ Rn with F(x∗ ) = 0 and det(F0 (x∗ )) , 0. Then the sequence {xk } generated by the iterative scheme MZ2 converges to x∗ with local order of convergence seven, and produces the following error equation ek+1 = Lek 7 + O(ek 8 ),

(18) p-times

z }| { where ek = xk − x∗ , ek p = (ek , ek , · · · , ek ) and L = −2060C62 − 618C3 C42 + 260/9C32 C4 + 26/3C3 C2 C4 − 30C3 C2 C3 C2 − p-times

6C3 C22 C3

−

100C32 C3 C2

−

20C42 C3

z }| { is a p-linear function i.e. L ∈ L (Rn , Rn , · · · , Rn ) and Lek p ∈ Rn .

Proof. Let F : Γ ⊆ R → Rn be sufficiently Frechet differentiable function in Γ. The qth Frechet derivative of F at q-times z }| { n (q) n n v ∈ R , q ≥ 1, is the q − linear function F (v) : R R · · · Rn such that F(q) (v)(u1 , u2 , · · · , uq ) ∈ Rn . The Taylor’s series expansion of F(xk ) around x∗ can be written as: n

F(xk ) = F(x∗ + xk − x∗ ) = F(x∗ + ek ),   1 1 = F(x∗ ) + F0 (x∗ )ek + F00 (x∗ )ek 2 + F (3) (x∗ )ek 3 + O e4k , 2! 3!    1 1 = F0 (x∗ ) ek + F0 (x∗ )−1 F00 (x∗ )ek 2 + F0 (x∗ )−1 F (3) (x∗ )ek 3 + O e4k , 2! 3!    = C1 ek + C2 ek 2 + C3 ek 3 + O e4k ,

(19) (20) (21) (22)

1 0 ∗ −1 (s) ∗ F (x ) F (x ) for s ≥ 2. From (22), we can calculate the Frechet derivative of F: s!    F0 (xk ) = C1 I + 2C2 ek + 3C3 ek 2 + 4C3 ek 3 + O e4k , (23)

where C1 = F0 (x∗ ) and C s =

where I is the identity matrix. Furthermore, we calculate the inverse of the Jacobian matrix       F0 (xk )−1 = I−2C2 ek + 4C22 −3C3 e2k + 6C3 C2 +6C2 C3 −8C32 −4C4 e3k + 8C4 C2 +9C23 +8C2 C4 −5C5 −   12C3 C22 − 12C2 C3 C2 − 12C22 C3 + 16C42 e4k + 24C3 C32 + 24C32 C3 + 24C22 C3 C2 + 24C2 C3 C22 +

(24)

10C5 C2 + 12C4 C3 + 12C3 C4 + 10C2 C5 − 6C6 − 16C4 C22 − 18C23 C2 − 18C3 C2 C3 − 16C2 C4 C2 −   18C2 C23 − 16C22 C4 − 32C52 e5k + 32C4 C32 + 64C62 − 48C3 C42 + 12C2 C6 + 16C24 + 15C3 C5 + 15C5 C3 + 12C6 C2 − 24C4 C2 C3 − 24C4 C3 C2 − 20C22 C5 − 24C2 C3 C4 − 24C2 C4 C3 + 32C32 C4 − 20C2 C5 C2 + 36C22 C23 − 20C5 C22 + 32C22 C4 C2 + 32C2 C4 C22 + 36C2 C23 C2 + 36C2 C3 C2 C3 + 36C23 C22 − 7C7 − 24C3 C2 C4 − 27C33 − 24C3 C4 C2 + 36C3 C2 C3 C2 + 36C3 C22 C3 − 48C22 C3 C22 −    48C32 C3 C2 − 48C42 C3 − 48C2 C3 C32 e6k + O e7k C−1 1 By multiplying F0 (xk )−1 and F(xk ), we obtain φ 1 :      φ 1 = ek −C2 e2k + 2C22 −2C3 e3k + −3C4 −4C32 +3C3 C2 +4C2 C3 e4k + −4C5 −6C3 C22 −6C2 C3 C2 −   8C22 C3 + 8C42 + 4C4 C2 + 6C23 + 6C2 C4 e5k + − 5C6 + 12C3 C32 + 16C32 C3 + 12C22 C3 C2 +

(25)

12C2 C3 C22 − 8C4 C22 − 9C23 C2 − 12C3 C2 C3 − 8C2 C4 C2 − 12C2 C23 − 12C22 C4 − 16C52 + 5C5 C2 +    8C4 C3 + 9C3 C4 + 8C2 C5 e6k + O e7k . 9

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The expression for φ 2 is the following:     φ 2 = 2C2 e2k + − 8C22 + 10/3C3 e3k + 26C32 − 38/3C3 C2 − 12C2 C3 + 100/27C4 e4k +  − 364/27C2 C4 − 18C23 − 416/27C4 C2 + 116/3C22 C3 + 36C2 C3 C2 + 122/3C3 C22 +   2500/729C5 −76C42 e5k + −106C2 C3 C22 −298/3C22 C3 C2 −344/3C32 C3 +1282/27C22 C4 +

(26)

140/3C2 C23 + 1106/27C2 C4 C2 − 118C3 C32 + 1364/27C4 C22 − 10664/729C2 C5 − 520/27C3 C4 −544/27C4 C3 −12290/729C5 C2 +54C23 C2 +184/3C3 C2 C3 +6250/2187C6 +    208C52 e6k + O e7k . The expressions for y1 , T , y2 and y3 in order are     y1 − x∗ = −2C2 e2k + 10C22 − 3C3 e3k + − 23/9C4 − 35C32 + 16C3 C2 + 14C2 C3 e4k + (−278/243C5 −   55C3 C22 −48C2 C3 C2 −50C22 C3 +106C42 +172/9C4 C2 +21/9C23 +128C2 C4 e5k + 147C2 C3 C22 +

(27)

137C22 C3 C2 +156C32 C3 −533/9C22 C4 −58C2 C23 −481/9C2 C4 C2 +165C3 C32 −610/9C4 C22 + 3388/243C2 C5 + 179/9C3 C4 + 200/9C4 C3 + 4930/243C5 C2 − 72C23 C2 − 80C3 C2 C3 +    520/729C6 − 296C52 e6k + O e7k .    T = I − 2C2 ek − 3C3 e2k + 6C3 C2 − 4C4 + 20C32 e3k + 12C3 C22 + 20C2 C3 C2 + 28C22 C3 − 110C42 +   8C4 C2 + 9C23 + 26/9C2 C4 − 5C5 e4k + − 180C3 C32 − 156C32 C3 − 136C22 C3 C2 − 134C2 C3 C22 +

(28)

18C3 C2 C3 + 200/9C2 C4 C2 ) + 24C2 C23 + 68/3C22 C4 + 432C52 + 10C5 C2 + 12C4 C3 + 12C3 C4 +   1874/243C2 C5 − 6C6 e5k + − 112C4 C32 − 1456C62 + 1050C3 C42 + 9788/729C2 C6 + 16C24 + 15C3 C5 +15C5 C3 +12C6 C2 −24C4 C3 C2 +3028/243C22 C5 +142/9C2 C3 C4 +184/9C2 C4 C3 − 1474/9C32 C4 +5000/243C2 C5 C2 −164C22 C23 −454/3C22 C4 C2 −1220/9C2 C4 C22 −144C2 C23 C2 − 196C2 C3 C2 C3 −222C23 C22 −7C7 +20/3C3 C2 C4 −26/3C3 C4 C2 −240C3 C2 C3 C2 −258C3 C22 C3 +    562C22 C3 C22 + 546C32 C3 C2 + 624C42 C3 + 690C2 C3 C32 e6k + O e7k .    y2 − x∗ = − 5C3 C2 + 13/9C4 − 103C32 − C2 C3 e4k + − 104/9C2 C4 − 21/2C23 − 80/9C4 C2 −   148C22 C3 −100C2 C3 C2 −109C3 C22 +937/243C5 +666C42 e5k + 869C2 C3 C22 +873C22 C3 C2 +

(29)

954C32 C3 − 1133/9C22 C4 − 124C2 (C23 ) − 895/9C2 C4 C2 + 1074C3 C32 − 1114/9C4 C22 − 715/27C2 C5 − 238/9C3 C4 − 178/9C4 C3 − 3575/243C5 C2 − 75C23 C2 − 158C3 C2 C3 +   4894/729C6 − 1990C52 e6k + 3632/3C4 C32 + 420C62 − 4958C3 C42 − 30616/729C2 C6 − 404/9C24 − 7343/162C3 C5 − 16001/486C5 C3 − 15620/729C6 C2 − 1580/9C4 C2 C3 − 580/9C4 C3 C2 − 18334/243C22 C5 − 761/9C2 C3 C4 − 847/9C2 C4 C3 + 1074C32 C4 − 19556/243C2 C5 C2 +1118C22 C23 −35410/243C5 C22 )+8924/9C22 C4 C2 +3038/3C2 C4 C22 + 1040C2 C23 C2 + 1262C2 C3 C2 C3 + 1390C23 C22 + 63418/6561C7 − 919/9C3 C2 C4 − 165/2C33 −589/9C3 C4 C2 +1331C3 C2 C3 C2 +1542C3 C22 C3 −2678C22 C3 C22 −2886C32 C3 C2 −    2881C42 C3 − 3871C2 C3 C32 e7k + O e8k .  y3 − x∗ = −2060C62 −618C3 C42 +260/9C32 C4 +26/3C3 C2 C4 −30C3 C2 C3 C2 −6C3 C22 C3 −100C32 C3 C2 −    20C42 C3 e7k + O e8k .

(30)

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Theorem 3.2. The multi-step iterative scheme MZ2 has the local convergence-order 3m − 2, using m(≥ 2) evaluations of a sufficiently differentiable function F, two first-order Frechet derivatives F0 and one second-order Frechet derivate F00 per full-cycle. Proof. The proof is established from mathematical induction. For m = 1, 2, 3 the convergence-orders are two, four and seven from (27), (29) and (30) respectively. Consequently our claim concerning the convergence-order 3m − 2 is true for m = 2, 3. We assume that our claim is true for m = q > 3, i.e., the convergence-order of MZ2 is 3q − 2. The qth-step and (q − 1)th-step of iterative scheme MZ2 can be written as: Frozen-factor = (2I − T)F0 (x)−1 , yq−1 = yq−2 − (Frozen-factor) F(yq−2 ), yq = yq−1 − (Frozen-factor) F(yq−1 ).

(31) (32) (33)

The enhancement in the convergence-order of MZ2 from (q − 1)th-step to qth-step is (3q − 2) − (3(q − 1) − 2) = 3 . Now we write the (q + 1)th-step of MZ2 : yq+1 = yq − (Frozen-factor) F(yq ).

(34)

The increment in the convergence-order of MZ2 , due to (q + 1)th-step, is exactly three, because the use of the Frozenfactor adds an additive constant in the convergence-order[19]. Finally the convergence-order after the addition of the (q + 1)th-step is 3q − 2 + 3 = 3q + 1 = 3(q + 1) − 2, which completes the proof. 4. Numerical Testing For the verification of convergence-order, we use the following definition for the computational convergence-order (COC):   log ||xq+2 − x∗ ||∞ /||xq+1 − x∗ ||∞ COC ≈ (35)   , log ||xq+1 − x∗ ||∞ /||xq − x∗ ||∞ where Max(|xq+2 − x∗ |) is the maximum absolute error. The number of solutions of systems of linear equations are same in both iterative methods when right hand side is a matrix so we will not mention it in comparison tables. The main benefit of multi-step iterative methods is that we invert Jacobian once and then use it again and again in multi-steps part to get better convergence-order for a single cycle of iterative method. We have conducted numerical tests for four different problems to show the accuracy and validity of our proposed multi-step iterative method MZ2 . For the purpose of comparison we adopt two ways (i) when both iterative methods have same number of function evaluations and solution of systems of linear equations (ii) when both schemes have same convergence order. Tables 5, 7 and 8 show that when we number of function evaluations and solutions of systems of linear equation are equal and the convergence order of MZ2 is higher than ten then our proposed scheme show better accuracy in less execution time. On the other hand if convergence-order of MZ2 is less than ten then the performance of HM is relatively better. For the second cases when we equate the convergence-orders the execution time of MZ2 are always less than that of HM because HM performs more steps to achieve the same convergence-order. Tables 6, 9 and 10 shows that MZ2 achieve better or almost equal accuracy with less execution time. We have also simulated one PDE Klein-Gordon and results are depicted in Table 11. As we have commented if the convergence-order is less ten the performance of HM is better and it is clearly evident in Table 11 but the accuracy of MZ2 is comparable with HM. The numerical error in solution due to MZ2 is shown in Figure 1 and Figure 2 corresponds to numerical solution of Klein-Gordon PDE. In the case of Klein-Gordon equation by keeping the mesh size fix, if we increase the number of iterations or either number of steps both iterative method can not improve the accuracy. 11

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Table 5: Comparison of performances for different multi-step methods in the case of the Bratu problem when number of function evaluations and number of solutions of systems of linear equations are equal in both iterative methods.

Iterative methods Number of iterations Size of problem Number of steps Theoretical convergence-order(CO) Number of function evaluations per iteration Solutions of system of linear equations per iteration Number of matrix vector multiplication per iteration λ 1 2 3

||xq − x∗ ||∞

Execution time

MZ2 1 200 32 94 34 32 32

HM 1 200 33 66 34 32 33

3.62e − 156 4.78e − 142 3.91e − 50 23.48

7.55e − 110 2.31e − 98 4.05e − 35 24.0

Table 6: Comparison of performances for different multi-step methods in the case of the Bratu problem when convergence orders are equal in both itrative methods.

Iterative methods Number of iterations Size of problem Number of steps Theoretical convergence-order(CO) Number of function evaluations per iteration Solutions of system of linear equations per iteration Number of matrix vector multiplication per iteration ||xq − x∗ ||∞ , (λ = 1) Execution time

MZ2 1 250 120 358 122 120 120 3.98e − 235 59.67

HM 1 250 179 358 180 178 179 3.98e − 235 70.22

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Table 7: Comparison of performances for different multi-step methods in the case of the Bratu problem when number of function evaluations and number of solutions of systems of linear equations are equal in both iterative methods.

Iterative methods Number of iterations Size of problem Number of steps Theoretical convergence-order(CO) Computational convergence-order(COC) Number of function evaluations per iteration Solutions of system of linear equations per iteration Number of matrix vector multiplication per iteration ||xq − x∗ ||∞ Execution time

MZ2 3 250 3 7 6.75 5 3 3 8.44e − 150 63.75

HM 3 250 4 8 7.81 5 3 4 3.92e − 161 64.66

Table 8: Comparison of performances for different multi-step methods in the case of the Frank Kamenetzkii problem when number of function evaluations and number of solutions of systems of linear equations are equal in both iterative methods.

Iterative methods Number of iterations Size of problem Number of steps Theoretical convergence-order(CO) Computational convergence-order(COC) Number of function evaluations per iteration Solutions of system of linear equations per iteration Number of matrix vector multiplication per iteration ||xq − x∗ ||∞ Execution time

MZ2 3 150 3 7 7.39 5 3 3 4.21e − 126 16.10

HM 3 150 4 8 8.64 5 3 4 3.21e − 149 16.68

Table 9: Comparison of performances for different multi-step methods in the case of the Frank Kamenetzkii problem when convergence orders are equal in both iterative methods.

Iterative methods Number of iterations Size of problem Number of steps Theoretical convergence-order(CO) Number of function evaluations per iteration Solutions of system of linear equations per iteration Number of matrix vector multiplication per iteration ||xk − x∗ ||∞ , (λ = 1) Execution time

MZ2 1 150 80 238 82 80 80 6.46e − 116 19.89

HM 1 150 119 238 120 118 119 3.95e − 99 28.21

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Table 10: Comparison of performances for different multi-step methods in the case of the Lane-Emden equation when convergence orders are equal.

Iterative methods Number of iterations Size of problem Number of steps Theoretical convergence-order(CO) Number of function evaluations per iteration Solutions of system of linear equations per iteration Number of matrix vector multiplication per iteration ||xq − x∗ ||∞ Execution time

MZ2 1 100 30 88 32 30 30 1.95e − 34 3.01

HM 1 100 44 88 45 43 44 2.64e − 37 3.53

Table 11: Comparison of performances for different multi-step methods in the case of the Klien Gordon equation , initial guess u(xi , t j ) = 0, s r k 2k , c = 1, γ = 1, ν = 0.5, k = 0.5, n x = 170, nt = 26, x ∈ [−22, 22], t ∈ [0, 0.5]. u(x, t) = δsech(κ(x − νt), κ = , δ = γ c2 − ν 2

Iterative methods Number of iterations Size of problem Number of steps Theoretical convergence-order(CO) Number of function evaluations per iteration Solutions of system of linear equations per iteration Number of matrix vector multiplication per iteration Steps 1 2 3 4

||xq − x∗ ||∞

Execution time

MZ2 1 4420 4 10 6 4 4

HM 1 4420 4 8 5 3 4

3.24e − 1 7.51e − 3 2.70e − 5 5.59e − 7 94.13

4.11e − 1 2.62e − 3 2.63e − 5 4.39e − 7 80.18

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−7

x 10 6 4

Error

2 0 −2 −4 40 20

0.5 0.4

0

0.3 0.2

−20

0.1 −40

x−axis

0

t−axis

Figure 1: Absolute error plot for multi-step method MZ2 in the case of the Klien Gordon equation , initial guess u(xi , t j ) = 0, u(x, t) = δsech(κ(x − νt), s r k 2k , c = 1, γ = 1, ν = 0.5, k = 0.5, n x = 170, nt = 26, x ∈ [−22, 22], t ∈ [0, 0.5]. κ= ,δ= γ c2 − ν 2

1.2 1

0.6 0.4

u

numerical

0.8

0.2 0 −0.2 40

0.4 20

0.3 0

0.2 −20

0.1 −40

0 t−axis

x−axis

Figure 2: Numerical solution of the Klien Gordon equation , x ∈ [−22, 22], t ∈ [0, 0.5].

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5. Conclusions The inversion of Jacobian is computationally expensive and multi-step iterative methods can provide remedy to it by offering good convergence-order with relatively less computational cost. The best way to construct a multi-step method is to reduce the number of Jacobian and function evaluations, inversion of Jacobian, matrix-vector and vector-vector multiplications. Higher-order Frechet derivatives are computationally expensive when use them for the solution of systems of nonlinear equations but for a particular of ODEs and PDEs we could use them because they are just diagonal matrices. Our proposed scheme MZ2 shows good accuracy when we perform more and more multi-steps and it also depends on the nature of problem sometime. The computational convergence-order of MZ2 is also calculated in some examples and it agrees with theoretical proved convergence-order. Acknowledgement This project was funded by the Deanship of Scientific Research (DSR) at King Abdulaziz University under grant no. HiCi-20-130-1433. The authors, therefore, acknowledge with thanks DSR for technical and financial support.

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References [1] Alicia Cordero Jose L. Hueso Eulalia Martnez, Juan R. Torregrosa, A modified Newton-Jarratts composition, Numer Algor (2010) 55:87-99 DOI 10.1007/s11075-009-9359-z [2] Bratu, G. ”Sur les equations integrales non-lineaires” Bulletins of the Mathematical Society of France. Volume 42. 1914. pp. 113-142. [3] Frank-Kamenetzkii, D.A. Diffusion and Heat Transfer in Chemical Kinetics. Plenum Press, New York (1969). [4] S. S. Motsa and S. Shateyi, New Analytic Solution to the Lane-Emden Equation of Index 2, Mathematical Problems in Engineering Volume 2012 (2012), Article ID 614796, 19 pages. [5] T. S. Jang, An integral equation formalism for solving the nonlinear Klein-Gordon equation, Applied Mathematics and Computation, 01/2014, 243, 322-338. [6] J.F. Traub, Iterative Methods for the Solution of Equations, Prentice-Hall, Englewood Cliffs, 1964. [7] J.M. Gutirrez, M.A. Hernndez, A family of Chebyshev-Halley type methods in Banach spaces, Bull. Aust. Math. Soc. 55 (1997) 113-130. [8] M. Frontini, E. Sormani, Some variant of Newtons method with third-order convergence, Appl. Math. Comput. 140 (2003) 419-426. [9] H.H.H. Homeier, A modified Newton method with cubic convergence: the multivariable case, J. Comput. Appl. Math. 169 (2004) 161-169. [10] A. Cordero, J.R. Torregrosa, Variants of Newtons method using fifth-order quadrature formulas, Appl. Math. Comput. 190 (2007) 686-698. [11] M. Grau-Snchez, . Grau, M. Noguera, Ostrowski type methods for solving systems of nonlinear equations, Appl. Math. Comput. 218 (2011) 2377-2385. [12] J.R. Sharma, R.K. Guha, R. Sharma, An efficient fourth order weighted-Newton method for systems of nonlinear equations, Numer. Algorithms 62 (2013) 307-323. [13] A.M. Ostrowski, Solution of Equations and Systems of Equations, Academic Press, New york, 1966. [14] P. Jarratt, Some fourth order multipoint iterative methods for solving equations, Math. Comp. 20 (1966) 434-437. [15] C.T. Kelley, Solving Nonlinear Equations with Newtons Method, SIAM, Philadelphia, 2003. [16] Eman S. Alaidarous, Malik zaka ullah, Fayyaz Ahmad, and A.S. Al-Fhaid, An Efficient Higher-Order Quasilinearization Method for Solving Nonlinear BVPs,Journal of Applied Mathematics Volume 2013 (2013), Article ID 259371, 11 pages. [17] M.S. Petkovic, On a general class of multipoint root-finding methods of high computational efficiency, SIAM J. Numer. Anal. 49 (2011) 1317-1319. [18] Malik zaka ullah, Fazlollah Soleymani, A. S. Al-Fhaid, Numerical solution of nonlinear systems by a general class of iterative methods with application to nonlinear PDEs, Numerical Algorithms September 2014, Volume 67, Issue 1, pp 223-242. [19] Fazlollah Soleymani, Taher Lotfi, Parisa Bakhtiari, A multi-step class of iterative methods for nonlinear systems, Optimization Letters, March 2014, Volume 8, Issue 3, pp 1001-1015. [20] H. Montazeri, F. Soleymani, S. Shateyi, S. S. Motsa , On a New Method for Computing the Numerical Solution of Systems of Nonlinear Equations,, Volume 2012, Article ID 751975, 15 pages.

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QUADRATIC ρ-FUNCTIONAL INEQUALITIES IN FUZZY NORMED SPACES JI-HYE KIM AND CHOONKIL PARK∗ Abstract. In this paper, we solve the following quadratic ρ-functional inequalities N (f (x + y) + f (x − y) − 2f (x) − 2f (y), t)         x−y x+y + 2f − f (x) − f (y) , t ≥ N ρ 2f 2 2 where ρ is a fixed real number with |ρ| < 1, and       x+y x−y N 2f + 2f − f (x) − f (y), t 2 2 ≥ N (ρ (f (x + y) + f (x − y) − 2f (x) − 2f (y)) , t)

(0.1)

(0.2)

where ρ is a fixed real number with |ρ| < 12 . Using the fixed point method, we prove the Hyers-Ulam stability of the quadratic ρfunctional inequalities (0.1) and (0.2) in fuzzy Banach spaces.

1. Introduction and preliminaries Katsaras [21] defined a fuzzy norm on a vector space to construct a fuzzy vector topological structure on the space. Some mathematicians have defined fuzzy norms on a vector space from various points of view [13, 24, 52]. In particular, Bag and Samanta [2], following Cheng and Mordeson [8], gave an idea of fuzzy norm in such a manner that the corresponding fuzzy metric is of Kramosil and Michalek type [23]. They established a decomposition theorem of a fuzzy norm into a family of crisp norms and investigated some properties of fuzzy normed spaces [3]. We use the definition of fuzzy normed spaces given in [2, 28, 29] to investigate the Hyers-Ulam stability of quadratic ρ-functional inequalities in fuzzy Banach spaces. Definition 1.1. [2, 28, 29, 30] Let X be a real vector space. A function N : X × R → [0, 1] is called a fuzzy norm on X if for all x, y ∈ X and all s, t ∈ R, (N1 ) N (x, t) = 0 for t ≤ 0; (N2 ) x = 0 if and only if N (x, t) = 1 for all t > 0; t ) if c 6= 0; (N3 ) N (cx, t) = N (x, |c| (N4 ) N (x + y, s + t) ≥ min{N (x, s), N (y, t)}; (N5 ) N (x, ·) is a non-decreasing function of R and limt→∞ N (x, t) = 1. (N6 ) for x 6= 0, N (x, ·) is continuous on R. The pair (X, N ) is called a fuzzy normed vector space. The properties of fuzzy normed vector spaces and examples of fuzzy norms are given in [27, 28]. 2010 Mathematics Subject Classification. Primary 46S40, 39B52, 47H10, 39B62, 26E50, 47S40. Key words and phrases. fuzzy Banach space; quadratic ρ-functional inequality; fixed point method; HyersUlam stability. ∗ Corresponding author: Choonkil Park (email: [email protected]).

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Definition 1.2. [2, 28, 29, 30] Let (X, N ) be a fuzzy normed vector space. A sequence {xn } in X is said to be convergent or converge if there exists an x ∈ X such that limn→∞ N (xn −x, t) = 1 for all t > 0. In this case, x is called the limit of the sequence {xn } and we denote it by N limn→∞ xn = x. Definition 1.3. [2, 28, 29, 30] Let (X, N ) be a fuzzy normed vector space. A sequence {xn } in X is called Cauchy if for each ε > 0 and each t > 0 there exists an n0 ∈ N such that for all n ≥ n0 and all p > 0, we have N (xn+p − xn , t) > 1 − ε. It is well-known that every convergent sequence in a fuzzy normed vector space is Cauchy. If each Cauchy sequence is convergent, then the fuzzy norm is said to be complete and the fuzzy normed vector space is called a fuzzy Banach space. We say that a mapping f : X → Y between fuzzy normed vector spaces X and Y is continuous at a point x0 ∈ X if for each sequence {xn } converging to x0 in X, then the sequence {f (xn )} converges to f (x0 ). If f : X → Y is continuous at each x ∈ X, then f : X → Y is said to be continuous on X (see [3]). The stability problem of functional equations originated from a question of Ulam [51] concerning the stability of group homomorphisms. The functional equation f (x + y) = f (x) + f (y) is called the Cauchy equation. In particular, every solution of the Cauchy equation is said to be an additive mapping. Hyers [17] gave a first affirmative partial answer to the question of Ulam for Banach spaces. Hyers’ Theorem was generalized by Aoki [1] for additive mappings and by Th.M. Rassias [40] for linear mappings by considering an unbounded Cauchy difference. A generalization of the Th.M. Rassias theorem was obtained by G˘avruta [14] by replacing the unbounded Cauchy difference by a general control function in the spirit of Th.M. Rassias’ approach. The functional equation f (x+y)+f (x−y) = 2f (x)+2f (y) is called the quadratic functional equation. In particular, every solution of the quadratic functional equation is said to be a quadratic mapping. The stability of quadratic functional equation was proved by Skof [50] for mappings f : E1 → E2 , where E1 is a normed space and E2 is a Banach space. Cholewa [9] noticed that the theorem of Skof is still true if the relevant domain E1 is replaced by an Abelian group. Czerwik [10] proved stability of the quadratic functional  Hyers-Ulam    the x−y 1 1 + f = equation. The functional equation f x+y 2 2 2 f (x) + 2 f (y) is called a Jensen type quadratic equation. The stability problems of several functional equations have been extensively investigated by a number of authors and there are many interesting results concerning this problem (see [4, 18, 20, 25, 36, 37, 38, 41, 42, 44, 45, 46, 47, 48, 49]). Gil´anyi [15] showed that if f satisfies the functional inequality k2f (x) + 2f (y) − f (x − y)k ≤ kf (x + y)k

(1.1)

then f satisfies the Jordan-von Neumann functional equation 2f (x) + 2f (y) = f (x + y) + f (x − y). See also [43]. Fechner [12] and Gil´anyi [16] proved the Hyers-Ulam stability of the functional inequality (1.1). Park, Cho and Han [35] investigated the Cauchy additive functional inequality kf (x) + f (y) + f (z)k ≤ kf (x + y + z)k

(1.2)

and the Cauchy-Jensen additive functional inequality

 

x+y

kf (x) + f (y) + 2f (z)k ≤ 2f +z

2

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(1.3)

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and proved the Hyers-Ulam stability of the functional inequalities (1.2) and (1.3) in Banach spaces. Park [33, 34] defined additive ρ-functional inequalities and proved the Hyers-Ulam stability of the additive ρ-functional inequalities in Banach spaces and non-Archimedean Banach spaces. We recall a fundamental result in fixed point theory. Let X be a set. A function d : X × X → [0, ∞] is called a generalized metric on X if d satisfies (1) d(x, y) = 0 if and only if x = y; (2) d(x, y) = d(y, x) for all x, y ∈ X; (3) d(x, z) ≤ d(x, y) + d(y, z) for all x, y, z ∈ X. Theorem 1.4. [5, 11] Let (X, d) be a complete generalized metric space and let J : X → X be a strictly contractive mapping with Lipschitz constant L < 1. Then for each given element x ∈ X, either d(J n x, J n+1 x) = ∞ for all nonnegative integers n or there exists a positive integer n0 such that (1) d(J n x, J n+1 x) < ∞, ∀n ≥ n0 ; n (2) the sequence {J x} converges to a fixed point y ∗ of J; (3) y ∗ is the unique fixed point of J in the set Y = {y ∈ X | d(J n0 x, y) < ∞}; 1 d(y, Jy) for all y ∈ Y . (4) d(y, y ∗ ) ≤ 1−L In 1996, G. Isac and Th.M. Rassias [19] were the first to provide applications of stability theory of functional equations for the proof of new fixed point theorems with applications. By using fixed point methods, the stability problems of several functional equations have been extensively investigated by a number of authors (see [6, 7, 22, 27, 31, 32, 38, 39]). In Section 2, we solve the quadratic ρ-functional inequality (0.1) and prove the Hyers-Ulam stability of the quadratic ρ-functional inequality (0.1) in fuzzy Banach spaces by using the fixed point method. In Section 3, we solve the quadratic ρ-functional inequality (0.2) and prove the Hyers-Ulam stability of the quadratic ρ-functional inequality (0.2) in fuzzy Banach spaces by using the fixed point method. Throughout this paper, assume that X is a real vector space and (Y, N ) is a fuzzy Banach space. 2. Quadratic ρ-functional inequality (0.1) In this section, we prove the Hyers-Ulam stability of the quadratic ρ-functional inequality (0.1) in fuzzy Banach spaces. Let ρ be a real number with |ρ| < 1. We need the following lemma to prove the main results. Lemma 2.1. Let f : X → Y be a mapping such that N (f (x + y) + f (x − y) − 2f (x) − 2f (y), t)         x−y x+y + 2f − f (x) − f (y) , t ≥ N ρ 2f 2 2 for all x, y ∈ X and all t > 0. Then f is quadratic.

(2.1)

Proof. Assume that f : X → Y satisfies (2.1).   t Letting x = y = 0 in (2.1), we get N (2f (0), t) ≥ N (ρ(2f (0)), t) = N 2f (0), |ρ| for all t > 0. By (N5 ) and (N6 ), N (f (0), t) = 1 for all t > 0. It follows from (N2 ) that f (0) = 0.

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Letting y = x in (2.1), we get N (f (2x) − 4f (x), t) ≥ N (0, t) = 1 and so f (2x) = 4f (x) for all x ∈ X. Thus   x 1 f = f (x) (2.2) 2 4 for all x ∈ X. It follows from (2.1) and (2.2) that N (f (x + y) + f (x − y) − 2f (x) − 2f (y), t)         x−y x+y + 2f − f (x) − f (y) , t ≥ N ρ 2f 2 2   1 =N ρ (f (x + y) + f (x − y) − 2f (x) − 2f (y)) , t 2   2t = N f (x + y) + f (x − y) − 2f (x) − 2f (y), |ρ| for all t > 0. By (N5 ) and (N6 ), N (f (x + y) + f (x − y) − 2f (x) − 2f (y), t) = 1 for all t > 0. It follows from (N2 ) that f (x + y) + f (x − y) = 2f (x) + 2f (y) for all x, y ∈ X.  Theorem 2.2. Let ϕ : X 2 → [0, ∞) be a function such that there exists an L < 1 with L ϕ (2x, 2y) 4 for all x, y ∈ X. Let f : X → Y be an even mapping satisfying f (0) = 0 and ϕ(x, y) ≤

N (f (x + y) + f (x − y) − 2f (x) − 2f (y), t)           x−y t x+y + 2f − f (x) − f (y) , t , ≥ min N ρ 2f 2 2 t + ϕ(x, y) for all x, y ∈ X and all t > 0. Then Q(x) := N -limn→∞ 4n f defines a quadratic mapping Q : X → Y such that N (f (x) − Q(x), t) ≥

x 2n




(2.3)

exists for each x ∈ X and

(4 − 4L)t (4 − 4L)t + Lϕ(x, x)

(2.4)

for all x ∈ X and all t > 0. Proof. Letting y = x in (2.3), we get N (f (2x) − 4f (x), t) ≥

t t + ϕ(x, x)

(2.5)

for all x ∈ X. Consider the set S := {g : X → Y } and introduce the generalized metric on S: 

t , ∀x ∈ X, ∀t > 0 , t + ϕ(x, x) 

d(g, h) = inf µ ∈ R+ : N (g(x) − h(x), µt) ≥

where, as usual, inf φ = +∞. It is easy to show that (S, d) is complete (see [26, Lemma 2.1]). Now we consider the linear mapping J : S → S such that   x Jg(x) := 4g 2 for all x ∈ X.

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Let g, h ∈ S be given such that d(g, h) = ε. Then N (g(x) − h(x), εt) ≥

t t + ϕ(x, x)

for all x ∈ X and all t > 0. Hence x x x L − 4h , Lεt = N g −h , εt 2 2 2 4 Lt Lt t 4 4 =
≥ Lt x x L t + ϕ(x, x) + ϕ 2, 2 4 + 4 ϕ(x, x)

x N (Jg(x) − Jh(x), Lεt) = N 4g 2

 



≥

Lt 4

 



  

 



for all x ∈ X and all t > 0. So d(g, h) = ε implies that d(Jg, Jh) ≤ Lε. This means that d(Jg, Jh) ≤ Ld(g, h) for all g, h ∈ S.  It follows from (2.5) that N f (x) − 4f

x 2






, L4 t ≥

t t+ϕ(x,x)

for all x ∈ X and all t > 0. So

d(f, Jf ) ≤ L4 . By Theorem 1.4, there exists a mapping Q : X → Y satisfying the following: (1) Q is a fixed point of J, i.e., x 2

 

Q

1 = Q(x) 4

(2.6)

for all x ∈ X. Since f : X → Y is even, Q : X → Y is a even mapping. The mapping Q is a unique fixed point of J in the set M = {g ∈ S : d(f, g) < ∞}. This implies that Q is a unique mapping satisfying (2.6) such that there exists a µ ∈ (0, ∞) satisfying t N (f (x) − Q(x), µt) ≥ t + ϕ(x, x) for all x ∈ X; (2) d(J n f, Q) → 0 as n → ∞. This implies the equality N - lim 4n f n→∞

for all x ∈ X; (3) d(f, Q) ≤

1 1−L d(f, Jf ),



x 2n



= Q(x)

which implies the inequality d(f, Q) ≤

L . 4 − 4L

This implies that the inequality (2.4) holds. By (2.3), x−y x y − 2f − 2f , 4n t 2n 2n 2n (     )         x+y x−y x y t n n
≥ min N ρ 4 2f + 2f −f −f ,4 t , 2n+1 2n+1 2n 2n t + ϕ 2xn , 2yn 

 

N 4n f

x+y 2n

















+f

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for all x, y ∈ X, all t > 0 and all n ∈ N. So            x+y x−y x y N 4n f + f − 2f − 2f ,t 2n 2n 2n 2n (             x+y x−y x y ≥ min N ρ 4n 2f + 2f − f − f ,t , n+1 n+1 n 2 2 2 2n for all x, y ∈ X, all t > 0 and all n ∈ N. Since limn→∞ t > 0,

t 4n t Ln + ϕ(x,y) 4n 4n

t 4n

+

t 4n Ln 4n ϕ (x, y)

)

= 1 for all x, y ∈ X and all

N (Q(x + y) + Q(x − y) − 2Q(x) − 2Q(y), t)         x−y x+y + 2Q − Q(x) − Q(y) , t ≥ N ρ 2Q 2 2 for all x, y ∈ X and all t > 0. By Lemma 2.1, the mapping Q : X → Y is quadratic, as desired.  Corollary 2.3. Let θ ≥ 0 and let p be a real number with p > 2. Let X be a normed vector space with norm k · k. Let f : X → Y be an even mapping satisfying N (f (x + y) + f (x − y) − 2f (x) − 2f (y), t)           x+y x−y t ≥ min N ρ 2f + 2f − f (x) − f (y) , t , 2 2 t + θ(kxkp + kykp ) for all x, y ∈ X and all t > 0. Then Q(x) := N -limn→∞ 4n f ( 2xn ) exists for each x ∈ X and defines a quadratic mapping Q : X → Y such that (2p − 4)t N (f (x) − Q(x), t) ≥ p (2 − 4)t + 2θkxkp for all x ∈ X. Proof. The proof follows from Theorem 2.2 by taking ϕ(x, y) := θ(kxkp + kykp ) for all x, y ∈ X. Then we can choose L = 22−p , and we get the desired result.  Theorem 2.4. Let ϕ : X 2 → [0, ∞) be a function such that there exists an L < 1 with   x y , ϕ(x, y) ≤ 4Lϕ 2 2 for all x, y ∈ X. Let f : X → Y be an even mapping satisfying f (0) = 0 and (2.3). Then Q(x) := N -limn→∞ 41n f (2n x) exists for each x ∈ X and defines a quadratic mapping Q : X → Y such that (4 − 4L)t N (f (x) − Q(x), t) ≥ (2.7) (4 − 4L)t + ϕ(x, x) for all x ∈ X and all t > 0. Proof. Let (S, d) be the generalized metric space defined in the proof of Theorem 2.2. It follows from (2.5) that   1 1 t N f (x) − f (2x), t ≥ 4 4 t + ϕ(x, x) for all x ∈ X and all t > 0. So d(f, Jf ) ≤ 14 . Hence d(f, Q) ≤ inequality (2.7) holds. The rest of the proof is similar to the proof of Theorem 2.2.

467

1 4−4L ,

which implies that the 

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QUADRATIC ρ-FUNCTIONAL INEQUALITIES IN FUZZY NORMED SPACES

Corollary 2.5. Let θ ≥ 0 and let p be a real number with 0 < p < 2. Let X be a normed vector space with norm k · k. Let f : X → Y be an even mapping satisfying N (f (x + y) + f (x − y) − 2f (x) − 2f (y), t)           x+y x−y t ≥ min N ρ 2f + 2f − f (x) − f (y) , t , 2 2 t + θ(kxkp + kykp ) for all x, y ∈ X and all t > 0. Then Q(x) := N -limn→∞ defines a quadratic mapping Q : X → Y such that N (f (x) − Q(x), t) ≥

1 n 4n f (2 x)

exists for each x ∈ X and

(4 − 2p )t (4 − 2p )t + 2θkxkp

for all x ∈ X. Proof. The proof follows from Theorem 2.4 by taking ϕ(x, y) := θ(kxkp + kykp ) for all x, y ∈ X. Then we can choose L = 2p−2 , and we get the desired result.  3. Quadratic ρ-functional inequality (0.2) In this section, we prove the Hyers-Ulam stability of the quadratic ρ-functional inequality (0.2) in fuzzy Banach spaces. Let ρ be a real number with |ρ| < 12 . We need the following lemma to prove the main results. Lemma 3.1. Let f : X → Y be a mapping such that       x−y x+y + 2f − f (x) − f (y), t N 2f 2 2 ≥ N (ρ (f (x + y) + f (x − y) − 2f (x) − 2f (y)) , t)

(3.1)

for all x, y ∈ X and all t > 0. Then f is quadratic. Proof. Assume that f : X → Y satisfies (3.1).   t Letting x = y = 0 in (3.1), we get N (2f (0), t) ≥ N (ρ(2f (0)), t) = N 2f (0), |ρ| for all t > 0. By (N5 ) and (N6 ), N (f (0), t) = 1 for all t > 0. It follows from (N2 ) that f (0) = 0.

Letting y = 0 in (3.1), we get N 4f x2 − f (x), t ≥ N (0, t) = 1 for all t > 0 and so x 2

 

f

1 = f (x) 4

(3.2)

for all x ∈ X. It follows from (3.1) and (3.2) that       x+y x−y + 2f − f (x) − f (y), t N 2f 2 2   1 1 =N f (x + y) + f (x − y) − f (x) − f (y), t 2 2 = N (f (x + y) + f (x − y) − 2f (x) − 2f (y), 2t) ≥ N (ρ(f (x + y) + f (x − y) − 2f (x) − 2f (y)), t)   t = N f (x + y) + f (x − y) − 2f (x) − 2f (y), |ρ| for all t > 0. By (N5 ) and (N6 ), N (f (x + y) + f (x − y) − 2f (x) − 2f (y), t) = 1 for all t > 0. It follows from (N2 ) that f (x + y) + f (x − y) = 2f (x) + 2f (y) for all x, y ∈ X. 

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Theorem 3.2. Let ϕ : X 2 → [0, ∞) be a function such that there exists an L < 1 with L ϕ (2x, 2y) 4 for all x, y ∈ X. Let f : X → Y be an even mapping satisfying       x+y x−y N 2f + 2f − f (x) − f (y), t 2 2 ϕ(x, y) ≤

(3.3)



≥ min N (ρ (f (x + y) + f (x − y) − 2f (x) − 2f (y)) , t) , for all x, y ∈ X and all t > 0. Then Q(x) := N -limn→∞ 4n f defines a quadratic mapping Q : X → Y such that N (f (x) − Q(x), t) ≥

x 2n




t t + ϕ(x, y)



exists for each x ∈ X and

(1 − L)t (1 − L)t + ϕ(x, 0)

(3.4)

for all x ∈ X and all t > 0. 



t Proof. Letting x = y = 0 in (3.3), we get N (2f (0), t) ≥ N (ρ(2f (0)), t) = N 2f (0), |ρ| for all t > 0. So f (0) = 0. Letting y = 0 in (3.3), we get

x 2

 



N 4f



− f (x), t ≥

t t + ϕ(x, 0)

(3.5)

for all x ∈ X. Consider the set S := {g : X → Y } and introduce the generalized metric on S: t , ∀x ∈ X, ∀t > 0 , t + ϕ(x, 0) 



d(g, h) = inf µ ∈ R+ : N (g(x) − h(x), µt) ≥

where, as usual, inf φ = +∞. It is easy to show that (S, d) is complete (see [26, Lemma 2.1]). Now we consider the linear mapping J : S → S such that   x Jg(x) := 4g 2 for all x ∈ X. Let g, h ∈ S be given such that d(g, h) = ε. Then t N (g(x) − h(x), εt) ≥ t + ϕ(x, 0) for all x ∈ X and all t > 0. Hence x x x L , Lεt = N g −h , εt 2 2 2 4 Lt Lt t 4 4 =
≥ Lt x L t + ϕ(x, 0) + ϕ 2,0 4 + 4 ϕ(x, 0)

≥

Lt 4

x 2

 



N (Jg(x) − Jh(x), Lεt) = N 4g

 



  

 



− 4h

for all x ∈ X and all t > 0. So d(g, h) = ε implies that d(Jg, Jh) ≤ Lε. This means that d(Jg, Jh) ≤ Ld(g, h) for all g, h ∈ S.

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It follows from (3.5) that t x ,t ≥ 2 t + ϕ(x, 0)

 



N f (x) − 4f



for all x ∈ X and all t > 0. So d(f, Jf ) ≤ 1. By Theorem 1.4, there exists a mapping Q : X → Y satisfying the following: (1) Q is a fixed point of J, i.e.,   1 x = Q(x) Q (3.6) 2 4 for all x ∈ X. Since f : X → Y is even, Q : X → Y is an even mapping. The mapping Q is a unique fixed point of J in the set M = {g ∈ S : d(f, g) < ∞}. This implies that Q is a unique mapping satisfying (3.6) such that there exists a µ ∈ (0, ∞) satisfying t N (f (x) − Q(x), µt) ≥ t + ϕ(x, 0) for all x ∈ X; (2) d(J n f, Q) → 0 as n → ∞. This implies the equality   x = Q(x) N - lim 4n f n→∞ 2n for all x ∈ X; 1 (3) d(f, Q) ≤ 1−L d(f, Jf ), which implies the inequality d(f, Q) ≤

1 . 1−L

This implies that the inequality (3.4) holds. By (3.3),            x+y x−y x y n N 4n 2f + 2f − f − f , 4 t 2n+1 2n+1 2n 2n (     )         x+y x−y x y t n n
≥ min N ρ 4 f +f − 2f − 2f ,4 t , 2n 2n 2n 2n t + ϕ 2xn , 2yn for all x, y ∈ X, all t > 0 and all n ∈ N. So            x−y x y x+y N 4n 2f + 2f − f − f ,t 2n+1 2n+1 2n 2n (             x+y x−y x y ≥ min N ρ 4n f + f − 2f − 2f ,t , n n n 2 2 2 2n for all x, y ∈ X, all t > 0 and all n ∈ N. Since limn→∞ t > 0,

t 4n t Ln + 4n ϕ(x,y) 4n

t 4n

+

t 4n Ln 4n ϕ (x, y)

)

= 1 for all x, y ∈ X and all

x+y x−y N 2Q +2 − Q(x) − Q(y), t 2 2 ≥ N (ρ (Q (x + y) + Q (x − y) − 2Q(x) − 2Q(y)) , t) 











for all x, y ∈ X and all t > 0. By Lemma 3.1, the mapping Q : X → Y is quadratic, as desired. 

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Corollary 3.3. Let θ ≥ 0 and let p be a real number with p > 2. Let X be a normed vector space with norm k · k. Let f : X → Y be an even mapping satisfying       x+y x−y N 2f + 2f − f (x) − f (y), t 2 2   t ≥ min N (ρ (f (x + y) + f (x − y) − 2f (x) − 2f (y)) , t) , t + θ(kxkp + kykp ) for all x, y ∈ X and all t > 0. Then Q(x) := N -limn→∞ 4n f ( 2xn ) exists for each x ∈ X and defines a quadratic mapping Q : X → Y such that (2p − 4)t N (f (x) − Q(x), t) ≥ p (2 − 4)t + 2p θkxkp for all x ∈ X. Proof. The proof follows from Theorem 3.2 by taking ϕ(x, y) := θ(kxkp + kykp ) for all x, y ∈ X. Then we can choose L = 22−p , and we get the desired result.  Theorem 3.4. Let ϕ : X 2 → [0, ∞) be a function such that there exists an L < 1 with   x y ϕ(x, y) ≤ 4Lϕ , 2 2 for all x, y ∈ X. Let f : X → Y be an even mapping satisfying f (0) = 0 and (3.3). Then Q(x) := N -limn→∞ 41n f (2n x) exists for each x ∈ X and defines a quadratic mapping Q : X → Y such that (1 − L)t N (f (x) − Q(x), t) ≥ (3.7) (1 − L)t + ϕ(x, 0) for all x ∈ X and all t > 0. Proof. Let (S, d) be the generalized metric space defined in the proof of Theorem 3.2. It follows from (3.5) that   t 1 N f (x) − f (2x), Lt ≥ 4 t + ϕ(x, 0) for all x ∈ X and all t > 0. So d(f, Jf ) ≤ L. Hence 1 d(f, Q) ≤ , 1−L which implies that the inequality (3.7) holds. The rest of the proof is similar to the proof of Theorem 3.2.



Corollary 3.5. Let θ ≥ 0 and let p be a real number with 0 < p < 2. Let X be a normed vector space with norm k · k. Let f : X → Y be an even mapping satisfying       x−y x+y +f − f (x) − f (y), t N 2f 2 2   t ≥ min N (ρ (f (x + y) + f (x − y) − 2f (x) − 2f (y)) , t) , t + θ(kxkp + kykp ) for all x, y ∈ X and all t > 0. Then Q(x) := N -limn→∞ 41n f (2n x) exists for each x ∈ X and defines a quadratic mapping Q : X → Y such that (4 − 2p )t N (f (x) − Q(x), t) ≥ (4 − 2p )t + 2p θkxkp for all x ∈ X.

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Proof. The proof follows from Theorem 3.4 by taking ϕ(x, y) := θ(kxkp + kykp ) for all x, y ∈ X. Then we can choose L = 2p−2 , and we get the desired result.  References [1] T. Aoki, On the stability of the linear transformation in Banach spaces, J. Math. Soc. Japan 2 (1950), 64–66. [2] T. Bag and S.K. Samanta, Finite dimensional fuzzy normed linear spaces, J. Fuzzy Math. 11 (2003), 687– 705. [3] T. Bag and S.K. Samanta, Fuzzy bounded linear operators, Fuzzy Sets and Systems 151 (2005), 513–547. [4] L. C˘ adariu, L. G˘ avruta, P. G˘ avruta, On the stability of an affine functional equation, J. Nonlinear Sci. Appl. 6 (2013), 60–67. [5] L. C˘ adariu and V. Radu, Fixed points and the stability of Jensen’s functional equation, J. Inequal. Pure Appl. Math. 4, no. 1, Art. ID 4 (2003). [6] L. C˘ adariu and V. Radu, On the stability of the Cauchy functional equation: a fixed point approach, Grazer Math. Ber. 346 (2004), 43–52. [7] L. C˘ adariu and V. Radu, Fixed point methods for the generalized stability of functional equations in a single variable, Fixed Point Theory and Applications 2008, Art. ID 749392 (2008). [8] S.C. Cheng and J.M. Mordeson, Fuzzy linear operators and fuzzy normed linear spaces, Bull. Calcutta Math. Soc. 86 (1994), 429–436. [9] P.W. Cholewa, Remarks on the stability of functional equations, Aequationes Math. 27 (1984), 76–86. [10] S. Czerwik, On the stability of the quadratic mapping in normed spaces, Abh. Math. Sem. Univ. Hamburg 62 (1992), 59–64. [11] J. Diaz and B. Margolis, A fixed point theorem of the alternative for contractions on a generalized complete metric space, Bull. Amer. Math. Soc. 74 (1968), 305–309. [12] W. Fechner, Stability of a functional inequalities associated with the Jordan-von Neumann functional equation, Aequationes Math. 71 (2006), 149–161. [13] C. Felbin, Finite dimensional fuzzy normed linear spaces, Fuzzy Sets and Systems 48 (1992), 239–248. [14] P. G˘ avruta, A generalization of the Hyers-Ulam-Rassias stability of approximately additive mappings, J. Math. Anal. Appl. 184 (1994), 431–436. [15] A. Gil´ anyi, Eine zur Parallelogrammgleichung a ¨quivalente Ungleichung, Aequationes Math. 62 (2001), 303– 309. [16] A. Gil´ anyi, On a problem by K. Nikodem, Math. Inequal. Appl. 5 (2002), 707–710. [17] D.H. Hyers, On the stability of the linear functional equation, Proc. Nat. Acad. Sci. U.S.A. 27 (1941), 222–224. [18] D.H. Hyers, G. Isac and Th.M. Rassias, Stability of Functional Equations in Several Variables, Birkh¨ auser, Basel, 1998. [19] G. Isac and Th.M. Rassias, Stability of ψ-additive mappings: Appications to nonlinear analysis, Internat. J. Math. Math. Sci. 19 (1996), 219–228. [20] S. Jung, Hyers-Ulam-Rassias Stability of Functional Equations in Mathematical Analysis, Hadronic Press lnc., Palm Harbor, Florida, 2001. [21] A.K. Katsaras, Fuzzy topological vector spaces II, Fuzzy Sets and Systems 12 (1984), 143–154. [22] J. Kim, G. A. Anastassiou and C. Park, Additive ρ-functional inequalities in fuzzy normed spaces, J. Comput. Anal. Appl. (to appear). [23] I. Kramosil and J. Michalek, Fuzzy metric and statistical metric spaces, Kybernetica 11 (1975), 326–334. [24] S.V. Krishna and K.K.M. Sarma, Separation of fuzzy normed linear spaces, Fuzzy Sets and Systems 63 (1994), 207–217. [25] J. Lee, C. Park and D. Shin, An AQCQ-functional equation in matrix normed spaces, Results Math. 27 (2013), 305–318. [26] D. Mihet¸ and V. Radu, On the stability of the additive Cauchy functional equation in random normed spaces, J. Math. Anal. Appl. 343 (2008), 567–572. [27] M. Mirzavaziri and M.S. Moslehian, A fixed point approach to stability of a quadratic equation, Bull. Braz. Math. Soc. 37 (2006), 361–376. [28] A.K. Mirmostafaee, M. Mirzavaziri and M.S. Moslehian, Fuzzy stability of the Jensen functional equation, Fuzzy Sets and Systems 159 (2008), 730–738. [29] A.K. Mirmostafaee and M.S. Moslehian, Fuzzy versions of Hyers-Ulam-Rassias theorem, Fuzzy Sets and Systems 159 (2008), 720–729.

472

JI-HYE KIM et al 462-473

J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 22, NO.3, 2017, COPYRIGHT 2017 EUDOXUS PRESS, LLC

J. KIM, C. PARK

[30] A.K. Mirmostafaee and M.S. Moslehian, Fuzzy approximately cubic mappings, Inform. Sci. 178 (2008), 3791–3798. [31] C. Park, Fixed points and Hyers-Ulam-Rassias stability of Cauchy-Jensen functional equations in Banach algebras, Fixed Point Theory and Applications 2007, Art. ID 50175 (2007). [32] C. Park, Generalized Hyers-Ulam-Rassias stability of quadratic functional equations: a fixed point approach, Fixed Point Theory and Applications 2008, Art. ID 493751 (2008). [33] C. Park, Additive ρ-functional inequalities and equations, J. Math. Inequal. 9 (2015), 17–26. [34] C. Park, Additive ρ-functional inequalities in non-Archimedean normed spaces, J. Math. Inequal. 9 (2015), 397–407. [35] C. Park, Y. Cho and M. Han, Stability of functional inequalities associated with Jordan-von Neumann type additive functional equations, J. Inequal. Appl. 2007, Art. ID 41820 (2007). [36] C. Park, K. Ghasemi, S. G. Ghaleh and S. Jang, Approximate n-Jordan ∗-homomorphisms in C ∗ -algebras, J. Comput. Anal. Appl. 15 (2013), 365-368. [37] C. Park, A. Najati and S. Jang, Fixed points and fuzzy stability of an additive-quadratic functional equation, J. Comput. Anal. Appl. 15 (2013), 452–462. [38] C. Park and Th.M. Rassias, Fixed points and generalized Hyers-Ulam stability of quadratic functional equations, J. Math. Inequal. 1 (2007), 515–528. [39] V. Radu, The fixed point alternative and the stability of functional equations, Fixed Point Theory 4 (2003), 91–96. [40] Th.M. Rassias, On the stability of the linear mapping in Banach spaces, Proc. Amer. Math. Soc. 72 (1978), 297–300. [41] Th.M. Rassias, Problem 16; 2, Report of the 27th International Symp. on Functional Equations, Aequationes Math. 39 (1990), 292–293; 309. [42] Th.M. Rassias and K. Shibata, Variational problem of some quadratic functionals in complex analysis, J. Math. Anal. Appl. 228 (1998), 234–253. [43] J. R¨ atz, On inequalities associated with the Jordan-von Neumann functional equation, Aequationes Math. 66 (2003), 191–200. ˇ ankov´ [44] L. Reich, J. Sm´ıtal and M. Stef´ a, Singular solutions of the generalized Dhombres functional equation, Results Math. 65 (2014), 251–261. [45] S. Schin, D. Ki, J. Chang and M. Kim, Random stability of quadratic functional equations: a fixed point approach, J. Nonlinear Sci. Appl. 4 (2011), 37–49. [46] S. Shagholi, M. Bavand Savadkouhi and M. Eshaghi Gordji, Nearly ternary cubic homomorphism in ternary Fr´echet algebras, J. Comput. Anal. Appl. 13 (2011), 1106–1114. [47] S. Shagholi, M. Eshaghi Gordji and M. Bavand Savadkouhi, Stability of ternary quadratic derivation on ternary Banach algebras, J. Comput. Anal. Appl. 13 (2011), 1097–1105. [48] D. Shin, C. Park and Sh. Farhadabadi, On the superstability of ternary Jordan C ∗ -homomorphisms, J. Comput. Anal. Appl. 16 (2014), 964–973. [49] D. Shin, C. Park and Sh. Farhadabadi, Stability and superstability of J ∗ -homomorphisms and J ∗ -derivations for a generalized Cauchy-Jensen equation, J. Comput. Anal. Appl. 17 (2014), 125–134. [50] F. Skof, Propriet` a locali e approssimazione di operatori, Rend. Sem. Mat. Fis. Milano 53 (1983), 113–129. [51] S. M. Ulam, A Collection of the Mathematical Problems, Interscience Publ. New York, 1960. [52] J.Z. Xiao and X.H. Zhu, Fuzzy normed spaces of operators and its completeness, Fuzzy Sets and Systems 133 (2003), 389–399. Ji-hye Kim Department of Mathematics, Hanyang University, Seoul 133-791, Korea E-mail address: [email protected] Choonkil Park Research Institute for Natural Sciences, Hanyang University, Seoul 133-791, Korea E-mail address: [email protected]

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õStieltjes

The Quadrature rules of the fuzzy Henstock integral on a infinite interval† Ling Wang∗

School of Information Engineering, Gansu Institute of Political Science and Law, Lanzhou, Gansu 730070, P.R. China

Abstract: In this paper, the calculating methods for the fuzzy Henstock-Stieltjes integral on a infinite interval are proposed. It includes quadrature rules and the error estimates such as the midpoint-type rule, trapezoidal-type rule, Simpson’s formula, δ−fine quadrature rules, their error estimates, and so on. Finally, an example is given to illuminate the effectiveness the methods proposed in this paper. Keywords: Fuzzy numbers; Fuzzy Henstock-Stieltjes integral; calculating methods AMS subject classifications. 26E50; 28E10. 1 Introduction It is well known that the notion of the Stieltjes integral for fuzzy-number-valued functions was originally proposed by Nanda [1] in 1989. Many generalizations of the fuzzy Riemann-Stieltjes integral were considered by scholars [2, 3, 4]. In 1998, Wu [5] proposed the concept of fuzzy Riemann-Stieltjes integral by means of the representation theorem of fuzzy-number-valued functions, whose membership function could be obtained by solving a nonlinear programming problem, but it is difficult to calculate and extend to the higher-dimensional space. In 2006, Ren et al. introduced the concept of two kinds of fuzzy Riemann-Stieltjes integral for fuzzy-number-valued functions [3, 4] and showed that a continuous fuzzy-number-valued function was fuzzy Riemann-Stieltjes integrable with respect to a real-valued increasing function. To overcome the limitations of the existing studies and to characterize continuous linear functionals on the space of Henstock integrable fuzzy-number-valued functions, the concept of the Henstock-Stieltjes integral for fuzzy-number-valued functions was defined and discussed in 2012, and some useful results for this integral were shown, such as the integrability, the continuity and the differentiability of the primitive, numerical calculus of the integration, the convergence theorems, and so on. The integral for fuzzy-number-valued functions on a infinite interval, as a expectation of fuzzy random variable, was originally investigated by Puri and Ralescu in 1986 [6]. In their opinion, a fuzzy random variable as a fuzzy-number-valued function and the expectation E(X) of a fuzzy random variable X equals to a R fuzzy integral E(X) = X or set-valued integral of Xλ . In 2007, the concept of the fuzzy Henstock integral on infinite interval was proposed and discussed in order to solve the expectation E(X) of a fuzzy random variable X which distribution function has some kinds of discontinuity or non-integrability by Gong and Wang [7]. After that, the Henstock-Stieltjes integral for fuzzy-number-valued functions on infinite interval which is an extension of the usual fuzzy Riemann-Stieltjes integral on infinite interval was investigated by Duan in 2014 [8], and several necessary and sufficient conditions of the integrability for fuzzy-number-valued functions are given by means of the Henstock-Stieltjes integral of real-valued functions on infinite interval and Henstock integral of fuzzy-number-valued functions on infinite interval. In this paper, we shall discuss the calculating methods for the fuzzy Henstock-Stieltjes integral on a infinite interval: one is to calculate directly by the fuzzy Henstock-Stieltjes integral on a infinite interval, including quadrature rules and the error estimates such as the midpoint-type rule, trapezoidal-type rule, Simpson’s formula, δ−fine quadrature rules and their error estimates; another is to calculate by using the equivalent characteristic of fuzzy Henstock-Stieltjes integrability, whose membership function could be obtained by solving a nonlinear programming problem. †

The work is supported by the Natural Scientific Fund of China (11161041) * Tel.:+8618693076970, E-mail: [email protected] 474

Ling Wang 474-483

J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 22, NO.3, 2017, COPYRIGHT 2017 EUDOXUS PRESS, LLC

Ling Wang: The Quadrature rules of the fuzzy Henstock-Stieltjes integrals on a infinite interval

2 Preliminaries Fuzzy set u ˜ ∈ E 1 is called a fuzzy number if u ˜ is a normal, convex fuzzy set, upper semi-continuous and supp u = {x ∈ R | u(x) > 0} is compact. Here A¯ denotes the closure of A. We use E 1 to denote the fuzzy number space [1-6]. Let u ˜, v˜ ∈ E 1 , k ∈ R, the addition and scalar multiplication are defined by [˜ u + v˜]λ = [˜ u]λ + [˜ v ]λ ,

[k˜ u]λ = k[˜ u]λ ,

+ respectively, where [˜ u]λ = {x : u(x) > λ} = [u− λ , uλ ], for any λ ∈ [0, 1]. We use the Hausdorff distance between fuzzy numbers given by D : E 1 × E 1 → [0, +∞) as follows [1-6]: − + + D(˜ u, v˜) = sup d([˜ u]λ , [˜ v ]λ ) = sup max{|u− λ − vλ |, |uλ − vλ |}, λ∈[0,1]

λ∈[0,1]

where d is the Hausdorff metric. D(˜ u, v˜) is called the distance between u ˜ and v˜. Recall, also, that a function f˜ : [a, b] → E 1 is said to be bounded if there exists M ∈ R such that kf˜(x)k = D(f˜(x), ˜0) 6 M for any x ∈ [a, b]. Notice that here kf˜(x0 )k does not stand for the norm of E 1 . Definition 2.1 [7,8,9]. R denote the generalized real line, for f˜ defined on [a, +∞], we define f˜(+∞) = ˜0, and ˜0 · (+∞) = ˜0. Let δ : [a, +∞] → R+ be a positive real function. A division P = {[xi−1 , xi ]; ξi } is said to be δ-fine, if the following conditions are satisfied: (1)a = x0 < x1 < ... < xn−1 = b < xn = +∞; (2)ξi ∈ [xi−1 , xi ] ⊂ O(ξi ), i = 1, 2, ..., n; where O(ξi ) = (ξi − δ(ξi ), ξi + δ(ξi )) for i = 1, 2, ..., n − 1, and O(ξn ) = [b, +∞). For brevity, we write T = {[u, v]; ξ}, where [u, v] denotes a typical interval in T and ξ is the associated point of [u, v]. Definition 2.2 [8]. Let α : [a, +∞] → R be an increasing function. A fuzzy-number-valued function ˜ f (x) is said to be fuzzy Henstock-Stieltjes integrable with respect to α on [a, +∞] if there exists a fuzzy ˜ ∈ E 1 such that for every ε > 0, there is a function δ(x) > 0 on [a, +∞] such that for any number H δ-fine division T = {[xi−1 , xi ]; ξi }ni=1 , we have n X ˜ < ε. D( [α(xi ) − α(xi−1 )]f˜(ξi ), H) i=1

R +∞ ˜ and (f˜, α) ∈ F HS[a, +∞]. We write (F HS) a f˜(x)dα = H The definition of f˜ ∈ F HS(−∞, a] is similar. Naturally, we define f˜ ∈ F HS(−∞, +∞) iff f˜ ∈ F HS(−∞, a] and f˜ ∈ F HS[a, +∞), and furthermore Z +∞ Z a Z +∞ ˜ ˜ (F HS) f (x)ddα = (F HS) f (x)ddα + (F HS) f˜(x)ddα. −∞

−∞

a

For brevity, we always assume that α : [a, +∞] → R is an increasing function. Lemma 2.1[8]. Let α : [a, +∞] → R be an increasing function and let f˜ : [a, +∞] → E 1 . Then the following statements are equivalent: R +∞ ˜ (1) (f˜, α) ∈ F HS[a, +∞] and (F HS) a f˜(x)dα = A; − + (2) for any λ ∈ [0, 1], fλ and fλ are Henstock-Stieltjes integrable with respect to α on [a, +∞] for any λ ∈ [0, 1] uniformly (δ(x) is independent of λ ∈ [0, 1]), and Z +∞ Z +∞ Z +∞ − ˜ [(F HS) f (x)dα]λ = [(HS) fλ (x)dα, (HS) fλ+ (x)dα]. a

a

(3) For any b > a, f˜ ∈ F HS[a, b], lim

Rb

b→+∞ a

Z lim

b→+∞ a

b

0

f˜(x)dα as a fuzzy number exists and

f˜(x)dα =

Z

+∞

f˜(x)dα.

a

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Ling Wang: The Quadrature rules of the fuzzy Henstock-Stieltjes integrals on a infinite interval

3 Quadrature rules of the Henstock-Stieltjes integral for fuzzy-number-valued functions on infinite interval We shall use the modulus of oscillation for a fuzzy-valued function to discuss the quadrature rules of expectations for fuzzy random variables in this section. For the numerical calculus of fuzzy integral, there were some discussions by the fuzzy Riemann integral, improper fuzzy Riemann integral, using the probabilistic Monte Carlo method, and the quadrature rules for fuzzy Henstock integral on a finite interval[1, 3, 4]. However, the calculus above will be restricted when the distribution function of a random variable on (−∞, +∞) or the distribution function of a random variable has some kind of discontinuity or non-integrability. Furthermore, fuzzy Henstock integral is convenient for numerical calculus since it is a Riemann-type integral. Since a fuzzy random variable is a measurable fuzzy-valued function f˜ : (−∞, +∞) → E 1 , therefore without loss of the generality, we only discuss the quadrature rules of Henstock integrals for the measurable fuzzy-valued functions on [a, +∞). For a fuzzy-valued function, since its Henstock integrability implies measurability, for brevity we always assume that the fuzzy-valued functions discussed are measurable throughout this section. Definition 3.1[7, 10]. Let f˜ : [a, +∞) → E 1 be a bounded mapping. Then the function ω[a,+∞) (f˜, ·) : S + R {0} → R+ , ω[a,+∞) (f˜, δ) = sup{D(f˜(x), f˜(y)) : x, y ∈ [a, +∞), |x − y| ≤ δ} is called the modulus of oscillation of f˜ on [a, +∞). Theorem 3.1[7] Obviously, the following statements hold: (i) D(f˜(x), f˜(y)) ≤ ω[a,+∞) (f˜, |x − y|), ∀x, y ∈ [a, +∞) for any x, y ∈ [a, +∞); (ii) ω[a,+∞) (f˜, δ) is nondecreasing mapping in δ and nonincreasing in a; (iii) ω[a,+∞) (f˜, 0) = 0; (iv) ω[a,+∞) (f˜, δ1 + δ2 ) ≤ ω[a,+∞) (f˜, δ1 ) + ω[a,+∞) (f˜, δ2 ) for any δ1 , δ2 ≥ 0; (v) ω[a,+∞) (f˜, nδ) ≤ nω[a,+∞) (f˜, δ) for any δ ≥ 0, n ∈ N ; (vi) ω[a,+∞) (f˜, λδ) ≤ (λ + 1)ω[a,+∞) (f˜, δ) for any δ ≥ 0, λ ≥ 0. Theorem 3.2 Let f˜ ∈ F HS[a, +∞) be a bounded function, and α : [a, +∞] → R an increasing function. Then for any division T : a = x0 < x1 < ... < xn−1 = b < xn = +∞ and any point ξi ∈ [xi−1 , xi ], i = 1, 2, 3, ..., n − 1, and ξn = +∞, we have Z D( a

+∞

f˜(x)dα,

n X i=1

[α(xi ) − α(xi−1 )]f˜(ξi )) ≤

n−1 X

[α(xi ) − α(xi−1 )]ω[xi−1 ,xi ] (f˜, xi − xi−1 ) + αb ,

i=1

R +∞ R +∞ where αb stands for k b f˜(x)dαkE 1 or b kf˜(x)kE 1 dα, and αb → 0 (b → +∞). Proof. The subinterval including +∞ is denoted by [b, +∞](xn−1 = b, xn = +∞), according to R +∞ Rb the additivity of interval for fuzzy Henstock-Stieltjes integral, we have a f˜(x)dα = a f˜(x)dα +

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Ling Wang: The Quadrature rules of the fuzzy Henstock-Stieltjes integrals on a infinite interval

R +∞ b

f˜(x)dα, and Z D(

+∞

f˜(x)dα,

a

n X

[α(xi ) − α(xi−1 )]f˜(ξi ))

i=1

Z b Z n−1 X ˜ ˜ ≤ D( f (x)dα, [α(xi ) − α(xi−1 )]f (ξi )) + D( a

≤

n−1 X

≤

f˜(x)dα,

b

i=1

Z

+∞

f˜(ξn )dα, )

b

Z ˜ [α(xi ) − α(xi−1 )]ω[xi−1 ,xi ] (f , xi − xi−1 ) + D(

+∞

f˜(x), ˜0)dα

b

i=1 n X

+∞

[α(xi ) − α(xi−1 )]ω[xi−1 ,xi ] (f˜, xi − xi−1 ) + αb ,

i=1

where f˜(ξn ) = ˜0. By Lemma 2.1, αb → 0 when b → +∞. The proof is complete. Taking in Theorem 3.2 n = 2, x1 = ξ1 = ξ2 = x; n = 2, x1 = x, ξ1 = u, ξ2 = v and n = 4, x1 = α, x2 = β, ξ1 = u, ξ2 = v, ξ3 = w respectively, we obtain the midpoint-type, trapezoidal-type and Simpson’s inequalities in some sense with its error estimations as follows. Corollary 3.1 Let f˜ ∈ F HS[a, +∞) be a bounded function. Then (i) Z +∞ D( f˜(x)dα, [α(b) − α(a)]f˜(x)) ≤ [α(x) − α(a)]ω[a,x] (f˜, x − a) + [α(b) − α(x)]ω[x,b] (f˜, b − x) + αb a

for any b ≥ a and x ∈ [a, b]; (ii) Z +∞ D( f˜(x)dα, [α(x) − α(a)]f˜(u) + [α(b) − α(x)]f˜(v)) a

≤ [α(x) − α(a)]ω[a,x] (f˜, x − a) + [α(b) − α(x)]ω[x,b] (f˜, b − x) + αb for any b ≥ a and x ∈ [a, b], u ∈ [a, x], v ∈ [x, b]; (iii) Z +∞ D( f˜(x)dα, [α(β1 ) − α(a)]f˜(u) + [α(β2 ) − α(β1 )]f˜(v) + [α(b) − α(β2 )]f˜(w)) a

≤ [α(β1 ) − α(a)]ω[a,α] (f˜, β1 − a) + [α(β2 ) − α(β1 )]ω[β1 ,β2 ] (f˜, β2 − β1 ) + [α(b) − α(β2 )]ω[β ,b] (f˜, b − β2 ) + αb 2

for any b ≥ a, α, β ∈ [a, b], and u ∈ [a, β1 ], v ∈ [β1 , β2 ], w ∈ [β2 , b], where αb stands for k R +∞ or b kf˜(x)kE 1 dα, and αb → 0 (b → +∞). Corollary 3.2 Let f˜ ∈ F HS[a, +∞) be a bounded function. Then (i) Z +∞ a+b )) D( f˜(x)dα, [α(b) − α(a)]f˜( 2 a b−a ≤ [α(b) − α(a)]ω[a,b] (f˜, ) + αb 2 b−a ≤ [α(b) − α(a)]ω[a,+∞) (f˜, ) + αb ; 2

477

R +∞ b

f˜(x)dαkE 1

Ling Wang 474-483

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Ling Wang: The Quadrature rules of the fuzzy Henstock-Stieltjes integrals on a infinite interval

(ii) Z D(

+∞

b+a b+a ˜ f˜(x)dα, [α( ) − α(a)]f˜(a) + [α(b) − α( )]f (b)) 2 2 a b−a b+a b−a b+a ) − α(a)]ω[a, b+a ] (f˜, ) + [α(b) − α( )]ω[ b+a ,b] (f˜, ) + αb ; ≤ [α( 2 2 2 2 2 2

(iii) Z +∞ 2a + b a + 2b 2a + b ˜ a + b a + 2b ˜ D( f˜(x)dα, [α( ) − α(a)]f˜(a) + [α( ) − α( )]f ( ) + [α(b) − α( )]f (b)) 3 3 3 2 3 a 2a + b b−a a + 2b 2a + b b−a ≤ [α( ) − α(a)]ω[a, 2a+b ] (f˜, ) + [α( ) − α( )]ω[ 2a+b ,[ a+2b ] (f˜, ) 3 3 3 3 3 3 3 3 b−a a + 2b + [α(b) − α( )]ω[ a+2b ,b] (f˜, ) + αb , 3 3 3 R +∞ R +∞ where αb stands for k b f˜(x)dαkE 1 or b kf˜(x)kE 1 dα, and αb → 0 (b → +∞). Using Theorem 3.2, we can also obtain another numerical calculous of Henstock-Stieltjes integrals with error estimations. Corollary 3.3 Let f˜ ∈ F HS[a, +∞) be a bounded function. Then (1) Z D(

+∞

f˜(x)dα,

a

n X

[α(xi ) − α(xi−1 )]f˜(ξi ))

i=1

≤ [α(b) − α(a)]ω[a,b] (f˜, kT k) + αb ≤ [α(b) − α(a)]ω[a,+∞) (f˜, kT k) + αb ; (2) Z D(

+∞

f˜(x)dα,

a

n X

[α(xi ) − α(xi−1 )]f˜(ξi ))

i=1

≤ kα(T )k

≤ kα(T )k

n−1 X i=1 n−1 X

ω[a,b] (f˜, xi − xi−1 ) + αb ω[a,+∞) (f˜, xi − xi−1 ) + αb ;

i=1

(3) If α : [a, b] → R is an increasing function satisfying α ∈ C 1 [a, +∞], then Z D(

+∞

f˜(x)dα,

a

≤ M kT k

n X

[α(xi ) − α(xi−1 )]f˜(ξi ))

i=1 n−1 X

ω[a,b] (f˜, xi − xi−1 ) + αb

i=1

for any division T : a = x0 < x1 < ... < xn−1 R +∞= b < xn = +∞ R +∞and any point ξi ∈ [xi−1 , xi ], i = ˜ 1, 2, ...n − 1 ξn = +∞, where αb stands for k b f (x)dαkE 1 or b kf˜(x)kE 1 dα, αb → 0 (b → +∞), kT k = max{xi −xi−1 : i = 1, 2, ...n−1} denotes the modulus of division T , kα(T )k = max{α(xi )−α(xi−1 ) : i = 1, 2, ...n − 1}, and M is the bound of α on [a, b].

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Ling Wang: The Quadrature rules of the fuzzy Henstock-Stieltjes integrals on a infinite interval

Proof. By using Theorem 3.2 and Theorem 3.1, (1) and (2) are obvious. We only prove that (3) holds. In fact, we have +∞

Z D(

f˜(x)dα,

n X

a

≤

=

n−1 X i=1 n−1 X

[α(xi ) − α(xi−1 )]f˜(ξi ))

i=1

[α(xi − α(xi−1 )]ω[xi−1 ,xi ] (f˜, xi − xi−1 ) + αb α0 (ζi )(xi − xi−1 )ω[xi−1 ,xi ] (f˜, xi − xi−1 ) + αb

i=1

≤ M kT k

≤ M kT k

n−1 X i=1 n−1 X

ω[a,b] (f˜, xi − xi−1 ) + αb ω[a,+∞) (f˜, xi − xi−1 ) + αb .

i=1

4. δ− fine quadrature rules of the Henstock-Stieltjes integral for fuzzy-number-valued functions on infinite interval Definition 4.1 Let Sn =

n P

[α(xi ) − α(xi−1 )]f˜(ξi ) be a quadrature rule and δ : [a, +∞] → R+ .

i=1

Sn is said to be a δ−fine quadrature rule, if ξi ∈ [xi−1 , xi ] ⊂ O(ξi ), i = 1, 2, ..., n, where O(ξi ) = (ξi − δ(ξi ), ξi + δ(ξi )) for i = 1, 2, ..., n − 1, and O(ξn ) = [b, +∞). We can deduce expressions for the remainder of δ−fine quadrature rules by using Theorem 3.2 and Theorem 3.1(ii,v) as follows. n P Theorem 4.1 Let f˜ ∈ F HS[a, +∞) be a bounded function. If Sn = [α(xi ) − α(xi−1 )]f˜(ξi ) is a i=1

δ−fine quadrature rule, then Z D(

+∞

f˜(x)dα, Sn ) ≤ 2

a

n−1 X

[α(xi ) − α(xi−1 )]δ(ξi )ω[xi−1 xi ] (f˜, δ(ξi )) + αb .

i=1

R +∞ R +∞ Here αb stands for k b f˜(x)dαkE 1 or b kf˜(x)kE 1 dα, and αb → 0 (b → +∞). Theorem 4.2 Let f˜ ∈ F HS[a, +∞) be a bounded function, and α : [a, b] → R be an increasing n P function such that α ∈ C 1 [a, +∞]. If Sn = [α(xi ) − α(xi−1 )]f˜(ξi ) is a δ−fine quadrature rule, then i=1

Z D( a

+∞

f˜(x)dα, Sn ) ≤ 4M

n−1 X

δ(ξi )ω[xi−1 xi ] (f˜, δ(ξi )) + αb .

i=1

Here M is the bound of α0 on [a, b], αb stands for k (b → +∞).

479

R +∞ b

f˜(x)dαkE 1 or

R +∞ b

kf˜(x)kE 1 dα, and αb → 0

Ling Wang 474-483

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Ling Wang: The Quadrature rules of the fuzzy Henstock-Stieltjes integrals on a infinite interval

Proof By using Theorem 3.2 and Theorem 3.1(ii,v), we have Z +∞ D( f˜(x)dα, Sn ) a n−1 X

≤2

≤2

i=1 n−1 X

[α(xi ) − α(xi−1 )]δ(ξi )ω[xi−1 xi ] (f˜, δ(ξi )) + αb α0 (ζi )(xi − xi−1 )δ(ξi )ω[xi−1 xi ] (f˜, δ(ξi )) + αb

i=1 n−1 X

≤ 4M

δ(ξi )ω[xi−1 xi ] (f˜, δ(ξi )) + αb .

i=1

Corollary 4.1 Let f˜ ∈ F HS[a, +∞) be a bounded function, α : [a, b] → R an increasing function such that α ∈ C 1 [a, +∞], M the bound of α0 on [a, b]. Then (i) Z +∞ D( f˜(x)dα, [α(b) − α(a)]f˜(x)) a

≤ 4M δ(x)ω[a,b] (f˜, δ(x)) + αb ; for any x ∈ [a, b] such that the quadrature rule [α(b) − α(a)]f˜(x) is δ−fine; (ii) Z +∞ D( f˜(x)dα, [α(x) − α(a)]f˜(u) + [α(b) − α(x)]f˜(v)) a

≤ 4M [δ(u)ω[a,x] (f˜, δ(u) + δ(v)ω[x,b] (f˜, δ(v))] + αb ; for any x ∈ [a, b], u ∈ [a, x] and v ∈ [x, b] such that the trapezoidal-type quadrature rule [α(x)−α(a)]f˜(u)+ [α(b) − α(x)]f˜(v)) is δ−fine; (iii) Z +∞ D( f˜(x)dα, [α(β1 ) − α(a)]f˜(u) + [α(β2 ) − α(β1 )]f˜(v) + [α(b) − α(β2 )]f˜(w)) a

≤ 4M [δ(u)ω[a,b] (f˜, δ(u)) + δ(v)ω[a,b] (f˜, δ(v)) + δ(w)ω[a,b] (f˜, δ(w))] + αb for any β1 , β2 ∈R[a, b], and u ∈ [a, β1R], v ∈ [β1 , β2 ], w ∈ [β2 , b], such that Simpson’s formula is δ−fine. Here +∞ +∞ αb stands for k b f˜(x)dαkE 1 or b kf˜(x)kE 1 dα, and αb → 0 (b → +∞). The following theorem shows that δ−fine quadrature rules converge for the bounded Henstock-Stieltjes integrable functions. Theorem 4.3 Let f˜ ∈ F HS[a, +∞) be a bounded function. Then there exist functions δn : m Pn [a, +∞] → R+ and a sequence of δn −fine quadrature rules Sn = [α(xi ) − α(xi−1 )]f˜(ξi ) such that i=1 R +∞ Sn converges to a f˜(x)dα. Proof. From the definition of Henstock-Stieltjes integrability on infinite interval for all ε > 0 there exists a function δ such that for any δ−fine division(which can be interpreted as a δ−fine quadrature rule), we have Z +∞

D(

f˜(x)dα, Sn ) < ε.

a

Taking ε =

1 n

in the inequality we obtain that the statement of the theorem holds. The proof is complete.

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Ling Wang: The Quadrature rules of the fuzzy Henstock-Stieltjes integrals on a infinite interval

Corollary 4.2 Let f˜ ∈ F HS[a, +∞) be a bounded function. Then for any natural number n, there exist functions δn : [a, +∞] → R+ , bn ≥ a, and a sequence of δn −fine quadrature rules Sn = m Pn [α(xi ) − α(xi−1 )]f˜(ξi ) such that i=1

+∞

Z D(

f˜(x)dα, Sn ) < ε.

a

5. Examples Example 5.1 Let f˜ : [1, +∞] → E 1 be given by  s, s ∈ [0, 1], x is rational,     0, s ∈ (−∞, 0) ∪ (1, +∞), x is rational,   2  1− s , s ∈ [0, e−x ], x is irrational, e−x2 f˜(x, s) = 2  0, s ∈ (−∞, 0) ∪ (e−x , +∞), x is irrational,     1, s = 0, x = +∞,   0, s ∈ (−∞, 0) ∪ (0, +∞), x = +∞, and α(x) = x. We could prove that f˜ is (FHS) integrable on [0, +∞) according to the R +∞equivalence of fuzzy (HS) + − ˜ and δ−fine integrability and uniform (HS) integrability of fλ and fλ . Furthermore, 0 f˜(x)dα = H, n P ˜ which relationship function quadrature rule Sn = [α(xi ) − α(xi−1 )]f˜(ξi ) converges to fuzzy number H i=1

is defined by ( H(s) =

1−

√2 s, π

0,

√

s ∈ [0, 2π ], x is others,

√

That is to say, Hλ− = 0, Hλ+ = (1 − λ) 2π . In fact,  we note that 1, x is rational,  λ, x is rational,  0, x = +∞, 0, x = +∞, fλ− (x) = fλ+ (x) = 2   −x 0, x is irrational, (1 − λ)e , x is irrational. 2 Since fλ+ (x) ≤ e−x , fλ− , fλ+ are Henstock integrable uniformly for λ ∈ [0, 1] and Z 0

+∞

fλ− (x)dα

+∞

Z

fλ+ (x)dα

= 0(+∞) = 0, 0

It follows that Z

+∞

Z = lim

b→+∞ 0

√

b

fλ+ (x)dα

= (1 − λ)

π . 2

˜ f˜(x)dα = H.

0

For any ε > 0, we define ( δ(ξ) =

ε , 2i+2 ε 4 ξ,

ξ = ri , otherwise,

where Q = {r1 , r2 , r3 , ...} stands for the set of all rational numbers on [0, +∞) and for any δ−fine division T : 1 = x0 < x1 < ... < xn−1 = b < xn = +∞(ξn = +∞, f˜(ξn ) = (0, 0, 0)), then Sn = n P [α(xi ) − α(xi−1 )]f˜(ξi ) a any δ−fine quadrature rule. Note that ω[xi−1 xi ] (f˜, δ(ξi )) = 1. Then we have i=1

the following results.

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Ling Wang: The Quadrature rules of the fuzzy Henstock-Stieltjes integrals on a infinite interval

(1) According to Theorem 3.2, we have ˜ D(Sn , H) ≤

=

n−1 X i=1 n−1 X

[α(xi ) − α(xi−1 )]ω[xi−1 ,xi ] (f˜, xi − xi−1 ) + αb [α(xi ) − α(xi−1 )] + αb

i=1

= b + αb , R +∞ R +∞ where αb = k b f˜(x)dαkE 1 or b kf˜(x)kE 1 dα, and αb → 0 (b → +∞). Indeed, Z +∞ Z +∞ 2 αb = (1 − λ)e−x }, kf˜(x)kE 1 dα = sup { b

λ∈[0,1]

b

and αb → 0. (2) According to Theorem 4.2, we have ˜ D(Sn , H) ≤

n−1 X

δ(ξi )ω[xi−1 xi ] (f˜, δ(ξi )) + k

Z

=≤ 4

f˜(x)dxkE 1

b

i=1 n−1 X

+∞

δ(ξi ) + αb .

i=1

(iii) According to Corollary 4.1, we have (i) ˜ ≤ 4δ(x) + αb D((b − 0)f˜(x), H) for any x ∈ [0, b] such that the quadrature rule (b − 0)f˜(x) is δ−fine; (ii) ˜ ≤ 4(δ(u) + δ(v)) + αb D((x − 0)f˜(u) + (b − x)f˜(v), H) for any x ∈ [0, b], u ∈ [0, x] and v ∈ [x, b] such that the trapezoidal-type quadrature rule (x − 0)f˜(u) + (b − x)f˜(v) is δ−fine; (iii) ˜ D((β1 − 0)f˜(u) + (β2 − β1 )f˜(v) + (b − β2 )f˜(w), H) ≤ 4(δ(u) + δ(v) + δ(w)) + αb for anyR α, β ∈ [0, b], and uR ∈ [0, α], v ∈ [α, β], w ∈ [β, b], such that Simpson’s formula is δ−fine, where +∞ +∞ αb = k b f˜(x)dαkE 1 or b kf˜(x)kE 1 dα, and αb → 0 (b → +∞). Indeed, Z +∞ Z +∞ 2 ˜ αb = kf (x)kE 1 dα = sup { (1 − λ)e−x }, b

λ∈[0,1]

b

and αb → 0. 5. Conclusion We have discussed the numerical calculus of the fuzzy Henstock-Stieltjes integral for fuzzy-valued functions on [a, +∞). It is well known that the quadrature rules and numerical calculus are restricted when the distribution function of a random variable is unbounded, defined on (−∞, +∞) or have some kind of non-integrability in the previous papers, however, applying the methods proposed in this paper, the problems mentioned above are solved. It includes quadrature rules and the error estimates, such as the midpoint-type rule, trapezoidal-type rule, Simpson’s rule, δ− fine formula and their error estimates, and so on. 482

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References [1] S. Nanda, On fuzzy integrals, Fuzzy Sets and Systems 32 (1989) 95-101. [2] Y.H. Feng, Fuzzy-valued mappings with finite variation, fuzzy-valued measures and fuzzy-valued Lebesgue-Stieltjes integrals, Fuzzy Sets and Systems 121 (2001) 227-236. [3] X.K. Ren, C.X. Wu, Z.G. Zhu, A new kind of fuzzy Riemann-Stieltjes integral, in: X.Z. Wang, D. Yeung, X.L. Wang (Eds.), Proc. 5th Int. Conf. on Machine Learning and Cybernetics, ICMLC 2006, Dalian, China, 2006, pp. 1885-1888. [4] X.K. Ren, The non-additive measure and the fuzzy Riemann-Stieltjes integral, Ph.D dissertation, Harbin Institute of Technology, 2008. [5] H.C. Wu, The fuzzy Riemann-Stieltjes integral. International Journal of Uncertainty, Fuzziness and Knowledge-Based Systems 6 (1998) 51-67. [6] D.A. Ralescu, Fuzzy random variables, Journal of Mathematics Analysis and Applications 114 (1986) 409-422. [7] Z.T. Gong, L. Wang, The numerical calculus of expectations of fuzzy random variables, Fuzzy Sets and Systems 158 (2007) 722-738. The [8] K.F. Duan, Henstock-Stieltjes integral for fuzzy-number-valued functions on a infinite interval, J. Comput. Anal. Appl.(accepted to publish) [9] P.Y. Lee, Lanzhou Lectures on Henstock Integration, World Scientific, Singapore, 1989. [10] B. Bede, S.G. Gal, Quadrature rules for integrals of fuzzy-number-valued functions, Fuzzy Sets and Systems 145 (2004) 359-380.

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CUBIC AND QUARTIC ρ-FUNCTIONAL INEQUALITIES IN FUZZY NORMED SPACES JOOHO ZHIANG, JEONGHUN CHU∗ , GEORGE A. ANASTASSIOU, AND CHOONKIL PARK∗ Abstract. In this paper, we solve the following cubic ρ-functional inequality N (f (2x + y) + f (2x − y) − 2f (x + y) − 2f (x − y) − 12f (x), t) (0.1)       y y + 4f x − − f (x + y) − f (x − y) − 6f (x) , t ≥ N ρ 4f x + 2 2 in fuzzy normed spaces, where ρ is a fixed real number with |ρ| < 2, and the following quartic ρ-functional inequality N (f (2x + y) + f (2x − y) − 4f (x + y) − 4f (x − y) − 24f (x) + 6f (y), t) (0.2)       y y ≥ N ρ 8f x + + 8f x − − 2f (x + y) − 2f (x − y) − 12f (x) + 3f (y) , t 2 2 in fuzzy normed spaces, where ρ is a fixed real number with |ρ| < 2. Using the fixed point method, we prove the Hyers-Ulam stability of the cubic ρ-functional inequality (0.1) and the quartic ρ-functional inequality (0.2) in fuzzy Banach spaces.

1. Introduction and preliminaries Katsaras [20] defined a fuzzy norm on a vector space to construct a fuzzy vector topological structure on the space. Some mathematicians have defined fuzzy norms on a vector space from various points of view [11, 24, 50]. In particular, Bag and Samanta [2], following Cheng and Mordeson [8], gave an idea of fuzzy norm in such a manner that the corresponding fuzzy metric is of Kramosil and Michalek type [23]. They established a decomposition theorem of a fuzzy norm into a family of crisp norms and investigated some properties of fuzzy normed spaces [3]. We use the definition of fuzzy normed spaces given in [2, 29, 30] to investigate the Hyers-Ulam stability of cubic ρ-functional inequalities and quartic ρ-functional inequalities in fuzzy Banach spaces. Definition 1.1. [2, 29, 30, 31] Let X be a real vector space. A function N : X × R → [0, 1] is called a fuzzy norm on X if for all x, y ∈ X and all s, t ∈ R, (N1 ) N (x, t) = 0 for t ≤ 0; (N2 ) x = 0 if and only if N (x, t) = 1 for all t > 0; t (N3 ) N (cx, t) = N (x, |c| ) if c 6= 0; (N4 ) N (x + y, s + t) ≥ min{N (x, s), N (y, t)}; (N5 ) N (x, ·) is a non-decreasing function of R and limt→∞ N (x, t) = 1. (N6 ) for x 6= 0, N (x, ·) is continuous on R. The pair (X, N ) is called a fuzzy normed vector space. The properties of fuzzy normed vector spaces and examples of fuzzy norms are given in [28, 29]. 2010 Mathematics Subject Classification. Primary 46S40, 39B52, 47H10, 39B62, 26E50, 47S40. Key words and phrases. fuzzy Banach space; cubic ρ-functional inequality; quartic ρ-functional inequality; fixed point method; Hyers-Ulam stability. ∗ Corresponding author.

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Definition 1.2. [2, 29, 30, 31] Let (X, N ) be a fuzzy normed vector space. A sequence {xn } in X is said to be convergent or converge if there exists an x ∈ X such that limn→∞ N (xn − x, t) = 1 for all t > 0. In this case, x is called the limit of the sequence {xn } and we denote it by N -limn→∞ xn = x. Definition 1.3. [2, 29, 30, 31] Let (X, N ) be a fuzzy normed vector space. A sequence {xn } in X is called Cauchy if for each ε > 0 and each t > 0 there exists an n0 ∈ N such that for all n ≥ n0 and all p > 0, we have N (xn+p − xn , t) > 1 − ε. It is well-known that every convergent sequence in a fuzzy normed vector space is Cauchy. If each Cauchy sequence is convergent, then the fuzzy norm is said to be complete and the fuzzy normed vector space is called a fuzzy Banach space. We say that a mapping f : X → Y between fuzzy normed vector spaces X and Y is continuous at a point x0 ∈ X if for each sequence {xn } converging to x0 in X, then the sequence {f (xn )} converges to f (x0 ). If f : X → Y is continuous at each x ∈ X, then f : X → Y is said to be continuous on X (see [3]). The stability problem of functional equations originated from a question of Ulam [49] concerning the stability of group homomorphisms. Hyers [15] gave a first affirmative partial answer to the question of Ulam for Banach spaces. Hyers’ Theorem was generalized by Aoki [1] for additive mappings and by Th.M. Rassias [41] for linear mappings by considering an unbounded Cauchy difference. A generalization of the Th.M. Rassias theorem was obtained by G˘avruta [12] by replacing the unbounded Cauchy difference by a general control function in the spirit of Th.M. Rassias’ approach. The stability problems of several functional equations have been extensively investigated by a number of authors and there are many interesting results concerning this problem (see [7, 16, 19, 21, 22, 25, 37, 38, 39, 43, 44, 45, 46, 47, 48]). In [18], Jun and Kim considered the following cubic functional equation f (2x + y) + f (2x − y) = 2f (x + y) + 2f (x − y) + 12f (x).

(1.1)

It is easy to show that the function f (x) = x3 satisfies the functional equation (1.1), which is called a cubic functional equation and every solution of the cubic functional equation is said to be a cubic mapping. In [26], Lee et al. considered the following quartic functional equation f (2x + y) + f (2x − y) = 4f (x + y) + 4f (x − y) + 24f (x) − 6f (y).

(1.2)

It is easy to show that the function f (x) = x4 satisfies the functional equation (1.2), which is called a quartic functional equation and every solution of the quartic functional equation is said to be a quartic mapping. Gil´anyi [13] showed that if f satisfies the functional inequality k2f (x) + 2f (y) − f (x − y)k ≤ kf (x + y)k

(1.3)

then f satisfies the Jordan-von Neumann functional equation 2f (x) + 2f (y) = f (x + y) + f (x − y). See also [42]. Fechner [10] and Gil´anyi [14] proved the Hyers-Ulam stability of the functional inequality (1.3). Park, Cho and Han [36] investigated the Cauchy additive functional inequality kf (x) + f (y) + f (z)k ≤ kf (x + y + z)k

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and the Cauchy-Jensen additive functional inequality

 

x+y

kf (x) + f (y) + 2f (z)k ≤ 2f +z

2

(1.5)

and proved the Hyers-Ulam stability of the functional inequalities (1.4) and (1.5) in Banach spaces. Park [34, 35] defined additive ρ-functional inequalities and proved the Hyers-Ulam stability of the additive ρ-functional inequalities in Banach spaces and non-Archimedean Banach spaces. We recall a fundamental result in fixed point theory. Let X be a set. A function d : X × X → [0, ∞] is called a generalized metric on X if d satisfies (1) d(x, y) = 0 if and only if x = y; (2) d(x, y) = d(y, x) for all x, y ∈ X; (3) d(x, z) ≤ d(x, y) + d(y, z) for all x, y, z ∈ X. Theorem 1.4. [4, 9] Let (X, d) be a complete generalized metric space and let J : X → X be a strictly contractive mapping with Lipschitz constant L < 1. Then for each given element x ∈ X, either d(J n x, J n+1 x) = ∞ for all nonnegative integers n or there exists a positive integer n0 such that (1) d(J n x, J n+1 x) < ∞, ∀n ≥ n0 ; n (2) the sequence {J x} converges to a fixed point y ∗ of J; (3) y ∗ is the unique fixed point of J in the set Y = {y ∈ X | d(J n0 x, y) < ∞}; 1 (4) d(y, y ∗ ) ≤ 1−L d(y, Jy) for all y ∈ Y . In 1996, G. Isac and Th.M. Rassias [17] were the first to provide applications of stability theory of functional equations for the proof of new fixed point theorems with applications. By using fixed point methods, the stability problems of several functional equations have been extensively investigated by a number of authors (see [5, 6, 28, 32, 33, 39, 40]). In Section 2, we solve the cubic ρ-functional inequality (0.1) and prove the Hyers-Ulam stability of the cubic ρ-functional inequality (0.1) in fuzzy Banach spaces by using the fixed point method. In Section 3, we solve the quartic ρ-functional inequality (0.2) and prove the Hyers-Ulam stability of the quartic ρ-functional inequality (0.2) in fuzzy Banach spaces by using the fixed point method. Throughout this paper, assume that ρ is a fixed real number with |ρ| < 2. 2. Cubic ρ-functional inequality (0.1) In this section, we solve and investigate the cubic ρ-functional inequality (0.1) in fuzzy Banach spaces. Lemma 2.1. Let (Y, N ) be a fuzzy normed vector space. Let f : X → Y be a mapping such that N ((f (2x + y) + f (2x − y) − 2f (x + y) − 2f (x − y) − 12f (x), t)       y y ≥ N ρ 4f x + + 4f x − − f (x + y) − f (x − y) − 6f (x) , t 2 2 for all x, y ∈ X and all t > 0. Then f : X → Y is cubic.

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Proof. Assume that f : X → Y satisfies (2.1). Letting x = y = 0 in (2.1), we get N (−14f (0), t) ≥ 1. So f (0) = 0. Letting y = 0 in (2.1), we get N (2f (2x) − 16f (x), t) ≥ 1 and so f (2x) = 8f (x) for all x ∈ X. Thus x 1 f = f (x) (2.2) 2 8 for all x ∈ X. It follows from (2.1) and (2.2) that N (f (2x + y) + f (2x − y) − 2f (x + y) − 2f (x − y) − 12f (x), t)       y y ≥ N ρ 4f x + + 4f x − − f (x + y) − f (x − y) − 6f (x) , t 2 2   2t = N f (2x + y) + f (2x − y) − 2f (x + y) − 2f (x − y) − 12f (x), |ρ| for all t > 0. By (N5 ) and (N6 ), f (2x + y) + f (2x − y) = 2f (x + y) + 2f (x − y) + 12f (x) for all x, y ∈ X, since |ρ| < 2. So f : X → Y is cubic.



We prove the Hyers-Ulam stability of the cubic ρ-functional inequality (2.1) in fuzzy Banach spaces. Theorem 2.2. Let ϕ : X 2 → [0, ∞) be a function such that there exists an L < 1 with L ϕ(x, y) ≤ ϕ(2x, 2y) 8 for all x, y ∈ X. Let f : X → Y be a mapping satisfying N (f (2x + y) + f (2x − y) − 2f (x + y) − 2f (x − y) − 12f (x), t) (2.3)         y y t ≥ min(N ρ 4f x + + 4f x − − f (x + y) − f (x − y) − 6f (x) , t , )) 2 2 t + ϕ(x, y)
for all x, y ∈ X and all t > 0. Then C(x) := N -limn→∞ 8n f 2xn exists for each x ∈ X and defines a cubic mapping C : X → Y such that (16 − 16L)t N (f (x) − C(x), t) ≥ (2.4) (16 − 16L)t + Lϕ(x, 0) for all x ∈ X and all t > 0. Proof. Letting y = 0 in (2.3), we get N (2f (2x) − 16f (x), t) ≥ and so N f (x) − 8f

x 2





, 2t ≥

t t+ϕ( x2 ,0)

t t + ϕ(x, 0)

(2.5)

for all x ∈ X.

Consider the set S := {g : X → Y } and introduce the generalized metric on S:   t d(g, h) = inf µ ∈ R+ : N (g(x) − h(x), µt) ≥ , ∀x ∈ X, ∀t > 0 , t + ϕ(x, 0) where, as usual, inf φ = +∞. It is easy to show that (S, d) is complete (see [27, Lemma 2.1]).

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Now we consider the linear mapping J : S → S such that x Jg(x) := 8g 2 for all x ∈ X. Let g, h ∈ S be given such that d(g, h) = ε. Then N (g(x) − h(x), εt) ≥

t t + ϕ(x, 0)

for all x ∈ X and all t > 0. Hence     x L  x N (Jg(x) − Jh(x), Lεt) = N 8g − 8h , Lεt = N g −h , εt 2 2 2 2 8 

≥

x

Lt 8 Lt 8

+ϕ

x 2,0

x


≥

Lt 8

+

Lt 8 L 8 ϕ(x, 0)

=

t t + ϕ(x, 0)

for all x ∈ X and all t > 0. So d(g, h) = ε implies that d(Jg, Jh) ≤ Lε. This means that d(Jg, Jh) ≤ Ld(g, h) for all g, h ∈ S. It follows from (2.5) that  N

f (x) − 8f

x L  t , t ≥ 2 16 t + ϕ(x, 0)

L for all x ∈ X and all t > 0. So d(f, Jf ) ≤ 16 . By Theorem 1.4, there exists a mapping C : X → Y satisfying the following: (1) C is a fixed point of J, i.e., x 1 C = C(x) 2 8 for all x ∈ X. The mapping C is a unique fixed point of J in the set

(2.6)

M = {g ∈ S : d(f, g) < ∞}. This implies that C is a unique mapping satisfying (2.6) such that there exists a µ ∈ (0, ∞) satisfying t N (f (x) − C(x), µt) ≥ t + ϕ(x, 0) for all x ∈ X; (2) d(J n f, C) → 0 as n → ∞. This implies the equality x N - lim 8n f n = C(x) n→∞ 2 for all x ∈ X; 1 (3) d(f, C) ≤ 1−L d(f, Jf ), which implies the inequality d(f, C) ≤

L . 16 − 16L

This implies that the inequality (2.4) holds.

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By (2.3),             x  2x + y 2x − y x+y x−y n n N 8 f +f − 2f − 2f − 12f n ,8 t 2n 2n 2n 2n 2              x  x − y2 x + y2 x+y x−y n n + 4f −f −f − 6f n ,8 t , ≥ min N 8 ρ 4f 2n 2n 2n 2n 2  t t + ϕ( 2xn , 2yn ) for all x, y ∈ X, all t > 0 and all n ∈ N. So            x   2x + y 2x − y x+y x−y n N 8 f ,t +f − 2f − 2f − 12f n 2n 2n 2n 2n 2             x   x + y2 x − y2 x+y x−y n ≥ min N 8 ρ 4f + 4f −f −f − 6f n ,t , 2n 2n 2n 2n 2 ) t 8n

Since limn→∞

t 8n t Ln + 8n ϕ(x,y) 8n

+

t 8n Ln 8n ϕ(x, y)

= 1 for all x, y ∈ X and all t > 0,

N (C(2x + y) + C(2x − y) − 2C(x + y) − 2C(x − y) − 12C(x), t) y y ≥ N (ρ(4C(x + ) + 4C(x − ) − C(x + y) − C(x − y) − 6C(x), t) 2 2 for all x, y ∈ X and all t > 0. By Lemma 2.1, the mapping C : X → Y is cubic, as desired.



Corollary 2.3. Let θ ≥ 0 and let p be a real number with p > 3. Let X be a normed vector space with norm k · k. Let f : X → Y be a mapping satisfying N (f (2x + y) + f (2x − y) − 2f (x + y) − 2f (x − y) − 12f (x), t) (2.7)    n    y y + 4f x − − f (x + y) − f (x − y) − 6f (x) , t , ≥ min N ρ 4f x + 2 2  t t + θ(kxkp + kykp ) for all x, y ∈ X and all t > 0. Then C(x) := N -limn→∞ 8n f ( 2xn ) exists for each x ∈ X and defines a cubic mapping C : X → Y such that N (f (x) − C(x), t) ≥

2(2p − 8)t 2(2p − 8)t + θkxkp

for all x ∈ X and all t > 0. Proof. The proof follows from Theorem 2.2 by taking ϕ(x, y) := θ(kxkp + kykp ) for all x, y ∈ X. Then we can choose L = 23−p , and we get the desired result.  Theorem 2.4. Let ϕ : X 2 → [0, ∞) be a function such that there exists an L < 1 with x y  ϕ(x, y) ≤ 8Lϕ , 2 2

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for all x, y ∈ X. Let f : X → Y be a mapping satisfying (2.3). Then C(x) := N -limn→∞ exists for each x ∈ X and defines a cubic mapping C : X → Y such that (16 − 16L)t N (f (x) − C(x), t) ≥ (16 − 16L)t + ϕ(x, 0) for all x ∈ X and all t > 0.

1 8n f

(2n x)

(2.8)

Proof. Let (S, d) be the generalized metric space defined in the proof of Theorem 2.2. It follows from (2.5) that   1 1 t N f (x) − f (2x), t ≥ 8 16 t + ϕ(x, 0) for all x ∈ X and all t > 0. Now we consider the linear mapping J : S → S such that 1 Jg(x) := g (2x) 8 1 for all x ∈ X. Then d(f, Jf ) ≤ 16 . Hence d(f, C) ≤

1 , 16 − 16L

which implies that the inequality (2.8) holds. The rest of the proof is similar to the proof of Theorem 2.2.



Corollary 2.5. Let θ ≥ 0 and let p be a real number with 0 < p < 3. Let X be a normed vector space with norm k · k. Let f : X → Y be a mapping satisfying (2.7). Then C(x) := N limn→∞ 81n f (2n x) exists for each x ∈ X and defines a cubic mapping C : X → Y such that N (f (x) − C(x), t) ≥

2(8 − 2p )t 2(8 − 2p )t + θkxkp

for all x ∈ X and all t > 0. Proof. The proof follows from Theorem 2.4 by taking ϕ(x, y) := θ(kxkp + kykp ) for all x, y ∈ X. Then we can choose L = 2p−3 , and we get the desired result.  3. Quartic ρ-functional inequality (0.2) In this section, we solve and investigate the quartic ρ-functional inequality (0.2) in fuzzy Banach spaces. Lemma 3.1. Let (Y, N ) be a fuzzy normed vector space. A mapping f : X → Y satisfies f (0) = 0 and N (f (2x + y) + f (2x − y) − 4f (x + y) − 4f (x − y) − 24f (x) + 6f (y), t) (3.1)       y y ≥ N ρ 8f x + + 8f x − − 2f (x + y) − 2f (x − y) − 12f (x) + 3f (y) , t 2 2 for all x, y ∈ X and all t > 0. Then f is quartic. Proof. Assume that f : X → Y satisfies (3.1). Letting y = 0 in (3.1), we get N (2f (2x) − 32f (x), t) ≥ N (0, t) = 1 and so x 1 f = f (x) 2 16 for all x ∈ X.

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It follows from (3.1) and (3.2) that N (f (2x + y) + f (2x − y) − 4f (x + y) − 4f (x − y) − 24f (x) + 6f (y), t)       y y ≥ N ρ 8f x + + 8f x − − 2f (x + y) − 2f (x − y) − 12f (x) + 3f (y) , t 2 2  ρ (f (2x + y) + f (2x − y) − 4f (x + y) − 4f (x − y) − 24f (x) + 6f (y)) , t =N 2  2t = N f (2x + y) + f (2x − y) − 4f (x + y) − 4f (x − y) − 24f (x) + 6f (y), |ρ| for all t > 0 and all x, y ∈ X. By (N5 ) and (N6 ), N (f (2x + y) + f (2x − y) − 4f (x + y) − 4f (x − y) − 24f (x) + 6f (y), t) = 1 for all t > 0 and all x, y ∈ X. It follows from (N2 ) that f (2x + y) + f (2x − y) = 4f (x + y) + 4f (x − y) + 24f (x) − 6f (y) for all x, y ∈ X.



We prove the Hyers-Ulam stability of the quartic ρ-functional inequality (3.1) in fuzzy Banach spaces. Theorem 3.2. Let ϕ : X 2 → [0, ∞) be a function such that there exists an L < 1 with L ϕ(x, y) ≤ ϕ (2x, 2y) 16 for all x, y ∈ X. Let f : X → Y be a mapping satisfying f (0) = 0 and N (f (2x + y) + f (2x − y) − 4f (x + y) − 4f (x − y) − 24f (x) + 6f (y), t) (3.3) n       y y ≥ min N ρ 8f x + + 8f x − − 2f (x + y) − 2f (x − y) − 12f (x) + 3f (y) , t , 2  2 t t + ϕ(x, 0)
for all x, y ∈ X and all t > 0. Then Q(x) := N -limn→∞ 16n f 2xn exists for each x ∈ X and defines a quartic mapping Q : X → Y such that N (f (x) − Q(x), t) ≥

(32 − 32L)t (32 − 32L)t + Lϕ(x, 0)

(3.4)

for all x ∈ X and all t > 0. Proof. Let (S, d) be the generalized metric space defined in the proof of Theorem 2.2. Letting y = 0 in (3.3), we get t N (2f (2x) − 32f (x) , t) = N (32f (x) − 2f (2x), t) ≥ t + ϕ(x, 0)

(3.5)

for all x ∈ X. Now we consider the linear mapping J : S → S such that x Jg(x) := 16g 2 for all x ∈ X. Let g, h ∈ S be given such that d(g, h) = ε. Then t N (g(x) − h(x), εt) ≥ t + ϕ(x, 0)

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for all x ∈ X and all t > 0. Hence     x L  x N (Jg(x) − Jh(x), Lεt) = N 16g − 16h , Lεt = N g −h , εt 2 2 2 2 16 x



≥

Lt 16 Lt 16

+ϕ

x 2,0

x


≥

Lt 16

+

Lt 16 L 16 ϕ(x, 0)

=

t t + ϕ(x, 0)

for all x ∈ X and all t > 0. So d(g, h) = ε implies that d(Jg, Jh) ≤ Lε. This means that d(Jg, Jh) ≤ Ld(g, h) for all g, h ∈ S. It follows from (3.5) that  x L  t N f (x) − 16f , t ≥ 2 32 t + ϕ(x, 0) L for all x ∈ X and all t > 0. So d(f, Jf ) ≤ 32 . By Theorem 1.4, there exists a mapping Q : X → Y satisfying the following: (1) Q is a fixed point of J, i.e., x 1 = Q(x) Q 2 16

(3.6)

for all x ∈ X. Since f : X → Y is even, Q : X → Y is an even mapping. The mapping Q is a unique fixed point of J in the set M = {g ∈ S : d(f, g) < ∞}. This implies that Q is a unique mapping satisfying (3.6) such that there exists a µ ∈ (0, ∞) satisfying t N (f (x) − Q(x), µt) ≥ t + ϕ(x, 0) for all x ∈ X; (2) d(J n f, Q) → 0 as n → ∞. This implies the equality x N - lim 16n f n = Q(x) n→∞ 2 for all x ∈ X; (3) d(f, Q) ≤

1 1−L d(f, Jf ),

which implies the inequality d(f, Q) ≤

L . 32 − 32L

This implies that the inequality (3.4) holds. By the same method as in the proof of Theorem 2.2, it follows from (3.3) that N (Q(2x + y) + Q(2x − y) − 4Q(x + y) − 4Q(x − y) − 24Q(x) + 6Q(y), t)       y y ≥ N ρ 8Q x + + 8Q x − − 2Q (x + y) − 2Q (x − y) − 12Q (x) + 3Q (y) , t 2 2 for all x, y ∈ X, all t > 0 and all n ∈ N. By Lemma 3.1, the mapping Q : X → Y is quartic.

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Corollary 3.3. Let θ ≥ 0 and let p be a real number with p > 4. Let X be a normed vector space with norm k · k. Let f : X → Y be a mapping satisfying f (0) = 0 and N (f (2x + y) + f (2x − y) − 4f (x + y) − 4f (x − y) − 24f (x) + 6f (y), t) (3.7) n       y y ≥ min N ρ 8f x + + 8f x − − 2f (x + y) − 2f (x − y) − 12f (x) + 3f (y) , t , 2 2  t t + θ(kxkp + kykp for all x, y ∈ X and all t > 0. Then Q(x) := N -limn→∞ 16n f ( 2xn ) exists for each x ∈ X and defines a quartic mapping Q : X → Y such that 2(2p − 16)t N (f (x) − Q(x), t) ≥ 2(2p − 16)t + θkxkp for all x ∈ X and all t > 0. Proof. The proof follows from Theorem 3.2 by taking ϕ(x, y) := θ(kxkp + kykp ) for all x, y ∈ X. Then we can choose L = 24−p , and we get the desired result.  Theorem 3.4. Let ϕ : X 2 → [0, ∞) be a function such that there exists an L < 1 with x y  ϕ(x, y) ≤ 16Lϕ , 2 2 for all x, y ∈ X. Let f : X → Y be a mapping satisfying f (0) = 0 and (3.3). Then Q(x) := N limn→∞ 161n f (2n x) exists for each x ∈ X and defines a quartic mapping Q : X → Y such that (32 − 32L)t (3.8) N (f (x) − Q(x), t) ≥ (32 − 32L)t + ϕ(x, 0) for all x ∈ X and all t > 0. Proof. Let (S, d) be the generalized metric space defined in the proof of Theorem 2.2. It follows from (3.5) that   1 1 t N f (x) − f (2x), t ≥ 16 32 t + ϕ(x, 0) for all x ∈ X and all t > 0. Now we consider the linear mapping J : S → S such that 1 Jg(x) := g (2x) 16 1 for all x ∈ X. Then d(f, Jf ) ≤ 32 . Hence d(f, Q) ≤

1 , 32 − 32L

which implies that the inequality (3.8) holds. The rest of the proof is similar to the proof of Theorem 3.2.



Corollary 3.5. Let θ ≥ 0 and let p be a real number with 0 < p < 4. Let X be a normed vector space with norm k · k. Let f : X → Y be a mapping satisfying f (0) = 0 and (3.7). Then Q(x) := N -limn→∞ 161n f (2n x) exists for each x ∈ X and defines a quartic mapping Q : X → Y such that 2(16 − 2p )t N (f (x) − Q(x), t) ≥ 2(16 − 2p )t + θkxkp

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for all x ∈ X. Proof. The proof follows from Theorem 3.4 by taking ϕ(x, y) := θ(kxkp + kykp ) for all x, y ∈ X. Then we can choose L = 2p−4 , and we get the desired result.  References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24] [25] [26] [27] [28] [29]

T. Aoki, On the stability of the linear transformation in Banach spaces, J. Math. Soc. Japan 2 (1950), 64–66. T. Bag and S.K. Samanta, Finite dimensional fuzzy normed linear spaces, J. Fuzzy Math. 11 (2003), 687–705. T. Bag and S.K. Samanta, Fuzzy bounded linear operators, Fuzzy Sets and Systems 151 (2005), 513–547. L. C˘ adariu and V. Radu, Fixed points and the stability of Jensen’s functional equation, J. Inequal. Pure Appl. Math. 4, no. 1, Art. ID 4 (2003). L. C˘ adariu and V. Radu, On the stability of the Cauchy functional equation: a fixed point approach, Grazer Math. Ber. 346 (2004), 43–52. L. C˘ adariu and V. Radu, Fixed point methods for the generalized stability of functional equations in a single variable, Fixed Point Theory and Applications 2008, Art. ID 749392 (2008). I. Chang and Y. Lee, Additive and quadratic type functional equation and its fuzzy stability, Results Math. 63 (2013), 717–730. S.C. Cheng and J.M. Mordeson, Fuzzy linear operators and fuzzy normed linear spaces, Bull. Calcutta Math. Soc. 86 (1994), 429–436. J. Diaz and B. Margolis, A fixed point theorem of the alternative for contractions on a generalized complete metric space, Bull. Amer. Math. Soc. 74 (1968), 305–309. W. Fechner, Stability of a functional inequalities associated with the Jordan-von Neumann functional equation, Aequationes Math. 71 (2006), 149–161. C. Felbin, Finite dimensional fuzzy normed linear spaces, Fuzzy Sets and Systems 48 (1992), 239–248. P. G˘ avruta, A generalization of the Hyers-Ulam-Rassias stability of approximately additive mappings, J. Math. Anal. Appl. 184 (1994), 431–436. A. Gil´ anyi, Eine zur Parallelogrammgleichung ¨ aquivalente Ungleichung, Aequationes Math. 62 (2001), 303– 309. A. Gil´ anyi, On a problem by K. Nikodem, Math. Inequal. Appl. 5 (2002), 707–710. D.H. Hyers, On the stability of the linear functional equation, Proc. Nat. Acad. Sci. U.S.A. 27 (1941), 222–224. D.H. Hyers, G. Isac and Th.M. Rassias, Stability of Functional Equations in Several Variables, Birkh¨ auser, Basel, 1998. G. Isac and Th.M. Rassias, Stability of ψ-additive mappings: Appications to nonlinear analysis, Internat. J. Math. Math. Sci. 19 (1996), 219–228. K. Jun and H. Kim, The generalized Hyers-Ulam-Rassias stability of a cubic functional equation, J. Math. Anal. Appl. 274 (2002), 867–878. S. Jung, Hyers-Ulam-Rassias Stability of Functional Equations in Mathematical Analysis, Hadronic Press lnc., Palm Harbor, Florida, 2001. A.K. Katsaras, Fuzzy topological vector spaces II, Fuzzy Sets and Systems 12 (1984), 143–154. H. Kim, M. Eshaghi Gordji, A. Javadian and I. Chang, Homomorphisms and derivations on unital C ∗ -algebras related to Cauchy-Jensen functional inequality, J. Math. Inequal. 6 (2012), 557–565. H. Kim, J. Lee and E. Son, Approximate functional inequalities by additive mappings, J. Math. Inequal. 6 (2012), 461–471. I. Kramosil and J. Michalek, Fuzzy metric and statistical metric spaces, Kybernetica 11 (1975), 326–334. S.V. Krishna and K.K.M. Sarma, Separation of fuzzy normed linear spaces, Fuzzy Sets and Systems 63 (1994), 207–217. J. Lee, C. Park and D. Shin, An AQCQ-functional equation in matrix normed spaces, Results Math. 27 (2013), 305–318. S. Lee, S. Im and I. Hwang, Quartic functional equations, J. Math. Anal. Appl. 307 (2005), 387–394. D. Mihet¸ and V. Radu, On the stability of the additive Cauchy functional equation in random normed spaces, J. Math. Anal. Appl. 343 (2008), 567–572. M. Mirzavaziri and M.S. Moslehian, A fixed point approach to stability of a quadratic equation, Bull. Braz. Math. Soc. 37 (2006), 361–376. A.K. Mirmostafaee, M. Mirzavaziri and M.S. Moslehian, Fuzzy stability of the Jensen functional equation, Fuzzy Sets and Systems 159 (2008), 730–738.

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[30] A.K. Mirmostafaee and M.S. Moslehian, Fuzzy versions of Hyers-Ulam-Rassias theorem, Fuzzy Sets and Systems 159 (2008), 720–729. [31] A.K. Mirmostafaee and M.S. Moslehian, Fuzzy approximately cubic mappings, Inform. Sci. 178 (2008), 3791– 3798. [32] C. Park, Fixed points and Hyers-Ulam-Rassias stability of Cauchy-Jensen functional equations in Banach algebras, Fixed Point Theory and Applications 2007, Art. ID 50175 (2007). [33] C. Park, Generalized Hyers-Ulam-Rassias stability of quadratic functional equations: a fixed point approach, Fixed Point Theory and Applications 2008, Art. ID 493751 (2008). [34] C. Park, Additive ρ-functional inequalities and equations, J. Math. Inequal. 9 (2015), 17–26. [35] C. Park, Additive ρ-functional inequalities in non-Archimedean normed spaces, J. Math. Inequal. 9 (2015), 397–407. [36] C. Park, Y. Cho and M. Han, Stability of functional inequalities associated with Jordan-von Neumann type additive functional equations, J. Inequal. Appl. 2007, Art. ID 41820 (2007). [37] C. Park, K. Ghasemi, S. G. Ghaleh and S. Jang, Approximate n-Jordan ∗-homomorphisms in C ∗ -algebras, J. Comput. Anal. Appl. 15 (2013), 365-368. [38] C. Park, A. Najati and S. Jang, Fixed points and fuzzy stability of an additive-quadratic functional equation, J. Comput. Anal. Appl. 15 (2013), 452–462. [39] C. Park and Th.M. Rassias, Fixed points and generalized Hyers-Ulam stability of quadratic functional equations, J. Math. Inequal. 1 (2007), 515–528. [40] V. Radu, The fixed point alternative and the stability of functional equations, Fixed Point Theory 4 (2003), 91–96. [41] Th.M. Rassias, On the stability of the linear mapping in Banach spaces, Proc. Amer. Math. Soc. 72 (1978), 297–300. [42] J. R¨ atz, On inequalities associated with the Jordan-von Neumann functional equation, Aequationes Math. 66 (2003), 191–200. ˇ ankov´ [43] L. Reich, J. Sm´ıtal and M. Stef´ a, Singular solutions of the generalized Dhombres functional equation, Results Math. 65 (2014), 251–261. [44] S. Schin, D. Ki, J. Chang and M. Kim, Random stability of quadratic functional equations: a fixed point approach, J. Nonlinear Sci. Appl. 4 (2011), 37–49. [45] S. Shagholi, M. Bavand Savadkouhi and M. Eshaghi Gordji, Nearly ternary cubic homomorphism in ternary Fr´echet algebras, J. Comput. Anal. Appl. 13 (2011), 1106–1114. [46] S. Shagholi, M. Eshaghi Gordji and M. Bavand Savadkouhi, Stability of ternary quadratic derivation on ternary Banach algebras, J. Comput. Anal. Appl. 13 (2011), 1097–1105. [47] D. Shin, C. Park and Sh. Farhadabadi, On the superstability of ternary Jordan C ∗ -homomorphisms, J. Comput. Anal. Appl. 16 (2014), 964–973. [48] D. Shin, C. Park and Sh. Farhadabadi, Stability and superstability of J ∗ -homomorphisms and J ∗ -derivations for a generalized Cauchy-Jensen equation, J. Comput. Anal. Appl. 17 (2014), 125–134. [49] S. M. Ulam, A Collection of the Mathematical Problems, Interscience Publ. New York, 1960. [50] J.Z. Xiao and X.H. Zhu, Fuzzy normed spaces of operators and its completeness, Fuzzy Sets and Systems 133 (2003), 389–399. Jooho Zhiang, Jeonghun Chu Mathematics Branch, Seoul Science High School, Seoul 110-530, Korea E-mail address: [email protected] George A. Anastassiou Department of Mathematical Sciences, University of Memphis, Memphis, TN 38152, USA E-mail address: [email protected] Choonkil Park Research Institute for Natural Sciences, Hanyang University, Seoul 133-791, Korea E-mail address: [email protected]

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A RIGHT PARALLELISM RELATION FOR MAPPINGS TO POSETS HEE SIK KIM, J. NEGGERS AND KEUM SOOK SO∗

Abstract. In this paper, we study mappings f, g : X → P , where P is a poset and X is a set, under the relation f || g, of right parallelism, f (a) ≤ f (b) implies g(a) ≤ g(b), which is reflexive and transitive but not necessarily symmetric. We prove several results of the type: if f has property P and f || g, then g has property P as well, or of the converse type. Doing so permits us to observe several conditions on mappings and/or groupoids (X, ∗), upon which mappings may act in particular ways, which are of interest in their own right also. The special case f (x) = x with f || g yielding increasing/non-decreasing mappings g : X → P brings into focus a number of well-known situations seen from a different perspective.

1. Introduction. Y. Imai and K. Is´eki introduced two classes of abstract algebras: BCK-algebras and BCI-algebras ([4, 5]). In the study of groupoids (X, ∗) defined on a set X, it has also proven useful to investigate the semigroups (Bin(X), 2) where Bin(X) is the set of all binary systems (groupoids) (X, ∗) along with an associative product operation (X, ∗)2(X, •) = (X, 2) such that x2y = (x ∗ y) • (y ∗x) for all x, y ∈ X. Thus, e.g., it becomes possible to recognize that the left-zero-semigroup (X, ∗) with x ∗ y = x for all x, y ∈ X acts as the identity of this semigroup ([2]). H. F. Fayoumi ([1]) introduced the notion of the center ZBin(X) in the semigroup Bin(X) of all binary systems on a set X, and showed that a groupoid (X, •) ∈ ZBin(X) if and only if it is a locally-zero groupoid.

2010 Mathematics Subject Classification. 20N02, 03G25, 06A06. Key words and phrases. (left, right)-parallel, right-parallel-property, left(shrinking, expanding), Bin(X), groupoid parallel. ∗ Correspondence: Tel.: +82 33 248 2011, Fax: +82 33 256 2011 (K. S. So). 1

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H. S. Kim, J. Neggers and K. S. So

2

In this paper, we study mappings f, g : X → P where P is a poset and relations of the type f || g, i.e., f is right-(left-) parallel to g provided that f (a) ≤ f (b) implies g(a) ≤ g(b) as well, for all a, b ∈ X, where this condition applies. Since no assumptions about any order relation on X are made, this relation is a generalization of the special case X = P and f (x) = x, where a ≤ b implies g(a) ≤ g(b), i.e., g is an order-preserving mapping. Even in this most general format it is possible to extract information concerning properties of ||, i.e., f || f , and f || g, g || h implies f || h and the fact that f || g does not imply g || f , to demonstrate the one-sided-ness of f || g. At the same time through the introduction of the groupoid structures (X, ∗) as elements of (Bin(X), 2), the semigroup of binary systems (groupoids) on X, mappings f may acquire many different kinds of properties, such as f (x ∗ y) ≤ f (x) for all x, y ∈ X (left shrinking), for example, which then implies g(x ∗ y) ≤ g(x) for all x, y ∈ X, so that this property is preserved by parallelism. If P = [0, 1] with the usual order, then f, g : X → P yields the mappings f, g as fuzzy subsets of X and then the condition f (x ∗ y) ≥ min{f (x), f (y)} implies that if f || g, then g(x ∗ y) ≥ min{g(x), g(y)} as well, i.e., if f is a fuzzy subgroupoid of (X, ∗) and f || g, then g is a fuzzy subgroupoid also. From these examples it should be clear that many other similar conclusions can be obtained in this setting, several of which we have provided in the following. 2. Preliminaries. Let (X, 0, Γ(q) 0 (t − s)1−q

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3

provided the integral exists. Definition 2.3 The Riemann-Liouville fractional derivative of order q for a continuous function g(t) is defined by  n Z t 1 d g(s) Dq g(t) = ds, n = [q] + 1, Γ(n − q) dt (t − s)q−n+1 0 provided the right hand side is pointwise defined on (0, ∞). Lemma 2.4 For a given g ∈ C([0, 1], R) the unique solution of the boundary value problem  c q D x(t) = g(t), 0 < t < 1, 1 < q ≤ 2, x(0) = βx(η), x(1) = αx(η), is given by 1 x(t) = Γ(q)

Z

t q−1

(t − s) 0

β + (α − β)t + ∆Γ(q)

Z

(β − 1)t − βη g(s)ds + ∆Γ(q)

Z

1

(1 − s)q−1 g(s)ds

0

(2.1)

η q−1

(η − s)

g(s)ds,

0 ≤ t ≤ 1,

0

where ∆ = 1 − β + (β − α)η 6= 0. Proof. For some constants c0 , c1 ∈ R, we have Z t (t − s)q−1 q g(s)ds − c0 − c1 t. (2.2) x(t) = I g(t) − c0 − c1 t = Γ(q) 0 Z η (η − s)q−1 We have x(0) = −c0 , x(η) = g(s)ds − c0 − c1 η and thus from the first boundary Γ(q) 0 condition we have Z η (η − s)q−1 (β − 1)c0 + βηc1 = β g(s)ds. (2.3) Γ(q) 0 Also from the second boundary condition we get Z η Z 1 (η − s)q−1 (1 − s)q−1 (α − 1)c0 + (αη − 1)c1 = α g(s)ds − g(s)ds. (2.4) Γ(q) Γ(q) 0 0 From (2.3), (2.4) we find c0 , c1 and substituting in (2.2) we obtain the solution (2.1).

3

2

Existence results-Differential Equations

In view of Lemma 2.4, we define an operator F : C → C, C = C([0, 1], R) by Z t 1 (F x)(t) = (t − s)q−1 f (t, x(s))ds Γ(q) 0 Z (β − 1)t − βη 1 + (1 − s)q−1 f (t, x(s))ds ∆Γ(q) 0 Z η β + (α − β)t + (η − s)q−1 f (t, x(s))ds, 0 ≤ t ≤ 1, ∆Γ(q) 0

509

(3.1)

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M. A. Darwish and S.K. Ntouyas For convenience, let us set λ1 =

1 sup |(β − 1)t − βη| , |∆| t∈[0,1]

and Λ=

3.1

λ2 =

1 sup |β + (α − β)t| |∆| t∈[0,1]

1 (1 + λ1 + λ2 η q ). Γ(q + 1)

(3.2)

Existence result via Banach’s fixed point theorem

Theorem 3.1 Assume that f : [0, 1] × R → R is a continuous function and satisfies the assumption (A1 ) |f (t, x) − f (t, y)| ≤ L|x − y|, ∀t ∈ [0, 1], L > 0, x, y ∈ R with L < 1/Λ, where Λ is given by (3.2). Then the boundary value problem (1.1) has a unique solution. ΛM , we show that F Bρ ⊂ Bρ , 1 − LΛ where Bρ = {x ∈ C : kxk ≤ ρ} and F defined in (3.1). For x ∈ Bρ , we have  Z t Z 1 1 λ1 q−1 k(F x)(t)k ≤ sup (t − s) |f (s, x(s))|ds + (1 − s)q−1 f (s, x(s))|ds Γ(q) 0 t∈[0,1] Γ(q) 0  Z η λ2 q−1 + (η − s) |f (s, x(s))|ds Γ(q) 0  Z t 1 (t − s)q−1 (|f (s, x(s)) − f (s, 0)| + |f (s, 0)|) kds ≤ sup Γ(q) 0 t∈[0,1] Z 1 λ1 + (1 − s)q−1 (|f (s, x(s)) − f (s, 0)| + |f (s, 0)|)ds Γ(q) 0  Z η λ2 q−1 + (η − s) (|f (s, x(s)) − f (s, 0)| + |f (s, 0)|)ds Γ(q) 0  Z t Z 1 1 λ1 (t − s)q−1 ds + (1 − s)q−1 ds ≤ (Lρ + M ) sup Γ(q) Γ(q) 0 0 t∈[0,1]  Z η λ2 + (η − s)q−1 ds Γ(q) 0 (Lρ + M ) (1 + λ1 + λ2 η q ) = (Lρ + M ) Λ ≤ ρ. ≤ Γ(q + 1)

Proof. Setting supt∈[0,1] |f (t, 0)| = M and choosing ρ ≥

Now, for x, y ∈ C and for each t ∈ [0, 1], we obtain  Z t 1 k(F x)(t) − (F y)(t)k ≤ sup (t − s)q−1 |f (t, x(s)) − f (s, y(s))|ds Γ(q) 0 t∈[0,1] Z 1 λ1 + (1 − s)q−1 |f (t, x(s)) − f (s, y(s))|ds Γ(q) 0

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5

η

 (η − s) |f (t, x(s)) − f (s, y(s))|ds 0  Z t Z 1 λ1 1 q−1 (t − s) ds + (1 − s)q−1 ds ≤ Lkx − yk sup Γ(q) 0 t∈[0,1] Γ(q) 0  Z η λ2 q−1 + (η − s) ds Γ(q) 0 L (1 + λ1 + λ1 η q ) kx − yk = LΛkx − yk, ≤ Γ(q + 1) λ2 + Γ(q)

Z

q−1

where Λ is given by (3.2). Observe that Λ depends only on the parameters involved in the problem. As L < 1/Λ, therefore F is a contraction. Thus, the conclusion of the theorem follows by the contraction mapping principle (Banach fixed point theorem). 2 Example 3.2 Consider the following generalized three-point fractional boundary value problem  |x| 1   , t ∈ [0, 1],  c D1/2 x(t) = 2 (t  +9) 1 + |x|   (3.3) 1 1 1   x(0) = x , x(1) = 2x .  2 4 4 Here, q = 3/2, β = 1/2, α = 2, η = 1/4, and f (t, x) =

1 |x| 1 . We find ∆ = , λ1 = 2 (t + 9) 1 + |x| 8

1 32 |x − y|, therefore, (A1 ) is satisfied with 5, λ2 = 16 and Λ = √ . As |f (t, x) − f (t, y)| ≤ 81 3 π 1 32 √ < 1. Thus, by the conclusion of Theorem 3.1, the boundary L= . Further, LΛ = 81 243 π value problem (3.3) has a unique solution on [0, 1].

3.2

Existence result via Krasnoselskii’s fixed point theorem

Theorem 3.3 (Krasnoselskii’s fixed point theorem)[24]. Let M be a closed, bounded, convex and nonempty subset of a Banach space X. Let A, B be the operators such that: (i) Ax+By ∈ M whenever x, y ∈ M ; (ii) A is compact and continuous; (iii) B is a contraction mapping. Then there exists z ∈ M such that z = Az + Bz. Theorem 3.4 Let f : [0, 1] × R → R be a continuous function and the assumption (A1 ) holds. In addition we assume that (A2 ) |f (t, x)| ≤ µ(t), ∀(t, x) ∈ [0, 1] × R, and µ ∈ C([0, 1], R+ ). If L (λ1 + λ2 η q ) < 1, Γ(q + 1)

(3.4)

then the boundary value problem (1.1) has at least one solution on [0, 1].

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Proof. Letting supt∈[0,1] |µ(t)| = kµk, we fix r≥

kµk (1 + λ1 + λ2 η q ), Γ(q + 1)

and consider Br = {x ∈ C : kxk ≤ r}. We define the operators P and Q on Br as Z t (t − s)q−1 (Px)(t) = f (s, x(s))ds, 0 ≤ t ≤ 1, Γ(q) 0 Z (β − 1)t − βη 1 (Qx)(t) = (1 − s)q−1 f (t, x(s))ds ∆Γ(q) 0 Z β + (α − β)t η + (η − s)q−1 f (t, x(s))ds, 0 ≤ t ≤ 1. ∆Γ(q) 0 kµk (1 + λ1 + λ2 η q ) ≤ r. Γ(q + 1) Thus, Px + Qy ∈ Br . It follows from the assumption (A1 ) together with (3.4) that Q is a contraction mapping. Continuity of f implies that the operator P is continuous. Also, kµk P is uniformly bounded on Br as kPxk ≤ . Now we prove the compactness of the Γ(q + 1) operator P. In view of (A1 ), we define sup(t,x)∈[0,1]×Br |f (t, x)| = f , and consequently we have

Z t1

1

[(t2 − s)q−1 − (t1 − s)q−1 ]f (t, x(s))ds k(Px)(t1 ) − (Px)(t2 )k = Γ(q) 0

Z t2

1 q−1 + (t2 − s) f (t, x(s))ds

Γ(q) t1 For x, y ∈ Br , we find that kPx + Qyk ≤

≤

f |2(t2 − t1 )q + tq1 − tq2 |, Γ(q + 1)

which is independent of x. Thus, P is equicontinuous. Hence, by the Arzel´a-Ascoli Theorem, P is compact on Br . Thus all the assumptions of Theorem 3.3 are satisfied. So the conclusion of Theorem 3.3 implies that the boundary value problem (1.1) has at least one solution on [0, 1]. 2

3.3

Existence result via Leray-Schauder Alternative

Theorem 3.5 (Nonlinear alternative for single valued maps)[19]. Let E be a Banach space, C a closed, convex subset of E, U an open subset of C and 0 ∈ U. Suppose that F : U → C is a continuous, compact (that is, F (U ) is a relatively compact subset of C) map. Then either (i) F has a fixed point in U , or (ii)]there is a u ∈ ∂U (the boundary of U in C) and λ ∈ (0, 1) with u = λF (u). Theorem 3.6 Let f : [0, 1] × R → R be a continuous function. Assume that: (A3 ) There exist a function p ∈ C([0, 1], R+ ), and ψ : R+ → R+ nondecreasing such that |f (t, x)| ≤ p(t)ψ(kxk), ∀(t, x) ∈ [0, 1] × R;

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(A4 ) There exists a constant M > 0 such that M > 1. kpkψ(M ) q {1 + λ1 + λ2 η } Γ(q + 1) Then the boundary value problem (1.1) has at least one solution on [0, 1]. Proof. Consider the operator F : C → C defined by (3.1). We show that F maps bounded sets into bounded sets in C([0, 1], R). For a positive number ρ, let Bρ = {x ∈ C([0, 1], R) : kxk ≤ ρ} be a bounded set in C([0, 1], R). Then, Z t Z 1 1 λ1 (t − s)q−1 |f (t, x(s))|ds + (1 − s)q−1 |f (t, x(s))|ds Γ(q) 0 Γ(q) 0 Z η λ2 + (η − s)q−1 |f (t, x(s))|ds Γ(q) 0 Z t Z 1 1 λ1 q−1 ≤ (t − s) p(s)ψ(kxk) ds + (1 − s)q−1 p(s)ψ(kxk) ds Γ(q) 0 Γ(q) 0 Z η λ2 + (η − s)q−1 p(s)ψ(kxk) ds Γ(q) 0 kpkψ(kxk) ≤ {1 + λ1 + λ2 η q } . Γ(q + 1)

|(F x)(t)| ≤

Hence kF xk ≤

kpkψ(ρ) {1 + λ1 + λ2 η q } . Γ(q + 1)

Next we show that F maps bounded sets into equicontinuous sets of C([0, 1], R). Let t0 , t00 ∈ [0, 1] with t0 < t00 and x ∈ Bρ , where Bρ is a bounded set of C([0, 1], R). Then we have Z t0 1 Z t00 1 |(F x)(t00 ) − (F x)(t0 )| = (t00 − s)q−1 f (t, x(s))ds − (t0 − s)q−1 f (t, x(s))ds Γ(q) 0 Γ(q) 0 Z 1 |(β − 1)||t00 − t0 | + (1 − s)q−1 f (t, x(s))ds ∆Γ(q) Z 0 |α − β||t00 − t0 | η + (η − s)q−1 f (t, x(s))ds ∆Γ(q) 0 Z 0 Z 00 kpkψ(ρ) t 00 kpkψ(ρ) t 00 q−1 0 q−1 ≤ |(t − s) − (t − s) |ds + (t − s)q−1 ds Γ(q) Γ(q) 0 t0 kpkψ(ρ)|(β − 1)||t00 − t0 | kpkψ(ρ)|α − β||t00 − t0 | + + . ∆Γ(q + 1) ∆Γ(q + 1) Obviously the right hand side of the above inequality tends to zero independently of x ∈ Bρ as t00 − t0 → 0. As F satisfies the above assumptions, therefore it follows by the Arzel´a-Ascoli theorem that F : C([0, 1], R) → C([0, 1], R) is completely continuous.

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Let x be a solution. Then for t ∈ [0, 1], and using the computations in proving that F is bounded, we have |x(t)| = |λ(F x)(t)| Z t Z 1 1 λ1 q−1 ≤ (t − s) |f (t, x(s))|ds + (1 − s)q−1 |f (t, x(s))|ds Γ(q) 0 Γ(q) 0 Z η λ2 + (η − s)q−1 |f (t, x(s))|ds Γ(q) 0 kpkψ(kxk) ≤ {1 + λ1 + λ2 η q } Γ(q + 1) and consequently kxk kpkψ(kxk) {1 + λ1 + λ2 η q } Γ(q + 1)

≤ 1.

In view of (A4 ), there exists M such that kxk 6= M . Let us set U = {x ∈ C([0, 1], R) : kxk < M + 1}. Note that the operator F : U → C([0, 1], R) is continuous and completely continuous. From the choice of U , there is no x ∈ ∂U such that x = λF x for some λ ∈ (0, 1). Consequently, by the nonlinear alternative of Leray-Schauder type (Theorem 3.5), we deduce that F has a fixed point x ∈ U which is a solution of the problem (1.1). This completes the proof. 2

3.4

Existence result via nonlinear contractions

Definition 3.7 Let E be a Banach space and let F : E → E be a mapping. F is said to be a nonlinear contraction if there exists a continuous nondecrasing function Ψ : R+ → R+ such that Ψ(0) = 0 and Ψ(ξ) < ξ for all ξ > 0 with the property: kF x−F yk ≤ Ψ(kx−yk), ∀x, y ∈ E. Lemma 3.8 (Boyd and Wong)[15]. Let E be a Banach space and let F : E → E be a nonlinear contraction. Then F has a unique fixed point in E. Theorem 3.9 Assume that: (A5 ) |f (t, x) − f (t, y)| ≤ h(t)

|x − y| , H ∗ + |x − y|

t ∈ [0, 1], x, y ≥ 0, where h : [0, 1] → R+ is

continuous and Z 1 Z 1 Z η 1 λ1 λ2 H∗ = (1 − s)q−1 h(s)ds + (1 − s)q−1 h(s)ds + (η − s)q−1 h(s)ds. Γ(q) 0 Γ(q) 0 Γ(q) 0 Then the boundary value problem (1.1) has a unique solution. Proof. Consider the operator F : C → C given by (3.1). Let the continuous nondecreasing function Ψ : R+ → R+ satisfying Ψ(0) = 0 and Ψ(ξ) < ξ for all ξ > 0 defined by Ψ(ξ) =

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9

∀ξ ≥ 0. Let x, y ∈ C([0, 1], R). Then |f (s, x(s)) − f (s, y(s))| ≤

h(s) Ψ(kx − yk) so H∗

Z t |x(s) − y(s)| 1 (t − s)q−1 h(s) ∗ |F x(t) − F y(t)| ≤ ds Γ(q) 0 H + |x(s) − y(s)| Z 1 |x(s) − y(s)| λ1 (1 − s)q−1 h(s) ∗ + ds Γ(q) 0 H + |x(s) − y(s)| Z η λ2 |x(s) − y(s)| + ds, (η − s)q−1 h(s) ∗ Γ(q) 0 H + |x(s) − y(s)| for t ∈ [0, 1]. Then kF x − F yk ≤ Ψ(kx − yk) and F is a nonlinear contraction and it has a unique fixed poitn in C([0, 1], R), by Lemma 3.8. 2 Example 3.10 Let us consider the boundary value problem    c D3/2 x(t) = t|x| , 0 < t < 1,  + |x|  1   1 1 1   , x(1) = 2x .  x(0) = x 2 4 4

(3.5)

t|x| Here, q = 3/2, β = 1/2, α = 2, η = 1/4 and f (t, x) = . We choose h(t) = 1 + t 1 + |x| t(|x| − |y|) (1 + t)|x − y| . and find that H ∗ = 7.97. Clearly f (t, x) − f (t, y) = ≤ 1 + |x| + |y| + |x||y| 7.97 + |x − y| Thus, the conclusion of Theorem 3.9 applies and problem (3.5) has a unique solution.

4

Existence results-Differential Inclusions

Definition 4.1 A function x ∈ C 2 ([0, 1], R) is a solution of the problem (1.2) if x(0) = βx(η), x(1) = αx(η), and exists a function f ∈ L1 ([0, 1], R) such that f (t) ∈ F (t, x(t)) a.e. on [0, 1] and x(t) =

1 Γ(q)

Z

t q−1

(t − s) 0

β + (α − β)t + ∆Γ(q)

4.1

Z

(β − 1)t − βη f (s)ds + ∆Γ(q)

Z

1

(1 − s)q−1 f (s)ds

0

η

(η − s)q−1 f (s)ds.

0

The Carath´ eodory case

In this subsection, we are concerned with the existence of solutions for the problem (1.2) when the right hand side has convex values. We first recall some preliminary facts. For a normed space (X, k · k), let Pcl (X) = {Y ∈ P(X) : Y is closed}, Pb (X) = {Y ∈ P(X) : Y is bounded}, Pcp (X) = {Y ∈ P(X) : Y is compact}, and Pcp,c (X) = {Y ∈ P(X) : Y is compact and convex}.

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Definition 4.2 A multi-valued map G : X → P(X) : (i) is convex (closed) valued if G(x) is convex (closed) for all x ∈ X; (ii) is bounded on bounded sets if G(B) = ∪x∈B G(x) is bounded in X for all B ∈ Pb (X) (i.e. supx∈B {sup{|y| : y ∈ G(x)}} < ∞); (iii) is called upper semi-continuous (u.s.c.) on X if for each x0 ∈ X, the set G(x0 ) is a nonempty closed subset of X, and if for each open set N of X containing G(x0 ), there exists an open neighborhood N0 of x0 such that G(N0 ) ⊆ N ; (iv) is said to be completely continuous if G(B) is relatively compact for every B ∈ Pb (X); (v) has a fixed point if there is x ∈ X such that x ∈ G(x). The fixed point set of the multivalued operator G will be denoted by Fix G. Remark 4.3 It is known that, if the multi-valued map G is completely continuous with nonempty compact values, then G is u.s.c. if and only if G has a closed graph, i.e., xn → x∗ , yn → y∗ , yn ∈ G(xn ) imply y∗ ∈ G(x∗ ). Definition 4.4 A multivalued map G : [0; 1] → Pcl (R) is said to be measurable if for every y ∈ R, the function t 7−→ d(y, G(t)) = inf{ky − zk : z ∈ G(t)} is measurable. Definition 4.5 A multivalued map F : [0, 1] × R → P(R) is said to be Carath´eodory if (i) t 7−→ F (t, x) is measurable for each x ∈ X; (ii) x 7−→ F (t, x) is upper semicontinuous for almost all t ∈ [0, 1]; Further a Carath´eodory function F is called L1 −Carath´eodory if (iii) for each α > 0, there exists ϕα ∈ L1 ([0, 1], R+ ) such that kF (t, x)k = sup{|v| : v ∈ F (t, x)} ≤ ϕα (t) for all kxk ≤ α and for a. e. t ∈ [0, 1]. For each y ∈ C([0, 1], R), define the set of selections of F by SF,y := {v ∈ L1 ([0, 1], R) : v(t) ∈ F (t, y(t)) for a.e. t ∈ [0, 1]}. The consideration of this subsection is based on the following fixed point theorem ([19]). Theorem 4.6 (Nonlinear alternative for Kakutani maps).[19]. Let E be a Banach space, C a closed convex subset of E, U an open subset of C and 0 ∈ U. Suppose that F : U → Pcp,c (C) is a upper semicontinuous compact map. Then either (i) F has a fixed point in U , or (ii) there is a u ∈ ∂U and λ ∈ (0, 1) with u ∈ λF (u).

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The following lemma will be used in the sequel. Lemma 4.7 ([25]) Let X be a Banach space. Let F : [0, T ] × X → Pcp,c (X) be an L1 − Carath´eodory multivalued map and let Θ be a linear continuous mapping from L1 ([0, 1], X) to C([0, 1], X). Then the operator Θ ◦ SF : C([0, 1], X) → Pcp,c (C([0, 1], X)), x 7→ (Θ ◦ SF )(x) = Θ(SF,x ) is a closed graph operator in C([0, 1], X) × C([0, 1], X). Theorem 4.8 Assume that (A4 ) holds. In addition we suppose that the following conditions (H1 ) F : [0, 1] × R → P(R) is Carath´eodory and has nonempty compact convex values; (H2 ) there exists a continuous nondecreasing function ψ : [0, ∞) → (0, ∞) and a function p ∈ C([0, 1], R+ ) such that kF (t, x)kP := sup{|y| : y ∈ F (t, x)} ≤ p(t)ψ(kxk) for each (t, x) ∈ [0, 1] × R, are satisfied. Then the boundary value problem (1.2) has at least one solution on [0, 1]. Proof. In order to transform boundary value problem (1.2) into a fixed point problem, consider the multivalued operator Ω : C([0, 1], R) → P(C([0, 1], R)) defined by   h ∈ C([0, 1], R) :      Z t       1  q−1       (t − s) f (s)ds    Γ(q)       0  Z 1  Ω(x) = (β − 1)t − βη h(t) =   + (1 − s)q−1 f (s)ds      ∆Γ(q)    0    Z η       β + (α − β)t    q−1    (η − s) f (s)ds, 0 ≤ t ≤ 1, +    ∆Γ(q) 0 for f ∈ SF,x . Clearly, according to Lemma 2.4, the fixed points of Ω are solutions to boundary value problem (1.2). We will show that Ω satisfies the assumptions of the nonlinear alternative of Leray-Schauder type. The proof consists of several steps. As a first step, we show that Ω is convex for each x ∈ C([0, 1], R). This step is obvious since SF,x is convex (F has convex values), and therefore we omit the proof. Next, we show that Ω maps bounded sets into bounded sets in C([0, 1], R). For a positive number ρ, let Bρ = {x ∈ C([0, 1], R) : kxk ≤ ρ} be a bounded set in C([0, 1], R). Then, for each h ∈ Ω(x), x ∈ Bρ , there exists f ∈ SF,x such that Z t Z (β − 1)t − βη 1 1 (t − s)q−1 f (s)ds + (1 − s)q−1 f (s)ds h(t) = Γ(q) 0 ∆Γ(q) 0 Z β + (α − β)t η + (η − s)q−1 f (s)ds, ∆Γ(q) 0 Then, as in Theorem 3.6, we have Z t Z 1 1 λ1 q−1 |h(t)| ≤ (t − s) |f (s)|ds + (1 − s)q−1 |f (s)|ds Γ(q) 0 Γ(q) 0

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M. A. Darwish and S.K. Ntouyas Z η λ2 (η − s)q−1 |f (s)|ds + Γ(q) 0 kpkψ(kxk) ≤ {1 + λ1 + λ2 η q } . Γ(q + 1)

Thus, khk ≤

kpkψ(ρ) {1 + λ1 + λ2 η q } . Γ(q + 1)

Now we show that Ω maps bounded sets into equicontinuous sets of C([0, 1], R). Let t0 , t00 ∈ [0, 1] with t0 < t00 and x ∈ Bρ , where Bρ is a bounded set of C([0, 1], R). For each h ∈ Ω(x), we obtain, as in Theorem 3.6, |h(t00 ) − h(t0 )|

Z t00 1 Z t0 1 [(t00 − s)q−1 − (t0 − s)q−1 ]f (s)ds + (t00 − s)q−1 f (s)ds ≤ Γ(q) 0 Γ(q) t0 +

kpkψ(ρ)|(β − 1)||t00 − t0 | kpkψ(ρ)|α − β||t00 − t0 | + . ∆Γ(q + 1) ∆Γ(q + 1)

Obviously the right hand side of the above inequality tends to zero independently of x ∈ Br0 as t00 − t0 → 0. As Ω satisfies the above three assumptions, therefore it follows by the Arzel´ aAscoli theorem that Ω : C([0, 1], R) → P(C([0, 1], R)) is completely continuous. In our next step, we show that Ω has a closed graph. Let xn → x∗ , hn ∈ Ω(xn ) and hn → h∗ . Then we need to show that h∗ ∈ Ω(x∗ ). Associated with hn ∈ Ω(xn ), there exists fn ∈ SF,xn such that for each t ∈ [0, 1], Z t Z 1 (β − 1)t − βη 1 q−1 hn (t) = (t − s) fn (s)ds + (1 − s)q−1 fn (s)ds Γ(q) 0 ∆Γ(q) 0 Z β + (α − β)t η q−1 + (η − s) fn (s)ds, t ∈ [0, 1]. ∆Γ(q) 0 Thus we have to show that there exists f∗ ∈ SF,x∗ such that for each t ∈ [0, 1], h∗ (t) =

Z t Z 1 (β − 1)t − βη 1 (t − s)q−1 f∗ (s)ds + (1 − s)q−1 f∗ (s)ds Γ(q) 0 ∆Γ(q) 0 Z β + (α − β)t η + (η − s)q−1 f∗ (s)ds. ∆Γ(q) 0

Let us consider the continuous linear operator Θ : L1 ([0, 1], R) → C([0, 1], R) given by Z t Z 1 (β − 1)t − βη 1 q−1 f→ 7 Θ(f ) = (t − s) f (s)ds + (1 − s)q−1 f (s)ds Γ(q) 0 ∆Γ(q) 0 Z β + (α − β)t η + (η − s)q−1 f (s)ds. ∆Γ(q) 0 Observe that

Z t

1

khn (t) − h∗ (t)k = (t − s)q−1 (fn (s) − f∗ (s))ds Γ(q) 0

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Z (β − 1)t − βη 1 (1 − s)q−1 (fn (s) − f∗ (s))ds + ∆Γ(q)

Z0

β + (α − β)t η q−1 + (η − s) (fn (s) − f∗ (s))ds

→ 0, ∆Γ(q) 0 as n → ∞. Thus, it follows by Lemma 4.7 that Θ ◦ SF is a closed graph operator. Further, we have hn (t) ∈ Θ(SF,xn ). Since xn → x∗ , therefore, we have Z t Z 1 (β − 1)t − βη 1 q−1 h∗ (t) = (t − s) f∗ (s)ds + (1 − s)q−1 f∗ (s)ds Γ(q) 0 ∆Γ(q) 0 Z β + (α − β)t η (η − s)q−1 f∗ (s)ds, + ∆Γ(q) 0 for some f∗ ∈ SF,x∗ . Finally, we discuss a priori bounds on solutions. Let x be a solution of (1.2). Then there exists f ∈ L1 ([0, 1], R) with f ∈ SF,x such that, for t ∈ [0, 1], we have h(t) =

Z t Z 1 (β − 1)t − βη 1 (t − s)q−1 f (s)ds + (1 − s)q−1 f (s)ds Γ(q) 0 ∆Γ(q) 0 Z β + (α − β)t η (η − s)q−1 f (s)ds. + ∆Γ(q) 0

In view of (H2 ), and using the computations in second step above, for each t ∈ [0, 1], we obtain Z t Z 1 1 λ1 q−1 |h(t)| ≤ (t − s) |f (s)|ds + (1 − s)q−1 |f (s)|ds Γ(q) 0 Γ(q) 0 Z η λ2 + (η − s)q−1 |f (s)|ds Γ(q) 0 kpkψ(kxk) {1 + λ1 + λ2 η q } . ≤ Γ(q + 1) Consequently, we have kxk kpkψ(kxk) {1 + λ1 + λ2 η q } Γ(q + 1)

≤ 1.

In view of (A4 ), there exists M such that kxk 6= M . Let us set U = {x ∈ C([0, 1], R) : kxk < M + 1}. Note that the operator Ω : U → P(C([0, 1], R)) is upper semicontinuous and completely continuous. From the choice of U , there is no x ∈ ∂U such that x ∈ µΩ(x) for some µ ∈ (0, 1). Consequently, by the nonlinear alternative of Leray-Schauder type (Theorem 4.6), we deduce that Ω has a fixed point x ∈ U which is a solution of the problem (1.2). This completes the proof. 2

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Example 4.9 Consider the following fractional boundary value problem  c 3/2   D x(t) ∈ F (t, x(t)), 0 < t < 1,     1 1 1   x(0) = x , x(1) = 2x . 2 4 4

(4.1)

Here, q = 3/2, β = 1/2, α = 2, η = 1/4 and F : [0, 1] × R → P(R) is a multivalued map |x|3 |x| given by x → F (t, x) = + 2t3 + 1, + t + 1 . For f ∈ F, we have |f | ≤ 3 |x| + 3 |x| + 1   |x|3 |x| max + 2t3 + 1, + t + 1 ≤ 4, x ∈ R. Thus, kF (t, x)kP := sup{|y| : y ∈ 3 |x| + 3 |x| + 1 F (t, x)} ≤ 4 = p(t)ψ(kxk), x ∈ R, with p(t) = 1, ψ(kxk) = 4. Further, using the condition (A4 ) we find that M > 21.092278. Clearly, all the conditions of Theorem 4.8 are satisfied. So there exists at least one solution of the problem (4.1) on [0, 1].

4.2

The lower semi-continuous case

As a next result, we study the case when F is not necessarily convex valued. Our strategy to deal with this problems is based on the nonlinear alternative of Leray Schauder type together with the selection theorem of Bressan and Colombo [16] for lower semi-continuous maps with decomposable values. Definition 4.10 Let X be a nonempty closed subset of a Banach space E and G : X → P(E) be a multivalued operator with nonempty closed values. G is lower semi-continuous (l.s.c.) if the set {y ∈ X : G(y) ∩ B 6= ∅} is open for any open set B in E. Definition 4.11 Let A be a subset of [0, 1] × R. A is L ⊗ B measurable if A belongs to the σ−algebra generated by all sets of the form J × D, where J is Lebesgue measurable in [0, 1] and D is Borel measurable in R. Definition 4.12 A subset A of L1 ([0, 1], R) is decomposable if for all x, y ∈ A and measurable J ⊂ [0, 1] = J, the function xχJ + yχJ−J ∈ A, where χJ stands for the characteristic function of J . Definition 4.13 Let Y be a separable metric space and let N : Y → P(L1 ([0, 1], R)) be a multivalued operator. We say N has a property (BC) if N is lower semi-continuous (l.s.c.) and has nonempty closed and decomposable values. Let F : [0, 1] × R → P(R) be a multivalued map with nonempty compact values. Define a multivalued operator F : C([0, 1] × R) → P(L1 ([0, 1], R)) associated with F as F(x) = {w ∈ L1 ([0, 1], R) : w(t) ∈ F (t, x(t)) for a.e. t ∈ [0, 1]}, which is called the Nemytskii operator associated with F. Definition 4.14 Let F : [0, 1]×R → P(R) be a multivalued function with nonempty compact values. We say F is of lower semi-continuous type (l.s.c. type) if its associated Nemytskii operator F is lower semi-continuous and has nonempty closed and decomposable values.

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Lemma 4.15 ([16]) Let Y be a separable metric space and let N : Y → P(L1 ([0, 1], R)) be a multivalued operator satisfying the property (BC). Then N has a continuous selection, that is, there exists a continuous function (single-valued) g : Y → L1 ([0, 1], R) such that g(x) ∈ N (x) for every x ∈ Y . Theorem 4.16 Assume that (A4 ), (H2 ) and the following condition holds: (H3 ) F : [0, 1] × R → P(R) is a nonempty compact-valued multivalued map such that (a) (t, x) 7−→ F (t, x) is L ⊗ B measurable, (b) x 7−→ F (t, x) is lower semicontinuous for each t ∈ [0, 1]. Then the boundary value problem (1.2) has at least one solution on [0, 1]. Proof. It follows from (H3 ) and (H2 ) that F is of l.s.c. type. Then from Lemma 4.15, there exists a continuous function f : C 2 ([0, 1], R) → L1 ([0, 1], R) such that f (x) ∈ F(x) for all x ∈ C([0, 1], R). Consider the problem  c q D x(t) = f (x(t)), 0 < t < 1, 1 < q ≤ 2, (4.2) x(0) = βx(η), x(1) = αx(η) in the space C 2 ([0, 1], R). It is clear that if x ∈ C 2 ([0, 1], R) is a solution of the problem (4.2), then x is a solution to the problem (1.2). In order to transform the problem (4.2) into a fixed point problem, we define the operator Ω as Z t Z 1 (β − 1)t − βη 1 (t − s)q−1 f (x(s))ds + (1 − s)q−1 f (x(s))ds Γ(q) 0 ∆Γ(q) 0 Z β + (α − β)t η q−1 + (η − s) f (x(s))ds, 0 ≤ t ≤ 1. ∆Γ(q) 0

Ωx(t) =

It can easily be shown that Ω is continuous and completely continuous. The remaining part of the proof is similar to that of Theorem 4.8. So we omit it. This completes the proof. 2

4.3

The Lipschitz case

Now we prove the existence of solutions for the problem (1.2) with a nonconvex valued right hand side by applying a fixed point theorem for multivalued map due to Covitz and Nadler [18]. Let (X, d) be a metric space induced from the normed space (X; k · k). Consider Hd : P(X) × P(X) → R ∪ {∞} given by Hd (A, B) = max{supa∈A d(a, B), supb∈B d(A, b)}, where d(A, b) = inf a∈A d(a; b) and d(a, B) = inf b∈B d(a; b). Then (Pb,cl (X), Hd ) is a metric space and (Pcl (X), Hd ) is a generalized metric space (see [23]). Definition 4.17 A multivalued operator N : X → Pcl (X) is called:

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(a) γ−Lipschitz if and only if there exists γ > 0 such that Hd (N (x), N (y)) ≤ γd(x, y) for each x, y ∈ X; (b) a contraction if and only if it is γ−Lipschitz with γ < 1. Lemma 4.18 (Covitz-Nadler) [18]. Let (X, d) be a complete metric space. If N : X → Pcl (X) is a contraction, then F ixN 6= ∅. Definition 4.19 A measurable multi-valued function F : [0, 1] → P(X) is said to be integrably bounded if there exists a function h ∈ L1 ([0, 1], X) such that for all v ∈ F (t), kvk ≤ h(t) for a.e. t ∈ [0, 1]. Theorem 4.20 Assume that the following conditions hold: (H4 ) F : [0, 1] × R → Pcp (R) is such that F (·, x) : [0, 1] → Pcp (R) is measurable for each x ∈ R; (H5 ) Hd (F (t, x), F (t, x ¯)) ≤ m(t)|x − x ¯| for almost all t ∈ [0, 1] and x, x ¯ ∈ R with m ∈ C([0, 1], R+ ) and d(0, F (t, 0)) ≤ m(t) for almost all t ∈ [0, 1]. Then the boundary value problem (1.2) has at least one solution on [0, 1] if kmk {1 + λ1 + λ2 η q } < 1. Γ(q + 1) Proof. We transform the problem (1.2) into a fixed point problem. Consider the set-valued map Ω : C([0, 1], R) → P(C([0, 1], R)) defined at the begining of the proof of Theorem 4.8. It is clear that the fixed point of Ω are solutions of the problem (1.2). Note that, by the assumption (H4 ), since the set-valued map F (·, x) is measurable, it admits a measurable selection f : [0, 1] → R (see Theorem III.6 [17]). Moreover, from assumption (H5 ) |f (t)| ≤ m(t) + m(t)|x(t)|, i.e. f (·) ∈ L1 ([0, 1], X). Therefore the set SF,x is nonempty. Also note that since SF,x 6= ∅, Ω(x) 6= ∅ for any x ∈ C([0, 1], R). Now we show that the operator Ω satisfies the assumptions of Lemma 4.18. To show that Ω(x) ∈ Pcl ((C[0, 1], R)) for each x ∈ C([0, 1], R), let {un }n≥0 ∈ Ω(x) be such that un → u (n → ∞) in C([0, 1], R). Then u ∈ C([0, 1], R) and there exists vn ∈ SF,x such that, for each t ∈ [0, 1], un (t) =

Z t Z 1 (β − 1)t − βη 1 (t − s)q−1 vn (s)ds + (1 − s)q−1 vn (s)ds Γ(q) 0 ∆Γ(q) 0 Z β + (α − β)t η + (η − s)q−1 vn (s)ds. ∆Γ(q) 0

As F has compact values, we may pass onto a subsequence (if necessary) to obtain that vn converges to v in L1 ([0, 1], R). Thus, v ∈ SF,x and for each t ∈ [0, 1], un (t) → u(t) =

1 Γ(q)

Z

t q−1

(t − s) 0

(β − 1)t − βη v(s)ds + ∆Γ(q)

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Z

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17

Z

η

(η − s)q−1 v(s)ds.

0

Hence, u ∈ Ω(x) and Ω(x) is closed. Next we show that Ω is a contraction on C([0, 1], R), i.e. there exists γ < 1 such that Hd (Ω(x), Ω(¯ x)) ≤ γkx − x ¯k for each x, x ¯ ∈ C([0, 1], R). Let x, x ¯ ∈ C([0, 1], R) and h1 ∈ Ω(x). Then there exists v1 (t) ∈ F (t, x(t)) such that, for each t ∈ [0, 1], Z t Z (β − 1)t − βη 1 1 (t − s)q−1 v1 (s)ds + (1 − s)q−1 v1 (s)ds h1 (t) = Γ(q) 0 ∆Γ(q) 0 Z β + (α − β)t η + (η − s)q−1 v1 (s)ds. ∆Γ(q) 0 By (H6 ), we have Hd (F (t, x), F (t, x ¯)) ≤ m(t)|x(t) − x ¯(t)|. So, there exists w ∈ F (t, x ¯(t)) such that |v1 (t) − w| ≤ m(t)|x(t) − x ¯(t)|, t ∈ [0, 1]. Define U : [0, 1] → P(R) by U (t) = {w ∈ R : |v1 (t) − w| ≤ m(t)|x(t) − x ¯(t)|}. Since the multivalued operator U (t) ∩ F (t, x ¯(t)) is measurable (Proposition III.4 [17]), there exists a function v2 (t) which is a measurable selection for U. So v2 (t) ∈ F (t, x ¯(t)) and for each t ∈ [0, 1], we have |v1 (t) − v2 (t)| ≤ m(t)|x(t) − x ¯(t)|. For each t ∈ [0, 1], let us define Z t Z 1 (β − 1)t − βη 1 q−1 h2 (t) = (t − s) v2 (s)ds + (1 − s)q−1 v2 (s)ds Γ(q) 0 ∆Γ(q) 0 Z β + (α − β)t η + (η − s)q−1 v2 (s)ds. ∆Γ(q) 0 Thus, Z t Z 1 1 λ1 q−1 |h1 (t) − h2 (t)| ≤ (t − s) |v1 (s) − v2 (s)|ds + (1 − s)q−1 |v1 (s) − v2 (s)|ds Γ(q) 0 Γ(q) 0 Z η λ2 + (η − s)q−1 |v1 (s) − v2 (s)|ds Γ(q) 0 kmkkx − xk ≤ {1 + λ1 + λ2 η q } . Γ(q + 1) Hence, kh1 − h2 k ≤

kmkkx − xk {1 + λ1 + λ2 η q } . Γ(q + 1)

Analogously, interchanging the roles of x and x, we obtain Hd (Ω(x), Ω(¯ x)) ≤ γkx − x ¯k ≤

kmkkx − xk {1 + λ1 + λ2 η q } . Γ(q + 1)

Since Ω is a contraction, it follows by Lemma 4.18 that Ω has a fixed point x which is a solution of (1.2). This completes the proof. 2

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Remark 4.21 The results of this paper can easily to be generalized to boundary value problems for fractional differential equations and inclusions with deviating arguments and generalized three point boundary conditions. Thus we can study, by similar methods and obvious modifications, the following boundary value problem for fractional differential equations  c q  D x(t) = f (t, x(σ(t))), 0 < t < 1, 1 < q ≤ 2, x(t) = βx(η), −r ≤ t ≤ 0 (4.3)  x(1) = αx(η), where c Dq denotes the Caputo fractional derivative of order q, f : [0, 1]×R → R is continuous, −r = mint∈[0,1] σ(t), σ : [0, 1] → [−r, 1] is continuous with σ(t) ≤ t, ∀t ∈ [0, 1] and α, β, η are constants with 0 < η < 1 and 1 − β + (β − α)η 6= 0, or the corresponding boundary value problem for fractional differential inclusions  c q  D x(t) ∈ F (t, x(σ(t))), 0 < t < 1, 1 < q ≤ 2, x(t) = βx(η), −r ≤ t ≤ 0 (4.4)  x(1) = αx(η), where c Dq denotes the Caputo fractional derivative of order q, and F : [0, 1] × R → P(R) is a multivalued map, P(R) is the family of all subsets of R. Remark 4.22 It is obvious that the methods used in this paper can be applied to other types of nonlocal boundary value problems. For example for the following four point boundary value problem  c q D x(t) = f (t, x(t)), 0 < t < 1, 1 < q ≤ 2, (4.5) x(t) = αx(ξ), x(1) = βx(η), where α, β, ξ, η are constants with 0 < ξ, η < 1 and ∆ := α(βη − 1) − (β − 1)(αξ − 1) 6= 0. The solution of the problem (4.5) is given by t

Z (t − s)q−1 α[(β − 1)t − βη + 1] ξ (ξ − s)q−1 f (s, x(s)) ds + f (s, x(s)) ds Γ(q) ∆ Γ(q) 0 0 Z β[αξ − 1 − αt] η (η − s)q−1 + f (s, x(s)) ds ∆ Γ(q) 0 Z αt − αξ + 1 1 (1 − s)q−1 + f (s, x(s)) ds, 0 ≤ t ≤ 1. ∆ Γ(q) 0 Z

x(t) =

Acknowledgment: This project was funded by the Deanship of Scientific Research (DSR), King Abdulaziz University, Jeddah, under grant no. (479/363/1432). The authors, therefore, acknowledge with thanks DSR for technical and financial support.

References [1] R. P. Agarwal, B. Andrade, C. Cuevas, Weighted pseudo-almost periodic solutions of a class of semilinear fractional differential equations, Nonlinear Anal. Real World Appl. 11 (2010), 3532-3554.

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[2] R. P. Agarwal, V. Lakshmikantham, J. J. Nieto, On the concept of solution for fractional differential equations with uncertainty, Nonlinear Anal. 72 (2010), 2859-2862. [3] R.P. Agarwal, Y. Zhou, Y. He, Existence of fractional neutral functional differential equations, Comput. Math. Appl. 59 (2010), 1095-1100. [4] B. Ahmad, S. Sivasundaram, On four-point nonlocal boundary value problems of nonlinear integro-differential equations of fractional order, Appl. Math. Comput. 217 (2010), 480-487. [5] B. Ahmad, Existence of solutions for irregular boundary value problems of nonlinear fractional differential equations, Appl. Math. Lett. 23 (2010), 390-394. [6] B. Ahmad, Existence of solutions for fractional differential equations of order q ∈ (2, 3] with anti-periodic boundary conditions, J. Appl. Math. Comput. 34 (2010), 385-391. [7] B. Ahmad, J.J. Nieto, Existence of solutions for nonlocal boundary value problems of higher order nonlinear fractional differential equations, Abstr. Appl. Anal., 2009 (2009), Article ID 494720, 9 pages. [8] B. Ahmad, A. Alsaedi, Existence and uniqueness of solutions for coupled systems of higher order nonlinear fractional differential equations, Fixed Point Theory Appl., 2010 (2010) Article ID 364560, 17 pages. [9] B. Ahmad, S.K. Ntouyas, A boundary value problem of fractional differential equations with anti-periodic type integral boundary conditions, J. Comput. Anal. Appl. 15 (2013), 1372-1380. [10] B. Ahmad, J.J. Nieto, Existence results for a coupled system of nonlinear fractional differential equations with three-point boundary conditions, Comput. Math. Appl. 58 (2009) 1838-1843. [11] Z.B. Bai, On positive solutions of a nonlocal fractional boundary value problem, Nonlinear Anal. 72 (2010), 916-924. [12] K. Balachandran, J. J. Trujillo, The nonlocal Cauchy problem for nonlinear fractional integrodifferential equations in Banach spaces, Nonlinear Anal. 72 (2010) 4587-4593. [13] M.Benchohra, S. Hamani, S.K. Ntouyas, Boundary value problems for differential equations with fractional order and nonlocal conditions, Nonlinear Anal. 71 (2009) 2391-2396. [14] M. Benchohra, J. Henderson, S. K. Ntouyas, I. K. Purnaras, Positive solutions for systems of generalized second order three-point boundary value problems on time scales, Commun. Appl. Nonlinear Anal. 15 (2008), 39-54. [15] D.W. Boyd, J.S.W. Wong, On nonlinear contractions, Proc. Am. Math. Soc. 20 (1969), 458-464. [16] A. Bressan, G. Colombo, Extensions and selections of maps with decomposable values, Studia Math. 90 (1988), 69-86.

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[17] C. Castaing, M. Valadier, Convex Analysis and Measurable Multifunctions, Lecture Notes in Mathematics 580, Springer-Verlag, Berlin-Heidelberg-New York, 1977. [18] H. Covitz, S. B. Nadler Jr., Multivalued contraction mappings in generalized metric spaces, Israel J. Math. 8 (1970), 5-11. [19] A. Granas, J. Dugundji, Fixed Point Theory, Springer-Verlag, New York, 2003. [20] J. Henderson, S. K. Ntouyas, I. K. Purnaras, Positive solutions for systems of generalized three-point nonlinear boundary value problems, Comment. Math. Univ. Carolin. 49 (2008), 79-91. [21] J. Henderson, S. K. Ntouyas, I. K. Purnaras, Positive solutions for systems of nonlinear generalized three-point boundary value problems with deviating arguments, Commun. Appl. Nonlinear Anal. 15 (2008), 59-76. [22] A.A. Kilbas, H.M. Srivastava, J.J. Trujillo, Theory and Applications of Fractional Differential Equations, North-Holland Mathematics Studies, 204. Elsevier Science B.V., Amsterdam, 2006. [23] M. Kisielewicz, Differential Inclusions and Optimal Control, Kluwer, Dordrecht, The Netherlands, 1991. [24] M.A. Krasnoselskii, Two remarks on the method of successive approximations, Uspekhi Mat. Nauk 10 (1955), 123-127. [25] A. Lasota, Z. Opial, An application of the Kakutani-Ky Fan theorem in the theory of ordinary differential equations, Bull. Acad. Polon. Sci. Ser. Sci. Math. Astronom. Phys. 13 (1965), 781-786. [26] R. Liang, J. Peng, J. Shen, Positive solutions to a generalized second order three-point boundary value problem, Appl. Math. Comput. 196 (2008), 931-940. [27] I. Podlubny, Fractional Differential Equations, Academic Press, San Diego, 1999. [28] J. Sabatier, O.P. Agrawal, J.A.T. Machado (Eds.), Advances in Fractional Calculus: Theoretical Developments and Applications in Physics and Engineering, Springer, Dordrecht, 2007. [29] S.G. Samko, A.A. Kilbas, O.I. Marichev, Fractional Integrals and Derivatives, Theory and Applications, Gordon and Breach, Yverdon, 1993.

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QUADRATIC ρ-FUNCTIONAL INEQUALITIES IN FUZZY BANACH SPACES CHOONKIL PARK AND SUN YOUNG JANG∗ Abstract. In this paper, we solve the following quadratic ρ-functional inequalities         x−y x+y + 2f − f (x) − f (y) , t N f (x + y) + f (x − y) − 2f (x) − 2f (y) − ρ 2f 2 2 t ≥ , (0.1) t + ϕ(x, y) where ρ is a fixed real number with ρ 6= 2, and       x+y x−y N 2f + 2f − f (x) − f (y) − ρ (f (x + y) + f (x − y) − 2f (x) − 2f (y)) , t 2 2 t ≥ , (0.2) t + ϕ(x, y) where ρ is a fixed real number with ρ 6= 21 . Using the fixed point method, we prove the Hyers-Ulam stability of the quadratic ρfunctional inequalities (0.1) and (0.2) in fuzzy Banach spaces.

1. Introduction and preliminaries Katsaras [21] defined a fuzzy norm on a vector space to construct a fuzzy vector topological structure on the space. Some mathematicians have defined fuzzy norms on a vector space from various points of view [13, 24, 52]. In particular, Bag and Samanta [2], following Cheng and Mordeson [8], gave an idea of fuzzy norm in such a manner that the corresponding fuzzy metric is of Kramosil and Michalek type [23]. They established a decomposition theorem of a fuzzy norm into a family of crisp norms and investigated some properties of fuzzy normed spaces [3]. We use the definition of fuzzy normed spaces given in [2, 28, 29] to investigate the Hyers-Ulam stability of quadratic ρ-functional inequalities in fuzzy Banach spaces. Definition 1.1. [2, 28, 29, 30] Let X be a real vector space. A function N : X × R → [0, 1] is called a fuzzy norm on X if for all x, y ∈ X and all s, t ∈ R, (N1 ) N (x, t) = 0 for t ≤ 0; (N2 ) x = 0 if and only if N (x, t) = 1 for all t > 0; t (N3 ) N (cx, t) = N (x, |c| ) if c 6= 0; (N4 ) N (x + y, s + t) ≥ min{N (x, s), N (y, t)}; (N5 ) N (x, ·) is a non-decreasing function of R and limt→∞ N (x, t) = 1. (N6 ) for x 6= 0, N (x, ·) is continuous on R. The pair (X, N ) is called a fuzzy normed vector space. The properties of fuzzy normed vector spaces and examples of fuzzy norms are given in [27, 28]. 2010 Mathematics Subject Classification. Primary 46S40, 39B52, 47H10, 39B62, 26E50, 47S40. Key words and phrases. fuzzy Banach space; quadratic ρ-functional inequality; fixed point method; HyersUlam stability. ∗ Corresponding author: Sung Young Jang (email: [email protected]).

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Definition 1.2. [2, 28, 29, 30] Let (X, N ) be a fuzzy normed vector space. A sequence {xn } in X is said to be convergent or converge if there exists an x ∈ X such that limn→∞ N (xn −x, t) = 1 for all t > 0. In this case, x is called the limit of the sequence {xn } and we denote it by N limn→∞ xn = x. Definition 1.3. [2, 28, 29, 30] Let (X, N ) be a fuzzy normed vector space. A sequence {xn } in X is called Cauchy if for each ε > 0 and each t > 0 there exists an n0 ∈ N such that for all n ≥ n0 and all p > 0, we have N (xn+p − xn , t) > 1 − ε. It is well-known that every convergent sequence in a fuzzy normed vector space is Cauchy. If each Cauchy sequence is convergent, then the fuzzy norm is said to be complete and the fuzzy normed vector space is called a fuzzy Banach space. We say that a mapping f : X → Y between fuzzy normed vector spaces X and Y is continuous at a point x0 ∈ X if for each sequence {xn } converging to x0 in X, then the sequence {f (xn )} converges to f (x0 ). If f : X → Y is continuous at each x ∈ X, then f : X → Y is said to be continuous on X (see [3]). The stability problem of functional equations originated from a question of Ulam [51] concerning the stability of group homomorphisms. The functional equation f (x + y) = f (x) + f (y) is called the Cauchy equation. In particular, every solution of the Cauchy equation is said to be an additive mapping. Hyers [17] gave a first affirmative partial answer to the question of Ulam for Banach spaces. Hyers’ Theorem was generalized by Aoki [1] for additive mappings and by Th.M. Rassias [40] for linear mappings by considering an unbounded Cauchy difference. A generalization of the Th.M. Rassias theorem was obtained by G˘avruta [14] by replacing the unbounded Cauchy difference by a general control function in the spirit of Th.M. Rassias’ approach. The functional equation f (x+y)+f (x−y) = 2f (x)+2f (y) is called the quadratic functional equation. In particular, every solution of the quadratic functional equation is said to be a quadratic mapping. The stability of quadratic functional equation was proved by Skof [50] for mappings f : E1 → E2 , where E1 is a normed space and E2 is a Banach space. Cholewa [9] noticed that the theorem of Skof is still true if the relevant domain E1 is replaced by an Abelian group. Czerwik [10] proved stability of the quadratic functional  Hyers-Ulam    the x−y x+y 1 equation. The functional equation f 2 + f 2 = 2 f (x) + 21 f (y) is called a Jensen type quadratic equation. The stability problems of several functional equations have been extensively investigated by a number of authors and there are many interesting results concerning this problem (see [4, 18, 20, 25, 36, 37, 38, 41, 42, 44, 45, 46, 47, 48, 49]). Gil´anyi [15] showed that if f satisfies the functional inequality k2f (x) + 2f (y) − f (x − y)k ≤ kf (x + y)k

(1.1)

then f satisfies the Jordan-von Neumann functional equation 2f (x) + 2f (y) = f (x + y) + f (x − y). See also [43]. Fechner [12] and Gil´anyi [16] proved the Hyers-Ulam stability of the functional inequality (1.1). Park, Cho and Han [35] investigated the Cauchy additive functional inequality kf (x) + f (y) + f (z)k ≤ kf (x + y + z)k

(1.2)

and the Cauchy-Jensen additive functional inequality

kf (x) + f (y) + 2f (z)k ≤

2f



x+y +z

2 

(1.3)

and proved the Hyers-Ulam stability of the functional inequalities (1.2) and (1.3) in Banach spaces. Park [33, 34] defined additive ρ-functional inequalities and proved the Hyers-Ulam stability of the additive ρ-functional inequalities in Banach spaces and non-Archimedean Banach spaces.

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We recall a fundamental result in fixed point theory. Let X be a set. A function d : X × X → [0, ∞] is called a generalized metric on X if d satisfies (1) d(x, y) = 0 if and only if x = y; (2) d(x, y) = d(y, x) for all x, y ∈ X; (3) d(x, z) ≤ d(x, y) + d(y, z) for all x, y, z ∈ X. Theorem 1.4. [5, 11] Let (X, d) be a complete generalized metric space and let J : X → X be a strictly contractive mapping with Lipschitz constant L < 1. Then for each given element x ∈ X, either d(J n x, J n+1 x) = ∞ for all nonnegative integers n or there exists a positive integer n0 such that (1) d(J n x, J n+1 x) < ∞, ∀n ≥ n0 ; (2) the sequence {J n x} converges to a fixed point y ∗ of J; (3) y ∗ is the unique fixed point of J in the set Y = {y ∈ X | d(J n0 x, y) < ∞}; 1 d(y, Jy) for all y ∈ Y . (4) d(y, y ∗ ) ≤ 1−L In 1996, G. Isac and Th.M. Rassias [19] were the first to provide applications of stability theory of functional equations for the proof of new fixed point theorems with applications. By using fixed point methods, the stability problems of several functional equations have been extensively investigated by a number of authors (see [6, 7, 22, 27, 31, 32, 38, 39]). In Section 2, we solve the quadratic ρ-functional inequality (0.1) and prove the Hyers-Ulam stability of the quadratic ρ-functional inequality (0.1) in fuzzy Banach spaces by using the fixed point method. In Section 3, we solve the quadratic ρ-functional inequality (0.2) and prove the Hyers-Ulam stability of the quadratic ρ-functional inequality (0.2) in fuzzy Banach spaces by using the fixed point method. Throughout this paper, assume that X is a real vector space and (Y, N ) is a fuzzy Banach space. 2. Quadratic ρ-functional inequality (0.1) In this section, we prove the Hyers-Ulam stability of the quadratic ρ-functional inequality (0.1) in fuzzy Banach spaces. Let ρ be a real number with ρ 6= 2. We need the following lemma to prove the main results. Lemma 2.1. Let f : X → Y be a mapping satisfying f (0) = 0 and 



f (x + y) + f (x − y) − 2f (x) − 2f (y) = ρ 2f

x+y 2





+ 2f

x−y 2





− f (x) − f (y)

(2.1)

for all x, y ∈ X. Then f : X → Y is quadratic. Proof. Replacing y by x in (2.1), we get f (2x) − 4f (x) = 0 and so f (2x) = 4f (x) for all x ∈ X. Thus       x−y x+y f (x + y) + f (x − y) − 2f (x) − 2f (y) = ρ 2f + 2f − f (x) − f (y) 2 2 ρ = (f (x + y) + f (x − y) − 2f (x) − 2f (y)) 2 and so f (x + y) + f (x − y) = 2f (x) + 2f (y) for all x, y ∈ X, as desired.  Theorem 2.2. Let ϕ : X 2 → [0, ∞) be a function such that there exists an L < 1 with ϕ(x, y) ≤

L ϕ (2x, 2y) 4

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for all x, y ∈ X. Let f : X → Y be a mapping satisfying f (0) = 0 and N (f (x + y) + f (x − y) − 2f (x) − 2f (y)       x+y x−y t −ρ 2f + 2f − f (x) − f (y) , t) ≥ 2 2 t + ϕ(x, y) x 2n

for all x, y ∈ X and all t > 0. Then Q(x) := N -limn→∞ 4n f defines a quadratic mapping Q : X → Y such that N (f (x) − Q(x), t) ≥




(2.2)

exists for each x ∈ X and

(4 − 4L)t (4 − 4L)t + Lϕ(x, x)

(2.3)

for all x ∈ X and all t > 0. Proof. Letting y = x in (2.2), we get t t + ϕ(x, x)

N (f (2x) − 4f (x), t) ≥

(2.4)

for all x ∈ X. Consider the set S := {g : X → Y } and introduce the generalized metric on S: t , ∀x ∈ X, ∀t > 0 , t + ϕ(x, x) 



d(g, h) = inf µ ∈ R+ : N (g(x) − h(x), µt) ≥

where, as usual, inf φ = +∞. It is easy to show that (S, d) is complete (see [26, Lemma 2.1]). Now we consider the linear mapping J : S → S such that x 2

 

Jg(x) := 4g for all x ∈ X. Let g, h ∈ S be given such that d(g, h) = ε. Then N (g(x) − h(x), εt) ≥

t t + ϕ(x, x)

for all x ∈ X and all t > 0. Hence x N (Jg(x) − Jh(x), Lεt) = N 4g 2

x x x L − 4h , Lεt = N g −h , εt 2 2 2 4 Lt Lt t 4 4 =
≥ Lt L x x t + ϕ(x, x) + ϕ 2, 2 4 + 4 ϕ(x, x)



≥

Lt 4

 

 



  

 



for all x ∈ X and all t > 0. So d(g, h) = ε implies that d(Jg, Jh) ≤ Lε. This means that d(Jg, Jh) ≤ Ld(g, h) for all g, h ∈ S.  It follows from (2.4) that N f (x) − 4f

x 2






, L4 t ≥

t t+ϕ(x,x)

for all x ∈ X and all t > 0. So

d(f, Jf ) ≤ L4 . By Theorem 1.4, there exists a mapping Q : X → Y satisfying the following: (1) Q is a fixed point of J, i.e., x 2

 

Q

1 = Q(x) 4

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for all x ∈ X. Since f : X → Y is even, Q : X → Y is a even mapping. The mapping Q is a unique fixed point of J in the set M = {g ∈ S : d(f, g) < ∞}. This implies that Q is a unique mapping satisfying (2.5) such that there exists a µ ∈ (0, ∞) satisfying t N (f (x) − Q(x), µt) ≥ t + ϕ(x, x) for all x ∈ X; (2) d(J n f, Q) → 0 as n → ∞. This implies the equality   x n = Q(x) N - lim 4 f n→∞ 2n for all x ∈ X; 1 (3) d(f, Q) ≤ 1−L d(f, Jf ), which implies the inequality d(f, Q) ≤

L . 4 − 4L

This implies that the inequality (2.3) holds. By (2.2),           x+y x−y x y n N 4 f +f − 2f − 2f n n n 2 2 2 2n            x−y x y x+y t n n
− ρ 4 2f + 2f −f −f ,4 t ≥ n+1 n+1 n n 2 2 2 2 t + ϕ 2xn , 2yn for all x, y ∈ X, all t > 0 and all n ∈ N. So           x−y x y x+y n +f − 2f − 2f N 4 f n n n 2 2 2 2n            x+y x−y x y − ρ 4n 2f + 2f − f − f ,t ≥ 2n+1 2n+1 2n 2n for all x, y ∈ X, all t > 0 and all n ∈ N. Since limn→∞ t > 0,

t 4n t Ln + 4n ϕ(x,y) 4n

t 4n

+

t 4n Ln 4n ϕ (x, y)

= 1 for all x, y ∈ X and all

x−y x+y Q(x + y) + Q(x − y) − 2Q(x) − 2Q(y) = ρ 2Q + 2Q − Q(x) − Q(y) 2 2 for all x, y ∈ X. By Lemma 2.1, the mapping Q : X → Y is quadratic, as desired. 













Corollary 2.3. Let θ ≥ 0 and let p be a real number with p > 2. Let X be a normed vector space with norm k · k. Let f : X → Y be a mapping satisfying f (0) = 0 and         x+y x−y N f (x + y) + f (x − y) − 2f (x) − 2f (y) − ρ 2f + 2f − f (x) − f (y) , t 2 2 t ≥ (2.6) t + θ(kxkp + kykp ) for all x, y ∈ X and all t > 0. Then Q(x) := N -limn→∞ 4n f ( 2xn ) exists for each x ∈ X and defines a quadratic mapping Q : X → Y such that (2p − 4)t N (f (x) − Q(x), t) ≥ p (2 − 4)t + 2θkxkp for all x ∈ X.

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Proof. The proof follows from Theorem 2.2 by taking ϕ(x, y) := θ(kxkp + kykp ) for all x, y ∈ X. Then we can choose L = 22−p , and we get the desired result.  Theorem 2.4. Let ϕ : X 2 → [0, ∞) be a function such that there exists an L < 1 with x y ϕ(x, y) ≤ 4Lϕ , 2 2 



for all x, y ∈ X. Let f : X → Y be a mapping satisfying f (0) = 0 and (2.2). Then Q(x) := N limn→∞ 41n f (2n x) exists for each x ∈ X and defines a quadratic mapping Q : X → Y such that (4 − 4L)t N (f (x) − Q(x), t) ≥ (2.7) (4 − 4L)t + ϕ(x, x) for all x ∈ X and all t > 0. Proof. Let (S, d) be the generalized metric space defined in the proof of Theorem 2.2. Now we consider the linear mapping J : S → S such that 1 Jg(x) := g (2x) 4 for all x ∈ X. It follows from (2.4) that 1 t 1 N f (x) − f (2x), t ≥ 4 4 t + ϕ(x, x) 



for all x ∈ X and all t > 0. So d(f, Jf ) ≤ 14 . Hence d(f, Q) ≤ inequality (2.7) holds. The rest of the proof is similar to the proof of Theorem 2.2.

1 4−4L ,

which implies that the 

Corollary 2.5. Let θ ≥ 0 and let p be a real number with 0 < p < 2. Let X be a normed vector space with norm k · k. Let f : X → Y be amapping satisfying f (0) = 0 and (2.6). Then Q(x) := N -limn→∞ 41n f (2n x) exists for each x ∈ X and defines a quadratic mapping Q : X → Y such that N (f (x) − Q(x), t) ≥

(4 − 2p )t (4 − 2p )t + 2θkxkp

for all x ∈ X. Proof. The proof follows from Theorem 2.4 by taking ϕ(x, y) := θ(kxkp + kykp ) for all x, y ∈ X. Then we can choose L = 2p−2 , and we get the desired result.  3. Quadratic ρ-functional inequality (0.2) In this section, we prove the Hyers-Ulam stability of the quadratic ρ-functional inequality (0.2) in fuzzy Banach spaces. Let ρ be a real number with ρ 6= 12 . We need the following lemma to prove the main results. Lemma 3.1. Let f : X → Y be a mapping satisfying f (0) = 0 and 

2f

x+y 2





+ 2f

x−y 2



− f (x) − f (y) = ρ (f (x + y) + f (x − y) − 2f (x) − 2f (y))

(3.1)

for all x, y ∈ X. Then f : X → Y is quadratic.

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Proof. Letting y = 0 in (3.1), we get 4f x2 − f (x) = 0 and so f (2x) = 4f (x) for all x ∈ X. Thus     x−y 1 x+y 1 + 2f − f (x) − f (y) f (x + y) − f (x − y) − f (x) − f (y) = 2f 2 2 2 2 = ρ(f (x + y) + f (x − y) − 2f (x) − 2f (y))


and so f (x + y) + f (x − y) = 2f (x) + 2f (y) for all x, y ∈ X, as desired.



Theorem 3.2. Let ϕ : X 2 → [0, ∞) be a function such that there exists an L < 1 with L ϕ (2x, 2y) 4 for all x, y ∈ X. Let f : X → Y be a mapping satisfying f (0) = 0 and ϕ(x, y) ≤



N 2f



x+y 2 ≥





+ 2f

x−y 2





− f (x) − f (y) − ρ (f (x + y) + f (x − y) − 2f (x) − 2f (y)) , t

t t + ϕ(x, y)

(3.2)

for all x, y ∈ X and all t > 0. Then Q(x) := N -limn→∞ 4n f defines a quadratic mapping Q : X → Y such that N (f (x) − Q(x), t) ≥

x 2n




exists for each x ∈ X and

(1 − L)t (1 − L)t + ϕ(x, 0)

(3.3)

for all x ∈ X and all t > 0. Proof. Letting y = 0 in (3.2), we get 

x 2

 

N 4f



− f (x), t ≥

t t + ϕ(x, 0)

(3.4)

for all x ∈ X. Consider the set S := {g : X → Y } and introduce the generalized metric on S: t d(g, h) = inf µ ∈ R+ : N (g(x) − h(x), µt) ≥ , ∀x ∈ X, ∀t > 0 , t + ϕ(x, 0) 



where, as usual, inf φ = +∞. It is easy to show that (S, d) is complete (see [26, Lemma 2.1]). Now we consider the linear mapping J : S → S such that   x Jg(x) := 4g 2 for all x ∈ X. Let g, h ∈ S be given such that d(g, h) = ε. Then t N (g(x) − h(x), εt) ≥ t + ϕ(x, 0) for all x ∈ X and all t > 0. Hence x x x L , Lεt = N g −h , εt 2 2 2 4 Lt Lt t 4 4 =
≥ Lt x L t + ϕ(x, 0) + ϕ 2,0 4 + 4 ϕ(x, 0)



N (Jg(x) − Jh(x), Lεt) = N 4g ≥

Lt 4

x 2

 

 



  

 



− 4h

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for all x ∈ X and all t > 0. So d(g, h) = ε implies that d(Jg, Jh) ≤ Lε. This means that d(Jg, Jh) ≤ Ld(g, h) for all g, h ∈ S. It follows from (3.4) that 

x t ,t ≥ 2 t + ϕ(x, 0)

 

N f (x) − 4f



for all x ∈ X and all t > 0. So d(f, Jf ) ≤ 1. By Theorem 1.4, there exists a mapping Q : X → Y satisfying the following: (1) Q is a fixed point of J, i.e.,   1 x = Q(x) (3.5) Q 2 4 for all x ∈ X. Since f : X → Y is even, Q : X → Y is an even mapping. The mapping Q is a unique fixed point of J in the set M = {g ∈ S : d(f, g) < ∞}. This implies that Q is a unique mapping satisfying (3.5) such that there exists a µ ∈ (0, ∞) satisfying t N (f (x) − Q(x), µt) ≥ t + ϕ(x, 0) for all x ∈ X; (2) d(J n f, Q) → 0 as n → ∞. This implies the equality   x = Q(x) N - lim 4n f n→∞ 2n for all x ∈ X; 1 (3) d(f, Q) ≤ 1−L d(f, Jf ), which implies the inequality d(f, Q) ≤

1 . 1−L

This implies that the inequality (3.4) holds. By (3.2),           x+y x−y x y n N 4 2f + 2f −f −f n+1 n+1 n 2 2 2 2n            x−y x y t x+y n n
−ρ 4 f +f − 2f − 2f ,4 t ≥ n n n n 2 2 2 2 t + ϕ 2xn , 2yn for all x, y ∈ X, all t > 0 and all n ∈ N. So           x+y x−y x y n N 4 2f + 2f −f −f n+1 n+1 n 2 2 2 2n            x−y x y x+y +f − 2f − 2f ,t ≥ −ρ 4n f 2n 2n 2n 2n for all x, y ∈ X, all t > 0 and all n ∈ N. Since limn→∞ t > 0,

t 4n t Ln + 4n ϕ(x,y) 4n

t 4n

+

t 4n Ln 4n ϕ (x, y)

= 1 for all x, y ∈ X and all

x+y x−y 2Q +2 − Q(x) − Q(y) = ρ (Q (x + y) + Q (x − y) − 2Q(x) − 2Q(y)) 2 2 for all x, y ∈ X=. By Lemma 3.1, the mapping Q : X → Y is quadratic, as desired. 







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Corollary 3.3. Let θ ≥ 0 and let p be a real number with p > 2. Let X be a normed vector space with norm k · k. Let f : X → Y be a mapping satisfying f (0) = 0 and     x+y x−y N (2f + 2f − f (x) − f (y) (3.6) 2 2 t −ρ (f (x + y) + f (x − y) − 2f (x) − 2f (y)) , t) ≥ t + θ(kxkp + kykp ) for all x, y ∈ X and all t > 0. Then Q(x) := N -limn→∞ 4n f ( 2xn ) exists for each x ∈ X and defines a quadratic mapping Q : X → Y such that N (f (x) − Q(x), t) ≥

(2p − 4)t (2p − 4)t + 2p θkxkp

for all x ∈ X. Proof. The proof follows from Theorem 3.2 by taking ϕ(x, y) := θ(kxkp + kykp ) for all x, y ∈ X. Then we can choose L = 22−p , and we get the desired result.  Theorem 3.4. Let ϕ : X 2 → [0, ∞) be a function such that there exists an L < 1 with   x y , ϕ(x, y) ≤ 4Lϕ 2 2 for all x, y ∈ X. Let f : X → Y be a mapping satisfying f (0) = 0 and (3.2). Then Q(x) := N limn→∞ 41n f (2n x) exists for each x ∈ X and defines a quadratic mapping Q : X → Y such that (1 − L)t N (f (x) − Q(x), t) ≥ (3.7) (1 − L)t + ϕ(x, 0) for all x ∈ X and all t > 0. Proof. Let (S, d) be the generalized metric space defined in the proof of Theorem 3.2. Now we consider the linear mapping J : S → S such that 1 Jg(x) := g (2x) 4 for all x ∈ X. It follows from (3.4) that   1 t N f (x) − f (2x), Lt ≥ 4 t + ϕ(x, 0) for all x ∈ X and all t > 0. So d(f, Jf ) ≤ L. Hence 1 d(f, Q) ≤ , 1−L which implies that the inequality (3.7) holds. The rest of the proof is similar to the proof of Theorem 3.2.



Corollary 3.5. Let θ ≥ 0 and let p be a real number with 0 < p < 2. Let X be a normed vector space with norm k · k. Let f : X → Y be a mapping satisfying f (0) = 0 and (3.6). Then Q(x) := N -limn→∞ 41n f (2n x) exists for each x ∈ X and defines a quadratic mapping Q : X → Y such that N (f (x) − Q(x), t) ≥

(4 − 2p )t (4 − 2p )t + 2p θkxkp

for all x ∈ X.

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Proof. The proof follows from Theorem 3.4 by taking ϕ(x, y) := θ(kxkp + kykp ) for all x, y ∈ X. Then we can choose L = 2p−2 , and we get the desired result.  Acknowledgments S. Y. Jang was supported by Basic Science Research Program through the National Research Foundation of Korea funded by the Ministry of Education, Science and Technology (2013007226). References [1] T. Aoki, On the stability of the linear transformation in Banach spaces, J. Math. Soc. Japan 2 (1950), 64–66. [2] T. Bag and S.K. Samanta, Finite dimensional fuzzy normed linear spaces, J. Fuzzy Math. 11 (2003), 687– 705. [3] T. Bag and S.K. Samanta, Fuzzy bounded linear operators, Fuzzy Sets and Systems 151 (2005), 513–547. [4] L. C˘ adariu, L. G˘ avruta, P. G˘ avruta, On the stability of an affine functional equation, J. Nonlinear Sci. Appl. 6 (2013), 60–67. [5] L. C˘ adariu and V. Radu, Fixed points and the stability of Jensen’s functional equation, J. Inequal. Pure Appl. Math. 4, no. 1, Art. ID 4 (2003). [6] L. C˘ adariu and V. Radu, On the stability of the Cauchy functional equation: a fixed point approach, Grazer Math. Ber. 346 (2004), 43–52. [7] L. C˘ adariu and V. Radu, Fixed point methods for the generalized stability of functional equations in a single variable, Fixed Point Theory and Applications 2008, Art. ID 749392 (2008). [8] S.C. Cheng and J.M. Mordeson, Fuzzy linear operators and fuzzy normed linear spaces, Bull. Calcutta Math. Soc. 86 (1994), 429–436. [9] P.W. Cholewa, Remarks on the stability of functional equations, Aequationes Math. 27 (1984), 76–86. [10] S. Czerwik, On the stability of the quadratic mapping in normed spaces, Abh. Math. Sem. Univ. Hamburg 62 (1992), 59–64. [11] J. Diaz and B. Margolis, A fixed point theorem of the alternative for contractions on a generalized complete metric space, Bull. Amer. Math. Soc. 74 (1968), 305–309. [12] W. Fechner, Stability of a functional inequalities associated with the Jordan-von Neumann functional equation, Aequationes Math. 71 (2006), 149–161. [13] C. Felbin, Finite dimensional fuzzy normed linear spaces, Fuzzy Sets and Systems 48 (1992), 239–248. [14] P. G˘ avruta, A generalization of the Hyers-Ulam-Rassias stability of approximately additive mappings, J. Math. Anal. Appl. 184 (1994), 431–436. [15] A. Gil´ anyi, Eine zur Parallelogrammgleichung a ¨quivalente Ungleichung, Aequationes Math. 62 (2001), 303– 309. [16] A. Gil´ anyi, On a problem by K. Nikodem, Math. Inequal. Appl. 5 (2002), 707–710. [17] D.H. Hyers, On the stability of the linear functional equation, Proc. Nat. Acad. Sci. U.S.A. 27 (1941), 222–224. [18] D.H. Hyers, G. Isac and Th.M. Rassias, Stability of Functional Equations in Several Variables, Birkh¨ auser, Basel, 1998. [19] G. Isac and Th.M. Rassias, Stability of ψ-additive mappings: Appications to nonlinear analysis, Internat. J. Math. Math. Sci. 19 (1996), 219–228. [20] S. Jung, Hyers-Ulam-Rassias Stability of Functional Equations in Mathematical Analysis, Hadronic Press lnc., Palm Harbor, Florida, 2001. [21] A.K. Katsaras, Fuzzy topological vector spaces II, Fuzzy Sets and Systems 12 (1984), 143–154. [22] J. Kim, G. A. Anastassiou and C. Park, Additive ρ-functional inequalities in fuzzy normed spaces, J. Comput. Anal. Appl. (to appear). [23] I. Kramosil and J. Michalek, Fuzzy metric and statistical metric spaces, Kybernetica 11 (1975), 326–334. [24] S.V. Krishna and K.K.M. Sarma, Separation of fuzzy normed linear spaces, Fuzzy Sets and Systems 63 (1994), 207–217. [25] J. Lee, C. Park and D. Shin, An AQCQ-functional equation in matrix normed spaces, Results Math. 27 (2013), 305–318. [26] D. Mihet¸ and V. Radu, On the stability of the additive Cauchy functional equation in random normed spaces, J. Math. Anal. Appl. 343 (2008), 567–572.

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[27] M. Mirzavaziri and M.S. Moslehian, A fixed point approach to stability of a quadratic equation, Bull. Braz. Math. Soc. 37 (2006), 361–376. [28] A.K. Mirmostafaee, M. Mirzavaziri and M.S. Moslehian, Fuzzy stability of the Jensen functional equation, Fuzzy Sets and Systems 159 (2008), 730–738. [29] A.K. Mirmostafaee and M.S. Moslehian, Fuzzy versions of Hyers-Ulam-Rassias theorem, Fuzzy Sets and Systems 159 (2008), 720–729. [30] A.K. Mirmostafaee and M.S. Moslehian, Fuzzy approximately cubic mappings, Inform. Sci. 178 (2008), 3791–3798. [31] C. Park, Fixed points and Hyers-Ulam-Rassias stability of Cauchy-Jensen functional equations in Banach algebras, Fixed Point Theory and Applications 2007, Art. ID 50175 (2007). [32] C. Park, Generalized Hyers-Ulam-Rassias stability of quadratic functional equations: a fixed point approach, Fixed Point Theory and Applications 2008, Art. ID 493751 (2008). [33] C. Park, Additive ρ-functional inequalities and equations, J. Math. Inequal. 9 (2015), 17–26. [34] C. Park, Additive ρ-functional inequalities in non-Archimedean normed spaces, J. Math. Inequal. 9 (2015), 397–407. [35] C. Park, Y. Cho and M. Han, Stability of functional inequalities associated with Jordan-von Neumann type additive functional equations, J. Inequal. Appl. 2007, Art. ID 41820 (2007). [36] C. Park, K. Ghasemi, S. G. Ghaleh and S. Jang, Approximate n-Jordan ∗-homomorphisms in C ∗ -algebras, J. Comput. Anal. Appl. 15 (2013), 365-368. [37] C. Park, A. Najati and S. Jang, Fixed points and fuzzy stability of an additive-quadratic functional equation, J. Comput. Anal. Appl. 15 (2013), 452–462. [38] C. Park and Th.M. Rassias, Fixed points and generalized Hyers-Ulam stability of quadratic functional equations, J. Math. Inequal. 1 (2007), 515–528. [39] V. Radu, The fixed point alternative and the stability of functional equations, Fixed Point Theory 4 (2003), 91–96. [40] Th.M. Rassias, On the stability of the linear mapping in Banach spaces, Proc. Amer. Math. Soc. 72 (1978), 297–300. [41] Th.M. Rassias, Problem 16; 2, Report of the 27th International Symp. on Functional Equations, Aequationes Math. 39 (1990), 292–293; 309. [42] Th.M. Rassias and K. Shibata, Variational problem of some quadratic functionals in complex analysis, J. Math. Anal. Appl. 228 (1998), 234–253. [43] J. R¨ atz, On inequalities associated with the Jordan-von Neumann functional equation, Aequationes Math. 66 (2003), 191–200. ˇ ankov´ [44] L. Reich, J. Sm´ıtal and M. Stef´ a, Singular solutions of the generalized Dhombres functional equation, Results Math. 65 (2014), 251–261. [45] S. Schin, D. Ki, J. Chang and M. Kim, Random stability of quadratic functional equations: a fixed point approach, J. Nonlinear Sci. Appl. 4 (2011), 37–49. [46] S. Shagholi, M. Bavand Savadkouhi and M. Eshaghi Gordji, Nearly ternary cubic homomorphism in ternary Fr´echet algebras, J. Comput. Anal. Appl. 13 (2011), 1106–1114. [47] S. Shagholi, M. Eshaghi Gordji and M. Bavand Savadkouhi, Stability of ternary quadratic derivation on ternary Banach algebras, J. Comput. Anal. Appl. 13 (2011), 1097–1105. [48] D. Shin, C. Park and Sh. Farhadabadi, On the superstability of ternary Jordan C ∗ -homomorphisms, J. Comput. Anal. Appl. 16 (2014), 964–973. [49] D. Shin, C. Park and Sh. Farhadabadi, Stability and superstability of J ∗ -homomorphisms and J ∗ -derivations for a generalized Cauchy-Jensen equation, J. Comput. Anal. Appl. 17 (2014), 125–134. [50] F. Skof, Propriet` a locali e approssimazione di operatori, Rend. Sem. Mat. Fis. Milano 53 (1983), 113–129. [51] S. M. Ulam, A Collection of the Mathematical Problems, Interscience Publ. New York, 1960. [52] J.Z. Xiao and X.H. Zhu, Fuzzy normed spaces of operators and its completeness, Fuzzy Sets and Systems 133 (2003), 389–399. Choonkil Park Research Institute for Natural Sciences, Hanyang University, Seoul 04763, Korea E-mail address: [email protected] Sung Young Jang Department of Mathematics, University of Ulsan, Ulsan 44610, Korea E-mail address: [email protected]

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Remarks on common fixed point results for cyclic contractions in ordered b-metric spaces Huaping Huang1 , Stojan Radenović2 , Tatjana Aleksic Lampert3 1

School of Mathematics and Statistics, Hubei Normal University, Huangshi, 435002, China 2 State University of Novi Pazar, Serbia 3 Faculty of Science, Department of Mathematics and Informatics Radoja Domanovica 12, 34 000 Kragujevac, Serbia

1

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Abstract. The purpose of this paper is to prove that some common fixed point theorems for

cyclic contractions are equivalent to the counterpart of noncyclic contractions in the same setting. Our results improve and complement several results for cyclic contractions established in [Fixed Point Theory Appl., 2013: 256]. Furthermore, an application to the existence and uniqueness of solution for a class of integral equations is given to illustrate the superiority of the obtained assertions. Keywords: (A, B)-weakly increasing, common fixed point, altering distance function, regular MSC: 47H10, 54H25.

————————————————————————————————————1. INTRODUCTION AND PRELIMINARIES Since Banach fixed point theorem (see [1]) appeared in the world, there have been overwhelming trend in mathematical activities. This theorem presents numerous applications. For instance, it gives the conditions under which maps (single or multivalued) have solutions. Fixed point theory itself is a beautiful mixture of analysis (pure and applied), topology, and geometry. It has been revealed as a very powerful and important tool in the study of nonlinear phenomena. Over the last several decades, scholars have generalized this theorem greatly from several directions. Whereas, one of most influential generalizations is from spaces. Wherein, the fact from usual metric spaces to b-metric spaces is very popular. b-metric spaces, also called metric type spaces, were introduced in [2] and [3]. Afterwards, a large number of fixed point theorems have been presented in such spaces (see [4-15]). Recently, scholars cultivate some interests in fixed point theorems for cyclic contractions (see [15-19]). However, the authors of this paper find that many fixed point results for cyclic contractions are actually equivalent to those of noncyclic contractions in the same spaces. Throughout this paper, we obtain some equivalences between cyclic contractions and noncyclic contractions in the setting of b-metric spaces. Moreover, we obtain some common fixed point theorems without considering cyclic contractions. Further, as an applications, we cope with the existence and uniqueness of solutions of integral equations. For the sake of the reader, we recall some well-known concepts and results as follows. Definition 1.1([9]) Let X be a (nonempty) set and s ≥ 1 a given real number. A function d : X × X → [0, ∞) is called a b-metric on X if, for all x, y, z ∈ X, the following conditions hold: (b1) d (x, y) = 0 if and only if x = y; (b2) d (x, y) = d (y, x); 1

E-mail addresses: [email protected] (H. Huang); [email protected], [email protected] (S. Radenović); [email protected] (T.-A. Lampert)

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(b3) d (x, z) ≤ s [d (x, y) + d (y, z)]. In this case, the pair (X, d) is called a b-metric space or metric type space. If (X, ) is still a partially ordered set, then (X, , d) is called an ordered b-metric space. Otherwise, for some other definitions in b-metric spaces such as convergence, Cauchy sequence, completeness, see [8-15] and the references therein. Definition 1.2([22]) A function ϕ : [0, ∞) → [0, ∞) is called an altering distance function if the following properties hold: (1) ϕ is continuous and nondecreasing; (2) ϕ (t) = 0 if and only if t = 0. Definition 1.3([21]) Let (X, ) be a partially ordered set, and let A and B be closed subsets of X with A ∪ B = X. Let f, g : X → X be two mappings. The pair (f, g) is said to be (A, B)-weakly increasing if f x  gf x for all x ∈ A and gy  f gy for all y ∈ B. In particular, (f, g) is said to be weakly increasing if f x  gf x and gx  f gx for all x ∈ X. Definition 1.4([13]) An ordered b-metric space (X, , d) is called regular if for any nondecreasing sequence {xn } in X such that xn → x (n → ∞), one has xn  x for all n ∈ N. Definition 1.5([16]) Let A and B be nonempty subsets of a metric space (X, d) and T : A ∪ B → A ∪ B. Then T is called a cyclic map if T (A) ⊆ B and T (B) ⊆ A. Shatanawi and Postolache proved the following common fixed point results for cyclic contractions in the framework of ordered metric spaces. Theorem 1.6([19]) Let (X, , d) be a complete ordered metric space, and let A, B be closed nonempty subsets of X with X = A ∪ B. Let f, g : X → X be (A, B)-weakly increasing mappings with respect to . Suppose that (a) X = A ∪ B is a cyclic representation of X with respect to the pair (f, g), i.e., f (A) ⊆ B and g (B) ⊆ A; (b) there exist 0 < δ < 1 and an altering distance function ψ such that for any comparable elements x, y ∈ X with x ∈ A and y ∈ B, we have    1 ; ψ (d (f x, gy)) ≤ δψ max d (x, y) , d (x, f x) , d (y, gy) , (d (x, gy) + d (y, f x)) 2 (c) f or g is continuous, or (c’) (X, , d) is regular. Then f and g have a common fixed point. It should be noted that cyclic contractions (unlike Banach-type contractions) need not to be continuous. This concept is an interesting increase in nonlinear analysis. In addition, Hussain et al. [15] introduced the notion of ordered cyclic weakly (ψ, ϕ, L, A, B)contraction and proved the following fixed point results. Definition 1.7 Let (X, , d) be an ordered b-metric space, let f, g : X → X be two mappings, and let A and B be nonempty closed subsets of X. The pair (f, g) is called an ordered cyclic weakly (ψ, ϕ, L, A, B)-contraction if (1) X = A ∪ B is a cyclic representation of X with respect to the pair (f, g); (2) there exist two altering distance functions ψ, ϕ and a constant L ≥ 0, such that for arbitrary comparable elements x, y ∈ X with x ∈ A and y ∈ B, we have
ψ s2 d (f x, gy) ≤ ψ (Ms (x, y)) − ϕ (Ms (x, y)) + Lψ (N (x, y)) , where   1 Ms (x, y) = max d (x, y) , d (x, f x) , d (y, gy) , (d (x, gy) + d (y, f x)) 2s 539

(1.1)

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and N (x, y) = min {d (y, gy) , d (x, gy) , d (y, f x)} .

(1.2)

Theorem 1.8 Let (X, , d) be a complete ordered b-metric space, and let A and B be closed subsets of X. Let f, g : X → X be (A, B)-weakly increasing mappings with respect to . Suppose that (a) the pair (f, g) is an ordered cyclic weakly (ψ, ϕ, L, A, B)-contraction; (b) f or g is continuous. Then f and g have a common fixed point u ∈ A ∩ B. Theorem 1.9 Let the hypothesis of Theorem 1.8 be satisfied, except that condition (b) is replaced by the following assumption: (b’) (X, , d) is regular. Then f and g have a common fixed point u ∈ A ∩ B. The following lemmas will be utilized in the proof of our main results. Lemma 1.10([20]) If some ordinary fixed point theorem in the setting of complete metric spaces has a true cyclic-type extension, then these both theorems are equivalent. Lemma 1.11([5]) Let {yn } be a sequence in a b-metric space (X, d) with s ≥ 1 such that d (yn , yn+1 ) ≤ λd (yn−1 , yn ) for some λ ∈ [0, 1s ), and each n = 1, 2, . . .. Then {yn } is a Cauchy sequence in (X, d). 2. MAIN RESULTS In this section, following the trend mentioned above, we extend such considerations to the simpler equivalent results so that we can enlarge, in a unified manner, the class of problems that can be investigated. Theorem 2.1 Let (X, , d) be a complete ordered metric space, and let f, g : X → X be the weakly increasing mappings. Suppose that (a) there exist 0 < δ < 1 and an altering distance function ψ such that for any comparable elements x, y ∈ X, we have that 1 ψ(d(f x, gy)) ≤ δψ(max{d(x, y), d(x, f x), d(y, gy), (d(x, gy) + d(y, f x))}); 2 (b) f or g is continuous, or (c) (X, , d) is regular. Then f and g have a common fixed point. The proof of Theorem 2.1 is trivial because we have the following: Theorem 2.2 Theorem 1.6 is equivalent with Theorem 2.1. Proof Putting A = B = X in Theorem 1.6, we obtain Theorem 2.1. In other words, Theorem 1.6 implies Theorem 2.1. The proof for the converse is same as in [20-21]. Namely, we depend on Lemma 1.10. In the sequel, we announce the following noncyclic case result. Theorem 2.3 Let (X, , d) be a complete ordered b-metric space, and let f, g : X → X be the weakly increasing mappings. Suppose that there exist altering distance function ψ and ϕ, and the constants ε > 1, L ≥ 0 such that ψ (s ε d (f x, gy)) ≤ ψ (Ms (x, y)) − ϕ (Ms (x, y)) + Lψ (N (x, y))

(2.1)

for all comparable x, y ∈ X, where Ms (x, y) and N (x, y) are given by (1.1) and (1.2), respectively. If either f or g is continuous, or the space (X, , d) is regular, then f and g have a common fixed point. 540

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Proof Choose x0 ∈ X and construct a sequence {xn } as follows: x2n+1 = f x2n ,

x2n+2 = gx2n+1 .

Since (f, g) is weakly increasing, then x1  x2  x3  · · ·  xn  xn+1  · · · . If x2n = x2n+1 or x2n+1 = x2n+2 for some n, then the proof is trivial and hence we omit it. Now we assume that xn 6= xn+1 for all n. We shall only prove that d (xn , xn+1 ) ≤ λd (xn−1 , xn ) ,

(2.2)

for all n = 1, 2, ..., where λ ∈ [0, 1s ). Indeed, by (2.1), it establishes that ψ (s ε d (x2n+1 , x2n+2 )) = ψ (s ε d (f x2n , gx2n+1 )) ≤ ψ (Ms (x2n , x2n+1 )) + Lψ (N (x2n , x2n+1 )) , where Ms (x2n , x2n+1 ) = max {d (x2n , x2n+1 ) , d (x2n+1 , x2n+2 )} and N (x2n , x2n+1 ) = 0. Hence, it is not hard to verify that s ε d (x2n+1 , x2n+2 ) ≤ d (x2n , x2n+1 ) .

(2.3)

s ε d (x2n , x2n+1 ) ≤ d (x2n−1 , x2n ) .

(2.4)

Similarly, we obtain that

Uniting (2.3) and (2.4), ones have (2.2). Now by Lemma 1.11, we demonstrate that {xn } is a Cauchy sequence and therefore there exists x ∈ X such that xn → x as n → ∞. Thus lim x2n+1 = lim f x2n = x.

n→∞

(2.5)

n→∞

In view of x2n → x, without loss of generality, assume that f is continuous. Then lim f x2n = f x.

(2.6)

n→∞

It follows immediately from (2.5) and (2.6) that x = f x. Further, by using x  x we can prove that the condition (2.1) implies the existence of common fixed point of f and g. Indeed, put x = y in (2.1) it follows that ψ (s ε d (f x, gx)) ≤ ψ (Ms (x, x)) − ϕ (Ms (x, x)) + Lψ (N (x, x)) . Now that Ms (x, x) = d(x, gx) and N (x, y) = 0, one has ψ (s ε d (f x, gx)) ≤ ψ (d (x, gx)) − ϕ (d (x, gx)) + L · 0 ≤ ψ (d (x, gx)) , which means that s ε d (f x, gx) = s ε d (x, gx) ≤ d (x, gx) , Consequently, x = gx (because ε > 1). The assumption of continuity of one of the mappings f or g can be replaced by the condition that b-metric space (X, , d) is regular. 541

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In fact, let (X, , d) be regular. Via the mentioned above, we can construct an increasing sequence {xn } in X such that xn → x (n → ∞) for some x ∈ X. Then xn  x for all n ∈ N. We shall have to show that f x = gx = x. First, we have 1 d (x, gx) ≤ d (x, x2n+1 ) + d (f x2n , gx) . (2.7) s By (2.1) we get ψ (s ε d (f x2n , gx)) ≤ ψ (Ms (x2n , x)) + Lψ (N (x2n , x)) , where   d (x2n , gx) + d (x, x2n+1 ) Ms (x2n , x) = max d (x2n , x) , d (x2n , x2n+1 ) , d (x, gx) , (2.8) 2s and N (x2n , x) = min {d (x, gx) , d (x2n , gx) , d (x, x2n+1 )} .

(2.9)

Letting n → ∞ in (2.8) and (2.9) and using d (x2n , x) + d (x, gx) d (x, x2n+1 ) d (x2n , gx) + d (x, x2n+1 ) ≤ + , 2s 2 2s we obtain limn→∞ Ms (x2n , x) = d (x, gx) and limn→∞ N (x2n , x) = 0. Further, we deduce that   ε lim ψ (s d (f x2n , gx)) ≤ ψ lim Ms (x2n , x) + L · ψ (0) = ψ (d (x, gx)) . n→∞

n→∞

Since ψ is nondecreasing, we arrive at lim s ε d (f x2n , gx) ≤ d (x, gx) .

n→∞

(2.10)

Now (2.7) and (2.10) imply that gx = x. Similarly, we claim that f x = x. Remark 2.4 Theorem 2.3 improves and generalizes the main results of [15] (also see Theorem 1.8 and Theorem 1.9 ) in several directions. For one thing, the constant ε > 1 is arbitrary and is not only limited to ε = 2 stated by Theorem 1.8 and Theorem 1.9. This probably brings us more convenience in applications. For another thing, Theorem 2.3 dismisses the cyclic representation. In addition, the proof Theorem 2.3 is much simpler than the one of Theorem 1.8 and Theorem 1.9. Finally we announce the main result of this paper: Theorem 2.5 Theorem 1.8 together with Theorem 1.9 is equivalent to Theorem 2.3 in case of ε = 2. Proof For all details and explanations see [20], [21] and the proof of Theorem 2.1. 3. APPLICATION By using Theorem 2.3, we shall consider the existence of solutions for the following integral equation with an unknown function u: Z T u (t) = G (t, s) f (s, u (s)) ds, t ∈ [0, T ] , (3.1) 0

where T > 0 is a constant, f : [0, T ] × R → R, G : [0, T ] × [0, T ] → [0, ∞) are given continuous functions. 542

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Denote X = C [0, T ] be the set of real continuous functions on [0, T ] and let d : X ×X → R be given by +

d (u, v) = max |u (t) − v (t)|2 , 0≤t≤T

∀u, v ∈ X.

It is easy to check that (X, d) is a complete b-metric space with parameter s = 2. We endow X with the partial order given by x  y ⇔ x (t) ≤ y (t) for all t ∈ [0, T ] . Validly, (X, , d) is regular. Define a mapping T : X → X by Z T G (t, z) f (z, u (z)) dz, t ∈ [0, T ] , T u (t) := 0

then u is a solution of the given equation (3.1) if and only if it is a fixed point of T . We shall prove that T has a fixed point under the following assumptions. (i) For all z ∈ [0, T ] , f (z, .) is a decreasing function, that is, x, y ∈ R, x ≥ y implies f (z, x) ≤ f (z, y); (ii) There exists a constant γ > 0 such that Z 0≤t≤T

T

G (t, z) dz ≤

max

0

10 √ ; 21 γ

(iii) For all z ∈ [0, T ] and for all comparable x, y ∈ X, 0 ≤ |f (z, x (z)) − f (z, y (z)) |   ≤ γ max |x (z) − y (z)|2 , |x (z) − T x (z)|2 , |y (z) − T y (z)|2 ,  12 |x (z) − T y (z)|2 + |y (z) − T x (z)|2 . (3.2) 4
2.1 (iv) There exists a constant ε ∈ 1, 2 ln . ln 2 Theorem 3.1 Under the conditions (i)-(iv), the equation (3.1) has a solution x∗ ∈ X. Proof First of all, if x  y, then by (i), we have Z T y (t) − T x (t) =

T

G (t, z) [f (z, y (z)) − f (z, x (z))] dz ≥ 0, t ∈ [0, T ] . 0

That is, T x  T y. This means that T is increasing. By virtue of (3.2), we have that [f (z, x) − f (z, y)]2  ≤ γ max d (x, y) , d (x, T x) , d (y, T y) ,  d (x, T y) + d (y, T x) . 4

543

(3.3)

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Then for all t ∈ [0, T ] and all comparable x, y ∈ X, by (ii) and (3.3), we speculate that d (T x, T y) = max |T x (t) − T y (t)|2 t∈[0,T ]

Z

G (t, z) [f (z, x (z)) − f (z, y (z))] dz

= max t∈[0,T ]

2

T

0

  100 d (x, T y) + d (y, T x) ≤ max d (x, y) , d (x, T x) , d (y, T y) , . 441 4 100 By (iv), it follows that 441 < 21ε = s1ε , thus all the conditions of Theorem 2.3 are satisfied where ψ, ϕ are identity mappings and T = f = g, L = 0. So T has a fixed point u(t) ∈ X, that is, the integral equation (3.1) has a solution u(t) ∈ X = C [0, T ]. Remark 3.2 In the above application we use ordinary fixed point theorem, while Corollary 2 of [15] uses cyclical-type fixed point result. Actually, these both results are equivalent, then our approach has an advantage because we use the conditions (i)-(iv), while in Corollary 2 of [15] authors utilize the conditions (4.2)-(4.7) as well as two subsets A1 and A2 . Also, our application shows that their main result is not applicable.

ACKNOWLEDGMENTS The research is partially supported by the science and technology research project of education department in Hubei Province of China (B2015137).

References [1] S. Banach, Sur les opérations dans les ensembles abstraits et leur application aux équations intégrales, Fundam. Math., 3 (1922), 133-181. [2] I. A. Bakhtin, The contraction principle in quasimetric spaces, Funct. Anal., 30 (1989) 26-37. [3] S. Czerwik, Contraction mappings in b-metric spaces, Acta Math. Inform., Univ. Ostrav., 1 (1993), 5-11. [4] N. Hussain, D. Ðorić, Z. Kadelburg, S. Radenović, Suzuki-type fixed point results in metric type spaces, Fixed Point Theory Appl., 2012, 2012: 126. [5] M. Jovanović, Z. Kadelburg, S. Radenović, Common fixed point results in metric-type spaces, Fixed Point Theory Appl., 2010, Article ID 978121, 15 pages. [6] M. A. Khamsi, N. Hussain, KKM mappings in metric type spaces, Nonlinear Anal. 73 (2010), 3123-3129. [7] D. Ðukić, Z. Kadelburg, S. Radenović, Fixed point of Geraghty-type mappings in various generalized metric spaces, Abstract Appl. Anal., 2011, Article ID 561245, 13 pages. [8] M. Kir, H. Kiziltunc, On some well known fixed point theorems in b-metric spaces, Turkish J. Anal. Number Theory, 1(1) (2013), 13-16. [9] V. Parvaneh, J. R. Roshan, S. Radenović, Existence of tripled coincidence points in ordered b-metric spaces and an application to a system of integral equations, Fixed Point Theory Appl., 2013, 2013: 130. 544

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[10] J. R. Roshan, V. Parvaneh, N. Shobkolaei, S. Sedghi, W. Shatanawi, Common fixed points of almost generalized (ψ, ϕ)s -contractive mappings in ordered b-metric spaces, Fixed Point Theory Appl., 2013, 2013: 159. [11] A. Amini-Harandi, Fixed point theory for quasi-contraction maps in b-metric spaces, Fixed Point Theory, 15(2) (2014), 351-358. [12] J. R. Roshan, V. Parvaneh, Z. Kadelburg, Common fixed point theorems for weakly isotone increasing mappings in ordered b-metric spaces, J. Nonlinear Sci. Appl., 7 (2014), 229-245. [13] H. Huang, S. Radenović, J. Vujaković, On some recent coincidence and immediate consequences in partially ordered b -metric spaces, Fixed Point Theory Appl., 2015, 2015: 63. [14] J. R. Roshan, V. Parvaneh, S. Radenović, M. Rajović, Some coincidence point results for generalized (ψ, ϕ)-weakly contractions in ordered b-metric spaces, Fixed Point Theory Appl., 2015, 2015: 68. [15] N. Hussain, V. Parvaneh, J. R. Roshan, Z. Kadelburg, Fixed points of cyclic (ψ, ϕ, L, A, B)-contractive mappings in ordered b-metric spaces with applications, Fixed Point Theory Appl., 2013, 2013: 256. [16] W. A. Kirk, P. S. Srinavasan, P. Veeramani, Fixed points for mapping satisfying cyclical contractive conditions, Fixed Point Theory, 4 (2003), 79-89. [17] G. Petrusel, Cyclic representations and periodic points, Studia Univ. Babes-Bolyai Math., 50 (2005), 107-112. [18] I. A. Rus, Cyclic representations and fixed points, Ann. Tiberiu Popovicu Semin. Funct. Equ. Approx. Convexity, 3 (2005), 171-178. [19] W. Shatanawi, M. Postolache, Common fixed point results of mappings under nonlinear contraction of cyclic form in ordered metric spaces, Fixed Point Theory Appl., 2013, 2013: 60. [20] S. Radenović, A note on fixed point theory for cyclic weaker Meir-Keeler function in complete metric spaces, Int. J. Anal. Appl., 7(1) (2015), 16-21. [21] S. Radenović, Some remarks on mappings satisfying cyclical contractive conditions, Afr. Mat., 2015, doi: 10.1007/s13370-015-0339-2, 7 pages. [22] M. S. Khan, M. Swaleh, S. Sessa, Fixed point theorems by altering distances between the points. Bul. Aust. Math. Soc., 30(1) (1984), 1-9.

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J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 22, NO.3, 2017, COPYRIGHT 2017 EUDOXUS PRESS, LLC

A FIXED POINT METHOD TO THE STABILITY OF A JENSEN FUNCTIONAL EQUATION IN INTUITIONISTIC FUZZY 2-BANACH SPACES CHOONKIL PARK, EHSAN MOVAHEDNIA∗ , GEORGE A. ANASTASSIOU, AND SUNGSIK YUN∗ Abstract. In this paper, we recall the notion of intuitionistic fuzzy 2-normed space introduced in [1] and using the fixed point method, we investigate the Hyers-Ulam stability of the following functional equation 2f

x + y

+f

x − y

2 2 in intuitionistic fuzzy 2-Banach spaces.

+f

y − x 2

= f (x) + f (y)

(1)

1. Introduction The concept of the stability for functional equations was introduced for the first time by Ulam in 1940 [2]. He proposed the famous Ulam stability problem for a metric group homomorphism. In 1941, Hyers [3] solved this stability problem for additive mappings subject to the Hyers condition on approximately additive mappings in Banach spaces. In 1951, Bourgin [4] treated the Ulam stability problem for additive mappings. Subsequently the result of Hyers was generalized by Rassias [5] for linear mapping by considering an unbounded Cauchy difference. The functional equation f (x + y) + f (x − y) = 2f (x) + 2f (y) is called a quadratic functional equation. In particular, every solution of the quadratic functional equation is said to be a quadratic mapping. The Hyers-Ulam stability problem for the quadratic functional equation was proved by Skof [6] for mappings f : X → Y, where X is a normed space and Y is a Banach space. In 1984, Katsaras [7] defined a fuzzy norm on a linear space to construct a fuzzy vector topological structure on the space. Later, some mathematicians have defined fuzzy norms on a linear space from various points of view [8, 9]. In particular, in 2003, Bag and Samanta [10], following Cheng and Mordeson [11], gave an idea of a fuzzy norm in such a manner that the corresponding fuzzy metric is of Kramosil and Michalek type [12]. They also established a decomposition theorem of a fuzzy norm into a family of crisp norms and investigated some properties of fuzzy normed spaces. Recently, considerable attention has been increasing to the problem of fuzzy stability of functional equations. Several various stability results concerning Cauchy, Jensen, simple quadratic, and cubic functional equations have been investigated [13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24]. 2010 Mathematics Subject Classification. 47S40, 54A40, 46S40, 39B52, 47H10. Key words and phrases. Intuitionistic fuzzy 2-normed space; Fixed point; Hyers-Ulam stability; Jensen functional equation, ∗ Corresponding author.

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CHOONKIL PARK et al 546-557

J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 22, NO.3, 2017, COPYRIGHT 2017 EUDOXUS PRESS, LLC

C. PARK, E. MOVAHEDNIA, G. A. ANASTASSIOU, AND S. YUN

Quite recently, the stability results in the setting of intuitionistic fuzzy normed space have been studied in [25, 26, 27, 28]; respectively, while the idea of intuitionistic fuzzy normed space was introduced in [29]. 2. Preliminaries Definition 2.1. Let X be a real linear space of dimension greater than one and let k·, ·k be a real-valued function on X × X satisfying the following condition: (1) kx, yk=ky, xk for all x, y ∈ X ; (2) kx, yk = 0 if and only if x, y are linearly dependent; (3) kαx, yk = |α|kx, yk for all x, y ∈ X and α ∈ R; (4) kx, y + zk ≤ kx, yk + kx, zk for all x, y, z ∈ X . Then the function k·, ·k is called a 2-norm on X and the pair (X, k·, ·k) is called a 2-normed linear space. Definition 2.2. A binary operation ∗ : [0, 1]×[0, 1] → [0, 1] is a continuous t-norm if ∗ satisfies the following conditions: (1) ∗ is commutative and associative; (2) ∗ is continuous; (3) a ∗ 1 = a for all a ∈ [0, 1]; (4) a ∗ b ≤ c ∗ d, whenever a ≤ c and b ≤ d for all a, b, c, d ∈ [0, 1]. Example 2.1. An example of continuous t-norm is a ∗ b = min{a, b}. Definition 2.3. A binary operation  : [0, 1] × [0, 1] → [0, 1] is a continuous t-conorm if  satisfies the following conditions: (1)  is commutative and associative; (2)  is continuous; (3) a  0 = a for all a ∈ [0, 1]; (4) a  b ≤ c  d, whenever a ≤ c and b ≤ d for all a, b, c, d ∈ [0, 1]. Example 2.2. An example of continuous t-conorm is a  b = max{a, b}. Definition 2.4. Let X be a real linear space. A fuzzy subset µ of X × X × R is called a fuzzy 2-norm on X if and only if for all x, y, z ∈ X , and t, s, c ∈ R, (1) µ(x, y, t) = 0 for all t ≤ 0. (2) µ(x, y, t) = 1 if and only if x, y are linearly dependent for all t > 0. (3) µ(x, y, t) is invariant under any permutation of x, y. t ) for all t > 0 and c 6= 0. (4) µ(x, cy, t) = µ(x, y, |c| (5) µ(x + z, y, t + s) ≥ µ(x, y, t) ∗ µ(z, y, s) for all t, s > 0. (6) µ(x, y, .) is a non-decreasing function on R and lim µ(x, y, t) = 1

t→∞

Then µ is said to be a fuzzy 2-norm on a linear space X , and the pair (X , µ) is called a fuzzy 2-normed linear space.

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CHOONKIL PARK et al 546-557

J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 22, NO.3, 2017, COPYRIGHT 2017 EUDOXUS PRESS, LLC

FUNCTIONAL EQUATION IN INTUITIONISTIC FUZZY 2-BANACH SPACES

Example 2.3. Let (X , k·, ·k) be a 2-normed linear space. Define  t   t+kx,yk if t > 0 µ(x, y, t) =   0 if t ≤ 0, where x, y ∈ X and t ∈ R. Then (X , µ) is a fuzzy 2-normed linear space. Definition 2.5. Let (X , µ) be a fuzzy 2-normed linear space. Let {xn } be a sequence in X . Then {xn } is said to be convergent if there exists x ∈ X such that lim µ(xn − x, y, t) = 1

n→∞

for all t > 0 and all y ∈ X . Definition 2.6. Let (X , µ) be a fuzzy 2-normed linear space. Let {xn } be a sequence in X . Then {xn } is said to be a Cauchy sequence if lim µ(xn+p − xn , y, t) = 1

n→∞

for all t > 0, all y ∈ X and p = 1, 2, 3, · · · . Let (X , µ) be a fuzzy 2-normed linear space and {xn } be a Cauchy sequence in X . If {xn } is convergent in X then (X , µ) is said to be a fuzzy 2-Banach space. Definition 2.7. Let X be a real linear space. A fuzzy subset ν of X × X × R such that for all x, y, z ∈ X , and t, s, c ∈ R, (1) ν(x, y, t) = 1 for all t ≤ 0. (2) ν(x, y, t) = 0 if and only if x, y are linearly dependent for all t > 0. (3) ν(x, y, t) is invariant under any permutation of x, y. t (4) ν(x, cy, t) = ν(x, y, |c| ) for all t > 0, c 6= 0. (5) ν(x, y + z, t + s) ≤ ν(x, y, t)  ν(x, z, s) for all s, t > 0 (6) ν(x, y, .) is a nonincreasing function and lim ν(x, y, t) = 0

t→∞

Then ν is said to be an anti fuzzy 2-norm on a linear space X and the pair (X , ν) is called an anti fuzzy 2-normed linear space. Definition 2.8. Let (X , ν) be an anti fuzzy 2-normed linear space and {xn } be a sequence in X . Then {xn } is said to be convergent if there exists x ∈ X such that lim ν(xn − x, y, t) = 0

n→∞

for all t > 0 and all y ∈ X . Definition 2.9. Let (X , ν) be an anti fuzzy 2-normed linear space and {xn } be a sequence in X . Then {xn } is said to be a Cauchy sequence if lim ν(xn+p − xn , y, t) = 0

n→∞

for all t > 0, all y ∈ X and p = 1, 2, 3, · · · .

548

CHOONKIL PARK et al 546-557

J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 22, NO.3, 2017, COPYRIGHT 2017 EUDOXUS PRESS, LLC

C. PARK, E. MOVAHEDNIA, G. A. ANASTASSIOU, AND S. YUN

Let (X , ν) be an anti fuzzy 2-normed linear space and {xn } be a Cauchy sequence in X . If {xn } is convergent in X then (X , ν) is said to be an anti fuzzy 2-Banach space. The following lemma is easy to prove and we will omit it. Lemma 2.1. Consider the set L∗ and operation ≤L∗ defined by L∗ = {(x1 , x2 ) : (x1 , x2 ) ∈ [0, 1]2 and x1 + x2 ≤ 1} (x1 , x2 ) ≤L∗ (y1 , y2 ) ⇐⇒ x1 ≤ y1 , x2 ≥ y2 for all (x1 , x2 ), (y1 , y2 ) ∈ L∗ . Then (L∗ , ≤L∗ ) is a complete lattice. Definition 2.10. A continuous t-norm τ on L = [0, 1]2 is said to be continuous t-representable if there exist a continuous t-norm ∗ and a continuous t-conorm  on [0, 1] such that, for all x = (x1 , x2 ), y = (y1 , y2 ) ∈ L, τ (x, y) = (x1 ∗ y1 , x2  y2 ). Definition 2.11. Let X be a set. A function d : X × X → [0, ∞] is called a generalized metric on X if and only if d satisfies: (M1 ) d(x, y) = 0 ⇔ x = y ∀x, y ∈ X (M2 ) d(x, y) = d(y, x) ∀x, y ∈ X (M3 ) d(x, z) ≤ d(x, y) + d(y, z) ∀x, y, z ∈ X Theorem 2.1. ([30]) Let (X ,d) be a complete generalized metric space and J : X → X be a strictly contractive mapping with Lipschitz constant L < 1. Then, for all x ∈ X , either d(J n x, J n+1 x) = ∞ for all nonnegative integers n or there exists a positive integer n0 such that (a) d(J n x, J n+1 x) < ∞ for all n ≥ n0 ; (b) the sequence {J n x} converges to a fixed point y ∗ of J ; (c) y ∗ is the unique fixed point of J in the set Y = {y ∈ X : d(J n0 x, y) < ∞}; 1 d(y, J y) for all y ∈ Y. (d) d(y, y ∗ ) ≤ 1−L This theorem was used by Cˇadariu and Radu (see [31, 32, 33, 34]) and then others to obtain the applications of fixed point theory in stability problems (cf. [24, 35, 36, 37, 38, 39, 40, 41, 42, 43]). Definition 2.12. A 3-tuple (X , ρµ,ν , τ ) is said to be an intuitionistic fuzzy 2-normed space(for short, IF2NS) if X is a real linear space, and µ and ν are a fuzzy 2-norm and an anti fuzzy 2-norm, respectively, such that ν(x, y, t) + µ(x, y, t) ≤ 1, τ is continuous t-representable, and ρµ,ν : X × X × R → L∗ ρµ,ν (x, y, t) = (µ(x, y, t), ν(x, y, t)) is a function satisfying the following conditions, for all x, y, z ∈ X , and t, s, α ∈ R, (1) ρµ,ν (x, y, t) = (0, 1) = 0L∗ for all t ≤ 0. (2) ρµ,ν (x, y, t) = (1, 0) = 1L∗ if and only if x, y are linearly dependent, for all t > 0. t (3) ρµ,ν (αx, y, t) = ρµ,ν (x, y, |α| ) for all t > 0 and α 6= 0 (4) ρµ,ν (x, y, t) is invariant under any permutation of x, y. (5) ρµ,ν (x + z, y, t + s) ≥L∗ τ (ρµ,ν (x, y, t), ρµ,ν (z, y, s)) for all t, s > 0.

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CHOONKIL PARK et al 546-557

J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 22, NO.3, 2017, COPYRIGHT 2017 EUDOXUS PRESS, LLC

FUNCTIONAL EQUATION IN INTUITIONISTIC FUZZY 2-BANACH SPACES

(6) ρµ,ν (x, y, .) is continuous and lim ρµ,ν (x, y, t) = 0L∗ and lim ρµ,ν (x, y, t) = 1L∗ t→∞

t→0

Then ρµ,ν is said to be an intuitionstic fuzzy 2-norm on a real linear space X . Example 2.4. Let (X , k·, ·k) be a 2-normed space, τ (a, b) = (a1 b1 , min(a2 + b2 , 1)) be continuous t-representable for all a = (a1 , a2 ), b = (b1 , b2 ) ∈ L∗ and µ, ν be a fuzzy and an anti fuzzy 2-norm, respectively. We define   t kx, yk ρµ,ν (x, y, t) = , t + mkx, yk t + mkx, yk for all t ∈ R+ and m > 1. Then (X , ρµ,ν , τ ) is an IF2NS. Definition 2.13. A sequence {xn } in an IF2NS (X , ρµ,ν , τ ) is said to be convergent to a point x ∈ X if lim ρµ,ν (xn − x, y, t) = 1L∗ n→∞

for all t > 0 and all y ∈ X . Definition 2.14. A sequence {xn } in an IF2NS (X , ρµ,ν , τ ) is said to be a Cauchy sequence if for any 0 <  < 1 and t > 0, there exists n0 ∈ N such that ρµ,ν (xn − xm , y, t) ≥L∗ (1 − , ) for all n, m ≥ n0 and all y ∈ X . Definition 2.15. An IF2NS space (X , ρµ,ν , τ ) is said to be complete if every Cauchy sequence in (X , ρµ,ν , τ ) is convergent. A complete intuitionistic fuzzy 2-normed space is called an intuitionistic fuzzy 2-Banach space. 3. Hyers-Ulam stability of the functional equation (1) in IF2NS: an odd mapping case Using the fixed point alternative approach, we prove the Hyers-Ulam stability of the functional equation (1) in intuitionistic fuzzy 2-Banach spaces for an odd mapping case. Let X , Y be real linear spaces. For a given mapping f : X → Y, we define       x+y x−y y−x Df (x, y) := 2f +f +f − f (x) − f (y). 2 2 2 Lemma 3.1. Let X , Y be real linear spaces. An odd mapping f : X → Y satisfies       x+y x−y y−x 2f +f +f = f (x) + f (y) 2 2 2

(2)

if and only if it is Jensen additive. Proof. Assume that f : X → Y satisfies  (2).Since f is odd, we have f (−x) = −f (x) for all x+y x, y ∈ X . It follows from (2) that 2f = f (x) + f (y) for all x, y ∈ X . 2 Conversely, assume that f : X → Y is Jensen additive. Then it is easy to show that f satisfies (2). 

550

CHOONKIL PARK et al 546-557

J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 22, NO.3, 2017, COPYRIGHT 2017 EUDOXUS PRESS, LLC

C. PARK, E. MOVAHEDNIA, G. A. ANASTASSIOU, AND S. YUN

Theorem 3.1. Let X be a real linear space, (Z, ρ0µ,ν , τ 0 ) an intuitionistic fuzzy 2-normed space and let φ : X × X → Z, ϕ : X × X → Z be mappings such that for some 0 < α2 < 2 ρ0µ,ν (φ(2x, 2y), ϕ(2x, 2y), t) ≥L∗ ρ0µ,ν (αφ(x, y), ϕ(x, y), t)

(3)

for all x, y ∈ X and t ∈ R+ . Let (Y, ρµ,ν , τ ) be a complete intuitionistic fuzzy 2-normed space. If ξ : X × X → Y is a mapping such that ξ(2x, 2y) = α1 ξ(x, y) for all x, y ∈ X and f : X → Y is an odd mapping such that ρµ,ν (Df (x, y), ξ(x, y), t) ≥L∗ ρ0µ,ν (φ(x, y), ϕ(x, y), t) for all x, y ∈ X , t > 0, then there is a unique additive mapping A : X → Y such that   2 − α2 0 t ρµ,ν (f (x) − A(x), ξ(x, 0), t) ≥L∗ ρµ,ν φ(x, 0), ϕ(x, 0), α2 Proof. Putting y = 0 in (4), we have  x  ρµ,ν 2f − f (x), ξ(x, 0), t ≥L∗ ρ0µ,ν (φ(x, 0), ϕ(x, 0), t) . 2

(4)

(5)

(6)

Replacing x by 2x in (6), we have ρµ,ν (2f (x) − f (2x), ξ(2x, 0), t) ≥L∗ ρ0µ,ν (φ(2x, 0), ϕ(2x, 0), t) .

(7)

It follows from (3), (7) and the property of ξ that     2 f (2x) ≥L∗ ρ0µ,ν φ(2x, 0), ϕ(2x, 0), t , ξ(x, 0), t ρµ,ν f (x) − 2 α  2  α 0 ≥L∗ ρµ,ν φ(x, 0), ϕ(x, 0), t 2 for all x ∈ X and t > 0. Consider the set Ω = {g : X → Y} and define a generalized metric d on Ω by  d(g, h) = inf c ∈ R+ : ρµ,ν (g(x) − h(x), ξ(x, 0), t) ≥L∗ ρ0µ,ν (cφ(x, 0), ϕ(x, 0), t) for all x ∈ X and t > 0 with inf ∅ = ∞. It is easy to show that (Ω, d) is complete (see [44]). g(2x) Define J : X → X by Jg(x) = for all x ∈ X . Now, we prove that J is strictly 2 2 contractive mapping of Ω with the Lipschitz constant α2 . Let g, h ∈ E be given such that d(g, h) < . Then ρµ,ν (g(x) − h(x), ξ(x, 0), t) ≥L∗ ρ0µ,ν (φ(x, 0), ϕ(x, 0), t) for all x ∈ X and t > 0. So   2 ρµ,ν (Jg(x) − Jh(x), ξ(x, 0), t) = ρµ,ν g(2x) − h(2x), ξ(2x, 0), t α    2  2 α 0 0 ≥L∗ ρµ,ν φ(2x, 0), ϕ(2x, 0), t =L∗ ρµ,ν φ(x, 0), ϕ(x, 0), t . α 2

551

CHOONKIL PARK et al 546-557

J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 22, NO.3, 2017, COPYRIGHT 2017 EUDOXUS PRESS, LLC

FUNCTIONAL EQUATION IN INTUITIONISTIC FUZZY 2-BANACH SPACES

Then d(Jg, Jh) ≤

α2 d(g, h) for all g, h ∈ Ω. It follows from (7) that 2

α2 0 and 0 < p < 21 , then there exists a unique additive mapping A : X → Y such that   2 − 22p 0 p ∗ t ρµ,ν (f (x) − A(x), ξ(x, 0), t) ≥L ρµ,ν kxk z0 , z1 , 22p for all x ∈ X and t > 0. Proof. Let φ, ϕ : X × X → Z be defined by φ(x, y) = (kxkp + kykp ) z0 and ϕ(x, y) = z1 . Then the result follows from Theorem 3.1 by taking α = 2p .  Corollary 3.2. Let X be a linear space, (Z, ρ0µ,ν , τ 0 ) be an IF2N-space, (Y, ρµ,ν , τ ), be a complete IF2N-space and let z0 , z1 ∈ Z. If f : X → Y is an odd mapping such that ρµ,ν (Df (x, y), ξ(x, y), t) ≥L∗ ρ0µ,ν (z0 , z1 , t) for all x, y ∈ X , t > 0, then there exists a unique additive mapping A : X → Y such that ρµ,ν (f (x) − A(x), ξ(x, 0), t) ≥L∗ ρ0µ,ν (z0 , z1 , t) for all x ∈ X and t > 0. Proof. Let φ, ϕ : X × X → Z be defined by φ(x, y) = z0 and ϕ(x, y) = z1 . Then the result follows from Theorem 3.1 by taking α = 1.  4. Hyers-Ulam stability of the functional equation (1) in IF2NS: an even mapping case Using the fixed point alternative approach, we prove the Hyers-Ulam stability of the functional equation (1) in intuitionistic fuzzy 2-Banach spaces for an even mapping case. Lemma 4.1. Let X , Y be real linear spaces. An even mapping f : X → Y satisfies       x−y y−x x+y +f +f = f (x) + f (y) 2f 2 2 2

(9)

if and only if it is Jensen quadratic. Proof. Assume that f : X → Y satisfies  (9). Sincef is even,  we have f (−x) = f (x) for all x+y x−y + 2f = f (x) + f (y) for all x, y ∈ X . x, y ∈ X . It follows from (9) that 2f 2 2 Conversely, assume that f : X → Y is Jensen quadratic. Then it is easy to show that f satisfies (9).  Theorem 4.1. Let X be a real linear space, (Z, ρ0µ,ν , τ 0 ) an intuitionistic fuzzy 2-normed space and let φ : X × X → Z, ϕ : X × X → Z be mappings such that for some 0 < α2 < 4 ρ0µ,ν (φ(2x, 2y), ϕ(2x, 2y), t) ≥L∗ ρ0µ,ν (αφ(x, y), ϕ(x, y), t)

(10)

for all x, y ∈ X and t ∈ R+ . Let (Y, ρµ,ν , τ ) be a complete intuitionistic fuzzy 2-normed space. If ξ : X × X → Y is a mapping such that ξ(2x, 2y) = α1 ξ(x, y) for all x, y ∈ X and f : X → Y

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J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 22, NO.3, 2017, COPYRIGHT 2017 EUDOXUS PRESS, LLC

FUNCTIONAL EQUATION IN INTUITIONISTIC FUZZY 2-BANACH SPACES

is an even mapping satisfying f (0) = 0 and (4), then there is a unique quadratic mapping Q : X → Y such that   4 − α2 ρµ,ν (f (x) − Q(x), ξ(x, 0), t) ≥L∗ ρ0µ,ν φ(x, 0), ϕ(x, 0), t (11) α2 for all x ∈ X and t > 0. Proof. Putting y = 0 in (4), we have  x  ρµ,ν 4f − f (x), ξ(x, 0), t ≥L∗ ρ0µ,ν (φ(x, 0), ϕ(x, 0), t) . 2 Replacing x by 2x in (12), we have ρµ,ν (4f (x) − f (2x), ξ(2x, 0), t) ≥L∗ ρ0µ,ν (φ(2x, 0), ϕ(2x, 0), t) .

(12)

(13)

It follows from (10), (13) and the property of ξ that     f (2x) 4 0 ρµ,ν f (x) − , ξ(x, 0), t ≥L∗ ρµ,ν φ(2x, 0), ϕ(2x, 0), t 4 α  2  α 0 ≥L∗ ρµ,ν φ(x, 0), ϕ(x, 0), t 4 for all x ∈ X and t > 0. Consider the set Ω = {g : X → Y} and define a generalized metric d on Ω as in Theorem 3.1. g(2x) Define J : X → X by Jg(x) = for all x ∈ X . Now, we prove that J is strictly 4 2 contractive mapping of Ω with the Lipschitz constant α4 . Let g, h ∈ E be given such that d(g, h) < . Then ρµ,ν (g(x) − h(x), ξ(x, 0), t) ≥L∗ ρ0µ,ν (φ(x, 0), ϕ(x, 0), t) for all x ∈ X and t > 0. So   4 ρµ,ν (Jg(x) − Jh(x), ξ(x, 0), t) = ρµ,ν g(2x) − h(2x), ξ(2x, 0), t α  2    α 4 φ(x, 0), ϕ(x, 0), t . ≥L∗ ρ0µ,ν φ(2x, 0), ϕ(2x, 0), t =L∗ ρ0µ,ν α 4 α2 α2 Then d(Jg, Jh) ≤ d(g, h) for all g, h ∈ Ω. It follows from (13) that d(f, Jf ) ≤ < ∞. 4 4 It follows from Theorem 2.1 that there exists a mapping Q : X → Y satisfying the following: (1) Q is a fixed point of J, that is, Q(2x) = 4Q(x)

(14)

(2) The mapping Q is a unique fixed point of J in the set ∆ = {h ∈ Ω : d(g, h) < ∞} This implies that Q is a unique mapping satisfying (14).

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CHOONKIL PARK et al 546-557

J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 22, NO.3, 2017, COPYRIGHT 2017 EUDOXUS PRESS, LLC

C. PARK, E. MOVAHEDNIA, G. A. ANASTASSIOU, AND S. YUN

(3) d(J n f, Q) → 0 as n → ∞. This implies that f (2n x) = Q(x) n→∞ 4n lim

for all x ∈ X. 1 (4) d(f, Q) ≤ d(f, Jf ) with f ∈ ∆, which implies the inequality d(f, Q) ≤ 1−L   4 − α2 0 ρµ,ν (f (x) − Q(x), ξ(x, 0), t) ≥L∗ ρµ,ν φ(x, 0), ϕ(x, 0), t . α2 This implies that the inequality (11) holds. The rest of the proof is similar to the proof of Theorem 3.1.

α2 . 4−α2

So



Corollary 4.1. Let X be a linear space, (Z, ρ0µ,ν , τ 0 ) be an IF2N-space, (Y, ρµ,ν , τ ) be a complete IF2N-space, p be real number and z0 , z1 ∈ Z. If ξ : X × X → Y is a mapping such that ξ(2x, 2y) = 21p ξ(x, y) for all x, y ∈ X and f : X → Y is an even mapping satisfying f (0) = 0 and ρµ,ν (Df (x, y), ξ(x, y), t) ≥L∗ ρ0µ,ν ((kxkp + kykp )z0 , z1 , t) for all x, y ∈ X , t > 0 and 0 < p < 1, then there exists a unique quadratic mapping Q : X → Y such that   4 − 4p 0 p ρµ,ν (f (x) − Q(x), ξ(x, 0), t) ≥L∗ ρµ,ν kxk z0 , z1 , t 4p for all x ∈ X and t > 0. Proof. Let φ, ϕ : X × X → Z be defined by φ(x, y) = (kxkp + kykp ) z0 and ϕ(x, y) = z1 . Then the result follows from Theorem 4.1 by taking α = 2p .  Corollary 4.2. Let X be a linear space, (Z, ρ0µ,ν , τ 0 ) be an IF2N-space, (Y, ρµ,ν , τ ), be a complete IF2N-space and let z0 , z1 ∈ Z. If f : X → Y is an even mapping satisfying f (0) = 0 and ρµ,ν (Df (x, y), ξ(x, y), t) ≥L∗ ρ0µ,ν (z0 , z1 , t) for all x, y ∈ X , t > 0, then there exists a unique quadratic mapping Q : X → Y such that ρµ,ν (f (x) − Q(x), ξ(x, 0), t) ≥L∗ ρ0µ,ν (z0 , z1 , 3t) for all x ∈ X and t > 0. Proof. Let φ, ϕ : X × X → Z be defined by φ(x, y) = z0 and ϕ(x, y) = z1 . Then the result follows from Theorem 4.1 by taking α = 1.  References [1] M. Mursaleen, Q. M. D. Lohani, Intuitionistic fuzzy 2-normed space and some related concepts, Chaos Solitons Fractals 42 (2009), 224–234. [2] S. M. Ulam, Problems in Modern Mathematics, Chapter VI, Science Editions, Wiley, New York, 1964. [3] D. H. Hyers, On the stability of the linear functional equation, Proc. Natl. Acad. Sci. USA 27 (1941), 222–224. [4] D. G. Bourgin, Classes of transformations and bordering transformations, Bull. Amer. Math. Soc. 57 (1951), 223-237.

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[5] Th. M. Rassias, On the stability of functional equations and a problem of Ulam, Acta Appl. Math. 62 (2000), 123–130. [6] F. Skof, Local properties and approximation of operators, Rend. Sem. Mat. Fis. Milano 53 (1983), 113–129. [7] A. K. Katsaras, Fuzzy topological vector spaces II, Fuzzy Sets and Systems 12 (1984), 143–154. [8] C. Felbin, Finite dimensional fuzzy normed linear space, Fuzzy Sets and Systems 48 (1992), 239–248. [9] J.-Z. Xiao, X.-H. Zhu, Fuzzy normed spaces of operators and its completeness, Fuzzy Sets and Systems 133 (2003), 389–399. [10] T. Bag, S. K. Samanta, Finite dimensional fuzzy normed linear spaces, J. Fuzzy Math. 11 (2003), 687–705. [11] S. C. Cheng, J. N. Mordeson, Fuzzy linear operators and fuzzy normed linear spaces, Bull. Calcutta Math. Soc. 86 (1994), 429–436. [12] I. Kramosil, J. Michalek, Fuzzy metric and statistical metric spaces, Kybernetica 11 (1975), 326–334. [13] L. C˘ adariu, L. G˘ avruta, P. G˘ avruta, On the stability of an affine functional equation, J. Nonlinear Sci. Appl. 6 (2013), 60–67. [14] C. Park, K. Ghasemi, S. G. Ghaleh, S. Jang, Approximate n-Jordan ∗-homomorphisms in C ∗ -algebras, J. Comput. Anal. Appl. 15 (2013), 365-368. [15] C. Park, A. Najati, S. Jang, Fixed points and fuzzy stability of an additive-quadratic functional equation, J. Comput. Anal. Appl. 15 (2013), 452–462. [16] S. Shagholi, M. Bavand Savadkouhi, M. Eshaghi Gordji, Nearly ternary cubic homomorphism in ternary Fr´echet algebras, J. Comput. Anal. Appl. 13 (2011), 1106–1114. [17] S. Shagholi, M. Eshaghi Gordji, M. Bavand Savadkouhi, Stability of ternary quadratic derivation on ternary Banach algebras, J. Comput. Anal. Appl. 13 (2011), 1097–1105. [18] D. Shin, C. Park, Sh. Farhadabadi, On the superstability of ternary Jordan C ∗ -homomorphisms, J. Comput. Anal. Appl. 16 (2014), 964–973. [19] D. Shin, C. Park, Sh. Farhadabadi, Stability and superstability of J ∗ -homomorphisms and J ∗ -derivations for a generalized Cauchy-Jensen equation, J. Comput. Anal. Appl. 17 (2014), 125–134. [20] A. K. Mirmostafaee, M. S. Moslehian, Fuzzy versions of Hyers-Ulam-Rassias theorem, Fuzzy Sets and Systems 159 (2008), 720–729. [21] A. K. Mirmostafaee, M. Mirzavaziri, M. S. Moslehian, Fuzzy stability of the Jensen functional equation, Fuzzy Sets and Systems 159 (2008), 730–738. [22] A. K. Mirmostafaee, M. S. Moslehian, Fuzzy almost quadratic functions, Results Math. 52 (2008), 161–177. [23] A. K. Mirmostafaee, M. S. Moslehian, Fuzzy approximately cubic mappings, Infor. Sci. 178 (2008), 3791– 3798. [24] J. Kim, G. A. Anastassiou, C. Park, Additive ρ-functional inequalities in fuzzy normed spaces, J. Comput. Anal. Appl. (to appear). [25] S. A. Mohiuddine, Stability of Jensen functional equation in intuitionistic fuzzy normed space, Chaos Solitons Fractals 42 (2009), 2989-2996. [26] S. A. Mohiuddine, M. Cancan, H. Sevli, Intuitionistic fuzzy stability of a Jensen functional equation via fixed point technique, Math. Comput. Modelling 54 (2011), 2403-2409. [27] S. A. Mohiuddine, H. Sevli, Stability of Pexiderized quadratic functional equation in intuitionistic fuzzy normed space, J. Comput. Appl. Math. 235 (2011), 2137-2146. [28] M. Mursaleen, S. A. Mohiuddine, On stability of a cubic functional equation in intuitionistic fuzzy normed spaces, Chaos Solitons Fractals 42 (2009), 2997-3005. [29] R. Saadati, J. Park, On the intuitionistic fuzzy topological spaces, Chaos, Solitons Fractals 27 (2006), 331-344. [30] J. B. Diaz, B. Margolis, A fixed point theorem of the alternative for contractions on a generalized complete metric space, Bull. Amer. Math. Soc. 74 (1968), 305–309.

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C. PARK, E. MOVAHEDNIA, G. A. ANASTASSIOU, AND S. YUN

[31] L. C˘ adariu, V. Radu, Fixed points and the stability of Jensen’s functional equation, J. Inequal. Pure Appl. Math. 4, no. 1, Art. ID 4 (2003). [32] L. C˘ adariu, V. Radu, On the stability of the Cauchy functional equation: a fixed point approach, Grazer Math. Ber. 346 (2004), 43–52 [33] L. C˘ adariu, V. Radu, Fixed point methods for the generalized stability of functional equations in a single variable, Fixed Point Theory Appl. 2008, Art. ID 749392 (2008). [34] V. Radu, The fixed point alternative and the stability of functional equations, Fixed Point Theory 4 (2003), 91–96. [35] M. Eshaghi Gordji, H. Khodaei, J.M. Rassias, Fixed point methods for the stability of general quadratic functional equation, Fixed Point Theory 12 (2011), 71–82. [36] M. Eshaghi Gordji, H. Khodaei, Th.M. Rassias, R. Khodabakhsh, J ∗ -homomorphisms and J ∗ -derivations on J ∗ -algebras for a generalized Jensen type functional equation, Fixed Point Theory 13 (2012), 481–494. [37] M. Eshaghi Gordji, C. Park, M.B. Savadkouhi, The stability of a quartic type functional equation with the fixed point alternative, Fixed Point Theory 11 (2010), 265–272. [38] C. Park, Fixed points and Hyers-Ulam-Rassias stability of Cauchy-Jensen functional equations in Banach algebras, Fixed Point Theory Appl. 2007, Art. ID 50175 (2007). [39] C. Park, Generalized Hyers-Ulam-Rassias stability of quadratic functional equations: a fixed point approach, Fixed Point Theory Appl. 2008, Art. ID 493751 (2008). [40] C. Park, A fixed point approach to the stability of additive functional inequalities in RN -spaces, Fixed Point Theory 11 (2011), 429–442. [41] C. Park, H.A. Kenary, S. Kim, Positive-additive functional equations in C ∗ -algebras, Fixed Point Theory 13 (2012), 613–622. [42] A. Rahimi, A. Najati, A strong quadratic functional equation in C ∗ -algebras, Fixed Point Theory 11 (2010), 361–368. [43] S. Schin, D. Ki, J. Chang, M. Kim, Random stability of quadratic functional equations: a fixed point approach, J. Nonlinear Sci. Appl. 4 (2011), 37–49. [44] D. Mihet , V. Radu, On the stability of the additive Cauchy functional equation in random normed spaces, J. Math. Anal. Appl. 343 (2008), 567–572. Choonkil Park Department of Mathematics, Research Institute for Natural Sciences, Hanyang University, Seoul 04763, Korea E-mail address: [email protected] Ehsan Movahednia Department of Mathematics, Behbahan Khatam Al-Anbia University of Technology, Behbahan, Iran E-mail address: [email protected] George A. Anastassiou Department of Mathematical Sciences, University of Memphis, Memphis, TN 38152, USA E-mail address: [email protected] Sungsik Yun Department of Financial Mathematics, Hanshin University, Gyeonggi-do 447-791, Korea E-mail address: [email protected]

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Characterization of modular spaces Manuel De la Sen1 , Donal O’Regan2 , Reza Saadati3 1 Institute

of Research and Development of Processes

University of the Basque Country Campus of Leioa Barrio Sarriena 48940 (Bizkaia), Spain [email protected] 2 School

of Mathematics Statistics and Applied Mathematics

National University of Ireland, Galway, Ireland. [email protected] 3 Department

of Mathematics

Iran University of Science and Technology, Tehran, Iran. [email protected]

Abstract In this paper we study the structure of modular spaces and random normed spaces and we show that a modular could induce a random norm 0

Tel/Fax: +981212263650

1

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2 and vice versa. Also we prove the topology generated by a modular (with a certain property) coincides with the topology generated by a random norm, and so in some situations the study of modular spaces reduces to the study of random normed spaces. AMS: 47H09; 47H10; 39B82. Keywords: modular spaces; random normed spaces; topology.

1

Introduction Orlicz and Birnbaum generalized the Lebesgue function spaces Lp and the

theory of Orlicz spaces inspired Nakano [1] to develop the theory of modular spaces. This was generalized by Musielak and Orlicz [2]. For a good introduction to the theory of Orlicz spaces we refer the reader to Krasnoselskii and Rutickii [3]. In this paper, we show that a modular could induce a random norm and vice versa and also we show that the topology generated by a modular (with a certain property) coincides with the topology generated by a random norm.

2

Modular spaces We start with a brief introduction to modular spaces (see [4–6, 8, 9]). Let X be a vector space over F (R or C). A functional ρ : X → [0, ∞] is

called a modular, if for f, g ∈ X, we have for any α ∈ F: 559

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3 (i) ρ(f ) = 0 if and only if f = 0; (ii) ρ(αf ) = ρ(f ) whenever |α| = 1; (iii) ρ(αf + βg) ≤ ρ(f ) + ρ(g) whenever α, β ≥ 0 and α + β = 1. If ρ is a modular in X, then the set defined by

Xρ = {h ∈ X : lim ρ(λh) = 0}

(2.1)

λ→0

is called a modular space. Definition 2.1. Let Xρ be a modular space. The sequence {fn }n∈N in Xρ is said to be ρ–convergent to f ∈ Xρ if ρ(fn − f ) → 0, as n → ∞. The following definition plays an important role in the theory of modular function spaces. Definition 2.2. Let Xρ be a modular space. We say that ρ has the Ω-property if ρ(xn ) → 0 implies ρ(λxn ) → 0 for λ > 0; here xn is a sequence in Xρ . For example it is easy to see that ρ(x) = ln(1 + kxk) and ρ(x) = exp(kxk) − 1 have the Ω-property (see [4]).

3

Random normed spaces Definition 3.1. A triangular norm (shorter t-norm) is a binary operation on

the unit interval [0, 1], i.e., a function T : [0, 1] × [0, 1] → [0, 1] such that for all 560

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4 a, b, c ∈ [0, 1] the following four axioms are satisfied: (T1) T (a, b) = T (b, a)

(: commutativity);

(T2) T (a, (T (b, c))) = T (T (a, b), c) (T3) T (a, 1) = a

(: associativity);

(: boundary condition);

(T4) T (a, b) ≤ T (a, c) whenever b ≤ c

(: monotonicity).

The commutativity of (T1), the monotonicity (T4), and the boundary condition (T3) imply that, for any t-norm T and x ∈ [0, 1], the following boundary conditions are also satisfied: T (x, 1) = T (1, x) = x, T (x, 0) = T (0, x) = 0, and so all the t-norms coincide on the boundary of the unit square [0, 1]2 . The monotonicity of a t-norm T in its second component (T4) is, together with the commutativity (T1), equivalent to the (joint) monotonicity in both components, i.e., to T (x1 , y1 ) ≤ T (x2 , y2 ) whenever x1 ≤ x2 and y1 ≤ y2 .

(3.1)

Basic examples are the Lukasiewicz t-norm TL : TL (a, b) = max(a + b − 1, 0), ∀a, b ∈ [0, 1] and the t-norms TP , TM , TD , where TP (a, b) := ab, 561

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5 TM (a, b) := min{a, b},     min(a, b), if max(a,b)=1; TD (a, b) :=    0, otherwise. If, for two t-norms T1 and T2 , the inequality T1 (x, y) ≤ T2 (x, y) holds for all (x, y) ∈ [0, 1]2 , then we say that T1 is weaker than T2 or, equivalently, that T2 is stronger than T2 . As a result of (3.1), we obtain T (x, y) ≤ T (x, 1) = x, T (x, y) ≤ T (1, y) = y for each (x, y) ∈ [0, 1]2 . Since trivially T (x, y) ≥ 0 = TD (x, y) for all (x, y) ∈ (0, 1)2 , for an arbitrary t-norm T , we get TD ≤ T ≤ TM , i.e., TD is weaker and TM is stronger than any other t-norm, and also since TL < TP we obtain the following ordering for the four basic t-norms TD < TL < TP < TM . Throughout this paper, ∆+ is the space of distribution functions that is, the space of all mappings F : R ∪ {−∞, ∞} −→ [0, 1], such that F is left-continuous, non-decreasing on R and F (0) = 0. Now D+ is a subset of ∆+ consisting of all functions F ∈ ∆+ for which l− F (+∞) = 1, where l− F (x) denotes the left limit of the function f at the point 562

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6 x, that is, l− F (x) = limt→x− F (t). In particular for any a ≥ 0, εa is the specific distribution function defined by

εa (t) =

     0

t≤a

    1

t > a.

Definition 3.2. [10] A random normed space (briefly, RN-space) is a triple (X, µ, T ), where X is a vector space, T is a continuous t-norm, and µ is a mapping from X into D+ such that, if µx denotes the value of µ at x ∈ X, the following conditions hold: (RN1) µx (t) = ε0 (t) for all t > 0 if and only if x = 0; (RN2) µαx (t) = µx



t |α|



for all x ∈ X, t > 0, α 6= 0;

(RN3) µx+y (t + s) ≥ T (µx (t), µy (s)) for all x, y ∈ X and t, s ≥ 0. Definition 3.3. Let (X, µ, T ) be an RN-space. A sequence {xn } in X is said to be convergent to x in X if, for every  > 0 and λ > 0, there exists a positive integer N such that µxn −x () > 1 − λ whenever n ≥ N . Definition 3.4. Let (X, µ, T ) be an RN-space. We say that µ has the Ω? property if µxn (1) → 1 implies µxn (t) → 1 for t > 0; here xn is a sequence in X.

Theorem 3.5. [11] If (X, µ, T ) is an RN-space and {xn } is a sequence such that xn → x, then limn→∞ µxn (t) = µx (t) almost everywhere.

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7 Example 3.6. [12] Let (X, k.k) be a normed linear space. Define

µx (t) =

        

0,

if t ≤ 0;

t , t+kxk

if t > 0.

Then (X, µ, TP ) is a random normed space. Example 3.7. [12] Let (X, k.k) be a normed linear space. Define

µx (t) =

    

if t ≤ 0;

0,

  kxk   e−( t ) , if t > 0. Then (X, µ, TP ) is a random normed space. Example 3.8. [13] Let (X, k · k) be a normed linear space. Define

µx (t) =

     max{1 −    

kxk , 0}, t

if t > 0; if t ≤ 0.

0,

Then (X, µ, TL ) is a RN-space (this was essentially proved by Musthari in [14]; see also [15]). Definition 3.9. Let (X, µ, T ) be an RN-space. We say that µ has the Ω1 property if µx (1) = 1 implies x = 0. It is easy to see that that the RN-spaces in Examples 3.6, 3.7, 3.8 have the Ω1 -property (and also the Ω? -property). For more results on RN-spaces and similar spaces refer [16]– [21]. 564

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8

4

Main results Theorem 4.1. Let (X, µ, T ) be a RN-space with the Ω1 -property. Define a

function ϕ : [0, 1] −→ [0, +∞] such that (1) ϕ is continuous and ϕ(0) = +∞ and ϕ(1) = 0; (2) ϕ is strictly decreasing on [0, 1]; (3) ϕ(T (a, b)) ≤ ϕ(a) + ϕ(b) for all a, b ∈ [0, 1]. Let ρ(x) = φ(µx (1)) for x ∈ X. Then, Xρ is a modular space. Proof. Let (X, µ, T ) be an RN-space with the Ω1 -property and let ϕ be a function satisfying (1)–(3). (i) Let x ∈ X. The Ω1 -property of µ together with (RN1) imply

0 = ρ(x) = ϕ(µx (1)) ⇐⇒ µx (1) = 1 ⇐⇒ x = 0.

(ii) is clear. (iii) Let x, y ∈ X and α, β ≥ 0 and α + β = 1. Then (note (RN3), (2) and then (RN2),(3))

ρ(αx + βy) = ϕ(µαx+βy (1)) ≤ ϕ[T (µαx (α), µβy (β))] ≤ ϕ(µx (1)) + ϕ(µy (1)) = ρ(x) + ρ(y). 565

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9 Now, for x ∈ X (note (RN2)), 



lim ρ(tx) = lim ϕ(µtx (1)) = lim ϕ µx t→0

t→0

t→0

1 |t|

 = ϕ(1) = 0,

so, Xρ is a modular space. Example 4.2. Let X be a normed linear space and let (X, µ, TP ) be the random normed space in Example 3.6. Let      +∞, if u = 0; ϕ(u) =     ln 1 , if 0 < u ≤ 1. u The function ϕ satisfies conditions (1)–(3) in Theorem 4.1. Now Theorem 4.1 guarantees that φ(µx (1)) = ln(1 + kxk) is a modular (note it is also easy to check this directly). Theorem 4.3. Let Xρ be a modular space. Let T be a continuous t-norm. Define a function ψ : [0, +∞] −→ [0, 1] such that (1) ψ is continuous and ψ(0) = 1 and ψ(+∞) = 0; (2) ψ is strictly decreasing on [0, +∞]; (3) ψ(a + b) ≥ T (ψ(a), ψ(b)) for all a, b ∈ [0, +∞). Let µx (t) =

         ψ ρ

if t ≤ 0;

0, x t





, if t > 0.

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10 Then, (X, µ, T ) is a RN-space. Proof. (RN1). For t > 0 and x ∈ X we have µx (t) = 1 iff ψ(ρ ρ

x t




x t




) = 1 iff

= 0 iff x = 0.

(RN2). For t > 0 and x ∈ X we have for α 6= 0 (note (ii))        αx  x t µαx (t) = ψ ρ =ψ ρ = µx . t t/|α| |α| (RN3). For t, s > 0 and x, y ∈ X we have (note (iii) and (3)) µx+y (t + s) = = ≥ ≥

   x+y ψ ρ t+s    1 y  1 x + ψ ρ 1 + st t 1 + st s h x  y i ψ ρ +ρ t s    x    y  ,ψ ρ T ψ ρ t s

= T (µx (t), µy (s)).

Example 4.4. Let X be a normed linear space. Consider the modular ρ (x) = ln (1 + kxk) , for x ∈ X. Let ψ(t) = exp(−t) for t ∈ (−∞, +∞). Then the function satisfies conditions (1)–(3) in Theorem 4.3. Consider the t-norm TP and      0, if λ ≤ 0; µx (λ) =  

  λ = ψ ρ x , if 0 < λ. λ+kxk λ Now Theorem 4.3 guarantees that (X, µ, TP ) is an RN-space. 567

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11 Now, we consider the topology induced by a modular. Theorem 4.5. (1). Let (X, µ, TP ) be a RN-space with the Ω? -property and the Ω1 -property. Let τµ be the topology induced by the random norm µ. Then, there exists a modular which induces a topology which coincides with τµ on X. (2). Let (Xρ , ρ) be a modular space with the Ω-property and let τρ be the topology induced by the modular ρ. Then there exists a random norm µ which induces a topology which coincides with τρ on X. Proof. (1). Let (X, µ, TP ) be a RN-space. Let ϕ be as in Example 4.2 and let ρ(x) = φ(µx (1)) for x ∈ X. Then, from Theorem 4.1, ρ is a modular. Now, let {xn } be a sequence in (X, µ, Tp ) converging to x in X, i.e., µxn −x (t) tends to 1 for t > 0 (so in particular µxn −x (1) tends to 1). Then, ρ(xn − x) = ϕ(µxn −x (1)) tends to 0, i.e., {xn } converges to x in the sense of Definition 2.1. Next let {xn } be a sequence converging to x in X in the sense of Definition 2.1 with modular ρ (here ϕ is as in Example 4.2 and ρ(x) = φ(µx (1)) for x ∈ X) i.e., ϕ(µxn −x (1)) tends to 0. Then µxn −x (1) tends to 1. Now since µ has the Ω? -property, then for t > 0 we have that µxn −x (t) tends to 1 i.e., {xn } converges to x in the sense of Definition 3.3. Now, let A be an open set in (X, µ, TP ). Put B = Ac . We show B is a closed set in (Xρ , ρ). Let x be an element in the closure of B in (Xρ , ρ). Then there exists a sequence {xn } in B with xn converging to x in the sense of Definition 2.1 with modular ρ. Now from the above xn converges to x in the sense of Definition 568

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12 3.3. Now since B is a closed set in (X, µ, TP ) then x ∈ B. Thus B is a closed set in (Xρ , ρ) so A is an open set in (Xρ , ρ). A similar argument show that if C is an open set in (Xρ , ρ) then C is an open set in (X, µ, TP ).

(2). Let (Xρ , ρ) be a modular space with the Ω-property. Let ψ be as in Example 4.4. Then Theorem 4.3 guarantees that (X, µ, TP ) is a RN-space (here µ is as in Theorem 4.3). Now, let {xn } be a sequence in (Xρ , ρ) converging to x in X, i.e., ρ(xn − x) tends to 0. Now since ρ has the Ω-property, then for t > 0 we have that µxn −x (t) = ψ ρ

xn −x t





tends to 1, i.e., {xn } converges to x in the sense of

Definition 3.3.

Next let {xn } be a sequence converging to x in X in the sense of Definition 3.3 i.e., µxn −x (t) = ψ ρ

xn −x t





tends to 1 for t > 0 (here ψ is as in Example 4.4

and µ is as in Theorem 4.3). Then ρ

xn −x t




tends to 0 for t > 0 so in particular

ρ(xn − x) tends to 0 i.e., {xn } converges to x in the sense of Definition 2.1.

Now, let A be an open set in (Xρ , ρ). Put B = Ac . We show B is a closed set in (X, µ, TP ). Let x be an element in the closure of B in (X, µ, TP ). Then there exists a sequence {xn } in B with xn converging to x in the sense of Definition 3.3 with random norm µ. Now from the above xn converges to x in the sense of Definition 2.1. Now since B is a closed set in (Xρ , ρ) then x ∈ B. Thus B is a closed set in (X, µ, TP ) so A is an open set in (X, µ, TP ). A similar argument show that if C is an open set in (X, µ, TP ) then C is an open set in (Xρ , ρ). 569

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13

Acknowledgements The first author is very grateful to the Spanish Ministry of Economy and Competitiveness for Grant DPI 2012-30651, to the Basque Government for Grant Grant IT378-10 and to the University of the Basque Country for Grant UFI 11/07.

References [1] H. Nakano, Modulared Semi-Ordered Linear Spaces, Maruzen, Tokyo, Japan, 1950.

[2] J. Musielak and W. Orlicz, On modular spaces, Studia Mathematica, 18 (1959), 49-65.

[3] M. A. Krasnoselskii and Ya. B. Rutickii, Convex Functions and Orlicz Spaces, Noordhoff, Groningen, The Netherlands, 1961.

[4] M. A. Khamsi, Fixed point theory in modular function spaces, in Recent Advances on Metric Fixed Point Theory (Seville, 1995), vol. 48 of Ciencias, pp. 31-57, University of Sevilla, Seville, Spain, 1996.

[5] M. A. Khamsi, Quasicontraction Mappings in Modular Spaces without ∆2 Condition, Fixed Point Theory and Applications Volume 2008, Article ID 916187, 6 pages. 570

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14 [6] N. Hussain and M. A. Khamsi, On asymptotic poinwise contractions in metric spaces, Nonlinear Analysis, 71 (2009), 4423-4429.

[7] P. Kumam, Fixed point theorems for nonexpansive mapping in modular spaces. Arch Math. 40 (2004), 345-353.

[8] K. Nourouzi, S. Shabanian, Operators defined on n-modular spaces, Mediterr. J. Math., 6 (2009), no. 4, 431–446.

[9] B. Azadifar, M. Maramaei, Gh. Sadeghi, On the modular G-metric spaces and fixed point theorems. J. Nonlinear Sci. Appl. 6 (2013), no. 4, 293–304.

[10] A. N. Sherstnev, On the notion of a random normed space, Dokl. Akad. Nauk SSSR 149 (1963), 280–283 (in Russian).

[11] B. Schweizer, A. Sklar, Probabilistic Metric Spaces, North-Holland, New York, 1983.

[12] R. P. Agarwal, Y. J. Cho, R. Saadati, On random topological structures. Abstr. Appl. Anal. 2011 (2011) Article ID 762361, 41 pages.

[13] D. Mihet¸, R. Saadati, S. M. Vaezpour, The stability of the quartic functional equation in random normed spaces, Acta Appl. Math. 110 (2010), 797-803.

[14] D.H. Mushtari, On the linearity of isometric mappings on random normed spaces, Kazan Gos. Univ. Uchen. Zap. 128 (1968), 86–90. 571

Manuel De la Sen et al 558-572

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15 [15] V. Radu, Some Remarks on Quasi-normed and Random Normed Structures, Seminar on Probability Theory and Applications (STPA) 159 (2003), West Univ. of Timi¸soara. [16] Y. J. Cho, Th. M. Rassias, R. Saadati, Stability of functional equations in random normed spaces. Springer Optimization and Its Applications, 86. Springer, New York, 2013. [17] C. Zaharia, On the probabilistic stability of the monomial functional equation. J. Nonlinear Sci. Appl. 6 (2013), no. 1, 51–59. [18] W. Xu, C. Zhu, Zh. Wu, Li Zhu, Fixed point theorems for two new types of cyclic weakly contractive mappings in partially ordered Menger PM-spaces. J. Nonlinear Sci. Appl. 8 (2015), no. 4, 412–422. [19] Y. J. Cho, C. Park, Y.O. Yang, Stability of derivations in fuzzy normed algebras. J. Nonlinear Sci. Appl. 8 (2015), no. 1, 1–7. [20] S. Chauhan, W. Shatanawi, S. Kumar, S. Radenovic, Existence and uniqueness of fixed points in modified intuitionistic fuzzy metric spaces. J. Nonlinear Sci. Appl. 7 (2014), no. 1, 28–41. [21] D. Mihet, Common coupled fixed point theorems for contractive mappings in fuzzy metric spaces. J. Nonlinear Sci. Appl. 6 (2013), no. 1, 35–40.

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Ulam-Hyers stability of 3-Jordan homomorphisms in C ∗ -ternary algebras Madjid Eshaghi Gordji1 , Vahid Keshavarz1 , Choonkil Park2∗ and Sun Young Jang3∗ 1

Department of Mathematics, Semnan University, P. O. Box 35195-363, Semnan, Iran 2

Research Institute for Natural Sciences, Hanyang University, Seoul 133-791, Korea 3

Department of Mathematics, University of Ulsan, Ulsan 680-749, Korea

e-mail: [email protected], [email protected], [email protected], [email protected] Abstract. In this paper, we investigate the Ulam-Hyers stability of C ∗ -ternary 3-Jordan homomorphisms for the functional equation X f (x1 + x2 , y1 + y2 , z1 + z2 ) = f (xi , yj , zk ) 1≤i,j,k≤2

in C ∗ -ternary algebras.

1. Introduction Ternary algebraic operations were considered in the 19th century by several mathematicians and physicists such as Cayley [8] who introduced the notion of cubic matrix which in turn was generalized by Kapranov, Gelfand and Zelevinskii [14]. As an application in physics, the quark model inspired a particular brand of ternary algebraic systems. The so-called Nambu mechanics which has been proposed by Nambu [11] in 1973, is based on such structures. There are also some applications, although still hypothetical, in the fractional quantum Hall effect, the non-standard statistics (the anyons), supersymmetric theories, Yang-Baxter equation, etc, (cf. [15, 16, 26]). The comments on physical applications of ternary structures can be found in [1, 6, 14]. A C ∗ -ternary algebra is a complex Banach space, equipped with a ternary product (x, y, z) → [x, y, z] of A3 into A, which h i is C-linear in the outer variables, conjugate C-linear in the middle variable, and associative in the sense that x, y, [z, u, v] = h i h i x, [y, z, u], v = [x, y, z], u, v , and satisfies k[x, y, z]k ≤ kxk · kyk · kzk, k[x, x, x]k = kxk3 (see [3, 28]). Every left Hilbert C ∗ -module is a C ∗ -ternary algebra via the ternary product [x, y, z] := hx, yiz. Let A and B be two Banach ternary algebras. An additive mapping H : (A, [ ]A ) → (B, [ ]B ) is called a ternary ring homomorphism if H([x, y, z]A ) = [H(x), H(y), H(z)]B for all x, y, z ∈ A. An additive mapping H : (A, [ ]A ) → (B, [ ]B ) is called a Jordan homomorphism if H([x, x, x]A ) = [H(x), H(x), H(x)]B for all x ∈ A. Definition 1.1. Let A and B be C ∗ -ternary algebras. A 3-linear mapping H : A × A × A → B over C is called a C ∗ -ternary 3-homomorphism if it satisfies H([x1 , y1 , z1 ], [x2 , y2 , z2 ], [x3 , y3 , z3 ])

= [H(x1 , x2 , x3 ), H(y1 , y2 , y3 ), H(z1 , z2 , z3 )]

0

2014 Mathematics Subject Classification. Primary 39B52; 39B82; 46B99; 17A40. Keywords: Ulam-Hyers stability; 3-additive mapping; 3-Jordan homomorphisms; C ∗ -ternary algebra. 0∗ Corresponding author. 0

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Ulam-Hyers stability of 3-Jordan homomorphisms in C ∗ -ternary algebras for all x1 , y1 , z1 , x2 , y2 , z2 , x3 , y3 , z3 ∈ A. A 3-linear mapping H : A × A × A → B over C is called a C ∗ -ternary algebra 3-Jordan homomorphism if it satisfies H([x, x, x], [y, y, y], [z, z, z]) = [H(x, x, x), H(y, y, y), H(z, z, z)] for all x, y, z ∈ A The study of stability problems originated from a famous talk given by Ulam [27] in 1940: “Under what condition does there exist a homomorphism near an approximate homomorphism?” In the next year 1941, Hyers [13] answered affirmatively the question of Ulam for additive mappings between Banach spaces. Then, Aoki [4] considered the stability problem with unbounded Cauchy differences. A generalized version of the theorem of Hyers for approximately additive maps was given by Rassias [20] in 1978. Let X and Y be real or complex vector spaces. For a mapping f : X × X × X → Y , consider the functional equation: f (x1 + x2 , y1 + y2 , z1 + z2 ) =

X

f (xi , yj , zk )

(1.1)

1≤i,j,k≤2

In 2006, Park and Bae [19] showed that a mapping f : X × X × X → Y satisfies the equation (1.1) if and only if the mapping f is 3-additive. We investigate the Ulam-Hyers stability in C ∗ -ternary algebras for the 3-additive mappings satisfying (1.1). The stability problems of several functional equations have been extensively investigated by a number of authors and there are many interesting results concerning this problem (see [2, 7, 9, 10, 17, 18, 21, 22, 23, 24, 25, 29, 30]). 2. Ulam-Hyers stability of C ∗ -ternary 3-Jordan homomorphisms The following lemma was proved in [5]. Lemma 2.1. Let X and Y be real or complex vector spaces. Let f : X × X × X → Y be a 3-additive mapping such that f (λx, µy, νz) = λµνf (x, y, z)for all λ, µ, ν ∈ T1 := {λ ∈ C : |λ| = 1} and all x, y, z ∈ X. Then f is 3-linear over C. Using the above lemma, one can obtain the following result. The following lemma was proved in [5]. Lemma 2.2. Let X and Y be complex vector spaces and let f : X × X × X → Y be a mapping such that X f (λx1 + λx2 , µy1 + µy2 , νz1 + νz2 ) = λµν f (xi , yj , zk )

(2.1)

1≤i,j,k≤2

for all λ, µ, ν ∈ T1 and all x1 , x2 , y1 , y2 , z1 , z2 ∈ X. Then f is 3-linear over C. Lemma 2.3. Let A and B be two Banach ternary algebras. Let f : A → B be an additive mapping. Then the following assertions are equivalent H([x, x, x], [y, y, y], [z, z, z]) = [H(x, x, x), H(y, y, y), H(z, z, z)]

(2.2)

 kf ([x1 , x2 , x3 ] + [x2 , x3 , x1 ] + [x3 , x1 , x2 ]), ([y1 , y2 , y3 ] + [y2 , y3 , y1 ] + [y3 , y1 , y2 ]),  ([z1 , z2 , z3 ] + [z2 , z3 , z1 ] + [z3 , z1 , z2 ])  = f ([x1 , x2 , x3 ] + [x2 , x3 , x1 ] + [x3 , x1 , x2 ]), f ([y1 , y2 , y3 ] + [y2 , y3 , y1 ] + [y3 , y1 , y2 ]),  f ([z1 , z2 , z3 ] + [z2 , z3 , z1 ] + [z3 , z1 , z2 ) kB ,

(2.3)

for all x, y, z ∈ A.

for all x1 , x2 , x3 , y1 , y2 , y3 , z1 , z2 , z3 ∈ A. 574

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M. Eshaghi Gordji, V. Keshavarz, C. Park, S.Y. Jang Proof. The proof is similar to the proof of [12, Lemma 2.1]. If we replace x, y, z by x1 + x2 + x3 , y1 + y2 + y3 , z1 + z2 + z3 in (2.2), respectively, then we can easily obtain (2.3). For the converse, if we replace x1 , x2 , x3 by x, y1 , y2 , y3 by y and z1 , z2 , z3 by z in (2.3), we can easily obtain (2.2).



From now on, assume that A is a C ∗ -ternary algebra with norm k.kA and that B is a C ∗ -ternary algebra with norm k.kB . For a given mapping f : A × A × A → B, we define Dλ,µ,ν f (x1 , x2 , y1 , y2 , z1 , z2 ) := f (λx1 + λx2 , µy1 + µy2 , νz1 + νz2 ) − λµν

X

f (xi , yj , zk ).

(2.4)

1≤i,j,k≤2

Theorem 2.4. Let p, q, r ∈ (0, ∞) with p + q + r < 3 and θ ∈ (0, ∞), and let f : A × A × A → B be a mapping such that

kDλ,µ,ν f (x1 , x2 , y1 , y2 , z1 , z2 )kB ≤ θ · max{kx1 kA , kx2 kA }p · max{ky1 kA , ky2 kA }q · max{kz1 kA , kz2 kA }r ,

(2.5)

  kf ([x1 , x2 , x3 ] + [x2 , x3 , x1 ] + [x3 , x1 , x2 ]), ([y1 , y2 , y3 ] + [y2 , y3 , y1 ] + [y3 , y1 , y2 ]), ([z1 , z2 , z3 ] + [z2 , z3 , z1 ] + [z3 , z1 , z2 ])  − f ([x1 , x2 , x3 ] + [x2 , x3 , x1 ] + [x3 , x1 , x2 ]), f ([y1 , y2 , y3 ] + [y2 , y3 , y1 ] + [y3 , y1 , y2 ]), (2.6) 3  X (kxi kpA · kyi kqA · kzi krA ) f ([z1 , z2 , z3 ] + [z2 , z3 , z1 ] + [z3 , z1 , z2 ) kB ≤ θ i=1

for all λ, µ, ν ∈ T1 and all x1 , x2 , x3 , y1 , y2 , y3 , z1 , z2 , z3 ∈ A. Then there exists a unique C ∗ -ternary 3-Jordan homomorphism H :A×A×A→B kf (x, y, z) − H(x, y, z)kB ≤

θ kxkpA · kykqA · kzkrA 8 − 2p+q+r

(2.7)

for all x, y, z ∈ A. Proof. By the same reasoning as in the proof of [5, Theorem 2.3], there exists a unique 3-additive mapping H : A×A×A → B satisfying (2.7). By Lemma 2.1, the 3-linear mapping H : A × A × A → B is given by H(λx, µy, νz) := lim

n→∞

1 1 f (2n λx, 2n µy, 2n νz) = lim λµν n f (2n x, 2n y, 2n z) = λµνH(x, y, z) n→∞ 8n 8

for all λ, µ, ν ∈ T1 and all x, y, z ∈ A. It follows from (2.6) that   kH ([x1 , x2 , x3 ] + [x2 , x3 , x1 ] + [x3 , x1 , x2 ]), ([y1 , y2 , y3 ] + [y2 , y3 , y1 ] + [y3 , y1 , y2 ]), ([z1 , z2 , z3 ] + [z2 , z3 , z1 ] + [z3 , z1 , z2 ])   − H([x1 , x2 , x3 ] + [x2 , x3 , x1 ] + [x3 , x1 , x2 ]), H([y1 , y2 , y3 ] + [y2 , y3 , y1 ] + [y3 , y1 , y2 ]), H([z1 , z2 , z3 ] + [z2 , z3 , z1 ] + [z3 , z1 , z2 ) kB  1 = lim n kf ([2n x1 , 2n x2 , 2n x3 ] + [2n x2 , 2n x3 , 2n x1 ] + [2n x3 , 2n x1 , 2n x2 ]), n→∞ 8  n ([2 y1 , 2n y2 , 2n y3 ] + [2n y2 , 2n y3 , 2n y1 ] + [2n y3 , 2n y1 , 2n y2 ]), ([2n z1 , 2n z2 , 2n z3 ] + [2n z2 , 2n z3 , 2n z1 ] + [2n z3 , 2n z1 , 2n z2 ])  − f ([2n x1 , 2n x2 , 2n x3 ] + [2n x2 , 2n x3 , 2n x1 ] + [2n x3 , 2n x1 , 2n x2 ]),  f ([2n y1 , 2n y2 , 2n y3 ] + [2n y2 , 2n y3 , 2n y1 ] + [2n y3 , 2n y1 , 2n y2 ]), f ([2n z1 , 2n z2 , 2n z3 ] + [2n z2 , 2n z3 , 2n z1 ] + [2n z3 , 2n z1 , 2n z2 ) kB ≤ lim

n→∞

3 θ X kxi kpA · kyi kqA · kzi krA = 0 8n i=1

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Ulam-Hyers stability of 3-Jordan homomorphisms in C ∗ -ternary algebras for all x1 , x2 , x3 , y1 , y2 , y3 , z1 , z2 , z3 ∈ A. So   H ([x1 , x2 , x3 ] + [x2 , x3 , x1 ] + [x3 , x1 , x2 ]), ([y1 , y2 , y3 ] + [y2 , y3 , y1 ] + [y3 , y1 , y2 ]), ([z1 , z2 , z3 ] + [z2 , z3 , z1 ] + [z3 , z1 , z2 ])   = H([x1 , x2 , x3 ] + [x2 , x3 , x1 ] + [x3 , x1 , x2 ]), H([y1 , y2 , y3 ] + [y2 , y3 , y1 ] + [y3 , y1 , y2 ]), H([z1 , z2 , z3 ] + [z2 , z3 , z1 ] + [z3 , z1 , z2 ) kB for all x1 , x2 , x3 , y1 , y2 , y3 , z1 , z2 , z3 ∈ A. Now, let T : A × A × A → B be another 3-additive mapping satisfying (2.7). Then we have 1 kH(2n x, 2n y, 2n z) − T (2n x, 2n y, 2n z)kB 8n 1 1 ≤ n kH(2n x, 2n y, 2n z) − f (2n x, 2n y, 2n z)kB + n kf (2n x, 2n y, 2n z) − T (2n x, 2n y, 2n z)kB 8 8 2(p+q+r−3)n+1 θ kxkpA · kykqA · kzkrA , ≤ 8 − 2p+q+r which tends to zero as n → ∞ for all x, y, z ∈ A. So we can conclude that H(x, y, z) = T (x, y, z) for all x, y, z ∈ A. This kH(x, y, z) − T (x, y, z)kB =

proves the uniqueness of H. Thus the mapping H : A → B is a unique C ∗ -ternary 3-Jordan homomorphism satisfying (2.7).



Putting p = q = r = 0 and θ = ε in Theorem 2.3, we obtain the Ulam stability for the 3-additive functional equation (1.1). Corollary 2.5. Let ε ∈ (0, ∞) and let f : A × A × A → B be a mapping satisfying kDλ,µ,ν f (x1 , x2 , y1 , y2 , z1 , z2 )kB ≤ ε,   kf ([x1 , x2 , x3 ] + [x2 , x3 , x1 ] + [x3 , x1 , x2 ]), ([y1 , y2 , y3 ] + [y2 , y3 , y1 ] + [y3 , y1 , y2 ]), ([z1 , z2 , z3 ] + [z2 , z3 , z1 ] + [z3 , z1 , z2 ])  − f ([x1 , x2 , x3 ] + [x2 , x3 , x1 ] + [x3 , x1 , x2 ]), f ([y1 , y2 , y3 ] + [y2 , y3 , y1 ] + [y3 , y1 , y2 ]),  f ([z1 , z2 , z3 ] + [z2 , z3 , z1 ] + [z3 , z1 , z2 ) kB ≤ 3ε for all λ, µ, ν ∈ T1 and all x1 , x2 , x3 , y1 , y2 , y3 , z1 , z2 , z3 ∈ A. Then there exists a unique C ∗ -ternary 3-Jordan homomorphism H :A×A×A→B kf (x, y, z) − H(x, y, z)kB ≤

ε 7

for all x, y, z ∈ A.

Theorem 2.6. Let p ∈ (0, 3) and θ ∈ (0, 8), and let f : A × A × A → B be a mapping such that kDλ,µ,ν f (x1 , x2 , y1 , y2 , z1 , z2 )kB ≤ θ

2 X

(kxi kpA + kyi kqA + kzi krA ),

(2.8)

i=1

  kf ([x1 , x2 , x3 ] + [x2 , x3 , x1 ] + [x3 , x1 , x2 ]), ([y1 , y2 , y3 ] + [y2 , y3 , y1 ] + [y3 , y1 , y2 ]), ([z1 , z2 , z3 ] + [z2 , z3 , z1 ] + [z3 , z1 , z2 ])   − f ([x1 , x2 , x3 ] + [x2 , x3 , x1 ] + [x3 , x1 , x2 ]), f ([y1 , y2 , y3 ] + [y2 , y3 , y1 ] + [y3 , y1 , y2 ]), f ([z1 , z2 , z3 ] + [z2 , z3 , z1 ] + [z3 , z1 , z2 ) kB ≤θ

3 X

(kxi kpA + kyi kqA + kzi krA )

i=1

(2.9) ∗

for all λ, µ, ν ∈ T1 and all x1 , x2 , x3 , y1 , y2 , y3 , z1 , z2 , z3 ∈ A. Then there exists a unique C -ternary 3-Jordan homomorphism H :A×A×A→B kf (x, y, z) − H(x, y, z)kB ≤

2θ (kxkpA + kykqA + kzkrA ) 8 − 2p

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M. Eshaghi Gordji, V. Keshavarz, C. Park, S.Y. Jang for all x, y, z ∈ A.

Proof. The proof is similar to the proof of Theorem 2.4.



Theorem 2.7. Let p, q, r ∈ (0, ∞) with p + q + r < 3, s ∈ (0, 3) and θ, η ∈ (0, ∞), and let f : A × A × A → B be a mapping such that kDλ,µ,ν f (x1 , x2 , y1 , y2 , z1 , z2 )kB ≤ θ · max{kx1 kA , kx2 kA }p · max{ky1 kA , ky2 kA }q · max{kz1 kA , kz2 kA }r +η

2 X

(2.10)

(kxi ksA + kyi ksA + kzi ksA ),

i=1

  kf ([x1 , x2 , x3 ] + [x2 , x3 , x1 ] + [x3 , x1 , x2 ]), ([y1 , y2 , y3 ] + [y2 , y3 , y1 ] + [y3 , y1 , y2 ]), ([z1 , z2 , z3 ] + [z2 , z3 , z1 ] + [z3 , z1 , z2 ])  − f ([x1 , x2 , x3 ] + [x2 , x3 , x1 ] + [x3 , x1 , x2 ]), f ([y1 , y2 , y3 ] + [y2 , y3 , y1 ] + [y3 , y1 , y2 ]), (2.11) 3 2  X X f ([z1 , z2 , z3 ] + [z2 , z3 , z1 ] + [z3 , z1 , z2 ) kB ≤ θ (kxi kpA · kyi kqA · kzi krA ) + η (kxi ksA + kyi ksA + kzi ksA ) i=1

i=1

for all λ, µ, ν ∈ T1 and all x1 , x2 , x3 , y1 , y2 , y3 , z1 , z2 , z3 ∈ A. Then there exists a unique C ∗ -ternary 3-Jordan homomorphism H :A×A×A→B kf (x, y, z) − H(x, y, z)kB ≤

2η θ kxkpA · kykqA · kzkrA + (kxi ksA + kyi ksA + kzi ksA ) 8 − 2p+q+r 8 − 2s

for all x, y, z ∈ A. Proof. The proof is similar to the proof of Theorem 2.4.



Theorem 2.8. Let p ∈ (0, 3) and θ ∈ (0, 8), and let f : A×A×A → B be a mapping satisfying (2.8), (2.9) and f (0, 0, 0) = 0. Then there exists a unique C ∗ -ternary 3-Jordan homomorphism H : A × A × A → B such that kf (x, y, z) − H(x, y, z)kB ≤

2θ (kxkpA + kykqA + kzkrA ) 2p − 8

for all x, y, z ∈ A.

Theorem 2.9. Letp, q, r ∈ (0, ∞) with p + q + r > 3, s ∈ (0, 3) and θ, η ∈ (0, ∞), and let f : A × A × A → B be a mapping satisfying (2.10), (2.11) and f (0, 0, 0) = 0. Then there exists a unique C ∗ -ternary 3-Jordan homomorphism H : A × A × A → B such that kf (x, y, z) − H(x, y, z)kB ≤

θ 2η kxkpA · kykqA · kzkrA + s (kxi ksA + kyi ksA + kzi ksA ) 2p+q+r−8 2 −8

for all x, y, z ∈ A.

Acknowledgments S. Y. Jang was supported by Basic Science Research Program through the National Research Foundation of Korea funded by the Ministry of Education, Science and Technology (2013007226). 577

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Ulam-Hyers stability of 3-Jordan homomorphisms in C ∗ -ternary algebras References [1] V. Abramov, R. Kerner, B. Le Roy, Hypersymmetry: A Z3 graded generalization of supersymmetry, J. Math. Phys. 38 (1997), 1650-1669. [2] M. Adam, On the stability of some quadratic functional equation, J. Nonlinear Sci. Appl. 4 (2011), 50-59. [3] M. Amyari, M.S. Moslehian, Approximately ternary semigroup homomorphisms, Lett. Math. Phys. 77 (2006), 1-9. [4] T. Aoki, On the stability of the linear transformation in Banach spaces, J. Math. Soc. Japan 2 (1950), 64-66. [5] J. Bae, W. Park, Generalized Ulam-Hyers stability of C ∗ -ternary algebra 3-homomorphisms for a functional equation, J. Chungcheong Math. Soc. 24 (2011), 147-162. [6] F. Bagarello, G. Morchio, Dynamics of mean-field spin models from basic results in abstract differential equations, J. Stat. Phys. 66 (1992), 849-866. [7] L. C˘ adariu, L. G˘ avruta, P. G˘ avruta, On the stability of an affine functional equation, J. Nonlinear Sci. Appl. 6 (2013), 60-67. [8] A. Cayley, On the 34concomitants of the ternary cubic, Amer. J. Math. 4 (1881). 1-15. [9] A. Chahbi, N. Bounader, On the generalized stability of d’Alembert functional equation, J. Nonlinear Sci. Appl. 6 (2013), 198-204. [10] Y. Cho, C. Park, M. Eshaghi Gordji, Approximate additive and quadratic mappings in 2-Banach spaces and related topics, Int. J. Nonlinear Anal. Appl. 3 (2012), No. 1, 75-81. [11] Y. L. Daletskii, L. A. Takhtajan, Leibniz and Lie algebra structures for Nambu algebra, Lett. Math. Phys.39 (1997), 127-141. [12] M. Eshaghi Gordji, V. Keshavarz, Ternary Jordan homomorphisms between unital ternary C ∗ -algebras (preprint). [13] D.H. Hyers, On the stability of the linear functional equation, Proc. Natl. Acad. Sci. U.S.A. 27 (1941), 222-224. [14] M. Kapranov, I. M. Gelfand, A. Zelevinskii, Discriminants, Resultants and Multidimensional Determinants, Birkh¨ auser, Berlin, 1994. [15] R. Kerner, Ternary Algebraic Structures and Their Applications in Physics (Pierre et Marie Curie University, Paris, 2000; Ternary algebraic structures and their applications in physics, Proc. BTLP, 23rd International Conference on Group Theoretical Methods in Physics, Dubna, Russia, 2000. [16] R. Kerner, The cubic chessboard: Geometry and physics, Class. Quantum Grav. 14, A203, 1997. [17] C. Park, K. Ghasemi, S. G. Ghaleh, S. Jang, Approximate n-Jordan ∗-homomorphisms in C ∗ -algebras, J. Comput. Anal. Appl. 15 (2013), 365-368. [18] C. Park, A. Najati, S. Jang, Fixed points and fuzzy stability of an additive-quadratic functional equation, J. Comput. Anal. Appl. 15 (2013), 452-462. [19] W. Park, J. Bae, On the solution of a multi-additive functional equation and its stability, J. Appl. Math. Computing 22 (2006), 517-522. [20] Th.M. Rassias, On the stability of the linear mapping in Banach spaces, Proc. Amer. Math. Soc. 72 (1978), 297-300. [21] S. Schin, D. Ki, J. Chang, M. Kim, Random stability of quadratic functional equations: a fixed point approach, J. Nonlinear Sci. Appl. 4 (2011), 37–49. [22] S. Shagholi, M. Bavand Savadkouhi, M. Eshaghi Gordji, Nearly ternary cubic homomorphism in ternary Fr´echet algebras, J. Comput. Anal. Appl. 13 (2011), 1106-1114. [23] S. Shagholi, M. Eshaghi Gordji, M. Bavand Savadkouhi, Stability of ternary quadratic derivation on ternary Banach algebras, J. Comput. Anal. Appl. 13 (2011), 1097-1105. [24] D. Shin, C. Park, Sh. Farhadabadi, On the superstability of ternary Jordan C ∗ -homomorphisms, J. Comput. Anal. Appl. 16 (2014), 964-973. [25] D. Shin, C. Park, Sh. Farhadabadi, Stability and superstability of J ∗ -homomorphisms and J ∗ -derivations for a generalized Cauchy-Jensen equation, J. Comput. Anal. Appl. 17 (2014), 125-134. [26] L. A. Takhtajan, On foundation of the generalized Nambu mechanics, Commun. Math. Phys. 160 (1994), 295-315. [27] S.M. Ulam, Problems in Modern Mathematics, Chapter VI, Science ed., Wiley, New York, 1940. [28] H. Zettl, A characterization of ternary rings of operators, Adv. Math. 48 (1983), 117-143. [29] C. Zaharia, On the probabilistic stability of the monomial functional equation, J. Nonlinear Sci. Appl. 6 (2013), 51-59. [30] S. Zolfaghari, Approximation of mixed type functional equations in p-Banach spaces, J. Nonlinear Sci. Appl. 3 (2010), 110-122.

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TABLE OF CONTENTS, JOURNAL OF COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 22, NO. 3, 2017

Periodic Orbits of Singular Radially Symmetric Systems, Shengjun Li, Wulan Li, and Yiping Fu,…............................................................................................................................................393 Approximate Ternary Jordan Ring Homomorphisms in Ternary Banach Algebras, M. Eshaghi Gordji, Vahid Keshavarz, Jung Rye Lee, Dong Yun Shin, and Choonkil Park,……………….402 Approximate Controllability of Fractional Impulsive Stochastic Functional Differential Inclusions with Infinite Delay and Fractional Sectorial Operators, Zuomao Yan, and Xiumei Jia,………………………………………………………………………………………………409 Hyers-Ulam Stability of General Additive Mappings in C*-Algebra, Gang Lu, Guoxian Cai, Yuanfeng Jin, and Choonkil Park,………………………………………………………..…….432 A Higher Order Multi-step Iterative Method for Computing the Numerical Solution of Systems of Nonlinear Equations Associated with Nonlinear PDEs and ODEs, Malik Zaka Ullah, S. SerraCapizzano, Fayyaz Ahmad, Arshad Mahmood, and Eman S. Al-Aidarous,………………….445 Quadratic 𝜌-Functional Inequalities in Fuzzy Normed Spaces, Ji-Hye Kim, Choonkil Park,..462

The Quadrature Rules of the Fuzzy Henstock-Stieltjes Integral on a Infinite Interval, Ling Wang,…………………………………………………………………………………………474 Cubic and Quartic 𝜌-Functional Inequalities in Fuzzy Normed Spaces, Jooho Zhiang, Jeonghun Chu, George A. Anastassiou, and Choonkil Park,……………………………………………484 A Right Parallelism Relation for Mappings to Posets, Hee Sik Kim, J. Neggers, and Keum Sook So,…………………………………………………………………………………………….496 Existence Results for Nonlinear Generalized Three-Point Boundary Value Problems for Fractional Differential Equations and Inclusions, Mohamed Abdalla Darwish, and Sotiris K. Ntouyas,………………………………………………………………………………………507 Quadratic 𝜌-Functional Inequalities in Fuzzy Banach Spaces, Choonkil Park, and Sun Young Jang,…………………………………………………………………………………………..527 Remarks On Common Fixed Point Results For Cyclic Contractions In Ordered b-Metric Spaces, Huaping Huang, Stojan Radenovic, and Tatjana Aleksic Lampert,…………………………..538

TABLE OF CONTENTS, JOURNAL OF COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 22, NO. 3, 2017 (continued) A Fixed Point Method to the Stability of a Jensen Functional Equation in Intuitionistic Fuzzy 2Banach Spaces, Choonkil Park, Ehsan Movahednia, George A. Anastassiou, Sungsik Yun,…546 Characterization of Modular Spaces, Manuel De la Sen, Donal O'Regan, and Reza Saadati,…558 Ulam-Hyers Stability of 3-Jordan Homomorphisms in C*-Ternary Algebras, Madjid Eshaghi Gordji, Vahid Keshavarz, Choonkil Park, and Sun Young Jang,………………………………573

Volume 22, Number 4 ISSN:1521-1398 PRINT,1572-9206 ONLINE

April 2017

Journal of Computational Analysis and Applications EUDOXUS PRESS,LLC

Journal of Computational Analysis and Applications ISSNno.’s:1521-1398 PRINT,1572-9206 ONLINE SCOPE OF THE JOURNAL An international publication of Eudoxus Press, LLC (fourteen times annually) Editor in Chief: George Anastassiou Department of Mathematical Sciences, University of Memphis, Memphis, TN 38152-3240, U.S.A [email protected] http://www.msci.memphis.edu/~ganastss/jocaaa The main purpose of "J.Computational Analysis and Applications" is to publish high quality research articles from all subareas of Computational Mathematical Analysis and its many potential applications and connections to other areas of Mathematical Sciences. Any paper whose approach and proofs are computational,using methods from Mathematical Analysis in the broadest sense is suitable and welcome for consideration in our journal, except from Applied Numerical Analysis articles. Also plain word articles without formulas and proofs are excluded. The list of possibly connected mathematical areas with this publication includes, but is not restricted to: Applied Analysis, Applied Functional Analysis, Approximation Theory, Asymptotic Analysis, Difference Equations, Differential Equations, Partial Differential Equations, Fourier Analysis, Fractals, Fuzzy Sets, Harmonic Analysis, Inequalities, Integral Equations, Measure Theory, Moment Theory, Neural Networks, Numerical Functional Analysis, Potential Theory, Probability Theory, Real and Complex Analysis, Signal Analysis, Special Functions, Splines, Stochastic Analysis, Stochastic Processes, Summability, Tomography, Wavelets, any combination of the above, e.t.c. "J.Computational Analysis and Applications" is a peer-reviewed Journal. See the instructions for preparation and submission of articles to JoCAAA. Assistant to the Editor: Dr.Razvan Mezei,Lenoir-Rhyne University,Hickory,NC

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Editorial Board Associate Editors of Journal of Computational Analysis and Applications Francesco Altomare Dipartimento di Matematica Universita' di Bari Via E.Orabona, 4 70125 Bari, ITALY Tel+39-080-5442690 office +39-080-3944046 home +39-080-5963612 Fax [email protected] Approximation Theory, Functional Analysis, Semigroups and Partial Differential Equations, Positive Operators.

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J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 22, NO.4, 2017, COPYRIGHT 2017 EUDOXUS PRESS, LLC

SOME NEW RESULTS ON PRODUCTS OF THE APOSTOL-GENOCCHI POLYNOMIALS YUAN HE

Abstract. We perform a further investigation for the Apostol-Genocchi polynomials numbers. By making use of the generating function methods and summation transform techniques, we establish some new formulae for products of any arbitrary number of the Apostol-Genocchi polynomials and numbers. The results presented here are the corresponding generalizations of some known formulae on the classical Genocchi polynomials and numbers.

1. Introduction The classical Bernoulli polynomials Bn (x) and the classical Genocchi polynomials Gn (x) are usually defined by means of the following generating functions: ∞ X text tn Bn (x) = t e − 1 n=0 n!

(|t| < 2π)

(1.1)

∞ X 2text tn = G (x) n et + 1 n=0 n!

(|t| < π).

(1.2)

and Gn = Gn (0)

(1.3)

and

The rational numbers Bn and Gn given by Bn = Bn (0)

are called the classical Bernoulli numbers and the classial Genocchi numbers, respectively. These polynomials and numbers play important roles in different areas of mathematics such as number theory, combinatorics, special functions and analysis. Numerous interesting properties for them can be found in many books and papers; see for example, [7, 14, 17, 18, 19, 26, 27, 29, 30, 31]. We now turn to some widely-investigated analogues of the classical Bernoulli polynomials Bn (x) and the classical Genocchi polynomials Gn (x), i.e., the ApostolBernoulli polynomials Bn (x; λ) and the Apostol-Genocchi polynomials Gn (x; λ). They are usually defined by means of the following generating functions (see, e.g., [20, 21, 24]): ∞ X tn text = B (x; λ) (1.4) n λet − 1 n=0 n! (|t| < 2π when λ = 1; |t| < | log λ| when λ 6= 1) 2010 Mathematics Subject Classification. 11B68; 05A19. Keywords. Apostol-Bernoulli polynomials; Apostol-Genocchi polynomials; Convolution formulae; Recurrence relations. 1

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YUAN HE 591-600

J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 22, NO.4, 2017, COPYRIGHT 2017 EUDOXUS PRESS, LLC

2

YUAN HE

and ∞ X 2text tn = Gn (x; λ) t λe + 1 n=0 n!

(1.5)

(|t| < π when λ = 1; |t| < | log(−λ)| when λ 6= 1) In particular, Bn (λ) and Gn (λ) given by Bn (λ) = Bn (0; λ)

and Gn (λ) = Gn (0; λ)

(1.6)

are called the Apostol-Bernoulli numbers and the Apostol-Genocchi numbers, respectively. Obviously, Bn (x; λ) and Gn (x; λ), respectively, reduces to Bn (x) and Gn (x) when λ = 1. It is worth mentioning that the Apostol-Bernoulli polynomials were firstly introduced by Apostol [3] (see also Srivastava [28] for a systematic study) in order to evaluate the value of the Hurwitz-Lerch zeta function. For some related results on the Apostol type polynomials and numbers, one can consult to [6, 8, 11, 16, 22, 24, 33]. The idea of the present paper stems from the work of Agoh [1, 2]. We establish some new formulae of products of any arbitrary number of the Apostol-Genocchi polynomials and numbers by making use of the generating function methods and summation transform techniques. It turns out that some results presented here are the corresponding generalizations of several known formulae including the recent ones discovered by Agoh [2] on the classical Genocchi polynomials and numbers. 2. The statement of results Let n be a positive integer and let m1 , . . . , mn be non-negative integers. In m1 mn mn 1 the following we denote by [tm 1 · · · tn ]f (t1 , . . . , tn ) the coefficients of t1 · · · tn in f (t1 , . . . , tn ). We first recall the elementary and beautiful idea contributed to Euler, namely (see, e.g., [4, 5]) (1 + x1 )(1 + x2 )(1 + x3 ) · · · = (1 + x1 ) + x2 (1 + x1 ) + x3 (1 + x1 )(1 + x2 ) + · · · . (2.1) Obviously, the finite form of (2.1) can be expressed as (1 + x1 )(1 + x2 ) · · · (1 + xn ) = (1 + x1 ) + x2 (1 + x1 ) + · · · + xn (1 + x1 )(1 + x2 ) · · · (1 + xn−1 ). (2.2) We shall make use of (2.2) to establish some new formulae for products of any arbitrary number of the Apostol-Genocchi polynomials and numbers. It is easily seen that for 1 ≤ r ≤ n, substituting xr − 1 for xr in (2.2) gives x1 · · · xn − 1 =

n X (xr − 1)x1 · · · xr−1 ,

(2.3)

r=1

where the product x1 · · · xr−1 is considered to be equal to 1 when r = 1. If we take xr = −λr etr for 1 ≤ r ≤ n in (2.3) then we have (−1)n λ1 · · · λn et1 +···+tn − 1 =

n r−1 X Y (−1)r (λr etr + 1) λ k e tk . r=1

592

(2.4)

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It follows from (2.4) that n Y 1 2ti exi ti = ti + 1 n λ · · · λ et1 +···+tn − 1 λ e (−1) i 1 n i=1

×

n r−1 n X Y Y 2ti exi ti . (2.5) (−1)r (λr etr + 1) λ k e tk λ eti + 1 r=1 i=1 i k=1

Observe that tr

(λr e + 1)

r−1 Y k=1

n Y 2ti exi ti λk e λ eti + 1 i=1 i tk

= 2tr exr (t1 +···+tn )

r−1 Y

λi

i=1

n 2ti e(xi −xr +1)ti Y 2ti e(xi −xr )ti . (2.6) λi eti + 1 λi eti + 1 i=r+1

Hence, by applying (2.6) to (2.5), we get n n X Y 2tr exr (t1 +···+tn ) 2ti exi ti r = (−1) λ eti + 1 r=1 (−1)n λ1 · · · λn et1 +···+tn − 1 i=1 i

×

r−1 Y i=1

λi

n 2ti e(xi −xr +1)ti Y 2ti e(xi −xr )ti , (2.7) λi eti + 1 λi eti + 1 i=r+1

which means  m1 Y  n n t1 tm 2ti exi ti n ··· m1 ! mn ! i=1 λi eti + 1 = m1 ! · · · mn !

n X mr−1 mr −1 mr+1 1 n (−1)r [tm tr+1 · · · tm n ] 1 · · · tr−1 tr r=1

 ×

r−1 Y 2ti e(xi −xr +1)ti 2exr (t1 +···+tn ) λi (−1)n λ1 · · · λn et1 +···+tn − 1 i=1 λi eti + 1

 n Y 2ti e(xi −xr )ti × . (2.8) λi eti + 1 i=r+1 It is trivial to get 

1 tm tmn 1 ··· n m1 ! mn !

Y n

 Y n 2ti exi ti = Gmi (xi ; λi ). λ e ti + 1 i=1 i i=1

(2.9)

We next consider the right hand side of (2.8). Since B0 (x; λ) = 0 when λ 6= 1 and G0 (x; λ) = 0 (see, e.g., [20, 23]) , then by (1.3) we have ∞ X Bn+1 (x; λ) tn ext = t λe − 1 n=0 n + 1 n!

and

(λ 6= 1),

∞ X 2ext Gn+1 (x; λ) tn = . λet + 1 n=0 n + 1 n!

593

(2.10)

(2.11)

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Notice that for non-negative integer N (see, e.g., [32]),   X N (t1 + · · · + tn )N = tk1 · · · tknn , k1 , . . . , k n 1

(2.12)

k1 +···+kn =N k1 ,...,kn ≥0

n r1 ,...,rk

where




denotes by the multinomials coefficient given by   n n! = (n, r1 , . . . , rk ≥ 0). r1 , . . . , rk r1 ! · · · rk !

(2.13)

So from (2.10) and (2.11), we obtain that for even integer n and λ1 · · · λn 6= 1, exr (t1 +···+tn ) (−1)n λ1 · · · λn et1 +···+tn − 1 ∞ X BN +1 (xr ; λ1 · · · λn ) = N +1 N =0

X k1 +···+kn =N k1 ,...,kn ≥0

tkn tk11 · · · n , (2.14) k1 ! kn !

and for odd integer n, 2exr (t1 +···+tn ) (−1)n λ1 · · · λn et1 +···+tn − 1 ∞ X GN +1 (xr ; λ1 · · · λn ) =− N +1 N =0

X k1 +···+kn =N k1 ,...,kn ≥0

tkn tk11 · · · n . (2.15) k1 ! kn !

It follows from (1.3), (1.4), (2.8), (2.9), (2.14) and (2.15) that if n is an even integer, then for positive integers m1 , . . . , mn and λ1 · · · λn 6= 1, Gm1 (x1 ; λ1 )Gm2 (x2 ; λ2 ) · · · Gmn (xn ; λn ) =2

n X (−1)r

X k1 ,··· ,kr−1 ,kr+1 ,··· ,kn ≥0

r=1

m1 ! · · · mn ! k1 ! · · · kr−1 ! · (mr − 1)! · kr+1 ! · · · kn !

Bk1 +···+kr−1 +(mr −1)+kr+1 +···+kn +1 (xr ; λ1 · · · λn ) × k1 + · · · + kr−1 + (mr − 1) + kr+1 + · · · + kn + 1 ×

r−1 Y i=1

λi

n Gmi −ki (xi − xr + 1; λi ) Y Gmi −ki (xi − xr ; λi ) . (2.16) (mi − ki )! (mi − ki )! i=r+1

and if n is an odd integer, then for positive integers m1 , . . . , mn , Gm1 (x1 ; λ1 )Gm2 (x2 ; λ2 ) · · · Gmn (xn ; λn ) =

n X (−1)r−1

X k1 ,··· ,kr−1 ,kr+1 ,··· ,kn ≥0

r=1

×

m1 ! · · · mn ! k1 ! · · · kr−1 ! · (mr − 1)! · kr+1 ! · · · kn !

Gk1 +···+kr−1 +(mr −1)+kr+1 +···+kn +1 (xr ; λ1 · · · λn ) k1 + · · · + kr−1 + (mr − 1) + kr+1 + · · · + kn + 1 r−1 Y

n Gmi −ki (xi − xr + 1; λi ) Y Gmi −ki (xi − xr ; λi ) × λi . (2.17) (mi − ki )! (mi − ki )! i=1 i=r+1

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Thus, by replacing ki by mi − ki for i 6= r in (2.16) and (2.17), we obtain the following formulae for products of an arbitrary number of the Apostol-Genocchi polynomials. Theorem 2.1. Let m1 , · · · , mn be n positive integers. If n is an even integer, then Gm1 (x1 ; λ1 )Gm2 (x2 ; λ2 ) · · · Gmn (xn ; λn ) X

=2

k1 +···+kn =m1 +···+mn k1 ,...,kn ≥0

×

n X mr (−1)r Bk (xr ; λ1 · · · λn ) kr r r=1

r−1 Y

 mi λi Gki (xi − xr + 1; λi ) ki i=1   n Y mi × Gki (xi − xr ; λi ) (λ1 · · · λn 6= 1). (2.18) ki i=r+1

If n is an odd integer, then Gm1 (x1 ; λ1 )Gm2 (x2 ; λ2 ) · · · Gmn (xn ; λn ) X

=

k1 +···+kn =m1 +···+mn k1 ,...,kn ≥0

×

r−1 Y i=1

n X mr Gk (xr ; λ1 · · · λn ) (−1)r−1 kr r r=1

   n Y mi mi λi Gki (xi − xr + 1; λi ) Gki (xi − xr ; λi ). (2.19) ki ki i=r+1

It follows that we show some special cases of Theorem 2.1. Since the ApostolGenocchi polynomials satisfy the following difference equation (see, e.g., [23]): λGn (x + 1; λ) + Gn (x; λ) = 2nxn−1

(n ≥ 0),

(2.20)

by taking n = 2 in Theorem 2.1, we get that for positive integers m, n and λµ 6= 1,

Gm (x; λ)Gn (y; µ) = 2n

m   X m k=0

k

{2k(x − y)k−1 − Gk (x − y; λ)}

− 2m

n   X n k=0

k

Gk (y − x; µ)

Bm+n−k (y; λµ) m+n−k

Bm+n−k (x; λµ) . (2.21) m+n−k

The identity (2.21) can be also found in [12] where it was further considered the case λµ = 1. We also refer to [9, 10, 35] for some similar formulae to (2.21). If we take n = 3 in Theorem 2.1, in light of the symmetric relation for the Apostol-Genocchi polynomials (see, e.g., [23]): λGn (1 − x; λ) = (−1)

n+1

595

  1 Gn x; (n ≥ 0), λ

(2.22)

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we obtain that for positive integers m1 , m2 , m3 , Gm1 (x1 ; λ1 )Gm2 (x2 ; λ2 )Gm3 (x3 ; λ3 )     X m1 m2 m3 = Gk1 (x1 ; µ)Gk2 (x2 − x1 ; λ2 )Gk3 (x3 − x1 ; λ3 ) k1 k2 k3 k1 +k2 +k3 =m1 +m2 +m3 k1 ,k2 ,k3 ≥0

+

   m2 m1 m3 (−1)k1 Gk2 (x2 ; µ)Gk1 (x2 − x1 ; 1/λ1 )Gk3 (x3 − x2 ; λ3 ) k2 k1 k3    m3 m1 m2 + (−1)k1 +k2 Gk3 (x3 ; µ)Gk1 (x3 − x1 ; 1/λ1 ) k3 k1 k2  × Gk2 (x3 − x2 ; 1/λ2 ) (µ = λ1 λ2 λ3 ). (2.23)

Remark 2.2. Note that (2.19) does not require the condition λ1 · · · λn 6= 1. However, we were unable to get the formula analogous (2.18) in the case λ1 · · · λn = 1. We next give some higher-order convolution formulae for the Apostol-Genocchi polynomials, which are the corresponding generalization of Agoh’s convolution formula on the classical Genocchi polynomials presented in [1, 12]. Clearly, by substituting k for n and ui t for ti with u1 + u2 + · · · + uk = 1 in (2.7), we discover that for positive integer k, n, 

tn n!

Y k

 X  n  k 2ui texi ui t 2ur texr t r t = (−1) u t k λ e i +1 n! (−1) λ1 · · · λk et − 1 r=1 i=1 i ×

r−1 Y i=1

λi

 k 2ui te(xi −xr +1)ui t Y 2ui te(xi −xr )ui t . (2.24) λi eui t + 1 λi eui t + 1 i=r+1

It is easy to see from (1.4) that the left hand side of (2.24) can be rewritten as 

tn n!

Y k

 2ui texi ui t = λ eui t + 1 i=1 i j

X

n! · uj11 uj22 · · · ujkk j1 ! · j2 ! · · · jk ! =n

1 +j2 ···+jk j1 ,j2 ,...,jk ≥0

× Gj1 (x1 ; λ1 )Gj2 (x2 ; λ2 ) · · · Gjk (xk ; λk ), (2.25) and the right hand side of (2.24) can be rewritten in the following ways: if k is an even integer then 

tn n!

Y k

ui texi ui t λ eui t − 1 i=1 i

= −2

k X



j

X

j

r−1 r+1 n! · uj11 uj22 · · · ur−1 ur ur+1 · · · ujkk Bjr (xr ; λ1 λ2 · · · λk ) j1 ! · j2 ! · · · jk ! =n

r=1 j1 +j2 ···+jk j1 ,j2 ,...,jk ≥0

×

r−1 Y

k Y

i=1

i=r+1

{−λi Gji (xi − xr + 1; λi )}

596

Gji (xi − xr ; λi ), (2.26)

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and if k is an odd integer then  n Y  k t ui texi ui t n! i=1 λi eui t − 1 =

j

k X

j

r+1 r−1 · · · ujkk ur ur+1 n! · uj11 uj22 · · · ur−1 Gjr (xr ; λ1 λ2 · · · λk ) j1 ! · j2 ! · · · jk ! =n

X

r=1 j1 +j2 ···+jk j1 ,j2 ,...,jk ≥0

×

r−1 Y

k Y

i=1

i=r+1

{−λi Gji (xi − xr + 1; λi )}

Gji (xi − xr ; λi ). (2.27)

Since for positive integer k ≥ 2 and complex numbers α1 , α2 , . . . , αk with Re(αj ) > −1 for j = 1, 2, . . . , k, (see, e.g., [2, 34]) Z 1 Z 1−u1 Z 1−u1 −···−uk−2 αk 1 α2 ··· uα 1 u2 · · · uk du1 d2 · · · duk−1 0

0

0

Γ(α1 + 1)Γ(α2 + 1) · · · Γ(αk + 1) = Γ(α1 + α2 + · · · + αk + k)

(u1 + u2 + · · · + uk = 1). (2.28)

by equating (2.25), (2.26) and (2.27) and making the above integral operation, with the help of (2.28), we get that if k is an even integer then X (n + k) Gj1 (x1 ; λ1 )Gj2 (x2 ; λ2 ) · · · Gjk (xk ; λk ) j1 +j2 ···+jk =n j1 ,j2 ,...,jk ≥0

= −2

k X



X

r=1 j1 +j2 ···+jk =n j1 ,j2 ,...,jk ≥0

 r−1 Y n+k {−λi Gji (xi − xr + 1; λi )} Bjr (xr ; λ1 λ2 · · · λk ) jr i=1 k Y

×

Gji (xi − xr ; λi ), (2.29)

i=r+1

and if k is an odd integer then X (n + k) Gj1 (x1 ; λ1 )Gj2 (x2 ; λ2 ) · · · Gjk (xk ; λk ) j1 +j2 ···+jk =n j1 ,j2 ,...,jk ≥0 k X

=

X

  r−1 Y n+k Gjr (xr ; λ1 λ2 · · · λk ) {−λi Gji (xi − xr + 1; λi )} jr =n i=1

r=1 j1 +j2 ···+jk j1 ,j2 ,...,jk ≥0

k Y

×

Gji (xi − xr ; λi ). (2.30)

i=r+1

Notice that from (2.20) we have r−1 Y

{−λi Gji (xi − xr + 1; λi )}

i=1

=

X

Y

Gji (xi − xr ; λi ) ×

T ⊆{1,...,r−1} i∈T

Y

{−2ji (xi − xr )ji −1 }. (2.31)

i∈T

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Thus, by applying (2.31) to the right hand sides of (2.29) and (2.30) and then taking x1 = x2 = · · · = xk = x, we immediately obtain the following result. Theorem 2.3. Let k, n be positive integers. If k is an even integer, then X (n + k) Gj1 (x; λ1 )Gj2 (x; λ2 ) · · · Gjk (x; λk ) j1 +j2 ···+jk =n j1 ,j2 ,...,jk ≥0

=

 k  X k (−2)k−r+1 r − 1 r=1



X j1 +j2 ···+jr =n−k+r j1 ,j2 ,...,jr ≥0

 n+k Bjr (x; λ1 λ2 · · · λk ) jr

× Gj1 (λ1 )Gj2 (λ2 ) · · · Gjr−1 (λr−1 ). (2.32) If k is an odd integer, then X Gj1 (x; λ1 )Gj2 (x; λ2 ) · · · Gjk (x; λk ) (n + k) j1 +j2 ···+jk =n j1 ,j2 ,...,jk ≥0

=

 k  X k (−2)k−r r − 1 r=1



X j1 +j2 ···+jr =n−k+r j1 ,j2 ,...,jr ≥0

 n+k Gjr (x; λ1 λ2 · · · λk ) jr

× Gj1 (λ1 )Gj2 (λ2 ) · · · Gjr−1 (λr−1 ). (2.33) It becomes obvious that setting k = 2 in Theorem 2.3 gives that for positive integer n,  n  n X 4 X n+2 Bk (x; λµ)Gn−k (λ) Gk (x; λ)Gn−k (x; µ) + n+2 k k=0

k=0

2n(n + 1) Bn−1 (x; λµ). (2.34) 3 Since the classical Genocchi polynomials can be expressed in terms of the classical Bernoulli polynomials, as follows,   x Gn (x) = 2Bn (x) − 2n+1 Bn (n ≥ 0), (2.35) 2 =

by B1 = −1/2, we have G0 = 0 and G1 = 1. Hence, the case λ = µ = 1 in (2.34) gives the convolution identity on the classical Genocchi polynomials due to Agoh [1, 12], namely  n−2  n−1 X 4 X n+2 Gk (x)Gn−k (x) + Bk (x)Gn−k = 0 (n ≥ 2). (2.36) n+2 k k=1

k=0

It is worth noticing that x = 0 in (2.36) can give the result (see, e.g., [1]): n−2 n−2 X X n + 1 Bk Gn−k 4 Gk Gn−k + 4 =− Gn (n ≥ 4), k−1 k n+2 k=2

(2.37)

k=2

which is very analogous to the convolution identity on the classical Bernoulli numbers due to Matiyasevich [25], in an equivalent form, as follows, n−2 n−2 X X n + 1 Bk Bn−k n(n + 1) Bk Bn−k − 2 = Bn (n ≥ 4). (2.38) k−1 k n+2 k=2

k=2

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For some similar convolution formulae to (2.37) and (2.38), one is referred to [12, 13, 15]. If we take k = 3 in Theorem 2.3, we obtain that for positive integer n ≥ 2, X Gj1 (x; λ1 )Gj2 (x; λ2 )Gj3 (x; λ3 ) j1 +j2 +j3 =n

3 − n+3j

X

1 +j2 +j3

  n+3 Gj1 (λ1 )Gj2 (λ2 )Gj3 (x; µ) j3 =n

 n−1  6 X n+3 + Gk (x; µ)Gn−1−k (λ1 ) n+3 k k=0   4 n+3 = Gn−2 (x; µ) (µ = λ1 λ2 λ3 ). (2.39) n+3 5 The case λ1 = λ2 = λ3 = 1 in (2.39) gives the corresponding formula of products of the classical Genocchi polynomials, as follows,   X X 3 n+3 Gj1 (x)Gj2 (x)Gj3 (x) − Gj1 Gj2 Gj3 (x) n + 3 j +j +j =n j3 j +j +j =n 1

2

3

1

+

6 n+3

n−1 X k=0

2

3



  n+3 4 n+3 Gk (x)Gn−1−k = Gn−2 (x), (2.40) k n+3 5

which is very analogous to the convolution identity on the classical Euler polynomials presented in [2, Corollary 3]. Acknowledgements This work was done when the author was visiting State University of New York at Stony Brook. The author is supported by the Foundation for Fostering Talents in Kunming University of Science and Technology (Grant No. KKSY201307047) and the National Natural Science Foundation of P.R. China (Grant No. 11326050, 11071194). References [1] T. Agoh, Convolution identities for Bernoulli and Genocchi polynomials, Electronic J. Combin., 21 (2014), Article ID P1.65. [2] T. Agoh, K. Dilcher, Higher-order convolutions for Bernoulli and Euler polynomials, J. Math. Anal. Appl., 419 (2014), 1235–1247. [3] T.M. Apostol, On the Lerch zeta function, Pacific J. Math., 1 (1951), 161–167. [4] G.E. Andrews, Euler’s Pentagonal Number Theorem, Math. Maga., 56 (1983), 279–284. [5] J. Bell, A summary of Euler’s work on the Pentagonal Number Theorem, Arch. Hist. Exact Sci., 64 (2010), 301–373. [6] K.N. Boyadzhiev, Apostol-Bernoulli functions, derivative polynomials and Eulerian polynomials, Adv. Appl. Discrete Math., 1 (2008), 109–122. [7] H. Cohen, Number Theory, Volume II, Analytic and Modern Tools, Graduate Texts in Math., vol. 240, Springer, New York, 2007. [8] R. Dere, Y. Simsek, H.M. Srivastava, A unified presentation of three families of generalized Apostol type polynomials based upon the theory of the umbral calculus and the umbral algebra, J. Number Theory, 133 (2013), 3245–3263. [9] Y. He, C.P. Wang, Some formulae of products of the Apostol-Bernoulli and Apostol-Euler polynomials, Discrete Dyn. Nat. Soc., 2012 (2012), Article ID 927953. [10] Y. He, C.P. Wang, Recurrence formulae for Apostol-Bernoulli and Apostol-Euler polynomials, Adv. Differ. Equ., 2012 (2012), Article ID 209.

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[11] Y. He, S. Araci, Sums of products of Apostol-Bernoulli and Apostol-Euler polynomials, Adv. Differ. Equ., 2014 (2014), Article ID 155. [12] Y. He, S. Araci, H.M. Srivastava, M. Acikgoz, Some new identities for the Apostol-Bernoulli polynomials and the Apostol-Genocchi polynomials, Appl. Math. Comput., 262 (2015), 31–41. [13] D.S. Kim, T. Kim, S.-H. Lee, Y.-H. Kim, Some identities for the products of two Bernoulli and Euler polynomials, Adv. Differ. Equ., 2012 (2012), Article ID 95. [14] D.S. Kim, T. Kim, Umbral calculus associated with Bernoulli polynomials, J. Number Theory, 147 (2015), 871–882. [15] D.S. Kim, T. Kim, Identities arising from higher-order Daehee polynomials bases, Open Math., 13 (2015), 196–208. [16] M.S. Kim, S. Hu, Sums of products of Apostol-Bernoulli numbers, Ramanujan J., 28 (2012), 113–123. [17] T. Kim, Symmetry p-adic invariant integral on Zp for Bernoulli and Euler polynomials, J. Differ. Equ. Appl., 14 (2008), 1267–1277. [18] T. Kim, L.C. Jang, Y.H. Kim, Some properties on the p-adic invariant integral on Zp associated with Genocchi and Bernoulli polynomials, J. Comput. Anal. Appl., 7 (2011), 1201–1207. [19] T. Kim, T. Mansour, Umbral calculus associated with Frobenius-type Eulerian polynomials, Russian J. Math. Physics, 21 (2014), 484–493. [20] Q.-M. Luo, H.M. Srivastava, Some generalizations of the Apostol-Bernoulli and Apostol-Euler polynomials, J. Math. Anal. Appl., 308 (2005), 290–302. [21] Q.-M. Luo, H.M. Srivastava, Some relationships between the Apostol-Bernoulli and ApostolEuler polynomials, Comput. Math. Appl., 51 (2006), 631–642. [22] Q.-M. Luo, Fourier expansions and integral representations for the Apostol-Bernoulli and Apostol-Euler polynomials, Math. Comput., 78 (2009), 2193–2208. [23] Q.-M. Luo, Extension for the Genocchi polynomials and its Fourier expansions and integral representations, Osaka J. Math., 48 (2011), 291–309. [24] Q.-M. Luo, H.M. Srivastava, Some generalizations of the Apostol-Genocchi polynomials and the Stirling numbers of the second kind, Appl. Math. Comput., 217 (2011), 5702–5728. [25] Y. Matiyasevich, Identities with Bernoulli numbers, http://logic.pdmi.ras.ru/ yumat/Journal/Bernoulli/bernulli.htm, 1997. [26] N. Nielsen, Trait´ e´ el´ ementaire des nombres de Bernoulli, Gauthier-Villars, Paris, 1923. [27] S.H. Rim, E.J. Moon, J.H. Jin, S.J. Lee, On the symmetric properties for the generalized Genocchi polynomials, J. Comput. Anal. Appl., 13 (2011), 1240–1245. [28] H.M. Srivastava, Some formulas for the Bernoulli and Euler polynomials at rational arguments, Math. Proc. Camb. Philos. Soc., 129 (2000), 77–84. ´ Pint´ [29] H.M. Srivastava, A. er, Remarks on some relationships between the Bernoulli and Euler polynomials, Appl. Math. Letters, 17 (2004), 375–380. [30] H.M. Srivastava, B. Kurt, Y. Simsek, Some families of Genocchi type polynomials and their interpolation functions, Integral Trans. Special Funct., 23 (2012), 919–938; see also Corrigendum, Integral Trans. Special Funct., 23 (2012), 939–940. [31] H.M. Srivastava, J. Choi, Zeta and q-Zeta Functions and Associated Series and Integrals, Elsevier Sci. Pub., Amsterdam, London and New York, 2012. [32] R.P. Stanley, Enumerative Combinatorics, Vol. 1, Cambridge Univ. Press, 1997. [33] R. Tremblay, S. Gaboury, B.-J. Fug` ere, A new class of generalized Apostol-Bernoulli polynomials and some analogues of the Srivastava-Pint´ er addition theorem, Appl. Math. Letters, 24 (2011), 1888–1893. [34] S. Waldron, A generalised beta integral and the limit of the Bernstein-Durrmeyer operator with Jacobi weights, J. Approx. Theory, 122 (2003), 141–150. [35] J.Z. Wang, New recurrence formulae for the Apostol-Bernoulli and Apostol-Euler polynomials, Adv. Differ. Equ., 2013 (2013), Article ID 247. Faculty of Science, Kunming University of Science and Technology, Kunming, Yunnan 650500, People’s Republic of China E-mail address: [email protected], [email protected]

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J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 22, NO.4, 2017, COPYRIGHT 2017 EUDOXUS PRESS, LLC

FUNCTIONAL INEQUALITIES IN FUZZY NORMED SPACES CHOONKIL PARK∗ , GEORGE A. ANASTASSIOU, REZA SAADATI AND SUNGSIK YUN∗ Abstract. In this paper, we solve the following additive functional inequality     1 1 x+y − f (x) − f (y), t N (f (x + y) − f (x) − f (y), t) ≥ N f 2 2 2 and the following quadratic functional inequality

(0.1)

N (f (x + y) + f (x − y) − 2f (x) − 2f (y), t) (0.2)       x+y x−y ≥ N 2f + 2f − f (x) − f (y), t 2 2 in fuzzy normed spaces. Using the fixed point method, we prove the Hyers-Ulam stability of the additive functional inequality (0.1) and the quadratic functional inequality (0.2) in fuzzy Banach spaces.

1. Introduction and preliminaries Katsaras [21] defined a fuzzy norm on a vector space to construct a fuzzy vector topological structure on the space. Some mathematicians have defined fuzzy norms on a vector space from various points of view [13, 24, 52]. In particular, Bag and Samanta [2], following Cheng and Mordeson [8], gave an idea of fuzzy norm in such a manner that the corresponding fuzzy metric is of Kramosil and Michalek type [23]. They established a decomposition theorem of a fuzzy norm into a family of crisp norms and investigated some properties of fuzzy normed spaces [3]. We use the definition of fuzzy normed spaces given in [2, 28, 29] to investigate the Hyers-Ulam stability of a quadratic functional inequality in fuzzy Banach spaces. Definition 1.1. [2, 28, 29, 30] Let X be a real vector space. A function N : X × R → [0, 1] is called a fuzzy norm on X if for all x, y ∈ X and all s, t ∈ R, (N1 ) N (x, t) = 0 for t ≤ 0; (N2 ) x = 0 if and only if N (x, t) = 1 for all t > 0; t ) if c 6= 0; (N3 ) N (cx, t) = N (x, |c| (N4 ) N (x + y, s + t) ≥ min{N (x, s), N (y, t)}; (N5 ) N (x, ·) is a non-decreasing function of R and limt→∞ N (x, t) = 1. (N6 ) for x 6= 0, N (x, ·) is continuous on R. The pair (X, N ) is called a fuzzy normed vector space. The properties of fuzzy normed vector spaces and examples of fuzzy norms are given in [27, 28]. Definition 1.2. [2, 28, 29, 30] Let (X, N ) be a fuzzy normed vector space. A sequence {xn } in X is said to be convergent or converge if there exists an x ∈ X such that limn→∞ N (xn −x, t) = 1 for all t > 0. In this case, x is called the limit of the sequence {xn } and we denote it by N limn→∞ xn = x. 2010 Mathematics Subject Classification. Primary 46S40, 39B52, 47H10, 39B62, 26E50, 47S40. Key words and phrases. fuzzy Banach space; additive functional inequality; quadratic functional inequality; fixed point method; Hyers-Ulam stability. ∗ Corresponding author.

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Definition 1.3. [2, 28, 29, 30] Let (X, N ) be a fuzzy normed vector space. A sequence {xn } in X is called Cauchy if for each ε > 0 and each t > 0 there exists an n0 ∈ N such that for all n ≥ n0 and all p > 0, we have N (xn+p − xn , t) > 1 − ε. It is well-known that every convergent sequence in a fuzzy normed vector space is Cauchy. If each Cauchy sequence is convergent, then the fuzzy norm is said to be complete and the fuzzy normed vector space is called a fuzzy Banach space. We say that a mapping f : X → Y between fuzzy normed vector spaces X and Y is continuous at a point x0 ∈ X if for each sequence {xn } converging to x0 in X, then the sequence {f (xn )} converges to f (x0 ). If f : X → Y is continuous at each x ∈ X, then f : X → Y is said to be continuous on X (see [3]). The stability problem of functional equations originated from a question of Ulam [51] concerning the stability of group homomorphisms. The functional equation f (x + y) = f (x) + f (y) is called the Cauchy equation. In particular, every solution of the Cauchy equation is said to be an additive mapping. Hyers [17] gave a first affirmative partial answer to the question of Ulam for Banach spaces. Hyers’ Theorem was generalized by Aoki [1] for additive mappings and by Th.M. Rassias [40] for linear mappings by considering an unbounded Cauchy difference. A generalization of the Th.M. Rassias theorem was obtained by G˘avruta [14] by replacing the unbounded Cauchy difference by a general  = control function in the spirit of Th.M. Rassias’ approach. The functional equation f x+y 2 1 2 f (x)

+ 12 f (y) is called the Jensen equation. The functional equation f (x+y)+f (x−y) = 2f (x)+2f (y) is called the quadratic functional equation. In particular, every solution of the quadratic functional equation is said to be a quadratic mapping. The stability of quadratic functional equation was proved by Skof [50] for mappings f : E1 → E2 , where E1 is a normed space and E2 is a Banach space. Cholewa [9] noticed that the theorem of Skof is still true if the relevant domain E1 is replaced by an Abelian group. Czerwik [10] proved stability of the quadratic functional  the  Hyers-Ulam   x−y 1 1 equation. The functional equation f x+y + f = 2 2 2 f (x) + 2 f (y) is called a Jensen type quadratic equation. The stability problems of several functional equations have been extensively investigated by a number of authors and there are many interesting results concerning this problem (see [4, 18, 20, 25, 36, 37, 38, 41, 42, 44, 45, 46, 47, 48, 49]). Gil´anyi [15] showed that if f satisfies the functional inequality k2f (x) + 2f (y) − f (x − y)k ≤ kf (x + y)k

(1.1)

then f satisfies the Jordan-von Neumann functional equation 2f (x) + 2f (y) = f (x + y) + f (x − y). See also [43]. Fechner [12] and Gil´anyi [16] proved the Hyers-Ulam stability of the functional inequality (1.1). Park, Cho and Han [35] investigated the Cauchy additive functional inequality kf (x) + f (y) + f (z)k ≤ kf (x + y + z)k

(1.2)

and the Cauchy-Jensen additive functional inequality



x+y +z (1.3)

2 and proved the Hyers-Ulam stability of the functional inequalities (1.2) and (1.3) in Banach spaces. Park [33, 34] defined additive ρ-functional inequalities and proved the Hyers-Ulam stability of the additive ρ-functional inequalities in Banach spaces and non-Archimedean Banach spaces. We recall a fundamental result in fixed point theory. 



kf (x) + f (y) + 2f (z)k ≤

2f

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Let X be a set. A function d : X × X → [0, ∞] is called a generalized metric on X if d satisfies (1) d(x, y) = 0 if and only if x = y; (2) d(x, y) = d(y, x) for all x, y ∈ X; (3) d(x, z) ≤ d(x, y) + d(y, z) for all x, y, z ∈ X. Theorem 1.4. [5, 11] Let (X, d) be a complete generalized metric space and let J : X → X be a strictly contractive mapping with Lipschitz constant L < 1. Then for each given element x ∈ X, either d(J n x, J n+1 x) = ∞ for all nonnegative integers n or there exists a positive integer n0 such that (1) d(J n x, J n+1 x) < ∞, ∀n ≥ n0 ; (2) the sequence {J n x} converges to a fixed point y ∗ of J; (3) y ∗ is the unique fixed point of J in the set Y = {y ∈ X | d(J n0 x, y) < ∞}; 1 (4) d(y, y ∗ ) ≤ 1−L d(y, Jy) for all y ∈ Y . In 1996, G. Isac and Th.M. Rassias [19] were the first to provide applications of stability theory of functional equations for the proof of new fixed point theorems with applications. By using fixed point methods, the stability problems of several functional equations have been extensively investigated by a number of authors (see [6, 7, 22, 27, 31, 32, 38, 39]). In Section 2, we solve the additive functional inequality (0.1) and prove the Hyers-Ulam stability of the additive functional inequality (0.1) in fuzzy Banach spaces by using the fixed point method. In Section 3, we solve the quadratic functional inequality (0.2) and prove the Hyers-Ulam stability of the quadratic functional inequality (0.2) in fuzzy Banach spaces by using the fixed point method. Throughout this paper, assume that X is a real vector space and (Y, N ) is a fuzzy Banach space. 2. Additive functional inequality (0.1) In this section, we prove the Hyers-Ulam stability of the additive functional inequality (0.1) in fuzzy Banach spaces. We need the following lemma to prove the main results. Lemma 2.1. Let f : X → Y be a mapping such that  

N (f (x + y) − f (x) − f (y), t) ≥ N f

x+y 2



1 1 − f (x) − f (y), t 2 2



(2.1)

for all x, y ∈ X and all t > 0. Then f is Cauchy additive, i.e., f (x + y) = f (x) + f (y) for all x, y ∈ X. Proof. Assume that f : X → Y satisfies (2.1). Letting x = y = 0 in (2.1), we get N (f (0), t) = N (0, t) = 1. So f (0) = 0. Letting y = x in (2.1), we get N (f (2x) − 2f (x), t) ≥ N (0, t) = 1 and so f (2x) = 2f (x) for all x ∈ X. Thus x 2

 

f

1 = f (x) 2

(2.2)

for all x ∈ X.

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It follows from (2.1) and (2.2) that  

N (f (x + y) − f (x) − f (y), t) ≥ N f

x+y 2



1 1 − f (x) − f (y), t 2 2



1 (f (x + y) − f (x) − f (y)), t 2 = N (f (x + y) − f (x) − f (y), 2t) 



= N

for all t > 0. By (N5 ) and (N6 ), N (f (x + y) − f (x) − f (y), t) = 1 for all t > 0. It follows from (N2 ) that f (x + y) = f (x) + f (y) for all x, y ∈ X.



Theorem 2.2. Let ϕ : X 2 → [0, ∞) be a function such that there exists an L < 1 with L ϕ (2x, 2y) 2 for all x, y ∈ X. Let f : X → Y be an odd mapping satisfying ϕ(x, y) ≤

N (f (x + y) − f (x) − f (y), t)       1 1 t x+y − f (x) − f (y), t , ≥ min N f 2 2 2 t + ϕ(x, y) for all x, y ∈ X and all t > 0. Then A(x) := N -limn→∞ 2n f defines an additive mapping A : X → Y such that N (f (x) − A(x), t) ≥

x 2n




(2.3)

exists for each x ∈ X and

(2 − 2L)t (2 − 2L)t + Lϕ(x, x)

(2.4)

for all x ∈ X and all t > 0. Proof. Letting y = x in (2.3), we get N (f (2x) − 2f (x), t) ≥

t t + ϕ(x, x)

(2.5)

for all x ∈ X. Consider the set S := {g : X → Y } and introduce the generalized metric on S: t d(g, h) = inf µ ∈ R+ : N (g(x) − h(x), µt) ≥ , ∀x ∈ X, ∀t > 0 , t + ϕ(x, x) 



where, as usual, inf φ = +∞. It is easy to show that (S, d) is complete (see [26, Lemma 2.1]). Now we consider the linear mapping J : S → S such that x 2

 

Jg(x) := 2g for all x ∈ X. Let g, h ∈ S be given such that d(g, h) = ε. Then N (g(x) − h(x), εt) ≥

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for all x ∈ X and all t > 0. Hence x N (Jg(x) − Jh(x), Lεt) = N 2g 2

x x x L − 2h , Lεt = N g −h , εt 2 2 2 2 Lt Lt t 2 2 =
≥ Lt x x L t + ϕ(x, x) + ϕ 2, 2 2 + 2 ϕ(x, x)



≥

Lt 2

 

 



  

 



for all x ∈ X and all t > 0. So d(g, h) = ε implies that d(Jg, Jh) ≤ Lε. This means that d(Jg, Jh) ≤ Ld(g, h) for all g, h ∈ S. It follows from (2.5) that 

x L t , t ≥ 2 2 t + ϕ(x, x)

 

N f (x) − 2f



for all x ∈ X and all t > 0. So d(f, Jf ) ≤ L2 . By Theorem 1.4, there exists a mapping A : X → Y satisfying the following: (1) A is a fixed point of J, i.e., 1 = A(x) 2

x 2

 

A

(2.6)

for all x ∈ X. Since f : X → Y is odd, A : X → Y is an odd mapping. The mapping A is a unique fixed point of J in the set M = {g ∈ S : d(f, g) < ∞}. This implies that A is a unique mapping satisfying (2.6) such that there exists a µ ∈ (0, ∞) satisfying t N (f (x) − A(x), µt) ≥ t + ϕ(x, x) for all x ∈ X; (2) d(J n f, A) → 0 as n → ∞. This implies the equality 

n

N - lim 2 f n→∞

for all x ∈ X; (3) d(f, A) ≤

1 1−L d(f, Jf ),

x 2n



= A(x)

which implies the inequality d(f, A) ≤

L . 2 − 2L

This implies that the inequality (2.4) holds. By (2.3), x+y 2n

y N 2 f −f −f , 2n t 2n (  )        x+y x y t n−1 n−1 n n
≥ min N 2 f −2 f −2 f ,2 t , 2n+1 2n 2n t + ϕ 2xn , 2yn 

n

 





x 2n







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for all x, y ∈ X, all t > 0 and all n ∈ N. So          x+y x y N 2n f − f − f ,t 2n 2n 2n (         x y x+y n−1 n−1 − 2 f − 2 f ,t , ≥ min N 2n f n+1 n 2 2 2n for all x, y ∈ X, all t > 0 and all n ∈ N. Since limn→∞ t > 0,

t 2n t Ln + ϕ(x,y) 2n 2n





+

)

= 1 for all x, y ∈ X and all

1 x+y − A(x) − N (A(x + y) − A(x) − A(y), t) ≥ N A 2 2 for all x, y ∈ X and all t > 0. By Lemma 2.1, the mapping A : X → Y 

t 2n

t 2n Ln 2n ϕ (x, y)

1 A(y), t 2 is Cauchy additive. 



Corollary 2.3. Let θ ≥ 0 and let p be a real number with p > 1. Let X be a normed vector space with norm k · k. Let f : X → Y be a mapping satisfying N (f (x + y) − f (x) − f (y), t)       1 x+y 1 t − f (x) − f (y), t , ≥ min N f 2 2 2 t + θ(kxkp + kykp ) for all x, y ∈ X and all t > 0. Then A(x) := N -limn→∞ 2n f ( 2xn ) exists for each x ∈ X and defines an additive mapping A : X → Y such that (2p − 2)t N (f (x) − A(x), t) ≥ p (2 − 2)t + 2θkxkp for all x ∈ X and all t > 0. Proof. The proof follows from Theorem 2.2 by taking ϕ(x, y) := θ(kxkp + kykp ) for all x, y ∈ X. Then we can choose L = 21−p , and we get the desired result.  Theorem 2.4. Let ϕ : X 2 → [0, ∞) be a function such that there exists an L < 1 with   x y , ϕ(x, y) ≤ 2Lϕ 2 2 for all x, y ∈ X. Let f : X → Y be a mapping satisfying (2.3). Then A(x) := N -limn→∞ exists for each x ∈ X and defines an additive mapping A : X → Y such that (2 − 2L)t N (f (x) − A(x), t) ≥ (2 − 2L)t + ϕ(x, x) for all x ∈ X and all t > 0. Proof. Let (S, d) be the generalized metric space defined in the proof of Theorem 2.2. It follows from (2.5) that   1 1 t N f (x) − f (2x), t ≥ 2 2 t + ϕ(x, x) for all x ∈ X and all t > 0. Now we consider the linear mapping J : S → S such that 1 Jg(x) := g (2x) 2 1 for all x ∈ X. So d(f, Jf ) ≤ 2 . The rest of the proof is similar to the proof of Theorem 2.2.

606

1 2n f

(2n x)

(2.7)



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Corollary 2.5. Let θ ≥ 0 and let p be a real number with 0 < p < 1. Let X be a normed vector space with norm k · k. Let f : X → Y be an odd mapping satisfying N (f (x + y) − f (x) − f (y), t)       x+y 1 1 t ≥ min N f − f (x) − f (y), t , 2 2 2 t + θ(kxkp + kykp ) for all x, y ∈ X and all t > 0. Then A(x) := N -limn→∞ defines an additive mapping A : X → Y such that N (f (x) − A(x), t) ≥

1 n 2n f (2 x)

exists for each x ∈ X and

(2 − 2p )t (2 − 2p )t + 2θkxkp

for all x ∈ X and all t > 0. Proof. The proof follows from Theorem 2.4 by taking ϕ(x, y) := θ(kxkp + kykp ) for all x, y ∈ X. Then we can choose L = 2p−1 , and we get the desired result.  3. Quadratic functional inequality (0.2) In this section, we prove the Hyers-Ulam stability of the quadratic functional inequality (0.2) in fuzzy Banach spaces. We need the following lemma to prove the main results. Lemma 3.1. Let f : X → Y be a mapping satisfying f (0) = 0 and N (f (x + y) + f (x − y) − 2f (x) − 2f (y), t)       x−y x+y + 2f − f (x) − f (y), t ≥ N 2f 2 2

(3.1)

for all x, y ∈ X and all t > 0. Then f is quadratic. Proof. Assume that f : X → Y satisfies (3.1). Letting y = x in (3.1), we get N (f (2x) − 4f (x), t) ≥ N (0, t) = 1 and so f (2x) = 4f (x) for all x ∈ X. Thus   x 1 = f (x) (3.2) f 2 4 for all x ∈ X. It follows from (3.1) and (3.2) that N (f (x + y) + f (x − y) − 2f (x) − 2f (y), t)       x−y x+y ≥ N 2f + 2f − f (x) − f (y), t 2 2   1 =N (f (x + y) + f (x − y) − 2f (x) − 2f (y)) , t 2 = N (f (x + y) + f (x − y) − 2f (x) − 2f (y), 2t) for all t > 0. By (N5 ) and (N6 ), N (f (x + y) + f (x − y) − 2f (x) − 2f (y), t) = 1 for all t > 0. It follows from (N2 ) that f (x + y) + f (x − y) = 2f (x) + 2f (y) for all x, y ∈ X.  Theorem 3.2. Let ϕ : X 2 → [0, ∞) be a function such that there exists an L < 1 with ϕ(x, y) ≤

L ϕ (2x, 2y) 4

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for all x, y ∈ X. Let f : X → Y be a mapping satisfying f (0) = 0 and N (f (x + y) + f (x − y) − 2f (x) − 2f (y), t)         x+y x−y t ≥ min N 2f + 2f − f (x) − f (y), t , 2 2 t + ϕ(x, y) for all x, y ∈ X and all t > 0. Then Q(x) := N -limn→∞ 4n f defines a quadratic mapping Q : X → Y such that N (f (x) − Q(x), t) ≥

x 2n




(3.3)

exists for each x ∈ X and

(4 − 4L)t (4 − 4L)t + Lϕ(x, x)

(3.4)

for all x ∈ X and all t > 0. Proof. Letting y = x in (3.3), we get N (f (2x) − 4f (x), t) ≥

t t + ϕ(x, x)

(3.5)

for all x ∈ X. Consider the set S := {g : X → Y } and introduce the generalized metric on S: t , ∀x ∈ X, ∀t > 0 , t + ϕ(x, x) 



d(g, h) = inf µ ∈ R+ : N (g(x) − h(x), µt) ≥

where, as usual, inf φ = +∞. It is easy to show that (S, d) is complete (see [26, Lemma 2.1]). Now we consider the linear mapping J : S → S such that   x Jg(x) := 4g 2 for all x ∈ X. Let g, h ∈ S be given such that d(g, h) = ε. Then t N (g(x) − h(x), εt) ≥ t + ϕ(x, x) for all x ∈ X and all t > 0. Hence x N (Jg(x) − Jh(x), Lεt) = N 4g 2

x x x L − 4h , Lεt = N g −h , εt 2 2 2 4 Lt Lt t 4 4 =
≥ Lt L x x t + ϕ(x, x) + ϕ 2, 2 4 + 4 ϕ(x, x)



≥

Lt 4

 

 



  

 



for all x ∈ X and all t > 0. So d(g, h) = ε implies that d(Jg, Jh) ≤ Lε. This means that d(Jg, Jh) ≤ Ld(g, h) for all g, h ∈ S.  It follows from (3.5) that N f (x) − 4f

x 2






, L4 t ≥

t t+ϕ(x,x)

for all x ∈ X and all t > 0. So

L 4.

d(f, Jf ) ≤ By Theorem 1.4, there exists a mapping Q : X → Y satisfying the following: (1) Q is a fixed point of J, i.e., x 2

 

Q

1 = Q(x) 4

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for all x ∈ X. The mapping Q is a unique fixed point of J in the set M = {g ∈ S : d(f, g) < ∞}. This implies that Q is a unique mapping satisfying (3.6) such that there exists a µ ∈ (0, ∞) satisfying t N (f (x) − Q(x), µt) ≥ t + ϕ(x, x) for all x ∈ X; (2) d(J n f, Q) → 0 as n → ∞. This implies the equality   x n N - lim 4 f = Q(x) n→∞ 2n for all x ∈ X; 1 (3) d(f, Q) ≤ 1−L d(f, Jf ), which implies the inequality d(f, Q) ≤

L . 4 − 4L

This implies that the inequality (3.4) holds. By (3.3),            x+y x−y x y n n N 4 f +f − 2f − 2f ,4 t 2n 2n 2n 2n (    )         x + y x − y x y t
≥ min N 4n 2f + 2f −f −f , 4n t , 2n+1 2n+1 2n 2n t + ϕ 2xn , 2yn for all x, y ∈ X, all t > 0 and all n ∈ N. So            x−y x y x+y n +f − 2f − 2f ,t N 4 f n n n 2 2 2 2n (            x+y x−y x y ≥ min N 4n 2f + 2f − f − f ,t , 2n+1 2n+1 2n 2n for all x, y ∈ X, all t > 0 and all n ∈ N. Since limn→∞ t > 0,

t 4n t Ln + 4n ϕ(x,y) 4n

t 4n

+

t 4n Ln 4n ϕ (x, y)

)

= 1 for all x, y ∈ X and all

N (Q(x + y) + Q(x − y) − 2Q(x) − 2Q(y), t)       x+y x−y ≥ N 2Q + 2Q − Q(x) − Q(y), t 2 2 for all x, y ∈ X and all t > 0. By Lemma 3.1, the mapping Q : X → Y is quadratic, as desired.  Corollary 3.3. Let θ ≥ 0 and let p be a real number with p > 2. Let X be a normed vector space with norm k · k. Let f : X → Y be a mapping satisfying f (0) = 0 and N (f (x + y) + f (x − y) − 2f (x) − 2f (y), t)         x−y t x+y ≥ min N 2f + 2f − f (x) − f (y), t , 2 2 t + θ(kxkp + kykp ) for all x, y ∈ X and all t > 0. Then Q(x) := N -limn→∞ 4n f ( 2xn ) exists for each x ∈ X and defines a quadratic mapping Q : X → Y such that (2p − 4)t N (f (x) − Q(x), t) ≥ p (2 − 4)t + 2θkxkp

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for all x ∈ X. Proof. The proof follows from Theorem 3.2 by taking ϕ(x, y) := θ(kxkp + kykp ) for all x, y ∈ X. Then we can choose L = 22−p , and we get the desired result.  Theorem 3.4. Let ϕ : X 2 → [0, ∞) be a function such that there exists an L < 1 with   x y ϕ(x, y) ≤ 4Lϕ , 2 2 for all x, y ∈ X. Let f : X → Y be a mapping satisfying f (0) = 0 and (3.3). Then Q(x) := N limn→∞ 41n f (2n x) exists for each x ∈ X and defines a quadratic mapping Q : X → Y such that (4 − 4L)t N (f (x) − Q(x), t) ≥ (3.7) (4 − 4L)t + ϕ(x, x) for all x ∈ X and all t > 0. Proof. Let (S, d) be the generalized metric space defined in the proof of Theorem 3.2. It follows from (3.5) that   1 1 t N f (x) − f (2x), t ≥ 4 4 t + ϕ(x, x) for all x ∈ X and all t > 0. Now we consider the linear mapping J : S → S such that 1 Jg(x) := g (2x) 4 1 for all x ∈ X. So d(f, Jf ) ≤ 4 . The rest of the proof is similar to the proof of Theorem 3.2.



Corollary 3.5. Let θ ≥ 0 and let p be a real number with 0 < p < 2. Let X be a normed vector space with norm k · k. Let f : X → Y be a mapping satisfying f (0) = 0 and N (f (x + y) + f (x − y) − 2f (x) − 2f (y), t)         x−y t x+y + 2f − f (x) − f (y), t , ≥ min N 2f 2 2 t + θ(kxkp + kykp ) for all x, y ∈ X and all t > 0. Then Q(x) := N -limn→∞ 41n f (2n x) exists for each x ∈ X and defines a quadratic mapping Q : X → Y such that (4 − 2p )t N (f (x) − Q(x), t) ≥ (4 − 2p )t + 2θkxkp for all x ∈ X. Proof. The proof follows from Theorem 3.4 by taking ϕ(x, y) := θ(kxkp + kykp ) for all x, y ∈ X. Then we can choose L = 2p−2 , and we get the desired result.  References [1] T. Aoki, On the stability of the linear transformation in Banach spaces, J. Math. Soc. Japan 2 (1950), 64–66. [2] T. Bag and S.K. Samanta, Finite dimensional fuzzy normed linear spaces, J. Fuzzy Math. 11 (2003), 687– 705. [3] T. Bag and S.K. Samanta, Fuzzy bounded linear operators, Fuzzy Sets and Systems 151 (2005), 513–547. [4] L. C˘ adariu, L. G˘ avruta, P. G˘ avruta, On the stability of an affine functional equation, J. Nonlinear Sci. Appl. 6 (2013), 60–67.

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[5] L. C˘ adariu and V. Radu, Fixed points and the stability of Jensen’s functional equation, J. Inequal. Pure Appl. Math. 4, no. 1, Art. ID 4 (2003). [6] L. C˘ adariu and V. Radu, On the stability of the Cauchy functional equation: a fixed point approach, Grazer Math. Ber. 346 (2004), 43–52. [7] L. C˘ adariu and V. Radu, Fixed point methods for the generalized stability of functional equations in a single variable, Fixed Point Theory and Applications 2008, Art. ID 749392 (2008). [8] S.C. Cheng and J.M. Mordeson, Fuzzy linear operators and fuzzy normed linear spaces, Bull. Calcutta Math. Soc. 86 (1994), 429–436. [9] P.W. Cholewa, Remarks on the stability of functional equations, Aequationes Math. 27 (1984), 76–86. [10] S. Czerwik, On the stability of the quadratic mapping in normed spaces, Abh. Math. Sem. Univ. Hamburg 62 (1992), 59–64. [11] J. Diaz and B. Margolis, A fixed point theorem of the alternative for contractions on a generalized complete metric space, Bull. Amer. Math. Soc. 74 (1968), 305–309. [12] W. Fechner, Stability of a functional inequalities associated with the Jordan-von Neumann functional equation, Aequationes Math. 71 (2006), 149–161. [13] C. Felbin, Finite dimensional fuzzy normed linear spaces, Fuzzy Sets and Systems 48 (1992), 239–248. [14] P. G˘ avruta, A generalization of the Hyers-Ulam-Rassias stability of approximately additive mappings, J. Math. Anal. Appl. 184 (1994), 431–436. [15] A. Gil´ anyi, Eine zur Parallelogrammgleichung a ¨quivalente Ungleichung, Aequationes Math. 62 (2001), 303– 309. [16] A. Gil´ anyi, On a problem by K. Nikodem, Math. Inequal. Appl. 5 (2002), 707–710. [17] D.H. Hyers, On the stability of the linear functional equation, Proc. Nat. Acad. Sci. U.S.A. 27 (1941), 222–224. [18] D.H. Hyers, G. Isac and Th.M. Rassias, Stability of Functional Equations in Several Variables, Birkh¨ auser, Basel, 1998. [19] G. Isac and Th.M. Rassias, Stability of ψ-additive mappings: Appications to nonlinear analysis, Internat. J. Math. Math. Sci. 19 (1996), 219–228. [20] S. Jung, Hyers-Ulam-Rassias Stability of Functional Equations in Mathematical Analysis, Hadronic Press lnc., Palm Harbor, Florida, 2001. [21] A.K. Katsaras, Fuzzy topological vector spaces II, Fuzzy Sets and Systems 12 (1984), 143–154. [22] J. Kim, G. A. Anastassiou and C. Park, Additive ρ-functional inequalities in fuzzy normed spaces, J. Comput. Anal. Appl. (to appear). [23] I. Kramosil and J. Michalek, Fuzzy metric and statistical metric spaces, Kybernetica 11 (1975), 326–334. [24] S.V. Krishna and K.K.M. Sarma, Separation of fuzzy normed linear spaces, Fuzzy Sets and Systems 63 (1994), 207–217. [25] J. Lee, C. Park and D. Shin, An AQCQ-functional equation in matrix normed spaces, Results Math. 27 (2013), 305–318. [26] D. Mihet¸ and V. Radu, On the stability of the additive Cauchy functional equation in random normed spaces, J. Math. Anal. Appl. 343 (2008), 567–572. [27] M. Mirzavaziri and M.S. Moslehian, A fixed point approach to stability of a quadratic equation, Bull. Braz. Math. Soc. 37 (2006), 361–376. [28] A.K. Mirmostafaee, M. Mirzavaziri and M.S. Moslehian, Fuzzy stability of the Jensen functional equation, Fuzzy Sets and Systems 159 (2008), 730–738. [29] A.K. Mirmostafaee and M.S. Moslehian, Fuzzy versions of Hyers-Ulam-Rassias theorem, Fuzzy Sets and Systems 159 (2008), 720–729. [30] A.K. Mirmostafaee and M.S. Moslehian, Fuzzy approximately cubic mappings, Inform. Sci. 178 (2008), 3791–3798. [31] C. Park, Fixed points and Hyers-Ulam-Rassias stability of Cauchy-Jensen functional equations in Banach algebras, Fixed Point Theory and Applications 2007, Art. ID 50175 (2007). [32] C. Park, Generalized Hyers-Ulam-Rassias stability of quadratic functional equations: a fixed point approach, Fixed Point Theory and Applications 2008, Art. ID 493751 (2008). [33] C. Park, Additive ρ-functional inequalities and equations, J. Math. Inequal. 9 (2015), 17–26. [34] C. Park, Additive ρ-functional inequalities in non-Archimedean normed spaces, J. Math. Inequal. 9 (2015), 397–407.

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[35] C. Park, Y. Cho and M. Han, Stability of functional inequalities associated with Jordan-von Neumann type additive functional equations, J. Inequal. Appl. 2007, Art. ID 41820 (2007). [36] C. Park, K. Ghasemi, S. G. Ghaleh and S. Jang, Approximate n-Jordan ∗-homomorphisms in C ∗ -algebras, J. Comput. Anal. Appl. 15 (2013), 365-368. [37] C. Park, A. Najati and S. Jang, Fixed points and fuzzy stability of an additive-quadratic functional equation, J. Comput. Anal. Appl. 15 (2013), 452–462. [38] C. Park and Th.M. Rassias, Fixed points and generalized Hyers-Ulam stability of quadratic functional equations, J. Math. Inequal. 1 (2007), 515–528. [39] V. Radu, The fixed point alternative and the stability of functional equations, Fixed Point Theory 4 (2003), 91–96. [40] Th.M. Rassias, On the stability of the linear mapping in Banach spaces, Proc. Amer. Math. Soc. 72 (1978), 297–300. [41] Th.M. Rassias, Problem 16; 2, Report of the 27th International Symp. on Functional Equations, Aequationes Math. 39 (1990), 292–293; 309. [42] Th.M. Rassias and K. Shibata, Variational problem of some quadratic functionals in complex analysis, J. Math. Anal. Appl. 228 (1998), 234–253. [43] J. R¨ atz, On inequalities associated with the Jordan-von Neumann functional equation, Aequationes Math. 66 (2003), 191–200. ˇ ankov´ [44] L. Reich, J. Sm´ıtal and M. Stef´ a, Singular solutions of the generalized Dhombres functional equation, Results Math. 65 (2014), 251–261. [45] S. Schin, D. Ki, J. Chang and M. Kim, Random stability of quadratic functional equations: a fixed point approach, J. Nonlinear Sci. Appl. 4 (2011), 37–49. [46] S. Shagholi, M. Bavand Savadkouhi and M. Eshaghi Gordji, Nearly ternary cubic homomorphism in ternary Fr´echet algebras, J. Comput. Anal. Appl. 13 (2011), 1106–1114. [47] S. Shagholi, M. Eshaghi Gordji and M. Bavand Savadkouhi, Stability of ternary quadratic derivation on ternary Banach algebras, J. Comput. Anal. Appl. 13 (2011), 1097–1105. [48] D. Shin, C. Park and Sh. Farhadabadi, On the superstability of ternary Jordan C ∗ -homomorphisms, J. Comput. Anal. Appl. 16 (2014), 964–973. [49] D. Shin, C. Park and Sh. Farhadabadi, Stability and superstability of J ∗ -homomorphisms and J ∗ -derivations for a generalized Cauchy-Jensen equation, J. Comput. Anal. Appl. 17 (2014), 125–134. [50] F. Skof, Propriet` a locali e approssimazione di operatori, Rend. Sem. Mat. Fis. Milano 53 (1983), 113–129. [51] S. M. Ulam, A Collection of the Mathematical Problems, Interscience Publ. New York, 1960. [52] J.Z. Xiao and X.H. Zhu, Fuzzy normed spaces of operators and its completeness, Fuzzy Sets and Systems 133 (2003), 389–399. Choonkil Park Research Institute for Natural Sciences, Hanyang University, Seoul 133-791, Korea E-mail address: [email protected] George A. Anastassiou Department of Mathematical Sciences, University of Memphis, Memphis, TN 38152, USA E-mail address: [email protected] Reza Saadati Department of Mathematics, Iran University of Science and Technology, Tehran, Iran E-mail address: [email protected] Sungsik Yun Department of Financial Mathematics, Hanshin University, Gyeonggi-do 447-791, Korea E-mail address: [email protected]

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A generalization of Simpson type inequality via differentiable functions using extended (s, m)ϕ-preinvex functions Yujiao Li 1

1

Tingsong Du

1∗

College of Science, China Three Gorges University, Yichang 443002 P. R. China. E-mail:

∗

[email protected], [email protected]

February 5, 2016 Abstract Some new class of preinvex functions which called (s, m)-preinvex, extended (s, m)-preinvex, (s, m)ϕ -preinvex and extended (s, m)ϕ -preinvex function are introduced respectively in this paper. An integral identity is established, and then we prove some Simpson type integral inequalities, dealing with the existing similar type integral inequalities in a relatively uniform frame. In particular, we also show some results obtained by these inequalities for extended (s, m)ϕ preinvex under some suitable conditions, which improve the previously known results. 2010 Mathematics Subject Classification: Primary 26D15; 26D20; Secondary 26A51, 26B12, 41A55, 41A99. Key words and phrases: Simpson’s inequality; H¨older’s inequality; (s, m)ϕ convex function.

1

Introduction

The following notations are used throughout this paper. I is an interval on the real line R, R0 = [0, ∞). Rn is used to denote a generic n-dimensional vector space, Rn0 denotes an n-dimensional nonegative vector space, and Rn+ denotes an n-dimensional positive vector space. For any subset K ⊆ Rn , L1 [a, b] is the set of integrable functions over the interval [a, b]. Let us firstly recall some definitions of various convex type functions.

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Definition 1.1 ([6]) A function f : I ⊆ R → R0 is said to be a Godunova-Levin function if f is nonnegative and for all x, y ∈ I, λ ∈ (0, 1) we have that ( ) f (x) f (y) f λx + (1 − λ)y ≤ + . λ 1−λ Definition 1.2 ([5]) For some (s, m) ∈ (0, 1]2 , a function f : [0, b] → R is said to be (s, m)-convex in the second sense if for every x, y ∈ [0, b] and λ ∈ (0, 1] we have that ( ) f λx + m(1 − λ)y ≤ λs f (x) + m(1 − λ)s f (y). Definition 1.3 ([36]) For some s ∈ [−1, 1] and m ∈ (0, 1], a function f : [0, b] → R0 is said to be extended (s, m)-convex if for all x, y ∈ [0, b] and λ ∈ (0, 1) we have that ( ) f λx + m(1 − λ)y ≤ λs f (x) + m(1 − λ)s f (y). Definition 1.4 ([1]) A set K ⊆ Rn is said to be invex with respect to the map η : K × K → Rn , if x + tη(y, x) ∈ K for every x, y ∈ K and t ∈ [0, 1]. Notice that every convex set is invex with respect to the map η(y, x) = y −x, but the converse is not necessarily true. For more details please refer to [1, 37] and the references therein. Definition 1.5 ([1]) Let K ⊆ Rn be an invex set with respect to η : K × K → Rn , for every x, y ∈ K, the η-path Pxν joining the points x and ν = x + η(y, x) is defined by { } Pxν = z|z = x + tη(y, x), t ∈ [0, 1] . Definition 1.6 ([27]) The function f defined on the invex set K ⊆ Rn is said to be preinvex with respect to η if for every x, y ∈ K and t ∈ [0, 1] we have that ( ) f x + tη(y, x) ≤ (1 − t)f (x) + tf (y). The concept of preinvexity is more general than convexity since every convex function is preinvex with respect to the map η(y, x) = y − x, but the converse is not true. Definition 1.7 ([13]) The function f defined on the invex set K ⊆ [0, b∗ ] with b∗ > 0 is said to be m-preinvex with respect to η if for all x, y ∈ K, t ∈ [0, 1] and for some fixed m ∈ (0, 1], we have that (y) ( ) f x + tη(y, x) ≤ (1 − t)f (x) + mtf . m y could be Remark 1.1 Notice that if y ∈ [0, b∗ ], then for any 0 < m < 1, m ∗ greater than b , which is not in the domain of f . Thus, the right hand side of the inequality in this definition could be meaningless. To fix this flaw, we suggest to replace [0, b∗ ] by the half real line R0 . 2

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Definition 1.8 ([14]) Let K ⊆ R0 be an invex set with respect to η. A function f : K → R is said to be s-preinvex with respect to η, if for all x, y ∈ K, t ∈ [0, 1] and some fixed s ∈ (0, 1] we have that ( ) f x + tη(y, x) ≤ (1 − t)s f (x) + ts f (y). Definition 1.9 ([21]) The set Kϕη ⊆ Rn is said to be ϕ-invex at u with respect to ϕ(.), if there exists a bifunction η(., .) : Kϕη × Kϕη → Rn , such that u + teiϕ η(v, u) ∈ Kϕη ,

∀u, v ∈ Kϕη , t ∈ [0, 1].

The ϕ-invex set Kϕη is also called ϕη-connected set. Note that the convex set with ϕ = 0 and η(v, u) = v − u is a ϕ-invex set, but the converse is not true (see [21]). Definition 1.10 ([22]) For some fixed s ∈ (0, 1], a function f on the set Kϕη is said to be sφ -preinvex function with respect to ϕ and η, if ( ) f u + teiϕ η(v, u) ≤ (1 − t)s f (u) + ts f (v), ∀u, v ∈ Kϕη , t ∈ [0, 1]. Definition 1.11 ([22]) A function f on the set Cϕ is said to be ϕ-convex function with respect to ϕ, if and only if ( ) f u + teiϕ (v − u) ≤ (1 − t)f (u) + tf (v), ∀u, v ∈ Cϕ , t ∈ [0, 1]. The following inequality is very remarkable and well known in the literature as Simpson type inequality, which plays an important role in analysis. Particularly, it is well applied in numerical integration. Theorem 1.1 ([4]) Let f : [a, b] → R be a four times continuously differentiable mapping on (a, b) and ||f (4) ||∞ = supx∈(a,b) |f (4) (x)| < ∞.Then the following inequality holds: [ ] ∫ b 1 f (a) + f (b) a+b 1 ≤ 1 ||f (4) ||∞ (b−a)4 . (1.1) +2f ( ) − f (x)dx 3 2880 2 2 b−a a In recent decades, a lot of inequalities of Simpson type and Hadamard type for various kinds of convex functions have been established and developed by many scholars, some of them may be reformulated as follows. Theorem 1.2 ([30]) Let f : I ⊆ R0 → R be a differentiable mapping on I ◦ such that f ′ ∈ L1 [a, b], where a, b ∈ I ◦ with a < b. If |f ′ | is s-convex on [a, b], for some fixed s ∈ (0, 1], then [ ∫ b )] ( 1 f (a) + f (b) + 4f a + b − 1 f (x)dx 6 2 b−a a [ ] (s − 4)6s+1 + 2 × 5s+2 − 2 × 3s+2 + 2 ≤ (b − a) |f ′ (a)| + |f ′ (b)| .(1.2) s+2 6 (s + 1)(s + 2) 3

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Theorem 1.3 ([4]) Let f : [a, b] → R is a differentiable mapping whose deriva∫b tive is continuous on (a, b) and ||f ′ ||1 = a |f ′ (x)|dx < ∞. Then we have the inequality: ∫ b ( a + b )] 1 b − a [ f (a) + f (b) ≤ ||f ′ ||1 (b − a)2 . + 2f (1.3) f (x)dx − 3 3 2 2 a Theorem 1.4 ([35]) Let f : I ⊆ R0 → R be a differentiable mapping on I ◦ , a, b ∈ I with a < b, f ′ ∈ L1 [a, b] and 0 ≤ λ, µ ≤ 1. If |f ′ (x)|q for q ≥ 1 is an extended s-convex on [a, b] for some s ∈ [−1, 1], specially, when q = 1 and s = −1, the following inequality holds: ( ∫ b ) ( a + b) 1 f ≤ (b − a) ln 2 |f ′ (a)| + |f ′ (b)| . − (1.4) f (x)dx 2 b−a a Theorem 1.5 ([2, 29]) Let K ⊆ R be an open invex subset with respect to η : K × K → R. Suppose that f : K → R is a differentiable function. If |f ′ | is preinvex on K, then, for every a, b ∈ K with η(b, a) ̸= 0 we have that: ( ) ∫ a+η(b,a) f (a) + f a + η(b, a) 1 − f (x)dx 2 η(b, a) a ) |η(b, a)| ( ′ |f (a)| + |f ′ (b)| (1.5) ≤ 8 and ( ) ∫ a+η(b,a) ) 2a + η(b, a) |η(b, a)| ( ′ 1 f ≤ − f (x)dx |f (a)| + |f ′ (b)| .(1.6) 2 η(b, a) a 8 Theorem 1.6 ([33]) Let A ⊆ R be an open invex subset with respect to η : A × A → R. Suppose that f : A → R is a differentiable function. If q > 1, q ≥ r, s ≥ 0 and |f ′ | is preinvex on A, then for every a, b ∈ A with η(a, b) ̸= 0, we have that ( ) ∫ b+η(a,b) 2b + η(a, b) 1 f − f (x)dx 2 η(a, b) b {( ) q1 ( )1− q1 [ ]1 q−1 (r + 1)|f ′ (a)|q + (r + 3)|f ′ (b)|q q |η(a, b)| 1 ≤ 4 r+1 2q − r − 1 2(r + 2) ( ) q1 ( )1− q1 [ ]1 } 1 q−1 (s + 3)|f ′ (a)|q + (s + 1)|f ′ (b)|q q + . s+1 2q − s − 1 2(s + 2) Corollary 1.1 ([33]) Under the conditions of Theorem 1.6, when r = s = 0,

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the following inequality holds: ( ) ∫ b+η(a,b) 2b + η(a, b) 1 f − f (x)dx 2 η(a, b) b ( ) q1 )1− q1 [( |η(a, b)| q−1 1 ′ 3 ≤ |f (a)|q + |f ′ (b)|q 2q − 1 4 4 4 ( ) q1 ] 3 1 + |f ′ (a)|q + |f ′ (b)|q . 4 4

(1.7)

Theorem 1.7 ([22]) Let I ⊆ R be an open ϕ-invex set with respect to η : I × I → R. Suppose that f : I → R is a differentiable function such that f ′ ∈ L1 [a, a + eiϕ η(b, a)]. If |f ′ | is ϕ-preinvex on I , then, for η(b, a) > 0, ( ) ∫ a+eiϕ η(b,a) f (a) + f a + eiϕ η(b, a) 1 − iϕ f (x)dx 2 e η(b, a) a ) |eiϕ η(b, a)| ( ′ ≤ |f (a)| + |f ′ (b)| . 8

(1.8)

Currently, the Simpson type inequalities concerning different kinds of preinvex and ϕ-convex functions are still interesting research topics to many researchers in the field of convex analysis. For more information please refer to [7–12, 15, 17–20, 23–26, 31, 32, 34] and references cited therein. Motivated by the inspiring idea in [3, 13, 21, 28] and based on our previous works [16, 38], in this paper we are mainly going to introduce the (s, m)ϕ preinvex function and the extended (s, m)ϕ -preinvex function, and then we will establish some Simpson type integral inequalities for extended (s, m)ϕ -preinvex functions. In Section 2, we will introduce new definitions and an integral identity. Section 3 will be devoted of presenting the main results.

2

New definitions and an integral identity

We now mainly introduce some new concepts about preinvex function. The class of (s, m)ϕ -preinvex function is quite a general and unifying one. This is one of the main motivation of this paper. Definition 2.1 Let K ⊆ Rn0 be an open invex set with respect to η : K × K → Rn+ . For f : K → R and some fixed (s, m) ∈ (0, 1] × (0, 1], if ( ) (y) f x + λη(y, x) ≤ (1 − λ)s f (x) + mλs f (2.1) m is valid for all x, y ∈ K, λ ∈ [0, 1], then we say that f (x) is an (s, m)-preinvex function with respect to η.

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Definition 2.2 Let K ⊆ Rn0 be an open invex set with respect to η : K × K → Rn+ . For f : K → R0 and some fixed (s, m) ∈ [−1, 1] × (0, 1], if ( ) (y) (2.2) f x + λη(y, x) ≤ (1 − λ)s f (x) + mλs f m is valid for all x, y ∈ K, λ ∈ [0, 1], then we say that f (x) is an extended (s, m)preinvex function with respect to η. Remark 2.1 In Definition 2.1, if s = 1 then one obtains the usual definition of m-preinvex function. If m = 1 then one obtains the usual definition of s-preinvex function. It is also worthwhile to note that every (s, m)-preinvex function is (s, m)-convex and every extended (s, m)-preinvex functions is extended (s, m)convex with respect to η(y, x) = y − x respectively. Definition 2.3 A function f on the set Kϕη ⊆ Rn0 is said to be (s, m)φ -preinvex function with respect to ϕ(.) and η(., .). For f : Kϕη → R and some fixed (s, m) ∈ (0, 1] × (0, 1], if ( ) (y) f x+λeiϕ η(y, x) ≤ (1−λ)s f (x)+mλs f , ∀x, y ∈ Kϕη , λ ∈ [0, 1]. (2.3) m Remark 2.2 In Definition 2.3, if ϕ = 0 then it reduces to the definition for (s, m)-preinvex function. If m = 1 then it reduces to the definition for sφ preinvex function. Also, it is obvious that Definition 2.3 is the ϕ-convex function when η(y, x) = y − x and s = m = 1. Definition 2.4 A function f on the set Kϕη ⊆ Rn0 is said to be extended (s, m)φ -preinvex function with respect to ϕ(.) and η(., .). For f : Kϕη → R0 and some fixed (s, m) ∈ [−1, 1] × (0, 1], if ( ) (y) f x+λeiϕ η(y, x) ≤ (1−λ)s f (x)+mλs f , ∀x, y ∈ Kϕη , λ ∈ [0, 1]. (2.4) m In order to establish some new Simpson type integral inequalities, we need the following key integral identity, which will be used in the sequel. Lemma 2.1 Let Kϕη ⊆ R be a ϕ-invex subset with respect to ϕ(.) and η : Kϕη × Kϕη ⊆ R, a, b ∈ Kϕη with a < a + η(b, a). If k, t ∈ R, f : Kϕη → R is a differentiable function and f ′ ∈ L[a, a + eiϕ η(b, a)] we have that ( ) ( ) eiϕ η(b, a) tf (a) + (1 − k)f a + eiϕ η(b, a) + (k − t)f a + 2 ∫ a+eiϕ η(b,a) 1 f (x)dx − iϕ e η(b, a) a [∫ 1 ( ) 2 iϕ = e η(b, a) (λ − t)f ′ a + λeiϕ η(b, a) dλ ∫

0

1

+ 1 2

] (λ − k)f a + λe η(b, a) dλ . ′

(

)

iϕ

(2.5)

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Proof. Set

[

( ) iϕ ′ a + λe η(b, a) dλ (λ − t)f 0 ( ) ] ∫1 + 1 (λ − k)f ′ a + λeiϕ η(b, a) dλ .

J = eiϕ η(b, a)

∫

1 2

2

Since a, b ∈ Kϕη and Kϕη is ϕ-invex subset with respect to ϕ and η, for every t ∈ [0, 1], we have a + λeiϕ η(b, a) ∈ Kϕη . Integrating by part, it yields that { [ ( ) 12 1 iϕ J = e η(b, a) iϕ (λ − t)f a + λeiϕ η(b, a) e η(b, a) 0 ] ∫ 21 ( ) − f a + λeiϕ η(b, a) dλ 0 } [ ( ) 1 ∫ 1 ( ) ] 1 iϕ iϕ (λ − k)f a + λe η(b, a) 1 − f a + λe η(b, a) dλ + iϕ 1 e η(b, a) 2 2 ) ∫ 21 ( (1 ) ) ( eiϕ η(b, a) = f a + λeiϕ η(b, a) dλ −t f a+ + tf (a) − 2 2 0 ) ( ) (1 ) ( eiϕ η(b, a) iϕ + (1 − k)f a + e η(b, a) − −k f a+ 2 2 ∫ 1 ( ) − f a + λeiϕ η(b, a) dλ 1 2

( ) ( ) eiϕ η(b, a) = tf (a) + (1 − k)f a + eiϕ η(b, a) + (k − t)f a + 2 ∫ 1 ( ) − f a + λeiϕ η(b, a) dλ. 0

Let x = a + λeiϕ η(b, a), then dx = eiϕ η(b, a)dλ and we have ( ) ( ) eiϕ η(b, a) J = tf (a) + (1 − k)f a + eiϕ η(b, a) + (k − t)f a + 2 ∫ a+eiϕ η(b,a) 1 − iϕ f (x)dx, e η(b, a) a which is required. 1 Remark 2.1 Clearly, applying Lemma 2.1 for ϕ = 0, η(b, a) = b − a, t = , 6 5 and k = , then we obtain the Lemma 2.1 in ([28], 2013). 6

3

Some Simpson type integral inequalities

In what follows, we establish another refinement of the Simpson’s inequality for extended (s, m)ϕ -preinvex functions in the second sense. 7

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Theorem 3.1 Let Aϕη ⊆ R0 be an open ϕ-invex subset with respect to ϕ(.) and η : Aϕη × Aϕη → R0 , a, b ∈ Aϕη with a < a + eiϕ η(b, a), a < b. Let k, t ∈ R. Suppose that f : Aϕη → R0 is a differentiable function and f ′ is integrable on [a, a + eiϕ η(b, a)]. If |f ′ | is extended (s, m)ϕ -preinvex on Aϕη for some fixed (s, m) ∈ [−1, 1] × (0, 1] then the following inequality holds: 1. when s ∈ (−1, 1], we have ( ) ( ) eiϕ η(b, a) iϕ tf (a) + (1 − k)f a + e η(b, a) + (k − t)f a + 2 iϕ ∫ a+e η(b,a) 1 − iϕ f (x)dx e η(b, a) a [ ) ] ( iϕ ′ b ′ ≤ e η(b, a) ν1 |f (a)| + mν2 f (3.1) , m where [ ] 1 2(1 − t)s+2 + 2(1 − k)s+2 + 2(k + t)(s + 2) − 2(s + 3) s+2 + (ts + 2t − 1) 2 v1 = (s + 1)(s + 2) and [ ] 1 2ts+2 + 2k s+2 + 2(s + 1) − 2(s + 2)(k + t) s+2 + (s + 1 − ks − 2k) 2 v2 = ; (s + 1)(s + 2) 2. when s = −1, t = 0 and k = 1, we have ( ) ∫ a+eiϕ η(b,a) iϕ e η(b, a) 1 f a + − iϕ f (x)dx 2 e η(b, a) a ) ] [ ( b ≤ eiϕ η(b, a) ln 2 |f ′ (a)| + m f ′ . m

(3.2)

Proof. 1. When −1 < s ≤ 1, by Lemma 2.1 and using the extended (s, m)ϕη preinvexity of |f ′ | on Aϕη , we have ( ) ( ) eiϕ η(b, a) tf (a) + (1 − k)f a + eiϕ η(b, a) + (k − t)f a + 2 ∫ a+eiϕ η(b,a) 1 − iϕ f (x)dx e η(b, a) a [∫ 1 ( ) 2 ≤ eiϕ η(b, a) |λ − t| f ′ a + eiϕ λη(b, a) dλ 0

∫

1

+ 1 2

( ) |λ − k| f ′ a + λeiϕ η(b, a) dλ

]

8

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{∫ iϕ ≤ e η(b, a)

∫

( b ) |λ − t|λs f ′ dλ m 0 0 } ∫ 1 ∫ 1 ( b ) |λ − k|(1 − λ)s |f ′ (a)|dλ + m |λ − k|λs f ′ + dλ 1 1 m 2 2 {[ ∫ 1 ] ∫ 1 2 iϕ s s |λ − t|(1 − λ) dλ + |λ − k|(1 − λ) dλ f ′ (a) = e η(b, a) [

1 2

|λ − t|(1 − λ)s |f ′ (a)|dλ + m

1 2

0

∫

1 2

+ m

∫ |λ − t|λ dλ + m

1

] ( ) } b |λ − k|λ dλ f ′ . m s

s

0

1 2

1 2

Using the fact that ∫

1 2

∫ |λ − t|(1 − λ) dλ +

1

|λ − k|(1 − λ)s dλ

s

0

1 2

[ ] 1 2(1 − t)s+2 + 2(1 − k)s+2 + 2(k + t)(s + 2) − 2(s + 3) s+2 + (ts + 2t − 1) 2 = (s + 1)(s + 2) and ∫

1 2

∫ |λ − t|λs dλ +

0

1 1 2

|λ − k|λs dλ

[ ] 1 2ts+2 + 2k s+2 + 2(s + 1) − 2(s + 2)(k + t) s+2 + (s + 1 − ks − 2k) 2 = , (s + 1)(s + 2) the desired inequality (3.1) is established. 2. When s = −1, t = 0, and k = 1, utilizing Lemma 2.1 again and the extended (−1, m)ϕη -preinvexity of |f ′ | on Aϕη , we have that ( ) ∫ a+eiϕ η(b,a) iϕ e η(b, a) 1 f a + − f (x)dx 2 eiϕ η(b, a) a [∫ 1 ( ) 2 |λ| f ′ a + λeiϕ η(b, a) dλ ≤ eiϕ η(b, a) 0

∫

1

+ 1 2

] ( ) ′ |λ − 1| f a + λeiϕ η(b, a) dλ

{∫ ≤ eiϕ η(b, a) 0

1 2

[

] λ λ ( b ) |f ′ (a)| + m f ′ dλ 1−λ λ m

9

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] } 1 − λ ′ ( b ) 1−λ ′ |f (a)| + m + f dλ 1 1−λ λ m 2 ( b ) ] [ = eiϕ η(b, a) ln 2 |f ′ (a)| + m f ′ . m ∫

1

[

This proves as required. Direct computation yields the following corollaries. Corollary 3.1 Under the conditions of Theorem 3.1 and s ∈ (−1, 1], 5 1 1. if t = and k = , we have 6 6 [ ( )] 1 ) ( eiϕ η(b, a) f (a) + f a + eiϕ η(b, a) + 4f a + 6 2 ∫ a+eiϕ η(b,a) 1 f (x)dx − iϕ e η(b, a) a [ ] (s − 4)6s+1 + 2 × 5s+2 − 2 × 3s+2 + 2 ≤ 6s+2 (s + 1)(s + 2) [ ( b ) ] iϕ ′ × e η(b, a) |f (a)| + m f ′ ; m

(3.3)

2. if ϕ = 0, η(b, a) = b − a, and m = 1 in inequality (3.3), we have [ ∫ b ( )] 1 f (a) + f (b) + 4f a + b − 1 f (x)dx 6 2 b−a a [ ] (s − 4)6s+1 + 2 × 5s+2 − 2 × 3s+2 + 2 ′ ′ − a) |f (a)| + |f (b)| ;(3.4) (b ≤ 6s+2 (s + 1)(s + 2) 1 and s = m = 1 in inequality (3.1), we have 2 ( ) ∫ a+eiϕ η(b,a) f (a) + f a + eiϕ η(b, a) 1 − f (x)dx iϕ 2 e η(b, a) a ) |eiϕ η(b, a)| ( ′ ≤ |f (a)| + |f ′ (b)| ; 8

3. if t = k =

4. if ϕ = 0 in inequality (3.5), we have ( ) ∫ a+η(b,a) f (a) + f a + η(b, a) 1 f (x)dx − 2 η(b, a) a ) |η(b, a)| ( ′ ≤ |f (a)| + |f ′ (b)| . 8

(3.5)

(3.6)

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Remark 3.1 Inequality (3.4) is the same as inequality of (1.2) presented by Sarikaya et al. in ([30], 2010). Inequality (3.5) is the same as inequality of (1.8) presented by Noor et al. in ([22], 2015). Inequality (3.6) is the same as inequality of (1.5) established by Barani et al. in ([2], 2012). Thus, inequality (3.1) is a generalization of these inequalities. Corollary 3.2 The upper bound of the midpoint inequality for the first derivative is developed as follows: ( ) ( ) iϕ in inequality (3.1), 1. By putting f (a) = f a + eiϕ η(b, a) = f a + e η(b,a) 2 we have: ∫ a+eiϕ η(b,a) ( eiϕ η(b, a) ) 1 − iϕ f (x)dx f a + 2 e η(b, a) a [ ] ( ) b ≤ eiϕ η(b, a) ν1 |f ′ (a)| + mν2 f ′ (3.7) , m where v1 and v2 are defined in Theorem 3.1. 1 5 2. Putting ϕ = 0, s = 1, m = 1, t = , and k = in the above inequality (3.7), 6 6 it yields that ( ) ∫ a+η(b,a) ) 2a + η(b, a) 5|η(b, a)| ( ′ 1 ′ f ≤ − f (x)dx |f (a)| + |f (b)| .(3.8) 2 η(b, a) a 72 Remark 3.2 It is noted that the above midpoint inequality (3.8) is better than the inequality (1.6) presented by Sarikaya et al. in ([29], 2012). Corollary 3.3 Under the conditions of Theorem 3.1 and s = −1, if ϕ = 0, η(b, a) = b − a, we have ( ∫ b ( ) ] [ a + b) 1 f ≤ (b − a) ln 2 |f ′ (a)| + m f ′ b . − f (x)dx 2 b−a a m

(3.9)

Remark 3.3 When applying m = 1 to inequality (3.9), then we get inequality (1.4). Thus, Theorem 3.1 and its consequences generalize the main result in ([35], 2015). Theorem 3.2 Let f be defined as in Theorem 3.1 with p1 + 1q = 1. If |f ′ |q for q > 1 is extended (s, m)ϕη -preinvex on Aϕη for some fixed (s, m) ∈ (−1, 1]×(0, 1]

11

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then the following inequality holds: ) ( ( ) eiϕ η(b, a) iϕ tf (a) + (1 − k)f a + e η(b, a) + (k − t)f a + 2 ∫ a+eiϕ η(b,a) 1 − iϕ f (x)dx e η(b, a) a {[ (1 )p+1 ] p1 |eiϕ η(b, a)| p+1 t + −t ≤ 1 1 2 (p + 1) p (s + 1) q [( ( 1 )s+1 ) ( 1 )s+1 ( b ) q ] q1 ′ ′ q × 1− (a) + m f f 2 2 m [( ] p1 1 )p+1 + k− + (1 − k)p+1 2 } [( ) ( ( 1 )s+1 ) ( b ) q ] q1 1 s+1 ′ q ′ × . (3.10) f (a) + m 1 − f 2 2 m Proof. By Lemma 2.1 and using the famous H¨older’s inequality, we have ) ( ( ) eiϕ η(b, a) tf (a) + (1 − k)f a + eiϕ η(b, a) + (k − t)f a + 2 ∫ a+eiϕ η(b,a) 1 − iϕ f (x)dx e η(b, a) a [∫ 1 ( ) 2 |λ − t| f ′ a + λeiϕ η(b, a) dλ ≤ eiϕ η(b, a) 0 ∫ 1 ( ) ] |λ − k| f ′ a + λeiϕ η(b, a) dλ + 1 2

{( ∫ iϕ ≤ e η(b, a)

1 2

|λ − t| dλ

0

(∫ +

) p1 [ ∫ 1 |λ − k|p dλ

1 2

p

) p1 [ ∫

1 2

( ) q ] q1 ′ iϕ f a + λe η(b, a) dλ

0

} 1 ( ) q ] q1 ′ iϕ . f a + λe η(b, a) dλ

1 2

Also, making use of the extended (s, m)ϕη -convexity of |f ′ |q , it follows that ( ) ( ) eiϕ η(b, a) iϕ tf (a) + (1 − k)f a + e η(b, a) + (k − t)f a + 2 ∫ a+eiϕ η(b,a) 1 − iϕ f (x)dx e η(b, a) a

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{( ∫ iϕ ≤ e η(b, a)

1 2

|λ − t|p dλ

) p1

0

( q ( ) q ) ] q1 s ′ s ′ b (1 − λ) f (a) + mλ f × dλ m 0 } (∫ 1 ) p1 [ ∫ 1 ( q ( b ) q ) ] q1 . + |λ − k|p dλ (1 − λ)s f ′ (a) + mλs f ′ dλ 1 1 m 2 2 [∫

1 2

Direct calculation yields that ∫

1 2

0

( )p+1 ∫ 1 ( )p+1 tp+1 + 21 − t k − 12 + (1 − k)p+1 p |λ − t| dλ = , |λ − k| dλ = , 1 p+1 p+1 2 p

similarly, we have ∫ 0

1 2

∫ (1 − λ) dλ =

1

s

( )s+1 ∫ 1 ( 1 )s+1 ∫ 1 2 1 − 12 s s λ dλ = λ dλ = (1 − λ) dλ = 2 , . 1 s+1 s+1 0 2 s

1 2

Therefore, combining the above four equalities can lead to the desired result. The statement in Theorem 3.2 is proved. Corollary 3.4 Under the condition of Theorem 3.2, 1. when s = 1, we have ( ) ( ) eiϕ η(b, a) iϕ tf (a) + (1 − k)f a + e η(b, a) + (k − t)f a + 2 ∫ a+eiϕ η(b,a) 1 − iϕ f (x)dx e η(b, a) a {[ b q ] q1 (1 )p+1 ] p1 [ 3|f ′ (a)|q m|f ′ ( m )| |eiϕ η(b, a)| p+1 t + −t + ≤ 1 1 2 4 4 2 q (p + 1) p 1[ 1} [( ] ′ b q ]q ′ q p 3m|f ( )| 1 )p+1 |f (a)| m + k− + (1 − k)p+1 + ; (3.11) 2 4 4 2. when ϕ = 0, k = 1, t = 0, and m = 1 in inequality (3.11), we can get ( ) ∫ a+η(b,a) η(b, a) 1 f (x)dx − f a + 2 η(b, a) a ) q1 ( ) p1 [( 1 |η(b, a)| 3 ′ 1 ≤ |f (a)|q + |f ′ (b)|q p+1 4 4 4 ( ) q1 ] 1 3 + |f ′ (a)|q + |f ′ (b)|q . (3.12) 4 4 13

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q , exchange a and b in inequality (3.12), then we can q−1 deduce the inequality (1.7). In the following corollary, we have the midpoint inequality for powers in terms of the first derivative. ) ( ) ( iϕ , Corollary 3.5 By substituting f (a) = f a + eiϕ η(b, a) = f a + e η(b,a) 2 1 5 t = , and k = in Theorem 3.2, we have 6 6 ( ) ∫ a+eiϕ η(b,a) iϕ 1 e η(b, a) f (x)dx − f a + eiϕ η(b, a) 2 a 1 [ ] |eiϕ η(b, a)| ( 1 )p+1 ( 1 )p+1 p + ≤ 1 1 6 3 2 q (p + 1) p {[ } [ ′ b q ] q1 b q ] q1 m|f ′ ( m )| 3m|f ′ ( m )| |f (a)|q 3|f ′ (a)|q + + + . (3.13) × 4 4 4 4 Remark 3.4 As p =

In the following theorem, we obtain another form of Simpson type inequality for powers in term of the first derivative. Theorem 3.3 Let f be defined as in Theorem 3.1. If the mapping |f ′ |q for q ≥ 1 is extended (s, m)ϕη -preinvex on Aϕη for some fixed (s, m) ∈ (−1, 1]×(0, 1] then ( ) ( ) eiϕ η(b, a) iϕ tf (a) + (1 − k)f a + e η(b, a) + (k − t)f a + 2 ∫ a+eiϕ η(b,a) 1 − iϕ f (x)dx e η(b, a) a {( )1− q1 [ ( b ) q ] q1 iϕ 1 1 ′ q 2 ξ1 |f (a)| + mξ2 f ′ ≤ e η(b, a) t − t+ 2 8 m } ( )1− q1 [ ( b ) q ] q1 3 5 ′ 2 ′ q + k − k+ ξ3 |f (a)| + mξ4 f , (3.14) 2 8 m where t(s + 2) − 1 + 2(1 − t)s+2 + (2ts + 4t − s − 3) ξ1 =

(s + 1)(s + 2) 1 2ts+2 + (s + 1 − 2ts − 4t) s+2 2 , ξ2 = (s + 1)(s + 2) 2(1 − k)s+2 + (2ks + 4k − s − 3) ξ3 =

(s + 1)(s + 2)

1 2s+2 ,

1 2s+2

,

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and 1 + (s + 1 − ks − 2k) 2s+2 . (s + 1)(s + 2)

2k s+2 + (s + 1 − 2ks − 4k) ξ4 =

Proof. By Lemma 2.1 and power-mean inequality, it follows that ( ) ( ) η(b, a) tf (a) + (1 − k)f a + eiϕ η(b, a) + (k − t)f a + eiϕ 2 ∫ a+eiϕ η(b,a) 1 − iϕ f (x)dx e η(b, a) a [∫ 1 ( ) 2 ≤ eiϕ η(b, a) |λ − t| f ′ a + λeiϕ η(b, a) dλ 0

∫

1

+ 1 2

( ) ] |λ − k| f ′ a + λeiϕ η(b, a) dλ

{( ∫ iϕ ≤ e η(b, a)

1 2

|λ − t|dλ

0

(∫

1

+ 1 2

|λ − k|dλ

)1− q1 [ ∫

)1− q1 [ ∫

1 2

( ) q ] q1 ′ iϕ |λ − t| f a + λe η(b, a) dλ

0

} ( 1 ) q ] q1 ′ iϕ |λ − k| f a + λe η(b, a) dλ .

1 2

Using the extended (s, m)ϕη -convexity of |f ′ |q , we have that ) ( ( ) iϕ iϕ η(b, a) tf (a) + (1 − k)f a + e η(b, a) + (k − t)f a + e 2 ∫ a+eiϕ η(b,a) 1 f (x)dx − iϕ e η(b, a) a {( ∫ 1 )1− q1 2 ≤ eiϕ η(b, a) |λ − t|dλ 0

( )1− q1 q ( ) q ) ] q1 ( ∫ 1 s ′ s ′ b × |λ − t| (1 − λ) f (a) + mλ f |λ − k|dλ dλ + 1 m 0 2 } [∫ 1 ( ( ) q ) ] q1 q s ′ s ′ b × |λ − k| (1 − λ) f (a) + mλ f . dλ 1 m 2 [∫

1 2

By simple calculations, we can get ∫ 0

1 2

1 1 |λ − t|dλ = t2 − t + , 2 8

∫

1 1 2

3 5 |λ − k|dλ = k 2 − k + , 2 8

(3.15)

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∫

1 2

t(s + 2) − 1 + 2(1 − t)s+2 + (2ts + 4t − s − 3)

|λ − t|(1 − λ) dλ = s

(s + 1)(s + 2)

0

∫

1 2

|λ − t|λ dλ = s

0

∫

1

|λ − k|(1 − λ) dλ = s

1 2

2ts+2 + (s + 1 − 2ts − 4t) (s + 1)(s + 2)

1 2s+2 ,

2(1 − k)s+2 + (2ks + 4k − s − 3) (s + 1)(s + 2)

1 2s+2(3.16) ,

(3.17)

1 2s+2 ,

(3.18)

and ∫

1

|λ − k|λ dλ = s

1 2

1 + (s + 1 − ks − 2k) 2s+2 . (3.19) (s + 1)(s + 2)

2k s+2 + (s + 1 − 2ks − 4k)

Thus, our desired result can be obtained by combining equalities (3.15)-(3.19), the proof is completed. 1 5 Corollary 3.6 Let f be defined as in Theorem 3.3, if s = 1, t = , and k = , 6 6 the inequality holds for m-convex functions: [ ( )] 1 ( ) eiϕ η(b, a) f (a) + f a + eiϕ η(b, a) + 4f a + 6 2 ∫ a+eiϕ η(b,a) 1 − iϕ f (x)dx e η(b, a) a [ 1 ( 5 )1− q ( 61 ′ 29m ′ ( b ) q ) q1 ≤ eiϕ η(b, a) |f (a)|q + f 72 1296 1296 m ] ( 29 61m ′ ( b ) q ) q1 + |f ′ (a)|q + . (3.20) f 1296 1296 m In particular, if m = 1, ϕ = 0, and η(b, a) = b − a in inequality (3.20), the inequality holds for convex function. If |f ′ (x)| ≤ Q, ∀x ∈ I, then we have [ ] ∫ b ( ) 1 5(b − a) f (a) + 4f b + a + f (b) − 1 f (x)dx ≤ Q. (3.21) 6 2 b−a a 36 Remark 3.5 It is observed that the inequality (3.21) is an improvement compared with inequality (1.3). Thus, Theorem 3.3 and its consequences generalize the main results in ([4], 2000). 16

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Acknowledgments This work was supported by the National Natural Science foundation of China under Grant 11301296, Hubei Province Key Laboratory of Systems Science in Metallurgical Process of China under Grant Z201402, and the Natural Science Foundation of Hubei Province, China under Grants 2013CFA131. Finally we thank the referees for their time and comments.

References [1] T. Antczak, Mean value in invexity Anal., 60, no. 8, 1473-1484 (2005); http://dx.doi.org/10.1016/j.na.2004.11.005.

analysis, Available

Nonlinear online at

[2] A. Barani, A. G. Ghazanfari, and S. S. Dragomir, Hermite-Hadamard inequality for functions whose derivatives absolute values are preinvex, J. Inequal. Appl., 2012:247, 9 pages (2012); Available online at http://dx.doi.org/10.1186/1029-242X-2012-247. [3] L. Chun and F. Qi, Inequalities of Simpson type for functions whose third derivatives are extended s-convex functions and applications to means, J. Comput. Anal. Appl., 19, no. 3, 555-569 (2015). [4] S. S. Dragomir, R. P. Agarwal, and P. Cerone, On Simpson’s inequality and applications, J. Inequal. Appl., 5, no. 6, 533-579 (2000); Available online at http://dx.doi.org/10.1155/S102558340000031X. [5] N. Eftekhari, Some remarks on (s, m)-convexity in the second sense, J. Math. Inequal., 8, no. 3, 489-495 (2014); Available online at http://dx.doi.org/10.7153/jmi-08-36. [6] E. K. Godunova and V. I. Levin, Inequalities for functions of a broad class that contains convex, monotone and some other forms of functions, Numer. Math. Math. Phys., 166, 138-142 (1985). ˙ I¸ ˙ scan, Hermite-Hadamard’s inequalities for preinvex function via fractional [7] I. integrals and related fractional inequalities, American J. Math. Anal., 1, no. 3, 33-38 (2013); Available online at http://arxiv.org/abs/1204.0272v2. [8] W.-D. Jing, D.-W. Niu, and F. Qi, Some inequalities of Hermite-Hadamard type for r-ϕ-preinvex functions, Tamkang J. Math., 45, no. 1, 31-38 (2014); Available online at http://dx.doi.org/10.5556/j.tkjm.45.2014.1261. [9] M. A. Latif, On Hermite-Hadamard type integral inequalities for n-times differentiable preinvex functions with applications, Stud. Univ. Babe¸s-Bolyai Math., 58, no. 3, 325-343 (2013).

17

629

Yujiao Li et al 613-632

J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 22, NO.4, 2017, COPYRIGHT 2017 EUDOXUS PRESS, LLC

[10] M. A. Latif, Some Hermite-Hadamard type inequalities for functions whose partial derivatives in absloute value are preinvex on the co-ordinates, Facta ˘ Ser. Math. Inform., 28, no. 3, 257-270 (2013). Universitatis (NIS) [11] M. A. Latif and S. S. Dragomir, Some weighted integral inequalities for differentiable preinvex and prequsiinvex functions with applications, J. Inequal. Appl., 2013:575, 19 pages (2013); Available online at http://dx.doi.org/10.1186/1029-242X-2013-575. [12] M. A. Latif, S. S. Dragomir, and E. Momoniat, Some weighted HermiteHadamard-Noor type inequalities for differentiable preinvex and quasi preinvex functions, Punjab Univ. J. Math., 47, no. 1, 57-72 (2015). [13] M. A. Latif and M. Shoaib, Hermite-Hadamard type integral inequalities for differentiable m-preinvex and (α, m)-preinvex functions, J. Egypt. Math. Soc., 23, 236-241(2015); Available online at http://dx.doi.org/10.1016/j.joems.2014.06.006. [14] J.-Y. Li, On Hadamard-type inequalities for s-preinvex functions, J. Chongqing Norm. Univ. (Natural Science) China, 27, no. 4, 58 (2010); Available online at http://dx.doi.org/10.3969/J.ISSN.16726693.2010.04.002. [15] T.-Y. Li and G.-H. Hu, On the strictly G-preinvex function, J. Inequal. Appl., 2014:210, 9 pages (2014); Available online at http://dx.doi.org/10.1186/1029-242X-2014-210. [16] Y.-J. Li and T.-S. Du, On Simpson type inequalities for functions whose derivatives are extended (s, m)-GA-convex functions, Pure. Appl. Math. China, 31, no. 5, 487-497 (2015); Available online at http://dx.doi.org/10.3969/j.issn.1008-5513.2015.05.008. [17] M. Matloka, On some Hadamard-type inequalities for (h1 , h2 )-preinvex functions on the co-ordinates, J. Inequal. Appl., 2013:227, 12 pages (2013); Available online at http://dx.doi.org/10.1186/1029-242X-2013-227. [18] M. Matloka, Inequalities for h-preinvex functions, Appl. Math. Comput., 234, 52-57 (2014); Available online at http://dx.doi.org/10.1016/j.amc.2014.02.030. [19] M. A. Noor, Hadamard integral inequalities for product of two preinvex functions, Nonlinear Anal. Forum, 14, 167-173 (2009); Available online at http://prof.ks.ac.kr/bslee/naf/table/vol-1400/NAF1418.pdf [20] M. A. Noor, Hermite-Hadamard integral inequalities for log-ϕ-convex functions, Nonlinear Anal. Forum, 13, no. 2, 119-124 (2008); Available online at http://prof.ks.ac.kr/bslee/naf/table/vol-1302/NAF130201.pdf

18

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Yujiao Li et al 613-632

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[21] M. A. Noor and K. I. Noor, Generalized preinvex functions and their properties, J. Appl. Math. Stochastic Anal., Article ID 12736, 13 pages (2006); Available online at http://dx.doi.org/10.1155/JAMSA/2006/12736. [22] M. A. Noor, K. I. Noor, M. U. Awan, and S. Khan, Hermite-Hadamard type inequalities for differentiable hϕ -preinvex functions, Arab. J. Math., 4, 63-76 (2015); Available online at http://dx.doi.org/10.1007/s40065-014-0124-3. [23] M. A. Noor, K. I. Noor, M. A. Ashraf, M. U. Awan, and B. Bashir, HermiteHadamard inequalities for hϕ -convex functions, Nonlinear Anal. Formum, 18, 65-76 (2013). [24] J. Park, Simpson-like and Hermite-Hadamard-like type integral inequalities for twice differentiable preinvex functions, International J. Pure. Appl. Math., 79, no. 4, 623-640 (2012). [25] J. Park, Hermite-Hadamard-like inequalities for functions whose derivatives of n-th order are preinvex and logarithmic preinvex, Appl. Math. Sci., 7, no. 122, 6053-6068 (2013); Available online at http://dx.doi.org/10.12988/ams.2013.39520. [26] J. Park, Hermite-Hadamard-like type integral inequalities for functions whose derivatives of n-th order are preinvex, Appl. Math. Sci., 7, no. 133, 6637-6650 (2013); Available online at http://dx.doi.org/10.12988/ams.2013.310569. [27] R. Pini, Invexity and generalized convexity, Optimization, 22, no.4, 513-525 (1991); Available online at http://dx.doi.org/10.1080/02331939108843693. [28] S. Qaisar, C.-J. He, and S. Hussain, A generalizations of Simpson’s type inequality for differentiable functions using (α, m)-convex functions and applications, J. Inequal. Appl., 2013:158, 13 pages (2013); Available online at http://dx.doi.org/10.1186/1029-242X-2013-158. [29] M. Z. Sarikaya, N. Alp, and H. Bozkurt, On Hermite-Hadamard type integral inequalities for preinvex and log-preinvex functions, Contemporary Anal. Appl. Math., 1, No. 2, 237-252(2013); Available online at http://caam.fatih.edu.tr/docs/articles/171.pdf ¨ [30] M. Z. Sarikaya, E. Set, and M. E. Ozdemir, On new inequalities of Simpson’s type for s-convex functions, Comput. Math. Appl., 60, no. 8, 2191-2199 (2010); Available online at http://dx.doi.org/10.1016/j.camwa.2010.07.033. [31] S.-H. Wang and F. Qi, Hermite-Hadamard type inequalities for n-times differentiable and preinvex functions, J. Inequal. Appl., 2014:49, 9 pages (2014); Available online at http://dx.doi.org/10.1186/1029-242X-2014-49. [32] Y. Wang, S.-H. Wang, and F. Qi, Simpson type integral inequalities in which the power of the absolute value of the first derivative of the integrand is s-preinvex, Facta Univ., Ser. Math. Inform., 28, no. 2, 151-159(2013). 19

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[33] Y. Wang, B.-Y. Xi, and F. Qi, Hermite-Hadamard type integral inequalities when the power of the absolute value of the first derivative of the integrand is preinvex, Le. Mat., 69, no. 1, 89-96 (2014); Available online at http://dx.doi.org/10.4418/2014.69.1.6. [34] Y. Wang, M.-M. Zheng, and F. Qi, Integral inequalities of Hermite-Hadamard type for functions whose derivatives are α-preinvex, J. Inequal. Appl., 2014:97, 10 pages (2014); Available online at http://dx.doi.org/10.1186/1029-242X-2014-97. [35] B.-Y. Xi and F. Qi, Inequalities of Hermite-Hadamard type for extended s-convex functions and applications to means, J. Nonlinear Convex Anal., 16, no. 5, 873-890 (2015). [36] B.-Y. Xi, Y. Wang, and F. Qi, Some integral inequalities of HermiteHadamard type for extended (s, m)-convex functions, Transylv. J. Math. Mech., 5, no. 1, 69-84 (2013). [37] X.-M. Yang, X.-Q. Yang, and K. L. Teo, Generalized invexity and generalized invariant monotonicity, J. Optimiz. Theory App., 117, no. 3, 607-625 (2003). [38] Z.-Q. Yang, Y.-J. Li, and T.-S. Du, A generalization of Simpson type inequality via differentiable functions using (s, m)-convex functions, Italian J. Pure. Appl. Maths., 35, 327-338(2015).

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ISOMETRIC EQUIVALENCE OF LINEAR OPERATORS ON SOME SPACES OF ANALYTIC FUNCTIONS LI-GANG GENG

∗

Abstract. In this paper, we are interested in the isometric equivalence problem for the weighted composition operator Wu,ϕ and the composition operator Cϕ on the Hardy and the Dirichlet space. We show what properties the operators must satisfy to insure that they are isometric equivalent.

1. introduction Let D be the unit disk in the complex plane, and S(D) be the set of analytic self-maps of D. The algebra of all holomorphic functions with domain D will be denoted by H(D). Let ϕ be an analytic self-map of D and u ∈ H(D), the multiplication operator Mu is defined by (Mu f )(z) = u(z)f (z), and the weighted composition operator Wu,ϕ induced by u and ϕ is defined by (Wu,ϕ f )(z) = u(z)f (ϕ(z)) for z ∈ D and f ∈ H(D). If let u ≡ 1, then Wu,ϕ = Cϕ , which is often called composition operator. As is well known, the weighted composition operator can be regarded as a generalization of a multiplication operator and a composition operator. Let X and Y be two Banach spaces and T1 and T2 are bounded linear operators on X and Y respectively . We say that T1 and T2 are isometrically equivalent if there exists surjective isometries UX and UY on X and Y respectively such that UX T1 = T2 UY . For X = Y , two operators T1 and T2 are said to be similar if there is a bounded invertible operator S such that ST2 = T1 S. If S could be chosen to be an isometry as well, then T1 and T2 are said to be isometrically isomorphic. If X is a Hilbert space as well as a Banach space, then isometric isomorphism on X is referred to as unitary equivalence. 2000 Mathematics Subject Classification. Primary: 47B35; Secondary: 46E15, 32A36, 32A37. Key words and phrases. isometric equivalence; weighted composition operator; composition operator; Hardy space; Dirichlet space. ∗ Corresponding author. Supported in part by the Chongqing Education Commission(KJ120704), the National Natural Science Foundation of China (Grand No. 11271388,11401059, Tianyuan fund for Mathematics, No.11426046)and Chongqing Technology and Business University (under Grant No. 20125606). 1

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2

L.G.GENG

The “unitary equivalence” problem for operators on a Hilbert space has received a lot of attention over the yeas. The analogous problem for operators on Banach spaces, due to the lack of inner product structure, requires different techniques directly related to the specific settings under consideration. The “isometric equivalence problem” arises as to what properties the operators must satisfy to insure that they are isometrically equivalent. There has been some recent work on the isometric equivalence problem in spaces of analytic functions. In [1], Wright investigated the isometric equivalence of composition operators for X = Y = H p (D) for 1 ≤ p < ∞ and p 6= 2, he obtained that if two composition operators Cϕ1 and Cϕ2 are isometrically equivalent on Hardy space H p , then ϕ1 (z) = eiθ ϕ2 (e−iθ z). In [2, 3], Hornor and Jamison studied isometric equivalence of composition operators on several important Banach spaces of analytic function spaces on the unit disk D. In [4], Jamison studied isometric equivalence of composition operators for X = Y = B, where B is a Bloch space. He obtained that if two composition operators Cϕ1 and Cϕ2 are isometrically equivalent on B, then there is an automorphism ϕ such that ϕ1 (ϕ(z)) = ϕ(ϕ2 (z)); he also investigated the isometric equivalence problem of certain operators on some specific types of Banach spaces. In [5], Nadia studied isometric equivalence of differentiated composition operators on some analytic function spaces. He obtained that two operators DCϕ1 and DCϕ2 : H p → H q (1 < p, q < ∞, and p, q 6= 2), then DCϕ1 Wp = Wq DCϕ2 if and only if ϕ1 (z) = e−iθp ϕ2 (eiθq z), here DCϕ : H p → H q is defined to be DCϕ f = (f ◦ ϕ)0 , Wp and Wq are surjective isometries on H p and H q . Building on those foundation, the present paper continues this line of research. More precisely, we first investigated the case of weighted composition operators on the Hardy spaces. 2. Isometric equivalence of weighted composition operators on Hardy spaces Let H ∞ (D) denote the space of bounded holomorphic functions f on the unit disk with the supremum kf k∞ = sup |f (z)|. z∈D

The surjective linear isometries of H ∞ (D) were determined in [6]. It was proven that, a surjective linear isometry T of H ∞ (D) is of the form: T f = αf (τ )

(1)

for every f ∈ H ∞ (D). Where τ is a conformal map of D and α is a unimodular complex number. The Hardy space H p (D) for 1 ≤ p < ∞ is defined to be the Banach space of analytic functions in D such that  Z 2π p 1/p 1 iθ kf kH p = sup < ∞. f (re ) dθ 2π 0 m0 > S(µ(a), µ(b)) ≥ max(µ(a), µ(b)). Hence a, b ∈ 2 L(µ; m0 ), but a ∗ b ∈ / L(µ; m0 ). This is a contradiction and so µ satisfies the inequality µ(x ∗ y) ≤ S(µ(x), µ(y)) for all x, y ∈ X. This completes the proof. □ The converse of Proposition 3.5 may not be true as seen in the following example. Example 3.6. In Example 3.2, define a fuzzy set ν in X by ν(1) = 0, ν(b) = ν(d) = 0.5 and ν(a) = ν(c) = 1. Let Sm be the s-norm in Example 3.2. Then it is easy to see that ν is an Sm -fuzzy subalgebra of X, but the lower level subset L(ν; 0.5) = {1, b, d} is not a subalgebra of X, since b ∗ d = c ∈ / L(ν; 0.5). Proposition 3.7. Let µ be an idempotent S-fuzzy subalgebra of X. Then the non-empty lower level subset L(µ; α) of µ is a subalgebra of X. Proof. Let x, y ∈ L(µ; α), where α ∈ [0, 1]. Then µ(x) ≤ α and µ(y) ≤ α. Hence µ(x ∗ y) ≤ S(µ(x), µ(y)) ≤ S(α, α) = α and so x ∗ y ∈ L(µ; α). Thus L(µ; α) is a subalgebra of X. □ Proposition 3.8. Let µ be an S-fuzzy subalgebra of X. If there is a sequence {xn } in X such that limn→∞ S(µ(xn ), µ(xn )) = 0, then µ(1) = 0. Proof. For any x ∈ X, we have µ(1) = µ(x∗x) ≤ S(µ(x), µ(x)). Therefore µ(1) ≤ S(µ(xn ), µ(xn )) for each n ∈ N and so 0 ≤ µ(1) ≤ limn→∞ S(µ(xn ), µ(xn )) = 0. It follows that µ(1) = 0. □ Let f be a mapping defined on X and let µ be a fuzzy set in f (X). The fuzzy set f −1 (µ) in X defined by [f −1 (µ)](x) := µ(f (x)) for all x ∈ X is called the preimage of µ under f . Theorem 3.9. Let f : X → Y be an epimorphism of BE-algebras and let µ be an S-fuzzy subalgebra of Y . Then the preimage f −1 (µ) of µ under f is also an S-fuzzy subalgebra of X. Proof. Assume that µ is an S-fuzzy subalgebra of Y . Let x, y ∈ X. Then [f −1 (µ)](x ∗ y)) =µ(f (x ∗ y)) = µ(f (x) ∗ f (y)) ≤S(µ(f (x)), µ(f (y))) = S([f −1 (µ)](x), [f −1 (µ)](y)). Hence f −1 (µ) is an S-fuzzy subalgebra of X.

□

Let µ be a fuzzy set in X and let f be a mapping defined on X. The fuzzy set µf in f (X) defined by µf (y) := inf x∈f −1 (y) µ(x) for all y ∈ f (X) is called the anti-image of µ under f .

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S-fuzzy subalgebras and their S-products in BE-algebras

Definition 3.10. An s-norm S on [0, 1] is said to be continuous if S is a continuous function from [0, 1] × [0, 1] to [0, 1] with respect to the usual topology. Theorem 3.11 Let S be a continuous s-norm and let f : X → Y be an epimorphism of BEalgebras. If µ is an S-fuzzy subalgebra of X, then anti-image µf is also an S-fuzzy subalgebra of Y. Proof. Let A1 := f −1 (y1 ), A2 := f −1 (y2 ) and A12 := f −1 (y1 ∗ y2 ), where y1 , y2 ∈ Y . Consider the set A1 ∗ A2 := {x ∈ X|x = a1 ∗ a2 for some a1 ∈ A1 , a2 ∈ A2 }. If x ∈ A1 ∗ A2 , then x = x1 ∗ x2 for some x1 ∈ A1 , x2 ∈ A2 and so f (x) = f (x1 ∗ x2 ) = f (x1 ) ∗ f (x2 ) = y1 ∗ y2 , i.e., x ∈ f −1 (y1 ∗ y2 ) = A12 . Hence A1 ∗ A2 ⊆ A12 . It follows that µf (y1 ∗ y2 ) = ≤ ≤

inf

x∈f −1 (y1 ∗y2 )

inf

x∈A1 ∗A2

inf

µ(x) = inf µ(x)

µ(x) =

x∈A1 ,x2 ∈A2

x∈A12

inf

x∈A1 ,x2 ∈A2

µ(x1 ∗ x2 )

S(µ(x1 ), µ(x2 )).

Since S is continuous, if ϵ is any positive number, then there exists a number δ > 0 such that S(x∗1 , x∗2 ) ⊆ S(inf x1 ∈A1 µ(x1 ), inf x2 ∈A2 µ(x2 )) + ϵ, whenever x∗1 ≤ inf x1 ∈A1 µ(x1 ) + δ and x∗2 ≤ inf x2 ∈A2 µ(x2 ) + δ. Choose a1 ∈ A1 , a2 ∈ A2 such that µ(a1 ) ≤ inf x1 ∈A1 µ(x1 ) + δ and µ(a2 ) ≤ inf x2 ∈A2 µ(x2 ) + δ. Then S(µ(a1 ), µ(a2 )) ≤ S(inf x1 ∈A1 µ(x1 ), inf x2 ∈A2 µ(x2 )) + ϵ. Hence we have µf (y1 ∗ y2 ) ≤

inf

x1 ∈A1 ,x2 ∈A2

S(µ(x1 ), µ(x2 ))

≤ S( inf µ(x1 ), inf µ(x2 )) x1 ∈A1

x2 ∈A2

= S(µf (y1 ), µf (y2 )). □

Thus µf is an S-fuzzy subalgebra of Y . Theorem 3.12. Let µ be an idempotent S-fuzzy subalgebra of X. Then the set Xµ := {x ∈ X|µ(x) = µ(1)} is a subalgebra of X.

Proof. Noticing that µ(1) ≤ µ(x) for all x ∈ X, we have L(µ; µ(1)) = {x ∈ X|µ(x) ≤ µ(1)} = {x ∈ X|µ(x) = µ(1)} = Xµ . By Proposition 3.7, Xµ is a subalgebra of X. □ Proposition 3.13. Let µ, ν be idempotent S-fuzzy subalgebras of X. If µ ⊂ ν and µ(1) = ν(1), then Xµ ⊂ Xν . Proof. Assume that µ ⊂ ν and µ(1) = ν(1). Let x ∈ Xµ . Then ν(x) > µ(x) = µ(1) = ν(1). Noticing ν(x) ≤ ν(1) for all x ∈ X, we have ν(x) = ν(1), i.e., x ∈ Xν . This completes the proof. □

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Theorem 3.14. Let SM be an s-norm defined in Example 3.2. Let µ be an SM -fuzzy subalgebra of X and let f : [µ(1), 1] → [0, 1] be an increasing function. Define a fuzzy set µf : X → [0, 1] by µf (x) := f (µ(x)) for all x ∈ X. Then µf is an SM -fuzzy subalgebra of X. Furthermore, if f (α) ≥ α for all α ∈ [µ(1), 1], then µ ⊆ µf . Proof. Let x, y ∈ X. Then µf (x ∗ y) =f (µ(x ∗ y)) ≤ f (SM (µ(x), µ(y))) ≤SM (f (µ(x)), f (µ(y))) = SM (µf (x), µf (y)). Hence µf is an SM -fuzzy subalgebra of X. Assume that f (α) ≥ α for all α ∈ [µ(1), 1]. Then µf (x) = f (µ(x)) ≥ µ(x) for all x ∈ X, which proves that µ ⊂ µf . □ 4. Direct products and s-normed products Definition 4.1. Let µ and ν be fuzzy sets of X and let S be an s-norm of X. Then the S-product of µ and ν is defined by [µ · ν]S (x) := S(µ(x), ν(x)) for all x ∈ X and we denote it by [µ · ν]S . Theorem 4.2. Let µ, ν be two S-fuzzy subalgebras of X and let S ∗ be an s-norm which dominates S, i.e., S ∗ (S(a, b), S(c, d)) ≤ S(S ∗ (a, c), S ∗ (b, d)) for all a, b, c and d ∈ [0, 1]. Then the S ∗ -product [µ · ν]S ∗ of µ and ν is an S-fuzzy subalgebra of X. Proof. For any x, y ∈ X, we have [µ · ν]S ∗ (x ∗ y) = S ∗ {µ(x ∗ y), ν(x ∗ y)} ≤ S ∗ {S{µ(x), µ(y)}, S{ν(x), ν(y)}} ≤ S{S ∗ {µ(x), ν(x)}, S ∗ {µ(y), ν(y)}} = S{[µ · ν]S ∗ (x), [µ · ν]S ∗ (y)}. Hence [µ · ν]S ∗ is an S-fuzzy subalgebra of X.

□

Let f : X → Y be an epimorphism of BE-algebras. If µ and ν are S-fuzzy subalgebras of Y , then the S ∗ -product [µ · ν]S ∗ of µ and ν is also an S-fuzzy subalgebra of Y whenever S ∗ dominates S. Since every epimorphic preimage of an S-fuzzy subalgebra is also an S-fuzzy subalgebra, the

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preimages f −1 (µ), f −1 (ν) and f −1 ([µ · ν]S ∗ ) are S-fuzzy subalgebras. The next theorem provides the relation between f −1 ([µ · ν]S ∗ ) and the S ∗ -product [f −1 (µ) · f −1 (ν)]S∗ of f −1 (µ) and f −1 (ν). Proposition 4.3. Assume that f : X → Y is an epimorphism of BE-algebras and S, S ∗ are s-norms such that S ∗ dorminates S. For any S-fuzzy subalgebras µ and ν of Y , we have f −1 ([µ · ν]S ∗ ) = [f −1 (µ) · f −1 (ν)]S ∗ . Proof. For any x ∈ X, we obtain {f −1 ([µ · ν]S∗ )}(x) =[µ · ν]S ∗ (f (x)) =S ∗ {µ(f (x)), ν(f (x))} =S ∗ ([f −1 (µ)](x), [f −1 (ν)](x)) =[f −1 (µ) · f −1 (ν)]S ∗ (x), □

completing the proof.

Let (X1 , ∗1 , 11 ) and (X2 , ∗2 , 12 ) be BE-algebras. Define a binary operation “ ∗ ” on X1 × X2 by (x1 , x2 ) ∗ (y1 , y2 ) := (x1 ∗1 x2 , y1 ∗2 y2 ) for all (x1 , x2 ), (y1 , y2 ) ∈ X. Then (X, ∗, 1) is a BE-algebra, where 1 = (11 , 12 ). Theorem 4.4. Let X = X1 × X2 be the direct product of BE-algebras X1 and X2 . If µ1 (resp., µ2 ) is an S-fuzzy subalgebra of X1 (resp., X2 ), then µ := µ1 × µ2 is an S-fuzzy subalgebra of X defined by µ(x1 , x2 ) = (µ1 × µ2 )(x1 , x2 ) = S(µ1 (x1 ), µ2 (x2 )) for all (x1 , x2 ) ∈ X1 × X2 . Proof. Let x = (x1 , x2 ), y = (y1 , y2 ) ∈ X. Then we have µ(x ∗ y) =µ((x1 , x2 ) ∗ (y1 , y2 )) =µ(x1 ∗ y1 , x2 ∗ y2 ) =S(µ1 (x1 ∗ y1 ), µ2 (x2 ∗ y2 )) ≤S(S(µ1 (x1 ), µ1 (y1 )), S(µ2 (x2 ), µ2 (y2 ))) =S(S(µ1 (x1 ), µ2 (x2 )), S(µ1 (y1 ), µ2 (y2 ))) =S(µ(x1 , x2 ), µ(y1 , y2 )) =S(µ(x), µ(y)). Hence µ = µ1 × µ2 is an S-fuzzy subalgebra of X.

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Sun Shin Ahn and Keum Sook So

Now, we generalize the idea to the product of S-fuzzy subalgebras. We first need to generalize ∏ the domain of an s-norm to ni=1 [0, 1] as follows. ∏ Definition 4.5. We define a map Sn : ni=1 [0, 1] → [0, 1] is defined by Sn (α1 , α2 , · · · , αn ) := S(αi , Sn−1 (α1 , · · · , αi−1 , αi+1 , · · · , αn )) for all 1 ≤ i ≤ n, where S2 = S and S1 = id[0,1] . Using the induction on n, we have following two lemmas: Lemma 4.6. For an s-norm S and every αi , βi , where 1 ≤ i ≤ n and n ≥ 2, we have Sn (S(α1 , β1 ),S(α2 , β2 ), · · · , S(αn , βn ) =S(Sn (α1 , α2 , · · · , αn ), Sn (β1 , β2 , · · · , βn )). Lemma 4.7. For an s-norm S and every α1 , α2 , · · · , αn ∈ [0, 1], where n ≥ 2, we have Sn (α1 , α2 , · · · , αn ) =S(· · · , S(S(S(α1 , α2 ), α3 , α4 ), · · · , αn ) =S(α1 , S(α2 , S(α3 , · · · , S(αn−1 , αn ) · · · ))). ∏ Theorem 4.8. Let X := ni=1 Xi be the direct product of BE-algebras {Xi }ni=1 . If µi is an ∏ S-fuzzy subalgebra of Xi , where 1 ≤ i ≤ n, then µ = ni=1 µi defined by µ(x1 , · · · , xn ) = (

n ∏

µi )(x1 , · · · , xn ) = S(µ(x1 ), · · · , µ(xn ))

i=1

is an S-fuzzy subalgebra of X. Proof. Let x = (x1 , · · · , xn ), y = (y1 , · · · , yn ) be any elements of X. Using Lemmas 4.6 and 4.7, we have µ(x ∗ y) =µ((x1 ∗ y1 ), (x2 ∗ y2 ), · · · , (xn ∗ yn )) =Sn (µ1 (x1 ∗ y1 ), · · · , µn (xn ∗ yn )) ≤Sn (S(µ1 (x1 ), µ1 (y1 )), · · · , S(µn (xn ), µn (yn ))) =S(Sn (µ1 (x1 ), µ2 (x2 ), · · · , µn (xn )), Sn (µ1 (y1 ), µ2 (y2 ), · · · , µn (yn ))) =S(µ((x1 , · · · , xn ) ∗ (y1 , · · · , yn ))) =S(µ(x ∗ y)). Hence µ =

∏n i=1

□

µi is an S-fuzzy subalgebra of X.

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S-fuzzy subalgebras and their S-products in BE-algebras

Theorem 4.9. Let S be a continuous s-norm and f : X → Y be an epimorphism of BE-algebras, and let µ and ν be S-fuzzy subalgebras of X. If an s-norm S ∗ dominates S, then ([µ · ν]S ∗ )f ⊆ [µf · ν f ]S ∗ . Proof. By Theorems 4.2 and 3.11, the S ∗ -product [µ · ν]S ∗ is an S-fuzzy subalgebra of X, and the S ∗ -product [µf · ν f ]S ∗ is an S-fuzzy subalgebra of Y . Moreover, for each y ∈ Y , we have ([µ · ν]S ∗ )f (y) = =

inf [µ · ν]S ∗ (x)

x∈f −1 (y)

inf

x∈f −1 (y)

S ∗ (µ(x), ν(x))

≤ S ∗ ( inf −1 x∈f

µ(x), (y)

inf

x∈f −1 (y)

ν(x))

= S ∗ (µf (y), ν f (y)) = ([µf · ν f ]S ∗ )(y), proving that ([µ · ν]S ∗ )f ⊆ [µf · ν f ]S ∗ .

□ References

[1] [2] [3] [4] [5] [6] [7] [8]

S. S. Ahn, Y. H. Kim and K. S. So, Fuzzy BE-algebras, J. Appl. Math. and Informatics 29 (2011), 1049-1057. S. S. Ahn and J. M. Ko, On vague filters in BE-algebras, Commun. Korean Math. Soc. 26 (2011), 417-425. S. S. Ahn and K. K. So, On ideals and upper sets in BE-algebras, Sci. Math. Jpn. 68 (2008), 279-285. S. S. Ahn and K. K. So, On generalized upper sets in BE-algebras, Bull. Korean Math. Soc. 46 (2009), 281-287. H. S. Kim and Y. H. Kim, On BE-algebras, Sci. Math. Jpn. 66 (2007), 113-116. E. P. Klement, R. Mesiar and E. Pap, Triangular norms. Position paper I: basic anaytical and algebraic propertis, Fuzzy Sets and Systems 143(2004), 5-26. Y. B. Jun and S. S. Ahn, Fuzzy implicative filters of BE-algebras with degrees in the interval (0, 1], J. Computational Analsis and Applications 15(2013), 1456-1466. L. A. Zadeh, Fuzzy sets, Inform. Control 56 (2008), 338-353.

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J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 22, NO.4, 2017, COPYRIGHT 2017 EUDOXUS PRESS, LLC

Fixed point results for generalized g-quasi-contractions of Perov-type in cone metric spaces over Banach algebras without the assumption of normality Shaoyuan Xu ∗ School of Mathematics and Statistics, Hanshan Normal University, Chaozhou, 521041, China Branislav Z. Popovi´c Faculty of Science, University of Kragujevac, Radoja Domanovi´ca 12, 34000 Kragujevac, Serbia Stojan Radenovi´c Faculty of Mechanical Engineering, University of Belgrade, Kraljice Marije 16, 11120 Beograd, Serbia, and Department of Mathematics, State University of Novi Pazar, Novi Pazar, Serbia Abstract: In this paper, we introduce the concept of generalized g-quasicontractions of Perov-type in the setting of cone metric spaces over Banach algebras. By omitting the assumption of normality of the cone we establish common fixed point theorems for generalized g-quasi-contractions of Perov-type with the spectral radius r(λ) of the g-quasi-contractive constant vector λ satisfying r(λ) ∈ [0, 1) in the setting of cone metric spaces over Banach algebras. The main results generalize, extend and unify several well-known comparable results in the ´ c fixed point theorem to the literature. As a result, we extend the famous Ciri´ version in the setting of cone metric spaces over Banach algebras. AMS Mathematics Subject Classification 2010: 54H25 47H10 Keywords: cone metric spaces over Banach algebras; non-normal cones; csequences; generalized g-quasi-contractions of Perov-type; fixed point theorems ∗

Corresponding author: Shaoyuan Xu. E-mail: [email protected]; [email protected] (S.

Xu)

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1

Introduction

It is known that the modern metric fixed point theory was motivated from Banach contraction principle (see, e.g., [1]) which plays an important role in various fields of applied mathematical analysis. In 1922, Polish mathematician proved the following classical Banach contraction principle: Theorem 1.1 ([1]) Let T : X → X be a contraction on a complete metric space (X, d). Then T possesses exactly one fixed point x∗ ∈ X. Moreover, for any point x ∈ X, the sequence {T n (x) : n = 0, 1, 2, ....} converges to x∗ ∈ X. That is limn→∞ T n (x) = x∗ , for each x ∈ X, where T n denotes the n-fold composition of T. Since 1922, many authors have obtained all kinds of versions to extend the famous Banach contraction principle. In general, people did such extensions by means of two methods. One is to extend Banach contraction to other more general mapping or mappings (for example, when two or more mappings are involved and discussed, the common fixed point(s) is(are) usually investigated). The other is to extend classical metric space to more general spaces (usually called abstract spaces). There are many generalizations of the concept metric space in the literature. In 1964, Perov [34] introduced vector valued metric space, instead of general metric space, and obtained a Banach type fixed point theorem on such a complete generalized metric space. Later on, following Perov, many authors studied fixed point results of Perov-type in more general abstract spaces, such as cone metric spaces, etc (see [35]-[39]). Among them, Cvetkovi´c and Rakoˇcevi´c [36] introduced the concept of f -quasi-contraction of Perov-type and obtained fixed point results for such ´ c mappings. Let (X, d) be a kind mappings, which is a generalization of the famous Ciri´ complete metric space. Recall that a mapping T : X → X is called a quasi-contraction if, for some k ∈ [0, 1) and for all x, y ∈ X, one has d(T x, T y) 6 k max{d(x, y), d(x, T x), d(y, T y), d(x, T y), d(y, T x)}. ´ c [13] introduced and studied quasi-contractions as one of the most general classes of Ciri´ contractive-type mappings. He proved the well-known theorem that any quasi-contraction T has a unique fixed point. Recently, many authors obtained various similar results on cone b-metric spaces (some authors call such spaces cone metric type spaces) and cone metric spaces. See, for instance, [7]-[15]. 2

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Since 2010, some authors have investigated the problem of whether cone metric spaces are equivalent to metric spaces in terms of the existence of the fixed points of the mappings involved. They used to establish the equivalence between some fixed point results in metric and in (topological vector spaces valued) cone metric spaces by means of the nonlinear scalarization function ξe where e denotes the vector in the internal of the underlying solid cone (see [16]-[19]). Very recently, based on the concept of cone metric spaces, Liu and Xu [21] studied cone metric spaces with Banach algebras, replacing Banach spaces by Banach algebras as the underlying spaces of cone metric spaces. In [21], the authors proved some fixed point theorems of quasi-contractions in cone metric spaces over Banach algebras, but the proof relied strongly on the assumption that the underlying cone is normal. We need state that it is significant to study cone metric spaces with Banach algebras (which we would like to call in this paper cone metric spaces over Banach algebras). This is because there are examples to show that one is unable to conclude that the cone metric space (X, d) over a Banach algebra A discussed is equivalent to the metric space (X, d∗ ), where the metric d∗ is defined by d∗ = ξe ◦ d, here the nonlinear scalarization function ξe : A → R (e ∈ intP ) is defined by ξe (y) = inf{r ∈ R : y ∈ re − P }.

(1.1)

See [20] for more details. In the present paper we introduce the concept of generalized g-quasi-contractions of Perov-type in cone metric spaces over Banach algebras and obtain common fixed point theorems for two weakly compatible self-mappings satisfying g-quasi-contractive condition in the case of g-quasi-contractive constant vector with r(λ) ∈ [0, 1/s) in cone metric spaces without the assumption of normality. Our main results extend the fixed point theorem of quasi-contractions of Das-Naik in metric spaces to the case in cone metric spaces over ´ c fixed point theorem and Banach algebras. As consequences, we obtain the versions of Ciri´ Banach contraction principle in the setting of cone metric spaces over Banach algebras. Our main results generalize and extend the relevant results in the literature (see, for example, [3]-[9], [13], [15], [21], [23], [25], [27]). In addition, we give an example to show that the main results are genuine generalizations of the corresponding results in the literature.

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2

Preliminaries

Let A always be a real Banach algebra. That is, A is a real Banach space in which an operation of multiplication is defined, subject to the following properties (for all x, y, z ∈ A, α ∈ R): 1. (xy)z = x(yz); 2. x(y + z) = xy + xz and (x + y)z = xz + yz; 3. α(xy) = (αx)y = x(αy); 4. ∥xy∥ 6 ∥x∥ ∥y∥. Throughout this paper, we shall assume that a Banach algebra A has a unit (i.e., a multiplicative identity) e such that ex = xe = x for all x ∈ A. An element x ∈ A is said to be invertible if there is an inverse element y ∈ A such that xy = yx = e. The inverse of x is denoted by x−1 . For more details, we refer to [28]. The following proposition is well known (see [28]). Proposition 2.1 Let A be a Banach algebra with a unit e, and x ∈ A. If the spectral radius r(x) of x is less than 1, i.e., 1

1

r(x) = lim ∥xn ∥ n = inf ∥xn ∥ n < 1, n>1

n→∞

then e − x is invertible. Actually, −1

(e − x)

=

∞ ∑

xi .

i=0

Now let us recall the concepts of cone and semi-order for a Banach algebra A. A subset P of A is called a cone if 1. P is non-empty closed and {θ, e} ⊂ P ; 2. αP + βP ⊂ P for all non-negative real numbers α, β; 3. P 2 = P P ⊂ P ; 4. P ∩ (−P ) = {θ}, 4

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where θ denotes the null of the Banach algebra A. For a given cone P ⊂ A, we can define a partial ordering ≼ with respect to P by x ≼ y if and only if y − x ∈ P . x ≺ y will stand for x ≼ y and x ̸= y, while x ≪ y will stand for y − x ∈ intP , where intP denotes the interior of P. The cone P is called normal if there is a number M > 0 such that for all x, y ∈ A, θ ≼ x ≼ y ⇒ ∥x∥ 6 M ∥y∥. The least positive number satisfying above is called the normal constant of P . In the following we always assume that P is a cone in Banach algebra A with intP ̸= ∅ and ≼ is the partial ordering with respect to P . Definition 2.1 (See [2], [3], [20], [21]) Let X be a non-empty set. Suppose the mapping d : X × X → A satisfies 1. 0 ≼ d(x, y) for all x, y ∈ X and d(x, y) = 0 if and only if x = y; 2. d(x, y) = d(y, x) for all x, y ∈ X; 3. d(x, y) ≼ d(x, z) + d(z, x) for all x, y, z ∈ X. Then d is called a cone metric on X, and (X, d) is called a cone metric space over a Banach algebra A. Definition 2.2 (See [2], [3], [20], [21]) Let (X, d) be a cone metric space with a solid cone P over a Banach algebra A, x ∈ X and {xn } a sequence in X. Then 1. {xn } converges to x whenever for each c ∈ A with θ ≪ c there is a natural number N such that d(xn , x) ≪ c for all n > N . We denote this by limn→∞ xn = x or xn → x. 2. {xn } is a Cauchy sequence whenever for each c ∈ A with θ ≪ c there is a natural number N such that d(xn , xm ) ≪ c for all n, m > N . 3. (X, d) is a complete cone metric space if every Cauchy sequence is convergent.

Now, we shall appeal to the following lemmas in the sequel.

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Lemma 2.1 (See [12]) If E is a real Banach space with a cone P and if a ≼ λa with a ∈ P and 0 ≤ λ < 1, then a = θ. Lemma 2.2 (See [27]) If E is a real Banach space with a solid cone P and if θ < u ≪ c for each θ ≪ c, then u = θ. Lemma 2.3 (See [27]) If E is a real Banach space with a solid cone P and if ∥xn ∥ → 0(n → ∞), then for any θ ≪ ϵ, there exists N ∈ N such that for any n > N , we have xn ≪ ϵ. Finally, let us recall the concept of quasi-contraction defining on the cone metric spaces over Banach algebras, which is introduced in [21]. Definition 2.3 (See [21]) Let (X, d) be a cone metric space over a Banach algebra A. A mapping T : X → X is called a quasi-contraction if for some k ∈ P with r(k) < 1 and for all x, y ∈ X, one has d(T x, T y) ≼ ku, where u ∈ {d(x, y), d(x, T x), d(y, T y), d(x, T y), d(y, T x)}.

Remark 2.1 (See [29]) If r(k) < 1, then ∥k m ∥ → 0(m → ∞). Lemma 2.4 (See [4]-[6], [23], [32]) Let ≼ be the partial ordering with respect to P , where P is the given solid cone P of the Banach algebra A. The following properties are often used while dealing with cone metric spaces where the underlying cone is solid but not necessarily normal. (1) If u ≪ v and v ≼ w, then u ≪ w. (2) (3) (4) (5)

If θ ≼ u ≪ c for each c ∈ intP , then u = θ. If a ≼ b + c for each c ∈ intP , then a ≼ b. If c ∈ intP and an → θ, then there exists n0 ∈ N such that an ≪ c for all n > n0 . Let (X, d) be a cone metric space over a Banach algebra A, x ∈ X and {xn } be a

sequence in X. If d(xn , x) ≼ bn and bn → θ, then xn → x. 6

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Lemma 2.5 (See [3]) The limit of a convergent sequence in cone metric space is unique. Definition 2.6 (See [4], [11]) The mappings f, g : X → X are called weakly compatible, if for every x ∈ X holds f gx = gf x whenever f x = gx. Definition 2.7 (See [4], [11], [14]) Let f and g be self-maps of a set X. If w = f x = gx for some x in X, then x is called a coincidence point of f and g, and w is called a point of coincidence of f and g. Lemma 2.6 (See (See [4], [11], [14]) Let f and g be weakly compatible self-maps of a set X. If f and g have a unique point of coincidence w = f x = gx, then w is the unique common fixed point of f and g. Definition 2.8 (See [23] Let (X, d) be a cone metric space. A mapping f : X → X such that, for some constant λ ∈ [0, 1) and for every x, y ∈ X, there exists an element u ∈ C(g; x, y) = {d(gx, gy), d(gx, f x), d(gy, f y), d(gx, f y), d(gy, f x)} for which d(f x, f y) ≼ λu is called a g-quasi-contraction, where g : X → X, f (X) ⊂ g(X). Definition 2.9 Let (X, d) be a cone metric space over a Banach algebra A. A mapping f : X → X is called a generalized g-quasi-contractions of Perov-type, if there exist a mapping g : X → X with f (X) ⊂ g(X) and some λ ∈ P with r(λ) ∈ [0, 1), for all x, y ∈ X, one has d(f x, f y) ≼ λu, (2.1) where u ∈ C(g; x, y) = {d(gx, gy), d(gx, f x), d(gy, f y), d(gx, f y), d(gy, f x)}.

Definition 2.10 (See [30], [31]) Let P be a solid cone in a Banach space A. A sequence {un } ⊂ P is a c-sequence if for each c ≫ θ there exists n0 ∈ N such that un ≪ c for n ≥ n0 . It is easy to show the following propositions.

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Proposition 2.2 (See [30]) Let P be a solid cone in a Banach space A and let {un } and {vn } be sequences in P. If {un } and {vn } are c-sequences and α, β > 0, then {αun + βvn } is a c-sequence. In addition to Proposition 2.2 above, the following propositions are crucial to the proof of our main results. Proposition 2.3 (See [30]) Let P be a solid cone in a Banach algebra A and let {un } be a sequence in P. Then the following conditions are equivalent. (1) {un } is a c-sequence. (2) For each c ≫ θ there exists n0 ∈ N such that un ≺ c for n ≥ n0 . (3) For each c ≫ θ there exists n1 ∈ N such that un ≼ c for n ≥ n1 . Proposition 2.4 (See [30]) Let P be a solid cone in a Banach algebra A and let {un } be a sequence in P. Suppose that k ∈ P is an arbitrarily given vector and {un } is a c-sequence in P. Then {kun } is a c-sequence. Proposition 2.5

Let A be a Banach algebra with a unit e, P be a cone in A and ≼ be

the semi-order be yielded by the cone P . Let λ ∈ P . If the spectral radius r(λ) of λ is less than 1, then the following assertions hold true. (i) We have (e − λ)−1 ≻ θ. In addition, we have θ ≼ λn ≼ (e − λ)−1 λn ≼ (e − λ)−1 λ for any integer n ≥ 1. (ii) For any u ≻ θ, we have u ̸≼ λu. Moreover, we have u ̸≼ λn u for any integer n ≥ 1. Proposition 2.6 (See [29])

Let (X, d) be a complete cone metric space over a Banach

algebra A and let P be the underlying solid cone in Banach algebra A. Let {xn } be a sequence in X. If {xn } converges to x ∈ X, then we have (i) {d(xn , x)} is a c-sequence; (ii) for any p ∈ N, {d(xn , xn+p )} is a c-sequence.

3

Main results

In this section, without the assumption of normality of the underlying cone, we give some common fixed point theorems for generalized g-quasi-contractions of Perov-type with 8

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the spectral radius r(λ) of the g-quasi-contractive constant vector λ satisfying r(λ) ∈ [0, 1) in the setting of cone metric spaces over Banach algebras. Theorem 3.1 Let (X, d) be a cone metric space over a Banach algebra A and the underlying solid cone P . Let the mapping f : X → X be the g-quasi-contractions of Perovtype with the spectral radius r(λ) of the g-quasi-contractive constant vector λ satisfying r(λ) ∈ [0, 1). If the range of g contains the range of f and g(X) or f (X) is a complete subspace of X, then f and g have a unique point of coincidence in X. Moreover, if f and g are weakly compatible, then f and g have a unique common fixed point in X. We begin the proof of Theorem 3.1 with a useful lemma. For each x0 ∈ X, set gx1 = f x0 and gxn+1 = f xn . We will prove that {gxn } is a Cauchy sequence. First, we shall show the following lemmas. Note that for these lemmas, we suppose that all the conditions in Theorem 3.1 are satisfied. Lemma 3.1 For any N ≥ 2 and 1 ≤ m ≤ N − 1, one has that d(gxN , gxm ) ≼ λ(e − λ)−1 d(gx1 , gx0 ).

(3.1)

Proof. We now prove Lemma 3.1 by induction. When N = 2, m = 1, since f : X → X is a generalized g-quasi-contractions of Perov-type satisfying (2.1), there exists u1 ∈ C(g; x1 , x0 ) = {d(gx1 , gx0 ), d(gx1 , gx2 ), d(gx0 , gx1 ), d(gx1 , gx1 ), d(gx0 , gx2 )} such that d(gx2 , gx1 ) ≼ λu1 . Hence, u1 = d(gx1 , gx0 ) or u1 = d(gx0 , gx2 ). (Note that it is obvious that u1 ̸= d(gx1 , gx2 ) since d(gx2 , gx1 )  λd(gx1 , gx2 ) and u1 ̸= d(gx1 , gx1 ) since d(gx1 , gx2 ) ̸= θ.) When u1 = d(gx1 , gx0 ), we have d(gx2 , gx1 ) ≼ λd(gx0 , gx1 ) ≼ λ(e − λ)−1 d(gx1 , gx0 ). When u1 = d(gx2 , gx0 ), then we have d(gx2 , gx1 ) ≼ λd(gx2 , gx0 ) ≼ λ[d(gx2 , gx1 ) + d(gx1 , gx0 )]. So we get (e − λ)d(gx2 , gx1 ) ≼ λd(gx1 , gx0 ), 9

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which implies that d(gx2 , gx1 ) ≼ λ(e − λ)−1 d(gx1 , gx0 ). Hence, (3.1) holds for N = 2 and m = 1. Suppose that for some N ≥ 2 and for any 2 ≤ p ≤ N and 1 ≤ n ≤ p, one has d(gxp , gxn ) ≼ λ(e − λ)−1 d(gx1 , gx0 ).

(3.2)

d(gxp , gx1 ) ≼ λ(e − λ)−1 d(gx1 , gx0 ),

(3.2.1)

d(gxp , gx2 ) ≼ λ(e − λ)−1 d(gx1 , gx0 ),

(3.2.2)

That is,

...... d(gxp , gxp−1 ) ≼ λ(e − λ)−1 d(gx1 , gx0 ).

(3.2.p − 1)

Then, we need to prove that for N + 1 ≥ 2 and any 1 ≤ n < N + 1, one has d(gxN +1 , gxn ) ≼ λ(e − λ)−1 d(gx1 , gx0 ).

(3.3)

d(gxN +1 , gx1 ) ≼ λ(e − λ)−1 d(gx1 , gx0 ),

(3.3.1)

d(gxN +1 , gx2 ) ≼ λ(e − λ)−1 d(gx1 , gx0 ),

(3.3.2)

That is,

...... d(gxN +1 , gxN −1 ) ≼ λ(e − λ)−1 d(gx1 , gx0 ),

(3.3.N − 1)

d(gxN +1 , gxN ) ≼ λ(e − λ)−1 d(gx1 , gx0 ).

(3.3.N )

In fact, since f : X → X is a g-quasi-contraction, there exists u1 ∈ C(g; xN , x0 ) = {d(gxN , gx0 ), d(gxN , gxN +1 ), d(gx0 , gx1 ), d(gxN , gx1 ), d(gx0 , gxN +1 )} such that d(gxN +1 , gx1 ) ≼ λu1 . If u1 = d(gxN , gx1 ), then by (3.2.1) we have d(gxN +1 , gx1 ) ≼ λ2 (e − λ)−1 d(gx1 , gx0 ) ≼ λ2 (e − λ)−1 d(gx1 , gx0 ) ≼ λ(e − λ)−1 d(gx1 , gx0 ). If u1 = d(gx0 , gx1 ), then we have d(gxN +1 , gx1 ) ≼ λd(gx1 , gx0 ) ≼ λd(gx1 , gx0 ) ≼ λ(e − λ)−1 d(gx1 , gx0 ). 10

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If u1 = d(gxN , gx0 ), then by (3.2.1) we have d(gxN +1 , gx1 ) ≼ λd(gxN , gx0 ) ≼ λ(d(gxN , gx1 ) + d(gx1 , gx0 )) ≼ λ(λ(e − λ)−1 d(gx1 , gx0 ) + d(gx1 , gx0 )) = λ(λ(e − λ)−1 + e)d(gx1 , gx0 ) = λ(e − λ)−1 d(gx1 , gx0 ). If u1 = d(gx0 , gxN +1 ), then we have d(gxN +1 , gx1 ) ≼ λd(gx0 , gxN +1 ) ≼ λ(d(gx0 , gx1 ) + d(gx1 , gxN +1 )). Hence, we see (e − λ)d(gxN +1 , gx1 ) ≼ λd(gx0 , gx1 ), which implies that d(gxN +1 , gx1 ) ≼ (e − λ)−1 λd(gx0 , gx1 ). Without loss of generality, suppose that u1 = d(gxN , gxN +1 ). Since f : X → X is a g-quasi-contraction, there exists u2 ∈ C(g; xN −1 , xN ) such that u1 = d(gxN , gxN +1 ) ≼ λu2 , where C(g; xN −1 , xN ) = {d(gxN −1 , gxN ), d(gxN −1 , gxN ), d(gxN , gxN +1 ), d(gxN −1 , gxN +1 ), d(gxN , gxN )}. So, we get d(gxN +1 , gx1 ) ≼ λu1 ≼ λ2 u2 . Similarly, it is easy to see that u2 ̸= d(gxN , gxN ) since u2 ̸= θ and u2 ̸= d(gxN , gxN +1 ) since d(gxN , gxN +1 )  λ2 d(gxN , gxN +1 ). If u2 = d(gxN −1 , gxN ), then by the induction assumption (3.2) we have d(gxN +1 , gx1 ) ≼ λ2 u2 ≼ λ3 (e − λ)−1 d(gx1 , gx0 ) ≼ λ3 (e − λ)−1 d(gx1 , gx0 ) ≼ λ(e − λ)−1 d(gx1 , gx0 ).

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Without loss of generality, suppose that u2 = d(gxN −1 , gxN +1 ). There exists u3 ∈ C(g; xN −2 , xN ) such that u2 = d(gxN −1 , gxN +1 ) ≼ λu3 , where C(g; xN −2 , xN ) = {d(gxN −2 , gxN ), d(gxN −2 , gxN −1 ), d(gxN , gxN +1 ), d(gxN −2 , gxN +1 ), d(gxN , gxN −1 )}. In general, suppose that ui−1 = d(gxN −i+2 , gxN +1 ). Since f : X → X is a g-quasicontraction, by the similar arguments above, there exists ui ∈ C(g; xN −i+1 , xN ) such that ui−1 = d(gxN −i+2 , gxN +1 ) ≼ λui , for which we obtain d(gxN +1 , gx1 ) ≼ λu1 ≼ λ2 u2 ≼ · · · ≼ λi ui , where C(g; xN −i+1 , xN ) = {d(gxN −i+1 , gxN ), d(gxN −i+1 , gxN −i+2 ), d(gxN , gxN +1 ), d(gxN −i+1 , gxN +1 ), d(gxN , gxN −i+2 )}. Similarly, it is easy to see that ui ̸= d(gxN , gxN +1 ). This is because by Proposition 2.5(iii) we have u1 = d(gxN , gxN +1 )  λi−1 d(gxN , gxN +1 ). So we know that if ui = d(gxN −i+1 , gxN ) or ui = d(gxN −i+1 , gxN −i+2 ) or ui = d(gxN , gxN −i+2 ) then by the induction assumption (3.2) we have ui ≼ λ(e − λ)−1 d(gx1 , gx0 ). Hence, d(gxN +1 , gx1 ) ≼ λi ui ≼ λi+1 (e − λ)−1 d(gx1 , gx0 ) ≼ (λ)i+1 (e − λ)−1 d(gx1 , gx0 ) ≼ λ(e − λ)−1 d(gx1 , gx0 ), which means that (3.3.1) holds true. Without loss of generality, suppose that ui = d(gxN −i+1 , gxN +1 ). Then by the similar arguments as above we have ui ≼ λui+1 , where ui+1 ∈ C(g; xN −i , xN ). Hence, there is a sequence {un } such that d(gxN +1 , gx1 ) ≼ λu1 ≼ λ2 u2 ≼ · · · ≼ λN −1 uN −1 ≼ λN uN , 12

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where uN −1 = d(gx2 , gxN +1 ) ≼ λuN and uN ∈ C(g; x1 , xN ) = {d(gx1 , gxN ), d(gx1 , gx2 ), d(gxN , gxN +1 ), d(gxN , gx2 ), d(gx1 , gxN +1 )}. Obviously, uN ̸= d(gx1 , gxN +1 ) and uN ̸= d(gxN , gxN +1 ). On the contrary, if uN = d(gx1 , gxN +1 ), then uN ≼ λN uN , a contradiction. If uN = d(gxN , gxN +1 ) = u1 , then we have u1 = d(gxN , gxN +1 ) ≼ λ2 u2 ≼ · · · ≼ λN −1 uN −1 ≼ λN −1 u1 , a contradiction. Hence, it follows that uN = d(gx1 , gxN ), uN = d(gx1 , gx2 ) or uN = d(gxN , gx2 ). By the induction assumption (3.2), in any case, we have uN ≼ λ(e − λ)−1 d(gx1 , gx0 ).

(3.4)

Therefore, we get d(gxN +1 , gx1 ) ≼ λu1 ≼ λ2 u2 ≼ · · · ≼ λN uN ≼ λN (e − λ)−1 λd(gx1 , gx0 ) ≼ (λ)N +1 (e − λ)−1 d(gx1 , gx0 ) ≼ λ(e − λ)−1 d(gx1 , gx0 ).

(3.5)

That is to say, (3.3.1) is true. By (3.5), we have u1 ≼ λN −1 λ(e − λ)−1 d(gx1 , gx0 ). Thus, d(gxN , gxN +1 ) = u1 ≼ λN −1 λ(e − λ)−1 d(gx1 , gx0 ) ≼ (λ)N (e − λ)−1 d(gx1 , gx0 ) ≼ λ(e − λ)−1 d(gx1 , gx0 ), which implies that (3.3.N) is true. Similarly, since u2 = d(gxN −1 , gxN +1 ), . . . , ui = d(gxN −i+1 , gxN +1 ), . . . , by (3.4) and (3.5) we get ui ≼ λN −i uN ≼ λn−i+1 (e − λ)−1 d(gx1 , gx0 ).

(3.6)

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Hence, it follows from (3.6) that (3.3.2)-(3.3.N − 1) are all true. That is, (3.3) is true. Therefore, we conclude that Lemma 3.1 holds true. By Lemma 3.1, we immediately obtain the following result. Lemma 3.2 For all i, j ∈ N+ , one has d(gxi , gxj ) ≼ λ(e − λ)−1 d(gx0 , gx1 ).

(3.7)

Now, we begin to prove Theorem 3.1. First, we need to show that {gxn } is a Cauchy sequence. For all n > m, there exists ν1 ∈ C(g; xn−1 , xm−1 ) ={d(gxn−1 , gxm−1 ), d(gxn−1 , gxn ), d(gxm−1 , gxm ), d(gxn−1 , gxm ), d(gxm−1 , gxn )} such that d(f xn−1 , f xm−1 ) ≼ λν1 . Using the g-quasi-contractive condition repeatedly, we easily show by induction that there must exist νk ∈ {d(gxi , gxj ) : 0 ≤ i < j ≤ n} (k = 2, 3, . . . , m) such that νk ≼ λνk+1 (k = 1, 2, . . . , m − 1).

(3.8)

For convenience, we write νm = d(gxi , gxj ) where 0 ≤ i < j ≤ n. Using the triangular inequality, we have d(gxi , gxj ) ≼ d(gxi , gx0 ) + d(gx0 , gxj ) (0 ≤ i, j ≤ n), and by Lemma 3.2 we obtain d(gxn , gxm ) = d(f xn−1 , f xm−1 ) ≼ λν1 ≼ λ2 ν2 ≼ · · · ≼ λm νm ≼ λm d(gxi , gxj ) = λm+1 (e − λ)−1 d(gx1 , gx0 ).

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Since r(λ) < 1, by Remark 2.1 we have that λm+1 (e − λ)−1 d(gx1 , gx0 ) → θ as m → ∞, so by Proposition 2.4, it is easy to see that for any c ∈ intP , there exists n0 ∈ N such that for all n > m > n0 , d(gxn , gxm ) ≼ λm+1 (e − λ)−1 d(gx1 , gx0 ) ≪ c. So {gxn } is a Cauchy sequence in g(X). If g(X) ⊂ X is complete, there exist q ∈ g(X) and p ∈ X such that gxn → q as n → ∞ and gp = q. Now, from (2.1) we get d(f xn , f p) ≼ λν where ν ∈ C(g; xn , p) = {d(gxn , gp), d(gxn , f xn ), d(gp, f p), d(gxn , f p), d(f xn , gp)}. Clearly at least one of the following five cases holds for infinitely many n. (1) d(f xn , f p) ≼ λd(gxn , gp) ≼ λd(gxn+1 , gp) + λd(gxn+1 , gxn ); (2) d(f xn , f p) ≼ λd(gxn , f xn ) = λd(gxn , gxn+1 ); (3) d(f xn , f p) ≼ λd(gp, f p) ≼ λd(gxn+1 , gp) + λd(gxn+1 , f p), that is, d(f xn , f p) ≼ λ(e − λ)−1 d(gxn+1 , gp); (4) d(f xn , f p) ≼ λd(gxn , f p) ≼ λd(gxn+1 , f p) + λd(gxn+1 , gxn ), that is, d(f xn , f p) ≼ λ(e − λ)−1 d(gxn+1 , gxn ); (5) d(f xn , f p) ≼ λd(f xn , gp) = λd(gxn+1 , gp). As λ ≼ λ(e − λ)−1 (since θ ≼ λ and r(λ) < 1 ), we obtain that d(gxn+1 , f p) ≼ λ(e − λ)−1 [d(gxn+1 , gxn ) + d(gxn+1 , q)]. Since gxn → q as n → ∞, we get that for any c ∈ intP , there exists n1 ∈ N such that for all n > n1 , one has d(gxn+1 , f p) ≪ c. By Lemmas 2.4 and 2.5, we have gxn → f p as n → ∞ and q = f p. Now if w is another point such that gu = f u = w, hence d(w, q) = d(f u, f p) ≼ λν, where r(λ) ∈ [0, 1) and ν ∈ C(g; u, p) = {d(gu, gp), d(gu, f u), d(gp, f p), d(gu, f p), d(f u, gp)}. 15

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It is obvious that d(w, q) = θ, i.e., w = q. Therefore, q is the unique point of coincidence of f and g in X. Moreover, the mappings f and g are weakly compatible, by Lemma 2.6 we know that q is the unique common fixed point of f and g. Similarly, if f (X) is complete, the above conclusion is also established. According to Das-Naik version of the known theorem in the setting of metric spaces from [33], we have following result similar to Theorem 3.1. Theorem 3.2 Suppose one of the following conditions holds: (1) As in Theorem 3.1, let (X, d) be a complete cone metric space over a Banach algebra A. Assume one of f (X) or g (X) is closed and the other conditions in Theorem 3.1 are not changeable; (2) As in Theorem 3.1, let (X, d) be a complete cone metric space over a Banach algebra A. Assume f, g are cone compatible and both continuous and the other conditions in Theorem 3.1 are not changeable; (3) As in Theorem 3.1, let (X, d) be a complete cone metric space over a Banach algebra A. Assume f commutes with g, f or g is continuous (see Theorem 3.2 in Cvetkovi´cRakoˇcevi´c [36]) and the other conditions in Theorem 3.1 are not changeable. Then the conclusions of Theorem 3.1 are also true. Proof. (1) The proof of this case is the same as that in Theorem 3.1. (2) The sequence yn = f xn = gxn+1 , yn ̸= yn+1 for all n ∈ N converges to some z ∈ X as n → ∞. Further, since f xn → z and gxn → z we get that d (f z, gz) ≤ d (f z, f gxn ) + d (f gxn , gf xn ) + d (gf xn , gz) → 0 + 0 + 0 = 0. Hence f z = gz = ω. Hence, f, g has (a unique) point of coincidence. Since f and g are compatible then they are weakly compatible. Therefore by standard result they have a unique common fixed point (in this case it is ω). (3) Let g be continuous. Then we get gyn → gz and f yn → gz since f commutes with g. Indeed, f yn = f gxn+1 = gf yn+1 → gz.

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So we get d (f z, gz) ≼ d (f z, f yn ) + d (f yn , gz) ≼ λu + d (f yn , gz) , where u ∈ {d (gz, gyn ) , d (gz, f z) , d (gyn , f yn ) , d (gz, f yn ) , d (gyn , f z) + d (f yn , gz)} . If u = d (gz, gyn ) or u = d (gyn , f yn ) or d (gz, f yn ), then we obtain that λu + d (f yn , gz) is a c-sequence. This mens that f z = gz. If u = d (gz, f z) or u = d (gyn , f z) ≼ d (gyn , gz) + d (gz, f z) we get d (f z, gz) ≼ λd (gz, f z) + d (f yn , gz) or d (f z, gz) ≼ λd (gz, f z) + λd (gyn , gz) + d (f yn , gz) . In both cases we have that d (f z, gz) ≼ (e − λ)−1 cn , where cn is a c-sequence. Hence, f, g have a unique point of coincidence. Since f commutes with g then they are weakly compatible and by known result have a unique fixed point. Now let f be continuous. Again, f yn → f z and gyn = gf xn = f gxn = f yn−1 → f z. Further we get d (f z., z) ≼ d (f z, f yn ) + d (f yn , yn ) + d (yn , y) . Since d (f z, f yn ) + d (yn , y) = cn is c-sequence it is sufficient to estimate d (f yn , yn ) . We have d (f yn , yn ) = d (f yn , f xn ) ≼ λu, where u ∈ {d (gyn , gxn ) , d (gyn , f yn ) , d (gxn , f xn ) , d (gyn , f xn ) , d (gxn , f yn )} = {d (f yn−1 , yn−1 ) , d (f yn−1 , f yn ) , d (f yn−1 , yn ) , d (yn−1 , f yn ) , d (yn−1 , yn )} . Now we get the following cases:

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I) u = d (f yn−1 , yn−1 ) . Then d (f z., z) ≼ cn + λd (f yn−1 , yn−1 ) ≼ cn + λ (d (f yn−1 , f y) + d (f y, y) + d (y, yn−1 )) or d (f z., z) ≼ (e − λ)−1 cn + (e − λ)−1 λd (f yn−1 , f y) + (e − λ)−1 λd (y, yn−1 ) = dn where dn is a new c-sequence. II) u = d (f yn−1 , f yn−1 ) III) u = d (f yn−1 , yn ) IV) u = d (yn−1 , f yn ) V) u = d (yn−1 , yn ) In all cases we obtained that f z = z. For details see Theorem 3.2 in Cvetkovi´c-Rakoˇcevi´c [36]. Corollary 3.1 Let (X, d) be a complete cone metric space over a Banach algebra A and let P be the underlying cone with k ∈ P . If the mapping T : X → X is a quasicontraction, then T has a unique fixed point in X. And for any x ∈ X, the iterative sequence {T n x} converges to the fixed point. Proof. Set g = IX , the identity mapping from X to X. It is obvious to see that Theorem 3.1 yields Corollary 3.1. Remark 3.1 Corollary 3.1 does not need to require the assumption of normality of the cone P . So Corollary 3.1 improves and generalizes Theorem 9 in [21]. Remark 3.2 From the proof of Lemma 3.1, we note that the technique of induction appearing in Theorem 3.1 is somewhat different from that in Theorem 9 from [21], and also different from that in Theorem 2.6 from [11], which is more interesting and easily to understood. In addition, the proof of Theorem 3.1 is a valuable addition to [9] since Theorem 3.1 is a generalization of Theorem 3 from [9] but some main results in the proof of Theorem 3 from [9] were not proved in general. Remark 3.3 Taking E = R, P = [0, +∞), ∥.∥ = |.| , λ ∈ [0, 1) in Theorem 3.1, we get ´ c’s result from [13], both in the setting of Das-Naik’s result from [33]; if g = IX we get Ciri´ 18

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metric spaces. The following corollary is the Jungck’s result in the setting of cone metric spaces over Banach algebras. Corollary 3.2 Let (X, d) be a cone metric space over a Banach algebra A with the underlying solid cone P . Let the mappings f, g : X → X satisfy the condition that for λ ∈ P with r(λ) ∈ [0, 1) and for every x, y ∈ X holds d(f x, f y) ≼ λd(gx, gy). If g(X) ⊂ f (X) and g(X) or f (X) is a complete subspace of X, then f and g have a unique point of coincidence in X. Moreover, if f and g are weakly compatible, then f and g have a unique common fixed point. The next result is the Banach contraction principle in the setting of cone metric spaces over Banach algebras. Corollary 3.3 (see [29]) Let (X, d) be a cone metric space over a Banach algebra A the underlying solid cone P . Let the mapping f : X → X satisfy the condition that for λ ∈ P with r(λ) ∈ [0, 1) and for every x, y ∈ X holds d(f x, f y) ≼ λd(x, y) (namely, f is a generalized Lipschitz contraction). If f (X) is a complete subspace of X, then f has a unique point in X. We will present an example to show that the results presented above are real generalizations of the corresponding results in the literature. Example 3.1 Let X = [1, ∞) and A be a set of all real valued function on [0, 1] which also have continuous derivates on [0, 1] with the norm ∥x∥ = ∥x∥∞ + ∥x′ ∥∞ and the usual multiplication. Let P = {x ∈ A : x (t) ≥ 0, t ∈ [0, 1]} . It is clear that P is a nonrmal cone and A is a Banach algebra with a unit e = 1. Define a mapping d:X ×X →A by d (x, y) (t) := |x − y| et . We make a conclusion that (X, d) is a complete cone metric space over Banach algebra A. Now define the mappings f, g : X → X by f (x) = 3x − 2, g (x) = 4x − 3. Choose 19

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1 λ (t) = 12 t + 43 . Since f (X) ⊆ g (X) and r (λ) = 56 , thus, all the conditions of Theorem 3.1 are satisfied and consequently f and g have a unique comon fixed point x = 1. Indeed, for x, y ∈ X we can putting u (x, y) = d (g (x) , g (y)) = 4 |x − y| . In this case we have ( ) 3 1 3 1 3 t d (f (x) , f (y)) = 3 |x − y| e ≤ t+ 4 |x − y| et ⇔ ≤ t + , 12 4 4 12 4

which is indeed true. On the other hand, we see that f (g (x)) = f (4x − 3) = 3 (4x − 3) − 2 = 12x − 11 = 4 (3x − 2) − 3 = g (f (x)) , that is, f commutes with g and other words f, g are weakly compatible. 1 Now let us estimate r (λ) = limn→∞ ∥λn ∥ n . Since )n ( )n−1 ( 3 n 3 1 1 ′ n n t+ , (λ (t)) = t+ , λ (t) = 12 4 12 12 4 we have (t = 1) ′

∥λ∥∞ +∥λ ∥∞

( )n ( )n−1 ( )n−1 ( ) ( )n−1 ( ) 5 12 5 n 5 n 5 n 5 10 = + = · +1 = 1+ . 6 12 6 12 6 n 6 12 6 n

Further we get

( )1 ( n ) n1 ( 5 ) n−1 n 10 n 5 ∥λ ∥ = 1+ → < 1. 12 6 n 6 n

1 n

However, both f and g are not quasi-contraction. Indeed, for x = 2, y = 1 and for all λ with r (λ) ∈ [0, 1), we get d (f 2, f 1) (t) = d (4, 1) (t) = 3et > λ (t) u, for all { } d (2, 1) et , d (2, f 2) et , d (1, f 1) et , d (2, f 1) et , d (1, f 2) et { } = et , 2et , 0, et , 3et ,

u ∈

and similarly d (g2, g1) (t) = d (5, 1) (t) = 4et > λ (t) u for all { } d (2, 1) et , d (2, g2) et , d (1, g1) et , d (2, g1) et , d (1, g2) et { } = et , 3et , 0, et , 4et .

u ∈

Hence, Theorem 3.1 is a genuine generalization of Theorem 9 from [21].

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Competing interests The authors declare that they have no competing interests.

Authors’ contributions The authors contribute equally and significantly in writing this paper. All the authors read and approve the final manuscript.

Authors details 1

School of Mathematics and Statistics, Hanshan Normal University, Chaozhou, 521041, China. 2 Faculty of Science, University of Kragujevac, Radoja Domanovi´ca 12, 34000 Kragujevac, Serbia. 3 Faculty of Mechanical Engineering, University of Belgrade, Kraljice Marije 16, 11120 Beograd, Serbia, and Department of Mathematics, State University of Novi Pazar, Novi Pazar, Serbia. Email address: [email protected]; [email protected] (S. Xu); [email protected] (B. Z. Popovi´c); [email protected] (S. Radenovi´c).

Acknowledgments The research is partially supported by the foundation of the research item of Strong Department of Engineering Innovation of Hanshan Normal University, China (2013), and by the Serbian Ministry of Science and Technological Developments (Project: Methods of Numerical and Nonlinear Analysis with Applications, grant number #174002).

References [1] S. Banach, Sur les operations dans les ensembles abstrait et leur application aux equations, integrals . Fundam. Math., (3)1922, 13. [2] P. P. Zabrejko, K-metric and K-normed linear spaces: survey. Collect. Math., 48(1997), 825-859. [3] L.-G. Huang and X. Zhang, Cone metric space and fixed point theorems of contractive mappings. J. Math. Anal. Appl., 332(2007), 1468-1476. 21

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[4] M. Abbas and G. Jungck, Common fixed point results for noncommuting mappings without continuity in cone metric spaces. J. Math. Anal. Appl. 341(2008), 416-420. [5] S. Radenovi´c, V. Rakoˇcevi´c and Sh. Rezapour, Common fixed points for (g, f ) type maps in cone metric spaces. Appl. Math. Comput., 218(2011), 480-491. [6] S. H. Cho and J. S. Bae, Common fixed points theorems for mappings satisfying property (E, A) on cone metric spaces. Math. Comput. Modell., 53(2011), 945-951. [7] D. Ili´c and V. Rakoˇcevi´c, Quasi-contraction on a cone metric space. Appl. Math. Lett., 22: 5(2009), 728-731. [8] Z. Kadelburg, S. Radenovi´c and V. Rakoˇcevi´c, Remarks on “Quasi-contraction on a cone metric space”. Appl. Math. Lett., 22: 11(2009), 1674-1679. [9] Lj. Gaji´c and V. Rakoˇcevi´c, Quasi-contractions on a nonnormal cone metric space. Funct. Anal. Appl., 46: 1(2012), 75-79. [10] N. Hussian and M. H. Shah, KKM mappings in cone b-metric spaces. Comput. Math. Appl., 62(2011), 1677-1684. [11] L. Shi and S. Xu, Common fixed point theorems for two weakly compatible selfmappings in cone b-metric spaces. Fixed Point Theory Appl., 2013, 2013: 120. [12] Z. Kadelburg, M. Ravlovi´c and S. Radenovi´c, Common Fixed point theorems for ordered contractions and quasi-contractions in ordered cone metric spaces, Comput. Math. Appl., 59(2010), 3148-3159. ´ c, A generalization of Banach’s contraction priciple. Proc. Amer. Math. [13] Lj. B. Ciri´ Soc., 45(1974), 267-273. [14] M.H. Shah, S. Simi´c, N. Hussain, A. Sretenovi´c and S. Radenovi´c, Common fixed points theorems for occasionally weakly compatible pairs on cone metric type spaces. J. Comput. Anal. Appl., 14(2) (2012), 290-297. [15] M.P. Stani´c, A.S. Cvetkovi´c, S. Simi´c and S. Dimitrijevi´c, Common fixed point under ´ c type on cone metric type spaces. Fixed Point Theory contractive condition of Ciri´ Appl., 2012, 2012:35.Volume 2011, Article ID 589725.

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[16] W. S. Du, A note on cone metric fixed point theory and its equivalence. Nonl. Anal., 72: 5 (2010) 2259-2261. [17] Y. Feng and W. Mao, The equivalence of cone metric spaces and metric spaces. Fixed Point Theory 11:2(2010), 259-264. [18] Z. Kadelburg, S. Radenovi´c and V. Rakoˇcevi´c, A note on the equivalence of some metric and cone metric fixed point results. Appl. Math. Lett., 24(2011), 370-374. [19] H. C ¸ akallı, A. S¨onmez and C ¸ . Gen¸c, On an equivalence of topological vector space valued cone metric spaces and metric spaces. Appl. Math. Lett., 25 (2012) 429-433. [20] H. Liu and S. Xu, Cone metric spaces with Banach algebras and fixed point theorems of generalized Lipschitz mappings. Fixed Point Theory Appl., 2013, 2013: 320. [21] H. Liu and S. Xu, Fixed point theorem of quasi-contractions on cone metric spaces with Banach algebras, Abstract Appl. Anal., Volume 2013, Article ID 187348, 5 pages. [22] I. Marian and A. Branga, Common fixed point results in b-cone metric spaces over topological vector spaces. General Mathematics., Vol. 20, No. 1(2012). [23] D. lli´c and V. Rakoˇcevi´c, Common fixed points for maps on cone metric space, J. Math. Anal. Appl., 341(2008), 876-882. [24] Sh. Rezapour and R. Hamlbarani, Some notes on the paper“Cone metric spaces and fixed point theorems of contractive mappings”, J. Math. Anal. Appl., 345(2008) 719-724. [25] S. Jankovi´c, Z. Golubovi´c and S. Radenovi´c,

Compatible and weakly compatible

mappings in cone metric spaces. Math. Comput. Modell., 52(2010), 1728-1738. [26] M. Jovanovi´c, Z. Kadelburg and S. Radenovi´c, Common Fixed Point Results in Metric-Type Spaces. Fixed Point Theory Appl., Volume 2010, Article ID 978121. [27] S. Radenovi´c and B. E. Rhoades, Fixed point theorem for two non-self mappings in cone metric spaces. Comput. Math. Appl., 57(2009), 1701-1707. [28] Walter Rudin, Functional Analysis, 2nd ed., McGraw-Hill, 1991.

23

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Shaoyuan Xu et al 648-671

J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 22, NO.4, 2017, COPYRIGHT 2017 EUDOXUS PRESS, LLC

[29] S. Xu and S. Radenovi´c, Fixed point theorems of generalized Lipschitz mappings on cone metric spaces over Banach algebras without assumption of normality. Fixed Point Theory Appl., 2014, 2014:102. [30] Z. Kadelburg, and S. Radenovi´c, A note on various types of cones and fixed point results in cone metric spaces. Asian J. Math. Appl., Volume 2013, Article ID ama0104, 7 pages. [31] M. Dordevi´c, D. Dori´c, Z. Kadelburg, S. Radenovi´c and D. Spasi´c, Fixed point results under c-distance in tvs-cone metric spaces. Fixed Point Theory Appl., 2011, 2011: 29. [32] S. Jankovi´c, Z. Kadelburg and S. Radenovi´c, On the cone metric space: A survey. Nonl. Anal., 74(2011), 2591-2601. [33] K. M. Das, K. Naik, Common fixed point theorems for commuting maps on a metric space. Proc. Am. Math. Soc., 77 (1979), 369-373. [34] A. I. Perov, On Cauchy problem for a system of ordinary differential equations, (in Russian) Priblizhen. Metody Reshen. Difer. Uravn. 2 (1964) 115-134. [35] A. Szilard, A note on Perov’s fixed point theorem, Fixed Point Theory, Volume 4, No. 1, 2003, 105-108. [36] M. Cvetkovi´c, V. Rakoˇcevi´c, Common fixed point results for mappings of Perov type, Math. Nachr. 1C18 (2015) / DOI 10.1002/mana.201400098. [37] M. Cvetkovi´c, V. Rakoˇcevi´c, Extensions of Perov theorem, Carpat. J. Math. 31 (2015), 2, 181-188. [38] M. Cvetkovi´c, V. Rakoˇcevi´c, Quasi-contraction of Perov type, Appl. Math. Comput. 235 (2014) 712-722. [39] Z. Li and S. Jing, Quasi-contractions restricted with linear bounded mappings. Fixed Point Theory Appl., 2014, 2014:87.

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On some inequalities of the Bateman’s G−function Mansour Mahmoud1 and Hanan Almuashi2 1,2

King Abdulaziz University, Faculty of Science, Mathematics Department, P. O. Box 80203, Jeddah 21589, Saudi Arabia. 1

[email protected]

. . Abstract In the paper, we prove that the Bateman’s G−function satisfies the double inequality 2m ∑ (2n − 1)B2n n=1

nx2n

2m−1 ∑ (2n − 1)B2n 1 < G(x) − < , x nx2n

m∈N

n=1

with best bounds, where Br′ s are the Bernoulli numbers and we study the monotonicity of some functions involving the function G(x). Also, we present some estimates for the error term of a class of the alternating series, which improve and generalize some recent resutls and we prove the increasing monotonicity of a sequence arising from computation of the intersecting probability between a plane couple and a convex body. 2010 Mathematics Subject Classification: 33B15, 26D15, 41A80. Key Words: Digamma function, Bateman’s G-function, sharp inequality, monotonicity, alternating series, sequence.

1

Introduction.

The ordinary gamma function is defined by [3] ∫ ∞ Γ(x) = tx−1 e−t dt,

x>0

0

and the derivative of log Γ(x) is called the digamma function and is denoted by ψ(x). We can considered to the gamma function, the digamma function and the the Riemann zeta function as the most important special functions [5]. For more details on bounding the gamma function and its logarithmic derivatives, please refer to the papers [2]-[5], [7], [8], [14]-[20] [22]-[26], [35]-[41] 1

Permanent address: Mansour Mahmoud, Department of Mathematics, Faculty of Science, Mansoura University, Mansoura 35516, Egypt.

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and plenty of references therein. The Bateman’s G−function is defined by Erd´elyi [6] as ( ) (x) x+1 , x ̸= 0, −1, −2, ... G(x) = ψ −ψ 2 2

(1)

which satisfies [6]: G(1 + x) + G(x) =

2 x

(2)

and G(1 − x) + G(x) = 2π csc(πx).

(3)

The function G(x) can be defined by the hypergeometric function as G(x) =

2 2 F1 (1, x; 1 + x; −1). x

From the integral representation of the function ψ(z) [3] ) ∫ ∞ ( −t e−xt e ψ(x) = − dt, t 1 − e−t 0 we obtain the following integral representation ∫ ∞ 2 e−xt G(x) = dt, 1 + e−t 0

x>0

x > 0.

(4)

The function G(x) is very useful in summing and estimating certain numerical and algebraic series [27] . For example: ∞ ∑ (−1)k 1 (u) = G , sk+u 2s s k=0

u ̸= 0, −s, −2s, ...

(5)

and its nth partial sum is given by n (u )] ∑ (−1)k 1 [ (u) = G + (−1)n G +n+1 , sk+u 2s s s k=0

u ̸= 0, −s, −2s, ... .

(6)

Qiu and Vuorinen [43] deduced the inequality 4(1.5 − log 4) 1 1 < G(x) − < 2 , 2 x x 2x

1 x> . 2

(7)

Mahmoud and Agarwal [16] presented an asymptotic formula for Bateman’s G-function G(x) and deduced the double inequality 2x2

1 1 1 < G(x) − < 2 , + 1.5 x 2x

x>0

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(8)

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which improve the lower bound of the inequality (7) and they posed a sharp double inequality of the function G(x) as a conjecture. Mortici [21] established the inequality 0 < ψ(x + u) − ψ(x) ≤ ψ(u) + γ +

1 −u u

x ≥ 1; u ∈ (0, 1),

(9)

where γ is the Euler constant, which also improves the result of Qiu and Vuorinen. Also, Alzer presented the double inequality [2] 1 1 − An (u; x) − δn (u; x) < ψ(x + u) − ψ(x) < − An (u; x), x x where n ≥ 0 be an integer, x > 0, u ∈ (0, 1), [ ] n−1 ∑ 1 1 + An (u; x) = (1 − u) u + n + 1 i=0 (x + i + 1)(x + i + u) and δn (u; x) =

(x + n)(x+n)(1−u) (x + n + 1)(x+n+1)u 1 log . x+n+u (x + n + u)x+n+u

In this paper, we prove the conjecture posed by Mahmoud and Agarwal [16] about a sharp double inequality of the function G(x). We will study the completely monotonicity property of some functions involving the Bateman’s G−function. Our results generalize and improve some inequalities about the error term of a class of alternating series and will prove the main result of [9] about the increasing monotonicity of a certain sequence .

2

Main results.

Theorem 1. The Bateman’s G−function satisfies 1 ∑ (22n − 1)B2n (22m+2 − 1)B2m+2 + + θ1 , x n=1 nx2n (m + 1)x2m+2 m

G(x) =

m = 1, 2, 3, ...

(10)

where Bi′ s are Bernoulli numbers, θ1 is independent of x and 0 < θ1 < 1. Proof. Using the integral representation of the function G(x) and the formula [1] ∫ ∞ 1 1 ts−1 e−xt dt, s∈N = xs (s − 1)! 0 we get 1 G(x) − = x

∫

∞

tanh(t/2) e−xt dt.

(11)

0

We will apply a technique which used later by Qi and Guo [34]. By the expansion [1] tanh(t/2) =

∞ ∑ k=1

4t t2 + π 2 (2k − 1)2

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and the identity ∑ 4(−1)n−1 t2n−1 4(−1)m t2m+1 1 4t = + , 2 2 2 2n 2n 2m 2m 2 2 t + π (2k − 1) π (2k − 1) π (2k − 1) t + π (2k − 1)2 n=1 m

we obtain 1 G(x) − = x Now

∫

∞ ∞∑

( m ∑ 4(−1)n−1 t2n−1

k=1

π 2n (2k − 1)2n

0

n=1

∞ ∑ m ∑ 4(−1)n−1 t2n−1

π 2n

k=1 n=1

4(−1)m t2m+1 1 + 2m 2m 2 2 π (2k − 1) t + π (2k − 1)2

m∈N

) e−xt dt

m ∈ N.

∑ 4(−1)n−1 t2n−1 1 = (1 − 2−2n )ζ(2n), 2n (2k − 1)2n π n=1 m

where ζ(t) is the Riemann zeta function which satisfies [3] ζ(2s) = Then

m ∞ ∑ ∑ 4(−1)n−1 t2n−1 k=1 n=1

π 2n

(−1)s−1 π 2s 22s−1 B2s , (2s)!

s ∈ N.

∑ 2(22n − 1)B2n 1 = t2n−1 , (2k − 1)2n (2n)! n=1 m

m ∈ N.

(12)

Also, ∞ ∞ ∑ 4(−1)m t2m+1 1 1 4(−1)m t2m+1 ∑ = ( 2m 2m 2 2 2 2m+2 π (2k − 1) t + π (2k − 1) π (2k − 1)2m+2 k=1 k=1

1 t π ( 2k−1)

m∈N

)2 +1

and hence ∞ ∞ ∑ ∑ 4(−1)m t2m+1 1 1 4(−1)m t2m+1 = θ(t) , 2m 2m 2 2 2 2m+2 π (2k − 1) t + π (2k − 1) π (2k − 1)2m+2 k=1 k=1

m∈N

where 0 < θ(t) < 1. Then ∞ ∑ 4(−1)m t2m+1 1 2(22m+2 − 1)t2m+1 B2m+2 = θ(t), π 2m (2k − 1)2m t2 + π 2 (2k − 1)2 (2m + 2)! k=1

0 < θ(t) < 1; m ∈ N. (13)

Now 1 ∑ 2(2n − 1)B2n G(x) − = x n=1 (2n)! m

∫

∞

t

2n−1 −xt

e

0

2(22m+2 − 1)B2m+2 dt + (2m + 2)!

∫

∞

θ(t)t2m+1 e−xt dt. (14)

0

Using the ordinary gamma function and its functional equation Γ(n + 1) = n! for n ∈ N, we get 1 ∑ (22n − 1)B2n (22m+2 − 1)B2m+2 G(x) − = + θ1 , x n=1 nx2n (m + 1)x2m+2 m

m∈N

where θ1 is independent of x and 0 < θ1 < 1. 4

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Theorem 2 ([16], Conjecture 1). The Bateman’s G−function satisfies the following double inequality 2m−1 2m ∑ (22n − 1)B2n ∑ (22n − 1)B2n 1 < G(x) < m = 1, 2, 3, ... − (15) 2n 2n nx x nx n=1 n=1 with sharp bounds, where Bi′ s are Bernoulli numbers. Proof. The inequality (15) satisfies from the relation (10) and the following property of Bernoulli constants [12]: B2r+2 < 0 and B2r+4 > 0 f or r = 1, 3, 5, ... . (16) Now, we will prove the sharpness of the inequality (15) using Mortici’s technique [25]. From the definition [11], the asymptotic expansion of a function T (x) of the form ∞ ∑ bk T (x) = K(x) + b0 + xk k=1

satisfies for every fixed r, that [

(

lim xr T (x) −

x→∞

r ∑ bk K(x) + b0 + xk k=1

)] = 0.

Using the relation (10), we have ] [ m−1 ∑ (22n − 1)B2n (22m − 1)B2m 1 = , lim x2m G(x) − − x→∞ x n=1 nx2n m

m = 1, 2, 3, ... .

(17)

If we have other constants h2 , h4 , h6 , ... satisfy 2 ∑ h2i i=1

x2i

4 ∑ h2i i=1

x2i

6 ∑ h2i i=1

x2i

1 ∑ h2i < G(x) − < , 2i x x i=1 1

< G(x) −

1 ∑ h2i < , 2i x x i=1

< G(x) −

1 ∑ h2i < , 2i x x i=1

3

5

etc. Then these inequalities give us that [ ] limx→∞ x2 G(x) − x1 = h2 , [ limx→∞ x4 G(x) − [ limx→∞ x6 G(x) −

1 x

1 x

−

h2 x2

−

h2 x2

−

]

= h4 ,

h4 x4

]

(18)

= h6 ,

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etc. Comparing the relations (17) and (18), gives us that (22j − 1)B2j , j

h2j =

∀ j ∈ N.

(19)

(22j −1)B

2j This means that the constants in the inequality (15) are the best. Also, the constant j 1 one in the function G(x) − x can not be improved whatsoever, see [16].

Remark 1. As a special case of the inequality (15), we get 1 1 1 1 1 1 − 4 < G(x) − < 2 − 4 + 6 , 2 2x 4x x 2x 4x 2x

(20)

which √ improve the right hand side of the inequality (8) for x > 0 and its left hand side for x > 32 . Lemma 2.1. For m ∈ N, the functions 1 ∑ (22n − 1)B2n Fm (x) = G(x) − − x n=1 nx2n 2m

and Hm (x) = −G(x) +

2m−1 ∑ (22n − 1)B2n 1 + x nx2n n=1

are strictly completely monotonic. Proof. Using the relation (14), we have 1 ∑ (22n − 1)B2n 2(22m+2 − 1)B2m+2 G(x) − − = x n=1 nx2n (2m + 2)! m

Then dk (−1) dxk k

(

1 ∑ (22n − 1)B2n G(x) − − x n=1 nx2n m

)

∫

∞

θ(t)t2m+1 e−xt dt.

0

2(22m+2 − 1)B2m+2 = (2m + 2)!

∫

∞

θ(t)t2m+k+1 e−xt dt.

0

Using the Bernoulli number’s property (16), we get ( ) 2m k 2n ∑ 1 d (2 − 1)B 2n (−1)k k G(x) − − >0 dx x n=1 nx2n and dk (−1)k k dx

(

2m−1 ∑ (22n − 1)B2n 1 G(x) − − x nx2n n=1

) < 0.

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Corollary 2.2. For odd k, we have 2m−1 ∑ n=1


0 for all r, is called an alternating series. By Leibnitz’s Theorem [11], the alternating series converges if ar decreases monotonically and ar → 0 as r → ∞. Moreover, let S denote the sum of the series and Sn its nth partial sum, then |Sn − S| < an+1 ,

n ∈ N.

For further details about finding estimates for the error |Sn − S|, please refer to [13], [28]-[33]. The alternating series [10] ∞ ∑ (−1)k−1 k=1

k

= ln 2

and

∞ ∑ (−1)k−1 k=1

2k − 1

=

π 4

presented early important results of the calculus. Kazarinoff [10] deduced the following error estimates n ∑ r 1 (−1) π 1 < − < , n∈N (21) 4n + 2 r=1 2r − 1 4 4n − 2 and

n ∑ (−1)r+1 1 1 < − ln 2 < , 2n 2(n + 1) r=1 r

by studying the function

∫

n∈N

(22)

π/4

tann θdθ,

En =

n ∈ N.

0

T´oth [32] improved Kazarinoff’s estimates by n ∑ r 1 1 (−1) π √ < − < , 4n + 2 19 − 8 r=1 2r − 1 4 4n

n∈N

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(23)

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n ∑ (−1)r+1 1 1 √ < − ln 2 < , 2n + 1 r 2n + 2 7 − 4 r=1

and

n ∈ N.

(24)

Also, T´oth and Bukor [33] shown that the best constants a and b such that the inequalities n ∑ (−1)r+1 1 1 ≤ − ln 2 < , n≥1 (25) 2n + b 2n + a r=1 r hold are a =

2 ln 2−1 1−ln 2

and b = 1.

Koumandos [13] refined Kazarinoff’s estimate (21) by n ∑ (−1)r 1 π 1 ≤ − < , 4n + c 2r − 1 4 4n + d

n∈N

(26)

r=1

where the constants c =

4 4−π

− 4 and d = 0 are the best possible.

In [16], Mahmoud and Agarwal presented the following generalization ∞ ∑ 4(l + n)2 + 10(l + n) + 9 (−1)r−1 2(l + n) + 3 < , < 2 2(l + n + 1) [4(l + n) + 8(l + n) + 7] r=n+1 r + l 4(l + n + 1)2

(27)

where l > −n − 1 and −l ∈ / N. The double inequality (27) improved the two inequalities (25) and (26) for n > 1. Now, using (5) and (6), we have ∞ ∑ r−1 1 (−1) (−1)n G (l + n + 1) = G (l + n + 1) , = r+l 2 2 r=n+1

−l ∈ / N.

∑ Then our double inequality (15) will give us sharp bounds of the the error ∞ r=n+1 /N. −l ∈

(28)

(−1)r−1 , r+l

for

Lemma 3.1.

2n ∞ 2n−1 r−1 2r ∑ ∑ (−1) 2 ∑ 2(22r − 1)B2r 2 2(2 − 1)B2r + < < + n r=1 r(l + n + 1)2r r=n+1 r + l n r(l + n + 1)2r r=1

n∈N

(29)

with sharp bounds, where l > −n − 1 and −l ∈ / N. Remark 2. The inequality (29) improve the inequalities (25) and (26) for special values of the parameter l. Also, it is a generalization of the inequality (27).

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3.2

New proof of the increasing monotonicity of a sequence arising from computation of the intersecting probability between a plane couple and a convex body

The increasing of the sequence k−1 Pk = 2

(∫

)2

π/2

sink−1 v dv

,

k∈N

0

was a question arises from computation of the intersecting probability between a plane couple and a convex body [9]. To prove the increasing monotonicity of the sequences Pk , Guo and Qi [9] studied equivalently the increasing monotonicity of the sequence ( ) 1 Γ2 k+1 (2) Qk = k ∈ N. k Γ2 k2 Qi, Mortici and Guo [42] investigated an asymptotic formula for the function ( ( ) ( )) t+1 t ϕ(t) = 2 log Γ − log Γ − log t t>0 2 2 and proved some properties of the sequence Qk . Also, Mahmoud [17] generalized some properties of the function ϕ(t) and answered about the two posed questions in [42] about the sequence Qk . The first derivative of the function ϕ(t) can be represented by ϕ′ (t) = G(t) −

1 t

and then the function ϕ′ (t) is strictly completely monotonic, that is (−1)r (ϕ′ (t))

(r)

> 0,

r = 0, 1, 2, ... .

Hence the function ϕ(t) is increasing and also the function ( ) 1 Γ2 t+1 (2) Q(t) = t>0 t Γ2 2t since Q′ (t) = Q(t)ϕ′ (t). Then Q(t) is increasing function and hence the sequence Qk is increasing sequence, which is the main result of [9].

References [1] M. Abramowitz, I. A. Stegun (Eds.), Handbook of Mathematical Functions with Formulas, Graphs and Mathematical Tables, Dover, New York, 1965. [2] H. Alzer, On some inequalities for the gamma and psi function, Math. Comput., 66, 217, 373-389, 1997. 9

680

Mansour Mahmoud et al 672-683

J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 22, NO.4, 2017, COPYRIGHT 2017 EUDOXUS PRESS, LLC

[3] G. E. Andrews, R.Askey and R.Roy, Special Functions, Cambridge Univ. Press, 1999. [4] N. Batir, Inequalities for the gamma function, Arch. Math (Basel), 91, 554-563, 2008. [5] N. Batir, An approximation formula for n!, Proyecciones Journal of Mathematics Vol. 32, No 2, 173-181, 2013. [6] A. Erd´ elyi et al., Higher Transcendental Functions Vol. I-III, California Institute of Technology - Bateman Manuscript Project, 1953-1955 McGraw-Hill Inc., reprinted by Krieger Inc. 1981. [7] B.-N. Guo and F. Qi, A class of completely monotonic functions involving divided differences of the psi and tri-gamma functions and some applications, J. Korean Math. Soc. 48, 655-667, 2011. [8] B.-N. Guo and F. Qi, Sharp inequalities for the psi function and harmonic numbers, Analysis–International mathematical journal of analysis and its applications 34 , no. 2, 201208, 2014. [9] B.-N. Guo and F. Qi, On the increasing monotonicity of a sequence originating from Computation of the probability of intersecting between a plane couple and a convex body, Turkish Journal of Analysis and Number Theory, vol. 3, no. 1, 21-23, 2015. [10] N. D. Kazarinoff, Analytic Inequalities, Holt, Rhinehart and Winston, New York, 1961. [11] K. Knopp, Theory and application of infinite series, Dover, New York, 1990. [12] S. Koumandos, Remarks on some completely monotonic functions, Journal of Mathematical Analysis and Applications 324, no. 2, 1458-1461, 2006. [13] S. Koumandos, Monotonicity of some functions involving the Psi function and sharp estimates for the remainders of certain series, Int. J. Contemp. Math. Sci., Vol. 2, no. 15, 713 720, 2007. [14] M. Mahmoud, M. A. Alghamdi and R. P Agarwal, New upper bounds of n!, Journal of Inequalities and Application, 2012, 2012:27. [15] M. Mahmoud, A new proof of an inequality for the logarithm of the gamma function and its sharpness, Turkish Journal of Analysis and Number Theory, Vol. 2, No. 4, 147-151, 2014. [16] M. Mahmoud and R. P. Agarwal, Bounds for Bateman’s G−function and its applications, Georgian Mathematical Journal, To appear 2016. [17] M. Mahmoud, Some properties of a function related to a sequence originating from computation of the probability of intersecting between a plane couple and a convex body, Submitted for publication. [18] M. Mansour and M. A. Obaid, Bounds of q-factorial [n]q!, Ars Combinatoria, Vol. CII, 313-319, 2011.

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681

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[19] M. Mansour, On quicker convergence towards Euler’s constant, J. Comput. Anal. Appl. 17, No. 4, 632-638, 2014. [20] M. Mansour, Refinement of some inequalities of q-gamma function, J. Computational Analysis and Applications, Vol. 18, No.4, 645-654, 2015. [21] C. Mortici, A sharp inequality involving the psi function, Acta Universitatis Apulensis, 41-45, 2010. [22] C. Mortici, A quicker convergence toward the γ constant with the logarithm term involving the constant e, Carpathian J. Math. 26(1), 86-91, 2010. [23] C. Mortici, Estimating gamma function in terms of digamma function, Math. Comput. Model., 52, no. 5-6, 942-946, 2010. [24] C. Mortici, New approximation formulas for evaluating the ratio of gamma functions. Math. Comp. Modell. 52(1-2), 425-433, 2010. [25] C. Mortici, A new representation formula for the factorial function, Thai Journal of Mathematics, Vol. 8, no. 2, 249-254, 2010. [26] C. Mortici, Accurate estimates of the gamma function involving the psi function, Numer. Funct. Anal. Optim., 32, no. 4, 469-476, 2011. [27] K. Oldham, J. Myland and J. Spanier, An Atlas of Functions, 2nd edition. Springer, 2008. [28] A. Sˆınt˘am˘arina, A new proof for estimating the remainder of the alternating harmonic series, Creat. Math. Inform., 21, 2, 221-225, 2012. [29] A. Sˆınt˘am˘arina, Sharp estimates regarding the remainder of the alternating harmonic series, Mathematical Inequalities & Applications, Volume 18, Number 1, 347352 2015. [30] V. Timofte, On Leibniz series defined by convex functions, J. Math. Anal. Appl. 300, no. 1, 160-171, 2004. [31] V. Timofte, Integral estimates for convergent positive series, J. Math. Anal. Appl. 303, no. 1, 90-102, 2005. [32] L. T´oth, On a class of Leibniz series, Rev. Anal. Num´er. Th´eor. Approx., 21 , 195-199, 1992. [33] L. T´oth and J. Bukor, On the alternating series 1 − 282, 1, 21-25, 2003.

1 2

+

1 3

−

1 4

+ ... , J. Math. Anal. Appl.

[34] F. Qi and Bai-Ni Guo, Monotonicity and convexity of ratio between gamma functions to different powers, Journal of the Indonesian Mathematical Society (MIHMI) 11, no. 1, 39-49, 2005. [35] F. Qi and H. M Srivastava, Some uniqueness results for the nontrivially complete monotonicity of a class of functions involving the polygamma and related functions, Integral Transforms Spec. Funct. 21 , 849-858, 2010. 11

682

Mansour Mahmoud et al 672-683

J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 22, NO.4, 2017, COPYRIGHT 2017 EUDOXUS PRESS, LLC

[36] F. Qi, R.-Q. Cui, C.-P. Chen and B.-N. Guo, Some completely monotonic functions involving polygamma functions and an application, Journal of Mathematical Analysis and Applications 310, no. 1, 303-308, 2005. [37] F. Qi and B.-N. Guo, Completely monotonic functions involving divided differences of the diand tri-gamma functions and some applications, Commun. Pure Appl. Anal. 8, 1975-1989, 2009. [38] F. Qi and B.-N. Guo, Necessary and sufficient conditions for functions involving the tri- and tetra-gamma functions to be completely monotonic, Adv. in Appl. Math. 44, 71-83, 2010. [39] F. Qi and Q.-M. Luo, Bounds for the ratio of two gamma functions—From Wendel’s and related inequalities to logarithmically completely monotonic functions, Banach Journal of Mathematical Analysis 6, no. 2, 132-158, 2012. [40] F. Qi and Q.-M. Luo, Bounds for the ratio of two gamma functions: from Wendel’s asymptotic relation to Elezovi´c-Giordano-Peˇcari´c’s theorem, Journal of Inequalities and Applications 2013, 2013:542, 20 pages. [41] F. Qi , Bounds for the ratio of two gamma functions, J. Inequal. Appl. 2010 (2010), Article ID 493058, 84 pages. [42] F. Qi, C. Mortici, and B.-N. Guo, Some properties of a sequence arising from computation of the intersecting probability between a plane couple and a convex body, ResearchGate Research, available online at http://dx.doi.org/10.13140/RG.2.1.1176.0165. [43] S.-L. Qiu and M. Vuorinen, Some properties of the gamma and psi functions with applications, Math. Comp., 74, no. 250, 723-742, 2004.

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3-VARIABLE ADDITIVE ρ-FUNCTIONAL INEQUALITIES IN FUZZY NORMED SPACES JOONHYUK JUNG, JUNEHYEOK LEE∗ , GEORGE A. ANASTASSIOU, AND CHOONKIL PARK∗ Abstract. In this paper, we introduce and investigate the following additive ρ-functional inequalities N (f (x + y + z) − f (x) − f (y) − f (z), t)      x + y + z − f (x) − f (y) − 2f (z) , t , ≥ N ρ 2f 2  x + y   N 2f + z − f (x) − f (y) − 2f (z), t 2   x + y + z    ≥ N ρ 2f − f (x) − f (y) − f (z) , t , 2 N (f (x + y + z) − f (x) − f (y) − f (z), t)   x + y + z    ≥ N ρ 2f − f (x) − f (y) − f (z) , t 2 in fuzzy normed spaces. Furthermore, we prove the Hyers-Ulam stability of the above additive ρ-functional inequalities in fuzzy Banach spaces.

1. Introduction and preliminaries Katsaras [16] defined a fuzzy norm on a vector space to construct a fuzzy vector topological structure on the space. Some mathematicians have defined fuzzy norms on a vector space from various points of view [10, 20, 44]. In particular, Bag and Samanta [2], following Cheng and Mordeson [8], gave an idea of fuzzy norm in such a manner that the corresponding fuzzy metric is of Kramosil and Michalek type [19]. They established a decomposition theorem of a fuzzy norm into a family of crisp norms and investigated some properties of fuzzy normed spaces [3]. We use the definition of fuzzy normed spaces given in [2, 24, 25] to investigate the Hyers-Ulam stability of additive ρ-functional inequalities in fuzzy Banach spaces. Definition 1.1. [2, 24, 25, 26] Let X be a real vector space. A function N : X × R → [0, 1] is called a fuzzy norm on X if for all x, y ∈ X and all s, t ∈ R, (N1 ) N (x, t) = 0 for t ≤ 0; (N2 ) x = 0 if and only if N (x, t) = 1 for all t > 0; t (N3 ) N (cx, t) = N (x, |c| ) if c 6= 0; (N4 ) N (x + y, s + t) ≥ min{N (x, s), N (y, t)}; (N5 ) N (x, ·) is a non-decreasing function of R and limt→∞ N (x, t) = 1. (N6 ) for x 6= 0, N (x, ·) is continuous on R. The pair (X, N ) is called a fuzzy normed vector space. We know that N (−x, t) = N (x, t) for all x ∈ X by (N3 ). 2010 Mathematics Subject Classification. Primary 46S40, 39B52, 47H10, 39B62, 26E50, 47S40. Key words and phrases. Hyers-Ulam stability; additive ρ-functional inequality; fuzzy normed space. ∗ Corresponding author.

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The other properties of fuzzy normed vector spaces and examples of fuzzy norms are given in [23, 24]. Definition 1.2. [2, 24, 25, 26] Let (X, N ) be a fuzzy normed vector space. A sequence {xn } in X is said to be convergent or converge if there exists an x ∈ X such that limn→∞ N (xn − x, t) = 1 for all t > 0. In this case, x is called the limit of the sequence {xn } and we denote it by N -limn→∞ xn = x. Definition 1.3. [2, 24, 25, 26] Let (X, N ) be a fuzzy normed vector space. A sequence {xn } in X is called Cauchy if for each ε > 0 and each t > 0 there exists an n0 ∈ N such that for all n ≥ n0 and all p > 0, we have N (xn+p − xn , t) > 1 − ε. It is well-known that every convergent sequence in a fuzzy normed vector space is Cauchy. If each Cauchy sequence is convergent, then the fuzzy norm is said to be complete and the fuzzy normed vector space is called a fuzzy Banach space. We say that a mapping f : X → Y between fuzzy normed vector spaces X and Y is continuous at a point x0 ∈ X if for each sequence {xn } converging to x0 in X, then the sequence {f (xn )} converges to f (x0 ). If f : X → Y is continuous at each x ∈ X, then f : X → Y is said to be continuous on X (see [3]). The stability problem of functional equations originated from a question of Ulam [43] concerning the stability of group homomorphisms. The functional equation f (x+y) = f (x)+f (y) is called the Cauchy equation. In particular, every solution of the Cauchy equation is said to be an additive mapping. Hyers [12] gave a first affirmative partial answer to the question of Ulam for Banach spaces. Hyers’ Theorem was generalized by Aoki [1] for additive mappings and by Th.M. Rassias [36] for linear mappings by considering an unbounded Cauchy difference. A generalization of the Th.M. Rassias theorem was obtained by G˘avruta [11] by replacing the unbounded Cauchy difference by a general function in the spirit of Th.M. Rassias’
control 1 1 = f (x) + approach. The functional equation f x+y 2 2 2 f (y) is called the Jensen equation. The stability problems of several functional equations have been extensively investigated by a number of authors and there are many interesting results concerning this problem (see [7, 13, 15, 17, 18, 21, 31, 32, 33, 34, 37, 38, 39, 40, 41, 42]). Park [29, 30] defined additive ρ-functional inequalities and proved the Hyers-Ulam stability of the additive ρ-functional inequalities in Banach spaces and non-Archimedean Banach spaces. We recall a fundamental result in fixed point theory. Let X be a set. A function d : X × X → [0, ∞] is called a generalized metric on X if d satisfies (1) d(x, y) = 0 if and only if x = y; (2) d(x, y) = d(y, x) for all x, y ∈ X; (3) d(x, z) ≤ d(x, y) + d(y, z) for all x, y, z ∈ X. Theorem 1.4. [4, 9] Let (X, d) be a complete generalized metric space and let J : X → X be a strictly contractive mapping with Lipschitz constant L < 1. Then for each given element x ∈ X, either d(J n x, J n+1 x) = ∞ for all nonnegative integers n or there exists a positive integer n0 such that (1) d(J n x, J n+1 x) < ∞, ∀n ≥ n0 ; (2) the sequence {J n x} converges to a fixed point y ∗ of J; (3) y ∗ is the unique fixed point of J in the set Y = {y ∈ X | d(J n0 x, y) < ∞}; 1 (4) d(y, y ∗ ) ≤ 1−L d(y, Jy) for all y ∈ Y . In 1996, G. Isac and Th.M. Rassias [14] were the first to provide applications of stability theory of functional equations for the proof of new fixed point theorems with applications. By using

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fixed point methods, the stability problems of several functional equations have been extensively investigated by a number of authors (see [5, 6, 23, 27, 28, 34, 35]). Lemma 1.5. Let N (x, t) ≥ N (λx, t) for all t > 0. Assume that λ is a fixed real with |λ| < 1. Then x = 0. Proof. Putting

t |λ|n−1

instead of t, we get       t t t N x, n−1 ≥ N |λ|x, n−1 ≥ N x, n |λ| |λ| |λ|

So we get  N (x, t) ≥ N

x,

t |λ|n



for all positive integers n. Passing the limit n → ∞, we get N (x, t) = 1 by (N5 ), and so x = 0 by (N2 ).  In this paper, we introduce and investigate additive ρ-functional inequalities associated with the following additive functional equations f (x + y + z) − f (x) − f (y) − f (z) = 0  x+y 2f + z − f (x) − f (y) − 2f (z) = 0 2   x+y+z − f (x) − f (y) − f (z) = 0 2f 2 

in fuzzy normed spaces. Furthermore, we prove the Hyers-Ulam stability of the additive ρ-functional inequalities in fuzzy Banach spaces. Throughout this paper, assume that X is a real fuzzy normed space with norm N (·, t) and that Y is a fuzzy Banach space with norm N (·, t). 2. Additive ρ-functional inequality I In this section, we investigate the additive ρ-functional inequality (2.1)

N (f (x + y + z) − f (x) − f (y) − f (z), t)       x+y ≥ N ρ 2f + z − f (x) − f (y) − 2f (z) , t 2

in fuzzy normed spaces. Assume that ρ is a fixed real number with |ρ| < 12 . Lemma 2.1. Let f : X → Y be a mapping satisfying (2.1) for all x, y, z ∈ X. Then f : X → Y is additive. Proof. Letting x = y = z = 0 in (2.1), we get N (2f (0), t) ≥ N (2ρf (0), t) and so f (0) = 0 by Lemma 1.5. Replacing y by x and z by −x in (2.1), we get N (f (x) + f (−x), t) ≥ N (2ρ(f (x) + f (−x)), t) and so f (−x) = −f (x) for all x ∈ X by Lemma 1.5. Replacing y by x and z by −2x in (2.1), we get N (f (2x) − 2f (x), t) ≥ N (2ρ(f (2x) − 2f (x)), t) and so f (2x) = 2f (x) for all x ∈ X by Lemma 1.5. Replacing z by −x − y in (2.1), we get N (f (x + y) − f (x) − f (y), t) ≥ N (ρ (f (x + y) − f (x) − f (y)) , t)

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and so f (x + y) = f (x) + f (y) for all x, y ∈ X by Lemma 1.5. Hence f : X → Y is additive.



We prove the Hyers-Ulam stability of the additive ρ-functional inequality (2.1) in fuzzy Banach spaces. Theorem 2.2. Let ϕ : X 3 → [0, ∞) be a function such that there exists an L < 1 satisfying L ϕ (2x, 2y, 2z) , ϕ(0, 0, 0) = 0 2 for all x, y, z ∈ X. Let f : X → Y be a mapping satisfying ϕ(x, y, z) ≤

N (f (x + y + z) − f (x) − f (y) − f (z), t)         t x+y + z − f (x) − f (y) − 2f (z) , t , ≥ min N ρ 2f 2 t + ϕ(x, y, z)
for all x, y, z ∈ X and all t > 0. Then A(x) := N -limn→∞ 2n f 2xn exists for each x ∈ X and defines an additive mapping A : X → Y such that (2.2)

(2.3)

N (f (x) − A(x), t) ≥

(2 − 2L)t (2 − 2L)t + Lϕ(x, x, 0)

for all x ∈ X and all t > 0. Proof. Letting x = y = z = 0 in (2.2), we get N (2f (0), t) ≥ N (2ρf (0), t) and so f (0) = 0 by Lemma 1.5. Replacing y by x and z by 0 in (2.2), we get (2.4)

N (f (2x) − 2f (x), t) ≥

t t + ϕ(x, x, 0)

for all x ∈ X. Consider the set S := {g : X → Y } and introduce the generalized metric on S:   t d(g, h) = inf µ ∈ R+ : N (g(x) − h(x), µt) ≥ , ∀x ∈ X, ∀t > 0 , t + ϕ(x, x, 0) where, as usual, inf φ = +∞. It is easy to show that (S, d) is complete (see [22, Lemma 2.1]). Now we consider the linear mapping J : S → S such that x Jg(x) := 2g 2 for all x ∈ X. Let g, h ∈ S be given such that d(g, h) = ε. Then N (g(x) − h(x), εt) ≥

t t + ϕ(x, x, 0)

for all x ∈ X and all t > 0. Hence     x L  x N (Jg(x) − Jh(x), Lεt) = N 2g − 2h , Lεt = N g −h , εt 2 2 2 2 2 

≥

Lt 2

+

x

x

Lt 2
ϕ x2 , x2 , 0

687

≥

Lt 2 Lt 2

+

L 2 ϕ(x, x, 0)

=

t t + ϕ(x, x, 0)

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for all x ∈ X and all t > 0. So d(g, h) = ε implies that d(Jg, Jh) ≤ Lε. This means that d(Jg, Jh) ≤ Ld(g, h) for all g, h ∈ S. It follows from (2.4) that N f (x) − 2f

x 2





, L2 t ≥

t t+ϕ(x,x,0)

for all x ∈ X and all t > 0. So

L 2.

d(f, Jf ) ≤ By Theorem 1.4, there exists a mapping
A : X → Y satisfying the following: (1) A is a fixed point of J, i.e., A x2 = 12 A(x) for all x ∈ X. Since f : X → Y is odd, A : X → Y is an odd mapping. The mapping A is a unique fixed point of J in the set M = {g ∈ S : d(f, g) < ∞}. This implies that A is a unique mapping satisfying (2.6) such that there exists a µ ∈ (0, ∞) t for all x ∈ X; satisfying N (f (x) − A(x), µt) ≥ t+ϕ(x,x,0) n (2) d(J f, A) → 0 as n → ∞. This implies the equality x N - lim 2n f n = A(x) n→∞ 2 for all x ∈ X; L 1 (3) d(f, A) ≤ 1−L d(f, Jf ), which implies the inequality d(f, A) ≤ 2−2L . This implies that the inequality (2.3) holds. By (2.2),      x y  z  x+y+z n − f − f − f , 2 t N 2n f 2n 2n 2n 2n         y  z  x + y + 2z x n n ≥ min N 2 ρ 2f − f n − f n − 2f n ,2 t , 2n+1 2 2 2 ) t
t + ϕ 2xn , 2yn , 2zn for all x, y ∈ X, all t > 0 and all n ∈ N. Replacing t by 2tn , we get     x y  z   x+y+z n N 2 f −f n −f n −f n ,t 2n 2 2 2      x y  z   x + y + 2z n ≥ min N 2 ρ 2f − f n − f n − 2f n ,t , 2n+1 2 2 2 ) t 2n

t 2n

+

Ln 2n

ϕ (x, y, z)

for all x, y ∈ X, all t > 0 and all n ∈ N. Since limn→∞ t > 0,

t 2n t Ln + 2n ϕ(x,y,z) 2n

= 1 for all x, y ∈ X and all

N (A(x + y + z) − A(x) − A(y) − A(z), t)       x+y ≥ N ρ 2A + z − A(x) − A(y) − 2A(z) , t 2 for all x, y ∈ X and all t > 0. By Lemma 2.1, the mapping A : X → Y is Cauchy additive, as desired. 

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Theorem 2.3. Let ϕ : X 3 → [0, ∞) be a function such that there exists an L < 1 satisfying x y z  ϕ(x, y, z) ≤ 2Lϕ , , , ϕ(0, 0, 0) = 0 2 2 2 for all x, y, z ∈ X. Let f : X → Y be a mapping satisfying (2.2). Then A(x) := N limn→∞ 21n f (2n x) exists for each x ∈ X and defines an additive mapping A : X → Y such that (2 − 2L)t (2.5) N (f (x) − A(x), t) ≥ (2 − 2L)t + ϕ(x, x, 0) for all x ∈ X and all t > 0. Proof. Let (S, d) be the generalized metric space defined in the proof of Theorem 2.2. It follows from (2.4) that f (0) = 0 and t N (f (2x) − 2f (x), t) ≥ t + ϕ(x, x, 0) for all x ∈ X. Consider the linear mapping J : S → S such that Jg(x) := 12 g(2x). N (Jf (x) − f (x), t) ≥

t t+

1 2 ϕ(x, x, 0)

So, we can get d(Jf, f ) ≥ 21 The rest of the proof is similar to the proof of Theorem 2.2.



Lemma 2.4. Let f : X → Y be a mapping satisfying     x+y (2.6) f (x + y + z) − f (x) − f (y) − f (z) = ρ 2f + z − f (x) − f (y) − 2f (z) 2 for all x, y, z ∈ X. Then f : X → Y is additive. Proof. Letting x = y = z = 0 in (2.6), we get −2f (0) = −2ρf (0) and so f (0) = 0. Replacing y by x and letting z = 0 in (2.6), we get f (2x) − 2f (x) = 0 and so f (2x) = 2f (x) for all x ∈ X. Letting z = 0 in (2.6), we get     x+y − f (x) − f (y) = ρ(f (x + y) − f (x) − f (y)) f (x + y) − f (x) − f (y) = ρ 2f 2 and so f (x + y) = f (x) + f (y) for all x, y ∈ X.



Now, we prove the Hyers-Ulam stability of an additive ρ-functional inequality associated with (2.6) in fuzzy Banach spaces. Theorem 2.5. Let ϕ : X 3 → [0, ∞) be a function such that there exists an L < 1 satisfying L ϕ (2x, 2y, 2z) , ϕ(0, 0, 0) = 0 2 for all x, y, z ∈ X. Let f : X → Y be a mapping satisfying ϕ(x, y, z) ≤

(2.7)

N ((f (x + y + z) − f (x) − f (y) − f (z)      x+y t −ρ 2f + z − f (x) − f (y) − 2f (z) , t ≥ 2 t + ϕ(x, y, z)

for all x, y, z ∈ X and all t > 0. Then there exists an unique additive mapping A : X → Y satisfying (2.3).

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Proof. Letting x = y = z = 0 in (2.7), we get N (2(1 − ρ)f (0), t) = 1. So f (0) = 0. Replacing y by x and z by 0 in (2.7), we get t N (f (2x) − 2f (x), t) ≥ (2.8) t + ϕ(x, x, 0) for all x ∈ X. So  x  t t ,t ≥ N f (x) − 2f ≥ x x L 2 t + ϕ( 2 , 2 , 0) t + 2 ϕ(x, x, 0)
for all x ∈ X. Consider the linear mapping J : S → S such that Jg(x) = 2g x2 . N (f (x) − Jf (x), t) ≥

t t+

L 2 ϕ(x, x, 0)

L 2

and so d(f, Jf ) ≤ The rest of the proof is similar to the proof of Theorem 2.2.



X3

→ [0, ∞) be a function such that there exists an L < 1 satisfying x y z  ϕ(x, y, z) ≤ 2Lϕ , , , ϕ(0, 0, 0) = 0 2 2 2 for all x, y, z ∈ X. Let f : X → Y be a mapping satisfying (2.7). Then there exists an unique additive mapping A : X → Y satisfying (2.5). Theorem 2.6. Let ϕ :

Proof. It follows from (2.8) that f (0) = 0 and N (f (2x) − 2f (x), t) ≥

t t + ϕ(x, x, 0)

for all x ∈ X. Consider the linear mapping J : S → S such that Jg(x) = 12 g(2x). N (Jf (x) − f (x), t) ≥

t t+

1 2 ϕ(x, x, 0)

So, we can get d(f, Jf ) ≥ 21 The rest of the proof is similar to the proof of Theorem 2.2.



3. Additive ρ-functional inequality II In this section, we investigate the additive ρ-functional inequality     x+y (3.1) N 2f + z − f (x) − f (y) − 2f (z), t 2       x+y+z ≥ N ρ 2f − f (x) − f (y) − f (z) , t 2 in fuzzy normed spaces. Assume that ρ is a fixed real with |ρ| < 1. Lemma 3.1. Let f : X → Y be a mapping satisfying (3.1) for all x, y, z ∈ X. Then f : X → Y is additive. Proof. Letting x = y = z = 0 in (3.1), we get N (2f (0), t) ≥ N (ρf (0), t) and so f (0) = 0 by Lemma 1.5.

Replacing z by x and letting y = 0 in (3.1), we get N 2f 3x − 3f (x), t = 1 and so 2 3 f ( 3x ) = f (x) for all x ∈ X by Lemma 1.5. 2 2 Replacing y by x and z by x in (3.1), we get N (2f (2x) − 4f (x), t) = 1 and so f (2x) = 2f (x) for all x ∈ X by Lemma 1.5.

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Replacing y by −x and z by y in (3.1), we get N (f (x) + f (−x), t) ≥ N (ρ (f (x) + f (−x)) , t) and so f (−x) = −f (x) for all x ∈ X by Lemma 1.5. Replacing z by −x − y in (3.1), we get N (f (x + y) − f (x) − f (y), t) ≥ N (ρ (f (x + y) − f (x) − f (y)) , t) and so f (x + y) = f (x) + f (y) for all x, y ∈ X by Lemma 1.5. So f : X → Y is additive.



We prove the Hyers-Ulam stability of the additive ρ-functional inequality (3.1) in fuzzy Banach spaces. Theorem 3.2. Let ϕ : X 3 → [0, ∞) be a function such that there exists an L < 1 satisfying   2 3 3 3 ϕ(x, y, z) ≤ Lϕ x, y, z , ϕ(0, 0, 0) = 0 3 2 2 2 for all x, y, z ∈ X. Let f : X → Y be a mapping satisfying     x+y (3.2) + z − f (x) − f (y) − 2f (z), t N 2f 2         x+y+z t ≥ min N ρ 2f − f (x) − f (y) − f (z) , t , 2 t + ϕ(x, y, z)
for all x, y, z ∈ X and all t > 0. Then A(x) := N -limn→∞ 2n f 2xn exists for each x ∈ X and defines an additive mapping A : X → Y such that (3.3)

N (f (x) − A(x), t) ≥

(3 − 3L)t (3 − 3L)t + Lϕ(x, 0, x)

for all x ∈ X and all t > 0. Proof. Letting x = y = z = 0 in (3.2), we get N (2f (0), t) ≥ N (ρf (0), t) and so f (0) = 0 by Lemma 1.5. Replacing y by 0 and z by x, we get     3 t N 2f (3.4) x − 3f (x), t ≥ 2 t + ϕ(x, 0, x) for all x ∈ X. Consider the set S := {g : X → Y } and introduce the generalized metric on S:  d(g, h) = inf µ ∈ R+ : N (g(x) − h(x), µt) ≥

 t , ∀x ∈ X, ∀t > 0 , t + ϕ(x, 0, x)

where, as usual, inf φ = +∞. It is known that (S, d) is complete. Now we consider the linear mapping J : S → S such that   3 2 x Jg(x) := g 2 3 for all x ∈ X.

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ADDITIVE ρ-FUNCTIONAL INEQUALITIES IN FUZZY NORMED SPACES t Let g, h ∈ S be given such that d(g, h) = ε. Then N (g(x) − h(x), εt) ≥ t+ϕ(x,0,x) for all x ∈ X and all t > 0. Hence             3 2 3 2 2 2 2 N (Jg(x) − Jh(x), Lεt) = N g x − h x , Lεt = N g x −h x , Lεt 2 3 2 3 3 3 3

≥

2Lt 3

+

2Lt 3
ϕ 23 x, 0, 32 x

≥

2Lt 3

+

2Lt 3 2L 3 ϕ(x, 0, x)

=

t t + ϕ(x, 0, x)

for all x ∈ X and all t > 0. So d(g, h) = ε implies that d(Jg, Jh) ≤ Lε. This means that d(Jg, Jh) ≤ Ld(g, h) for all g, h ∈ S. It follows from (3.4) that     2 L t 3 x , t ≥ N f (x) − f 2 3 3 t + ϕ(x, 0, x) for all x ∈ X and all t > 0. So d(f, Jf ) ≤ L3 . By Theorem 1.4, there exists a mapping
A : X → Y satisfying the following: (1) A is a fixed point of J, i.e., A 23 x = 23 A(x) for all x ∈ X. Since f : X → Y is odd, A : X → Y is an odd mapping. The mapping A is a unique fixed point of J in the set M = {g ∈ S : d(f, g) < ∞}. This implies that A is a unique mapping satisfying (2.6) such that there exists a µ ∈ (0, ∞) t satisfying N (f (x) − A(x), µt) ≥ t+ϕ(x,0,x) for all x ∈ X; (2) d(J n f, A) → 0 as n → ∞. This implies the equality  n  n  3 2 N - lim f x = A(x) n→∞ 2 3 for all x ∈ X; 1 d(f, Jf ), which implies the inequality d(f, A) ≤ (3) d(f, A) ≤ 1−L inequality (3.3) holds. By (3.2),

L 3−3L .

This implies that the

 n   n    n   n   n  3 2 x+y 2 2 2 2f +z −f x −f y − 2f z , 2 3 2 3 3 3  n  3 t 2   n   n   n   n  3 2 x+y+z 2 3 ≥ min N ρ 2f −f x , t , 2 3 2 3 2 ) t
n
n
n
t + ϕ 32 x, 23 y, 23 z N

for all x, y ∈ X, all t > 0 and all n ∈ N.

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Replacing t by 2tn , we get that  n   n    n   n    n  2 x+y 2 2 3 2 2f x −f y − 2f z ,t N +z −f 2 3 2 3 3 3 ) (     
  n   2 n t 3 n 2 n x+y+z 2 3
≥ min N ρ 2f −f x , t , 2
n 2L n 2 3 2 3 t + ϕ(x, y, z) 3 3 for all x, y ∈ X, all t > 0 and all n ∈ N. Since limn→∞

2 n t 3 2 n 2L n t+ ϕ(x,y,z) 3 3

( )

( ) ( )

= 1 for all x, y ∈ X

and all t > 0,     x+y N 2A + z − A(x) − A(y) − 2A(z), t 2       x+y+z ≥ N ρ 2A − A(x) − A(y) − A(z) , t 2 for all x, y ∈ X and all t > 0. By Lemma 2.1, the mapping A : X → Y is Cauchy additive, as desired. The rest of the proof is similar to the proof of Theorem 2.2.  Theorem 3.3. Let ϕ : X 3 → [0, ∞) be a function such that there exists an L < 1 satisfying   2 2 2 3 x, y, z , ϕ(0, 0, 0) = 0 ϕ (x, y, z) ≤ Lϕ 2 3 3 3 for all x, y,
nz ∈ X.
n
Let f : X → Y be a mapping satisfying (3.2). Then A(x) := N limn→∞ 23 f 32 x exists for each x ∈ X and defines an additive mapping A : X → Y such that (3 − 3L) t (3.5) N (f (x) − A(x), t) ≥ (3 − 3L) t + ϕ (x, 0, x) for all x ∈ X and all t > 0. Proof. Let (S, d) be the generalized metric space defined in the proof of Theorem 3.2. It follows from (3.4) that f (0) = 0 and     t 3 x − 3f (x), t ≥ N 2f 2 t + ϕ(x, 0, x)
for all x ∈ X. Consider the linear mapping J : S → S such that Jg(x) := 23 g 23 x . N (Jf (x) − f (x), t) ≥

t t+

1 3 ϕ(x, x, 0)

So, we can get d(Jf, f ) ≥ 31 The rest of the proof is similar to the proof of Theorem 3.2.



From now on, we investigate another additive ρ-functional inequality     x+y+z (3.6) N 2f − f (x) − f (y) − f (z), t 2       x+y ≥ N ρ 2f + z − f (x) − f (y) − 2f (z) , t 2 in fuzzy normed spaces. Assume that ρ is a fixed real with |ρ| < 12 .

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Lemma 3.4. Let f : X → Y be a mapping satisfying (3.6) for all x, y, z ∈ X. Then f : X → Y is additive. Proof. Letting x = y = z = 0 in (3.6), we get N (f (0), t) ≥ N (2|ρ|f (0), t). and so f (0) = 0 by Lemma 1.5.

Letting x = y = 0 in (3.6), we get N 2f z2 − f (z), t = 1 and so f ( x2 ) = 12 f (x) for all x ∈ X. Replacing z by −x − y in (3.6), we get N (f (−x − y) + f (x) + f (y), t) ≥ N (ρ (f (−x − y) + f (x) + f (y)) , t) and so f (−x − y) = −f (x) − f (y) for all x, y ∈ X. Letting y = 0 in (3.6), we get f (−x) = −f (x) for all x ∈ X. Thus f (x) + f (y) = −f (−x − y) = f (x + y) for all x, y ∈ X. Hence f : X → Y is additive.  4. Additive ρ-functional inequality III In this section, we investigate the additive ρ-functional inequality (4.1)

N (f (x + y + z) − f (x) − f (y) − f (z), t)       x+y+z ≥ N ρ 2f − f (x) − f (y) − f (z) , t 2

in fuzzy normed spaces. Assume that ρ is a fixed real with |ρ| < 1. Lemma 4.1. Let f : X → Y be a mapping satisfying (4.1) for all x, y, z ∈ X. Then f : X → Y is additive. Proof. Letting x = y = z = 0 in (4.1), we get N (2f (0), t) ≥ N (ρf (0), t). and so f (0) = 0 by Lemma 1.5. Replacing z by −x − y in (4.1), we get N (f (x) + f (y) + f (−x − y), t) ≥ N (ρ(f (x) + f (y) + f (−x − y)), t) and so (4.2)

f (x) + f (y) + f (−x − y) = 0

for all x, y ∈ X. Letting y = −x in (4.2), we get f (−x) = −f (x) for all x ∈ X. Thus f (x) + f (y) = −f (−x − y) = f (x + y) for all x, y ∈ X. So f : X → Y is additive.



We prove the Hyers-Ulam stability of the additive ρ-functional inequality (4.1) in fuzzy Banach spaces. Theorem 4.2. Let ϕ : X 3 → [0, ∞) be a function such that there exists an L < 1 satisfying L ϕ (2x, 2y, 2z) , ϕ(0, 0, 0) = 0 2 for all x, y, z ∈ X. Let f : X → Y be a mapping satisfying ϕ(x, y, z) ≤

(4.3)

N (f (x + y + z) − f (x) − f (y) − f (z), t)         x+y+z t ≥ min N ρ 2f − f (x) − f (y) − f (z) , t , 2 t + ϕ(x, y, z)

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for all x, y, z ∈ X and all t > 0. Then A(x) := N -limn→∞ 2n f defines an additive mapping A : X → Y such that (4.4)

N (f (x) − A(x), t) ≥

x 2n




exists for each x ∈ X and

(2 − 2L)t (2 − 2L)t + Lϕ(x, x, 0)

for all x ∈ X and all t > 0. Proof. Letting x = y = z = 0 in (4.3), we get N (2f (0), t) ≥ N (ρf (0), t) and so f (0) = 0 by Lemma 1.5. Replacing y by x and letting z = 0 in (4.3), we get (4.5)

N (f (2x) − 2f (x), t) ≥

t t + ϕ(x, x, 0)

for all x ∈ X. Let (S, d) be the generalized metric space defined in the proof of
Theorem 2.2. Consider the linear mapping J : S → S such that Jg(x) = 2g x2 . Then N (f (x) − Jf (x), t) ≥

t t+ϕ

x x 2, 2,0


≥

t t+

L 2 ϕ(x, x, 0)

So d(f, Jf ) ≤ L2 . The rest of the proof is similar to the proof of the Theorem 2.2.



Theorem 4.3. Let ϕ : X 3 → [0, ∞) be a function such that there exists an L < 1 satisfying x y z  , , , ϕ(0, 0, 0) = 0 ϕ(x, y, z) ≤ 2Lϕ 2 2 2 for all x, y, z ∈ X. Let f : X → Y be a mapping satisfying (4.3). Then A(x) := N limn→∞ 21n f (2n x) exists for each x ∈ X and defines an additive mapping A : X → Y such that (2 − 2L)t (4.6) N (f (x) − A(x), t) ≥ (2 − 2L)t + ϕ(x, x, x) for all x ∈ X and all t > 0. Proof. Let (S, d) be the generalized metric space defined in the proof of Theorem 2.2. It follows from (4.5) that f (0) = 0 and N (f (2x) − 2f (x), t) ≥

t t + ϕ(x, x, 0)

for all x ∈ X. Consider the linear mapping J : S → S such that Jg(x) = 21 g(2x). Then N (f (x) − Jf (x), t) ≥

2t t ≥ 1 2t + ϕ(x, x, 0) t + 2 ϕ(x, x, 0)

So d(f, Jf ) ≤ 21 . The rest of the proof is similar to the proof of the Theorem 4.2.



Lemma 4.4. Let f : X → Y be a mapping satisfying     x+y+z (4.7) f (x + y + z) − f (x) − f (y) − f (z) = ρ 2f − f (x) − f (y) − f (z) 2 for all x, y, z ∈ X. Then f : X → Y is additive.

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Proof. Letting x = y = z = 0 in (4.7), we get −2f (0) = −ρf (0) and so f (0) = 0. Replacing y by x and letting z = 0 in (4.7), we get f (2x) − 2f (x) = 0 and so f (2x) = 2f (x) for all x ∈ X. Letting z = 0 in (4.7), we get     x+y − f (x) − f (y) = ρ(f (x + y) − f (x) − f (y)) f (x + y) − f (x) − f (y) = ρ 2f 2 and so f (x + y) = f (x) + f (y) for all x, y ∈ X.



Now, we prove the Hyers-Ulam stability of an additive ρ-functional inequality associated with (4.7) in fuzzy Banach spaces. Theorem 4.5. Let ϕ : X 3 → [0, ∞) be a function such that there exists an L < 1 satisfying L ϕ(x, y, z) ≤ ϕ (2x, 2y, 2z) , ϕ(0, 0, 0) = 0 2 for all x, y, z ∈ X. Let f : X → Y be a mapping satisfying N ((f (x + y + z) − f (x) − f (y) − f (z)      t x+y+z − f (x) − f (y) − f (z) , t ≥ −ρ 2f 2 t + ϕ(x, y, z) for all x, y, z ∈ X and all t > 0. Then there exists an unique additive mapping A : X → Y satisfying (4.4). (4.8)

Proof. Letting x = y = z = 0 in (4.8), we get N ((2 − ρ)f (0), t) = 1 and so f (0) = 0. Replacing y by x and letting z = 0 in (4.8), we get t (4.9) N (f (2x) − 2f (x), t) ≥ t + ϕ(x, x, 0) for all x ∈ X. Let (S, d) be the generalized metric space defined in the proof of Theorem 2.2.
Let Jg(x) = 2g x2 . Then t t
≥ N (f (x) − Jf (x), t) ≥ x x L t + ϕ 2, 2,0 t + 2 ϕ(x, x, 0) So d(f, Jf ) ≤ L2 . The rest of the proof is similar to the proof of Theorem 4.2.



Theorem 4.6. Let ϕ : X 3 → [0, ∞) be a function such that there exists an L < 1 satisfying x y z  , , , ϕ(0, 0, 0) = 0 ϕ(x, y, z) ≤ 2Lϕ 2 2 2 for all x, y, z ∈ X. Let f : X → Y be a mapping satisfying (4.8). Then there exists an unique additive mapping A : X → Y satisfying (4.6). Proof. It follows from (4.9) that f (0) = 0 and N (f (2x) − 2f (x), t) ≥

t t + ϕ(x, x, 0)

for all x ∈ X. Let (S, d) be the generalized metric space defined in the proof of Theorem 2.2. Consider the linear mapping J : S → S such that Jg(x) = 21 g(2x). Then 2t t N (f (x) − Jf (x), t) ≥ ≥ 1 2t + ϕ(x, x, 0) t + 2 ϕ(x, x, 0)

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So d(f, Jf ) ≤ 21 . The rest of the proof is similar to the proof of Theorem 4.2.



References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24] [25] [26] [27] [28] [29] [30]

T. Aoki, On the stability of the linear transformation in Banach spaces, J. Math. Soc. Japan 2 (1950), 64–66. T. Bag and S.K. Samanta, Finite dimensional fuzzy normed linear spaces, J. Fuzzy Math. 11 (2003), 687–705. T. Bag and S.K. Samanta, Fuzzy bounded linear operators, Fuzzy Sets and Systems 151 (2005), 513–547. L. C˘ adariu and V. Radu, Fixed points and the stability of Jensen’s functional equation, J. Inequal. Pure Appl. Math. 4, no. 1, Art. ID 4 (2003). L. C˘ adariu and V. Radu, On the stability of the Cauchy functional equation: a fixed point approach, Grazer Math. Ber. 346 (2004), 43–52. L. C˘ adariu and V. Radu, Fixed point methods for the generalized stability of functional equations in a single variable, Fixed Point Theory and Applications 2008, Art. ID 749392 (2008). I. Chang and Y. Lee, Additive and quadratic type functional equation and its fuzzy stability, Results Math. 63 (2013), 717–730. S.C. Cheng and J.M. Mordeson, Fuzzy linear operators and fuzzy normed linear spaces, Bull. Calcutta Math. Soc. 86 (1994), 429–436. J. Diaz and B. Margolis, A fixed point theorem of the alternative for contractions on a generalized complete metric space, Bull. Amer. Math. Soc. 74 (1968), 305–309. C. Felbin, Finite dimensional fuzzy normed linear spaces, Fuzzy Sets and Systems 48 (1992), 239–248. P. G˘ avruta, A generalization of the Hyers-Ulam-Rassias stability of approximately additive mappings, J. Math. Anal. Appl. 184 (1994), 431–436. D.H. Hyers, On the stability of the linear functional equation, Proc. Nat. Acad. Sci. U.S.A. 27 (1941), 222–224. D.H. Hyers, G. Isac and Th.M. Rassias, Stability of Functional Equations in Several Variables, Birkh¨ auser, Basel, 1998. G. Isac and Th.M. Rassias, Stability of ψ-additive mappings: Appications to nonlinear analysis, Internat. J. Math. Math. Sci. 19 (1996), 219–228. S. Jung, Hyers-Ulam-Rassias Stability of Functional Equations in Mathematical Analysis, Hadronic Press lnc., Palm Harbor, Florida, 2001. A.K. Katsaras, Fuzzy topological vector spaces II, Fuzzy Sets and Systems 12 (1984), 143–154. H. Kim, M. Eshaghi Gordji, A. Javadian and I. Chang, Homomorphisms and derivations on unital C ∗ -algebras related to Cauchy-Jensen functional inequality, J. Math. Inequal. 6 (2012), 557–565. H. Kim, J. Lee and E. Son, Approximate functional inequalities by additive mappings, J. Math. Inequal. 6 (2012), 461–471. I. Kramosil and J. Michalek, Fuzzy metric and statistical metric spaces, Kybernetica 11 (1975), 326–334. S.V. Krishna and K.K.M. Sarma, Separation of fuzzy normed linear spaces, Fuzzy Sets and Systems 63 (1994), 207–217. J. Lee, C. Park and D. Shin, An AQCQ-functional equation in matrix normed spaces, Results Math. 27 (2013), 305–318. D. Mihet¸ and V. Radu, On the stability of the additive Cauchy functional equation in random normed spaces, J. Math. Anal. Appl. 343 (2008), 567–572. M. Mirzavaziri and M.S. Moslehian, A fixed point approach to stability of a quadratic equation, Bull. Braz. Math. Soc. 37 (2006), 361–376. A.K. Mirmostafaee, M. Mirzavaziri and M.S. Moslehian, Fuzzy stability of the Jensen functional equation, Fuzzy Sets and Systems 159 (2008), 730–738. A.K. Mirmostafaee and M.S. Moslehian, Fuzzy versions of Hyers-Ulam-Rassias theorem, Fuzzy Sets and Systems 159 (2008), 720–729. A.K. Mirmostafaee and M.S. Moslehian, Fuzzy approximately cubic mappings, Inform. Sci. 178 (2008), 3791– 3798. C. Park, Fixed points and Hyers-Ulam-Rassias stability of Cauchy-Jensen functional equations in Banach algebras, Fixed Point Theory and Applications 2007, Art. ID 50175 (2007). C. Park, Generalized Hyers-Ulam-Rassias stability of quadratic functional equations: a fixed point approach, Fixed Point Theory and Applications 2008, Art. ID 493751 (2008). C. Park, Additive ρ-functional inequalities and equations, J. Math. Inequal. 9 (2015), 17–26. C. Park, Additive ρ-functional inequalities in non-Archimedean normed spaces, J. Math. Inequal. 9 (2015), 397–407.

697

JOONHYUK JUNG et al 684-698

J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 22, NO.4, 2017, COPYRIGHT 2017 EUDOXUS PRESS, LLC

ADDITIVE ρ-FUNCTIONAL INEQUALITIES IN FUZZY NORMED SPACES

[31] C. Park, Y. Cho and M. Han, Stability of functional inequalities associated with Jordan-von Neumann type additive functional equations, J. Inequal. Appl. 2007, Art. ID 41820 (2007). [32] C. Park, K. Ghasemi, S. G. Ghaleh and S. Jang, Approximate n-Jordan ∗-homomorphisms in C ∗ -algebras, J. Comput. Anal. Appl. 15 (2013), 365-368. [33] C. Park, A. Najati and S. Jang, Fixed points and fuzzy stability of an additive-quadratic functional equation, J. Comput. Anal. Appl. 15 (2013), 452–462. [34] C. Park and Th.M. Rassias, Fixed points and generalized Hyers-Ulam stability of quadratic functional equations, J. Math. Inequal. 1 (2007), 515–528. [35] V. Radu, The fixed point alternative and the stability of functional equations, Fixed Point Theory 4 (2003), 91–96. [36] Th.M. Rassias, On the stability of the linear mapping in Banach spaces, Proc. Amer. Math. Soc. 72 (1978), 297–300. ˇ ankov´ [37] L. Reich, J. Sm´ıtal and M. Stef´ a, Singular solutions of the generalized Dhombres functional equation, Results Math. 65 (2014), 251–261. [38] S. Schin, D. Ki, J. Chang and M. Kim, Random stability of quadratic functional equations: a fixed point approach, J. Nonlinear Sci. Appl. 4 (2011), 37–49. [39] S. Shagholi, M. Bavand Savadkouhi and M. Eshaghi Gordji, Nearly ternary cubic homomorphism in ternary Fr´echet algebras, J. Comput. Anal. Appl. 13 (2011), 1106–1114. [40] S. Shagholi, M. Eshaghi Gordji and M. Bavand Savadkouhi, Stability of ternary quadratic derivation on ternary Banach algebras, J. Comput. Anal. Appl. 13 (2011), 1097–1105. [41] D. Shin, C. Park and Sh. Farhadabadi, On the superstability of ternary Jordan C ∗ -homomorphisms, J. Comput. Anal. Appl. 16 (2014), 964–973. [42] D. Shin, C. Park and Sh. Farhadabadi, Stability and superstability of J ∗ -homomorphisms and J ∗ -derivations for a generalized Cauchy-Jensen equation, J. Comput. Anal. Appl. 17 (2014), 125–134. [43] S. M. Ulam, A Collection of the Mathematical Problems, Interscience Publ. New York, 1960. [44] J.Z. Xiao and X.H. Zhu, Fuzzy normed spaces of operators and its completeness, Fuzzy Sets and Systems 133 (2003), 389–399. Joonhyuk Jung, Junehyeok Lee Mathematics Branch, Seoul Science High School, Seoul 110-530, Korea E-mail address: [email protected]; [email protected] George A. Anastassiou Department of Mathematical Sciences, University of Memphis, Memphis, TN 38152, USA E-mail address: [email protected] Choonkil Park Research Institute for Natural Sciences, Hanyang University, Seoul 04763, Korea E-mail address: [email protected]

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An Approach to Separability of Integrable Hamiltonian System Hai Zhang a a School

of Mathematical and Computational Sciences, Anqing Normal University, Anqing, 246133, P.R.China E-mail: [email protected]

Abstract Directly from Benenti’s theorem, which characterizes separability with one single Killing tensor, we adopt an algorithm to execute the task of separability test. The algorithm is applied to generalized quartic and quintic polynomial potentials as well as some multi-separable systems on (pseudo)-Euclidean spaces. It yields many well-known integrable systems in a unified and straight manner, in contrast to some complicated techniques employed in the literature to derive them. Keywords: completely integrable system; separable system; Killing two-tensor; H´enon-Heiles systems

1. Introduction Finite dimensional completely integrable system has always attracted much attention. Recently authors adopted various methods or perspectives to investigate them. Prominent of all, are the separability theory of Hamilton-Jacobi equation [1], the approach of Lax representations [2], and the bi-Hamiltonian theory [3, 4] among others (see e.g. the references above and therein). For a given Hamilton system finding canonical separation coordinates is very non-trivial. The above approaches can partly solve this problem. Sklyanin developed a method based on a Lax pair [5]. The separable coordinates are obtained from the spectrum of the Lax operator. Another approach is based on the existence of a bi-Hamiltonian representation [6, 4]. The separable variables, called Darboux-Nijenhuis coordinates, are related with the recursion operator constructed from the Poisson pencil. For a generic system there is no intrinsic criterion of the existence of a Lax or bi-Hamiltonian formulation. Benenti [7, 8] has developed an intrinsic characterization for a Hamiltonian system being separable (see Theorem 1). It is based on geometric properties of the Killing tensors corresponding to the first integrals of the system. We can make a comparison of these approaches to the separability of the Hamilton-Jacobi equation. While the technique based on the Lax or bi-Hamiltonian formulation may be more effective in studying particular examples (for which such formulation has been found beforehand), the Benenti approach is more rigorous from the mathematical point of view. Though integrability does not necessarily imply separability, the separable class constitute the vital examples among all integrable systems. Directly from Benenti’s theorem, we can adopt a strategy to cope with the problem of separability test. We present this method as an executable algorithm, which are especially applicable to families of Hamiltonian systems containing some numeric constants. In this paper we employ this algorithm to test several natural systems, recovering some known models obtained by other approaches such as Painlev´e analysis or differential Galois theory [9]. This paper is organised as follows. In Section 2, some basic concepts in Killing tensors method of H-J separability are reviewed, then based on it we suggest an executable algorithm to make concrete separability test. The algorithm is, in Section 3, applied to test several potentials, including inhomogeneous quartic polynomial potential, homogeneous quintic potential, as well as some multi-separable systems on Euclidean and Minkowski planes. These will yield many well-known integrable systems in a unified and straight manner, in contrast to complicated techniques employed in the literature. The last section is devoted to some concluding remarks.

Preprint submitted to Journal of Computational Analysis and Applications

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2. The Geometric Method to Variables Separation and an Executable Algorithm We briefly recall some necessary facts about the separation of variables method, considered in the framework of symplectic geometry. Let (M, ω) be a symplectic manifold with symplectic form ω. Then the Poisson bivector is P = ω −1 . Note here we view ω (and P ) as transformation taking vector field to one-form (and vice-versa, respectively). It is well known that the Schouten bracket vanishes, [P, P ] = 0. Let (q, p), q = (q1 ,∑ . . . , qn ), p = (p1 , . . . , pn ) be the (local) canonical coordinates on M, then the n ∂ . The Hamiltonian vector field corresponding to a smooth function Poisson bivector is P = i=1 ∂q∂ i ∧ ∂p i H = H(q, p) is defined as XH = P dH. The triple (M, P, H) is called a Hamiltonian system. In this paper we will focus on Hamiltonian system in the setting of Riemannian geometry. That is to say, the phase space is the cotangent bundle T ∗ M of some (pseudo)-Riemannian manifold (M, g). We remind that we are finding separable coordinates related to the original physics coordinates (q, p) via a point transformation. In the setting of Riemannian geometry a natural Hamiltonian H = T + V takes the form as follows n ∑ 1 ij G (q) pi pj + V (q) (2.1) H= 2 i,j=1 where Gij is the inverse of metric g and V (q) the potential. The classical Hamilton-Jacobi equation reads n ∑ 1 ij G ∂i W ∂j W + V = E 2 i,j=1

(2.2)

where E is the constant of conserved energy (Hamiltonian). It is a first-order partial differential equation of the unknown W . Definition 1. The Hamiltonian H (2.1) is separable in the canonical variable (q, p), if the Hamilton-Jacobi equation (2.2) admits a complete integral of additive form ∑n W (q, α) = Wi (q i , α), (2.3) i=1

where α = (α1 , . . . , αn ) are integration constants, such that det separable variables.

[

∂2W ∂qi ∂αj

]

̸= 0. The variables (q, p) are called

It is well known that the n first integrals obtained by solving the Hamilton-Jacobi equation (2.2) are either quadratic or linear in momenta and thus correspond to Killing 2-tensors or Killing vectors, respectively. Definition 2. A Killing tensor K of valence p defined on (M, g) is a symmetric (p, 0)-tensor satisfying the Killing tensor equation [K, G] = 0, (2.4) where [ , ] denotes the Schouten bracket. All Killing p-tensors constitute a vector space Kp (M ). For manifold of constant curvature its dimension attains the maximum, see e.g. [10]. Separability of the natural Hamiltonian H = T + V depends on the Killing 2-tensor of the underlying manifold (M, g). This idea, due to Eisenhart, has been extensively exploited by many authors, see e.g. [11, 12]. The intrinsic criterion given by Benenti [7, 8] allows one to characterize separability by a single Killing 2-tensor (orthogonal case), or a Killing 2-tensor together with an abelian algebra of Killing vectors (non-orthogonal case). Here we focus on the orthogonal case since it is more common. Theorem 1 (Benenti). A Hamiltonian H = T + V is separable in some orthogonal coordinates if and only if there exists a Killing 2-tensor K with pointwise simple and real eigenvalues, orthogonally integrable eigenvectors and such that d (K dV ) = 0. (2.5)

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The (0, 2)-tensor K is called characteristic Killing tensor. On a Riemannian manifold it can be viewed as (2, 0)- or (1, 1)-tensor by lowering or raising indices via metric g or G. Here K in (2.5) is seen as a (1, 1)-tensor which takes a one-form to another. In local coordinates (2.5) entails the following one-form is closed, ∑ K dV = gij Kjl ∂l V dq i . (2.6) i,j,l

This theorem elegantly and intrinsically characterizes the orthogonal separability of the natural Hamiltonian (2.1). Often in the literature one is faced with a general system with some parameters involved in the Hamiltonian. Many sophisticated methods have to be invented and applied to identify the rare cases which are separable, or integrable. We will revisit some of these examples in later sections. From the Benenti’s theorem 1 we come up with an approach to deal with the problem of searching for separable case of parameters, presented by an Algorithm as below: Algorithm. Let H be a natural Hamiltonian with potential V (q; ai ) defined on some pseudo-Riemannian manifold (M, g), where ai ’s are some constant parameters. The special values of parameters, that guarantees the system H is H-J separable, are achieved during execution of the algorithm. Begin. Step 1. For the pseudo-Riemannian manifold (M, g), using the Killing tensor equation (2.4) to calculate the general Killing 2-tensor K. All of them constitute a vector space K2 (M ) of dimension d. The ∑d expression of a general Killing 2-tensor is K = i=1 Ci Ki , where (Ki ) is the basis, Ci ∈ R. If d < dim(M ), then by theorems due to Kalnins & Miller [11] there exists no separable potential — Stop. Essentially, this step is a pure problem of differential geometry. Step 2. The Killing tensor K obtained above is of covariant (2, 0)-type. Using metric g, transform it to a b which can be regarded as an endomorphism of the cotangent bundle T ∗ M . (1, 1)-tensor K By abuse of notation we use K to denote the new tensor below. Note that in matrix form K is always b is not so in general. symmetric, K Step 3. Insert the (1, 1)-tensor into the core equation (2.5). The vanishing of form d(KdV ) entails the vanishing of all its coefficients. Thus a system of equations involving variables qk and constants ai , Cj follows, which are usually (or can be transformed to) polynomials of qk . Simplify this system of equations, eventually we obtain algebraic equations of the parameters ai , Cj only. Solve the system of algebraic equations. The obtained solutions are candidates of separable cases. Step 4. Substitute the Cj ’s back into the general expression of Killing tensor. Calculate its eigenvalues and eigenvectors. Check whether they satisfy the additional condition in Benenti’s theorem. These gives the complete set of all separable cases. Step 5 (Optional). For a separable case, by using the eigenvalues and eigenvectors obtained in step 4, we can figure out which concrete coordinate system permits the separability of the corresponding system. (see e.g. [12]) End.

3. Applications In this section, we first review the Killing vectors (tensors) of E2 (see e.g. [13]), then we use them to make a detailed analysis of several systems involved with some constants.

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In the situation of E2 we will write the familiar (x, y) for (q 1 , q 2 ), and (px , py ) for (p1 , p2 ), here (x, y) is the usual Cartesian coordinate. The space of Killing vectors has dim K1 (E2 ) = 3 with a basis [13] being ∂x , ∂y (two translations), where we adopt the notation ∂x = K1 = ∂x2 , K4 = K6 =

∂ ∂x ,

∂y =

K2 = ∂y2 ,

∂ ∂y .

y∂x − x∂y (rotation)

(3.1)

The second space K2 (E2 ) has a basis [13] K3 = ∂x ∂y + ∂y ∂x (= G) ,

−2y ∂x2 + x ∂x ∂y + x ∂y ∂x , K5 = −2x ∂y2 y 2 ∂x2 + x2 ∂y2 − xy ∂x ∂y − xy ∂y ∂x .

+ y ∂ x ∂y + y ∂ y ∂x ,

(3.2)

So the expression for a general Killing 2-tensor is K=

∑6 i=1

Ci Ki = (C6 y 2 − 2C4 y + C1 ) ∂x2 + (C6 x2 − 2C5 x + C2 ) ∂y2 + + (−C6 xy + C4 x + C5 y + C3 ) (∂x ∂y + ∂y ∂x )

(3.3)

or, in matrix form ( (Kij ) =

C6 y 2 − 2C4 y + C1 −C6 xy + C4 x + C5 y + C3

−C6 xy + C4 x + C5 y + C3 C6 x2 − 2C5 x + C2

) (3.4)

where Ci are constants. At last, we mention a special non-separable situation, that is, (NS)

C1 = C2 ,

C3 = C4 = C5 = C6 = 0.

In such a case the matrix K = C1 I2 , where I2 denotes the identity matrix. It is not simple as it admits two coincident eigenvalues. This means the characteristic tensor K does not exist, hence the system is not separable. Such a special case arises several times during our arguments later. 3.1. System with a General Quartic Potential We shall use these general results to several specified systems defined on E2 . In this section we consider a system with a quartic polynomial potential, whose Hamiltonian is H=

1 2 (p + p2y ) + (λ x2 + µ y 2 ) + (c x4 + b x2 y 2 + a y 4 ) 2 x

(3.5)

in which a, b, c, λ, µ are constants. Note that the system (3.5) is called Yang-Mills-type system in [16]. In general, the system with arbitrary parameters are non-integrable and display chaotic behaviour. After applying the celebrated Painlev´e analysis or differential Galois theory, several special cases for values of constants are identified, which turn out to be integrable (see e.g. [14, 15, 16]). These cases are given by (i) (ii) (iii) (iv) (v)

b = 0, all other parameters are arbitrary, a : b : c = 1 : 2 : 1, λ and µ arbitrary, a : b : c = 1 : 6 : 1, λ = µ, a : b : c = 1 : 12 : 16, λ = 4µ, a : b : c = 1 : 6 : 8, λ = 4µ, (proved to be the only non-separable case below)

Cases (ii)–(v) are well-known integrable H´enon-Heiles systems [17]. For each of these cases there exists a second integral of motion K independent of H [17]. Remark 1. One may note that the four cases given above are not symmetric for λ, µ whereas the Hamiltonian is so. Actually there are not only five integrable cases as above, but more. The additional cases

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(iv)′ a : b : c = 16 : 12 : 1, (v)′ a : b : c = 8 : 6 : 1,

λ = µ/4, λ = µ/4,

are symmetric (thus equivalent) to the cases (iv) and (v), respectively. We can make an assumption a ≤ c to eliminate these isomorphisms. Our main result in this subsection is the following Theorem 2. For Hamiltonian system with quartic potential (3.5), there exist exact four cases of values of constants, which guarantee the corresponding H-J equation being additively separable. These cases are exactly the first four cases (i)—(iv) in the list above. This shows case (v) is the only integrable, but non-separable case. ∑2 Proof. Notice now (2.6) reads K dV = i,j=1 Kij ∂j V dq i , whose explicit expressions is messy. After taking exterior differentiation d(K dV ) = Z dx ∧ dy, its coefficient is a polynomial of x and y, which after collecting the same entries reads Z = (24aC6 − 12bC6 ) y 3 x + (12bC6 − 24cC6 ) yx3 + (−12aC4 + 16bC4 ) y 2 x +(−16bC5 + 12cC5 ) yx2 + (−2bC4 + 24cC4 ) x3 + (−24aC5 + 2bC5 ) y 3 +(−12aC3 + 2bC3 ) y 2 + (−4bC1 + 4bC2 + 8µC6 − 8λC6 ) xy + (−2bC3 + 12cC3 ) x2

(3.6)

+(−2µC4 + 8λC4 ) x + (−8µC5 + 2λC5 ) y − 2µC3 + 2λC3 . The vanishing of two-form d(KdV ) means its coefficient Z vanishes. All the parameters a, b, c, λ, µ, Ci in (3.6) are constants. In turn Z vanishes identically entails all its coefficients of x, y vanishes. A system of algebraic equations follows, C3 (λ − µ) = C4 (4λ − µ) = C5 (λ − 4µ) = 0

(3.7a)

C3 (6a − b) = C3 (b − 6c) = 0 C4 (b − 12c) = C4 (3a − 4b) = 0

(3.7b) (3.7c)

C5 (12a − b) = C5 (4b − 3c) = 0 C6 (2a − b) = C6 (b − 2c) = 0 (C1 − C2 )b + 2C6 (λ − µ) = 0

(3.7d) (3.7e) (3.7f)

This is the system of algebraic equations we want to solve in detail. First we notice when b = 0, the parameters C1 − C2 ̸= 0, Ci = 0, i = 3, 4, 5, 6, directly solve the above system. The matrix corresponding to Killing tensor K has distinct eigenvalues C1 , C2 . Hence this is a separable case, which corresponds to case (i) in our list. To proceed we will always assume b ̸= 0 below. It can be seen b ̸= 0 implies a, c ̸= 0. Otherwise constants Ci are exactly in the non-separable situation (NS). To solve the system (3.7), we observe that (3.7b) and (3.7e) imply C3 (a − c) = C6 (a − c) = 0 (3.8) Based on this observation one can make classification as below: • a ̸= c. Equations (3.8) imply that C3 = C6 = 0. Substituting this into (3.7), one has C4 (4λ − µ) = C5 (λ − 4µ) = 0 C4 (b − 12c) = C4 (3a − 4b) = 0 C5 (12a − b) = C5 (4b − 3c) = 0

(3.9)

C1 − C2 = 0

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The second equation implies C4 (a − 16c) = 0. As a − 16c ̸= 0 (otherwise c/a = 1/16 < 1), it holds that C4 = 0. We claim C5 ̸= 0, otherwise the constants Ck are in non-separable situation (NS). Combing all the results above, we arrive at a : b : c = 1 : 12 : 16,

λ = 4µ,

which recovers the case (iv) in the list. • a = c. Again we have C4 (a − 16c) = 0, which reduces to C4 a = 0. As a ̸= 0 hence C4 = 0. Similarly C5 = 0. Substituting C4 = C5 = 0, a = c into the system (3.7), it reduces to C3 (λ − µ) = 0 C3 (6a − b) = C6 (2a − b) = 0 (C1 − C2 )b + 2C6 (λ − µ) = 0

(3.10)

We discuss its possible solutions: – a = c, λ = µ. The condition λ = µ reduces the system (3.10) further to C3 (6a − b) = C6 (2a − b) = 0,

C1 − C2 = 0.

One of C3 and C6 should be non-zero (otherwise the situation (NS) arise again), which implies a : b : c = 1 : 6 : 1, or a : b : c = 1 : 2 : 1. They corresponds to the case (iii) and (ii), respectively. – a = c, λ ̸= µ. The system (3.10) is reduced to C3 = C6 (2a − b) = (C1 − C2 )b + 2C6 (λ − µ) = 0 Once more C6 ̸= 0 to avoid the situation (NS), which gives a : b : c = 1 : 2 : 1. It is case (ii) in the list. This completes the proof of Theorem 2.

Remark 2. One could use the Killing tensor to figure out the coordinate system in which the H-J equation separates. For example, let us consider the case (ii). According to Step 4 in Algorithm, taking b = 2a, c = a ̸= 0 back into the original system (3.7) one find the following to be a solution of the system (3.7): C1 =

µ−λ , C2 = C3 = C4 = C5 = 0, C6 = 1 a

Note that (µ − λ) is in general not zero. The characteristic tensor (3.4) turns out to be ( 2 µ−λ ) y + a −xy K= . −xy x2 Its characteristic equation is Λ2 − (x2 + y 2 +

µ−λ 2 µ−λ )Λ + x =0 a a

(3.11)

(3.12)

or in equivalent form x2 ay 2 + = 0. Λ aΛ − (µ − λ)

(3.13)

For the case of λ ̸= µ, the above equation (3.13) defines just well-known elliptic-hyperbolic coordinates in the Euclidean plane. The eigenvalues λ1 , λ2 , i.e. the solutions of the equation (3.12) or (3.13), are the variables of separation for the dynamical system. Hence we conclude the system is separable in the elliptic coordinates (λ1 , λ2 ), determined by µ−λ a .

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For the case of λ = µ, the solutions of (3.12) are λ1 = 0, λ2 = x2 + y 2 , one of which is constant. This implies the system separates in degenerated elliptic coordinates. The Hamiltonian is H=

1 2 (p + p2y ) + λ(x2 + y 2 ) + a(x4 + 2x2 y 2 + y 4 ) 2 x

with potential V = λ r2 + a r4 depending on r only. It is easy to see the system separates in the standard polar coordinates (r, θ). 3.2. System with a Homogeneous Quintic Potential Next we consider a family of Hamiltonian systems with a quintic polynomial potential, H=

1 2 (p + p2y ) + ay 5 + b y 3 x2 + c yx4 , 2 x

(3.14)

where a, b, c are scalar constants. Note they are not the most general quintic potential. Our main result regarding this potential is the following Theorem 3. For the Hamiltonian (3.14), there exists exact 3 cases of parameters for the corresponding H-J equation to be additively separable. These cases are: (i) b = c = 0, a arbitrary;

(ii) a : b : c = 16 : 16 : 3;

(iii) a : b : c = 1 : 10 : 5. (p2x

Remark 3. Case (i) is trivial in that it corresponds to Hamiltonian H = + p2y )/2 + ay 5 , which trivially separates in Cartesian coordinates. Case (ii) has already appeared in literature [14] where the authors obtained it by using the (weak) Painlev´e method. Case (iii) appeared in Perelomov’s book [18, p.81] Proof of Theorem 3. We apply the algorithm again, now to the potential (3.14). Using the general Killing tensor (3.4), it follows that the 2-form d(KdV ) = Zdx ∧ dy, with coefficient Z given by Z

= (−6bC4 + 32cC4 )yx3 + (−20aC4 + 20bC4 )y 3 x + (21bC6 − 28cC6 )y 2 x3 + (35aC6 − 14bC6 )y 4 x + (−6bC1 + 6bC2 )y 2 x + (−6bC3 + 12cC3 )yx2 + (−27bC5 + 12cC5 )y 2 x2 + (−4cC1 + 4cC2 )x3 + (−35aC5 + 2bC5 )y 4 + 7cx5 C6 + (−20aC3 + 2bC3 )y 3 − 11cx4 C5 .

The form d(KdV ) vanishes entails Z also vanishes. Again, Z is a polynomial of variable x, y, hence all of its coefficients are zero. Thus we arrive at a system of algebraic equations c(C1 − C2 ) = b(C1 − C2 ) = 0, C3 (b − 2c) = C3 (10a − b) = C4 (3b − 16c) = 0, C4 (a − b) = C5 (9b − 4c) = C5 (35a − 2b) = 0, C6 (3b − 4c) = C6 (5a − 2b) = 0, cC5 = cC6 = 0,

(3.15)

We analyze the solution of this family: • c = 0. Substitute this into the system (3.15) to produce b(C1 − C2 ) = 0 C3 (10a − b) = C4 (a − b) = 0, C5 (35a − 2b) = C6 (5a − 2b) = 0, C3 b = C4 b = C5 b = C6 b = 0,

(3.16)

One easily sees that b is also zero. (Otherwise the new system leads to the (NS) case). Hence the above system (3.16) further reduces to C3 a = C4 a = C5 a = C6 a = 0. For any a, it admits a solution C1 − C2 ̸= 0, C3 = C4 = C5 = C6 = 0. Thus we have a separability corresponding to the case (i). In fact we can obviously see this from the original potential (3.14). In the special case of b = c = 0, the Hamiltonian is additively itself, implying it separates in the canonical Cartesian system.

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• c ̸= 0.

The system (3.15) can be simplified to be C1 − C2 = C5 = C6 = 0, C3 (b − 2c) = C3 (10a − b) = 0, C4 (3b − 16c) = C4 (a − b) = 0,

One can see one of C3 , C4 is not zero. (Otherwise C1 − C2 = C3 = C4 = C5 = C6 = 0 — dissatisfies the basic theorem 1). So there exists two possibilities: – C3 ̸= 0, which gives a : b : c = 1 : 10 : 5 corresponding to case (iii). – C4 = ̸ 0, which gives a : b : c = 16 : 16 : 3 corresponding to case (ii). We thus reproduce all the cases in the theorem.

3.3. Multi-Separable Potentials on Euclidean and Minkowski Planes We now apply our Algorithm to identify some multi-separable systems which are defined on (pseudo)Euclidean spaces. We remind that a Hamiltonian system is multi-separable if it separates in several distinct coordinate systems. Theorem 4. For the system 1 2 b (p + p2y ) + x2 + ay 2 + 2 (3.17) 2 x x with potential defined on Euclidean plane E2 , where a, b are two constants, there exists exact three cases of parameters such that H is multi-separable. They are given by H=

(i) a = 1/4, b = 0;

(ii) a = 1, b arbitrary;

(iii) a = 4, b arbitrary. nd

Remark 4. Here we consider the multi-separability, i.e. 2 -order super-integrability for the system (3.17). For any integer a = k 2 , k ∈ N, it admits an additional first integral which is a k th -order polynomial in momenta [19], implying its (higher order) super-integrability. For such potentials there exist much more super-integrable cases than multi-separable ones. Note that case (iii) is the celebrated Smorodinsky-Winternitz I potential [20], thus by using our Algorithm we reproduce this system quite straightforwardly. Proof of Theorem 4. According to the algorithm we apply K (3.4) to dV where V = x2 + ay 2 + b/x2 . After exterior derivative one has d (K dV ) = Z dx ∧ dy (3.18) with the coefficient Z given by 2 ( · 4(a − 1)x5 y C6 + (1 − 4a)x4 y C5 x4 ) +(4 − a)x5 C4 + (1 − a)x4 C3 + 3byC5 + 3bC3 Z=

(3.19)

In Z’s expression, Ck , a, b are constants. One notice Z is not polynomial in (x, y), but rational functions. Nevertheless, we can transform it to be a polynomial as below. The form d(KdV ) vanishes equivalents to Z ≡ 0, which, in turn, equivalents to the vanishing of the polynomial Z˜ = Z · x4 /2. So we obtain a system of algebraic equations bC3 = bC5 = 0, C4 (a − 4) = C3 (a − 1) = 0, C6 (a − 1) = C5 (4a − 1) = 0

(3.20)

Since C1 , C2 do not arise in the equations, all of Ck = 0 except that C1 − C2 ̸= 0 solves the system above. This implies the Hamiltonian is separable in the Cartesian coordinates. For the system to be multi-separability, it suffices to find another solution linearly independent of the trivial solution given above. A new solution to equations (3.20) exists if and only if one of the following cases occurs:

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• C6 ̸= 0 ⇒ a = 1, b arbitrary; • C4 ̸= 0 ⇒ a = 4, b arbitrary; • C5 ̸= 0 ⇒ a = 14 , b = 0; • C3 ̸= 0 ⇒ a = 1, b = 0. Observe that the last case is only a subcase of the first case. Thus we obtain exact three multi-separable cases, corresponding to cases (ii), (iii), (i) in the theorem, respectively. The next configuration space we consider is a Minkowski case M2 whose metric is g = dx2 − dy 2 . We compare the two planes M2 and E2 . Both are of constat curvature (zero), hence the dimensions of vector spaces of their Killing tensor attain the maxima: K1 (M2 ) , K2 (M2 ) are of dimension three and six, respectively. Nevertheless, the basis of Killing tensors (hence the entire spaces) are not identical. The Minkowski M2 has the basis of Killing vectors (compare with (3.1)) ∂x , ∂y (two translations),

y∂x + x∂y

(Minkowski “rotation”)

(3.21)

The basis of Killing 2-tensors are the following (compare with (3.2)) K1 = ∂x2 , K4 = K6 =

K2 = ∂y2 ,

K3 = ∂x ∂y + ∂y ∂x ,

2y ∂x2 + x ∂x ∂y + x ∂y ∂x , y 2 ∂x2 + xy ∂x ∂y + xy ∂y ∂x

K5 = 2x ∂y2 + y ∂x ∂y + y ∂y ∂x , +

(3.22)

x2 ∂y2 ,

Carrying out an analysis similar to that for the Euclidean plane (Theorem 4), we arrive at Theorem 5. For the system H=

1 2 b (p − p2y ) + x2 + ay 2 + 2 2 x x

(3.23)

with potential defined on Minkowski M2 , a, b are constants, there exists exact three cases such that H is multi-separable. They are given by (i) a = −1/4, b = 0;

(ii) a = −1, b arbitrary;

(iii) a = −4, b arbitrary.

4. Concluding Discussions Based on Benenti’s classical theorem 1, we have suggested an Algorithm and applied it to detect H-J separability of several families of two-dimensional natural systems. This method has the advantage of having a clear procedure and not depending on intricate techniques which can be seen in lots of literatures, thus executable in a computer-like environment. However, the applications we make here are merely preliminary. There are several directions one can take into account to improve and extend its scope. For example, one may consider some nontrivial (pseudo)Riemannian spaces such as spaces of constant curvature S n , H n etc., or surfaces of revolution. These manifolds are easy to handle as their Killing tensor are much investigated. The crucial task in step 1 in our Algorithm is thus solved. Another line is to generalize the potentials under discussion to more general ones, which may be involved some arbitrary functions. This can greatly enlarge the families of separable systems. Proceeding the analysis as above may yield some well-known or novel models. It is natural that the calculations are much more complicated, with the aid of computer symbolic system sometimes being a necessity. Acknowledgments. The author would like to thank Profs. Qing Chen and Dafeng Zuo for encouragement and support.

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References [1] V. I. Arnol’d, Mathematical Methods of Classical Mechanics, Springer-Verlag, New-York 1988. [2] J. Moser, Finitely many mass points on the line under the influence of an exponential potential—an integrable system, in Dynamical Systems, Theory and Applications (1974, Rencontres, BattelleRes. Inst., Seattle, Wash.), Lecture Notes in Phys., Vol. 38, Berlin, Springer, 1975, 467–497. [3] M. Blaszak, Multi-Hamiltonian Theory of Dynamical Systems, Springer-Verlag, New York 1998. [4] A. Ibort, F. Magri and G. Marmo, Bihamiltonian structures and St¨ ackel separability, J. Geom. Phys. 33 (2000) 210–228. [5] E. K. Sklyanin, Separation of variables. New trends, Progr. Theor. Phys. Suppl. 118 (1995), 35–60. [6] G. Falqui, F. Magri, M. Pedroni and J. P. Zubelli, A Bi-Hamiltonian Theory for Stationary KdV Flows and Their Separability, Regular and Chaotic Dynamics, 2000, V.5, 33–51. [7] S. Benenti. Orthogonal separable dynamical systems. In: O.Kowalski, D.Krupka (Eds.) Proceedings of the 5th International Conference on Differential Geometry and Its Applications, Silesian University at Opava, August 24–28, 1992. Diff. Geom. Appl. 1993. V. 1. P. 163–184. [8] S. Benenti, Intrinsic characterization of the variable separation in the Hamilton-Jacobi equation, J. Math. Phys. 38 (1997), 6578–6602. [9] J. J. Morales-Ruiz, Differential Galois Theory and Non-Integrability of Hamiltonian Systems, Progress in Mathematics, vol. 179, Birkh¨ auser Verlag, Basel (1999) [10] G. Thompson, Killing tensors in spaces of constant curvature, J. Math. Phys. 27 (1986), 2693–2699. [11] E. G. Kalnins and W. Miller, Jr., Killing tensors and variable separation for Hamilton-Jacobi and Helmholtz equations, SIAM J. Math. Anal. 11 (1980), 1011–1026. [12] R. G. McLenaghan, R. G. Smirnov and D. The, An extension of the classical theory of algebraic invariants to pseudo-Riemannian geometry and Hamiltonian mechanics, Journal of Mathematical Physics 45, 1079 (2004) [13] A. T. Bruce, R. G. McLenaghan and R. G. Smirnov, A systematic study of the Toda lattice in the context of the Hamilton-Jacobi theory, Z. angew. Math. Phys. ZAMP 52 (2001) 171–190 [14] A. Ramani, B. Dorozzi, and B. Grammaticos, Painlev´e Conjecture Revisited, Phys. Rev. Lett., (1982), 49 1539–1541. [15] T. Bountis, H. Segur, and F. Vivaldi, Integrable Hamiltonian systems and the Painlev´e property, Phys. Rev. A 25 1257–1264, (1982) [16] M. E. Sansaturio, I. Vigo-Aguiar and J. M. Ferr´ andiz, Non-integrability of some Hamiltonian systems in polar coordinates, J. Phys. A: Math. Gen. 30 (1997) 5869 [17] R. Conte, M. Musette and C. Verhoeven, Explicit integration of the H´enon-Heiles Hamiltonians. J. of Nonlin. Math. Phys., (2005) 12(Supplement 1): 212–227. [arXiv:nlin.SI/0412057] [18] A.M. Perelomov, Integrable Systems of Classical Mechanics and Lie Algebras, Birk¨ auser Verlag, Basel (1990) [19] J. F. Cari˜ nena, G. Marmo and M. F. Ra˜ nada, Non-symplectic symmetries and bi-Hamiltonian structures of the rational harmonic oscillator, J. Phys. A 35, L679–L686 (2002). [20] T.I. Friˇs, V. Mandrosov, Y.A. Smorodinsky, M. Uhl´ıˇr and P. Winternitz, On higher symmetries in quantum mechanics, Phys. Lett., 16 (1965), 354-356

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Cross-entropy for generalized hesitant fuzzy sets and their use in multi-criteria decision making Jin Han Park, Hee Eun Kwark Department of Applied Mathematics, Pukyong National University, Pusan 608-737, South Korea [email protected] (J.H. Park), [email protected] (H.E. Kwark) Young Chel Kwun∗ Department of Mathematics, Dong-A University, Pusan 604-714, South Korea [email protected]

Abstract In this paper, the cross-entropy for generalized hesitant fuzzy sets (GHFSs) is developed by integrating the cross-entropy for intuitionistic fuzzy sets (IFSs) and hesitant fuzzy sets (HFSs). First, several measurement formulas are discussed and their properties are studied. Then, two approaches, which are based on the developed generalized hesitant fuzzy cross-entropy, are proposed for solving multi-criteria decision making (MCDM) problems under an generalized hesitant fuzzy environment. Finally, an example is provided to illustrate the practicality and effectiveness of the developed approaches.

1

Introduction

The cross-entropy measures are mainly used to measure the discrimination information, and then it is an important measure in decision making, pattern recognition and other real-world problems. Lots of studies on this issue have been extended and developed to fuzzy and its extended environments. For instance, Vlachos and Sergiadis [14] introduced the concepts of discrimination information and cross-entropy for intuitionistic fuzzy sets (IFSs), and revealed the connection between the notions of entropies for fuzzy sets and IFSs in terms of fuzziness and intuitionism. Hung and Yang [6] constructed J-divergence of IFSs and introduced some useful distance and similarity measures between two IFSs, and applied them to clustering analysis and pattern recognition. Based ∗ Corresponding

author: [email protected] (Y.C. Kwun)

1

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on which, Xia and Xu [17] proposed some cross-entropy and entropy formulas for IFSs and applied them to group decision making. Ye [21] proposed a method of fault diagnosis based on the vague cross-entropy. He [22] also introduced the cross-entropy for IFSs and interval-valued intuitionistic fuzzy sets (IVIFSs) and utilized then to solve multi-criteria decision making (MCDM) problems. Wang and Li [15] provided two improved methods for solving MCDM problems, which were based on the cross-entropy for IFSs. Hung et al. [5] introduced the discrimination information and cross-entropy for IFSs and also used them to improve the fault diagnosis of turbine problems. Mao et al. [9] introduced the crossentropy and entropy measures for IFSs. Zang and Yu [28] constructed a series of mathematical programming models, which were based on an interval-valued intuitionistic fuzzy cross-entropy, in order to determine the criteria weights and applied them to MCDM problems. Xia and Xu [17] proposed two methods for determining the optimal weights of criteria and developed two pairs of entropy and cross-entropy measures for intuitionistic fuzzy values. The relationships among the entropy, cross-entropy and similarity measures have also attracted many attentions. For example, Liu [8] gave the axiomatic definitions of entropy, distance measure, and similarity measure of fuzzy sets and discussed their basic relations. Zeng and Li [25] discussed the relationship between the similarity measure and the entropy of interval-valued fuzzy sets. Zang and Jiang [27] proposed the entropy and cross-entropy for IVIFSs and discussed the connections among some important information measures. Xu and Xia [19] introduced the concepts of entropy and cross-entropy for hesitant fuzzy sets (HFSs), analyzed the relationships among the entropy, cross-entropy and similarity measures, and developed two multi-attribute decision making methods. Qjan et al. [10], recently, introduced the concept of generalized hesitant fuzzy sets (GHFSs), extending the element of HFSs from real numbers to intuitionistic fuzzy values, which can arise in group decision making problem. GHFS is fit for the situation when decision maker have a hesitation among several possible memberships with uncertainties. GHFS can reflect the human’s hesitance more objectively than other extensions of fuzzy set (IFS, IVIFS and HFS), and thus it is necessary to develop some theories about GHFSs. In this paper, we discuss the cross-entropy for generalized hesitant information. To do this, Section 2 reviews some related preliminaries such as IFSs, HFSs and GHFSs. In Section 3, we propose some cross-entropy formulas for generalized hesitant fuzzy elements, obtain some important conclusions, and provide an example to illustrate the application of cross-entropy in MCDM problem. Finally, Section 4 gives the concluding remarks.

2

Basic concepts

Intuitionistic fuzzy sets introduced by Atanassov [1] have been proven to be highly useful to deal with uncertainty and vagueness.

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Definition 1. [1] Let X be ordinary non-empty set. An intuitionistic fuzzy set (IFS) A in X is defined as A = {⟨x, µA (x), νA (x)⟩|x ∈ X},

(1)

where µA , νA : X → [0, 1] denote, respectively, the membership and nonmembership functions of A with the condition: 0 ≤ µA (x) + νA (x) ≤ 1 for all x ∈ X. For an IFS A, πA (x) = 1−µA (x)−νA (x) represents the degree of hesitation or intuitionistic index of x to A. For a fuzzy set, the degree of hesitation πA (x) = 0. Thus for each x, µA (x) and νA (x) define an interval [µA (x), 1 − νA (x)]. This interval is the vague value of value set by Gau and Buethrer [4] (Bustince and Burillo [3] proved that vague sets are equivalent to IFSs). Further, the interval can also represent an interval-valued fuzzy set [10]. Hence Xu [18] concluded that IFSs are also equivalent to interval-valued fuzzy sets, and replaced Eq. (1) with A = {⟨x, [µA (x), 1 − νA (x)]⟩|x ∈ X}.

(2)

The ordered pair α(x) = (µα (x), να (x)) is referred to an intuitionistic fuzzy value (IFV) [18], where µα (x), να (x) ∈ [0, 1] and µα (x) + να (x) ≤ 1. Associated with the degree of hesitation, an IFV can also be equivalently denoted by α(x) = (µα (x), να (x), πα (x)), where µα (x), να (x), πα (x) ∈ [0, 1] and µα (x) + να (x) + πα (x) = 1. In the rest of this paper, for a certain x in X, IFV a = (µ, ν, π) is abbreviated as a = (µ, ν) when no misunderstanding raises. Since an IFV represent an interval, an interval [µ, 1 − ν] in [0, 1] will be directly transformed into (µ, ν). Definition 2. [5, 6, 14, 17, 22] Let α1 = (µα1 , να1 ) and α2 = (µα2 , να2 ) be IFVs, then the cross-entropy α1 and α2 , denoted as CE(α1 , α2 ), should satisfy the following properties: (1) CE(α1 , α2 ) ≥ 0; (2) CE(α1 , α2 ) = 0 if α1 = α2 ; (3) CE(α1c , α2c ) = CE(α1 , α2 ), where αic = (ναi , µαi ) is the complement of αi (i = 1, 2). In the following, some intuitionistic fuzzy cross-entropy and symmetric intuitionistic fuzzy cross-entropy formulas are reviewed. Vlachos and Sergiadis [14] developed CE1 (α1 , α2 ) = µα1 ln and

2µα1 2να1 + να1 ln , µα1 + µα2 να1 + να2

(3)

(

µα1 ln µα1 + µα2 ln µα2 µα + µα2 µα + µα2 − 1 ln 1 2 2 2 ) να1 ln να1 + να2 ln να2 να1 + να2 να1 + να2 + − ln . (4) 2 2 2

CE2 (α1 , α2 ) = 2

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Hung and Yang [6] defined ( µα + µα2 µα + µα2 µα1 ln µα1 + µα2 ln µα2 − 1 ln 1 CE3 (α1 , α2 ) = 2 2 2 2 να1 ln να1 + να2 ln να2 να1 + να2 να1 + να2 + − ln 2 2 2 ) πα1 ln να1 + να2 ln να2 να1 + να2 να1 + να2 + − ln . (5) 2 2 2 Ye [22] proposed µα1 + 1 − να1 2(µα1 + 1 − να1 ) log2 2 2 + µα1 − να1 + µα2 − να2 2(να1 + 1 − µα1 ) να1 + 1 − µα1 log2 . + 2 2 − µα1 + να1 − µα2 + να2

CE4 (α1 , α2 ) =

(6)

Hung et al. [5] developed 2µα1 2να1 + να1 log2 µα1 + µα2 να1 + να2 2πα1 +πα1 log2 . π α1 + π α2

CE5 (α1 , α2 ) = µα1 log2

(7)

Xia and Xu [17] proposed ( q ( )q µα1 + µqα2 ναq 1 + ναq 2 1 µα1 + µα2 CE6 (α1 , α2 ) = − + 1 − 21−q 2 2 2 )q ( )q ) ( q q π + πα2 π α1 + π α2 να1 + να2 + α1 − , (8) − 2 2 2 where 1 < q ≤ 2. For the symmetric property, it is necessary to modify Eqs. (3)-(8) to obtain a symmetric discrimination information measures for IFVs ([11, 26]): CE∗k (α1 , α2 ) = CEk (α1 , α2 ) + CEk (α2 , α1 ), k = 1, 2, . . . , 6.

(9)

The hesitant fuzzy set [12, 13], as a generalization of fuzzy set, permits the membership degree of an element to a set presented as several possible values between 0 and 1, which can better describe the situations where people have hesitancy in providing their preferences over objects in process of decision making. Definition 3. [12, 13] Given a fixed set X, a hesitant fuzzy set (HFS) on X in terms of function h is that when applied to X returns a subset of [0, 1], which can be represented as the following mathematical symbol: E = {⟨x, h(x)⟩|x ∈ X},

(10)

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where h(x) is a set of the some values in [0, 1], denoting the possible membership degrees of the element x ∈ X to the set E. For convenience, Xia and Xu [16] called h(x) a hesitant fuzzy element (HFE) and the set of all HFEs is denoted by HFES. Definition 4. [19] Let h1 and h2 be two HFEs, then the cross-entropy of h1 and h2 , denoted as CE(h1 , h2 ), should satisfy the following properties: (1) CE(h1 , h2 ) ≥ 0; σ(i) σ(i) (2) CE(h1 , h2 ) = 0 if and only if h1 = h2 for all i = 1, 2, . . . , l. Based on Definition 4, l = l(h1 ) = l(h2 ) and denote the number of elements in h1 and h2 . The elements are arranged in increasing order in h1 and h2 , σ(i) σ(i) respectively, and h1 (i = 1, 2, . . . , l(h1 )) and h2 (i = 1, 2, . . . , l(h2 )) are the ith smallest values in h1 and h2 , respectively. Xu and Xia [19] constructed several cross-entropy for HFEs: CE1 (h1 , h2 ) l 1 ∑ = lT i=1

(

σ(i)

(1 + qh1

σ(i)

σ(i)

) ln(1 + qh1

σ(i)

σ(i)

σ(i)

) + (1 + qh2 2 σ(i)

σ(i)

) ln(1 + qh2

)

σ(l−i+1)

)) (1 + q(1 − h1 2 2 2 σ(l−i+1) σ(l−i+1) )) )) ln(1 + q(1 − h2 (1 + q(1 − h2 σ(l−i+1) )) + × ln(1 + q(1 − h1 2 σ(l−i+1) σ(l−i+1) ) 2 + q(1 − h1 + 1 − h2 − 2 ) σ(l−i+1) σ(l−i+1) + 1 − h2 ) 2 + q(1 − h1 × ln , (11) 2 −

2 + qh1

+ qh2

ln

2 + qh1

+ qh2

+

where T = (1 + q) ln(1 + q) − (2 + q)(ln(2 + q) − ln 2) and q > 0. CE2 (h1 , h2 )

( l σ(i) σ(l−i+1) p σ(i) σ(l−i+1) p ∑ 1 ) + (1 − h2 ) (h1 )p + (h2 )p (1 − h1 = + (1 − 21−p )l i=1 2 2 ( ) ) ) ( σ(i) p σ(i) σ(l−i+1) σ(l−i+1) p h1 + h2 + 1 − h2 1 − h1 − (12) + , p > 1. 2 2

For the symmetric property, it is necessary to modify Eqs. (11) and (12) to obtain a symmetric discrimination information measure for HFEs: CE∗k (h1 , h2 ) = CEk (h1 , h2 ) + CEk (h2 , h1 ), k = 1, 2.

(13)

Note that Eqs. (11) and (12) are all defined under the assumption that two HFEs are of the same length. If the corresponding HFEs are not equal in length, 5

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then the shorter one should be extended to be the same size as the longer one by adding the same value repeatedly.

3

Generalized hesitant fuzzy sets and their crossentropy measures

3.1

Generalized hesitant fuzzy sets

During the evaluating process, several possible memberships of an alternative satisfying a certain criterion may be not only crisp values but also interval values in [0, 1]. In order to handle this kind of assessment in decision making, Qjan et al. [10] extended HFSs by using IFSs to modify Definition 3. Definition 5. [10] Given a set of N membership functions: M = {αi = (µαi , ναi )|0 ≤ µαi , ναi ≤ 1, µαi + ναi ≤ 1, i = 1, 2, . . . , N }

(14)

˜ M , is the generalized hesitant fuzzy set (GHFS) associated with M , that is h defined as follows: ˜ M (x) = ∪(µ ,ν )∈M {(µα (x), να (x))}. h i i αi αi

(15)

Note that HFSs, IFSs and fuzzy sets are special cases of GHFSs redefined here. In fact, if µαi + ναi = 1, for i = 1, 2, . . . , N , then GHFSs reduce to HFSs. If N = 1 or union of N IFSs, i.e. ∪N i=1 αi , in Eq. (14) is convex set in [0, 1], then GHFSs reduce to IFSs. If N = 1 and µαN + ναN = 1, then GHFSs reduce to FSs. Thus GHFSs are not only the generalization of HFSs but also the generalized representation of fuzzy sets, IFSs and HFSs. For the sake of ˜ convenience, given a certain x ∈ X, α represents an IFS in h(x). Notice that α ˜ M (x), abbreviated as h(x), ˜ is represented an interval as well. Similar to [16], h is called a generalized hesitant fuzzy element (GHFE) and the set of all GHFEs is denoted by GHFES. ˜ be the number of elements of a GHFE h. ˜ In most cases of two Let l(h) ˜ ˜ ˜ ˜ 2 may be different, GHFEs h1 and h2 , the numbers of elements of h1 and h ˜ ˜ ˜ ˜ 2 )}. To operate i.e. l(h1 ) ̸= l(h2 ), and for convenience, let l = max{l(h1 ), l(h correctly, we should extend the shorter ones, until both of them have the same length when we compare them. To extend the shorter one, the best way is to add the same values several times in it. In fact, we can extend the shorter one by adding any values in it. The selection of this value mainly depends on the decision makers’ risk preferences. Optimists anticipate desirable outcomes and may add the maximum value, while pessimists expect unfavorable outcomes and ˜ 1 and h ˜2 may add the minimum value. In this paper, we assume the GHFEs h should have the same length l when we compare them. Some useful operations on GHFEs are as follows: ˜ h ˜ 1 and h ˜ 2 be three GHFEs and λ > 0, then Definition 6. [10] Let h, 6

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˜ 1 ∪h ˜2 = ∪ ˜ (1) h ˜ 2 {α1 ∪α2 } = ∪α1 ∈h ˜ 1 ,α2 ∈h ˜ 2 {(max{µα1 , µα2 }, min{να1 , να2 })}; α1 ∈h1 ,α2 ∈h ˜ ˜ (2) h1 ∩h2 = ∪α1 ∈h˜ 1 ,α2 ∈h˜ 2 {α1 ∩α2 } = ∪α1 ∈h˜ 1 ,α2 ∈h˜ 2 {(min{µα1 , µα2 }, max{να1 , να2 })}; ˜ c = ∪ ˜ {αc } = ∪ ˜ {(να , µα )}; (3) h α∈h α∈h ˜ ˜2 = ∪ ˜ (4) h1 ⊕h ˜ 2 {α1 ⊕α2 } = ∪α1 ∈h ˜ 1 ,α2 ∈h ˜ 2 {(µα1 +µα2 −µα1 µα2 , να1 να2 )}; α1 ∈h1 ,α2 ∈h ˜ ˜ (5) h1 ⊗ h2 = ∪α1 ∈h˜ 1 ,α2 ∈h˜ 2 {α1 ⊗ α2 } = ∪α1 ∈h˜ 1 ,α2 ∈h˜ 2 {(µα1 µα2 , να1 + να2 − να1 να2 )}; ˜ = ∪ ˜ {λα} = ∪ ˜ {(1 − (1 − µα )λ , ν λ )}; (6) λh α α∈h α∈h λ ˜ = ∪ ˜ {αλ } = ∪ ˜ {(µλ , 1 − (1 − να )λ )}. (7) h α α∈h α∈h ˜ i (i = 1, 2, . . . , n) be a collection of GHFEs, and let Definition 7. Let h n GHFWA : GHFES → GHFES, if ˜ 1, h ˜ 2, . . . , h ˜ n ) = w1 h ˜ 1 ⊕ w2 h ˜ 2 ⊕ · · · ⊕ wn h ˜ n, GHFWAw (h

(16)

˜ i (i = 1, 2, . . . , n) with where w = ∑ (w1 , w2 , . . . , wn )T is the weight vector of h n wi ≥ 0 and i=1 wi = 1, then GHFWA is called the generalized hesitant fuzzy weighted averaging (GHFWA) operator. Based on operations (4)-(7) of GHFEs described in Definition 6, we can derive the following result. ˜ i = ∪ ˜ {αi } (i = 1, 2, . . . , n) be a collection of GHFEs, Theorem 1. Let h αi ∈hi T ˜ i (i = 1, 2, . . . , n) with and w = (w∑ be the weight vector of h 1 , w2 , . . . , wn ) n wi ≥ 0 and i=1 wi = 1. Then the aggregated value, by using the GHFWA operator, is also a GHFE, and ˜1, h ˜ 2, . . . , h ˜ n) GHFWAw (h {( )} n n ∪ ∏ ∏ wi wi = 1− (1 − µαi ) , ναi . i=1

˜ 1 ,α2 ∈h ˜ 2 ,...,αn ∈h ˜n α1 ∈ h

(17)

i=1

Theorem 1 can be proved by using the mathematical induction and then the process is omitted here. ˜ i (i = 1, 2, . . . , n) be a collection of GHFEs, and let Definition 8. Let h n GHFWG : GHFES → GHFES, if ˜ wn , ˜1, h ˜ 2, . . . , h ˜ n) = h ˜ w1 ⊗ h ˜ w2 ⊗ · · · ⊗ h GHFWGw (h n 1 2

(18)

˜ i (i = 1, 2, . . . , n) with where w = ∑ (w1 , w2 , . . . , wn )T is the weight vector of h n wi ≥ 0 and i=1 wi = 1, then GHFWG is called the generalized hesitant fuzzy weighted geometric (GHFWG) operator. ˜ i = ∪ ˜ {αi } (i = 1, 2, . . . , n) be a collection of GHFEs, Theorem 2. Let h αi ∈hi T ˜ i (i = 1, 2, . . . , n) with and w = (w∑ be the weight vector of h 1 , w2 , . . . , wn ) n wi ≥ 0 and i=1 wi = 1. Then the aggregated value, by using the GHFWG

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operator, is also a GHFE, and ˜ 1, h ˜2, . . . , h ˜ n) GHFWGw (h {( n )} n ∪ ∏ ∏ i µw (1 − ναi )wi . = αi , 1 − i=1

˜ 1 ,α2 ∈h ˜ 2 ,...,αn ∈h ˜n α1 ∈ h

(19)

i=1

Theorem 2 can be also proved by using the mathematical induction and then the process is omitted here.

3.2

Cross-entropy measures of GHFEs

˜ 1, h ˜ 2 ∈ GHFES and CE : GHFES × GHFES → R, then the Definition 9. Let h ˜ ˜ 2 , denoted as CE(h ˜ 1, h ˜ 2 ), should satisfy the following cross-entropy of h1 and h properties: ˜1, h ˜ 2 ) ≥ 0; (1) CE(h ˜ ˜ 2 , then CE(h ˜ 1, h ˜ 2 ) = 0; (2) If h1 = h c ˜c ˜ i defined in ˜ ˜ ˜ ˜ c is the complement of h (3) CE(h1 , h2 ) = CE(h1 , h2 ), where h i Definition 6. On the basis of Definition 9, we can construct several cross-entropy for GHFEs:   ) ∑ ∑ ( 1 1 2µ α 1 ˜1, h ˜ 2) =   CE1 (h µα1 log2 ˜ 1) ˜ 2) µα1 + µα2 l(h l( h ˜1 ˜2 α1 ∈ h α2 ∈ h   ) ∑ ∑ ( 1 1 2ν α 1  ; (20)  + να1 log2 ˜1) ˜ 2) να1 + να2 l(h l( h ˜ ˜ α1 ∈h1

α2 ∈h2

v   u ) p ( u ∑ ∑ 1 2µα1 p ˜ 1, h ˜ 2) = u   1 t CE2 (h µα1 log2 ˜ 1) ˜ 2) µα1 + µα2 l(h l( h ˜ ˜ α1 ∈ h 1 α2 ∈h2 v   u ( ) p u ∑ ∑ 1 1 2ν u α p 1  , (21)  +t να1 log2 ˜1) ˜ 2) να1 + να2 l(h l( h ˜ ˜ α1 ∈h1

α2 ∈h2

where p ≥ 1; ˜ 1, h ˜2) CE3 (h

  ) ∑ ∑ ( µα + 1 − να 1 1 2(µ + 1 − ν ) α1 α1 1 1   = log2 ˜ 1) ˜2) 2 2 + µα1 − να1 + µα2 − να2 l(h l( h ˜1 ˜2 α1 ∈h α2 ∈ h   ) ∑ ∑ ( 1 − µα + να 2(1 − µ + ν ) 1 1 α1 α1 1 1   ; (22) + log2 ˜ 1) ˜ 2) 2 2 − µα1 + να1 − µα2 + να2 l(h l(h ˜1 α1 ∈ h

˜2 α2 ∈h

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˜ 1, h ˜2) CE4 (h v   u ( ) p u ∑ ∑ 1 µα1 + 1 − να1 2(µα1 + 1 − να1 ) u p  1  log2 =t ˜ 1) ˜ 2) 2 2 + µα1 − να1 + µα2 − να2 l(h l( h ˜ ˜ α1 ∈ h 1 α2 ∈h2 v   u ( ) p u ∑ ∑ 1 1 − µ + ν 1 2(1 − µ + ν ) u α1 α1 α1 α1 p   , (23) +t log2 ˜ 1) ˜2) 2 2 − µα1 + να1 − µα2 + να2 l(h l(h ˜1 α1 ∈ h

˜2 α2 ∈ h

where p ≥ 1; ˜ 1, h ˜2) CE5 (h

   ( )q ) ∑ ∑ ( µqα + µqα 1 µ + µ 1 1 α α 1 2 1 2    = − ˜1) ˜ 2) 1 − 21−q l(h 2 2 l( h ˜1 ˜2 α1 ∈ h α2 ∈h   ( ( )q ) ∑ ∑ ναq + ναq 1 1 ν + ν α α 1 2 1 2   , (24) + − ˜ 1) ˜ 2) 2 2 l(h l( h ˜ ˜ α1 ∈h1

α2 ∈h2

where 1 < q ≤ 2; ˜ 1, h ˜2) CE6 (h v   u ( q ( )q ) p u q ∑ ∑ + µ 1 µ 1 1 µ + µ u α2 α1 α1 α2 p   =t − ˜ 1) ˜2) 1 − 21−q l(h 2 2 l(h ˜i ˜2 α1 ∈ h α2 ∈ h v  p u u ∑ ( ναq 1 + ναq 2 ( να + να )q ) 1 ∑ 1 1 u p 1 2   , (25) +t − ˜1) ˜ 2) 1 − 21−q l(h 2 2 l(h ˜ ˜ α1 ∈hi

α2 ∈h2

where p ≥ 1 and 1 < q ≤ 2. For the symmetric property, it is necessary to modify Eqs. (20)-(25) to a symmetric discrimination information measure for GHFEs as follows: ˜ 1, h ˜ 2 ) = CEk (h ˜1, h ˜ 2 ) + CEk (h ˜2, h ˜ 1 ), k = 1, 2, . . . , 6. CE∗k (h

(26)

˜ i (i = 1, 2, 3) and h ˜ be three patterns and a sample. Example 1. Let h ˜ 1 = {(0.5, 0.4), (0.6, 0.3)}, h ˜2 = They are denoted by GHFEs as follows: h ˜ ˜ {(0.4, 0.5), (0.8, 0.1)}, h3 = {(0.3, 0.4), (0.7, 0.2)} and h = {(0.5, 0.4), (0.7, 0.2)}. ˜ which pattern does this sample h ˜ most probably belong to Given the sample h, ? 9

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For convenience, let p = q = 2. By (20)-(25), we have ˜ 1 , h) ˜ = 0.0275, CE∗1 (h ∗ ˜ ˜ CE2 (h1 , h) = 0.2601, ˜ 1 , h) ˜ = 0.0241, CE∗3 (h ˜ 1 , h) ˜ = 0.2609, CE∗ (h

˜ 2 , h) ˜ = 0.0976, CE∗1 (h ∗ ˜ ˜ CE2 (h2 , h) = 0.4285, ˜ 2 , h) ˜ = 0.0838, CE∗3 (h ˜ 2 , h) ˜ = 0.4298, CE∗ (h

˜ 3 , h) ˜ = 0.0693; CE∗1 (h ∗ ˜ ˜ CE2 (h3 , h) = 0.3982; ˜ 3 , h) ˜ = 0.0541; CE∗3 (h ˜ 3 , h) ˜ = 0.3856; CE∗ (h

= 0.0300,

= 0.1000,

= 0.0800;

4 ˜ 1 , h) ˜ CE∗5 (h ∗ ˜ ˜ CE6 (h1 , h)

= 0.0239,

4 ˜ 2 , h) ˜ CE∗5 (h ∗ ˜ ˜ CE6 (h2 , h)

= 0.0707,

4 ˜ 3 , h) ˜ CE∗5 (h ∗ ˜ ˜ CE6 (h3 , h)

= 0.0620.

From this data, the proposed symmetric discrimination information measures CE∗k (k = 1, 2, 3, 4, 5, 6) show the same classification according to the principle of the minimum degree of symmetric discrimination information measure for ˜ belongs to the pattern h ˜ 1. GHFEs. Thus, the sample h

4

Two MCDM approaches based on the crossentropy measures of GHFEs

For a MCDM problem, let X = {x1 , x2 , . . . , xm } be a set of m alternatives, and Y = {y1 , y2 , . . . , yn } be a set of n criteria, whose∑weight vector is w = n (w1 , w2 , . . . , wn )T , satisfying wj > 0, j = 1, 2, . . . , n and j=1 wj = 1, where wj denotes the importance degree of the criterion yj . Decision makers evaluate the performance of alternatives with respect to criteria based on their knowledge and experience. One decision maker could give several evaluation values. However, in the case where two or more decision makers give the same value, it is counted only once. The performance of the alternative xi with respect to the criterion yj is measured by a GHFE e˜ij = {βij = (µβij , νβij )|βij ∈ e˜ij }, where µβij indicates the degree that the alternative xi satisfies the criterion yj , νβij indicates the degree that the alternative xi does not satisfy the criterion yj , such that µβij ∈ [0, 1], νβij ∈ [0, 1], µβij + νβij ≤ 1 (i = 1, 2, . . . , m; j = 1, 2, . . . , n). All e˜ij (i = 1, 2, . . . , m; j = 1, 2, . . . , n) are contained in the generalized hesitant fuzzy decision matrix E = (˜ eij )m×n (see Table 1). Table 1: The generalized hesitant fuzzy decision matrix x1 x2 .. . xm

y1 e˜11 e˜21 .. . e˜m1

y2 e˜12 e˜22 .. . e˜m2

··· ··· ··· .. . ···

yn e˜in e˜2n .. . e˜mn

In what follows, we develop two approaches to multi-criteria decision making under generalized hesitant fuzzy environment. 10

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Approach I. Step 1. Normalize the performance values and then construct the normalized generalized hesitant fuzzy decision matrix. If all the criteria yj (j = 1, 2, . . . , n) are of the same type, then the performance values do not need normalization. Whereas there are, generally, benefit criteria (the bigger the performance values the better) and cost criteria (the smaller the performance values the better) in multi-criteria decision making, in such case, we may transform the performance values of the cost type into the performance values of the benefit type. Then, E = (˜ eij )m×n can be transformed ˜ ij )m×n , where into the matrix F = (h { ∪βij ∈d˜ij {βij }, for benefit criterion yj ; ˜ ij = ∪ h ˜ ij {αij } = c αij ∈h ∪βij ∈d˜ij {βij }, for cost criterion yj , i = 1, 2, . . . , m; j = 1, 2, . . . , n,

(27)

c c where βij is the complement of βij = (µβij , νβij ) such that βij = (νβij , µβij ).

˜ ij to positive Step 2. Calculate the separation degree of each component h ideal solution and negative ideal solution. The positive ideal solution (PIS) and negative ideal solution (NIS) can be ˜ + = {(1, 0)} and h ˜ − = {(0, 1)}, respectively, within the generalized denoted as h hesitant fuzzy environment. The separation between alternatives can be calculated by cross-entropies. For the convenience of both calculation and analysis, − only one cross-entropy (21) is selected. The separation degrees, G+ ij and Gij , of ˜ ij (i = 1, 2, . . . , m; j = 1, 2, . . . , n) to the PIS h ˜ + and NIS h ˜ − , respectively, each h are derived from Eq. (21): ∗ ˜ ˜+ ˜ ˜+ ˜+ ˜ G+ ij = CE2 (hij , h ) = CE2 (hij , h ) + CE2 (h , hij ) v )p v u u 1 ∑ ( ∑ p 2µαij u 1 u p =t µαij log2 +t ναij p ˜ ˜ µαij + 1 l(hij ) l(hij ) ˜ ˜ αij ∈hij

v u u u p  +t

αij ∈hij

 ) p ∑ ( 2 1  log2 ˜ ij ) 1 + µαij l(h ˜

(28)

αij ∈hij

and ∗ ˜ ˜− ˜ ˜− ˜− ˜ G− ij = CE1 (hij , h ) = CE1 (hij , h ) + CE1 (h , hij ) v v )p u u 1 ∑ p ∑ ( 2ναij u 1 u p =t µαij + t ναij log2 p ˜ ij ) ˜ ij ) ναij + 1 l(h l(h ˜ ˜ αij ∈hij

v u u u p  +t

1 ˜ l(hij )

αij ∈hij

∑ ( log2 ˜ ij αij ∈h

2 1 + ναij

 ) p  .

(29)

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Step 3. Calculate the closeness degree of the alternatives to the NIS. The closeness degree G(xi ) of each alternative xi (i = 1, 2, . . . , m) to the NIS can be obtained by following: G(xi ) =

n ∑

wj Gij , where Gij =

j=1

G− ij − G+ ij + Gij

.

(30)

Step 4. Rank the alternatives. The bigger the closeness degree G(xi ), the better the alternative xi will be, ˜ + . Therefore, the alternatives xi as the alternative xi is closer to the PIS h (i = 1, 2, . . . , m) can be ranked in descending order according to the closeness degrees so that the best alternative can be selected. Approach II. Step 1. For this step, see Approach I. Step 2. Calculate the overall aggregated values of each alternative. Utilize the GHFWA operator (17) (or the GHFWG operator (18)): ˜ i = GHFWAw (h ˜ i1 , h ˜ i2 , . . . , h ˜ in ) h   n n   ∪ ∏ ∏ 1 − = (1 − µαij )wj , ναwijj  (31)   j=1

˜ i1 ,αi2 ∈h ˜ i2 ,...,αin ∈h ˜ in αi1 ∈h

j=1

or ˜ i = GHFWGw (h ˜ i1 , h ˜ i2 , . . . , h ˜ ) h in  n n  ∏  ∪ ∏ j  = µw (1 − ναij )wj  (32) αij , 1 −   j=1

˜ i1 ,αi2 ∈h ˜ i2 ,...,αin ∈h ˜ in αi1 ∈h

j=1

˜ ij (j = 1, 2, . . . , n) of the ith line and to aggregate all the performance values h ˜ get the overall performance value hi corresponding to the alternatives xi . Step 3. Calculate the closeness degree of the alternatives to the PIS. ˜ i (i = Utilize the cross-entropy (21) between the overall performance value h ˜ + = {(1, 0)} to get closeness degree of each alternative 1, 2, . . . , m) and the PIS h ˜ +: xi (i = 1, 2, . . . , m) to the PIS h ˜ +, h ˜i) ˜ i, h ˜ + ) = CE2 (h ˜ i, h ˜ + ) + CE∗ (h G(xi ) = CE∗2 (h 2 v v ( ) u p u 1 ∑ 2µαi u 1 ∑ u p =t µαi log2 +t ναp i p ˜ ˜ i) µαi + 1 l(hi ) l( h ˜i ˜i αi ∈ h αi ∈ h v  u ( ) p u 1 ∑ 2 u p   . +t log2 (33) ˜ i) 1 + µαi l(h ˜ αi ∈ h i

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Step 4. Rank the alternatives. The smaller the closeness degree G(xi ), the better the alternative xi will ˜ + . Therefore, the alternatives xi be, as the alternative xi is closer to the PIS h (i = 1, 2, . . . , m) can be ranked in ascending order according to the closeness degrees so that the best alternative can be selected.

5

An illustrative example

In this section, a generalized hesitant fuzzy MCDM problem of selecting an investment is used to illustrate the proposed methods. A city is planning to build a municipal library. One of the problems facing the city development commissioner is to determine what kind of air-conditioning system should be installed in the library (adapted from [20]). The contractor offers five feasible alternatives xi (i = 1, 2, 3, 4, 5), which might be adapted to the physical structure of the library. Suppose that three criteria y1 (economic), y2 (functional), and y3 (operational) are taken into consideration in the installation problem, and the weight vector of the criteria yj (j = 1, 2, 3) is w = (0.3, 0.5, 0.2)T . Assume that the characteristics of the alternatives xi (i = 1, 2, 3, 4, 5) with respect to the criterion yj (j = 1, 2, 3) are represented ˜ ij = {αij = (µα , να )|αij ∈ h ˜ ij }, where µα indicates by the GHFEs h ij ij ij the degree that the alternative xi satisfies the criterion yj and ναij indicates the degree that the alternative xi does not satisfy the criterion yj , such that ˜ ij = {αij = (µα , να )|αij ∈ h ˜ ij } µαij , ναij ∈ [0, 1] and µαij + ναij ≤ 1. All h ij ij (i = 1, 2, 3, 4, 5; j = 1, 2, 3) are contained in the generalized hesitant fuzzy ˜ ij )5×3 (see Table 2). decision matrix E = (h Table 2: The generalized hesitant fuzzy decision matrix x1 x2 x3 x4 x5

y1 {(0.3, 0.2), (0.3, 0.4)} {(0.5, 0.2), (0.6, 0.2)} {(0.3, 0.4), (0.4, 0.5)} {(0.2, 0.6), (0.2, 0.7)} {(0.8, 0.1), (0.7, 0.2)}

y2 {(0.7, 0.2), (0.5, 0.2)} {(0.3, 0.1), (0.4, 0.2)} {(0.7, 0.2), (0.8, 0.1)} {(0.8, 0.1), (0.7, 0.2)} {(0.6, 0.3), (0.7, 0.2)}

y3 {(0.5, 0.2), (0.6, 0.3)} {(0.7, 0.1), (0.8, 0.1)} {(0.4, 0.3), (0.4, 0.4)} {(0.7, 0.2), (0.8, 0.1)} {(0.2, 0.5), (0.2, 0.6)}

To select the best air-conditioning system, we utilize above-mentioned two approaches to find the decision result(s). Approach I. Step 1. Considering that all the attributes yj (j = 1, 2, 3) are benefit type attributes, the performance values of the alternatives xi (i = 1, 2, 3, 4, 5) do not need normalization. Step 2. Utilize Eqs. (28) and (29) (let p = 2) to calculate the separation ˜ ij (i = 1, 2, 3, 4, 5; j = 1, 2, 3) to PIS h ˜+ = degree Gij of each component h 13

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˜ − = {(0, 1)} and then we get the following results: {(1, 0)} and NIS h + + + + G+ 11 = 1.2724, G12 = 0.7737, G13 = 0.8951, G21 = 0.8401, G22 = 1.0549, + + + + G+ 23 = 0.4620, G31 = 1.3496, G32 = 0.5201, G33 = 1.1911, G41 = 1.7059, + + + + G+ 42 = 0.5201, G43 = 0.5201, G51 = 0.5201, G52 = 0.7573, G53 = 1.6062, − − − − G− 11 = 1.2458, G12 = 1.6622, G13 = 1.5574, G21 = 1.6062, G22 = 1.4369, − − − − G− 23 = 1.8601, G31 = 1.1264, G32 = 1.8351, G33 = 1.2969, G41 = 0.7023, − − − − − G42 = 1.8351, G43 = 1.8351, G51 = 1.8351, G52 = 1.6571, G53 = 0.8401.

Step 3. Utilize the weight vector w = (0.3, 0.5, 0.2)T of the criteria yj (j = 1, 2, 3) and Eq. (30) to calculate the closeness degree G(xi ) of the alternatives xi (i = 1, 2, 3, 4, 5) to the NIS: G(x1 ) = 0.6166, G(x2 ) = 0.6455, G(x3 ) = 0.6303, G(x4 ) = 0.6329, G(x5 ) = 0.6456. Using this, we rank all alternatives xi (i = 1, 2, 3, 4, 5) in descending order in accordance with the values G(xi ) (i = 1, 2, 3, 4, 5): x5 ≻ x2 ≻ x4 ≻ x3 ≻ x1 . Therefore, the best alternative is x5 . Approach II. Step 1. For this step, see Approach I. Step 2. Utilize the GHFWA operator (31) to aggregate all the performance ˜ ij (i = 1, 2, 3, 4, 5; j = 1, 2, 3) of the ith line and get the overall perforvalues h ˜ i corresponding to the alternatives xi (i = 1, 2, 3, 4, 5); mance value h ˜ 1 = {(0.5716, 0.2000), (0.5903, 0.2169), (0.4469, 0.2000), (0.4710, 0.2169), h (0.5716, 0.2462), (0.5903, 0.2670), (0.4469, 0.2462), (0.4710, 0.2670)}; ˜ 2 = {(0.4658, 0.1231), (0.5075, 0.1231), (0.5055, 0.1741), (0.5440, 0.1741), h (0.5004, 0.1231), (0.5393, 0.1231), (0.5375, 0.1741), (0.5735, 0.1741)}; ˜ h3 = {(0.5557, 0.2670), (0.5557, 0.2828), (0.6372, 0.1888), (0.6372, 0.2000), (0.5757, 0.2855), (0.5757, 0.3024), (0.6536, 0.2018), (0.6536, 0.2138)}; ˜ h4 = {(0.6712, 0.1966), (0.6969, 0.1711), (0.5974, 0.2781), (0.6287, 0.2421), (0.6712, 0.2059), (0.6969, 0.1793), (0.5974, 0.2912), (0.6287, 0.2535)}; ˜ h5 = {(0.6268, 0.2390), (0.6268, 0.2479), (0.6768, 0.1951), (0.6768, 0.2024), (0.5785, 0.2942), (0.5785, 0.3051), (0.6350, 0.2402), (0.6350, 0.2491)}. Step 3. Utilize Eq. (33) to calculate the closeness degree G(xi ) of each ˜ + = {(1, 0)}: alternative xi (i = 1, 2, 3, 4, 5) to the PIS h G(x1 ) = 0.9146, G(x2 ) = 0.8291, G(x3 ) = 0.8103, G(x4 ) = 0.7350, G(x5 ) = 0.7799. 14

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Using this, we rank all alternatives xi (i = 1, 2, 3, 4, 5) in ascending order according to the values G(xi ) (i = 1, 2, 3, 4, 5): x4 ≻ x5 ≻ x3 ≻ x2 ≻ x1 . Therefore, the best alternative is x4 . If we utilize the GHFWG operator (32) in Step 2 of Approach II, then the closeness degree G(xi ) of each alternative xi (i = 1, 2, 3, 4, 5) to the PIS is calculated: G(x1 ) = 0.9835, G(x2 ) = 0.9117, G(x3 ) = 0.9700, G(x4 ) = 1.0543, G(x5 ) = 0.9597. and so the ranking of all alternatives xi (i = 1, 2, 3, 4, 5) in ascending order according to the values G(xi ) (i = 1, 2, 3, 4, 5) is obtained: x2 ≻ x5 ≻ x3 ≻ x1 ≻ x4 . Therefore, the best alternative is x2 . From the above analysis, we know that the results obtained by the proposed approaches are different. Each of methods has its advantages and disadvantages and none of them can always perform better than the others in any situations. It perfectly depends on how we look at things, and not on how they are themselves. As we can see, in Approach II, depending on aggregation operators used, the ranking of the alternatives is different. Therefore, the results may lead to different decisions.

6

Conclusions

In this paper, we developed cross-entropy under generalized hesitant fuzzy environment. Axiomatic definition about this information measure have been given for GHFEs. Two approaches, based on the developed generalized hesitant fuzzy cross-entropy, of generalized hesitant fuzzy MCDM are developed which permits the decision maker to provide several possible IFVs for an alternative under the given criterion, which is consistent with humans’ hesitant thinking. The illustrative example demonstrated the validity and practicability of the developed approaches.

Acknowledgement This work was supported by a Research Grant of Pukyong National University (2015).

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References [1] K. Atanassov, Intuitionistic fuzzy sets, Fuzzy Sets and Systems 20 (1986), 87-96. [2] K. Atanassov and G. Gargov, Interval-valued intuitionistic fuzzy sets, Fuzzy Sets and Systems 31 (1989), 343-349. [3] P. Burillo and H. Bustince, Vague sets are intuitionistic fuzzy sets, Fuzzy Sets and Systems 79 (1996), 403-405. [4] W.L. Gau and D.J. Bueherer, Vague sets, IEEE Transactions on Systems, Man and Cybernetics 23 (1994), 610-614. [5] K.C. Hung, K.P. Lin and C.C. Weng, Fault diagnosis of turbine using an improved intuitionistic fuzzy croos entropy approach, in: 2011 International Conference on Fuzzy System, Taipei, Taiwan, 2011, pp. 590-594. [6] W.L. Hung and M.S. Yang, On the J-divergence og intuitionistic fuzzy sets with its application to pattern recognition, Information Sciences 178 (2008), 1641-1650. [7] J.Q. Li, G.N. Deng, H.X. Li and W.Y. Zeng, The relationship between similarity measure and entropy of intuitionisticfuzzy sets, Information Sciences 188 (2012), 314-321. [8] X.C. Liu, Entropy, distance measure and similarity measure of fuzzy sets and their relations, Fuzzy Sets and Systems 52 (1992), 305-318. [9] J.J. Mao, D.B. Yao and C.C. Wang, A novel cross-entropy and entropy measures of IFSs and their applications, Knowledge-Based Systems 48 (2013), 37-45. [10] G. Qjan, H. Wang, X.Q. Feng, Generalized hesitant fuzzy sets and their application in decision support system, Knowledge-Based Systems 37 (2013), 357-365. [11] X.G. Shang and W.S. Jiang, A note on fuzzy information measure, Pattern Recognition Letter 18 (1997), 425-432. [12] V. Torra, Hesitant fuzzy sets, International Journal of Intelligent Systems 25 (2010), 529-539. [13] V. Torra, Y. Narukawa, On hesitant fuzzy sets and decision, in: The 18th IEEE International Conference on Fuzzy Systems, Jeju Island, Korea, 2009. pp. 1378-1382. [14] I.K. Vlachos and Sergiadis, Intuitionistic fuzzy information application to pattern recognition, Pattern Recognition Letter 28 (2007), 197-206.

16

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[15] J.Q. Wang and J.J. Li, Multi-criteria fuzzy decision making method based on cross entropy and score functions, Expert Systems with Applications 38 (2011), 1032-1038. [16] M.M. Xia, Z.S. Xu, Hesitant fuzzy information aggregation in decision making, International Journal of Approximate Reasoning 52 (2011), 395407. [17] M.M. Xia and Z.S. Xu, Entropy/cross entropy based group decision making under intuitionistic fuzzy environment, Information Fusion 13 (2012), 3147. [18] Z.S. Xu, Intuitionistic fuzzy aggregation operators, IEEE Transactions on Fuzzy Systems 15 (2007), 1179-1187. [19] Z.S. Xu and M.M. Xia, Hesitant fuzzy entropy and cross-entropy and their use in multiattribute decision-making, International Journal of IntelligentSystems 27 (2012), 799-822. [20] Z.S. Xu and R.R. Yager, Intuitionistic fuzzy Bonferroni means, IEEE Transactions on Systems, Man and Cybernetics 41 (2011), 568-578. [21] J. Ye, Fault diagnosis of turbine based on fuzzy cross entropy of vague sets, Expert Systems with Applications 36 (2009), 3103-3106. [22] J. Ye, Multicriteria fuzzy decision-making method based on intuitionistic fuzzy cross-entropy, in: Proceedings of the 2009 International Conference on Intelligent Human-Machine and Cybernetics, Hangzhou, China, 2009, pp. 59-61. [23] J. Ye, Fuzzy cross entropy of interval-valued intuitionistic fuzzy sets and its optimal decision-making method based on the weights of alternatives, Expert Systems with Applications 38 (2011), 6179-6183. [24] L.A. Zadeh, Fuzzy sets, Information and Control 8 (1965), 338-353. [25] Q.S. Zeng and H.X. Li, Relationship between simialrity measure and entropy of interval-valued fuzzy sets, Fuzzy Sets and Systems 157 (2006), 1477-1484. [26] Q.S. Zhang and S.Y. Jiang, A note on information entropy measure for vague sets and its application, Information Sciences 178 (2008), 4184-4191. [27] Q.S. Zhang and S.Y. Jiang, Relationships between entropy and simialrity measure of interval-valued intuitionistic fuzzy sets, International Journal of Intelligent Systems 25 (2010), 1121-1140. [28] H.M. Zhang and L.Y. Yu, MADM method based on cross-entropy and extended TOPSIS with interval-valued intuitionistic fuzzy sets, KnowledgeBased Systems 30 (2012), 115-120.

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ON HARMONIC QUASICONFORMAL MAPPINGS WITH FINITE AREA HONG-PING LI AND JIAN-FENG ZHU Abstract. In this paper, we study the class of harmonic K-quasiconformal mappings of the unit disk U with finite Euclidean areas |f (U)|euc . We first give the Schwarz-pick lemma (cf. [8]) for this class of mappings as follows: √ |f (U)|euc 1 |fz (z)| ≤ , z ∈ U, π(1 − k 2 ) 1 − |z| where k = K−1 K+1 . Furthermore, we obtain the sharp coefficient estimates of this 0 class of mappings. As an application, for harmonic mappings f ∈ SH with finite |f (U)|euc we obtain sharp coefficient estimates.

1. Introduction Let U = {z ∈ C : |z| < 1} be the unit disk in the complex plane C. The classical Schwarz lemma says that if an analytic function φ of U satisfies that |φ(z)| < 1 and φ(0) = 0. Then |φ(z)| ≤ |z| and |φ′ (0)| ≤ 1. The equality occurs if and only if φ(z) = eiα z, where α is a real constant. The classical Schwarz lemma is a cornerstone in complex analysis and attracts one to give various versions of its generalization. A complex-valued function f (z) of class C 2 is said to be a harmonic mapping if it satisfies fzz¯ = 0. It is known that every harmonic mapping f (z) defined in U admits a canonical decomposition f (z) = h(z) + g(z), where h(z) and g(z) are analytic in U with g(0) = 0. One can refer to [5] and the references therein for more details about harmonic mappings. For z ∈ U, let (1)

Λf (z) = max |fz (z) + e−2iθ fz¯(z)| = |fz (z)| + |fz¯(z)|, 0≤θ≤2π

and (2)

λf (z) = min |fz (z) + e−2iθ fz¯(z)| = ||fz (z)| − |fz¯(z)||. 0≤θ≤2π

2000 Mathematics Subject Classification. Primary: 30C62; Secondary: 30C20, 30F15. Key words and phrases. Harmonic mappings, harmonic quasiconformal mappings, coefficients estimate, Ahlfors-Schwarz lemma. File: LiZhu.tex, printed: 19-8-2015, 10.36. The authors of this work are supported by NNSF of China (11501220) and the NSFF of China (11471128).

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The Lewy theorem [7] tells us that a harmonic mapping f is locally univalent and sense-preserving in U if and only if its Jacobian satisfies the following condition Jf (z) = λf (z)Λf (z) = |fz (z)|2 − |fz¯(z)|2 > 0 for z ∈ U. Suppose that f (z) is a sense-preserving univalent harmonic mapping of U. Then f (z) is a K-quasiconformal mapping if and only if |fz (z)| + |fz¯(z)| ≤ K. z∈U |fz (z)| − |fz¯(z)|

K(f ) := sup

Harmonic quasiconformal mappings are natural generalizations of conformal mappings. Recently, many mathematicians have studied such an active topic and obtained many interesting results (cf.[1], [6], [10], [8], [12], [13], [14], [15] ). In 2007, M.Kne˘ z evi´c and M.Mateljevi´c [8] proved the following Schwarz-Pick lemma for harmonic quasiconformal mappings. Theorem A. Let f be a harmonic K-quasiconformal mapping of U into itself. Then |fz (z)| ≤

(K + 1)(1 − |f (z)|2 ) 2(1 − |z|2 )

holds for all z ∈ U, and dλ (f (z1 ), f (z2 )) ≤ Kdλ (z1 , z2 ) holds for any z1 , z2 ∈ U, where dλ is the hyperbolic distance. Furthermore, they obtained the opposite inequalities in Theorem A as |fz (z)| ≥ and dλ (f (z1 ), f (z2 )) ≥ K1 dλ (z1 , z2 ) by assuming that f is onto. Such an assumption is necessary since that |fz (z)| will bounded below by a positive constant (see [8] for more details). In 2010, X. Chen and A. Fang [2] improved the above results as follows. (K+1)(1−|f (z)|2 ) 2K(1−|z|2 )

Theorem B. Let Ω ⊂ C be a simply connected convex domain of hyperbolic type and λΩ be its hyperbolic metric density with the Gaussian curvature −4. If f is a harmonic K-quasiconformal mapping of U onto Ω, then the inequalities (K + 1)λU (z) (K + 1)λU (z) ≤ |fz (z)| ≤ 2KλΩ (f (z)) 2λΩ (f (z)) hold for all z ∈ U. Moreover, the above estimates are sharp. We point out that the composition of a harmonic mapping f with a conformal mapping φ is still harmonic. Hence we can fix the defined domain as U for harmonic mappings. However, φ ◦ f is not harmonic in general. This implies that the Schwarz-Pick lemma for harmonic quasiconformal mappings will closely relate to its target domain. Instead of the assumption that harmonic quasiconformal mappings have convex or bounded ranges, we study the class of harmonic mappings with finite Euclidean areas. Example 1 shows there exists a harmonic mapping with an unbounded range but finite Euclidean area.

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J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 22, NO.4, 2017, COPYRIGHT 2017 EUDOXUS PRESS, LLC

On harmonic quasiconformal mappings with finite area

3

Assume that f (z) = h(z) + g(z) is a harmonic K-quasiconformal mapping of U with a finite Euclidean area |f (U)|euc , where h(z) =

∞ ∑

an z

n

and g(z) =

n=0

∞ ∑

bn z n

n=1

are analytic in U. Under the assumption of finite Euclidean areas, we obtain a new version of the Schwarz-Pick lemma for the class of harmonic K-quasiconformal mappings as follows √ |f (U)|euc 1 , z ∈ U, (3) |fz (z)| ≤ π(1 − k 2 ) 1 − |z| where k = K−1 . See Theorem 1 for details. K+1 Furthermore, we obtain the sharp coefficient estimates for f (z) (K + 1/K)|f (U)|euc (n = 1, 2, . . .). 2nπ This result is given at Theorem 2. Denote by SH the family of all sense-preserving univalent harmonic mappings defined in U which admit a canonical representation f = h + g, where ∞ ∞ ∑ ∑ n (5) h(z) = z + an z and g(z) = bn z n |an |2 + |bn |2 ≤

(4)

n=2

n=1

0 are analytic in U. The class SH is the subclass of SH with g ′ (0) = 0. A well-known result of the classical analytic univalent functions is the Bieberbach conjecture which was posed by Ludwig Bieberbach in 1916 and was finally proven by Louis de Branges [4]. This result has many important geometric applications. In 1984, T.Sheil-Small [3] published a landmark paper which pointed out that many classical results of conformal mappings have analogues of harmonic mappings. One 0 of the famous results is the coefficients conjecture of SH . As an application of 0 Theorem 2, we obtain the coefficients estimate for f ∈ SH which is given by (9).

2. Main results and their proofs Theorem 1. Let K ≥ 1 be a constant. If f (z) = h(z) + g(z) is a harmonic Kquasiconformal mapping of the unit disk U such that its Euclidean area |f (U)|euc is finite, then √ |fz (z)| ≤ where k =

|f (U)|euc 1 , z ∈ U, π(1 − k 2 ) 1 − |z|

K−1 . K+1

Proof. Since f (z) is a harmonic K-quasiconformal mapping, we obtain that ′ g (z) K −1 sup ′ ≤ k = . K +1 z∈U h (z)

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Then

∫ ∫

(|h′ (z)|2 − |g ′ (z)|2 ) dσ ( ) ∫ ∫U |g ′ (z)|2 ′ 2 = |h (z)| 1 − ′ dσ |h (z)|2 U ∫ ∫ 2 ≥ (1 − k ) |h′ (z)|2 dσ.

|f (U)|euc =

U

This implies that

∫ ∫

|h′ (z)|2 dσ ≤

(6) U

|f (U)|euc . 1 − k2

By [11, Corollary 2.6.4 and Theorem 2.4.1], we obtain |h′ (z)|2 is subharmonic in U. Thus, for r ∈ [0, 1 − |z|), it follows ∫ 2π 1 ′ 2 (7) |h (z)| ≤ |h′ (z + reiθ )|2 dθ. 2π 0 Utilizing the inequality (6), we get ′

π(1 − |z|) |h (z)| 2

∫ 2

≤

2π

∫

∫ ∫ 0

1−|z|

|h′ (ζ)|2 dσ

= ∫ ∫

r|h′ (z + reiθ )|2 dr dθ

0

D(z)

|h′ (ζ)|2 dσ ≤

≤ U

|f (U)|euc , 1 − k2

where D(z) := {ζ ∈ C, |ζ − z| < 1 − |z|} ⊆ U. Then |h′ (z)|2 ≤ This completes the proof.

|f (U)|euc . π(1−|z|)2 (1−k2 )



Remark 1. The Euclidean area of f (U) is finite doesn’t imply that f is bounded. The following Example 1 shows that Theorem 1 is not covered by Theorem A and Theorem B. Furthermore, let f (z) = eiα z be a conformal mapping of U onto itself, where α is a real constant. Then |fz (z)| = 1 and |f (U)|euc = π. This shows that (3) is sharp at z = 0. Example 1. Let Ω1 = {ζ ∈ C : 0 ≤ Im ζ ≤ 1, Re ζ ≥ 0} and φ1 (z) : U 7→ Ω1 be 1+ζi be a conformal mapping of Ω1 onto a conformal mapping. Let φ2 (ζ) := 2i1 ln 1−ζi Ω2 . Then w(z) = φ2 ◦ φ1 (z) is a conformal mapping of U onto Ω2 . Here Ω2 is an unbounded (and not convex) domain with the boundary curves {c1 : Im w = 0, 0 ≤ ti , t ∈ [0, ∞)} Re w ≤ π2 }, {c2 : Re w = 0, Im w ∈ R} and {c3 : w(t + i) = 2i1 ln 2−ti π ln 2 which is shown by figure 1. Then |w(U)|euc = 8 is finite. This shows that Theorem 1 is not covered by Theorem A and Theorem B.

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J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 22, NO.4, 2017, COPYRIGHT 2017 EUDOXUS PRESS, LLC

On harmonic quasiconformal mappings with finite area

5

15

10

5

0.5

Π

1

2

Figure 1. Image of the domain Ω2 Theorem 2. Let K ≥ 1 be a constant. If f (z) = h(z) + g(z) is a harmonic Kquasiconformal mapping of U such that |f (U)|euc is finite, where (8)

h(z) =

∞ ∑

an z n and

g(z) =

n=0

∞ ∑

bn z n

n=1

are analytic in U. Then |an |2 + |bn |2 ≤

(K + 1/K)|f (U)|euc (n = 1, 2, . . .). 2nπ

The above coefficient estimates are sharp for all n = 1, 2, . . ., with the extremal functions fn (z) = √an z n + √kan z n where k = K−1 and a ∈ C is a constant. K+1 Proof. For every z = reiθ ∈ U, iθ

f (re ) =

∞ ∑

n inθ

an r e

+

n=0

Hence h′ (reiθ ) =

∞ ∑

∞ ∑

bn rn e−inθ .

n=1 ∞ ∑

nan rn−1 ei(n−1)θ and g ′ (reiθ ) =

n=1

nbn rn−1 ei(n−1)θ . Applying the

n=1

Parseval identity, we obtain ∫ ∫ ∞ ∑ (|h′ (z)|2 + |g ′ (z)|2 )dσ = π n(|an |2 + |bn |2 ). U

n=1

Since f (z) is a K-quasiconformal mapping, we have ( ) 1 |h′ (z)|2 + |g ′ (z)|2 ≤ (K + 1/K) |h′ (z)|2 − |g ′ (z)|2 . 2

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This implies that π

∞ ∑

∫ ∫

n(|an | + |bn | ) ≤ 2

2

U

n=1

1 (K + 1/K)(|h′ (z)|2 − |g ′ (z)|2 )dσ 2

1 (K + 1/K)|f (U)|euc . 2

= Then

(K + 1/K)|f (U)|euc (n = 1, 2, . . .). 2nπ ∫∫ Let fn (z) = √an z n + √kan z n , then |fn (z)|euc = J (z)dσ = (1 − k 2 )π|a|2 . This U fn |an |2 + |bn |2 ≤

2

2

shows that |an |2 + |bn |2 = (1+kn )|a| = sharp. This completes the proof.

(K+1/K)|f (U)|euc . 2nπ

Hence, the estimates are 

0 Theorem 3. If f = h + g¯ ∈ SH satisfies that |f (U)|euc is finite, where h, g are given by (5) with b1 = 0. Then

|an |2 + |bn |2 ≤ s(n, t0 ), (n = 2, 3, . . .), where s(n, t0 ) is given by (10). 0 Proof. Let F (ζ) := f (tζ) , where f ∈ SH , ζ ∈ U and 0 < t < 1. Then ΛF (ζ) = Λf (tζ) t holds for all ζ ∈ U. According to (5) we see that

F (ζ) = ζ +

∞ ∑

an tn−1 ζ n +

n=2

= ζ+

∞ ∑

∞ ∑

bn tn−1 ζ n

n=2

An ζ n +

n=2

∞ ∑

Bn ζ n ,

n=2

where An = tn−1 an and Bn = tn−1 bn . Let ω(z) = ffzz¯(z) . Then ω(z) is holomorphic (z) in U satisfying ω(0) = 0 and |ω(z)| < 1. By the Schwarz lemma we know that |ω(z)| ≤ |z| for z ∈ U. Therefore, for any 0 < t < 1 and Ut := {z : |z| < t} we have 1+t 1 + |ω(z)| Λf (z) ≤ = := Kt . λf (z) 1 − |ω(z)| 1−t This implies that F is a Kt -quasiconformal mapping of U. Furthermore, ∫ ∫ |F (ζ)|euc = Jf (ζ)dσ ≤ |f (U)|euc . Ut

Applying Theorem 2, we have |An |2 + |Bn |2 ≤

(1 + t2 )|f (U)|euc , (n = 2, 3, . . .). nπ(1 − t2 )

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On harmonic quasiconformal mappings with finite area

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Hence, (9)

|an |2 + |bn |2 ≤

(1 + t2 )|f (U)|euc := s(n, t), (n = 2, 3, . . .). nπt2n−2 (1 − t2 )

Since lim s(n, t) = ∞ = lim s(n, t), we see that min s(n, t) exists. Choose the t→0 t→1 0 0 is said to be (α, m)-convex, where (α, m) ∈ [0, 1]2 , if we have f (tx + m(1 − t)y) ≤ tα f (x) + m(1 − tα )f (y)

(1.4)

for all x, y ∈ [0, b] and t ∈ [0, 1]. In [1], S. S. Dragomir and G. Toader proved the following Hadamard type inequalities for m-convex functions. Theorem 1.1 ([1]). Let f : R0 → R be an m-convex function with m ∈ (0, 1]. if 0 ≤ a < b < ∞ and f ∈ L1 [a, b], then the following inequality holds (b) ( a )} { ∫ b f (b) + mf m f (a) + mf m 1 f (x) dx ≤ min , . (1.5) b−a a 2 2 In [2], S. S. Dragomir established new Hadamard-type inequalities for m-convex functions. Theorem 1.2 ([2]). Let f : R0 → R be an m-convex function with m ∈ (0, 1]. if 0 ≤ a < b < ∞ and f ∈ L1 [am, b], then the following inequality holds [ ] ∫ mb ∫ b 1 1 f (a) + f (b) 1 f (x) dx + f (x) dx ≤ . (1.6) m + 1 mb − a a b − ma ma 2 In [10], E. Set et al. proved the following Hadamard type inequalities for (α, m)-convex functions. Theorem 1.3 ([10]). Let f : R0 → R be an (α, m)-convex function with (α, m) ∈ (0, 1]2 . if 0 ≤ a < b < ∞ and f ∈ L1 [a, b], then the following inequality holds ( ) ∫ b[ ( x )] a+b 1 f ≤ α f (x) + m(2α − 1)f dx 2 2 (b − a) a m { [ (a) ( b )] 1 ≤ α+1 f (a) + f (b) + m(α + 2α − 1) f +f 2 (α + 1) m m } [ ( a )] b + αm2 (2α − 1) f ) + f( 2 dx. (1.7) m2 m Some generalizations of this result can be found in [12] and [13]. In recent years, several extensions and generalizations have been considered for classical convexity. A significant generalization of convex functions is that of operator convex functions introduced by S. S. Dragomir in [5]. Now we review the operator order in B(H) and the continuous functional calculus for a bounded self-adjoint operator. For self-adjoint operators A, B ∈ B(H), we write A ≤ B if ⟨Ax, x⟩ ≤ ⟨Bx, x⟩ for every vector x ∈ H, we call it the operator order. Let A be a bounded self-adjoint linear operator on a complex Hilbert space (H; ⟨., .⟩). The Gelfand map establishes a ∗-isometrically isomorphism Φ between the set C(Sp(A)) of all continuous complex-valued functions defined on the spectrum of A, denoted Sp(A), and the C ∗ -algebra C ∗ (A) generated by A and the identity operator 1H on H as follows (see for instance [6], p.3). For any f, g ∈ C(Sp(A)) and any α, β ∈ C, we have (i) (ii)

Φ(αf + βg) = αΦ(f ) + βΦ(g); Φ(f g) = Φ(f )Φ(g)

(iii) ∥Φ(f ) ∥=∥ f ∥ :=

and sup

Φ(f ∗ ) = Φ(f )∗ ; | f |;

t∈Sp(A)

(iv) Φ(f0 ) = 1H

and

Φ(f1 ) = A,

where f0 (t) = 1

745

and

f1 (t) = t for t ∈ Sp(A).

SHUHONG WANG 744-753

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HERMITE-HADAMARD TYPE INEQUALITIES

3

With this notation, we define f (A) := Φ(f )

for all

f ∈ C(Sp(A))

(1.8)

and we call it the continuous functional calculus for a bounded self-adjoint operator A. If A is a bounded self-adjoint operator and f is a real-valued continuous function on Sp(A), then f (t) ≥ 0 for any t ∈ Sp(A) implies that f (A) ≥ 0, i.e. f (A) is a positive operator on H. Moreover, if both f and g are real-valued functions on Sp(A) such that f (t) ≤ g(t) for any t ∈ Sp(A), then f (A) ≤ f (B) in the operator order in B(H). We denoted by B(H)+ the set of all positive operators in B(H) and C(H) := {A ∈ B(H)+ : AB + BA ≥ 0 for all B ∈ B(H)+ }.

(1.9)

It is obvious that C(H) is a closed convex cone in B(H). A real valued continuous function f on an interval I ⊆ R is said to be operator convex (operator concave) if the operator inequality f ((1 − t)A + tB) ≤ (≥)(1 − t)f (A) + tf (B)

(1.10)

holds in the operator order in B(H), for all t ∈ [0, 1] and for every bounded self-adjoint operators A and B in B(H) whose spectra are contained in I. In [5], S. S. Dragomir gave the operator version of the Hermite-Hadamard inequality for operator convex functions. Theorem 1.4. Let f : I ⊆ R → R be an operator convex function on the interval I. Then for any self-adjoint operators A and B with spectra in I, we have the inequality ( ) [ ( ) ( )] A+B 1 3A + B A + 3B f ≤ f +f 2 2 4 4 [ ( ) ] ∫ 1 1 A+B f (A) + f (B) f (A) + f (B) ≤ f + ≤ . (1.11) f (tA + (1 − t)B) dt ≤ 2 2 2 2 0 For recent results related to Hermite-Hadamard type inequalities are given in [3], [4], [6], [7], and plenty of references therein. The goal of this paper is to obtain new inequalities like those given in Theorems 1.1, 1.2, 1.3, but now for operator m-convex and (α, m)-convex functions. 2. Operator m-convex and (α, m)-convex functions In order to verify our main results, the following preliminary definitions and lemmas are necessary. Definition 2.1. Let [0, b] ⊆ R0 with b > 0 and K be a convex set of B(H)+ . A continuous function f : [0, b] → R is said to be operator m-convex on [0, b] for operators in K, if f (tA + m(1 − t)B) ≤ tf (A) + m(1 − t)f (B)

(2.1)

in the operator order in B(H), for all t ∈ [0, 1] and every positive operators A and B in K whose spectra are contained in [0, b] and for some fixed m ∈ [0, 1]. Remark 2.1. For m = 1, we recapture the concept of operator convex functions defined on [0, b] and for m = 0 we get the concept of operator starshaped functions on [0, b], namely, we call f : [0, b] → R to be operator starshaped if f (tA) ≤ tf (A)

(2.2)

for all t ∈ [0, 1] and every positive operators A in B(H) whose spectra are contained in [0, b]. +

Lemma 2.1. If f is operator m-convex, then f (0) ≤ 0, where 0 is the zero operator on H.

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Proof. Taking A = 0 and B = 0 in the inequality (2.1), then (1 − t)(1 − m)f (0) ≤ 0. Also by t, m ∈ [0, 1], we get f (0) ≤ 0.



Lemma 2.2. If f is operator m-convex, then f is operator starshaped. Proof. For all t ∈ [0, 1] and positive operators A ∈ B(H)+ whose spectra is contained in [0, b], we write f (tA) = f (tA + m(1 − t)0) ≤ tf (A) + m(1 − t)f (0) ≤ tf (A).  Lemma 2.3. If f is operator m1 -convex and 0 ≤ m2 < m1 ≤ 1, then f is operator m2 -convex. Proof. For all t ∈ [0, 1] and positive operators A, B ∈ B(H)+ whose spectra are contained in [0, b], we drive ( (m ) ) (m ) 2 2 f (tA + m2 (1 − t)B) = f tA + m1 (1 − t) B ≤ tf (A) + m1 (1 − t)f B m1 m1 m2 f (B) = tf (A) + m2 (1 − t)f (B). ≤ tf (A) + m1 (1 − t) m1  Definition 2.2. Let [0, b] ⊆ R0 with b > 0 and K be a convex set of B(H)+ . A continuous function f : [0, b] → R is said to be operator (α, m)-convex on [0, b] for operators in K, if f (tA + m(1 − t)B) ≤ tα f (A) + m(1 − tα )f (B)

(2.3)

in the operator order in B(H), for all t ∈ [0, 1] and every positive operators A and B in K whose spectra are contained in [0, b] and for some fixed (α, m) ∈ [0, 1]2 . Remark 2.2. It can be easily seen that for (α, m) ∈ {(0, 0), (1, 1), (1, m)} one obtains the following classes of functions: operator increasing, operator convex and operator m-convex functions respectively. Similarly to the proof of Lemma 2.1, the following result is valid. Lemma 2.4. If f is operator (α, m)-convex, then f (0) ≤ 0, where 0 is the zero operator on H. Lemma 2.5 ([9]). Let A, B ∈ B(H)+ . Then AB + BA is positive if and only if f (A + B) ≤ f (A) + f (B) for all non-negative operator monotone functions f on R0 . Now, we give an example of operator m-convex function. Example 2.1. Since for every positive operator A, B ∈ C(H), AB + BA ≥ 0. Utilizing Lemma 2.5, we obtain [tA + m(1 − t)B]s ≤ ts As + ms (1 − t)s B s ≤ tAs + m(1 − t)B s . Therefore, the continuous function f (t) = ts (0 < s ≤ 1) is operator m-convex on R0 for operators in C(H). Remark 2.3. We can consider the same continuous function f (t) = ts (0 < s ≤ 1) as an example of operator (α, m)-convex function for α = 1.

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HERMITE-HADAMARD TYPE INEQUALITIES

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3. Some new Hermite-Hadamard type inequalities We will now point out some new results of the Hermite-Hadamard type. Theorem 3.1. Let the continuous function f : R0 → R be operator (α, m)-convex for operators in K ⊆ B(H)+ with (α, m) ∈ [0, 1] × (0, 1]. Then for all positive operator A, B ∈ K with spectra in R0 , the following inequality holds: { (B) ( A )} ∫ 1 f (B) + αmf m f (A) + αmf m , . (3.1) f (tA + (1 − t)B) dt ≤ min α+1 α+1 0 Proof. For x ∈ H with ∥x∥ = 1 and m, t ∈ (0, 1], we have ⟨ ⟩ (tA + m(1 − t)B)x, x = t⟨Ax, x⟩ + m(1 − t)⟨Bx, x⟩ ∈ R0 ,

(3.2)

since ⟨Ax, x⟩ ∈ Sp(A) ⊆ R0 and ⟨Bx, x⟩ ∈ Sp(B) ⊆ R0 . ∫1 Continuity of f and (3.2) imply that the operator-valued integral 0 f (tA + m(1 − t)B) dt exists. Since f is operator (α, m)-convex, therefore for (α, m) ∈ [0, 1] × (0, 1] and A, B ∈ K, we show ( ) B α α f (tA + (1 − t)B) ≤ t f (A) + m(1 − t )f m (

and f (tB + (1 − t)A) ≤ tα f (B) + m(1 − tα )f

A m

)

for all t ∈ [0, 1]. Integrating over t on [0, 1], we obtain (B) ∫ 1 f (A) + αmf m f (tA + (1 − t)B) dt ≤ α+1 0 and

∫ 0

However

∫

1

f (B) + αmf f (tB + (1 − t)A) dt ≤ α+1 ∫

1

f (tA + (1 − t)B) dt = 0

(A) m

.

1

f (tB + (1 − t)A) dt, 0

and the inequality (3.1) is obtained, which completes the proof.



Corollary 3.1.1. Under the assumptions of Theorem 3.1, choosing α = 1, we get the inequality for operator m-convex functions: { (B) ( A )} ∫ 1 f (A) + mf m f (B) + mf m . (3.3) f (tA + (1 − t)B) dt ≤ min , 2 2 0 Furthermore, for α, m = 1 we have ∫ 1 f (A) + f (B) f (tA + (1 − t)B) dt ≤ . 2 0

(3.4)

Theorem 3.2. Let the continuous function f : R0 → R be operator (α, m)-convex for operators in K ⊆ B(H)+ with (α, m) ∈ [0, 1] × (0, 1]. Then for all positive operator A, B ∈ K with spectra in R0 , the following inequalities hold: ) )] ( ( ∫ 1[ 1 (1 − t)A + tB A+B α ≤ α f (tA + (1 − t)B) + m(2 − 1)f dt f 2 2 0 m

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≤

{ [ ( ) ( )] B 1 A α f (A) + f + f (B) + m(α + 2 − 1) f 2α+1 (α + 1) m m ) ( )]} [ ( B A + f . + αm2 (2α − 1) f m2 m2

Proof. By operator (α, m)-convexity of f , we give ( ) ( ) ( ) A+B 1 (1 − t)A + tB 1 f ≤ α f (tA + (1 − t)B) + m 1 − α f 2 2 2 m [ )] ( 1 (1 − t)A + tB , = α f (tA + (1 − t)B) + m(2α − 1)f 2 m where (α, m) ∈ [0, 1] × (0, 1], t ∈ [0, 1] and A, B ∈ K with spectra in R0 . Integrating over t ∈ [0, 1], we drive the first inequality in (3.5). Next, from operator (α, m)-convexity of f , we also deduce [ ( )] 1 (1 − t)A + tB α f (tA + (1 − t)B) + m(2 − 1)f 2α m { ( ) [ ( ) ( )]} B A 1 B α α α α α + m(2 − 1) t f + m(1 − t )f . ≤ α t f (A) + m(1 − t )f 2 m m m2 Integrating over t on [0, 1], we get ( )] ∫ 1[ (1 − t)A + tB 1 α f (tA + (1 − t)B) + m(2 − 1)f dt 2α 0 m { ( ) ( )] 1 B A α 2 α = α f (A) + m(α + 2 − 1)f + αm (2 − 1)f . 2 (α + 1) m m2 Similarly, taking into account that ∫ 1 ∫ f (tA + (1 − t)B) dt = 0

(3.5)

(3.6)

(3.7)

(3.8)

1

f (tB + (1 − t)A) dt

0

and changing the roles of A and B, we obtain ( )] ∫ 1[ (1 − t)A + tB 1 α + m(2 − 1)f dt f (tA + (1 − t)B) 2α 0 m ( ) { ( )] A 1 B = α f (B) + m(α + 2α − 1)f + αm2 (2α − 1)f . 2 (α + 1) m m2

(3.9)

Summing the inequalities (3.8) and (3.9) and dividing by 2, we get the second inequality in (3.5). The proof thus is complete.  Corollary 3.2.1. With the conditions of Theorem 3.2, taking α = 1, we obtain the inequalities for operator m-convex functions: ( ) )] ( ∫ [ A+B 1 1 (1 − t)A + tB f ≤ f (tA + (1 − t)B) + mf dt 2 2 0 m { [ ( ) ( )] [ ( ) ( )]} 1 A B A B 2 ≤ f (A) + f (B) + 2m f +f +m f +f . (3.10) 8 m m m2 m2 In addition, if α, m = 1, we have ( ) ∫ 1 A+B f (A) + f (B) f ≤ f (tA + (1 − t)B) dt ≤ . 2 2 0

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7

Theorem 3.3. Let the continuous function f : R0 → R be operator (α, m)-convex for operators in K ⊆ B(H)+ with (α, m) ∈ [0, 1] × (0, 1]. Then for all positive operator A, B ∈ K with spectra in R0 , the following inequality holds: [ (A) ( B )] ∫ 1 f (A) + f (B) + αm f m +f m f (tA + (1 − t)B) dt ≤ . (3.12) 2(α + 1) 0 Proof. Using operator (α, m)-convexity of f , we can write

(

B f (tA + (1 − t)B) ≤ t f (A) + m(1 − t )f m α

)

α

(

) A f (tB + (1 − t)A) ≤ t f (B) + m(1 − t )f m for all t ∈ [0, 1] and some fixed (α, m) ∈ [0, 1] × (0, 1]. Adding the above inequalities and integrating over t on [0, 1], we have [ (A) ( B )] ∫ 1 ∫ 1 f (A) + f (B) + αm f m +f m f (tA + (1 − t)B) dt + f (tB + (1 − t)A) dt ≤ . α+1 0 0 As it is easy to see that ∫ 1 ∫ 1 f (tA + (1 − t)B) dt = f (tB + (1 − t)A) dt, and

α

α

0

0



we deduce the desired result. The proof of Theorem 3.3 is complete.

Corollary 3.3.1. Under the assumptions of Theorem 3.3, letting α = 1, we get the inequality for operator m-convex functions: [ (A) ( B )] ∫ 1 f (A) + f (B) + m f m +f m f (tA + (1 − t)B) dt ≤ . (3.13) 4 0 In addition, for α, m = 1, we have ∫ 1 f (A) + f (B) f (tA + (1 − t)B) dt ≤ . (3.14) 2 0 Theorem 3.4. Let the continuous function f : R0 → R be operator (α, m)-convex for operators in K ⊆ B(H)+ with (α, m) ∈ [0, 1] × (0, 1]. Then for all positive operator A, B ∈ K with spectra in R0 , the following inequality holds: ∫ 1 [ ] (1 + mα)[f (A) + f (B)] f (tA + m(1 − t)B) + f (tB + m(1 − t)A) dt ≤ . (3.15) α+1 0 Proof. By operator (α, m)-convexity of f , we can obtain f (tA + m(1 − t)B) ≤ tα f (A) + m(1 − tα )f (B), f ((1 − t)A + mtB) ≤ (1 − t)α f (A) + m(1 − (1 − t)α )f (B), f (tB + m(1 − t)A) ≤ tα f (B) + m(1 − tα )f (A), and f ((1 − t)B + mtA) ≤ (1 − t)α f (B) + m(1 − (1 − t)α )f (A) for all t ∈ [0, 1] and some fixed (α, m) ∈ [0, 1] × (0, 1]. Adding the above inequalities with each other, we get f (tA + m(1 − t)B) + f ((1 − t)A + mtB) + f (tB + m(1 − t)A) + f ((1 − t)B + mtA) [α ][ ] ≤ t + (1 − t)α + m(1 − tα ) + m(1 − (1 − t)α ) f (A) + f (B) .

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Now integrating over t ∈ [0, 1] and taking into account that ∫ 1 ∫ 1 f (tA + m(1 − t)B) dt = f ((1 − t)A + mtB) dt 0

and

∫

0

∫

1

f (tB + m(1 − t)A) dt = 0

1

f ((1 − t)B + mtA) dt 0



we obtain the inequality (3.15). The proof of Theorem 3.4 is complete.

Corollary 3.4.1. Under the assumptions of Theorem 3.4, choosing α = 1, we get the inequality for operator m-convex functions: ∫ 1 [ ] (1 + m)[f (A) + f (B)] f (tA + m(1 − t)B) + f (tB + m(1 − t)A) dt ≤ . (3.16) 2 0 Moreover, for α, m = 1, we have ∫ 1 f (A) + f (B) . (3.17) f (tA + (1 − t)B) dt ≤ 2 0 Theorem 3.5. Let the continuous function f : R0 → R be operator (α, m)-convex for operators in K ⊆ B(H)+ with (α, m) ∈ [0, 1] × (0, 1]. Then for all positive operator A, B ∈ K with spectra in R0 , the following inequalities hold: ( ) 2−m m f B + (mA) 2 2 ( )] ∫ 1[ ( ) 1 (1 − t)(2 − m)B + tm2 A α 2 ≤ α dt f t(2 − m)B + (1 − t)m A + m(2 − 1)f 2 0 m [ ( ) 1 ≤ α f ((2 − m)B) + m α + 2α − 1 f (mA) 2 (α + 1) ( )] (2 − m)B + m2 α(2α − 1)f . (3.18) m2 Proof. From operator (α, m)-convexity of f , we can deduce ( ) 2−m m f B + (mA) 2 2 ( ) ( ) ) 1 ( 1 (1 − t)(2 − m)B + tm2 A ≤ α f t(2 − m)B + (1 − t)m2 A + m 1 − α f 2 2 m [ ( )] ( ) 1 (1 − t)(2 − m)B + tm2 A = α f t(2 − m)B + (1 − t)m2 A + m(2α − 1)f , 2 m where (α, m) ∈ [0, 1] × [0, 1], t ∈ (0, 1] and A, B ∈ K with spectra in R0 . Integrating over t ∈ [0, 1], we drive the first inequality in (3.18). Next, by operator (α, m)-convexity of f , we also write ( ) f t(2 − m)B + (1 − t)m2 A ≤ tα f ((2 − m)B) + m(1 − tα )f (mA) ) ( ) (2 − m)B (1 − t)(2 − m)B + tm2 A α α f ≤ t f (mA) + m(1 − t )f . m m2 Submitting the inequalities (3.20) and (3.21) into the inequality (3.19), we get [ )] ( ( ) 1 (1 − t)(2 − m)B + tm2 A 2 α f t(2 − m)B + (1 − t)m A + m(2 − 1)f 2α m

and

(3.19)

(3.20)

(

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≤

{ [ ] 1 tα f ((2 − m)B) + m 1 − tα + tα (2α − 1) f (mA) α 2 )} ( (2 − m)B . + m2 (2α − 1)(1 − tα )f m2

9

(3.22)

Integrating over t on [0, 1], we deduce the second inequality in (3.18). This completes the proof of the Theorem 3.5.  Corollary 3.5.1. Under the assumptions of Theorem 3.5, letting α = 1, we get the inequalities for operator m-convex functions: ( ) 2−m m f B + (mA) 2 2 [ ( )] ∫ 1 ( ) 1 (1 − t)(2 − m)B + tm2 A 2 f t(2 − m)B + (1 − t)m A + mf dt ≤ 2 0 m [ )] ( 1 (2 − m)B ≤ f ((2 − m)B) + 2mf (mA) + m2 f . (3.23) 4 m2 In addition, for α, m = 1, we drive ) ∫ 1 ( A+B f (A) + f (B) ≤ f f (tA + (1 − t)B) dt ≤ . (3.24) 2 2 0

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Acknowledgements. This work was partially supported by the National Natural Science Foundation of China under Grant No. 11361038 and by the Inner Mongolia Autonomous Region Natural Science Foundation Project under Grant No. 2015MS0123, China. Competing interests. The authors declare that they have no competing interests. References [1] S. S. Dragomir and G. Toader,Some inequalities for m-convex functions, Studia Univ. Babes-Bolyai, Mathematica, 38:1, 21–28 (1993). [2] S. S. Dragomir, On some new inequalities of Hermite-Hadamard type for m-convex functions, Tamkang J. Math., 3:1, 45–55 (2002). [3] S. S. Dragomir, An inequality improving the first Hermite-Hadamard inequality for convex functions defined on linear spaces and applications for semi-inner products, J. Inequal. Pure Appl. Math., 3:2, 775–788 (2002); Available online at http://jipam.vu.edu.cn. [4] S. S. Dragomir, An inequality improving the second Hermite-Hadamard inequality for convex functions defined on linear spaces and applications for semi-inner products, J. Inequal. Pure Appl. Math., 3:3, Article 35 (2002); Available online at http://jipam.vu.edu.cn. [5] S. S. Dragomir, Hermite-Hadamard’s type inequalities for operator convex functions, Applied Mathematics and Computation, 218, 766–772 (2011); Available online at http://www.elsevier.com/locate/amc. [6] T. Furuta, J. M. Hot, J. Peˇ cari´ c, and Y. Seo, Mond-Peˇ cari´ c method in operator inequalities, Inequalities for Bounded Selfadjoint Operators on a Hilbert Space, Element, Zagreb, 2005. [7] A. G. Ghazanfari,The Hermite-Hadamard type inequalities for operator s-convex functions, Journal of Advanced Research in Pure Mathematics, 6:3, 52–61 (2014); Available online at http://dx.doi.org/10.5373/ jarpm.1876.110613. [8] V. G. A. Mihesan, A generalization of the convexity, Seminar on Functional Equations, Approx. and Convex., Cluj-Napoca, Romania (1993). [9] M. S. Moslehian and H. Najafi, Around operator monotone functions, Integr. Equ. Oper. Theory., 71, 575–582 (2011); Available online at http://dx.doi.org/10.1007/s00020-011-1921-0. [10] E. Set, M. Z. Sardari, M. E. Ozdemir, and J. Rooin, On generalizations of the Hadamard inequality for (α, m)-convex functions, RGMIA Res. Rep. Coll., 12:4, Article 4 (2009). [11] G. Toader,Some generalizations of the convexity, Proc. Colloq. Approx. Opt. ClujNapoca, 329–338 (1984) [12] S. H. Wang, B. Y. Xi, and F. Qi, Some new inequalities of Hermite-Hadamard type n-time differentiable functions which are m-convex, Analysis, 32, 247–262 (2012); Available online at http://dx.doi.org/10. 1524/anly.2012.1167. [13] S. H. Wang, B. Y. Xi, and F. Qi, On Hermite-Hadamard type inequalities for (α; m)-Convex Functions, Int. J. Open Problems Comput. Math., 5:4, 46–57 (2012); Available online at http://www.i-csrs.org.

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Non-periodic Multivariate Stochastic Fourier Sine Approximation and Uncertainty Analysis ∗ Zhihua Zhang1 and Palle E. T. Jorgensen2 1. College of Global Change and Earth System Science, Beijing Normal University, Beijing, China, 100875. 2. Department of Mathematics, University of Iowa, Iowa City, IA, USA, 52242 Email: [email protected],

[email protected]

Abstract. In data analysis, one needs to study Fourier sine analysis on the unit cube. However, for this kind of non-periodic case, no exact result is available. In this paper, firstly, based on our multivariate function decomposition, we deduce an asymptotic formula of Fourier sine coefficients of continuously differentiable functions f on [0, 1]d . Secondly, we deduce an asymptotic formula of hyperbolic cross approximations of Fourier sine series of f on [0, 1]d . By this way we can reconstruct high-dimensional signals by using fewest Fourier sine coefficients. Thirdly, we extend these results to Fourier sine analysis of stochastic processes and give uncertainty of stochastic Fourier sine approximation, i.e., we obtain expectations and variances of stochastic Fourier sine coefficients and stochastic Fourier sine approximation errors. Finally, we discuss some known stochastic processes.

Key words: asymptotic behavior, multivariate decomposition, stochastic approximation, hyperbolic cross truncation

1. Introduction It is well known that Fourier sine analysis on [0, 1]d is an important tool for signal processing. Based our decomposition of multivariate continuous functions on the cube [11], we first deduce an asymptotic formula of Fourier sine coefficients of continuously differentiable function f on [0, 1]d and obtain a necessary and sufficient condition: ¶ µ 1 cn (f ) = o n1 · · · nd for each nk → ∞. Next we deduce an asymptotic formula of hyperbolic cross truncations of the Fourier sine series of f . Thirdly, we extend these results to the case of stochastic processes. In detail, we will obtain the following three asymptotic behaviors of stochastic Fourier sine analysis. Suppose that ξ is a continuously differentiable stochastic process on [0, 1]d . ∗ Zhihua Zhang is a full professor at Beijing Normal University, China. He has published more than 40 first-authored papers in applied mathematics, signal processing and climate change. His research is supported by National Key Science Program No.2013CB956604; Fundamental Research Funds for the Central Universities (Key Program) No.105565GK; Beijing Young Talent fund and Scientific Research Foundation for the Returned Overseas Chinese Scholars, State Education Ministry. Zhihua Zhang is an associate editor of “EURASIP Journal on Advances in Signal Processing” (Springer, SCI-indexed) and an editorial board member in applied mathematics of “SpringerPlus” (Springer, SCI-indexed). Palle E. T. Jorgensen is a full professor at the University of Iowa, USA. He has published more than 100 papers and is an editorial board member of “Acta Applicandae Mathematicae” (Springer, SCI-indexed).

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(i) The expectation of Fourier sine coefficients cn (ξ) satisfy   d Y 2  (αn (ξ) + o(1)) E[ cn (ξ) ] =  πn j j=1

(n = (n1 , ..., nd ) ∈ Zd+ )

for each nk → ∞, where αn (ξ) is an algebraic sum of expectation of ξ at vertexes of the cube [0, 1]d . (ii) The variance of Fourier sine coefficients cn (ξ) satisfy   d Y 4  (θn (ξ) + o(1)) Var (cn (ξ)) =  2 n2 π j j=1 for each nk → ∞, where θn (ξ) is an algebraic sum of covariance of ξ(λ) and ξ(λ0 ), where λ and λ0 are any two vertexes of [0, 1]d . (h) (iii) The mean square error of hyperbolic cross truncations SN (ξ) (see (6.1)) of the Fourier sine series of ξ satisfies (h) E[k SN (ξ) − ξ k22 ] = WN (1 + o(1)) (N → ∞), the principal part WN is equal to µ WN =

1 π2

¶d

 

X p∈ΘN

 X 1  2 2 2 p1 p2 · · · p d

 E[|ξ(λ)|2 ]

λ∈{0,1}d

and WN ∼

logd−1 N , N

where {0, 1}d is the set of vertexes of the cube [0, 1]d . The number of Fourier sine coefficients in the N th (h) hyperbolic cross truncation SN (ξ) satisfies Nc ∼ N logd−1 N . So (h)

E[k ξ − SN (ξ) k22 ] ∼

log2d−2 Nc . Nc

However, the number of Fourier sine coefficients in the N th ordinary partial sum satisfies Nc = N d . So E[k ξ − SN (ξ) k22 ] ∼

1 1

.

Ncd

Finally, we discuss some known stochastic processes. 2. Preliminaries Denote the set of vertexes of the unit cube [0, 1]d by {0, 1}d and the boundary of [0, 1]d by ∂([0, 1]d ), and Zd+ = {(n1 , ..., nd )| each nk ∈ Z+ } and Z+ is the set of natural numbers. d f If f is a function defined on [0, 1]d and ∂t1∂···∂t continuous on [0, 1]d , we say f ∈ W ([0, 1]d ). If ξ is a stochastic d d

ξ continuous on [0, 1]d , we say ξ ∈ SW ([0, 1]d ). Denote the expectation and process defined on [0, 1]d and ∂t1∂···∂t d variance of a stochastic variable η by E[η] and Var(η), respectively. Denote the covariance and correlation of two stochastic variable ξ, η by Cov(ξ, η) and R(ξ, η), respectively. 2.1. Projection operators and fundamental polynomials

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We always assume e1 and e2 are two disjoint subsets of the set {1, 2, ..., d}. Define a projection operator Qe1 ,e2 from [0, 1]d to ∂([0, 1]d ) as Qe1 ,e2 (t1 , ..., td ) = (v1 , ..., vd ), (2.1) where

  0, k ∈ e1 , 1, k ∈ e2 , vk = S  tk , k ∈ e (e = {1, ..., d} \ (e1 e2 )).

The fundamental polynomial P (e1 ,e2 ) (t) is defined as Y Y P (e1 ,e2 ) (t) = (1 − tk ) tk . k∈e1

(2.2)

k∈e2

For example, consider the case d = 3. If e1 = {1, 3} and e2 = 2, then Qe1 ,e2 (t) = (0, 1, 0), P (e1 ,e2 ) (t) = t2 (1 − t1 )(1 − t3 ), where t = (t1 , t2 , t3 ) ∈ [0, 1]3 . If e1 = ∅ and e2 = {1, 2}, then Qe1 ,e2 (t) = (1, 1, t3 ), P (e1 ,e2 ) (t) = t1 t2 , where t = (t1 , t2 , t3 ) ∈ [0, 1]3 . 2.2. Decompositions of continuous functions on [0, 1]d Any continuous function f on the cube [0, 1]d can be decomposed into [11] f=

d+1 X

hν ,

(2.3)

ν=1

where h1 =

X

f (Qe1 ,e2 )P (e1 ,e2 ) .

|e1 |+|e2 |=d

hν =

X

fν−1 (Qe1 ,e2 )P (e1 ,e2 )

(2 ≤ ν ≤ d),

|e1 |+|e2 |=d−ν+1

hd+1 = f − h1 − · · · − hd , and f0 = f, fν−1 = fν−2 − hν−1

(2 ≤ ν ≤ d),

fd = fd−1 − hd . and the cardinality of a set F is denoted by |F |, and X Ae1 ,e2 := |e1 |+|e2 |=k

X

Ae1 ,e2 .

e1 ,e2 ∈{1,...,d} T e1 e2 =∅

|e1 |+|e2 |=k

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The following proposition shows the structure of each hν . Proposition 2.1 [11]. If f is a d−variate continuous function on the unit cube [0, 1]d , then (i) h1 is a d−variate polynomial and each f (Qe1 ,e2 t) is a constant and it is the value of f at a vertex of the cube [0, 1]d . Precisely say, f (Qe1 ,e2 t) = f (λ1 , ..., λd ), where

½ λk =

0, k ∈ e1 , S 1, k ∈ e2 (e1 e2 = {1, ..., d}).

(ii) for each 2 ≤ ν ≤ d, hν is a sum of products of a (ν − 1)−variate function fν−1 (Qe1 ,e2 ·) on [0, 1]ν−1 and (d − ν + 1)−variate polynomial P (e1 ,e2 ) , where each product is of separation of variables. Moreover, on ∂([0, 1]ν−1 );

fν−1 (Qe1 ,e2 ·) = 0

(iii) hd+1 is a d−variate function on [0, 1]d and hd+1 (·) = 0 on ∂([0, 1]d ). If ξ is a d-variate continuous stochastic process on [0, 1]d , then the above decomposition and Proposition 2.1 are still valid. For example, if ξ is a bivariate continuous function on [0, 1]2 , then ξ = h1 + h2 + h3 and h1 (t) = ξ(0, 0)(1 − t1 )(1 − t2 ) + ξ(0, 1)(1 − t1 )t2 + ξ(1, 0)t1 (1 − t2 ) + ξ(1, 1)t1 t2 , h2 (t) = ξ1 (1, t2 )t1 + ξ1 (0, t2 )(1 − t1 ) + ξ1 (t1 , 1)t2 + ξ1 (t1 , 0)(1 − t2 )

(ξ1 = ξ − h1 ),

h3 (t) = ξ1 (t) − h2 (t). We see from this decomposition that h1 is a polynomial determined by values of f at four vertexes of [0, 1]2 , h2 is a sum of products of separation of variables, and h2 (t) = 0 at four vertexes of [0, 1]2 , and the bivariate function h3 (t) vanishes on the boundary of [0, 1]2 . 2.3. Fourier sine series of stochastic processes Let a probability space (Ω, F, P ) be given. A stochastic variable ξ is defined as a function ξ from Ω to R or C. In this paper we always assume that ξ satisfies E[|ξ|2 ] < ∞, i.e., assume that ξ is a second-order stochastic variable. For a stochastic process ξ(t) on [0, 1]d , its autocorrelation function and covariance function are defined respectively as: Rξ (t, s) := E[ξ(t)ξ(s)] Cov(ξ(t), ξ(s)) := E[(ξ(t) − E[ξ(t)])(ξ(s) − E[ξ(s)])] We recall some known concepts in stochastic calculus [14, 15]. Let {ξn }∞ 1 be a sequence of second-order stochastic variables and ξ be a second-order stochastic variable. If lim E[|ξn − ξ|2 ] = 0, then we say {ξn }∞ 1 converges to ξ in the mean square sense. Based on the above concepts, n→∞ one can derive the concept of continuous and the concepts of the derivatives and the integrals of stochastic processes [3]. For a stochastic process ξ on [0, 1]d , the derivative and the expectation can be exchanged, the integral and the expectation can be exchanged, and Newton-Leibnitz formula holds. For a product of a stochastic process and a deterministic function, differential formula of products holds and the integration by parts also holds. 4

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Let ξ(t) be a stochastic process on [0, 1]d and Z E[ξ 2 (t)]dt < ∞. [0,1]d

Then ξ(t) can be expanded into the stochastic Fourier sine series ξ(t) =

X

cn (ξ) Tn (t)

( Tn (t) =

d Y

sin πnj tj )

(2.4)

j=1

n∈Zd +

in mean square sense, where the Fourier sine coefficients are stochastic variables and Z d cn (ξ) = 2 ξ(t) Tn (t) dt (n ∈ Zd+ ). [0,1]d

3. Asymptotic behavior of Fourier sine coefficients Let f be a continuous function on the unit cube [0, 1]d . By (2.3), f can be decomposed as f=

d+1 X

hν ,

ν=1

where {hν }d+1 are stated in Section 2.1 and 1 f0 = f, fν−1 = fν−2 − hν−1

(2 ≤ ν ≤ d).

If f ∈ W ([0, 1]d ), we easily prove fν−1 ∈ W ([0, 1]d )

(ν = 2, ..., d).

(3.1)

Based on this decomposition, we give an asymptotic representation of Fourier sine coefficients of f . Theorem 3.1. If f is a continuous function on [0, 1]d and f ∈ W ([0, 1]d ), then its Fourier sine coefficients cn (f ) possess the asymptotic behavior:   d Y 2  d (Kn (f ) + η1 + · · · + ηd ), cn (f ) =  πn j j=1 where ηk → 0 as nk → ∞ (k = 1, ., , , d) and X

Knd (f ) =

f (λ)²n (λ),

λ∈{0,1}d

( ²n (λ) =

Q

(−1)nj +1 ,

Gλ 6= ∅,

(3.2)

j∈Gλ

1,

Gλ = ∅,

where λ = (λ1 , ..., λd ) ∈ {0, 1}d and n = (n1 , ..., nd ), and Gλ = {j ∈ {1, ..., d}, λj = 1}.

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From this, we see that ²n (λ) = ±1 and Knd (f ) is an algebraic sum of values of f at vertexes of [0, 1]d . From Theorem 3.1, we deduce the following corollary. This corollary plays an important role in the proof of Theorem 4.1. Corollary 3.2. If f is a continuous function on [0, 1]d and f ∈ W ([0, 1]d ), then its Fourier sine coefficients cn (f ) satisfy à ! ¡ ¢ P P d (i) q∈{0,1}d |c2p+q (f )|2 = π22 p2 p21···p2 |f (λ)|2 + η10 + · · · + ηd0 ; 1 2 d λ∈{0,1}d à ! d Q 1 (ii) cn (f ) = (η1 + · · · + ηd ) if and only if f (λ) = 0 (λ ∈ {0, 1}d ) (ηk → 0 as nk → ∞). nj j=1 ´ ³ 1 for each nk → ∞ if and only if From Corollary 3.2, we deduce immediately that cn (f ) = o n1 ···n d f (λ) = 0 (λ ∈ {0, 1}d ). For example, consider the case d = 3. If f ∈ W ([0, 1]3 ), then Fourier sine coefficients cn1 ,n2 ,n3 (f ) possess the asymptotic behavior: cn1 ,n2 ,n3 (f ) =

8 ( f (0, 0, 0) − (−1)n1 f (1, 0, 0) − (−1)n2 f (0, 1, 0) n1 n2 n3 π 3 −(−1)n3 f (0, 0, 1) + (−1)n1 +n2 f (1, 1, 0) + (−1)n1 +n3 f (1, 0, 1) +(−1)n2 +n3 f (0, 1, 1) − (−1)n1 +n2 +n3 f (1, 1, 1) + η1 + η2 + η3 ),

where ηk → 0 as nk → ∞ (k = 1, 2, 3). Proof of Theorem 3.1. By (2.3), the Fourier sine coefficients cn (f ) satisfy cn (f ) =

d+1 X

cn (hν ),

(3.3)

ν=1

where cn (hν ) = 2d

R

hν (t)Tn (t)dt.

[0,1]d

First, we compute cn (h1 ). By (2.4), we have Z

X

cn (h1 ) = 2d

f (Qe1 ,e2 t)P (e1 ,e2 ) (t)Tn (t)dt.

|e1 |+|e2 |=d [0,1]d

By Proposition 2.1 (i), f (Qe1 ,e2 t) = f (λ) is a constant independent of t. So Z X d cn (h1 ) = 2 P (e1 ,e2 ) (t)Tn (t)dt, f (λ1 , ..., λd ) |e1 |+|e2 |=d

½ where λk =

[0,1]d

0, k ∈ e1 , S 1, k ∈ e2 (e1 e2 = {1, ..., d}). P (e1 ,e2 ) (t) =

Since Y

(1 − tj )

j∈e1

Tn (t) =

d Y

Y

tj ,

j∈e2

sin(πnj tj ),

t = (t1 , ..., td ),

j=1

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(3.4)

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a direct computation shows that Ã

R

P

(e1 ,e2 )

(t)Tn (t)dt

=

Q R1 j∈e1 0

[0,1]d

à =

1 πnj

j∈e1

j∈(e1

=

d Q

j=1

j∈e2

S

e2 )

! 1 πnj

1 πnj

Q

Q R1 j∈e2 0

! tj sin(πnj tj )dtj

!

Q !

Q

= Ã

(1 − tj ) sin(πnj tj )dtj !Ã

Q

Ã

!Ã

(−1)nj +1 πnj

Q

(−1)nj +1

j∈e2

(−1)nj +1

(e1

j∈e2

S

e2 = {1, ..., d}).

From this and (3.4), we get  cn (h1 ) = 2d 

 X

f (λ1 , ..., λd )

Since e1

(−1)nj +1 

j∈e2

|e1 |+|e2 |=d

S

Y

d Y 1 . πn j j=1

e2 = {1, ..., d} and e1 = {j ∈ {1, ..., d}, λj = 0}, and e2 = {j ∈ {1, ..., d}, λj = 1} =: Gλ ,   d Y 1  cn (h1 ) = 2d Knd (f )  , πn j j=1

where Knd (f ) =

P

f (λ)

λ∈{0,1}d

Q

(3.5)

(−1)nj +1 .

j∈Gλ

Next, we compute cn (hν ) (2 ≤ ν ≤ d). By (2.3) and (2.4), Z X d cn (hν ) = 2 fν−1 (Qe1 ,e2 t) P (e1 ,e2 ) (t)Tn (t)dt.

(3.6)

|e1 |+|e2 |=d−ν+1 [0,1]d

S Since |e1 | + |e2 | = d − ν + 1, we may denote e1 e2 = {β1 , ..., βd−ν+1 }. By (2.2), the fundamental polynomial P (e1 ,e2 ) (t) only depends on tβ1 , ..., tβd−ν+1 , write P (e1 ,e2 ) (t) = P (e1 ,e2 ) (tβ1 , ..., tβd−ν+1 ). S Since e = {1, ..., d} \ (e1 e2 ) and |e| = ν − 1, we may denote e = {α1 , ..., αν−1 }. Then fν−1 (Qe1 ,e2 t) only depends on tα1 , ..., tαν−1 , write e1 ,e2 (tα1 , ..., tαν−1 ). (3.7) fν−1 (Qe1 ,e2 t) = fν−1 So each product fν−1 (Qe1 ,e2 t)P (e1 ,e2 ) (t) in (3.6) is of separated variable type, and so Z (2) fν−1 (Qe1 ,e2 t) P (e1 ,e2 ) (t)Tn (t)dt = L(1) ν,n (e1 , e2 )Lν,n (e1 , e2 ), [0,1]d

where

Z L(1) ν,n (e1 , e2 ) =

P (e1 ,e2 ) (tβ1 , ..., tβd−ν+1 )

d−ν+1 Y

sin(πnβj tβj )dtβ1 · · · dtβd−ν+1 ,

j=1

[0,1]d−ν+1

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(3.8)

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Z e1 ,e2 fν−1 (tα1 , ..., tαν−1 )

L(2) ν,n (e1 , e2 ) =

ν−1 Y

sin(πnαj tαj )dtα1 · · · dtαν−1 .

j=1

[0,1]ν−1 (2)

We compute Lν,n (e1 , e2 ). We rewrite it as follows:  1  Z Z ν−1 Y e ,e 1 2  fν−1 sin(πnαj tαj )dtα2 · · · dtαν−1 . L(2) (tα1 , ..., tαν−1 ) sin(πnα1 tα1 )dtα1  ν,n (e1 , e2 ) =

(3.9)

j=2

0

[0,1]ν−2

e1 ,e2 From (3.1) and (3.7), we know that fν−1 ∈ W ([0, 1]ν−1 ). Using integration by parts, the part inside brackets:

Z1 e1 ,e2 e1 ,e2 fν−1 (tα1 , ..., tαν−1 ) sin(πnα1 tα1 )dtα1 = −fν−1 (tα1 , ..., tαν−1 ) 0

Z1

1 + πnα1

0

¯1 cos(πnα1 tα1 ) ¯¯ ¯ πnαk tα

1

=0

e1 ,e2 ∂fν−1 (tα1 , ..., tαν−1 ) cos(πnα1 tα1 )dtα1 . ∂tα1

By (3.7) and Proposition 2.1 (ii), we have ¯1 e1 ,e2 fν−1 (tα1 , ..., tαν−1 )¯t

αk =0

=0

(k = 1, ..., ν − 1).

(3.10)

Therefore, Z1 e1 ,e2 fν−1 (tα1 , ..., tαν−1 ) sin(πnα1 tα1 )dtα1 0

1 = πnα1

Z1 0

e1 ,e2 ∂fν−1 (tα1 , ..., tαν−1 ) cos(πnα1 tα1 )dtα1 . ∂tα1

From this and (3.9), we get L(2) ν,n (e1 , e2 ) =

1 πnα1



Z

Z1



e1 ,e2 ∂fν−1 (tα1 , ..., tαν−1 )

∂tα1

0

[0,1]ν−2

 sin(πnα2 tα2 ) dtα2 

ν−1 Y

sin(πnαj tαj )

j=3

cos(πnα1 tα1 ) dtα1 dtα3 · · · dtαν−1 .

(3.11)

Using integration by parts, the part inside brackets: R1 0

∂ ∂tα1

e1 ,e2 fν−1 (tα1 , ..., tαν−1 ) sin(πnα2 tα2 ) dtα2

¯1

∂f 1 2 (tα ,...,tαν−1 ) cos(πα2 tα2 ) ¯ ¯ − ν−1 ∂t1α πnα2 ¯ 1 e ,e

=

+ πn1α

R 2

[0,1]ν−1

tα2 =0

e ,e

1 2 (t ∂ 2 fν−1 α1 ,...,tαν−1 ) ∂tα1 ∂tα2

cos(πnα2 tα2 ) dtα2 .

By Proposition 2.1 (ii), fν−1 (tα1 , 0, tα3 , ..., tαν−1 ) = fν−1 (tα1 , 1, tα3 , ..., tαν−1 ) = 0, 8

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and so

e1 ,e2 ∂fν−1 (tα1 , 0, ..., tαν−1 ) ∂f e1 ,e2 (tα1 , 1, ..., tαν−1 ) = ν−1 = 0. ∂tα1 ∂tα1

Therefore,

R1 0

=

∂ ∂tα1

1 πnα2

e1 ,e2 (tα1 , ..., tαν−1 ) sin(πnα2 tα2 ) dtα2 fν−1

R

e ,e

[0,1]ν−1

1 2 (t ∂ 2 fν−1 α1 ,...,tαν−1 ) ∂tα1 ∂tα2

cos(πnα2 tα2 ) dtα2 .

From this and (3.11), we get 1 L(2) ν,n (e1 , e2 ) = (πnα1 )(πnα2 )

•

ν−1 Y



Z



Z1 ∂ 2 f e1 ,e2 (t α1 ,...,tαν−1 ) ν−1 ∂tα1 ∂tα2

0

[0,1]ν−2

 sin(πnα3 tα3 ) dtα3 

sin(πnαj tαj ) cos(πnα1 tα1 ) cos(πnα2 tα2 ) dtα1 dtα2 dtα4 · · · dtαν−1 .

j=4

Continuing this procedure, we deduce finally that     Z ν−1 ν−1 ν−1 e1 ,e2 Y Y 1 ∂ f (t , ..., t ) α1 αν−1 ν−1    cos(πnαj tαj ) dtα1 · · · dtαν−1 . L(2) ν,n (e1 , e2 ) = πn ∂t · · · ∂t α α α j 1 ν−1 j=1 j=1 [0,1]ν−1

Since

e1 ,e2 (tα1 , · · · , tαν−1 ) ∂ ν−1 fν−1 ∈ C([0, 1]ν−1 ) ∂tα1 · · · ∂tαν−1

and e = {α1 , ..., αν−1 }, applying the Riemann-Lebesgue lemma, we get     ν−1 Y 1 Y 1 =  ²e ,  L(2) ν,n (e1 , e2 ) = o n n j=1 αj j∈e j

(3.12)

where ²e → 0 as nj → ∞ (j ∈ e). S (1) We compute Lν,n (e1 , e2 ). Notice that e1 e2 = {β1 , ..., βd−ν+1 }. We assume in which e1 = {γ1 , ..., γm1 }, e2 = {δ1 , ..., δm2 }, where m1 + m2 = d − ν + 1. By (2.2), we get (1)

Lν,n (e1 , e2 ) R

=

P (e1 ,e2 ) (tβ1 , ..., tβd−ν+1 )

[0,1]d−ν+1

=

R

m Q1

[0,1]m1 j=1

d−ν+1 Q j=1

sin(πnβj tβj )dtβ1 · · · dtβd−ν+1 ,

(1 − tγj ) sin(πnγj tγj ) dtγ1 · · · dtγm1

R

m Q2

[0,1]m2 j=1

tδj sin(πnδj tδj ) dtδ1 · · · dtδm2 .

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A direct computation shows that 

   m2 nδj +1 Y 1 (−1)    = O L(1) ν,n (e1 , e2 ) = πn πn γ δ j j j=1 j=1 m1 Y

Y j∈e1

S

e2

 1 . nj

By (3.12) and (3.6), we have  d Y 1  ²e fν−1 (Qe1 ,e2 t) P (e1 ,e2 ) (t)Tn (t)dt =  n j j=1 

Z [0,1]d

and



 d Y 1  cn (hν ) = 2d  n j=1 j

where e = {1, ..., d} \ (e1

S

 X

²e  ,

|e1 |+|e2 |=d−ν+1

e2 ) and ²e → 0 as nj → ∞ (j ∈ e). From this, we can deduce that   d Y 1  (η1ν + · · · + ηdν ) (ν = 2, ..., d), cn (hν ) =  n j j=1

(3.13)

where ηk00 → 0 as nk → ∞ (k = 1, ..., d). Finally, we compute cn (hd+1 ). Since Z cn (hd+1 ) = 2

d

hd+1 (t1 , ..., td )

d Y

sin(πnj tj ) dt1 · · · dtd ,

j=1

[0,1]d

by Proposition 2.1 (iii) and (3.3), applying the integration by parts and the Riemann-Lebesgue lemma, we get cn (hd+1 ) =

d Q

2d

j=1

à = o

d Q

j=1

1 πnj

R [0,1]d

! 1 nj

∂ d hd+1 (t1 ,...,td ) ∂t1 ···∂td

à =

d Q

j=1

d Q j=1

cos(πnj tj ) dt1 · · · dtd

! 1 nj

²,

where ² → 0 as nj → ∞ (j = 1, ..., d). From this and (3.5), and (3.13), it follows by (3.3) that   d d X Y ¡ ¢ 1  Knd (f ) + η1 + · · · + ηd , cn (f ) = cn (h1 ) + cn (hν ) + cn (hd+1 ) = 2d  πnj ν=2 j=1 where Knd (f ) is stated in (3.5) and ηk → 0 as nk → ∞ (k = 1, ..., d). Theorem 3.1 is proved. ¤ d Proof of Corollary 3.2. Let n = 2 p + q (p ∈ Z+ , q ∈ {0, 1}d ). Then, for each q = (q1 , ..., qd ) and p = (p1 , ..., pd ), by Theorem 3.1, we have   d Y 1 d  (K2p+q c2p+q (f ) = 2d  (f ) + o(η1 + · · · + ηd )) π(2p + q ) j j j=1

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and

X

d K2p+q (f ) =

Y

f (λ)

(−1)qj +1 =: Kqd (f ).

j∈Gλ

λ∈{0,1}d

d From this, we see that K2p+q only depends on q. So, for q ∈ {0, 1}d , we have

 d Y 1  (Kqd (f ) + ηˆ1 + · · · + ηˆd , c2p+q (f ) =  πp j j=1 

where ηˆk → 0 as pk → ∞ (k = 1, ..., d) and  d X Y |c22p+q (f )| = 

 1  π 2 p2j j=1

q∈{0,1}d

X

¡

(3.14)

 ¢ 2 Kqd (f ) + η10 + · · · + ηd0  ,

(3.15)

q∈{0,1}d

where ηk0 → 0 as pk → ∞ (k = 1, ..., d). By (3.2), we have X

(Kqd (f ))2 =

q∈{0,1}d

where

Y

0

²q (λ)²q (λ ) =

X

X

λ∈{0,1}d

λ0 ∈{0,1}d

qj +1

(−1)

j∈Gλ

X

f (λ)f (λ0 )

q∈{0,1}d P

Y

²q (λ)²q (λ0 ).

qj +1

(−1)

qj +

j∈Gλ

P

j∈G 0 λ

= (−1)

qj +|Gλ |+|Gλ0 |

.

j∈Gλ0

When λ 6= λ0 , without loss of generality, we assume that i ∈ Gλ , i 6∈ Gλ0 . So we have P

X

0

²q (λ)²q (λ ) =

1 X

···

q1 =0

q∈{0,1}d

When λ = λ0 . Then

1 X

1 X

···

qi−1 =0 qi+1 =0

X

0

²q (λ)²q (λ ) =

1 X

P

j∈G 0 λ

(−1)

qj +|Gλ |+|Gλ0 |

1 X

1 X

(−1)qi = 0.

(3.16)

qi =0

qd =0

···

q1 =0

q∈{0,1}d

qj +

j∈Gλ j6=i

1 X

2

(−1)

P

qj +2|Gλ |

j∈Gλ

= 2d .

(3.17)

qd =0

From this, we get 

 X q∈{0,1}d

 (Kqd (f ))2 =  

X

+

λ,λ0 ∈{0,1}d λ=λ0

X

  f (λ)f (λ0 ) 

X

²q (λ)²q (λ0 ) = 2d

q∈{0,1}d

λ, λ0 ∈{0,1}d λ6=λ0

X

f 2 (λ).

λ∈{0,1}d

From this and (3.15), we get (i). If  d Y 1  (η1 + · · · + ηd ), cn (f ) =  n j j=1 

we have

(3.18)

 d Y 1  (η1 + · · · + ηd ). |c2p+q |2 =  2 p d j=1 j 

X q∈{0,1}

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By (i), the later is equivalent to

X

f 2 (λ) = 0

λ∈{0,1}d

which is equivalent to f (λ) = 0 (λ ∈ {0, 1}d ). Conversely, if f (λ) = 0 (λ ∈ {0, 1}d ), then, by Theorem 3.1, we deduce (3.18). So we get (ii). Corollary 3.2 is proved. ¤ 4. Asymptotic behaviors of hyperbolic cross truncation approximation Approximation rate of multivariate functions by partial sum of Fourier sine series deteriorates rapidly as the dimension d increases. Let f be a continuous function on [0, 1]d . Denote the partial sums of its Fourier sine series by SN (f ): N X

SN (f ; t) =

cn (f ) Tn (t)

n = (n1 , ..., nd ),

n1 ,...,nd =1

where cn (f ) = 2d

R

f (t)Tn (t) dt and Tn (t) is stated in (2.4).

[0,1]d

If f is a continuous function on [0, 1]d and f ∈ W ([0, 1]d ), then, by the Parseval identity, the partial sum SN (f ) of the Fourier sine series of f satisfies à ! ∞ N P P d 2 2 k SN (f ) − f k2 = − c2n1 ,...,nd (f ) n1 ,...,nd =1

Ã

n1 ,...,nd =1

∞ P

= O(1)

−

n1 ,...,nd =1

à = O(1)

d P

ν=1

à = O(1)

d P

ν=1

n1 ,...,nd =1

à d! ν!(d−ν)!

Ã

!

N P

1 n21 n22 ···n2d

!Ã

∞ P

n1 ,...,nν =N +1

!ν µ

∞ P

k=N +1

1 k2

N P k=1

1 n21 ···n2ν

nν+1 ,...,nd =1

¶d−ν+1 ! 1 k2

!!

N P

=O

¡1¢ N

(4.1)

1 n2ν+1 ···n2d

.

In the partial sum of Fourier sine series, the number of its Fourier sine coefficients: Nc = N d . So, by (4.1), it follows that for f ∈ W ([0, 1]d ), the partial sums SN (f ) satisfy à ! µ ¶ 1 1 2 =O k f − SN (f ) k2 = O . 1 N Ncd Consider hyperbolic cross truncations of the Fourier sine series of f on [0, 1]d . The Fourier sine series of f can be rewritten in the form f (t) =

∞ X

X

c2p+q T2p+q (t)

(p = (p1 , ..., pd ) ∈ Z+ ).

p1 ,...,pd =1 q∈{0,1}d

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Define the hyperbolic cross truncations of Fourier sine series of f are X

X

1≤|p|≤N −1

q∈{0,1}d

(h)

SN (f ; t) =

1≤p1 ,...,pd ≤N −1

where |p| =

d Q

c2p+q T2p+q (t),

pk .

k=1

Based on asymptotic formula, we deduce that the asymptotic formula of hyperbolic cross truncations of Fourier sine series Denote ( ) d Y ΘN = p = (p1 , ..., pd ) ∈ Zd+ p1 ∈ Z+ , 1 ≤ p2 , ..., pd ≤ N − 1, pk ≥ N . (4.2) k=1 (h)

The difference f (t) − SN (f ; t) is equal to X

Ã

X

∞ X

c2p+q T2p+q (t) +

∞ X

−

p1 ,...,pd =1

p∈ΘN q∈{0,1}d

!

N X

p1 =1 p2 ,...,pd =1

X

c2p+q T2p+q (t).

q∈{0,1}d

By the Parseval identity, we deduce that (h)

2d k f − SN (f ) k22

P

P

p∈ΘN

q∈{0,1}d

=

Ã

∞ P

+

∞ P

−

p1 ,...,pd =1

c22p+q !

N P

p1 =1 p2 ,...,pd =1

P q∈{0,1}d

c22p+q

(4.3)

= PN (f ) + QN (f ). By Corollary 3.2 and (4.1), Ã QN (f ) ≤

∞ P

−

p1 ,...,pd =1

à = O(1)

!

N P p1 ,...,pd =1

∞ P

−

p1 ,...,pd =1

P q∈{0,1}d

c22p+q (4.4)

!

N P p1 ,...,pd =1

1 p21 ···p2d

=O

¡1¢ N

.

By Corollary 3.2, it follows that PN (f ) =

¡

2 π2

¢d P p∈ΘN

(1)

à 1 p21 ···p2d

!

P

λ∈{0,1}d

|f (λ)|2 + η10 + · · · + ηd0 (4.5)

(2)

= PN + PN , where

µ (1)

PN =

2 π2

¶d X p∈ΘN

p21

1 · · · p2d

X

|f (λ)|2 ,

λ∈{0,1}d

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X

(2)

PN = O(1)

1 (η 0 + · · · ηd0 ), p21 · · · p2d 1

p∈ΘN

and ηk → 0 as pk → ∞ (k = 1, ..., d). (1) We estimate the order of PN → 0 as N → ∞. Notice that Z X 1 dx1 · · · dxd (1) ∼ PN ∼ 2 =: RN 2 p21 · · · p2d x1 ,...,xd ≥N x1 · · · xd p∈ΘN

and

Z

Z

N

1

Z

N x1

dx1

RN =

dx2 · · ·

1

N x1 ···xk

Z dxk+1 · · ·

1

∞ N x1 ···xd−1

dxd . x21 · · · x2d

A direct computation shows that R∞ N x1 ···xd−1

R

dxd x21 ···x2d

N x1 ···xd−2

dxd−1

1

and

R

R∞

=

1 x21 ···x2d−1

=

1 N x1 ···xd−1 ,

R∞

d

xd x21 ···x2d

N x1 ···xd−1

N x1 ···xd−3

1

=

1 N x1 ···xd−3

=

1 N x1 ···xd−3

∼

1 N x1 ···xd−3

dxd−2 R

R

1

N x1 ···xd−3

N x1 ···xd−3

1

log2

1 xd−2

dxd x2d

=

1 N x1 ···xd−2

=

1 N x1 ···xd−2

N x1 ···xd−2

1

R

N x1 ···xd−1

dxd−1

N x1 ···xd−2

1

N x1 ···xd−1

dxd−1 xd−1

N x1 ···xd−2 ,

log

R∞

N x1 ···xd−2

log

R

dxd x21 ···x2d

dxd−2

log u u du

N x1 ···xd−3 .

Continuing this procedure, we deduce that (1)

PN ∼ RN ∼

1 N

Z

N

1

1 N logd−1 N . logd−2 dx1 ∼ x1 x1 N

(4.6)

(2)

We estimate PN . Let (k)

SN =

X p∈ΘN

p21

ηk0 · · · p2d

(k = 1, ..., d).

From this and (4.6), we deduce that (1) SN

=

X p∈ΘN

η10 =O p21 · · · p2d

µ

1 N

¶Z 1

N

1 N 0 logd−2 η dx1 . x1 x1 1

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Since η10 → 0 as x1 → ∞ and

R∞ 1

1 x1

logd−2 µ

(1)

SN = o

N x1 dx1

¶Z

1 N

= ∞, by a known result in Calculus and (4.6), Ã ! 1 logd−1 N d−2 N log dx1 = o . x1 x1 N

N

1

(1)

An argument similar to SN shows that for each k, Ã (k) SN

=o

logd−1 N N

! .

From this and (4.2)-(4.6), we get (h)

(1)

2d k f − SN k22 = PN (1 + o(1)). (h)

Theorem 4.1. Let f ∈ W ([0, 1]d ). Then hyperbolic cross truncations SN (f ; t) satisfy (h) (1) k f − SN (f ) k22 = PeN (1 + o(1))

where

µ (1) PeN =

1 π2

¶d



(N → ∞),

 X



p∈ΘN

1  p21 · · · p2d

 X

|f (λ)|2 

λ∈{0,1}d

and ΘN = {p = (p1 , ..., pd ) :

p1 ∈ Z+ ,

1 ≤ p2 , ..., pd ≤ N − 1,

p1 , ..., pd ≥ N }

(4.7)

and

logd−1 N . N We easily see that the number of Fourier sine coefficients in hyperbolic cross truncations satisfies (1)

PN ∼

Nc ∼ N logd−1 N. In fact,

P

Nc =

P

1

p∈ΘN q∈{0,1}d

∼ ∼

RN 1

RN 1

dx1 dx1

R

N x1

1

R

N x1

1

dx2 · · · dx2 · · ·

R

N x1 ···xd−1

1

R

N x1 ···xd−2

1

dxd N x1 ···xd−1 dxd−1

∼ N logd−1 N. Therefore, by Theorem 4.1, (h)

k f − SN (f ) k22 ∼

log2d−2 Nc . Nc

From this, we see that for f ∈ W ([0, 1]d ), the hyperbolic cross approximation of Fourier sine series is a better approximation tool than ordinary partial sum approximation. 5. Asymptotic behaviors of stochastic Fourier sine coefficients 15

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We extend the results in Sections 3-4 to stochastic processes. Let ξ(t) be a continuous stochastic process on [0, 1]d . Then ξ(t) can be expanded into the stochastic Fourier sine series ξ(t) =

X

cn (ξ) Tn (t)

(5.1)

n∈Zd +

in mean square sense, where

Z cn (ξ) = 2d

ξ(t)Tn (t) dt

[0,1]d

and Tn (t) is stated in (2.4). The Fourier sine coefficients are stochastic variables. We discuss their expectations, second-order moments, and variances. Theorem 5.1. If ξ is a stochastic process on [0, 1]d and ξ ∈ SW ([0, 1]d ), then the expectations, second-order moments, and variances of its ! Fourier sine coefficients possess the following asymptotic behaviors: Ã d Q 2 (i) E[ cn (ξ) ] = (αn (ξ) + r1 + · · · + rd ) and πnj j=1

αn (ξ) =

X

( E[ ξ(λ) ]²n (λ) ) ,

λ∈{0,1}d

where αn (ξ) is an algebraic sum of expectation of ξ at vertexes of the cube [0, 1]d and rk → 0 as nk → ∞, and ²n (λ) is stated in (3.2). ! Ã d Q 4 2 (βn (ξ) + r10 + · · · + rd0 ) and (ii) E[ cn (ξ) ] = π 2 n2 j=1

j

X

βn (ξ) =

X

E[ ξ(λ)ξ(λ0 ) ]²n (λ)²n (λ0 ),

λ∈{0,1}d λ0 ∈{0,1}d

where rk0 → 0 as nk →Ã∞. d Q (iii) Var [ cn (ξ) ] =

j=1

! 4 π 2 n2j

(θn (ξ) + r100 + · · · + rd00 ), where

θn (ξ) =

X

X

λ∈{0,1}d

λ0 ∈{0,1}d

Cov(ξ(λ), ξ(λ0 )) ²n (λ) ²n (λ0 ).

and rk00 → 0 as nk → ∞. For example, consider the case d = 2. Assume that a stochastic process ξ ∈ SW ([0, 1]2 ). Then E[ cn (ξ) ] = and

4 ( E[ ξ(0, 0) ] − (−1)n1 E[ ξ(1, 0) ] − (−1)n2 E[ ξ(0, 1) ] + E[ ξ(1, 1) ] + r1 + r2 ) n1 n2 π 2

Var (cn (ξ)) =

16 (η0,0 n21 n22 π 4

− (−1)n1 η0,1 − (−1)n2 η0,2 + η0,3 − (−1)n1 η1,0

+ η1,1 + (−1)n1 +n2 η1,2 − (−1)n1 η1,3 − (−1)n2 η2,0 + (−1)n1 +n2 η2,1 + η2,2 − (−1)n2 η2,3 + η3,0 − (−1)n1 η3,1 − (−1)n2 η3,2 + η3,3 + r10 + r20 ),

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where ηλ1 +2λ2 ,λ01 +2λ02 = Cov (ξ(λ1 , λ2 ), ξ(λ01 , λ02 ))

(λ1 , λ2 , λ01 , λ02 = 0 or 1)

and r1 , r10 → 0 as n1 → ∞ and r2 , r20 → 0 as n2 → ∞. Proof of Theorem 5.1. Exchanging the expectation and integral, we deduce from (5.1) that Z d E[ cn (ξ) ] = 2 E[ ξ(t) ] Tn (t) dt = c n ( E [ ξ(t) ] ), [0,1]d

i.e., E[ cn (ξ) ] is the Fourier sine coefficients of the deterministic function E[ ξ(t) ]. Exchanging the expectation and partial derivative, we deduce from ξ ∈ SW ([0, 1]d ) that E[ ξ(t) ] ∈ W ([0, 1]d ). Using Theorem 3.1, we get (i). R By (5.1), we get |cn (ξ)|2 = 22d

R

ξ(t)ξ(s) Tn (t)Tn (s)dt ds, and so

[0,1]d [0,1]d

Z

Z 2

E[ |cn (ξ)| ] = 2

Rξ (e t) Tn (t)Tn (s) dt ds,

2d [0,1]d

(5.2)

[0,1]d

where the autocorrelation function Rξ (e t) = E[ ξ(t)ξ(s) ] is a 2d−variate deterministic function and t = (t1 , ..., td ),

s = (s1 , ..., sd ),

e t = (t1 , ..., t2d )

(td+i = si , i = 1, ..., d).

Let nd+j = nj (j = 1, ..., d). Then (5.2) can be rewritten into Z 2

E[ |cn1 ,...,nd (ξ)| ] = 2

Rξ (e t)

2d

2d Y

sin(πnj tj )de t.

j=1

[0,1]2d

From the definition of Fourier sine coefficients, we see that E[ |cn1 ,...,nd (ξ)|2 ] is the Fourier sine coefficient of 2d−variate function Rξ , that is, E[ |cn1 ,...,nd (ξ)|2 ] = cn1 ,...,n2d (Rξ )

(nd+j = nj , j = 1, ..., d).

(5.3)

By the assumption ξ ∈ SW ([0, 1]d ), we deduce that Rξ ∈ W ([0, 1]2d ). In Theorem 3.1, replacing f by Rξ and d by 2d and letting nd+j = nj (j = 1, ..., d), we obtain   2d Y 2  cn1 ,...,n2d (Rξ ) =  ( βn1 ,...,nd (ξ) + r1 + · · · + rd ) , (5.4) πn j j=1 where

 X

βn1 ,...,nd (ξ) = Kn2d1 ,...,n2d (Rξ ) =

2d e λ∈{0,1}

e  Rξ (λ)

 Y

(−1)nj +1 

j∈Gλ e

17

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and e = (λ1 , ..., λ2d ) = (λ1 , ..., λd , λ0 , ..., λ0 ), λ 1 d Gλe = {j ∈ {1, ..., 2d}, λj = 1}. By nd+j = nj (j = 1, ..., d), we have Q

à nj +1

(−1)

=

j∈Gλ e

j∈(Gλ e

Q

=

!Ã

Q T

(−1)

{1,...,d})

(−1)nj +1

j∈Gλ

j∈(Gλ e

Q

!

Q

nj +1

T

nj +1

(−1)

{d+1,...,2d})

(5.5)

(−1)nj +1 ,

j∈Gλ0

where Gλ = {j ∈ {1, ..., d}, λj = 1}, Gλ0 = {j ∈ {1, ..., d}, λ0j = 1}. Since Rξ (λ1 , ..., λ2d ) = E[ξ(λ1 , ..., λd )ξ(λ01 , ..., λ0d )] by (5.5), we deduce by (5.4) that X βn1 ,...,nd (ξ) =

X

E[ ξ(λ1 , ..., λd )ξ(λ01 , ..., λ0d ) ]

(λd+j = λ0j ), Y

(−1)nj +1

j∈Gλ

λ∈{0,1}d λ0 ∈{0,1}d

Y

(−1)nj +1 .

(5.6)

j∈Gλ0

From this and (5.3)-(5.4), we get (ii). From (i), E[c2n (ξ)] =

d Y

4

π 2 n2j j=1

(αn2 (ξ) + re1 + · · · + red ),

where each rek → 0 as nk → ∞. Again, by (ii), Var(cn (ξ)) = E[c2n (ξ)] − |E[cn (ξ)]|2 = where rk00 → 0 as nk → ∞. Noticing that βn (ξ) − αn2 (ξ)

=

P

d Y

4

π 2 n2j j=1

(βn (ξ) − αn2 (ξ) + r100 + · · · + rd00 ),

(E[ξ(λ)ξ(λ0 )] − E[ξ(λ)]E[ξ(λ0 )]) ²n (λ)²n (λ0 )

λ,λ0 ∈{0,1}d

=

P

Cov(ξ(λ), ξ(λ0 ))²n (λ)²n (λ0 ),

λ,λ0 ∈{0,1}d

we get (iii). Theorem 5.1 is proved. ¤ 6. Asymptotic behavior of hyperbolic cross hyperbolic cross approximations of stochastic Fourier sine series Let ξ(t) be a continuous stochastic process on [0, 1]d . The hyperbolic cross truncation of its Fourier sine series is X X (h) SN (ξ, t) = c2p+q T2p+q (t), (6.1) |p|≤N −1 q∈{0,1}d

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where |p| =

d Q k=1

pk (p = (p1 , ..., pd ) ∈ Zd+ ), is a stochastic sine polynomial. We give an asymptotic behavior of

the hyperbolic cross approximation. Theorem 6.1. Let ξ be a stochastic process on [0, 1]d . If ξ ∈ SW ([0, 1]d ). Then the hyperbolic cross (h) truncations SN (ξ) of the stochastic Fourier sine series of ξ satisfy (h)

E[ k SN (ξ) − ξ k22 ] = WN (ξ)(1 + o(1)) where

µ WN (ξ) =

1 π2

¶d X p∈ΘN

p21

1 · · · p2d

(N → ∞),

X

E[|ξ(λ)|2 ]

λ∈{0,1}d

and ΘN is stated in (4.2) and

logd−1 N . N Proof. By using an argument similar to Section 4, we deduce that (4.3) is still valid when f is replaced by ξ. Taking expectation on both sides, (1)

PN (ξ) ∼

(h)

2d E[k ξ − SN (ξ) k22 ] P

P

p∈ΘN

q∈{0,1}d

=

Ã

∞ P

+

E[c22p+q (ξ)] ∞ P

−

p1 ,...,pd =1

!

N P

p1 =1 p2 ,...,pd =1

P q∈{0,1}d

E[c22p+q (ξ)]

= PN (ξ) + QN (ξ). By Theorem 5.1,

µ E[c22p+q (ξ)] = O

This implies that QN (ξ) = O

¡1¢ N

1 p21 · · · p2d

¶ .

. By Theorem 5.1 (ii), it follows that (h)

2d E[ k ξ − SN (ξ) k22 ] =

1 π 2d

P p∈ΘN

(1)

à 1 p21 ···p2d

!

P

q∈{0,1}d

(2)

=: Pn (ξ) + PN (ξ) + O

β2p+q + r100 + · · · + rd00

¡1¢ N

,

where ΘN is stated in (4.7) and each rk00 → 0 as nk → ∞, and  X 1 X 1 (1)  PN (ξ) = 2d π p21 · · · p2d p∈ΘN

(6.2)

 β2p+q  ,

q∈{0,1}d

 X 1 1 (2)  PN (ξ) = 2d π p21 · · · p2d p∈ΘN

 X

(r100 + · · · + rd00 ) .

q∈{0,1}d

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By (3.17), P

β2p+q

=

q∈{0,1}d

=

1 π 2d

2d π 2d

So

P

E[ξ(λ)ξ(λ0 )]

λ,λ0 ∈{0,1}d

P

! ²2p+q (λ)²2p+q (λ0 )

q∈{0,1}d

E[ξ 2 (λ)].

λ∈{0,1}d

µ (1)

PN (ξ) =

Ã

P

2 π2

¶d X

1 · · · p2d

p21

p∈ΘN

X

E[|ξ(λ)|2 ].

λ∈{0,1}d

Similar to the argument of Theorem 4.1, (1)

PN (ξ) ∼

X p∈ΘN

p21

and

logd−1 N 1 ∼ 2 · · · pd N

à (2) PN (ξ)

=o

logd−1 N N

! .

From this and (6.2), we deduce the desired result. ¤ 7. Examples In data analysis, the following three stochastic processes are often used [16]. (i) Gaussian stochastic process ξSE (t) with mean 0 and square exponential covariance function: ¶ µ kt−t0 k2 − 2l2 2

0

KSE (t, t ) = e where t = (t1 , ..., td ) and t0 = (t01 , ..., t0d ), and k t k22 =

d P k=1

,

(7.1)

t2k , and l > 0.

(ii) Gaussian stochastic process ξRQ (t) with mean 0 and rational quadratic covariance function: KRQ (t, t0 ) = (1+ k t − t0 k2 )−α , where t = (t1 , ..., td ) and t0 = (t01 , ..., t0d ), and α ≥ 0. (iii) Gaussian stochastic process ξL (t) with mean 0 and linear covariance function: KL (t, t0 ) =< t, t0 >, where t = (t1 , ..., td ) and t0 = (t01 , ..., t0d ), and < t, t0 >=

d P k=1

tk t0k .

Since these stochastic processes are differentiable [14, 15], we can use the theorems in Sections 5-6 to research their Fourier sine expansions, including variance estimates of Fourier sine coefficients and asymptotic formulas of hyperbolic cross truncation approximation. We expand ξSE into a Fourier sine series on [0, 1]d as follows: X cn (ξSE )Tn (t), ξSE (t) = n∈Zd +

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R

where cn (ξSE ) = 2d

ξSE (t)Tn (t)dt.

[0,1]d

Since E[ξSE (t)] = 0 (t ∈ [0, 1]d ), by Theorem 5.1 (i), we have µ ¶ 1 E[cn (ξSE )] = o as each nk → ∞ n1 · · · nd

(n = (n1 , ..., nd )).

Let t = (t1 , ..., td ) and t0 = (t01 , ..., t0d ). Then d X

k t − t0 k22 =

(tk − t0k )2 ,

k=1

and by (7.1), Cov(ξSE (t), ξSE (t0 )) =

d Y

0

1

2

e− 2l2 (tk −tk ) .

k=1

By Theorem 5.1 (iii), we obtain that the Fourier sine coefficients cn (ξSE ) satisfy, Var(cn (ξSE )) =

d Y

4 (θ (ξ ) + r100 + · · · + rd00 ) 2 n2 n SE π j j=1

where θn (ξSE ) =

Ã

X

X

d Y

and rk00 → 0 as nk → ∞. !

− 2l12 (λk −λ0k )2

e

εn (λ)εn (λ0 ),

k=1

λ∈{0,1}d λ0 ∈{0,1}d

0

and εn (λ) is stated in (3.2) and λ = (λ1 , ..., λd ) and λ = (λ01 , ..., λ0d ). Next we find the asymptotic formula of the hyperbolic cross truncation approximation of Fourier sine series of ξSE . The hyperbolic cross truncation is X X (h) SN (ξSE , t) = c2p+q (ξSE )T2p+q (t) (pi ∈ Z+ , i = 1, ..., d). p1 ···pd ≤N −1 q∈{0,1}d

P

Note that E[|ξSE (t)|2 ] = RSE (0) = 1 and

1 = 2d . By theorem 6.1, we have

λ∈{0,1}d (h)

E[k SN (ξSE ) − ξSE k22 ] = WN (ξSE )(1 + o(1)) where

µ WN (ξSE ) =

1 π2

¶d X p∈ΘN

1 p21 · · · p2d

So we get the asymptotic formula as follows: µ E[k

(h) SN (ξSE )

− ξSE k22 ] =

2 π2

¶d

X λ∈{0,1}d

 

X

p∈ΘN

µ 1=

4 π2

(N → ∞), ¶d X p∈ΘN

 1  (1 + o(1)) p21 · · · p2d

1 . p21 · · · p2d

(N → ∞),

where ΘN is stated in (4.2) and the number Nc of Fourier sine coefficients in the hyperbolic cross truncation (h) SN (ξSE ) is equivalent to N logd−1 N . Similarly, for the stochastic processes ξRQ and ξL , using the same method as above, we can give the variance estimates of their Fourier sine coefficients and asymptotic formulas of hyperbolic cross truncation approximations.

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References [1] V. Barthelmann, E. Novak, and K. Ritter, High dimensional polynomial interpolation on sparse grids, Advances in Computational Mathematics, 12(4) (2000), 273-288. [2] M. Griebel and J. Hamaekers, Sparse grids for the Schr¨ odinger equation, ESALM Mathematical Modelling and Numerical Analysis, 41 (2007), 215-247. [3] E. C. Klebaner, Introduction to stochastic calculus with application, World scientific publishing, 2012. [4] C. Lanczos, Discourse on Fourier series, Hafner Pub. Co. New York, 1966. [5] J. Shen and L. Wang, Sparse spectral approximations of high-dimensional problems based on hyperbolic cross, SIAM Journal on Numerical Analysis, 48 (3) (2010), 1087-1109. [6] J. Shen and H. Yu, Efficient spectral sparse grid methods and applications to high-dimensional elliptic problems, SIAM Journal on Scientific Computing, 32(6) (2010), 3228-3250. [7] E. M. Stein and G. Weiss, Introduction to Fourier analysis on Euclidean spaces, Princeton Univ. Press, 1971. [8] T. Szabados, An elementary introduction to the Wiener process and stochastic integrals. Studia Sci. Math. Hung. 31 (1996), 249-297. [9] A. F. Timan, Theory at approximation of functions of a real variable, Pergamon, 1963. [10] L. Villafuerte and B. M. Chen-Charpentier, A random differential transform method: Theory and applications, Appl. Math. Letters, 25(10) (2012), 1490-1494. [11] Z. Zhang, Decomposition and approximation of multivariate functions on the cube, Acta. Math. Sinica, 29(1) (2013), 119-136. [12] A. Zygmund, Trigonometric series, I, II, 2nd Edition, Cambridge, 1968. [13] Z. Zhang, Approximation of bivariate functions via smooth extensions, The Scientific World Journal, vol. 2014, Article ID 102062, pp 1-16, 2014. [14] B. Hajek, An Exploration of Random Processes for Engineers, 2012, see: jek/Papers/randomprocDec11.pdf

http://www.ifp.illinois.edu/ Ha-

[15] J.-P. Kahane, Some random series of functions, Cambridge Studies in Advanced Mathematics, Volume 5, 2rd, Cambridge University Press, Cambridge, 1994 [16] C. E. Rasmussen, C.K.I. Williams, Gaussian Processes for Machine Learning, MIT Press, 2006.

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TABLE OF CONTENTS, JOURNAL OF COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 22, NO. 4, 2017

Some New Results on Products of the Apostol-Genocchi Polynomials, Yuan He,…………591 Functional Inequalities in Fuzzy Normed Spaces, Choonkil Park, George A. Anastassiou, Reza Saadati, and Sungsik Yun,…………………………………………………………………..601 A Generalization of Simpson Type Inequality via Differentiable Functions Using Extended (𝑠, 𝑚)𝜙 -preinvex functions, Yujiao Li, and Tingsong Du,………………………………….613

Isometric Equivalence of Linear Operators on Some Spaces of Analytic Functions, Li-Gang Geng,………………………………………………………………………………………..633 S-Fuzzy Subalgebras and Their S-Products in BE-Algebras, Sun Shin Ahn,Keum Sook So,639 Fixed Point Results for Generalized g-Quasi-Contractions of Perov-type in Cone Metric Spaces Over Banach Algebras Without the Assumption of Normality, Shaoyuan Xu, Branislav Z. Popovic, Stojan Radenovic,.…………………………………………………………………648 On Some Inequalities of the Bateman's G-Function, Mansour Mahmoud, Hanan Almuashi,672 3-Variable Additive 𝜌-Functional Inequalities in Fuzzy Normed Spaces, Joonhyuk Jung, Junehyeok Lee, George A. Anastassiou, and Choonkil Park,…………………......................684 An Approach to Separability of Integrable Hamiltonian System, Hai Zhang,……………….699 Cross-Entropy for Generalized Hesitant Fuzzy Sets and Their Use in Multi-Criteria Decision Making, Jin Han Park, Hee Eun Kwark, and Young Chel Kwun,……………………… 709 On Harmonic Quasiconformal Mappings with Finite Area, Hong-Ping Li, and Jian-Feng Zhu,726 Weak Estimates of the Multidimensional Finite Element and Their Applications, Yinsuo Jia, and Jinghong Liu,…………………………………………………………………………………..734 New Integral Inequalities of Hermite-Hadamard Type for Operator m-Convex and (𝛼, 𝑚)-Convex Functions, Shuhong Wang,………………………………………………...….744 Non-periodic Multivariate Stochastic Fourier Sine Approximation and Uncertainty Analysis, Zhihua Zhang, and Palle E. T. Jorgensen,……………………………………………………..754

Volume 22, Number 5 ISSN:1521-1398 PRINT,1572-9206 ONLINE

May 2017

Journal of Computational Analysis and Applications EUDOXUS PRESS,LLC

Journal of Computational Analysis and Applications ISSNno.’s:1521-1398 PRINT,1572-9206 ONLINE SCOPE OF THE JOURNAL An international publication of Eudoxus Press, LLC (fourteen times annually) Editor in Chief: George Anastassiou Department of Mathematical Sciences, University of Memphis, Memphis, TN 38152-3240, U.S.A [email protected] http://www.msci.memphis.edu/~ganastss/jocaaa The main purpose of "J.Computational Analysis and Applications" is to publish high quality research articles from all subareas of Computational Mathematical Analysis and its many potential applications and connections to other areas of Mathematical Sciences. Any paper whose approach and proofs are computational,using methods from Mathematical Analysis in the broadest sense is suitable and welcome for consideration in our journal, except from Applied Numerical Analysis articles. Also plain word articles without formulas and proofs are excluded. The list of possibly connected mathematical areas with this publication includes, but is not restricted to: Applied Analysis, Applied Functional Analysis, Approximation Theory, Asymptotic Analysis, Difference Equations, Differential Equations, Partial Differential Equations, Fourier Analysis, Fractals, Fuzzy Sets, Harmonic Analysis, Inequalities, Integral Equations, Measure Theory, Moment Theory, Neural Networks, Numerical Functional Analysis, Potential Theory, Probability Theory, Real and Complex Analysis, Signal Analysis, Special Functions, Splines, Stochastic Analysis, Stochastic Processes, Summability, Tomography, Wavelets, any combination of the above, e.t.c. "J.Computational Analysis and Applications" is a peer-reviewed Journal. See the instructions for preparation and submission of articles to JoCAAA. Assistant to the Editor: Dr.Razvan Mezei,Lenoir-Rhyne University,Hickory,NC

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Editorial Board Associate Editors of Journal of Computational Analysis and Applications Francesco Altomare Dipartimento di Matematica Universita' di Bari Via E.Orabona, 4 70125 Bari, ITALY Tel+39-080-5442690 office +39-080-3944046 home +39-080-5963612 Fax [email protected] Approximation Theory, Functional Analysis, Semigroups and Partial Differential Equations, Positive Operators.

Fractional Differential Equations Nonlinear Analysis, Fractional Dynamics Carlo Bardaro Dipartimento di Matematica e Informatica Universita di Perugia Via Vanvitelli 1 06123 Perugia, ITALY TEL+390755853822 +390755855034 FAX+390755855024 E-mail [email protected] Web site: http://www.unipg.it/~bardaro/ Functional Analysis and Approximation Theory, Signal Analysis, Measure Theory, Real Analysis.

Ravi P. Agarwal Department of Mathematics Texas A&M University - Kingsville 700 University Blvd. Kingsville, TX 78363-8202 tel: 361-593-2600 [email protected] Differential Equations, Difference Equations, Inequalities

Martin Bohner Department of Mathematics and Statistics, Missouri S&T Rolla, MO 65409-0020, USA [email protected] web.mst.edu/~bohner Difference equations, differential equations, dynamic equations on time scale, applications in economics, finance, biology.

George A. Anastassiou Department of Mathematical Sciences The University of Memphis Memphis, TN 38152,U.S.A Tel.901-678-3144 e-mail: [email protected] Approximation Theory, Real Analysis, Wavelets, Neural Networks, Probability, Inequalities.

Jerry L. Bona Department of Mathematics The University of Illinois at Chicago 851 S. Morgan St. CS 249 Chicago, IL 60601 e-mail:[email protected] Partial Differential Equations, Fluid Dynamics

J. Marshall Ash Department of Mathematics De Paul University 2219 North Kenmore Ave. Chicago, IL 60614-3504 773-325-4216 e-mail: [email protected] Real and Harmonic Analysis

Luis A. Caffarelli Department of Mathematics The University of Texas at Austin Austin, Texas 78712-1082 512-471-3160 e-mail: [email protected] Partial Differential Equations George Cybenko Thayer School of Engineering

Dumitru Baleanu Department of Mathematics and Computer Sciences, Cankaya University, Faculty of Art and Sciences, 06530 Balgat, Ankara, Turkey, [email protected]

781

Dartmouth College 8000 Cummings Hall, Hanover, NH 03755-8000 603-646-3843 (X 3546 Secr.) e-mail:[email protected] Approximation Theory and Neural Networks

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J .A. Goldstein Department of Mathematical Sciences The University of Memphis Memphis, TN 38152 901-678-3130 [email protected]

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Hrushikesh N. Mhaskar Department Of Mathematics California State University Los Angeles, CA 90032 626-914-7002 e-mail: [email protected] Orthogonal Polynomials, Approximation Theory, Splines, Wavelets, Neural Networks

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Burkhard Lenze Fachbereich Informatik Fachhochschule Dortmund University of Applied Sciences Postfach 105018 D-44047 Dortmund, Germany e-mail: [email protected] Real Networks, Fourier Analysis, Approximation Theory

Vassilis Papanicolaou Department of Mathematics

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National Technical University of Athens Zografou campus, 157 80 Athens, Greece tel:: +30(210) 772 1722 Fax +30(210) 772 1775 [email protected] Partial Differential Equations, Probability

Approximation Theory, Banach spaces, Classical Analysis T. E. Simos Department of Computer Science and Technology Faculty of Sciences and Technology University of Peloponnese GR-221 00 Tripolis, Greece Postal Address: 26 Menelaou St. Anfithea - Paleon Faliron GR-175 64 Athens, Greece [email protected] Numerical Analysis

Choonkil Park Department of Mathematics Hanyang University Seoul 133-791 S. Korea, [email protected] Functional Equations

H. M. Srivastava Department of Mathematics and Statistics University of Victoria Victoria, British Columbia V8W 3R4 Canada tel.250-472-5313; office,250-4776960 home, fax 250-721-8962 [email protected] Real and Complex Analysis, Fractional Calculus and Appl., Integral Equations and Transforms, Higher Transcendental Functions and Appl.,q-Series and q-Polynomials, Analytic Number Th.

Svetlozar (Zari) Rachev, Professor of Finance, College of Business, and Director of Quantitative Finance Program, Department of Applied Mathematics & Statistics Stonybrook University 312 Harriman Hall, Stony Brook, NY 11794-3775 tel: +1-631-632-1998, [email protected] Alexander G. Ramm Mathematics Department Kansas State University Manhattan, KS 66506-2602 e-mail: [email protected] Inverse and Ill-posed Problems, Scattering Theory, Operator Theory, Theoretical Numerical Analysis, Wave Propagation, Signal Processing and Tomography

I. P. Stavroulakis Department of Mathematics University of Ioannina 451-10 Ioannina, Greece [email protected] Differential Equations Phone +3-065-109-8283 Manfred Tasche Department of Mathematics University of Rostock D-18051 Rostock, Germany [email protected] Numerical Fourier Analysis, Fourier Analysis, Harmonic Analysis, Signal Analysis, Spectral Methods, Wavelets, Splines, Approximation Theory

Tomasz Rychlik Polish Academy of Sciences Instytut Matematyczny PAN 00-956 Warszawa, skr. poczt. 21 ul. Śniadeckich 8 Poland [email protected] Mathematical Statistics, Probabilistic Inequalities Boris Shekhtman Department of Mathematics University of South Florida Tampa, FL 33620, USA Tel 813-974-9710 [email protected]

Roberto Triggiani Department of Mathematical Sciences University of Memphis Memphis, TN 38152 PDE, Control Theory, Functional

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Analysis, [email protected]

USA. Tel 334-844-6557 office 678-642-8703 home Fax 334-844-6555 [email protected] Approximation Theory, Chebychev Systems, Wavelet Theory

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Ahmed I. Zayed Department of Mathematical Sciences DePaul University 2320 N. Kenmore Ave. Chicago, IL 60614-3250 773-325-7808 e-mail: [email protected] Shannon sampling theory, Harmonic analysis and wavelets, Special functions and orthogonal polynomials, Integral transforms

Ram Verma International Publications 1200 Dallas Drive #824 Denton, TX 76205, USA [email protected] Applied Nonlinear Analysis, Numerical Analysis, Variational Inequalities, Optimization Theory, Computational Mathematics, Operator Theory

Ding-Xuan Zhou Department Of Mathematics City University of Hong Kong 83 Tat Chee Avenue Kowloon, Hong Kong 852-2788 9708,Fax:852-2788 8561 e-mail: [email protected] Approximation Theory, Spline functions, Wavelets

Xiang Ming Yu Department of Mathematical Sciences Southwest Missouri State University Springfield, MO 65804-0094 417-836-5931 [email protected] Classical Approximation Theory, Wavelets

Xin-long Zhou Fachbereich Mathematik, Fachgebiet Informatik Gerhard-Mercator-Universitat Duisburg Lotharstr.65, D-47048 Duisburg, Germany e-mail:[email protected] Fourier Analysis, Computer-Aided Geometric Design, Computational Complexity, Multivariate Approximation Theory, Approximation and Interpolation Theory

Lotfi A. Zadeh Professor in the Graduate School and Director, Computer Initiative, Soft Computing (BISC) Computer Science Division University of California at Berkeley Berkeley, CA 94720 Office: 510-642-4959 Sec: 510-642-8271 Home: 510-526-2569 FAX: 510-642-1712 [email protected] Fuzzyness, Artificial Intelligence, Natural language processing, Fuzzy logic Richard A. Zalik Department of Mathematics Auburn University Auburn University, AL 36849-5310

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J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 22, NO.5, 2017, COPYRIGHT 2017 EUDOXUS PRESS, LLC

HIGHER-ORDER DEGENERATE BERNOULLI POLYNOMIALS DAE SAN KIM AND TAEKYUN KIM

Abstract. Carlitz introduced the degenerate Bernoulli polynomials and derived, among other things, the so-called degenerate Staudt-Clausen theorem for the degenerate Bernoulli numbers as an analogue of the classical StaudtClausen theorem. In this paper, we consider the higher-order Carlitz’s degenerate Bernoulli polynomials with umbral calculus viewpoint and derive new identities and properties of those polynomials associated with special polynomials which are derived from umbral calculus.

1. Introduction The degenerate Bernoulli polynomials βn (λ, x) (λ ̸= 0) are defined by Carlitz to be t

(1.1)

x

(1 + λt) λ =

1 λ

(1 + λt) + 1

∞ ∑

βn (λ, x)

n=0

tn , n!

(λ ̸= 0) ,

(see [3, 4]) .

Ustinov rediscovered these polynomials in [18], which are called Korobov poly(λ) nomials of the second kind and denoted by kn (x). When x = 0, βn (λ) = βn (λ, 0) are called the degenerate Bernoulli numbers. Now, we observe that (1.2)

lim βn (λ, x) = βn (0, x) = Bn (x) ,

λ→0

lim λ−n βn (λ, λx) = bn (x) ,

λ→∞

where Bn (x) and bn (x) are the Bernoulli polynomials of the first kind and of the second kind. The first few degenerate Bernoulli polynomials are given by β0 (λ, x) = 1, β1 (λ, x) = x − 12 + 12 λ, β2 (λ, x) = x2 − x + 16 − 16 λ2 , β3 (λ, x) = x3 − 32 x2 + 12 x − 23 λx2 + 3 1 3 1 2 λx + 4 λ − 4 λ, . . . . As an analogue of the classical Staudt-Clausen theorem for Bernoulli numbers, Carlitz proved the so called degenerate Staudt-Clausen theorem for βn (λ), (λ a rational number) (see [3, 19, 20]). The generalized falling factorials (x|λ)n for any λ ∈ C are defined as (x|λ)0 = 1,

(x|λ)n = x (x − λ) · · · (x − λ (n − 1)) ,

(for n > 0) .

Carlitz also found in [4] the following relation expressing sums of generalized falling factorals in terms of degenerate Bernoulli polynomials: for integers l, m with l ≥ 1,m ≥ 0,

(1.3)

l−1 ∑ i=0

(i|λ)m =

1 (βm+1 (λ, l) − βm+1 (λ)) , m+1

2000 Mathematics Subject Classification. 05A19, 05A40, 11B83. Key words and phrases. Higher-order degenerate Bernoulli polynomial, Umbral calculus. 1

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DAE SAN KIM AND TAEKYUN KIM

which, by letting λ → 0, becomes the familiar relation l−1 ∑

(1.4)

im =

i=0

1 (Bm+1 (l) − Bm+1 ) . m+1

For r ∈ N, the Bernoulli polynomials of the second kind of order r are defined by the generating function to be )r ( ∞ ∑ t tn x (1.5) (1 + t) = bn(r) (x) , (see [16]) , log (1 + t) n! n=0 and the Bernoulli polynomials of order r are given by ( )r ∞ ∑ t tn xt (1.6) e = Bn(r) (x) , (see [2, 5–7, 9]) . t e −1 n! n=0 (r)

(r)

(r)

(r)

When x = 0, Bn = Bn (0), bn = bn (0) are called the Bernouli numbers of the first kind of order r and of the second kind of order r. For µ ∈ C with µ ̸= 1, the Frobenius-Euler polynomials with order s ∈ N are defined by the generating function to be )s ( ∞ ∑ tn 1−µ xt (s) e = (x|µ) , (see [1, 10–12]) . H (1.7) n et − µ n! n=0 (s)

(s)

When x = 0, Hn (µ) = Hn (0|µ) are called the Frobenius-Euler numbers of order s. As is well known, the Stirling number of the second kind is defined by the generating function to be ∞ ∑ ( t )n tl (1.8) e − 1 = n! S2 (l, m) , (n ∈ Z≥0 ) , (see [16, 17]) . l! l=n

For n ≥ 0, the Stirling number of the first kind is given by n ∑ (x)n = x (x − 1) · · · (x − (n − 1)) = S1 (n, l) xl , (see [13, 15, 16, 21]) . l=0

Let F be the set of all formal power series in the variable t: { } ∞ ∑ tk (1.9) F = f (t) = ak ak ∈ C . k! k=0

∗

Let P = C [x] and let P be the vector space of all linear functionals on P.⟨ L| p (x)⟩ denotes the action of the linear functional L on p (x) which satisfies ⟨ L + M | p (x)⟩ = ⟨ L| p (x)⟩+⟨ M | p (x)⟩, and ⟨ cL| p (x)⟩ = c ⟨ L| p (x)⟩, where c is a complex constant. The linear functional ⟨ f (t)| ·⟩ on P is defined as (1.10)

⟨ f (t)| xn ⟩ = an ,

(n ≥ 0) ,

where f (t) ∈ F .

Thus, by (1.9) and (1.10), we get ⟨ k n⟩ (1.11) t x = n!δn,k , (n, k ≥ 0) ,

(see [14, 16]) ,

where δn,k is the Kronecker symbol. ∑∞ ⟨ L|xk ⟩ Let fL (t) = k=0 k! tk . Then, by (1.11), we get ⟨ fL (t)| xn ⟩ = ⟨ L| xn ⟩. So, the map L 7→ fL (t) is a vector space isomorphism from P∗ onto F. Henceforth, F denotes both the algebra of formal power series in t and the vector space of all

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J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 22, NO.5, 2017, COPYRIGHT 2017 EUDOXUS PRESS, LLC

HIGHER-ORDER DEGENERATE BERNOULLI POLYNOMIALS

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linear functionals on P, and so an element f (t) of F will be thought of as both a formal power series and a linear functional. We call F the umbral algebra and the umbral calculus is the study of umbral algebra. The order o (f (t)) of a power series f (t) ̸= 0 is the smallest integer k for which the coefficient of tk does not vanish (see [14, 16]). If o (f (t)) = 0, then f (t) is called an invertible series; if o (f (t)) = 1, then f (t) is called a delta series. Let f (t) , g (t) be a delta series and an invertible series, respectively. Then there ⟨ ⟩ exists a unique sequence sn (x)(deg sn (x) = n) k such that g (t) f (t) sn (x) = n!δn,k , for k ≥ 0. Such a sequence sn (x) is called the Sheffer sequence for (g (t) , f (t))which is denoted by sn (x) ∼ (g (t) , f (t)) (see [14, 16]). The sequence sn (x) is Sheffer for (g (t) , f (t)) if and only if ∞

(1.12)

∑ sk (y) 1 ) eyf (t) = tk , k! g f (t) k=0 (

(y ∈ C) ,

(see [11, 17]) ,

( ) where f (t) is the compositional inverse of f (t) with f (f (t)) = f f (t) = t. Let f (t) , g (t) ∈ F and p (x) ∈ P. Then we see that (1.13)

f (t) =

∞ ∑ ⟨

f (t)| xk

k=0

⟩ tk , k!

p (x) =

∞ ∑ ⟨ k ⟩ xk t p (x) . k!

k=0

From (1.13), we have (1.14)

tk p (x) = p(k) (x) =

dk p (x) , dxk

eyt p (x) = p (x + y) .

By (1.14), we get ⟨ eyt | p (x)⟩ = p (y) . For sn (x) ∼ (g (t) , f (t)), we have the following equations ([16]): n ( ) ∑ n (1.15) f (t) sn (x) = nsn−1 (x) , (n ≥ 1) , sn (x + y) = sj (x) pn−j (y) , j j=0 where pn (x) = g (t) sn (x), (1.16) ( ) g ′ (t) 1 sn+1 (x) = x − sn (x) , g (t) f ′ (t)

sn (x) =

n ⟩ ∑ )−1 1 ⟨ ( j g f (t) f (t) xn xj , j! j=0

and (1.17)

⟨ f (t)| xp (x)⟩ = ⟨ ∂t f (t)| p (x)⟩ , n−1 ∑ ( n) ⟨ ⟩ d sn (x) = f (t) xn−l sl (x) , dx l

(n ≥ 1) .

l=0

In particular, for pn (x) ∼ (1, f (t)), qn (x) ∼ (1, g (t)), we note that )n ( f (t) x−1 pn (x) , (n ≥ 1) . (1.18) qn (x) = x g (t) Let us assume that sn (x) ∼ (g (t) , f (t)), rn (x) ∼ (h (t) , l (t)). Then we have (1.19)

sn (x) =

n ∑

Cn,m rm (x) ,

(n ≥ 0) ,

m=0

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DAE SAN KIM AND TAEKYUN KIM

where (1.20)

Cn,m

1 = m!

⟩ ⟨ ( ) ))m n h f (t) ( ( ( ) l f (t) , x g f (t)

(see [16]) .

In this paper, we consider, for any positive integer r, the degenerate Bernoulli (r) polynomials βn (λ, x) of order r which are defined by the generating function to be ( )r ∞ ∑ x t tn (1.21) βn(r) (λ, x) , (r ∈ Z≥0 ) . (1 + λt) λ = 1 n! (1 + λt) λ − 1 n=0 From (1.20) and (1.21), we note that (( )r ) ) 1 ( λt λ (et − 1) , . (1.22) βn(r) (λ, x) ∼ e − 1 eλt − 1 λ (r)

That is, βn (λ, x) is the Sheffer polynomial for the pair ( ( )r ) ) λ (et − 1) 1 ( λt , f (t) = e −1 . g (t) = eλt − 1 λ The purpose of this paper is to give new identities and properties of the higherorder degenerate Bernoulli polynomials associated with special polynomials which are derived from umbral calculus. 2. Higher-order degenerate Bernoulli polynomials For n ≥ 0, we note that ( ) ( )r ) 1 ( λt λ (et − 1) (r) (λ, x) ∼ 1, e − 1 , xn ∼ (1, t) , β n eλt − 1 λ From (1.18), we can derive the following equation: ( )r ( )n ( )n λ (et − 1) λt λt (r) −1 n (2.1) βn (λ, x) = x λt x x = x λt xn−1 eλt − 1 e −1 e −1 n−1 ∑ (n − 1 ) (n) = λl Bl xn−l , (n ≥ 1) . l l=0

Thus, by (2.1), we get (2.2)

βn(r) (λ, x) ( λt )r n−1 ∑ (n − 1) e −1 (n) = λl Bl xn−l l λ (et − 1) l=0 )r ( λt )r ( n−1 ∑ (n − 1) t e −1 (n) = λl Bl xn−l l et − 1 λt l=0 ) )r ( ∑ ( ∞ n−1 ∑ (n − 1) λk t l (n) k = r! S2 (k + r, r) t xn−l λ Bl l et − 1 (k + r)! l=0 k=0 ( )r ∑ n−1 n−l (n−l) ∑ (n − 1) t (n) k = λl Bl (k+r ) S2 (k + r, r) λk xn−l−k l et − 1 r l=0

k=0

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=

n−1 n−l ∑∑ l=0 k=0

=

n−1 n−l ∑∑

(n−1)(n−l)

( (n)

(k+r)k

S2 (k + r, r) λk+l Bl

l

r

t t e −1

5

)r xn−l−k

(n−1)(n−l) (n)

(k+r)k

(r)

S2 (k + r, r) λk+l Bl Bn−l−k (x) .

l

r

l=0 k=0

Therefore, by (2.2), we obtain the following theorem. Theorem 2.1. For n ≥ 1, we have n−1 n−l (n−1)(n−l) ∑∑ (n) (r) l βn(r) (λ, x) = (k+r)k S2 (k + r, r) λk+l Bl Bn−k−l (x) . r

l=0 k=0

Remark. When x = 0 and r = 1, we get ( )( ) n−1 n−l ∑∑ n − 1 n − l k+l (n) 1 λ Bl Bn−k−l . (2.3) βn (λ) = k+1 l k l=0 k=0

From (1.18), (1.22) and (x|λ)n = λn We note that

(x) λ

n ∑

= n

S1 (n, m) λn−m xm ∼

m=0

( ) ) 1 ( λt 1, e −1 . λ

(

)r eλt − 1 xm t − 1) λ (e m=0 )r ( λt )r ( n ∑ e −1 t n−m = xm S1 (n, m) λ t−1 e λt m=0 )r ∑ ( m (m) n ∑ t n−m k = S1 (n, m) λ (k+r ) S2 (k + r, r) λk xm−k t−1 e r m=0 k=0 n ∑ m ( m) ∑ (r) k = λn (k+r ) S1 (n, m) S2 (k + r, r) λk−m Bm−k (x) .

(2.4) βn(r) (λ, x) =

n ∑

S1 (n, m) λn−m

m=0 k=0

r

Therefore, by (2.4), we obtain the following theorem. Theorem 2.2. For n ≥ 0, we have n ∑ m ( m) ∑ (r) (r) n k βn (λ, x) = λ (k+r ) S1 (n, m) S2 (k + r, r) λk−m Bm−k (x) . m=0 k=0

r

Remark. For r = 1 and x = 0, we get an expression for the degenerate Bernoulli numbers:

(2.5)

n

βn (λ) = λ

n ∑ m ∑ m=0 k=0

( ) m 1 S1 (n, m) λk−m Bm−k . k+1 k

Here we use the conjugation representation.

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(r)

For βn (λ, x) ∼

( ( t )r λ(e −1) g (t) = , f (t) = eλt −1

) ( λt ) e − 1 , we observe that

1 λ

⟩ ⟨ ( )−1 j g f (t) f (t) xn ⟨( )r ( )j ⟩ t 1 = log (1 + λt) xn 1 λ λ (1 + λt) − 1 ⟨( )r ∞ ⟩ ∑ l λ t S1 (l, j) tl xn =λ−j j! 1 l! (1 + λt) λ − 1 l=j ⟨ ⟩ n ( ) ∞ m ∑ ∑ n t (r) S1 (l, j) λl =j!λ−j βm (λ) xn−l l m! m=0 l=j n ( ) ∑ n (r) =j!λ−j S1 (l, j) λl βn−l (λ) . l

(2.6)

l=j

Therefore, by (1.16) and (2.6), we obtain the following theorem. Theorem 2.3. For n ≥ 0, r ≥ 1, we have   n n ( ) ∑ ∑ n (r) S1 (l, j) λl βn−l (λ) xj . βn(r) (λ, x) = λ−j  l j=0 l=j

Remark. Recall that ( )r ( ) ) λ (et − 1) 1 ( λt (r) (2.7) βn (λ, x) ∼ 1, e −1 , eλt − 1 λ Thus, by (2.7), we get ( )r λ (et − 1) (2.8) βn(r) (λ, x) = (x|λ)n , eλt − 1 From (2.8), we have ( t )r e − 1 βn(r) (λ, x) =

(2.9)

( {

= By (2.9), we get (2.10)

tr βn(r)

(λ, x) =

and

eλt − 1 λ

( ) ) 1 ( λt (x|λ)n ∼ 1, e −1 . λ

eλt − 1 (x|λ)n = n (x|λ)n−1 . λ

)r (x|λ)n

(n)r (x|λ)n−r 0

( )r ( ) { x t (n)r λn−r et −1 λ n−r

, if r ≤ n , if r > n.

, if r ≤ n

0 , if r > n { ∑ (r) n−r (n)r λn−r m=0 S1 (n − r, m) λ−m Bm (x) = 0

, if r ≤ n , if r > n.

Therefore, from (1.14) and (2.10), we have (2.11) { ∑n−r ( )r (r) d (n)r λn−r m=0 S1 (n − r, m) λ−m Bm (x) (r) βn (λ, x) = dx 0

, if r ≤ n , if r > n.

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In particular, (2.12)

d βn (λ, x) = dx

{

nλn−1 0

∑n−1 m=0

S1 (n − 1, m) λ−m Bm (x)

7

, if r ≤ n , if r > n. (r)

To proceed further, we recall that the λ-Daehee polynomials Dn,λ (x) of order r are given by ( )r ∞ ∑ λ log (1 + t) tn (r) x (2.13) (1 + t) = Dn,λ (x) , (see [12, 16]) . λ n! (1 + t) − 1 n=0 From (1.5), (1.11) and (2.13), we have (2.14)

βn(r) (λ, y) ⟨(

⟩ = (1 + λt) xn 1 λ (1 + λt) − 1  r ⟨ ⟩ n ( ) ∑ n−l log (1 + λt) n l (r) ( y )   ( ) x = λ bl 1 λ l λ (1 + λt) λ − 1 l=0 ⟨ ⟩ ∞ n ( ) ∑ n l (r) ( y ) ∑ (r) m tm n−l Dm, 1 λ = λ bl x λ λ m! l m=0 l=0 n ( ) ∑ n l (r) ( y ) (r) = Dn−l, 1 λn−l λ bl λ λ l l=0 n ( ) ( ) ∑ n (r) y (r) =λn Dn−l, 1 bl , λ l λ t

)r

y λ

l=0

and (2.15)

βn(r) (λ, y) ⟩ ⟨ ∞ ∑ (r) tl n = βl (λ, y) x l! l=0 ⟩ )r ⟨( y t (1 + λt) λ xn = 1 (1 + λt) λ − 1 r  ⟨( ⟩ )r y λt log (1 + λt) n  ( )  (1 + λt) λ x = log (1 + λt) λ (1 + λt) λ1 − 1 ⟨( ⟩ )r ∑ ∞ (r) ( y ) l tl n λt = λ x Dl,λ−1 log (1 + λt) λ l! l=0 ⟩ ⟨ n ( ) ∞ ∑ n l (r) ( y ) ∑ (r) m tm n−l = λ Dl,λ−1 bm λ x l λ m! m=0 l=0 n ( ) ∑ n l (r) ( y ) (r) n−l = λ Dl,λ−1 b λ l λ n−l l=0

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DAE SAN KIM AND TAEKYUN KIM n ( ) ∑ n (r) ,(r) ( y ) =λ b D −1 . l n−l l,λ λ n

l=0

Therefore, by (2.14) and (2.15), we obtain the following theorem. Theorem 2.4. For n ≥ 0, we have n ( ) n ( ) ( ) ∑ ∑ n n (r) (r) ( x ) (r) x (r) Dn−l, 1 bl = b D −1 = λ−n βn(r) (λ, x) . λ l λ l n−l l,λ λ l=0

l=0

Recalling that βn(r)

( ( )r ) ) λ (et − 1) 1 ( λt (λ, x) ∼ g (t) = , f (t) = e −1 , eλt − 1 λ

we observe that (2.16) ⟩ ⟨ ( )−1 j g f (t) f (t) xn )r ⟩ ( ) ⟨( n ∑ n l t n−l −j =j!λ S1 (l, j) λ x 1 l (1 + λt) λ − 1 l=j r  ⟩ ( )r ( ) ⟨ n ∑ (1 + λt) n log λt n−l −j l   ( ) x =j!λ S1 (l, j) λ 1 l log (1 + λt) λ (1 + λt) λ − 1 l=j  r ⟩ ∑ ( ) ⟨ n ∑ ∞ (r) λm m n−l n l  log (1 + λt) −j ( )  =j!λ S1 (l, j) λ t x bm 1 l m! m=0 λ (1 + λt) λ − 1 l=j r ⟨ ⟩ ( ) ∑ ) n n−l ( ∑ n−l−m n n − l log (1 + λt) −j l m (r)   ( ) x =j!λ S1 (l, j) λ λ bm 1 l m λ (1 + λt) λ − 1 m=0 l=j =j!λ

−j

n ∑ l=j

) ( ) ∑ n−l ( n − l m (r) (r) n l λ bm Dn−l−m,λ−1 λn−l−m S1 (l, j) λ m l m=0

) n ∑ n−l ( )( ∑ n n−l (r) n =j!λ S1 (l, j) λ−j b(r) m Dn−l−m,λ−1 . l m m=0 l=j

From (1.16) and (2.16), we have (2.17)   ) n ∑ n ∑ n−l ( )(  ∑ n n − l (r) βn(r) (λ, x) = λn S1 (l, j) λ−j b(r) xj . m Dn−l−m,λ−1   l m m=0 j=0 l=j

Remark. We have lim βn(r) (λ, x) = βn(r) (0, x) = Bn(r) (x) ,

λ→0

(r)

lim Dn,λ (x) = (x)n ,

λ→0

lim λ−n βn(r) (λ, λx) = b(r) n (x) ,

λ→∞

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9

lim λ−n Dn,λ (λx) = Bn(r) (x) . (r)

λ→∞

where r > 0. From (1.22), we note that ( ( )r ) ) λ (et − 1) 1 ( λt (r) (2.18) pn (x) = β (λ, x) = (x|λ) ∼ 1, . e − 1 n n eλt − 1 λ By (2.18) and (1.15), we get (2.19)

βn(r)

(λ, x + y) =

n ( ) ∑ n j=0

j

(r)

βj (λ, x) (y|λ)n−j ,

and, by (1.14) and (1.15), we get ) 1 ( λt (r) (2.20) e − 1 βn(r) (λ, x) = nβn−1 (λ, x) . λ From (2.20), we have (2.21)

(r)

βn(r) (λ, x + λ) − βn(r) (λ, x) = nλβn−1 (λ, x) .

Therefore, by (2.17), (2.19) and (2.20), we obtain the following theorem. Theorem 2.5. For n ≥ 0, we have   ) n ∑ n ∑ n−l ( )(  ∑ n n−l (r) βn(r) (λ, x) = λn xj , S1 (l, j) λ−j b(r) m Dn−l−m,λ−1   l m m=0 j=0 l=j

and βn(r)

n ( ) ∑ n (r) (λ, x + λ) = β (λ, x) (λ|λ)n−j j j j=0 (r)

= nλβn−1 (λ, x) + βn(r) (λ, x) . ( ( t )r ) ( λt ) λ(e −1) (r) 1 , f (t) = e − 1 , we note that For βn (λ, x) ∼ g (t) = λ eλt −1 g ′ (t) g (t)

(2.22)

′

= (log g (t)) ( ( ) ( ))′ =r log λ + log et − 1 − log eλt − 1 (∞ ) ∞ r ∑ tl ∑ λl tl = Bl (1) − Bl (1) t l! l! l=0

l=0

∞ ( ) tl r∑ = Bl (1) 1 − λl t l! l=1

=r

∞ ∑

( ) Bl+1 (1) 1 − λl+1

l=0

tl . (l + 1)!

By (2.22), we get (2.23)

g ′ (t) (r) β (λ, x) g (t) n

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=r

∞ ∑

( ) Bl+1 (1) 1 − λl+1

l=0 n ∑ m ∑

×

( m)

(r)

k (k+r ) S1 (n, m) S2 (k + r, r) λk−m Bm−k (x)

m=0 k=0

=λn r

tl λn (l + 1)!

r

n ∑ m m−k ∑ ∑

S1 (n, m) S2 (k + r, r) λk−m

m=0 k=0 l=0 m−k ∑

( ) Bl+1 (1) 1 − λl+1

1 (r) (m − k)l Bm−k−l (x) (l + 1)! l=0 (m)(m−k) n ∑ m m−k ∑ ∑ ( k−m ) 1 k =λn r λ − λ1−l (k+rl) m−k−l+1 r m=0 ×

k=0 l=0

(r)

× S1 (n, m) S2 (k + r, r) Bm−k−l+1 (1) Bl (x) (m)(k) n ∑ n ∑ m ∑ ( ) 1 k l =r (m−k+r ) λn−k − λn−l+1 S1 (n, m) k−l+1 r l=0 m=l k=l

(r)

× S2 (m − k + r, r) Bk−l+1 (1) Bl

(x) .

From (1.16) and (2.23), we have (2.24)

(r)

βn+1 (λ, x) g ′ (t) (r) β (λ, x) g (t) n ( n n ∑ m ∑ ∑ 1

=xβn(r) (λ, x − λ) − e−λt =xβn(r) (λ, x − λ) − r

l=0

m=l k=l

× S1 (n, m) S2 (m − k +

k−l+1

(m)(k) ( ) k l (m−k+r ) λn−k − λn−l+1

(r) r, r) Bk−l+1 (1)) Bl

r

(x − λ) .

Therefore, by (2.24), we obtain the following theorem. Theorem 2.6. For n ≥ 0, we have (r)

βn+1 (λ, x) =xβn(r)

(λ, x − λ) − r

n ∑ l=0

(

n ∑ m ∑ m=l k=l

(m)(k) ( ) 1 k l (m−k+r ) λn−k − λn−l+1 k−l+1 r (r)

×S1 (n, m) S2 (m − k + r, r) Bk−l+1 (1)) Bl (x − λ) . )r ) ( ( t ( λt ) λ(e −1) (r) 1 By βn (λ, x) ∼ g (t) = , f (t) = λ e − 1 , we get eλt −1 (2.25)

⟩ f (t) xn−l ⟨ ⟩ 1 = log (1 + λt) xn−l λ ⟨ ∞ ⟩ m ∑ t n−l m−1 m −1 =λ (−1) λ (m − 1)! x m! m=1 ⟨

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=λ−1 (−1)

n−l−1

n−l−1

= (−λ)

11

λn−l (n − l − 1)!

(n − l − 1)!.

From (1.17) and (2.25), we have ∑ (−λ) d (r) (r) βn (λ, x) = n! β (λ, x) . dx l! (n − l) l n−1

(2.26)

n−l−1

l=0

Let n ≥ 1. Then, by (1.11) and (1.17), we get (2.27)

βn(r) (λ, y) ⟩ ⟨ ∞ ∑ (r) tl n = βl (λ, y) x l! l=0 ⟩ )r ⟨( y t n λ (1 + λt) = x 1 (1 + λt) λ − 1 ) ⟨ (( )r ⟩ y t = ∂t (1 + λt) λ xn−1 1 (1 + λt) λ − 1 ⟨( )r ⟩ y t n−1 = ∂t (1 + λt) λ x 1 (1 + λt) λ − 1 ⟨( ( )r ) ⟩ y t n−1 λ ∂t + (1 + λt) x . 1 (1 + λt) λ − 1

The first term of (2.27) is ⟨( )r ⟩ y−λ t n−1 (r) λ (2.28) y = yβn−1 (λ, y − λ) . (1 + λt) x 1 λ (1 + λt) − 1 For the second term of (2.27), we observe that ( )r ( )r−1 ( ) t t t (2.29) ∂t =r ∂t , 1 1 1 (1 + λt) λ − 1 (1 + λt) λ − 1 (1 + λt) λ − 1 where (2.30)

( ∂t

)

t 1

(1 + λt) λ − 1 1

1

−1

(1 + λt) λ − 1 − t (1 + λt) λ = ( )2 1 (1 + λt) λ − 1 { } 1 1 −1 −1 −1 − t (1 + λt) (1 + λt) λ − 1 − t (1 + λt) λ − (1 + λt) = ( )2 1 (1 + λt) λ − 1 (2.31)

=−

1 t 1 + 1 1 + λt (1 + λt) λ − 1 t

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 

t

1 × −  (1 + λt) λ1 − 1 (1 + λt)

(

)2  

t



1

(1 + λt) λ − 1

.

Thus, by (2.29) and (2.30), we get ( )r t (2.32) ∂t 1 (1 + λt) λ − 1 )r ( r t =− (1 + λt) (1 + λt) λ1 − 1 ( )r ( )r+1   t t r 1 . + − 1 1  t  (1 + λt) λ − 1 1 + λt (1 + λt) λ − 1 From (2.32), we note that the second term of (2.27) is (2.33)

⟨(

−r

t

)r y−λ λ

(1 + λt) 1 (1 + λt) λ − 1 ( ⟨  y 1 t + r (1 + λt) λ 1  t (1 + λt) λ − 1

⟩ n−1 x )r

1 − 1 + λt

(

)r+1  

t 1

(1 + λt) λ − 1



⟩ n−1

x

(r)

= − rβn−1 (λ, y − λ) ( ⟨ ( )r )r+1  ⟩   y r t 1 t + (1 + λt) λ − xn 1 1  n 1 + λt (1 + λt) λ − 1  (1 + λt) λ − 1 (r)

= − rβn−1 (λ, y − λ) ⟩ ⟨( )r y r t λ + (1 + λt) xn 1 n (1 + t) λ − 1 ⟩ ⟨( )r+1 y−λ r t λ n − (1 + λt) 1 x n (1 + λt) λ − 1 r (r) r β (λ, y) − βn(r+1) (λ, y − λ) . n n n By (2.27), (2.28) and (2.33), we get ( r ) (r) r (r) (2.34) 1− β (λ, x) = (x − r) βn−1 (λ, x − λ) − βn(r+1) (λ, x − λ) . n n n Therefore, by (2.34), we obtain the following theorem. (r)

= − rβn−1 (λ, y − λ) +

Theorem 2.7. For n ≥ 1, we have (n ) ( n ) (r) (r) βn(r+1) (λ, x − λ) = 1 − βn (λ, x) + x − n βn−1 (λ, x − λ) . r r Here we compute ⟨( )r ( )m ⟩ t 1 (2.35) log (1 + λt) xn 1 λ (1 + λt) λ − 1

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13

in two different ways. On one hand, it is equal to ⟩ )r ⟨( t m (log (1 + λt)) xn (2.36) λ−m 1 (1 + λt) λ − 1 )r ⟨( ⟩ ∞ ∑ λl l n t −m S1 (l, m) t x =λ m! 1 l! (1 + λt) λ − 1 l=m ⟨ ( )r ⟩ n ( ) ∑ n t n−l −m l =m!λ S1 (l, m) λ x 1 l (1 + λt) λ − 1 l=m ⟨ ∞ ⟩ n ( ) k ∑ ∑ (r) n t S1 (l, m) λl =m!λ−m βk (λ) xn−l l k! l=m k=0 n ( ) ∑ n (r) =m!λ−m S1 (l, m) λl βn−l (λ) . l l=m

On the other hand, it is equal to ⟨ (( ⟩ )r ( )m ) t 1 n−1 (2.37) ∂t log (1 + λt) x 1 λ (1 + λt) λ − 1 ⟩ ⟨( )r ( )m t 1 n−1 = log (1 + λt) x ∂t 1 λ (1 + λt) λ − 1 ⟨( ( ⟩ )r ) ( )m t 1 n−1 + ∂t log (1 + λt) x . 1 λ (1 + λt) λ − 1 The first term of (2.37) is (2.38) ⟨( m

)r (

t 1

(1 + λt) λ − 1 ⟨(

1 log (1 + λt) λ )r

)m−1

⟩ xn−1

−1

(1 + λt)

⟩ m−1 n−1 (1 + λt) (log (1 + λt)) x =mλ 1 (1 + λt) λ − 1 ⟨( )r ⟩ ∞ ∑ t λl tl n−1 −1 −(m−1) (1 + λt) (m − 1)! x =mλ S1 (l, m − 1) 1 l! (1 + λt) λ − 1 l=m−1 ⟨( )r ⟩ n−1 ∑ (n − 1) λ t −λ −(m−1) l n−1−l (1 + λt) x =m!λ S1 (l, m − 1) λ 1 l (1 + λt) λ − 1 −(m−1)

l=m−1

=m!λ

−(m−1)

n−1 ∑ l=m−1

t

−1

( ) n−1 (r) S1 (l, m − 1) λl βn−1−l (λ, −λ) . l

For the second term of (2.37), we recall that ( )r t (2.39) ∂t 1 (1 + λt) λ − 1

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DAE SAN KIM AND TAEKYUN KIM

)r ( r t =− (1 + λt) (1 + λt) λ1 − 1 ( ( )r )r+1    r t t 1 + . −  t  (1 + λt) λ1 − 1 1 + λt (1 + λt) λ1 − 1 Now, the second term of (2.37) is (2.40)

)r ⟩ m n−1 λ ∂t (log (1 + λt)) x 1 (1 + λt) λ − 1 ⟨ ( )r ⟩ n−1 ∑ (n − 1) t =m!λ−m S1 (l, m) λl ∂t xn−1−l 1 l λ (1 + λt) − 1 l=m { ⟨ ( ) ⟩ r n−1 ∑ (n − 1) −λ t −m l n−l−1 =m!λ S1 (l, m) λ −r (1 + λt) λ x 1 l (1 + λt) λ − 1 l=m ⟨( )r ⟩ t r n−l + x 1 n−l λ (1 + λt) − 1 ⟨( )r+1 ⟩  λ r t − − (1 + λt) λ xn−l 1  n−l (1 + λt) λ − 1 n−1 ∑ (n − 1) −m =m!λ S1 (l, m) λl l l=m { } r r (r) (r) (r+1) × −rβn−1−l (λ, −λ) + β (λ) − β (λ, −λ) . n − l n−l n − l n−l ⟨

(

t

−m

From (2.35), (2.36), (2.37), and (2.40), we have n ( ) ∑ n (r) m!λ S1 (l, m) λl βn−l (λ) l l=m n−1 ∑ (n − 1) (r) −(m−1) =m!λ S1 (l, m − 1) λl βn−l−1 (λ, −λ) l l=m−1 ( n−1 ∑ (n − 1) r (r) (r) −m l β + m!λ S1 (l, m) λ −rβn−1−l (λ, −λ) + (λ) n − l n−l l l=m ) r (r+1) − βn−l (λ, −λ) , n−l −m

where n − 1 ≥ m ≥ 1. After simplification and modification, we get: for n − 1 ≥ m ≥ 1, (2.41)

n−m ∑( l=0

) n (r) S1 (n − l, m) λn−l βl (λ) l

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15

n−m ∑(

) n−1 (r) =λ S1 (n − l − 1, m − 1) λn−l−1 βl (λ, −λ) l l=0 n−m−1 ∑ (n − 1) + S1 (n − l − 1, m) λn−l−1 l l=0 ( ) r r (r) (r) (r+1) × −rβl (λ, −λ) + β (λ) − β (λ, −λ) l + 1 l+1 l + 1 l+1 n−m ∑ (n − 1) (r) =λ S1 (n − l − 1, m − 1) λn−l−1 βl (λ, −λ) l l=0 n−m−1 ∑ (n − 1 ) (r) −r S1 (n − l − 1, m) λn−l−1 βl (λ, −λ) l l=0 ( ) n−m−1 n r ∑ (r) S1 (n − l − 1, m) λn−l−1 βl+1 (λ) + n l+1 l=0 ( ) n−m−1 r ∑ n (r+1) − S1 (n − l − 1, m) λn−l−1 βl+1 (λ, −λ) l+1 n l=0 n−m ∑ (n − 1) (r) =λ S1 (n − l − 1, m − 1) λn−l−1 βl (λ, −λ) l l=0 n−m−1 ∑ (n − 1 ) (r) −r S1 (n − l − 1, m) λn−l−1 βl (λ, −λ) l l=0 ( ) n−m ∑ r n (r) + S1 (n − l, m) λn−l βl (λ) l n l=0 ( ) n−m ∑ r n (r+1) − S1 (n − l, m) λn−l βl (λ, −λ) . l n l=0

Therefore, by (2.41), we obtain the following theorem. Theorem 2.8. For n − 1 ≥ m ≥ 1, we have n−m ( ) r) ∑ n (r) 1− S1 (n − l, m) λn−l βl (λ) l n l=0 n−m ∑ (n − 1 ) (r) =λ S1 (n − l − 1, m − 1) λn−l−1 βl (λ, −λ) l l=0 n−m−1 ∑ (n − 1) (r) −r S1 (n − l − 1, m) λn−l−1 βl (λ, −λ) l l=0 ( ) n−m ∑ n r (r+1) S1 (n − l, m) λn−l βl (λ, −λ) . − n l

(

l=0

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For r > s ≥ 1, by (1.19), (1.20) and (1.22), we get βn(r) (λ, x) =

(2.42)

n ∑

(s) Cn,m βm (λ, x) ,

m=0

where (2.43)

Cn,m

⟩ m n t x 1 λ (1 + λt) − 1 ⟨( )r−s ⟩ m n 1 t t x = 1 m! (1 + λt) λ − 1 ⟨ ( ) ⟩ r−s ( ) n−m t n x = 1 m (1 + λt) λ − 1 ⟩ ( ) ⟨∑ ∞ tl n−m n (r−s) βl (λ) x = m l! l=0 ( ) n (r−s) = β (λ) . m n−m

1 = m!

⟨(

)r−s

t

Therefore, by (2.42)and (2.43), we obtain the following theorem. Theorem 2.9. For r > s ≥ 1, we have n ( ) ∑ n (r−s) (r) (s) βn (λ, x) = βn−m (λ) βm (λ, x) . m m=0 Remark. Replacing x by x + λ in Theorem 2.7, we have (n ) ( n n ) (r) (r) βn (λ, x + λ) + x + λ − n βn−1 (λ, x) . (2.44) βn(r+1) (λ, x) = 1 − r r r From Theorem 2.5, we note that n ( ) ∑ n (r) (r) (2.45) βn (λ, x + λ) = β (λ, x) (λ|λ)n−j j j j=0 (r)

= nλβn−1 (λ, x) + βn(r) (λ, x) . Substituting (2.45) into (2.44), we get (2.46) ( n ) (r) n (r) βn(r+1) (λ, x) = 1 − βn (λ, x) + (x + (λ − 1) r − (n − 1) λ) βn−1 (λ, x) . r r By using this and induction on r, it is shown in [[19]] that ( )∑ r−1 n βn−k (λ, x) r−1−k (2.47) βn(r) (λ, x) = r (−1) σr−1,k (λ, x, n) , r n−k k=0

where σr,k (λ, x, n) =

∑

k ∏

(x + (λ − 1) ij − (n − j) λ) .

1≤ik s ≥ 1, we have ⟨( t )r ( )s ⟩ n−l−k e −1 t x t et − 1

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⟨( = ⟨ =

et − 1 t

⟩ )r−s n−l−k x

⟩ ∞ ∑ S2 (j + r − s, r − s) j! tj n−l−k (r − s)! x (j + r − s)! j! j=0

S2 (n − l − k + r − s, r − s) (n − l − k)! (n − l − k + r − s)! /(n − l − k + r − s) =S2 (n − l − k + r − s, r − s) r−s = (r − s)!

Case 2. (2.59)

For r = s ≥ 1, we get ⟨( t )r ( )s ⟩ n−l−k ⟨ ⟩ t e −1 x = 1| xn−l−k = δ0,n−l−k = δk,n−l . t t e −1

Case 3. For s > r ≥ 1, we have ⟩ ⟨( t )r ( )s ⟩ ⟨( )s−r n−l−k t t e −1 n−l−k (s−r) x = (2.60) = Bn−l−k . x t et − 1 et − 1 Therefore, by (2.56), (2.57), (2.58), (2.59) and (2.60), we obtain the following theorem. Theorem 2.12. Let n ≥ 0. Then we have  n { (n)(n−l) n ∑ n−l ∑ ∑   −m l k  λ (n−l−k+r−s ) S2 (l, m)     r−s m=0 l=m k=0  }   (r) k+l (r)  × S (n − l − k + r − s, r − s) λ B βm (λ, x) ,  2 k   { }  n n ( )  ∑ ∑ n (r) n −m (r) (s) λ S2 (l, m) Bn−l βm (λ, x) , Bn (x) = λ  l  m=0 { l=m   )  n n ∑ n−l ( )(  ∑ ∑ n n−l  −m   S2 (l, m) λ   l k   m=0 l=m k=0 }    ×λk+l B (r) B (s−r) β (r) (λ, x) , m k n−l−k

if r > s ≥ 1, if r = s ≥ 1,

if s > r ≥ 1.

Remark. Let r > s ≥ 1. Then we get (2.61) { } ) n n ∑ n−l ( )( ∑ ∑ n n−l (r) −m k+l (s) (r−s) (s) (x) . λ βn (λ, x) = S1 (l, m) λ bk βn−l−k (λ) Bm l k m=0 l=m k=0

For r = s ≥ 1, we have (2.62)

βn(r)

n

(λ, x) = λ

n ∑

{ −m

λ

m=0

n ( ) ∑ n (s) S1 (l, m) bn−l l

} (s) Bm (x) .

l=m

If s > r ≥ 1, then we note that (2.63)

βn(r) (λ, x) { (n)(n−l) n n ∑ n−l n−l−k ∑ ∑ ∑ n −m l k =λ λ (n−l−k+s−r ) S1 (l, m) m=0

l=m k=0

i=0

s−r

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× S1 (n − l − k + s − r, i + s − r) S2 (i + s − r, s − r) λ−i bk

(s)

From (1.7) and (1.12), we get

((

Hn(s) (x|µ) ∼

(2.64)

et − µ 1−µ

)s

21

}

(s) Bm (x) .

) ,t .

By (1.19), (1.20), (1.22) and (2.64), we have βn(r) (λ, x) =

(2.65)

n ∑

(s) Cn,m Hm (x|µ) ,

m=0

where (2.66) Cn,m

)s / 1 )m ⟩ ( e λ log(1+λt) − µ 1 − µ 1 ( ( 1 )/ )r log (1 + λt) xn λ λ e λ log(1+λt) − 1 elog(1+λt) − 1 ⟩ ( ⟨ )r )s ( 1 1 t m n λ = − µ (1 + λt) (log (1 + λt)) x s 1 m!λm (1 − µ) (1 + λt) λ − 1 ⟨ )r ⟩ ( ( ) n ( )s ∑ n 1 1 t l n−l S1 (l, m) λ (1 + λt) λ − µ = m x s 1 l λ (1 − µ) (1 + λt) λ − 1 l=m ) n ( ) n−l ( ⟨( )s ⟩ ∑ ∑ 1 n n − l (r) 1 n−l−k l λ S (l, m) λ β (λ) (1 + λt) = m − µ . x 1 s k l k λ (1 − µ) 1 = m!

⟨

(

l=m

k=0

It is easy to show that ⟨( )s ⟩ 1 (1 + λt) λ − µ xn−l−k (2.67) ⟩ ⟨ ∞ s ( ) ∑ ∑ s tj n−l−k s−i = (−µ) (i|λ)j x j! i i=0 j=0 =

s ( ) ∑ s i=0

i

(−µ)

s−i

(i|λ)n−l−k .

From (2.66) and (2.67), we have (2.68) Cn,m ) n ( ) n−l ( s ( ) ∑ ∑ ∑ 1 n n − l (r) s s−i l = m S1 (l, m) λ βk (λ) (−µ) (i|λ)n−l−k s l k i λ (1 − µ) i=0 l=m k=0 )( ) n ∑ n−l ∑ s ( )( ∑ n n−l s 1 s−i (r) S1 (l, m) λl (−µ) βk (λ) (i|λ)n−l−k . = m s l k i λ (1 − µ) i=0 l=m k=0

Therefore, by (2.65) and (2.68), we obtain the following theorem.

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Theorem 2.13. For µ ∈ C with µ ̸= 1, n ≥ 0, we have βn(r) (λ, x)

{ )( ) n n ∑ n−l ∑ s ( )( ∑ ∑ n n−l s 1 s−i −m S1 (l, m) λl (−µ) = λ s l k i (1 − µ) m=0 l=m k=0 i=0 } (r) (s) × βk (λ) (i|λ)n−l−k Hm (x|µ) . Remark. For n ≥ 0, we have Hn(s) (x|µ) { } ) n n−m n−l ( )( ∑ 1 ∑ ∑ n n−l k+l (r) (r) (r) S2 (k, m) λ Bl Hn−l−k−j (µ) βm = (λ, x) . m λ l k m=0 l=0 k=m

ACKNOWLEDGEMENTS.The work reported in this paper was conducted during the sabbatical year of Kwangwoon University in 2014.

References 1. S. Araci and M. Acikgoz, A note on the Frobenius-Euler numbers and polynomials associated with Bernstein polynomials, Adv. Stud. Contemp. Math. (Kyungshang) 22 (2012), no. 3, 399–406. MR 2976598 2. A. Bayad and T. Kim, Identities for the Bernoulli, the Euler and the Genocchi numbers and polynomials, Adv. Stud. Contemp. Math. (Kyungshang) 20 (2010), no. 2, 247–253. MR 2656975 (2011d:11274) 3. L. Carlitz, A degenerate Staudt-Clausen theorem, Arch. Math. (Basel) 7 (1956), 28–33. MR 0074436 (17,586a) 4. , Degenerate Stirling, Bernoulli and Eulerian numbers, Utilitas Math. 15 (1979), 51–88. MR 531621 (80i:05014) 5. D. Ding and J. Yang, Some identities related to the Apostol-Euler and ApostolBernoulli polynomials, Adv. Stud. Contemp. Math. (Kyungshang) 20 (2010), no. 1, 7–21. MR 2597988 (2011k:05030) 6. S. Gaboury, R. Tremblay, and B.-J. Fug`ere, Some explicit formulas for certain new classes of Bernoulli, Euler and Genocchi polynomials, Proc. Jangjeon Math. Soc. 17 (2014), no. 1, 115–123. MR 3184467 7. Y. He and W. Zhang, A convolution formula for Bernoulli polynomials, Ars Combin. 108 (2013), 97–104. MR 3060257 8. F. T. Howard, Explicit formulas for degenerate Bernoulli numbers, Discrete Math. 162 (1996), no. 1-3, 175–185. MR 1425786 (97m:11024) 9. K.-W. Hwang, D. V. Dolgy, D. S. Kim, T. Kim, and S. H. Lee, Some theorems on Bernoulli and Euler numbers, Ars Combin. 109 (2013), 285–297. MR 3087218 10. Hassan Jolany, R. Eizadi Alikelaye, and S. Sharif Mohamad, Some results on the generalization of Bernoulli, Euler and Genocchi polynomials, Acta Univ. Apulensis Math. Inform. (2011), no. 27, 299–306. MR 2896405 (2012k:11026) 11. D. S. Kim and T. Kim, Some identities of Bernoulli and Euler polynomials arising form umbral calculus, Adv. Stud. Contemp. Math. (Kyungshang) 23 (2013), no. 1, 159–171. MR 3059325

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HIGHER-ORDER DEGENERATE BERNOULLI POLYNOMIALS

23

12. D. S. Kim, T. Kim, S.-H. Lee, and J.-J. Seo, A note on the lambda-Daehee polynomials, Int. J. Math. Anal. (Ruse) 7 (2013), no. 61-64, 3069–3080. MR 3162167 13. T. Kim, Some identities for the Bernoulli, the Euler and the Genocchi numbers and polynomials, Adv. Stud. Contemp. Math. (Kyungshang) 20 (2010), no. 1, 23–28. MR 2597989 (2011a:11211) 14. , Identities involving Laguerre polynomials derived from umbral calculus, Russ. J. Math. Phys. 21 (2014), no. 1, 36–45. MR 3182545 15. S.-H. Rim and J. Jeong, Identities on the modified q-Euler and q-Bernstein polynomials and numbers with weight, J. Comput. Anal. Appl. 15 (2013), no. 1, 39–44. MR 3076716 16. S. Roman, The umbral calculus, Pure and Applied Mathematics, vol. 111, Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], New York, 1984. MR 741185 (87c:05015) 17. E. S¸en, Theorems on Apostol-Euler polynomials of higher order arising from Euler basis, Adv. Stud. Contemp. Math. (Kyungshang) 23 (2013), no. 2, 337– 345. MR 3088764 18. A. V. Ustinov, Korobov polynomials and umbral analysis, Chebyshevski˘ı Sb. 4 (2003), no. 4(8), 137–152. MR 2097912 (2005m:33018) 19. M. Wu and H. Pan, Sums of products of the degenerate Euler numbers, Advances in Difference Equations (2014), 2014:40. 20. P. T. Young, Degenerate Bernoulli polynomials, generalized factorial sums, and their applications, J. Number Theory 128 (2008), no. 4, 738–758. MR 2400037 (2009b:11045) 21. Z. Zhang and H. Yang, Some closed formulas for generalized Bernoulli-Euler numbers and polynomials, Proc. Jangjeon Math. Soc. 11 (2008), no. 2, 191–198. MR 2482602 (2010a:11036) Department of Mathematics, Sogang University, Seoul 121-742, Republic of Korea E-mail address: [email protected] Department of Mathematics, Kwangwoon University, Seoul 139-701, Republic of Korea E-mail address: [email protected]

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KOROBOV POLYNOMIALS OF THE SEVENTH KIND AND OF THE EIGHTH KIND DAE SAN KIM, TAEKYUN KIM, TOUFIK MANSOUR, AND JONG-JIN SEO

Abstract. In this paper, we consider the Korobov polynomials of the seventh kind and of the eighth kind. We present several explicit formulas and recurrence relations for these polynomials. In addition, we establish connections between our polynomials and several known families of polynomials.

1. Introduction The degenerate Bernoulli polynomials are the degenerate version of Bernoulli polynomials introduced by Calitz [3, 4]. On the other hand, the Korobov polynomials of the first kind are the first degenerate version of the Bernoulli polynomials of the second kind, see [13, 14]. In recent years, many researchers studied various kinds of degenerate versions of families polynomials like Bernoulli polynomials, Euler polynomials, falling factorial polynomials, Bell polynomials and their variants, see [6–10] and references therein. Along this line of research, we introduced in [8, 9] four kinds of new degenerate versions of Bernoulli polynomials of the second kind, called the Korobov polynomials of the third, fourth, fifth, and sixth kind. Here, we will discuss two other degenerate versions of Bernoulli polynomials of the second kind, namely, the Korobov polynomials of the seventh and eighth kind. We will investigate some properties, explicit expressions, recurrence relations, and connections with other families polynomials with the help of umbral calculus (see [10, 15, 16]). To do that, we recall some families polynomials. The Bernoulli polynomials of the second kind bn (x) are given by the generating function ∑ t tn (1.1) (1 + t)x = bn (x) . log(1 + t) n! n≥0 For x = 0, bn = bn (0) are called the Bernoulli numbers of the second kind. The Daehee polynomials Dn (x) are defined by the generating function (1.2)

∑ tn log(1 + t) (1 + t)x = Dn (x) . t n! n≥0

2010 Mathematics Subject Classification. 05A19, 05A40, 11B83. Key words and phrases. Korobov polynomials of the seventh kind and of the eighth kind, Umbral calculus. 1

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When x = 0, Dn = Dn (0) are called the Daehee numbers. The Krobov polynomials Kn (λ, x) of the first kind are given by ∑ λt tn x (1 + t) = K (λ, x) . n (1 + t)λ − 1 n! n≥0

(1.3)

When x = 0, Kn (λ) = Kn (λ, 0) are called the Korobov numbers of the first kind. The degenerate falling factorial polynomials (x)n,λ were defined in [7] by the generating function λ ∑ x (1+t) −1 tn (1.4) (1 + λ) λ λ = (x)n,λ . n! n≥0 Clearly, limλ→0 (x)n,λ = (x)n , the nth falling factorial polynomial. These polynomials can be defined as (x)n,λ ∼ (1, f (t)), where (

) λ1

log(1 + λ) (1 + t)λ − 1 − 1 and f¯(t) = . λ λ ∑ n ¯ Note that we write sn (x) ∼ (g(t), f (t)) if n≥0 sn (x) tn! = g(f¯1(t)) exf (t) , where f¯(t) is the compositional inverse of f (t), see [15, 16]. The degenerate Stirling numbers of the first kind S1 (n, k | λ), n ≥ k ≥ 0, were given in [7] by the generating function (1.5)

(1.6)

f (t) =

λ2 t 1+ log(1 + λ)

1 k!

(

(1 + t)λ − 1 λ

)k =

∑

S1 (n, k | λ)

n≥k

so that, in the notation of umbral calculus, S1 (n, k | λ) =

1 k!

tn , n! ⟨(

(1+t)λ −1 λ

)k

⟩ |x . Then, n

it was shown in [7] that (x)n,λ =

)k n ( ∑ log(1 + λ) λ

k=0

S1 (n, k | λ)xk

with S1 (n, k | λ) =

n ∑

S1 (n, m)S2 (m, k)λm−k ,

m=k

where limλ→0 S1 (n, k | λ) = S1 (n, k) is the Stirling number of the first kind. Here, we introduce Korobov polynomials of the seventh kind Kn,7 (λ, x) and of the eighth kind Kn,9 (λ, x), respectively given by (1.7) (1.8)

λ ∑ x (1+t) −1 log(1 + λt) tn = Kn,7 (λ, x) , (1 + λ) λ λ λ log(1 + t) n! n≥0

λ ∑ x (1+t) −1 log(1 + λt) tn λ λ (1 + λ) = K (λ, x) . n,8 (1 + t)λ − 1 n! n≥0

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3

When x = 0, Kn,7 (λ) = Kn,7 (λ, 0) and Kn,8 (λ) = Kn,8 (λ, 0) are called the Korobov numbers of the seventh kind and of the eighth kind, respectively. We observe that λ x (1+t) −1 λt log(1 + λt) x λ λ (1 + t) = lim (1 + λ) λ→0 (1 + t)λ − 1 λ→0 λ log(1 + t) λ x (1+t) −1 t log(1 + λt) λ λ (1 = = lim + λ) (1 + t)x , λ→0 (1 + t)λ − 1 log(1 + t)

lim

which implies that limλ→0 Kn (λ, x) = limλ→0 Kn,7 (λ, x) = limλ→0 Kn,8 (λ, x) = bn (x). It is immediate to see that (see [15,16]) ( Kn,7 (λ, x) and)Kn,8 (λ, ( x) are Sheffer sequences )

log(1+f (t)) (1+f (t)) −1 for the respective pairs λlog(1+λf , f (t) and log(1+λf , f (t) , where f (t) is given (t)) (t)) in (1.5). Thus, (1.7) and (1.8) can be presented as ( )   λ2 t ( ) log 1 + log(1+λ) λ log(1 + f (t)) Kn,7 (λ, x) ∼ (1.9) , f (t) =  , f (t) , log(1 + λf (t)) log(1 + λf (t)) ) ( ) ( λ2 t (1 + f (t))λ − 1 log(1+λ) (1.10) , f (t) = , f (t) . Kn,8 (λ, x) ∼ log(1 + λf (t)) log(1 + λf (t)) λ

In the next two sections, we will use umbral calculus in order to study some properties, explicit formulas, recurrence relations and identities about the Korobov polynomials of the seventh kind and of the eighth kind. In last section, we present connections between our polynomials and several known families of polynomials.

2. Explicit expressions In this section, we present several explicit formulas for the Korobov polynomials of the seventh kind and of the eighth kind, namely Kn,7 (λ, x) and Kn,8 (λ, x). Theorem 2.1. For all n ≥ 0, n ∑ n ( ) ∑ n logk (1 + λ) Kn,7 (λ, x) = S1 (ℓ, k|λ)Kn−ℓ,7 (λ)xk k ℓ λ k=0 ℓ=k ) n ∑ n ∑ n−ℓ ( )( ∑ n n − ℓ logk (1 + λ) = S1 (ℓ, k|λ)bm Dn−ℓ−m λn−ℓ−m xk , k λ ℓ m k=0 ℓ=k m=0 ( ) n ∑ n ∑ n logk (1 + λ) Kn,8 (λ, x) = S1 (ℓ, k|λ)Kn−ℓ,8 (λ)xk k λ ℓ k=0 ℓ=k ) n ∑ n ∑ n−ℓ ( )( ∑ n n − ℓ logk (1 + λ) = S1 (ℓ, k|λ)Km (λ)Dn−ℓ−m λn−ℓ−m xk . k λ ℓ m k=0 ℓ=k m=0 Proof. We proceed the proof ∑ by using the conjugation representation for Sheffer sequences (see [15,16]): sn (x) = nk=0 k!1 ⟨g(f¯(t))−1 f¯(t)k |xn ⟩xk , for any sn (x) ∼ (g(t), f (t)).

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Thus, by (1.9), we have ⟩ ⟨ n ∑ 1 log(1 + λt) logk (1 + λ)((1 + t)λ − 1)k n Kn,7 (λ, x) = |x xk 2k k! λ log(1 + t) λ k=0 ⟨ ⟩ n ∑ logk (1 + λ) log(1 + λt) ((1 + t)λ − 1)k n = | x xk , k k λ λ log(1 + t) k!λ k=0 which, by (1.6) and (1.7), implies

⟩ log(1 + λt) ∑ tℓ n Kn,7 (λ, x) = | S1 (ℓ, k|λ) x xk k λ λ log(1 + t) ℓ! k=0 ℓ≥k ⟨ ⟩ n ∑ n ( ) ∑ n logk (1 + λ) log(1 + λt) n−ℓ = S1 (ℓ, k|λ) |x xk k ℓ λ λ log(1 + t) k=0 ℓ=k n ∑ n ( ) ∑ n logk (1 + λ) S1 (ℓ, k|λ)Kn−ℓ,7 (λ)xk . = k ℓ λ k=0 ℓ=k n ∑ logk (1 + λ)

(2.1)

⟨

On the other hand, by (2.1), we have ⟨ ⟩ n ( ) n ∑ ∑ n logk (1 + λ) log(1 + λt) t n−ℓ S1 (ℓ, k|λ) | x xk , Kn,7 (λ, x) = k ℓ λ λt log(1 + t) k=0 ℓ=k which, by (1.1) and (1.2), we obtain Kn,7 (λ, x)

⟩ log(1 + λt) ∑ tm n−ℓ = S1 (ℓ, k|λ) | bm x xk k ℓ λ λt m! m≥0 k=0 ℓ=k ( )( ) ⟩ ⟨ n−ℓ n n k ∑ ∑ ∑ n n − ℓ log (1 + λ) log(1 + λt) n−ℓ−m = S1 (ℓ, k|λ)bm |x xk k ℓ m λ λt k=0 ℓ=k m=0 ⟨ ⟩ ) n−ℓ ( )( n ∑ n ∑ j ∑ ∑ n n − ℓ logk (1 + λ) t = S1 (ℓ, k|λ)bm Dj λj |xn−ℓ−m xk k ℓ m λ j! j≥0 k=0 ℓ=k m=0 ) n ∑ n ∑ n−ℓ ( )( ∑ n n − ℓ logk (1 + λ) = S1 (ℓ, k|λ)bm Dn−ℓ−m λn−ℓ−m xk , k ℓ m λ k=0 ℓ=k m=0 n ( ) n ∑ ∑ n logk (1 + λ)

⟨

which completes the proof of formulas for kn,7 (λ, x). Now let us deal with the case Kn,8 (λ, x). Similarly, by using the conjugation representation for Sheffer sequences, (1.10) and (1.6), we obtain ⟨ ⟩ n ∑ n ( ) ∑ log(1 + λt) n−ℓ n logk (1 + λ) S1 (ℓ, k|λ) |x (2.2) Kn,8 (λ, x) = xk k λ−1 λ (1 + t) ℓ k=0 ℓ=k n ∑ n ( ) ∑ n logk (1 + λ) S1 (ℓ, k|λ)Kn−ℓ,8 (λ)xk . = k λ ℓ k=0 ℓ=k

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On the other hand, by (2.2), we have ⟨ ⟩ n ∑ n ( ) ∑ λt n logk (1 + λ) log(1 + λt) n−ℓ S1 (ℓ, k|λ) | x Kn,8 (λ, x) = xk , k λ λ λt (1 + t) − 1 ℓ k=0 ℓ=k which, by (1.3) and (1.2), we obtain Kn,8 (λ, x)

⟩ tm n−ℓ log(1 + λt) ∑ = S1 (ℓ, k|λ) | Km (λ) x xk k ℓ λ λt m! m≥0 k=0 ℓ=k ⟨ ⟩ ( )( ) n n n−ℓ ∑ ∑ ∑ n n − ℓ logk (1 + λ) log(1 + λt) n−ℓ−m = S1 (ℓ, k|λ)Km (λ) |x xk k ℓ m λ λt k=0 ℓ=k m=0 ⟨

n ∑ n ( ) ∑ n logk (1 + λ)

(2.3) =

) n ∑ n ∑ n−ℓ ( )( ∑ n n − ℓ logk (1 + λ) k=0 ℓ=k m=0

ℓ

m

λk

S1 (ℓ, k|λ)Km (λ)Dn−ℓ−m λn−ℓ−m xk , 

which completes the proof.

Now, we express our polynomials in terms of the degenerate falling factorial polynomials. Theorem 2.2. For all n ≥ 0, ( n−ℓ ( ) n ( ) ∑ ∑ n − ℓ) n Kn,7 (λ, x) = λn−ℓ−m bm Dn−ℓ−m (x)ℓ,λ , ℓ m m=0 ℓ=0 ( ) ( ) ) n n−ℓ ( ∑ ∑ n n − ℓ n−ℓ−m Kn,8 (λ, x) = λ Km (λ)Dn−ℓ−m (x)ℓ,λ . ℓ m m=0 ℓ=0 Proof. By (1.9), we have ⟨ ⟩ ⟨ ⟩ y (1+t)λ −1 y (1+t)λ −1 log(1 + λt) log(1 + λt) n n Kn,7 (λ, y) = (1 + λ) λ λ |x = |(1 + λ) λ λ x , λ log(1 + t) λ log(1 + t) which, by (1.4), implies ⟨ ⟩ ⟨ ⟩ n ( ) ∑ tℓ n n log(1 + λt) n−ℓ log(1 + λt) ∑ | (y)ℓ,λ x (y)ℓ,λ |x . Kn,7 (λ, y) = = λ log(1 + t) ℓ≥0 ℓ! ℓ λ log(1 + t) ℓ=0 Therefore, by (2.1), we obtain ( n−ℓ ( ) ) n ( ) ∑ ∑ n n − ℓ n−ℓ−m Kn,7 (λ, y) = λ bm Dn−ℓ−m (y)ℓ,λ , ℓ m m=0 ℓ=0 which completes the proof for Kn,7 (λ, y). By using similar arguments as above together with (1.10) and (1.4), we obtain ⟨ ⟩ n ( ) ∑ n log(1 + λt) n−ℓ Kn,8 (λ, y) = (y)ℓ,λ |x . λ−1 ℓ (1 + t) ℓ=0

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Therefore, by (2.2) and (2.3), we have ( n−ℓ ( ) n ( ) ∑ ∑ n − ℓ) n Kn,8 (λ, y) = λn−ℓ−m Km (λ)Dn−ℓ−m (y)ℓ,λ , ℓ m m=0 ℓ=0 

which completes the proof.

In the next theorem, we find explicit formulas for the coefficient of xj in Kn,7 (λ, x) and Kn,8 (λ, x). Theorem 2.3. For all n ≥ 0 and s = 7, 8, Kn,s (λ, x) ( n k ℓ ) n ∑ ∑ ∑ ∑ (−1)ℓ−m ( ℓ )(k ) logj (1 + λ) S1 (n, k|λ)Kℓ,s (λ) xj . = (m|λ)k−j j ℓ! m j λ j=0 k=j ℓ=0 m=0 Proof. By (1.4) and (1.9), we have ∑ logk (1 + λ) λ log(1 + f (t)) Kn,7 (λ, x) = (x)n,λ = S1 (n, k|λ)xk ∼ (1, f (t)). k log(1 + λf (t)) λ k=0 n

Thus, Kn,7 (λ, x) =

n ∑ logk (1 + λ) k=0

=

(2.4)

S1 (n, k|λ)

k n ∑ ∑ logk (1 + λ) k=0 ℓ=0

Note that

λk

λk

log(1 + λf (t)) k x λ log(1 + f (t))

S1 (n, k|λ)

Kℓ,7 (λ) (f (t))ℓ xk . ℓ!

ℓ ( ) ∑ ( )m/λ k ℓ (f (t)) x = (−1)ℓ−m 1 + λ2 t/ log(1 + λ) x m m=0 ( ) ℓ ∑ k ( ) ∑ ℓ j k ℓ−m = (−1) (m|λ)j (λ/ log(1 + λ)) xk−j . m j m=0 j=0 ℓ k

Therefore, Kn,7 (λ, x)

( )( ) Kℓ,7 (λ) ℓ k j logj (1 + λ) = S1 (n, k|λ) x (−1) (m|λ)k−j j λ ℓ! m j k=0 ℓ=0 m=0 j=0 ( n k ℓ ) n j ∑ ∑ ∑ ∑ (−1)ℓ−m ( ℓ )(k ) log (1 + λ) (m|λ)k−j = S1 (n, k|λ)Kℓ,7 (λ) xj . j ℓ! m j λ j=0 k=j ℓ=0 m=0 n ∑ k ∑ ℓ ∑ k ∑

ℓ−m

By (1.4) and (1.10), we have ∑ logk (1 + λ) (1 + f (t))λ − 1 Kn,8 (λ, x) = (x)n,λ = S1 (n, k|λ)xk ∼ (1, f (t)). k log(1 + λf (t)) λ k=0 n

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Thus, by using the above arguments, we obtain Kn,8 (λ, x) ( n k ℓ ) n ∑ ∑ ∑ ∑ (−1)ℓ−m ( ℓ )(k ) logj (1 + λ) = S1 (n, k|λ)Kℓ,8 (λ) xj , (m|λ)k−j j ℓ! m j λ j=0 k=j ℓ=0 m=0 

which completes the proof.

In the next theorem, we express Korobov polynomials of seventh and eighth kinds in terms of Korobov polynomials of fifth and sixth kinds. Theorem 2.4. For all n ≥ 0 and s = 7, 8, n ( ) ∑ n Kn,s (λ, x) = Dn−ℓ λn−ℓ Kℓ,s−2 (λ, x). ℓ ℓ=0 Proof. Recall that Korobov polynomials of the fifth kind (see [9]) is given by λ ∑ x (1+t) −1 t tn (1 + λ) λ λ = Kn,5 (λ, x) . log(1 + t) n! n≥0 So, by (1.7), we have

⟨

y (1+t)λ −1 log(1 + λt) t Kn,7 (λ, y) = | (1 + λ) λ λ xn λt log(1 + t) ( ) ⟨ ⟩ n ∑ n log(1 + λt) n−ℓ = Kℓ,5 (λ, y) |x , ℓ λt ℓ=0 ∑ ( ) which, by (1.2), implies Kn,7 (λ, y) = nℓ=0 nℓ Kℓ,5 (λ, y)Dn−ℓ λn−ℓ .

⟩

Recall that Korobov polynomials of the sixth kind (see [9]) is defined by λ ∑ x (1+t) −1 λt tn λ,y λ (1 + λ) = K (λ, x) . n,6 (1 + t)λ − 1 n! n≥0 ∑ ( ) Similarly, by (1.2) and (1.8), we obtain Kn,8 (λ, y) = nℓ=0 nℓ Kℓ,6 (λ, y)Dn−ℓ λn−ℓ , as claimed.  In the next theorem, we express our polynomials Kn,7 (λ, x) and Kn,8 (λ, x) in terms (n) of degenerate Bernoulli numbers βℓ (λ) of order n, which are given by the generating function ∑ (n) tℓ tn (2.5) = βℓ (λ) . ((1 + λt)1/λ − 1)n ℓ! ℓ≥0 Theorem 2.5. For all n ≥ 1 and s = 7, 8, Kn,s (λ, x) ( n k ℓ ( )( ℓ )(k) n ∑ ∑ ∑ ∑ (−1)ℓ−m n−1 k−1 m j j=0

k=j ℓ=0 m=0

ℓ!

( (m|λ)k−j

818

log(1 + λ) λ

)j

) (n) βn−k (λ)Kℓ,s (λ)

xj .

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n

∼ (1, λt/ log(1 + λ)). Thus, by (1.9), we Proof. It is not hard to see that log (1+λ)x λn have ( )n λt n n λ log(1 + f (t)) log(1+λ) −1 log (1 + λ)x Kn,7 (λ, x) = x x log(1 + λf (t)) (1 + λ2 t/ log(1 + λ))1/λ − 1 λn ( )n logn (1 + λ) r = x |r=λt/ log(1+λ) xn−1 , λn (1 + λr)1/λ − 1 which, by (2.5), implies ( )k λ log(1 + f (t)) 1 logn (1 + λ) ∑ (n) λt x βk (λ) xn−1 Kn,7 (λ, x) = n log(1 + λf (t)) λ k! log(1 + λ) k≥0 ) ( )n−k n−1 ( ∑ n − 1 (n) log(1 + λ) xn−k = βk (λ) k λ k=0 ) ( )k n ( ∑ n − 1 (n) log(1 + λ) = βn−k (λ) xk . k − 1 λ k=0 On the other hand, by (2.4), we have log(1 + λf (t)) k x λ log(1 + f (t)) ( )( ) ℓ ∑ k k ∑ ∑ Kℓ,7 (λ) ℓ k λk−j = xj . (−1)ℓ−m (m|λ)k−−j k−j ℓ! m j log (1 + λ) ℓ=0 m=0 j=0 Therefore, the polynomials Kn,7 (λ, x) is given by ( n k ℓ ) ( )( ℓ )(k) )j ( n ∑ ∑ ∑ ∑ (−1)ℓ−m n−1 log(1 + λ) k−1 m j (n) (m|λ)k−j βn−k (λ)Kℓ,7 (λ) xj . ℓ! λ j=0 k=j ℓ=0 m=0 By using similar argument as above with using (1.10), we obtain the formula for the nth Korobov polynomial kn,8 (λ, x) of the eighth kind (we leave the details for the interested reader).  3. Recurrences In this section, we present several recurrences for the Korobov polynomials of the seventh kind and of the eighth kind. Note that, ∑ (by) (1.9), (1.10) and the fact that (x)n,λ ∼ (1, f (t)), we obtain Kn,d (λ, x + y) = nj=0 nj Kj,d (λ, x)(y)n−j,λ , for d = 7, 8. Proposition 3.1. For all n ≥ 1 and s = 7, 8, Kn,s (λ, x) + nKn−1,s (λ, x) ( n n ( )( ) ) n ∑ ∑∑ n k logm (1 + λ) S1 (ℓ, k|λ)Kn−ℓ,s (λ) xm . = (1|λ)k−m m λ ℓ m m=0 k=m ℓ=k

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Proof. It is well-known that if sn (x) ∼ (g(t), f (t)), ( then we have f (t)sn (x) ) = nsn−1 (x) 1 ) ( λ λ2 t − 1 Kn,s (λ, x) = (see [15, 16]). Thus, by (1.9) and (1.10), we obtain 1 + log(1+λ) ( ) λ1 λ2 t nKn−1,s (λ, x), which implies Kn,s (λ, x) + nKn−1,s (λ, x) = 1 + log(1+λ) Kn,s (λ, x). By Theorem 2.1 we have Kn,s (λ, x) + nKn−1,s (λ, x) ( ) λ1 n ∑ n ( ) ∑ n logk (1 + λ) λ2 t S1 (ℓ, k|λ)Kn−ℓ,s (λ) 1 + xk = k λ log(1 + λ) ℓ k=0 ℓ=k n ∑ n ∑ k ( ) ∑ n logk−m (1 + λ) tm k = S (ℓ, k|λ)K (λ)(1|λ) x 1 n−ℓ,s m ℓ λk−m m! k=0 ℓ=k m=0 n ∑ n ∑ k ( )( ) ∑ n k logk−m (1 + λ) S1 (ℓ, k|λ)Kn−ℓ,s (λ)(1|λ)m xk−m = k−m λ ℓ m k=0 ℓ=k m=0 ( n n ( )( ) ) n ∑ ∑∑ n k logm (1 + λ) (1|λ)k−m = S1 (ℓ, k|λ)Kn−ℓ,s (λ) xm . m ℓ m λ m=0 k=m ℓ=k 

which completes the proof. In the next result, we express Kn,8 (λ, x), respectively.

d K (λ, x) dx n,7

and

d K (λ, x) dx n,8

in terms of Kn,7 (λ, x) and

Proposition 3.2. For all n ≥ 0 and s = 7, 8,

n−1 ( ) d log(1 + λ) ∑ n Kn,s (λ, x) = (λ)n−ℓ Kℓ,s (λ, x). dx λ2 ℓ ℓ=0

(n ) ∑ d n−ℓ ¯ Proof. Note that dx sn (x) = n−1 ⟩sℓ (x), for all sn (x) ∼ (g(t), f (t)), see ℓ=0 ℓ ⟨f (t)|x [15, 16]. So, for sn (x) = Kn,s (λ, x), it remains to compute A = ⟨f¯(t)|xn−ℓ ⟩. By (1.9) ∑ j and (1.10), we have A = log(1+λ) ⟨ j≥1 (λ)j tj! |xn−ℓ ⟩ = log(1+λ) (λ)n−ℓ , which completes λ2 λ2 the proof.  Theorem 3.3. For all n ≥ 1 and s = 7, 8, ) n−1 ( x log(1 + λ) ∑ n − 1 (λ − 1)n−1−ℓ Kℓ,s (λ, x) Kn,s (λ, x) = λ ℓ ℓ=0 ( ) n n−ℓ 1 ∑∑ n + (n − ℓ)n−ℓ−m us (ℓ, m), n ℓ=0 m=0 ℓ where

{ } u7 (ℓ) = bℓ (−λ)n−ℓ−m (x)m,λ − (−1)n−ℓ−m Km,7 (λ, x) , { ( ) } λ−1 n−ℓ−m u8 (ℓ) = Kℓ (λ) (−λ) (x)m,λ − Km,8 (λ, x) . n−ℓ−m

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Proof. Since the similarity between Kn,7 (λ, x) and Kn,8 (λ, x) (see (1.9) and (1.10)), we omit the proof of the case Kn,8 (λ, x) and give only the details of the case Kn,7 (λ, x). By (1.9), we have ⟨ ( ) ⟩ y (1+t)λ −1 d log(1 + λt) n−1 (3.1) Kn,7 (λ, y) = (1 + λ) λ λ |x = A + B, dt λ log(1 + t) ⟩ ⟨ ⟩ ⟨ y (1+t)λ −1 y (1+t)λ −1 log(1+λt) d d log(1+λt) n−1 n−1 λ λ λ λ |x and A = λ log(1+t) dt (1 + λ) |x . where B = dt λ log(1+t) (1 + λ) First, we compute the term B. ⟨ ⟩ y (1+t)λ −1 log(1 + λ) log(1 + λt) λ−1 n−1 y(1 + t) |x B= (1 + λ) λ λ λ log(1 + t) λ ⟨ ⟩ y (1+t)λ −1 y log(1 + λ) log(1 + λt) λ−1 n−1 = (1 + λ) λ λ |(1 + t) x λ λ log(1 + t) ) ⟨ ⟩ n−1 ( y (1+t)λ −1 y log(1 + λ) ∑ n − 1 log(1 + λt) n−1−ℓ λ λ = (1 + λ) |x (λ − 1)ℓ λ ℓ λ log(1 + t) ℓ=0 ) n−1 ( y log(1 + λ) ∑ n − 1 = (λ − 1)ℓ Kn−1−ℓ,7 (λ, y) λ ℓ ℓ=0 ( ) n−1 y log(1 + λ) ∑ n − 1 = (λ − 1)n−1−ℓ Kℓ,7 (λ, y). λ ℓ ℓ=0 Now, we compute the first term A, ⟨ } ⟩ { y (1+t)λ −1 t 1 1 log(1 + λt) −1 n−1 A= (1 + t) (1 + λ) λ λ |x − log(1 + t) t 1 + λt λ log(1 + t) ⟨{ ⟩ } y (1+t)λ −1 1 1 log(1 + λt) t n −1 = − x (1 + t) (1 + λ) λ λ | n 1 + λt λ log(1 + t) log(1 + t) ⟨{ ⟩ } ℓ y (1+t)λ −1 ∑ t 1 1 log(1 + λt) bℓ xn . = − (1 + t)−1 (1 + λ) λ λ | n 1 + λt λ log(1 + t) ℓ! ℓ≥0 Note that

1 1+λt

−

log(1+λt) (1 λ log(1+t)

+ t)−1 has order at least one. Thus,

⟩ n ( ) {⟨ y (1+t)λ −1 1∑ n 1 n−ℓ A= bℓ (1 + λ) λ λ | x n ℓ=0 ℓ 1 + λt ⟨ ⟩} y (1+t)λ −1 log(1 + λt) 1 n−ℓ − (1 + λ) λ λ | x λ log(1 + t) 1+t ⟨ ⟩ n ( ) {∑ n−ℓ k ∑ 1∑ n t = bℓ (−λ)m (n − ℓ)m (y)k,λ |xn−ℓ−m n ℓ=0 ℓ k! m=0 k≥0 ⟨ ⟩} n−ℓ ∑ ∑ tk n−ℓ−m m , − (−1) (n − ℓ)m Kk,7 (λ, y) |x k! m=0 k≥0

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which implies } n n−ℓ ( ) { 1 ∑∑ n m m A= bℓ (−λ) (n − ℓ)m (y)n−ℓ−m,λ − (−1) (n − ℓ)m Kn−ℓ−m,7 (λ, y) , n ℓ=0 m=0 ℓ Hence, by substituting the expressions of A and B in (3.1), we complete the proof.  4. Connections with families of polynomials In this section, we present some examples on the connections with families of polynomials. To do that, we recall for any two ∑Sheffer sequences sn (x) ∼ (g(t), f (t)) and rn (x) ∼ (h(t), ℓ(t)), we have that sn (x) = nm=0 Cn,m rm (x), where (see [15, 16]) ⟩ ⟨ 1 h(f¯(t)) ¯ m n (4.1) Cn,m = (ℓ(f (t))) |x . m! g(f¯(t)) (s)

We start with the connection to Bernoulli polynomials Bn (x) of order s. Recall that (s) the Bernoulli polynomials Bn (x) of order s are defined by the generating function ( t )s xt ∑ n (s) e = n≥0 Bn (x) tn! , equivalently, et −1 (( t )s ) e −1 (s) (4.2) Bn (x) ∼ ,t t (see [2, 5, 15]). In the next result, we express our polynomials in terms of Bernoulli polynomials of order s. ∑ (s) Theorem 4.1. Let d = 7, 8. For all n ≥ 0, Kn,d (λ, x) = nk=0 Cn,m Bm (x), where ( )( ) n−m ∑ n k+m ∑ n−ℓ−m logk+m (1 + λ) ℓ m (k+s) Kℓ,d (λ)S2 (k + s, s) Cn,m = S1 (n − ℓ, k + m|λ). k+m λ s ℓ=0 k=0 Proof. Since the similarity between Kn,7 (λ, x) and Kn,8 (λ, x) (see (1.9) and (1.10)), we omit the proof of the case Kn,8 (λ, x) and give only the details of the case Kn,7 (λ, x). ∑ (s) Let Kn,7 (λ, x) = nm=0 Cn,m Bm (x). So, by (1.9) and (4.2), we have ⟨ ¯ ⟩ 1 (ef (t) − 1)s log(1 + λt) ¯m Cn,m = f (t)|xn m! f¯s (t) λ log(1 + t) ⟩ ⟨ ¯ 1 (ef (t) − 1)s ¯m log(1 + λt) n = f (t)| x m! λ log(1 + t) f¯s (t) ⟨ ¯ ⟩ ∑ 1 (ef (t) − 1)s ¯m tℓ n = f (t)| Kℓ,7 (λ) x m! ℓ! f¯s (t) ℓ≥0 ⟨ ⟩ n ( ) ∑ 1 ∑ n f¯k+m (t) n−ℓ Kℓ,7 (λ) s! S2 (k + s, s) |x . = m! ℓ=0 ℓ (k + s)! k≥0

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12

DAE SAN KIM, TAEKYUN KIM, TOUFIK MANSOUR, AND JONG-JIN SEO

Thus, ⟩ ⟨ ¯k+m n−m n−ℓ−m ( ) s! ∑ ∑ n f (t) n−ℓ Cn,m = |x Kℓ,7 (λ)S2 (k + s, s) m! ℓ=0 k=0 (k + s)! ℓ ⟨ ⟩ n−m n−ℓ−m ( ) s! ∑ ∑ n logk+m (1 + λ) ((1 + t)λ − 1)k+m n−ℓ |x = Kℓ,7 (λ)S2 (k + s, s) m! ℓ=0 k=0 (k + s)!λk+m λk+m ℓ n−m n−ℓ−m ( ) s! ∑ ∑ n logk+m (1 + λ) = (k + m)!S1 (n − ℓ, k + m|λ). Kℓ,7 (λ)S2 (k + s, s) m! ℓ=0 k=0 (k + s)!λk+m ℓ Therefore, Cn,m =

n−m ∑ n−ℓ−m ∑ ℓ=0

k=0

(n)(k+m) ℓ

(k+sm) Kℓ,7 (λ)S2 (k + s, s) s

logk+m (1 + λ) S1 (n − ℓ, k + m|λ), λk+m 

as required.

Similar techniques as in the proof of the previous theorem, we can express our polynomials Kn,7 (λ, x), Kn,8 (λ, x) in terms of other families. Below we present two examples, where we leave the proofs to the interested reader. In the first example, we express our polynomials in terms of Frobenius-Euler polynomials. Note that the (s) Frobenius-Euler polynomials Hn (x|µ) of order s are defined by the generating ( (( t function )s )s ) ∑ n (s) (s) 1−µ e −µ t xt (x|µ) H (x|µ) ∼ e = , (µ = ̸ 1), or equivalently, H ,t n n t n≥0 e −µ n! 1−µ (see [1, 11, 12]). ∑ (s) Theorem 4.2. For all n ≥ 0 and d = 7, 8, Kn,d (λ, x) = nm=0 Cn,m Hm (x|µ), where ( )( )( ) s n−ℓ−m n−m ∑ k! j+m n s logj+m (1 + λ) ∑∑ m ℓ k Cn,m = S1 (n − ℓ, j + m|λ)S2 (j, k)Kℓ,d (λ). k j+m (1 − µ) λ ℓ=0 k=0 j=k For what follows, we define the associated sequence for 1 − (1 + λ2 t/ log(1 + λ))−1/λ , namely (x)(n,λ) . Thus, (x)(n,λ) ∼ (1, 1 − (1 + λ2 t/ log(1 + λ))−1/λ ). Recall here that (x)n ∼ (1, et − 1), (x)(n) ∼ (1, 1 − e−t ), (x)n,λ ∼ (1, (1 + λ2 t/ log(1 + λ))1/λ − 1) and (1 + λ2 t/ log(1 + λ))1/λ − 1 → et − 1, as λ → 0. Now, we ready to present our second example. ∑ Theorem 4.3. For all n ≥ 0 and d = 7, 8, Kn,d (λ, x) = nm=0 Cn,m (x)(m,λ) , where ( )( ) n ∑ n−ℓ n−ℓ−m n Cn,m = (−1) (n − 1 − ℓ)n−ℓ−m Kℓ,d (λ). ℓ m ℓ=0 ACKNOWLEDGEMENTS.The work reported in this paper was conducted during the sabbatical year of Kwangwoon University in 2014.

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KOROBOV POLYNOMIALS OF THE SEVENTH KIND AND OF THE EIGHTH KIND

13

References [1] S. Araci and M. Acikgoz, A note on the Frobenius-Euler numbers and polynomials associated with Bernstein polynomials, Adv. Stud. Contemp. Math. 22(3) (2012) 399–406. [2] A. Bayad and T. Kim, Identities involving values of Bernstein, q-Bernoulli, and q-Euler polynomials, Russ. J. Math. Phys. 18(2) (2011) 133–143. [3] L. Carlitz, A degenerate Staudt-Clausen theorem, Arch. Math. 7 (1956) 28–33. [4] L. Carlitz, Degenerate Stirling, Bernoulii and Eulerian numbers, Utilitas Math. 15 (1979) 51-88. [5] D. Ding and J. Yang, Some identities related to the Apostol-Euler and Apostol-Bernoulli polynomials, Adv. Stud. Contemp. Math. 20(1) (2010) 7–21. [6] D.S. Kim and T. Kim, On degenerate Bell numbers and polynomials, preprint. [7] D.S. Kim and T. Kim, Degenerate falling factorial polynomials, preprint. [8] D.S. Kim and T. Kim, Korobov polynomials of the third kind and of the fourth kind, preprint. [9] D.S. Kim, T. Kim, H.I. Kwon and T. Mansour, Korobov polynomials of the fifth kind and of the sixth kind, preprint. [10] D. S. Kim and T. Kim, Some identities of Korobov-type polynomials associated with p-adic integrals on Zp , Adv. Difference Equ. 2015 (2015), DOI: 10.1186/s13662-015-0602-8. [11] T. Kim, Identities involving Laguerre polynomials derived from umbral calculus, Russ. J. Math. Phys. 21 (2014) 36–45. [12] T. Kim and T. Mansour, Umbral calculus associated with Frobenius-type Eulerian polynomials, Russ. J. Math. Phys. 21 (2014) 484–493 . [13] N.M. Korobov, On some properties of special polynomials, Proceedings of the IV International Conference ”Modern Problems of Number Theory and its Applications” (Russian)(Tula, 2001), Vol. 1, 2001, 40–49. [14] N.M. Korobov, Speical polynomials and their applications, in ”Diophantine Approcimations”, Mathematical Notes (Russian), Vol. 2, Izd. Moskov. Univ., Moscow, 1996, 77–89. [15] S. Roman, The umbral calculus, Pure and Applied Mathematics 111, Academic Press, New York, 1984. [16] S. Roman, More on the umbral calculus, with emphasis on the q-umbral calculus, J. Math. Anal. Appl. 107 (1985) 222–254. [17] A.V. Ustinov, Korobov polynomials and umbral calculus, Chebyshevski˘ı Sb. 4:8 (2003) 137–152. Department of Mathematics, Sogang University, Seoul 121-742, South Korea E-mail address: [email protected] Department of Mathematics, Kwangwoon University, Seoul 139-701, South Korea E-mail address: [email protected] University of Haifa, Department of Mathematics, 3498838 Haifa, Israel E-mail address: [email protected] Department of Applied Mathematics, Pukyong National University, Busan 48513, Republic of Korea. E-mail address: [email protected]

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Some identities on the higher-order twisted q-Euler numbers and polynomials C. S. Ryoo Department of Mathematics, Hannam University, Daejeon 306-791, Korea

Abstract : In this paper we investigate some interesting symmetric identities for twisted q-Euler polynomials of higher order in complex field. Key words : Symmetric properties, power sums, Euler numbers and polynomials, twisted q-Euler numbers and polynomials. 2000 Mathematics Subject Classification : 11B68, 11S40, 11S80. 1. Introduction Euler numbers and polynomials possess many interesting properties and arising in many areas of mathematics, mathematical physics and statistical physics. Many mathematicians have studied in the area of the q- extension of Euler numbers and polynomials(see [1, 2, 3, 5, 6, 7, 8, 9, 11, 13]). Recently, Y. Hu studied several identities of symmetry for Carlitz’s q-Bernoulli numbers and polynomials in complex field(see [3]). D. Kim et al.[4] derived some identities of symmetry for (h, q)-extension of higher-order Euler numbers and polynomials. D. V. Dolgy et al.[2] derived some identities of symmetry for higher-order generalized q-Euler polynomials. In this paper, we establish some interesting symmetric identities for twisted q-Euler polynomials of higher order in complex field. The purpose of this paper is to present a systemic study of the twisted q-Euler numbers and polynomials of higher-order by using the multiple q-Euler zeta function. Throughout this paper, the notations N, Z, R, and C denote the sets of positive integers, integers, real numbers, and complex numbers, respectively, and Z+ := N ∪ {0}. We assume that q ∈ C with |q| < 1. Throughout this paper we use the notation: 1 − qx (cf. [1, 2, 3, 5]) . [x]q = 1−q Note that limq→1 [x] = x. Let ε be the pN -th root of unity(see [10, 12, 13]). In [5], T. Kim introduced the multiple q-Euler zeta function which interpolates higher-order q-Euler polynomials at negative integers as follows: ζq,r (s, x) =

[2]rq

∞ ∑ m1 ,··· ,mr

∑r

∑r

(−1) j=1 mj q j=1 mj , [m1 + · · · + mr + x]sq =0

(1)

where s ∈ C and x ∈ R, with x ̸= 0, −1, −2, . . .. Recently, D. V. Dolgy et al.[2] considered some symmetric identities for higher-order generalized q-Euler polynomials. The Euler polynomials of order r ∈ N attached to χ are also defined by the generating function: )r ( d−1 ∞ ∑ χ(l)(−1)l e(x+l)t ∑ tm (r) (x) . 2 = Em,χ (2) dt e +1 m! m=0 l=0

When x =

(r) 0, En,χ

=

(r) En,χ (0)

(r)

are called the Euler numbers En,χ attached to χ(see [2, 4]).

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For h ∈ Z, α, k ∈ N, and n ∈ Z+ , we introduced the higher order twisted q-Euler polynomials with weight α as follows(see [7]): n ( ) ∑ [2]kq n q αlx (α) l e En,q,ε (k|x) = (−1) . (1 − q α )n (1 + εq αl+h ) · · · (1 + εq αl+h−k+1 ) l l=0

(α) (α) en,q,w en,q,w In the special case, x = 0, E (k|0) = E (k) are called the higher-order twisted q-Euler numbers with weight α. We consider the higher order q-Euler polynomials of order r attached to χ twisted by ramified

roots of unity as follows(see [10]): ∞ ∑

∞ ∑ tn (r) En,χ,ε,q (x) = 2r n! n=0 m ,...,m

(−ε)

(

∑r j=0

mj

r =0

1

r ∏

) χ(mi ) e[x+

∑r j=1

mj ] q t

.

i=1

(r)

(r)

In the special case x = 0, the sequence En,χ,ε,q (0) = En,χ,ε,q are called the n-th q-Euler numbers of order r attached to χ twisted by ramified roots of unity. (k)

As is well known, the higher-order twisted q-Euler polynomials En,q,ε (x) are defined by the following generating function to be ∞ ∑

(k) (t, x) = [2]kq Feq,ε

(−1)m1 +···+mk εm1 +···+mk e[m1 +···+mk +x]q t

m1 ,··· ,mk =0

=

∞ ∑

(3)

tn (k) En,q,ε (x) ,

n=0 (k)

n!

(k)

where k ∈ N. When x = 0, En,q,ε = En,q,ε (0) are called the higher-order twisted q-Euler numbers (k) (k) (k) (k) (k) En,q,ε . Observe that if q → 1, ε → 1, then En,q,ε → En and En,q,ε (x) → En,χ (x). By using (3) and Cauchy product, we have n ( ) ∑ n lx (k) (k) En,q,ε (x) = q El,q,ε [x]n−l q l (4) l=0 (k) + [x]q )n , = (q x Eq,ε (k)

(k)

with the usual convention about replacing (Eq,ε )n by En,q,ε . By using complex integral and (3), we can also obtain the multiple twisted q-l-function as follows: ∫ ∞ 1 (k) (k) lq,ε (s, x) = Feq,ε (−t, x)ts−1 dt Γ(s) 0 ∑k ∑k (5) ∞ ∑ (−1) j=1 mj ε j=1 mj k = [2]q , [m1 + · · · + mk + x]sq m ,··· ,m =0 1

k

where s ∈ C and x ∈ R, with x ̸= 0, −1, −2, . . .. By using Cauchy residue theorem, the value of multiple twisted q-l-function at negative integers is given explicitly by the following theorem: Theorem 1. Let k ∈ N and n ∈ Z+ . We obtain (k) (k) lq,ε (−n, x) = En,q,ε (x).

The purpose of this paper is to obtain some interesting identities of the power sums and (k) the higher-order twisted q-Euler polynomials En,q,ε (x) using the symmetric properties for multiple

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twisted q-l-function. In this paper, if we take ε = 1 in all equations of this article, then [2] are the special case of our results. 2. Symmetry identities for multiple twisted q-l-function In this section, by using the similar method of [2, 3, 4], expect for obvious modifications, we (k)

investigate some symmetric identities for higher-order twisted q-Euler polynomials En,q,ε (x). We assume that ε be the pN -th root of unity. Let w1 , w2 ∈ N with w1 ≡ 1 (mod 2), w2 ≡ 1 (mod 2). For h ∈ Z, k ∈ N and n ∈ Z+ , we obtain certain symmetry identities for multiple twisted q-l-function. Observe that [xy]q = [x]qy [y]q for any x, y ∈ C. In (5), we derive next result by substitute w2 (j1 + · · · + jk ) for x in and replace q and ε by q w1 and εw1 , respectively. w2 x + w1 1 (k) w2 (j1 + · · · + jk )) lqw1 ,εw1 (s, w2 x + k w1 [2]qw1 =

∑k

∞ ∑

(−1)

j=1

mj w 1

ε

[m1 + · · · + mk + w2 x +

m1 ,··· ,mk =0

∑k

∞ ∑

∑k

j=1

mj

w2 (j1 + · · · + jk )]sqw1 w1 ∑k

(−1) j=1 mj εw1 j=1 mj [ ]s = w1 (m1 + · · · + mk ) + w1 w2 x + w2 (j1 + · · · + jk ) m1 ,··· ,mk =0 w1 q w1 =

∑k

∞ ∑ m1 ,··· ,mk =0

=

∞ ∑

[w1 ]sq

m1 ,··· ,mk ∞ ∑

= [w1 ]sq

∑k

∞ ∑

∑k

(−1) j=1 mj εw1 j=1 mj [w1 (m1 + · · · + mk ) + w1 w2 x + w2 (j1 + · · · + jk )]sq =0 ∑k

w∑ 2 −1

m1 ,··· ,mk =0 i1 ,··· ,ik

= [w1 ]sq

∑k

(−1) j=1 mj εw1 j=1 mj [w1 (m1 + · · · + mk ) + w1 w2 x + w2 (j1 + · · · + jk )]sq [w1 ]sq

(6)

∑k

(−1) j=1 mj εw1 j=1 mj [w1 (m1 + · · · + mk ) + w1 w2 x + w2 (j1 + · · · + jk )]sq =0

w∑ 2 −1

(−1)

∑k

j=1 (dw2 mj +ij )

m1 ,··· ,mk =0 i1 ,··· ,ik =0 ∑k j=1 (dw2 mj +ij )

×ε ( )−1 × [w1 (dw2 m1 + i1 ) + · · · + w1 (dw2 mk + ik ) + w1 w2 x + w2 (j1 + · · · + jk )]sq w1

∞ ∑

= [w1 ]sq

w∑ 2 −1

(−1)

∑k j=1

mj

(−1)

∑k

j=1 ij

m1 ,··· ,mk =0 i1 ,··· ,ik =0 ∑k ∑k j=1 mj w1 j=1 ij

×ε ε ( )−1 × [w1 w2 (x + dm1 + · · · + dmk ) + w1 (i1 + · · · + ik ) + w2 (j1 + · · · + jk )]sq dw1 w2

Thus, from (6), we can derive the following equation. [w2 ]sq [2]kqw1

w∑ 1 −1

(−1)

∑k l=1

jl w2

ε

∑k l=1

jl

j1 ,··· ,jk =0 (k)

w2 (j1 + · · · + jk )) w1 w∑ w∑ 2 −1 1 −1 ∑k (−1) l=1 (jl +il +ml )

× lqw1 ,εw1 (s, w2 x + = [w1 ]sq [w2 ]sq

∞ ∑

(7)

m1 ,··· ,mk =0 i1 ,··· ,ik =0 j1 ,··· ,jk =0 ∑k ∑k ∑k l=1 jl l=1 il w2 l=1 ml w1

ε ε ×ε ( )−1 × [w1 w2 (x + dm1 + · · · + dmk ) + w1 (i1 + · · · + ik ) + w2 (j1 + · · · + jk )]sq dw1 w2

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By using the same method as (7), we have w∑ 2 −1

[w1 ]sq [2]kqw2

(−1)

∑k l=1

∑k

jl w1

ε

l=1

jl

j1 ,··· ,jk =0

w1 (j1 + · · · + jk )) w2 w∑ w∑ 1 −1 2 −1 ∑k (−1) l=1 (jl +il +ml )

(k)

× lqw2 ,εw2 (s, w1 x + ∞ ∑

= [w1 ]sq [w2 ]sq

(8)

m1 ,··· ,mk =0 j1 ,··· ,jk =0 i1 ,··· ,ik =0 ∑k ∑k ∑k l=1 ml w2 l=1 il w1 l=1 jl

×ε ε ε ( )−1 × [w1 w2 (x + dm1 + · · · + dmk ) + w1 (j1 + · · · + jk ) + w2 (i1 + · · · + ik )]sq dw1 w2

Therefore, by (7) and (8), we have the following theorem. Theorem 2. Let w1 , w2 ∈ N with w1 ≡ 1 (mod 2), w2 ≡ 1 (mod 2). For h ∈ Z , we obtain ( ) w∑ 1 −1 ∑k ∑k w2 (k) (j1 + · · · + jk ) [w2 ]sq [2]kqw2 (−1) l=1 jl εw2 l=1 jl lqw1 ,εw1 s, w2 x + w1 j1 ,··· ,jk =0 (9) ( ) w∑ 2 −1 ∑k ∑k w 1 (k) s k j w j 1 l=1 l l w (−1) l=1 l ε = [w1 ]q [2]qw1 s, w1 x + (j1 + · · · + jk ) q 2 ,εw2 w2 j ,··· ,j =0 1

k

By (9) and Theorem 1, we obtain the following theorem. Theorem 3. Let w1 , w2 ∈ N with w1 ≡ 1 (mod 2), w2 ≡ 1 (mod 2). For h ∈ Z, k ∈ N and n ∈ Z+ , we obtain w∑ 1 −1

[w2 ]sq [2]kqw2

∑k

(−1)

l=1

jl w2

ε

∑k l=1

jl

(k) En,qw1 ,εw1

j1 ,··· ,jk =0

= [w1 ]sq [2]kqw1

w∑ 2 −1

(−1)

∑k l=1

jl w 1

ε

) ( w2 (j1 + · · · + jk ) w2 x + w1 (

∑k

(k) l=1 jl E n,q w2 ,εw2

j1 ,··· ,jk =0

) w1 w1 x + (j1 + · · · + jk ) . w2

(10)

From (4), we note that (k) (k) En,q,ε (x + y) = (q x+y En,q,ε + [x + y]q )n n ( ) ∑ n xi (k) = q Ei,q,ε (y)[x]n−i . q i i=0 (k)

(11)

(k)

with the usual convention about replacing (Eq,ε )n by En,q,ε . By (11), we have w∑ 1 −1

(−1)

∑k l=1

jl w2

ε

∑k l=1

jl

(k)

En,qw1 ,εw1

j1 ,··· ,jk =0

=

w∑ 1 −1

(−1)

∑k l=1

jl w2

ε

∑k l=1

( ) w2 w2 x + (j1 + · · · + jk ) w1

jl

j1 ,··· ,jk =0

× =

[ ]n−i n ( ) ∑ n w2 i(j1 +···+jk ) (k) w2 q Ei,qw1 ,εw1 (w2 x) (j1 + · · · + jk ) i w1 q w1 i=0 w∑ 1 −1

(−1)

∑k l=1

jl w2

ε

∑k l=1

(12)

jl

j1 ,··· ,jk =0

]i [ n ( ) ∑ n w2 (n−i) ∑kl=1 jl (k) w2 En−i,qw1 ,εw1 (w2 x) (j1 + · · · + jk ) × q w1 i q w1 i=0

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Hence we have the following theorem. Theorem 4. Let w1 , w2 ∈ N with w1 ≡ 1 (mod 2), w2 ≡ 1 (mod 2). For k ∈ N and n ∈ Z+ , we obtain w∑ 1 −1

(−1)

∑k l=1

jl w 2

ε

∑k l=1

jl

(k) En,qw1 ,εw1

j1 ,··· ,jk =0

=

n ( ) ∑ n i=0

i

( ) w2 w2 x + (j1 + · · · + jk ) w1 w∑ 1 −1

[w2 ]iq [w1 ]−i q En−i,q w1 ,εw1 (w2 x) (k)

(−1)

∑k l=1

jl w 2

ε

∑k l=1

jl

[j1 · · · + jk ]iqw2 .

j1 ,··· ,jk =0

For each integer n ≥ 0, let w−1 ∑

(k)

Sn,i,q,ε (w) =

(−1)

∑k l=1

jl

ε

∑k l=1

jl

[j1 · · · + jk ]iq .

j1 ,··· ,jk =0 (k)

The above sum Sn,i,q,ε (w) is called the alternating q-power sums. By Theorem 4, we have w∑ 1 −1

[2]kqw2 [w1 ]nq

(−1)

∑k l=1

jl w2

ε

∑k l=1

( jl

(k)

En,qw1 ,εw1

w2 x +

j1 ,··· ,jk =0

= [2]kqw2

) w2 (j1 + · · · + jk ) w1 (13)

n ( ) ∑ n (k) (k) [w2 ]iq [w1 ]n−i En−i,qw1 ,εw1 (w2 x)Sn,i,qw2 ,εw2 (w1 ) q i i=0

By using the same method as in (13), we have [2]kqw1 [w2 ]nq = [2]kqw1

w∑ 2 −1

(−1)

∑k l=1

jl w1

ε

∑k l=1

j1 ,··· ,jk =0 n ( ) ∑ i=0

( jl

(k) En,qw2 ,εw2

) w1 (j1 + · · · + jk ) w1 x + w2 (14)

n (k) (k) [w1 ]iq [w2 ]n−i En−i,qw2 ,εw2 (w1 x)Sn,i,qw1 ,εw1 (w2 ) q i

Therefore, by (13) and (14) and Theorem 3, we have the following theorem. Theorem 5. Let w1 , w2 ∈ N with w1 ≡ 1 (mod 2), w2 ≡ 1 (mod 2). For k ∈ N and n ∈ Z+ , we obtain [2]kqw2

n ( ) ∑ n i=0

= [2]kqw1

i

(k)

(k)

En−i,qw1 ,εw1 (w2 x)Sn,i,qw2 ,εw2 (w1 ) [w2 ]iq [w1 ]n−i q

n ( ) ∑ n (k) (k) [w1 ]iq [w2 ]n−i En−i,qw2 ,εw2 (w1 x)Sn,i,qw1 ,εw1 (w2 ). q i i=0

By Theorem 5, we obtain the interesting symmetric identity for the higher-order twisted q-Euler (k) numbers En,q,ε in complex field. Corollary 6. Let w1 , w2 ∈ N with w1 ≡ 1 (mod 2), w2 ≡ 1 (mod 2). For k ∈ N and n ∈ Z+ , we obtain n ( ) ∑ n (k) (k) k [2]qw2 [w2 ]iq [w1 ]n−i Sn,i,qw2 ,εw2 (w1 )En−i,qw1 ,εw1 q i i=0 n ( ) ∑ n (k) (k) = [2]kqw1 [w1 ]iq [w2 ]n−i Sn,i,qw1 ,εw1 (w2 )En−i,qw2 ,εw2 . q i i=0

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REFERENCES 1. M. Cenkci, The p-adic generalized twisted (h, q)-Euler-l-function and its applications, Adv. Stud. Contemp. Math., 15(2007), 34-47. 2. D. V. Dolgy, D.S. Kim, T.G. Kim, J.J. Seo, Identities of Symmetry for Higher-Order Generalized q-Euler Polynomials, Abstract and Applied Analysis, 2014(2014), Article ID 286239, 6 pages. 3. Yuan He,

Symmetric identities for Carlitz’s q-Bernoulli numbers and polynomials,

Adv.

Difference Equ., 246(2013), 10 pages. 4. D. Kim, T. Kim, J.-J. Seo, Identities of symmetric for (h, q)-extension of higher-order Euler polynomials, Applied Mathemtical Sciences 8 (2014), 3799-3808. 5. T. Kim, New approach to q-Euler polynomials of higher order, 17(2010), 218-225.

Russ. J. Math. Phys.

6. T. Kim, Barnes type multiple q-zeta function and q-Euler polynomials, J. phys. A : Math. Theor. 43(2010) 255201(11pp). 7. H. Y. Lee, N. S. Jung, J. Y. Kang, C. S. Ryoo, Some identities on the higher-order-twisted q-Euler numbers and polynomials with weight α, Adv. Difference Equ., 2012:21(2012), 10pp. 8. E.-J. Moon, S.-H. Rim, J.-H. Jin, S.-J. Lee, On the symmetric properties of higher-order twisted q-Euler numbers and polynomials, Adv. Difference Equ., 2010, Art ID 765259, 8pp. 9. H. Ozden, Y. Simsek, I. N. Cangul, Euler polynomials associated with p-adic q-Euler measure, Gen. Math., 15(2007), 24-37. 10. C. S. Ryoo, On the generalized Barnes type multiple q-Euler polynomials twisted by ramified roots of unity, Proc. Jangjeon Math. Soc. 13(2010), 255-263. 11. C. S. Ryoo, A note on the weighted q-Euler numbers and polynomials, Adv. Stud. Contemp. Math., 21(2011), 47-54. 12. Y. Simsek, q-analogue of twisted l-series and q-twisted Euler numbers, Journal of Number Theory, 110(2005), 267-278. 13. Y. Simsek, Twisted (h, q)-Bernoulli numbers and polynomials related to twisted (h, q)-zeta function and L-function, J. Math. Anal. Appl., 324(2006), 790-804.

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UMBRAL CALCULUS ASSOCIATED WITH NEW DEGENERATE BERNOULLI POLYNOMIALS DAE SAN KIM, TAEKYUN KIM, AND JONG-JIN SEO

Abstract. In this paper, we introduce new degenerate Bernoulli polynomials which are derived from umbral calculus and investigate some interesting properties of those polynomials.

1. Introduction The Bernoulli polynomials are defined by the generating function ∞ ∑ t tn xt (1.1) e = , (see [1–14]) . B (x) n et − 1 n! n=0 When x = 0, Bn = Bn (0) are called the ordinary Bernoulli numbers. From (1.1), we note that n ( ) ∑ n (1.2) Bn (x) = Bl xn−l , (n ≥ 0) , (see [13]) . l l=0

Thus, by (1.2), we get d Bn (x) = nBn−1 (x) , (n ∈ N) . (1.3) dx In [2], L. Carlitz introduced the degenerate Bernoulli polynomials which are given by the generating function ∞ ∑ x t tn λ (1.4) (1 + λt) = βn (x | λ) . 1 n! (1 + λt) λ − 1 n=0 When x = 0, βn (0 | λ) = βn (λ) are called Carlitz’s degenerate Bernoulli numbers (see [2]). Thus, by (1.4), we get n ( ) ∑ n (1.5) βn (x | λ) = βl (λ) (x | λ)n−l , (n ≥ 0) , l l=0

where (x | λ)n = x (x − λ) · · · (x − λ (n − 1)). Let C be the field of complex numbers and let F be the set of all formal power series in the variable t over C with } { ∞ ∑ tk F = f (t) = ak ak ∈ C . k! k=0

2010 Mathematics Subject Classification. 11B83, 11B75, 05A19, 05A40. Key words and phrases. Degenerate Bernoulli polynomial, Higher-order degenerate Bernoulli polynomial, Umbral calculus. 1

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Let P = C [x] and P∗ denotes the vector space of all linear functionals on P. The action of the linear functional L ∈ P∗ on a polynomial p (x) is denoted by ⟨ L| p (x)⟩, and linearly extended as ⟨ cL + c′ L′ | p (x)⟩ = c ⟨ L| p (x)⟩ + c′ ⟨ L′ | p (x)⟩, where c and c′ ∈ C. ∑∞ k For f (t) = k=0 ak tk! ∈ F , we define a linear functional on P by setting ⟨ f (t)| xn ⟩ = an

(1.6)

for all n ≥ 0, (see [1, 5, 13]). Thus, by (1.6), we get ⟨ k n⟩ (1.7) t x = n!δn,k ,

(n, k ≥ 0) ,

(see [7, 13]) ,

where δn,k is the Kronecker’s symbol. ⟩ k ∑∞ ⟨ Let fL (t) = k=0 L| xk tk! . Then we have ⟨ fL (t)| xn ⟩ = ⟨ L| xn ⟩ (n ≥ 0). The mapping L 7→ fL (t) is a vector space isomorphism from P∗ onto F. Henceforth, F will denote both the algebra of formal power series in t and the vector space of all linear functionals on P, and so an element f (t) of F will be thought of as both a formal power seires and a linear functional. We shall call F the umbral algebra. The umbral calculus is the study of umbral algebra and can be also desribed as a systematic study of the class of Sheffer sequences. The order o (f ) of the non-zero power series f (t) is the smallest integer k for which the coefficient of tk does not vanish (see [12, 13]). For f (t) , g (t) ∈ F with o (f ) = 1 and o (g) = 0, there exists a unique sequence sn (x) of polynomials such that ⟨ ⟩ k g (t) f (t) sn (x) = n!δn,k , (n, k ≥ 0) . The sequence sn (x) is called the Sheffer sequence for (g (t) , f (t)) which is denoted by sn (x) ∼ (g (t) , f (t))(see [10, 13]). Let f (t) ∈ F and p (x) ∈ P. Then by (1.7), we get ⟨ yt ⟩ (1.8) e p (x) = p (y) , ⟨ f (t) g (t)| p (x)⟩ = ⟨ g (t)| f (t) p (x)⟩ , and (1.9)

f (t) =

∞ ∑ ⟨

f (t)| xk

k=0

⟩ tk , k!

p (x) =

∞ ∑ ⟨ k ⟩ xk t p (x) , k!

(see [13]) .

k=0

By (1.9), we easily get (1.10)

⟩ ⟨ ⟩ ⟨ p(k) (0) = tk p (x) = 1| p(k) (x) ,

(k ≥ 0) ,

where p(k) (0) denotes the k-th derivative of p (x) with respect to x at x = 0. From (1.10), we have tk p (x) = p(k) (x) .

(1.11) In [13], it is known that

∞ ∑ 1 tn ( ) exf (t) = sn (x) , n! g f (t) n=0 ( ) where f (t) is the compositional inverse of f (t) such that f f (t) = f (f (t)) = t. From (1.7), we can easily derive

(1.12)

sn (x) ∼ (g (t) , f (t)) ⇐⇒

(1.13)

eyt p (x) = p (x + y) ,

where p (x) ∈ P = C [x] .

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UMBRAL CALCULUS ASSOCIATED WITH NEW DEGENERATE BERNOULLI POLYNOMIALS 3

For p (x) ∈ P, we have ⟨ yt ⟩ ˆ y e − 1 p (x) = p (u) du, t 0

⟨ f (t)| xp (x)⟩ = ⟨ ∂t f (t)| p (x)⟩ .

Let f1 (t) , f2 (t) , . . . , fm (t) ∈ F . Then we have ) ∑( ⟨ ⟩ ⟨ ⟩ n n (1.14) ⟨ f1 (t) f2 (t) · · · fm (t)| x ⟩ = f1 (t)| xi1 · · · fm (t)| xim i1 , . . . , im where the sum is over all nonnegative integers i1 , . . . , im such that i1 + · · · + im = n. In this paper, we introduce new degenerate Bernoulli polynomials which are different Carlitz’s degenerate Bernoulli polynomials and investigate some interesting properties of those polynomials. 2. Umbral calculus and degenerate Bernoulli polynomials From (1.1) and (1.13), we have

(

Bn (x) ∼

(2.1)

) et − 1 ,t , t

(n ≥ 0) .

Now, we introduce the new degenerate Bernoulli polynomials which are derived from Sheffer sequence as follows: ( ) 1 (1 + λt) λ − 1 (2.2) βn,λ (x) ∼ , t , (n ≥ 0) . t From (1.12) and (2.2), we have ∞ ∑

(2.3)

βn,λ (x)

n=0

t tn = ext . 1 n! (1 + λt) λ − 1

When x = 0, βn,λ = βn,λ (0) are called the degenerate Bernoulli numbers. Note that ∞ ∑ tn t (2.4) lim βn,λ (x) = lim ext 1 λ→0 λ→0 (1 + λt) λ − 1 n! n=0 t ext et − 1 ∞ ∑ tn = Bn (x) . n! n=0 =

Thus, by (2.4), we get (2.5)

lim βn,λ (x) = Bn (x) ,

λ→0

From (2.3), we have (2.6)

∞ ∑

)( ∞ ) ∑ xm tl m βl,λ t l! m! m=0 l=0 ( n ( ) ) ∞ ∑ ∑ n tn = βl,λ xn−l . l n! n=0

tn βn,λ (x) = n! n=0

(

(n ≥ 0) .

∞ ∑

l=0

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Thus, by (2.6), we get βn,λ (x) =

n ( ) ∑ n

l

l=0

and (2.7)

βl,λ xn−l ,

(n ≥ 0) ,

n ( ) ∑ d n βn,λ (x) = βl,λ (n − l) xn−l−1 dx l l=1 n−1 ∑ (n − 1) =n βl,λ xn−1−l l l=0

= nβn−1,λ (x) . From (1.11) and (2.3), we have (2.8)

t 1

(1 + λt) λ − 1

xn = βn,λ (x) ,

(n ≥ 0) ,

and (2.9)

tβn,λ (x) =

d βn,λ (x) = nβn−1,λ (x) , dx

(n ≥ 1) .

Thus, by (2.8) and (2.9), we get ˆ x+y (2.10) βn,λ (u) du x

1 {βn+1,λ (x + y) − βn+1,λ (x)} n+1 eyt − 1 = βn,λ (x) t ∞ ∑ y k k−1 t βn,λ (x) . = k! =

k=1

From (2.9), we have (2.11)

{ βn (x) = t

} 1 βn+1,λ (x) . n+1

Thus, by (2.11), we get ⟩ ⟨ ⟩ ⟨ yt 1 e − 1 yt (2.12) β (x) = e − 1 β (x) n,λ n+1,λ t n+1 ˆ y = βn,λ (u) du. 0

Therefore, by (2.12), we obtain the following theorem. Theorem 1. For n ≥ 0, we have ⟩ ˆ y ⟨ yt e − 1 βn,λ (x) = βn,λ (u) du. t 0

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For r ∈ N, the degenerate Bernoulli polynomials of order r are defined by the generating function ( )r ∞ ∑ t tn (r) xt (2.13) e = βn,λ (x) . 1 n! (1 + λt) λ − 1 n=0 (r)

(r)

When x = 0, βn,λ = βn,λ (0) are called the higher order degenerate Bernoulli numbers. (r) (r) (r) Indeed, limλ→0 βn,λ (x) = Bn (x), where Bn (x) are the higher-order Bernoulli polynomials which are defined by the generating function ( )r ∞ ∑ t tn xt e = Bn(r) (x) . t e −1 n! n=0 From (2.13), we have (2.14)

(r) βn,λ

n ( ) ∑ n (r) n−l (x) = β x , l l,λ

(n ≥ 0) ,

l=0

and (2.15)

n ( ) ∑ n (r) d (r) β (x) = β (n − l) xn−l−1 dx n,λ l l,λ l=0 n−1 ∑ (n − 1) (r) =n βl,λ xn−l−1 l

=

l=0 (r) nβn−1,λ

(x) ,

(n ≥ 1) .

By (2.8) and (2.13), we easily get ( ) ∑ n (r) (2.16) βn,λ = βl1 ,λ · · · βlr ,λ . l1 , . . . , lr l1 +···+lr =n

(r)

Thus, by (2.14) and (2.16), we see that βn,λ (x) is a monic polynomial of degree n with coefficients in Q (λ). From (2.14) and (2.15), we can derive ˆ x+y } 1 { (r) (r) (r) (2.17) βn+1,λ (x + y) − βn+1,λ (x) βn,λ (u) du = n+1 x eyt − 1 (r) = βn,λ (x) . t If sn (x) ∼ (g (t) , t), then sn (x) is called an Appell sequence. From (1.12) and (2.13), we have )r ) (( 1 (1 + λt) λ − 1 (r) , t , (n ≥ 0) . (2.18) βn,λ (x) ∼ t ( )r 1 (r) (1+λt) λ −1 Thus, by (2.18), we note that βn,λ (x) is the Appell sequence for . t From (2.18), we have ( )r 1 (1 + λt) λ − 1 (r) (2.19) βn,λ (x) ∼ (1, t) , t

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Thus, by (2.19), we get ( n

(2.20)

x =

1

(1 + λt) λ − 1 t

)r (r)

(n ≥ 0) .

βn,λ (x) ,

We observe that ( )r 1 (1 + λt) λ − 1 (2.21) t ( )r 1 1 = r e λ log(1+λt) − 1 t ∞ l 1 ∑ (log (1 + λt)) = r r! S2 (l, r) λ−l t l! l=r

=

1 r! tr

∞ ∑

S2 (l + r, r) λ−(l+r)

l=0 ∞ ∑

∞ ∑ λn tn 1 (l + r)! S1 (n, l + r) (l + r)! n! n=l+r

∞ ∑

λn+r n+r 1 −(l+r) r! S (l + r, r) λ S (n + r, l + r) t 2 1 tr (n + r)! l=0 n=l ) ( n ∞ ∑ ∑ 1 tn = S2 (l + r, r) S1 (n + r, l + r) λn−l (n+r) n! r n=0 =

l=0

By (2.20) and (2.21), we get (2.22) (m) m ∑ n ∑ m n−l n ( ) (r) x = S2 (l + r, r) S1 (n + r, l + r) λ n+r βm−n,λ (x) ,

(m ≥ 0) .

r

n=0 l=0

Therefore, by (2.22), we obtain the following theorem. Theorem 2. For m ≥ 0, we have m

x

=

m ∑ n ∑

( m) n−l

S2 (l + r, r) S1 (n + r, l + r) λ

(r) n (n+r ) βm−n,λ (x) , r

n=0 l=0

where S1 (m, n) and S2 (m, n) are the Stirling numbers of the first kind and of the second kind defined by (x)n =

n ∑

S1 (n, l) xl ,

l=0

xn =

n ∑

S2 (n, l) (x)l .

l=0

From (1.11) and (2.18), we have { } 1 (r) (r) (2.23) tβn,λ (x) = t βn+1,λ (x) , n+1 and (2.24)

⟨

(n ≥ 0) ,

⟩ ⟨ ⟩ 1 eyt − 1 (r) (r) yt β (x) = e − 1 β (x) t n,λ n + 1 n+1,λ

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ˆ = 0

y

(r)

βn,λ (u) du.

Moreover, (r)

(2.25) βn,λ ⟨(

)r ⟩ n = x 1 λ (1 + λt) − 1 ⟩ ⟨ ⟩ )⟨ ( ∑ t t n i1 ir ··· = x x 1 1 . , i i , . . r 1 (1 + λt) λ − 1 (1 + λt) λ − 1 n=i1 +···+ir t

and

⟨

(2.26)

βn,λ =

⟩ n , x 1 (1 + λt) λ − 1 t

(n ≥ 0) .

Therefore, by (2.24), (2.25) and (2.26), we obtain the following theorem. Theorem 3. For n ≥ 0, we have ⟨ yt ⟩ ˆ y e − 1 (r) (r) β (x) = βn,λ (u) du, t n,λ 0 and ( ) ∑ n (r) βi1 ,λ · · · βir ,λ . βn,λ = i1 , . . . , ir n=i +···+i 1

r

Let Pn = { p (x) ∈ C [x]| deg p (x) ≤ n} , (n ≥ 0). For p (x) ∈ Pn , we assume that n ∑ (2.27) p (x) = bk βk,λ (x) . k=0

From (2.2), we have ⟨ ⟩ 1 (1 + λt) λ − 1 k (2.28) t βn,λ (x) = n!δn,k , t

(n, k ≥ 0) .

Thus, by (2.27) and (2.28), we get ⟨ ⟩ ⟩ ⟨ 1 1 n ∑ (1 + λt) λ − 1 k (1 + λt) λ − 1 k (2.29) t p (x) = t βl,λ (x) bl t t l=0

=

n ∑

bl l!δl,k = k!bk .

l=0

Hence,

⟩ 1 (1 + λt) λ − 1 k t p (x) t ⟨ ⟩ 1 1 (1 + λt) λ − 1 (k) = p (x) , k! t

1 bk = k!

(2.30)

⟨

k

d where p(k) (x) = dx k p (x). Therefore, by (2.27) and (2.30), we obtain the following theorem.

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Theorem 4. Let p (x) ∈ Pn . Then we have p (x) =

n ∑

bk βk,λ (x) ,

k=0

where 1 bk = k!

⟨

⟩ 1 (1 + λt) λ − 1 (k) p (x) . t

(r)

Let p (x) ∈ Pn with p (x) = βn,λ (x). Then, we have ( p(k) (x) =

(2.31)

d dx

)k (r)

βn,λ (x) = k!

( ) n (r) β (x) . k n−k,λ

Let us assume that (r)

(2.32)

p (x) = βn,λ (x) =

n ∑

bk βk,λ (x) .

k=0

Then, by Theorem 5, we get ⟨ ⟩ 1 1 (1 + λt) λ − 1 (k) bk = (2.33) p (x) k! t ⟩ 1 ( )⟨ n (1 + λt) λ − 1 (r) = βn−k,λ (x) t k ( ⟨ )r ⟩ 1 ( ) n (1 + λt) λ − 1 t n−k = x (1 + λt) λ1 − 1 k t )r−1 ⟩ ( ) ⟨ ( n t n−k = 1 x 1 k (1 + λt) λ − 1 ( ) n (r−1) = β . k n−k,λ Therefore, by (2.32) and (2.33), we obtain the following theorem. Theorem 5. For r ∈ N and n ≥ 0, we have n ( ) ∑ n (r−1) (r) βn,λ (x) = β βk,λ (x) . k n−k,λ k=0

Let p (x) ∈ Pn with p (x) = (2.34) ⟨(

1

(1 + λt) λ − 1 t

)r

∑n

(r) (r) k=0 bk βk,λ

(x). By (2.18), we get

⟩ ⟨( )r ⟩ 1 n ∑ (1 + λt) λ − 1 (r) k (r) bl t p (x) = t βl,λ (x) t k

l=0

=

n ∑

(r)

(r)

bl l!δl,k = k!bk .

l=0

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Thus, by (2.34), we get (r) bk

(2.35)

1 = k!

⟨(

)r

1

(1 + λt) λ − 1 t

⟩ t p (x) . k

Theorem 6. For p (x) ∈ Pn , we have n ∑ (r) (r) p (x) = bk βk,λ (x) , k=0

where (r) bk

where p(k) (x) =

(

) d k dx

⟩ t p (x) ⟨( ) ⟩ r 1 1 (1 + λt) λ − 1 (k) = p (x) , k! t 1 = k!

⟨(

1

(1 + λt) λ − 1 t

)r

k

p (x). References

1. S. Araci, M. Acikgoz, and A. Kilicman, Extended p-adic q-invariant integrals on Zp associated with applications of umbral calculus, Adv. Difference Equ. (2013), 2013:96, 14. MR 3055847 2. L. Carlitz, Degenerate Stirling, Bernoulli and Eulerian numbers, Utilitas Math. 15 (1979), 51–88. MR 531621 (80i:05014) 3. R. Dere and Y. Simsek, Applications of umbral algebra to some special polynomials, Adv. Stud. Contemp. Math. (Kyungshang) 22 (2012), no. 3, 433–438. MR 2976601 4. Q. Fang and T. Wang, Umbral calculus and invariant sequences, Ars Combin. 101 (2011), 257–264. MR 2828068 (2012e:05045) 5. S. Gaboury, R. Tremblay, and B.-J. Fug`ere, Some explicit formulas for certain new classes of Bernoulli, Euler and Genocchi polynomials, Proc. Jangjeon Math. Soc. 17 (2014), no. 1, 115–123. MR 3184467 6. Y. He and W. Zhang, A convolution formula for Bernoulli polynomials, Ars Combin. 108 (2013), 97–104. MR 3060257 7. D. S. Kim and T. Kim, Some identities of Bell polynomials, Science China Mathematics (2015). 8. , Some identities of degenerate special polynomials, Open Math. 13 (2015), Art. 37. MR 3353617 9. D. S. Kim, T. Kim, S.-H. Lee, and S.-H. Rim, Umbral calculus and Euler polynomials, Ars Combin. 112 (2013), 293–306. MR 3112583 10. T. Kim, Identities involving Laguerre polynomials derived from umbral calculus, Russ. J. Math. Phys. 21 (2014), no. 1, 36–45. MR 3182545 , Barnes’ type multiple degenerate Bernoulli and Euler polynomials, 11. Appl. Math. Comput. 258 (2015), 556–564. MR 3323091 12. T. Kim and T. Mansour, Umbral calculus associated with Frobenius-type Eulerian polynomials, Russ. J. Math. Phys. 21 (2014), no. 4, 484–493. MR 3284958 13. S. Roman, The umbral calculus, Pure and Applied Mathematics, vol. 111, Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], New York, 1984. MR 741185 (87c:05015)

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14. Y. Simsek, Generating functions of the twisted Bernoulli numbers and polynomials associated with their interpolation functions, Adv. Stud. Contemp. Math. (Kyungshang) 16 (2008), no. 2, 251–278. MR 2404639 (2009f:11021) ACKNOWLEDGEMENTS.The work reported in this paper was conducted during the sabbatical year of Kwangwoon University in 2014. Department of Mathematics, Sogang University, Seoul 121-742, Republic of Korea E-mail address: [email protected] Department of Mathematics, Kwangwoon University, Seoul 139-701, Republic of Korea E-mail address: [email protected] Department of Applied Mathematics,Department of Applied Mathematics, Pukyong National University, Busan 48513, Re- public of Korea. E-mail address: [email protected]

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Regularization Smoothing Approximation of Fuzzy Parametric Variational Inequality Constrained Stochastic Optimization Heng-you Lan Department of Mathematics, Sichuan University of Science & Engineering, Zigong, Sichuan 643000, PR China E-mail: [email protected]

Abstract. This work is motivated by the fact that very little is known about the fuzzy parametric variational inequalities constrained stochastic optimization problems in finite dimension real numeral spaces, which are studied more difficult because of the existence of random variable and fuzzified version. Based on the notion of quasi-Monte Carlo estimate and method of centres with entropic regularization, we develop a class of new regularization smoothing approximation approaches to discretize the stochastic optimization problem with continuous random variable, and construct a centre iterative algorithm for approximating the optimal solutions of the stochastic optimization problems. Further, we give some comprehensive convergence theorems of optimal solutions for the resulting optimization problem. Finally, a numerical illustration is analyzed. Key Words and Phrases. Regularization smoothing approximation, fuzzy parametric variational inequality, Stochastic optimization problem, centre iterative algorithm with quasi-Monte Carlo estimate, comprehensive convergence. AMS Subject Classification. 49J40, 65K05, 90C30, 90C33

1

Introduction

As all we know, mathematical program with equilibrium constraints is a constrained optimization problem in which the essential constraints are defined by a parametric variational inequality. This class of problems can be regarded as a generalization of a bilevel programming problem and it therefore plays an important role in many fields such as transportation, communication networks, structural mechanics, economic equilibrium, multilevel game, and mathematical programming itself. See, for example, [1–7] and the reference therein. Moreover, in order to describe the uncertainties, Monica [5] considered the Bochner integrability setting, a measure space of indices and use random fuzzy mappings, and presented random fixed point theorems with random fuzzy mappings, extensions of the ones with random data. In this paper, we study approximation of optimal solutions for the following fuzzy parametric variational inequality constrained stochastic optimization problem in n-dimension real numeral set Rn : min Eω [f (x, y(ω), ω)] x,y(·)

s.t.

x ∈ U ⊂ Rn , y(ω) ∈ C(x, ω) ⟨F (x, y(ω), ω), z(ω) − y(ω)⟩ ≥ ∼ 0, for all z(ω) ∈ C(x, ω) and a.e. ω ∈ Ω,

(1.1)

where Eω denotes the mathematical expectation with respect to the random variable ω ∈ Ω on probability space (Ω, A, Γ), f : Rn+m × Ω → R and F : Rn+m × Ω → Rm are two nonlinear random m functions, C : Rn × Ω → 2R is a multi-valued random function, ⟨F (x, y(ω), ω), z(ω) − y(ω)⟩ ≥ ∼ 0 are fuzzy inequalities (also called fuzzy stochastic variational inequality problems, in short, Vg Iω ), “ ≥ ∼” 1

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denotes the fuzzified version of “≥” with the linguistic interpretation “approximately greater than or equal to”, and “a.e.” is the abbreviation for almost everywhere”. Remark 1.1. Problem (1.1) is brand new in the literature including and can be thought as a generalized version of some problems, includes a number of stochastic mathematical program with equilibrium constraints (SMPEC), mathematical programs with fuzzy equilibrium constraints (MPFEC) mathematical program with equilibrium constraints (MPEC) and mathematical program with complementarity constraints (MPCC) have been studied by many authors as special cases. See, for example, [1–4, 6, 8–14] and the references therein, and the following examples. Example 1.1 If Ω is a singleton, then problem (1.1) reduces to the following MPFEC: min s.t.

g(x, u) x ∈ U, u solves VfI (G(x, ·), D(x)),

(1.2)

where g : Rn+m → R and G : Rn+m → Rm is a continuously differentiable function, D : Rn → 2R is a set valued function, and u solves VfI(G(x, ·), D(x)) if and only if u ∈ D(x) and ⟨G(x, u), z − u⟩ ≥ ∼0 for all z ∈ D(x). Problem (1.2) was introduced and studied by Hu and Liu [12] and Lan et al. [13]. Moreover, Hu and Liu [12] pointed out “although a powerful theory has been developed for variational inequalities, the parameterized setting in MPEC makes these problems very difficult to solve, and due to the vagueness involved in real world problems, the MPEC problem in a fuzzy environment becomes an important problem both in theory and in practice”, and “problem (1.2) is a constrained optimization problem whose constraints include some fuzzy parametric variational inequalities. In 2013, inspired by the works of Hu and Liu [12] and other researchers, Lan et al. [13] constructed an iterative algorithm for finding a solution of a class of mathematical program problems with fuzzy parametric variational inequality constraints by using a new smoothing approach based on a version of the method of centres with entropic regularization techniques. In fact, the tolerance approach and entropic regularization technique have been successfully proposed in solving various problems, which are important numerical methods for solving fuzzy variational inequalities in a fuzzy environment and nonlinear semi-infinite programming problems. See, for example, [3, 8, 14–21] and the references therein. Example 1.2. Since a solution satisfying a fuzzy inequality system to a membership degree close to 1 is a near optimal solution to the corresponding regular inequality problem [22], if y(ω) ≡ v for all ω ∈ Ω and the degree for the fuzzy inequalities in (1.1) is close to 1, then problem (1.1) is equivalent to the following SMPEC: m

min Eω [f (x, v, ω)] s.t. x ∈ U, ω ∈ Ω v solves V I(F (x, ·, ω), C(x, ω)),

(1.3)

where V I(F (x, ·, ω), C(x, ω)) denotes the variational inequality problem defined by the pair (F (x, ·, ω), C(x, ω)) for all x ∈ Rn and ω ∈ Ω. In 2003, Lin et al. [9] considered problem (1.3) and showed that SMPEC can be thought as a generalization of MPEC, and proposed a smoothing implicit programming method to establish a comprehensive convergence theory for the lower-level waitand-see model. Further, there are many stochastic formulations of MPEC proposed in the recent discussions. For related works, we refer readers to [1, 3, 6, 8, 10, 11]. However, there has been very little study on applications of these theories and approaches to (1.1). Over years of development, optimization approaches have become one of the most promising techniques for engineering applications and an MPEC is a hard problem because its constraints fail to satisfy a standard constraint qualification at any feasible point [23]. However, since the existence of the random variable ω and the fuzzified version “ ≥ ∼ ” mean that (1.1) involves multiple complementarity-type constraints, it is more difficult to solve problem (1.1) than to solve an ordinary MPCC, MPEC, MPFEC or SMPEC generally. Therefore, our focus in this paper is to develop a class of new regularization smoothing approximation approaches to define some parameters of the objective function fuzzy yielded by fuzzy constraints, and consider the approximation-solvability for an equivalent stochastic parametric optimization problem of problem (1.1). 2

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Motivated and inspired by the above works, we shall give some preliminaries needed throughout the whole paper in Section 2. Specially, by using the notion of tolerance approach and the fuzzy set theory, we show that the fuzzy parametric variational inequality constrained stochastic optimization problem (1.1) and a fuzzy complementarity constrained optimization problem can be converted to a regular nonlinear parametric optimization problem. In Sections 3, we will construct a centre iterative algorithm and develop a class of new regularization smoothing approximation approach for solving the stochastic fuzzy optimization based on quasi-Monte Carlo estimate, and establish comprehensive convergence theorems of the solution. We also report some numerical simulation analysis results in Section 4.

2

Preliminaries

Throughout in this paper, we assumption that (Ω, A, Γ) is a complete σ-finite measure space and the probability measure Γ of our considered space (Ω, A, Γ) is non-atomic. Let B(Rm ) be the class of Borel σ-fields in Rm and P (U ) denote the power set of a vector space U . Definition 2.1. (i) A function y : Ω → Rm is said to be measurable, if for any B ∈ B(Rm ), {ω ∈ Ω : y(ω) ∈ B} ∈ A). (ii) The multi-valued function Ψ : Ω → P (U ) is called said to be measurable, if for any B ∈ B(U ), Ψ−1 (B) = {ω ∈ Ω : Ψ(ω) ∩ B ̸= ∅} ∈ A. m (iii) A multi-valued random function Φ : Rn × Ω → 2R is said to be measurable, if for any x ∈ Rn , Φ(x, ·) is measurable. (iv) F : Rn+m × Ω → Rm is called a random and continuously differentiable function, when F (x, z, ω) = ζ(ω) is measurable for any x ∈ Rn and z ∈ Rm , and F (·, ·, ω) is continuously differentiable for all ω ∈ Ω. m Definition 2.2. Let C, C ∗ : Rn × Ω → 2R be two multi-valued random function. Then (i) C(x, ω) is said to be convex cone, if C(x, ·) is convex cone for every x ∈ Rn , that is, αy(·) + βw(·) ∈ C(x, ·) for any positive scalars α, β and all measurable function y(·), w(·) ∈ C(x, ·); (ii) C ∗ (x, ω) is called polar (dual) cone of C(x, ω) ⊂ Rm for x ∈ Rn and ω ∈ Ω, if C ∗ (x, ·) is polar (dual) cone for every x ∈ Rn , i.e. ⟨ξ, ν(·)⟩ ≥ 0 ∀ξ ∈ Rm and for each measurable function ν(·) ∈ C(x, ·). In other words, the polar (dual) cone C ∗ (x, ω) can be expressed as follows: C ∗ (x, ω) = {ξ ∈ Rm ∥ ⟨ξ, ν(ω)⟩ ≥ 0 ∀ν(ω) ∈ C(x, ω)} . Definition 2.3. Let σ > 0 and ς > 0 be constants. a function M : Rn → Rm is said to be H¨older continuous on K ⊂ Rm with order σ and H¨older constant ς if ∥M (u) − M (v)∥ ≤ ς∥u − v∥σ ,

∀u, v ∈ K.

holds for all u and v in K. Remark 2.1. If σ = 1, then the definition of H¨older continuity reduces to definition of Lipschitz continuity. We note that for two different positive numbers σ and σ ′ , H¨older continuous functions with order√σ and those with order σ ′ constitute different subclasses. For example, the function M (u) := ∥u∥ for all u ∈ K ⊂ Rm is H¨older continuous with order σ = 12 , but not Lipschitz continuous. In the sequel, we give some preparations needed later to approximating the optimal solutions of problem (1.1). First, we propose discretization of the stochastic objective function in (1.1) with continuous random variable.

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Lemma 2.1. Let ζ : Ω → [0, +∞) be the continuous probability density function of ω. Then the objective function in (1.1) can be represented as Eω (f (x, y(ω), ω) =

1 ∑ f (x, y(ω), ω)ζ(ω), L

(2.1)

ω∈ΩL

where ΩL := {ω1 , ω2 , · · · , ωL } is a uniformly distributed sample set from Ω. Proof. Let Ω be a sample space, which is usually denoted using set notation, and the possible outcomes are listed as elements in the set. If Ω is unbounded, under some mild conditions, we can approximate the problem by a sequence of programs with bounded sampling spaces (see [11]) for more details. In the sequel, let Ω be a bounded rectangle. In particular, without loss of generality, we assume that Ω = [0, 1]κ . Let ζ : Ω → [0, +∞) be the continuous probability density function of ω. Then the objective function in (1.1) can be represented as ∫ Eω (f (x, y(ω), ω) = f (x, y(ω), ω)ζ(ω)dω. Ω

Based on quasi-Monte Carlo method in [24], now we estimate numerical integration to the objective function in problem (1.1). Roughly speaking, given a function ϕ : Ω → R, the quasiMonte Carlo estimate for Eω [ϕ(ω)] is obtained by taking a uniformly distributed sample set ΩL := ∑ {ω1 , ω2 , · · · , ωL } from Ω and letting Eω [ϕ(ω)] ≈ L1 ω∈ΩL ϕ(ω). This implies that (2.1) holds. 2 Next, we consider the random membership functions of each fuzzy stochastic inequality and stochastic fuzzy objective yielded by the fuzzy constraints in (1.1). Let the membership function for each fuzzy stochastic inequality ⟨F (x, y(ω), ω), z − y(ω)⟩ ≥ ∼ 0 as follows: for all x ∈ Rn and any z ∈ C(x, ω),  if ⟨F (x, y(ω), ω), z − y(ω)⟩ ≥ 0,  1, µz (⟨F (x, y(ω), ω), z − y(ω)⟩), if ⟨F (x, y(ω), ω), z − y(ω)⟩ ∈ [−tz , 0), (2.2) µΩ˜ z (x, y(ω), ω) =  0, if ⟨F (x, y(ω), ω), z − y(ω)⟩ < −tz , ˜ z is specify the degree to which the regular inequality ⟨F (x, y(ω), ω), z−y(ω)⟩ ≥ 0 is satisfied, where Ω n+m a fuzzy set actually determined by the fuzzy stochastic inequality in R ×Ω, tz ≥ 0 is the tolerance level which can be tolerated by decision makers in the accomplishment of the fuzzy stochastic inequality ⟨F (x, y(ω), ω), z − y(ω)⟩ ≥ ∼ 0. We usually assume that µz (⟨F (x, y(ω), ω), z − y(ω)⟩) ∈ [0, 1] and it is continuous and strictly increasing over [−tz , 0). Fig. 1 shows different shapes of such membership functions. µΩ˜ z (x, y(ω), ω)

1

µz (hF(x, y(ω), ω), z − y(ω)i)

α

−tz

0

µ−1 ˜ (α) Ω

hF(x, y(ω), ω), z − y(ω)i

z

Figure 1: The membership function µΩ˜ z (x, y(ω), ω).

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Similarly, the random membership function of the objective, µS˜0 (x, y(ω), ω), is defined as follows:    1, µ0 (Eω [f (x, y(ω), ω)]), µS˜0 (x, y(ω), ω) =   0,

if Eω [f (x, y(ω), ω)] < f , if Eω [f (x, y(ω), ω)] ∈ [f , f¯), if Eω [f (x, y(ω), ω)] ≥ f¯,

(2.3)

where f and f¯ are two parameters defined as follows: f¯ = min s.t.

Eω [f (x, y(ω), ω)] x ∈ U, ω ∈ Ω, ⟨F (x, y(ω), ω), z − y(ω)⟩ ≥ 0,

(2.4) ∀z ∈ C(x, ω)

and f = min s.t.

Eω [f (x, y(ω), ω)] x ∈ U, ω ∈ Ω, ⟨F (x, y(ω), ω), z − y(ω)⟩ ≥ −tz ,

(2.5) ∀z ∈ C(x, ω).

By [22, 25], one can know that studying such a problem (1.1) is related to finding “almost optimal” solutions for a general convex minimization problem (see also [13, 14, 17]). Thus, we extend the idea and have the following result. Lemma 2.2. Let C(x, ω) be a convex cone for all x ∈ Rn and ω ∈ Ω. Then the problem Vg Iω , i.e., finding y(ω) ∈ C(x, ω) such that ⟨F (x, y(ω), ω), z(ω) − y(ω)⟩ ≥ ∼ 0,

∀z(ω) ∈ C(x, ω),

(2.6)

is equivalent to the fuzzy complementarity problem of finding y(ω) ∈ Rm such that y(ω) ∈ C(x, ω),

⟨F (x, y(ω), ω), y(ω)⟩ = ∼ 0,

∗ F (x, y(ω), ω) ∈ ∼ C (x, ω),

(2.7)

where “= ∼ ” denotes the fuzzified version of “=” with the linguistic interpretation “approximately equal to”, “∈ ∼ ” denotes the fuzzified version of “∈” with the linguistic interpretation “approximately in” and C ∗ (x, ω) is a polar (dual) cone of C(x, ω) ⊂ Rm for all x ∈ Rn and ω ∈ Ω. Proof. We start by showing that problem (2.7)⊂ problem (2.6). For any x ∈ Rn and ω ∈ Ω, suppose that y ∗ (ω) is a solution of problem (2.7), then we have ⟨F (x, y ∗ (ω), ω), y ∗ (ω)⟩ = ∼0 and

⟨F (x, y ∗ (ω), ω), υ(ω)⟩ ≥ ∼ 0,

(2.8)

∀υ(ω) ∈ C(x, ω).

(2.9)

Combining (2.8) and (2.9), we have ⟨F (x, y ∗ (ω), ω), υ(ω) − y ∗ (ω)⟩ ≥ ∼ 0 for all υ(ω) ∈ C(x, ω). Thus, y ∗ (ω) is also a solution of problem (2.6) for all ω ∈ Ω. Now we show that problem (2.6) ⊂ problem (2.7). Let y ∗ (ω) be the solution of problem (2.6) with the membership degree α ∈ [0, 1] for every ω ∈ Ω. According to the tolerance approach [15, 21], by (2.2), we have ⟨F (x, y ∗ (ω), ω), υ(ω) − y ∗ (ω)⟩ ≥ µ−1 ˜ (α) ≥ −tz , Ω z

∀υ(ω) ∈ C(x, ω),

(2.10)

where for all υ(ω) ∈ C(x, ω) and any ω ∈ Ω, µ−1 ˜ z (x, ·, ω) and tz > 0 is ˜ z is the inverse functions of µΩ Ω the tolerance level which a decision maker can tolerate in the accomplishment of the fuzzy inequality ¯ ˆ ⟨F (x, y(ω), ω), υ(ω) − y(ω)⟩ ≥ ∼ 0. Suppose that for tz ≥ 0 and tz < 0, either ⟨F (x, y ∗ (ω), ω), y ∗ (ω)⟩ > t¯z

or ⟨F (x, y ∗ (ω), ω), y ∗ (ω)⟩ < tˆz

is true. For any x ∈ Rn and each ω ∈ Ω, since C(x, ω) is a convex cone, we have ⟨F (x, y ∗ (ω), ω), y ∗ (ω)⟩ ≥ −tz ∗ ∗ ∗ λ−1 , tz ≥ 0 when υ(ω) = λy (ω) with λ > 1, and ⟨F (x, y (ω), ω), y (ω)⟩ ≤ tz , when υ(ω) = 0. If

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⟨F (x, y ∗ (ω), ω), y ∗ (ω)⟩ > t¯z for t¯z ≥ 0, This leads to a contradiction. If ⟨F (x, y ∗ (ω), ω), y ∗ (ω)⟩ < tˆz for tˆz < 0, then tz ≤ tˆz . There lies a contradiction. Therefore, tˆz ≤ ⟨F (x, y ∗ (ω), ω), y ∗ (ω)⟩ ≤ t¯z , for t¯z ≥ 0 and tˆz < 0, that is, ⟨F (x, y ∗ (ω), ω), y ∗ (ω)⟩ = ∼ 0. Furthermore, from (2.10), we have for any υ(ω) ∈ C(x, ω), ⟨F (x, y ∗ (ω), ω), υ(ω)⟩ ≥ ⟨F (x, y ∗ (ω), ω), y ∗ (ω)⟩ − tz ≥ tˆz − tz . ∗ ∈ This implies that ⟨F (x, y ∗ (ω), ω), υ(ω)⟩ ≥ ∼ 0 for all υ(ω) ∈ C(x, ω). Hence, we have F (x, y (ω), ω) ∼ ∗ ∗ C (x, ω). Therefore, y (ω) for any ω ∈ Ω is also a solution of problem (2.7). This completes the proof. 2 Based on Lemma 2.2 and the work of [21], we have the following results. Lemma 2.3. Let C(x, ω) is a convex cone with polar (dual) cone C ∗ (x, ω) for all (x, ω) ∈ Rn ×Ω and α be a new variable. Then the stochastic optimization problem (1.1) can eventually be expressed as the following regular semi-infinite optimization problem with finitely many variables x, y(ω), ω and α: max α s.t. µS˜0 (x, y(ω), ω) ≥ α, µΩ˜ z (x, y(ω), ω) ≥ α, (2.11) (x, y(ω), ω) ∈ S, 0 ≤ α ≤ 1, ∗ where S = {(x, y(ω), ω) ∈ Rn+m × Ω| x ∈ U, ω ∈ Ω, F (x, y(ω), ω) ∈ ∼ C (x, ω)}, and the random membership functions µS˜0 and µΩ˜ z are the same as in (2.3) and (2.2), respectively. Proof. It follows from Lemma 2.1 that, in order to find a solution to the stochastic optimization problem (1.1) with C(x, ω) being a convex cone for (x, ω) ∈ Rn × Ω, we should consider the following stochastic fuzzy complementarity constrained optimization problem:

min s.t.

Eω [f (x, y(ω), ω)] x ∈ U, ω ∈ Ω, y(ω) ∈ C(x, ω), ⟨F (x, y(ω), ω), y(ω)⟩ = ∼ 0, ∗ C (x, ω). F (x, y(ω), ω) ∈ ∼

(2.12)

Since a global minimum is often required for practical problems, by the work of [21] and the description of the fuzzy stochastic inequalities (2.6), and a solution of problem (2.12) can be taken as the solution with the highest membership in the fuzzy decision set and eventually obtained by solving the following regular nonlinear parametric optimization problem: { } max min µS˜0 (x, y(ω), ω), µΩ˜ z (x, y(ω), ω) , (x,y(ω),ω)∈S

which implies that the result holds for the new variable α. 2 Remark 2.2. Moreover, if the membership functions µS˜0 and µΩ˜ z in Lemma 2.3 are invertible, then from (2.11), we get max α s.t.

(x, y(ω), ω)0 ≥ µ−1 ˜0 (α), S −1 ∗ (x, y(ω), ω)C ≥ µΩ ˜ z (α), (x, y(ω), ω) ∈ S, 0 ≤ α ≤ 1,

(2.13)

where (x, y(ω), ω)0 and (x, y(ω), ω)C ∗ can be followed by (2.3) and (2.2), respectively.

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3

Regularization smoothing approximation algorithms

In this section, based on the “method of centres” with entropic regularization, we develop a class of new smoothing approach and construct a centre iterative algorithm for solving the stochastic fuzzy optimization (1.1), and give the solution theorems. In the sequel, we first give the following assumption (HC ) for convenience: Define C(x, ω) :=

{υ(ω) ∈ Rm | D(x)υ(ω) ≥ 0, D(x) = [di (x)] is an l × m matrix, di (x) is the ith row of D(x), ∀i = 1, 2, · · · , l}.

(3.1)

STEP I. The random membership function of the fuzzy stochastic inequalities in (1.1) can be specified under condition (HC ). From (3.1), it is easy to see that the multi-valued operator C(x, ω) is a convex cone for any (x, ω) ∈ Rn × Ω, and can be shown that F (x, y(ω), ω) ∈ C ∗ (x, ω) if and only if there exists a nonnegative random vector r(ω) = (r1 (ω), r2 (ω), · · · , rl (ω))T ∈ Rl such that F (x, y(ω), ω) = r1 (ω)dT1 (x) + r2 (ω)dT2 (x) + · · · + rl (ω)dTl (x) = DT (x)r(ω),

(3.2)

that is, for every i = 1, 2, · · · , l, d′i (x)F (x, y(ω), ω) ≥ 0, where d′i (x) is normal to di (x) (see [17, 26]). It follows that the fuzzy stochastic optimization problem (2.12) can be rewritten as the following generalized stochastic optimization problem with fuzzy stochastic inequality constraints: min Eω [f (x, y(ω), ω)] s.t. x ∈ U, ω ∈ Ω, di (x)y(ω) ≥ 0, i = 1, 2, · · · , l, ⟨F (x, y(ω), ω), y(ω)⟩ ≥ ∼ 0, ⟨−F (x, y(ω), ω), y(ω)⟩ ≥ ∼ 0, 0, i = 1, 2, · · · , l, d′i (x)F (x, y(ω), ω) ≥ ∼

(3.3)

and each fuzzy stochastic inequality in (3.3) can be represented by a fuzzy set S˜j (i.e., represent ˜ z in (2.11) or (2.13)) with corresponding random membership function µ ˜ (x, y(ω), ω) for j = of Ω Sj 1, 2, · · · , l + 2. To specify the membership functions µS˜j , j = 1, 2, · · · , l + 2, similar treatment to (2.2), we define the membership functions as follows:  if ⟨F (x, y(ω), ω), y(ω)⟩ ≥ 0,  1, µ1 (⟨F (x, y(ω), ω), y(ω)⟩), if ⟨F (x, y(ω), ω), y(ω)⟩ ∈ [−t1 , 0), µS˜1 (x, y(ω), ω) =  0, if ⟨F (x, y(ω), ω), y(ω)⟩ < −t1 ,  if ⟨−F (x, y(ω), ω), y(ω)⟩ ≥ 0,  1, µ2 (⟨−F (x, y(ω), ω), y(ω)⟩), if ⟨−F (x, y(ω), ω), y(ω)⟩ ∈ [−t2 , 0), (3.4) µS˜2 (x, y(ω), ω) =  0, if ⟨−F (x, y(ω), ω), y(ω)⟩ < −t2 ,  if d′i F (x, y(ω), ω) ≥ 0,  1, ′ µi+2 (di F (x, y(ω), ω)), if d′i F (x, y(ω), ω) ∈ [−ti+2 , 0), µS˜i+2 (x, y(ω), ω) =  0, if d′i F (x, y(ω), ω) < −ti+2 , where ti+2 ≥ 0 for i = 1, 2, · · · , l, is the tolerance level which one decision maker can tolerate in the accomplishment of the fuzzy stochastic inequalities in (3.3). STEP II. Discrete approximation of problem (1.1) with condition (HC ) need to be given. By Lemma 2.1 and STEP I, we have the following problem as an appropriate discrete approximation of problem (3.3): ∑ min L1 f (x, y(ω), ω)ζ(ω) ω∈ΩL

s.t.

x ∈ U ⊂ Rn , ω ∈ ΩL , di (x)y(ω) ≥ 0, i = 1, 2, · · · , l, ⟨F (x, y(ω), ω), y(ω)⟩ ≥ ∼ 0, ⟨−F (x, y(ω), ω), y(ω)⟩ ≥ ∼ 0, d′i (x)F (x, y(ω), ω) ≥ 0, i = 1, 2, · · · , l. ∼

(3.5)

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We note that the sample set ΩL is chosen to be asymptotically dense in Ω. Especially, it follows from (2.4), (2.5) and (3.5) that the appropriate discrete approximation of two parameter f¯ and f can be shown as follows, respectively: ∑ f¯ = min L1 f (x, y(ω), ω)ζ(ω) ω∈ΩL

x ∈ U ⊂ Rn , ω ∈ ΩL , di (x)y(ω) ≥ 0, i = 1, 2, · · · , l, ⟨F (x, y(ω), ω), y(ω)⟩ ≥ 0, ⟨−F (x, y(ω), ω), y(ω)⟩ ≥ 0, d′i (x)F (x, y(ω), ω) ≥ 0, i = 1, 2, · · · , l

s.t.

and f = min s.t.

1 L

∑

(3.6)

f (x, y(ω), ω)ζ(ω)

ω∈ΩL

x ∈ U ⊂ Rn , ω ∈ ΩL , di (x)y(ω) ≥ 0, i = 1, 2, · · · , l, ⟨F (x, y(ω), ω), y(ω)⟩ ≥ −t1 , ⟨−F (x, y(ω), ω), y(ω)⟩ ≥ −t2 , d′i (x)F (x, y(ω), ω) ≥ −ti+2 , i = 1, 2, · · · , l.

(3.7)

STEP III. A new centre iterative method for solving problem (3.8) should be adopt. It follows from (2.13), (2.3) and (3.4)-(3.7) that an optimal solution of the stochastic optimization problem (1.1) can be obtained by approximating for the following stochastic parametric optimization problem:  max α   ∑   1  s.t. f (x, y(ω), ω)ζ(ω) ≥ 0, µS−1  ˜0 (α) − L   ω∈ΩL    µS−1  ˜1 (α) − ⟨F (x, y(ω), ω), y(ω)⟩ ≥ 0,   −1 (3.8) µS˜ (α) + ⟨F (x, y(ω), ω), y(ω)⟩ ≥ 0,  2    ′  −µ−1 j = 3, 4, · · · , l + 2,  ˜j (α) + dj−2 F (x, y(ω), ω) ≥ 0, S      di (x)y(ω) ≥ 0, i = 1, 2, · · · , l,    0 ≤ α ≤ 1, x ∈ U, ω ∈ ΩL . It is interested in developing an efficient algorithm to solve (3.8) based on a framework of centre iterations. This iterative approach can be traced back to Huard’s work [28]. The basic concepts are easy to understand and very adaptive to new developments. To describe the approach, we denote the feasible domain of (3.8) by a set V and define some terminologies. A general assumption for this approach is that V is bounded and convex, and the interior of V is nonempty. Definition 3.1 For any given point (x, y(ω), ω, α) in the convex domain V , we define the “distance L of (x, y(ω), ω, α) to the boundary of V ” by a continuous function L((x, y(ω), ω, α), V )

{ =

min i=1,2,··· ,l j=3,4,··· ,l+2

α, 1 − α, µ−1 ˜ (α) − S 0

1 ∑ f (x, y(ω), ω)ζ(ω), L ω∈ΩL

µ−1 ˜1 (α) S

− ⟨F (x, y(ω), ω), y(ω)⟩, µ−1 ˜2 (α) + ⟨F (x, y(ω), ω), y(ω)⟩, S } ′ −µ−1 ˜ (α) + dj−2 F (x, y(ω), ω), di (x)y(ω) . S j

Definition 3.2 Let a distance function L((x, y(ω), ω, α), V ) be defined on a convex domain V . Then a point (¯ x, y¯(ω), ω, α) ¯ ∈ V is called the “centre of V ”, if it maximizes the distance function L((x, y(ω), ω, α), V ), i.e., (¯ x, y¯(ω), ω, α ¯) :

L((¯ x, y¯(ω), ω, α ¯ ), V ) = max {L((x, y(ω), ω, α), V )| (x, y(ω), ω, α) ∈ V } .

Thus, a new centre iterative method for problem (3.8) could be described as follows.

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Algorithm 3.1. Step 1. Taking a point (xk , y k (ω), ω, αk ) in V , then we consider the distance L in convex domain Wk = V ∩ {(x, y(ω), ω, α)|α ≥ αk }. Step 2. Solving the maximal problem max{L((x, y(ω), ω, α), Wk )| (x, y(ω), ω, α) ∈ Wk } and denoting the new iterative point (xk+1 , y k+1 (ω), ω, αk+1 ) as a centre of Wk , then we have (xk+1 , y k+1 (ω), ω, αk+1 ) : L((xk+1 , y k+1 (ω), ω, αk+1 ), Wk ) = max{L((x, y(ω), ω, α), Wk )| (x, y(ω), ω, α) ∈ Wk }, where L((x, y(ω), ω, α), Wk )

=

{ 1 ∑ α − αk , α, 1 − α, µ−1 f (x, y(ω), ω)ζ(ω), ˜0 (α) − S i=1,2,··· ,l L j=3,4,··· ,l+2 min

ω∈ΩL

µ−1 ˜1 (α) S

− ⟨F (x, y(ω), ω), y(ω)⟩,

µ−1 ˜2 (α) S

+ ⟨F (x, y(ω), ω), y(ω)⟩, } ′ −µ−1 F (x, y(ω), ω), d (x)y(ω) (α) + d i j−2 ˜ S j

is the distance function defined on the convex domain Wk . Step 3. Start working again with (xk+1 , y k+1 (ω), ω, αk+1 ) instead of (xk , y k (ω), ω, αk ) and go to Step 1. It follows from the properties introduced in [28, Lemma 2.2] and Algorithm 3.1 that the major computational work lies in the determination of the centres required, i.e., at the kth iteration, the following “min-max problem” should be solved: { − min L((x, y(ω), ω, α), Wk ) = min max i=1,2,··· ,l αk − α, −α, α − 1, x,y(ω),ω,α x,y(ω),ω,α ∑ j=3,4,··· ,l+2 1 f (x, y(ω), ω)ζ(ω) − µ−1 ˜ (α), L S 0

ω∈ΩL

⟨F (x, y(ω), ω), y(ω)⟩ − µ−1 ˜1 (α), S −⟨F (x, y(ω), ω), y(ω)⟩ − µ−1 ˜ (α), S

(3.9)

2 } −d′j−2 F (x, y(ω), ω) + µ−1 (α), −d (x)y(ω) . i ˜ S j

STEP IV A class of new regularization smoothing approximation algorithms is developed under condition (HC ). Since the maximal membership function (see [12]) in the “min-max” problem (3.9) is nondifferentiability, it is easy to see that one major difficulty encountered is to develop a class of new smoothing approximation methods, which are based on the notion of newly proposed “entropic regularization procedure” (see [18]). Algorithm 3.2. Step 1. Set k = 0, give the initial iterate (x0 , y 0 (ω), ω, α0 ) which is an interior point of V defined by (3.8), a sufficiently small constant ϵ > 0, and an upper bound Q which is the maximum number of unconstrained minimizations to be performed. Step 2. Starting from (xk , y k (ω), ω, αk ), apply a standard quasi-Newton line search of MATLAB software to solve the unconstrained smooth convex program (3.6), (3.7) and the following unconstrained smooth convex program: −

min Lγ ((x, y(ω), ω, α), Wk ) x,y(ω),ω,α { = γ1 ln exp[γ(αk − α)] + exp[γ(−α)] + exp[γ(α − 1)] ∑ + exp[γ( L1 f (x, y(ω), ω)ζ(ω) − µ−1 ˜0 (α))] S ω∈ΩL + exp[γ(⟨F (x, y(ω), ω), y(ω)⟩ − µ−1 ˜1 (α))] S + exp[γ(−⟨F (x, y(ω), ω), y(ω)⟩ − µ−1 ˜2 (α))] S l+2 ∑ + exp[γ(−d′j−2 F (x, y(ω), ω) + µ−1 ˜j (α))] S j=3 } l ∑ +

(3.10)

exp[γ(−di (x)y)]

i=1

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with a sufficiently large γ. Denote its solution by (xk+1 , y k+1 (ω), ω, αk+1 ) in the light of Algorithm 3.1. Step 3. If k > 1 and ∥(xk+1 , y k+1 (ω), ω, αk+1 ) − (xk , y k (ω), ω, αk )∥2 ≤ ϵ, then the computation terminates with (xk+1 , y k+1 (ω), ω, αk+1 ) as the solution. If k > Q, then the computation terminates with a failure. Step 4. k ← k + 1 and go to Step 2. From Algorithm 3.2, it follows that minx,y(ω),ω,α Lγ ((x, y(ω), ω, α), Wk ) provides a centre of Wk , as γ → ∞. By using a moderately large γ, we can obtain an accurate approximation. Also because of the special “log-exponential” form of Lγ ((x, y(ω), ω, α), Wk ), we can avoid most overflow problems in computation. Moreover, since problem (3.10) is an unconstrained, smooth, and convex optimization program, the commonly used solution methods, such as the quasi-Newton line search of MATLAB software, can be readily applied. Remark 3.1. We note that Algorithm 3.1 appears in Step 2 of Algorithm 3.2. It is the fuzzy constraints in (1.1) that yields a fuzzy objective. Hence, a class of new and interesting regularization smoothing approximation approaches must be chosen to define two parameters in (3.6) and (3.7), and to employ for solving problem (3.10) which is equivalent to the stochastic parametric optimization problem (3.8). STEP V Comprehensive convergence theorems based on Algorithm 3.2 should be proved. In the sequel, we first give the following lemmas and results. Lemma 3.1. Let the function φ : Ω → R be continuous. Then we have ∫ 1 ∑ lim φ(ω)ζ(ω) = φ(ω)ζ(ω)dω. L→∞ L Ω ω∈ΩL

Proof. Taking N = L, I¯s = Ω, J = ΩL , xi = ωi (i = 1, 2, · · · ) and f = φζ, then from the results (2.2) and (2.3) given in Chapter 2 of [24, pp. 13-14], the result holds. This completes the proof. 2 Remark 3.2. By Lemma 3.1, we know immediately that ∫ 1 ∑ lim f (x, y(ω), ω)ζ(ω) = f (x, y(ω), ω)ζ(ω)dω (3.11) L→∞ L Ω ω∈ΩL

and particularly,

∫ 1 ∑ lim ζ(ω) = ζ(ω)dω = 1. L→∞ L Ω

(3.12)

ω∈ΩL

Lemma 3.2. If ψ(x) is continuous, strictly increasing and linear over a convex set U in Rn , then its inverse ψ −1 is linear. Theorem 3.1. Suppose that condition (HC ) holds, the set U ⊂ Rn is nonempty and bounded, F : Rn+m × Ω → Rm is continuously differentiable, and f : Rn+m × Ω → R is H¨older continuous in (x, y(·)) on U × Rm with order σ > 0 and H¨older constant ς(ω) > 0 for all ω ∈ ΩL satisfying ∫ ς(ω)dζ(ω) < +∞. Ω

Then (i) problem (3.6) has at least one optimal solution when L is large enough; (ii) (x∗ , y ∗ (·)) is an optimal solution of problem (2.4) when x∗ is an accumulation point of the sequence {xL } and y ∗ (·) is defined by y ∗ (ω) :=

max {−F (x∗ , 0, ω), −d′i (x∗ )F (x∗ , 0, ω), 0},

i=1,2,··· ,l

ω ∈ Ω.

(3.13)

Proof. (i) Let FL be the feasible region of problem (3.6). It is not difficult to see that FL is a nonempty and closed set and the objective function of problem (3.6) is bounded below on FL . Thus,

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there exists a sequence {(xk , y k (ω))ω∈ΩL } ⊂ FL such that 1 ∑ f (xk , y k (ω), ω)ζ(ω) k→∞ L ω∈ΩL 1 ∑ = inf f (x, y(ω), ω)ζ(ω). (x,y(ω),ω)ω∈ΩL L lim

(3.14)

ω∈ΩL

It follows from the boundedness of U and the H¨older continuity of f that the sequence {xk } and the function f are bounded. On the other hand, noting that (xk , y k (ω))ω∈ΩL ∈ FL for every k, we have 0 ≤ y k (ω) ⊥

1 ∑ F (xk , y k (ω), ω)ζ(ω) ≥ 0, L

(3.15)

ω∈ΩL

where the symbol ⊥ means the two vectors are perpendicular to each other. Assume that the sequence {y k (ω)}ω∈ΩL is unbounded. Taking a subsequence if necessary, let lim ∥y k (ω)∥ = +∞, lim

k→∞ ω∈ΩL

k→∞ ω∈ΩL

y k (ω) = y¯(ω), ∥¯ y (ω)∥ = 1. ∥y k (ω)∥

(3.16)

Then, for all ω ∈ ΩL , dividing (3.15) by ∥y k (ω)∥ and letting k → +∞, we have for any x ∈ U , 0 ≤ y¯(ω) ⊥

1 ∑ F (x, y¯(ω), ω)ζ(ω) ≥ 0. L ω∈ΩL

This contradicts (3.16) by the continuous differentiability of F , and so {y k (ω)} is bounded for each ω ∈ ΩL with ζ(ω) > 0. For any ω ∈ ΩL with ζ(ω) = 0, we redefine y k (ω) by y k (ω) :=

max {−F (xk , 0, ω), −d′i (xk )F (xk , 0, ω), 0}.

i=1,2,··· ,l

Hence, the sequence {(xk , y k (ω))ω∈ΩL } is bounded and (3.14) remains valid. Therefore, the closeness of FL implies that any accumulation point of {(xk , y k (ω))ω∈ΩL } must be an optimal solution of problem (3.6). (ii) By the assumptions, the sequence {xL } contains a subsequence converging to x∗ . Without loss of generality, we suppose limL→∞ xL = x∗ . Firstly, we prove that (x∗ , y ∗ (·)) is feasible to problem (3.6). To this end, we define { } yˆL (ω) := max −F (xL , 0, ω), −d′i (xL )F (xL , 0, ω), 0 , ω ∈ Ω. (3.17) i=1,2,··· ,l

It is obvious that (x∗ , yˆL (ω))ω∈ΩL is feasible to problem (3.6) for every L. Since F (x∗ , y ∗ (ω), ω) ≥ 0 by the definition (3.13), it is sufficient to show that (y ∗ (ω))T F (x∗ , y ∗ (ω), ω) = 0,

ω ∈ Ω.

(3.18)

Let ω ¯ ∈ Ω be fixed. Since the sample set ΩL is chosen to be asymptotically dense in Ω, there exists a sequence {¯ ωL } of samples such that ω ¯ L ∈ ΩL for each L and limL→∞ ω ¯L = ω ¯ . Thus, we obtain (ˆ y L (¯ ωL ))T F (xL , yˆL (¯ ωL ), ω ¯ L ) = 0,

L = 1, 2, · · · .

Letting L → +∞ and taking the continuity of the functions F (x, y(·), ·) on the compact set Ω into account, we have (y ∗ (¯ ω ))T F (x∗ , y ∗ (¯ ω ), ω ¯ ) = 0. By the arbitrariness of ω ¯ ∈ Ω, now we know that (3.18) immediately holds. This completes the proof of the feasibility of (x∗ , y ∗ (·)) in (3.6). 11

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Next, let (x, y(·)) be an arbitrary feasible solution of (3.6). It follows from the results of (i) and obvious that (x, y(ω), ω)ω∈ΩL is feasible to problem (3.6) for any L. Moreover, from the H¨ocontinuity of f , we have ] 1 ∑ [ f (xL , y L (ω), ω) − f (xL , yˆL (ω), ω) ζ(ω) L ω∈ΩL ] 1 ∑ [ = f (xL , y L (ω), ω) − f (xL , yˆL (ω), ω) ζ(ω) L ω∈ΩL ] 1 ∑ [ L ≤ ∥y (ω) − yˆL (ω)∥ς(ω) · ζ(ω) → 0 as k → ∞, L

w.p.1.

ω∈ΩL

which along with Lemma 3.1 yields 1 ∑ 1 ∑ f (xk , y k (ω), ω) = lim f (x∗ , y ∗ (ω), ω) = Eω [f (x∗ , y ∗ (ω), ω)] L→∞ L L→∞ L lim

ω∈ΩL

w.p.1,

ω∈ΩL

which indicates that (x∗ , y ∗ (·)) is an optimal solution of problem (1.1) with probability one and the feasibility of (xL , y L (ω), ω)ω∈ΩL in (3.6) that (xL , yˆL (ω), ω)ω∈ΩL is also an optimal solution of problem (3.6). Thus, since f is H¨older continuous in (x, y(·)) on U × Rm , we obtain ] 1 ∑ [ f (x∗ , y ∗ (ω), ω) − f (x, y(ω), ω) ζ(ω) L ω∈ΩL ] 1 ∑ [ f (x∗ , y ∗ (ω), ω) − f (xL , yˆL (ω), ω) ζ(ω) ≤ L ω∈ΩL 1 ∑ ≤ f (x∗ , y ∗ (ω), ω) − f (xL , yˆL (ω), ω) ζ(ω) L ω∈ΩL [ ] 1 ∑ ≤ ζ(ω) · ∥xL − x∗ ∥ + ∥ˆ y L (ω) − y ∗ (ω)∥ ς(ω). L

(3.19)

ω∈ΩL

It follows from (3.12) that the sequence

{1 ∑ L

ω∈ΩL

} ζ(ω) is bounded. This yields

1 ∑ f (x∗ , y ∗ (ω), ω) − f (xL , yˆL (ω), ω) ζ(ω) = 0. L→∞ L lim

(3.20)

ω∈ΩL

Thus, by letting L → +∞ in (3.19) and taking (3.11) and (3.20) into account, we have ∫ ∫ ∗ ∗ f (x , y (ω), ω)ζ(ω)dω ≤ f (x, y(ω), ω)ζ(ω)dω, Ω

Ω

which implies that x∗ together with y ∗ (·) constitutes an optimal solution of problem (3.6). This completes the proof. 2 Similarly, by Lemma 3.1, (3.11), (3.12) and proof of Theorem 3.1, we have the following result. Theorem 3.2. Assume that condition (HC ) holds, and f , F and U are the same as in Theorem 3.1. Then (i) problem (3.7) has at least one optimal solution when L is large enough; (ii) (x∗ , y ∗ (·)) is an optimal solution of problem (2.5) when x∗ is an accumulation point of the sequence {xL } and y ∗ (·) is defined by y ∗ (ω) :=

max {−t1 − F (x∗ , 0, ω), −t2 + F (x∗ , 0, ω), −ti+2 − d′i (x∗ )F (x∗ , 0, ω), 0},

i=1,2,··· ,l

ω ∈ Ω.

Now, consider the case that the membership function of each fuzzy stochastic inequality and the objective function Eω [f (x, y(ω), ω)] in (3.3) is continuous, strictly increasing, and linear over the 12

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corresponding tolerance interval. A commonly used example in fuzzy set theory is that ψ(x) = 1−bxβ with b > 0 and β > 1. In this case, from the theory of convex analysis [27], Lemma 3.2, and Theorems 3.1 and 3.2, we have the following simple result. Theorem 3.3. Suppose that condition (HC ) holds. If F : Rn+m × Ω → Rm is monotone in the second variable, and µΩ˜ z (x, y(ω), ω) is continuous, strictly increasing and linear for all z ∈ C(x, ω) and any (x, y(ω), ω) ∈ Rn+m ×Ω, then we can find an optimal solution (x∗ , y ∗ (·)) of the stochastic optimization problem (1.1) by solving the following stochastic parametric optimization problem: (3.8), which can be readily approximated by the iterative sequence {(xk+1 , y k+1 (ω), ω, αk+1 )} generated by Algorithm 3.2.

4

Simulation analysis

In this section, we shall give an example to illustrate the validity of our approaches. Taking n = 2, m = 3, l = 2, U = [1, 14] × [1, 14], d1 (x) = (−1, −1, 3), d2 (x) = (−n2, 1, −1),   −2x1 + y1 − 3y2 + ω f (x, y(ω), ω) = (x1 − y1 )2 + x2 y2 + 2ω, F (x, y(ω), ω) =  x1 + x2 + 3y1 − y3 − 2ω  , −2x2 + y2 + 2y3 − ω ( { ) } d1 (x) C(x, ω) = y(ω) = (y1 , y2 , y3 )T ∈ R3 y(ω) ≥ 0 , d2 (x) and Letting d′1 (x) = (0, 3, 1) and d′2 (x) = (1, 2, 0) in (3.3), and ζ(ωℓ ) = pℓ (ℓ = 1, 2, · · · , L) in (3.5), then we have ∑L φ(x, y(ω), ω) := Eω [(x1 − y1 )2 + x2 y2 + 2ω] = L1 ℓ=1 [(x1 − y1 )2 + x2 y2 + 2ωℓ ]pℓ , f1 (x, y(ω), ω) = ⟨F (x, y(ω), ω), y(ω)⟩ = −2x1 y1 + x1 y2 + x2 y2 − 2x2 y3 + y12 +ωy1 − 2ωy2 + 2y32 − ωy3 , f2 (x, y(ω), ω) = ⟨−F (x, y(ω), ω), y(ω)⟩ = 2x1 y1 − x1 y2 − x2 y2 + 2x2 y3 − y12 −ωy1 + 2ωy2 − 2y32 + ωy3 , ′ f3 (x, y(ω), ω) = d1 (x)F (x, y(ω), ω) = 3x1 + x2 + 9y1 + y2 − y3 − 7ω, f4 (x, y(ω), ω) = d′2 (x)F (x, y(ω), ω) = 2x2 + 7y1 − 3y2 − 2y3 − 3ω. Thus, problem (3.5) is equivalent to the following generalized fuzzy stochastic inequality constrained optimization program: min φ(x, y(ω), ω) s.t. 1 ≤ x1 , x2 ≤ 14, y1 , y2 , y3 ≥ 0, −y1 − y2 + y3 ≥ 0, −2y1 + y2 − y3 ≥ 0, ι = 1, 2, 3, 4, fι (x, y(ω), ω) ≥ ∼ 0,

(4.1)

with the membership function µS˜τ (x, y(ω), ω) (τ = 1, 2, 3, 4), being specified as t1 = 9, t2 = 2, t3 = 6, t4 = 10,  if f1 (x) ≥ 0,  1, µS˜1 (x, y(ω), ω) = 1 − f1 (x,y(ω),ω) , if f1 (x, y(ω), ω) ∈ [−9, 0), 9  0, if f1 (x, y(ω), ω) < −9,  if f2 (x, y(ω), ω) ≥ 0,  1, µS˜2 (x, y(ω), ω) = , if f2 (x, y(ω), ω) ∈ [−2, 0), 1 − f2 (x,y(ω),ω) 2  0, if f2 (x) < −2,  if f3 (x, y(ω), ω) ≥ 0,  1, f3 (x,y(ω),ω) µS˜3 (x, y(ω), ω) = , if f3 (x, y(ω), ω) ∈ [−6, 0), 1− 6  0, if f3 (x, y(ω), ω) < −6,  if f4 (x) ≥ 0,  1, µS˜4 (x, y(ω), ω) = , if f4 (x, y(ω), ω) ∈ [−10, 0), 1 − f4 (x,y(ω),ω) 10  0, if f4 (x) < −10, 13

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and µS˜0 (x, y(ω), ω) =

 1,    ¯   

if

φ(x, y(ω), ω) < f ,

f −φ(x,y(ω),ω) , f¯−f

if

φ(x, y(ω), ω) ∈ [f , f¯),

0,

if

⟨φ(x, y(ω), ω) ≥ f¯,

where f¯ = min φ(x, y(ω), ω) s.t. 1 ≤ x1 , x2 ≤ 14, y1 , y2 , y3 ≥ 0, −y1 − y2 + 3y3 ≥ 0, −2y1 + y2 − y3 ≥ 0, fι (x, y(ω), ω) ≥ 0, ι = 1, 2, 3, 4,

(4.2)

f = min φ(x, y(ω), ω) s.t. 1 ≤ x1 , x2 ≤ 14, y1 , y2 , y3 ≥ 0, , −y1 − y2 + 3y3 ≥ 0, −2y1 + y2 − y3 ≥ 0, f1 (x, y(ω), ω) ≥ −9, f2 (x, y(ω), ω) ≥ −2, f3 (x, y(ω), ω) ≥ −6, f4 (x, y(ω), ω) ≥ −10.

(4.3)

and

By Bellman and Zadeh’s method of fuzzy decision making [15] and Algorithm 3.2, now we know that the conditions of Theorem 3.3 hold, and so an optimal solution of the problem (4.1) can be obtained by solving the following unconstrained and smooth nonlinear parametric optimization problem: { minx,y(ω),ω,α γ1 ln exp[γ(αk − α)] + exp[γ(−α)] + exp[γ(α − 1)] + exp[γ(φ(x, y(ω), ω) − (f¯ − α(f¯ − f )))] + exp[γ(f1 (x, y(ω), ω) − 9(1 − α))] + exp[γ(f2 (x, y(ω), ω) − 2(1 − α))] (4.4) + exp[γ(−f3 (x, y(ω), ω) + 6(1 − α))] + exp[γ(−f4 (x, y(ω), ω) + 10(1 − α))] + exp[γ(y1 + y2 − 3y3 )] + exp[γ(2y1 − y2 + y3 )] + exp[γ(1 − x1 )] + exp[γ(1 − x2 )] + exp[γ(x1 − 14)] } + exp[γ(x2 − 14)] + exp[γ(−y1 )] + exp[γ(−y2 )] + exp[γ(−y3 )]

with γ being sufficiently large, where the optimal values of f¯ and f are obtained by computing (4.2) and (4.3), respectively. Choosing x0 = (4.0000, 2.0000), y 0 (ω) = (1.0547, 1.0564, 0.1574) and α0 = 0.2 and setting L = 3 with the probability p1 = 0.1590, p2 = 0.6821, p3 = 0.1589, and ϵ = 10−5 , Q = 106 and fixed γ = 12, then for each iteration of Algorithm 3.2, we first generate the random variable ω by using normrnd function (that is, normal distribution function) of MATLAB 7.0 software. Secondly, we solve from problems (4.2) and (4.3) to problem (4.4) in turn by the commonly used quasi-Newton line search of MATLAB software 7.0. Here, the first layer iteration searching optimization is to solve problems (4.2) and (4.3), respectively. And the second optimizing process is to find the optimal solution of problem (4.1) via solving the unconstrained and smooth nonlinear parametric optimization problem (4.4). We only present four optimal solution (x∗ , y ∗ (·)) with respect to the random variable ω and the corresponding membership degree α∗ for whole stochastic optimization problem, which is listed in Table 1. Further, Table 2 show that each iteration calculation results including iterative solutions with the random variable ω = 0.128808 for the second optimizing process to this problem. The results for every step in Table 2 (i.e., k = 0, 1, 2, · · · , 11) come from the first layer iteration process, which are too much and so they are omitted.

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Table 1: The optimal solution with the random variable and membership degree k 1 2 3 4

(x∗ , y ∗ (ω), ω) (0.984674, 1.158197, (0.987781, 1.219403, (0.984337, 1.156955, (0.988056, 1.226168,

0.036645, 0.024577, 0.036586, 0.023580,

0.228989, 0.187961, 0.232004, 0.184104,

0.128690, 0.097964, 0.127346, 0.095786,

α∗ 0.128808) 0.954363 0.345629) 0.949652 0.191080) 0.953185 0.353731) 0.949391

Table 2: Data for Computational results with the random variable ω = 0.128808 k (xk , y k ) αk Iterations No. 0.2 46 0 (4.0000, 2.0000, 1.0547, 1.0564, 0.1574) 1 (0.989779,1.252773,0.027679,0.173089,0.106607) 0.871896 16 2 (0.985831,1.166544,0.034618,0.217474,0.125437) 0.924577 13 3 (0.984928,1.159731,0.036190,0.226463,0.128032) 0.945141 15 4 (0.984726,1.158504,0.036552,0.228473,0.128555) 0.951749 14 5 (0.984687,1.158268,0.036623,0.228867,0.128658) 0.953643 9 6 (0.984677,1.158241,0.036639,0.228955,0.128681) 0.954166 4 7 (0.984676,1.158237,0.036638,0.228959,0.128678) 0.954309 3 8 (0.984676,1.158233,0.036638,0.228962,0.128678) 0.954348 5 9 (0.984674,1.158198,0.036645,0.228989,0.128690) 0.954359 1 10 (0.984674,1.158197,0.036645,0.228989,0.128690) 0.954362 1 11 (0.984674,1.158197,0.036645,0.228989,0.128690) 0.954363

5

Concluding remarks

In this paper, by developing a class of new regularization smoothing approximation approaches, we investigated approximation solvability of the following fuzzy parametric variational inequality constrained stochastic optimization problems in n-dimension real numeral set Rn : min x,y(·)

s.t.

Eω [f (x, y(ω), ω)] x ∈ U, y(ω) ∈ C(x, ω), ⟨F (x, y(ω), ω), z(ω) − y(ω)⟩ ≥ ∼ 0,

(5.1) ∀z(ω) ∈ C(x, ω),

which has been very little studied by right of the known theories and approaches in the literature. It is because the existence of the random variable and the fuzzified version mean that (5.1) involves multiple complementarity-type constraints, and solving problem (5.1) is more difficult than solving an ordinary mathematical program with (fuzzy) equilibrium constraints or stochastic mathematical program with equilibrium constraints. Based on the notion of tolerance approach with entropic regularization and fuzzy set theory, we first showed that solving the stochastic optimization problem with fuzzy parametric variational inequality constraints is equivalent to solving a fuzzy complementarity constrained stochastic optimization problem, which can be converted to a regular nonlinear parametric optimization problem with continuous random variables. Then, we constructed a centre iterative algorithm and developed a class of new regularization smoothing approximation approaches for solving a problem with continuous random variables based on quasi-Monte Carlo estimate and entropic regularization technique, and discussed a comprehensive convergence theory for approximating the resulting optimization problem. Finally, numerical example was provided to illustrate our main results applying quasiNewton line search of MATLAB software. We remark that in the paper, based on the concept that fuzzy constraints should yield a fuzzy 15

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objective, we must choose a class of new regularization smoothing approximation approaches to define the objective function value of two optimization problems as the parameters in an equivalent stochastic parametric optimization problem. Hence, the problem presented in this paper is brand new and the method is also new and interesting. Whether the corresponding results of Theorem 3.3 hold when the objective function is a fuzzy stochastic function, the constraints are fuzzy implicit variational inequalities (such as in [14, 17]) or elliptic inequalities subject to physical phenomenon, and the numerical testing is some large-scale applications, which are still open questions to be solved in further research.

Acknowledgments This work has been partially supported by Sichuan Province Cultivation Fund Project of Academic and Technical Leaders, and the Scientific Research Project of Sichuan University of Science & Engineering (2015RC07).

References [1] S.I. Birbil, G. G¨ urkan and O. Listes, Solving stochastic mathematical programs with complementarity constraints using simulation, Math. Oper. Res. 31(4) (2006), 739–760. [2] Y. Chen and Q.J. Hu, A SQP algorithm based on a smoothing lower order penalty function for inequality constrained optimization, J. Comput. Anal. Appl. 11(3) (2009), 481–491. [3] X.J. Chen, H.L. Sun and Roger J.B. Wets, Regularized mathematical programs with stochastic equilibrium constraints: estimating structural demand models, SIAM J. Optim. 25(1) (2015), 53–75. [4] Q.J. Hu, W.Y. Chen and Y.H. Xiao, An improved active set feasible SQP algorithm for the solution of inequality constrained optimization problems, J. Comput. Anal. Appl. 11(1) (2009), 54–63. [5] M. Patriche, Bayesian abstract fuzzy economies, random quasi-variational inequalities with random fuzzy mappings and random fixed point theorems, Fuzzy Sets and Systems 245 (2014), 125–136. [6] M. Patriksson, On the applicability and solution of bilevel optimization models in transportation science: a study on the existence, stability and computation of optimal solutions to stochastic mathematical programs with equilibrium constraints, Transportation. Res. B 42 (2008), 843–860. [7] F. Toyasaki, P. Daniele and T. Wakolbinger, A variational inequality formulation of equilibrium models for end-of-life products with nonlinear constraints, European J. Oper. Res. 236(1) (2014), 340–350. [8] J. Zhang, Y.Q. Zhang and L.W. Zhang, A sample average approximation regularization method for a stochastic mathematical program with general vertical complementarity constraints, J. Comput. Appl. Math. 280 (2015), 202–216. [9] G.H. Lin, X.J. Chen, M. Fukushima, Smoothing Implicit Programming Approaches for Stochastic Mathematical Programs with Linear Complementarity Constraints, Technical Report 2003-006, Department of Applied Mathematics and Physics, Graduate School of Informatics, Kyoto University, Kyoto, Japan, 2003. [10] G.H. Lin, X.J. Chen and M. Fukushima, Solving stochastic mathematical programs with equilibrium constraints via approximation and smoothing implicit programming with penalization, Math. Program. Ser. B 116 (2009), 343–368. [11] H. Xu, An implicit programming approach for a class of stochastic mathematical programs with linear complementarity constraints, SIAM J. Optim. 16 (2006), 670–696. [12] C.F. Hu and F.B. Liu, Solving mathematical programs with fuzzy equilibrium constraints, Comput. Math. Appl. 58 (2009), 1844–1851. [13] H.Y. Lan, C.J. Liu and T.X. Lu, Method of centres for solving mathematical programs with fuzzy parametric variational inequality constraints, in: X.S. Zhang et al. (Eds.), 11th International Symposium on Operations Research and its Applications in Engineering, Technology and Management, Vol. 2013, Issue 644CP, 2013, pp. 194–199. [14] H.Y. Lan and J.J. Nieto, Solving implicit mathematical programs with fuzzy variational inequality constraints based on the method of centres with entropic regularization, Fuzzy Optim. Decis. Mak. (in press, March 2015) http://dx.doi.org/10.1007/s10700-015-9207-7

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[15] R. Bellman and L.A. Zadeh, Decision making in a fuzzy environment, Management Sci. 17B (1970), 141–164. [16] Z.T. Gong, X.X. Liu and X. Feng, Almost ideal statistical convergence and strongly almost ideal lacunary convergence of sequences of fuzzy numbers with respect to the Orlicz functions, J. Comput. Anal. Appl. 19(3) (2015), 418–425. [17] H.Y. Lan, An approach for solving fuzzy implicit variational inequalities with linear membership functions, Comput. Math. Appl. 55(3) (2008), 563–572. [18] E.Y. Pee and J.O. Royset, On solving large-scale finite minimax problems using exponential smoothing, J. Optim. Theory Appl. 148(2) (2011), 390–421. [19] D.M. Yuan and X.L. Cheng, Method of fundamental solutions with an optimal regularization technique for the Cauchy problem of the modified Helmholtz equation, J. Comput. Anal. Appl. 14(1) (2012), 54–66. [20] Z.H. Zhang, Asymptotic representations in stochastic process approximations, J. Comput. Anal. Appl. 18(4) (2015), 672–684. [21] H.J. Zimmermann, Fuzzy Set Theory and Its Applications (2nd edition), Kluwer Academic, Dordrecht, 1991. [22] C.F. Hu, Solving Systems of Fuzzy Inequalities, Ph.D thesis, North Carolina State University, ProQuest LLC, Ann Arbor, MI, 1997. [23] Y. Chen and M. Florian, The nonlinear bilevel programming problem, formulations, regularity and optimality conditions, Optimization 32 (1995), 193–209. [24] H. Niederreiter, Random number generation and quasi-Monte Carlo methods, in: CBMS-NSF Regional Conference Series in Applied Mathematics, vol. 63, Society for Industrial and Applied Mathematics, Philadelphia, PA, vi+241 pp, 1992. [25] O.G. Mancino and G. Stampacchia, Convex programming and variational inequalities, J. Optim. Theory Appl. 9 (1972), 3–23. [26] H.F. Wang and H.L. Liao, Variational inequality with fuzzy convex cone, J. Global Optim. 14(4) (1999), 395–414. [27] R.T. Rockafellar, Convex Analysis, Princeton, NJ: Princeton Univ. Press, 1970. [28] P. Huard, Resolution of mathematical programming with nonlinear constraints by the method of centres in nonlinear programming, in: Nonlinear Programming (eds. J. Abadie), North-Holland, Amsterdam, 1967, pp. 207–219.

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The Split Common Fixed Point Problem for Demicontractive Mappings in Banach Spaces Li Yang1 , Fuhai Zhao2 and Jong Kyu Kim3 1 School

of Science, South West University of Science and Technology, Mianyang, Sichuan 621010, P. R. China e-mail: [email protected]

2 School

of Science, South West University of Science and Technology, Mianyang, Sichuan 621010, P. R. China e-mail: [email protected]

3 Department

of Mathematics Education, Kyungnam University, Changwon, Gyeongnam 51767, Korea e-mail: [email protected]

Abstract. In this paper, based on the work by Moudafi and inspired by Takahashi and Xu, we try to investigate the split common fixed point problems for the class of demicontractive mappings in the setting of two Banach spaces, and obtain the strong and weak convergence theorems. The results presented in the paper improve and extend some recent well-known corresponding results. Keywords: split common fixed point problem; demicontractive mapping; Demiclosed principle, weak and strong convergence theorems. 2010 AMS Subject Classification: 47H09, 49J25.

1

Introduction and Preliminaries

The split common fixed point problem was introduced by Moudafi [1] in 2010. Moudafi proposed an iteration scheme and obtained a weak convergence theorem of the split common fixed point problem for demicontractive mappings in the setting of two Hilbert spaces. Since then, many authors investigated the split common fixed point problems of other nonlinear mappings in the setting of two Hilbert spaces (see [2-7]). At the beginning of 2015, Takahashi [8] first attempted to introduce and consider the split feasibility problem and split common null point problem in the setting of one Hilbert space and one Banach space. By using hybrid methods and Halperns type methods under suitable conditions, some strong and weak convergence theorems for such problems are obtained. The results presented in [8] seem to be the first outside Hilbert spaces. This naturally brings us to solve the split common fixed point problem for demicontractive mappings in the setting of two Banach space. Let E1 and E2 be two real Banach spaces, and A : E1 → E2 be a bounded linear operator such that A 6= 0. The split common fixed point problem (SCFP) for nonlinear mappings S and T is to find a point x ∈ E1 such thst x ∈ F (S) and Ax ∈ F (T ), 0 Corresponding

(1.1)

author: Jong Kyu Kim([email protected])

1

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where F (S) and F (T ) denote the sets of fixed points of S and T , respectively. We use Γ to denote the set of solutions of SCFP for mappings S and T , that is, Γ = {x ∈ F (S)| Ax ∈ F (T )}. In this paper, we use the following algorithm to approximate a split common fixed point of demicontractive mappings in the setting of two Banach spaces. Algorithm: Let E1 and E2 be two real Banach spaces, A : E1 → E2 be a bounded linear operator, A∗ be the adjoint operator of A and Ji be the normalized duality ∗ mapping from Ei to 2Ei , i = 1, 2. Now, we define the iterative scheme {xn }: Let x1 ∈ E1 be arbitrary, for all n ≥ 1, set yn = xn + γJ1−1 A∗ J2 (T − I)Axn ,

(1.2)

xn+1 = (1 − αn )yn + αn S(yn ),

(1.3)

where S : E1 → E1 and T : E2 → E2 are two demicontractive mappings. Under some suitable conditions, the iterative scheme {xn } is shown to converge weakly and strongly to a split common fixed point of demicontractive mappings T and S. Our result extends the split common fixed point problem from Hilbert spaces to Banach spaces. In order to solve this problem mentioned above, we recall the following concepts and results. Let E be a real Banach space with norm k · k and let E ∗ be the dual space of E, We denote the value of y ∗ ∈ E at x ∈ E by hx, y ∗ i. When {xn } is a sequence in E, we denote the strong convergence of {xn } to x ∈ E by xn → x and the weak convergence by xn * x. We recall that T : E → E is demicontractive (see for example [9]) if there exists a constant η ∈ [0, 1) such that kT x − qk2 ≤ kx − qk2 + ηkx − T xk2 , ∀(x, q) ∈ E × F (T ).

(1.4)

An operator satisfying (1.4) will be referred to as a η-demicontractive mapping. It is worth noting that the class of demicontractive maps contains important operators such as the quasi-nonexpansive maps and the strictly pseudocontractive maps with fixed points. A mapping T : E → E is called quasi-nonexpansive, if kT x − qk ≤ kx − qk for all (x, q) ∈ E × F (T ). A mapping T : E → E is strictly pseudocontractive, if kT x − T yk2 ≤ kx − yk2 + βkx − y − (T x − T y)k2 for all (x, y) ∈ E × E and for some β ∈ [0, 1). A mapping T : E → E is called demiclosed at zero, if for any sequence {xn } ⊂ E and x ∈ E, we have xn * x, (I − T )(xn ) → 0 ⇒ x ∈ F (T ). A mapping S : E → E is said to be semi-compact, if for any sequence {xn } in E such that kxn − Sxn k → 0, (n → ∞), there exists subsequence {xnj } of {xn } such that {xnj } converges strongly to x∗ ∈ E. 2

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∗

The normalized duality mapping J from E to 2E is defined by Jx = {x∗ ∈ E ∗ : hx, x∗ i = kxk2 = kx∗ k2 }, ∀x ∈ E. Let U = {x ∈ E : kxk = 1}. The norm of E is said to be Gˆateaux differentiable if for each x, y ∈ U , the limit kx + tyk − kxk exists. t In the case, E is called smooth. E is smooth if and only if J is single-valued. We denote the single-valued normalized duality mapping by J. lim

t→0

The modulus of convexity of E is defined by n o kx + yk δE () = inf 1 − : kxk ≤ 1, kyk ≤ 1, kx − yk ≥  , 2 for every  with 0 ≤  ≤ 2. A Banach space E is said to be uniformly convex if δ() > 0 for every  > 0. E is said to be p-uniformly convex, if there exists a constant a > 0 such that δE () ≥ ap for all 0 <  ≤ 2. Let ρE : [0, ∞) → [0, ∞) be the modulus of smoothness of E defined by 1 ρE (t) = sup{ (kx + yk + kx − yk) − 1 : x ∈ U, kyk ≤ t}. 2 A Banach space E is said to be uniformly smooth if ρEt(t) → 0 as t → 0. Let q be a fixed real number with q > 1. Then a Banach space E is said to be q-uniformly smooth if there exists a constant b > 0 such that ρE (t) ≤ btq for all t > 0. It is well known that every q-uniformly smooth Banach space is uniformly smooth. A Banach space E is said to satisfy the Opial’s condition [10] if for any sequence {xn } ⊂ E, xn * x implies lim sup kxn − xk < lim sup kxn − yk, n→∞

n→∞

for all y ∈ E with y 6= x. Lemma 1.1. [11] Let E be a 2-uniformly convex Banach space. Then the following inequality holds: kλx + (1 − λ)yk2 ≤ λkxk2 + (1 − λ)kyk2 − λ(1 − λ)ckx − yk2 , ∀x, y ∈ E,

(1.5)

where 0 ≤ λ ≤ 1, c = µ(1) > 0, µ(t) :

=

inf

n λkxk2 + (1 − λ)kyk2 − kλx + (1 − λ)yk2 λ(1 − λ)

0 < λ < 1, x, y ∈ E and kx − yk = t

:

o

> 0.

Lemma 1.2. [11] Let E be a 2-uniformly smooth Banach space with the best smoothness constants κ > 0. Then the following inequality holds: kx + yk2 ≤ kxk2 + 2hy, Jyi + 2kκyk2 , for all x, y ∈ E. 3

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2

Main Results

Lemma 2.1. Let E1 be a real 2-uniformly convex and 2-uniformly smooth Banach spaces with the best smoothness constant κ satisfying 0 < κ < √12 , E2 be a real Banach space, and A : E1 → E2 be a bounded linear operator. Let S : E1 → E1 be βdemicontractive and T : E2 → E2 be η-demicontractive with F (S) 6= ∅ and F (T ) 6= ∅. Then the sequence {xn } generated by algorithm (1.2)-(1.3) is F´ ejer-monotone with respect to Γ = {x ∈ F (S)|Ax ∈ F (T )}, that is, for every z ∈ Γ, kxn+1 − zk ≤ kxn − zk, ∀n ∈ N, where 0 < γ < min kAk2 , 1−2κ and αn ∈ (0, 1 − βc ], β < c = µ(1). kAk2  1−η

Proof. Let z ∈ Γ. Then z ∈ F (S) and Az ∈ F (T ). It follows from Lemma 1.1 and (1.3) that kxn+1 − zk2

= k(1 − αn )yn + αn S(yn ) − zk2 = k(1 − αn )(yn − z) + αn (S(yn ) − z)k2 ≤ (1 − αn )kyn − zk2 + αn kS(yn ) − zk2 −αn (1 − αn )ckS(yn ) − yn k2 (2.1) 2 2 2 ≤ (1 − αn )kyn − zk + αn kyn − zk + αn βkS(yn ) − yn k −αn (1 − αn )ckS(yn ) − yn k2 ≤ kyn − zk2 − αn (c − β − cαn )kS(yn ) − yn k2 ,

where c = µ(1). On the other hand, It follows from (1.2) and Lemma 1.2 that kyn − zk2

= kxn + γJ1−1 A∗ J2 (T − I)Axn − zk2 = kxn − z + γJ1−1 A∗ J2 (T − I)Axn k2 ≤ kγJ1−1 A∗ J2 (T − I)Axn k2 + 2γhxn − z, A∗ J2 (T − I)Axn i +2κ2 kxn − zk2 ≤ γ 2 kAk2 k(T − I)Axn k2 + 2γhAxn − Az, J2 (T − I)Axn i +2κ2 kxn − zk2 = γ 2 kAk2 k(T − I)Axn k2 + 2κ2 kxn − zk2 2γhAxn − T Axn + T Axn − Az, J2 (T − I)Axn i ≤ γ 2 kAk2 k(T − I)Axn k2 + 2κ2 kxn − zk2 −2γk(T − I)Axn k2 + 2γhT Axn − Az, J2 (T − I)Axn i ≤ (γ 2 kAk2 − 2γ)k(T − I)Axn k2 + 2κ2 kxn − zk2 +γ(kT Axn − Azk2 + k(T − I)Axn k2 ) ≤ 7(γ 2 kAk2 − 2γ)k(T − I)Axn k2 + 2κ2 kxn − zk2 +γ(kAxn − Azk2 + η|(Axn − T Axn k2 + k(T − I)Axn k2 ) ≤ (2κ2 + γkAk2 )kxn − zk2 − γ(1 − η − γkAk2 )k(T − I)Axn k2 ,

where A∗ is the adjoint operator of A and Ji is the normalized duality mapping from ∗ Ei to 2Ei , i = 1, 2. 2 2 2 In addition, since 0 < κ < √12 and 0 < γ < 1−2κ kAk2 , 0 < γkAk + 2κ < 1, so we have kyn − zk2 ≤ kxn − zk2 − γ(1 − η − γkAk2 )k(T − I)Axn k2 . (2.2) 4

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It follows from (2.1) and (2.2) that kxn+1 − zk2

≤

kxn − zk2 − γ(1 − η − γkAk2 )k(T − I)Axn k2 −αn (c − β − cαn )kS(yn ) − yn k2 .

(2.3)

Finally, by the assumptions on γ and αn , we obtain the desired result. Theorem 2.2. Let E1 be a real 2-uniformly convex and 2-uniformly smooth Banach space satisfying Opial’s condition with the best smoothness constant κ satisfying 0 < κ < √12 , and E2 be a real Banach space. Let A : E1 → E2 be a bounded linear operator, S : E1 → E1 and T : E2 → E2 be two demicontractive mappings with constants β and η with F (S) 6= ∅ and F (T ) 6= ∅, respectively. Assume that I − S and I − T are demiclosed at zero. If Γ 6= ∅, then the sequence {xn } generated by algorithm (1.2)-(1.3) converges weakly to a split common fixed point x ∈ Γ, for 0 < 1−η 1−2κ , αn ∈ (δ, 1 − βc − δ), β < c = µ(1), and for small enough δ > 0. γ < min kAk 2 , kAk2  1−η 1−2κ Proof. From (2.3) and the fact that 0 < γ < min kAk and αn ∈ (δ, 1− βc −δ), 2 , kAk2 we obtain that the sequence {kxn −zk} is monotonically decreasing and thus converges to some positive real limit l(z). From (2.3), we have γ(1 − η − γkAk2 )k(I − T )Axn k2 ≤ kxn − zk2 − kxn+1 − zk2 . Therefore, lim k(I − T )Axn k = 0.

n→∞

(2.4)

From the F´ ejer-monotonicity of {xn }, it follows that the sequence is bounded. Denoting by x a weak-cluster point of {xn }. Let k = 0, 1, 2, ... be the sequence of indices, such that xnk * x, as k → ∞. Then from (2.4) and demiclosedness of I − T at zero, we obtain T (Ax) = Ax, that is, Ax ∈ F (T ). Now, by setting yn = xn + γJ1−1 A∗ J2 (I − T )Axn , it follows that ynk * x. Again from (2.3), we obtain αn (c − β − cαn )kyn − S(yn )k2 ≤ kxn − zk2 − kxn+1 − zk2 . Using the convergence of the sequence {kxn − zk}, we get lim kyn − S(yn )k = 0,

n→∞

(2.5)

which combined with the demiclosedness of I − S at zero and the weak convergence of {ynk } to y yields S(x) = x. Hence, x ∈ F (S) and therefore x ∈ Γ. Since E1 satises Opial’s condition, we know that {xn } converges weakly to x ∈ Γ. Theorem 2.3. Let E1 be a real 2-uniformly convex and 2-uniformly smooth Banach space satisfying Opial’s condition with the best smoothness constant κ satisfying 0 < κ < √12 , and E2 be a real Banach space. Let A : E1 → E2 be a bounded linear operator, S : E1 → E1 and T : E2 → E2 be two demicontractive mappings with constants β and η with F (S) 6= ∅ and F (T ) 6= ∅, respectively. Assume that I − S and I − T are demiclosed at zero. If Γ 6= ∅ and S is semi-compact, then the sequence {xn } generated by algorithm (1.2)-(1.3) converges strongly to a split common fixed 1−η 1−2κ β point x ∈ Γ, for 0 < γ < min{ kAk β < c = µ(1), and 2 , kAk2 }, αn ∈ (δ, 1 − c − δ), for a small enough δ > 0. Proof. It follows from (1.2) that kxn − yn k = kJ1 (xn − yn )k = kγA∗ J2 (I − T )Axn k, 5

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and so, from (2.4) we have lim kxn − yn k = 0.

n→∞

(2.6)

Since S is semi-compact, from (2.5), there exist subsequence {ynj } of {yn } such that {ynj } converges strongly to x∗ ∈ E1 . Using (2.6), we know that {xnj } converges strongly to x∗ . By Theorem 2.2, we know that {xn } converges weakly to x, so we have x∗ = x. Since lim kxn − xk exists and lim kxnj − xk = 0, we know that {xn } n→∞

j→∞

converges strongly to x ∈ Γ. Acknowledgements: The first author was Supported by Scientific Reserch Fund of Sichuan Provincial Education Department (No.15ZA0112) and third author was supported by the Basic Science Research Program through the National Research Foundation(NRF) Grant funded by Ministry of Education of the republic of Korea(2014046293).

References [1] A. Moudafi, The split common fixed point problem for demi-contractive mappings, Inverse Problem, Vol. 26(2010), Article ID 055007, 2010. [2] S.S. Chang, L. Wang, Y.K. Tang and L. Yang, The split common fixed foint problem for total asymptotically strictly pseudocontractive mappings, Journal of Applied Mathematics, Vol. 2012, Article ID 385638, 2012. [3] S.S. Chang, J.K. Kim, Y.J. Cho and J.Y. Sim,, Weak and strong convergence theorems of solution to split feasibility problem for nonspreading type mapping in Hilbert spaces, Fixed Point Theory and Appl., Vol. 2014:11, 2014. [4] X.F. Zhang, L. Wang, Z.L. Ma and W. Duan, The strong convergence theorems for split common fixed point problems of asymptotically nonexpansive mappings in Hilbert spaces, Journal of Inequalities and Appl., Vol. 2015:1, 2015. [5] J. Quan, S.S. Chang and X. Zhang, Multiple-set split feasibility problems for k-strictly pseudononspreading mapping in Hilbert spaces, Abstract and Applied Analysis, Vol. 2013, Article ID 342545, doi:10.1155/ 2013/342545. [6] S.S. Chang, Y.J. Cho, J.K. Kim, W.B. Zhang and L. Yang, Multiple-set split feasibility problems for asymptotically strict pseudocontractions, Abstract and Applied Analysis, Vol. 2012, Article ID 491760, doi:10.1155/2012/491760. [7] A. Moudafi, A note on the split common fixed point problem for quasinonexpansive operators, Nonlinear Anal.TMA, Vol. 74(2011), 4083-4087. [8] W. Takahashi, The split feasibility problem in Banach spaces, Nonlinear Convex Anal., Vol. 15(6)(2015), 1349-1355. [9] S. Maruster and C. Popirlan On the Mann-type iteration and convex feasibility problem J. Comput. Appl. Math., Vol. 212(2008), 390396. [10] Z. Opial, Weak convergence of the sequence of successive approximations for nonexpansive mappings, Bull. Amer. Math. Soc., Vol. 73(1967), 591-597. [11] H.K. Xu, Inequalities in Banach spaces with applications, Nonlinear Anal. TMA, Vol. 16(1991), 1127-1138.

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Iterated binomial transform of the k-Lucas sequence Nazmiye Yilmaz and Necati Taskara Department of Mathematics, Faculty of Science, Selcuk University, Campus, 42075, Konya - Turkey [email protected] and [email protected]

Abstract In this study, we apply ”r” times the binomial transform to k-Lucas sequence. Also, the Binet formula, summation, generating function of this transform are found using recurrence relation. Finally, we give the properties of iterated binomial transform with classical Lucas sequence. Keywords: k-Lucas sequence, iterated binomial transform, Pell sequence. Ams Classification: 11B65, 11B83.

1

Introduction and Preliminaries

There are so many studies in the literature that concern about the special number sequences such as Fibonacci, Lucas and generalized Fibonacci anad Lucas numbers (see, for example [1]-[3], and the references cited therein). In Fibonacci and Lucas numbers, there clearly exists the term Golden ratio which is defined √ as the ratio of two consecutive of these numbers that converges to 1+ 5 α = 2 . It is also clear that the ratio has so many applications in, specially, Physics, Engineering, Architecture, etc.[4]. Also, many generalizations of the Fibonacci sequence have been introduced and studied matrix applications of this sequence in [13]-[16]. For n ≥ 1, k-Lucas sequence is defined by the recursive equation: Lk,n+1 = kLk,n + Lk,n−1 ,

Lk,0 = 2 and Lk,1 = k.

(1.1)

In addition, some matrix-based transforms can be introduced for a given sequence. Binomial transform is one of these transforms and there are also other ones such as rising and falling binomial transforms(see [5]-[12]). Given 1

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an integer sequence X = {x0 , x1 , x2 , . . .}, the binomial transform B of the sequence X, B (X) = {bn } , is given by n   X n bn = xi . i i=0

In [10], authors gave the application of the several class of transforms to the k-Lucas sequence. For example, for n ≥ 1, authors obtained recurrence relation of the binomial transform for k-Lucas sequence bk,n+1 = (2 + k) bk,n − kbk,n−1 , bk,0 = 2 and bk,1 = k + 2. Falcon [11] studied the iterated application of some Binomial transforms to the k-Fibonacci sequence. For example, author obtained recurrence relation of the iterated binomial transform for k-Fibonacci sequence
(r) (r) (r) (r) (r) ck,n+1 = (2r + k) ck,n − r2 + kr − 1 ck,n−1 , ck,0 = 0 and ck,1 = 1. Motivated by [11, 12], the goal of this paper is to apply iteratively the binomial transform to the k-Lucas sequence. Also, the properties of this transform are found by recurrence relation. Finally, the relation of between the transform and the iterated binomial transform of k-Fibonacci sequence by deriving new formulas are illustrated.

2

Iterated Binomial Transform of k-Lucas Sequences

In this section, we will mainly focus on iterated binomial transforms of k Lucas sequences to get some important results. In fact, we will also present the recurrence relation, Binet formula, summation, generating function of the transform and relationships betweeen of the transform and iterated binomial transform of k-Fibonacci sequence. The iterated n obinomial transform of the k-Lucas sequences is demonstrated (r) (r) (r) by Bk = bk,n , where bk,n is obtained by applying ”r” times the binomial (r)

(r)

transform to k-Lucas sequence. It is obvious that bk,0 = 2 and bk,1 = 2r + k. The following lemma will be the key proof of the next theorems. Lemma 2.1 For n ≥ 0 and r ≥ 1, the following equality hold: n   X n (r−1) (r) (r) bk,n+1 = bk,n + b . j k,j+1 j=0

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Proof. By using definition of binomial transform and the well known binomial equality       n+1 n n = + , i i i−1 we obtain (r) bk,n+1

=

n+1 X j=0

=

n+1 X j=1

(r) bk,n+1

=

 n + 1 (r−1) bk,j j  n + 1 (r−1) (r−1) bk,j + bk,0 j

n+1 X j=1

  n+1  n (r−1) X n (r−1) (r−1) b + b + bk,0 j k,j j − 1 k,j j=1

n+1 X

  n+1  n (r−1) X n (r−1) = bk,j + b j j − 1 k,j j=0 j=0 n   n   X X n (r−1) n (r−1) bk,j + b = j k,j+1 j j=−1 j=0 n   X n (r−1) (r) = bk,n + b j k,j+1 j=0

which is desired result. In [10], the authors obtained the following equality for binomial transform of k-Lucas sequences. However, in here, we obtain the equality in terms of iterated binomial transform of the k-Lucas sequences as a consequence of Lemma 2.1. To do that we take r = 1 in Lemma 2.1: n   X n bk,n+1 = bk,n + Lk,j+1 . j j=0

Theorem 2.1 For n ≥ 0 and r ≥ 1, the recurrence relation of sequence n o (r) bk,n is
(r) (r) (r) bk,n+1 = (2r + k) bk,n − r2 + kr − 1 bk,n−1 , (2.1) (r)

(r)

with initial conditions bk,0 = 2 and bk,1 = 2r + k. 3

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Proof. The proof will be done by induction steps on r and n. First of all, for r = 1, from the equality 2.2 in [10], it is true bk,n+1 = (2 + k) bk,n − kbk,n−1 . Let us consider definition of iterated binomial transform, then we have (r)

bk,2 = k 2 + 2rk + 2r2 + 2. The initial conditions are (r)

(r)

bk,0 = 2 and bk,1 = 2r + k.
(r) (r) (r) Hence, for n = 1, the Eq. (2.1) is true, that is bk,2 = (2r + k) bk,1 − r2 + kr − 1 bk,0 . Actually, by assuming the Eq. (2.1) holds for all (r − 1, n) and (r, n − 1), that is,   (r−1) (r−1) (r−1) bk,n+1 = (2r − 2 + k) bk,n − (r − 1)2 + k (r − 1) − 1 bk,n−1 , and
(r) (r) (r) bk,n = (2r + k) bk,n−1 − r2 + kr − 1 bk,n−2 . Now, by taking into account Lemma 2.1, we obtain n   X n (r−1) (r) (r) bk,n+1 = bk,n + b j k,j+1 j=0 n   n   X n (r−1) X n (r−1) = b + b j k,j j k,j+1 j=0 j=0 n    X n (r−1) (r−1) (r−1) (r−1) bk,j + bk,j+1 + bk,0 + bk,1 . = j j=1

By reconsidering our assumption, we write n   X
(r−1)  n (r) (r−1) (r−1) (r−1) (r−1) bk,n+1 = bk,j + (2r − 2 + k) bk,j − r2 − 2r + kr − k bk,j−1 + bk,0 + bk,1 j j=1 n   n   X
X n (r−1) n (r−1) (r−1) (r−1) 2 = (2r + k − 1) bk,j − r − 2r + kr − k bk,j−1 + bk,0 + bk,1 j j j=1 j=1     n n X
X n (r−1) n (r−1) (r−1) (r−1) 2 = (2r + k − 1) b − r − 2r + kr − k b + bk,0 + bk,1 j k,j j k,j−1 j=0

− (2r + k − = (2r + k −

j=1

(r−1) 1) bk,0

(r) 1) bk,n

n  
X n (r−1) (r−1) (r−1) − r − 2r + kr − k bk,j−1 + (2 − 2r − k) bk,0 + bk,1 . j 2

j=1

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Then we have (r) bk,n+1

− (2r + k −

(r) 1) bk,n

n  
X n (r−1) = − r − 2r + kr − k b + 4 − 2r − k. j k,j−1 2

j=1

(2.2) By taking n → n − 1, it is (r) bk,n

X n − 1 (r−1)
n−1 = (2r + k − − r − 2r + kr − k bk,j−1 + 4 − 2r − k j j=1   n  
X n n−1 (r) (r−1) 2 = (2r + k − 1) bk,n−1 − r − 2r + kr − k − bk,j−1 + 4 − 2r − k j j−1 j=1   n
X n (r−1) (r) = (2r + k − 1) bk,n−1 − r2 − 2r + kr − k b j k,j−1 j=1  n 
X n − 1 (r−1) 2 + r − 2r + kr − k b + 4 − 2r − k j − 1 k,j−1 (r) 1) bk,n−1

2

j=1

(r) bk,n

= (2r + k −

(r) 1) bk,n−1

n  
X n (r−1) − r − 2r + kr − k b j k,j−1 2

j=1

n−1 X


n − 1 (r−1) 2 + r − 2r + kr − k bk,j + 4 − 2r − k j j=0 n  
X n (r−1) (r) 2 = (2r + k − 1) bk,n−1 − r − 2r + kr − k b j k,j−1 j=1
(r) + r2 − 2r + kr − k bk,n−1 + 4 − 2r − k n  
(r)
X n (r−1) 2 2 = r + kr − 1 bk,n−1 − r − 2r + kr − k b + 4 − 2r − k. j k,j−1 j=1

Hence, we have (r) bk,n

2

− r + kr − 1




(r) bk,n−1

n  
X n (r−1) b + 4 − 2r − k. = − r − 2r + kr − k j k,j−1 2

j=1

If last expression put in place in the equation (2.2), then we get
(r) (r) (r) (r) bk,n+1 = (2r + k − 1) bk,n + bk,n − r2 + kr − 1 bk,n−1
(r) (r) = (2r + k) bk,n − r2 + kr − 1 bk,n−1 5

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which completed the proof of this theorem. The characteristic equation of sequence

n o (r) bk,n

in

(2.1)

is

λ2 − (2r + k) λ + r2 + kr − 1 = 0. Let and λ2 be the roots of this equation. n λ1 o (r) Then, Binet’s formulas of sequence bk,n can be expressed as (r) bk,n

=

k+

√

k2 + 4 +r 2

!n +

k−

√

k2 + 4 +r 2

!n .

(2.3)

In here, we obtain the equalities given in [10] in terms of iterated binomial transform of the k-Lucas sequences as a consequence of Theorem 2.1. To do that we take r = 1 in Theorem 2.1 and the Eq. (2.3): bk,n+1 = (2 + k) bk,n − kbk,n−1 , and bk,n =

!n √ k + 2 + k2 + 4 + 2

!n √ k + 2 − k2 + 4 . 2

Now, we give the sum of iterated binomial transform for k-Lucas sequences. n o (r) Theorem 2.2 Sum of sequence bk,n is n−1 X

(r) bk,i

=


(r) (r) r2 + kr − 1 bk,n−1 − bk,n − k − 2r + 2 r2 + kr − k − 2r

i=0

.

Proof. By considering Eq. (2.3), we have n−1 X

(r)

bk,i =

i=0

Then we obtain

n−1 X i=0

(r) bk,i


λi1 + λi2 .

i=0

 =

n−1 X

λn1 − 1 λ1 − 1



 +

λn2 − 1 λ2 − 1

 .

Afterward, by taking into account equations λ1 .λ2 = r2 + kr − 1 and λ1 + λ2 = k + 2r, we conclude n−1 X i=0

(r) bk,i

=


(r) (r) r2 + kr − 1 bk,n−1 − bk,n − k − 2r + 2 r2 + kr − k − 2r

.

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Note that, if we take r = 1 in Theorem 2.2, we obtain the summation of binomial transform for k-Lucas sequence: n−1 X

bk,i = bk,n − kbk,n−1 + k

i=0

Theorem 2.3 The generating function of the iterated binomial transform for {Lk,n } is ∞ X 2 − (2r + k) x (r) bk,i xi = . 1 − (2r + k) x + (r2 + kr − 1) x2 i=0

Proof. Assume that b (k, x, r) =

∞ P i=0

(r)

bk,i xi is the generating function of the

iterated binomial transform for {Lk,n }. From Theorem 2.1, we obtain (r)

(r)

b (k, x, r) = bk,0 + bk,1 x +

∞  X


(r)  (r) (2r + k) bk,i−1 − r2 + kr − 1 bk,i−2 xi

i=2 (r)

(r)

(r)

= bk,0 + bk,1 x − (2r + k) bk,0 x + (2r + k) x

∞ X

(r)

bk,i xi

i=0 ∞
X (r) − r2 + kr − 1 x2 bk,i xi i=0

=

 (r) − (2r + k) bk,0 x + (2r + k) xb (k, x, r)
− r2 + kr − 1 x2 b (k, x, r) .

(r) bk,0



+

(r) bk,1

Now rearrangement of the equation implies that   (r) (r) (r) bk,0 + bk,1 − (2r + k) bk,0 x , b (k, x, r) = 1 − (2r + k) x + (r2 + kr − 1) x2 which equals to the

∞ P i=0

(r)

bk,i xi in theorem. Hence, the result.

In here, we obtain the generating function given in [10] in terms of iterated binomial transform of the k-Lucas sequences as a consequence of Theorem 2.3. To do that we take r = 1 in Theorem 2.3: ∞ X i=0

bk,i xi =

2 − (2 + k) x . 1 − (2 + k) x + kx2 7

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In the following theorem, we present the relationship between the iterated binomial transform of k-Lucas sequence and iterated binomial transform of k-Fibonacci sequence. n o (r) Theorem 2.4 For n > 0, the relationship of between the transforms bk,n n o (r) and ck,n is illustrated by following way:
(r) (r) (r) bk,n = ck,n+1 − r2 + kr − 1 ck,n−1 , (r)

(2.4) (r)

where bk,n is the iterated binomial transform of k-Lucas sequence and ck,n is the iterated binomial transform of k-Fibonacci sequence. Proof. By using the Eq.(2.4), let be (r)

(r)

(r)

bk,n = Xck,n+1 + Y ck,n−1 . If we take n = 1 and 2, we have the system ( (r) (r) (r) bk,1 = Xck,2 + Y ck,0 , (r)

(r)

(r)

bk,2 = Xck,3 + Y ck,1 . By considering definition of the iterated binomial transforms for k-Lucas, kFibonacci sequence and Cramer rule for the system, we obtain  2r + k = (2r + k) X,
k 2 + 2rk + 2r2 + 2 = 3r2 + 3rk + k 2 + 1 X + Y and X = 1 and Y = − r2 + kr − 1




which is completed the proof of this theorem. Note that, if we take r = 1 in Theorem 2.4, we obtain the relationship of between the binomial transform for k-Lucas sequence and the binomial transform for k-Fibonacci sequence: bk,n = ck,n+1 − kck,n−1 . Corollary 2.1 We should note that choosing k = 1 in the all results of section 2, it is actually obtained some properties of the iterated binomial transform for classical Lucas sequence such that the recurrence relation, Binet formula, summation, generating function and relationship of between binomial transforms for Fibonacci and Lucas sequences. 8

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Corollary 2.2 We should note that choosing k = 2 in the all results of section 2, it is actually obtained some properties of the iterated binomial transform for classical Pell-Lucas sequence such that the recurrence relation, Binet formula, summation, generating function and relationship of between binomial transforms for Pell and Pell-Lucas sequences. Conclusion 2.1 In this paper, we define the iterated binomial transform for k-Lucas sequence and present some properties of this transform. By the results in Sections 2 of this paper, we have a great opportunity to compare and obtain some new properties over this transform. This is the main aim of this paper. Thus, we extend some recent result in the literature. In the future studies on the iterated binomial transform for number sequences, we expect that the following topics will bring a new insight. (1) It would be interesting to study the iterated binomial transform for Fibonacci and Lucas matrix sequences, (2) Also, it would be interesting to study the iterated binomial transform for Pell and Pell-Lucas matrix sequences. Acknowledgement 2.1 A part of this study presented at Sharjah-BAE ”The Second International Conference on Mathematics and Statistics (AUS-ICMS’15)”. This research is supported by TUBITAK and Selcuk University Scientific Research Project Coordinatorship (BAP).

References [1]

T. KOSHY, Fibonacci and Lucas Numbers with Applications, John Wiley and Sons Inc, NY, 2001.

[2]

S. FALCON and A. PLAZA, On k-Fibonacci numbers of arithmetic indexes, Applied Mathematics and Computation, 208, 2009, 180-185.

[3]

S. FALCON, On the k-Lucas number, International Journal of Contemporary Mathematical Sciences, Vol. 6, no. 21, 2011, 1039 - 1050.

[4]

L. MAREK-CRNJAC, The mass spectrum of high energy elementary particles via El Naschie’s golden mean nested oscillators, the DunkerlySouthwell eigenvalue theorems and KAM, Chaos, Solutions & Fractals, 18(1), 2003, 125-133.

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[5]

˙ H. PRODINGER, Some information about the binomial transform, The Fibonacci Quarterly, 32 (5), 1994, 412-415.

[6]

K.W. CHEN, Identities from the binomial transform, Journal of Number Theory, 124, 2007, 142-150.

[7]

S. FALCON and A. PLAZA, Binomial Transforms of k-Fibonacci Sequence, International Journal of Nonlinear Sciences and Numerical Simulation, 10(11-12), 2009, 1527-1538.

[8]

N. YILMAZ and N. TASKARA, Binomial Transforms of the Padovan and Perrin Matrix Sequences, Abstract and Applied Analysis, Article Number: 497418 Published: 2013.

[9]

Y. YAZLIK, N. YILMAZ, N. TASKARA, The Generalized (s, t)-matrix Sequence’s Binomial Transforms, Gen. Math. Notes, Vol. 24, No. 1, 2014, pp.127-136.

[10] P. BHADOURIA, D. JHALA, B. SINGH, Binomial Transforms of the kLucas Sequences and its Properties, Journal of mathematics and computer science, 8, 2014, 81-92. [11] S. FALCON, Iterated Binomial Transforms of the k-Fibonacci Sequence, British Journal of Mathematics & Computer Science, 4(22), 2014, 31353145. [12] N. YILMAZ, N. TASKARA, On the Properties of Iterated Binomial Transforms for the Padovan and Perrin Matrix Sequences, Mediterranean Journal of Mathematics, DOI 10.1007/s00009-015-0612-5, 2015. [13] Y. YAZLIK and N. TASKARA, A note on generalized k-Horadam sequence, Computers & Mathematics with Applications, 63, 2012, 36-41. [14] Y. YAZLIK and N. TASKARA, On the Inverse of Circulant Matrix via Generalized k-Horadam Numbers, Applied Mathematics and Computation, 223, 2013, 191-196. [15] Y. YAZLIK and N. TASKARA, Spectral norm, Eigenvalues and Determinant of Circulant Matrix involving the Generalized k-Horadam numbers, Ars Combinatoria, 104, 2012, 505-512. [16] Y. YAZLIK and N. TASKARA, On the norms of an r-circulant matrix with the generalized k-Horadam numbers, Journal of Inequalities and Applications, 2013.1, 2013, 1-8.

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Nielsen xed point theory for digital images 1

Ozgur EGE ,

1

Ismet KARACA

2

Celal Bayar University, Department of Mathematics, 45140 Manisa, Turkey E-mail: [email protected]

2

Ege University, Department of Mathematics, 35100 Izmir, Turkey E-mail: [email protected] September 1, 2015

Abstract In this paper, we introduce the Nielsen xed point theory in digital images. We also deal with some important properties of the Nielsen number and calculate the Nielsen number of some digital images. We get some new results using digital covering maps and Nielsen number. Keywords: Fixed point, Nielsen number, digital homotopy.

2010 Mathematics Subject Classication.

1

47H10, 54H25, 68U10.

Introduction

Digital topology is often used in computer graphics, pattern recognition and image processing. This topic has been studied by important researchers such as Rosenfeld, Kong, Kopperman, Boxer, Karaca, Han, etc. Their goal is to determine not only similarities but also dierences between digital images and topology. Fixed point theory with applications is an important area in topology. This theory continues to develop with new computations and come out of new invariants. Nielsen xed point theorem is a notable theorem in this theory because it gives a way to count xed points. One of the main goals in digital topology is to classify digital images. For this reason, we use the Nielsen number which is a powerful invariant for digital images. In 1920s, Jakop Nielsen introduced the Nielsen theory and the Nielsen number. He focused on both the existence problem of xed points and the problem of determining the minimal number of xed points in the homotopy classes. He did this by introducing the Nielsen number of a self map. This number is a homotopy invariant lower bound for the number of xed points of the map. In this area, there are signicant works [10, 11, 12, 16, 17, 18, 19]. Boxer [6] introduces the digital covering space and showed that the existence of digital universal covering spaces. Boxer and Karaca [7] classify digital covering spaces using the conjugacy class corresponding to a digital covering space. Boxer and Karaca [8] study digital versions of some properties of covering spaces from algebraic topology. Karaca and Ege [20] get some results related to the simplicial homology groups of 2D digital images. Ege and Karaca [13] give characteristic properties of the simplicial homology groups. This paper is organized as follows. The second section provides the general notions of digital images, digital homotopy, digital covering spaces and digital homology groups. In Section 3 we present the Nielsen xed point theorem for digital images, give some examples and properties. In Section 4 we discuss about the relation between Nielsen theory and digital universal covering space. We nally make some conclusions about this topic.

2

Preliminaries

A digital image consists of a pair (X, κ), where Z is the set of integers, X ⊂ Zn for some positive integer n, and κ indicates an adjacency relation for the members of X .

Denition 2.1. [3]. For a positive integer l with 1 ≤ l ≤ n and two distinct points p = (p1 , p2 , . . . , pn ), q = (q1 , q2 , . . . , qn ) ∈ Zn , p and q are cl -adjacent, if (1) there are at most l indices i such that |pi − qi | = 1, and (2) for all other indices j such that |pj − qj | 6= 1, pj = qj .

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The notation cl represents the number of points q ∈ Zn that are adjacent to a given point p ∈ Zn . Thus, in Z, we have c1 = 2-adjacency; in Z2 , we have c1 = 4-adjacency and c2 = 8-adjacency; in Z3 , we have c1 = 6-adjacency, c2 = 18-adjacency, and c3 = 26-adjacency [5]. A κ-neighbor of p ∈ Zn [3] is a point of Zn that is κ-adjacent to p. A digital interval [2] is dened by [a, b]Z = {z ∈ Z | a ≤ z ≤ b} where a, b ∈ Z and a < b. A digital image X ⊂ Zn is κ-connected [15] if and only if for every pair of dierent points x, y ∈ X , there is a set {x0 , x1 , . . . , xr } of points of a digital image X such that x = x0 , y = xr and xi and xi+1 are κ-neighbors where i = 0, 1, . . . , r − 1.

Denition 2.2.

[3]. Let X ⊂ Zn0 and Y ⊂ Zn1 be digital images with κ0 -adjacency and κ1 -adjacency, respectively. A function f : X −→ Y is said to be (κ0 , κ1 )-continuous if for every κ0 -connected subset U of X , f (U ) is a κ1 -connected subset of Y . We say that such a function is digitally continuous.

Proposition 2.3. [3]. Let X ⊂ Zn0 and Y

⊂ Zn1 be digital images with κ0 -adjacency and κ1 -adjacency, respectively. The function f : X → Y is (κ0 , κ1 )-continuous if and only if for every κ0 -adjacent points {x0 , x1 } of X , either f (x0 ) = f (x1 ) or f (x0 ) and f (x1 ) are κ1 -adjacent in Y .

In a digital image X , if there is a (2, κ)-continuous function f : [0, m]Z → X such that f (0) = x and f (m) = y , then there exists a digital κ-path [6] from x to y . If f (0) = f (m), then we say that f is digital κ-loop and the point f (0) is the base point of the loop f . When a digital loop f is a constant function, it is said to be a trivial loop.

Denition 2.4.

Let (X, κ0 ) ⊂ Zn0 and (Y, κ1 ) ⊂ Zn1 be digital images. A function f : X → Y is called a (κ0 , κ1 )isomorphism [2] if f is (κ0 , κ1 )-continuous and bijective and f −1 : Y → X is (κ1 , κ0 )-continuous.

Denition 2.5.

[3]. Let (X, κ0 ) ⊂ Zn0 and (Y, κ1 ) ⊂ Zn1 be digital images. We say that two (κ0 , κ1 )-continuous functions f, g : X → Y are digitally (κ0 , κ1 )-homotopic in Y if there is a positive integer m and a function H : X × [0, m]Z → Y such that • for all x ∈ X , H(x, 0) = f (x) and H(x, m) = g(x); • for all x ∈ X , the induced function Hx : [0, m]Z → Y dened by for all t ∈ [0, m]Z ,

Hx (t) = H(x, t)

is (2, κ1 )-continuous; and • for all t ∈ [0, m]Z , the induced function Ht : X → Y dened by for all x ∈ X,

Ht (x) = H(x, t)

is (κ0 , κ1 )-continuous. The function H is called a digital (κ0 , κ1 )-homotopy between f and g . If these functions are digitally (κ0 , κ1 )-homotopic, it is denoted f '(κ0 ,κ1 ) g . The digital (κ0 , κ1 )-homotopy relation [3] is equivalence among digitally continuous functions f : (X, κ0 ) → (Y, κ1 ). If f : [0, m1 ]Z → X and g : [0, m2 ]Z → X are digital κ-paths with f (m1 ) = g(0), then dene the product (f ∗ g) : [0, m1 + m2 ]Z → X [3] by  f (t), t ∈ [0, m1 ]Z (f ∗ g)(t) = g(t − m1 ), t ∈ [m1 , m1 + m2 ]Z . 0

0

Let f and f be κ-loops in a digital image (X, x0 ). We say f is a trivial extension of f [3] if there are sets of κ-paths {f1 , . . . , fr } and {F1 , . . . , Fp } in X such that: (1) r ≤ p, (2) f = f1 ∗ . . . ∗ fr , 0 (3) f = F1 ∗ . . . ∗ Fp , (4) There are indices 1 ≤ i1 < i2 < . . . < ir ≤ p such that Fij = fj , 1 ≤ j ≤ r and i 6= {i1 , . . . , ir } implies Fi is a trivial loop. If f, g : [0, m]Z → X are κ-paths such that f (0) = g(0) and f (m) = g(m), then a homotopy H : [0, m]Z × [0, M ]Z → X

between f and g such that for all t ∈ [0, M ]Z , H(0, t) = f (0) and H(m, t) = f (m), holds the endpoints xed. Two loops f, f0 with the same base point x0 ∈ X belong to the same loop class [f ]X if they have trivial extensions that can be joined by a homotopy that holds the endpoints xed (see [4]). Let (E, κ) be a digital image and let ε be a positive integer. The κ-neighborhood of e0 ∈ E with radius ε is the set Nκ (e0 , ε) = {e ∈ E | lκ (e0 , e) ≤ ε} ∪ {e0 },

where lκ (e0 , e) is the length of a shortest κ-path from e0 to e in E (see [14]).

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Denition 2.6.

[6]. Let (E, κ0 ) and (B, κ1 ) be digital images. A map p : E → B is called a (κ0 , κ1 )-covering map if the followings are true: 1. p is a (κ0 , κ1 )-continuous surjection. 2. for each b ∈ B , there exists an indexing set M such that p−1 (b) can be indexed as p−1 (b) = {ei |i ∈ M } and the following conditions hold: S • p−1 (Nκ1 (b, 1)) = i∈M Nκ0 (ei , 1), • if i, j ∈ M , i 6= j , then Nκ0 (ei , 1) ∩ Nκ0 (ej , 1) = ∅, • the restriction map p|Nκ0 (ei ,1) : Nκ0 (ei , 1) → Nκ1 (b, 1) is a (κ0 , κ1 )-isomorphism for all i ∈ M . Let (E, κ0 ), (B, κ1 ) and (X, κ2 ) be digital images, let p : E → B be a (κ0 , κ1 )-covering map, and f : X → B be (κ2 , κ1 )-continuous. A lifting of f with respect to p is a (κ2 , κ0 )-continuous function f˜ : X → E such that p ◦ f˜ = f (see [14]).

Denition 2.7.

[6]. Let (E0 , p0 , B) be a (κ0 , κB )-covering. Suppose C is a set of (κE , κB )-coverings of B such that for every (E, p, B) ∈ C , there is a (κ0 , κE )-covering (E0 , pE , E). Then the pair (E0 , p0 ) is a universal covering space of B for the set C .

Denition 2.8.

[21]. Let S be a set of nonempty subset of a digital image (X, κ). Then the members of S are called

simplexes of (X, κ), if the followings hold:

• If p and q are distinct points of s ∈ S , then p and q are κ-adjacent, • If s ∈ S and ∅ = 6 t ⊂ s, then t ∈ S . 0

An m-simplex is a simplex S such that |S| = m + 1. For a digital m-simplex P , if P is a nonempty proper subset of 0 P , then P is called a face of P .

Denition 2.9. [1]. Let (X, κ) be a nite collection of digital m-simplices, 0 ≤ m ≤ d for some non-negative integer d. If the followings hold, then (X, κ) is called a nite digital simplicial complex : • If P belongs to X , then every face of P also belongs to X . • If P, Q ∈ X , then P ∩ Q is either empty or a common face of P and Q. Denition 2.10.

[1]. Let (X, κ) ⊂ Zn be a digital oriented simplicial complex with m-dimension. Cqκ (X) is a free abelian κ group with basis all digital (κ, q)-simplices in X . A homomorphism ∂q : Cqκ (X) −→ Cq−1 (X) called the boundary operator. If σ = [v0 , . . . , vq ] is an oriented simplex with 0 < q ≤ m, ∂q is dened by ∂q σ = ∂q [v0 , . . . , vq ] =

q X

(−1)i [v0 , . . . , vˆi , . . . , vq ]

i=0

where vˆi means the vertex vi is to be deleted from the array. We remark that for q < 0, m < q , since Cqκ (X) is the trivial group, the operator ∂q is the trivial homomorphism for q ≤ 0, m < q . We notice that ∂q−1 ◦ ∂q = 0 [1] for q ≥ 0.

Denition 2.11. • • •

[1]. Let (X, κ) ⊂ Zn be a digital oriented simplicial complex with m-dimension.

Zqκ (X) = Ker ∂q is called the group of digital simplicial q -cycles. Bqκ (X) = Im ∂q+1 is called the group of digital simplicial q -boundaries. Hqκ (X) = Zqκ (X)/Bqκ (X) is called the q th digital simplicial homology group.

3

Nielsen Theory for Digital Images

Let (X, κ) be a digital image and let f : X → X be a digital map. The xed point set of f is F ix(f ) = {x ∈ X : f (x) = x}. The main object of study in topological xed point theory is the minimum number of xed points which is denoted by M [f ] among all digital maps (κ, κ)-homotopic to f . For example, M [f ] = 0 means that there is a digital map g which is (κ, κ)-homotopic to f such that g(x) 6= x for all x ∈ X . To calculate M [f ] we have to examine the xed point sets of every map homotopic to f . In the xed point theory, it is made use of a homotopy invariant, called the Nielsen number of f . Its computation requires only a knowledge of the map f itself.

Denition 3.1.

Let f : (X, κ1 ) −→ (Y, κ2 ) be a (κ1 , κ2 )-continuous map where (X, κ1 ) and (Y, κ2 ) are digital images. Then f induces homomorphisms f∗ : H∗κ1 (X) −→ H∗κ2 (Y ) and f∗ can be thought of as a homomorphisms of the integers. The integer deg(f ) to which the number 1 gets sent is called the degree of the map f .

Denition 3.2.

Let (X, κ) be a digital image, A ⊂ X and f : A → X a digital map. We dene the xed point index of f as ind(f ) = deg(F ) where F (x) = x − f (x) and x ∈ X .

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Some properties of xed point index can be given as follows. We don't prove these because they are proved similarly in Algebraic Topology. 1. (Homotopy invariance) Let A ⊂ X × [0, m]Z be digital image with κ-adjacency and F : A → X be a digital map such that F ix(F ) = {(x, t) ∈ A : F (x, t) = x}. Then ind(f0 ) = ind(fm ), where ft = F (−, t) for 0 ≤ t < m and a positive integer m. 2. (Commutativity) Let (X, κ1 ) and (Y, κ2 ) be digital images and let f : A → Y and g : B → X be digital (κ1 , κ2 )continuous maps, respectively, where A ⊂ X , B ⊂ Y . Then F ix(gf ) = F ix(f g) and ind(f g) = ind(gf ). Now we dene Nielsen number for digital images.

Denition 3.3.

Let (X, κ) be a digital image and f : (X, κ) → (X, κ) a self-map. Two xed points x, y ∈ F ix(f ) are Nielsen related if and only if there is a κ-path c : [0, m]Z → X satisfying c(0) = x, c(m) = y and the κ-paths c, f ◦ c are xed end point homotopic, i.e. there is a digital map H : [0, n]Z × [0, m]Z → X satisfying H(t, 0) = c(t), H(t, m) = f ◦ c(t), H(0, s) = x, H(n, s) = y.

This is an equivalence relation, hence F ix(f ) splits into disjoint Nielsen classes. A xed point class F is essential if its index is nonzero. The number of essential xed point classes is called the Nielsen number of f , denoted N (f ). We give some characteristic examples about the Nielsen number.

Example 3.4.

Let (X, κ) be a digital image. If f : X → X is a constant digital map, then N (f ) = 1.

Since the boundary Bd(I n+1 ) of an (n+1)-cube I n+1 is homeomorphic to n-sphere S n , we can represent a digital sphere by using the boundary of a digital cube. Boxer [5] denes sphere-like digital image as Sn = [−1, 1]n+1 \ {0n+1 } ⊂ Zn+1 , Z n where 0n denotes the origin of Z .

Example 3.5. S1 = {c0 = (1, 0), c1 = (1, 1), c2 = (0, 1), c3 = (−1, 1), c4 = (−1, 0), c5 = (−1, −1), c6 = (0, −1), c7 = (1, −1)} is digital 1-sphere with 4-adjacency in Z2 . Let f : (S1 , 4) → (S1 , 4) be a digital map of degree 1. Then f can be considered as identity map and is (4, 4)-homotopic to a xed point free map. Thus we have N (f ) = 0. Let's give some important properties of Nielsen number for digital images.

Theorem 3.6. Let (X, κ) be any digital image. If f

'(κ,κ) g : X → X , then N (f ) = N (g).

Proof. We must show that there is a bijection between sets of essential classes of f and g . Let H(t, s) be a digital (κ, κ)0

homotopy from f and g . For every Nielsen class A ⊂ F ix(f ), there is one A ⊂ F ix(H) containing A. Let 0

B = {x ∈ X | (x, m) ∈ A }.

So B is a Nielsen class of g or is empty. From homotopy invariance index property, we have ind(f, A) = ind(g, B). If A is essential, then B is essential. As a result, we nd a map from the set of essential classes of f to the set of essential classes of g . On the other hand, H(x, m − t) gives the inverse map. Consequently, we get N (f ) = N (g).

Theorem 3.7. Let (X, κ) be a digital image and f to f has at least N (f ) xed points.

: X → X be a digital map. Any digital map g digital (κ, κ)-homotopic

Proof. Using Theorem 3.6, we have N (f ) = N (g). Since each essential Nielsen class of g is nonempty, we get M [g] ≥ N (g) where

M [g] = min{#F ixg | g '(κ,κ) f : X → X}.

Theorem 3.8. Let (X, κ) and (Y, κ ) be any digital images, f 0

N (f ◦ g).

: X → Y and g : Y → X be digital maps. Then N (g ◦ f ) =

Proof. For digital maps f and g , if we use commutativity property of xed point index, i.e. F ix(f ◦ g) = F ix(g ◦ f ) and ind(f ◦ g) = ind(g ◦ f ), we have a bijection which preserves index between the sets of essential Nielsen classes. As a result, we have N (g ◦ f ) = N (f ◦ g).

Lemma 3.9. Let A ⊂ X be digital image with κ-adjacency and f : X → X be digital map such that f (X) ⊂ A, where X is any digital image with κ-adjacency. If fA : A → A is the restriction of f , then N (fA ) = N (f ).

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Proof. Let i : A → X be inclusion map. Assume that g : X → A is given by g(x) = f (x). Using Theorem 3.8, we conclude N (f ) = N (i ◦ g) = N (g ◦ i) = N (fA )

because i ◦ g = f and g ◦ i = fA .

Theorem 3.10. Let

0

0

(X, κ) and (Y, κ ) be any two digital images. Suppose that h : X → Y is a digital (κ, κ )-homotopy

equivalence and let the diagram

X h

f



Y

/X h



g

/Y

0

be digital (κ, κ )-homotopy commutative, i.e. h ◦ f '(κ,κ0 ) g ◦ h. Then N (f ) = N (g). 0

Proof. Assume that the digital (κ , κ)-homotopy inverse of h is m : Y → X . Then m ◦ h '(κ,κ) 1X and h ◦ m '(κ0 ,κ0 ) 1Y .

By Theorem 3.6 and Theorem 3.8, we have

N (f ) = N (f (mh)) = N ((f m)h) = N (h(f m)) = N ((hf )m) = N ((gh)m) = N (g(hm)) = N (g).

Theorem 3.11. Let (X, κ) be a digital image and f : (X, κ) → (X, κ) be a digital map. N (f ) is a lower bound for the number of xed points in the homotopy class of f , i.e. 0 ≤ N (f ) ≤ M [f ] := min{#F ixg | g '(κ,κ) f : X → X}.

Proof. By the denition of Nielsen number, we have N (f ) ≥ 0. On the other hand, since we know that each Nielsen class contains at least one xed point of f , we conclude that 0 ≤ N (f ) ≤ M [f ].

4

Nielsen Theory and Digital Universal Covering Spaces

Since there is a connection between the digital fundamental group and the digital universal covering of a space, same results can be also obtained by the lifts of the considered maps to the digital universal coverings. Let (X, κ) be a digital image and ˜ →X ˜ be the digital universal covering of X , with group π of covering transformations. Let f˜ : X ˜ →X ˜ be a lifting of p:X f , i.e., have a commutative diagram ˜ X

f˜

˜ /X

p

p



X



f

/X

0 0 If f˜ is another lifting of f˜, then f˜ = α ◦ f˜ for some α ∈ π . The set of all liftings of f is {α ◦ f˜ | α ∈ π}. 0 0 For any α ∈ π , f˜ ◦ α is a lifting of f and so we have α ◦ f˜ = f˜ ◦ α for some α ∈ π . This denes a homomorphism 0 ϕ : π → π given by ϕ(α) = α . Dene the Reidemeister action of π on π as follows:

π×π →π (γ, α) 7→ γαϕ(γ)−1 ,

where γ, α ∈ π . This denes an equivalence relation. We say that the Reidemeister classes of its equivalence classes. The set of the Reidemeister classes determined by ϕ is denoted by R[ϕ] = {[α] | α ∈ π}.

Theorem 4.1. Let (X, κ) be a digital image, f

˜ →X ˜ be a lifting of f . Then [α] = [α0 ] : X → X be a digital map and f˜ : X 0 if and only if p(F ix(α ◦ f˜)) = p(F ix(α ◦ f˜)), where p : X˜ → X is a digital covering map of f and α, α ∈ π . 0

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Proof. For necessary condition, since the xed point sets of any two digital homotopic maps are same, we conclude that 0

[α] = [α ] ⇒ α '(˜κ,˜κ) α

0 0

⇒ α ◦ f˜ '(˜κ,˜κ) α ◦ f˜ 0

⇒ F ix(α ◦ f˜) = F ix(α ◦ f˜) 0 ⇒ p(F ix(α ◦ f˜)) = p(F ix(α ◦ f˜)). 0

For sucient condition, let a ∈ p(F ix(α ◦ f˜)) = p(F ix(α ◦ f˜)). Then we have p(˜ a) = a and p(a˜0 ) = a. Moreover, we get α ◦ f˜(˜ a) = a ˜ = 1X˜ (˜ a)

Finally, we can say

0

α(f˜(˜ a)) = α (f˜(˜ a))

0 α ◦ f˜(˜ a) = a˜0 = 1X˜ (a˜0 )

and ⇒

α=α

0

0

⇒

α '(˜κ,˜κ) α ,

˜ . As a result, we have [α] = [α0 ]. where a ˜ is any point of X

Corollary 4.2. If p(F ix(αf˜)) is any xed point class, then F ix(f ) =

a

p(F ix(αf˜)),

[α]∈R[ϕ]

where [α] is a Reidemeister class.

Lemma 4.3. Let

˜ κ ˜ →X ˜ be any lifting of f . Then (X, κ), (X, ˜ ) be digital images, f : X → X be a digital map and f˜ : X ˜ any two points in p(F ix(f )) ⊂ F ix(f ) are Nielsen related. We have also [ F ix(f ) = p(F ix(f˜0 )) ˜0 f

where f˜0 is a lifting of f . Proof. Let a and b be any two points in p(F ix(f˜)). We say that p(˜ a) = a ˜,

p(˜b) = ˜b,

f˜(˜ a) = a ˜,

f˜(˜b) = ˜b,

˜ by θ˜. There is a digital homotopy H ˜ such that for some a ˜, ˜b ∈ F ix(f˜). We denote a κ ˜ -path from a ˜ to ˜b in X ˜ :X ˜ × [0, m]Z → X ˜ H ˜ x, 0) = θ˜ and H(˜ ˜ x, m) = f˜ ◦ θ˜ H(˜ ˜ is κ ˜ = H : X × [0, m0 ]Z → X is a digital homotopy between two because X ˜ -connected digital image. Then we have p ◦ H κ-paths ˜ θ = p ◦ θ˜ and f ◦ θ = p ◦ (f˜ ◦ θ)

which join two points a, b ∈ F ix(f ). As a result, a and b are Nielsen related points. Now we prove the latter statement. Let u ∈ F ix(f ) and u ˜ ∈ p−1 (u). Then f (u) = u and p−1 (u) = u ˜. Moreover, 0 0 f˜ (˜ u) = u ˜ because f˜ is a lifting of f . We can say the following result. f˜0 (˜ u) = u ˜ ⇒ p ◦ f˜0 (˜ u) = p(˜ u) = u [ Consequently, we have F ix(f ) = p(F ix(f˜0 )).

⇒

u ∈ p(F ix(f˜0 )).

˜0 f

˜ → X be a digital universal covering map. Let Let p : X ˜ →X ˜ : p ◦ α = p} OX = {α ∈ X

denote the group of deck transformations of this digital covering map.

Lemma 4.4. Let C be the set of liftings of f and f˜, f˜0

that α ◦ f˜ = f˜0 ◦ α.

∈ C . If p(F ix(f˜)) = p(F ix(f˜0 )) 6= ∅, then there is an α ∈ OX such

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Proof. If p(F ix(f˜)) = p(F ix(f˜0 )) 6= ∅, then there are two points x˜, x˜0 such that p(˜x) = p(x˜0 ) where x ˜ ∈ F ix(f˜) ⇒ f˜(˜ x) = x ˜

and

x˜0 ∈ F ix(f˜0 ) ⇒ f˜0 (x˜0 ) = x˜0 .

Since α ∈ OX , i.e. p ◦ α = p, we have α(˜ x) = x˜0 . We conclude that f˜0 ◦ α(˜ x) = f˜0 (x˜0 ) = x˜0 = α(˜ x) = α(f˜(˜ x)) = α ◦ f˜(˜ x).

As a result, we have αf˜ = f˜0 α.

Lemma 4.5. Let f˜0

= αf˜α−1 for an α ∈ OX . Then p(F ix(f˜)) = p(F ix(f˜0 )). Proof. By assumption, we have p(α(˜x)) = p(˜x). If u ∈ p(F ix(f˜)), then p(˜x) = u and f˜(˜x) = x˜. Since p ◦ f˜(˜ x) = u

⇒

p ◦ α−1 ◦ f˜0 ◦ α(˜ x) = p ◦ f˜0 (x˜0 ) = u,

we have u ∈ p(F ix(f˜0 )). As a result, p(F ix(f˜)) = p(F ix(f˜0 )).

5

Conclusion

The essential aim of this paper is to determine xed point properties for a digital image. This work can play an important role in digital images because Nielsen theory gives an information about the number of xed points of a map. Since the Nielsen number is a powerful invariant in digital images, we think that this work will be useful for xed point theory, especially Nielsen theory.

References [1] H. Arslan, I. Karaca and A. Oztel, Homology groups of n-dimensional digital images, XXI. Turkish National Mathematics Symposium, B, 113 (2008). [2] L. Boxer, Digitally continuous functions, Pattern Recognition Letters, 15, 833839 (1994). [3] L. Boxer, A classical construction for the digital fundamental group, Journal of Mathematical Imaging and Vision, 10, 5162 (1999). [4] L. Boxer, Properties of digital homotopy, Journal of Mathematical Imaging and Vision, 22, 1926 (2005). [5] L. Boxer, Homotopy properties of sphere-like digital images, Journal of Mathematical Imaging and Vision, 24, 167175 (2006). [6] L. Boxer, Digital products, wedges and covering spaces, Journal of Mathematical Imaging and Vision, 25, 159171 (2006). [7] L. Boxer and I. Karaca, The classication of digital covering spaces, Journal of Mathematical Imaging and Vision, 32, 2329 (2008). [8] L. Boxer and I. Karaca, Some properties of digital covering spaces, Journal of Mathematical Imaging and Vision, 37, 1726 (2010). [9] L. Boxer, I. Karaca, A. Oztel, Topological invariants in digital images, Journal of Mathematical Sciences: Advances and Applications, 11(2), 109140 (2011). [10] R.F. Brown, Retraction methods in Nielsen xed point theory, Pacic Journal of Mathematics, 115(2), 277297 (1984). [11] R.F. Brown, Nielsen xed point theory on manifolds, Banach Center Publications, 49(1), 1927 (1999). [12] R.F. Brown, M. Furi, L. Gorniewicz and B. Jiang, Handbook of Topological Fixed Point Theory, Springer (2005). [13] O. Ege and I. Karaca, Fundamental properties of simplicial homology groups for digital images, American Journal of Computer Technology and Application, 1(2), 2542 (2013). [14] S.E. Han, Non-product property of the digital fundamental group, Information Sciences, 171, 7391 (2005). [15] G.T. Herman, Oriented surfaces in digital spaces, CVGIP:Graphical Models and Image Processing, 55, 381396 (1993). [16] J. Jezierski, The Nielsen number product formula for coincidence, Fund. Math., 134, 183212 (1989). [17] J. Jezierski and W. Marzantowicz, Homotopy Methods in Topological Fixed and Periodic Points Theory, Springer (2006). [18] B. Jiang, Estimation of the Nielsen numbers, Chinese Math., 5, 330339 (1964). [19] B. Jiang, Lectures on Nielsen xed point theory, Contemp. Math., 14 (1983). [20] I. Karaca and O. Ege, Some results on simplicial homology groups of 2D digital images, International Journal of Information and Computer Science, 1(8), 198203 (2012). [21] E. Spanier, Algebraic Topology, McGraw-Hill, New York (1966).

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A FIXED POINT THEOREM AND STABILITY OF ADDITIVE-CUBIC FUNCTIONAL EQUATIONS IN MODULAR SPACES CHANG IL KIM, GILJUN HAN∗ , AND SEONG-A SHIM

Abstract. In this paper, we investigate a fixed point theorem for a mapping without the condition of bounded orbit in a modular space, whose induced modular is lower semi-continunous. Using this fixed point theorem, we prove the generalized Hyers-Ulam stability for an additive-cubic functional equation in modular spaces without 42 -conditions and the convexity.

1. Introduction and preliminaries The question of stability for a generic functional equation was originated in 1940 by Ulam [14]. Concerning a group homomorphism, Ulam posted the question asking how likely to an automorphism a function should behave in order to guarantee the existence of an automorphism near such functions. Hyers [3] gave the first affirmative partial answer to the question of Ulam for Banach spaces. Hyers’ theorem was generalized by Aoki [1] for additive mappings and by Rassias [12] for linear mappings by considering an unbounded Cauchy difference, the latter of which has influenced many developments in the stability theory. This area is then referred to as the generalized Hyers-Ulam stability. In 1994, P. Gˇavruta [2] generalized these theorems for approximate additive mappings controlled by the unbounded Cauchy difference with regular conditions. A problem that mathematicians has dealt with is ”how to generalize the classical function space Lp ”. A first attempt was made by Birnhaum and Orlicz in 1931. This generalization found many applications in differential and intergral equations with kernls of nonpower types. The more abstract generalization was given by Nakano [10] in 1950 based on replacing the particular integral form of the functional by an abstract one that satisfies some good properties. This functional was called modular. This idea was refined, generalized by Musielak and Orlicz [8] in 1959 and studied by many authors ([4], [7], [11], [19]). Recently, Sadeghi [13] presented a fixed point method to prove the generalized Hyers-Ulam stability of functional equations in modular spaces with the 42 condition and Wongkum, Chaipunya, and Kumam [15] proved the fixed point theorem and the generalized Hyers-Ulam stability for quadratic mappings in a modular space whose modular is convex, lower semi-continuous but do not satisfy the 42 condition. 2010 Mathematics Subject Classification. 39B52, 39B72, 47H09. Key words and phrases. fixed point theorem, stability, additive-cubic functional equation, modular space. * Corresponding author. 1

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In this paper, we investigate a fixed point theorem in modular spaces, whose induced modular is lower semi-continuous, for a mapping with some conditions in place of the condition of bounded orbit and using this fixed point theorem, we will prove the generalized Hyers-Ulam stability for the following additive-cubic functional equation (1.1)

f (2x + y) + f (2x − y) − 2f (x + y) − 2f (x − y) − 2f (2x) + 4f (x) = 0

in modular spaces without 42 -conditions and the convexity. In fact, the equation (1.1) has been studied in various spaces. For example, in quasi-Banach spaces ([9]), in F-spaces ([16]), in non-Archimedean fuzzy normed spaces ([17]), and in intuitionistic fuzzy normed spaces ([18]), etc. Unlike Banach spaces and F-spaces, due to the absence of the triangle inequality in modular spaces, we need subtle caculations in the proofs of Lemma 1.4 and Theorem 2.2. Definition 1.1. Let X be a vector space over a field K(R or C). (1) A generalized functional ρ : X −→ [0, ∞] is called a modular if (M1) ρ(x) = 0 if and only if x = 0, (M2) ρ(αx) = ρ(x) for every scalar α with |α| = 1, and (M3) ρ(αx + βy) ≤ ρ(x) + ρ(y) for all x, y ∈ X and for all nonnegative real numbers α, β with α + β = 1. (2) If (M3) is replaced by (M4) ρ(αx + βy) ≤ αρ(x) + βρ(y) for all x, y ∈ X and for all nonnegative real numbers α, β with α + β = 1, then we say that ρ is a convex modular. Remark 1.2. Let ρ be a modular on a vector space X. Then by (M1) and (M3), we can easily show that for any positive real number δ with δ < 1, ρ(δx) ≤ ρ(x) for all x ∈ X. For any modular ρ on X, the modular space Xρ is defined by Xρ := {x ∈ X | ρ(λx) → 0 as λ → 0}. Let Xρ be a modular space and let {xn } be a sequence in Xρ . Then (i) {xn } is called ρ-convergent to a point x ∈ Xρ if ρ(xn − x) → 0 as n → ∞, (ii) {xn } is called ρ-Cauchy if for any  > 0, there is a k ∈ N such that ρ(xn − xm ) <  for all m, n ∈ N with n, m ≥ k, and (iii) a subset K of Xρ is called ρ-complete if each ρ-Cauchy sequence is ρ-convergent to an element of K. Another unnatural behavior one usually encounter is that the convergence of a sequence {xn } to x does not imply that {cxn } converges to cx for some c ∈ K. Thus, many mathematicians imposed some additional conditions for a modular to meet in order to make the multiples of {xn } converge naturally. Such preferences are referred to mostly under the term related to the 42 -conditions. A modular space Xρ is said to satisfy the 42 -condition if there exists k ≥ 2 such that ρ(2x) ≤ kρ(x) for all x ∈ Xρ . Some authors varied the notion so that only k > 0 is required and called it the 42 -type condition. In fact, one may see that these two notions coincide. There are still a number of equivalent notions related to the 42 -conditions.

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In [5], Khamsi proved a series of fixed point theorems in modular spaces where the modulars do not satisfy 42 -conditions. His results exploit one unifying hypothesis in which the boundedness of an orbit is assumed. Lemma 1.3. (see [5]) Let Xρ be a modular space whose induced modular is lower semi-continuous and let C ⊆ Xρ be a ρ-complete subset. If T : C −→ C is a ρ-contraction, that is, there is a constant L ∈ [0, 1) such that ρ(T x − T y) ≤ Lρ(x − y), ∀x, y ∈ C and T has a bounded orbit at a point x0 ∈ C, that is, sup{ρ(T n x0 − T m x0 ) | n, m ∈ N ∪ {0}} < ∞ then the sequence {T n x0 } is ρ-convergent to a point w ∈ C. Now, we will prove a fixed point theorem in modular spaces where the map T do not assume to be the boundedness of an orbit. Our results exploit one unifying hypothesis in which some conditions are assumed. Lemma 1.4. Let Xρ be a modular space whose induced modular is lower semicontinuous and let C ⊆ Xρ be a ρ-complete subset. Let T : C −→ C be a mapping such that 2T x = T 2x, ∀x ∈ C.

(1.2)

Suppose that there is a constant L ∈ [0, 1) with (1.3)

ρ(2T x − 2T y) ≤ Lρ(x − y), ∀x, y ∈ C

and ρ(T xo − xo ) < ∞ at xo ∈ C. Then the sequence {T n x40 } is ρ-convergent to some point w ∈ C and (1.4)

ρ(

2 x0 − w) ≤ ρ(T x0 − x0 ). 4 1−L

Proof. By (M1) and (M3), we have ρ(T x − T y) ≤ ρ(2T x − 2T y) and so, by (1.3), T is a ρ-contration. Hence we have 1 1 ρ( T 2 x0 − x0 ) ≤ ρ(T 2 x0 − T x0 ) + ρ(T x0 − x0 ) 2 2 ≤ (L + 1)ρ(T x0 − x0 ). Let Gx = 2T x for all x ∈ C. By (1.3), we have 1 1 ρ( T n x0 − x0 ) ≤ ρ(T n x0 − T x0 ) + ρ(T x0 − x0 ) 2 2 1  1 = ρ G(T n−1 x0 ) − Gx0 + ρ(T x0 − x0 ) 2 2 1 1  n−1 ≤ Lρ T x0 − x0 + ρ(T x0 − x0 ) 2 2 for all n ∈ N with n ≥ 2 and by induction, we have 1 1 1 k ρ( T n x0 − x0 ) ≤ Σn−1 ρ(T x0 − x0 ) k=0 L ρ(T x0 − x0 ) ≤ 2 2 1−L for all n ∈ N. For any non-negative integers m, n with m > n,

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1 1 1 1 1 1 ρ( T n x0 − T m x0 ) ≤ ρ( T n x0 − x0 ) + ρ( T m x0 − x0 ) 4 4 2 2 2 2 (1.5) 2 ≤ ρ(T x0 − x0 ). 1−L By (1.2), T has a bounded orbit at a point x40 ∈ C and thus by Lemma 1.3, {T n x40 } is ρ-convergent to a point ω ∈ C. Since ρ is lower semi-continuous, by taking n = 0 and m → ∞ in (1.5), we have (1.4).  If ρ is convex, then Lemma 1.4 can be replaced by the following lemma. Lemma 1.5. All conditions in Lemma 1.4 are assumed. Suppose that ρ is convex and 0 ≤ L < 2. Then the sequence {T n x40 } is ρ-convergent to some point w ∈ C and 1 x0 ρ(T x0 − x0 ). (1.6) ρ( − w) ≤ 4 2−L Proof. By (M1) and (M4), we have ρ(T x − T y) ≤ 21 ρ(2T x − 2T y) and since 0 ≤ L < 2, by (1.3), T is a ρ-contration. Hence by (M4), we have 1 1 1 1 ρ( T 2 x0 − x0 ) ≤ ρ(T 2 x0 − T x0 ) + ρ(T x0 − x0 ) 2 2 2 2 1 1 ≤ ( L + )ρ(T x0 − x0 ). 4 2 Let Gx = 2T x for all x ∈ C. By (1.3), we have 1 1 1 1 ρ( T n x0 − x0 ) ≤ ρ(T n x0 − T x0 ) + ρ(T x0 − x0 ) 2 2 2 2 1  1 1 = ρ G(T n−1 x0 ) − Gx0 + ρ(T x0 − x0 ) 2 2 2 1  1 n−1 1  1 ≤ Lρ T x0 − x0 + ρ(T x0 − x0 ) 2 2 2 2 for all n ∈ N with n ≥ 2 and by induction, we have 1 Lk 1 1 ρ(T x0 − x0 ) ρ( T n x0 − x0 ) ≤ Σn−1 k=0 k+1 ρ(T x0 − x0 ) ≤ 2 2 2 2−L for all n ∈ N. For any non-negative integers m, n with m > n, 1 1 1 1 1 1 1 1 ρ( T n x0 − T m x0 ) ≤ ρ( T n x0 − x0 ) + ρ( T m x0 − x0 ) 4 4 2 2 2 2 2 2 1 ≤ ρ(T x0 − x0 ). 2−L The rest of the proof is similar to Lemma 1.4.



Let ρ be a modular on X, V a linear space. Define a set M by M := {g : V −→ Xρ | g(0) = 0} and a generalized function ρe on M by ρe(g) := inf{c > 0 | ρ(g(x)) ≤ cψ(x, x), ∀x ∈ V }, for each g ∈ M, where ψ : V 2 −→ [0, ∞) a mapping. Then M is a linear space, ρe is a modular on M. Furthermore, if ρ is convex, then ρe is also convex([15]).

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Lemma 1.6. Let V be a linear space, Xρ a ρ-complete modular space, where ρ is lower semi-continuous and f : V −→ Xρ a mapping with f (0) = 0. Let ψ : V 2 −→ [0, ∞) be a mapping. Then we have the following : (1) Mρe = M and Mρe is ρe-complete. (2) ρe is lower semi-continuous. Proof. (1) By the definition of Mρe, Mρe = M. Let  > 0 be given. Take any ρeCauchy sequence {gn } in Mρe. Then there is an l ∈ N such that for n, m ∈ N with n, m ≥ l, ρ(gn (x) − gm (x)) ≤ ψ(x, x)

(1.7)

for all x ∈ V . Hence {gn (x)} is a ρ-Cauchy sequence in Xρ for all x ∈ X. Since Xρ is ρ-complete, there is a mapping g : V −→ Xρ such that ρ(gn (x) − g(x)) −→ 0 as n −→ ∞ for all x ∈ X. Then there is an m ∈ N such that ρ(gm (0) − g(0)) = ρ(g(0)) ≤  and hence g ∈ Mρe. Since ρ is a lower semi-continuous, by (1.7), we have ρ(gn (x) − g(x)) ≤ lim inf ρ(gn (x) − gm (x)) ≤ ψ(x, x) m−→∞

for all x ∈ X. Hence Mρe is ρe-complete. (2) Suppose that {gn } is a sequence in Mρe which is ρe-convergent to g ∈ Mρe. Let  > 0. Then for any n ∈ N, there is a positive real number cn such that ρe(gn ) ≤ cn ≤ ρe(gn ) +  and so ρ(g(x)) ≤ lim inf ρ(gn (x)) ≤ lim inf cn ψ(x, x) ≤ n→∞

n→∞



 lim inf ρ(gn (x)) +  ψ(x, x) n→∞

for all x ∈ X. Hence ρe is lower semi-continuous.



2. The generalized Hyers-Ulam stability for (1.1) in modular spaces Throughout this section, we assume that every modular is lower semi-continuous. In this section, we will prove the generalized Hyers-Ulam stability for (1.1) by using our fixed point theorem. We can easily show the following lemma. Lemma 2.1. Let X and Y be vector spaces. Let f : X −→ Y satisfies (1.1) and f (0) = 0. Then we have : (1) f is additive if and only if f (2x) = 2f (x) for all x ∈ X. (2) f is cubic if and only if f (2x) = 8f (x) for all x ∈ X. For any mapping g : X −→ Y , let ga (x) =

1 (g(2x) − 8g(x)), gc (x) = g(2x) − 2g(x) 4

and Dg(x, y) = g(2x + y) + g(2x − y) − 2g(x + y) − 2g(x − y) − 2g(2x) + 4g(x).

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Theorem 2.2. Let V be a linear space, Xρ a ρ-complete modular space. Suppose that f : V −→ Xρ satisfies f (0) = 0 and ρ(Df (x, y)) ≤ φ(x, y)

(2.1) 2

for all x, y ∈ V , where φ : V −→ [0, ∞) is a mapping such that (2.2)

φ(2x, 2y) ≤ Lφ(x, y)

for some L with 0 ≤ L < 1 and for all x, y ∈ V . Then there exists a unique additive-cubic mapping F : V −→ Xρ such that 3 4 (2.3) ρ(F (x) − f (x)) ≤ ψ(x, x) 16 1−L for all x ∈ V , where ψ(x, y) = φ(x, 2y) + φ(x, y) + φ(0, y). Proof. Let ψ(x, y) = φ(x, 2y) + φ(x, y) + φ(0, y). Then by Lemma 1.6, Mρe = M is ρe-complete and ρe is lower semi-continuous. Define Ta : Mρe −→ Mρe by Ta g(x) = 12 g(2x) for all g ∈ Mρe and all x ∈ V . Let g, h ∈ Mρe. Suppose that ρe(g − h) ≤ c for some positive real number c. Then by (2.3) and Remark 1.2, we have ρ(Ta g(x) − Ta h(x)) ≤ ρ(g(2x) − h(2x)) ≤ Lcψ(x, x) for all x ∈ V and so ρe(Ta g − Ta h) ≤ Le ρ(g − h). Hence Ta is a ρe-contraction. By (2.1), we get (2.4)

ρ(f (x) + f (−x)) ≤ φ(0, x),

(2.5)

ρ(f (3x) − 4f (2x) + 5f (x)) ≤ φ(x, x),

and (2.6)

ρ(f (4x) − 2f (3x) − 2f (2x) − 2f (−x) + 4f (x)) ≤ φ(x, 2x),

for all x ∈ V . By (2.4), (2.5), and (2.6), we obtain 1 ρ(Ta fa (x) − fa (x)) = ρ( fa (2x) − fa (x)) ≤ φ(x, 2x) + φ(x, x) + φ(0, x) = ψ(x, x) 2 for all x ∈ V and hence we have (2.7)

ρe(Ta fa − fa ) ≤ 1.

Let Gg = 2Ta g for all g ∈ Mρe. Then Gg(x) = g(2x) for all g ∈ Mρe and for all x ∈ V . Suppose that ρe(g − h) ≤ c for some positive real number c, where g, h ∈ Mρe. Then ρ(g(x) − h(x)) ≤ cψ(x, x) for all x ∈ V and by (2.2), we have ρ(Gg(x) − Gh(x)) = ρ(g(2x) − h(2x)) ≤ cψ(2x, 2x) ≤ cLψ(x, x) for all x ∈ V . Hence ρe(Gg − Gh) ≤ cL and so ρe(Gg − Gh) ≤ Le ρ(g − h).

Since Ta is linear, by Lemma 1.4, there is an A ∈ Mρe such that {Tan f4a } is ρeconvergent to A. In fact, we get

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1 fa (2n x) − A(x)) = 0 2n+2 for all x ∈ V . Since ρe is lower semi-continuous, we get fa fa ρ(A − Tan ) = 0 ρe(Ta A − A) ≤ lim inf ρe(Ta A − Tan+1 ) ≤ lim inf Le n→∞ n→∞ 4 4 and hence A is a fixed point of Ta in Mρe. Replacing x and y by 2n x and 2n y in (2.1), respectively, by (2.2), we have 1 ρ( n+2 Dfa (2n x, 2n y)) 2 1 1 ≤ ρ( n+3 Df (2n+1 x, 2n+1 y)) + ρ( n Df (2n x, 2n y)) 2 2 ≤ Ln+1 φ(x, y) + Ln φ(x, y)

(2.8)

lim ρ(

n→∞

for all x, y ∈ V and for all n ∈ N. Hence we get  1  (2.9) lim ρ n+2 Dfa (2n x, 2n y) = 0 n→∞ 2 for all x, y ∈ V . Note that  1  1 ρ n+10 Dfa (2n x, 2n y) − 8 DA(x, y) 2 2  1   1  1 1 ≤ ρ n+9 fa (2n (2x + y)) − 7 A(2x + y) + ρ n+8 fa (2n (2x − y)) − 6 A(2x − y) 2 2 2 2  1   1  1 1 + ρ n+7 2fa (2n (x + y)) − 5 2A(x + y) + ρ n+6 2fa (2n (x − y)) − 4 2A(x − y) 2 2 2 2  1   1  1 1 + ρ n+5 2fa (2n 2x) − 3 2A(2x) + ρ n+5 2fa (2n (x − y)) − 3 4A(x − y) 2 2 2 2 for all x, y ∈ V and for all n ∈ N. Hence we have (2.10)

lim ρ

n→∞

1

 2

Dfa (2n x, 2n y) − n+10

 1 DA(x, y) =0 28

for all x, y ∈ V . Since   1   1  1 1 DA(x, y) ≤ ρ n+10 Dfa (2n x, 2n y)− 8 DA(x, y) +ρ n+10 Dfa (2n x, 2n y) 9 2 2 2 2 for all x, y ∈ V and for all n ∈ N, by (2.9) and (2.10), we get ρ

(2.11)

DA(x, y) = 0

for all x, y ∈ V . By (1.4) in Lemma 1.4, we get 1 2 ρe(A − fa ) ≤ . 4 1−L Define Tc : Mρe −→ Mρe by Tc g(x) = 81 g(2x) for all g ∈ Mρe and all x ∈ V . By (2.4), (2.5), and (2.6), we obtain (2.12)

ρ(

1 fc (2x) − fc (x)) ≤ ψ(x, x) 23

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for all x ∈ V and hence ρe(Tc fc − fc ) ≤ 1.

(2.13)

Let Hg = 2Tc g for all g ∈ Mρe. Then 1 g(2x). 4 for all g ∈ Mρe and for all x ∈ V . Let g, h ∈ Mρe. Suppose that ρe(g − h) ≤ c for some positive real number c. Then ρ(g(x) − h(x)) ≤ cψ(x, x) for all x ∈ V and by (2.3), we get Hg(x) =

1 1 ρ(Hg(x) − Hh(x)) = ρ( g(2x) − h(2x)) ≤ cψ(2x, 2x) ≤ cLψ(x, x) 4 4 for all x ∈ V . Hence ρe(Hg − Hh) ≤ cL and so ρe(Hg − Hh) ≤ Le ρ(g − h). Since Tc is linear, by Lemma 1.4, there is a C ∈ Mρe such that {Tcn 41 fc } is ρeconvergent to C. Since ρe is lower semi-continuous, we get 1 1 ρe(Tc C − C) ≤ lim inf ρe(Tc C − Tcn+1 fc ) ≤ lim inf Le ρ(C − Tcn fc ) = 0 n→∞ n→∞ 4 4 and hence C is a fixed point of Tc in Mρe. Replacing x and y by 2n x and 2n y in (2.1), respectively, by (2.2), we have 1 ρ( 3n+2 Dfc (2n x, 2n y)) 2 1 1 ≤ ρ( 3n+1 Df (2n+1 x, 2n+1 y)) + ρ( 3n Df (2n x, 2n y)) 2 2 ≤ Ln+1 φ(x, y) + Ln φ(x, y) for all x, y ∈ V . Hence we get

(2.14)



1

 n n Df (2 x, 2 y) =0 c n→∞ 23n+2 for all x, y ∈ V . Similar to A, we have lim ρ

DC(x, y) = 0

for all x, y ∈ V and by (1.4) in Lemma 1.4, we get 1 2 ρ(C(x) − fc (x)) ≤ ψ(x, x) 4 1−L for all x ∈ X. Hence we have (2.15)

1 2 ρe(C − fc ) ≤ . 4 1−L

Let F = 18 C − 21 A. Since A is a fixed point of Ta , A(2x) = 2A(x) for all x ∈ X and similarly, C(2x) = 8C(x) for all x ∈ X. By Lemma 2.1, A is additive and C is cubic. Hence F is an additive-cubic mapping. Since f (x) = 16 fc (x) − 23 fa (x), we have 3 1 1 1 1 1 ρe(F − f ) ≤ ρe(A − fa ) + ρe( C − fc ) ≤ ρe(A − fa ) + ρe(C − fc ), 16 4 4 16 4 4

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and hence by (2.12) and (2.15), we have (2.3). To prove the uniquness of F , let K : V −→ Xρ be another additive-cubic mapping with (2.3). By (2.3), we get 1 1 3 3 ρ( K(x) − F (x)) ≤ ρ(K(x) − f (x)) + ρ(F (x) − f (x))) 4 4 16 16 8 ≤ ψ(x, x) 1−L for all x ∈ V and so 1 1 1 1 1 1 ρ( Ka (x) − Fa (x)) ≤ ρ( K(2x) − F (2x)) + ρ( K(x) − F (x))) 16 16 32 32 4 4 8(1 + L) ≤ ψ(x, x) 1−L for all x ∈ V . Since Fa and Ka are fixed points of Ta , we have 1 1 1 1 ρ( Ka (x) − Fa (x)) = ρ( Tan Ka (x) − Tan Fa (x)) 16 16 16 16 8(1 + L) n ≤ L ψ(x, x) 1−L for all x ∈ V and for all n ∈ N. Letting n → ∞ in the last inequality, we have Fa = Ka and similarly, we have Fc = Kc . Thus F = K.  Comparing the results in a modular and a convex modular, we may see that the coefficient in the case of convex modular is smaller. Theorem 2.3. Suppose that every assumption of Theorem 2.2 holds, ρ is convex and 0 ≤ L < 2. Then there exists a unique additive-cubic mapping F : V −→ Xρ such that 5 3 ψ(x, x) (2.16) ρ(F (x) − f (x)) ≤ 16 32(2 − L) for all x ∈ V , where ψ(x, y) = φ(x, 2y) + φ(x, y) + φ(0, y). Proof. Define Ta : Mρe −→ Mρe by Ta g(x) = 12 g(2x) for all g ∈ Mρe and all x ∈ V . Let g, h ∈ Mρe. Suppose that ρe(g − h) ≤ c for some positive real number c. Then by (2.3) and (M4), we have 1 1 ρ(g(2x) − h(2x)) ≤ Lcψ(x, x) 2 2 1 for all x ∈ V and so ρe(Ta g − Ta h) ≤ 2 Le ρ(g − h). Hence Ta is a ρe-contraction. By (2.1), we get ρ(Ta g(x) − Ta h(x)) ≤

(2.17)

ρ(f (x) + f (−x)) ≤ φ(0, x),

(2.18)

ρ(f (3x) − 4f (2x) + 5f (x)) ≤ φ(x, x),

and (2.19)

ρ(f (4x) − 2f (3x) − 2f (2x) − 2f (−x) + 4f (x)) ≤ φ(x, 2x),

for all x ∈ V . By (2.17), (2.18), and (2.19), we obtain 1 1 1 1 1 ρ( fa (2x) − fa (x)) ≤ φ(x, 2x) + φ(x, x) + φ(0, x) ≤ ψ(x, x) 2 8 4 4 4

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for all x ∈ V and hence ρ(Ta fa (x) − fa (x)) ≤

1 ψ(x, x) 4

for all g ∈ Mρe and all x ∈ V . Hence we have ρe(Ta fa − fa ) ≤

(2.20)

1 . 4

Let Gg = 2Ta g for all g ∈ Mρe. Then Gg(x) = g(2x) for all x ∈ V . Suppose that ρe(g − h) ≤ c for some positive real number c. Then ρ(g(x) − h(x)) ≤ cψ(x, x) for all x ∈ V and by (2.2), we have ρ(Gg(x) − Gh(x)) = ρ(g(2x) − h(2x)) ≤ cψ(2x, 2x) ≤ cLψ(x, x) for all x ∈ V . Hence ρe(Gg − Gh) ≤ cL and so ρe(Gg − Gh) ≤ Le ρ(g − h).

Since Ta is linear, by Lemma 1.5, there is an A ∈ Mρe such that {Tan f4a } is ρeconvergent to A. In fact, we get lim ρ(

n→∞

1 fa (2n x) n+2 2

− A(x)) = 0

for all x ∈ V . Since ρe is lower semi-continuous, we get fa fa ρ(A − Tan ) = 0 ρe(Ta A − A) ≤ lim inf ρe(Ta A − Tan+1 ) ≤ lim inf Le n→∞ n→∞ 4 4 and hence A is a fixed point of Ta in Mρe. Similar to Theorem 2.2, we have (2.21)

DA(x, y) = 0

for all x, y ∈ V and by (1.6) in Lemma 1.5 and (2.20), we get 1 1 ρe(A − fa ) ≤ . 4 4(2 − L)

(2.22)

Define Tc : Mρe −→ Mρe by Tc g(x) = 18 g(2x) for all g ∈ Mρe and all x ∈ V . By (2.17), (2.18), and (2.19), we obtain ρ(

1 1 fc (2x) − fc (x)) ≤ ψ(x, x) 23 4

for all x ∈ V and hence (2.23)

ρe(Tc fc − fc ) ≤

1 . 4

Let Hg = 2Tc g for all g ∈ Mρe. Then 1 g(2x). 4 for all g ∈ Mρe and for all x ∈ V . Let g, h ∈ Mρe. Suppose that ρe(g − h) ≤ c for some positive real number c. Then ρ(g(x) − h(x)) ≤ cψ(x, x) for all x ∈ V and by (2.3), we get Hg(x) =

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1 1 ρ(Hg(x) − Hh(x)) = ρ( g(2x) − h(2x)) ≤ cψ(2x, 2x) ≤ cLψ(x, x) 4 4 for all x ∈ V . Hence ρe(Gg − Gh) ≤ cL and so ρe(Hg − Hh) ≤ Le ρ(g − h). Since Tc is linear, by Lemma 1.5, there is a C ∈ Mρe such that {Tcn 41 fc } is ρeconvergent to C. Since ρe is lower semi-continuous, we get 1 1 ρe(Tc C − C) ≤ lim inf ρe(Tc C − Tcn+1 fc ) ≤ lim inf Le ρ(C − Tcn fc ) = 0 n→∞ n→∞ 4 4 and hence C is a fixed point of Tc in Mρe. Similar to Theorem 2.2, we get (2.24)

DC(x, y) = 0

for all x, y ∈ V and by (1.6) in Lemma 1.5, we get 1 1 ψ(x, x) ρ(C(x) − fc (x)) ≤ 4 4(2 − L) for all x ∈ X. Hence we have 1 1 . ρe(C − fc ) ≤ 4 4(2 − L)

(2.25)

Let F = 18 C − 21 A. Since A is a fixed point of Ta , A(2x) = 2A(x) for all x ∈ X and similarly, C(2x) = 8C(x) for all x ∈ X. By Lemma 2.1, A is additive and C is cubic. Hence F is an additive-cubic mapping. Since f (x) = 16 fc (x) − 23 fa (x), by (2.22) and (2.25), we have 1 1 1 1 1 1 1 1 1 3 ρe(F − f ) ≤ ρe(A − fa ) + ρe( C − fc ) ≤ ρe(A − fa ) + ρe(C − fc ). 16 2 4 2 4 16 2 4 8 4 and hence we have (2.16). The rest of the proof is similar to Theorem 2.2.  Remark 2.4. Sadeghi [13] proved the generalized Hyers-Ulam stability of functional equations in modular spaces with the 42 -condition and in [15], authors proved the stability of mappings f : V −→ Xρ and φ : V 2 −→ [0, ∞) satisfying f (0) = 0, (2.26)

φ(2n x, 2n y) = 0, φ(2x, 2x) ≤ 4Lφ(x, x), ∀x, y ∈ V, n→∞ 4n lim

and ρ(4f (x + y) + 4f (x − y) − 8f (x) − 8f (y)) ≤ φ(x, y), ∀x, y ∈ V for some real number L with 0 ≤ L < 21 whose codomain is equipped with a convex and lower semi-continuous modular without 42 -conditions. Our results guarantee the stability of an additive-cubic mapping, whose induced modular is lower semicontinuous without the convexity and 42 -conditions if 0 ≤ L < 41 . Further, in [15], authors left whether the multiple of 4 on the left side of the inequality (6) can be dropped as a problem. We can solve the problem by using Lemma 1.4 and its proof is similar to the proof in Theorem 2.2. In fact, suppose that φ : V 2 −→ [0, ∞) is a mapping with (2.26) and that 0 ≤ L < 41 . Let f : V −→ Xρ be a mapping such that f (0) = 0 and (2.27)

ρ(f (x + y) + f (x − y) − 2f (x) − 2f (y)) ≤ φ(x, y)

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for all x, y ∈ X. Let ψ(x, y) = φ(x, y) for all x, y ∈ V , f0 (x) = 4f (x), and T g(x) = 14 g(2x). Then by (2.27), we have 1 ρ(T f0 (x) − f0 (x)) = ρ( f0 (2x) − f0 (x)) ≤ φ(x, x) 4 for all x ∈ V and so ρe(T f0 − f0 ) ≤ 1.

(2.28) Moreover, by (2.26), we have

ρe(2T g − 2T h) ≤ 4Le ρ(g − h).

(2.29)

Since 0 ≤ L < by Lemma 1.4, there is a fixed point Q ∈ Mρe such that {T n f40 } = {T n f } converges to Q in Xρ and 1 4,

2 φ(x, x) 1 − 4L for all x ∈ V . We can show that Q is a quadratic mapping ([15]). Further, suppose that ρ is convex and 0 ≤ L < 1. Then (2.28) and (2.29) can be replaced by (2.30)

ρ(Q(x) − f (x)) ≤

ρe(T f0 − f0 ) ≤

1 4

and ρe(2T g − 2T h) ≤ 2Le ρ(g − h), respectively. By Lemma 1.5 and (1.6), 1 1 ρ(Q(x) − f (x)) ≤ φ(x, x) = φ(x, x) 4(2 − 2L) 8(1 − L) for all x ∈ V . Conflict of Interests The authors declare that there is no conflict of interests regarding the publication of this paper. Acknowledgment The corresponding author was supported by the research fund of Dankook University in 2014. References [1] T. Aoki, On the stability of the linear transformation in Banach spaces, J. Math. Soc. Japan 2(1950), 64-66. [2] P. Gˇ avruta, A generalization of the Hyer-Ulam-Rassias stability of approximately additive mappings, J. Math. Anal. Appl. 184(1994), 431-436. [3] D. H. Hyers, On the stability of the linear functional equation, Proc. Nat. Acad. Sci. 27(1941), 222-224. [4] M. A. Kamasi and W. M. Kozlowski, Fixed point thoery in modular function spaces, Birkhauser, 2015. [5] M. A. Khamsi, Quasicontraction mappings in modular spaces without 42 -condition., Fixed Point Theory and Applications, 2008(2008), 1-6. [6] M. A. Khamsi, W. M. Kozlowski, and S. Reich, Fixed point theory in Modular spaces, Nonlinear analysis, Theory, Method and Applications, 14(1990), 935-953.

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[7] W. A. Luxemburg, Banach function spaces, Ph. D. thesis, Delft Univrsity of technology, Delft, The Netherlands, 1959. [8] J. Musielak and W. Orlicz, On modular spaces, Studia Mathematica, 18(1959), 591-597. [9] A. Najati and G. Z. Eskandani, Stability of a mixed additive and cubic functional equation in quasi-Banach spaves, J. Math. Anal. Appl., 342(2008), 1318-1331. [10] H. Nakano, Modular semi-ordered spaces, Tokyo, Japan, 1959. [11] W. Orlicz, Collected Papers, Vols. I, II, PWN, Warszawa, 1988. [12] Th. M. Rassias, On the stability of the linear mapping in Banach sapces, Proc. Amer. Math. Sco. 72(1978), 297-300. [13] G. Sadeghi, A fixed point approach to stability of functional equations in modular spaces, Bulletin of the Malaysian Mathematical Sciences Society. Second Series, 37(2014), 333-344. [14] S. M. Ulam, Problems in Modern Mathematics, Wiley, New York; 1964. [15] K. Wongkum, P. Chaipunya, and P. Kumam, On the generalized Ulam-Hyers-Rassias stability of quadratic mappings in modular spaces without 42 -conditions, 2015(2015), 1-6. [16] T. Z. Xu, J. M. Rassias, and W. X. Xu, Stability of a general mixed additive-cubic equation in F-spaces, J. Comp. Anal. Appl., 14(2012), 1026-1037. [17] T. Z. Xu, J. M. Rassias, and W. X. Xu, Stability of a general mixed additive-cubic functional equation in non-Archimedean fuzzy normed spaces, J. Math. Phys. 51(2010). [18] T. Z. Xu, J. M. Rassias, and W. X. Xu, Intuitionistic fuzzy stability of a general mixed additive-cubic equation, J. Math. Phys. 51(2010). [19] S. Yamamuro, On conjugate spaces of Nakano spaces, Trans. Amer. Math. Soc. 90(1959) 291-311. Department of Mathematics Education, Dankook University, 152, Jukjeon-ro, Sujigu, Yongin-si, Gyeonggi-do, 448-701, Korea E-mail address: [email protected] Department of Mathematics Education, Dankook University, 152, Jukjeon-ro, Sujigu, Yongin-si, Gyeonggi-do, 448-701, Korea E-mail address: [email protected] Department of Mathematics, Sungshin Women’s University, 249-1, Dongseon-Dong 3-Ga, SeongBuk-Gu, Seoul 136-742, Korea E-mail address: [email protected]

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Results on value-shared of admissible function and non-admissible function in the unit disc ∗ Hong-Yan Xua a

Department of Informatics and Engineering, Jingdezhen Ceramic Institute, Jingdezhen, Jiangxi, 333403, China

Abstract In this paper, we consider the uniqueness problem of admissible functions and non-admissible functions sharing some values in the unit disc. We obtain: If f1 is admissible and f2 is inadmissible satisfying lim T (r, f2 ) = ∞, aj (j = 1, 2, . . . ; q) r→1−

be q distinct complex numbers. Then (i) f1 (z), f2 (z) can share at most three values a1 , a2 , a3 IM ; (ii) f1 (z), f2 (z) can share at most five values aj (j = 1, 2, . . . ; 5) with reduced weight 1. Our results of this paper are improvement of the uniqueness theorems of meromorphic functions sharing some values on the whole complex plane which given by Yi and Cao. Key words: uniqueness; meromorphic function; admissible; non-admissible. Mathematical Subject Classification (2010): 30D 35.

1

Introduction and Main Results

In what follows, we shall assume that reader is familiar with the fundamental results and the standard notations of the Nevanlinna value distribution theory of meromorphic functions such as the proximity function m(r, f ), counting function N (r, f ), characteristic function T (r, f ), the first and second main theorems, lemma on the logarithmic derivatives etc. of Nevalinna theory, (see Hayman [7] , Yang [16] and Yi and Yang [19]). For a meromorphic function f , S(r, f ) denotes any quantity satisfying S(r, f ) = o(T (r, f )) for all r outside a possible exceptional set of finite logarithmic measure. b := C S{∞} to denote the extended We use C to denote the open complex plane, C complex plane, and D = {z : |z| < 1} to denote the unit disc. b If E(a, D, f ) = Let f, g be two non-constant meromorphic functions in D and a ∈ C. E(a, D, g), we say f and g share a CM (counting multiplicities) in D. If E(a, D, f ) = ∗ This work was supported by NSFC(11561033, 11301233, 61202313), the Natural Science Foun- dation of Jiangxi Province in China 20132BAB211001, 20151BAB201008), and the Foundation of Education Department of Jiangxi (GJJ14644) of China.

1

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E(a, D, g), we say f and g share a IM (ignoring multiplicities) in D. If D is replaced by C, we give the simple notation as before, E(a, f ), E(a, f ) and so on(see [16]). R.Nevanlinna [12] proved the following well-known theorems. Theorem 1.1 (see [12]) If f and g are two non-constant meromorphic functions that share five distinct values a1 , a2 , a3 , a4 , a5 IM in C, then f (z) ≡ g(z). Theorem 1.2 (see [12]) If f and g are two distinct non-constant meromorphic functions that share four distinct values a1 , a2 , a3 , a4 CM in C, then f is a M¨ obius transformation of g , two of the shared values, say a1 and a2 are Picard values, and the cross ratio (a1 , a2 , a3 , a4 ) = −1. After their very work, the uniqueness of meromorphic functions with shared values in the whole complex plane attracted many investigations (see [16]). In 1987 and 1988, Yi [17, 18] dealt with the problems of multiple values and uniquness of meromorphic functions sharing some values in the whole complex plane by adopting L.Yang’s method and obtained some results which improved the concerning theorems due to Gopalakrishna and Bhoosnurmath’s [6], Ueda [14]. To state the theorems, we will explain some notations as follows. Let f (z) be a non-constant meromorphic function, an arbitrary complex number ˆ and k be a positive integer. We use E k) (a, f ) to denote the set of zeros of f − a, a ∈ C, with multiplicities no greater than k, in which each zero counted only once. We say that f (z) and g(z) share the value a with reduced weight k, if E k) (a, f ) = E k) (a, g). In 1987, Yi [17] obtained the uniqueness theorems concerning multiple values of meromorphic functions as follows. Theorem 1.3 (see [17, 19, Theorem 3.15]). Let f (z) and g(z) be two non-constant meromorphic functions, aj (j = 1, 2, . . . , q) be q distinct complex numbers, and let kj (j = 1, 2, . . . , q) be positive integers or ∞ satisfying (1)

k1 ≥ k2 ≥ · · · ≥ kq ≥ 1.

If E kj ) (aj , f ) = E kj ) (aj , g)

(j = 1, 2, . . . , q)

and (2)

q X j=3

kj > 2, kj + 1

then f (z) ≡ g(z). In recent, it is an interesting topic to investigate the uniqueness with shared values in the subregion of the complex plane such as the unit disc, an angular domain, see [1, 2, 9, 10, 11, 15, 20, 21, 22]. In 1999, Fang [5] studied the uniqueness problem of admissible meromorphic functions in the unit disc D sharing two sets and three sets. Later, there were some results of uniqueness of meromorphic function in the unit disc concerning admissible functions. To state some uniqueness theorems of meromorphic functions in the unit disc D, we need the following basic notations and definitions of meromorphic functions in D(see [3], [4], [8]). 2

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Definition 1.1 Let f be a meromorphic function in D and limr→1− T (r, f ) = ∞. Then α(f ) := lim sup r→1−

T (r, f ) − log(1 − r)

is called the index of inadmissibility of f . If α(f ) = ∞, f is called admissible. Definition 1.2 Let f be a meromorphic function in D and limr→1− T (r, f ) = ∞. Then ρ(f ) := lim sup r→1−

log+ T (r, f ) − log(1 − r)

is called the order (of growth) of f . For admissible functions, the following theorem plays a very important role in studies the uniqueness problems of meromorphic functions in the unit disc. Theorem 1.4 (see [13, Theorem 3]). Let f be an admissible meromorphic function in D, q be a positive integer and a1 , a2 , . . . , aq be pairwise distinct complex numbers. Then, for r → 1− , r 6∈ E,   q X 1 N r, + S(r, f ), (q − 2)T (r, f ) ≤ f − aj j=1 R dr < ∞. If the order of where E ⊂ (0, 1) is a possibly occurring exceptional set with E 1−r   1 f is finite, the remainder S(r, f ) is a O log 1−r without any exceptional set. In 2005, Titzhoff [13] investigated the uniqueness of two admissible functions in the unit disc D by using the Second Main Theorem for admissible functions (Theorem 1.4) and obtained the five values theorem in the unit disc D as follows. Theorem 1.5 (see [13]). If two admissible function f, g share five distinct values, then f ≡ g. In 2009, Mao and Liu [11] gave a different method to investigate the uniqueness problem of meromorphic functions in unit disc and obtained the following results. b = Theorem 1.6 (see [11]). Let f, g be two meromorphic functions in D, aj ∈ C(j 1, 2, . . . , 5) be five distinct values, and ∆(θ0 , δ)(0 < δ < π) be an angular domain such b that for some a ∈ C, (3)

lim sup r→1−

log n(r, ∆(θ0 , δ/2), f (z) = a) = τ > 1. 1 log 1−r

If f and g share aj (j = 1, 2, . . . , 5) IM in ∆(θ0 , δ), then f (z) ≡ g(z). Remark 1.1 In fact, the condition (3) implies that f is admissible in the unit disc. Therefore, Theorem 1.6 is one result of uniqueness of admissible functions in the unit disc. 3

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For admissible functions in the unit disc D, from Theorem 1.4, using the same argument as in the proofs of Theorem 1.3, we can easily get the following results. Theorem 1.7 Let f1 (z) and f2 (z) be two admissible meromorphic functions in D, aj (j = 1, 2, . . . , q) be q distinct complex numbers, and let kj (j = 1, 2, . . . , q) be positive integers or ∞ satisfying (1). If f1 (z) and f2 (z) satisfy (4)

E kj ) (aj , D, f1 ) = E kj ) (aj , D, f2 )

(j = 1, 2, . . . , q)

and (2), then f1 (z) ≡ f2 (z), where E k) (a, D, f ) to denote the set of zeros of f − a in D, with multiplicities no greater than k, in which each zero counted only once. b and a positive integer k, we can say that f1 (z), f2 (z) share the Remark 1.2 For a ∈ C value a in D with reduced weight k, if E k) (a, D, f1 ) = E k) (a, D, f2 ). Similar to the corollary of Theorem 1.3 (see [19, Corollary,pp.181.]), we can get the following corollary. Corollary 1.1 Let f1 (z) and f2 (z) be two admissible meromorphic functions in D, aj (j = 1, 2, . . . , q) be q distinct complex numbers, and let kj (j = 1, 2, . . . , q) be positive integers or ∞ satisfying (1) and (4). (i) if q = 7, then f1 (z) ≡ f2 (z); (ii) if q = 6 and k3 ≥ 2, then f1 (z) ≡ f2 (z); (iii) if q = 5, k3 ≥ 3 and k5 ≥ 2, then f1 (z) ≡ f2 (z); (iv) if q = 5 and k4 ≥ 4, then f1 (z) ≡ f2 (z); (v) if q = 5, k3 ≥ 5 and k4 ≥ 3, then f1 (z) ≡ f2 (z); (vi) if q = 5, k3 ≥ 6 and k4 ≥ 2, then f1 (z) ≡ f2 (z). Remark 1.3 In Theorem 1.5, the conclusion f (z) ≡ g(z) holds when q = 5 and kj = ∞ (j = 1, 2, . . . , 5). From Corollary 1.1, we can get that f1 (z) ≡ f2 (z) when q = 5 and kj (j = 1, 2, . . . , 5) satisfy any of the four conditions (i)-(iv). Hence, Corollary 1.1 is an improvement of Theorem 1.5. For non-admissible functions, the following theorem also plays a very important role in studies theirs uniqueness problems. Theorem 1.8 (see [13, Theorem 2]). Let f be a meromorphic function in D and limr→1− T (r, f ) = ∞, q be a positive integer and a1 , a2 , . . . , aq be pairwise distinct complex numbers. Then, for r → 1− , r 6∈ E,   q X 1 1 (q − 2)T (r, f ) ≤ N r, + log + S(r, f ). f − a 1 − r j j=1 1 Remark 1.4 (i) In contrast to admissible functions, the term log 1−r in Theorem 1.8 does not necessarily enter the remainder S(r, f ) because the non-admissible function f   1 may have T (r, f ) = O log 1−r .   1 (ii) If 0 < α(f ) < ∞, we can see that S(r, f ) = o log 1−r holds in Theorem 1.8 without a possible exception set.

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From Theorem 1.8 and Remark 1.4, we can see that the uniqueness of non-admissible functions is more intricate than the case of admissible functions. In this paper, we will deal with the uniqueness problem of non-admissible functions in D. We use Υα to denote the class of non-admissible functions satisfying the condition: α(f ) = α(0 < α < ∞) for f ∈ Υα . For the class Υα , we get the following results Theorem 1.9 Let f (z) ∈ Υα , aj (j = 1, 2, . . . , q) are q distinct complex numbers. If q = 5 + [ k2 + k+1 kα ], then there does not exist g(z)(6≡ f (z)) ∈ Υα satisfying (5)

E k) (aj , D, f ) = E k) (aj , D, g),

(j = 1, 2, . . . , q),

where [x] denotes the largest integer less than or equal to x. Corollary 1.2 Let f (z) ∈ Υα . Then f (z) is uniquely determined in Υα by one of the following cases: (i) if f have seven point-sets E 1) (aj , D, f )(j = 1, 2, . . . , 7) and α > 1; (ii) if f have six point-sets E 2) (aj , D, f )(j = 1, 2, . . . , 6) and α > 32 ; (iii) if f have five point-sets E 3) (aj , D, f )(j = 1, 2, . . . , 5) and α > 4. Remark 1.5 For Corollary 1.1, we can see that the conclusion (iii) in Corollary 1.1 is an improvement of Theorem 1.6. In fact, the conclusion of Theorem 1.6 is that non-constant meromorphic function f is uniquely determined in D by five point-sets E ∞) (aj , D, f )(j = 1, 2, . . . , 5) and α(f ) = ∞. Theorem 1.10 Let α > 12 and f (z) ∈ Υα , aj (j = 1, 2, . . . , 5) be five distinct complex numbers. Then f (z) is uniquely determined in Υ by three point-sets E 3) (aj , D, f )(j = 1, 2, 3) and two point-sets E 2) (aj , D, f )(j = 4, 5). Furthermore, for the uniqueness of regular inadmissibility functions we obtain the following theorems Theorem 1.11 Let aj (j = 1, 2, . . . , q) be q distinct complex numbers, and let kj (j = 1, 2, . . . , q) be positive integers or ∞ satisfying (1). If f1 (z), f2 (z) be non-constant regular inadmissibility functions satisfying 0 < α(f1 ), α(f2 ) < ∞, (4) and q X

(6)

j=3

kj 2 −2> , kj + 1 α(f1 ) + α(f2 )

then f1 (z) ≡ f2 (z). From Theorem 1.11, similar to Corollary 1.1, we can get the following results easily. Corollary 1.3 Let aj (j = 1, 2, . . . , q) be q distinct complex numbers, and let kj (j = 1, 2, . . . , q) be positive integers or ∞ satisfying (1), α := min{α(f1 ), α(f2 )}. And let f1 (z), f2 (z) be non-constant regular inadmissibility functions satisfying 0 < α(f1 ), α(f2 ) < ∞ and (4), (i) if α > 1, q = 7 and k7 ≥ 2, then f1 (z) ≡ f2 (z); (ii) if α > 1, q = 6 and k6 ≥ 4, then f1 (z) ≡ f2 (z); 5

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(iii) if α > 2 and q = 7, then f1 (z) ≡ f2 (z); (iv) if α > 3, q = 6 and k3 ≥ 2, then f1 (z) ≡ f2 (z); (v) if α > 6, q = 5, k3 ≥ 3 and k5 ≥ 2, then f1 (z) ≡ f2 (z); (vi) if α > 10, q = 5 and k4 ≥ 4, then f1 (z) ≡ f2 (z); (vii) if α > 12, q = 5, k3 ≥ 5 and k4 ≥ 3, then f1 (z) ≡ f2 (z); (viii) if α > 42, q = 5, k3 ≥ 6 and k4 ≥ 2, then f1 (z) ≡ f2 (z). Remark 1.6 In Corollary 1.1, f1 (z), f2 (z) are all admissible functions, that is, α(f1 ) = ∞ and α(f2 ) = ∞. From the conclusions of Corollary 1.3, we see that f1 (z) ≡ f2 (z) holds when f1 (z), f2 (z) are non-constant regular inadmissibility functions with min{α(f1 ), α(f2 )} > ζ and ζ a positive constant. Hence, Corollary 1.3 is an improvement of Corollary 1.1. The following theorem will show that an admissible function can share sufficiently many values concerning multiple values with another inadmissible function. Theorem 1.12 If f1 is admissible and f2 is inadmissible satisfying limr→1− T (r, f2 ) = ∞, aj (j = 1, 2, . . . , q) be q distinct complex numbers, and let kj (j = 1, 2, . . . , q) be positive integers or ∞ satisfying (1). Then q X

(7)

j=2

kj −2>0 kj + 1

and (4) do not hold at same time. Corollary 1.4 If f1 is admissible and f2 is inadmissible satisfying limr→1− T (r, f2 ) = ∞, aj (j = 1, 2, . . . , q) be q distinct complex numbers. Then (i) f1 (z), f2 (z) can share at most three values a1 , a2 , a3 IM ; (ii) f1 (z), f2 (z) can share at most five values aj (j = 1, 2, . . . , 5) with reduced weight 1; And any one of the following cases can not hold (iii) q = 4 and E k1 ) (a1 , D, f1 ) = E k1 ) (a1 , D, f2 ) (k1 ≥ 6), E 6) (a2 , D, f1 ) = E 6) (a2 , D, f2 ), E 2) (a3 , D, f1 ) = E 2) (a3 , D, f2 ) and E 1) (a4 , D, f1 ) = E 1) (a4 , D, f2 ); (iv) q = 4 and E k1 ) (a1 , D, f1 ) = E k1 ) (a1 , D, f2 ) (k1 ≥ 3), E 3) (a2 , D, f1 ) = E 3) (a2 , D, f2 ), E 2) (a3 , D, f1 ) = E 2) (a3 , D, f2 ) and E 2) (a4 , D, f1 ) = E 2) (a4 , D, f2 ); (v) q = 5 and E k) (ai , D, f1 ) = E k) (a, D, f2 ) (k ≥ 2, i = 1, 2), E 1) (aj , D, f1 ) = E 1) (aj , D, f2 ) (j = 3, 4, 5).

2

Some Lemmas

To prove our results, we will require the following lemmas. Lemma 2.1 (see [13, Lemma 1]). Let f (z), g(z) satisfy limr→1− T (r, f ) = ∞ and limr→1− T (r, g) = ∞. If there is a K ∈ (0, ∞) with T (r, f ) ≤ KT (r, g) + S(r, f ) + S(r, g), then each S(r, f ) is also an S(r, g). 6

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Lemma 2.2 (see [19, Lemma 3.4]). Let f (z) be a non-constant meromorphic function, a be an arbitrary complex number, and k be a positive integer. Then       1 k 1 1 1 N r, ≤ N k) r, + N r, , f −a k+1 f −a k+1 f −a and

 N r,

1 f −a



  k 1 1 ≤ N k) r, + T (r, f ) + O(1), k+1 f −a k+1

  1 are denoted by the zeros of f − a in |z| ≤ r, whose multiplicities are where N k) r, f −a not greater than k and are counted only once. From Lemma 2.2 and Theorems 1.4 and 1.8, we can get the following Lemma Lemma 2.3 Let f (z) be a meromorphic function in D and limr→1− T (r, f ) = ∞, aj (j = 1, 2, . . . , q) be q distinct complex numbers, and kj (j = 1, 2, . . . , q) be positive integers or ∞. If f is an admissible function, then     q q X X 1 1 kj q − 2 −  T (r, f ) ≤ N kj ) r, + S(r, f ); k +1 k +1 f − aj j=1 j j=1 j If f is a non-admissible function, then     q q X X 1 1 kj 1 q − 2 −  T (r, f ) ≤ N kj ) r, + S(r, f ), + log k + 1 k + 1 f − a 1 − r j j=1 j j=1 j where S(r, f ) is stated as in Theorem 1.4.

3 3.1

Proofs of Theorems 1.9 and 1.10 The Proof of Theorem 1.9

Suppose that there exists g(z) ∈ Υα satisfying (5) and f (z) 6≡ g(z). Without loss of generality, we can assume that aj (j = 1, 2, . . . , q) are all finite numbers, otherwise, a suitable linear transformation will be done. Since f (z), g(z) ∈ Υα , from Lemma 2.3, we have     q q k X 1 1 (8) q−2− T (r, f ) ≤ N k) r, + log + S(r, f ). k+1 k + 1 j=1 f − aj 1−r If follows form (5) that (9)

q X j=1

 N k)

1 r, f − aj



 ≤ N r,

1 f −g

 ≤ T (r, f ) + T (r, g) + O(1).

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From (8) and (9), we have   qk 3k + 2 k 1 − T (r, f ) ≤ T (r, g) + log + S(r, f ). k+1 k+1 k+1 1−r Similarly, we have   qk 3k + 2 k 1 − T (r, g) ≤ T (r, f ) + log + S(r, g). k+1 k+1 k+1 1−r Combining the above two inequalities, we get   qk 4k + 2 1 (10) − {T (r, f ) + T (r, g)} ≤ 2 log + S(r, f ) + S(r, g). k+1 k+1 1−r Since f (z), g(z) ∈ Υα and 0 < α < ∞, from the definition of index and q = 5 + [ k2 + k+1 kα ], k+1 we have for 0 < ε < α − qk−4k−2 , there exists a sequence {rm } → 1− such that (11)

T (rm , f ) > (α − ε) log

1 , 1 − rm

T (rm , g) > (α − ε) log

1 , 1 − rm

for all m → ∞.  From f (z), of Theorem 1.9, we can see  g(z) ∈ Υα and the assumptions  1 1 that S(r, f ) = o log 1−r and S(r, g) = o log 1−r . From this fact and (10)-(11), we have       qk 4k + 2 1 1 (12) 2 − (α − ε) − 2 log < o log . k+1 k+1 1 − rm 1 − rm   qk − 4k+2 (α − ε) − 2 > 0, we can get a contradiction. Hence, we From (12) and 2 k+1 k+1 have f (z) ≡ g(z). Thus, this completes the proof of Theorem 1.9.

3.2

The Proof of Theorem 1.10

Suppose that there exists g(z) ∈ Υα satisfying f (z) 6≡ g(z) and (13)

E 3) (aj , D, f ) = E 3) (aj , D, g),

(j = 1, 2, 3)

E 2) (aj , D, f ) = E 2) (aj , D, g),

(j = 4, 5).

Without loss of generality, we can assume that aj (j = 1, 2, . . . , 5) are all finite numbers, otherwise, a suitable linear transformation will be done. Since f (z), g(z) ∈ Υα , from Lemma 2.3, we have   3 2 (14) T (r, f ) 5−2− − 4 3     3 5 3X 1 2X 1 1 ≤ N 3) r, + N 2) r, + log + S(r, f ) 4 j=1 f − aj 3 j=4 f − aj 1−r     X   3 5 1 1 3 X  + log 1 + S(r, f ). N 3) r, + N 2) r, ≤ 4 j=1 f − aj f − aj 1−r j=4

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From (13), we have 3 X

 N 3) r,

j=1

1 f − aj

 +

5 X j=4

 N 2) r,

1 f − aj



 ≤N

r,

1 f −g

 ≤ T (r, f ) + T (r, g) + O(1).

From this inequality and (14), we have 5 3 1 T (r, f ) ≤ T (r, g) + log + S(r, f ). 6 4 1−r

(15) Similarly, we have

3 1 5 T (r, g) ≤ T (r, f ) + log + S(r, g). 6 4 1−r

(16)

Since f (z), g(z) ∈ Υα and α > 12, from the definition of index, we have for any ε(0 < ε < α − 12), there exists a sequence {rm } → 1− satisfying (11) for all m → ∞. From this fact and (15)-(16), we have     1 1 1 (α − ε) − 2 log < o log . (17) 6 1 − rm 1 − rm Since α > 12 and 0 < ε < α − 12, we have 16 (α − ε) − 2 61 (α − ε) − 2 > 0, a contradiction. Hence, we have f (z) ≡ g(z). Thus, this completes the proof of Theorem 1.10.

4

Proof of Theorem 1.11

Without loss of generality, we may assume that all aj (j = 1, 2, . . . , q) are finite, otherwise, a suitable M¨ obius transformation will be done. From Lemma 2.3, we have     q q X X 1  kj 1 1 q − 2 − (18) T (r, f1 ) ≤ N kj ) r, + log k + 1 k + 1 f − a 1 − r j j 1 j j=1 j=1 + S(r, f1 ). From (1), we have (19)

1 kq k2 k1 ≤ ≤ ··· ≤ ≤ ≤ 1. 2 kq + 1 k2 + 1 k1 + 1

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From (18) and (19), we have   q X k j  (20) − 2 T (r, f1 ) k + 1 j j=1 ≤

   X   q 2  1 1 k3 X k3 kj N kj ) r, N kj ) r, + − k3 + 1 j=1 f1 − aj kj + 1 k3 + 1 f1 − aj j=1

1 + S(r, f1 ) 1−r   X  q 2  k3 X 1 k3 kj ≤ N kj ) r, + − T (r, f1 ) k3 + 1 j=1 f1 − aj kj + 1 k3 + 1 j=1 + log

+ log

1 + S(r, f1 ). 1−r

If f1 (z) 6≡ f2 (z), from the assumptions of Theorem 1.11, we have     q X 1 1 N kj ) r, ≤ N r, ≤ T (r, f1 ) + T (r, f2 ) + O(1). (21) f1 − a j f1 − f2 j=1 From this inequality, we have   q X k3 k3 1 kj  + − 2 T (r, f1 ) ≤ T (r, f2 ) + log + S(r, f1 ). (22) k + 1 k + 1 k + 1 1 − r 3 3 j=3 j Similarly, we have   q X k k k3 1 j 3  (23) + − 2 T (r, f2 ) ≤ T (r, f1 ) + log + S(r, f2 ). k + 1 k3 + 1 k3 + 1 1−r j=3 j    1 Since 0 < α(f1 ), α(f2 ) < ∞, we have S(r, f1 ) = o log 1−r , S(r, f2 ) = o log from the definition of index, for any ε satisfying     2 0 < 2ε < min α(f1 ), α(f2 ), α(f1 ) + α(f2 ) − Pq , kj  

1 1−r



. And

j=3 kj +1

there exists a sequence {rm } → 1− such that (24)

T (rm , f1 ) > (α(f1 ) − ε) log

1 , 1 − rm

T (rm , f2 ) > (α(f2 ) − ε) log

1 , 1 − rm

for all m → ∞. From (22)-(24), we have     q X k 1 1 j (α(f1 ) + α(f2 ) − 2ε) (25) − 2 log < o log . k +1 1 − rm 1 − rm j=3 j 10

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Since 0 < 2ε < α(f1 ) + α(f2 ) −

2

, we have (α(f1 ) + α(f2 ) − 2ε)

kj Σqj=3 k +1 j

kj j=3 kj +1

Pq

− 2 > 0,

a contradiction. Hence, we have f1 (z) ≡ f2 (z). Thus, this completes the proof of Theorem 1.11.

5 5.1

Proofs of Theorem 1.12 and Corollary 1.4 The Proof of Theorem 1.12

We will employ the proof by contradiction, that is, suppose that (4) and (7) hold at the same time. Since f1 (z) is admissible, from Lemma 2.3, and by using the same argument as in Theorem 1.11, we can easily get   q X 2k k2 k 2 j  + − 2 T (r, f1 ) ≤ (T (r, f1 ) + T (r, f2 )) + S(r, f1 ), k + 1 k + 1 k 2 2+1 j=3 j that is,  q X 

(26)

j=2

Set K =

kj j=2 kj +1

Pq

 k2 kj − 2 T (r, f1 ) ≤ T (r, f2 ) + S(r, f1 ). kj + 1 k2 + 1

− 2. If K > 0, from (26), we have T (r, f1 ) ≤ K 0 T (r, f2 ) + S(r, f1 ),

(27)

1 k2 0 where K 0 = K k2 +1 . Since kj > 0(j = 1, 2, . . . , q), we have K > 0 as K > 0. From this and Lemma 2.1, we can get that each S(r, f1 ) is also an S(r, f2 ). Since f1 (z) is admissible and f2 (z) is non-admissible, we can get T (r, f2 ) = S(r, f1 ). Thus, we have

(28)

T (r, f2 ) = S(r, f1 ) = S(r, f2 ) = o(T (r, f2 )).

Since limr→1− T (r, f2 ) = ∞ and (28), we can get a contradiction. Hence, we prove that (4) and (7) do not hold at the same time.

6

The Proof of Corollary 1.4

(i) Suppose that f1 (z), f2 (z) share four values aj (j = 1, 2, 3, 4) IM , that is, kj = ∞(j = 1, 2, 3, 4). Since f1 (z) is admissible, from Theorem 1.4, we have 2T (r, f1 ) ≤

(29)

4 X

 N r,

j=1

1 f1 − aj

 + S(r, f1 ).

Since f1 (z), f2 (z) share four values aj (j = 1, 2, 3, 4) IM , we have (30)

4 X j=1

N

 r,

1 f1 − aj



 ≤N

r,

1 f1 − f2

 ≤ T (r, f1 ) + T (r, f2 ) + O(1).

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From (29) and (30), we have T (r, f1 ) ≤ T (r, f2 ) + S(r, f1 ).

(31)

By Lemma 2.1, similar to the proof of Theorem 1.12, we have T (r, f2 ) = S(r, f1 ) = S(r, f2 ) = o(T (r, f2 )). From this and limr→1− T (r, f2 ) = ∞, we can get a contradiction. Thus, this completes (i) of Corollary 1.4. Similar to the proof of Corollary 1.4 (i), we can prove (iii),(iv) and (v) of Corollary 1.4 easily. Here we omit the detail. (ii) Suppose that f1 , f2 share six values aj (j = 1, 2, . . . , 6) with reduced weight 1, that is, (32)

E 1) (aj , D, f1 ) = E 1) (aj , D, f2 ),

(j = 1, 2, . . . , 6),

and k1 = k2 = · · · = k6 = 1. Then, we can deduce that 6 X j=2

1 1 kj − 2 = 5 × − 2 = > 0. kj + 1 2 2

From this and the conclusion of Theorem 1.12, we get a contradiction. Thus, this completes the proof of Corollary 1.4.

Competing interests The authors declare that they have no competing interests.

Author’s contributions HYX, LZY and CFY completed the main part of this article, HYX corrected the main theorems. All authors read and approved the final manuscript.

References [1] T. B. Cao, H. X. Yi, Analytic functions sharing three values DM in one angular domain, J. Korean Math. Soc. 45(6)(2008), 1523-1534. [2] T. B. Cao, H. X. Yi, On the uniqueness of meromorphic functions that share four values in one angular domain, J. Math. Anal. Appl. 358 (2009), 81-97. [3] T. B. Cao, H. X. Yi, The growth of solutions of linear differential equations with coefficients of iterated order in the unit disc, J. Math. Anal. Appl. 319 (2006), 278-294.

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[4] Z. X. Chen, K. H. Shon, The growth of solutions of differential equations with coefficients of small growth in the disc, J. Math. Anal. Appl. 297 (2004), 285-304. [5] M. L. Fang, On the uniqueness of admissible meromorphic functions in the unit disc, Sci. China A 42 (1999), 367-381. [6] H. S. Gopalakrishna, S. S. Bhoosnurmath, Uniqueness theorem for meromorphic functions, Math. Scand. 39 (1976), 125-130. [7] W. K. Hayman, Meromorphic Functions, Oxford Univ. Press, London, 1964. [8] J. Heittokangas, On complex differential equations in the unit disc, Ann. Acad. Sci. Fenn. Math. Diss. 122 (2000), 1-54. [9] W. C. Lin, S. Mori, K. Tohge, Uniqueness theorems in an angular domain, Tohoku Math. J. 58 (2006), 509-527. [10] W. C. Lin, S. Mori, H. X. Yi, Uniqueness theorems of entire functions with shared-set in an angular domain, Acta Mathematica Sinica 24 (2008), 1925-1934. [11] Z. Q. Mao, H. F. Liu, Meromorphic functions in the unit disc that share values in an angular domain, J. Math. Anal. Appl. 359 (2009), 444-450. [12] R. Nevanlinna, Le th´eor`eme de Picard-Borel et la th´eorie des fonctions m´eromorphes, Reprinting of the 1929 original, Chelsea Publishing Co. New York, 1974(in Frech). [13] F. Titzhoff, Slowly growing functions sharing values, Fiz. Mat. Fak. Moksl. Semin. Darb. 8 (2005), 143-164. [14] H. Ueda, Unicity theorems for meromorphic or entire functions, Kodai Math. J. 6 (1983), 26-36. [15] Z. J. Wu, A remark on uniqueness theorems in an angular domain, Proc. Japan Acad. Ser. A 64 (6) (2008), 73-77. [16] L. Yang, Value Distribution Theory, Springer/Science Press, Berlin/Beijing, 1993/1982. [17] H. X. Yi, A note on the questions of multiple values and uniquness of meromorphic functions, Chin. Quart. J. Math. 2 (1) (1987), 99-104. [18] H. X. Yi, Notes on a theorem of H. Ueda, J. Shandong Univ. 23 (4) (1988), 71-77. [19] H. X. Yi, C. C. Yang, Uniqueness Theory of Meromorphic Functions, Science Press/ Kluwer., Beijing, 2003. [20] Q. C. Zhang, Meromorphic functions sharing values in an angular domain, J. Math. Anal. Appl. 349 (2009), 100-112. [21] J. H. Zheng, On uniqueness of meromorphic functions with shared values in some angular domains, Canad J. Math. 47 (2004), 152-160. [22] J. H. Zheng, On uniqueness of meromorphic functions with shared values in one angular domains, Complex Var. Elliptic Equ. 48 (2003), 777-785.

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COMPOSITIONS INVOLVING SCHUR HARMONICALLY CONVEX FUNCTIONS HUAN-NAN SHI AND JING ZHANG†

Abstract. The decision theorem of the Schur harmonic convexity for the compositions involving Schur harmonically convex functions is established and used to determine the Schur harmonic convexity of some symmetric functions. 2010 Mathematics Subject Classification: Primary 26D15; 05E05; 26B25 Keywords: Schur harmonically convex function; harmonically convex function; composite function; symmetric function

1. Introduction Throughout the article, R denotes the set of real numbers, x = (x1 , x2 , . . . , xn ) denotes n-tuple (n-dimensional real vectors), the set of vectors can be written as Rn = {x = (x1 , x2 , . . . , xn ) : xi , ∈ R, i = 1, 2, . . . , n} , Rn++ = {x = (x1 , x2 , . . . , xn ) : xi > 0, i = 1, 2, . . . , n}, Rn+ = {x = (x1 , x2 , . . . , xn ) : xi ≥ 0, i = 1, 2, . . . , n}. In particular, the notations R, R++ and R+ denote R1 , R1++ and R1+ , respectively. The following conclusion is proved in reference [1, p. 91], [2, p. 64-65].

Theorem A. Let the interval [a, b] ⊂ R, ϕ : Rn → R, f : [a, b] → R and ψ(x1 , x2 , . . . , xn ) = ϕ(f (x1 ), f (x2 ), . . . , f (xn )) : [a, b]n → R. † J. Zhang: Correspondence author. 1

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(i) If ϕ is increasing and Schur-convex and f is convex, then ψ is Schur-convex. (ii) If ϕ is increasing and Schur-concave and f is concave, then ψ is Schurconcave. (iii) If ϕ is decreasing and Schur-convex and f is concave, then ψ is Schurconvex. (iv) If ϕ is increasing and Schur-convex and f is increasing and convex, then ψ is increasing and Schur-convex. (v) If ϕ is decreasing and Schur-convex and f is decreasing and concave, then ψ is increasing and Schur-convex. (vi) If ϕ is increasing and Schur-convex and f is decreasing and convex, then ψ is decreasing and Schur-convex. (vii) If ϕ is decreasing and Schur-convex and f is increasing and concave, then ψ is decreasing and Schur-convex. (viii) If ϕ is decreasing and Schur-concave and f is decreasing and convex, then ψ is increasing and Schur-concave.

Theorem A is very effective for determine of the Schur-convexity of the composite functions. The Schur harmonically convex functions were proposed by Chu et al. [3, 4, 5] in 2009. The theory of majorization was enriched and expanded by using this concepts. Regarding the Schur harmonically convex functions, the aim of this paper is to establish the following theorem which is similar to Theorem A.

Theorem 1. Let the interval [a, b] ⊂ R++ , ϕ : Rn++ → R++ , f : [a, b] → R++ and ψ(x1 , x2 , . . . , xn ) = ϕ(f (x1 ), f (x2 ), . . . , f (xn )) : [a, b]n → R++ .

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(i) If ϕ is increasing and Schur harmonically convex and f is harmonically convex, then ψ is Schur harmonically convex. (ii) If ϕ is increasing and Schur harmonically concave and f is harmonically concave, then ψ is Schur harmonically concave. (iii) If ϕ is decreasing and Schur harmonically convex and f is harmonically concave, then ψ is Schur harmonically convex. (iv) If ϕ is increasing and Schur harmonically convex and f is increasing and harmonically convex, then ψ is increasing and Schur harmonically convex. (v) If ϕ is decreasing and Schur harmonically convex and f is decreasing and harmonically concave, then ψ is increasing and Schur harmonically convex. (vi) If ϕ is increasing and Schur harmonically convex and f is decreasing and harmonically convex, then ψ is decreasing and Schur harmonically convex. (vii) If ϕ is decreasing and Schur harmonically convex and f is increasing and harmonically concave, then ψ is decreasing and Schur harmonically convex. (viii) If ϕ is decreasing and Schur harmonically concave and f is decreasing and harmonically convex, then ψ is increasing and Schur harmonically concave.

2. Definitions and lemmas In order to prove our results, in this section we will recall useful definitions and lemmas.

Definition 1. [1, 2] Let x = (x1 , x2 , . . . , xn ) and y = (y1 , y2 , . . . , yn ) ∈ Rn . (i) x ≥ y means xi ≥ yi for all i = 1, 2, . . . , n. (ii) Let Ω ⊂ Rn , ϕ: Ω → R is said to be increasing if x ≥ y implies ϕ(x) ≥ ϕ(y). ϕ is said to be decreasing if and only if −ϕ is increasing.

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Definition 2. [1, 2] Let x = (x1 , x2 , . . . , xn ) and y = (y1 , y2 , . . . , yn ) ∈ Rn . We say y majorizes x (x is said to be majorized by y), denoted by x ≺ y, if

Pk

i=1

x[i] ≤

Pk

i=1

y[i] for k = 1, 2, . . . , n − 1 and

Pn

i=1

xi =

Pn

i=1

yi , where

x[1] ≥ x[2] ≥ · · · ≥ x[n] and y[1] ≥ y[2] ≥ · · · ≥ y[n] are rearrangements of x and y in a descending order.

Definition 3. [1, 2] Let x = (x1 , x2 , . . . , xn ) and y = (y1 , y2 , . . . , yn ) ∈ Rn . (i) A set Ω ⊂ Rn is said to be a convex set if

αx + (1 − α)y = (αx1 + (1 − α)y1 , αx2 + (1 − α)y2 , . . . , αxn + (1 − α)yn ) ∈ Ω

for all x, y ∈ Ω, and α ∈ [0, 1]. (ii) Let Ω ⊂ Rn be convex set. A function ϕ: Ω → R is said to be a convex function on Ω if

ϕ (αx + (1 − α)y) ≤ αϕ(x) + (1 − α)ϕ(y)

holds for all x, y ∈ Ω, and α ∈ [0, 1]. ϕ is said to be a concave function on Ω if and only if −ϕ is convex function on Ω. (iii) Let Ω ⊂ Rn . A function ϕ: Ω → R is said to be a Schur-convex function on Ω if x ≺ y on Ω implies ϕ (x) ≤ ϕ (y) . A function ϕ is said to be a Schur-concave function on Ω if and only if −ϕ is Schur-convex function on Ω.

Lemma 1. (Schur-convex function decision theorem)[1, 2] : Let Ω ⊂ Rn be symmetric and have a nonempty interior convex set. Ω0 is the interior of Ω. ϕ : Ω → R is continuous on Ω and differentiable in Ω0 . Then ϕ is the Schur −

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convex (or Schur − concave, respectively) f unction if and only if ϕ is symmetric on Ω and  (x1 − x2 )

∂ϕ ∂ϕ − ∂x1 ∂x2

 ≥ 0(or ≤ 0, respectively)

(1)

holds for any x ∈ Ω0 .

Definition 4. [6] Let Ω ⊂ Rn++ . xy ∈ Ω λx + (1 − λ)y Pn 1 = for every x, y ∈ Ω and λ ∈ [0, 1], where xy = i=1 xi yi and x 1 1 1
. , ,..., x1 x2 xn

(i) A set Ω is said to be a harmonically convex set if

(ii) Let Ω ⊂ Rn++ be a harmonically convex set. A function ϕ : Ω → R++ be a continuous function, then ϕ is called a harmonically convex (or concave, respectively) function, if

ϕ

α x

1 + 1−α y

! ≤ (or ≥, respectively)

α ϕ(x)

1 +

1−α ϕ(y)

holds for any x, y ∈ Ω, and α ∈ [0, 1]. (iii) A function ϕ : Ω → R++ is said to be a Schur harmonically convex (or concave, respectively) function on Ω if

1 1 ≺ implies ϕ(x) ≤ (or ≥, x y

respectively) ϕ(y).

By Definition 4, it is not difficult to prove the following propositions.

Proposition 1. Let Ω ⊂ Rn++ be a set, and let

1 1 1 1
={ , ,..., : Ω x1 x2 xn

(x1 , x2 , . . . , xn ) ∈ Ω}. Then ϕ : Ω → R++ is a Schur harmonically convex (or 1 concave, respectively) function on Ω if and only if ϕ( ) is a Schur-convex (or x 1 concave, respectively) function on . Ω

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1 1 1 , there exist x, y ∈ Ω such that u = , v = . Ω x y 1 1 Let u ≺ v, that is ≺ , if ϕ : Ω → R++ is a Schur harmonically convex x y In fact, for any u, v ∈

(or concave, respectively) function on Ω, then ϕ(x) ≤ (or ≥, respectively)ϕ(y), 1 1 1 namely, ϕ( ) ≤ (or ≥, respectively)ϕ( ), this means that ϕ( ) is a Schur-convex u v x 1 (or concave, respectively) function on . The necessity is proved. The sufficiency Ω can be similar to proof. Proposition 2. f : [a, b](⊂ R++ ) → R++ is harmonically convex (or concave, 1 is concave (or convex, respectively) on respectively) if and only if g(x) = f ( x1 )   1 1 , . b a   1 1 1 1 In fact, for any x, y ∈ , , then , ∈ [a, b]. If f : [a, b](⊂ R++ ) → R++ is b a x y harmonically convex (or concave, respectively), then  f

1 αx + (1 − α)y

 ≤ (or ≥, respectively)

α 1 f( x )

1 +

1−α f ( y1 )

,

this is 1 α 1−α ≥ (or ≤, respectively) 1 + , 1 f ( αx+(1−α)y ) f(x) f ( y1 ) 1 this means that g(x) = is concave (or convex, respectively) on f ( x1 )



 1 1 , . The b a

necessity is proved. The sufficiency can be similar to proof. Lemma 2. (Schur harmonically convex function decision theorem)[5] Let Ω ⊂ Rn++ be a symmetric and harmonically convex set with inner points, and let ϕ : Ω → R++ be a continuously symmetric function which is differentiable on interior Ω0 . Then ϕ is Schur harmonically convex (or Schur harmonically concave, respectively) on Ω if and only if   2 ∂ϕ(x) 2 ∂ϕ(x) (x1 − x2 ) x1 − x2 ≥0 ∂x1 ∂x2

912

(or ≤ 0, respectively),

x ∈ Ω0 .

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3. Proof of main results Proof of Theorem 1. We only give the proof of Theorem 1 (vi) in detail. Similar argument leads to the proof of the rest part. If ϕ is increasing and Schur harmonically convex and f is decreasing and harmon1 1 1 , ,..., ) is decreasing x1 x2 xn 1 and Schur convex, and by Proposition 2, it follows that g(x) = 1 is decreasing f ( x)   1 1 and concave on , . And then from Theorem A (iii), it follows that b a     1 1 1 1 1 1 , ,..., = ϕ f ( ), f ( ), . . . , f ( ) ϕ g(x1 ) g(x2 ) g(xn ) x1 x2 xn ically convex, then by Proposition 1, it follows that ϕ(

is increasing and Schur-convex. Again by Proposition 1, it follows that ψ(x1 , x2 , . . . , xn ) = ϕ(f (x1 ), f (x2 ), . . . , f (xn )) is decreasing and Schur harmonically convex. 4. Applications Let x = (x1 , x2 , . . . , xn ) ∈ Rn . Its elementary symmetric functions are X

Er (x) = Er (x1 , x2 , . . . , xn ) =

r Y

xij , r = 1, 2, . . . , n,

1≤i1 0. The solutions of model (1)-(5) are limited to Λ, the largest dt ≤ 0  dV2 2 invariant subset of (x, y1 , ..., yn , v, z) : dV dt = 0 . It is easy to see that dt = 0 occurs at Q2 . The global asymptotic stability of Q2 follows from LaSalle’s invariance principle. 

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Example and numerical simulations

In this section, we introduce an example and perform some numerical simulations to confirm our theoretical results. By using the Lyapunov direct method, we have established a set of conditions on the functions g(x, v) and φi (yi ) and on the parameters R0M and R1M ensuring the global asymptotic stability of the steady states of model (1)-(5). We consider the following model with two stages (i.e. n = 2): πxv , (1 + γx) (1 + δv) πxv y˙ 1 = − a1 y1 , (1 + γx) (1 + δv) x˙ = λ − dx −

(33) (34)

y˙ 2 = a ˜1 y1 − a2 y2 ,

(35)

v˙ = a ˜2 y2 − pzv − uv,

(36)

z˙ = rzv − bz,

(37)

where π ∈ (0, ∞) and γ, δ ∈ [0, ∞). In this example we have φi (yi ) = yi , i = 1, ..., n, g(x, v) =

πxv , (1 + γx) (1 + δv)

which guarantee that Condition C3 holds true. Now, we verify Conditions C1, C2 and C4. Clearly, g(x, v) > 0, g(0, v) = g(x, 0) = 0 for all x, v ∈ (0, ∞), and ∂g(x, v) πv = , ∂x (1 + γx)2 (1 + δv)

∂g(x, v) πx = 2, ∂v (1 + γx) (1 + δv)

∂g(x, 0) πx = . ∂v 1 + γx

Then, for all x, v ∈ (0, ∞), we have ∂g(x,v) > 0, ∂g(x,v) > 0 and ∂g(x,0) > 0. Therefore Condition C1 is satisfied. ∂x ∂v ∂v We have also πxv πxv ∂g(x, 0) g(x, v) = ≤ =v , (1 + γx) (1 + δv) 1 + γx ∂v  0 ∂g(x, 0) π > 0 for all x > 0. = ∂v (1 + γx)2 It follows that, C2 is satisfied. Moreover,  1−

g(x, vi ) g(x, v)



g(x, v) v − g(x, vi ) vi

2

 =−

δ (v − vi ) < 0 for all v, vi ∈ (0, ∞), i = 1, 2. vi (1 + δv) (1 + δvi )

Thus, C4 is satisfied and the global stability results demonstrated in Theorems 1-3 are guaranteed. The parameters R0M and R1M are given by: R0M =

a ˜1 a ˜ 2 π x0 , a1 a2 u 1 + γx0

R1M =

x2 a ˜1 a ˜2 π . a1 a2 u (1 + γx2 ) (1 + δv2 )

(38)

Now, we will perform some numerical simulations for the model (33)-(37). The values of some parameters of the example are listed in Table 1. The other parameters π, r and γ will be varied. All computations are carried out by MATLAB. We are interested to study the following cases: Case (A): Effect of π and r on the stability of steady states: In this case, we have chosen three different initial conditions: IC(1): x(0) = 400, y1 (0) = y2 (0) = 1, v(0) = 0.2 and z(0) = 0.5, IC(2): x(0) = 600, y1 (0) = y2 (0) = 2, v(0) = 0.5 and z(0) = 1, IC(3): x(0) = 800, y1 (0) = 5, y2 (0) = 3, v(0) = 0.9 and z(0) = 1.5. The evolution of the dynamics of model (33)-(37) was observed over a time interval [0, 500]. We fix the value of γ = 0.5 and change the values of parameters π and r to get three sets as follows:

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Table 1: The values of the parameters of model (33)-(37). Parameter

Value

Parameter

Value

Parameter

Value

λ

10

a1

1

p

0.5

d

0.01

a2

1.5

r

Varied

β

Varied

a ˜1

0.5

b

0.3

γ

Varied

a ˜2

1

δ

0.1

u

3

Set (I): We choose, π = 4 and r = 0.3. Using the values of the parameters given in Table 1, we compute = 0.89 < 1 and R1M = 0.80 < 1, which means that the system has a disease-free steady state Q0 and it is GAS based on Theorem 1. Evidently, Figures 1-5 show that, the states of the system eventually approach Q0 = (1000, 0, 0, 0, 0) for the three initial conditions IC(1)-IC(3). This case corresponds to the healthy state where the viruses are cleared. Set (II): We take π = 5 and r = 0.3. With such choice we have, R1M = 0.99 < 1 < R0M = 1.11. Consequently, Lemma 1 and Theorem 2 state that, Q1 exists and it is GAS. Figures 1-5 show that the numerical simulations illustrate our theoretical results given in Theorem 2. We observe that, the trajectory of the system will converge to Q1 = (140.43, 8.60, 2.87, 0.96, 0) for the three initial conditions IC(1)-IC(3). This case corresponds to a chronic infection but with inactive immune response. Set (III): We choose, π = 5 and r = 1. Then, we calculate R0M = 1.11 > 1 and R1M = 1.08 > 1, this means that, the system has three steady states Q0 , Q1 and Q2 . Thus, from Theorem 3, Q2 is GAS. From Figures 1-5, we observe a consistency between the numerical results and theoretical results of Theorem 3. We observe that, the trajectory of the system show oscillating behavior for a period before reaching Q2 = (709.56, 2.90, 0.97, 0.3, 0.45), in the same time frame for the three initial conditions IC(1)-IC(3). Case (B): Effect of γ on the stability of the steady states Let us consider π and r be fixed. In this case, we take the values of π = 5 and r = 1, and consider different values of γ. Here we take the initial condition as given in IC(1), while the evolution of the dynamics of model (33)-(37) was observed over a time interval [0, 600]. Table 2 contains the values of the bifurcation parameters R0M and R1M with different values of γ of model (33)-(37). R0M

Table 2: The values of the threshold parameters R0M and R1M with different values of γ of model (33)-(37). Different values of γ

R0M

R1M

The equilibria

0.30

1.85

1.79

Q2 = (517.67, 4.82, 1.61, 0.3, 4.72)

0.40

1.39

1.34

Q2 = (637.35, 3.63, 1.21, 0.3, 2.06)

0.54

1.03

0.996

Q1 = (763.16, 2.37, 0.79, 0.26, 0)

0.55

1.01

0.98

Q1 = (926.89, 0.73, 0.24, 0.08, 0)

0.60

0.92

0.90

Q0 = (1000, 0, 0, 0, 0)

0.70

0.79

0.77

Q0 = (1000, 0, 0, 0, 0)

Table 2 and Figures 6-10 show that, when γ is increased, the infection rate is decreased which leads to an increase in the concentration of the uninfected cells and a decrease on the concentrations of the (first/second) stage of infected cells, free viruses and B cells. Case (C): Effect of the multiple stages of infected cells on the dynamics of virus dynamics: To show the effect of multiple stages of infected cells on the dynamical behavior of the virus, we consider

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the following model with single stage of infected cells and compare it with model (33)-(37): πxv , (1 + γx) (1 + δv) πxv − a1 y1 , y˙ 1 = (1 + γx) (1 + δv)

x˙ = λ − dx −

(39) (40)

v˙ = a ˜1 y1 − pzv − uv,

(41)

z˙ = rzv − bz.

(42)

Consequently, the bifurcation parameters for this system are given by: R0single =

a ˜ 1 π x0 , a1 u 1 + γx0

R1single =

a ˜1 π x2 . a1 u (1 + γx2 ) (1 + δv2 )

(43)

Since a ˜i < ai , then from Eqs. (38) and (43) we have a ˜ 1 π x0 a ˜1 a ˜ 2 π x0 < = R0single , a1 a2 u 1 + γx0 a1 u 1 + γx0 a ˜1 a ˜2 π x2 a ˜1 π x2 = < = R1single . a1 a2 u (1 + γx2 ) (1 + δv2 ) a1 u (1 + γx2 ) (1 + δv2 )

R0M = R1M

Here we consider the following initial condition: x(0) = 400, y1 (0) = 0.5, y2 (0) = 1, v(0) = 0.2 and z(0) = 0.5. The evolution of the dynamics of models (33)-(37) and (39)-(42) was observed over a time interval [0, 600]. Let us consider the values of parameters listed in Table 1 and choose the values π = 3.5, r = 1.5 and γ = 0.5. By calculating the bifurcation parameters for systems (33)-(37) and (39)-(42), we obtain R0M = 0.78 < 1.16 = R0single ,

R1M = 0.76 < 1.14 = R1single .

Therefore, with the same values of the parameters, the steady state Q0 is stable for system (33)-(37) but unstable for system (39)-(42). The presence of multiple stages of infected cells reduces the infection progress. Figures 11-14 show a comparison between the evolution of the uninfected cells, infected cells, free virus particles and B cells of the two systems (33)-(37) and (39)-(42). We observe that, the concentration of uninfected cells of the model with three stages of infected cells is larger than that of system with only one single stage of infected cells (see Figures 11), while the concentrations of first stage of infected cells, viruses and B cells with three stages are less than that of system with a single stage of infected cells (see Figures 12-14). From a biological point of view, the multiple stages of infected cells plays a similar role as antiviral treatment in eliminating the virus. We observe that, if the number of stages of infected cells is increased, then the viral replication is suppressed and the viruses can be cleared from the body. This give us some suggestions on new drugs to increase the number of stages of infected cells.

7

Conclusion

We have studied a general virus dynamics model with humoral immunity. We have assumed that the infected cells passes through n-stages to produce mature viruses. We have obtained two bifurcation parameters, the basic reproduction number and the humoral immunity number. We have established a set of sufficient conditions which guarantee the global stability of the model. The global asymptotic stability of the three steady states, Q0 , Q1 and Q2 has been investigated by constructing Lyapunov functionals and using LaSalle’s invariance principle. To support our theoretical results, we have presented an example and conducted some numerical simulations.

8

Acknowledgment

This article was funded by the Deanship of Scientific Research (DSR), King Abdulaziz University, Jeddah. The authors, therefore, acknowledge with thanks DSR technical and financial support.

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1000 Set (I) 900 800

Uninfected target cells

700 Set (III) 600 500 400 300 200 100 0

Set (II)

50

100

150

200

250 Time

300

350

400

450

500

Figure 1: The uninfected cells for model (33)-(37).

10 Set (II)

9

First stage of infected cells

8 7 6 5 4 Set (III) 3 2 1 Set (I) 0 0

50

100

150

200

250 Time

300

350

400

450

500

Figure 2: The first stage infected cells for model (33)-(37).

3 Set (II)

Second stage of infected cells

2.5

2

1.5 Set (III) 1

0.5 Set (I) 0 0

50

100

150

200

250 Time

300

350

400

450

500

Figure 3: The second stage infected cells for model (33)-(37).

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Set (II)

1

Free virus particles

0.8

0.6

0.4 Set (III)

0.2 Set (I) 0 0

50

100

150

200

250 Time

300

350

400

450

500

Figure 4: The free virus particles for model (33)-(37).

3.5

3

2.5

B cells

2

1.5

1 Set (III)

Sets (I) & (II) 0.5

0 0

50

100

150

200

250 Time

300

350

400

450

500

Figure 5: The B cells for model (33)-(37).

1000

Uninfected target cells

900

800

700

600

500 γ=0.30 γ=0.54 γ=0.60

400 0

100

200

300 Time

400

500

600

Figure 6: The uninfected target cells for model (33)-(37) under different values of γ.

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14 γ=0.30 γ=0.54 γ=0.60

First stage of infected cells

12

10

8

6

4

2

0 0

100

200

300 Time

400

500

600

Figure 7: The first stage infected cells for model (33)-(37) under different values of γ.

4.5 γ=0.30 γ=0.54 γ=0.60

4

Second stage of infected cells

3.5 3 2.5 2 1.5 1 0.5 0 0

100

200

300 Time

400

500

600

Figure 8: The second stage infected cells for model (33)-(37) under different values of γ.

1 0.9

γ=0.30 γ=0.54 γ=0.60

0.8

Free virus particles

0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 0

100

200

300 Time

400

500

600

Figure 9: The free virus particles for model (33)-(37) under different values of γ.

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12 γ=0.30 γ=0.54 γ=0.60

10

B cells

8

6

4

2

0 0

100

200

300 Time

400

500

600

Figure 10: The B cells for model (33)-(37) under different values of γ.

1000

Uninfected target cells

900

800

700

600

500

400 0

For two stages of infected For single stage of infected

100

200

300 Time

400

500

600

Figure 11: Comparison on the concentration of the uninfected cells for systems (33)-(37) and (39)-(42).

2.5 For two stages of infected For single stage of infected

First stage of infected cells

2

1.5

1

0.5

0 0

100

200

300 Time

400

500

600

Figure 12: Comparisons on the concentration of the first stage of infected cells for systems (33)-(37) and (39)-(42).

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0.35 For two stages of infected For single stage of infected

0.3

Free virus particles

0.25

0.2

0.15

0.1

0.05

0 0

100

200

300 Time

400

500

600

Figure 13: Comparisons on the concentration of the free virus particles for systems (33)-(37) and (39)-(42).

1.8 For two stages of infected For single stage of infected

1.6 1.4

B cells

1.2 1 0.8 0.6 0.4 0.2 0 0

100

200

300 Time

400

500

600

Figure 14: Comparisons on the concentration of the B cells for systems (33)-(37) and (39)-(42).

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References [1] M. A. Nowak and R. M. May, Virus dynamics: Mathematical Principles of Immunology and Virology, Oxford Uni., Oxford, (2000). [2] C. Connell McCluskey, Yu Yang, Global stability of a diffusive virus dynamics model with general incidence function and time delay, Nonlinear Anal. Real World Appl., 25 (2015), 64-78. [3] P. K. Roy, A. N. Chatterjee, D. Greenhalgh, and Q. J.A. Khan, Long term dynamics in a mathematical model of HIV-1 infection with delay in different variants of the basic drug therapy model, Nonlinear Anal. Real World Appl., 14 (2013), 1621-1633. [4] A. M. Elaiw, and N. H. AlShamrani, Stability analysis of general viral infection models with humoral immunity, Journal of Nonlinear Science and Applications, 9 (2016), 684-704. [5] M. Y. Li and L. Wang, Backward bifurcation in a mathematical model for HIV infection in vivo with anti-retroviral treatment, Nonlinear Anal. Real World Appl., 17 (2014), 147-160. [6] G. Huang, W. Ma, Y. Takeuchi, Global properties for virus dynamics model with Beddington–DeAngelis functional response, Appl. Math. Lett., 22 (2009), 1690-1693. [7] A.M. Elaiw and S.A. Azoz, Global properties of a class of HIV infection models with Beddington-DeAngelis functional response, Math. Methods Appl. Sci., 36 (2013), 383-394. [8] A.M. Elaiw, Global properties of a class of HIV models, Nonlinear Anal. Real World Appl., 11 (2010), 2253-2263. [9] A. M. Elaiw, Global properties of a class of virus infection models with multitarget cells, Nonlinear Dynam., 69 (2012), 423-435. [10] A. M. Elaiw, Global stability analysis of humoral immunity virus dynamics model including latently infected cells, Journal of Biological Dynamics, DOI:10.1080/17513758.2015.1056846. [11] L. Wang and M. Y. Li, Mathematical analysis of the global dynamics of a model for HIV infection of CD4 + T cells, Math. Biosci., 200(1) (2006), 44-57. [12] K. Hattaf, N. Yousfi and A. Tridane, Mathematical analysis of a virus dynamics model with general incidence rate and cure rate, Nonlinear Anal. Real World Appl., 13(4) (2012), 1866-1872. [13] K. Wang, A. Fan and A. Torres, Global properties of an improved hepatitis B virus model, Nonlinear Anal. Real World Appl., 11 (2010), 3131-3138. [14] X. Song and A. Neumann, Global stability and periodic solution of the viral dynamics, J. Math. Anal. Appl., 329 (2007), 281-297. [15] A. U. Neumann, N. P. Lam, H. Dahari, D. R. Gretch, T. E. Wiley, T. J, Layden and A. S. Perelson, Hepatitis C viral dynamics in vivo and the antiviral efficacy of interferon-alpha therapy, Science, 282 (1998), 103-107. [16] L. Wang, M. Y. Li, and D. Kirschner, Mathematical analysis of the global dynamics of a model for HTLV-I infection and ATL progression, Math. Biosci., 179 (2002) 207-217. [17] A. M. Elaiw, and N. A. Almuallem, Global properties of delayed-HIV dynamics models with differential drug efficacy in co-circulating target cells, Applied Mathematics and Computation, 265 (2015), 1067-1089. [18] A. M. Elaiw, and N. A. Almuallem, Global dynamics of delay-distributed HIV infection models with differential drug efficacy in cocirculating target cells, Mathematical Methods in the Applied Sciences, 39 (2016), 4-31.

966

A. M. Elaiw et al 949-967

J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 22, NO.5, 2017, COPYRIGHT 2017 EUDOXUS PRESS, LLC

[19] A. Korobeinikov, Global properties of basic virus dynamics models, Bull. Math. Biol., 66 (2004), 879-883. [20] P. Georgescu and Y.H. Hsieh, Global stability for a virus dynamics model with nonlinear incidence of infection and removal, SIAM J. Appl. Math., 67 (2) (2006), 337-353. [21] G. Huang, Y. Takeuchi and W. Ma, Lyapunov functionals for delay differential equations model of viral infections, SIAM J. Appl. Math., 70(7) (2010), 2693-2708. [22] Z. Grossman, M. Polis, M. B. Feinberg, Z. Grossman, I. Levi, S. Jankelevich, R. Yarchoan, J. Boon, F. de Wolf, J. M.A. Lange, J. Goudsmit, D. S. Dimitrov and W. E. Paul, Ongoing HIV dissemination during HAART, Nat. Med., 5 (1999), 1099-1104. [23] J. Wang and S. Liu, The stability analysis of a general viral infection model with distibuted delays and multi-staged infected progression, Commun. Nonlinear Sci. Numer. Simul., 20(1) (2015), 263–272. [24] J. A. Deans and S. Cohen, Immunology of malaria, Ann. Rev. Microbiol. 37 (1983), 25-49. [25] A. Murase, T. Sasaki and T. Kajiwara, Stability analysis of pathogen-immune interaction dynamics, J. Math. Biol., 51 (2005), 247-267. [26] M. A . Obaid and A. M. Elaiw, Stability of virus infection models with antibodies and chronically infected cells, Abstr. Appl. Anal, (2014) Article ID 650371. [27] S. Wang and D. Zou, Global stability of in host viral models with humoral immunity and intracellular delays, Appl. Math. Model., 36 (2012), 1313-1322. [28] T. Wang, Z. Hu and F. Liao, Stability and Hopf bifurcation for a virus infection model with delayed humoral immunity response, J. Math. Anal. Appl., 411 (2014) 63-74. [29] A. M. Elaiw and N. H. AlShamrani, Global properties of nonlinear humoral immunity viral infection models, Int. J. Biomath., 8(5) (2015), 1550058, 53 pages. [30] A. M. Elaiw and N. H. AlShamrani, Global stability of humoral immunity virus dynamics models with nonlinear infection rate and removal, Nonlinear Anal. Real World Appl., 26 (2015), 161-190. [31] T. Wang, Z. Hu, F. Liao and W. Ma, Global stability analysis for delayed virus infection model with general incidence rate and humoral immunity, Math. Comput. Simulation, 89 (2013), 13-22. [32] R. Larson and B. H. Edwards, Calculus of a single variable, Cengage Learning, Inc., USA, (2010). [33] J.K. Hale and S. Verduyn Lunel, Introduction to functional differential equations, Springer-Verlag, New York, (1993). [34] A.S. Perelson and P.W. Nelson, Mathematical analysis of HIV-1 dynamics in vivo, SIAM Rev., 41 (1999), 3–44. [35] P. De. Leenheer and H. L. Smith, Virus dynamics: A global analysis, SIAM J. Appl. Math., 63 (2003), 1313-1327. [36] G. Huang, W. Ma, Y. Takeuchi, Global properties for virus dynamics model with Beddington–DeAngelis functional response, Appl. Math. Lett. 22 (2009), 1690–1693.

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On the dynamics of a certain four-order fractional difference equations Chang-you Wang1,2, Xiao-jing Fang1,2, Rui Li1* 1. College of Automation, Chongqing University of Posts and Telecommunications, Chongqing 400065 P. R. China 2. Key Laboratory of Industrial Internet of Things & Networked Control of Ministry of Education, Chongqing University of Posts and Telecommunications, Chongqing 400065 P.R. China Abstract: This paper is concerned with the following rational recursive sequences x x y y xn +1 = n −1 n − 2 ，yn +1 = n −1 n − 2 , n = 0, 1, , A + Byn −3 C + Dxn −3 where the parameters A, B, C , D are positive constants. The initial condition x−3 , x−2 ,

x−1 , x0 and y−3 , y−2 , y−1 , y0 are arbitrary nonnegative real numbers. We give sufficient conditions under which the equilibrium (0, 0) of the system is globally asymptotically stable, which extends and includes corresponding results obtained in the cited references [12-17]. Moreover, the asymptotic behavior of others equilibrium points is also studied. Our approach to the problem is based on new variational iteration method for the more general nonlinear difference equations and inequality skills as well as the linearization techniques. Keywords: recursive sequences; equilibrium point; asymptotical stability; positive solutions.

1. Introduction Nonlinear Difference equations have been studied because they model numerous real life problems in biology, ecology, physics, economics and so forth [1-5]. Today, with the dramatically development of computer-based computational techniques, difference equations are found to be much appropriate mathematical representations for computer simulation, experiment and computation, which play an important role in realistic applications [6]. Therefore, recently there has been an increasing interest in the study of qualitative analysis of rational difference equations. And the present cardinal problem of *

Corresponding author at: College of Automation, Chongqing University of Posts and Telecommunications, Chongqing 400065 P. R. China, Chongqing 400065, PR China. Email address: [email protected]

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asymptotic behavior of solutions for a rational difference equation has received extensive attention from researchers (see, e.g., [7-11] and the references therein). Elabbasy [12] obtained the form of the solutions of the following rational difference system xn +1 =

xn −1 yn −1 ， yn +1 = ±1 + xn -1 yn ∓1 + yn -1 xn

(1.1)

with nonzero real number initial conditions. In particular, Clark and Kulenovic [13, 14] discussed the global stability properties and asymptotic behavior of solutions for the recursive sequence xn yn , , (1.2) xn +1 = yn +1 = n = 0,1, , a + cyn b + dxn where a, b, c, d ∈ (0, ∞) and the initial conditions x0 and y0 are arbitrary nonnegative numbers. In 2012, Zhang et al. [15] investigated the stability character and asymptotic behavior of the solution for the system of difference equations xn -2 yn -2 xn +1 = , yn +1 = , n = 0,1, , (1.3) B + yn − 2 yn −1 yn A + xn − 2 xn −1 xn

where A, B ∈ (0, ∞) , and the initial conditions x−3 , x−2 , x−1 , x0 , y−3 , y−2 , y−1 , y0 ∈ (0, ∞). Recently, the following nonlinear two-dimensional difference systems

xn +1 = ϕ ( xn −t1 , yn − s1 ),

yn +1 = ψ ( yn − s2 , xn −t2 ) ,

(1.4)

where t1 , s1 , s2 , t2 are all positive integers, was studied by Liu et al. [16], in which they gave some sufficient conditions such that every positive solution of this equation converges to the unique equilibrium point. More recently, in [17] the authors studied analogous results for the system of difference equations xn +1 = axn + byn −1e − xn ,

yn +1 = cyn + dxn −1e− yn ,

(1.5)

where a, b, c, d are positive constants and the initial values x1 , x0 , y1 , y0 are positive numbers. For more related work, one can refer to [18-22] and references therein. Inspired by the above works, the essential problem we consider in this paper is the asymptotic behavior of the solution for the difference equation x x y y (1.6) xn +1 = n −1 n − 2 ， yn +1 = n −1 n − 2 , n = 0, 1, , A + Byn −3 C + Dxn −3 where the initial conditions x−3 , x−2 , x−1 , x0 ∈ (0, ∞) , y−3 , y−2 , y−1, y0 ∈(0, ∞) and A, B, C , D are positive constants. This paper proceeds as follows. In Section 2, we introduce some definitions and preliminary results. The main results and their proofs are given in Section 3.

2. Preliminaries 2 969

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Let I x , I y be some intervals of real numbers and f : I x4 × I y4 → I x , g : I x4 × I y4 → I y be continuously differentiable functions. Then for every initial conditions ( xi , yi ) ∈ I x × I y , (i = −3, −2, −1, 0) , the system of difference equations ⎧ xn +1 = f (xn ,xn -1 ,xn -2 ,xn -3 ,yn ,yn -1 ,yn -2 ,yn -3 ), ⎨ ⎩ yn +1 = g (xn ,xn -1 ,xn -2 ,xn -3 ,yn ,yn -1 ,yn -2 ,yn -3 ),

n = 0, 1, 2,

(2.1)

,

has a unique solution {(xn ,yn )}∞n =−3 . A point (x , y ) ∈ I x × I y is called an equilibrium point of (2.1) if x = f ( x , x , x , x , y , y , y , y ), y = g ( x .x , x , x , y , y , y , y ) , i. e., ( xn , yn ) = ( x , y ) for all n ≥ 0 . Interval I x × I y is called invariant for system (2.1) if, for all n > 0 , xn ∈ I x , yn ∈ I y when the initial conditions x−3 , x−2 x−1 , x0 ∈ I x , y−3 , y−2 , y−1 , y0 ∈ I y . Definition 2.1 Assume that ( x , y ) is a fixed point of (2.1). Then

(i) ( x , y ) is said to be stable relative to I x × I y if for every ε > 0 , there exits δ > 0 such that for any initial conditions ( xi, yi ) ∈ I x × I y (i = −3, −2, −1,0) , with

∑

0 i =−3

∑

0 i =−3

xi − x < δ ,

yi − y < δ , implies xn − x < ε , yn − y < ε .

(ii) ( x , y ) is called an attractor relative to I x × I y if for all ( xi , yi ) ∈ I x × I y

(i = −3, −2, −1, 0) , limn→∞ xn = x , limn →∞ yn = y . (iii) ( x , y ) is called asymptotically stable relative to I x × I y if it is stable and an attractor. (iv) Unstable if it is not stable. Theorem 2.1 Assume that X (n + 1) = F ( X ( n)) , n = 0,1,

, is a system of difference

equations and X is the equilibrium point of this system i.e., F ( X ) = X . (i) If all eigenvalues of the Jacobian matrix J F , evaluated at X lie inside the open unit disk λ < 1 , then X is locally asymptotically stable. (ii) If all eigenvalues of the Jacobian matrix J F , evaluated at X has modulus greater than one then X is unstable. Definition 2.2 Let p , q , s , t be four nonnegative integers such that p + q = s + t = n . Splitting ( x , y ) = ( x1 , x 2 ,

, x n , y1 , y 2 ,

, y n ) into ( x , y ) = ( [ x ] p , [ x ] q , [ y ] s , [ y ] t ) ,

where [ x]σ denotes a vector with σ -components of x , we say that the function f ( x1 , x2 , R 2 n if

, xn , y1 , y2 ,

, yn ) possesses a mixed monotone property in subsets I 2 n of

f ([ x] p ,[ x]q ,[ y ]s ,[ y ]t ) is monotone nondecreasing in each component of

[ x ] p , [ y ] s and is monotone nonincreasing in each component of [ x ] q , [ y ] t for ( x, y ) ∈ I 2 n . In particular, if q = t = 0 , then it is said to be monotone nondecreasing 3 970

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in I 2 n .

3. The Main Results In this section, we investigate the asymptotic behavior of the equilibrium points of the systems (1.6). It is easy to know that the systems (1.6) have four equilibrium points (0, 0) , (0, C ) , ( A, 0) , and (( A + BC) / (1− BD), (C + AD) / (1− BD)) . Theorem 3.1 The equilibrium point (0, 0) of (1.6) is locally asymptotically stable. Proof. We can easily obtain that the linearized system of (1.6) about the equilibrium point (0, 0) is

ϕn +1 = Dϕ n

(3.1)

where ⎡ xn ⎤ ⎢x ⎥ ⎢ n −1 ⎥ ⎢ xn − 2 ⎥ ⎢ ⎥ xn −3 ⎥ ⎢ ϕ n= ⎢ , yn ⎥ ⎢ ⎥ ⎢ yn −1 ⎥ ⎢y ⎥ ⎢ n−2 ⎥ ⎢⎣ yn −3 ⎥⎦

⎡0 ⎢1 ⎢ ⎢0 ⎢ 0 D=⎢ ⎢0 ⎢ ⎢0 ⎢0 ⎢ ⎣⎢0

0 0 0 0 0 0 0⎤ 0 0 0 0 0 0 0 ⎥⎥ 1 0 0 0 0 0 0⎥ ⎥ 0 1 0 0 0 0 0⎥ . 0 0 0 0 0 0 0⎥ ⎥ 0 0 0 1 0 0 0⎥ 0 0 0 0 1 0 0⎥ ⎥ 0 0 0 0 0 1 0 ⎦⎥

(3.2)

Thus, the characteristic equation of (3.2) is f (λ ) = λ 8 = 0 . This shows that all the roots of characteristic equation lie inside unit disk. So the equilibrium （0, 0） is locally asymptotically stable. Theorem 3.2 Let ⎧ xn+1 = f ( xn , xn−1 , ⎨ ⎩ yn+1 = g ( xn , xn−1 ,

, xn−k , yn , yn−1 ,

, yn−k ),

, xn−k , yn , yn−1 ,

, yn−k ),

n = 0,1,

,

(3.3)

[a, b] be an interval of real numbers and assume that f :[a, b]k +1 ×[c, d ]k +1 → [a, b] and g :[a, b]k +1 × [c, d ]k +1 → [c, d ] are two continuous functions satisfying the mixed monotone

property. If there exit m0 ≤ min{x− k , x− k +1 ,

, x0 } ≤ max{x− k , x− k +1 ,

, x0 } ≤ M 0 ,

n0 ≤ min{ y− k , y− k +1 ,

, y0 } ≤ max{ y− k , y− k +1 ,

, y0 } ≤ N 0

and such that m0 ≤ f ([m0 ] p ,[ M 0 ]q ,[n0 ]s ,[ N 0 ]t ) ≤ f ([ M 0 ] p ,[m0 ]q ,[ N 0 ]s ,[n0 ]t ) ≤ M 0 ,

(3.4)

n0 ≤ g ([m0 ] p ,[ M 0 ]q ,[n0 ]s ,[ N 0 ]t ) ≤ g ([ M 0 ] p ,[m0 ]q ,[ N 0 ]s ,[n0 ]t ) ≤ N 0 ,

(3.5)

and

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then there exit (m, M ) ∈ [m0 , M 0 ]2 and (n, N ) ∈ [n0 , N 0 ]2 satisfying M = f ([ M ] p ,[m]q ,[ N ]s ,[n]t ), m = f ([ m] p ,[ M ]q ,[ n]s ,[ N ]t ) ,

(3.6)

N = g ([ M ] p ,[m]q ,[ N ]s ,[n]t ), n = g ([m] p ,[ M ]q ,[n]s ,[ N ]t ) .

(3.7)

and Moreover, if m = M and n = N , then the system (3.3) has a unique equilibrium point ( x , y ) ∈ [m0 , M 0 ] × [n0 , N 0 ] and every solution of (3.3) converges to ( x , y ) . Proof. Using m0 , M 0 and n0 , N 0 as two couples of initial iteration, we construct four

sequences {mi },{M i },{ni } and {N i } (i = 1, 2, ) from the following equations mi = f ([mi −1 ] p ,[ M i −1 ]q ,[ni −1 ]s ,[ N i −1 ]t ), M i = f ([ M i −1 ] p ,[mi −1 ]q ,[ N i −1 ]s ,[ni −1 ]t ) , and ni = g ([mi −1 ] p ,[ M i −1 ]q ,[ni −1 ]s ,[ N i −1 ]t ), N i = g ([ M i −1 ] p ,[mi −1 ]q ,[ N i −1 ]s ,[ni −1 ]t ) . It is obvious from the mixed monotone property of functions f and g that the sequences{mi },{M i },{ni } and{N i } (i = 1, 2, ) possess the following monotone property

m0 ≤ m1 ≤

≤ mi ≤

≤ Mi ≤

≤ M1 ≤ M 0 ,

(3.8)

≤ ni ≤

≤ Ni ≤

≤ N1 ≤ N 0 ,

(3.9)

and

n0 ≤ n1 ≤ where i =0,1,2,

.

Moreover, one has

mi ≤ xl ≤ M i

for l ≥ (k + 1)i + 1, i = 0,1, 2,

.

(3.10)

and

ni ≤ yl ≤ N i

for l ≥ (k + 1)i + 1, i = 0,1, 2,

.

(3.11)

m = lim mi , M = lim M i , n = lim ni , N = lim N i ,

(3.12)

m ≤ lim inf xi ≤ lim sup xi ≤ M , n ≤ lim inf yi ≤ lim sup yi ≤ N .

(3.13)

Set i →∞

i →∞

i →∞

i →∞

then i →∞

i →∞

i →∞

i →∞

By the continuity of f and g , we have M = f ([ M ] p ,[m]q ,[ N ]s ,[n]t ), m = f ([m] p ,[ M ]q ,[n]s ,[ N ]t ) ,

(3.14)

N = g ([ M ] p ,[m]q ,[ N ]s ,[n]t ), n = g ([m] p ,[ M ]q ,[n]s ,[ N ]t ) .

(3.15)

and Moreover, if m = M , n = N , then m = M = lim xi = x , n = N = lim yi = y , and then the i →∞

i →∞

proof is complete. Theorem 3.3 If A = C , B = D , the equilibrium point （0, 0） of the systems (1.6) is a global attractor for any initial conditions ( x−3 , x−2 , x−1 , x0 , y−3 , y−2 , y−1 , y0 ) ∈ (0, A)8 . 5 972

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Proof. Let ( f , g ) : (0, ∞) 4 × (0, ∞) 4 → (0, ∞) × (0, ∞) be a function defined by

f ( x−3 , x−2 , x−1 , x0 , y−3 , y−2 , y−1 , y0 ) =

xn −1 xn − 2 , A + Byn −3

g ( x−3 , x−2 , x−1 , x0 , y−3 , y−2 , y−1 , y0 ) =

yn −1 yn − 2 . A + Bxn −3

and

We can easily see that the functions f and g possess a mixed monotone property in subsets (0, A)8 of R8 . Let M 0 = N 0 = max{x−3 , x−2 , x−1 , x0 , y−3 , y−2 , y−1 , y0 },

M0 − A < n0 = m0 < 0 . B

We have m0 ≤

m02 M 02 ≤ ≤ M0, A + BN 0 A + Bn0

(3.16)

n0 ≤

n02 N 02 ≤ ≤ N0 , A + BM 0 A + Bm0

(3.17)

Then from (1.6) and Theorem 3.2, there exit m, M ∈ [m0 , M 0 ] , n, N ∈ [n0 , N 0 ] satisfying m=

m2 , A + BN

M2 , A + Bn

(3.18)

N2 N= . A + Bm

(3.19)

M=

n2 n= , A + BM In view of

m < M < M 0 < A + Bn0 < A + Bn < A + BN , and n < N < N 0 < A + Bm0 < A + Bm < A + BM , thus, one has M =m= N =n=0. (3.20) It follows by Theorem 3.2 that the equilibrium point （0, 0） of (1.6) is a global attractor.

The proof is complete. Theorem 3.4 The equilibrium point （0, 0） of the system (1.6) is global asymptotically stability for any initial conditions ( x−3 , x−2 , x−1 , x0 , y−3 , y−2 , y−1 , y0 ) ∈ (0, A)8 . Proof. The result follows from Theorems 3.1 and 3.3. Theorem 3.5 The equilibrium point (0, C ) , ( A, 0) of the system (1.6) is unstable. Proof. We can easily obtain that the linearized system of the system (1.6) about the equilibrium (0, C ) is 6 973

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ϕn +1 = D*ϕn ,

(3.21)

where ⎡ xn ⎤ ⎡0 0 0 ⎢x ⎥ ⎢1 0 0 ⎢ n −1 ⎥ ⎢ ⎢ xn − 2 ⎥ ⎢0 1 0 ⎢ ⎥ ⎢ xn −3 ⎥ ⎢ * ⎢0 0 1 , D ϕ n= ⎢ = ⎢0 0 0 y ⎥ ⎢ n ⎥ ⎢ ⎢ yn −1 ⎥ ⎢0 0 0 ⎢y ⎥ ⎢0 0 0 ⎢ n−2 ⎥ ⎢ ⎢⎣ yn −3 ⎥⎦ ⎢⎣0 0 0

0 0 0 0 -D 0 0 0

0 0 0 0 0 1 0 0

0 0 0 0 1 0 1 0

0 0 0 0 1 0 0 1

0⎤ 0 ⎥⎥ 0⎥ ⎥ 0⎥ 0⎥ ⎥ 0⎥ 0⎥ ⎥ 0 ⎥⎦

The characteristic equation of the systems (3.20) is P (λ )=λ 5 (λ 3 − λ − 1).

(3.22)

It is obvious that P (1) = −1, P(2) = 160 . It follows by the intermediate value theorem for continuous function that there exists λ > 1 so that P (λ ) = 0 . Therefore, one of the roots of characteristic equation (3.22) lies outside unit disk. According to Theorem 2.1, the equilibrium (0, C ) is unstable. Similarly, we can obtain that the unique equilibrium ( A, 0) is unstable.

A + BC C + AD , ) is locally 1 − BD 1 − BD asymptotically stable. If BD > 1 , the equilibrium point ( x , y ) is unstable. Theorem 3.6 If BD < 1 , the equilibrium point ( x , y ) = (

Proof. We can easily obtain that the linearized system of (1.6) about the equilibrium ( x , y ) is

ϕn +1 = D*ϕn ,

(3.23)

where ⎡ xn ⎤ ⎡0 0 0 0 ⎢x ⎥ ⎢1 0 0 0 ⎢ n −1 ⎥ ⎢ ⎢ xn − 2 ⎥ ⎢0 1 0 0 ⎢ ⎥ ⎢ xn −3 ⎥ ⎢ * ⎢0 0 1 0 , D = ϕ n= ⎢ ⎢0 0 0 -D y ⎥ ⎢ n ⎥ ⎢ ⎢ yn −1 ⎥ ⎢0 0 0 0 ⎢y ⎥ ⎢0 0 0 0 ⎢ n−2 ⎥ ⎢ ⎢⎣ yn −3 ⎥⎦ ⎢⎣0 0 0 0

0 0 0 0 0 1 0 0

0 0 0 0 1 0 1 0

0 0 0 0 1 0 0 1

-B ⎤ 0 ⎥⎥ 0⎥ ⎥ 0⎥ 0⎥ ⎥ 0⎥ 0⎥ ⎥ 0 ⎥⎦

The characteristic equation of (3.23) is P (λ ) = λ 8 − BD = 0.

(3.24)

In view of BD < 1 , this shows that all the roots of characteristic equation lie inside unit disk, so the unique equilibrium ( x , y ) is locally asymptotically stable. If BD > 1 , one of the roots of characteristic equation lie outside unit disk, so the unique equilibrium 7 974

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J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 22, NO.5, 2017, COPYRIGHT 2017 EUDOXUS PRESS, LLC

( x , y ) is unstable.

4. Conclusions This paper presents the use of a variational iteration method for systems of nonlinear difference equations. This technique is a powerful tool for solving various difference equations and can also be applied to other nonlinear differential equations in mathematical physics. The variational iteration method provides an efficient method to handle the nonlinear structure. We have dealt with the problem of global asymptotic stability analysis for a class of nonlinear difference equations. The general sufficient conditions have been obtained to ensure the existence, uniqueness and global asymptotic stability of the equilibrium point (0, 0) for the nonlinear difference equation. These criteria generalize and improve some known results in [12-17]. Moreover, the asymptotic behavior of others equilibrium points is also studied. In addition, the sufficient conditions that we obtained are very simple, which provide flexibility for the application and analysis of nonlinear difference equation. Remark: Our model and results are different from the existence ones such as those of

References [12-17]. In particular, the new variational iteration method can be applied to the models of References [12-17] and the more general nonlinear difference equations. In some sense, we enrich the theoretical results of the difference equations.

Acknowledgements This work is supported by Science Fund for Distinguished Young Scholars (cstc2014jc yjjq40004) of China, the National Nature Science Fund (Project nos.11372366 and 61503053) of China, the Science and Technology Project of Chongqing Municipal Education Committee (Grants no. kj1400423) of China, and the excellent talents project of colleges and universities in Chongqing of China.

References 1. Elsayed EM, Iricanin BD: On a max-type and a min-type difference equation. Appl. Math. Comput. 215, 608-614 (2009) 2. Li WT, Zhang YH, Su YH: Global attractivity in a higher-order nonlinear difference equation. Acta Mathematica Scientia 25, 59-66 (2005) 3. Yang ZG, Xu, DY: Mean square exponential stability of impulsive stochastic difference equations. Appl. Math. Lett. 20, 938-945 (2007) 4. Jia XM, Hu LX, Li WT: Dynamics of a rational difference equation. Advances in Difference Equations. 2010, Article ID 970720 (2010) 5. Li WT, Sun HR: Global attractivity in a rational recursive sequence. Dynamic Systems and Applications 11, 339-345 (2002) 6. Elaydi S: An Introduction to Difference Equations, third ed., Springer, New York 8 975

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(2005) 7. Muroya Y, Ishiwata E, Guglielmi N: Global stability for nonlinear difference equations with variable coefficients. J. Math. Anal. Appl. 334, 232-247 (2007) 8. Stević S: Solutions of a max-type system of difference equations. Appl. Math. Comput. 218, 9825-9830 (2012) 9. Papaschinopoulos G, Radin M, Schinas CJ: Study of the asymptotic behavior of the solutions of three systems of difference equations of exponential form. Appl. Math. Comput. 218, 5310-5318 (2012) 10. Migda J: Asymptotically polynomial solutions of difference equations, Advances in Difference Equations 2013, 92 (2013) 11. Hu LX, He WS, Xia HM: Global asymptotic behavior of a rational difference equation. Appl. Math. Comput. 218, 7818-7828 (2012) 12. Elsayed EM: Solutions of rational difference systems of order two. Mathematical and Computer Modelling 55, 378-384 (2012) 13. Clark D, Kulenovic MRS: A coupled system of rational difference equations. Computers and Mathematics with Applications 43, 849-867(2002) 14. Clark D, Kulenovic MRS, Selgrade JF: Global asymptotic behavior of a twodimensional difference equation modelling competition. Nonlinear Anal. 52, 17651776 (2003) 15. Zhang QH, Yang LH, Liu JZ: Dynamics of a system of rational third-order difference equation. Advances in Difference Equations 2012, 136 (2012) 16. Liu WP, Yang XF, Liu XZ: Dynamics of a family of two-dimensional difference systems. Appl. Math. Comput. 219, 5949-5955 (2013) 17. Papaschinopoulos G, Ellina G, Papadopoulos KB: Asymptotic behavior of the positive solutions of an exponential type system of difference equations. Appl. Math. Comput. 245, 181-190 (2014) 18. Elsayed EM: Behavior and Expression of the Solutions of Some Rational Difference Equations. Journal of Computational Analysis and Applications 15, 73-81 (2013) 19. Zhou LQ: Global asymptotic stability of cellular neural networks with proportional delays. Nonlinear Dyn. 77, 41-47(2014) 20. Elsayed EM, Cinar C: On the solutions of some systems of difference equations. Utilitas Mathematica 93, 279-289 (2014) 21. Hien LV: A novel approach to exponential stability of nonlinear non-autonomous difference equations with variable delays. Appl. Math. Lett. 38, 7-13 (2014) 22. Kulenovic MRS, Pilav E, Silic E: Local dynamics and global attractivity of a certain second-order quadratic fractional difference equation. Advances in Difference Equations 2014, 68 (2014)

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TABLE OF CONTENTS, JOURNAL OF COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 22, NO. 5, 2017

Higher-Order Degenerate Bernoulli Polynomials, Dae San Kim, and Taekyun Kim,………789 Korobov Polynomials of the Seventh Kind and of the Eighth Kind, Dae San Kim, Taekyun Kim, Toufik Mansour, and Jong-Jin Seo,………………………………………………………….812 Some Identities on the Higher-Order Twisted q-Euler Numbers and Polynomials, C. S. Ryoo,825 Umbral Calculus Associated With New Degenerate Bernoulli Polynomials, Dae San Kim, Taekyun Kim, and Jong-Jin Seo,………………………………………………………………831 Regularization Smoothing Approximation of Fuzzy Parametric Variational Inequality Constrained Stochastic Optimization, Heng-you Lan,…………………………………………841 The Split Common Fixed Point Problem for Demicontractive Mappings in Banach Spaces, Li Yang, Fuhai Zhao, and Jong Kyu Kim,……………………………………………………….858 Iterated Binomial Transform of the k-Lucas Sequence, Nazmiye Yilmaz and Necati Taskara,864 Nielsen Fixed Point Theory for Digital Images, Ozgur Ege, and Ismet Karaca,……………..874 A Fixed Point Theorem and Stability of Additive-Cubic Functional Equations in Modular Spaces, Chang Il Kim, Giljun Han, and Seong-A Shim,………………………………………881 Results on Value-Shared of Admissible Function and Non-Admissible Function in the Unit Disc, Hong-Yan Xu,…………………………………………………………………………………894 Compositions Involving Schur Harmonically Convex Functions, Huan-Nan Shi, and Jing Zhang,…………………………………………………………………………………………907 A Note on Degenerate Generalized q-Genocchi Polynomials, Jongkyum Kwon, Jin-Woo Park, and Sang Jo Yun,……………………………………………………………………………..923 Cubic Soft Ideals in BCK/BCI-Algebras, Young Bae Jun, G. Muhiuddin, Mehmet Ali Ozturk, and Eun Hwan Roh,………………………………………………………………………….929 Hyers-Ulam Stability of the Delayed Homogeneous Matrix Difference Equation with Constructive Method, Soon-Mo Jung, and Young Woo Nam,………………………………941 Mathematical Analysis of (n + 3)-Dimensional Virus Dynamics Model, A. M. Elaiw, and N. H. AlShamrani,…………………………………………………………………………………..949

TABLE OF CONTENTS, JOURNAL OF COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 22, NO. 5, 2017 (continued)

On the Dynamics of a Certain Four-Order Fractional Difference Equations, Chang-you Wang, Xiao-jing Fang, and Rui Li,…………………………………………………………………968

Volume 22, Number 6 ISSN:1521-1398 PRINT,1572-9206 ONLINE

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J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 22, NO.6, 2017, COPYRIGHT 2017 EUDOXUS PRESS, LLC

A new result on the almost increasing sequences ¨ H. S. OZARSLAN and A. KARAKAS¸ Department of Mathematics, Erciyes University, 38039 Kayseri, TURKEY E-mail:[email protected]; [email protected]

Abstract ¯ , pn |k summability factors of In this paper, we have generalized a known theorem on |N infinite series to the ϕ − |A, pn |k summability by using an almost increasing sequence. This new theorem also includes several new results.

1. INTRODUCTION A positive sequence (bn ) is said to be almost increasing if there exists a positive increasing sequence (cn ) and two positive constants A and B such that Acn ≤ bn ≤ Bcn (see [1]). Let

P

an be a given infinite series with partial sums (sn ) and let A = (anv ) be a normal

matrix, i.e., a lower triangular matrix of nonzero diagonal entries. Then A defines the sequence-to-sequence transformation, mapping the sequence s = (sn ) to As = (An (s)), where An (s) =

n X

anv sv ,

n = 0, 1, ...

(1)

v=0

The series

P

an is said to be summable |A|k , k ≥ 1, if (see [13]) ∞ X

¯ n (s) k < ∞, nk−1 ∆A



(2)

n=1

2010 AMS Subject Classification: 40D15, 40F05, 40G99. Key Words: Summability factors, absolute matrix summability, almost increasing sequence, infinite series. 1

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OZARSLAN et al 989-998

J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 22, NO.6, 2017, COPYRIGHT 2017 EUDOXUS PRESS, LLC

where ¯ n (s) = An (s) − An−1 (s). ∆A

(3)

Let (pn ) be a sequence of positive numbers such that Pn =

n X

pv → ∞ as

n → ∞,

(P−i = p−i = 0, i ≥ 1).

(4)

v=0

The sequence-to-sequence transformation un =

n 1 X pv sv Pn v=0

(5)

¯ , pn ) mean of the sequence (sn ), generated by the defines the sequence (un ) of the (N sequence of coefficients (pn ) (see [8]). The series

P

¯ , pn | , an is said to be summable | N k

k ≥ 1, if (see [2]) ∞ X

(Pn /pn )k−1 | ∆un−1 |k < ∞,

(6)

n=1

and it is said to be summable |A, pn |k , k ≥ 1, if (see [12])  ∞  X k Pn k−1 ¯ ∆An (s) < ∞, n=1

(7)

pn

where ¯ n (s) = An (s) − An−1 (s). ∆A Let (ϕn ) be any sequence of positive real numbers. The series

P

an is summable ϕ −

|A, pn |k , k ≥ 1, if (see [11]) ∞ X

k ¯ ϕk−1 n |∆An (s)| < ∞.

(8)

n=1

If we take ϕn =

Pn pn ,

then ϕ − |A, pn |k summability reduces to |A, pn |k summability (see

[10]). Also, if we take ϕn = pv Pn , anv = Ppvn

Pn pn

and anv =

pv Pn ,

¯ , pn |k summability. If we then we get |N

take ϕn = n and anv =

then we get |R, pn |k summability (see [5]). Furthermore, if we

take ϕn = n and

and pn = 1 for all values of n, then ϕ − |A, pn |k summability

reduces to |C, 1|k summability (see [7]). 2

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¯ , pn summability factors of infinite In [6], Bor has proved the following theorem for N k



series. Theorem 1.1. Let (Xn ) be an almost increasing sequence and let there be sequences (βn ) and (λn ) such that | ∆λn |≤ βn ,

βn → 0 as ∞ X

(9)

n → ∞,

(10)

n | ∆βn | Xn < ∞,

(11)

n=1

| λn | Xn = O(1)

(12)

and n X | t v |k v=1

v

= O(Xn )

as

n → ∞,

(13)

where (tn ) is the n-th (C, 1) mean of the sequence (nan ). Suppose further, the sequence (pn ) is such that

then the series

P∞

P n λn n=1 an npn

Pn = O(npn ),

(14)

Pn ∆pn = O(pn pn+1 ),

(15)

¯ , pn | , k ≥ 1. is summable | N k

Remark 1.2. It should be noted that, from the hypotheses of the Theorem 1.1, (λn ) is bounded and ∆λn = O(1/n) (see [3]). 2. THE MAIN RESULT The aim of this paper is to generalize Theorem 1.1 for absolute matrix summability.

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Before stating the main theorem we must first introduce some further notations. Given a normal matrix A = (anv ), we associate two lover semimatrices A¯ = (¯ anv ) and Aˆ = (ˆ anv ) as follows: a ¯nv =

n X

ani ,

n, v = 0, 1, ...

(16)

i=v

and a ˆ00 = a ¯00 = a00 ,

a ˆnv = a ¯nv − a ¯n−1,v ,

n = 1, 2, ...

(17)

It may be noted that A¯ and Aˆ are the well-known matrices of series-to-sequence and series-to-series transformations, respectively. Then, we have An (s) =

n X

anv sv =

n X

a ¯nv av

(18)

v=0

v=0

and ¯ n (s) = ∆A

n X

a ˆnv av .

(19)

v=0

Now, we shall prove the following theorem. Theorem 2.1. Let A = (anv ) be a positive normal matrix such that ano = 1, n = 0, 1, ...,

(20)

an−1,v ≥ anv , f or n ≥ v + 1,

(21)

ann = O(

pn ), Pn

(22)

bn,v+1 |= O(v | ∆v (a bnv ) |) |a

(23)

Let (Xn ) be an almost increasing sequence and ( ϕPnnpn ) be a non-increasing sequence. If conditions (9)-(15) of the Theorem 1.1 and m X n=1

ϕk−1 n (

pn k k ) |tn | = O(Xm ) as Pn

m→∞

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are satisfied, then the series

P∞

Pn λn n=1 an npn

is summable ϕ − |A, pn |k , k ≥ 1.

We need the following lemmas for the proof of our theorem. Lemma 2.2. ([9]) If (Xn ) an almost increasing sequence, then under the conditions (10)-(11) we have that nXn βn = O(1), ∞ X

(25)

βn Xn < ∞.

(26)

n=1

Lemma 2.3. ([4]) If the conditions (14) and (15) are satisfied, then ∆(Pn /pn n2 ) = O(1/n2 ). 3. PROOF OF THEOREM 2.1 Let (Tn ) denotes A-transform of the series ¯ n = ∆T

P∞

n=1

an Pn λn npn .

a ˆnv

av Pv λ v . vpv

n X v=1

Then we have by (18) and (19)

Applying Abel’s transformation to this sum, we get that ¯ n ∆T

= = =

n X

a ˆnv

v=1 n−1 X

v n a ˆnv Pv λv X a ˆnn Pn λn X ∆v ( 2 rar + rar ) v pv n2 pn r=1 v=1 r=1

n−1 X

∆v (

v=1

=

a ˆnv Pv λv ann Pn λn )(v + 1)tv + (n + 1)tn v 2 pv n2 p n

n−1 X (v + 1) Pv λv ann Pn λn (n + 1)t + ∆v (ˆ anv ) tv n n2 pn v2 pv v=1

+

n−1 X v=1

=

vav Pv λv v 2 pv

X a ˆn,v+1 Pv (v + 1) n−1 Pv ∆λv tv + a ˆn,v+1 λv+1 ∆( 2 )tv (v + 1) pv v2 v pv v=1

Tn,1 + Tn,2 + Tn,3 + Tn,4 ,

say.

Since |Tn,1 + Tn,2 + Tn,3 + Tn,4 |k ≤ 4k (|Tn,1 |k + |Tn,2 |k + |Tn,3 |k + |Tn,4 |k ) 5

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to complete the proof of Theorem 2.1, it is sufficient to show that m X

| Tn,r |k < ∞, ϕk−1 n

f or

r = 1, 2, 3, 4.

(27)

n=1

Firstly, by using Abel’s transformation, we have that m X

ϕk−1 | Tn,1 |k = O(1) n

n=1

= O(1) = O(1) = O(1) = O(1) = O(1)

m X n=1 m X n=1 m X

|tn |k Pn k ) |λn |k k pn n

k ϕk−1 n ann (

ϕk−1 n (

pn k ) |λn |k−1 |λn ||tn |k Pn

ϕk−1 n (

pn k ) |λn ||tn |k Pn

n=1 m−1 X n=1 m−1 X n=1 m−1 X

∆|λn |

n X

ϕk−1 v (

v=1

m X pn k k pv k k ) |tv | + O(1)|λm | ϕk−1 ) |tn | n ( Pv Pn n=1

|∆λn |Xn + O(1)|λm |Xm βn Xn + O(1)|λm |Xm

n=1

= O(1) as

m → ∞,

by virtue of the hypotheses of Theorem 2.1 and Lemma 2.2. Now, using the fact that Pv = O(vpv ) by (14), we have that m X

ϕk−1 n

k

| Tn,2 |

= O(1)

n=1

m+1 X

ϕk−1 n

n=2

n−1 X

!k

|∆v (ˆ anv )| |λv | |tv |

v=1

1 Now, applying H¨older’s inequality with indices k and k’, where k > 1 and k1 + k’ = 1, as in Tn,1 , we have that m X

ϕk−1 n

k

| Tn,2 |

=

n=1

O(1)

m+1 X

ϕk−1 n

n=2 n−1 X

=

! k

k

|∆v (ˆ anv )| |λv | |tv |

v=1

|∆v (ˆ anv )|)k−1

×( =

n−1 X

v=1 m+1 X

O(1)

k−1 ϕk−1 n ann

n=2 m+1 X

ϕn pn k−1 ) O(1) ( Pn n=2

n−1 X

! k

k

|∆v (ˆ anv )| |λv | |tv |

v=1 n−1 X

! k

k

|∆v (ˆ anv )| |λv | |tv |

v=1

6

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=

O(1)

=

O(1)

=

O(1)

=

O(1)

m X ϕv pv

(

v=1 m X v=1 m X

(

Pv

k−1

)

k

|λv | |tv |

m+1 X

k

|∆v (ˆ anv )|

n=v+1

ϕv pv k−1 ) |λv |k−1 |λv ||tv |k avv Pv

ϕk−1 v (

v=1

pv k ) |λv | |tv |k Pv

m → ∞,

as

by virtue of the hypotheses of Theorem 2.1 and Lemma 2.2. Now, using H¨older’s inequality we have that m+1 X

ϕk−1 n

| Tn,3 |

k

= O(1)

n=2

m+1 X

n−1 X

ϕk−1 n

n=2

= O(1) = O(1) = O(1)

! k

|ˆ an,v+1 |βv |tv |

×

v=1 k−1 ϕk−1 n ann (

(

n=2

= O(1)

n−1 X

ϕk−1 n

n=2 m+1 X

m X

|ˆ an,v+1 ||∆λv ||tv |

v=1

m+1 X

n=2 m+1 X

!k

n−1 X

!k−1

|ˆ an,v+1 |βv

v=1 n−1 X

|ˆ an,v+1 |βv |tv |k )

v=1 n−1 X

ϕn pn k−1 ) ( |ˆ an,v+1 |βv |tv |k ) Pn v=1

βv |tv |k

m+1 X

(

n=v+1

v=1 m X

ϕn pn k−1 ) |ˆ an,v+1 | Pn

m+1 X ϕv pv k−1 k ( = O(1) |ˆ an,v+1 | ) βv |tv | Pv v=1 n=v+1

= O(1) = O(1) = O(1) = O(1)

m X

ϕk−1 v (

v=1 m−1 X v=1 m−1 X v=1 m−1 X

pv k ) vβv |tv |k Pv

∆(vβv )

v X

ϕk−1 ( r

r=1

|∆(vβv )|Xv + O(1)mβm Xm v|∆βv |Xv + O(1)

v=1

= O(1) as

m X pr k k pv k k ϕk−1 ) |tr | + O(1)mβm ) |tv | v ( Pr P v v=1

m−1 X

βv+1 Xv+1 + O(1)mβm Xm

v=1

m → ∞,

by virtue of the hypotheses of Theorem 2.1 and Lemma 2.2. 7

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Finally, since ∆( vP2 pvv ) = O( v12 ), as in Tn,1 , we have that m+1 X

k ϕk−1 n |Tn (4)|

= O(1)

n=2

m+1 X

n−1 X

|tv | |ˆ an,v+1 ||λv+1 | v v=1

ϕk−1 n

n=2

=

O(1)

m+1 X

n−1 X

ϕk−1 n

n=2

= = =

O(1) O(1)

m+1 X n=2 m+1 X

n−1 X

ϕk−1 n

k−1 ϕk−1 n ann

n=2 m+1 X

= O(1) = O(1) = O(1)

m X

|λv+1 |

v=1 m X

(

v=1 m X v=1 m X

(

k k |tv |

|ˆ an,v+1 ||λv+1 | n−1 X

!

n−1 X

!

n−1 X

v

!k−1

!k−1

|∆v a ˆnv |

v=1 k−1

|ˆ an,v+1 ||λv+1 |

v=1 n−1 X

1 |ˆ an,v+1 | v v=1

v

k k |tv |

v=1

ϕn pn k−1 O(1) ( ) Pn n=2

= O(1)

|ˆ an,v+1 ||λv+1 |

v=1

!k

|tv |k |λv+1 | v

|tv |k |ˆ an,v+1 ||λv+1 | v v=1

!

!

X ϕn pn |tv | k m+1 ( )k−1 |ˆ an,v+1 | v n=v+1 Pn

X ϕv pv k−1 |tv | k m+1 |ˆ an,v+1 | ) |λv+1 | Pv v n=v+1

ϕv pv k−1 |tv | k ) |λv+1 | Pv v

ϕk−1 v (

v=1

pv k ) |λv+1 ||tv |k Pv

m → ∞.

= O(1) as

by virtue of hypotheses of Theorem 2.1 and Lemma 2.3 Therefore we get m X

ϕk−1 | Tn,r |k = O(1) n

as

m → ∞,

f or

r = 1, 2, 3, 4.

n=1

This completes the proof of Theorem 2.1 Corollary 3.1. If we take ϕn =

Pn pn ,

then we get a theorem dealing with |A, pn |k summa-

Pn pn

and anv =

bility. Corollary 3.2. If we take ϕn = Corollary 3.3.

If we take anv =

pv Pn ,

pv Pn ,

then we get Theorem 1.1.

then we have another a result dealing with

¯ , pn |k summability. ϕ − |N 8

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Corollary 3.4. If we take anv =

pv Pn

and pn = 1 for all values of n, then we get a result

dealing with ϕ − |C, 1|k summability. Corollary 3.5. If we take ϕn = n, anv =

pv Pn

and pn = 1 for all values of n, then we get

a result for |C, 1|k summability. Corollary 3.6. If we take k = 1 and anv = bility and in this case the condition ”



ϕn pn Pn



pv Pn ,

¯ , pn summathen we get a result for N



is a non-increasing sequence” is not needed.

References [1] N. K. Bari and S. B. Steˇ ckin, Best approximations and differential properties of two conjugate functions, (Russian) Trudy Moskov. Mat. Obˇ scˇ. 5, 483-522 (1956). [2] H. Bor, On two summability methods, Math. Proc. Camb. Philos Soc. 97, 147-149 (1985). ¯ , pn |k summability factors of infinite series, Indian J. Pure Appl. [3] H. Bor, A note on |N Math. 18, 330-336 (1987). [4] H. Bor, Absolute summability factors of infinite series, Indian J. Pure Appl. Math. 19, 664-671 (1988). [5] H. Bor, On the relative strength of two absolute summability methods, Proc. Amer. Math. Soc. 113, 1009-1012 (1991). [6] H. Bor, A note on absolute Riesz summability factors, Math. Inequal. Appl. 10, 619625 (2007). [7] T. M. Flett, On an extension of absolute summability and some theorems of Littlewood and Paley, Proc. London Math. Soc. 7, 113-141 (1957). [8] G. H. Hardy, Divergent Series, Oxford Univ. Press, Oxford, 1949. [9] S. M. Mazhar, A note on absolute summability factors, Bull. Inst. Math. Acad. Sinica. 25, 233-242 (1997). 9

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¨ [10] H. S. Ozarslan , A new application of almost increasing sequences, Miskolc Math. Notes. 14, 201-208 (2013). ¨ [11] H. S. Ozarslan and A. Keten, A new application of almost increasing sequences, An. S¸tiint¸. Univ. Al. I. Cuza Ia¸si. Mat. (N.S.) 61, 153-160 (2015). [12] W. T. Sulaiman, Inclusion theorems for absolute matrix summability methods of an infinite series (IV), Indian J.Pure Appl. Math. 34 (11) , 1547-1557 (2003). [13] N. Tanovi˘ c-Miller, On strong summability, Glas. Mat. 34 (14), 87-97 (1979).

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Certain Chebyshev type inequalities involving the generalized fractional integral operator Zhen Liu1 , Wengui Yang2∗and Praveen Agarwal3 1 2

Department of Mathematics, Kashgar University, Kashi, Xinjiang 844000, China

Ministry of Public Education, Sanmenxia Polytechnic, Sanmenxia, Henan 472000, China 3

Anand International College of Engineering, Jaipur, Rajasthan 303012, India

Abstract: In this paper, we establish certain new Chebyshev type fractional integral inequalities involving the Gauss hypergeometric function. Several special cases as Chebyshev type fractional integral inequalities involving Saigo, Erd´elyi-Kober, and Riemann-Liouville type fractional integral operators are presented. Furthermore, we also consider their relevance with other related known results. An example is also given to show the applications of obtained results. Keywords: Chebyshev type inequalities; fractional integral inequalities; hypergeometric fractional integrals; synchronous (asynchronous) functions 2010 Mathematics Subject Classification: 26D10; 26A33; 33C05

1

Introduction and preliminaries

Due to the fact that the tools of fractional integral inequalities have many applications in establishing uniqueness of solutions in fractional boundary value problems and in fractional partial differential equations, fractional integral inequalities involving the fractional operators (like Saigo, Erd´elyi-Kober, Riemann-Liouville type fractional integral operators, etc.) has gained considerable attention, attracting the interest of several researchers. For some recent developments on fractional integral inequalities, we refer the reader to [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12] and the references cited therein. Belarbi and Dahmani [13] gave the following integral inequality, using the Riemann-Liouville fractional integrals: if f and g are two synchronous functions (see Definition 1.4) on C[0, ∞), then J α (f g)(t) ≥ and

Γ(α + 1) α J f (t)J α g(t), tα

(1.1)

tα tβ J β (f g)(t) + J α (f g)(t) ≥ J α f (t)J β g(t) + J β f (t)J α g(t), Γ(α + 1) Γ(β + 1)

(1.2)

¨ gu ¨ for all t > 0, α > 0, and β > 0. O˘ ¨nmez and Ozkan [14], Chinchane and Pachpatte [15] and Purohit and Raina [16] obtained the Riemann-Liouville fractional q-integral inequalities, the Hadamard fractional integral inequalities and the Saigo fractional integral and q-integral inequalities similar to the inequalities (1.1) and (1.2), respectively. Dahmani in [17] established the following fractional integral inequalities which are generalizations of the inequalities (1.1) and (1.2), by using the Riemann-Liouville fractional integrals. Let f and g be two synchronous functions on [0, ∞) and let u, v : [0, ∞) → [0, ∞). Then J α u(t)J α (vf g)(t) + J α v(t)J α (uf g)(t) ≥ J α (uf )(t)J α (vg)(t) + J α (vf )(t)J α (ug)(t),

(1.3)

J α u(t)J β (vf g)(t) + J β v(t)J α (uf g)(t) ≥ J α (uf )(t)J β (vg)(t) + J β (vf )(t)J α (ug)(t),

(1.4)

and for all t > 0, α > 0 and β > 0. Yang [18], Brahim and Taf [19] and Chinchane and Pachpatte [20] and Agarwal et al. [21] gave the fractional q-integral inequalities, the fractional integral inequalities with two parameters of deformation q1 and q2 , the Hadamard fractional integral inequalities and generalized Erd´elyi-Kober fractional q-integral inequalities similar to inequalities (1.3) and (1.4), respectively. ∗ Corresponding author. Email:[email protected] (Z. Liu), [email protected] (W. Yang) and [email protected] (P. Agarwal)

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Let us consider the celebrated Chebyshev functional (see [22]) 1 T (f, g) = b−a

b

Z a

1 f (x)g(x)dx − b−a

Z a

b

1 f (x)dx · b−a

Z

b

g(x)dx a

where f and g are two integrable functions on [a, b]. In [23], Gr¨ uss proved the well known inequality: |T (f, g)| ≤

1 (Φ − φ)(Ψ − ψ), 4

(1.5)

where f and g are two integrable functions on [a, b] satisfying the conditions φ ≤ f (x) ≤ Φ,

ψ ≤ g(x) ≤ Ψ,

φ, Φ, ψ, Ψ ∈ R, x ∈ [a, b].

(1.6)

The inequality (1.5) is known as Gr¨ uss inequality. By using the Riemann-Liouville fractional integral and qintegral operators, Dahmani et al. [26] and Zhu et al. [27] gave the fractional integral and q-integral inequality similar to inequality (1.5) satisfying the conditions (1.6), respectively. Wang et al. [29] and Baleanu [30] et al. obtained some q-integral inequality of Gr¨ uss type for the Saigo fractional q-integral operator, respectively. Throughout the present paper, we shall investigate a fractional integral over the space Cλ introduced in [31] and defined as follows. Definition 1.1. For each real number λ, let Cλ define the space of all functions f : R+ → R that can be represented in the form f (x) = xp f1 (x) with p > λ and f1 ∈ C[0, ∞), where C[0, ∞) denotes the set of all continuous real functions defined in [0, ∞). We give the generalized fractional integral operator Ktα,β,η,µ associated with the Gauss hypergeometric function as follows. Definition 1.2. [28] Consider λ ∈ R and f ∈ Cλ . For α > max{0, −(µ + η + 1)}, β < 1, µ > −1 and β − 1 < η < 0, we define the fractional integral Ktα,β,η,µ f (x) =

Γ(1 − β)Γ(α + µ + η + 1) β+µ α,β,η,µ x It {f (x)}, Γ(η − β + 1)Γ(µ + 1)

(1.7)

where Itα,β,η,µ is the Gauss hypergeometric fractional integral of order α and is defined in the following. Definition 1.3. Let α > 0, µ > −1, β, η ∈ R. Then the generalized fractional integral Itα,β,η,µ (in terms of the Gauss hypergeometric function) of order α for real-valued continuous function f (t) is defined by [31] (see also [32])   Z x−α−β−2µ x µ t α,β,η,µ α−1 It {f (x)} = t (x − t) f (t)dt, (1.8) 2 F1 α + β + µ, −η; α; 1 − Γ(α) x 0 where the function 2 F1 (·) appearing as a kernel for the operator (1.7) is the Gaussian hypergeometric function defined by ∞ X (a)n (b)n tn , 2 F1 (a, b; c; t) = (c)n n! n=0 and (a)n is the Pochhammer symbol defined by (a)0 = 1;

(a)n = a(a + 1) · · · (a + n − 1),

for n ∈ N.

Here N denotes the set of positive integers. The above integral (1.8) has the following commutative property(see also [32, 33]): Itα,β,η,µ Itγ,δ,ζ,ν f (x) = Itγ,δ,ζ,ν Itα,β,η,µ f (x). Definition 1.4. Two functions f and g are said to be synchronous (asynchronous) functions on [0, ∞) if A(u, v) = (f (u) − f (v))(g(u) − g(v)) ≥ (≤)0, 2

1000

u, v ∈ [0, ∞).

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In [31], Baleanu et al. obtained the following fractional integral inequalities involving the Gauss hypergeometric function: Let f and g be two synchronous functions on [0, ∞). Then Itα,β,η,µ {f (t)g(t)} ≥

Γ(1 − β)Γ(1 + µ + α + η)tβ+µ α,β,η,µ It {f (t)}Itα,β,η,µ {g(t)}, Γ(1 + µ)Γ(1 − β + η)

for all t > 0, where α, β, η, µ are real constants satisfying α > max{0, −β, −µ}, β < 1, µ > −1 and β −1 < η < 0, and also Γ(1 + ν)Γ(1 − δ + ζ) Γ(1 + µ)Γ(1 − β + η) I γ,δ,ζ,ν {f (t)g(t)} + I α,β,η,µ {f (t)g(t)} Γ(1 − β)Γ(1 + µ + α + η)tβ+µ t Γ(1 − δ)Γ(1 + ν + γ + ζ)tδ+ν t ≥ Itα,β,η,µ {f (t)}Itγ,δ,ζ,ν {g(t)} + Itγ,δ,ζ,ν {f (t)}Itα,β,η,µ {g(t)}, for all t > 0, where α, β, η, µ satisfies the above inequalities and the constants γ, δ, ζ, ν satisfies γ > max{0, −δ, −ν}, δ < 1, ν > −1, δ − 1 < ζ < 0. In [28], Wang et al. gave the following integral inequalities by using the generalized fractional integral operator: Let f and g be two integrable functions with f, g ∈ Cλ and satisfying the condition (1.6) on [0, ∞). Thus we have 1 |Ktα,β,η,µ (f g)(x) − Ktα,β,η,µ f (x)Ktα,β,η,µ g(x)| ≤ (Φ − φ)(Ψ − ψ), 4 for all x ∈ [0, ∞), where α, β, η, µ are real constants with α > 0, µ > −1, η ≤ 0 and α + β + µ ≥ 0. And Let f and g be two synchronous functions on [0, ∞). Then the following inequality holds: Ktα,β,η,µ (f g)(x) ≥ Ktα,β,η,µ f (x)Ktα,β,η,µ g(x), for all x ∈ [0, ∞), where α, β, η, µ are real constants such that α > 0, µ > −1, η ≤ 0 and α + β + µ ≥ 0. Motivated by the results mentioned above and using the generalized fractional integral operator, we establish certain new Chebyshev type fractional integral inequalities and some related inequalities. Furthermore, several special cases as Chebyshev type fractional integral inequalities involving Saigo, Erd´elyi-Kober, and RiemannLiouville type fractional integral operators are given. Then we present an example to show the applications of obtained results. At last, concluding remarks are also given.

2

Generalized fractional integral inequalities

In this section, we establish certain new Chebyshev type fractional integral inequalities and some related inequalities involving the generalized fractional integral operator. For the sake of simplicity, we always assume that Ktα,β,η,µ u denotes Ktα,β,η,µ u(x) and all of the generalized fractional integral operator holds in this work. Lemma 2.1. Let f and g be two synchronous functions on [0, ∞) and let u and v be two nonnegative functions on [0, ∞). Then we have Ktα,β,η,µ uKtα,β,η,µ (vf g) + Ktα,β,η,µ vKtα,β,η,µ (uf g) ≥ Ktα,β,η,µ (vf )Ktα,β,η,µ (ug) + Ktα,β,η,µ (uf )Ktα,β,η,µ (vg), (2.1) for all x ∈ [0, ∞), and real constants α, β, η, µ with α > 0, µ > −1, η ≤ 0 and α + β + µ ≥ 0. Proof. Since f and g are two synchronous functions on [0, ∞), for all τ > 0 and ρ > 0, then we have (f (τ ) − f (ρ))(g(τ ) − g(ρ)) ≥ 0.

(2.2)

f (τ )g(τ ) + f (ρ)g(ρ) ≥ f (τ )g(ρ) + f (ρ)g(τ ).

(2.3)

Rewriting (2.2), we obtain µ

α−1

) τ Multiplying both side of (2.3) by v(τ ) τ (x−τ 2 F1 (α + µ + β, −η; α; 1 − x ), where x > 0 and τ ∈ (0, x), when Γ(α) we integrate the inequality with respect to τ from 0 to x, we obtain by Definition 1.2 that

Ktα,β,η,µ (vf g)(x) + f (ρ)g(ρ)Ktα,β,η,µ v(x) ≥ g(ρ)Ktα,β,η,µ (vf )(x) + f (ρ)Ktα,β,η,µ (vg)(x). 3

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µ

α−1

ρ Again, by multiplying both side of (2.4) by u(ρ) ρ (x−ρ) 2 F1 (α + µ + β, −η; α; 1 − x ), where x > 0 and Γ(α) ρ ∈ (0, x), and integrating the resulting identity with respect to ρ from 0 to x, and then applying Definition 1.2, we conclude

Ktα,β,η,µ u(x)Ktα,β,η,µ (vf g)(x) + Ktα,β,η,µ v(x)Ktα,β,η,µ (uf g)(x) ≥ Ktα,β,η,µ (vf )(x)Ktα,β,η,µ (ug)(x) + Ktα,β,η,µ (uf )(x)Ktα,β,η,µ (vg)(x), which implies (2.1). Theorem 2.2. Let f and g be two synchronous functions on [0, ∞) and let p, q and r be three nonnegative functions on [0, ∞). Then we have   α,β,η,µ α,β,η,µ α,β,η,µ α,β,η,µ α,β,η,µ 2Kt p Kt qKt (rf g) + Kt rKt (qf g) + 2Ktα,β,η,µ qKtα,β,η,µ rKtα,β,η,µ (pf g)    α,β,η,µ α,β,η,µ α,β,η,µ α,β,η,µ α,β,η,µ α,β,η,µ ≥ Kt p Kt (qf )Kt (rg) + Kt (rf )Kt (qg) + Kt q Ktα,β,η,µ (pf )Ktα,β,η,µ (rg)    α,β,η,µ α,β,η,µ α,β,η,µ α,β,η,µ α,β,η,µ α,β,η,µ α,β,η,µ + Kt (rf )Kt (pg) + Kt r Kt (pf )Kt (qg) + Kt (qf )Kt (pg) , (2.5) for all x ∈ [0, ∞), and real constants α, β, η, µ with α > 0, µ > −1, η ≤ 0 and α + β + µ ≥ 0. Proof. Putting u = q, v = r and using Lemma 2.1, we can write Ktα,β,η,µ qKtα,β,η,µ (rf g)+Ktα,β,η,µ rKtα,β,η,µ (qf g) ≥ Ktα,β,η,µ (rf )Ktα,β,η,µ (qg)+Ktα,β,η,µ (qf )Ktα,β,η,µ (rg). (2.6) Multiplying both sides of (2.6) by Ktα,β,η,µ p, we obtain   Ktα,β,η,µ p Ktα,β,η,µ qKtα,β,η,µ (rf g) + Ktα,β,η,µ rKtα,β,η,µ (qf g)   ≥ Ktα,β,η,µ p Ktα,β,η,µ (rf )(x)Ktα,β,η,µ (qg) + Ktα,β,η,µ (qf )Ktα,β,η,µ (rg) . (2.7) Putting u = p, v = r and using Lemma 2.1, we can state that Ktα,β,η,µ pKtα,β,η,µ (rf g)+Ktα,β,η,µ rKtα,β,η,µ (pf g) ≥ Ktα,β,η,µ (rf )Ktα,β,η,µ (pg)+Ktα,β,η,µ (pf )Ktα,β,η,µ (rg). (2.8) α,β,η Multiplying both sides of (2.8) by I0,t y(t), one verifies that   Ktα,β,η,µ q Ktα,β,η,µ pKtα,β,η,µ (rf g) + Ktα,β,η,µ r(x)Ktα,β,η,µ (pf g)   ≥ Ktα,β,η,µ q Ktα,β,η,µ (rf )Ktα,β,η,µ (pg) + Ktα,β,η,µ (pf )Ktα,β,η,µ (rg) . (2.9)

With the same arguments as before, we can get   α,β,η,µ α,β,η,µ α,β,η,µ α,β,η,µ α,β,η,µ Kt r Kt pKt (qf g) + Kt q(x)Kt (pf g)   α,β,η,µ α,β,η,µ α,β,η,µ α,β,η,µ α,β,η,µ ≥ Kt r Kt (qf )Kt (pg) + Kt (pf )Kt (qg) . (2.10) The required inequality (2.5) follows on adding the inequalities (2.7), (2.9) and (2.10). Lemma 2.3. Let f and g be two synchronous functions on [0, ∞) and let u and v be two nonnegative functions on [0, ∞). Then we have Ktα,β,η,µ u(x)Ktγ,δ,ζ,ν (vf g)(x) + Ktγ,δ,ζ,ν v(x)Ktα,β,η,µ (uf g)(x) ≥ Ktα,β,η,µ (uf )(x)Ktγ,δ,ζ,µ (vg)(x) + Ktγ,δ,ζ,µ (vf )(x)Ktα,β,η,ν (ug)(x), (2.11) for all x ∈ [0, ∞), and real constants α, γ, β, δ, η, ζ, µ, ν satisfying α, γ > 0, µ, ν > −1, η, ζ ≤ 0 and α + β + µ, γ + δ + ζ ≥ 0. 4

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ν

γ−1

ρ Proof. Multiplying both sides of (2.3) by v(ρ) ρ (x−ρ) 2 F1 (γ + ν + δ, −ζ; γ; 1 − x ), where x > 0 and ρ ∈ (0, x), Γ(γ) when we integrate the inequality with respect to ρ from 0 to x, we obtain by Definition 1.2 that

f (τ )g(τ )Ktγ,δ,ζ,ν v(x) + Ktγ,δ,ζ,ν (vf g)(x) ≥ f (τ )Ktγ,δ,ζ,ν (vg)(x) + g(τ )Ktγ,δ,ζ,ν (vf )(x). µ

(2.12)

α−1

) τ Again, by multiplying both side of (2.12) by u(τ ) τ (x−τ 2 F1 (α + µ + β, −η; α; 1 − x ), where x > 0 and Γ(α) τ ∈ (0, x), and integrating the resulting identity with respect to τ from 0 to x, and then applying Definition 1.2, we obtain

Ktα,β,η,µ u(x)Ktγ,δ,ζ,ν (vf g)(x) + Ktγ,δ,ζ,ν v(x)Ktα,β,η,µ (uf g)(x) ≥ Ktα,β,η,µ (uf )(x)Ktγ,δ,ζ,µ (vg)(x) + Ktγ,δ,ζ,µ (vf )(x)Ktα,β,η,ν (ug)(x), which implies (2.11). Theorem 2.4. Let f and g be two synchronous functions on [0, ∞) and let p, q and r be three nonnegative functions on [0, ∞). Then we have   α,β,η,µ α,β,η,µ γ,δ,ζ,ν α,β,η,µ γ,δ,ζ,ν γ,δ,ζ,ν α,β,η,µ Kt p Kt rKt (qf g) + 2Kt qKt (rf g) + Kt rKt (qf g)   γ,δ,ζ + Ktα,β,η,µ qI0,t r + Ktγ,δ,ζ,ν qKtα,β,η,µ r Ktα,β,η,µ (pf g)    ≥ Ktα,β,η,µ p Ktα,β,η,µ (qf )Ktγ,δ,ζ,ν (rg) + Ktγ,δ,ζ,ν (rf )Ktα,β,η,µ (qg) + Ktγ,δ,ζ,ν q Ktα,β,η,µ (pf )Ktγ,δ,ζ,ν (rg)    + Ktγ,δ,ζ,ν (rf )(t)Ktα,β,η,µ (pg) + Ktγ,δ,ζ,ν r Ktα,β,η,µ (pf )Ktγ,δ,ζ,ν (qg) + Ktγ,δ,ζ,ν (qf )Ktα,β,η,µ (pg) , (2.13) for all x ∈ [0, ∞), and real constants α, γ, β, δ, η, ζ, µ, ν satisfying α, γ > 0, µ, ν > −1, η, ζ ≤ 0 and α + β + µ, γ + δ + ζ ≥ 0. Proof. Putting u = q, v = r and using Lemma 2.3, we can write Ktα,β,η,µ qKtγ,δ,ζ,ν (rf g) + Ktγ,δ,ζ,ν rKtα,β,η,µ (qf g) ≥ Ktα,β,η,µ (qf )Ktγ,δ,ζ,µ (rg) + Ktγ,δ,ζ,µ (rf )Ktα,β,η,ν (qg). (2.14) Multiplying both sides of (2.14) by Ktα,β,η,µ p, we obtain   α,β,η,µ α,β,η,µ γ,δ,ζ,ν γ,δ,ζ,ν α,β,η,µ Kt p Kt qKt (rf g) + Kt rKt (qf g)   α,β,η,µ α,β,η,µ γ,δ,ζ,µ γ,δ,ζ,µ α,β,η,ν ≥ Kt p Kt (qf )Kt (rg) + Kt (rf )Kt (qg) . (2.15) Putting u = p, v = r and using Lemma 2.3, we can state that Ktα,β,η,µ pKtγ,δ,ζ,ν (rf g) + Ktγ,δ,ζ,ν rKtα,β,η,µ (pf g) ≥ Ktα,β,η,µ (pf )Ktγ,δ,ζ,µ (rg) + Ktγ,δ,ζ,µ (rf )Ktα,β,η,ν (pg). Multiplying both sides of (2.14) by Ktα,β,η,µ q, one verifies that   Ktα,β,η,µ q Ktα,β,η,µ pKtγ,δ,ζ,ν (rf g) + Ktγ,δ,ζ,ν rKtα,β,η,µ (pf g)   ≥ Ktα,β,η,µ q Ktα,β,η,µ (pf )Ktγ,δ,ζ,µ (rg) + Ktγ,δ,ζ,µ (rf )Ktα,β,η,ν (pg) . (2.16) With the same arguments as before, we can get   Ktα,β,η,µ r Ktα,β,η,µ qKtγ,δ,ζ,ν (pf g) + Ktγ,δ,ζ,ν pKtα,β,η,µ (qf g)   ≥ Ktα,β,η,µ r Ktα,β,η,µ (qf )Ktγ,δ,ζ,µ (pg) + Ktγ,δ,ζ,µ (pf )Ktα,β,η,ν (qg) . (2.17) The required inequality (2.13) follows on adding the inequalities (2.15), (2.16) and (2.17). 5

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Remark 2.5. The inequalities (2.5) and (2.13) are reversed in the following cases: (a) The functions f and g are synchronous on [0, ∞). (b) The functions p, q and r are negative on [0, ∞). (c) Two of he functions p, q and r are positive and the third one is negative on [0, ∞). Theorem 2.6. Let f, g and h be three synchronous functions on [0, ∞) and let u be a nonnegative function on [0, ∞). Then we have Ktα,β,η,µ uKtγ,δ,ζ,ν (uf gh) + Ktα,β,η,µ (uh)Ktγ,δ,ζ,ν (uf g) + Ktα,β,η,µ (uf g)Ktγ,δ,ζ,ν (uh) + Ktα,β,η,µ (uf gh)Ktγ,δ,ζ,ν u ≥ Ktα,β,η,µ (uf )Ktγ,δ,ζ,ν (ugh) + Ktα,β,η,µ (ug)Ktγ,δ,ζ,ν (uf h) + Ktα,β,η,µ (ugh)Ktγ,δ,ζ,ν (uf ) + Ktα,β,η,µ (uf h)Ktγ,δ,ζ,ν (ug), (2.18) for all x ∈ [0, ∞), and real constants α, γ, β, δ, η, ζ, µ, ν satisfying α, γ > 0, µ, ν > −1, η, ζ ≤ 0 and α + β + µ, γ + δ + ζ ≥ 0. Proof. Let f, g and h be three synchronous functions on [0, ∞), Then, for all τ, ρ ≥ 0, we have (f (τ ) − f (ρ))(g(τ ) − g(ρ))(h(τ ) + h(ρ)) ≥ 0, which implies that f (τ )g(τ )h(τ ) + f (ρ)g(ρ)h(ρ) + f (τ )g(τ )h(ρ) + f (ρ)g(ρ)h(τ ) ≥ f (τ )g(ρ)h(τ ) + f (τ )g(ρ)h(ρ) + f (ρ)g(τ )h(τ ) + f (ρ)g(τ )h(ρ). (2.19) ν

γ−1

) τ Multiplying both side of (2.19) by u(τ ) τ (x−τ 2 F1 (γ + ν + δ, −ζ; γ; 1 − x ), where x > 0 and τ ∈ (0, x), and Γ(γ) integrating the resulting identity with respect to τ from 0 to x, and then applying Definition 1.2, we obtain

Ktγ,δ,ζ,ν (uf gh) + f (ρ)g(ρ)h(ρ)Ktγ,δ,ζ,ν u + h(ρ)Ktγ,δ,ζ,ν (uf g) + f (ρ)g(ρ)Ktγ,δ,ζ,ν (uh) ≥ g(ρ)Ktγ,δ,ζ,ν (uf h) + g(ρ)h(ρ)Ktγ,δ,ζ,ν (uf ) + f (ρ)Ktγ,δ,ζ,ν (ugh) + f (ρ)h(ρ)Ktγ,δ,ζ,ν (ug). (2.20) µ

α−1

ρ Again, by multiplying both sides of (2.20) by u(ρ) ρ (x−ρ) 2 F1 (α + µ + β, −η; α; 1 − x ) where x > 0 and Γ(α) ρ ∈ (0, x), when we integrate the inequality with respect to ρ from 0 to x, we obtain by Definition 1.2 that

Ktα,β,η,µ uKtγ,δ,ζ,ν (uf gh) + Ktα,β,η,µ (uh)Ktγ,δ,ζ,ν (uf g) + Ktα,β,η,µ (uf g)Ktγ,δ,ζ,ν (uh) + Ktα,β,η,µ (uf gh)Ktγ,δ,ζ,ν u ≥ Ktα,β,η,µ (uf )Ktγ,δ,ζ,ν (ugh) + Ktα,β,η,µ (ug)Ktγ,δ,ζ,ν (uf h) + Ktα,β,η,µ (ugh)Ktγ,δ,ζ,ν (uf ) + Ktα,β,η,µ (uf h)Ktγ,δ,ζ,ν (ug), which implies (2.18). Theorem 2.7. Let f, g and h be three synchronous functions on [0, ∞) and let u and v be two nonnegative functions on [0, ∞). Then we have Ktα,β,η,µ uKtγ,δ,ζ,ν (vf gh) + Ktα,β,η,µ (uh)Ktγ,δ,ζ,ν (vf g) + Ktα,β,η,µ (uf g)Ktγ,δ,ζ,ν (vh) + Ktα,β,η,µ (uf gh)Ktγ,δ,ζ,ν v ≥ Ktα,β,η,µ (uf )Ktγ,δ,ζ,ν (vgh) + Ktα,β,η,µ (ug)Ktγ,δ,ζ,ν (vf h) + Ktα,β,η,µ (ugh)Ktγ,δ,ζ,ν (vf ) + Ktα,β,η,µ (uf h)Ktγ,δ,ζ,ν (vg), (2.21) for all x ∈ [0, ∞), and real constants α, γ, β, δ, η, ζ, µ, ν satisfying α, γ > 0, µ, ν > −1, η, ζ ≤ 0 and α + β + µ, γ + δ + ζ ≥ 0. ν

γ−1

) τ Proof. Multiplying both side of (2.19) by v(τ ) τ (x−τ 2 F1 (γ + ν + δ, −ζ; γ; 1 − x ), where x > 0 and τ ∈ (0, x), Γ(γ) and integrating the resulting identity with respect to τ from 0 to x, and then applying Definition 1.2, we obtain

Ktγ,δ,ζ,ν (vf gh) + f (ρ)g(ρ)h(ρ)Ktγ,δ,ζ,ν v + h(ρ)Ktγ,δ,ζ,ν (vf g) + f (ρ)g(ρ)Ktγ,δ,ζ,ν (vh) ≥ g(ρ)Ktγ,δ,ζ,ν (vf h) + g(ρ)h(ρ)Ktγ,δ,ζ,ν (vf ) + f (ρ)Ktγ,δ,ζ,ν (vgh) + f (ρ)h(ρ)Ktγ,δ,ζ,ν (vg). (2.22) 6

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µ

α−1

ρ Again, by multiplying both sides of (2.22) by u(ρ) ρ (x−ρ) 2 F1 (α + µ + β, −η; α; 1 − x ) where x > 0 and Γ(α) ρ ∈ (0, x), when we integrate the inequality with respect to ρ from 0 to x, we obtain by Definition 1.2 that

Ktα,β,η,µ uKtγ,δ,ζ,ν (vf gh) + Ktα,β,η,µ (uf gh)Ktγ,δ,ζ,ν v + Ktα,β,η,µ (uh)Ktγ,δ,ζ,ν (vf g) + Ktα,β,η,µ (uf g)Ktγ,δ,ζ,ν (vh) ≥ Ktα,β,η,µ (ug)Ktγ,δ,ζ,ν (vf h) + Ktα,β,η,µ (ugh)Ktγ,δ,ζ,ν (vf ) + Ktα,β,η,µ (uf )Ktγ,δ,ζ,ν (vgh) + Ktα,β,η,µ (uf h)Ktγ,δ,ζ,ν (vg), which implies (2.21). Remark 2.8. It may be noted that the inequalities in (2.18) and (2.21) are reversed if functions f, g and h are asynchronous. It is also easily seen that the special case u = v of (2.21) in Theorem 2.7 reduces to Theorem 2.6. Lemma 2.9. Let f and u be two functions defined on [0, ∞) satisfying the condition (1.6). Then we have    2  Ktα,β,η,µ uKtα,β,η,µ (uf 2 ) − Ktα,β,η,µ (uf ) = ΦKtα,β,η,µ u − Ktα,β,η,µ (uf ) Ktα,β,η,µ (xf )(t) − φKtα,β,η,µ u   − Ktα,β,η,µ uKtα,β,η,µ u(x)(Φ − f (x))(f (x) − φ) , (2.23) for all x ∈ [0, ∞), and real constants α, β, η, µ with α > 0, µ > −1, η ≤ 0 and α + β + µ ≥ 0. Proof. Let f be a function defined on [0, ∞) satisfying the condition (1.6) on [0, ∞). For any ρ, τ ∈ [0, ∞), we have (Φ − f (ρ))(f (τ ) − φ) + (Φ − f (τ ))(f (ρ) − φ) − (Φ − f (τ ))(f (τ ) − φ) − (Φ − f (ρ))(f (ρ) − φ) = f 2 (τ ) + f 2 (ρ) − 2f (ρ)f (τ ). (2.24) µ

α−1

ρ Multiplying both sides of (2.24) by u(ρ) ρ (x−ρ) 2 F1 (α + µ + β, −η; α; 1 − x ) where x > 0 and ρ ∈ (0, x), when Γ(α) we integrate the inequality with respect to ρ from 0 to x, we obtain by Definition 1.2 that     (f (τ ) − φ) ΦKtα,β,η,µ u − Ktα,β,η,µ (uf ) + (Φ − f (τ )) Ktα,β,η,µ (uf ) − φKtα,β,η,µ u   − (Φ − f (τ ))(f (τ ) − φ)Ktα,β,η,µ u − Ktα,β,η,µ u(x)(Φ − f (x))(f (x) − φ)

= f 2 (τ )Ktα,β,η,µ u + Ktα,β,η,µ (uf 2 ) − 2f (τ )Ktα,β,η,µ (uf ). (2.25) µ

α−1

ρ Again, by multiplying both sides of (2.25) by u(ρ) ρ (x−ρ) 2 F1 (α + µ + β, −η; α; 1 − x ) where x > 0 and Γ(α) ρ ∈ (0, x), when we integrate the inequality with respect to ρ from 0 to x, we obtain by Definition 1.2 that    α,β,η,µ α,β,η,µ α,β,η,µ α,β,η,µ Kt (uf ) − φKt u ΦKt u − Kt (uf )    α,β,η,µ α,β,η,µ α,β,η,µ α,β,η,µ + ΦKt u − Kt (uf ) Kt (uf ) − φKt u     − Ktα,β,η,µ u(x)(Φ − f (x))(f (x) − φ) Ktα,β,η,µ u − Ktα,β,η,µ uKtα,β,η,µ u(x)(Φ − f (x))(f (x) − φ)

= Ktα,β,η,µ (uf 2 )Ktα,β,η,µ u + Ktα,β,η,µ uKtα,β,η,µ (uf 2 ) − 2Ktα,β,η,µ (uf )Ktα,β,η,µ (uf ), which gives (2.23) and proves the lemma. Theorem 2.10. Let f and g be two functions defined satisfying the condition (1.6) on [0, ∞) and let u be a nonnegative function on [0, ∞). Then we have  2 α,β,η,µ 1 α,β,η,µ α,β,η,µ α,β,η,µ α,β,η,µ Kt uK (uf g) − K (uf )K (ug) ≤ (Φ − φ)(Ψ − ψ) K u , (2.26) t t t t 4 for all x ∈ [0, ∞), and real constants α, β, η, µ with α > 0, µ > −1, η ≤ 0 and α + β + µ ≥ 0. 7

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Proof. Let f and g be two functions satisfying the conditions of Theorem 2.10. Let H(τ, ρ) be defined by H(τ, ρ) = (f (τ ) − f (ρ))(g(τ ) − g(ρ)),

τ, ρ ∈ (0, x),

x > 0.

(2.27)

Multiplying both sides of (2.27) by u(τ )F (x, τ )u(ρ)F (x, ρ), where F (x, τ ) =

τ Γ(1 − β)Γ(α + µ + η + 1) α+β x−α−β−2µ µ x τ (x − τ )α−1 2 F1 (α + µ + β, −η; α; 1 − ), Γ(η − β + 1)Γ(µ + 1) Γ(α) x

(2.28)

where x > 0 and τ ∈ (0, x), and integrating the resulting inequality obtained with respect to τ and ρ from 0 to x, we have Z xZ x u(τ )F (x, τ )u(ρ)F (x, ρ)H(τ, ρ)dτ dρ = 2Ktα,β,η,µ uKtα,β,η,µ (uf g) − 2Ktα,β,η,µ (uf )Ktα,β,η,µ (ug). (2.29) 0

0

Thanks to the weighted Cauchy-Schwartz integral inequality for double integrals, we can write that Z

x

Z

2

x

u(τ )F (x, τ )u(ρ)F (x, ρ)H(τ, ρ)dτ dρ 0 Z

0 xZ x

≤

 Z

x

Z

u(τ )F (x, τ )u(ρ)F (x, ρ)(f (τ ) − f (ρ))dτ dρ 0

0

x

 u(τ )F (x, τ )u(ρ)F (x, ρ)(g(τ ) − g(ρ))dτ dρ

0

0

  2   2  = 4 Ktα,β,η,µ uKtα,β,η,µ (uf 2 ) − Ktα,β,η,µ (uf ) Ktα,β,η,µ uKtα,β,η,µ (ug 2 ) − Ktα,β,η,µ (ug) . (2.30) Since (Φ − f (τ ))(f (τ ) − φ) ≥ 0 and (Ψ − g(τ ))(g(τ ) − ψ) ≥ 0, we have   Ktα,β,η,µ uKtα,β,η,µ u(x)(Φ − f (x))(f (x) − φ) ≥ 0, and

(2.31)

  Ktα,β,η,µ uKtα,β,η,µ u(x)(Ψ − g(x))(g(x) − ψ) ≥ 0.

(2.32)

Thus, from (2.31), (2.32) and Lemma 2.9,we get  2    Ktα,β,η,µ uKtα,β,η,µ (uf 2 ) − Ktα,β,η,µ (uf ) ≤ ΦKtα,β,η,µ u − Ktα,β,η,µ (uf ) Ktα,β,η,µ (uf ) − φKtα,β,η,µ u , (2.33) and Ktα,β,η,µ uKtα,β,η,µ (ug 2 )

 −

Ktα,β,η,µ (ug)

2

 ≤

ΨKtα,β,η,µ u

−

  α,β,η,µ α,β,η,µ Kt (ug) − φKt u .

Ktα,β,η,µ (ug)

(2.34) Combining (2.29), (2.30), (2.33) and (2.34), we deduce that  2   Ktα,β,η,µ uKtα,β,η,µ (uf g) − Ktα,β,η,µ (uf )Ktα,β,η,µ (ug) ≤ ΦKtα,β,η,µ u − Ktα,β,η,µ (uf )     × Ktα,β,η,µ (uf ) − φKtα,β,η,µ u ΨKtα,β,η,µ u − Ktα,β,η,µ (ug) Ktα,β,η,µ (ug) − φKtα,β,η,µ u . (2.35) Now using the elementary inequality 4xy ≤ (x + y)2 , x, y ∈ R, we can state that     2 α,β,η,µ α,β,η,µ α,β,η,µ α,β,η,µ α,β,η,µ 4 ΦKt u − Kt (uf ) Kt (uf ) − φKt u ≤ (Φ − φ)Kt u , and

    2 α,β,η,µ α,β,η,µ α,β,η,µ α,β,η,µ α,β,η,µ 4 ΨKt u − Kt (ug) Kt (ug) − φKt u ≤ (Ψ − ψ)Kt u .

(2.36)

(2.37)

From (2.35)-(2.37), we abtain (2.26). This complete the proof of Theorem 2.10. 8

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Lemma 2.11. Let f and g be two functions defined on [0, ∞) and let u and v be two nonnegative functions on [0, ∞). Then we have  2 α,β,η,µ γ,δ,ζ,ν γ,δ,ζ,ν α,β,η,µ α,β,η,µ γ,δ,ζ,ν γ,δ,ζ,ν α,β,η,µ Kt uKt (vf g) + Kt vKt (uf g) − Kt (uf )Kt (vg) − Kt (vf )Kt (ug)   ≤ Ktα,β,η,µ uKtγ,δ,ζ,ν (vf 2 ) + Ktγ,δ,ζ,ν vKtα,β,η,µ (uf 2 ) − 2Ktα,β,η,µ (uf )Ktγ,δ,ζ,ν (vf )   × Ktα,β,η,µ uKtγ,δ,ζ,ν (vg 2 ) + Ktγ,δ,ζ,ν uKtα,β,η,µ (ug 2 ) − 2Ktα,β,η,µ (ug)Ktγ,δ,ζ,ν (vg) , (2.38) for all x ∈ [0, ∞), and real constants α, γ, β, δ, η, ζ, µ, ν satisfying α, γ > 0, µ, ν > −1, η, ζ ≤ 0 and α + β + µ, γ + δ + ζ ≥ 0. Proof. Multiplying (2.27) by u(τ )F (t, τ )v(ρ)G(t, ρ), where F (t, τ ) is defined by (2.28), and G(x, ρ) =

ρ Γ(1 − δ)Γ(γ + ν + ζ + 1) γ+δ x−γ−δ−2ν ν x ρ (x − ρ)γ−1 2 F1 (γ + ν + δ, −ζ; γ; 1 − ), Γ(ζ − δ + 1)Γ(ν + 1) Γ(γ) x

(2.39)

where x > 0 and ρ ∈ (0, x), and integrating the resulting inequality obtained with respect to τ and ρ from 0 to x, we have x

Z 0

Z

x

u(τ )F (x, τ )v(ρ)G(t, ρ)H(τ, ρ)dτ dρ = Ktα,β,η,µ uKtγ,δ,ζ,ν (vf g) + Ktγ,δ,ζ,ν vKtα,β,η,µ (uf g)

0

− Ktα,β,η,µ (uf )Ktγ,δ,ζ,ν (vg) − Ktγ,δ,ζ,ν (vf )Ktα,β,η,µ (ug). (2.40) Then, thanks to the weighted Cauchy-Schwartz integral inequality for double integrals, we can obtain (2.38). Lemma 2.12. Let f be a function defined on [0, ∞) and let u and v be two nonnegative functions on [0, ∞). Then we have   Ktα,β,η,µ uKtγ,δ,ζ,ν (vf 2 ) + Ktγ,δ,ζ,ν vKtα,β,η,µ (uf 2 ) − 2Ktγ,δ,ζ,ν (vf )Ktα,β,η,µ (uf ) = ΦKtα,β,η,µ u − Ktα,β,η,µ (uf )      × Ktγ,δ,ζ,ν (vf ) − φKtγ,δ,ζ,ν v + Ktα,β,η,µ (uf ) − φKtα,β,η,µ u ΦKtγ,δ,ζ,ν v − Ktγ,δ,ζ,ν (vf )     − Ktα,β,η,µ uKtγ,δ,ζ,ν v(x)(Φ − f (x))(f (x) − φ) − Ktγ,δ,ζ,ν vKtα,β,η,µ u(x)(Φ − f (x))(f (x) − φ) , (2.41) for all x ∈ [0, ∞), and real constants α, γ, β, δ, η, ζ, µ, ν satisfying α, γ > 0, µ, ν > −1, η, ζ ≤ 0 and α + β + µ, γ + δ + ζ ≥ 0. Proof. Multiplying both sides of (2.25) by v(τ )G(t, τ ) (G(t, τ ) defined by (2.39)), and integrating the resulting inequality obtained with respect to τ from 0 to x, we have 

  α,β,η,µ α,β,η,µ − ΦKt u − Kt (uf )    γ,δ,ζ,ν γ,δ,ζ,ν α,β,η,µ α,β,η,µ + ΦKt v − Kt (vf ) Kt (uf ) − φKt u     γ,δ,ζ,ν α,β,η,µ γ,δ,ζ,ν α,β,η,µ − Kt v(x)(Φ − f (x))(f (x) − φ) Kt u − Kt vKt u(x)(Φ − f (x))(f (x) − φ)

Ktγ,δ,ζ,ν (vf )

φKtγ,δ,ζ,ν v

= Ktγ,δ,ζ,ν (vf 2 )Ktα,β,η,µ u + Ktγ,δ,ζ,ν vKtα,β,η,µ (uf 2 ) − 2Ktγ,δ,ζ,ν (vf )Ktα,β,η,µ (uf ), (2.42) which gives (2.41) and proves the lemma.

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Theorem 2.13. Let f and g be two functions satisfying the condition (1.6) on [0, ∞) and let u and v be two nonnegative functions on [0, ∞). Then we have  2 α,β,η,µ γ,δ,ζ,ν γ,δ,ζ,ν α,β,η,µ α,β,η,µ γ,δ,ζ,ν γ,δ,ζ,ν α,β,η,µ Kt uKt (vf g) + Kt vKt (uf g) − Kt (uf )Kt (vg) − Kt (vf )Kt (ug)      ≤ ΦKtα,β,η,µ u − Ktα,β,η,µ (uf ) Ktγ,δ,ζ,ν (vf ) − φKtγ,δ,ζ,ν v + Ktα,β,η,µ (uf ) − φKtα,β,η,µ u     × ΦKtγ,δ,ζ,ν v − Ktγ,δ,ζ,ν (vf ) ΨKtα,β,η,µ u − Ktα,β,η,µ (ug) Ktγ,δ,ζ,ν (vg) − ψKtγ,δ,ζ,ν v    + Ktα,β,η,µ (ug) − ψKtα,β,η,µ u ΨKtγ,δ,ζ,ν v − Ktγ,δ,ζ,ν (vg) , (2.43) for all x ∈ [0, ∞), and real constants α, γ, β, δ, η, ζ, µ, ν satisfying α, γ > 0, µ, ν > −1, η, ζ ≤ 0 and α + β + µ, γ + δ + ζ ≥ 0. Proof. Since (Φ − f (τ ))(f (τ ) − φ) ≥ 0 and (Ψ − g(τ ))(g(τ ) − ψ) ≥ 0, we have     −Ktα,β,η,µ uKtγ,δ,ζ,ν v(x)(Φ − f (x))(f (x) − φ) − Ktγ,δ,ζ,ν vKtα,β,η,µ u(x)(Φ − f (x))(f (x) − φ) ≤ 0, (2.44) and −Ktα,β,η,µ uKtγ,δ,ζ,ν



   γ,δ,ζ,ν α,β,η,µ v(x)(Φ − g(x))(g(x) − φ) − Kt vKt u(x)(Φ − g(x))(g(x) − φ) ≤ 0, (2.45)

Applying Lemma 2.12 to f and g, and using Lemma 2.11 and the formulas (2.44), (2.45), we obtain (2.43). Theorem 2.14. Let u be a nonnegative function on [0, ∞) and let f, g and h be three functions defined on [0, ∞), satisfying the following condition φ ≤ f (x) ≤ Φ,

ψ ≤ g(x) ≤ Ψ,

ω ≤ h(x) ≤ Ω,

φ, Φ, ψ, Ψ, ω, Ω ∈ R, x ∈ [0, ∞).

(2.46)

Then we have α,β,η,µ Kt (uf gh)Ktγ,δ,ζ,ν u + Ktα,β,η,µ (uh)Ktγ,δ,ζ,ν (uf g) + Ktα,β,η,µ (ug)Ktγ,δ,ζ,ν (uf h) + Ktα,β,η,µ (uf )Ktγ,δ,ζ,ν (ugh) − Ktα,β,η,µ (ugh)Ktγ,δ,ζ,ν (uf ) − Ktα,β,η,µ (uf h)Ktγ,δ,ζ,ν (ug) γ,δ,ζ,ν α,β,η,µ γ,δ,ζ,ν α,β,η,µ (uf gh) ≤ Ktα,β,η,µ uKtγ,δ,ζ,ν u(Φ − φ)(Ψ − ψ)(Ω − ω), (uf g)Kt (uh) − Kt uKt − Kt for all x ∈ [0, ∞), and real constants α, γ, β, δ, η, ζ, µ, ν satisfying α, γ > 0, µ, ν > −1, η, ζ ≤ 0 and α + β + µ, γ + δ + ζ ≥ 0. Proof. From the condition (2.46), we have |f (τ ) − f (ρ)| ≤ Φ − φ,

|g(τ ) − g(ρ)| ≤ Ψ − ψ,

|h(τ ) − h(ρ)| ≤ Ω − ω,

τ, ρ ∈ [0, ∞),

which implies that |(f (τ ) − f (ρ))(g(τ ) − g(ρ))(h(τ ) − h(ρ))| ≤ (Φ − φ)(Ψ − ψ)(Ω − ω).

(2.47)

Let us define a function A(τ, ρ) = (f (τ ) − f (ρ))(g(τ ) − g(ρ))(h(τ ) − h(ρ)) = f (τ )g(τ )h(τ ) + f (ρ)g(ρ)h(τ ) + f (τ )g(ρ)h(ρ) + f (ρ)g(τ )h(ρ) − f (τ )g(ρ)h(τ ) − f (ρ)g(ρ)h(ρ) − f (τ )g(τ )h(ρ) − f (ρ)g(τ )h(τ ). (2.48)

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Multiplying (2.48) by u(τ )F (t, τ ), where F (t, τ ) is defined by (2.28), and integrating the resulting inequality obtained with respect to τ from 0 to x, we have Z

x

u(τ )F (x, τ )A(τ, ρ)dτ = Ktα,β,η,µ (uf gh) + f (ρ)g(ρ)Ktα,β,η,µ (uh) + f (ρ)h(ρ)Ktα,β,η,µ (ug)

0

+ g(ρ)h(ρ)Ktα,β,η,µ (uf ) − h(ρ)Ktα,β,η,µ (uf g) − g(ρ)Ktα,β,η,µ (uf h) − f (ρ)Ktα,β,η,µ (ugh) − f (ρ)g(ρ)h(ρ)Ktα,β,η,µ u. (2.49) Again, by multiplying (2.49) by u(ρ)G(t, ρ), where G(t, τ ) is defined by (2.39), and integrating the resulting inequality obtained with respect to ρ from 0 to x, we have Z 0

x

Z

x

u(τ )F (x, τ )u(ρ)G(t, ρ)A(τ, ρ)dτ dρ = Ktα,β,η,µ (uf gh)Ktγ,δ,ζ,ν u + Ktα,β,η,µ (uh)Ktγ,δ,ζ,ν (uf g)

0

+ Ktα,β,η,µ (ug)Ktγ,δ,ζ,ν (uf h) + Ktα,β,η,µ (uf )Ktγ,δ,ζ,ν (ugh) − Ktα,β,η,µ (ugh)Ktγ,δ,ζ,ν (uf ) − Ktα,β,η,µ (uf h)Ktγ,δ,ζ,ν (ug) − Ktα,β,η,µ (uf g)Ktγ,δ,ζ,ν (uh) − Ktα,β,η,µ uKtγ,δ,ζ,ν (uf gh).

(2.50)

Finally, by using (2.47) on to (2.50), we arrive at the desired result (??), involved in Theorem 2.14, after a little simplification. This concludes the proof. Theorem 2.15. Let u and v be two nonnegative functions on [0, ∞) and let f, g and h be three functions defined on [0, ∞), satisfying the condition (2.46). Then we have α,β,η,µ Kt (uf gh)Ktγ,δ,ζ,ν v + Ktα,β,η,µ (uh)Ktγ,δ,ζ,ν (vf g) + Ktα,β,η,µ (ug)Ktγ,δ,ζ,ν (vf h) + Ktα,β,η,µ (uf )Ktγ,δ,ζ,ν (vgh) − Ktα,β,η,µ (ugh)Ktγ,δ,ζ,ν (vf ) − Ktα,β,η,µ (uf h)Ktγ,δ,ζ,ν (vg) − Ktα,β,η,µ (uf g)Ktγ,δ,ζ,ν (vh) − Ktα,β,η,µ uKtγ,δ,ζ,ν (vf gh) ≤ Ktα,β,η,µ uKtγ,δ,ζ,ν v(Φ − φ)(Ψ − ψ)(Ω − ω), (2.51) for all x ∈ [0, ∞), and real constants α, γ, β, δ, η, ζ, µ, ν satisfying α, γ > 0, µ, ν > −1, η, ζ ≤ 0 and α + β + µ, γ + δ + ζ ≥ 0. Proof. Multiplying (2.49) by v(ρ)G(t, ρ), where G(t, τ ) is defined by (2.39), and integrating the resulting inequality obtained with respect to ρ from 0 to x, and then applying (2.47) on the resulting inequality, we get the desired result (2.51). This concludes the proof. Remark 2.16. It is not difficult to notice that the spacial case u = v of (2.51) in Theorem 2.15 reduces to Theorem 2.14. Theorem 2.17. Let f and g be two integrable functions satisfying the condition M -g-Lipschitzian on [0, ∞), i.e., |f (x) − f (y)| ≤ M |g(x) − g(y)|, M > 0, x, y ∈ R, and let u and v be two nonnegative continuous functions on [0, ∞). Then we have α,β,η,µ γ,δ,ζ,ν γ,δ,ζ,ν α,β,η,µ α,β,η,µ γ,δ,ζ,ν γ,δ,ζ,ν α,β,η,µ Kt uKt (vf g) + Kt vKt (uf g) − Kt (uf )Kt (yg) − Kt (vf )Kt (xg)   α,β,η,µ γ,δ,ζ,ν γ,δ,ζ,ν α,β,η,µ α,β,η,µ γ,δ,ζ,ν 2 2 ≤ M Kt uKt (vg ) + Kt vKt (ug ) − 2Kt (ug)Kt (vg) , (2.52) for all x ∈ [0, ∞), and real constants α, γ, β, δ, η, ζ, µ, ν satisfying α, γ > 0, µ, ν > −1, η, ζ ≤ 0 and α + β + µ, γ + δ + ζ ≥ 0. Proof. Let us define the following relations |f (τ ) − f (ρ)| ≤ M |g(τ ) − g(ρ)| τ, ρ ∈ [0, ∞),

(2.53)

|H(τ, ρ)| = |f (τ ) − f (ρ)||g(τ ) − g(ρ)| ≤ M (g(τ ) − g(ρ))2 .

(2.54)

which implies that

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Multiplying (2.27) by u(τ )F (t, τ )u(ρ)G(t, ρ), where F (t, τ ) and G(t, ρ) are defined by (2.28) and (2.39), respectively, and integrating the resulting inequality obtained with respect to τ and ρ from 0 to x, then applying (2.40) and (2.54) on the resulting inequality, we get the desired result (2.52). This concludes the proof of the theorem. Theorem 2.18. Let u and v be two nonnegative functions on [0, ∞) and let f and g be two Lipschitzian functions defined on [0, ∞) with the constants L1 and L2 , respectively. Then we have α,β,η,µ γ,δ,ζ,ν γ,δ,ζ,ν α,β,η,µ α,β,η,µ γ,δ,ζ,ν γ,δ,ζ,ν α,β,η,µ Kt uKt (vf g) + Kt vKt (uf g) − Kt (uf )Kt (yg) − Kt (vf )Kt (xg)   α,β,η,µ γ,δ,ζ,ν γ,δ,ζ,ν α,β,η,µ 2 α,β,η,µ γ,δ,ζ,ν 2 ≤ L1 L2 Kt uKt (x v(x)) + Kt vKt (x u(x)) − 2Kt (xu(x))Kt (xv(x)) , (2.55) for all x ∈ [0, ∞), and real constants α, γ, β, δ, η, ζ, µ, ν satisfying α, γ > 0, µ, ν > −1, η, ζ ≤ 0 and α + β + µ, γ + δ + ζ ≥ 0. Proof. From the conditions of Theorem 2.18, we have |f (τ ) − f (ρ)| ≤ L1 |τ − ρ|,

|g(τ ) − g(ρ)| ≤ L2 |τ − ρ|,

τ, ρ ∈ [0, ∞),

which implies that |H(τ, ρ)| = |f (τ ) − f (ρ)||g(τ ) − g(ρ)| ≤ L1 L2 (τ − ρ)2 .

(2.56)

Multiplying (2.27) by u(τ )F (t, τ )v(ρ)G(t, ρ), where F (t, τ ) and G(t, ρ) are defined by (2.28) and (2.39), respectively, and integrating the resulting inequality obtained with respect to τ and ρ from 0 to x, then applying (2.40) and (2.56), on the resulting inequality, we get the desired result (2.55). This completes the proof. Corollary 2.19. Let u and v be two nonnegative functions on [0, ∞) and let f and g be two differentiable functions on [0, ∞) with supt≥0 |f 0 (t)|, supt≥0 |g 0 (t)| < ∞. Then we have α,β,η,µ γ,δ,ζ,ν γ,δ,ζ,ν α,β,η,µ α,β,η,µ γ,δ,ζ,ν γ,δ,ζ,ν α,β,η,µ Kt uK (vf g) + K vK (uf g) − K (uf )K (yg) − K (vf )K (xg) t t t t t t t   ≤ kf 0 k∞ kg 0 k∞ Ktα,β,η,µ uKtγ,δ,ζ,ν (x2 v(x)) + Ktγ,δ,ζ,ν vKtα,β,η,µ (x2 u(x)) − 2Ktα,β,η,µ (xu(x))Ktγ,δ,ζ,ν (xv(x)) , for all x ∈ [0, ∞), and real constants α, γ, β, δ, η, ζ, µ, ν satisfying α, γ > 0, µ, ν > −1, η, ζ ≤ 0 and α + β + µ, γ + δ + ζ ≥ 0. Rτ Rτ Proof. We have f (τ ) − f (ρ) = ρ f 0 (t)dt and g(τ ) − g(ρ) = ρ g 0 (t)dt. That is, |f (τ ) − f (ρ)| ≤ kf 0 k∞ |τ − ρ|, |g(τ ) − g(ρ)| ≤ kg 0 k∞ |τ − ρ|, τ, ρ ∈ [0, ∞), and the result follows from Theorem 2.18. This ends the proof.

3

An example

In this section we present a way for constructing the four bounding functions, and use them to give some estimates of Chebyshev type inequalities involving the generalized fractional integral operator of two unknown functions. For 0 = x0 < x1 < x2 < · · · < xn < xn+1 = T , we define a notation of sub-integrals of generalized fractional integral Ixα,β,η,µ as Ixα,β,η,µ {f (T )} = j ,xj+1

x−α−β−2µ Γ(α)

Z

xj+1

xj

  t tµ (T −t)α−1 2 F1 α + β + µ, −η; α; 1 − f (t)dt, T

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j = 0, 1, . . . , n.

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Note that α,β,η,µ I0,T {f (T )}

=

n X j=0

+

x−α−β−2µ Γ(α)

x−α−β−2µ = Γ(α)

x1



 t t (T − t) f (t)dt 2 F1 α + β + µ, −η; α; 1 − T 0   Z x2 t tµ (T − t)α−1 2 F1 α + β + µ, −η; α; 1 − f (t)dt + · · · T x1   Z t x−α−β−2µ T µ α−1 t (T − t) f (t)dt. (3.2) + 2 F1 α + β + µ, −η; α; 1 − Γ(α) T xn

Ixα,β,η,µ {f (T )} j ,xj+1

Z

µ

α−1

So, from (3.2), we can rewrite (1.7) as Γ(1 − β)Γ(α + µ + η + 1) β+µ α,β,η,µ T I0,T {f (T )} Γ(η − β + 1)Γ(µ + 1) n Γ(1 − β)Γ(α + µ + η + 1) β+µ X α,β,η,µ Γ(1 − β)Γ(α + µ + η + 1) β+µ = T x Ixj ,xj+1 {f (T )} = Γ(η − β + 1)Γ(µ + 1) Γ(η − β + 1)Γ(µ + 1) j=0    −α−β−2µ Z x1 t T tµ (T − t)α−1 2 F1 α + β + µ, −η; α; 1 − f (t)dt × Γ(α) T 0   Z t x−α−β−2µ x2 µ t (T − t)α−1 2 F1 α + β + µ, −η; α; 1 − f (t)dt Γ(α) T x1 )   Z x−α−β−2µ T µ t α−1 t (T − t) f (t)dt . (3.3) ··· + 2 F1 α + β + µ, −η; α; 1 − Γ(α) T xn

α,β,η,µ K0,T f (T ) =

Let ϕ be a unit step function defined by ( 1, ϕ(x) = 0,

x > 0, x ≤ 0,

and let ϕa (x) the Heaviside unit step function defined by ( 1, ϕa (x) = ϕ(x − a) = 0,

x > a, x ≤ a.

Let u be a piecewise continuous function on [0, T ] defined by u(x) = U1 (ϕ0 (x) − ϕx1 (x)) + U2 (ϕx1 (x) − ϕx2 (x)) + U3 (ϕx2 (x) − ϕx3 (x)) + · · · + Um+1 ϕxm (x) = U1 ϕ0 (x) m X + (U2 − U1 )ϕx1 (x) + (U3 − U2 )ϕx2 (x) + · · · + (Um+1 − Um )ϕxm (x) = (Uj+1 − Uj )ϕxj (x), (3.4) j=0

where U0 ≡ 0 and 0 = x0 < x1 < x2 < · · · < xm < xm+1 = T . Similarly, we have v(x) =

m X (Vj+1 − Vj )ϕxj (x).

(3.5)

j=0

where constants U0 = V0 ≡ 0. Proposition 3.1. Let f and g be two synchronous functions on [0, T ). Assume that let u and v defined by (3.4) and (3.5), respectively. Then for α > 0, µ > −1, η ≤ 0 and α + β + µ ≥ 0, the following inequality holds:       m m m m X X X X  Uj+1   Vj+1 Kxα,β,η,µ (f g)(T ) +  Vj+1   Uj+1 Kxα,β,η,µ (f g)(T ) j ,xj+1 j ,xj+1 j=0

j=0

j=0

j=0

      m m m m X X X X ≥ Uj+1 Kxα,β,η,µ g(T )  Vj+1 Kxα,β,η,µ f (T )+ Vj+1 Kxα,β,η,µ g(T )  Uj+1 Kxα,β,η,µ f (T ) . j ,xj+1 j ,xj+1 j ,xj+1 j ,xj+1 j=0

j=0

j=0

j=0

(3.6) 13

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Proof. By using the definition (3.1) and (3.3), we have α,β,η,µ K0,T u(T )

=

m X

Uj+1 Kxα,β,η,µ (1)(T ) j ,xj+1

=

j=0

m X

Uj+1 ,

j=0

and α,β,η,µ K0,T v(T ) =

m X

Vj+1 Kxα,β,η,µ (1)(T ) = j ,xj+1

j=0

m X

Vj+1 ,

j=0

where Kxα,β,η,µ (1)(T ) = 1. Similarly, we have j ,xj+1 α,β,η,µ K0,T (uf g)(T ) =

m X

α,β,η,µ Uj+1 Kxα,β,η,µ (f g)(T ), K0,T (vf g)(T ) = j ,xj+1

j=0 α,β,η,µ K0,T (uf )(T ) =

m X

α,β,η,µ K0,T (ug)(T ) =

Vj+1 Kxα,β,η,µ (f g)(T ), j ,xj+1

j=0

α,β,η,µ f (T ), K0,T (vf )(T ) = Uj+1 Kxα,β,η,µ j ,xj+1

j=0 m X

m X

m X

f (T ), Vj+1 Kxα,β,η,µ j ,xj+1

j=0 α,β,η,µ Uj+1 Kxα,β,η,µ g(T ), K0,T (vg)(T ) = j ,xj+1

j=0

m X

Vj+1 Kxα,β,η,µ g(T ), j ,xj+1

j=0

By applying Lemma 2.1, the desired inequality (3.6) is established.

4

Concluding remarks

In this section, we consider some consequences of the main results derived in the previous section. Following Curiel and Galue [33], the operator would reduce immediately to the extensively investigated Saigo, Erd´elyiKober, and Riemann-Liouville type fractional integral operators, respectively, given by the following relationships (see also [32, 34]): α,β,η I0,x {f (x)}

=

Ixα,β,η,0 {f (x)}

x−α−β = Γ(α)

x

Z

α−1

(x − t)

 2 F1

0

t α + β, −η; α; 1 − x

 f (τ )dt, (α > 0; β, η ∈ R), (4.1)

I

α,η

{f (x)} =

Ixα,0,η,0 {f (x)}

x−α−η = Γ(α)

and J α {f (x)} = Ixα,−α,η,0 {f (x)} =

Z

x

(x − t)α−1 tα−1 f (t)dt, (α > 0; η ∈ R),

(4.2)

0

1 Γ(α)

Z

x

(x − t)α−1 f (t)dt, (α > 0).

(4.3)

0

By setting µ = 0, µ = β = 0, and µ = 0 and β = −α in (1.7), Definition 1.2 would immediately reduce to the Saigo, Erd´elyi-Kober, and Riemann-Liouville type fractional integral operators, respectively, given as follows: Kxα,β,η f (x) =

Γ(1 − β)Γ(α + η + 1) β α,β,η x I0,x {f (x)}, Γ(η − β + 1)

Kxα,η f (x) =

Γ(η + α + 1) α,η I {f (x)}, Γ(1 + η)

and Kxα f (x) =

Γ(α + 1) α J {f (x)}, xα

(4.4)

(4.5)

(4.6)

α,β,η where I0,x {f (x)}, I α,η {f (x)} and J α {f (x)} are given by (4.1), (4.2), and (4.3), respectively. Similar to main results in the preceding section, by using the fractional integral operators (4.1)-(4.6), we obtain various fractional integral inequalities involving such relatively more familiar fractional integral operators (4.1)-(4.6). Therefore, we omit the further details. For example, by (4.1), Theorem 2.2 and 2.4 yield the known

14

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results in [24, 25]. If we consider u = v = 1 and make use of fractional integral operator Ixα,β,η,µ {f (x)}, Lemma 2.1 and 2.3 provides respectively, the known fractional integral inequalities due to Baleanu et al. [31]. Let u = 1, Theorem 2.10 corresponds to the known results due to Wang et al. [28]. Taking u = 1, µ = 0 and β = −α in Theorem 2.10 yields the known result due to Dahmani et al. [26]. Make use of fractional integral operator (4.3), Lemma 2.1 and 2.3 provides respectively, the known fractional integral inequalities due to Dahmani [17]. At the end of this paper, generalized fractional integral inequalities obtained in the previous section are expected to find more applications, for example, applications for establishing the solutions in fractional differential equations and fractional integral equations boundary value problems. Authors’ contributions. ZL and WY equally participated in the design of the study and drafted the manuscript. PA gave an example to show the applications. All authors read and approved the final manuscript.

References [1] Z. Dahmani, On Minkowski and Hermite-Hadamard integral inequalities via fractional integration, Annals of Functional Analysis, vol. 1, no. 1, pp. 51-58, 2010. [2] Z. Dahmani, New inequalities in fractional integrals, International Journal of Nonlinear Science, vol. 9, no. 4, pp. 493-497, 2010. [3] Z. Dahmani, O. Mechouar, and S. Brahami, Certain inequalities related to the Chebyshev’s functional involving a type Riemann-Liouville operator. Bulletin of Mathematical Analysis and Applications, vol. 3, no. 4, pp. 38-44, 2011. [4] J. Choi, and P. Agarwal, Some new Saigo type fractional integral inequalities and their q-analogues, Abstract and Applied Analysis, Vol. 2014, Article ID 579260, 11 pages, 2014. [5] M.Z. Sarikaya, and H. Ogunmez, On new inequalities via Riemann-Liouville fractional integration, Abstract and Applied Analysis, Vol. 2012, Article ID 428983, 10 pages, 2014. [6] J. Tariboon, S.K. Ntouyas, and W. Sudsutad, Some new Riemann-Liouville fractional integral inequalities, International Journal of Mathematics and Mathematical Sciences, Vol. 2014, Article ID 869434, 6 pages, 2014. [7] W. Sudsutad, S.K. Ntouyas, and J. Tariboon, fractional integral inequalities via Hadamard’s fractional integral, Abstract and Applied Analysis, Vol. 2014, Article ID 563096, 11 pages, 2014. [8] S.K. Ntouyas, S.D. Purohit, and J. Tariboon, Certain Chebyshev type integral inequalities involving Hadamard’s fractional operators, Abstract and Applied Analysis, Vol. 2014, Article ID 249091, 7 pages, 2014. [9] D. Baleanu, and P. Agarwal, Certain inequalities involving the fractional q-integral operators, Abstract and Applied Analysis, Vol. 2014, Article ID 371274, 10 pages, 2014. [10] G.A. Anastassiou, Fractional differentiation inequalities, Springer, NewYork, NY,USA, 2009. [11] G.A. Anastassiou, Fractional Polya type integral inequality, Journal of Computational Analysis and Applications, vol. 17, no. 4, 736-742, 2014. [12] W. Liu, Some Ostrowski type inequalities via Riemann-Liouville fractional integrals for h-convex functions, Journal of Computational Analysis and Applications, vol. 16, no. 4, 998-1004, 2014. [13] S. Belarbi, and Z. Dahmani, On some new fractional integral inequalities, Journal of Inequalities in Pure and Applied Mathematics, vol. 10, no. 3, Art. 86, 5, pages, 2009. ¨ gu ¨ [14] H. O˘ ¨nmez, and U. Ozkan, Fractional quantum integral inequalities, Journal of Inequalities and Applications, vol. 2011 Article ID 787939, 7 pages, 2011. [15] V. Chinchane, and D. Pachpatte, A note on some fractional integral inequalities via Hadamard integral, Journal Fractional Calculus and Applications, vol. 4, no. 1, pp. 125-129, 2013. 15

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[16] S. Purohit, and R. Raina, Chebyshev type inequalities for the Saigo fractional integrals and their qanalogues, Journal of Mathematical Inequalities, vol.7, no. 2, pp. 239-249, 2013. [17] Z. Dahmani, New inequalities in fractional integrals, International Journal of Nonlinear Sciences, vol. 9, no. 4, pp. 493-497, 2010. [18] W. Yang, Some new fractional quantum integral inequalities, Applied Mathematics Letters, vol. 25, no. 6, 963-969, 2012. [19] K. Brahim, and S. Taf, Some fractional integral inequalities in quantum calculus, Journal Fractional Calculus and Applications, vol. 4, no. 2, pp. 245-250, 2013. [20] V. Chinchane, and D. Pachpatte, On some integral inequalities using Hadamard fractional integral, Malaya Journal of Matematik, vol. 1, no. 1, pp. 62-66, 2012. [21] Agarwal, P, Salahshour, S, Ntouyas, and Tariboon, J: Certain inequalities involving generalized Erd´elyiKober fractional q-integral operators, Sci. World J. 2014, Article ID 174126, 2014. ˇ [22] P.L. Cebyˇ sev, Sur les expressions approximatives des int´egrales d´efinies par les autres prises entre les mˆemes limites, Proceedings of the Mathematical Society of Kharkov, vol. 2, pp. 93-98, 1882. Rb Rb Rb 1 1 ¨ [23] G. Gr¨ uss, Uber das maximum des absoluten Betrages von b−a f (x)g(x)dx − (b−a) f (x)dx a g(x)dx, 2 a a Mathematische Zeitschrift, vol. 39, no. 1, pp. 215-226, 1935. [24] W. Yang, Some new Chebyshev and Gr¨ uss-type integral inequalities for Saigo fractional integral operators and their q-analogues. Filomat, vol. 29, no. 6, 1269-1289, 2015. [25] V. Chinchane, and D. Pachpatte, Certain inequalities using Saigo fractional integral operator. Facta Universitatis, Series: Mathematics and Informatics, vol. 29, no. 4, 343-350, 2014. [26] Z. Dahmani, L. Tabharit, and S. Taf, New generalisations of Gr¨ uss inequality using Riemann-Liouville fractional integrals, Bulletin of Mathematical Analysis and Applications, vol. 2, no. 3, pp. 93-99, 2010. [27] C. Zhu, W. Yang, and Q. Zhao, Some new fractional q-integral Gr¨ uss-type inequalities and other inequalities, Journal of Inequalities and Applications, vol. 2012, Article 299, 15 pages, 2012. [28] G. Wang, P. Agarwal, and Mehar Chand, Certain Gr¨ uss type inequalities involving the generalized fractional integral operator, Journal of Inequalities and Applications, vol. 2014, no. 147, pp.1-8, 2014. [29] D.Baleanu, S.D. Purohit, and F. Ucar, On Gr¨ uss type integral inequality involving the Saigo’s fractional integral operators, Journal of Computational Analysis and Applications, vol. 19, no. 3, 480-489, 2015. [30] G. Wang, P. Agarwal, and D. Baleanu, Certain new Gr¨ uss type inequalities involving Saigo fractional q-integral operator, Journal of Computational Analysis and Applications, vol. 19, no. 5, 862-873, 2015. [31] D. Baleanu, S.D. Purohit, and P. Agarwal, On fractional integral inequalities involving Hypergeometric operators, Chinese Journal of Mathematics, Vol. 2014, Article ID 609476, 5 pages, 2014. [32] V.S. Kiryakova, Generalized fractional calculus and applications, Pitman Research Notes in Mathematics Series no. 301, Longman Scientific and Technical, Harlow, UK, 1994. [33] L. Curiel and L. Galu´e, A generalization of the integral operators involving the Gauss’ hypergeometric function, Revista T´ecnica de la Facultad de Ingenieria Universidad del Zulia, vol. 19, no. 1, pp. 17-22, 1996. [34] M. Saigo, A remark on integral operators involving the Gauss hypergeometric functions, Mathematical Reports, Kyushu University, vol. 11, pp. 135-143, 1978.

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Estimates for the Green’s Function of 3D Elliptic Equations Jinghong Liu∗and Yinsuo Jia† This article will first introduce the definition of the Green’s function of 3D elliptic equations, which plays important roles in local superconvergence estimates for the finite element approximation. Then, using the weighted-norm methods, we derive some estimates for the 3D Green’s function.

1

Introduction

It is well known that estimates for the Green’s function play very important roles in the study of the superconvergence (especially, pointwise superconvergence) of the finite element method (see [1–9]). For dimensions three and up, we have obtained the estimates for discrete Green’s functions and discrete derivative Green’s functions, which were used to the global superconvergence estimates of the finite element approximation. However, the fact is that the high generalization conditions to the true solution is difficult to satisfy for the global superconvergence estimates. Thus the global superconvergence results is only theoretical. In order to study local superconvergence properties of the finite element approximation, we need to introduce a Green’s function, which will play important roles in the study of local superconvergence properties. we shall use the symbol C to denote a generic constant, which is independent from the discretization parameter h and which may not be the same in each occurrence and also use the standard notations for the Sobolev spaces and their norms. In this article, we consider the following elliptic equation: Lu ≡ −

3 ∑

∂j (aij ∂i u) + a0 u = f in Ω,

u = 0 on ∂Ω,

(1.1)

i,j=1

where Ω ⊂ R3 is a bounded polytopic domain. The weak formulation of (1.1) reads, { Find u ∈ H01 (Ω) satisfying a(u , v) = (f , v) for all v ∈ H01 (Ω), ∗ School of Information Science and Engineering, Ningbo Institute of Technology, Zhejiang University, Ningbo 315100, China, email: [email protected] † School of Mathematics and Computer Science, Shangrao Normal University, Shangrao 334001, China, email: [email protected]

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LIU, JIA: ESTIMATES FOR THE 3D GREEN’S FUNCTION

where ∫ a(u , v) ≡

(

∫

3 ∑

aij ∂i u∂j v + a0 uv) dxdydz, (f , v) ≡

Ω i,j=1

f v dxdydz. Ω

We assume that the given functions aij ∈ W 1, ∞ (Ω), aij = aji , a0 ∈ L∞ (Ω), ∂u ∂u and f ∈ L2 (Ω). In addition, we write ∂1 u = ∂u ∂x , ∂2 u = ∂y , and ∂3 u = ∂z , h which are usual partial derivatives. Let {T } be a regular family of partitions ¯ Denote by S h (Ω) a continuous finite elements space of degree m(m ≥ 1) of Ω. regarding this kind of partitions and let S0h (Ω) = S h (Ω) ∩ H01 (Ω). Discretizing the above weak formulation using S0h (Ω) as approximating space means, { Find uh ∈ S0h (Ω) satisfying a(uh , v) = (f , v) for all v ∈ S0h (Ω). h For every Z ∈ Ω, we define the discrete δ function δZ ∈ S0h (Ω), the dish crete derivative δ function ∂Z,ℓ δZ ∈ S0h (Ω), the regularized Green’s function G∗Z ∈ H 2 (Ω) ∩ H01 (Ω), the regularized derivative Green’s function ∂Z,ℓ G∗Z ∈ H 2 (Ω) ∩ H01 (Ω), the discrete Green’s function GhZ ∈ S0h (Ω), the discrete derivative Green’s function ∂Z,ℓ GhZ ∈ S0h (Ω), and the L2 -projection Ph u ∈ S0h (Ω) such that (see [9]) h (v, δZ ) = v(Z) ∀ v ∈ S0h (Ω), (1.2) h (v, ∂Z,ℓ δZ ) = ∂ℓ v(Z) ∀ v ∈ S0h (Ω),

a(G∗Z ,

v) =

h , (δZ

v) ∀ v ∈

(1.3)

H01 (Ω),

(1.4)

h a(∂Z,ℓ G∗Z , v) = (∂Z,ℓ δZ , v) ∀ v ∈ H01 (Ω),

(1.5)

v) = v(Z) ∀ v ∈

(1.6)

a(GhZ ,

S0h (Ω),

a(∂Z,ℓ GhZ , v) = ∂ℓ v(Z) ∀ v ∈ S0h (Ω), (u − Ph u, v) = 0 ∀ v ∈

(1.7)

S0h (Ω).

(1.8)

h Here, for any direction ℓ ∈ R3 , |ℓ| = 1, ∂Z,ℓ δZ , ∂Z,ℓ GhZ , and ∂ℓ v(Z) stand for the following onesided directional derivatives, respectively.

h ∂Z,ℓ δZ =

h h δZ+∆Z − δZ GhZ+∆Z − GhZ , ∂Z,ℓ GhZ = lim , |∆Z| |∆Z| |∆Z|→0 |∆Z|→0

lim

∂ℓ v(Z) =

lim

|∆Z|→0

v(Z + ∆Z) − v(Z) , ∆Z = |∆Z|ℓ. |∆Z|

As for G∗Z , ∂Z,ℓ G∗Z , GhZ , and ∂Z,ℓ GhZ , we have obtained some optimal estimates (see [4–6]), which will be used in next section. From (1.4)–(1.7), we easily find GhZ and ∂Z,ℓ GhZ are the finite element approximations to G∗Z and ∂Z,ℓ G∗Z , respectively. For the L2 -projection operator Ph , we have (see [4])

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Lemma 1.1. For Ph w the L2 -projection of w ∈ Lp (Ω), we have the following stability estimate: ∥Ph w∥0, p, Ω ≤ C t ∥w∥0, p, Ω , (1.9) where t = 1 − p2 , and 1 ≤ p ≤ ∞. Further, by Lemma 1.1, we easily obtain the following result: ∥w − Ph w∥0, p, Ω

≤ ≤

(1 + C t ) inf v∈S0h Ω ∥w − v∥0, p, Ω C∥w − Πw∥0, p, Ω ≤ Chm+1 ∥w∥m+1, p, Ω ,

(1.10)

where 1 ≤ p ≤ ∞. In addition, we also assume the following a priori estimate holds. Lemma 1.2. For the true solution u of (1.1), there exists a q0 (1 < q0 ≤ ∞) such that for every 1 < q < q0 , ∥u∥2, q, Ω ≤ C(q)∥Lu∥0, q, Ω .

2

(1.11)

Definition of the 3D Green’s Function

For Z ∈ Ω, we introduce the definition of the 3D Green’s function GZ as follows a(GZ , v) = v(Z) ∀ v ∈ C0∞ (Ω). In the following, we will prove the existence and uniqueness of the Green’s function. Lemma 2.1. For G∗Z and GhZ defined by (1.4) and (1.6), respectively, we have

∗

2

GZ − GhZ ≤ Ch |ln h| 3 . 1,1

(2.1)

This result can be seen in [4]. Theorem 2.1. There exists a unique GZ ∈ W01,1 (Ω) such that a(GZ , v) = v(Z) ∀ v ∈ W01,∞ (Ω).

(2.2)

Proof. We first prove the uniqueness of GZ . Suppose there exists another Green’s function HZ ∈ W01,1 (Ω) satisfying (2.2). Set EZ = GZ − HZ , thus a(EZ , v) = 0 ∀ v ∈ W01,∞ (Ω).

(2.3)

1

Let w ∈ W 2,4 (Ω) ∩ W01,4 (Ω) and Lw = sgnEZ |EZ | 4 . We have 5

1

4 4 ∥EZ ∥0, 5 = (EZ , sgnEZ |EZ | ) = a(EZ , w),

(2.4)

4

By the Sobolev Embedding Theorem [10], W 2,4 (Ω) ,→ W 1,∞ . Thus w ∈ W01,∞ (Ω). From (2.3) and (2.4), EZ = 0, i.e., GZ = HZ . The proof of the uniqueness is completed.

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Next, we prove the existence of GZ . We give a series of finite element spaces h S0hi (Ω), i = 0, 1, 2, · · · satisfying S0hi (Ω) ⊂ S0 j (Ω) when i < j, where h0 ≡ h and 1 1 ∗ 4 hi−1 ≤ hi ≤ 2 hi−1 . Let GZ,i be the regularized Green’s function for the finite hi element space S0 (Ω), and GhZi the discrete Green’s function. Their definitions can be seen in Section 1. Obviously, we have a(GhZi , v) = v(Z), a(G∗Z,i+1 , v) = v(Z), ∀ v ∈ S0hi (Ω). Thus,

a(G∗Z,i+1 − GhZi , v) = 0 ∀ v ∈ S0hi (Ω).

(2.5)

Similar to the proof of Lemma 2.1, we have

2

∗

GZ,i+1 − GhZi ≤ Chi |ln hi | 3 .

(2.6)

1,1

In addition, from (2.1),



∗

GZ,i − GhZi

1,1

2

≤ Chi |ln hi | 3 .

(2.7)

By (2.6), (2.7), and the triangular inequality, we immediately obtain

∗ 2

GZ,i+1 − G∗Z,i ≤ Chi |ln hi | 3 . 1,1 Thus,

2 ∞ ∞ ∑ ∑

∗

2 h h 3

GZ,i+1 − G∗Z,i ≤ C ln i ≤ Ch |ln h| 3 . i 1,1 2 2 i=0 i=0

Set GZ ≡ G∗Z +

(2.8)

∞ ∑ (G∗Z,i+1 − G∗Z,i ). i=0

Thus we have GZ ∈

W01,1 (Ω).

From (2.8), 2

∥GZ − G∗Z ∥1,1 ≤ Ch |ln h| 3 . Thus, we have

(2.9)

G∗Z,i −→ GZ in W 1,1 (Ω) when i → ∞.

Hence, for v ∈ W01,∞ (Ω), we have a(GZ , v) = lim a(G∗Z,i , v) = lim Phi v(Z).

(2.10)

lim Phi v(Z) = v(Z).

(2.11)

i→∞

i→∞

From (1.10), i→∞

Combining (2.10) and (2.11) yields the result (2.2). Finally, we show GZ is independent of h. Suppose there exists a Green’s ˜ i−1 ≤ h ˜i ≤ 1 h ˜ i−1 and h ˜ 0 = h. ˜ ˜ In addition, 1 h ˜ Z for the mesh-size h. function G 4

2

Thus, for every f ∈ L∞ (Ω), we choose v ∈ W 2,∞ (Ω) ∩ W01,∞ (Ω) such that Lv = ˜ Z , f ) = a(G ˜ Z , v) = v(Z). f . Then we get (GZ , f ) = a(GZ , v) = v(Z) and (G ˜ ˜ ˜ Z . The Thus, (GZ , f ) = (GZ , f ), i.e., (GZ − GZ , f ) = 0. So we get GZ = G proof of Theorem 2.1 is completed.

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LIU, JIA: ESTIMATES FOR THE 3D GREEN’S FUNCTION

3

Estimates for the 3D Green’s Function

Lemma 3.1. Suppose 1 < p < min{2, q0 } and GhZ , and ∂Z,ℓ GhZ defined by (1.4)–(1.7), we have

1 p

+

1 q

= 1. For G∗Z , ∂Z,ℓ G∗Z ,



∗

GZ − GhZ + h ∂Z,ℓ G∗Z − ∂Z,ℓ GhZ ≤ Ch2− p3 . 0,q 0,q

(3.1)

Proof. Obviously, by the interpolation error estimate and the a priori estimate (1.11), we have

∗

G − Gh ≤ C inf v∈S h (Ω) ∥G∗ − v∥ ≤ ∥G∗ − ΠG∗ ∥ Z Z 1 Z Z 1 0

Z 1 3 3 (3.2) ≤ Ch2.5− p ∥G∗ ∥ ≤ Ch2.5− p δ h . Z 2,p

For φ ∈ Lp (Ω), we choose have ∗ (G − Gh , φ) = Z Z ≤

Z 0,p

Φ ∈ W 2,p (Ω) ∩ W01,p (Ω) such that LΦ = φ. Then we a(G∗ − Gh , Φ) = a(G∗ − Gh , Φ − ΠΦ) Z Z Z

Z∗ C GZ − GhZ 1 ∥Φ − ΠΦ∥1 .

(3.3)

From (3.2), (3.3), and the interpolation error estimate, we get ∗

h (GZ − GhZ , φ) ≤ Ch5− p6 δZ

∥φ∥ . 0,p 0,p

(3.4)

h

∗

.

GZ − GhZ ≤ Ch5− p6 δZ 0,p 0,q

(3.5)

Thus

In addition, for 1 ≤ p ≤ ∞, we easily prove

h

3 h

δZ + h ∂Z,ℓ δZ ≤ Ch−3+ p . 0,p 0,p From (3.5) and (3.6), Similarly, we have

(3.6)

∗

GZ − GhZ ≤ Ch2− p3 . 0,q

∂Z,ℓ G∗Z − ∂Z,ℓ GhZ ≤ Ch1− p3 . 0,q

The result (3.1) is proved. We now introduce a weight function defined by ( ) 3 ¯ 2 + θ2 − 2 ∀ X ∈ Ω, ¯ ϕ ≡ ϕ(X) = |X − X| ¯ ∈Ω ¯ is a fixed point, θ = γh, and γ ∈ [3, +∞) is a suitable real number. where X As for the function ϕ, it is easy to prove the following properties hold. ∫ ϕk (X)dX ≤ C(k − 1)−1 θ−3(k−1) ∀ k > 1, (3.7) Ω

∫ ϕk (X)dX ≤ Ω

C ∀ 0 < k < 1, 1−k

1019

(3.8)

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∫ ϕ(X)dX ≤ C(β)| ln θ|, θ ≤ β < 1.

(3.9)

Ω

Similar to the arguments of Lemma 2.4 in [4], we can get the following Lemma 3.2. h h Lemma 3.2. For δZ and ∂Z,ℓ δZ , the discrete δ function and the discrete derivative δ function defined by (1.2) and (1.3), respectively, we have the following weighted-norm estimate:

h

h 3(α−1) h

−α + h ∂Z,ℓ δZ

δZ −α + h ∇δZ ≤ Ch 2 ∀α > 0. ϕ ϕ ϕ−α

(3.10)

h Lemma 3.3. For δZ and G∗Z , the discrete δ function and the regularized Green’s function defined by (1.2) and (1.4), respectively, we have the following weightednorm estimate:

h

−α− 2 + C ∥G∗Z ∥ −α+ 2 ∀α ∈ R. ∥∇G∗Z ∥ϕ−α ≤ C δZ (3.11) 3 3 ϕ

ϕ

Proof. First, we find ∥∇G∗Z ∥ϕ−α ≤ a(G∗Z , ϕ−α G∗Z ) + C ∥G∗Z ∥

2 2 ϕ−α+ 3

2

.

(3.12)

Moreover, a(G∗Z , ϕ−α G∗Z )

h = (δZ , ϕ−α G∗Z ) h ≤ ∥δZ ∥ −α− 23 ∥G∗Z ∥ −α+ 32 ϕ ϕ h 2 ≤ 21 (∥δZ ∥ −α− 2 + ∥G∗Z ∥2 −α+ 2 ). ϕ

3

ϕ

(3.13)

3

Combining (3.12) and (3.13) immediately yields the result (3.11). Theorem 3.1. Suppose q0 > 32 , 32 < p < min{2, q0 }, and p1 + 1q = 1, then we have 3−q 3 ∥GZ − G∗Z ∥0,q ≤ Ch2− p = Ch q . (3.14) Remark 1. Similar to the arguments of (2.9) and with the result (3.1), we easily obtain the result (3.14). Obviously, we have max{2, q0′ } < q < 3 and 1 1 q0 + q ′ = 1. 0

Theorem 3.2. Suppose q0 > 32 . For GZ , the Green’s function defined by (2.2), and the weight function τ = |X − Z|−3 , we have ∥GZ ∥0,q ≤ C(q), 1 ≤ q ≤ 3.

(3.15)

1 < ϵ < ∞. (3.16) 3 3 ∥GZ ∥1,q ≤ C(q), 1 ≤ q < . (3.17) 2 Proof. Obviously, from (3.14), GZ ∈ Lq (Ω) and 1 ≤ q < 3. In addition, we have proved ∥G∗Z ∥0,3 ≤ C in [4]. Moreover, L3 (Ω) is a reflexive space. Thus, {G∗Z,i } ∥GZ ∥1,τ −ϵ ≤ C(ϵ),

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is weakly convergent to QZ ∈ L3 (Ω) ⊂ Lq (Ω), where max{2, q0′ } < q < 3. From (3.14), G∗Z,i −→ GZ in Lq (Ω) when i → ∞. Thus GZ = QZ ∈ L3 (Ω). So we have GZ ∈ Lq (Ω)(1 ≤ q ≤ 3). When max{2, q0′ } < q < 3, we have 23 < p < min{2, q0 }, where p1 + 1q = 1. For every φ ∈ C0∞ (Ω), we can find a function φ˜ ∈ C0∞ (Ω) such that Lφ˜ = φ. Moreover, by the Sobolev Embedding Theorem [10] and the a priori estimate (1.11), we get (GZ , φ) = a(GZ , φ) ˜ = φ(Z) ˜ ≤ ∥φ∥ ˜ 0,∞ ≤ C(q) ∥φ∥ ˜ 2,p ≤ C(q) ∥φ∥0,p . Thus, ∥GZ ∥0,q ≤ C(q).

∗ Since GZ,i 0,3 ≤ C, and {G∗Z,i } is weakly convergent to GZ ∈ L3 (Ω), thus, ∥GZ ∥0,3 ≤ C. In addition, when 1 ≤ q ≤ max{2, q0′ }, we have ∥GZ ∥0,q ≤ C(q) ∥GZ ∥0,3 ≤ C(q). Thus we have finished the proof of the result (3.15). Now we prove the result (3.16). We have obtained the result ∥G∗Z ∥ 31 ≤ ϕ

1

C |ln h| 6 in [4]. When 0 < r < 13 , we have by (3.8) and ∥G∗Z ∥0,3 ≤ C, ∥G∗Z ∥ϕr =

∫

2

ϕr |G∗Z | dX ≤

(∫

2

ϕ3r dX

Ω

) 31

Ω

∥G∗Z ∥0, 3 ≤ C(r) ∥G∗Z ∥0, 3 ≤ C(r). 2

2

Namely, ∥G∗Z ∥ϕr ≤ C(r) ∀ 0 < r < 13 . Obviously, when s < t, we have ϕs ≤ Cϕt . Thus, ∥G∗Z ∥ϕr ≤ C(r) ∀ r ≤ 0. So we have ∥G∗Z ∥ϕr ≤ C(r) ∀ r
By the H¨older inequality, we have for 1 ≤ q < ∥∇G∗Z ∥0,q =

∫

qϵ

qϵ

ϕ 2 ϕ− 2 |∇G∗Z |q dX ≤

q

Ω

Choosing a suitable ϵ such that

(3.20)

3 2

(∫

qϵ

ϕ 2−q dX Ω

qϵ 2−q

1 . 3 ) 2−q 2

∥∇G∗Z ∥ϕ−ϵ . q

< 1, we have by (3.8) and (3.20),

∥∇G∗Z ∥0,q ≤ C(q).

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(3.21)

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Obviously, 1 . (3.22) 3 Since G∗Z is bounded according to the weighted-norm ∥ · ∥1,τ −ϵ , thus, {G∗Z,i } is weakly convergent to a function FZ with ∥FZ ∥1,τ −ϵ < ∞. Further, we have ∥FZ ∥1,1,τ −ϵ < ∞. From (2.9), ∥G∗Z ∥1,τ −ϵ ≤ ∥G∗Z ∥1,ϕ−ϵ ≤ C(ϵ) ∀ ϵ >

2

∥GZ − G∗Z ∥1,1,τ −ϵ ≤ C(ϵ) ∥GZ − G∗Z ∥1,1 ≤ C(ϵ)h |ln h| 3 , which shows {G∗Z,i } is convergent to the function GZ with ∥GZ ∥1,1,τ −ϵ < ∞. Thus, FZ = GZ . Namely, 1 . 3 Up to now, the result (3.16) is thoroughly proved. Similar to the arguments of (3.16), from (3.21), we can obtain the result (3.17). ∥GZ ∥1,τ −ϵ ≤ C(ϵ) ∀ ϵ >

Acknowledgments This work was supported by the National Natural Science Foundation of China Grant 11161039, the Zhejiang Provincial Natural Science Foundation Grant LY13A010007 and the Natural Science Foundation of Ningbo City Grant 2015A610163. References 1. C. M. Chen, Construction theory of superconvergence of finite elements (in Chinese), Hunan Science and Technology Press, Changsha, China, 2001. 2. C. M. Chen and Y. Q. Huang, High accuracy theory of finite element methods (in Chinese), Hunan Science and Technology Press, Changsha, China, 1995. 3. G. Goodsell, Pointwise superconvergence of the gradient for the linear tetrahedral element, Numer. Meth. Part. Differ. Equ. 10 (1994), 651–666. 4. J. H. Liu, B. Jia, and Q. D. Zhu, An estimate for the three-dimensional discrete Green’s function and applications, J. Math. Anal. Appl. 370 (2010), 350-363. 5. J. H. Liu and Q. D. Zhu, The estimate for the W 1,1 -seminorm of discrete derivative Green’s function in three dimensions (in Chinese), J. Hunan Univ. Arts Sci. 16 (2004), 1-3. 6. J. H. Liu and Q. D. Zhu, Pointwise supercloseness of tensor-product block finite elements, Numer. Meth. Part. Differ. Equ. 25 (2009), 990-1008. 7. J. H. Liu and Q. D. Zhu, The W 1,1 -seminorm estimate for the four-dimensional discrete derivative Green’s function, J. Comp. Anal. Appl. 14 (2012), 165-172. 8. J. H. Liu and Y. S. Jia, Five-dimensional discrete Green’s function and its estimates, J. Comp. Anal. Appl. 18 (2015), 620-627. 9. Q. D. Zhu and Q. Lin, Superconvergence theory of the finite element methods (in Chinese), Hunan Science and Technology Press, Changsha, China, 1989. 10. R. A. Adams, Sobolev Spaces, Academic Press, New York, 1975.

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The structure of the zeros and fixed point for Genocchi polynomials J. Y. Kang, C. S. Ryoo Department of Mathematics, Hannam University, Daejeon 306-791, Korea Abstract We find the behavior of complex roots and fixed point for Genocchi polynomials by using numerical investigation. By means of numerical experiments, we display a remarkably regular structure of the complex roots and fixed point for the Genocchi polynomials. 2000 Mathematics Subject Classification - 11B83, 37N30, 41A10 Key words- Genocchi polynomials, Newton method, complex roots, fixed point 1.

Introduction

Mathematicians have studied various kinds of the Euler, Bernoulli, Tangent, and Genocchi polynomials. Recently, many authors have studied the relations between these polynomials and Stirling numbers of the second kind(see [1-24]). Numerical experiments of Bernoulli, Euler, and Genocchi polynomials also have been made the subject of extensive research. The computing environment will be making more and more rapid advance and this environment has been increasing the interest in solving mathematical problems with the aid of computers. The zeros of Genocchi polynomials Gn (x) is very interesting a realistic study by using computer(see [2,16-20,23]). The Genocchi numbers Gn and Genocchi polynomials Gn (x) are usually defined by the following generating functions. Definition 1.1.[5,14,17] Let n ∈ N0 . Then we define ∞ X

tn 2t = t , |t| < π, n! e +1 n=0   ∞ X tn 2t Gn (x) = etx , t+1 n! e n=0 Gn

1

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n

where we use the notation by replacing G(x) by Gn (x) symbolically. Clearly, Gn = Gn (0). In general, it satisfies G3 = G5 = G7 = G9 = · · · = 0, and even coefficients are given Gn = 2nE2n−1 = 2(1−22n )B2n , where En are the Euler numbers and Bn are the Bernoulli numbers(see [4-5, 6, 8, 12, 15]). These polynomials and numbers play important roles in many different areas of mathematics such as combinatorics, number theory, special function and analysis, and numerous interesting results for them have been explored. The following elementary properties of Genocchi polynomials Gn (x) are readily derived from the Definition 1.1. Therefore we choose to omit the details involved. More studies and results in this subject we may see references(see [5-6,14-20]). Throughout this paper, we always make use of the following notations: N = {1, 2, 3, · · · } denotes the set of natural numbers, N0 = {0, 1, 2, 3, · · · } denotes the set of nonnegative integers, Z denotes the set of integers, R denotes the set of real numbers, and C denotes the set of complex numbers, and C∞ = C ∪ {∞}. Theorem 1.2.[5,6,17,19] For n ∈ N0 , we know Gn (x) =

n   X n Gk xn−k . k

k=0

Theorem 1.3.[5,6,15] Let x ∈ N0 . Then we have  2 if n = 1 n (G + 1) + Gn = . 0 if n 6= 1 From the Theorem 1.2 and Theorem 1.3, it is easy to deduce that Gn (x) are polynomials of degree n. The Genocchi polynomials are as follows. G1 (x) = 1, G2 (x) = 2x − 1, G3 (x) = 3x2 − 3x, G4 (x) = 4x3 − 6x2 + 1, G5 (x) = 5x4 − 10x3 + 5x, G6 (x) = 6x5 − 15x4 + 15x2 − 3, G7 (x) = 7x6 − 21x5 + 35x3 − 21x, G8 (x) = 8x7 − 28x6 + 70x4 − 84x2 + 17, ··· . Definition 1.4. Let f : D → D be a complex function, with D a subset of C. We define the iterated maps of the complex function as the following: fn : z0 7→ f (f (· · · (f (z0 ) · · · )) | {z } n−times

2

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The iterates of f are the functions f, f ◦ f, f ◦ f ◦ f, ..., which are denoted f 1 , f 2 , f 3 , .... If z ∈ C, then the orbit of z0 under f is the sequence (z0 , f (z0 ), f (f (z0 )), · · · ). We consider the Newton’s dynamical system as the follows:   S(x) C∞ : R(x) = x − 0 . S (x) R is called the Newton iteration function of S. It can be shown that the fixed points of R are zeros of S and all fixed points of R are attracting. R may also have one or more attracting cycles(see [2, 23-24]). This paper is organized as follows. In Section 2, we study some properties of zeros for Genocchi polynomials from Newtons’method. In section 3, we find some distributions and properties of fixed point for Genocchi polynomials by using iterating map. 2.

The observation for scattering of zeros of the Genocchi polynomials

In this section, we can see the several conjecture from the Tables. we also find the approximate zeros of the Genocchi polynomials. Using the Mathematica software, we can see the structure of the zeros of the Genocchi polynomials in various viewpoints. From the Definition of Genocchi polynomials, we get ∞ X n=0

Gn (1 − x)

∞ X (−t)n −2t −t(1−x) 2t tx tn = −t e =− t e =− Gn (x) . n! e +1 e −1 n! n=0

From the above equation, we find the following theorem. Theorem 2.1.[14,-15,17,19-20]. For n ∈ N0 , we have Gn (x) = (−1)n+1 Gn (1 − x). Conjecture 2.2. Gn (x) = 0 has n distinct solutions. We find a counterexample of the conjecture 2.2. When n = 6, there exist five num1 bers, xi (i = 1, 2, 3, 4, 5) such that G6 (xi ) = 0. That is, we can find x1 = , x2 = 2 √
√
√
√
1 1 1 1 1 − 5 , x3 = 1 − 5 , x4 = 1 + 5 , x5 = 1 + 5 . Therefore, the conjec2 2 2 2 ture 2.3 is not true for all n. Using computers, many more values of n have been checked. It still remains unknown if the conjecture fails or holds for any value n 6= 6. See Table 1 for tabulated values of RGn (x) and CGn (x) , where RGn (x) denote the numbers of real zeros and CGn (x) denotes the numbers of complex zeros. Our numerical results, that is the numbers of real and complex zeros of Gn (x) for 29 ≤ n ≤ 60 are displayed in the Table 1.

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Table 1. Numbers of real and complex zeros of Gn (x) degree n 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44

RGn (x) 8 9 10 11 8 9 10 11 12 9 10 11 12 13 11 11

CGn (x) 20 20 20 20 24 24 24 24 24 28 28 28 28 28 32 32

degree n 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

RGn (x) 12 13 14 15 12 13 14 15 16 13 14 15 16 17 14 15

CGn (x) 32 32 32 32 36 36 36 36 36 40 40 40 40 40 44 44

If we consider Gn (x) for 2 ≤ n ≤ 100, we then find the Figure 1. The x-axis means the numbers of real zeros and the y-axis means the numbers of complex zeros in the Genocchi polynomials in Figure 1. From Table 1 and Figure 1, we can suggest a below conjecture. 70 60 50 40 30 20 10

5

10

15

20

25

Figure 1: Numbers of real and complex zeros of Gn (x) for 2 ≤ n ≤ 100 Conjecture 2.3. When Im(x) 6= 0, we find that (1) the numbers of RGn (x) of Gn (x): RGn (x) = n − 1 − CGn (x) . (2) the numbers of CGn (x) of Gn (x):   n−1−α CGn (x) = 4 , 5



 n + 19 α= , 21

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where [x] is the greatest integer not exceeding x. By using the Theorem 2.1, we also have the following theorem.   1 Theorem 2.4. For n ∈ N0 , if n ≡ 0 (mod 2), then Gn = 0. 2 By Theorem 2.4, we can know the center of the structure of zeros in Genocchi polynomials is 12 (see the Figure2). The forms of 3D structure which is stacks of zeros of Gn (x) for 2 ≤ n ≤ 60 are presented in the top-left of Figure 2. We can draw the top-right figure and bottom-left figure when we look at the top-left Figure 2 in the above position and left orthographic viewpoint, respectively.

Figure 2: Stacks of zeros of Gn (x) for 2 ≤ n ≤ 60 From Definition of Genocchi polynomials, we get ∞ X

(Gn (x + 1) + Gn (x))

n=0

tn 2t t(x+1) 2t tx = t e + t e n! e +1 e +1 tx

= 2te

=2

∞ X

(n + 1)xn

n=0

tn . n!

tn By comparing the coefficients of on both sides of the above equation, we find the then! orem 2.5. Theorem 2.5. For n ∈ N0 we find Gn (x + 1) + Gn (x) = 2nxn−1 .

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Substituting x = 0 in the Theorem 2.5, we find the following corollary 2.6. Corollary 2.6. For n ∈ N, one has Gn = −Gn (1). We consider the Newton’s dynamical system at numbers of roots in G10 (x). We can obtain roots in the G10 (x), that is, x1 x3 x5 x7 x9

= −1.31362 − 0.876373i, = −1.21973, = 0.5, = 2.21973, = 2.31362 + 0.876373i.

x2 = −1.31362 + 0.876373i, x4 = −0.50008, x6 = 1.50008, x8 = 2.31362 − 0.876373i,

The orbit of x0 from the Newton method appears by calculating until 30 iterations or the absolute difference value of the last two iterations is within 10−6 . We hope to determine whether the orbit of x0 under the action of Newton’s dynamical system converges to one of roots when it is given a point x0 in the complex plane.

Figure 3: General structure of orbits for {−1.5 ≤ x ≤ 2.5}, {−1.5 ≤ y ≤ 2.5}

The output of Figure 3 is the orbit values by using the above method. We plot the blue, brown, yellow, skyblue, green, ocher, navy blue, red, or gray to x0 in the Figure 3, when an

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orbit of x0 converge to −1.31362−0.876373i, −1.31362+0.876373i, −1.21973, −0.50008, 0.5, 1.50008, 2.21973, 2.31362 − 0.876373i, 2.31362 + 0.876373i, respectively. From the top-left figure, we can observe general structure for {−1.5 ≤ x ≤ 2.5}, {−1.5 ≤ y ≤ 2.5}. Moreover, we can observe property of complex conjugate from the top-right figure and bottomfigures in the right part of general structure by narrowing range. The interesting result is the fact that each boundaries of range parts have every colors and self-similarity. 3.

The fixed points of Genocchi polynomials

In this section, we present distributions of fixed points and period points from iterating map. From definition and property of fixed point, we find it and construct structure of this points in the complex plane. By expanding method of previous section we can discuss the fixed points and period points of the Genocchi polynomials. Definition 3.1. The orbit of the point z0 ∈ C under the action of the function f is said to be bounded if there exists M ∈ R such that |f n (z0 )| < M for all n ∈ N. If the orbit is not bounded, it is said to be unbounded. Definition 3.2. Let f : D → D be a transformation on a metric space. A point z0 ∈ D such that f (z0 ) = z0 is called a fixed point of the transformation. Suppose that the complex function f is analytic in a region D of C, and f has a fixed point at z0 ∈ D. Then z0 is said to be: an attracting fixed point if |f 0 (z0 )| < 1; a repelling fixed point if |f 0 (z0 )| > 1; a neutral fixed point if |f 0 (z0 )| = 1. For example, G4 (x) − 1.01 − 0.1i have three points satisfying G4 (x) − 1.01 − 0.1i = x. That is, x0 = −0.174314+0.0695883i, 0.0220059−0.0779681i, 1.65231+0.00837978i. Since d G4 (0.0220059 − 0.0779681i) − 1.01 − 0.1i = 0.953792 < 1, dz we obtain the following theorem. Theorem 3.3. The Genocchi polynomials G4 (x) − 1.01 − 0.1i has the only one attracting fixed point at α = 0.0220059 − 0.0779681i. We can separate the numerical results for fixed point of Gn (x) by using Mathematica software. In the Table 2, we can look for numbers of fixed points of Gn (x) for 3 ≤ n ≤ 10 and find property of their points.

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Table 2. Numbers of attracting, repelling, and neutral fixed points of Gn (x) degree n 3 4 5 6 7 8 9 10

attractor 0 0 0 0 0 0 0 0

repellor 2 3 4 5 6 7 8 9

neutral 0 0 0 0 0 0 0 0

Conjecture 3.4. The Genocchi polynomials Gn (x) has no attracting and neutral fixed point except for infinity. In the Table 3, we consider Gr4 (x) by using iterating map. We can know the numbers of real roots of Gr4 (x) using iterated function are less than 3r . In addition, we observe the numbers of real roots will be 2r+1 − 1 for r ≥ 1 and find there is no the real number which is related to fixed point. Table 3. Numbers of roots and fixed points of Gr4 (x) for 1 ≤ r ≤ 9 r G14 (x) G24 (x) G34 (x) G44 (x) G54 (x) G64 (x) G74 (x) G84 (x) G94 (x) ···

numbers of real roots 3 7 15 31 63 127 255 511 1023 ···

numbers of real numbers in fixed points 3 5 15 51 0 0 0 0 0 ···

In the top-left Figure 4, we can see the forms of 3D structure which is related to stacks of fixed points of iterated Gr4 (x) for 1 ≤ r ≤ 6. We can draw the top-right figure when we look at the top-left Figure 4 in the below position. The bottom-left of Figure 4 represent that image and n axes are exist but there is no real axis. The bottom-right of Figure 4 is the right orthographic viewpoint for the top-left figure, that is, there exist real and n axes but don’t exist image axis.

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Figure 4: Stacks of fixed points of Gr4 (x) for 1 ≤ n ≤ 6

We consider G24 (x) for x ∈ C. This polynomial has nine distinct complex numbers, ai (i = 1, 2, 3, 4, 5, 6, 7, 8, 9) such that G24 (ai ) = ai . We obtain a1 = −0.430403, a2 = −0.244653, a3 = −0.0322871−0.240632i, a4 = −0.0322871+0.240632i, a5 = 0.372949, a6 = 0.582294, a7 = 1.36347 − 0.0405081i, a8 = 1.36347 + 0.0405081i, a9 = 1.55745. By combining Newton’s method in the G24 (x), we have   G24 (x) e C∞ : R(x) =x− . (G24 (x))0 e The general expectation is a typical orbit {R(x)} will converge to one of the fixed points of G24 (x) for x0 ∈ C. If we choose x0 close enough to ai then it is readily proved that e 0 ) = ai , for i = 1, 2, 3, 4, 5, 6, 7, 8, 9. lim R(x

n→∞

Given a point x0 in the complex plane, we want to find out if the orbit of x0 under the e action of R(x) does or does not converge to one of the fixed points, and if so, which one. e e When R(x) is applied to x0 , the orbit of x0 under the action of R(x) is calculated until the absolute value of the last 2 iterations differs by an amount less than 10−6 , or until 30 iteration have been carried out.

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The Figure 5 is the last orbit value calculated. We construct a function which ase signs one of nine colors to each point in the plane, according to the outcome of R. We allocate the red, violet, yellow, skyblue, green, ocher, blue, navy blue, or gray to x0 if its orbit converges to −0.430403, −0.244653, −0.0322871 − 0.240632i, −0.0322871 + 0.240632i, 0.372949, 0.582294, 1.36347 − 0.0405081i, 1.36347 + 0.0405081i, 1.55745, respectively. We make the range which is {(x, y) : −4 ≤ x ≤ 4, −4 ≤ y ≤ 4}. For example, the red region represents part of the attracting basin of a1 = −0.430403

e for G24 (x) Figure 5: Orbit of x0 under the action of R The Figure 6 express the coloring of the next Figure 7. Points which escape after 1 to 30 iterations are colored red to green.

0

5

10

15

20

25

30

Figure 6: Palette for escaping points

In the Figure 7, the above-mentioned rapid change can be illustrated by applying the three-dimensional structure to the escape-time function. We consturct the range of left figure which is {(x, y) : −3 ≤ x ≤ 3, −3 ≤ y ≤ 3} and the range of right figure which is {(x, y) : −4 ≤ x ≤ 4, −4 ≤ y ≤ 4}. From this figure, we can see the same color regions which are the orbit of point, z0 , approached an one of fixed points at the equivalent iterated step.

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e Figure 7: Escape-time map of R(x) for G24 (x)

Acknowledgements This work was supported by NRF(National Research Foundation of Korea) Grant funded by the Korean Government(NRF-2013-Fostering Core Leaders of the Future Basic Science Program). References [1] M. Alkan and Y. Simsek, Generating function for q-Eulerian polynomials and their decomposition and applications, Fixed Point Theory and Applications, 2013(2013), 72. [2] Kathleen T. Alligood, Tim D. Sauer, James A. Yorke, Chaos:An introduction to dynamical systems, Springer, 1996. [3] R. Ayoub, Euler zeta function, Amer. Math, Monthly, 81 (1974), 1067-1086. [4] A. Bayad, Modular properties of elliptic Bernoulli and Euler functions, Advanced Studies in Contemporary Mathematics, 20 (2010), 389-401. [5] A. F. Horadam, Genocchi Polynomials, Applications of Fibonacci Numbers, (1991), 145-166. [6] L. C. Jang, A study on the distribution of twisted q-Genocchi polynomials, Advanced Studies in Contemporary Mathematics, 18(2) (2009), 182-189. [7] J. Y. Kang, H. Y. Lee, N. S. Jung, Some relations of the twisted q-Genocchi numbers and polynomials with weight α and weak Weight β, Abstract and Applied Analysis, 2012, Article ID 860921, 9 pages, 2012. [8] M. S. Kim, S. Hu, On p-adic Hurwitz-type Euler Zeta functions, Journal of Number Theory, (132) (2012), 2977-3015.

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[9] T. Kim, C. S. Ryoo, L. C. Jang, S. H. Rim, Exploring the q-Riemann Zeta function and q-Bernoulli polynomials, Discrete Dynamics in Nature and Society, (2) (2005), 171-181. [10] T. Kim, S. H. Rim, Generalized Carlitz’s Euler Numbers in the p-adic number field , Advanced Studies in Contemporary Mathematics, 2 (2000), 9-19 . [11] T. Kim, Euler numbers and polynomials associated with zeta functions, Abstract and Applied Analysis, 2008(2008), Article ID 581582. [12] H. Ozden, Y. Simsek, A new extension of q-Euler numbers and polynomials related to their interpolation functions, Appl. Math. Letters, 21 (2008), 934-938. [13] H. Ozden, Y. Simsek, Interpolation function of the (h, q)-extension of twisted Euler numbers, Comput. Math. Appl., 56(4) (2008), 898-908. [14] K. H. Park, S. H. Rim, E. J. Moon, On Genocchi numbers and polynomials, Abstract and Applied Analysis, 2008 (2008), Article ID 898471. [15] Seog-Hoon Rim, Joohee Jeong, Sun-Jung Lee, Identities on the Bernoulli and Genocchi numbers and polynomials, International Journal of Mathematics and Mathematical Sciences, 2012 (2012), Article ID 184649, 9 pages. [16] C. S. Ryoo, T. Kim, R. P. Agarwal, A numerical investigation of the roots of qpolynomials, Inter. J. Comput. Math., 83(2) (2006), 223-234. [17] C. S. Ryoo, A mumerical investigation on the zeros of the Genocchi polynomials, Journal of Applied Mathematics and Computing, 22 (2006), 125-132. [18] C. S. Ryoo, Calculating zeros of the twisted Genocchi polynomials, Advanced Studies in Contemporary Mathematics, 17 (2008), 147-159. [19] C. S. Ryoo, A note on the reflection symmetries of the Genocchi polynomials, Journal of Applied Mathematics and Informatics, 27(5-6) (2009), 1394-1404. [20] C. S. Ryoo, A note on the reflection symmetries of the Genocchi polynomials, Journal of Applied Mathematics and Informatics, 27(5-6) (2009), 1394-1404. [21] Y. Simsek, Generating functions of the twisted Bernoulli numbers and polynomials, Adv. Stud. Contemp. Math., 16 (2008), 251-257. [22] Y. Simsek, Twisted (h, q)-Bernoulli numbers and polynomials related to twisted (h, q)-zeta function and L-function, J.Math. Anal. Appl., 324 (2006), 790-804. [23] Steven H. Strogatz, Nonlinear dynamics and chaos, Perseus Books, 1994. [24] C. Getz, J. M. Helmstedt, Graphics with Mathematica: Fractals, Julia Sets, Patterns and Natural Forms, Elsevier Science, 2004.

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ADDITIVE ρ-FUNCTIONAL EQUATIONS CHOONKIL PARK AND SUN YOUNG JANG∗ Abstract. In this paper, we solve the additive ρ-functional equations ( ( ) ) x+y − f (x) − f (y) , (0.1) f (x + y) − f (x) − f (y) = ρ 2f 2 ( ) x+y 2f − f (x) − f (y) = ρ (f (x + y) − f (x) − f (y)) , (0.2) 2 where ρ is a fixed non-Archimedean number or a fixed real or complex number with ρ ̸= 1. Using the direct method, we prove the Hyers-Ulam stability of the additive ρfunctional equations (0.1) and (0.2) in non-Archimedean Banach spaces and in Banach spaces.

1. Introduction and preliminaries A valuation is a function | · | from a field K into [0, ∞) such that 0 is the unique element having the 0 valuation, |rs| = |r| · |s| and the triangle inequality holds, i.e., |r + s| ≤ |r| + |s|,

∀r, s ∈ K.

A field K is called a valued field if K carries a valuation. The usual absolute values of R and C are examples of valuations. Let us consider a valuation which satisfies a stronger condition than the triangle inequality. If the triangle inequality is replaced by |r + s| ≤ max{|r|, |s|},

∀r, s ∈ K,

then the function | · | is called a non-Archimedean valuation, and the field is called a non-Archimedean field. Clearly |1| = | − 1| = 1 and |n| ≤ 1 for all n ∈ N. A trivial example of a non-Archimedean valuation is the function | · | taking everything except for 0 into 1 and |0| = 0. Throughout this paper, we assume that the base field is a non-Archimedean field, hence call it simply a field. 2010 Mathematics Subject Classification. Primary 46S10, 39B62, 39B52, 47S10, 12J25. Key words and phrases. Hyers-Ulam stability; additive ρ-functional equation; non-Archimedean normed space; Banach space. ∗ Corresponding author: Sun Young Jang (email: [email protected]). 1

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Definition 1.1. ([12]) Let X be a vector space over a field K with a non-Archimedean valuation | · |. A function ∥ · ∥ : X → [0, ∞) is said to be a non-Archimedean norm if it satisfies the following conditions: (i) ∥x∥ = 0 if and only if x = 0; (ii) ∥rx∥ = |r|∥x∥ (r ∈ K, x ∈ X); (iii) the strong triangle inequality ∥x + y∥ ≤ max{∥x∥, ∥y∥},

∀x, y ∈ X

holds. Then (X, ∥ · ∥) is called a non-Archimedean normed space. Definition 1.2. (i) Let {xn } be a sequence in a non-Archimedean normed space X. Then the sequence {xn } is called Cauchy if for a given ε > 0 there is a positive integer N such that ∥xn − xm ∥ ≤ ε for all n, m ≥ N . (ii) Let {xn } be a sequence in a non-Archimedean normed space X. Then the sequence {xn } is called convergent if for a given ε > 0 there are a positive integer N and an x ∈ X such that ∥xn − x∥ ≤ ε for all n ≥ N . Then we call x ∈ X a limit of the sequence {xn }, and denote by limn→∞ xn = x. (iii) If every Cauchy sequence in X converges, then the non-Archimedean normed space X is called a non-Archimedean Banach space. The stability problem of functional equations originated from a question of Ulam [17] concerning the stability of group homomorphisms. The functional equation f (x + y) = f (x) + f (y) is called the Cauchy equation. In particular, every solution of the Cauchy equation is said to be an additive mapping. Hyers [11] gave a first affirmative partial answer to the question of Ulam for Banach spaces. Hyers’ Theorem was generalized by Aoki [1] for additive mappings and by Rassias [15] for linear mappings by considering an unbounded Cauchy difference. A generalization of the Rassias theorem was obtained by G˘avruta [8] by replacing the unbounded Cauchy difference by a general ( )control function in the spirit of Rassias’ x+y approach. The functional equation f 2 = 12 f (x) + 12 f (y) is called the Jensen equation. See [2, 3, 4, 5, 6, 7, 9, 10, 13, 14, 16, 18] for more information on functional equations. In Section 2, we solve the additive ρ-functional equation (0.1) in vector spaces and prove the Hyers-Ulam stability of the additive ρ-functional equation (0.1) in nonArchimedean Banach spaces.

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3

In Section 3, we solve the additive ρ-functional equation (0.2) in vector spaces and prove the Hyers-Ulam stability of the additive ρ-functional equation (0.2) in nonArchimedean Banach spaces. In Section 4, we prove the Hyers-Ulam stability of the additive ρ-functional equation (0.1) in Banach spaces. In Section 5, we prove the Hyers-Ulam stability of the additive ρ-functional equation (0.2) in Banach spaces. 2. Additive ρ-functional equation (0.1) in non-Archimedean Banach spaces Throughout Sections 2 and 3, assume that X is a non-Archimedean normed space and that Y is a non-Archimedean Banach space. Let |2| ̸= 1 and let ρ be a fixed non-Archimedean number with ρ ̸= 1. We solve the additive ρ-functional equation (0.1) in vector spaces. Lemma 2.1. Let X and Y be vector spaces. If a mapping f : X → Y satisfies (

(

)

x+y f (x + y) − f (x) − f (y) = ρ 2f − f (x) − f (y) 2

)

(2.1)

for all x, y ∈ X, then f : X → Y is additive. Proof. Assume that f : X → Y satisfies (2.1). Letting x = y = 0 in (2.1), we get −f (0) = 0. So f (0) = 0. Letting y = x in (2.1), we get f (2x)−2f (x) = 0 and so f (2x) = 2f (x) for all x ∈ X. Thus ( ) x 1 f = f (x) (2.2) 2 2 for all x ∈ X. It follows from (2.1) and (2.2) that (

(

)

x+y − f (x) − f (y) 2 = ρ(f (x + y) − f (x) − f (y))

f (x + y) − f (x) − f (y) = ρ 2f

)

and so f (x + y) = f (x) + f (y) for all x, y ∈ X.

□

We prove the Hyers-Ulam stability of the additive ρ-functional equation (2.1) in non-Archimedean Banach spaces.

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Theorem 2.2. Let φ : X 2 → [0, ∞) be a function and let f : X → Y be a mapping such that ) ( ∞ ∑ x y j < ∞, (2.3) Ψ(x, y) := |2| φ j , j 2 2 j=1

) ) ( (

x+y

f (x + y) − f (x) − f (y) − ρ 2f − f (x) − f (y)

≤ φ(x, y) (2.4)

2 for all x, y ∈ X. Then there exists a unique additive mapping A : X → Y such that 1 ∥f (x) − A(x)∥ ≤ Ψ(x, x) (2.5) |2| for all x ∈ X. Proof. Letting y = x in (2.4), we get ∥f (2x) − 2f (x)∥ ≤ φ(x, x) for all x ∈ X. So

(2.6)

( ) ) (

x

x y

f (x) − 2f , ≤ φ

2

2 2

for all x ∈ X. Hence

( ) ( )

l x

x m

2 f −2 f m (2.7)

2l 2

( { ( ) ) ( ) } ( )

x x x x ≤ max

2l f l − 2l+1 f l+1

, · · · ,

2m−1 f m−1 − 2m f m

2 2 ) 2 2 ) }

(

( ) ( ) ( {



x x x x ≤ max |2|l

f l − 2f l+1

, · · · , |2|m−1

f m−1 − 2f m

2 2 2 2 ( ) ∞ ∑ x x ≤ |2|j φ j+1 , j+1 2 2 j=l for all nonnegative integers m and l with m > l and all x ∈ X. It follows from (2.7) that the sequence {2k f ( 2xk )} is Cauchy for all x ∈ X. Since Y is a non-Archimedean Banach space, the sequence {2k f ( 2xk )} converges. So one can define the mapping A : X → Y by ( ) x k A(x) := lim 2 f k k→∞ 2 for all x ∈ X. Moreover, letting l = 0 and passing the limit m → ∞ in (2.7), we get (2.5). Now, let T : X → Y be another additive mapping satisfying (2.5). Then we have

( ) ( )

q x

x q

∥A(x) − T (x)∥ = 2 A q − 2 T 2 2q ( ) { ( ) ( ) ( ) }

q x x



q x x q q

≤ max 2 A q − 2 f q , 2 T − 2 f q

q 2 ) 2 2 2 ( x x ≤ |2|q−1 Ψ q , q , 2 2

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5

which tends to zero as q → ∞ for all x ∈ X. So we can conclude that A(x) = T (x) for all x ∈ X. This proves the uniqueness of A. It follows from (2.3) and (2.4) that

) ) ( (

x+y

A(x + y) − A(x) − A(y) − ρ 2A − A(x) − A(y)

2

( ( ) ( ) ( ) ( ( ) ( ) ( )))

n

x + y x y x+y x y

= lim

2 f − f − f − ρ 2f − f − f

n n n n+1 n n n→∞ 2 2 2 2 2 2 ) (

≤ lim |2|n φ n→∞

x y , 2n 2n

=0

for all x, y ∈ X. So (

(

)

x+y A(x + y) − A(x) − A(y) = ρ 2A − A(x) − A(y) 2

)

for all x, y ∈ X. By Lemma 2.1, the mapping A : X → Y is additive.

□

Corollary 2.3. Let r < 1 and θ be nonnegative real numbers, and let f : X → Y be a mapping such that

( ( ) )

x+y

f (x + y) − f (x) − f (y) − ρ 2f − f (x) − f (y)

≤ θ(∥x∥r + ∥y∥r ) (2.8)

2

for all x, y ∈ X. Then there exists a unique additive mapping A : X → Y such that ∥f (x) − A(x)∥ ≤

2θ ∥x∥r − |2|

|2|r

for all x ∈ X. Theorem 2.4. Let φ : X 2 → [0, ∞) be a function and let f : X → Y be a mapping satisfying (2.4) and Ψ(x, y) :=

∞ ∑ 1 j j=0 |2|

φ(2j x, 2j y) < ∞

for all x, y ∈ X. Then there exists a unique additive mapping A : X → Y such that ∥f (x) − A(x)∥ ≤

1 Ψ(x, x) |2|

(2.9)

for all x ∈ X. Proof. It follows from (2.6) that



1 1

f (x) − f (2x) ≤ φ(x, x)

2

|2|

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for all x ∈ X. Hence



1 1 m

f (2l x) − f (2 x) (2.10)

l

2 2m

}

{ ( )

1 ( l )

1 1 ( l+1 )

1 m−1 m

≤ max l f 2 x − l+1 f 2 x , · · · , m−1 f 2 x − m f (2 x) 2 2 2 2 {

}

1

( l ) 1 ( l+1 )

1

( m−1 ) 1 m ≤ max f 2 x − f 2 x f 2 x − f (2 x) , · · · ,



|2|l 2 |2|m−1 2 ∞ ∑ 1 ≤ φ(2j x, 2j x) j+1 |2| j=l for all nonnegative integers m and l with m > l and all x ∈ X. It follows from (2.10) that the sequence { 21n f (2n x)} is a Cauchy sequence for all x ∈ X. Since Y is complete, the sequence { 21n f (2n x)} converges. So one can define the mapping A : X → Y by 1 A(x) := lim n f (2n x) n→∞ 2 for all x ∈ X. Moreover, letting l = 0 and passing the limit m → ∞ in (2.10), we get (2.9). The rest of the proof is similar to the proof of Theorem 2.2. □ Corollary 2.5. Let r > 1 and θ be nonnegative real numbers, and let f : X → Y be a mapping satisfying (2.8). Then there exists a unique additive mapping h : X → Y such that 2θ ∥x∥r ∥f (x) − h(x)∥ ≤ |2| − |2|r for all x ∈ X. 3. Additive ρ-functional equation (0.2) in non-Archimedean Banach spaces We solve the additive ρ-functional equation (0.2) in vector spaces. Lemma 3.1. Let X and Y be vector spaces. If a mapping f : X → Y satisfis f (0) = 0 and ( ) x+y 2f − f (x) − f (y) = ρ (f (x + y) − f (x) − f (y)) (3.1) 2 for all x, y ∈ X, then f : X → Y is additive. Proof. Assume that f : X → Y satisfies (3.1). Letting y = 0 in (3.1), we get ( ) x 2f − f (x) = 0 2 and so f

( ) x 2

(3.2)

= 12 f (x) for all x ∈ X.

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It follows from (3.1) and (3.2) that

(

7

)

x+y f (x + y) − f (x) − f (y) = 2f − f (x) − f (y) 2 = ρ(f (x + y) − f (x) − f (y)) and so f (x + y) = f (x) + f (y) for all x, y ∈ X.

□

Now, we prove the Hyers-Ulam stability of the additive ρ-functional equation (3.1) in non-Archimedean Banach spaces. Theorem 3.2. Let φ : X 2 → [0, ∞) be a function and let f : X → Y be a mapping satisfying f (0) = 0 and ∞ ∑

(

x y Ψ(x, y) := |2| φ j , j 2 2 j=0

)

j

< ∞,

(

)

x+y

2f − f (x) − f (y) − ρ (f (x + y) − f (x) − f (y))

≤ φ(x, y) (3.3) 2 for all x, y ∈ X. Then there exists a unique additive mapping A : X → Y such that ∥f (x) − A(x)∥ ≤ Ψ(x, 0)

(3.4)

for all x ∈ X. Proof. Letting y = 0 in (3.3), we get

( ) ( )

x



x

f (x) − 2f − f (x)

≤ φ(x, 0) (3.5)

= 2f

2 2 for all x ∈ X. So

( ) ( )

l x

x m

2 f −2 f m (3.6)

2l 2

{ ( ) ( ) ( ) } ( )

x x x x ≤ max

2l f l − 2l+1 f l+1

, · · · ,

2m−1 f m−1 − 2m f m

2 2 ) 2 2 ) }

(

( ) { ( ) (



x x x x ≤ max |2|l

f l − 2f l+1

, · · · , |2|m−1

f m−1 − 2f m

2 2 2 2 ( ) ∞ ∑ x ≤ |2|j φ j , 0 2 j=l for all nonnegative integers m and l with m > l and all x ∈ X. It follows from (3.6) that the sequence {2k f ( 2xk )} is Cauchy for all x ∈ X. Since Y is a non-Archimedean Banach space, the sequence {2k f ( 2xk )} converges. So one can define the mapping A : X → Y by ( ) x k A(x) := lim 2 f k k→∞ 2

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C. PARK AND S. Y. JANG

for all x ∈ X. Moreover, letting l = 0 and passing the limit m → ∞ in (3.6), we get (3.4). The rest of the proof is similar to the proof of Theorem 2.2. □ Corollary 3.3. Let r < 1 and θ be nonnegative real numbers, and let f : X → Y be a mapping satisfying f (0) = 0 and

(

)

x+y

2f − f (x) − f (y) − ρ(f (x + y) − f (x) − f (y))

≤ θ(∥x∥r + ∥y∥r ) (3.7)

2 for all x, y ∈ X. Then there exists a unique additive mapping A : X → Y such that ∥f (x) − A(x)∥ ≤

|2|r θ ∥x∥r r |2| − |2|

for all x ∈ X. Theorem 3.4. Let φ : X 2 → [0, ∞) be a function and let f : X → Y be a mapping satisfying f (0) = 0, (3.3) and Ψ(x, y) :=

∞ ∑ 1 j j=1 |2|

φ(2j x, 2j y) < ∞

for all x, y ∈ X. Then there exists a unique additive mapping A : X → Y such that ∥f (x) − A(x)∥ ≤ Ψ(x, 0)

(3.8)

for all x ∈ X. Proof. It follows from (3.5) that



1 1

f (x) − f (2x) ≤ φ(2x, 0)

|2|

2

for all x ∈ X. Hence



1 1 m

f (2l x) − f (2 x) (3.9)

l 2 2m

} { ( )

1 ( l )

1

1 ( l+1 )

1 m−1 m

≤ max l f 2 x − l+1 f 2 x , · · · , m−1 f 2 x − m f (2 x) 2 2 2 2 {

(

(

} ) ( ) )



1 1 1 1 l+1 m l m−1

,··· ,

f 2

f 2 x − f 2 x x − f (2 x) ≤ max



|2|l 2 |2|m−1 2 ∞ ∑ 1 ≤ φ(2j x, 0) j |2| j=l+1 for all nonnegative integers m and l with m > l and all x ∈ X. It follows from (3.10) that the sequence { 21n f (2n x)} is a Cauchy sequence for all x ∈ X. Since Y is complete, the sequence { 21n f (2n x)} converges. So one can define the mapping A : X → Y by A(x) := n→∞ lim

1 f (2n x) 2n

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9

for all x ∈ X. Moreover, letting l = 0 and passing the limit m → ∞ in (3.10), we get (3.9). The rest of the proof is similar to the proof of Theorem 2.2. □ Corollary 3.5. Let r > 1 and θ be positive real numbers, and let f : X → Y be a mapping satisfying f (0) = 0 and (3.7). Then there exists a unique additive mapping A : X → Y such that |2|r θ ∥f (x) − A(x)∥ ≤ ∥x∥r |2| − |2|r for all x ∈ X. 4. Additive ρ-functional equation (0.1) in Banach spaces Throughout Sections 4 and 5, assume that X is a normed space and that Y is a Banach space. Let ρ be a fixed real or complex number with ρ ̸= 1. We prove the Hyers-Ulam stability of the additive ρ-functional equation (2.1) in Banach spaces. Theorem 4.1. Let φ : X 2 → [0, ∞) be a function and let f : X → Y be a mapping such that ( ) ∞ ∑ x y j Ψ(x, y) := 2 φ j, j < ∞, (4.1) 2 2 j=1

) ) ( (

x+y

f (x + y) − f (x) − f (y) − ρ 2f

≤ φ(x, y) (4.2) − f (x) − f (y)

2 for all x, y ∈ X. Then there exists a unique additive mapping A : X → Y such that 1 ∥f (x) − A(x)∥ ≤ Ψ(x, x) (4.3) 2 for all x ∈ X. Proof. Letting y = x in (4.2), we get ∥f (2x) − 2f (x)∥ ≤ φ(x, x) for all x ∈ X. So

(4.4)

( ) ( )

x

x y

f (x) − 2f ≤ φ ,

2

for all x ∈ X. Hence

( ) ( )

l x x

m

2 f −2 f m

l

2

2

≤

2 2

m−1 ∑

j

2 f

(

j=l

≤

m−1 ∑

( j

2φ

j=l

1043

x 2j x

)

(

−2

j+1

x

, 2j+1 2j+1

f

)

j+1

x

2

)

(4.5)

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for all nonnegative integers m and l with m > l and all x ∈ X. It follows from (4.5) that the sequence {2k f ( 2xk )} is Cauchy for all x ∈ X. Since Y is a Banach space, the sequence {2k f ( 2xk )} converges. So one can define the mapping A : X → Y by (

A(x) := lim 2k f k→∞

x 2k

)

for all x ∈ X. Moreover, letting l = 0 and passing the limit m → ∞ in (4.5), we get (4.3). Now, let T : X → Y be another additive mapping satisfying (4.3). Then we have

( ) ( )

q x x

q

∥A(x) − T (x)∥ = 2 A q − 2 T 2 2q( )

( ) ( ) ( )

q x



q x x x q q

− 2 f q

≤ 2 A q − 2 f q + 2 T q 2 2 2 2 ) (

≤ 2q Ψ

x x , , 2q 2q

which tends to zero as q → ∞ for all x ∈ X. So we can conclude that A(x) = T (x) for all x ∈ X. This proves the uniqueness of A. It follows from (4.1) and (4.2) that

) ) ( (

x+y

A(x + y) − A(x) − A(y) − ρ 2A − A(x) − A(y)

2

( ( ) ( ) ( ) ( ( ) ( ) ( )))

n x y x+y x y x + y

− f − f − ρ 2f − f − f = lim 2 f

n n n n+1 n n n→∞ 2 2 2 2 2 2 ( )

≤ lim 2n φ n→∞

x y , 2n 2n

=0

for all x, y ∈ X. So (

(

)

x+y A(x + y) − A(x) − A(y) = ρ 2A − A(x) − A(y) 2

)

for all x, y ∈ X. By Lemma 2.1, the mapping A : X → Y is additive.

□

Corollary 4.2. Let r > 1 and θ be nonnegative real numbers, and let f : X → Y be a mapping such that

) ) ( (

x+y

f (x + y) − f (x) − f (y) − ρ 2f − f (x) − f (y)

≤ θ(∥x∥r + ∥y∥r ) (4.6)

2

for all x, y ∈ X. Then there exists a unique additive mapping A : X → Y such that ∥f (x) − A(x)∥ ≤

2θ ∥x∥r −2

2r

for all x ∈ X.

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ADDITIVE ρ-FUNCTIONAL EQUATIONS

11

Theorem 4.3. Let φ : X 2 → [0, ∞) be a function and let f : X → Y be a mapping satisfying (4.2) and Ψ(x, y) :=

∞ ∑ 1 j j=0 2

φ(2j x, 2j y) < ∞

for all x, y ∈ X. Then there exists a unique additive mapping A : X → Y such that 1 ∥f (x) − A(x)∥ ≤ Ψ(x, x) 2

(4.7)

for all x ∈ X. Proof. It follows from (4.4) that



1 1

f (x) − f (2x) ≤ φ(x, x)

2

2

for all x ∈ X. Hence



1 1 m

f (2l x) − f (2 x)

l

m

2

2

≤

m−1 ∑

j=l

≤

1 ( j ) 1 ( j+1 )

f 2 x − f 2 x 2j 2j+1

m−1 ∑

1

j=l

2j+1

φ(2j x, 2j x)

(4.8)

for all nonnegative integers m and l with m > l and all x ∈ X. It follows from (4.8) that the sequence { 21n f (2n x)} is a Cauchy sequence for all x ∈ X. Since Y is complete, the sequence { 21n f (2n x)} converges. So one can define the mapping A : X → Y by 1 f (2n x) 2n for all x ∈ X. Moreover, letting l = 0 and passing the limit m → ∞ in (4.8), we get (4.7). The rest of the proof is similar to the proof of Theorem 4.1. □ A(x) := n→∞ lim

Corollary 4.4. Let r < 1 and θ be nonnegative real numbers, and let f : X → Y be a mapping satisfying (4.6). Then there exists a unique additive mapping A : X → Y such that 2θ ∥f (x) − A(x)∥ ≤ ∥x∥r 2 − 2r for all x ∈ X. 5. Additive ρ-functional equation (0.2) in Banach spaces In this section, we prove the Hyers-Ulam stability of the additive ρ-functional equation (3.1) in Banach spaces.

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C. PARK AND S. Y. JANG

Theorem 5.1. Let φ : X 2 → [0, ∞) be a function and let f : X → Y be a mapping satisfying f (0) = 0 and (

∞ ∑

x y Ψ(x, y) := 2 φ j, j 2 2 j=0

)

j

( )

x+y

2f − f (x) − f (y) − ρ (f (x + y) − f (x) − f (y))

2

< ∞, ≤ φ(x, y)

(5.1)

for all x, y ∈ X. Then there exists a unique additive mapping A : X → Y such that ∥f (x) − A(x)∥ ≤ Ψ(x, 0)

(5.2)

for all x ∈ X. Proof. Letting y = 0 in (5.1), we get

( ) ( )

x



x

f (x) − 2f

≤ φ(x, 0) − f (x) = 2f



2

for all x ∈ X. So

( ) ( )

l x

x m

2 f −2 f m

l

2

2

(5.3)

2

≤

m−1 ∑

j

2 f

(

j=l

≤

m−1 ∑ j=l

x 2j

)

−2

j+1

(

f

)

j+1

x

2

)

(

x 2 φ j,0 2 j

(5.4)

for all nonnegative integers m and l with m > l and all x ∈ X. It follows from (5.4) that the sequence {2k f ( 2xk )} is Cauchy for all x ∈ X. Since Y is a Banach space, the sequence {2k f ( 2xk )} converges. So one can define the mapping A : X → Y by (

x A(x) := lim 2 f k k→∞ 2

)

k

for all x ∈ X. Moreover, letting l = 0 and passing the limit m → ∞ in (5.4), we get (5.2). The rest of the proof is similar to the proof of Theorem 4.1. □ Corollary 5.2. Let r > 1 and θ be nonnegative real numbers, and let f : X → Y be a mapping satisfying f (0) = 0 and

(

)

x+y

2f − f (x) − f (y) − ρ(f (x + y) − f (x) − f (y))

≤ θ(∥x∥r + ∥y∥r )

2

(5.5)

for all x, y ∈ X. Then there exists a unique additive mapping A : X → Y such that ∥f (x) − A(x)∥ ≤

2r θ ∥x∥r 2r − 2

for all x ∈ X.

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ADDITIVE ρ-FUNCTIONAL EQUATIONS

13

Theorem 5.3. Let φ : X 2 → [0, ∞) be a function and let f : X → Y be a mapping satisfying f (0) = 0, (5.1) and Ψ(x, y) :=

∞ ∑ 1 j j=1 2

φ(2j x, 2j y) < ∞

for all x, y ∈ X. Then there exists a unique additive mapping A : X → Y such that ∥f (x) − A(x)∥ ≤ Ψ(x, 0)

(5.6)

for all x ∈ X. Proof. It follows from (5.3) that



1 1

f (x) − f (2x) ≤ φ(2x, 0)

2

2

for all x ∈ X. Hence



1 1 m

f (2l x) − f (2 x)

l m

2

2

≤ ≤

m ∑

1 ( j )

f 2 x −

j j=l+1 m ∑

2

1 2j+1

(

j+1

f 2

)

x

1 φ(2j x, 0) j j=l+1 2

(5.7)

for all nonnegative integers m and l with m > l and all x ∈ X. It follows from (5.7) that the sequence { 21n f (2n x)} is a Cauchy sequence for all x ∈ X. Since Y is complete, the sequence { 21n f (2n x)} converges. So one can define the mapping A : X → Y by 1 f (2n x) n→∞ 2n

A(x) := lim

for all x ∈ X. Moreover, letting l = 0 and passing the limit m → ∞ in (5.7), we get (5.6). The rest of the proof is similar to the proof of Theorem 4.1. □ Corollary 5.4. Let r < 1 and θ be positive real numbers, and let f : X → Y be a mapping satisfying f (0) = 0 and (5.5). Then there exists a unique additive mapping A : X → Y such that ∥f (x) − A(x)∥ ≤

2r θ ∥x∥r r 2−2

for all x ∈ X. Acknowledgments S. Y. Jang was supported by University of Ulsan, Research Program 2014.

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C. PARK AND S. Y. JANG

References [1] T. Aoki, On the stability of the linear transformation in Banach spaces, J. Math. Soc. Japan 2 (1950), 64–66. [2] A. Bahyrycz and M. Piszczek, Hyperstability of the Jensen functional equation, Acta Math. Hungar. 142 (2014), 353–365. [3] M. Balcerowski, On the functional equations related to a problem of Z. Boros and Z. Dar´ oczy, Acta Math. Hungar. 138 (2013), 329–340. [4] A. Chahbi and N. Bounader, On the generalized stability of d’Alembert functional equation, J. Nonlinear Sci. Appl. 6 (2013), 198–204. [5] Z. Dar´oczy and Gy. Maksa, A functional equation involving comparable weighted quasi-arithmetic means, Acta Math. Hungar. 138 (2013), 147–155. [6] G. Z. Eskandani and P. Gˇavruta, Hyers-Ulam-Rassias stability of pexiderized Cauchy functional equation in 2-Banach spaces, J. Nonlinear Sci. Appl. 5 (2012), 459–465. [7] W. Fechner, On some functional inequalities related to the logarithmic mean, Acta Math. Hungar. 128 (2010), 31–45. [8] P. Gˇavruta, A generalization of the Hyers-Ulam-Rassias stability of approximately additive mappings, J. Math. Anal. Appl. 184 (1994), 431–43. [9] A. Gil´anyi, Eine zur Parallelogrammgleichung ¨ aquivalente Ungleichung, Aequationes Math. 62 (2001), 303–309. [10] A. Gil´anyi, On a problem by K. Nikodem, Math. Inequal. Appl. 5 (2002), 707–710. [11] D. H. Hyers, On the stability of the linear functional equation, Proc. Natl. Acad. Sci. U.S.A. 27 (1941), 222–224. [12] M.S. Moslehian and Gh. Sadeghi, A Mazur-Ulam theorem in non-Archimedean normed spaces, Nonlinear Anal.–TMA 69 (2008), 3405–3408. [13] C. Park, Orthogonal stability of a cubic-quartic functional equation, J. Nonlinear Sci. Appl. 5 (2012), 28–36. [14] W. Prager and J. Schwaiger, A system of two inhomogeneous linear functional equations, Acta Math. Hungar. 140 (2013), 377–406. [15] Th. M. Rassias, On the stability of the linear mapping in Banach spaces, Proc. Amer. Math. Soc. 72 (1978), 297–300. [16] K. Ravi, E. Thandapani and B. V. Senthil Kumar, Solution and stability of a reciprocal type functional equation in several variables, J. Nonlinear Sci. Appl. 7 (2014), 18–27. [17] S. M. Ulam, A Collection of the Mathematical Problems, Interscience Publ. New York, 1960. [18] C. Zaharia, On the probabilistic stability of the monomial functional equation, J. Nonlinear Sci. Appl. 6 (2013), 51–59. Choonkil Park Research Institute for Natural Sciences Hanyang University Seoul 04763 Republic of Korea E-mail address: [email protected] Sun Young Jang Department of Mathematics University of Ulsan Ulsan 44610 Republic of Korea E-mail address: [email protected]

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HYPERSTABILITY OF A GENERALIZED CAUCHY FUNCTIONAL EQUATION ABBAS NAJATI, DARYOUSH MOLAEE, AND CHOONKIL PARK Abstract. The aim of this paper is to present some results concerning the hyperstability of the generalized Cauchy functional equation f (ax + by) = Af (x) + Bf (y) + C Namely, we show, under some assumptions, that a function satisfying the equation approximately must be actually a solution to it.

1. Introduction and preliminaries Throughout the paper F and K denote the fields of real or complex numbers. Let X and Y be linear spaces over F and K, respectively. In this paper we give some hyperstability results for the generalized Cauchy functional equation f (ax + by) = Af (x) + Bf (y) + C

(1.1)

where f : X → Y and a, b ∈ F\{0}, A, B ∈ K, C ∈ Y . In [10], Piszczek proved the hyperstability of the generalized Cauchy functional equation (1.1). Theorem 1.1. [10] Let X be a normed space over a field F, Y be a Banach space over K, a, b ∈ F \ {0}, A, B ∈ K, p < 0 and g : X → Y satisfy kg(ax + by) − Ag(x) − Bg(y)k 6 ε(kxkp + kykp ) for all x, y ∈ X \ {0}. Then g satisfies g(ax + by) = Ag(x) + Bg(y) for all x, y ∈ X \ {0}. The method of the proof used in Theorem 1.1 is based on a fixed point theorem in [3]. Let us recall that the study of stability problems of functional equations was motivated by a question of Ulam [15] asked in 1940. The first result of stability proved by Hyers [6] in 1941. For more details about various results concerning such problems the reader is referred to [4, 5, 8, 9, 11, 12, 13, 14]. It seems the first hyperstability result was published in [1] and concerned ring homomorphisms. However the term hyperstability was used for the first time in [7]. 2000 Mathematics Subject Classification. Primary 39B82, 39B62; Secondary 47H14, 47H10. Key words and phrases. Hyperstability, generalized Cauchy functional equation. ∗Corresponding author: Choonkil Park (email: [email protected]).

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2. Hyperstability results In this part, we will prove a general version of Theorem 1.1. Let us start with a result. A version of the next result was proved in [2]. But we give another simple proof. Proposition 2.1. Assume that X and Y are linear spaces over F and K, respectively. Let a, b ∈ F \ {0}, A, B ∈ K, C ∈ Y and f : X → Y satisfy f (ax + by) = Af (x) + Bf (y) + C

(2.1)

for all x, y ∈ X \ {0}. Then f satisfies f (ax + by) = Af (x) + Bf (y) + C for all x, y ∈ X . Proof. Let x ∈ X \ {0}. Then in view of (2.1), we get f (0) = Af (bx) + Bf (−ax) + C h i h i = A Af (2a−1 bx) + Bf (−x) + C + B Af (−2x) + Bf (ab−1 x) + C + C h i h i = A Af (2a−1 bx) + Bf (−2x) + C + B Af (−x) + Bf (ab−1 x) + C + C = Af (0) + Bf (0) + C. Therefore we have f (0) = Af (bx) + Bf (−ax) + C for all x ∈ X . Consequently, by (2.1) and (2.2), we get

(2.2)

f (2a2 bx) = Af (abx + b2 y) + Bf (a2 x − aby) + C h i h i = A Af (bx) + Bf (by) + C + B Af (ax) + Bf (−ay) + C + C h i h i = A Af (bx) + Bf (by) + C + B Af (ax) + f (0) − Af (by) + C h i = A Af (bx) + Bf (ax) + C + Bf (0) + C = Af (2abx) + Bf (0) + C Hence f (2a2 bx) = Af (2abx) + Bf (0) + C for all x ∈ X \ {0}. Replacing x by (2ab)−1 x, we infer that f (ax) = Af (x) + Bf (0) + C holds for x ∈ X by (2.2). Similarly, one can prove that f (by) = Af (0) + Bf (y) + C holds for y ∈ X . Thus we have proved that f satisfies f (ax + by) = Af (x) + Bf (y) + C for all x, y ∈ X .  In the following results we assume that X is a vector space over F and Y is a normed space over K. Theorem 2.2. Let a, b ∈ F \ {0} and ϕ : X × X → [0, +∞) be a function such that lim ϕ(a−1 (m + 1)x, −b−1 mx) = 0,

m→∞

lim ϕ(mx, my) = 0

m→∞

(2.3)

for all x, y ∈ X \ {0}. Let A, B ∈ K, C ∈ Y and f : X → Y satisfy kf (ax + by) − Af (x) − Bf (y) − Ck 6 ϕ(x, y)

(2.4)

for all x, y ∈ X \ {0}. Then f satisfies f (ax + by) = Af (x) + Bf (y) + C,

(2.5)

(A + B)f (0) = Af (x) + Bf (−ab−1 x)

(2.6)

for all x, y ∈ X. Moreover,

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HYPERSTABILITY OF A GENERALIZED CAUCHY FUNCTIONAL EQUATION

for all x ∈ X. Proof. Replacing x by a−1 (m + 1)x and y by −b−1 mx in (2.4), we get



−1 −1 f (x) − Af (a (m + 1)x) − Bf (−b mx) − C

6 ϕ(a−1 (m + 1)x, −b−1 mx),

(2.7)

for all x ∈ X \ {0} and positive integers m. Letting m → ∞ in (2.7) and using (2.3), we obtain h i f (x) = lim Af (a−1 (m + 1)x) + Bf (−b−1 mx) + C (2.8) m→∞

for all x ∈ X \ {0}. If x ∈ X \ {0}, then we get from (2.3) and (2.8)



(A + B)f (0) − Af (x) − Bf (−ab−1 x)

= lim (A + B)f (0) − A2 f (a−1 (m + 1)x) − ABf (−b−1 mx) − AC m→∞

− ABf (−b−1 (m + 1)x) − B 2 f (ab−2 mx) − BC



6 |A| lim f (0) − Af (a−1 (m + 1)x) − Bf (−b−1 (m + 1)x) − C m→∞



+ |B| lim f (0) − Af (−b−1 mx) − Bf (ab−2 mx) − C m→∞

6 |A| lim ϕ(a−1 (m + 1)x, −b−1 (m + 1)x) + |B| lim ϕ(−b−1 mx, ab−2 mx) = 0. m→∞

m→∞

Hence we get (A + B)f (0) = Af (x) + Bf (−ab−1 x) for all x ∈ X. If we replace x by bmx and y by −amx in (2.4), we get

f (0) − Af (bmx) − Bf (−amx) − Ck 6 ϕ(bmx, −amx), for all x ∈ X \ {0} and positive integers m. Thus h i f (0) = lim Af (bmx) + Bf (−amx) + C

(2.9)

(2.10)

m→∞

for all x ∈ X \ {0}. Replacing x by bmx in (2.9) and letting m → ∞, we get from (2.10) (1 − A − B)f (0) = C. Therefore (2.8) holds for all x ∈ X. To prove (2.5), let x, y ∈ X \ {0}. Then kf (ax + by) − Af (x) − Bf (y) − Ck

= lim Af (a−1 (m + 1)(ax + by)) + Bf (−b−1 m(ax + by)) m→∞

− A2 f (a−1 (m + 1)x) − ABf (−b−1 mx) − AC

− ABf (a−1 (m + 1)y) − B 2 f (−b−1 my) − BC



6 |A| lim f (a−1 (m + 1)(ax + by)) − Af (a−1 (m + 1)x) − Bf (a−1 (m + 1)y) − C m→∞



+ |B| lim f (−b−1 m(ax + by)) − Af (−b−1 mx) − Bf (−b−1 my) − C m→∞

6 |A| lim ϕ(a−1 (m + 1)x, −a−1 (m + 1)y) + |B| lim ϕ(−b−1 mx, −b−1 my) = 0. m→∞

m→∞

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Therefore f satisfies (2.5) for all x, y ∈ X \ {0}. Hence f satisfies (2.5) for all x, y ∈ X by Proposition 2.1.  Remark 2.3. If f satisfies (2.4) with A + B = 1, then C = 0 and f satisfies f (ax + by) = Af (x) + Bf (y) for all x, y ∈ X. When X is a normed linear space, Theorem 1.1 is a corollary of Theorem 2.2. In the following results, we assume that X and Y are normed linear spaces. Corollary 2.4. Let ε > 0 and p, q < 0. If a, b ∈ F \ {0}, A, B ∈ K, C ∈ Y and f : X → Y satisfies kf (ax + by) − Af (x) − Bf (y) − Ck 6 ε(kxkp + kykq ) for all x, y ∈ X \ {0}. Then f satisfies (2.5) and (2.6) for all x, y ∈ X. Corollary 2.5. Let ε > 0 and p, q be real numbers such that p + q < 0. If a, b ∈ F \ {0}, A, B ∈ K, C ∈ Y and f : X → Y satisfies kf (ax + by) − Af (x) − Bf (y) − Ck 6 εkxkp kykq for all x, y ∈ X \ {0}. Then f satisfies (2.5) and (2.6) for all x, y ∈ X. Corollary 2.6. Let δ, ε > 0, p, q < 0 and l, r, s be real numbers such that l > 0 and r + s < 0. If a, b ∈ F \ {0}, A, B ∈ K, C ∈ Y and f : X → Y satisfies kf (ax + by) − Af (x) − Bf (y) − Ck 6 ε(kxkp + kykq )l + δkxkr kyks for all x, y ∈ X \ {0}. Then f satisfies (2.5) and (2.6) for all x, y ∈ X. Corollary 2.7. Let θ, δ, ε > 0, p, q < 0 and r, s be real numbers such that r + s < 0. If a, b ∈ F \ {0}, A, B ∈ K, C ∈ Y and f : X → Y satisfies kf (ax + by) − Af (x) − Bf (y) − Ck 6 εkx + ykp + δkx − ykq + θkxkr kyks

(2.11)

for all x, y ∈ X \ {0} with x ± y 6= 0. Then we have (i) if a 6= ±b, then f satisfies (2.5) and (2.6) for all x, y ∈ X; (ii) if a = ±b and A, B ∈ K \ {0}, then f satisfies (2.5) for all x, y ∈ X \ {0} with x ± y 6= 0. Proof. Let ϕ(x, y) = kx + ykp + δkx − ykq + θkxkr kyks . If a 6= ±b, then ϕ satisfies (2.3). Therefore the result follows from Theorem 2.2. If a = ±b, then (2.11) implies that h i Af (x) = lim f ((a + bm)x) − Bf (mx) − C m→∞

for all x ∈ X \ {0}. Therefore



f (ax + by) − Af (x) − Bf (y) − C

= |A|−1 lim f ((a + bm)(ax + by)) − Bf (m(ax + by)) − C m→∞

− Af ((a + bm)x) + ABf (mx) − Bf ((a + bm)y) + B 2 f (my) + BC



6 |A|−1 lim f ((a + bm)(ax + by)) − Af ((a + bm)x) − Bf ((a + bm)y) − C m→∞



+ |B||A|−1 lim f (m(ax + by)) − Af (mx) − Bf (my) − C m→∞

−1

6 |A|

lim ϕ((a + bm)x, (a + bm)y) + |B||A|−1 lim ϕ(mx, my) = 0.

m→∞

m→∞

Hence f (ax + by) = Af (x) + Bf (y) + C for all x, y ∈ X \ {0} with x ± y 6= 0.

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HYPERSTABILITY OF A GENERALIZED CAUCHY FUNCTIONAL EQUATION

In the next result we will derive from Theorem 2.2 a hyperstability result for the inhomogeneous version of the generalized Cauchy functional equation. Theorem 2.8. Let a, b ∈ F \ {0}, A, B ∈ K and ϕ : X × X → [0, +∞) be a function satisfy (2.3) for all x, y ∈ X \ {0}. Assume that d : X × X → Y and f : X → Y satisfy the inequality kf (ax + by) − Af (x) − Bf (y) − d(x, y)k 6 ϕ(x, y)

(2.12)

for all x, y ∈ X \ {0}. If the functional equation g(ax + by) = Ag(x) + Bg(y) + d(x, y),

x, y ∈ X

(2.13)

has a solution f0 : X → Y , then f is a solution to (2.13). Proof. It follows from (2.12) that h := f − f0 satisfies (2.4) with C = 0. Consequently, Theorem 2.2 implies that h is a solution to (2.5) with C = 0, which means that f is a solution to (2.13).  In the following results, we assume that a, b ∈ F \ {0}, A, B ∈ K, X and Y are normed linear spaces. Corollary 2.9. Let ε > 0 and p, q < 0. Assume that d : X × X → Y and f : X → Y satisfy kf (ax + by) − Af (x) − Bf (y) − d(x, y)k 6 ε(kxkp + kykq ) for all x, y ∈ X \ {0}. If the functional equation (2.13) has a solution f0 : X → Y , then f is a solution to (2.13). Corollary 2.10. Let ε > 0 and p, q be real numbers such that p + q < 0. Assume that d : X × X → Y and f : X → Y satisfy kf (ax + by) − Af (x) − Bf (y) − d(x, y)k 6 εkxkp kykq for all x, y ∈ X \ {0}. If the functional equation (2.13) has a solution f0 : X → Y , then f is a solution to (2.13). Corollary 2.11. Let δ, ε > 0, p, q < 0 and l, r, s be real numbers such that l > 0 and r + s < 0. Assume that d : X × X → Y and f : X → Y satisfy kf (ax + by) − Af (x) − Bf (y) − d(x, y)k 6 ε(kxkp + kykq )l + δkxkr kyks for all x, y ∈ X \ {0}. If the functional equation (2.13) has a solution f0 : X → Y , then f is a solution to (2.13). Corollary 2.12. Let θ, δ, ε > 0, p, q < 0 and r, s be real numbers such that r + s < 0. Assume that the functional equation (2.13) has a solution f0 : X → Y . Let d : X × X → Y and f : X → Y satisfy kf (ax + by) − Af (x) − Bf (y) − d(x, y)k 6 εkx + ykp + δkx − ykq + θkxkr kyks for all x, y ∈ X \ {0} with x ± y 6= 0. Then we have (i) if a 6= ±b, then f satisfies (2.13) for all x, y ∈ X; (ii) if a = ±b and A, B ∈ K\{0}, then f satisfies (2.13) for all x, y ∈ X \{0} with x±y 6= 0.

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A. NAJATI, D. MOLAEE, AND C. PARK

References [1] D. G. Bourgin, Approximately isometric and multiplicative transformations on continuous function rings, Duke Math. J. 16 (1949), 385–397. [2] J. Brzd¸ek, Remark on stability of some inhomogeneous functional equations, Aequationes Math. 89 (2015), 83–96. [3] J. Brzd¸ek, J. Chudziak, d Zs. P´ ales, A fixed point approach to stability of functional equations, Nonlinear Anal.–TMA 74 (2011), 6728–6732. [4] L. C˘ adariu, L. G˘ avruta, P. G˘ avruta, On the stability of an affine functional equation, J. Nonlinear Sci. Appl. 6 (2013), 60–67. [5] A. Chahbi, N. Bounader, On the generalized stability of d’Alembert functional equation, J. Nonlinear Sci. Appl. 6 (2013), 198–204. [6] D. H. Hyers, On the stability of the linear functional equation, Proc. Natl. Acad. Sci. USA 27 (1941), 222–224. [7] Gy. Maksa, Zs. P´ ales, Hyperstability of a class of linear functional equations. Acta Math. Acad. Paedag. Ny´ıregyh´ aziensis 17 (2001), 107–112. [8] C. Park, K. Ghasemi, S. G. Ghaleh, S. Jang, Approximate n-Jordan ∗-homomorphisms in C ∗ -algebras, J. Comput. Anal. Appl. 15 (2013), 365–368. [9] C. Park, A. Najati, S. Jang, Fixed points and fuzzy stability of an additive-quadratic functional equation, J. Comput. Anal. Appl. 15 (2013), 452–462. [10] M. Piszczek, Remark on hyperstability of the general linear equation, Aequationes Math., 88 (2014), 163– 168. [11] S. Shagholi, M. Bavand Savadkouhi, M. Eshaghi Gordji, Nearly ternary cubic homomorphism in ternary Fr´echet algebras, J. Comput. Anal. Appl. 13 (2011), 1106–1114. [12] S. Shagholi, M. Eshaghi Gordji, M. Bavand Savadkouhi, Stability of ternary quadratic derivation on ternary Banach algebras, J. Comput. Anal. Appl. 13 (2011), 1097–1105. [13] D. Shin, C. Park, Sh. Farhadabadi, On the superstability of ternary Jordan C ∗ -homomorphisms, J. Comput. Anal. Appl. 16 (2014), 964–973. [14] D. Shin, C. Park, Sh. Farhadabadi, Stability and superstability of J ∗ -homomorphisms and J ∗ -derivations for a generalized Cauchy-Jensen equation, J. Comput. Anal. Appl. 17 (2014), 125–134. [15] S. M. Ulam, Problems in Modern Mathematics, Science Editions, Wiley, New York, 1964. Abbas Najati Department of Mathematics, University of Mohaghegh Ardabili, Ardabil, Iran E-mail address: [email protected] Daryoush Molaee Department of Mathematics, University of Mohaghegh Ardabili, Ardabil, Iran Choonkil Park Research Institute for Natural Sciences, Hanyang University, Seoul 04763, Korea E-mail address: baak@@hanyang.ac.kr

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Stability analysis and optimal control of a cholera model with time delay Shu Liao a

Fang Fang a

a

School of Mathematics and Statistics Chongqing Technology and Business University, Chongqing, 400067,China Abstract An optimal control method for cholera epidemic with time delay is developed in this paper. We first explore the local stability of both the disease-free and endemic equilibria of ODE model by analyzing the corresponding characteristic equations, whose global stability is established by constructing two suitable Lyapunov functionals. Furthermore, in order to, we use optimal control theory via the Pontryagin’s Maximum Principle and genetic algorithm based on the forward and backward difference approximation to minimize the infected populations and the costs. Numerical simulations demonstrate that the time delay and multiple optimal controls can bring different effects on the dynamics behaviors of the proposed cholera model.

Cholera; optimal control; time delay; global asymptotical stability; Pontryagin’s Maximum Principle.

1

Introduction

Cholera, a waterborne gastroenteric infection, caused by a number of types of Vibrio cholerae, remains a significant threat to public health for most of the developing countries in the past few years. Since 1961, cholera has become an acute disease throughout the world, according to the World Health Organization (WHO) report (2010), with an estimated 3-5 million cases worldwide and causes 58,000-130,000 deaths a year, children and the senior are being most affected. It was found in Congo (2008), in Iraq (2008), in Zimbabwe (2008-2009), in Vietnam (2009), in Kenya (2010), in Nigeria (2010), in Haiti (2010), in Mexico (2013), and most recently in South Sudan (2014). In the last few decades, enormous attention is being paid to the cholera disease and a number of mathematical models have been contributed to a better understanding of the transmission of cholera. In 2001, Code¸co [1] put an emphasis on the decisive importance of the environmental component and proposed a SIRB epidemic model in which B represents the V. cholerae concentration in water. Meanwhile, according to the laboratory results, Hartley Morris and Smith [2] in 2006 discovered a representitive hyperinfectious state of the pathogen-the explosive infectivity of freshly shed V. cholerae. Tien and Earn later [3] proposed a water-borne disease model with multiple transmission pathways, accounting both direct human-to-human and indirect water-to-human transmissions, they identified how these transmission routes influence disease dynamics. Mukandavire et al. [4] in 2011 simplified Hartley’s model to understand transmission dynamics of cholera outbreak in Zimbabwe. Liao and Wang [5] conducted a dynamical analysis of the Hartley’s model to study the stability of both the disease-free and endemic equilibria so as to explore the complex epidemic and endemic dynamics of the disease. 1 1055

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These epidemiological models above often take the form of a system of ordinary differential equations and ignore the time delay by assuming that the infectious process is instantaneous. However, it may make these models more biologically reasonable and mathematically challenging to consider incorporating suitable delay terms. Time delay plays an important role to reflect the real dynamical behaviors of models, many researchers have proposed and analyzed more realistic models including delays to model different mechanisms in the dynamics of epidemics. Wei et al. [6] considered a differential delay model of a vector-borne disease which has direct mode of transmission in addition to the vector-mediated transmission. The delay in their model accounts for the incubation time the vectors need to become infectious. They studied the effect of that delay on the stability of the equilibria and investigated that the introduction of a time delay in the host-to-vector transmission term can destabilize the system. McCluskey [7] in 2010 studied two SIRS models with distributed delay and with discrete delay, respectively. They solved the global stability of the endemic equilibrium for larger delay when R0 > 1. Misra et al. [8] in 2012 proposed a delay model to explore the dynamics of water borne diseases like cholera by using disinfectants to control the disease. Their analysis showed that under certain conditions, the cholera disease can be controlled by using disinfectants but a longer delay in their use may destabilize the system. Misra et al. [9] in 2013 analyzed a nonlinear delay mathematical model for the control of carrier-dependent infectious diseases, they suggested that as delay in using insecticides exceeds some critical value, the system loses its stability and Hopf-bifurcation occurs. Wang and Wei [10] investigated the global dynamics of a cholera model with delay to demonstrate the impact of the time lag. Optimal control method [11] as a powerful tool has been applied to control various kinds of diseases [12–16]. Sunmi et al. [17] in 2010 studied a model for the transmission dynamics of influenza to evaluate the impact of isolation and/or antiviral drug delivery measures. They compared five control strategies to show the optimal control strategy involving antiviral treatment and/or isolation measures can reduce significantly the number of clinical cases of influenza. Ding et al. [18] studied the control problem of maximizing the total payoff in the conservation of a single species with a fixed amount of resource. The existence of an optimal control was established while its uniqueness and characterization was investigated as well. Okosun et al. [19] in 2011 derived and analyzed a deterministic model for the transmission of malaria disease with mass action form of infection. They obtained the conditions under which it is optimal to eradicate the disease and examined the impact of a possible combination of vaccination and treatment strategy on the disease transmission by using optimal control theory and the Pontryagin’s Maximum Principle. Kar and Jana [20] in 2013 proposed an epidemic model and used the optimal control strategy to minimize both the infected populations and the associated costs. They compared the numerical results with no controls, with only vaccination control, with only treatment control and with both vaccination as well as treatment controls. It is observed that the best result comes out from the application of both vaccination and treatment controls in this case that the number of infected individuals would be the least in number. Wang and Modnak [21] presented a cholera epidemiological model with three control measures. Equilibrium analysis was conducted in the cases with constant controls and with optimal controls, respectively. According to the above collection of works, an optimal control model including time delay in the context has been not completely understood yet. There are only few papers that tackle 2 1056

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this problem. In recent years, Laarabi et al. [22] studied an epidemic model with optimal control strategies and time delay, the optimality system was numerically solved by using an algorithm based on the forward and backward difference approximation in their work. Mohamed et al. [23] investigated an optimally controlled SIR epidemic model with time delay in state and control variables, they used optimal control approach via Pontryagin’s Maximum Principle to minimize the number of susceptible and infected individuals and to maximize the number of recovered individuals during the course of an epidemic. In this paper, we will consider an optimally controlled cholera model with time delay based on the model originally suggested by Wang and Modnak [21], which involves both the environment-to-human and human-to-human transmission modes. Our main aim is to explore the role of time delay and optimal control on the spread of cholera in the model. Note most of the delay epidemic models mentioned above are only concerned with local stability of equilibria, we will pay attention to global stability of our model in this paper. The rest of the paper is organized as follows. In the next section, we formulate the mathematical model and determine the basic reproductive number R0 . Section 3 is devoted to the local and global stability analysis of both the disease-free and endemic equilibria of our model. The analysis of optimization problem is presented in Section 4. In Section 5 we present genetic algorithm based on the forward and backward difference approximation and carry out the numerical study of the model, which confirms our theoretical results. Finally, the conclusions are summarized in Section 6.

2

The model formulation

Cholera has been found in multiple transmission pathways including both direct human-tohuman and indirect environment-to-human transmissions pathways, which distinct cholera from many other infectious diseases. It is important to notice that, it takes a period for the infected individual to affect the bacterial concentration of cholera, and its size may be very influential in controlling the outbreak of cholera. Thus the delay τ is used to describe the period during the person being infected to his pathogenic bacteria of V. cholera being given off to the aquatic environment. Motivated by the works of Wang and Modnak [21], the deterministic model is given by the following system of ODE: dS dt dI dt dW dt dR dt

= µN − βW = βW

SW − βI SI − µS − u1 S, κ+W

SW − βI SI − (γ + µ)I − u2 I, κ+W

(1) (2)

= ξI(t − τ ) − δW − u3 W,

(3)

= γI − µR + u2 I + u1 S.

(4)

In the equations above, let N be the total population which is divided into three epidemiological compartments, susceptible compartment S, infectious compartment I, recovered compartment R. Let W be the density of V. cholerae in the aquatic environment. The parameter κ is the concentration of vibrios in contaminated water in the environment, 3 1057

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βW and βI are rates of ingesting vibrios from the contaminated environment and through human-to-human interaction, respectively. µ represents the natural human birth/death rate, ξ the shedding rate, γ the recovery rate, δ the bacterial death rate. All the parameters are strictly positive constants. Intervention strategies are modeled by the control variables ui (t) (i = 1, 2, 3), which are bounded, Lebesgue integrable functions. The control u1 (t) represents the rate of vaccination, u2 (t) represents the rate of therapeutic treatment, water sanitation leads to the death of vibrios at a rate u3 (t). Based on biological assumption, we assume that for θ ∈ [−τ, 0], S(θ), I(θ) and R(θ) are non negative real valued functions. Let C = C([−τ, 0], R3 ) be the Banach space of continuous functions mapping the interval [−τ, 0] into R3 with the topology of uniform convergence. For ecological reasons, we assume that the initial conditions for system (1-4) satisfies: S0 (θ) ≥ 0 , I0 (θ) ≥ 0 , R0 (θ) ≥ 0 , θ ∈ [−τ, 0].

(5)

In order to determine the dynamics of each class, we only need to study the first three equations in model (1-4), thereby reducing the order of the system through eliminating R to obtain the following system: dS SW = µN − βW − βI SI − µS − u1 S, dt κ+W dI SW = βW − βI SI − (γ + µ)I − u2 I, dt κ+W dW = ξI(t − τ ) − δW − u3 W. dt

(6) (7) (8)

As the study of model system (1-4) is equivalent to study model system (6-8), so we study model system (6-8). Based on the next-generation matrix approach [25], we define the basic reproduction number R0 , representing the average number of secondary infections that occurs when one infective is introduced into a completely susceptible host population, as: R0 =

3

µN [ξβW + (δ + u3 )κβI ] . κ(µ + u1 )(δ + u3 )(γ + µ + u2 )

(9)

Mathematical analysis of the epidemic model

In particular, when the time delay is set to zero, i.e. τ = 0, the above system (6-8) is reduced to the original model developed in Wang and Modnak [21]. Based on their work, the results below directly follows: )T ( µN , 0 , 0 , 0 , Theorem 1 The disease-free equilibrium (DFE) of the model (6-8) E0 = µ+u 1 is both locally and globally asymptotically stable if R0 < 1 with τ = 0. Theorem 2 The endemic equilibrium of the model (6-8) E ∗ = (S ∗ , I ∗ , W ∗ ) is locally asymptotically stable and globally asymptotically stable if R0 > 1 with τ = 0.

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3.1

The stability of the disease-free equilibrium

Our primary focus is on the stability analysis of the model when τ ̸= 0 in this section. First, we prove the local and global stability of the disease-free equilibrium E0 with τ > 0. Theorem 3 The disease-free equilibrium (DFE) of the model (6-8) is locally asymptotically stable if R0 < 1 with τ > 0. Proof After linearizing the ODE system (6-8) around the disease-free equilibrium E0 , we obtain one negative characteristic solution λ = −µ − u1 and the following transcendental characteristic equation is: λ2 + a1 λ + a2 + b1 e−λτ = 0,

(10)

where µN , µ + u1 µN = (δ + u3 )(γ + µ + u2 − βI ), µ + u1 ξβW µN = − . κ µ + u1

a1 = δ + γ + µ + u2 + u3 − βI a2 b1

We can rearrange equation (10) in the form: λ2 + a1 λ = (δ + u3 )(γ + µ + u2 )[( +

µN κβI − 1) κ(µ + u1 )(γ + µ + u2 )

µN ξβW e−λτ ]. κ(µ + u1 )(δ + u3 )(γ + µ + u2 ) (11)

Let the left-hand side and right-hand side of equation (11) be F (λ) and H(λ), respectively. It is easy to see that F (0) = 0 and limλ→∞ F (λ) = ∞, therefore, F (λ) is an increasing function of λ. On the other hand, H(λ) is a decreasing function of λ and H(0) = (δ + u3 )(γ + µ + u2 )(R0 − 1) is less than zero when R0 < 1. Thus, equation (11) has no nonnegative real roots. If equation (10) has roots with non-negative real parts, they must be complex and obtained from a pair of complex conjugate roots which cross the imaginary axis. As a result, a pair of purely imaginary solution may come out from the equation (10) for τ > 0. Assume that iω (ω > 0) is the root of equation (10) and ω satisfies the following equation: −ω 2 + a1 iω + a2 + b1 (cos(ωτ ) − isin(ωτ )) = 0.

(12)

Separating the real and imaginary parts of equation (12) gives −ω 2 + a2 = −b1 cos(ωτ ) ,

−a1 ω = −b1 sin(ωτ ).

(13)

To eliminate the trigonometric functions, we add up the squares of equation (13) above, and obtain the following forth order equation in ω: ω 4 + (a21 − 2a2 )ω 2 + a22 − b21 = 0.

(14)

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We can solve that ω

2

1 [−(a21 − 2a2 ) ± = 2

√

(a21 − 2a2 )2 − 4(a22 − b22 )].

(15)

This implies equation (14) has no positive roots, which leads to the conclusion that there is no ω such that iω is a solution of equation (10) for time delay τ > 0. Based on Rouche’s theorem [26], E0 is locally asymptotically stable if R0 < 1. Next, we will analyze the global stability of the disease-free equilibrium of the model system (6-8) for time delay τ > 0. Theorem 4 The disease-free equilibrium (DFE) of the model (6-8) is globally asymptotically stable with time delay τ > 0 if R0 < 1. Proof Adding equations (1) and (2), we obtain ′

′

S + I = µN − (µ + u1 )S − (γ + µ + u2 )I ≤ µN − η(S + I),

(16)

and equation (3) yields ′

W = ξI(t − τ ) − (δ + u3 )W ≤ ξ

µN − (δ + u3 )W, η

(17)

where η = min{(µ + u1 ), (γ + µ + u2 )}. These imply µN . η

(18)

ξµN . η(δ + u3 )

(19)

lim sup I(t) ≤ t→∞

and lim sup W (t) ≤ t→∞

We consider the following Lyapunov function: µN S(t) V1 (t) = ξ[S(t) − ln µN ] + ξIt (0) + (γ + µ + u2 )W (t) + ξ(γ + µ + u2 ) µ + u1 µ+u 1

∫

0

−τ

It (θ)dθ. (20)

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Here, It (θ) = I(t+θ) for θ ∈ [−τ, 0], therefore, It (0) = I(t) in this equation (20). Calculating the time derivative of V1 (t) along solutions of system (6-8), ∫ t ′ dV1 (t) µN S (t) ′ ′ ′ ′ = ξ(S (t) − ) + ξI (t) + (γ + µ + u2 )W (t) + ξ(γ + µ + u2 )[ I(t)dS] dt µ + u1 S(t) t−τ S(t)W (t) = ξ[µN − βW − βI S(t)I(t) − (µ + u1 )S(t) κ + W (t) µN βW W (t) µN S(t)W (t) + ( + βI I(t) + µ + u1 − )] + ξβW µ + u1 κ + W (t) S(t) κ + W (t) +ξβI S(t)I(t) − ξ(γ + µ + u2 )I(t) + (γ + µ + u2 )ξI(t − τ ) −(γ + µ + u2 )(δ + u3 )W (t) + ξ(γ + µ + u2 )I(t) − (γ + µ + u2 )ξI(t − τ ) ξµN βW W (t) µN = 2ξµN − ξ(µ + u1 )S(t) + ( + βI I(t) − ) µ + u1 κ + W (t) S(t) −(γ + µ + u2 )(δ + u3 )W (t) µN 1 µ + u1 ξµN βW W (t) = ξµN (2 − − S(t)) + [ ( + βI I(t)) µ + u1 S(t) µN µ + u1 κ + W (t) −(γ + µ + u2 )(δ + u3 )W (t)]. (21) µN µN 1 1 − µ+u S(t) ≤ 0, thus, dVdt1 (t) = 0 if and only if S = µ+u . In addition, Obviously, 2 − µ+u µN 1 S(t) 1 if R0 < 1, it is sufficient to verify that the second term of equation (21) is less than 0 by combining equations (18) and (19). Therefore, dVdt1 (t) ≤ 0. This completes the proof.

3.2

The stability of the endemic equilibrium

To study the stability of the endemic equilibrium E ∗ (S ∗ , I ∗ , W ∗ ), we linearize the system (6-8) at the point E ∗ by Letting S = S ∗ + s, I = I ∗ + i, W = W ∗ + w, here s, i and w are small perturbations around the equilibrium E ∗ . To make the algebraic manipulation ∗ ∗ WW simpler, we set P ∗ = βκ+W ∗ + βI I . When τ > 0, the characteristic polynomial for linearized equation is obtained as: λ3 + a1 λ2 + a2 λ + a3 + (b1 λ + b2 )e−λτ = 0,

(22)

where a1 = −βI S ∗ + P ∗ + γ + 2µ + δ + u1 + u2 + u3 , a2 = (P ∗ + µ + u1 )(−βI S ∗ + γ + µ + u2 ) + P ∗ S ∗ βI + (δ + u3 )× (−βI S ∗ + P ∗ + γ + 2µ + u1 + u2 ), a3 = (δ + u3 )(P ∗ + µ + u1 )(−βI S ∗ + γ + µ + u2 ) + βI (δ + u3 )P ∗ S ∗ , κ b1 = −ξβW S ∗ , (κ + W ∗ )2 κ b2 = −ξ(µ + u1 )βW S ∗ . (κ + W ∗ )2 7 1061

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Now we suppose λ is a root of equation (22), and substitute λ = iω (ω > 0) into equation (22), after separating real and imaginary parts, we finally obtain the following two transcendental equations: −a1 ω 2 + a3 = −b2 cos(ωτ ) − b1 ωsin(ωτ ), −ω 3 + a2 ω = −b1 ωcos(ωτ ) + b2 sin(ωτ ).

(23) (24)

By adding up the squares of both the equations (23) and (24), the following sixth degree equation for ω is obtained: ω 6 + ω 4 (a21 − 2a2 ) + ω 2 (a22 − 2a1 a3 − b21 ) + a23 − b22 = 0.

(25)

Letting ω 2 = x gives: F (x) = x3 + B1 x2 + B2 x + B3 = 0,

(26)

where B1 = a21 − 2a2 , B2 = a22 − 2a1 a3 − b21 , B3 = a23 − b22 . Here, we establish the following theorem. Theorem 5 When R0 > 1, the endemic equilibrium E ∗ of ODE system (6-8) is locally asymptotically stable for the delay τ > 0 if B1 ≥ 0, B3 ≥ 0 and B2 > 0. Proof In order to show that the endemic equilibrium E ∗ is locally stable, we have to show that equation (26) does not have a positive real root. In fact, if we take the derivative of ′ ′ F (x) with respect to x, F√ (x) = 3x2 + 2B1 x + B2 . The roots of equation F (x) = 0 can √ −B1 ± B12 −3B2 . If B > 0, then be solved as x1,2 = B12 − 3B2 < B1 . Hence, neither x1 2 3 ′ nor x2 is positive, it follows that equation F (x) = 0 has no positive roots. Also, a simple assumption that F (0) = B3 ≥ 0, implies that equation (26) will have no positive real roots. Therefore, there is no ω such that iω is an eigenvalue of the characteristic equation (22). By Rouch’s theorem [26], the real parts of all the eigenvalues of (22) are negative for time delay τ ≥ 0. This completes the proof. Next, we turn our attention to the global stability of the ODE system (6-8) if R0 > 1 for all values of the delay τ > 0. Theorem 6 When R0 > 1, the positive endemic equilibrium E ∗ of ODE system (6-8) is globally asymptotically stable for all delay τ > 0. Proof We consider the following Lyapunov function: S(t) S(t) It (0) It (0) γ + µ + u2 ∗ − 1 − ln ∗ ) + I ∗ ( ∗ − 1 − ln ∗ ) + W × ∗ S S I I ξ ∫ 0 It (s) W (t) W (t) It (s) ∗ ( ∗ − 1 − ln ∗ )ds. ( − 1 − ln ) + (γ + µ + u2 )I ∗ ∗ W W I I −τ

V2 (t) = S ∗ (

(27)

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Differentiating V2 (t) along solutions of (6-8), we can obtain: dV2 (t) µN βW S ∗ S(t)W (t) = µN − µS(t) − u1 S(t) − S ∗ + S ∗ P + 2µS ∗ + 2u1 S ∗ − dt S(t) κ + W (t) (γ + µ + u )(δ + u )W (t) 2 3 −βI S(t)I ∗ + 2(γ + µ + u2 )I ∗ − ξ (γ + µ + u2 )W ∗ I(t − τ ) (γ + µ + u2 )(δ + u3 )W ∗ − + W (t) ξ I(t − τ ) I(t) +(γ + µ + u2 )I ∗ (ln − ln ) I∗ I∗ S(t) S∗ S∗ S(t) = µS ∗ (2 − ∗ − ) + u1 S ∗ (2 − − ∗ ) + (γ + µ + u2 )I ∗ × S S(t) S(t) S ∗ ∗ P W (t) S∗ P (t) ∗ S [( ∗ − 1)(1 − )] − (γ + µ + u )I ( − 1 − ln ) 2 P P (t) W ∗ S(t) S(t) P (t) I ∗ S(t) P (t) I ∗ S(t) −(γ + µ + u2 )I ∗ [ ∗ ∗ − 1 − ln( ∗ ∗ )] P S I(t) P S I(t) W ∗ I(t − τ ) W ∗ I(t − τ ) − 1 − ln( )]. (28) −(γ + µ + u2 )I ∗ [ W (t) I ∗ W (t) I ∗ ∗

S Clearly, 2− S(t) − S(t) ≤ 0 for S(t) > 0. Furthermore, note that at the endemic equilibrium S∗ ∗ E , the right-hand side of equation (8) becomes 0, which yields ξI ∗ = (δ + u3 )W ∗ , and P ∗ W (t) combine the facts (18) and (19), we can get ( PP(t) ∗ − 1)(1 − P (t) W ∗ ) < 0 if R0 > 1. Also, for all t ≥ 0, the function g(t) = t − 1 − lnt is always non-negative, and g(t) = 0 if and only if t = 1, then the fourth term, the fifth term and the last term in (28) are non-negative. Therefore, we can finally show dVdt2 (t) ≤ 0. This completes the proof.

4

Optimal control analysis

In this section, we seek to minimize the objective functional defined by decreasing the number of infected and the costs of time-related controls,the method is described in [28]. We choose a linear function for the cost on infection I, and quadratic forms for the cost on the controls u1 , u2 and u3 . The objective function subject to the differential equations (1-4) is constructed as follows: ∫ tf J= (A0 I + A1 u21 + A2 u22 + A3 u23 )dt. 0

We assume tf is the fixed final time, the parameters A0 , A1 , A2 and A3 are weight parameters describing the comparative importance of the all terms on control cost. The optimal control problem is that of finding optimal functions u∗1 , u∗2 and u∗3 such that J(u∗1 , u∗2 , u∗3 ) =

min

u1 ,u2 ,u3 ∈Θ

J(u1 , u2 , u3 ),

(29)

where Θ is measurable on [0, 1] and Θ = {ui |0 ≤ ui ≤ 1} for the controls. 9 1063

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The Lagrangian of this object is given by L(I, u1 , u2 , u3 ) = A0 I + A1 u21 + A2 u22 + A3 u23 ,

(30)

and the Hamiltonian H for the control problem is: H(S, I, W, R, u1 , u2 , u3 , λ1 , λ2 , λ3 , λ4 ) = L + λ1 (t)

dI dW dR dS + λ2 (t) + λ3 (t) + λ4 (t) , dt dt dt dt (31)

where λi (t) for i = 1, 2, 3, 4 are the adjoint variables, which determine the adjoint system, and can be solved by the following system: ∂H ∂H λ˙1 (t) = − − χ[0,tf −τ ] (t + τ ) ∂S ∂Sτ βW W βW W = λ1 ( + βI + µ + u1 ) − λ2 ( + βI ) − λ4 µ, κ+W κ+W ∂H ∂H λ˙2 (t) = − − χ[0,tf −τ ] (t + τ ) ∂I ∂Iτ = −A0 + λ1 βI S − λ2 [βI S − (γ + µ + u2 )] − λ4 (γ + u2 ) − λ2 (t + h)ξ, ∂H ∂H λ˙3 (t) = − − χ[0,tf −τ ] (t + τ ) ∂W ∂Wτ βW Sκ βW Sκ − λ2 + λ3 (δ + u3 ), = λ1 2 (κ + W ) (K + W )2 ∂H ∂H λ˙4 (t) = − − χ[0,tf −τ ] (t + τ ) ∂R ∂Rτ = λ4 µ.

(32)

(33)

(34)

(35)

Satisfying the transversality conditions: λi (tf ) = 0,

i = 1, 2, 3, 4.

(36)

The combination of the ODE system (1-4) and the state system (32-35) is the optimality system, which describes how the system behaves minimize J under the control applications. By applying Pontryagin’s Maximum theory and the existence result for the optimal control [27], we thus establish the following theorem: Theorem 7 There is a triplet of optimal control (u∗1 , u∗2 , u∗3 ) such that J(u∗1 , u∗2 , u∗3 ) = minu1 ,u2 ,u3 ∈Θ J(u1 , u2 , u3 ) subject to the optimality control system. Theorem 8 There is a triplet of optimal control (u∗1 , u∗2 , u∗3 ) which minimizes J over the region Θ given by u∗1 = min{max{0, u1 }, 1} , u∗2 = min{max{0, u2 }, 1} , u∗3 = min{max{0, u3 }, 1}, (37) where u1 =

(λ1 (t) − λ4 (t))S ∗ (λ2 (t) − λ4 (t))I ∗ λ3 (t)W ∗ , u2 = , u3 = . 2A1 2A2 2A3

(38)

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Proof The optimal controls u∗1 , u∗2 and u∗3 can be solved by setting the partial derivatives of H equal to zero, ∂H = 2A1 u1 − λ1 (t)S ∗ + λ4 (t)S ∗ = 0, ∂u1 ∂H = 2A2 u2 − λ2 (t)I ∗ + λ4 (t)I ∗ = 0, ∂u2 ∂H = 2A3 u3 − λ3 (t)W ∗ = 0. ∂u3

(39) (40) (41)

After a simple manipulation, the optimal control pair (u∗1 , u∗2 , u∗3 ) is characterized as (37) and (38). By standard control arguments involving the bounds on the controls, we conclude  (λ (t)−λ (t))S ∗ ∗ 1 4 4 (t))S  if 0 < (λ1 (t)−λ < 1,  2A1 2A1 (λ1 (t)−λ4 (t))S ∗ ∗ u1 = 0 ≤ 0, if 2A1   (λ1 (t)−λ4 (t))S ∗ 1 if ≥ 1. 2A1  (λ (t)−λ (t))I ∗ ∗ 2 4 4 (t))I  if 0 < (λ2 (t)−λ < 1,  2A2 2A2 (λ2 (t)−λ4 (t))I ∗ ∗ u2 = 0 ≤ 0, if 2A2   (λ2 (t)−λ4 (t))I ∗ 1 if ≥1 2A2  λ (t)W ∗ (t)W ∗ 3  if 0 < λ32A < 1,  2A3 3 λ3 (t)W ∗ ∗ u3 = 0 if 2A3 ≤ 0,   (t)W ∗ ≥ 1. 1 if λ32A 3

5

Numerical results

In this section, we work out the optimality system which is combined by the ODE system (1-4) and the adjoint system (32-35) by using the data regarding the course of the cholera in Zimbabwe (2008-2009). It began in August 2008, not only swept to all of Zimbabwe’s ten provinces but also spread to Botswana, Mozambique, South Africa and Zambia quickly. The principal cause of the outbreak was the collapse of Zimbabwe’s public health system. By the end of November 2008, three of Zimbabwe’s four major hospitals had shut down, and many places had no basic drugs, medicines and water supply for such a long enough period during the outbreak period. On 4 December 2008, the Zimbabwe government declared the outbreak to be a national emergency. By March 2009, the World Health Organization (WHO) estimated that 4,011 people had succumbed to this waterborne disease and 91,164 cases were infected. The total population in Zimbabwe is 12,347,240, in order to make the calculation simpler, we scale down all data numbers by a factor of 1,200. All epidemiological parameter values for cholera in literature are given as N = 10000, µ = 0.000442, γ = 1.4, ξ = 70, δ = 0.023, βW = 0.12, βI = 0.00075. We use the initial values as S0 = 9999, I0 = 1, W0 = 0, R0 = 0. The weight constants are set as A0 = A1 = A2 = A3 = 10. We note that the optimality system is a two-point boundary value problem, with separated boundary conditions at initial time t = 0 and final time t = tf . Solving this optimality 11 1065

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system requires an iterative scheme which is combination of forward and backward difference approximation developed by [22, 24], we show this procedure in the following algorithm. In the programming, let there exist a uniform step size h > 0 and (n, m) ∈ N 2 , τ = mh and tf = nh. We can obtain the following partition by setting m knots to left of 0 and right of tf . ∆ = (t−m = −τ < · · · < t−1 < 0 < t1 < · · · < tn = tf < · · · < tn+m ). Therefore, ti = ih(−m ≤ i ≤ n + m). The state and adjoint variables and control variables, such as S(t), I(t), W (t), R(t), λi and ui in terms of nodal points Si , Ii , Wi , Ri , λii and ui . Fig.1 (a) represents the number of infected individuals as a function of time when τ = 5, epidemic outbreak increases rapidly and reaches the peak at t = 22 weeks with value 40, the controls take some time to react with the infected individuals, it then starts to gradually drop to almost zero, meaning the disease is gradually eradicated from the population. Fig.1 (b) shows the susceptible population S vs. time (weeks), we observe that there is a significant decrease in the number of susceptible after around 40 weeks. In order to clearly see the effect of the time lag on the dynamical behavior of the system, we take a smaller time delay as τ = 1 in Fig.2. By comparison with Fig.1, we can observe the smaller the time delay, the shorter it takes the equilibrium points to settle to their state value, which implies that the disease will be more serious if the delay lag is shorter. 45

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Algorithm Step1 for i = −m, ..., 0, do Si = S(0), Ii = I(0), Wi = W (0), Ri = R(0), ui1 = 0, ui2 = 0, ui3 = 0, end for for i = n, ..., n + m, do λi1 = 0, λi2 = 0, λi3 = 0, end for Step2 for i = 0, ..., n − 1, do Si +hµN βW Wi+1 +βI Ii+1 +µ+u1 ) κ+Wi+1 Si+1 Wi+1 Ii +hβW κ+W i+1

Si+1 =

1+h(

Ii+1 =

1+h(γ+µ+u2 −βI Si+1 )

,

,

Wi+1 =

Wi +hξIi−m , 1+h(δ+u3 )

Ri+1 =

Ri +h(γIi+1 +u2 Ii+1 +u1 Si+1 ) , 1+hµ

λn−i−1 1

=

λn−i−1 = 2 λn−i−1 3

=

λn−i−1 = 4

βW Wi+1 +βI Ii+1 )λn−i +hµλn−i 2 4 κ+Wi+1 βW Wi+1 1+h( κ+W +βI Ii+1 +µ+u1 ) i+1 n−i−1 n−i λn−i +h−hλ β (γ+u2 )+hλn−i+m χ[0,tf −τ ] (tn−i )ξ I Si+1 +hλ4 2 1 2

λn−i +h( 1

,

1+h[βI Si+1 −(γ+µ+u2 )] λn−i −hλ1n−i−1 3

βW κSi+1 βW κSi+1 +hλn−i−1 2 (κ+Wi+1 )2 (κ+Wi+1 )2

1+h(δ+u3 )

,

,

λ4n−i , 1+hµ

T1i+1 =

(λ1n−i −λn−i )Si+1 4 , 2A1

T2i+1 =

(λ2n−i −λn−i )Ii+1 4 , 2A2

T3i+1 =

λ3n−i Wi+1 , 2A1

ui+1 = min(max(0, T1i+1 ), 1), 1 ui+1 = min(max(0, T2i+1 ), 1), 2 ui+1 = min(max(0, T3i+1 ), 1), 2 Step3 for i = 0, ..., n, write S ∗ (ti ) = Si , I ∗ (ti ) = Ii , W ∗ (ti ) = Wi , R∗ (ti ) = Ri , u∗1 (ti ) = ui1 , u∗2 (ti ) = ui2 , u∗3 (ti ) = ui3 , end for 13 1067

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different cases: τ = 6 and τ = 3, respectively. From Fig.3, it is apparent that a larger value of optimal control variables is necessary in case of smaller time delay. It is also clear to see that the control u2 in both cases always needs to be the maximal while the other two controls u1 and u3 , which need not to be the maximal at very first, increase gradually and reach the maximal until certain weeks. Hence, we can firstly apply more of the therapeutic treatment in order to effectively reduce the number of infectious individuals. 1

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in the number of infected individuals and susceptible individuals controlled compared with optimal controlled, so that the infected population is affected very much due to the lack of all the three controls. Compared with Fig.6, Fig.7 and Fig.8, the number of infectious does not differ significantly by applying either the strategies with control u1 only or with control u3 only, but does make greater significance when only treatment control u2 is employed, thus the application of therapeutic treatment control gives better result than the application of u1 or u3 only. This simulation indicates that therapeutic treatment is more effective in reducing the infection level, which highlights the effectiveness of treatment measure in controlling the diseases. In a word, the use of a single optimal control method does not make a significant impact, while the use of multi-strategies is more efficient. However, if the budget is limited, it is much better to apply the treatment well before the occurrence of the outbreak. 140

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6

Conclusions and discussions

In this paper, we have presented a cholera epidemiological model by incorporating three types of intervention strategies and time delay inspired by the work in Wang and Modnak [21]. We have mainly investigated that by applying both an optimal control and a time delay to a 15 1069

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cholera model in order to eliminate the infectious disease. First of all, both the disease-free equilibrium E0 and endemic equilibrium E ∗ of the model were obtained. By analyzing the corresponding characteristic equations, the local stability of E0 and E ∗ was investigated. In particular, we have established the global stability analysis of the disease-free and endemic equilibria of ODE system by constructing two suitable Lyapunov functionals. Moreover, we used the Pontryagins Maximum Principle with delay to characterize optimal controls and derived the optimality system at the same time. Finally, we presented an efficient numerical simulation based on a specific algorithm to show that the optimal control strategy is much more effective for reducing the number of infected individuals than using of any single control, which highlights the effectiveness of treatment measure in controlling the diseases. However, if the budget is limited, it is much better to apply the therapeutic treatment well before the occurrence of the outbreak. Since the choice of the weights Ai reflects the different scales of the costs for different controls, it is important to notice that the ideal weights are very difficult to obtain in the real world. We only use theoretical weights to propose the simulations in this paper, thus the appropriate data is a difficult problem and it still remains for our further work. We also need to pay attention to that different choices of final time tf lead to different results, because there is an opposite time orientations for the optimality system when we carry out the simulations. Mathematically speaking, the control is very sensitive to the final time. In the work of [19] in 2011, it was mentioned that the shorter the period of control programme is, the smaller the marginal cost of control will be.

7

Acknowledgments

This work was partially supported by the Natural Science Foundation of China (NO.11271388, NO. 11401059), National Social Science Foundation of China (NO.13CTJ016), Natural Science Foundation of CQ (NO. cstc2015jcyjA00024).

References References [1] Code¸co CT, Endemic and epidemic dynamics of cholera: the role of the aquatic reservoir, BMC Infectious Diseases, 1. 1, 2001. [2] Hartley DM, Morris JG and Smith DL, Hyperinfectivity: A critical element in the ability of V. cholerae to cause epidemics?, PLoS Medicine, 3: 63-69, 2006. [3] Tien JH and Earn DJD, Multiple transmission pathways and disease dynamics in a waterborne pathogen model, Bulletin of Mathematical Biology, 72(6): 1502-1533, 2010. [4] Mukandavire Z, Liao S, Wang J, Gaff H, Smith DL and Morris JG, Estimating the reproductive numbers for the 2008–2009 cholera outbreaks in Zimbabwe, Proceedings of

17 1071

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the National Academy of Sciences of the United States of America, 108(21): 8767-8772, 2011. [5] Liao S and Wang J, Stability analysis and application of a mathematical cholera model, Mathematical Biosciences and Engineering, 8(3): 733-752, 2011. [6] Wei H, Li X and Martcheva M, An epidemic model of a vector-borne disease with direct transmission and time delay, Journal of Mathematical Analysis and Applications, 342(2): 895-908, 2008. [7] McCluskey CC, Global stability of an sir epidemic model with delay and general nonlinear incidence, Mathematical biosciences and engineering, 7(4): 837-850, 2010. [8] Misra AK, Mishra SN, Pathak AL, Chandra P and Naresh R, Modeling the effect of time delay in controlling the carrier dependent infectious disease-Cholera, Applied Mathematics and Computation, 218: 11547-11557, 2012. [9] Misra AK, Mishra SN, Pathak AL, Srivastava PK and Chandra P, A mathematical model for the control of carrier-dependent infectious diseases with direct transmission and time delay, Chaos, Solitons and Fractals, 57: 41-53, 2013. [10] Wang Y and Wei J, Global dynamics of a cholera model with time delay, International Journal of Biomathematics, 232: 436-444, 2014. [11] Morton IK and Nancy LS, Dynamics Optimization The calculus of Variations and Optimal Control in Economics and Management, Elsevier Science, The Netherlands, 2000. [12] Joshi HR, Optimal control of an HIV immunology model, Optimal Control Applications & Methods, 23(4): 199-213, 2002. [13] Karrakchou J, Rachik M and Gourari S, Optimal control and infectiology: application to an HIV/AIDS model, Applied Mathematics and Computation, 177(2): 807-818, 2006. [14] Nanda S, Moore H and Lenhart S, Optimal control of treatment in a mathematical model of chronic myelogenous leukemia, Mathematical Biosciences, 210(1): 143-156, 2007. [15] Blayneh K, Cao Y and Hee-Dae K, Optimal control of vector-borne diseases: treatment and prevention, Discrete and Continuous Dynamical Systems-series B, 11(3): 587-611, 2009. [16] Lashari AA, Hattaf K, Zaman G and Li XZ, Backward bifurcation and optimal control of a vector borne disease, Applied Mathematics & Information Sciences, 7(1): 301-309, 2013. [17] Sunmi L, Chowell G and Castillo-Chavez C, Optimal control for pandemic influenza: The role of limited antiviral treatment and isolation, Journal of Theoretical Biology, 265(2): 136-150, 2010.

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[18] Ding W, Finotti H, Lenhart S, Lou Y and Ye Q, Optimal control of growth coefficient on a steady-state population model, Nonlinear Analysis: Real World Applications, 11(2): 688-704, 2010. [19] Okosun KO, Ouifki R and Marcus N, Optimal control analysis of a malaria disease transmission model that includes treatment and vaccination with waning immunity, BioSystems, 106(2-3): 136-145, 2011. [20] Kar TK and Jana S, A theoretical study on mathematical modelling of an infectious disease with application of optimal control, BioSystems, 111(1): 37-50, 2013. [21] Wang J and Modnak C, Modeling odeling cholera dyanmics with controls, Canadian applied mathematics quarterly, 19(3): 255-273, 2011. [22] Laarabi H, Abta A and Hattaf K, Optimal Control of a Delayed SIRS Epidemic Model with Vaccination and Treatment, Acta Biotheor, 63(2): 87-97, 2015. [23] Mohamed E, Mostafa R and Elhabib B, Optimal control of an SIR model with delay in state and control variables, ISRN Biomathematics, 2013, 2013. [24] Hattaf K and Yousfi N, Optimal control of a delayed HIV infection model with immune response using an efficient numerical method. ISRN Biomathematics, 2012. [25] Driessche PVD and Watmough J, Reproduction numbers and sub-threshold endemic equilibria for compartmental models of disease transmission, Mathematical Biosciences, 180(1-2): 29-48, 2002. [26] LaSalle JP, The Stability of Dynamical Systems, Regional Conference Series in Applied Mathematics, SIAM, Philadelphia, PA, 1976. [27] Fleming W and Rishel R, Deterministic and Stochastic Optimal Control, SpringerVerlag, New York, 1975. [28] Lenhart S and Workman J, Optimal Control Applied to Biological Models, Chapman Hall/CRC, 2007.

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Effect of antibodies and latently infected cells on HIV dynamics with differential drug efficacy in cocirculating target cells A. M. Shehataa,b , A. M. Elaiwc and E. Kh. Elnaharye a Department of Electrical, Electronic and Computer Engineering, University of Pretoria, South Africa b Department of Mathematics, Faculty of Science, Al-Azhar University, Assiut, Egypt. c Department of Mathematics, Faculty of Science, King Abdulaziz University, P.O. Box 80203, Jeddah 21589, Saudi Arabia. e Department of Mathematics, Faculty of Science, Sohag University, Sohag, Egypt. Abstract In this paper, we investigate the qualitative behaviors of three viral infection models with two types of cocirculating target cells. The models take into account both antibodies and latently infected cells. The incidence rate is represented by bilinear, saturation and general function. For the first two models, we have derived two threshold parameters, R0 and R1 which completely determined the global properties of the models. Lyapunov functions are constructed and LaSalle’s invariance principle is applied to prove the global asymptotic stability of all equilibria of the models. For the third model, we have established a set of conditions on the general incidence rate function which are sufficient for the global stability of the equilibria of the model. Theoretical results have been checked by numerical simulations. Keywords: Virus infection; Global stability; Latently infected cells; cocirculating target cells; Lyapunov function.

1

Introduction

Mathematical modeling and model analysis of virus infection in vivo have attracted the interests of mathematicians during the recent years. Such virus infection models can be very useful in the control of epidemic diseases and provide insights into the dynamics of viral load in vivo. Therefore, mathematical analysis of the virus infection models can play a significant role in the development of a better understanding of diseases and various drug therapy strategies.Many authors have formulated mathematical models to describe the population dynamics of several viruses such as, human immunodeficiency virus (HIV) (see e.g. [1]-[10]), hepatitis B virus (HBV) [11]-[13], hepatitis C virus (HCV) [14]-[15], human T cell leukemia HTLV [16] and dengue virus [17], etc. During viral infections, the host immune system reacts with antigen-specific immune response. The immune system has two main responses to viral infections. The first is based on the Cytotoxic T Lymphocyte (CTL) cells which are responsible to attack and kill the infected cells. The second immune response is based on the antibodies that are produced by the B cells. The function of the antibodies is to attack the viruses [1]. In some infections such as in malaria, the CTL immune response is less effective than the antibody immune response [18]. Several mathematical models have been proposed to consider the antibody immune response into the viral infection models ([19]-[24]). The basic model of viral infection with antibody immune response has been Emails: ah [email protected] (A. M. Shehata), a m [email protected] (A.M.Elaiw), e [email protected] (E. Kh. Elnahary).

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introduced by Murase et. al. [19] and Wang and Zou [21] as: ¯ x˙ = λ − dx − βxv, ¯ − ay, y˙ = βxv

(1)

v˙ = ky − cv − rzv,

(3)

z˙ = gzv − µz,

(4)

(2)

where x, y, v and z represent, respectively, the concentrations of uninfected cells, infected cells, free viruses and the antibody immune cells. Parameters λ, k and g represent respectively, the rate of new uninfected cells that are generated from sources within the body, the rate of free virus production and the proliferation rate constant of the antibody immune cells. Parameters d, a, c and µ are the natural death rate constant of uninfected cells, infected cells, free virus particles and the antibody immune cells respectively. Parameter β¯ is the infection rate constant at which a target cell becomes infected via contacting with virus and r is the removal rate constant of the virus due to the antibodies. Model (1)-(4) is based on the assumption that the infection could occur and that the viruses are produced from infected cells instantaneously, once the uninfected cells are contacted by the virus particles. Other accurate models incorporate the latently infected cells which are due to the delay between the time of infection and the time when the infected cell becomes active to produce infectious viruses. In [26], model (1)-(4) was extended to take into consideration both latently and actively infected cells as: ¯ x˙ = λ − dx − βxv, ¯ − (e + b)w, w˙ = (1 − α)βxv

(5)

¯ + bw − ay, y˙ = αβxv

(7)

v˙ = ky − cv − rvz,

(8)

z˙ = gvz − µz,

(9)

(6)

where w and y are the concentrations of latently infected and actively infected cells, respectively. Eq. (6) describes the population dynamics of the latently infected cells and show that they are converted to actively infected cells with rate constant b. The parameters e and a are the death rate constants of the latently and actively infected cells, respectively. The fractions (1 − α) where, 0 < α < 1 are the probabilities that upon infection, an uninfected cell will become either latently infected or actively infected. Model (5)-(9) it have been assumed that, the HIV has one class of target cells, CD4+ T cells. However, Perelson et al. in [25] have shown that, HIV infects the macrophages in addition to the CD4+ T cells. Recently, many efforts have been devoted to study various mathematical models of HIV dynamics with two classes of target cells (see e.g. [3]). Our primary goal of the present paper is to propose the global stability analysis of three viral infection models with two types of target cells, CD4+ T cells and macrophages taking into consideration the latently, actively infected cells and antibody immune response. The infection rate is represented by bilinear incidence and saturated incidence in the first and the second models, respectively, while it is given by a general function in the third one. The global stability of the three models is established using Lyapunov functionals.

2

HIV model with bilinear incidence rate

In this section, we introduce an HIV dynamics model which describes two cocirculation populations of target cells, CD4+ T cells and macrophages and takes into account the antibody immune response. We consider two types of infected cells, the latently infected and actively infected cells. x˙ i = λi − di xi − βi xi v,

i = 1, 2,

(10)

w˙ i = (1 − αi )βi xi v − (ei + bi )wi ,

i = 1, 2,

(11)

y˙ i = αi βi xi v + bi wi − ai yi ,

i = 1, 2,

(12)

v˙ =

2 X

ki yi − cv − rvz,

(13)

i=1

z˙ = gvz − µz.

(14)

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Here i = 1, 2 correspond to the CD4+ T cells and macrophages and β1 = (1 − ε)β¯1 , β2 = (1 − εf )β¯2 . The model incorporates RTI drug therapy where in the CD4+ T cells, the drug efficacy is ε and 0 6 ε < 1, while in the macrophages the drug efficacy εf is reduced by a factor f and 0 < f < 1. All the parameters and variables of the model have the same meanings as given in (5)-(9).

2.1

Properties of solutions

One can easily show that the non-negative orthant R8 ≥ 0 by model (10)-(14). Proposition 1. There exist positive numbers Lj , j = 1, 2, 3, 4 such that the compact set Ω = © ª (xi , wi , yi , v, z) ∈ R8 ≥ 0 : 0 ≤ xi , wi , yi ≤ Li , 0 ≤ v ≤ L3 , 0 ≤ z ≤ L4 , i = 1, 2 is positively invariant. Proof. To show the boundedness of the solutions of system (10)-(14) we let Ti (t) = xi (t) + wi (t) + yi (t), then T˙i (t) = λi − di xi (t) − ei wi (t) − ai yi (t) ≤ λi − ρi Ti (t), λi where ρi = min{di , ai , ei }, i = 1, 2. Hence Ti (t) ≤ Li , if Ti (0) ≤ Li , where Li = . Since xi (t), wi (t) and y(t) ρi are all non-negative, then 0 ≤ xi (t), wi (t), yi (t) ≤ Li , for all t ≥ 0, if 0 ≤ xi (0) + wi (0) + yi (0) ≤ Li , i = 1, 2. r On the other hand, let G(t) = v(t) + z(t), then g µ ¶ X 2 2 2 X X r rµ ˙ z≤ ki Li − δ v + z = ki Li − δG(t), = ki yi − cv − G(t) g g i=1 i=1 i=1 2 1 P ki Li . Since v(t) ≥ 0 and z(t) ≥ 0, then δ i=1 r gL3 . 0 ≤ v(t) ≤ L3 and 0 ≤ z(t) ≤ L4 if 0 ≤ v(0) + z(0) ≤ L3 , where L4 = g r

where δ = min{c, µ}. Hence G(t) ≤ L3 , if G(0) ≤ L3 , where L3 =

2.2

Equilibria and biological thresholds ◦

Let Ω be the interior of Ω. Lemma 1. For system (10)-(14) we have (i) There exist only one uninfected equilibrium E0 = (x01 , x02 , 0, 0, 0, 0, 0, 0) ∈ Ω, when R0 ≤ 1. (ii) There exist E0 and a chronic-infection equilibrium without antibody immune response E1 = (˜ x1 , x ˜2 , w ˜1 , w ˜2 , y˜1 , y˜2 , v˜, 0, ) ∈ Ω, when R1 ≤ 1 < R0 . (iii) There exist E0 , E1 and a chronic-infection equilibrium with antibody immune response E2 = (¯ x1 , x ¯2 , w ¯1 , w ¯2 , y¯1 , y¯2 , v¯, z¯) ∈ ˚ Ω, when R1 > 1. Proof. The equilibria of (10)-(14) satisfy the following equations: λi − di xi − βi xi v = 0,

(15)

(1 − αi )βi xi v − (ei + bi )wi = 0,

(16)

αi βi xi v + bi wi − ai yi = 0,

(17)

2 X

ki yi − cv − rvz = 0,

(18)

gvz − µz = 0.

(19)

i=1

Eq. (19) has two possible solutions z = 0 or v = xi = where x0i =

x0i , (1 + ηi v)

wi =

µ . If z = 0, then from Eqs.(15)-(17) we get g

(1 − αi )βi x0i v, (ei + bi )(1 + ηi v)

βi λi , ηi = , i = 1, 2. From Eq. (18) we obtain di di à 2 X (ei αi + bi )ki βi x0 i

i=1

ai c(ei + bi )(1 + ηi v)

1076

yi =

(ei αi + bi )βi x0i v, ai (ei + bi )(1 + ηi v)

(20)

! − 1 cv = 0.

(21)

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We note that v = 0 is a solution for Eq. (21) which leads to the disease-free equilibrium E0 = (x01 , x02 , 0, 0, 0, 0, 0, 0). If v 6= 0, we have 2 X Φi = 1. (22) 1 + ηi v i=1 where Φi =

(ei αi + bi )ki βi x0i . Equation (22) can be written as: ai c(ei + bi ) Av 2 + Bv − C = 0,

(23)

where A = η1 η2 ,

B = η1 Φ1 + η2 Φ2 + (1 − Φ1 − Φ2 )(η1 + η2 ),

C = Φ1 + Φ 2 − 1

The solutions of Eq. (23) is given by

√ B 2 + 4AC . v = 2A We have A > 0, therefore if C > 0, then v + > 0 and v − < 0. Let v˜ = v + , then from Eq. (20) we get ±

x ˜i =

x0i , 1 + ηi v˜

w ˜i =

−B ±

(1 − αi )βi x0i v˜, (ei + bi )(1 + ηi v˜)

y˜i =

(ei αi + bi )βi x0i v˜, ai (ei + bi )(1 + ηi v˜)

i = 1, 2.

(24)

Therefore, a chronic-infection equilibrium without antibody immune response E1 = (˜ x1 , x ˜2 , w ˜1 , w ˜2 , y˜1 , y˜2 , v˜, 0) exists when C > 0 or (Φ1 + Φ2 > 1). Now we are ready to define the basic infection reproduction number R0 as 2 2 X X ki βi x0i (ei αi + bi ) . R0 = Φ1 + Φ2 = R0i = ai c(ei + bi ) i=1 i=1 µ , then we obtain the chronic-infection equilibrium with antibody immune response E2 = g (¯ x1 , x ¯2 , w ¯1 , w ¯2 , y¯1 , y¯2 , v¯, z¯), where

If v =

(1 − αi )λi βi µ (ei αi + bi )λi βi µ gλi , w ¯i = , y¯i = , gdi + µβi (ei + bi )(gdi + µβi ) ai (ei + bi )(gdi + µβi ) Ã 2 ! c X gki βi λi (ei αi + bi ) µ z¯ = v¯ = , −1 . g r i=1 ai c(ei + bi )(gdi + µβi )

x ¯i =

We note that E2 exists when activation number as

i = 1, 2,

2 P

gki βi λi (ei αi + bi ) > 1. Let us define the antibody immune response i=1 ai c(ei + bi )(gdi + µβi ) R1 =

2 X i=1

2

X gki βi λi (ei αi + bi ) = ai c(ei + bi )(gdi + µβi ) i=1

R0i , µβi 1+ gdi

which determines whether or not a persistent antibody immune response can be established. Then we can write c z¯ = (R1 − 1). Clearly R1 < R0 . r Now, we show that E0 , E1 ∈ Ω and E2 ∈ ˚ Ω. Clearly, E0 ∈ Ω. Let R0 > 1, then from Eq. (20) we have 0 x ˜i < xi , then λi λi ≤ = Li . 0 0. Thus, the solutions of system (10)-(14) converge to Ω, the largest dt ©≤ 0 for all ª i dW0 0 0 invariant subset of dt = 0 [27]. Clearly, it follows from Eq. (26) that dW dt = 0 if and only if xi = xi , v = 0 and z = 0. The set Ω is invariant and for any element belongs to Ω satisfies v = 0 and z = 0, then v˙ = 0. We 2 P can see from Eq. (13) that 0 = v˙ = ki yi , and thus yi = 0. Moreover, from Eq. (12) we get wi = 0. Hence

dW0 dt

i=1

= 0 occurs at E0 . From LaSalle’s invariance principle, E0 is GAS. Theorem 2. The chronic-infection equilibrium without antibody immune response E1 of system (10)-(14) is GAS when R1 ≤ 1 < R0 .

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Proof. We construct the following Lyapunov function W1 =

µ ¶ µ ¶¸ · ³v´ r bi wi ei + bi yi xi w ˜i F + y˜i F +˜ vF + z. γi x ˜i F ( ) + x ˜i ei αi + bi w ˜i ei αi + bi y˜i v˜ g i=1

2 X

dW1 along the trajectories of (10)-(14) we get dt "µ ¶ µ ¶ 2 X x ˜i w ˜i dW1 bi = γi 1− (λi − di xi − βi xi v) + 1− ((1 − αi )βi xi v − (ei + bi )wi ) dt xi ei αi + bi wi i=1 # µ ! µ ¶ ¶ ÃX 2 y˜i v˜ ei + bi r 1− (αi βi xi v + bi wi − ai yi ) + 1 − ki yi − cv − rvz + (gvz − µz) . (28) + ei αi + bi yi v g i=1

Calculating

Collecting terms of Eq. (28) we get "µ ¶ 2 X x ˜i ˜i ei + bi αi (ei + bi ) βi xi v y˜i dW1 bi (1 − αi ) βi xi v w = γi 1− (λi − di xi ) + βi x + bi w ˜i − ˜i v − dt x e α + b w e α + b ei αi + bi yi i i i i i i i i i=1 # 2 ei + bi v˜ X bi (ei + bi ) wi y˜i rµ + ai y˜i − cv − ki yi + c˜ v + r˜ vz − z. − (29) ei αi + bi yi ei αi + bi v i=1 g µ Using the value of x ˜i given in Eq. (24) we get "µ

2 P

i=1

¶ γi β i x ˜i − c v = 0. Applying λi = di x ˜i + βi x ˜i v˜, we obtain

¶ µ bi (1 − αi ) βi xi v w ˜i ei + b i x ˜i (di x ˜i − di xi ) + βi x − + bi w ˜i ˜i v˜ 1 − xi ei α i + b i wi ei αi + bi # 2 bi (ei + bi ) wi y˜i ei + bi v˜ X αi (ei + bi ) βi xi v y˜i rµ − + ai y˜i − ki yi + c˜ v + r˜ vz − z. − ei αi + bi yi ei αi + bi yi ei αi + bi v i=1 g 2

X dW1 = γi dt i=1

x ˜i 1− xi

¶

(30)

Using the equilibrium condition for E1 (1 − αi )βi x ˜i v˜ = (ei + bi )w ˜i ,

αi βi x ˜i v˜ + bi w ˜i = ai y˜i ,

c˜ v=

2 X i=1

ki y˜i =

2 X

γi βi x ˜i v˜,

i=1

bi (1 − αi ) (ei + bi )αi ei + bi ai y˜i = βi x ˜i v˜ = βi x ˜i v˜ + βi x ˜i v˜. ei αi + bi ei α i + b i ei αi + bi we have " µ ¶µ ¶ 2 X (xi − x ˜i )2 x ˜i bi (1 − αi ) xi w ˜i v dW1 bi (1 − αi ) (ei + bi )αi = γi − di + βi x ˜i v˜ 1 − + − βi x ˜i v˜ dt xi xi ei α i + b i ei αi + bi ei αi + bi x ˜i wi v˜ i=1 (ei + bi )αi xi y˜i v bi (1 − αi ) wi y˜i bi (1 − αi ) (ei + bi )αi bi (1 − αi ) βi x ˜i v˜ − βi x ˜i v˜ − βi x ˜i v˜ + βi x ˜i v˜ + βi x ˜i v˜ ei αi + bi ei αi + bi x ˜i yi v˜ ei αi + bi w ˜i yi ei αi + bi ei αi + bi # ¶ µ ¶ µ yi v˜ bi (1 − αi ) (ei + bi )αi bi (1 − αi ) (ei + bi )αi + βi x ˜i v˜ + + βi x ˜i v˜ + (˜ − v − v¯) rz. ei αi + bi ei αi + bi y˜i v ei αi + bi ei α i + b i " µ ¶ 2 X bi (1 − αi ) x ˜i xi w ˜i v yi v˜ wi y˜i (xi − x ˜i )2 + βi x ˜i v˜ 4 − − − − = γi − di xi ei αi + bi xi x ˜i wi v˜ y˜i v w ˜i yi i=1 # ¶ µ x ˜i yi v˜ xi y˜i v (ei + bi )αi βi x ˜i v˜ 3 − − − + (˜ + v − v¯) rz. (31) ei αi + bi xi y˜i v x ˜i yi v˜

+

We have xi , wi , yi , v > 0 when R0 > 1. Since the geometrical mean is less than or equal to the arithmetical mean, the second and the third terms are less than or equal to zero. Now we show that if R1 ≤ 1 then v˜ ≤ µg = v¯.

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Using the steady state conditions for E1 we have

R1 − 1 =

2 X i=1

=

2 X i=1

2 P

ki βi λi (ei αi + bi ) = 1, then a cd ˜) i=1 i i (ei + bi )(1 + ηi v 2

X ki βi λi (ei αi + bi ) gki βi λi (ei αi + bi ) − ai c(ei + bi )(gdi + µβi ) i=1 ai di c(ei + bi )(1 + ηi v˜) 2

X ki βi λi (ei αi + bi ) ki βi λi (ei αi + bi ) − = (˜ v − v¯)χ, ai di c(ei + bi )(1 + ηi v¯) i=1 ai di c(ei + bi )(1 + ηi v˜)

(32)

2 P

ki βi λi ηi (ei αi + bi ) 1 . It follows that, if R1 ≤ 1 then dW dt ≤ 0 for all xi , wi , yi , v, z > 0. a d c(e ¯)(1 + ηi v˜) i + bi )(1 + ηi v i=1 i i ª © 1 Thus, the solutions of system (10)-(14) limit to Ω, the largest invariant subset of dW dt = 0 [27]. It can be 1 seen that, dW dt = 0 occurs at E1 . Applying LaSalle’s invariance principle we obtain that E1 is GAS. Theorem 3. The chronic-infection equilibrium with antibody immune response E2 of system (10)-(14) is GAS when R1 > 1. Proof. Consider the following Lyapunov function

where χ =

2 X

·

bi xi w ¯i F W2 = γi x ¯i F ( ) + x ¯ e α i i i + bi i=1

µ

wi w ¯i

¶

ei + bi + y¯i F ei αi + bi

µ

yi y¯i

¶¸ ³z ´ ³v´ r + z¯F . +¯ vF v¯ g z¯

Calculating the derivative of W2 along the trajectories of (10)-(14) we get "µ ¶ µ ¶ 2 X x ¯i w ¯¯ı bi dW2 = γi 1− (λi − di xi − βi xi v) + 1− ((1 − αi )βi xi v − (ei + bi )wi ) dt xi ei α i + b i wi i=1 # µ ¶ 2 ³ y¯i v¯ ´ X z¯ ´ ei + bi r³ 1− (αi βi xi v + bwi − ai yi ) + 1 − ( ki yi − cv − rvz) + 1− (gvz − µz) . + ei αi + bi yi v i=1 g z (33) Collecting terms of Eq. (33) we get "µ ¶ 2 X x ¯i ¯¯ı ei + bi dW2 bi (1 − αi ) βi xi v w = γi 1− (λi − di xi ) + βi x + bi w ¯¯ı ¯i v − dt xi ei αi + bi wi ei αi + bi i=1 # 2 bi (ei + bi ) wi y¯i ei + bi v¯ X αi (ei + bi ) βi xi v y¯i rµ − + ai y¯i − cv − ki yi + c¯ v − rv¯ z¯. − z+ ei αi + bi yi ei αi + bi yi ei αi + bi v i=1 g

(34)

Applying λi = di x ¯i + β x ¯i v¯ , we get "µ ¶ ¶ µ 2 X x ¯i bi (1 − αi ) βi xi v w ¯¯ı ei + bi x ¯i dW2 = γi 1− (di x ¯i − di xi ) + βi x + βi x ¯i v − + bi w ¯¯ı ¯i v¯ 1 − dt xi xi ei α i + b i wi ei αi + bi i=1 # 2 bi (ei + bi ) wi y¯i ei + bi v¯ X αi (ei + bi ) βi xi v y¯i rµ − + ai y¯i − cv − ki yi + c¯ v − rv¯ z¯. − z+ (35) ei αi + bi yi ei αi + bi yi ei αi + bi v i=1 g Using the equilibrium conditions for E2 (1 − αi )βi x ¯i v¯ = (ei + bi )w ¯i ,

αi βi x ¯i v¯ + bi w ¯i = ai y¯i ,

c¯ v + r¯ v z¯ =

2 X i=1

bi (1 − αi ) (ei + bi )αi ei + bi ai y¯i = βi x ¯i v¯ = βi x ¯i v¯ + βi x ¯i v¯, ei αi + bi ei αi + bi (ei αi + bi )

1080

2 X

ki y¯i =

2 X

γi βi x ¯i v¯,

i=1

γi β i x ¯i v¯ − c¯ v − r¯ v z¯ = 0,

i=1

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we have

" µ ¶µ ¶ 2 X (xi − x ¯i )2 x ¯i bi (1 − αi ) (ei + bi )αi bi (1 − αi ) xi w ¯i v dW2 = γi − di + βi x ¯i v¯ 1 − + − βi x ¯i v¯ dt x x e α + b e α + b e α + b x ¯ w ¯ i i i i i i i i i i i i iv i=1 (ei + bi )αi xi y¯i v bi (1 − αi ) wi y¯i bi (1 − αi ) (ei + bi )αi bi (1 − αi ) βi x ¯i v¯ − βi x ¯i v¯ − βi x ¯i v¯ + βi x ¯i v¯ + βi x ¯i v¯ ei αi + bi ei αi + bi x ¯i yi v¯ ei αi + bi w ¯i yi ei αi + bi ei αi + bi # ¶ µ ¶ µ yi v¯ bi (1 − αi ) (ei + bi )αi bi (1 − αi ) (ei + bi )αi + βi x ¯i v¯ + + βi x ¯i v¯ − ei αi + bi ei αi + bi y¯i v ei αi + bi ei α i + b i " · ¸ 2 X bi (1 − αi ) x ¯i xi w ¯i v yi v¯ wi y¯i (xi − x ¯i )2 + βi x ¯i v¯ 4 − − − − = γi − di xi ei αi + bi xi x ¯i wi v¯ y¯i v w ¯ i yi i=1 · ¸# x ¯i yi v¯ xi y¯i v (ei + bi )αi βi x ¯i v¯ 3 − − − . + (ei αi + bi ) xi y¯i v x ¯i yi v¯

+

Thus, if R1 > 1, then x ¯i , w ¯i , y¯i , v¯, z¯ > 0. Using the relation between arithmetical and geometrical means, we dW2 dW2 ≤ 0. Clearly, = 0 if and only if xi = x get ¯i , wi = w ¯i , yi = y¯i and v = v¯. If v = v¯, then v˙ = 0 and dt dt 2 P dW2 equal to zero at E2 . The from Eq. (13) we have 0 = ki y¯i − c¯ v − r¯ v z¯, which give z = z¯. Therefore, dt i=1 global stability of E2 follows from LaSalle’s invariance principle.

3

Model with saturation functional response

In this section, we modify model (10)-(14) by taking into account the saturation functional response as: βi xi v , 1 + σi v (1 − αi )βi xi v w˙ i = − (ei + bi )wi , 1 + σi v αi βi xi v + bi wi − ai yi , y˙ i = 1 + σi v 2 X v˙ = ki yi − cv − rvz, x˙ i = λi − di xi −

i = 1, 2,

(36)

i = 1, 2,

(37)

i = 1, 2,

(38) (39)

i=1

z˙ = gvz − µz,

(40)

where σi > 0, i = 1, 2, is the saturation constant, and all the variables and parameters of the model have the same definition as given in (10)-(14). We mention that the compact set Ω given in Section 2 is also positively invariant with respect to system (36)-(40).

3.1

Equilibria

Lemma 2. For system (36)-(40) we have (i) There exist only one uninfected equilibrium E0 = (x01 , x02 , 0, 0, 0, 0, 0, 0) ∈ Ω, when R0 ≤ 1. (ii) There exist E0 and a chronic-infection equilibrium without antibody immune response E1 = (˜ x1 , x ˜2 , w ˜1 , w ˜2 , y˜1 , y˜2 , v˜, 0, ) ∈ Ω, when R1 ≤ 1 < R0 . (iii) There exist E0 , E1 and a chronic-infection equilibrium with antibody immune response E2 = (¯ x1 , x ¯2 , w ¯1 , w ¯2 , y¯1 , y¯2 , v¯, z¯) ∈ ˚ Ω, when R1 > 1. Proof. We let the right-hand side of Eqs.(36)-(40) equal zero, then we obtain the following: µ Eq. (40) has two possible solutions z = 0 or v = . g If z = 0, then from Eqs.(36)-(38) we have xi =

x0i (1 + σi v) , (1 + ξi v)

wi =

(1 − αi )βi x0i v, (ei + bi )(1 + ξi v)

1081

yi =

(ei αi + bi )βi x0i v, ai (ei + bi )(1 + ξi v)

(41)

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where x0i =

βi λi , ξi = σi + , i = 1, 2. From Eq. (39) we find di di à 2 X (ei αi + bi )ki βi x0 i

ai c(ei + bi )(1 + ξi v)

i=1

! (42)

− 1 cv = 0.

(ei αi + bi )ki βi x0i − 1 = 0. i=1 ai c(ei + bi )(1 + ξi v) If v = 0, then substituting it in Eq. (41) we get the disease-free equilibrium E0 = (x01 , x02 , 0, 0, 0, 0, 0, 0). If v 6= 0, we have 2 X Ψi = 1. (43) (1 + ξi v) i=1

Eq. (42) has also two possible solutions v = 0 or

where Ψi =

2 P

(ei αi + bi )ki βi x0i . Eq. (43) can be written as: ai c(ei + bi ) A1 v 2 + B1 v − C1 = 0

(44)

where A1 = ξ1 ξ2 ,

B1 = ξ1 Ψ1 + ξ2 Ψ2 + (1 − Ψ1 − Ψ2 )(ξ1 + ξ2 ),

C1 = Ψ1 + Ψ2 − 1

The solutions of Eq. (23) is given by: ±

v =

−B1 ±

p B12 + 4A1 C1 . 2A

We have A1 > 0, therefore v + > 0 and v − < 0 when C1 > 0. Let v˜ = v + , then from Eq. (41) we get x ˜i =

x0i (1 + σi v˜) > 0, (1 + ξi v˜)

w ˜i =

(1 − αi )βi x0i v˜ > 0, (ei + bi )(1 + ξi v˜)

y˜i =

(ei αi + bi )βi x0i v˜ > 0, ai (ei + bi )(1 + ξi v˜)

i = 1, 2.

Therefore, an endemic equilibrium E1 = (˜ x1 , x ˜2 , w ˜1 , w ˜2 , y˜1 , y˜2 , v˜, 0, ) exists when C1 > 0 or (Ψ1 + Ψ2 > 1). Now we are ready to define the basic reproduction number R0 as R0 =

2 X i=1

R0i =

2 X

Ψi =

i=1

2 X (ei αi + bi )ki βi x0 i

i=1

ai c(ei + bi )

.

µ , then we obtain the chronic-infection equilibrium with antibody immune response E2 = g (¯ x1 , x ¯2 , w ¯1 , w ¯2 , y¯1 , y¯2 , v¯, z¯), where

If v =

(1 − αi )βi µx0i (ei αi + bi )βi µx0i (g + µσi )x0i , w ¯i = , y¯i = , g + µξi (ei + bi )(g + µξi ) ai (ei + bi )(g + µξi ) Ã 2 ! c X (ei αi + bi )ki βi gx0i µ z¯ = −1 . v¯ = , g r i=1 ai c(ei + bi )(g + µξi )

x ¯i =

i = 1, 2,

2 P (ei αi + bi )ki βi gx0i > 1. This equilibrium represents the state that both the i=1 ai c(ei + bi )(g + µξi ) viruses and antibodies are present. Let us define the antibody immune response activation number as

We note that E2 exists when

R1 =

2 2 X X R0i (ei αi + bi )ki βi gx0i ¶, µ = µ a c(ei + bi )(g + µξi ) i=1 i=1 i 1 + ξi g

which determines whether a persistent antibody immune response can be established. Then we can write c z¯ = (R1 − 1). Clearly R1 < R0 . Similar to Section 2.2, one can show that, E0 , E1 ∈ Ω and E2 ∈ ˚ Ω r

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3.2

Global stability

Theorem 4. The disease-free equilibrium E0 of system (36)-(40) is GAS when R0 ≤ 1. Proof. We define a Lyapunov function W0 as: W0 =

¸ · µ ¶ bi ei + bi r xi + w + y + v + z. γi x0i F i i 0 xi ei α i + b i ei αi + bi g i=1

2 X

(45)

dW0 along the trajectories of (36)-(40) dt "µ ¶µ µ ¶ ¶ 2 X x0i (1 − αi )βi xi v βi xi v bi dW0 = γi 1− λi − d i x i − + − (ei + bi )wi dt xi 1 + σi v ei αi + bi 1 + σi v i=1 µ ¶¸ X 2 αi βi xi v ei + bi r + bi wi − ai yi + ki yi − cv − rvz + (gvz − µz). + ei αi + bi 1 + σi v g i=1

We calculate

(46)

Collecting terms of Eq. (46) we get · µ ¶ ¸ 2 X x0 βi x0i v rµ dW0 = γi di 1 − i (x0i − xi ) + z − cv − dt x 1 g + σ v i i i=1 2 X

2

(xi − x0i )2 X (ei αi + bi )ki βi x0i rµ + z v − cv − xi a (e + bi )(1 + σi v) g i=1 i i i=1 Ã 2 ! 2 X X R0i (xi − x0i )2 rµ + z =− γi di − 1 cv − x (1 g + σ v) i i i=1 i=1 =−

=−

2 X

γi di

γi di

i=1

2 X (xi − x0i )2 rµ cσi R0i v 2 + (R0 − 1)cv − z. − xi (1 g + σ v) i i=1

(47)

dW0 0 If R0 ≤ 1 then dW dt ≤ 0 for all xi , v, z > 0. Similar to the proof of Theorem 1, one can easily show that dt = 0 at E0 . Then using LaSalle’s invariance principle, we can show the global stability of E0 . Next, we show that the endemic equilibrium E1 is GAS. Theorem 5. The chronic-infection equilibrium without antibody immune response E1 of system (36)-(40) is GAS when R1 ≤ 1 < R0 . Proof. We consider the following Lyapunov function

W1 =

2 X i=1

· γi x ˜i F

µ

xi x ˜i

¶

bi + w ˜i F ei αi + bi

µ

wi w ˜i

¶

ei + bi + y˜i F ei αi + bi

µ

yi y˜i

¶¸ +˜ vF

³v´

r + z. v˜ g

dW1 along the solutions of (36)-(40) we get dt "µ ¶µ µ ¶µ ¶ ¶ 2 X x ˜i w ˜i βi xi v bi (1 − αi )βi xi v dW1 = γi 1− 1− λi − di xi − + − (ei + bi )wi dt xi 1 + σi v ei αi + bi wi 1 + σi v i=1 µ ¶µ ¶ 2 ¶# µ y˜i ei + b i r αi βi xi v v˜ X 1− ki yi − cv − rvz) + (gvz − µz) . (48) ( + + bi wi − ai yi + 1− ei αi + bi yi 1 + σi v v g i=1

Calculating

Collecting terms of Eq. (48) we have: "µ ¶ µ ¶ 2 X x ˜i (1 − αi )βi xi v w ˜i βi x ˜i v bi dW1 1− (λi − di xi ) + − = γi + + (ei + bi )w ˜i dt xi 1 + σ i v ei α i + b i (1 + σi v)wi i=1 µ ¶# 2 αi βi xi v y˜i v˜ X ei + b i bi wi y˜i µr − + ai y˜i − cv − ki yi + c˜ v + r˜ vz − z. + + ei αi + bi (1 + σi v)yi yi v i=1 g

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Using the equilibrium condition for E1 : (1 − αi )βi x ˜i v˜ ei αi + bi βi x ˜i v˜ βx ˜i v˜ αi βi x ˜i v˜ , = (ei + bi )w + bi w ˜i = , ˜i , ai y˜i = 1 + σi v˜ 1 + σi v˜ 1 + σi v˜ ei + bi 1 + σi v˜ 2 2 2 2 2 X X X βi x ˜i v˜ yi v˜ βi x ˜i v˜ βi x ˜i v˜ v˜ X vX ki y i = γi , cv = γi c˜ v= ki y˜i = γi , , 1 v 1 y ˜ v v ˜ 1 + σ v ˜ + σ v ˜ + σi v˜ i i i i=1 i=1 i=1 i=1 i=1 λi = d i x ˜i +

˜i v˜ ˜i v˜ bi (1 − αi ) βi x (ei + bi )αi βi x βi x ˜i v˜ = + , 1 + σi v˜ (ei αi + bi ) (1 + σi v˜) (ei αi + bi ) (1 + σi v˜) we obtain "µ ¶µ µ ¶ ¶ 2 X x ˜i (1 − αi )βi xi v w ˜i βi x ˜i v˜ βi x ˜i v bi (1 − αi )βi x dW1 ˜i v˜ = γi 1− − di x ˜i + − di x i + + + dt xi 1 + σi v˜ 1 + σi v ei αi + bi (1 + σi v)wi 1 + σi v˜ i=1 # µ ¶ αi βi xi v y˜i ei αi + bi βi x ˜i v˜ ˜i v˜ ˜i v˜ ei + bi bi wi y˜i w ˜i yi v βi x v βi x βi x ˜i v˜ µr − + z. + + − − + + r˜ vz − ei αi + bi (1 + σi v)yi yi w ˜i ei + bi 1 + σi v˜ y˜i v˜ 1 + σi v˜ v˜ 1 + σi v˜ 1 + σi v˜ g " µ ¶ 2 X βi x ˜i v˜ (xi − x ˜i )2 v(1 + σi v˜) v 1 + σi v + = γi − di −1 + − + xi 1 + σi v˜ v˜(1 + σi v) v˜ 1 + σi v˜ i=1 µ ¶ ˜i v˜ xi w ˜i v(1 + σi v˜) yi v˜ wi y˜i 1 + σi v x ˜i bi (1 − αi ) βi x − − − 5− − + (ei αi + bi ) (1 + σi v˜) xi x ˜i wi v˜(1 + σi v) y˜i v w ˜i yi 1 + σi v˜ ¶ µ ¶# µ ˜i v˜ xi y˜i v(1 + σi v˜) yi v˜ 1 + σi v (ei + bi )αi βi x x ˜i µ − − rz + 4− − + v˜ − (ei αi + bi ) (1 + σi v˜) xi x ˜i yi v˜(1 + σi v) y˜i v 1 + σi v˜ g · 2 X βi x ˜i v˜ ˜i )2 (xi − x σi (v − v˜)2 − = γi −di xi (1 + σi v˜) (1 + σi v)(1 + σi v˜)˜ v i=1 µ ¶ ˜i v˜ xi w ˜i v(1 + σi v˜) yi v˜ wi y˜i 1 + σi v x ˜i bi (1 − αi ) βi x − − − 5− − + (ei αi + bi ) (1 + σi v˜) xi x ˜i wi v˜(1 + σi v) y˜i v w ˜i yi 1 + σi v˜ ¶ µ ¶¸ µ ˜i v˜ xi y˜i v(1 + σi v˜) yi v˜ 1 + σi v (ei + bi )αi βi x x ˜i µ − − rz. + 4− − + v˜ − (49) (ei αi + bi ) (1 + σi v˜) xi x ˜i yi v˜(1 + σi v) y˜i v 1 + σi v˜ g (32) we can show that (˜ v − v¯) = ω1 (R1 − 1), where ω = As the same proof of Eq. 2 P ki βi λi ξi (ei αi + bi ) . So, if R1 ≤ 1 then v˜ ≤ µg = v¯. We have xi , wi , yi , v > 0 when R0 > 1. ¯)(1 + ξi v˜) i=1 ai di c(ei + bi )(1 + ξi v Since the geometrical mean is less than or equal to the arithmetical mean, then the third and fourth terms of dW1 1 Eq. (49) are less than or equal zero, then if R1 ≤ 1 then dW dt ≤ 0 for all xi , wi , yi , v, z > 0. Clearly, dt = 0 occurs at E1 . LaSalle’s invariance principle implies global stability of E1 . Theorem 6. The chronic-infection equilibrium with antibody immune response E2 of system (36)-(40) is GAS when R1 > 1. Proof. Define Lyapunov function W2 as: " µ ¶ µ ¶# µ ¶ 2 ³v´ r ³z ´ X bi wi ei + b i yi xi + w ¯i F + y¯i F + v¯F + z¯F . W2 = γi x ¯i F x ¯i ei αi + bi w ¯i ei αi + bi y¯i v¯ g z¯ i=1 The time derivative of W2 along the trajectories of (36)-(40) is given by "µ ¶µ µ ¶µ ¶ ¶ 2 X x ¯i w ¯¯ı dW2 βi xi v bi (1 − αi )βi xi v = γi 1− 1− λi − d i x i − + − (ei + bi )wi dt xi 1 + σi v ei αi + bi wi 1 + σi v i=1 à 2 ! µ ¶µ ¶# ³ y¯i z¯ ´ ei + bi αi βi xi v v¯ ´ X r³ 1− ki yi − cv − rvz + 1− (gvz − µz) . + + bwi − ai yi + 1− ei αi + bi yi 1 + σi v v g z i=1 (50) Collecting terms of Eq. (50) and using the equilibrium condition for E2 2

λi = di x ¯i +

X αi βi x ¯i v¯ βx ¯i v¯ (1 − αi )βi x ¯i v¯ ¯i , , = (ei + bi )w + bi w ¯i = ai y¯i , c¯ v + r¯ v z¯ = ki y¯i , 1 + σi v¯ 1 + σi v¯ 1 + σi v¯ i=1

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βi x ¯i v¯ bi (1 − αi ) βi x ¯i v¯ (ei + bi )αi βi x ¯i v¯ ei + bi ai y¯i = = + ei αi + bi 1 + σi v¯ (ei αi + bi ) (1 + σi v¯) (ei αi + bi ) (1 + σi v¯) Eq. (50) becomes

" 2 X (xi − x ¯i )2 βi x ¯i v¯ dW2 σi (v − v¯)2 = γi − di − dt xi (1 + σi v¯) v¯(1 + σi v)(1 + σi v¯) i=1 µ ¶ ¯i v¯ xi w ¯i v(1 + σi v¯) yi v¯ wi y¯i 1 + σi v x ¯i bi (1 − αi ) βi x − − − 5− − + (ei αi + bi ) (1 + σi v¯) xi x ¯i wi v¯(1 + σi v) y¯i v w ¯ i yi 1 + σi v¯ µ ¶# ¯i v¯ xi y¯i v(1 + σi v¯) yi v¯ 1 + σi v (ei + bi )αi βi x x ¯i − − + 4− − (ei αi + bi ) (1 + σi v¯) xi x ¯i yi v¯(1 + σi v) y¯i v 1 + σi v¯

Thus, if R1 > 1 then xi , wi , yi , v and z > 0. Similar to the proof of Theorem 3, one can show that E2 is GAS.

4

Model with general incidence rate

In this section, we propose a viral infection model with latently infected cells and antibody immune response. The incidence rate of infection is represented by a general function of the populations of the uninfected target cells and free viruses. x˙ i = λi − di xi − fi (xi , v),

i = 1, 2,

(51)

w˙ i = (1 − αi )fi (xi , v) − (ei + bi )wi ,

i = 1, 2,

(52)

y˙ i = αi fi (xi , v) + bi wi − ai yi ,

i = 1, 2,

(53)

v˙ =

2 X

ki yi − cv − rvz,

(54)

i=1

z˙ = gvz − µz,

(55)

where the function fi (xi , v) represents the rate of the uninfected target cells to be infected by the viruses. Assumption A1 For i = 1, 2, function fi satisfies: (i) fi (xi , v) is positive, continuous, and differentiable, ∂fi (xi , v) ∂fi (xi , v) ∂fi (xi , 0) > 0 and > 0 for any xi ,v > 0. Furthermore, > 0 for any xi > 0, (ii) ∂v ∂xi ∂v (iii) fi (xi , 0) = fi (0, v) = 0, for all xi > 0 and v > 0. Assumption A2 For i = 1, 2, function fi satisfies: ∂fi (xi , 0) , for all v > 0. (i) fi (xi , v) ≤ v ∂v µ ¶ ∂f (x , 0) i i (ii) dxd i >0 ∂v

4.1

Equilibria and biological thresholds

We define the basic infection reproduction number of system (51)-(55) as: R0 =

2 X ki (ei αi + bi ) ∂fi (x0 , 0) i

i=1

ai c(ei + bi )

∂v

.

The equilibria of (51)-(55) satisfy the following equations: λi − di xi − fi (xi , v) = 0,

(56)

(1 − αi )fi (xi , v) − (ei + bi )wi = 0,

(57)

αi fi (xi , v) + bi wi − ai yi = 0,

(58)

2 X

ki yi − cv − rvz = 0,

(59)

(gv − µ)z = 0.

(60)

i=1

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Equation (60) has two possible solutions, z = 0 or v = µ/g. When z = 0, we obtain two equilibria, the infectionλi free equilibrium E0 = (x01 , x02 , 0, 0, 0, 0, 0, 0), where x0i = , i = 1, 2 and the infected steady state without di antibody immune response E1 = (˜ x1 , x ˜2 , w ˜1 , w ˜2 , y˜1 , y˜2 , v˜, 0, ), where the coordinates satisfy the equalities: λi = d i x ˜i + fi (˜ xi , v˜), (1 − αi )fi (˜ xi , v˜) = (ei + bi )w ˜i , αi fi (˜ xi , v˜) + bi w ˜i = ai y˜i ,

2 X

ki y˜i = c˜ v.

(61)

i=1

The other possibility of Eq. (60) z 6= 0 leads to v¯ =

µ . Substitute the value of v¯ in Eq. (56) and let g

Π(xi ) = λi − di xi − fi (xi , v¯) = 0. According to Assumptions A1, Π is a strictly decreasing function of xi . Besides, Π(0) = λi > 0 and Π(x0i ) = −fi (x0i , v¯) < 0. Thus, there exists a unique x ¯i ∈ (0, x0i ) such that Π(¯ xi ) = 0. From Eqs. (57)-(59) we have " 2 # (ei αi + bi )fi (¯ c X ki (ei αi + bi )fi (¯ xi , v¯) xi , v¯) (1 − αi )fi (¯ xi , v¯) , y¯i = , z¯ = −1 . w ¯i = (ei + bi ) ai (ei + bi ) r i=1 ai c(ei + bi )¯ v Thus w ¯i > 0 and y¯i > 0, moreover, z¯ > 0 when immune response activation number as: R1 =

2 k (e α + b )f (¯ P ¯) i i i i i xi , v > 1. Now we define the antibody ai c(ei + bi )¯ v i=1

2 X ki (ei αi + bi )fi (¯ xi , v¯) i=1

ai c(ei + bi )¯ v

.

Hence, z¯ can be rewritten as z¯ = rc (R1 − 1). It follows that, there exists a chronic-infection equilibrium with antibody immune response E2 = (¯ x1 , w ¯1 , y¯1 , x ¯2 , w ¯2 , y¯2 , v¯, z¯) when R1 > 1. Clearly from Assumptions A1 and A2, we have R1 =

2 X ki (ei αi + bi )fi (¯ xi , v¯) i=1

5

ai c(ei + bi )¯ v


0,

(65)

(fi (˜ xi , v¯) − fi (˜ xi , v˜)) (¯ v − v˜) > 0,

(fi (¯ xi , v¯) − fi (¯ xi , v˜)) (¯ v − v˜) > 0.

(66)

Using Assumption A3 with xi = x ˜i and v = v¯, we get (fi (˜ xi , v¯)˜ v − fi (˜ xi , v˜)¯ v ) (fi (˜ xi , v¯) − fi (˜ xi , v˜)) ≤ 0 It follows from inequality (66) that ((fi (˜ xi , v¯)˜ v − fi (˜ xi , v˜)¯ v )) (˜ v − v¯) > 0.

(67)

Suppose that, sgn (¯ xi − x ˜i ) = sgn (¯ v − v˜). Using the conditions of the equilibria E1 and E2 we have (λi − di x ¯i ) − (λi − di x ˜i ) = fi (¯ xi , v¯) − fi (˜ xi , v˜) = fi (¯ xi , v¯) − fi (¯ xi , v˜) + fi (¯ xi , v˜) − fi (˜ xi , v˜), and from inequalities (65) and (66) we get sgn (˜ xi − x ¯i ) = sgn (¯ xi − x ˜i ), which leads to contradiction. Thus, 2 P ki (ei αi +bi )fi (˜ xi ,˜ v) sgn (¯ xi − x ˜i ) = sgn (˜ v − v¯) . Using the equilibrium conditions for E1 we have = 1, then ai c(ei +bi )˜ v i=1

R1 − 1 =

µ 2 X ki (ei αi + bi ) fi (¯ xi , v¯) i=1

=

ai c(ei + bi )

2 X ki (ei αi + bi ) i=1

ai c(ei + bi )

v¯ µ

−

fi (˜ xi , v˜) v˜

¶

¶ 1 1 (fi (¯ (fi (˜ xi , v¯) − fi (˜ xi , v¯)) + xi , v¯)˜ v − fi (˜ xi , v˜)¯ v) . v¯ v˜v¯

From inequalities (65) and (67) we get sgn (R1 − 1) = sgn (˜ v − v¯). It follows that, if R1 ≤ 1 then v˜ ≤ µr = v¯. 1 Therefore, if R1 ≤ 1 then dW dt ≤ 0 for all xi , wi , yi , v, z > 0, where the equality occurs at the equilibrium E1 . LaSalle’s invariance principle implies the global stability of E1 . Theorem 9. Let Assumptions A1-A3 be hold true and R1 > 1, then chronic-infection equilibrium with antibody immune response E2 for system (51)-(55) is GAS. Proof. We construct the following Lyapunov functional W2 =

2 X

·

Zxi

γi xi − x ¯i −

i=1

x ¯i

bi fi (¯ xi , v¯) dsi + w ¯i F fi (si , v¯) ei αi + bi

µ

wi w ¯i

¶ +

ei + bi y¯i F ei αi + bi

µ

yi y¯i

¶¸ ³z ´ ³v´ r + z¯F . +¯ vF v¯ g z¯

We calculate the time derivative of W2 along the trajectories of (51)-(55) as: "µ ¶ 2 ³ X fi (¯ w ¯i ´ xi , v¯) bi dW2 = γi 1− (λi − di xi − fi (xi , v)) + 1− ((1 − αi )fi (xi , v) − (ei + bi )wi ) dt fi (xi , v¯) ei αi + bi w i=1 # µ ¶ 2 ³ y¯i z¯ ´ ei + bi v¯ ´ X r³ 1− (αi fi (xi , v) + bi wi − ai yi ) + 1 − ( ki yi − cv − rvz) + 1− (gvz − µz) . + ei αi + bi y v i=1 g z (68) Collecting terms of Eq. (68) and using the equilibrium conditions for E2 λi = d i x ¯i + fi (¯ xi , v¯),

(1 − αi )fi (¯ xi , v¯) = (ei + bi )w ¯i ,

ai y¯i = αi fi (¯ xi , v¯) + bi w ¯i , c¯ v=

2 X

γi fi (¯ xi , v¯) − r¯ v z¯,

i=1 2

cv =

vX γi fi (¯ xi , v¯) − rv¯ z, v¯ i=1

bi (1 − αi ) (ei + bi )αi ei + bi ai y¯i = fi (¯ fi (¯ fi (¯ xi , v¯) = xi , v¯) + xi , v¯), ei αi + bi (ei αi + bi ) (ei αi + bi )

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we get " µ ¶ µ ¶µ ¶ 2 X dW2 fi (¯ xi , v¯) xi fi (xi , v¯) fi (xi , v) v = γi d i x ¯i 1 − (1 − ) + fi (¯ xi , v¯) 1 − − dt fi (xi , v¯) x ¯i fi (xi , v) fi (xi , v¯) v¯ i=1 ¶ µ ¯i fi (xi , v) y¯i wi yi v¯ vfi (xi , v¯) bi (1 − αi ) fi (¯ xi , v¯) w fi (¯ − − − + xi , v¯) 5 − − (ei αi + bi ) fi (xi , v¯) wi fi (¯ ¯i y¯i v v¯fi (xi , v) xi , v¯) yi w µ ¶# fi (¯ xi , v¯) y¯i fi (xi , v) yi v¯ vfi (xi , v¯) (ei + bi )αi fi (¯ − − xi , v¯) 4 − − + (ei αi + bi ) fi (xi , v¯) yi fi (¯ xi , v¯) y¯i v v¯fi (xi , v)

(69)

Thus, if R1 > 1 then x ¯i , w ¯i , y¯i , v¯ and z¯ > 0. From Assumptions A1 and A3, we get that the first and second terms of Eq. (69) are less than or equal zero. Since the arithmetical mean is greater than or equal to the dW2 dW2 ≤ 0. It can be seen that, = 0 if and only if xi = x ¯i , wi = w ¯i and v = v¯. geometrical mean, then dt dt 2 P From Eq. (54), if v = v¯ and yi = y¯i then v˙ = 0 and 0 = k¯ yi − c¯ v − r¯ v z¯ = 0, which yields z = z¯ and hence i=1

dW2 equal to zero at E2 . LaSalle’s invariance principle implies global stability of E2 . dt

5.1

Special forms of the incidence rate

By using the Lyapunov direct method, we have established a set of conditions on fi (xi , v), i = 1, 2 ensuring the global asymptotic stability of the equilibria of model (51)-(55). Now we introduce some forms of the incidence rate and verify A1-A3. (1) Bilinear incidence rate: fi (xi , v) = βi xi v, βi xi v , (2) Saturation functional response: fi (xi , v) = 1+η iv βi xi v 1+γi xi +ηi v , βi xi v (1+γi xi )(1+ηi v) ,

(3) Beddington-DeAngelis functional response: fi (xi , v) = (4) Crowley-Martin functional response: fi (xi , v) = β xn v

i (5) Hill type incidence rate: fi (xi , v) = γ ni +x n , where βi , γi , n > 0. i i One can easily show that A1-A3 for the functions fi , i = 1, 2 given above. β xn i v Now we verify Assumptions A1-A3 for the function fi (xi , v) = γ ni +x n , i = 1, 2. We have fi (xi , v) > 0 for all i i xi > 0, v > 0, fi (0, v) = fi (xi , 0) = 0 and

∂fi (xi , v) nβi γin xn−1 v i , = ∂xi (γin + xni )2 Then, for all xi > 0, v > 0, we have A1 is satisfied. We have also

∂fi (xi ,v) ∂xi

> 0,

∂fi (xi , v) ∂fi (xi , 0) βi xn = n i n = . ∂v γi + xi ∂v ∂fi (xi ,v) ∂v

> 0 and

∂fi (xi ,0) ∂v

> 0 if n > 0. Therefore Assumptions

βi xn ∂fi (xi , 0) βi xn v , fi (xi , v) = n i n = v n i n = v γi + xi γi + xi ∂v µ ¶ ∂fi (x0i , 0)/∂v d nγ n (x0 )n = − n i 0 in n+1 < 0, dxi ∂fi (xi , 0)/∂v (γi + (xi ) )xi then, Assumptions A2 is satisfied. Moreover, ¶µ µ ¶ µ ¶ ³ v˜ fi (xi , v) v fi (xi , v˜) v v´ − − 1− = 0. 1− = fi (xi , v˜) v˜ fi (xi , v) v˜ v˜ v Thus, Assumptions A3 is satisfied. In this case, R0 and R1 are given by R0 =

2 X ki (ei αi + bi ) ∂fi (x0 , 0) i

i=1

R1 =

ai c(ei + bi )

∂v

2 X ki (ei αi + bi )fi (¯ xi , v¯) i=1

ai c(ei + bi )¯ v

=

1089

=

2 X ki (ei αi + bi ) βi (x0i )n , ai c(ei + bi ) γin + (x0i )n i=1

2 X ¯ni ki (ei αi + bi ) βi x . ai c(ei + bi ) γin + x ¯ni i=1

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Numerical simulations

In this section, we will perform some numerical simulations to confirm our theoretical results. Let us consider β xn i v model (51)-(55) with the incidence rate fi (xi , v) = γ ni +x n , i = 1, 2. In Table 1, we provide the values of some i i parameters of model (51)-(55) with the incidence rate given by the function fi . The effect of the parameter ε on the dynamical behavior of the system will be discussed below in details.In order to investigate the theoretical

P arameter V alue

λ1 6.03198

P arameter V alue

γ2 0.5

Table 1: The values of the parameters of model (51)-(55). β¯1 β¯2 λ2 d1 d2 α1 α2 e1 e2 0.03198 0.05 0.08 0.01 0.01 0.5 0.5 0.02 0.02 k1 10

k2 5

a1 0.3

a2 0.1

f 0.3

r 0.5

c 3

µ 0.07

g 0.1

b1 0.2

b2 0.2

n 1

ε Varied

γ1 0.1

results involved in Theorems 7-9, we shall study the following cases: Case (I): R0 ≤ 1. Choosing ε = 0.85 and using the data in Table 1, we have R0 = 0.899 and R1 = 0.641. Since R0 < 1, then according to Theorem 7, the infection-free equilibrium E0 is GAS. Evidently, Figures 1-8 show that, the numerical results are consistent with the theoretical results of Theorem 7. We can see that, the concentration of uninfected target cells tends to its normal value λd11 = 603.198, λd22 = 3.198, respectively, while the concentrations of latently infected cells, actively infected cells, free virus particles and antibody immune cells are decreasing and tend to zero. In this case, the treatment succeeded to eliminate the HIV viruses from the blood. Case (II): R1 ≤ 1. By taking ε = 0.40, we have R1 = 0.915 < 1 and E1 exists where E1 = (601.504, 0.780, 0.038, 0.055, 0.054, 0.231, 0.565, 0.000). Based on Theorem 8, E1 is GAS. Figures 1-8 show that the numerical simulations confirm our theoretical result presented in Theorem 8. We observe that, the trajectory of the system will converge to the chronic-infection equilibrium without antibody immune response E1 . In such situation, the infection becomes chronic but without antibody immune response. Case (III): R1 > 1.We choose, ε = 0.0. Then, we calculate R0 = 1.631 and R1 = 1.149 > 1, this means that, E2 exists and it is GAS. From Figures 1-8, we can see that, our simulation results are consistent with the theoretical results of Theorem 9. We observe that, the trajectory of the system tend to the chronic-infection equilibrium with antibody immune response E2 = (599.699, 0.474, 0.079, 0.062, 0.111, 0.260, 0.700, 0.896). In this case, the infection becomes chronic but with persistent antibody immune response. Figures 1 and 7 demonstrate that, when R1 > 1, the antibody immune response is activated and it reduces the concentration of free virus particles and increases the concentration of uninfected cells. In case (i) we calculate the critical drug efficacy (i.e, the efficacy needed in order stabilize the system around the disease-free equilibrium). For system (51)-(55), E0 is GAS when R0 ≤ 1 i.e. ½ ¯0 − 1 ¾ R crit crit ε1 ≤ ε < 1, ε1 = max 0, ¯ ¯ 02 , R01 + f R ¯ 0 = R0 |ε=0 and R ¯ 0i = R0i |ε=0 , i = 1, 2. Using the data in Table 1, we have εcrit = 0.7332. Also, in where, R 1 = 0.2566. case (ii) we can calculate the critical drug efficacy εcrit 2

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0.12

R0≤1 R1≤11

0.08

0.06

0.04

0.02 600 599.5

0

100

200

300 Time

400

500

0

600

Figure 1: The evolution of uninfected CD4+T cells for model (51)-(55).

0

100

200

300 Time

400

500

600

Figure 2: The evolution of uninfected macrophage cells for model (51)-(55).

0.14

3.5

0.12

3

Uinfected macrophage cells

Actively infected CD4 T cells

R0≤1 0.1 R0≤1 R1≤11 0.06

0.04

0.02

0

R1≤11 2

1.5

1

0.5

0

100

200

300 Time

400

500

0

600

Figure 3: The evolution of actively infected CD4+T cells for model (51)-(55). 0.1

0

100

200

300 Time

400

0.35

R ≤1 0

0

R1≤11

R >1 1

Actively infected macrophage cells

Latently infected macrophage cells

R1≤11

1

Antibody immune cells

0.7 Free virus particles

R1≤1 0. Then lim k%n+1 − %n k = 0.

n→∞

Lemma 1.4. ([1]) Assume that, (i) t2n+1 ≤ atn for all n ≥ 0 and a > 0, (ii) lim inf n→∞ tn > 0. Then the sequence {%n } generated by Algorithm 1.1 satisfies the property lim k%n − T %n k = 0.

n→∞

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Theorem 1.5. ([1]) Let H be a Hilbert space and T : H → H be a nonexpansive mapping with F ix(T ) 6= ∅. Assume that {%n } is generated by Algorithm 1.1, where the sequence {tn } of parameters satisfies the conditions: (i) t2n+1 ≤ atn for all n ≥ 0 and a > 0, (ii) lim supn→∞ tn > 0. Then {%n } converges weakly to a fixed point of T . We establish the convergence properties of the implicit midpoint iterative scheme for solving the nonlinear equation T % = % for asymptotically nonexpansive mappings in Hilbert and more general uniformly convex Banach, spaces.

2

Preliminaries

Throughout this section we always assume that H is a Hilbert space with the inner product h·, ·i and the norm k · k and that T : H → H is a nonexpansive mapping with a fixed point. We use F ix(T ) to denote the set of fixed points of T . We establish the strong convergence of a new implicit midpoint iterative scheme for nonexpansive mappings under the setting of Hilbert and more general uniformly convex Banach spaces. We need the following well known results: Lemma 2.1. ([5]) Let {σn } and {βn } be sequences of nonnegative real numbers satisfying the following inequality βn+1 ≤ (1 + σn ) βn , n ≥ 0. P∞ If n=1 σn < ∞, then limn→∞ βn exists.

Lemma 2.2. ([3]) For all %, ς ∈ H and λ ∈ [0, 1], the following well-known identity holds: k(1 − λ)% + λςk2 = (1 − λ)k%k2 + λkςk2 − λ(1 − λ)k% − ςk2 . For every ε with 0 ≤ ε ≤ 2, we define the modulus δ(ε) of convexity of E by   k% + ςk δ(ε) = inf 1 − : k%k ≤ 1, kςk ≤ 1, k% − ςk ≥ ε, %, ς ∈ E . 2 The space E is said to be uniformly convex if δ(ε) > 0 for every ε > 0. If E is uniformly convex, then for each r, ε with r ≥ ε > 0, we have δ( rε ) > 0 and

  

% + ς

≤r 1−δ ε

2 r

for every %, ς ∈ E with k%k ≤ r, kςk ≤ r and k% − ςk ≥ ε. The space E is said to be strictly convex if

% + ς

2 1, r > 0 be two fixed numbers. Then X is uniformly convex if and only if there exists a continuous, strictly increasing and convex function g : [0, ∞) → [0, ∞), g(0) = 0, such that kλ% + (1 − λ)ςkp ≤ λ k%kp + (1 − λ) kςkp − wp (λ)g (k% − ςk) for all %, ς in Br = {% ∈ X : k%k ≤ r}, λ ∈ [0, 1], where wp (λ) = λ(1 − λ)p + λp (1 − λ).

3

Main results

Algorithm 3.1. Initialize %0 ∈ H arbitrarily and define   %n−1 + %n %n−1 + %n %n = (1 − tn ) + tn T n , 2 2

n ≥ 0,

where tn ∈ (0, 1) for all n, and T is asymptotically nonexpansive, that is, kT n % − T n ςk ≤ kn k% − ςk, P {kn } ∈ [0, ∞) satisfying ∞ n=1 (kn − 1) < ∞.

%, ς ∈ H;

Remark 3.2. The Algorithm 3.1 can be rewritten as   n %n−1 + %n %n = en %n−1 + (1 − en )T , 2 where en =

n ≥ 0,

1−tn 1+tn .

Remark 3.3. The Algorithm 3.1 is well defined. Indeed, for each fixed u ∈ H and t ∈ (0, 1), the mapping   n u+% % 7→ Tu % = tu + (1 − t)T , n ≥ 0, 2 is asymptotically nonexpansive with coefficient 1−t 2 kn ∈ [0, ∞). That is,

   

n u+%

n u+ς

kTu% − Tu ςk = (1 − t) T −T

2 2 1−t ≤ kn k% − ςk, %, ς ∈ H. 2 P q Remark 3.4. Since kn ≥ 1, it is obvious that for any q > 0, ∞ n=1 (kn − 1) < ∞ implies P∞ n=1 (kn − 1) < ∞. Now we prove our main results.

Lemma 3.5. The sequence {%n } defined by the Algorithm 3.1, where {tn } ∈ (0, 1) satisfying {tn } ∈ [δ, 1 − δ] , is bounded. 4 1097

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Proof. For %∗ ∈ F ix(T ), consider

 

%n−1 + %n n %n−1 + %n ∗ ∗

+ tn T −% k%n − % k = (1 − tn ) 2 2

     

%n−1 + %n ∗ n %n−1 + %n ∗

− % + tn T = (1 − tn ) −% 2 2







%n−1 + %n ∗ n %n−1 + %n ∗

− % + tn T ≤ (1 − tn ) −% 2 2





%n−1 + %n % + % n−1 n ∗ ∗ ≤ (1 − tn ) −% + tn kn −%



2 2

%n−1 + %n

∗ − % = (1 − tn + tn kn )

2

1

1 ∗ ∗

= (1 − tn + tn kn )

2 (%n−1 − % ) + 2 (%n − % )   1 1 ∗ ∗ k%n−1 − % k + k%n − % k , ≤ (1 − tn + tn kn ) 2 2 which implies that k%n − % k ≤ ∗

Let 1

By

P∞

n=1 (kn

1

1 2 (1 − tn + tn kn ) − 12 (1 − tn + tn kn )

1 2 (1 − tn + tn kn ) − 12 (1 − tn + tn kn )

= 1+

k%n−1 − %∗ k .

tn (kn − 1) 1 2 (1 − tn + tn kn )

1− 2tn (kn − 1) = 1+ . 1 − tn (kn − 1)

− 1) < ∞, there exists n0 ∈ N such that for all n ≥ n0 , kn − 1 ≤ 1 and 1 − tn (kn − 1) ≥ δ,

which implies that 1 1 ≤ . 1 − tn (kn − 1) δ Thus

  δ k%n − % k ≤ 1 + 2 (kn − 1) k%n−1 − %∗ k . 1−δ ∗

Hence according to Lemma 2.1, the sequence {%n } is bounded. This completes the proof.

Lemma 3.6. Let {%n} be the sequence generated by Algorithm 3.1 where {tn } ∈ (0, 1) satisfying {tn } ∈ [δ, 1 − δ]. Then (i) limn→∞ k% − %nk = 0,

%n−1

n−1 +%n

(ii) limn→∞ − T n ( %n−12+%n ) = 0. 2 5 1098

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Proof. According to Lemma 2.2,

2  

%n−1 + %n ∗ 2 n %n−1 + %n ∗

k%n − % k = (1 − tn ) + tn T −% 2 2

     2 

%n−1 + %n ∗ n %n−1 + %n ∗

= (1 − tn ) − % + tn T −% 2 2

2



2 

%n−1 + %n

n %n−1 + %n

∗ ∗

= (1 − tn ) − % + tn T −% 2 2

  2

%n−1 + %n %n−1 + %n

− tn (1 − tn ) − Tn

2 2

2

2



%n−1 + %n ∗ 2 %n−1 + %n ∗

− % + tn kn −% ≤ (1 − tn ) 2 2

  2

%n−1 + %n

n %n−1 + %n

− tn (1 − tn ) −T

2 2

2

%n−1 + %n

− %∗ = (1 − tn + tn kn2 )

2

  2

%n−1 + %n n %n−1 + %n

−T − tn (1 − tn )

2 2   1 1 1 2 2 ∗ 2 ∗ 2 ≤ (1 − tn + tn kn ) k%n−1 − % k + k%n − % k − k%n−1 − %n k 2 2 4

  2

%n−1 + %n n %n−1 + %n − tn (1 − tn ) − T

,

2 2 which implies that

k%n − %∗ k2 ≤

  2

%n−1 + %n

tn (1 − tn ) n %n−1 + %n

− − T

. 2 2 1 − 12 (1 − tn + tn kn2 )

Let us assume that 1

By

2 n=1 (kn

P∞

1 2 2 (1 − tn + tn kn ) k%n−1 − %∗ k2 1 − 12 (1 − tn + tn kn2 ) 1 (1 − tn + tn kn2 ) − k%n−1 − %n k2 4 1 − 12 (1 − tn + tn kn2 )

1 2 2 (1 − tn + tn kn ) − 12 (1 − tn + tn kn2 )

= 1+

tn (kn2 − 1) 1 − 21 (1 − tn + tn kn2 )

= 1+

2tn (kn2 − 1) . 1 − tn (kn2 − 1)

− 1) < ∞, there exists n0 ∈ N such that for all n ≥ n0 , kn2 − 1 ≤ 1 and

which implies that

1 − tn (kn2 − 1) ≥ δ, 1 1 ≤ . 2 1 − tn (kn − 1) δ 6 1099

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Also 1 − tn + tn kn2 = 1 + tn (kn2 − 1) ≥ 1 and


1 1 1 − (1 − tn + tn kn2 ) = 1 − 1 + tn (kn2 − 1) 2 2
1 1 − tn (kn2 − 1) = 2 1 ≤ , 2

which yields that 1−

1 2 (1

1 ≥ 2. − tn + tn kn2 )

Thus for M > 0, 

 δ 1 1+2 (kn2 − 1) k%n−1 − %∗ k2 − k%n−1 − %n k2 1−δ 2

  2

%n−1 + %n %n−1 + %n

− 2δ 2 − Tn

2 2 δ 1 ≤ k%n−1 − %∗ k2 + 2M 2 (kn2 − 1) − k%n−1 − %n k2 1−δ 2

  2

%n−1 + %n %n−1 + %n − 2δ 2 − Tn

, 2 2

k%n − %∗ k2 ≤

which implies that

2

1 %n−1 + %n 2 2 %n−1 + %n k%n−1 − %n k + 2δ − T( )

2 2 2 δ (k2 − 1). ≤ k%n−1 − %∗ k2 − k%n − %∗ k2 + 2M 2 1−δ n Thus

  2 m m X

%j−1 + %j

1X 2 2 n %j−1 + %j

k%j−1 − %j k + 2δ − T

2 2 2 j=1 j=1  m  X ∗ 2 2 δ 2 ∗ 2 ≤ k%j−1 − % k − k%j − % k + 2M (k − 1) . 1−δ j j=1

Hence

∞ X

k%n−1 − %n k2 < +∞

j=1

and

  2 ∞ X

%n−1 + %n

n %n−1 + %n

−T

< +∞. 2 2 j=1

It implies that

lim k%n−1 − %n k = 0

n→∞

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and

 

%n−1 + %n n %n−1 + %n − T lim

= 0.

n→∞ 2 2

This completes the proof.

Lemma 3.7. Let {%n } be the sequence generated by Algorithm 3.1, where {tn } ∈ (0, 1) satisfying {tn } ∈ [δ, 1 − δ] . Then limn→∞ k%n − T %n k = 0. Proof. Consider

 

%n−1 + %n

%n−1 + %n n %n−1 + %n

k%n − T %n k ≤ %n − − T +



2 2 2





n %n−1 + %n

+ − T n %n

T

2



 

%n−1 + %n %n−1 + %n n %n−1 + %n

− T ≤ %n − +

2 2 2

%n−1 + %n

+ kn

%n −

2

 

%n−1 + %n %n−1 + %n n %n−1 + %n

−T = (1 + kn ) %n − +

2 2 2

 

%n−1 + %n 1 + kn %n−1 + %n

= k%n−1 − %n k + − Tn

2 2 2 n

→ 0 as n → ∞

and k%n − T %n k ≤ k%n − T n %n k + kT n %n − T n %n−1 k + kT n %n−1 − T %n k ≤ k%n − T n %n k + kn k%n − %n−1 k + k1 kT n−1 %n−1 − %n k ≤ k%n − T n %n k + kn k%n − %n−1 k + k1 kT n−1 %n−1 − %n−1 k + k%n−1 − %n k →0

as n → ∞.




This completes the proof. Theorem 3.8. Let T : H → H be asymptotically nonexpansive. For arbitrary %0 ∈ K, generate the sequence {%n} by the Algorithm 3.1. If T is completely continuos, then {%n } converges strongly to some fixed point of T in H. Proof. From Lemma 3.7, lim k%n − T %n k = 0. Therefore, there exists a subsequence n→∞

{%nj } of {%n } such that limj→∞ %nj − T %nj = 0. Since {%nj } is bounded and T is completely continuous, then {T %nj } has a subsequence {T %njk } which converges strongly. Hence {%njk } converges strongly.

Let limj→∞ %njk = p. Then limj→∞ T %njk = T p. Thus

we have limj→∞ %njk − T %njk = kp − T pk = 0. Hence p ∈ F (T ). From Lemma 2.1 and Lemma 3.7 it follows that limn→∞ k%n − pk = 0. This completes the proof. 8 1101

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Lemma 3.9. Let E be the uniformly convex Banach space and T : E → E be asymptotically nonexpansive mapping. Let {%n} ∈ E be the sequence generated by Algorithm 3.1 and {tn } ∈ (0, 1) satisfying {tn } ∈ [δ, 1 − δ] . Then (i) limn→∞ k%n−1 − %nk = 0,

(ii) limn→∞ %n−12+%n − T n ( %n−12+%n ) = 0.

Proof. According to Lemma 2.3,

p  

%n−1 + %n ∗ p n %n−1 + %n ∗

k%n − % k = (1 − tn ) + tn T −% 2 2

      p

%n−1 + %n ∗ n %n−1 + %n ∗ − t ) − % + t T − % = (1 n n

2 2

p

p





n %n−1 + %n

%n−1 + %n − %∗ − %∗ + tn T ≤ (1 − tn )



2 2    

%n−1 + %n n %n−1 + %n − T − wp(tn )g

2 2

p

p



%n−1 + %n

%n−1 + %n ∗ ∗ p −% −% ≤ (1 − tn ) + tn kn



2 2    

%n−1 + %n %n−1 + %n

− Tn − wp(tn )g

2 2

p

%n−1 + %n

= (1 − tn + tn knp ) − %∗

2    

%n−1 + %n n %n−1 + %n

−T − wp(tn )g

, 2 2 where

p

p

%n−1 + %n

1 1 ∗ ∗ ∗

− % (% − % ) + (% − % ) = n−1 n



2 2 2 p  1 1 ∗ ∗ k%n−1 − % k + k%n − % k ≤ 2 2 1 1 ≤ k%n−1 − %∗ kp + k%n − %∗ kp . 2 2

Thus   1 1 k%n − %∗ kp ≤ (1 − tn + tn knp ) k%n−1 − %∗ kp + k%n − %∗ kp 2 2    

%n−1 + %n

n %n−1 + %n

−T − wp(tn )g

, 2 2

which implies that

k%n − %∗ kp ≤

1

p 1 2 (1 − tn + tn kn ) p − 12 (1 − tn + tn kn )

k%n−1 − %∗ kp

   

%n−1 + %n

wp(tn ) % + % n−1 n n

. − −T p g 1

2 2 1 − 2 (1 − tn + tn kn ) 9

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Let us assume that 1

p 1 2 (1 − tn + tn kn ) − 12 (1 − tn + tn knp )

tn (knp − 1) = 1+ 1 − 21 (1 − tn + tn knp ) 2tn (knp − 1) = 1+ . 1 − tn (knp − 1)

By

p n=1 (kn

P∞

− 1) < ∞, there exists n0 ∈ N such that for all n ≥ n0 , knp − 1 ≤ 1, and 1 − tn (knp − 1) ≥ δ,

which implies that

1 1 ≤ . p 1 − tn (kn − 1) δ

Also

1 1 1 − (1 − tn + tn knp ) = 1 − (1 + tn (knp − 1)) 2 2 1 = (1 − tn (knp − 1)) 2 1 ≤ , 2

which yields that

Hence

1 ≥ 2. 1 − 21 (1 − tn + tn knp )

For M > 0,



 δ p k%n − % k ≤ 1 + 2 (k − 1) k%n−1 − %∗ kp 1−δ n    

%n−1 + %n

p+1 n %n−1 + %n

− 4δ g −T

. 2 2 ∗ p

δ k%n − %∗ kp ≤ k%n−1 − %∗ kp + 2M p (knp − 1) 1 − δ    

%n−1 + %n

n %n−1 + %n − T − 4δ p+1 g

, 2 2

which implies that

 

%n−1 + %n

% + % n−1 n n 4δ g −T ( )

2 2 δ ≤ k%n−1 − %∗ kp − k%n − %∗ kp + 2M p (kp − 1). 1−δ n p+1

Thus 4δ

p+1

≤

  m X

%n−1 + %n

n %n−1 + %n

g −T ( ) 2 2

j=1 m X

k%j−1 − %∗ kp − k%j − %∗ kp + 2M p

j=1

 δ (knp − 1) . 1−δ

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Hence

  ∞ X

%n−1 + %n

n %n−1 + %n

g −T ( ) < +∞. 2 2 j=1

It implies that

 

%n−1 + %n

n %n−1 + %n

lim −T

= 0. n→∞ 2 2

From this, it can be easily see that

lim k%n−1 − %n k = 0.

n→∞

This completes the proof. Lemma 3.10. Let E and T as in Lemma 3.9. Let {%n } be the sequence generated by Algorithm 3.1, where {tn } ∈ (0, 1) satisfying {tn } ∈ [δ, 1 − δ] . Then limn→∞ k%n − T %n k = 0. Theorem 3.11. Let E and T as in Lemma 3.9. For arbitrary %0 ∈ K, generate the sequence {%n } by the Algorithm 3.1. If T is completely continuos, then {%n} converges strongly to some fixed point of T in E.

Acknowledgment This study was supported by research funds from Dong-A University.

References [1] M. A. Alghamdi, M. A. Alghamdi, N. Shahzad and H. K. Xu, The implicit midpoint rule for nonexpansive mappings, Fixed Point Theory Appl. 96 (2014), 9 pages. [2] K. Goebel and W. A. Kirk, A fixed point theorem for asymptotically nonexpansive mappings, Proc. Amer. Math. Soc. 35 (1972), 171–174. [3] S. Ishikawa, Fixed point by a new iteration method, Proc. Amer. Math. Soc. 44 (1974), 147–150. [4] J. Schu, Iterative construction of fixed points of asymptotically nonexpansive mappings, J. Math. Anal. Appl. 158 (1991), 407–413. [5] K. K. Tan and H. K. Xu, Approximating fixed points of nonexpansive mapping by the Ishikawa iteration process, J. Math. Anal. Appl. 178 (1993), 301–308. [6] H. K. Xu, Inequalities in Banach spaces with applications, Nonlinear Anal. 16 (1991), 1127-1138. [7] H. K. Xu and R. Ori, An implicit iterative process for nonexpansive mappings, Numer. Funct. Anal. Optim. 22 (2001), 767–773.

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Hesitant fuzzy filters in lattice implication algebras G. Muhiuddin1 , Eun Hwan Roh2 , Sun Shin Ahn3,∗ and Young Bae Jun4 1

2

Department of Mathematics, University of Tabuk, Tabuk 71491, Saudi Arabia Department of Mathematics Education, Chinju National University of Education, Jinju 52673, Korea 3 Department of Mathematics Education, Dongguk University, Seoul 04620, Korea 4

Department of Mathematics Education, Gyeongsang National University, Jinju 52828, Korea

Abstract. The notion of hesitant fuzzy filters in lattice implication algebras is introduced, and several properties are investigated. Characterizations of hesitant fuzzy filters are discussed.

1. Introduction In the field of many-valued logic, lattice-valued logic plays an important role for two aspects: One is that it extends the chain-type truth-value field of some well-known presented logic [1] to some relatively general lattices. The other is that the incompletely comparable property of truth value characterized by general lattice can more efficiently reflect the uncertainty of people’s thinking, judging and decision. Hence, lattice-valued logic is becoming a research field which strongly influences the development of Algebraic Logic, Computer Science and Artificial Intelligence Technology. Therefore Goguen, Novak and Pavelka researched on this lattice-valued logic formal systems (see [2, 10, 11]). In order to research the logical system whose propositional value is given in a lattice, Xu [12] proposed the concept of lattice implication algebras, and discussed their some properties. For the general development of lattice implication algebras, filter theory and its fuzzification play an important role. Xu and Qin [14] introduced the notion of (implicative) filters in a lattice implication algebra, and investigated their properties. Jun (together with Xu and Qin) [3, 9] discussed positive implicative and associative filters of a lattice implication algebra, and Jun [4] considered the fuzzification of positive implicative and associative filters of a lattice implication algebra. In [13], Xu and Qin considered the fuzzification of (implicative) filters. Torra [16] introduced the hesitant fuzzy set which is a useful generalization of the fuzzy set that is designed for situations in which it is difficult to determine the membership of an element to a set owing to ambiguity between a few different values. The hesitant fuzzy set permits the 0

2010 Mathematics Subject Classification: 03G10; 06B10; 06D72. Keywords: hesitant fuzzy filter; hesitant level set. ∗ The corresponding author. 0 E-mail: [email protected] (G. Muhiuddin); [email protected] (E. H. Roh); [email protected] (S. S. Ahn); [email protected] (Y. B. Jun) 0

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membership degree of an element to a set to be represented by a set of possible values between 0 and 1 (see [16] and [17]). Jun et al. applied the notion of hesitant fuzzy sets to semigroups, MTL-algebras and EQ-algebras (see [5, 6, 7, 8]). In this paper, we apply the notion of hesitant fuzzy sets to the filter theory in lattice implication algebras. We introduce the concept of hesitant fuzzy filters in lattice implication algebras, and investigate several properties. We discuss characterizations of hesitant fuzzy filters. 2. Preliminaries By a lattice implication algebra we mean a bounded lattice L := (L, ∨, ∧, 0, 1) with orderreversing involution “ ′ ” and a binary operation “ → ” satisfying the following axioms: (I1) (I2) (I3) (I4) (I5) (L1) (L2)

x → (y → z) = y → (x → z), x → x = 1, x → y = y ′ → x′ , x → y = y → x = 1 ⇒ x = y, (x → y) → y = (y → x) → x, (x ∨ y) → z = (x → z) ∧ (y → z), (x ∧ y) → z = (x → z) ∨ (y → z),

for all x, y, z ∈ L. We define a relation ≤ on a lattice implication algebra L by x ≤ y if and only if x → y = 1. In a lattice implication algebra L, the following hold (see [12]): (a1) (a2) (a3) (a4) (a5) (a6) (a7)

0 → x = 1, 1 → x = x and x → 1 = 1. x → y ≤ (y → z) → (x → z). x ≤ y implies y → z ≤ x → z and z → x ≤ z → y. x′ = x → 0. x ∨ y = (x → y) → y. ((y → x) → y ′ )′ = x ∧ y = ((x → y) → x′ )′ . x ≤ (x → y) → y

where x ≤ y means x → y = 1. A subset F of a lattice implication algebra L is called a filter of L (see [14]) if it satisfies: (F1) 1 ∈ F , (F2) x ∈ F and x → y ∈ F imply y ∈ F for all x, y ∈ L. Let L be a reference set. Then we define hesitant fuzzy set on L in terms of a function H that when applied to X returns a subset of [0, 1] (see [16]).

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For a hesitant fuzzy set H on L and x, y, z ∈ L, we use the notations Hx := H(x), Hxy := H(x) ∩ H(y), Hx (ε) := H(x) ∩ ε and Hxy (ε) := H(x) ∩ H(y) ∩ ε where ε ∈ P([0, 1]). It is clear that Hxy = Hyx , Hxy (ε) ⊆ Hx (ε) and Hx = Hy ⇔ Hx ⊆ Hy , Hy ⊆ Hx for all x, y ∈ L. For a hesitant fuzzy set H on L and a subset ε of [0, 1], the set L(H; ε) := {x ∈ L | ε ⊆ Hx }, is called the hesitant level set of H. 3. Hesitant fuzzy filters In what follows, we take a lattice implication algebra L as a reference set unless otherwise specified. Definition 3.1. A hesitant fuzzy set H on L is a hesitant fuzzy filter of L if it satisfies the following assertions. (∀x ∈ L) (H1 ⊇ Hx ) , ( ) x (∀x, y ∈ L) Hy ⊇ Hx→y ) .

(3.1) (3.2)

Example 3.2. Let L = {0, a, b, c, d, 1} be a set with the following Hasse diagram and Cayley tables: x x′ → 0 a b c d 1 0 1 0 1 1 1 1 1 1 @ b r @r a c a c 1 b c b 1 @ a c r @r b d b d a 1 b a 1 @ d c a c a a 1 1 a 1 @r d b d b 1 1 b 1 1 0 1 0 1 0 a b c d 1 Then L is a lattice implication algebra (see [15]). Let H be a hesitant fuzzy set on L which is given as follows: { [0.2, 0.8] if x ∈ {a, 1}, H : L → P([0, 1]), x 7→ [0.3, 0.7] otherwise. 1r

Then H is a hesitant fuzzy filter of L. Theorem 3.3. A hesitant fuzzy set H on L is a hesitant fuzzy filter of L if and only if the hesitant level set L(H; ε) of H is a filter of L for all ε ∈ P([0, 1]) with L(H; ε) ̸= ∅.

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Proof. Assume that H is a hesitant fuzzy filter of L. Let ε ∈ P([0, 1]) be such that L(H; ε) ̸= ∅. Then there exists a ∈ L(H; ε), and so Ha ⊇ ε. It follows from (3.1) that H1 ⊇ Ha ⊇ ε and so that 1 ∈ L(H; ε). Let x, y ∈ L be such that x ∈ L(H; ε) and x → y ∈ L(H; ε). Then ε ⊆ Hx and x ε ⊆ Hx→y . Using (3.2), we get Hy ⊇ Hx→y ⊇ ε. Thus y ∈ L(H; ε), and hence L(H; ε) is a filter of L for all ε ∈ P([0, 1]) with L(H; ε) ̸= ∅. Conversely, suppose that the nonempty hesitant level set L(H; ε) of H is a filter of L for all ε ∈ P([0, 1]). For any x ∈ L, let Hx = εx . Then x ∈ L(H; εx ), and so L(H; εx ) ̸= ∅. Hence x = δ. Then 1 ∈ L(H; εx ), and thus H1 ⊇ εx = Hx for all x ∈ L. For any x, y ∈ L, let Hx→y Hx ⊇ δ and Hx→y ⊇ δ, that is, x ∈ L(H; δ) and x → y ∈ L(H; δ). It follows from (F2) that x for all x, y ∈ L. Therefore H is a hesitant fuzzy filter of y ∈ L(H; δ) and so that Hy ⊇ δ = Hx→y L. □ Proposition 3.4. Every hesitant fuzzy filter H of L satisfies: (∀x, y ∈ L) (x ≤ y ⇒ Hx ⊆ Hy ) .

(3.3)

Proof. Let x, y ∈ L satisfy x ≤ y. Then x → y = 1, and so x Hy ⊇ Hx→y = H1x = Hx

□

by (3.2) and (3.1).

Theorem 3.5. A hesitant fuzzy set H on L is a hesitant fuzzy filter of L if and only if it satisfies (3.1) and ( ) x→y (∀x, y, z ∈ L) Hx→z ⊇ Hy→z . (3.4) Proof. Assume that H is a hesitant fuzzy filter of L. Since x → y ≤ (y → z) → (x → z) for all x, y, z ∈ L, it follows from (3.3) that Hx→y ⊆ H(y→z)→(x→z) and so from (3.2) that y→z y→z Hx→z ⊇ H(y→z)→(x→z) ⊇ Hx→y

for all x, y, z ∈ L. Conversely, let H satisfy (3.1) and (3.4). Taking x = 1 in (3.4) and using (a1), we have y 1→y = Hy→z Hz = H1→z ⊇ Hy→z

for all y, z ∈ L. Therefore H is a hesitant fuzzy filter of L.

□

Theorem 3.6. For any hesitant fuzzy set H on L, the following assertions are equivalent. (1) H is a hesitant fuzzy filter of L. ) ( (2) (∀x, y, z ∈ L) x ≤ y → z ⇒ Hz ⊇ Hyx .

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Proof. Suppose that H is a hesitant fuzzy filter of L. Let x, y, z ∈ L satisfy x ≤ y → z. Using y (3.2) and (3.3) implies that Hz ⊇ Hy→z ⊇ Hxy . Assume that the second condition is valid. Since x ≤ x → 1 for all x ∈ L, we have H1 ⊇ Hxx = y Hx for all x ∈ L. Note that y ≤ (y → x) → x for all x, y ∈ L. Hence Hx ⊇ Hy→x for all x, y ∈ L. Therefore H is a hesitant fuzzy filter of L. □ Theorem 3.7. A hesitant fuzzy set H on L is a hesitant fuzzy filter of L if and only if it satisfies (3.1), (3.3) and ( ) (∀x, y ∈ L) H(x→y′ )′ ⊇ Hyx . (3.5) Proof. Assume that H is a hesitant fuzzy filter of L. Then the conditions (3.1) and (3.3) are valid by Definition 3.1 and Proposition 3.4. Using (3.1), (3.2) and (I2), we have y y ′ ′ H(x→y′ )′ ⊇ Hy→(x→y ′ )′ ⊇ Hx (x → (y → (x → y ) ))

= Hxy ((x → y ′ )′ → (x → y ′ )′ ) = Hyx (1) = Hyx for all x, y ∈ L. Hence (3.5) is valid. Conversely, let H satisfy conditions (3.1), (3.3) and (3.5). Note that (x → (x → y)′ )′ ≤ y for all x, y ∈ L. It follows from (3.3) and (3.5) that x Hy ⊇ H(x→(x→y)′ )′ ⊇ Hx→y

for all x, y ∈ L. Therefore H is a hesitant fuzzy filter of L by Theorem 3.3.

□

Theorem 3.8. A hesitant fuzzy set H on L is a hesitant fuzzy filter of L if and only if it satisfies (3.1) and ( ) y (∀x, y, z ∈ L) Hz→x ⊇ H(z→y)→x . (3.6) Proof. Suppose that H is a hesitant fuzzy filter of L. Let x, y, z ∈ L. Since x ≤ z → x and y ≤ z → y, we have (z → y) → x ≤ (z → y) → (z → x) ≤ y → (z → x). It follows from (3.2) and (3.3) that y y Hz→x ⊇ Hy→(z→x) ⊇ H(z→y)→x .

Hence (3.6) is valid.

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Conversely, let H satisfy conditions (3.1) and (3.6). If we take z = 1 in (3.6) and use (a1), then y y Hx = H1→x ⊇ H(1→y)→x = Hy→x for all x, y ∈ L. Therefore H is a hesitant fuzzy filter of L.

□

Let H be a hesitant fuzzy set on L and a ∈ L. We consider the set Ha→ := {x ∈ L | Ha ⊆ Hx } . Obviously, a ∈ Ha→ . If H is a hesitant fuzzy filter of L , then 1 ∈ Ha→ since H1 ⊇ Hx for all x ∈ L. Let H satisfy the condition (3.1). Then there exists a ∈ L such that Ha→ is not a filter of L as seen in the following example. Example 3.9. Consider the set L = {ai | i = 1, 2, · · · , n}. For any 1 ≤ j, k ≤ n, define aj ∨ ak = amax{j,k} , aj ∧ ak = amin{j,k} , (aj )′ = an−j+1 , aj → ak = amin{n−j+k,n} . Then (L, ∨, ∧,′ , →) is a lattice implication algebra which is called the Lukasiewicz implication algebra (of order n) (see [15]). The Lukasiewicz implication algebra L = {0, a, b, c, 1} of order 5 is represented by r1 x x′ → 0 a b c 1 0 1 0 1 1 1 1 1 rc a c 1 1 1 1 a c rb b b b b c 1 1 1 ra c a b c 1 1 c a 1 0 1 0 a b c 1 r0 Let H be a hesitant fuzzy set on L defined by  (0.2, 0.3) ∪ (0.6, 0.8]    [0.1, 0.3) ∪ (0.5, 0.9) H : L → P([0, 1]), x 7→ [0.2, 0.3) ∪ [0.6, 0.9)    [0.1, 0.3] ∪ [0.5, 0.9]

if if if if

x ∈ {0, c}, x = a, x = b, x = 1.

Then Hb→ = {a, b, 1} is not a filter of L since a → c = 1 ∈ Hb→ and a ∈ Hb→ , but c ∈ / Hb→ . We provide conditions for the set Ha→ to be a filter of L for a ∈ L. Theorem 3.10. Let a ∈ L. If H is a hesitant fuzzy filter of L , then Ha→ is a filter of L.

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Proof. Obviously 1 ∈ Ha→ by (3.1). Let x, y ∈ L satisfy x → y ∈ Ha→ and x ∈ Ha→ . Then Hx→y ⊇ Ha and Hx ⊇ Ha . It follows from (3.2) that x Hy ⊇ Hx→y ⊇ Ha .

Thus y ∈ Ha→ and Ha→ is a filter of L.

□

Theorem 3.11. For any a ∈ L and a hesitant fuzzy set H on L, we have the following assertions: (1) If Ha→ is a filter of L, then H satisfies the following implication. ( ) x (∀x, y ∈ L) Ha ⊆ Hx→y ⇒ Ha ⊆ H y .

(3.7)

(2) If H satisfies (3.1) and (3.7), then Ha→ is a filter of L. Proof. (1) Assume that Ha→ is a filter of L for a ∈ L. Let x, y ∈ L be such that x Ha ⊆ Hx→y .

Then x → y ∈ Ha→ and x ∈ Ha→ . Since Ha→ is a filter of L, it follows that y ∈ Ha→ , that is, H a ⊆ Hy . (2) Suppose that H satisfies (3.1) and (3.7). Let x, y ∈ L be such that x → y ∈ Ha→ and x x ∈ Ha→ . Then Ha ⊆ Hx→y and Ha ⊆ Hx , which implies that Ha ⊆ Hx→y . It follows from (3.7) → → that Ha ⊆ Hy , i.e., y ∈ Ha . Since H satisfies (3.1), we have 1 ∈ Ha . Therefore Ha→ is a filter of L. □ For a fixed element a ∈ L and a hesitant fuzzy set H on L, let [aH] be a hesitant fuzzy set on L given as follows: { ε1 if a ≤ x, [aH] : L → P([0, 1]), x 7→ ε2 otherwise where ε1 , ε2 ∈ P([0, 1]) with ε1 ⊋ ε2 . Let L = {0, a, b, c, 1} be the lattice implication algebra in Example 3.9. For b ∈ L, the hesitant fuzzy set [bH] on L which is given by { [0.2, 0.7] if b ≤ x, [bH] : L → P([0, 1]), x 7→ [0.3, 0.6] otherwise is not a hesitant fuzzy filter of L since [bH]a = [0.3, 0.6] ⊉ [0.2, 0.7] = [bH]cc→a . Given a ∈ L, we provide conditions for the hesitant fuzzy set [aH] to be a hesitant fuzzy filter of L. Theorem 3.12. Given a ∈ L, the hesitant fuzzy set [aH] is a hesitant fuzzy filter of L if and only if the following assertion is valid. (∀x, y ∈ L) (a ≤ y → x, a ≤ y ⇒ a ≤ x) .

1111

(3.8)

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Proof. Suppose that [aH] is a hesitant fuzzy filter of L and let x, y ∈ L satisfy a ≤ y → x and a ≤ y. Then [aH]y→x = ε1 = [aH]y , and so [aH]x ⊇ [aH]yy→x = ε1 . Thus a ≤ x, which satisfies the condition (3.8). Conversely, assume that the condition (3.8) is valid. Note that  if ε ⊆ ε2 ,  L L([aH]; ε) = {x ∈ L | a ≤ x} if ε2 ⊊ ε ⊆ ε1 ,  ∅ otherwise For the case of ε2 ⊊ ε ⊆ ε1 , obviously 1 ∈ L([aH]; ε). Let x, y ∈ L be such that x ∈ L([aH]; ε) and x → y ∈ L([aH]; ε). Then a ≤ x and a ≤ x → y, which imply from the hypothesis that a ≤ y, that is, y ∈ L([aH]; ε). Hence L([aH]; ε) is a filter of L whenever it is nonempty. Therefore [aH] is a hesitant fuzzy filter of L. □ Theorem 3.13. For a subset J of L, let G be a hesitant fuzzy set on L given as follows: { ε1 if x ∈ J, G : L → P([0, 1]), x 7→ ε2 otherwise where ε1 , ε2 ∈ P([0, 1]) with ε1 ⊋ ε2 . Then G is a hesitant fuzzy filter of L if and only if the following assertion is valid. (∀x, y ∈ J)(∀z ∈ L) (x, y ∈ J, y ≤ x → z ⇒ z ∈ J) . Proof. Note that

(3.9)

  L if ε ⊆ ε2 , L(G; ε) = J if ε2 ⊊ ε ⊆ ε1 ,  ∅ otherwise

Assume that G is a hesitant fuzzy filter of L. Then J = L(G; ε) for ε2 ⊊ ε ⊆ ε1 , and J is a filter of L. Let x, y, z ∈ L be such that x, y ∈ J and y ≤ x → z. Then y → (x → z) = 1 ∈ J, and so z ∈ J. Conversely, let G be a hesitant fuzzy set on L and suppose that (3.9) is valid. Since y ≤ 1 = x → 1 for all x, y ∈ L, we have 1 ∈ J by (3.9), and so 1 ∈ L(G; ε) for ε2 ⊊ ε ⊆ ε1 . Let x, y ∈ L be such that y ∈ J = L(G; ε) and y → x ∈ J = L(G; ε) for ε2 ⊊ ε ⊆ ε1 . Since y ≤ (y → x) → x, it follows from (3.9) that x ∈ J = L(G; ε). Hence L(G; ε) is a filter of L for all ε ∈ P([0, 1]) with L(G; ε) ̸= ∅. Therefore G is a hesitant fuzzy filter of L. □ References [1] L. Bolc and P. Borowik, Many-Valued Logic, Springer, Berlin, 1994. [2] J. A. Goguen, The logic of inexact concepts, Synthese 19 (1969), 325–373.

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[3] Y. B. Jun, Implicative filters of lattice implication algebras, Bull. Korean Math. Soc. 34 (1997), no. 2, 193–198. [4] Y. B. Jun, Fuzzy positive implicative and fuzzy associative filters of lattice implication algebras, Fuzzy Sets and Systems 121 (2001), 353–357. [5] Y. B. Jun, K. J. Lee and S. Z. Song, Hesitant fuzzy bi-ideals in semigroups, Commun. Korean Math. Soc. 30(3) (2015) 143–154. [6] Y. B. Jun and S. Z. Song, Hesitant fuzzy set theory applied to filters in MTL-algebras, Honam Math. J. 36(4) (2014) 813–830. [7] Y. B. Jun and S. Z. Song, Hesitant fuzzy prefilters and filters of EQ-algebras, Appl. Math. Sci. 9 (2015) 515–532. [8] Y. B. Jun, S. Z. Song and G. Muhiuddin, Hesitant fuzzy semigroups with a frontier, J. Intell. Fuzzy Systems (in press). [9] Y. B. Jun, Y. Xu and K. Y. Qin, Positive implicative and associative filters of lattice implication algebras, Bull. Korean Math. Soc. 35 (1998), no. 1, 53–61. [10] V. Novak, First order fuzzy logic, Studia Logica 46 (1982), no. 1, 87–109. [11] J. Pavelka, On fuzzy logic I, II, III, Zeit. Math. Logik u. Grundl. Math. 25 (1979), 45–52, 119–134, 447–464. [12] Y. Xu, Lattice implication algebras, J. Southwest Jiaotong Univ. 1 (1993), 20–27. [13] Y. Xu and K. Y. Qin, Fuzzy lattice implication algebras, J. of Southwest Jiaotong University 30 (1995), no. 2, 121–127. [14] Y. Xu and K. Y. Qin, On filters of lattice implication algebras, J. Fuzzy Math. 1 (1993), no. 3, 251–260. [15] Y. Xu, D. Ruan, K. Y. Qin and J. Liu, Lattice-Valued Logic, Springer-Verlag, Berlin, Heidelberg 2003. [16] V. Torra, Hesitant fuzzy sets, Int. J. Intell. Syst. 25 (2010), 529–539. [17] V. Torra and Y. Narukawa, On hesitant fuzzy sets and decision, in: The 18th IEEE International Conference on Fuzzy Systems, Jeju Island, Korea, 2009, pp. 1378–1382.

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3D Green’s Function and Its Finite Element Error Estimates Jinghong Liu∗and Yinsuo Jia† In our previous article, we introduced the definition of the 3D Green’s function, and gave some estimates for this function. In this article, we will give the finite element approximation to the 3D Green’s function. Moreover, some error estimates between 3D Green’s function and its finite element approximation are derived, which will be used to the local superconvergence analysis.

1

Introduction

Superconvergence study is still an important topic in the finite element method, and the Green’s function plays very important roles in the study of the superconvergence (especially, pointwise superconvergence) of the finite element method (see [1–9]). As for the global superconvergence, we know that the discrete Green’s function and the discrete derivative Green’s function are usually used. However, as for the local superconvergence, we need to use the Green’s function which is independent of the mesh-size h. In our recent articles, we have introduced the definition of the 3D Green’s function and its some estimates. This article will focus on the finite element approximation to the 3D Green’s function. we shall use the symbol C to denote a generic constant, which is independent of the mesh-size h and which may not be the same in each occurrence and also use the standard notations for the Sobolev spaces and their norms. In this article, we consider the following Poisson equation: Lu ≡ −∆u = f in Ω,

u = 0 on ∂Ω,

where Ω ⊂ R3 is a bounded polytopic domain. The weak formulation of the above equation reads, { Find u ∈ H01 (Ω) satisfying (1.1) a(u , v) = (f , v) for all v ∈ H01 (Ω), ∗ School of Information Science and Engineering, Ningbo Institute of Technology, Zhejiang University, Ningbo 315100, China, email: [email protected] † School of Mathematics and Computer Science, Shangrao Normal University, Shangrao 334001, China, email: [email protected]

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∫

where

∫

a(u , v) ≡

∇u · ∇v dX, (f , v) ≡ Ω

f v dX. Ω

¯ Denote by S h (Ω) a continLet {T h } be a regular family of partitions of Ω. uous finite elements space of degree m(m ≥ 1) regarding this kind of partitions and let S0h (Ω) = S h (Ω) ∩ H01 (Ω). ¯ we define the discrete δ function δ h ∈ S h (Ω), the disFor every Z ∈ Ω, 0 Z h crete derivative δ function ∂Z,ℓ δZ ∈ S0h (Ω), the regularized Green’s function G∗Z ∈ H 2 (Ω) ∩ H01 (Ω), the regularized derivative Green’s function ∂Z,ℓ G∗Z ∈ H 2 (Ω) ∩ H01 (Ω), the discrete Green’s function GhZ ∈ S0h (Ω), the discrete derivative Green’s function ∂Z,ℓ GhZ ∈ S0h (Ω), and the L2 -projection Ph u ∈ S0h (Ω) such that (see [9]) h (v, δZ ) = v(Z) ∀ v ∈ S0h (Ω), (1.2) h (v, ∂Z,ℓ δZ ) = ∂ℓ v(Z) ∀ v ∈ S0h (Ω),

(1.3)

h a(G∗Z , v) = (δZ , v) ∀ v ∈ H01 (Ω),

(1.4)

a(∂Z,ℓ G∗Z ,

v) =

h (∂Z,ℓ δZ ,

v) ∀ v ∈

H01 (Ω),

(1.5)

a(GhZ , v) = v(Z) ∀ v ∈ S0h (Ω), a(∂Z,ℓ GhZ ,

v) = ∂ℓ v(Z) ∀ v ∈

(1.6)

S0h (Ω),

(1.7)

(u − Ph u, v) = 0 ∀ v ∈ S0h (Ω).

(1.8)

Here, for any direction ℓ ∈ R , |ℓ| = 1, and ∂ℓ v(Z) stand for the following onesided directional derivatives, respectively. 3

h ∂Z,ℓ δZ =

h ∂Z,ℓ δZ ,

∂Z,ℓ GhZ ,

h h δZ+∆Z − δZ GhZ+∆Z − GhZ , ∂Z,ℓ GhZ = lim , |∆Z| |∆Z| |∆Z|→0 |∆Z|→0

lim

∂ℓ v(Z) =

v(Z + ∆Z) − v(Z) , ∆Z = |∆Z|ℓ. |∆Z| |∆Z|→0 lim

As for G∗Z , ∂Z,ℓ G∗Z , GhZ , and ∂Z,ℓ GhZ , we have obtained some optimal estimates (see [4–6]), which will be used in next section. From (1.4)–(1.7), we easily find GhZ and ∂Z,ℓ GhZ are the finite element approximations to G∗Z and ∂Z,ℓ G∗Z , respectively. For the L2 -projection operator Ph , we have (see [4]) Lemma 1.1. For Ph w the L2 -projection of w ∈ Lp (Ω), we have the following stability estimate: ∥Ph w∥0, p, Ω ≤ C t ∥w∥0, p, Ω , (1.9) where t = 1 − p2 , and 1 ≤ p ≤ ∞. Further, by Lemma 1.1, we easily obtain the following result: ∥w − Ph w∥0, p, Ω ≤ (1 + C t ) inf ∥w − v∥0, p, Ω , v∈S0h Ω

1115

(1.10)

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where 1 ≤ p ≤ ∞. Using the result (1.10), we easily obtain ∥Ph w∥1, p, Ω ≤ C∥w∥1, p, Ω , for 3 < p ≤ ∞.

(1.11)

In addition, we also assume the following a priori estimate holds. Lemma 1.2. For the true solution u of (1.1), there exists a q0 (1 < q0 ≤ ∞) such that for every 1 < q < q0 , ∥u∥2, q, Ω ≤ C(q)∥Lu∥0, q, Ω .

2

(1.12)

Regularized Green’s Function and Its Finite Element Approximation

We introduce two weight functions defined by ( )− 3 ¯ ϕ = |X − Z|2 + θ2 2 and τ = |X − Z|−3 ∀ X ∈ Ω, ¯ is a fixed point, θ = γh, and γ ∈ [3, +∞) is a suitable real number. where Z ∈ Ω They will be used in this section and next section. In [4], we derived the following Lemma 2.1 (see (2.62) and (2.63) in [4]). Lemma 2.1. Suppose q0 > 3. For G∗Z and GhZ defined by (1.4) and (1.6), respectively, we have

∗

1

GZ − GhZ −1 ≤ Ch ∇2 G∗Z −1 ≤ Ch |ln h| 6 . 1,ϕ ϕ

(2.1)

Lemma 2.2. For G∗Z and GhZ defined by (1.4) and (1.6), respectively, we have { 5 2

∗

GZ − GhZ −α ≤ C(α)h ∀ 1 < α < 3 − q0 when 3 < q0 < 6, (2.2) 4 1,ϕ ∀ 1 < α < 3 when q0 ≥ 6. Proof. Similar to the proof of the result (2.43) in [4], we have



∗

GZ − GhZ 2 −α ≤ Ch2 ∇2 G∗Z 2 −α + C G∗Z − GhZ 2 −α+ 2 . 3 1, ϕ ϕ ϕ

(2.3)

We easily obtain

= ≤ ≤ ≤ ≤

( )

∗

G − Gh 2 −α+ 2 = ϕ−α+ 23 (G∗ − Gh ), G∗ − Gh Z Z ϕ Z Z Z Z 3 ∗ h ∗ h a(v, − G ) = a(v − Πv, G − G ) G Z Z ∗ Z h Z G − G −α · |v − Πv| α Z Z 1, ϕ 1, ϕ 2 2 ε G∗Z − GhZ 1, ϕ−α + C(ε) |v − Πv|1, ϕα ∗ 2 2 ε GZ − GhZ 1, ϕ−α + C(ε)h2 ∇2 v ϕα 2 2 2 ε G∗Z − GhZ 1, ϕ−α + C(ε)h2 θ−2 ∇(ϕ−α+ 3 (G∗Z − GhZ )) α− 4 , ϕ

(2.4)

3

where Lv = ϕ−α+ 3 (G∗Z − GhZ ). 2

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2 2 2 Note that the result ∇2 v ϕα ≤ Cθ−2 ∇(ϕ−α+ 3 (G∗Z − GhZ )) α− 4 in (2.4) ϕ

3

should satisfy one of the following two conditions: (1) 1 < α < 53 − q20 when 3 < q0 < 6; (2) 1 < α < 34 when q0 ≥ 6. In addition, 2 2 ∇(ϕ−α+ 3 (G∗Z − GhZ )) α− 4 3 ϕ ∫ 2 4 2 2 = ϕα− 3 ∇ϕ−α+ 3 · (G∗Z − GhZ ) + ϕ−α+ 3 · ∇(G∗Z − GhZ ) dX Ω ∫ ( ) 4 2 2 ≤ C ϕα− 3 |∇ϕ−α+ 3 |2 |G∗Z − GhZ |2 + (ϕ−α+ 3 )2 |∇(G∗Z − GhZ )|2 dX (Ω ) 2

2 ≤ C G∗Z − GhZ 1, ϕ−α + G∗Z − GhZ ϕ−α+ 23 . Combining (2.4) and the

∗

G − Gh 2 −α+ 2 ≤ Z Z ϕ 3

above result, we have 2 ε G∗Z − GhZ 1, ϕ−α ) ( 2

2 +C(ε)h2 θ−2 G∗Z − GhZ 1, ϕ−α + G∗Z − GhZ ϕ−α+ 32 2 = ε G∗Z − GhZ 1, ϕ−α ( ) 2

2 +C(ε)γ −2 G∗ − Gh −α + G∗ − Gh −α+ 2 . Z

Z 1, ϕ

Z

Z ϕ

3

(2.5) Choosing γ ∈ [3, +∞) in (2.5) such that 0 < C(ε)γ −2 < min(ε, 21 ), we have



∗

GZ − GhZ 2 −α+ 2 ≤ 4ε G∗Z − GhZ 2 −α . (2.6) 3 1, ϕ ϕ Taking a suitable ε ∈ (0, +∞), from (2.3) and (2.6), we obtain

∗

GZ − GhZ −α ≤ Ch ∇2 G∗Z −α . 1, ϕ ϕ We can prove

2 ∗

h

∇ GZ −α ≤ C δZ

−α + C ∥G∗Z ∥ ϕ ϕ

4

ϕ−α+ 3

≤ Ch

3(α−1) 2

(2.7)

+ C ∥G∗Z ∥

. (2.8)

4

ϕ−α+ 3

Further, from (1.4), (1.8), (1.9), (1.12), and the Sobolev Embedding Theorem [10], we have ∥G∗Z ∥ =

2 4 ϕ−α+ 3

4

4

Ph w(Z) ≤ ∥Ph w∥0,∞ ≤ C ∥w∥0,∞ ≤ C ∥w∥2,p ≤ C ϕ−α+ 3 G∗Z (∫

=

= (G∗Z , ϕ−α+ 3 G∗Z ) = a(G∗Z , w)

ϕ( 3 −α)p |G∗Z |p dX 4

C

) p1

(∫ ≤C

Ω

ϕ

( 4 −α)p 3 2−p

) 2−p 2p dX

Ω

6 Here we choose p such that 32 < p < 7−3α < 2 and 0 < to prove ∫ ( 4 −α)p 3 ϕ 2−p dX ≤ C(α).

( 43 −α)p 2−p

0,p

∥G∗Z ∥

4

ϕ−α+ 3

.

< 1. It is easy

Ω

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Thus we have

∥G∗Z ∥

≤ C(α).

4

ϕ−α+ 3

(2.9)

From (2.7)–(2.9), the result (2.2) is obtained. Lemma 2.3. For ∂Z,ℓ G∗Z and ∂Z,ℓ GhZ defined by (1.5) and (1.7), respectively, we have

4−3α 3(α−1)

∂Z,ℓ G∗Z − ∂Z,ℓ GhZ −α ≤ Ch 2 |ln h| 6 , (2.10) 1,ϕ where 1 < α < 53 − q20 when 3 < q0 < 6 and 1 < α < 34 when q0 ≥ 6. Proof. Similar to the result (2.7), we have



∂Z,ℓ G∗Z − ∂Z,ℓ GhZ −α ≤ Ch ∇2 ∂Z,ℓ G∗Z −α . 1, ϕ ϕ In addition

2

∇ ∂Z,ℓ G∗ −α Z ϕ

h + C ∥∂Z,ℓ G∗Z ∥ ≤ C ∂Z,ℓ δZ ϕ−α ≤ Ch

3α−5 2

+C

∥∂Z,ℓ G∗Z ∥ −α+ 34 ϕ

(2.11)

4

ϕ−α+ 3

(2.12)

.

Further, from (1.5), (1.8), (1.11), (1.12), the inverse inequality, the Sobolev Embedding Theorem [10], and the H¨older inequality, we have ∥∂Z,ℓ G∗Z ∥

2 4 ϕ−α+ 3

= (∂Z,ℓ G∗Z , ϕ−α+ 3 ∂Z,ℓ G∗Z ) = a(∂Z,ℓ G∗Z , w) = ∂Z,ℓ Ph w(Z) 4

|Ph w|1,∞ ≤ Ch− q |Ph w|1,q ≤ Ch− q |w|1,q ≤ Ch− q ∥w∥2,s (∫ ) 1s

3 3 4 4

≤ Ch− q ϕ 3 −α ∂Z,ℓ G∗Z = Ch− q ϕ( 3 −α)s |∂Z,ℓ G∗Z |s dX 3

≤

3

0,s

≤ Ch

− q3

(∫ ϕ

( 4 −α)s 3 2−s

) 2−s 2s dX

3

Ω

∥∂Z,ℓ G∗Z ∥

4

ϕ−α+ 3

Ω

.

6 Here we choose s = 7−3α and 1q = 1s − 13 . Obviously, 3q0 (A) 32 < s < 3+q and 3 < q < q0 when 3 < q0 < 6. 0 3 (B) 2 < s < 2 and 3 < q < 6 when q0 ≥ 6. ( 43 −α)s 2−s

In the meantime, we have get

(∫ ϕ

( 4 −α)s 3 2−s

= 1. By the result (2.14) in [4], we then ) 2−s 2s

dX

≤ C |ln h|

4−3α 6

3α−5 2

4−3α 6

.

Ω

Thus we have

∥∂Z,ℓ G∗Z ∥

4

ϕ−α+ 3

≤ Ch

|ln h|

.

(2.13)

4−3α 3(α−1) From (2.11)–(2.13), ∂Z,ℓ G∗Z − ∂Z,ℓ GhZ 1, ϕ−α ≤ Ch 2 |ln h| 6 . The proof of the result (2.10) is completed. Lemma 2.4. For G∗Z and GhZ defined by (1.4) and (1.6), respectively, we have { 3−2p 1

∗

Ch p |ln h| 6 , 1 < p < 23 ,

GZ − GhZ ≤ (2.14) 2 1,p Ch |ln h| 3 , p = 1.

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Proof. When p = 1, the result can be seen in [4]. Thus we only need to prove the case of 1 < p < 32 . By the H¨older inequality, we have (∫

∗

GZ − GhZ ≤ 1,p From (2.13) in [4],

ϕ

p 2−p

) 2−p 2p dX

Ω

∫

∗

GZ − GhZ −1 . 1,ϕ

6−6p

p

ϕ 2−p dX ≤ Ch 2−p .

(2.15)

(2.16)

Ω

3−2p 1 Combining (2.1), (2.15), and (2.16) yields G∗Z − GhZ 1,p ≤ Ch p |ln h| 6 . The proof of the result (2.14) is completed.

3

Finite Element Approximation to the 3D Green’s Function

In this section, we discuss the 3D Green’s function and its finite element approximation. We call GZ Green’s function which satisfies the following Theorem 3.1. Theorem 3.1. There exists a unique GZ ∈ W01,p (Ω) (1 ≤ p < 32 ) such that ′

a(GZ , v) = v(Z) ∀ v ∈ W01,p (Ω),

1 1 + ′ = 1. p p

(3.1)

Proof. We first prove the uniqueness of GZ . Suppose there exists another Green’s function G′Z ∈ W01,p (Ω) satisfying (3.1). Set EZ = GZ − G′Z , thus ′

a(EZ , v) = 0 ∀ v ∈ W01,p (Ω).

(3.2)

′

′

′

When 1 < p < 32 , for each φ ∈ Lp (Ω), there exists a w ∈ W 2,p ∩ W01,p (Ω) ′ such that Lw = φ. Obviously, sgnEZ |EZ |p−1 ∈ Lp (Ω), thus we can find ′ ′ w ∈ W 2,p ∩ W01,p (Ω) such that Lw = v. Then we have p

∥EZ ∥0,p = (EZ , sgnEZ |EZ |p−1 ) = a(EZ , w),

(3.3)

From (3.2) and (3.3), ∥EZ ∥0,p = 0, i.e., GZ = G′Z . Similarly, when p = 1, we can also prove GZ = G′Z . Thus we have completed the proof of the uniqueness. Next, we prove the existence of GZ . We give a series of finite element spaces h hi S0 (Ω), i = 0, 1, 2, · · · satisfying S0hi (Ω) ⊂ S0 j (Ω) when i < j, where h0 ≡ h and 1 1 ∗ 4 hi−1 ≤ hi ≤ 2 hi−1 . Let GZ,i be the regularized Green’s function for the finite hi element space S0 (Ω), and GhZi the discrete Green’s function. Their definitions can be seen in Section 1. Obviously, we have a(G∗Z,i+1 − GhZi , v) = 0 ∀ v ∈ S0hi (Ω). Similar to the proof of the result (2.14), we have for 1 < p < 23

∗

GZ,i+1 − GhZi

1,p

1119

3−2p p

≤ Chi

1

|ln hi | 6 ,

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which combined with (2.14), we get 3−2p

∗

1

GZ,i+1 − G∗Z,i ≤ Ch p |ln hi | 6 . i 1,p

(3.4)

Thus ) ∞ ∞ ( ∑ ∑

∗

h

GZ,i+1 − G∗Z,i ≤ C 1,p 2i i=0 i=0 Set GZ ≡ G∗Z +

1 h 6 3−2p 1 ln ≤ Ch p |ln h| 6 . 2i

3−2p p

(3.5)

∞ ∑ (G∗Z,i+1 − G∗Z,i ). i=0

Thus we have GZ ∈

W01,p (Ω).

From (3.5),

∥GZ − G∗Z ∥1,p ≤ Ch

3−2p p

1

|ln h| 6 .

(3.6)

Similarly, when p = 1, we have 2

∥GZ − G∗Z ∥1,1 ≤ Ch |ln h| 3 .

(3.7)

Therefore, for 1 ≤ p < 32 , we have G∗Z,i −→ GZ in W 1,p (Ω) when i → ∞. Using (1.10) and the interpolation error estimate, we obtain ∥v − Ph v∥0, ∞, Ω ≤ C∥v − Πv∥0, ∞, Ω ≤ Ch

1− p3′

∥v∥1,p′ , Ω ,

(3.8)

′

where 3 < p′ ≤ ∞. Thus, for every v ∈ W01,p (Ω), we have by (3.6)–(3.8) a(GZ , v) = lim a(G∗Z,i , v) = lim Phi v(Z) = v(Z). i→∞

i→∞

The proof of Theorem 3.1 is completed. Now we show GZ is independent of h. ˜ In addition, ˜ Z for the mesh-size h. Suppose there exists a Green’s function G 1˜ 1˜ p′ ˜ ˜ ˜ 4 hi−1 ≤ hi ≤ 2 hi−1 and h0 = h. Thus, for every f ∈ L (Ω), we choose v ∈ ′

′

W 2,p (Ω) ∩ W01,p (Ω) such that Lv = f . Then we get (GZ , f ) = a(GZ , v) = v(Z) ˜ Z , f ) = a(G ˜ Z , v) = v(Z). Thus, (GZ , f ) = (G ˜ Z , f ), i.e., (GZ − G ˜Z , f ) = and (G ˜ 0. So we get GZ = GZ . Namely, GZ is independent of h. In addition, we find ′

a(GZ , v) = v(Z) ∀ v ∈ S0h (Ω) ⊂ W 1,p (Ω).

(3.9)

Combining (1.6) and (3.9), we have a(GZ − GhZ , v) = 0 ∀ v ∈ S0h (Ω). Thus GhZ is the finite element approximation to GZ . Further, we have the following error estimates. Theorem 3.2. For GZ and GhZ defined by (3.1) and (1.6), respectively, we have { 3−2p 1

Ch p |ln h| 6 , 1 < p < 23 ,

GZ − GhZ ≤ (3.10) 2 1,p Ch |ln h| 3 , p = 1,

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where C is independent of h and Z. Proof. From (2.14), (3.6), (3.7), and the triangular inequality, we immediately obtain the result (3.10). Theorem 3.3. Suppose q0 = ∞, for GZ and GhZ defined by (3.1) and (1.6), respectively, we have

5

GZ − GhZ ≤ Ch2 |ln h| 3 , (3.11) 0,1 where C is independent of h and Z. Proof. For every φ ∈ L∞ (Ω), there exists a unique v ∈ W 2,∞ (Ω) ∩ H01 (Ω) such that Lv = φ and (GZ − GhZ , φ) = a(GZ − GhZ , v) = a(GZ , v − vh ) = v(Z) − vh (Z),

(3.12)

where vh is the finite element approximation to v. From (1.10), 3

|v(Z) − Ph v(Z)| ≤ ∥v − Ph v∥0,∞ ≤ C ∥v − Πv∥0,∞ ≤ Ch2− q ∥v∥2,q ,

(3.13)

where 1 < q < q0 . In addition, by (2.14), the H¨older inequality, and the interpolation error estimate, we have |Ph v(Z) − vh (Z)| = |a(v − vh , G∗Z )| = |a(v − vh , G∗Z − GhZ )| = |a(v − Πv, G∗Z − GhZ )| ≤ C G∗Z − GhZ 1,1 ∥v − Πv∥1,∞ ≤ Ch

2− q3

(3.14)

2 3

|ln h| ∥v∥2,q .

From (3.12)–(3.14), and the triangular inequality, 3

2

|(GZ − GhZ , φ)| = |v(Z) − vh (Z)| ≤ Ch2− q |ln h| 3 ∥v∥2,q . From (1.12), 3

2

|(GZ − GhZ , φ)| ≤ C(q)h2− q |ln h| 3 ∥φ∥0,q .

(3.15)

Because of q0 = ∞, we can take q = | ln h| < q0 in (3.15), and we have C(q) ≤ Cq. Thus, 5 |(GZ − GhZ , φ)| ≤ Ch2 |ln h| 3 ∥φ∥0,∞ . (3.16) From (3.16), we know the result (3.11) holds. So, the proof of the result (3.11) is completed. Theorem 3.4. For GZ and GhZ defined by (3.1) and (1.6), respectively, we have

1

GZ − GhZ −1 ≤ Ch |ln h| 6 , 1,τ

GZ − GhZ −α ≤ C(α)h 1,τ

{

∀1 < α < ∀1 < α
0. Thus from (2.1) and (2.2),

∗

1

GZ − GhZ −1 ≤ Ch |ln h| 6 , 1,τ

1121

(3.18)

(3.19)

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LIU, JIA: ERROR ESTIMATES FOR THE 3D GREEN’S FUNCTION

∗

GZ − GhZ −α ≤ C(α)h 1,τ

{

∀1 < α < ∀1 < α
0 with b > a > 0 are defined by Z x Z b 1 1 α−1 α α (x − t) f (t) d t and Jb− f (x) = (t − x)α−1 f (t) d t Ja+ f (x) = Γ(α) a Γ(α) x for x ∈ R(a, b) respectively, where Γ is the classical Euler gamma function defined for Re(z) > 0 by ∞ Γ(z) = 0 e−u uz−1 d u. Moreover, define Jb0− f (x) = Ja0+ f (x) = f (x). In the case α = 1, the fractional integral reduces to the classical and usual integral. Very recently, Hermite–Hadamard’s inequality was extended in [9] to the case of Riemann– Liouville fractional integrals. Theorem 1.6 ([9, Theorem 2]). Let f : [a, b] → R be a positive function with 0 ≤ a < b and x ∈ [a, b]. If f is a convex function on [a, b], then    f (a) + f (b) Γ(α + 1)  α a+b ≤ , α > 0. J + f (b) + Jbα− f (a) ≤ f 2 2(b − a)α a 2 Theorem 1.7 ([9, Theorem 3]). Let f : [a, b] → R be a differentiable mapping on (a, b) and a < b. If |f 0 | is convex on [a, b], then   f (a) + f (b)   Γ(α + 1)  α 1  0 α ≤ b−a − 1 − J f (b) + J f (a) |f (a)|+|f 0 (b)| , α > 0. + − a b α α 2 2(b − a) 2(α + 1) 2 Theorem 1.8 ([10, Theorem 7]). Let f : [a, b] ⊆ R0 → R be a differentiable mapping on (a, b) with a < b such that f 0 ∈ L1 ([a, b]). If |f 0 | is s-convex on [a, b] for some fixed s ∈ (0, 1] and |f 0 (x)|≤ M , then (x − a)α + (b − x)α  Γ(α + 1)  α α f (x) − J + f (b) + Jx− f (a) b−a (b − a)α x   Γ(α + 1)Γ(s + 1) (x − a)α+1 + (b − x)α+1 M , α > 0, x ∈ [a, b]. 1+ ≤ b−a Γ(α + s + 1) α+s+1 For recent development on fractional calculus, one can see the monographs [9, 10, 11] and the references therein. Motivated by the above results, we establish a Riemann–Liouville fractional integral identity involving a n-times differentiable mapping and give some new Hermite–Hadamard type inequalities involving Riemann–Liouville fractional integrals for s-convex functions.

2

A lemma

In order to obtain our main results, we need the following lemma. Lemma 2.1. For n ∈ N and a < b, let f : [a, b] ⊆ R0 → R be an n-times differentiable mapping on (a, b) and α > 0. If f (n) ∈ L1 ([a, b]), then

3

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X Γ(α + n)(b − a)k   n−1  Γ(α + n)  α α J f (k) (a) + (−1)k f (k) (b) + f (b) + Jb− f (a) = a 2(b − a)α 2Γ(α + k + 1) k=0 Z  (b − a)n 1  − (−1)n−1 (1 − t)α+n−1 − tα+n−1 f (n) (ta + (1 − t)b) d t. 2 0 Proof. When n = 1, by integrating by part in the right-hand side of (2.1), we have b−a 2

Z

1

 (1 − t)α − tα f 0 (ta + (1 − t)b) d t

0

f (a) + f (b) α = − 2 2 where Z 1 α (1 − t)α−1 f (ta + (1 − t)b) d t = 0

α b−a

1

Z

 (1 − t)α−1 + tα−1 f (ta + (1 − t)b) d t, (2.1)

0

Z b a

x−a b−a

α−1 f (x) d x =

Γ(α + 1) α J − f (a) (b − a)α b

(2.2)

and Z

1

α

tα−1 f (ta + (1 − t)b) d t =

0

α b−a

Z b a

b−x b−a

α−1 f (x) d x =

Γ(α + 1) α J + f (b). (2.3) (b − a)α a

Substituting (2.2) and (2.3) into (2.1) yields the identity (2.1) for n = 1. When n = m − 1 and m ≥ 2, suppose that the identity (2.1) is valid. When n = m, by the hypothesis, we have Z  (b − a)m 1  (−1)m−1 (1 − t)α+m−1 − tα+m−1 f (m) (ta + (1 − t)b) d t 2 0   (b − a)m−1  (m−1) = f (a) + (−1)m−1 f (m−1) (b) 2  Z 1   + (α + m − 1) (−1)m−2 (1 − t)α+m−2 − tα+m−2 f (m−1) (ta + (1 − t)b) d t 0

 (b − a)m−1  (m−1) = f (a) + (−1)m−1 f (m−1) (b) 2 Z  (α + m − 1)(b − a)m−1 1  + (−1)m−2 (1 − t)α+m−2 − tα+m−2 f (m−1) (ta + (1 − t)b) d t 2 0  (b − a)m−1  (m−1) = f (a) + (−1)m−1 f (m−1) (b) 2 m−2 X (α + m − 1)Γ(α + m − 1)(b − a)k   + f (k) (a) + (−1)k f (k) (b) 2Γ(α + k + 1) k=0

 (α + m − 1)Γ(α + m − 1)  α − Ja+ f (b) + Jbα− f (a) α 2(b − a) =

m−1 X k=0

 Γ(α + m)  α  Γ(α + m)(b − a)k  (k) f (a) + (−1)k f (k) (b) − Ja+ f (b) + Jbα− f (a) . α 2Γ(α + k + 1) 2(b − a) 4

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Therefore, when n = m, the identity (2.1) holds. By induction, the proof of Lemma 2.1 is complete. Remark 2.1. When n = 1 in (2.1), we obtain the identity  b−a f (a) + f (b) Γ(α + 1)  α − Ja+ f (b) + Jbα− f (a) = α 2 2(b − a) 2

Z

1

 (1 − t)α − tα f 0 (ta + (1 − t)b) d t,

0

which is the identity established in [9].

3

Hermite–Hadamard type inequalities involving Riemann– Liouville fractional integrals

Now we start out to establish some new Hermite–Hadamard type inequalities involving Riemann– Liouville fractional integrals for s-convex functions. Theorem 3.1. For n ∈ N and a, b ∈ R0 with a < b, let f : R0 → R be an n-times differentiable function on R0 such that f (n) ∈ L1 ([a, b]). If |f (n) |q is s-convex on [a, b] for q ≥ 1 and some fixed s ∈ (0, 1], then n−1 Γ(α + n)  X Γ(α + n)(b − a)k    α α (k) k (k) J f (a) + (−1) f (b) + f (b) + Jb− f (a) − a 2(b − a)α 2Γ(α + k + 1) k=0   (n) q (n) q 1/q (b − a)n 1 f (a) + f (b) B(s + 1, α + n) ≤ α+n+s 2(α + n)1−1/q    (n) q (n) q 1/q 1 f (a) +B(s + 1, α + n) f (b) , + α+n+s where α > 0 and B is the classical Beta function which may be defined for Re(x) > 0 and Re(y) > 0 R1 by B(x, y) = 0 tx−1 (1 − t)y−1 d t. Proof. By Lemma 2.1, s-convexity of |f (n) |q , and H¨older’s inequality, we obtain n−1 Γ(α + n)  X Γ(α + n)(b − a)k    α α (k) k (k) J f (a) + (−1) f (b) + f (b) + Jb− f (a) − a 2(b − a)α 2Γ(α + k + 1) k=0 Z  Z 1 1 (b − a)n α+n−1 (n) α+n−1 (n) ≤ (1 − t) f (ta + (1 − t)b) d t + t f (ta + (1 − t)b) d t 2 0 0 Z 1 1−1/q Z 1 1/q q (b − a)n (1 − t)α+n−1 d t (1 − t)α+n−1 f (n) (ta + (1 − t)b) d t ≤ 2 0 0 Z 1 1−1/q Z 1 1/q  q + tα+n−1 d t tα+n−1 f (n) (ta + (1 − t)b) d t 0

0 n

(b − a) ≤ 2(α + n)1−1/q

Z 1 

α+n−1 s (n)

(1 − t)



t f

q q  (a) +(1 − t)α+n+s−1 f (n) (b) d t

1/q

0

5

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Z

1 α+n+s−1 (n)

t

+



f

q q
(a) +tα+n−1 (1 − t)s f (n) (b) d t

1/q 

0

=

  (n) q (n) q 1/q (b − a)n 1 f (a) + f (b) B(s + 1, α + n) α+n+s 2(α + n)1−1/q    (n) q (n) q 1/q 1 + f (a) +B(s + 1, α + n) f (b) . α+n+s

Theorem 3.1 is proved. Corollary 3.1.1. Under the assumptions of Theorem 3.1, 1. when s = 1, we have n−1 Γ(α + n)  X Γ(α + n)(b − a)k    α α (k) k (k) J f (a) + (−1) f (b) + f (b) + Jb− f (a) − a 2(b − a)α 2Γ(α + k + 1) k=0  h i1/q (b − a)n f (n) (a) q + (α + n) f (n) (b) q ≤ 2(α + n)(α + n + 1)1/q  h (n) q (n) q i1/q ; + (α + n) f (a) + f (b) 2. when n = 1, we have ( f (a) + f (b) q  Γ(α + 1)  α b−a α B(s + 1, α + 1) f 0 (a) − Ja+ f (b) + Jb− f (a) ≤ α 1−1/q 2(α + 1) 2 2(b − a) 1/q   ) 0 q 1/q 0 q 0 q 1 1 ; + f (b) f (a) + B(s + 1, α + 1) f (b) + α+s+1 α+s+1 3. when q = 1, we have Γ(α + n)  X Γ(α + n)(b − a)k   n−1  α (k) k (k) α f (a) + (−1) f (b) J + f (b) + Jb− f (a) − 2(b − a)α a 2Γ(α + k + 1) k=0   h i (b − a)n 1 f (n) (a) + f (n) (b) ; B(s + 1, α + n) + ≤ 2 α+n+s 4. when s = n = q = 1, we have f (a) + f (b)  Γ(α + 1)  α b − a h 0 0 i α − J f (a) + f (b) . + f (b) + Jb− f (a) ≤ a 2 2(b − a)α 2(α + 1) Theorem 3.2. For n ∈ N and a, b ∈ R0 with a < b, let f : R0 → R be an n-times differentiable function on R0 such that f (n) ∈ L1 ([a, b]). If |f (n) |q is s-convex on [a, b] for q > 1 and some fixed 6

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s ∈ (0, 1], then Γ(α + n)  X Γ(α + n)(b − a)k   n−1  α α (k) k (k) J + f (b) + Jb− f (a) − f (a) + (−1) f (b) 2(b − a)α a 2Γ(α + k + 1) k=0  1−1/q   (n) q 1/q q (b − a)n q−1 1 f (b) ≤ B(s + 1, r + 1) f (n) (a) + 2 q(α + n) − r − 1 r+s+1    (n) q 1/q (n) q 1 + f (a) + B(s + 1, r + 1) f (b) r+s+1 for α > 0 and 0 ≤ r ≤ q(α + n − 1). Proof. From Lemma 2.1, s-convexity of |f (n) |q , and the H¨older’s inequality, it follows that Γ(α + n)  X Γ(α + n)(b − a)k   n−1  α α (k) k (k) J + f (b) + Jb− f (a) − f (a) + (−1) f (b) 2(b − a)α a 2Γ(α + k + 1) k=0  Z Z 1 1 (b − a)n ≤ tα+n−1 f (n) (ta + (1 − t)b) d t (1 − t)α+n−1 f (n) (ta + (1 − t)b) d t + 2 0 0 Z 1 1−1/q Z 1 1/q q (b − a)n ≤ (1 − t)[q(α+n−1)−r]/(q−1) d t (1 − t)r f (n) (ta + (1 − t)b) d t 2 0 0 Z 1 1−1/q Z 1 1/q  q [q(α+n−1)−r]/(q−1) r (n) + t dt t f (ta + (1 − t)b) d t 0

0 n

≤



1−1/q Z 1 

q q  q−1 (b − a) (1 − t)r ts f (n) (a) + (1 − t)r+s f (n) (b) d t 2 q(α + n) − r − 1 0 Z 1  1/q  q  q + tr+s f (n) (a) + tr (1 − t)s f (n) (b) d t

1/q

0

 1−1/q   (n) q 1/q (n) q q−1 1 (b − a)n = f (b) B(s + 1, r + 1) f (a) + 2 q(α + n) − r − 1 r+s+1    (n) q 1/q q 1 f (b) . + B(1, r + s + 1) f (n) (a) + r+s+1 Theorem 3.2 is proved. Corollary 3.2.1. Under the assumptions of Theorem 3.2, 1. if s = 1, then Γ(α + n)  X Γ(α + n)(b − a)k   n−1  α α (k) k (k) J + f (b) + Jb− f (a) − f (a) + (−1) f (b) 2(b − a)α a 2Γ(α + k + 1) k=0   1−1/q (b − a)n q−1 ≤
1/q q(α + n) − r − 1 2 (r + 1)(r + 2) h  (n) q (n) q i1/q h (n) q (n) q i1/q × f (a) + (r + 1) f (b) + (r + 1) f (a) + f (b) ; 7

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2. if n = 1, then  1−1/q f (a) + f (b)  b − a Γ(α + 1)  α q−1 α J + f (b) + Jb− f (a) ≤ − 2 2(b − a)α a 2 q(α + 1) − r − 1 (  0 q 1/q q 1 f (b) × B(s + 1, r + 1) f 0 (a) + r+s+1   ) 0 q 0 q 1/q 1 + f (a) + B(s + 1, r + 1) f (b) ; r+s+1 3. is s = n = 1, then f (a) + f (b)  b−a Γ(α + 1)  α α − J + f (b) + Jb− f (a) ≤  1/q 2 2(b − a)α a 2 (r + 1)(r + 2)  1−1/q h  0 q i1/q h q q i1/q q−1 f (a) + (r + 1) f 0 (b) q × + (r + 1) f 0 (a) + f 0 (b) . q(α + 1) − r − 1 Corollary 3.2.2. Under the assumptions of Theorem 3.2, 1. when r = 0, we have n−1 Γ(α + n)  X Γ(α + n)(b − a)k    α α (k) k (k) J f (a) + (−1) f (b) + f (b) + Jb− f (a) − a 2(b − a)α 2Γ(α + k + 1) k=0  1−1/q h (n) q (n) q i1/q q−1 (b − a)n f (a) + f (b) ≤ ; (s + 1)1/q q(α + n) − 1 2. when r = 0 and s = n = 1, we have f (a) + f (b)  Γ(α + 1)  α α − J + f (b) + Jb− f (a) 2 2(b − a)α a  1−1/q " 0 q 0 q #1/q f (a) + f (b) q−1 ≤ (b − a) ; q(α + 1) − 1 2 3. when r = q, we have Γ(α + n)  X Γ(α + n)(b − a)k   n−1  α α (k) k (k) J + f (b) + Jb− f (a) − f (a) + (−1) f (b) 2(b − a)α a 2Γ(α + k + 1) k=0 (    1−1/q (n) q (n) q 1/q (b − a)n q−1 1 ≤ B(s + 1, q + 1) f (a) + f (b) 2 q(α + n − 1) − 1 q+s+1   ) (n) q (n) q 1/q 1 + f (a) + B(s + 1, q + 1) f (b) ; q+s+1 8

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4. when r = q and s = n = 1, we have  1−1/q f (a) + f (b)  b−a q−1 Γ(α + 1)  α α − J + f (b) + Jb− f (a) ≤  1/q qα − 1 2 2(b − a)α a 2 (q + 1)(q + 2) h  0 q 0 q i1/q h 0 q 0 q i1/q × f (a) + (q + 1) f (b) + (q + 1) f (a) + f (b) ; 5. when r = q(α + n − 1), we have Γ(α + n)  X Γ(α + n)(b − a)k   n−1  α α (k) k (k) J + f (b) + Jb− f (a) − f (a) + (−1) f (b) 2(b − a)α a 2Γ(α + k + 1) k=0 (  (n) q 1/q (n) q (b − a)n 1 ≤ f (b) B(s + 1, q(α + n − 1) + 1) f (a) + 2 q(α + n − 1) + s + 1   ) (n) q (n) q 1/q 1 f (a) + B(s + 1, q(α + n − 1) + 1) f (b) + ; q(α + n − 1) + s + 1 6. when r = q(α + n − 1) and s = n = 1, we have f (a) + f (b)  Γ(α + 1)  α b−a α − J + f (b) + Jb− f (a) ≤  1/q 2 2(b − a)α a 2 (qα + 1)(qα + 2) h  0 q 0 q i1/q h 0 q 0 q i1/q f (a) + (qα + 1) f (b) × + (qα + 1) f (a) + f (b) . Theorem 3.3. For n ∈ N and a, b ∈ R0 with a < b, let f : R0 → R be an n-times differentiable function on R0 such that f (n) ∈ L1 ([a, b]). If |f (n) |q is s-concave on [a, b] for q > 1 and some fixed s ∈ (0, 1], then Γ(α + n)  X Γ(α + n)(b − a)k   n−1  α (k) k (k) α f (a) + (−1) f (b) J + f (b) + Jb− f (a) − 2(b − a)α a 2Γ(α + k + 1) k=0     1−1/q (n) a + b (b − a)n q−1 , ≤ (1−s)/q f q(α + n) − 1 2 2

α > 0.

Proof. Using Lemma 2.1 and the well-known H¨older’s inequality yields Γ(α + n)  X Γ(α + n)(b − a)k   n−1  α α (k) k (k) J + f (b) + Jb− f (a) − f (a) + (−1) f (b) 2(b − a)α a 2Γ(α + k + 1) k=0 Z 1  Z 1 n (b − a) ≤ (1 − t)α+n−1 f (n) (ta + (1 − t)b) d t + tα+n−1 f (n) (ta + (1 − t)b) d t 2 0 0 Z 1 1−1/q Z 1 1/q n (n) q (b − a) q(α+n−1)/(q−1) (1 − t) dt f (ta + (1 − t)b) d t ≤ 2 0 0 9

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Z

q(α+n−1)/(q−1)

1/q  1−1/q Z 1 (n) q f (ta + (1 − t)b) d t dt



1−1/q Z

1

t

+

0

0

= (b − a)n

q−1 q(α + n) − 1

1

f (n) (ta + (1 − t)b) q d t

1/q .

0

Since |f (n) |q is s-concave, we have   q Z 1 (n) f (ta + (1 − t)b) q d t ≤ 2s−1 f (n) a + b . 2 0 Combining the above two inequalities yields (3.3). The proof of Theorem 3.3 is complete. Corollary 3.3.1. Under the assumptions of Theorem 3.3, 1. if s = 1, then Γ(α + n)  X Γ(α + n)(b − a)k   n−1  α α (k) k (k) J + f (b) + Jb− f (a) − f (a) + (−1) f (b) 2(b − a)α a 2Γ(α + k + 1) k=0   1−1/q  (n) a + b q−1 n ; ≤ (b − a) f q(α + n) − 1 2 2. if n = 1, then  1−1/q   f (a) + f (b) 0 a+b  Γ(α + 1)  α b−a q−1 α ; − J + f (b) + Jb− f (a) ≤ (1−s)/q f 2 2(b − a)α a q(α + 1) − 1 2 2 3. if s = n = 1, then  1−1/q   f (a) + f (b) 0 a+b  Γ(α + 1)  α q−1 α . − J + f (b) + Jb− f (a) ≤ (b − a) f 2 2(b − a)α a q(α + 1) − 1 2

Acknowledgements This work was partially supported by the National Natural Science Foundation of China under Grant No. 11361038 and by the Inner Mongolia Autonomous Region Natural Science Foundation Project under Grant No. 2015MS0123, China.

References [1] M. Alomari, M. Darus, S. S. Dragomir, and P. Cerone, Ostrowski type inequalities for functions whose derivatives are s-convex in the second sense, Appl. Math. Lett., 23 (2010), no. 9, 1071–1076; Available online at http://dx.doi.org/10.1016/j.aml.2010.04.038. [2] S. S. Dragomir and R. P. Agarwal, Two inequalities for differentiable mappings and applications to special means of real numbers and to trapezoidal formula, Appl. Math. Lett., 11 (1998), no. 5, 91–95; Available online at http://dx.doi.org/10.1016/S0893-9659(98)00086-X. 10

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[3] S. S. Dragomir and S. Fitzpatrik, The Hadamard’s inequality for s-convex functions in the second sense, Demonstratio Math. 32 (1999), no. 4, 687–696. [4] H. Hudzik and L. Maligranda, Some remarks on s-convex functions, Aequationes Math., 48 (1994), no. 1, 100–111; Available online at http://dx.doi.org/10.1007/BF01837981. [5] U. S. Kirmaci, Inequalities for differentiable mappings and applications to special means of real numbers to midpoint formula, Appl. Math. Comput., 147 (2004), no. 1, 137–146; Available online at http://dx.doi.org/10.1016/S0096-3003(02)00657-4. [6] D. S. Mitrinovi´c and I. B. Lackovi´c, Hermite and convexity, Aequationes Math., 28 (1985), no. 1, 229–232; Available online at http://dx.doi.org/10.1007/BF02189414. [7] C. E. M. Pearce and J. Peˇcariˇc, Inequalities for differentiable mappings with application to special means and quadrature formulae, Appl. Math. Lett., 13 (2000), no. 2, 51–55; Available online at http://dx.doi.org/10.1016/S0893-9659(99)00164-0. ¨ [8] M. Z. Sarikaya, E. Set, and M. E. Ozdemir, On new inequalities of Simpson’s type for sconvex functions, Comput. Math. Appl., 60(2010), no. 8, 2191–2199; Available online at http: //dx.doi.org/10.1016/j.camwa.2010.07.033. [9] M. Z. Sarikaya, E. Set, H. Yaldiz, and N. Ba¸sak, Hermite–Hadamard’s inequalities for fractional integrals and related fractional inequalities, Math. Comput. Modelling, 57 (2013), no. 9-10, 2403–2407; Available online at http://dx.doi.org/10.1016/j.mcm.2011.12.048. [10] E. Set, New inequalities of Ostrowski type for mappings whose derivatives are s-convex in the second sense via fractional integrals, Comput. Math. Appl., 63 (2012), no. 7, 1147–1154; Available online at http://dx.doi.org/10.1016/j.camwa.2011.12.023. [11] J. Wang, X. Li, M. Feˇckan, and Y. Zhou, Hermite–Hadamard-type inequalities for Riemann– Liouville fractional integrals via two kinds of convexity, Appl. Anal., 92 (2013), no. 11, 2241–2253; Available online at http://dx.doi.org/10.1080/00036811.2012.727986.

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A Monotone Hybrid Projection Algorithm for Solving Fixed Point and Equilibrium Problems in a Banach Space Xiaoying Gong1 , Sun Young Cho2,∗

1

Department of Mathematics and Sciences, Shijiazhuang University of Economics, Shijiazhuang, China 2 Department of Mathematics, Gyeongsang National University, Jinju, Korea Abstract. In this paper, an uncountable infinite family of nonlinear mappings are investigated. Strong convergence theorems of common solutions are established in a strictly convex and uniformly smooth Banach space which also has the Kadec-Klee property. The results obtained in this paper unify and improve many corresponding results announced recently. Keywords: quasi-φ-nonexpansive mapping; equilibrium problem; fixed point; projection. 2010 AMS Subject Classification: 65J15, 90C33.

1

Introduction

Recently, common solution problems have been intensively investigated based on iterative methods. The so called common solution problems which capture lots of applications in multi-disciplines such as image restoration, and radiation therapy treatment planning are to find a special point in the intersection of a family of convex sets, which are usually considered as solution sets of nonlinear problems; see [1]-[15] and the references therein. Mean-valued iterative processes, in particular, Mann iterative process and Ishikawa iterative process, are efficient and powerful for studying fixed points of Lipschitz continuous nonlinear operators. However, in the framework of infinite-dimensional Hilbert spaces, ∗ Corresponding

author.

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they are only weakly convergent; see [16], [17] and the references therein. In many modern disciplines, including image recovery, economics, control theory, and quantum physics, problems arises in the framework of infinite dimension spaces. In such nonlinear problems, strong convergence is often much more desirable than the weak convergence; see [18] and the references therein. To guarantee the strong convergence of mean-valued iteration processes, many authors use different regularization methods. The projection method which was first introduced by Haugazeau [19] has been considered for the approximation of fixed points of nonexpansive mappings. The advantage of projection methods is that strong convergence of iterative sequences can be guaranteed without compact restrictions imposed on operators. In this paper, we study a common solution problem via projection methods. Strong convergence theorems of common solutions are established with the aid of a generalized projection in a Banach space. The results obtained in this paper mainly unify and improve the corresponding results in [20]-[30].

2

Preliminaries

Let E be a real Banach space and let E ∗ be the dual space of E. Let BE be the unit sphere of E. Recall that E is said to be a strictly convex space if for all x, y ∈ BE and x 6= y, kx + yk < 2. It is said to be uniformly convex if for any  ∈ (0, 2] there exists δ > 0 such that for any x, y ∈ BE , kx − yk ≥  implies kx + yk ≤ 2 − 2δ. It is known that a uniformly convex Banach space is reflexive and strictly convex; see [31] and the references therein. Recall that E is said to have a Gˆateaux differentiable norm if for all x, y ∈ BE . limt→0 (k xt + yk − k xt k). In this case, we also say that E is a smooth space. E is said to have a uniformly Gˆateaux differentiable norm if for each y ∈ BE , the limit is attained uniformly for all x ∈ BE . E is also said to have a uniformly Fr´echet differentiable norm if the above limit is attained uniformly for x, y ∈ BE . In this case, we say that E is uniformly smooth. It is known that a uniformly smooth Banach space is reflexive and smooth. ∗ Recall that normalized duality mapping J from E to 2E is defined by Jx = {y ∈ E ∗ : kxk2 = hx, yi = kyk2 }. It is known if E is uniformly smooth, then J is uniformly norm-to-norm continuous on every bounded subset of E; if E is a strictly convex Banach space, then 2

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J is strictly monotone; if E is a smooth Banach space, then J is single-valued and demicontinuous, i.e.,continuous from the strong topology of E to the weak star topology of E; if E is a reflexive and strictly convex Banach space with a strictly convex dual E ∗ and J ∗ : E ∗ → E is the normalized duality mapping in E ∗ , then J −1 = J ∗ ; if E is a smooth, strictly convex and reflexive Banach space, then J is single-valued, one-to-one and onto. Recall that E has the Kadec-Klee Property (KKP) if limm→∞ kxm − xk = 0, for any sequence {xm } ⊂ E, and x ∈ E with {xn } converges weakly to x, and {kxn k} converges strongly to kxk. It is known that every uniformly convex Banach space has the KKP; see [31] and the references therein. Let C be a nonempty closed and convex subset of E and let B : C × C → R be a function. Recall that the following equilibrium problem in the terminology of Blum and Oettli [32]. Find x ¯ ∈ C such that B(¯ x y) ≥ 0, ∀y ∈ C. We use Sol(B) to denote the solution set of the equilibrium problem. That is, Sol(B) = {x ∈ C : B(x, y) ≥ 0, ∀y ∈ C}. The following restrictions are essential for solving the equilibrium problem in this paper. (R-1) B(a, a) ≡ 0, ∀a ∈ C; (R-2) B(b, a) + B(a, b) ≤ 0, ∀a, b ∈ C; (R-3) B(a, b) ≥ lim supt↓0 B(tc + (1 − t)a, b), ∀a, b, c ∈ C; (R-4) b 7→ B(a, b) is convex and weakly lower semi-continuous, ∀a ∈ C. Let T be a self mapping on C. T is said to be closed if for any sequence {xn } ⊂ C such that limn→∞ xn = x ¯ and limm→∞ T xn = y¯, then y¯ = T x ¯. Let B be a bounded subset of C. Recall that T is said to be uniformly asymptotically regular on C if and only if lim supn→∞ supx∈B {kT n x − T n+1 xk} = 0. From now on, we use → and * to stand for the strong convergence and weak convergence, respectively. and use F ix(T ) to denote the fixed point set of mapping T . Recall that a point p is said to be an asymptotic fixed point of mapping T if and only if subset C contains a sequence {xm } which converges weakly to p such g that limm→∞ kT xm − xm k = 0. We use F ix(T ) to stand for the asymptotic fixed point set in this paper. Next, we assume that E is a smooth Banach space which means duality mapping J is single-valued. Study the functional φ(x, y) := kxk2 + kyk2 − 2hx, Jyi,

∀x, y ∈ E.

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In [33], Alber studied a generalized projection P rojC : E → C, which is a mapping assigning to an arbitrary point x ∈ E the minimum point of φ(x, y), which implies from the definition of φ φ(x, y)+2kxkkyk ≥ kxk2 +kyk2 , ∀x, y ∈ E. T is said to be relatively nonexpansive iff φ(p, x) ≥ φ(p, T x),

g ∀x ∈ C, ∀p ∈ F ix(T ) = F ix(T ) 6= ∅.

T is said to be relatively asymptotically nonexpansive iff φ(p, x) + ξn φ(p, x) ≥ φ(p, T n x),

g ∀x ∈ C, ∀p ∈ F ix(T ) = F ix(T ) 6= ∅, ∀n ≥ 1,

where {ξn } ⊂ [0, ∞) is a sequence such that µn → 0 as n → ∞. Remark 2.1. The class of relatively asymptotically nonexpansive mappings, which was first considered in [34], covers the class of relatively nonexpansive mappings [35]. T is said to be quasi-φ-nonexpansive iff φ(p, x) ≥ φ(p, T x),

∀x ∈ C, ∀p ∈ F ix(T ) 6= ∅.

T is said to be asymptotically quasi-φ-nonexpansive if and only if there exists a sequence {ξn } ⊂ [0, ∞) with µn → 0 as n → ∞ such that φ(p, x) + ξn φ(p, x) ≥ φ(p, T n x),

∀x ∈ C, ∀p ∈ F ix(T ) 6= ∅, ∀n ≥ 1.

Remark 2.2. The class of quasi-φ-nonexpansive mappings [26] and the class of asymptotically quasi-φ-nonexpansive mappings [27] cover the class of relatively nonexpansive mappings and the class of relatively asymptotically nonexpansive mappings. Quasi-φ-nonexpansive mappings and asymptotically quasiφ-nonexpansive mappings do not require the strong restriction that the fixed point set equals the asymptotic fixed point set. Remark 2.3. The class of quasi-φ-nonexpansive mappings and the class of asymptotically quasi-φ-nonexpansive mappings are generalizations of the class of quasi-nonexpansive mappings and the class of asymptotically quasi-nonexpansive p mappings in Banach spaces because of φ(x, y) = kx − yk. The following lemmas also play an important role in this paper. Lemma 2.4. [33] Let E be a strictly convex, reflexive, and smooth Banach space and let C be a nonempty, closed, and convex subset of E. Let x ∈ E. Then φ(y, ΠC x) ≤ φ(y, x) − φ(ΠC x, x), ∀y ∈ C, 4

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hy − x0 , Jx − Jx0 i ≤ 0, ∀y ∈ C if and only if x0 = ΠC x. Lemma 2.5. ([26], [32]) Let E be a strictly convex, smooth, and reflexive Banach space and let C be a closed convex subset of E. Let B be a function with the restrictions (R-1), (R-2), (R-3) and (R-4), from C ×C to R. Let x ∈ E and let r > 0. Then there exists z ∈ C such that rB(z, y) + hz − y, Jz − Jxi ≤ 0, ∀y ∈ C Define a mapping K B,r by K B,r x = {z ∈ C : rB(z, y) + hy − z, Jz − Jxi ≥ 0,

∀y ∈ C}.

The following conclusions hold: (1) K B,r is single-valued quasi-φ-nonexpansive; (2) Sol(B) = F ix(K B,r ) is closed and convex. Lemma 2.6 [36] Let E be a strictly convex and uniformly smooth Banach space which also has the KKP. Let C be a convex and closed subset of E and let T be an asymptotically quasi-φ-nonexpansive mapping on C. F ix(T ) is convex. Lemma 2.7 [37] Let r be a positive real number and let E be uniformly convex. Then there exists a convex, strictly increasing and continuous function cof : [0, 2r] → R such that cof (0) = 0 and tkak2 + (1 − t)kbk2 ≥ k(1 − t)b + tak2 + t(1 − t)cof (kb − ak) for all t ∈ [0, 1] and a, b ∈ B r := {a ∈ E : kak ≤ r}.

3

Main results

Theorem 3.1. Let E be a strictly convex and uniformly smooth Banach space which also has the KKP. Let C be a convex and closed subset of E and let Λ be an arbitrary index set. Let Bi be a bifunction with (R-1), (R-2), (R-3) and (R4). Let Ti be an asymptotically quasi-φ-nonexpansive mapping on C for every i ∈ Λ. Assume that Ti is uniformly asymptotically regular and closed for every T i ∈ Λ and ∩i∈Λ Sol(Bi ) ∩i∈Λ F ix(Ti ) is nonempty and bounded. Let {xj } be a

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sequence generated by   x0 ∈ E chosen arbitrarily,       C(1,i) = C, ∀i ∈ Λ,     C = ∩ C 1 i∈Λ (1,i) , x1 = P rojC1 x0 , Jy(j,i) = α(j,i) JT j xj + (1 − α(j,i) )Ju(j,i) ,  i      C(j+1,i) = {z ∈ C(j,i) : φ(z, y(j,i) ) − φ(z, xj ) ≤ α(j,i) ξ(j,i) D(j,i) },     C j+1 = ∩i∈Λ C(j+1,i) , xj+1 = P rojCj+1 x1 , where u(j,i) is such that r(j,i) Bi (u(j,i) , µ) ≥ hu(j,i) − µ, Ju(j,i) − Jxj i, ∀µ ∈ T Cj , D(j,i) = sup{φ(z, xj ) : z ∈ ∩i∈Λ F ix(Ti ) ∩i∈Λ Sol(Bi )}, {α(j,i) } is a real sequence in (0, 1) such that lim inf j→∞ α(j,i) (1 − α(j,i) ) > 0 and {r(j,i) } ⊂ [r, ∞) is a real sequence, where r is some positive real number. Then {xj } converges strongly to P roj∩i∈Λ F ix(Ti ) T ∩i∈Λ Sol(Bi ) x1 . T Proof. First, we prove ∩i∈Λ Sol(Bi ) ∩i∈Λ F ix(Ti ) is convex and closed. Using Lemma 2.5 and 2.6, we find that Sol(Bi ) is convex and closed and F ix(Ti ) is convex for every i ∈ Λ. Since Ti is closed, we find that F ix(Ti ) is also closed. So, P roj∩i∈Λ Sol(Bi ) T ∩i∈Λ F ix(Ti ) x is well defined, for any element x in E. Next, we prove that Cj is convex and closed. It is obvious that C(1,i) = C is convex and closed. Assume that C(m,i) is convex and closed for some m ≥ 1. Let p1 , p2 ∈ C(m+1,i) . It follows that p = sp1 + (1 − s)p2 ∈ C(m,i) , where s ∈ (0, 1). Notice that φ(p1 , y(m,i) ) − φ(p1 , xm ) ≤ α(m,i) ξ(m,i) D(m,i) , and φ(p2 , y(m,i) ) − φ(p2 , xm ) ≤ α(m,i) ξ(m,i) D(m,i) . Hence, one has 2hp1 , Jxm − Jy(m,i) i − kxm k2 + ky(m,i) k2 ≤ α(m,i) ξ(m,i) D(m,i) , and 2hp2 , Jxm − Jy(m,i) i − kxm k2 + ky(m,i) k2 ≤ α(m,i) ξ(m,i) D(m,i) . Using the above two inequalities, one has φ(p, y(m,i) )−φ(p, xm ) ≤ α(m,i) ξ(m,i) D(m,i) . This shows that C(m+1,i) is closed and convex. Hence, Cj = ∩i∈Λ C(j,i) is a convex and closed set. This proves that P rojCj+1 x1 is well defined. T On the other hand, we find that ∩i∈Λ Sol(Bi ) ∩i∈Λ F ix(Ti ) ⊂ C1 = C T is clear. Suppose that ∩i∈Λ Sol(Bi ) ∩i∈Λ F ix(Ti ) ⊂ C(m,i) for some positive

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integer m. For any w ∈ ∩i∈Λ Sol(Bi )

T

∩i∈Λ F ix(Ti ) ⊂ C(m,i) , we see that

φ(z, y(m,i) ) = kzk2 + kα(m,i) JTim xm + (1 − α(m,i) )Ju(m,i) k2 − 2hz, α(m,i) JTim xm + (1 − α(m,i) )Ju(m,i) i ≤ kzk2 + α(m,i) kTim xm k2 + (1 − α(m,i) )ku(m,i) k2 − 2α(m,i) hz, JTim xm i − 2(1 − α(m,i) )hz, Ju(m,i) i ≤ φ(z, xm ) + α(m,i) ξ(m,i) D(m,i) , T where D(m,i) = sup{φ(z, xm ) : z ∈ ∩i∈Λ F ix(Ti ) ∩i∈Λ Sol(Bi )}. This shows T that z ∈ C(m+1,i) . This implies that ∩i∈Λ Sol(Bi ) ∩i∈Λ F ix(Ti ) ⊂ ∩i∈Λ C(j,i) = Cj . Using Lemma 2.4, one has hz −xj , Jx1 −Jxj i ≤ 0, for any z ∈ Cj . It follows that \ hz − xj , Jx1 − Jxj i ≤ 0, ∀z ∈ ∩i∈Λ Sol(Bi ) ∩i∈Λ F ix(Ti ) ⊂ Cj . (3.1) Using Lemma 2.4 yields that φ(xj , x1 ) ≤ φ(P roj∩i∈Λ F ix(Ti ) T ∩i∈Λ Sol(Bi ) x1 , x1 ) − φ(P roj∩i∈Λ F ix(Ti ) T ∩i∈Λ Sol(Bi ) x1 , xj ), which shows that {φ(xj , x1 )} is bounded. Hence, {xj } is also bounded. Without loss of generality, we assume xj * x ¯ ∈ Cj . Hence φ(xj , x1 ) ≤ φ(¯ x, x1 ). This implies that φ(¯ x, x1 ) ≤ lim inf (kxj k2 + kx1 k2 − 2hxj , Jx1 i) = lim sup φ(xj , x1 ) ≤ φ(¯ x, x1 ). j→∞

j→∞

It follows that limj→∞ φ(xj , x1 ) = φ(¯ x, x1 ). Hence, we have limj→∞ kxj k = k¯ xk. Using the KKP, one obtains that {xj } converges strongly to x ¯ as j → ∞. On the other hand, we find that φ(xj+1 , x1 ) ≥ φ(xj , x1 ), which shows that {φ(xj , x1 )} is nondecreasing. It follows that limj→∞ φ(xj , x1 ) exists. Since φ(xj+1 , x1 ) − φ(xj , x1 ) ≥ φ(xj+1 , xj ), one has limj→∞ φ(xj+1 , xj ) = 0. Since xj+1 ∈ Cj+1 , one sees that φ(xj+1 , y(j,i) ) − φ(xj+1 , xj ) ≤ α(j,i) ξ(j,i) D(j,i) . It follows that limj→∞ φ(xj+1 , y(j,i) ) = 0. Hence, one has limj→∞ (ky(j,i) k−kxj+1 k) = 0. This implies that limj→∞ kJy(j,i) k = limj→∞ ky(j,i) k = k¯ xk = kJ x ¯k. This implies that {Jy(j,i) } is bounded. Without loss of generality, we assume that {Jy(j,i) } converges weakly to y (∗,i) ∈ E ∗ . In view of the reflexivity of E, we see that J(E) = E ∗ . This shows that there exists an element y i ∈ E such that Jy i = y (∗,i) . It follows that φ(xj+1 , y(j,i) )+2hxj+1 , Jy(j,i) i = kxj+1 k2 +kJy(j,i) k2 . Taking lim inf j→∞ , one has 0 ≥ k¯ xk2 − 2h¯ x, y (∗,i) i + ky (∗,i) k2 = k¯ xk2 + kJy i k2 − 2h¯ x, Jy i i = φ(¯ x, y i ) ≥ 0. That is, x ¯ = y i , which in turn implies that J x ¯ = y (∗,i) . 7

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Hence, Jy(j,i) * J x ¯ ∈ E ∗ . Since E ∗ is uniformly convex. Hence, it has the KKP, we obtain limi→∞ Jy(j,i) = J x ¯. Since J −1 : E ∗ → E is demi-continuous and E has the KKP, one gets that y(j,i) → x ¯, as j → ∞. Using the fact φ(z, xj ) − φ(z, y(j,i) ) ≤ (kxj k + ky(j,i) k)ky(j,i) − xj k + 2hz, Jy(j,i) − Jxj i, we find
lim φ(z, xj ) − φ(z, y(j,i) ) = 0.

(3.2)

j→∞

On the other hand, one sees from Lemma 2.7 φ(z, y(j,i) ) = kzk2 + kα(j,i) JTij xj + (1 − α(j,i) )Ju(j,i) k2 − 2hz, α(j,i) JTij xj + (1 − α(j,i) )Ju(j,i) i ≤ kzk2 + α(j,i) kTij xj k2 + (1 − α(j,i) )ku(j,i) k2 − α(j,i) (1 − α(j,i) )cof (k|Ju(j,i) − JTij xj k) − 2α(j,i) hz, JTij xj i − 2(1 − α(j,i) )hz, Ju(j,i) i ≤ φ(z, xj ) + α(j,i) ξ(j,i) D(j,i) − α(j,i) (1 − α(j,i) )cof (k|Ju(j,i) − JTij xj k). This implies α(j,i) (1 − α(j,i) )cof (k|Ju(j,i) − JTij xj k) ≤ φ(z, xj ) − φ(z, y(j,i) ) + α(j,i) ξ(j,i) D(j,i) . Using the restriction imposed on the sequence {α(j,i) } and (3.2), one has lim k|Ju(j,i) − JTij xj k = 0.

j→∞

It follows that JTij xj → J x ¯ as j → ∞. Since J −1 : E ∗ → E is demi-continuous, one has Tij xj * x ¯. Using the fact |kTij xj k − k¯ xk| = |kJTij xj k − kJ x ¯k| ≤ j j kJTi xj − J x ¯k, one has kTi xj k → k¯ xk as j → ∞. Since E has the KKP, one has limj→∞ k|¯ x − Tij xj k = 0. Since Ti is also uniformly asymptotically regular, one has limj→∞ k¯ x − Tij+1 xj k = 0. That is, Ti (Tij xj ) → x ¯. Using the closedness of Ti , we find Ti x ¯=x ¯. This proves x ¯ ∈ F ix(Ti ), that is, x ¯ ∈ ∩i∈Λ F ix(Ti ). Next, we show that x ¯ ∈ ∩i∈Λ Sol(Bi ). Since Bi is monotone, we find that r(j,i) Bi (µ, u(j,i) ) ≤ kµ − u(j,i) kkJu(j,i) − Jxj k. Therefore, one sees Bi (µ, x ¯) ≤ 0. For 0 < ti < 1, define µ(t,i) = (1 − ti )¯ x + ti µ. This implies that 0 ≥ Bi (µ(t,i) , x ¯). Hence, we have 0 = Bi (µ(t,i) , µ(t,i) ) ≤ ti Bi (µ(t,i) , µ). It follows that Bi (¯ x, µ) ≥ 0, ∀µ ∈ C. This implies that x ¯ ∈ Sol(Bi ) for every i ∈ Λ.

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Finally, we prove x ¯ = P roj∩i∈Λ (F ix(Ti )∩Sol(Bi )) x1 . Using (3.1), one has h¯ x− z, Jx1 − J x ¯i ≥ 0 z ∈ ∩i∈Λ (F ix(Ti ) ∩ Sol(Bi )). Using Lemma 2.4, we find that x ¯ = P roj∩i∈Λ (F ix(Ti )∩Sol(Bi )) x1 . This completes the proof. For the class of quasi-φ-nonexpansive mappings, the boundedness of the common solution set is not required. Indeed, we have the following result. Corollary 3.2. Let E be a strictly convex and uniformly smooth Banach space which also has the KKP. Let C be a convex and closed subset of E and let Λ be an arbitrary index set. Let Bi be a bifunction with (R-1), (R-2), (R-3) and (R-4). Let Ti be a quasi-φ-nonexpansive mapping on C for every i ∈ Λ. Assume T that Ti is closed for every i ∈ Λ and ∩i∈Λ Sol(Bi ) ∩i∈Λ F ix(Ti ) is nonempty. Let {xj } be a sequence generated by   x0 ∈ E chosen arbitrarily,       C(1,i) = C, ∀i ∈ Λ,     C = ∩ C , x = P roj x , 1

i∈Λ

(1,i)

1

C1 0

  Jy(j,i) = α(j,i) JTi xj + (1 − α(j,i) )Ju(j,i) ,      C(j+1,i) = {z ∈ C(j,i) : φ(z, y(j,i) ) ≤ φ(z, xj )},     C j+1 = ∩i∈Λ C(j+1,i) , xj+1 = P rojCj+1 x1 , where u(j,i) is such that r(j,i) Bi (u(j,i) , µ) ≥ hu(j,i) − µ, Ju(j,i) − Jxj i, ∀µ ∈ T Cj , D(j,i) = sup{φ(z, xj ) : z ∈ ∩i∈Λ F ix(Ti ) ∩i∈Λ Sol(Bi )}, {α(j,i) } is a real sequence in (0, 1) such that lim inf j→∞ α(j,i) (1 − α(j,i) ) > 0 and {r(j,i) } ⊂ [r, ∞) is a real sequence, where r is some positive real number. Then {xj } converges strongly to P roj∩i∈Λ F ix(Ti ) T ∩i∈Λ Sol(Bi ) x1 . From Theorem 3.1, we also have the following result. Corollary 3.3. Let E be a strictly convex and uniformly smooth Banach space which also has the KKP. Let C be a convex and closed subset of E and let B be a bifunction with (R-1), (R-2), (R-3) and (R-4). Let T be an asymptotically quasi-φ-nonexpansive mapping on C. Assume that T is uniformly asymptotically regular and closed and Sol(B) ∩ F ix(T ) is nonempty and bounded. Let {xj } be

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a sequence generated by   x0 ∈ E chosen arbitrarily,        C1 = C, x1 = P rojC1 x0 ,

Jyj = αj JT j xj + (1 − αj )Juj ,     Cj+1 = {z ∈ Cj : φ(z, yj ) − φ(z, xj ) ≤ αj ξj Dj },      xj+1 = P rojCj+1 x1 ,

where uj is such that rj B(uj , µ) ≥ huj − µ, Juj − Jxj i, ∀µ ∈ Cj , Dj = sup{φ(z, xj ) : z ∈ F ix(T ) ∩ Sol(B)}, {αj } is a real sequence in (0, 1) such that lim inf j→∞ αj (1 − αj ) > 0 and {rj } ⊂ [r, ∞) is a real sequence, where r is some positive real number. Then {xj } converges strongly to P rojF ix(T )∩Sol(B) x1 .

4

Applications

In this section, we consider common solutions of a family of variational inequalities in the framework Banach spaces. we give some deduced results of our main results in the framework of Hilbert spaces. Let A : C → E ∗ be a single valued monotone operator which is continuous along each line segment in C with respect to the weak∗ topology of E ∗ (hemicontinuous). Recall the the following variational inequality. Finding a point x ∈ C such that hx − y, Axi ≤ 0, ∀y ∈ C. The symbol N c(x) stand for the normal cone for C at a point x ∈ C; that is, N c(x) = {x∗ ∈ E ∗ : hx − y, x∗ i ≥ 0, ∀y ∈ C}. From now on, we use V I(C, A) to denote the solution set of the variational inequality. Theorem 4.1. Let E be a strictly convex and uniformly smooth Banach space which also has the KKP. Let C be a convex and closed subset of E. Let Λ be an index set and let Ai : C → E ∗ be a single valued, monotone and hemicontinuous operator. Let Bi be a bifunction with (R-1), (R-2), (R-3) and (R-4). Assume that ∩i∈Λ V I(C, Ai ) is not empty. Let {xn } be a sequence generated in the

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following process.   x0 ∈ E chosen arbitrarily,       C(1,i) = C, ∀i ∈ Λ,      C1 = ∩i∈Λ C(1,i) , x1 = P rojC1 x0 ,   u(n,i) = V I(C, Ai + r1i (J − Jxn )),     Jy(j,i) = α(j,i) Jxj + (1 − α(j,i) )Ju(j,i) ,      C(j+1,i) = {z ∈ C(j,i) : φ(z, y(j,i) ) ≤ φ(z, xj )},     C j+1 = ∩i∈Λ C(j+1,i) , xj+1 = P rojCj+1 x1 , where {α(j,i) } is a real sequence in (0, 1) such that lim inf j→∞ α(j,i) (1 − α(j,i) ) > 0. Then {xj } converges strongly to P roj∩i∈Λ V I(C,Ai ) x1 . Proof. Define a new operator Mi by Mi x = Ai x + N c(x), x ∈ C, Mi x = ∅, x ∈ / −1 −1 C. Hence, Mi is maximal monotone and Mi (0) = V I(C, Ai ), where Mi (0) stand for the zero point set of Mi . For each ri > 0, and x ∈ E, we see that there exists a unique xri in the domain of Mi such that Jx ∈ Jxri + ri Mi (xri ), where xri = (J + ri Mi )−1 Jx. Notice that uj,i = V I(C, r1i (J − Jxj ) + Ai ), which is equivalent to huj,i − y, Ai zj,i + r1i (Jzj,i − Jxj )i ≤ 0, ∀y ∈ C, that is,
1 −1 Jxj . ri Jxj − Juj,i ∈ N c(uj,i ) + Ai zj,i . This implies that uj,i = (J + ri Mi ) From [26], we find that (J + ri Mi )−1 J is closed quasi-φ-nonexpansive with F ix((J + ri Mi )−1 J) = Mi−1 (0). Using Theorem 3.1, we find the desired conclusion immediately. Theorem 4.2. Let E be a Hilbert. Let C be a convex and closed subset of E and let Λ be an arbitrary index set. Let Bi be a function with (R-1), (R-2), (R-3) and (R-4). Let Ti be an asymptotically quasi-nonexpansive mapping on C for every i ∈ Λ. Assume that Ti is uniformly asymptotically regular and closed T for every i ∈ Λ and ∩i∈Λ Sol(Bi ) ∩i∈Λ F ix(Ti ) is nonempty and bounded. Let {xj } be a sequence generated by   x0 ∈ E chosen arbitrarily,       C(1,i) = C, ∀i ∈ Λ,     C = ∩ C ,x = P x , 1

i∈Λ

(1,i)

1

C1 0

  y(j,i) = α(j,i) Tij xj + (1 − α(j,i) )u(j,i) ,      C(j+1,i) = {z ∈ C(j,i) : kz − y(j,i) k2 − kz − xj k2 ≤ α(j,i) ξ(j,i) D(j,i) },     C j+1 = ∩i∈Λ C(j+1,i) , xj+1 = PCj+1 x1 , where u(j,i) is such that r(j,i) Bi (u(j,i) , µ) ≥ hu(j,i) − µ, u(j,i) − xj i, ∀µ ∈ Cj , T D(j,i) = sup{kz − xj k2 : z ∈ ∩i∈Λ F ix(Ti ) ∩i∈Λ Sol(Bi )}, {α(j,i) } is a real 11

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sequence in (0, 1) such that lim inf j→∞ α(j,i) (1 − α(j,i) ) > 0 and {r(j,i) } ⊂ [r, ∞) is a real sequence, where r is some positive real number. Then {xj } converges strongly to P∩i∈Λ F ix(Ti ) T ∩i∈Λ Sol(Bi ) x1 . p Proof. In the framework of Hilbert spaces, we see that φ(x, y) = kx − yk, ∀x, y ∈ E. The generalized projection is reduced to the metric projection and the asymptotically-φ-nonexpansive mapping is reduced to the asymptotically quasi-nonexpansive mapping. Using Theorem 3.1, we find the desired conclusion immediately.

References [1] B.A.B. Dehaish, et al., Weak and strong convergence of algorithms for the sum of two accretive operators with applications, J. Nonlinear Convex Anal., 16 (2015), 1321–1336. [2] C. Wu, Strong convergence theorems for common solutions of variational inequality and fixed point problems, Adv. Fixed Point Theory, 4 (2014), 229-244. [3] M. Zhang, S.Y. Cho, A monotone projection algorithm for solving fixed points of nonlinear mappings and equilibrium problems, J. Nonlinear Sci. Appl., 9 (2016), 1453–1462. [4] S.Y. Cho, X. Qin, L. Wang, Strong convergence of a splitting algorithm for treating monotone operators, Fixed Point Theory Appl., 2014 (2014), Article ID 94. [5] S.Y. Cho, Generalized mixed equilibrium and fixed point problems in a Banach space, J. Nonlinear Sci. Appl., 9 (2016), 1083–1092. [6] Z. Wang, Y. Su, D. Wang, Y. Dong, A modified Halpern-type iteration algorithm for a family of hemi-relatively nonexpansive mappings and systems of equilibrium problems in Banach spaces, J. Comput. Appl. Math., 235 (2011), 2364–2371. [7] Z.M. Wang, X. Zhang, Shrinking projection methods for systems of mixed variational inequalities of Browder type, systems of mixed equilibrium problems and fixed point problems, J. Nonlinear Funct. Anal., 2014 (2014), Article ID 15.

12

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[8] S.Y. Cho, X. Qin, S.M. Kang, Iterative processes for common fixed points of two different families of mappings with applications, J. Global Optim., 57 (2013), 1429–1446. [9] Y. Zhang, Q. Yuan, Iterative common solutions of fixed point and variational inequality problems, J. Nonlinear Sci. Appl., 9 (2016), 1882–1890. [10] B.A.B. Dehaish, A. Latif, H.O. Bakodah, X. Qin, A regularization projection algorithm for various problems with nonlinear mappings in Hilbert spaces J. Inequal. Appl., 2015 (2015), Article ID 51. [11] G. Wang, S. Sun, Hybrid projection algorithms for fixed point and equilibrium problems in a Banach space, Adv. Fixed Point Theory, 3 (2013), 578–594. [12] X. Qin, S.Y. Cho, L. Wang, Convergence of splitting algorithms for the sum of two accretive operators with applications, Fixed Point Theory Appl., 2014 (2014), Article ID 75. [13] J. Zhao, Strong convergence theorems for equilibrium problems, fixed point problems of asymptotically nonexpansive mappings and a general system of variational inequalities, Nonlinear Funct. Appl. Appl., 16 (2011), 447–464. [14] X. Qin, S.Y. Cho, L. Wang, Iterative algorithms with errors for zero points of m-accretive operators, Fixed Point Theory App., 2013 (2013), Article ID 148. [15] X. Qin, S.Y. Cho, J.K. Kim, On the weak convergence of iterative sequences for generalized equilibrium problems and strictly pseudocontractive mappings, Optimization, 61 (2012), 805–821. [16] A. Genel, J. Lindenstruss, An example concerning fixed points, Israel J. Math., 22 (1975), 81–86. [17] J. Schu, Iterative construction of fixed points of asymptotically nonexpansive mappings, J. Math. Anal. Appl., 158 (1991), 407–413. [18] O. G¨ uler, On the convergence of the proximal point algorithm for convex minimization, SIAM J. Control Optim., 29 (1991), 403–409. [19] Y. Haugazeau, Sur les inequations variationnelles et la minimization de fonctionnelles convexes, These, Universite de Paris, France, (1968).

13

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[20] Y. Hao, Some results on a modified Mann iterative scheme in a reflexive Banach space, Fixed Point Theory Appl., 2013 (2013), Article ID 227. [21] Y. Hao, On generalized quasi-φ-nonexpansive mappings and their projection algorithms, Fixed Point Theory Appl., 2013 (2013), Article ID 204. [22] J.S. Jung, A general composite iterative method for equilibrium problems and fixed point problems, J. Comput. Anal. Appl., 12 (2010), 124–140. [23] J.K. Kim, Strong convergence theorems by hybrid projection methods for equilibrium problems and fixed point problems of the asymptotically quasiφ-nonexpansive mappings, Fixed Point Theory Appl., 2011 (2011), Article ID 10. [24] J.K. Kim, Convergence theorems of iterative sequences for generalized equilibrium problems involving strictly pseudocontractive mappings in Hilbert spaces, J. Comput. Anal. Appl., 18 (2015), 454–471. [25] B. Liu, C. Zhang, Strong convergence theorems for equilibrium problems and quasi-φ-nonexpansive mappings, Nonlinear Funct. Anal. Appl., 16 (2011), 365–385. [26] X. Qin, Y.J. Cho, S.M. Kang, Convergence theorems of common elements for equilibrium problems and fixed point problems in Banach spaces, J. Comput. Appl. Math., 225 (2009), 20–30. [27] X. Qin, S.Y. Cho, S.M. Kang, On hybrid projection methods for asymptotically quasi-φ-nonexpansive mappings, Appl. Math. Comput., 215 (2010), 3874–3883. [28] Q. Yuan, S. Lv, A strong convergence theorem for solutions of equilibrium problems and asymptotically quasi-φ-nonexpansive mappings in the intermediate sense, Fixed Point Theory Appl., 2013 (2013), Articl ID 305. [29] Q.N. Zhang, H. Wu, Hybrid algorithms for equilibrium and common fixed point problems with applications, J. Inequal. Appl., 2014 (2014), Article ID 221. [30] Y.J. Cho, X. Qin, S.M. Kang, Some results for equilibrium problems and fixed point problems in Hilbert spaces, J. Comput. Anal. Appl., 11 (2009), 294–316. [31] I. Cioranescu, Geometry of Banach Spaces, Duality Mappings and Nonlinear Problems, Kluwer, Dordrecht, (1990). 14

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[32] E. Blum, W. Oettli, From optimization and variational inequalities to equilibrium problems, Math. Stud., 63 (1994), 123–145. [33] Y.I. Alber, Metric and generalized projection operators in Banach spaces: properties and applications, in: A.G. Kartsatos (Ed.), Theory and Applications of Nonlinear Operators of Accretive and Monotone Type, Marcel Dekker, New York, 1996. [34] R.P. Agarwal, Y.J. Cho, X. Qin, Generalized projection algorithms for nonlinear operators, Numer. Funct. Anal. Optim., 28 (2007), 1197–1215. [35] D. Butnariu, S. Reich, A.J. Zaslavski, Asymptotic behavior of relatively nonexpansive operators in Banach spaces, J. Appl. Anal., 7 (2001), 151– 174. [36] X. Qin, R.P. Agarwal, S.Y. Cho, L. Wang, Convergence of algorithms for fixed points of generalized asymptotically quasi-φ-nonexpansive mappings with applications, Fixed Point Theory Appl., 2012 (2012), Article ID 58. [37] T. Takahashi, Nonlinear Functional Analysis, Yokohama-Publishers, 2000.

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Inner-outer factorization on Besov-type spaces Ruishen Qian and Songxiao Li∗ Abstract. In this paper, motivated by some results of Dyakonov, we give an inner-outer factorization on Besov-type spaces. MSC 2000: 30H25, 30J05. Keywords: Inner function, outer function, BM OA space, Besov-type spaces.

1

Introduction

We denote the unit disc {z ∈ C : |z| < 1} by D and its boundary by ∂D. Let H(D) be the space of all analytic functions in D. For 0 < p < ∞, the Hardy space H p is the set of f ∈ H(D) for which 1 0 '(t) for all t > 0 where 2 g. We now introduce generalized rational contraction mappings as follows: De…nition 6. Let f; g; S and T be selfmaps of a metric space (X; d), and (f; g) be an ST -admissible pair: We say that (f; g) is a generalized ( ; ; ')(S;T ) -rational contraction if (Sx; T y) for all x; y 2 X; where M (x; y)

=

1 implies 2

;'2

(d (f x; gy))

' (M (x; y))

(1.1)

and

d (Sx; gy) + d (f x; T y) ; 2 d (T y; gy) [1 + d (Sx; f x)] d (f x; T y) [1 + d (Sx; gy)] ; : 1 + d (Sx; T y) 1 + d (Sx; T y)

max d (Sx; T y) ; d (Sx; f x) ; d (T y; gy) ;

In this paper, we prove some common …xed point results of generalized ( ; ; ')(S;T ) rational contractions for a quadruple of self-mappings de…ned on ordinary as well as ordered metric spaces. Our results extend, generalize and unify comparable results in the existing literature. Applying these results, we deduce …xed point results on metric spaces endowed with graph. An example is presented to support the results obtained herein. As an application of o¤ered results, the existence of the common solution for a system of integral equations are also investigated.

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GENERALIZED RATIONAL CONTRACTIONS ENDOW ED W ITH A GRAPH

3

2. Main Results We start with the following …rst result. Theorem 1. Let f; g; S and T be selfmaps of a complete metric space (X; d) with f (X) T (X), g(X) S(X) and (f; g) be a generalized ( ; ; ')(S;T ) -rational contraction pair. Suppose that: (a) there exists x0 2 X such that (Sx0 ; f x0 ) 1; (b) (Sxn ; T xn+1 ) 1 for all n even implies that (Sxn ; T xj ) 1 for all n even and j > n odd; (c) (Sxn ; T xn+1 ) 1 for all n even and, Sxn and T xn+1 converge to an x 2 X as n ! 1 implies that (Sxn ; x) 1 and (x; T xn+1 ) 1 for all n even. Then the pairs (f; S) and (g; T ) have a point of coincidence in X. Moreover, if (i) ff; Sg and fg; T g are weakly compatible, (ii) (Su; T v) 1 whenever u 2 C (f; S) and v 2 C (g; T ) : Then f; g; S and T have a common …xed point. Proof. Let x0 2 X such that (Sx0 ; f x0 ) 1: Since f X T X; there exists an x1 2 X such that f x0 = T x1 : Again since gX SX; there exists an x2 2 X such that gx1 = T x2 : Continuing this process, we can construct the sequences fxn g and fyn g in X de…ned by y2n = f x2n = T x2n+1 ;

y2n+1 = gx2n+1 = Sx2n+2 ;

n 2 N0 ;

(2.1)

where N0 = N [ f0g : As (f; g) is an ST -admissible pair and (Sx0 ; f x0 ) = (Sx0 ; T x1 ) 1, we have (f x0 ; gx1 ) 1 and (gx0 ; f x1 ) 1 which implies that (T x1 ; Sx2 ) 1: Again, since (T x1 ; Sx2 ) 1; we have (f x1 ; gx2 ) 1 and (gx1 ; f x2 ) 1 which gives that (Sx2 ; T x3 ) 1: Continuing this way, we obtain

(Sx2n ; T x2n+1 )

1 and

(T x2n+1 ; Sx2n+2 )

1 for all n 2 N0 :

(2.2)

Suppose that y2n 6= y2n+1 for all n 2 N0 : Now we show that lim d (yn ; yn+1 ) = 0:

(2.3)

n!1

Putting x = x2n and y = x2n+1 in (1.1) and using (2.1) and (2.2), we get (d (y2n ; y2n+1 ))

=

(d (f x2n ; gx2n+1 )) ' (M (x2n ; x2n+1 )) ;

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(2.4)

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4

where M (x2n ; x2n+1 )

=

max d (Sx2n ; T x2n+1 ) ; d (Sx2n ; f x2n ) ; d (T x2n+1 ; gx2n+1 ) ; d (Sx2n ; gx2n+1 ) + d (f x2n ; T x2n+1 ) ; 2 d (T x2n+1 ; gx2n+1 ) [1 + d (Sx2n ; f x2n )] ; 1 + d (Sx2n ; T x2n+1 ) d (f x2n ; T x2n+1 ) [1 + d (Sx2n ; gx2n+1 )] 1 + d (Sx2n ; T x2n+1 )

=

max d (y2n d (y2n

1 ; y2n ) ; d (y2n 1 ; y2n ) ; d (y2n ; y2n+1 ) ;

+ d (y2n ; y2n ) ; 2 d (y2n ; y2n+1 ) [1 + d (y2n 1 ; y2n )] ; 1 + d (y2n 1 ; y2n ) d (y2n ; y2n ) [1 + d (y2n 1 ; y2n+1 )] 1 + d (y2n 1 ; y2n )

= If d (y2n

1 ; y2n )

1 ; y2n+1 )

max d (y2n

1 ; y2n ) ; d (y2n ; y2n+1 ) ;

max (d (y2n

1 ; y2n ) ; d (y2n ; y2n+1 )) :

d (y2n

1 ; y2n )

+ d (y2n ; y2n+1 ) 2

d (y2n ; y2n+1 ) for some n 2 N; then by (2.4); we have (d (y2n ; y2n+1 ))

' (d (y2n ; y2n+1 )) ;

a contradiction to the fact that y2n 6= y2n+1 : So for all n 2 N; we have d (y2n ; y2n+1 ) < d (y2n 1 ; y2n ) : From (2.4); we also obtain (d (y2n ; y2n+1 )) Again, putting x = x2n those given above, we get

1

' (d (y2n

1 ; y2n )) :

(2.5)

and y = x2n in (1.1) and following arguing similar to

(d (y2n

1 ; y2n ))

' (d (y2n

2 ; y2n 1 )) :

(2.6)

1 ; yn )) :

(2.7)

From (2.5) and (2.6), we conclude (d (yn ; yn+1 ))

' (d (yn

It follows that the sequence fd (yn ; yn+1 )g is decreasing and bounded below. Hence, there exists r 0 such that limn!1 d (yn ; yn+1 ) = r: If r > 0, then taking limit as n ! 1 on both sides of (2.7), we have (r)

lim

n!1

(d (yn ; yn+1 ))

lim ' (d (yn

n!1

1 ; yn ))

' (r) ;

a contradiction and hence r = 0; that is, the equation (2.3) holds. Now, we prove that fyn g is a Cauchy sequence. To this end, it is su¢ cient to verify that fy2n g is a Cauchy sequence. Suppose, to the contrary, that fy2n g is not a Cauchy sequence. Then, there exists an " > 0 for which we can …nd two

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GENERALIZED RATIONAL CONTRACTIONS ENDOW ED W ITH A GRAPH

5

subsequences fy2mk g and fy2nk g of fy2n g such that mk is the smallest index for which mk > nk > k and d (y2mk ; y2nk )

"

and

d (y2mk

1 ; y2nk )

< ":

(2.8)

Using the triangular inequality and (2.8), we have "

d (y2mk ; y2nk ) d (y2mk ; y2mk < d (y2mk ; y2mk 1 ) + ":

1)

+ d (y2mk

1 ; y2nk )

Taking k ! 1 on both sides of above inequality and using (2.3), we obtain lim d (y2mk ; y2nk ) = ":

(2.9)

k!1

Again, using the triangular inequality, we get jd (y2nk ; y2mk +1 )

d (y2nk ; y2mk )j

d (y2mk ; y2mk +1 ) :

Letting k ! 1 in the above inequality and using (2.3) and (2.9), we have lim d (y2nk ; y2mk +1 ) = ":

(2.10)

k!1

Similarly, one can easily show that lim d (y2nk

k!1

1 ; y2mk )

= lim d (y2nk k!1

1 ; y2mk +1 )

= ":

(2.11)

Since (Sx2nk ; T x2mk +1 ) 1 from (2.2) and the hypothesis (b) ; putting x = x2nk and y = x2mk +1 in (1.1); we get (d (y2nk ; y2mk +1 ))

=

(d (f x2nk ; gx2mk +1 )) ' (M (x2nk ; x2mk +1 )) ;

(2.12)

where M (x2nk ; x2mk +1 )

=

max d (Sx2nk ; T x2mk +1 ) ; d (Sx2nk ; f x2nk ) ; d (T x2mk +1 ; gx2mk +1 ) ; d (Sx2nk ; gx2mk +1 ) + d (f x2nk ; T x2mk +1 ) ; 2 d (T x2mk +1 ; gx2mk +1 ) [1 + d (Sx2nk ; f x2nk )] ; 1 + d (Sx2nk ; T x2mk +1 ) d (f x2nk ; T x2mk +1 ) [1 + d (Sx2nk ; gx2mk +1 )] 1 + d (Sx2nk ; T x2mk +1 )

=

max d (y2nk d (y2nk

1 ; y2mk ) ; d (y2nk 1 ; y2nk ) ; d (y2mk ; y2mk +1 ) ;

1 ; y2mk +1 )

+ d (y2nk ; y2mk ) ; 2 d (y2mk ; y2mk +1 ) [1 + d (y2nk 1 ; y2nk )] ; 1 + d (y2nk 1 ; y2mk ) d (y2nk ; y2mk ) [1 + d (y2nk 1 ; y2mk +1 )] : 1 + d (y2nk 1 ; y2mk )

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Now, from the properties of k ! 1 in (2.12); we obtain (")

and ' and using (2.3), (2.9), (2.10) and (2.11) as lim

k!1

(d (y2nk ; y2mk +1 ))

lim ' (M (x2nk ; x2mk +1 ))

k!1

' (max ("; 0; 0; "; 0; ")) = ' (") ; which implies that " = 0; a contradiction with " > 0: Thus fy2n g is a Cauchy sequence in X and hence fyn g is a Cauchy sequence. From the completeness of (X; d) ; there exists z 2 X such that lim yn = z:

(2.13)

n!1

From (2.1) and (2.13), we get f x2n ! z;

T x2n+1 ! z;

gx2n+1 ! z;

Sx2n+2 ! z

as n ! 1: (2.14)

Now we shall prove that z is a common …xed point of f; g; S and T . Since g(X) S(X), we can choose a point u in X such that z = Su. Suppose that d(z; f u) 6= 0. By (2.2), (2.14) and the condition (c) ; we have (Su; T x2n+1 ) 1: Then, substituting x = u and y = x2n+1 in (1.1), we deduce (d (f u; gx2n+1 ))

' (M (u; x2n+1 )) ;

(2.15)

where M (u; x2n+1 )

=

max d (Su; T x2n+1 ) ; d (Su; f u) ; d (T x2n+1 ; gx2n+1 ) ; d (Su; gx2n+1 ) + d (f u; T x2n+1 ) ; 2 d (T x2n+1 ; gx2n+1 ) [1 + d (Su; f u)] ; 1 + d (Su; T x2n+1 ) d (f u; T x2n+1 ) [1 + d (Su; gx2n+1 )] : 1 + d (Su; T x2n+1 )

Letting k ! 1 in (2.15), we have (d (f u; z))

lim

n!1

(d (f u; gx2n+1 ))

lim ' (M (u; x2n+1 ))

n!1

' max 0; d (z; f u) ; 0;

d (f u; z) ; 0; d (f u; z) 2

= ' (d (f u; z)) ; a contradiction and hence d (f u; z) = 0; that is f u = z; and so u 2 C (f; S) : Similarly, since f (X) T (X), we can choose a point v in X such that z = T v. Suppose that d(z; gv) 6= 0. By (2.2), (2.14) and the condition (c) ; we have (Sx2n ; T v) 1: Then, putting x = x2n and y = v in (1.1), we obtain (d (f x2n ; gv))

' (M (x2n ; v)) ;

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(2.16)

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7

where M (x2n ; v)

=

max d (Sx2n ; T v) ; d (Sx2n ; f x2n ) ; d (T v; gv) ; d (Sx2n ; gv) + d (f x2n ; T v) ; 2 d (T v; gv) [1 + d (Sx2n ; f x2n )] ; 1 + d (Sx2n ; T v) d (f x2n ; T v) [1 + d (Sx2n ; gv)] : 1 + d (Sx2n ; T v)

Taking limit on (2.16), we get (d (z; gv))

lim

n!1

(d (f x2n ; gv))

lim ' (M (x2n ; v))

n!1

' max 0; 0; d (z; gv) ;

d (z; gv) ; d (z; gv) ; 0 2

= ' (d (z; gv)) ; a contradiction and hence d (z; gv) = 0; that is z = gv; and so v 2 C (g; T ) : Thus, z = f u = Su = gv = T v: By the weak compatibility of the pairs (f; S) and (g; T ), we deduce that f z = Sz and gz = T z: Since z 2 C (f; S) and v 2 C (g; T ) ; by (ii) ; we have (Sz; T v) 1 and so, from (1.1) (2.17) (d (f z; z)) = (d (f z; gv)) ' (M (z; v)) ; where M (z; v)

=

max d (Sz; T v) ; d (Sz; f z) ; d (T v; gv) ;

=

d (Sz; gv) + d (f z; T v) d (T v; gv) [1 + d (Sz; f z)] ; ; 2 1 + d (Sz; T v) d (f z; T v) [1 + d (Sz; gv)] 1 + d (Sz; T v) max (d (f z; z) ; 0; 0; d (f z; z) ; 0; d (f z; z)) = d (f z; z)

By (2.17), we get (d (f z; z))

' (d (f z; z)) ;

which implies that z = f z; and so z = f z = Sz: Similarly, it can be shown that z = gz = T z: This completes the proof. Corollary 1. Let f; g; S and T be selfmaps of a complete metric space (X; d) with f (X) T (X), g(X) S(X) and (f; g) be an ST -admissible pair such that (Sx; T y)

(d (f x; gy))

' (M (x; y)) ;

(2.18)

for all x; y 2 X; where 2 and ' 2 : Assume that the following conditions are satis…ed: (a) there exists x0 2 X such that (Sx0 ; f x0 ) 1; (b) (Sxn ; T xn+1 ) 1 for all n even implies that (Sxn ; T xj ) 1 for all n even and j > n odd;

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(c)

(Sxn ; T xn+1 ) 1 for all n even and, Sxn and T xn+1 converge to an x 2 X as n ! 1 implies that (Sxn ; x) 1 and (x; T xn+1 ) 1 for all n even. Then the pairs (f; S) and (g; T ) have a point of coincidence in X. Moreover, if (i) ff; Sg and fg; T g are weakly compatible, (ii) (Su; T v) 1 whenever u 2 C (f; S) and v 2 C (g; T ) : Then f; g; S and T have a common …xed point. Proof. Let

1 for x; y 2 X: Then by (2.18), we have

(Sx; T y)

(d (f x; gy))

' (M (x; y)) :

This implies that the inequality (1.1) holds. Therefore, the proof follows from Theorem 1. If we take (Sx; T y) = 1 in Corollary 1, we have a generalized version of Theorem 2.3 in [29]: Theorem 2. Let f; g; S and T be selfmaps of a complete metric space (X; d) with f (X) T (X) and g(X) S(X). Suppose that (d (f x; gy))

' (M (x; y)) ;

(2.19)

for all x; y 2 X; where 2 and ' 2 : Then the pairs (f; S) and (g; T ) have a point of coincidence in X. Moreover, if ff; Sg and fg; T g are weakly compatible, then f; g; S and T have a common …xed point. If we take in [28]:

(t) = t in Corollary 1, we have a generalized version of Theorem 2.2

Theorem 3. Let f; g; S and T be selfmaps of a complete metric space (X; d) with f (X) T (X), g(X) S(X) and (f; g) be an ST -admissible pair such that (Sx; T y) d (f x; gy)

' (M (x; y)) ;

(2.20)

for all x; y 2 X; where ' 2 : Assume that the following conditions are satis…ed: (a) there exists x0 2 X such that (Sx0 ; f x0 ) 1; (b) (Sxn ; T xn+1 ) 1 for all n even implies that (Sxn ; T xj ) 1 for all n even and j > n odd; (c) (Sxn ; T xn+1 ) 1 for all n even and, Sxn and T xn+1 converge to an x 2 X as n ! 1 implies that (Sxn ; x) 1 and (x; T xn+1 ) 1 for all n even. Then the pairs (f; S) and (g; T ) have a point of coincidence in X. Moreover, if (i) ff; Sg and fg; T g are weakly compatible, (ii) (Su; T v) 1 whenever u 2 C (f; S) and v 2 C (g; T ) : Then f; g; S and T have a common …xed point. If we take ' (t) =

(t)

(t) in Corollary 1, we have the following result.

Corollary 2. Let f; g; S and T be selfmaps of a complete metric space (X; d) with f (X) T (X), g(X) S(X) and (f; g) be an ST -admissible pair such that (Sx; T y) for all x; y 2 X; where are satis…ed:

(d (f x; gy)) 2

and

(M (x; y)) 2

1165

(M (x; y)) ;

(2.21)

: Assume that the following conditions

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(a) there exists x0 2 X such that (Sx0 ; f x0 ) 1; (b) (Sxn ; T xn+1 ) 1 for all n even implies that (Sxn ; T xj ) 1 for all n even and j > n odd; (c) (Sxn ; T xn+1 ) 1 for all n even and, Sxn and T xn+1 converge to an x 2 X as n ! 1 implies that (Sxn ; x) 1 and (x; T xn+1 ) 1 for all n even. Then the pairs (f; S) and (g; T ) have a point of coincidence in X. Moreover, if (i) ff; Sg and fg; T g are weakly compatible, (ii) (Su; T v) 1 whenever u 2 C (f; S) and v 2 C (g; T ) : Then f; g; S and T have a common …xed point. Let us give the following hypothesis for the uniqueness of the common …xed point in Theorem 1. (H) For all x; y 2 F(f; g; S; T ); we have (Sx; T y) 1: Theorem 4. Adding condition (H) to the hypotheses of Theorem 1, we obtain the uniqueness of the common …xed point of f; g; S and T: Proof. Suppose that x = f x = gx = Sx = T x and y = f y = gy = Sy = T y: Then, from (H) ; we have (Sx; T y) 1: Then, applying (1.1), we obtain (d (x; y)) =

(d (f x; gy))

' (M (x; y)) ;

(2.22)

where M (x; y)

=

max d (Sx; T y) ; d (Sx; f x) ; d (T y; gy) ;

=

d (Sx; gy) + d (f x; T y) d (T y; gy) [1 + d (Sx; f x)] ; ; 2 1 + d (Sx; T y) d (f x; T y) [1 + d (Sx; gy)] 1 + d (Sx; T y) max (d (x; y) ; 0; 0; d (x; y) ; 0; d (x; y)) = d (x; y) :

From (2.22), we have (d (x; y)) ' (d (x; y)) ; which implies that d (x; y) = 0; that is, x = y: Remark 1. Adding condition (H) to the hypotheses of Corollaries 1 and 2, we obtain the uniqueness of the common …xed point: If we choose S = T = IX in Corollary 1, we have the following corollary. Corollary 3. Let f and g be selfmaps of a complete metric space (X; d) and (f; g) be an -admissible pair such that (x; y) for all x; y 2 X; where

2

(d (f x; gy)) ,'2

' (Mf g (x; y)) ;

(2.23)

and

d (x; gy) + d (f x; y) ; 2 d (y; gy) [1 + d (x; f x)] d (f x; y) [1 + d (x; gy)] ; : 1 + d (x; y) 1 + d (x; y) Assume that the following conditions are satis…ed: Mf g (x; y)

=

max d (x; y) ; d (x; f x) ; d (y; gy) ;

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HUSEYIN ISIK y , NAWAB HUSSAIN, AND M ARWAN A. KUTBI

(a) there exists x0 2 X such that (x0 ; f x0 ) 1; (b) (xn ; xn+1 ) 1 for all n implies that (xn ; xj ) 1 for all j > n; (c) (xn ; xn+1 ) 1 for all n and, xn ! x 2 X as n ! 1 implies that (xn ; x) 1 for all n: Then f and g have a common …xed point. Moreover, if (x; y) x; y 2 F (f; g) ; then f and g have a unique common …xed point.

1 whenever

Now, we furnish the following example which illustrates Theorem 1 as well as Theorem 4. Example 1. Let X = R+ with the usual metric d (x; y) = jx yj for all x; y 2 X and ; ' : R+ ! R+ be de…ned by (t) = t and ' (t) = 2t . De…ne the mappings f; g; S and T on X by ( ( x x if x 2 [0; 1] ; if x 2 [0; 1] ; fx = 6 and gx = 4 3x if x > 1; 6x if x > 1; Sx =

(

x 2

3x

if x 2 [0; 1] ; if x > 1;

and

Tx =

(

x 3

2x

if x 2 [0; 1] ; if x > 1:

Note that f (X) T (X) and g(X) S(X), ff; Sg and fg; T g are weakly compatible. Also, we de…ne the mapping : S(X) [ T (X) S(X) [ T (X) ! R+ by ( 1 if x; y 2 0; 12 ; (x; y) = 0 otherwise. Now, let x; y 2 X such that (Sx; T y) 1: Then Sx; T y 2 0; 12 and this implies that x; y 2 [0; 1] : By the de…nitions of f; g and ; we have f x; gy 2 0; 12 and gx; f y 2 0; 12 which implies that (f x; gy) 1 and (gx; f y) 1: In case of (T x; Sy) 1; analogously to the above proof, one can easily obtain that (f x; gy) 1 and (gx; f y) 1: Then (f; g) is ST -admissible. Moreover, the condition (Sx0 ; f x0 ) 1 is satis…ed with x0 = 0: Let fxn g be a sequence in X such that (Sxn ; T xn+1 ) 1 for all n even: Then, by the de…nition of ; we get xn 2 [0; 1] for all n even: Thus, xj 2 [0; 1] for all j > n odd; and so (Sxn ; T xj ) 1: Similarly, if fxn g is any sequence in X such that (Sxn ; T xn+1 ) 1 for all n even and, Sxn and T xn+1 converge to an x 2 X as n ! 1; then by the de…nition of ; we have Sxn 2 0; 12 and T xn+1 2 0; 12 for all n even and so x 2 0; 12 which implies that (Sxn ; x) 1 and (x; T xn+1 ) 1: Now, we prove that (f; g) is a generalized ( ; ; ')(S;T ) -rational contraction. Let (Sx; T y) 1: Then, x; y 2 [0; 1] ; and so (d (f x; gy))

= jf x

gyj =

x 6

y 4

1 x = jSx f xj 6 2 1 M (x; y) = ' (M (x; y)) : 2

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Obviously, assumption (ii) of Theorem 1 and condition (H) are satis…ed. Consequently, by Theorems 1 and 4, f; g; S and T have a unique common …xed point which is 0: 3. Fixed Point Results on Partially Ordered Metric Spaces The existence of …xed points of nonlinear contraction mappings in metric spaces endowed with a partial ordering has been considered recently by Ran and Reurings [26] in order to obtain a solution of a matrix equation in 2004. Nieto and Lopez [24] extended the results in [26] by removing the continuity condition of the mapping. They applied their result to get a solution of a boundary value problem (see also [4, 13, 14] and references mentioned therein). Let X be a non-empty set. If d is a complete metric on X and is a partial order on the set X; then (X; d; ) is called complete partially ordered metric space. Let (X; ) be a partially ordered set and f; g; S and T be self-mappings on X. Then, (f; g) is called a (S; T )-nondecreasing mapping pair if f x gy and gx f y whenever Sx T y or T x Sy for all x; y 2 X. From Theorem 1, in the setting of complete partially ordered metric spaces, we obtain the following theorem. Theorem 5. Let (X; d; ) be a complete partially ordered metric space and let f; g; S and T be self-mappings on X such that f (X) T (X), g(X) S(X): Let (f; g) be a (S; T )-nondecreasing pair such that (d (f x; gy))

' (M (x; y)) ;

(3.1)

for all x; y 2 X such that Sx T y; where 2 and ' 2 : Assume that the following conditions are satis…ed: (a) there exists x0 2 X such that Sx0 f x0 ; (b) Sxn T xn+1 for all n even implies that Sxn T xj for all n even and j > n odd; (c) Sxn T xn+1 for all n even and, Sxn and T xn+1 converge to an x 2 X as n ! 1 implies that Sxn x and x T xn+1 for all n even. Then the pairs (f; S) and (g; T ) have point of coincidence in X. Moreover, if (i) ff; Sg and fg; T g are weakly compatible, (ii) Su T v whenever u 2 C (f; S) and v 2 C (g; T ) : Then f; g; S and T have common …xed point. Moreover, if Sx T y whenever x; y 2 F(f; g; S; T ); then f; g; S and T have a unique common …xed point. Proof. De…ne the function

Let

(Sx; T y)

X ! R+ by ( 1 if x y; (x; y) = 0 otherwise.

:X

1: Then Sx

T y:

(3.2)

From (3.1), we obtain that (d (f x; gy))

' (M (x; y)) :

Also, since (f; g) is (S; T )-nondecreasing, by (3.2) we have f x gy and gx f y; which gives us that (f x; gy) 1 and (gx; f y) 1: Then (f; g) is ST -admissible.

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On the other hand, one can easily show that the hypotheses (a) ; (b) ; (c) and (ii) imply the conditions (a) ; (b) ; (c) and (ii) of Theorem 1. Now, let x; y 2 F(f; g; S; T ): Then, Sx T y and so (Sx; T y) 1: Therefore, the uniqueness of the common …xed point follows from condition (H). If we take ' (t) =

(t)

(t) in Theorem 5, we have the following result.

Corollary 4. Let (X; d; ) be a complete partially ordered metric space and let f; g; S and T be self-mappings on X such that f (X) T (X), g(X) S(X): Let (f; g) be a (S; T )-nondecreasing pair such that (d (f x; gy))

(M (x; y))

(M (x; y)) ;

(3.3)

for all x; y 2 X such that Sx T y; where 2 and ' 2 : Assume that the following conditions are satis…ed: (a) there exists x0 2 X such that Sx0 f x0 ; (b) Sxn T xn+1 for all n even implies that Sxn T xj for all n even and j > n odd; (c) Sxn T xn+1 for all n even and, Sxn and T xn+1 converge to an x 2 X as n ! 1 implies that Sxn x and x T xn+1 for all n even.

Then the pairs (f; S) and (g; T ) have point of coincidence in X. Moreover, if (i) ff; Sg and fg; T g are weakly compatible, (ii) Su T v whenever u 2 C (f; S) and v 2 C (g; T ) :

Then f; g; S and T have common …xed point. Moreover, if Sx T y whenever x; y 2 F(f; g; S; T ); then f; g; S and T have a unique common …xed point. If we take result.

(t) = t and

(t) = (1

k) t in Corollary 4, we have the following

Corollary 5. Let (X; d; ) be a complete partially ordered metric space and let f; g; S and T be self-mappings on X such that f (X) T (X), g(X) S(X): Let (f; g) be a (S; T )-nondecreasing pair such that d (f x; gy)

kM (x; y) ;

(3.4)

for all x; y 2 X such that Sx T y; where k 2 [0; 1): Assume that the following conditions are satis…ed: (a) there exists x0 2 X such that Sx0 f x0 ; (b) Sxn T xn+1 for all n even implies that Sxn T xj for all n even and j > n odd; (c) Sxn T xn+1 for all n even and, Sxn and T xn+1 converge to an x 2 X as n ! 1 implies that Sxn x and x T xn+1 for all n even.

Then the pairs (f; S) and (g; T ) have point of coincidence in X. Moreover, if (i) ff; Sg and fg; T g are weakly compatible, (ii) Su T v whenever u 2 C (f; S) and v 2 C (g; T ) :

Then f; g; S and T have common …xed point. Moreover, if Sx T y whenever x; y 2 F(f; g; S; T ); then f; g; S and T have a unique common …xed point.

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4. Some Results for Graphic Contractions Consistent with Jachymski [18], let (X; d) be a metric space and let := f(x; x) : x 2 Xg be a diagonal of the Cartesian product X X. Consider a graph G such that the set V (G) of its vertices coincides with X and the set E (G) of its edges contains all loops; that is, E (G) . We assume G has no parallel edges, so we can identify G with the pair (V (G) ; E (G)). Moreover, we may treat G as a weighted graph by assigning to each edge the distance between its vertices. If x and y are vertices in a graph G, then a path in G from x to y of length N (N 2 N) is a N sequence fxi gi=0 of N + 1 vertices such that x0 = x, xN = y and (xi 1 ; xi ) 2 E (G) for i = 1; : : : ; N . A graph G is connected if there is a path between any two vertices. G is weakly connected if G is connected (see for more details [2, 9, 10]). In this section, we give the existence and uniqueness of …xed point theorems on a metric space endowed with graph. Before presenting our results, we give the following notions and de…nitions. De…nition 7 ([18]). Let (X; d) be a metric space endowed with a graph G and T : X ! X be a mapping. One says that T preserves edges of G if 8x; y 2 X;

(x; y) 2 E (G) ) (T x; T y) 2 E (G) :

(4.1)

De…nition 8. Let f; g; S and T be selfmaps of a metric space (X; d) endowed with a graph G: One says that (f; g) preserves edges of G with respect to (S; T ) if for all x; y 2 X; (Sx; T y) 2 E (G) ) (f x; gy) 2 E (G) and (gx; f y) 2 E (G) :

(4.2)

De…nition 9. Let (X; d) be a metric space endowed with a graph G and f; g; S and T be selfmaps on X such that (f; g) preserves edges of G with respect to (S; T ) : We say that (f; g) is a generalized ( ; ; ')(S;T ) -graphic contraction involving rational expressions if (d (f x; gy)) ' (M (x; y)) ; (4.3) for all x; y 2 X for which (Sx; T y) 2 E (G) ; where M (x; y)

=

2

,'2

and

d (Sx; gy) + d (f x; T y) ; 2 d (T y; gy) [1 + d (Sx; f x)] d (f x; T y) [1 + d (Sx; gy)] ; : 1 + d (Sx; T y) 1 + d (Sx; T y)

max d (Sx; T y) ; d (Sx; f x) ; d (T y; gy) ;

Theorem 6. Let f; g; S and T be selfmaps of a metric space (X; d) endowed with a graph G, and f (X) T (X), g(X) S(X) and (f; g) be a generalized ( ; ; ')(S;T ) graphic contraction involving rational expressions. Assume that the following conditions are satis…ed: (a) there exists x0 2 X such that (Sx0 ; f x0 ) 2 E (G); (b) (Sxn ; T xn+1 ) 2 E (G) for all n even implies that (Sxn ; T xj ) 2 E (G) for all n even and j > n odd; (c) (Sxn ; T xn+1 ) 2 E (G) for all n even and, Sxn and T xn+1 converge to an x 2 X as n ! 1 implies that (Sxn ; x) 2 E (G) and (x; T xn+1 ) 2 E (G) for all n even. Then the pairs (f; S) and (g; T ) have a point of coincidence in X. Moreover, if (i) ff; Sg and fg; T g are weakly compatible,

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(ii) (Su; T v) 2 E (G) whenever u 2 C (f; S) and v 2 C (g; T ) : Then f; g; S and T have common …xed point. Moreover, if (Sx; T y) 2 E (G) whenever x; y 2 F(f; g; S; T ); then f; g; S and T have a unique common …xed point. Proof. De…ne the function

Let

(Sx; T y)

X ! R+ by ( 1 if (x; y) 2 E (G) ; (x; y) = 0 otherwise. :X

1: Then

From (4.3), we obtain that

(Sx; T y) 2 E (G) : (d (f x; gy))

(4.4)

' (M (x; y)) :

Also, since (f; g) preserves edges of G with respect to (S; T ), by (4.4) we have 1 and (f x; gy) 2 E (G) and (gx; f y) 2 E (G) ; which gives us that (f x; gy) (gx; f y) 1: Then (f; g) is ST -admissible. On the other hand, it is easy to see that the hypotheses (a) ; (b) ; (c) and (ii) imply the conditions (a) ; (b) ; (c) and (ii) of Theorem 1. 1: Now, let x; y 2 F(f; g; S; T ): Then, (Sx; T y) 2 E (G) and so (Sx; T y) Therefore, the uniqueness of the common …xed point follows from condition (H). If we take ' (t) =

(t)

(t) in Theorem 6, we have the following result.

Corollary 6. Let f; g; S and T be selfmaps of a metric space (X; d) endowed with a graph G, and f (X) T (X), g(X) S(X). Assume that (f; g) preserves edges of G with respect to (S; T ) such that (d (f x; gy))

(M (x; y))

(M (x; y)) ;

(4.5)

for all x; y 2 X for which (Sx; T y) 2 E (G) ; where 2 and 2 . Suppose also that the following conditions are satis…ed: (a) there exists x0 2 X such that (Sx0 ; f x0 ) 2 E (G); (b) (Sxn ; T xn+1 ) 2 E (G) for all n even implies that (Sxn ; T xj ) 2 E (G) for all n even and j > n odd; (c) (Sxn ; T xn+1 ) 2 E (G) for all n even and, Sxn and T xn+1 converge to an x 2 X as n ! 1 implies that (Sxn ; x) 2 E (G) and (x; T xn+1 ) 2 E (G) for all n even. Then the pairs (f; S) and (g; T ) have a point of coincidence in X. Moreover, if (i) ff; Sg and fg; T g are weakly compatible and, (ii) (Su; T v) 2 E (G) whenever u 2 C (f; S) and v 2 C (g; T ) : Then f; g; S and T have common …xed point. Moreover, if (Sx; T y) 2 E (G) whenever x; y 2 F(f; g; S; T ); then f; g; S and T have a unique common …xed point. If we take result.

(t) = t and

(t) = (1

k) t in Corollary 6, we have the following

Corollary 7. Let f; g; S and T be selfmaps of a metric space (X; d) endowed with a graph G, and f (X) T (X), g(X) S(X). Assume that (f; g) preserves edges of G with respect to (S; T ) such that d (f x; gy)

kM (x; y) ;

1171

(4.6)

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GENERALIZED RATIONAL CONTRACTIONS ENDOW ED W ITH A GRAPH

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for all x; y 2 X for which (Sx; T y) 2 E (G) ; where 2 and 2 . Suppose also that the following conditions are satis…ed: (a) there exists x0 2 X such that (Sx0 ; f x0 ) 2 E (G); (b) (Sxn ; T xn+1 ) 2 E (G) for all n even implies that (Sxn ; T xj ) 2 E (G) for all n even and j > n odd; (c) (Sxn ; T xn+1 ) 2 E (G) for all n even and, Sxn and T xn+1 converge to an x 2 X as n ! 1 implies that (Sxn ; x) 2 E (G) and (x; T xn+1 ) 2 E (G) for all n even. Then the pairs (f; S) and (g; T ) have a point of coincidence in X. Moreover, if (i) ff; Sg and fg; T g are weakly compatible and, (ii) (Su; T v) 2 E (G) whenever u 2 C (f; S) and v 2 C (g; T ) : Then f; g; S and T have common …xed point. Moreover, if (Sx; T y) 2 E (G) whenever x; y 2 F(f; g; S; T ); then f; g; S and T have a unique common …xed point. 5. An Application Consider the following integral equations: Z b x (s) = H1 (s; r; x (r)) dr;

(5.1)

a

and

x (s) =

Z

b

H2 (s; r; x (r)) dr;

(5.2)

a

where s; r 2 I = [a; b] ; H1 ; H2 : I I R ! R and b > a 0. In this section, we present an existence and uniqueness theorem for a common solution to (5.1) and (5.2) that belongs to X := C(I; R) (the set of continuous functions de…ned on I) by using the obtained result in Corollary 3. We consider the operators f; g : X ! X given by for all x 2 X Z b f x(s) = H1 (s; r; x (r)) dr; s 2 I; a

and

gx(s) =

Z

b

H2 (s; r; x (r)) dr;

a

s 2 I:

Then the existence of a common solution to (5.1) and (5.2) are equivalent to the existence of a common …xed point of f and g. Meanwhile, X endowed with the metric d de…ned by d(x; y) = sup jx (s) s2I

y (s) j

for all x; y 2 X; is a complete metric space. Suppose that the following conditions hold. (A1) H1 ; H2 : I I R ! R are continuous; (A2) there exist : X X ! R such that if (x; y) every s; r 2 I, we have jH1 (s; r; x (r))

2

H2 (s; r; y (r))j

1172

0 for all x; y 2 X; then for

(s; r) ln 1 + jx (r)

2

y (r)j

HUSEYIN ISIK et al 1158-1175

J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 22, NO.6, 2017, COPYRIGHT 2017 EUDOXUS PRESS, LLC

HUSEYIN ISIK y , NAWAB HUSSAIN, AND M ARWAN A. KUTBI

16

where : I I ! R+ is a continuous function satisfying sups2I 1= (b a) ; (A3) for every s 2 I there exist x0 2 X such that (x0 (s) ; f x0 (s)) (A4) for all s 2 I and x; y 2 X; (x(s); y(s))

0 )

(f x(s); gy(s))

Rb

(s; r)

a

0 and (gx(s); f y(s))

0;

0;

(A5)

(xn (s) ; xn+1 (s)) 0 for all n and s 2 I implies that (xn (s) ; xj (s)) 0 for all j > n; (A6) (xn (s) ; xn+1 (s)) 0 for all n and s 2 I and, xn ! x 2 X as n ! 1 implies that (xn (s) ; x (s)) 0 for all n.

Theorem 7. Assume that the conditions (A1) (A6) are satis…ed. Then, integral equations (5.1) and (5.2) have a common solution in X. Proof. Let x; y 2 X such that (x; y) jf x(s)

2

gy (s)j

Z

b

a

Z

jH1 (s; r; x (r))

b

12 dr

a

(b

0: Then, by (A2), for all s; r 2 I, we deduce !2

a)

Z

Z

b

jH1 (s; r; x (r))

a b

a)

Z

2

H2 (s; r; y (r))j dr 2

(s; r) ln 1 + jx (r)

a

(b

H2 (s; r; y (r))j dr

b

y (r)j

dr

2

(s; r) ln 1 + d (x; y) dr ! Z b 2 a) (s; r) dr ln 1 + d (x; y) a

=

(b

a

2

ln 1 + d (x; y)

2

ln 1 + Mf g (x; y)

;

where Mf g (x; y)

=

max d (x (s) ; y (s)) ; d (x (s) ; f x (s)) ; d (y (s) ; gy (s)) ; d (x (s) ; gy (s)) + d (f x (s) ; y (s)) ; 2 d (y (s) ; gy (s)) [1 + d (x (s) ; f x (s))] ; 1 + d (x (s) ; y (s)) d (f x (s) ; y (s)) [1 + d (x (s) ; gy (s))] : 1 + d (x (s) ; y (s))

Therefore, we obtain 2

sup jf x(s)

2

gy (s)j

ln 1 + Mf g (x; y)

:

s2I

Now, de…ne

X ! R+ by ( 1 if (x; y) 0 where x; y 2 X; (x; y) = 0 otherwise.

:X

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J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 22, NO.6, 2017, COPYRIGHT 2017 EUDOXUS PRESS, LLC

GENERALIZED RATIONAL CONTRACTIONS ENDOW ED W ITH A GRAPH

Also, de…ne ; ' : R+ ! R+ by using the last inequality, we have (x; y)

17

(t) = t2 and ' (t) = ln 1 + t2 : Therefore,

(d (f x; gy))

' (Mf g (x; y)) :

It easily shows that all the hypotheses of Corollary 3 are satis…ed. Therefore f and g have a common …xed point, that is, integral equations (5.1) and (5.2) have a common solution. References [1] M. Abbas, D. Doric, Common …xed point theorem for four mappings satisfying generalized weak contractive condition, Filomat, 24 (2) (2010) 1-10. [2] M. Abbas, T. Nazir, Common …xed point of a power graphic contraction pair in partial metric spaces endowed with a graph, Fixed Point Theory Appl., (2013) 2013:20. [3] T. Abdeljawad, Meir-Keeler -contractive …xed and common …xed point theorems, Fixed Point Theory Appl., 2013, (2013) 2013:19. [4] R.P. Agarwal, N. Hussain, M.A. Taoudi, Fixed point theorems in ordered Banach spaces and applications to nonlinear integral equations, Abstr. Appl. Anal., 2012 (2012), Article ID 245872, 15 pp. [5] S. Alizadeh, F. Moradlou, P. Salimi, Some …xed point results for ( ; )-( ; ')-contractive mappings, Filomat, 28 (3) (2014) 635-647. [6] A.H. Ansari, H. Isik, S. Radenovi´c, Coupled …xed point theorems for contractive mappings involving new function classes and applications, Filomat, to appear. [7] H. Aydi, M. Abbas, C. Vetro, Common …xed points for multivalued generalized contractions on partial metric spaces, RACSAM, 108 (2) (2014) 483-501. [8] S. Banach, Sur les opérations dans les ensembles abstraits et leur application aux équations intégrales, Fundam. Math., 3 (1922) 133-181. [9] I. Beg, A.R. Butt, S. Radojevic, The contraction principle for set valued mappings on a metric space with a graph, Comput. Math. Appl., 60 (5) (2010) 1214–1219. [10] F. Bojor, Fixed point theorems for Reich type contractions on metric spaces with a graph, Nonlinear Anal., 75 (9) (2012) 3895–3901. [11] B.S. Choudhury, N. Metiya, C. Bandyopadhyay, Fixed points of multivalued -admissible mappings and stability of …xed point sets in metric spaces, Rend. Circ. Mat. Palermo, (2015) 64:43–55. [12] J. Esmaily, S.M. Vaezpour, B.E. Rhoades, Coincidence point theorem for generalized weakly contractions in ordered metric spaces, Appl. Math. Comput., 219 (4) (2012) 1536–1548. [13] N. Hussain, S. Al-Mezel and P. Salimi, Fixed points for -graphic contractions with application to integral equations, Abstr. Appl. Anal., Vol. 2013, Article ID 575869. [14] N. Hussain, M.A. Taoudi, Krasnosel’skii-type …xed point theorems with applications to Volterra integral equations, Fixed Point Theory Appl., 2013, 2013:196. [15] H. Isik, B. Samet, C. Vetro, Cyclic admissible contraction and applications to functional equations in dynamic programming, Fixed Point Theory Appl., 2015 (2015), 19 pages. [16] H. Isik, D. Turkoglu, Fixed point theorems for weakly contractive mappings in partially ordered metric-like spaces, Fixed Point Theory Appl., 2013 (2013), 12 pages. [17] H. Isik, D. Turkoglu, Generalized weakly -contractive mappings and applications to ordinary di¤erential equations, Miskolc Mathematical Notes, to appear. [18] J. Jachymski, The contraction principle for mappings on a metric space with a graph, Proc. Amer. Math. Soc., 136 (4) (2008) 1359–1373. [19] G. Jungck, B.E. Rhoades, Fixed points for set valued functions without continuity, Indian J. Pure Appl. Math., 29 (1998) 227–238. [20] M.S. Khan, M. Swaleh, S. Sessa, Fixed point theorems by altering distances between the points, Bulletin of the Australian Mathematical Society, 30 (1) (1984) 1–9. [21] P. Kumam, C. Vetro, F. Vetro, Fixed points for weak - -contractions in partial metric spaces, Abstr. Appl. Anal., 2013, 986028, 9 pp., 2013. [22] V. La Rosa, P. Vetro, Common …xed points for - -'-contractions in generalized metric spaces, Nonlinear Anal. Model. Control, 19 (1) (2014) 43-54.

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HUSEYIN ISIK y , NAWAB HUSSAIN, AND M ARWAN A. KUTBI

[23] A. Latif, H. Isik, A.H. Ansari, Fixed points and functional equation problems via cyclic admissible generalized contractive type mappings, J. Nonlinear Sci. Appl., 9 (2016), 11291142. [24] J.J. Nieto, R. Rodríguez-López, Contractive mapping theorems in partially ordered sets and applications to ordinary di¤erential equations, Order 22 (2005) 223–239. [25] D.K. Patel, T. Abdeljawad, D. Gopal, Common …xed points of generalized Meir-Keeler contractions, Fixed Point Theory Appl., (2013) 2013:260. [26] A.C.M. Ran, M.C.B. Reurings, A …xed point theorem in partially ordered sets and some applications to matrix equations, Proc. Amer. Math. Soc., 132 (2004) 1435–1443. [27] P. Salimi, C. Vetro, P. Vetro, Fixed point theorems for twisted ( ; )- -contractive type mappings and applications, Filomat, 27(4) (2013) 605-615. [28] B. Samet, C. Vetro, P. Vetro, Fixed point theorems for - -contractive type mappings, Nonlinear Anal., 75 (2012) 2154–2165. [29] Y. Su, Contraction mapping principle with generalized altering distance function in ordered metric spaces and applications to ordinary di¤erential equations, Fixed Point Theory Appl., (2014) 2014:227. Huseyin Isik, Department of Mathematics, Faculty of Science, Gazi University, 06500Teknikokullar, Ankara, Turkey, Department of Mathematics, Faculty of Science and Arts, MuS¸ Alparslan University, MuS¸ 49100, Turkey E-mail address : [email protected] Nawab Hussain, Department of Mathematics, King Abdulaziz University, P.O. Box 80203, Jeddah, 21589, Saudi Arabia E-mail address : [email protected] Marwan A. Kutbi, Department of Mathematics, King Abdulaziz University, P.O. Box 80203, Jeddah, 21589, Saudi Arabia E-mail address : [email protected]

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TABLE OF CONTENTS, JOURNAL OF COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 22, NO. 6, 2017

A New Result on the Almost Increasing Sequences, H. S. Ozarslan and A. Karakas,………989 Certain Chebyshev Type Inequalities Involving the Generalized Fractional Integral Operator, Zhen Liu, Wengui Yang, and Praveen Agarwal,………………………………………………999 Estimates for the Green's Function of 3D Elliptic Equations, Jinghong Liu and Yinsuo Jia,..1015 The Structure of the Zeros Fixed Point for Genocchi Polynomials, J. Y. Kang, C. S. Ryoo,.1023 Additive 𝜌-Functional Equations, Choonkil Park and Sun Young Jang,…………………….1035

Hyperstability of a Generalized Cauchy Functional Equation, Abbas Najati, Daryoush Molaee, and Choonkil Park,……………………………………………………………………………1049 Stability Analysis and Optimal Control of a Cholera Model with Time Delay, Shu Liao and Fang Fang,…………………………………………………………………………………………..1055 Effect of Antibodies and Latently Infected Cells on HIV Dynamics with Differential Drug Efficacy in Cocirculating Target Cells, A. M. Shehata, A. M. Elaiw, and E. Kh. Elnahary,…1074 A New Implicit Midpoint Iterative Scheme Involving Asymptotically Nonexpansive Mappings in Abstract Spaces, Shin Min Kang, Arif Rafiq, Faisal Ali, and Young Chel Kwun,………………………………………………………………………………………….1094 Hesitant Fuzzy Filters in Lattice Implication Algebras, G. Muhiuddin, Eun Hwan Roh, Sun Shin Ahn, and Young Bae Jun,…………………………………………………………………….1105 3D Green's Function and Its Finite Element Error Estimates, Jinghong Liu and Yinsuo Jia,..1114 Hermite-Hadamard Type Inequalities for s-Convex Functions via Riemann-Liouville Fractional Integrals, Shu-Hong Wang and Feng Qi,……………………………………………………..1124 Monotone Hybrid Projection Algorithm for Solving Fixed Point and Equilibrium Problems in a Banach Space, Xiaoying Gong and Sun Young Cho,…………………………………………1135 Inner-Outer Factorization on Besov-Type Spaces, Ruishen Qian and Songxiao Li,…………1150 Generalized Rational Contractions Endowed With a Graph and an Application to a System of Integral Equations, Huseyin Isik, Nawab Hussain, and Marwan A. Kutbi,………………….1158

Volume 22, Number 7 ISSN:1521-1398 PRINT,1572-9206 ONLINE

Journal of Computational Analysis and Applications EUDOXUS PRESS,LLC

June 15, 2017

Journal of Computational Analysis and Applications ISSNno.’s:1521-1398 PRINT,1572-9206 ONLINE SCOPE OF THE JOURNAL An international publication of Eudoxus Press, LLC (fourteen times annually) Editor in Chief: George Anastassiou Department of Mathematical Sciences, University of Memphis, Memphis, TN 38152-3240, U.S.A [email protected] http://www.msci.memphis.edu/~ganastss/jocaaa The main purpose of "J.Computational Analysis and Applications" is to publish high quality research articles from all subareas of Computational Mathematical Analysis and its many potential applications and connections to other areas of Mathematical Sciences. Any paper whose approach and proofs are computational,using methods from Mathematical Analysis in the broadest sense is suitable and welcome for consideration in our journal, except from Applied Numerical Analysis articles. Also plain word articles without formulas and proofs are excluded. The list of possibly connected mathematical areas with this publication includes, but is not restricted to: Applied Analysis, Applied Functional Analysis, Approximation Theory, Asymptotic Analysis, Difference Equations, Differential Equations, Partial Differential Equations, Fourier Analysis, Fractals, Fuzzy Sets, Harmonic Analysis, Inequalities, Integral Equations, Measure Theory, Moment Theory, Neural Networks, Numerical Functional Analysis, Potential Theory, Probability Theory, Real and Complex Analysis, Signal Analysis, Special Functions, Splines, Stochastic Analysis, Stochastic Processes, Summability, Tomography, Wavelets, any combination of the above, e.t.c. "J.Computational Analysis and Applications" is a peer-reviewed Journal. See the instructions for preparation and submission of articles to JoCAAA. Assistant to the Editor: Dr.Razvan Mezei,Lenoir-Rhyne University,Hickory,NC

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Journal of Computational Analysis and Applications(JoCAAA) is published by EUDOXUS PRESS,LLC,1424 Beaver Trail Drive,Cordova,TN38016,USA,[email protected] http://www.eudoxuspress.com. Annual Subscription Prices:For USA and Canada,Institutional:Print $700, Electronic OPEN ACCESS. Individual:Print $350. For any other part of the world add $130 more(handling and postages) to the above prices for Print. No credit card payments. Copyright©2017 by Eudoxus Press,LLC,all rights reserved.JoCAAA is printed in USA. JoCAAA is reviewed and abstracted by AMS Mathematical Reviews,MATHSCI,and Zentralblaat MATH. It is strictly prohibited the reproduction and transmission of any part of JoCAAA and in any form and by any means without the written permission of the publisher.It is only allowed to educators to Xerox articles for educational purposes.The publisher assumes no responsibility for the content of published papers.

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Editorial Board Associate Editors of Journal of Computational Analysis and Applications Francesco Altomare Dipartimento di Matematica Universita' di Bari Via E.Orabona, 4 70125 Bari, ITALY Tel+39-080-5442690 office +39-080-3944046 home +39-080-5963612 Fax [email protected] Approximation Theory, Functional Analysis, Semigroups and Partial Differential Equations, Positive Operators.

Fractional Differential Equations Nonlinear Analysis, Fractional Dynamics Carlo Bardaro Dipartimento di Matematica e Informatica Universita di Perugia Via Vanvitelli 1 06123 Perugia, ITALY TEL+390755853822 +390755855034 FAX+390755855024 E-mail [email protected] Web site: http://www.unipg.it/~bardaro/ Functional Analysis and Approximation Theory, Signal Analysis, Measure Theory, Real Analysis.

Ravi P. Agarwal Department of Mathematics Texas A&M University - Kingsville 700 University Blvd. Kingsville, TX 78363-8202 tel: 361-593-2600 [email protected] Differential Equations, Difference Equations, Inequalities

Martin Bohner Department of Mathematics and Statistics, Missouri S&T Rolla, MO 65409-0020, USA [email protected] web.mst.edu/~bohner Difference equations, differential equations, dynamic equations on time scale, applications in economics, finance, biology.

George A. Anastassiou Department of Mathematical Sciences The University of Memphis Memphis, TN 38152,U.S.A Tel.901-678-3144 e-mail: [email protected] Approximation Theory, Real Analysis, Wavelets, Neural Networks, Probability, Inequalities.

Jerry L. Bona Department of Mathematics The University of Illinois at Chicago 851 S. Morgan St. CS 249 Chicago, IL 60601 e-mail:[email protected] Partial Differential Equations, Fluid Dynamics

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Dumitru Baleanu Department of Mathematics and Computer Sciences, Cankaya University, Faculty of Art and Sciences, 06530 Balgat, Ankara, Turkey, [email protected]

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Dartmouth College 8000 Cummings Hall, Hanover, NH 03755-8000 603-646-3843 (X 3546 Secr.) e-mail:[email protected] Approximation Theory and Neural Networks

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Instructions to Contributors Journal of Computational Analysis and Applications An international publication of Eudoxus Press, LLC, of TN.

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J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 22, NO.7, 2017, COPYRIGHT 2017 EUDOXUS PRESS, LLC

ON QUADRATIC ρ-FUNCTIONAL INEQUALITIES IN FUZZY NORMED SPACES CHOONKIL PARK, SUN YOUNG JANG, AND SUNGSIK YUN∗ Abstract. In this paper, we solve the following quadratic ρ-functional inequalities         x−y x+y + 2f − f (x) − f (y) , t N f (x + y) + f (x − y) − 2f (x) − 2f (y) − ρ 2f 2 2 t ≥ , (0.1) t + ϕ(x, y) where ρ is a fixed real number with ρ 6= 2, and       x+y x−y N 2f + 2f − f (x) − f (y) − ρ (f (x + y) + f (x − y) − 2f (x) − 2f (y)) , t 2 2 t ≥ , (0.2) t + ϕ(x, y) where ρ is a fixed real number with ρ 6= 21 . Using the direct method, we prove the Hyers-Ulam stability of the quadratic ρ-functional inequalities (0.1) and (0.2) in fuzzy Banach spaces.

1. Introduction and preliminaries Katsaras [14] defined a fuzzy norm on a vector space to construct a fuzzy vector topological structure on the space. Some mathematicians have defined fuzzy norms on a vector space from various points of view [9, 16, 37]. In particular, Bag and Samanta [2], following Cheng and Mordeson [6], gave an idea of fuzzy norm in such a manner that the corresponding fuzzy metric is of Kramosil and Michalek type [15]. They established a decomposition theorem of a fuzzy norm into a family of crisp norms and investigated some properties of fuzzy normed spaces [3]. We use the definition of fuzzy normed spaces given in [2, 19, 20] to investigate the Hyers-Ulam stability of quadratic ρ-functional inequalities in fuzzy Banach spaces. Definition 1.1. [2, 19, 20, 21] Let X be a real vector space. A function N : X × R → [0, 1] is called a fuzzy norm on X if for all x, y ∈ X and all s, t ∈ R, (N1 ) N (x, t) = 0 for t ≤ 0; (N2 ) x = 0 if and only if N (x, t) = 1 for all t > 0; t (N3 ) N (cx, t) = N (x, |c| ) if c 6= 0; (N4 ) N (x + y, s + t) ≥ min{N (x, s), N (y, t)}; (N5 ) N (x, ·) is a non-decreasing function of R and limt→∞ N (x, t) = 1. (N6 ) for x 6= 0, N (x, ·) is continuous on R. The pair (X, N ) is called a fuzzy normed vector space. The properties of fuzzy normed vector spaces and examples of fuzzy norms are given in [18, 19]. Definition 1.2. [2, 19, 20, 21] Let (X, N ) be a fuzzy normed vector space. A sequence {xn } in X is said to be convergent or converge if there exists an x ∈ X such that limn→∞ N (xn − x, t) = 1 2010 Mathematics Subject Classification. Primary 46S40, 39B52, 39B62, 26E50, 47S40. Key words and phrases. fuzzy Banach space; quadratic ρ-functional inequality; Hyers-Ulam stability. ∗ Corresponding author.

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J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 22, NO.7, 2017, COPYRIGHT 2017 EUDOXUS PRESS, LLC

C. PARK, S. Y. JANG, AND S. YUN

for all t > 0. In this case, x is called the limit of the sequence {xn } and we denote it by N -limn→∞ xn = x. Definition 1.3. [2, 19, 20, 21] Let (X, N ) be a fuzzy normed vector space. A sequence {xn } in X is called Cauchy if for each ε > 0 and each t > 0 there exists an n0 ∈ N such that for all n ≥ n0 and all p > 0, we have N (xn+p − xn , t) > 1 − ε. It is well-known that every convergent sequence in a fuzzy normed vector space is Cauchy. If each Cauchy sequence is convergent, then the fuzzy norm is said to be complete and the fuzzy normed vector space is called a fuzzy Banach space. We say that a mapping f : X → Y between fuzzy normed vector spaces X and Y is continuous at a point x0 ∈ X if for each sequence {xn } converging to x0 in X, then the sequence {f (xn )} converges to f (x0 ). If f : X → Y is continuous at each x ∈ X, then f : X → Y is said to be continuous on X (see [3]). The stability problem of functional equations originated from a question of Ulam [36] concerning the stability of group homomorphisms. The functional equation f (x + y) = f (x) + f (y) is called the Cauchy equation. In particular, every solution of the Cauchy equation is said to be an additive mapping. Hyers [11] gave a first affirmative partial answer to the question of Ulam for Banach spaces. Hyers’ Theorem was generalized by Aoki [1] for additive mappings and by Th.M. Rassias [27] for linear mappings by considering an unbounded Cauchy difference. A generalization of the Th.M. Rassias theorem was obtained by G˘avruta [10] by replacing the unbounded Cauchy difference by a general control function in the spirit of Th.M. Rassias’ approach. The functional equation f (x + y) + f (x − y) = 2f (x) + 2f (y) is called the quadratic functional equation. In particular, every solution of the quadratic functional equation is said to be a quadratic mapping. The stability of quadratic functional equation was proved by Skof [35] for mappings f : E1 → E2 , where E1 is a normed space and E2 is a Banach space. Cholewa [7] noticed that the theorem of Skof is still true if the relevant domain E1 is replaced by an Abelian group. Czerwik [8] proved Hyers-Ulam stability of the quadratic functional equation. The  the   x+y x−y 1 functional equation f 2 +f 2 = 2 f (x)+ 12 f (y) is called a Jensen type quadratic equation. The stability problems of several functional equations have been extensively investigated by a number of authors and there are many interesting results concerning this problem (see [5, 12, 13, 17, 24, 25, 26, 28, 29, ?, 30, 31, 32, 33, 34]). Park [22, 23] defined additive ρ-functional inequalities and proved the Hyers-Ulam stability of the additive ρ-functional inequalities in Banach spaces and non-Archimedean Banach spaces. In Section 2, we solve the quadratic ρ-functional inequality (0.1) and prove the Hyers-Ulam stability of the quadratic ρ-functional inequality (0.1) in fuzzy Banach spaces by using the direct method. In Section 3, we solve the quadratic ρ-functional inequality (0.2) and prove the Hyers-Ulam stability of the quadratic ρ-functional inequality (0.2) in fuzzy Banach spaces by using the direct method. Throughout this paper, assume that X is a real vector space and (Y, N ) is a fuzzy Banach space. 2. Quadratic ρ-functional inequality (0.1) In this section, we prove the Hyers-Ulam stability of the quadratic ρ-functional inequality (0.1) in fuzzy Banach spaces. Let ρ be a real number with ρ 6= 2. We need the following lemma to prove the main results.

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J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 22, NO.7, 2017, COPYRIGHT 2017 EUDOXUS PRESS, LLC

QUADRATIC ρ-FUNCTIONAL INEQUALITIES IN FUZZY NORMED SPACES

Lemma 2.1. Let f : X → Y be a mapping satisfying f (0) = 0 and       x+y x−y f (x + y) + f (x − y) − 2f (x) − 2f (y) = ρ 2f + 2f − f (x) − f (y) 2 2 for all x, y ∈ X. Then f : X → Y is quadratic.

(2.1)

Proof. Replacing y by x in (2.1), we get f (2x) − 4f (x) = 0 and so f (2x) = 4f (x) for all x ∈ X. Thus       x+y x−y f (x + y) + f (x − y) − 2f (x) − 2f (y) = ρ 2f + 2f − f (x) − f (y) 2 2 ρ = (f (x + y) + f (x − y) − 2f (x) − 2f (y)) 2 and so f (x + y) + f (x − y) = 2f (x) + 2f (y) for all x, y ∈ X, as desired.  Theorem 2.2. Let ϕ : X 2 → [0, ∞) be a function such that ∞ X

x y Φ(x, y) := 4 ϕ j, j 2 2 j=1 

j



0. Then Q(x) := N -limn→∞ 4n f defines a quadratic mapping Q : X → Y such that t N (f (x) − Q(x), t) ≥ t + 41 Φ(x, x)

x 2n




(2.3)

exists for each x ∈ X and (2.4)

for all x ∈ X and all t > 0. Proof. Letting y = x in (2.3), we get t t + ϕ(x, x)

N (f (2x) − 4f (x), t) ≥ and so N f (x) − 4f 

l



N 4f

x 2l





x 2







,t ≥

x ,t 2m

x 2l



−4 f

≥ min N 4l f





x 2l

 

= min N f ≥ min



m



 

t t+ϕ( x2 , x2 )



for all x ∈ X. Hence



(2.6)

− 4l+1 f



x



x



− 4f



2l+1 

4l+1

t , l 4

l

 t +ϕ 4l  

x

x , 2l+1

, · · · ,





, t , · · · , N 4m−1 f



t

4

(2.5)

,··· ,N f

2m−1

+ϕ

x x 2m , 2m

x



2m−1





− 4f

− 4m f



x ,t 2m

x t , m−1 m 2 4 







 

t 4m−1 t 4m−1

x

 






   t t , · · · , 
= min  t + 4l ϕ x , x t + 4m−1 ϕ 2xm , 2xm  2l+1 2l+1 2l+1

t

≥ t+

1 4

Pm

j j=l+1 4 ϕ



x x , 2j 2j



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J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 22, NO.7, 2017, COPYRIGHT 2017 EUDOXUS PRESS, LLC

C. PARK, S. Y. JANG, AND S. YUN

for all nonnegative integers m and l with m > l and all x ∈ X and all t > 0. It follows from (2.2) and (2.6) that the sequence {4n f ( 2xn )} is a Cauchy sequence for all x ∈ X. Since Y is complete, the sequence {4n f ( 2xn )} converges. So one can define the mapping Q : X → Y by x ) n→∞ 2n for all x ∈ X. Moreover, letting l = 0 and passing the limit m → ∞ in (2.6), we get (2.4). By (2.3), Q(x) := N - lim 4n f (



n

N 4

 

f 

−ρ 4

n

x+y 2n





2f





+f x+y 2n+1

x−y 2n







+ 2f



− 2f x−y 2n+1

x 2n





− 2f





−f

x 2n





x 2n

y 2n

 

−f

y 2n



n

t



,4 t ≥

t+ϕ


x y 2n , 2n

for all x, y ∈ X, all t > 0 and all n ∈ N. So 

n

 

N 4

f

x+y 2n





+f

x−y 2n



− 2f





− 2f

x−y − ρ 4 2f + 2f −f 2n+1 t t 4n

= ≥ t x y x y n t + 4 ϕ 2n , 2n 4n + ϕ 2n , 2n 

n





x+y 2n+1







for all x, y ∈ X, all t > 0 and all n ∈ N. Since limn→∞ t > 0, 



Q(x + y) + Q(x − y) − 2Q(x) − 2Q(y) = ρ 2Q



x 2n

y 2n







−f

t t+4n ϕ( 2xn , 2yn )

x+y 2





+ 2Q

y 2n





,t

= 1 for all x, y ∈ X and all

x−y 2





− Q(x) − Q(y)

for all x, y ∈ X. By Lemma 2.1, the mapping Q : X → Y is quadratic, as desired.



Corollary 2.3. Let θ ≥ 0 and let p be a real number with p > 2. Let X be a normed vector space with norm k · k. Let f : X → Y be a mapping satisfying f (0) = 0 and 



N f (x + y) + f (x − y) − 2f (x) − 2f (y) − ρ 2f ≥



x+y 2





+ 2f

t t + θ(kxkp + kykp )

x−y 2







− f (x) − f (y) , t (2.7)

for all x, y ∈ X and all t > 0. Then Q(x) := N -limn→∞ 4n f ( 2xn ) exists for each x ∈ X and defines a quadratic mapping Q : X → Y such that N (f (x) − Q(x), t) ≥

(2p − 4)t (2p − 4)t + 2θkxkp

for all x ∈ X. Proof. The proof follows from Theorem 2.2 by taking ϕ(x, y) := θ(kxkp + kykp ) for all x, y ∈ X, as desired.  Theorem 2.4. Let ϕ : X 2 → [0, ∞) be a function such that Φ(x, y) :=

∞ X 1 j=0

4





ϕ 2j x, 2j y < ∞ j

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J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 22, NO.7, 2017, COPYRIGHT 2017 EUDOXUS PRESS, LLC

QUADRATIC ρ-FUNCTIONAL INEQUALITIES IN FUZZY NORMED SPACES

for all x, y ∈ X. Let f : X → Y be a mapping satisfying f (0) = 0 and (2.3). Then Q(x) := N limn→∞ 41n f (2n x) exists for each x ∈ X and defines a quadratic mapping Q : X → Y such that 1 N (f (x) − Q(x), t) ≥ 1 t + 4 Φ(x, x) for all x ∈ X and all t > 0. 



Proof. It follows from (2.5) that N f (x) − 14 f (2x), 41 t ≥

t t+ϕ(x,x)

and so

1 4t t N f (x) − f (2x), t ≥ = 1 4 4t + ϕ(x, x) t + 4 ϕ(x, x) 



for all x ∈ X and all t > 0. The rest of the proof is similar to the proof of Theorem 2.2.



Corollary 2.5. Let θ ≥ 0 and let p be a real number with 0 < p < 2. Let X be a normed vector space with norm k · k. Let f : X → Y be a mapping satisfying f (0) = 0 and (2.7). Then Q(x) := N -limn→∞ 41n f (2n x) exists for each x ∈ X and defines a quadratic mapping Q : X → Y such that (4 − 2p )t N (f (x) − Q(x), t) ≥ (4 − 2p )t + 2θkxkp for all x ∈ X. Proof. The proof follows from Theorem 2.4 by taking ϕ(x, y) := θ(kxkp + kykp ) for all x, y ∈ X, as desired.  3. Quadratic ρ-functional inequality (0.2) In this section, we prove the Hyers-Ulam stability of the quadratic ρ-functional inequality (0.2) in fuzzy Banach spaces. Let ρ be a real number with ρ 6= 21 . We need the following lemma to prove the main results. Lemma 3.1. Let f : X → Y be a mapping satisfying f (0) = 0 and 

2f

x+y 2





+ 2f

x−y 2



− f (x) − f (y) = ρ (f (x + y) + f (x − y) − 2f (x) − 2f (y))

(3.1)

for all x, y ∈ X. Then f : X → Y is quadratic. Proof. Letting y = 0 in (3.1), we get 4f x2 − f (x) = 0 and so f (2x) = 4f (x) for all x ∈ X. Thus     1 1 x+y x−y f (x + y) − f (x − y) − f (x) − f (y) = 2f + 2f − f (x) − f (y) 2 2 2 2 = ρ(f (x + y) + f (x − y) − 2f (x) − 2f (y))


and so f (x + y) + f (x − y) = 2f (x) + 2f (y) for all x, y ∈ X, as desired.



Theorem 3.2. Let ϕ : X 2 → [0, ∞) be a function such that ∞ X

x y Φ(x, y) := 4 ϕ j, j 2 2 j=0 j



1193



0. Then Q(x) := N -limn→∞ 4n f defines a quadratic mapping Q : X → Y such that N (f (x) − Q(x), t) ≥

x 2n




exists for each x ∈ X and

t t + Φ(x, 0)

(3.4)

for all x ∈ X and all t > 0. Proof. Letting y = 0 in (3.3), we get x , t = N 4f 2

 



N f (x) − 4f





x 2

 



− f (x), t ≥

t t + ϕ(x, 0)

(3.5)

for all x ∈ X. Hence 

l



N 4f

x 2l





m

x ,t 2m

x 2l



−4 f

≥ min N 4l f





x 2l



 

= min N f  

≥ min





(3.6)

− 4l+1 f



t l 4



x



− 4f

x





2l+1 

2l+1

t , l 4





, t , · · · , N 4m−1 f



,··· ,N f

2m−1

x



2m−1





− 4f

− 4m f



x ,t 2m

x t , m−1 m 2 4 







 

t 4m−1 

, · · · ,

x

 



 x t   t + ϕ x ,0 + ϕ , 0 4m−1 2m−1 4l 2l     t t   , · · · ,  = min x  t + 4l ϕ x , 0 ,0  t + 4m−1 ϕ 2m−1 2l

t

≥ t+

Pm−1 j=l

4j ϕ



x ,0 2j



for all nonnegative integers m and l with m > l and all x ∈ X and all t > 0. It follows from (3.2) and (3.6) that the sequence {4n f ( 2xn )} is a Cauchy sequence for all x ∈ X. Since Y is complete, the sequence {4n f ( 2xn )} converges. So one can define the mapping Q : X → Y by Q(x) := N - lim 4n f ( n→∞

x ) 2n

for all x ∈ X. Moreover, letting l = 0 and passing the limit m → ∞ in (3.6), we get (3.4). By (3.2), x−y x y x+y + 2f −f −f N 4 2f n+1 n+1 n 2 2 2 2n            x+y x−y x y t n n
−ρ 4 f +f − 2f − 2f ,4 t ≥ n n n n 2 2 2 2 t + ϕ 2xn , 2yn 

n















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J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 22, NO.7, 2017, COPYRIGHT 2017 EUDOXUS PRESS, LLC

QUADRATIC ρ-FUNCTIONAL INEQUALITIES IN FUZZY NORMED SPACES

for all x, y ∈ X, all t > 0 and all n ∈ N. So           x−y x y x+y n + 2f −f −f N 4 2f n+1 n+1 n 2 2 2 2n            x+y x−y x y −ρ 4n f +f − 2f − 2f ,t n n n 2 2 2 2n t t 4n

= ≥ t x y nϕ x , y t + 4 + ϕ , 2n 2n 4n 2n 2n for all x, y ∈ X, all t > 0 and all n ∈ N. Since limn→∞ t > 0,

t t+4n ϕ( 2xn , 2yn )

= 1 for all x, y ∈ X and all

x−y x+y +2 − Q(x) − Q(y) = ρ (Q (x + y) + Q (x − y) − 2Q(x) − 2Q(y)) 2 2 for all x, y ∈ X=. By Lemma 3.1, the mapping Q : X → Y is quadratic, as desired. 







2Q



Corollary 3.3. Let θ ≥ 0 and let p be a real number with p > 2. Let X be a normed vector space with norm k · k. Let f : X → Y be a mapping satisfying f (0) = 0 and 

N (2f

x+y 2





+ 2f

x−y 2



− f (x) − f (y)

−ρ (f (x + y) + f (x − y) − 2f (x) − 2f (y)) , t) ≥

(3.7) t t + θ(kxkp + kykp )

for all x, y ∈ X and all t > 0. Then Q(x) := N -limn→∞ 4n f ( 2xn ) exists for each x ∈ X and defines a quadratic mapping Q : X → Y such that N (f (x) − Q(x), t) ≥

(2p − 4)t (2p − 4)t + 2p θkxkp

for all x ∈ X. Proof. The proof follows from Theorem 3.2 by taking ϕ(x, y) := θ(kxkp + kykp ) for all x, y ∈ X, as desired.  Theorem 3.4. Let ϕ : X 2 → [0, ∞) be a function such that Φ(x, y) :=

∞ X 1 j=1

4





ϕ 2j x, 2j y < ∞ j

for all x, y ∈ X. Let f : X → Y be a mapping satisfying f (0) = 0 and (3.3). Then Q(x) := N limn→∞ 41n f (2n x) exists for each x ∈ X and defines a quadratic mapping Q : X → Y such that t N (f (x) − Q(x), t) ≥ t + Φ(x, 0) for all x ∈ X and all t > 0. 



Proof. It follows from (3.5) that N f (x) − 14 f (2x), 4t ≥

t t+ϕ(2x,0)

and so

1 4t t N f (x) − f (2x), t ≥ = 1 4 4t + ϕ(2x, 0) t + 4 ϕ(2x, 0) 



for all x ∈ X and all t > 0. The rest of the proof is similar to the proof of Theorem 3.2.

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C. PARK, S. Y. JANG, AND S. YUN

Corollary 3.5. Let θ ≥ 0 and let p be a real number with 0 < p < 2. Let X be a normed vector space with norm k · k. Let f : X → Y be a mapping satisfying f (0) = 0 and (3.7). Then Q(x) := N -limn→∞ 41n f (2n x) exists for each x ∈ X and defines a quadratic mapping Q : X → Y such that (4 − 2p )t N (f (x) − Q(x), t) ≥ (4 − 2p )t + 2p θkxkp for all x ∈ X. Proof. The proof follows from Theorem 3.4 by taking ϕ(x, y) := θ(kxkp + kykp ) for all x, y ∈ X, as desired.  Acknowledgments This research was supported by Hanshin University Research Grant. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24]

T. Aoki, On the stability of the linear transformation in Banach spaces, J. Math. Soc. Japan 2 (1950), 64–66. T. Bag and S.K. Samanta, Finite dimensional fuzzy normed linear spaces, J. Fuzzy Math. 11 (2003), 687–705. T. Bag and S.K. Samanta, Fuzzy bounded linear operators, Fuzzy Sets and Systems 151 (2005), 513–547. L. C˘ adariu and V. Radu, Fixed point methods for the generalized stability of functional equations in a single variable, Fixed Point Theory and Applications 2008, Art. ID 749392 (2008). I. Chang and Y. Lee, Additive and quadratic type functional equation and its fuzzy stability, Results Math. 63 (2013), 717–730. S.C. Cheng and J.M. Mordeson, Fuzzy linear operators and fuzzy normed linear spaces, Bull. Calcutta Math. Soc. 86 (1994), 429–436. P.W. Cholewa, Remarks on the stability of functional equations, Aequationes Math. 27 (1984), 76–86. S. Czerwik, On the stability of the quadratic mapping in normed spaces, Abh. Math. Sem. Univ. Hamburg 62 (1992), 59–64. C. Felbin, Finite dimensional fuzzy normed linear spaces, Fuzzy Sets and Systems 48 (1992), 239–248. P. G˘ avruta, A generalization of the Hyers-Ulam-Rassias stability of approximately additive mappings, J. Math. Anal. Appl. 184 (1994), 431–436. D.H. Hyers, On the stability of the linear functional equation, Proc. Nat. Acad. Sci. U.S.A. 27 (1941), 222–224. D.H. Hyers, G. Isac and Th.M. Rassias, Stability of Functional Equations in Several Variables, Birkh¨ auser, Basel, 1998. S. Jung, Hyers-Ulam-Rassias Stability of Functional Equations in Mathematical Analysis, Hadronic Press lnc., Palm Harbor, Florida, 2001. A.K. Katsaras, Fuzzy topological vector spaces II, Fuzzy Sets and Systems 12 (1984), 143–154. I. Kramosil and J. Michalek, Fuzzy metric and statistical metric spaces, Kybernetica 11 (1975), 326–334. S.V. Krishna and K.K.M. Sarma, Separation of fuzzy normed linear spaces, Fuzzy Sets and Systems 63 (1994), 207–217. J. Lee, C. Park and D. Shin, An AQCQ-functional equation in matrix normed spaces, Results Math. 27 (2013), 305–318. M. Mirzavaziri and M.S. Moslehian, A fixed point approach to stability of a quadratic equation, Bull. Braz. Math. Soc. 37 (2006), 361–376. A.K. Mirmostafaee, M. Mirzavaziri and M.S. Moslehian, Fuzzy stability of the Jensen functional equation, Fuzzy Sets and Systems 159 (2008), 730–738. A.K. Mirmostafaee and M.S. Moslehian, Fuzzy versions of Hyers-Ulam-Rassias theorem, Fuzzy Sets and Systems 159 (2008), 720–729. A.K. Mirmostafaee and M.S. Moslehian, Fuzzy approximately cubic mappings, Inform. Sci. 178 (2008), 3791– 3798. C. Park, Additive ρ-functional inequalities and equations, J. Math. Inequal. 9 (2015), 17–26. C. Park, Additive ρ-functional inequalities in non-Archimedean normed spaces, J. Math. Inequal. 9 (2015), 397–407. C. Park, K. Ghasemi, S. G. Ghaleh and S. Jang, Approximate n-Jordan ∗-homomorphisms in C ∗ -algebras, J. Comput. Anal. Appl. 15 (2013), 365-368.

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[25] C. Park, A. Najati and S. Jang, Fixed points and fuzzy stability of an additive-quadratic functional equation, J. Comput. Anal. Appl. 15 (2013), 452–462. [26] C. Park and Th.M. Rassias, Fixed points and generalized Hyers-Ulam stability of quadratic functional equations, J. Math. Inequal. 1 (2007), 515–528. [27] Th.M. Rassias, On the stability of the linear mapping in Banach spaces, Proc. Amer. Math. Soc. 72 (1978), 297–300. [28] Th.M. Rassias, Problem 16; 2, Report of the 27th International Symp. on Functional Equations, Aequationes Math. 39 (1990), 292–293; 309. [29] Th.M. Rassias and K. Shibata, Variational problem of some quadratic functionals in complex analysis, J. Math. Anal. Appl. 228 (1998), 234–253. [30] S. Schin, D. Ki, J. Chang and M. Kim, Random stability of quadratic functional equations: a fixed point approach, J. Nonlinear Sci. Appl. 4 (2011), 37–49. [31] S. Shagholi, M. Bavand Savadkouhi and M. Eshaghi Gordji, Nearly ternary cubic homomorphism in ternary Fr´echet algebras, J. Comput. Anal. Appl. 13 (2011), 1106–1114. [32] S. Shagholi, M. Eshaghi Gordji and M. Bavand Savadkouhi, Stability of ternary quadratic derivation on ternary Banach algebras, J. Comput. Anal. Appl. 13 (2011), 1097–1105. [33] D. Shin, C. Park and Sh. Farhadabadi, On the superstability of ternary Jordan C ∗ -homomorphisms, J. Comput. Anal. Appl. 16 (2014), 964–973. [34] D. Shin, C. Park and Sh. Farhadabadi, Stability and superstability of J ∗ -homomorphisms and J ∗ -derivations for a generalized Cauchy-Jensen equation, J. Comput. Anal. Appl. 17 (2014), 125–134. [35] F. Skof, Propriet` a locali e approssimazione di operatori, Rend. Sem. Mat. Fis. Milano 53 (1983), 113–129. [36] S. M. Ulam, A Collection of the Mathematical Problems, Interscience Publ. New York, 1960. [37] J.Z. Xiao and X.H. Zhu, Fuzzy normed spaces of operators and its completeness, Fuzzy Sets and Systems 133 (2003), 389–399. Choonkil Park Research Institute for Natural Sciences, Hanyang University, Seoul 04763, Korea E-mail address: [email protected] Sun Young Jang Department of Mathematics, University of Ulsan, Ulsan 44610, Korea E-mail address: [email protected] Sungsik Yun Department of Financial Mathematics, Hanshin University, Gyeonggi-do 18101, Korea E-mail address: [email protected]

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J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 22, NO.7, 2017, COPYRIGHT 2017 EUDOXUS PRESS, LLC

ON A DOUBLE INTEGRAL EQUATION INCLUDING A SET OF TWO VARIABLES POLYNOMIALS SUGGESTED BY LAGUERRE POLYNOMIALS M. AL I· ÖZARSLAN AND CEMAL I·YE KÜRT

Abstract. In this paper, we introduce general classes of bivariate and Mittag( ; ; ; ; ) ( ; ; ; ; ) Le- er functions E 1 ; 2 (x; y) and Laguerre polynomials Ln;m (x; y). We investigate double fractional integrals and derivative properties of the above mentioned classes. We further obtain linear generating function for ( ; ; ; ; ) ( ; ; ; ; ) Ln;m (x; y) in terms of E 1 ; 2 (x; y). Finally, we calculate double Laplace transforms of the above mentioned classes and then we consider a gen( ; ; ; ; ) eral singular integral equation with Ln;m (x; y) in the kernel and obtain the solution in terms of E

( ; ; ; ; ) (x; y). 1; 2

1. Introduction The special function of the form [7] (1.1)

E (z) =

1 X

k=0

zk ( k + 1)

( 2 C; Re( ) > 0; z 2 C) and more general function [12] of (1.1)

(1.2)

E

;

(z) =

1 X

k=0

( ;

zk ( k+ )

2 C; Re( ); Re( ) > 0; z 2 C)

are known as Mittag-Le- er functions the …rst of which was introduced by Swedish mathematician G. Mittag-Le- er and the second one by Wiman. Setting = = 1, the equation (1.2) becomes the exponential function ez . When 0 < < 1, it bridges an interpolation between the pure exponential function ez and a geometric function 1 1

z

=

1 X

zn:

n=0

(jzj < 1)

Key words and phrases. Double fractional integrals and derivatives, Bivariate Mittag-Le- er function, Bivariate Laguerre polynomials, Double generating functions, Singular double integral equation, Double Laplace integral. 2010 Mathematics Subject Classi…cation. 33E12, 33C45, 45E10. 1

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M . AL I· ÖZARSLAN AND CEM AL I·YE KÜRT

2

A further generalization of (1.2) was introduced by Prabhakar (see [9]) as E

(1.3)

(z) =

;

1 X

( )n zn ( n + ) n!

n=0

( ; ;

2 C; Re( ); Re( ) > 0)

where the Pochammer symbol [11], ( )n , is de…ned as ( + n) = ( )

( )n =

1 ( +n

( + 1)

; n = 0; 6= 0 ; n = 1; 2;

1)

In the special case, we have the polynomials Zn (x; k) (see [6],[10]) which were de…ned by (kn + + 1) Ek;n+1 (xk ) . n! (Re( ) > 0; k 2 Z0+ )

Zn (x; k) =

Note that, in [6] and [10], generating functions, integrals and recurrence relations were developed for the polynomials Zn (x; k) of degree n in xk , which form one set of the biorthogonal pair corresponding to the weight function e x x over the interval (0; 1). For k = 1, we have Zn (x; 1) = Ln (x) where Ln (x) is the usual Laguerre polynomial which were given as follows Ln (x) =

(1 + )n 1 F1 ( n; 1 + ; x) n!

where 1 F1 (

n; 1 + ; x) =

n X ( n)k xk : (1 + )k k!

k=0 ( ) Zn1 ;

; xj ; 1 ; ; Very recently, a class of polynomials ;nj x1 ; suggested by the multivariate Laguerre polynomials were de…ned by

j

(see [8])

(1.4) ( )

Zn1 ;

;nj

1 n1

= ( ;

1;

;

j

;nj

= j = 1, where [2] given by ;nj (x1 ;

+

+

1;

j nj

n1 !

( )

( )

; xj ;

;

+

x1 ; ( ) Ln1 ;

nj !

; xj ) =

; xj ; ;nj

1;

(x1 ;

(n1 +

j

+1

n1 ;

k1 ;

2 C; Re( i ) > 0 (i = 1;

Obviously Zn1 ;

Ln1 ;

x1 ;

; j)) ;

X;nj

( n1 )k1

( )

j

1 k1

;kj =0

gives Ln1 ;

;nj

(x1 ;

+

( nj )kj x11 +

j kj

+

; xj ) when

k1

xj j

kj

+ 1 k1 !

1

kj !

=

; xj ) is the multivariable Laguerre polynomial

+ nj + n1 ! nj !

1199

+ 1)

n1 ;

k1 ;

X;nj

;kj =0

( n1 )k1 (k1 +

( nj )kj xk11 + kj +

+ 1) k1 !

k

xj j kj !

:

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:

J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 22, NO.7, 2017, COPYRIGHT 2017 EUDOXUS PRESS, LLC

DOUBLE INTEGRAL EQUATION INCLUDING TWO VARIABLES LAGUERRE POLYNOM IALS3

It is known that the multivariate Mittag-Le- er functions were de…ned by the multiple series as [13] (1.5)

E

(

1; 1;

; ;

j)

j;

(x1 ;

; xj ) = k1 ;

( ;

1;

;

j;

1;

;

j

1 X

(

1 )k1

1 k1

;kj =0

j kj

+

+

2 C; Re( i ) > 0 (i = 1;

j kj

k

xk11 +

xj j k1 !

kj !

:

; j))

Note that the function in (1.5) is a special case of the generalized Lauricella series in several variables introduced and investigated by Srivastava and Daoust [16] (see also see [14],[15]). Also, when j = 1; 1 = ; = ; 1 = ; x1 = z, the function (1.5) reduces to (1.3). ( ) The polynomials Zn1 ; ;nj x1 ; ; xj ; 1 ; ; j can be represented in terms of the multivariate Mittag-Le- er functions as follows (see [8]): ( )

(1.6)

Zn1 ;

1 n1

=

( ) ;nj

+

; xj ; +

1

(x1 ;

=

2

=

; xj ) =

1;

;

j nj + nj !

n1 !

Clearly, setting Ln1 ;

x1 ;

;nj

=

j

+1

j

E

( n1 ; ; 1;

; nj ) 1 +1 (x1 ;

; xj j ):

j;

= 1 in (1.6) gives

(n1 + n1 !

+ nj + nj !

+ 1)

( n1 ; ; nj ) (x1 ; ;1; +1

E1;

; xj ):

Very recently, a slight motivated form of the multivariate Mittag-Le- er functions were introduced and investigated in [3]. On the other hand, a nontrivial two variables Mittag-Le- er functions were de…ned in [4] by x 1; 1; 2; 1 y ; ; ; ; ; ; 1 2 3 3 2 2 3 1 1 X X ( 1) 1m( 2) n xm 1 = ( 1 + 2 m + 2 n) ( 2 + 3 m) ( m=0 n=0

E1 (x; y) = E1

(

1;

2 ; 1 ; 2 ; 3 ; x; y

2 C; minf

1;

2;

3;

1;

2;

3g

yn 3+

3 n)

:

> 0g)

Motivated essentially by the above de…nitions and investigations, in this paper, we introduce a class of bivariate Mittag-Le- er function E ( 1 ;;

(1.7)

; ; ; ) 2

(x; y) =

1 X 1 X

( 1 )k1 ( 2 )k2 xk1 y k2 ( k1 + k2 + ) ( k2 + )k1 !k2 !

k1 =0 k2 =0

where 1 ; 2 ; ; ; ; ; 2 C; Re( + ) > 0 and Re( ) > 0: According to the convergence conditions investigated by Srivastava and Daoust ([15], p. 155) for the generalized Lauricella series in two variables, the series in (1.7) converges absolutely for Re( + ) > 0 and Re( ) > 0. We also introduce a general class of bivariate Laguerre polynomials ; ; L(n;m

(1.8) =

; ; )

(x; y)

n m ( n + m + + 1) X X ( + m)

k1 =0 k2 =0

( n)k1 ( m)k2 x k1 y k2 ( k1 + k2 + + 1) ( k2 + )k1 !k2 !

where ; ; ; ; 2 C; Re( ); Re( ); Re( ); Re( ) > 0; Re( ) >

1200

1:

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M . AL I· ÖZARSLAN AND CEM AL I·YE KÜRT

4

Comparing (1.7) and (1.8), we see that (1.9)

; ; L(n;m

; ; )

( n + m + + 1) ( ; ; ; ; ) E n; m (x ; y ): ( + m)

(x; y) =

This paper is organized as follows. In section 2, we calculate the double fractional integrals and derivatives of the above mentioned classes (1.7) and (1.8). Linear ( ; ; ; ; ) ( ; ; ; ; ) generating functions for Ln;m (x; y) are given in terms of E 1 ; 2 (x; y) in Section 3. In the last section, we …rst investigate double Laplace transforms of the above mentioned classes and then we consider a general singular integral ( ; ; ; ; ) equation with Ln;m (x; y) in the kernel and obtain the solution by means of ( ; ; ; ; ) E 1; 2 (x; y).

2. Fractional integrals and derivatives This section aims to provide the fractional integral formulas of the functions ( ; ; ; ; ) ( ; ; ; ; ) (x; y) : Throughout this section, we assume that (x; y) and Ln;m E 1; 2 Re( ); Re( ) > 0; Re( ); Re( ) > 0; Re( ) > 1: De…nition 2.1. ([1],[8])Let = [a; b] be a …nite interval of the real axis. The Riemann-Liouville fractional integral of order 2 C (Re ( ) > 0) is de…ned by x Ia+

Z

1 ( )

[f ] =

x

a

f (t) dt (x t)1

: (x > a; Re ( ) > 0)

Similarly, the partial fractional integrals of a function f (x; t), where (x; t) 2 R is de…ned as follows: x Ia+ f (x; t)

=

1 ( )

Z

1 ( )

Z

t Ia+ f (x; t) =

=

t Ib+ x Ia+ f (x; t) Z tZ x

1 ( ) ( )

b

(t

x

(x

)

1

)

1

f ( ; t)d ; (x > a; Re ( ) > 0)

a

t

(t

f (x; )d ; (t > b; Re ( ) > 0)

b

1

)

(x

1

)

f ( ; )d d : (x > a; y > b; Re ( ) > 0; Re ( ) > 0)

a

De…nition 2.2. ([1],[8])The Riemann-Liouville fractional derivative of order C (Re ( ) 0) is de…ned by x Da+ [f ] =

d dx

R

n

1 (n

)

Z

2

x

(x

)

n 1

f ( )d ; (n = [Re( )] + 1; x > a)

a

where, as usual, [Re( )] means the integral part of Re( ):

1201

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DOUBLE INTEGRAL EQUATION INCLUDING TWO VARIABLES LAGUERRE POLYNOM IALS5

R

Similarly, the partial fractional derivatives of a function f (x; t), where (x; t) 2 R is de…ned as follows: d dx

n

x Da+ f (x; t) =

d dt

m

t Db+ f (x; t) =

1 (n

)

1 (m

)

t Db+ x Da+ f (x; t) m n

Z

x

(x

)n

1

)m

1

f ( ; t)d ; (n = [Re( )] + 1; x > a)

a

Z

t

(t

f (x; )d ; (m = [Re( )] + 1; t > b)

b

Z tZ x d d 1 1 = (t dt dx (n ) (m ) b a (n = [Re( )] + 1; m = [Re( )] + 1; t > b; x > a)

)m

1

(x

)n

1

f ( ; )d d :

Theorem 2.1. We have for Re( + ) > 0 and Re( ) > 0 and ( ) > 0, that h i 1 y 1 E ( 1 ;; 2; ; ; ) x ; x y = x + 1 y + 1 E ( 1 ;; 2; ; + ; + ) x ; x y y I0+ x I0+ x Proof. Because of the hypothesis of the Theorem, we have a right to interchange of the order of series and fractional integral operators, which yields h i 1 y 1 E ( 1 ;; 2; ; ; ) x ; x y y I0+ x I0+ x Z yZ x (y ) 1 (x t) 1 1 ( ; ; ; ; ) t 1 E 1; 2 t ;t dtd = ( ) ( ) 0 0 1 X 1 X 1 ( 1 )k1 ( 2 )k2 = ( ) ( ) ( k1 + k2 + ) ( k2 + )k1 !k2 ! k1 =0 k2 =0 Z y Z x (y ) 1 k2 + 1 d (x t) 1 t k1 + k2 + 1 dt 0

=

0

1 X 1 X ( 1 )k1 ( 2 )k2 x 1 ( ) ( ) ( k1 + k2 + k1 =0 k2 =0

=x

+

1

y

+

1

E ( 1 ;;

; ; + ; + )

2

k1 + k2 + +

1

y k2 + + 1 + ) ( k2 + + )k1 !k2 !

x ;x y

In a similar manner, we have the following corollary: Corollary 2.2. For Re( ) > 0 and Re( ) > 0, that h i 1 ( ; ; ; ; ) Ln;m x; xy y I0+ x I0+ x y =

( n + m + + 1) ( + + m) x ( + m) ( n + m + + + 1)

+

y

+

1

; ; L(n;m

Theorem 2.3. For Re( + ) > 0,Re( ) > 0 and ( ) > 0, that h i 1 1 1 ( ; y 1 E ( 1 ;; 2; ; ; ) x ; x y =x y E 1; y D0+ x D0+ x

1202

+ ; ; + )

; ; 2

;

x; xy

)

x ;x y

:

:

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J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 22, NO.7, 2017, COPYRIGHT 2017 EUDOXUS PRESS, LLC

M . AL I· ÖZARSLAN AND CEM AL I·YE KÜRT

6

Proof. Because of the hypothesis of the Theorem, we have a right to interchange of the order of series and fractional derivate operators, which yields h

y D0+ x D0+

=

=

y D0+ x D0+ 1 X

1 X

1

x

(

"

1

y

1

x

E ( 1 ;; 1

y

1 ) (m

)m

k2 +

1

1

d

0

=x

k1 + k 2 +

1

y

k2 +

1

#

]

) Z

1 X 1 X

( 1 )k1 ( 2 )k2 ( k1 + k2 + ) ( k2 + )k1 !k2 !

k1 =0 k2 =0

x

)n

(x

k1 + k2 +

1

1

d

0

1

1

y

1 X 1 X

k1 =0 k2 =0

=x

k2

( 1 )k1 ( 2 )k2 x k1 x k2 y ( k1 + k2 + ) ( k2 + )k1 !k2 !

( k1 + k2 + ) ( k2 + )k1 !k2 !

y

(y

1 X 1 X

1 )k1 ( 2 )k2 x D0+ y D0+ [x

d m d n 1 ) ( ) dy dx (n

Z

i

x ;x y

k1 =0 k2 =0

k1 =0 k2 =0

=(

; ; ; ) 2

1

1

y

E ( 1 ;;

; ;

(

1 )k1 ( 2 )k2 x

y ) ( k2 +

( k1 + k2 + ;

)

2

x ;x y

k1 + k 2

k2

)k1 !k2 !

:

In a similar manner, we have the following corollary: Corollary 2.4. For Re( + ) > 0 and Re( ) > 0, that y D0+ x D0+ [x

=

y

1

; ; L(n;m

; ; )

x; xy

]

( n + m + + 1) ( + m) x ( + m) ( n+ m+ + 1)

y

1

; ; L(n;m

; ;

)

x; xy

:

3. Linear generating function In this section, we provide a linear generating function for the polynomials (x; y) by means of two variables analogue of Mittag-Le- er functions de…ned in (1.7): ( ; ; ; ; ) Ln;m

Theorem 3.1. For jt1 j < 1 and jt2 j < 1, 1 X 1 X (

( ; ; ; ; ) 1 )n ( 2 )m Ln;m

t1 )

1

(1

t2 )

2

E ( 1 ;;

2

2 C and

; ; ; ; 2 C, we have

(x; y) ( + m) n m t 1 t2 + 1)n!m!

( n+ m+

n=0 m=0

= (1

1;

; ; ; +1) 2

1203

1

x t1 y t2 ; t1 1 t2

:

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J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 22, NO.7, 2017, COPYRIGHT 2017 EUDOXUS PRESS, LLC

DOUBLE INTEGRAL EQUATION INCLUDING TWO VARIABLES LAGUERRE POLYNOM IALS7

Proof. Direct calculations yield that 1 X 1 X (

( ; ; ; ; ) 1 )n ( 2 )m Ln;m

n=0 m=0 1 X

n;m X

=

( 1 )n ( 2 )m ( n)k1 ( m)k2 x k1 y k2 t n tm ( k1 + k2 + + 1) ( + k2 )k1 !k2 !n!m! 1 2

n;m=0 k1 ;k2 =0

=

(x; y) ( + m) n m t1 t2 + 1)n!m!

( n+ m+

n;m X

1 X

( 1)k1 +k2 ( 1 )n ( 2 )m x k1 y ( k1 + k2 + + 1) ( + k2 )k1 !k2 !(n

n;m=0 k1 ;k2 =0

k2

k1 )!(m

k2 )!

tn1 tm 2 :

Letting n ! n + k1 and m ! m + k2 , we get 1 X

1 X

n;m=0 k1 ;k2 =0

Since (

1 )n+k1

1 X

(

=(

( 1)k1 +k2 ( 1 )n+k1 ( 2 )m+k2 x k1 y k2 2 tn+k1 tm+k : 2 ( k1 + k2 + + 1) ( + k2 )k1 !k2 !(n)!(m)! 1 1

+ k1 )n (

1 )k1 ( 2 )k2 (

( k1 + k2 +

k1 ;k2 =0

= (1

t1 )

1

(1

t2 )

1 )k1

and (

2 )m+k2

=(

2

1 X x t1 )k1 ( y t2 )k2 ( + 1) ( + k2 )k1 !k2 ! n;m=0 2

E ( 1 ;;

; ; ; +1) 2

+ k2 )m ( 1

x t1 y t2 ; 1 t 1 1 t2

2 )k2 ,

we have

+ k1 )n ( 2 + k2 )m n m t1 t2 (n)!(m)! :

Note that, because of the uniform converge of the series under the conditions jt1 j < 1 and jt2 j < 1; we have interchanged the order of summations. 4. Singular double integral equation In this section, we …rst obtain the double Laplace transform of the functions ( ; ; ; ; ) ( ; ; ; ; ) (x; y) : Then, we compute the double integral in(x; y) and Ln;m E 1; 2 ( ; ; ; ; ) (x; y) functions in the integrand. Finally, we volving the product of two E 1 ; 2 ( ; ; ; ; ) (x; y) in the kernel, in terms of the solve a double integral equation with Ln;m ( ; ; ; ; ) two E 1 ; 2 (x; y) functions: As usual [5], Z 1 Z 1 px (4.1) L2 [f (x; t)] = e e st f (x; t)dtdx 0

0

(x; t > 0;

p; s 2 C)

denotes the double Laplace transform of f: Lemma 4.1. For Re( and

L2 [x

1

s1

1

;

y

2

s1 s2 1

1 ); Re( 2 ); Re(

+ ) > 0; Re( ) > 0; Re(s1 ); Re(s2 ) > 0

< 1, we have

E ( 1 ;;

; ; ; ) 2

((

1 x)

;(

2x

y ))](s1 ; s2 ) =

1204

1 1 (1 1 ) s1 s1 s2

1

(1

2

s1 s2

)

2

:

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M . AL I· ÖZARSLAN AND CEM AL I·YE KÜRT

8

Proof. Using de…nition (4.1) and taking into account that we get 1

L2 [x

1

y

E ( 1 ;;

1 X 1 X

=

((

1 x)

;(

2x

x

0

0

1

s1

)

1

We deduce the following result from Lemma 4:1 by setting equation (1.9). Corollary 4.2. For Re( s1

< 1,

y ))](s1 ; s2 )

1 1 1 1 X ( 1 )k1 1 k1 X ( 2 )k2 1 1 = ( ) ( 2 )k2 = (1 k2 ! s1 s2 s1 s2 k =0 k1 ! s1 s1 s2 k2 =0 1

1

2

s1 s2

( 1 )k1 ( 2 )k2 1 k1 2 k2 ( k1 + k2 + ) ( k2 + )k1 !k2 ! Z 1 k2 + 1 s1 x e dx y k2 + 1 e s2 y dy

k1 =0 k2 =0 1 k1 +

Z

; ; ; ) 2

< 1 and

1

s1

;

1 ); Re( 2 ); Re(

s1 s2

1

L2 [t =

1

1

+1

s2

s1

Theorem 4.3. Let y

0

Z

s1 s2

1=

)

2

:

and using

); Re( ); Re( ); Re(s1 ); Re(s2 ) > 0 and

< 1; we have

2

Z

2

(1

x

0

t

1

=x

+

h

(x

t)

1

E ( 3;;

y

+

1;

1

2

; ; )

((

1 t); ( 2 t

))](s1 ; s2 )

( n + m + + 1) (1 ( m+ )

1 n

)m :

2

) (1

s1

s1 s2

2 C; Re( + ) > 0 and Re( ) > 0: Then

(y

1

)

; ; ; ) 4

E ( 1 ;;

; ; L(n;m

(

; ; ; ) 2

1t

(

E ( 1 ;;

;

1x

; ; ; ) 2

2t

)dtd

2x

;

( 1 (x i

y )E ( 3 ;;

t) ;

; ; ; )

4

(

2 (x

1x

;

t) (y

2x

) )

y ):

Proof. Using the convolution theorem for the Laplace transform we have,

L2

Z

y

0

Z

x

(x

=

1 1 s1 s2

(y

1

y

1

1

;

E ( 1;; 1

s1

E ( 1 ;; 2; ; ; ) ( i )dtd (s1 ; s2 ) )

0

E ( 3 ;; 4; ; ; ) ( 1 t = L2 [x

1

t)

2t ; ; ; ) 2

(

1

1x

;

1

1

2

s1 s2

2x

!

1 (x

1

y )]L2 [x

2

1 1 s1 s2

1205

t) ;

1

1

y 1

s1

2 (x

E ( 3 ;;

t) (y

; ; ; ) 4

(

) )t

1t

3

1

2

s1 s2

; !

2t

1

1

)](s1 ; s2 )

4

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DOUBLE INTEGRAL EQUATION INCLUDING TWO VARIABLES LAGUERRE POLYNOM IALS9

We have for Re(s1 ); Re(s2 ) > 0 Z yZ x (4.2) L2 (x t) 1 (y 1

t 1

=

1

)

0

0

1

E ( 1 ;;

1

; ; ; )

2

+

1t

;

E ( 1 ;;

2t 2

1

s1 ; ; ; ) 2

(

s1 s2

1x

; ; ; ) 2

(

1 (x

t) ;

i

1

1

1

s1 + s2+ h = L2 x + y

(

E ( 1 ;;

;

)dtd (s1 ; s2 ) ! 2

s1

y )E ( 3 ;;

2x

; ; ; 4

t) (y

2

1

) )

!

3

1

1

2 (x

4

s1 s2 i ) ( 1 x ; 2 x y ) (s1 ; s2 ):

Taking inverse Laplace transform on both sides of (4.2), the result follows.

The next assertation follows from T heorem 4:5 by letting into account (1.9). Corollary 4.4. Let 1 ; Z yZ x (x t) (y 0

2

1

=x

L(n2;;m; 2 2 ;

1 + 2 +1

;

1+ 2

y

2)

1

and taking

2 C; Re( ); Re( ); Re( ); Re( ) > 0: Then )

1

(

1 t;

0

t

1=

L(n1;;m; 1 1 ; 2t

Ln( 1;;m; 1 1 ;

;

;

1)

(

1 (x

t);

2 (x

t)(y

) ))

)dtd 1)

(

1 x;

)Ln( 2;;m; 2 2 ;

2 xy

;

2)

(

1 x;

2 xy

):

Now, we consider the following double convolution equation: (4.3) Z yZ 0

x

(x

1

t)

(y

1

)

; ; L(n;m

; ; )

((

1 x)

;(

2x

y )) (t; )dtd =

(x; y)

0

where Re( ) > 1: For the solution of the integral equation (4.3), we have the following theorem: Theorem 4.5. The singular double integral equation (4.3) admits a locally integrable solution ( m+ ) ( n + m + + 1)

(t; ) = Z

y

0

Z

x

(x

t)

2

1

(y

)

1

2

0

E ( 1 ;;

; ; ; ) 2

((

1 x)

;(

2x

y ))[I0+ 1 I0+

2

(t; )]dtd :

Proof. Applying double Laplace transform on both sides of (4.3), then using double convolution theorem, we get 1

1

+1

s2

s1

( n + m + + 1) 2 (1 1 )n (1 )m L2 [ (t; )](s1 ; s2 ) = L2 [ (t; )](s1 ; s2 ) ( m+ ) s1 s1 s2

Therefore, we have, L2 [ (t; )](s1 ; s2 ) = (s1 )

1 +1

(s2 )

2

(1

1

s1

)

n

(1

( m+ ) ( n + m + + 1) 2

s1 s2

1206

)

m

fs1 1 s2 2 L2 [ (t; )](s1 ; s2 )g

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10

Finally taking inverse Laplace transform on both sides and using Lemma 3:2 of [1] and Lemma 4:1, we get ( m+ ) ( n + m + + 1)

(t; ) = Z

0

y

Z

x

(x

t)

1

2

(y

)

0

2

1

E ( 1 ;;

; ; ; ) 2

((

1 x)

;(

2x

y ))[I0+ 1 I0+

2

(t; )]dtd

and the proof is completed. References

[1] Anwar, AMO, Jarad, F, Baleanu, D, Ayaz, F: Fractional Caputo Heat equation within the double Laplace transform, Rom. Journ. Phys., 58, 15-22 (2013). [2] Carlitz, L: Bilinear generating functions for Laguerre and Lauricella polinomials in several variables, Rend. Sem. Mat. Univ. Padova, 43, 269-276 (1970). [3] Gaboury, S, Özarslan, MA: Singular integral equation involving a multivariable analog of Mittag-Le- er function, Advances in Di¤erence Equations, 252 (2014) [4] Garg, M, Manohar, P, Kalla, SL: A Mittag-Le- er-type function of two variables, Int. Trans. Special Funct., 24 (11), 934-944 (2013). [5] K¬l¬cman, A, Gadain, HE: On the applications of Laplace and Sumudu transforms, Journal of the Franklin Institude, 347 (5), 848-862 (2010). [6] Konhauser, JDE: Biorthogonal polynomials suggested by the Laguerre polynomials, Pasi…c J. Math., 21, 303-314 (1967). [7] Mittag-Le- er, GM:Sur la nouvelle function E (x), C. R. Acad. Sci. Paris, 137, 554-558 (1903). [8] Ozarslan MA: On a singular integral equation including a set of multivariate polynomials suggested by Laguerre polynomials, Appl. Math. Comput., 229, 350-358 (2014). [9] Prabhakar, TR: A singular integral equation with a generalized Mittag-Le- er function in the kernel, Yokohama Math. J., 19, 7-15 (1971). [10] Prabhakar, TR: On a set of polynomials suggested by Laguerre polynomials, Paci…c J. Math. 35 (1), 213-219 (1970). [11] Rainville, ED: Special Functions, Macmillan Company, New York, (1960). [12] Saxena, RK, Mathai, AM, Haubold, HJ: On fractional kinetic equations. Astrophys. Space Sci. 282(1), 281-287 (2002). [13] Saxena, RK, Kalla, SL, Saxena, R: Multivariate analogue of generalized Mittag-Le- er function, Int. Trans. Special Funct., 22 (7), 533-548 (2011). [14] Srivastava, HM, Karlsson, PW: Multiple Gaussian Hypergeometric Series, Ellis Horwood Limited, New York, (1985). [15] Srivastava, HM, Daust, MC: A note on the convergence of Kampé de Feriét’s double hypergeometric series, Math. Nachr., 53, 151-159 (1972) . [16] Srivastava, HM, Daust, MC: Certain generalized Neumann expansion associated with Kampé de Feriet function, Nederl. Akad., Wetensch. Proc. Ser. A, 72 31 (1969) 449-457 (Indag. Math.). Eastern Mediterranean University, Department of Mathematics, Gazimagusa, TRNC, Mersin 10, Turkey, Email: [email protected], [email protected]

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Generalized Inequalities of the type of Hermite-Hadamard-Fejer with Quasi-Convex Functions by way of k-Fractional Derivatives A. Ali(1) , G. Gulshan(1) , R. Hussain(1) , A. Latif(1) , M. Muddassar(2 ∗) (1)

Department of Mathematics, Mirpur University of Science & Technology, Mirpur, Pakistan, [email protected], [email protected], [email protected], [email protected]

(2)

Department of Mathematics,University of Engineering & Technology, Taxila, Pakistan, [email protected] (∗ for correspondence)

Abstract: In this article, Hermite-Hadamard-Fejer type inequalities are discussed with quasi-convex functions and obtained the generalized results of the type using k-fractional derivatives. And proposed some new bounds in terms of some special means. Keywords: Hermite-Hadamard inequality, Hermite-Hadamard-Fejer inequality, quasi convex functions, k-Riemann-Liouville fractional derivatives, H¨older’s integral inequality, Power mean inequality.

1. I NTRODUCTION The function f : I ⊂ R −→ R is said to be convex on I if for every x, y ∈ I and t ∈ [0, 1], we get f (tx + (1 − t)y) ≤ tf (x) + (1 − t)f (y). Let f : I ⊂ R −→ R be a convex function on the interval I of real numbers and a, b ∈ I with a < b, f satisfies the following well-known Hermite-Hadamard type inequality

 f

a+b 2

 ≤

1 b−a

Z

b

f (t)dt ≤ a

f (a) + f (b) . 2

Definition 1. The function f : I ⊂ R −→ R is said to be quasi-convex if f (tx + (1 − t)y) ≤ max {f (x), f (y)} , for every x, y ∈ I and t ∈ [0, 1] (see [4]). In [3] Mubeen and Habibullah introduced the following class of fractional derivatives. 1

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A. Ali, G. Gulshan, R. Hussain, A. Latif, M. Muddassar and J. Park

Definition 2. Let f ∈ L[a, b], then k-Riemann-Liouville fractional derivatives k Jaα+ f (x) and k Jbα− f (x) of order α > 0 are defined by Z x α 1 α (x − t) k −1 f (t)dt, k Ja+ f (x) = kΓk (α) a and Z b α 1 α (t − x) k −1 f (t)dt, k Jb− f (x) = kΓk (α) x

(0 ≤ a < x < b)

(0 ≤ a < x < b)

respectively, where k > 0 and Γk (α) is the k-gamma function given as Γk (α) = Furthermore Γk (α + k) = αΓk (α) and

0 k Ja+ f (x)

0 k Jb− f (x)

=

R∞ 0

tα−1 e

−tk k

dt.

= f (x).

In [1] Fej´er established the following inequality. Lemma 1. Let f : [a, b] ⊂ R −→ R be a convex function, the inequality

 f

a+b 2

Z

b

Z g(x)dx ≤

a

b

f (x)g(x)dx ≤ a

f (a) + f (b) 2

Z

b

g(x)dx a

holds, where g : [a, b] ⊂ R −→ R is non-negative integrable and symmetric to

a+b 2 .

This

inequality is called Hermite-Hadamard-Fejer inequality. Lemma 2. ([7]) For 0 < t ≤ 1 and 0 ≤ a < b, we get |at − bt | ≤ (b − a)t . E. Set et al. established the following Lemma in [6]. Lemma 3. Let f : [a, b] ⊂ R −→ R be a differentiable mapping on (a, b) and g : [a, b] ⊂ 0

R −→ R. If f , g ∈ L[a, b], the following identity for fractional derivatives holds      a+b J αa+b − g(a) + J αa+b + g(b) − J αa+b − (f g)(a) + J αa+b + (f g)(b) f ( 2 ) ( 2 ) ( 2 ) ( 2 ) 2 Z b 0 1 = m(t)f (t)dt (1.1) Γ(α) a where

m(t) =

 R   t (s − a)α−1 g(s)ds a

R  − b (b − s)α−1 g(s)ds t


t ∈ a, a+b 2   t ∈ a+b 2 ,b .

Iscan obtained the following lemma in [2].

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Generalized Inequalities of the type of Hermite-Hadamard-Fejer with Quasi-Convex Functions by way of k-Fractional Derivatives 3 0

Lemma 4. Let f : [a, b] ⊂ R −→ R be a differentiable mapping on (a, b) and f ∈ L[a, b]. If g : [a, b] −→ R is integrable and symmetric to

a+b 2 ,

the following identity for fractional

derivatives holds f (a) + f (b) α [Ja+ g(b) + Jbα− g(a)] − [Jaα+ (f g)(b) + Jbα− (f g)(a)] 2 ! Z b Z t Z b 0 1 α−1 α−1 (b − s) = g(s)ds − (s − a) g(s)ds f (t)dt Γ(α) a a t

(1.2)

where α > 0.

In the present paper motivated by the recent results given in [5] we established some Hermite-Hadamard-Fej´er type inequalities for quasi-convex functions via k-fractional derivatives. 2. M AIN F INDINGS Throughout this paper, let I be an interval on R and let ||g||[a,b],∞ = supt∈[a,b] g(t) for continuous function g : [a, b]R −→ R. The following identity is the generalization of identity (1.1) in Lemma 3 for k-fractional derivatives. Lemma 5. Let f : [a, b] ⊂ R −→ R be a differentiable mapping on (a, b) and g : [a, b] ⊂ 0

R −→ R. If f , g ∈ L[a, b], the following identity for k-fractional derivatives holds      a+b α α α α f k J a+b − g(a) + k J a+b + g(b) − k J a+b − (f g)(a) + k J a+b + (f g)(b) ( 2 ) ( 2 ) ( 2 ) ( 2 ) 2 Z b 0 1 = m(t)f (t)dt Γk (α) a where

m(t) =

 R   t (s − a) αk −1 g(s)ds a

R  − b (b − s) αk −1 g(s)ds t


t ∈ a, a+b 2   t ∈ a+b 2 ,b .

Here the identity (1.2) of Lemma 4 is also generalized for k-fractional derivatives. 0

Lemma 6. Let f : [a, b] ⊂ R −→ R be a differentiable mapping on (a, b) and f ∈ L[a, b]. If g : [a, b] ⊂ R −→ R is integrable and symmetric to

1210

a+b 2 ,

the following for k-fractional

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A. Ali, G. Gulshan, R. Hussain, A. Latif, M. Muddassar and J. Park

derivatives holds f (a) + f (b) [k Jaα+ g(b) + k Jbα− g(a)] − [k Jaα+ (f g)(b) + k Jbα− (f g)(a)] 2 ! Z b Z t Z b 0 α α 1 −1 −1 (b − s) k g(s)ds − (s − a) k g(s)ds f (t)dt = Γk (α) a a t

where

α k

> 0.

0

Theorem 1. Let f : I ⊂ R −→ R be a differentiable mapping on I o and f ∈ L[a, b] and 0

g : [a, b] ⊂ R −→ R is continuous. If |f |q is quasi-convex function on [a, b], q ≥ 1, the following inequality for k-fractional derivatives holds      α α α α f a + b k J a+b − g(a) + k J a+b + g(b) − k J a+b − (f g)(a) + k J a+b + (f g)(b) ( 2 ) ( 2 ) ( 2 ) ( 2 ) 2 α n 0 o q1 (b − a) k +1 ||g||[a,b],∞  0
≤ α α max |f (a)|q , |f (b)|q 2 k k + 1 Γk (α + k) where

α k

> 0. 0

Proof. Since |f |q is quasi-convex on [a, b], we know that for t ∈ [a, b]   q n 0 o 0 b−t 0 t − a |f (t)| = f a+ b ≤ max |f (a)|q , |f (b)|q . b−a b−a 0

q

0

Using lemma 5, power mean inequality and the fact that |f |q is quasi-convex function on [a, b], it follows that      α α α α f a + b J g(a) + J g(b) − J (f g)(a) + J (f g)(b) k k k k a+b − a+b + a+b − a+b + ( 2 ) ( 2 ) ( 2 ) ( 2 ) 2 ! q1 Z t !1− q1 Z a+b Z t Z a+b 2 2 0 α α 1 −1 −1 q (s − a) k g(s)ds dt (s − a) k g(s)ds |f (t)| dt ≤ Γk (α) a a a a !1− q1 Z ! q1 Z b Z b 0 b Z b α α 1 + (b − s) k −1 g(s)ds dt (b − s) k −1 g(s)ds |f (t)|q dt a+b a+b Γk (α) t t 2 2

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Generalized Inequalities of the type of Hermite-Hadamard-Fejer with Quasi-Convex Functions by way of k-Fractional Derivatives 5

||g||[a, a+b ],∞ 2 ≤ Γk (α) +

||g||[ a+b ,b],∞ 2 Γk (α)

a+b 2

Z

b a+b 2

α

(b − a) k +1
α 2 k +1 αk + 1

Z

b a+b 2

! q1

! q1 Z t 0 α (s − a) k −1 ds |f (t)|q dt a

a

Z !1− q1 b α −1 (b − s) k ds dt t

!1− q1 α 1 (b − a) k +1
≤ α Γk (α + k) 2 k +1 αk + 1  n 0 o q1 0 max |f (a)|q , |f (b)|q

a+b 2

Z

a

a

Z

Z t !1− q1 α (s − a) k −1 ds dt



Z ! q1 b 0 α −1 q (b − s) k ds |f (t)| dt t

||g||[a, a+b ],∞ + ||g||[ a+b ,b],∞ 2 2



α

≤

o q1 n 0 (b − a) k +1 ||g||[a,b],∞  0 q q
max |f (a)| , |f (b)| α 2 k αk + 1 Γk (α + k)

where Z

a+b 2

a

Z t Z b (s − a) αk −1 ds dt = a+b a

2

Z α b α (b − a) k +1 −1
α (b − s) k ds dt = α +1 α t 2k k +1 k

Which completes the proof.

Corollary 1. If we choose g(x) = 1 and α = k in Theorem 1, we get   1 Z b n 0 o q1 0 a + b b − a  . f (x)dx − f max |f (a)|q , |f (b)|q ≤ b − a a 2 4 0

Theorem 2. Let f : I ⊂ R −→ R be a differentiable mapping on I o and f ∈ L[a, b] and 0

g : [a, b] ⊂ R −→ R is continuous. If |f |q is quasi-convex function on [a, b], q > 1, the following inequality for k-fractional derivatives holds

     α α α α f a + b k J a+b − g(a) + k J a+b + g(b) − k J a+b − (f g)(a) + k J a+b + (f g)(b) ( ) ) ( ) ) ( ( 2 2 2 2 2 α

≤

α

2k where

1 p

+

1 q

 n 0 o q1 0 (b − a) k +1 ||g||∞ q q max |f (a)| , |f (b)|
p1 α Γk (α + k) kp+1

= 1. 0

Proof. Using Lemma 5, H¨older’s inequality and the fact that |f |q is quasi-convex function on [a, b], it follows that

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A. Ali, G. Gulshan, R. Hussain, A. Latif, M. Muddassar and J. Park

     α α α α f a + b g(a) + J g(b) − J (f g)(a) + J (f g)(b) J − + − + k k k k a+b a+b a+b a+b ( 2 ) ( 2 ) ( 2 ) ( 2 ) 2 Z t Z a+b 2 1 (s − a) αk −1 g(s)ds |f 0 (t)|dt ≤ Γk (α) a a ! Z b Z b 0 α −1 + (b − s) k g(s)ds |f (t)|dt a+b t 2

a+b 2

Z

1 ≤ Γk (α)

1 = Γk (α)  Z 

b

Z

a+b 2 a+b 2

Z

Z

Z p ! p1 b α (b − s) k −1 g(s)ds dt t

Z

a

a

1 + Γk (α)

p ! p1 Z t α (s − a) k −1 g(s)ds dt

! q1 0

|f (t)|q dt

a b

! q1 0

q

|f (t)| dt a+b 2

Z t p ! p1 (s − a) αk −1 g(s)ds dt a

a

! q1

a+b 2

0

|f (t)|q dt

Z

b

+ a+b 2

a

! q1  |f (t)|q dt  0

! p1 α ||g||∞ (b − a) k p+1

≤ α Γk (α) 2 k p+1 αk p + 1 αk p  ! q1 Z a+b n 0 o 2 0  + max |f (a)|q , |f (b)|q dt a

=

a+b 2

Z

b a+b 2

! q1  n 0 o 0 max |f (a)|q , |f (b)|q dt 

α  n 0 o q1 0 (b − a) k +1 ||g||∞ . max |f (a)|q , |f (b)|q 1
α p Γk (α + k) kp+1

α

2k

Where Z a

a+b 2

Z t p α k p+1 (s − a) αk −1 ds dt = α (b − a)
2 k p+1 αk p + 1 a


. α p k

Corollary 2. If we choose g(x) = 1 and α = k in Theorem 2, then we get   1 Z b n 0 o q1 0 a + b b−a  f (x)dx − f max |f (a)|q , |f (b)|q . ≤ 1 b − a a 2(p + 1) p 2 0

Theorem 3. Let f : I ⊂ R −→ R be a differentiable mapping on I o and f ∈ L[a, b]. If 0

|f | is quasi-convex function on [a, b] and g : [a, b] ⊂ R −→ R is continuous and symmetric to

a+b 2 ,

the following inequality for k-fractional derivatives holds

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f (a) + f (b) α α α α [k Ja+ g(b) + k Jb− g(a)] − [k Ja+ (f g)(b) + k Jb− (f g)(a)] 2   α n 0 o 0 2(b − a) k +1 ||g||∞ 1
≤ α 1 − α max |f (a)|, |f (b)| 2k k + 1 Γk (α + k) α k

where

> 0.

Proof. From Lemma 6, we get f (a) + f (b) α α α α [ J g(b) + J g(a)] − [ J (f g)(b) + J (f g)(a)] + − + − k a k b k a k b 2 Z b Z t Z b 0 α α 1 ≤ (s − a) k −1 g(s)ds |f (t)|dt. (b − s) k −1 g(s)ds − Γk (α) a a t 0

Since |f | is quasi-convex on [a, b], we know that for t ∈ [a, b]

  n 0 o 0 b−t 0 0 t − a |f (t)| = f a+ b ≤ max |f (a)|, |f (b)| b−a b−a and since g : [a, b] ⊂ R −→ R is continuous and symmetric to Z

b

α

(s − a) k −1 g(s)ds =

Z

t

Z

a+b−t

we can write

α

(b − s) k −1 g(a + b − s)ds

a a+b−t

=

a+b 2

α

(b − s) k −1 g(s)ds

a

therefore we get Z Z b t α α −1 −1 ! − (s − a) k g(s)ds (b − s) k g(s)ds a t Z a+b−t α = (b − s) k −1 g(s)ds t

 R   a+b−t (b − s) αk −1 g(s) ds, t ≤ R  (b − s) αk −1 g(s) ds,  t a+b−t

1214

  t ∈ a, a+b 2   t ∈ a+b 2 ,b .

(2.3)

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Therefore we get f (a) + f (b) α α α α (f g)(a)] (f g)(b) + J g(a)] − [ J g(b) + J [ J k b− k a+ k b− k a+ 2 # "Z a+b Z Z b Z t a+b−t 2 α α 1 −1 −1 (b − s) k g(s) dsdt + (b − s) k g(s) dsdt ≤ a+b Γk (α) a t a+b−t 2  n 0 o 0 max |f (a)|, |f (b)| ! Z a+b Z b 2   α α α α ||g||∞ ≤ (b − t) k − (t − a) k dt + (t − a) k − (b − t) k dt a+b Γk (α + k) a 2  n 0 o 0 max |f (a)|, |f (b)|   α n 0 o 0 1  2(b − a) k +1 ||g||∞
1− α = α max |f (a)|, |f (b)| 2k k + 1 Γk (α + k) since Z

a+b 2

a


α α (b − a) k +1 2 k +1 − 1
(b − t) dt = (t − a) dt = α a+b 2 k +1 αk + 1 2 α k

Z

b

Z

b

α k

and Z

a+b 2

α

(t − a) k dt =

α

α

(b − t) k dt = a+b 2

a

(b − a) k +1
. α 2 k +1 αk + 1

Corollary 3. In Theorem 3, if we take g(x) = 1, we get the inequality f (a) + f (b) Γk (α + k) α α − α [k Ja+ f (b) + k Jb− f (a)] k 2 2(b − a)   n 0 o 0 1 b−a
1 − α max |f (a)|, |f (b)| . ≤ α 2k k +1 . 0

Theorem 4. Let f : I ⊂ R −→ R be a differentiable mapping on I o and f ∈ L[a, b]. If 0

|f |q , q ≥ 1 is quasi-convex function on [a, b] and g : [a, b] ⊂ R −→ R is continuous and symmetric to

a+b 2 ,

the following inequality for k-fractional derivatives holds

f (a) + f (b) α α α α [k Ja+ g(b) + k Jb− g(a)] − [k Ja+ (f g)(b) + k Jb− (f g)(a)] 2   α n 0 o q1 0 1  2(b − a) k +1 ||g||∞
1− α max |f (a)|q , |f (b)|q ≤ α 2k k + 1 Γk (α + k) where

α k

> 0.

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Proof. From Lemma 6, power mean inequality, inequality (2.3) and the quasi-convexity 0

of |f |q , we get

f (a) + f (b) α α α α [k Ja+ g(b) + k Jb− g(a)] − [k Ja+ (f g)(b) + k Jb− (f g)(a)] 2 !1− q1 Z Z ! q1 Z b Z a+b−t 0 b a+b−t α α 1 ≤ (b − s) k −1 g(s)ds dt (b − s) k −1 g(s)ds |f (t)|q dt Γk (α) a t a t "Z

"Z

#

a+b−t

|(b − s)

×

α k −1

0

Z

q

α





α k −1

b

Z



1 1− α 2k

Z

t

α k −1

! q1  0 q g(s)|ds |f (t)| dt

a+b−t

Z

t

|(b − s) a+b 2

2(b − a) k +1
= α α k k +1

a+b−t

|(b − s)

g(s)|ds dt +

t α

b

 !1− q1 α −1 (b − s) k g(s) ds dt

t

n 0 o q1 0 max |f (a)|q , |f (b)|q

#

a+b−t

|(b − s) a

1 α 2k

1−

Z

a+b 2

a+b 2

2(b − a) k +1 ||g||∞
α k + 1 Γk (α + k)

where "Z Z a+b 2

b

g(s)|ds |f (t)| dt +

t

a

≤

# Z α −1 (b − s) k g(s) ds dt +

a+b−t

t

a

a+b 2

Z

a+b 2

Z

1 ≤ Γk (α)

α k −1

 g(s)|ds dt

a+b−t

 .

0

Theorem 5. Let f : I ⊂ R −→ R be a differentiable mapping on I o and f ∈ L[a, b]. 0

If |f |q , q > 1 is quasi-convex function on [a, b], and g : [a, b] −→ R is continuous and symmetric to

a+b 2 ,

the following inequality for k-fractional derivatives holds

f (a) + f (b) α α α α [k Ja+ g(b) + k Jb− g(a)] − [k Ja+ (f g)(b) + k Jb− (f g)(a)] 2   p1  1 α n 0 o q1 0 1 2 p (b − a) k +1 ||g||∞ q q 1 − max |f (a)| , |f (b)| , ≤ α
p1 α 2kp p + 1 Γ (α + k) k k where

α k

> 0.

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Proof. 0

From Lemma 6, H¨older’s inequality, inequality (2.3) and the quasi-convexity of

q

|f | , we get

f (a) + f (b) α α α α [k Ja+ g(b) + k Jb− g(a)] − [k Ja+ (f g)(b) + k Jb− (f g)(a)] 2 p ! p1 Z ! q1 Z b Z a+b−t b 0 α 1 (b − s) k −1 g(s)ds dt ≤ |f (t)|q dt Γk (α) a t a ||g||∞ ≤ Γk (α + k) Z

b

Z

a+b 2

 α α p (b − t) k − (t − a) k dt +

Z

b

! p1  α α p (t − a) k − (b − t) k dt

a+b 2

a

! q1 o n 0 0 q q max |f (a)| , |f (b)| dt

a

! p1 Z 1 Z 12 α   α α α p α p ||g||∞ (b − a) k +1 (1 − t) k − t k dt + ≤ t k − (1 − t) k dt 1 Γk (α + k) 0 2  n 0 o q1 0 max |f (a)|q , |f (b)|q ! p1 Z 12 Z 1 α   αp α α  α  ||g||∞ (b − a) k +1 p p p ≤ (1 − t) k − t k dt + t k − (1 − t) k dt 1 Γk (α + k) 0 2  n 0 o q1 0 max |f (a)|q , |f (b)|q 1

α

2 p (b − a) k +1 ||g||∞ ≤
p1 α Γk (α + k) kp+1

 p1   n 0 o q1 0 1 max |f (a)|q , |f (b)|q . 1 − αp 2k

Where  α α p α α (1 − t) k − t k ≤ (1 − t) k p − t k p ,

  1 for t ∈ 0, 2

and  α α α p α t k − (1 − t) k ≤ t k p − (1 − t) k p ,



1 for t ∈ ,1 2



which follows from (A − B)q ≤ Aq − B q , for any A > B ≥ 0 and q ≥ 1. Hence the proof is complete. 0

Theorem 6. Let f : I ⊂ R −→ R be a differentiable mapping on I o and f ∈ L[a, b]. If 0

|f |q , q > 1 is quasi-convex function on [a, b], and g : [a, b] ⊂ R −→ R is continuous and symmetric to

a+b 2 ,

the following inequality for k-fractional derivatives holds

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f (a) + f (b) α α α α [k Ja+ g(b) + k Jb− g(a)] − [k Ja+ (f g)(b) + k Jb− (f g)(a)] 2 α o q1 n 0 0 (b − a) k +1 ||g||∞  ≤ max |f (a)|q , |f (b)|q 1
α p Γk (α + k) kp+1 where 0
3k + m + 8 and P (z) be defined as in Theorem 1.4. If (f n P (f ))(k) and (g n P (g))(k) share h(z) CM, then one of the following three cases holds: (i) f = tg for a constant t such that td = 1, where d =GCD(n + m, . . . , n + m − i, . . . , n), am−i ̸= 0 for some i = 0, 1, . . . , m;

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(ii) f and g satisfying the algebraic function equation R(f, g) = 0, where R(w1 , w2 ) = w1n (am w1m +am−1 w1m−1 +· · ·+a0 )−w2n (am w2m +am−1 w2m−1 + · · · + a0 ); (iii) (f n P (f ))(k) (g n P (g))(k) = h2 . In 2011 Dyavanal [6] considered the uniqueness problem of meromorphic function related to the value sharing of two nonlinear differential polynomials in which the multiplicities of zeros and poles of f and g are taken into account. In 2013, Bhoosnurmath and Kabbur [3] proved the following uniqueness theorem by using the idea from Dyavanal [6]. Theorem 1.6. Let f and g be two nonconstant meromorphic functions, whose zeros and poles are of multiplicities at least s, where s is a positive integer. Let n and m be two positive integers with (n − m − 1)s ≥ max{10, 2m + 3}, and let P (z) be defined as in Theorem 1.4. If f n P (f )f ′ and g n P (g)g ′ share 1 CM, then either f = tg for a constant t such that td = 1, where d =GCD(n + m + 1, · · · , n + m + 1 − i, · · · , n + 1), am−i ̸= 0 for some i = 0, 1, · · · , m or f and g satisfy the algebraic function equation a0 am m−1 m−1 xm + an+m x + · · · + n+1 )− R(f, g) = 0, where R(x, y) = xn+1 ( n+m+1 am−1 m−1 a0 am n+1 m y ( n+m+1 y + n+m y + · · · + n+1 ). Similar Theorem 1.5 in which a small function and kth derivative are considered, what can we say when the condition sharing 1 and the first derivative in Theorem 1.6 are replaced with sharing a small function and kth derivative respectively? In this paper, we will study the problem and establish the following uniqueness theorem. Theorem 1.7. Let f and g be two transcendental meromorphic functions, whose zeros and poles are of multiplicities at least s, where s is a positive integer. Let n and m be two positive integers with n − m > max{2 + 2m (n+2)(k+4) , }, Θ(∞, f ) + Θ(∞, g) > n4 and let P (z) be defined as in Theos ns rem 1.4. If (f n P (f ))(k) and (g n P (g))(k) share h(z) CM, where h(z)(̸≡ 0, ∞) is a small function of f and g, then one of the following three cases hold: (i) (f n P (f ))(k) (g n P (g))(k) = h2 ; (ii) f = tg for a constant t such that td = 1, where d = GCD(n + m, n + m − 1, . . . , n + m − i, . . . , n + 1, n), am−i ̸= 0 for i = 0, 1, . . . , m; (iii) f and g satisfy the algebraic equation R(f, g) = 0, where R(f, g) = f n P (f ) − g n P (g). The possibility (f n P (f ))(k) (g n P (g))(k) = h2 does not occur for k = 1.

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2. Auxiliary results For the proof of our result, we need the following lemmas and definitions. Definition 2.1. [11] Let a ∈ C. We use N (r, a; f | = 1) to denote the counting function of simple a−points of f . For a positive integer p we denote by N (r, a; f | ≤ p) the counting function of those a−points of f (counted with proper multiplicities) whose multiplicities are not greater than p. By N (r, a; f | ≤ p) we denote the corresponding reduced counting function. Similarly, we can define N (r, a; f | ≥ p) and N (r, a; f | ≥ p). Definition 2.2. [1] Let a ∈ C, and let k be a nonnegative integer. We denote 1 by Nk (r, f −a ) the counting function of a−points of f , where an a−point of multiplicity m is counted m times if m ≤ k and k times if m > k. Then (2.1) Nk (r,

1 1 ) = N (r, ) + N (r, a; f | ≥ 2) + · · · + N (r, a; f | ≥ k). f −a f −a

1 1 Obviously N1 (r, f −a ) = N (r, f −a ).

Lemma 2.1. [15] Let f and g be two nonconstant meromorphic functions that share 1 CM. Then one of the following cases hold: (i) T (r) ≤ N2 (r, f1 ) + N2 (r, g1 ) + N2 (r, f ) + N2 (r, g) + S(r); (ii) f = g; (iii) f g = 1. Lemma 2.2. [17] Let f be a nonconstant meromorphic function, and p, k be positive integers. Then 1 1 (2.2) Np (r, (k) ) ≤ T (r, f (k) ) − T (r, f ) + Np+k (r, ) + S(r, f ), f f (2.3)

Np (r,

1

1 ) ≤ kN (r, f ) + N (r, ) + S(r, f ). p+k f (k) f

Lemma 2.3. [13] Let f be a nonconstant meromorphic function, and let n ∑ Pn (f ) = aj f j be a polynomial in f , where an ̸= 0, an−1 , · · · , a1 , a0 j=0

satisfying T (r, aj ) = S(r, f ). Then (2.4)

T (r, Pn ) = nT (r, f ) + S(r, f ).

Lemma 2.4. Let f and g be two nonconstant meromorphic functions such that 4 Θ(∞, f ) + Θ(∞, g) > n

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for all integer n ≥ 3. Then f n (af + b) = g n (ag + b) implies f = g, where a and b are two finite nonzero complex constants. Proof. By using similar way in [12], we can obtain the lemma.



Lemma 2.5. Let f and g be two nonconstant meromorphic functions, whose zeros and poles are of multiplicities at least s, where s is a positive integer and let n, k be two positive integers. Let F = (f n P (f ))(k) and G = (g n P (g))(k) , where P (z) be defined as in Theorem 1.4. If there ex1 ist two nonzero constants b1 and b2 such that N (r, F1 ) = N (r, G−b ) and 1 (k+1)(n+2) 1 1 N (r, G ) = N (r, F −b2 ), then n − m ≤ . ns Proof. By the second fundamental theorem of Nevanlinna’s theory, 1 1 T (r, F ) ≤ N (r, ) + N (r, F ) + N (r, ) + S(r, F ) F F − b2 1 1 (2.5) ≤ N (r, ) + N (r, F ) + N (r, ) + S(r, F ). F G Combining (2.2), (2.3), (2.5) and Lemma 2.3, we get 1 1 ) + S(r, f ) (n + m)T (r, f ) ≤ T (r, F ) − N (r, ) + Nk+1 (r, n F f P (f ) 1 1 ≤ N (r, ) + N (r, f ) + Nk+1 (r, n ) + S(r, f ) G f P (f ) 1 1 ≤ Nk+1 (r, n ) + Nk+1 (r, n ) + N (r, f ) f P (f ) g P (g) + kN (r, g) + S(r, f ) + S(r, g) k + 1 + nk k+1+n ≤( + m)T (r, f ) + ( + m)T (r, g) ns ns + S(r, f ) + S(r, g) (k + 1)(n + 2) + 2m)T (r) + S(r). ns Similarly, for the case of g, (k + 1)(n + 2) (2.7) + 2m)T (r) + S(r). (n + m)T (r, g) ≤ ( ns It follows from (2.6) and (2.7) that (2.6)

≤(

(2.8)

(n −

which gives n − m ≤

(k + 1)(n + 2) − m)T (r) ≤ S(r), ns

(k+1)(n+2) . ns

This completes the proof.



Lemma 2.6. Let f and g be two transcendental meromorphic functions, whose zeros and poles are of multiplicities at least s, where s is a positive

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integer. Let P (z) be defined as in Theorem 1.4, and n, m, k be three positive integers, and h(̸≡ 0, ∞) be small function of f and g. Then (f n P (f ))(k) (g n P (g))(k) ̸≡ h2

(2.9)

holds for k = 1 and (n + m − 2)p > 2m(1 + 1s ), where p is the number of distinct roots of P (z) = 0. Proof. If (2.9) is possible for k = 1, i.e., (f n P (f ))′ (g n P (g))′ = h2 . Then f n−1 Q(f )f ′ g n−1 Q(g)g ′ = h2 ,

(2.10) where Q(z) =

m ∑

bj z j , bj = (n + j)aj , j = 0, 1, . . . , m. Denote Q(z) as

j=0

Q(z) = bm (z − d1 )l1 (z − d2 )l2 · · · (z − dp )lp , where

p ∑

li = m, 1 ≤ p ≤ m, di ̸= dj , i ̸= j, 1 ≤ i, j ≤ p, di are nonzero

i=1

constants and li are positive integers, i = 1, 2, . . . p. Suppose that z1 ̸∈ S0 is a zero of f with multiplicity s1 (≥ s), where S0 is a set defined as S0 = {z : h(z) = 0} ∪ {z : h(z) = ∞}. Then z1 is a pole of g with multiplicity q1 (≥ s). We deduce from (2.10) that ns1 − 1 = (n + m)q1 + 1 and so mq1 + 2 = n(s1 − q1 ).

(2.11) From (2.11) we get q1 ≥

n−2 , m

so s1 ≥

n+m−2 . m

Hence, (2.12)

1 m 1 N (r, ) ≤ N (r, ) + S(r, f ). f n+m−2 f

Suppose that z2 ̸∈ S0 is a zero of Q(f ) with multiplicity s2 and is a zero of f − di of order qi , i = 1, 2, . . . p. Then s2 = li qi , i = 1, 2, · · · p. Then z2 is a pole of g with multiplicity q(≥ s). It follows from (2.10) that qi li + qi − 1 = (n + m)q + 1 ≥ (n + m)s + 1.

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So qi ≥

(n + m)s + 2 , li + 1

i = 1, 2, . . . p.

Hence, N (r,

li + 1 1 )≤ N (r, di , f ) + S(r, f ), f − di (n + m)s + 2

i = 1, 2, . . . p.

By this and the first fundamental theorem of Nevanlinna’s theory, we have (2.13)

p ∑ i=1

N (r,

1 m+p )≤ T (r, f ) + S(r, f ). f − di (n + m)s + 2

Suppose that z3 ∈ ̸ S0 is a pole of f . Then we know that z3 is either a n−1 zero of g Q(g) or a zero of g ′ by (2.10). Therefore, ∑ 1 1 1 N (r, f ) ≤ N (r, ) + N (r, ) + N0 (r, ′ ) + S(r, f ) + S(r, g) g g − di g i=1 p

m 1 m+p + )T (r, g) + N0 (r, ′ ) n + m − 2 (n + m)s + 2 g + S(r, f ) + S(r, g), ≤(

(2.14)

where N0 (r, g1′ ) denote the reduce counting function of those zeros of g ′ which are not the zeros of gQ(g). By (2.12)-(2.14), and the second fundamental theorem of Nevanlinna’s theory, ∑ 1 1 1 pT (r, f ) ≤ N (r, f ) + N (r, ) + N (r, ) − N0 (r, ′ ) + S(r, f ) f f − di f i=1 p

m m+p 1 + )(T (r, f ) + T (r, g)) + N0 (r, ′ ) n + m − 2 (n + m)s + 2 g 1 − N0 (r, ′ ) + S(r, f ) + S(r, g). f ≤(

(2.15)

Similarly, for the case of g, m m+p 1 + )(T (r, f ) + T (r, g)) + N0 (r, ′ ) n + m − 2 (n + m)s + 2 f 1 − N0 (r, ′ ) + S(r, f ) + S(r, g). g

pT (r, g) ≤ ( (2.16)

It follows from (2.15) and (2.16) that (p −

2m 2(m + p) − )(T (r, f ) + T (r, g)) ≤ S(r, f ) + S(r, g). n + m − 2 (n + m)s + 2

This is a contradiction with our assumption that (n + m − 2)p > 2m(1 + 1s ), and hence the proof is complete. 

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3. Proof of Theorem 1.7 n

(k)

(f P (f )) Let F = h Applying Lemma 2.3,

(3.1)

(g n P (g))(k) , G = . Then F and G share 1 CM. h

T (r, h) = o(T (r, F )) = S(r, f ),

T (r, h) = o(T (r, G)) = S(r, g).

It follows from (2.2) and (3.1) that

N2 (r,

1 1 ) ≤ N2 (r, n ) + N2 (r, h) + S(r, f ) F (f P (f ))(k) 1 ) + S(r, f ) ≤ N2 (r, n (f P (f ))(k) ≤ T (r, (f n P (f )) − (n + m)T (r, f ) + Nk+2 (r,

(3.2)

≤ T (r, F ) − (n + m)T (r, f ) + Nk+2 (r,

1 f n P (f )

1 f n P (f )

) + S(r, f )

) + S(r, f ).

We deduce from (2.3) that

N2 (r,

1 1 ) ≤ N2 (r, n ) + S(r, f ) F (f P (f ))(k) ≤ kN (r, f n P (f ))(k) ) + Nk+2 (r,

(3.3)

≤ kN (r, f ) + Nk+2 (r,

1 f n P (f )

1 f n P (f )

) + S(r, f )

) + S(r, f ).

It follows from (3.2) that

(3.4) (n + m)T (r, f ) ≤ T (r, F ) + Nk+2 (r,

1 1 ) − N (r, ) + S(r, f ). 2 f n P (f ) F

Suppose that (i) of Lemma 2.1 holds, i.e., 1 1 ) + N2 (r, ) + N2 (r, F ) + N2 (r, G) F G + S(r, f ) + S(r, g).

max{T (r, F ), T (r, G)} ≤ N2 (r,

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9

Combining this, (3.3) and (3.4), 1 1 ) − N (r, ) + S(r, f ) + S(r, g) 2 f n P (f ) F 1 1 ) ≤ N2 (r, ) + N2 (r, F ) + N2 (r, G) + Nk+2 (r, n G f P (f ) + S(r, f ) + S(r, g) 1 1 ) + Nk+2 (r, n ) + kN (r, g) ≤ Nk+2 (r, n f P (f ) g P (g)

(n + m)T (r, f ) ≤ T (r, F ) + Nk+2 (r,

+ 2N (r, f ) + 2N (r, g) + S(r, f ) + S(r, g) k + 2 + 2n (k + 2)(n + 1) + m)T (r, f ) + ( + m)T (r, g) ns ns + S(r, f ) + S(r, g) ≤(

k(n + 2) + 4(n + 1) + 2m)T (r) + S(r). ns Similarly, for the case of g, (3.5)

≤(

k(n + 2) + 4(n + 1) + 2m)T (r) + S(r). ns It follows from (3.5) and (3.6) that (3.6)

(n + m)T (r, g) ≤ (

(n + m)T (r) ≤ (

k(n + 2) + 4(n + 1) + 2m)T (r) + S(r). ns

This implies that (3.7)

(n − m −

k(n + 2) + 4(n + 1) )T (r) ≤ S(r). ns

This contradicts with our assumption that (n−m) > max{2+ 2m , (n+2)(k+4) }. s ns So, we conclude that either F G = 1 or F = G by Lemma 2.1. Suppose that F G = 1, then (f n P (f ))(k) (g n P (g))(k) = h2 . This is a contradiction when k = 1 by Lemma 2.6. So F = G, this implies that (3.8)

(f n P (f ))(k) = (g n P (g))(k) .

Integrating for (3.8), we have (3.9)

(f n P (f ))(k−1) = (g n P (g))(k−1) + bk−1 ,

where bk−1 is constant. If bk−1 ̸= 0, we obtain n−m ≤ (k+1)(n+2) < (k+4)(n+2) ns ns by Lemma 2.5. This is a contradiction with our assumption that (n − m) > , (n+2)(k+4) }. Thus bk−1 = 0. By repeating k−times, max{2 + 2m s ns (3.10)

f n P (f ) = g n P (g).

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10

J. R. LONG

If m = 1 in (3.10), then f = g by Lemma 2.4. Suppose that m ≥ 2 and b = fg . If b is a constant, putting f = bg in (3.10), we get (3.11) am g n+m (bn+m − 1) + am−1 g n+m−1 (bn+m−1 − 1) + · · · + a0 g n (bn − 1) = 0, which implies bd = 1, where d = GCD(n + m, n + m − 1, . . . , n + 1, n). Hence f = tg for a constant t such that td = 1, d = GCD(n + m, n + m − 1, . . . , n + m − i, . . . , n + 1, n), i = 0, 1, . . . , m. If b is not a constant, then we can see that f and g satisfy the algebraic function equation R(f, g) = 0 by (3.10), where R(f, g) = f n P (f ) − g n P (g). This completes the proof of theorem. Acknowledgements. This research work is supported in part by the Foundation of Science and Technology of Guizhou Province (Grant No. [2015]2112), the National Natural Science Foundation of China (Grant No. 11501142). References [1] T. C. Zlzahary and H. X. Yi, Weighted sharing three values and uniqueness of meromorphic functions, J. Math. Aanl. Appl. 295 (2004), no. 1, 247-257. [2] W. Bergweiler and A. Eremenko, On the singularities of the inverse to a meromorphic function of finite order, Rev. Mat. Iberoamericana 11 (1995), no. 2, 355-373. [3] S. S. Bhoosnurmath and S. R. Kabbur, Uniqueness and value sharing of meromorphic functions with regard to multiplicity, Tamkang J. Math. 44 (2013), 11-22. [4] H. H. Chen and M. L. Fang, The value distribution of f n f ′ , Sci. China Ser. A 38 (1995), no. 7, 789-798. [5] J. Clunie, On a result of Hayman, J. London Math. Soc. 42 (1967), 389-392. [6] R. S. Dyavanal, Uniqueness and value sharing of differnetial polynomials of meromorphic functions, J. Math. Anal. Appl. 374 (2011), 335345. [7] M. L. Fang, Uniqueness and value-sharing of entire functions, Comput. Math. Appl. 44 (2002), 823-831. [8] M. L. Fang and X. H. Hua, Entire functions that share one value, Nanjing Daxue Xuebao Shuxue Banniankan 13 (1996), no. 1, 44-48. [9] W. K. Hayman, Meromorphic Functions, Clarendon Press, Oxford, 1964.

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NONLINEAR DIFFERENTIAL POLYNOMIALS SHARING A SMALL FUNCTION 11

[10] W. K. Hayman, Picard values of meromorphic functions and their derivatives, Ann. Math. 70 (1959), 9-42. [11] I. Lahiri, Value distribution of certain differential polynomials, Int. J. Math. Soc. 28 (2001), 83-91. [12] I. Lahiri, On a question of Hong Xun Yi, Arch. Math. (Brno) 38 (2002), 119-128. [13] C. C. Yang, On the deficiencies of differential polynomials II, Math. Z. 125 (1972), 107-112. [14] C. C. Yang and H. X. Yi, Uniqueness Theory of Meromorphic Functions, Mathematics and its Applications, 557, Kluwer Acad. Publ., Dordrecht, 2003. [15] C. C. Yang and X. H. Hua, Uniqueness and value sharing of meromorphic functions, Ann. Acad. Sci. Fenn. Math. 22 (1997), no. 2, 395-406. [16] L. Yang, Value Distribution Theory, Springer-Verlag, New York, 1993. [17] J. L. Zhang and L. Z. Yang, Some results related to a conjecture of R. Br¨ uck, J. Inequal. Pure Appl. Math. 8 (2007), Article 18, 11 PP. [18] X. Y. Zhang, J. F. Chen and W. C. Lin, Entire or meromorphic functions sharing one value, Comput. Math. Appl. 56 (2008), 1876-1883. Jianren Long School of Mathematics and Computer Science, Guizhou Normal University, Guiyang, 550001, P.R. China. School of Mathematical Sciences, Xiamen University, Xiamen, 361005, P.R. China. E-mail address: [email protected], [email protected]

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J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 22, NO.7, 2017, COPYRIGHT 2017 EUDOXUS PRESS, LLC

Impulsive hybrid fractional quantum difference equations Bashir Ahmada , Sotiris K. Ntouyasb,a , Jessada Tariboonc , Ahmed Alsaedia , Wafa Shammakha a

Nonlinear Analysis and Applied Mathematics (NAAM)-Research Group, Department of Mathematics, Faculty of Science, King Abdulaziz University, P.O. Box 80203, Jeddah 21589, Saudi Arabia e-mail: bashirahmad− [email protected], [email protected], [email protected] b

Department of Mathematics, University of Ioannina, 451 10 Ioannina, Greece e-mail: [email protected] c

Nonlinear Dynamic Analysis Research Center, Department of Mathematics, Faculty of Applied Science, King Mongkut’s University of Technology North Bangkok, Bangkok, 10800 Thailand e-mail: [email protected] Abstract This paper is concerned with the existence of solutions for impulsive hybrid fractional q-difference equations involving a q-shifting operator of the type a Φq (m) = qm + (1 − q)a. A hybrid fixed point theorem for two operators in a Banach algebra due to Dhage [29] is applied to obtain the existence result. An example illustrating the main result is also presented.

Key words and phrases: Quantum calculus; impulsive fractional q-difference equations; hybrid differential equations; existence; fixed point theorem AMS (MOS) Subject Classifications: 34A08; 34A12; 34A37

1

Introduction

Fractional differential equations have been extensively investigated by several researchers in the recent years. The overwhelming interest in this branch of mathematics is due to the application of fractionalorder operators in the mathematical modelling of several phenomena occurring in a variety of disciplines of applied sciences and engineering such as biomathematics, signal and image processing, control theory, dynamical systems, etc. Hybrid fractional differential equations dealing with the fractional derivative of an unknown function hybrid with the nonlinearity depending on it is another interesting field of research. For some recent works on this topic, we refer the reader to a series of papers ([1]-[6]). The subject of q-difference calculus or quantum calculus dates back to the beginning of the 20th century, when Jackson [7] introduced the concept of q-difference operator. Afterwards, this field of research flourished with the contributions of researchers from different parts of the world, for instance, see ([8]-[15]). The intensive development of fractional calculus motivated several investigators to consider fractional q-difference calculus. Now a great deal of work on initial and boundary value problems involving nonlinear fractional q-difference equations is available, for example, see [16]-[24] and the references therein. The quantum calculus, known as the calculus without limits, provides a descent approach to study nondifferentiable functions in terms of difference operators. Quantum difference operators appear in different areas of mathematics such as orthogonal polynomials, basic hyper-geometric functions, combinatorics, the calculus of variations, mechanics and the theory of relativity. For the fundamental concepts of quantum calculus, we refer the reader to a text by Kac and Cheung [25]. More recently, the topic of qk -calculus has also gained consideration attention. The notions of qk derivative and qk -integral for a function f : Jk := [tk , tk+1 ] → R, together with their properties can

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B. Ahmad, S.K. Ntouyas, J. Tariboon, A. Alsaedi and W. Shammakh

be found in [26, 27]. In [28], new concepts of fractional quantum calculus were defined via a q-shifting operator of the form: a Φq (m) = qm + (1 − q)a. The purpose of the present work is to study the following impulsive hybrid fractional quantum difference equations:    x(t)  c αk    tk Dqk f (t, x(t)) = g(t, x(t)), t ∈ Jk ⊆ [0, T ], t 6= tk ,  (1) ∆x(tk ) = ϕk (x(tk )) , k = 1, 2, . . . , m,      x(0) = µ, where 0 = t0 < t1 < · · · < tm < tm+1 = T , ctk Dqαkk denotes the Caputo fractional qk -derivative of order αk on intervals Jk , J0 = [0, t1 ], Jk = (tk , tk+1 ], 0 < αk ≤ 1, 0 < qk < 1, k = 0, 1, . . . , m, J = [0, T ], f ∈ C(J × R, R \ {0}), ϕk ∈ C(R, R), k = 1, 2, . . . , m, µ ∈ R and ∆x(tk ) = x(t+ k ) − x(tk ), + x(t+ ) = lim x(t + θ), k = 1, 2, . . . , m. Here, we emphasize that the above initial value problem k θ→0 k contains the new q-shifting operator a Φq (m) = qm + (1 − q)a [28]. The papers is organized as follows. In Section 2, we recall some preliminary concepts and present an auxiliary lemma which is used to convert the impulsive problem (1) into an equivalent integral equation. An existence result for the problem (1) obtained by means of a hybrid fixed point theorem due to Dhage [29] is presented in Section 3, which is well illustrated with the aid of an example.

2

Preliminaries

For the convenience of the reader, we recall some preliminary concepts from [28]. First of all, we define a q-shifting operator as a Φq (m)

= qm + (1 − q)a

(2)

such that k a Φq (m)

= a Φk−1 (a Φq (m)) q

and

0 a Φq (m)

= m,

for any positive integer k. The power law for q-shifting operator is a (n

− m)(0) q = 1,

a (n

k−1 Y

− m)(k) q =


n − a Φiq (m) ,

k ∈ N ∪ {∞}.

i=0

In case γ ∈ R, the above power law takes the form (γ) (γ) a (n − m)q = n

∞ Y 1 − na Φiq (m/n) i=0

1 − na Φγ+i q (m/n)

.

The q-derivative of a function h on interval [a, b] is defined by (a Dq h)(t) =

h(t) − h(a Φq (t)) , t 6= a, and (a Dq h)(a) = lim (a Dq h)(t), t→a (1 − q)(t − a)

while the higher order q-derivative is given by the formula (a Dq0 f )(t) = f (t) and (a Dqk f )(t) = a Dqk−1 (a Dq f )(t), k ∈ N. The product and quotient formulas for q-derivative are a Dq (h1 h2 )(t)

= h1 (t)a Dq h2 (t) + h2 (a Φq (t))a Dq h1 (t) = h2 (t)a Dq h1 (t) + h1 (a Φq (t))a Dq h2 (t),  a Dq

h1 h2

 (t) =

h2 (t)a Dq h1 (t) − h1 (t)a Dq h2 (t) , h2 (t)h2 (a Φq (t))

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Impulsive hybrid fractional q-difference equations

where h1 and h2 are well defined on [a, b] with h2 (t)h2 (a Φq (t)) 6= 0. The q-integral of a function h defined on the interval [a, b] is given by t

Z

h(s)a ds = (1 − q)(t − a)

(a Iq h)(t) =

∞ X

a

q i h(a Φqi (t)), t ∈ [a, b],

(3)

i=0

with (a Iq0 h)(t) = h(t) and (a Iqk h)(t) = a Iqk−1 (a Iq h)(t), k ∈ N. The fundamental theorem of calculus applies to the operator a Dq and a Iq , that is, (a Dq a Iq h)(t) = h(t), and if h is continuous at t = a, then (a Iq a Dq h)(t) = h(t) − h(a). The q-integration by parts formula on the interval [a, b] is Z b b Z b f (s)a Dq g(s)a dq s = (f g)(t) − g(a Φq (s))a Dq f (s)a dq s. a

a

a

Let us now define Riemann-Liouville fractional q-derivative and q-integral on interval [a, b] and outline some of their properties [28]. Definition 2.1 The fractional q-derivative of Riemann-Liouville type of order ν ≥ 0 on interval [a, b] is defined by (a Dq0 h)(t) = h(t) and (a Dqν h)(t) = (a Dql a Iql−ν h)(t), ν > 0, where l is the smallest integer greater than or equal to ν. Definition 2.2 Let α ≥ 0 and h be a function defined on [a, b]. The fractional q-integral of RiemannLiouville type is given by (a Iq0 h)(t) = h(t) and (a Iqα h)(t) =

1 Γq (α)

Z

t a (t

− a Φq (s))(α−1) h(s)a dq s, α > 0, t ∈ [a, b]. q

a

From [28], we have the following formulas Γq (β + 1) (t − a)β−α , Γq (β − α + 1) Γq (β + 1) α β (t − a)β+α . a Iq (t − a) = Γq (β + α + 1)

α a Dq (t

− a)β =

(4) (5)

Lemma 2.3 Let α, β ∈ R+ and f be a continuous function on [a, b], a ≥ 0. The Riemann-Liouville fractional q-integral has the following semi-group property β α a Iq a Iq h(t)

= a Iqα a Iqβ h(t) = a Iqα+β h(t).

Lemma 2.4 Let h be a q-integrable function on [a, b]. Then the following equality holds α α a Dq a Iq h(t)

= h(t),

for α > 0, t ∈ [a, b].

Lemma 2.5 Let α > 0 and p be a positive integer. Then for t ∈ [a, b] the following equality holds α p p α a Iq a Dq h(t) = a Dq a Iq h(t) −

p−1 X k=0

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J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 22, NO.7, 2017, COPYRIGHT 2017 EUDOXUS PRESS, LLC

B. Ahmad, S.K. Ntouyas, J. Tariboon, A. Alsaedi and W. Shammakh

We define Caputo fractional q-derivative as follows. Definition 2.6 The fractional q-derivative of Caputo type of order α ≥ 0 on interval [a, b] is defined by (ca Dq0 f )(t) = h(t) and (ca Dqα h)(t) = (a Iqn−α a Dqn h)(t), α > 0, where n is the smallest integer greater than or equal to α. Lemma 2.7 Let α > 0 and n be the smallest integer greater than or equal to α. Then for t ∈ [a, b] the following equality holds αc α a Iq a Dq h(t)

= h(t) −

n−1 X k=0

(t − a)k k a D h(a). Γq (k + 1) q

Proof. From Lemma 2.5, for α = p = m, where m is a positive integer, we have m m a Iq a Dq h(t)

=

m m a Dq a Iq h(t)

−

m−1 X

m−1 X (t − a)k (t − a)k k k a Dq h(a) = h(t) − a D h(a). Γq (k + 1) Γq (k + 1) q

k=0

k=0

Then, by Definition 2.6, we have αc α α n−α n n n a Iq a Dq h(t) = a Iq a Iq a Dq h(t) = a Iq a Dq h(t) = h(t) −

n−1 X k=0

(t − a)k k a D h(a). Γq (k + 1) q 

Now we present a lemma which plays a pivotal role in the forthcoming analysis. x is injection for each t ∈ J. x ∈ P C(J, R) is the solution f (t, x) of (1) if and only if x is a solution of the impulsive integral equation Lemma 2.8 Assume that the map x 7→

x(t)

= f (t, x(t))

k k X Y µ Y f (ti , x(ti )) + + f (0, µ) i=1 f (ti , x(ti )) i=1

αi−1 ti−1 Iqi−1 g(ti , x(ti ))

i≤j≤k

+

k X

Y

i=1 i 0 , t + φ(x, x)

where, as usual, inf ϕ = +∞. It is easy to show that (S, d) is complete (see [28, Lemma 2.1]). Now we consider the linear mapping J : S → S such that ( ) x Jg(x) := 4g 2 for all x ∈ X. Let g, h ∈ S be given such that d(g, h) = ε. Then t N (g(x) − h(x), εt) ≥ t + φ(x, x) for all x ∈ X and all t > 0. Hence

(

( )

( )

x x − 4h , Lεt 2 2 ( ( ) ( ) ) x x L = N g −h , εt 2 2 4

)

N (Jg(x) − Jh(x), Lεt) = N 4g

≥ =

Lt 4

(x

Lt 4

) ≥ x

+ φ 2, 2 t t + φ(x, x)

Lt 4

+

Lt 4 L 2 φ(x, x)

for all x ∈ X and all t > 0. So d(g, h) = ε implies that d(Jg, Jh) ≤ Lε. This means that d(Jg, Jh) ≤ Ld(g, h) for all g, h ∈ S. It follows from (2.4) that

(

N f (x) − 4f for all x ∈ X and all t > 0. So d(f, Jf ) ≤

( )

)

L t x , t ≥ 2 4 t + φ(x, x)

L 4.

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By Theorem 1.4, there exists a mapping Q : X → Y satisfying the following: (1) Q is a fixed point of J, i.e., ( ) x 1 Q = Q(x) 2 4 for all x ∈ X. The mapping Q is a unique fixed point of J in the set

(2.5)

M = {g ∈ S : d(f, g) < ∞}. This implies that Q is a unique mapping satisfying (2.5) such that there exists a µ ∈ (0, ∞) satisfying t N (f (x) − Q(x), µt) ≥ t + φ(x, x) for all x ∈ X; (2) d(J n f, Q) → 0 as n → ∞. This implies the equality ( ) x n N - lim 4 f = Q(x) n→∞ 2n for all x ∈ X; 1 (3) d(f, Q) ≤ 1−L d(f, Jf ), which implies the inequality d(f, Q) ≤

L . 4 − 4L

This implies that the inequality (2.3) holds. By (2.2), ( ( ( ) ( ) ( ) ( )) ) x y x+y x−y t n −2 n (x y) N 4 2f + 2f −f −α f α n ,4 t ≥ n n n 2 2 2 2 t + φ 2n , 2n for all x, y ∈ X, all t > 0 and all n ∈ N. So (

N 4

(

n

(

2f

x 2n

)

(

+ 2f

y 2n

)

(

−f

x+y 2n

)

−α

−2

(

x−y f α n 2

for all x, y ∈ X, all t > 0 and all n ∈ N. Since limn→∞ t > 0,

))

)

,t ≥

t 4n Ln t + 4n φ(x,y) 4n

t 4n

+

t 4n n L 4n φ (x, y)

= 1 for all x, y ∈ X and all

2Q(x) + 2Q(y) − Q(x + y) − α−2 Q(α(x − y)) = 0 for all x, y ∈ X. By Lemma 2.1, the mapping Q : X → Y is quadratic, as desired.



Corollary 2.3. Let θ ≥ 0 and let p be a real number with p > 2. Let X be a normed vector space with norm ∥ · ∥. Let f : X → Y be a mapping satisfying f (0) = 0 and ( ) t (2.6) N 2f (x) + 2f (y) − f (x + y) − α−2 f (α(x − y)), t ≥ t + θ(∥x∥p + ∥y∥p ) for all x, y ∈ X and all t > 0. Then Q(x) := N -limn→∞ 4n f ( 2xn ) exists for each x ∈ X and defines a quadratic mapping Q : X → Y such that (2p − 4)t N (f (x) − Q(x), t) ≥ p (2 − 4)t + 2θ∥x∥p for all x ∈ X and all t > 0.

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C. PARK, J. R. LEE, AND D. Y. SHIN

Proof. The proof follows from Theorem 2.2 by taking φ(x, y) := θ(∥x∥p +∥y∥p ) for all x, y ∈ X. Then we can choose L = 22−p , and we get the desired result.  Theorem 2.4. Let φ : X 2 → [0, ∞) be a function such that there exists an L < 1 with (

x y φ(x, y) ≤ 4Lφ , 2 2

)

for all x, y ∈ X. Let f : X → Y be a mapping satisfying f (0) = 0 and (2.2). Then Q(x) := N limn→∞ 41n f (2n x) exists for each x ∈ X and defines a quadratic mapping Q : X → Y such that N (f (x) − Q(x), t) ≥

(4 − 4L)t (4 − 4L)t + φ(x, x)

(2.7)

for all x ∈ X and all t > 0. Proof. Let (S, d) be the generalized metric space defined in the proof of Theorem 2.2. Now we consider the linear mapping J : S → S such that 1 Jg(x) := g (2x) 4 for all x ∈ X. It follows from (2.4) that (

)

1 1 t N f (x) − f (2x), t ≥ 4 4 t + φ(x, x) for all x ∈ X and all t > 0. So d(f, Jf ) ≤ 14 . Hence d(f, A) ≤

1 , 4 − 4L

which implies that the inequality (2.7) holds. The rest of the proof is similar to the proof of Theorem 2.2.



Corollary 2.5. Let θ ≥ 0 and let p be a real number with 0 < p < 2. Let X be a normed vector space with norm ∥ · ∥. Let f : X → Y be a mapping satisfying f (0) = 0 and (2.6). Then Q(x) := N -limn→∞ 41n f (2n x) exists for each x ∈ X and defines a quadratic mapping Q : X → Y such that N (f (x) − Q(x), t) ≥

(4 − 2p )t (4 − 2p )t + 2θ∥x∥p

for all x ∈ X and all t > 0. Proof. The proof follows from Theorem 2.4 by taking φ(x, y) := θ(∥x∥p +∥y∥p ) for all x, y ∈ X. Then we can choose L = 2p−2 , and we get the desired result. 

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J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 22, NO.7, 2017, COPYRIGHT 2017 EUDOXUS PRESS, LLC

QUADRATIC α-FUNCTIONAL EQUATION IN FUZZY BANACH SPACES

References [1] T. Aoki, On the stability of the linear transformation in Banach spaces, J. Math. Soc. Japan 2 (1950), 64–66. [2] T. Bag and S.K. Samanta, Finite dimensional fuzzy normed linear spaces, J. Fuzzy Math. 11 (2003), 687– 705. [3] T. Bag and S.K. Samanta, Fuzzy bounded linear operators, Fuzzy Sets and Systems 151 (2005), 513–547. [4] A. Bahyrycz and J. Brzdek, On functions that are approximate fixed points almost everywhere and Ulam’s type stability, J. Fixed Point Theory Appl. (in press). [5] L. C˘ adariu and V. Radu, Fixed points and the stability of Jensen’s functional equation, J. Inequal. Pure Appl. Math. 4, no. 1, Art. ID 4 (2003). [6] L. C˘ adariu and V. Radu, On the stability of the Cauchy functional equation: a fixed point approach, Grazer Math. Ber. 346 (2004), 43–52. [7] L. C˘ adariu and V. Radu, Fixed point methods for the generalized stability of functional equations in a single variable, Fixed Point Theory Appl. 2008, Art. ID 749392 (2008). [8] I. Chang and Y. Lee, Additive and quadratic type functional equation and its fuzzy stability, Results Math. 63 (2013), 717–730. [9] S.C. Cheng and J.M. Mordeson, Fuzzy linear operators and fuzzy normed linear spaces, Bull. Calcutta Math. Soc. 86 (1994), 429–436. [10] J. Diaz and B. Margolis, A fixed point theorem of the alternative for contractions on a generalized complete metric space, Bull. Amer. Math. Soc. 74 (1968), 305–309. [11] M. Eshaghi Gordji and Th.M. Rassias, Fixed points and generalized stability for quadratic and quartic mappings in C ∗ -algebras, J. Fixed Point Theory Appl. (in press). [12] W. Fechner, Stability of a functional inequalities associated with the Jordan-von Neumann functional equation, Aequationes Math. 71 (2006), 149–161. [13] C. Felbin, Finite dimensional fuzzy normed linear spaces, Fuzzy Sets and Systems 48 (1992), 239–248. [14] P. G˘ avruta, A generalization of the Hyers-Ulam-Rassias stability of approximately additive mappings, J. Math. Anal. Appl. 184 (1994), 431–436. [15] A. G. Ghazanfari and Z. Alizadeh, On approximate ternary m-derivations and σ-homomorphisms, J. Fixed Point Theory Appl. (in press). [16] A. Gil´ anyi, Eine zur Parallelogrammgleichung ¨ aquivalente Ungleichung, Aequationes Math. 62 (2001), 303– 309. [17] A. Gil´ anyi, On a problem by K. Nikodem, Math. Inequal. Appl. 5 (2002), 707–710. [18] D.H. Hyers, On the stability of the linear functional equation, Proc. Nat. Acad. Sci. U.S.A. 27 (1941), 222–224. [19] D.H. Hyers, G. Isac and Th.M. Rassias, Stability of Functional Equations in Several Variables, Birkh¨ auser, Basel, 1998. [20] G. Isac and Th.M. Rassias, Stability of ψ-additive mappings: Appications to nonlinear analysis, Internat. J. Math. Math. Sci. 19 (1996), 219–228. [21] S. Jung, Hyers-Ulam-Rassias Stability of Functional Equations in Mathematical Analysis, Hadronic Press lnc., Palm Harbor, Florida, 2001. [22] A.K. Katsaras, Fuzzy topological vector spaces II, Fuzzy Sets and Systems 12 (1984), 143–154. [23] H. Kim, M. Eshaghi Gordji, A. Javadian and I. Chang, Homomorphisms and derivations on unital C ∗ algebras related to Cauchy-Jensen functional inequality, J. Math. Inequal. 6 (2012), 557–565. [24] H. Kim, J. Lee and E. Son, Approximate functional inequalities by additive mappings, J. Math. Inequal. 6 (2012), 461–471. [25] I. Kramosil and J. Michalek, Fuzzy metric and statistical metric spaces, Kybernetica 11 (1975), 326–334. [26] S.V. Krishna and K.K.M. Sarma, Separation of fuzzy normed linear spaces, Fuzzy Sets and Systems 63 (1994), 207–217. [27] J. Lee, C. Park and D. Shin, An AQCQ-functional equation in matrix normed spaces, Results Math. 27 (2013), 305–318. [28] D. Mihet¸ and V. Radu, On the stability of the additive Cauchy functional equation in random normed spaces, J. Math. Anal. Appl. 343 (2008), 567–572. [29] M. Mirzavaziri and M.S. Moslehian, A fixed point approach to stability of a quadratic equation, Bull. Braz. Math. Soc. 37 (2006), 361–376.

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C. PARK, J. R. LEE, AND D. Y. SHIN

[30] A.K. Mirmostafaee, M. Mirzavaziri and M.S. Moslehian, Fuzzy stability of the Jensen functional equation, Fuzzy Sets and Systems 159 (2008), 730–738. [31] A.K. Mirmostafaee and M.S. Moslehian, Fuzzy versions of Hyers-Ulam-Rassias theorem, Fuzzy Sets and Systems 159 (2008), 720–729. [32] A.K. Mirmostafaee and M.S. Moslehian, Fuzzy approximately cubic mappings, Inform. Sci. 178 (2008), 3791–3798. [33] C. Park, Fixed points and Hyers-Ulam-Rassias stability of Cauchy-Jensen functional equations in Banach algebras, Fixed Point Theory Appl. 2007, Art. ID 50175 (2007). [34] C. Park, Generalized Hyers-Ulam-Rassias stability of quadratic functional equations: a fixed point approach, Fixed Point Theory Appl. 2008, Art. ID 493751 (2008). [35] C. Park, Additive ρ-functional inequalities and equations, J. Math. Inequal. 9 (2015), 17–26. [36] C. Park, Additive ρ-functional inequalities in non-Archimedean normed spaces, J. Math. Inequal. 9 (2015), 397–407. [37] C. Park, Y. Cho and M. Han, Stability of functional inequalities associated with Jordan-von Neumann type additive functional equations, J. Inequal. Appl. 2007, Art. ID 41820 (2007). [38] C. Park, K. Ghasemi, S. G. Ghaleh and S. Jang, Approximate n-Jordan ∗-homomorphisms in C ∗ -algebras, J. Comput. Anal. Appl. 15 (2013), 365-368. [39] C. Park, A. Najati and S. Jang, Fixed points and fuzzy stability of an additive-quadratic functional equation, J. Comput. Anal. Appl. 15 (2013), 452–462. [40] C. Park and Th.M. Rassias, Fixed points and generalized Hyers-Ulam stability of quadratic functional equations, J. Math. Inequal. 1 (2007), 515–528. [41] V. Radu, The fixed point alternative and the stability of functional equations, Fixed Point Theory 4 (2003), 91–96. [42] Th.M. Rassias, On the stability of the linear mapping in Banach spaces, Proc. Amer. Math. Soc. 72 (1978), 297–300. [43] J. R¨ atz, On inequalities associated with the Jordan-von Neumann functional equation, Aequationes Math. 66 (2003), 191–200. ˇ ankov´ [44] L. Reich, J. Sm´ıtal and M. Stef´ a, Singular solutions of the generalized Dhombres functional equation, Results Math. 65 (2014), 251–261. [45] S. Schin, D. Ki, J. Chang and M. Kim, Random stability of quadratic functional equations: a fixed point approach, J. Nonlinear Sci. Appl. 4 (2011), 37–49. [46] S. Shagholi, M. Bavand Savadkouhi and M. Eshaghi Gordji, Nearly ternary cubic homomorphism in ternary Fr´echet algebras, J. Comput. Anal. Appl. 13 (2011), 1106–1114. [47] S. Shagholi, M. Eshaghi Gordji and M. Bavand Savadkouhi, Stability of ternary quadratic derivation on ternary Banach algebras, J. Comput. Anal. Appl. 13 (2011), 1097–1105. [48] D. Shin, C. Park and Sh. Farhadabadi, On the superstability of ternary Jordan C ∗ -homomorphisms, J. Comput. Anal. Appl. 16 (2014), 964–973. [49] D. Shin, C. Park and Sh. Farhadabadi, Stability and superstability of J ∗ -homomorphisms and J ∗ -derivations for a generalized Cauchy-Jensen equation, J. Comput. Anal. Appl. 17 (2014), 125–134. [50] S. M. Ulam, A Collection of the Mathematical Problems, Interscience Publ. New York, 1960. [51] J.Z. Xiao and X.H. Zhu, Fuzzy normed spaces of operators and its completeness, Fuzzy Sets and Systems 133 (2003), 389–399. Choonkil Park Research Institute for Natural Sciences, Hanyang University, Seoul 04763, Korea E-mail address: [email protected] Jung Rye Lee Department of Mathematics, Daejin University, Kyeonggi 11159, Korea E-mail address: [email protected] Dong Yun Shin Department of Mathematics, University of Seoul, Seoul 02504, Korea E-mail address: [email protected]

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J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 22, NO.7, 2017, COPYRIGHT 2017 EUDOXUS PRESS, LLC

Four-point impulsive multi-orders fractional boundary value problems N. I. Mahmudov and H. Mahmoud Eastern Mediterranean University Gazimagusa, TRNC, Mersiin 10, Turkey Email: [email protected]; [email protected] Abstract Four-point boundary value problem for impulsive multi-orders fractional di¤erential equation is studied. The existence and uniqueness results are obtained for impulsive multi-orders fractional di¤erential equation with four-point fractional boundary conditions by applying standard …xed point theorems. An example for the illustration of the main result is presented.

Keywords: fractional di¤erential equations, …xed point theorems, multi-orders, impulse.

1

Introduction

Impulsive di¤erential equations have extensively been studied in the past two decades. Impulsive di¤erential equations are used to describe the dynamics of processes in which sudden, discontinuous jumps occur. Such processes are naturally seen in harvesting, earthquakes, diseases, and so forth. Recently, fractional impulsive di¤erential equations have attracted the attention of many researchers. For the general theory and applications of such equationswe refer the interested reader to see [1]-[18] and the references therein. In [8], Kosmatov considered the following two impulsive problems: 8 C < D u (t) = f (t; u (t)) ; 1 < < 2; t 2 [0; 1] n ft1 ; t2 ; :::; tp g ; C C D u t+ D u tk = Ik u tk ; tk 2 (0; 1) ; k = 1; :::; p; k : 0 u (0) = u0 ; u (0) = u0 ; 0 < < 1; and

8
> < u t(t ; u0 (tk ) = Jk u tk ; tk 2 (0; T ) ; k = 1; :::; p; k ) = Ik u tk Pp > u (0) = k=0 k Itkk u ( k ) ; tk < k < tk+1 ; > : 0 u (0) = 0: 1249

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Yukunthorn et.al. [18] studied the similar problem for multi-order Caputo–Hadamard fractional di¤erential equations equipped with nonlinear integral boundary conditions. Motivated by the above works, in this paper, we study the existence of solutions for the four-point nonlocal boundary value problems of nonlinear impulsive equations of fractional order 8 C k Dtk u (t) = f (t; u (t) ; u0 (t)) ; 1 + 2; t 2 [tk ; tk+1 ) ; k > > < u (t 0 0 ; u (tk ) = Jk u tk ; tk 2 (0; T ) ; k = 1; :::; p; k ) = Ik u tk (1) C u (0) + D u (0) = u ( 1 ) ; 0 < 1 < t1 < T; > + 1 1 0 > : u (T ) + 2 C Dtp u (T ) = 2 u ( 2 ) ; 0 < tp < 2 < T; 0 < < 1;

where C Dt is the Caputo derivative, f : [0; T ] R R ! R is a continuous function, Ik ; Jk : R ! R; u (tk ) = u t+ u tk ; u0 (tk ) = u0 t+ u 0 tk ; u t+ and u tk represent the right hand limit k k k and the left hand limit of the function u (t) at t = tk ; and the sequence ftk g satis…es that 0 = t0 < t1 < ::: < tp < tp+1 = T . To the best of our knowledge, there is no paper that consider the four-point impulsive boundary value problem involving nonlinear di¤erential equations of fractional order (1). The main di¢ culty of this problem is that the corresponding integral equation is very complex because of the impulse e¤ects. In this paper, we study the existence and uniqueness of solutions for four-point impulsive boundary value problem (1). By use of Banach’s …xed point theorem and Schauder’s …xed point theorem, some existence and uniqueness results are obtained.

2

Preliminaries

Let [0; T ] = [0; T ] n ft1 ; t2 ; :::; tp g and P C ([0; T ] ; R) = fx : [0; T ] ! R : x (t) is continuous everywhere except for some tk at which x t+ k ; x tk

exist and x tk = x (tk ) ; k = 1; :::; p ;

and P C 1 ([0; T ] ; R) = fx 2 P C ([0; T ] ; R) : x0 (t) is continuous everywhere except for some 0 tk at which x0 t+ k ; x tk

exist and x0 tk = x0 (tk ) ; k = 1; :::; p :

P C ([0; T ] ; R) and P C 1 ([0; T ] ; R) are Banach spaces with the norms kxkP C = sup fjx (t)j : t 2 [0; T ]g and

kxkP C 1 = max fkxkP C ; kx0 kP C g ; respectively. Let X = P C 1 ([0; T ] ; R)\C 2 [0; T ] ; R : A function x 2 X is called a solution of problem (1) if it satis…es (1). Throughout the paper we will use the following notations. = A0 = Ap =

1 1

+ T+

1

1 (1

2

1

1 1

1

1 1)

;

)

(1

B0 =

1) ; 1

;

1

:

1

1 1

1

T1 (2

;

Bp =

1

1

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Fk (y; u; u ) (t) = +

k X

k X

t

(t

tk

(

j 1)

(t

(

)

k

t1 (2

+

)

t1 (2

+

Fk0

1

)

(y; u; u ) (t) = +

(tj

Z

1)

j 1

Z

k X

(tj

tj

(t

1 (

k X

Ij u tj

1)

j 1

J j u 0 tj

j

s)

1

2

y (s) ds +

1

1

k

s)

tk

j=1

y (s) ds +

tj

t

k X

1

1

j=1

1 (

j

s)

1

k X

tj ) J j u 0 tj

(t

;

j=1

y (s) ds

Z

tj

(tj

tj

s)

j

2

1

y (s) ds

1

:

j=1

1

0

y (s) ds

tj

tj

tj )

j=1

Gk (y; u; u0 ) (t) =

Z

1

k

s)

1

j=1

+

Z

1 ( k)

0

(

1)

k

k X

Z

t

(t

tk

1 (

j=1

1)

j 1

Z

2

k

s)

y (s) ds

tj

(tj

tj

j

s)

1

2

y (s) ds +

1

k X

J j u 0 tj

:

j=1

Lemma 1 Let y 2 C [0; T ]. A function u 2 P C 1 [0; T ] is a solution of the boundary value problem 8 C k Dtk u (t) = y (t) ; 1 + < k 2; t 2 [0; T ] n ft1 ; t2 ; :::; tp g ; > > < u (t )=I u t ; u0 (t ) = J u0 t ; t 2 (0; T ) ; k = 1; :::; p; k

if and only if

u (0) + > > : u (T ) +

1

2

k C

k

k

D0+ u (0) = C Dtp u (T ) =

k

u (t) = Fk (y; u) (t) + 1 1

1

1 +

2

1

< T; 0
0; l2 > 0; 0

t

T; x; y 2 R:

For convenience, we will give some notations: T = max fT 1

=

p X

(tj

j=1

3

=

k

T1 (2

:0

tj (

1)

j 1

)

k j

+ 1)

p X (tj j=1

pg ;

= min f (

1

;

2

=

p X

(T

tj ) (tj (

j=1

tj (

1)

j

j 1)

1253

1

1

;

k)

4

=

:0 tj

1)

pg ; j

1

1

;

j 1)

p X (tj j=1

k

tj (

1)

j

j 1)

1

1

:

N. I. Mahmudov et al 1249-1260

J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 22, NO.7, 2017, COPYRIGHT 2017 EUDOXUS PRESS, LLC

F

:= lf

G

:= lf

F0

:= lf

T T T

+ lf

1

+ lf

+ lf

1

T1 (2

+ lf

4

+ l2 p:

2

+ pl1 + l2 pT; )

+ l2 p

T1 (2

)

;

Lemma 2 Fk (f; u; u0 ) and Gk (f; u; u0 ) are Lipschitzian operators. jFk (f; u; u0 )

Fk (f; v; v 0 )j

0

jGk (f; u; u )

ku

F

0

Gk (f; v; v )j

vkP C 1 ; LFk > 0; vkP C 1 ; LGk > 0; u; v 2 P C 1 ([0; T ] ; R) :

ku

G

Proof. For u; v 2 P C 1 ([0; T ] ; R), we have jFk (f; u; u0 ) (t) Z t 1 (t ( k ) tk +

k X

1 (

j=1

+

k X

j 1)

Fk (f; v; v 0 ) (t)j

Z

1

k

s)

jf (s; u (s) ; u0 (s))

tj

(tj

tj

j

s)

1

1

1

Ij u tj

f (s; v (s) ; v 0 (s))j ds

jf (s; u (s) ; u0 (s))

f (s; v (s) ; v 0 (s))j ds

Ij v tj

j=1

+

k X j=1

+

k X

Z

(t tj ) ( j 1 1)

tj

(tj

tj

s)

j

2

1

1

(t

tj ) J j u 0 tj

lf

1 ( k)

jf (s; u (s) ; u0 (s))

f (s; v (s) ; v 0 (s))j ds

J j v 0 tj

j=1

+ lf

k X j=1

+ l1

k X

Z

t

(t

tk

1 (

j 1)

s) Z

u tj

1

k

v (s)j + ju0 (s)

(ju (s)

tj

(tj

tj

s)

j

+ lf

v tj

k X

1 (

j=1

tj

(tj

tj

+ l2

s)

j

1

2

(ju (s)

1

k X

(t

v (s)j + ju0 (s)

(ju (s)

1

j=1

Z

1

1

tj ) u 0 tj

j 1

v 0 (s)j) ds

1)

(t

v 0 (s)j) ds

tj )

v (s)j + ju0 (s)

v 0 (s)j) ds

v 0 tj

j=1

F

ku

vkP C 1 :

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J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 22, NO.7, 2017, COPYRIGHT 2017 EUDOXUS PRESS, LLC

Similarly, jGk (f; u; u0 ) (t) Gk (f; v; v 0 ) (t)j Z t 1 1 (t s) k jf (s; u (s) ; u0 (s)) f (s; v (s) ; v 0 (s))j ds ( k ) tk Z tj k X T1 1 2 + (tj s) j 1 jf (s; u (s) ; u0 (s)) f (s; v (s) ; v 0 (s))j ds (2 ) j=1 ( j 1 1) tj 1 +

t1 (2 0

@lf (T ( G

ku

)

k X

J j v 0 tj

j=1

tk ) k

J j u 0 tj k

+ 1)

+ lf

T1 (2

)

p X (tj

tj (

j=1

j

1)

1

1

T1 (2

+

j 1)

vkP C 1 :

)

1

l2 A ku

vkP C 1

Also, we have jFk0 (f; u; u0 ) (t)

Fk0 (f; v; v 0 ) (t)j

In view of Lemma 1 we de…ne an operator

ku

F0

vkP C 1 :

: X ! X by

( u) (t) = Fk (f; u) (t) (A0 B0 t) F0 (f; u) ( 1 ) + 2 (Ap + Bp t) Fp (f; u) ( 2 ) (Ap + Bp t) Fp (f; u) (T ) 2 (Ap + Bp t) Gp (f; u) (T ) ; where A0 = Ap =

1

1 (1

1 1

1

1

1 1)

1 1

1

;

B0 =

1

;

1

:

1

;

Bp =

1

1

Let := max f

F;

G;

F0g :

Theorem 3 Suppose that the assumption (A1 ), (A2 ) are satis…ed. If := max f(1 + jA0 j + jB0 j T + (j 2 j + j ; (1 + jB0 j + (j 2 j + j 2 j + 1) jBp j)g < 1;

2j

+ 1) (jAp j + jBp j T ))

then the boundary value problem (1) has a unique solution. Proof. Let u; v 2 P C 1 ([0; T ] ; R) : For u; v 2 (tk ; tk+1 ] ; k = 0; :::; p; we have j( u) (t)

( v) (t)j

jFk (f; u; u0 ) (t) Fk (f; v; v 0 ) (t)j + jA0 B0 tj jF0 (f; u; u0 ) ( 1 ) F0 (f; v; v 0 ) ( 1 )j + j 2 j jAp + Bp tj jFp (f; u; u0 ) ( 2 ) Fp (f; v; v 0 ) ( 2 )j + jAp + Bp tj jFp (f; u; u0 ) (T ) Fp (f; v; v 0 ) (T )j + j 2 j jAp + Bp tj jGp (f; u; u0 ) (T ) Gp (f; v; v 0 ) (T )j (1 + jA0 j + jB0 j T + (j 2 j + j 2 j + 1) (jAp j + jBp j T )) ku 1255

vkP C 1 :

N. I. Mahmudov et al 1249-1260

J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 22, NO.7, 2017, COPYRIGHT 2017 EUDOXUS PRESS, LLC

Similarly, for u; v 2 (tk ; tk+1 ] we have 0

( u) (t)

0

( v) (t)

jFk0 (f; u; u0 ) (t) Fk0 (f; v; v 0 ) (t)j + jB0 j jF0 (f; u; u0 ) ( 1 ) F0 (f; v; v 0 ) ( 1 )j + j 2 j jBp j jFp (f; u; u0 ) ( 2 ) Fp (f; v; v 0 ) ( 2 )j + jBp j jFp (f; u; u0 ) (T ) Fp (f; v; v 0 ) (T )j + j 2 j jBp j jGp (f; u; u0 ) (T ) Gp (f; v; v 0 ) (T )j (1 + jB0 j + (j 2 j + j 2 j + 1) jBp j) ku vkP C 1 :

It follows that k u

vkP C 1

ku

vkP C 1 :

Since < 1; is a contraction. According to the Banach …xed point theorem that is the problem (1) has a unique solution.

4

has a unique …xed point,

Existence

In this section, we assume that (A3 ) f : [0; T ]

R ! R is continuous function and there exists h 2 C ([0; T ] ; R+ ) such that %

jf (t; u; v)j

h (t) + b1 juj + b2 jvj ;

(t; u; v) 2 [0; T ]

R; 0 < ; % < 1:

R

(A4 ) Ik ; Jk : R ! R are continuous functions and there L2 > 0; L3 > 0 such that jIk (x)j

L2 ; jJk (x)j

L3 ;

x 2 R:

For convenience, we will give some notations: C1 := (1 + jA0 j + jB0 j T + (j +j

2 j (jAp j

T (2

+ jBp j T )

C2 := (1 + jA0 j + jB0 j T + (j +j

2 j (jAp j

2j

+ 1) (jAp j + jBp j T )) (pL2 + pT L3 ) khk )

2j

L3 khk ;

+ 1) (jAp j + jBp j T ))

+

3

T

+

1

+

2

:

n o 1 1 max 3C1 ; (3b1 C2 ) 1 ; (3b1 C2 ) 1 % ;

R then

T

+ jBp j T )

Lemma 4 If

1

maps B (0; R) := u 2 P C 1 ([0; T ] ; R) : kukP C 1

Proof. Assume that

R into itself.

n o 1 1 max 3C1 ; (3b1 C2 ) 1 ; (3b1 C2 ) 1 % :

R

Then for t 2 (tk ; tk+1 ] ; k = 0; :::; p; we have

jFk (f; u; u0 ) (t)j Z t 1 1 (t s) k jf (s; u (s) ; u0 (s))j ds ( k ) tk Z tj k k X X 1 1 (tj s) j 1 jf (s; u (s) ; u0 (s))j ds + Ij u tj + ( j 1 ) tj 1 j=1 j=1 +

k X j=1

(t tj ) ( j 1 1)

Z

tj

tj

(tj 1

s)

j

1

2

jf (s; u (s) ; u0 (s))j ds +

1256

k X

(t

tj ) J j v 0 tj

;

j=1

N. I. Mahmudov et al 1249-1260

J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 22, NO.7, 2017, COPYRIGHT 2017 EUDOXUS PRESS, LLC

jFk (f; u; u0 ) (t)j Z t 1 % 1 (t s) k h (s) + b1 ju (s)j + b2 ju0 (s)j ds ( k ) tk Z tj k X 1 % 1 (tj s) j 1 h (s) + b1 ju (s)j + b2 ju0 (s)j ds + ( ) j 1 t j 1 j=1 +

k X

Ij u tj

j=1

+

k X

(t tj ) ( j 1 1)

j=1

+

k X

(t

Z

tj

(tj

tj

j

s)

2

1

1

%

(h (s) + b1 ju (s)j + b2 jv (s)j ) ds

tj ) J j u tj

j=1

T

+

+

k

khk + b1 kuk + b2 ku0 k

( k + 1) p X (tj tj

j=1 p X

(

(t

+

1

(

%

khk + b1 kuk + b2 ku0 k

)

1

khk + b1 kuk + b2 ku0 k

+ 1) p X (tj

%

+ pT L3

+ pL2 + pT L3 ;

khk + b1 kuk + b2 ku0 k tj (

j=1

1)

j

1

1

%

khk + b1 kuk + b2 ku0 k

j 1)

%

+

khk + b1 kuk + b2 ku0 k

3

0

jFk0 (y; u; u0 ) (t)j

+ pL2

%

k

T

1

khk + b1 kuk + b2 ku0 k

2

k

t (2

j

%

+

1

+

1

1 + 1)

j

T

jGk (y; u; u0 ) (t)j

j

tj ) (tj tj 1 ) ( j 1)

j=1

T

1)

%

@T

1

k

(

k)

+

k X (tj

tj (

j=1

+ L3 :

1)

j

j 1)

1

1

+

t1 (2

)

+

t1 (2

)

L3

L3 ;

1

A khk + b1 kuk + b2 ku0 k%

It follows that j( u) (t)j

(1 + jA0 j + jB0 j T + (j T

+j

+

2 j (jAp j

1

+

2j

+ 1) (jAp j + jBp j T )) %

2

+ jBp j T )

khk + b1 kuk + b2 ku0 k T

+

3

+ pL2 + pT L3

khk + b1 kuk + b2 ku0 k

%

+

T1 (2

)

L3

%

C1 + C2 b1 kuk + C2 b2 ku0 k ; 1257

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and 0

@T

0

( u) (t)

1

k

(

k)

+ (jB0 j + j

+

tj (

j=1

2 j jBp j

T

+ j j jBp j

k X (tj

1)

j

1

1

j 1)

T

+ jBp j)

+

1

A khk + b1 kuk + b2 ku0 k% + L3

%

+

1

khk + b1 kuk + b2 ku0 k

2

%

+

khk + b1 kuk + b2 ku0 k

3

T1 (2

+

)

+ pL2 + pT L3

L3

%

C1 + C2 b1 kuk + C2 b2 ku0 k : Thus

R R R + + = R: 3 3 3

C1 + C2 b1 R + C2 b2 R%

k( u)kP C 1

Theorem 5 Assume that the conditions (A3 ) and (A4 ) are satis…ed. Then the problem (1) has at least one solution. Proof. Firstly, we prove that : P C 1 ([0; T ] ; R) ! P C 1 ([0; T ] ; R) is completely continuous operator. It is clear that, the continuity of functions f; Ik and Jk implies the continuity of the operator . Let P C 1 ([0; T ] ; R) be bounded. Then there exist positive constants such that jf (t; u; u0 )j

L1 ; jIk (u)j

L2 ; jJk (u)j

L3 ;

for all u 2 . Thus, for any u 2 ,we have 0

Fk (f; u; u )

T

L1

+

1

+

2

+ pL2 + L3 pT;

Similarly, Gk f; u; u

0

(t)

L1

T

+ L1

T1 (2

1

)

T1 (2

+

)

pL3 :

It follows that 1

j( u) (t)j

(constant):

In a like manner, Fk0 f; u; u

0

(t)

T

L1

+

4

+ L3 p:

It follows that 0

( u) (t) + (j

0j

+j

2 j jBp j

+ jBp j)

+j

F

2 j jBp j

G

L1 =:

T

+

4

+ L3 p

2

Thus 1

k ukP C 1 On the other hand, for

1

;

2

2 [tk ; tk+1 ] with

j( u)( 1 )

( u) 2 j

+

1

Z

2 2

2

= constant:

and we have 0

( u) (s) ds

(

2

1) :

1

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Similarly 0

( u) ( 2 )

0

( u) ( 1 )

(

2

1 );

where is a constant. This implies that u is equicontinuous on all (tk ; tk+1 ], k = 0; 1; :::; p. Consequently, Arzela-Ascoli theorem ensures us that the operator is a completely continuous operator and by Lemma 4 : B(0; R) ! B(0; R). Hence, we conclude that : B(0; R) ! B(0; R) is completely continuous. It follows from the Schauder …xed point theorem that has at least one …xed point. That is problem (1) has at least one solution. 1 Example 1. For p = 1; t1 = 14 ; T = 1; = 21 ; 1 = 2; 1 = 12 ; 2 = 3; 1 = 10 ; 1 = 15 ; 2 = 23 ; 0 = 3 3 k = 2 ; we consider the following impulsive multi-orders fractional di¤erential equation: 2; 8 > > > > < It is clear that

> > > > :

ju0 (t)j 1 Dtkk u (t) = 100 cos u (t) + ju0 (t)j+100 + t; ju( 1 )j ju0 ( 1 )j u0 14 = u0 1 4 +70 ; u 41 = u 1 4 +50 ; j ( 4 )j j ( 4 )j u (0) + 2 C D0+ u (0) = 12 u 15 ; u (1) + 2 C D0+ u (1) = 12 u 23 : C

jf (t; x; x1 )

f (t; y; y1 )j

0:02 (jx

yj + jx1

y1 j) ; 0

0 < t < 1; t 6= 14 ; (6)

t

1; x; y; x1 ; y1 2 R:

One can easily calculate that = 0:2178 < 1: Therefore, all the assumptions of Theorem 3 hold. Thus, by Theorem 3, the impulsive multi-orders fractional boundary value problem (6) has a unique solution on [0; 1].

References [1] B. Ahmad and S. Sivasundaram, “Existence of solutions for impulsive integral boundary value problems of fractional order,” Nonlinear Analysis: Hybrid Systems, vol. 4, no. 1, pp. 134–141, 2010. [2] C. Bai, “Existence result for boundary value problem of nonlinear impulsive fractional di¤erential equation at resonance,” Journal of Applied Mathematics and Computing, vol. 39, no. 1-2, pp. 421–443, 2012. [3] J. Cao and H. Chen, “Some results on impulsive boundary value problem for fractional di¤erential inclusions,” Electronic Journal of Qualitative Theory of Di¤erential Equations, vol. 2011, no. 11, pp. 1–24, 2011. [4] Feµckan, Y. Zhou, and J. R. Wang, “On the concept and existence of solution for impulsive fractional di¤erential equations,”Communications in Nonlinear Science and Numerical Simulation, vol. 17, no. 7, pp. 3050–3060, 2012. [5] T. L. Guo and J. Wei, “Impulsive problems for fractional di¤erential equations with boundary value conditions,” Computers & Mathematics with Applications, vol. 64, no. 10, pp. 3281–3291, 2012. [6] Z. He and J. Yu, “Periodic boundary value problem for …rstorder impulsive ordinary di¤erential equations,” Journal of Mathematical Analysis and Applications, vol. 272, no. 1, pp. 67–78, 2002. [7] R. W. Ibrahim, Stability of sequential fractional di¤erential equations, Appl. Comput. Math., V.14, N.2, 2015, pp.141-149.

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[8] N. Kosmatov, “Initial value problems of fractional order with fractional impulsive conditions,” Results in Mathematics, vol. 63, no. 3-4, pp. 1289–1310, 2013. [9] V. Lakshmikantham, D. D. Bainov, and P. S. Simeonov, Theory of Impulsive Di¤erential Equations, vol. 6 of Series in Modern Applied Mathematics,World Scienti…c, Singapore, 1989. [10] Z. Luo and J. Shen, “Stability of impulsive functional di¤erential equations via the Liapunov functional,” Applied Mathematics Letters, vol. 22, no. 2, pp. 163–169, 2009. [11] S. Tiwari, M. Kumari, An initial value tecgnique to solve two-point nonlinear singularly perturbed boundary value problems, Appl. Comput. Math., V.14, N.2, 2015, pp.150-157. [12] H.Wang, “Existence results for fractional functional di¤erential equations with impulses,” Journal of Applied Mathematics and Computing, vol. 38, no. 1-2, pp. 85–101, 2012. [13] G. Wang, B. Ahmad, and L. Zhang, “Impulsive anti-periodic boundary value problem for nonlinear di¤erential equations of fractional order,” Nonlinear Analysis: Theory, Methods & Applications, vol. 74, no. 3, pp. 792–804, 2011. [14] G. Wang, B. Ahmad, and L. Zhang, “Some existence results for impulsive nonlinear fractional di¤erential equations with mixed boundary conditions,”Computers & Mathematics with Applications, vol. 62, no. 3, pp. 1389–1397, 2011. [15] X.Wang, “Existence of solutions for nonlinear impulsive higher order fractional di¤erential equations,” Electronic Journal of QualitativeTheory of Di¤erential Equations, vol. 2011,no. 80, pp. 1–12, 2011. [16] J.R.Wang, Y. Zhou, andM. Feµckan, “On recent developments in the theory of boundary value problems for impulsive fractional di¤erential equations,” Computers &Mathematics with Applications, vol. 64, no. 10, pp. 3008–3020, 2012. [17] G. Wang, S. Liu, D. Baleanu, and L. Zhang, A New Impulsive Multi-Orders Fractional Di¤erential Equation Involving Multipoint Fractional Integral Boundary Conditions, Abstract and Applied Analysis Volume 2014, Article ID 932747, 10 pages, http://dx.doi.org/10.1155/2014/932747. [18] W. Yukunthorna, B. Ahmad, S. K. Ntouyas, J. Tariboon, On Caputo–Hadamard type fractional impulsive hybrid systems with nonlinear fractional integral conditions, Nonlinear Analysis: Hybrid Systems 19 (2016) 77–92.

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Convergence of modification of the Kantorovich-type q-Bernstein-Schurer operators Qing-Bo Caia,∗and Guorong Zhoub a

School of Mathematics and Computer Science, Quanzhou Normal University, Quanzhou 362000, China b

School of Applied Mathematics, Xiamen University of Technology, Xiamen 361024, China E-mail: [email protected], [email protected]

Abstract. In this paper, we introduce a new modification of Kantorovich-type ∗ (f ; x) based on the concept of q-integers. We Bernstein-Schurer operators Kn,p,q investigate statistical approximation properties, establish a local approximation theorem, give a convergence theorem for the Lipschitz continuous functions and obtain a Voronovskaja-type theorem. Furthermore, we also give some illustrative graphics and some numerical examples for comparisons for the convergence of operators to some function. 2000 Mathematics Subject Classification: 41A10, 41A25, 41A36. Key words and phrases: q-integers, Bernstein-Schurer operators, A-statistical convergence, rate of convergence, Lipschitz continuous functions.

1

Introduction

In 2015, Agrawal, Finta and Kumar [1] introduced a new Kantorovich-type generalization of the q-Bernstein-Schurer operators, they gave the basic convergence theorem, obtained the local direct results, estimated the rate of convergence and so on. The operators are defined as Z [k+1]q n+p X [n+1]q −k Kn,p,q (f ; x) = [n + 1]q bn+p,k (q; x)q f (t)dR (1) q t, [k] q [n+1]q

k=0

where bn+p,k (q; x) is defined by " bn+p,k (q; x) =

n+p k

# xk (1 − x)n+p−k . q

(2)

q

They obtained the following lemma of the moments. ∗

Corresponding author.

1 1261

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Q. -B. CAI AND G. ZHOU

Lemma 1.1. (See [1], Lemma 2.1) The following equalities hold Kn,p,q (1; x) = 1; (3) 2q[n + p]q 1 x+ ; (4) Kn,p,q (t; x) = [2]q [n + 1]q [2]q [n + 1]q
q 2 (1 + q + 4q 2 )[n + p]q [n + p − 1]q 2 q(3 + 5q + 4q 2 )[n + p]q Kn,p,q t2 ; x = x + x [2]q [3]q [n + 1]2q [2]q [3]q [n + 1]2q 1 + . (5) [3]q [n + 1]2q Apparently, these operators reproduce only constant functions. In present paper, we will introduce a new modification of Kantorovich-type q-Bernstein-Schurer operators ∗ Kn,p,q (f ; x) which will be defined in (7). The advantage of these new operators is that they reproduce not only constant functions but also linear functions. We will investigate statistical approximation properties, establish a local approximation theorem, give a convergence theorem for the Lipschitz continuous functions and obtain a Voronovskaja-type theorem. Furthermore, we will give some illustrative graphics and some numerical examples for comparisons for the convergence of operators to some function. We may observe ∗ that the new operators Kn,p,q (f ; x) give a better approximation to f (x) than Kn,p,q (f ; x). Before introducing the operators, we mention certain definitions based on q-integers, detail can be found in [4, 5]. For any fixed real number 0 < q ≤ 1 and each nonnegative integer k, we denote q-integers by [k]q , where ( 1−q k 1−q , q 6= 1; [k]q = k, q = 1. Also q-factorial and q-binomial coefficients are defined as follows: ( " # [n]q ! [k]q [k − 1]q ...[1]q , k = 1, 2, ...; n [k]q ! = , = , [k]q ![n − k]q ! 1, k = 0, k

(n ≥ k ≥ 0).

q

For x ∈ [0, 1] and n ∈ N0 , we recall that ( (1 − x)nq =

Qn−1 j=0

1, n = 0;

. 1 − q j x = (1 − x)(1 − qx)... 1 − q n−1 x , n = 1, 2, ...

The Riemann-type q-integral is defined by Z b ∞ X
f (t)dR t = (1 − q)(b − a) f a + (b − a)q j q j , q a

(6)

j=0

where the real numbers a, b and q satisfy that 0 ≤ a < b and 0 < q < 1. For f ∈ C(I), I = [0, 1+p], p ∈ N0 , q ∈ (0, 1) and n ∈ N, we introduce the modification of Kantorovich-type q-Bernstein-Schurer operators as follows: ∗ Kn,p,q (f ; x)

= [n + 1]q

n+p X

bn+p,k (q; u(x)) q

k=0

1262

−k

Z

[k+1]q [n+1]q [k]q [n+1]q

f (t)dR q t,

(7)

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J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 22, NO.7, 2017, COPYRIGHT 2017 EUDOXUS PRESS, LLC

CONVERGENCE OF MODIFICATION OF THE KANTOROVICH-TYPE q-BERNSTEIN-SCHURER OPERATORS where bn+p,k (q; x) is defined by (2), and   [2]q [n + 1]q x − 1 1 + 2q[n + p]q 1 u(x) = , ≤x≤ . 2q[n + p]q [2]q [n + 1]q [2]q [n + 1]q

2

(8)

Auxiliary Results In order to obtain the approximation properties, We need the following lemmas:

Lemma 2.1. For the modification of Kantorovich-type q-Bernstein-Schurer operators (7), using lemma 1.1, by some easily computations we have ∗ Kn,p,q (1; x) = 1,

(9)

∗ Kn,p,q (t; x) = x, ∗ Kn,p,q t2 ; x




=

(10) 4q 2

x2




4q 2




3 + 5q + x [2]q 1 + q + [n + p − 1]q + 4[3]q [n + p]q 2[3]q [n + 1]q

2 1 + q + 4q [n + p − 1]q x 1 + q + 4q 2 [n + p − 1]q − + 2[3]q [n + 1]q [n + p]q 4[2]q [3]q [n + 1]2q [n + p]q −

3 + 5q + 4q 2 1 + . 2 2[2]q [3]q [n + 1]q [3]q [n + 1]2q

(11)

Remark 2.2. Let {qn } denotes a sequence such that 0 < qn < 1. Then, by Bohman and ∗ Korovkin Theorem, for any f ∈ C(I), operators Kn,p,q (f ; x) converge uniformly to f (x), if and only if limn→∞ qn = 1. Lemma 2.3. For the modification of Kantorovich-type q-Bernstein-Schurer operators (7), we have ∗ Kn,p,q (t − x; x) = 0, ∗ Kn,p,q

∗ Kn,p,q

(12)

q 2 + 4q 3 − 2q − 3 x2 (1 + 2q)x (3 + 5q + 4q 2 )x (t − x)2 ; x ≤ + + (13) 4[3]q [3]q [n + 1]q 2[3]q [n + 1]q [n + p]q (3 + 5q + 4q 2 )x (1 + 2q)x + , (14) ≤ [3]q [n + 1]q 2[3]q [n + 1]q [n + p]q  
1 4 (t − x) ; x ≤ O . (15) [n]2q

Proof. By (9) and (10), we get (12). Using (10), (11) and some computations, we have
∗ Kn,p,q (t − x)2 ; x ∗ ∗ = Kn,p,q (t2 ; x) − 2xKn,p,q (t; x) + x2
q 2 + 4q 3 − 2q − 3 x2 (1 + 2q)[n + p − 1]q x q n+p−1 (3 + 5q + 4q 2 )x + + ≤ 4[3]q [3]q [n + 1]q [n + p]q 2[3]q [n + 1]q [n + p]q
2 2 3 q + 4q − 2q − 3 x (1 + 2q)x (3 + 5q + 4q 2 )x ≤ + + . 4[3]q [3]q [n + 1]q 2[3]q [n + 1]q [n + p]q

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Q. -B. CAI AND G. ZHOU


Indeed, using the similar method for estimate Kn,p,q (t − x)4 ; x in [1, P. 229], we have ∗ Kn,p,q (t − x)4 ; x




4 4  n+p X [k]q [k]q [k]q − −x + 64 bn+p,k (q; u(x)) ≤ 64 bn+p,k (q; u(x)) [n + 1]q [n + p]q [n + p]q k=0 k=0  4 n+p qk 8 X bn+p,k (q; u(x)) + [5]q [n + 1]q k=0  4  n  n+p X [k]q q ([p]q − 1) 4 bn+p,k (q; u(x)) ≤ 64 [n + p]q [n + 1]q k=0 4  n+p X [2]q [n + 1]q x − 1 [2]q [n + 1]q x − 1 [k]q − + −x +64 bn+p,k (q; u(x)) [n + p]q 2q[n + p]q 2q[n + p]q k=0  4 n+p qk 8 X bn+p,k (q; u(x)) + [5]q [n + 1]q n+p X



k=0

n+p X
([p]q − 1)4 ≤ C1 + 512 bn+p,k (q; u(x)) (t − u(x))4 ; x 2 [n]q k=0 4  n+p X [2]q [n + 1]q x − 1 C2 − x + 2, +512 bn+p,k (q; u(x)) 2q[n + p]q [n]q k=0

where u(x) is defined in (8), C1 and C2 are some positive constants. Thus, ∗ Kn,p,q (t − x)4 ; x




  ([p]q − 1)4 [2]q ([n]q + q n ) x − 1 − 2q ([n]q + q n [p]q ) x 4 C3 C2 + 512 + 512 + 2 [n]2q [n]2q 2q[n + p]q [n]q " #4
  1 + q n+1 − 2q n+1 [p]q x − 1 ([p]q − 1)4 C2 C3 1 + 512 2 + 512 + 2 =O , = C1 [n]2q [n]q 2q[n + p]q [n]q [n]2q ≤ C1

where C3 is a positive constant, lemma 2.3 is proved.

3

Statistical approximation properties In this section, we present the statistical approximation properties of the operator

∗ Kn,p,q (f ; x).

Let K be a subset of N, the set of all natural numbers. The density of K is defined P by δ(K) := limn n1 nk=1 χK (k) provided the limit exists, where χK is the characteristic function of K. A sequence x := {xn } is called statistically convergent to a number L if, for every ε > 0, δ{n ∈ N : |xn − L| ≥ ε} = 0. Let A := (ajn ), j, n = 1, 2, ... be an infinite summability matrix. For a given sequence x := {xn }, the A−transform of x, denoted by P Ax := ((Ax)j ), is given by (Ax)j = ∞ k=1 ajn xn provided the series converges for each 1264

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CONVERGENCE OF MODIFICATION OF THE KANTOROVICH-TYPE q-BERNSTEIN-SCHURER OPERATORS

j. We say that A is regular if limn (Ax)j = L whenever lim x = L. Assume that A is a non-negative regular summability matrix. A sequence x = {xn } is called A-statistically P convergent to L provided that for every ε > 0, limj n:|xn −L|≥ε ajn = 0. We denote this limit by stA − limn xn = L. For A = C1 , the Ces` aro matrix of order one, A-statistical convergence reduces to statistical convergence. It is easy to see that every convergent sequence is statistically convergent but not conversely. We consider a sequence q := {qn } for 0 < qn < 1 satisfying stA − lim qn = 1.

(16)

n

If ei = ti , t ∈ R+ , i = 0, 1, 2, ... stands for the ith monomial, then we have Theorem 3.1. Let A = (ank ) be a non-negative regular summability matrix and q := {qn } be a sequence satisfying (16), then for all f ∈ C(I), x ∈ [0, 1], we have ∗ stA − lim ||Kn,p,q f − f ||C(I) = 0.

(17)

∗ stA − lim ||Kn,p,q (ei ) − ei ||C(I) = 0. (i = 0, 1) n

(18)

n

Proof. Obviously n

By (11) and (13), we have ∗ Kn,p,q (e2 ; x) − e2 (x) ≤ n

1 + 2qn 3 + 5qn + 4qn2 + . [3]qn [n + 1]qn 2[3]qn [n + 1]qn [n + p]qn

Now for a given ε > 0, let us define the following sets:   o n 1 + 2qk ε ∗ , U := k : Kn,p,qk (e2 ) − e2 C(I) ≥ ε , U1 := k : ≥ [3]qk [n + 1]qk 2   3 + 5qk + 4qk2 ε U2 := k : . ≥ 2[3]qk [n + 1]qk [n + p]qk 2 Then one can see that U ⊆ U1 ∪ U2 , so we have    1 + 2qk ε ∗ δ k ≤ n : ||Kn,p,qk (e2 ) − e2 ||C(I) ≤ δ k≤n: ≥ [3]qk [n + 1]qk 2   3 + 5qk + 4qk2 ε +δ k ≤ n : ≥ , 2[3]qk [n + 1]qk [n + p]qk 2 since stA − lim qn = 1, we have n

stA − lim n

1 + 2qn = 0, [3]qn [n + 1]qn

stA − lim n

3 + 5qn + 4qn2 = 0, 2[3]qn [n + 1]qn [n + p]qn

which implies that the right-hand side of the above inequality is zero, thus we have ∗ stA − lim ||Kn,p,q (e2 ) − e2 ||C(I) = 0. n n

(19)

Combining (18) and (19), theorem 3.1 follows from the Korovkin-type statistical approximation theorem established in [3], the proof is completed. 1265

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4

Local approximation properties

Let f ∈ C(I), endowed with the norm ||f || = supx∈I |f (x)|. The Peetre’s K−functional is defined by  K2 (f ; δ) = inf ||f − g|| + δ||g 00 || , g∈C 2

where δ > 0 and C 2 = {g ∈ C(I) : g 0 , g 00 ∈ C(I)} . By [2, p.177, Theorem 2.4], there exits an absolute constant C > 0 such that √ K2 (f ; δ) ≤ Cω2 (f ; δ),

(20)

where ω2 (f ; δ) = sup

sup

|f (x + 2h) − 2f (x + h) + f (x)|

00 . Throughout the paper we assume that a = a1 , . . . , ar and b = b1 , b2 , . . . , bs . The Barnes-type degenerate Bernoulli and Euler mixed-type polynomials βE n (λ, x|a; b) with a1 , . . . , ar ; b1 , . . . , bs ̸= 0 are defined by the generating function )∏ ) r ( s ( ∏ ∑ t 2 tn x/λ (1) . (1 + λt) = βE (λ, x|a; b) n n! (1 + λt)ai /λ − 1 i=1 (1 + λt)bi /λ + 1 i=1 n≥0 If x = 0, βE n (λ|a; b) = βE n (λ, 0|a; b) are called the Barnes-type degenerate Bernoulli and Euler mixed-type numbers. Here, we recall that the polynomial βn (λ, x|a) with a1 , . . . , ar ̸= 0 are given by ) r ( ∏ ∑ t tn x/λ (2) (1 + λt) = β (λ, x|a) n n! (1 + λt)ai /λ − 1 i=1

n≥0

are called the Barnes-type degenerate Bernoulli polynomials and studied in [7]. We note here that lim βn (λ, x|a) = Bn (x|a),

λ→0

lim λ−n βn (λ, λx|a) = (a1 a2 · · · ar )−1 b(r) n (x),

λ→∞

) ∏ ( where Bn (x|a) are the Barnes-type Bernoulli polynomials given by ri=1 eai tt −1 etx = ∑ (r) tn n≥0 Bn (x|a) n! and bn (x) are the Bernoulli polynomials of the second kind of order 2000 Mathematics Subject Classification. 05A40, 11B83. Key words and phrases. Euler polynomials, Bernoulli polynomials, Umbral calculus. 1

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( )r ∑ n (r) t r given by log(1+t) (1 + t)x = n≥0 bn (x) tn! (see [12, 22]). Also, we recall that the polynomial En (λ, x|b) with b1 , . . . , bs ̸= 0 are given by ) s ( ∏ ∑ 2 tn x/λ (3) (1 + λt) = E (λ, x|b) n n! (1 + λt)bi /λ + 1 i=1 n≥0 are called the Barnes-type degenerate Euler polynomials and studied in [11, 17, 25]. We denote En (λ, 0|b) by En (λ|b). We note here that lim En (λ, x|b) = En (x|b),

λ→0

lim λ−n En (λ, λx|a) = (x)n = x(x − 1) · · · (x − n + 1),

λ→∞

where En (x|a) are the Barnes-type Euler polynomials given by (see [3]) ) s ( ∏ ∑ 2 tn tx e = E (x|b) . n ebi t + 1 n! i=1

n≥0

In order to study the Barnes-type degenerate Bernoulli and Euler mixed-type polynomials, we use the umbral calculus technique. We denote the algebra of polynomials in a single variable x over C by Π. Let Π∗ be the vector space of all linear functionals on Π. Let ⟨L|p(x)⟩ be the action of a linear functional L ∈ Π∗ on a polynomial p(x), where we extend it as ⟨cL + c′ L′ |p(x)⟩ = c⟨L|p(x)⟩ + c′ ⟨L′ |p(x)⟩, where c, c′ ∈ C (see [22, 23]). Define     ∑ tk ak | ak ∈ C (4) H = f (t) =   k! k≥0

to be the algebra of formal power series in a single variable t. The formal power series in the variable t defines a linear functional on Π by setting ⟨f (t)|xn ⟩ = an , for all n ≥ 0 (see [22, 23]). By (4), we have ⟨tk |xn ⟩ = n!δn,k , for all n, k ≥ 0, (see [22, 23]), ∑ n where δn,k is the Kronecker’s symbol. For fL (t) = n≥0 ⟨L|xn ⟩ tn! , by (5), we have that ⟨fL (t)|xn ⟩ = ⟨L|xn ⟩. Thus, the map L 7→ fL (t) is a vector space isomorphism from Π∗ onto H, namely H is thought of as set of both formal power series and linear functionals. We call H the umbral algebra. The umbral calculus is the study of umbral algebra. (5)

The order O(f (t)) of the non-zero power series f (t) is the smallest integer ℓ for which the coefficient of tℓ does not vanish (see [22, 23]). If O(f (t)) = 1 (O(f (t)) = 0) then f (t) is called a delta (an invertable) series. If O(f (t)) = 1 and O(g(t)) = 0, then there exists a unique sequence sn (x) of polynomials such that ⟨g(t)(f (t))k |sn (x)⟩ = n!δn,k , where n, k ≥ 0. The sequence sn (x) is called the Sheffer sequence for (g(t), f (t)), and we write sn (x) ∼ yt |p(x)⟩ = p(y), (g(t), f (t)) (see [22, 23]). For f (t) ∈ H and we have that ⟨e ∑ ∑ p(x) ∈ Π, n n ⟨f (t)g(t)|p(x)⟩ = ⟨g(t)|f (t)p(x)⟩, f (t) = n≥0 ⟨f (t)|xn ⟩ tn! and p(x) = n≥0 ⟨tn |p(x)⟩ xn! . Thus, (6)

⟨tk |p(x)⟩ = p(k) (0),

⟨1|p(k) (x)⟩ = p(k) (0),

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3

where p(k) (0) denotes the k-th derivative of p(x) with respect to x at x = 0. So, by (6), we dk get that tk p(x) = p(k) (x) = dx k p(x), for all k ≥ 0, (see [22, 23]). Let sn (x) ∼ (g(t), f (t)). Then we have ∑ 1 tn y f¯(t) (7) , e = s (y) n n! g(f¯(t)) n≥0

for all y ∈ C, where f¯(t) is the compositional inverse of f (t) (see [22, 23]). For sn (x) ∼ ∑ (g(t), f (t)) and rn (x) ∼ (h(t), ℓ(t)), let sn (x) = nk=0 cn,k rk (x), then we have ⟨ ⟩ 1 h(f¯(t)) k n (8) (ℓ(f¯(t))) |x , cn,k = k! g(f¯(t)) (see [22, 23]). By the theory of Sheffer sequences, it is immediate that the Barnes-type degenerate Bernoulli and Euler mixed-type polynomial is the Sheffer sequence for the pair g(t) = ( )r ∏ ( at ) ∏s ( ebi t +1 ) r λ i − 1 and f (t) = λ1 (eλt − 1). Thus i=1 e i=1 2 eλt −1 ((

(9)

βE n (λ, x|a; b) ∼

λ eλt − 1

)r ∏ r

(

e

ai t

i=1

s )∏ −1 i=1

(

ebi t + 1 2

)

) 1 λt , (e − 1) . λ

The aim of the present paper is to present several new identities for Barnes-type degenerate Bernoulli and Euler mixed-type polynomials by the use of umbral calculus.

2. Explicit Expressions In this section we suggest several explicit formulas for the Barnes-type degenerate Bernoulli and Euler mixed-type polynomials. To∑ do so, we recall that the Stirling numbers S1 (n, m) of the first kind are defined as (x)n = nm=0 S1 (n, m)xm ∼ (1, et − 1) or j!1 (log(1 + t))j = ∑ tℓ ℓ≥j S1 (ℓ, j) ℓ! . Also, we recall that the Stirling numbers S2 (n, m) of the second kind t k ∑ tℓ n are defined by (e −1) = ℓ≥k S2 (ℓ, k) ℓ! . Define (x|λ)n = λ (x/λ)n to be (x|λ)n = k! x(x − λ)(x − 2λ) · · · (x − (n − 1)λ) with (x|λ)0 = 1. Also, we define )∏ ) r ( s ( ∏ t 2 Pr,s (t) = (1 + λt)ai /λ − 1 i=1 (1 + λt)bi /λ + 1 i=1 and Qr,s (t) =

r ( ∏ i=1

Theorem 2.1. For all n ≥ 0, βE n (λ, x|a; b) =

n ∑ j=0

t a t i e −1

)∏ s ( i=1

2 b t i e +1

) .

  n ( ) ∑ n  S1 (ℓ, j)λℓ−j βE n−ℓ (λ|a; b) xj . ℓ ℓ=j

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Proof. By applying the conjugation representation for sn (x) ∼ (g(t), f (t)), that is, n ∑ 1 sn (x) = ⟨g(f¯(t))−1 f¯(t)j |xn ⟩xj , j! j=0

(see [22, 23]) we obtain ⟨ ⟨g(f¯(t))−1 f¯(t)j |xn ⟩ =

( Pr,s (t) ⟨

= λ−j

log(1 + λt) λ

Pr,s (t)|j!

∑ ℓ≥j

⟩

)j |xn

⟨ ⟩ = λ−j Pr,s (t)| (log(1 + λt))j xn

tℓ S1 (ℓ, j)λℓ xn ℓ!

⟩ .

Thus, n ( ) ⟨ ⟩ ∑ n 1 −1 ¯ j n ¯ ⟨g(f (t)) f (t) |x ⟩ = S1 (ℓ, j)λℓ−j Pr,s (t)|xn−ℓ j! ℓ ℓ=j ⟨ ⟩ n ( ) m ∑ ∑ n t S1 (ℓ, j)λℓ−j βE m (λ|a; b) |xn−ℓ = ℓ m! m≥0 ℓ=j n ( ) ∑ n = S1 (ℓ, j)λℓ−j βE n−ℓ (λ|a; b), ℓ ℓ=j



which completes the proof. Theorem 2.2. For all n ≥ 0, m n ∑ ∑

n

βE n (λ, x|a; b) = λ

(m) k (k+r ) S1 (n, m)S2 (k + r, r)λk−m BEm−k (x|a; b), r

m=0 k=0

where BEn (x|a; b) are the Barnes-type Bernoulli and Euler mixed-type polynomials with r ( ∏ i=1

t eai t − 1

)∏ s ( i=1

2 ebi t + 1

) e

xt

=

∞ ∑ n=0

BEn (x|a; b)

tn n!

(see [26]). Proof. By (9), we have ( (10)

λ λt e −1

)r ∏ r i=1

(

e

ai t

s )∏ −1 i=1

(

ebi t + 1 2

1276

)

( ) 1 λt βE n (λ, x|a; b) ∼ 1, (e − 1) , λ

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5

which implies (

)r eλt − 1 βE n (λ, x|a; b) = S1 (n, m)λ Qr,s (t) xm λt m=0  n ∑ ∑ = S1 (n, m)λn−m Qr,s (t) r! S2 (k + r, r) n ∑

n−m

m=0

= λn = λn

n ∑ m ∑ m=0 k=0 n ∑ m ∑

k≥0

(m)

 λk tk  x m (k + r)!

k (k+r ) S1 (n, m)S2 (k + r, r)λk−m Qr,s (t)xm−k r

(m) k (k+r ) S1 (n, m)S2 (k + r, r)λk−m BEm−k (x|a; b), r

m=0 k=0



which completes the proof. Theorem 2.3. For all n ≥ 1, βE n (λ, x|a; b) =

n−1 n−ℓ ∑∑

(n−1)(n−ℓ) (n)

λk+ℓ S2 (k + r, r)Bℓ BEn−ℓ−k (x|a; b).

(k+r)k

ℓ

r

ℓ=0 k=0

Proof. We proceed the proof by invoking the following transfer (see (7) and (8)): ( )formula n f (t) −1 for pn (x) ∼ (1, f (t)) and qn (x) ∼ (1, g(t)), then qn (x) = x g(t) x pn (x), for all n ≥ 1. In our case, by xn ∼ (1, t) and (10), we have )r ∏ ) ( s ( bi t r ( at )∏ e +1 λ i e −1 βE n (λ, x|a; b) 2 eλt − 1 i=1 i=1 ( )n n−1 ∑ (n − 1) ∑ (n) λℓ λt (n) ℓ n−1 n−1 = x λt Bℓ tx = λℓ Bℓ xn−ℓ . x =x e −1 ℓ! ℓ ℓ≥0

ℓ=0

Thus, n−1 ∑(

) ( ( λt )r ) n − 1 ℓ (n) e −1 n−ℓ βE n (λ, x|a; b) = λ Bℓ Qr,s (t) x ℓ λt ℓ=0   n−1 k tk ∑ (n − 1) ∑ r!λ (n) S2 (k + r, r) xn−ℓ  = λℓ Bℓ Qr,s (t) (k + r)! ℓ ℓ=0

=

=

n−1 n−ℓ ∑∑

k≥0

(n−1)(n−ℓ)

(n)

λk+ℓ S2 (k + r, r)Bℓ Qr,s (t)xn−ℓ−k

(k+r)k

ℓ

r

ℓ=0 k=0 n−1 n−ℓ ∑∑

(n−1)(n−ℓ)

ℓ=0 k=0

r

(k+r)k

ℓ

(n)

λk+ℓ S2 (k + r, r)Bℓ BEn−ℓ−k (x|a; b), 

as claimed.

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Note that the Barnes-type Daehee polynomials with λ-parameter Dn,λ (x|a) with a1 , . . . , ar ̸= 0 was defined as r ∑ ∏ log(1 + λt) tn x/λ (11) (1 + λt) = , D (x|a) n,λ n! λ(1 + λt)ai /λ − 1 i=1

n≥0

see [15, 16]. When x = 0 we write Dn,λ (a) = Dn,λ (0|a); the Barnes-type Daehee numbers. Theorem 2.4. For all n ≥ 0,

) n ∑ n−ℓ ( )( ∑ n n − ℓ ℓ (r) βE n (λ, x|a; b) = λ bℓ (x/λ)Dk,λ (a)En−ℓ−k (λ|b) ℓ k ℓ=0 k=0 ) n ∑ n−ℓ ( )( ∑ n n − ℓ ℓ (r) = λ bℓ Dk,λ (x|a)En−ℓ−k (λ|b) ℓ k ℓ=0 k=0 ) n ∑ n−ℓ ( )( ∑ n n − ℓ ℓ (r) = λ bℓ Dk,λ (a)En−ℓ−k (x, λ|b). ℓ k ℓ=0 k=0

Proof. By (9) we have ⟨ ⟩ βE n (λ, x|a; b) = Pr,s (t)(1 + λt)x/λ |xn ⟨ r ( ⟩ )∏ )r )( s ( ∏ log(1 + λt) 2 λt (1 + λt)x/λ |xn = ai /λ − 1) bi /λ + 1 log(1 + λt) λ((1 + λt) (1 + λt) i=1 i=1 ⟩ ⟨ r ( ) ) ∑ s ( ∏ ∏ 2 λℓ ℓ n log(1 + λt) (r) bℓ (x/λ) t x = | ai /λ − 1) bi /λ + 1 ℓ! λ((1 + λt) (1 + λt) i=1 i=1 ℓ≥0 ⟨ r ( ⟩ )∏ ) n ( ) s ( ∏ ∑ n (r) log(1 + λt) 2 ℓ n−ℓ b (x/λ)λ = |x . ai /λ − 1) bi /λ + 1 ℓ ℓ λ((1 + λt) (1 + λt) i=1 i=1 ℓ=0 Thus, by (11), we obtain ⟨ s ( ⟩ ) ∑ n ( ) k ∏ ∑ n (r) 2 t b (x/λ)λℓ | Dk,λ (a) xn−ℓ βE n (λ, x|a; b) = bi /λ + 1 ℓ ℓ k! (1 + λt) i=1 ℓ=0 k≥0 ⟨ s ( ⟩ ) ) n ∑ n−ℓ ( )( ∑ ∏ n n − ℓ ℓ (r) 2 n−ℓ−k λ bℓ (x/λ)Dk,λ (a) = |x bi /λ + 1 ℓ k (1 + λt) i=1 ℓ=0 k=0 ) n ∑ n−ℓ ( )( ∑ n n − ℓ ℓ (r) = λ bℓ (x/λ)Dk,λ (a)En−ℓ−k (λ|b), ℓ k ℓ=0 k=0

which proves the first formula. Similar techniques show the second and the third formulas.  3. Recurrence relations In this section, we present several recurrence relations for the Barnes-type degenerate Bernoulli and Euler mixed-type polynomials.

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7

Theorem 3.1. For all n ≥ 0,

n ( ) ∑ n βE n (λ, x + y|a; b) = βE j (λ, x|a; b)(y|λ)n−j . j j=0

( Proof. By (9) we have the result.

λt

)r

( λt ) βE (λ, x|a; b) = (x|λ) ∼ 1, e λ−1 , which implies n n Qr,s (t)  1

eλt −1

Theorem 3.1 with x = 0, gives the following result. Corollary 3.2. For all n ≥ 0,

n ( ) ∑ n βE n−j (λ|a; b)(x|λ)j . j

βE n (λ, x|a; b) =

j=0

Theorem 3.3. For all n ≥ 1, βE n (λ, x + λ|a; b) = βE n (λ, x|a; b) + nλβE n−1 (λ, x|a; b). Proof. By (7) we have that f (t)sn (x) = nsn−1 (x) when sn (x) ∼ (g(t), f (t)). In our case, from (9), we have eλt − 1 βE n (λ, x|a; b) = nβE n−1 (λ, x|a; b), λ which implies that βE n (λ, x + λ|a; b) − βE n (λ, x|a; b) = nλβE n−1 (λ, x|a; b), as required.  Theorem 3.4. For all n ≥ 1, ∑ (−λ)n−1−ℓ d βE n (λ, x|a; b) = n! βE ℓ (λ, x|a; b) dx ℓ!(n − ℓ) n−1 ℓ=0

n−1

= nλ

n−1 ∑

S1 (n − 1, ℓ)λ−ℓ BEℓ (x|a; b).

ℓ=0 d Proof. By (7) we have dx sn (x) = our case, from (9), we have

∑n−1 (n) ¯ n−ℓ ⟩s (x) when s (x) ∼ (g(t), f (t)). In n ℓ ℓ=0 ℓ ⟨f (t)|x

∑ (−1)m−1 λm tm 1 ⟨f¯(t)|xn−ℓ ⟩ = ⟨ log(1 + λt)|xn−ℓ ⟩ = λ−1 ⟨ |xn−ℓ ⟩ λ m m≥1

−1

=λ Thus

d dx βE n (λ, x|a; b)

= n!

n−ℓ−1 n−ℓ

(−1)

∑n−1 ℓ=0

λ

(n − ℓ − 1)! = (−λ)n−ℓ−1 (n − ℓ − 1)!.

(−λ)n−1−ℓ ℓ!(n−ℓ) βE ℓ (λ, x|a; b),

as required.

( λt ) ∑ To show the second formula, we note that (x|λ)n = nℓ=0 S1 (n, ℓ)λn−ℓ xℓ ∼ 1, e λ−1 , ( λt )r λt which shows that e λ−1 (x|λ)n = n(x|λ)n−1 . Thus e λ−1 (x|λ)n = (n)r (x|λ)n−r , for all

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n ≥ r. Thus, by (10), we have )r eλt − 1 (x|λ)n λ n−r ∑ = (n)r λn−r S1 (n − r, m)λ−m BEm (x|a; b),

dr βE (λ, x|a; b) = tr βE n (λ, x|a; b) = Qr,s (t) dxr n = (n)r Qr,s (t)(x|λ)n−r

(

m=0



which completes the proof. Theorem 3.5. For all n ≥ 1,  (1 − r/n)βE n (λ, x|a; b) = x −

r ∑

ai −

i=1

1 − n

r ∑ i=1

s ∑

 bj  βE n−1 (λ, x − λ|a; b)

j=1

1∑ ai βE n (λ, x − λ|ai , a; b) + bi βE n−1 (λ, x − λ|a; bi , b). 2 s

i=1

Proof. Let n ≥ 1. By (9), we have ⟨ ⟩ βE n (λ, y|a; b) = Pr,s (t)(1 + λt)y/λ |xn ⟨ ( r ( ) ⟩ )∏ ) s ( t 2 d ∏ y/λ n−1 = (1 + λt) |x ai /λ − 1 bi /λ + 1 dt (1 + λt) (1 + λt) i=1 i=1 ⟨ ⟩ )∏ ) r ( s ( ∏ t 2 d (1 + λt)y/λ |xn−1 (12) = ai /λ − 1 bi /λ + 1 dt (1 + λt) (1 + λt) i=1 i=1 ⟩ ⟨ r ( ) ∏ ) s ( ∏ 2 d t (13) (1 + λt)y/λ |xn−1 + ai /λ − 1 bi /λ + 1 dt (1 + λt) (1 + λt) i=1 i=1 ⟩ ⟨ r ( ) ) s ∏( ∏ 2 t d y/λ n−1 (14) + (1 + λt) |x . (1 + λt)ai /λ − 1 (1 + λt)bi /λ + 1 dt i=1

i=1

The term in (14) is given by ⟨ ⟩ (15) y Pr,s (t)(1 + λt)y/λ−1 |xn−1 = yβE n−1 (λ, y − λ|a; b). In order to find the first term, namely (12), we note that d ∏ dt r

=

(

i=1 r ( ∏ i=1

t (1 + λt)ai /λ − 1 t (1 + λt)ai /λ − 1

) )∑ r ( − i=1

ai 1 + 1 + λt t

1280

( 1−

ai t 1 + λt (1 + λt)ai /λ − 1

)) ,

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where the order of 1 − −

r ∑

ai t 1+λt (1+λt)ai /λ −1

9

is at least 1. Thus the term in (12) is given by

⟨ ⟩ ai Pr,s (t)(1 + λt)y/λ−1 |xn−1

i=1

⟨ +

⟩ ) r ( ∑ a t 1 i 1− Pr,s (t)(1 + λt)y/λ | xn−1 t 1 + λt (1 + λt)ai /λ − 1 i=1

which equals −

r ∑

ai βE n−1 (λ, y − λ|a; b) +

i=1

1∑ − ai n r

⟨

i=1

=−

(16)

r ∑

⟩ r⟨ Pr,s (t)(1 + λt)y/λ |xn n

t Pr,s (t)(1 + λt)y/λ−1 |xn (1 + λt)ai /λ − 1

⟩

1∑ r βE n (λ, y|a; b) − ai βE n (λ, y − λ|ai , a; b). n n r

ai βE n−1 (λ, y − λ|a; b) +

i=1

i=1

In order to find the second term, namely (13), we note that d ∏ dt s

=

(

i=1 s ( ∏

2 (1 + λt)bi /λ + 1

i=1

2 (1 + λt)bi /λ + 1

) )∑ s ( − i=1

bi bi 2 + 1 + λt 2(1 + λt) (1 + λt)bi /λ + 1

) .

Thus the term in (13) is given by s ∑

⟨( bi −1 +

i=1

=−

(17)

s ∑

1 (1 + λt)bi /λ + 1

)

⟩ y/λ−1

Pr,s (t)(1 + λt)

|x

n−1

1∑ bi βE n−1 (λ, y − λ|a; bi , b). 2 s

bi βE n−1 (λ, y − λ|a; b) +

i=1

i=1

Altogether, namely by (15), (16) and (17), we complete the proof.



Theorem 3.6. For n ≥ 0, βE n+1 (λ, x|a; b) = xβE n (λ, x − λ|a; b) n ∑ m ∑ k ∑ λ−k S1 (n, m)S2 (m − k + r, r) −λ (m−k+r) n

m=0 k=0 ℓ=0

( ·

Bk−ℓ+1 (1) k−ℓ+1

(

r r ∑ i=1

)

ak−ℓ+1 − rλk−ℓ+1 i

(m)(k) k

ℓ

·

  ) r Ek−ℓ (1) ∑ k−ℓ+1  + bj BEℓ (x − λ|a; b). 2 j=1

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( Proof. By (7), we have that sn+1 (x) = x − our case, see (9), we have

g ′ (t) g(t)

)

1 f ′ (t) sn (x)

βE n+1 (λ, x|a; b) = xβE n (λ, x − λ|a; b) − e−λt

when sn (x) ∼ (g(t), f (t)). In

g ′ (t) βE (λ, x|a; b), g(t) n

where g ′ (t) = (log g(t))′ g(t)  = r log λ − r log(eλt − 1) +

r ∑

log(eai t − 1) +

i=1 r ∑

s ∑

′ log(ebj t + 1) − s log 2

j=1

s ∑

bj ebj t ai eai t rλeλt + + eλt − 1 ea i t − 1 ebj t + 1 i=1 j=1 ( ) r s ∑ ai teai t 1 rλteλt 1 ∑ 2bj ebj t = + − λt + t e −1 ea i t − 1 2 ebj t + 1 i=1 j=1   r s ℓ bℓ+1 1 ∑ λ ℓ tℓ ∑ ∑ aℓi tℓ  1 ∑ ∑ j t = −r Bℓ (1) + Bℓ (1) + Eℓ (1) t ℓ! ℓ! 2 ℓ! i=1 ℓ≥0 j=1 ℓ≥0 ℓ≥0   ( r ) s ∑ Bℓ+1 (1) ∑ ∑ Eℓ (1) ∑ 1   tℓ . = aℓ+1 bℓ+1 − rλℓ+1 tℓ + i j (ℓ + 1)! 2 ℓ! =−

i=1

ℓ≥0

j=1

ℓ≥0

Therefore, by Theorem 2.2, we obtain g ′ (t) βE (λ, x|a; b) g(t) n

( r ) ( ) m−k n ∑ m ∑ ∑ Bℓ+1 (1) ∑ λk−m S1 (n, m)S2 (k + r, r) m ℓ+1 ℓ+1 k =λ ai − rλ tℓ BEm−k (x|a; b) (k+r) (ℓ + 1)! r m=0 k=0 i=1 ℓ=0   ( ) n m m−k s m k−m n ∑ ∑ ∑ ∑ λ S1 (n, m)S2 (k + r, r) k λ Eℓ (1)   tℓ BEm−k (x|a; b) + bℓ+1 (k+r) j 2 ℓ! r n

m=0 k=0

ℓ=0

( )(k) n ∑ m ∑ k ∑ λ−k S1 (n, m)S2 (m − k + r, r) m n k ℓ =λ · (m−k+r) m=0 k=0 ℓ=0

( ·

Bk−ℓ+1 (1) k−ℓ+1

(

r r ∑

)

ak−ℓ+1 − rλk−ℓ+1 i

i=1

j=1

  ) r Ek−ℓ (1) ∑ k−ℓ+1  + bj BEℓ (x|a; b). 2 j=1

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Thus, βE n+1 (λ, x|a; b) = xβE n (λ, x − λ|a; b)

( )(k) n ∑ m ∑ k ∑ λ−k S1 (n, m)S2 (m − k + r, r) m k ℓ −λ · (m−k+r) n

m=0 k=0 ℓ=0

( ·

Bk−ℓ+1 (1) k−ℓ+1

(

r r ∑



)

ak−ℓ+1 − rλk−ℓ+1 i

+

i=1

Ek−ℓ (1)  2

r ∑

 ) k−ℓ+1  BEℓ (x − λ|a; b), bj

j=1



as claimed. 4. Connections with families of polynomials (α)

The Bernoulli polynomials Bn (x) of order α are defined by the generating function ( )α ∑ t tn xt (α) e = B (x) , n et − 1 n! n≥0 (( t )α ) (α) e −1 equivalently, Bn (x) ∼ , t (see [3, 9, 10]). In the next result, we express our t polynomials βE n (λ, x|a; b) in terms of Bernoulli polynomials of order α. Theorem 4.1. For n ≥ 0, βE n (λ, x|a; b) =

n ∑

(α) λ−m dn,m Bm (x),

m=0

where dn,m

) n ∑ n−ℓ [( )( ∑ n n − ℓ k+ℓ (α) = λ S1 (ℓ, m)bk · ℓ k ℓ=m k=0 (n−ℓ−k) ] q n−ℓ−k ∑ ∑ q p (q+α) S1 (q + α, q − p + α)S2 (q − p + α, α)λ βE n−ℓ−k−q (λ|a; b) . · q=0 p=0

α

∑ (α) Proof. Let βE n (λ, x|a; b) = nm=0 cn,m Bm (x). By (8) and (9), we have ⟨ ( )α ( ⟩ )α 1 (1 + λt)1/λ − 1 λt cn,m = Pr,s (t) (log(1 + λt))m |xn m!λm t log(1 + λt) ⟨ ( )α ( ⟩ )α ∑ λℓ tℓ n (1 + λt)1/λ − 1 λt 1 |m! S1 (ℓ, m) Pr,s (t) x = m!λm t log(1 + λt) ℓ! ℓ≥m ⟨ ( )α ( ⟩ )α n ( ) 1/λ ∑ n (1 + λt) − 1 λt = λ−m λℓ S1 (ℓ, m) Pr,s (t) | xn−ℓ ℓ t log(1 + λt) ℓ=m )α ⟩ ⟨ ( ) n ∑ n−ℓ ( )( 1/λ − 1 ∑ n n − ℓ (1 + λt) (α) = λ−m λk+ℓ S1 (ℓ, m)bk Pr,s (t)| xn−ℓ−k . ℓ k t ℓ=m k=0

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TAEKYUN KIM, DAE SAN KIM, HYUCKIN KWON, AND TOUFIK MANSOUR

Before proceeding further, we note that ( )α ( 1 )α (1 + λt)1/λ − 1 e λ log(1+λt) − 1 = t t ) q ( ∑∑ q + α −1 λp tq = S1 (q + α, q − p + α)S2 (q − p + α, α) . α q! q≥0 p=0

Thus, cn,m

) n ∑ n−ℓ [( )( ∑ n n − ℓ k+ℓ (α) =λ λ S1 (ℓ, m)bk · ℓ k ℓ=m k=0 (n−ℓ−k) q n−ℓ−k ⟨ ⟩] ∑ ∑ q p n−ℓ−k−q (q+α) S1 (q + α, q − p + α)S2 (q − p + α, α)λ Pr,s (t)|x · , −m

q=0 p=0

α

which gives ) n ∑ n−ℓ [( )( ∑ n n − ℓ k+ℓ (α) λ S1 (ℓ, m)bk · ℓ k ℓ=m k=0 (n−ℓ−k) ] q n−ℓ−k ∑ ∑ q p (q+α) S1 (q + α, q − p + α)S2 (q − p + α, α)λ βE n−ℓ−k−q (λ|a; b) , ·

cn,m = λ−m

q=0 p=0

α



which completes the proof. (α)

The degenerate Bernoulli polynomials βn (λ, x) of order α are defined by the generating function ( )α ∑ t tn x/λ (α) + λt) = β (λ, x) , (1 n n! (1 + λt)1/λ − 1 n≥0 (( t )α ) (α) λ(e −1) equivalently, Bn (λ, x) ∼ , λ1 (eλt − 1) . Then by using similar arguments as eλt −1 in the proof of Theorem 4.1, we obtain the following result. Theorem 4.2. For n ≥ 0,

n ( ) ∑ n (α) βE n (λ, x|a; b) = dn,m βm (λ, x), m m=0

where dn,m =

q n−m ∑∑ q=0 p=0

(n−m) q (q+α ) S1 (q + α, q − p + α)S2 (q − p + α, α)λp βE n−m−q (λ|a; b). α

The Frobenius-Euler polynomials of order α are defined by the generating function ) ( tn 1 − µ α xt ∑ (α) e = H (x|µ) , n et − µ n! n≥0 (( t )α ) (α) e −µ equivalently, Hn (x|µ) ∼ , t (see [2, 13]). In the next result, we express our 1−µ polynomials βE n (λ, x|a; b) in terms of Frobenius-Euler polynomials.

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13

Theorem 4.3. For n ≥ 0, βE n (λ, x|a; b) = (1 − µ)−α

n ∑

(α) λ−m dn,m Hm (x|µ),

m=0

where dn,m

)( ) n ∑ n−ℓ ∑ α ( )( ∑ n n−ℓ α = S1 (ℓ, m)λℓ (−µ)α−p βE k (λ|a; b)(p|λ)n−ℓ−k . ℓ k p ℓ=m k=0 p=0

∑ (α) Proof. Let βE n (λ, x|a; b) = nm=0 cn,m Hm (x|µ). Then ⟨ ⟩ 1 1/λ α m n cn,m = ((1 + λt) − µ) P (t)|(log(1 + λt)) x r,s m!(1 − µ)α λm ⟨ ⟩ ℓ tℓ ∑ λ 1 ((1 + λt)1/λ − µ)α Pr,s (t)|m! xn = S1 (ℓ, m) m!(1 − µ)α λm ℓ! ℓ≥m ( ) n ⟨ ⟩ ∑ n 1 ℓ 1/λ α n−ℓ λ S (ℓ, m) ((1 = + λt) − µ) |P (t)x 1 r,s ℓ (1 − µ)α λm ℓ=m ) n−ℓ ( )( n ∑ ⟨ ⟩ ∑ n n−ℓ ℓ 1 1/λ α n−ℓ−k λ S (ℓ, m)βE (λ|a; b) ((1 + λt) − µ) |x . = 1 k ℓ k (1 − µ)α λm ℓ=m k=0

Note that ⟨

((1 + λt)1/λ − µ)α |xn−ℓ−k

⟩

⟨ α ( ) ⟩ ∑ α = (−µ)α−p (1 + λt)p/λ |xn−ℓ−k p p=0 ⟩ ⟨ α q ∑ (α) ∑ t = (−µ)α−p (p|λ)q |xn−ℓ−k q! p p=0 q≥0 ( ) α ∑ α = (−µ)α−p (p|λ)n−ℓ−k . p p=0

Thus, cn,m

)( ) n ∑ n−ℓ ∑ α ( )( ∑ 1 n n−ℓ α ℓ = λ (−µ)α−p S1 (ℓ, m)βE k (λ|a; b)(p|λ)n−ℓ−k , (1 − µ)α λm ℓ k p ℓ=m k=0 p=0



which completes the proof. (α)

The degenerate Euler polynomials En (λ, x) of order α are defined by the generating function )α ( ∑ tn 2 x/λ (α) (1 + λt) = E (λ, x) , n n! (1 + λt)1/λ + 1 n≥0 (( t )α λt ) (α) e +1 equivalently, En (λ, x) ∼ , e λ−1 . Then by using similar arguments as in the 2 proof of Theorems 4.1 and 4.3, we obtain the following result.

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TAEKYUN KIM, DAE SAN KIM, HYUCKIN KWON, AND TOUFIK MANSOUR

Theorem 4.4. For n ≥ 0, βE n (λ, x|a; b) = 2

−α

n ( ) ∑ n (α) dn,m Em (λ, x), m

m=0

where dn,m =

n−m α ( ∑∑ q=0 p=0

n−m q

)( ) α βE n−m−q (λ|a; b)(p|λ)q . p

Acknowledgemts. The first author is appointed as a chair professor at Tianjin Polytechnic University by Tianjin City in China from August 2015 to August 2019.

References [1] Agapito Ruiz, J., Riordan arrays from an umbral symbolic viewpoint, Bol. Soc. Port. Mat., Special Issue (2012), 5-8 [2] Araci, S. and Acikgoz, M., A note on the Frobenius-Euler numbers and polynomials associated with Bernstein polynomials, Adv. Stud. Contemp. Math. 22(3) (2012) 399-406. [3] Bayad A. and Kim, T., Identities involving values of Bernstein, q-Bernoulli, and q-Euler polynomials, Russ. J. Math. Phys. 18(2) (2011) 133–143. [4] Biedenharn, L.C., Gustafson, R.A., Lohe, M.A., Louck, J.D. and Milne, S.C., Special functions and group theory in theoretical physics. In Special functions: group theoretical aspects and applications, Math. Appl. (1984) 129–162, Reidel, Dordrecht. [5] Biedenharn, L.C., Gustafson, R.A. and Milne, S.C., An umbral calculus for polynomials characterizing U (n) tensor products, Adv. Math. 51 (1984) 36–90. [6] Di Bucchianico, A. and Loeb, D., A selected survey of umbral calculus, Electron. J. Combin. 2 (2000) #DS3. [7] Carlitz, L., Degenerate Stirling, Bernoulli and Eulerian numbers, Utilitas Math. 15 (1979) 51–88. [8] Dattoli, G., Levi, D. and Winternitz, P., Heisenberg algebra, umbral calculus and orthogonal polynomials, J. Math. Phys. 49 (2008), no. 5, 053509. [9] Ding, D. and Yang, J., Some identities related to the Apostol-Euler and Apostol-Bernoulli polynomials, Adv. Stud. Contemp. Math. 20(1) (2010) 7–21. [10] Kim, T., A note on q-Bernstein polynomials, Russ. J. Math. Phys. 18 (2011) 73–82. [11] Kim, D.S. and Kim, T., Higher-order degenerate Euler polynomials, Applied Mathematical Sciences 9:2 (2015) 57–73. [12] Kim, D.S. and Kim, T., q-Bernoulli polynomials and q-umbral calculus, Sci. China Math. 57:9 (2014) 1867–1874. [13] Kim, T. and Mansour, T., Umbral calculus associated with Frobenius-type Eulerian polynomials, Russ. J. Math. Phys. 21(4) (2014) 484–493. [14] Kim, T., Identities involving Laguerre polynomials derived from umbral calculus, Russ. J. Math. Phys. 21 (2014), no. 1, 36–45. [15] Kim, D.S. and Kim, T., Higher-order Cauchy of the first kind and poly-Cauchy of the first kind mixed type polynomials, Adv. Stud. Contemp. Math. 23:4 (2013) 621–636. [16] Kim, D.S. and Kim, T., Daehee numbers and polynomials, Appl. Math. Sci. 7:120 (2013) 5969–5976. [17] Kim, D.S., Kim, T., Kwon, H.I., T. Mansour and J. Seo, Barnes-type Peters polynomial with umbral calculus viewpoint, J. Inequal. Appl. 2014 (2014) 324. [18] Kim, D.S., Kim, T., Lee, S.-H. and Rim, S.-H., Frobenius-Euler polynomials and umbral calculus in the p-adic case, Adv. Difference Equ. 2012 (2012) 222. [19] Kim, T., Kim, D.S., Mansour, T., Rim, S.-H. and Schork, M., Umbral calculus and Sheffer sequences of polynomials, J. Math. Phys. 54 (2013), no. 8, 083504. [20] Kwa´sniewski, A.K., q-quantum plane, ψ(q)-umbral calculus, and all that, Quantum groups and integrable systems (Prague, 2001), Czechoslovak J. Phys. 51 (2001), no. 12, 1368–1373.

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[21] Levi, D., Tempesta, P. and Winternitz, P., Umbral calculus, difference equations and the discrete Schr¨ odinger equation, J. Math. Phys. 45 (2004), no. 11, 4077–4105. [22] Roman, S., More on the umbral calculus, with emphasis on the q-umbral calculus, J. Math. Anal. Appl. 107 (1985) 222–254. [23] Roman, S., The umbral calculus, Dover Publ. Inc. New York, 2005. [24] Wilson, B.G. and Rogers, F.G., Umbral calculus and the theory of multispecies nonideal gases, Phys. A 139 (1986) 359–386. [25] Kim, D.S. and Kim, T., Some identities of degenerate Euler polynomials arising from p-adic fermionic integrals on Zp , Integral Transforms Spec. Funct., to appear. [26] Kim, D.S., Kim, T., Kwon, H.I. and Seo, J.-J., Identities of some special mixed-type polynomials, Adv. Stud. Theor. Phys. 8:17 (2014) 745–754. Department of Mathematics, College of Science, Tianjin Polytechnic University, Tianjin City, 300387, China, Department of Mathematics, Kwangwoon University, Seoul, S. Korea E-mail address: [email protected], [email protected] Department of Mathematics, Sogang University, Seoul 121-742, S. Korea E-mail address: [email protected] (corresponding) Department of Mathematics, Kwangwoon University, Seoul 139-701, S. Korea Department of Mathematics, University of Haifa, 3498838 Haifa, Israel

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Ground state solutions for second order nonlinear p-Laplacian difference equations with periodic coefficients Ali Mai Guowei Sun

∗

Department of Applied Mathematics, Yuncheng University Shanxi, Yuncheng 044000, China

Abstract We study the existence of homoclinic solutions for nonlinear p-Laplacian difference equations with periodic coefficients. The proof of the main result is based on the critical point theory in combination with the Nehari manifold approach. Under rather weaker conditions, we obtain the existence of ground state solutions and considerably improve some existing ones even for some special cases. Key words: P-Laplacian Difference equations; Nehari manifold; Ground state solutions; Critical point theory.

1

Introduction

Difference equations represent the discrete counterpart of ordinary differential equations, have been widely used in many fields such as computer science, economics, neural network, ecology, cybernetics, etc. In the past decades, the existence of homoclinic solutions for difference equations with p-Laplacian has been extensively studied, The classical method used is fixed point theory, to mention a few, see [1–3] and references therein for details. As it is well known, the critical point theory is used to deal with the existence of solutions of difference equations [4–10]. Here we mention the works of Cabada, Iannizzotto and Tersian [4], Jiang and Zhou [5], Long and Shi [6]. In these papers, critical point theory is applied on bound discrete intervals, which leads to the study of critical points of an energy functional defined on a finite-dimensional Banach space. For unbounded discrete intervals such as the whole set of integers Z, Ma and Guo used critical point theory in combination with periodic approximation to deal with such problems [7]. In the present paper, under convenient assumption, ∗

Corresponding author. E-mail address: [email protected]

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without periodic approximation and without verifying Palais-Smale condition, we not only prove the existence of homoclinic solution, but also obtain the ground state solution. we extend [11] to the case of the p-lapacian difference equation with periodic coefficients. In this paper, our work focus on the existence of homoclinic solution for the following second order nonlinear difference equations with p-Laplacian −∆[a(k)φp (∆u(k − 1))] + b(k)φp (u(k)) = f (k, u(k)),

k ∈ Z,

(1.1)

where φp (t) = |t|p−2 t for all t ∈ R, p > 1. a(k), b(k) are positive and T −periodic sequences, T is a fixed positive integer. f (k, u) : Z × R → R is a continuous function on u and T −periodic on k. The forward difference operator ∆ is defined by ∆u(k − 1) = u(k) − u(k − 1),

for all k ∈ Z.

where Z and R denote the set of all integers and real numbers, respectively. In addition, we are interested in the existence of nontrivial homoclinic solution for (1.1), that is, solutions that are not equal to 0 identically. We call that a solution u = {u(k)} of (1.1) is homoclinic (to 0) if lim u(k) = 0.

(1.2)

|k|→∞

Throughout this paper, we always suppose that the following conditions hold. (A) a(k) > 0 and a(k + T ) = a(k) for all k ∈ Z. (B) b(k) > 0 and b(k + T ) = b(k) for all k ∈ Z. (f1 ) f ∈ C(Z × R, R), and there exist C > 0, q ∈ (p, ∞) such that |f (k, u)| ≤ C(1 + |u|q−1 ), for all k ∈ Z, u ∈ R. (f2 ) lim f (k, u)/|u|p−1 = 0 uniformly for k ∈ Z. |u|→0

(f3 ) lim F (k, u)/|u|p = +∞ uniformly for k ∈ Z, where F (k, u) is the primitive function |u|→∞

of f (k, u), i.e.,

u

Z F (k, u) =

f (k, s)ds. 0

(f4 ) u 7→ f (k, u)/|u|p−1 is strictly increasing on (−∞, 0) and (0, ∞). The main result in this paper is the following theorem: Theorem 1.1. Suppose conditions (A), (B) and (f1 )−(f4 ) are satisfied. Then equation (1.1) has at least a nontrivial ground state solution. Remark 1.1. In [7], Ma and Guo considered the special case of (1.1) with p = 2 and obtained the following theorem:

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Theorem A Suppose conditions (A), (B), (f2 ) and the following generalized AmbrosettiRabinowitz superlinear condition are satisfied: (GAR) there exists a constant µ > p such that 0 < µF (k, u) ≤ f (k, u)u, for all k and u 6= 0,

(1.3)

Then equation (1.1) has a nontrivial ground state solution. It is easy to see that (1.3) implies (f3 ). There exists a p-superlinear function, such as f (k, u) = |u|p−2 u ln(1 + |u|), does not satisfy (1.3). However, it satisfies the condition (f1 ) − (f4 ). So our conditions are weaker than conditions in [7]. And we do not need periodic approximation technique to obtain homoclinic solutions. Furthermore, we obtain the existence of a ground state solution. Therefore, our result not only extends the main result in [7] to difference equations with p-Laplacian but also improves it. Remark 1.2. In [12], the authors considered the following second order nonlinear difference equations with p-Laplacian −∆φp (∆u(k − 1)) + b(k)φp (u(k)) = f (k, u(k)),

k ∈ Z,

(1.4)

without any periodic assumption, they obtained the homoclinic solutions of the equation. However, PS condition need to be proved in [12], in this paper, we only prove the coercive condition (below Lemma 3.2) is satisfied. Example 1.1. Let  f (k, u) =

p−2

|u|

0, u = 0, u ln(1 + |u|), u = 6 0,

for all k ∈ Z, If (A) and (B) are satisfied, then it is easy to check that all the conditions of our Theorem 1.1 are satisfied. Therefore, the nontrivial homoclinic solution is obtained at once. The rest of the paper is organized as follows: In Section 2, we establish the variational framework associated with (1.1), then present the main results of this paper. Section 3 is devoted to prove the main result.

2

Preliminaries

In this section, we shall establish the corresponding variational framework associated with (1.1). We are going to define a suitable space E and an energy functional J ∈ E, such that critical points of J in E are exactly solutions of (1.1).

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Consider the real sequence spaces   lp ≡ lp (Z) = u = {u(k)}k∈Z : ∀ k ∈ Z, u(k) ∈ R, kuklp = 

! p1 X

|u(k)|p

 

0 such that sw w ∈ N . Proof.

Let I(u) =

P

F (k, u(k)). By (f2 ), we have

k∈Z

I 0 (u) = o(kukp−1 ) as u → 0.

(3.1)

From (f4 ), for all u 6= 0 and s > 0, we have s 7→ I 0 (su)u/sp−1 is strictly increasing.

(3.2)

Let W ⊂ E \ {0} be a weakly compact subset and s > 0, we claim that I(su)/sp → ∞ uniformly for u on W, as s → ∞.

(3.3)

Indeed, let {un } ⊂ W . It suffices to show that if sn → ∞,

I(sn un )/(sn )p → ∞.

as n → ∞. Passing to a subsequence if necessary, un * u ∈ E \ {0} and un (k) → u(k) for every k, as n → ∞. Note that from (f2 ) and (f4 ), it is easy to get that F (k, u) > 0, for all u 6= 0.

(3.4)

Since |sn un (k)| → ∞ and un 6= 0, by (f3 ) and (3.4), we have I(sn un ) X F (k, sn un (k)) = |un (k)|p → ∞ as n → ∞. p p (sn ) |sn un (k)| k∈Z Therefore, (3.3) holds. Let g(s) := J(sw), s > 0. Then g 0 (s) = J 0 (sw)w = sp−1 (kwkp − s1−p I 0 (sw)w), from (3.1)-(3.3), then there exists a unique sw , such that g 0 (s) > 0 whenever 0 < s < sw , g 0 (s) < 0 whenever s > sw and g 0 (sw ) = J 0 (sw w)w = 0. So sw w ∈ N .  1292

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Lemma 3.2. J is coercive on N , i.e., J(u) → ∞ as kuk → ∞, u ∈ N . Proof. Suppose by contradiction, there exists a sequence {un } ⊂ N such that kun k → ∞ and J(un ) ≤ d. Let vn = kuunn k , then there exists a subsequence, still denoted by the same notation, such that vn * v and vn (k) → v(k) for every k, as n → ∞. First we know that there exist δ > 0 and kj ∈ Z such that |vn (kj )| ≥ δ.

(3.5)

Indeed, if not, then vn → 0 in l∞ as n → ∞. For r > p, p kvn krlr ≤ kvn kr−p l∞ kvn klp

we have vn → 0 in all lr , r > p. Note that by (f1 ) and (f2 ), for any ε > 0, there exists cε > 0 such that |f (k, u)| ≤ ε|u|p−1 + cε |u|q−1

and |F (k, u)| ≤ ε|u|p + cε |u|q .

(3.6)

Then for each s > 0, we have X F (k, svn (k)) ≤ εsp kvn kplp + cε sq kvn kqlq k∈Z

which implies that

P

F (k, svn (k)) → 0 as n → ∞. So

k∈Z

sp (k) p X sp d ≥ J(un ) ≥ J(svn ) = kv k − F (k, svn (k)) → , p p k∈Z as n → ∞. This is a contradiction with s >

√ p

(3.7)

pd.

Due to periodicity of coefficients, we know J and N are both invariant under T-translation. Making such shifts, we can assume that 1 ≤ kj ≤ T − 1 in (3.5). Moreover, passing to a subsequence, we can assume that kj = k0 is independent of j. Next we may extract a subsequence, still denoted by {vn }, such that vn (k) → v(k) for all k ∈ Z. Specially, for k = k0 , inequality (3.5) shows that |v(k0 )| ≥ δ, so v 6= 0. Since |un (k)| → ∞ as n → ∞, it follows again from (f3 ) that 0≤

J(un ) 1 X F (k, un (k)) = − (vn (k))p → −∞ kun kp p k∈Z (un (k))p

as n → ∞,

a contradiction again.  Proof of Theorem 1.1. The proof consists of five steps. The proof of step 1-3 is similar to [12], for readers’ convenience, we give the proof.

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step 1. we claim that N is homeomorphic to the unit sphere S in E. By (3.1) and (3.3), g(s) > 0 for s > 0 small and g(s) < 0 for s > 0 large. So sw is a unique maximum of g(s) and sw w is the unique point on the ray s 7→ sw (s > 0) which intersects N . That is, u ∈ N is the unique maximum of J on the ray. Therefore, by Lemma 3.1, we may define the mapping m ˆ : E \ {0} → N by setting m(w) ˆ := sw w. Next we show the mapping m ˆ is continuous. Indeed, suppose wn → w 6= 0. Since m(tu) ˆ = m(u) ˆ for each t > 0, we may assume wn ∈ S for all n. Write m(w ˆ n ) = swn wn . Then {swn } is bounded. If not, swn → ∞ as n → ∞. Note that by (f4 ), for all u 6= 0, Z u 1 1 f (k, u)u − F (k, u) = f (k, u)u − f (k, s)ds p p 0 Z 1 f (k, u) u p−1 > f (k, u)u − p−1 s ds p u 0 = 0. Therefore, for all u ∈ N , we have X 1 J(u) = J(u) − J 0 (u)u = p k∈Z



 1 f (k, u(k))u(k) − F (k, u(k)) > 0. p

(3.8)

Combining with (f3 ) and Lemma 3.1, we have 0
0 after passing to a subsequence if needed. Since N is closed and m(w ˆ n ) = swn wn → sw, sw ∈ N . Hence sw = sw w = m(w) ˆ by the uniqueness of sw of Lemma 3.1. Therefore, m ˆ is continuous. Then we define a mapping m : S → N by setting m := m| ˆ S , then m is a homeomorphism u between S and N , and the inverse of m is given by m−1 (u) = kuk . ˆ : E \ {0} → R and Ψ : S → R by step 2. now we define the functional Ψ ˆ ˆ S. Ψ(w) := J(m(w)) ˆ and Ψ(w) := Ψ| Then we have ˆ ∈ C 1 (E \ {0}, R) and Ψ ∈ C 1 (S, R). Moreover, Ψ ˆ ˆ 0 (w)z = km(w)k Ψ J 0 (m(w))z ˆ kwk

for all w, z ∈ E, w 6= 0.

1294

(3.9)

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Ψ0 (w)z = km(w)kJ 0 (m(w))z

for all z ∈ Tw (S) = {v ∈ E : (w, v) = 0}.

(3.10)

In fact, let w ∈ E \ {0} and z ∈ E. By Lemma 3.1 and the mean value theorem, we obtain ˆ + tz) − Ψ(w) ˆ Ψ(w = J(sw+tz (w + tz)) − J(sw w) ≤ J(sw+tz (w + tz)) − J(sw+tz (w)) = J 0 (sw+tz (w + τt tz))sw+tz tz, where |t| is small enough and τt ∈ (0, 1). Similarly, ˆ + tz) − Ψ(w) ˆ Ψ(w = J(sw+tz (w + tz)) − J(sw w) ≥ J(sw (w + tz)) − J(sw (w)) = J 0 (sw (w + ηt tz))sw tz, where ηt ∈ (0, 1). Combining these two inequalities and the continuity of function w 7→ sw , we have ˆ + tz) − Ψ(w) ˆ Ψ(w km(w)k ˆ = sw J 0 (sw w)z = J 0 (m(w))z. ˆ t→0 t kwk

lim

ˆ is bounded linear in z and continuous in w. It follows Hence the Gˆ ateaux derivative of Ψ 1 ˆ is a class of C and (3.9) holds. Note only that since w ∈ S, m(w) = m(w), that Ψ ˆ so (3.10) is clear. step 3. {wn } is a Palais-Smale sequence for Ψ if and only if {m(wn )} is a Palais-Smale sequence for J. Let {wn } be a Palais-Smale sequence for Ψ, and let un = m(wn ) ∈ N . Since for every wn ∈ S we have an orthogonal splitting E = Twn S ⊕ Rwn , we have kΨ0 (wn )k = sup Ψ0 (wn )z = km(wn )k sup J 0 (m(wn ))z = kun k sup J 0 (un )z. z∈Twn S kzk=1

z∈Twn S kzk=1

z∈Twn S kzk=1

Then kΨ0 (wn )k ≤ kun kkJ 0 (un )k = kun k

sup

J 0 (un )(z + tw) kz + twk t∈R

z∈Twn S, z+tw6=0

≤ kun k

J 0 (un )(z) = kΨ0 (wn )k, kzk z∈Twn S\{0} sup

Therefore kΨ0 (wn )k = kun kkJ 0 (un )k.

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By (3.8), for un ∈ N , J(un ) > 0, so there exists a constant c0 > 0 such that J(un ) > c0 . √ And since c0 ≤ J(un ) = p1 kun kp − I(un ) ≤ p1 kun kp , kun k ≥ p pc0 . Together with Lemma √ 3.2, p pc0 ≤ kun k ≤ supn kun k < ∞. Hence {wn } is a Palais-Smale sequence for Ψ if and only if {un } is a Palais-Smale sequence for J. step 4. by (3.11), Ψ0 (w) = 0 if and only if J 0 (m(w)) = 0. So w is a critical point of Ψ if and only if m(w) is a nontrivial critical point of J. Moreover, the corresponding values of Ψ and J coincide and inf S Ψ = inf N J. If u0 ∈ N satisfies J(u0 ) = c := inf u∈N J(u), then m−1 (u0 ) ∈ S is a minimizer of Ψ and therefore a critical point of Ψ, so u0 is a critical point of J. It remains to show that there exists a minimizer u ∈ N of J|N . Let {wn } ⊂ S be a minimizing sequence for Ψ. By Ekeland’s variational principle we may assume Ψ(wn ) → c, Ψ0 (wn ) → 0 as n → ∞, hence J(un ) → c, J 0 (un ) → 0 as n → ∞, where un := m(wn ) ∈ N . We know that {un } is bounded in N by Lemma 3.2, then there exists a subsequence, still denoted by the same notation, such that un weakly converges to some u ∈ E. We claim that there exist δ > 0 and kj ∈ Z such that |un (kj )| ≥ δ.

(3.12)

Indeed, if not, then un → 0 in l∞ as n → ∞. From the simple fact that, for r > p, p kun krlr ≤ kun kr−p l∞ kun klp

we have un → 0 in all lr , r > p. By (3.6), we know X X X f (k, un (k))un (k) ≤ ε |un (k)|p−1 · |un (k)| + cε |un (k)|q−1 · |un (k)| k∈Z

k∈Z

≤ which implies that

P

εkun kplp

k∈Z

+

cε kun kq−1 lq

f (k, un (k))un (k) = o(kun k) as n → ∞. Then

k∈Z

o(kun k) = (J 0 (un ), un ) = kun kp −

X

f (k, un (k))un (k) = kun kp − o(kun k).

k∈Z p

So kun k → 0, as n → ∞, which contradicts with un ∈ N . Since J and J 0 are both invariant under T -translation. Making such shifts, we can assume that 1 ≤ kj ≤ T −1 in (3.12). Moreover passing to a subsequence, we can assume that kj = k0 is independent of j. Extract a subsequence, still denoted by {un }, we have un * u and un (k) → u(k) for all k ∈ Z. Specially, for k = k0 , inequality (3.12) shows that |u(k0 )| ≥ δ, so u 6= 0. Hence u ∈ N . step 5. we need to show that J(u) = c. By Fatou’s lemma, we have    X 1 1 0 f (k, un (k))un (k) − F (k, un (k)) c = lim J(un ) − J (un )un = lim n→∞ n→∞ 2 2 k∈Z  X 1 1 ≥ f (k, u(k))u(k) − F (k, u(k)) = J(u) − J 0 (u)u = J(u) ≥ c. 2 2 k∈Z 1296

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Hence J(u) = c. The proof of Theorem 1.1 is completed.  Acknowledgments This work is Supported by National Natural Science Foundation of China(11526183, 11371313, 11401121), the Natural Science Foundation of Shanxi Province (2015021015) and Foundation of Yuncheng University(YQ-2014011,XK-2014035).

References [1] R. Avery and J. Henderson, Existence of three positive pseudo-symmetric solutions for a one dimensional discrete p-Laplacian, J. Difference Equ. Appl. 10 (2004) 529C539. [2] Z. He, On the existence of positive solutions of p-Laplacian difference equations, J. Comput. Appl. Math. 161 (2003) 193C201. [3] Y. Li and L. Lu, Existence of positive solutions of p-Laplacian difference equations, Appl. Math. Lett. 19 (2006) 1019C1023. [4] A. Cabada, A. Iannizzotto, S. Tersian, Multiple solutions for discrete boundary value problems, J. Math. Anal. Appl. 356 (2009) 418-428. [5] L. Jiang, Z. Zhou, Three solutions to Dirichlet boundary value problems for p-Laplacian difference equations, Adv. Differential Equations 2008 (2008) 1-11. [6] Y. Long, H. Shi, Multiple solutions for the discrete p-Laplacian boundary value problems, Discrete Dyn. Nat. Soc. 2014 (2014) 1-6. [7] M. Ma, Z. Guo, Homoclinic orbits for nonliear second order difference equations, Nonlinear Anal. 67 (2007) 1737-1745. [8] A. Mai, Z. Zhou, Ground state solutions for the periodic discrete nonlinear Schr¨odinger equations with superlinear nonlinearities. Abstr. Appl. Anal. 2013(2013) 1-11. [9] A. Mai, Z. Zhou, Discrete solitons for periodic discrete nonlinear Schr¨odinger equations, Appl. Math. Comput, 222 (2013): 34-41. [10] A. Mai, Z. Zhou, Homoclinic solutions for a class of nonlinear difference equations, J. Appl. Math., 2014: 1-8. [11] G. Sun, On Standing Wave solutions for discrete nonlinear Schr¨oinger equations, Abstr. Appl. Anal. 2013(2013) 1-6. [12] G. Sun, Ali Mai, Infinitely many homoclinic solutions for second order nonlinear difference equations with p-Laplacian, Sci. World J. 2014:1-6. [13] P. Rabinowitz, Minimax Methods in Critical Point Theory with Applications to Differential Equations, CBMS Regional Conference Series in Mathematics vol 65, American Mathematical Society, Providence, RI, 1986.

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On a solutions of fourth order rational systems of difference equations E. M. Elsayed1,2 , Abdullah Alotaibi1 , and Hajar A. Almaylabi1 1 King AbdulAziz University, Faculty of Science, Mathematics Department, P. O. Box 80203, Jeddah 21589, Saudi Arabia. 2 Mathematics Department, Faculty of Science, Mansoura University, Mansoura 35516, Egypt. E-mail: [email protected], [email protected], [email protected]. ABSTRACT In this paper, we get the form of the solutions of the following difference equation systems of order four xn+1 =

yn xn−2 , yn + yn−3

yn+1 =

xn yn−2 , ±xn ± xn−3

n = 0, 1, 2, · · · ,

where the initial conditions x−3 , x−2 , x−1 , x0 , y−3 , y−2 , y−1 , y0 are arbitrary non zero real numbers.

Keywords: difference equations, recursive sequences, system of difference equations, stability, periodicity, boundedness. Mathematics Subject Classification: 39A10. –––––––––––––––––––

1. INTRODUCTION Difference equations enter as approximations of continuous problems and as models describing life situations in many directions. Recently there has been a great interest in studying difference equations, see, for instance [4], [11], [30] and references cited therein, as well as in studying systems of difference equations (see, e.g. [1], [3], [6], [8]-[10]). Some of the systems of difference equations that are of considerable interest nowadays are symmetric or those obtained from symmetric ones by modifications of their parameters or the sequence coefficients appearing in them (for the case of nonautonomous systems of difference equations). Such systems are studied, for example, in the following papers: Clark et al. [2] has investigated the global stability properties and asymptotic behavior of solutions of the system xn yn , yn+1 = . xn+1 = a + cyn b + dxn Din and Elsayed [5] investigated the boundedness character, persistence, local and global behavior of positive solutions of following two directional interactive and invasive species model xn+1 = α + βxn + γxn−1 e−yn , yn+1 = δ + yn + ζyn−1 e−xn . Halim et al. [13] deal with the form of the solutions of the two following systems of rational difference equations xn+1

=

xn+1

=

yn (xn−2 + yn−3 ) xn−1 (xn−1 + yn−2 ) , yn+1 = , yn−3 + xn−2 − yn 2xn−1 + yn−2 (yn−3 − xn−2 )yn (yn−2 − xn−1 )xn−1 , yn+1 = . yn−3 − xn−2 + yn yn−2

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Kurbanli [21] investigated the behavior of the solution of the difference equation system xn+1 =

xn−1 xn−1 yn −1 ,

yn+1 =

yn−1 yn−1 xn −1 ,

zn+1 =

1 zn yn .

The authors in [27] have got the form of the solutions of some systems of the following rational difference equations xn−1 yn−1 , yn+1 = . xn+1 = α − xn−1 yn β + γyn−1 xn In [29] Papaschinnopoulos and Schinas studied the oscillatory behavior, the boundedness of the solutions, and the global asymptotic stability of the positive equilibrium of the system of nonlinear difference equations xn+1 = A +

yn xn , yn+1 = A + . xn−p yn−q

In [39], Yalcinkaya et al. studied the periodic character of the following two systems of difference equations (1)

xn+1 = and

(1)

xn+1 =

x(2) n (2)

(2)

xn −1 x(k) n (k)

, xn+1 = (2)

xn −1

, xn+1 =

x(3) n (3)

x(1) n (1)

(k)

xn −1

, . . . , xn+1 =

x(1) n (1)

, . . . , xn+1 =

(1)

(2)

(k)

xn −1

xn −1

,

x(k−1) n (k−1)

−1

xn

,

(k)

where the initial values are nonzero real numbers for x0 , x0 , . . . , x0 6= 1.

In [42]-[43] Zhang et al. studied the boundedness, the persistence and global asymptotic stability of the positive solutions of the systems of difference equations xn+1 = A +

yn−m , xn

and xn = A +

1 yn−p

,

yn+1 = A +

yn = A +

xn−m , yn

yn−1 . xn−r yn−s

Similar to difference equations and nonlinear systems of rational difference equations were investigated see [12][45]. In this paper, we obtain the expressions of the solutions of the following nonlinear systems of difference equations yn xn−2 xn yn−2 , yn+1 = , n = 0, 1, 2, · · · , xn+1 = yn + yn−3 ±xn ± xn−3

where the initial values x−3 , x−2 , x−1 , x0 , y−3 , y−2 , y−1 , y0 are arbitrary non zero real numbers, moreover, we take some numerical examples for the equation to illustrate the results.

2. ON THE SYSTEM XN +1 =

YN XN −2 YN +YN −3 ,

YN+1 =

XN YN −2 XN +XN −3

In this section,we study the solutions of the following system of difference equations xn+1 =

yn xn−2 , yn + yn−3

yn+1 =

xn yn−2 , xn + xn−3

(1)

where the initial values x−3 , x−2 , x−1 , x0 , y−3 , y−2 , y−1 , y0 are arbitrary nonzero real numbers. Theorem 1. Suppose that {xn , yn } are solutions of the system (1). Then for n = 0, 1, 2, · · · , we have the following formula x6n−3

=

Qn−1 i=0

x6n−1

=

Qn−1 i=0

adn hn (e + (6i + 3)h)(a + (6i)d)

x6n−2 = Qn−1

,

cdn hn (e + (6i + 5)h)(a + (6i + 2)d)

i=0

x6n = Qn−1

,

i=0

1299

bdn hn (e + (6i + 1)h)(a + (6i + 4)d) dn+1 hn

(e + (6i + 3)h)(a + (6i + 6)d)

,

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J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 22, NO.7, 2017, COPYRIGHT 2017 EUDOXUS PRESS, LLC

x6n+1

=

y6n−3

=

y6n−1

=

(e + h)

(e + (6i + 7)h)(a + (6i + 4)d) edn hn , Qn−1 i=0 (e + (6i)h)(a + (6i + 3)d) Qn−1 i=0

y6n+1

=

bdn hn+1

Qn−1 i=0

gdn hn

(e + (6i + 2)h)(a + (6i + 5)d)

(a + d)

, x6n+2 =

Qn−1 i=0

y6n−2

(a + 2d)

(e + (6i + 5)h)(a + (6i + 8)d) f dn hn = Qn−1 , i=0 (e + (6i + 4)h)(a + (6i + 1)d)

y6n = Qn−1

,

i=0

f dn+1 hn (e + (6i + 4)h)(a + (6i + 7)d)

, y6n+2 =

cdn+1 hn

Qn−1

,

i=0

dn hn+1

(e + (6i + 6)h)(a + (6i + 3)d) gdn hn+1

Qn−1

(e + 2h)

,

(e + (6i + 8)h)(a + (6i + 5)d)

i=0

,

where x−3 = a, x−2 = b, x−1 = c, x0 = d, y−3 = e, y−2 = f, y−1 = g, y0 = h.

Proof. By using mathematical induction. The result holds for n = 0. Suppose that the result holds for n − 1 x6n−7

=

Qn−2 i=0

x6n−5 y6n−7

= = =

(e + (6i + 5)h)(a + (6i + 2)d)

(e + h) Qn−2 i=0

y6n−5

cdn−1 hn−1

Qn−2 i=0

(e + (6i + 7)h)(a + (6i + 4)d)

(e + (6i + 2)h)(a + (6i + 5)d)

(a + d)

i=0

=

=

Qn−2 i=0

=

y6n−3

= = =

i=0

=

Qn−2 Qn−2 i=0

d h

i=0

Tn−2 i=0

(e + (6i + 3)h)(a + (6i + 6)d)

(a+2d)

´³

´

+

Qn−2 i=0

Qn−2 i=0

Tn−2

³

i=0

(e + (6i + 5)h)(a + (6i + 8)d) dn−1 hn

(e + (6i + 8)h)(a + (6i + 5)d)

i=0

= Qn−1 i=0

dn hn 

(a+(6i+8)d)(e+(6i+6)h)(a+(6i+3)d) n dn h

+

Tn−2 i=0

´

adn hn

(e + (6i + 3)h)(a + (6i)d)

n n−1 dn−1 hn Tn−2 cd h Tn−2 i=0 (e+(6i+5)h)(a+(6i+8)d) i=0 (e+(6i+6)h)(a+(6i+3)d)

cdn hn−1 T (a+2d) n−2 (e+(6i+5)h)(a+(6i+8)d) i=0

´

gdn−1 hn−1 (e+(6i+2)h)(a+(6i+5)d)

cdn−1 hn−1 (e+(6i+5)h)(a+(6i+2)d)

d + Tn−2 1 Tn−2 (a+(6i+2)d) (a+2d) (a+(6i+8)d) i=0 i=0



,

´

 Tn−2 (a+2d) (a+(6i+8)d) i=0 Tn−2 (a+(6i+2)d) i=0

dn hn

³ (e + (6i + 6)h)(a + (6i + 3)d) d + dn hn

´ T (a+2d) n−2 (a+(6i+8)d) Tn−2i=0 i=0 (a+(6i+2)d)

(e + (6i + 6)h)(a + (6i + 3)d) (a + (6n − 3)d)

1300

= Qn−1 i=0

,

,

gdn−1 hn

dn hn−1 (e+(6i+3)h)(a+(6i+6)d)

Tn−2

,

cdn hn−1

´ T (e+2h) n−2 (e+(6i+8)h) Tn−2i=0 i=0 (e+(6i+2)h)

dn hn

(e+(6i+6)h)(a+(6i+3)d) d+

i=0

=

gdn−1 hn T (e+2h) n−2 (e+(6i+8)h)(a+(6i+5)d) i=0 n n

Qn−2

x6n−4 y6n−6 =³ x6n−4 + x6n−7

Tn−2

i=0

(e + 2h)

gdn−1 hn (e+(6i+8)h)(a+(6i+5)d)

³ (e + (6i + 3)h)(a + (6i + 6)d) h +

(e + (6n − 3)h)

(a+2d)

(e+2h)

Tn−2

(e + (6i + 3)h)(a + (6i + 6)d)

(e + (6i + 6)h)(a + (6i + 3)d)

i=0

, y6n−4 =

dn hn−1

(a + 2d)

y6n−6 = Qn−2

(e + (6i + 4)h)(a + (6i + 7)d)

y6n−4 x6n−6 =³ y6n−4 + y6n−7

, x6n−4 =

,

f dn hn−1

From system (1) we can prove as follow ³ x6n−3

i=0

bdn−1 hn

gdn−1 hn−1

Qn−2

x6n−6 = Qn−2

,

edn hn

(e + (6i)h)(a + (6i + 3)d)

.

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The other relations can be proved similarly, this completes the proof. Lemma 1. Every positive solution of system (1) is bounded and lim xn = lim yn = 0. n→∞

n→∞

Proof: It follows from system (1), that xn+1 =

yn xn−2 yn xn−2 < = xn−2 , yn + yn−3 yn

yn+1 =

xn yn−2 xn yn−2 < = yn−2 . xn + xn−3 xn

∞ ∞ ∞ ∞ ∞ Then the subsequences {x3n−2 }∞ n=0 , {x3n−1 }n=0 , {x3n }n=0 , {y3n−2 }n=0 , {y3n−1 }n=0 , {y3n }n=0 are decreasing and so are bounded from above by M, N respectively since M = max{x−3 , x−2 , x−1 , x0 } , N = max{y−3 , y−2 , y−1 , y0 }.

3. ON THE SYSTEM XN +1 =

YN XN −2 YN +YN −3 ,

YN+1 =

XN YN −2 XN −XN −3

We study, in this section, the solutions formulas of the system of rational difference equations xn+1 =

yn xn−2 , yn + yn−3

yn+1 =

xn yn−2 , xn − xn−3

(2)

where the initial values x−3 , x−2 , x−1 , x0 , y−3 , y−2 , y−1 , y0 are arbitrary nonzero real numbers. Theorem 2. Assume that {xn , yn } are solutions of system (2) with x−3 6= x0 , x−3 6= 2x0 and y−3 6= ±y0 . Then for n = 0, 1, 2, ...,

x12n−3

=

x12n−2

=

x12n−1

=

x12n

=

x12n+1

=

x12n+2

=

x12n+3

=

x12n+4

=

x12n+5

=

x12n+6

=

x12n+7

=

x12n+8

=

d2n h2n , an−1 (h + e)n (h − e)n (2d − a)n bd2n h2n , an (h + e)n (h − e)n (2d − a)n cd2n h2n , an (h + e)n (h − e)n (2d − a)n d2n+1 h2n , an (h + e)n (h − e)n (2d − a)n bd2n h2n+1 , an (h + e)n+1 (h − e)n (2d − a)n cd2n+1 h2n , an (h + e)n (h − e)n (2d − a)n+1 d2n+1 h2n+1 , an (h + e)n (h − e)n+1 (2d − a)n bd2n+1 h2n+1 , an+1 (h + e)n+1 (h − e)n (2d − a)n cd2n+1 h2n+1 , an (h + e)n+1 (h − e)n (2d − a)n+1 d2n+2 h2n+1 , an (h + e)n (h − e)n+1 (2d − a)n+1 bd2n+1 h2n+2 , an+1 (h + e)n+1 (h − e)n+1 (2d − a)n cd2n+2 h2n+1 , an+1 (h + e)n+1 (h − e)n (2d − a)n+1

1301

h2n d2n , e2n−1 (d − a)2n f h2n d2n y12n−2 = 2n , e (d − a)2n gh2n d2n y12n−1 = 2n , e (d − a)2n h2n+1 d2n y12n = 2n , e (d − a)2n f h2n d2n+1 y12n+1 = 2n , e (d − a)2n+1 −gh2n+1 d2n y12n+2 = 2n+1 , e (d − a)2n −h2n+1 d2n+1 y12n+3 = 2n , e (d − a)2n+1 f h2n+1 d2n+1 y12n+4 = 2n+1 , e (d − a)2n+1 −gh2n+1 d2n+1 y12n+5 = 2n+1 , e (d − a)2n+1 h2n+2 d2n+1 y12n+6 = 2n+1 , e (d − a)2n+1 −f h2n+1 d2n+2 y12n+7 = 2n+1 , e (d − a)2n+2 −gh2n+2 d2n+1 y12n+8 = 2n+2 . e (d − a)2n+1 y12n−3 =

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Proof. By using mathematical induction. The result holds for n = 0. Suppose that the result holds for n − 1 x12n−7

=

x12n−6

=

x12n−5

=

x12n−4

=

cd2n−1 h2n−1 , an−1 (h + e)n (h − e)n−1 (2d − a)n d2n h2n−1 , n−1 n−1 a (h + e) (h − e)n (2d − a)n bd2n−1 h2n , n n a (h + e) (h − e)n (2d − a)n−1 cd2n h2n−1 , n n a (h + e) (h − e)n−1 (2d − a)n

−gh2n−1 d2n−1 , e2n−1 (d − a)2n−1 h2n d2n−1 y12n−6 = 2n−1 , e (d − a)2n−1 −f h2n−1 d2n y12n−5 = 2n−1 , e (d − a)2n −gh2n d2n−1 y12n−4 = 2n , e (d − a)2n−1 y12n−7 =

From system (2) we have

x12n−3

=

= y12n−3

=

x12n−2

= =

y12n−2

= =

d2n h2n−1 −gh2n d2n−1 2n−1 n−1 n−1 y12n−4 x12n−6 − a) a (h + e) (h − e)n (2d − a)n = 2n 2n−1 2n−1 2n−1 −gh d −gh d y12n−4 + y12n−7 + e2n (d − a)2n−1 e2n−1 (d − a)2n−1 d2n h2n h2n d2n = , an−1 (h + e)n−1 (h − e)n (2d − a)n (h + e) an−1 (h + e)n (h − e)n (2d − a)n e2n (d

x12n−4 y12n−6 = x12n−4 − x12n−7 y12n−3 x12n−5 y12n−3 + y12n−6

cd2n h2n−1 h2n d2n−1 an (h+e)n (h−e)n−1 (2d−a)n e2n−1 (d−a)2n−1 cd2n h2n−1 cd2n−1 h2n−1 an (h+e)n (h−e)n−1 (2d−a)n − an−1 (h+e)n (h−e)n−1 (2d−a)n

d2n h2n bd2n−1£ h2n d2n h2n e2n−1 (d−a)2n an (h+e)n (h−e)n (2d−a)n−1 e2n + (d−a)2n

x12n−3 y12n−5 x12n−3 − x12n−6

d2n−1 h2n e2n−1 (d−a)2n−1

2n 2n 2n 2n−1 −h £ d fd h

an−1 (h+e)n (h−e)n (2d−a)n e2n−1 (d−a)2n

h2n d2n an−1 (h+e)n (h−e)n (2d−a)n

−

¤=

=

h2n d2n , e2n−1 (d − a)2n

bd2n h2n an (h+e)n (h−e)n (2d−a)n ,

h2n−1 d2n an−1 (h+e)n−1 (h−e)n (2d−a)n

¤=

f d2n h2n e2n (d−a)2n .

So, we can prove the other relations and the proof is completed. yn xn−2 Lemma 2. Every positive solution of the equation xn+1 = is bounded and lim xn = 0. n→∞ yn + yn−3 The following cases can be proved similarly.

4. ON THE SYSTEM XN+1 =

YN XN −2 YN +YN −3 ,

YN +1 =

In this section, we study the solutions of the system of the difference equations yn xn−2 xn yn−2 , yn+1 = , xn+1 = yn + yn−3 −xn + xn−3

XN YN −2 −XN +XN −3

(3)

where the initial values x−3 , x−2 , x−1 , x0 , y−3 , y−2 , y−1 , y0 are arbitrary nonzero real numbers with x−3 6= x0 , and y−3 6= −y0 .

Theorem 3. Let {xn , yn }+∞ n=−3 be solutions of system (3). Then for n = 0, 1, 2, ..., x6n−3

=

x6n−2

=

x6n−1

=

hn dn , + e)n bhn dn , an (h + e)n chn dn , an (h + e)n an−1 (h

1302

hn dn , + a)n f hn dn = n , e (−d + a)n ghn dn = n , e (−d + a)n

y6n−3 = y6n−2 y6n−1

en−1 (−d

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x6n

=

x6n+1

=

x6n+2

=

hn dn+1 , + e)n bdn hn+1 , n a (h + e)n+1 chn dn+1 , an+1 (h + e)n

dn hn+1 , + a)n f hn dn+1 y6n+1 = n , e (−d + a)n+1 gdn hn+1 y6n+2 = n+1 , e (−d + a)n y6n =

an (h

en (−d

Lemma 3. The system (3) has a periodic solutions of period 6 iff hd = e(a − d) = a(h + e).

Proof. First, if hd = e(a − d) = a(h + e), then from the form of the solutions of system (3), we see that x6n−3 y6n−3

= an (h + e)n an−1 (h + e)n = a , x6n−2 = b, x6n−1 = c, x6n = d, x6n+1 = hh + e, = e, y6n−2 = f, y6n−1 = g, y6n = h, y6n+1 = f da − d, y6n+2 = hge.

x6n+2 = cda,

Thus system (3) has a periodic solution with period 6. Second:if we have a period 6 then x6n−3

=

x6n

=

y6n−3

=

y6n−1

=

y6n+1

=

hn dn bhn dn chn dn = x = a, x = = x = b, x = = x−1 = c, −3 6n−2 −2 6n−1 an−1 (h + e)n an (h + e)n an (h + e)n hn dn+1 bdn hn+1 bh chn dn+1 cd , x6n+2 = n+1 = x0 = d, x6n+1 = n = x1 = = x2 = , n n n+1 n a (h + e) a (h + e) h+e a (h + e) a hn dn f hn dn = y−3 = e, y6n−2 = n = y−2 = f, en−1 (−d + a)n e (−d + a)n ghn dn dn hn+1 = y = g, y = = y0 = h, −1 6n en (−d + a)n en (−d + a)n f hn dn+1 fd gdn hn+1 gh , y , = y = = = y2 = 1 6n+2 n n+1 n+1 n e (−d + a) a−d e (−d + a) e

Then we get hd = a(h + e),

hd = e(a − d), and the proof is completed.

5. ON THE SYSTEM XN+1 =

YN XN −2 YN +YN −3 ,

YN +1 =

XN YN −2 −XN −XN −3

In this section,we study the solutions of the system of the difference equations xn+1 =

yn xn−2 , yn + yn−3

yn+1 =

xn yn−2 , −xn − xn−3

(4)

where the initial values x−3 , x−2 , x−1 , x0 , y−3 , y−2 , y−1 , y0 are arbitrary nonzero real numbers. Theorem 4. If {xn , yn } are solutions of difference equation system (4). Then for n = 0, 1, 2, ..., we have x12n−3

=

x12n−2

=

x12n−1

=

x12n

=

d2n h2n , a2n−1 (h + e)2n bd2n h2n , a2n (h + e)2n cd2n h2n , a2n (h + e)2n d2n+1 h2n , a2n (h + e)2n

(−1)n d2n h2n , en−1 (d + a)n (d − a)n (2h + e)n (−1)n f d2n h2n y12n−2 = n , e (d + a)n (d − a)n (2h + e)n (−1)n gd2n h2n y12n−1 = n , e (d + a)n (d − a)n (2h + e)n (−1)n d2n h2n+1 y12n = n , e (d + a)n (d − a)n (2h + e)n y12n−3 =

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x12n+1

=

x12n+2

=

x12n+3

=

x12n+4

=

x12n+5

=

x12n+6

=

x12n+7

=

x12n+8

=

bd2n h2n+1 , a2n (h + e)2n+1 −cd2n+1 h2n , a2n+1 (h + e)2n −d2n+1 h2n+1 , a2n (h + e)2n+1 bd2n+1 h2n+1 , 2n+1 a (h + e)2n+1 −cd2n+1 h2n+1 , 2n+1 a (h + e)2n+1 d2n+2 h2n+1 , 2n+1 a (h + e)2n+1 −bd2n+1 h2n+2 , 2n+1 a (h + e)2n+2 −cd2n+2 h2n+1 , 2n+2 a (h + e)2n+1

(−1)n+1 f d2n+1 h2n , en (d + a)n+1 (d − a)n (2h + e)n (−1)n+1 gd2n h2n+1 y12n+2 = n , e (d + a)n (d − a)n (2h + e)n+1 (−1)n+1 d2n+1 h2n+1 y12n+3 = n , e (d + a)n (d − a)n+1 (2h + e)n (−1)n+1 f d2n+1 h2n+1 y12n+4 = n+1 , e (d + a)n+1 (d − a)n (2h + e)n (−1)n gd2n+1 h2n+1 y12n+5 = n , e (d + a)n+1 (d − a)n (2h + e)n+1 (−1)n d2n+1 h2n+2 y12n+6 = n , e (d + a)n (d − a)n+1 (2h + e)n+1 (−1)n f d2n+2 h2n+1 y12n+7 = n+1 , e (d + a)n+1 (d − a)n+1 (2h + e)n (−1)n gd2n+1 h2n+2 y12n+8 = n+1 . e (d + a)n+1 (d − a)n (2h + e)n+1 y12n−3 =

6. NUMERICAL EXAMPLES Here, we consider interesting numerical examples in order to illustrate the results of the previous sections and to support our theoretical discussions. Example 1. We consider numerical example for the difference system (1) with the initial conditions x−3 = 2, x−2 = 14, x−1 = 6, x0 = 7, y−3 = 5, y−2 = 9, y−1 = 7 and y0 = −8. (See Fig. 1). plot of x(n+1)=x(n−2)y(n)/y(n)+y(n−3),y(n+1)=x(n)y(n−2)/x(n)+x(n−3)

40 x(n) y(n)

35 30 25

x(n),y(n)

20 15 10 5 0 −5 −10

0

5

10

15 n

20

25

30

Figure 1. Example 2. Assume for the system (2) with the initial conditions x−3 = 4, x−2 = 5, x−1 = 6, x0 = 3, y−3 = 1.8, y−2 = 9, y−1 = 2 and y0 = 1.9. See Figure (2).

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4

1.5

plot of x(n+1)=x(n−2)y(n)/y(n)+y(n−3),y(n+1)=x(n)y(n−2)/x(n)−x(n−3)

x 10

x(n) y(n) 1

x(n),y(n)

0.5

0

−0.5

−1

0

5

10

15

20 n

25

30

35

40

Figure 2. Example 3. Figure (3) shows the behavior of the solution of the difference system (3) with the initial conditions x−3 = 4, x−2 = 5, x−1 = 6, x0 = 10, y−3 = 8, y−2 = 9, y−1 = 2 and y0 = 2. plot of x(n+1)=x(n−2)y(n)/y(n)+y(n−3),y(n+1)=x(n)y(n−2)/x(n−3)−x(n)

15 x(n) y(n) 10

x(n),y(n)

5

0

−5

−10

−15

0

10

20

30

40

50

60

70

n

Figure 3. Example 4. We take the initial conditions, for the system (4), as follows x−3 = 3, x−2 = 5, x−1 = −9, x0 = 6, y−3 = 2, y−2 = 1.7, y−1 = 2.8 and y0 = 4. See Figure (4).

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plot of x(n+1)=x(n−2)y(n)/y(n)+y(n−3),y(n+1)=x(n)y(n−2)/−x(n−3)−x(n)

250 x(n) y(n)

200

x(n),y(n)

150 100 50 0 −50 −100

0

10

20

30 n

40

50

60

Figure 4.

Acknowledgements This article was funded by the Deanship of Scientific Research (DSR), King Abdulaziz University, Jeddah. The authors, therefore, acknowledge with thanks DSR technical and financial support.

References [1] M. Alghamdi, E. M. Elsayed and M. M. El-Dessoky, On the Solutions of Some Systems of Second Order Rational Difference Equations, Life Sci J., 10 (3) (2013), 344-351. [2] D. Clark, and M. R. S. Kulenovic, A coupled system of rational difference equations, Comput. Math. Appl., 43 (2002), 849-867. [3] C. Cinar, I. Yalçinkaya and R. Karatas, On the positive solutions of the difference equation system xn+1 = m/yn , yn+1 = pyn /xn−1 yn−1 , J. Inst. Math. Comp. Sci., 18 (2005), 135-136. [4] R. DeVault, E. A. Grove, G. Ladas, R. Levins, and C. Puccia, Oscillation and stability in models of a perennial grass, in Proceedings of the Dynamic Systems and Applications, vol. 1, pp. 87—93, Dynamic, Atlanta, Ga, USA, 1994. [5] Q. Din, and E. M. Elsayed, Stability analysis of a discrete ecological model, Computational Ecology and Software, 4 (2) (2014), 89—103. [6] Q. Din, M. N. Qureshi, and A. Q. Khan, Dynamics of a fourth-order system of rational difference equations, Advances in Difference Equations, 2012; 1: 1—15. [7] E. M. Elabbasy, H. El-Metwally and E. M. Elsayed, On the solutions of a class of difference equations systems, Demonstratio Mathematica, 41(1), (2008), 109-122.

1306

E. M. Elsayed et al 1298-1308

J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 22, NO.7, 2017, COPYRIGHT 2017 EUDOXUS PRESS, LLC

[8] H. El-Metwally and E. M. Elsayed, Form of solutions and periodicity for systems of difference equations, Journal of Computational Analysis and Applications, 15(5) (2013), 852-857. [9] E. M. Elsayed, Solutions of Rational Difference System of Order Two, Mathematical and Computer Modelling, 55 (2012), 378—384. [10] E. M. Elsayed, Behavior and Expression of the Solutions of Some Rational Difference Equations, Journal of Computational Analysis and Applications, 15 (1) (2013), 73-81. [11] E. M. Elsayed, On the solutions and periodic nature of some systems of difference equations, International Journal of Biomathematics, 7 (6) (2014), 1450067, (26 pages). [12] E. M. Elsayed, Solution for systems of difference equations of rational form of order two, Computational and Applied Mathematics, 33 (3) (2014), 751-765. [13] Y. Halim, N. Touafek and E. M. Elsayed, Closed form solutions of some systems of rational difference equations in terms of Fibonacci numbers, Dynamics of Continuous, Discrete and Impulsive Systems, Series A: Mathematical Analysis, 21 (2014), 473-486. [14] T. F. Ibrahim, and N. Touafek, Max-type system of difference equations with positive two-periodic sequences, Math. Meth. Appl. Sci. 2014, 37 2541—2553. [15] S. Kalabusic, M.R.S. Kulenovic and E. Pilav, Dynamics of a two-dimensional system of rational difference equations of Leslie—Gower type, Advances in Difference Equations, 2011, doi:10.1186/16871847-2011-29. [16] A. Q. Khan, M. N. Qureshi, and Q. Din, Qualitative behavior of two systems of second-order rational difference equations, Journal of Computational Analysis and Applications, 18 (6) (2015), 1027-1041. [17] V. Khuong, and T. Thai, Asymptotic Behavior of the Solutions of System of Difference Equations of Exponential Form, Journal of Difference Equations, Volume 2014, Article ID 936302, 6 pages. [18] M. R. S. Kulenovic and M. Pilling, Global Dynamics of a Certain Two-dimensional Competitive System of Rational Difference Equations with Quadratic Terms, Journal of Computational Analysis and Applications, 19 (1) (2015), 156-166. [19] A. S. Kurbanli, On the behavior of solutions of the system of rational difference equations, Disc. Dyn. Nat. Soc., Volume 2011, Article ID 932362, 12 pages. [20] A. S. Kurbanli, On the behavior of solutions of the system of rational difference equations, World Appl. Sci. J., 10 (11) (2010), 1344-1350. [21] A. S. Kurbanli, On the behavior of solutions of the system of rational difference equations, Adv. Differ. Equ., 2011, 2011:40. [22] A. S. Kurbanli, C. Cinar and M. Erdoan, On the periodicity of solutions of the system of rational difference equations, Appl. Math., 2(4) (2011), 410-413. [23] A. S. Kurbanli, C. Cinar and M. Erdo˘ gan, On the behavior of solutions of the system of rational xn−1 yn−1 xn , yn+1 = , zn+1 = , Applied Mathematdifference equations xn+1 = xn−1 yn − 1 yn−1 xn − 1 zn−1 yn ics, 2 (2011), 1031-1038. [24] A. S. Kurbanli, C. Cinar and I. Yalçınkaya, On the behavior of positive solutions of the system of rational difference equations, Math. Comput. Mod., 53 (2011), 1261-1267. [25] H. Ma and H. Feng, On Positive Solutions for the Rational Difference Equation Systems, International Scholarly Research Notices, Volume 2014, Article ID 857480, 4 pages. [26] M. Mansour, M. M. El-Dessoky and E. M. Elsayed, The form of the solutions and periodicity of some systems of difference equations, Discrete Dynamics in Nature and Society, Volume 2012, Article ID 406821, 17 pages. doi:10.1155/2012/406821. [27] M. Mansour, M. M. El-Dessoky and E. M. Elsayed, On the solution of rational systems of difference equations, J. Comp. Anal. Appl., 15 (5) (2013), 967-976.

1307

E. M. Elsayed et al 1298-1308

J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 22, NO.7, 2017, COPYRIGHT 2017 EUDOXUS PRESS, LLC

[28] A. Y. Ozban, On the system of rational difference equations xn+1 = a/yn−3 , yn+1 = byn−3 /xn−q yn−q , Appl. Math. Comp., 188(1) (2007), 833-837. [29] G. Papaschinopoulos and C. J. Schinas, On a system of two nonlinear difference equations, Journal of Mathematical Analysis and Applications, 219 (1998), 415—426. [30] M. N. Qureshi A. Q. Khan, and Q. Din, Global behavior of third order system of rational difference equations, International Journal of Engineering Research & Technology (IJERT), 2 (5) (2013), 2182-2191. [31] D. Simsek, and C. Cinar, Ono the solutions of the system of difference equations n B. Demir n o A yn A xn xn+1 = max xn , xn , yn+1 = max yn , yn , Discrete Dynamics in Nature and Society, Volume 2009, Article ID 325296, 11 pages. [32] S. Stevic, M. A. Alghamdi, A. Alotaibi, and E. M. Elsayed, Solvable product—type system of difference equations of second order, Electronic Journal of Differential Equations, Vol. 2015 (2015) (169), 1—20. [33] D. Tollu, Y. Yazlik, and N. Taskara, The Solutions of Four Riccati Difference Equations Associated with Fibonacci Numbers, Balkan J. Math., 2 (2014), 163-172. [34] N. Touafek and E. M. Elsayed, On the solutions of systems of rational difference equations, Math. Comput. Mod., 55 (2012), 1987—1997. [35] N. Touafek and E. M. Elsayed, On the periodicity of some systems of nonlinear difference equations, Bull. Math. Soc. Sci. Math. Roumanie, 55 (103) (2) (2012), 217—224. [36] N. Touafek and E. M. Elsayed, On a second order rational systems of difference equation, Hokkaido Mathematical Journal, 44 (1) (2015), 29—45. [37] I. Yalcinkaya, On the global asymptotic stability of a second-order system of difference equations, Disc. Dyn. Nat. Soc., Vol. 2008, Article ID 860152 (2008), 12 pages. [38] I. Yalçınkaya, On the global asymptotic behavior of a system of two nonlinear difference equations, ARS Combinatoria, 95 (2010), 151-159. [39] I. Yalcinkaya, C. Çinar and M. Atalay, On the solutions of systems of difference equations, Advances in Difference Equations, Vol. 2008, Article ID 143943: 9 pages. [40] Y. Yazlik, D. T. Tollu, and N .Taskara, On the Behaviour of Solutions for Some Systems of Difference Equations, Journal of Computational Analysis and Applications, 18 (1) (2015), 166-178. [41] Q. Zhang, J. Liu, and Z. Luo, Dynamical Behavior of a System of Third-Order Rational Difference Equation, Discrete Dynamics in Nature and Society, Volume 2015, Article ID 530453, 6 pages. [42] Y. Zhang, X. Yang, D. J. Evans and C. Zhu, On the nonlinear difference equation system, Comp. Math. Appl., 53 (2007), 1561—1566. [43] Y. Zhang, X. Yang, G. M. Megson and D. J. Evans, On the system of rational difference equations, Appl. Math. Comp., 176 (2006), 403—408. [44] Q. Zhang and W. Zhang, On a system of two high-order nonlinear difference equations, Advances in Mathematical Physics, 2014 (2014), Article ID 729273, 8 pages. [45] Q. Zhang, and W. Zhang, On the System of Nonlinear Rational Difference Equations, International Journal of Mathematical, Computational, Natural and Physical Engineering, 8 (4) (2014), 692-695.

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On the dynamics of higher Order di¤erence ®xn xn¡l equations xn+1 = axn + ¯xn+°x n¡k

M. M. El-Dessoky1;2 1 King Abdulaziz University, Faculty of Science, Mathematics Department, P. O. Box 80203, Jeddah 21589, Saudi Arabia. 2 Department of Mathematics, Faculty of Science, Mansoura University, Mansoura 35516, Egypt. E-mail: [email protected] Abstract The main objective of this paper is to study the global stability of the positive solutions and the periodic character of the di¤erence equation xn+1 = axn +

®xn xn¡l ; ¯xn + °xn¡k

n = 0; 1; :::;

where the parameters ®; ¯; ° and a are positive real numbers and the initial conditions x¡t ; x¡t+1 :::; x¡1 and x0 are positive real numbers where t = maxfl; kg. Numerical examples to the di¤erence equation are given to explain our results.

Keywords: di¤erence equations, stability, global stability, boundedness, periodic solutions. Mathematics Subject Classi…cation: 39A10 —————————————————

1

Introduction and Preliminaries

Our object in this paper is to study some qualitative behavior of the positive solutions of the di¤erence equation xn+1 = axn +

®xn xn¡l ; ¯xn +°xn¡k

n = 0; 1; :::;

(1)

1

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where the parameters ®; ¯; ° and ± are positive real numbers and the initial conditions x¡t ; x¡t+1 :::; x¡1 and x0 are positive real numbers where t = maxfl; kg. In addition, we obtain the solutions of some special cases of this equation. Many researchers have Many researchers have studied the behavior of the solution of di¤erence equations for example: Kalabuši´c et al. [1] studied the global character of the solution of the nonlinear rational di¤erence equation xn+1 =

¯xn¡l +±xn¡k ; Bxn¡l +Dxn¡k

n = 0; 1; :::;

with positive parameters and non-negative initial conditions. Cinar [2] studied the solutions of the following di¤erence equation xn+1 =

axn¡1 ; 1+bxn xn¡1

n = 0; 1; :::;

where a; b; x¡1 and x0 are non-negative real numbers. Yang et al. [3] studied the invariant intervals, the asymptotic behavior of the solutions, and the global attractivity of equilibrium points of the recursive sequence xn+1 =

axn¡1 +bxn¡2 ; c+dxn¡1 xn¡2

n = 0; 1; :::;

where a ¸ 0; b; c; d > 0. In [4] kenneth et al. got the global asymptotic stability for positive solutions to the di¤erence equation yn+1 =

yn¡k + yn¡m ; 1+yn¡k yn¡m

n = 0; 1; :::;

with y¡m ; y¡m+1 ; :::; y¡1 2 (0; 1) and 1 · k · m. Raafat [5] investigated the global asymptotic stability of all solutions of the difference equation n¡2 ; n = 0; 1; :::; xn+1 = B+CxAx n xn¡1 xn¡2 where A; B; C are positive real numbers and the initial conditions x¡2 ; x¡1 ; x0 are real numbers. Also, Raafat [6] introduced an explicit formula and discuss the global behavior of solutions of the di¤erence equation xn+1 =

axn¡3 ; b+cxn¡1 xn¡3

n = 0; 1; :::;

where a; b; c are positive real numbers and the initial conditions x¡3 ; x¡2 ; x¡1 ; x0 are real numbers. In [7] Elsayed studied the behavior of the solutions of the di¤erence equation xn+1 = axn¡1 +

bxn xn¡1 ; cxn +dxn¡2

n = 0; 1; :::;

where a; b; c are positive constant and the initial conditions x¡2 ; ; x¡1 ; x0 are arbitrary positive real numbers. 2

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Zayed et al. [8] investigated some qualitative behavior of the solutions of the di¤erence equation, xn+1 = °xn¡k +

axn +bxn¡k ; cxn ¡dxn¡k

n = 0; 1; :::;

where the coe¢cients °; a; b; c and d are positive constants and the initial conditions x¡k ; :::; x¡1 ; x0 are arbitrary positive real numbers, while k is a positive integer number. Other related results on rational di¤erence equations can be found in refs. [11] [24]. Let I be some interval of real numbers and let F : I t+1 ! I; be a continuously di¤erentiable function. Then for every set of initial conditions x¡t ; x¡t+1 ; :::; x0 2 I; the di¤erence equation (2)

xn+1 = F (xn ; xn¡1; :::; xn¡t ); n = 0; 1; :::; has a unique solution fxn g1 n=¡t .

De…nition 1 The linearized equation of the di¤erence equation (2) about the equilibrium x is the linear di¤erence equation yn+1 =

t X @F (x; x; :::; x) i=0

@xn¡i

(3)

yn¡i :

Now, assume that the characteristic equation associated with (3) is p(¸) = p0 ¸t + p1 ¸t¡1 + ::: + pt¡1 ¸+ pt = 0; where pi =

(4)

@F (x; x; :::; x) : @xn¡i

Theorem 1 [9]: Assume that pi 2 R; i = 1; 2; :::; t and t is non-negative integer. Then t X jpi j < 1; i=1

is a su¢cient condition for the asymptotic stability of the di¤erence equation xn+t + p1 xn+t¡1 + ::: + pt xn = 0; n = 0; 1; ::: : 3

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Theorem 2 [10, 11]: Let g : [a; b]t+1 ! [a; b], be a continuous function, where t is a positive integer, and where [a; b] is an interval of real numbers. Consider the di¤erence equation xn+1 = g(xn ; xn¡1 ; :::; xn¡t );

(5)

n = 0; 1; ::: :

Suppose that g satis…es the following conditions. (1) For each integer i with 1 · i · t + 1; the function g(z1 ; z2 ; :::; zt+1 ) is weakly monotonic in zi for …xed z1 ; z2 ; :::; zi¡1 ; zi+1 ; :::; zt+1 : (2) If m; M is a solution of the system m = g(m1 ; m2 ; :::; mt+1 );

M = g(M1 ; M2 ; :::; Mt+1 );

then m = M , where for each i = 1; 2; :::; t + 1; we set ½ ¾ m; if g is non-decreasing in zi ; ; mi = M; if g is non-increasing in zi ; and Mi =

½

M; m;

¾ if g is non-decreasing in zi ; : if g is non-increasing in zi .

Then there exists exactly one equilibrium point x¹ of Equation (5), and every solution of Equation (5) converges to x¹.

2 2.1

Stability of the Equilibrium Point of Eq. (1) Local stability

In this subsection, we study the local stability character of the equilibrium point of Eq. (1). Eq. (1) has equilibrium point and is given by x¹ = a¹ x+

®¹ x2 ; ¯¹ x+°¹ x

or

((1 ¡ a) (¯+ °) ¡ ®) x¹2 = 0;

if (1 ¡ a) (¯+ °) 6= ®, then the unique equilibrium point is x = 0: 2® Theorem 3 Assume that a + ¯+° < 1;then equilibrium x of Eq. (1) is locally asymptotically stable.

Proof: Let f : (0; 1)3 ¡! (0; 1) be a continuous function de…ned by f (v0 ; v1 ; v2 ) = av0 +

®v0 v1 : ¯v0 + °v2

(6)

4

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Therefore, it follows that @f (v0 ; v1 ; v2 ) @v0

= a+

®v1 (¯v0 +°v2 )¡®¯v0 v1 (¯v0 +°v2 )2

@f (v0 ; v1 ; v2 ) @v1 @f (v0 ; v1 ; v2 ) @v2

=

®v0 ; ¯v0 +°v2

=

¡®v0 v1 (¯v0 +°v2 )2

=a+

®¯v12 ; (¯v0 +°v2 )2

0 v1 = ¡ (¯v®°v : +°v )2 0

2

Then, we see that @f (¹ x; x ¹; x ¹) @v0

= a+

@f (¹ x; x ¹; x ¹) @v1

®¯ ; (¯+°)2

=

@f(¹ x; x ¹; x ¹) @v2

® ; ¯+°

®° = ¡ (¯+°) 2:

and the linearized equation of Eq. (1) about x¹; is ³ ³ ³ ´ ´ ´ ®¯ ¡®° ® yn+1 = a + (¯+°) + + y y yn¡k ; 2 n n¡l ¯+° (¯+°)2

Under the conditions, we get ¯ ¯ ¯a +

and so

¯

®¯ ¯ ¯ (¯+°)2

¯ ¯ ¯ ¯ ¯ ® ¯ ¯ ¡®° ¯ + ¯ ¯+°¯ + ¯ (¯+°)2 ¯ < 1;

a+

2® ¯+°

< 1:

According to Theorem 1, the proof is complete. Example 1. The solution of the di¤erence equation (1) is local stability if l = 2; k = 3; ® = 0:1; ¯ = 0:2; ° = 1; a = 0:2 and the initial conditions x¡3 = 0:6; x¡2 = 0:3; x¡1 = 0:4 and x0 = 0:8 (See Fig. 1). plot of x(n+1)= a X(n)+(alpha X(n) X(n-l)/(beta X(n)+gamma X(n-k))) 0.8

0.7

0.6

x(n)

0.5

0.4

0.3

0.2

0.1

0

0

5

10

15

20

25 n

30

35

40

45

50

Fig. 1. Plot the behavior of the solution of equation (1).

5

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Example 2. See Figure (2) when we take the di¤erence equation (1) with l = 2; k = 3; ® = 1; ¯ = 0:2; ° = 0:4; a = 0:5 and the initial conditions x¡3 = 0:6; x¡2 = 0:3; x¡1 = 0:4 and x0 = 0:8. 12

x 10

7

plot of x(n+1)= a X(n)+(alpha X(n) X(n-l)/(beta X(n)+gamma X(n-k)))

10

8

x(n)

6

4

2

0

-2

0

5

10

15

20

25 n

30

35

40

45

50

Fig. 2. Draw the behavior of the solution of equation (1).

2.2

Global Stability

In this subsection we study the global stability of the positive solutions of Eq. (1). Theorem 4 The equilibrium point x¹ is a global attractor of equation (1) if (1 ¡ a) (¯¡ °) 6= ®: Proof. Let r; s be nonnegative real numbers and assume that h : [r; s]3 ! [r; s] be a function de…ned by h(v0 ; v1 ; v2 ) = av0 +

®v0 v1 : ¯v0 +°v2

Then @h(v0 ; v1 ; v2 ) @v0

=a+

®¯v12 ; v1 ; v2 ) ; @h(v0@v (¯v0 +°v2 )2 1

=

®v0 ¯v0 +°v2

and

@h(v0 ; v1 ; v2 ) @v2

0 v1 = ¡ (¯v®°v : +°v )2 0

2

We can see that the function h(v0 ; v1 ; v2 ) increasing in v0 ; v1 and decreasing in v2 . Suppose that (m; M ) is a solution of the system M = h(M; M; m)

and

m = h(m; m; M):

Then from Equation (1), we see that M = aM +

®M 2 ; ¯M +°m

m = am +

®m2 ; ¯m+°M

then ¯(1 ¡ a) M + °(1 ¡ a) m = ®M; ¯(1 ¡ a) m + °(1 ¡ a) M = ®m; 6

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Subtracting this two equations, we obtain ((1 ¡ a) (¯¡ °) ¡ ®) (M ¡ m) = 0; under the condition (1 ¡ a) (¯¡ °) 6= ®; we see that M = m: It follows from Theorem 2 that x¹ is a global attractor of Equation (1). Example 3. The solution of the di¤erence equation (1) is global stability if l = 2; k = 3; ® = 0:01; ¯ = 0:2; °= 0:4; a = 0:1 and the initial conditions x¡3 = 0:6; x¡2 = 0:3; x¡1 = 0:4 and x0 = 0:8 (See Fig. 3). plot of x(n+1)= a X(n)+(alpha X(n) X(n-l)/(beta X(n)+gamma X(n-k))) 0.8

0.7

0.6

x(n)

0.5

0.4

0.3

0.2

0.1

0

0

10

20

30 n

40

50

60

Fig. 3. Sketch the behavior of the solution of Eq. (1).

3

Boundedness of Solutions of Equation (1)

In this section we investigate the boundedness nature of the solutions of Equation (1). Theorem 5 Every solution of Equation (1) is bounded if a < 1: Proof. Let fxn g1 n=¡m be a solution of Equation (1). It follows from Equation (1) that ³ ´ ®xn xn¡l n xn¡l ® xn+1 = axn + ¯x®x xn¡l : · ax + = ax + n n ¯xn ¯ n +°xn¡k By using a comparison, we can right hand side as follows ³ ´ tn+1 = atn + ®¯ tn¡l :

and this equation is locally asymptotically stable if a < 1; and converges to the equilibrium point t = 0: Therefore lim sup xn 6 0:

n!1

7

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Example 4. Figure (4) shows that l = 4; k = 3; ® = 0:1; ¯ = 0:2; °= 0:4; a = 1:3; the solution of the di¤erence equation (1) with initial conditions x¡3 = 0:6; x¡2 = 0:3; x¡1 = 0:4 and x0 = 0:8 is unbounded. 7

x 10

8

plot of x(n+1)= a X(n)+(alpha X(n) X(n-l)/(beta X(n)+gamma X(n-k)))

6

5

x(n)

4

3

2

1

0

0

10

20

30 n

40

50

60

Fig. 4. Polt the behavior of the solution of equation (1) when a > 1.

4

Existence of Periodic Solutions

In this section we investigate the existence of periodic solutions of Eq. (5). Theorem 6 Equation (1) has no prime period two solutions if l and k are even when a + ® 6= 0 and ¯+ °6= 0. Proof. Suppose that there exists a prime period two solution :::p; q; p; q; :::; of Equation (1). We see from Equation (1) when l and k are even that p = aq +

®q2 ; ¯q + °q

q = ap +

®p2 : ¯p + °p

(¯+ °) pq = a (¯+ °) q2 + ®q 2 ; (¯+ °) pq = a (¯+ °) p2 + ®p2

(7) (8)

Subtracting (7) from (8) gives (a + ®) (¯+ °) (p2 ¡ q2 ) = 0; Since a + ® 6= 0 and ¯+ °6= 0, then p = q. This is a contradiction. Thus, the proof is completed. Theorem 7 Equation (1) has no prime period two solutions if l and k are odd when °6= a¯. Theorem 8 Equation (1) has no prime period two solutions if l is an even and k is an odd when ®+ °6= a¯. Theorem 9 Equation (1) has no prime period two solutions if l is an odd and k is an even when a (¯+ °) = 6 0. 8

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5

Special Cases of Equation (1)

5.1

First Equation When l = k = 1; a = 0 and ® = ¯ = ° = 1:

In this subsection we study the following special case of Eq. (1) xn+1 =

xn xn¡1 ; xn +xn¡1

(9)

where the initial conditions are arbitrary non zero real numbers. Theorem 10 Let fxn g1 n=¡1 be a solution of Eq. (9). Then for n = 0; 1; 2; ::: xn =

cb ; fn b+fn+1 c

where x¡1 = c; x0 = b; ffn g1 n=1 = f1; 1; 2; 3; 5; 8; 13; :::g f0 = 0 and f¡1 = 1. Proof: For n = 0 the result holds. Now suppose that n > 0 and that our assumption holds for n ¡ 1 and n. Now, it follows xn¡2 =

cb fn¡2 b+fn¡1 c

and xn¡1 =

cb : fn¡1 b+fn c

Now, it follows from Eq. (9) that xn+1 = =

xn xn¡1 xn +xn¡1

=

³

´³

´

cb cb f b+fn+1 c f b+fn c ³ n ´ ³ n¡1 ´ cb cb + fn b+fn+1 c fn¡1 b+fn c

c 2 b2 cb(fn¡1 b+fn c)+cb(fn b+fn+1 c)

=

=

µ

µ

c2 b 2 (fn b+fn+1 c)(fn¡1 b+fn c)

¶

cb(fn¡1 b+fn c)+cb(fn b+fn+1 c)

(fn b+fn+1 c)(fn¡1 b+fn c)

cb (fn¡1 +fn )b+(fn +fn+1 )c

=

¶

cb : fn+1 b+fn+2 c

Thus, the proof is completed.

5.2

Second Equation When l = k = 1; a = 0; ® = ¯ = 1 and °= ¡1:

In this subsection we study the following special case of Eq. (1) xn+1 =

xn xn¡1 ; xn ¡xn¡1

(10)

where the initial conditions are arbitrary non zero real numbers. Theorem 11 Let fxn g1 n=¡1 be a solution of Eq. (10). Then for n = 0; 1; 2; ::: xn =

(¡1)n+1 cb ; fn b¡fn+1 c

where x¡1 = c; x0 = b; and ffn g1 n=¡1 = f1; 0; 1; 1; 2; 3; 5; 8; 13; :::g: 9

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Proof: For n = 0 the result holds. Now suppose that n > 0 and that our assumption holds for n ¡ 1 and n. Now, it follows xn¡2 =

(¡1)n¡1 cb fn¡2 b¡fn¡1 c

(¡1)n cb : fn¡1 b¡fn c

and xn¡1 =

Now, it follows from Eq. (10) that xn+1 = =

xn xn¡1 xn ¡xn¡1

=

µ

¶³

(¡1)n+1 cb fn b¡fn+1 c ³ ´ (¡1)n+1 cb fn b¡fn+1 c

(¡1)n cb fn¡1 b¡fn c

³ ¡ f

(¡1)2n+2 c2 b2 cb(fn¡1 b¡fn c)+cb(fn b¡fn+1 c)

´

(¡1)n cb n¡1 b¡fn c

=

=

´

µ

µ

(¡1)2n+1 c2 b2 (fn b¡fn+1 c)(fn¡1 b¡fn c)

¶

¡cb(fn¡1 b¡fn c)¡cb(fn b¡fn+1 c)

(fn b¡fn+1 c)(fn¡1 b¡fn c)

(¡1)n+2 cb (fn¡1 +fn )b¡(fn+1 +fn )c

=

(¡1)n+2 cb : fn+1 b¡fn+2 c

¶

Thus, the proof is completed.

5.3

Third Equation When l = k = 1; a = 0; ¯ = ¡1:

® = ° = 1 and

In this subsection we study the following special case of Eq. (1) xn+1 =

xn xn¡1 ; ¡xn +xn¡1

(11)

where the initial conditions are arbitrary non zero real numbers. Theorem 12 Let fxn g1 n=¡1 be a solution of Eq. (11). Then for n = 0; 1; 2; ::: x3n¡1 = (¡1)n c; x3n = (¡1)n b; and x3n+1 =

(¡1)n+1 bc ; b¡c

where x¡1 = c; x0 = b: Proof: For n = 0 the result holds. Now suppose that n > 0 and that our assumption holds for n ¡ 1 and n. Now, it follows x3n¡4 = (¡1)n¡1 c; x3n¡3 = (¡1)n¡1 b; and x3n¡2 =

(¡1)n bc : b¡c

Now, it follows from Eq. (11) that x3n+2 = x3n =

x3n+1 x3n ¡x3n+1 +x3n

=

x3n¡1 x3n¡2 ¡x3n¡1 +x3n¡2

µ

(¡1)n+1 bc b¡c

¶

((¡1)n b) ´ ³ (¡1)n+1 bc +(¡1)n b ¡ b¡c ´ ³ (¡1)n bc ((¡1)n c) b¡c

=

n

¡(¡1) c+(

and x3n+4 =

x3n x3n¡1 ¡x3n +x3n¡1

=

(¡1)n bc b¡c

)

= =

(¡1)n+1

(

³

bc2 b¡c bc ¡c+ b¡c

((¡1)n b)((¡1)n c) ¡(¡1)n b+(¡1)n c

=

)

´

´

=

b2 c b¡c

bc +b b¡c

(¡1)n

(

³

)

(¡1)n bc ¡(b¡c)

=

(¡1)n+1 b2 c b2

(¡1)n bc2 c2

=

= (¡1)n+1 c;

= (¡1)n b;

(¡1)n+1 bc : b¡c

Thus, the proof is completed. 10

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Theorem 13 Let fxn g1 n=¡1 be a solution of Eq. (11). Then every solution of Eq. (11) is a periodic with period six. Moreover fxn g1 n=¡1 takes the form form © ª bc bc bc bc fxn g = c; b; ¡ b¡c ; ¡ c; ¡ b; b¡c ; c; b; ¡ b¡c ; ¡ c; ¡ b; b¡c ; ::: : where x¡1 = c; x0 = b:

Example 5. Figure (5) shows the solution of Eq. (11) when the initial conditions x¡1 = 0:3 and x0 = 0:6. plot of x(n+1)= X(n) X(n-1)/( - X(n)+ X(n-1)) 0.6

0.4

0.2

x(n)

0

-0.2

-0.4

-0.6

-0.8

0

10

20

30 n

40

50

60

Fig. 5. Draw the solution of equation (11) has a periodic with period six.

5.4

Fourth Equation When l = k = 1; a = 0; ¯ = ° = 1 and ® = ¡1:

In this subsection we study the following special case of Eq. (1) xn¡1 xn+1 = ¡ xxnn+x ; n¡1

(12)

where the initial conditions are arbitrary non zero real numbers. Theorem 14 Let fxn g1 n=¡1 be a solution of Eq. (12). Then for n = 0; 1; 2; ::: bc x3n¡1 = c; x3n = b; and x3n+1 = ¡ b+c ;

where x¡1 = c; x0 = b: Proof: For n = 0 the result holds. Now suppose that n > 0 and that our assumption holds for n ¡ 1 and n. Now, it follows bc x3n¡4 = c; x3n¡3 = b; and x3n¡2 = ¡ b+c :

11

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Now, it follows from Eq. (12) that 3n+1 x3n ¡ xx3n+1 +x3n

x3n+2 =

3n¡1 x3n¡2 ¡ xx3n¡1 +x3n¡2

x3n =

³

´

2

b c ¡ b+c (¡ bc )(b) ³ ´ = = ¡ ¡ b+c = ¡ bc ¡bc+b2 +bc ( b+c )+b b+c

=¡

bc (c)(¡ b¡c )

c+( ¡bc b¡c )

=

³

´

bc2 b¡c ³ ´ bc+c2 ¡bc b¡c

=

b2 c b2

bc2 c2

= c;

= b;

and x3n+4 =

x3n x3n¡1 ¡x3n +x3n¡1

bc = ¡ b+c :

Thus, the proof is completed. Theorem 15 Let fxn g1 n=¡1 be a solution of Eq. (12). Then every solution of Eq. (12) is a periodic with period three. Moreover fxn g1 n=¡1 takes the form form © ª bc bc bc fxn g = c; b; ¡ b+c ; c; b; ¡ b+c ; c; b; ¡ b+c ; ::: ;

where x¡1 = c; x0 = b:

Example 6. The solution of Eq. (12) when the initial conditions x¡1 = 0:3 and x0 = 0:6 (See Fig. 6). plot of x(n+1)= - X(n) X(n-1) / ( X(n) + X(n-1)) 0.6

0.5

0.4

x(n)

0.3

0.2

0.1

0

-0.1

-0.2

0

10

20

30 n

40

50

60

Fig. 6. Polt the solution of equation (12) has a periodic with period three.

Acknowledgements This article was funded by the Deanship of Scienti…c Research (DSR), King Abdulaziz University, Jeddah. The authors, therefore, acknowledge with thanks DSR technical and …nancial support.

12

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References [1] S. Kalabuši´c, M. R. S. Kulenovi´c and C. B. Overdeep, Dynamics of the Recursive ¯xn¡l + ±xn¡k Sequence xn+1 = , J. Di¤erence Equ. Appl. 10(10), (2004), 915Bxn¡l + Dxn¡k 928. [2] C. Cinar, On the positive solutions of the di¤erence equation xn+1 = axn¡1 ; Appl. Math. Comp., 156 (2004), 587-590. 1 + bxn xn¡1 [3] X. Yang, W. Su, B. Chen, G. M. Megson and D. J. Evans, On the recursive axn¡1 + bxn¡2 sequence xn+1 = , Appl. Math. Comp., 162 (2005), 1485–1497. c + dxn¡1 xn¡2 [4] Kenneth S. Berenhaut, John D. Foley, Stevo Stevi´c, the global attractivity of the n¡k + yn¡m , Appl. Math. Lett., 20, (2007), rational di¤erence equation yn+1 = y1+y n¡k yn¡m 54-58. [5] R. Abo-Zeid, On the oscillation of a third order rational di¤erence equation, J. Egypt. Math. Soc., 23, (2015), 62–66. [6] R. Abo-Zeid, Global Behavior of the di¤erence equation xn+1 = Archivun Mathematicum (BRNO), Tomus 51, (2015), 77-85.

axn¡3 , b+cxn¡1 xn¡3

[7] E. M. Elsyayed, Solution and attractivity for a rational recuursive Sequence, Discrete Dyn. Nat. Soc., Vol. 2011, (2011), Article ID 982309, 17 pages. [8] E. M. E. Zayed, M. A. El-Moneam, On the Rational Recursive Sequence Xn+1 = n +bXn¡k °Xn¡k + aX , Bulletin of the Iranian Mathematical Society, Vol. 36 (1), cXn ¡dXn¡k (2010), 103-115. [9] V. L. Kocic and G. Ladas, Global Behavior of Nonlinear Di¤erence Equations of Higher Order with Applications, Kluwer Academic Publishers, Dordrecht, 1993. [10] E. A. Grove, G. Ladas, Periodicities in Nonlinear Di¤erence Equations, Chapman & Hall/CRC, London/Boca Raton, 2005. [11] M. R. S. Kulenovic and G. Ladas, Dynamics of Second Order Rational Di¤erence Equations with Open Problems and Conjectures, Chapman & Hall / CRC Press, 2002. [12] X. Yang, On the global asymptotic stability of the di¤erence equation xn+1 = xn¡1xn¡2 + xn¡3 + a , Appl. Math. Comp., 171(2) (2005), 857-861. xn¡1 + xn¡2 xn¡3 + a

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[13] Bratislav D. Iriµcanin, On a Higher-Order Nonlinear Di¤erence Equation, Abstr. Appl. Anal., Vol. 2010, (2010), Article ID 418273, 8 pages. [14] M. M. El-Dessoky, Qualitative behavior of rational di¤erence equation of big Order, Discrete Dyn. Nat. Soc., Vol. 2013, (2013), Article ID 495838, 6 pages. [15] Mehmet G½um½u¸s, The Periodicity of Positive Solutions of the Nonlinear Di¤erence xp Equation xn+1 = ® + n¡k p , Discrete Dyn. Nat. Soc., Vol. 2013, (2013), Article xn ID 742912, 3 pages. [16] R. Abo-Zeid, Attractivity of two nonlinear third order di¤erence equations, J. Egypt. Math. Soc., 21(3), (2013), 241–247. [17] Wanping Liu and Xiaofan Yang, Quantitative Bounds for Positive Solutions of a Stevi´c Di¤erence Equation, Discrete Dyn. Nat. Soc., Vol. 2010, (2010), Article ID 235808, 14 pages. [18] S. Ebru Das, Dynamics of a nonlinear rationaldi¤erence equation, Hacettepe Journal of Mathematics and Statistics, Vol. 42(1), (2013), 9-14. [19] R. Karatas, C. Cinar and D. Simsek, On positive solutions of the di¤erence xn¡5 ; Int. J. Contemp. Math. Sci., Vol. 1, 2006, no. equation xn+1 = 1 + xn¡2 xn¡5 10, 495-500. [20] R. Abo-Zeid, Global asymptotic stability of a second order rational di¤erence equation, J. Appl. Math. & Inform. 2 (3) (2010), 797–804. ½ ½ [21] Mehmet G½um½u¸s and Ozkan Ocalan, Global Asymptotic Stability of a Nonautonomous Di¤erence Equation, J. Appl. Math., Vol. 2014, (2014), Article ID 395954, 5 pages. [22] E. M. Elsayed and M. M. El-Dessoky, Dynamics and behavior of a higher order rational recursive sequence, Adv. Di¤er. Equ., 2012, (2012), 69. [23] M. A. El-Moneam, On the dynamics of the higher order nonlinear rational difference equation, Math. Sci. Lett., 3(2), (2014), 121–129. [24] X. Yan and W. Li , Global attractivity in the recursive sequence xn+1 = ®¡ ¯xn , Appl. Math. Comp., 138(2-3) (2003), 415-423. °¡ xn¡1 [25] E. Camouzis, G. Ladas and H. D. Voulov, On the dynamics of xn+1 = ®+ °xn¡1 + ±xn¡2 , J. Di¤er Equations Appl., 9 (8) (2003), 731-738. A + xn¡2 [26] Art¯ uras Dubickas, Rational di¤erence equations with positive equilibrium point, Bull. Korean Math. Soc. 47(3), (2010), 645–651. 14

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Applications of soft sets in BF -algebras Jeong Soon Han1 and Sun Shin Ahn2,∗ 1 2

Department of Applied Mathematics, Hanyang University, Ahnsan 15588, Korea

Department of Mathematics Education, Dongguk University, Seoul 04620, Korea

Abstract. The aim of this article is to lay a foundation for providing a soft algebraic tool in considering many problems that contain uncertainties. In order to provide these soft algebraic structures, the notion of an intersectional soft subalgebra and an intersectional soft normal subalgebra of a BF -algebra are introduced, and related properties are investigated. A quotient structure of a BF -algebra using an intersectional soft normal subalgebra is constructed. The fundamental homomorphism of a quotient BF -algebra is established.

1. Introduction The real world is inherently uncertain, imprecise and vague. Various problems in system identification involve characteristics which are essentially non-probabilistic in nature [14]. In response to this situation Zadeh [15] introduced fuzzy set theory as an alternative to probability theory. Uncertainty is an attribute of information. In order to suggest a more general framework, the approach to uncertainty is outlined by Zadeh [16]. To solve complicated problem in economics, engineering, and environment, we can’t successfully use classical methods because of various uncertainties typical for those problems. There are three theories: theory of probability, theory of fuzzy sets, and the interval mathematics which we can consider as mathematical tools for dealing with uncertainties. But all these theories have their own difficulties. Uncertainties can’t be handled using traditional mathematical tools but may be dealt with using a wide range of existing theories such as probability theory, theory of (intuitionistic) fuzzy sets, theory of vague sets, theory of interval mathematics, and theory of rough sets. However, all of these theories have their own difficulties which are pointed out in [11]. Maji et al. [10] and Molodtsov [11] suggested that one reason for these difficulties may be due to the inadequacy of the parametrization tool of the theory. To overcome these difficulties, Molodtsov [11] introduced the concept of soft set as a new mathematical tool for dealing with uncertainties that is free from the difficulties that have troubled the usual theoretical approaches. Molodtsov pointed out several directions for the applications of soft sets. Worldwide, there has been a rapid growth in interest in soft set theory and its applications in recent years. Evidence of this can be found in the increasing number of high-quality articles on soft sets and related topics that have been published in a variety of international journals, symposia, workshops, and international conferences in recent years. 0

2010 Mathematics Subject Classification: 06F35; 03G25; 06D72. Keywords: γ-inclusive set, int-soft (normal) subalgebra, BF -algebra. The corresponding author. Tel: +82 2 2260 3410, Fax: +82 2 2266 3409 0 E-mail: [email protected] (J. S. Han); [email protected] (S. S. Ahn) 0

∗

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2

Maji et al. [10] described the application of soft set theory to a decision making problem. Maji et al. [9] also studied several operations on the theory of soft sets. Akta¸s and C ¸ a˘gman [2] studied the basic concepts of soft set theory, and compared soft sets to fuzzy and rough sets, providing examples to clarify their differences. They also discussed the notion of soft groups. Jun [7] discussed the union soft sets with applications in BCK/BCI-algebras. We refer the reader to the papers [1, 3, 5, 6, 13] for further information regarding algebraic structures/properties of soft set theory. In this paper, we discuss applications of the an intersectional soft sets in a (normal) subalgebra of a BF -algebra. We introduce the notion of an intersectional (normal) soft subalgebra of a BF -algebra, and investigated related properties. We consider a new construction of a quotient BF -algebra induced by an int-soft normal subalgebra. Also we establish the fundamental homomorphism of a quotient BF -algebra. 2. Preliminaries We review some definitions and properties that will be useful in our results (see [12]). By a BF -algebra we mean an algebra (X, ∗, 0) of type (2,0) satisfying the following conditions: (B1) x ∗ x = 0, (B2) x ∗ 0 = x, (B3) 0 ∗ (x ∗ y) = y ∗ x for all x, y ∈ X. A BF -algebra (X, ∗, 0) is called a BF1 -algebra if it satisfies the following identity: (BG) x = (x ∗ y) ∗ (0 ∗ y) for all x, y ∈ X. A BF -algebra (X, ∗, 0) is called a BF2 -algebra if it satisfies the following identity: (BH) x ∗ y = y ∗ x = 0 imply x = y for all x, y ∈ X. For brevity, we also call X a BF -algebra. If we can define a binary operation “ ≤ ” by x ≤ y if and only if x ∗ y = 0. A non-empty subset A of a BF -algebra X is called a subalgebra of X if x ∗ y ∈ A for any x, y ∈ A. A non-empty subset A of a BF -algebra X is said to be normal (or normal subalgebra) ([8]) of X if (x ∗ a) ∗ (y ∗ b) ∈ A for any x ∗ y, a ∗ b ∈ A. Note that any normal subalgebra A of a BF -algebra X is a subalgebra of X, but the converse need not be true. A mapping f : X → Y of BF -algebras is called a homomorphism if f (x∗y) = f (x)∗f (y) for all x, y ∈ X. Lemma 2.1. If X is a BF -algebra, then (i) 0 ∗ (0 ∗ x) = x, for all x ∈ X. (ii) 0 ∗ x = 0 ∗ y implied x = y for any x, y ∈ X. (iii) if x ∗ y = 0, then y ∗ x = 0 for any x, y ∈ X. Lemma 2.2. Let X be a BF -algebra and let N be a subalgebra of X. If x ∗ y ∈ N for any x, y ∈ N , then y ∗ x ∈ N.

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A BG-algebra (X; ∗, 0) is an algebra of type (2, 0) satisfying (B1), (B2) and (BG). Theorem 2.3 Let X be a BF1 -algebra. Then (i) X is a BG-algebra. (ii) x ∗ y = 0 implies x = y for any x, y ∈ X. (iii) The right cancellation law holds in X, i.e., if x ∗ y = z ∗ y, then x = z for any x, y, z ∈ X. (iv) The left cancellation law holds in X, i.e., if y ∗ x = y ∗ z, then x = z for any x, y, z ∈ X. Molodtsov [11] defined the soft set in the following way: Let U be an initial universe set and let E be a set of parameters. We say that the pair (U, E) is a soft universe. Let P(U ) denotes the power set of U and A, B, C, · · · ⊆ E. A fair (f˜, A) is called a soft set over U , where f˜ is a mapping given by f˜ : X → P(U ). In other words, a soft set over U is parameterized family of subsets of the universe U . For ε ∈ A, f˜(ε) may be considered as the set of ε-approximate elements of the set (f˜, A). A soft set over U can bd represented by the set of ordered pairs: (f˜, A) = {(x, f˜(x))|x ∈ A, f˜(x) ∈ P(U )}, where f˜ : X → P(U ) such that f˜(x) = ∅ if x ∈ / A. Clearly, a soft set is not a set. ˜ For a soft set (f , A) of X and a subset γ of U , the γ-inclusive set of (f˜, A), defined to be the set iA (f˜; γ) := {x ∈ A|γ ⊆ f˜(x)}.

3. Intersectional soft subalgebras In what follows let X denote a BF -algebra X unless otherwise specified. Definition 3.1. A soft set (f˜, X) over U is called an intersectional soft subalgebra (briefly, int-soft subalgebra of X if it satisfies: (3.1) f˜(x) ∩ f˜(y) ⊆ f˜(x ∗ y) for all x, y ∈ X. Proposition 3.2. Every int-soft subalgebra (f˜, X) of a BF -algebra X satisfies the following inclusion: (3.2) f˜(x) ⊆ f˜(0) for all x ∈ X. Proof. Using (3.1) and (B1), we have f˜(x) = f˜(x) ∩ f˜(x) ⊆ f˜(x ∗ x) = f˜(0) for all x ∈ X. Example 3.3. Let (U = Z, X) where X = {0, 1, 2, 3} is a BF -algebra ([12]) with the following Cayley table: ∗ 0 1 2 3

0 0 1 2 3

1 1 0 1 1

2 2 1 0 1

3 3 1 1 0

Let (f˜, X) be a soft set over U defined as follows:

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Jeong Soon Han and Sun Shin Ahn   Z ˜ f : X → P(U ), x 7→ 2Z  3Z

if x = 0 if x ∈ {1, 2} if x = 3.

It is easy to check that (f˜, X) is an int-soft subalgebra over U . Theorem 3.4. A soft set (f˜, X) of a BF -algebra X over U is an int-soft subalgebra of X over U if and only if the γ-inclusive set iX (f˜; γ) is a subalgebra of X for all γ ∈ P(U ) with iX (f˜; γ) ̸= ∅. The subalgebra iX (f˜; γ) in Theorem 3.4 is called the inclusive subalgebra of X. Proof. Assume that (f˜, X) is an int-soft subalgebra over U . Let x, y ∈ X and γ ∈ P(U ) be such that x, y ∈ iX (f˜; γ). Then γ ⊆ f˜(x) and γ ⊆ f˜(y). It follows from (3.1) that γ ⊆ f˜(x) ∩ f˜(y) ⊆ f˜(x ∗ y) Hence x ∗ y ∈ iX (f˜; γ). Thus iX (f˜, X) is a subalgebra of X. Conversely, suppose that iX (f˜; γ) is a subalgebra X for all γ ∈ P(U ) with iX (; γ) ̸= ∅. Let x, y ∈ X, be such that f˜(x) = γx and f˜(y) = γy . Take γ = γx ∩ γy . Then x, y ∈ iX (f˜; γ) and so x ∗ y ∈ iX (f˜; γ) by assumption. Hence f˜(x) ∩ f˜(y) = γx ∩ γy = γ ⊆ f˜(x ∗ y). Thus (f˜, X) is an int-soft subalgebra over U . □ Theorem 3.5. Every subalgebra of a BF -algebra can be represented as a γ-inclusive set of an int-soft subalgebra. Proof. Let A be a subalgebra of a BF -algebra X. For a subset γ of U , define a soft set (f˜, X) over U by { γ if x ∈ A ˜ f : X → P(U ), x 7→ ∅ if x ∈ /A Obviously, A = iX (f˜; γ). We now prove that (f˜; γ) is an int-soft subalgebra over U . Let x, y ∈ X. If x, y ∈ A, then x ∗ y ∈ A because A is a subalgebra of X. Hence f˜(x) = f˜(y) = f˜(x ∗ y) = γ, and so f˜(x) ∩ f˜(y) ⊆ f˜(x ∗ y). If x ∈ A and y ∈ / A, then f˜(x) = γ and f˜(y) = ∅ which imply that f˜(x) ∩ f˜(y) = γ ∩ ∅ = ∅ ⊆ f˜(x ∗ y). Similarly, if x ∈ / A and y ∈ A, then f˜(x) ∩ f˜(y) ⊆ f˜(x ∗ y). Obviously, if x ∈ / A and y ∈ / A, then f˜(x) ∩ f˜(y) ⊆ f˜(x ∗ y). Therefore (f˜, X) is an int-soft subalgebra over U . □ Any subalgebra of a BF -algebra X may not be represented as a γ-inclusive set of an int-soft subalgebra (f˜, X) over U in general (see the following example). Example 3.6. Let E = X be the set of parameters, and let U = X be the initial universe set where where X = {0, 1, 2, 3} is a BF -algebra ([12]) with the following Cayley table: ∗ 0 1 2 3

0 0 1 2 3

1 1 0 3 0

2 2 3 0 2

3 3 0 2 0

Consider a soft set (f˜, X) which is given by

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{0, 3} {3}

5

if x = 0 if x ∈ {1, 2, 3}

Then (f˜, X) is an int-soft subalgebra over U . The γ-inclusive set of (f˜, X) are described as follows:   if γ ∈ {∅, {3}}  X iX (f˜; γ) = {0} if γ ∈ {{0}, {0, 3}}   ∅ otherwise. The subalgebra {0, 2} cannot be a γ-inclusive set iX (f˜; γ) since there is no γ ⊆ U such that iX (f˜; γ) = {0, 2}. We make a new int-soft subalgebra from old one. Theorem 3.7. Let (f˜, X) be a soft set of a BF -algebra X over U . Define a soft set (f˜∗ , X) of X over U by { f˜(x) if x ∈ iX (f˜; γ) ∗ ˜ f : X → P(U ), x 7→ ∅ otherwise where γ is a non-empty subset subset of U . If (f˜, X) is an int-soft subalgebra of X, then so is (f˜∗ , X). Proof. If (f˜, X) is an int-soft subalgebra over U , then iX (f˜; γ) is a subalgebra of X for all γ ⊆ U by Theorem 3.6. Let x, y ∈ X. If x, y ∈ iX (f˜; γ), then x ∗ y ∈ iX (f˜; γ). Hence we have f˜∗ (x) ∩ f˜∗ (y) = f˜(x) ∩ f˜(y) ⊆ f˜(x ∗ y) = f˜∗ (x ∗ y). If x ∈ / iX (f˜; γ) or y ∈ / iX (f˜; γ), then f˜∗ (x) = ∅ or f˜∗ (y) = ∅. Thus f˜∗ (x) ∩ f˜∗ (y) = ∅ ⊆ f˜∗ (x) ∗ f˜∗ (y). Therefore (f˜∗ , X) is an int-soft subalgebra over U .

□

Definition 3.8. A soft set (f˜, X) over U is called an intersectional soft normal subalgebra (briefly, int-soft normal subalgebra of X if it satisfies: (3.3) f˜(x ∗ y) ∩ f˜(a ∗ b) ⊆ f˜((x ∗ a) ∗ (y ∗ b)) for all x, y, a, b ∈ X. Proposition 3.9. Every int-soft subalgebra (f˜, X) of a BF -algebra X satisfies the following inclusion: (3.4) f˜(x ∗ y) ⊆ f˜(y ∗ x) for all x, y ∈ X. Proof. Using (B3), (3.1) and (3.2), we have f˜(y ∗ x) = f˜(0 ∗ (x ∗ y)) ⊇ f˜(0) ∩ f˜(x ∗ y) = f˜(x ∗ y), ∀x, y ∈ X. □ Proposition 3.10. Every int-soft normal subalgebra (f˜, X) of a BF -algebra X is an int-soft subalgebra of X. Proof. Put y := 0, b := 0 and a := y in (3.3). Then f˜(x ∗ 0) ∩ f˜(y ∗ 0) ⊆ f˜((x ∗ y) ∗ (0 ∗ 0)) for any x, y ∈ X. Using (B2) and (B1), we have f˜(x) ∩ f˜(y) ⊆ f˜(x ∗ y). Hence (f˜, X) is an int-soft subalgebra of X. □ The converse of Proposition 3.10 may not be true in general (see Example 3.11).

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Example 3.11 Let E = X be the set of parameters where where X = {0, 1, 2, 3} is a BF -algebra with the following Cayley table: ∗ 0 1 2 3

0 0 1 2 3

1 2 0 2 2

2 1 1 0 1

3 3 1 2 0

Let (f˜, X) be a soft set over U defined as follows:   γ3 ˜ f : X → P(U ), x 7→ γ  2 γ1

if x = 0 if x = 3 if x ∈ {1, 2}.

where γ1 , γ2 and γ3 are subsets of U with γ1 ⊊ γ2 ⊊ γ3 . It is easy to check that (f˜, X) is an int-soft normal subalgebra over U . Let (˜ g , X) be a soft set over U defined as follows:   α3 g˜ : X → P(U ), x 7→ α  2 α1

if x = 0 if x ∈ {1, 2} if x = 3.

where α1 , α2 and α3 are subsets of U with α1 ⊊ α2 ⊊ α3 . It is easy to check that (f˜, X) is an int-soft subalgebra over U . But it is not an int-soft normal subalgebra over U since ˜g(2 ∗ 3) ∩ g˜(2 ∗ 0) = g˜(2) ∩ g˜(2) = α2 ⊈ α1 = g˜(3) = g˜((2 ∗ 2) ∗ (3 ∗ 0)). Theorem 3.12. A soft set (f˜, X) of X over U is an int-soft normal subalgebra of X over U if and only if the γ-inclusive set iX (f˜; γ) is a normal subalgebra of X for all γ ∈ P(U ) with iX (f˜; γ) ̸= ∅. □

Proof. Similar to Theorem 3.4. The normal subalgebra iX (f˜; γ) in Theorem 3.12 is called the inclusive normal subalgebra of X.

4. Quotient BF -algebras induces by soft sets Let (f˜, X) be an int-soft normal subalgebra of a BF -algebra X. For any x, y ∈ X, we define a binary operation ˜

“ ∼f ” on X as follows: ˜

x ∼f y ⇔ f˜(x ∗ y) = f˜(0). ˜

Lemma 4.1. The operation ∼f is an equivalence relation on a BF -algebra X. ˜ Proof. Obviously, it is reflexive. Let x ∼f y. Then f˜(x ∗ y) = f˜(0). It follows from (3.4) and (3.2) that ˜ f˜(0) = f˜(x ∗ y) ⊆ f˜(y ∗ x) ⊆ f˜(0). Hence f˜(y ∗ x) = f˜(0). Hence ∼f is symmetric. Let x, y, z ∈ X be such that

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˜ ˜ x ∼f y and y ∼f z. Then f˜(x ∗ y) = f˜(0) and f˜(y ∗ z) = f˜(0). Using (3.4), (3.3), (B1), (B2) and (3.2), we have

f˜(0) = f˜(x ∗ y)∩f˜(y ∗ z) ⊆ f˜(x ∗ y) ∩ f˜(z ∗ y) ⊆f˜((x ∗ z) ∗ (y ∗ y)) =f˜((x ∗ z) ∗ 0) = f˜(x ∗ z) ⊆ f˜(0). ˜ ˜ Hence f˜(x ∗ z) = f˜(0), i.e., ∼f is transitive. Therefore “ ∼f ” is an equivalence relation on X. ˜

˜

□

˜

Lemma 4.2. For any x, y, p, q ∈ X, if x ∼f y and p ∼f q, then x ∗ p ∼f y ∗ q. ˜

˜

Proof. Let x, y, p, q ∈ X be such that x ∼f y and p ∼f q. Then f˜(x ∗ y) = f˜(y ∗ x) = f˜(0) and f˜(p ∗ q) = f˜(q ∗ p) = f˜(0). Using (3.3) and (3.2), we have f˜(0) =f˜(x ∗ y) ∩ f˜(p ∗ q) ⊆f˜((x ∗ p) ∗ (y ∗ q)) ⊆ f˜(0). ˜ Hence f˜((x ∗ p) ∗ (y ∗ q)) = f˜(0). By similar way, we get f˜((y ∗ q) ∗ (x ∗ p)) = f˜(0). Therefore x ∗ p ∼f y ∗ q. Thus ˜

“ ∼f ” is a congruence relation on X.

□

Denote f˜x and X/f˜ the set of all equivalence classes containing x and the set of all equivalence classes of X, respectively, i.e., ˜ f˜x := {y ∈ X|y ∼f x} and X/f˜ := {f˜x |x ∈ X}.

Define a binary relation • on X/f˜ as follows: f˜x • f˜y = f˜x∗y for all f˜x , f˜y ∈ X/f˜. Then this operation is well-defined by Lemma 4.2. Theorem 4.3. If (f˜, X) is an int-soft normal subalgebra of a BF -algebra X, then the quotient X/f˜ := (X/f˜, •, f˜0 ) is a BF -algebra. Proof. Let f˜x , f˜y , f˜z ∈ X/f˜. Then we have f˜x • f˜x = f˜x∗x = f˜0 , f˜x • f˜0 = f˜x∗0 = f˜x , f˜0 • (f˜x • f˜y ) = f˜0∗(x∗y) = f˜y∗x = f˜y • f˜x . Therefore X/f˜ = (X/f˜, •, f˜0 ) is a BF -algebra. □ Corollary 4.4. If (f˜, X) is an int-soft normal subalgebra of a BF2 -algebra X, then the quotient X/f˜ := (X/f˜, •, f˜0 ) is a BF2 -algebra. Proof. It is enough to show that X/f˜ satisfies (BH). If f˜x • f˜y = f˜0 and f˜y • f˜x = f˜0 for any f˜x , f˜y ∈ X/f˜, then ˜ f˜x∗y = f˜0 = f˜y∗x . Hence f˜(x ∗ y) = f˜(0) = f˜(y ∗ x) and so x ∼f y. Hence f˜x = f˜y . Therefore X/f˜ = (X/f˜, •, f˜0 ) □

is a BF2 -algebra.

Proposition 4.5. Let µ : X → Y be a homomorphism of BF -algebras. If (f˜, Y ) is an int-soft normal subalgebra of Y , then (f˜ ◦ µ, X) is an int-soft normal subalgebra of X.

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8

Proof. For any x, y, a, b ∈ X, we have (f˜ ◦ µ)((x ∗ a) ∗ (y ∗ b)) =f˜(µ((x ∗ a) ∗ (y ∗ b)) =f˜((µ(x) ∗ µ(a)) ∗ (µ(y) ∗ µ(b))) ⊇f˜(µ(x) ∗ µ(y)) ∩ f˜(µ(a) ∗ µ(b)) =f˜(µ(x ∗ y)) ∩ f˜(µ(a ∗ b)) =(f˜ ◦ µ)(x ∗ y) ∩ (f˜ ◦ µ)(a ∗ b). Hence f˜ ◦ µ is an int-soft normal subalgebra. Therefore (f˜ ◦ µ, X) is an int-soft normal subalgebra of X.

□

Theorem 4.6. Let X := (X; ∗X , 0X ) be a BF -algebra and Y := (Y ; ∗Y , 0Y ) be a BF2 -algebra and let µ : X → Y be an epimorphism. If (f˜, Y ) is an int-soft normal subalgebra of Y , then the quotient algebra X/(f˜ ◦ µ) := (X/(f˜ ◦ µ), •X , (f˜ ◦ µ)0X ) is isomorphic to the quotient algebra Y /f˜ := (Y /f˜, •Y , f˜0Y ). Proof. By Theorem 4.3, Corollary 4.4, and Proposition 4.5, X/f˜ ◦ µ : (X/(f˜ ◦ µ), •X , (f˜ ◦ µ)0X ) is a BF -algebra and Y /f˜ := (Y /f˜, •Y , f˜0Y ) is a BF2 -algebra. Define a map η : X/(f˜ ◦ µ) → Y /f˜, (f˜ ◦ µ)x 7→ f˜µ(x) for all x ∈ X. Then the function η is well-defined. In fact, assume that (f˜ ◦ µ)x = (f˜ ◦ µ)y for all x, y ∈ X. Then we have f˜(µ(x) ∗Y µ(y)) =f˜(µ(x ∗X y)) = (f˜ ◦ µ)(x ∗X y) =(f˜ ◦ µ)(0X ) = f˜(µ(0X )) = f˜(0Y ) and f˜(µ(y) ∗Y µ(x)) =f˜(µ(y ∗X x)) = (f˜ ◦ µ)(y ∗X x) =(f˜ ◦ µ)(0X ) = f˜(µ(0X )) = f˜(0Y ). Hence f˜µ(x) = f˜µ(y) . For any (f˜ ◦ µ)x , (f˜ ◦ µ)X ∈ X/(f˜ ◦ µ), we have η((f˜ ◦ µ)x •X (f˜ ◦ µ)y ) =η((f˜ ◦ µ)x∗y ) = f˜µ(x∗X y) =f˜µ(x)∗Y µ(y) = f˜µ(x) • f˜µ(y) =η((f˜ ◦ µ)x ) •Y η((f˜ ◦ µ)y )). Therefore η is a homomorphism. Let f˜a ∈ Y /f˜. Then there exists x ∈ X such that µ(x) = a since µ is surjective. Hence η((f˜◦ µ)X ) = f˜µ(x) = f˜a and so η is surjective. Let x, y ∈ X be such that f˜µ(x) = f˜µ(y) . Then we have (f˜ ◦ µ)(x ∗X y) =f˜(µ(x ∗X y)) = f˜(µ(x) ∗Y µ(y)) =f˜(0Y ) = f˜(µ(0X )) = (f˜ ◦ µ)(0X )

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Applications of soft sets in BF -algebras

9

and (f˜ ◦ µ)(y ∗X x) =f˜(µ(y ∗X x)) = f˜(µ(y) ∗Y µ(x)) =f˜(0Y ) = f˜(µ(0X )) = (f˜ ◦ µ)(0X ). It follows that (f˜ ◦ µ)X = (f˜ ◦ µ)y . Thus η is injective. This completes.

□

The homomorphism π : X → X/f˜, x → f˜X , is called the natural homomorphism of X onto X/f˜. In Theorem 4.6, if we define natural homomorphisms πX : X → X/f˜ ◦ µ and πY : Y → Y /f˜ then it is easy to show that η ◦ πX = πY ◦ µ, i.e., the following diagram commutes. X   πX y

µ

−−−−→

Y   πY y

η

X/(f˜ ◦ µ) −−−−→ Y /f˜.

References [1] U. Acar, F. Koyuncu and B. Tanay, Soft sets and soft rings, Comput. Math. Appl. 59(2010) 3458-3463. [2] H. Akta¸s and N. C ¸ a˘gman, Soft sets and soft groups, Inform. Sci. 177(2007) 2726-2735. [3] A. O. Atag¨ un and A. Sezgin, Soft substructures of rings, fields and modules, Comput. Math. Appl. 61 (2011) 592-601. [4] J. C. Endam and J. P. Vilela, On BF -algebras, Math. Slovaca 64(2014), 13-20. [5] F. Feng, Y. B. Jun and X. Zhao, Soft semirings, Comput. Math. Appl. 56 (2008) 2621-2628. [6] Y. B. Jun, Soft BCK/BCI-algebras, Comput. Math. Appl. 56 (2008) 1408-1413. [7] Y. B. Jun, Union soft sets with applications in BCK/BCI-algebras, Bull. Korean Math. Soc. 50 (2013), 1937-1956. [8] J. M. Ko and S. S. Ahn, Structure of BF -algebras, Applied Mathematical Sciences 15 (2015), 6369-6374. [9] P. K. Maji, R. Biswas and A. R. Roy, Soft set theory, Comput. Math. Appl. 45 (2003) 555-562. [10] P. K. Maji, A. R. Roy and R. Biswas, An application of soft sets in a decision making problem, Comput. Math. Appl. 44 (2002) 1077-1083. [11] D. Molodtsov, Soft set theory - First results, Comput. Math. Appl. 37 (1999) 19-31. [12] A. Walmendiziak, On BF -algebras, Mat. Solvoca 57(2007), 119-128. [13] K. S. Yang and S. S. Ahn, Union soft q-ideals in BCI-algebras, Applied Mathematical Scineces 8(2014), 2859-2869. [14] L. A. Zadeh, From circuit theory to system theory, Proc. Inst. Radio Eng. 50 (1962) 856-865. [15] L. A. Zadeh, Fuzzy sets, Inform. Control 8 (1965) 338-353. [16] L. A. Zadeh, Toward a generalized theory of uncertainty (GTU) - an outline, Inform. Sci. 172 (2005) 1-40.

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Symmetric solutions for hybrid fractional differential equations Jessada Tariboona,∗, Sotiris K. Ntouyasb,c , and Suthep Suantaid a

Nonlinear Dynamic Analysis Research Center, Department of Mathematics, Faculty of Applied Science, King Mongkut’s University of Technology North Bangkok, Bangkok, Thailand E-mail: [email protected] b

Department of Mathematics, University of Ioannina, 451 10 Ioannina, Greece

c

Nonlinear Analysis and Applied Mathematics (NAAM)-Research Group, Department of Mathematics, Faculty of Science, King Abdulaziz University, P.O. Box 80203, Jeddah 21589, Saudi Arabia E-mail: [email protected] d

Department of Mathematics, Faculty of Science, Chiang Mai University, Chiang Mai, 50200 Thailand E-mail: [email protected] Abstract In this paper we introduce a new class of symmetric functions and study the existence of symmetric solutions for hybrid Caputo fractional differential equations. A fixed point theorem in Banach algebra for two operators is used. An example is presented to illustrate our result.

Keywords: Caputo fractional derivative; hybrid fractional differential equation; symmetric solution; fixed point theorem 2010 Mathematics Subject Classifications: 34A08; 34A12.

1

Introduction

The aim of this manuscript is to study the existence at least one symmetric solution for hybrid Caputo fractional differential equation subject to initial and symmetric conditions  [ ] x(t)   Dα + g(t, x(t)) = 0, t ∈ J := [0, T ], f (t, x(t)) (1.1)   x(0) = β, x(t) = x(T − t), where Dα denotes the Caputo fractional derivative of order α, 1 < α ≤ 2, f ∈ C(J × R, R \ {0}), g ∈ C(J × R, R), β ∈ R. A function x ∈ C([0, T ], R) satisfying the relation x(t) = x(T − t), t ∈ [0, T ], is called symmetric on [0, T ]. Fractional differential equations have been of great interest recently. It is caused both by the intensive development of the theory of fractional calculus itself and by the applications of such constructions in various science such as physics, mechanics, chemistry, and engineering. There have appeared lots of works, in which fractional derivatives are used for a better description of considered material properties. For details, and some recent results on the subject we refer to [1]-[17] and references cited therein. Recently, many authors have focused on the existence of symmetric solutions for ordinary differential equation boundary value problems; for example, see [18]-[21] and the references therein. In [22] ∗ Corresponding

author

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J. TARIBOON , S. K. NTOUYAS AND S. SUANTAI

the existence and uniqueness of symmetric solutions for a boundary value problem for nonlinear fractional differential equations with multi-order fractional integral boundary conditions was studied, by using a variety of fixed point theorems (such as Banach contraction principle, nonlinear contractions, Krasnoselskii fixed point theorem and Leray-Schauder nonlinear alternative). Hybrid fractional differential equations have also been studied by several researchers. This class of equations involves the fractional derivative of an unknown function hybrid with the nonlinearity depending on it. Some recent results on hybrid differential equations can be found in a series of papers ([23]-[28]). In this paper we prove the existence of symmetric solutions for the hybrid Caputo fractional boundary value problem (1.1). One new result is proved by using a hybrid fixed point theorem for two operators in a Banach algebra due to Dhage [29]. The rest of this paper is organized as follows: In Section 2 we present some preliminary notations, definitions and lemmas that we need in the sequel. Also we introduce a new class of symmetric functions and prove some interesting properties, which are used to establish the Green function. In Section 3 we establish the existence of symmetric solutions for the boundary value problem (1.1). An example illustrating the obtained result is also presented.

2

Preliminaries

In this section, we introduce some notations and definitions of fractional calculus [1, 2] and present preliminary results needed in our proofs later. In addition, a new definition of α-symmetric function is presented and also some properties are proved. Definition 2.1 The Riemann-Liouville fractional integral of order α > 0 of a function g : (0, ∞) → R is defined by ∫ t (t − s)α−1 α I g(t) = g(s)ds, Γ(α) 0 provided the right-hand side is point-wise defined on (0, ∞), where Γ is the Gamma function. Definition 2.2 The Caputo fractional derivative of order α > 0 for an at least n-times differentiable function g : (0, ∞) → R is defined by 1 Γ(n − α)

Dα g(t) =

∫ 0

t

g (n) (s) ds, (t − s)α−n+1

n − 1 < α < n,

where n = [α] + 1, [α] denotes the integer part of real number α. From the definition of the Caputo fractional derivative, we can obtain the following lemmas. Lemma 2.3 (see [1]) Let α > 0, the general solution of the fractional differential equation Dα y(t) = 0 is given by y(t) = c0 + c1 t + · · · + cn−1 tn−1 , where ci ∈ R, i = 0, 1, 2, . . . , n − 1, n = [α] + 1. Definition 2.4 A function y ∈ C 2 (J, R) is called symmetric, if it satisfies the relation y(t) = y(T − t). From Definition 2.4 we have y 0 (t) = −y 0 (T − t), y 00 (t) = y 00 (T − t) and ∫

∫

T −t

y(s)ds −

y(s)ds = 0

∫

T 0

t

y(s)ds.

(2.1)

0

Lemma 2.5 Let f ∈ L2 (J, R) be symmetric function. Then we have I 1 f (T ) =

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2 2 I f (T ). T

(2.2)

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SYMMETRIC SOLUTIONS FOR HYBRID FRACTIONAL DIFFERENTIAL EQUATIONS

Proof. Since f is symmetric on [0, T ], we have ∫

T

1

I f (T ) =

f (s)ds

=

0

= =

1 T 1 T 2 T

∫

T

(T − s + s)f (s)ds ∫

0

∫

0

T

∫

1 (T − s)f (s)ds + T

T

(T − s)f (s)ds = 0

T

sf (s)ds 0

2 2 I f (T ). T 2

Therefore, (2.2) holds. Now, we define a new class of symmetric functions as follows:

Definition 2.6 A function f ∈ C 1 (J, R) is called α-symmetric if D2−α f (t) is symmetric function on [0, T ], where 1 < α ≤ 2. Example 2.7 Let f : [0, 1] → R be defined as 4 3 f (t) = √ t 2 3 π

(

) 4 1− t . 5

It easy to verify that 3

D2− 2 f (t)

1

= D 2 f (t) 1 3 1 5 4 16 √ D2 t2 − √ D2 t2 = 3 π 15 π = t(1 − t).

Therefore, f is 32 -symmetric function. Remark 2.8 If α = 2, then the class of α-symmetric functions is reduced to the class of usual symmetric functions. Lemma 2.9 Let z ∈ C 1 (J, R) be an α-symmetric function. Then the symmetric solution of linear fractional differential equation Dα y(t) = z(t), 1 < α ≤ 2, t ∈ J, y(t) = y(T − t),

(2.3) (2.4)

is given by y(t) = I α z(t) −

t α I z(T ) + c0 , T

(2.5)

where c0 ∈ R. Proof. By Lemma 2.3, we have y(t) = I α z(t) + c1 t + c0 ,

(2.6)

where c0 , c1 ∈ R. We apply symmetric condition to obtain I α z(t) + c1 t + c0 = I α z(T − t) + c1 (T − t) + c0 .

(2.7)

Evidently, (2.7) becomes c1 (2t − T )

= I α z(T − t) − I α z(t) ∫ t ∫ T −t (t − s)α−1 (T − t − s)α−1 z(s)ds − z(s)ds. = Γ(α) Γ(α) 0 0

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(2.8)

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Taking the first-order usual derivative with respect to t in (2.8), we get 2c1

= −I α−1 z(T − t) − I α−1 z(t) ( ( ) ) = −I 1 D2−α z (T − t) − I 1 D2−α z (t).

Since D2−α z(t) is symmetric on J, and z is symmetric, by (2.1), we have ( ) ( ) ( ) I 1 D2−α z (T − t) = I 1 D2−α z (T ) − I 1 D2−α z (t), which leads to 2c1

( ) −I 1 D2−α z (T ) ) 2 ( − I 2 D2−α z (T ), T

= =

by using Lemma 2.5. Therefor, we obtain the constant c1 as 1 1 c1 = − I α z(T ) = − T T

∫

T 0

(T − s)α−1 z(s)ds. Γ(α)

Substituting the constant c1 in (2.6), we get the result in (2.5) as desired.

2

In the following we present the Green function of the hybrid fractional boundary value problem (1.1). Lemma 2.10 Let h ∈ C 1 (J, R) be the α-symmetric function and f ∈ C(J × R, R \ {0}). Then the unique solution of [ ] x(t) α D + h(t) = 0, t ∈ J, (2.9) f (t, x(t)) x(0) = β, x(t) = x(T − t), (2.10) is given by (∫

T

x(t) = f (t, x(t)) 0

where

β G(t, s)h(s)ds + f (0, β)

 α−1 α−1 t (T − s) − T (t − s)   ,  T Γ(α) G(t, s) = α−1   t (T − s)  , T Γ(α)

) ,

(2.11)

0 ≤ s ≤ t ≤ T, (2.12) 0 ≤ t ≤ s ≤ T.

Proof. Applying Lemma 2.9, the equation (2.9) can be written as x(t) t = −I α h(t) + I α h(T ) + c0 , f (t, x(t)) T

(2.13)

where c0 ∈ R. The condition x(0) = 0 implies that c0 =

β . f (0, β)

Therefore, the unique solution of problem (2.9)-(2.10) is ( ∫ t 1 α−1 (t − s) h(s)ds x(t) = f (t, x(t)) − Γ(α) 0

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SYMMETRIC SOLUTIONS FOR HYBRID FRACTIONAL DIFFERENTIAL EQUATIONS

t + T Γ(α)

∫

)

T

α−1

(T − s) ∫

h(s)ds +

0 T

= f (t, x(t))

G(t, s)h(s)ds + 0

β f (t, x(t)) f (0, β)

β f (t, x(t)). f (0, β) 2

The proof is completed.

Remark 2.11 The Green’s function G(t, s) defined by (2.12), is not positive for all t, s ∈ J. For √ example, if T = 5, t = 2, s = 1 and α = 3/2, then we have G(2, 1) = −2/(5 π). Lemma 2.12 The Green’s function G(t, s) in (2.12) satisfies the following inequalities G(t, s) ≤ G(s, s) ≤

((α − 1)T )α−1 αα−1 Γ(α + 1)

for all

s, t ∈ J.

(2.14)

Proof. Let us define two functions by g1 (t, s) = t (T − s)

α−1

− T (t − s)

α−1

, 0 ≤ s ≤ t ≤ T,

and α−1

g2 (t, s) = t (T − s)

, 0 ≤ t ≤ s ≤ T.

Obviously, for 0 ≤ t ≤ s ≤ T , the function g2 (t, s) satisfies g2 (t, s) ≤ g2 (s, s) = s(T − s)α−1 . Let s ∈ [0, T ) be fixed. Differentiating with respect to t the function g1 (t, s), we have ∂ α−1 α−2 g1 (t, s) = (T − s) − (α − 1)T (t − s) , s < t. ∂t We can find that ∂g1 /∂t = 0 if and only if α−1

t = t∗ = s +

(T − s) α−2 1

((α − 1)T ) α−2

.

It follows from ∂g1 /∂t > 0 on (0, t∗ ) and ∂g1 /∂t < 0 on (t∗ , T ) that g1 (t, s) ≤ g1 (t∗ , s). Simplifying the above inequality, we get g1 (t, s) ≤

g1 (t∗ , s)

=

s(T − s)α−1 − (2 − α)T ·

≤

s(T − s)α−1 = g1 (s, s),

(T − s)

(α−1)2 α−2 α−1

((α − 1)T ) α−2

which implies the first inequality. Next, we will prove the second inequality. Taking the first derivative for g2 (s, s) with respect to s on [0, T ), we have g20 (s, s) = (T − s)α−2 (T − αs). Thus g20 (s, s) has a unique zero at the point s = s∗ = T /α such that s∗ ∈ (0, T ). Observe that g20 (s, s) > 0 on (0, s∗ ) and g20 (s, s) < 0 on (s∗ , T ). Hence ) ( (α − 1)α−1 α T T , = T . g2 (s, s) ≤ g2 α α αα

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Then the second inequality is proved.

Let E = C([0, T ], R) be the Banach space endowed with the supremum norm k · k. Define a multiplication in E by (xy)(t) = x(t)y(t), ∀t ∈ J. Clearly E is a Banach algebra with respect to above supremum norm and the multiplication in it. The main result is based on the following fixed point theorem for two operators in Banach algebra due to Dhage [29]. Lemma 2.13 Let S be a non-empty, closed convex and bounded subset of the Banach algebra E, let A : E → E and B : S → E be two operators such that: (a) A is Lipschitzian with a Lipschitz constant δ, (b) B is completely continuous, (c) x = AxBy ⇒ x ∈ S for all y ∈ S, and (d) M δ < 1, where M = kB(S)k = sup{kB(x)k : x ∈ S}. Then the operator equation x = AxBx has a solution in S.

3

Main Result

Now, we are in the position to prove the existence of symmetric solutions for hybrid fractional problem (1.1). Theorem 3.1 Assume that the following conditions are satisfied: (H1 ) The functions f ∈ C(J × R, R \ {0}) and g ∈ C 1 (J × R, R) are symmetric and α-symmetric on J, respectively. (H2 ) There exists a bounded function φ(t), with bound kφk, such that |f (t, x) − f (t, y)| ≤ kφk · |x − y| for t ∈ J and x, y ∈ R. (H3 ) There exist a function p ∈ C(J, R+ ) and a continuous nondecreasing function Ψ : [0, ∞) → (0, ∞) such that |g(t, x)| ≤ p(t)Ψ(|x|), (t, x) ∈ J × R. (H4 ) There exist a number r > 0 such that ] [ (α − 1)α−1 T α |β| kpkΨ(r) + F0 α−1 α Γ(α + 1) |f (0, β)| ] [ r≥ (α − 1)α−1 T α |β| 1 − kφk α−1 kpkΨ(r) + α Γ(α + 1) |f (0, β)|

(3.1)

where F0 = supt∈J |f (t, 0)| and [

] (α − 1)α−1 T α |β| kpkΨ(r) + < 1. kφk α−1 α Γ(α + 1) |f (0, β)|

(3.2)

Then the problem (1.1) has at least one symmetric solution on J.

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SYMMETRIC SOLUTIONS FOR HYBRID FRACTIONAL DIFFERENTIAL EQUATIONS

Proof. To prove our main result, we first define a subset S of E by S = {x ∈ E : kxk ≤ r}, where r satisfies (3.1). Clearly S is closed, convex and bounded subset of the Banach space E. By Lemma 2.10, we define two operators A : E → E by Ax(t) = f (t, x(t)), and

∫

T

Bx(t) =

G(t, s)g(s, x(s))ds + 0

t ∈ J,

(3.3)

β , f (0, β)

t ∈ J.

(3.4)

Hence, the problem (1.1) is transformed into an operator equation as x = AxBx.

(3.5)

Next, we shall show that the operators A and B satisfy all the conditions of Lemma 2.13 under our assumptions. This will be achieved in the series of following steps. Step 1. We first show that A is Lipschitzian on E. Let x, y ∈ E. Then by (H2 ), for t ∈ J we have |Ax(t) − Ay(t)|

= |f (t, x(t)) − f (t, y(t))| ≤ φ(t)|x(t) − y(t)| ≤ kφkkx − yk,

which implies that kAx − Ayk ≤ kφkkx − yk for all x, y ∈ E. Therefore, A is a Lipschitzian on E with Lipschitz constant δ = kφk. Step 2. The operator B is completely continuous on S. We first show that the operator B is continuous on S. Let {xn } be a sequence in S converging to a point x ∈ S. Then by Lebesgue dominated convergence theorem, for all t ∈ J, we have ∫ lim Bxn (t)

n→∞

=

T

G(t, s)g(s, xn (s))ds +

β f (0, β)

G(t, s) lim g(s, xn (s))ds +

β f (0, β)

lim

n→∞

∫ = ∫

0

T n→∞

0 T

=

G(t, s)g(s, x(s))ds + 0

β f (0, β)

= Bx(t). This shows that {Bxn } converges to Bx pointwise on J. Next, we will show that {Bxn } is an equicontinuous sequence of functions in S. Let τ1 , τ2 ∈ J be arbitrary with τ1 < τ2 . Then ¯∫ ¯ ∫ T ¯ T ¯ ¯ ¯ |Bxn (τ2 ) − Bxn (τ1 )| = ¯ G(τ2 , s)g(s, xn (s))ds − G(τ1 , s)g(s, xn (s))ds¯ ¯ 0 ¯ 0 ¯∫ ¯ ∫ ¯ T ¯ T ¯ ¯ ≤ kpkΨ(r)¯ G(τ2 , s)ds − G(τ1 , s)ds¯ ¯ 0 ¯ 0 ¯∫ ¯ τ2 τ (T − s)α−1 − T (τ − s)α−1 ¯ 2 2 ds ≤ kpkΨ(r)¯ ¯ 0 T Γ(α)

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J. TARIBOON , S. K. NTOUYAS AND S. SUANTAI ∫

τ2 (T − s)α−1 ds − T Γ(α)

∫

τ1 (T − s)α−1 ds T Γ(α) τ2 τ1 ¯ ∫ τ1 τ1 (T − s)α−1 − T (τ1 − s)α−1 ¯¯ − ds¯ ¯ T Γ(α) 0 ¯∫ ∫ τ2 ¯ T τ (T − s)α−1 (τ2 − s)α−1 ¯ 2 ds − ds = kpkΨ(r)¯ ¯ 0 T Γ(α) Γ(α) 0 ¯ ∫ τ1 ∫ T (τ1 − s)α−1 ¯¯ τ1 (T − s)α−1 ds + ds¯ − ¯ T Γ(α) Γ(α) 0 0 ∫ T (τ2 − τ1 )(T − s)α−1 ds ≤ kpkΨ(r) T Γ(α) 0 ∫ τ1 (τ2 − s)α−1 − (τ1 − s)α−1 + kpkΨ(r) ds Γ(α) 0 ∫ τ2 (τ2 − s)α−1 + kpkΨ(r) ds. Γ(α) τ1 +

T

T

Consequently |Bxn (τ2 ) − Bxn (τ1 )| → 0 as

τ2 → τ1

uniformly for all n ∈ N. This shows that the convergence Bxn → Bx is uniformly and hence B is a continuous operator on S. Now we will prove that the set B(S) is a uniformly bounded in S. For any x ∈ S and using Lemma 2.12, we have ¯∫ ¯ ¯ T β ¯¯ ¯ |Bx(t)| = ¯ G(t, s)g(s, x(s))ds + ¯ ¯ 0 f (0, β) ¯ ∫ T |β| ≤ |G(t, s)|p(s)Ψ(r)ds + |f (0, β)| 0 (α − 1)α−1 T α |β| ≤ kpkΨ(r) + := K1 , α−1 α Γ(α + 1) |f (0, β)| for all t ∈ J. Therefore, kBxk ≤ K1 which shows that B is uniformly bounded on S. Next, we will show that B(S) is an equicontinuous set in E. Let τ1 , τ2 ∈ J with τ1 < τ2 and x ∈ S. Then, as above, we have ¯ ¯∫ ∫ T ¯ ¯ T ¯ ¯ G(τ1 , s)g(s, x(s))ds¯ G(τ2 , s)g(s, x(s))ds − |Bx(τ2 ) − Bx(τ1 )| = ¯ ¯ ¯ 0 0 ¯∫ ¯ ∫ T ¯ T ¯ ¯ ¯ ≤ kpkΨ(r)¯ G(τ2 , s)ds − G(τ1 , s)ds¯ ¯ 0 ¯ 0 ∫ T α−1 (τ2 − τ1 )(T − s) ≤ kpkΨ(r) ds T Γ(α) 0 ∫ τ1 (τ2 − s)α−1 − (τ1 − s)α−1 + kpkΨ(r) ds Γ(α) 0 ∫ τ2 (τ2 − s)α−1 + kpkΨ(r) ds, Γ(α) τ1 which is independent of x ∈ S. As τ1 → τ2 , the right-hand side of the above inequality tends to zero. Therefore, it follows from the Arzel´ a-Ascoli theorem that B is a completely continuous operator on S. Step 3. The hypothesis (c) of Lemma 2.13 is satisfied.

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SYMMETRIC SOLUTIONS FOR HYBRID FRACTIONAL DIFFERENTIAL EQUATIONS Let x ∈ E and y ∈ S be arbitrary elements such that x = AxBy. Then we have |x(t)|

≤ = ≤ ≤

|Ax(t)||By(t)| ¯∫ ¯ ¯ T β ¯¯ ¯ |f (t, x(t))| ¯ G(t, s)g(s, y(s))ds + ¯ ¯ 0 f (0, β) ¯ ( ) (α − 1)α−1 T α |β| (|f (t, x(t)) − f (t, 0)| + |f (t, 0)|) kpkΨ(r) + αα−1 Γ(α + 1) |f (0, β)| ( ) α−1 α (α − 1) T |β| kpkΨ(r) + , (|x(t)| · kφk + F0 ) αα−1 Γ(α + 1) |f (0, β)|

which leads to

( kxk ≤

(kxk · kφk + F0 )

≤ r.

(α − 1)α−1 T α |β| kpkΨ(r) + αα−1 Γ(α + 1) |f (0, β)|

)

Therefore, x ∈ S. Step 4. Finally we show that δM < 1, that is (d) of Lemma 2.13 holds. Since M

= kB(S)k { } = sup sup |Bx(t)| x∈S

≤ by (3.2) we have

( δkM ≤ kφk

t∈J α−1

(α − 1) Tα |β| kpkΨ(r) + , α−1 α Γ(α + 1) |f (0, β)|

(α − 1)α−1 T α |β| kpkΨ(r) + αα−1 Γ(α + 1) |f (0, β)|

(3.6)

) < 1,

with δ = kφk. Thus all the conditions of Lemma 2.13 are satisfied and hence the operator equation x = AxBx has a solution in S. In consequence, the problem (1.1) has a symmetric solution on J. This completes the proof. 2 Next, we present an example to illustrate our result. Example 3.2 Consider the following hybrid fractional differential equation with initial and symmetric conditions    ( ( 3 (  )))    2  3  x(t) 1 8t 2  +  √ 1 + sin2 1− t D2     5 3 π x2 (t) + 2|x(t)| 1  24   +   2  2(5 + (t − 1) )(|x(t)| + 1) 2 (3.7) ( ) 2  x (t)   × + 1 = 0, t ∈ [0, 2],    1 + |x(t)|        x(0) = 1 , x(t) = x(2 − t). 3 √ 3 Here α = 3/2, T = 2 and β = 1/3. Since (t − 1)2 is symmetric on [0, 2] and (8t 2 )(1 − (2/5)t)/(3 π) is 3/2-symmetric by ( 3 ( )) 1 8t 2 2 2 √ D 1− t = t(2 − t), t ∈ [0, 2], 5 3 π

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J. TARIBOON , S. K. NTOUYAS AND S. SUANTAI then we get that f (t, ·) and g(t, ·) are symmetric and 3/2-symmetric functions on [0, 2], respectively. With the above information, we find that ¯ ¯ ¯ ¯ x2 + 2|x| y 2 + 2|y| ¯ |f (t, x) − f (t, y)| = ¯¯ − 2 2 2(5 + (t − 1) )(|x| + 1) 2(5 + (t − 1) )(|y| + 1) ¯ 1 ≤ |x − y|, 5 + (t − 1)2 and |g(t, x)| ≤

1 (|x| + 1), 12

and F0 = supt∈[0,2] |f (t, 0)| = 1/2. Choosing φ(t) = 1/(5+(t−1)2 ), p(t) = 1/12, we have kφk = 1/5 and kpk = 1/12. Setting Ψ(|x|) = |x| + 1, we can find that there exists 0.06962115393 < r < 45.01973321 which is satisfied (3.1)-(3.2). Thus all the conditions of Theorem 3.1 are satisfied. Therefore, by the conclusion of Theorem 3.1, the problem (3.7) has at least one symmetric solution on [0, 2].

Acknowledgements This paper was supported by the Thailand Research Fund under the project RTA5780007.

References [1] A.A. Kilbas, H.M. Srivastava, J.J. Trujillo, Theory and Applications of Fractional Differential Equations, North-Holland Mathematics Studies, 204. Elsevier Science B.V., Amsterdam, 2006. [2] I. Podlubny, Fractional Differential Equations, Academic Press, San Diego, 1999. [3] S. G. Samko, A.A. Kilbas and O.I. Marichev, Fractional Integrals and Derivatives, Theory and Applications, Gordon and Breach, Yverdon, 1993. [4] K.S. Miller, B. Ross, An introduction to the Fractional Calculus and Fractional Differential Equations, Wiley, New York, 1993. [5] J. Sabatier, O.P. Agrawal and J.A.T. Machado (Eds.), Advances in Fractional Calculus: Theoretical Developments and Applications in Physics and Engineering, Springer, Dordrecht, 2007. [6] V. Lakshmikantham, S. Leela and J. Vasundhara Devi, Theory of Fractional Dynamic Systems, Cambridge Academic Publishers, Cambridge, 2009. [7] R.P. Agarwal, Y. Zhou, Y. He, Existence of fractional neutral functional differential equations, Comput. Math. Appl. 59 (2010), 1095-1100. [8] B. Ahmad, J.J. Nieto, Riemann-Liouville fractional integro-differential equations with fractional nonlocal integral boundary conditions, Bound. Value Probl. 2011, 2011:36, 9 pp. [9] P. Thiramanus, J. Tariboon, S. K. Ntouyas, Average value problems for nonlinear second-order impulsive q-difference equations, J. Comput. Anal. Appl. 18 (2015), 590-611. [10] B. Ahmad, S.K. Ntouyas, A. Alsaedi, New existence results for nonlinear fractional differential equations with three-point integral boundary conditions, Adv. Differ. Equ. (2011) Art. ID 107384, 11pp. [11] B. Ahmad, S.K. Ntouyas, Fractional differential inclusions with fractional separated boundary conditions, Fract. Calc. Appl. Anal. 15 (2012), 362-382.

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SYMMETRIC SOLUTIONS FOR HYBRID FRACTIONAL DIFFERENTIAL EQUATIONS

[12] B. Ahmad, S. K. Ntouyas, A. Alsaedi, A study of nonlinear fractional differential equations of arbitrary order with Riemann-Liouville type multistrip boundary conditions, Math. Probl. Eng. (2013), Art. ID 320415, 9 pp. [13] B. Ahmad, S.K. Ntouyas, Nonlocal fractional boundary value problems with slit-strips integral boundary conditions, Fract. Calc. Appl. Anal. 18 (2015), 261-280. [14] S. Choudhary, V. Daftardar-Gejji, Nonlinear multi-order fractional differential equations with periodic/anti-periodic boundary conditions, Fract. Calc. Appl. Anal. 17 (2014), 333-347. [15] X. Liu, M. Jia, W. Ge, Multiple solutions of a p-Laplacian model involving a fractional derivative, Adv. Differ. Equ. 2013, 2013:126. [16] D. O’Regan, S. Stanek, Fractional boundary value problems with singularities in space variables, Nonlinear Dynam. 71 (2013), 641-652. [17] L. Zhang, B. Ahmad, G. Wang, R.P. Agarwal, Nonlinear fractional integro-differential equations on unbounded domains in a Banach space, J. Comput. Appl. Math. 249 (2013), 51–56. [18] Y. Sun, X. Zhang, Existence of symmetric positive solutions for an m-point boundary value problem, Bound. Value Probl. 2007, Article ID 79090 (2007). [19] N. Kosmatov, Symmetric solutions of a multi-point boundary value problem, J. Math. Anal. Appl. 309 (2005), 25-36. [20] J. Zhao, C. Miao, W. Ge, J. Zhang, Multiple symmetric positive solutions to a new kind of four point boundary value problem, Nonlinear Anal. 71 (2009), 9-18. [21] H. Pang, Y. Tong, Symmetric positive solutions to a second-order boundary value problem with integral boundary conditions, Bound. Value Probl. 2013, 2013:150. [22] A. Aphithana, S.K. Ntouyas, J. Tariboon, Existence and uniqueness of symmetric solutions for fractional differential equations with multi-order fractional integral conditions, Bound. Value Probl. (2015), 2015:68. [23] Y. Zhao, S. Sun, Z. Han, Q. Li, Theory of fractional hybrid differential equations, Comput. Math. Appl. 62 (2011), 1312-1324. [24] S. Sun, Y. Zhao, Z. Han, Y. Li, The existence of solutions for boundary value problem of fractional hybrid differential equations, Commun. Nonlinear Sci. Numer. Simul., 17 (2012), 4961-4967. [25] B. Ahmad, S.K. Ntouyas, An existence theorem for fractional hybrid differential inclusions of Hadamard type with Dirichlet boundary conditions, Abstr. Appl. Anal. 2014, Art. ID 705809, 7 pages, [26] B.C. Dhage, S.K. Ntouyas, Existence results for boundary value problems for fractional hybrid differential inclucions, Topol. Methods Nonlinar Anal. 44 (2014), 229-238. [27] Y. Zhao, Y. Wang, Existence of solutions to boundary value problem of a class of nonlinear fractional differential equations, Adv. Differ. Equ. 2014, 2014: 174. [28] B. Ahmad, S.K. Ntouyas, A. Alsaedi, Hybrid boundary value problems of q-difference equations and inclusions, J. Comput. Anal. Appl. 19 (2015), 984-993. [29] B. C. Dhage, On a fixed point theorem in Banach algebras, Appl. Math. Lett. 18 (2005), 273-280.

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ON THE k-TH DEGENERATION OF THE GENOCCHI POLYNOMIALS LEE-CHAE JANG, C.S. RYOO, JEONG GON LEE, HYUCK IN KWON

Graduate School of Education, Konkuk University, Seoul 143-701, Republic of Korea E-mail : [email protected] Department of Mathematics, Hannam University, Daejeon 306-791, Republic of Korea E-mail : [email protected] Division of Mathematics and Informational Statistics, and Nanoscale Science and Technology Institute, Wonkwang University, Iksan 570-749, Republic of Korea E-mail : [email protected] Department of Mathematics, Kwangwoon University, Seoul 139-50, Republic of Korea E-mail : [email protected]

Abstract. Jeong-Rim-Kim(2015) studied the degenerate Cauchy numbers and polynomials and investigated some properties of these k-times degenerate Cauchy numbers and polynomials. In this paper, we define the degenerate Genocchi polynomials and the k-th degeneration of Genocchi polynomials , and investigate some properties of these polynomials.

1. Introduction Let p be a fixed odd prime number. Throughout this paper, Zp , Qp , and Cp will, respectively, denote the rings of p-adic integers, the field of p-adic rational numbers, and the completion of algebraic closure of Qp . The p-adic norm | · | is normalized by |p|p = p1 . Let U D(Zp ) be the space of uniformly differentiable functions on Zp . For f ∈ U D(Zp ), the fermionic p-adic integral on Zp is defined by Kim as Z I−1 (f ) =

f (x)dµ−1 (x) = lim

N →∞

Zp

N pX −1

f (x)(−1)x

(1)

x=0

(see [7,8,11,13,14,16,17,20,22]). Then, by (1), we get I−1 (f ) = −I−1 (f ) + 2f (0),

(2)

where f1 (x) = f (x + 1). 1991 Mathematics Subject Classification. 11B68, 11S40. Key words and phrases. Genocchi polynomials, degenerate Genocchi polynomials, fermionic p-adic integral, Higher order Daehee polynomials. 1

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LEE-CHAE JANG, C.S. RYOO, JEONG GON LEE, HYUCK IN KWON

From (2), we can derive the following integral equation I−1 (fn ) = (−1)n I−1 (f ) + 2

n−1 X

(−1)n−1−l f (l),

(3)

l=0

where fn (x) = f (x + n), (n ∈ N). As is well known, the Euler polynomials are also defined by the generating function to be   ∞ X tn 2 xt e (see [1, 2, 4 − 22]). (4) = En (x) t e +1 n! n=0 When x = 0, En = En (0) are called the Euler numbers. The degenerate Euler polynomials are also defined by the degenerating function to be 2

x

1 λ

(1 + λt) + 1

(1 + λt) λ =

∞ X

En (λ, x)

n=0

tn n!

(see [1, 4, 8, 11, 13, 14, 16, 17, 20, 22]).

(5)

When x = 0, En (λ) = En (λ, 0) are called the degenerate Euler number. Note that limx→0 En (λ, x) = En (x). We recall that the Genocchi polynomials are defined by the generating function to be ∞ X tn 2 = G (x) n et + 1 n=0 n!

(see [18, 20, 22]).

(6)

In recent years, many researchers have studied various types of special polynomials, for examples, Barnes-type degenerate Euler polynomials, the degenerate Frobenius-Euler polynomials, the degenerate Frobenius-Genocchi polynomials, and degenerate Bernoulli polynomials (see [2,3,6,9,10,12,13,15]). In particular, recently, Jeong-Rim-Kim ([5]) studied finite times degenerate Cauchy polynomials and investigated some properties of them. Thus, our motivation in this paper is to define the degenerate Genocchi polynomials, to define the k-th degeneration of Genocchi polynomials, and to investigate some properties of these k-th degeneration of Genocchi polynomials.

2. The k-th degeneration of Genocchi polynomials In this section, we define the degenerate Genocchi polynomials which are defined by the generating function to be 2t

x

1 λ

(1 + λt) + 1 (0)

(1 + λt) λ =

∞ X n=1

gn(0) (x|λ)

tn . n!

(7)

(0)

When x = 0, gn (λ) = gn (0|λ) are called the degenerate Genocchi number. From (2), we easily obtain Z x+y x 2t λ = t (1 + λt) λ dµ−1 (y). (1 + λt) 1 (1 + λt) λ + 1 Zp

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We note that the Stirling number of the first kind is defined as (x)n =

n X

S1 (n, l)xl

(n ≥ 0)

(9)

l=0

where (x)n = x(x − 1) · · · (x − n + 1) and (x)0 = 1, and (log(1 + t))n = n!

∞ X

S1 (m, n)

m=n

tm m!

(10)

and the Stirling number of the second kind is defined as t

n

(e − 1) = n!

∞ X l=n

tl S2 (l, n) . l!

(11)

From (7), we get ∞ X

gn(0) (x|λ)

n=1

tn n!

=

∞ X

(0)

gn+1 (x|λ)

n=0 ∞ X

= t

tn+1 (n + 1)!

(0)

gn+1 (x|λ) tn . n + 1 n! n=0

(12)

From (8), we get 2t

x

1 λ

(1 + λt) + 1

(1 + λt) λ

Z

x+y

= t

(1 + λt) λ dµ−1 (y) Zp  Z  ∞ X tn x+y dµ−1 = λ−n λ n! Zp n n=0 Z ∞ n X t = (x + y|λ)n dµ−1 . n! n=0 Zp

(13)

Thus, by (7), (12), and (13), we get (0)

gn+1 (x|λ) = (n + 1)

Z (x + y|λ)n dµ−1 .

(14)

Zp

In the viewpoint of (7), we consider the first degeneration of Genocchi polynomials which are defined by the generating function to be ∞ X

1

2 log(1 + λt) λ

x

1 λ

(1 + log(1 + λt)) + 1

(1 + log(1 + λt) λ =

m=1

(1) gm (x|λ)

tm . m!

(15)

1

By replacing t by log(1 + λt) λ in (8), we get 1

2 log(1 + λt) λ 1 λ

x

(1 + log(1 + λt) λ

(1 + log(1 + λt)) +1 Z x+y 1 log(1 + λt) (1 + log(1 + λt)) λ dµ−1 (y) = λ Zp Z ∞ X 1 1 = log(1 + λt) λ−n (x + y|λ)n dµ−1 (log(1 + λt))n λ n! Zp n=0 Z ∞ X 1 = (x + y|λ)n λ−n−1 dµ−1 (log(1 + λt))n+1 n! Z p n=0

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LEE-CHAE JANG, C.S. RYOO, JEONG GON LEE, HYUCK IN KWON ∞ (n + 1)! X m tm λ S1 (m, n + 1) n! m! n=0 Zp m=n+1 Z ∞ m−1 X X tm = (n + 1)λm−n−1 S1 (m, n + 1) (x + y|λ)n dµ−1 (y) . m! Zp m=1 n=0

=

∞ Z X

(x + y|λ)n λ−n−1 dµ−1

(16)

Thus, by (14), (15), and (16), we get (1) gm (x|λ)

= =

m−1 X

(n + 1)λ

n=0 m−1 X

m−n−1

Z S1 (m, n + 1)

(x + y|λ)n dµ−1 (y) Zp

(0)

λ−m−n−1 S1 (m, n + 1)gn+1 (x|λ).

(17)

n=0

By (17), we obtain the following theorem. Theorem 2.1. Let m ∈ N. Then we have (1) gm (x|λ)

=

m−1 X

(0)

λm−n−1 S1 (m, n + 1)gn+1 (x|λ).

(18)

n=0

Now, we consider the second degeneration of Genocchi polynomials as follows: 1

2 log(1 + log(1 + λt)) λ

1

1 λ

(1 + log(1 + log(1 + λt))) + 1 ∞ X tm (2) = gm (x|λ) . m! m=1

(1 + log(1 + log(1 + λt))) λ (19)

From (19), we get 2 λ

log(1 + log(1 + λt)) 1 λ

=

x

(1 + log(1 + log(1 + λt))) λ

(1 + log(1 + log(1 + λt))) +1 Z x+y 1 log(1 + log(1 + λt)) (1 + log(1 + log(1 + λt))) λ dµ−1 (y). λ Zp

(20)

From (20), we get

= = = =

Z x+y 1 log(1 + log(1 + λt)) (1 + log(1 + log(1 + λt))) λ dµ−1 (y) λ Zp Z ∞ X 1 1 log(1 + log(1 + λt)) λ−n (x + y|λ)n dµ−1 (y)(log(1 + log(1 + λt)))n λ n! Z p n=0 Z ∞ X 1 −n−1 (x + y|λ)n λ dµ−1 (log(1 + log(1 + λt)))n n! Zp n=0 Z ∞ ∞ X X 1 (log(1 + λt))m (x + y|λ)n λ−n−1 dµ−1 (y)(n + 1)! S1 (m, n + 1) n! Zp m! n=0 m=n+1 ∞ Z ∞ ∞ X X X tl (x + y|λ)n λ−n−1 dµ−1 (y)(n + 1) S1 (m, n + 1) S1 (l, m)λl l! n=0 Zp m=n+1 l=m

=

n3 nX ∞ X 2 −1 X n3 =0 n2 =0 n1

tn3 (0) λn3 −n1 −1 S1 (n3 , n2 )S1 (n2 , n1 + 1)gn1 +1 (x|λ) . n3 ! =0

(21)

From (20) and (21), we obtain the following theorem.

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Theorem 2.2. Let n3 ∈ N. Then we have n3 nX 2 −1 X

gn(2) (x|λ) = 3

(0)

λn3 −n1 −1 S1 (n3 , n1 )S1 (n2 , n1 + 1)gn1 +1 (x|λ).

(22)

n2 =0 n1 =0

Inductively, we have the k-th degeneration of Genocchi polynomials as follows: Theorem 2.3. Let k, nk ∈ N. Then we have gn(k−1) (x|λ) k

X

=

···

nX 2 −1

(0)

λnk −n1 −1 S1 (nk , nk−1 ) · · · S1 (n2 , n1 + 1)gn1 +1 (x|λ).

(23)

n1 −0

nk−1 =0

By replacing t by λ1 (eλt − 1) in (19) and (20). We have ∞ ∞ ∞ X X X 1 (eλt − 1)n tl = gn(2) (x|λ) n gn(2) (x|λ) λl−n S2 (l, n) λ n! l! n=1 n=1 l=n

∞ X l X tl = ( gn(2) (x|λ)λl−n S2 (l, n)) . l! n=0

(24)

l=0

By (14) and (23), we obtain the following theorem. Theorem 2.4. Let l ∈ N. Then we have l X (p) gl (x|λ) = gn(2) (x|λ)λl−n S2 (l, n).

(25)

n=0

We note the Daehee polynomials of order r is defined by the generating function to be r  ∞ X tn log(1 + t) (26) (1 + t)x = Dn(r) (x) t n! n=0 (r)

(r)

When x = 0, Dn = Dn (0) are called the Daehee numbers of order r. 1 By replacing t by log(1 + λt) λ in (7), we get 1 1 ∞ X x (log(1 + λt) λ )n 2(log(1 + λt) λ ) gn(0) (x|λ) = − (1 + log(1 + λt) λ 1 n! (1 + log(1 + λt)) λ + 1 n=1 ∞ X tn gn(1) (λ|x) . = n! n=1

(27)

and ∞ X n=1

gn(0) (x|λ)



log(1 + λt) t

n

−n t

λ

n

n!

=

∞ X

gn(0) (x|λ)

n=1 ∞ X ∞ X

∞ X

(n) t Dl

l

l!

l=0

! λ−n

tn n!

tl+n l!n! n=1 l=0  ! m ∞ m X X n t (n) = gn(0) (x|λ)Dm−n λ−n . (28) m m! m=1 n=0 =

(n)

gn(0) (x|λ)Dl λ−n

Thus, by (27) and (28), we obtain the following theorem. Theorem 2.5. Let m ∈ N. Then we have m   X m −n (0) (n) (1) gm (x|λ) = λ gn (x|λ)Dm−n . n n=0

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6

LEE-CHAE JANG, C.S. RYOO, JEONG GON LEE, HYUCK IN KWON

By replacing t by

1 λt λ (e

∞ X

− 1) in (15), we get  λt m e

(1) gm (x|λ)

m=1

−1 λ

x 2t (1 + λt) λ 1 (1 + λt) λ + 1 ∞ X tl (0) = gl (x|λ) . l!

=

m!

(30)

l=1

and ∞ X

(1) gm (x|λ)λ−m

m=1

eλt − 1 m!


m

∞ X

∞ X

(λt)l l! m=1 l=m ! ∞ ∞ X X tl l−m (1) = λ gm (x|λ)S2 (l, m) l! m=1 l=m ! ∞ l X X tl (1) = λl−m gm (x|λ)S2 (l, m) . l! m=1 =

(1) gm (x|λ)λ−m

S2 (l, m)

(31)

l=0

Thus, by (30) and (31), we obtain the following theorem. Theorem 2.6. Let l ∈ N. Then we have (0)

gl (x|λ) =

l X

(1) λl−m gm (x|λ)S2 (l, m).

(32)

m=1

References [1] A. Bayad, T. Kim, Identities involving values of Bernstein, q-Bernoulli, and q-Euler polynomials, Russ. J. Math. Phys. 18(2) (2011) 133-143. [2] L. Carlitz, Degenerate Stirling, Bernoulli and Eulerian numbers, Utilitas Math. 15 (1979) 51-88. [3] D. V. Dolgy, D. S. Kim, T. Kim, T. Mansour, Degenerate poly-Cauchy polynomials, Appl. Math. Comput. 269 (2015) 637-646. [4] S. Gaboury, R. Tremblay, B. J. Fug` ere, Some explicit formulas for certain new classes of Bernoulli, Euler and Genocchi polynomials, Proc. Jangjeon Math. Soc. 17(1) (2014) 115-123. [5] J. Jeong, S.-H. Rim, B. M. Kim, On finite-times degenerate Cauchy numbers and polynomials, Adv. Differece Equ. 2015 (2015) 2015:321. [6] D. S. Kim, T. Kim, Higher-order Cauchy of the first kind and poly-Cauchy of the first kind mixed type polynomials, Adv. Stud. Contemp. Math. 24 (2013) 512-527. [7] D. S. Kim, Identities associatied with generalized twisted Euler polynomials twisted by ramified roots of unity, Adv. Stud. Contemp. Math. 22(3) (2012) 363-377. [8] D. S. Kim, Symmetry identities for generlaized twisted Euler polynomials twisted by unramified roots of unity, Proc. Jangjeon Math. Soc. 15(3) (2012) 303-316. [9] T. Kim, On the degenerate higher-order Cauchy numbers and polynomials, Adv. Stud. Contemp. Math. 25(3) (2015) 417-421. [10] T. Kim, Degenerate Bernoulli polynomials associated with p-adic invariant integral on Zp , Adv. Stud. Contemp. Math. 25(3) (2015) 273-279. [11] T. Kim, D. S. Kim, D. V. Dolgy, Degenerate q-Euler polynomials, Adv. Difference Equ. 2015 (2015) 2015:246 [12] T. Kim, On the degenerate Cauchy numbers and polynomials, Proc. Jangjeon Math. Soc. 18(3) (2015) 307-312. [13] T. Kim, J. Y. Choi, J. Y. Sug, Extended q-Euler numbers and polynomials associated with fermionic p-adic q-integral on Zp , Russ. J. Math. Phys. 14(2) (2007) 160-163. [14] T. Kim, Barnes’ type multiple degenerate Bernoulli and Euler polynomials, Appl. Math. Comput. 258 (2015) 556-564. [15] T. Kim, A note on q-Bernstein polynimials, Russ. J. Math. Phys. 18(1) (2011) 73-82.

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[16] T. Kim, Some identities on the q-Euler polynomials of higher order and q-Stirling numbers by the fermionic p-adic integral on Zp , Russ. J. Math. Phys. 16(4) (2009) 484-491. [17] T. Kim, New approach to q-Euler polynomials of higher order, Russ. J. Math. Phys. 17(2) (2010) 218225. [18] T. Kim, On the multiple q-Genocchi and Euler numbers, Russ. J. Math. Phys. 15(4) (2008) 481-486. [19] T. Kim, q-Volkenborn integration Russ. J. Math. Phys. 9(3) (2002) 288-299. [20] T. Kim, On Euler-Barnes multiple zeta functions, Russ. J. Math. Phys. 10(3) (2003) 261-267. [21] C. S. Ryoo, Calculating zeros of the twisted Genocchi polynomials, Adv. Stud. Contemp. Math. 17(2) (2008) 147-159. [22] J. J. Seo, T. Kim, Some symmetric identities of the modified q-Euler polynomials under symmetric group of degree four, Proc. Jangjeon Math. Soc. 18(2) (2015) 211-216.

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J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 22, NO.7, 2017, COPYRIGHT 2017 EUDOXUS PRESS, LLC

Regularization solutions of ill-posed Helmholtz-type equations with fuzzy mixed boundary value† Hong Yanga,b , Zeng-Tai Gonga,∗ a College of Mathematics and Statistics, Northwest Normal University, Lanzhou 730070, China b College of Technology and Engineering, Lanzhou University of Technology, Lanzhou 730050, China

Abstract In this study, we discuss the solutions of fuzzy Helmholtz-type equations(FHTEs) and their ill-posedness. A regularization method is required to recover the numerical stability. Moreover, the error estimates and convergence of the method is considered. To support our study, one numerical example is illustrated. Keywords: Fuzzy numbers; Ill-posed problem; Helmholtz equation; Regularization method; Convergence estimate. 1. Introduction The study of fuzzy partial differential equations (FPDEs) provides a suitable setting for the mathematical modeling of real-world problems that include uncertainty or vagueness. As a new and powerful mathematical tool, FPDEs have been studied using several approaches. The first definition of an FPDE was presented by Buckley and Feuring in [1]. In [2], the authors considered the application of FPDEs obtained using fuzzy rule-based systems. Furthermore, Oberguggenberger described weak and fuzzy solutions for FPDEs [3] and Chen et al. presented a new inference method with applications to FPDEs [4]. In [5], an interpretation was provided of the use of FPDEs for modeling hydrogeological systems. Studies of heat, wave, and Poisson equations with uncertain parameters were provided in [6]. Fuzzy solutions for heat equations based on generalized Hukuhara differentiability were considered in [7]. Several numerical methods have been proposed to solve FPDEs. Such as Allahviranloo ([8]) proposed a difference method for solving FPDEs. The Adomian decomposition method was studied for finding the approximate solution of the fuzzy heat equation in [9]. Solving FPDEs by the differential transformation method was considered in [10]. In this paper, we proposed a numerical method to solve ill-posed problems for the fuzzy Helmholtztype equation (FHTEs). The Helmholtz equation arises in many areas, especially in practical physical applications, such as acoustic, wave propagation and scattering, vibration of the structure, electromagnetic scattering and so on, see [11, 12, 13, 14]. The direct problems, i.e. Dirichlet, Neumann or mixed boundary value problems for the Helmholtz equation have been studied extensively in the past century. However, in some practical problems, the boundary data on the whole boundary cannot be obtained. We only know the noisy data on a part of the boundary or at some interior points of the concerned domain, which will lead to some inverse problems and severely ill-posed problems. In 1923, Hadamard [15] introduced the concept of a well-posed problem from philosophy where the mathematical model of a physical problem must have properties where the solution exhibits uniqueness, existence, and stability. If one of the properties fails to hold, the problem is known as ill-posed. Numerical computation is difficult due to the ill-posedness of the problem. That means the solution does not depend continuously on the given Cauchy data and any small perturbation in the given data may cause large change to the solution [15, 16, 17]. The present study addresses two issues. First, we consider the ill-posedness of FHTEs using the decomposition theorem. Second, we use the regularization method to recover the numerical stability. The remainder of this paper is organized as follows. In Section2, we briefly introduce the necessary notions related to fuzzy numbers and differentiability properties for fuzzy set-valued mappings. In Section3, we define the solution and ill-posedness of FHTEs. The regularization method and convergence † ∗

Supported by the Natural Scientific Fund of China (11461062, 61262022). Corresponding Author:Zeng-Tai Gong. Tel.: +869317971430. E-mail addresses: [email protected] 1350

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estimates for the initial-boundary value problems of FHTEs are considered in Section4. In Section5, we present some numerical results and our conclusions are given in Section6. 2. Definitions and preliminaries Let Pk (Rn ) denote the family of all nonempty compact convex subset of Rn and define the addition and scalar multiplication in Pk (Rn ) as usual. Let A and B be two nonempty bounded subset of Rn . The distance between A and B is defined by the Hausdorff metric { } dH (A, B) = max sup inf ||a − b||, sup inf ||b − a|| , (2.1) a∈A b∈B

b∈B a∈A

where || · || denotes the usual Euclidean norm in Rn [18]. Then (Pk (Rn ); dH ) is a metric space. Denote E n = {u : Rn → [0, 1]|u satisfies (1)-(4) below} is a fuzzy number space, where (1) u is normal, i.e. there exists an x0 ∈ Rn such that u(x0 ) = 1, (2) u is fuzzy convex, i.e. u(λx + (1 − λ)y) ≥ min{u(x), u(y)} for any x, y ∈ Rn and 0 ≤ λ ≤ 1, (3) u is upper semi-continuous, (4) [u]0 = cl{x ∈ Rn |u(x) > 0} is compact. Here, cl(X) denotes the closure of set X. For 0 < α ≤ 1, the α-level set of u (or simply the α-cut) is defined by [u]α = {x ∈ Rn |u(x) ≥ α}. The core of u is the set of elements of Rn having membership grade 1, i.e., [u]1 = {x|x ∈ Rn , u(x) = 1}. Then from above (1)-(4), it follows that the α-level set [u]α ∈ Pk (Rn ) for all 0 < α ≤ 1. According to Zadeh’s extension principle, we have addition and scalar multiplication in fuzzy number space E n as follows: [u + v]α = [u]α + [v]α = {x + y|x ∈ [u]α , y ∈ [v]α }, [ku]α = k[u]α = {kx|x ∈ [u]α }, [0]α = {0}. where u, v ∈ E n and 0 < α ≤ 1. The distance between two fuzzy numbers u and v is defined by D(u, v) = sup dH ([u]α , [v]α ).

(2.2)

α∈[0,1]

We recall some differentiability properties for fuzzy set-valued mappings. Definition 2.1 (See[19]) Let KC denote the family of all bounded closed intervals in R, the generalized Hukuhara difference of two intervals A, B ∈ KC (gH-difference, for short), denoted by A⊖gH B, is defined by { (i)A = B + C; A ⊖gH B = C ⇐⇒ (2.3) or(ii)B = A + (−C). Definition 2.2 (See[20]) The generalized Hukuhara difference of two fuzzy numbers u, v ∈ E 1 (gHdifference, for short) is the fuzzy number ω, if it exists, such that { (i)u = v + ω; u ⊖gH v = ω ⇐⇒ (2.4) or(ii)v = u + (−ω). It is easy to show that (i) and (ii) are both valid if and only if w is a crisp number. It may happen that the gH-difference of two fuzzy numbers does not exist (see, for example, [21]). In order to overcome this shortcoming, in [20, 21], a new difference between fuzzy numbers was proposed, which always exists. 1351

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Henceforth, T=]a, b[denotes an open interval in R. A function F : T → FC is said to be a fuzzy function. For each α ∈ [0, 1], associated to F, we define the family of interval-valued functions Fα : T → KC given by Fα (x) = [F (x)]α , for x ∈ T . For any ∈ [0, 1], we denote Fα (x) = [f α (x), f α (x)].

(2.5)

Here, for each α ∈ [0, 1], the endpoint functions f α , f α : T → R are called upper and lower functions of F , respectively. Next, we present the concept of gH-differentiable fuzzy functions based on the gH-difference of fuzzy intervals. Definition 2.3 (See [21]) The gH-derivative of a fuzzy function F : T → FC at x0 ∈ T is defined as 1 [F (x0 + h) ⊖gH F (x0 )]. h→0 h

F ′ (x0 ) = lim

(2.6)

If F (x0 ) ∈ FC satisfying (2.5) exists, we say that F is generalized Hukuhara differentiable (gH-differentiable, for short) at x0 . Theorem 2.1(See [22]) If F : T → FC is gH-differentiable at x0 ∈ T , then Fα is gH-differentiable at x0 uniformly in α ∈ [0, 1] and Fα′ (x0 ) = [F ′ (x0 )]α , (2.7) for all α ∈ [0, 1]. Theorem 2.2 (See [22]) Let F: T → FC be a fuzzy function and x ∈ T . Then F is gH-differentiable at x if and only if one of the following four cases holds: (a) f α and f α are differentiable at x, uniformly in α ∈ [0, 1], (f α )′ (x) is monotonic increasing and (f α )′ (x) is monotonic decreasing as functions of α and (f α )′ (x) ≤ (f α )′ (x). In this case, Fα′ (x) = [(f α )′ (x), (f α )′ (x)].

(2.8),

for all α ∈ [0, 1]. (b) f α and f α are differentiable at x, uniformly in α ∈ [0, 1], (f α )′ (x) is monotonic increasing and (f α )′ (x) is monotonic decreasing as functions of α and (f α )′ (x) ≤ (f α )′ (x). In this case, Fα′ (x) = [f α )′ (x), (f α )′ (x)].

(2.9),

for all α ∈ [0, 1]. (c) (f α )′+/− (x) and (f α )′+/− (x) exist uniformly in α ∈ [0, 1], (f α )′+ (x) = (f α )′− (x) is monotonic

increasing and (f α )′+ (x) = (f α )′− (x) is monotonic decreasing as functions of α and (f α )′+ (x) ≤ (f α )′+ (x). In this case, Fα′ (x) = [(f α )′+ (x), f α )′+ (x)] = [f α )′− (x), (f α )′− (x)]. (2.10), for all α ∈ [0, 1]. (d) (f α )′+/− (x) and (f α )′+/− (x) exist uniformly in α ∈ [0, 1], (f α )′+ (x) = (f α )′− (x) is monotonic

increasing and (f α )′+ (x) = (f α )′− (x) is monotonic decreasing as functions of α and (f α )′+ (x) ≤ (f α )′+ (x). In this case, Fα′ (x) = [f α )′+ (x), (f α )′+ (x)] = [(f α )′− (x), f α )′− (x)]. (2.11), for all α ∈ [0, 1]. Theorem 2.3(Decomposition Theorem[23]) If u ∈ E n , then 1352

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u=

∪

(α · [u]α ).

(2.3)

α∈[0,1]

The following well-known characterization theorem makes the connection between a fuzzy interval and its LU-representation. Theorem 2.4 (See[24]) Let u be a fuzzy number. Then the functions u, u: [0, 1] → R, defining the endpoints of the α-level sets of u, satisfy the following conditions: (i) u is a bounded, non-decreasing, left-continuous function in (0, 1] and it is right-continuous at 0. (ii) u is a bounded, non-increasing, left-continuous function in (0, 1] and it is right-continuous at 0. (iii) u(1) ≤ u(1). Reciprocally, given two functions that satisfy the above conditions, they uniquely determine a fuzzy number. 3. Solutions of FHTEs and Ill-posedness Now, we consider a Cauchy problem for the Helmholtz-type equation with fuzzy initial-boundary value in a rectangle domain as follows  2 ∂ u e ∂2u e   + 2 + k2 u e=e 0, 0 < x < π, 0 < y < 1,  2  ∂x ∂y     u e(x, 0) = φ(x), e 0 ≤ x ≤ π, (3.1)  ∂e u  e  (x, 0) = 0, 0 ≤ x ≤ π,   ∂y    u e(0, y) = u e(π, y) = e 0, 0 ≤ y ≤ 1, 2 2 e where where constant k > 0 is the wave number. u e, ∂∂xue2 , ∂∂yue2 , ∂∂yue , φ(x), e 0 are fuzzy-number-valued functions and their α-cut sets are shown as follows:

[e u(x, y)]α = [u(x, y, α), u(x, y, α)], [ [

∂2u e ∂x

(x, y)]α = [ 2

∂2u ∂2u (x, y, α), (x, y, α)], ∂x2 ∂x2

∂2u e ∂2u ∂2u (x, y)] = [ (x, y, α), (x, y, α)], α ∂y 2 ∂y 2 ∂y 2 [

∂e u ∂u ∂u (x, y)]α = [ (x, y, α), (x, y, α)], ∂y ∂y ∂y

[φ(x)] e 0]α = [0(α), 0(α)]. α = [φ(x, α), φ(x, α)], [e From Theorem 2.1 and Theorem 2.2, in order to investigate the solution of (3.1), we consider the following two systems of two partial differential equations  2 ∂ u ∂2u   (x, (x, y, α) + k 2 u(x, y, α) = 0, 0 < x < π, 0 < y < 1, y, α) +  2 2  ∂x ∂y     u(x, 0) = φ(x, α), 0 ≤ x ≤ π, (3.2)  ∂u   0 ≤ x ≤ π, (x, 0, α) = 0(α),   ∂y    u(0, y, α) = u(π, y) = 0(α), 0 ≤ y ≤ 1,

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 2 ∂ u ∂2u   (x, y, α) + (x, y, α) + k 2 u(x, y, α) = 0, 0 < x < π, 0 < y < 1,  2 2  ∂x ∂y     u(x, 0, α) = φ(x, α), 0 ≤ x ≤ π,  ∂u   (x, 0, α) = 0(α), 0 ≤ x ≤ π,   ∂y    u(0, y) = u(π, y, α) = 0(α), 0 ≤ y ≤ 1,

(3.3)

Definition 3.1 (see [1]) Let u(x, y, α) and u(x, y, α) be solutions of equations (3.2) and (3.3), respectively. If [u(x, y, α), u(x, y, α)] defines the α-cut of a fuzzy number, for all (x, y) ∈ [0, π] × [0, 1], then u e(x, y) is a solution for (3.1). By the method of separation of variables, it is easy to derive a solution of the direct problem (3.2) and (3.3), respectively as follows: u(x, y, α) =

[k] ∑

cn sin(nx) cos(

√

k 2 − n2 y) +

n=1

where

u(x, y, α) =

cn sin(nx) cos(

√

∫

√

n2 − k 2 y)

(3.4)

π

(3.5)

φ(x, α) sin(nx)dx 0

k2

−

n2 y)

n=1

where

cn sin(nx) cosh(

n=[k]+1

2 cn = π [k] ∑

∞ ∑

+

∞ ∑

cn sin(nx) cosh(

√

n2 − k 2 y)

(3.6)

n=[k]+1

2 cn = π

∫

π

φ(x, α) sin(nx)dx

(3.7)

0

Obviously, for the solutions u(x, y, α) of the equations (3.2) and the solutions u(x, y, α) of the equations (3.3), [u(x, y, α), u(x, y, α)] satisfies the conditions of Theorem 2.2, [u(x, y, α), u(x, y, α)] determines a solution of problem (3.1) as follows: ∪ u= (α · [u(x, y, α), u(x, y, α)]). (3.8) α∈[0,1]

Remark 3.1 If 0 < k < 1, the first term in Equations (3.4) and (3.6) is vanished. In the following, we discuss the ill-posedness of problem (3.1). Definition 3.2 (Hadamard’s definition of well-posedness [15]) If a deterministic solution problem of FPDE satisfies the following properties (3.9-3.11), then it is well-posed. For all admissible date, a solution exists. (3.9) For all admissible date, the solution is unique. (3.10) The solution depends continuously on the date. (3.11) Conversely, if one of the properties (3.9-3.11) does not satisfy for a deterministic solution problem of FPDE, then it is ill-posed. Next, we are always suppose that (3.9) and (3.10) hold for the convenience of discussion, (3.11) does not hold. Definition 3.3 Problem of FHTEs (3.1) is said to be ill-posed if both problems of PDE (3.2) and PDE (3.3) are ill-posed. 1354

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The the systems of PDE (3.2) and (3.3) are highly ill-posed, see[16]. Thus, the systems (3.1) is ill-posed. Ill-posed problem means the solution does not depend continuously on the given Cauchy data and any small perturbation in the given data may cause large change to the solution. Thus regularization techniques are required to stabilize numerical computations. In general terms, regularization is the approximation of an ill-posed problem by a family neighbouring well-posed problems. 4. Regularization and Convergence estimates In this section, we use the solution of perturbation problems to approach the solution of problems (3.2) and (3.3). Thus the regularization solution of problems (3.1) be derived by (3.4). For 0 < k < 1, we consider the following problem  0 < x < π, 0 < y < 1, ∆v(x, y) + k 2 v(x, y) = 0,      v(x, 0) + βv(x, 1) = φδ1 (x, α), 0 ≤ x ≤ π,  v y (x, 0) = 0, 0 ≤ x ≤ π,     v(0, y) = v(π, y) = 0, 0 ≤ y ≤ 1,   ∆v(x, y) + k 2 v(x, y) = 0, 0 < x < π, 0 < y < 1,     v(x, 0) + βv(x, 1) = φδ2 (x, α), 0 ≤ x ≤ π,  v y (x, 0) = 0, 0 ≤ x ≤ π,     v(0, y) = v(π, y) = 0, 0 ≤ y ≤ 1,

(4.1)

(4.2)

where 0 < α ≤ 1 is α-level set parameter, and β > 0 is a regularization parameter. The measured data of equations (3.1) is fuzzy-number-valued function φ(x), e and its α-level set is defined as [φ(x)] e α = [φ(x, α), φ(x, α)]. φδ1 ∈ L2 (0, π), φδ2 ∈ L2 (0, π) satisfies ∥φδ1 − φ∥ ≤ δ1 ,

(4.3)

∥φδ2 − φ∥ ≤ δ2 ,

(4.4)

in which the constant δ1 > 0 and δ2 > 0 is called an error level and ∥ · ∥ denotes the L2 -norm. Further assume that there exists a constant E > 0 such that the following a priori bound exists ∥u(·, 1)∥ ≤ E.

(4.5)

By the method of separation of variables, it is easy to derive a solution of direct problem (4.1) and (4.2) as follows, respectively √ ∞ ∑ cosh( n2 − k 2 y) δ1 √ cn sin(nx), (4.6) v(x, y, α) = 2 − k2 ) 1 + β cosh( n n=1 where cδn1 v(x, y, α) =

2 = π ∞ ∑

∫

φδ1 (x, α) sin(nx)dx.

cδn2

cδn2

2 = π

(4.7)

0

n=1

where

π

∫

√ cosh( n2 − k 2 y) √ sin(nx), 1 + β cosh( n2 − k 2 )

(4.8)

π

φδ2 (x, α) sin(nx)dx.

(4.9)

0

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For k ≥ 1, we define a regularized solution v as follows: v(x, y, α) =

[k] ∑

cδn1

cosh(

√

n2

−

k 2 y)

∞ ∑

+

n=1

cδn1

√ cosh( n2 − k 2 y) √ sin(nx), 1 + β cosh( n2 − k 2 )

(4.10)

cδn2

√ cosh( n2 − k 2 y) √ sin(nx), 1 + β cosh( n2 − k 2 )

(4.11)

n=[k]+1

where cδn1 is defined by Equation (4.7).

v(x, y, α) =

[k] ∑

cδn2

cosh(

√

n2

−

k 2 y)

n=1

∞ ∑

+

n=[k]+1

where cδn2 is defined by Equation (4.9). Remark 4.1 (see[25]) For k ≥ 1, the regularized solution (4.10) and (4.11) be not an exact solution of the problem (4.1) and (4.2), respectively, but a modified solution. This is done to avoid the case √ 1 + β cos( n2 − k 2 ) = 0 for k ≥ 1 and n < k and prove a convergence result. In the following results shall show that the regularization solution v given by Equation (4.6)and(4.10), and v given by Equation (4.8) and (4.11) are a stable approximation to the exact solution u and u given by Equation (3.4) and (3.6),respectively. The regularization solution v and v depends continuously on the measured data φδ1 and φδ2 for a fixed parameter β > 0, respectively. Theorem 4.1 (see[25]) Suppose that u and u is defined by Equation (3.4) and (3.6) with the exact data φ and φ, respectively. Suppose that v is defined by Equation (4.6) for the case 0 < k < 1 or Equation (4.10)for the case k ≥ 1 with the measured data φδ1 , v is defined by Equation (4.8) for the case 0 < k < 1 or Equation (4.11) for the case k ≥ 1 with the measured data φδ2 . Let the measured data φδ1 and φδ2 satisfy Equation (4.3) and (4.4), respectively. Let the exact solution u at y = 1 satisfy Equation (4.5). If the regularization parameter β is chosen as, respectively β=

δ1 , E

δ2 , E then for fixed 0 < y < 1, we have the following convergence estimate β=

where Cy =

(4.11) (4.12)

∥v(·, y) − u(·, y)∥ ≤ δ1 + 2Cy E y δ11−y .

(4.13)

∥v(·, y) − u(·, y)∥ ≤ δ2 + 2Cy E y δ21−y .

(4.14)

1−y y. 2y ( (1−y) )

However, the convergence estimate in Equation (4.13) and (4.14) is not useful for y = 1. In order to obtain a convergence rate at y = 1, we need a stronger a priori assumption ∥

∂ p u(·, 1) ∥ ≤ E, ∂y p

(4.15)

where p ≥ 1 is an integer. We have the following convergence estimate. Theorem 4.2 (see[25]) Suppose that u and u is defined by Equation (3.4) and (3.6) with the exact data φ and φ, respectively. Suppose that v is defined by Equation (4.6) for the case 0 < k < 1 or Equation (4.10) for the case k ≥ 1 with the measured data φδ1 , v is defined by Equation (4.8) for the case 0 < k < 1 or Equation (4.11) for the case k ≥ 1 with the measured data φδ2 . Let the measured data φδ1 and φδ2 1356

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satisfy Equation (4.3) and (4.4), respectively. Let the exact solution u at y = 1 satisfy Equation (4.15). If the regularization parameter β is chosen as, respectively β=

δ1 , E

(4.16)

δ2 , E then we have the following convergence estimate at y = 1, β=

∥v(·, 1) − u(·, 1)∥ ≤ δ1 + ∥v(·, 1) − u(·, 1)∥ ≤ δ2 + where K =

√

√ √

(4.17)

δ1 E +

2E δ1 1 1 E max{K −p ( ) 3 , 2( ln )−p }. −2k E 6 δ1 1−e

(4.18)

δ2 E +

2E δ2 1 1 E max{K −p ( ) 3 , 2( ln )−p }. −2k E 6 δ2 1−e

(4.19)

([k] + 1)2 − k 2 and [·] denotes the integer part of a real number.

Theorem 4.3 Suppose that u e defined by Equation (3.8) is a solution of problem (3.1) and ve is its regularization solution. If u is defined by Equation (3.4) and v is its regularization solution defined by Equation (4.6) for the case 0 < k < 1 or Equation (4.10) for the case k ≥ 1, while u is defined by Equation (3.6) and v is its regularization solution defined by Equation (4.8) for the case 0 < k < 1 or Equation (4.11) for the case k ≥ 1. then ve is a stable approximation to u e, where ∪ ve = (α · [v(x, y, α), v(x, y, α)]). (4.20) α∈[0,1]

Proof By Equation(2.2), since D(e u, ve) = sup dH ([e u]α , [e v ]α ) α∈[0,1]

= sup max{|u(α) − v(α)|, |u(α) − v(α)|},

(4.21)

α∈[0,1]

from Theorem 4.1 and 4.2, v(α) is a stable approximation to u(α) and v(α) is a stable approximation to u(α). Hence, From (4.21) we have that ve is a stable approximation to u e. The proof is complete. 5. Numerical examples Consider the following direct problem for the Helmholtz equation with fuzzy mixed boundary value  2 ∂ u e ∂2u e    + 2 + k2 u e=e 0, 0 < x < π, 0 < y < 1,  2  ∂x ∂y    u e(x, 1) = fe(x), 0 ≤ x ≤ π, (5.1)  ∂e u  e  (x, 0) = 0, 0 ≤ x ≤ π,   ∂y     u e(0, y) = u e(π, y) = e 0, 0 ≤ y ≤ 1, in which fe : [0, π] → E 1 .

fe = ve · 2x(π − x), x ∈ [0, π].

(5.2)

where ve ∈ E 1 is given by a triangular fuzzy number  t ∈ (−1, 0),   t + 1, − t + 1, t ∈ (0, 1), ve(t) =   0, t ∈ (−∞, −1] ∪ [1, +∞). 1357

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The α-cut set of fe(x) is given by [fe(x)]α = [2x(π − x)v(t, α), 2x(π − x)v(t, α)] = [2x(π − x)(α − 1), 2x(π − x)(1 − α)].

(5.4)

In order to investigate the numerical solution of (5.1), we consider the following two systems of two partial differential equations  2 ∂ u ∂2u   + 2 + k 2 u = 0, 0 < x < π, 0 < y < 1,  2  ∂x ∂y     u(x, 1) = 2x(π − x)(α − 1), 0 ≤ x ≤ π, (5.5)  ∂u   (x, 0) = 0, 0 ≤ x ≤ π,    ∂y   u(0, y) = u(π, y) = 0, 0 ≤ y ≤ 1,  2 ∂ u ∂2u   + 2 + k 2 u = 0, 0 < x < π, 0 < y < 1,  2  ∂x ∂y     u(x, 1) = 2x(π − x)(1 − α), 0 ≤ x ≤ π,  ∂u   (x, 0) = 0, 0 ≤ x ≤ π,   ∂y    u(0, y) = u(π, y) = 0, 0 ≤ y ≤ 1,

(5.6)

By the method of separation of variables, the solution of the direct problem (5.5) and (5.6) can be obtained as follows, respectively. u(x, y, α) =

[k] ∑

cn sin(nx) cos(

√

k2

−

n2 y)

+

n=1

u(x, y, α) =

[k] ∑

∞ ∑

cn sin(nx) cosh(

√

n2 − k 2 y),

(5.7)

n2 − k 2 y),

(5.8)

n=[k]+1

cn sin(nx) cos(

√

k2

−

n2 y)

+

n=1

∞ ∑

cn sin(nx) cosh(

√

n=[k]+1

∫π ∫π 2 2 where φn = π cosh(n) dn , dn = 0 2x(π − x)(α − 1) sin(nx)dx, φn = π cosh(n) dn , dn = 0 2x(π − x)(1 − α) sin(nx)dx, and they can be computed by the Simpson formulation, respectively. Remark 5.1 If 0 < k < 1, the first term in Equations (5.7) and (5.8) is vanished. Then we choose the initial data φ(x) for equation (3.2) and φ(x) for equation (3.3) as follows, φ(x) = u(x, 0) ≈

25 ∑

φn sin(nx).

(5.9)

φn sin(nx).

(5.10)

n=1

φ(x) = u(x, 0) ≈

25 ∑ n=1

The measured data φδ and φδ2 is given by φδ1 (xi ) = φ(xi ) + ε · rand(i), and φδ2 (xi ) = φ(xi ) + ε · rand(i), 1 respectively, where ε is an error level, ( δ1 := ∥φδ − φ∥l2 = 1

N1 2 1 ∑ φδ1 (xi ) − φ(xi ) N1

)1/2 .

(5.11)

i=1

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( δ2 := ∥φδ2 − φ∥l2 =

N1 1 ∑ φδ (xi ) − φ(xi ) 2 2 N1

)1/2 .

(5.12)

i=1

The function rand(·) denotes a random number uniformly distributed in the interval [0, 1]. In our numerical computations, we always take N1 = 31. The regularization parameter β is chosen by (4.10),(4.11) and (4.15),(4.16) respectively. The numerical results for u(·, y) and uδβ (·, y) with k = 12 , ε = 0.0001, α = 12 are shown in Figure1.

0 regularization solution

0

exact solution

−10

−20

−30

−40 1

−10

−20

−30

−40 1 4

4

3

0.5

3

0.5

2

2

1 y

0

0

1 y

x

0

x

(b) regularization solution uδβ .

(a) exact solution u.

40 regularization solution

40

30 exact solution

0

20

10

0 1

30

20

10

0 1 4

4

3

0.5

3

0.5

2

2

1 y

0

0

1 y

x

0

0

x

(d) regularization solution uδβ .

(c) exact solution u.

Figure 1: ε = 1 × 10−4 , α = 12 , k = 21 . 6. Conclusion In this paper, we investigate a new numerical method of solution for inverse problem of FHTEs. We defined the ill-posedness for deterministic solution problem of FHTEs and the regularization method is proposed to solve a Cauchy problem for the ill-posed FHTEs. The convergence and stability estimates for 0 < y < 1, y = 1 have been obtained under a-priori bound assumptions for the exact solution. Finally, one example shows that our proposed numerical method is effective.

References [1] J.J. Buckley, T. Feuring, Introduction to fuzzy partial differential equations, Fuzzy Sets Syst. 105 (1999) 241-248. [2] G.Q. Zhang, Fuzzy continuous function and its properties, Fuzzy Sets Syst. 43 (2) (1991) 159-171. 1359

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[3] M. Oberguggenberger, Fuzzy and weak solutions to differential equations, in: Proceedings of the 10th International IPMU Conference, 2004. [4] Y.Y. Chen, Y.T. Chang, B.S. Chen, Fuzzy solutions to partial differential equations: adaptive approach, IEEE Trans. Fuzzy Syst. 17 (1) (2009) 116-127. [5] B.A. Faybishenko, Introduction to modeling of hydrogeologic systems using fuzzy partial differential equation, in: M. Nikravesh, L. Zadeh, V. Korotkikh (Eds.), Fuzzy Partial Differential Equations and Relational Equations: Reservoir Characterization and Modeling, Springer, 2004. [6] A.M. Bertone, R.M. Jafelice, L.C. de Barros, R.C. Bassanezi, On fuzzy solutions for partial differential equations, Fuzzy Sets Syst. 219 (2013) 68-80. [7] T. Allahviranlooa, Z. Gouyandeha., A. Armanda, A. Hasanoglub, On fuzzy solutions for heat equation based on generalized Hukuhara differentiability, Fuzzy Sets Syst. 265 (2015) 1-23. [8] T. Allahviranloo, Difference methods for fuzzy partial differential equations, Comput. Methods Appl. Math. 2 (3) (2002) 233-242 [9] T. Allahviranloo, N. Taheri, An analytic approximation to the solution of fuzzy heat equation by adomian decomposition method, Int. J. Contemp. Math. Sci. 4 (3) (2009). [10] M.B. Ahmadi, N.A. Kiani, Solving fuzzy partial differential equation by differential transformation method, Journal of Applied Mathematics, Islamic Azad University of Lahijan, 7 (2011) 1-16. [11] J.T. Chen, F.C.Wong, Dual formulation of multiple reciprocity method for the acoustic mode of a cavity with a thin partition, J. Sound Vib. 217 (1) (1998) 75-95. [12] W.S. Hall, X.Q. Mao, Boundary element investigation of irregular frequencies in electromagnetic scattering, Eng. Anal. Bound. Elem. 16 (3) (1995) 245-252. [13] I. Harari, P.E. Barbone, M. Slavutin, R. Shalom, Boundary infinite elements for the Helmholtz equation in exterior domains, Int. J. Numer. Meth. Eng. 41 (6) (1998) 1105-1131. [14] L. Marin, L. Elliott, P.J. Heggs, D.B. Ingham, D. Lesnic, X. Wen, An alternating iterative algorithm for the Cauchy problem associated to the Helmholtz equation, Comput. Meth. Appl. Mech. Eng. 192 (5-6) (2003) 709-722. [15] J. Hadamard, Lectures on Cauchy Problems in Linear PDE, New Haven: Yale University Press, 1923. [16] V. Isakov, Inverse problems for partial differential equations, Applied Mathematical Sciences, vol. 127, Springer-Verlag, New York, 1998. [17] A.N. Tikhonov, V.Y. Arsenin, Solutions of Ill-posed Problems, V.H. Winston and Sons, John Wiley and Sons, Washington, DC, New York, 1977. [18] P. Diamond, P. Kloeden, Characterization of compact subsets of fuzzy sets, Fuzzy Sets Syst. 29 (1989) 341-348. [19] L. Stefanini, B. Bede, Generalized Hukuhara differentiability of interval-valued functions and interval differential equations, Nonlinear Anal. 71 (2009) 1311–1328. [20] L. Stefanini, A generalization of Hukuhara difference and division for interval and fuzzy arithmetic, Fuzzy Sets Syst. 161 (2010) 1564–1584. [21] B. Bede, L. Stefanini, Generalized differentiability of fuzzy-valued functions, Fuzzy Sets Syst. 230 (2013) 119–141. [22] Y.Chalco-Canoa,., R.Rodriguez-Lopezb, M.D.Jimenez-Gameroc, Characterizations of generalized differentiable fuzzy functions. Fuzzy Sets Syst. Accepted 4 September 2015. [23] O. Kaleva, Fuzzy differential equations, Fuzzy Sets Syst. 24 (1987) 301-317. [24] R. Goestschel, W. Voxman, Elementary fuzzy calculus, Fuzzy Sets Syst. 18 (1986) 31-34. [25] H.W. Zhang, T. Wei, A quasi-reversibility regularization method for the Cauchy problem of the Helmholtz equation, Int J Comput Math. 88 (4) (2011) 839-850.

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Behavior of the Difference Equations xn+1 = xn xn−1 − 1 Keying Liu1, 2 ,

Peng Li2 ,

Fei Han3 ,

Weizhou Zhong1, 4

1. School of Economics and Finance, Xi’an Jiaotong University, Xi’an 710061, China 2. School of Mathematics and Information Sciences, North China University of Water Resources and Electric Power, Zhengzhou 450045, China 3. Nutrition Data Laboratory, United States Department of Agriculture, Beltsville, Maryland 20705, USA 4. College of Business Administration, Huaqiao University, Quanzhou 362021, China

Correspondence should be addressed to Zhong Weizhou, [email protected].

Abstract In this paper, the behavior of solutions of a kind of nonlinear difference equations was studied. According to the first initial value, the regions of the second initial values was partitioned by zeroes of auxiliary functions such that the asymptotical behavior of the equation was determined, which was convergent or unbounded. Key words:

Nonlinear difference equations;

AMS 2000 Subject Classification:

1

Convergent;

Unbounded

39A10, 39A11

Introduction In 2011, Kosmala[1] proposed a kind of nonlinear difference equations xn+1 = xn−k xn−l − 1,

n = 1, 2, . . .

(1)

with k, l ∈ N and the initial values being real numbers. It stems from investigating periodic difference equations. Stevi´ c and Iriˇcanin [2] presented the first general result on the behavior of solutions of (1), by describing the long-term behavior of the solutions of (1) for all values of parameters k and l, where the initial values satisfy a special condition. Moreover, some particular cases of (1) were investigated in [3–7]. Paper [3] investigated the case where k = 1, l = 2; paper [4] and [7] investigated the case where k = 0, l = 1; paper [5] investigated the case where k = 0, l = 2; paper [6] investigated the case where k = 0, l = 3. The relatively simple appearance of (1) is deceiving in that its behavior changes significantly for different choices of k and l. These results of (1) were mainly about the periodicity, unboundedness and stability for particular choices of k and l. In this paper, we consider a special case of (1), which was investigated in [4] and [7], xn+1 = xn xn−1 − 1,

n = 0, 1, 2, . . .

(2)

1 1361

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with the initial values x−1 and x0 being real numbers. Note that the equilibria x ¯ of (2) are √ √ 1+ 5 1− 5 , x ¯2 = . x ¯1 = 2 2 Furthermore, x ¯1 was locally asymptotically stable and x ¯2 is unstable[4]. We first summarize the main results[4, 7] on the solutions of (2). (1) (C) If −1 < x−1 , x0 < 0, then every solution of (2) converges to x ¯1 . (2) (UB) If one of the following holds, then the solution of (2) is unbounded. (i) (ii) (iii) (iv)

x−1 > x ¯2 , x0 > x ¯2 ; x−1 < −1, x0 < −1; x−1 < 0, x0 > 0; 0 < x−1 < 1, 0 < x0 < 1, x20 x2−1 − 2x0 x−1 + 1 − x−1 > 0.

(3) (UB or C) – If 1 < x−1 , x0 < x ¯2 , then one of the following occurs. (i) The solution of (2) is unbounded. (ii) There exists n0 ≥ 1 such that xn ∈ (−1, 0) for all n ≥ n0 . – If x−1 > 0, x0 < 0, then the solution of (2) in certain cases is bounded and in other cases is unbounded. – If 0 < x−1 , x0 < x ¯2 , then the solutions of (2) exhibit somewhat chaotic behavior relative to the initial values. A little change in the initial conditions can cause a drastic difference in the long-term behavior of the solutions. For simplicity, we show them in Figure 1. For the initial values (x−1 , x0 ) in different colored regions, the solution of (2) is of three kinds: being convergent(C) and being unbounded(UB), being unbounded or convergent(UB or C). y 3

UB

2

UB UB/C

1

UB −3

−2

−1

1

2

3

x

C −1

UB/C UB

−2

−3

Figure 1: Different regions of the initial values of (2) From the above results, one can see that these regions were presented from the perspective of the relation of two initial values of (2). For the initial values in the green regions, the corresponding solution is bounded and convergent. For the initial values in the red regions, the solution is unbounded. As far as the initial values in the blue regions is concerned, the solution was either unbounded or convergent and such a conclusion was not concise. 2 1362

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Specially, for the initial values in the blank regions, the behavior of the solution is unknown. It is interesting to investigate the evolution of the solution according to the initial values in the plane. In the following, we try to use a new method to consider the behavior of (2). Different from the method in [4], we construct auxiliary functions and then use the zeroes of them to create new partitions of the second initial value. In this way, for the first initial value which is arbitrarily chosen, the corresponding solution is convergent only for the second initial value in some intervals which are determined by the zeroes of auxiliary functions. And the lengths of these intervals are decreasing to zero.

2

Main Results In this section, we present the main result by investigating the behavior of solutions of (2). First of all, from the results in [4], we made a little generalization.

Theorem 2.1. (I) If there is an N ≥ 0 such that −1 < xN−1 , xN < 0, then {xn } of (2) converges to x ¯1 . (II) If there is an N ≥ 0 such that one of the following five conditions holds, then the solution of (2) is unbounded. 1) 2) 3) 4) 5)

xN −1 > x ¯2 , xN > x ¯2 ; xN −1 < −1, xN < −1; xN −1 < 0, xN > 0; 0 < xN −1 < 1, 0 < xN < 1, x2N x2N −1 − 2xN xN −1 +1 − xN −1 > 0; xN −1 > 0, xN < −1.

It is worth pointing out that the last case 5) is a direct result of the case 2) and it is crucial for our main result. Thus, the behavior of solutions of (2) depends on the location of its two consecutive terms of xN −1 and xN being less than −1, greater than x ¯2 or in the interval (−1, 0). However, it is still complicated in terms of the boundedness of solutions of (2) for other cases. By Remark 2.6 in [4], if the solution of (2) is not periodic or eventually periodic with minimal period three, then the solution is either bounded, while inside (−1, 0), or unbounded. Now, we present a necessary and sufficient condition on the existence of eventually prime period-three solutions of (2). Lemma 2.1. The solution {xn } of (2) is an eventually prime period-three solution if and only if there is an N ≥ 1 such that xN = 0. Proof. By Theorem 2.1 in [4], if the solution {xn } is an eventually prime period-three solution, then there is an N ≥ 1 such that xN = 0. On the other hand, if there is an N ≥ 1 such that xN = 0, then we have xN +1 = −1 and xN +2 = −1 from (2). Thus, it is an eventually prime period-three solution. In the following, letting the first initial value x−1 being fixed, we consider the behavior of the solution for the second initial value x0 , mainly on the convergence and unboundedness of the corresponding solution of (2). For simplicity, we could assume that x−1 = a and x0 = b, where a and b are real numbers. 3 1363

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Now, we introduce auxiliary functions Fi (b) = xi for i ≥ 1, from (2), which are F1 (b) = ab − 1,

(3)

F2 (b) = b F1 (b) − 1 = ab − b − 1, 2

(4)

F3 (b) = F2 (b) F1 (b) − 1 =

b (F12 (b)

F4 (b) = F3 (b) F2 (b) − 1 =

F1 (b) (F22 (b) F2 (b) (F32 (b)

F5 (b) = F4 (b) F3 (b) − 1 =

− a),

(5)

− b),

(6)

− F1 (b)),

(7)

and by induction, for i ≥ 5, we have 2 Fi+1 (b) = Fi (b) Fi−1 (b) − 1 = Fi−2 (b)(Fi−1 (b) − Fi−3 (b)),

(8)

from which we know that Fi (b) is a higher-degree polynomial of b. By listing the roots of Fi (b) = 0 for each i ≥ 1, we consider the behavior of Fi (b) with b in the intervals between these adjacent roots, which describes the long term behavior of the solution of (2) with the second initial value x0 in such intervals for the first one x−1 being fixed. In the following, we investigate the roots of Fi (b) = 0 step by step. It is obvious that r11 = 1/a is the root of F1 (b) = 0 if a ̸= 0. If a ≥ −0.25 and a ̸= 0, then F2 (b) = 0 has two roots which are √ √ 1 − 1 + 4a 1 + 1 + 4a r21 = , r22 = 2a 2a and they satisfy r21 < r11 < r22 for a > 0. It is noted that 0 is always a root of F3 (b) = 0 (for convenience, denoted by itself) and for a > 0, the other two roots are √ √ 1− a 1+ a r31 = , r32 = a a satisfying 0 < r31 < r11 < r22 < r32 for 0 < a < 1 and r31 < 0 < r11 < r22 < r32 for a > 1. From (6), we know that F4 (b) = 0 is equivalent to F1 (b) = 0 or F22 (b) = b. Thus r11 is always a root of F4 (b) = 0. From F22 (b) = b, in view of the strict monotonicity of F2 (b) for b > r11 , there are only two roots of F4 (b) = 0, satisfying r41 < r22 < r42 for a > 0 and b > 0. Similarly, the other two roots of F5 (b) = 0 satisfy r51 < r32 < r52 for a > 0 and b > r11 , which are different from r21 and r22 . Here and after, we only focus on these ”new” roots of Fi (b) = 0, which have not been labeled by other smaller indices. Now, we conclude the existence of two roots of Fi+1 (b) = 0 for i ≥ 5. Lemma 2.2. Fi+1 (b) = 0 has only two roots for a > 0 and b > r(i−3)2 for i ≥ 5. Proof. Letting rij be the roots of Fi (b) = 0 for i > 1 and j = 1, 2, from (8), we have ′ ′ Fi+1 (b) = Fi′ (b)Fi−1 (b) + Fi (b)Fi−1 (b) > 0

(9)

for b > ri2 and thus Fi+1 (b) is strictly increasing for b > ri2 . 2 From (8), we have Fi−1 (b) = Fi−3 (b) for i ≥ 5. Hence, in view of the monotonicity of Fi−1 (b) for b > r(i−2)2 and the positivity of Fi−3 (b) for b > r(i−3)2 , by induction, there are only two roots of Fi+1 (b) = 0 satisfying r(i+1)1 < r(i−1)2 < r(i+1)2 for a > 0 and b > r(i−3)2 . +∞ Furthermore, we could conclude that {ri1 }+∞ i=2 and {ri2 }i=2 are convergent.

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Lemma 2.3. lim ri1 = lim ri2 .

i→+∞

(10)

i→+∞

Proof. First, from the strict monotonicity of Fi+1 (b) for b > ri2 , we have ri2 < r(i+1)2 and thus r22 < r32 < r42 < . . .. The convergence of {ri2 }+∞ i=2 is guaranteed by Fi (b) being a higher-degree +∞ polynomial of b. Similarly, {ri1 }i=2 is convergent. Denote lim ri1 = b = b(a), lim ri2 = ˆb = ˆb(a). i→+∞

i→+∞

In order to prove (10), we suppose that b < ˆb. Then there exists N such that rN 2 > b. From the above, there exists a root r(N +2)1 of FN +2 (b) = 0 near rN 2 . In view of {ri2 }+∞ i=2 being increasing, it is enough to choose such a rN 2 that the corresponding r(N +2)1 > b, which ˆ contradicts the convergence of {ri1 }+∞ i=2 . The case of b < b is similar. In fact, we can find such ˆ an M that b < rM 1 < rM 2 and they are roots of FM (b) = 0, which contradicts the convergence of {ri2 }+∞ i=2 . Hence, (10) is true. From the above, for the first initial value x−1 = a being fixed, we have obtained that +∞ Fi (b) = 0 has only two ”new” roots for i > 3 and the sequences {ri1 }+∞ i=2 and {ri2 }i=2 converge to a same number. To investigate the behavior of the solution of (2) with initial values in these intervals which are partitioned by the adjacent roots rij , we consider three cases. Case 1

a=0 If −1 < b < 0, it follows that both F2 (b) = −b − 1 and F3 (b) = b are in the interval (−1, 0). Thus {xn } of (2) converges to x ¯1 by Theorem 2.1.

Case 2

a 0, {xn } of (2) is unbounded. Thus, the dynamics of (2) is clear.

Case 3

a>0 For this case, it is complicated to arrange these roots rij . We divide it into three cases. 3.1

0 0

which is guaranteed by 1 − F1 (r22 ) =

2(2 − a) √ >0 3 + 1 + 4a

(15)

for 0 < a < 2. Thus, the conclusion is true. In a similar way, from r61 < r42 < r62 , we conclude r51 < r61 < r32 . In fact, in view of F6 (r32 ) = 0 and F6 (r51 ) = −1, we have F6′ (r51 ) = F5′ (r51 )F4 (r51 ) > 0, F6′ (r32 ) = F3′ (r22 ) (1 − F2 (r32 )) < 0 which is guaranteed by 1 − F2 (r32 ) =

(16)

a−1 √ 0. Thus, {xn } is unbounded by Theorem 2.1. It is also true for b ∈ (0, r31 )∪(r41 , r11 )∪(r22 , r51 )∪(r61 , r32 ) which are listed in Table 1. (2) For b ∈ (r21 , 0), we have −1 < F2 (b), F3 (b) < 0. Thus, {xn } converges to x ¯1 by Theorem 2.1. It is also true for b ∈ (r31 , r41 ) ∪ (r11 , r22 ) ∪ (r51 , r61 ) which are listed in Table 2. Table 1: Intervals of x0 such that {xn } is unbounded for 0 < x−1 < 1 Intervals of x0 (−∞, r21 ) (0, r31 ) (r41 , r11 ) (r22 , r51 ) (r61 , r32 )

Reasons F1 (b) < 0, F2 (b) > 0 F2 (b) < 0, F3 (b) > 0 F3 (b) < 0, F4 (b) > 0 F4 (b) < 0, F5 (b) > 0 F5 (b) < 0, F6 (b) > 0

{xn } is unbounded unbounded unbounded unbounded unbounded

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Table 2: Intervals of x0 such that {xn } is convergent for 0 < x−1 < 1 Intervals of x0 (r21 , 0) (r31 , r41 ) (r11 , r22 ) (r51 , r61 )

Reasons −1 < F2 (b), F3 (b) < 0 −1 < F3 (b), F4 (b) < 0 −1 < F4 (b), F5 (b) < 0 −1 < F5 (b), F6 (b) < 0

{xn } converges to converges to converges to converges to

x ¯1 x ¯1 x ¯1 x ¯1

3.2 1≤a 0, (19) a−1 √ F6′ (r32 ) = F3′ (r22 ) a+ > 0. a It follows that r11 < r41 and r32 < r61 . And r22 < r51 follows from (14) and (15). Thus, (18) holds for 1 ≤ x−1 < 2. It is worth pointing out that {xn } of (2) converges to x ¯1 for 1 ≤ x−1 < 2 and x0 ∈ (r21 , r31 ) ∪ (0, r11 ) ∪ (r41 , r22 ) ∪ (r51 , r32 ) which are listed in Table 3. Table 3: Intervals of x0 such that {xn } is convergent for 1 ≤ x−1 < 2 Intervals of x0 (r21 , r31 ) (0, r11 ) (r41 , r22 ) (r51 , r32 )

Reasons −1 < F2 (b), F3 (b) < 0 −1 < F3 (b), F4 (b) < 0 −1 < F4 (b), F5 (b) < 0 −1 < F5 (b), F6 (b) < 0

{xn } converges to converges to converges to converges to

x ¯1 x ¯1 x ¯1 x ¯1

3.3 a≥2 In this case, we prove that r21 < r31 < 0 < r11 < r41 < r51 ≤ r22 < r32 < r61 < r42 < r52 < r62 .

(20)

Compared with (18), we only need to prove r22 ≥ r51 for a ≥ 2. In fact, from (14) and (15), for a ≥ 2, we have that r22 ≥ r51 . It is worth pointing out that {xn } of (2) converges to x ¯1 for x−1 ≥ 2 and x0 ∈ (r21 , r31 ) ∪ (0, r11 ) ∪ (r41 , r51 ) ∪ (r22 , r32 ) which are listed in Table 4. From the above, we derive such intervals of x0 for x−1 such that {xn } of (2) is convergent. It is worth pointing out that we couldn’t continue such a procedure because there are no explicit expressions of r42 and so on. From the above procedures, we know that the key is how to compare ri2 with rj1 where j = i + 3 for i ≥ 4. In fact, for such an interval Ii = (ri2 , rj1 ) (or (rj1 , ri2 )) where j = i + 3 for i ≥ 4, in view of auxiliary functions Fj (b), we have Fj−1 (b) < 0 and Fj (b) > 0. Thus, for x−1 being fixed and ∪ x0 ∈ Ii (the union of Ii for i ≥ 4), {xn } of (2) is unbounded by Theorem 2.1. 7 1367

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Table 4: Intervals of x0 such that {xn } is convergent for x−1 ≥ 2 Intervals of x0 (r21 , r31 ) (0, r11 ) (r41 , r51 ) (r22 , r32 )

Reasons −1 < F2 (b), F3 (b) < 0 −1 < F3 (b), F4 (b) < 0 −1 < F4 (b), F5 (b) < 0 −1 < F5 (b), F6 (b) < 0

{xn } converges to converges to converges to converges to

x ¯1 x ¯1 x ¯1 x ¯1

In view of Lemma 2.3, we obtain that the lengths of these open intervals Ii for i ≥ 4 tend to zero as i tends to +∞. For x0 > ˆb, the increasing property of {xn } of (2) leads to its divergence. Therefore, we generalize the above results into the following theorem. Theorem 2.2. The solution {xn } of (2) is unbounded only for its second initial value x0 in such open intervals depending on the first initial valuex0 , which are listed in Table 5, where the endpoints rij are the roots of auxiliary functions Fi (b) = xi = 0 with x0 = b and x−1 = a for i ≥ 1. And {xn } of (2) is an eventually prime period-three solution just at x0 = rij or x0 = 0. For x0 belongs to the complementary set of such intervals except those endpoints, {xn } of (2) is convergent to the negative equilibrium x ¯1 .

Table 5: Intervals of x0 for x−1 such that {xn } is unbounded x−1 (−∞, −0.25) [−0.25, 0) 0 (0, 1) [1, +∞)

(−∞, (−∞, (−∞, (−∞, (−∞,

Intervals of x0 r11 ) ∪ (0, +∞) r11 ) ∪ (r21 , r22 ) ∪ (0, +∞) −1) ∪ (0, +∞) ∪ r21 ) ∪ (0, r31 ) ∪ (r41 , r11 ) ∪ (ˆb, +∞) ∪ ( Ii ) ∪ r21 ) ∪ (0, r31 ) ∪ (r41 , r11 ) ∪ (ˆb, +∞) ∪ ( Ii )

From Theorem 2.2 and Table 5, for x−1 and x0 greater than zero, solutions of (2) would exhibit somewhat chaotic behavior[4], that is, {xn } is either unbounded or convergent alternately for x0 depending on x−1 , which is more concise from Table 5. Now, we give some examples for particular x−1 which are listed in Table 6. Here, we only present the former six intervals of x0 such that the solution {xn } of (2) is convergent. It is noted that the numerical values of these endpoints of these intervals are approximated to the values of the solutions of the auxiliary equations Fi (b) = 0. From Table 6, for x−1 = 1.5, it is shown the former six intervals of x0 such that the solution {xn } of (2) is convergent, which are on both sides of zero. If x0 = 1.6 in (1.4975, 1.6073), then the solution of (2) enters and then remains in the interval (−1, 0), and hence is bounded and convergent. Whereas if x0 = 1.61, then the solution is unbounded. It is clear for the third case that the solution is UB or C.

3

Conclusion

The existence of prime period-three solutions of (2) is proved in [4] and the convergence of (2) in its invariant interval (−1, 0) is proved in [7]. In this paper, we present a new method to partition the intervals of x0 depending on x−1 to describe the behavior of solutions of (2) and explain in detail that the solution of (2) exhibits somewhat chaotic behavior relative to the

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Table 6: Intervals of x0 for x−1 > 0 such that {xn } is convergent x−1 0.1 0.618 1 1.5 2.5 10

Intervals of x0 (−0.9161, 0), (6.8377, 7.6946), (10, 10.9161), (12.4540, 12.8553), (13.1623, 13.4675), (13.6396, 13.7755) (−0.6985, 0), (0.3461, 1.4048), (1.6181, 2.3166), (2.5350, 2.8614), (2.8902, 3.0690), (3.0996, 3.1756) (−0.618, 0), (0, 1), (1, 1.6180), (1.7121, 2), (2, 2.1479), (2.1637, 2.2237) (−0.5486, −0.1498), (0, 0.6667), (0.7717, 1.2153), (1.2447, 1.4832), (1.4975, 1.6073), (1.6149, 1.6633) (−0.4633, −0.2325), (0, 0.4), (0.5711, 0.8476), (0.8633, 1.0325), (1.0558, 1.1302), (1.1316, 1.1680) (−0.2702, −0.2162), (0, 0.1), (0.2740, 0.3327), (0.3702, 0.4162), (0.4383, 0.4596), (0.4630, 0.4752)

initial values. Compared with the known results[4], our results are much more accurate and easy to obtain by computers to describe the evolution of (2) for the initial values in the plane. We conclude that the solution of (2) is bounded and convergent only for x0 in particular intervals depending on x−1 , which are partitioned by the zeroes of auxiliary functions presented in this paper. Specially, it is unbounded only for x0 in such open intervals listed in Table 5 which depend on x−1 . It is of great interest to continue the investigation of the monotonicity, periodicity, and boundedness nature of solutions of (1) for different choices of parameters k and l and other equations presented in [4]. We believe that prime-period solutions and the negative equilibrium are crucial for the dynamics of difference equations (1). The future work is to extend our study to a more generalized equation (1).

Conflict of Interests The authors declare that they have no competing interests.

Acknowledgement This work is financially supported by the Natural Science Foundation of China (No. 71271086, 71172184) and the Education Department of Henan Province (No. 12A110014).

References [1] W. Kosmala,A period 5 difference equation, International Journal of Nonlinear Analysis and Applications, 2(1)( 2011): 82-84. [2] S. Stevi´ c, B. Iriˇcanin, Unbounded solutions of the difference equation xn+1 = xn−l xn−k − 1 , Abstract and Applied Analysis, Article ID 561682, 2011. [3] C. M. Kent, W. Kosmala, S. Stevi´c, Long-term behavior of solutions of the difference equation xn+1 = xn−1 xn−2 − 1 , Abstract and Applied Analysis, Article ID 152378, 2010.

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[4] C.M. Kent, W. Kosmala, M. Radin, S. Stevi´c, Solutions of the difference equation xn+1 = xn xn−1 − 1 , Abstract and Applied Analysis, Article ID 469683, 2010. [5] C.M. Kent, W. Kosmala, S. Stevi´c, On the difference equation xn+1 = xn xn−2 −1 , Abstract and Applied Analysis, Article ID 815285, 2011. [6] C.M. Kent, W. Kosmala, On the nature of solutions of the difference equation xn+1 = xn xn−3 − 1, International Journal of Nonlinear Analysis and Applications, 2(2) (2011): 24-43. [7] Yitao Wang, Yong Luo, Zhengyi Lu, Convergence of solutions of xn+1 = xn xn−1 − 1, Appl. Math. E-Notes, 12 (2012): 153-157.

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TABLE OF CONTENTS, JOURNAL OF COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 22, NO. 7, 2017

On Quadratic 𝜌-Functional Inequalities in Fuzzy Normed Spaces, Choonkil Park, Sun Young Jang, and Sungsik Yun, ………………………………………………………………………1189 On a Double Integral Equation Including a Set of Two Variables Polynomials Suggested by Laguerre Polynomials, M. Ali Özarslan and Cemaliye Kürt,………………………………...1198 Generalized Inequalities of the type of Hermite-Hadamard-Fejer with Quasi-Convex Functions by way of k-Fractional Derivatives, A. Ali, G. Gulshan, R. Hussain, A. Latif, and M. Muddassar,…………………………………………………………………………………….1208 Nonlinear Differential Polynomials of Meromorphic Functions with Regard to Multiplicity Sharing a Small Function, Jianren Long,……………………………………………………...1220 Impulsive Hybrid Fractional Quantum Difference Equations, Bashir Ahmad, Sotiris K. Ntouyas, Jessada Tariboon, Ahmed Alsaedi, and Wafa Shammakh,…………………………………...1231 A Fixed Point Alternative to the Stability of a Quadratic 𝛼-Functional Equation in Fuzzy Banach Spaces, Choonkil Park, Jung Rye Lee, and Dong Yun Shin,…………………………………1241 Four-Point Impulsive Multi-Orders Fractional Boundary Value Problems, N. I. Mahmudov and H. Mahmoud,…………………………………………………………………………………1249 Convergence of Modification of the Kantorovich-Type q-Bernstein-Schurer Operators, Qing-Bo Cai and Guorong Zhou,………………………………………………………………………..1261 Barnes-Type Degenerate Bernoulli and Euler Mixed-Type Polynomials, Taekyun Kim, Dae San Kim, Hyuckin Kwon, and Toufik Mansour,…………………………………………………..1273 Ground State Solutions for Second Order Nonlinear p-Laplacian Difference Equations with Periodic Coefficients, Ali Mai and Guowei Sun,……………………………………………..1288 On a Solutions of Fourth Order Rational Systems of Difference Equations, E. M. Elsayed, Abdullah Alotaibi, and Hajar A. Almaylabi,………………………………………………….1298 𝛼𝑥𝑛 𝑥𝑛−𝑙

On the Dynamics of Higher Order Difference Equations 𝑥𝑛+1 = 𝑎𝑥𝑛 + 𝛽𝑥

𝑛 +𝛾𝑥𝑛−𝑘

, M. M. El-

Dessoky,………………………………………………………………………………………1309 Applications of Soft Sets in BF-Algebras, Jeong Soon Han and Sun Shin Ahn,…………….1323

TABLE OF CONTENTS, JOURNAL OF COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 22, NO. 7, 2017 (continued)

Symmetric Solutions for Hybrid Fractional Differential Equations, Jessada Tariboon, Sotiris K. Ntouyas, and Suthep Suantai,…………………………………………………………………1332 On the k-th Degeneration of the Genocchi Polynomials, Lee-Chae Jang, C.S. Ryoo, Jeong Gon Lee, and Hyuck In Kwon,……………………………………………………………………..1343 Regularization Solutions of Ill-Posed Helmholtz-Type Equations with Fuzzy Mixed Boundary Value, Hong Yang and Zeng-Tai Gong,………………………………………………………1350 Behavior of the Difference Equations 𝑥𝑛+1 = 𝑥𝑛 𝑥𝑛−1 − 1, Keying Liu, Peng Li, Fei Han, and Weizhou Zhong,……………………………………………………………………………….1361