JOURNAL OF COMPUTATIONAL ANALYSIS AND APPLICATIONS VOLUME 20- 2016

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Volume 20, Number 1 ISSN:1521-1398 PRINT,1572-9206 ONLINE

January 2016

Journal of Computational Analysis and Applications EUDOXUS PRESS,LLC

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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©2016 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

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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

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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. 20, NO.1, 2016, COPYRIGHT 2016 EUDOXUS PRESS, LLC

ON THE λ-DAEHEE POLYNOMIALS WITH q-PARAMETER JIN-WOO PARK

Abstract. In this paper, we consider the generalization of Daehee polynomials with q-parameter and investigate some properties of those polynomials.

1. Introduction Let p be a fixed prime number. Throughout this paper, Zp , Qp , and Cp will respectively denote the ring of p-adic rational integers, the field of p-adic rational numbers and the completions of algebraic closure of Qp . The p-adic norm is defined |p|p = p1 . When one talks of q-extension, q is variously considered as an indeterminate, a complex q ∈ C, or p-adic number q ∈ Cp . If q ∈ C, one normally assumes that 1 |q| < 1. If q ∈ Cp , then we assume that |q − 1|p < p− p−1 so that q x = exp(x log q) for each x ∈ Zp . Throughout this paper, we use the notation : [x]q =

1 − qx . 1−q

Note that limq→1 [x]q = x for each x ∈ Zp . Let U D(Zp ) be the space of uniformly differentiable functions on Zp . For f ∈ U D(Zp ), the p-adic invariant integral on Zp is defined by Kim as follows : Z pn −1 1 X I (f ) = f (x) dµ0 (x) = lim n f (x) , (see [4, 5, 6]). (1.1) n→∞ p Zp x=0 Let f1 be the translation of f with f1 (x) = f (x + 1) . Then, by (1.1), we get df (x) I (f1 ) = I (f ) + f 0 (0) , where f 0 (0) = . (1.2) dx x=0

As it is well-known fact, the Stirling number of the first kind is defined by (x)n = x (x − 1) · · · (x − n + 1) =

n X

S1 (n, l) xl ,

(1.3)

l=0

and the Stirling number of the second kind is given by the generating function to be ∞ X
m tl (1.4) et − 1 = m! S2 (l, m) , l! l=m

(see [1, 10]). 1991 Mathematics Subject Classification. 05A19, 11B65, 11B83. Key words and phrases. Bernoulli polynomials, Daehee polynomials with q-parameter, p-adic invariant integral. 1

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JIN-WOO PARK 11-20

J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 20, NO.1, 2016, COPYRIGHT 2016 EUDOXUS PRESS, LLC

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JIN-WOO PARK

Unsigned Stirling numbers of the first kind is given by xn = x(x + 1) · · · (x + n − 1) =

n X

|S1 (n, l)|xl .

(1.5)

l=0

Note that if we replace x to −x in (1.3), then (−x)n =(−1)n xn =

n X

S1 (n, l)(−1)l xl

l=0 n

=(−1)

n X

(1.6) l

|S1 (n, l)|x .

l=0

Hence S1 (n, l) = |S1 (n, l)|(−1)n−l . For r ∈ N, the Bernoulli polynomials of order r are defined by the generating function to be r  ∞ X t tn xt (r) e = , (see [7, 8, 11]). (1.7) B (x) n et − 1 n! n=0 (r)

(r)

When x = 0, Bn = Bn (0) are called the Bernoulli numbers of order r, and in the (1) special case, r = 1, Bn (x) = Bn (x) are called the ordinary Bernoulli polynomials. For n ∈ N, let Tp be the p-adic locally constant space defined by Tp = ∪ Cpn = lim Cpn , n→∞

n≥1



pn

where Cpn = ω|ω = 1 is the cyclic group of order pn . 1 We assume that q is an indeterminate in Cp with |1 − q|p < p− p−1 . Then we define the q-analogue of falling factorial sequence as follows : (x)n,q = x(x − q)(x − 2q) · · · (x − (n − 1)q), (n ≥ 1), (x)0,q = 1. Note that lim (x)n,q = (x)n =

q→1

n X

S1 (n, l)xl .

l=0

Recently, D. S. Kim and T. Kim introduced the Daehee polynomials as follows : Z Dn (x) = (x + y)n dµ0 (y), (n ≥ 0), (see [2, 5, 9]). (1.8) Zp

When x = 0, Dn = Dn (0) are called the n’s Daehee numbers. From (1.8), we can derive the generating function to be   ∞ X log(1 + t) tn (1 + t)x = Dn (x) , (see [2]). (1.9) t n! n=0 In addition, D. S. Kim et. al. consider the Daehee polynomials with q-parameter which is defined by the generating function to be ∞ X n=0

Dn,q

x tn log(1 + qt)  , (see [3]). = (1 + qt) q  1 n! q (1 + qt) q − 1

(1.10)

When x = 0, Dn,q = Dn,q (0) are called the Daehee numbers with q-parameter.

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JIN-WOO PARK 11-20

J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 20, NO.1, 2016, COPYRIGHT 2016 EUDOXUS PRESS, LLC

ON THE λ-DAEHEE POLYNOMIALS WITH q-PARAMETER

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In the viewpoint of generalization of the Daehee polynomials with q-parameter, we consider the λ-Daehee polynomials with q-parameter are defined to be ∞ X n=0

Dn,q (λ|x)

x tn λ log(1 + qt) . = (1 + qt) q  λ n! q (1 + qt) q − 1

(1.11)

When x = 0, Dn,q (λ) = Dn,q (λ|0) are called the λ-Daehee numbers with qparameter. In particular, the case λ = 1 is the Daehee polynomials with qparameter. In this paper, we give a p-adic integral representation of the λ- Daehee polynomials with q-parameter, which are called the Witt-type formula for the λ-Daehee polynomials with q-parameter. We can derive some interesting properties related to the λ-Daehee polynomials with q-parameter.

2. Witt-type formula for the n-th λ-Daehee polynomials with q-parameter 1

In this section, we assume that t, q ∈ Cp with |t|p < |q|p p− p−1 and λ ∈ Zp . First, we consider the following integral representation associated with falling factorial sequences : Z (x + λy)n,q dµ0 (y), where n ∈ Z+ = N ∪ {0} .

(2.1)

Zp

By (2.1), ∞ Z X n=0

∞

(x + λy)n,q dµ0 (y)

Zp

tn X n = q n! n=0

 tn x + λy dµ0 (y) q n! Zp n   Z ∞ x+λy X q = qn dµ0 (y)tn n Zp n=0 Z x+λy = (1 + qt) q dµ0 (y) Z



(2.2)

Zp 1

1

where t ∈ Cp with |t|p < |q|p p− p−1 . For t ∈ Cp with |t|p < |q|p p− p−1 , we get Z x+λy x λ log(1 + qt)  (1 + qt) q dµ0 (y) =(1 + qt) q  λ Zp q (1 + qt) q − 1 ∞ X

tn = Dn,q (λ|x) . n! n=0

(2.3)

By (2.2) and (2.3), we obtain the following theorem. Theorem 2.1. For n ≥ 0, we have Z Dn,q (λ|x) =

(x + λy)n,q dµ0 (y). Zp

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JIN-WOO PARK 11-20

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JIN-WOO PARK 1 q

In (2.3), by replacing t by

(et − 1), we have

λ ∞ n X x Dn,q (λ|x) (et − 1) qt qt = e λ n q n! eqt − 1 n=0

=

∞ X

Bn

 x  λn tn

n=0

λ

q n n!

(2.4) ,

and ∞ ∞ ∞ X
n X Dn,q (λ|x) 1 t tm Dn,q (λ|x) X e S (m, n) − 1 = 2 qn n! qn m! m=n n=0 n=0 ! ∞ m X X Dm,q (λ|x) tn S (n, m) = . 2 qm n! n=0 m=0

(2.5)

By (2.4) and (2.5), we obtain the following corollary. Corollary 2.2. For n ≥ 0, we have n x X Bn = Dm,q (λ|x)q n−m λ−n S2 (n, m). λ m=0 By the Theorem 2.1, Z Dn,q (λ|x) =

(x + λy)n,q dµ0 (y) Zp



Z

=q n

Zp n X

=q n

x + λy q



1 S1 (n, l) ql

l=0

dµ0 (y)

(2.6)

n

Z

(x + λy)l dµ0 (y).

Zp

By (1.2), we can derive easily that Z e(x+λy)t dµ0 (y) =

∞  x  (λt)n X λt xt e = Bn λt e −1 λ n! n=0 Z ∞ X tl = (x + λy)l dµ0 (y) , l! Zp

Zp

(2.7)

l=0

and so Bn

x λ

Z =

x λ

Zp

+y

n

dµ0 (y), (n ≥ 0).

(2.8)

By (1.6), (2.7) and (2.8), we obtain the following corollary. Corollary 2.3. For n ≥ 0, we have Dn,q (λ|x) = =

n X l=0 n X

q n−l S1 (n, l)λl Bl

x λ

|S1 (l, n)|(−q)n−l λl Bl

l=0

14

x λ

.

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ON THE λ-DAEHEE POLYNOMIALS WITH q-PARAMETER

5

From now on, we consider λ-Daehee polynomials of order k(∈ N) with q-parameter. λ-Daehee polynomials of order k with q-parameter are defined by the multivariant p-adic invariant integral on Zp : Z Z (k) Dn,q (λ|x) = ··· (λ(x1 + · · · + xk ) + x)n,q dµ0 (x1 ) · · · dµ0 (xk ) (2.9) Zp

Zp (k)

where n is an nonnegative integer and k ∈ N. In the special case, x = 0, Dnq (λ) = (k) Dn,q (λ|0) are called the λ-Daehee numbers of order k with q-parameter. (k) From (2.9), we can derive the generating function of Dn,q (x) as follows: ∞ X

=

n=0 ∞ X

tn n! Z  λ(x1 +···+xk )+x 

(k) Dn,q (λ|x)

q

n

Z

n=0

Z Zp

n

Zp

Z ···

=

q

··· Zp

(1 + qt)

λ(x1 +···+xk )+x q

dµ0 (x1 ) · · · dµ0 (xk )tn

dµ0 (x1 ) · · · dµ0 (xk )

(2.10)

Zp

Z

x

Z ···

= (1 + qt) q

(1 + qt)

Zp

λ(x1 +···+xk ) q

dµ0 (x1 ) · · · dµ0 (xk )

Zp

k λ log(1 + qt)  . = (1 + qt)   λ q (1 + qt) q − 1 

x q

Note that, by (2.9), (k) Dn,q (λ|x)

Z Z n X (2.11) S1 (n, m) ··· (λ(x1 + · · · + xk ) + x)m dµ0 (x1 ) · · · dµ0 (xk ). =q m q Zp Zp m=0 n

Since Z

Z

e(x1 +···+xk +x)t dµ0 (x1 ) · · · dµ0 (xk )

··· Zp

 = we can derive easily Z Bn(k) (x) = Zp

Zp

t et − 1

k

ext =

∞ X n=0

Z

Bn(k) (x)

tn , n!

(x1 + · · · + xk + x)n dµ0 (x1 ) · · · dµ0 (xk ).

···

(2.12)

Zp

Thus, by (2.11) and (2.12), we have (k) Dn,q (λ|x) =q n

= =

n X S1 (n, m) m (k)  x  λ Bm qm λ m=0

n X m=0 n X

(k) q n−m S1 (n, m)Bm

x

(k) |S1 (n, m)|(−q)n−m Bm

m=0

15

(2.13)

λ x λ

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In (2.10), by replacing t by 1q (et − 1), we get ∞ X

x (et − 1)n (k) =e q t Dn,q (λ|x) n n! q n=0

=

∞ X n=0

e

λt q λ qt

!k

−1

(2.14)

(k) x
n n Bn λ t , λ n q n!

and ∞ ∞ ∞ (k) (k) X X Dn,q (λ|x) 1 t tl Dn,q (λ|x) X n (e S2 (l, n) − 1) = n n q n! q l! n=0 n=0 l=n

∞ X

=

! m (k) X Dn,q (λ|x) tm S (m, n) . 2 qn m! n=0

m=0

(2.15)

By (2.13), (2.14) and (2.15), we obtain the following theorem. Theorem 2.4. For n ≥ 0 and k ∈ N, we have n   X (k) (k) x Dn,q (λ|x) = q n−m S1 (n, m)Bm λ m=0 =

n X

(k) |S1 (n, m)|(−q)n−m Bm

m=0

and Bn(k)

x λ

n X

= λ−n

x λ

,

(k) Dm,q (λ|x)q n−m S2 (n, m).

m=0

Now, we consider the λ-Daehee polynomials of the second kind with q-parameter as follows : Z b n,ξ,q (λ|x) = D (−λy + x)n,q dµ0 (y), (n ≥ 0). (2.16) Zp

b n,q (λ) = D b n,q (λ|0) are called the λ-Daehee numbers of In the special case, x = 0, D the second kind with q-parameter. By (2.16), we have  Z  −λy + x n b Dn,q (λ|x) = q dµ0 (y), (2.17) q Zp n b n,q (x) by (1.1) as follows : and so we can derive the generating function of D   Z ∞ ∞ X tn X n −λy + x tn b Dn,q (λ|x) = q dµ0 (y) n! n=0 q n! Zp n n=0   Z ∞ −λy+x X q = qn dµ0 (y)tn n Z p n=0 (2.18) Z −λy+x = (1 + qt) q dµ0 (y) Zp

= (1 + qt)

16

x+λ q

λ log(1 + qt)  . λ q (1 + qt) q − 1

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J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 20, NO.1, 2016, COPYRIGHT 2016 EUDOXUS PRESS, LLC

ON THE λ-DAEHEE POLYNOMIALS WITH q-PARAMETER

From (1.3), (1.6) and (2.17), we get  Z  −λy + x b n,q (λ|x) =q n D dµ0 (y) q Zp n Z X n S1 (n, l) (−λy + x)l dµ0 (y) =q n l q Zp l=0 Z  n X x l l dµ0 (y)q n−l = S1 (n, l)(−λ) y− λ Zp =

l=0 n X l=0

7

(2.19)

 x q n−l S1 (n, l)(−λ)l Bl − λ

=(−1)n

n X l=0

 x |S1 (n, l)|λl Bl − q n−l . λ

By replacing qt to et − 1 in the equation (2.18), we have ∞ X

λ t
(x+λ)t b n,q (λ|x) 1 et − 1 n =  q e q D λ t n! q eq − 1 n=0

=

∞ X n=0

Bn



(2.20)

x  n −n tn 1+ , λ q λ n!

and, by (1.4), ∞ X

∞ X
b n,q (λ|x) 1 et − 1 n = D n! n=0 n=0

n X

! b m,q (λ|x)S2 (n, m) D

m=0

tm . m!

(2.21)

Note that , by (1.10), it is easy to show that Bn (−x) = (−1)n Bn (x + 1). Thus, from (2.19), (2.20) and (2.21), we have the following theorem. Theorem 2.5. For n ≥ 0, we have b n,q (λ|x) = D

n X l=0

 x S1 (n, l)(−λ)l Bl − q n−l λ

=(−1)n

n X l=0

 x |S1 (n, l)|λl q n−l Bl − . λ

and n  X x b m,q (λ|x)S2 (n, m). = qn D λn Bn 1 + λ m=0

By Theorem 2.5, we obtain the following corollary. Corollary 2.6. For n ≥ 0, b n,q (λ|x) = q n D

n X l X

b m,q (λ|x)S1 (n, l)S2 (l, m). D

(2.22)

l=0 m=0

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Now, we observe that q −n (−1)n

Z  x+λy  Dn,q (λ|x) q =(−1)n dµ0 (y) n! n Zp  Z  x+λy − q +n−1 dµ0 (y) = n Zp  Z  −x−λy  n  X n−1 q = dµ0 (y) n − m m Z p m=1  n  X b m,q (λ| − x) n − 1 q −n D = , m! n−m m=1

and, by the similar method to (2.23), we have  n  X b n,q (λ|x) D n − 1 Dn,q (λ| − x) −n q −n (−1)n = q . n! m! n−m m=1

(2.23)

(2.24)

Hence, by (2.23) and (2.24), we obtain the following theorem. Theorem 2.7. For n ≥ 1, we have q −n (−1)n

b n  X n−1 D Dn,q (λ|x) m,q (λ| − x) −n = q n! m! n−m m=1

q −n (−1)n

 n  X b n,q (λ|x) D n − 1 Dn,q (λ| − x) −n = q . n−m n! m! m=1

and

Now, we consider higher-order λ-Daehee polynomials of second kind with qparameter. Higher-order λ-Daehee polynomials of second kind with q-parameter are defined by the multivariant p-adic invariant integral on Zp : Z Z b (k) (λ|x) = D · · · (−λ(x1 + · · · + xk ) + x)n,q dµ0 (x1 ) · · · dµ0 (xk ) (2.25) n,ξ,q Zp

Zp (k)

b n,q (λ) = where n is an nonnegative integer and k ∈ N. In the special case, x = 0, D (k) b Dn,q (λ|0) are called the higher-order λ-Daehee numbers of second kind with qparameter. (k) b n,q (λ|x) as follows: From (2.25), we can derive the generating function of D ∞ X

n

t (k) b n,q D (λ|x) n! n=0  −λ(x1 +···+xk )+x  Z Z ∞ X n q = q ··· dµ0 (x1 ) · · · dµ0 (xk )tn n Z Z p p n=0 Z Z −λ(x1 +···+xk )+x q = ··· (1 + qt) dµ0 (x1 ) · · · dµ0 (xk ) Zp

(2.26)

Zp

k λ log(1 + qt)    . λ q (1 + qt) q − 1 

= (1 + qt)

x+λk q

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ON THE λ-DAEHEE POLYNOMIALS WITH q-PARAMETER

9

By (2.25), (k) b n,q D (λ|x)

Z Z n X S1 (n, m) · · · (−λ(x1 + · · · + xk ) + x)m dµ0 (x1 ) · · · dµ0 (xk ) m q Zp Zp m=0 Z Z  n X x m S1 (n, m) m n (−λ) · · · x + · · · + x − dµ0 (x1 ) · · · dµ0 (xk ) =q 1 k qm λ Zp Zp m=0 =q n

=q n

n  x X S1 (n, m) m (k) (−λ) B − m qm λ m=0

=(−1)n

n X

 x (k) q n−m λm |S1 (n, m)|Bm − . λ m=0 (2.27) (k)

(k)

From (1.10), we know that Bn (−x) = (−1)n Bn (k + x). Hence, by (2.27), we obtain the following theorem. Theorem 2.8. For n ≥ 0, we have (k) b n,q D (λ|x) =

n X

 x (k) S1 (n, m)q n−m (−λ)m Bm − λ m=0

=(−1)n

n X

 x (k) (−λ)m q n−m |S1 (n, m)|Bm k+ . λ m=0

In (2.26), by replacing t by 1q (et − 1), we get ∞ X

t n b (k) (λ|x) (e − 1) =e qt (x+λk) D n,q n q n! n=0

λt q

!k

λt

e q −1


(k) ∞ X λn Bn λx + k tn = , qn n! n=0

(2.28)

and ∞ b (k) ∞ b (k) ∞ X
n X Dn,q (λ|x) 1 t Dn,q (λ|x) X tl e − 1 = S2 (l, n) n n q n! q l! n=0 n=0 l=n

=

∞ X m=0

! m b (k) X Dn,q (λ|x) tm S (m, n) . 2 qn m! n=0

(2.29)

By (2.28) and (2.29), we obtain the following theorem. Theorem 2.9. For n ≥ 0 and k ∈ N, we have Bn(k)

x λ

n  X b (k) (λ|x)q n−m S2 (n, m). + k = λ−n D m,q m=0

By Theorem 2.8 and Theorem 2.9, we obtain the following corollary.

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Corollary 2.10. For n ≥ 0, we have n X m X b (k) (λ|x) = b (k) (λ|x)q n−l S1 (n, m)S2 (m, l). D D n,q l,q m=0 l=0

References [1] L. Comtet, Advanced Combinatorics, Reidel, Dordrecht, 1974. [2] D. S. Kim and T. Kim, Daehee numbers and polynomials, Appl. Math. Sci., 7 (2013), no. 120, 5969-5976. [3] D. S. Kim, T. Kim, H. I. Kwon and J. J. Seo, Daehee polynomials with q-parameter, Adv. Studies Theor. Phys., 8 (2014), no. 13, 561-569. [4] T. Kim, On q-analogye of the p-adic log gamma functions and related integral, J. Number Theory, 76 (1999), no. 2, 320-329. [5] T. Kim, An invariant p-adic integral associated with Daehee numbers, Integral Transforms Spec. Funct., 13 (2002), no. 1, 65-69. [6] T. Kim, q-Volkenborn integration, Russ. J. Math. Phys., 9 (2002), no. 3, 288-299. [7] Q. L. Luo, Some recursion formulae and relations for Bernoulli numbers and Euler numbers of higher order, Adv. Stud. Contemp. Math. 10 (2005), no. 1, 63-70. [8] H. Ozden, I. N. Cangul and Y. Simsek, Remarks on q -Bernoulli numbers associated with Daehee numbers, Adv. Stud. Contemp. Math., 18 (2009), no. 1, 41-48. [9] J. W. Park, S. H. Rim and J. Kim, The twisted Daehee numbers and polynomials, Adv. Difference Equ., 2014, 2014:1. [10] S. Roman, The umbral calculus, Dover Publ. Inc. New York, 2005. [11] Y. Simsek,Generating functions of the twisted Bernoulli numbers and polynomials associated with their interpolation functions, Adv. Stud. Contemp. Math. 16 (2008), no. 2, 251-278. Department of Mathematics Education, Daegu University, Jillyang, Gyeongsan, Gyeongbuk 712-714, Republic of Korea. E-mail address: [email protected]

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Stability of ternary quadratic derivation on ternary Banach algebras: revisited Choonkil Park Research Institute for Natural Sciences, Hanyang University, Seoul 133-791, Korea Abstract. In [6], Shagholi et al. defined ternary quadratic derivations on ternary Banach algebras and proved the HyersUlam stability of ternary quadratic derivations on ternary Banach algebras. But the definition is not well-defined and so the proofs of the main results are wrong. In this paper, we correct the definition of ternary quadratic derivation and the proofs of the main results.

1. Introduction The study of stability problems for functional equations is related to a question of Ulam [7] concerning the stability of group homomorphisms and affirmatively answered for Banach spaces by Hyers [3]. Subsequently, the result of Hyers was generalized by Aoki [1] for additive mappings and by Th. M. Rassias [4] for linear mappings by considering an unbounded Cauchy difference. The functional equation f (x + y) + f (x − y) = 2f (x) + 2f (y)

(1.1)

is called a quadratic functional equation. In particular, every solution of the quadratic equation (1.1) is said to be a quadratic mapping. The Hyers-Ulam stability problem for the quadratic functional equation (1.1) was proved by Skof for mappings f : A → B, where A is a normed space and B is a Banach space (see [5]). In [6], Shagholi et al. defined a ternary quadratic derivation D from a ternary Banach algebra A into a ternary Banach algebra B such that D[x, y, z] = [D(x), y 2 , z 2 ] + [x2 , D(y), z 2 ] + [x2 , y 2 , D(z)] for all x, y, z ∈ A. But x2 , y 2 , z 2 are not defined and the brackets of the right side are not defined, since A is not an algebra and D(x) ∈ B and y 2 , z 2 ∈ A. So we correct them as follows. Definition 1.1. Let A be an algebra and ternary Banach algebra with norm k · k. A mapping D : A → A is called a ternary quadratic derivation if (1) D is a quadratic mapping, (2) D[x, y, z] = [D(x), y 2 , z 2 ] + [x2 , D(y), z 2 ] + [x2 , y 2 , D(z)] for all x, y, z ∈ A. In this paper, the proofs of the main results given in [6] are corrected. 2. Stability of ternary quadratic derivations Let A be an algebra and ternary Banach algebra with norm k · k. Theorem 2.1. Let f : A → A be a mapping for which there exists a function φ : A × A × A → [0, ∞) such that ˜ y, z) := φ(x,

∞ X 1 φ(2j x, 2j y, 2j z) < ∞ j 4 j=0

kf (x + y) + f (x − y) − 2f (x) − 2f (y)k ≤ φ(x, y, 0), 2

2

2

2

2

2

(2.1) (2.2)

kf ([x, y, z]) − [f (x), y , z ] − [x , f (y), z ] − [x , y , f (z)])k ≤ φ(x, y, z)

(2.3)

for all x, y, z ∈ A. Then there exists a unique ternary quadratic derivation D : A → A such that 1˜ kf (x) − D(x)k ≤ φ(x, x, 0), 4 for all x ∈ A.

(2.4)

0

2010 Mathematics Subject Classification: 39B52, 13N15, 47B47. Keywords: Hyers-Ulam stability; quadratic functional equation; ternary Banach algebra; ternary quadratic derivation. 0 E-mail: [email protected] 0

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C. Park Proof. Putting x = y = 0 in (2.2), we get f (0) = 0. If we replace y in (2.2) by x and multiply both sides of (2.2) by 14 , we get φ(x, x, 0) f (2x) k − f (x)k ≤ (2.5) 4 4 for all x ∈ A. Now we use the Rassias’ method on inequality (2.5) (see [2]). One can use induction on n to show that k

n−1 f (2n x) 1 X φ(2j x, 2j x, 0) − f (x)k ≤ 2n 2 4 j=0 4j

(2.6)

for all x ∈ A and all nonnegative integers n. Hence k

f (2n+m x) f (2m x) 1 − k≤ 22m 4 22(n+m)

n+m−1 X j=m

φ(2j x, 2j x, 0) 4j n

x) } is Cauchy. for all nonnegative integers n and m with n ≥ m and all x ∈ A. It follows from (2.1) that the sequence { f (2 22n Due to the completeness of A, this sequence is convergent. So one can define the mapping D : A → A by

D(x) := lim

n→∞

f (2n x) 22n

for all x ∈ A. Replacing x, y by 2n x, 2n y, respectively, in (2.2) and multiplying both sides of (2.2) by

(2.7) 1 , 22n

we get

kD(x + y) + D(x − y) − 2D(x) − 2D(y)k 1 kf (2n (x + y)) + f (2n (x − y)) − 2f (2n x) − 2f (2n y)k 22n φ(2n x, 2n y, 0) ≤ lim =0 n→∞ 22n for all x, y ∈ A and all nonnegative integers n. So = lim

n→∞

D(x + y) + D(x − y) = 2D(x) + 2D(y) for all x, y ∈ A. Moreover, it follows from (2.6) and (2.7) that kf (x) − D(x)k ≤

1˜ φ(x, x, 0) 4

for all x ∈ A. It follows from (2.3) we get kD([x, y, z]) − [D(x), y 2 , z 2 ] − [x2 , D(y), z 2 ] − [x2 , y 2 , D(z)]k 1 kf ([2n x, 2n y, 2n z]) − [f (2n x), (2n y)2 , (2n z)2 ] − [(2n x)2 , f (2n y), (2n z)2 ] − [(2n x)2 , (2n y)2 , f (2n z)]k 43n φ(2n x, 2n y, 2n z) φ(2n x, 2n y, 2n z) ≤ lim ≤ lim =0 3n n→∞ n→∞ 4 4n for all x, y, z ∈ A. So D[x, y, z] = [D(x), y 2 , z 2 ] + [x2 , D(y), z 2 ] + [x2 , y 2 , D(z)] ≤ lim

n→∞

for all x, y, z ∈ A. Now, let D0 : A → A be another ternary quadratic derivation satisfying (2.4). Then we have 1 kD(2n x) − D0 (2n x)k 22n 1 ≤ 2n (kD(2n x) − f (2n x)kB + kf (2n x) − D0 (2n x)k) 2 2 ≤ 2n φ(2n x, 2n x, 0) 2 which tends to zero as n → ∞ for all x ∈ A. So we can conclude that D(x) = D0 (x) for all x ∈ A. This proves the uniqueness of D. Thus, the mapping D : A → A is a unique ternary quadratic derivation satisfying (2.4).  kD(x) − D0 (x)k =

Theorem 2.2. Let f : A → A be a mapping for which there exists a function φ : A × A × A → [0, ∞) satisfying (2.2), (2.3) and ∞ X x y z 43j φ( j , j , j ) < ∞ (2.8) 2 2 2 j=0 22

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Ternary quadratic derivation on ternary Banach algebras for all x, y, z ∈ A. Then there exists a unique ternary quadratic derivation D : A → A such that ˜ x , x , 0), kf (x) − D(x)k ≤ φ( 2 2 for all x ∈ A. Here, ∞ X x y z ˜ y, z) := φ(x, 4j φ( j , j , j )9 2 2 2 j=0

(2.9)

for all x, y, z ∈ A. Proof. It follows from (2.5) that

x x x kf (x) − 4f ( )k ≤ φ( , , 0) 2 2 2

for all x ∈ A. By the same reasoning as in the proof od Theorem 2.1, one can define a quadartic unique mapping D : A → A by x (2.10) D(x) := lim 22n f ( n ) n→∞ 2 for all x ∈ A. It follows from (2.8) and (2.10) that kD([x, y, z]) − [D(x), y 2 , z 2 ] − [x2 , D(y), z 2 ] − [x2 , y 2 , D(z)]k x y z x y z x y z x y z ≤ lim 43n kf ([ n , n , n ]) − [f ( n ), ( n )2 , ( n )2 ] − [( n )2 , f ( n ), ( n )2 ] − [( n )2 , ( n )2 , f ( n )]k n→∞ 2 2 2 2 2 2 2 2 2 2 2 2 x y z ≤ lim 43n φ( n , n , n ) = 0 n→∞ 2 2 2 for all x, y, z ∈ A. So D[x, y, z] = [D(x), y 2 , z 2 ] + [x2 , D(y), z 2 ] + [x2 , y 2 , D(z)] for all x, y, z ∈ A. Thus the mapping D : A → A is a unique ternary quadratic derivation satisfying (2.9).



From Theorems 2.1 and 2.2, we obtain the following corollary concerning the Hyers-Ulam stability of the functional equation (1.1). Corollary 2.3. Let p and θ be nonnegative real numbers with p 6= 2, and let f : A → A be a mapping such that kf (x + y) + f (x − y) − 2f (x) − 2f (y)k ≤ θ(kxkp + kykp ), kf ([x, y, z]) − [f (x), y 2 , z 2 ] − [x2 , f (y), z 2 ] − [x2 , y 2 , f (z)])k ≤ θ(kxkp + kykp + kzkp ), for all x, y, z ∈ A. Then there exists a unique ternary quadratic derivation D : A → A such that 2θ kf (x) − D(x)k ≤ kxkp 4 − 2p holds for all x ∈ X, where p < 2, and the inequality 2θ kf (x) − D(x)k ≤ p kxkp 2 −4 holds for all x ∈ X, where p > 6.

References [1] [2] [3] [4] [5] [6]

T. Aoki, On the stability of the linear transformation in Banach spaces, J. Math. Soc. Japan 2 (1950), 64–66. Z. Gajda, On stability of additive mappings, Internat. J. Math. Math. Sci. 14 (1991), 431–434. D. H. Hyers, On the stability of the linear functional equation, Proc. Natl. Acad. Sci, 27 (1941), 222–224. Th. M. Rassias, On the stability of the linear mapping in Banach spaces, Proc. Amer. Math. Soc. 72 (1978), 297–300. F. Skof, Propriet locali e approssimazione di operatori, Rend. Sem. Mat. Fis. Milano 53 (1983), 113–129. S. Shagholi, M. Eshaghi Gordji and M. B. Savadkouhi, Stability of ternary quadratic derivation on ternary Banach algebras. J. Comput. Anal. Appl. 13 (2011), 1097–1105. [7] S. M. Ulam, Problems in Modern Mathematics, Chapter V I, Science ed., Wiley, New York, 1940.

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SOME PROPERTIES OF MODULAR S -METRIC SPACES AND ITS FIXED POINT RESULTS MELTEM ERDEN EGE AND CIHANGIR ALACA

Abstract. In this paper, we introduce modular

†

S -metric spaces and deal with their some properties. S -metric spaces.

We also prove some xed point theorems on complete modular

1.

Introduction

Fixed point theory in metric spaces begins with the Banach Contraction Principle which is published in 1922 [6]. Since it is simple and useful, it has become a very popular tool to solve existence problems in mathematical analysis. There are some authors introduced the generalization of metric spaces such as Gahler [16], which is called 2-metric space, and Dhage [14], which is called D-metric space. In 2013, Mustafa and Sims [24] found that the fundamental topology properties of the metric spaces are incorrect. They [25] introduced a generalization of metric spaces which is called G-metric spaces. The concept of S -metric spaces was rstly introduced by Sedghi et al. [28] in 2012. Sedghi and Dung [29] proved a general xed point theorem in S -metric spaces which is a generalization [[28], Theorem 3.1]. Gupta [17] introduced the concepts of cyclic contraction on S -metric space and proved some xed point theorems on S -metric spaces. Chouhan [12] proved a common unique xed point theorem for expansive mappings in S -metric space. Hieu et al. [18] gave a xed point theorem for a class of maps depending on another map on S -metric spaces. The notion of modular space was rstly introduced by Nakano [26] and developed by Koshi, Shimogaki, Yamamuro (see [22, 30]) and others. Recently, many researchers have been interested in xed point of modular space. In 2008, Chistyakov [7] introduced the notion of modular metric space generated by F -modular and developed the theory of this space. He also dened the notion of a modular on an arbitrary set and the modular metric spaces in 2010 [8]. Abdou [1] studied and proved some new xed points theorems for pointwise and asymptotic pointwise contraction mappings in modular metric spaces. Azadifer et. al. [3] introduced the notion of modular G-metric spaces and proved some xed point theorems of contractive in this space. Many authors studied on modular metric spaces [4],[5],[10],[11],[19],[20],[21]. In this paper we introduce the concept of modular S -metric spaces and their properties. Then we give xed point theorems for self mappings on complete modular S -metric spaces. 2.

Preliminaries

Denition 2.1. [27]. A modular on a real linear space X is a functional ρ : X −→ [0, ∞] satisfying the followings: (A1) ρ(0) = 0; (A2) If x ∈ X and ρ(αx) = 0 for all numbers α > 0, then x = 0; (A3) ρ(−x) = ρ(x) for all x ∈ X ; (A4) ρ(αx + βy) ≤ ρ(x) + ρ(y) for all α, β ≥ 0 with α + β = 1 and x, y ∈ X . Let X be a non-empty set and λ ∈ (0, ∞). We remark that the function ω : (0, ∞)×X ×X −→ [0, ∞] is denoted by ωλ (x, y) = ω(λ, x, y) for all λ > 0 and x, y ∈ X . Mathematics Subject Classication. 46A80, 47H10, 54E35. Key words and phrases. modular S -metric space, s-contraction, xed point. 2010

† :Corresponding

Author.

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Denition 2.2. [8]. Let X be a non-empty set, a function ω : (0, ∞) × X × X

−→ [0, ∞] is said to

be a metric modular on X if satisfying, for all x, y, z ∈ X the following conditions hold: (i) ωλ (x, y) = 0 for all λ > 0 ⇔ x = y ; (ii) ωλ (x, y) = ωλ (y, x) for all λ > 0; (iii) ωλ+µ (x, y) ≤ ωλ (x, z) + ωµ (z, y) for all λ, µ > 0.

Denition 2.3. [28] Let

X be a non-empty set. An S -metric on X is a function S : X 3 → [0, ∞) that satises the following conditions, for each x, y, z, a ∈ X , (i) S(x, y, z) ≥ 0; (ii) S(x, y, z) = 0 if and only if x = y = z ; (iii) S(x, y, z) ≤ S(x, x, a) + S(y, y, a) + S(z, z, a). The pair (X, S) is called an S -metric space.

3.

Modular

S -metric

spaces

We dene a new concept combining with S -metric and modular metric space.

Denition 3.1. Let X be a non-empty set. An modular S -metric on X is a function sλ : (0, ∞) × X × X × X → [0, ∞]

that satises the following conditions for all x, y, z ∈ X and λ > 0 : (S1) sλ (x, y, z) ≥ 0; (S2) sλ (x, y, z) = 0 if and only if x = y = z ; (S3) sλ+µ+ν (x, y, z) ≤ sλ (x, x, a) + sµ (y, y, a) + sν (z, z, a) for all λ, µ, ν > 0 and a ∈ X .

Example 3.2. (1) sλ (x, y, z) = 0 if x = y = z and sλ (x, y, z) = ∞ if x 6= y 6= z . (2) If S is an modular S -metric on X , we can get:

(a) sλ (x, y, z) = 0 if λ > S(x, y, z) and sλ (x, y, z) = ∞ if λ ≤ S(x, y, z). (b) sλ (x, y, z) = 0 if λ ≥ S(x, y, z) and sλ (x, y, z) = ∞ if λ < S(x, y, z). S(x, y, z) ; where ϕ : (0, ∞) → (0, ∞) is non-decreasing function. (c) sλ (x, y, z) = ϕ(λ)

Lemma 3.3. If the function have sλ (x, x, y) = sλ (y, y, x).

0 < λ → sλ (x, y, z) is continuous on (0, ∞) where x, y, z ∈ X , then we

Proof. There exists ε > 0 such that sλ (x, x, y) ≤ sε (x, x, x) + sε (x, x, x) + sλ−2ε (y, y, x).

If we take limit as ε → 0, we get sλ (x, x, y) ≤ sλ (y, y, x). Similarly sλ (y, y, x) ≤ sλ (x, x, y). So we get sλ (x, x, y) ≤ sλ (y, y, x) ≤ sλ (x, x, y)

and sλ (x, x, y) = sλ (y, y, x). 

Remark 3.4. The function sλ (x, y, z) for λ > 0 is non-increasing on (0, ∞) where x, y, z ∈ X , if it is continuous on (0, ∞). In fact if 0 < ν < µ < λ, (S3) implies sλ (x, x, y) ≤ sλ−µ (x, x, x) + sµ−ν (x, x, x) + sν (y, y, x)

and we have sλ (x, x, y) ≤ sν (y, y, x)

from (S2). From Lemma 3.3, we conclude that sλ (x, x, y) ≤ sν (x, x, y). From that inequality the function sλ (x, y, z) is non-increasing on (0, ∞). It follows that at each point λ > 0 the right limit sλ+0 (x, y, z) = lim sµ (x, y, z) µ→λ+0

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and the left limit sλ−0 (x, y, z) = lim sλ−ε (x, y, z) ε→0

exist in [0, ∞] and the following two inequalities hold: sλ+0 (x, y, z) ≤ sλ (x, y, z) ≤ sλ−0 (x, y, z).

Denition 3.5. Let sλ be a modular S -metric on X . The binary relation ∼s on X dened for x, y ∈ X

by

(3.1)

s

x ∼ y ⇔ lim sλ (x, x, y) = 0 λ→∞

is an equivalence relation. Indeed x ∼ x is clear by virtue of (S2). From Lemma 3.3, we have s

s

s

x ∼ y ⇔ lim sλ (x, x, y) = 0 = lim sλ (y, y, x) ⇔ y ∼ x. λ→∞

λ→∞

If x ∼ y and y ∼ z , we get lim sλ (x, x, y) = 0 and lim sλ (y, y, z) = 0. By (S3) and Lemma 3.3, λ→∞ λ→∞ we conclude that s

s

lim s3λ (x, x, z) ≤ lim sλ (x, x, y) + lim sλ (x, x, y) + lim sλ (y, y, z)

λ→∞

λ→∞

λ→∞

λ→∞

=0 + 0 + 0.

It is clear that

s

lim s3λ (x, x, z) = 0 ⇔ x ∼ z

λ→∞

by (S1). The equivalence class of the element x ∈ X in the quotient set X ∼ is dened by s

s

Xs ≡ Xs (x) = {y ∈ X : y ∼ x}.

For x0 ∈ X , the set Xs∗ is dened as follows: Xs∗ ≡ Xs∗ (x0 ) = {x ∈ X : ∃λ = λ(x) > 0 such that sλ (x, x, x0 ) < ∞}.

From Remark 3.4, the function ∼

s

s

s

S : (X ∼) × (X ∼) × (X ∼) → [0, ∞]

given by

∼

S(Xs (x), Xs (x), Xs (y)) = sλ (x, x, y) is well-dened and satises the axioms of S -metric.

Theorem 3.6. If

sλ is a modular S -metric on X , then the modular set Xs is an modular S -metric space with S -metric given by S ◦ (x, x, y) = inf{λ > 0 : sλ (x, x, y) ≤ λ},

for all x, y ∈ Xs . Proof. Since x ∼ y , there exists λ0 > 0 such that s

sλ (x, x, y) ≤ 1

for all λ ≥ λ0 by (3.1). Taking λ1 = max{1, λ0 }, we get sλ1 (x, x, y) ≤ 1 ≤ λ1

which together with the denition of S (x, x, y) gives ◦

S ◦ (x, x, y) ≤ λ1 < ∞.

Given x ∈ Xs , (S2) implies that sλ (x, x, x) = 0 for all λ > 0, and so, S ◦ (x, x, x) = 0. Let sλ satisfy (S2), x, y ∈ Xs and S ◦ (x, x, y) = 0. Then sµ (x, x, y) does not exceed µ for all µ > 0. Hence for any λ > 0 and 0 < µ < λ, from Remark 3.4 we have sλ (x, x, y) ≤ sµ (x, x, y) ≤ µ → 0 as µ → +0. It follows that sλ (x, x, y) = 0 for all λ > 0. Thus axiom (S2) implies x = y .

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It is clear from (S1) that S ◦ (x, x, y) ≥ 0. Now we show the triangle inequality: S ◦ (x, x, y) ≤ 2S ◦ (x, x, z) + S ◦ (y, y, z)

for some z ∈ Xs . In fact by the denition of S ◦ for any λ > S ◦ (x, x, z) and µ > S ◦ (y, y, z), we nd sλ (x, x, z) ≤ λ and sµ (y, y, z) ≤ µ. As a result, we get s2λ+µ (x, x, y) ≤ 2sλ (x, x, z) + sµ (y, y, z) ≤ 2λ + µ

by the axiom (S3). It follows from the denition of S ◦ that S ◦ (x, x, y) ≤ 2λ + µ and it remains to pass limit as λ → S ◦ (x, x, z) and µ → S ◦ (y, y, z). 

Theorem 3.7. Let sλ be a modular S -metric on a set X and S 1 (x, x, y) = inf{λ + sλ (x, x, y) : λ > 0}

be dened for all x, y ∈ Xs . Then S 1 is an S -metric on Xs such that S ◦ ≤ S 1 ≤ 2S ◦ . Proof. Since, for x, y ∈ Xs , the value sλ (x, x, y) is nite due to (3.1) for λ > 0 large enough, then the set {λ + sλ (x, x, y) : λ > 0} ⊂ R+ is non-empty and bounded from below, therefore S 1 (x, x, y) ∈ R+ . Since sλ (x, x, x) = 0, then from the denition of S 1 , S 1 (x, x, x) = inf{λ + sλ (x, x, x) : λ > 0} = 0. |

{z 0

}

Let sλ satisfy (S2), x, y ∈ Xs and S 1 (x, x, y) = 0. The equality x = y will follow from (S2) if we show that sλ (x, x, y) = 0 for all λ > 0. On the contrary, suppose that sλ0 (x, x, y) > 0 for some λ0 > 0. Then for λ ≥ λ0 we nd λ + sλ (x, x, y) ≥ λ0 , and if 0 < λ < λ0 , then 0 < sλ0 (x, x, y) ≤ sλ (x, x, y) ≤ λ + sλ (x, x, y)

from Remark 3.4. Thus, λ + sλ (x, x, y) ≥ λ1 = min{λ0 , sλ0 (x, x, y)} for all λ > 0. By the denition of S 1 , S 1 (x, x, y) ≥ λ1 > 0, which contradicts the assumption. Now let us show that triangle inequality: S 1 (x, x, y) ≤ 2S 1 (x, x, z) + S 1 (y, y, z). For any ε > 0 we nd λ = λ(ε) > 0 and µ = µ(ε) > 0 such that λ + sλ (x, x, z) ≤ S 1 (x, x, z) + ε and µ + sµ (y, y, z) ≤ S 1 (y, y, z) + ε

from the denition of S 1 . Applying axiom (S3), S 1 (x, x, y) ≤(2λ + µ) + s2λ+µ (x, x, y) ≤ 2λ + µ + 2sλ (x, x, z) + sµ (y, y, z) ≤2S 1 (x, x, z) + 2ε + S 1 (y, y, z)

and it remains to take into account the arbitrariness of ε > 0. Let us prove that metrics S ◦ and S 1 are equivalent on Xs . In order to obtain the left-hand side inequality, suppose that λ > 0 is arbitrary. If sλ (x, x, y) ≤ λ, then the denition of S ◦ implies S ◦ (x, x, y) ≤ λ. If sλ (x, x, y) > λ, then S ◦ (x, x, y) ≤ sλ (x, x, y). Setting µ = sλ (x, x, y) we nd µ > λ. Thus it follows from Remark 3.4 that sµ (x, x, y) ≤ sλ (x, x, y) = µ.

Hence

S ◦ (x, x, y) ≤ µ = sλ (x, x, y).

Therefore for any λ > 0 we have S ◦ (x, x, y) ≤ max{λ, sλ (x, x, y)} ≤ λ + sλ (x, x, y).

Taking the inmum over all λ > 0, we get the inequality S ◦ (x, x, y) ≤ S 1 (x, x, y)

To obtain the right-hand side inequality, we note that given λ > 0 such that S ◦ (x, x, y) < λ by the denition of S ◦ . We get sλ (x, x, y) ≤ λ. So S 1 (x, x, y) ≤ λ + sλ (x, x, y) ≤ 2λ. Passing to the limit as λ → S ◦ (x, x, y), we get S 1 (x, x, y) ≤ 2S ◦ (x, x, y).



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Theorem 3.8. Let sλ be a modular S -metric on a set X , x, y ∈ Xs and λ > 0. We have (a) (b) (c) (d)

If S ◦ (x, x, y) < λ, then sλ (x, x, y) ≤ S ◦ (x, x, y) < λ. If sλ (x, x, y) = λ, then S ◦ (x, x, y) = λ. If λ = S ◦ (x, x, y) > 0, then sλ+0 (x, x, y) ≤ λ ≤ sλ−0 (x, x, y). If the function µ → sµ (x, x, y) is continuous from the right on (0, ∞), then along with (a) − (c) we have: S ◦ (x, x, y) ≤ λ ⇔ sλ (x, x, y) ≤ λ.

(e) If the function µ → sµ (x, x, y) is continuous from the left on (0, ∞), then along with (a) − (c) we have: S ◦ (x, x, y) < λ ⇔ sλ (x, x, y) < λ.

(f) If the function µ → sµ (x, x, y) is continuous on (0, ∞), then along with (a) − (e) we have: S ◦ (x, x, y) = λ ⇔ sλ (x, x, y) = λ.

Proof. (a) For any µ > 0 such that S ◦ (x, x, y) < µ < λ by the denition of S ◦ and Remark 3.4, we have sµ (x, x, y) ≤ µ and sλ (x, x, y) ≤ sµ (x, x, y). Hence sλ (x, x, y) ≤ µ and it remains to pass to the limit as µ → S ◦ (x, x, y). (b) By the denition, S ◦ (x, x, y) ≤ λ and item (a) implies S ◦ (x, x, y) = λ. (c) For any µ > λ = S ◦ (x, x, y), the denition of S ◦ implies sµ (x, x, y) ≤ µ and so sλ+0 (x, x, y) = lim sµ (x, x, y) ≤ lim µ = λ. µ→λ+0

µ→λ+0

For any 0 < µ < λ we nd sµ (x, x, y) > µ and so sλ−0 (x, x, y) = lim sµ (x, x, y) ≥ lim µ = λ. µ→λ−0

µ→λ−0

(d) The sucient condition follows from the denition of S ◦ . Let us prove the reverse implication. If S ◦ (x, x, y) < λ, then by virtue of item (a), sλ (x, x, y) < λ and if S ◦ (x, x, y) = λ, then sλ (x, x, y) = sλ+0 (x, x, y) ≤ λ

which is a consequence of the continuity from the right of the function µ → sµ (x, x, y) and item (c). (e) By item (a), it suces to prove the sucient condition. The denition of S ◦ gives S ◦ (x, x, y) ≤ λ but if S ◦ (x, x, y) = λ, then by item (c) we would have sλ (x, x, y) = sλ−0 (x, x, y) ≥ λ

which contradicts the assumption. (f ) Sucient condition follows from (b). For the reverse asertion the two inequalities sλ (x, x, y) ≤ λ ≤ sλ (x, x, y)

follows from (c).



Denition 3.9. Let sλ be a modular S -metric on a set X . (1) A sequence (xn ) ⊂ Xs∗ converges to x ∈ Xs∗ if sλ (xn , xn , x) → 0 as n → ∞. That is, for each ε > 0, there exists n0 ∈ N such that for all n ≥ n0 we have sλ (xn , xn , x) < ε. We write xn → x. (2) A sequence (xn ) ⊂ Xs∗ is a s-Cauchy if sλ (xn , xn , xm ) → 0 as m, n → ∞. That is, for each ε > 0, there exists n0 ∈ N such that for all n ≥ n0 we have sλ (xn , xn , xm ) < ε. (3) The modular S -metric space Xs∗ is s-complete if every s-Cauchy is a s-convergent in Xs∗ .

Lemma 3.10. Let sλ be a modular S -metric on a set X . If xn → x and yn → y, then sλ (xn , xn , yn ) → sλ (x, x, y).

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Proof. Since xn → x and yn → y , then for each ε > 0 there exist n1 , n2 ∈ N such that ∀n ≥n1 , sλ (xn , xn , x) < ε ∀n ≥n2 , sλ (yn , yn , y) < ε.

Without loss of generality we can assume ε 4 ε ∀n ≥n2 , sδ (yn , yn , y) < ε(δ) = . 4 If we set n0 = max{n1 , n2 }, therefore for every n ≥ n0 we get ∀n ≥n1 , sδ (xn , xn , x) < ε(δ) =

sλ (xn , xn , yn ) ≤2sδ (xn , xn , x) + sλ−2δ (yn , yn , x) ≤2sδ (xn , xn , x) + 2sδ (yn , yn , y) + sλ−4δ (x, x, y)

for λ > δ > 0 by triangle inequality. If we take δ → 0, we have ε ε sλ (xn , xn , yn ) ≤ + + sλ (x, x, y) 2 2 sλ (xn , xn , yn ) ≤ε + sλ (x, x, y) sλ (xn , xn , yn ) − sλ (x, x, y) ≤ ε.

On the other hand we get sλ (x, x, y) ≤2sδ (x, x, xn ) + sλ−2δ (y, y, xn ) ≤2sδ (x, x, xn ) + 2sδ (y, y, yn ) + sλ−4δ (xn , xn , yn ).

From Lemma 3.3 and taking the limit as δ → 0 we have: ε ε sλ (x, x, y) ≤ + + sλ (xn , xn , yn ) 2 2 ≤ε + sλ (xn , xn , yn ) sλ (x, x, y) − sλ (xn , xn , yn ) ≤ ε.

So we get from that inequalities |sλ (xn , xn , yn ) − sλ (x, x, y)| < ε, that is, sλ (xn , xn , yn ) → sλ (x, x, y). 

4.

Fixed Point Theorems

In this section we introduce some xed point theorems on modular S -metric space.

Denition 4.1. Let

sλ be a modular S -metric on a set X . A map T : Xs∗ → Xs∗ is said to be a s-contraction if there exists a constant 0 ≤ k < 1 such that sλ (T x, T x, T y) ≤ ksλ (x, x, y)

for all x, y ∈ X .

Corollary 4.2. Let Xs∗ , Ys∗ modular S -metric spaces and f at x ∈ Xs∗ if and only if f (xn ) → f (x) where xn → x.

Theorem 4.3. Let Xs∗ be a s-complete and T point u ∈

Xs∗

.

: Xs∗ → Ys∗ be a map. Then f is continuous

: Xs∗ → Xs∗ be s-contraction. Then T has a unique xed

Proof. First, we show uniqueness. Suppose that there exist x, y ∈ Xs∗ with x = T x and y = T y . Then sλ (x, x, y) = sλ (T x, T x, T y) ≤ ksλ (x, x, y) .

Therefore sλ (x, x, y) = 0.

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To show the existence, we select x ∈ Xs∗ and show that (T n x) is a Cauchy sequence. For n = 0, 1, 2, · · · , we get by induction sλ (T n x, T n x, T n+1 x) ≤ksλ (T n−1 x, T n−1 x, T n x)

.. . ≤k n sλ (x, x, T x).

Taking the limit as n → ∞, we get lim sλ (T n x, T n x, T n+1 x) = 0.

n→∞

Thus there exists ε > 0 such that sλ (T n x, T n x, T n+1 x) ≤ ε.

Without loss of generality, we can assume that there exists m−2 X

sλ (T n x, T n x, T m x) ≤2

s

λ m−n

ε m−n

(T i x, T i x, T i+1 x) + s

for λ m−n

λ m−n

such that

(T m−1 x, T m−1 x, T m x)

i=n m−2 X

ki s

≤2

λ m−n

(x, x, T x) + k m−1 s

λ m−n

(x, x, T x)

i=n

ε ε +···+ ) m−n m−n ≤2ε.

≤2(

That is for m > n, sλ (T n x, T n x, T m x) ≤ 2ε.

This shows that (T n x) is a Cauchy sequence and since Xs∗ is s-complete, there exists u ∈ Xs∗ with lim T n x = u. n→∞ From the continuity of T , we get u = lim T n+1 x = lim T (T n x) = T u. n→∞

n→∞

Therefore u is a xed point of T .



Let M be the family of all continuous functions of ve variables M : R5+ → R+ . For some k ∈ [0, 1), we consider the following conditions: (C1) For all x, y, z ∈ R+ , if y ≤ M (x, x, 0, z, y) with z ≤ 2x + y , then y ≤ kx. (C2) If y ≤ M (y, 0, y, y, 0) for all y ∈ R+ , then y = 0.

Theorem 4.4. Let T be a self-map on s-complete Xs∗ and (4.1)

sλ (T x, T x, T y) ≤ M (sλ (x, x, y), sλ (T x, T x, x), sλ (T x, T x, y), s3λ (T y, T y, x), sλ (T y, T y, y))

for all x, y, z ∈ Xs∗ and some M ∈ M. Then we have (1) If M satises the condition (C1), then T has a xed point. (2) If M satises the condition (C2) and T has a xed point x, then the xed point is unique. Proof. (1) For each x0 ∈ Xs∗ and n ∈ N, we take xn+1 = T xn . It follows from (4.1) and Lemma 3.3 that sλ (xn+1 , xn+1 , xn+2 ) =sλ (T xn , T xn , T xn+1 ) ≤M (sλ (xn , xn , xn+1 ), sλ (xn+1 , xn+1 , xn ), sλ (xn+1 , xn+1 , xn+1 ), s3λ (xn+2 , xn+2 , xn ), sλ (xn+2 , xn+2 , xn+1 )) =M (sλ (xn , xn , xn+1 ), sλ (xn , xn , xn+1 ), 0, s3λ (xn , xn , xn+2 ), sλ (xn+1 , xn+1 , xn+2 )).

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By triangle inequality and Lemma 3.3, we have (4.2)

s3λ (xn , xn , xn+2 ) ≤ 2sλ (xn , xn , xn+1 ) + sλ (xn+1 , xn+1 , xn+2 )

From (4.2), we see that z ≤ 2x + y . Since M satises the condition (C1), there exists k ∈ [0, 1) such that (4.3)

sλ (xn+1 , xn+1 , xn+2 ) ≤ ksλ (xn , xn , xn+1 ) ≤ · · · ≤ k n+1 sλ (x0 , x0 , x1 ).

Taking the limit as n → ∞, we get lim sλ (xn , xn , xn+1 ) = 0.

n→∞

Hence there exists ε > 0 for λ > 0 such that sλ (xn , xn , xn+1 ) ≤ ε.

Without loss of generality, we can assume that there exists s

λ m−n

(xn , xn , xn+1 ) ≤

ε m−n

for

λ m−n

> 0 such that

ε . m−n

Thus for all n < m by using (S3), Remark 3.4 and (4.3) we have sλ (xn , xn , xm ) ≤2s λ (xn , xn , xn+1 ) + s λ (xm , xm , xn+1 ) 3

3

≤2s λ (xn , xn , xn+1 ) + s λ (xn+1, xn+1 , xm )

.. .

3

3

ε ε ε + +···+ ) m−n m−n m−n ≤2ε.

≤2(

This proves that (xn ) is s-Cauchy in the s-complete Xs∗ . Then (xn ) converges an x ∈ Xs∗ . Now we prove that x is a xed point of T . By using (4.1), we get sλ (xn+1 , xn+1 , T x) =sλ (T xn , T xn , T x) ≤M (sλ (xn , xn , x), sλ (T xn , T xn , xn ), sλ (T xn , T xn , x), s3λ (T x, T x, xn ), sλ (T x, T x, x)).

Since M ∈ M, then using Lemma 3.10 and taking the limit as n → ∞, we obtain sλ (x, x, T x) ≤ M (0, 0, 0, s3λ (T x, T x, x), sλ (T x, T x, x)).

From Remark 3.4, we can rewrite s3λ (T x, T x, x) ≤ sλ (T x, T x, x).

Then the inequality can be written as follows: sλ (x, x, T x) ≤ M (0, 0, 0, sλ (T x, T x, x), sλ (T x, T x, x)).

Since M satises the condition (C1), then sλ (x, x, T x) ≤ k.0 = 0. This proves that x = T x.

(2) Let x, y be xed points of T . We prove that x = y. It follows from (4.1) that sλ (x, x, y) =sλ (T x, T x, T y)

≤M (sλ (x, x, y), sλ (T x, T x, x), sλ (T x, T x, y), s3λ (T y, T y, x), sλ (T y, T y, y)) ≤M (sλ (x, x, y), 0, sλ (x, x, y), s3λ (y, y, x), 0).

From Remark 3.4 and Lemma 3.3, we get sλ (x, x, y) ≤ M (sλ (x, x, y), 0, sλ (x, x, y), sλ (x, x, y), 0).

Since M satises the condition (C2), sλ (x, x, y) = 0 ⇐⇒ x = y. 

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SOME PROPERTIES OF MODULAR

S -METRIC

SPACES AND ITS FIXED POINT RESULTS

Remark 4.5. Theorem 4.3 is a corollary of Theorem 4.4 when we take k ∈ [0, 1) and x, y, z, s, t ∈ R+ .

9

M (x, y, z, s, t) = k.x for

Now we will give a new corollary of Theorem 4.4.

Corollary 4.6. Let T be a self map on s-complete Xs∗ and sλ (T x, T x, T y) ≤ a(sλ (T x, T x, x) + sλ (T y, T y, y))

for some a ∈ [0,

1 2)

and all x, y ∈ Xs∗ . Then T has a unique xed point in Xs∗ .

Proof. We must show that M (x, y, z, s, t) = a(y + t) satises conditions (C1) and (C2). First, we have M (x, x, 0, z, y) = a(x + y).

So, if y ≤ M (x, x, 0, z, y) with z ≤ 2x + y , then y ≤M (x, x, 0, z, y) = a(x + y) y ≤ax + ay a y≤ x 1−a a

∈ [0, 1). Therefore, M satises condition (C1). with 1−a If y ≤ M (y, 0, y, y, 0) = 0, then y = 0. Therefore, M satises the condition (C2). Since sλ (T x, T x, T y) ≤M (sλ (x, x, y), sλ (T x, T x, x), sλ (T x, T x, y), sλ (T y, T y, x), sλ (T y, T y, y)) =a(sλ (T x, T x, x) + sλ (T y, T y, y)), T has a unique xed point in Xs∗ by Theorem 4.4.



Open problems How can obtain some similar results for the papers (see [2, 15]) in fuzzy metric spaces with the help of this technique? References [1] A.A.N. Abdou, On asymptotic pointwise contractions in modular metric spaces, Abstract and Applied Analysis, Article ID 501631, 2013, 1-7. [2] C. Alaca, Fixed point results for mappings satisfying a general contractive condition of operator type in dislocated fuzzy quasi-metric spaces, J. Computational Analysis and Applications, 12 (1-b), 361-368 (2010). [3] B. Azadifar, M. Maramaei, Gh. Sadeghi, On the modular G-metric spaces and xed point theorems, J. Nonlinear Sci. Appl. 6, 293-304 (2013). [4] B. Azadifar, M. Maramaei and Gh. Sadeghi, Common xed point theorems in modular G-metric spaces, Nonlinear Anal. Appl (2013) 1-9. [5] B. Azadifar, Gh. Sadeghi, R. Saadati and C. Park, Integral type contractions in modular metric spaces. Journal of Inequalities and Applications, 2013(1), 483. [6] S. Banach, Sur les operations dans les ensembles abstraits et leurs applications aux equations integrales, Fund. Math., 3, 133-181 (1922). [7] V.V. Chistyakov, Modular metric spaces generated by

F -modulars,

Folia Math., 14, 3-25 (2008).

[8] V.V. Chistyakov, Modular metric spaces I. basic conceps, Nonlinear Anal., 72, 1-14 (2010). [9] V.V. Chistyakov, Fixed points of modular contractive maps, Dokl. Math., 86, 515-518 (2012). [10] V.V. Chistyakov, Modular contractions and their application, In Models, Algorithms, and Technologies for Network Analysis (pp. 65-92), Springer New York, (2013). [11] Y.J. Cho, R. Saadati and G. Sadeghi, Quasi-contractive mappings in modular metric spaces, J. Appl. Math., 907951 (2012). [12] P. Chouhan, A common unique xed point theorem for expansive type mappings in

S -metric

spaces, International

Mathematical Forum, 8, 1287-1293 (2013). [13] H. Dehghan, M.E. Gordji and A. Ebadian, Comment on xed point theorems for contraction mappings in modular metric spaces, Fixed Point Theory and Applications, 144 (2012). [14] B.C. Dhage, Generalized metric spaces and mappings with xed point, Bulletin Calcutta Mathematical Society, 84(4), 329-336 (1992). [15] H. Efe, C. Alaca, C. Yldz, Fuzzy multi-metric spaces, J. Computational Analysis and Applications, 10 (3), 367-375 (2008).

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†

10

MELTEM ERDEN EGE AND CIHANGIR ALACA

[16] S. Gahler, 2-metrische Raume und ihre topologische struktur, Mathematische Nachrichten, 26, 115-148 (1963). [17] A. Gupta, Cyclic contraction on

S -metric

space, International Journal of Analysis and Applications, 3(2), 119-130

(2013). [18] N.T. Hieu, N.T.T. Ly and N.V. Dung, A generalization of Ciric quasi-contractions for maps on

S -metric

spaces,

Thai Journal of Mathematics (2014). [19] N. Hussain and P. Salimi, Implicit contractive mappings in modular metric and fuzzy metric spaces, The Scientic World Journal 2014 (2014). [20] E. Kilinc and C. Alaca, A xed point theorem in modular metric spaces, Adv. Fixed Point Theory, 4(2), 199-206 (2014). [21] E. Kilinc, C. Alaca, Fixed point results for commuting mappings in modular metric spaces, J. Applied Functional Analysis, 10(3-4), 204-210 (2015). [22] S. Koshi, T. Shimogaki, On

F -norms

of quasi-modular spaces, J. Fac. Sci. Hokkaido Univ. Ser. I 15(3), 202-218

(1961). [23] C. Mongkolkeha, W. Sintunavarat and P. Kumam, Fixed point theorems for contraction mappings in modular metric spaces, Fixed Point Theory and Applications 2011:93. [24] Z. Mustafa and B. Sims, Some remarks concerning

D-metric

spaces, in Proceedings of the International Conference

on Fixed Point Theory and Applications, Valencia, Spain, 189-198 (2004). [25] Z. Mustafa and B. Sims, A new approach to generalized metric spaces, Journal of Nonlinear and convex Analysis 7(2), 289-297 (2006). [26] H. Nakano, Modulared semi-ordered linearspace, In Tokyo Math. Book Ser, Maruzen Co, Tokyo, 1 (1950). [27] W. Orlicz, A note on modular spaces, Bull. Acad. Pol. Sci. Ser. Sci. Math., Astron. Phys., 9, 157-162 (1961). [28] S. Sedghi, N. Shobe and A. Aliouche, A generalization of xed point theorem in

S -metric

spaces, Mat. Vesnik 64,

258-266 (2012). [29] S. Sedghi and N.V. Dung, Fixed point theorems on

S -metric

spaces, 66, 113-124 (2014).

[30] S. Yamamuro, On conjugate spaces of Nakano spaces, Trans. Amer. Math. Soc. 90, 291-311 (1959).

Department of Mathematics, Institute of Natural and Applied Sciences, Celal Bayar University, Muradiye Campus 45140 Manisa, Turkey

E-mail address : [email protected]

Department of Mathematics, Faculty of Science and Arts, Celal Bayar University, Muradiye Campus 45140 Manisa, Turkey

E-mail address, Corresponding author: [email protected]

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The strong converse inequality for de la Vall´ee Poussin means on the sphere ∗ Chunmei Ding

Ruyue Yang

Feilong Cao†

Department of Mathematics, China Jiliang University, Hangzhou 310018, Zhejiang Province, P R China

Abstract This paper discusses the approximation by de la Vall´ee Poussin means Vn f on the unit sphere. Especially, the lower bound of approximation is studied. As a main result, the strong converse inequality for the means is ( ) established. Namely, it is( proved ) that there are constants 1 1 √ √ C1 and C2 such that C1 ω f, n ≤ ∥Vn f − f ∥p ≤ C2 ω f, n for any p-th Lebesgue p

p

integrable or continuous function f defined on the sphere, where ω(f, t)p is the modulus of smoothness of f . MSC(2000): 41A25, 42C10 Keywords: sphere; de la Vall´ee Poussin means; approximation; modulus of smoothness; lower bound

1

Introduction

Motivated by geoscience, meteorology, and oceanography, sphere-oriented mathematics has gained increasing attention in recent decades. As main tools, spherical positive polynomial operators play prominent roles in the approximation and the interpolation on the sphere by means of orthonormal spherical harmonics. Several authors such as Ditzian [5], Dai and Ditzian [4], Bernes and Li [3], Wang and Li [16], Nikol’skiˇı and Lizorkin [10, 8] introduced and studied some spherical versions of some known one-dimensional polynomial operators, for example, spherical Jackson operators [8], spherical de la Vall´ee Poussin operators [3, 16], spherical delay mean operators [13] and best approximation operators [5, 4, 16] etc.. The main aim of the present paper is to study the approximation by the de la Vall´ee Poussin means on the unit sphere. For to formulate our results, we first give some notations. Let Rd , d ≥ 3, be ∑dthe Euclidean space of the points x := (x1 , x2 , . . . , xd ) endowed with the scalar product x · x′ = j=1 xj x′j (x, x′ ∈ Rd ) and let σ := σ d−1 be the unit sphere in Rd consisting of the points x satisfying x2 = x · x = 1. We shall denote the points of σ by µ, and the elementary surface piece on σ by dσ. If it is necessary, we shall write dσ :≡ dσ(µ) referring to the variable of integration. The surface area of d ∫ 2π 2 σ d−1 is denoted by |σ d−1 |, and it is easy to deduce that |σ d−1 | = σ dσ = Γ( d . ) 2

By C(σ) and Lp (σ), 1 ≤ p < +∞, we denote the space of continuous, real valued functions and the space of (the equivalence classes of ) p-integrable functions defined on σ endowed with (∫ )1/p (1 ≤ p < ∞). In the the respective norms ∥f ∥∞ := maxµ∈σ |f (µ)|, ∥f ∥p := σ |f (µ)|p dσ(µ) following, Lp (σ) will always be one of the spaces Lp (σ) for 1 ≤ p < ∞, or C(σ) for p = ∞. Now we state some properties of spherical harmonics (see [16], [7], [9]). For integer k ≥ 0, the restriction of a homogeneous harmonic polynomial of degree k on the unit sphere is called a spherical harmonic of degree k. The class of all spherical harmonics of degree k will be denoted by Hk , and the class of all spherical harmonics of degree k ≤ n will be denoted by Πdn . Of course, ∗ The

research was supported by the National Natural Science Foundation of China (No. 61272023) author: Feilong Cao, E-mail: [email protected]

† Corresponding

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C. M. Ding et al.: The strong converse inequality for de la Vall´ee Poussin means on the sphere ⊕n Πdn = k=0 Hk , and it comprises the restriction to σ of all algebraic polynomials in d variables of total degree not exceeding n. The dimension of Hk is given by { 2k+d−2 (k+d−2) , k ≥ 1; k+d−2 k ddk := dim Hkd := 1, k = 0, ∑ n and that of Πdn is k=0 ddk . The spherical harmonics have an intrinsic characterization. To describe this, we first introduce the Laplace-Beltrami operators (see [9]) to sufficiently smooth functions f defined on σ, which ∑d ∂ 2 is the restriction of Laplace operator ∆ := i=1 ∂x 2 on the sphere σ, and can be expressed as i ( ) µ Df (µ) := ∆f |µ| . Clearly, the operator D is an elliptic, (unbounded) selfadjoint operator µ∈σ

on L2 (σ), is invariant under arbitrary coordinate changes, and its spectrum comprises distinct eigenvalues λk := −k(k + d − 2), k = 0, 1, . . . , each having finite multiplicity. The space Hk can be characterized intrinsically as the eigenspace corresponding to λk , i.e. Hk = {Ψ ∈ C ∞ (σ) : DΨ = −k(k + d − 2)Ψ}. Since the λk ’s are distinct, ⊕ and the operator is selfadjoint, the spaces Hk are mutually orthogonal; also, L2 (σ) = closure { k Hk }. Hence, if we choose an orthogonal basis {Yk,l : l = 1, . . . , ddk } for each Hk , then the set {Yk,l : k = 0, 1, . . . , l = 1, . . . , ddk } is an orthogonal basis for L2 (σ). The orthogonal projection Yk : L1 (σ) → Hk is given by ∫ Γ(λ)(k + λ) Yk (f ; µ) := Pkλ (µ · ν)f (ν)dσ(ν), 2π λ+1 σ (or Gegenbauer) polynomials defined by the where 2λ = d − 2, and Pkλ are the ultraspherical ∑∞ generating equation (1 − 2r cos θ + r2 )−λ = k=0 rk Pkλ (cos θ)(0 ≤ θ ≤ π). The further details for the ultraspherical polynomials can be found in [15]. For an arbitrary number θ, 0 < θ < π, we define the spherical translation operator of the function f ∈ Lp (σ) with a step θ by the aid of the following equation (see [12], [2]): ∫ 1 Sθ (f ) := Sθ (f ; µ) := f (ν)dσ(ν), (1.1) |σ d−2 | sind−2 θ µ·ν=cos θ where |σ d−2 | means the (d − 2)-dimensional surface area of the unit sphere of Rd−1 . Here we integrate over the family of points ν ∈ σ whose spherical distance from the given point µ ∈ σ (i.e. the length of minor arc between µ and ν on the great circle passing through them) is equal to θ. Thus Sθ (f ; µ) can be interpreted as the mean value of the function f on the surface of (d − 2)-dimensional sphere with radius sin θ. The properties of spherical translation operator (1.1) are well known; see e.g., [2]. In particular, it can be expressed as the following series Sθ (f ; µ) =

∞ ∑ P λ (cos θ) k

k=0

Pkλ (1)

Yk (f ; µ) :=

∞ ∑

Qλk (cos θ)Yk (f ; µ)

k=0

Pkλ (cos θ) Pkλ (1)

where Qλk (cos θ) := , and for any f ∈ Lp (σ), ∥Sθ (f )∥p ≤ ∥f ∥p , limθ→0 ∥Sθ f − f ∥p = 0. We usually apply the translation operator to define spherical modulus of smoothness of a function f ∈ Lp (σ), i.e. (see [16]) ω(f, t)p := sup0 n.

)

∞ ∑ P λ (cos θ) k

k=0

Pkλ (1)

) Yk (f ; µ) sin2λ θdθ

sin2λ θdθ Yk (f ; µ),

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C. M. Ding et al.: The strong converse inequality for de la Vall´ee Poussin means on the sphere

it is sufficient to prove

∫π 0

vn (θ)

Pkλ (cos θ) Pkλ (1)

(λ)

sin2λ θ dθ = ωn,k (k ≥ 0). Indeed, when k = 1, one has

) ( ∫ π P λ (cos θ) θ vn (θ) 1 λ sin2λ θdθ sin2λ θdθ = vn (θ) 1 − 2 sin2 2 P1 (1) 0 0 (∫ ( ) )2(n+λ) )2(n+λ) ∫ π( π 22λ θ θ 2λ θ 2(λ+1) θ cos sin dθ − 2 cos sin dθ In,d 2 2 2 2 0 0 ( ( ) ( )) 22λ+1 1 1 1 1 1 B λ + ,n + λ + − B λ + 1 + ,n + λ + In,d 2 2 2 2 2 n n!(n + 2λ)! (λ) = = ωn,1 , n + 2λ + 1 (n − 1)!(n + 1 + 2λ)!

∫

= = =

π

where B(a, b) is Beta function. Now, we suppose for k ≤ n that the relation (see page 81 of [15])

∫π 0

vn (θ)

Pkλ (cos θ) Pkλ (1)

(λ)

sin2λ θdθ = ωn,k . Then for k + 1 we first recall

λ λ (k + 1)Pk+1 (x) − 2(λ + k)xPkλ (x) + (2λ + k − 1)Pk−1 (x) = 0

i.e., λ (cos θ) = Pk+1

Then, ∫

(k ≥ 1),

) 1 ( λ (cos θ) . 2(λ + k) cos θPkλ (cos θ) − (2λ + k − 1)Pk−1 k+1

( ∫ π λ Pk+1 (cos θ) 1 2λ sin θdθ = 2(λ + k) vn (θ) cos θPkλ (cos θ) sin2λ θdθ λ (1) λ (1)(k + 1) Pk+1 Pk+1 0 0 ) ∫ π 1 2λ λ −(2λ + k − 1) (2(λ + k)J2 − J1 ) . vn (θ)Pk−1 (cos θ) sin θdθ := λ P (1)(k + 1) 0 k+1 π

vn (θ)

By the assumption, we obtain ∫ π P λ (cos θ) (2λ + k − 1)!n!(n + 2λ)! λ J1 = (2λ + k − 1)Pk−1 (1) vn (θ) k−1 sin2λ θdθ = . λ (1) Γ(2λ)(k − 1)!(n − k + 1)!(n + k − 1 + 2λ)! P 0 k−1 For J2 we have J2

( )2n ( ) θ 2 θ cos 2 cos − 1 Pkλ (cos θ) sin2λ θdθ In,d 0 2 2 ∫ π ∫ π 2In+1,d λ P λ (cos θ) P λ (cos θ) Pk (1) vn+1 (θ) k λ sin2λ θdθ − Pkλ (1) vn (θ) k λ sin2λ θdθ In,d Pk (1) Pk (1) 0 0 J21 − J22 , 1

= = :=

∫

π

which implies from the assumption that J22 = J21 =

Γ(k+2λ) n!(n+2λ)! k!Γ(2λ) (n−k)!(n+k+2λ)! ,

and

2Γ(n + 1 + λ + 1/2)Γ(n + 2λ + 1) Γ(k + 2λ) (n + 1)!(n + 1 + 2λ)! . Γ(n + λ + 1/2)Γ(n + 1 + 2λ + 1) k!Γ(2λ) (n + 1 − k)!(n + 1 + k + 2λ)!

Therefore, J2 =

Γ(k + 2λ) n!(n + 2λ)! (n(n + 1) + k(2λ + k)) . k!Γ(2λ) (n + 1 − k)!(n + 1 + k + 2λ)!

vn (θ)

λ (cos θ) Pk+1 n!(n + 2λ)! (λ) sin2λ θdθ = = ωn,k+1 . λ (n − k − 1)!(n + k + 1 + 2λ)! Pk+1 (1)

So, ∫ 0

π

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C. M. Ding et al.: The strong converse inequality for de la Vall´ee Poussin means on the sphere

(λ)

On the other hand, it is clear that for k > n, ωn,k = 0. Hence, de la Vall´ee Poussin means Vn (f ; µ) have the form of multiplier expression given in (2.1). Now we give some properties for the de la Vall´ee Poussin kernel vn . Lemma 2.1. Let vn (t) be the kernel of de la Vall´ee Poussin defined by (1.4), 2λ = d − 2 and d ≥ 3. Then there hold ∫ π λ θ−λ vn (θ) sin2λ θdθ ≤ C(d)n 2 , (2.2) 0

∫

and

π

θ− m vn (θ) sin2λ θdθ ≤ C(d)n m , 2

1

m = 1, 2, . . . .

(2.3)

0

Proof. We only prove (2.2). The proof of (2.3) is similar. First, a direct computation implies In,d = C(d) Then,

∫π 0

θ−λ vn (θ) sin2λ θdθ =

( )2n sin2λ dθ = θ−λ cos θ2

1

∫π

In,d

0

(−λ)

=

∫ Jn,d

d−1 (2n + d − 3)!! ≈ n− 2 . (2n + 2d − 4)!!

π 2

θ−

d−2 2

0

(−λ)

Jn,d In,d

, where

θ sind−2 θ cos2n dθ. 2

So, we have d

(−λ)

∫

π 2

≤ 22

Jn,d

sin

d−2 2

t cos2n+d−2 tdt = 2 2 −1 B d

0

( d−2 + 1 2n + d − 2 + 1 ) 2 , 2 2

Γ(n + d−1 d d d Γ(n + d−1 2 ) 2 ) = C(d) ≈ n− 4 . = 2 2 −1 Γ( ) 3d−2 4 Γ(n + 4 ) Γ(n + 3d−2 ) 4 Therefore

∫

(−2)

π

θ

−λ

2λ

vn (θ) sin

θdθ =

Jn,d

0

In,d

n− 4

d

≤ C(d)

d−1 n− 2

= C(d)n

d−2 4

.

The proof of Lemma 2.1 is completed.  Lemma 2.2. For the kernel of de la Vall´ee Poussin vn (t) defined by (1.4), we have ∫ π θ4 vn (θ) sin2λ θdθ ≤ C(d)n−2 . 0

Proof. Since ∫ (4) Jn,d

π 4

θ cos

= 0

{ =

∫

θ sin2λ θdθ = 2d−1 π 4 2

∫

π 2

sind+2 θ cos2n+d−2 θdθ

0

2d−2 π 5 (2n+d−3)!!(d+1)!! , (2n+2d)!!

if d is even;

2d−1 π 4 (2n+d−3)!!(d+1)!! , (2n+2d)!!

if d is odd

= C(d) we have

2n

(2n + d − 3)!! , (2n + 2d)!! (4)

π

θ4 vn (θ) sin2λ θdθ = 0

This finishes the proof of Lemma 2.2.

Jn,d In,d

= C(d)

(2n + 2d − 4)!! ≤ C(d)n−2 . (2n + 2d)!!



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C. M. Ding et al.: The strong converse inequality for de la Vall´ee Poussin means on the sphere

3

Lower bound of approximation for de la Vall´ ee Poussin means

In this section we prove the main result of this paper, which can be stated as follows. Theorem 3.1. Let Vn (f ; µ) be de la Vall´ee Poussin means on the sphere given by (1.5). Then for f ∈ Lp (σ), 1 ≤ p ≤ +∞, there exists a constant C which is independent of f and n, such that ( ) 1 ω f, √ ≤ C∥Vn f − f ∥p . n p In order to prove the result, we first prove the following lemma. Lemma 3.1. For any g, Dg, D2 g ∈ Lp (σ), 1 ≤ p ≤ ∞, there exist the constants A, B and C2 which are are independent of n and g, such that ∥Vn g − g − α(n)Dg∥p ≤ C2 n−2 ∥D2 g∥p , where B 0< A n ≤ α(n) ≤ n . ∫θ ∫t Proof. Since (see (3.6) of [11]) Sθ (g; µ) − g(µ) = 0 sin−2λ t dt 0 sin2λ u Su (Dg; µ)du, we have ∫ u ∫ γ −2λ Su (Dg; µ) − Dg(µ) = sin γdγ sin2λ ν Sν (D2 g; µ)dν. 0

0

Observing that ∫ Vn (g; µ) − g(µ)

∫

π

θ

vn (θ) sin2λ θdθ

= 0

=

0

∫

2λ

vn (θ) sin 0

vn (θ) sin

∫

θ

t

sin2λ u Su (Dg; µ)du 0

θ

−2λ

sin 0

θdθ

0

:=

θdθ

∫

π 2λ

+

∫

π

Dg(µ) ∫

sin−2λ tdt

−2λ

sin

∫

∫

sin2λ udu 0

t

tdt

0

t

tdt

( ) sin2λ u Su (Dg; µ) − Dg(µ) du

0

Dg(µ)α(n) + Ψ(g; µ),

where α(n) = C(d)n−1 satisfies 0 < An−1 ≤ C(d)n−1 ≤ Bn−1 , we obtain that from the H¨olderMinkowski’s inequality and the contractility of translation operator ∫ π ∫ θ ∫ t ∫ u ∫ γ ∥Ψg∥p ≤ ∥D2 g∥p vn (θ) sin2λ θdθ sin−2λ tdt sin2λ udu sin−2λ γdγ sin2λ νdν 0 0 0 0 0 ∫ π 2λ 2 4 ≤ C3 ∥D g∥p vn (θ)θ sin θdθ. 0

Thus, from Lemma 2.2 it follows that ∥Ψg∥p ≤ C4 n−2 ∥D2 g∥p . The Lemma 3.1 has been proved.  Now we turn to the proof of Theorem 3.1. We first introduce an operator Vnm given by Vnm (f ; µ)

=

n (∫ ∑ k=0

π

vn (θ)Qλk (cos θ) sin2λ θdθ

)m Yk (f ; µ).

0

Then, form the orthogonality of projection operator Yk , it follows that Vnm+l f

= =

n (∫ ∑

π

vn (θ)Qλk (cos θ) sin2λ θdθ

)m

0 k=0 m l Vn (Vn f ).

Yk

n (∫ (∑ s=0

π

)l ) vn (θ)Qλs (cos θ) sin2λ θdθ Ys f

0

Thus, we take g = Vnm f and obtain that ∥f − g∥p = ∥f − Vnm f ∥p ≤

m ∑

∥Vnk−1 f − Vnk f ∥p ≤ m∥f − Vn f ∥p ,

k=1

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where Vn0 f = f. A Next, we prove the estimate: ∥DVnm f ∥p ≤ 2C C1 n∥f ∥p , where A and C2 are the same as that 2 in Lemma 3.1. In fact, we have

n

∑

(∫ π )m



∥DVnm f ∥p ≤ k(k + d − 2) vn (θ) Qλk (cos θ) sin2λ θdθ Yk (f ) .

0 k=0

Since (see [1])

p

P λ (cos θ) ( ) |Qλk (cos θ)| ≡ k λ ≤ C5 min (kθ)−λ , 1 , Pk (1)

we use (2.2) and obtain for kθ ≥ 1 and θ ≤ π2 , that

n

∑

(∫ π )m

2λ −λ m − d−2 m k(k + d − 2)k 2 vn (θ)θ sin θdθ Yk (f ) ∥DVn f ∥p ≤ C6

0 k=0

≤ C7 n

d−2 4 m

∥f ∥p

∞ ∑

k

2− d−2 2 m

p

.

k=0

For 2 −

d−2 2 m

< −1, i.e. m >

6 d−2 ,

it is clear that the series ∥DVnm f ∥p ≤ C8 n

d−2 4 m

∑∞ k=0

k 2−

d−2 2 m

is convergence. Thus

∥f ∥p .

For kθ ≤ 1, then (2.3) implies that

n ∫

∑ ( π

)m 1 2



2λ m 2 −m λ ∥DVn f ∥p ≤ (θ k(k + d − 2)) m Qk (cos θ) sin θdθ Yk (f ) vn (θ)θ

0 k=0 p

n ∫

∞

∑ ( π

) m 2 A

∑

C1 n∥f ∥p , ≤ C9 vn (θ)θ− m sin2λ θdθ Yk (f ) ≤ C10 n Yk (f ) =



2C2 0 k=0

k=0

p

p

where A and C2 are the same as that in Lemma 3.1. Therefore, when m > ∥DVnm f ∥p ≤

6 d−2 ,

we have

A C1 n∥f ∥p . 2C2

In the next, without loss generality, we assume m1 > Lemma 3.1 we see that

6 d−2 ,

and m >

6 d−2

α(n)∥DVnm f ∥p ≤ ∥Vnm f − f ∥p + C2 n−2 ∥D2 Vnm f ∥p ≤ m∥Vn f − f ∥p +

+ m1 . According to

AC1 −1 n ∥DVnm−m1 f ∥p 2

AC1 −1 AC1 −1 n ∥DVnm f ∥p + n ∥DVnm−m1 (Vnm1 f − f )∥p 2 2 AC1 C11 m1 AC1 ≤ m∥Vn f − f ∥p + ∥DVnm f ∥p + ∥Vn f − f ∥p 2n 2 AC1 = C12 ∥Vn f − f ∥p + ∥DVnm f ∥p . 2n ≤ m∥Vn f − f ∥p +

Taking α(n) =

AC1 n ,

one has 1 2C12 ∥DVnm f ∥p ≤ ∥Vn f − f ∥p . n AC1

So from the definition of K-functional it follows ( ) ( 1 )2 1 K f, √ ≤ ∥f − Vnm f ∥p + √ ∥DVnm f ∥p n n 2C12 ≤ m∥f − Vn f ∥p + ∥f − Vn f ∥p ≤ C14 ∥f − Vn f ∥p , AC1

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which together with (1.3) implies ( ) 1 ω f, √ ≤ C∥f − Vn f ∥p . n p This finishes the proof of Theorem 3.1.  From (1.6) and Theorem 3.1, the following Corollary 3.1 follows directly. Corollary 3.1. For any f ∈ Lp (σ), 1 ≤ p ≤ ∞, there holds ( ) 1 ∥Vn f − f ∥p ≈ ω f, √ . n p

References [1] E. Belinsky, F. Dai, Z. Ditzian, Multivariate approximation averages, J. Approx. Theory, 125 (2003), 85-105. [2] H. Berens, P. L. Butzer, S. Pawelke, Limitierungsverfahren von Reihen mehrdimensionaler Kugelfunktionen und deren Saturationsverhalten, Publ. RIMS, Kyoto Univ., Ser. A, 4 (1968), 201-268. [3] H. Berens, L. Q. Li, On the de la Vall´ee Poussin means on the sphere, Results in Math., 24 (1993), 12-26. [4] F. Dai, Z. Ditzian, Jackson theorem in Lp , 0 < p < 1, for functions on the sphere, J. Approx. Theory, (2009) doi: 10. 1016/jat. 2009. 06. 003. [5] Z. Ditzian, Jackson-type inequality on the sphere, Acta Math. Hungar, 102 (1-2) (2004), 1-35. [6] Z. Ditzian, K. G. Ivanov, Strong converse inequalities, Jour. D’Analyse Math., 61 (1993), 61-111. [7] W. Freeden, T. Gervens, M. Schreiner, Constructive approximation on the sphere, Oxford University Press Inc., New York, 1998. [8] P. I. Lizorkin, S. M. Nikol’skiˇl, A theorem concerning approximation on the sphere, Anal. Math., 9 (1983), 207-221. [9] C. M¨ uller, Spherical harmonics, Lecture Notes in Mathematics, Vol. 17, Springer, Berlin, 1966. [10] S. M. Nikol’skiˇl, P. I. Lizorkin, Approximation theory on the sphere, Proc. Steklov Inst. Math. 172 (1985), 295-302. ¨ [11] S. Pawelke, Uber die approximationsordnung bei kugelfunktionen und algebraischen polynomen, Tˆohoku Math. J., 24 (3) (1972), 473-486. [12] W. Rudin, Uniqueness theory for Laplace series, Trans. Amer. Math. Soc., 68 (1950), 287-303. [13] E. M. Stein, Interpolation in polynomial classes and Markoff’s inequality, Duke Math. J., 24 (1957), 467-476. [14] E. M. Stein G. Weiss, Introduction of Functions of Real Variable, Princeton University Press, Princeton N. J., 1971. [15] G. Szeg¨o, Orthogonal polynomials, Amer. Math. Soc. Coll. Publ., Vol. 23, 2003. [16] K. Wang, L. Li, Harmonic analysis and approximation on the unit sphere, Science Press, Beijing, 2000.

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On the fixed point method for stability of a mixed type AQ-functional equation Ick-Soon Changa and Yang-Hi Leeb a

Department of Mathematics, Chungnam National University, Daejeon 305-764, Korea.

b

Department of Mathematics Education, Gongju National University of Education, Gongju 314-711, Korea.

Abstract In this article, we take into account the stability for the following functional equation of additive-quadratic type f (x − y) − f (−x + y) − 4f (x) + f (2x) − f (−y) + f (y) = 0 with the fixed point method. Keywords : Stability ; Fixed point method ; Additive-quadratic mapping. AMS Mathematics Subject Classification (2000) : 39B52, 39B82, 47H10.

1

Introduction

Ulam [9] proposed the following question concerning the stability of homomorphisms : Let G1 be a group and let G2 be a metric group with the metric d(·, ·). Given ε > 0, does there exist a δ > 0 such that if a function 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)) < ε for all x ∈ G1 ? Hyers [5] answers the problem of Ulam under the assumption that the groups are Banach spaces. A generalized version of the theorem of Hyers for approximately additive mappings was given by Aoki [1], and for approximately linear mappings was presented by Rassias [7] by considering an unbounded Cauchy difference. Thereafter, many interesting results of the stability of several functional equation have been extensively investigated. On the contrary, C˘adariu and Radu [2] observed that the existence of the solution for a functional equation and the estimation of the difference with the given mapping can be obtained from the fixed point alternative. This method is called a fixed point method. In particular, they [3, 4] applied this method to prove the stability theorems of the additive functional equation and the quadratic functional equation by using the fixed point method. Now we consider the stability of the following mixed type additive-quadratic functional equation (briefly, AQ–functional equation) f (x − y) − f (−x + y) − 4f (x) + f (2x) − f (−y) + f (y) = 0.

(1.1)

by using the fixed point method. In this case, every solution of the functional equation (1.1) is said to be an additive-quadratic mapping. a Corresponding

author. E-mail address : [email protected] (I.-S. Chang), [email protected] (Y.-H. Lee) 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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2

2

Stability of Eq. (1.1) and its applications

Throughout this article, let V be a real or complex linear space and Y a Banach space. For a given mapping f : V → Y, we use the following abbreviation Df (x, y) := f (x − y) − f (−x + y) − 4f (x) + f (2x) − f (−y) + f (y) for all x, y ∈ V. We first prove the following lemma. Lemma 2.1 Let f : V → Y be a mapping with f (0) = 0 such that Df (x, y) = 0 for all x, y ∈ V \{0}. Then f is an additive-quadratic mapping. Proof. Since f (0) = 0, we get Df (x, 0) = Df (x, x) = 0 for all x ∈ V \{0}, and Df (0, y) = 0 for all y ∈ V . This completes the proof.  For explicitly later use, we state the following theorem : Theorem 2.2 (The alternative of fixed point) ([6] or [8]) Suppose that a complete generalized metric space (X, d), which means that the metric d may assume infinite values, and a strictly contractive mapping J : X → X with the Lipschitz constant 0 < L < 1 are given. Then, for each given element x ∈ X, either d(J n x, J n+1 x) = +∞, ∀n ∈ N ∪ {0}, or there exists a nonnegative integer k such that : (1) d(J n x, J n+1 x) < +∞ for all n ≥ k ; (2) the sequence {J n x} is convergent to a fixed point y ∗ of J ; (3) y ∗ is the unique fixed point of J in Y := {y ∈ X, d(J k x, y) < +∞} ; (4) d(y, y ∗ ) ≤ (1/(1 − L))d(y, Jy) for all y ∈ Y. Now, by the use of fixed point method, we obtain the main results as follow. Theorem 2.3 Let φ : (V \{0})2 → [0, ∞) be a function with φ(x, y) = φ(−x, −y) for all x, y ∈ V \{0}. Suppose that a mapping f : V → Y satisfies ∥Df (x, y)∥ ≤ φ(x, y)

(2.1)

for all x, y ∈ V \{0} with f (0) = 0. If there exists a constant 0 < L < 1 such that a function φ has the property φ(2x, 2y) ≤ 2Lφ(x, y)

(2.2)

for all x, y ∈ V \{0}, then there exists a unique additive-quadratic mapping F : V → Y such that ∥f (x) − F (x)∥ ≤

φ(x, x) 2(1 − L)

(2.3)

for all x ∈ V \{0}. In particular, F is represented by ( ) f (2n x) + f (−2n x) f (2n x) − f (−2n x) F (x) = lim + n→∞ 2 · 4n 2n+1

(2.4)

for all x ∈ V. Proof. Consider the set S := {g : g : V → Y, g(0) = 0} and introduce a generalized metric on S by d(g, h) = inf{K ∈ R+ : ∥g(x) − h(x)∥ ≤ Kφ(x, x) for all x ∈ V \{0}}. It is easy to see that (S, d) is a generalized complete metric space.

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3

Now we define a mapping J : S → S by g(2x) − g(−2x) g(2x) + g(−2x) + 4 8

Jg(x) := for all x ∈ V. Note that

g(2n x) − g(−2n x) g(2n x) + g(−2n x) + 2n+1 2 · 4n for all n ∈ N and all x ∈ V. Let g, h ∈ S and let K ∈ [0, ∞] be an arbitrary constant with d(g, h) ≤ K. From the definition of d, we have J n g(x) =

1 3 ∥Jg(x) − Jh(x)∥ = ∥g(2x) − h(2x)∥ + ∥g(−2x) − h(−2x)∥ 8 8 1 ≤ Kφ(2x, 2x) 2 ≤ KLφ(x, x) for all x ∈ V \{0}, which implies that

d(Jg, Jh) ≤ Ld(g, h)

for any g, h ∈ S, that is, J is a strictly contractive self-mapping of S with the Lipschitz constant L. Moreover, by (2.1), we see that ∥f (x) − Jf (x)∥ =

1 φ(x, x) ∥ − 3Df (x, x) + Df (−x, −x)∥ ≤ 8 2

for all x ∈ V \{0}. It means that d(f, Jf ) ≤ 12 < ∞ by the definition of d. Therefore, according to Theorem 2.2, the sequence {J n f } converges to the unique fixed point F : V → Y of J in the set T = {g ∈ S : d(f, g) < ∞}, which is represented by (2.4). Note that 1 1 d(f, F ) ≤ d(f, Jf ) ≤ , 1−L 2(1 − L) which implies (2.3). By the definition of F, together with (2.1) and (2.4), we find that

Df (2n x, 2n y) − Df (−2n x, −, 2n y)

∥DF (x, y)∥ = lim n→∞ 2n+1 n n Df (2 x, 2 y) + Df (−2n x, −2n y)

+

2 · 4n n 2 +1 ≤ lim (φ(2n x, 2n y) + φ(−2n x, −2n y)) n→∞ 2 · 4n =0 for all x, y ∈ V \{0}. By Lemma 2.1, we have proved that DF (x, y) = 0 for all x, y ∈ V. This completes the proof.  We continue our investigation with the following theorem. Theorem 2.4 Let φ : (V \{0})2 → [0, ∞) with φ(x, y) = φ(−x, −y) for all x, y ∈ V \{0}. Suppose that f : V → Y satisfies the inequality ∥Df (x, y)∥ ≤ φ(x, y) for all x, y ∈ V \{0} with f (0) = 0. If there exists 0 < L < 1 such that the mapping φ has the property Lφ(2x, 2y) ≥ 4φ(x, y)

(2.5)

for all x, y ∈ V \{0}, then there exists a unique additive-quadratic mapping F : V → Y such that ∥f (x) − F (x)∥ ≤

Lφ(x, x) 4(1 − L)

(2.6)

for all x ∈ V \{0}. In particular, F is given by ( ( (x) ( x )) 4n ( ( x ) ( x ))) F (x) = lim 2n−1 f n − f − n + f n +f − n n→∞ 2 2 2 2 2

(2.7)

for all x ∈ V.

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4

Proof. Let (S, d) be the set as in the proof of Theorem 2.3, and we consider the mapping J : S → S defined by (x) ( x) ( (x) ( x )) Jg(x) := g −g − +2 g +g − 2 2 2 2 for all g ∈ S and all x ∈ V. Observe that ( (x) ( x )) 4n ( ( x ) ( x )) J n g(x) = 2n−1 g n − g − n + g n +g − n 2 2 2 2 2 and J 0 g(x) = g(x) for all x ∈ V. Let g, h ∈ S and let K ∈ [0, ∞] be an arbitrary constant with d(g, h) ≤ K. The definition of d yields

(x) (x) ( x) ( x)



−h −h − ∥Jg(x) − Jh(x)∥ = 3 g

+ g −

2 2 2 2 (x x) ≤ 4Kφ , 2 2 ≤ LKφ(x, x) for all x ∈ V \{0}. So we get

d(Jg, Jh) ≤ Ld(g, h)

for any g, h ∈ S, that is, J is a strictly contractive self-mapping of S with the Lipschitz constant L. Also we see that

( x x ) (x x) L

∥f (x) − Jf (x)∥ = Df , , ≤ φ(x, x)

≤φ 2 2 2 2 4 for all x ∈ V \{0}, which implies that d(f, Jf ) ≤ L4 < ∞. Therefore, according to Theorem 2.2, the sequence {J n f } converges to the unique fixed point F of J in the set T := {g ∈ S : d(f, g) < ∞}, which is given by (2.7). Since 1 L d(f, F ) ≤ d(f, Jf ) ≤ 1−L 4(1 − L) the inequality (2.6) holds. From the definition of F with (2.1) and (2.5), we have

( (x y ) ( x y ))

∥DF (x, y)∥ = lim 2n−1 Df n , n − Df − n , − n n→∞ 2 2 2 2 (x y ) ( x y )) 4n (

Df n , n + Df − n , − n +

2 2 2 2 2 ) ( n n ( ( 2 +4 x y x y )) ≤ lim φ n, n + φ − n,− n n→∞ 2 2 2 2 2 =0 for all x, y ∈ V \{0}. So, by Lemma 2.1, F is an additive-quadratic mapping, which completes the proof.  From now on, given a mapping f : V → Y, we set 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 ∈ V. Using Theorem 2.3 and Theorem 2.4, we will prove the stability of the additive functional equation Af ≡ 0, and the quadratic functional equation Qf ≡ 0 in the following results. Corollary 2.5 Let fi : V → Y, i = 1, 2, be mappings for which there exist functions ϕi : (V \{0})2 → [0, ∞), i = 1, 2, such that ∥Afi (x, y)∥ ≤ ϕi (x, y)

45

(2.8)

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5

for all x, y ∈ V \{0}. If fi (0) = 0, ϕi (x, y) = ϕi (−x, −y), i = 1, 2, for all x, y ∈ V \{0}, and there exists 0 < L < 1 such that 1 ϕ1 (x, y) ≤ ϕ1 (2x, 2y) ≤ 2Lϕ1 (x, y), L ϕ2 (2x, 2y) ≤ Lϕ2 (2x, 2y)

(2.9) (2.10)

for all x, y ∈ V \{0}, then there exist unique additive mappings Fi : V → Y, i = 1, 2, such that ϕ1 (x, x) + 3ϕ1 (x, −x) , 2(1 − L)

(2.11)

L(ϕ2 (x, x) + 3ϕ2 (x, −x)) 4(L − 1)

(2.12)

∥f1 (x) − F1 (x)∥ ≤

∥f2 (x) − F2 (x)∥ ≤

for all x ∈ V \{0}. In particular, the mappings Fi , i = 1, 2, are represented by f1 (2n x) , n→∞ 2n( ) x F2 (x) = lim 2n f2 n n→∞ 2 F1 (x) = lim

(2.13) (2.14)

for all x ∈ V. Proof. We first note that Dfi (x, y) = Afi (x, −y) − Afi (−x, y) + Afi (x, x) + Afi (x, −x) for all x, y ∈ V and i = 1, 2. Put φi (x, y) := ϕi (x, −y) + ϕi (−x, y) + ϕi (x, x) + ϕi (x, −x) for all x, y ∈ V \{0} and i = 1, 2, then φ1 satisfies (2.2) and φ2 fulfills (2.5). Therefore ∥Dfi (x, y)∥ ≤ φi (x, y) for all x, y ∈ V \{0} and i = 1, 2. According to Theorem 2.3, there exists a unique mapping F1 : V → Y satisfying (2.11), which is represented by (2.4). Observe that, by virtue of (2.8) and (2.9),



f1 (2n x) + f1 (−2n x)

f1 (2n x) + f1 (−2n x) − f1 (0)



lim lim

= n→∞

n→∞ 2n+1 2n+1 1 = lim n+1 ∥Af1 (2n x, −2n x)∥ n→∞ 2 1 ≤ lim n+1 ϕ1 (2n x, −2n x) n→∞ 2 Ln ≤ lim ϕ1 (x, −x) = 0 n→∞ 2 and

n n

f1 (2n x) + f1 (−2n x)

≤ lim 2 L ϕ1 (x, −x) = 0 lim

n n→∞ n→∞ 2 · 4n 2·4

for all x ∈ V \{0}. This inequality and (2.4) guarantees (2.13). Moreover, we have

Af1 (2n x, 2n y) ϕ1 (2n x, 2n y)

≤ ≤ Ln ϕ1 (x, y)

2n 2n for all x, y ∈ V \{0}. Sending the limit as n → ∞ in the above inequality, and using F1 (0) = 0, we get AF1 (x, y) = 0 for all x, y ∈ V. On the other hand, according to Theorem 2.4, we see that there exists a unique mapping F2 : V → Y satisfying (2.12), which is given by (2.7).

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6

Notice that, by (2.8) and (2.11), ( )

(x)

(x −x x )

2n−1 2n−1 , − lim 2 = lim 2

f2 n + f2

Af

2 n→∞ n→∞ 2 2n 2n 2n ) (x x ≤ lim 22n−1 ϕ2 n , − n n→∞ 2 2 Ln ϕ2 (x, −x) = 0. ≤ lim n→∞ 2 as well as ( )

(x) −x Ln

lim 2n−1 f2 n + f2 ϕ2 (x, −x) = 0 ≤ lim

n→∞ n→∞ 2n+1 2 2n for all x ∈ V \{0}. From these and (2.7), we obtain (2.14). Moreover, we have

( x y ) ( x y ) Ln

n

2 Af2 n , n ≤ 2n ϕ2 n , n ≤ n ϕ2 (x, y) 2 2 2 2 2 for all x, y ∈ V \{0}. Taking the limit as n → ∞ in the above inequality, and using F2 (0) = 0, we fee that AF2 (x, y) = 0 for all x, y ∈ V. The proof is ended.  Corollary 2.6 Let fi : V → Y, i = 1, 2, be mappings for which there exist functions ϕi : (V \{0})2 → [0, ∞), i = 1, 2, such that ∥Qfi (x, y)∥ ≤ ϕi (x, y) for all x, y ∈ V \{0}. If fi (0) = 0, ϕi (x, y) = ϕi (−x, −y), i = 1, 2, for all x, y ∈ V \{0}, and there exists 0 < L < 1 such that the mapping ϕ1 satisfies (2.9) and ϕ2 satisfies (2.10) for all x, y ∈ V \{0}, then there exist unique quadratic mappings Fi : V → Y, i = 1, 2, such that 3ϕ1 (x, x) + 5ϕ1 (x, −x) , 4(1 − L)

(2.15)

L(3ϕ2 (x, x) + 5ϕ2 (x, −x)) 8(1 − L)

(2.16)

∥f1 (x) − F1 (x)∥ ≤

∥f2 (x) − F2 (x)∥ ≤

for all x ∈ V \{0}. In particular, the mappings Fi , i = 1, 2, are given by f1 (2n x) , n→∞ 4n( ) x F2 (x) = lim 4n f2 n n→∞ 2 F1 (x) = lim

(2.17) (2.18)

for all x ∈ V. Proof. Note that 1 1 Dfi (x, y) = Qfi (x, y) − Qfi (y, −x) + fi (x, −x) + Qfi (y, −y) − Qfi (y, y) 2 2 for all x, y ∈ V and i = 1, 2. Put φi (x, y) := ϕi (x, y) + ϕi (y, −x) + ϕi (x, −x) + 12 ϕi (y, y) + 12 ϕi (y, −y) for all x, y ∈ V \{0} and i = 1, 2, then φ1 (resp. φ2 ) satisfies (2.2) (resp. (2.5)). Moreover, ∥Dfi (x, y)∥ ≤ φi (x, y) for all x, y ∈ V \{0} and i = 1, 2. It follows from Theorem 2.3 that there exists a unique mapping F1 : V → Y satisfying (2.15), which is represented by (2.4). Observe that

f (2n x) − f (−2n x)

1

1 1 lim

= lim n+1 Qf1 (2n−1 x, −2n−1 x) − Qf1 (−2n−1 x, 2n−1 x) n+1 n→∞ 2 n→∞ 2 ) 1 ( ≤ lim n+1 ϕ1 (2n−1 x, −2n−1 x) + ϕ1 (−2n−1 x, 2n−1 x) n→∞ 2 ( x x )) Ln ( ( x x ) ≤ lim ϕ1 ,− + ϕ1 − , n→∞ 2 2 2 2 2 =0

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and

7

(x x) ( x x )) n (

f1 (2n x) − f1 (−2n x)

≤ lim L lim ϕ , − + ϕ − , =0 1 1

n→∞ 2n+1 n→∞ 2 · 4n 2 2 2 2

for all x ∈ V \{0}. Due to this fact and (2.4), we get (2.17). Moreover, we have

Qf1 (2n x, 2n y) ϕ1 (2n x, 2n y) Ln

≤ ≤ n ϕ1 (x, y)

n n 4 4 2 for all x, y ∈ V \{0}. As n → ∞ in the above inequality, we see that QF1 (x, y) = 0 for all x, y ∈ V \{0}. By using F1 (0) = 0, then we have (y y) ( y y) QF1 (x, 0) = 0, QF1 (0, y) = − QF1 ,− + QF1 − , =0 2 2 2 2 for all x, y ∈ V \{0}. Therefore, QF1 (x, y) = 0 for all x, y ∈ V. On the other hand, Theorem 2.4 guarantees that there exists a unique mapping F2 : V → Y satisfying (2.16), which is represented by (2.7). Observe that

(x)

( x ) ( ( x x ) x x )



4n f2 n − f2 − n =4n Qf2 n+1 , − n+1 − Qf2 − n+1 , n+1 2 2 2 2 2 2 ( ( x ( x ) x x )) ≤4n ϕ2 n+1 , − n+1 + ϕ2 − n+1 , n+1 2 2 ( (2x x )2 ( x x )) n ≤L ϕ2 ,− + ϕ2 − , 2 2 2 2 for all x ∈ V \{0}. It leads us to get ( (x) ( x )) ( (x) ( x )) lim 4n f2 n − f2 − n = 0, lim 2n f2 n − f2 − n =0 n→∞ n→∞ 2 2 2 2 for all x ∈ V \{0}. Based on these facts and (2.7), we obtain (2.18). Moreover, we have

( x y ) (x y )

n

4 Qf2 n , n ≤ 4n ϕ2 n , n ≤ Ln ϕ2 (x, y) 2 2 2 2 for all x, y ∈ V \{0}. Going the limit as n → ∞ in the previous inequality, and using F2 (0) = 0, we get QF2 (x, y) = 0 for all x, y ∈ V, which complete the proof. Now, we obtain the stability in the framework of normed spaces using Theorem 2.3 and Theorem 2.4. Corollary 2.7 Let X be a normed space and Y a Banach space. Suppose that the mapping f : X → Y satisfies the inequality ∥Df (x, y)∥ ≤ θ(∥x∥p + ∥y∥p ) for all x, y ∈ X\{0} with f (0) = 0, where θ ≥ 0 and p ∈ (−∞, 1) ∪ (2, ∞). Then there exists a unique quadratic-additive mapping F : X → Y such that { 2θ p if p > 2, 2p −4 ∥x∥ ∥f (x) − F (x)∥ ≤= 2θ p ∥x∥ if p < 1, 2−2p for all x ∈ X\{0}. Proof. This follows from Theorem 2.3 and Theorem 2.4 by putting φ(x, y) := θ(∥x∥p + ∥y∥p ) for all x, y ∈ X\{0} with L = 2p−1 < 1 if p < 1 and L = 22−p < 1 if p > 2. Corollary 2.8 Let X be a normd space and Y a Banach space. Suppose that the mapping f : X → Y satisfies the inequality ∥Df (x, y)∥ ≤ θ∥x∥p ∥y∥q

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8

for all x, y ∈ X\{0} with f (0) = 0, where θ ≥ 0 and p + q ∈ (−∞, 1) ∪ (2, ∞). Then there exists a unique quadratic-additive mapping F : X → Y such that { θ∥x∥p+q if p + q > 2, 2p+q −4 ∥f (x) − F (x)∥ ≤= θ∥x∥p+q if p + q < 1 2(2−2p+q ) for all x ∈ X\{0}. Proof. By considering φ(x, y) := θ∥x∥p ∥y∥q for all x, y ∈ X\{0} with L = 2p+q−1 < 1 if p + q < 1 and L = 22−p−q < 1 if p + q > 2, then by Theorem 2.3 and Theorem 2.4, we arrive at the conclusion of the corollary.

References [1] T. Aoki, On the stability of the linear mapping in Banach spaces, J. Math. Soc. Japan 2 (1950), 64–66. [2] L. C˘adariu and V. Radu, Fixed points and the stability of Jensen’s functional equation, J. Inequal. Pure Appl. Math. 4 (2003), Art. 4. [3] L. C˘adariu and V. Radu, Fixed points and the stability of quadratic functional equations, An. Univ. Timisoara Ser. Mat.-Inform. 41 (2003), 25–48. [4] L. C˘adariu and V. Radu, On the stability of the Cauchy functional equation : a fixed point approach in iteration theory, Grazer Mathematische Berichte, Karl-Franzens-Universit¨aet, Graz, Graz, Austria 346 (2004), 43–52. [5] D.H. Hyers, On the stability of the linear functional equation, Proc. Natl. Acad. Sci. USA 27 (1941), 222–224. [6] B. Margolis and J.B. Diaz, A fixed point theorem of the alternative for contractions on a generalized complete metric space, Bull. Amer. Math. Soc. 74 (1968), 305–309. [7] Th.M. Rassias, On the stability of the linear mapping in Banach spaces, Proc. Amer. Math. Soc. 72 (1978), 297–300. [8] I.A. Rus, Principles and applications of fixed point theory, Ed. Dacia, Cluj-Napoca, (1979) (in Romanian). [9] S.M. Ulam, A collection of mathematical problems, Interscience, New York, (1968).

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DIFFERENCES OF COMPOSITION OPERATORS FROM LIPSCHITZ SPACE TO WEIGHTED BANACH SPACES IN POLYDISK CHANG-JIN WANG SCHOOL OF SCIENCE, JIMEI UNIVERSITY, XIAMEN FUJIAN 361021, P.R. CHINA. [email protected] YU-XIA LIANG∗ SCHOOL OF MATHEMATICAL SCIENCES, TIANJIN NORMAL UNIVERSITY, TIANJIN 300387, P.R. CHINA. [email protected]

Abstract. Let ϕ and ψ be holomorphic self-maps of the unit polydisk Dn in the n-dimensional complex space C n , denote by Cϕ and Cψ the induced composition operators. In this paper, we estimate the essential norm of the differences of composition operators Cϕ − Cψ from Lipschitz space to weighted Banach space in the unit polydisk.

1. Introduction The algebra of all holomorphic functions on domain Ω will be denoted by H(Ω), where Ω is a bounded domain in C n , where n ≥ 1 is a fixed integer. Let Dn = {z = (z1 , ..., zn ) ∈ C n , |zi | < 1, 1 ≤ i ≤ n} be the open unit polydisk of the complex ndimensional Euclidean space C n and H(Dn ) be the space of all holomorphic functions on Dn . For z = (z1 , ..., zn ) and w = (w1 , ..., wn ) in C n , the inner product of z and w isg hz, wi = z1 w1 + ... + zn wn , where h., .i denotes the inner product. Moreover, g z = maxj {|zj |} stands for the supremum norm on Dn . For z, w ∈ D, the pseudo-hyperbolic distance between z and w is defined by ρ(z, w) = |(z − w)/(1 − wz)|. It is well known that if f ∈ H(D), then ρ(f (z), f (w)) ≤ ρ(z, w). The Bergman metric on the unit polydisk is given by n X uj vj Hz (u, v) = . (1 − |zj |2 )2 j=1 The Kobayashi distance kDn on Dn is defined by g g 1 + φz (w) 1 g g, kDn (z, w) = log 2 1 − φz (w) where φz : Dn → Dn is the automorphism of Dn given by  w −z wn − zn  1 1 φz (w) = , ..., . 1 − z1 w1 1 − zn wn

(1.1)

The work was supported in part by the National Natural Science Foundation of China (Grant Nos. 11371276; 11301373; 11201331). ∗ Corresponding author. 2010 Mathematics Subject Classification.Primary: 47B33; Secondary: 47B38, 32A37, 32H02. Key words and phrases. composition operator, Lipschitz space, weighted Banach space, polydisk. 1

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Wang and Liang:: Differences of composition operators Let v be a strictly positive bounded continuous function (weight) on the open unit polydisk Dn in C n , n ≥ 1. We first introduce the weighted Banach spaces of analytic functions of the following form: o n Hv∞ := f ∈ H(Dn ); kf kv = sup v(z)|f (z)| < ∞ z∈Dn

endowed with the sup-norm k.kv . Spaces of this type appear in the study of growth conditions of analytic functions and have been studied in various articles, see, e.g. [2, 8, 10]. For 0 ≤ α < 1, an f ∈ H(Dn ) belongs to the Lipschitz space Lipα (Dn ), if n X ∂f (z) (1 − |zl |2 )1−α < ∞. ∂zl z∈Dn

kf kα = |f (0)| + sup

(1.2)

l=1

It is easy to show that Lipα (Dn ) is a Banach space endowed with the norm k.kα (see, e.g.[13, 14]). Let ϕ = (ϕ1 (z), ..., ϕn (z)) and ψ(z) = (ψ1 (z), ..., ψn (z)) be holomorphic self-maps of Dn . The composition operator Cϕ induced by ϕ is defined by (Cϕ )f (z) = f (ϕ(z)) for z ∈ Dn and f ∈ H(Dn ) (see, e.g.[3]). The essential norm of a continuous linear operator T is the distance from T to the set of all compact operators, that is, kT ke = inf{kT − Kk : K is compact }. Notice that kT ke = 0 if and only if T is compact, so estimates on kT ke lead to conditions for T to be compact (see, e.g.[6, 14, ?]). In the past few years, many authors have been interested in studying the mapping properties of the differences of two composition operators, that is, an operator of the form T = Cϕ − Cψ . The primary motivation for this has been the desire to understand the topological structure of the whole set of composition operators. Most papers in this area have focused on the classical reflexive spaces, but some classical nonreflexive spaces in the unit disc in the complex plane have also recently been discussed. We refer the readers to the recent papers [1, 4, 5, 6, 7, 9, 12] to learn more about the propertied about the differences. Building on the above foundations we estimate the essential norm for the differences of composition operators induced by ϕ and ψ acting from Lipschitz space to weighted Banach space in the unit polydisk Dn , where ϕ and ψ are two holomorphic self-maps of the unit polydisk in n-dimensional complex space C n . The paper is organized as following: Some lemmas are given in section 2. Section 3 is devoted to the main results. 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. 2. Some Lemmas Lemma 1. Assume that f ∈ Lipα (Dn ), then |f (z) − f (w)| ≤ nkf kα kDn (z, w) for any z, w ∈ Dn .

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Wang and Liang:: Differences of composition operators Proof. Empolying the definitions in (1.1) and (1.2) we have that n Z 1 Z 1 0, and we denote the second term in (3.5) by I. Empolying lemma 1, it follows that I

≤

sup v(z)(|f (ϕ(z)) − f (ψ(z))| + |f (rϕ(z)) − f (rψ(z))|)

sup

kf kα ≤1 z∈Eδ

≤

sup v(z)nkf kα (kDn (ϕ(z), ψ(z)) + kDn (rϕ(z), rψ(z)))

sup

kf kα ≤1 z∈Eδ

≤ 2n sup v(z)kDn (ϕ(z), ψ(z)),

(3.6)

z∈Eδ

the last inequality is obtained from kDn (rϕ(z), rψ(z)) ≤ kDn (ϕ(z), ψ(z)). Firstly letting r → 1 and then δ → 0, the upper estimate yeilds. The lower estimate. For l = 1, 2, .., n, set Eδl = {z ∈ Dn : max(|ϕl (z)|, |ψ1 (z)|) > 1 − δ}. Sn It is easy to see that Eδ = l=1 Eδl . For a fixed l (1 ≤ l ≤ n), define al = lim sup v(z)(1 − |ϕl (z)|2 )α |ϕψl (z) (ϕl (z))|. δ→0 z∈E l

δ

If we put δm = 1/m, then δm → 0 as m → ∞. For the case kϕl k∞ = 1 or kψl k∞ = 1, then for large enough m with Eδl m 6= ∅, there exists z m ∈ Eδl m such that lim v(z m )(1 − |ϕl (z m )|2 )α |ϕψl (zm ) (ϕl (z m ))| = al .

m→∞

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(3.7)

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Wang and Liang:: Differences of composition operators Since z m ∈ Eδl m implies that |ϕl (z m )| > 1 − δm or |ψl (z m )| > 1 − δm , without loss of generality we assume that |ϕl (z m )| → 1. Set fm (z) =

1 − |ϕl (z m )|2 (1 − ϕl (z m )zl )1−α

·

hϕψl (zm ) (z), ϕψl (zm ) (ϕl (z m ))i . |ϕψl (zm ) (ϕl (z m ))|

We can easily obtain that (fm )m∈N converges to zero uniformly on compact subsets of Dn as m → ∞ and sup kfm kα ≤ C. Thus for any compact operator K : Lipα → Hv∞ , k∈N

we get kKfm kHv∞ → 0, m → ∞. Moreover, it is obvious that fm (ϕ(z m )) = (1 − |ϕl (z m )|2 )α |ϕψl (zm ) (ϕl (z m ))|, fm (ψ(z m )) = 0.

(3.8)

Thus using the above results, (3.7) and (3.8), it is clear that kCϕ − Cψ − KkLipα →Hv∞ ≥ C lim sup k(Cϕ − Cψ − K)fm kHv∞ m→∞

≥ C lim sup(k(Cϕ − Cψ )fm kHv∞ − kKfm kHv∞ ) m→∞

= C lim sup k(Cϕ − Cψ )fm kHv∞ m→∞

= C lim sup sup v(z)|fm (ϕ(z)) − fm (ψ(z))| m→∞ z∈Dn m

≥ C lim sup v(z )|fm (ϕ(z m )) − fm (ψ(z m ))| m→∞

= C lim sup v(z m )|(1 − |ϕl (z m )|2 )α |ϕψl (zm ) (ϕl (z m ))| m→∞

= Cal = C lim sup v(z)(1 − |ϕl (z)|2 )α |ϕψl (z) (ϕl (z))| δ→0 z∈E l

δ

From the above inequality we obtain that kCϕ − Cψ ke,Lipα →Hv∞ ≥ C lim sup v(z)(1 − |ϕl (z)|2 )α |ϕψl (z) (ϕl (z))|. (3.9) δ→0 z∈E l

δ

If both kϕl k∞ < 1 and kψl k∞ < 1, in this condition, when δ is small enough, Eδl is empty, and without loss of generality we may assume that lim sup v(z)(1 − |ϕl (z)|2 )α |ϕψl (z) (ϕl (z))| = 0.

(3.10)

δ→0 z∈E l

δ

Since the above inequality (3.9) and (3.10) holds for every 1 ≤ l ≤ n, thus we obtain that kCϕ − Cψ ke,Lipα →Hv∞ ≥ C max lim sup v(z)|(1 − |ϕl (z)|2 )α |ϕψl (z) (ϕl (z))|. (3.11) 1≤l≤n δ→0 z∈E l

δ

Now for each l = 1, 2, ..., n, we define bl = lim sup v(z)(1 − |ϕl (z)|2 )α |ϕψl (z) (ϕl (z))|. δ→0 z∈Eδ

Then for any ε > 0, there exists a δ0 with 0 < δ0 < 1 such that v(z)(1 − |ϕl (z)|2 )α |ϕψl (z) (ϕl (z))| > bl − ε

(3.12)

whenever z ∈ Eδ0 and l = 1, 2, ..., n. From the above definition we know that z ∈ Eδl 0 implies that z ∈ Eδ0 , then by (3.11) and (3.12) we obtain that kCϕ − Cψ ke,Lipα →Hv∞ ≥ C max (bl − ε) 1≤l≤n

= C max lim sup v(z)|(1 − |ϕl (z)|2 )α |ϕψl (z) (ϕl (z))| − Cε. 1≤l≤n δ→0 z∈Eδ

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Wang and Liang:: Differences of composition operators Now letting ε → 0 in the above inequality we obtain that kCϕ − Cψ ke,Lipα →Hv∞ ≥ C max lim sup v(z)(1 − |ϕl (z)|2 )α |ϕψl (z) (ϕl (z))|. (3.13) 1≤l≤n δ→0 z∈Eδ

Using the similar proof of (3.13) we can get kCϕ − Cψ ke,Lipα →Hv∞ ≥ C max lim sup v(z)|(1 − |ψl (z)|2 )α |ψϕl (z) (ψl (z))|. (3.14) 1≤l≤n δ→0 z∈Eδ

Combining (3.13) and (3.14), we get the lower estimate for the essential norm of the differences.  References [1] J. Bonet, M.Lindstr¨ o, E. Wolf, Differences of composition operators between weighted Banach spaces of holomorphic functions, J. Austral. Math. Soc. 84 (2008) 9-20. [2] K.D. Bierstedt, W.H. Summers, Biduals of weighted Banach spaces of analytic functions, J. Austral. Math. Soc. (Series A) 54 (1993) 70-79. [3] C. C. Cowen, B. D. MacCluer, Composition Operators on Spaces of Analytic Functions, CRC Press, Boca Raton, FL, 1995. [4] Z. S. Fang, H. Z. Zhou, Differences of composition operators on the Bloch space in the polydisc, Bull. Aust. Math. Soc. 79 (2009) 465-471. [5] Z. S. Fang, H .Z. Zhou, Differences of composition operators on the space of bounded analytic functions in the polydisc, Abstr. Appl. Anal. Volume 2008, Article ID 983132, 10 pages. [6] P. Gorkin and B. D. MacCluer, Essential norms of composition operators, Integral Equations Operator Theory, 48 (2004) 27-40. [7] T. Hosokawa and S. Ohno, Topologicial structures of the set of composition operators on the Bloch space, J. Math. Anal. Appl. 34 (2006) 736-748. [8] W. Lusky, On weighted spaces of harmonic and holomorphic functions, J. London Math. Soc. 51(1995) 309-320. [9] J. Moorhouse, Compact differences of composition operators, J. Funct. Anal. 219 (2005) 70-92. [10] A. L. Shields, D. L. Williams, Bounded projections and the growth of harmonic conjugates in the disc, Michigan Math. J. 29 (1982) 3-25. [11] S. Li, S. Stevi´ c, Riemann-Stieltjes operators on Hardy spaces in the unit ball of Cn , Bull. Belg. Math. Soc. Simon Stevin 14 (2007) 621-628. [12] S. Stevi´ c , E. Wolf, Differences of composition operators between weighted-type spaces of holomorphic functions on the unit ball of C N , Appl. Math. Comput. 215 (2009) 1752-1760. [13] Z. Zhou, Composition operators on the Lipschitz space in polydiscs, Sci. China Ser. A, vol. 46 (1) 33-38. [14] Z. Zhou and Y. Liu, The essential norms of composition operators between generalized Bloch spaces in the polydisc and their applications, J. Inequal. Appl. vol. 2006, Article ID 90742, pages 1-22.

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THE PATH COMPONENT OF THE SET OF GENERALIZED COMPOSITION OPERATORS ON THE BLOCH TYPE SPACES LIU YANG Abstract. In this note, we give a characterization of the path component of the set of generalized composition operator on Bloch type spaces. Keywords: Path component, composition operator, Bloch type spaces

1. INTRODUCTION Let D be the unit disk of the complex plane C, and H(D) be the space of all analytic functions in D. f ∈ H(D) belongs to the Bloch type space Bα , if ∥f ∥Bα = |f (0)| + sup (1 − |z|2 ) |f ′ (z)| < ∞, α

z∈D

where 0 < α < ∞. It is known that B is a Banach space under the ∥ · ∥Bα norm. If α = 1, B α is just the well-known Bloch space. More details about properties on Bloch type space are given in [4], [32] and [16]. We denote S(D) be the set of analytic self-maps of D. Every analytic self-map φ ∈ S(D) induces a linear composition operator Cφ from H(D) to itself. A general and concerning problem in the investigation of composition operator is to characterize operator theoretic properties of Cφ in terms of function theoretic properties of φ. To learn more conclusions about the composition operator, see [6]. For φ ∈ S(D) and g ∈ H(D), Li and Stevic [10] defined the generalized composition operator Cφg as follows: ∫ z g Cφ (f )(z) = f ′ (φ(w))g(w)dw, f ∈ H(D). α

0

The boundedness and compactness of the generalized composition operator from Zygmund spaces to Bloch-type spaces were considered in [10]. Lindstrom and Sanatpour [15] gave the characterization of the generalized composition operator between Zygmund spaces. We can also refer to [11–14], [21–30] for the study of the operator Cφg and its generalizations. The composition operators between Bloch type spaces have been studied by several authors, for example [1–3, 5, 17, 19]. Recently, lots of researchers are interested in the difference of two composition operators, that is, an operator of the form T = Cφ − Cψ , where φ, ψ ∈ S(D). For example, Shapiro The work is supported by NSF of China (No. 11471202). Department of Mathematics, Shantou University, Guangdong Shantou 515063, P. R. China. e-mail:[email protected]. 1

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2

LIU YANG

and Sundberg [20] studied the difference of composition operators on Hardy spaces. In [18], MacCluer, Ohno and Zhao considered it on H ∞ . In [7] and [8], Hosokawa and Ohno investigated it on Bloch spaces. The purpose of studying the difference of composition operators is to investigate the topological structure of the set of composition operators acting on a given function space. Li [9] gave the sufficient and necessary conditions for the boundedness and compactness of the differences of generalized composition operator on the Bloch space. Yang, Luo, and Zhu [31] generalized Li’s results between Bloch type spaces, which help us to study the topological structure of the set of generalized composition operators on the Bloch type spaces. In fact, we give a sufficient condition for the path component of the set of generalized composition operator on Bloch type spaces.

2. NOTATIONS AND AUXILIARY RESULTS For w, z ∈ D, the pseudo-hyperbolic distance between z and w is defined by w−z ρ(w, z) = | |. 1 − wz ¯ Let us (z, w) = (1 − s)z + sw, ϕs (φ(z), ψ(z)) = (1 − s)φ(z) + sψ(z), where s ∈ [0, 1], w ∈ D, φ, ψ ∈ S(D) and simply denote ϕs (φ(z), ψ(z)) by ϕs (z). Let Γ(φ) = {{zn } ∈ D : |φ(zn )| → 1}, Γ(ψ) = {{zn } ∈ D : |ψ(zn )| → 1}. Obviously, Γ(ϕs ) ⊂ Γ(φ) ∩ Γ(ψ). Define β g(z) (1 − |z|2 ) φ,g Dαφ,g (z) = , D (z) = α g(z), α,β (1 − |φ(z)|2 )α (1 − |φ(z)|2 ) β

Dαψ,h (z)

h(z) (1 − |z|2 ) ψ,h = , D (z) = α h(z), α,β (1 − |ψ(z)|2 )α (1 − |ψ(z)|2 )

and ∫ Cϕs f (z) =

Dαϕs (z) z

1 − |z|α = [(1 − s)g(z) + sh(z)], (1 − |ϕ(z)|2 )α

f ′ ((1 − s)φ(w) + sψ(w))[(1 − s)g(w) + sh(w)]dw, f ∈ B α .

0

Let φ,g I1 (z) = Dα,β ρ(φ(z), ψ(z)), ψ,h I2 (z) = Dα,β ρ(φ(z), ψ(z)),

and φ,g ψ,h I3 (z) = Dα,β (z) − Dα,β (z).

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THE PATH COMPONENT OF THE SET OF GENERALIZED COMPOSITION OPERATORS

3

Lemma 2.1. ( [7, Lemma 4.1] ) Let z, w ∈ D and ρ(z, w) = λ < 1. Then the map s 7→ ρ(us , w) is continuous and decreasing on [0, 1]. Lemma 2.2. ( [31, Theorem 1.] ) The following statements are equivalent: (1) Cφg − Cψh : B α → B β is bounded. (2) supz∈D |I1 (z)| < ∞ and supz∈D |I3 (z)| < ∞. (3) supz∈D |I2 (z)| < ∞ and supz∈D |I3 (z)| < ∞. Lemma 2.3. ([31, Theorem 4.] ) Let 0 < α, β < ∞ and φ, ψ ∈ S(D), g, h ∈ H(D), if Cφg − Cψh : B α → B β is bounded, and Cφg , Cψh : B α → B β are not compact, then Cφg − Cψh : B α → B β is compact if and only if the following two conditions hold. (1) D(g, φ) = D(h, ψ) ̸= ∅, D(g, φ) ⊂ Γ(ψ). (2) For arbitrary {zn } ⊂ Γ(φ) ∩ Γ(ψ), lim |I1 (zn )| = lim |I2 (zn )| = lim |I3 (zn )| = 0.

n→∞

n→∞

n→∞

Lemma 2.4. If t < 0 or t > 1, then 1 − x ≤ t(1 − x). t

Proof. Let f (x) = 1 − xt − t(1 − x), then f ′ (x) = −tx( t − 1) + t, f ′′ (x) = −t(t − 1)x(t − 2). Obviously, f ′ (1) = 0, f ′′ (1) ̸= 0, f ′′ (x) > 0 for t < 0, f ′′ (x) < 0 for t > 1.



Lemma 2.5. Let φ, ψ be analytic self maps of the unit disk D, then (1) For any z ∈ D, when α < 1, we have φ,g Dα (z) − Dαϕs (z) ≤ Dαφ,g (z) − Dαψ,h (z) + (2 − α) Dαφ,g (z) ρ2 (φ(z), ψ(z)). (2) For any z ∈ D, when α ≥ 1, we have φ,g ) ( Dα (z) − Dαϕs (z) ≤ Dαφ,g (z) − Dαψ,h (z) + α Dαφ,g (z) + Dαψ,h (z) ρ2 (φ(z), ψ(z)). Proof. (1) The lemma is trivially for s = 0 or 1. In the following, we assume 0 < s < 1. For 1−|φ(z)|2 1−|ψ(z)|2 φ,g ψ,h arbitrary z ∈ D, denote ζ = 1−|ϕ 2 and ξ = 1−|ϕ (z)|2 . By the definition of Dα (z), Dα (z) (z)| s s and Dαϕs (z), it is easy to see 1 − |z|α [(1 − s)g(z) + sh(z)] (1 − |ϕ(z)|2 )α (1 − |φ(z)|2 )α φ,g (1 − |ψ(z)|2 )α ψ,h = (1 − s) Dα (z) + s D (z) 2 α (1 − |ϕs (z)| ) (1 − |ϕs (z)|2 )α α

Dαϕs (z) =

= Dαφ,g − (1 − s)ζ α Dαφ,g (z) − sξ α Dαψ,h (z)

and 2 α 2 α φ,g Dα (z) − Dαϕs (z) = Dαφ,g (z) − (1 − s) (1 − |φ(z)| ) Dαφ,g (z) − s (1 − |ψ(z)| ) Dαψ,h (z) (1 − |ϕs (z)|2 )α (1 − |ϕs (z)|2 )α = Dαφ,g (z) − (1 − s)ζ α Dαφ,g (z) − sξ α Dαψ,h (z) = Dαφ,g (z)(1 − (1 − s)ζ α ) − Dαψ,h (1 − (1 − s)ζ α ) + Dαψ,h (1 − (1 − s)ζ α ) − sξ α Dαψ,h (z) ≤ Dαφ,g (z) − Dαψ,h (1 − (1 − s)ζ α ) + Dαψ,h (1 − (1 − s)ζ α ) − sξ α ≤ Dαφ,g (z) − Dαψ,h sζ α + Dαφ,g (1 − (1 − s)ζ α ) − sξ α .

(2.1)

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φ,g Dα − Dαϕs (z) ≤ Dαφ,g − Dαψ,h sζ α + Dαφ,g (1 − (1 − s)ζ α ) − sξ α .

(2.2)

By simply calculating and the proving process of Proposition 4.2 in [7], we get 0≤

s(1 − s)|φ(z) − ψ(z)|2 = 1 − (1 − s)ζ − sξ ≤ ρ2 (φ(z), ψ(z)). 1 − |ϕs (t)|2

(2.3)

Firstly, we consider the case 0 < α < 1. 1−|φ(z)|2 Since sζ = s 1−|ϕ 2 ≤ 1, then s (z)| sζ α ≤ s1−α ≤ 1.

(2.4)

Now, we estimate (1 − (1 − s)ζ α ) − sξ α . Choosing f (ζ) = 1 − (1 − s)ζ α − sξ α − (1 − (1 − s)ζ − sξ),

(2.5)

then f (ζ) = (1 − s)ζ(1 − ζ α−1 ) + sξ(1 − ξ α−1 ) ≤ (α − 1)((1 − s)ζ(1 − ζ) + sξ(1 − ξ))

(2.6)

= (α − 1)((1 − s)ζ 2 − sξ 2 ) − (α − 1)(1 − (1 − s)ζ − sξ). The last inequality above is obtained by Lemma 2.4. Uniting (2.5) and (2.6), we obtain 1 − (1 − s)ζ α − sξ α − (1 − (1 − s)ζ − sξ) ≤(1 − (1 − sζ)α − sξ) − (α − 1)(1 − (1 − s)ζ 2 − sξ 2 ) − (α − 1)(1 − (1 − sζ) − sξ)

(2.7)

=(2 − α)(1 − (1 − s)ζ − sξ) + (α − 1)(1 − (1 − s)ζ − sξ ). 2

2

and 1 − (1 − s)ζ 2 − sξ 2 =

s|ψ(z)|2 (1 − |ψ(z)|2 ) + (1 − s)|φ(z)|2 (1 − |φ(z)|2 )) + s(1 − s)|φ(z) − ψ(z)|2 >0 (1 − |ϕs (z)|2 )2

(2.8) Hence, 1 − (1 − s)ζ α − sξ α ≤ (2 − α)(1 − (1 − s)ζ − sξ) ≤ (2 − α)ρ2 (φ(z), ψ(z)).

(2.9)

Combining (2.1), (2.4) and (2.9), we get φ,g Dα (z) − Dαϕs (z) ≤ Dαφ,g (z) − Dαψ,h (z) + (2 − α) Dαφ,g (z) ρ2 (φ(z), ψ(z)). This complete the proof of (1). Next, we are going to prove (2). If α = 1, then by (2.3), we have 1 − (1 − s)ζ α − sξ α = 1 − (1 − s)ζ − sξ ≤ ρ2 (φ(z), ψ(z)) = αρ2 (φ(z), ψ(z).

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(2.10)

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If α > 1, then 1 − (1 − s)ζ α − sξ α = 1 − s − (1 − s)ζ α + s − sξ α = (1 − s)(1 − ζ α ) + s(1 − ξ α ) ≤ α(1 − s)(1 − ζ) + s(1 − ξ))

(2.11)

= α(1 − (1 − s)ζ − sξ) ≤ αρ2 (φ(z), ψ(z)). The first inequality in (2.11) above is obtained by Lemma 2.4. If ξ ≤ 1, using (2.1) and (2.11), we obtain φ,g Dα (z) − Dαϕs (z) ≤ Dαφ,g (z) − Dαψ,h (z) + α Dαφ,g (z) ρ2 (φ(z), ψ(z)).

(2.12)

If ξ ≥ 1, for any s ∈ (0, 1), we have |ψ(z)| ≤ |ϕs (z)| ≤ (1 − s)|φ(z)| + s|ψ(z)| and 1−|φ(z)|2 |ψ(z)| ≤ |φ(z)|. Then |ϕs (z)| ≤ |φ(z)| and 1−|ϕ 2 = ζ ≤ 1. Combing (2.1), (2.10) with s (z)| (2.11), it is obvious that φ,g ) Dα (z) − Dαϕs (z) ≤ Dαφ,g (z) − Dαψ,h (z) + α Dαψ,h (z) ρ2 (φ(z), ψ(z)). (2.13) Due to (2.11), (2.12) and (2.13) above, we infer that φ,g ) ( Dα (z) − Dαϕs (z) ≤ Dαφ,g (z) − Dαψ,h (z) + α Dαφ,g (z) + Dαψ,h (z) ρ2 (φ(z), ψ(z)).  3. MAIN RESULTS Proposition 3.1. Let φ, ψ be analytic self maps of the unit disk D, g, h ∈ H(D). Suppose that Cφg and Cψh are bounded but not compact on B α . For any s ∈ [0, 1], when Cφg − Cψh is compact on B α , then we have (1) Dϕαs ⊂ Γ(φ) ∩ Γ(ψ), where Dϕαs = {{zn } ⊂ D : |φ(zn )| → 1, |Dϕαs (zn )| ̸→ 1}. (2) For any {z}n ⊂ Γ(φ) ∩ Γ(ψ), we have lim (Dαφ,g (zn ) − Dαϕs (zn )) = lim (Dαφ,g (zn )ρ(φ(zn ), ϕs (zn )) = 0.

n→∞

n→∞

Proof. (1) It is trivial. (2) For any {zn } ⊂ Γ(φ) ∩ Γ(ψ), it follows from Lemma 2.3 that lim Dαφ,g (zn ) − Dαϕs (zn ) = lim (Dαφ,g (zn ) ρ(φ(zn ), ϕs (zn )) n→∞ n→∞ = lim Dαψ,h (zn ) ρ(φ(zn ), ϕs (zn )) n→∞

= 0. Applying Lemma 2.5, lim |Dαφ,g (zn ) − Dαϕs (zn )| = 0,

n→∞

then by Lemma 2.1, |Dαφ,g (zn )|ρ(φ(zn ), ϕs (zn )) ≤ |Dαφ,g |ρ(φ(zn ), ψ(zn )) → 0.

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Equivalently, lim (Dαφ,g (zn ) − Dαϕs (zn )) = lim (Dαφ,g (zn )ρ(φ(zn ), ϕs (zn )) = 0.

n→∞

n→∞

 Theorem 3.2. Let φ, ψ be analytic self maps of the unit disk D, g, h ∈ H(D). Suppose that Cφg and Cψh are bounded but not compact on B α . If Cφg − Cψh is compact on Bα , then the following two conclusions are equivalent: (1) For any {zn } ⊂ Γ(ψ)\Γ(φ), Dαφ,g (zn ) → 0 as n → ∞ and for any {zn } ⊂ Γ(φ)\Γ(ψ), Dαϕ,h (zn ) → 0 as n → ∞ . (2) The map s 7→ Cϕs : [0, 1] → Cϕs (Bα ) is continous. Proof. (1) =⇒ (2) We only need to prove the continuity at s = 0. Let t(s) = sup |Dαφ,g (z) − Dαϕs (z)| + sup |Dαφ,g (z)|ρ(φ(z)), ψ(z)). z∈D

Then, it is easy to see that we have

z∈D

∥Cφg

− Cϕs ∥

Bα

≤ t(s). By lemma 2.3 and the conditions of (1),

lim |Dαφ,g (zn ) − Dαψ,h (zn )| = lim |Dαφ,g (zn )|ρ(φ(zn ), ψ(zn )) n→∞

n→∞

= lim |Dαψ,h (zn )|ρ(φ(zn ), ψ(zn )) n→∞

=0

Hence, for any ε > 0, there exists r1 ∈ (0, 1) such that for every z ∈ Γr1 (φ) = {z ∈ D : |φ(z)| > r1 }, ε |Dαφ,g (z) − Dαψ,h (z)| < , 2 and ε |Dαφ,g (z)|ρ(φ(z), ψ(z) < . 2 Applying Lemma 2.5, we obtain that ε 1 |Dαφ,g (z) − Dαψ,h (z)| < + αε = ( + α)ε. (3.1) 2 2 If z ∈ D\Γr1 (φ), Dαφ,g − Dαϕs is uniformly convergence to 0 when s approaches to 0, then there exists an s1 very close to 0 such that for any s < s1 , sup |Dαφ,g (z) − Dαϕs (z)| < ε. z∈D\Γr1

(3.2)

For any s < s1 , uniting (3.1) and (3.2), we get sup |Dαφ,g (z) − Dαϕs (z)| < ε.

(3.3)

sup |Dαφ,g (z) − Dαϕs (z)| → 0 as s → 0.

(3.4)

z∈D

Hence, z∈D

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Next, we are going to prove that sup |Dαφ,g (z)|ρ(φ(z)), ψ(z)) → 0 as s → 0. z∈D

For any {zn } ⊂ Γ(φ), applying Proposition 3.1 and Lemma 2.3, we have lim (Dαφ,g ρ(φ(zn ), ϕs (zn )) = 0.

n→∞

This implies that there exists an r2 ∈ (0, 1), such that for any z ∈ Γr2 (φ) = {z ∈ D : |φ(z)| > r2 }, |Dαφ,g (z)|ρ(φ(z), ψ(z)) ≤ |Dαφ,g (z)|ρ(φ(z), ψ(z)) < ε.

And because ρ(φ(z), ψ(z)) uniformly converges to 0 on D\Γr2 (φ), we can find a sufficiently small positive number s2 , such that for any s < s2 , |Dαφ,g (z)|ρ(φ(z), ϕs (z)) < ε.

sup z∈D\Γr2 (φ)

Then,

sup |Dαφ,g (z)|ρ(φ(z), ϕs (z)) → 0 as s → 0.

(3.5)

z∈D

Combing (3.4) with (3.5), we obtain that t(s) converges to 0 as s approaches to 0, which finishes the proof of continuity. (2) =⇒ (1) Assume there is a sequence {zn } ⊂ Γ(ψ)\Γ(φ), such that Dαφ,g (zn ) → δ ̸= 0 as n → ∞. Let λ ∈ D and λ ̸= 0, define the test function fλ and gλ respectively as follows: fλ (z) = gλ (z) =

1 2α+1

1 − |λ|2 ¯ ¯ α, αλ(1 − λz)

1 − |λ|2 λ−z 1 (¯ + ¯2 α+1 α+1 ¯ ¯ α+1 ). (α + 1)2 λ(1 − λz) αλ (1 − λz)

Then ∥fλ ∥Bα ≤ 1, ∥gλ ∥Bα ≤ 1, ∥Cφg − Cϕs ∥ ≥ ∥(Cφg − Cϕs )gφ(zn ) ∥Bα 1 (1 − |φ(zn )|2 )(1 − |φs (zn )|2 )α ≥ α+1 Dαϕs (zn ) ρ(φ(zn ), ϕs (zn )) . α+1 2 (1 − φ(zn )ϕs (zn ))

(3.6)

Because zn ∈ Γ(ψ)\Γ(φ), then ϕs (zn ) ̸→ 1 and limn→∞ ρ(φ(zn ), ϕs (zn )) ̸= 0. And s 7→ Cϕs is continous at 0, then by (3.6), we have (1 − |φ(zn )|2 )(1 − |φs (zn )|2 )α ϕs ρ(φ(zn ), ϕs (zn ) → 0, n → ∞, s → 0. Dα (zn ) (1 − φ(zn )ϕs (zn ))α+1

By the compactness of Cφg − Cψh , it is bounded. It follows from Lemma 2.1, Lemma 2.2 and lemma 2.5 that Cφg − Cϕs is bounded. So ∥Cφg − Cϕs ∥ ≥ ∥(Cφg − Cϕs )gφ(zn ) ∥Bα (1 − |φ(z )|2 )(1 − |φ (z )|2 )α ) 1 ( n s n ≥ α+1 Dαφ,g (zn ) − Dαϕs (zn ) . α+1 2 (1 − φ(zn )ϕs (zn ))

(3.7)

Letting n → ∞ and s → 0, we have ∥Cφg − Cϕs ∥ ≥

62

δ 2α+1

> 0.

(3.8)

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For φ(zn ) ≡ 0, suppose λ ∈ D, λ ̸= 0 and hλ (z) =

1 2α+1

1 ¯ ¯ α. αλ(1 − λz)

(3.9)

Then hλ ∈ B α and ∥hλ ∥Bα ≤ 1. If s ̸= 0, then ϕs (zn ) → s ̸= 0. Choosing λ = ϕs (zn ), we have ∥(Cφg − Cϕs )hϕs (zn ) ∥Bα ≥ ∥(Cφg − Cϕs )hϕs (zn ) ∥Bα 1 ( Dαϕs (zn ) ) ≥ α+1 |(1 − |zn |2 )α φ′ (zn )|Dαϕs (zn ) − . 2 1 − |ϕs (zn )|α

For Γ(ψ)\Γ(φ), Proposition 3.1 implies that Dαϕs (zn ) → 0. Letting n → ∞ and s → 0, we get ∥Cφg − Cϕs ∥ ≥ δ > 0. (3.10) It follows from (3.8) and (3.10) that the map s 7→ Cϕs is not continuous at 0, which is a contradiction. So we complete the proof.  Corollary 3.3. Let φ, ψ be two analytic self maps of the unit disk D, g, h ∈ H(D). Suppose Cφg and Cψh are bounded but not compact on Bα . If Cφg − Cψh is compact on B α , then Cφg and Cψh are in the same path component of B α . References [1] R. Allen and F. Colonna, Compact composition operators on the Bloch space, Trans. Amer. Math. Soc. 347, 2679-2687 (1995). [2] R. Allen and F. Colonna, On the isometric composition operators on the Bloch space in Cn , J. Math. Anal. Appl. 355, 675-688 (2009). [3] R. Allen and F. Colonna, Weighted composition operators from H ∞ to the Bloch space of a bounded homogeneous domain, Integr. Equ. Oper. Theory 66, 21-40 (2010). [4] Anderson, J. M., Clunie, J., and Pommerenke, Ch., On Bloch functions and normal functions, J. Reine Angew. Math. 270, 12-37 (1974). [5] F. Colonna, Characterisation of the isometric composition operators on the Bloch space, Bull. Austral. Math. Soc. 72, 283-290 (2005). [6] C. Cowen and B. MacCluer, Composition Operators on Spaces of Analytic Functions, CRC Press, Boca Raton, 1995. [7] T. Hosokawa and S. Ohno, Topological structures of the sets of composition operators on the Bloch spaces, J. Math. Anal. Appl. 314, 736-748 (2006). [8] T. Hosokawa and S. Ohno, Differences of composition operators on the Bloch spaces, J. Oper. Theory. 57, 229-242 (2007). [9] S Li, Differences of generalized composition operators on the Bloch space, J. Math. Anal. Appl. 394,706-711 (2012). [10] S. Li and S. Stevi´c, Generalized composition operators on Zygmund spaces and Bloch type spaces, J. Math. Anal. Appl. 338, 1282-1295 (2008). [11] S. Li and S. Stevi´c, Products of Volterra type operator and composition operator from H ∞ and Bloch spaces to the Zygmund space, J. Math. Anal. Appl. 345, 40-52 (2008).

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[12] S. Li and S. Stevi´c, Products of composition and integral type operators from H ∞ to the Bloch space, Complex Var. Elliptic Equ. 53, 463-474 (2008). [13] S. Li and S. Stevi´c, Products of integral-type operators and composition operators between Bloch-type spaces, J. Math. Anal. Appl. 349, 596-610 (2009). [14] S. Li and S. Stevi´c, On an integral-type operator from iterated logarithmic Bloch spaces into Bloch-type spaces, Appl. Math. Comput. 215, 3106-3115 (2009). [15] M. Lindstrom and A. Sanatpour, Derivative-free characterization of compact generalized composition operators between Zygmund type spaces, Bull. Austral. Math. Soc. 81, 398-408 (2010). [16] Z. Lou, Bloch Type Spaces of Analytic Functions, PhD Thesis, Institute of Mathematics, Academia Sinica, 1998. [17] Z. Lou, Composition operators on Bloch type spaces, Analysis (Munich). 23, 81-95 (2003). [18] B. MacCluer, S. Ohno and R. Zhao, Topological structure of the space of composition operators on H ∞ , Integr. Equ. Oper. Theory 40, 481-494 (2001). [19] S. Ohno, K. Stroethoff and R. Zhao, Weighted composition operators between Bloch type spaces, Rocky Mountain J. Math. 33, 191-215 (2003). [20] J. Shapiro and C. Sundberg, Isolation amongst the composition operators, Pacific J. Math. 145, 117-152 (1990). [21] S. Stevi´c, Generalized composition operators from logarithmic Bloch spaces to mixednorm spaces, Util. Math. 77, 167-172 (2008). [22] S. Stevi´c, On an integral operator from the Zygmund space to the Bloch-type space on the unit ball, Glasg. J. Math. 51, 275-287 (2009). [23] S. Stevi´c, Products of integral-type operators and composition operators from the mixed norm space to Bloch-type spaces, Siberian Math. J. 50, 726-736 (2009). [24] S. Stevi´c, Integral-type operators from a mixed norm space to a Bloch-type space on the unit ball, Siberian Math. J. 50, 1098-1105 (2009). [25] S. Stevi´c, On an integral operator between Bloch-type spaces on the unit ball, Bull. Sci. Math. 134, 329-339 (2010). [26] S. Stevi´c, On an integral-type operator from logarithmic Bloch-type spaces to mixed norm spaces on the unit ball, Appl. Math. Comput. 215, 3817-3823 (2010). [27] S. Stevi´c, On some integral-type operators between a general space and Bloch-type spaces, Appl. Math. Comput. 218, 2600-2618 (2011). [28] S. Stevi´c and A. Sharma, Composition operators from weighted Bergman-Privalov spaces to Zygmund type spaces on the unit disk, Ann. Polon. Math. 105, 77-86 (2012). [29] S. Stevi´c and A. Sharma, Generalized composition operators on weighted Hardy spaces, Appl. Math. Comput. 218, 8347-8352 (2012). [30] S. Stevi´c and S. Ueki, Integral-type operators acting between weighted-type spaces on the unit ball, Appl. Math. Comput. 215, 2464-2471 (2009). [31] W. Yang, Y. Luo and X. Zhu, Differences of generalized composition operators between Bloch type spaces, Math. Inequal. Appl. 17, 977-987 (2014). [32] K. Zhu, Bloch type spaces of analytic functions, Rocky Mountain J. Math. 23, 11431177 (1993).

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THE GENERALIZED HYERS-ULAM STABILITY OF QUADRATIC FUNCTIONAL EQUATIONS ON RESTRICTED DOMAINS CHANG IL KIM AND CHANG HYEOB SHIN*

Abstract. In this paper, we prove the generalized Hyers-Ulam stability for the functional equation f (ax + by) + abf (x − y) = a(a + b)f (x) + b(a + b)f (y) for some real numbers a, b with 2a + b = 1 on a restricted domain using the fixed point theorem. Key words. Generalized Hyers-Ulam stability, Quadratic functional equation, Banach space, Restricted domains, Fixed point theorem

1. Introduction In 1940, S. M. Ulam [15] proposed the following stability problem : “Let G1 be a group and G2 a metric group with the metric d. Given a constant δ > 0, does there exist a constant c > 0 such that if a mapping f : G1 −→ G2 satisfies d(f (xy), f (x)f (y)) < c for all x, y ∈ G1 , then there exists a unique homomorphism h : G1 −→ G2 with d(f (x), h(x)) < δ for all x ∈ G1 ?” In 1941, Hyers [7] answered this problem under the assumption that the groups are Banach spaces. Aoki [1] and Rassias [11] generalized the result of Hyers. Rassias [11] solved the generalized Hyers-Ulam stability of the functional inequality kf (x + y) − f (x) − f (y)k ≤ (kxkp + kykp ) for some  ≥ 0 and p with 0 < p < 1 and all x, y ∈ X, where f : X −→ Y is a function between Banach spaces. A generalization of the Rassias theorem was obtained by Gˇ avruta [6] by replacing the unbounded Cauchy difference by a general control function in the spirit of Rassis approach. The functional equation (1.1)

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

is called a quadratic functional equation and a solution of a quadratic functional equation is called quadratic. The generalized Hyers-Ulam stability problem for a quadratic functional equation was proved by Skof [13] for mappings f : X −→ Y , where X is a normed space and Y is a Banach space. Cholewa [2] noticed that the theorem of Skof is still true if the relevant domain X is replaced by an Abelian group. Czerwik [3] proved the generalized Hyers-Ulam stability for a quadratic functional equation. 2010 Mathematics Subject Classification. 39B52, 39B82. *Corresponding Author. 1

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Skof [14] was the first author to slove the Hyers-Ulam problem for additive mappings on a restricted domain and in 1998, Jung [8] investigated the HyersUlam stability for additive and quadratic mappimgs on resticted domains. In 2002, Rassias [12] proved that if f : X −→ Y satisfies the following inequality (1.2)

kf (x + y) + f (x − y) − 2f (x) − 2f (y)k ≤ δ,

then there exists a unique quadratic mapping which is approximately. Recently, Najati and Jung [9] showed that the functional equation (1.3)

f (ax + by) + abf (x − y) = af (x) + bf (y)

is equivalent to (1.1) if a, b are non-zero real numbers with a+b = 1 and proved that the Hyers-Ulam stability for the functional equation (1.3) on a resticted domain if f is even. Elhoucien and Youssef [5] showed the results in [9] by removing the Najati-Jung’s assumption that f is even. In this paper, we consider the functional equation (1.4)

f (ax + by) + abf (x − y) = a(a + b)f (x) + b(a + b)f (y)

for fixed non-zero real numbers a, b with 2a + b = 1, a 6= 1 and we prove the generalized Hyers-Ulam stability of it on a restricted domain. Throughout this paper, we assume that X is a normed space and Y is a Banach space. 2. Solutions of (1.4) Najati and Jung [9] showed that if an even mapping f : X −→ Y satisfies (1.3), then f is quadratic and that if a, b are rational numbers, then f satisfies (1.3) if and only if f is quadratic. Elhoucien and Youssef [5] showed that if a mapping f : X −→ Y satisfies (1.3), then f is additive-quadratic. In this section, we will show that if a mapping f : X −→ Y satisfies (1.4), then f is quadratic. Theorem 2.1. Let f : X −→ Y be a mapping satisfying (1.4). Then f is a quadratic mapping. Proof. Letting x = y = 0 in (1.4), since 2a + b = 1, we have (a2 + ab + b2 − 1)f (0) = 3a(a − 1)f (0) = 0. Since a 6= 0, 1, f (0) = 0. Letting y = 0 in (1.4), we have f (ax) = a2 f (x)

(2.1)

for all x ∈ X. Letting x = 0 in (1.4), we have (2.2)

f (by) = b(a + b)f (y) − abf (−y)

for all y ∈ X. Let fo (x) = hence by (2.2), we have

f (x)−f (−x) . 2

(2.3)

Then fo satisfies (1.4), (2.1) and (2.2) and

fo (bx) = bfo (x)

for all x ∈ X. By (1.4), we have (2.4)

fo (ax + by) + fo (ax − by) = 2a(a + b)fo (x) − ab[fo (x + y) + fo (x − y)]

for all x, y ∈ X. Letting y = ay in (2.4), by (2.1), we have (2.5)

a[fo (x + by) + fo (x − by)] = 2(a + b)fo (x) − b[fo (x + ay) + fo (x − ay)]

for all x, y ∈ X and letting x = bx in (2.5), by (2.3), we have (2.6)

fo (bx + ay) + fo (bx − ay) = 2(a + b)fo (x) − a[fo (x + y) + fo (x − y)]

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for all x, y ∈ X. Interchanging x and y in (1.4), we have (2.7)

fo (bx + ay) + fo (bx − ay) = 2b(a + b)fo (x) + ab[fo (x + y) + fo (x − y)]

for all x, y ∈ X. By (2.6) and (2.7), since a(a + b) 6= 0, we have fo (x + y) + fo (x − y) − 2fo (x) = 0 for all x, y ∈ X and hence fo is additive. By (2.1), we have a2 fo (x) = afo (x) and since a 6= 0, 1, fo (x) = 0 for all x ∈ X. (−x) Let fe (x) = f (x)+f . Then fe : X −→ Y is an even mapping satisfying (1.4) 2 and so fe satisfies (2.1) and (2.2). Replacing x and y by 2x and x + y in (1.4), we have (2.8)

fe (x + by) + abfe (x − y) − a(a + b)fe (2x) − b(a + b)fe (x + y) = 0

for all x, y ∈ X. Since a(a + b) 6= 0 and fe is even, by (2.8), we have (2.9)

fe (2x) = 4fe (x), fe (bx) = b2 f (x)

for all x ∈ X. Letting x = bx in (2.8), by (2.9), we have (2.10)

bfe (x + y) + afe (bx − y) − 4ab(a + b)fe (x) − (a + b)fe (bx + y) = 0

for all x, y ∈ X. Interchanging x and y in (2.10), we have (2.11)

bfe (x + y) + afe (x − by) − 4ab(a + b)fe (y) − (a + b)fe (x + by) = 0

for all x, y ∈ X. Letting y = −y in (2.8), we have (2.12)

fe (x − by) + abfe (x + y) − 4a(a + b)fe (x) − b(a + b)fe (x − y) = 0

for all x, y ∈ X. Since b(1 − 2a2 − 2ab − b2 ) = 2ab(a + b), by (2.8), (2.11), and (2.12), we have fe (x + y) + fe (x − y) = 2fe (x) + 2fe (y) for all x, y ∈ X and so fe is quadraric. Since f = fo + fe = fe , f is quadratic.



Corollary 2.2. Let f : X −→ Y be a mapping. If a, b are rational numbers, then f is quadratic if and only if f satisfies (1.4). 3. Stability of (1.4) In this section, we investigate the generalized Hyers-Ulam stability of (1.4) on a restricted domain. Jung [8] proved the Hyers-Ulam stability for additive and quadratic mappings on a resticted domain and Najati and Jung [9] proved the Hyers-Ulam syability of (1.3) on a resticted domain if f is an even mapping. Rahimi, Najati and Bae [10] investigated the generalized Hyers-Ulam syability of (1.1) with the bounded function δ + (kxk2p + kyk2p ) + θkxkp kykp on a resticted domain. Theorem 3.1. Let φ : X 2 −→ [0, ∞) be a mapping and M a non-negative real number. Let f : X −→ Y be a mapping with f (0) = 0. Suppose that f satisfies the following inequality (3.1)

kf (ax + by) + abf (x − y) − a(a + b)f (x) − b(a + b)f (y)k ≤ δ + φ(x, y)

for all x, y ∈ X with kxk + kyk ≥ M and for some non-negative real number δ. Then we have (3.2)

kf (2x) − 4f (x)k ≤ Φ(x, y)

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for all x, y ∈ X with kyk ≥ M , where Φ(x, y) = {φ(2x − 2by, x + (1 − b)y) + φ(2x − 2by, x − (1 + b)y) + φ(2x + 2by, x + (1 + b)y) + φ(2x + 2by, x − (1 − b)y) + |b|[φ(2x + 2y, x + 2y) + φ(2x + 2y, x) + φ(2x − 2y, x) + φ(2x − 2y, x − 2y)] + φ(2x, x + 2y) + φ(2x, x − 2y) + 4(|b| + 2)δ} × |2a(a + b)|−1 . Proof. Let x, y ∈ X with kxk + kyk ≥ M . Then k2xk + kx + yk ≥ kxk + kyk ≥ M . Hence by (3.1), we have (3.3)

kf (x + by) + abf (x − y) − a(a + b)f (2x) − b(a + b)f (x + y)k ≤ δ + φ(2x, x + y)

and letting y = −y in (3.3), we have (3.4)

kf (x − by) + abf (x + y) − a(a + b)f (2x) − b(a + b)f (x − y)k ≤ δ + φ(2x, x − y).

By (3.3) and (3.4), we have (3.5)

kf (x + by) − f (x − by) + bf (x − y) − bf (x + y)k ≤ 2δ + φ(2x, x + y) + φ(2x, x − y).

Let x, y ∈ X with kyk ≥ M . Since kx − byk + kyk ≥ M and kx + byk + kyk ≥ M , by (3.5), we have (3.6)

kf (x) − f (x − 2by) + bf (x − (1 + b)y) − bf (x + (1 − b)y)k ≤ 2δ + φ(2x − 2by, x + (1 − b)y) + φ(2x − 2by, x − (1 + b)y)

and (3.7)

kf (x + 2by) − f (x) + bf (x − (1 − b)y) − bf (x + (1 + b)y)k ≤ 2δ + φ(2x + 2by, x + (1 + b)y) + φ(2x + 2by, x − (1 − b)y).

Since kx + yk + kyk ≥ M and kx − yk + k − yk ≥ M , by (3.5), we have (3.8)

kf (x + (1 + b)y) − f (x + (1 − b)y) + bf (x) − bf (x + 2y)k ≤ 2δ + φ(2x + 2y, x + 2y) + φ(2x + 2y, x)

and (3.9)

kf (x − (1 + b)y) − f (x − (1 − b)y) + bf (x) − bf (x − 2y)k ≤ 2δ + φ(2x − 2y, x) + φ(2x − 2y, x − 2y).

Since kxk + k2yk ≥ M , by (3.3) and (3.4), we have (3.10)

kf (x + 2by) + abf (x − 2y) − a(a + b)f (2x) − b(a + b)f (x + 2y)k ≤ δ + φ(2x, x + 2y).

and (3.11)

kf (x − 2by) + abf (x + 2y) − a(a + b)f (2x) − b(a + b)f (x − 2y)k ≤ δ + φ(2x, x − 2y).

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Note that 2a(a + b)[f (2x) − 4f (x)] = −[f (x) − f (x − 2by) + bf (x − (1 + b)y) − bf (x + (1 − b)y)] + [f (x + 2by) − f (x) + bf (x − (1 − b)y) − bf (x + (1 + b)y)] (3.12)

+ b[f (x + (1 + b)y) − f (x + (1 − b)y) + bf (x) − bf (x + 2y)] + b[f (x − (1 + b)y) − f (x − (1 − b)y) + bf (x) − bf (x − 2y)] − [f (x + 2by) + abf (x − 2y) − a(a + b)f (2x) − b(a + b)f (x + 2y)] − [f (x − 2by) + abf (x + 2y) − a(a + b)f (2x) − b(a + b)f (x − 2y)]

for all x, y ∈ X with kyk ≥ M . By (3.6), (3.7), (3.8), (3.9), (3.10), (3.11), we have (3.2).  We apply the fixed point method to investigate the generalized Hyers-Ulam stability for the functional equation (1.4). Definition 3.2. Let X be a set. A function d : X × X −→ [0, ∞] is called a generalized metric on X if d satisfies (i) d(x, y) = 0 if and only if x = y, (ii) d(x, y) = d(y, x) for all x, y ∈ X, and (iii) d(x, z) ≤ d(x, y) + d(y, z) for all x, y, z ∈ X. Now, we consider the following fixed point theorem : Theorem 3.3. [4] Let (X, d) be a complete generalized metric space and J : X −→ X a strictly contractive mapping with a Lipschitz constant L with 0 < L < 1. Then for each element x ∈ X, either (3.13)

d(J n x, J n+1 x) = ∞

for all nonnegative integers n or there is a nonnegative integer k such that (1) d(J n x, J n+1 x) < ∞ for all n ≥ k, (2) a 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 k x, y) < ∞}, and 1 d(y, Jy) for all y ∈ Y . (4) d(y, y ∗ ) ≤ 1−L Now, we will prove the stability of (1.4) on a restrcted domain. Theorem 3.4. Let φ : X 2 −→ [0, ∞) be a function such that (3.14)

φ(2x, 2y) ≤ Lφ(x, y)

for all x, y ∈ X for some positive real number L with L < 1. Let f : X −→ Y be a mapping with (3.1). Then there exists a unique quadratic mapping Q : X −→ Y such that f satisfies (1.4) and 1 Φ(x, y) (3.15) kQ(x) − f (x)k ≤ 4(1 − L) for all x ∈ X and y ∈ X with kyk ≥ M . Proof. By Theorem 3.1, the following inequality (3.16)

kf (x) − 2−2 f (2x)k ≤ 2−2 Φ(x, y)

holds for all x, y ∈ X with kyk ≥ M .

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Let Ω = {g : X −→ Y | g(0) = 0}. Define a generalized metric d on Ω by d(g, h) = inf {C ∈ [0, ∞) | kg(x) − h(x)k ≤ CΦ(x, y), ∀x, y ∈ X with kyk ≥ M }. We claim that (Ω, d) is a complete metric space. Let {gn } be a Cauchy sequence in (Ω, d) and  > 0. Then there is a positive integer k such that d(gn , gm ) ≤  for all n, m ≥ k. Pick y0 ∈ X with ky0 k ≥ M and let x ∈ X. Since kgn (x) − gm (x)k ≤ Φ(x, y0 ) for all n, m ≥ k, {gn (x)} is a Cauchy sequence in Y and hence we can define a mapping g : X −→ Y by g(x) = limn−→∞ gn (x). Clearly, g ∈ Ω and limn−→∞ gn = g. Thus (Ω, d) is a complete metric space. Define a map J : Ω −→ Ω by Jh(x) = 41 h(2x) for all x ∈ X. Let g, h ∈ Ω. Suppose that C is a positive real number such that kg(x) − h(x)k ≤ CΦ(x, y) for all x, y ∈ X with kyk ≥ M . By (3.14), we have 1 1 1 kJg(x) − Jh(x)k = kg(2x) − h(2x)k ≤ CΦ(2x, 2y) ≤ CLΦ(x, y) 4 4 4 for all x, y ∈ X with kyk ≥ M and hence we have L d(Jg, Jh) ≤ d(g, h) 4 for all g, h ∈ Ω. Since 0 < L < 4, J is a strictly contractive mapping and by (3.16), we have 1 d(Jf, f ) ≤ . 4 By Theorem 3.3, {J n f } converges to the unique fixed element Q of J in Y = {h ∈ Ω | d(f, h) < ∞} and (3.15) holds. Further, we have Q(x) = lim J n f (x) = lim 2−2n f (2n x) n−→∞

n−→∞

for all x ∈ X and we have (3.15). Moreover, Q(0) = 0, because f (0) = 0. Now, we claim that Q satisafies (1.4). First, suppose that x 6= 0 or y 6= 0. Replacing x and y by 2n x and 2n y in (3.1), respectively and deviding both sides of (3.1) by 22n , we have k2−2n f (2n (ax + by)) + 2−2n abf (2n (x − y)) (3.17)

1 n [L φ(x, y) + δ] 4n for all x, y ∈ X and sufficiently large positive integer n. Letting n −→ ∞ in (3.17), Q satisfies (1.4). Clealy, if x = 0 and y = 0, then Q satisfies (1.4). By Theorem 2.1, Q is quadratic. Assume that Q1 : X −→ Y is another quadratic mapping satisfying (1.4) and (3.15). Then we have 1 kQ1 (x) − f (x)k ≤ Φ(x, y) 4(1 − L) − a(a + b)2−2n f (2n x) − b(a + b)2−2n f (2n y)k ≤

for all x ∈ X and y ∈ X with kyk ≥ M and so 1 d(Q1 , f ) ≤ < ∞. 4(1 − L) By (3) of Theorem 3.3, Q = Q1 .



Skof [13](Jung [8], resp.) proved an asymptotic property of aditive (quadratic, resp.) mappings. We consider such property for (1.4).

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Corollary 3.5. A mapping f : X −→ Y satisfies (1.4) if and only if the asymptotic condition kf (ax + by) + abf (x − y) − a(a + b)f (x) − b(a + b)f (y)k −→ 0 as kxk + kyk −→ ∞ holds. Acknowledgements The first 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, 64-66(1950). [2] P. W. Cholewa, Remarks on the stability of functional equations, Aequationes Math. 27, 76-86(1984). [3] S. Czerwik, On the stability of the quadratic mapping in normed spaces, Abh. Math. Sem. Univ. Hamburg. 62, 59-64(1992). [4] 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, 305-309(1968). [5] E. Elhoucien and M. Youssef, On the Paper by A. Najati and S.-M. Jung: The Hyers-Ulam Stability of Approximately Quadratic, Journal of Nonlinear Analysis and application 2012, 1-10(2012). [6] P. Gˇ avruta, A generalization of the Hyers-Ulam-Rassias stability of approximately additive mappings, J. Math. Anal. Appl. 18,4 431-436(1994). [7] D. H. Hyers, On the stability of the linear functional equation, Proc. Natl. Acad. Sci. USA 27, 222-224(1941). [8] S.-M. Jung, On the Hyers-Ulam stability of the functional equations that have the quadratic property, J. Math. Anal. Appl. 222, 126-137(1998). [9] A. Najati and S. M. Jung, Approximately quadratic mappings on restricted domains, J. Ineq. Appl. 2010, 1-10(2010). http://dx.doi.org/10.1155/2010/503458. [10] A. Rahimi, A. Najati, and J. H. Bae, On the Asymptoticity Aspect of Hyers-Ulam stability of quadratic mappings, J. Ineq. Appl. 2010, 1-14(2010). [11] Th. M. Rassias, On the stability of linear mapping in Banach spaces, Proc. Amer. Math. Soc. 72, 297-300(1978). [12] J. M. Rassias, On the Ulam stability of mixed type mappings on restricted domains, J. Math. Anal. Appl. 276, 747-762(2002). [13] F. Skof, Propriet´ a locali e approssimazione di operatori, Rend. Sem. Mat. Fis. Milano 53, 113-129(1983). [14] F. Skof, Sull’ approssinazione delle applicazioni localmente δ-additive, Atti Accad. Sc. Torino 117, 377-389(1983). [15] S. M. Ulam, A Collection of Mathematical Problems, Interscience Publ. New York, 1961. Problems in Modern Mathematics, Wiley, New York, 1964. Department of Mathematics Education, Dankook University, Yongin 448-701, Korea E-mail address: [email protected] Department of Mathematics, Soongsil University, Seoul 156-743, Korea E-mail address: [email protected]

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Hesitant fuzzy soft set and its lattice structures a

Xiaoqiang Zhoua , Qingguo Lib∗ College of Mathematics, Hunan Institute of Science and Technology Yueyang, 414006, P.R.China b College of Mathematics and Econometrics, Hunan University Changsha, 410082, P.R.China

Abstract: Hesitant fuzzy set and soft set were introduced by Torra and Molodtsov, respectively. The two sets have been used successfully as effective mathematical tools for dealing with vagueness and uncertainties. By combining hesitant fuzzy set and soft set, in this paper, we propose a new model named hesitant fuzzy soft set, which can be regarded as an extension of many models, such as hesitant fuzzy set, soft set, fuzzy soft set, interval-valued fuzzy soft set and multi-fuzzy soft set. Some basic operations of hesitant fuzzy soft set are defined and some desirable properties of those operations are investigated. Furthermore, the lattice structures of hesitant fuzzy soft set are discussed. Keywords: Hesitant fuzzy set; soft set; fuzzy soft set; hesitant fuzzy soft set; lattice

1

Introduction

Soft set was firstly proposed by Molodtsov [1], it is a new mathematical tool for modeling vagueness and uncertainty. Since its appearance, soft set theory has attracted more and more attention from many researchers and many important results on soft set have been achieved in theory and application. Maji and Biswas et al. [2] defined some basic operations. Ali et al. [3, 4] gave some new operations on soft sets and studied some algebraic structures of soft sets. Yang and Guo [5] introduced some kernels and closures of soft set relations. Many authors applied soft sets to some algebraic structures such as groups, rings, fields and modules [6–8]. The applications of soft set in decision making and other areas could be found in [9–12]. At the same time, in order to extend the application ranges of soft set, fuzzy extension of soft set theory has become a hot research topic. Maji et al. [13] introduced the notions of fuzzy soft set. Jiang et al. [14] and Majumdar and Samanta [15] further generalized fuzzy soft set to intuitionistic fuzzy soft set and generalised fuzzy soft set, respectively. Yang et al. [16] proposed the concept of interval-valued fuzzy soft set by combining the interval-valued fuzzy set and soft set. Some other generalized models of soft set could be seen in [17–19] Recently, Torra [20] introduced hesitant fuzzy set which is a new extension of fuzzy set. It permits the membership degree of an element to a set to be represented as some possible values between 0 and 1. Presently, work on hesitant fuzzy set is making progress rapidly and lots of results on hesitant fuzzy set have been obtained [21–25]. The main goal of this paper is to combine the hesitant fuzzy set and soft set and obtain a new hybrid model named hesitant fuzzy soft set. It can be viewed as a hesitant fuzzy extension of the soft set or a generalization of the hesitant fuzzy set. The rest of this paper is structured as follows. The following section briefly reviews some basic notions of soft set, fuzzy soft set and hesitant fuzzy set. Two new operations on hesitant fuzzy element are defined, and some of their properties are investigated. In Section 3, the concept of ∗

Corresponding author. Tel./fax: +86 13789003995/+86 731 88822755. E-mail address: [email protected], [email protected]. Mailing address: College of Mathematics, Hunan Institute of Science and Technology, Yueyang, Hunan, 414006, P.R.China

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2

hesitant fuzzy soft set is first proposed by combining hesitant fuzzy set and soft set. Some operations on hesitant fuzzy soft set are given and some of their properties are studied. In Section 4, we discuss the lattice structures of hesitant fuzzy soft set. The conclusion is finally reached in Section 5.

2

Preliminary

Let U be an initial universe of objects and E the set of parameters in relation to objects in U . Parameters are often attributes, characteristics, or properties of objects. Let P (U ) denote the power set of U and A ⊆ E. Molodtsov [1] first gave the definition of soft set as follows. Definition 2.1. [1] A pair (F, A) is called a soft set over U , where A ⊆ E and F is a set valued mapping given by F : A → P (U ). Maji [13] introduced fuzzy soft set which is an fuzzy extension of soft set. Definition 2.2. [13] Let P(U ) be the set of all fuzzy subsets of U . A pair (F, A) is called a fuzzy soft set over U , where F is a set valued mapping given by F : A → P(U ). As a generalization form of fuzzy set, hesitant fuzzy set (HF S) was first introduced by Torra [20] as follows. Definition 2.3. [20] Let X be a reference set, an HF S on X is in terms of{ a function that } when hH (x) applied to X returns a subset of [0, 1], which can be represented as H = x |x ∈ X , where hH (x) is a set of some values in [0, 1], denoting the possible membership degrees of the element x ∈ X to the set H. For convenience, Xu and Xia [21,22] called hH (x) an hesitant fuzzy element (HF E) with respect to x of H. It is worth noting that the number of values of different HF Es may be different, in this paper, let l(hH (x)) denote the number of values of hH (x). We arrange the values of hH (x) in σ(j) increasing order, and let hH (x) be the jth largest value of hH (x). { } Definition 2.4. [20] Let H = hHx(x) |x ∈ X be an HF S. Then (1) H is said to be an empty hesitant set, denoted by Φ, if hH (x) = 0 for all x ∈ X; (2) H is said to be a full hesitant set, denoted by I, if hH (x) = 1 for all x ∈ X; (3) H is said to be a complete ignorance set, denoted by W, if hH (x) = [0, 1] for all x ∈ X. Definition 2.5. [20] Let λ > 0, h, h1 and h2 be three HF Es, some operations on them are given as follows: (1) h1 ∪ h2 = ∪γ1 ∈h1 ,γ2 ∈h2 {max(γ1 , γ2 )}; (2) h1 ∩ h2 = ∪γ1 ∈h1 ,γ2 ∈h2 {min(γ1 , γ2 )}; (3) hc = ∪γ∈h {1 − γ}. We further define the strict union and the strict intersection for HF Es h1 and h2 , which will be useful in the sequel. + Definition 2.6. Let h1 and h2 be two HF Es, h− i = min{γi |γi ∈ hi } and hi = max{γi |γi ∈ hi }(i = 1, 2). The strict union and the strict intersection of h1 and h2 are defined as follows: + (1) h1 ⊔ h2 = ∪γi ∈hi ,i=1,2 {γi |γi > min(h+ 1 , h2 ) or γ1 = γ2 }; − (2) h1 ⊓ h2 = ∪γi ∈hi ,i=1,2 {γi |γi < max(h− 1 , h2 ) or γ1 = γ2 };

For example, let h1 = {0.2, 0.3, 0.6, 0.8} and h2 = {0.4, 0.5, 0.8, 0.9}, then h1 ⊔ h2 = {0.8, 0.9} ̸= {0.4, 0.5, 0.6, 0.8, 0.9} = h1 ∪ h2 , h1 ⊓ h2 = {0.2, 0.3} ̸= {0.2, 0.3, 0.4, 0.5, 0.6, 0.8} = h1 ∩ h2 . In fact, all the above operations on HF Es can be suitable for HF Ss. Some relationships can be further established for these operations on HF Es. 73

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Theorem 2.7. For three HF Es h, h1 and h2 , the following is valid: (1) hc1 ⊔ hc2 = (h1 ⊓ h2 )c ; (2) hc1 ⊓ hc2 = (h1 ⊔ h2 )c . − Proof. (1) Since h1 ⊓ h2 = ∪γi ∈hi ,i=1,2 {γi |γi < max(h− 1 , h2 ) or γ1 = γ2 }, then − (h1 ⊓ h2 )c = ∪γi ∈hi ,i=1,2 {1 − γi |γi < max(h− 1 , h2 ) or γ1 = γ2 }

Since hc1 = ∪γ1 ∈h1 {1 − γ1 } and hc2 = ∪γ2 ∈h2 {1 − γ2 }, then hc1 ⊔ hc2 = {∪γ1 ∈h1 {1 − γ1 }} ⊔ {∪γ2 ∈h2 {1 − γ2 }}

− = ∪γi ∈hi ,i=1,2 {1 − γi |1 − γi < min(1 − h− 1 , 1 − h2 ) or 1 − γ1 = 1 − γ2 }

− = ∪γi ∈hi ,i=1,2 {1 − γi |γi < 1 − min(1 − h− 1 , 1 − h2 ) or γ1 = γ2 } − = ∪γi ∈hi ,i=1,2 {1 − γi |γi < max(h− 1 , h2 ) or γ1 = γ2 }

Theorem 2.8. For three HF Es h1 , h2 and h3 , the following is valid: (1) (h1 ∪ h2 ) ∪ h3 = h1 ∪ (h2 ∪ h3 ); (2) (h1 ∩ h2 ) ∩ h3 = h1 ∩ (h2 ∩ h3 ); (3) h1 ∪ (h2 ∩ h3 ) = (h1 ∪ h2 ) ∩ (h1 ∪ h3 ); (4) h1 ∩ (h2 ∪ h3 ) = (h1 ∩ h2 ) ∪ (h1 ∩ h3 ). Proof. (2) and (4) are similar to (1) and (3), respectively, so we only prove (1) and (3). (1) Since (h1 ∪ h2 ) = ∪γi ∈hi ,i=1,2 {max(γ1 , γ2 )}, then (h1 ∪ h2 ) ∪ h3 = {∪γi ∈hi ,i=1,2 {max(γ1 , γ2 )}} ∪ h3 = ∪γi ∈hi ,i=1,2,3 {max(max(γ1 , γ2 ), γ3 )} = ∪γi ∈hi ,i=1,2,3 {max(γ1 , γ2 , γ3 )} = ∪γi ∈hi ,i=1,2,3 {max(γ1 , max(γ2 , γ3 ))} = h1 ∪ (h2 ∪ h3 ). (3) Since (h2 ∩ h3 ) = ∪γi ∈hi ,i=2,3 {min(γ2 , γ3 )}, then h1 ∪ (h2 ∩ h3 ) = h1 ∪ {∪γi ∈hi ,i=2,3 {min(γ2 , γ3 )}} = ∪γi ∈hi ,i=1,2,3 {max(γ1 , min(γ2 , γ3 ))} = ∪γi ∈hi ,i=1,2,3 {min(max(γ1 , γ2 ), max(γ2 , γ3 ))} = {∪γi ∈hi ,i=1,2 {max(γ1 , γ2 )}} ∩ {∪γi ∈hi ,i=1,3 {max(γ1 , γ3 )}} = (h1 ∪ h2 ) ∩ (h1 ∪ h3 )

Theorem 2.9. For three HF Es h1 , h2 and h3 , the following is valid: (1) (h1 ⊓ h2 ) ⊓ h3 = h1 ⊓ (h2 ⊓ h3 ); (2) (h1 ⊔ h2 ) ⊔ h3 = h1 ⊔ (h2 ⊔ h3 ); (3) (h1 ⊔ h2 ) ⊓ h1 = h1 ; (4) (h1 ⊓ h2 ) ⊔ h1 = h1 . Proof. (2) and (4) are similar to (1) and (3), respectively, so we only prove (1) and (3). − (1) Since h1 ⊓ h2 = ∪γi ∈hi ,i=1,2 {γi |γi < max(h− 1 , h2 ) or γ1 = γ2 }, then − (h1 ⊓ h2 ) ⊓ h3 = {∪γi ∈hi ,i=1,2 {γi |γi < max(h− 1 , h2 ) or γ1 = γ2 }} ⊓ h3

− − = ∪γi ∈hi ,i=1,2,3 {γi |γi < max(max(h− 1 , h2 ), h3 ) or γ1 = γ2 = γ3 }

− − = ∪γi ∈hi ,i=1,2,3 {max(h− 1 , h2 , h3 ) or γ1 = γ2 = γ3 }

− − = ∪γi ∈hi ,i=1,2,3 {γi |γi < max(h− 1 , max(h2 , h3 )) or γ1 = γ2 = γ3 }

− = h1 ⊓ {∪γi ∈hi ,i=2,3 {γi |γi < max(h− 2 , h3 ) or γ2 = γ3 }}

= h1 ⊓ (h2 ⊓ h3 ). 74

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+ (3) Since h1 ⊔ h2 = ∪γi ∈hi ,i=1,2 {γi |γi > min(h+ 1 , h2 ) or γ1 = γ2 }, + + + + + i) If h1 ≤ h2 , then min(h1 , h2 ) = h1 . It follows that h1 ⊔h2 = ∪γi ∈hi ,i=1,2 {γ2 |γ2 > h+ 1 or γ2 = γ1 }. By Definition 2.6, we have (h1 ⊔ h2 ) ⊓ h1 = h1 . + + + + + ii) If h+ 1 > h2 , then min(h1 , h2 ) = h2 . It follows that h1 ⊔h2 = ∪γi ∈hi ,i=1,2 {γ1 |γ1 > h2 or γ1 = γ2 }. By Definition 2.6, we have (h1 ⊔ h2 ) ⊓ h1 = h1 .

3

Hesitant fuzzy soft set

In this section, we present an extended soft set model which is called hesitant fuzzy soft set by combining the hesitant fuzzy set and soft set. Some operations and their properties on hesitant fuzzy soft set will also be discussed. Definition 3.1. Let HF (U ) be the class of all HF Ss of the universe U , A ⊆ E. A pair (Fe, A) is called a hesitant fuzzy soft set (HF SS), where Fe : A → HF (U ) is a mapping. In other words, a hesitant fuzzy soft set over U is a parameterized family of hesitant fuzzy set of the universe U . To illustrate this idea, let us consider the following example. Example 3.2. Let U = {u1 , u2 , u3 } be a set of mobile telephones and A = {e1 , e2 , e3 } ⊆ E be a set of parameters. The ei (i = 1, 2, 3) stands for the parameters “expensive”, “beautiful” and “multifunctional”, respectively. Let Fe : A → HF (U ) be a function given as follows: } { {0.2, 0.7, 0.8} {0.5, 0.8} {0.4, 0.6, 0.8} e , , , F (e1 ) = u1 u2 u3 { } {0.3, 0.5, 0.7} {0.4, 0.6, 0.9} {0.5, 0.7} e F (e2 ) = , , , u1 u2 u3 } { {0.5, 0.8} {0.3, 0.5, 0.8} {0.5, 0.6, 0.9} , , . Fe(e3 ) = u1 u2 u3 Then (Fe, A) is a hesitant fuzzy soft set. Remark 3.3. (1) If A has only an element, i.e. A = {e}, then hesitant fuzzy soft set becomes hesitant fuzzy set [20]; (2) If hFe(e) (u) has only one value for all e ∈ A and u ∈ U , then hesitant fuzzy soft set degenerates to traditional fuzzy soft set [13]; (3) If hFe(e) (u) is a subinterval of [0, 1] for all e ∈ A and u ∈ U , then hesitant fuzzy soft set reduces to interval-valued fuzzy soft set [17]; (4) For all e ∈ A, if hFe(e) (u) has the same number of values with respect to u ∈ U , then hesitant fuzzy soft set transforms to multi-fuzzy soft set [19]. Definition 3.4. The complement of an HF SS (Fe, A) is denoted by (Fe, A)c {and is defined } by hFe c (e) (u) (Fe, A)c = (Fec , A), where Fec : A → HF (U ) is a mapping given by Fec (e) = |u ∈ U , u ∪ where hFec (e) (u) = γ∈h e (u) {1 − γ}. F (e)

Example 3.5. (continued) The complement of (Fe, A) is following as: { } {0.2, 0.3, 0.8} {0.2, 0.5} {0.2, 0.4, 0.6} Fec (e1 ) = , , , u1 u2 u3 { } {0.3, 0.5, 0.7} {0.1, 0.4, 0.6} {0.3, 0.5} c e , , , F (e2 ) = u1 u2 u3 { } {0.2, 0.5} {0.2, 0.5, 0.7} {0.1, 0.4, 0.5} c e F (e3 ) = , , . u1 u2 u3 75

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Definition 3.6. Let (Fe, A) be an HF SS over U . Then e A , if hF (e) (u) = 0 for all u ∈ U and (1) (Fe, A) is said to be an empty hesitant soft set, denoted by Φ e ∈ A; (2) (Fe, A) is said to be a full hesitant soft set, denoted by IeA , if hF (e) (u) = 1 for all u ∈ U and e ∈ A; fA , if hF (e) (u) = [0, 1] for all u ∈ U (3) (Fe, A) is said to be a complete hesitant soft set, denoted by W and e ∈ A. Proposition 3.7. Let A ⊆ E. Then e c = IeA ; (1) Φ A c =Φ e A; (2) IeA fc = W fA . (3) W A e B) be two HF SSs over U and A, B ⊆ E. We define a mapping Definition 3.8. Let (Fe, A) and (G, e H : A ∪ B → HF (U ) such that for all e ∈ A ∪ B ̸= ∅,    Fe(e), if e ∈ A − B, e e H(e) = G(e), if e ∈ B − A,   H(e), e if e ∈ A ∩ B. e e e A ∪ B) is called the extended union of (Fe, A) and (G, e B), (1) If H(e) = Fe(e) ∪ G(e), then (H, e e e (G, B). denoted by (F , A)∪ e e e e A ∪ B) is called the extended intersection of (Fe, A) and (G, e B), (2) If H(e) = F (e) ∩ G(e), then (H, e e e (G, B). denoted by (F , A)∩ e e e A ∪ B) is called the extended-strict union of (Fe, A) and (G, e B), (3) If H(e) = Fe(e) ⊔ G(e), then (H, e e e denoted by (F , A)⊔(G, B). e e e A ∪ B) is called the extended-strict intersection of (Fe, A) and (4) If H(e) = Fe(e) ⊓ G(e), then (H, e B), denoted by (Fe, A)⊓ e B). e (G, (G, e e B) = Φ e ∅ , (Fe, A)∩ e B) = Φ e ∅ , (Fe, A)⊔ e B) = Φ e ∅ and e (G, e (G, e (G, If A ∪ B = ∅, then (F , A)∪ e B) = Φ e ∅. e (G, (Fe, A)⊓ e B) be two HF SSs over U and A, B ⊆ E. We define a mapping Definition 3.9. Let (Fe, A) and (G, e : A ∩ B → HF (U ) such that for all e ∈ A ∩ B ̸= ∅, H e e e A ∩ B) is called the strict union of (Fe, A) and (G, e B), denoted (1) If H(e) = Fe(e) ∪ G(e), then (H, e e e (G, B). by (F , A)d e e e A ∩ B) is called the strict intersection of (Fe, A) and (G, e B), (2) If H(e) = Fe(e) ∩ G(e), then (H, e e e denoted by (F , A) (G, B). e e e A ∩ B) is called the strict-strict union of (Fe, A) and (G, e B), (3) If H(e) = Fe(e) ⊔ G(e), then (H, e B). e (G, denoted by (Fe, A)⊎ e e e e A ∩ B) is called the strict-strict intersection of (Fe, A) and (4) If H(e) = F (e) ⊓ G(e), then (H, e B), denoted by (Fe, A)C e B). e (G, (G, e e B) = Φ e ∅ , (Fe, A) e (G, e B) = Φ e ∅ , (Fe, A)⊎ e B) = Φ e ∅ and e (G, If A ∩ B = ∅, then (F , A) d (G, e e e e (G, B) = Φ∅ . (F , A)C e }, θ2 ∈ {∩ e, ⊓ e }, Proposition 3.10. Let A ⊆ E, (Fe, A) be an HF SS over (U, E), θ1 ∈ { e , C e } and θ4 ∈ {∪ e, ⊔ e }. Then e, ⊎ θ3 ∈ {d e e e (1) (F , A) θ1 IE = (F , A) θ2 IeA = (Fe, A); (2) (Fe, A) θ3 IeE = (Fe, A) θ4 IeA = IeA ; e E = (Fe, A) θ2 Φ eE = Φ e A; (3) (Fe, A) θ1 Φ e e e e (4) (F , A) θ3 ΦE = (F , A) θ4 ΦA = (Fe, A); e ϕ = (Fe, A) θ3 Φ eϕ = Φ e ϕ; (5) (Fe, A) θ1 Φ e ϕ = (Fe, A) θ4 Φ e ϕ = (Fe, A). (6) (Fe, A) θ2 Φ

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e B) and (H, e C) be HF SSs e, ⊔ e, C e, ⊓ e, ∩ e, e, ∪ e, d e },A, B, C ⊆ E, (Fe, A), (G, Theorem 3.11. Let α ∈ {⊎ over (U, E). Then the following holds: (1) (Fe, A) α (Fe, A) = (Fe, A); e B) = (G, e B) α (Fe, A); (2) (Fe, A) α (G, e B) α (H, e C)) = ((Fe, A) α (G, e B)) α (H, e C). (3) (Fe, A) α ((G, e , the others can be proved Proof. (1) and (2) are trivial. We only prove (3). For example, let α = ⊎ analogously. e B)⊎ e C)) = (J, e M ) and ((Fe, A)⊎ e B))⊎ e C) = (K, e N ), thus e ((G, e (H, e (G, e (H, Suppose that (Fe, A)⊎ e e e e e e C). e ((G, B) ⊎ e (H, C)) = Φϕ = ((F , A) ⊎ e (G, B)) ⊎ e (H, M = N = A ∩ B ∩ C. If M = ϕ, then (F , A) ⊎ If M ̸= ϕ, then by (2) in Theorem 2.9, we have hF (e) (u) ⊔ (hG(e) (u) ⊔ hH(e) (u)) = (hF (e) (u) ⊔ e ⊔ H(e)) e e hG(e) (u)) ⊔ hH(e) (u) for all e ∈ M and u ∈ U . It follows that Fe(e) ⊔ (G(e) = (Fe(e) ⊔ G(e)) ⊔ e e B)⊎ e C)) = e , we have (Fe, A)⊎ e ((G, e (H, H(e) for all e ∈ M . By the definition of the operation ⊎ e e e e e ((F , A)⊎(G, B))⊎(H, C). e , e, ∪ e, d e, ⊔ e, C e and ⊓ e are idempotent, e, ⊎ Remark 3.12. Theorem 3.11 shows that the operations ∩ commutative and associative, respectively. e B) be HF SSs over (U, E). Then the following Theorem 3.13. Let A, B ⊆ E, (Fe, A) and (G, holds: e B))c = (Fe, A)c ⊔ e B)c ; e (G, e (G, (1) ((Fe, A)⊓ e B))c = (Fe, A)c ⊓ e B)c ; e (G, e (G, (2) ((Fe, A)⊔ c c e e e e B)c ; e (G, B)) = (F , A) ⊎ e (G, (3) ((F , A)C c c e B)) = (Fe, A) C e B)c ; e (G, e (G, (4) ((Fe, A)⊎ e B))c = (Fe, A)c d e B)c ; e (G, (5) ((Fe, A) e (G, c c e e e e B)c ; e (G, B)) = (F , A) e(G, (6) ((F , A)d e B))c = (Fe, A)c ∪ e B)c ; e (G, e (G, (7) ((Fe, A)∩ c c e e e e B)c , e (G, B)) = (F , A) ∩ e (G, (8) ((F , A)∪ Proof. We only prove (1). By using a similar technique, (2)-(8) can be proved, too. e B) = (H, e C). Then C = A ∪ B, e (G, Suppose that (Fe, A)⊓ e B))c = Φ e ϕ =(Fe, A)c ⊓ e B)c . e (G, e (G, (i) if C = ϕ, then A = ϕ and B = ϕ. Hence ((Fe, A)⊓ (ii) if C ̸= ϕ, then for each e ∈ C and u ∈ U , we have  if e ∈ A − B,   hFe(e) (u), hG(e) if e ∈ B − A, hH(e) (u) = e (u), e   h (u) ⊓ hG(e) e (u), if e ∈ A ∩ B. Fe(e) Then

 if e ∈ A − B,   hFec (e) (u), h (u), if e ∈ B − A, hHe c (e) (u) = e c (e) G   (h c (u) ⊓ hG(e) e (u)) , if e ∈ A ∩ B. Fe(e)

e B)c = (J, e D). Then D = A ∪ B and for each e ∈ D and u ∈ U , we e (G, Again suppose that (Fe, A)c ⊔ have  if e ∈ A − B,   hFec (e) (u), hGec (e) (u), if e ∈ B − A, hJ(e) e (u) =   h (u) ⊔ hGec (e) (u), if e ∈ A ∩ B. Fec (e) c By Theorem 2.7, we have hFec (e) (u) ⊔ hGec (e) (u) = (hFe(e) (u) ⊓ hG(e) e (u)) , i.e., hJ(e) e (u) = hH e c (e) (u) for all e ∈ A and u ∈ U . e C) and (J, e D) are the same HF SSs. It follows that ((Fe, A)⊓ e B))c = e (G, Therefore, (H, c c e e e (G, B) . (F , A) ⊔

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7

Lattice structures of hesitant fuzzy soft set

In this section, we first recall briefly the necessary definitions and notations. For convenience, we give the following axioms on an algebra Q = (X, ∨, ∧): (1) x ∨ x = x, x ∧ x = x; (2) x ∨ y = y ∨ x, x ∧ y = y ∧ x; (3) (x ∨ y) ∨ z = x ∨ (y ∨ z), (x ∧ y) ∧ z = x ∧ (y ∧ z); (4) (x ∨ y) ∧ x = x, (x ∧ y) ∨ x = x; (5) x ∧ (y ∨ z) = (x ∧ y) ∨ (x ∧ z), x ∨ (y ∧ z) = (x ∨ y) ∧ (x ∨ z), where x, y, z ∈ X. The algebra Q is called a quasilattice, if it satisfies the axioms (1),(2) and (3). If a quasilattice further satisfies the axiom (4), then it is called a lattice. If a quasilattice (or lattice ) further satisfies the axiom (5), then it is called a distributive quasilattice (or lattice ). e e For convenience, let S(U, E) denote the set of all HF SSs over U , i.e., S(U, E) = {(Fe, A)|A ⊆ E, Fe : A → HF (U )}. Then based on Theorem 3.11, we have the following property. e e , e, C e, ⊓ e } and β ∈ {∪ e, d e, ⊔ e }, then (S(U, e, ⊎ Proposition 4.1. Let α ∈ {∩ E), α, β) is a quasilattice. e , the distributive laws hold. For the operations e and ∪ e e B), (H, e C) ∈ S(U, Theorem 4.2. Let (Fe, A), (G, E). Then e e e e e e e C)); e e e e (1) ((F , A) (G, B))∪(H, C) = ((F , A) (G, B))∪((F , A) e (H, e e e e e e e e (G, B))e(H, C) = ((F , A)∪ e (G, B))e((F , A)∪ e (H, C)). (2) ((F , A)∪ Proof. we only prove (1). (2) can be proved by using a similar technique. Suppose that e B)∪ e C)) = (J, e M ) and ((Fe, A)e(G, e B))∪ e C)) = (K, e N ). Then M = e (H, e ((Fe, A)e(H, (Fe, A)e((G, A ∩ (B ∪ C) = (A ∩ B) ∪ (A ∩ C) = N . For each e ∈ M , it follows that e ∈ A and e ∈ B ∪ C. e = Fe(e) ∩ H(e) e e (i) if e ∈ A, e ∈ / B, e ∈ C, then J(e) = K(e). e = Fe(e) ∩ G(e) e e (ii) if e ∈ A, e ∈ B, e ∈ / C, then J(e) = K(e). (iii) if e ∈ A, e ∈ B, e ∈ C, then by (4) in Theorem 2.8, we have hFe(e) (u) ∩ (hG(e) (u)) = e (u) ∪ hH(e) e e e e e e (h e (u) ∩ h e (u) ∪ (F (e) ∩ h e (u)) for all u ∈ U . It follows that J(e) = F (e) ∩ (G(e) ∪ H(e)) = F (e)

G(e)

H(e)

e ∪ (Fe(e) ∩ (H, e C)) = K(e). e (Fe(e) ∩ G(e) e e e B)∪ e C)) e (H, Thus, (J, M ) and (K, N ) are the same HF SS, i.e., (Fe, A)e((G, e B))∪ e C)). e ((Fe, A)e(H, ((Fe, A)e(G,

=

e e ) is a distributive quasilattice. Corollary 4.3. (S(U, E), e, ∪ e and d e. e have the similar properties with the operations e and ∪ The operations ∩ e e B), (H, e C) ∈ S(U, Theorem 4.4. Let (Fe, A), (G, E). Then e e e e e e e C)); e e e (H, e e (1) ((F , A)∩(G, B))d(H, C) = ((F , A)∩(G, B))d((F , A)∩ e e e e e e e e (H, C) = ((F , A)d e ((F , A)d e (G, B))∩ e (G, B))∩ e (H, C)). (2) ((F , A)d e e ) is a distributive quasilattice. Corollary 4.5. (S(U, E), e , ∪ e and ⊎ e hold. The following theorem shows that the absorption laws with respect to operations ⊓ e e B) ∈ S(U, Theorem 4.6. Let (Fe, A), (G, E). Then e B))⊎ e (G, e (Fe, A) = (Fe, A); (1) ((Fe, A)⊓ e B))⊓ e (G, e (Fe, A) = (Fe, A). (2) ((Fe, A)⊎ e B) = (J, e M) e (G, Proof. We only prove (1) since (2) can be proved similarly. Suppose that (Fe, A)⊓ e e e e e e and ((F , A) ⊓ (G, B)) C (F , A) = (K, N ). Then M = A ∪ B, N = (A ∪ B) ∩ A = A, and for all e ∈ A and u ∈ U , (i) if e ∈ / B, then hJ(e) (u) = hJ(e) e (u) = hFe(e) (u) and hK(e) e e (u) ⊓ hFe(e) (u) = hFe(e) (u). 78

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(ii) if e ∈ B, then hJ(e) (u) = hJ(e) e (u) = hFe(e) (u) ⊔ hG(e) e (u) and hK(e) e e (u) ⊓ hFe(e) (u) = (hFe(e) (u) ⊔ hG(e) e (u)) ⊓ hFe(e) (u). By (3) in Theorem 2.9, we have (hFe(e) (u) ⊔ hG(e) e (u)) ⊓ hFe(e) (u) = hFe(e) (u), i.e. hK(e) (u) = h (u). e Fe(e) e e B))⊎ e (G, e (Fe, A) = (Fe, A). Thus (K, N ) = (Fe, A), i.e. ((Fe, A)⊓ e e, ⊎ e ) is a bounded lattice. Theorem 4.7. (S(U, E), ⊓ e e, ⊎ e ) is a lattice. It is clear that Proof. By Theorem 3.11 and Theorem 4.6, we get that (S(U, E), ⊓ e e ϕ are the maximum element and the minimum element in (S(U, IeE and Φ E), respectively. e and ⊎ e , the operations ⊔ e and C e have also the following properties. Similar to ⊓ e e B) ∈ S(U, Theorem 4.8. Let (Fe, A), (G, E). Then e e e e e (G, B))C e (F , A) = (F , A); (1) ((F , A)⊔ e B))⊔ e (G, e (Fe, A) = (Fe, A). (2) ((Fe, A)C e e, C e ) is a bounded lattice. Theorem 4.9. (S(U, E), ⊔ e e e e, ⊓ e ), (S(U, e, C e ) and (S(U, Remark 4.10. It is worth noting that (S(U, E), ⊔ E), ⊎ E), α, β) are not e , e} and β ∈ {∪ e, d e }. To lattices, as the absorption laws do not hold necessarily, where α ∈ {∩ illustrate this idea, we give an example below. Example 4.11. Let U = {u1 , u2 , u3 } be the universe, E = {e1 , e2 , e3 } the set of parameters, e B) over U are given as: A = {e1 , e2 } and B = {e2 , e3 }. The HF SSs (Fe, A) and (G, { } {0.2, 0.3, 0.7, 0.8} {0.5, 0.8} {0.4, 0.5, 0.6} e F (e1 ) = , , , u1 u2 u3 } { {0.3, 0.4, 0.7} {0.5, 0.7} {0.1, 0.2, 0.4, 0.7} e , , , F (e2 ) = u1 u2 u3 { } e 2 ) = {0.5, 0.6} , {0.4, 0.8, 0.9} , {0.3, 0.5, 0.7, 0.8} , G(e u1 u2 u3 { } {0.1, 0.3, 0.5} {0.5, 0.6, 0.8} {0.6, 0.9} e G(e3 ) = , , . u1 u2 u3 e B))⊓ e M ), then M = A ∪ B = {e1 , e2 , e3 } ̸= A. So (J, e M ) ̸= e (G, e (Fe, A) = (J, (1) Let ((Fe, A)⊔ e B))⊓ e (G, e (Fe, A) ̸= (Fe, A). (Fe, A), i.e. ((Fe, A)⊔ e e e N ), then N = A ∩ B = {e2 } ̸= A, Therefore, (K, e N ) ̸= e e (2) Let ((F , A)⊎ (G, B))C(Fe, A) = (K, e e e e e e (G, B))C e (F , A) ̸= (F , A). (F , A), i.e. ((F , A)⊎ (3) If e2 ∈ A ∩ B, then (hFe(e2 ) (u1 ) ∩ hG(e e 2 ) (u1 )) ∪ hFe(e2 ) (u1 ) = ({0.3, 0.4, 0.7} ∩ {0.5, 0.6}) ∪ {0.3, 0.4, 0.7} = {0.3, 0.4, 0.5, 0.6} ∪ {0.3, 0.4, 0.7} = {0.3, 0.4, 0.5, 0.6, 0.7} ̸= e 2 )) ∪ Fe(e2 ) ̸= Fe(e2 ). Consequently, {0.3, 0.4, 0.7} = hFe(e2 ) (u1 ). It follows that (Fe(e2 ) ∩ G(e e B)) β (Fe, A) ̸= (Fe, A), where α ∈ {∩ e , e} and β ∈ {∪ e, d e }. ((Fe, A) α (G,

5

Conclusion

Considering that soft set and its existing extension models cannot deal with the situations in which the evaluations of parameters have many possible values, in this paper, we have introduced the notion of HF SS as an new extension to the HF S or the soft set model. We have also defined some basic operations on the HF SS and discussed their properties. Finally, The lattice structures of HF SS have been studied in detail based on the proposed operations.

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References [1] Molodtsov D. Soft set theory–First results. Comput Math Appl 1999; 37: 19-31. [2] Maji PK, Biswas R, Roy AR. Soft set theory, Comput Math Appl 2003; 45: 555-562. [3] Ali MI, Feng F, Liu X et al. On some new operations in soft set theory. Comput Math Appl 2009; 57: 1547-1553. [4] Ali MI, Shabir M, Naz M. Algebraic structures of soft sets associated with new operations. Comput Math Appl 2011; 61; 2647-2654. [5] Yang HL, Guo ZL Kernels and closures of soft set relations, and soft set relation mappings. Comput Math Appl 2011; 61: 651-662. [6] Aktas H, Cagman N. Soft sets and soft groups, inform Sciences 2007; 177: 2726-2735. [7] Acar U, Koyuncu F, Tanay B. Soft sets and soft rings. Comput Math Appl 2010; 59: 3458-3463. [8] Atagn AO, Sezgin A. Soft substructures of rings, fields and modules. Comput Math Appli 2011; 61: 592-601. [9] Maji PK, Roy AR, Biswas R. An application of soft sets in a decision making problem. Comput Math Appl 2002; 44: 1077-1083. [10] Cagman N, Enginoglu S. Soft set theory and uni-int decision making. Eur J Oper Res 2010; 207: 848-855. [11] Zou Y, Xiao Z. analysis approaches of soft sets under incomplete information. Knowl-Based Syst 2008; 21: 941-945. [12] Herawan T, Deris MM. A soft set approach for association rules mining. Knowl-Based Syst 2011; 24: 186-195. [13] Maji PK et al. Fuzzy soft sets. J Fuzzy Math 2001; 9: 589-602. [14] Jiang Y, Tang Y, Chen Q. An adjustable approach to intuitionistic fuzzy soft sets based decision making. Appl Math Model 2011; 35: 824-836. [15] Majumder P, Samanta SK. Generalised fuzzy soft sets. Comput Math Appl 2010; 59: 1425-1432. [16] Yang XB, Lin TY, Yang JY et al. Combination of interval-valued fuzzy set and soft set. Comput Math Appl 2009; 58: 521-527. [17] Xu W, Ma J, Wang S et al. Vague soft sets and their properties. Comput Math Appl 2010; 59: 787-794. [18] Y. Jiang, Y. Tang, Q. Chen, H. Liu, and J. Tang, Interval-valued intuitionistic fuzzy soft sets and their properties, Comput. Math. Appl. 60 (2010) 906-918. [19] Yang Y, Tan X, Meng C. The multi-fuzzy soft set and its application in decision making. Appl Math Model 2013; 37: 4915-4923. [20] Torra V. Hesitant fuzzy sets. Int J Intell Syst 2010; 25: 529-539. [21] Xu ZS, Xia MM. Distance and similarity measures for hesitant fuzzy sets. Inform Sciences 2011; 181: 2128-2138, . [22] Xia MM, Xu ZS. Hesitant fuzzy information aggregation in decision making. Int J Approx Reason 2011; 52: 395-407. [23] Farhadinia B. A Novel Method of Ranking Hesitant Fuzzy Values for Multiple Attribute DecisionMaking Problems. Int J Intell Syst 2013; 28: 752-767. [24] Wei G. Hesitant fuzzy prioritized operators and their application to multiple attribute decision making. Knowl-Based Syst 2012; 31: 176-182. [25] Zhu BZ, Xu ZS, Xia MM. Hesitant fuzzy geometric Bonferroni means. Inform Sciences 2012; 205: 72-85.

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INCLUSION PROPERTIES FOR CERTAIN SUBCLASSES OF ANALYTIC FUNCTIONS ASSOCIATED WITH BESSEL FUNCTIONS N. E. CHO1,∗ , G. MURUGUSUNDARAMOORTHY2 AND T. JANANI3

1

Department of Applied Mathematics Pukyong National University Busan 608-737, KOREA. E-mail: [email protected].

2,3

School of Advanced Sciences, VIT University Vellore - 632014, INDIA. E-mail: [email protected]; [email protected] ∗

Corresponding Author

Abstract: The purpose of the present paper is to investigate some characterization for generalized Bessel functions of first kind to be in the new subclasses G(λ, α) and K(λ, α) of analytic functions. 2010 Mathematics Subject Classification: 30C45. Keywords and Phrases: Starlike functions, Convex functions, Starlike functions of order α, Convex functions of order α, Hadamard product, Bessel function. 1. Introduction Let A be the class of functions f normalized by f(z) = z +

∞ X

an z n

(1.1)

n=2

which are analytic in the open disk U = {z : z ∈ C and |z| < 1}. As usual, we denote by S the subclass of A consisting of functions which are normalized by f(0) = 0 = f 0 (0) − 1 and also univalent in U. Denote by T [16] the subclass of A consisting of functions of the form ∞ X f(z) = z − an z n , an ≥ 0, n = 2, 3, . . . . (1.2) n=2

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Also, for functions f ∈ A given by (1.1) and g ∈ A given by g(z) = z + define the Hadamard product (or convolution) of f and g by (f ∗ g)(z) = z +

∞ X n=2

2

P∞

n=2 bn z

an bn z n , (z ∈ U).

n

, we

(1.3)

The class S ∗(α) of starlike functions of order α (0 ≤ α < 1) may be defined as   0   zf (z) ∗ S (α) = f ∈ A : < > α, z ∈ U . f(z) The class S ∗(α) and the class K(α) of convex functions of order α (0 ≤ α < 1)     zf 00 (z) K(α) = f ∈ A : < 1 + 0 > α, z ∈ U f (z) = {f ∈ A : zf 0 ∈ S ∗ (α)}

were introduced by Robertson in [14]. We also write S ∗(0) = S ∗, where S ∗ denotes the class of functions f ∈ A that f(U) is starlike with respect to the origin. Further, K(0) = K is the well-known standard class of convex functions. It is an established fact that f ∈ K(α) ⇐⇒ zf 0 ∈ S ∗(α). A function f ∈ A is said to be in the class α, (z ∈ U). < (1.9) f(z) and also let K(λ, α) the subclass of functions f ∈ A which satisfy the condition   z[zf 0 (z) + λz 2 f 00(z)]0 < (1.10) > α, (z ∈ U). zf 0 (z)

Also denote G ∗(λ, α) = G(λ, α) ∩ T and K∗(λ, α) = K(λ, α) ∩ T The study of the generalized Bessel function is a recent interesting topic in geometric function theory (e.g. see the work of [1, 2, 3, 4] and [9]). In this paper, due to Ramesha et al. [13], Padmanabhan [12], and motivated by the works of Srivastava et al. [17], Murugusundaramoorthy and Magesh [11],(see [6, 8, 10, 15]) and by work of Baricz [1, 2, 3, 4], we obtain sufficient conditions for function z(2 − up (z)) in G(λ, α) and K(λ, α) and connections between Rτ (A, B). Remark 1. It is of interest to note that for λ = 0, we have G(λ, α) ≡ S ∗(α) and K(λ, α) ≡ K(α) To prove the main results, we need the following Lemmas. Lemma 1. [18] A function f ∈ A belongs to the class G(λ, α) if ∞ X n=2

(n + λn(n − 1) − α)|an | ≤ 1 − α.

Lemma 2. [18] A function f ∈ A belongs to the class K(λ, α) if ∞ X n=2

n(n + λn(n − 1) − α)|an | ≤ 1 − α.

Further we can easily prove that the conditions are also necessary if f ∈ T . Lemma 3. [18] A function f ∈ T belongs to the class G ∗(λ, α) if and only if ∞ X n=2

(n + λn(n − 1) − α)|an | ≤ 1 − α.

Lemma 4. [18] A function f ∈ T belongs to the class K∗(λ, α) if and only if ∞ X n=2

n(n + λn(n − 1) − α)|an | ≤ 1 − α.

Lemma 5. [4] If b, p, c ∈ C and m 6= 0, −1, −2, . . . then the function up satisfies the recursive relation 4mu0p(z) = −cup+1(z) for all z ∈ C.

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2. Main Results Theorem 1. If c < 0 and m > 0, then z(2 − up(z)) is in G(λ, α) if λu00p (1) + [1 + 2λ]u0p (1) + (1 − α)up(1) ≤ 2(1 − α).

(2.1)

Proof. Since z(2 − up (z)) = z −

∞ X n=2

(−c/4)n−1 zn (m)n−1 (n − 1)!

and by virtue of Lemma 1, it suffices to show that ∞ X L(c, m, λ, α) = (n + λn(n − 1) − α) n=2

(−c/4)n−1 ≤ 1 − α. (m)n−1 (n − 1)!

Writing n2 = (n− 1)(n− 2) + 3(n− 1) + 1 and n = (n− 1) + 1, and by simple computation, we get ∞ X (−c/4)n−1 L(c, m, λ, α) = (n2 λ + n(1 − λ) − α) (m)n−1 (n − 1)! n=2 ∞ X

∞ X (−c/4)n−1 (−c/4)n−1 ≤ λ(n − 1)(n − 2) + (1 + 2λ) (n − 1) (m)n−1 (n − 1)! (m)n−1 (n − 1)! n=2 n=2

+ (1 − α) =λ =λ

∞ X

n=3 ∞ X n=1

∞ X n=2

(−c/4)n−1 (m)n−1 (n − 1)!

∞ ∞ X X (−c/4)n−1 (−c/4)n−1 (−c/4)n−1 + (1 + 2λ) + (1 − α) (m)n−1 (n − 3)! (m)n−1 (n − 2)! (m)n−1 (n − 1)! n=2 n=2 ∞ ∞ X X (−c/4)n+1 (−c/4)n+1 (−c/4)n+1 + (1 + 2λ) + (1 − α) (m)n+1 (n − 1)! (m)n+1 (n)! (m)n+1 (n + 1)! n=0 n=0

∞ ∞ (−c/4)2 X (−c/4)n (−c/4) X (−c/4)n =λ + (1 + 2λ) m(m + 1) n=0 (m + 2)n n! m (m + 1)n n! n=0

+ (1 − α) 2

=λ

∞ X n=0

(−c/4)n+1 (m)n+1 (n + 1)!

(−c/4) (−c/4) up+2 (1) + (1 + 2λ) up+1(1) + (1 − α)[up(1) − 1] m(m + 1) m

= λu00p (1) + (1 + 2λ)u0p (1) + (1 − α)[up (1) − 1]. By a simplification, we see that the last expression is bounded above by 1 − α if (2.1) is satisfied.  By taking λ = 0, we state the following corollary.

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Corollary 1. If c < 0 and m > 0, then z(2 − up(z)) is in S ∗(α) if u0p (1) + (1 − α)up (1) ≤ 2(1 − α).

(2.2)

Remark 2. In particular, when c = −1 and b = 1, the condition (2.1) becomes 2p−2 Γ(p + 1) [λIp+2 (1) + [1 + 2λ]Ip+1 (1) + 2(1 − α)Ip (1)] ≤ 1 − α,

(2.3)

which is necessary and sufficient condition for z(2 − ζp (z 1/2) to be in G ∗ (λ, α), where up(z 1/2) = 2p Γ(p + 1)z −p/2 Ip (z 1/2). Theorem 2. If c < 0 and m > 0, then z(2 − up(z)) is in K(λ, α) if 00 0 λu000 p (1) + (1 + 5λ)up (1) + (3 + 4λ − α)up (1) + (1 − α)up (1) ≤ 2(1 − α).

(2.4)

Proof. Since z(2 − up (z)) = z −

∞ X n=2

(−c/4)n−1 zn (m)n−1 (n − 1)!

and by virtue of Lemma 2, it suffices to show that L(c, m, λ, α) =

∞ X n=2

(n3 λ + n2 (1 − λ) − nα)

(−c/4)n−1 ≤ 1 − α. (m)n−1 (n − 1)!

Writing n3 = (n − 1)(n − 2)(n − 3) + 6(n − 1)(n − 2) + 7(n − 1) + 1, n2 = (n − 1)(n − 2) + 3(n − 1) + 1 and n = (n − 1) + 1, we can rewrite the above terms as L(c, m, λ, α) ≤ λ

∞ X

n=2 ∞ X

(n − 1)(n − 2)(n − 3)

(−c/4)n−1 (m)n−1 (n − 1)!

∞ X (−c/4)n−1 (−c/4)n−1 + (1 + 5λ) (n − 1)(n − 2) + (3 + 4λ − α) (n − 1) (m)n−1 (n − 1)! (m)n−1 (n − 1)! n=2 n=2

+ (1 − α) =λ

∞ X n=4

=λ

n=2

(−c/4)n−1 (m)n−1 (n − 1)!

∞ X

(−c/4)n−1 (m)n−1 (n − 1)!

n=2

∞ ∞ X X (−c/4)n−1 (−c/4)n−1 (−c/4)n−1 + (1 + 5λ) + (3 + 4λ − α) (m)n−1 (n − 4)! (m)n−1 (n − 3)! (m)n−1 (n − 2)! n=3 n=2

+ (1 − α) ∞ X

∞ X

n=2

∞ X (−c/4)n+1 (−c/4)n+1 + (1 + 5λ) (m)n+1 (n − 2)! (m)n+1 (n − 1)! n=1

∞ ∞ X X (−c/4)n+1 (−c/4)n+1 + (3 + 4λ − α) + (1 − α) (m)n+1 (n)! (m)n+1 (n + 1)! n=0 n=0

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7

 ∞ ∞  X (−c/4)3 (−c/4)n−2 (−c/4)2 X (−c/4)n−1 =λ + (1 + 5λ) m(m + 1)(m + 2) n=2 (m + 3)n−2 (n − 2)! m(m + 1) n=1 (m + 2)n−1 (n − 1)!  ∞ ∞  X (−c/4)n (−c/4) X (−c/4)n+1 + (3 + 4λ − α) + (1 − α) m (m + 1)n (n)! (m)n+1 (n + 1)! n=0 n=0 (−c/4)2 (−c/4)3 up+2 (1) up+3(1) + (1 + 5λ) m(m + 1)(m + 2) m(m + 1) (−c/4) + (3 + 4λ − α) up+1 (1) + (1 − α)[up(1) − 1] m 00 0 = λu000 p (1) + (1 + 5λ)up (1) + (3 + 4λ − α)up (1) + (1 − α)[up (1) − 1]. =λ

By a simplification, we see that the last expression is bounded above by 1 − α if (2.4) is satisfied.  By taking λ = 0, we state the following corollary. Corollary 2. If c < 0 and m > 0, then z(2 − up(z)) is in ∈ K(α) if u00p (1) + (3 − α)u0p (1) + (1 − α)up (1) ≤ 2(1 − α).

(2.5)

Remark 3. We also note that the function z(2 − up(z)) is in K∗(λ, α) if and only if the condition (2.4) is satisfied. 3. Inclusion Properties Making use of the following lemma, we will study the action of the Bessel function on the classes K(λ, α). Lemma 6. [7] A function f ∈ 0. If f ∈ r + s + t or q1 , · · · , qp < r + s + t. Let f : A → B be a mapping satisfying (2.3) and q
kf (xr x∗ s xt ) − f (x)r f (x)∗ s f (x)t kB ≤ θ kxkqA1 + · · · + kxkAp (2.5) for all x ∈ A. Then the mapping f : A → B is an (r, s, t)-J ∗ -homomorphism. q
Proof. The proof follows from Theorem 2.4 by taking ϕ(x1 , · · · , xp ) := θ kx1 kqA1 + · · · + kxp kAp with b > 1 for the case q1 , · · · , qp > r +s+t and with b < 1 for the case q1 , · · · , qp < r +s+t. 

Corollary 2.6. Let θ be a nonnegative real number and q1 , · · · , qp be positive real numbers such that q1 + · · · + qp 6= r + s + t. Let f : A → B be a mapping satisfying (2.3) and q +···+qp

kf (xr x∗ s xt ) − f (x)r f (x)∗ s f (x)t kB ≤ θkxkA1

for all x ∈ A. Then the mapping f : A → B is an (r, s, t)-J ∗ -homomorphism. q
Proof. The proof follows from Theorem 2.4 by taking ϕ(x1 , · · · , xp ) := θ kx1 kqA1 · · · kxp kAp with b > 1 for the case q1 + · · · + qp > r + s + t and with b < 1 for the case q1 + · · · + qp < r + s + t. 

3. Hyers-Ulam stability of (r, s, t)-J ∗ -homomorphisms: fixed point method In this section, by using the fixed point method, we prove the Hyers-Ulam stability of (r, s, t)J ∗ -homomorphisms associated with the functional equation (0.2). For a given mapping f : A → B, we define  Pp   Pp  X  Pp  p j=1 xj − pxi j6=i i=1 xi i=2 xi − x1   %µ f (x1 , · · · , xp ) := f µ + f µ +f µ − µf (x1 ) p−1 p−1 p−1 i=2

for all µ ∈ T1 and all x1 , · · · , xp ∈ A. Lemma 3.1. The mapping f : A → B is a C-linear mapping if and only if %µ f (x1 , · · · , xp ) = 0 for all µ ∈ T1 and all x1 , · · · , xp ∈ A. Proof. The proof is easy and thus omitted.



In the following theorems, we will except the case p = 3. This case will be considered individually. Theorem 3.2. Let ϕ : Ap → [0, ∞) be a function with ϕ(0, · · · , 0) = 0 and p 6= 3 such that there exists an L < 1 with L (3.1) ϕ(x1 , · · · , xp ) < ϕ(kx1 , · · · , kxp ) k 2 for all x1 , · · · , xp ∈ A, where k = p−1 . Suppose that f : A → B is an odd mapping satisfying (2.4) and k%µ f (x1 , · · · , xp )kB ≤ ϕ(x1 , · · · , xp ) (3.2)

for all µ ∈ T1 and all x1 , · · · , xp ∈ A. Then there exists a unique (r, s, t)-J ∗ -homomorphism H : A → B such that L kf (x) − H(x)kB ≤ ϕ (0, x, · · · , x) (3.3) 2(1 − L) for all x ∈ A.

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Proof. We first consider the set S := {g : A → B} and introduce the generalized metric d as follows:  d(g, h) = inf C ∈ R+ : kg(x) − h(x)kB ≤ Cϕ (0, x, · · · , x) . x∈A

It is easy to show that (S, d) is complete (see the proof of [35, Lemma 2.1]). Now we define the linear mapping J : S → S such that x J (g(x)) := kg k for all x ∈ A. From (3.2), we can get f (0) = 0. By letting µ = 1, x1 = 0 and x2 = · · · = xp = x in (3.2) and the fact that f (−x) = −f (x), (f is an odd mapping) and then by (3.1), we have

 

−2

2f (x) + (p − 1)f

≤ ϕ(0, x, · · · , x), x

p − 1 B

x k  x x L

− f (x) ≤ ϕ 0, , · · · , ≤ ϕ (0, x, · · · , x)

kf k 2 k k 2 B for all x ∈ A. This means that L (3.4) d(J (f ), f ) ≤ 2 Assume that g, h ∈ S are given with d(g, h) = ε. Then we have

x  x   x x

kJ (g(x)) − J (h(x))kB = k g −h ≤ kεϕ 0, , · · · ,

k k B k k < Lεϕ (0, x, · · · , x) for all x ∈ A. This implies that d (J (g), J (h)) < Lε = Ld(g, h), which means J is a strictly contractive mapping. By Theorem 1.4, we have the following: (1) J has a fixed point, i.e., there exists a mapping H : A → B, such that J (H) = H. So x (3.5) H(x) = kH k for all x ∈ A. The mapping H is also the unique fixed point of J in the set M = {g ∈ S : d(f, g) < ∞}. This signifies that H is a unique mapping satisfying (3.5), moreover there exists a C ∈ (0, ∞) such that kf (x) − H(x)kB ≤ Cϕ (0, x, · · · , x) for all x ∈ A; (2) The sequence {J n (g)} converges to H, for each given g ∈ S. Thus d (J n (f ), H) → 0, as n → ∞. This implies the equality x H(x) = lim k n f n→∞ kn for all x ∈ A; 1 (3) d(g, H) ≤ 1−L d (g, J (g)), for all g ∈ M. Therefore (3.4) shows us that d(f, H) ≤

1 L d (f, J (f )) ≤ . 1−L 2(1 − L)

By this, we get the inequality (3.3).

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It follows from (3.1) that

x x xp  xp 

1 1 n k%µ h(x1 , · · · , xp )kB = k n %µ f , · · · , ≤ k ϕ , · · · ,

kn kn B kn kn n < L ϕ (x1 , · · · , xp ) for all x1 , · · · , xp ∈ A, in which the right-hand side tends to zero as n → ∞. Hence by Lemma 3.1, we deduce that H is C-linear. By (3.1) and (2.4), we obtain

h(xr x∗ s xt ) − h(x)r h(x)∗ s h(x)t

   B     x r  x ∗ s  x t

x r x ∗s x t (r+s+t)n

=k −f f f

f

n n n n n n k k k k k k B  x x (r+s+t)n (r+s+t)n ϕ (x, · · · , x) ≤k ϕ n,··· , n < L k k for all x ∈ A. The right-hand side tends to zero as n → ∞, and so the mapping H : A → B is an (r, s, t)-J ∗ -homomorphism, as desired.  Theorem 3.3. Let ϕ : Ap → [0, ∞) be a function with ϕ(0, · · · , 0) = 0 and p 6= 3 such that there exists an L < 1 with x xp  1 ϕ(x1 , · · · , xp ) < kLϕ ,··· , (3.6) k k 2 . Suppose that f : A → B is an odd mapping satisfying for all x1 , · · · , xp ∈ A, where k = p−1 (3.2) and (2.4). Then there exists a unique (r, s, t)-J ∗ -homomorphism H : A → B such that  x L x kf (x) − H(x)kB ≤ ϕ 0, , · · · , (3.7) (1 − L)(p − 1) k k

for all x ∈ A. Proof. Let S be the defined set in the proof of Theorem 3.2. Consider the following generalized metric d: n  x x o d(g, h) = inf C ∈ R+ : kg(x) − h(x)kB ≤ Cϕ 0, , · · · , . x∈A k k It is easy to show that (S, d) is complete (see the proof of [35, Lemma 2.1]). we define the linear mapping J : S → S such that 1 J (g(x)) := g(kx) k for all x ∈ A. By the same argument as in the proof of Theorem 3.2, we can obtain the mapping H : A → B, as the unique fixed point of J such that H(x) := lim

n→∞

for all x ∈ A. By (3.2) and (3.6), we



f (x) − 1 f (kx) ≤

k B

have  x 1 L x ϕ (0, x, · · · , x) ≤ ϕ 0, , · · · , 2 (p − 1) k k

for all x ∈ A. This means that d(f, J (f )) ≤ d(f, H) ≤

1 f (k n x) kn

L (p−1) .

Hence

1 L d (f, J (f )) ≤ 1−L (1 − L)(p − 1)

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which implies that the inequality (3.7) holds. The rest of the proof is similar to the proof of Theorem 3.2.



Theorem 3.4. Let ϕ : A3 → [0, ∞) be a function such that there exists an L < 1 with ϕ(x, y, z)
l. By (4.4), for m − l > 0 and k l x, we have



1

1 1 1 m l m−l l l

k x) − f (k x)

k m f (k x) − k l f (k x) = k l k m−l f (k B

B

≤

≤

1 p−1 1 p−1

m−1 X s=l ∞ X

k −(s+1) ϕ(0, k s x, · · · , k s x)

k −(s+1) ϕ(0, k s x, · · · , k s x)

s=l

for all x ∈ A. By (4.1), the right-hand side tends to zero as l → ∞. Therefore the sequence { k1n f (k n x)} is Cauchy. Since A is a complete space, the sequence { k1n f (k n x)} is convergent and we can define for all x ∈ A, the mapping h : A → B by 1 h(x) := lim n f (k n x). n→∞ k Passing the limit n → ∞ in (4.4) and then by (4.1), we obtain (4.3). It follows from (4.1) and (3.2) that 1 k%µ f (k n x1 , · · · , k n xp )kB n→∞ k n 1 ≤ lim n ϕ (k n x1 · · · , k n xp ) = 0 n→∞ k

k%µ h(x1 , · · · , xp )kB =

lim

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for all µ ∈ T1 and all x1 , · · · , xp ∈ A. So by Lemma 3.1 we deduce that h is C-linear. By (4.2) and substituting x by k n x in (2.4), we obtain

h(xr x∗ s xt ) − h(x)r h(x)∗ s h(x)t B


1 s = lim (r+s+t)n f (k n x)r (k n x)∗ s (k n x)t − f (k n x)r f (k n x)∗ f (k n x)t B n→∞ k 1 ≤ lim (r+s+t)n ϕ (k n x, · · · , k n x) = 0 n→∞ k for all x ∈ A. Hence h(xr x∗ s xt ) = h(x)r h(x)∗ s h(x)t for all x ∈ A. Let g : A → B be another (r, s, t)-J ∗ -homomorphism satisfying (4.3). Then we have kh(x) − g(x)kB ≤ ≤ =

1 1 kf (k n x) − h(k n x)kB + n kf (k n x) − g(k n x)kB n k k   1 2 n n φ (0, k x, · · · , k x) kn p − 1 ∞ 2 X −(s+1) k ϕ (0, k s x, · · · , k s x) p − 1 s=n

for all x ∈ A. By (4.1), the right-hand side tends to zero as n → ∞, which means h is unique.  Theorem 4.2. Let ϕ : Ap → [0, ∞) be a function with ϕ(0, · · · , 0) = 0 and p ≥ 4. Denote by φ a function such that ∞   X φ(x1 , · · · , xp ) := k n ϕ k −(n+1) x1 , · · · , k −(n+1) xp < ∞ (4.5) n=0 2 for all x1 , · · · , xp ∈ A, where k = p−1 . Suppose that f : A → B be an odd mapping satisfying (3.2) and (2.4). Then there exists a unique (r, s, t)-J ∗ -homomorphism h : A → B satisfying (4.3).

Proof. It follows from (3.2) that

x

 x 1 x

− f (x) ≤ ϕ 0, · · · ,

kf k p−1 k k B for all x ∈ A. By the same method which was done in the
proof of Theorem 4.1, we can get the unique and C-linear mapping h(x) := limn→∞ k n f k1n x satisfying (4.3). By (2.4), (4.5) and the fact that k < 1, we have

h(xr x∗ s xt ) − h(x)r h(x)∗ s h(x)t B

     x r  x  ∗ s  x t r

x x  ∗ s  x t (r+s+t)n

f f = lim k − f f

n→∞ kn kn kn kn kn k n B x x x x ≤ lim k (r+s+t)n ϕ n , · · · , n ≤ lim k n ϕ n , · · · , n = 0 n→∞ n→∞ k k k k for all x ∈ A. Hence h(xr x∗ s xt ) = h(x)r h(x)∗ s h(x)t for all x ∈ A.



Corollary 4.3. Let θ be a nonnegative real number and q1 , · · · , qp be positive real numbers such that q1 , · · · , qp > r + s + t or q1 , · · · , qp < 1. Let f : A → B be an odd mapping satisfying (2.5) and q k%µ f (x1 , · · · , xp )kB ≤ θ(kx1 kqA1 + · · · + kxp kAp )

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for all µ ∈ T1 and all x1 , · · · , xp ∈ A. Then there exists a unique (r, s, t)-J ∗ -homomorphism h : A → B such that p q X θkxkAj kf (x) − h(x)kB ≤ 2 |1 − k qj −1 | j=2

for all x ∈ A. q
Proof. Defining ϕ(x1 , · · · , xp ) = θ kx1 kqA1 + · · · + kxp kAp and applying Theorem 4.1 for the case q1 , · · · , qp > r + s + t, and Theorem 4.2 for the case q1 , · · · , qp < 1, we get the result. 

Theorem 4.4. Let ϕ : A2 → [0, ∞) be a function with ϕ(0, 0) = 0. Denote by φ a function such that ∞ X φ(x, y) := 2−(n+1) ϕ (2n x, 2n y) < ∞ n=0

for all x, y ∈ A. Suppose that f : A → B is an odd mapping satisfying kf (µx + µy) + f (µx − 2µy) + f (µy − µx) − µf (x)kB ≤ ϕ(x, y), r ∗s t

(4.6)

∗s

kf (x x x ) − f (x)r f (x) f (x)t kB ≤ ϕ(x, x) for all µ ∈ such that

T1

and all x, y ∈ A. Then there exists a unique

(4.7)

(r, s, t)-J ∗ -homomorphism

h:A→B

kf (x) − h(x)kB ≤ φ(0, x)

(4.8)

for all x ∈ A. Proof. From (4.6), it follows that

1

f (2x) − f (x) ≤ 1 ϕ(0, x)

2

2 B for all x ∈ A. Using the same method as in the proof of Theorem 4.1, we conclude that the mapping h(x) := limn→∞ 21n f (2n x) is a unique (r, s, t)-J ∗ -homomorphism satisfying (4.8).  Theorem 4.5. Let ϕ : A2 → [0, ∞) be a function with ϕ(0, 0) = 0. Denote by φ a function such that ∞   X φ(x, y) := 2n ϕ 2−(n+1) x, 2−(n+1) y < ∞, n=0


lim 2(r+s+t)n ϕ 2−n x, 2−n x = 0

n→∞

for all x, y ∈ A. Suppose that f : A → B is an odd mapping satisfying (4.6) and (4.7). Then there exists a unique (r, s, t)-J ∗ -homomorphism h : A → B satisfying (4.8). Proof. The proof is similar to the proof of Theorem 4.4.



Corollary 4.6. Let θ be a nonnegative real number and q1 , q2 be positive real numbers such that q1 , q2 < 1 or q1 , q2 > r + s + t. Let f : A → B be an odd mapping satisfying kf (µx + µy) + f (µx − 2µy) + f (µy − µx) − µf (x)kB ≤ θ(kxkqA1 + kykqA2 ), kf (xr x∗ s xt ) − f (x)r f (x)∗ s f (x)t kB ≤ θ(kxkqA1 + kxkqA2 ) for all µ ∈ T1 and all x, y ∈ A. Then there exists a unique (r, s, t)-J ∗ -homomorphism h : A → B such that θkxkqA2 kf (x) − h(x)kB ≤ |2 − 2q2 | for all x ∈ A.

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Proof. Defining ϕ(x, y) = θ kxkqA1 + kykqA2 and applying Theorem 4.4 for the case q1 , q2 < 1, and Theorem 4.5 for the case q1 , q2 > r + s + t, we get the result.  Theorem 4.7. Let ϕ : A3 → [0, ∞) be a function. Denote by φ a function such that φ(x, y, z) :=

∞ X

2−n ϕ (2n x, 2n y, 2n z) < ∞

n=1

for all x, y, z ∈ A. Suppose that f : A → B is an odd mapping satisfying (3.9) and (3.10). Then there exists a unique (r, s, t)-J ∗ -homomorphism h : A → B such that kf (x) − h(x)kB ≤ φ(x, 0, 0)

(4.9)

for all x ∈ A.

Proof. By (3.9), we get 21 f (2x) − f (x) B ≤ 12 ϕ(2x, 0, 0) for all x ∈ A. The same method as in the proof of Theorem 4.1, leads us to the unique (r, s, t)-J ∗ -homomorphism h(x) := limn→∞ 21n f (2n x)satisfying (4.9).  Theorem 4.8. Let ϕ : A3 → [0, ∞) be a function. Denote by φ a function such that φ(x, y, z) :=

∞ X


2n ϕ 2−n x, 2−n y, 2−n z < ∞,

n=0


lim 2(r+s+t)n ϕ 2−n x, 2−n x, 2−n x = 0

n→∞

for all x, y, z ∈ A. Suppose that f : A → B is an odd mapping satisfying (3.9) and (3.10). Then there exists a unique (r, s, t)-J ∗ -homomorphism h : A → B satisfying (4.9). Proof. The proof is similar to the proof of Theorem 4.7.



Corollary 4.9. Let θ be a nonnegative real number and q1 , q2 , q3 be positive real numbers such that q1 , q2 , q3 < 1 or q1 , q2 , q3 > r + s + t. Let f : A → B be an odd mapping satisfying

     

f µ x + y + z + f µ x + z − 3y + f µ x + y − 3z

2 2 2

 

y+z−x q1 q2 q3
+f µ − µf (x) ≤ θ kxk + kyk + kzk A A A ,

2 B
kf (xr x∗ s xt ) − f (x)r f (x)∗ s f (x)t kB ≤ θ kxkqA1 + kxkqA2 + kxkqA3 for all µ ∈ T1 and all x, y, z ∈ A. Then there exists a unique (r, s, t)-J ∗ -homomorphism h : A → B such that 2 q1 kf (x) − h(x)kB ≤ θkxkqA1 |2 − 2q1 | for all x ∈ A.
Proof. Defining ϕ(x, y, z) = θ kxkqA1 + kykqA2 + kzkqA3 and applying Theorem 4.7 for the case q1 , q2 , q3 < 1, and Theorem 4.8 for the case q1 , q2 , q3 > r + s + t, we get the result.  Remark 4.10. The obtained results in this paper, could be more remarkable and interesting. In other words, as a consequence including simpler and better results, one can set q1 = · · · = qp = q, as well as r = s = t = 1 (or a fixed n ∈ N) in all the statements. Furthermore, all the obtained results do also hold for (r, s, t)-J ∗ -derivations similarly. The reader can directly verify this point just with a little difference in details.

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Acknowledgments C. Park and D. Y. Shin were supported by Basic Science Research Program through the National Research Foundation of Korea funded by the Ministry of Education, Science and Technology (NRF-2012R1A1A2004299) and (NRF-2010-0021792), respectively.

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Differential subordinations obtained by using a generalization of Marx-Strohhäcker theorem Georgia Irina Oros1 , Gheorghe Oros2 , Alina Alb Lupa¸s3 , Vlad Ionescu4 1,2,3 University of Oradea, Department of Mathematics Str. Universita˘¸tii, No.1, 410087 Oradea, Romania 1 [email protected], 2 [email protected], 3 [email protected], 4 [email protected] Abstract In [1] and [6] Marx and Strohhäcker have proved that if f ∈ A is a convex function, then it has the property of starlikeness of order 12 . In [5, Theorem 9.5.6], P. T. Mocanu extended this result to the class A2 for a convex function of order − 12 . In this paper we extend the results proven by Marx and Strohhäcker and by P. T. Mocanu and we’ll prove that, if the function f ∈ An , n ≥ 3, is a close-to-convex function, then it is starlike of order 12 .

Keywords: Analytic function, univalent function, integral operator, close-to-convex function. 2000 Mathematical Subject Classification: 30C45, 30A20, 34A40.

1

Introduction and preliminaries

Let U be the unit disc of the complex plane U = {z ∈ C : |z| < 1}. Let H(U ) be the class of holomorphic functions in U . Also, let An = {f ∈ H(U ) : f (z) = z + an+1 z n+1 + . . . , z ∈ U }, with A1 = A. Let S = {f ∈ A : f univalent in U } be the class of holomorphic and univalent functions in the open unit disc U , with conditions f (0) = 0, f 0 (0) = 1, that is the holomorphic and univalent functions with the following power series development f (z) = z + a2 z 2 + . . . , z ∈ U.o n 00 (z) Denote by K = f ∈ A : Re zff 0 (z) + 1 > 0, z ∈ U the class of normalized convex functions in U and by n o 0 f (z) C = f ∈ A : ∃ ϕ ∈ K, Re ϕ the class of normalized close-to-convex functions in U . 0 (z) > 0, z ∈ U An equivalent formulation would involve the h (not necessarily normalized) ³ existence o ´ n of a starlike function 0 00 (z) (z) 1 1 such that Re zfh(z) > 0, z ∈ U. We consider K − 2γ + 1 > − 2γ , z ∈ U, γ ≥ 1 . = f ∈ An : Re zff 0 (z) n o 0 (z) Let S ∗ = f ∈ A : Re zff (z) > 0, z ∈ U denote the class of starlike functions in U , and n o 0 (z) S ∗ (α) = f ∈ A : Re zff (z) > α, z ∈ U , denote the class of starlike functions of order α, with 0 ≤ α < 1. In order to prove our original results, we use the following lemmas: Lemma 1.1 [2], [3], [4, Theorem 2.3.i, p. 35] Let ψ : C3 × U → C, satisfy the condition Re ψ(is, t) ≤ 0, z ∈ U, for s, t ∈ R, t ≤ − n2 (1 + s2 ). If p(z) = 1 + pn z n + pn+1 z n+1 + . . . satisfies Re [p(z), zp0 (z); z] > 0, then Re p(z) > 0, z ∈ U. More general forms of this lemma can be found in [6]. 0 Lemma 1.2 [5, h Theorem i 4.6.3, p. 84] The function f ∈ A, with f (z) 6= 0, z ∈ U , is close-to-convex if and R θ2 zf 00 (z) only if θ1 Re 1 + f 0 (z) dθ > −π, z = reiθ , for all θ1 , θ2 , with 0 ≤ θ1 < θ2 ≤ 2π and r ∈ (0, 1).

Definition 1.1 [4, Definition ½ 2.2.b, p. 21] We denote ¾ by Q the set of functions q that are analytic and injective on U \ E(q), where E(q) = ζ ∈ ∂U : lim q(z) = ∞ and are such that q 0 (ζ) 6= 0, for ζ ∈ ∂U \ E(q). The set z→ζ

E(q) is called exception set. The subclass of Q for which f (0) = a is denoted by Q(a). 135

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Definition 1.2 [4, Definition 2.3.a, p. 27] Let Ω be a set in C, q ∈ Q and n be a positive integer. The class of admissible functions Ψn [Ω, q] consists of those functions ψ : C × U → C that satisfy the admissibility condition (A)

ψ(r, s, t) 6∈ Ω h i ¡ ¢ 00 (ζ) + 1 , z ∈ U , ζ ∈ ∂U \ E(q), m ≥ n. We write Ψ1 [Ω, q] where r = q(ζ), s = mζq 0 (ζ), R st + 1 ≥ mRe ζqq0 (ζ) as Ψ[Ω, q]. In the special case when Ω is a simply connected domain, Ω 6= C, and h is a conformal mapping of U onto Ω, we denote this class by Ψn [h, q]. If ψ : C2 × U → C, then the admissibility condition (A) reduces to ψ(q(ζ), mζq 0 (ζ); z) 6∈ Ω,

(A0 )

where z ∈ U , ζ ∈ ∂U \ E(q) and m ≥ n. If ψ : C × U → C, then the admissibility condition (A) reduces to (A00 )

ψ(q(ζ); z) 6∈ Ω

where z ∈ U and ζ ∈ ∂U \ E(q). Definition 1.3 [4, p. 36] Let f and F be members of H(U ). The function f is said to be subordinate to F , written f ≺ F or f (z) ≺ F (z), if there exists a function w analytic in U , with w(0) = 0 and |w(z)| < 1, and f (0) = F (0) and f (U ) ⊂ F (U ). Definition 1.4 [4, p. 16] Let ψ : C3 × U → C and let h be univalent in U . If p is analytic in U and satisfies the (second-order) differential subordination (i) ψ(p(z), zp0 (z), z 2 p00 (z); z) ≺ h(z), then p is called a solution of the differential subordination. The univalent function q is called a dominant of the solutions of the differential subordination, or more simply a dominant, if p ≺ q for all p satisfying (i). A dominant qe that satisfies qe ≺ q for all dominant q of (i) is said to be the best dominant of (i). (Note that the best dominant is unique up to a rotation of U ). If we require the more restrictive condition p ∈ [a, n], then p will be called an (q, n)-solution, q an (a, n)dominant, and qe the best (a, n)-dominant,

Lemma 1.3 [4, Theorem 2.3.c, p. 30] Let ψ ∈ Ψn [h, q] with q(0) = a. If p ∈ H[a, n], ψ(p(z), zp0 (z), z 2 p00 (z); z) is analytic in U , and (ii) ψ(p(z), zp0 (z), z 2 p00 (z); z) ≺ h(z), then p(z) ≺ q(z), z ∈ U. ´ ³ 00 (z) + 1 > 0, then Theorem 1.1 [1, 6, Marx-Strohhacker] If f ∈ A and satisfy the condition Re zff 0 (z) £ ¡ ¢¤ 0 (z) (a) Re zff (z) > 12 i.e., f ∈ S ∗ 12 and 1 (b) Re f (z) z > 2 , for z ∈ U .

In [5] has shown that the odd and convex functions of order − 12 are starlike functions of order 12 . ³ 00 ´ (z) +1 Theorem 1.2 [5, Marx-Strohhscker, Theorem 9.5.6, p. 218] If f ∈ A2 and satisfy the condition Re zff 0 (z) £ ¡ ¢¤ 0 (z) > − 12 , then Re zff (z) > 12 i.e., f ∈ S ∗ 12 , for z ∈ U .

2

Main results

We’ll extend the theorem Marx-Strohhäcker for the functions f ∈ An , n ≥ 3, which are close-to-convex functions.

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Theorem 2.1 Let n ≥ 3, γ ≥ 1, f ∈ An , satisfy the condition Re

1 zf 00 (z) +1>− , 0 f (z) 2γ

(2.1)

0

(z) then Re zff (z) > 12 .

Proof. According to Lemma 1.2 we obtain ¸ ∙ Z θ2 Z θ2 Z θ2 1 π zf 00 (z) 1 1 2π dθ ≥ = − > −π, λ ≥ 1. Re 1 + 0 − dθ = − dθ = − (θ2 − θ1 ) > − f (z) 2γ 2γ 2γ 2γ γ θ1 θ1 θ1

(2.2)

From (2.2) we have f ∈ C, hence it is univalent. 0 (z) Let p(z) = 2 · zff (z) − 1. Since f ∈An and f is close-to-convex function (univalent), the function p is analytic in U and p(0) = 1. A simple computation leads to zf 0 (z) p(z) + 1 = . (2.3) 2 f (z) By differentiating (2.3), we obtain zf 00 (z) zf 0 (z) zp0 (z) =1+ 0 − . p(z) + 1 f (z) f (z)

(2.4)

Using (2.3) in (2.4), we have zf 00 (z) p(z) + 1 zp0 (z) + = 1+ 0 . 2 p(z) f (z) h i zp0 (z) 1 + p(z)+1 , which is equivalent to Using (2.1) in (2.5), we obtain Re p(z)+1 > − 2γ 2

(2.5)

¸ zp0 (z) 1 p(z) + 1 + + > 0. Re 2 p(z) + 1 2γ ∙

(2.6)

0

zp (z) 1 s 1 Let ψ : C2 × U → C, ψ(p(z), zp0 (z); z) = p(z)+1 + p(z)+1 + 2γ , where ψ(r, s) = r+1 2 2 + r+1 + 2γ . Then (2.6) is equivalent to Re ψ(p(z), zp0 (z); z) > 0, z ∈ U. ³ ´ t 1 In order to prove Theorem 2.1, we use Lemma 1.1. For that we calculate Re ψ(is, t) = Re is+1 2 + 1+is + 2γ ³ ´ t(1−is) n(1+s2 ) (1−n)γ+1 1 t 1 1 1 1−n 1 = Re is+1 + + ≤ 0. Since n ≥ 3, = 12 + 1+s 2 + 2γ ≤ 2 − 2(1+s2 ) + 2γ = 2 1+s2 2γ 2 + 2γ = 2γ 0

(z) γ ≥ 1. Now, using Lemma 1.1, we get that Re p(z) > 0, z ∈ U , i.e., Re zff (z) > 12 , z ∈ U.

Remark 2.1 Each of the four conditions in the Marx-Strohhäcker theorem can be rewritten in terms of subordination. This leads to the following equivalent form of the theorem. Theorem 2.2 Let n ≥ 3, γ ≥ 1, f ∈ An , satisfies the condition

zf 00 (z) f 0 (z)

Theorem 2.3 Let n ≥ 3, γ ≥ 1, f ∈ An satisfies the conditions ¶ µ 00 1 zf (z) +1 >− Re f 0 (z) 2γ and Re

1 zf 0 (z) > f (z) 2

+1≺

1−( γ1 +1)z , 1+z

then

zf 0 (z) f (z)

≺

1 1+z .

(2.7)

(2.8)

1 then Re f (z) z > 2 , for z ∈ U.

Proof. In order to prove Theorem 2.1, we saw that, if f ∈An , n ≥ 3 and satisfies the condition (2.1) or (2.7), then the function f is close-to-convex (univalent).

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Let p(z) = 2fz(z) − 1. Since f ∈ An , n ≥ 3 and f is close-to-convex function (univalent) then the function p is analytic in U and p(0) = 1. A simple computation leads to f (z) p(z) + 1 = . 2 z

(2.9)

zf 0 (z) zp0 (z) = − 1. p(z) + 1 f (z)

(2.10)

By differentiating (2.9), we obtain

Using (2.8) in (2.10), we have Re

µ

zp0 (z) 1 + p(z) + 1 2 0

¶

> 0, z ∈ U.

(2.11)

zp (z) s + 12 , where ψ(r, s) = 12 + 1+r . Then (2.11) is equivalent to Let ψ : C2 × U → C, ψ(p(z), zp0 (z); z) = 1+p(z) Re ψ(p(z), zp0 (z); z) > 0, z ∈ U. h i t In order to prove Theorem 2.1, we use Lemma 1.1. For that we calculate Re ψ(is, t) = Re 12 + 1+is = h i 2 n(1+s ) t 1 1−n Re 12 + t(1−is) < 0, since n ≥ 3. Therefore, by applying Lemma 1.1 we = 12 + 1+s 2 ≤ 2 − 2(1+s2 ) = 1+s2 2

1 conclude that p satisfies Re p(z) > 0. This is equivalent to Re f (z) z > 2 , z ∈ U. For 0 < γ < 1, n ≥ 3, Theorem 2.2 can be written as the following corollary. h 00 i 0 (z) (z) Corollary 2.4 Let n ≥ 3, 0 < γ < 1, f ∈ An satisfy the conditions Re zff 0 (z) + 1 > − γ2 and Re zff (z) > 12 , 1 then Re f (z) z > 2 , z ∈ U.

Theorem 2.5 Let n ≥ 3, γ ≥ 1, f ∈ An satisfy differential subordination ³ ´ 1 − γ1 + 1 z zf 00 (z) +1≺ , f 0 (z) 1+z and

then

1 zf 0 (z) ≺ f (z) 1+z f (z) z

≺

1 1+z ,

(2.12)

(2.13)

z ∈ U.

Proof. Consider

2f (z) − 1. (2.14) z Since f ∈ An , and f is close-to-convex function (univalent) then the function p is analytic in U , and p(0) = 1. By differentiating (2.14), we obtain zf 0 (z) zp0 (z) +1= . (2.15) p(z) + 1 z p(z) =

Using (2.13) in (2.15), we have 1 zp0 (z) +1≺ . p(z) + 1 1+z

(2.16)

1 Since Re 1+z ≥ 12 , differential subordination (2.16) is equivalent to

Re

µ

1 zp0 (z) + p(z) + 1 2

¶

> 0,

z ∈ U.

(2.17)

zp0 (z) p(z)+1

+ 12 , then (2.17) becomes Re ψ(p(z), zp0 (z); z) > 0, z ∈ U. ³ ´ t In order to prove Theorem 2.5, we use Lemma 1.3. For that we calculate Re ψ(is, t) = Re 1+is + 12 = h i −n(1+s2 ) 1 t 1 1 1−n Re t(1−is) + 2 1+s 2 = 1+s2 + 2 ≤ 2(1+s2 ) + 2 = 2 < 0. Using Definition 1.2, we have ψ ∈ Ψn [h, q]. Therefore Let ψ : C2 × U → C, ψ(p(z), zp0 (z); z) =

by Lemma 1.3, we conclude that p(z) ≺ q(z), i.e.,

f (z) z

≺

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1 1+z ,

for z ∈ U.

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³ 00 ´ p (z) 1 Theorem 2.6 If f ∈ An , n ≥ 3, γ ≥ 1 and satisfy the condition Re zff 0 (z) + 1 > − 2γ , then Re f 0 (z) > 12 , for z ∈ U. p Proof. Consider p(z) = 2 f 0 (z) − 1, z ∈ U. Since f ∈ An , n ≥ 3, and f is close-to-convex function (univalent) then the function p is analytic in U and p(0) = 1. A simple computation leads to

By differentiating (2.18), we have

2zp0 (z) 1+p(z)

p(z) + 1 p 0 = f (z). 2 +1= Re

If we let ψ : C2 × U → C, ψ(p(z), zp0 (z)) =

∙

zf 00 (z) f 0 (z)

(2.18)

+ 1. Using (2.1), we have

¸ 1 2zp0 (z) +1+ > 0. 1 + p(z) 2γ

2zp0 (z) 1+p(z)

(2.19)

1+2γ 2γ ,

then (2.19) becomes Re ψ(p(z), zp0 (z)) > 0. ³ ´ 2t In order to prove Theorem 2.6, we use Lemma 1.1. For that, we calculate Re ψ(is, t) = Re 1+is + 1+2γ 2γ ³ ´ −n(1+s2 ) 1+2γ 1+2γ 2t = Re 2t(1−is) + + 1+2γ = −2γn+1+2γ = 2γ(1−n)+1 ≤ 0, since n ≥ 3, γ ≥ 1. = 1+s 2 2 + 1+s 2γ 2γ ≤ 1+s2 2γ 2γ p 2γ 1 Using Lemma 1.1, we have Re p(z) > 0, i.e., Re f 0 (z) > 2 . For 0 < γ < 1, n ≥ 3, Theorem 2.6 can be written as the following corollary. ³ 00 ´ p (z) Corollary 2.7 If f ∈ An , n ≥ 3, 0 < γ < 1, satisfy the condition Re zff 0 (z) + 1 > − γ2 , then Re f 0 (z) > 12 , for z ∈ U . +

In differential subordination language Theorem 2.6 can be written as Theorem 2.8 If f ∈ An , n ≥ 3, γ ≥ 1, and satisfy the differential subordination ³ ´ 1 1 − 00 + 1 z γ zf (z) + 1 ≺ , f 0 (z) 1+z p 1 then f 0 (z) ≺ 1+z , for z ∈ U .

(2.20)

References [1] A. Marx, Untersuchungen über schlichte Abbildungen, Math. Ann., 107(1932-1933), 40-67. [2] S. S. Miller, P. T. Mocanu, Differential subordinations and univalent functions, Michig. Math. J., 28(1981), 157-171. [3] S. S. Miller, P. T. Mocanu, Differential subordinations and inequalities in the complex plane, J. of Diff. Eqs., 2(1987), 192-211. [4] S. S. Miller, P. T. Mocanu, Differential subordinations. Theory and applications, Pure and Applied Mathematics, Marcel Dekker, Inc., New York, Basel, 2000. [5] P. T. Mocanu, T. Bulboac˘ a, G. S ¸ t. S˘ al˘ agean, Teoria Geometrica˘ a Func¸tiilor Univalente, Casa C˘ ar¸tii de S ¸ tiin¸ta˘, Cluj-Napoca, 1999. [6] E. Strohhäcker, Beiträge zür Theorie der schhlichten Functionen, Math. Z., 37(1933), 356-380.

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A finite difference method for Burgers’ equation in the unbounded domain using artificial boundary conditions

∗

Quan Zheng†, Yufeng Liu, Lei Fan College of Sciences, North China University of Technology, Beijing 100144, China

Abstract: This paper discusses the numerical solution of one-dimensional Burgers’ equation in the infinite domain. The original problem is converted by Hopf-Cole transformation to the heat equation in the infinite domain, the latter is reduced to an equivalent problem in a finite computational domain with two artificial integral boundary conditions, a finite difference method is constructed for last problem by the method of reduction of order, and therefore the numerical solution of Burgers’ equation is obtained. The method is proved and verified to be uniquely solvable, unconditionally stable and convergent with the order 2 in space and the order 3/2 in time for solving the heat equation as well as Burgers’ equation in the computational domain. Keywords: Burgers’ equation; infinite domain; Hopf-Cole transformation; Artificial boundary condition; Finite difference method

1

Introduction When an analytic solution is not available, or the analytic one is not suitable to be used, a

numerical method is necessary for solving partial differential equations. Therefore, several kinds of exterior problems in the areas of heat transfer, fluid dynamics and other applications were solved numerically by using artificial boundary conditions [1-5]. The artificial boundary methods were established on bounded computational domains for various problems of heat equation on unbounded domains and the feasibility and effectiveness of the methods were shown by the numerical examples [6, 7]. Moreover, for the heat equation in ∗ †

The research is supported by National Natural Science Foundation of China (11471019). E-mail: [email protected] (Q. Zheng).

1

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a semi-unbounded domain [−1, ∞) × [0, ∞), by using an artificial integral boundary condition 1 ux (0, t) = − √ π

∫ 0

t

uλ (0, λ) √ dλ, t−λ

Sun and Wu [8] firstly proved that the finite difference scheme is uniquely solvable, unconditionally stable and convergent with the order 2 in space and the order 3/2 in time under an energy norm. Wu and Zhang [9] also obtained the high-order artificial boundary conditions for the heat equation in unbounded domains, but only proved that the reduced initial-boundary-value problems were stable. Furthermore, Han, Wu and Xu [10] started to consider the nonlinear Burgers’ equation in the unbounded domain as follows: wt + wwx − νwxx = F (x, t), w(x, 0) = f (x), w(x, t) → 0, where ν =

1 Re ,

∀(x, t) ∈ R × (0, T ],

(1.1)

∀x ∈ R,

(1.2)

when |x| → +∞,

∀t ∈ [0, T ],

(1.3)

Re is the Reynolds number, and the given functions F and f are sufficiently

smooth with compact supports supp{F (x, t)} ⊂ [xl , xr ] × [0, T ] and supp{f (x)} ⊂ [xl , xr ]. They obtained nonlinear artificial boundary conditions, constructed a nonlinear difference method with no theoretical convergence analysis, and supported it by numerical examples. Recently, Sun and Wu [11] introduced a function transformation to reduce nonlinear Burgers’ equation to a linear initial boundary value problem, deduced a linear finite difference scheme, and also proved that the finite difference scheme is uniquely solvable, unconditionally stable and convergent with the order 2 in space and 3/2 in time. In this paper, we consider the problem (1.1)-(1.3) with F ≡ 0 and convert it into an initial value problem of heat equation by using Hopf-Cole transformation in the following. Let ∫ ω(x, t) = −

∞

w(y, t)dy,

∀(x, t) ∈ R × (0, T ],

x

we obtain 1 ωt + ωx2 − νωxx = 0, 2 ∫ ∞ ω(x, 0) = − f (y)dy, ∀x ∈ R, x

ω(x, t) → 0,

when |x| → +∞,

∀t ∈ [0, T ].

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Let u = exp(−ω/2ν) − 1, then we have the initial value problem of heat equation: ut − νuxx = 0,

∀(x, t) ∈ R × (0, T ], ∫ ∞ 1 u(x, 0) = ϕ(x) := exp( f (y)dy) − 1, 2ν x u(x, t) → 0,

when |x| → +∞,

(1.4) (1.5)

∀t ∈ [0, T ],

(1.6)

where the sufficiently smooth given function ϕ(x) has compact support supp{ϕ(x)} ⊂ [xl , xr ]. By using artificial linear integral boundary conditions similar to that in [8], we reduce the problem (1.4)-(1.6) to a problem in the bounded computational domain: ut − νuxx = 0,

∀(x, t) ∈ [xl , xr ] × [0, T ],

(1.7)

u(x, 0) = ϕ(x), ∀x ∈ [xl , xr ], ∫ t uλ (xl , λ) 1 √ ux (xl , t) = √ dλ, ∀t ∈ [0, T ], πν 0 t−λ ∫ t 1 uλ (xr , λ) √ ux (xr , t) = − √ dλ, ∀t ∈ [0, T ]. πν 0 t−λ

(1.8) (1.9) (1.10)

In section 2, we construct a finite difference scheme for solving the problem (1.7)-(1.10). Then a new solution of Burgers’ equation is obtained and the difficulty for solving the nonlinear problem is avoided. In section 3, we prove that the finite difference scheme is uniquely solvable, unconditionally stable and convergent with the order 2 in space and 3/2 in time. In section 4, a numerical example confirms the stability and convergence of the finite difference method.

2

The construction of the difference scheme In order to construct the finite difference method, the bounded computational domain is

divided into an M × N uniform mesh. Let h = (xr − xl )/M , xi = xl + ih for 0 ≤ i ≤ M , τ = T /N , tn = nτ for 0 ≤ n ≤ N , r =

ντ , h2

and uni be the numerical solution of u(x, t) at (xi , tn ).

Introduce the notations: 1 uni− 1 = (uni + uni−1 ), 2 2

δx uni− 1 = 2

1 n (u − uni−1 ), h i

n− 21

ui

1 = (uni + un−1 ), i 2

1 n 1 (ui − un−1 ), δx2 uni = 2 (uni+1 − 2uni + uni−1 ), i τ h v v u M u M u ∑ u ∑ n n t n 2 ∥u ∥A = h (ui− 1 ) , ∥δx u ∥ = th (δx uni− 1 )2 . n− 12

δt ui

=

i=1

2

i=1

2

3

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Lemma 2.1 Suppose f (t) ∈ C 2 [0, tn ], then ∫ tn ∫ n ∑ √ 3 f (tk ) − f (tk−1 ) tk dt 1 dt ′ √ − | ≤ (20 2 − 23) max |f ′′ (t)|τ 2 . | f (t) √ 0≤t≤tn τ 12 tn − t tn − t tk−1 0 k=1

Proof

Lemma 2.1 is proved by using

√ √ tn − t − ( tkτ−t tn − tk−1 +

t−tk−1 √ tn τ

− tk ) = 81 (tn −

ξk )− 2 (t − tk−1 )(tk − t) to correct (2.2) and thereupon (2.1) in [8], as corrected in [12]. 3

By introducing a new variable v =

∂u ∂x



to reduce the order of heat equation, the problem

(1.7)-(1.10) is equivalent to the problem of first-order differential equations: ∂u ∂v =ν , ∂x ∂x

∀(x, t) ∈ [xl , xr ] × [0, T ],

(2.1)

∂u = 0, ∂x

∀(x, t) ∈ [xl , xr ] × [0, T ],

(2.2)

v−

u(x, 0) = ϕ(x), xl ≤ x ≤ xr , ∫ t ∂u(xl , λ) 1 1 √ √ dλ, v(xl , t) = ∂λ πν 0 t−λ ∫ t 1 ∂u(xr , λ) 1 √ v(xr , t) = − √ dλ. ∂λ πν 0 t−λ

(2.3) (2.4) (2.5)

Define the grid functions: Uin = u(xi , tn ),

Vin = v(xi , tn ),

0 ≤ i ≤ M,

n ≥ 0.

Using Lemma 2.1, it follows from (2.5) that n ∫ 1 ∑ tk ∂u(xr , λ) dλ n √ VM = − √ ∂λ πν tn − λ k=1 tk−1 ∫ n tk k − U k−1 3 1 ∑ UM dλ M √ = −√ + O(τ 2 ) τ πν tn − λ tk−1 k=1 n 3 2 ∑ k k−1 (UM − UM )an−k + O(τ 2 ) = −√ πν k=1

∑ 3 2 k 0 n (an−k−1 − an−k )UM − an−1 UM ] + O(τ 2 ), = − √ [a0 UM − πν n−1

n = 1, 2, · · · .

k=1

Therefore, we have n− 12

VM

∑ 3 1 n−1 2 n− 1 k− 1 0 n = (VM ] + O(τ 2 ), + VM ) = − √ [a0 UM 2 − (an−k−1 − an−k )UM 2 − an−1 UM 2 πν n−1 k=1

and similarly, n− 1 V0 2

∑ 3 1 2 n− 1 k− 1 = (V0n−1 + V0n ) = √ [a0 U0 2 − (an−k−1 − an−k )U0 2 − an−1 U00 ] + O(τ 2 ). 2 πν n−1 k=1

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Using Taylor expansion, we have n− 1

n− 1

n− 1

δt Ui− 12 − νδx Vi− 1 2 = pi− 12 , 2

2

n− 1

n− 1

n− 1

Vi− 1 2 − δx Ui− 12 = qi− 12 , 2

1 ≤ i ≤ M,

n ≥ 1,

(2.6)

2

2

1 ≤ i ≤ M,

n ≥ 1,

(2.7)

2

0 ≤ i ≤ M,

Ui0 = ϕ(xi ),

(2.8)

∑ 1 2 n− 1 k− 1 = √ [a0 U0 2 − (an−k−1 − an−k )U0 2 − an−1 U00 ] + sn− 2 , πν

n ≥ 1,

(2.9)

∑ 1 2 n− 1 k− 1 0 = − √ [a0 UM 2 − (an−k−1 − an−k )UM 2 − an−1 UM ] + tn− 2 , πν

n ≥ 1,

(2.10)

n− 1 V0 2

n−1 k=1

n− 21

VM

n−1 k=1

where n− 1

|pi− 12 | ≤ c(τ 2 + h2 ), 2

1

n− 1

|qi− 12 | ≤ c(τ 2 + h2 ),

1 ≤ i ≤ M,

n ≥ 1,

(2.11)

2

3

|tn− 2 | ≤ cτ 2 ,

1

3

|sn− 2 | ≤ cτ 2 ,

n ≥ 1,

(2.12)

and c is a constant. Thus, we construct a difference scheme for (2.1)-(2.5) in the following: n− 1

n− 1

δt ui− 12 − νδx vi− 12 = 0, 2

n− 1

n− 1

vi− 12 − δx ui− 12 = 0,

1 ≤ i ≤ M,

u0i = ϕ(xi ),

0 ≤ i ≤ M,

2

n− 12

v0

1 ≤ i ≤ M,

n ≥ 1,

(2.13)

2

n ≥ 1,

(2.14)

2

(2.15)

∑ 2 k− 1 n− 1 (an−k−1 − an−k )u0 2 − an−1 u00 ], = √ [a0 u0 2 − πν n−1

n ≥ 1,

(2.16)

k=1

n− 1 vM 2

∑ 2 k− 1 n− 1 = − √ [a0 uM 2 − (an−k−1 − an−k )uM 2 − an−1 u0M ]. n ≥ 1, πν n−1

(2.17)

k=1

Theorem 2.2 The difference scheme (2.13)-(2.17) is equivalent to the following (2.18)-(2.22): 0 ≤ i ≤ M,

u0i = ϕ(xi ),

1 n− 1 n− 1 n− 1 (δt ui− 12 + δt ui+ 12 ) − νδx2 ui 2 = 0, 2 2 2 n− 12

δt u 1

1 ≤ i ≤ M − 1,

(2.18) n ≥ 1,

∑ 2ν 2 n− 1 k− 1 n− 1 [ √ (a0 u0 2 − (an−k−1 − an−k )u0 2 − an−1 u00 ) − δx u 1 2 ], h πν 2

(2.19)

n−1

+

2

n ≥ 1, (2.20)

k=1

n− 1

δt uM −21 + 2

2ν 2 n− 1 [ √ (a0 uM 2 − h πν

n−1 ∑

k− 12

(an−k−1 − an−k )u0

k=1

n− 1

− an−1 u0M ) + δx uM −21 ],

n ≥ 1, (2.21)

2

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where am = √ Proof

1 1 √ =√ √ √ , tm+1 + tm τ ( m + 1 + m)

m = 0, 1, 2, · · · .

(2.22)

Multiplying (2.13) by 21 h and using (2.14) we obtain n− 21

vi

n− 21

vi

n− 1

= δx ui− 12 + 2

n− 1

= δx ui+ 12 − 2

h n− 1 δt ui− 12 , 2ν 2

h n− 1 δt ui+ 12 , 2ν 2

1 ≤ i ≤ M, 0 ≤ i ≤ M − 1,

n ≥ 1,

(2.23)

n ≥ 1,

(2.24)

From (2.23) and (2.24) for i from 1 to M − 1 we obtain n− 1

δx ui− 12 + 2

h h n− 1 n− 1 n− 1 δt ui− 12 = δx ui+ 12 − δt ui+ 12 , 2ν 2ν 2 2 2

1 ≤ i ≤ M − 1,

n ≥ 1,

or 1 n− 1 n− 1 n− 1 (δt ui− 12 + δt ui+ 12 ) − νδx2 ui 2 = 0, 2 2 2

1 ≤ i ≤ M − 1,

n ≥ 1,

which is (2.19). When i = 0, from (2.16) and (2.24), we know that √ n−1 ∑ h n− 1 2 ν k− 1 n− 1 n− 12 √ [a0 u0 (an−k−1 − an−k )u0 2 − an−1 u00 ] = νδx u 1 2 − δt u 1 2 . − 2 π 2 2 k=1

Dividing by h/2 on the both sides we obtain (2.20). Similarly, when i = M , from (2.17) and (2.23), we know that √ n−1 ∑ 2 ν h n− 1 k− 1 n− 1 n− 1 (an−k−1 − an−k )uM 2 − an−1 u0M ] = νδx uM −21 + δt uM −21 , − √ [a0 uM 2 − 2 π 2 2 k=1

Dividing by h/2 on the both sides we obtain (2.21).



The difference scheme (2.18)-(2.21) can be sorted as the following: 1 1 1 1 n−1 ( −r)uni+1 +(1+2r)uni +( −r)uni−1 = ( +r)un−1 +( +r)un−1 i+1 +(1−2r)ui i−1 , 1 ≤ i ≤ M − 1, 2 2 2 2 (2.25) √ √ 4 r n 4 r (1 + 2r + √ )u0 + (1 − 2r)un1 = (1 − 2r − √ )un−1 + (1 + 2r)un−1 1 π π 0 √ n−1 √ 4 rτ ∑ 8 rτ k−1 k + √ (an−k−1 − an−k )(u0 + u0 ) + √ an−1 u00 , (2.26) π π k=1 √ √ 4 r 4 r (1 + 2r + √ )unM + (1 − 2r)unM −1 = (1 − 2r − √ )un−1 + (1 + 2r)un−1 M −1 π π M √ n−1 √ 4 rτ ∑ 8 rτ √ + √ (an−k−1 − an−k )(ukM + uk−1 ) + an−1 u0M . (2.27) M π π k=1

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3

The error estimate of the difference scheme

Lemma 3.1 For any F = {F1 , F2 , F3 , · · · }, we have n l−1 n ∑ ∑ 1 ∑ 2 [a0 Fl − (al−k−1 − al−k )Fk ]Fl ≥ √ Fl , 2 tn l=1

k=1

n = 1, 2, · · · ,

l=1

where am is defined in (2.22). Proof

Let bm = am−1 − am =

1 √ √1 ( √ τ m+ m−1

−

√

1 √ ), m+1+ m

m ≥ 1, then bm > 0, and

n l−1 ∑ ∑ [a0 Fl − (al−k−1 − al−k )Fk ]Fl

= ≥

l=1 n ∑ l=1 n ∑

a0 Fl2 − a0 Fl2 −

l=1

=

=

n ∑ l=1 n ∑

a0 Fl2 − a0 Fl2 −

l=1

≥

n ∑

k=1 n ∑ l−1 ∑

(am−1 − am )Fl−m Fl

l=1 m=1 n ∑ l−1 ∑

1 2

l=1 m=1

n l−1 1∑∑

2 1 2

a0 Fl2 − (

1∑∑ bm Fl2 2 n

2 bl−m Fm −

l=1 m=1 n n ∑ ∑

2 bl−m Fm −

bm )

n ∑

l−1

l=1 m=1 n ∑ l−1 ∑

m=1 l=m+1

n−1 ∑

m=1

l=1

2 bm (Fl−m + Fl2 )

1 2

bm Fl2

l=1 m=1

Fl2

l=1

n ∑ 1 1 1 √ = [ √ − √ (1 − √ )] Fl2 τ τ n + n − 1 l=1

≥

n 1 ∑ 2 √ Fl . 2 tn



l=1

Lemma 3.2 Suppose {uni } be the solution of n− 1

n− 1

n− 1

δt ui− 12 − νδx vi− 12 = Pi− 12 , 2

2

n− 1

n− 1

n− 1

vi− 12 − δx ui− 12 = Qi− 12 , 2

2

n ≥ 1,

(3.1)

1 ≤ i ≤ M,

n ≥ 1,

(3.2)

2

0 ≤ i ≤ M,

u0i = ϕ(xi ), n− 1 v0 2

1 ≤ i ≤ M,

2

(3.3)

∑ 1 2 n− 1 k− 1 = √ [a0 u0 2 − (an−k−1 − an−k )u0 2 − an−1 u00 ] + S n− 2 , πν n−1

n ≥ 1,

(3.4)

k=1

n− 1 vM 2

∑ 1 2 n− 1 k− 1 = − √ [a0 uM 2 − (an−k−1 − an−k )uM 2 − an−1 u0M ] + T n− 2 . πν n−1

n ≥ 1,

(3.5)

k=1

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where Supp{ϕ(x)} ⊂ [x0 , xM ], then

2T ∥un ∥2A ≤ exp( 4−τ )·

+2τ Proof

1 1− τ4

∑n

l=1 (∥P

{∥u0 ∥2A +

l− 21

√ πνtn 2 τ l− 12

∥2A + ∥Q

∑n

l=1 [(T

∥2A )},

l− 12 2 )

1

+ (S l− 2 )2 ] (3.6)

n = 1, 2, · · · .

n− 1

n− 1

Multiplying (3.1) by 2ui− 12 and multiplying (3.2) by 2vi− 12 , then adding the results, 2

2

we have 1 n 2 τ [(ui− 1 ) 2

n− 21

≤ h2 (ui

n− 21 n− 12 vi

n− 1

n− 1 n− 1

− (un−1 )2 ] + 2(vi− 12 )2 = h2 (ui i− 1 2

n− 12

vi

2

n− 12

n− 12

− ui−1 vi−1 ) + 12 (u

n− 12 i− 21

)2 + 2(P

n− 1

n− 1

n− 1

n− 1

− ui−12 vi−12 ) + 2ui− 12 Pi− 12 + 2vi− 12 Qi− 12 2

n− 12 i− 12

n− 21 i− 12

)2 + 12 (v

)2 + 2(Q

2

n− 21 i− 12

2

2

)2 ,

1 ≤ i ≤ M, n ≥ 1.

(3.7)

Multiplying the above inequality by τ h and summing up for i from 1 to M , we obtain n− 1 n− 12

1

(∥un ∥2A − ∥un−1 ∥2A ) + 2τ ∥v n− 2 ∥2A ≤ 2τ (uM 2 vM

n− 21 n− 12 v0 )

− u0

1

1 1 τ τ + ∥un− 2 ∥2A + ∥v n− 2 ∥2A 2 2

1

+2τ ∥P n− 2 ∥2A + 2τ ∥Qn− 2 ∥2A ,

n ≥ 1.

(3.8)

1

Noticing τ2 ∥un− 2 ∥2A ≤ τ4 (∥un ∥2A + ∥un−1 ∥2A ), thus τ + (∥ul ∥2A + ∥ul−1 ∥2A ) 4 l− 12 2 l− 12 2 +2τ ∥P ∥A + 2τ ∥Q ∥A , l = 1, 2, . . . , n. l− 1 l− 1

l− 21 l− 12 v0 )

∥ul ∥2A − ∥ul−1 ∥2A ≤ 2τ (uM 2 vM 2 − u0

Summing up for l from 1 to n, we have ∥un ∥2A

≤

∥u0 ∥2A

+ 2τ

n ∑

l− 1 l− 1

l− 12 l− 12 v0 )

(uM 2 vM 2 − u0

l=1 n−1 n ∑ 1 1 τ∑ l 2 τ + ∥un ∥2A + ∥u ∥A + 2τ (∥P l− 2 ∥2A + ∥Ql− 2 ∥2A ). 4 2 l=0

l=1

Substituting (3.4) and (3.5) into the above inequality, and using Lemma 3.1, we have ∥un ∥2A ≤

∑ l− 1 l− 1 ∑ 1 1 1 l− 12 l− 12 0 2 2 2 [∥u ∥ + 2τ (u v − u v ) + 2τ (∥P l− 2 ∥2A + ∥Ql− 2 ∥2A ) A 0 0 τ M M 1− 4 +

τ 2

n−1 ∑

n

n

l=1

l=1

∥ul ∥2A ]

l=0

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∑ 1 2τ 2 ∑ l− 1 k− 1 l− 1 0 2 ) [a0 uM 2 − (al−k−1 − al−k )uM 2 ]uM 2 τ ∥u ∥A + τ · (− √ 1− 4 1− 4 πν n

=

l−1

l=1

+

+ ≤

2τ 1 − τ4 2τ 1 − τ4

n ∑

l− 1

1

uM 2 T l− 2 −

l=1 n ∑

l− 12

u0

1

S l− 2 +

l=1

k=1

2 2τ · (√ ) 1 − τ4 πν 2τ 1 − τ4

n ∑

l− 12

[a0 u0

−

l−1 ∑

k− 21

(al−k−1 − al−k )u0

l− 12

]u0

k=1

l=1

n n−1 ∑ 1 1 2τ ∑ l 2 (∥P l− 2 ∥2A + ∥Ql− 2 ∥2A ) + ∥u ∥A 4−τ l=1

l=0

n ∑

1 2τ 2 1 2 τ l− 1 0 2 · √ (uM 2 )2 + τ ∥u ∥A − τ · √ τ (√ 1− 4 1− 4 1− 4 πν 2 tn πνtn l=1 √ n n 2 πνtn ∑ l− 1 2 2τ 1 ∑ l− 21 2 √ √ (T 2 ) ) − · (u0 ) · + 2 1 − τ4 πν 2 tn l=1 l=1 √ n n ∑ ∑ 1 1 τ 2 πνtn l− + (u0 2 )2 + (S l− 2 )2 ) τ (√ 1− 4 2 πνtn l=1

n ∑

l− 1

(uM 2 )2

l=1

l=1

n n−1 2τ ∑ 2τ ∑ l 2 l− 12 2 l− 12 2 + (∥P ∥ + ∥Q ∥ ) + ∥u ∥A A A 1 − τ4 4−τ l=1 l=0 √ n 1 τ πνtn ∑ l− 1 2 1 0 2 ∥u ∥ + [(T 2 ) + (S l− 2 )2 ] ≤ A τ τ 1− 4 1− 4 2 l=1

2τ + 1 − τ4

n ∑

1 (∥P l− 2 ∥2A

+

1 ∥Ql− 2 ∥2A )

l=1

n−1 2τ ∑ l 2 ∥u ∥A , + 4−τ

n = 1, 2, · · · .

l=0



Using Gronwall’s lemma, we can obtain (3.6).

Theorem 3.3 The difference scheme (2.18)-(2.22) is uniquely solvable. Proof

By Theorem 2.2, it suffices to prove that the difference scheme (2.13)-(2.17) is solvable u-

niquely. When initial value is homogeneous, by Lemma 3.2, we have ∥un ∥2A = 0, n = 1, 2, · · · .



Theorem 3.4 Let {uni |0 ≤ i ≤ M, n ≥ 1} be the solution of (2.18)-(2.22), then ∥un ∥2A ≤ Proof

2T exp( 4−τ ) 0 2 τ ∥u ∥A , 1− 4

n = 1, 2, · · · .

(3.9)

From Theorem 2.2, it suffices to prove that (3.9) hold for the difference scheme (2.13)-

(2.17). Therefore, (3.9) follows directly from Lemma 3.2.



4,3 Theorem 3.5 Suppose (1.4)-(1.6) have solution u(x, t) ∈ Cx,t (R × [0, T ]). Let {uni } be the

solution of (2.18)-(2.22), and let u ˜ni = Uin − uni , then ∥˜ un ∥2A ≤

3 2T CT √ ( πνT + 4) exp( )(τ 2 + h2 )2 , 4−τ 4−τ

n = 1, 2, · · · , [T /τ ],

(3.10)

where C is a constant independent of τ and h.

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Proof

Subtracting (2.13)-(2.17) from (2.6)-(2.10), respectively, we obtain the error equations: n− 1

n− 1

n− 1

δt u ˜i− 12 − νδx v˜i− 12 = pi− 12 , 2

2

n− 1

n− 1

n− 1

v˜i− 12 − δx u ˜i− 12 = qi− 12 , 2

2

n− 12

n ≥ 1,

(3.11)

1 ≤ i ≤ M,

n ≥ 1,

(3.12)

2

0 ≤ i ≤ M,

u ˜0i = 0, v˜0

1 ≤ i ≤ M,

2

(3.13)

∑ 1 2 n− 1 k− 1 = √ [a0 u ˜0 2 − (an−k−1 − an−k )˜ u0 2 − an−1 u ˜00 ] + sn− 2 , πν n−1

n ≥ 1,

(3.14)

k=1

n− 21

v˜M

∑ 1 2 n− 1 k− 1 = − √ [a0 u ˜M 2 − (an−k−1 − an−k )˜ uM 2 − an−1 u ˜0M ] + tn− 2 , πν n−1

n ≥ 1.

(3.15)

k=1

Using Lemma 3.2 and applying (2.11), (2.12) and (3.13), we obtain √ n 1 2T 1 πνtn ∑ l− 1 2 0 2 n 2 · {∥˜ u ∥ + ∥˜ u ∥A = exp( )· τ [(t 2 ) + (sl− 2 )2 ] A τ 4−τ 1− 4 2 l=1

+2τ

n ∑

1

1

(∥pl− 2 ∥2 + ∥q l− 2 ∥2 )}

l=1

≤

3 CT √ 2T ( πνT + 4) exp( )(τ 2 + h2 )2 , 4−τ 4−τ

n = 1, 2, · · · , [T /τ ]. 

Theorem 3.5 shows that the convergence order of (2.18)-(2.21) is 2 in space and 3/2 in time for the problem (1.7)-(1.10). Finally, the numerical solution of Burgers’ equation is obtained by win = −

ν uni+1 − uni−1 , h 1 + uni

(3.16)

which keeps the corresponding unique solvability, unconditional stability and convergence.

4

The numerical example 8νx(x −9) For the problem of Burgers’ equation with an initial condition f (x) = − (x 2 −9)2 +1 in the 2

support [xl , xr ] = [−3, 3], the exact solution is w(x, t) = 2ν

∫3

(x−ξ)2 x−ξ 2 2 −3 2νt (ξ −9) exp(− 4νt )dξ ∫3 (x−ξ)2 1+ √1 (ξ 2 −9)2 exp(− 4νt )dξ 2 πνt −3

√1 2 πνt

. The

numerical solutions are obtained by the proposed scheme, then the convergence order w.r.t h is shown in Table 1, and the convergence order w.r.t τ is shown in Table 2. Table 1. Convergence w.r.t. h of the problem for T = 1, ν = 0.1, τ = 0.01 and τ = h4/3 respectively. M 50 100 200 400 800

N 100 100 100 100 100

L∞ -error 2.2705e-3 6.0651e-4 1.6444e-4 5.0024e-5 3.0569e-5

order — 1.9044 1.8830 1.7169 0.7106

L2 -error 2.0737e-3 5.5643e-4 1.4962e-4 4.5653e-5 1.9714e-5

order — 1.8979 1.8949 1.7125 1.2115

N 9 22 54 137 345

L∞ -error 3.1455e-3 7.6893e-4 1.8419e-4 4.5577e-5 1.1295e-5

order — 2.0324 2.0617 2.0148 2.0126

L2 -error 2.5729e-3 6.5174e-4 1.6620e-4 4.1607e-5 1.0393e-5

order — 1.9810 1.9714 1.9980 2.0012

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Table 2. Convergence w.r.t. τ of the problem for T = 1, ν = 0.1, h = 0.002 and h = τ 3/4 respectively. N 20 40 80 160 320

5

M 3000 3000 3000 3000 3000

L∞ -error 1.0398e-3 3.6910e-4 1.0386e-4 2.6518e-5 1.5322e-5

order — 1.4942 1.8294 1.9696 0.7914

L2 -error 2.1342e-4 6.2138e-5 1.8884e-5 6.3713e-6 2.6822e-6

order — 1.7801 1.7183 1.5675 1.2482

M 95 159 267 450 757

L∞ -error 8.7265e-4 2.9735e-4 1.0258e-4 3.5936e-5 1.2623e-5

order — 1.5532 1.5354 1.5132 1.5094

L2 -error 7.2610e-4 2.6197e-4 9.3238e-5 3.2868e-5 1.1614e-5

order — 1.4708 1.4904 1.5042 1.5008

Conclusions In this works, a new finite difference method for Burgers’ equation in the unbounded domain

is presented by (2.18), (2.25)-(2.27) and (3.16) succinctly. The inequality in Lemma 2.1 is slightly stronger than Lemma 1 in [8]. Lemma 3.2 is proved by using Gronwall’s lemma, but for heat equation in the semi-infinite domain, similar Lemma 4 in [8], i.e. Lemma 3.2.4 in [12], was incorrectly proved by not using Gronwall’s lemma, and the lemma can be modified and proved as Lemma 3.2. Finally, the proposed method is clearly proved and verified to be uniquely solvable, unconditionally stable and convergent with the order 2 in space and the order 3/2 in time to solve Burgers’ equation in the unbounded domain.

References [1] B. Enquist, A. Majda, Absorbing boundary conditions for numerical simulation of waves, Math. Comput. 31 (1977) 629-651. [2] K. Feng, Asymptotic radiation conditions for reduced wave equations, J. Comp. Math. 2 (1984) 130-138. [3] D.-H. Yu, Natural Boundary Integral Method and Its Applications, Beijing/Dordrecht/New York/London: Kluwer Academic Publisher/Science Press, 2002. [4] J.M. Strain, Fast adaptive methods for the free-space heat equation, SIAM J. Sci. Comput. 15 (1992) 185-206. [5] D. Givoli, Numerical Methods for Problem in Infinite Domains, Elsevier, Amsterdam, 1992. [6] H.-D. Han, Z.-Y. Huang, A class of artificial boundary conditions for heat equation in unbounded domains, Comput. Math. Appl. 43 (2002) 889-900. [7] H.-D. Han, Z.-Y. Huang, Exact and approximating boundary conditions for the parabolic problems on unbounded domains, Comput. Math. Appl. 44 (2002) 655-666. [8] X.-N. Wu, Z.-Z. Sun, Convergence of difference scheme for heat equation in unbounded domains using artificial boundary conditions, Appl. Numer. Math. 50 (2004) 261-277. [9] X.-N. Wu, J.-W. Zhang, High-order local absorbing boundary conditions for heat equation in unbounded domains, J. Comput. Math. 29 (2011) 74-90. [10] H.-D. Han, X.-N. Wu, Z.-L. Xu, Artificial boundary method for Burgers’ equation using nonlinear boundary conditions, J. Comput. Math. 24 (2006) 295-304. [11] Z.-Z. Sun, X.-N. Wu, A difference scheme for Burgers equation in an unbounded domain, Appl. Math. Comput. 209 (2009) 285-304. [12] H.-D. Han, X.-N. Wu, Artificial Boundary Method, Beijing: Tsinghua University Press/Springer Press, 2012.

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Barnes-type Peters polynomials associated with poly-Cauchy polynomials of the second kind Dae San Kim Department of Mathematics, Sogang University Seoul 121-742, Republic of Korea [email protected]

Taekyun Kim Department of Mathematics, Kwangwoon University Seoul 139-701, Republic of Korea [email protected]

Takao Komatsu Graduate School of Science and Technology, Hirosaki University Hirosaki 036-8561, Japan [email protected]

Hyuck In Kwon Department of Mathematics, Kwangwoon University Seoul 139-701, Republic of Korea [email protected]

Sang-Hun Lee Division of General Education, Kwangwoon University Seoul 139-701, Republic of Korea [email protected] MR Subject Classifications: 05A15, 05A40, 11B68, 11B75, 65Q05

Abstract In this paper, by considering Barnes-type Peters polynomials of the second kind as well as poly-Cauchy polynomials of the second kind, we define and investigate the mixed-type polynomials of these polynomials. From the properties of Sheffer sequences of these polynomials arising from umbral calculus, we derive new and interesting identities.

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1

Introduction

In this paper, we consider the polynomials sb(k) b(k) b(k) n (x) = s n (x|λ; µ) = s n (x|λ1 , . . . , λr ; µ1 , . . . , µr ) called the Barnes-type Peters of the second kind and poly-Cauchy of the second kind mixed-type polynomials, whose generating function is given by µj r  ∞ Y X
(1 + t)λj tn x (k) , (x|λ , . . . , λ ; µ , . . . , µ ) Lif − ln(1 + t) (1 + t) = s b 1 r 1 r k n λj 1 + (1 + t) n! n=0 j=1 (1) where λ1 , . . . , λr , µ1 , . . . , µr ∈ C with λ1 , . . . , λr 6= 0. Here, Lif k (x) (k ∈ Z) is the polyfactorial function ([8]) defined by ∞ X xm . Lif k (x) = m!(m + 1)k m=0 (k)

(k)

(k)

(k)

When x = 0, sbn = sbn (0) = sbn (0|λ; µ) = sbn (0; λ1 , . . . , λr ; µ1 , . . . , µr ) are called the the Barnes-type Peters of the second kind and poly-Cauchy of the second kind mixed-type numbers. Recall that the Barnes-type Peters polynomials of the second kind, denoted by sbn (x|λ1 , . . . , λr ; µ1 , . . . , µr ), are given by the generating function as µj ∞ r  X Y tn (1 + t)λj x (1 + t) = s b (x|λ , . . . , λ ; µ , . . . , µ ) . n 1 r 1 r 1 + (1 + t)λj n! n=0 j=1 If r = 1, then sbn (x|λ; µ) are the Peters polynomials of the second kind. Peters polynomials were mentioned in [12, p.128] and have been investigated in e.g. [7]. (k) The poly-Cauchy polynomials of the second kind, denoted by b cn (x) ([6, 9]), are given by the generating function as ∞ X
tn x b c(k) (x) . Lif k − ln(1 + t) (1 + t) = n n! n=0 The generalized Barnes-type Euler polynomials En (x|λ1 , . . . , λr ; µ1 , . . . , µr ) are defined by the generating function µj r  ∞ Y X tn 2 xt e = E (x|λ , . . . , λ ; µ , . . . , µ ) . n 1 r 1 r λj t 1 + e n! n=0 j=1 If µ1 = · · · = µr = 1, then En (x|λ1 , . . . , λr ) = En (x|λ1 , . . . , λr ; 1, . . . , 1) are called (r) the Barnes-type Euler polynomials. If further λ1 = · · · = λr = 1, then En (x) = En (x|1, . . . , 1; 1, . . . , 1) are called the Euler polynomials of order r. In this paper, by considering Barnes-type Peters polynomials of the second kind as well as poly-Cauchy polynomials of the second kind, we define and investigate the mixedtype polynomials of these polynomials. From the properties of Sheffer sequences of these polynomials arising from umbral calculus, we derive new and interesting identities. 2

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2

Umbral calculus

Let C be the complex number field and let F be the set of all formal power series in the variable t: ) ( ∞ X ak k t ak ∈ C . (2) F = f (t) = k! k=0

∗

Let P = C[x] and let P be the vector space of all linear functionals on P. hL|p(x)i is the action of the linear functional L on the polynomial p(x), and we recall that the vector space operations on P∗ are defined by hL + M |p(x)i = hL|p(x)i + hM |p(x)i, hcL|p(x)i = c hL|p(x)i, where c is a complex constant in C. For f (t) ∈ F, let us define the linear functional on P by setting hf (t)|xn i = an , (n ≥ 0). (3) In particular,

k n t |x = n!δn,k

(n, k ≥ 0),

(4)

where δn,k is the Kronecker’s symbol. P hL|xk i k For fL (t) = ∞ t , we have hfL (t)|xn i = hL|xn i. That is, L = fL (t). The map k=0 k! 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 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)(6
= 0) is the smallest integer k k for which the coefficient of t does not vanish. If O f (t) = 1, then f (t) is called a delta
series;
if O f (t) = 0, then f (t) is called an invertible series. For f (t), g(t) ∈ F with
O f (t) = 0, there exists a unique sequence sn (x) (deg sn (x) = n) such

= 1 and O g(t)  that g(t)f (t)k |sn (x)
= n!δn,k , for n, k ≥ 0. Such a sequence s
n (x) is called the Sheffer sequence for g(t), f (t) which is denoted by sn (x) ∼ g(t), f (t) , (see [1, 4-12]). For f (t), g(t) ∈ F and p(x) ∈ P, we have hf (t)g(t)|p(x)i = hf (t)|g(t)p(x)i = hg(t)|f (t)p(x)i and

∞ X

 tk f (t)|xk , f (t) = k! k=0

p(x) =

∞ X

tk |p(x)

k=0

(5)

 xk k!

(6)

([12, Theorem 2.2.5]). Thus, by (6), we get tk p(x) = p(k) (x) =

dk p(x) dxk

and eyt p(x) = p(x + y).

(7)

Sheffer sequences are characterized in the generating function ([12, Theorem 2.3.4]).
Lemma 1 The sequence sn (x) is Sheffer for g(t), f (t) if and only if ∞

X sk (y) 1 ¯
eyf (t) = tk k! g f¯(t) k=0

(y ∈ C) ,

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where f¯(t) is the compositional inverse of f (t).
For sn (x) ∼ g(t), f (t) , we have the following equations ([12, Theorem 2.3.7, Theorem 2.3.5, Theorem 2.3.9]): f (t)sn (x) = nsn−1 (x) (n ≥ 1), n X 1 D ¯
−1 ¯ j n E j g f (t) f (t) |x x , sn (x) = j! j=0 n   X n sn (x + y) = sj (x)pn−j (y) , j j=0

(8) (9) (10)

where pn (x) = g(t)sn (x).

Assume that pn (x) ∼ 1, f (t) and qn (x) ∼ 1, g(t) . Then the transfer formula ([12, Corollary 3.8.2]) is given by n  f (t) x−1 pn (x) (n ≥ 1). qn (x) = x g(t)

For sn (x) ∼ g(t), f (t) and rn (x) ∼ h(t), l(t) , assume that sn (x) =

n X

Cn,m rm (x) (n ≥ 0) .

m=0

Then we have ([12, p.132]) Cn,m

3

1 = m!

*

h g

+

m n f¯(t)
l f¯(t) x . f¯(t)

(11)

Main results (k)

From the definition (1), sbn (x|λ1 , . . . , λr ; µ1 , . . . , µr ) is the Sheffer sequence for the pair g(t) =

µ r  Y 1 + e λj t j j=1

eλj t

1 Lif k (−t)

and f (t) = et − 1.

So, sb(k) n (x|λ1 , . . . , λr ; µ1 , . . . , µr ) ∼

µ r  Y 1 + eλj t j j=1

e λj t

! 1 , et − 1 . Lif k (−t)

(12)

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3.1

Explicit expressions

Let (n)j = n(n − 1) · · · (n − j + 1) (j ≥ 1) with (n)0 = 1. The (signed) Stirling numbers of the first kind S1 (n, m) are defined by (x)n =

n X

S1 (n, m)xm .

m=0

Theorem 1 Let λµ =

Pr

j=1

λj µj . Then, we have

sb(k) n (x|λ1 , . . . , λr ; µ1 , . . . , µr )
n X m X P (−1)l ml − rj=1 µj S1 (n, m)Em−l (x + λµ|λ1 , . . . , λr ; µ1 , . . . , µr ) =2 k (l + 1) m=0 l=0 n X n   X n (k) = S1 (l, j)b sn−l xj l j=0 l=j n−j l    n X X X n l (k) = S1 (n − l, j)b ci sbl−i (λ1 , . . . , λr ; µ1 , . . . , µr )xj l i j=0 l=0 i=0   n X n (k) = sbn−l (λ1 , . . . , λr ; µ1 , . . . , µr )b cl (x) , l l=0   n X n (k) = sbl (x|λ1 , . . . , λr ; µ1 , . . . , µr )b cn−l . l l=0

(13) (14)

(15) (16) (17)

Proof. Since µ r  Y 1 + eλj t j j=1

e λj t

1 sb(k) (x|λ1 , . . . , λr ; µ1 , . . . , µr ) ∼ (1, et − 1) Lif k (−t) n

(18)

and (x)n ∼ (1, et − 1) ,

(19)

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we have sb(k) n (x|λ1 , . . . , λr ; µ1 , . . . , µr ) µj r  Y eλj t = Lif k (−t)(x)n λj t 1 + e j=1 µj n r  X Y e λj t Lif k (−t)xm = S1 (n, m) λj t 1 + e m=0 j=1 µj X n r  m X Y e λj t (−1)l tl m = S1 (n, m) x λj t k 1 + e l!(l + 1) m=0 j=1 l=0
r  µj n m l m Y X X (−1) l eλj t = S1 (n, m) xm−l λj t k (l + 1) 1 + e m=0 j=1 l=0
µj n m r  X X Pr (−1)l ml λµt Y 2 − j=1 µj S1 (n, m) =2 e xm−l λj t k (l + 1) 1 + e m=0 j=1 l=0
n m m l X X Pr (−1) l Em−l (x + λµ|λ1 , . . . , λr ; µ1 , . . . , µr ) . = 2− j=1 µj S1 (n, m) k (l + 1) m=0 l=0 So, we get (13). By (9) with (12), we get E D
−1 j n ¯ ¯ g f (t) f (t) |x * r  + µj Y

j n (1 + t)λj = Lif k − ln(1 + t) ln(1 + t) x 1 + (1 + t)λj j=1 + * r  µj ∞ Y
X tl n (1 + t)λj S1 (l, j) x Lif k − ln(1 + t) j! = λj 1 + (1 + t) l! j=1 l=j * + µj n   r  λj X Y
n (1 + t) = j! S1 (l, j) Lif k − ln(1 + t) xn−l λ j l 1 + (1 + t) j=1 l=j + * n   ∞ i X X n (k) t n−l = j! S1 (l, j) sbi x i! l i=0 l=j n   X n (k) = j! S1 (l, j)b sn−l . l l=j

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On the other hand, D E
−1 j n ¯ ¯ g f (t) f (t) |x * r  + µj n   X Y
n−l n (1 + t)λj = j! S1 (l, j) Lif k − ln(1 + t) x 1 + (1 + t)λj l j=1 l=j + * r  µj  n   n−l  λj X X Y n n − l (k) (1 + t) n−l−i = j! S1 (l, j) b ci x λ j 1 + (1 + t) l i i=0 j=1 l=j + * ∞  n   n−l  X X n n − l (k) X tm n−l−i = j! S1 (l, j) b ci sbm (λ1 , . . . , λr ; µ1 , . . . , µr ) x m! l i m=0 i=0 l=j  n X n−l   X n n−l (k) = j! S1 (l, j)b ci sbn−l−i (λ1 , . . . , λr ; µ1 , . . . , µr ) . l i l=j i=0 Thus, we obtain sb(k) n (x)

=

n X n   X n j=0 l=j

l

(k)

S1 (l, j)b sn−l xj

n−j l    n X X X n l (k) = S1 (n − l, j)b ci sbl−i (λ1 , . . . , λr ; µ1 , . . . , µr )xj , l i j=0 l=0 i=0

which are the identities (14) and (15).

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Next, sb(k) n (y|λ1 , . . . , λr ; µ1 , . . . , µr ) =

*∞ X

ti (k) sbi (y|λ1 , . . . , λr ; µ1 , . . . , µr ) xn

+

i! * i=0 + µj r  Y
(1 + t)λj = Lif k − ln(1 + t) (1 + t)y xn λj 1 + (1 + t) j=1 * r  + µj Y
(1 + t)λj = Lif k − ln(1 + t) (1 + t)y xn λj 1 + (1 + t) j=1 + * r  µj X ∞ Y (1 + t)λj tl n (k) = b cl (y) x 1 + (1 + t)λj l! j=1 l=0 + * µj n   r  X Y n (k) (1 + t)λj n−l = b cl (y) x λj 1 + (1 + t) l j=1 l=0 *∞ +   n X n (k) X ti n−l = b cl (y) sbi (λ1 , . . . , λr ; µ1 , . . . , µr ) x i! l i=0 l=0   n X n (k) = b cl (y)b sn−l (λ1 , . . . , λr ; µ1 , . . . , µr ) . l l=0

Thus, we obtain (16). Finally, we obtain that + i t (k) sb(k) sbi (y|λ1 , . . . , λr ; µ1 , . . . , µr ) xn n (y|λ1 , . . . , λr ; µ1 , . . . , µr ) = i! * i=0 + µj r  Y
(1 + t)λj Lif k − ln(1 + t) (1 + t)y xn = λj 1 + (1 + t) j=1 + * µj r 
Y (1 + t)λj (1 + t)y xn = Lif k − ln(1 + t) λ j 1 + (1 + t) j=1 * + ∞
X tl n = Lif k − ln(1 + t) sbl (y|λ1 , . . . , λr ; µ1 , . . . , µr ) x l! l=0   n X
n−l E n D = sbl (y|λ1 , . . . , λr ; µ1 , . . . , µr ) Lif k − ln(1 + t) x l l=0 +   *X n ∞ i X n (k) t n−l = sbl (y|λ1 , . . . , λr ; µ1 , . . . , µr ) b ci x l i! i=0 l=0 n   X n (k) = sbl (y|λ1 , . . . , λr ; µ1 , . . . , µr )b cn−l . l l=0 *

∞ X

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Thus, we get the identity (17).

3.2

Sheffer identity

Theorem 2 sb(k) n (x

+ y|λ1 , . . . , λr ; µ1 , . . . , µr ) =

n   X n j=0

j

(k)

sbj (x|λ1 , . . . , λr ; µ1 , . . . , µr )(y)n−j .

(20)

Proof. By (12) with pn (x) =

µ r  Y 1 + eλj t j j=1

eλj t

1 sbn (x|λ1 , . . . , λr ; µ1 , . . . , µr ) Lif k (−t)

= (x)n ∼ (1, et − 1) , using (10), we have (20).

3.3

Difference relations

Theorem 3 sb(k) b(k) n (x + 1|λ1 , . . . , λr ; µ1 , . . . , µr ) − s n (x|λ1 , . . . , λr ; µ1 , . . . , µr ) (k)

= nb sn−1 (x|λ1 , . . . , λr ; µ1 , . . . , µr ) . (21) Proof. By (8) with (12), we get (k)

(et − 1)b s(k) sn−1 (x|λ1 , . . . , λr ; µ1 , . . . , µr ) . n (x|λ1 , . . . , λr ; µ1 , . . . , µr ) = nb By (7), we have (21).

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3.4

Recurrence

Theorem 4 (k)

sbn+1 (x|λ1 , . . . , λr ; µ1 , . . . , µr ) = (x + λµ)b s(k) n (x − 1|λ1 , . . . , λr ; µ1 , . . . , µr )
n X m X r X P (−1)m−l ml −1− rj=1 µj −2 S1 (n, m) µ λ E (x + λ(µ + ei ) − 1|λ; µ + ei ) k i i l (m − l + 1) m=0 l=0 i=1
n X m m−l m X Pr (−1) l S (n, m)El (x + λµ − 1|λ; µ) (22) − 2− j=1 µj k 1 (m − l + 2) m=0 l=0 = (x + µλ)b s(k) n (x − 1|λ1 , . . . , λr ; µ1 , . . . , µr )   n−j r   n 1 XXX n x + λi − 1 (k) j+1 µi λi S1 (n − l, j)b s l Ej − 2 j=0 l=0 i=1 l λi
n X m X P (−1)m−l ml − rj=1 µj −2 S (n, m)El (x + λµ − 1|λ; µ) , k 1 (m − l + 2) m=0 l=0 λµ =

Pr

j=1

(23)

λi µi .

Remark. Comparing (22) and (23),
r n X m X X
(−1)m−l ml 2 µ λ S (n, m)E x + λ(µ + e ) − 1|λ; µ + e i i 1 l i i (m − l + 1)k m=0 l=0 i=1   n−j r   n X X X n x + λi − 1 (k) j+1 . µi λi S1 (n − l, j)b s l Ej = λi l j=0 l=0 i=1 −

Pr

j=1 µj

Proof. By applying

  g 0 (t) 1 sn+1 (x) = x − sn (x) 0 g(t) f (t)

(24)

([12, Corollary 3.7.2]) with (12), we get (k)

sbn+1 (x|λ1 , . . . , λr ; µ1 , . . . , µr ) −t = xb s(k) n (x − 1|λ1 , . . . , λr ; µ1 , . . . , µr ) − e

g 0 (t) (k) sb (x|λ1 , . . . , λr ; µ1 , . . . , µr ) . g(t) n

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Since
0 g 0 (t) = ln g(t) g(t) !0 X  r r X = µi ln(1 + eλi t ) − µi λi t − ln Lif k (−t) i=1

=

i=1

r X µi λi eλi t i=1

1 + e λi t

−

r X

µi λi +

i=1

Lif 0k (−t) , Lif k (−t)

by (13), we have g 0 (t) (k) sb (x) g(t) n ! r r X µi λi eλi t X Lif 0k (−t) sb(k) = − µi λi + n (x) λ t i 1 + e Lif (−t) k i=1 i=1
r µj n X m r  X Y Pr (−1)l ml X 2 2 −1− j=1 µj (λµ+λi )t S1 (n, m) =2 µi λi e xm−l λj t k λi t (l + 1) 1 + e 1 + e m=0 l=0 i=1 j=1 µj r  n λj t Y X e S1 (n, m) Lif 0k (−t)xm . (25) − λµb s(k) n (x) + λj t 1 + e m=0 j=1 The first term in (25) is P −1− rj=1 µj

2

n X m X r X m=0 l=0 i=1



(−1)l ml S1 (n, m) µ λ E x + λ(µ + e )|λ; µ + e , i i m−l i i (l + 1)k

where λ = (λ1 , . . . , λr ), µ = (µ1 , . . . , µr ) and ei = (0, . . . , 0, 1, 0, . . . , 0) (i = 1, 2, . . . , r). | {z } | {z } i−1

Since

 Lif k−1 (−t) − Lif k (−t) =

1 1 − k−1 k 2 2

r−i

 t + ··· ,

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the third term in (25) is −

2

Pr

j=1 µj

n X

S1 (n, m)e

λµt

Lif 0k (−t)

m=0

= 2−

Pr

j=1

= −2− −

Pr

×

j=1

n X

S1 (n, m)eλµt

m=0 n X j=1 µj

Pr

= −2

µj

l=0

2 1 + eλj t

µj

xm

Lif k−1 (−t) − Lif k (−t) Em (x|λ; µ) −t


Em+1 (x|λ; µ) S1 (n, m)eλµt Lif k−1 (−t) − Lif k (−t) m+1 m=0

j=1

m+1 X

r  Y

µj

n X S1 (n, m) λµt e m+1 m=0

! m+1 X (−1)l tl (−1)l tl Em+1 (x|λ; µ) − Em+1 (x|λ; µ) l!(l + 1)k−1 l!(l + 1)k l=0

n X S1 (n, m) λµt e m + 1 m=0 !

m+1 m+1 X (−1)l m+1 X (−1)l m+1 l l × Em+1−l (x|λ; µ) − Em+1−l (x|λ; µ) k−1 k (l + 1) (l + 1) l=0 l=0
n m+1 l X S1 (n, m) X (−1) m+1 l P − rj=1 µj λµt l = −2 e Em+1−l (x|λ; µ) m+1 (l + 1)k m=0 l=1
m n X X P (−1)m−l ml − rj=1 µj S1 (n, m) =2 E (x + λµ|λ; µ) . k l (m + 2 − l) m=0 l=0

= −2−

Pr

j=1

µj

Thus, we obtain (k)

sbn+1 (x) = (x + λµ)b s(k) n (x − 1)
r n m X m−l m XX Pr
(−1) l − 2−1− j=1 µj S1 (n, m) µi λi El x + λ(µ + ei ) − 1|λ; µ + ei k (m + 1 − l) m=0 l=0 i=1
n X m X P (−1)m−l ml − rj=1 µj −2 S1 (n, m)El (x + λµ − 1|λ; µ) , (m + 2 − l)k m=0 l=0 which is (22).

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On the other hand, by (14) with (22), we have g 0 (t) (k) sb (x) g(t) n ! r r X µi λi eλi t X Lif 0k (−t) = sb(k) − µi λi + n (x) λi t 1 + e Lif (−t) k i=1 i=1 n−j   r n X X 2 n 1X (k) λi t S1 (n − l, j)b s l xj = µi λi e λ t i 2 i=1 1 + e j=0 l=0 l
n X m m−l m X Pr (−1) − j=1 µj l − µλb s(k) S (n, m)El (x + λµ|λ; µ) . n (x) + 2 k 1 (m + 2 − l) m=0 l=0

(26)

The first term in (26) is r n n−j   X 2 1 XX n (k) S1 (n − l, j)b sl µi λi eλi t xj λi t 2 j=0 l=0 l 1 + e i=1     n n−j r X 1 XX n x (k) λi t j = S1 (n − l, j)b sl µi λi e λi Ej 2 j=0 l=0 l λi i=1   n n−j   r X 1 XX n x + λi (k) j+1 = S1 (n − l, j)b sl µi λi Ej . 2 j=0 l=0 l λi i=1

Thus, we obtain (k)

s(k) sbn+1 (x) = (x + µλ)b n (x − 1)     n n−j r 1 XXX n x + λi − 1 (k) j+1 − sl E j µi λi S1 (n − l, j)b 2 j=0 l=0 i=1 l λi
n X m X P (−1)m−l ml − rj=1 µj −2 S1 (n, m)El (x + λµ − 1|λ; µ) . (m + 2 − l)k m=0 l=0 which is (23).

3.5

Differentiation

Theorem 5 n−1

X (−1)n−l−1 (k) d (k) sbn (x|λ1 , . . . , λr ; µ1 , . . . , µr ) = n! sb (x|λ1 , . . . , λr ; µ1 , . . . , µr ) . dx l!(n − l) l l=0

(27)

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Proof. We shall use n−1   X  d n ¯ sbn (x) = f (t)|xn−l sbl (x) dx l l=0

(Cf. [12, Theorem 2.3.12]). Since



 f¯(t)|xn−l = ln(1 + t)|xn−l * ∞ + X (−1)m−1 tm = xn−l m m=1 n−l X (−1)m−1 m n−l  t |x = m m=1 n−l X (−1)m−1 = (n − l)!δm,n−l m m=1

= (−1)n−l−1 (n − l − 1)! , with (12), we have d (k) sb (x|λ1 , . . . , λr ; µ1 , . . . , µr ) dx n n−1   X n (k) = (−1)n−l−1 (n − l − 1)!b sl (x|λ1 , . . . , λr ; µ1 , . . . , µr ) l l=0 = n!

n−1 X (−1)n−l−1 l=0

l!(n − l)

(k)

sbl (x|λ1 , . . . , λr ; µ1 , . . . , µr ) ,

which is the identity (27).

3.6

A more relation

The classical Cauchy numbers cn of the first kind are defined by ∞ X tn t cn = ln(1 + t) n=0 n! (see e.g. [3, 8]). Theorem 6 sb(k) n (x|λ1 , . . . , λr ; µ1 , . . . , µr ) =

(k) xb sn−1 (x

+

r X

n  
1X n (k−1) (k) − 1|λ1 , . . . , λr ; µ1 , . . . , µr ) + cn−l sbl (x − 1) − sbl (x − 1) n l=1 l (k)

µi λi sbn−1 (x − λi − 1|λ; µ + ei ) .

(28)

i=1

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Proof. For n ≥ 1, we have sb(k) n (y|λ1 , . . . , λr ; µ1 , . . . , µr ) + *∞ X (k) tl n = sbl (y|λ1 , . . . , λr ; µ1 , . . . , µr ) x l! * l=0 + µj r  Y
(1 + t)λj = Lif k − ln(1 + t) (1 + t)y xn λ j 1 + (1 + t) j=1 * ! + µj r  Y
(1 + t)λj = ∂t Lif k − ln(1 + t) (1 + t)y xn−1 λj 1 + (1 + t) j=1 * + µj ! r  Y
(1 + t)λj = ∂t Lif k − ln(1 + t) (1 + t)y xn−1 λj 1 + (1 + t) j=1 * r  + µj   λj Y
(1 + t) + ∂t Lif k − ln(1 + t) (1 + t)y xn−1 λ j 1 + (1 + t) j=1 + * r  µj Y
(1 + t)λj Lif k − ln(1 + t) (∂t (1 + t)y ) xn−1 . + λj 1 + (1 + t) j=1 The third term is * y

µj r  Y
(1 + t)λj y−1 n−1 Lif k − ln(1 + t) (1 + t) x 1 + (1 + t)λj j=1

+

(k)

= yb sn−1 (y − 1|λ1 , . . . , λr ; µ1 , . . . , µr ) . Since Lif k−1





− ln(1 + t) − Lif k − ln(1 + t) =



1 1 − k−1 k 2 2

 t + ··· ,

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the second term is * r  +

µj Y Lif k−1 − ln(1 + t) − Lif k − ln(1 + t) (1 + t)λj (1 + t)y xn−1 λj 1 + (1 + t) (1 + t) ln(1 + t) j=1 * r  µj Y (1 + t)λj = 1 + (1 + t)λj j=1 +

Lif k−1 − ln(1 + t) − Lif k − ln(1 + t) t y−1 n−1 (1 + t) x t ln(1 + t) * r  µj Y (1 + t)λj = 1 + (1 + t)λj j=1 +

∞ X l Lif k−1 − ln(1 + t) − Lif k − ln(1 + t) t (1 + t)y−1 cl xn−1 t l! l=0 *  µj n−1  r  X Y n−1 (1 + t)λj = cl l 1 + (1 + t)λj j=1 l=0 +

Lif − ln(1 + t) − Lif − ln(1 + t) k k−1 (1 + t)y−1 xn−1−l t

  *Y µj r  1 n−1 (1 + t)λj = cl n − l l 1 + (1 + t)λj j=1 l=0 E


(1 + t)y−1 Lif k−1 − ln(1 + t) − Lif k − ln(1 + t) xn−l * r  +   µj n−1 Y X
n (1 + t)λj 1 Lif k−1 − ln(1 + t) (1 + t)y−1 xn−l cl = λj n − l l 1 + (1 + t) j=1 l=0 * r  +! µj Y
(1 + t)λj y−1 n−l − Lif k − ln(1 + t) (1 + t) x 1 + (1 + t)λj j=1 n−1  
1X n (k−1) (k) cl sbn−l (y − 1|λ1 , . . . , λr ; µ1 , . . . , µr ) − sbn−l (y − 1|λ1 , . . . , λr ; µ1 , . . . , µr ) = n l=0 l n  
1X n (k−1) (k) = cn−l sbl (y − 1|λ1 , . . . , λr ; µ1 , . . . , µr ) − sbl (y − 1|λ1 , . . . , λr ; µ1 , . . . , µr ) . n l=1 l n−1 X

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Since µj r  Y (1 + t)λj ∂t 1 + (1 + t)λj j=1 =

r X

µi λi (1 + t)

i=1

−λi −1

µj r  (1 + t)λj (1 + t)λi Y , (1 + (1 + t)λi j=1 1 + (1 + t)λj

the first term is * + µj r r  λi λj X Y
(1 + t) (1 + t) µi λi Lif k − ln(1 + t) (1 + t)y−λi −1 |xn−1 λ j (1 + (1 + t)λi j=1 1 + (1 + t) i=1 =

r X

(k)

µi λi sbn−1 (y − λi − 1|λ; µ + ei ) .

i=1

Therefore, we obtain sb(k) n (x|λ1 , . . . , λr ; µ1 , . . . , µr ) =

(k) xb sn−1 (x

+

r X

n  
1X n (k−1) (k) cn−l sbl (x − 1) − sbl (x − 1) − 1|λ1 , . . . , λr ; µ1 , . . . , µr ) + n l=1 l (k)

µi λi sbn−1 (x − λi − 1|λ; µ + ei ) ,

i=1

which is the identity (28).

3.7

A relation including the Stirling numbers of the first kind

Theorem 7 For n − 1 ≥ m ≥ 1, we have n−m X

 n (k) m S1 (n − l, m)b sl (λ1 , . . . , λr ; µ1 , . . . , µr ) l l=0 n−m X n − 1 (k) = (m − 1) S1 (n − l − 1, m − 1)b sl (−1|λ1 , . . . , λr ; µ1 , . . . , µr ) l l=0   n−m X n−1 (k−1) (−1|λ1 , . . . , λr ; µ1 , . . . , µr ) + S1 (n − l − 1, m − 1)b sl l l=0  n−m−1 r  X X n−1 (k) S1 (n − l − 1, m)µi λi sbl (−λi − 1|λ; µ + ei ) . +m l i=1 l=0

(29)

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Proof. We shall compute * r  + µj λj Y

(1 + t) m Lif k − ln(1 + t) ln(1 + t) xn λ j 1 + (1 + t) j=1 in two different ways. On the one hand, it is equal to + * r  µj Y

m n (1 + t)λj Lif k − ln(1 + t) ln(1 + t) x λj 1 + (1 + t) j=1 * r  + µj ∞ Y
X (1 + t)λj tl n = Lif k − ln(1 + t) m! S1 (l, m) x λj 1 + (1 + t) l! j=1 l=0 * r  + µj n   X Y
n−l n (1 + t)λj = m! S1 (l, m) Lif k − ln(1 + t) x 1 + (1 + t)λj l j=1 l=m *∞ + n   X X (k) n ti n−l = m! S1 (l, m) sbi (λ1 , . . . , λr ; µ1 , . . . , µr ) x i! l i=0 l=m n   X n (k) = m! S1 (l, m)b sn−l (λ1 , . . . , λr ; µ1 , . . . , µr ) l l=m n−m X n (k) = m! S1 (n − l, m)b sl (λ1 , . . . , λr ; µ1 , . . . , µr ) . l l=0 On the other hand, it is equal to ! + * µj r  Y

m n−1 (1 + t)λj Lif k − ln(1 + t) ln(1 + t) ∂t x λj 1 + (1 + t) j=1 * + µj ! r  Y

m n−1 (1 + t)λj = ∂t Lif k − ln(1 + t) ln(1 + t) x λj 1 + (1 + t) j=1 + * r  µj Y


m n−1 (1 + t)λj ∂t Lif k − ln(1 + t) ln(1 + t) x + 1 + (1 + t)λj j=1 * r  + µj Y

m
n−1 (1 + t)λj + Lif k − ln(1 + t) ∂t ln(1 + t) . x λj 1 + (1 + t) j=1

(30)

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The third term of (30) is equal to * r  + µj Y

m−1 n−1 (1 + t)λj −1 m Lif k − ln(1 + t) (1 + t) ln(1 + t) x 1 + (1 + t)λj j=1 * r  µj Y
(1 + t)λj −1 Lif − ln(1 + t) (1 + t) =m k λj 1 + (1 + t) j=1 + ∞ X tl n−1 (m − 1)! S1 (l, m − 1) x l! l=m−1  n−1  X n−1 = m! S1 (l, m − 1) l l=m−1 * r  + µj Y
(1 + t)λj × Lif k − ln(1 + t) (1 + t)−1 xn−1−l λj 1 + (1 + t) j=1  n−1  X n−1 (k) = m! S1 (l, m − 1)b sn−1−l (−1|λ1 , . . . , λr ; µ1 , . . . , µr ) l l=m−1 n−m X n − 1 (k) = m! S1 (n − l − 1, m − 1)b sl (−1|λ1 , . . . , λr ; µ1 , . . . , µr ) . l l=0 The second term of (30) is equal to + * r 

! µj Y
m n−1 Lif k−1 − ln(1 + t) − Lif k − ln(1 + t) (1 + t)λj ln(1 + t) x λj 1 + (1 + t) (1 + t) ln(1 + t) j=1 * r  + µj Y

m−1 n−1 (1 + t)λj −1 = Lif k−1 − ln(1 + t) (1 + t) ln(1 + t) x 1 + (1 + t)λj j=1 * r  + µj Y

(1 + t)λj m−1 Lif k − ln(1 + t) (1 + t)−1 ln(1 + t) − xn−1 λj 1 + (1 + t) j=1 n−m X n − 1 (k−1) = (m − 1)! S1 (n − l − 1, m − 1)b sl (−1|λ1 , . . . , λr ; µ1 , . . . , µr ) l l=0 n−m X n − 1 (k) − (m − 1)! S1 (n − l − 1, m − 1)b sl (−1|λ1 , . . . , λr ; µ1 , . . . , µr ) . l l=0

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The first term of (30) is equal to * + µj ! r  Y

m n−1 (1 + t)λj ∂t Lif k − ln(1 + t) ln(1 + t) x 1 + (1 + t)λj j=1 * µj r  r X (1 + t)λj (1 + t)λi Y = µi λi 1 + (1 + t)λi j=1 1 + (1 + t)λj i=1
m n−1 E
−λi −1 Lif k − ln(1 + t) (1 + t) ln(1 + t) x * µj r r  X (1 + t)λi Y (1 + t)λj = µi λi 1 + (1 + t)λi j=1 1 + (1 + t)λj i=1 + ∞ X l
t Lif k − ln(1 + t) (1 + t)−λi −1 m! S1 (l, m) xn−1 l! l=m  r n−1  X X n−1 = m! µi λi S1 (l, m) l l=m + *i=1 µj r 
(1 + t)λj (1 + t)λi Y Lif k − ln(1 + t) (1 + t)−λi −1 xn−1−l × 1 + (1 + t)λi j=1 1 + (1 + t)λj  r n−1  X X n−1 (k) = m! µi λi S1 (l, m)b sn−1−l (−λi − 1|λ; µ + ei ) l i=1 l=m  r  n−m−1 X X n−1 (k) S1 (n − 1 − l, m)µi λi sbl (−λi − 1|λ; µ + ei ) . = m! l i=1 l=0 Therefore, we get, for n − 1 ≥ m ≥ 1, n−m X

 n (k) m! S1 (n − l, m)b sl (λ1 , . . . , λr ; µ1 , . . . , µr ) l l=0 n−m X n − 1 (k) = m! S1 (n − l − 1, m − 1)b sl (−1) l l=0 n−m X n − 1 (k−1) + (m − 1)! S1 (n − l − 1, m − 1)b sl (−1) l l=0 n−m X n − 1 (k) − (m − 1)! S1 (n − l − 1, m − 1)b sl (−1) l l=0  n−m−1 r  X X n−1 (k) + m! S1 (n − l − 1, m)µi λi sbl (−λi − 1|λ; µ + ei ) . l i=1 l=0

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Dividing both sides by (m − 1)!, we obtain, for n − 1 ≥ m ≥ 1, n−m X n (k) m S1 (n − l, m)b sl (λ1 , . . . , λr ; µ1 , . . . , µr ) l l=0 n−m X n − 1 (k) = (m − 1) S1 (n − l − 1, m − 1)b sl (−1|λ1 , . . . , λr ; µ1 , . . . , µr ) l l=0   n−m X n−1 (k−1) + S1 (n − l − 1, m − 1)b sl (−1|λ1 , . . . , λr ; µ1 , . . . , µr ) l l=0  n−m−1 r  X X n−1 (k) +m S1 (n − l − 1, m)µi λi sbl (−λi − 1|λ; µ + ei ) . l i=1 l=0 Thus, we get (29).

3.8

A relation with the falling factorials

Theorem 8 sb(k) n (x|λ1 , . . . , λr ; µ1 , . . . , µr )

n   X n (k) (x)m . sb = m n−m m=0

(31)

P (k) Proof. For (12) and (19), assume that sbn (x|λ1 , . . . , λr ; µ1 , . . . , µr ) = nm=0 Cn,m (x)m . By (11), we have + *
m n 1 1  λ ln(1+t) µj Lif k − ln(1 + t) t x Cn,m = 1+e j m! Qr j=1 eλj ln(1+t) * r  + µj
m n 1 Y (1 + t)λj = Lif k − ln(1 + t) t x m! j=1 1 + (1 + t)λj +   *Y µj r 
n−m n (1 + t)λj = Lif k − ln(1 + t) x m 1 + (1 + t)λj j=1   n (k) = sb . m n−m Thus, we get the identity (31).

3.9

A relation with higher-order Frobenius-Euler polynomials (r)

For α ∈ C with α 6= 1, the Frobenius-Euler polynomials of order r, Hn (x|α) are defined by the generating function  r ∞ X 1−α tn xt (r) e = H (x|α) n et − α n! n=0 21

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(see e.g. [10]). Theorem 9 sb(k) n (x|λ1 , . . . , λr ; µ1 , . . . , µr )

=

n X

n−m X n−m−j X

m=0

j=0

−j

×(1 − α) S1 (n − j −

l=0

(k) l, m)b sl



   s n−j (n)j j l (s) Hm (x|α) .

(32)

Proof. For (12) and

s  et − α ,t , (33) ∼ 1−α P (k) (s) assume that sbn (x|λ1 , . . . , λr ; µ1 , . . . , µr ) = nm=0 Cn,m Hm (x|α). By (11), similarly to the proof of (29), we have  ln(1+t) s + * e −α

m n 1−α 1  λ ln(1+t) µj Lif k − ln(1 + t) ln(1 + t) x Cn,m = 1+e j m! Qr j=1 eλj ln(1+t) * r  + µj Y

m 1 (1 + t)λj = Lif k − ln(1 + t) ln(1 + t) (1 − α + t)s xn m!(1 − α)s j=1 1 + (1 + t)λj Hn(s) (x|α)



1 m!(1 − α)s * r  + µj min{s,n}   Y

m X (1 + t)λj s Lif k − ln(1 + t) ln(1 + t) × (1 − α)s−i ti xn λj 1 + (1 + t) i j=1 i=0   n−m X s 1 = (1 − α)s−i (n)i m!(1 − α)s i=0 i + * r  µj Y

m n−i (1 + t)λj Lif k − ln(1 + t) ln(1 + t) x × 1 + (1 + t)λj j=1   n−m n−m−i X s X 1 n−i (k) s−i = (1 − α) (n)i m! S1 (n − i − l, m)b sl m!(1 − α)s i=0 i l l=0    n−m n−m−i X X s n−i (k) = (n)i (1 − α)−i S1 (n − i − l, m)b sl . i l i=0 l=0

=

Thus, we get the identity (32).

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3.10

A relation with higher-order Bernoulli polynomials (r)

Bernoulli polynomials Bn (x) of order r are defined by r  ∞ (r) X t Bn (x) n xt t e = et − 1 n! n=0 (r)

(see e.g. [12, Section 2.2]). In addition, Cauchy numbers of the first kind Cn of order r are defined by  r X ∞ (r) t Cn n = t ln(1 + t) n! n=0 (see e.g. [2, (2.1)], [11, (6)]). Theorem 10 sb(k) n (x|λ1 , . . . , λr ; µ1 , . . . , µr ) ! n n−m X X n−m−i X nn − i (s) (k) = Ci S1 (n − i − l, m)b sl B(s) m (x) . i l m=0 i=0 l=0 Proof. For (12) and B(s) n (x) (k)

et − 1 t Pn

 ∼

s

 ,t ,

(34)

(35)

(s)

assume that sbn (x|λ1 , . . . , λr ; µ1 , . . . , µr ) = m=0 Cn,m Bm (x). By (11), similarly to the proof of (29), we have  ln(1+t) s * + e −1

m n ln(1+t) 1  λ ln(1+t) µj Lif k − ln(1 + t) ln(1 + t) x Cn,m = 1+e j m! Qr j=1 eλj ln(1+t) * r  µj s + 

m 1 Y (1 + t)λj t Lif k − ln(1 + t) ln(1 + t) = xn m! j=1 1 + (1 + t)λj ln(1 + t) * r  + µj ∞ i

m X 1 Y (1 + t)λj t (s) = Lif k − ln(1 + t) ln(1 + t) Ci xn m! j=1 1 + (1 + t)λj i! i=0 * r  +    n−m µ j λ X Y j

1 (1 + t) m (s) n = Ci Lif k − ln(1 + t) ln(1 + t) xn−i λ j m! i=0 i 1 + (1 + t) j=1   n−m−i   n−m 1 X (s) n X n−i (k) = C m! S1 (n − i − l, m)b sl m! i=0 i i l l=0 n−m X n−m−i X nn − i (s) (k) = Ci S1 (n − i − l, m)b sl . i l i=0 l=0 Thus, we get the identity (34). 23

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Acknowledgements The work reported in this paper was conducted during the sabbatical year of Kwangwoon University in 2014.

References [1] S. Araci,M. Acikgoz, A. Kilicman, Extended p-adic q-invariant integrals on Zp associated with applications of umbral calculus, Adv. Difference Equ. 2013 (2013), 96, 14 pp. [2] L. Carlitz, A note on Bernoulli and Euler polynomials of the second kind, Scripta Math. 25 (1961), 323–330. [3] L. Comtet, Advanced Combinatorics, Reidel, Dordrecht, 1974. [4] R. Dere, Y. Simsek, Applications of umbral algebra to some special polynomials, Adv. Stud. Contemp. Math., 22 (2012), 433-438. [5] Q. Fang, T. Wang, Umbral calculus and invariant sequences, Ars Combinatoria, 101 (2011),257–264. [6] D. S. Kim, T. Kim, Higher-order Cauchy of the second kind and poly-Cauchy of the second kind mixed type polynomials, Ars Combinatoria, 115 (2014), 435-451. [7] D. S. Kim and T. Kim, Poly-Cauchy and Peters mixed-type polynomials, Adv. Difference Equ. 2014, (2014), #4. [8] 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. 23 (2013), 621–636. [9] D. S. Kim, T. Kim, S. H. Lee, Poly-Cauchy numbers and polynomials with umbral calculus viewpoint, Int. J. Math. Anal. (Ruse) 7 (2013), 2235–2253. [10] T. Kim, Identities involving Laguerre polynomials derived from umbral calculus , Russ. J. Math. Phys. 21 (2014), 36–45. [11] H. Liang and Wuyungaowa, Identities involving generalized harmonic numbers and other special combinatorial sequences, J. Integer Seq. 15 (2012), Article 12.9.6, 15 pp. [12] S. Roman, The umbral Calculus, Dover, New York, 2005.

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On the solution for a system of two rational difference equations Chang-you Wang, Xiao-jing Fang, Rui Li 1. Key Laboratory of Industrial Internet of Things & Networked Control of Ministry of Education, Chongqing University of Posts and Telecommunications, Chongqing 400065 P.R. China 2. Institute of Applied Mathematics, Chongqing University of Posts and Telecommunications, Chongqing 400065 P. R. China

Abstract: This paper is concerned with the dynamical behavior and the expression of the solution for a system of two rational difference equations xn -3 yn -3 ， yn +1 = ， n = 0,1, xn +1 = A + xn -3 yn -1 B + yn -3 xn -1

,

where the parameters A, B and the initial conditions x−3 , x−2 , x−1 , x0 , y−3 , y−2 , y−1 , y0 are positive real numbers.

Keywords: difference equations; expression of solutions; recursive sequences, equilibrium

point; asymptotical stability.

1. Introduction Rational difference equations that are one of the most important and practical classes of nonlinear difference equations have applications in various scientific branches such as biology, ecology, physiology, physics, engineering and economics, etc [1-4]. Although difference equations are very simple in form, it is extremely difficult to understand thoroughly the behaviors of their solutions. So recently there has been an increasing interest in the study of qualitative analysis of rational difference equation and systems of difference equations [5-7]. In particular, Papaschinopoulos and Schinas [8] studied the oscillatory behavior, the boundedness of the solutions, and the global asymptotic stability of the positive equilibrium of the system of two nonlinear difference equations

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xn +1 = A +

yn , xn − p

yn +1 = A +

xn , n = 0,1, yn − q

(1.1)

,

where p, q are positive integers. Clark and Kulenovic [9, 10] investigated the global stability properties and asymptotic behavior of solutions of the recursive sequences xn yn xn +1 = yn +1 = n = 0,1, . , , a + cyn b + dxn

(1.2)

where a, b, c, d ∈ (0, ∞) and the initial conditions x0 and y0 are arbitrary nonnegative numbers. The periodicity of the positive solutions of the system of rational difference equations yn 1 , xn +1 = , yn +1 = n = 0,1, , (1.3) yn xn −1 yn −1 was studied by Cinar in [11]. Yalcinkaya [12] has obtained the sufficient conditions for the global asymptotic stability of the system of two nonlinear difference equations x + yn −1 y +x xn +1 = n yn +1 = n n −1 , n = 0,1, . , (1.4) xn yn −1 − 1 yn xn −1 − 1 More recently, Din et al. [13] studied the equilibrium points, local asymptotic stability of an equilibrium point, instability of equilibrium points, periodicity behavior of positive solutions, and global character of an equilibrium point of the following fourth-order system of rational difference equations xn +1 =

α xn −3

β + γ yn yn −1 yn − 2 yn −3

, yn +1 =

α1 yn − 3

β1 + γ 1 xn xn −1 xn − 2 xn −3

, n = 0,1,

.

(1.5)

In [14], Elsayed deals with the form of the solutions of the following rational difference system xn +1 =

xn -1 yn -1 ， yn +1 = ， n = 0, 1, ±1 + xn -1 yn ∓1 + yn -1 xn

,

(1.6)

with nonzero real number initial conditions. Other related results on the difference equation can be found in references [15-28] and references therein. Based on the above results, we are mainly interested in study the asymptotic behavior and the expression of the solution for the following nonlinear rational difference equations xn -3 yn -3 xn +1 = ， yn +1 = ， n = 0,1, , (1.7) A + xn -3 yn -1 B + yn -3 xn -1 where the parameters A, B and the initial conditions x−3 , x−2 , x−1 , x0 , y−3 , y−2 , y−1 , y0 are positive real numbers.

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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 and notations In this section we prepare some materials used throughout this paper, namely notations, the basic definitions and preliminary results. We refer to the monographs of Kocic et al. [5, 29, 30]. Lemma 2.1 Let

Ix , 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 ), n = 0, 1, 2, ⎨ ⎩ yn +1 = g (xn ,xn -1 ,xn -2 ,xn -3 ,yn ,yn -1 ,yn -2 ,yn -3 ),

(2.1)

,

has a unique solution {(xn ,yn )}∞n =−3 . Definition 2.1 A point (x , y ) ∈ I x × I y is called an equilibrium point of system (2.1) if x = f (x, x, x, x, y, y, y, y), y = g ( x.x, x, x, y, y, y, y ) .

That is, ( xn , yn ) = ( x , y ) for all n ≥ 1 when the initial conditions ( x0 , x−1 , x−2 , x−3 , y0 , y−1 , y−2 , y−3 ) = ( x , x , x , x , y , y , y , y ) . Definition 2.2 Let ( x , y ) be an equilibrium point of system (2.1). Then

(1) The equilibrium ( x , y ) of system (2.1) is said to be stable relative to I x × I y if for every ε > 0 ,

there

exits

( xi , yi ) ∈ I x × I y (i =−3, − 2, −1, 0) ,

δ >0 with

such

∑

0

i =−3

that

for

any

initial

xi − x < δ ,

∑

yi − y < δ

0

i =−3

conditions implies

xn − x < ε , yn − y < ε .

(2) The equilibrium ( x , y ) of system (2.1) is called an attractor relative to I x × I y if for all ( xi , yi ) ∈ I x × I y (i = −3, −2, −1,0) , lim n→∞ xn = x and lim n→∞ yn = y hold. (3) The equilibrium ( x , y ) of system (2.1) is called asymptotically stable relative to I x × I y if it is stable and an attractor. (4) The equilibrium ( x , y ) of system (2.1) is called unstable if it is not stable. Definition 2.3 Let ( x , y ) be an equilibrium point of the system (2.1), and f and g are

continuously differentiable functions at ( x , y ) . The linearized system of system (2.1) about

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the equilibrium point ( x , y ) is X n +1 = F ( X n ) = FJ X n where X n = ( xn , xn −1 , xn − 2 , xn −3 , yn , yn −1 , yn − 2 , yn −3 )T and FJ is a Jacobian matrix of the system (2.1) about the equilibrium point ( x , y ) . Lemma 2.2 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 . If all eigenvalues of the Jacobian matrix FJ

about

X

lie inside the open unit disk

λ < 1 , then X

is locally

asymptotically stable. If one of them has a modulus greater than one, then X is unstable.

3. Main results and their proofs It is obviously, if A > 1, B ≠ 1 or B > 1, A ≠ 1 , then （0, 0） is the unique equilibrium point of the system (1.7). Theorem 3.1 Let {xn , yn }∞n =−3 be positive solutions of system (1.7), then for all k ≥ 0 ,

(1)

⎧ x−3 ⎪ Ak +1 , ⎪ ⎪ x−2 , ⎪ k +1 0 ≤ xn ≤ ⎨ A ⎪ x−1 , ⎪ Ak +1 ⎪ x ⎪ k0+1 , ⎩A

n = 4k + 1, n = 4k + 2, (2) n = 4k + 3, n = 4k + 4.

⎧ y−3 ⎪ B k +1 , ⎪ ⎪ y−2 , ⎪ k +1 0 ≤ yn ≤ ⎨ B ⎪ y−1 , ⎪ B k +1 ⎪ y ⎪ k0+1 , ⎩B

n = 4k + 1, n = 4k + 2, (3.1) n = 4k + 3, n = 4k + 4.

Proof. This assertion is true for k = 0 , Assume that it is true for k = m , then for k = m + 1 , we have

x4(m+1)-3 x4m+1 1 x-3 ⎧ x−3 x , n = 4(m +1) +1; ≤ = ≤ = ⎪ 4(m+1)+1 m+1 ( m+1)+1 A A A A A ⎪ x4(m+1)+1-3 x4m+2 1 x-2 x−2 ⎪ x , n = 4(m +1) + 2, ≤ = ≤ = 4( m + 1) + 2 m+1 ( m+1)+1 ⎪⎪ A A A A A xn = ⎨ x4(m+1)+2-3 x4m+3 1 x-1 x ⎪x ≤ = ≤ = (m−+11)+1 , n = 4(m +1) + 3, 4( m+1)+3 + 1 m ⎪ A A AA A ⎪ x4(m+1)+3-3 x4m+4 1 x0 x0 ⎪x = ≤ = , n = 4(m +1) + 4. 4( m+1)+4 ≤ ⎩⎪ A A A Am+1 A(m+1)+1

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y4( m +1)-3 y4 m +1 1 y-3 ⎧ y = ≤ = ( m−+31) +1 , n = 4(m + 1) + 1; ⎪ y4( m +1) +1 ≤ m +1 B B BB B ⎪ y4( m +1) +1-3 y4 m + 2 1 y-2 y ⎪ = ≤ = ( m−+21) +1 , n = 4(m + 1) + 2, m +1 ⎪⎪ y4( m +1) + 2 ≤ B B BB B yn = ⎨ y y 1 y-1 y ⎪y ≤ 4( m +1) + 2-3 = 4 m +3 ≤ = ( m−+11)+1 , n = 4(m + 1) + 3, m +1 ⎪ 4( m +1) +3 B B BB B ⎪ y4( m +1) +3-3 y4 m + 4 1 y0 y ⎪y = ≤ = ( m +01) +1 , n = 4(m + 1) + 4. 4( m +1) + 4 ≤ m +1 ⎪⎩ B B BB B This completes our inductive proof. Corollary 3.1 If A > 1, B > 1 , then by Theorem 3.1 {( xn , yn )}∞n =−3 the solutions of the system

(1.7) exponentially converges to the equilibrium poin（ t 0, 0）. Theorem 3.2 For the equilibrium point （0, 0） of the system (1.7), the following results

hold: (1) If A > 1, B > 1 , then the equilibrium point （0, 0） of the system (1.7) is locally asymptotically stable. (2) If A < 1 or B < 1 , then the equilibrium point （0, 0） of the system (1.7) is unstable. Proof. We can easily obtain that the linearized system of (1.7) about the equilibrium point （0, 0） is

ϕn +1 = Dϕ n

(3.2)

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 ⎣

1 A 0 0 0 1 0 0 0 1 0 0 0

0 0 0 0 0 0 0 0 0 0 0 0

⎤ 0 0 0 0⎥ ⎥ 0 0 0 0⎥ 0 0 0 0⎥ ⎥ 0 0 0 0⎥ 1⎥ ⎥ 0 0 0 B⎥ 1 0 0 0⎥ ⎥ 0 1 0 0⎥ 0 0 1 0 ⎥⎦

the characteristic equation of (3.2) is

1 1 (3.3) f (λ ) = (λ 4 − )(λ 4 − ) = 0 . A B 1 1 (1) If A > 1, B > 1 , then we have | |< 1, | |< 1 , this shows that all the roots of A B characteristic equation (3.3) lie inside unit disk. So the unique equilibrium （0, 0） is

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locally asymptotically stable. (2) It is easy to see that if A < 1 or B < 1 , then there exists at least one root λ of the characteristic equation (3.3) such that λ > 1 . Thus, the equilibrium （0, 0） of the system (1.7) is unstable when A < 1 or B < 1 . By Corollary 3.1 and Theorem 3.2, we have the following result. Corollary 3.2 If A > 1, B > 1 , then the equilibrium point （0, 0） is globally asymptotically stable. Theorem 3.3 If A = B = 1 , then every solution of the system (1.7) is bounded when the initial conditions x−3 , x−2 , x−1 , x0 , y−3 , y−2 , y−1 and y0 are positive real numbers. Proof. It follows from Eq. (1.7) that xn -3 yn -3 xn +1 = ≤ xn -3 , yn +1 = ≤ yn -3 . 1 + xn -3 yn -1 1 + yn -3 xn -1

Then the subsequences { x4 n −3 } ∞n =0 , { x4 n − 2 } ∞n =0 , { x4 n −1 } ∞n =0 , { x4 n } ∞n =0 are decreasing and so are bounded from above by M =max{ x−3 , x−2 , x−1 , x0 }, also, the subsequences { y4 n −3 } ∞n =0 , { y4 n − 2 } ∞n =0 , { y4 n −1 } ∞n =0 , { y4 n } ∞n =0 are decreasing and so are bounded from above by m =max{ y−3 , y−2 , y−1 , y0 }. Hence, every solution of the system (1.7) is bounded for any positive initial conditions. In next section, we study the expressions of the solutions for the systems (1.7) with the parameters A = B . Theorem 3.4 If A = B , suppose that {( xn , yn )}∞n =−3 are solutions of the system (1.7). Also,

assume that x−3 , x−2 , x−1 , x0 , y−3 , y−2 , y−1 and y0 are arbitrary positive numbers and let x−3 = a, x−2 = b, x−1 = c, x0 = d , y−3 = e, y−2 = f , y−1 = g , y0 = h . Then x4 n −3

A2i + A2i −1ag + + Aag + ag , = a∏ 2i +1 + A2i ag + A2i −1ag + + Aag + ag i =0 A

x4 n − 2

A2i + A2i −1bh + + Abh + bh , = b∏ 2i +1 + A2i bh + A2i −1bh + + Abh + bh i =0 A

n −1

n −1

A2i +1 + A2i ce + + Ace + ce , x4 n −1 = c∏ 2i + 2 + A2i +1ce + A2i ce + + Ace + ce i =0 A n −1

(3.4)

A2i +1 + A2i df + + Adf + df , x4 n = d ∏ 2i + 2 + A2i +1df + A2i df + + Adf + df i =0 A n −1

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n −1

y4 n −3 = e∏ i =0

A2i + A2i −1ce + + Ace + ce , A2i +1 + A2i ce + A2i −1ce + + Ace + ce

A2i + A2i −1df + + Adf + df , = f ∏ 2i +1 + A2i df + A2i −1df + + Adf + df i =0 A n −1

y4 n − 2

A2i +1 + A2i ag + + Aag + ag , y4 n −1 = g ∏ 2i + 2 + A2i +1ag + A2i ag + + Aag + ag i =0 A n −1

A2i +1 + A2i bh + + Abh + bh , A2i + 2 + A2i +1bh + A2i bh + + Abh + bh

n −1

y4 n = h∏ i =0

where n = 1, 2,

(3.5)

.

Proof. If A = B , then the system (1.7) is reduced to xn -3 yn -3 ， yn +1 = . xn +1 = A + xn -3 yn -1 A + yn -3 xn -1

(3.6)

It is easy to prove that Eqs. (3.4) and (3.5) hold for n = 1 . Now suppose that k ∈ N , k > 1 and that Eqs. (3.4) and (3.5) hold for n = k − 1 . That is, x4 k −7

A2i + A2i −1ag + + Aag + ag , = a∏ 2i +1 + A2i ag + A2i −1ag + + Aag + ag i =0 A

x4 k −6

A2i + A2i −1bh + + Abh + bh , = b∏ 2i +1 + A2i bh + A2i −1bh + + Abh + bh i =0 A

x4 k −5

A2i +1 + A2i ce + + Ace + ce , = c∏ 2 i + 2 + A2i +1ce + A2i ce + + Ace + ce i =0 A

x4 k − 4

A2i +1 + A2i df + + Adf + df , = d ∏ 2i + 2 + A2i +1df + A2i df + + Adf + df i =0 A

k −2

k −2

k −2

k −2

A2i + A2i −1ce + + Ace + ce , A2i +1 + A2i ce + A2i −1ce + + Ace + ce

k −2

y4 k −7 = e∏ i =0

k −2

y4 k − 6 = f ∏ i =0

k −2

y4 k − 5 = g ∏ i =0

k −2

y4 k − 4 = h∏ i =0

A2i + A2i −1df + + Adf + df , A2i +1 + A2i df + A2i −1df + + Adf + df A2i +1 + A2i ag + + Aag + ag , A2i + 2 + A2i +1ag + A2i ag + + Aag + ag

A2i +1 + A2i bh + + Abh + bh . A2i + 2 + A2i +1bh + A2i bh + + Abh + bh

Then, it follows from Eq. (3.6) and our assumptions that

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x4 k − 7 A + x4 k −7 y4 k −5

x4 k −3 =

A2i + A2i −1ag + + Aag + ag 2 i +1 + A2i ag + A2i −1ag + + Aag + ag i =0 A = k −2 k −2 A2i + A2i −1ag + + Aag + ag A2i +1 + A2i ag + + Aag + ag A + a∏ 2i +1 g ∏ + A2i ag + A2i −1ag + + Aag + ag i = 0 A2i + 2 + A2i +1ag + A2i ag + + Aag + ag i =0 A k −2

a∏

A2i + A2i −1ag + + Aag + ag 2 i +1 + A2i ag + A2i −1ag + + Aag + ag i =0 A 1 A + ag 2 k − 2 2 k −1 A + A ag + + Aag + ag

k −2

=

a∏

k −2

= a∏ i =0

k −1

= a∏ i =0

⎛ A2 k − 2 + A2 k −1ag + A2i + A2i −1ag + + Aag + ag ⎜ A2i +1 + A2i ag + A2i −1ag + + Aag + ag ⎝ A2 k −1 + A2 k − 2 ag +

+ Aag + ag ⎞ ⎟ + Aag + ag ⎠

A2i + A2i −1ag + + Aag + ag . A2i +1 + A2i ag + A2i −1ag + + Aag + ag

That is k −1

x4 k −3 = a∏ i =0

A2i + A2i −1ag + + Aag + ag . A2i +1 + A2i ag + A2i −1ag + + Aag + ag

In addition to, by Eq. (3.6) and our assumptions one has

y4k −3 =

y4k −7 A + y4k −7 x4k −5

A2i + A2i−1ce + + Ace + ce e∏ 2i+1 2i + A ce + A2i−1ce + + Ace + ce i =0 A = k −2 k −2 A2i + A2i−1ce + + Ace + ce A2i+1 + A2i ce + + Ace + ce A + e∏ 2i+1 2i c ∏ + A ce + A2i−1ce + + Ace + ce i=0 A2i+2 + A2i+1ce + A2i ce + + Ace + ce i =0 A k −2

A2i + A2i−1ce + + Ace + ce e∏ 2i+1 2i A + A ce + A2i−1ce + + Ace + ce = i=0 1 A + ce 2n−2 2n−1 A + A ce + + Ace + ce k −2 ⎛ A2k −2 + A2k −1ce + A2i + A2i−1ce + + Ace + ce = e∏ 2i+1 2i ⎜ + A ce + A2i−1ce + + Ace + ce ⎝ A2k −1 + A2k −2ce + i =0 A k −2

+ Ace + ce ⎞ ⎟ + Ace + ce ⎠

A2i + A2i−1ce + + Ace + ce . = e∏ 2i+1 2i + A ce + A2i−1ce + + Ace + ce i =0 A k −1

That is,

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A2i + A2i −1ce + + Ace + ce . = e∏ 2i +1 + A2i ce + A2i −1ce + + Ace + ce i =0 A k −1

y4 k − 3

Similarly, one can prove k −1

x4 k − 2 = b∏ i =0

A2i + A2i −1bh + + Abh + bh , A2i +1 + A2i bh + A2i −1bh + + Abh + bh

A2i +1 + A2i ce + + Ace + ce x4 k −1 = c∏ 2i + 2 , + A2i +1ce + A2i ce + + Ace + ce i =0 A k −1

A2i +1 + A2i df + + Adf + df x4 k = d ∏ 2i + 2 , + A2i +1df + A2i df + + Adf + df i =0 A k −1

k −1

y4 k − 2 = f ∏ i =0

k −1

y4 k −1 = g ∏ i =0

k −1

y 4 k = h∏ i =0

A2i + A2i −1df + + Adf + df , A2i +1 + A2i df + A2i −1df + + Adf + df A2i +1 + A2i ag + + Aag + ag , A2i + 2 + A2i +1ag + A2i ag + + Aag + ag

A2i +1 + A2i bh + + Abh + bh . A2i + 2 + A2i +1bh + A2i bh + + Abh + bh

Hence, Eqs. (3.4) and (3.5) hold for n = k . The proof is complete according to the mathematical induction. Corollary 3.3 If A = B = 1 , suppose that {( xn , yn )}∞n =−3 are solutions of the system (1.7).

Also, assume that x−3 , x−2 , x−1 , x0 , y−3 , y−2 , y−1 and y0 are arbitrary positive numbers and let x−3 = a, x−2 = b, x−1 = c, x0 = d , y−3 = e, y−2 = f , y−1 = g , y0 = h , then one has n −1 1 + 2iag 1 + 2ibh , x4 n − 2 = b∏ , i = 0 1 + (2i + 1) ag i = 0 1 + (2i + 1)bh n −1

x4 n −3 = a∏

n −1 1 + (2i + 1)ce 1 + (2i + 1)df , x4 n = d ∏ , i = 0 1 + (2i + 2)ce i = 0 1 + (2i + 2) df n −1

x4 n −1 = c∏

1 + 2ice , i = 0 1 + (2i + 1)ce

1 + 2idf , i = 0 1 + (2i + 1) df

n −1

n −1

y4 n −3 = e∏

y4 n − 2 = f ∏

1 + (2i + 1)ag , i = 0 1 + (2i + 2) ag

n −1

y4 n −1 = g ∏ where n = 1, 2,

1 + (2i + 1)bh , i = 0 1 + (2i + 2)bh

n −1

y4 n = h∏

.

4. Conclusions It is obvious that the system of two rational difference equations (1.7) is the extension of the models in [9, 10, 13, 14]. In this paper, we investigated the globally asymptotically stable of the equilibrium point （0, 0）for the difference equation (1.7) with the parameters A > 1, B > 1 , and the unstable of the equilibrium point (0, 0) with the parameter A < 1 or B < 1 using

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linearization method. Moreover, the expressions of solutions of the system (1.7) with the parameters A = B are obtained according to the mathematical induction. This paper presents the use of a variational iteration method and mathematical induction 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. In addition, the sufficient conditions that we obtained are very simple, which provide flexibility for the application and analysis of nonlinear difference equation. In addition, the system can be used to analyze and describe the pier buffering isolation system.

Acknowledgements This work is supported by the Chongqing Natural Science Fund (Nos. cstc2012jjA20016 and cstc2012jjA40035), the National Nature Science Fund of People’s Republic of China (Nos. 11372366 and 11101298), and Chongqing Outstanding Youth Fund (No. cstc2014jcyjjq 40004).

References [1] W. Li, H. Sun, Global attractivity in a rational recursive sequence. Dynamic Systems and Applications, 11, 339-346 (2002). [2] M. Agop, L. Rusu, El Naschie’s self-organization of the patterns in a plasma discharge: Experimental and theoretical results. Chaos, Solitons & Fractals. 34, 172-186 (2007). [3] M. Shojaei, R. Saadati, H. Adibi, Stability and periodic character of a rational third order difference equation. Chaos, Solitons and Fractals, 39, 1203-1209 (2009). [4] C. Cinar, On the difference equation xn +1 = xn -1 / (1 + xn xn -1 ) . Appl. Math. Comput., 158, 813-816 (2004). [5] V. L. Kocic, G. Ladas, Global Behavior of Nonlinear Difference Equations of Higher Order with Applications. Kluwer Academic, Dordrecht, 1993. [6] M. R. S. Kulenovic, G. Ladas, N. R. Prokup, A rational difference equation. Computers & Mathematics with Applications, 41, 671-678 (2001). [7] X. Yan, W. Li, H. Sun, Global attractivity in a higher order nonlinear difference equation. Applied Mathematics E-Notes, 2, 51-58 (2002). [8] G. Papaschinopoulos, C. J. Schinas, On a system of two nonlinear difference equations. J. Math. Anal. Appl., 219, 415-426 (1998). [9] D. Clark, M. R. S. Kulenovic, A coupled system of rational difference equations. Computers & Mathematics with Applications, 43, 849-867 (2002). [10] D. Clark, M. R. S. Kulenovic, J. F. Selgrade, Global asymptotic behavior of a two-dimensional difference equation modelling competition. Nonlinear Analysis, 52, 1765-1776 (2003).

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[11] C. Cinar, On the positive solutions of the difference equation system xn +1 = 1/ yn , yn +1 = yn / xn −1 yn −1 . Appl. Math. Comput., 158, 303-305 (2004). [12] I. Yalcinkaya, On the global asymptotic behavior of a system of two nonlinear difference equations. ARS Combinatoria, 95, 151-159 (2010). [13] Q. Din, M. N. Qureshi, A. Q. Khan, Dynamics of a fourth-order system of rational difference equations. Adv. Differ. Equ., 2012, 2012: 215. [14] E. M. Elsayed, Solutions of rational difference systems of order two. Math. Comput. Model., 55, 378-384 (2012). [15] C. Y. Wang, S. Wang, W. Wang, Global asymptotic stability of equilibrium point for a family of rational difference equations. Appl. Math. Lett., 24, 714-718 (2011). [16] C. Y. Wang, S. Wang, Z. W. Wang, F. Gong, R. F. Wang, Asymptotic stability for a class of nonlinear difference equation. Dis. Dyn. Nat.Soc., Volume 2010, Article ID 791610, 10pages. [17] C. Y. Wang, F. Gong, S. Wang, L. R. LI, Q. H. Shi, Asymptotic behavior of equilibrium point for a class of nonlinear difference equation. Adv. Differ. Equ., Volume 2009, Article ID 214309. 8pages. [18] E. M. Elabbasy, H. El-Metwally, E. M. Elsayed, Global behavior of the solutions of difference equation, Adv. Differ. Equ., 2011, 2011:28. [19] E. M. Elsayed, Solution and attractivity for a rational recursive sequence, Dis. Dyn. Nat. Soc., Volume 2011, Article ID 982309, 17 pages. [20] Q. Zhang, L. Yang, J. Liu, Dynamics of a system of rational third order difference equation. Adv. Differ. Equ., 2012, 2012: 136. [21] M. Mansour, M. M. El-Dessoky, E. M. Elsayed, The form of the solutions and periodicity of some systems of difference equations, Dis. Dyn. Nat. Soc., Volume 2012, Article ID 406821, 17 pages. [22] Q. H. Shi, Q. Xiao, G. Q. Yuan, X. J. Liu, Dynamic behavior of a nonlinear rational difference equation and generalization. Adv. Diff. Equ., 2011, 2011:36. [23] A. S. Kurbanli, On the behavior of solutions of the system of rational difference equations, Adv. Differ. Equ., 2011, 2011:40. [24] E. M. Elsayed, Behavior and expression of the solutions of some rational difference equations. J. Comput. Anal. Appl., 15, 73-81 (2013). [25] O. Zkan, A. S. Kurbanli, On a system of difference equation. Dis. Dyn. Nat. Soc., Volume 2013, Article ID 970316, 7 pages. [26] L. Alsedà, M. Misiurewicz, A note on a rational difference equation. Journal of

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Difference Equations and Applications, 17, 1711-1713 (2011). [27] T. F. Ibrahim, Periodicity and Global Attractivity of Difference Equation of Higher Order. J. Comput. Anal. Appl., 16, 552-564 (2014). [28] E. M. Elsayed, H. El-Metwally, Stability and Solutions for Rational Recursive Sequence of Order Three, J. Comput. Anal. Appl., 17, 305-315 (2014). [29] H. Sedaghat, Nonlinear Difference Equations: Theory with Applications to Social Science Models, Kluwer Academic, Dordrecht, 2003. [30] M. R. Kulenovic, G. Ladas, Dynamics of Second Order Rational Difference Equations with Open Problems and Conjectures, Chapman Hall/CRC, Boca Raton, 2001.

Conflict of Interests The authors declare that there is no conflict of interests regarding the publication of this article.

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On Distributions of Discrete Order Statistics Y. Bulut1, M. Güngör2, B. Yüzbaşı3 , F. Özbey4 and E. Canpolat5 Department of Econometrics, Inonu University, 44280 Malatya, Turkey 4 Department of Statistics, Bitlis Eren University, 13000 Bitlis, Turkey 1 [email protected], [email protected], [email protected], [email protected] and 5 [email protected] 1,2,3,5

Abstract. In this study, the joint distributions of order statistics of innid discrete random variables are expressed in the form of an integral. Then, the results related to pf and df are given. 2010 Mathematics Subject Classification: 62G30, 62E15. Key words and phrases: Order statistics, discrete random variable, probability function, distribution function.

1. Introduction The joint probability density function(pdf) and marginal pdf of order statistics of independent but not necessarily identically distributed(innid) random variables was derived by Vaughan and Venables[22]

by means of permanents. In addition,

Balakrishnan[3], and Bapat and Beg[8] obtained the joint pdf and distribution function(df) of order statistics of innid random variables by means of permanents. In the first of two papers, Balasubramanian et al.[5] obtained the distribution of single order statistic in terms of distribution functions of the minimum and maximum order statistics of some subsets of { X 1 , X 2 ,..., X n } where X i ’s are innid random variables. Later, Balasubramanian et al.[6] generalized their previous results[5] to the case of the joint distribution function of several order statistics. Recurrence relationships among the distribution functions of order statistics arising from innid random variables were obtained by Cao and West[10]. Using multinomial arguments, the pdf of X r:n1 (1  r  n+1) was obtained by Childs and Balakrishnan[11] by adding another independent random variable to the original n variables X 1, X 2 ,..., X n . Also, Balasubramanian et

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al.[7] established the identities satisfied by distributions of order statistics from nonindependent non-identical variables through operator methods based on the difference and differential operators. In a paper published in 1991, Beg[9] obtained several recurrence relations and identities for product moments of order statistics of innid random variables using permanents. Recently, Cramer et al.[13] derived the expressions for the distribution and density functions by Ryser’s method and the distributions of maxima and minima based on permanents. A multivariate generalization of classical order statistics for random samples from a continuous multivariate distribution was defined by Corley[12]. Guilbaud[17] expressed the probability of the functions of innid random vectors as a linear combination of probabilities of the functions of independent and identically distributed(iid) random vectors and thus also for order statistics of random variables. Expressions for generalized joint densities of order statistics of iid random variables in terms of RadonNikodym derivatives with respect to product measures based on df were derived by Goldie and Maller[16]. Several identities and recurrence relations for pdf and df of order statistics of iid random variables were established by numerous authors including Arnold et al.[1], Balasubramanian and Beg[4], David[14], and Reiss[21]. Furthermore, Arnold et al.[1], David[14], Gan and Bain[15], and Khatri[18] obtained the probability function(pf) and df of order statistics of iid random variables from a discrete parent. Balakrishnan[2] showed that several relations and identities that have been derived for order statistics from continuous distributions also hold for the discrete case. In a paper published in 1986, Nagaraja[19] explored the behavior of higher order conditional probabilities of order statistics in a attempt to understand the structure of discrete order statistics. Later, Nagaraja[20] considered some results on order statistics of a random sample taken from a discrete population. In general, the distribution theory for order statistics is complex when the parent distribution is discrete. In this study, the joint distributions of p order statistics of innid discrete random variables are obtained as an p fold integral.

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As far as we know, these approaches have not been considered in the framework of order statistics from innid discrete random variables. From now on, the subscripts and superscripts are defined in the first place in which they are used and these definitions will be valid unless they are redefined. Let X 1 , X 2 ,..., X n be innid discrete random variables and X 1:n  X 2:n  ...  X n:n be the order statistics obtained by arranging the n X i ’s in increasing order of magnitude. Let Fi and f i be df and pf of X i (i =1, 2,…, n), respectively. For notational convenience we write

x1



,

z1 , z 2 ,..., z p

n  rp



m p 0

rp 1 r p1



r3 1 r2

...

k p 0



m2  0

r2 1 r1



k2 0



,



m p , k p ..., m1 , k1

r2 1 r1

r1 1

 

m1  0

k1  0

and

instead of



Fir ( x2 )





1

, 1

Fir ( x p )

2

   ...  Fi ( x1 ) Fi ( x2 ) r1

p

2

Fir ( x1  ) Fir ( x2  )

xp

x3

z1 0 z2  z1 z3  z2

V

Fir ( x1 )

x2

...



and

Fir ( x p  ) p

 0

r2

 vi(1) r1

,

z p  z p1

Fir ( x p ) p

...



in

vi( p1) r p1

the expressions below, respectively ( xi  0 , 1, 2 , . . .) ( z0  0) .

2. Theorems for distribution and probability functions In this section, the theorems related to pf and df of X r1 :n , X r2 :n ,..., X r p :n (1  r1  r2  ...  rp  n , p=1, 2,…, n) will be given. We will now express the following theorem for the joint pf of order statistics of innid discrete random variables.

Theorem 2.1.

 p 1 rw 1  p f r1 , r2 ,...,rp :n ( x1 , x2 ,..., x p )  D      [vi(l w )  vi(l w1) ]  dvi(rw ) , w P  w1 l rw1 1  w1

(2.1)

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where x1  x2  ...  x p ,



denotes the sum over all n! permutations (i1 , i2 ,...,in ) of

P

p 1

(1,2,…,n),

D   [ (rw  rw 1  1)!]1 , r0  0 ,

rp 1  n  1 , vi(l0)  0 , vi(l p 1)  1 and

w1

vi(lw )  [vi(rww)  Firw (xw )]

f il ( xw ) f irw ( xw )

 Fil (xw ) .

Proof. Consider the event { X r1 :n  x1 , X r2 :n  x2 ,..., X rp :n  x p } .

The above event can be realized mutually exclusive as follows: r1  1  k1 observations are less than x1 , kw  1  mw (w=1, 2,…, p) observations are equal to xw , r  1  k  m 1  r 1 (   2, 3,..., p ) observations are in interval ( x 1, x ) and n  m p  rp observation exceed x p . The probability function of the above event can be

written as f r1 , r2 ,..., r p :n ( x1 , x2 ,..., x p )  P{ X r1 :n  x1, X r2 :n  x2 ,..., X r p :n  x p } .

(2.2)

(2.2) can be expressed as f r1 , r2 ,...,rp :n ( x1 , x2 ,..., x p )





mp , k p ...,m1 ,k1

C

 P

 p1 rw 1kw  p rw mw [ Fil ( xw ) Fil ( xw1 ) ]    fi j ( xw ) ,     w1 l rw1  mw1 1  w1 j rw kw

(2.3)

 p1  p where C   [(rw 1 kw mw1 rw1 )!]1  [(kw1 mw )!]1, m0  0, k p 1  0, Fil ( x0 )  0 ,  w1  w1 Fil ( x p1 )  1 , Fil ( xw )  P ( X il  xw ) and mw 1  k w  rw  rw 1  1 (w =1, 2,…, p+1).

(2.3) can be written as

f r1 , r2 ,...,rp :n ( x1 , x2 ,..., x p ) 



m p ,k p ..., m1 , k1

C

 P

 p 1 rw 1 kw  [ Fil ( xw ) Fil ( xw1 ) ]      w 1 l  rw1  mw1 1 

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p

 w1

  rw  mw  1 kw (k w  1  mw )!  rw 1 m   fi j ( xw )  firw ( xw )   f i j ( xw )   yw (1  yw ) w dyw . k w ! mw !  j rw kw   j  rw 1 0

(2.4)

Also, (2.4) can be clearly written as 1 1

 

f r1 ,r2 ,...,rp:n ( x1 , x2 ,..., x p ) 

m p ,k p ...,m1 , k1

P

1

  ... 0 0

0



1 Fi ( x1 ) Fi2 ( x1 )...Fir1 1k1 ( x1 ) y1 f ir1 k1 ( x1 ) y1 f ir1 k11 ( x1 )... y1 f ir1 1 ( x1 )dy1 f ir1 ( x1 ) ( r1  1  k1 )!k1! 1



1 (1  y1 ) f ir11 ( x1 )(1  y1 ) f ir1 2 ( x1 )...(1  y1 ) f ir1 m1 ( x1 )[ Fir1  m11 ( x2 )  Fir1 m11 ( x1 ) ]... ( r2 r1m1k 2 1)! m1! k 2 !

 [ Fir2  k2 1 ( x2 )  Fir2  k 2 1 ( x1 ) ] y2 f ir2 k 2 ( x2 ) y 2 f ir2  k 2 1 ( x2 )...y 2 f ir2 1 ( x2 ) dy 2 f ir2 ( x2 )...



1 (1 y p1 ) f irp1 1 ( x p1 )(1y p1 ) f irp 1 2 ( x p1 )...(1y p1 ) f irp 1 m p1 ( x p1 ) (rp  rp1  mp1  k p  1)!mp1!k p !

 [ Firp1m p 1 1 ( xrp )  Firp1m p11 ( xrp 1 ) ]...[Firp k p 1 ( xrp )  Firp k p 1 ( xrp 1 ) ]  y p f irp k p ( x p ) y p f irp k p 1 ( x p )...y p f irp 1 ( x p )dy p firp ( x p )



1 (1  y p ) f irp 1 ( x p )(1  y p ) f irp  2 ( x p )...(1  y p ) f irp  m p ( x p )[1  Firp  m p 1 ( x p ) ]...[1  Fin ( x p ) ]. (n  rp  m p )!m p !

The following expression can be written from the last identity.

1 1

f r1 , r2 ,..., rp :n ( x1 , x2 ,..., x p ) 

 

m p , k p ..., m1 , k1 P

1  p 1 1 ... 0 0 0  w 1 ( rw  1  k w  mw 1  rw 1 )! mw 1! kw!

 rw1  m w1  rw 1 k w     (1  yw 1 ) f i1 ( xw 1 )  [ Fi 2 ( xw ) Fi 2 ( xw 1 ) ]     1  rw1 1   2  rw1  mw1 1   rw 1   p    yw fi ( xw )  fir ( xw )dyw .   3 w  3 rw kw   w 1

(2.5)

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In (2.5), if vi(jw)  y w f i j ( x w )  Fi j ( xw ) , the following identity is obtained. Fir ( x1 ) Fir ( x2 )



fr1 ,r2 ,...,rp :n ( x1 , x2 ,..., x p ) 

m p ,k p ...,m1 , k1

Fir ( x p )

 p 1 1 P   ...   ( rw  1  kw  mw1 rw1 )! mw1 ! kw ! Fir ( x1  ) Fir ( x2  ) Fir ( x p  )  w 1 1 2 p 1

2

p

 rw1 mw1  rw1k w  ( w 1)     [Fi1 ( xw 1 )  v i ]  [ Fi 2 ( xw) Fi2 ( xw 1 )]  1  1  rw11   2  rw1 mw11   rw 1   p ( w)    [vi Fi ( xw )]  dvi(rw ) .   3 w 3  3 rw kw   w1

(2.6)

By considering n

n

   0

 0

  (1)  n  (2)  n 1 Gi ( x )  Gi ( x)   Gi(3) ( x) P  !(n     )! !    3  1 2  1 1   2  1  3  n 1 

n 1 [Gi(l1) ( x )  Gi(l 2) ( x)  Gi(l3) ( x)] ,  n ! P l 1

(2.7)

where     n and using (2.7) for each mw 1 and kw in (2.6), we get f r1 ,r2 ,..., rp :n ( x1 , x2 ,..., x p )  p 1  1     w1 ( rw  rw1 1)!  P

rw 1

 p 1    w1

 p (w) ( w 1)        [ F ( x ) F ( x ) v F ( x ) F ( x ) v ] dvi(rw) .    il w il w 1 il w il w 1 il il  w l  rw11  w1

Thus, the proof is completed.

Specially,

in

Theorem

vi(32)  [vi(22)  Fi2 (x2 )] Fi1 ( x1 )

f1,2:3 ( x1 , x2 ) 

 

fi3 ( x2 ) f i2 ( x2 )

2.1,

by

taking

p  2,

n  3,

r1  1 ,

r2  2 ,

 Fi3 (x2 ) and for x1  x2 ,

Fi2 ( x2 )



[1  vi(2) ] dvi(2) dvi(1) 3 2 1

P Fi ( x1  ) Fi ( x2  ) 1 2

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  fi ( x2 ) 1   fi1 ( x1 )  fi2 ( x2 )  [ fi2 ( x2 )  Fi2 (x2 )  Fi2 (x2 )   fi2 ( x2 ) Fi2 (x2 )] 3  fi2 ( x2 ) Fi3 (x2 )  2 f i2 ( x2 ) P   1 1   = f1 ( x1 )  f 2 ( x2 )  f3 ( x2 ) F2 ( x2 )  f3 ( x2 ) F2 ( x2 )  f 2 ( x2 ) F3 ( x2 )  2 2   1 1    f1 ( x1 )  f 3 ( x2 )  f 2 ( x2 ) F3 ( x2  )  f 2 ( x2 ) F3 ( x2 )  f3 ( x2 ) F2 ( x2  )  2 2   1 1    f 2 ( x1 )  f3 ( x2 )  f1 ( x2 ) F3 ( x2 )  f1 ( x2 ) F3 ( x2 )  f3 ( x2 ) F1 ( x2 )  2 2   1 1    f 2 ( x1 )  f1 ( x2 )  f3 ( x2 ) F1 ( x2 )  f 3 ( x2 ) F1 ( x2 )  f1 ( x2 ) F3 ( x2 )  2 2   1 1    f3 ( x1 )  f1 ( x2 )  f 2 ( x2 ) F1 ( x2 )  f 2 ( x2 ) F1 ( x2 )  f1 ( x2 ) F2 ( x2 )  2 2   1 1    f 3 ( x1 )  f 2 ( x2 )  f1 ( x2 ) F2 ( x2 )  f1 ( x2 ) F2 ( x2 )  f 2 ( x2 ) F1 ( x2 )  . 2 2  

Morever, the above identity in the iid case can be expressed as f1,2:3 ( x1 , x2 )  6 f ( x1 ) f ( x2 )  6 f ( x1 ) f ( x2 ) F ( x2 )  3 f ( x1 ) f 2 ( x2 ) .

This result is obtained, if i  1 , j  2 and n  3 in equation (6) in [18]. In case x1  x2  ...  x p , vi(1)r  vi(r2)  ...  vi(rp ) is automatically satisfied because of 1

p

2

Fir1  x1   vi(1)r  Fir1  x1  , Fir2  x2   vi(r2)  Fir2  x2  , …, Firp  x p   vi(r p )  Firp  x p  . 1

2

p

Also, in case x1  x2  ...  x p  x , the integration region is over Fir1  x   vi(1)r  vi(r2)  ...  vi(rp )  Firp  x , Fir1  x   vi(1)r  Fir1  x  , 1

2

1

p

Fir2  x   vi(r2)  Fir2  x  , …, Firp  x   vi(rp )  Firp  x  . 2

p

7 193

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So, if x1  x2  ...  x p , it should be written



 ...

Fir ( x1 ) 1

instead of

Fir ( x p )

Fir ( x2 )

p

2





...

Fir ( x1  ) Fir ( x2  ) 1

in (2.1), where 

 ...



Fir ( x p  ) p

2

is to be carried out over the region: vi(1)r  vi(r2)  ...  vi(r p ) , 1

p

2

Fir1  x1   vi(1)r  Fir1  x1  , Fir2  x2   vi(r2)  Fir2  x2  , …, Firp  x p   vi(r p )  Firp  x p  . 1

2

p

The proof was given only in case x1  x2  ...  x p , the proof for case x1  x2  ...  x p is omitted. Specially,

in

Theorem

vi(32)  [vi(22)  Fi2 (x2 )] Fi1 ( x )

f1,2:3 ( x, x) 

 

P Fi ( x  ) 1

fi3 ( x2 ) f i2 ( x2 )

2.1,

by

taking

p  2,

n  3,

r1  1 ,

r2  2 ,

 Fi3 (x2 ) and for x1  x2  x ,

Fi2 ( x )



[1  vi(2) ] dvi(2) dvi(1) 3 2 1

vi(1) 1

 1 1    Fi2 ( x) f i1 ( x)   Fi1 ( x)  Fi1 ( x )  fi1 ( x )  Fi22 ( x) f i1 ( x) 2 2 P   fi ( x ) fi3 ( x ) 1  Fi2 (x) Fi2 (x ) 1   Fi1 ( x )  Fi1 ( x )  fi1 ( x ) Fi2 (x ) fi2 ( x ) 2

fi3 ( x)

f i ( x) 1   Fi13 ( x)  Fi13 ( x )  3 fi2 ( x) 6 f i2 ( x)

fi3 ( x ) f i2 ( x)

 Fi2 ( x) Fi3 (x ) fi1 ( x )

1    Fi1 ( x )  Fi1 ( x )  fi1 ( x ) Fi3 (x )  2   f ( x) 1 3 f ( x) 1 1   F2 ( x) f1 ( x)   F1 ( x)  F1 ( x )  f1 ( x )  F22 ( x ) f1 ( x) 3   F1 ( x)  F13 ( x )  3 2 2 f 2 ( x) 6 f 2 ( x)  f ( x ) f 3 ( x) 1 f ( x)  F2 (x) F2 (x ) 1   F1 ( x )  F1 ( x )  f1 ( x) F2 (x ) 3  F2 ( x ) F3 (x ) f1 ( x ) f 2 ( x) 2 f 2 ( x) 1    F1 ( x)  F1 ( x ) f1 ( x ) F3 (x )  2 

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 1 1 f ( x) 1 3 f ( x)   F3 ( x) f1 ( x )   F1 ( x)  F1 ( x )  f1 ( x)  F32 ( x) f1 ( x) 2   F1 ( x )  F13 ( x )  2 2 2 f 3 ( x) 6 f 3 ( x)  f ( x) f 2 ( x) 1 f ( x)  F3 (x ) F3 (x ) 1   F1 ( x )  F1 ( x )  f1 ( x ) F3 (x ) 2  F3 ( x) F2 (x ) f1 ( x) f3 ( x) 2 f 3 ( x) 1    F1 ( x)  F1 ( x ) f1 ( x ) F2 (x )  2   f ( x) 1 3 f ( x) 1 1   F1 ( x) f 2 ( x )   F2 ( x)  F2 ( x )  f 2 ( x )  F12 ( x) f 2 ( x) 3   F2 ( x)  F23 ( x )  3 2 2 f1 ( x) 6 f1 ( x)  f ( x) f 3 ( x ) 1 f ( x)  F1 (x) F1 (x ) 2   F2 ( x )  F2 ( x )  f 2 ( x ) F1 (x ) 3  F1 ( x ) F3 (x ) f 2 ( x) f1 ( x) 2 f1 ( x ) 1    F2 ( x )  F2 ( x )  f 2 ( x ) F3 (x )  2   1 1 f ( x) 1 3 f ( x)   F3 ( x ) f 2 ( x)   F2 ( x)  F2 ( x )  f 2 ( x)  F32 ( x) f 2 ( x) 1   F2 ( x)  F23 ( x )  1 2 2 f 3 ( x) 6 f 3 ( x)  f ( x) f1 ( x) 1 f ( x)  F3 (x) F3 (x ) 2   F2 ( x )  F2 ( x )  f 2 ( x ) F3 (x ) 1  F3 ( x) F1 (x ) f 2 ( x) f3 ( x) 2 f3 ( x) 1    F2 ( x )  F2 ( x )  f 2 ( x ) F1 (x )  2   1 1 f ( x) 1 3 f ( x)   F2 ( x) f3 ( x)   F3 ( x )  F3 ( x )  f3 ( x )  F22 ( x ) f3 ( x) 1   F3 ( x)  F33 ( x )  1 2 2 f 2 ( x) 6 f 2 ( x)  f ( x ) f1 ( x ) 1 f ( x)  F2 (x) F2 (x ) 3   F3 ( x )  F3 ( x )  f3 ( x) F2 (x ) 1  F2 ( x) F1 (x ) f3 ( x) f2 ( x) 2 f2 ( x) 1    F3 ( x)  F3 ( x )  f3 ( x) F1 (x )  2   1 1 f ( x) 1 3 f ( x)   F1 ( x) f3 ( x)   F3 ( x)  F3 ( x )  f3 ( x)  F12 ( x) f 3 ( x) 2   F3 ( x)  F33 ( x )  2 2 2 f1 ( x) 6 f1 ( x )  f ( x) f 2 ( x ) 1 f ( x)  F1 (x) F1 (x ) 3   F3 ( x )  F3 ( x  )  f 3 ( x ) F1 (x ) 2  F1 ( x) F2 (x ) f 3 ( x ) f1 ( x) 2 f1 ( x )



1  F3 ( x)  F3 ( x)  f3 ( x) F2 (x)  . 2 

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Morever, the above identity in the iid case can be expressed as  6 F ( x ) f ( x )  3 F ( x )  F ( x)  f ( x)  3F 2 ( x) f ( x )   F 3 ( x )  F 3 ( x)  6 F ( x) F ( x) f ( x )

3 F ( x )  F ( x)  F ( x) f ( x )  6 F ( x ) F ( x) f ( x)  3 F ( x )  F ( x)  f ( x ) F ( x)  6 F ( x ) f ( x )  3 F ( x ) f ( x )  3 F ( x) f ( x)  3 F 2 ( x) f ( x )  F 3 ( x )  F 3 ( x)  3 f 2 ( x )  3F 2 ( x ) f ( x )  f ( x) 3F 2 ( x)  3F ( x ) f ( x )  f 2 ( x)  f 3 ( x)  3 f 2 ( x) 1 F ( x ) .

This result is obtained, if r  1 , s  2 and n  3 in equation (2.4.3) in [14].

We will now express the following theorem to obtain the joint df of order statistics of innid discrete random variables. Theorem 2.2.

 p 1 rw 1 Fr1 ,r2 ,..., rp :n ( x1 , x2 ,..., x p )  D      [vi(l w )  vi(l w1) ] P V  w 1 l  rw1 1

 p ( w)  dvirw .  w1

(2.8)

Proof. We have

Fr1 , r2 ,...,rp :n ( x1 , x2 ,..., x p ) 



f r1 , r2 ,...,rp :n ( z1 , z 2 ,..., z p ) .

(2.9)

z1 , z 2 ,..., z p

Using (2.1) in (2.9), (2.8) is obtained. 3. Results for distribution and probability functions

In this section, the results related to pf and df of

X r1 :n , X r2 :n ,..., X r p :n will be

given. We will express the following result for pf of the rth order statistic of innid discrete random variables. Result 3.1. F (x )

ir1 1  r1 1 1  n 1   vil  [1  vil1 ] f r1:n ( x1 )     (r1  1)!(n  r1 )! P Fi ( x1  ) l 1  l  r1 1 r1

 1 dvi .  r1 

(3.1)

Proof. In (2.1), if p  1 , (3.1) is obtained.

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In Result 3.2 and Result 3.3, the pf ’s of minimum and maximum order statistics of innid discrete random variables are given, respectively. Result 3.2. Fi ( x1 )

1  n 1 f1:n ( x1 )    [1  vil1 ]  (n  1)! P Fi (x1  ) l  2 1

 1 dvi1 . 

(3.2)

Proof. Putting r1  1 in (3.1), one will get (3.2).

Result 3.3. Fi ( x1 )

n  n 1 1 1 f n:n ( x1 )    vi  (n  1)! P Fi (x1  ) l 1 l n

 1  dvin . 

(3.3)

Proof. On taking r1  n in (3.1), one will get (3.3).

In the following result, we will give the joint pf of X 1:n , X 2:n ,..., X p:n . Result 3.4. If x1  x2  ...  x p ,  n  p ( p) ( w) ... [1  v ]   dviw , P    l il   p 1  w1 (1) is to be carried out over the region: vi  vi( 2)  ...  vi( p ) ,

f1,2,..., p :n ( x1 , x2 ,..., x p ) 

where

  ...

1 (n  p)!

1

2

(3.4)

p

Fi1  x1   vi(1)  Fi1  x1  , Fi2  x2   vi( 2)  Fi2  x2  , …, Fi p  x p   vi(pp )  Fi p  x p  . 1

2

Proof. On taking rw  w for w  1, 2, ..., p and

  ...

instead of



in (2.1), one

will get (3.4). We will now give three results for the df of single order statistic of innid discrete random variables.

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Result 3.5. 1 Fr1 :n ( x1 )   (r1  1)!(n  r1 )! P

Fir ( x1 ) 1

 0

 r1 1 1  n   vil   [1  vil1 ]   l 1  l  r1 1

 1 dvi .  r1 

(3.5)

Proof. On taking p  1 in (2.8), one will get (3.5).

Result 3.6.

1 F1:n ( x1 )   (n  1)! P

Fi1 ( x1 )

 0

 n   [1  vil1 ]  l 2

 1 dvi1 . 

(3.6)

Proof. Putting r1  1 in (3.5), one will get (3.6). Result 3.7.

1 Fn:n ( x1 )   (n  1)! P

Fin ( x1 )

 0

 n 1 1   vil  l 1

 1 dvin . 

(3.7)

Proof. On taking r1  n in (3.5), one will get (3.7). Specially, in (3.7), by taking n=2 and vi11  [vi21  Fi2 (x1 )]

f i1 ( x1 ) f i2 ( x1 )

 Fi1 (x1 ) , the

following identity is obtained. Fi2 ( x1 )

F2:2 ( x1 ) 

  P

vi11 dvi21

0

 v1 i2     P  2 

 

2

Fi ( x1 )

 2  f ( x ) i 1  vi21 Fi2 ( x1 )  1  vi21 Fi1 ( x1 )   fi2 ( x1 )  0 

 Fi 2 ( x1 )   f i ( x1 )    2  Fi2 ( x1 ) Fi2 ( x1 )  1  Fi2 ( x1 ) Fi1 ( x1 )   2  f i ( x1 ) P    2   F22 ( x1 )  f (x )   F2 ( x1 ) F2 ( x1 )  1 1  F2 ( x1 ) F1 ( x1 )  2  f 2 ( x1 )  F 2 (x )  f (x )   1 1  F1 ( x1 ) F1 ( x1 )  2 1  F1 ( x1 ) F2 ( x1 ).  2  f1 ( x1 ) 12 198

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Morever, the above identity for iid case can be expressed as F2:2 ( x1 )  F 2 ( x1 ).

Also, the above identity for x1  1 can be written as F2:2 (1)  F 2 (1) 2

  f (0)  f (1) . In the following result, we will give the joint df of X 1:n , X 2:n ,..., X p:n . Result 3.8.

1 F1,2,..., p:n ( x1 , x2 ,..., x p )  (n  p )!

Fi p ( x p )

Fi1 ( x1 ) Fi2 ( x2 )

  P

0



...

vi(1) 1



vi( p1) p1

 n  p ( p) ( w) [1  ] v    dviw . il  l  p 1  w1

(3.8)

Proof. On considering rw  w for w  1, 2, ..., p from (2.8), one will get (3.8). References B. C. Arnold, N. Balakrishnan and H. N. Nagaraja, A first course in order statistics, John Wiley and Sons Inc., New York, 1992. [2] N. Balakrishnan, Order statistics from discrete distributions, Commun. Statist. Theory Meth. 15 (1986), no.3, 657-675. [3] N. Balakrishnan, Permanents, order statistics, outliers and robustness, Rev. Mat. Complut. 20 (2007), no.1, 7-107. [4] K. Balasubramanian and M. I. Beg, On special linear identities for order statistics, Statistics 37 (2003), no.4, 335-339. [5] K. Balasubramanian, M. I. Beg and R. B. Bapat, On families of distributions closed under extrema, Sankhyā Ser. A 53 (1991), no.3, 375-388. [6] K. Balasubramanian, M. I. Beg and R. B. Bapat, An identity for the joint distribution of order statistics and its applications, J. Statist. Plann. Inference 55 (1996), no.1, 13-21. [7] K. Balasubramanian, N. Balakrishnan and H. J. Malik, Identities for order statistics from nonindependent non- identical variables, Sankhyā Ser. B 56 (1994), no.1, 67-75. [8] R. B. Bapat and M. I. Beg, Order statistics for nonidentically distributed variables and permanents, Sankhyā Ser. A 51 (1989), no.1, 79-93. [9] M. I. Beg, Recurrence relations and identities for product moments of order statistics corresponding to nonidentically distributed variables, Sankhyā Ser. A 53 (1991), no.3, 365-374. [10] G. Cao and M. West, Computing distributions of order statistics, Commun. Statist. Theory Meth. 26 (1997), no.3, 755-764. [11] A. Childs and N. Balakrishnan, Relations for order statistics from non-identical logistic random variables and assessment of the effect of multiple outliers on bias of linear estimators, J. Statist. Plan. Inference 136 (2006), no.7, 2227-2253. [12] H. W. Corley, Multivariate order statistics, Commun. Statist. Theory Meth. 13 (1984), no.10, 1299-1304. [1]

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[13] E. Cramer, K. Herle and N. Balakrishnan, Permanent Expansions and Distributions of Order Statistics in the INID Case, Commun. Statist. Theory Meth. 38 (2009), no.12, 2078-2088. [14] H. A. David, Order statistics, John Wiley and Sons Inc., New York, 1970. [15] G. Gan and L. J. Bain, Distribution of order statistics for discrete parents with applications to censored sampling, J. Statist. Plann. Inference 44 (1995), no.1, 37-46. [16] C. M. Goldie and R. A. Maller, Generalized densities of order statistics, Statist. Neerlandica 53 (1999), no.2, 222-246. [17] O. Guilbaud, Functions of non-i.i.d. random vectors expressed as functions of i.i.d. random vectors, Scand. J. Statist. 9 (1982), no.4, 229-233. [18] C. G. Khatri, Distributions of order statistics for discrete case, Ann. Inst. Statist. Math. 14 (1962), no.1, 167-171. [19] H. N. Nagaraja, Structure of discrete order statistics, J. Statist. Plann. Inference 13 (1986), no.1, 165-177. [20] H. N. Nagaraja, Order statistics from discrete distributions, Statistics 23 (1992), no.3, 189-216. [21] R. -D. Reiss, Approximate distributions of order statistics, Springer-Verlag, New York, 1989. [22] R. J. Vaughan and W. N. Venables, Permanent expressions for order statistics densities, J. Roy. Statist. Soc. Ser. B 34 (1972), no.2, 308-310.

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TABLE OF CONTENTS, JOURNAL OF COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 20, NO. 1, 2016

On the λ-Daehee Polynomials With q-Parameter, Jin-Woo Park, ……………………………11 Stability of Ternary Quadratic Derivation on Ternary Banach Algebras: revisited, Choonkil Park,……………………………………………………………………………………………21 Some Properties of Modular S-Metric Spaces and its Fixed Point Results, Meltem Erden Ege and Cihangir Alaca,…………………………………………………………………………………24 The Strong Converse Inequality for de la Vallee Poussin Means on the Sphere, Chunmei Ding, Ruyue Yang, and Feilong Cao,…………………………………………………………………34 On the Fixed Point Method for Stability of a Mixed Type, AQ-Functional Equation, Ick-Soon Chang, and Yang-Hi Lee,……………………………………………………………………..…42 Differences of Composition Operators from Lipschitz Space to Weighted Banach Spaces in Polydisk, Chang-Jin Wang, and Yu-Xia Liang,…………………………………………………50 The Path Component of the Set of Generalized Composition Operators on the Bloch Type Spaces, Liu Yang,……………………………………………………………………………….56 The Generalized Hyers-Ulam Stability of Quadratic Functional Equations on Restricted Domains, Chang Il Kim, and Chang Hyeob Shin,………………………………………………65 Hesitant Fuzzy Soft Set and its Lattice Structures, Xiaoqiang Zhou, and Qingguo Li,…………72 Inclusion Properties for Certain Subclasses of Analytic Functions Associated With Bessel Functions, N. E. Cho, G. Murugusundaramoorthy, and T. Janani,…………………………….81 Barnes-type Narumi of the Second Kind and Poly-Cauchy of the Second Kind Mixed-Type Polynomials, Dae San Kim, Taekyun Kim, Takao Komatsu, Jong-Jin Seo, and Seog-Hoon Rim,……………………………………………………………………………………………...91 Superstability and Stability of (r,s,t)-J*-Homomorphisms: Fixed Point and Direct Methods, Shahrokh Farhadabadi, Choonkil Park, and Dong Yun Shin,…………………………………121 Differential Subordinations Obtained by Using a Generalization of Marx-Strohhäcker Theorem, Georgia Irina Oros, Gheorghe Oros, Alina Alb Lupas, and Vlad Ionescu,……………………135

TABLE OF CONTENTS, JOURNAL OF COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 20, NO. 1, 2016 (continued) A Finite Difference Method for Burgers’ Equation in the Unbounded Domain Using Artificial Boundary Conditions, Quan Zheng, Yufeng Liu, and Lei Fan,………………………………140 Barnes-Type Peters Polynomials Associated with Poly-Cauchy Polynomials of the Second Kind, Dae San Kim, Taekyun Kim, Takao Komatsu, Hyuck In Kwon, and Sang-Hun Lee,…………151 On the Solution for a System of two Rational Difference Equations, Chang-you Wang, Xiao-jing Fang, and Rui Li,………………………………………………………………………………175 On Distributions of Discrete Order Statistics, Y. Bulut, M. Güngör, B. Yüzbaşı, F. Özbey, and E. Canpolat,…………………………………………………………………………………..…..187

Volume 20, Number 2 ISSN:1521-1398 PRINT,1572-9206 ONLINE

February 2016

Journal of Computational Analysis and Applications EUDOXUS PRESS,LLC

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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©2016 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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Editor in Chief: George Anastassiou Department of Mathematical Sciences University of Memphis Memphis, TN 38152-3240, U.S.A.

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A Recurrent Neural Fuzzy Network George A. Anastassiou Department of Mathematical Sciences, University of Memphis Memphis, TN 38152, U.S.A. [email protected] Iuliana F. Iatan Department of Mathematics and Computer Science Technical University of Civil Engineering Bucharest [email protected] Abstract Besides the feedforward neural networks, there are the recurrent networks, where the impulses can be transmitted in both directions due to some reaction connections in these networks. Recurrent neural networks are linear or nonlinear dynamic systems. The dynamic behavior presented by the recurrent neural networks can be described both in continuous time, by differential equations and at discrete times by the recurrence relations (difference equations). The distinction between recurrent (or dynamic) neural networks and static neural networks is due to recurrent connections both between the layers of neurons of these networks and within the same layer, too. The aim of this paper is to describe a Recurrent Fuzzy Neural Network (RFNN) model, whose learning algorithm is based on the Improved Particle Swarm Optimization (IPSO) method. Each particle (candidate solution), which is moving permanently includes the parameters of the membership function and the weights of the recurrent neuralfuzzy network; initially, their values are randomly generated. The RFNN presented in this paper is unlike the others variants of RFNN models, by the number of the evolution directions that they use: in this paper, we update the velocity and the position of all particles along three dimensions, while in [8] are used two dimensions. Keywords: recurrent networks; Improved Particle Swarm Optimization method; fuzzy rules; Wavelet Neural Network; feedback weight; delayed operator.

1

Introduction

Neural network (NN) is one of the important components in Artificial Intelligence (AI). NN architectures used in modelling of the nervous systems can be classified into three categories, each with a different philosophy: feedforward, recurrent (feedback), self-organizing map. Neural networks (NNs) are used in many different application domains in order to solve various information processing problems. For several years now, neural network models have enjoyed wide popularity [4], being applied to problems of regression, classification, computational science, computer vision, data processing and time series analysis. 1

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Figure 1: Schematic diagram of the WNN. The main drawback of the feedforward neural networks is that the updating of the weights can fall [17] in a local minimum. An other major drawback of the feedforward neural networks consists in the fact that their application domain is limited to static problems due their inherent feedforward structure. Since recurrent networks incorporate feedback, they have powerful representation capability and can [17] successfully overcome disadvantages of feedforward networks. This feedback implies that the network has [12] local memory characteristics that is able to store activity patterns and present those patterns to the network more than once, allowing the layer with feedback connections to use its own past activation in its preceding behavior. The Recurrent Neural Network (RNN) has the feedforward and feedback connections contrasted which provides it with nonlinear mapping capacity and dynamical characteristics, so it can be used [22] to simulate dynamical system and solve dynamic problems. Different architectures can be created [12] by adding recurrent connections at different points in the basic feedforward architecture. Recently some researchers have proposed several recurrent neuro- fuzzy networks. Kumar et al. 2004 compares the traditional feedforward approach of RNNs to forecast monthly river flows. Lin & Hsu, 2007 has proposed [10] a recurrent wavelet-based neuro- fuzzy system with the reinforcement hybrid evolutionary learning algorithm for solving various control problems. Carcano et al., 2008 has simulated [3] daily river flows for water resource purposes using the Jordan Recurrent Neural Network. Maraqua et al., 2012 has proposed [12] the use of a recurrent network architecture as a classification engine for automatic ˇ Arabic Sign Language recognition system. Ster, 2013 has introduced [18] an extended architecture of recurrent neural networks (called Selective Recurrent Neural Network ) for dealing with long term dependencies.

1.1

Wavelet Neural Networks

Neural networks employing wavelet neurons are referred to as Wavelet Neural Networks(WNNs) [10]; they are characterized by weights and wavelet bases. Lin & Chin, 2004 was proposed a Recurrent Neural Fuzzy Network (RNFN) where each fuzzy rule corresponding to a WNN (see Figure 1) consists (see [11], [8]) of single-scaling wavelets. The shape and position of the wavelet bases are shown [11] in Figure 2. An ordinary wavelet neural network model is often used to normalize input vectors in the interval [0, 1]. The functions φa.b (xi ) are used to input vectors 2

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Figure 2: Wavelet bases are over-complete and compactly supported. to fire up the wavelet interval; a such value is given in the following equation, which gives the shape of the M wavelet bases φ0.0 , φ1.0 , . . . , φm.m :   φ(xi ) = cos(xi ), −0.5 ≤ xi ≤ 0.5 (1)  0 otherwise, φa.b (xi ) = cos(axi − b), b = 1, a, a = 1, m, b being a shifting parameter and a meaning a scaling parameter corresponding to the maximum value of b. A crisp value ϕa.b can be obtained as follows: Pn j=1 φa.b (xi ) ϕa.b = , (2) |X| where |X| represents the number of input dimensions and n is the dimension of the input vector to the model.

1.2

Z- transform

The Z- transform is [20] the discrete- time counterpart of the Laplace transform. The Z- transform can be considered to be an extension of the discrete- time Fourier transform as the Laplace transform can be considered an extension of the Fourier transform. The bilateral Z- transform of a discrete- time sequence x(n) is: ∞ X

Z{x(n)} = X(z) =

x(n)z −n .

(3)

n=−∞

For causal sequences (n ≥ 0) the Z- transform becomes: Z{x(n)} = X(z) =

∞ X

x(n)z −n .

(4)

n=0

The equation (4) is called the unilateral Z- transform; it exists only if the power series from its expression converges. 3

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There are several methods for computing the inverse Z- transform, namely the sequence x(n), given X(z): 1. using the inversion integral : 1 x(n) = 2πj

I

X(z)z n−1 dz,

(5)

Γ

H where Γ means the integration along the closed contour Γ in the counterclockwise closed contour in the region of convergence of X(z); 2. by a power series expansion: expressing X(z) in a power series in z −1 , x(n) can be achieved by identifying it with the coefficient of z −n in the power series expansion; 3. by partial fraction expansion: for a rational functions, can be obtained a partial fraction expansion of X(z) over its poles and the table of Ztransform helps to identify the sequences corresponding to the terms in that partial fraction expansion.

1.3

Application of Genetic Algorithms

The specialists think that the Genetic Algorithms are a computational intelligence application as well as the expert systems, fuzzy systems, neural networks, the intelligent agents, hybrid intelligent systems, electronic voice. The genetic algorithms are some adaptive techniques of heuristic search, based on the genetic and selection natural principles, enunciated by Darwin (the best adapted will survive). The mechanism is similar to the evolutionary biological process. This process has a feature through that only the species which one adapt better to the environment are capable to survive and to develop into generations, while that those less adapted fail to survive and they disappear in time, as a result of the natural selection. The main notions that allow the analogy between the solution of the search problems and the natural evolution are: 1. Population. A population consists in some individuals (chromosomes) that have to live in an environment to which they must adapt. 2. Fitness. Each of the population individuals is adapted more or less to the environment. The fitness is a measure of the degree of adaptation to the environment. 3. Chromosome. It is a ordered set of elements, named genes, whose values establish the individual features. 4. Generation. A stage in a population evolution. If we see evolution as an iterative process in which a population turns to another population, then the generation is an iteration in this process. 5. Selection. The process of natural selection has the survival of individuals with a high environmental fitness (high fitness) as effect.

4

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Figure 3: The RNFN architecture. 6. Reproduction. It is the process through which one passes from one generation to another. The individuals of the new generation inherit some features from their precursors (parents) but they can also get some new features as a result of some processes of mutation that have a random character. In the case when in the reproduction process at least two parents occur, the inherited features of the survivor (son) are obtained by combining (crossover) of the parent features. The remainder of the paper is organized as follows. In Section 2 is discussed and analyzed the RNFN. We follow with the learning algorithm of the recurrent model in Section 3. We conclude in Section 4.

2

RNFN Architecture

The network construction is based on fuzzy rules, each corresponding to a Wavelet Neural Network (WNN). The figure Figure 3 illustrates the RNFN model, whose training algorithm is based on Improved Particle Swarm Optimization (IPSO) method. The nodes from the first layer constitute some input nodes; hence they only pass the input signal to the next layer, namely: (1)

Oi

(1)

= xi .

(6)

The neurons in the second layer act as a membership function, meaning that they determine how an input value belongs to a fuzzy set. The following Gaussian function is chosen as the membership function:

5

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Figure 4: Delayed cell.

2

(2) Oij

−

(2) −mij ) (Iij σ2 ij

=e

,

(7)

where: • mij and σij are the mean and standard deviation, respectively; (2)

• Iij denotes the input of this layer for the discrete time scan: (2)

(2)

Iij = Oi

(f )

+ Oij ,

(8)

where (f )

(2)

Oij = Oij (t − 1)θij .

(9) (2)

The inputs of this layer contain the terms of memory Oij (t − 1), that store network information at a previous time; this information, which is an additional input of the network will be reintroduced at the entrance of the second layer. The weight θij constitutes the feedback weight of the network and z −1 signifies the delayed operator. Figure 4 represents [14] a delayed cell, X(z) being the Z- transform of the signal x[n]. The neurons of the third layer achieve the product operation of their input signals: 2

Oj3

=

n Y

(2) Oij

i=1

=

n Y

−

(2) −mij ) (Iij

e

σ2 ij

,

(10)

i=1

where n is the number of external dimensions. The neurons of the fourth layer receive both the output of a WNN, denoted yˆj and of a neuron from the third layer, namely Oj3 . The mathematical function of each node j is: Oj4 = yˆjp · Oj3 , yˆjp

(11)

being the local output of the WNN for the output yp and the j-th rule: yˆjp =

M X

p wjk ϕa.b ,

(12)

k=1

with ϕa.b from (2), where: • M = m+1 denotes the number of wavelet bases, which equals the number of existing fuzzy rules in the considered model, 6

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p • the link wjk is the output action strength associated with in the p output, j-th rule and k-th ϕa.b .

The fifth layer acts as a defuzzifier namely it provides the nonfuzzy outputs yp of the fuzzy recurrent neural network: 1

yp = 1+e

PM O4 j −λ· Pj=1 M O3 j=1 j

1

= −λ·

1+e

PM p y ˆ ·O 3 j=1 j j PM O3 j=1 j

,

(13)

namely: 1

yp = 1+e

3

PM p p p (w ϕ1.1 +w ϕ2.1 +...+w ϕ )·O 3 j2 j jM m.m −λ· j=1 j1 PM O3 j=1 j

, λ ∈ Fv then  Cv = Co     C  R = Cv Cm = CR   Cr = Cm    Cu = Cr . (2) Else if Fo > FR and Fo < Fv then  CR = Co    Cm = CR Cr = Cm    Cu = Cr . (3) Else if Fo > Fm and Fo < FR then   Cm = Co Cr = Cm  Cu = Cr . (4) Else if Fo > Fr and Fo < Fm then  Cr = Co Cu = Cr . (5) Else if Fo > Fu and Fo < Fr then Cu = Co . (6) Else if Fo = Fu = Fr = Fm = FR = Fv then Co = Co + Nr (Nr ∈ [0, 1]). (7) Else if Fo 0 and that our assumption holds for n ¡ 1. That is; x6n¡8 = x6n¡6 = x6n¡4 = x6n¡3 =

x¡2 xn¡1 xn¡1 0 ¡3 n¡1 xn¡1 ¡1 x¡4

n¡2 Y

(1+(3i+1)x¡1 x¡4 ) ; (1+(3i+2)x0 x¡3 )

x6n¡7 =

n¡1 xn ¡1 x¡4

xn¡1 xn¡1 0 ¡3

n¡2 Y

(1+(3i+1)x0 x¡3 ) ; (1+(3i+3)x¡1 x¡4 )

i=0 i=0 n¡2 n¡2 Y 1+(3i+2)x x n¡1 Y n xn xn 1+(3i+2)x¡1 x¡4 0 x¡3 ¡1 x¡4 0 ¡3 ; x = ; n¡1 n¡1 n¡1 n¡1 6n¡5 1+(3i+3)x0 x¡3 1+(3i+4)x¡1 x¡4 x¡1 x¡4 x¡2 x0 x¡3 (1+x¡1 x¡4 ) i=0 i=0 n¡2 Y ³ (1+(3i+3)x x ) ´ n xn ¡1 ¡4 0 x¡3 ; n¡1 (1+(3i+4)x0 x¡3 ) xn ¡1 x¡4 (1+x0 x¡3 ) i=0 n¡2 Y ³ (1+(3i+3)x x ) ´ n xn 0 ¡3 ¡1 x¡4 : n¡1 n (1+(3i+5)x¡1 x¡4 ) x0 x¡3 (1+2x¡1 x¡4 ) i=0

Now, it follows from Eq.(2) that x6n¡2 =

x6n¡4 x6n¡7 x6n¡5 (1+x6n¡4 x6n¡7 )

3

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Q

µ

= µ =

1+

0 @

³

=

³

=

x¡2 xn¡1 xn¡1 0 ¡3 (1+x¡1 x¡4 ) n¡2 Q³

n xn 0 x¡3

n¡1 xn ¡1 x¡4 (1+x0 x¡3 )

i=0

xn xn ¡1 ¡4 n¡1 (1+x x¡2 xn¡1 x ¡1 x¡4 ) 0 ¡3

x¡2 xn¡1 xn¡1 0 ¡3 (1+x¡1 x¡4 ) n xn ¡1 x¡4

(1+(3i+3)x¡1 x¡4 ) (1+(3i+4)x0 x¡3 )

´ n¡2 Q³ i=0

´ n¡2 Q³

n x¡2 xn 0 x¡3 (1+x¡1 x¡4 ) n xn ¡1 x¡4

´

n¡2 Q³

n¡1 xn ¡1 x¡4

xn¡1 xn¡1 0 ¡3

i=0

(1+(3i+4)x¡1 x¡4 ) (1+(3i+2)x0 x¡3 )

(1+(3i+4)x¡1 x¡4 ) (1+(3i+2)x0 x¡3 )

n x¡2 xn 0 x¡3 n xn x ¡1 ¡4

Similarly x6n¡1 =

i=0

n¡1 Y³ i=0

´

´

´¶

(1+(3i+1)x0 x¡3 ) (1+(3i+3)x¡1 x¡4 )

Q

n¡2 µ i=0

´¶

1 ¶ (1+(3i+1)x0 x¡3 ) A (1+(3i+4)x0 x¡3 )

x0 x¡3 µ (1+(3n¡2)x0 x¡3 ) ¶ x0 x¡3 1+ (1+(3n¡2)x0 x¡3 )

1 : (1+(3n¡1)x0 x¡3 )

(1+(3i+1)x¡1 x¡4 ) (1+(3i+2)x0 x¡3 )

´

:

x6n¡3 x6n¡6 x6n¡4 (1 + x6n¡3 x6n¡6 )

Q

Q

¶ n n¡1 n¡2µ ¶ n¡2µ n x0 x xn (1+(3i+3)x0 x¡3 ) (1+(3i+2)x¡1 x¡4 ) ¡1 x¡4 ¡3 n¡1 n¡1 (1+(3i+5)x¡1 x¡4 ) xn¡1 (1+(3i+3)x0 x¡3 ) xn 0 x¡3 (1+2x¡1 x¡4 ) i=0 ¡1 x¡4 i=0

= µ

=

i=0

(1+(3i+2)x0 x¡3 ) (1+(3i+4)x¡1 x¡4 )

n¡2 Q µ (1+(3i+1)x0 x¡3 ) ¶ x0 x¡3 (1+x0 x¡3 ) (1+(3i+4)x0 x¡3 ) i=0 10 n¡2 Q µ (1+(3i+2)x0 x¡3 ) ¶ x0 x¡3 A@1+ 1+(3i+4)x¡1 x¡4 ) (1+x0 x¡3 ) ( i=0

x6n¡2 =

=

n¡2 Q³

n xn ¡1 x¡4

Hence, we have

=

Q

¶ n n¡1 n¡2µ ¶ n¡2 µ n xn (1+(3i+3)x¡1 x¡4 ) x¡1 x¡4 (1+(3i+1)x0 x¡3 ) 0 x¡3 n¡1 n¡1 n¡1 n (1+(3i+4)x0 x¡3 ) x0 x¡3 i=0 (1+(3i+3)x¡1 x¡4 ) x¡1 x¡4 (1+x0 x¡3 ) i=0

1+

n xn ¡1 x¡4

n xn 0 x¡3

n¡1 xn ¡1 x¡4 (1+x0 x¡3 )

n¡1 xn 0 x¡3 (1+2x¡1 x¡4 )

n¡2 Q³ i=0

n xn 0 x¡3 @ n¡1 (1+x x xn x 0 ¡3 ) ¡1 ¡4

³ xn

n¡1 ¡1 x¡4 (1+x0 x¡3 n xn 0 x¡3

i=0

(1+(3i+3)x¡1 x¡4 ) (1+(3i+4)x0 x¡3 )

Q

n¡2 µ

´

n¡1 xn 0 x¡3

n¡1 xn¡1 ¡1 x¡4

¶

(1+(3i+2)x¡1 x¡4 ) (1+(3i+5)x¡1 x¡4 ) 10 µ ¶ n¡2 Q (1+(3i+3)x¡1 x¡4 ) x¡1 x¡4 A@1+ (1+2x¡1 x¡4 ) (1+(3i+4)x0 x¡3 ) i=0

´ n¡2 Q³ )

n¡1 xn ¡1 x¡4 (1+x0 x¡3 ) n xn 0 x¡3

Hence, we have

n¡2 Q³

(1+(3i+3)x0 x¡3 ) (1+(3i+5)x¡1 x¡4 )

x¡1 x¡4 (1+2x¡1 x¡4 )

0

³

µ

i=0

´ n¡2 Q³ i=0

i=0

(1+(3i+4)x0 x¡3 ) (1+(3i+3)x¡1 x¡4 ) (1+(3i+4)x0 x¡3 ) (1+(3i+3)x¡1 x¡4 )

x6n¡1 =

n xn+1 ¡1 x¡4 n n x0 x¡3

n¡1 Q³ i=0

´ ´

´¶

n¡2 Q³ i=0

Q

n¡2µ i=0

(1+(3i+2)x¡1 x¡4 ) (1+(3i+3)x0 x¡3 )

´¶

1 ¶ (1+(3i+2)x¡1 x¡4 ) A (1+(3i+5)x¡1 x¡4 )

x¡1 x¡4 µ (1+(3n¡1)x¡1 x¡4 ) ¶ x¡1 x¡4 1+ (1+(3n¡1)x¡1 x¡4 ) x¡1 x¡4 : (1+(3n)x¡1 x¡4 )

(1+(3i+1)x0 x¡3 ) (1+(3i+3)x¡1 x¡4 )

´

:

Similarly, we can easily obtain the other relations. Thus, the proof is completed. 4

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Theorem 2 Eq.(2) has x = 0 as a unique equilibrium point and it is unstable. Proof: For the equilibrium points of Eq.(2), set x= Then

x2 : x (1 + x2 )

¢ ¡ ¢ ¡ x2 1 + x2 = x2 ; ) x2 1 + x2 ¡ 1 = 0; ) x4 = 0:

Thus the equilibrium point of Eq.(2) is x = 0: Let f : (0; 1)3 ¡! (0; 1) be a function de…ned by f (t; u; v; w) =

vw : u(1+vw)

Thus the linearized equation of Eq.(2) about the equilibrium point x is given by yn+1 =

4 P

i=0

@f (x;x;x;x) : @xn¡i

The proof follows by Theorem A. Numerical examples For con…rming the results of this section, we consider some numerical examples which represent di¤erent types of solutions to Eq.(2). Example 1. Consider Eq.(2) with x¡4 = 0:21; x¡3 = 2; x¡2 = 0:5; x¡1 = 7; x0 = 0:3: See Fig. 1. Example 2. Consider Eq.(2) with x¡4 = 9; x¡3 = 2; x¡2 = 6; x¡1 = 7; x0 = 3. See Fig. 2. plot of x(n+1)= x(n-1)x(n-4)/(x(n-2)(1+x(n-1)x(n-4))

plot of x(n+1)= x(n-1)x(n-4)/(x(n-2)(1+x(n-1)x(n-4))

7

9 8

6 7 5 6 4 x(n)

x(n)

5 4

3

3 2 2 1 1 0

0

10

20

30

40

50

60

70

80

0

90

n

0

10

20

30

40

50

60

70

80

90

n

Figure 1.

Figure 2.

5

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3

On the Equation xn+1 =

xn¡1xn¡4 xn¡2 (¡1+xn¡1 xn¡4 )

In this section we obtain the solution of the di¤erence equation xn¡1 xn¡4 xn+1 = ; n = 0; 1; :::; xn¡2(¡1 + xn¡1 xn¡4 )

(3)

where the initial values are arbitrary non zero real numbers with x¡1 x¡4 6= 1; x¡3 x0 6= 1: Theorem 3 Every solution fxn g1 n=¡4 of Eq.(3) has the form 2n 2n

2n

x0 x¡3 (¡1+x¡1 x¡4 )n n ; 2n¡1 (¡1+x x2n ¡3 x0 ) ¡1 x¡4 2n 2n x x0 x¡3 (¡1+x¡1 x¡4 )n x12n¡2 = ¡2 ; 2n (¡1+x¡3 x0 )n x2n ¡1 x¡4 2n+1 2n n x x ¡1 x¡4 ) x12n = x02n x2n¡3 (¡1+x ; (¡1+x¡3 x0 )n ¡1 ¡4 2n+1 2n+1 x x (¡1+x¡1 x¡4 )n ; x12n+2 = x0 2n+1 x¡32n (¡1+x x )n+1 ¡1

x12n+4 =

x12n¡3 =

¡3 0

¡4

2n+1 x¡2 x02n+1 x¡3 (¡1+x¡1 x¡4 )n+1 ; 2n+1 2n+1 (¡1+x¡3 x0 )n x¡1 x¡4 2n+2 2n+1 n x0 x¡3 (¡1+x¡1 x¡4 )

x12n+6 =

2n+1 2n+1 x¡1 x¡4 (¡1+x¡3 x0 )n+1

2n

x¡1 x¡4 (¡1+x¡3 x0 )n n; 2n¡1 (¡1+x x2n ¡1 x¡4 ) 0 x¡3 2n+1 2n x x (¡1+x¡3 x0 )n x12n¡1 = x¡12n x2n¡4 (¡1+x n; ¡1 x¡4 ) 0 ¡3 2n+1 2n+1 x x¡4 (¡1+x¡3 x0 )n x12n+1 = x¡1 2n x2n n+1 ; x ¡2 0 ¡3 (¡1+x¡1 x¡4 ) 2n+1 2n+1 x x (¡1+x¡3 x0 )n x12n+3 = x¡12n+1 x¡42n (¡1+x n; ¡1 x¡4 ) ¡3 0 2n+2 2n+1 n+1 x x¡4 (¡1+x¡3 x0 ) x12n+5 = x¡1 ; 2n+1 2n+1 x (¡1+x x )n+1

x12n¡4 =

0

;

x12n+7 =

¡3

2n+2 2n+2 x¡1 x¡4

2n+1 x¡2 x2n+1 x¡3 0

¡1 ¡4

(¡1+x¡3 x0 )n : (¡1+x¡1 x¡4 )n+1

Proof: For n = 0 the result holds. Now suppose that n > 0 and that our assumption holds for n ¡ 1. That is; x12n¡16 = x12n¡14 = x12n¡12 = x12n¡10 = x12n¡8 = x12n¡6 =

2n¡2 x02n¡2 x¡3 (¡1+x¡1 x¡4 )n¡1 ; 2n¡2 2n¡3 x¡1 x¡4 (¡1+x¡3 x0 )n¡1

x12n¡15 =

2n¡2 x¡2 x2n¡2 x¡3 (¡1+x¡1 x¡4 )n¡1 0 ; 2n¡2 2n¡2 (¡1+x¡3 x0 )n¡1 x¡1 x¡4 2n¡2 x02n¡1 x¡3 (¡1+x¡1 x¡4 )n¡1 ; 2n¡2 2n¡2 x¡1 x¡4 (¡1+x¡3 x0 )n¡1

x12n¡13 =

x12n¡9 =

2n¡1 x¡2 x2n¡1 x¡3 (¡1+x¡1 x¡4 )n 0 ; 2n¡1 2n¡1 x¡1 x¡4 (¡1+x¡3 x0 )n¡1 2n¡1 x2n (¡1+x¡1 x¡4 )n¡1 0 x¡3 ; 2n¡1 2n¡1 (¡1+x¡3 x0 )n x¡1 x¡4

x12n¡7 =

2n¡1 2n¡2 x¡1 x¡4 (¡1+x¡3 x0 )n¡1 ; 2n¡2 (¡1+x¡1 x¡4 )n¡1 x02n¡2 x¡3 2n¡1 2n¡1 x¡1 x¡4

x12n¡11 =

2n¡1 x02n¡1 x¡3 (¡1+x¡1 x¡4 )n¡1 ; 2n¡1 2n¡2 (¡1+x¡3 x0 )n x¡1 x¡4

2n¡2 2n¡2 x¡1 x¡4 (¡1+x¡3 x0 )n¡1 ; 2n¡3 x02n¡2 x¡3 (¡1+x¡1 x¡4 )n¡1

(¡1+x¡3 x0 )n¡1 n 2n¡2 2n¡2 (¡1+x x¡2 x0 x¡3 ¡1 x¡4 ) 2n¡1 2n¡1 x¡1 x¡4 (¡1+x¡3 x0 )n¡1 2n¡1 2n¡2 x0 x¡3 (¡1+x¡1 x¡4 )n¡1

;

;

2n¡1 x2n ¡1 x¡4

(¡1+x¡3 x0 )n n; 2n¡1 (¡1+x x02n¡1 x¡3 ¡1 x¡4 )

x12n¡5 =

2n x2n ¡1 x¡4

(¡1+x¡3 x0 )n¡1 n : 2n¡1 (¡1+x x¡2 x02n¡1 x¡3 ¡1 x¡4 )

Now, it follows from Eq.(3) that x12n¡4 = =

=

x12n¡6 x12n¡9 x12n¡7 (¡1+x12n¡6 x12n¡9 ) Ã

2n¡1 x2n (¡1+x¡1 x¡4 )n¡1 0 x¡3 2n¡1 2n¡1 (¡1+x¡3 x0 )n x¡1 x¡4 Ã 2n¡1 2n¡1 x2n x2n (¡1+x¡3 x0 )n ¡1 x¡4 0 x¡3 ¡1+ 2n¡1 n 2n¡1 2n¡1 (¡1+x¡1 x¡4 ) x x x x2n¡1 0 ¡3 ¡1 ¡4

Ã

2n¡1 x2n ¡1 x¡4 2n¡1 2n¡1 x0 x¡3

µ

!Ã

!

n¡1 x2n¡1 x2n¡1 (¡1+x ¡3 x0 ) ¡1 ¡4 2n¡1 2n¡2 x0 x¡3 (¡1+x¡1 x¡4 )n¡1 ! 2n¡1 2n¡1 (¡1+x¡1 x¡4 )n¡1 x¡1 x¡4 (¡1+x¡3 x0 )n¡1 2n¡2 (¡1+x¡3 x0 )n x02n¡1 x¡3 (¡1+x¡1 x¡4 )n¡1

¶ x0 x¡3 (¡1+x¡3 x0 ) !µ µ ¶¶ (¡1+x¡3 x0 )n x0 x¡3 ¡1+ n (¡1+x¡1 x¡4 ) (¡1+x¡3 x0 )

=

2n n x2n 0 x¡3 (¡1+x¡1 x¡4 ) n ; 2n¡1 (¡1+x 2n x¡1 x¡4 ¡3 x0 )

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x12n¡3 = = =

x12n¡5 x12n¡8 x12n¡6 (¡1+x12n¡5 x12n¡8 ) 2n¡1 x2n¡1 2n x2n (¡1+x¡3 x0 )n¡1 x¡2 x0 (¡1+x¡1 x¡4 )n ¡1 x¡4 ¡3 n 2n¡1 2n¡1 2n¡1 2n¡1 (¡1+x¡1 x¡4 ) x¡2 x x x x (¡1+x¡3 x0 )n¡1 0 ¡1 ¡4 á3 ! n¡1 2n 2n n¡1 n 2n¡1 2n¡1 x2n¡1 ¡1+x x¡1 x¡4 (¡1+x¡3 x0 ) x2n x¡2 x0 ¡1 x¡4 ) 0 x¡3 (¡1+x¡1 x¡4 ) ¡3 ( ¡1+ 2n¡1 2n¡1 x2n¡1 ¡1+x n n¡1 2n¡1 x2n¡1 ¡1+x x¡1 x¡2 x2n¡1 x¡3 (¡1+x¡1 x¡4 )n x¡1 ¡3 x0 ) ¡3 x0 ) 0 ¡4 ( ¡4 (

2n¡1 2n¡1 x¡1 x¡4 2n¡1 x2n 0 x¡3

(¡1+x¡3 x0 )n x¡1 x¡4 (¡1+x¡1 x¡4 )n¡1 (¡1+x¡1 x¡4 )

=

2n x2n ¡1 x¡4

(¡1+x¡3 x0 )n n; 2n¡1 (¡1+x x2n x ¡1 x¡4 ) 0 ¡3

Similarly, we can easily obtain the other relations. Thus, the proof is completed. p Theorem 4 Eq.(3) has three equilibrium points which are x = 0 and x = § 2and all of them are unstable. Proof: The proof is similar to Theorem 2 and will be omitted. Lemma 1. It is easy to see that every solution of Eq.(3) is unbounded except in the case x¡3 x0 = x¡1 x¡4 . Theorem 5 Eq.(3) has a periodic solution of period twelve i¤ x¡3 x0 = x¡1 x¡4 : Moreover the periodic solution has the following form ( ) x¡1 x¡4 x0 x¡3 x¡4 ; x¡3 ; x¡2 ; x¡1 ; x0 ; x¡2 (¡1+x ; ; x ; x (¡1 + x x ) ; ¡3 ¡2 ¡1 ¡4 ¡1 x¡4 ) x¡1 (¡1+x¡3 x0 ) : x¡1 x¡4 x¡1 ; (¡1+xx0¡3 x0 ) ; x¡2 (¡1+x ; x¡4 ; x¡3 ; ::: ¡1 x¡4 ) Proof: First suppose that there exists a prime period twelve solution of Eq.(3) of the following form ) ( x¡1 x¡4 x0 x¡3 ; ; x ; x (¡1 + x x ) ; x¡4 ; x¡3 ; x¡2 ; x¡1 ; x0 ; x¡2 (¡1+x ¡3 ¡2 ¡1 ¡4 ¡1 x¡4 ) x¡1 (¡1+x¡3 x0 ) : x¡1 x¡4 x¡1 ; (¡1+xx0¡3 x0 ) ; x¡2 (¡1+x ; x ; x ; ::: ¡4 ¡3 ¡1 x¡4 ) Then we see from Theorem 3 that x12n¡4 = x12n¡2 = x12n = x12n+2 = x12n+3 = x12n+4 = x12n+5 =

2n n x2n 0 x¡3 (¡1+x¡1 x¡4 ) n 2n¡1 (¡1+x 2n x ) x¡1 x¡4 ¡3 0

2n n x¡2 x2n 0 x¡3 (¡1+x¡1 x¡4 ) n 2n (¡1+x x ) x2n x ¡3 0 ¡1 ¡4 n x02n+1 x2n ¡3 (¡1+x¡1 x¡4 ) n 2n 2n x¡1 x¡4 (¡1+x¡3 x0 )

x12n¡3 =

=x ;

x

= x0 ; x

2n+1 x02n+1 x¡3 (¡1+x¡1 x¡4 )n 2n+1 2n x¡1 x¡4 (¡1+x¡3 x0 )n+1 2n+1 2n+1 x¡1 x¡4 (¡1+x¡3 x0 )n n (¡1+x x02n+1 x2n ¡1 x¡4 ) ¡3

2n x2n ¡1 x¡4

(¡1+x¡3 x0 )n n 2n¡1 (¡1+x 2n x0 x¡3 ¡1 x¡4 ) 2n+1 2n x¡1 x¡4 (¡1+x¡3 x0 )n n 2n ¡2 12n¡1 x2n 0 x¡3 (¡1+x¡1 x¡4 ) 2n+1 2n+1 x¡1 x¡4 (¡1+x¡3 x0 )n n+1 12n+1 2n x¡2 x0 x2n ¡3 (¡1+x¡1 x¡4 )

= x¡4 ;

=

= x¡3 ;

=

=

=

= x¡1 ; x¡1 x¡4 ; x¡2 (¡1+x¡1 x¡4 )

x0 x¡3 ; x¡1 (¡1+x¡3 x0 )

= x¡3 ;

2n+1 x¡2 x2n+1 x¡3 (¡1+x¡1 x¡4 )n+1 0 2n+1 2n+1 (¡1+x¡3 x0 )n x¡1 x¡4 2n+2 2n+1 x¡1 x¡4 (¡1+x¡3 x0 )n+1 2n+1 2n+1 x0 x¡3 (¡1+x¡1 x¡4 )n+1

= x¡2 (¡1 + x¡1 x¡4 ) ;

= x¡1 ;

x12n+6 =

2n+1 x02n+2 x¡3 (¡1+x¡1 x¡4 )n 2n+1 2n+1 x¡1 x¡4 (¡1+x¡3 x0 )n+1

x12n+7 =

x¡1 x¡4 (¡1+x¡3 x0 )n 2n+1 n+1 x¡2 x2n+1 x (¡1+x ¡1 x¡4 ) 0 ¡3

=

x0 ; (¡1+x¡3 x0 )

2n+2 2n+2

=

x¡1 x¡4 : x¡2 (¡1+x¡1 x¡4 )

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Then we get (¡1 + x¡3 x0 ) = (¡1 + x¡1 x¡4 ) : Second assume that (¡1 + x¡3 x0 ) = (¡1 + x¡1 x¡4 ) : Then we see from the form of the solution of Eq.(3) that x12n¡4 = x¡4 ; x12n¡3 = x¡3 ; x12n¡2 = x¡2 ; x12n¡1 = x¡1 ; x12n = x0 ; x¡1 x¡4 x0 x¡3 x12n+1 = x¡2 (¡1+x ; x12n+2 = x¡1 (¡1+x ; x12n+3 = x¡1xx0 ¡4 = x¡3 ; ¡1 x¡4 ) ¡3 x0 ) x12n+4 = x¡2 (¡1 + x¡1 x¡4 ) ; x12n+5 = x¡1 ; x¡1 x¡4 x0 x12n+6 = ¡1+x ; x12n+7 = x¡2 (¡1+x : ¡3 x0 ¡1 x¡4 )

Thus we have a periodic solution of period twelve and the proof is complete. Theorem 6 n Eq.(3) has a periodic solution of period six oi¤ x¡1 x¡4 = x¡3 x0 = 2 and 2 has the form x¡4 ; x¡3 ; x¡2 ; x¡1 ; x0 ; x¡2 ; x¡4 ; x¡3 ; x¡2 ; ::: :

Proof: The proof is consequently from the previous Theorems and will be omitted. In the following we present some …gures illustrate the behavior of the solutions of Eq.(3) under some di¤erent initial values. plot of x(n+1)= x(n-1)x(n-4)/(x(n-2)(-1+x(n-1)x(n-4))

plot of x(n+1)= x(n-1)x(n-4)/(x(n-2)(-1+x(n-1)x(n-4))

60

3

50 2.5

40

x(n)

x(n)

2 30

1.5 20

1

10

0

0

10

20

30 n

40

50

0.5

60

Figure 3. x¡4 = 3; x¡3 = 2; x¡2 = 5; x¡1 = 4; x0 = 6:

0

10

20

30 n

40

50

60

Figure 4. x¡4 = 3; x¡3 = 2; x¡2 = 3; x¡1 = 2=3; x0 = 1:

The following cases can be treated similarly.

4

On the Equation xn+1 =

xn¡1xn¡4 xn¡2 (1¡xn¡1 xn¡4 )

In this section we get the solution of the third following equation xn+1 =

xn¡1 xn¡4 ; xn¡2 (1¡xn¡1 xn¡4 )

n = 0; 1; :::;

(4)

where the initial values are arbitrary positive real numbers.

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Theorem 7 Assume that fxn g1 n=¡4 be a solution of Eq.(4). Then for n = 0; 1; ::: x6n¡2 = x6n = x6n+2 = x6n+3 =

n x¡2 xn 0 x¡3 n n x¡1 x¡4

xn+1 xn ¡3 0 n x¡1 xn ¡4

n¡1 Y³

i=0 n¡1 Y³ i=0

xn+1 xn+1 0 ¡3

(1¡(3i+1)x¡1 x¡4 ) (1¡(3i+2)x0 x¡3 )

1¡(3i+2)x¡1 x¡4 1¡(3i+3)x0 x¡3

n xn+1 ¡1 x¡4 (1¡x0 x¡3 )

n¡1 Y³

; x6n¡1 =

i=0

´

n xn+1 ¡1 x¡4 n n x0 x¡3

n¡1 Y³ i=0

(1¡(3i+1)x0 x¡3 ) (1¡(3i+3)x¡1 x¡4 )

n+1 xn+1 ¡1 x¡4 n n x¡2 x0 x¡3 (1¡x¡1 x¡4 )

; x6n+1 =

(1¡(3i+3)x¡1 x¡4 ) (1¡(3i+4)x0 x¡3 )

i=0 n¡1 Y³

n+1 xn+1 ¡1 x¡4 n+1 n x0 x¡3 (1¡2x¡1 x¡4 )

´

´

n¡1 Y³ i=0

´

;

1¡(3i+2)x0 x¡3 1¡(3i+4)x¡1 x¡4

´

;

;

(1¡(3i+3)x0 x¡3 ) (1¡(3i+5)x¡1 x¡4 )

´

:

Theorem 8 Eq.(4) has the unique equilibrium point x = 0 and it is unstable. Example 3. Consider Eq.(4) with x¡4 = 3; x¡3 = 5; x¡2 = 2; x¡1 = 2=3; x0 = 0:4: See Fig. 5. plot of x(n+1)= x(n-1)x(n-4)/(x(n-2)(1-x(n-1)x(n-4)) 5

4

3

x(n)

2

1

0

-1

-2

-3

0

10

20

30

40

50 n

60

70

80

90

100

Figure 5.

5

On the Equation xn+1 =

xn¡1 xn¡4 xn¡2 (¡1¡xn¡1 xn¡4)

Here we obtain the analytical form of the solutions of the equation xn+1 =

xn¡1 xn¡4 ; xn¡2 (¡1¡xn¡1 xn¡4 )

(5)

n = 0; 1; :::;

where the initial values are arbitrary non zero real numbers with x¡1 x¡4 6= ¡1; x¡3 x0 6= ¡1: Theorem 9 Let fxn g1 n=¡4 be a solution of Eq.(5). Then for n = 0; 1; 2; ::: the solution of Eq.(5) is given by x12n¡4 = x12n¡2 =

2n n x2n 0 x¡3 (¡1¡x¡1 x¡4 ) n ; 2n¡1 (¡1¡x 2n x¡1 x¡4 ¡3 x0 )

x12n¡3 =

2n n x¡2 x2n 0 x¡3 (¡1¡x¡1 x¡4 ) n ; 2n (¡1¡x x ) x2n x ¡3 0 ¡1 ¡4

x12n¡1 =

2n x2n ¡1 x¡4

(¡1¡x¡3 x0 )n n 2n¡1 (¡1¡x 2n x0 x¡3 ¡1 x¡4 ) 2n+1 2n x¡1 x¡4 (¡1¡x¡3 x0 )n n 2n (¡1¡x x2n x ¡1 x¡4 ) 0 ¡3

;

;

9

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x12n = x12n+2 = x12n+4 = x12n+6 =

n x2n+1 x2n 0 ¡3 (¡1¡x¡1 x¡4 ) n ; 2n (¡1¡x x ) x2n x ¡3 0 ¡1 ¡4

x12n+1 =

2n+1 x2n+1 x¡3 (¡1¡x¡1 x¡4 )n 0 ; 2n+1 2n x¡1 x¡4 (¡1¡x¡3 x0 )n+1

2n+1 2n+1 x¡1 x¡4 (¡1¡x¡3 x0 )n 2n n+1 ; x¡2 x2n x 0 ¡3 (¡1¡x¡1 x¡4 )

x12n+3 =

x¡2 x02n+1 x2n+1 (¡1¡x¡1 x¡4 )n+1 ¡3 ; 2n+1 2n+1 (¡1¡x¡3 x0 )n x¡1 x¡4 2n+1 x2n+2 x¡3 (¡1¡x¡1 x¡4 )n 0 ; 2n+1 2n+1 x¡1 x¡4 (¡1¡x¡3 x0 )n+1

2n+1 x2n+1 (¡1¡x¡3 x0 )n ¡1 x¡4 n; 2n+1 2n (¡1¡x x0 x¡3 ¡1 x¡4 )

x12n+5 =

x12n+7 =

2n+2 2n+1 x¡1 x¡4 (¡1¡x¡3 x0 )n+1 ; 2n+1 2n+1 x0 x¡3 (¡1¡x¡1 x¡4 )n+1

2n+2 x2n+2 ¡1 x¡4

2n+1 x¡2 x02n+1 x¡3

(¡1¡x¡3 x0 )n : (¡1¡x¡1 x¡4 )n+1

Theorem 10 Eq.(5) has x = 0 as a unique equilibrium point which is unstable. Lemma 2. It is easy to see that every solution of Eq.(5) is unbounded except in the case x¡3 x0 = x¡1 x¡4 . Theorem 11 Eq.(5) has a periodic solution of period twelve i¤ x¡3 x0 = x¡1 x¡4 . x¡1 x¡4 Moreover the periodic solution has the form fx¡4 ; x¡3 ; x¡2 ; x¡1 ; x0 ; x¡2 (¡1¡x ; ¡1 x¡4 ) x0 x¡3 ; x¡3 ; x¡2 x¡1 (¡1¡x¡3 x0 )

x¡1 x¡4 x0 (¡1 ¡ x¡1 x¡4 ) ; x¡1 ; ¡1¡x ; ; x¡4 ; :::g ¡3 x0 x¡2 (¡1¡x¡1 x¡4 )

Theorem 12 Eq.(5) has a periodic solution of period six i¤ x¡3 x0 = x¡1 x¡4 = ¡2 and will be taken the form fx¡4 ; x¡3 ; x¡2 ; x¡1 ; x0 ; x¡2 ; x¡4 ; :::g: ¡2 Example 4. Fig. 6 below shows the behavior of the solution of Eq.(5) whenever x¡4 = 3; x¡3 = 5; x¡2 = ¡7; x¡1 = 4; x0 = 2. plot of x(n+1)= x(n-1)x(n-4)/(x(n-2)(-1-x(n-1)x(n-4)) 80 70 60 50

x(n)

40 30 20 10 0 -10

0

50

100

150

n

Figure 6.

Acknowledgements This Project was funded by the Deanship of Scienti…c Research (DSR), King Abdulaziz University, Jeddah, Saudi Arabia under grant no. (145/662/1434). The authors, therefore, acknowledge with thanks DSR technical and …nancial support. Last, but not least, sincere appreciations are dedicated to all our colleagues in the Faculty of Science, Rabigh branch for their nice wishes. 10

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References [1] R. P. Agarwal and E. M. Elsayed, On the solution of fourth-order rational recursive sequence, Advanced Studies in Contemporary Mathematics, 20 (4) (2010), 525–545. axn¡1 [2] C. Cinar, On the positive solutions of the di¤erence equation xn+1 = 1+bx ; n xn¡1 Appl. Math. Comp., 156 (2004) 587-590. [3] E. M. Elabbasy , H. El-Metwally and E. M. Elsayed, Global behavior of the solutions of di¤erence equation, Advances in Di¤erence Equations 2011, 2011:28 doi:10.1186/1687-1847-2011-28. [4] E. M. Elabbasy and E. M. Elsayed, Global attractivity and periodic nature of a di¤erence equation, World Applied Sciences Journal, 12 (1) (2011), 39–47. [5] H. El-Metwally, On the dynamics of a higher order di¤erence equation, Discrete Dynamics in Nature and Society, Volume 2012 (2012), Article ID 263053, 8 pages. [6] H. El-Metwally and E. M. Elsayed, Form of solutions and periodicity for systems of di¤erence equations, Journal of Computational Analysis and Applications, 15(5) (2013), 852-857. [7] H. El-Metwally and E. M. Elsayed, Solution and Behavior of a Third Rational Di¤erence Equation, Utilitas Mathematica, 88 (2012), 27–42. [8] E. M. Elsayed, Dynamics of recursive sequence of order two, Kyungpook Mathematical Journal, 50 (2010), 483-497. [9] E. M. Elsayed, Solutions of rational di¤erence system of order two, Mathematical and Computer Modelling, 55 (2012), 378–384. [10] R. Karatas and A. Geli¸sken, Qualitative behavior of a rational di¤erence equation, Ars Combinatoria, 100 (2011), 321–326. [11] R. Karatas, C. Cinar and D. Simsek, On positive solutions of the di¤erence n¡5 equation xn+1 = 1+xxn¡2 ; Int. J. Contemp. Math. Sci., 1(10) (2006), 495-500. xn¡5 [12] V. L. Kocic and G. Ladas, Global Behavior of Nonlinear Di¤erence Equations of Higher Order with Applications, Kluwer Academic Publishers, Dordrecht, 1993. [13] M. R. S. Kulenovic and G. Ladas, Dynamics of Second Order Rational Di¤erence Equations with Open Problems and Conjectures, Chapman & Hall / CRC Press, 2001. [14] C. Wang, F. Gong, S. Wang, L. Li and Q. Shi, Asymptotic behavior of equilibrium point for a class of nonlinear di¤erence equation, Advances in Di¤erence Equations, Volume 2009, 2009, Article number 214309 [15] I. Yalç¬nkaya, On the di¤erence equation xn+1 = ® + xn¡m , Discrete Dynamics xkn in Nature and Society, Vol. 2008, Article ID 805460, 8 pages, doi: 10.1155/2008/ 805460. [16] I. Yalç¬nkaya, C. Cinar and M. Atalay, On the solutions of systems of di¤erence equations, Advances in Di¤erence Equations, Vol. 2008, Article ID 143943, 9 pages, doi: 10.1155/2008/ 143943. [17] E. M. E. Zayed and M. A. El-Moneam, On the rational recursive sequence xn+1 = ®+¯xn +°xn¡1 ; Communications on Applied Nonlinear Analysis, 12 (4) (2005), 15–28. A+Bxn +Cxn¡1

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Worse-Case Conditional Value-at-Risk for Asymmetrically Distributed Asset Scenarios Returns Zhifeng Dai1,2 1

Donghui Li3 ,

Fenghua Wen2

∗

College of Mathematics and Computational Science, Changsha University of Science and Technology, Changsha 410114, China. 2

3

College of business, Central South University, Hunan, 410083, China.

School of Mathematical Sciences, South China Normal University Guangzhou, 510631, China.

Abstract: Many studies have reported empirical evidence of asymmetries in asset return distributions. Meanwhile, optimal solutions to the Conditional Value-at-Risk (CVaR) minimization are highly susceptible to estimation error of the risk measure because the estimate depends on only a small portion of sampled scenarios. In this paper, based on the robust optimization techniques Chen et al.(2007)[19], we propose a computationally tractable worst-case Conditional Value-at-Risk (CVaR). In the situation, the sampled scenario returns are generated by a factor model with some asymmetric affine uncertainty set. The remarkable characteristic of the new method is that the robust optimization model retains the complexity of original portfolio optimization problem, i.e., the robust counterpart problem is still a linear programming problem. Moreover, it takes into consideration asymmetries in the distributions of scenarios returns used for defining CVaR. We present some numerical experiments with simulated and real market data to illustrate the behavior of the robust optimization model. Keywords: Portfolio optimization, Conditional value at risk(CVaR), Robust optimization, Linear programming(LP).

1.

Introduction

Portfolio optimization problem is an attractive and important research topic since the pioneering Markowitz work on optimal portfolio selection [1]. It is now well known that while mean-variance optimization is appropriate for symmetrically distributed portfolio returns, it results in unsatisfactory asset allocations when returns are asymmetrically distributed, or when downside risk is more weighted than upside risk. ∗

E-mail: [email protected].

1

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Since the middle of 1990s, Value-at-Risk (VaR, [4]), a new measure of downside risk, has become popular in financial risk management. It has even been recommended as a standard on banking supervision by the Basel Committee. However, Critics have pointed out numerous shortcomings of VaR [5]. On the other hand, Conditional Value-at-Risk (CVaR), defined as the mean of the tail distribution exceeding VaR, has attracted much attention in recent years. As a measure of risk, CVaR exhibits some better properties than VaR. First, it can deal with the asymmetric distribution of asset return better than mean-variance analysis, especially for assets with returns that are heavy-tailed. Secondly, minimizing CVaR usually results in solving a convex programming problem, such as a linear programming problem, which allows the decision maker to deal with a large-scale portfolio problem efficiently [6, 7]. Finally, Artzner et al.[5] demonstrate that CVaR is a coherent measure of risk, which has been widely accepted as a benchmark to evaluate risk measures. All these stimulate the application of CVaR in practice, and CVaR is getting more and more popular in financial management. In fact, it is noted that in the process of portfolio selection, the original data brought to the model are not always accurate, i.e., it may be subject to some errors. Thus the result may be influenced by perturbations in the parameters. As pointed out by Black and Litterman [8], in the classical mean-variance model, the portfolio decision is very sensitive to the mean and the covariance matrix, especially to the mean. Chopra and Ziemba [9] showed that small changes in the input parameters can result in large changes in the optimal portfolio allocation. Thus, the modeling risk arises due to the uncertainty of the underlying probability distribution. Being aware of the importance of robustness in recent years, researchers from both finance and operations research have paid increasing attentions to the robust version of portfolio selection problems. Lobo and Boyd (2000)[10], Goldfarb and Iyengar (2003)[11] studied the robust portfolio problem under the mean-variance framework. Instead of assuming precise information on the mean and the covariance matrix of asset returns, they introduced some types of uncertainties, such as polyhedral uncertainty, box uncertainty and ellipsoidal uncertainty, in the parameters in determining the mean and the covariance matrix, and they then transformed the problem into semidefinite programs(SDP) or second-order cone programs(SOCP), which can be efficiently solved by interior-point algorithms developed in recent years. Halld´orsson and T¨ ut¨ uncu (2004) [12] applied their interior-point method for saddle-point problems to the robust mean-variance portfolio selection under the box uncertainty of the elements in the mean vector and the covariance matrix. El Ghaoui, Oks and Oustry (2003)[13] investigated the robust portfolio optimization problem using worst-case VaR, where only the first- and second-moment information on the distribution is available. Several formulations corresponding to various structures of partial information have been extensively exploited to derive the resulting portfolio selection problems in a form of a semidefinite program(SDP). Natarajan, Pachamanova, and Sim, (2008) [14] proposed a computationally tractable approximation method for minimizing the VaR of a portfolio based on robust optimization techniques in Chen et al.(2007)[19]. The method results in the optimization of a modified VaR

2

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measure, Asymmetry-Robust VaR, that takes into consideration asymmetries in the distributions of returns and is coherent. Zhu and Fukushima (2009)[15] further investigated the worst-case CVaR risk measure with several structures of uncertainty in the underlying distribution. They focus on the uncertainty in the probability distribution used for defining CVaR. Such a modeling is called distributionally robust modeling. It is true that the probability estimation itself is under uncertainty and we cannot know the true one. However, it is not easy to imagine what form of uncertainty set is proper for the probability measure. In this sense, employing the uncertainty of probability distribution may not provide investors with a satisfactory solution. On the other hand, since the estimate of CVaR is computed by using only an upper tail part of the loss distribution, a large number of samples are required for assuring the statistical reliability of the estimate. Especially when CVaR is employed as the objective of a portfolio optimization, a much larger number of samples are required for ensuring the accuracy of the optimal portfolio. In practice, however, the number of samples which is available for the estimation is limited, and the estimated CVaR and the resulting optimal portfolio may contain considerable estimation error. Meanwhile, many studies have reported empirical evidence of asymmetries and large kurtosis in asset return distributions. Empirically, however, there is evidence that both short- and longhorizon stock returns can be skewed and highly leptokurtic (Fama 1976 [22], Duffee 2002 [23]). Furthermore, the returns of portfolios involving derivatives or credit risky assets can have extremely left-skewed distributions (Sch¨onbucher 2000 [24]). More recently, Ang and Chen (2002)[25] find that the asymmetries in the data reject the null hypothesis of multivariate normal distributions. Conine and Tamarkin (1981) [26] also claim that though diversification can change skewness exposure, the remaining idiosyncratic skewness is relevant in asset pricing and thus portfolio optimization under asymmetric distribution is a significant topic for research. In this paper, we further study the Worse-Case Conditional Value-at-Risk by supposing the sampled scenario returns are generated by a factor model with some asymmetric affine uncertainty set in order to Mitigate the fragility of CVaR-based portfolio optimization problem. Motivated by the works in Chen et al.(2007)[19], we provide a computationally tractable robust optimization method for minimizing the Worse-Case CVaR of a portfolio. Moreover, it takes into consideration asymmetries in the distributions of returns used for defining CVaR. Notations: Throughout this paper, we use boldface letter such as x for vector to distinguish it from scalar x.

2.

Conditional value-at-risk (CVaR)

The conditional value-at-risk (CVaR) has gained growing popularity in financial risk management due to the coherence property and tractability in its optimization. Let f (x , y ) be the loss associated with the decision vector x , to be chosen from a certain subset X of Rn , and the random vector y in Rm . For convenience, the underling probability of y will be

3

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assumed to have a density function p(.). The probability of f (x , y ) not exceeding a threshold α is then given by Z

Ψ(x , α) =

f (x ,y )≤α

p(y )dy .

(2.1)

As a function of α for fixed x , Ψ(x , α) is the cumulative distribution function for the loss associated with x . For a confidence level β and a fixed x ∈ X the value-at-risk, denoted by VaRβ (x ) is defined as VaRβ (x ) = min{α ∈ R : Ψ(x , α) ≥ β}.

(2.2)

The conditional value-at-risk, denoted by CVaRβ (x ), is defined as the expected value of the loss that exceeds VaRβ (x ), that is, CVaRβ (x ) = (1 − β)

−1

Z f (x ,y )≥VaRβ (x )

f (x , y )p(y )dy .

(2.3)

The CVaR is a coherent risk measure [5]. We note that the problem involved CVaRβ (x ) is difficult to proceed due to its convoluted and implicit version. Rockafellar and Uryasev made a remarkable contribution in [6] by introducing a simpler auxiliary function Fβ on X × R, defined by Fβ (x , α) = α + (1 − β)−1

Z

y

[f (x , y ) − α]+ p(y )dy ,

(2.4)

∈Rm

In practice, the probability density function p(y ) is often not available, or is very difficult to estimate. Instead, we might have T different scenarios Y = (y [1] , y [2] , . . . , y [T ] ) that are sampled from the probability distribution or that have been obtained from computer simulations. Evaluating the auxiliary function F˜β (x , α) using the scenarios Y, we have F˜β (x , α) = α + (1 − β)−1

T X

πt [f (x , y [t] ) − α]+ ,

(2.5)

t=1

where y [t] denotes the tth sample (the subscript [t] is used to distinguish a vector from a scalar) generated by simple random sampling with respect to x according to its density function p(.), and T denotes the number of samples, where πt are probabilities of scenarios y [t] . If πt is equal to T−1 for all t, then (2.5) reduces to F˜α (x , α) = α +

T X 1 [f (x , y [t] ) − α]+ . T (1 − β) t=1

(2.6)

Obviously, F˜α (x , α) is convex and piecewise linear with respect to α. Further, F˜α (x , α) is convex for (x , α) if f (x , y ) is convex (see Theorem 2 in [6]). Replacing [f (x , y [t] ) − α]+ by the auxiliary variables dt along with appropriate constraints, we obtain the equivalent optimization problem min n T (x ,d ,α)∈R ×R ×R s.t.

α+

T X 1 dt , T (1 − β) t=1

x ∈X dt ≥ f (x , y [t] ) − α,

t = 1, . . . , T,

(2.7)

d ≥ 0. 4

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Generally, the loss and return functions of portfolio allocation are chosen by: f (x , y ) = −x T y ,

Rp (x) = Ep [x T y ] = x T Ep [y ] = x T r ,

(2.8)

in which y is the vector of the assets’ return, r is the vector of the expected assets’ return, and x T r is the mean return of the portfolio. Hence, adding an auxiliary variable θ ∈ R, the minimization model of CVaR (2.9) becomes the following linear programming (LP) problem with variables (x , d , α, θ) ∈ Rn × RT × R × R. min

θ

s.t.

x ∈X α+

T X 1 dt ≤ θ, T (1 − β) t=1

dt ≥ −x T y [t] − α,

(2.9)

t = 1, . . . , T,

d ≥ 0. Portfolio optimization tries to find an optimal trade-off between the risk and the return according to the investor’s preference. Thus, the portfolio selection problem using CVaR as a risk measure can be represented as min CVaRβ (x ) x ∈X where X denotes the constraint on the portfolio position, which usually includes the budget constraint and no short sales constraint x T 1 = 1,

x ≥ 0.

(2.10)

Let µ be the smallest expected return of the portfolio required by the investor. From (2.8), this return requirement can be represented as x T r ≥ µ.

(2.11)

Therefore, the feasible decision set of portfolios can be denoted as X = {x |

x T 1 = 1,

x T r ≥ µ}.

x ≥ 0,

(2.12)

From (2.9) and 2.12, the mean-CVaR Portfolio optimization can be be written as the following linear program min

θ

s.t.

α+

T X 1 dt ≤ θ, T (1 − β) t=1

dt ≥ −x T y [t] − α,

t = 1, . . . , T,

(2.13)

d ≥ 0. x T 1 = 1,

x ≥ 0, 5

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3.

Worst-Case Conditional value-at-risk (CVaR)

However, optimal solutions to the CVaR minimization are highly susceptible to estimation error of the risk measure because the estimate depends on only a small portion of sampled scenarios, for example Y = (y [1] , y [2] , . . . , y [T ] ). A practical way to alleviate the effect of such a perturbation is to employ a statistical model. For example, Konno, Waki and Yuuki (2002) replace the observed returns Y = (y [1] , y [2] , . . . , y [T ] ) in 2.6 with values estimated by a regression approach. Based on the robust optimization techniques in Chen et al.(2007)[19], we suppose that future asset returns r˜ are generated by the following factor model r = r 0 + ∆r z , z ∈ C

(3.1)

in which r 0 is a vector of expected returns, and ∆r is a matrix of factor loadings. The factors z are stochastically independent with following support set n

o

−1 −1 C = z : ∃v, w ∈ RN + , z = v − w , kP v + Q wk ≤ Ω ,

(3.2)

and P = diag(p1 , . . . , pN ), Q = diag(q1 , . . . , qN ). The parameters pj > 0 and qj > 0 are the ”forward”and the ”backward” deviations of random variable zj , j = 1, . . . , N , respectively. The uncertainty set C is convex, and its size is controlled by Ω. Intuitively speaking, the uncertain factors z are decomposed into two random variables: v = max{z , 0} and w = max{−z , 0}, so that z = v − w . The multipliers

1 pj

and

1 qj

normalize the effective perturbation contributed by

both v and w such that the norm of the aggregated values falls within the budget of uncertainty. Therefore, considered sampling error of the samples, we present the Sample-based Worst-Case CVaR, its mathematical definition is as follows: WSCVaRβ (x ) =

sup

CVaRβ (x ), )

(3.3)

(r 1 ,...,r T )∈SΩ

where n

o

SΩ = r t : r t = r 0t + ∆r t z t , z t ∈ Ct ) , n

(3.4) o

−1 −1 Ct = z t : ∃v, w ∈ RN + , z t = v t − w t , kPt vt + Qt wt k ≤ Ω .

(3.5)

Next, we prove the WSCVaR 3.3 is a coherent risk measure. Theorem 3.1 If (r1 , . . . , rT ) ∈ SΩ §then WSCVaR is a coherent risk measure. Proof. Letting ρ(x ) = CVaRβ (x ), ρw (x ) = WSCVaRβ (x ), we have ρw (x ) =

sup

ρ(x ).

(r 1 ,...,r T )∈SΩ

As CVaRβ (x ) is a coherent risk measure, so ρ(x ) satisfies four axioms of Coherent risk measure. In what following, we prove ρw (x ) also satisfies four axioms of Coherent risk measure. 6

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• Monotonicity: if x < y , then ρ(x ) < ρ(y ). Therefore ρw (x ) =

sup

ρ(x )
0, we have ρw (λx ) =

sup

ρ(λx ) = λ

(r 1 ,...,r T )∈SΩ

sup

ρ(x ) = λρw (x );

(r 1 ,...,r T )∈SΩ

• translation invariance: for any constant a ∈ R, we have ρw (x + a) =

sup

ρ(x + a) =

(r 1 ,...,r T )∈SΩ

sup

ρ(x ) + a = ρw (x ) + a.

(r 1 ,...,r T )∈SΩ

Therefore, the theorem is true. Chen, Sim and Sun [19] stated the uncertainty set SΩ is convex, and its size is determined by Ω. Therefore, SΩ is a compact convex set. Let f (x , y ) = −x T r be the loss associated with the decision vector x , to be chosen from a certain subset X of Rn , and the random vector r in Rm . So, from 2.6, WSCVaR can be converted to the following form: WSCVaRβ (x ) =

T n o X 1 max min α + max{−r Tt x − α, 0} . T (1 − β) t=1 (r 1 ,...,r T )∈SΩ

(3.6)

Next, we will show the WSCVaR enjoys an important nature, in the process the dual-norm kuk∗ , (see Bertsimas and Sim [18]) is required. It is defined as: kuk∗ = max u T x . {kx k≤1} Theorem 3.2 If (r1 , . . . , rT ) ∈ SΩ , we have T X Ω WSCVaRβ (x) = CVaRβ (x) + kut k∗ . T (1 − β) t=1

(3.7)

Proof. From 3.6, we can obtain T o X 1 WSCVaRβ (x ) = max min α + max{−r Tt x − α, 0} T (1 − β) t=1 (r 1 ,...,r T )∈SΩ

n

T o X 1 = max min α + max{−(r 0t )T x − (∆r t z t )T x − α, 0} z t ∈Ct T (1 − β) t=1

n

T n o X 1 (∆r t z t )T x = CVaRβ (x ) + max max z t ∈Ct T (1 − β) t=1

= CVaRβ (x ) +

T n o X 1 max z Tt y t , y t = ∆r Tt x . T (1 − β) t=1 z t ∈Ct

7

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Observe that max z Tt y t

{z t ∈Ct }

= {vt ,wt ∈R

(v t − w t )T y t max −1 −1 Pt vt +Qt wt k≤Ω}

N + :k

max (Pt y t )T v t − (Qt y t )T w t {vt ,wt ∈R v t +w t k≤Ω} ∗ = Ωkut k =

N + :k

where ut = max{Pt y t , −Qt y t , 0} = max{Pt y t , −Qt y t } Note: Theorem 3.7 indicates that the WSCVaR can be seen as the original CVaR plus a regular item. It is easy to know that CVaRβ (x ) ≤ WSCVaRβ (x ). Obvious, WSCVaR is more cautious than the original CVaR.

4.

Computing WSCVaR and its application in portfolio management

By the Chen, Sim and Sun [19] Theorem 2 and Theorem 3.2, adding an auxiliary variable ht ∈ R, t = 1, 2, . . . , T , the WSCVaR (3.7) can be transformed into the following form T T X X 1 Ω dt + + ht , T (1 − β) t=1 T (1 − β) t=1

min

α+

s.t.

ku t k∗ ≤ ht , t = 1, 2, . . . T, ut ≥ −Pt ∆r Tt x , t = 1, 2, . . . T,

(4.1)

ut ≥ Qt ∆r Tt x , t = 1, 2, . . . T, dt ≥ (r 0t )T x − α,

t = 1, . . . , T,

d ≥ 0. The complete formulation and complexity class of the robust counterpart depends on the representation of the dual norm constraint, ku t k∗ ≤ ht , t = 1, 2, . . . T . Table 1 lists the common choices of norms, the representation of their dual norms which is come from reference [18](See page 14, Table 2). Table 1: Representation of the dual norm for u ≥ 0. Norms

ktk

kuk∗ ≤ h

l2

ktk2

kuk2 ≤ h

l1

ktk1

uj ≤ h, ∀j = {1, . . . , N }

l∞

ktk∞

N P

uj ≤ h

j=1

l1

T

l∞

max{ Ω1 ktk1 , ktk∞ }

Ωδ +

N P j=1

N vj ≤ h; vj + δ ≥ uj , ∀j ∈ N ; δ ∈ R+ , v ∈ R+

8

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In [18], Bertsimas and Sim discussed the nature and size of the proposed robust conic problem. In terms of keeping the model linear and simplicity in size, the l1 norm also is an attractive choice. In this paper, we adopt l1 norm. So under l1 norm, the constraints ku t k∗ ≤ ht , t = 1, 2, . . . T in (4.1) is equivalent to ujt ≤ ht , ∀j = {1, . . . , N }, t = 1, 2, . . . T.

(4.2)

Hence, the resulting problem (4.2) is still a linear constraint. For the constraint term ut ≥ −Pt ∆r Tt x , t = 1, 2, . . . T in (4.1), as discussed in [18], when all the data entries of the problem have independent random perturbation, we can further reduce the size of the robust model. In this article, we assume that the dimension of x and u is identical (n=N), that is, zjt in (3.4)is the independent random variable associated with the j-th data element, and ∆r j contains mostly zeros except at the entries corresponding to the data element, such as ∆r jt = (0, . . . , 0, ∆rtj , 0, . . . , 0)T . Then ujt ≥ −pjt (∆r jt )T x will reduce to ujt ≥ −pjt ∆rtj · xj . Then, the constraint term ut ≥ −Pt ∆r Tt x , t = 1, 2, . . . T in (4.1) can be transformed into the following form ujt ≥ −pjt ∆rtj · xj , j = 1, . . . , n, t = 1, 2, . . . T.

(4.3)

Based on investor preferences, portfolio optimization try to find the balance between risk and return. Therefore, the WSCVaR-based portfolio problem can be expressed as min x ∈X

WSCVaRβ (x ),

where X denotes the constraint on the portfolio position, which usually includes the budget constraint, no short sales constraint, and the return requirement. Therefore, the feasible decision set of portfolios can be denoted as X = {x |

x T 1 = 1,

x ≥ 0,

x T r ≥ µ}.

(4.4)

From (4.1) (4.2) and (4.3), adding an auxiliary variable θ ∈ R, the AWCVaR-based robust portfolio selection problem can be written as the following linear programming problem with variables (x , d , u t , ht , θ, α) min

θ

s.t.

α+

T T X X Ω 1 dt + + ht ≤ θ, T (1 − β) t=1 T (1 − β) t=1

ujt ≤ ht , ∀j = {1, . . . , N }, t = 1, 2, . . . T, ujt ≥ −pjt ∆rtj · xj , j = 1, . . . , n, t = 1, 2, . . . T,

(4.5)

ujt ≥ qtj ∆rtj · xj , j = 1, . . . , n, t = 1, 2, . . . T, dt ≥ (r 0t )T x − α,

t = 1, . . . , T,

d ≥ 0, u t ≥ 0, x T 1 = 1,

x ≥ 0, 9

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5.

Computational Experiments

We compare the performance of minimizing-portfolio WSCVaR under our approach with the initial CVaR method [6]. Firstly, we use simulated asset returns and show that our WSCVaR approach performs well for negatively-skewed returns. Secondly, we compare initial CVaR method and the robust portfolio optimization methods by employing a widely available data set of Hedge Funds returns, from http://www.hedgeindex.com. In our numerical experiments, the methods have the following meanings: • ”CVaR” stands for the initial mean-CVaR Portfolio optimization model (2.13)[6]; • ”WSCVaR” stands for the robust mean-WSCVaR Portfolio optimization model (4.5). We utilize Matlab2012 to solve models CVaR and WSCVaR, which are linear programming problems.

5.1.

Experiments with Simulated Data

Consider a portfolio of n = 20 assets with uncertain returns r˜it , i = 1, . . . , n, t = 1, . . . , T . Each return r˜it is determined by a simple single factor model r˜it = rˆit + z˜(ωit ), where rˆit = 1. The factors z˜t (ωi ) are independent and distributed as follows:  √ t ωi (1−ωit )   , with probability t √ωit z˜(ωit ) = t) (1−ω ω  i i  − , with probability 1−ω t i

ωit ,

(5.1)

1 − ωit .

Note that the mean and the standard deviation of z˜(ωit ) are the same for all ωit ∈ (0, 1) - they are 0 and 1, respectively. However, the degree of symmetry of z˜t (ωit ) can be different. Higher values for ωit (e.g., ωit = 0.9) result in larger negative skew. We generate values for ωit as follows: ωit =

i  1 1+ , i = 1, . . . , n, t = 1, . . . , T. 2 N +t

(5.2)

Therefore, the return distributions for stocks with high index numbers in the portfolio are more negatively skewed than those for stocks with low index numbers. We use exact values for the parameters in the CVaR and WSCVaR optimization problems. These parameters include the standard deviation and average returns for the CVaR approaches, and the backward and forward deviations for the WSCVaR approach are set to ptj = 1.5, qjt = 2. ∆rjt is set to the vector of standard deviation of asset returns estimated by the T samples. We use a training set of 1,000 simulated returns from the above distributions that is T = 1000. The optimal portfolio allocations resulting from the five approximate CVaR optimization approaches for β = 1% are shown in Figure 1.

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0.1 WSCVaR CVaR

0.09 0.08

Asset weight

0.07 0.06 0.05 0.04 0.03 0.02 0.01

0

5

10 Asset number

15

20

Figure 1-Optimal portfolio weights (as proportions) for assets numbered 1 through 20 resulting from different optimization formulations. The behavior of the CVaR approach is erratic. In fact, the optimal weights for the portfolios found by the CVaR approach vary widely from sample to sample. WSCVaR is able to detect the asymmetry in the distributions, and allocates less in assets with more negatively skewed return distributions (those with high index numbers).

5.2.

Experiments with Hedge Funds

We select 12 Credit Suisse/Tremont Hedge Fund Indices (listed in Table 2) as the candidates for constructing hedge fund portfolios. Monthly returns of these indices, from January 1994 to December 2012 (240 samples in total) are used as the data set, which can be freely downloaded from http://www.hedgeindex.com. Table 2: Credit Suisse/Tremont Hedge Fund Indices 1

Convertible Arbitrage

2

Dedicated Short Bias

3

Emerging Markets

4

Equity Market Neutral

5

Event Driven

6

Distressed

7

Multi-Strategy

8

Risk Arbitrage

9

Fixed Income Arbitrage

10

Global Macro

11

Long/Short Equity

12

Managed Futures

11

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To construct an optimal portfolio with an accuracy to certain degree, we need to generate adequate scenarios with the given 240 samples. A question we face first in scenario generation is which distribution the asset returns follow. Statistic test shows that most of the distributions of returns of these hedge fund indices are skewed and exhibit a high kurtosis. Thus, the returns should not be modeled by a normal distribution. Table 3 shows the means and standard deviations of these 12 asset returns within three different but overlapped time periods. Each of these three time periods covers 100 months. The beginning and the end dates for each time period are specified in Table 3. We find that, for most assets, there exist remarkable differences among three periods for both the mean and the standard deviation, especially for the mean. For example, the mean of asset 4 during the time period of 1/31/1994-4/30/2002 is 15 times of that during the time period of 6/30/2002-9/30/2010. Table 3: Mean and standard deviation of asset returns within different time periods Time

1/31/1994-4/30/2002

Asset

Mean

Std

Mean

Std

Mean

Std

1

0.0084

0.0143

0.0066

0.0147

0.0046

0.0254

2

0.0005

0.0534

-0.0003

3

0.0061

0.0554

0.0048

0.0448

0.0091

4

0.0090

0.0076

0.0070

0.0006

5

0.0094

0.0178

0.0080

0.0178

0.0072

0.0175

6

0.0110

0.0202

0.0092

0.0193

0.0070

0.0182

7

0.0086

0.0193

0.0073

0.0192

0.0074

0.0184

8

0.0078

0.0130

0.0056

0.0134

0.0041

0.0109

9

0.0057

0.0117

0.0039

0.0117

0.0029

0.0214

10

0.0117

0.0381

0.0088

0.0270

0.0087

0.0160

11

0.0107

0.0342

0.0096

0.0320

0.0062

0.0226

12

0.0038

0.0332

0.0062

0.0358

0.0075

0.0347

0.0094

8/31/1997-11/30/2005

0.0534

6/30/2002-9/30/2010

-0.0036

0.0454 0.0296 0.0424

Since the distribution of asset returns is unknown, we adopt a distribution free method to generate scenarios given in Topaloglou et al. (2002)[20] and Zhu et al. (2013)[16]. We use back test method to check the performances of the robust approaches and the traditional approach in portfolio management, and the initial wealth is set at 1. Firstly, asset returns of the first N=162 (from 1/31/1994 to 7/31/2007) months are used to generate T=500 scenarios. Portfolio optimization models of the CVaR, and WSCVaR are then, respectively, solved to generate the traditional and the robust portfolio strategies. In month N+1, the two portfolios are constructed according to the derived strategies. At the beginning of month N+2, the scenarios are reproduced using the data from month 2 up to month N+1. The portfolio models are then re-solved, respectively, using the updated scenarios to generate new portfolio strategies for month N+1. The above procedure repeats until the end of the data set. In this experiments, we also use exact values for the parameters in the CVaR, WSCVaR optimization problems. These parameters include the standard deviation and average returns for the CVaR, and the backward and forward deviations for the WSCVaR approach are set to 12

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pjt = 1.5, qtj = 2. ∆rtj is set to the vector of standard deviation of asset returns estimated by the i − th T samples. 1.5

1.4

Portfolio Values

1.3 WSCVaR

1.2

1.1

CVaR

1

0.9

0.8

0

10

20

30 40 Time Period

50

60

70

Figure 2-Portfolio Values for Out-of-Sample Observations When a Simple Buy-and-Hold Strategy is Employed From Figure 2, we can see the optimal portfolio allocation based on the WSCVaR approach tends to result in stable returns, whereas, for example, the behavior of the optimal portfolio obtained with the CVaR approach is some erratic. In addition, the portfolio Values for generated by the WSCVaR model is better than the initial CVaR model at the end of investment period. But, during the gradually declining period from June to October, 2008, robust portfolio strategies perform better than the traditional ones in most cases.

6.

Conclusion

With an asymmetric affine uncertainty set based on the factor model, which is often employed in practice for estimating the asset return distribution, we propose a computationally tractable robust optimization method for minimizing the Worse-Case CVaR of a portfolio. The remarkable characteristic of the new method is that the robust optimization model retains the complexity of original portfolio optimization problem, i.e., the robust counterpart problem is still a linear programming problem. Specially in the new method, we incorporate information about asymmetries in the distributions of uncertainties. We present some numerical experiments with simulated and real market data to illustrate the behavior of robust optimization model. Acknowledgments This work is supported by the NSF of China Grants 11301041, 11371154, 71371065, and 71371195, Project funded by China Postdoctoral Science Foundation(2014M560654), Natural Science Foundation of Hunan Province(2015JJ3015), A Project Supported by Scientific Research Fund of Hunan Provincial Education Department. 13

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References [1] H.M. Markowitz, Portfolio selection, Journal of Finance. 7 (1952) 77-91. [2] H.M. Markowitz, Portfolio Selection: Efficient diversification of investment, New York: John Wiley & Sons, 1959. [3] Levy, H. 1992. Stochastic dominance and expected utility: Survey and analysis. Management Sci. 38(4) 555õ593. [4] T.J. Linsmeier and N. D. Pearson, Risk Measurement: An introduction to value-at-risk. Technical report 96-04, OFOR, University of Illinois, Urbana-Champaign, IL, 1996. [5] P. Artzner, F. Delbaen, J. M. Eber and D. Heath, Coherence measures of risk, Math. Finance, 9 (1999) 203-228. [6] R.T. Rockafellar and S. Uryasev, Optimization of conditional Valueat-Risk, J. Risk, 2 (2000) 21-41. [7] R.T. Rockafellar and S. Uryasev, Conditional Value-at-Risk for general loss distributions, J. Banking and Finance, 26 (2002) 1443-1471. [8] F. Black and R. Litterman, Global portfolio optimization, J. Financial Analysts, 48 (1992) 28-43. [9] V.K. Chopra and W.T. Ziemba, The effect of errors in means, variance and covariances on optimal portfolio choice, Journal Portfolio Management, 19 (1993) 6-11. [10] M.S. Lobo and S. Boyd. The worst-case risk of a portfolio,

Technical Report,

http://faculty.fuqua.duke.edu/∼ mlobo/bio/researchfiles/rsk-bnd.pdf, 2000. [11] D. Goldfarb and G. Iyengar, Robust portfolio selection problems, Mathematics of Operations Research, 28 (2003) 1-38. [12] R.H. T¨ ut¨ unc¨ u and M. Koenig, Robust asset allocation, Annals of Operations Research, 132 (2004) 157-187. [13] L. El Ghaoui, M. Oks, and F. Oustry, Worst-Case Value-at-Risk and Robust Portfolio Optimization: A Conic Programming Approach. Operations Research, 51 (2003) 543-556. [14] K. Natarajan, D. Pachamanova and M. Sim, Incorporating Asymmetric Distributional Information in Robust Value-at-Risk Optimization, Management Science, 54 (2008) 573-585. [15] S.S. Zhu and M. Fukushima, Worst-Case Conditional Value-at-Risk with application to robust portfolio management, Operations Research, 57 (2009) 1155-1168.

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[16] S. S. Zhu, X. D. Ji and D. Li, Robust set-valued scenario approach for handling modeling risk in portfolio optimization, Technical Report, Sun Yat-Sen Business School, Sun Yat-Sen University , 2013. [17] H. Konno, A. Yuuki, H. Waki. Portfolio Optimization under Lower Partial Risk Measures. Asia-Pacific Financial Markets 9 (2002) 127õ140. [18] D. Bertsimas and M. Sim, Tractable approximations to robust conic optimization problems, Math. Program, 107 (2006) 5-36. [19] X. Chen, M. Sim and P. Sun, A robust optimization perspective of stochastic programming, Operations Research, 55 (2007) 1058-1077. [20] N. Topaloglou, H. Vladimirou and S.A. Zenios, CVaR models with selective hedging for international asset allocation, Journal of Banking and Finance, 26 (2002) 1535-1561. [21] L.Y. Han and C. L. Zheng, Fuzzy options with application to default risk analysis for municipal bonds in China, Nonlinear Analysis, Theory, Methods and Applications, 2005, 63, 2353-2365. [22] E. Fama, Foundations of Finance. Basic Books, New York, 1976. [23] G. Duffee, The long-run behavior of firms. stock returns: Evidence and interpretations. Working paper, Haas School of Business, University of California at Berkeley, Berkeley, CA, 2002. [24] P. Sch¨onbucher, Factor models for portfolio credit risk. Working paper, University of Bonn, Germany. http://www.gloriamundi.org, 2000. [25] A. Andrew and J. Chen, Asymmetric Correlations of Equity Portfolios, Journal of Financial Economics, 63(3) (2002)443-494. [26] T.E. Conine, and M. J. Tamarkin, On diversification given asymmetry in returns, Journal of Finance 36 (1981)1143-1155. [27] Z.F. Dai, D.H. Li, F.H. Wen, Robust Conditional value-at-risk optimization for Asymmetrically Distributed Asset Returns, Pacific Journal of Optimization, 8 (2012) 429-445. [28] Z.F. Dai, F.H. Wen, Robust CVaR-based portfolio optimization under a genal affine data perturbation uncertainty set, Journal of Computational Analysis and Application, 16(2014) 93-102.

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A NOTE ON THE INTERVAL-VALUED SIMILARITY MEASURE AND THE INTERVAL-VALUED DISTANCE MEASURE INDUCED BY THE CHOQUET INTEGRAL WITH RESPECT TO AN INTERVAL-VALUED CAPACITY JEONG GON LEE AND LEE-CHAE JANG

Division of Mathematics and Informational Statistics, and Nanoscale Science and Technology Institute, Wonkwang University, Iksan 570-749, Republic of Korea E-mail : [email protected], Phone:082-63-850-6189 General Education Institute, Konkuk University, Chungju 138-701, Republic of Korea E-mail : [email protected], Phone:082-43-840-3591 Abstract. In this paper, we introduce an interval-valued capacity which is motivated by the goal to represent reasonable capacity and to define the Choquet integral with respect to an interval-valued capacity. We also investigate some properties of the Choquet integral with respect to an interval-valued capacity on the space of fuzzy sets and discuss their applications, for examples, interval-valued similarity measure and interval-valued distance measure induced by the Choquet integral with respect to an interval-valued capacity.

1. Introduction The theory of fuzzy sets defined by Zadeh (1965) has been researching many new approaches and theories, for examples, entropy, similarity measures, distance measures, Choquet integrals, fuzzy sets, and intuitionistic fuzzy sets which are applied to theories treating reasonability and uncertainty. Note that measuring the similarity between fuzzy sets is important in pattern recognition research and decision making. Balopoulos-Hatzimichailidis-Papadopoulos [2], Fan-Ma-Xie [5], Hong-Lee [6], Li-Sheng [13], Liu [11], Turksen [22], Wang-Li [23], Wei-Chen [25], Xu-Xia [26], Zeng-Li [27], Zeng-Guo [28], and Zhang-Zhang-Mei [29] have studied some properties and applications of similarity measures, entropy, and distance measures on interval-valued fuzzy sets (or fuzzy set), and Choquet [3], Murofushi-Sugeno [15,16], and Narukawa-Murofushi-Sugeno [18,19] have studied the theory of fuzzy measures(or capacity) and Choquet integrals. Couso-Montes-Gil [4], Jang [12], Murofushi-Sugen0-Suzaki [17], Pedrycz-Yang-Ha [20], and Wang [24] have studied various convergence properties of the Choquet integral with respect to a capacity. By using interval-valued functions, we have studied the Choquet integral with respect to a fuzzy measure of interval-valued functions which are able to better handle the representation of decision making and information theory (see [7-11]). Recently, we studied some convergence properties of the Choquet integral with respect to an interval-valued capacity functional (see 1991 Mathematics Subject Classification. 28E10, 28E20, 03E72, 26E50 11B68. Key words and phrases. Choquet integral, fuzzy set, interval-valued capacity, interval-valued similarity measure. 1

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2

LEE-CHAE JANG

[12]). Main purpose of this paper is to provide some applications of the Choquet integral with respect to an interval-valued capacity on the space of all fuzzy sets. In section 2, we define an interval-valued similarity measure and an interval-valued distance measure, and discuss some basic properties of them. In section 3, we define an interval-valued capacity and the Choquet integral with respect to an interval-valued capacity of a fuzzy set, and discuss some properties of them. In section 4, we prove that an interval-valued mapping induced by the Choquet integral with respect to a continuous from below intervalvalued capacity is an interval-valued similarity measure on the space of fuzzy sets, and discuss their applications, for examples, the interval-valued similarity measure and the interval-valued distance measure. In section 5, we discuss various convergence properties of the interval-valued distance measure induced by the Choquet integral with respect to an interval-valued capacity. In section 6, we give a brief summary results and some conclusions. 2. Choquet integrals and interval-valued similarity measures In this section, we consider the Choquet integral with respect to a capacity and discuss their properties. Let [0, 1] be the unit interval in the set of real numbers and Ω be a σ-algebra on a set X. Definition 2.1. ([14-17]) (1) A real-valued set function µ : Ω −→ [0, 1] is called a capacity if it satisfies the following properties: (i) µ(∅) = 0 and µ(X) = 1, and (ii) µ(E1 ) ≤ µ(E2 ) whenever E1 , E2 ∈ Ω and E1 ⊂ E2 . (2) A capacity µ is said to be continuous from below if for each increasing sequence {En } ⊂ Ω, µ(∪∞ n=1 En ) = limn→∞ µ(En ). (3) A capacity µ is said to be continuous from above if for each decreasing sequence {En } ⊂ Ω, µ(∩∞ n=1 En ) = limn→∞ µ(En ). (4) A capacity µ is said to be continuous if it is continuous from above and continuous from below. (5) A capacity µ is said to be subadditive if µ(E1 ∪ E2 ) ≤ µ(E1 ) + µ(E2 ) whenever E1 , E2 ∈ Ω and E1 ∩ E2 = ∅. We consider the Choquet integral with respect to a capacity which was introduced by Murofushi at el ([15-17]). Throughout this paper, we assume that the membership function of a fuzzy set A is a measurable function ηA from X to [0, 1]. Definition 2.2. ([14-17]) (1) The Choquet integral with respect to a capacity µ of a fuzzy set A is defined by ∫ ∫ 1 (C) Adµ = µηA (r)dr (1) 0

where µηA (r) = µ({x ∈ X|ηA (x) > r}) for all r ∈ [0, 1] and the integral on the right-hand side is the Lebesgue integral of µηA . (2) A fuzzy set A is said to be µ-integrable if the Choquet integral of A on X exists. We note that if A, B are fuzzy sets on X, then A ≤ B means ηA (x) ≤ ηB (x) for all x ∈ X and that ηA∨B (x) = ηA (x) ∨ ηB (x) and ηA∧B (x) = ηA (x) ∧ ηB (x) for all x ∈ X. Theorem 2.1. ([14-17]) Let ∫ A and B be ∫ µ-integrable fuzzy sets. (1) If A ≤ B, then (C) Adµ ≤ (C) Bdµ. ∫ ∫ (2) If E1 , E2 ∈ Ω and E1 ⊂ E2 , then (C) E1 Adµ ≤ (C) E2 Adµ.

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3

(3) If we define ηA∨B = ηA (x) ∨ ηB (x) and ηA∧B (x) = ηA (x) ∧ ηB (x) for all x ∈ X, then ∫ ∫ ∫ (C) A ∨ Bdµ ≥ (C) Adµ ∨ (C) Bdµ, and

∫ (C)

∫ A ∧ Bdµ ≤ (C)

∫ Adµ ∧ (C)

Bdµ.

Let [[0, 1]] is the set of all closed intervals in [0, 1] as follows: [[0, 1]] = {¯ a = [a− , a+ ]|a− , a+ ∈ [0, 1] and a− ≤ a+ }. For any a ∈ [0, 1], we define a = [a, a]. Obviously, a ∈ [[0, 1]](see [7-13, 21-223, 25, 27-29]). + Definition 2.3. Let I be an index set. If a ¯ = [a− , a+ ], ¯b = [b− , b+ ], a ¯n = [a− n , an ] ∈ [[0, 1]] for all n ∈ N and k ∈ [0, 1], then we define arithmetic, minimum, maximum, order, and inclusion operations as follows: (1) k¯ a = [ka− , ka+ ], ¯ (2) a ¯b = [a− b− , a+ b+ ], (3) a ¯ ∧ ¯b = [a− ∧ b− , a+ ∧ b+ ], (4) a ¯ ∨ ¯b = [a− ∨ b− , a+ ∨ b+ ], (5) a ¯ ≤ ¯b if and only if a− ≤ b− and a+ ≤ b+ , (6) a ¯ < ¯b if and only if a ¯ ≤ ¯b and a ¯ ̸= ¯b, − − ¯ (7) a ¯ ⊂ b if and only if b ≤ a and a+ ≤ b+ , (8) 1 − a ¯ = [1 − a+ , 1 − a− ], + (9) supn∈I a ¯n = [supn∈I a− n , supn∈I an ], and + − (10) inf n∈I a ¯n = [inf n∈I an , inf n∈I an ].

Theorem 2.2. For a ¯, ¯b, c¯ ∈ [[0, 1]], we have (1) idempotent law: a ¯∧a ¯=a ¯ and a ¯∨a ¯=a ¯, (2) commutative law: a ¯ ∧ ¯b = ¯b ∧ a ¯ and a ¯ ∨ ¯b = ¯b ∨ a ¯, (3) associative law: (¯ a ∧ ¯b) ∧ c¯ = a ¯ ∧ (¯b ∧ c¯) and (¯ a ∨ ¯b) ∨ c¯ = a ¯ ∨ (¯b ∨ c¯), ¯ ¯ (4) absorptive law: a ¯ ∧ (¯ a ∨ b) = a ¯ ∨ (¯ a ∧ b) = a ¯, and (5) distributive law: a ¯ ∧ (¯b ∨ c¯) = (¯ a ∧ ¯b) ∨ (¯ a ∧ c¯) and a ¯ ∨ (¯b ∧ c¯) = (¯ a ∨ ¯b) ∧ (¯ a ∨ c¯). Let F(X) be the family of all fuzzy sets A of X with the membership measurable function ηA : X → [0, 1]. Recall that for A, B ∈ F(X), A ≡ B means µ({x ∈ X|ηA (x) ̸= ηB (x)}) = 0, where µ is a capacity on X. We introduce the definitions of similarity measures and distance measures on F(X), and some characterizations of them(see [2,5,6,14,26-29]). Definition 2.4. (1) A real-valued function s : F(X) × F(X) −→ [0, 1] is called a similarity measure if it satisfies the following properties: (i) s(A, Ac ) = 0 if A is a crisp set, (ii) for A, B ∈ F(X), s(A, B) = 1 if and only if A ≡ B, (iii) for A, B ∈ F(X), s(A, B) = s(B, A), and (iv) if A, B, C ∈ F(X) and A ≤ B ≤ C, then s(A, C) ≤ s(A, B) and s(A, C) ≤ s(B, C). (2) A real-valued function d : F(X) × F(X) −→ [0, 1] is called a distance measure if it satisfies the following properties: (i) d(A, Ac ) = 1 if A is a crisp set, (ii) for A, B ∈ F(X), d(A, B) = 0 if and only if A ≡ B, (iii) for A, B ∈ F(X), d(A, B) = d(B, A), and (iv) if A, B, C ∈ F(X) and A ≤ B ≤ C, then d(A, C) ≥ d(A, B) and d(A, C) ≥ d(B, C).

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It is easy to see that if s is a similarity measure and we define l1 = 1 − s, then l1 is a distance measure and that if d is a distance measure and we define l2 = 1 − d, then l2 is a similarity measure. Definition 2.5. (1) An interval-valued function S = [s− , s+ ] : F(X) × F(X) −→ [[0, 1]] is called an interval-valued similarity measure if it satisfies the following properties: (i) S(A, Ac ) = 0 if A is a crisp set, (ii) for A, B ∈ F(X), S(A, B) = 1 if and only if A ≡ B, (iii) for A, B ∈ F(X), S(A, B) = S(B, A), and (iv) if A, B, C ∈ F(X) and A ≤ B ≤ C, then S(A, C) ≤ S(A, B) and S(A, C) ≤ S(B, C). (2) An interval-valued function D = [d− , d+ ] : F(X) × F(X) −→ [[0, 1]] is called a distance measure if it satisfies the following properties: (i) D(A, Ac ) = 1 if A is a crisp set, (ii) for A, B ∈ F(X), D(A, B) = 0 if and only if A ≡ B, (iii) for A, B ∈ F(X), D(A, B) = D(B, A), and (iv) if A, B, C ∈ F(X) and A ≤ B ≤ C, then D(A, C) ≥ D(A, B) and D(A, C) ≥ D(B, C). By the definitions of an interval-valued similarity measure and an interval-valued distance measure, we can obtain the following theorem. Theorem 2.3. (1) An interval-valued function S = [s− , s+ ] is an interval-valued similarity measure if and only if real-valued functions s− and s+ are real-valued similarity measures, and 0 ≤ s− ≤ s+ ≤ 1. (2) An interval-valued function D = [d− , d+ ] is an interval-valued distance measure if and only if real-valued functions d− and d+ are real-valued distance measures, and 0 ≤ d− ≤ d+ ≤ 1. (3) If S is an interval-valued similarity measure and we define H = 1−S = [1−s+ , 1−s− ], then H is an interval-valued distance measure. (4) If D is an interval-valued distance measure and we define L = 1 − D = [1 − d+ , 1 − d− ], then L is an interval-valued similarity measure. Proof. (1) (=⇒) Suppose that S is an interval-valued similarity measure. If A is a crisp set, then 0 = S(A, Ac ) = [s− (A, Ac ), s+ (A, Ac )]. Thus s− (A, Ac ) = 0 and s+ (A, Ac ) = 0. Since S(A, B) = S(B, A) for all A, B ∈ F(X), s− (A, B) = s− (B, A) and s+ (A, B) = S + (B, A). Let A, B, C ∈ F(X) and A ≤ B ≤ C. Then we have S(A, C) ≤ S(A, B) and S(A, C) ≤ S(B, C). Thus, we have

s− (A, C) ≤ s− (A, B) and s− (A, C) ≤ s− (B, C),

and s+ (A, C) ≤ s+ (A, B) and s+ (A, C) ≤ s+ (B, C), Therefore, we obtain that s− and s+ are real-valued similarity measures and 0 ≤ s− ≤ s+ ≤ 1. (⇐=) The proof is similar to the proof of (=⇒). (2) The proof is similar to the proof of (1). (3) Let S be an interval-valued similarity measure and we define H = 1−S = [1−s+ , 1−s− ]. If A is a crisp set, then S(A, Ac ) = 0. Thus, H(A, Ac ) = 1 − S(A, Ac ) = 1 − 1 = 0. Let A, B ∈ F(X). Then, A ≡ B if and only if S(A, B) = 1, that is, H(A, B) = 1 − S(A, B) = 0. If A, B ∈ F(X), then S(A, B) = S(B, A). Then, H(A, B) = 1 − S(A, B) = 1 − S(B, A) = H(B, A).

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If A, B, C ∈ F(X) and A ≤ B ≤ C, then S(A, C) ≤ S(A, B) and S(A, C) ≤ S(B, C). Thus, we have H(A, C) = 1 − S(A, C) ≥ 1 − S(A, B) = H(A, B) and H(A, C) = 1 − S(A, C) ≥ 1 − S(B, C) = H(B, C). Therefore, H is an interval-valued distance measure. (4) The proof is similar to the proof of (3). 3. The Choquet integral with respect to an interval-valued capacity In this section, we define an interval-valued capacity and the Choquet integral with respect to an interval-valued capacity of a fuzzy set. Note that a mapping dH : [[0, 1]] × [[0, 1]] −→ [0, ∞) is the Hausdorff metric defined by { } dH (A, B) = max sup inf |x − y|, sup inf |x − y| (2) x∈A y∈B

y∈B x∈A

for all A, B ∈ [[0, 1]], and ([[0, 1]], dH ) is a metric space. By the definition of the Hausdorff metric, it is easy to see that for any a ¯ = [a− , a+ ], ¯b = [b− , b+ ] ∈ [[0, 1]], we have { } dH (¯ a, ¯b) = max |a− − b− |, |a+ − b+ | . (3) We recall that for any {¯ an } ⊂ [[0, 1]] and a ¯ ∈ [[0, 1]], dH − lim a ¯n = a ¯ means n→∞

lim dH (¯ an , a ¯) = 0.

(4)

n→∞

We define an interval-valued capacity µ ¯ = [µ− , µ+ ] : Ω −→ [[0, 1]] on a measurable space (X, Ω) as follows: Definition 3.1. (1) An interval-valued set function µ ¯ : Ω −→ [[0, 1]] is called an intervalvalued capacity if it satisfies the following properties: (i) µ ¯(∅) = 0 and µ ¯(X) = 1, and (ii) µ ¯(E1 ) ≤ µ ¯(E2 ) whenever E1 , E2 ∈ Ω and E1 ⊂ E2 . (2) An interval-valued capacity µ ¯ is said to be continuous from above if for each increasing ¯(En ). sequence {En } ⊂ Ω, µ ¯(∪∞ n=1 En ) = dH − limn→∞ µ (3) An interval-valued capacity µ ¯ is said to be continuous from below if for each decreasing sequence {En } ⊂ Ω, µ ¯(∩∞ ¯(En ). n=1 En ) = dH − limn→∞ µ (4) An interval-valued capacity µ ¯ is said to be continuous if it is continuous from above and continuous from below. (5) An interval-valued capacity µ ¯ is said to be subadditive if µ ¯(E1 ∪ E2 ) ≤ µ ¯(E1 ) + µ ¯(E2 ), whenever E1 , E2 ∈ Ω and E1 ∩ E2 = ∅. It is easy to see that for each increasing sequence {En } ⊂ Ω with E = ∪∞ n=1 En , lim dH (¯ µ(En ), µ ¯(E)) = 0 if and only if lim µ− (En ) = µ− (E) and lim µ+ (En ) = µ+ (E), (5)

n→∞

n→∞

n→∞

and for each decreasing sequence {En } ⊂ Ω with F =

∩∞ n=1 En ,

lim dH (¯ µ(En ), µ ¯(F )) = 0 if and only if lim µ− (En ) = µ− (F ) and lim µ+ (En ) = µ+ (F ). (6)

n→∞

n→∞

n→∞

By (5) and (6), we can directly derive the following theorem.

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Theorem 3.1. (1) An interval-valued set function µ ¯ = [µ− , µ+ ] : Ω −→ [[0, 1]] is an interval− + valued capacity if and only if µ and µ are capacities and µ− ≤ µ+ . (2) An interval-valued capacity µ ¯ = [µ− , µ+ ] is continuous from below if and only if µ− + and µ are continuous from below and µ− ≤ µ+ . (3) An interval-valued capacity µ ¯ = [µ− , µ+ ] is continuous from above if and only if µ− + and µ are continuous from above and µ− ≤ µ+ . (4) An interval-valued capacity µ ¯ = [µ− , µ+ ] is continuous if and only if µ− and µ+ are − + continuous and µ ≤ µ . (5) An interval-valued capacity µ ¯ = [µ− , µ+ ] is subadditive if and only if µ− and µ+ are − + subadditive and µ ≤ µ . Recall that if ([0, 1], M, m) is the Lebesgue measure space and C([0, 1]) is the family of all closed subsets of I, then the Aumann integral of a closed set-valued function G : [0, 1] −→ C([0, 1]) is defined by {∫ } ∫ (A) Gdm = gdm| g ∈ S(G) , (7) where S(G) is the set of all integrable selections of G, that is, ∫ S(G) = {g : [0, 1] −→ [0, 1]| gdm < ∞ and g(r) ∈ G(r) m − a.e.}.

(8)

We note that m − a.e. means almost everywhere in the Lebesgue measure m (see[1,16]). Then, we introduce the following theorems which are used to define the Choquet integral with respect to an interval-valued capacity of a fuzzy set. Theorem 3.2. ([13, Lemma 2.1]) If a closed set-valued function G : [0, 1] −→ C([0, 1]) is ∫ M-measurable, then (A) Gdm is convex in [0, 1]. Theorem 3.3. ([13, Lemma 2.2]) If a closed set-valued function G : [0, 1] −→ C([0, 1]) is Mmeasurable and integrably bounded, that is, there exists a integrable function φ : [0, 1] −→ [0, 1] such that sup x ≤ φ(r)

for r ∈ [0, 1],

(9)

x∈G(r)

∫ then (A) Gdm is nonempty compact convex in [0, 1]. ∫ From Theorem 3.3, we can see that (A) Gdm is a nonempty bounded and closed subset in [0, 1] under the same assumption of G. Thus, we obtain the following corollary (see [12,13,21]). Corollary 3.4. If an interval-valued function G = [g − , g + ] : I −→ [[0, 1]] is M-measurable and integrably bounded, then g − , g + ∈ S(F ) and [∫ ] ∫ ∫ − + (A) Gdm = g dm, g dm , (10) where the integrals on the right-hand side are the Lebesgue integral with respect to m. ∫ ∫1 We write gdm = 0 g(r)dm(r) for all measurable functions g. By using an interval-valued capacity, we define the Choquet integral with respect to an interval-valued capacity of a fuzzy set A. Definition 3.2. (1) The Choquet integral with respect to an interval-valued capacity µ ¯ of a fuzzy set A ∈ F is defined by ∫ ∫ 1 (C) Ad¯ µ = (A) µ ¯A (r)dr, (11) 0

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where ηA is the membership measurable function of A, µ ¯A (r) = µ ¯({x ∈ X|ηA (x) > r}) for all r ∈ [0, 1], and the integral on the right-hand side is the Aumann integral in (7). ∫ (2) A fuzzy set A ∈ F is said to be µ ¯-integrable if (C) Ad¯ µ ∈ [[0, 1]]. Note that if an interval-valued capacity µ ¯ is continuous from below and A ∈ F(X), then µ ¯A : I −→ [[0, 1]] is continuous from below on [0, 1]. Thus, we obtain that µ ¯A is M-measurable and integrably bounded on [0, 1]. Thus, by Definition 3.2 and Corollary 3.4, we can easily obtain the following theorem. Theorem 3.5. If an interval-valued capacity µ ¯ is continuous from below and A ∈ F, then we have [ ] ∫ ∫ ∫ (C) Ad¯ µ = (C) Adµ− , (C) Adµ+ , (12) where the integrals on the right-hand side are Choquet integrals. Proof. By Definition 3.2 and Corollary 3.4, we can derive ∫ ∫ 1 (C) Ad¯ µ = (A) µ ¯A (r)dr ∫0 1 + = (A) [µ− A (r), µA (r)]dr [∫ 1 0 ] ∫ 1 − + = µA (r)dr, µA (r)dr 0 [ 0 ∫ ] ∫ = (C) Adµ− , (C) Adµ+ . By Theorem 3.5, we can easily obtain the following basic properties of the Choquet integrals with respect to a continuous from below interval-valued capacity of a fuzzy set. Theorem 3.6. Let (X, Ω) be a measurable space. Assume that an interval-valued µ ¯ is continuous from below. (1) If A, B ∈ F(X) and A ≤ B, then ∫ ∫ (C) Ad¯ µ ≤ (C) Bd¯ µ. (2) If A, B ∈ F(X) and we define η(A∨B) (x) = ηA (x) ∨ ηB (x) for all x ∈ X, then ∫ ∫ ∫ (C) A ∨ Bd¯ µ ≥ (C) Ad¯ µ ∨ (C) Ad¯ µ. (3) If A, B ∈ F(X) and we define η(A∧B) (x) = ηA (x) ∧ ηB (x) for all x ∈ X, then ∫ ∫ ∫ (C) A ∧ Bd¯ µ ≤ (C) Ad¯ µ ∧ (C) Ad¯ µ. 4. Interval-valued similarity measures induced by the Choquet integral In this section, we discuss some applications of the Choquet integral with respect to a continuous from below interval-valued capacity of a fuzzy set. Theorem 4.1. Assume that an interval-valued µ ¯ is continuous from below and µ ¯(X) = ¯{µ}(X) = 1. If we define an interval-valued function Sµ¯ : F × F −→ [[0, 1]] as following ∫ Sµ¯ (A, B) = 1 − (C) |ηA − ηB |d¯ µ (13) for all A, B ∈ F(X), then Sµ¯ is an interval-valued similarity measure.

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Proof. (i) If A is a crisp measurable set, then the membership measurable function ηA of a fuzzy set A is defined by { 1 if x ∈ A ηA (x) = 0 if x ∈ Ac = I \ A. We note that if the membership measurable function ηAc of the complement of a fuzzy set A, then { 0 if x ∈ A ηAc (x) = 1 if x ∈ Ac = I \ A. Thus, we have |ηA (x) − ηAc (x)| = 1 for all x ∈ X. Therefore, we have ∫ Sµ¯ (A, Ac ) = 1 − (C) |ηA − ηAc |d¯ µ ∫ 1 = 1− µ ¯({x ∈ X| |ηA (x) − ηAc (x)| > r})dr 0 ∫ 1 = 1− µ ¯(X)dr = 0. 0

(ii) If A ≡ B, then ηA = ηB µ ¯ − a.e. on X. Thus, we have ∫ Sµ¯ (A, B) = 1 − (C) |ηA − ηB |d¯ µ ∫ 1 = 1− µ ¯({x ∈ X| |ηA (x) − ηB (x)| > r})dr ∫0 1 = 1− µ ¯(∅)dr = 1. 0

If Sµ¯ (A, B) = 1, then

∫

1

µ ¯({x ∈ X| |ηA (x) − ηB (x)| > r})dr = 0. 0

Then, it is easy to see that µ ¯({x ∈ X| |ηA (x) − ηB (x)| > r}) = 0 m − a.e. on I.

(14)

From (14), we have µ ¯({x ∈ X| |ηA (x) − ηB (x)| ̸= 0}) = 0, that is, ηA = ηB µ ¯ − a.e. on X and hence A ≡ B. (iii) If A, B ∈ F(X), then we have ∫ Sµ¯ (A, B) = 1 − (C) |ηA − ηB |d¯ µ ∫ = 1 − (C) |ηB − ηA |d¯ µ = Sµ¯ (B, A). (iv) If A, B, C ∈ F(X) and A ≤ B ≤ C, then ηA ≤ ηB ≤ ηC . Thus, we have |ηA (x) − ηB (x)| ≤ |ηA (x) − ηC (x)| and |ηB (x) − ηC (x)| ≤ |ηA (x) − ηC (x)|,

(15)

for all x ∈ X. By (15) and Theorem 2.2 (1), we have ∫ Sµ¯ (A, C) = 1 − (C) |ηA − ηC |d¯ µ ∫ ≤ 1 − (C) |ηA − ηB |d¯ µ = Sµ¯ (A, B),

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and

∫ Sµ¯ (A, C)

=

1 − (C)

≤ 1 − (C)

∫

|ηA − ηC |d¯ µ |ηB − ηC |d¯ µ = Sµ¯ (B, C),

By (i),(ii),(iii), and (iv), we see that Sµ¯ is an interval-valued similarity measure. By Theorem 4.1 and Theorem 2.3(3), we can easily obtain the following corollary. Corollary 4.2. Assume that an interval-valued µ ¯ is continuous from below and µ ¯(X) = ¯{µ}(X) = 1. If we define an interval-valued function Dµ¯ = 1 − Sµ¯ = (C) ∫ |ηA − ηB |d¯ µ for all A, B ∈ F(X), then Dµ¯ is an interval-valued distance measure. In order to illustrate the proposed similarity measure are reasonable, we give the following example. Example 4.1. Let X = {x1 , x2 , x3 } and Ω = ℘(X) be the power set of X. Suppose that µ ¯ : Ω −→ [[0, 1]] is defined by µ ¯(E) = [µ− (E), µ+ (E)], (16) )2 ( m(E) m(E) where m(E) is the cardinality of E ∈ Ω, µ− (E) = m(X) , and µ+ (E) = m(X) . Since X is a finite set, clearly, we see that µ ¯ is a continuous from below interval-valued capacity on a measurable space (X, Ω) and µ ¯(X) = ¯{µ}(X) = 1. . The three patterns are denoted as follows: A1 = {(x1 , 0.3), (x2 , 0.2), (x3 , 0.1)}, A2 = {(x1 , 0.2), (x2 , 0.2), (x3 , 0.2)}, and A3 = {(x1 , 0.4), (x2 , 0.4), (x3 , 0.4)}. Assume that a sample B = {(x1 , 0.3), (x2 , 0.2), (x3 , 0.1)} is given. In order to interpret the measure of similarity of B with these patterns, we calculate the proposed interval-valued similarity measure Sµ¯ as follows: Sµ¯ (A1 , B) = 1 −

3 ∑ (|ηA1 (x(i)) − ηB (x(i)) |)(¯ µ(A(i) ) = 1,

(17)

i=1

[ ] 3 ∑ 14 43 (|ηA2 (x(i)) − ηB (x(i)) |)(¯ µ(A(i) ) = , , and 15 45 i=1 ] [ 3 ∑ 4 38 , . Sµ¯ (A3 , B) = 1 − (|ηA3 (x(i)) − ηB (x(i)) |)(¯ µ(A(i) ) = 5 45 i=1

Sµ¯ (A2 , B) = 1 −

(18)

(19)

By (17), (18), and (19), we interpret that B is equal(or, absolutely similar) to A1 and B is more similar to A2 than similar to A3 . Example 4.2. Let X = {x1 , x2 , x3 } and Ω = ℘(X) be the power set of X. Suppose that ν¯ : Ω −→ [I] is defined by ν¯(E) = [ν − (E), ν + (E)], (20) ( )3 ( )2 m(E) m(E) where m(E) is the cardinality of E ∈ Ω, ν − (E) = m(X) , and ν + (E) = m(X) . The three patterns are denoted as follows: A1 = {(x1 , 0.3), (x2 , 0.2), (x3 , 0.1)}, A2 = {(x1 , 0.2), (x2 , 0.2), (x3 , 0.2)}, and A3 = {(x1 , 0.4), (x2 , 0.4), (x3 , 0.4)}.

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Assume that a sample B = {(x1 , 0.3), (x2 , 0.2), (x3 , 0.1)} is given. In order to interpret the measure of similarity of B with these patterns, we calculate the proposed interval-valued similarity measure Sν¯ as follows: [ ] [ ] 38 13 43 131 , , and Sν¯ (A3 , B) = , . (21) Sν¯ (A1 , B) = 1, Sν¯ (A2 , B) = 45 135 45 15 Thus, we can see that there is an interpretation of the notions of these patterns under two different interval-valued capacity µ ¯ and ν¯ as follows: Sµ¯ (A1 , B) = 1 = Sν¯ (A1 , B), [ ] [ ] 14 43 43 131 Sµ¯ (A2 , B) = , < , = Sν¯ (A2 , B), and 15 45 45 135 ] [ ] [ 38 13 4 38 , < , = Sν¯ (A3 , B). Sµ¯ (A3 , B) = 5 45 45 15 Therefore, this means that ν¯ has more positive sense than µ ¯. 5. Convergence in the interval-valued distance measure Throughout this section, we assume that µ ¯ = [µ− , µ+ ] is continuous from below. At first, we introduce uniformly µ-integrability and convergence in the interval-valued distance measure on F(X). Definition 5.1. ([26]) Let µ be a capacity on a measurable space (X, Ω), {An } be a sequence of fuzzy sets and A be a fuzzy set. (1) A sequence {An } converges to A almost everywhere on X if there exist a null set N ∈ Ω with µ(N ) = 0 such that ηA (x) = lim ηAn (x), n→∞

for all x ∈ N c .

(22)

(2) A sequence {An } converges in the distance measure dµ to A if lim dµ (ηAn , ηA ) = 0,

where dµ (ηAn , ηA ) = (C)

∫

(23)

n→∞

|ηAn (x) − ηA (x)|dµ for all n ∈ N.

Remark that convergence in the distance measure dµ is equal to convergence in µ-mean(see [4 ]) Definition 5.2. ([4]) Let µ be a capacity on a measurable space (X, Ω) and I ⊂ N be an index set. A class {An }n∈I of fuzzy sets is said to be uniform µ-integrable if (i) sup dµ (An , 0) < ∞,

(24)

n∈I

(ii) ∀ε > 0, ∃δ(ε) > 0 such that dE,µ (An , 0) < ε if E ∈ Ω and µ(E) < δ(ε), ∫ where dE,µ (An , 0) = (C) E |ηAn |dµ for all n ∈ N.

(25)

We also introduce various convergence properties of the Choquet integral on F(X) as follows: Theorem 5.1. ([4]) Let a capacity µ be subadditive and {An } a sequence of fuzzy sets in F(X). Then {An } is an uniformly µ-integrable if and only if lim sup d[|ηAn |>a],µ (An , 0) = 0.

a→∞ n∈N

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Theorem 5.2. ([4]) Let a capacity µ be subadditive and a sequence {An } of fuzzy sets in F(X) converges to a fuzzy set A in F(X) µ-almost everywhere on X and An ≤ B for some µ-integrable fuzzy set B, then we have (1) An and A are µ-integrable for all n ∈ N, and (2) {An } converges to A in the distance measure dµ , that is, lim dµ (An , 0) = 0.

(27)

n→∞

We assume that an interval-valued capacity µ ¯(X) = [µ− , µ+ ] is continuous from below. Then we define convergence in the interval-valued distance measure Dµ¯ and uniform µ ¯integrability on f(X). It is easy to see that Dµ¯ (A, B) = [dµ− (A, B), dµ+ (A, B)], for all A, B ∈ F(X).

(28)

Definition 5.3. Let I ⊂ N be an index set. (1) A sequence {An } converges in the interval-valued distance measure Dµ¯ to A if dH − lim Dµ¯ (An , A) = 0,

(29)

n→∞

where dH − lim Dµ¯ (An , A) = lim dH {Dµ¯ (An , A), 0} n→∞

n→∞

and Dµ¯ (An , A) = [dµ− (An , A), dµ− (An , A)] for all n ∈ N. (2) A class {An }n∈I of fuzzy sets in F(X) is said to be µ ¯-integrable if (i) sup Dµ¯ (An , 0) < ∞,

(30)

n∈I

(ii) ∀ε > 0, ∃δ(ε) > 0 such that DE,¯µ (An , 0) < ε if E ∈ Ω and µ ¯(E) < δ(ε), ∫ where DE,¯µ (An , 0) = (C) E |ηAn |d¯ µ for all n ∈ N.

(31)

By (3), it is easy to see that (29) holds if and only if lim max{dµ− (An , A), dµ+ (An , A)}) = 0,

(32)

n→∞

By Definition 5.1 and Definition 5.3, we obtain various convergence properties of the intervalvalued distance measure Dµ¯ as follows: Theorem 5.3. Let I ⊂ N be an index set. (1) A class {An }n∈I is uniformly µ ¯-integrable if and only if it is uniformly µ− -integrable + − and uniformly µ -integrable, and µ ≤ µ+ . (2) A sequence {An } of fuzzy sets in F(X) converges to a fuzzy set A ∈ F(X) in the intervalvalued distance measure Dµ¯ if and only if {An } converges to A in the distance measures dµ− and dµ+ , and dµ− ≤ dµ+ . Proof. (1) Let {An } be a sequence of fuzzy sets in F(X). If {An } converges to A in the interval-valued distance measure Dµ¯ , then, by (12) and (29), lim dµ− (An , A)

n→∞

≤ =

lim (max{dµ− (An , A), dµ+ (An , A)})

n→∞

lim dH (Dµ¯ (An , A), 0) = 0.

(33)

n→∞

As in the same method with (33), we obtain lim dµ+ (An , A) = 0.

(34)

n→∞

Thus, by (33) and (34), {An } converges to A in the distance measure dµ− and dµ+ .

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Conversely, if we take an interval-valued distance measure µ ¯ = [µ− , µ+ ], then, similarly, we can obtain the converse result. (2) Suppose that {An }n∈I is uniformly µ ¯-integrable and µ ¯ is continuous from below. By (12) and Definition2.3 (9), we have sup Dµ¯ (An , 0)

=

sup[dµ− (An , 0), dµ+ (An , 0)]

=

[sup dµ− (An , 0), sup dµ+ (An , 0)] < ∞,

n∈I

n∈I

n∈I

(35)

n∈I

and for arbitrary ε > 0 and E ∈ Ω, there exists δ(ε) > 0 such that sup DE,¯µ (An , 0)

=

sup[dE,µ− (An , 0), dE,µ+ (An , 0)]

=

[sup dE,µ− (An , 0), sup dE,µ+ (An , 0)] < ε,

n∈I

n∈I n∈I

(36)

n∈I

if µ ¯ < δ(ε). By (35) and (36), {An } converges to A in the distance measures dµ− and dµ+ , nd dµ− ≤ dµ+ . Conversely, if we take an interval-valued distance measure µ ¯ = [µ− , µ+ ], then, similarly, we can obtain the converse result. Theorem 5.4. Let an interval-valued capacity µ ¯ be subadditive and {An } a sequence of fuzzy sets in F(X). Then, {An } is an uniformly µ ¯-integrable if and only if lim sup D[|ηAn |>a],¯µ (An , 0) = 0.

(37)

a→∞ n∈N

Proof. Since an interval-valued capacity µ ¯ = [µ− , µ+ ] is subadditive, by Theorem 3.1(5), + µ and µ are subadditive. From Theorem 5.3 (1), {An } is an uniformly µ ¯-integrable if and only if {An } is an uniformly µ− -integrable and an uniformly µ+ -integrable. Thus, by Theorem 5.1, {An } is an uniformly µ− -integrable if and only if −

lim sup d[|ηAn |>a],µ− (An , 0) = 0

(38)

a→∞ n∈N

and {An } is an uniformly µ+ -integrable if and only if lim sup d[|ηAn |>a],µ+ (An , 0) = 0

(39)

a→∞ n∈N

By (38) and (39), and (12), we have lim sup dH (D|ηAn |>a],¯µ (An , 0), 0)

a→∞ n∈N

=

lim sup max{d[|ηAn |>a],µ− (An , 0), d[|ηAn |>a],µ+ (An , 0)} = 0.

a→∞ n∈N

(40)

Conversely, by the similar method of the above proof, we can obtain the converse result. Lemma 5.5. Assume that an interval-valued capacity µ ¯ = [µ− , µ+ ] is continuous from below. Then {An } is µ ¯-integrable if and only if {An } is µ− -integrable and µ+ -integrable Proof. The proof is trivial. Theorem 5.6. Let an interval-valued capacity µ be subadditive. If a sequence {An } of fuzzy sets in F(X) converges to a fuzzy set A in F(X) µ-almost everywhere on X and An ≤ B for some µ ¯-integrable fuzzy set B, then we have (1) An and A are µ ¯-integrable for all n ∈ N, and (2) {An } converges to A in the interval-valued distance measure Dµ¯ , that is, dH − lim Dµ¯ (An , 0) = 0. n→∞

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Proof. Since B is µ ¯-integrable fuzzy set and An ≤ B, by Theorem 5.3 (1), we have (i)An and A are µ− -integrable and µ− -integrable for all n ∈ N, and (ii) {An } converges to A in the distance measure dµ− and in the distance measure dµ+ . Thus, by Lemma 5.5 and Theorem 5.3 (1) and (12), we obtain (1) An and A are µ ¯-integrable for all n ∈ N, and (2) {An } converges to A in the interval-valued distance measure Dµ¯ , that is, dH − lim Dµ¯ (An , 0) = 0.

(42)

n→∞

6. Conclusions In this paper, we define the concept of interval-valued capacity which means reasonable capacity. By using Aumann integral of integrably bounded interval-valued functions in Corollary 3.4, we consider the Choquet integral with respect to a continuous interval-valued capacity of a fuzzy set. From Definitions 2.3, 3.1, 3.2 and Theorems 3.5, 3.6, we discuss interval-valued similarity measures induced by the Choquet integral with respect to a continuous interval-valued capacity on F(X). By Examples 4.1 and 4.2, it is possible that we interpret the interval-valued measure of similarity of a sample with the three patterns. From Definitions 5.1, 5.2, 5.3, and Theorems 5.3, 5.4, and 5.6, we can provide the concept of convergence in the interval-valued distance measure and discuss various convergence properties of the interval-valued distance3 measure on the space of fuzzy sets for the Choquet integral. In the future, by using these results of this paper, we can develop various problems and models for representing uncertain similarity measures and uncertain distance measures in pattern recognition research, information theory, decision making, and fuzzy risk analysis, etc. Acknowledgement This paper was supported by Wonkwang University in 2013.

References [1] R.J. Aumann, Integrals of set-valued functions, J. Math. Anal. Appl. 12 (1965) 1-12. [2] V. Balopoulos, A.G. Hatzimichailidis, B.K. Papadopoulos, Distance and similarity measures for fuzzy operators, J. Math. Anal. Appl. 12 (1965) 1-12. [3] G. Choquet, Theory of capacities, Ann. Inst. Fourier 5 (1953) 131-295. [4] I. Couso, S. Montes, P. Gil, Stochastic convergence, uniform integrability and convergence in mean on fuzzy measure spaces, Fuzzy Sets and Systems 129 (2002) 95-104. [5] Jin-Lum Fan, Yuan-Liang Ma, and Wei-Xin Xie, On some properties of distance measures, Fuzzy Sets and Systems , 117 (2001), 355-361. [6] D.H. Hong, S.H. Lee, Some algebraic properties and distance measures for interval-valued fuzzy numbers, Information Sciences, 148 (2002), 1-10. [7] L.C. Jang, B.M. Kil, Y.K. Kim, J.S. Kwon, Some properties of Choquet integrals of set-valued functions, Fuzzy Sets and Systems, 91 (1997), 61-67. [8] L.C. Jang, J.S. Kwon, On the representation of Choquet integrals of set-valued functions and null sets, Fuzzy Sets and Systems, 112 (2000), 233-239. [9] L.C. Jang, Interval-valued Choquet integrals and their apllications, J. Appl. Math. and Computing, 16(12) (2004), 429-445. [10] L.C. Jang, A note on the monotone interval-valued set function defined by the interval-valued Choquet integral, Commun. Korean Math. Soc., 22 (2007), 227-234. [11] L.C. Jang, On properties of the Choquet integral of interval-valued functions, Journal of Applied Mathematics, 2011 (2011), Article ID 492149, 10pages. [12] L.C. Jang, A note on convergence properties of interval-valued capacity functionals and Choquet integrals, Information Sciences, 183 (2012), 151-158.

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[13] L.S. Li, Z. Sheng, The fuzzy set-valued measures generated by fuzzy random variables, Fuzzy Sets and Systems, 97 (1998), 203-209. [14] X. Liu, Entropy, distance measure and similarity measure of fuzzy sets and their relations, Fuzzy Sets and Systems, 52 (1992), 305-318. [15] T. Murofushi, M. Sugeno, An interpretation of fuzzy measures and the Choquet integral as an integral with respect to a fuzzy measure, Fuzzy Sets and Systems, 29 (1989), 201-227. [16] T. Murofushi, M. Sugeno, A theory of fuzzy measures: representations, the Choquet integral, and null sets, J. Math. Anal. Appl., 159 (1991), 532-549. [17] T. Murofushi, M. Sugeno,M. Suzaki, Autocontinuity, convergence in measure, and convergence in distribution, Fuzzy Sets and Systems 92(2)(1997) 197-203. [18] Y. Narukawa, T. Murofushi, M. Sugeno, Regular fuzzy measure and representation of comonotonically additive functional, Fuzzy sets and Systems, 112 (2000), 177-186. [19] Y. Narukawa, T. Murofushi, M. Sugeno, Extension and representation of comonotonically additive functional, Fuzzy sets and Systems, 121 (2001), 217-226. [20] W. Pedrycz, L. Yang, M. Ha, On the fundamental convergence in the (C) mean in problems of information fusion, J. Math. Anal. Appl. 358 (2009) 203-222. [21] P. Pucci, G. Vitillaro, A representation theroem for Aumann integrals, J. Math. Anal. Appl. , 102 (1984), 86-101. [22] I.B. Turksen, Non-specificity and interval-valued fuzzy sets, Fuzzy sets and Systems, 80 (1996), 87-100. [23] G. Wang and X. Li, The applications of interval-valued fuzzy numbers and interval-distribution numbers, Fuzzy Sets and Systems, 98 (1998), 331-335. [24] Z. Wang, Convergence theorems for sequences of Choquet integral, Int. Gen. Syst. 26 (1997) 133-143. [25] S.H. Wei, S.M. Chen, Fuzzy risk analysis based on interval-valued fuzzy sets, Expert Systems with Applications, 36(2009), 2285-2299. [26] Z. Xu, M. Xia, Distance and similarity measures for hesitant fuzzy sets, Information Sciences 181(2011), 2128-2138. [27] W. Zeng and H. Li, Relationship between similarity measure and entropy of interval-valued fuzzy sets, Fuzzy Sets and Systems, 157(2004), 1447-1484. [28] W. Zeng and P. Guo, Normalized distance, similarity measure, inclusion measure and entropy of intervalvalued fuzzy sets and their relationship, Information Sciences 179(2008), 1334-1342. [29] H. Zhang, W. Zhang, C. Mei, Entropy of interval-valued fuzzy sets based on distance and its relationship with similarity mesaure, Knowledge-Based Systems, 22(2009), 449-454.

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n-JORDAN ∗-DERIVATIONS ON INDUCED FUZZY C ∗ -ALGEBRAS GANG LU, YANDUO WANG, AND PENGYU YE Abstract. Using the fixed point alternative theorem, we investigate the Hyers-Ulam stability of of n-Jordan ∗-derivations on induced fuzzy C ∗ -algebras associated with the following functional equation f (y − x) + f (x − z) + f (3x − y + z) = f (3x).

1. Introduction and preliminaries The stability problem of functional equations originated from a question of Ulam [43] concerning the stability of group homomorphisms. Hyers [22] 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 [38] for linear mappings by considering an unbounded Cauchy difference. Those results have been recently complemented in [9]. A generalization of the Aoki and Rassias theorem was obtained by G˘avruta [21], who used a more general function controlling the possibly unbounded Cauchy difference in the spirit of Rassias’ approach. The stability problems for several functional equations or inequalities have been extensively investigated by a number of authors and there are many interesting results concerning this problem (see [8, 15], [23]–[31], [39]–[41]). 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 [14, 18]). 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) < ∞, for all 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 . By using the fixed point method, the stability problems of several functional equations have been extensively investigated by a number of authors (see [10, 13, 14, 17, 19, 28, 33, 34, 37, 46]). 2010 Mathematics Subject Classification. Primary 39B62, 39B52, 46B25. Key words and phrases. Fuzzy normed space; additive functional equation; Hyers-Ulam stability; induced fuzzy C ∗ -algebra. 1

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In 1984, Katsaras [27] defined a fuzzy norm on a linear space and at the same year Wu and Fang [44] also introduced a notion of fuzzy normed space and gave the generalization of the Kolmogoroff normalized theorem for fuzzy topological linear space. In [7], Biswas defined and studied fuzzy inner product spaces in linear space. Since then some mathematicians have defined fuzzy metrics and norms on a linear space from various points of view [6, 20, 30, 42, 45]. In 1994, Cheng and Mordeson introduced a definition of fuzzy norm on a linear space in such a manner that the corresponding induced fuzzy metric is of Kramosil and Michalek type [29]. In 2003, Bag and Samanta [6] modified the definition of Cheng and Mordeson [16] by removing a regular condition. They also established a decomposition theorem of a fuzzy norm into a family of crisp norms and investigated some properties of fuzzy norms (see [3]). Following [2], we give the employing notion of a fuzzy norm. Let X be a real linear space. A function N : X × R → [0, 1](the so-called fuzzy subset) is said to be a fuzzy norm on X if for all x, y ∈ X and all a, b ∈ R: (N1 ) N (x, a) = 0 for a ≤ 0; (N2 ) x = 0 if and only if N (x, a) = 1 for all a > 0; b ) if a 6= 0; (N3 ) N (ax, b) = N (x, |a| (N4 ) N (x + y, a + b) ≥ min{N (x, a), N (y, b)}; (N5 ) N (x, .) is a non-decreasing function on R and lima→∞ N (x, a) = 1; (N6 ) For x 6= 0, N (x, .) is (upper semi) continuous on R. The pair (X, N ) is called a fuzzy normed linear space. One may regard N (x, a) as the 0 truth value of the statement the norm of x is less than or equal to the real number a . Definition 1.2. Let (X, N ) be a fuzzy normed linear space. Let xn be a sequence in X. Then xn is said to be convergent if there exists x ∈ X such that limn→∞ N (xn − x, a) = 1 for all a > 0. In that case, x is called the limit of the sequence xn and we denote it by N -limn→∞ xn = x. Definition 1.3. A sequence xn in X is called Cauchy if for each  > 0 and each a > 0 there exists n0 such that for all n ≥ n0 and all p > 0, we have N (xn+p − xn , a) > 1 − . It is known that every convergent sequence in fuzzy normed space is Cauchy. If each Cauchy sequence is convergent, then the fuzzy norm is said to be complete and the fuzzy normed space is called a fuzzy Banach space. We say that a mapping f : X → Y between fuzzy normed vector space X, Y is continuous at 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 [2]) Definition 1.4. [36] Let X be a ∗-algebra and (X, N ) a fuzzy normed space. (1) The fuzzy normed space (X, N ) is called a fuzzy normed ∗-algebra if N (xy, st) ≥ N (x, s) · N (y, t) and N (x∗ , t) = N (x, t). (2) A complete fuzzy normed ∗-algebra is called a fuzzy Banach ∗-algebra. Example 1.5. Let (X, k.k) be a normed ∗-algebras. Let  a , a > 0 , x ∈ X, a+kxk N (x, a) = 0, a ≤ 0, x ∈ X Then N (x, t) is a fuzzy norm on X and (X, N (x, t)) is a fuzzy normed ∗-algebra.

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Definition 1.6. Let (X, k · k) be a C ∗ -algebra and N a fuzzy norm on X. (1) The fuzzy normed ∗-algebra (X, N ) is called an induced fuzzy normed ∗-algebra. (2) The fuzzy Banach ∗-algebra (X, N ) is called an induced fuzzy C ∗ -algebra. Definition 1.7. Let (X, k · k) be an induced fuzzy normed ∗-algebra. Then a C-linear mapping D : (X, N ) → (X, N ) is called a fuzzy n-Jordan ∗-derivation if D(xn ) = D(x)xn−1 + xD(x)xn−2 + · · · + xn−2 D(x)x + xn−1 D(x), D(x∗ ) = D(x)∗ for all x ∈ X. Throughout this paper, assume that (X, N ) is an induced fuzzy C ∗ -algebra. 2. Main results Lemma 2.1. Let (Z, N ) be a fuzzy normed vector space and f : X → Z be a mapping such that   t (2.1) N (f (y − x) + f (x − z) + f (3x − y + z) , t) ≥ N f (3x) , 2 for all x, y, z ∈ X and all t > 0. Then f is additive. Proof. Letting x = y = z = 0 in (2.1), we get     t t N (3f (0), t) = N f (0), ≥ N f (0), 3 2 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 x = z = 0 in (2.1), we get   t N (f (y) + f (0) + f (−y), t) ≥ N f (0), =1 2 for all t > 0. It follows from (N2 ) that f (−y) + f (y) = 0 for all y ∈ X. Thus f (−y) = −f (y) for all y ∈ X. Letting x = 0 and replacing z by −z in (2.1), we get   t N (f (y) + f (z) + f (−y − z), t) ≥ N f (0), =1 2 for all t > 0. It follows from (N2 ) that f (y) + f (z) + f (−y − z) = 0 for all y, z ∈ X. Thus f (y + z) = f (y) + f (z) for all y, z ∈ X, as desired.



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Theorem 2.2. Let φ : X 3 → [0, ∞) be a function such that there exists an L < 1 with x y z  L φ , , ≤ φ(x, y, z) (2.2) 3 3 3 3 for all x, y, z ∈ X. Let f : X → X be a mapping such that N (f (µ(y − x)) + f (µ(x − z)) + f (µ(3x − y + z)) − µf (3x) , t) t , ≥ t + φ(x, y, z)

(2.3)

N f (wn ) − f (w)wn−1 − wf (w)wn−2 − · · · − wn−2 f (w)w − wn−1 f (w) t +f (v ∗ ) − f (v)∗ , t) ≥ t + φ(w, v, 0)

(2.4)

for all x, y, z, w, v ∈ X, all t
> 0 and all µ ∈ T1 := {c ∈ C : |c| = 1}. Then the limit A(x) = N − limn→∞ 3n f 3xn exists for each x ∈ X and the mapping A : X → X is a fuzzy n-Jordan ∗-derivation satisfying N (f (x) − A(x), t) ≥

3(1 − L)t 3(1 − L)t + Lφ (x, 2x, 0)

(2.5)

for all x ∈ X and all t > 0. Proof. Letting µ = 1, y = 2x , z = 0 in (2.3), we have t N (3f (x) − f (3x), t) ≥ t + φ (x, 2x, 0)

(2.6)

and so  x  t t
= N 3f − f (x), t ≥ L x 2x 3 t + 3 φ (x, 2x, 0) t + φ 3, 3 , 0 for all x ∈ X. Thus     x L N 3f − f (x), t ≥ 3 3

L t 3 L t 3

+

L φ (x, 2x, 0) 3

=

t t + φ (x, 2x, 0)

(2.7)

for all x ∈ X. Consider the set G := {g : X → X} and introduce the generalized metric on G: d(g, h) := inf{a ∈ R+ : N (g(x) − h(x), at) ≥

t } t + φ (x, 2x, 0)

for all x ∈ X and all t > 0, where inf φ = +∞. It is easy to show that (S, d) is complete (see the proof of [32, Lemma 2.1] Now, we consider the linear mapping Q : G → G such that x Qg(x) := 3g 3 for all x ∈ X.

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Let g, h ∈ G be given such that d(g, h) = ε. Then N (g(x) − h(x), εt) ≥

t t + φ (x, 2x, 0)

for all x ∈ X and all t > 0. Hence     x x  x L  x N (Qg(x) − Qh(x), Lεt) = N 3g − 3h , Lεt = N g −h , εt 3 3 3 3 3 ≥ =

Lt 3

+φ

Lt 3
x 2x , ,0 3 3

≥

Lt 3 Lt 3

+ L3 φ (x, 2x, 0)

t t + φ (x, 2x, 0)

for all x ∈ X and all t > 0. Thus d(g, h) = ε implies that d(Qg, Qh) ≤ Lε. This means that d(Qg, Qh) ≤ Ld(g, h) for all g, h ∈ G. It follows from (2.7) that d(f, Qf ) ≤ L3 . By Theorem 1.1, there exists a mapping A : X → X satisfying the following: (1) A is a fixed point of Q, i.e., x 1 A = A(x) 3 3

(2.8)

for all x ∈ X. The mapping A is a unique fixed point of Q in the set M = {g ∈ G : d(f, g) < ∞}. This implies that A is a unique mapping satisfying (2.8) such that there exists an a ∈ (0, ∞) satisfying N (f (x) − A(x), at) ≥

t t + φ (x, 2x, 0)

for all x ∈ X. (2) d(Qk f, A) → 0 as k → ∞. This implies the equality x N − lim 3k f k = A(x) k→∞ 3 for all x ∈ X; (3) d(f, A) ≤

1 d(f, Qf ), 1−L

which implies the inequality d(f, A) ≤

L . 3(1 − L)

This implies that the inequality (2.5) holds.

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Next we show that A is additive. It follows from (2.2) that ∞ X

3k φ

k=0

x

x y z  x y z y z 2 = φ(x, y, z) + 3φ , , + 3 φ + ··· , , , , 3k 3k 3k 3 3 3 32 32 32 ≤ φ(x, y, z) + Lφ(x, y, z) + L2 φ(x, y, z) + · · · 1 = φ(x, y, z) < ∞ 1−L

for all x, y, z ∈ X. By (2.3),           x−z 3x − y + z 3 y−x k k k k +3 f µ k +f µ − 3 µf N 3 f µ k x ,3 t 3 3 3k 3k t
≥ x y t + φ 3k , 3k , 3zk and so           y−x x−z 3x − y + z 3 k k k k x ,t N 3 f µ k +3 f µ k +3 f µ − 3 µf 3 3 3k 3k ≥

t 3k

+φ

t 3k
x y z , , k k k 3 3 3

=

t t + 3k φ

x y . , z 3k 3k 3k




for all x, y, z ∈ X, all t > 0 and all µ ∈ T1 . Since limk→∞ x, y, z ∈ X and all t > 0,

t t+3k φ(

x y , , z 3k 3k 3k

)

= 1 for all

N (A (µ(y − x)) + A (µ(x − z)) + A (µ(3x − y + z)) − µA (3x) , t) = 1 for all x, y, z ∈ X, all t > 0 and all µ ∈ T1 . So A (µ(y − x)) + A (µ(x − z)) + A (µ(3x − y + z)) = µA (3x)

(2.9)

for all x, y, z ∈ X, all t > 0 and all µ ∈ T1 . Letting x = y = z = 0 in (2.9), we have A(0) = 0. Let µ = 1, x = 0 and replace z by −z in (2.9). By the same reasoning as in the proof of Lemma 2.1, one can easily show that A is additive. Letting y = 2x, z = 0 in (2.9), we get  x µA(x) = 3A µ = A(µx) 3 for all x ∈ X and µ ∈ T1 . The mapping A : X → X is C-linear by [35, Theorem 2.1]. By (2.4) and letting v = 0 in (2.4), we get   n  w   w n−1  w   w n−2 w nk nk nk w N 3 f − 3 f − 3 f − ··· 3nk 3k 3k 3k 3k 3k   w n−2  w   w n−1  w  t nk nk nk f k w−3 f k ,3 t ≥ −3 k k 3 3 3 3 t + φ( 3wk , 0, 0)

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for all w ∈ X and all t > 0. Thus   n  w   w n−1  w   w n−2 w nk nk nk w N 3 f − 3 f − 3 f − ··· 3nk 3k 3k 3k 3k 3k t  w n−1  w    w n−2  w  nk nk 3nk w − 3 , t ≥ −3 f f t 3k 3k 3k 3k + φ( 3wk , 0, 0) 3nk t ≥ n−1 t + (3 L)k φ(w, 0, 0) for all w ∈ X and all t > 0. Since limk→∞ t > 0, we get

t t+(3n−1 L)k φ(w,0,0)

= 1 for all w ∈ X and all

N (D(wn ) − D(w)wn−1 − wD(w)wn−2 − · · · − wn−2 D(w)w − wn−1 D(w), t) = 1 for all x ∈ X and all t > 0. So D(wn ) − D(w)wn−1 − wD(w)wn−2 − · · · − wn−2 D(w)w − wn−1 D(w) = 0 for all w ∈ X. Letting w = 0 in (2.4), similarly, we get D(v ∗ ) − D(v)∗ = 0 for all v ∈ X. Therefore, the mapping D : X → X is a fuzzy n-Jordan ∗-derivation.



Corollary 2.3. Let p be a real number with p > 1 , θ ≥ 0, and X be a normed vector space with norm k · k. Let f : X → X be a mapping satisfying N (f (µ(y − x)) + f (µ(x − z)) + f (µ(3x − y + z)) − µf (3x) , t) t , ≥ p t + θ(kxk + kykp + kzkp )

(2.10)

N f (wn ) − f (w)wn−1 − wf (w)wn−2 − · · · − wn−2 f (w)w − wn−1 f (w) t +f (v ∗ ) − f (v)∗ , t) ≥ t + θ(kwkp + kvkp )

(2.11)


for all x, y, w, v ∈ X, all t > 0 and all µ ∈ T1 . Then the limit A(x) = N −limn→∞ 3n f 3xn exists for each x ∈ X and the mapping A : X → X is a fuzzy n-Jordan ∗-derivation satisfying (3p − 3)t N (f (x) − A(x), t) ≥ p (3 − 3)t + θ(1 + 2p )kxkp for all x ∈ X and all t > 0. Proof. The proof follows from Theorem 2.2 by taking φ(x, y, z) = θ(kxkp + kykp + kzkp ) and L = 31−p .



Theorem 2.4. Let φ : X 3 → [0, ∞) be a function such that there exists an L < 1 with x y z  3Lφ , , ≤ φ(x, y, z) (2.12) 3 3 3

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for all x, y, z ∈ X. Let f : X → X be a mapping satisfying (2.3) and (2.4). Then the limit A(x) = N − limn→∞ 31n f (3n x) exists for each x ∈ X and the mapping A : X → X is a fuzzy n-Jordan ∗-derivation satisfying N (f (x) − A(x), t) ≥

3(1 − L)t 3(1 − L)t + φ (x, 2x, 0)

(2.13)

for all x ∈ X and all t > 0. Proof. Let (G, d) be generalized metric space defined in the proof of Theorem 2.2. Consider the linear mapping Q : G → G such that 1 Qg(x) := g(3x) 3 for all x ∈ X. It follow from (2.6) that   1 1 t N f (x) − f (3x), t ≥ 3 3 t + φ (x, 2x, 0) for all x ∈ X and all t > 0. Thus d(f, Qf ) ≤ 13 . Hence d(f, A) ≤

1 , 3(1 − L)

which implies that the inequality (2.13) 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 positive real number with p < 1. Let X be a normed vector space with normed k · k. Let f : X → X be a mapping satisfying (2.10) and (2.11). Then A(x) = N − limn→∞ 31n f (3n x) exists for each x ∈ X and defines a fuzzy n-Jordan ∗-derivation A : X → X such that N (f (x) − A(x), t) ≥

(3 − 3p )t (3 − 3p )t + θ(1 + 2p )kxkp

for every x ∈ X and all t > 0. Proof. The proof follows from Theorem 2.4 by taking φ(x, y, z) = θ(kxkp + kykp + kzkp ) and L = 3p−1 .



Acknowledgments G. Lu was supported by Doctoral Science Foundation of Liaoning Province, China, by Hall of Liaoning Province Science and Technology (No. 2012-1055), Shengyang University of Technology(No.521101302) and the Project Sponsored by the Scientific Research Foundation for the Returned Overseas Chinese Scholars, State Education Ministry. C. Park was supported by Basic Science Research Program through the National Research Foundation of Korea funded by the Ministry of Education, Science and Technology (NRF-2012R1A1A2004299).

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[27] A. K. Katsaras, Fuzzy topological vector spaces II, Fuzzy Sets Syst. 12 (1984), 143–154. [28] H. Khodaei, R. Khodabakhsh and M. Eshaghi Gordji, Fixed points, Lie ∗-homomorphisms and Lie ∗-derivations on Lie C ∗ -algebras, Fixed Point Theory 14 (2013), 387–400. [29] I. Kramosil and J. Michalek, Fuzzy metric and statistical metric spaces, Kybernetica 11 (1975), 326–334. [30] S. V. Krishna and K. K. M. Sarma, Separation of fuzzy normed linear spaces, Fuzzy Sets Syst. 63 (1994), 207–217. [31] G. Lu and C. Park, Hyers-Ulam stability of additive set-valued functional equations, Appl. Math. Lett. 24 (2011), 1312–1316. [32] 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. [33] F. Moradlou and M. Eshaghi Gordji, Approximate Jordan derivations on Hilbert C ∗ -modules, Fixed Point Theory 14 (2013), 413–425. [34] C. Park, Fixed points and Hyers-Ulam-Rassias stability of Cauchy-Jensen functional equations in Banach algebras, Fixed Point Theory Appl. 2007, Article ID 50175 (2007). [35] C.Park, Homomorphisms between Poisson JC ∗ -algebras, Bull. Braz. Math. Soc. 36 (2005), 79–97. [36] C. Park, K. Ghasemi and S. Ghaleh, Fuzzy n-Jordan ∗-derivations on induced fuzzy C ∗ -algebras, J. Comput. Anal. Appl. 16 (2014), 494–502. [37] C. Park and J. M. Rassias, Stability of the Jensen-type functional equation in C ∗ -algebras: A fixed point approach, Abs. Appl. Anal. 2009, Article ID 360432 (2009). [38] Th. M. Rassias, On the stability of the linear mapping in Banach spaces, Proc. Amer. Math. Soc. 72 (1978), 297–300. [39] Th. M. Rassias (Ed.), Functional Equations and Inequalities, Kluwer Academic, Dordrecht, 2000. [40] Th. M. Rassias, On the stability of functional equations in Banach spaces, J. Math. Anal. Appl. 251 (2000), 264–284. [41] Th .M. Rassias, On the stability of functional equations and a problem of Ulam, Acta Math. Appl. 62 (2000), 23–130. [42] B. Shieh, Infinite fuzzy relation equations with continuous t-norms, Inform. Sci. 178 (2008), 1961– 1967. [43] S.M. Ulam, Problems in Modern Mathematics, Chapter VI, Science ed., Wiley, New York, 1940. [44] C. Wu and J. Fang, Fuzzy generalization of Klomogoroff ’s theorem, J. Harbin Inst. Technol. 1 (1984), 1–7. [45] J. Z. Xiao and X.-H. Zhu, Fuzzy normed spaces of operators and its completeness, Fuzzy Sets Syst. 133 (2003), 389–399. [46] 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, Zhejiang University, Hangzhou 310027, People’s Republic of China 2.Department of Mathematics, School of Science, ShenYang University of Technology, Shenyang 110178, P. R. China E-mail address: [email protected] Yanduo Wang Department of Mathematics, School of Science, ShenYang University of Technology, Shenyang 110178, P. R. China E-mail address: [email protected] Pengyu Ye Department of Mathematics, School of Science, ShenYang University of Technology, Shenyang 110178, P. R. China E-mail address: yuxiang1 [email protected]

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Global stability analysis of a delayed viral infection model with antibodies and general nonlinear incidence rate A. M. Elaiw, N. H. AlShamrani and M. A. Alghamdi Department of Mathematics, Faculty of Science, King Abdulaziz University, P.O. Box 80203, Jeddah 21589, Saudi Arabia. Email: [email protected] (A. Elaiw).

Abstract In this paper, we study the global properties of a viral infection model with antibody immune response. The incidence rate is given by a general function of the populations of the uninfected target cells, infected cells and free viruses. The model contains two types of intracellular discrete time delays to describe the time required for viral contacting an uninfected target cell and viral emission. We have established a set of conditions on the general incidence rate function and determined two threshold parameters R0 (the basic infection reproduction number) and R1 (the antibody immune response activation number) which are su¢ cient to determine the global behavior of the model. The global asymptotic stability of the equilibria of the model has been proven by using direct Lyapunov method and applying LaSalle’s invariance principle.

Keywords: Virus dynamics; Intracellular delay; global stability; antibody immune response; Lyapunov functional.

1

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1

Introduction

In recent years, several works have been devoted to study and develop mathematical models of the virus dynamics such as human immunode…ciency virus (HIV) (see e.g. [1]-[14]), hepatitis B virus (HBV) [15]-[18], hepatitis C virus (HCV) [19]-[21] and human T cell leukemia HTLV [22], etc. Mathematical models of viral infection can help for understanding the viral dynamics and developing antiviral drug therapies. In reality, the immune response needs an indispensable components to do its job such as antibodies, cytokines, natural killer cells, and T cells. The antibody immune response is a part of the adaptive system in which the body responds to pathogens by primarily using antibodies that produced from the B cells. While the other part is the Cytotoxic T Lymphocytes (CTL) immune response where the CTL attacks and kills the infected cells [4]. In some infections such as in malaria, the CTL immune response is less e¤ective than the antibody immune response [23]. Mathematical models of viral infection with antibody immune response have been proposed and analyzed in ([24]-[29]). The basic model of viral infection with antibody immune response has introduced by Murase et. al. [24] and Shi… Wang [29] as: x(t) _ =s

dx(t)

y(t) _ = v(t)x(t) v(t) _ = ky(t)

v(t)x(t); ay(t);

bz(t)v(t)

z(t) _ = rz(t)v(t)

(1) (2)

cv(t);

(3)

z(t);

(4)

where x(t), y(t), v(t) and z(t) denote the populations of uninfected target cells, infected cells, free virus particles and antibody immune cells at time t, respectively. Parameters s, k and r represent, respectively, the rate at which new healthy cells are generated from the source within the body, the generation rate constant of free viruses produced from the infected cells and the proliferation rate constant of antibody immune cells. Parameters d, a, c and

are the natural death rate constants of the uninfected cells, infected

cells, free virus particles and antibody immune cells, respectively. Parameter

is the infection rate constant

and b is the removal rate constant of the virus due to the antibodies. All the parameters given in model (1)-(4) are positive. 2

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The intracellular time delay between the time of the virus contacting the target cells and the time of generating new infectious viruses has been neglected in system (1)-(4). In fact, the intracellular delay in the infection process is actually exists (see e.g. [8]-[12]). Note that, the infection rate in model (1)-(4) is presented to be bilinear in x and v, which can not be completely describe the interaction between the uninfected target cells and viruses. Nevertheless, there are many types of improved incidence rates which are more commonly used due to their bene…t for helping us gain the uni…cation theory through passing over the unessential details (see e.g. [30] and [31]). Variety of viral infection models with antibody immune response have been considered with di¤erent forms of the incidence rate such as saturated incidence rate, 0, [27], Beddington-DeAngelis functional response,

xv 1+ x+ v ,

;

xv 1+ v

0 [26], and general form,

where (x; v)v

[28]. In [28], a discrete time delay has been incorporated within the model. However, the infection rate does not depend on the infected cells y. In some viral infections such as HBV, the infection rate depends on x, y and v [17], [16]. In [32], the infection rate is given by (x; y; v)v, however the antibody immune response has been neglected. Our aim in this paper is to investigate the global stability analysis of a viral infection model with general incidence rate function and antibody immune response taking into consideration two types of discrete time delays. The rest of the paper is designed as follows. In the next section, we introduce the model and discuss the non-negativity and boundedness of the solutions. In Section 3, we de…ne two threshold parameters and discuss the existence of the model’s equilibria. In Section 4, we study the global asymptotic stability of the equilibria using suitable Lyapunov functional and applying LaSalle’s invariance principle. Finally, conclusion is given in Section 5.

3

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2

The mathematical model

In this section, we consider the following viral infection model with general incidence rate taking into consideration the antibody immune response. x(t) _ =s

dx(t)

y(t) _ =e

1 1

v(t) _ = ke

2 2

(x(t); y(t); v(t))v(t);

(x(t y(t

2)

z(t) _ = rz(t)v(t) where time t

1

and 1,

2

1 ); y(t

1 ); v(t

bz(t)v(t)

1)

ay(t);

(6)

cv(t);

(7) (8)

are the delay parameters. We assume that, the virus contacts an uninfected target cell at

the cell becomes infected at time t. The term e

addition, we assume that a cell infected at time t 2 2

1 ))v(t

z(t);

contacted cell during the time delay interval, where

term e

(5)

2

1

1 1

represents the probability of surviving the

is the death rate constant of the contacted cells. In

starts to generate new infectious viruses at time t. The

denotes the probability of surviving the infected cell during the time delay interval, where

2

is a constant. The de…nitions of all variables and parameters are identical to those given in Section 1. The incidence rate of infection is presented by a general function in the form (x; y; v)v, where

is continuously

di¤erentiable and satis…es the following assumptions [28] and [32]: Assumption A1.

(0; y; v) = 0 for all y; v

0 and

(x; y; v) > 0 for all x > 0, y

0, v

0.

@ (x; y; v) > 0 for all x > 0, y 0 and v 0: @x @ (x; y; v) @ (x; y; v) Assumption A3. < 0, < 0 for all x; y; v > 0: @y @v @ ( (x; y; v)v) Assumption A4. > 0 for all x; y; v > 0: @v Assumption A2.

Let the initial states of system (5)-(8) be given as: x( ) = j(

)

j (0)

where

= maxf 1 ;

2 g;

( 1 ( );

1(

0;

); y( ) = 2[

2(

); v( ) =

3(

); z( ) =

4(

);

; 0); j = 1; :::; 4;

> 0; j = 1; :::; 4; 2(

);

3(

);

4(

(9)

)) 2 C([

; 0]; R4 0 ). We denote by C = C([

; 0]; R4 0 ) the

4

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Banach space of continuous functions mapping the interval [ for

; 0] into R4 0 ; with norm k k =

sup j ( )j 0

2 C. We note that the system (5)-(8) with initial states (9) has a unique solution [33].

2.1

Non-negativity and boundedness of solutions

In this section, we show that the solutions of model (5)-(8) with initial states (9) are non-negative and ultimately bounded. Proposition 1. Assume that Assumption A1 is satis…ed. Then the solutions of (5)-(8) with the initial states (9) are non-negative and ultimately bounded. Proof. At the beginning, we show that x(t) is positive for all t x(t)

0 on the time interval [0; ] where

0. Let us assume in contrary that

is a constant, and let where t 2 [0; ] be such that x(t) = 0,

= s > 0. Thus, for su¢ ciently small " > 0, we have x(t) > 0 for some . Then from Eq. (5) we get x(t) _ t 2 t; t + " . This contradicts our assumption and then x(t) > 0, 8 t y(t) = y(0)e

at

+e

1 1

Zt

e

a(t

)

(x(

1 ); y(

0: Now from Eqs. (6)-(8) we get

1 ); v(

1 ))v(

1 )d

;

0

v(t) = v(0)e

Rt

z(t) = z(0)e

Rt

(c+bz( ))d 0

0

Zt

Rt

e

0

(

which yield y(t), v(t), z(t) all t

+ ke

2 2

rv( ))d

(c+bz( ))d

y(

2 )d

;

;

0 for all t 2 [0; ]. By a recursive argument, we get that y(t); v(t); z(t)

0 for

0:

Next we prove the ultimate bound of the solutions of system (5)-(8). From Eq. (5) we get x(t) _ s d.

and thus lim supt!1 x(t) T_1 (t) = e

1 1

+e

1 1

(s

= se

1 1

= se

1 1

Let T1 (t) = e dx(t

(x(t de

1)

1 ); y(t 1 1

1 T1 (t)

x(t s

1 1

x(t

1)

(x(t

1)

+ y(t), then

1 ); y(t

1 ); v(t

1 ))v(t

ay(t)

s dx(t)

se

1 ); v(t 1) 1 1

1 ))v(t

1 ))

ay(t); 1

e

1 1

x(t

1)

+ y(t)

1 T1 (t);

5

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where

1

= minfd; ag. Hence lim supt!1 T1 (t)

then lim supt!1 y(t)

where

2

L1 , where L1 =

s

. Since x(t) and y(t) are non-negative,

1

L1 . Moreover, let T2 (t) = v(t) + rb z(t), then

T_2 (t) = ke

2 2

y(t

2)

= ke

2 2

L1

2 T2 (t)

b z(t) r

cv(t)

kL1

ke

L1

2 (v(t)

b + z(t)) r

2 T2 (t);

= minfc; g. It follows that, lim supt!1 T2 (t)

non-negative, then lim supt!1 v(t)

2 2

L2 , where L2 =

kL1

. Since v(t) and z(t) are

2

L2 and lim supt!1 z(t)

L3 , where L3 =

r b L2 .

Therefore, all the

state variables of the model are ultimately bounded.

2.2

The equilibria and threshold parameters

At any equilibrium we have s e

dx

(x; y; v)v

1 1

ke

(x; y; v)v = 0;

2 2

y

bvz

(10)

ay = 0;

(11)

cv = 0;

(12)

)z = 0:

(13)

(rv

From Eq. (13), either z = 0 or z 6= 0. If z = 0, then from Eqs. (10)-(12) we get s dx c = v; 1 1 ae ke 2 2

y=

v=

k(s ace 1

dx)

:

(14)

v = 0:

(15)

1+ 2 2

Substituting from Eq. (14) into Eq. (11) we get: x;

s dx k(s ; ae 1 1 ace 1

ac e k

dx) 1+ 2 2

1 1+ 2 2

Eq. (15) has two possible solutions v = 0 or v 6= 0. If v = 0; then from Eqs. (10) and (11), we get x = s=d and y = 0 which leads to the infection-free equilibrium E0 (x0 ; 0; 0; 0) where x0 = s=d. If v 6= 0; then we have x;

s dx k(s ; ae 1 1 ace 1

dx)

ac e k

1+ 2 2

1 1+ 2 2

= 0:

Let 1

(x) =

x;

s dx k(s ; ae 1 1 ace 1

dx) 1+ 2 2

ac e k

1 1+ 2 2

= 0;

6

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then, we have 0 1

@ @x

(x) =

@ @y

d ae

1 1

0 1

Because of Assumptions A2 and A3, we have

kd ace

1 1+ 2 2

@ : @v

(x) > 0 which implies that function

1 (x)

is strictly

increasing w.r.t. x. Moreover, 1 (0)

=

1 (x0 )

=

Therefore, if

0;

s ae

1 1

;

ac e k

(x0 ; 0; 0)

k (x0 ; 0; 0) e ac

ks

1 1

2 2

ac e k

1 1+ 2 2

ace

1 1+ 2 2

ac e k

=

1 1+ 2 2

1 1+ 2 2

ac e k

=

1 1+ 2 2

k (x0 ; 0; 0) e ac

< 0;

1 1

2 2

1 :

> 1; then there exist a unique x1 2 (0; x0 ) such that

It follows from (12) and (14) that y1 =

d(x0 ae

x1 )

> 0 and v1 =

1 1

1 (x1 )

kd(x0 x1 ) > 0. ace 1 1 + 2 2

= 0.

It means

that, a chronic-infection equilibrium without antibody immune response E1 (x1 ; y1 ; v1 ; 0) exists when k (x0 ; 0; 0) e ac

1 1

2 2

> 1. Let us de…ne the basic infection reproduction number as: R0 =

k (x0 ; 0; 0) e ac

1 1

2 2

:

The parameter R0 determines whether a chronic-infection can be established. The other possibility of Eq. (13) is z 6= 0 which leads to v2 =

r

. From Eq. (10) we let 2 (x)

=s

dx

(x;

According to Assumptions A2 and A3, we know that and

2 (x0 )

=

2

s dx ; v2 )v2 = 0: ae 1 1 is a decreasing function of x. Clearly,

(x0 ; 0; v2 )v2 < 0. Thus, there exists a unique x2 2 (0; x0 ) such that

from Eq. (14) that, y2 =

d(x0 ae

x2 ) 1 1

> 0 and z2 =

k (x2 ; y2 ; v2 ) abe 1 1 + 2 2

c c = b b

2 (x2 )

k (x2 ; y2 ; v2 ) ace 1 1 + 2 2

2 (0)

=s>0

= 0. It follows 1 . Then, if

k (x2 ; y2 ; v2 ) > 1 then z2 > 0. Now we de…ne the antibody immune response activation number as ace 1 1 + 2 2 R1 =

k (x2 ; y2 ; v2 ) ; ace 1 1 + 2 2

which determines whether a persistent antibody immune response can be established. Hence, z2 can be c rewritten as z2 = (R1 b

1). It follows that, there is a chronic-infection equilibrium with antibody immune

response E2 (x2 ; y2 ; v2 ; z2 ) when R1 > 1. Clearly from Assumptions A2 and A3, we have R1 =

k (x2 ; y2 ; v2 ) k (x0 ; y2 ; v2 ) k (x0 ; 0; 0) < < = R0 : + + 1 1 2 2 1 1 2 2 ace ace ace 1 1 + 2 2 7

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2.3

Global stability analysis

In this section, the global asymptotic stability of the three equilibria of model (5)-(8) will be established by using direct Lyapunov method and applying LaSalle’s invariance principle. In the remaining parts of the paper we shall use the following function: H : (0; 1) ! [0; 1), H(u) = u

1

ln u:

Theorem 1. Let Assumptions A1-A3 be hold true and R0

1; then the infection-free equilibrium E0 is

globally asymptotically stable (GAS). Proof. We construct a Lyapunov functional as: U0 = x

Z

x0

x

(x0 ; 0; 0) d +e ( ; 0; 0)

1 1

y+

(x( ); y( ); v( ))v( )d + ae

1 1

x0

+ t dU0 dt

We calculate dU0 = dt

1 1

t

1

1 1+ 2 2

ab e rk

v+

1 1+ 2 2

z

y( )d :

(16)

2

along the solutions of model (5)-(8) as: (x0 ; 0; 0) (x; 0; 0)

1

+ ae

Zt

a e k Zt

y(t

2)

+ (x; y; v)v

(s ac e k

(x(t

dx

(x; y; v) v) + (x(t

1 1+ 2 2

ab e k

v

1 ); y(t

1 1+ 2 2

1 ); v(t

=s 1

(x0 ; 0; 0) (x; 0; 0)

1

x x0

+

=s 1

(x0 ; 0; 0) (x; 0; 0)

1

x x0

+

From Assumptions A2-A3 we know that

ac e k

zv +

1 ))v(t

(x; y; v)

1 ); y(t

ab e k

1)

1 1+ 2 2

+ ae

(x0 ; 0; 0) (x; 0; 0)

1 ); v(t

1 1

ac e k

(x; y; v) R0 (x; 0; 0)

1 1+ 2 2

1 ))v(t

ab e rk

zv

(y

y(t

1 1+ 2 2

1 v

1)

1 1+ 2 2

ae

1 1

y

z

2 ))

ab e rk

v ab e rk

1 1+ 2 2

1 1+ 2 2

z:

z (17)

(x; y; v) is an increasing function of x and decreasing function of

y and v. Then, the …rst term of Eq. (17) is less than or equal zero and (x; y; v) < (x; 0; 0),

x; y; v > 0:

It follows that dU0 dt

s 1

(x0 ; 0; 0) (x; 0; 0)

1

x x0

+

ac e k

1 1+ 2 2

(R0

1) v

ab e rk

1 1+ 2 2

z:

(18)

8

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Therefore, if R0 to

1, then

dU0 dt

0 for all x; y; v; z > 0. We note that the solutions of system (5)-(8) converge

, the largest invariant subset of

The set

dU0 dt

= 0 [33]. From (18), we have

is invariant and for any element belongs to

dU0 dt

= 0 i¤ x = x0 , v = 0 and z = 0.

satis…es v = 0 and z = 0. We can see from Eq. (7)

that v_ = 0 = ke It follows that, y = 0. Hence

dU0 dt

2 2

y(t

2 ):

= 0 i¤ x = x0 and y = v = z = 0. Using LaSalle’s invariance principle, we

derive that E0 is GAS. Assumption A5. 1

(x; y; v) (x; yi ; vi )

(x; yi ; vi ) (x; y; v)

v vi

0; i = 1; 2 for all x; y; v > 0:

Theorem 2. Let Assumptions A1-A5 be hold true and R1

1 < R0 , then the chronic-infection equilibrium

without antibody immune response E1 is GAS. Proof. De…ne: U1 = x

x1

Z

x

(x1 ; y1 ; v1 ) d +e ( ; y1 ; v1 )

x1

+

a e k

1 1+ 2 2

v v1

v1 H

+ (x1 ; y1 ; v1 )v1 t

Zt

H

+

ab e rk

1 1

y1 H

1 1+ 2 2

y y1

z

(x( ); y( ); v( ))v( ) (x1 ; y1 ; v1 )v1

1

d + ae

1 1

y1 t

Zt

H

y( ) y1

d :

(19)

2

9

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Calculating the time derivative of U1 along the trajectories of system (5)-(8), we obtain dU1 = dt +e +

(x1 ; y1 ; v1 ) (x; y1 ; v1 )

1

a e k

y1 y

1

1 1

1 1+ 2 2

e

=

y

1 1

y1 (x(t y ac e k ab + e k

y(t

v

1 1+ 2 2

v1 z

y1 ln

(s

1 ); v(t

2)

cv

1 ); v(t

1 ); v(t (x; y; v)v

y(t

2)

bvz +

1)

ab e rk

ay

1 1+ 2 2

(rvz

z)

1)

1 ))v(t

1)

2)

y

1 ))v(t

v1 e v

1 ))v(t

1 ))v(t

dx) + (x1 ; y1 ; v1 )

ab e rk (x(t 2)

y(t

1 ); y(t

1 ); v(t

y

1 ); y(t

2 2

+ y1 ln

ay(t

y(t

(x; y; v) v)

1 ); y(t

1 ); y(t

1 1+ 2 2

1 1

ke

2)

+ (x1 ; y1 ; v1 )v1 ln + ae

(x(t

(x(t

(x1 ; y1 ; v1 ) (x; y1 ; v1 )

1

1 1

(x(t

+ (x1 ; y1 ; v1 )v1 ln + ae

dx

v1 v

1

+ (x; y; v)v

(s

1 1

1 1+ 2 2

+

1)

ac e k

(x; y; v)v (x; y1 ; v1 )

+ ay1 e

1 1+ 2 2

1 1

v1

z

1 ); y(t

1 ); v(t (x; y; v)v

1 ))v(t

1)

:

(20)

Using the equilibrium conditions for E1 : s = dx1 + ae

1 1

y1 ;

(x1 ; y1 ; v1 )v1 = ae

1 1

y1 =

ac e k

1 1+ 2 2

v1 ;

10

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we obtain dU1 = dx1 1 dt

(x1 ; y1 ; v1 ) (x; y1 ; v1 ) (x1 ; y1 ; v1 ) + ae (x; y1 ; v1 )

ae

1 1

y1

ae

1 1

y1

ae

1 1

y1

+ ae

1 1

y1 ln

+ ae

1 1

y1 ln

y1 (x(t v v1

x x1

1 1 1

+ 3ae

1 ); v(t y (x1 ; y1 ; v1 )v1

1 1

y1

(x(t

v1 y(t vy1 1 ); y(t

y(t

2)

+

y

y1

(x; y; v)v (x; y1 ; v1 )v1

y1

1 ); y(t

ae

1 1

1 ))v(t

1)

2)

1 ); v(t (x; y; v)v

ab e k

1 ))v(t

1 1+ 2 2

v1

r

1)

z:

(21)

Using the following equalities: ln

(x(t

1 ); y(t

1 ); v(t

1 ))v(t

1)

y1 (x(t

= ln

(x; y; v)v

y(t

2)

y

1 ); v(t

1 ))v(t

1)

y (x1 ; y1 ; v1 )v1 (x1 ; y1 ; v1 ) (x; y1 ; v1 )

+ ln ln

1 ); y(t

= ln

vy1 v1 y

y1

(x; y; v)v (x; y1 ; v1 )v1

+ ln

(x; y1 ; v1 ) (x; y; v)

+ ln v1 y(t vy1

2)

+ ln

v1 y vy1

;

;

we get dU1 = dx1 1 dt ae + + +

1 1

(x1 ; y1 ; v1 ) (x; y1 ; v1 ) (x1 ; y1 ; v1 ) (x; y1 ; v1 )

y1

y1 (x(t v1 y(t vy1

1

x x1

1

1 ); v(t y (x1 ; y1 ; v1 )v1

(x; y1 ; v1 ) (x; y; v)

1

ln

1

ln

v1 y(t vy1

1 1

v v1

1+

(x; y1 ; v1 ) (x; y; v)

(x1 ; y1 ; v1 ) (x; y1 ; v1 )

ln

1 ); y(t

2)

+ ae

1 ))v(t

1)

1

ln

y1 (x(t

1 ); y(t

1 ); v(t y (x1 ; y1 ; v1 )v1

1 ))v(t

1)

2)

(x; y1 ; v1 ) (x; y; v)

+

ab e k

1 1+ 2 2

v1

r

z:

(22)

11

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Eq. (22) can be simpli…ed as: dU1 = dx1 1 dt

(x1 ; y1 ; v1 ) (x; y1 ; v1 )

+ ae

1 1

y1 1

(x; y; v) (x; y1 ; v1 )

ae

1 1

y1 H

(x1 ; y1 ; v1 ) (x; y1 ; v1 )

y1 (x(t

+H +

ab e k

1 1+ 2 2

x x1

1

(x; y1 ; v1 ) (x; y; v) v1 y(t vy1

+H

1 ); y(t

1 ); v(t y (x1 ; y1 ; v1 )v1

v1

r

v v1

1 ))v(t

2)

1)

+H

(x; y1 ; v1 ) (x; y; v)

z:

(23)

From Assumptions A1 and A5, we get that the …rst and second terms of Eq. (23) are less than or equal zero. Now we show that if R1

1 then v1

sgn(x2

r

x1 ) = sgn(v1

= v2 . Let R0 > 1, then we want to show that v2 ) = sgn(y1

y2 ) = sgn(R1

1):

From Assumptions A2-A4, for x1 ; x2 ; y1 ; y2 ; v1 ; v2 > 0, we have ( (x2 ; y2 ; v2 )

(x1 ; y2 ; v2 ))(x2

x1 ) > 0;

(24)

( (x1 ; y1 ; v1 )

(x1 ; y2 ; v1 ))(y2

y1 ) > 0;

(25)

( (x1 ; y1 ; v1 )

(x1 ; y1 ; v2 ))(v2

v1 ) > 0;

(26)

( (x2 ; y2 ; v2 )v2 First, we claim sgn(x2

x1 ) = sgn(v1

(x2 ; y2 ; v1 )v1 )(v2

v1 ) > 0:

v2 ). Suppose this is not true, i.e., sgn(x2

(27) x1 ) = sgn(v2

v1 ).

Using the conditions of the equilibria E1 and E2 we have (s

dx2 )

(s

dx1 ) = (x2 ; y2 ; v2 )v2 = ae

1 1

(y2

(x1 ; y1 ; v1 )v1

y1 ):

(28)

Then, sgn(x2

x1 ) = sgn(y1

y2 )

(29)

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Moreover, (s

dx2 )

(s

dx1 ) = (x2 ; y2 ; v2 )v2

(x1 ; y1 ; v1 )v1

= ( (x2 ; y2 ; v2 )v2

(x2 ; y2 ; v1 )v1 ) + ( (x2 ; y2 ; v1 )v1

+ ( (x1 ; y2 ; v1 )v1

(x1 ; y1 ; v1 )v1 ):

(x1 ; y2 ; v1 )v1 )

Therefore, from inequalities (24) and (29) we get: sgn (x1 which leads to contradiction. Thus, sgn (x2 we have

k (x1 ;y1 ;v1 ) ace 1 1 + 2 2

x2 ) = sgn (x2 x1 ) = sgn (v1

k (x2 ; y2 ; v2 ) k (x1 ; y1 ; v1 ) ace 1 1 + 2 2 ace 1 1 + 2 2 k = e 1 1 2 2 [ (x2 ; y2 ; v2 ) (x2 ; y2 ; v1 ) + (x2 ; y2 ; v1 ) ac

1=

(x1 ; y2 ; v1 ) + (x1 ; y2 ; v1 ) 1) = sgn(v1

(x1 ; y1 ; v1 )] :

v2 ): Hence, if R0 > 1; then x1 ; y1 ; v1 > 0, and if R1

follows from the above discussion that to

v2 ) : Using the equilibrium conditions for E1

= 1, then R1

We get sgn(R1

x1 ) ;

dU1 dt

1, then v1

v2 =

r.

It

0 for all x; y; v; z > 0. The solutions of system (5)-(8) converge

, the largest invariant subset of (x; y; v; z) :

dU1 dt

= 0 [33]. We have

dU1 dt

= 0 i¤ x = x1 ; v = v1 ; z = 0

and H = 0 i.e. y1 (x(t

1 ); y(t

1 ); v(t y (x1 ; y1 ; v1 )v1

1 ))v(t

1)

From Eq. (30), if v = v1 then y = y1 and hence

=

v1 y(t vy1

dU1 dt

2)

= 1 for almost all

i

2 [0; ]; i = 1; 2:

= 0 i¤ x = x1 ; y = y1 ; v = v1 and z = 0. So

(30)

contains a

unique point, that is E1 . Thus, the global asymptotic stability of the chronic-infection equilibrium without antibody immune response E1 follows from LaSalle’s invariance principle. Theorem 3. Let Assumptions A1-A5 be hold true and R1 > 1, then the chronic-infection equilibrium with antibody immune response E2 is GAS.

13

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Proof. We construct a Lyapunov functional as follows: U2 = x

Z

x2

x

(x2 ; y2 ; v2 ) d +e ( ; y2 ; v2 )

x2

+

a e k

1 1+ 2 2

v v2

v2 H Zt

+ (x2 ; y2 ; v2 )v2 t

ab e rk

+

1 1

y2 H

1 1+ 2 2

y y2 z z2

z2 H

(x( ); y( ); v( ))v( ) (x2 ; y2 ; v2 )v2

H

d + ae

1 1

y2 t

1

Zt

H

y( ) y2

1

z2 (rvz z

d :

(31)

2

Function U2 satis…es: dU2 = dt +e +

(x2 ; y2 ; v2 ) (x; y2 ; v2 )

1 1 1

a e k

y2 y

1

1 1+ 2 2

e 1

+ (x; y; v)v

1 1

dU2 =d 1 dt

+ ae +

1 1

ab e k

y2

(x(t ke

y(t

1 1

1 ); y(t

y(t

1 ); y(t

2)

1 ); v(t

2)

cv

1 ); v(t

1 ); y(t

1 ))v(t

bvz +

1 ))v(t

1 ); v(t (x; y; v)v

y(t

+ y2 ln

2)

ab e rk

1)

ay

1 1+ 2 2

1)

1 ))v(t

1)

:

y

z)

(32)

y2 , we get (x2

(x2 ; y2 ; v2 ) (x; y2 ; v2 ) ac e k

1 1+ 2 2

(x; y; v)v)

2 2

(x(t

(x2 ; y2 ; v2 ) (x; y2 ; v2 )

+ (x; y; v)v

1 1

(x(t

y

Applying s = dx2 + ae

dx

v2 v

+ (x2 ; y2 ; v2 )v2 ln + ae

(s

1 1+ 2 2

v2 z

+ (x2 ; y2 ; v2 )v2 ln

x) + ae

1 1

y2

(x2 ; y2 ; v2 )v2 v

ab e rk (x(t

ae

1 1

y(t

ae

y2 (x(t

2)

(x2 ; y2 ; v2 ) (x; y2 ; v2 )

y2

1 ); y(t

1 ); v(t y (x2 ; y2 ; v2 )v2

ac v2 + e v k

ab 1 1 + e k 1 ); y(t 1 ); v(t (x; y; v)v

1 1+ 2 2

1 1

z

2 2

1 1+ 2 2

z2 v +

1 ))v(t

1 ))v(t

1)

v2

ab e rk 1)

1 1+ 2 2

+ ae

1 1

z2

y2 ln

y(t

2)

y

(33)

By using the equilibrium conditions of E2 (x2 ; y2 ; v2 )v2 = ae

1 1

y2 ;

cv2 = ke

2 2

y2

bv2 z2 ;

= rv2 ;

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and the following equalities cv = cv2 ln

(x(t

1 ); y(t

1 ); v(t (x; y; v)v

1 ))v(t

1)

v = ke v2 y2 (x(t

= ln

y(t

2)

vy2 v2 y

= ln

y

+ ln

y2

bv2 z2

v ; v2

1 ); y(t

1 ); v(t y (x2 ; y2 ; v2 )v2

(x2 ; y2 ; v2 ) (x; y2 ; v2 )

+ ln ln

2 2

1 ))v(t

(x; y2 ; v2 ) (x; y; v)

+ ln v2 y(t vy2

2)

1)

v2 y vy2

+ ln

;

;

we obtain dU2 =d 1 dt ae + +

(x2 ; y2 ; v2 ) (x; y2 ; v2 )

1 1

(x2

(x2 ; y2 ; v2 ) (x; y2 ; v2 )

y2

y2 (x(t v2 y(t vy2

x) + ae 1

1 ); v(t y (x2 ; y2 ; v2 )v2

2)

1

ln

v2 y(t vy2

(x; y; v)v (x; y2 ; v2 )v2

y2

v v2

1+

(x; y2 ; v2 ) (x; y; v)

(x2 ; y2 ; v2 ) (x; y2 ; v2 )

ln

1 ); y(t

1 1

1 ))v(t

2)

1)

1

y2 (x(t

ln

(x; y2 ; v2 ) (x; y; v)

+

1

ln

1 ); y(t

1 ); v(t y (x2 ; y2 ; v2 )v2

(x; y2 ; v2 ) (x; y; v)

1 ))v(t

1)

:

(34)

We can rewrite (34) as dU2 = dx2 1 dt ae +H

1 1

(x2 ; y2 ; v2 ) (x; y2 ; v2 )

(x2 ; y2 ; v2 ) (x; y2 ; v2 )

y2 H

v2 y(t vy2

1

2)

+H

x x2

+ ae

1 1

y2 (x(t

+H

(x; y2 ; v2 ) (x; y; v)

(x; y2 ; v2 ) (x; y; v)

(x; y; v) (x; y2 ; v2 )

y2 1 1 ); y(t

1 ); v(t y (x2 ; y2 ; v2 )v2

1 ))v(t

v v2

1)

:

(35)

We note that from Assumptions A2 and A5, the …rst and second terms of Eq. (35) are less than or equal zero. Noting that x; y; v; z > 0, we have that largest invariant subset of (x; y; v; z) : y2 (x(t

1 ); y(t

1 ); v(t y (x2 ; y2 ; v2 )v2

dU2 dt

1 ))v(t

dU2 dt

0. The solutions of model (5)-(8) converge to

= 0 [33]. We have 1)

=

v2 y(t vy2

If v = v2 , then from Eq. (36) we get y = y2 . The set v = v2 =

r.

2)

dU2 dt

= 0 i¤ x = x2 ; v = v2 and H = 0 i.e.,

= 1 for almost all

i

2 [0; ]; i = 1; 2:

is invariant and for any element belongs to

From Eq. (7) we get z = z2 . Therefore,

dU2 dt

, the

(36) satis…es

= 0 i¤ x = x2 ; y = y2 ; v = v2 and z = z2 . The

global asymptotic stability of the chronic-infection equilibrium with antibody immune response E2 follows from LaSalle’s invariance principle. 15

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3

Conclusion

In this paper, we have proposed a delayed viral infection model with general incidence rate function and antibody immune response. The model has been incorporated with two kinds of discrete time delays representing the time needed for infecting an uninfected target cell and viral production. We have derived a set of conditions on the general functional response and have determined two threshold parameters R0 and R1 to prove the existence and the global stability of the model’s equilibria. The global asymptotic stability of the three equilibria, infection-free, chronic-infection without antibody immune response and chronic-infection with antibody immune response has been proven by using direct Lyapunov method and LaSalle’s invariance principle.

4

Con‡ict of Interests

The authors declare that there is no con‡ict of interests regarding the publication of this article.

5

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.

References [1] M.A. Nowak, C.R.M. Bangham, Population dynamics of immune responses to persistent viruses, Science, 272 (1996), 74-79. [2] A.S. Perelson, and P.W. Nelson, Mathematical analysis of HIV-1 dynamics in vivo, SIAM Rev., 41 (1999), 3-44. [3] H. Zhu, X. Zou, Impact of delays in cell infection and virus production on HIV-1 dynamics, Math Med Biol, 25 (2008), 99–112. 16

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[4] M.A. Nowak, and R.M. May, “Virus dynamics: Mathematical Principles of Immunology and Virology,” Oxford Uni., Oxford, 2000. [5] D.S. Callaway, and A.S. Perelson, HIV-1 infection and low steady state viral loads, Bull. Math. Biol., 64 (2002), 29-64. [6] P.W. Nelson, J. Murray, and A. Perelson, A model of HIV-1 pathogenesis that includes an intracellular delay, Math. Biosci., 163 (2000), 201-215. [7] P.W. Nelson, and A.S. Perelson, Mathematical analysis of delay di¤ erential equation models of HIV-1 infection, Math. Biosci., 179 (2002), 73-94. [8] R.V. Culshaw, and S. Ruan, A delay-di¤ erential equation model of HIV infection of CD4 + T-cells, Math. Biosci., 165 (2000), 27-39. [9] A.M. Elaiw, I. A. Hassanien, and S. A. Azoz, Global stability of HIV infection models with intracellular delays, J. Korean Math. Soc., 49 (2012), 779-794. [10] A.M. Elaiw, Global dynamics of an HIV infection model with two classes of target cells and distributed delays, Discrete Dyn. Nat. Soc., 2012, Article ID 253703. [11] A.M. Elaiw and A. S. Alsheri, Global Dynamics of HIV Infection of CD4+ T Cells and Macrophages, Discrete Dyn. Nat. Soc., 2013, Article ID 264759. [12] N.M. Dixit, and A.S. Perelson, Complex patterns of viral load decay under antiretroviral therapy: In‡uence of pharmacokinetics and intracellular delay, J. Theoret. Biol., 226 (2004), 95-109. [13] A. M. Elaiw, Global properties of a class of virus infection models with multitarget cells, Nonlinear Dynam., 69 (2012) 423-435. [14] A.M. Elaiw, Global properties of a class of HIV models, Nonlinear Anal. Real World Appl., 11 (2010), 2253–2263.

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[15] M.A. Nowak, C.R.M. Bangham, Population dynamics of immune responses to persistent viruses, Science, 272 (1996), 74-79. [16] S. Eikenberry, S. Hews, J. D. Nagy and Y. Kuang, The dynamics of a delay model of HBV infection with logistic hepatocyte growth, Math. Biosc. Eng., 6, (2009), 283-299. [17] S. A. Gourley,Y. Kuang and J. D. Nagy, Dynamics of a delay di¤ erential equation model of hepatitis B virus infection, J. Biological Dynamics, 2, (2008), 140-153 [18] J. Li, K. Wang, Y. Yang, Dynamical behaviors of an HBV infection model with logistic hepatocyte growth, Mathematical and Computer Modelling, 54 (2011), 704-711. [19] R. Qesmi, J. Wu, J. Wu and J.M. He¤ernan, In‡uence of backward bifurcation in a model of hepatitis B and C viruses, Math. Biosci. 224 (2010) 118–125. [20] R. Qesmi, S. ElSaadany, J.M. He¤ernan and J. Wu, A hepatitis B and C virus model with age since infection that exhibit backward bifurcation, SIAM J. Appl. Math., 71 (4) (2011) 1509–1530. [21] A. U. Neumann, N. P. Lam, H. Dahari, D. R. Gretch, T. E. Wiley, T. J, Layden, A. S. Perelson, Hepatitis C viral dynamics in vivo and the antiviral e¢ cacy of interferon-alpha therapy, Science, 282 (1998), 103-107. [22] M. Y. Li, H. Shu, Global dynamics of a mathematical model for HTLV-I infection of CD4+ T cells with delayed CTL response, Nonlinear Anal. Real World Appl., 13 (2012), 1080-1092. [23] J.A. Deans, S. Cohen, Immunology of malaria, Ann. Rev. Microbiol. 37 (1983), 25-49. [24] A. Murase, T. Sasaki, and T. Kajiwara, Stability analysis of pathogen-immune interaction dynamics, J. Math. Biol., 51 (2005), 247-267. [25] W. Dominik, R. M. May, M. A. Nowak, T he role of antigen-independent persistence of memory cytotoxic T lymphocytes, Int. Immunol. 12 (4) (2000), 467–477.

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[26] A. M. Elaiw, A. Alhejelan, and M. A. Alghamdi, A delayed viral infection model with antibody immune response, Life Science Journal 10(4) (2013) 695-700. [27] A. M. Elaiw, A. Alhejelan, and M. A. Alghamdi, Global dynamics of virus infection model with antibody immune response and distributed delays, Discrete Dynamics in Nature and Society, 2013, Article ID 781407, 2013. [28] T. Wang, Z. Hu, F. Liao, Wanbiao, Global stability analysis for delayed virus infection model with general incidence rate and humoral immunity, Math. Comput. Simulation, 89 (2013), 13-22. [29] S. Wang, D. Zou, Global stability of in host viral models with humoral immunity and intracellular delays, J. Appl. Math. Mod., 36 (2012), 1313-1322. [30] A. Korobeinikov, Global properties of infectious disease models with nonlinear incdence, Bull. Math. Biol., 69 (2007), 1871-1886. [31] G. Huang, Y. Takeuchi, and W. Ma, Lyapunov functionals for delay di¤ erential equations model of viral infection, SIAM J. Appl. Math., 70 (2010), 2693-2708. [32] K. Hattaf, N. Yous…, A. Tridane, Stability analysis of a virus dynamics model with general incidence rate and two delays, Applied Mathematics and Computation, 221 (2013) 514-521. [33] J.K. Hale, and S. Verduyn Lunel, Introduction to functional di¤ erential equations, Springer-Verlag, New York, 1993.

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STABILITY OF GENERALIZED CUBIC SET-VALUED FUNCTIONAL EQUATIONS DONGSEUNG KANG Abstract. We will show the general solution of the functional equation f (ax + by) + f (bx − ay) + (a + b)2 (a − b)f (y) 2

= a bf (x + y) + ab2 f (x − y) + (a + b)(a − b)2 f (x) and investigate the Hyers-Ulam stability of cubic set-valued functional equation when b = 1 .

1. Introduction The theory of set-valued functions in Banach spaces is connected to the control theory and the mathematical economics. Aumann [4] and Debreu [8] wrote papers that were motivated from the topic. We refer the reader to the papers by [1], [18], [10], [3], [17], [7] and [9]. The stability problem of functional equations originated from a question of Ulam [25] concerning the stability of group homomorphisms. Hyers [11] gave a first affirmative partial answer to the question of Ulam. Afterwards, the result of Hyers was generalized by Aoki [2] for additive mapping and by Rassias [23] for linear mappings by considering a unbounded Cauchy difference. Later, the result of Rassias has provided a lot of influence in the development of what we call Hyers-Ulam stability or Hyers-Ulam-Rassias stability of functional equations. For further information about the topic, we also refer the reader to [13], [12], [5] and [6]. Jun and Kim [15] introduced the following cubic functional equation: f (2x + y) + f (2x − y) = 2f (x + y) + 2f (x − y) + 12f (x) and established a general solution. Najati [20] investigated the following generalized cubic functional equation: (1.1)

f (ax + y) + f (ax − y) = af (x + y) + af (x − y) + 2(a3 − a)f (x) .

In this paper, we deal with the following functional equation: (1.2)

f (ax + by) + f (bx − ay) + (a + b)2 (a − b)f (y) = a2 bf (x + y) + ab2 f (x − y) + (a + b)(a − b)2 f (x)

2000 Mathematics Subject Classification. 39B55; 47B47; 39B72. Keywords : Hyers-Ulam-Rassias Stability, Cubic Mapping, Set-Valued Functional Equation, Closed and Convex Subset, Cone, Fixed Point. 1

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for all x , y ∈ X and integers a, b (a > b ≥ 1) . We will show the general solution of the functional equation (1.2) and investigate the Hyers-Ulam stability of cubic set-valued functional equation when b = 1 . 2. A generalized cubic functional equation In this section let X and Y be vector spaces and we investigate the general solution of the functional equation (1.2). Theorem 2.1. A function f : X → Y satisfies the functional equation (1.1) if and only if it satisfies the functional equation (2.1)

f (ax + y) + f (x − ay) − a2 f (x + y) − af (x − y) = (a − 1)(a2 − 1)f (x) − (a + 1)(a2 − 1)f (y)

Proof. See [16, Theorem 2.1].



Theorem 2.2. A function f : X → Y satisfies the functional equation (1.1) if and only if it satisfies the functional equation (1.2). Proof. Suppose that f satisfies the equation (1.1). Since f satisfies the equation (1.1), it is easy to show f (0) = 0 , f (x) = −f (−x) and f (ax) = a3 f (x) for all x ∈ X and integer a (a 6= 0 , ±1). Replacing x and y in the equation (1.1), we obtain (2.2)

f (x + ay) − f (x − ay) = a[f (x + y) − f (x − y)] + 2a(a2 − 1)f (y)

for all x , y ∈ X and an integer a (a 6= 0 , ±1). By letting x = ax in the equation (2.2), we have (2.3)

f (ax + y) − f (ax − y) = a2 [f (x + y) − f (x − y)] + 2(1 − a2 )f (y)

for all x , y ∈ X and an integer a (a 6= 0 , ±1). By replacing x and y in the equation (2.3), we get (2.4)

f (x + ay) + f (x − ay) = a2 [f (x + y) + f (x − y)] + 2(1 − a2 )f (x)

for all x , y ∈ X and an integer a (a 6= 0 , ±1). Replacing a by b in the equation (1.1), we have (2.5)

f (bx + y) + f (bx − y) = bf (x + y) + bf (x − y) + 2(b3 − b)f (x)

Letting y = by in the equation (1.1), (2.6) f (ax + by) + f (ax − by) = af (x + by) + af (x − by) + 2(a3 − a)f (x) Letting y = ay in equation (2.5), (2.7) f (bx + ay) + f (bx − ay) = bf (x + ay) + bf (x − ay) + 2(b3 − b)f (x) Replacing x and y in the equation (2.7), (2.8) f (ax + by) − f (ax − by) = bf (ax + y) − bf (ax − by) + 2(b3 − b)f (y) Replacing x and y in equation (2.6), (2.9) f (bx + ay) − f (bx − ay) = af (bx + y) − af (bx − y) + 2(a3 − a)f (y)

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Adding two equations (2.6) and (2.8), we obtain (2.10)

2f (ax + by) = af (x + by) + af (x − by) + 2(a3 − a)f (x) +bf (ax + y) − bf (ax − y) + 2(b3 − b)f (y)

Subtracting (2.9) from (2.7), we have (2.11)

2f (bx − ay) = bf (x + ay) + bf (x − ay) + 2(b3 − b)f (x) −af (bx + y) + af (bx − y) − 2(a3 − a)f (y)

Now, adding two equations (2.10) and (2.11), we get (2.12)

2[f (ax + by) + f (bx − ay)] = a[f (x + by) + f (x − by)] +b[f (ax + y) − f (ax − y)] + 2(a3 − a)f (x) + 2(b3 − b)f (y) +b[f (x + ay) + f (x − ay)] − a[f (bx + y) − f (bx − y)] +2(b3 − b)f (x) − 2(a3 − a)f (y)

The desired result is obtained from the equation (2.12) by using the equations (2.3) and (2.4). Conversely, suppose that f satisfies the equation (1.2). Letting b = 1 in the equation (1.2), we have the equation (2.1). The remains follow from Theorem 2.1.  If f satisfies the equation (1.2), we call f a generalized cubic mapping. 3. Stability of the generalized cubic set-valued functional equation In this section, we first introduce some definitions and notations which are needed to prove the main theorems. Let Y be a Banach space. The family of all closed subsets, containing 0 , of Y will be denoted by Cz (Y ) . Let A , B be nonempty subsets of a real vector space X and λ a real number. We define A + B = {a + b ∈ X | a ∈ A , b ∈ B} λA = {λa ∈ X | a ∈ A} . Lemma 3.1 ( [21]). Let λ and µ be real numbers. If A and B are nonempty subset of a real vector space, then λ(A + B) = λA + λB (λ + µ)A ⊆ λA + µA . Moreover, if A is a convex set and λµ ≥ 0 , then we have (λ + µ)A = λA + µA .

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A subset A ⊆ X is said to be a cone if A + A ⊆ A and λA ⊆ A for all λ > 0 . If the zero vector in X belongs to A , then we say that A is a cone with zero. Let Cb (Y ) be the set of all closed bounded subsets of Y , Cc (Y ) the set of all closed convex subsets of Y and Ccb (Y ) the set of all closed bounded convex subsets of Y . For elements A , B of Cc (Y ) and positive real values λ , µ , we denote A⊕B =A+B. For a subset A of Y , the distance function d(· , A) and the support function s(· , A) are defined by d(x , A) := inf {||x − y|| | y ∈ A} for all x ∈ Y s(x∗ , A) := sup {hx∗ , xi|| | x ∈ A} for all x∗ ∈ Y ∗ . For A , A0 ∈ Cb (Y ) , the Hausdorff distance h(A , A0 ) between A and A0 is defined by h(A , A0 ) := inf {α ≥ 0 | A ⊆ A0 + αBY , A0 ⊆ A + αBY } , where BY is the closed unit ball in Y . Castaing and Valadier [7] proved that (Ccb (Y ) , ⊕ , h) is a complete metric semigroup. Radstr¨ ˙ om [22] showed that (Ccb (Y ) , ⊕ , h) is isometrically embedded in a Banach space. The following remark is directly obtained from the notion of the Hausdorff distance. Remark 3.2. Let A , A0 , B , B 0 , C ∈ Ccb (Y ) and α > 0 . Then the following properties hold: (1) h(A ⊕ A0 , B ⊕ B 0 ) ≤ h(A , B) + h(A0 , B 0 ) (2) h(αA , αB) = αh(A , B) (3) h(A , B) = h(A ⊕ C , B ⊕ C) . First, let X be a real vector space , A ⊂ X a cone with zero and Y a Banach space. Theorem 3.3. If f : A + (−1)A → Cz (Y ) is a set-valued mapping with f (0) = {0} satisfying (3.1)

f (ax + y) + f (x − ay) + (a2 − 1)(a + 1)f (y) ⊆ a2 f (x + y) + af (x − y) + (a2 − 1)(a − 1)f (x)

and sup{diam(f (x)) | x ∈ A} < ∞ for all x , y ∈ A and an integer a (a ≥ 2) , then there exists a unique generalized cubic mapping C : A + (−1)A → Y such that C(x) ∈ f (x) for all x ∈ A. Proof. Letting y = 0 in (3.1), we have (3.2)

f (ax) ⊆ a3 f (x)

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for all x ∈ A and an integer a (a ≥ 2) . Replacing x by an x , n ∈ N in (3.2) , we get f (an+1 x) ⊆ a3 f (an x) and 1 1 f (an+1 x) ⊆ 3n f (an x) 3(n+1) a a 1 for all x ∈ A and an integer a (a ≥ 2) . Let fn (x) = a3n f (an x) for each x ∈ A , n ∈ N . Then {fn (x)}n≥0 is a decreasing sequence of closed subsets of the Banach space Y . Also, we obtain 1 diam(fn (x)) = 3n diam(f (an x)) . a Since sup{diam(f (x)) | x ∈ A} < ∞ , we have lim diam(fn (x)) = 0 .

n→∞

Using the Cantor theorem for the sequence {fn (x)}n≥0 , we get that ∩n≥0 fn (x) is a singleton set and we denote this intersection by C(x) for all x ∈ A . Hence we obtain a map C : A + (−1)A → Y and C(x) ∈ f0 (x) = f (x) for all x ∈ A . We claim that C is generalized cubic. We note that fn (ax + y) + fn (x − ay) + (a2 − 1)(a + 1)fn (y) f (an (ax + y)) f (an (x − ay)) (a2 − 1)(a + 1)f (an y) = + + a3n a3n a3n n 2 2 n a f (a (x + y)) af (a (x − y)) (a − 1)(a − 1)f (an x) + + ⊆ a3n a3n a3n 2 2 = a fn (x + y) + afn (x − y) + (a − 1)(a − 1)fn (x) for all x ∈ A and an integer a (a ≥ 2) . By the definition of C , we obtain C(ax + y) + C(x − ay) + (a2 − 1)(a + 1)C(y)   2 = ∩∞ f (ax + y) + f (x − ay) + (a − 1)(a + 1)f (y) n n n n=0   2 2 ⊆ ∩∞ a f (x + y) + af (x − y) + (a − 1)(a − 1)f (x) n n n n=0 for all x ∈ A and an integer a (a ≥ 2) . Hence we have ||C(ax + y) + C(x − ay) + (a2 − 1)(a + 1)C(y) −a2 C(x + y) − aC(x − y) − (a2 − 1)(a − 1)C(x)|| ≤ a2 diam(fn (x + y)) + a diam(fn (x − y)) + (a2 − 1)(a − 1)diam(fn (x)) , which tends to zero as n → ∞ . Thus C satisfies the equality (1.2). Hence C is a generalized cubic, as claimed. Next, let us prove the uniqueness of C . Assume f has two generalized cubic functional equations C1 and C2 from A + (−1)A into Y . Then we have (an)3 Ci (x) = Ci (anx) ∈ f (anx)

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for all x ∈ X , n ∈ N and i ∈ {1 , 2} . Then we have (an)3 ||C1 (x) − C2 (x)|| = ||(an)3 C1 (x) − (an)3 C2 (x)|| = ||(C1 (anx) − C2 (anx)|| ≤ diam(f (anx)) for all x ∈ X , n ∈ N . Since sup{diam(f (x)) | x ∈ A} < ∞ , C1 (x) = C2 (x) , for all x ∈ X .  Definition 3.4. Let f : X → Ccb (Y ) . The generalized cubic set-valued functional equation is defined by (3.3)

f (ax + y) ⊕ f (x − ay) ⊕ (a2 − 1)(a + 1)f (y)

= a2 f (x + y) ⊕ af (x − y) ⊕ (a2 − 1)(a − 1)f (x) for all x ∈ A and an integer a (a ≥ 2) . Every solution of the generalized cubic set-valued functional equation is called a generalized cubic set-valued mapping. Theorem 3.5. Let φ : X × X → [0, ∞) be a function such that ∞ X 1 e y) := (3.4) φ(x, φ(aj x, aj y) < ∞ a3j j=0

for all x, y ∈ X and an integer a (a ≥ 2) . Suppose that f : X → (Ccb (Y ), h) is a mapping with f (0) = {0} satisfying  (3.5) h f (ax + y) ⊕ f (x − ay) ⊕ (a2 − 1)(a + 1)f (y),  a2 f (x + y) ⊕ af (x − y) ⊕ (a2 − 1)(a − 1)f (x) ≤ φ(x, y) for all x, y ∈ X and an integer a (a ≥ 2) . Then there exists a unique generalized cubic set-valued mapping C : X → (Ccb (Y ), h) such that 1 e 0) (3.6) h(f (x), C(x)) ≤ 3 φ(x, a for all x, y ∈ X and an integer a (a ≥ 2) . Proof. Let y = 0 in the inequality (3.5). Since f (x) is convex, we have   h f (ax) ⊕ f (x), a2 f (x) ⊕ af (x) ⊕ (a2 − 1)(a − 1)f (x) ≤ φ(x, 0) , that is,   1 1 h f (x), 3 f (ax) ≤ 3 φ(x, 0) a a for all x ∈ X . Replacing x by ak x , k ∈ N , we get   1 1 h f (ak x), 3 f (ak+1 x) ≤ 3 φ(ak x, 0) a a and  1  1 1 h 3k f (ak x), 3(k+1) f (ak+1 x) ≤ 3(k+1) φ(ak x, 0) a a a

(3.7)

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for all x ∈ X . Using the induction on k , we obtain (3.8)

n−1  1 1 X 1 n h f (x), 3n f (a x) ≤ 3 φ(ak x, 0) a a a3k



k=0

for all x ∈ X and n ∈ N . Dividing the inequality (3.8) by a3m and replacing x by am x , we have (3.9)

n−1  1  1 1 X 1 1 h 3m f (am x), 3(n+m) f (an+m x) ≤ 3 m φ(am+k x, 0) a a a a3k a k=0

for all x ∈ X and n , m ∈ N . Since the right-hand side of the inequality (3.9) 1 tends to zero as m → ∞ , the sequence { a3n f (an x)} is a Cauchy sequence in (Ccb (Y ), h) . By the completeness of Ccb (Y ) , we can define C(x) := lim

n→∞

1 f (an x) a3n

for all x ∈ X and an integer a (a ≥ 2) . We note that  f (an (ax + y)) f (an (x − ay)) (a2 − 1)(a + 1)f (an y) h ⊕ ⊕ , a3n a3n a3n a2 f (an (x + y)) af (an (x − y)) (a2 − 1)(a − 1)f (an x)  ⊕ ⊕ a3n a3n a3n 1 ≤ 3n φ(an x, an y) a for all x, y ∈ X and an integer a (a ≥ 2) . By the definition of C , we have  h C(ax + y) ⊕ C(x − ay) ⊕ (a2 − 1)(a + 1)C(y),  a2 C(x + y) ⊕ aC(x − y) ⊕ (a2 − 1)(a − 1)C(x) 1 φ(an x, an y) = 0 . a3n Hence C is a generalized cubic set-valued mapping. Now, by taking n → ∞ in the inequality (3.8), we have the inequality (3.6). It remains to show the uniqueness of C . Assume C 0 : X → (Ccb (Y ), h) is another generalized cubic set-valued mapping satisfying the inequality (3.6). Then    1  n 0 n h C(a x), C (a x) h C(x), C 0 (x) = a3n   1  1  n n n 0 n ≤ h C(a x), f (a x) + h f (a x), C (a x) a3n a3n 2 e n ≤ φ(a x, 0) a3(n+1) 2 e n x, 0) → 0 as n → ∞ , we may conclude that for all x ∈ X . Since a3(n+1) φ(a the generalized cubic set-valued mapping C is unique.  ≤ lim

n→∞

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Corollary 3.6. Let 0 < p < 3 , θ ≥ 0 be real numbers and let X be a real normed space. Suppose that f : X → (Ccb (Y ), h) is a mapping with f (0) = {0} satisfying  h f (ax + y) ⊕ f (x − ay) ⊕ (a2 − 1)(a + 1)f (y),  a2 f (x + y) ⊕ af (x − y) ⊕ (a2 − 1)(a − 1)f (x) ≤ θ(||x||p + ||y||p ) for all x, y ∈ X and an integer a (a ≥ 2) . Then there exists a unique generalized cubic set-valued mapping C : X → (Ccb (Y ), h) satisfying h(f (x), C(c)) ≤

a3

θ ||x||p − ap

for all x, y ∈ X and an integer a (a ≥ 2) . Proof. It follows from Theorem 3.5 by letting φ(x, y) = θ(||x||p + ||y||p ) for all x, y ∈ X .  4. Stability of set-valued functional equation by the fixed point method Now, we will investigate the stability of the given functional equation (3.3) using the alternative fixed point method. Before proceeding the proof, we will state the theorem, the alternative of fixed point; see [19] and [24]. Definition 4.1. 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 4.2. [ The alternative of fixed point [19], [24] ] Suppose that we are given a complete generalized metric space (Ω, d) and a strictly contractive mapping T : Ω → Ω with Lipschitz constant L . Then for each given x ∈ Ω , either d(T n x, T n+1 x) = ∞ for all n ≥ 0 , or there exists a natural number 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 the set 4 = {y ∈ Ω|d(T n0 x, y) < ∞} ; (4) d(y, y ∗ ) ≤

1 1−L

d(y, T y) for all y ∈ 4 .

Theorem 4.3. Suppose that f : X → (Ccb (Y ), h) is a mapping with f (0) = {0} satisfying  (4.1) h f (ax + y) ⊕ f (x − ay) ⊕ (a2 − 1)(a + 1)f (y),

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9

 a2 f (x + y) ⊕ af (x − y) ⊕ (a2 − 1)(a − 1)f (x) ≤ φ(x, y) for all x, y ∈ X and an integer a (a ≥ 2) and there exists a constant L with 0 < L < 1 for which the function φ : X 2 → R+ satisfies φ(ax, 0) ≤ a3 Lφ(x, 0)

(4.2)

for all x ∈ X . Then there exists a unique generalized cubic set-valued mapn x) ping C : X → (Ccb (Y ), h) given by C(x) = limn→∞ f (a such that a3n (4.3)

h(f (x), C(x)) ≤

1 e 0) φ(x, − L)

a3 (1

for all x, y ∈ X and an integer a (a ≥ 2) . Proof. Consider the set Ω = {g | g : X → Ccb (Y ) , g(0) = {0}} and introduce the generalized metric on Ω defined by d(g1 , g2 ) = inf {µ ∈ (0, ∞) | h(g1 (x) , g2 (x)) ≤ µφ(x, 0) , for all x ∈ X} . We note that inf ∅ := ∞ . It is easy to show that (Ω, d) is complete; see [14]. Now we define a function T : Ω → Ω by 1 (4.4) T (g)(x) = 3 g(ax) a for all x ∈ X . Note that for all g1 , g2 ∈ Ω , let µ ∈ (0, ∞) be an arbitrary constant with d(g1 , g2 ) = µ . Then 1 µ 1 (4.5) h( 3 g1 (ax) , 3 g2 (ax)) ≤ 3 φ(ax, 0) a a a for all x ∈ X . By using (4.2), we have 1 1 (4.6) h( 3 g1 (ax) , 3 g2 (ax)) ≤ µ Lφ(x, 0) a a for all x ∈ X . Hence we obtain d(T g1 , T g2 ) ≤ Ld(g1 , g2 ) for all g1 , g2 ∈ Ω , that is, T is a strictly self-mapping of Ω with the Lipschitz constant L . Letting y = 0 in the inequality (4.1), we get 1 1 h( 3 f (ax) , f (x)) ≤ 3 φ(x, 0) a a for all x ∈ X . This means that 1 d(T f, f ) ≤ 3 . a By Theroem 4.2, there exists a fixed point C : X → (Ccb (Y ), h) of T in {g ∈ Ω | d(g1 , g2 ) < ∞} such that {T k f } → 0 ad k → ∞ . Hence we have (4.7)

f (an x) , n→∞ a3n

C(x) = lim

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DONGSEUNG KANG

for all x ∈ X . Also, we have 1 1 1 d(T f, f ) ≤ 3 . 1−L a 1−L This implies that the inequality (4.3) holds for all x ∈ X . By the inequalities (4.1) and (4.2), we have  h C(ax + y) ⊕ C(x − ay) ⊕ (a2 − 1)(a + 1)C(y),  a2 C(x + y) ⊕ aC(x − y) ⊕ (a2 − 1)(a − 1)C(x) d(f, C) ≤

≤ lim Ln φ(an x, an y) = 0 n→∞

for all x, y ∈ X and an integer a (a ≥ 2) . Thus C is a unique generalized cubic set-valued mapping.  Corollary 4.4. Let 0 < p < 3 and θ ≥ 0 be real numbers and let X be a real normed space. Suppose that f : X → (Ccb (Y ), h) is a mapping with f (0) = {0} satisfying  (4.8) h f (ax + y) ⊕ f (x − ay) ⊕ (a2 − 1)(a + 1)f (y),  a2 f (x + y) ⊕ af (x − y) ⊕ (a2 − 1)(a − 1)f (x) ≤ θ(||x||p + ||y||p ) for all x, y ∈ X and an integer a (a ≥ 2) . Then there exists a unique generalized cubic set-valued mapping C : X → (Ccb (Y ), h) such that (4.9)

h(f (x), C(x)) ≤

a3

θ ||x||p − ap

for all x ∈ X and an integer a (a ≥ 2) . Proof. It follows from Theorem 4.3 by letting φ(x, y) = θ(||x||p + ||y||p ) for all x, y ∈ X . Then we can choose L = ap−3 and hence we have the desired result.  References [1] K.J. Arrow and G. Debreu, Existence of anequilibrium for a competitive economy, Econometrica 22 (1954), 265–290. [2] T. Aoki, On the stability of the linear transformation in Banach spaces, J. Math. Soc. Japan 2 (1950), 64–66. [3] J.P. Aubin and H. Frankowska, Set-Valued Analysis, Birkhauser, Boston (1990). [4] R.J. Aumann, Integrals of set-valued functions, J. Math. Anal.Appl. 12 (1965) 1–12. [5] N. Brillou¨et-Belluot, J. Brzd¸ek and K. Ciepli´ nski, Fixed Point Theory and the Ulam Stability, Abstract and Applied Analysis 2014, Article ID 829419, 16 pages (2014). [6] J. Brzd¸ek, L. Cˇ adariu and K. Ciepli´ nski, On Some Recent Developments in Ulam’s Type Stability, Abstract and Applied Analysis 2012, Article ID 716936, 41 pages (2012). [7] C. Castaing, M. Valadier, Convex Analysis and Measurable Multifunctions, in: Lect. Notes in Math., 580, Springer, Berlin (1977).

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[8] G. Debreu, Integration of correspondences, in: Proceedings of Fifth Berkeley Symposium on Mathematical Statistics and Probability, vol. II, 1966, pp. 351–372. Part I. [9] C. Hess, Set-valued Integration and Set-valued Probability Theory: an Overview, in: Handbook of Measure Theory, vols. I, II, North-Holland, Amsterdam (2002). [10] W. Hindenbrand, Core and Equilibria of a Large Economy, Princeton Univ. Press, Princeton (1974). [11] D. H. Hyers, On the stability of the linear equation, Proc. Nat. Acad. Sci. U.S.A. 27 (1941), 222–224. [12] D. H. Hyers, G. Isac, and T. M. Rassias, Stability of Functional Equations in Several Variables, Birkhauser, Boston, Mass, USA (1998). [13] S.-M. Jung, Hyers-Ulam-Rassias Stability of Functional Equations in Nonlinear Analysis, vol. 48 of Springer Optimization and Its Applications, Springer, New York, NY, USA, (2011). [14] S.-M. Jung and Z.-H. Lee, A fixed point approach to the stability of quadratic functional equation with involution, Fixed Point Theory Appl. 2008 Article ID 732086, 11 pages (2008). [15] 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. [16] D. Kang, On the Stability of Lie ∗-Derivations of Cubic Functional Equations, Abstract and Applied Analysis 2014, Article ID 808042, 6 pages (2014). [17] E. Klein and A. Thompson, Theory of Correspondence, Wiley, New York (1984). [18] L.W. McKenzie, On the existence of general equilibrium for a competitive market, Econometrica 27 (1959) 54–71. [19] B. Margolis and J.B. Diaz, A fixed point theorem of the alternative for contractions on a generalized complete metric space, Bull. Amer. Math. Soc. 126, 74(1968), 305–309. [20] A. Najati, The Generalized Hyers-Ulam-Rassias Stability of a Cubic Functional Equation, Turk. J. Math. 31 (2007) 395–408. [21] K. Nikodem, K-Convex and K-Concave Set-Valued Functions, Zeszyty Naukowe Nr., 559, Lodz (1989). [22] H. Radstr¨ ˙ om, An embedding theorem for spaces of convex sets, Proc. Amer. Math. Soc. 3 (1952) 165–169. [23] Th. M. Rassias, On the stability of the linear mapping in Banach spaces, Proc. Amer. Math. Soc. 72 (1978) 297–300. [24] I.A. Rus, Principles and Appications of Fixed Point Theory, Ed. Dacia, ClujNapoca, 1979 (in Romanian). [25] S. M. Ulam, Problems in Morden Mathematics, Wiley, New York (1960). Department of Mathematical Education, Dankook University, 152, Jukjeon, Suji, Yongin, Gyeonggi, South Korea 448-701 E-mail address: [email protected] (D. Kang)

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A new regularity (p-regularity) of stratified L-generalized convergence spaces Lingqiang Lia,b , Qingguo Lia∗ a College

of Mathematics and Econometrics, Hunan University Changsha, 410082, P.R.China

b Department

of Mathematics, Liaocheng University,

Liaocheng, 252059, P.R.China

Abstract: In the classical theory of convergence spaces, both regularity (p-regularity) and topologicallness (p-topologicallness) are important notions. It is well known that topologicallness (p-topologicallness) can be described by a sophisticated Fischer-type diagonal condition, and regularity (p-regularity) can be described by dualizing that diagonal condition. Additionally, regularity (p-regularity) can also be characterized by the notion of closures of filters. In this paper, for stratified L-generalized convergence spaces, a new regularity (p-regularity) is defined by duzlizing a Fischer-type diagonal condition, which is used to describe the L-topologicallness of stratified L-convergence spaces (a subcategory of stratified L-generalized convergence spaces). Additionally, a characterization on this new regularity (p-regularity) by a notion of closures of stratified L-filters, is also presented. Keywords: Topology; Lattice-valued topology; Lattice-valued convergence space; regularity; Diagonal condition

1

Introduction

p-topologicallness [17] and p-regularity [11] are dual notions in the classical theory of convergence spaces [16]. For a set X, let F(X) denote the set of all filters on X. Let q and p be convergence structures on a set X. Then the space (X, q) is called p-topological if it satisfies either of the two conditions below. q q (1) Up (F) → x whenever F → x, where Up (F) is the neighborhood of F w.r.t p. ∗

Corresponding author. Tel./fax: +86 15206506635/+86 635 8258028. E-mail address: [email protected], [email protected]. Mailing address: Department of Mathematics, Liaocheng University, Liaocheng, 252059, P.R.China

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(2) (Fischer-type diagonal condition) Let J be any set, ψ : J −→ X, and let p σ : J −→ F(X) have the condition that σ(j) → ψ(j), for all j ∈ J. If F ∈ F(J) is S T q q such that ψ(F) → x, then kσF → x. Here, kσF = σ(j) ∈ F(X) is called the F ∈F j∈F

compression of F relative to σ. The space (X, q) is called p-regular if it satisfies either of the two conditions below. q q (1) Fp → x whenever F → x, where Fp is the closure of F w.r.t p. (2) (Dual Fischer-type diagonal condition) Let J be any set, ψ : J −→ X, and let p σ : J −→ F(X) have the condition that σ(j) → ψ(j), for all j ∈ J. If F ∈ F(J) is such q q that kσF → x, then ψ(F) → x. When p = q, p-topologicallness and p-regularity are refereed to topologicallness and regularity [1, 3, 12], respectively. Stratified L-generalized convergence spaces defined by J¨ager [7] are lattice- valued extensions of convergence spaces. In [9], J¨ager studied a regularity of stratified Lgeneralized convergence spaces both by a dual Fischer-type diagonal condition and a notion of α-lever closures of stratified L-filters. Later, Li and Jin [14] generalized J¨ager’s regularity to p-regularity. Quite recently, by modifying J¨ager’s Fischer-type diagonal condition, the first author and his co-author [15] introduced a new Fischertype diagonal condition, and proved that this condition happens to characterize the topologicallness of stratified L-convergence spaces [4, 13] (a subcategory of stratified L-generalized convergence spaces). In this paper, by dualizing that diagonal condition, a new regularity (p-regularity) of stratified L-generalized convergence spaces is defined, and a characterization on this new regularity (p-regularity) by the notion of closures of stratified L-filters, is also presented. The contents are arranged as follows. Section 2 fixes some notions and notations used in this note. Section 3 recalls the Fischer-type diagonal notion such that stratified L-convergence spaces are L-topological. Section 4 presents the main results. That is, by dualizing a Fischer-type diagonal condition in Section 3, we define a new regularity (p-regularity) of stratified L-generalized convergence spaces and then present a characterization on that regularity (p-regularity) by a notion of closures of stratified L-filters. In this paper, if not otherwise specified, L = (L, ≤) is always a complete lattice with a top element 1 and a bottom element 0, which satisfies the distributive law W W α ∧ ( i∈I βi ) = i∈I (α ∧ βi ). A lattice with these conditions is called a complete W Heyting algebra or a frame. The operation →: L × L −→ L given by α → β = {γ ∈ L : α ∧ γ ≤ β}, is called the residuation with respect to ∧. For the properties of ∧ and →, please refer to the literatures [6, 7, 14]. For a set X, the set LX of functions from X to L with the pointwise order becomes a complete lattice. Each element of LX is called an L-set (or a fuzzy subset) of X. And

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we make no difference between a constant function and its value since no confusion will arise. Let f : X −→ Y be a function. We definef ← : LY −→ LX [6] by f ← (µ) = µ ◦ f for µ ∈ LY . Let X be a set. A fuzzy partial order (or, an L-partial order) on X [2] is a reflexive, transitive and antisymmetric fuzzy relation on X. The pair (X, R) is called an L-partially ordered set. Let [LX ] : LX × LX −→ L be a function defined by V [LX ](λ, µ) = x∈X (λ(x) → µ(x)). Then [LX ] is an L-partial order on LX [2, 19]. The value [LX ](λ, µ) ∈ L is interpreted as the degree that λ is contained in µ. In the sequel, we use the symbol [λ, µ] to denote [LX ](λ, µ) for simplicity. The following lemma is useful to the subsequent section. Lemma 1.1. [14] Let f : X −→ Y be an function. For any λ, µ, ν ∈ LX and any {λi }i∈I , {µi }i∈I ⊆ LX , we have (1) λ ≤ µ implies [λ, ν] ≥ [µ, ν]; (2) [λ, ∧i∈I µi ] = ∧i∈I [λ, µi ]; (3) λ ∧ [λ, µ] ≤ µ; (4) [∨i∈I λi , µ] = ∧i∈I [λi , µ]; (5) [λ, µ] ≤ [f → (λ), f → (µ)]. A stratified L-filter [6] on a set X is a function F : LX −→ L such that: (F0) F(0) = 0, (F1) F(1) = 1, (F2) ∀λ, µ ∈ LX , F(λ) ∧ F(µ) = F(λ ∧ µ), (Fs) ∀α ∈ L, F(α) ≥ α. The set FLs (X) of all stratified L-filters on X is ordered by F ≤ G ⇔ ∀λ ∈ LX , F(λ) ≤ G(λ). There is a natural fuzzy partial order on FLs (X) inherited X X from L(L ) . Precisely, for all F, G ∈ FLs (X), if we let [FLs (X)](F, G) = [LL ](F, G) = V s λ∈LX (F(λ) → G(λ)), then [FL (X)] is an L-partially order. Example 1.2. (1) For each point x in a set X, the function [x] : LX −→ L, [x](λ) = V λ(x) is a stratified L-filter on X. (2) If {Fj |j ∈ J} ⊆ FLs (X), then j∈J Fj ∈ FLs (X). (3) Let f : X −→ Y be a function. If F ∈ FLs (X), then the function f ⇒ (F) ∈ FLs (Y ), where f ⇒ (F) : LY −→ L is defined by λ 7→ F(λ ◦ f ) = F(f ← (λ)).

2

Fischer-type diagonal condition of stratified Lconvergence spaces

In this section, we shall recall the Fischer-type diagonal condition such that a stratified L-convergence space is L-topological. Definition 2.1. A stratified L-generalized convergence structure [7, 18] on a set X is a function limq : FLs (X) −→ LX satisfying (LC1) ∀x ∈ X, limq [x](x) = 1; and (LC2) ∀F, G ∈ FLs (X), F ≤ G =⇒ limq F ≤ limq G. The pair (X, limq ) is called a stratified Lgeneralized convergence space. The pair (X, limq ) is called a stratified L-convergence space [13] (or, a stratified L-ordered convergence space in [4]) if lim : FLs (X) −→ LX is a function satisfying (LC1) and (LC20 ) ∀F, G ∈ FLs (X), [FLs (X)](F, G) ≤ [LX ](limq F, limq G). Because (LC2)0 ⇒(LC2), a stratified L-convergence space is a

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stratified L-generalized convergence space. A function f : X −→ X 0 between two 0 stratified L-generalized convergence spaces (X, limq ), (X 0 , limq ) is called continuous if 0 for all F ∈ FLs (X) and all x ∈ X we have limq F(x) ≤ limq f ⇒ (F)(f (x)). fi

For a given source (X −→ (Xi , limqi ))i∈I , the initial structure, limq on X is defined V by ∀F ∈ FLs (X), ∀x ∈ X, limq F(x) = limqi fi⇒ (F)(fi (x)). i∈I

qi

fi

For a given sink ((Xi , lim ) −→ X)i∈I , the final structure, limq on X is defined by ( 1, F ≥ [x]; W limq F(x) = qi i∈I,xi ∈Xi ,Gi ∈F s (Xi ),fi (xi )=x,f ⇒ (Gi )≤F lim Gi (xi ), F 6≥ [x]. i

L

When X = ∪i∈I fi (Xi ), the final structure limq can be simplified as [14] _ limq F(x) = limqi Gi (xi ). s (X ),f (x )=x,f ⇒ (G )≤F i∈I,xi ∈Xi ,Gi ∈FL i i i i i

In the theory of convergence spaces, Fischer-type diagonal condition is formulated by the aid of the notion of compression. The situation with lattice-valued convergence is similar. In [8], J¨ager introduced an lattice-valued version of compression, which first appeared in [5] with a slightly different formalization. Let σ : J −→ FLs (X) be a function and F ∈ FLs (X). Then the function kL σF : LX −→ L defined by ∀λ ∈ LX , kL σF(λ) := F(b σ (λ)), where σ b(λ) = σ(−)(λ) ∈ LJ forms a stratified L-filter on X; and it is called the compression of F w.r.t σ. In [15], the first author and his co-author modified J¨ager’s compression and introduced a Fischer-type diagonal condition. It was proved that a stratified L-convergence space with this diagonal condition is L-topological. Note that when a function σ : J −→ FLs (X) being given, that means an L-filter σ(j) is selected for each j ∈ J. In this sense, we call σ : J −→ FLs (X) an L-filter select function. The definition below generalizes that notion. Definition 2.2. [15] A function σ = (σ1 , σ2 ) : J −→ FLs (X)×L0 , where L0 = L−{0}, is said to be an L-filter select degree function. For any j ∈ J, the value σ2 (j) ∈ L is interpreted as the degree to which the stratified L-filter σ1 (j) is selected. Obviously, an L-filter select function can be regarded as an L-filter select degree function with σ2 ≡ 1. Definition 2.3. [15] Let σ : J −→ FLs (X) × L0 be an L-filter select degree function and F ∈ FLs (J). If the function kL σF : LX −→ L defined by ∀λ ∈ LX , kL σF(λ) := F(b σ (λ)), where σ b(λ) = σ2 (−) → σ1 (−)(λ) ∈ LJ

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forms a stratified L-filter on X, then we call such F compressible w.r.t σ and call kL σF as the compression of F w.r.t σ. It is easily seen that kL σF satisfies (F1), (F2) and (Fs) for any F ∈ FLs (J). If σ : J −→ FLs (X) × L0 is an L-filter select function, then kL σF ∈ FLs (X) for any F ∈ FLs (J). In this case, kL σF coincides with J¨ager’s compression. Thus, Definition 2.3 generalizes J¨ager’s compression. Theorem 2.4. [15] Let (X, limq ) be a stratified L-convergence spaces. Then (X, limq ) is L-topological if and only if it satisfies the following condition (Lf ). (Lf ) Let J be any set, ψ : J −→ X, and let σ : J −→ FLs (X) × L0 . If F ∈ FLs (J) is compressible w.r.t σ, then for each x ∈ X, limq ψ ⇒ (F)(x) ∗

^

limq σ(j)(ψ(j)) ≤ limq kL σF(x),

j∈J

where limq σ(j)(ψ(j)) := σ2 (j) → limq σ1 (j)(ψ(j)). Obviously, the condition (Lf ) implies the following condition (Lfw). (Lfw): Let J be any set, ψ : J −→ X, and let σ : J −→ FLs (X) × L0 have the condition ∀j ∈ J, σ2 (j) = limq σ1 (j)(ψ(j)) (which means that limq σ(j)(ψ(j)) ≡ 1). If F ∈ FLs (J) is compressible w.r.t σ, then limq ψ ⇒ (F)(x) ≤ limq kL σF(x) for each x ∈ X. Note that in the proof of the sufficiency of Theorem 2.4, the selected σ, ψ satisfies the condition σ2 (j) = limq σ1 (j)(ψ(j)) (see Theorem 4.9 in [15]). It follows immediately that (Lf )⇔(Lfw). In addition, the characterization on L-topologicallness of stratified L-convergence spaces by the notion of neighborhoods of stratified L-filters, was presented in [10].

3

regularity

and

p-regularity

of

stratified

L-

generalized convergence spaces In this section, by dualizing the condition (Lfw) we define a new regularity (pregularity) of stratified L-generalized convergence spaces. Then we also present a characterization on that regularity (p-regularity) by a notion of closures of stratified Lfilters. Let (X, limp , limq ) be a pair of stratified L-generalized convergence spaces. p-(DLfw): Let J be any set, ψ : J −→ X, and let σ : J −→ FLs (X) × L0 have the condition ∀j ∈ J, σ2 (j) = limp σ1 (j)(ψ(j)). If F ∈ FLs (J) is compressible w.r.t σ, then limq kL σF(x) ≤ limq ψ ⇒ (F)(x) for each x ∈ X.

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When limp = limq , the condition p-(DLfw) is denoted as (DLfw). Obviously, the condition (DLfw) is obtained by dualizing the condition (Lfw). It is easily seen that when L = {0, 1}, the condition p-(DLfw) is equivalent to the crisp dual Fischer-type diagonal condition. Definition 3.1. Let (X, limp , limq ) be a pair of stratified L-generalized convergence spaces. Then (X, limq ) is called p-regular if it satisfies the dual Fischer-type diagonal condition p-(DLfw). When limp = limq , then (X, limq ) is called regular if it is pregular. In the following, we shall give a characterization on regularity (p-regularity) by the notion of closures of stratified L-filters. Definition 3.2. Let (X, limp ) be a stratified L-generalized convergence space, and let λ ∈ LX . Then the L-set λp ∈ LX defined by _ ∀x ∈ X, λp (x) = (limp F(x) → F(λ)) s (X): limp F (x)6=0 F∈FL

is called the closure of λ w.r.t (X, limp ). Remark 3.3. When L = {0, 1}, a stratified L-generalized convergence space reduces p to a convergence space. Then it is easily seen that x ∈ λp ⇔∃F → x s.t. λ ∈ F. This shows that closure is precisely the crisp closure in [16] when L = {0, 1}. Lemma 3.4. Let (X, limp ) be a stratified L-generalized convergence space. Then for all λ, µ ∈ LX and all α ∈ L we get (1) λ ≤ λp ; (2) λ ≤ µ implies λp ≤ µp ; (3) αp ≥ α. Proof. (1) For each x ∈ X, by limp [x](x) = 1 we get λp (x) ≥ [x](λ) = λ(x). So, λ ≤ λp . Take λ = 1 in (1), we obtain 1p = 1. (2) It follows from the property (F2) of stratified L-filters. (3) For each x ∈ X we have αp (x) =

_

_

(Fs)

(limp F(x) → F(α)) ≥

lim F (x)6=0

(limp F(x) → α) ≥ α.

lim F (x)6=0

Theorem 3.5. Let (X, limp ) be a stratified L-generalized convergence space. For each W F ∈ FLs (X), the function F p : LX −→ L, defined by ∀λ ∈ LX , F p (λ) = µ∈LX (F(µ) ∧ [µp , λ]), is a stratified L-filter, called the closure of F w.r.t (X, limp ). Proof. (F1) That F p (1) = 1 is obvious. That F p (0) = 0 follows by F p (λ) =

_ µ∈LX

_

(F(µ) ∧ [µp , λ]) ≤

(F(µ) ∧ [µ, λ]) ≤ F(λ).

µ∈LX

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(F2) Obviously, F p (λ ∧ µ) ≤ F p (λ) ∧ F p (µ). Conversely, _ _ F p (λ) ∧ F p (µ) = (F(a) ∧ [ap , λ]) ∧ (F(b) ∧ [bp , µ]) a∈LX

b∈LX

_

=

(F(a) ∧ F(b) ∧ [ap , λ] ∧ [bp , µ])

a,b∈LX

_

≤

(F(a ∧ b) ∧ [(a ∧ b)p , λ ∧ µ])

a,b∈LX

≤

_

(F(c) ∧ [cp , λ ∧ µ]) = F p (λ ∧ µ).

c∈LX

(Fs) By 1p = 1, it follows that F p (α) =

W

µ∈LX (F(µ)

∧ [µp , α]) ≥ F(1) ∧ α = α.

Remark 3.6. When L = {0, 1}, a stratified L-generalized convergence space reduces to a convergence space. It is easily seen that F p is precisely the filter generated by {A : A ∈ F} as a filterbasis [16]. Lemma 3.7. Let J, X, σ, ψ satisfy the condition in p-(DLfw). Then for any λ, µ ∈ LX ˆ we have [µp , λ] ≤ [φ(µ), ψ ← (λ)]. Proof. Note that ∀j ∈ J, σ2 (j) = limp σ1 (j)(ψ(j)) 6= 0. Then ^ _ [µp , λ] = ( (limp G(x) → G(µ)) → λ(x)) s (X):limp G(x)6=0 x∈X G∈FL

=

^

^

((limp G(x) → G(µ)) → λ(x))

s (X):limp G(x)6=0 x∈X G∈FL

≤

^

(limp σ1 (j)(ψ(j)) → σ1 (j)(µ)) → λ(ψ(j)))

j∈J

≤

^

(σ2 (j) → σ1 (j)(µ)) → ψ ← (λ)(j))

j∈J

=

^

(ˆ σ (µ)(j) → ψ ← (λ)(j)) = [ˆ σ (µ), ψ ← (λ)].

j∈J

Lemma 3.8. Let J, X, σ, ψ satisfy the condition in p-(DLfw), and let F ∈ FLs (X). W Then the function F σ : LJ −→ L, defined by F σ (λ) = µ∈LX (F(µ)∧[ˆ σ (µ), λ]), satisfies σ (F1), (F2), (Fs) and kL σF ≥ F. Proof. (F1): It is obvious. (F2): Obviously, F σ (λ ∧ µ) ≤ F σ (λ) ∧ F σ (µ). Conversely, _ _ F σ (λ) ∧ F σ (µ) = (F(a) ∧ [b σ (a), λ]) ∧ (F(b) ∧ [b σ (b), µ]) a∈LX

=

b∈LX

_

(F(a) ∧ F(b) ∧ [b σ (a), λ] ∧ [b σ (b), µ])

a,b∈LX

≤

_

(F(a ∧ b) ∧ [b σ (a ∧ b), λ ∧ µ])

a,b∈LX

≤

_

(F(c) ∧ [b σ (c), λ ∧ µ]) = F σ (λ ∧ µ).

c∈LX

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8

(Fs): For any β ∈ L, we have _ F σ (β) = (F(µ) ∧ [b σ (µ), β]) ≥ F(1) ∧ [b σ (1), β] = 1 ∧ β = β. µ∈LX

It follows by the following inequality that kL σF σ ≥ F. For any λ ∈ LX , _ kL σF σ (λ) = F σ (ˆ σ (λ)) = (F(µ) ∧ [ˆ σ (µ), σ ˆ (λ)]) ≥ F(λ). µ∈LX

Theorem 3.9. Let (X, limp , limq ) be a pair of stratified L-generalized convergence spaces. Then (X, limq ) is p-regular if and only if limq F ≤ limq F p for any F ∈ FLs (X). Proof. Necessity: Let J = {(G, y) ∈ FLs (X) × X|limp G(y) 6= 0}, ψ : J −→ X, (G, y) 7→ y, σ : J −→ FLs (X) × L0 , (G, y) 7→ (G, limp G(y)). Then (1) σ2 (j) = limp σ1 (j)(ψ(j)) 6= 0. (2) For any F ∈ FLs (X) we have F σ ∈ FLs (J). Indeed, by Lemma 3.8, we need only to check that F σ (0) = 0. ^ _ _ σ (µ)(j) → 0)) (F(µ) ∧ (ˆ (F(µ) ∧ [ˆ σ (µ), 0]) = F σ (0) = µ∈LX

≤

_

(F(µ) ∧ (

_

(F(µ) ∧ (

=

(F(µ) ∧ (

_

^

((limp [y](y) → [y](µ)) → 0))

^

(µ(y) → 0)) =

y∈X

µ∈LX

≤

(ˆ σ (µ)([y], y) → 0))

y∈X

µ∈LX

_

j∈J

µ∈LX

y∈X

µ∈LX

=

^

_

(F(µ) ∧ [µ, 0])

µ∈LX

F(µ ∧ [µ, 0]) ≤ F(0) = 0.

µ∈LX

(3) ψ ⇒ (F σ ) = F p . For any λ, µ ∈ LX , ^ _ ( [µp , λ] =

(limp G(x) → G(µ)) → λ(x))

s (X):limp G(x)6=0 x∈X G∈FL

=

^

^

((limp G(x) → G(µ)) → λ(x))

s (X):limp G(x)6=0 x∈X G∈FL

=

^

(limp σ1 (j)(ψ(j)) → σ1 (j)(µ)) → λ(ψ(j)))

j∈J

=

^

(σ2 (j) → σ1 (j)(µ)) → ψ ← (λ)(j))

j∈J

=

^

(ˆ σ (µ)(j) → ψ ← (λ)(j)) = [ˆ σ (µ), ψ ← (λ)].

j∈J

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A new regularity (p-regularity) of stratified L-generalized convergence spaces

9

It follows that _

ψ ⇒ (F σ )(λ) = F σ (ψ ← (λ)) =

(F(µ)∧[ˆ σ (µ), ψ ← (λ)]) =

µ∈LX

_

(F(µ)∧[µp , λ]) = F p (λ).

µ∈LX

(4) F σ is compressible w.r.t. σ. For any λ, µ ∈ LX , ^ [b σ (λ), σ b(µ)] = (b σ (λ)(j) → σ b(µ)(j)) j∈J

^

=

((σ2 (j) → σ1 (j)(λ)) → (σ2 (j) → σ1 (j)(µ))

p

(G,y):lim G(y)6=0

^

≤

((limp [y](y) → [y](λ)) → (limp [y](y) → [y](µ)))

([y],y):y∈X

=

^

(λ(y) → µ(y)) = [λ, µ].

y∈X

Therefore, for any λ ∈ LX , kL σF σ (λ) = F σ (ˆ σ (λ)) =

_

(F(µ) ∧ [ˆ σ (µ), σ ˆ (λ)]) ≤

_

(F(µ) ∧ [µ, λ]) ≤ F(λ).

µ∈LX

µ∈LX

By Lemma 3.8, we have kL σF σ = F ∈ FLs (X). Thus kL σF σ is compressible w.r.t. σ. Applying (1)-(4) in p-(DLfw) we have limq F ≤ limq F p . Sufficiency: Let J, X, σ, ψ satisfy the condition in (DLfw). Then for any F ∈ FLs (J), by (X, limq ) is p-regular we have that limq kL σF ≤ limq kL σF p (x). For any λ ∈ LX , by Lemma 3.7 we have _ _ kL σF p (λ) = (kL σF(µ) ∧ [µp , λ]) = (F(ˆ σ (µ)) ∧ [µp , λ]) µ∈LX

≤

_

µ∈LX

(F(ˆ σ (µ)) ∧ [ˆ σ (µ), ψ ← (λ)]) ≤ F(ψ ← (λ)) = ψ ⇒ (F)(λ).

µ∈LX

So, kL σF p ≤ ψ ⇒ (F), and hence limq ψ ⇒ (F) ≥ limq kL σF p ≥ limq kL σF, i.e., the condition p-(DLfw) holds. The next two theorems show that p-regularity behave reasonably well relative to initial and final structures. Definition 3.10. Let f : (X, limq ) −→ (Y, limp ) be a function between stratified Lgeneralized convergence spaces. Then f is said to be a closure function if f → (λq ) ≥ f → (λ)p for all λ ∈ LX . Lemma 3.11. Let f : (X, limq ) −→ (Y, limp ) be a function between stratified Lgeneralized convergence spaces, and let F ∈ FLs (X). (1) If f is continuous, then f ⇒ (F q ) ≥ f ⇒ (F)p . (2) If f is a closure function, then f ⇒ (F q ) ≤ f ⇒ (F)p .

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A new regularity (p-regularity) of stratified L-generalized convergence spaces 10

Proof. (1) Let f be a continuous function. Then for each λ ∈ LY we check below that (f ← (λ))q ≤ f ← (λp ). Indeed, for each x ∈ X, _ ¡ q ¢ (f ← (λ))q (x) = lim G(x) → G(f ← (λ)) s (X):limq G(x)6=0 G∈FL

_

≤

¡

s (X):limq G(x)6=0 G∈FL

_

≤

¢ limp f ⇒ (G)(f (x)) → f ⇒ (G)(λ)

¡

¢ limp H(f (x)) → H(λ) = f ← (λp )(x),

s (Y ):limp H(x)6=0 H∈FL

where the first inequality holds for the continuity of f . Then for each F ∈ FLs (X) and each λ ∈ LY _ ([µq , f ← (λ)] ∧ F(µ)) f ⇒ (F q )(λ) = F q (f ← (λ)) = ≥

_

µ∈LX

ν∈LY

≥

_

_

([(f ← (ν))q , f ← (λ)] ∧ F(f ← (ν))) ≥

([f ← (ν p ), f ← (λ)] ∧ F(f ← (ν)))

ν∈LY

([ν p , λ] ∧ f ⇒ (F)(ν)) = f ⇒ (F)p (λ).

ν∈LY

Thus f ⇒ (F q ) ≥ f ⇒ (F)p . (2) Let f be a closure function. Then for each λ ∈ LY , _ _ f ⇒ (F)p (λ) = (f ⇒ (F)(µ) ∧ [µp , λ]) = (F(f ← (µ)) ∧ [µp , λ]) µ∈LY

≥

_

µ∈LY

(F(f ← f → (ν)) ∧ [f → (ν)p , λ]) ≥

ν∈LX

≥

_

(F(ν) ∧ [f → (ν q ), λ]) =

ν∈LX

_

_

(F(ν) ∧ [f → (ν)p , λ])

ν∈LX

(F(ν) ∧ [ν q , f ← (λ)])

ν∈LX

= F q (f ← (λ)) = f ⇒ (F q )(λ), where the third inequality holds for f being a closure function, and the third equality follows from Lemma 1.1 (7). By the arbitrariness of λ, we get f ⇒ (F q ) ≤ f ⇒ (F)p . Theorem 3.12. Let {(Xi , limqi , limpi )}i∈I be pairs of stratified L-generalized convergence spaces and let limq (resp., limp ) be the initial structure on X relative to the source fi fi (X −→ (Xi , limqi ))i∈I (resp., (X −→ (Xi , limpi ))i∈I ). If each limqi is pi -regular, then (X, limq ) is p-regular. Proof. Let F ∈ FLs (X) and x ∈ X. Then by Lemma 3.11 (1) we have fi⇒ (F p ) ≥ fi⇒ (F)pi for all i ∈ I. It follows by each (Xi , limqi ) being pi -regular that ^ ^ limq F p (x) = limqi fi⇒ (F)pi (fi (x)) limqi fi⇒ (F p )(fi (x)) ≥ i∈I

≥

^

i∈I qi

lim

fi⇒ (F)(fi (x))

= limq F(x).

i∈I q

Thus (X, lim ) is p-regular.

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A new regularity (p-regularity) of stratified L-generalized convergence spaces 11

Theorem 3.13. Let {(Xi , limqi , limpi )}i∈I be pairs of stratified L-generalized converfi gence spaces, and let limq be the final structure on X w.r.t. the sink ((Xi , limqi ) −→ X)i∈I with X = ∪i∈I fi (Xi ). If each limqi is pi -regular and limp is a stratified Lgeneralized convergence structure on X such that each fi : (Xi , limpi ) −→ (X, limp ) is a closure function, then (X, limq ) is p-regular. Proof. Let F ∈ FLs (X) and x ∈ X. Then _

limq F(x) =

limqi Gi (xi )

s (X ),f (x )=x,f ⇒ (G )≤F i∈I,xi ∈Xi ,Gi ∈FL i i i i i

_

≤

limqi Gipi (xi )

s (X ),f (x )=x,f ⇒ (G )≤F i∈I,xi ∈Xi ,Gi ∈FL i i i i i

_

≤

limqi Gipi (xi )

s (X ),f (x )=x,f ⇒ (G ) ≤F i∈I,xi ∈Xi ,Gi ∈FL p i p i i i i

_

=

limqi Gipi (xi )

⇒ s (X ),f (x )=x,f ⇒ (G i∈I,xi ∈Xi ,Gi ∈FL ipi )≤fi (Gi )p ≤F p i i i i

_

≤

limqi Hi (xi ) = limq F p (x),

s (X ),f (x )=x,f ⇒ (H )≤F i∈I,xi ∈Xi ,Hi ∈FL p i i i i i

where the first inequality holds for pi -regularity of limqi , the second equality follows from Lemma 3.11 (2). Then limq is p-regular by limq F ≤ limq F p . The regularity has similar characterization and properties, we omit them here.

4

Conclusion

In this paper, by dualizing the Fischer-type diagonal condition (Lfw), which is used to describe the L-topologicallness of stratified L-convergence spaces, we define a new regularity (p-regularity) of stratified L-generalized convergence spaces. Then we also present a characterization on that regularity (p-regularity) by the notion of closures of stratified L-filters. The regularity (p-regularity) is proved to behave reasonably well relative to initial and final structures.

References [1] H.J. Biesterfeld, Regular convergence spaces, Indag. Math. 28 (1966) 605–607. [2] R. Bˇelohl´avek, Fuzzy Relational Systems: Foundations and Principles, Kluwer Academic Publishers, New York, 2002. [3] C.H. Cook, H.R. Fischer, Regular Convergence Spaces, Math. Annalen 174 (1967) 1–7.

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A new regularity (p-regularity) of stratified L-generalized convergence spaces 12

[4] J.M. Fang, Stratified L-ordered convergence structures, Fuzzy Sets and Systems 161 (2010) 2130–2149. [5] W. G¨ahler, Monadic convergence structures, In: Topological and Algebraic Structures in Fuzzy Sets, (S.E. Rodabaugh, E.P. Klement, eds.), Kluwer Academic Publishers, Dordrecht, 2003. [6] U. H¨ohle, S.E. Rodabaugh (Eds.), Mathematics of Fuzzy Sets: Logic, Topology and Measure Theory, The Handbooks of Fuzzy Sets Series, Vol.3, Kluwer Academic Publishers, Boston, Dordrecht, London, 1999. [7] G. J¨ager, A category of L-fuzzy convergence spaces, Quaestiones Mathematicae 24 (2001) 501–517. [8] G. J¨ager, Fischer’s diagonal condition for lattice-valued convergence spaces, Quaestiones Mathematicae 31 (2008) 11–25. [9] G. J¨ager, Lattice-valued convergence spaces and regularity, Fuzzy Sets and Systems 159 (2008) 2488–2502. [10] G. J¨ager, G¨ahler’s neighbourhood condition for lattice-valued convergence spaces, Fuzzy Sets and Systems 204 (2012) 27–39. [11] D.C. Kent, G.D. Richardson, p-regular convergence spaces, Math. Nachr. 149 (1990) 215–222. [12] D.C. Kent, G.D. Richardson, Convergence spaces and diagonal conditions, Topology and its Applications 70 (1996) 167–174. [13] L. Li, Many-Valued Convergence, Many-Valued Topology, and Many-Valued Order Structure, PhD Thesis, Sichuan University, 2008 (In Chinese). [14] L. Li, Q. Jin, p-Topologicalness and p-Regularity for Lattice-Valued Convergence Spaces, Fuzzy Sets and Systems 238 (2014) 26–45. [15] L. Li, Q. Jin, K. Hu, On Stratified L-Convergence Spaces: Fischer’s Diagonal Axiom, DOI: 10.1016/j.fss.2014.09.001. [16] G. Preuss, Fundations of Topology, Kluwer Academic Publishers, London, 2002. [17] S.A. Wilde, D.C. Kent, p-topological and p-regular: dual notions in convergence theory, Internat. J. Math. & Math. Sci. 22 (1999) 1–12. [18] W. Yao, On many-valued stratified L-fuzzy convergence spaces, Fuzzy Sets and Systems 159 (2008) 2503-2519. [19] D. Zhang, An enriched category approach to many valued topology, Fuzzy Sets and Systems 158 (2007) 349–366.

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Uni-soft filters and uni-soft G-filters in residuated lattices Young Bae Jun Department of Mathematics Education, Gyeongsang National University, Jinju 660-701, Korea e-mail: [email protected] Seok Zun Song∗ Department of Mathematics, Jeju National University, Jeju 690-756, Korea e-mail: [email protected] Abstract The notions of uni-soft filters and uni-soft G-filters in residuated lattices are introduced, and their relations, properties and characterizations are investigated. Conditions for a uni-soft filter to be a uni-soft G-filter are provided. Keywords: Residuated lattice, Uni-soft filter, Uni-soft G-filter. 2010 Mathematics Subject Classification. 06F35, 03G25, 06D72.

1

Introduction

Non-classical logic has become a formal and useful tool in dealing with fuzzy and uncertain informations. Various logical algebras have been proposed as the semantical systems of non-classical logic systems. Residuated lattices are important algebraic structures which are basic of BL-algebras, M V -algebras, M T L-algebras, G¨odel algebras, R0 -algebras, lattice implication algebras, and so forth. The (fuzzy) filter theory in the logical algebras has an important role in studying these algebras and completeness of the corresponding non-classical logics, and it is studied in the papers [1], [2], [3], [9], [12], [13] and [14]. Filter theory, which is an important notion, in residuated lattices is studied by Shen and Zhang [11] and Zhu and Xu [16]. Uncertainty is an attribute of information. As a new mathematical tool for dealing with uncertainties, Molodtsov [10] introduced the concept of soft sets. Since then several authors studied (fuzzy) algebraic structures based on soft *Corresponding author.

1

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set theory in several algebraic structures. In order to lay a foundation for providing a soft algebraic tool in considering many problems that contain uncertainties, Jun [7] discussed the union soft sets with applications in BCK/BCI-algebras. Also, Jun et al. [8] discussed uni-soft sets applied to commutative BCI-ideals. In this paper, we introduce uni-soft filters and uni-soft G-filters in residuated lattices, and investigate their properties. We consider characterizations of uni-soft filters and unisoft G-filters. We provide conditions for a uni-soft filter to be a uni-soft G-filter.

2

Preliminaries

Definition 2.1 ([1, 5, 6]). A residuated lattice is an algebra L := (L, ∨, ∧, ⊙, →, 0, 1) of type (2, 2, 2, 2, 0, 0) such that (1) (L, ∨, ∧, 0, 1) is a bounded lattice. (2) (L, ⊙, 1) is a commutative monoid. (3) ⊙ and → form an adjoint pair, that is, (∀x, y, z ∈ L) (x ≤ y → z ⇔ x ⊙ y ≤ z) . In a residuated lattice L, the ordering ≤ is defined as follows: (∀x, y ∈ L) (x ≤ y ⇔ x ∧ y = x ⇔ x ∨ y = y ⇔ x → y = 1) and x′ will be reserved for x → 0, and x′′ = (x′ )′ , etc. for all x ∈ L. Proposition 2.2 ([1, 5, 6, 12, 13]). In a residuated lattice L, the following properties are valid. 1 → x = x, x → 1 = 1, x → x = 1, 0 → x = 1, x → (y → x) = 1.

(2.1)

x → (y → z) = (x ⊙ y) → z = y → (x → z).

(2.2)

x ≤ y ⇒ z → x ≤ z → y, y → z ≤ x → z.

(2.3)

z → y ≤ (x → z) → (x → y), z → y ≤ (y → x) → (z → x).

(2.4)

Definition 2.3 ([11]). A nonempty subset F of a residuated lattice L is called a filter of L if it satisfies the conditions: (∀x, y ∈ L) (x, y ∈ F ⇒ x ⊙ y ∈ F ) .

(2.5)

(∀x, y ∈ L) (x ∈ F, x ≤ y ⇒ y ∈ F ) .

(2.6)

2

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Proposition 2.4 ([11]). A nonempty subset F of a residuated lattice L is a filter of L if and only if it satisfies: 1 ∈ F.

(2.7)

(∀x ∈ F ) (∀y ∈ L) (x → y ∈ F ⇒ y ∈ F ) .

(2.8)

Definition 2.5 ([15]). A nonempty subset F of L is called a G-filter of L if it is a filter of L that satisfies the following condition: (∀x, y ∈ L) ((x ⊙ x) → y ∈ F ⇒ x → y ∈ F ) .

(2.9)

A soft set theory is introduced by Molodtsov [10], and C ¸ aˇgman et al. [4] provided new definitions and various results on soft set theory. In what follows, let U be an initial universe set and E be a set of parameters. Let P(U ) denotes the power set of U and A, B, C, · · · ⊆ E. Definition 2.6 ([4, 10]). A soft set (f˜, A) over U is defined to be the set of ordered pairs { } ˜ ˜ ˜ (f , A) := (x, fA (x)) : x ∈ E, fA (x) ∈ P(U ) , where f˜A : E → P(U ) such that f˜(x) = ∅ if x ∈ / A. The soft set (f˜, A) is simply denoted by f˜A . For a soft set f˜A over U and a subset τ of U, the τ -exclusive set of f˜A , denoted by e(f˜A ; τ ), is defined to be the set { } e(f˜A ; τ ) := x ∈ A | f˜A (x) ⊆ τ .

3

Uni-soft filters

In what follows, we take a residuated lattice L as a set of parameters. Definition 3.1. A soft set f˜L over U is called a uni-soft filter of L if it satisfies: ( ) (∀x, y ∈ L) x ≤ y ⇒ f˜L (x) ⊇ f˜L (y) , ( ) (∀x, y ∈ L) f˜L (x) ∪ f˜L (y) ⊇ f˜L (x ⊙ y) .

(3.1) (3.2)

3

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Proposition 3.2. Every uni-soft filter f˜L of L satisfies: ( ) (∀x ∈ L) f˜L (x) ⊇ f˜L (1) . ( ) (∀x, y ∈ L) f˜L (x) ∪ f˜L (x → y) ⊇ f˜L (y) .

(3.3) (3.4)

Proof. Let x, y ∈ L. Since x ≤ 1, we have f˜L (x) ⊇ f˜L (1) by (3.1). Since x ⊙ (x → y) ≤ y, it follows from (3.2) and (3.1) that f˜L (x) ∪ f˜L (x → y) ⊇ f˜L (x ⊙ (x → y)) ⊇ f˜L (y). This completes the proof. Lemma 3.3. If a soft set f˜L over U satisfies two conditions (3.3) and (3.4), then ( ) (∀x, y, z ∈ L) x ≤ y → z ⇒ f˜L (x) ∪ f˜L (y) ⊇ f˜L (z) , (3.5) ( ) (∀x, y, z ∈ L) x ⊙ y ≤ z ⇒ f˜L (x) ∪ f˜L (y) ⊇ f˜L (z) . (3.6) Proof. Assume that x ≤ y → z for all x, y, z ∈ L. Then x → (y → z) = 1, and so ( ) ˜ ˜ ˜ ˜ fL (x) ∪ fL (y) = fL (x) ∪ fL (1) ∪ f˜L (y) ( ) = f˜L (x) ∪ f˜L (x → (y → z)) ∪ f˜L (y) ⊇ f˜L (y) ∪ f˜L (y → z) ⊇ f˜L (z). Since x ≤ y → z ⇔ x ⊙ y ≤ z, we know that (3.5) induces (3.6). We consider characterizations of uni-soft filters. Theorem 3.4. A soft set f˜L over U is a uni-soft filter of L if and only if it satisfies two conditions (3.3) and (3.4). Proof. The necessity is from Proposition 3.2. Conversely, let f˜L be a soft set over U that satisfies (3.3) and (3.4). Let x, y ∈ L be such that x ≤ y. Then x → y = 1 and so f˜L (x) = f˜L (x) ∪ f˜L (1) = f˜L (x) ∪ f˜L (x → y) ⊇ f˜L (y). Since x ⊙ y ≤ x ⊙ y for all x, y ∈ L, it follows from (3.6) that f˜L (x) ∪ f˜L (y) ⊇ f˜L (x ⊙ y) for all x, y ∈ L. Therefore f˜L is a uni-soft filter of L. 4

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Theorem 3.5. A soft set f˜L over U is a uni-soft filter of L if and only if it satisfies the condition (3.5). Proof. The necessity is from Lemma 3.3 and Theorem 3.4. Conversely let f˜L be a soft set over U satisfying (3.5). Since x ≤ x → 1 and x → y ≤ x → y for all x, y ∈ L, it follows from (3.5) that f˜L (x) = f˜L (x) ∪ f˜L (x) ⊇ f˜L (1) and f˜L (x) ∪ f˜L (x → y) ⊇ f˜L (y) for all x, y ∈ L. Hence f˜L is a uni-soft filter of L by Theorem 3.4. Proposition 3.6. Every uni-soft filter f˜L of L satisfies the following condition: ( ) (∀x, y, z ∈ L) f˜L (x → (y → z)) ∪ f˜L (x → y) ⊇ f˜L (x → (x → z)) .

(3.7)

Proof. Let x, y, z ∈ L. Using (2.2) and (2.4), we have x → (y → z) = y → (x → z) ≤ (x → y) → (x → (x → z)). It follows from Theorem 3.5 that f˜L (x → (y → z)) ∪ f˜L (x → y) ⊇ f˜L (x → (x → z)). This completes the proof. Theorem 3.7. A soft set f˜L over U is a uni-soft filter of L if and only if f˜L satisfies the condition (3.3) and ( ) ˜ ˜ ˜ (∀x, y, z ∈ L) fL (x → (y → z)) ∪ fL (y) ⊇ fL (x → z) . (3.8) Proof. Assume that f˜L is a uni-soft filter of L. Then the condition (3.3) is valid. Using (3.4) and (2.2), we have f˜L (x → z) ⊆ f˜L (y) ∪ f˜L (y → (x → z)) = f˜L (y) ∪ f˜L (x → (y → z)) for all x, y, z ∈ L. Conversely, let f˜L be a soft set over U satisfying (3.3) and (3.8). Taking x := 1 in (3.8) and using (2.1), we have f˜L (z) = f˜L (1 → z) ⊆ f˜L (1 → (y → z)) ∪ f˜L (y) = f˜L (y → z) ∪ f˜L (y) for all y, z ∈ L. Thus f˜L is a uni-soft filter of L by Theorem 3.4. 5

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Proposition 3.8. Every uni-soft filter f˜L of L satisfies the following condition: ( ) (∀a, x ∈ L) f˜L (a) ⊇ f˜L ((a → x) → x) .

(3.9)

Proof. If we take y := (a → x) → x and x := a in (3.4), then f˜L ((a → x) → x) ⊆ f˜L (a) ∪ f˜L (a → ((a → x) → x)) = f˜L (a) ∪ f˜L ((a → x) → (a → x)) = f˜L (a) ∪ f˜L (1) = f˜L (a). This completes the proof. Theorem 3.9. A soft set f˜L over U is a uni-soft filter of L if and only if it satisfies the following conditions: ( ) (∀x, y ∈ L) f˜L (x) ⊇ f˜L (y → x) , (3.10) ( ) (∀x, a, b ∈ L) f˜L (a) ∪ f˜L (b) ⊇ f˜L ((a → (b → x)) → x) . (3.11) Proof. Assume that f˜L is a uni-soft filter of L. Using (3.4), (2.1) and (3.3), we have f˜L (y → x) ⊆ f˜L (x) ∪ f˜L (x → (y → x)) = f˜L (x) ∪ f˜L (1) = f˜L (x) for all x, y ∈ L. Using (3.8) and (3.9), we get f˜L ((a → (b → x)) → x) ⊆ f˜L ((a → (b → x)) → (b → x)) ∪ f˜L (b) ⊆ f˜L (a) ∪ f˜L (b) for all a, b, x ∈ L. Conversely, let f˜L be a soft set over U satisfying two conditions (3.10) and (3.11). If we take y := x in (3.10), then f˜L (x) ⊇ f˜L (x → x) = f˜L (1) for all x ∈ L. Using (3.11) induces f˜L (y) = f˜L (1 → y) = f˜L (((x → y) → (x → y)) → y) ⊆ f˜L (x → y) ∪ f˜L (x) for all x, y ∈ L. Therefore f˜L is a uni-soft filter of L by Theorem 3.4. Theorem 3.10. A soft set f˜L over U is a uni-soft filter of L if and only if the nonempty τ -exclusive set of f˜L is a filter of L for all τ ∈ P(U ).

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Proof. Assume that f˜L is a uni-soft filter of L and let τ ∈ P(U ) be such that e(f˜L ; τ ) ̸= ∅. Let x, y ∈ L be such that x ∈ e(f˜L ; τ ) and x → y ∈ e(f˜L ; τ ). Then τ ⊇ f˜L (x) and τ ⊇ f˜L (x → y). It follows from (3.3) and (3.4) that f˜L (1) ⊆ f˜L (x) ⊆ τ and f˜L (y) ⊆ f˜L (x) ∪ f˜L (x → y) ⊆ τ . Hence 1 ∈ e(f˜L ; τ ) and y ∈ e(f˜L ; τ ), and therefore e(f˜L ; τ ) is a filter of L by Proposition 2.4. Conversely, suppose that e(f˜L ; τ ) is a filter of L for all τ ∈ P(U ) with e(f˜L ; τ ) ̸= ∅. For any x ∈ L, let f˜L (x) = δ. Then x ∈ e(f˜L ; δ) and e(f˜L ; δ) is a filter of L. Hence 1 ∈ e(f˜L ; δ) and so f˜L (x) = δ ⊇ f˜L (1). For any x, y ∈ L, let f˜L (x) = δx and f˜L (x → y) = δx→y . If we take δ = δx ∪ δx→y , then x ∈ e(f˜L ; δ) and x → y ∈ e(f˜L ; δ) which imply that y ∈ e(f˜L ; δ). Thus f˜L (x) ∪ f˜L (x → y) = δx ∪ δx→y = δ ⊇ f˜L (y). Therefore f˜L is a uni-soft filter of L by Theorem 3.4. Theorem 3.11. For a soft set f˜L over U , let f˜L∗ be a soft set over U which is given as follows: { f˜L (x) if x ∈ e(f˜L ; τ ), f˜L∗ : L → P(U ), x 7→ U otherwise, where τ ∈ P(U ) with τ ̸= U . If f˜L is a uni-soft filter of L, then so is f˜L∗ . Proof. Suppose that f˜L is a uni-soft filter of L. Then e(f˜L ; τ ) is a filter of L for all τ ∈ P(U ) with e(f˜L ; τ ) ̸= ∅ by Theorem 3.10. Thus 1 ∈ e(f˜L ; τ ), and so f˜L∗ (1) = f˜L (1) ⊆ f˜L (x) ⊆ f˜L∗ (x) for all x ∈ L. Let x, y ∈ L. If x ∈ e(f˜L ; τ ) and x → y ∈ e(f˜L ; τ ), then y ∈ e(f˜L ; τ ). Hence f˜L∗ (x) ∪ f˜L∗ (x → y) = f˜L (x) ∪ f˜L (x → y) ⊇ f˜L (y) = f˜L∗ (y). If x ∈ / e(f˜L ; τ ) or x → y ∈ / e(f˜L ; τ ), then f˜L∗ (x) = U or f˜L∗ (x → y) = U . Thus f˜L∗ (x) ∪ f˜L∗ (x → y) = U ⊇ f˜L∗ (y). Therefore f˜L∗ is a uni-soft filter of L. Theorem 3.12. If f˜L is a uni-soft filter of L, then the set La := {x ∈ L | f˜L (a) ⊇ f˜L (x)} is a filter of L for every a ∈ L. 7

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Proof. Since f˜L (1) ⊆ f˜L (a) for all a ∈ L, we have 1 ∈ La . Let x, y ∈ L be such that x ∈ La and x → y ∈ La . Then f˜L (x) ⊆ f˜L (a) and f˜L (x → y) ⊆ f˜L (a). Since f˜L is a uni-soft filter of L, it follows from (3.4) that f˜L (a) ⊇ f˜L (x) ∪ f˜L (x → y) ⊇ f˜L (y) so that y ∈ La . Hence La is a filter of L by Proposition 2.4. Theorem 3.13. Let a ∈ L and let f˜L be a soft set over U. Then (1) If La is a filter of L, then f˜L satisfies the following condition: (∀x, y ∈ L) (f˜L (a) ⊇ f˜L (x) ∪ f˜L (x → y) ⇒ f˜L (a) ⊇ f˜L (y)).

(3.12)

(2) If f˜L satisfies (3.3) and (3.12), then La is a filter of L. Proof. (1) Assume that La is a filter of L. Let x, y ∈ L be such that f˜L (a) ⊇ f˜L (x) ∪ f˜L (x → y). Then x → y ∈ La and x ∈ La . Using (2.8), we have y ∈ La and so f˜L (a) ⊇ f˜L (y). (2) Suppose that f˜L satisfies (3.3) and (3.12). Then 1 ∈ La by (3.3). Let x, y ∈ L be such that x ∈ La and x → y ∈ La . Then f˜L (a) ⊇ f˜L (x) and f˜L (a) ⊇ f˜L (x → y), which imply that f˜L (a) ⊇ f˜L (x) ∪ f˜L (x → y). Thus f˜L (a) ⊇ f˜L (y) by (3.12), and so y ∈ La . Therefore La is a filter of L by Proposition 2.4.

4

Uni-soft G-filters

Definition 4.1. A soft set f˜L over U is called a uni-soft G-filter of L if it is a uni-soft filter of L that satisfies: ( ) ˜ ˜ (∀x, y ∈ L) fL ((x ⊙ x) → y) ⊇ fL (x → y) . (4.1) Note that the condition (4.1) is equivalent to the following condition: ( ) (∀x, y ∈ L) f˜L (x → (x → y)) ⊇ f˜L (x → y) .

(4.2)

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Example 4.2. Let L := [0, 1] (unit interval). For any a, b ∈ L, define a ∨ b = max{a, b}, a ∧ b = min{a, b}, { 1 if a ≤ b, a→b= and a ⊙ b = min{a, b}. b otherwise, Then L := (L, ∨, ∧, ⊙, →, 0, 1) is a residuated lattice. Let f˜L be a soft set over U defined by { τ if x ∈ [ 21 , 1], f˜L : L → P(U ), x 7→ U otherwise, where τ ∈ P(U ) with τ ̸= U . Then f˜L is a uni-soft G-filter of L. Theorem 4.3. Let f˜L be a soft set over U . Then f˜L is a uni-soft G-filter of L if and only if it is a uni-soft filter of L that satisfies the following condition: ( ) ˜ ˜ ˜ (∀x, y, z ∈ L) fL (x → (y → z)) ∪ fL (x → y) ⊇ fL (x → z) . (4.3) Proof. Assume that f˜L is a uni-soft G-filter of L. Then f˜L is a uni-soft filter of L. Note that x ≤ 1 = (x → y) → (x → y), and thus x → y ≤ x → (x → y) for all x, y ∈ L. It follows from (3.1) that f˜L (x → y) ⊇ f˜L (x → (x → y)). Combining this and (4.2), we have f˜L (x → y) = f˜L (x → (x → y))

(4.4)

for all x, y ∈ L. Using (3.7) and (4.4), we have f˜L (x → (y → z)) ∪ f˜L (x → y) ⊇ f˜L (x → z) for all x, y, z ∈ L. Conversely, let f˜L be a uni-soft filter of L that satisfies the condition (4.3). If we put y = x and z = y in (4.3) and use (2.1) and (3.3), then f˜L (x → y) ⊆ f˜L (x → (x → y)) ∪ f˜L (x → x) = f˜L (x → (x → y)) ∪ f˜L (1) = f˜L (x → (x → y)) for all x, y ∈ L. Therefore f˜L is a uni-soft G-filter of L.

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Theorem 4.4. Let f˜L be a soft set over U that satisfies the condition (3.3) and ( ) (∀x, y, z ∈ L) f˜L (x) ∪ f˜L ((y → z) → (x → y)) ⊇ f˜L (y) .

(4.5)

Then f˜L is a uni-soft G-filter of L. Proof. If we take z := 1 in (4.5) and use (2.1), then f˜L (x) ∪ f˜L (x → y) = f˜L (x) ∪ f˜L (1 → (x → y)) = f˜L (x) ∪ f˜L ((y → 1) → (x → y)) ⊇ f˜L (y). Hence f˜L is a uni-soft filter of L by Theorem 3.4. Let x, y, z ∈ L. Since x → (y → z) ≤ (x → y) → (x → (x → z)) by (2.2) and (2.4), we have f˜L (x → (y → z)) ⊇ f˜L ((x → y) → (x → (x → z))) by (3.1). It follows from (3.1), (3.3), (3.4), (2.4) and (4.5) that f˜L (x → y) ∪ f˜L (x → (y → z)) ⊇ f˜L (x → y) ∪ f˜L ((x → y) → (x → (x → z))) ⊇ f˜L (x → (x → z)) ⊇ f˜L (((x → z) → z) → (x → z)) = f˜L (((x → z) → z) → (1 → (x → z))) ⊇ f˜L (x → z). Therefore f˜L is a uni-soft G-filter of L by Theorem 4.3. The following example shows that any uni-soft G-filter may not satisfy the condition (4.5). Example 4.5. The uni-soft G-filter f˜L of L in Example 4.2 does not satisfy the condition (4.5) since f˜L ( 32 ) ∪ f˜L (( 31 → 14 ) → ( 32 → 31 )) = f˜L ( 23 ) ∪ f˜L (1) = τ ⊉ U = f˜L ( 13 ). Proposition 4.6. For a uni-soft filter f˜L of L, the condition (4.5) is equivalent to the following condition. ( ) (∀x, y ∈ L) f˜L ((x → y) → x) ⊇ f˜L (x) . (4.6) 10

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Proof. Assume that the condition (4.5) is valid. It follows from (3.3) and (2.1) that f˜L ((x → y) → x) = f˜L (1) ∪ f˜L ((x → y) → x) = f˜L (1) ∪ f˜L ((x → y) → (1 → x)) ⊇ f˜L (x) for all x, y ∈ L. Conversely, suppose that the condition (4.6) is valid. It follows from (2.2) and (3.4) that f˜L (x) ∪ f˜L ((y → z) → (x → y)) = f˜L (x) ∪ f˜L (x → ((y → z) → y)) ⊇ f˜L ((y → z) → y) ⊇ f˜L (y) for all x, y ∈ L. Combining Theorem 4.4 and Proposition 4.6, we have the following result. Theorem 4.7. Every uni-soft filter satisfying the condition (4.6) is a uni-soft G-filter. Proposition 4.8. Every uni-soft filter f˜L of L with the condition (4.5) satisfies the following condition. ( ) (∀x, y ∈ L) f˜L ((x → y) → y) ⊇ f˜L ((y → x) → x) . (4.7) Proof. Let f˜L be a uni-soft filter of L that satisfies the condition (4.5) and let x, y ∈ L. Since x → ((y → x) → x) = (y → x) → (x → x) = (y → x) → 1 = 1, that is, x ≤ (y → x) → x, we have ((y → x) → x) → y ≤ x → y by (2.3). It follows from (2.4), (2.2) and (2.3) that (x → y) → y ≤ (y → x) → ((x → y) → x) = (x → y) → ((y → x) → x) ≤ (((y → x) → x) → y) → ((y → x) → x). Using (3.1), (3.3), (2.1), (2.2) and (4.5), we have f˜L ((x → y) → y) ⊇ f˜L ((((y → x) → x) → y) → ((y → x) → x)) = f˜L (1) ∪ f˜L (1 → ((((y → x) → x) → y) → ((y → x) → x))) = f˜L (1) ∪ f˜L ((((y → x) → x) → y) → (1 → ((y → x) → x))) ⊇ f˜L ((y → x) → x). Hence the condition (4.7) is valid. 11

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Corollary 4.9. Every uni-soft filter f˜L of L with the condition (4.6) satisfies the condition (4.7). Proposition 4.10. Every uni-soft G-filter f˜L of L with the condition (4.7) satisfies the condition (4.5). Proof. Let f˜L be a uni-soft G-filter of L that satisfies the condition (4.7). For any x, y, z ∈ L, we have f˜L (z) ∪ f˜L ((x → y) → (z → x)) = f˜L (z) ∪ f˜L (z → ((x → y) → x)) ⊇ f˜L ((x → y) → x) ⊇ f˜L ((x → y) → ((x → y) → y)) ⊇ f˜L ((x → y) → y) ⊇ f˜L ((y → x) → x) by (2.2), (3.4), (3.1), (2.4), (4.2) and (4.7). Since (x → y) → x ≤ y → x ≤ z → (y → x), it follows from (3.1) that f˜L ((x → y) → x) ⊇ f˜L (z → (y → x)) and so from (3.4) that f˜L (z) ∪ f˜L ((x → y) → (z → x)) ⊇ f˜L (z) ∪ f˜L ((x → y) → x) ⊇ f˜L (z) ∪ f˜L (z → (y → x)) ⊇ f˜L (y → x). Therefore f˜L (z) ∪ f˜L ((x → y) → (z → x)) ⊇ f˜L (y → x) ∪ f˜L ((y → x) → x) ⊇ f˜L (x). Hence the condition (4.5) is valid. Theorem 4.11. Let f˜L be a uni-soft filter of L. Then f˜L is a uni-soft G-filter of L if and only if the following condition holds: ( ) (∀x ∈ L) f˜L (x → (x ⊙ x)) = f˜L (1) . (4.8) Proof. Suppose that f˜L is a uni-soft G-filter of L. Since x → (x → (x ⊙ x)) = 1 for all x ∈ L, we have f˜L (x → (x → (x ⊙ x))) = f˜L (1). It follows from (4.3) and (2.1) that f˜L (x → (x ⊙ x)) ⊆ f˜L (x → (x → (x ⊙ x))) ∪ f˜L (x → x) = f˜L (1) and so from (3.3) that f˜L (x → (x ⊙ x)) = f˜L (1) for all x ∈ L. 12

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Conversely, let f˜L be a uni-soft filter of L which satisfies the condition (4.8) and let x, y ∈ L. Since x → (x → y) = (x ⊙ x) → y ≤ (x → (x ⊙ x)) → (x → y) by (2.2) and (2.4), it follows from (3.1) that f˜L (x → (x → y)) ⊇ f˜L ((x → (x ⊙ x)) → (x → y)). Hence, we have f˜L (x → y) ⊆ f˜L ((x → (x ⊙ x)) → (x → y)) ∪ f˜L (x → (x ⊙ x)) ⊆ f˜L (x → (x → y)) ∪ f˜L (x → (x ⊙ x)) = f˜L (x → (x → y)) ∪ f˜L (1) = f˜L (x → (x → y)) by using (3.4), (4.8) and (3.3). Hence f˜L is a uni-soft G-filter of L. Theorem 4.12. A soft set f˜L over U is a uni-soft G-filter of L if and only if it is a uni-soft filter of L with an additional condition: ( ) ˜ ˜ (∀x, y ∈ L) fL (x → y) = fL (x → (x → y)) . (4.9) Proof. Suppose that f˜L is a uni-soft G-filter of L. Then f˜L is a uni-soft filter of L. Let x, y ∈ L. Since x → y ≤ x → (x → y), we have f˜L (x → y) ⊇ f˜L (x → (x → y)) by (3.1). Hence f˜L (x → y) = f˜L (x → (x → y)) by using (4.2). Conversely, let f˜L be a uni-soft filter of L with the condition (4.9). It follows from Proposition 3.6 that f˜L (x → (y → z)) ∪ f˜L (x → y) ⊇ f˜L (x → (x → z)) = f˜L (x → z) for all x, y, z ∈ L. Therefore f˜L is a uni-soft G-filter of L by Theorem 4.3. Proposition 4.13. Every uni-soft G-filter f˜L of L satisfies the following conditions: ( ) ˜ ˜ (∀x, y, z ∈ L) fL (x → (y → z)) ⊇ fL ((x → y) → (x → z)) . (4.10) ( ) (∀x, y, z ∈ L) f˜L (x → (y → z)) = f˜L ((x → y) → (x → z)) . (4.11)

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Proof. Let f˜L be a uni-soft G-filter of L. Using (2.2), (4.3), (2.4) and (3.3), we have f˜L ((x → y) → (x → z)) = f˜L (x → ((x → y) → z)) ⊆ f˜L (x → (y → z)) ∪ f˜L (x → ((y → z) → ((x → y) → z))) = f˜L (x → (y → z)) ∪ f˜L ((y → z) → ((x → y) → (x → z))) = f˜L (x → (y → z)) ∪ f˜L (1) = f˜L (x → (y → z)) for all x, y, z ∈ L. Thus (4.10) holds. Since (x → y) → (x → z) ≤ x → (y → z) for all x, y, z ∈ L, it follows from (3.1) that f˜L ((x → y) → (x → z)) ⊇ f˜L (x → (y → z)) and so that f˜L (x → (y → z)) = f˜L ((x → y) → (x → z)) for all x, y, z ∈ L by using (4.10). Proposition 4.14. Assume that L satisfies the divisibility, that is, x ∧ y = x ⊙ (x → y) for all x, y ∈ L. If f˜L is a uni-soft G-filter of L satisfying (4.11), then the following equality is true. ( ) (∀x, y, z ∈ L) f˜L ((x ⊙ y) → z) = f˜L ((x ∧ y) → z) . (4.12) Proof. Using the divisibility and (2.2), we have (x ∧ y) → z = (x ⊙ (x → y)) → z = (x → y) → (x → z) for all x, y, z ∈ L. It follows from (2.2) and (4.11) that f˜L ((x ⊙ y) → z) = f˜L (x → (y → z)) = f˜L ((x → y) → (x → z)) = f˜L ((x ∧ y) → z) for all x, y, z ∈ L. Theorem 4.15. Let L satisfy the divisibility, that is, x ∧ y = x ⊙ (x → y) for all x, y ∈ L. Then every uni-soft filter f˜L of L satisfying the condition (4.12) is a uni-soft G-filter of L. Proof. Using Proposition 3.6, (2.2) and (4.12), we have f˜L (x → (y → z)) ∪ f˜L (x → y) ⊇ f˜L (x → (x → z)) = f˜L ((x ⊙ x) → z) = f˜L ((x ∧ x) → z) = f˜L (x → z) for all x, y, z ∈ L. Therefore f˜L is a uni-soft G-filter of L by Theorem 4.3. 14

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Theorem 4.16. Let f˜L and g˜L be uni-soft filters of L such that f˜L (1) = g˜(1) and f˜L ⊇ g˜L , i.e., f˜L (x) ⊇ g˜L (x) for all x ∈ L. If f˜L is a uni-soft G-filter of L, then so is g˜L . Proof. Assume that f˜L is a uni-soft G-filter of L. Using (2.2) and (2.1), we have x → (x → ((x → (x → y)) → y)) = (x → (x → y)) → (x → (x → y)) = 1 for all x, y ∈ L. Thus g˜(x → ((x → (x → y)) → y)) ⊆ f˜L (x → ((x → (x → y)) → y)) = f˜L (x → (x → ((x → (x → y)) → y))) = f˜L (1) = g˜(1) by hypotheses and (4.4), and so g˜(x → ((x → (x → y)) → y)) = g˜(1) for all x, y ∈ L by (3.3). Since g˜L is a uni-soft filter of L, it follows from (3.4), (2.2) and (3.3) that g˜(x → y) ⊆ g˜(x → (x → y)) ∪ g˜((x → (x → y)) → (x → y)) = g˜(x → (x → y)) ∪ g˜(x → ((x → (x → y)) → y)) = g˜(x → (x → y)) ∪ g˜(1) = g˜(x → (x → y)) for all x, y ∈ L. Therefore g˜L is a uni-soft G-filter of L.

References [1] R. Belohlavek, Some properties of residuated lattices, Czechoslovak Math. J. 53(123) (2003) 161–171. [2] K. Blount and C. Tsinakis, The structure of residuated lattices, Internat. J. Algebra Comput. 13(4) (2003) 437–461. [3] R. A. Borzooei, S. Khosravi Shoar and R. Americ, Some types of filters in MTLalgebras, Fuzzy Sets and Systems 187 (2012) 92–102. [4] N. C ¸ aˇgman and S. Engino˘glu, Soft set theory and uni-int decision making, Eur. J. Oper. Res. 207 (2010) 848–855. 15

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[5] F. Esteva and L. Godo, Monoidal t-norm based logic: towards a logic for leftcontinuous t-norms, Fuzzy Sets and Systems 124 (2001) 271–288. [6] P. H´ajek, Metamathematics of Fuzzy Logic, Kluwer Academic Press, Dordrecht, 1998. [7] Y. B. Jun, Union soft sets with applications in BCK/BCI-algebras, Bull. Korean Math. Soc. 50 (2013) 1937–1956. [8] Y. B. Jun, S. Z. Song and S. S. Ahn, Union soft sets applied to commutative BCIideals, J. Comput. Anal. Appl. 16 (2014) 468–477. [9] K. H. Kim, Q. Zhang and Y. B. Jun, On fuzzy filters of MTL-algebras, J. Fuzzy Math. 10 (2002), no. 4, 981–989. [10] D. Molodtsov, Soft set theory - First results, Comput. Math. Appl. 37 (1999) 19–31. [11] J. G. Shen and X. H. Zhang, Filters of residuated lattices, Chin. Quart. J. Math. 21 (2006) 443–447. [12] E. Turunen, BL-algebras of basic fuzzy logic, Mathware & Soft Computing 6 (1999), 49–61. [13] E. Turunen, Boolean deductive systems of BL-algebras, Arch. Math. Logic 40 (2001) 467–473. [14] X. H. Zhang, On filters in MTL-algebras, Adv. Syst. Sci. Appl. 7 (2007) 32–38. [15] X. H. Zhang and W. H. Li, On fuzzy logic algebraic system MTL, Adv. Syst. Sci. Appl. 5 (2005) 475–483. [16] Y. Q. Zhu and Y. Xu, On filter theory of residuated lattices, Inform. Sci. 180 (2010) 3614–3632.

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Mathematical analysis of a general viral infection model with immune response N. H. AlShamrani, A. M. Elaiw and M. A. Alghamdi Department of Mathematics, Faculty of Science, King Abdulaziz University, P.O. Box 80203, Jeddah 21589, Saudi Arabia. Emails: [email protected]. (N. AlShamrani), [email protected] (A. Elaiw).

Abstract In this paper, we study the global dynamics of a viral infection model with antibody immune response. The incidence rate is given by a general function of the population of the uninfected target cells, infected cells and free viruses. We have established a set of conditions on the general incidence rate function and determined two threshold parameters R0 (the basic infection reproduction number) and R1 (the antibody immune response activation number) which are su¢ cient to determine the global behavior of the model. The global asymptotic stability of the equilibria of the model has been proven by using direct Lyapunov method and applying LaSalle’s invariance principle.

Keywords: Virus dynamics; global stability; antibody immune response; Lyapunov functional. Mathematics Subject Classi…cation: 34D20; 34D23; 37N25; 92D30

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1

Introduction

Several works have been devoted to propose mathematical models of viral infectious dynamics such as human immunode…ciency virus (HIV) (see, for example, [1]-[22]), hepatitis B virus (HBV) [23][26], hepatitis C virus (HCV) [27]-[29] and human T cell leukemia HTLV [30], etc. Mathematical models of viral infection can help for understanding the viral dynamics and developing antiviral drug therapies. In reality, the immune response needs an indispensable components to do its job such as antibodies, cytokines, natural killer cells, and T cells. The antibody immune response is a part of the adaptive system in which the body responds to pathogens by primarily using antibodies that produced from the B cells. While the other part is the Cytotoxic T Lymphocytes (CTL) immune response where the CTL attacks and kills the infected cells [7]. In some infections such as malaria, the CTL immune response is less e¤ective than the antibody immune response [31]. Mathematical models of viral infection with antibody immune response have been proposed and analyzed in ([32]-[39]). The basic model of viral infection with antibody immune response has introduced by Murase et. al. [32] and Shi… Wang [39] as:

x_ = s

dx

y_ = vx v_ = ky z_ = rzv

vx;

(1)

ay; bzv

(2) cv;

(3)

z;

(4)

where x, y, v and z denote the populations of uninfected target cells, infected cells, free virus particles and antibody immune cells at time t, respectively. Parameters s, k and r represent, respectively, the rate at which new healthy cells are generated from the source within the body, the generation rate constant of free viruses produced from the infected cells and the proliferation

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rate constant of antibody immune cells. Parameters d, a, c and

are the natural death rate

constants of the uninfected target cells, infected cells, free virus particles and antibody immune cells, respectively. Parameter

is the infection rate constant and b is the removal rate constant of

the viruses due to the antibodies. All the parameters given in model (1)-(4) are positive. Note that, the infection rate in model (1)-(4) is presented to be bilinear in x and v, which can not be completely describe the interaction between the uninfected target cells and viruses. Nevertheless, there are many types of an improved incidence rate which are more commonly used due to their bene…t for helping us gain the uni…cation theory through passing over the unessential details (see e.g. [40] and [41]). Variety of viral infection models with antibody immune response have been considered di¤erent forms of the incidence rate such as saturated incidence rate, where

0 [42], [37], [35], Beddington-DeAngelis functional response,

general form,

xv 1+ x+ v ,

;

xv 1+ v

0 [36], and

(x; v)v [38].

However the infection rate does not depend on the infected cells y. In some viral infections such as HBV, the infection rate depends on x, y and v [25], [24]. In [43], the infection rate is given by (x; y; v)v, however the antibody immune response has been neglected. Our aim in this paper is to investigate the global stability analysis of the viral infection model with general incidence rate function and antibody immune response. The rest of the paper is designed as follows. In the next section, we introduce the model and discuss the non-negativity and boundedness of the solutions. In Section 3, we de…ne two threshold parameters and discuss the existence of the model’s equilibria. In Section 4, we study the global asymptotic stability of the equilibria using suitable Lyapunov functional and applying LaSalle’s invariance principle. Finally, conclusion is given in Section 5.

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2

The mathematical model

In this section, we consider the following viral infection model with general incidence rate taking into consideration the antibody immune response. x_ = s

dx

(x; y; v)v;

y_ = (x; y; v)v v_ = ky

bzv

z_ = rzv

(5)

ay;

(6)

cv;

(7)

z:

(8)

The de…nitions of all variables and parameters are identical to those given in Section 1. The incidence rate of infection is presented by a general function in the form

(x; y; v)v, where

is

continuously di¤erentiable and satis…es the following assumptions (see [38] and [43]): Assumption A1. v

(x; y; v) > 0 for all x > 0, y

0, v

0, and

(0; y; v) = 0 for all y

0,

0. @ (x; y; v) > 0 for all x > 0, y 0 and v 0: @x @ (x; y; v) @ (x; y; v) < 0, < 0 for all x > 0, y > 0 and v > 0: Assumption A3. @y @v @ ( (x; y; v)v) Assumption A4. > 0 for all x > 0, y > 0 and v > 0: @v Assumption A2.

2.1

Positive invariance

In the following proposition, we show that the non-negative orthant R4 0 is the positively invariant and there exists a compact set which is positively invariant for model (5)-(8). Proposition 1. Assume that Assumption A1 is satis…ed. Then there exist positive numbers Li , i = 1; 2; 3, such that the compact set = (x; y; v; z) 2 R4 0 : 0

x; y

338

L1 ; 0

v

L2 ; 0

z

L3

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is positively invariant. Proof. First, we prove that the orthant R4 0 is positively invariance for system (5)-(8). We have

x_ jx=0 = s > 0; y_ jy=0 = (x; 0; v)v v_ jv=0 = ky

0 for all x > 0; v

0 for all y

0;

0;

z_ jz=0 = 0: Hence, all the solutions are nonnegative. Next we show that the solutions of system are bounded. Let T1 (t) = x(t) + y(t), then T_1 (t) = (s =s

where

1

x(t); y(t)

= minfd; ag. Hence 0 L1 for all t

dx dx

(x; y; v)v) + (x; y; v)v ay

T1 (t)

0 if x(0) + y(0)

s s 1

1 (x

+ y) = s

for all t

L1 , where L1 =

ay; 1 T1 (t);

0 if T1 (0) s 1

s 1

. It follows that, 0

. Moreover, let T2 (t) = v(t) + rb z(t),

then T_2 (t) = ky where 0

v(t)

2

cv

= minfc; g. Hence 0 L2 and 0

z(t)

b z r T2 (t)

L3 for all t

kL1

2 (v

b + z) = kL1 r

L2 for all t 0 if v(0) + rb z(0)

2 T2 (t);

0 when T2 (0)

L2 . It follows that

L2 , where L2 =

kL1 2

and L3 = rb L2 .

Therefore, x(t); y(t); v(t) and z(t) are all bounded.

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2.2

The equilibria and threshold parameters

At any equilibrium we have

s

dx

(x; y; v)v = 0;

(9)

(x; y; v)v

ay = 0;

(10)

ky

bzv

cv = 0;

(11)

rzv

z = 0:

(12)

From Eq. (12), either z = 0 or z 6= 0. If z = 0, then from Eqs. (9)-(11) we get s

y=

dx

=

a

c v; k

v=

k(s

dx) ac

:

(13)

Substituting from Eq. (13) into Eq. (10) we get:

x;

s a

dx k(s ;

dx) ac

ac v = 0: k

(14)

Eq. (14) has two possible solutions v = 0 or v 6= 0. If v = 0; then from Eqs. (9) and (10), we get x = s=d and y = 0 which leads to the infection-free equilibrium E0 (x0 ; 0; 0; 0) where x0 = s=d. If v 6= 0; then we have x;

s a

dx k(s ;

dx)

ac = 0: k

ac

Let 1 (x)

=

x;

s a

dx k(s ;

dx) ac

ac = 0: k

Then, we have 0 1 (x)

=

@ @x

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Because of Assumptions A2 and A3, we have

0 (x) 1

> 0 which implies that function

1 (x)

is

strictly increasing w.r.t. x. Moreover, 1 (0)

=

1 (x0 )

=

ac = k

s ks 0; ; a ac

ac < 0; k k (x0 ; 0; 0) ac

ac ac = k k

(x0 ; 0; 0)

1 :

k (x0 ; 0; 0) > 1; then there exists a unique x1 2 (0; x0 ) such that 1 (x1 ) = 0. ac d(x0 x1 ) kd(x0 x1 ) Therefore from Eq. (13) we obtain y1 = > 0 and v1 = > 0. It follows a ac k (x0 ; 0; 0) > 1, then there exists a chronic-infection equilibrium without antibody immune that, if ac Therefore, if

response E1 (x1 ; y1 ; v1 ; 0). Let us de…ne the basic reproduction number as: R0 =

k (x0 ; 0; 0) : ac

The parameter R0 determines whether a chronic-infection can be established. The other possibility of Eq. (12) is z 6= 0 which leads to v2 = 2 (x)

=s

2 (x0 )

=

. From Eq. (9) we let

dx

Assumptions A2 and A3 provide that and

r

2

x;

k (x2 ; y2 ; v2 ) ac

dx a

; v2 v2 = 0:

is a decreasing function of x. Clearly,

2 (0)

(x0 ; 0; v2 )v2 < 0. Thus, there exists a unique x2 2 (0; x0 ) such that

It follows from Eqs. (11) and (13) that, y2 = c b

s

d(x0

x2 )

> 0 and z2 =

= s > 0

2 (x2 )

k (x2 ; y2 ; v2 ) ab

= 0. c = b

a k (x2 ; y2 ; v2 ) 1 . Then if > 1 then z2 > 0. Now we De…ne the antibody ac

immune response activation number as: R1 =

k (x2 ; y2 ; v2 ) ; ac

which determines whether a persistent antibody immune response can be established. Hence, z2 c can be rewritten as z2 = (R1 b

1). It follows that, there is a chronic-infection equilibrium with

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antibody immune response E2 (x2 ; y2 ; v2 ; z2 ) i¤ R1 > 1. Clearly from Assumptions A2 and A3, we have

R1 =

2.3

k (x2 ; y2 ; v2 ) k (x0 ; y2 ; v2 ) k (x0 ; 0; 0) < < = R0 : ac ac ac

Global stability analysis

In this section, the global asymptotic stability of the three equilibria of model (5)-(8) will be established by using direct Lyapunov method and applying LaSalle’s invariance principle. Let us de…ne the function H : (0; 1) ! [0; 1) as H(w) = w

1

ln w:

Theorem 1. Let Assumptions A1-A3 be hold true and R0

1; then the infection-free equilibrium

E0 is globally asymptotically stable (GAS). Proof. We construct a Lyapunov functional as:

U0 = x

x0

Z

x

x0

We calculate

dU0 dt

(x0 ; 0; 0) a ab d + y + v + z: ( ; 0; 0) k rk

(15)

along the solutions of model (5)-(8) as:

dU0 =d 1 dt

(x0 ; 0; 0) (x; 0; 0)

(x0

=s 1

(x0 ; 0; 0) (x; 0; 0)

1

x) + x x0

(x; y; v) +

ac k

(x0 ; 0; 0) (x; 0; 0)

(x; y; v) R0 (x; 0; 0)

ac k 1 v

v

ab z rk

ab z: rk

(16)

From Assumptions A2 and A3 we know that (x; y; v) is an increasing function of x and decreasing function of y and v. Then the …rst term of Eq. (16) is less than or equal zero and

(x; y; v) < (x; 0; 0),

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It follows that dU0 dt Therefore, if R0

1, then

(5)-(8) converge to

(x0 ; 0; 0) (x; 0; 0)

s 1

x x0

1

+

ac (R0 k

ab z: rk

1) v

(17)

dU0 dt

0 for all x; y; v; z > 0. We note that the solutions of system n o 0 0 , the largest invariant subset of dU = 0 [44]. From (17), we have dU dt dt = 0

i¤ x = x0 , v = 0 and z = 0. The set

is invariant and for any element belong to

satis…es v = 0

and z = 0. We can see from Eq. (7) that v_ = 0 = ky: dU0 dt

It follows that, y = 0. Hence

= 0 i¤ x = x0 and y = v = z = 0. Using LaSalle’s invariance

principle, we derive that E0 is GAS. Assumption A5 (x; yi ; vi ) (x; y; v)

(x; y; v) (x; yi ; vi )

1

v vi

0; i = 1; 2 for all x; y; v > 0:

Theorem 2. Assume that Assumptions A1-A5 are satis…ed and R1

1 < R0 , then the chronic-

infection equilibrium without antibody immune response E1 is GAS. Proof. De…ne a Lyapunov functional as: U1 = x

x1

Z

x

x1

(x1 ; y1 ; v1 ) d + y1 H ( ; y1 ; v1 )

y y1

a + v1 H k

v v1

+

ab z: rk

Calculating the time derivative of U1 along the trajectories of system (5)-(8), we obtain dU1 = dt

1

+

a 1 k

=

1

(x1 ; y1 ; v1 ) (x; y1 ; v1 )

(s

dx

(x; y; v) v) + 1

y1 y

( (x; y; v)v

ay)

v1 ab z) (ky bzv cv) + (rzv v rk (x1 ; y1 ; v1 ) (x; y; v)v (s dx) + (x1 ; y1 ; v1 ) (x; y1 ; v1 ) (x; y1 ; v1 )

y1 (x; y; v)v + ay1 y

ac v k

ay

v1 ac ab + v1 + v1 z v k k

343

ab z: rk

(18)

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Using the equilibrium conditions for E1 : s = dx1 + ay1 ;

(x1 ; y1 ; v1 )v1 = ay1 =

ac v1 ; k

we obtain dU1 =d 1 dt ay1

(x1 ; y1 ; v1 ) (x; y1 ; v1 )

(x1

y1 (x; y; v)v y (x1 ; y1 ; v1 )v1

ay1

x) + 3ay1 v v1

ay1

ay1

(x1 ; y1 ; v1 ) (x; y; v)v + ay1 (x; y1 ; v1 ) (x; y1 ; v1 )v1

v1 y ab + v1 vy1 k

r

z:

(19)

Collecting terms of Eq. (19) we get (x1 ; y1 ; v1 ) (x; y1 ; v1 )

dU1 = dx1 1 dt + ay1

v v1

(x; y; v)v (x; y1 ; v1 )v1 (x1 ; y1 ; v1 ) (x; y1 ; v1 )

+ ay1 4 +

1

ab v1 k

r

x x1 1+

(x; y1 ; v1 ) (x; y; v)

y1 (x; y; v)v y (x1 ; y1 ; v1 )v1

v1 y vy1

(x; y1 ; v1 ) (x; y; v)

z:

(20)

Eq. (20) can be simpli…ed as: dU1 = dx1 1 dt

(x1 ; y1 ; v1 ) (x; y1 ; v1 )

1

+ ay1 1

(x; y; v) (x; y1 ; v1 )

(x; y1 ; v1 ) (x; y; v)

+ ay1 4

(x1 ; y1 ; v1 ) (x; y1 ; v1 )

+

ab v1 k

r

x x1 v v1

y1 (x; y; v)v y (x1 ; y1 ; v1 )v1

v1 y vy1

(x; y1 ; v1 ) (x; y; v)

z:

(21)

From Assumptions A1 and A5, we get that the …rst and second terms of Eq. (21) is less than or equal zero. Since the geometrical mean is less than or equal to the arithmetical mean, then the third term of Eq. (21) is also less than or equal zero. Now we show that if R1 sgn(x2

1 then v1 x1 ) = sgn(v1

r

= v2 . Let R0 > 1, then we want to show that v2 ) = sgn(y1

344

y2 ) = sgn(R1

1):

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From Assumptions A2-A4, for x1 ; x2 ; y1 ; y2 ; v1 ; v2 > 0, we have ( (x2 ; y2 ; v2 )

(x1 ; y2 ; v2 ))(x2

x1 ) > 0;

(22)

( (x1 ; y1 ; v1 )

(x1 ; y2 ; v1 ))(y2

y1 ) > 0

(23)

( (x1 ; y1 ; v1 )

(x1 ; y1 ; v2 ))(v2

v1 ) > 0;

(24)

( (x2 ; y2 ; v2 )v2

(x2 ; y2 ; v1 )v1 )(v2

v1 ) > 0:

(25)

First, we claim sgn(x2 x1 ) = sgn(v1 v2 ). Suppose this is not true, i.e., sgn(x2 x1 ) = sgn(v2 v1 ). Using the conditions of the equilibria E1 and E2 we have (s

dx2 )

(s

dx1 ) = (x2 ; y2 ; v2 )v2 = a(y2

then sgn(x1 (s

x2 ) = sgn(y2

dx2 )

(s

(x1 ; y1 ; v1 )v1

y1 );

(26)

y1 ). Moreover

dx1 ) = (x2 ; y2 ; v2 )v2

(x1 ; y1 ; v1 )v1

= ( (x2 ; y2 ; v2 )v2

(x2 ; y2 ; v1 )v1 ) + ( (x2 ; y2 ; v1 )v1

+ ( (x1 ; y2 ; v1 )v1

(x1 ; y1 ; v1 )v1 ):

(x1 ; y2 ; v1 )v1 )

Therefore, from inequalities (22)-(26) we get: sgn (x1 which leads to contradiction. Thus, sgn (x2 for E1 we have

k (x1 ;y1 ;v1 ) ac

R1

x2 ) = sgn (x2 x1 ) = sgn (v1

x1 ) ; v2 ) : Using the equilibrium conditions

= 1, then k (x2 ; y2 ; v2 ) k (x1 ; y1 ; v1 ) ac ac k = ( (x2 ; y2 ; v2 ) (x2 ; y2 ; v1 ) + (x2 ; y2 ; v1 ) ac

1=

(x1 ; y2 ; v1 ) + (x1 ; y2 ; v1 )

345

(x1 ; y1 ; v1 )):

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We get sgn(R1 v1

v2 =

r.

1) = sgn(v1

v2 ): Hence, if R0 > 1; then x1 ; y1 ; v1 > 0, and if R1

It follows from the above discussion that

i¤ x = x1 ; y = y1 ; v = v1 and z = 0. So

dU1 dt

1, then

0 for all x; y; v; z > 0 and

dU1 dt

=0

contains a unique point, the equilibrium E1 . Thus, we

prove the global asymptotic stability of the chronic-infection equilibrium without antibody immune response E1 by using LaSalle’s invariance principle. Theorem 3. Let Assumptions A1-A5 be hold true and R1 > 1, then the chronic-infection equilibrium with antibody immune response E2 is GAS. Proof. We construct a Lyapunov functional as follows: U2 = x

x2

Z

x

x2

(x2 ; y2 ; v2 ) d + y2 H ( ; y2 ; v2 )

y y2

a + v2 H k

v v2

+

ab z2 H rk

y2 y

( (x; y; v)v

z z2

:

(27)

Function U2 satis…es: dU2 = dt +

(x2 ; y2 ; v2 ) (x; y2 ; v2 )

1 a 1 k

v2 (ky v

(s

bzv

dx

(x; y; v)v) + 1

cv) +

ab 1 rk

z2 (rzv z

z):

ay) (28)

Applying s = dx2 + ay2 , we get dU2 =d 1 dt ay2

(x2 ; y2 ; v2 ) (x; y2 ; v2 )

x) + ay2

(x2 ; y2 ; v2 ) (x2 ; y2 ; v2 ) + (x; y; v)v (x; y2 ; v2 ) (x; y2 ; v2 )

(x2 ; y2 ; v2 )v2 +

(x2

y2 (x; y; v)v + ay2 y (x2 ; y2 ; v2 )v2

ac ab v2 + v2 z k k

ab z rk

ac v k

ay

v2 v

ab ab z2 v + z2 : k rk

(29)

By using the equilibrium conditions of E2 (x2 ; y2 ; v2 )v2 = ay2 ;

cv2 = ky2

bv2 z2 ;

= rv2 ;

and the following equality cv = cv2

v v = (ky2 v2 v2

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we obtain dU2 =d 1 dt

(x2 ; y2 ; v2 ) (x; y2 ; v2 )

(x2

(x; y; v)v (x; y2 ; v2 )v2

x) + ay2

(x2 ; y2 ; v2 ) (x; y2 ; v2 )

y2 (x; y; v)v y (x2 ; y2 ; v2 )v2

(x2 ; y2 ; v2 ) (x; y2 ; v2 )

1

+ ay2 4

v2 y vy2

v v2

1+

(x; y2 ; v2 ) (x; y; v)

(x; y2 ; v2 ) : (x; y; v)

(30)

We can simplify (30) as: dU2 = dx2 1 dt

(x2 ; y2 ; v2 ) (x; y2 ; v2 )

+ ay2 4

x x2

+ ay2 1 v2 y vy2

y2 (x; y; v)v y (x2 ; y2 ; v2 )v2

(x; y; v) (x; y2 ; v2 ) (x; y2 ; v2 ) : (x; y; v)

(x; y2 ; v2 ) (x; y; v)

v v2 (31)

We note that from assumptions A2, A5 and the relationship between the arithmetical and geometrical means, we have

dU2 dt

0. One can easily see that

dU2 dt

= 0 at E2 . The global asymptotic stability

of the chronic-infection equilibrium with antibody immune response E2 follows from LaSalle’s invariance principle.

3

Conclusion

In this paper, we have proposed a viral infection model with general incidence rate function and antibody immune response. We have derived a set of conditions on the general functional response and have determined two thresholds parameters R0 and R1 to prove the existence and global stability of the model’s equilibria. The global asymptotic stability of the three equilibria, infection-free, chronic-infection without antibody immune response and chronic-infection with antibody immune response has been proven by using direct Lyapunov method and LaSalle’s invariance principle.

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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.

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NEWTON’S METHOD FOR COMPUTING THE FIFTH ROOTS OF p-ADIC NUMBERS Y.H. KIM, H.M. KIM, AND J. CHOI

Abstract We consider Newton’s method to compute the fifth root of a p-adic number in Qp . We have the sufficient conditions for the convergence of Newton’s method and the speed of its convergence. We also calculate the number of iterations to obtain a number of corrected digits in the approximation.

1. Introduction Let p be a prime and Qp be the field of p-adic numbers. The theory of the field of p-adic numbers introduced by Hensel has been related to several areas of mathematics including number theory, analysis and other modern mathematics, and recently to physics. The study of this field has been an important area of research in mathematics([9]). The application of classical methods in numerical analysis to padic numbers and polynomials and the analysis of their convergence in Qp have been a recent development([2-3], [5], [7], [10-11]). Newton’s method is the most often used method to find zeros of polynomials. In [7], the authors applied Newton’s method to compute the cubic root of a p-adic number. In [2-3], the authors also used Newton-Raphson method to compute square and cube roots of p-adic numbers in Qp . Computing the q-th root of a p-adic number is useful in the field of computer science and cryptography, specially when q is a prime. In [6], Kim-Choi give the conditions for the existence of the q-th roots of p-adic numbers in Qp when (p, q) = 1, and also have the condition for the existence the fifth roots including p = q. In this paper, we use Newton’s method to compute the fifth root of a p-adic number in Qp . We have the sufficient conditions for the convergence of Newton’s method and the speed of its convergence. We also calculate the number of iterations to obtain a number of corrected digits in the approximation. 2010 Mathematics Subject Classification: 11E95, 26E30, 65H04 Key words and phrases: Newton’s method, p-adic roots Correspondence should be addressed to Jongsung Choi, [email protected]. The present research has been conducted by the Research Grant of Kwangwoon University in 2014. 1

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2. Preliminaries The following definitions and results are needed for our discussion. See [4] and [8] for details. Definition 1. Let p ∈ N be a prime number and x ∈ Q (x 6= 0). The p-adic order of x, ordp x, is defined by  the highest power of p which divides x, if x ∈ Z, ordp x = ordp a − ordp b, if x = ab , a, b ∈ Z, b 6= 0. Consider a map | · |p : Q → R+ as follows. Definition 2. Let p ∈ N be a prime number and x ∈ Q. The p-adic norm | · |p of x is defined by  −ord x p p , if x 6= 0, |x|p = 0, if x = 0. The field of p-adic numbers Qp is the completion of Q with respect to the p-adic norm |·|p of Definition 2. The elements of Qp are equivalence classes of Cauchy sequences in Q with respect to the extension of the p-adic norm defined by |a|p = lim |an |p , n→∞

where {an } is a Cauchy sequence in Q representing a ∈ Qp . Theorem 1. Every equivalence class a in Qp satisfying |a|p ≤ 1 has exactly one representative Cauchy sequence {ai } such that (1) ai ∈ Z, 0 ≤ ai < pi for i = 1, 2, . . ., (2) ai ≡ ai+1 (mod pi ) for i = 1, 2, . . . . From this, every p-adic number a ∈ Qp has a unique representation a=

∞ X

an p n ,

n=−m

where a−m 6= 0 and an ∈ {0, 1, 2, . . . , p − 1} for n ≥ −m. We represent the given p-adic number a as a fraction in the base p as follows: a = . . . an . . . a2 a1 a0 .a−1 . . . a−m . This representation is called the canonical p-adic expansion of a. P∞ i Definition 3. Let Zp = {a ∈ QP p| a = i=0 ai p } be the set of p-adic ∞ × i integers and Zp = {a ∈ Qp | a = i=0 ai p , a0 6= 0} be the set of p-adic units.

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3

From Definition 3, it is easy to see that Zp = {a ∈ Qp | |a|p ≤ 1} and Z× p = {a ∈ Qp | |a|p = 1}. Hence the following theorem follows. Theorem 2. Let a be a p-adic number of norm p−n . Then a = pn u for some u ∈ Z× p. From now, we discuss the conditions for the existence of p-adic roots. Definition 4. A p-adic number x ∈ Qp is said to be a q-th root of a ∈ Qp of order k ∈ N if and only if xq ≡ a (mod pk ). When q = 5, the q-th root of a ∈ Qp is called the fifth root of a. The following lemmata are essential for our discussions([4]). Lemma 3. Let a, b ∈ Qp . Then a and b are congruent modulo pk and write a ≡ b (mod pk ) if and only if |a − b|p ≤ 1/pk . Lemma 4. Let a, b ∈ Qp . If |a − b|p < |b|p , then |a|p = |b|p . The next theorem is the basis for the existence of p-adic roots([8]). Theorem 5. (Hensel’s lemma) Let F (x) = c0 + c1 x + · · · + cn xn be a polynomial whose coefficients are p-adic integers. Let F 0 (x) = c1 + c2 x + 3c3 x2 + · · · + ncn xn be the derivative of F (x). Let a0 be a p-adic integer such that F (a0 ) ≡ 0 (mod p) and F 0 (a0 ) 6≡ 0 (mod p). Then there exists a unique p-adic integer a such that F (a) = 0 and

a ≡ a0 (mod p).

The following theorem follows from Theorem 5, and provides the condition between p-adic numbers and congruence([4]). Theorem 6. A polynomial with integer coefficients has a root in Zp if and only if it has an integer root modulo pk for any k ≥ 1. Some results of the existence of square roots of p-adic numbers are obtained from Theorem 6([4]). In [6], we have the conditions for the existence of the fifth roots of p-adic numbers in Qp as followings. Theorem 7. A rational integer a not divisible by p has a fifth root in Zp (p 6= 5) if and only if a is a fifth residue modulo p. From Theorem 7, we have the following theorem([6]). Theorem 8. Let p be a prime number. Then we have: (1) If p 6= 5, then a = pordp a u ∈ Qp for some u ∈ Z× p has a fifth root in Qp if and only if ordp a = 5m for m ∈ Z and u = v 5 for some unit v ∈ Z× p. (2) If p = 5, then a = 5ord5 a u ∈ Q5 for some u ∈ Z× 5 has a fifth root in Q5 if and only if ord5 a = 5m for m ∈ Z and u ≡ 1 (mod 25) or u ≡ k (mod 5) for some k (2 ≤ k ≤ 4).

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Y.H. KIM, H.M. KIM, AND J. CHOI

3. Newton’s Method Newton’s method is a well known numerical method to find zeros of a polynomial f (x) in R([1]). The iterative formula for this method is given by f (xn ) xn+1 = xn − 0 , n = 0, 1, 2, . . . . (3.1) f (xn ) To seek the fifth root of a is to find the zero of f (x) = x5 − a. The iteration (3.1) for Newton’s method becomes the recurrence relation xn+1 =

4x5n + a , 5x4n

n = 0, 1, 2, . . . .

(3.2)

Like for real numbers, we can show that Newton’s method also converges quadratically for convergence. Let a(6= 0) ∈ Qp be a p-adic number such that |a|p = p−ordp a = p−5m ,

m ∈ Z.

The following theorem is the result when p 6= 5. Theorem 9. Let p 6= 5 and {xn } be the sequence of p-adic numbers obtained from the Newton’s iteration (3.2). If x0 is a fifth root of a of order r with |x0 |p = p−m and r > 5m, then (1) |xn |p = p−m , n = 1, 2, . . . , n n (2) x5n ≡ a (mod p2 r−5m(2 −1) ), (3) {xn } converges to the fifth root of a. Proof. We will prove (1) and (2) by induction. (i) First, we prove it when p > 5. Let n = 1. By assumption, we have x50 = a + bpr

(0 < b < p).

(3.3)

From (3.2), (3.3) and Lemma 4, we have |x1 |p =

|4x50 + a|p |5a + 4bpr |p max{|5a|p , |4bpr |p } = = = p−m . (3.4) 4 4 4 |5x0 |p |5x0 |p |5x0 |p

Also by (3.2), we have x51 − a =

(x50 − a)2 3 10 2 5 (1024x15 0 + 203ax0 + 22a x0 + a ). 3125x20 0

(3.5)

To calculate the p-adic norm of x51 − a, we let h(x) = 1024x15 + 203ax10 + 22a2 x5 + a3 .

(3.6)

From (3.3), we have h(x0 ) = 1250a3 + 3500a2 bpr + 3275ab2 p2r + 1024b3 p3r .

356

(3.7)

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Using the strong triangle inequality, we have from (3.7) that |h(x0 )|p  ≤ max |2 · 54 a3 |p , |22 53 7a2 bpr |p , |52 131ab2 p2r |p , |210 b3 p3r |p  = max p−15m , p−10m−r , p−5m−2r , p−3r

(3.8)

= p−15m . Also the p-adic norm of the denominator of the right hand of (3.5) is −20m 5 20 . |3125x20 0 |p = |5 x0 |p = p

(3.9)

Since x0 is a fifth root of a of order r, we have |(x50 − a)2 |p = p−2r .

(3.10)

By (3.5), (3.8), (3.9) and (3.10), we have |x51 − a|p ≤ p5m−2r . By Lemma 3, x51 − a ≡ 0 (mod p2r−5m ). Hence (1) and (2) is true when n = 1. Now assume that |xn−1 |p = p−m , (3.11) n−1 r−5m(2n−1 −1)

x5n−1 = a (mod p2

),

(3.12)

and so n−1

n−1 −1)

x5n−1 = a + bp2 r−5m(2 From (3.2), (3.11) and (3.13), we have

(0 < b < p).

n−1

|4x5n−1 + a|p |5a + 4bp2 r−5m(2 |xn |p = = |5x4n−1 |p |5x4n−1 |p n−1

n−1 −1)

max{|5a|p , |4bp2 r−5m(2 = |5x4n−1 |p

|p }

n−1 −1)

=p

(3.13)

|p (3.14)

−m

.

Thus (1) is proved by (3.4), (3.11) and (3.14). Also from (3.2), it follows that (x5 − a)2 x5n − a = n−1 h(xn−1 ). (3.15) 55 x20 n−1 Let Q = p2

n−1 r−5m(2n−1 −1)

for simplicity. From (3.13),

h(xn−1 ) = 2 · 54 a3 + 22 · 53 · 7a2 bQ + 52 · 131ab2 Q2 + 210 b3 Q3 . (3.16) Since r > 5m, the p-adic norm of h(xn−1 ) in (3.16) is n−1 (r−5m)

|h(xn−1 )|p ≤ max{p−15m , p−15m−2 p

−15m−2n (r−5m)

,p

,

−15m−3·2n−1 (r−5m)

}

(3.17)

= p−15m .

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Y.H. KIM, H.M. KIM, AND J. CHOI

Since xn−1 is a fifth root of a of order 2n−1 r − 5m(2n−1 − 1), we have from (3.12), (3.15) and (3.17) that n r+5m(2n −1)

|x5n − a|p ≤ p−2

.

n r−5m(2n −1)

By Lemma 3, we have x5n − a ≡ 0 (mod p2 for all n ∈ N.

). Thus (2) is true

(ii) When p < 5, there are two cases, p = 3 and p = 2. The proof is the same with (i) when the first case p = 3, because 3 is no factor of any coefficients of terms of h(x0 ) in (3.7). It means that |h(x0 )|p ≤ p−15m , and so x51 ≡ a (mod p2r−5m ). By assuming x5n−1 ≡ a n−1 n−1 n n (mod p2 r−5m(2 −1) ), we have x5n ≡ a (mod p2 r−5m(2 −1) ) using the same process of (i). Moreover we can check easily |xn |3 = 3−m by induction. The other case is p = 2. Let n = 1, |x1 |p = p−m is obtained easily from (3.4). And we have x51 − a =

(x50 − a)2 h(x0 ), 3125x20 0

(3.18)

where h(x) is the polynomial in (3.6). Since r > 5m, we have |h(x0 )|p ≤ max{p−15m−1 , p−10m−r−2 , p−5m−2r , p−3r−10 } ≤ p−15m . (3.19) In (3.18), we have −20m |3125x20 , (3.20) 0 |p = p and, by assumption, |(x50 − a)2 |p = p−2r .

(3.21) −2r+5m

− a|p ≤ p , and so x51 ≡ Also (3.19), (3.20) and (3.21) imply a (mod p2r−5m ). Thus (1) and (2) are true when n = 1 if p = 2. n−1 n−1 Assume that |xn−1 |p = p−m and x5n−1 ≡ a (mod p2 r−5m(2 −1) ). That is, n−1 n−1 x5n−1 = a + bp2 r−5m(2 −1) (0 < b < p). (3.22) |x51

It follows (3.15) and (3.16), and so we have n−1 (r−5m)

|h(xn−1 )|p ≤ max{p−15m−1 , p−15m−2−2 p

−15m−2n (r−5m)

, n−1 (r−5m)

, p−15m−10−3·2

}

(3.23)

≤ p−15m . By (3.15), (3.17), (3.20) and (3.23), we have n r+5m(2n −1)

|x5n − a|p ≤ p−2

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n

Hence we have that for all n ∈ N, x5n ≡ a (mod p2 r−5m(2 −1) ). We note that |xn |2 = 2−m is obtained easily from (3.14). So we complete the proof of (1) and (2). From (2), we have |x5n − a|p ≤ p−2

n r+5m(2n −1)

(3.24)

for each prime p(6= 5). (3) follows immediately from the inequality (3.24) as n → ∞.  When p = 5, we have the following theorem. Theorem 10. Let p = 5 and {xn } be the sequence of p-adic numbers obtained from the Newton’s iteration (3.2). If x0 is a fifth root of a of order r with |x0 |p = p−m and r > 5m + 1, then (1) |xn |p = p−m , n = 1, 2, . . . , n n (2) x5n ≡ a (mod p2 r−(5m+1)(2 −1) ), (3) {xn } converges to the fifth root of a. Proof. (1) and (2) will be proved by induction. Let n = 1. By assumption x50 ≡ a (mod pr ), and from (3.2) and Lemma 4, we have |x1 |p =

|5a + 4bpr |p max{|5a|p , |4bpr |p } p−5m−1 = p−m . = = |5x40 |p |5x40 |p p−4m−1

By calculating the p-adic norms of h(x0 ) in (3.7), we have |h(x0 )|p ≤ max{p−15m−4 , p−10m−r−3 , p−5m−2r−2 , p−3r } = p−15m−4 , −20m−5 . Thus since r > 5m + 1. Also we have |3125x20 0 |p = p

|x51 − a|p ≤ p−2r+5m+1 , and so x51 ≡ a (mod p2r−(5m+1) ) by Lemma 3. Hence it is true when n = 1. Now we assume that |xn−1 |p = p−m and n−1 r−(5m+1)(2n−1 −1)

x5n−1 ≡ a (mod p2

).

In the similar manner as (3.14), (3.16) and (3.17), we have n−1

n−1 −1)

|4x5n−1 + a|p |5a + 4bp2 r−(5m+1)(2 |xn |p = = |5x4n−1 |p |5x4n−1 |p n−1

n−1 −1)

max{|5a|p , |4bp2 r−(5m+1)(2 = |5x4n−1 |p

359

|p }

=

|p

p−5m−1 = p−m p−4m−1

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and n−1 [r−(5m+1)]

|h(xn−1 )|p ≤ max{p−15m−4 , p−15m−4−2 n [r−(5m+1)]

p−15m−4−2 =p

−15m−4

, p−15m−3−3·2

n−1 [r−(5m+1)]

}

.

And so we have |x5n − a|p ≤ p−2

n r+(5m+1)(2n −1)

.

(3.25)

It follows that (1) and (2) are true for all n ∈ N. (3) follows from the inequality (3.25) as n → ∞.



To determine the rate of convergence of the sequence {xn } given by (3.2), we consider the sequence {en } defined by en = xn+1 − xn ,

∀n ∈ N.

(3.26)

From Theorem 9 and Theorem 10, we obtain the following theorem. Theorem 11. If x0 is the fifth root of a of order r, then the sequence {en } in (3.26) is en ≡ 0 (mod pαn ), where  2n r − 5m · 2n + m, if p 6= 5, αn = n n 2 r − (5m + 1) · 2 + m, if p = 5. Proof. (i) First, let p 6= 5. Then, from the Newton’s iteration formula (3.2), we have 1 en = xn+1 − xn = 4 (a − x5n ), ∀n ∈ N. (3.27) 5xn By computing the p-adic norms of each side of the equation (3.27), we have from Theorem 8 that 1 n n |en |p = |xn+1 − xn |p = 4 · |a − xn |p ≤ p−2 r+5m·2 −m . 5x n p

Hence en ≡ 0 (mod pαn ) by Lemma 3. (ii) Let p = 5. By a similar way as (i), we have from Theorem 9 that 1 n n |en |p = 4 · |a − xn |p ≤ p−2 r+(5m+1)·2 −m . 5x n p

αn

Hence en ≡ 0 (mod p ) by Lemma 3. This completes the proof.



From Theorem 11, we have that the rate of convergence of the sequence {xn } is of order αn . Thus the number of correct digits in the approximation increases by αn for every iteration. We can compute the number of iterations to obtain certain finite digits. From Theorem 9 and Theorem 10, we have the following corollary.

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Corollary 12. (1) For p 6= 5, let {xn } be the sequence of approximation in Theorem 9. Then the number of iterations to obtain at least M correct digits is "
# −4m ln Mr−5m n= . (3.28) ln 2 (2) Let p = 5 and {xn } be the sequence of approximation in Theorem 10. Then the number of iterations to obtain at least M correct digits is   M −(4m+1)   ln r−(5m+1) . n= (3.29) ln 2 Proof. (1) Since we need M correct digits in the approximation, we must set the order to M + m to find the number of iterations with M correct digits. That is, 2n r − 5m(2n − 1) = M + m.

(3.30)

From (3.30), we have M − 4m . r − 5m Since {xn } converges to the fifth root of a by Theorem 8 (3) and r > 5m, we have the equation (3.28). (2) As in the proof of (1), we set 2n =

2n r − (5m + 1)(2n − 1) = M + m.

(3.31)

From (3.31), we have 2n =

M − 4m − 1 . r − (5m + 1)

Since r > 5m + 1, the result follows from (3.32).

(3.32) 

The numbers in (3.28) and (3.29) are sufficient numbers of iterations to provide at least M correct digits in the approximation. References [1] R. L. Burden, J. D. Faires, Numerical analysis (5th ed.), PWS Publishing, 1993. [2] P. S. Ignacio, On the square and cube roots of p-adic numbers, J. Math. Comput. Sci., 3 (2013), No. 4, 993–1003. [3] P. S. Ignacio, J. M. Addawe, W. V. Alangui, J. A. Nable, Computation of square and cube roots of p-adic numbers via Newton-Raphson method, J. Math. Research, 5 (2013), No. 2, 31–38. [4] S. Katok, p-Adic analysis compared with real, American Math. Soc., 2007

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[5] M. Keicies, T.Zerzaihi, General approach of the root of a p-adic number, Filomat, 27 (2013), No. 3, 431–436. [6] Y.-H. Kim, J. Choi, On the existence of p-adic roots, accepted in J. of Chungcheong Math. Soc. 28 (2015), No. 2. [7] M. Knapp, C. Xenophontos, Numerical analysis meets number theory using rootfinding method to calculate inverses mod pn , Appl. Anal. Discrete Math., 4 (2010), 23–31. [8] N. Koblitz, p-Adic numbers, p-adic analysis and zeta functions (2nd ed.), Springer-Verlag, 1984. [9] V. S. Vladimirov, I. V. Volvich, E. I. Zelenov, p-Adic analysis and mathematical physics, Norld Scientific, 1994. [10] T. Zerzaihi, M. Kecies, Computation of the cubic root of a p-adic number, J. Math. Research, 3 (2011), No. 3, 40–47. [11] T. Zerzaihi, M. Kecies, M. Knapp, Hensel codes of square roots of p-adic numbers, Appl. Anal. Discrete Math., 4 (2010), 32–44. Young-Hee Kim. Division of General Education-Mathematics, Kwangwoon University, Seoul 139-701, Republic of Korea, Hyun-Mee Kim. Mathematics Education Major, Graduate School of Education, Kookmin University, Seoul 136-702, Republic of Korea, Jongsung Choi. Division of General Education-Mathematics, Kwangwoon University, Seoul 139-701, Republic of Korea,

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Solution of the Ulam stability problem for Euler-Lagrange (α, β; k)-quadratic mappings S.A. Mohiuddine1, John Michael Rassias2 and Abdullah Alotaibi1 1

Department of Mathematics, Faculty of Science, King Abdulaziz University, P.O. Box 80203, Jeddah 21589, Saudi Arabia 2 National and Capodistrian University of Athens, Pedagogical Department, Mathematics and Informatics, 4, Agamemnonos Str., Aghia Paraskevi, Attikis 15342, Greece Email: 1 [email protected]; 2 [email protected]; 1 [email protected]

Abstract. In 1940 S. M. Ulam proposed at the University of Wisconsin the problem: “Give conditions in order for a linear mapping near an approximately linear mapping to exist”. In 1982-2013, the second author solved the above Ulam problem for a variety of quadratic mappings. Interesting stability results have been achieved by S. A. Mohiuddine et al., since 2009. In this paper, we solve the Ulam stability problem for Euler-Lagrange (α, β; k) quadratic mapping. The other authors of this research area have established important results also on functional inequalities. Keywords and phrases: Quartic functional equations and inequalities; Various normed spaces; Ulam stability. AMS subject classification (2000): 39B.

1. Introduction In 1940 S. M. Ulam [36] proposed the famous “Ulam stability problem”, which was solved by D. H. Hyers [4], in 1941, for additive mappings. In 1950 T. Aoki [3] solved this Ulam problem for weaker additive mappings. In 1978 Th. M. Rassias [33] generalized the theorem of Hyers for linear mappings. In 1982-1999, J. M. Rassias ( [23–30]) generalized this problem. For more detail of Ulam stability problem, we refer to [5, 6, 8–11, 19, 20, 32, 34] and references therein. In 1992, the second author [23, 24] introduced the term “Euler-Lagrange functional equation” and “Euler-Lagrange quadratic mappings”, of satisfying   Q(x + y) + Q(x − y) = 2 Q(x) + Q(y)

(1.1)

and then solved the Ulam stability problem of the Euler-Lagrange quadratic functional equation (1.1). In 1996, J. M. Rassias [30] established the Ulam stablity of the general Euler-Lagrange quadratic functional equation   Q(αx + βy) + Q(βx − αy) = (α2 + β 2 ) Q(x) + Q(y) . (1.2) In 2009-2014, S. A. Mohiuddine et al. ( [1,2,12–18]) solved this problem in several normed spaces. In

2008-2012 J. M. Rassias et al. ( [21, 22, 31,37]) solved the generalized Ulam problem via various methods. In 2010, M. E. Gordji et al [7] established Ulam stabilities on Banach algebras. Also J. R¨atz [35] results are interesting on orthogonal mappings. In this paper, we solve the Ulam stability problem for the Euler-Lagrange (α, β; k) quadratic mapping satisfying   kQ(αx + βy) + Q(kβx − αy) = (α2 + kβ 2 ) kQ(x) + Q(y) . 363

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(1.3)

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Let us note that Q(x) = |x|2 satisfies equation (1.3) because the following Euler-Lagrange quadratic identity   k|αx + βy|2 + |kβx − αy|2 = (α2 + kβ 2 ) k|x|2 + |y|2 (1.4) holds with any fixed reals α, β and k.

Definition 1.1. Let X be a normed linear space and let Y be a real complete normed linear space. Then a non-linear mapping Q : X → Y is called Euler-Lagrange quadratic if equation (1.3) holds for all 2-dimensional vectors (x, y) ∈ X 2 , and any fixed reals α, β and k. We note that Q may be called quadratic because the above Euler-Lagrange identity (1.4) holds and because the functional equation Q(mn x) = (mn )2 Q(x)

(1.5)

m = α2 + kβ 2 .

(1.6)

holds for all x ∈ X, all n ∈ N :

Assume m ∈ R − {0, 1} and k ∈ R − {−1, 0}. In fact, substitution of x = y = 0 in equation (1.3) yields (k + 1)(1 − m)Q(0) = 0, or Q(0) = 0,

m 6= 1 (and k 6= −1).

(1.7)

Substituting x = x, y = 0 in (1.3), one gets that kQ(αx) + Q(kβx) = kmQ(x) + mQ(0),

(1.8)

or

1 m Q(kβx) = mQ(x) + Q(0), k k holds for all x ∈ X, and any fixed real k 6= 0. Employing (1.7), we obtain from (1.8) that Q(αx) +

Q(αx) + Q(kβx) = kmQ(x).

(1.9)

(1.10)

Moreover, substitution x → αx, y = kβx in (1.3), we find that

or

  kQ(mx) + Q(0) = m kQ(αx) + Q(kβx) , kQ(αx) + Q(kβx) = km−1 Q(mx) +

1 Q(0), m

(1.11)

or

1 1 (1.12) Q(kβx) = m−1 Q(mx) + Q(0) k km holds for all x ∈ X, and any fixed reals k 6= 0, m 6= 0. Functional Equations (1.8) and (1.11), or (1.9) and (1.12) yield 1 km−1 Q(mx) + Q(0) = kmQ(x) + mQ(0), m or     1 −2 km Q(x) − m Q(mx) = − m Q(0), m Q(αx) +

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or

 1 − m2 Q(0), m   1 1 − m2 Q(0). Q(x) − m−2 Q(mx) = k m2

  km Q(x) − m2 Q(mx) =

or



(1.13)

Employing (1.7), one gets Q(x) = m−2 Q(mx),

(1.14)

Q(mx) = m2 Q(x)

(1.15)

or

Replaying x → mx in (1.15), we find Q(m2 x) = m2 Q(mx), or Q(m2 x) = m4 Q(x)

(1.16)

Then by induction on n ∈ N with x → mn−1 x yields equation (1.5). Definition 1.2. Let X be a normed linear space and let Y be a real complete normed linear space. Then ¯ : X → Y, a 2-dimensional quadratic weighted mean if we call the non-linear mapping Q kQ(αx) + Q(kβx) ¯ Q(x) = km

(1.17)

holds for all x ∈ X and any fixed reals k, m 6= 0. Let us note that from (1.8) and (1.17), one get kmQ(x) + mQ(o) ¯ Q(x) = , km or

1 ¯ = Q(x) + Q(o), Q(x) k for all x ∈ X, and any fixed real k = 6 0. From (1.7) and (1.18), we obtain

(1.18)

¯ Q(x) = Q(x),

(1.19)

for all x ∈ X.

2. Stability for Euler-Lagrange quadratic mappings Let us introduce the Euler-Lagrange (α, β; k) quadratic functional inequality

 

kf(αx + βy) + f(kβx − αy) − (α2 + kβ 2 ) kf(x) + f(y) ≤ c,

(2.1)

for all 2-dimensional vectors (x, y) ∈ X 2 and any fixed reals α, β and k as well as m = α2 + kβ 2 , with m ∈ R − {0, 1} (k ∈ R − {−1, 0}), and c(:= constant inde of x, y) ≥ 0. Then we prove the following theorem. Theorem 2.1. Let X be a normed linear space and let Y be a real complete normed linear space. Let us denote, kf(αx) + f(kβx) f¯(x) = (2.2) km 365

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holds for all x ∈ X and any fixed reals k, m 6= 0. Also let us assume m : |m| > 1. Then the limit Q(x) = lim m−2n f(mn x),

(2.3)

n→∞

exists for all x ∈ X, all n ∈ N, and any fixed real m : |m| > 1 and Q : X → Y is the unique quadratic mapping satisfying functional equation (1.3) such that

f(x) − Q(x) ≤ c3 =

where

c2 = m2 c1 =

c2 , |m| > 1, m2 − 1

(2.4)


|k + 1| 1 + |m| + |1 + m| c, |k| |k + 1| c2 c3 = 2 . m −1

Moreover , identity Q(x) = m−2n Q(mn x)

(2.5)

holds for all x ∈ X all n ∈ N, and any fixed reals: α, β; k, m : |m| > 1 with m ∈ R − {0, 1}, (k ∈ R − {−1, 0}). Proof of Existence in Theorem 2.1. In fact, substitution of x = y = 0 in equality (2.1) yields

|k + 1| |1 − m| f(0) ≤ c, or



f(0) ≤

c , k 6= −1, m 6= 1. |k + 1| |1 − m|

(2.6)

Substituting x = x, y = 0 in (2.1), one gets that

kmf(x) − [kf(αx) + f(kβx)] + mf(0) ≤ c, or



f(x) − f¯(x) + 1 f(0) ≤ c , k 6= 0, m 6= 0, |m| > 1 k |k||m|

(2.7)

from (2.2). Moreover substitution x → αx, y = kβx in (2.1), we find that

kf(mx) + f(0) − m[kf(αx) + f(kβx)] ≤ c, or

or



kf(αx) + f(kβx) − km−1 f(mx) − 1 f(0) ≤ c , m |m|

f(x) ¯ − m−2 f(mx) −

1 c f(0) ≤ , 2 km |k| m2

(2.8)

Functional inequalities (2.6),(2.7),(2.8) and triangle inequality yields

f(x) − m−2 f(mx)



1 1 ≤ f(x) − f¯(x) + f(0) + f¯(x) − m−2 f(mx) − f(0) 2 k km

1

1

+ f(0) − f(0) km2 k

c c |1 − m2 |

f(0) ≤ + + 2 2 |k| |m| |k| m |k|m 2

1 + |m| |1 − m |

= c + f(0) |k| m2 |k|m2 366

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≤ = = or where

 1 + |m| |1 − m2 | 1 + c |k| m2 |k|m2 |k + 1||1 − m|   |1 + m| 1 + |m| + c |k| m2 |k||k + 1|m2
|k + 1| 1 + |m| + |1 + m| c1 = c, |k||k + 1|m2





f(x) − m−2 f(mx) ≤ c1 = c2 , m2
|k + 1| 1 + |m| + |1 + m| 2 c, c2 = m c1 = |k||k + 1|

(2.9)

(2.10)

holds for fixed k, m 6= 0, m 6= 1, m > 1. Replacing x → mx in (2.9) and then multiplying by m−2 , we find

−2

m f(mx) − m−4 f(m2 x) ≤ m−2 c1 , m 6= 0 (2.11) From (2.9) and (2.11), one gets





f(x) − m−4 f(m2 x) ≤ f(x) − m−2 f(mx) + m−2 f(mx) − m−4 f(m2 x) ≤ 1 + m−2 c1 , or



f(x) − m−4 f(m2 x) ≤ (1 + m−2 )c1 , m 6= 0.

(2.12)

Employing (2.9) and (2.12) without induction, we obtain





f(x) − m−2n f(mn x) ≤ f(x) − m−2 f(mx) + m−2 f(mx) − m−4 f(m2 x) + · · ·

+km−2(n−1)f(mn−1 x) − m−2n f(mn x)
≤ 1 + m−2 + · · · + m−2(n−1) c1 , or

−2n


m2

f(x) − m−2n f(mn x) ≤ 1 − m c = 1 − m−2n c1 , 1 −2 2 1−m m −1 or the general inequality:



f(x) − m−2n f(mn x) ≤

where |m| > 1, c2 = m2 c1 . Claim now that the sequence



m2


1 1 − m−2n c2 , −1

 fn (x) , fn (x) = m−2n f(mn x)

(2.13)

(2.14)

(2.15)

converges. Note that from the general inequality (2.14) and the completeness of Y , one proves that the above sequence (2.15) is a Cauchy sequence. In fact, if i > j > 0, then



fi (x) − fj (x) = m−2i f(mi x) − m−2j f(mj x)

= m−2j m−2(i−j)f(mi x) − f(mj x)

= m−2j f(mj x) − m−2(i−j)f(mi−j · mj x) 1 ≤ m−2j · 2 (1 − m−2(i−j))c2 , m −1 or

fi (x) − fj (x) ≤

m2

1 (m−2j − m−2i )c2 , |m| > 1, −1

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or 0≤ or

lim fi (x) − fj (x) ≤ 0,

i>j→∞

lim fi (x) − fj (x) = 0,

(2.17)

i>j→∞

 completing the proof that the sequence fn (x) converges. Hence Q = Q(x) is well-defined via the formula (2.3). This means that the limit (2.3) exists for all x ∈ X. In addition claim that mapping Q satisfies the functional equation (1.3) for all vectors (x, y) ∈ X 2 . In fact, it is clear from functional inequality (2.1) and the limit (2.3) that inequality

   

k lim m−2n f mn (αx + βy) + lim m−2n f mn (kβx − αy) n→∞ n→∞  

−(α2 + kβ 2 ) k lim m−2n f(mn x) + lim m−2n f(mn y) n→∞

≤

c( lim m−2n ) = 0, n→0

n→∞

|m| > 1,

(2.18)

or

kQ(αx + βy) + Q(kβx − αy) − (α2 + kβ 2 )[kQ(x) + Q(y) = 0,

or mapping Q satisfies the functional equation (1.3) for all x, y ∈ X, and |m| > 1. Thus Q is a 2dimensional quadratic mapping. It is now clear from general inequality (2.14), n → ∞, and the formula (2.3) that inequality (2.4) holds in X, completing the existence proof of this Theorem 2.1. Proof of Uniqueness in Theorem 2.1. Let Q0 : X → Y be another 2-dimensional quadratic mapping satisfying equation (1.3), such that  

f(x) − Q0 (x) ≤ c3 = c2 , (2.4)0 m2 − 1

for all x ∈ X, and any fixed real m : |m| > 1. To prove the above-mentioned uniqueness employ (2.5) for Q and Q0 , as well, so that Q0 (x) = m−2n Q0 (mn x)

(2.5)0

holds for all x ∈ X, all n ∈ N, and any fixed real m : |m| > 1. Moreover, the triangle inequality and functional inequalities (2.4)-(2.4)0 yield

or





Q(mn x) − Q0 (mn x) ≤ Q(mn x) − f(mn x) + f(mn x) − Q0 (mn x) ,

Q(mn x) − Q0 (mn x) ≤ 2c3 ,

(2.19)

for all x ∈ X, all n ∈ N, and any fixed real m : |m| > 1. Then from (2.5)-(2.5)0, and (2.19), one proves that

or



Q(x) − Q0 (x) = m−2n Q(mn x) − m−2n Q0 (mn x) ,

Q(x) − Q0 (x) ≤ 2m−2n c3 ,

(2.20)

holds for all x ∈ X, all n ∈ N, and any fixed real m : |m| > 1. Therefore from (2.20), and n → ∞, one establishes  

0 ≤ lim Q(x) − Q0 (x) ≤ 2 lim m−2n c3 = 0, |m| > 1, n→∞

n→∞

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or

Q(x) − Q0 (x) = 0,

or

Q(x) = Q0 (x),

|m| > 1,

(2.21)

for all x ∈ X, completing the proof of uniqueness and thus the stability of Theorem 2.1. Theorem 2.2. Let X be a normed linear space and let Y be a real complete normed linear space. Let us denote      1 k m ¯ f(x) = m2 f¯(m−1 x) = (2.2)0 kf αx +f βx k m m holds for all x ∈ X and any fixed reals k, m 6= 0. Also let us assume |m| < 1. Then the limit Q(x) = lim m2n f(m−n x),

(2.3)0

n→∞

exists for all x ∈ X, all n ∈ N, and any fixed real m : |m| < 1, and Q : X → Y is the unique quadratic mapping satisfying functional equation (2.3)0 , such that

f(x) − Q(x) ≤ c4 = c1 . 1 − m2 Moreover, identity Q(x) = m2n Q(m−n x) (2.5)0 holds for all x ∈ X, n ∈ N and |m| < 1, m 6= 0. From (2.7) with x → m−1 x(m 6= 0, |m| < 1) and multiplying by m2 , one find

2

2 ¯ + m f(0)

m f(m−1 x) − f(x)

≤ |m| c, (2.22)

|k| k where

   ¯ = m2 f¯(m−1 x) = m kf m−1 αx
+ f k βx , m 6= 0, |m| < 1. f(x) k m

(2.23)

From (2.8) with x → m−1 x (m 6= 0, |m| < 1), one obtains



c

f¯(m−1 x) − m−2 f(x) − 1 f(0) ≤ .

2 km |k| m2

Multiplying by m2 , we get



¯

f(x) − f(x) − 1 f(0) ≤ c .

|k| k

(2.24)

Functional inequalities (2.6),(2.23),(2.24) and triangle inequality yield





¯

m2 ¯ + 1 f(0) 2 −1

f(x) − m2 f(m−1 x) ≤ f(x) − f(x)

+ f(x) − m f(m x) − f(0)



k k

2

m

1

+

k f(0) − k f(0)

c |m| |m2 − 1|

f(0) ≤ + c+ |k| |k| |k| = ≤ =

1 + |m| |1 − m2 | c+ kf(0)k |k| |k|   1 + |m| |1 + m| c + |k| |k||k + 1|
|k + 1| 1 + |m| + |1 + m| c = c2 , |k||k + 1| 369

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or

f(x) − m2 f(m−1 x) ≤ c2 ,

where c2 =


|k + 1| 1 + |m| + |1 + m| c, |k||k + 1|

(2.25)

|m| < 1, k 6= 0, k 6= −1, m 6= 0.

Replacing x → m−1 x in (2.25) and multiplying by m2 , we get m2 f(m−1 x) − m4 f(m−2 x)k ≤ m2 c2 ,

(2.26)

From (2.25)-(2.26), one finds





f(x) − m4 f(m−2 x) ≤ f(x) − m2 f(m−1 x) + m2 f(m−1 x) − m4 f(m−2 x) ≤ (1 + m2 )c2 , or




f(x) − m4 f(m−2 x) ≤ 1 + m2 c2 , m 6= 0.

(2.27)

Employing (2.25) and (2.27), without induction, we get





f(x) − m2n f(m−n x) ≤ f(x) − m2 f(m−1 x) + m2 f(m−1 x) − m4 f(m−2 x) + · · ·

+ m2(n−1)f(m−(n−1) x) − m2n f(m−n x)
≤ 1 + m2 + · · · + m2(n−1) c2 or

2(n−1)

c2

f(x) − m2n f(m−n x) ≤ 1 − m c2 = (1 − m2(n−1)), 1 − m2 1 − m2 or the general inequality:

f(x) − m2n f(m−n x) ≤ c2 , 1 − m2 where |m| < 1, m 6= 0. Rest of the proof is similar to the proof of Theorem 2.1.

(2.28) (2.29)

Assume the following condition on f: f(0) = 0.

(2.30)

From (2.30) and (2.7)-(2.8), we get

f(x) − f¯(x) ≤

and

¯ − m−2 f(mx) ≤

f(x)

c , |k| |m|

(2.31)

c , k 6= 0, m 6= 0, |m| > 1. |k| m2

(2.32)

From (2.31)-(2.32), one obtains





f(x) − m−2 f(mx) ≤ f(x) − f¯(x) + f¯(x) − m−2 f(mx) , or

Thus



f(x) − m−2 f(mx) ≤ c0 = |m| + 1 c, k 6= 0, m 6= 0, |m| > 1. 1 |k| m2



f(x) − m−2n f(mn x) ≤ ≤

(2.33)



f(x) − m−2 f(mx) + m−2 f(mx) − m−4 f(m2 x)

+ · · · + m−2(n−1)f(mn−1 x) − m−2n f(mn x)
1 + m−2 + · · · + m−2(n−1) c01 , 370

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or

−2n


m2

f(x) − m−2n f(mn x) ≤ 1 − m c01 = 2 1 − m−2n c01 , −2 1−m m −1

or



f(x) − m−2n f(mn x) ≤

where

m2


1 1 − m−2n c02 , −1

|m| > 1, with c02 = m2 c01 =

(2.34)

|m| + 1 c. |k|

Therefore the following Theorem 2.1a holds. Theorem 2.1a. Let X be a normed linear space and let Y be a real complete normed linear space. Then the limit (2.3) exists for all x ∈ X, all n ∈ N, |m| > 1 and Q : X → Y is the unique quadratic mapping satisfying equation (1.3), such that

f(x) − Q(x) ≤

c02 |m| + 1 1 = 2 c, k 6= 0, |m| > 1. −1 m − 1 |k|

(2.35)

m2

The proof of this Theorem 2.1a is similar to the proof of the previous Theorem 2.1. Alternatively: |m| < 1, f(0) = 0: From (2.30) and (2.22) , (2.24), we get

¯

f(x) − f(x)

≤ c , |k|

and

(2.36)

¯

f − m2 f(m−1 x) ≤ |m| c, |k|

(2.37)

k 6= 0, m 6= 0, |m| < 1. From (2.36)-(2.37), one obtains

or



¯

¯

f(x) − m2 f(m−1 x) ≤ f(x) − f(x)

+ f(x) − m2 f(m−1 x)

k 6= 0, m 6= 0, |m| < 1. Thus



f(x) − m2 f(m−1 x) ≤ c0 = |m| + 1 c 2 |k|



f(x) − m2n f(m−n x) ≤ ≤ or

(2.38)



f(x) − m−2 f(mx)

+ · · · + m2(n−1)f(m−(n−1) x) − m2n f(m−n x)
1 + m2 + · · · + m2(n−1) c02 ,



f(x) − m2n f(m−n x) ≤


1 1 − m2n c02 , 2 1−m

(2.39)

where |m| < 1, m 6= 0. Therefore the following Theorem 2.2a (analogous to Theorem 2.1a) holds for |m| < 1, m 6= 0. Theorem 2.2a. Let X be a normed linear space, and Y a real complete normed linear space. Then the limit (2.3)0 exists for all x ∈ X, n ∈ N, |m| < 1; m 6= 0, and Q : X → Y is the unique quadratic mapping satisfying equation (1.3), such that

f(x) − Q(x) ≤

c02 1 + |m| 1 = c, 1 − m2 1 − m2 |k|

371

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(2.40)

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k 6= 0, |m| < 1; m 6= 0. Special case: Replacing α = β = 1 in equation (1.3) and (2.1), one gets   kf(x + y) + f(kx − y) = (k + 1) kf(x) + f(y) , k ∈ R − {−1, 0}.

Thus

(2.41)

m = k + 1 ∈ R − {0, 1}. Also

 

kf(x + y) + f(kx − y) − (k + 1) kf(x) + f(y) ≤ c, k ∈ R − {−1, 0}.

(2.42)

Acknowledgement. The authors gratefully acknowledge the financial support from King Abdulaziz University, Jeddah, Saudi Arabia.

References [1] A. Alotaibi, S.A. Mohiuddine, On the stability of a cubic functional equation in random 2-normed spaces, Adv. Difference Equ. 2012, 2012:39. [2] A.S. Al-Fhaid, S.A. Mohiuddine, On the Ulam stability of mixed type QA mappings in IFN-spaces, Adv. [3] [4] [5] [6]

Difference Equ. 2013, 2013:203. T. Aoki, On the stability of the linear transformation in Banach spaces, J. Math. Soc. Japan, 2 (1950) 64-66. D.H. Hyers, On the stability of the linear functional equation, Proc. Nat. Acad. Sci., USA, 27 (1941) 222-224. Z. Gajda, On stability of additive mappings, Int. J. Math. Math. Sci. 14 (1991) 431-434. P. Gˇ avruta, A generalization of the Hyers-Ulam-Rassias stability of approximately additive mappings, J.

Math. Anal. Appl. 184 (1994) 431-436. [7] M.E. Gordji, N. Ghobadipour, Generalized Ulam-Hyers stabilities of quartic derivations on Banach algebras, Proyecciones J. Math., 29 (2010) 209-226. [8] K.W. Jun, H.M. Kim, The generalized Hyers-Ulam-Rassias stability of a cubic functional equation, J. Math. Anal. Appl. 274 (2002) 867-878. [9] S.-M. Jung, Hyers-Ulam-Rassias Stability of Functional Equations in Mathematical Analysis, Hadronic Press, Palm Harbor (2001) [10] S.-M. Jung, On the Hyers-Ulam stability of the functional equations that have the quadratic property, J. Math. Anal. Appl. 222 (1998) 126-137. [11] M. Mirmostafaee, M.S. Moslehian, Fuzzy versions of Hyers-Ulam-Rassias theorem, Fuzzy Sets Syst. 159 (2008) 720-729. [12] S.A. Mohiuddine, H. Sevli, Stability of Pexiderized quadratic functional equation in intuitionistic fuzzy normed space, J. Comp. Appl. Math. 235 (2011) 2137-2146. [13] S.A. Mohiuddine, M. Cancan, H. S ¸ evli, Intuitionistic fuzzy stability of a Jensen functional equation via fixed point technique, Math. Comput. Modelling 54 (2011) 2403-2409. [14] S.A. Mohiuddine, Stability of Jensen functional equation in intuitionistic fuzzy normed space, Chaos, Solitons Fract. 42 (2009) 2989-2996. [15] S.A. Mohiuddine, A. Alotaibi, Fuzzy stability of of a cubic functional equation via fixed point technique, Adv. Difference Equ. 2012, 2012:48. [16] S.A. Mohiuddine, A. Alotaibi, M. Obaid, Stability of of various functional equations in non-Archimedean intuitionistic fuzzy normed spaces, Discrete Dynamics Nature Soc. Volume 2012, Article ID 234727, 16 pages. [17] S.A. Mohiuddine, M.A. Alghamdi, Stability of a functional equation obtained through a fixed-point alternative, Adv. Difference Equ. 2012, 2012:141.

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[18] M. Mursaleen, S.A. Mohiuddine, On stability of a cubic functional equation in intuitionistic fuzzy normed spaces, Chaos, Solitons Fract. 42 (2009) 2997-3005. [19] M. Mursaleen, K.J. Ansari, Stability results in intuitionistic fuzzy normed spaces for a cubic functional equation, Appl. Math. Inf. Sci. 7(5) (2013) 1685-1692. [20] A. Najati, C. Park, On the stability of an n-dimensional functional equation originating from quadratic forms, Taiwan. J. Math. 12 (2008) 1609-1624. [21] C. Park, J.M. Rassias, Cubic derivations and quartic derivations on Banach modules, in: “Functional Equations, Difference Inequalities and Ulam Stability Notions” (F. U. N.), Editor: J.M. Rassias, 2010, 119-129, ISBN 978-1-60876-461-7, Nova Science Publishers, Inc. [22] M.M. Pourpasha, J.M. Rassias, R. Saadati, S.M. Vaezpour, A fixed point approach to the stability of Pexider quadratic functional equation with involution, J. Inequal. Appl. 2010, Art. ID 839639, 18 pp. [23] J.M. Rassias, On the stability of the Euler-Lagrange functional equation, C. R. Acad. Bulgare Sci. 45 (1992) 17-20. [24] J.M. Rassias, On the stability of the Euler-Lagrange functional equation, Chinese J. Math. 20 (1992) 185-190. [25] J.M. Rassias, Solution of a problem of Ulam, J. Approx. Theory 57 (1989) 268-273. [26] J.M. Rassias, On a new approximation of approximately linear mappings by linear mappings, Discuss, Math. 7 (1985) 193-196. [27] J.M. Rassias, On approximation of approximately linear mappings by linear mappings, Bull. Sci. Math. (2) 108 (1984) 445-446. [28] J.M. Rassias, On approximation of approximately linear mappings by linear mappings, J. Funct. Anal. 46 (1982) 126-130. [29] J.M. Rassias, Solution of the Ulam stability problem for quartic mappings, Glas. Mat. Ser. III 34(54) (1999) 243-252. [30] J.M. Rassias, On the stability of the general Euler-Lagrange functional equation, Demonstratio Math. 29 (1996) 755-766. [31] J.M. Rassias, H.-M. Kim, Approximate homomorphisms and derivations between C ∗ -ternary algebras. J. Math. Phys. 49 (2008), no. 6, 063507, 10 pp. [32] T.M. Rassias, On the stability of the linear mapping in Banach spaces, Proc. Am. Math. Soc. 72 (1978) 297-300. [33] T.M. Rassias, On a modified Hyers-Ulam sequence, J. Math. Anal. Appl. 158 (1) (1991) 106-113. [34] T.M. Rassias, On the stability of functional equations and a problem of Ulam, Acta Appl. Math. 62 (2000) 123-130. [35] J. R¨ atz, On the orthogonal additive mappings, Aequationes Math. 28 (1985) 35-49. [36] S.M. Ulam, A Collection of Mathematical Problems, Interscience Tracts in Pure and Applied Mathematics, No.8, Interscience. Publ., New York , 1960; “Problems in Modern Mathematics”, Ch. VI, Science Ed., Wiley, 1940. [37] T.Z. Xu, J.M. Rassias, W.X. Xu, A generalized mixed Quadratic-Quartic functional equation, Bull. Malays. Math. Sci. Soc. 35(3) (2012) 633-649.

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Some integral inequalities via (h − (α, m))−logarithmically convexity Jianhua Chen, Xianjiu Huang∗ Department of Mathematics, Nanchang University, Nanchang, 330031, P. R. China Abstract. In this paper, we introduce the concept of (h − (α, m))−logarithmically convex functions and establish some new integral inequalities of these classes of functions.

Keywords:

Hermite’s inequalities; m−logarithmically convex; (α, m)−logarithmically

convex; (h − (α, m))−logarithmically convex; MR(2010) Subject Classification: Primary 26D15, Secondary 26A51

1

Introduction and preliminaries The mathematical inequalities play an important role in the mathematical branches and their

enormous application can not be underestimated. Afterwards, many researchers[1-13] studied the properties of convexity and achieve some different integral inequalities. The purpose of this paper is to introduce the definition of (h − (α, m))−logarithmically convex functions and establish some new integral inequalities of these classes of functions. Before stating our results, we need recall some notions. Throughout this paper, by ℜ, we denote the set of all real numbers. Definition 1.1 Let f : I ⊂ ℜ → ℜ be a function define on interval I of real numbers. Then f is called convex (see[4]) if f (tx + (1 − t)y) ≤ tf (x) + (1 − t)f (y) †

To whom correspondence should be addressed. E-mail:[email protected](J. Chen), [email protected] (X.

Huang). † This work has been supported by the National Natural Science Foundation of China (11461043, 11361042 and 11326099) and supported partly by the Provincial Natural Science Foundation of Jiangxi, China (20114BAB201003 and 20142BAB201005) and the Science and Technology Project of Educational Commission of Jiangxi Province, China (GJJ11346).

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for all x, y ∈ I and t ∈ [0, 1]. In [2] , Toader gave the definition of m−convexity as follows. Definition 1.2 The function f : [a, b] → ℜ, 0 ≤ a < b is said to be m−convex , where m ∈ [0, 1], if f (tx + m(1 − t)y) ≤ tf (x) + m(1 − t)f (y) holds for all x, y ∈ [0, 1] and t ∈ [0, 1]. We say that f is m−concave if −f is m−convex. In [3] , Mihesan gave the definition of (α, m)−convexity as follows. Definition 1.3 The function f : [a, b] → ℜ, 0 ≤ a < b is said to be (α, m)−convex , where (α, m) ∈ [0, 1]2 , if f (tx + m(1 − t)y) ≤ tα f (x) + m(1 − tα )f (y) holds for all x, y ∈ [0, 1] and t ∈ [0, 1]. ¨ In [1], Ozedemir et al. gave the definition of (h − (α, m))−convexity as follows. Definition 1.4 Let h : K ⊂ ℜ → ℜ be a nonnegative function, h ̸= 0. The function f : L ⊂ ℜ → ℜ is said to be (h − (α, m))−convex function if f is non-negative and for all x, y ∈ [0, 1] and t ∈ (0, 1) for (α, m) ∈ [0, 1]2 , we have f (tx + m(1 − t)y) ≤ hα (t)f (x) + m(1 − hα (t))f (y). In [5], Bai gave the definition of m− and (α, m)−logarithmically convex functions as follows. Definition 1.5 The function f : [a, b] → (0, ∞),0 ≤ a < b is said to be m−logarithmically convex, where m ∈ (0, 1], if f (tx + m(1 − t)y) ≤ [f (x)]t [f (y)]m(1−t) holds for all x, y ∈ [0, 1] and t ∈ [0, 1]. Definition 1.6 The function f : [a, b] → (0, ∞),0 ≤ a < b is said to be (α, m)−logarithmically convex, where (α, m) ∈ (0, 1]2 , if α

f (tx + m(1 − t)x) ≤ [f (x)]t [f (x)]m(1−t

α)

holds for all x, y ∈ [0, 1] and t ∈ [0, 1].

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2

Main results In this section, we will introduce the concept of (h − (α, m))−logarithmically convex functions. We

give some new integral inequalities of these classes of functions. First, we present the definition of (h − (α, m))−logarithmically convex functions as follow. Definition 2.1 Let h : K ⊂ ℜ → ℜ be a nonnegative function, h ̸= 0. The function f : L ⊂ ℜ → ℜ is said to be (h − (α, m))−logarithmically convex function if f is nonnegative and for all x, y ∈ L and t ∈ (0, 1) for (α, m) ∈ (0, 1]2 , we have hα (t)

f (tx + m(1 − t)y) ≤ [f (x)]

(

)

m 1−hα (t)

[f (y)]

.

Obviously, if h(t) = t, then (h−(α, m))−logarithmically convex function is a (α, m)−logarithmically convex function; if h(t) = t,α = 1, then (h−(α, m))−logarithmically convex function is a m−logarithmically convex function. ¨ Before giving our results, we need the following lemma which is proved by Ozdemir et al. [13]. Lemma 2.1 Let f : [a, b] → ℜ, 0 ≤ a < b be continuous on [a, b] such that f ∈ L([a, b]). Then the equality

∫

b

∫

1

(x − a)p (x − b)q f (x)dx = (b − a)p+q+1

a

(1 − t)p tq f (tx + (1 − t)y)dt

0

holds for some fixed p, q > 0. Theorem 2.1 Let f : [a, b] → ℜ, 0 ≤ a < b be continuous on [a, b] such that f ∈ L([a, b]). If the mapping f is (h − (α, m))−logarithmically convex on [a, b] for all t ∈ (0, 1) and (α, m) ∈ (0, 1]2 , then ∫

b

a

q p (x − a)p (x − b)q f (x)dx ≤ (b − a)p+q+1 [β( + 1, + 1)]1−m 1 − m 1 − m {∫ 1 }m hα (t) b α [f (a)] m f ( )1−h (t) dt × m 0

∫1

where β(x, y) =

0

(2.1)

(t)x−1 (1 − t)y−1 dt.

Proof. Using Lemma 2.1 , we have ∫

b

∫ (x − a) (x − b) f (x)dx = (b − a) p

q

p+q+1

a

1

(1 − t)p tq f (ta + (1 − t)b)dt.

(2.2)

0

Since f is (h − (α, m))−logarithmically convex on [a, b], we know that for every t ∈ (0, 1) ( ) α b b α f (ta + (1 − t)b) = f (ta + m(1 − t)( )) ≤ [f (a)]h (t) [f ( )]m 1−h (t) . m m

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(2.3)

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From (2.1), (2.2), (2.3) and H¨older inequality, we can conclude that ∫

b

(x − a)p (x − b)q f (x)dx a ∫ 1 p+q+1 = (b − a) (1 − t)p tq f (ta + (1 − t)b)dt ∫0 1 b p+q+1 = (b − a) (1 − t)p tq f (ta + m(1 − t) )dt m( ∫0 1 ) b m 1−hα (t) p+q+1 p q hα (t) ≤ (b − a) (1 − t) t [f (a)] [f ( )] dt m{ ∫ }1−m }m {0∫ 1 1{ 1 b m(1−hα (t)) } m1 hα (t) p+q+1 p q 1−m dt [f (a)] [f ( ) ] dt ≤ (b − a) [(1 − t) t ] m 0 0 }1−m { ∫ 1 }m {∫ 1 { } q p hα (t) b α [f (a)] m [f ( )1−h (t) ] dt ≤ (b − a)p+q+1 [(1 − t) 1−m t 1−m dt m 0 0 {∫ }m 1{ hα (t) b 1−hα (t) } q p p+q+1 1−m ≤ (b − a) [β( + 1, + 1)] [f (a)] m [f ( ) ] dt . 1−m 1−m m 0 Hence, the proof of theorem 2.1 is completed. Remark 2.1 If α = 1, then we can conclude the following inequality: ∫

b

a

p q + 1, + 1)]1−m (x − a)p (x − b)q f (x)dx ≤ (b − a)p+q+1 [β( 1 − m 1 − m }m {∫ 1 { h(t) b 1−h(t) } m [f (a)] [f ( ) ] dt . × m 0

Theorem 2.2 Let f : [a, b] → ℜ, 0 ≤ a < b be continuous on [a, b] such that f ∈ L([a, b]). If k

the mapping |f | k−1 (k > 1) is (h − (α, m))−logarithmically convex on [a, b] for all t ∈ (0, 1) and (α, m) ∈ (0, 1]2 , then ∫

b

(x − a) (x − b) f (x)dx ≤ (b − a) p

q

p+q+1

a

[∫ × 0

where β(x, y) =

∫1 0

[β(kq + 1, kp + 1)]

1 k

[∫

1

|f (a)|

k2 hα (t) k−1

] k−1 2 dt

k

0

1

k2 m (1−hα (t)) (k−1)2

b |f ( )| m

(2.4)

2 ] (k−1) 2 k

(t)x−1 (1 − t)y−1 dt.

Proof. Using Lemma 2.1 , we have ∫

b

∫ (x − a) (x − b) f (x)dx = (b − a) p

q

p+q+1

a

1

(1 − t)p tq f (ta + (1 − t)b)dt.

(2.5)

0 k

Taking into account that |f | k−1 is (h − (α, m))−logarithmically convex on [a, b], we deduce that k

|f (ta + (1 − t)b)| k−1 = |f (ta + m(1 − t)(

k k k α α b k−1 b ))| ≤ |f (a)| k−1 h (t) |f ( )| k−1 m(1−h (t)) . m m

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(2.6)

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Hence, from (2.4), (2.5), (2.6) and H¨older inequality, we can achieve the following inequality: ∫ b (x − a)p (x − b)q f (x)dx a ∫ 1 p+q+1 = (b − a) (1 − t)p tq f (ta + (1 − t)b)dt 0 k } k−1 [∫ 1 ]1 {∫ 1 k k k−1 b p+q+1 kp kq f (ta + m(1 − t) ) dt ≤ (b − a) (1 − t) t dt m 0 k } k−1 { ∫ 10 k 1 b k−1 p+q+1 k f (ta + m(1 − t) = (b − a) [β(kq + 1, kp + 1)] ) dt m [ ∫ 01 ] k−1 k k k α (t) α (t)) 1 b h m(1−h p+q+1 . ≤ (b − a) [β(kq + 1, kp + 1)] k |f (a)| k−1 |f ( )| k−1 dt m 0

(2.7)

Using H¨older inequality again, we have [∫

] k−1 k k b k−1 m(1−hα (t)) |f (a)| |f ( )| dt m 0 {[ ]1 [∫ 1 ] k−1 } k−1 ∫ 1 k k k [ k2 k α α (t)) ] k b h (t) m(1−h k−1 |f (a)| k−1 ≤ |f ( )| k−1 dt dt m {[ ∫0 1 ] 1 [ ∫0 1 ] k−1 } k−1 2 k k k [ k k α (t) 2 m(1−hα (t)) ] b h ( ) ≤ |f (a)| k−1 dt |f ( )| k−1 dt m 0 0 1

k hα (t) k−1

(2.8)

Combining with (2.7) and (2.8), we can conclude that (2.4) holds. Hence, the proof of theorem 2.2 is completed. Remark 2.2 If α = 1, then we can conclude the following inequality: ∫

b

(x − a) (x − b) f (x)dx ≤ (b − a) p

q

p+q+1

a

[∫

1

× 0

[β(kq + 1, kp + 1)]

k2 m b (1−h(t)) |f ( )| (k−1)2 m

]

1 k

[∫

(k−1)2 k2

1

|f (a)|

k2 h(t) k−1

] k−1 k2 dt

0

.

Theorem 2.3 Let f : [a, b] → ℜ, 0 ≤ a < b be continuous on [a, b] such that f ∈ L([a, b]). If the mapping |f |l (l ≥ 1) is (h − (α, m))−logarithmically convex on [a, b] for all t ∈ (0, 1) and (α, m) ∈ (0, 1]2 , then ∫

b

(x − a) (x − b) f (x)dx ≤ (b − a) p

q

a

p+q+1

[∫ ×

1

0

|f (

[β(q + 1, p + 1)]

b )| m

l2 m(1−hα (t)) (l−1)2

l−1 l

[ ] 12 [ ∫ l β(ql + 1, pl + 1)

2 ] (l−1) 3

dt

1

|f (a)|

l3 hα (t) l−1

] l−1 3 dt

0

l

(2.9) where β(x, y) =

∫1 0

(t)x−1 (1 − t)y−1 dt.

Proof. Using Lemma 2.1 , we have ∫ b ∫ 1 p q p+q+1 (x − a) (x − b) f (x)dx = (b − a) (1 − t)p tq f (ta + (1 − t)b)dt. a

(2.10)

0

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Since, |f |l is (h − (α, m))−logarithmically convex on [a, b], we have |f (ta + (1 − t)b)|l = |f (ta + m(1 − t)(

b l b α α ))| ≤ |f (a)|lh (t) |f ( )|lm(1−h (t)) . m m

From (2.9), (2.10), (2.11) and H¨older inequality, we can achieve the following inequality: ∫ b (x − a)p (x − b)q f (x)dx a ∫ 1 b p+q+1 = (b − a) (1 − t)p tq f (ta + m(1 − t)( ))dt m ∫0 1 l−1 1 b ≤ (b − a)p+q+1 [(1 − t)p (t)q ] l [(1 − t)p (t)q ] l f (ta + m(1 − t)( ))dt m 0 [∫ ] l−1 { ∫ 1 }1 1 l l b l p+q+1 p q p q ≤ (b − a) (1 − t) (t) dt [(1 − t) (t) ]|f (ta + m(1 − t)( ))| dt m 0 {∫ 1 0 }1 l l−1 b l p+q+1 p q = (b − a) [β(q + 1, p + 1)] l [(1 − t) (t) ]|f (ta + m(1 − t)( ))| dt . m 0 Using H¨older inequality again, we have {∫ 1 }1 l b l p q [(1 − t) (t) ]|f (ta + m(1 − t)( ))| dt m {0 ∫ 1 }1 l b lm(1−hα (t)) p q lhα (t) [(1 − t) (t) ]|f (a)| ≤ |f ( )| dt m { 0∫ 1 ] l }1 ∫ 1[ l 1 { } { b lm(1−hα (t)) l−1 } l−1 p q l lhα (t) l l ≤ [(1 − t) (t) ] dt dt |f (a)| |f ( )| m 0 [ ] 12 [ ∫ 1 0 ] l−1 2 m(1−hα (t)) l2 hα (t) l l l2 b l−1 ≤ β(ql + 1, pl + 1) |f (a)| l−1 |f ( )| dt m 0 2 l−1 [ ∫ ] ] (l−1) [ ] 12 [ ∫ 1 1 l3 m(1−hα (t)) l3 hα (t) l l3 l3 b |f (a)| l−1 dt |f ( )| (l−1)2 dt . ≤ β(ql + 1, pl + 1) m 0 0

(2.11)

(2.12)

(2.13)

By (2.12) and (2.13), we can achieve that (2.9) holds. Hence, the proof of theorem 2.3 is completed. Remark 2.3 If α = 1, then we can conclude the following inequality: ∫

b

(x − a) (x − b) f (x)dx ≤ (b − a) p

a

q

p+q+1

[∫ × 0

1

|f (

[β(q + 1, p + 1)]

b )| m

l2 m(1−h(t)) (l−1)2

l−1 l

[ ] 12 [ ∫ l β(ql + 1, pl + 1)

l

|f (a)|

l3 h(t) l−1

] l−1 3 dt

l

0

2 ] (l−1) 3

dt

1

.

References ¨ [1] M. E. Ozdemir, H. Kavurmaci and M. Avci, Hermite-Hadamard Type Inequalities for (h − (α, m))−convex Functions, RGMIA Research Report Collection, 14(2011)Article 31. [ONLINE: http://http://rgmia.org/papers/v14/v14a31.pdf] [2] G. H. Toader, Some generalizations of the convexity, Proceedings of the Colloquium on Approximation and Optimization, 1984: 329-338. 6 379

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[3] V. G.Mihesan, A generalization of the convexity, Seminar on Functional Equations, Approx. and Convex. 1993. [4] D. S. Mitrinovi´c, J. E. Pe˘car´c and A. M. Fink, Classical and new inequalities in analysis, Mathematics and its Applications (East European Series), 61, Kluwer Acad. Publ., Dordrecht, 1993. [5] R. F. Bai, F. Qi, and B. Y. Xi, Hermite-Hadamard type inequalities for the m- and (α, m)− logarithmically convex functions, Filomat 27 (2013), no.1, 1-7. [6] W. J. Liu, New integral inequalities via (α, m)−convexity and quasi-convexity, arXiv:1201.6226v1 [math.FA] [7] Z. P. Ji, T. Y. Zhang, Integral inqualities of Hermite-Hadamard type for (α, m)−GA-convex functions,http://arxiv.org/abs/1306.0852v1[math.FA] [8] S. S. Dragomir, On some new inequalities of Hermite-Hadamard type for m-convex functions, Tamkang J. Math. 33 (2002) 45-55. [9] T. Y. Zhang, A. P. Ji, F. Qi, On integral inequalities of Hermite-Hadamard type for sgeometrically convex functions, Abstr. Appl. Anal. 2012 (2012), Article ID 560586, 14 pages; Available online at http://dx.doi.org/10.1155/2012/560586 [10] S. H. Wang, B. Y. Xi, F. Qi, Some new inequalities of Hermite-Hadamard type for n-time differentiable functions which are m-convex, Analysis (Munich) 32 (2012) 247-262; Available online at http://dx.doi.org/10.1524/anly.2012.1167 [11] M. Iqbal, M. I. Bahtti and M. Muddassar, Hadamard-type inequalities for h-Convex functions, Pakistan Journal of Science (ISSN 1016-2526), Vol. 63 No. 3 September 2011 pp. 170-175. [12] M. Muddassar, M. I. Bhatti and M. Iqbal, Some New s-Hermite Hadamard Type Ineqalities for Differentiable Functions and Their Applications, Proceedings of the Pakistan Academy of Sciences 49(1) (2012), 9-17. ¨ [13] M. E. Ozdemir, E. Set and M. Alomari, Integral inequalities via several kinds of convexity, Creat. Math. Inform. 20 (2011), no. 1, 62-73.

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On Gosper’s q-Trigonometric Function Mahmoud Jafari Shah Belaghi Bah¸ce¸sehir University, Istanbul, Turkey [email protected] Nuri Kuruo˘ glu ˙ Istanbul Geli¸sim University, Istanbul, Turkey [email protected] Abstract. In this paper, we study about periodicity of q-trigonometric function which was introduced by Gosper and also we rewrite the q-analogue of Legendres duplication formula with the same bases. Furthermore, we modify some identities involving q-shifted factorial. Keywords. Gosper’s q-trigonometric function, q-Gamma function, Legendres duplication formula. Mathematics Subject Classification. 11B65, 33D05.

1

Introduction

The q-shifted factorial [1, 3] is defined by ( 1 (a; q)n = Qn−1 m m=0 (1 − aq )

n = 0, n = 1, 2, ....

(1)

and it is assumed that a 6= q −m , m = 0, 1, .... The q-shifted factorial [1, 3] is also defined for any complex number α, (a; q)α =

(a; q)∞ , (aq α ; q)∞

(2)

Qn where (a; q)∞ := limn→∞ m=0 (1 − aq m ) and the principal value of q α is taken and it is assumed that 0 < q < 1. The q-Gamma function was introduced by Thomae [6] and Jackson [5], (see [3], page 20) (q; q)∞ Γq (x) = x (1 − q)1−x , 0 < q < 1. (3) (q ; q)∞ A q-analogue of Legendre’s duplication formula [5, 7] has the form 1 1 Γq (2x)Γq2 ( ) = (1 + q)2x−1 Γq2 (x)Γq2 (x + ). 2 2 Gosper [4] defined q-trigonometric functions as follows: (q 2z ; q 2 )∞ (q 2−2z ; q 2 )∞ , 0 < q < 1, (q; q 2 )2∞ 1+2z 2 2 (q ; q )∞ (q 1−2z ; q 2 )∞ cosq (πz) := q z , 0 < q < 1. (q; q 2 )2∞ 2

sinq (πz) := q (z−1/2)

(4)

(5) (6)

1

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It can be seen [4] that cosq (z) = sinq (

π − z). 2

(7)

By using (5), (6) and (7), one can see that, for the cases x = 0 and x = sinq (x) and cosq (x) are; sinq ( π2 ) = 1, cosq ( π2 ) = 0.

sinq (0) = 0, cosq (0) = 1,

π 2,

(8)

There are many identities involving q-shifted factorial [1, 3], but in this paper we are using the following identities; For all a ∈ C and n ∈ N, following equations hold (q 2a ; q 2 )n = (q a ; q)n (−q a ; q)n , 2

2

(a; q)2n = (a; q )n (aq; q )n ,

(10) n 2

(q 1−a−n ; q)n = (q a ; q)n (−1)n q −( )−an .

2

(9)

(11)

Main result

In the next lemma we show that the equations (9) and (10) are also valid for any complex number α, Lemma 1. For all a, α ∈ C, the following equations hold (q 2a ; q 2 )α = (q a ; q)α (−q a ; q)α , 2

2

(a; q)2α = (a; q )α (aq; q )α .

(12) (13)

Proof. To prove (12) we use (2), then we have (q 2a ; q 2 )α =

(q 2a ; q 2 )∞ . (q 2a+2α ; q 2 )∞

By using the definition of q-shifted factorial (1), we obtain (q 2a ; q 2 )∞ (q 2a+2α ; q 2 )∞ Q∞ (1 − q 2a+2i ) = Q∞i=0 2a+2α+2i ) i=0 (1 − q Q∞ (1 − q a+i )(1 + q a+i ) = Q∞ i=0 a+α+i )(1 + q a+α+i ) i=0 (1 − q Q Q∞ ∞ a+i (1 − q a+i ) ) i=0 (1 + q Q = Q∞i=0 ∞ a+α+i a+α+i (1 − q ) (1 + q ) i=0 i=0 a a (q ; q)∞ (−q ; q)∞ = a+α (q ; q)∞ (−q a+α ; q)∞ = (q a ; q)α (−q a ; q)α .

(q 2a ; q 2 )α =

2

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The proof of (12) is complete. To Prove the next equation, we use (1) and (2), then we have (a; q 2 )α (aq; q 2 )α (a; q 2 )∞ (aq; q 2 )∞ (aq 2α ; q)∞ = (a; q)2α (aq 2α ; q 2 )∞ (aq 2α+1 ; q 2 )∞ (a; q)∞ (aq 2α ; q)∞ (a; q 2 )∞ (aq; q 2 )∞ , = 2α (a; q)∞ (aq ; q 2 )∞ (aq 2α+1 ; q 2 )∞ 1

1

each fraction in the last line is equal 1, since (c; q)∞ (cq 2 ; q)∞ = (c; q 2 )∞ (see [8] , page 13). The proof of (13) is complete. In the next lemma, we want to modify the equation (11). Lemma 2. For all α and β ∈ C, the following equation holds (q 1−α−β ; q)α = (q β ; q)α

sin√q π(α + β) −(α)−αβ q 2 , sin√q π(β)

(14)

where sinq is defined as in (5). Proof. After applying the equation (2) for both numerator and denominator of the left hand side of the following equation, we obtain that (q 1−α−β ; q)α (q 1−α−β ; q)∞ (q α+β ; q)∞ = (q β ; q)α (q 1−β ; q)∞ (q β ; q)∞ and by using the definition of sinq which is written in (5), we have sin√q π(α + β) −(α)−αβ (q 1−α−β ; q)∞ (q α+β ; q)∞ . q 2 = 1−β β (q ; q)∞ (q ; q)∞ sin√q π(β) Therefore proof is complete. Theorem 1. For all n ∈ N and x ∈ C, the following equations hold sinq (x + nπ) = (−1)n sinq (x), cosq (x + nπ) = (−1)n cosq (x), tanq (x + nπ) = tanq (x), cotq (x + nπ) = cotq (x).

(15) (16) (17) (18)

Proof. We use lemma 2 for prove the equation (15). Taking any arbitrary n ∈ N and a ∈ C, then we have (q 1−n−a ; q)n = (q a ; q)n

sin√q π(a + n) −(n)−na q 2 . sin√q π(a)

(19)

3

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By comparing the equations (11) and (19), we obtain sin√q π(a + n) = (−1)n . sin√q π(a) √ Substituting q with q and x with aπ completes the proof of equation (15). By using (7) and (15), one can shows that (16) is valid for all n ∈ N and sin (x) cos (x) x ∈ C, and the last two equations (17) and (18) comes from cosqq (x) and sinqq (x) , respectively. Remark 1. The cosq (x) is an even function, its come from the definition (6) directly. And the sinq (x) is an odd function, since by using (7), we can write sinq (x) = cosq ( π2 −x) and also we know that cosq (x) is an even function then we have sinq (x) = cosq (x− π2 ), again apply (7), we obtain cosq (x− π2 ) = sinq (π−x). Now by using the Theorem 1, we obtain sinq (π − x) = −sinq (−x). Therefore sinq (x) = −sinq (−x). Lemma 3. For all k ∈ Z, zeroes of q-sine and q-cosine functions are kπ and (2k+1)π , respectively. 2 Proof. Since sinq is an odd function, therefore its enough to prove the lemma for positive value of k. We prove the lemma for positive value of k by induction. For k = 1 and using (15), we have sinq (π) = sinq (0 + π) = sinq (0) = 0, since sinq (0) = 0 comes from definition of sinq . Then lemma is valid for k = 1. Assume that sinq (nπ) = 0 is true. We need to show that sinq ((n + 1)π) = 0 is also true. By using (15), we have sinq ((n + 1)π) = sinq (nπ + π) = (−1)sinq (nπ) = 0. Therefore zeroes of sinq (x) are kπ, for all k ∈ Z. About the zeroes of cosq (x), take any arbitrary k ∈ Z, and by using the (7), we have cosq (

(2k + 1)π π ) = cosq (kπ + ) = sinq (−kπ) = 0. 2 2

Therefore zeroes of cosq (x) are

(2k+1)π , 2

for all k ∈ Z.

Lemma 4. For all z ∈ C, the following equation holds 1

1

1

(q z+1 ; q)z = (−q 2 ; q 2 )2z (q 2 ; q)z .

4

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1

Proof. Taking a = 1, α = 2z and substituting q with q 2 in (12), and applying 1 1 to (−q 2 ; q 2 )2z , we obtain 1

1

(q 2 ; q)z (q; q)2z (q 2 ; q)z = 1 1 . (−q ; q )2z z+1 (q ; q)z (q 2 ; q 2 )2z (q z+1 ; q)z 1 2

1 2

(20)

By using (2), the right hand side of (20) can be written as 1

(q; q)2z (q 2 ; q)z = (q z+1 ; q)z (q 21 ; q 12 )2z

(q;q)∞ (q 2z+1 ;q)∞ (q z+1 ;q)∞ (q 2z+1 ;q)∞

1

(q 2 ; q)z 1 2

1

(q ; q 2 )2z

,

1

=

(q 2 ; q)z (q; q)∞ , z+1 (q ; q)∞ (q 12 ; q 12 )2z 1

= (q; q)z

(q 2 ; q)z 1

1

(q 2 ; q 2 )2z

1

.

(21)

1

By substituting q with q 2 and then taking a = q 2 in equation (13), one can see that the right hand side of (21) is equal 1 and this completes the proof. Lemma 5. For all z ∈ C, the following equation holds (q

1 2 −z

; q)z = (q

z+1

; q)z

q−

z2 2 1

1

(−q 2 ; q 2 )2z

cos√q (πz).

(22)

Proof. By using the Lemma 4 the equation (22) can be written as 1

1

(q 2 −z ; q)z = (q 2 ; q)z q −

z2 2

cos√q (πz).

(23)

The equation (23) is a special case of lemma 2 when β = 21 , since cosq (z) = sinq ( π2 − z) and also cosq is an even function. Corollary 1. For the positive integers value of n, Lemma 5 deduce to q−

1

(q 2 −n ; q)n = (−1)n (q n+1 ; q)n

1

n2 2 1

(−q 2 ; q 2 )2n

.

Proof. The result is obtained by using Theorem 1. Euler (see [2], page 271 or [3], page 222) found the following formula in connection with partitions, (−q; q)∞ (q; q 2 )∞ = 1. In the next lemma, we want to generalize this Euler’s formula. 5

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Theorem 2. For all z ∈ C, the following equation holds (q z+1 ; q)z = (−q; q)z (q; q 2 )z . 1

1

Proof. By substituting q with q 2 and then taking a = −q 2 in equation (13), we have 1 1 1 (−q 2 ; q 2 )2z = (−q 2 ; q)z (−q; q)z , now, we apply the result to Lemma 5 and obtain 1

1

(q z+1 ; q)z = (−q 2 ; q)z (−q; q)z (q 2 ; q)z .

(24)

Taking a = 12 in the equation (12) and then applying to the right hand side of the equation (24) completes the proof. Theorem 3. For all x ∈ C, the following equation holds, 1 1 1 1 Γq (2x)Γq ( ) = Γq (x)Γq (x + )(−q 2 ; q 2 )2x−1 . 2 2

Proof. By using the definition of q-Gamma function (3) and then applying the equation (2), we can write 1

Γq (2x)Γq ( 21 ) (q x ; q)∞ (q x+ 2 ; q)∞ (q x ; q)x = = , 1 1 Γq (x)Γq (x + 21 ) (q 2x ; q)∞ (q 2 ; q)∞ (q 2 ; q)x the last equation holds since (q x ; q)x =

(q x ;q)∞ (q 2x ;q)∞

1

and (q 2 ; q)x =

(25) 1

(q 2 ;q)∞ 1 (q 2 +x ;q)∞

. Tak-

1 2

ing β = in Lemma 2 and applying for the denominator of the last fraction in (25), we get (q x ; q)x 1

(q 2 ; q)x

= =

(q x ; q)x sin√q π( 12 + x) − x2 q 2, 1 (q 2 −x ; q)x sin√q π( 12 ) (q x ; q)x (q

1 2 −x

; q)x

cos√q (πx) q −

x2 2

.

Now, by using Lemma 4, we have (q x ; q)x (q

1 2 −x

; q)x

cos√q (πx) q −

x2 2

=

1 1 (q x ; q)x (−q 2 ; q 2 )2x . (q x+1 ; q)x

Making use of (2), we have 1 1 1 1 (q x ; q)x (q x ; q)∞ (q 2x+1 ; q)∞ 2;q2) (−q = (−q 2 ; q 2 )2x . 2x (q x+1 ; q)x (q 2x ; q)∞ (q x+1 ; q)∞

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After expanding the first and second fractions and then a simplification, yields 1 1 1 1 (q x ; q)∞ (q 2x+1 ; q)∞ 1 − qx (−q 2 ; q 2 )2x , (−q 2 ; q 2 )2x = 2x x+1 2x (q ; q)∞ (q ; q)∞ 1−q 1 1 1 (−q 2 ; q 2 )2x , = 1 + qx 1

1

= (−q 2 ; q 2 )2x−1 .

References [1] Ernst, T., A Comprehensive Treatment of Q-calculus, Springer, 2012. [2] Euler, L., Introductio in Analysin Infinitorum, Vol. 1, Lausanne, 1748. [3] Gasper, G., Rahman, M., Basic hypergeometric series. Vol. 96, Cambridge university press, 2004. [4] Gosper, R. W., Experiments and discoveries in q-trigonometry, in Symbolic Computation, Number Theory, Special Functions, Physics and Combinatorics, Editors: F. G. Garvan and M. E. H. Ismail. Kluwer, Dordrecht, Netherlands, pp. 79–105, 2001. [5] Jackson, F. H., A Generalisation of the Functions Γ(n) and xn , Proceedings of the Royal Society of London, pp. 64–72, 1904. [6] Thomae, J., Beitr¨ age zur Theorie der durch die Heinesche Reihe:... darstellbaren Functionen, Journal f¨ ur die reine und angewandte Mathematik,Vol. 70, pp. 258–281, 1869. [7] Jackson, F. H., The basic gamma-function and the elliptic functions, Proceedings of the Royal Society of London. Series A, Vol. 76, no. 508, pp. 127– 144, 1905. [8] Berndt, B. C., Ramanujan’s Notebooks, Part III, Springer-Verlag, New York, 1991.

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APPROXIMATE QUADRATIC FORMS ON RESTRICTED DOMAINS WON-GIL PARK AND JAE-HYEONG BAE* Abstract. Let r, s be nonzero real numbers with r + s = 1. In [9], Najati and Jung investigated a quadratic functional equation g(rx + sy) + rs g(x − y) = rg(x) + sg(y). We introduce a functional equation f (rx + sy, rz + sw) + rsf (x − y, z − w) = rf (x, z) + sf (y, w) and investigate the relation between the above two functional equations. And we find out the general solution and the Hyers-Ulam stability of the latter on restricted domains.

1. Introduction In 1940 and in 1968, Ulam [12] proposed the general Ulam stability problem: “When is it true that by slightly changing the hypotheses of a theorem one can still assert that the thesis of the theorem remains true or approximately true?” In 1941, Hyers [7] solved this problem for linear mappings. In 1950, Aoki [2] provided a generalization of the Hyers’ theorem for additive mappings. This stability concept is also applied to the case of other functional equations. For more results on the stability of functional equations (see [5, 6, 11]). In 1998, S.-M. Jung [8] investigated the Hyers-Ulam stability for additive and quadratic mappings on restricted domains. Let X and Y be real vector spaces. For a mapping g : X → Y , consider the quadratic functional equation: (1.1)

g(x + y) + g(x − y) = 2g(x) + 2g(y).

In 1989, J. Aczel [1] solved the solution of the equation (1.1). Later, many different quadratic functional equations were solved by numerous authors [3, 8, 10]. In recent, A. Najati and S.-M. Jung [9] introduced a generalized quadratic functional equation (1.2)

g(rx + sy) + rs g(x − y) = rg(x) + sg(y),

where r, s are nonzero real numbers with r + s = 1. In 2007, the authors [4] solved the solution of the 2-variable quadratic functional equation (1.3)

f (x + y, z + w) + f (x − y, z − w) = 2f (x, z) + 2f (y, w).

Consider a generalized 2-variable quadratic functional equation (1.4)

f (rx + sy, rz + sw) + rsf (x − y, z − w) = rf (x, z) + sf (y, w),

where r, s are nonzero real numbers with r + s = 1. 2000 Mathematics Subject Classification. Primary 39B52, 39B72. Key words and phrases. Solution, Stability, Approximate quadratic form. * Corresponding author. 1

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WON-GIL PARK AND JAE-HYEONG BAE

2

In this paper, we investigate the relation between (1.2) and (1.4) by the same method as the proofs of Theorem 1 and Theorem 2 in [4]. And we find out the general solution and the Hyers-Ulam stability of (1.4) in the spirit of Najati and Jung [9]. 2. Relation between (1.2) and (1.4) The functional equation (1.4) induces the quadratic functional equation (1.2) as follows. Theorem 2.1. Let f : X × X → Y be a mapping satisfying (1.4) and let g : X → Y be the mapping given by (2.1)

g(x) := f (x, x)

for all x ∈ X, then g satisfies (1.2). Proof. By (1.4) and (2.1), we obtain g(rx + sy) + rsg(x − y) = f (rx + sy, rx + sy) + rsf (x − y, x − y) = rf (x, x) + sf (y, y) = rg(x) + sg(y) for all x, y ∈ X.  Example 1. Let X be a real algebra and D : X → X a derivation on X. Define a mapping f : X × X → X by f (x, y) := D(xy) = xD(y) + D(x)y for all x, y ∈ X. Then we see that f (rx + sy, rz + sw) + rsf (x − y, z − w) = D[(rx + sy)(rz + sw)] + rsD[(x − y)(z − w)] = (rx + sy)D(rz + sw) + D(rx + sy)(rz + sw) + rs[(x − y)D(z − w) + D(x − y)(z − w)] = (rx + sy)[rD(z) + sD(w)] + [rD(x) + sD(y)](rz + sw) ( ) + rs (x − y)[D(z) − D(w)] + [D(x) − D(y)](z − w) = r2 xD(z) + s2 yD(w) + r2 D(x)z + s2 D(y)w + rsxD(z) + rsyD(w) + rsD(x)z + rsD(y)w = r[xD(z) + D(x)z] + s[yD(w) + D(y)w] = rD(xz) + sD(yw) = rf (x, z) + sf (y, w) for all x, y, z, w ∈ X. Thus f satisfies (1.4). Define a mapping g : X → X by g(x) := D(x2 ) = xD(x) + D(x)x for all x ∈ X. Then g satisfies (2.1). By Theorem 2.1, g satisfies (1.2). The quadratic functional equation (1.2) induces the functional equation (1.4) with an additional condition. Theorem 2.2. Let a, b, c ∈ R and g : X → Y be a mapping satisfying (1.2). If f : X × X → Y is the mapping given by b (2.2) f (x, y) := ag(x) + [g(x + y) − g(x − y)] + cg(y) 4

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for all x, y ∈ X, then f satisfies (1.4). Furthermore, (2.1) holds if r is a rational number and a + b + c = 1. Proof. By (1.2) and (2.2), we see that f (rx + sy, rz + sw) + rsf (x − y, z − w) ) ( )] b[ ( = ag(rx + sy) + g r(x + z) + s(y + w) − g r(x − z) + s(y − w) + cg(rz + sw) 4 ) ( ] b[ +rs ag(x − y) + g(x − y + z − w) − g(x − y − z + w) + cg(z − w) 4 ) ( )] b[ ( = ag(rx + sy) + rsag(x − y) + g r(x + z) + s(y + w) + rsg (x + z) − (y + w) 4 ) ( )] b[ ( − g r(x − z) + s(y − w) + rsg (x − z) − (y − w) + cg(rz + sw) + rscg(z − w) 4 [ ] b[ ] = a g(rx + sy) + rsg(x − y) + rg(x + z) + sg(y + w) 4 ] [ ] b[ − rg(x − z) + sg(y − w) + c g(rz + sw) + rsg(z − w) 4 ] [ ]) b( [ = a[rg(x) + sg(y)] + r g(x + z) − g(x − z) + s g(y + w) − g(y − w) + c[rg(z) + sg(w)] 4 = rf (x, z) + sf (y, w) for all x, y, z, w ∈ X. Let r be a rational number. Since g satisfies (1.2), it also satisfies (1.1) (see Theorem 2.3. in [9]). Letting x = y = 0 and y = x in (1.1), respectively, g(0) = 0 and g(2x) = 4g(x) for all x ∈ X. By (2.2) and the above two equalities, b f (x, x) = ag(x) + [g(2x) − g(0)] + cg(x) 4 = (a + b + c)g(x) = g(x) for all x ∈ X.  Example 2. Consider the function g : R2 → R given by g(x) := xT Ax for all x ∈ R2 , where A is a 2 × 2 real matrix. Then we see that g(rx + sy) + rs g(x − y) = (rx + sy)T A(rx + sy) + rs(x − y)T A(x − y) = (rxT + syT )A(rx + sy) + rs(xT − yT )A(x − y) = r2 xT Ax + rs(xT Ay + yT Ax) + s2 yT Ay + rs(xT Ax − xT Ay − yT Ax + yT Ay) = r(r + s)xT Ax + s(r + s)yT Ay = rg(x) + sg(y)

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for all x, y ∈ R2 , where r, s are nonzero real numbers with r + s = 1. Thus g satisfies (1.2). Let a, b, c ∈ R and define f (x, y) := ag(x) + 4b [g(x + y) − g(x − y)] + cg(y) for all x, y ∈ R2 . By Theorem 2.2, the function f satisfies (1.4). In fact, ( )T ( )( ) x a 2b Ax f (x, y) = b y Ay c 2 for all x, y ∈ R2 . Example 3. Let Mn be the algebra of n × n real matrices. Consider the mapping g : Mn → Mn given by g(A) := A2 for all A ∈ Mn . Then we see that g(rA + sB) + rs g(A − B) = (rA + sB)2 + rs(A − B)2 = r2 A2 + rs(AB + BA) + s2 B 2 + rs(A2 − AB − BA + B 2 ) = r2 A2 + rsAB + rsBA + s2 B 2 + rsA2 − rsAB − rsBA + rsB 2 = r(r + s)A2 + s(r + s)B 2 = rg(A) + sg(B) for all A, B ∈ R2 , where r, s are nonzero real numbers with r + s = 1. Thus g satisfies (1.2). Let a, b, c ∈ R and define f (A, B) := aA2 + bA ◦ B + cB 2 , where A ◦ B the Jordan product 21 (AB + BA) of A and B for all A, B ∈ Mn . Then the mapping f : Mn × Mn → Mn satisfies (2.2). By Theorem 2.2, the mapping f satisfies (1.4). 3. Solution of the equation (1.4) We recall that r, s are nonzero real numbers with r + s = 1. In the following theorem, we find out the general solution of the functional equation (1.4). Theorem 3.1. Let f : X × X → Y be a mapping such that f (x, y) = f (−x, −y) for all x, y ∈ X. Then f satisfies (1.3) if it satisfies (1.4). If r and s are rational numbers and f satisfies (1.3), then it also satisfies (1.4). Proof. Letting x = y = z = w = 0 in (1.4), we gain f (0, 0) = 0. Putting y = w = 0 in (1.4), we get f (rx, rz) = r2 f (x, z) for all x, z ∈ X. Replacing x by x + y and z by z + w in (1.4), we have (3.1)

f (rx + y, rz + w) = rf (x + y, z + w) + sf (y, w) − rsf (x, z)

for all x, y, z, w ∈ X. Replacing y by −y and w by −w in (3.1), we obtain f (rx − y, rz − w) = rf (x − y, z − w) + sf (y, w) − rsf (x, z) for all x, y, z, w ∈ X. Adding (3.1) to the above equation, we see that (3.2) f (rx + y, rz + w) + f (rx − y, rz − w) = r[f (x + y, z + w)+ f (x − y, z − w)] + 2sf (y, w) − 2rsf (x, z) for all x, y, z, w ∈ X. Replacing y by x + ry and w by z + rw in (3.1), we obtain (3.3)

f (r(x + y) + x, r(z + w) + z) = rf (2x + ry, 2z + rw) + sf (x + ry, z + rw) − rsf (x, z)

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for all x, y, z, w ∈ X. Replacing x, y, z, w by 2x, ry, 2z, rw in (3.1), respectively, we obtain (3.4)

rf (2x + ry, 2z + rw) = r2 f (2x + y, 2z + w) − r2 sf (y, w) + rsf (2x, 2z)

for all x, y, z, w ∈ X. Replacing y by ry and w by rw in (3.1), we obtain (3.5)

sf (x + ry, z + rw) = rsf (x + y, z + w) − rs2 f (y, w) + s2 f (x, z)

for all x, y, z, w ∈ X. Replacing x, y, z, w by x + y, x, z + w, z in (3.1), respectively, we obtain (3.6)

f (r(x + y) + x, r(z + w) + z) = rf (2x + y, 2z + w) + sf (x, z) − rsf (x + y, z + w)

for all x, y, z, w ∈ X. By (3.3), (3.4), (3.5) and (3.6), we see that (3.7)

f (2x + y, 2z + w) + 2f (x, z) + f (y, w) = 2f (x + y, z + w) + f (2x, 2z)

for all x, y, z, w ∈ X. Putting y = −x and w = −z in (3.7), we get f (2x, 2z) = 4f (x, z) for all x, z ∈ X. Therefore, it follows from (3.7) that f (2x + y, 2z + w) + f (y, w) = 2f (x + y, z + w) + 2f (x, z) for all x, y, z, w ∈ X. Replacing y by y − x and w by w − z in the above equation, we have f (x + y, z + w) + f (y − x, w − z) = 2f (x, z) + 2f (y, w) for all x, y, z, w ∈ X. Hence f satisfies (1.3). Conversely, let r and s be rational numbers and let f satisfy (1.3). Then there exist two symmetric bi-additive mappings S1 , S2 : X × X → Y and a bi-additive mapping B : X × X → Y such that f (x, y) = S1 (x, x) + B(x, y) + S2 (y, y) for all x, y ∈ X (see [4]). Since r and s are rational numbers, rf (x, z) + sf (y, w) − rsf (x − y, z − w) = r2 S1 (x, x) + 2rsS1 (x, y) + s2 S1 (y, y) + r2 B(x, z) + rsB(x, w) + rsB(y, z) + s2 B(y, w) + r2 S2 (z, z) + 2rsS2 (z, w) + s2 S2 (w, w) = S1 (rx, rx) + 2S1 (rx, sy) + S1 (sy, sy) + B(rx, rz) + B(rx, sw) + B(sy, rz) + B(sy, sw) + S2 (rz, rz) + 2S2 (rz, sw) + S2 (sw, sw) = S1 (rx + sy, rx + sy) + B(rx + sy, rz + sw) + S2 (rz + sw, rz + sw) = f (rx + sy, rz + sw) for all x, y, z, w ∈ X. Therefore f satisfies (1.4).



4. Stability of the equation (1.4) From now on, let X be a real normed space and Y a Banach space. The authors proved a generalized Hyers-Ulam stability theorem on a functional equation (1.3). The following theorem is a particular case of Theorem 4 in [4]. Theorem 4.1 Let δ ≥ 0 be fixed. If a mapping f : X × X → Y satisfies the inequality (4.1)

∥f (x + y, z + w) + f (x − y, z − w) − 2f (x, z) − 2f (y, w)∥ ≤ δ

for all x, y, z, w ∈ X, then there exists a unique 2-variable quadratic mapping F : X × X → Y such that ∥f (x, y) − F (x, y)∥ ≤ 31 δ for all x, y ∈ X.

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Using a similar method used in the paper [8], we obtain the following theorem. Theorem 4.2 Let d > 0 and δ ≥ 0 be fixed and let X ̸= {0}. If a mapping f : X × X → Y satisfies the inequality (4.1) for all x, y, z, w ∈ X with ∥x + z∥ + ∥y + w∥ ≥ d, then there exists a unique 2-variable quadratic mapping F : X × X → Y such that 5 ∥f (x, y) − F (x, y)∥ ≤ δ 3

(4.2) for all x, y ∈ X.

Proof. Assume that ∥x + z∥ + ∥y + w∥ < d. Let ) d 1( t= 1+ (x + z) 2 ∥x + z∥ ) 1( d 1+ (y + w) t= 2 ∥y + w∥

if if

∥x + z∥ ≥ ∥y + w∥; ∥x + z∥ < ∥y + w∥.

If x + z = y + w = 0, then one can choose a t ∈ X with ∥t∥ = d2 . Note that 2∥t∥ = ∥x + z∥ + d ≥ d

if

∥x + z∥ ≥ ∥y + w∥;

2∥t∥ = ∥y + w∥ + d > d

if

∥x + z∥ < ∥y + w∥.

Clearly, we see that ∥x + z − 2t∥ + ∥y + w + 2t∥ ≥ 4∥t∥ − (∥x + z∥ + ∥y + w∥) ≥ 2d − (∥x + z∥ + ∥y + w∥) ≥ 2d > d, ∥x + z − y − w∥ + 4∥t∥ ≥ ∥x + z − y − w∥ + 2d ≥ 2d > d, ∥x + z + 2t∥ + ∥ − y − w + 2t∥ ≥ max{∥x + z + 2t∥, ∥ − y − w + 2t∥} ≥ d, (4.3)

∥x + z∥ + 2∥t∥ ≥ 2∥t∥ ≥ d,

2∥t∥ + ∥y + w∥ ≥ 2∥t∥ ≥ d,

4∥t∥ ≥ 2d > d.

These inequalities (4.3) come from the corresponding substitutions attached between the right-hand sided parentheses of the following functional identity. Besides from (4.1) with x = y = z = w = 0 we get ∥f (0, 0)∥ ≤ 2δ . Therefore from (4.1), (4.3) and the new functional identity 2[f (x + y, z + w) + f (x − y, z − w) − 2f (x, z) − 2f (y, w) − f (0, 0)] = [f (x + y, z + w) + f (x − y − 2t, z − w − 2t) − 2f (x − t, z − t) − 2f (y + t, w + t)] − [f (x − y − 2t, z − w − 2t) + f (x − y + 2t, z − w + 2t) − 2f (x − y, z − w) − 2f (2t, 2t)] + [f (x − y + 2t, z − w + 2t) + f (x + y, z + w) − 2f (x + t, z + t) − 2f (−y + t, −w + t)] + 2[f (x + t, z + t) + f (x − t, z − t) − 2f (x, z) − 2f (t, t)] + 2[f (t + y, t + w) + f (t − y, t − w) − 2f (t, t) − 2f (y, w)] − 2[f (2t, 2t) + f (0, 0) − 4f (t, t)],

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we get 2∥f (x + y, z + w) + f (x − y, z − w) − 2f (x, z) − 2f (y, w) − f (0, 0)∥ ≤ δ + δ + δ + 2δ + 2δ + 2δ = 9δ, or 9 ∥f (x + y, z + w) + f (x − y, z − w) − 2f (x, z) − 2f (y, w)∥ ≤ δ + ∥f (0, 0)∥ ≤ 5δ. 2 Applying now Theorem 4.1 and the above inequality, there exists a unique 2-variable quadratic mapping F : X × X → Y satisfying (4.2) such that F (x, y) = limn→∞ 2−2n f (2n x, 2n y), completing the proof.  (4.4)

We recall that r, s are nonzero real numbers with r + s = 1. Theorem 4.3. Let d > 0 and δ ≥ 0 be given. Assume that a mapping f : X × X → Y such that f (x, y) = f (−x, −y) and (4.5)

∥f (rx + sy, rz + sw) + rsf (x − y, z − w) − rf (x, z) − sf (y, w)∥ ≤ δ

for all x, y, z, w ∈ X with ∥x + z∥ + ∥y + w∥ ≥ d. Then there exists K > 0 such that f satisfies (4.6)

∥f (x + y, z + w) + f (x − y, z − w) − 2f (x, z) − 2f (y, w)∥ ≤

4(2 + |r| + |s|) δ |rs|

for all x, y, z, w ∈ X with ∥x + z∥ + ∥y + w∥ ≥ K. Proof. Let x, y, z, w ∈ X with ∥x+z∥+∥y+w∥ ≥ 2d. Since 2∥y+w∥ = ∥x+y+z+w+y+w−x−z∥ ≤ ∥x + y + z + w∥ + ∥y + w∥ + ∥x + z∥, we get 2∥y + w∥ − ∥x + z∥ ≤ ∥x + y + z + w∥ + ∥y + w∥. Since ∥x + z∥ = ∥x + y + z + w − y − w∥ ≤ ∥x + y + z + w∥ + ∥y + w∥, we have (4.7)

max{∥x + z∥, 2∥y + w∥ − ∥x + z∥} ≤ ∥x + y + z + w∥ + ∥y + w∥.

If ∥x + z∥ < d, then, since ∥x + z∥ + ∥y + w∥ ≥ 2d, we get 2∥y + w∥ > 2d = d + d > d + ∥x + z∥ and 2∥y + w∥ − ∥x + z∥ > d. So we have (4.8)

max{∥x + z∥, 2∥y + w∥ − ∥x + z∥} ≥ d.

By (4.7) and (4.8), we have ∥x + y + z + w∥ + ∥y + w∥ ≥ d. So it follows from (4.5) that (4.9)

∥f (rx + y, rz + w) + rsf (x, z) − rf (x + y, z + w) − sf (y, w)∥ ≤ δ

for all x, y, z, w ∈ X with ∥x + z∥ + ∥y + w∥ ≥ 2d. So (4.10)

∥f (ry + x, rw + z) + rsf (y, w) − rf (x + y, z + w) − sf (x, z)∥ ≤ δ

for all x, y, z, w ∈ X with ∥x + z∥ + ∥y + w∥ ≥ 2d.( ) Let x, y, z, w ∈ X with ∥x + z∥ + ∥y + w∥ ≥ 4d 1/|r| + 1 − 1/|r| . If ∥y + w∥ > 2d/|r|, then (4.11)

∥x + z∥ + ∥x + ry + z + rw∥ ≥ |r|(∥y + w∥) ≥ 2d.

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If ∥y + w∥ ≤ 2d/|r|, then ∥x + z∥ ≥ 2d(1/|r| + 2|1 − 1/|r||) and (4.12)

∥x + z∥ + ∥x + ry + z + rw∥ ≥ 2∥x + z∥ − |r| · ∥y + w∥ ≥

(

) 1 2 + 4 1 − − 1 ≥ 2d. |r| |r|

Therefore we get that ∥x + z∥ + ∥x + ry + z + rw∥ ≥ 2d from (4.11) and (4.12). Hence by (4.9) we have (4.13) ∥f (r(x + y) + x, r(z + w) + z) + rsf (x, z) − rf (2x + ry, 2z + rw) − sf (x + ry, z + rw)∥ ≤ δ ) ) ( ( for all x, y, z, w ∈ X with ∥x + z∥ + ∥y + w∥ ≥ 4d 1/|r| + 1 − 1/|r| . Set M := 4d 1/|r| + 1 − 1/|r| . Then M ≥ 2d, ∥2x + 2z∥ + ∥y + w∥ ≥ M ≥ 4d (4.14) ∥x + y + z + w∥ + ∥x + z∥ ≥ 2 for all x, y, z, w ∈ X with ∥x + z∥ + ∥y + w∥ ≥ M . From (4.9) and (4.10), we get the following inequalities: ∥f (r(x + y) + x, r(z + w) + z) + rsf (x + y, z + w) − rf (2x + y, 2z + w) − sf (x, z)∥ ≤ δ, ∥rf (ry + 2x, rw + 2z) + r2 sf (y, w) − r2 f (2x + y, 2z + w) − rsf (2x, 2z)∥ ≤ δ|r|, ∥sf (ry + x, rw + z) + rs2 f (y, w) − rsf (x + y, z + w) − s2 f (x, z)∥ ≤ δ|s|. Using (4.13) and the above three inequalities, we get (4.15)

∥f (2x + y, 2z + w) + 2f (x, z) + f (y, w) − 2f (x + y, z + w) − f (2x, 2z)∥ ≤

2 + |r| + |s| δ |rs|

for all x, y, z, w ∈ X with ∥x + z∥ + ∥y + w∥ ≥ M . If x, y, z, w ∈ X with ∥x + z∥ + ∥y + w∥ ≥ 2M , then ∥x + z∥ + ∥y + w − x − z∥ ≥ M . So it follows from (4.15) that (4.16)

∥f (x + y, z + w) + 2f (x, z) + f (y − x, w − z) − 2f (y, w) − f (2x, 2z)∥ ≤

2 + |r| + |s| δ |rs|

for all x, y, z, w ∈ X with ∥x + z∥ + ∥y + w∥ ≥ 2M . Letting y = 0 and w = 0 in (4.16), we get ∥4f (x, z) − f (2x, 2z) − 2f (0, 0)∥ ≤

(4.17)

2 + |r| + |s| δ |rs|

for all x, z ∈ X with ∥x+z∥ ≥ 2M . Letting x = 0 and z = 0 (and y, w ∈ X with ∥y∥ ≥ 2M, ∥w∥ ≥ 2M ) in (4.16), we get ( ) (4.18) ∥f (0, 0)∥ ≤ (2 + |r| + |s|)/|rs| δ. Therefore it follows from (4.16), (4.17) and (4.18) that ∥f (x + y,z + w) + f (y − x, w − z) − 2f (x, z) − 2f (y, w)∥ ≤ ∥f (x + y, z + w) + 2f (x, z) + f (y − x, w − z) − 2f (y, w) − f (2x, 2z)∥ + ∥4f (x, z) − f (2x, 2z) − 2f (0, 0)∥ + 2∥f (0, 0)∥ ≤

4(2 + |r| + |s|) δ |rs|

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for all x, y, z, w ∈ X with ∥x + z∥ ≥ 2M . Since f (x, y) = f (−x, −y) for all x, y ∈ X, the above inequality holds for all x, y, z, w ∈ X with ∥y + w∥ ≥ 2M . Therefore ∥f (x + y, z + w) + f (y − x, w − z) − 2f (x, z) − 2f (y, w)∥ ≤

4(2 + |r| + |s|) δ |rs|

for all x, y, z, w ∈ X with ∥x + z∥ + ∥y + w∥ ≥ 4M . This completes the proof by letting K := 4M .



Theorem 4.4 Let d > 0 and δ ≥ 0 be given. Assume that a mapping f : X × X → Y such that f (x, y) = f (−x, −y) and (4.5) for all x, y, z, w ∈ X with ∥x + z∥ + ∥y + w∥ ≥ d. Then there exists K > 0 such that f satisfies ∥f (x + y, z + w) + f (x − y, z − w) − 2f (x, z) − 2f (y, w)∥ ≤

19(2 + |r| + |s|) δ |rs|

for all x, y, z, w ∈ X. Proof. By Theorem 4.3, there exists K > 0 such that f satisfies (4.6) for all x, y, z, w ∈ X with ∥x + z∥ + ∥y + w∥ > K. By (4.4) and (4.18), we get that 18(2 + |r| + |s|) δ + ∥f (0, 0)∥ |rs| 19(2 + |r| + |s|) δ ≤ |rs|

∥f (x + y, z + w) + f (x − y, z − w) − 2f (x, z) − 2f (y, w)∥ ≤

for all x, y, z, w ∈ X.



Theorem 4.5 Let d > 0 and δ ≥ 0 be given. Assume that a mapping f : X × X → Y such that (4.5) and f (x, y) = f (−x, −y) for all x, y, z, w ∈ X with ∥x + z∥ + ∥y + w∥ ≥ d. Then there exists a unique quadratic mapping F : X × X → Y such that F (x, y) = limn→∞ 4−n f (2n x, 2n y) and ∥f (x, y) − Q(x, y)∥ ≤

19(2 + |r| + |s|) δ 3|rs|

for all x, y ∈ X. 

Proof. The result follows from Theorem 4.1 and Theorem 4.4.

Corollary 4.6. Let r and s be rational numbers and a mapping f : X × X → Y satisfy f (x, y) = f (−x, −y) for all x, y ∈ X. Then f is quadratic if and only if the asymptotic condition (4.19) ∥f (rx + sy, rz + sw) + rsf (x − y, z − w) − rf (x, z) − sf (y, w)∥ → 0 as ∥x + z∥ + ∥y + w∥ → ∞ holds true. Proof. The asymptotic condition (4.19) is equivalent to the condition that there exists a sequence {δn } monotonically decreasing to 0 such that (4.20)

∥f (rx + sy, rz + sw) + rsf (x − y, z − w) − rf (x, z) − sf (y, w)∥ ≤ δn

for all x, y, z, w ∈ X with ∥x + z∥ + ∥y + w∥ ≥ n.

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J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 20, NO.2, 2016, COPYRIGHT 2016 EUDOXUS PRESS, LLC

WON-GIL PARK AND JAE-HYEONG BAE

10

It follows from (4.20) and Theorem 4.4 that there exists a unique quadratic mapping Qn : X × X → Y such that 19(2 + |r| + |s|) (4.21) ∥f (x, y) − Qn (x, y)∥ ≤ δn |rs| for all x, y ∈ X. Since {δn } is monotonically decreasing, the quadratic mapping Qm satisfies (4.21) for all m ≥ n. The uniqueness of Qn implies Qm = Qn for all m ≥ n. By letting n → ∞ in (4.21), we conclude that f is quadratic. The converse is trivial.  Acknowledgment This research was supported by Basic Science Research Program through the National Research Foundation of Korea(NRF) funded by the Ministry of Education, Science and Technology(grant number 2014014135) References [1] J. Acz´el, and J. Dhombres, Functional Equations in Several Variables, Cambridge Univ. Press, Cambridge, 1989. [2] T. Aoki, On the stability of the linear transfomation in Banach spaces, J. Math. Soc. Japan 2 (1950), 64–66. [3] J.-H. Bae and K.-W. Jun, On the generalized Hyers-Ulam-Rassias stability of an n-dimensional quadratic functional equation, J. Math. Anal. Appl. 258 (2001), 183–193. [4] J.-H. Bae and W.-G. Park, A functional equation originating from quadratic forms, J. Math. Anal. Appl. 326 (2007), 1142–1148. [5] D. G. Bourgin, Classes of transformations and bordering transformations, Bull. Amer. Math. Soc. 57 (1951), 223– 237. [6] P. M. Gruber, Stability of isometries, Trans. Amer. Math. Soc. 245 (1978), 263–277. [7] D. H. Hyers, On the stability of the linear functional equation, Proc. Nat. Acad. Sci. 27 (1941), 222–224. [8] S.-M. Jung , On the Hyers-Ulam stability of the functional equations that have the quadratic property, J. Math. Anal. Appl. 222 (1998), 126–137. [9] A. Najati and S.-M. Jung , Approximate quadratic mappings on restricted domains, J. Inequal. Appl. 2010 (2010), Art. No. 503458. [10] W.-G. Park and J.-H. Bae, On a bi-quadratic functional equation and its stability, Nonlinear Anal. 62 (2005), 643–654. [11] Th. M. Rassias, On the stability of linear mappings in Banach spaces, Proc. Amer. Math. Soc. 72 (1978), 297–300. [12] S. M. Ulam, A Collection of Mathematical Problems, Interscience Publishers, New York, 1968, p.63. Won-Gil Park, Department of Mathematics Education, College of Education, Mokwon University, Daejeon 302-729, Republic of Korea E-mail address: [email protected] Jae-Hyeong Bae, Humanitas College, Kyung Hee University, Yongin 446-701, Republic of Korea E-mail address: [email protected]

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TABLE OF CONTENTS, JOURNAL OF COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 20, NO. 2, 2016

A Recurrent Neural Fuzzy Network, George A. Anastassiou, and Iuliana F. Iatan,…………213 Qualitative Behavior of some Rational Difference Equations, H. El-Metwally, and E. M. Elsayed,……………………………………………………………………………………….226 Worse-Case Conditional Value-at-Risk for Asymmetrically Distributed Asset Scenarios Returns, Zhifeng Dai, Donghui Li, and Fenghua Wen,………………………………………..……….237 A Note on the Interval-Valued Similarity Measure and the Interval-Valued Distance Measure Induced by the Choquet Integral with Respect to an Interval-Valued Capacity, Jeong Gon Lee and Lee-Chae Jang,……………………………………………………………………………252 n-Jordan *-Derivations on Induced Fuzzy C*-Algebras, Gang Lu, Yanduo Wang, and Pengyu Ye,………………………………………………………………………………………………266 Global Stability Analysis of a Delayed Viral Infection Model With Antibodies and General Nonlinear Incidence Rate, A. M. Elaiw, N. H. AlShamrani, and M. A. Alghamdi,……………277 Stability of Generalized Cubic Set-Valued Functional Equations, Dongseung Kang,…………296 A New Regularity (p-Regularity) of Stratified L-Generalized Convergence Spaces, Lingqiang Li, and Qingguo Li,………………………………………………………………………………..307 Uni-Soft Filters and Uni-Soft G-Filters in Residuated Lattices, Young Bae Jun, and Seok Zun Song,……………………………………………………………………………………………319 Mathematical Analysis of a General Viral Infection Model With Immune Response, N. H. AlShamrani, A. M. Elaiw and M. A. Alghamdi,……………………………………………….335 Newton's Method for Computing the Fifth Roots of p-Adic Numbers, Y.H. Kim, H.M. Kim, and J. Choi,………………………………………………………………………………………….353 Solution of the Ulam Stability Problem for Euler-Lagrange (α, β; k)-Quadratic Mappings, S.A. Mohiuddine, John Michael Rassias, and Abdullah Alotaibi,………………………………….363 Some Integral Inequalities via (h−(α,m))−Logarithmically Convexity, Jianhua Chen, and Xianjiu Huang,…………………………………………………………………………………………374 On Gosper's q-Trigonometric Function, Mahmoud Jafari Shah Belaghi, and Nuri Kuruoglu,.381

TABLE OF CONTENTS, JOURNAL OF COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 20, NO. 2, 2016 (continued) Approximate Quadratic Forms on Restricted Domains, Won-Gil Park and Jae-Hyeong Bae,.388

Volume 20, Number 3 ISSN:1521-1398 PRINT,1572-9206 ONLINE

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Journal of Computational Analysis and Applications EUDOXUS PRESS,LLC

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FIXED POINTS IN TOPOLOGICAL VECTOR SPACE(tvs)VALUED CONE METRIC SPACES Muhammd Arshad([email protected]) Department of mathematics, International Islamic University, H-10, Islamabad-44000, Pakistan. Abstract: We use the notion of topological vector space valued cone metric space and generalized a common …xed point theorem of a pair of mappings satisfying a generalized contractive type condition. Our results extend some well-known recent results in the literature. _____________________________ 2010 Mathematics Subject Classi…cation: 47H10; 54H25. Keywords and Phrases: Topological vector space valued;cone metric space; nonnormal cones; …xed point; common …xed point. ______________________________

1

Introduction and Preliminaries

Many authors [1, 3, 4, 6, 17, 10, 11, 12, 13, 14, 15, 16, 17, 18, 21] studied …xed points results of mappings satisfying contractive type condition in Banach space valued cone metric spaces. The class of tvs-cone metric spaces is bigger than the class of cone metric spaces studied in [2, 7, 8, 19, 20]. Recently Azam et al. [5] obtain common …xed points of mappings satisfying a generalized contractive type condition in tvs-cone metric spaces. In this paper we continue these investigations to generalize the results in [1, 10]. Let (E; ) be always a topological vector space (tvs) and P a subset of E. Then, P is called a cone whenever (i) P is closed, non-empty and P 6= f0g, (ii) ax + by 2 P for all x; y 2 P and non-negative real numbers a; b, (iii) P \ ( P ) = f0g. For a given cone P E, we can de…ne a partial ordering with respect to P by x y if and only if y x 2 P . x < y will stand for x y and x 6= y, while x y will stand for y x 2 intP , where intP denotes the interior of P . De…nition 1 Let X be a non-empty set. Suppose the mapping d : X X ! E satis…es (d1 ) 0 d(x; y) for all x; y 2 X and d(x; y) = 0 if and only if x = y, (d2 ) d(x; y) = d(y; x) for all x; y 2 X, (d3 ) d(x; y) d(x; z) + d(z; y) for all x; y; z 2 X. Then d is called a topological vector space-valued cone metric on X and (X; d) is called a topological vector space-valued cone metric space. 1

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If E is a real Banach space then (X; d) is called (Banach space valued) cone metric space [1, 6, 17, 10, 21] De…nition 2 [7] Let (X; d) be a tvs-cone metric space, x 2 X and fxn gn 1 a sequence in X. Then (i) fxn gn 1 converges to x whenever for every c 2 E with 0 c there is a natural number N such that d(xn ; x) c for all n N . We denote this by limn!1 xn = x or xn ! x. (ii) fxn gn 1 is a Cauchy sequence whenever for every c 2 E with 0 c there is a natural number N such that d(xn ; xm ) c for all n; m N . (iii) (X; d) is a complete cone metric space if every Cauchy sequence is convergent. Lemma 3 [7] Let (X; d) be a tvs-cone metric space, P be a cone. Let fxn g be a sequence in X and fan g be a sequence in P converging to 0. If d(xn ; xm ) an for every n 2 N with m > n, then fxn g is a Cauchy sequence. The …xed point theorems and other results, in the case of cone metric spaces with non-normal solid cones, cannot be proved by reducing to metric spaces. Further, the vector valued function cone metric is not continuous in the general case. Remark 4 [7] Let A; B; C; D; E be non negative real numbers with A + B + C + D + E < 1; B = C or D = E: If = (A + B + D)(1 C D) 1 and = (A + C + E)(1 B E) 1 , then < 1.

2

Common Fixed Points

The following theorem improves/generalizes the results in [1, 7]. Theorem 5 Let (X; d) be a complete topological vector space-valued cone metric space, P be a cone and m; n be positive integers. If mappings F; G : X ! X satis…es: d(F x; Gy)

A d(x; y)+B d(x; F x)+Cd(y; Gy)+D d(x; Gy)+E d(y; F x) (2.1)

for all x; y 2 X, where A; B; C; D; E are non negative real numbers with A + B + C + D + E < 1; B = C or D = E: Then F and G have a unique common …xed point. Proof. For x0 2 X and k

0, de…ne x2k+1 x2k+2

= F x2k = Gx2k+1 :

2

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Then, d(x2k+1 ; x2k+2 )

= d(F x2k ; Gx2k+1 ) Ad(x2k ; x2k+1 ) + Bd(x2k ; F x2k ) + Cd(x2k+1 ; Gx2k+1 ) +Dd(x2k ; Gx2k+1 ) + Ed(x2k+1 ; F x2k ) [A + B] d(x2k ; x2k+1 ) + Cd(x2k+1 ; x2k+2 ) + D d(x2k ; x2k+2 ) [A + B + D] d(x2k ; x2k+1 ) + [C + D] d(x2k+1 ; x2k+2 ):

It implies that [1

C

D]d(x2k+1 ; x2k+2 )

[A + B + D] d(x2k ; x2k+1 ):

That is, d(x2k+1 ; x2k+2 ) where

=

d(x2k ; x2k+1 );

A+B+D . Similarly, 1 C D

d(x2k+2 ; x2k+3 )

= d(F x2k+2 ; Gx2k+1 ) Ad(x2k+2 ; x2k+1 ) + B d(x2k+2 ; F x2k+2 ) + Cd(x2k+1 ; Gx2k+1 ) +Dd(x2k+2 ; Gx2k+1 ) + E d(x2k+1 ; F x2k+2 ) A d(x2k+2 ; x2k+1 ) + B d(x2k+2 ; x2k+3 ) + Cd(x2k+1 ; x2k+2 ) +D d(x2k+2 ; x2k+2 ) + E d(x2k+1 ; x2k+3 ) [A + C + E] d(x2k+1 ; x2k+2 ) + [B + E] d(x2k+2 ; x2k+3 );

which implies d(x2k+2 ; x2k+3 ) with

=

d(x2k+1 ; x2k+2 )

A+C +E . Now by induction, we obtain for each k = 0; 1; 2; : : : 1 B E d(x2k+1 ; x2k+2 )

d(x2k ; x2k+1 ) ( ) d(x2k 1 ; x2k ) ( ) d(x2k 2 ; x2k 1 ) ( )k d(x0 ; x1 )

and d(x2k+2 ; x2k+3 )

d(x2k+1 ; x2k+2 ) (

k+1

)

d(x0 ; x1 ):

3

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For p < q and by Remark 1.4, we have d(x2p+1 ; x2q+1 )

d(x2p+1 ; x2p+2 ) + d(x2p+2 ; x2p+3 ) + d(x2p+3 ; x2p+4 ) + + d(x2q ; x2q+1 ) 2 3 q 1 q X X i i 4 ( ) + ( ) 5 d(x0 ; x1 ) i=p

i=p+1

p

( )p+1 d(x0 ; x1 ) 1 1 ( )p (1 + ) d(x0 ; x1 ): 1 (

)

+

In analogous way, we deduce d(x2p ; x2q+1 ) d(x2p ; x2q )

(1 + )

(1 + )

and d(x2p+1 ; x2q )

(1 + )

)p

( 1

)p

( 1

d(x0 ; x1 ); d(x0 ; x1 )

)p

( 1

d(x0 ; x1 ):

Hence, for 0 < n < m d(xn ; xm )

an

p

( ) d(x0 ; x1 ) with p the integer part of n=2: Fix 0 c 1 and choose a symmetric neighborhood V of 0 such that c + V intP . Since an ! 0 as n ! 1, by Lemma 1.3, we deduce that fxn g is a Cauchy sequence. Since X is a complete, there exist u 2 X such that xn ! u: Fix 0 c and choose n0 2 N be such that where an = (1 + )

d(u; x2n ) for all n

c ; 3K

d(x2n

1 ; x2n )

c ; 3K

d(u; x2n

c 3K

1)

n0 , where K = max

1+D A+E ; ; 1 B E 1 B E 1

C B

E

:

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Now, d(u; F u)

d(u; x2n ) + d(x2n ; F u) d(u; x2n ) + d(Gx2n 1 ; F u) d(u; x2n ) + A d(u; x2n 1 ) + B d(u; F u) + Cd(x2n 1 ; Gx2n 1 ) +D d(u; Gx2n 1 ) + E d(x2n 1 ; F u) d(u; x2n ) + A d(u; x2n 1 ) + B d(u; F u) + Cd(x2n 1 ; x2n ) +D d(u; x2n ) + E d(x2n 1 ; u) + E d(u; F u)] (1 + D) d(u; x2n ) + (A + E) d(u; x2n 1 ) + Cd(x2n 1 ; x2n ) +(B + E) d(u; F u):

So, d(u; F u)

K d(u; x2n ) + K d(u; x2n c c c + + =c 3 3 3

Hence d(u; F u)

1)

+ K d(x2n

1 ; x2n )

c p

for every p 2 N. From

c d(u; F u) 2 intP; p being P closed, as p ! 1, we deduce d(u; F u) 2 P and so d(u; F u) = 0. This implies that u = F u: Similarly, by using the inequality, d(u; Gu)

d(u; x2n+1 ) + d(x2n+1 ; Gu);

we can show that u = Gu; which in turn implies that u is a common …xed point of F; G and, that is u = F u = Gu: For uniqueness, assume that there exists another point u in X such that u = T u = Gu for some u in X: From d(u; u )

= d(F u; Gu ) Ad(u; u ) + Bd(u; F u) + Cd(u ; Gu ) +Dd(u; Gu ) + Ed(u ; F u) Ad(u; u ) + Bd(u; u) + Cd(u ; u ) +D d(u; u ) + Ed(u; u ) (A + D + E)d(u; u );

we obtain that u = u: By substituting D = E = 0 in the Theorem 2.1, we obtain the following result. 5

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Corollary 6 Let (X; d) be a complete topological vector space-valued cone metric space, P be a cone and m; n be positive integers. If mappings F; G : X ! X satis…es: d(F x; Gy)

A d(x; y) + B d(x; F x) + Cd(y; Gy)

(2.2)

for all x; y 2 X, where A; B; C are non negative real numbers with A+B+C < 1: Then F and G have a unique common …xed point. By substituting B = C = 0 in the Theorem 2.1, we obtain the following result. Corollary 7 Let (X; d) be a complete topological vector space-valued cone metric space, P be a cone and m; n be positive integers. If mappings F; G : X ! X satis…es: d(F x; Gy)

A d(x; y) + D d(x; Gy) + E d(y; F x)

(2.3)

for all x; y 2 X, where A; D; E are non negative real numbers with A+D+E < 1: Then F and G have a unique common …xed point. By substituting F = T m ; G = T n in the Theorem 2.1, we obtain the following result. Corollary 8 [7] Let (X; d) be a complete topological vector space-valued cone metric space, P be a cone and m; n be positive integers. If a mapping T : X ! X satis…es: d(T m x; T n y)

A d(x; y)+B d(x; T m x)+Cd(y; T n y)+D d(x; T n y)+E d(y; T m x) (2.4) for all x; y 2 X, where A; B; C; D; E are non negative real numbers with A + B + C + D + E < 1; B = C or D = E: Then T has a unique …xed point.

Corollary 9 [1] Let (X; d) be a complete Banach space-valued cone metric space, P be a cone. If a mapping F; G : X ! X satis…es: d(F x; Gy)

pd(x; y) + q [d(x; F x) + d(y; Gy)] + r [d(x; Gy) + E d(y; F x)] (2.5) for all x; y 2 X, where p; q; r are non negative real numbers with p + 2q + 2r < 1: Then F and G have a unique common …xed point.

3

Multivalued Fixed point results in tvs-valued cone metric spaces

In the sequel, let E be a locally convex Hausdor¤ tvs with its zero vector , P be a proper, closed and convex pointed cone in E with int P 6= ; and 4 denotes the induced partial ordering with respect to P . 6

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According to [5] let (X; d) be a tvs-valued cone metric space with a solid cone P and CB(X) be a collection of nonempty closed and bounded subsets of X. Let T : X ! CB(X) be a multi-valued mapping. For any x 2 X; A 2 CB(X), de…ne a set Wx (A) as follows: Wx (A) = fd(x; a) : a 2 Ag: Thus, for any x; y 2 X, we have Wx (T y) = fd(x; u) : u 2 T yg: De…nition 10 [9] Let (X; d) be a cone metric space with the solid cone P . A multi-valued mapping S : X ! 2E is said to be bounded from below if, for any x 2 X, there exists z(x) 2 E such that Sx

z(x)

P:

De…nition 11 [9] Let (X; d) be a cone metric space with the solid cone P . A cone P is said to be complete if, for any bounded from above and nonempty subset A of E, sup A exists in E. Equivalently, a cone P is complete if, for any bounded from below and nonempty subset A of E, inf A exists in E. De…nition 12 [5] Let (X; d) be a tvs-valued cone metric space with the solid cone P: A multi-valued mapping T : X ! CB(X) is said to have the lower bound property ( l.b. property) on X if, for any x 2 X, the multi-valued mapping Sx : X ! 2E de…ned by Sx (y) = Wx (T y) is bounded from below, that is, for any x; y 2 X, there exists an element `x (T y) 2 E such that Wx (T y) `x (T y) P: `x (T y) is called the lower bound of T associated with (x; y) : De…nition 13 [5] Let (X; d) be a tvs-valued cone metric space with the solid cone P: A multi-valued mapping T : X ! CB(X) is said to have the greatest lower bound property (for short, g.l.b. property) on X if the greatest lower bound of Wx (T y) exists in E for all x; y 2 X: We denote d(x; T y) by the greatest lower bound of Wx (T y); that is, d(x; T y) = inffd(x; u) : u 2 T yg: According to [20], we denote s (p) = fq 2 E : p 4 qg for all q 2 E and s (a; B) = [ s (d (a; b)) = [ fx 2 E : d (a; b) 4 xg b2B

b2B

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for all a 2 X and B 2 CB(X). For any A; B 2 CB(X), we denote s (A; B) =

\ s (a; B) \

a2A

\ s (b; A) :

b2B

Remark 14 [20] Let (X; d) be a tvs-valued cone metric space. If E = R and P = [0; +1); then (X; d) is a metric space. Moreover, for any A; B 2 CB(X), H(A; B) = inf s(A; B) is the Hausdor¤ distance induced by d: Now we present the following theorem regarding the common …xed point of multivalued mapping with g.l.b property. Theorem 15 Let (X; d) be a complete tvs-valued cone metric space with the solid (normal or non-normal) cone P and let S; T : X ! CB(X) be multivalued mappings with g.l.b property such that A d(x; y)+B d(x; Sx)+Cd(y; T y)+Dd(x; T y)+Ed(y; Sx)) 2 s (Sx; T y) (2.6) ¤or all x; y 2 X, where A; B; C; D; E are non negative real numbers with A+ B + C + D + E < 1: Then S and T have common …xed point. Proof. Let x0 be an arbitrary point in X and x1 2 Sx0 : From (2.6), we have Ad (x0 ; x1 )+B(x0 ; Sx0 )+Cd(x1 ; T x1 )+Dd(x0 ; T x1 )+Ed(x1 ; Sx0 ) 2 s (Sx0 ; T x1 ) : This implies that Ad (x0 ; x1 )+B(x0 ; Sx0 )+Cd(x1 ; T x1 )+Dd(x0 ; T x1 )+Ed(x1 ; Sx0 ) 2

\

x2Sx0

s (x; T x1 )

and Ad (x0 ; x1 )+B(x0 ; Sx0 )+Cd(x1 ; T x1 )+Dd(x0 ; T x1 )+Ed(x1 ; Sx0 ) 2 s (x; T x1 ) f or all x 2 Sx0 : Since x1 2 Sx0 ; so we have Ad (x0 ; x1 )+B(x0 ; Sx0 )+Cd(x1 ; T x1 )+Dd(x0 ; T x1 )+Ed(x1 ; Sx0 ) 2 s (x1 ; T x1 ) and Ad (x0 ; x1 )+B(x0 ; Sx0 )+Cd(x1 ; T x1 )+Dd(x0 ; T x1 )+Ed(x1 ; Sx0 ) 2 s (x1 ; T x1 ) =

[ s (d (x1 ; x)) :

x2T x1

So there exists some x2 2 T x1 ; such that Ad (x0 ; x1 )+B(x0 ; Sx0 )+Cd(x1 ; T x1 )+Dd(x0 ; T x1 )+Ed(x1 ; Sx0 ) 2 s (d(x1 ; x2 )): That is d(x1 ; x2 )

Ad (x0 ; x1 )+B(x0 ; Sx0 )+Cd(x1 ; T x1 )+Dd(x0 ; T x1 )+Ed(x1 ; Sx0 ):

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By using the greatest lower bound property (g.l.b property) of S and T; we get d (x1 ; x2 )

Ad (x0 ; x1 ) + B(x0 ; x1 ) + Cd(x1 ; x2 ) + Dd(x0 ; x2 ) + Ed(x1 ; x1 );

which implies that d (x1 ; x2 )

(A + B + D)d (x0 ; x1 ) + (C + D)d(x1 ; x2 )

which further implies that d (x1 ; x2 )

A+B+D d (x0 ; x1 ) : 1 C D

Similarly from (2.6), we get Ad (x1 ; x2 )+B(x2 ; Sx2 )+Cd(x1 ; T x1 )+Dd(x2 ; T x1 )+Ed(x1 ; Sx2 ) 2 s (T x1 ; Sx2 ) : This implies that Ad (x1 ; x2 )+B(x2 ; Sx2 )+Cd(x1 ; T x1 )+Dd(x2 ; T x1 )+Ed(x1 ; Sx2 ) 2

\

x2T x1

s (x; Sx2 )

and Ad (x1 ; x2 )+B(x2 ; Sx2 )+Cd(x1 ; T x1 )+Dd(x2 ; T x1 )+Ed(x1 ; Sx2 ) 2 s (x; Sx2 ) for all x 2 T x1 : Since x2 2 T x1 ; so we have Ad (x1 ; x2 )+B(x2 ; Sx2 )+Cd(x1 ; T x1 )+Dd(x2 ; T x1 )+Ed(x1 ; Sx2 ) 2 s (x2 ; Sx2 ) and Ad (x1 ; x2 )+B(x2 ; Sx2 )+Cd(x1 ; T x1 )+Dd(x2 ; T x1 )+Ed(x1 ; Sx2 ) 2 s (x2 ; Sx2 ) =

[ s (d (x2 ; x)) :

x2Sx2

So there exists some x3 2 Sx2 ; such that Ad (x1 ; x2 )+B(x2 ; Sx2 )+Cd(x1 ; T x1 )+Dd(x2 ; T x1 )+Ed(x1 ; Sx2 ) 2 s (d(x2 ; x3 )): That is d(x2 ; x3 )

Ad (x1 ; x2 )+B(x2 ; Sx2 )+Cd(x1 ; T x1 )+Dd(x2 ; T x1 )+Ed(x1 ; Sx2 ):

By using the greatest lower bound property (g.l.b property) of S and T; we get d(x2 ; x3 )

Ad (x1 ; x2 ) + B(x2 ; x3 ) + Cd(x1 ; x2 ) + Dd(x2 ; x2 ) + Ed(x1 ; x3 ):

which implies that d(x2 ; x3 )

(A + C + E)d (x1 ; x2 ) + (B + E)(x2 ; x3 ):

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This further implies A+C +E d (x1 ; x2 ) : 1 B E

d(x2 ; x3 )

A+C+E Let = maxf A+B+D < 1: Thus inductively, one can easily 1 C D ; 1 B E g. Then construct a sequence fxn g in X such that

x2n+1 2 Sx2n ; and

x2n+2 2 T x2n+1

d(x2n ; x2n+1 ) 4 d(x2n

1 ; x2n ):

for each n 0. We assume that xn 6= xn+1 for each n 0. Otherwise, there exists n such that x2n = x2n+1 : Then x2n 2 Sx2n and x2n is a …xed point of S and hence a …xed point of T: Similarly, if x2n+1 = x2n+2 for some n, then x2n+1 is a common …xed point of T and S: Similarly, one can show that d(x2n+1 ; x2n+2 ) 4 d(x2n ; x2n+1 ): Thus we have d(xn ; xn+1 ) 4 d(xn for each n

1 ; xn )

4

2

d(xn

2 ; xn 1 )

4

4

n

d(x0 ; x1 )

0. Now, for any m > n; consider

d(xm ; xn ) 4 d(xn ; xn+1 ) + d(xn+1 ; xn+2 ) + + d(xm n n+1 m 1 4 + + + d(x0 ; x1 )

for all n

xm )

n

4 Let c+V

1;

d(x0 ; x1 ):

1

c be given and choose a symmetric neighborhood h Vn iof such that intP . Also, choose a natural number k1 such that 1 d(x0 ; x1 ) 2 V k1 . Then

n

1

d(x1 ; x0 )

d(xm ; xn ) 4

c for all n

k1 . Thus we have

n

d(x0 ; x1 )

1

c

for all m > n. Therefore, fxn g is a Cauchy sequence. Since X is complete, there exists 2 X such that xn ! : Choose a natural number k2 such that 1+E d( ; x2n+1 ) 1 C for all n

c ; 3

A 1

k2 . Then, for all n

C

d(x2n ; v)

c B and d(x2n ; x2n ) 3 1 C

c 3 (2.7)

k2 , we have

Ad(x2n ; v)+Bd(x2n ; Sx2n )+Cd(v; T v)+Dd(x2n ; T )+Ed( ; Sx2n ) 2 s (Sx2n ; T ) :

10

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This implies that Ad(x2n ; v)+Bd(x2n ; Sx2n )+Cd(v; T v)+Dd(x2n ; T )+Ed( ; Sx2n ) 2

\

x2Sx2n

s (x; T v)

and we have Ad(x2n ; v)+Bd(x2n ; Sx2n )+Cd(v; T v)+Dd(x2n ; T )+Ed( ; Sx2n ) 2 s (x; T v) f or all x 2 Sx2n : Since x2n+1 2 Sx2n ; so we have Ad(x2n ; v)+Bd(x2n ; Sx2n )+Cd(v; T v)+Dd(x2n ; T )+Ed( ; Sx2n ) 2 s x2n+1 ; T v : By de…nition, we obtain Ad(x2n ; v)+Bd(x2n ; Sx2n )+Cd(v; T v)+Dd(x2n ; T )+Ed( ; Sx2n ) 2 s x2n+1 ; T v = There exists some

n

[ s d x2n+1 ; u0

u0 2T u

2 T v such that

Ad(x2n ; v)+Bd(x2n ; Sx2n )+Cd(v; T v)+Dd(x2n ; T )+Ed( ; Sx2n ) 2 s x2n+1 ; T v 2 s d(x2n+1 ; that is d(x2n+1 ;

n)

Ad(x2n ; v)+Bd(x2n ; Sx2n )+Cd(v; T v)+Dd(x2n ; T )+Ed( ; Sx2n ):

By using the greatest lower bound property (g.l.b property) of S and T; we have d(x2n+1 ; vn )

Ad(x2n ; v)+Bd(x2n ; x2n )+Cd(v;

n )+Dd(x2n ;

n )+Ed(

; x2n+1 ):

Now by using the triangular inequality, we get d (x2n+1 ;

n)

Ad(x2n ; v)+Bd(x2n ; x2n+1 )+Cd(v; x2n+1 )+Dd(x2n ;

n )+Ed(

; x2n+1 )

and it follows that d (x2n+1 ;

n)

A 1

C

d(x2n ; v) +

B 1

C

d(x2n ; x2n )) +

C +E d( ; x2n+1 ): 1 C

By using again triangular inequality, we get d( ;

n)

d( ; x2n+1 ) + d(x2n+1 ; n ) A B C +E d( ; x2n+1 ) + d(x2n ; v) + d(x2n ; x2n )) + d( ; x2n+1 ) 1 C 1 C 1 C 1+E A B d( ; x2n+1 ) + d(x2n ; v) + d(x2n ; x2n ) 1 C 1 C 1 C c c c + + =c 3 3 3

Thus, we get

c m c c for all m 1 and so m d(v; vn ) 2 P for all m 1. Since m ! as m ! 1 and P is closed, it follows that d(v; vn ) 2 P: But d(v; vn ) 2 P . Therefore, d(v; vn ) = and vn ! v 2 T v; since T v is closed. This implies that v is a common point of S and T . This completes the proof. d(v; vn )

11

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Corollary 16 [5] Let (X; d) be a complete tvs-valued cone metric space with the solid (normal or non-normal) cone P and let S; T : X ! CB(X) be multivalued mappings with g.l.b property such that B d(x; Sx) + Cd(y; T y) 2 s (Sx; T y) ¤or all x; y 2 X, where B; C are non negative real numbers with B + C < 1: Then S and T have common …xed point. Theorem 17 [5] Let (X; d) be a complete tvs-valued cone metric space with the solid (normal or non-normal) cone P and let S; T : X ! CB(X) be multivalued mappings with g.l.b property such that Dd(x; T y) + Ed(y; Sx)) 2 s (Sx; T y) ¤or all x; y 2 X, where D; E are non negative real numbers with D + E < 1: Then S and T have common …xed point.

References [1] M. Abbas and B.E. Rhoades, Fixed and periodic point results in cone metric spaces, Appl. Math. Lett., 22 (2009) 511–515. [2] M. Abbas, Y.J. Cho and T. Nazir, Common …xed point theorems for four mappings in tvs-valued cone metric spaces, J. Math. Inequal., 5(2011), 287–299. [3] M. Abbas and G. Jungck, Common …xed point results for non-commuting mappings without continuity in cone metric spaces, J. Math. Anal. Appl. 341(2008), 416–420. [4] M. Arshad, A. Azam and P. Vetro, Some common …xed point results in cone metric spaces, Fixed Point Theory Appl., 2009, Article ID 493965 (2009), 11 pp. [5] A. Azam, N. Mehmood, Multivalued Fixed Point Theorems in tvs-Cone Metric Spaces, Fixed Point Theory and Appl., 2013, 2013:184. DOI: 10.1186/1687-1812-2013-184. [6] A. Azam, M. Arshad and I. Beg, Common …xed points of two maps in cone metric spaces, Rend. Circ. Mat. Palermo, 57(2008), 433–441. [7] A. Azam, I. Beg and M. Arshad, Fixed point in topological vector spacevalued cone metric spaces, Fixed Point Theory and Appl., 2010, Article ID 604084 (2010), 9 pp. [8] I. Beg, A. Azam and M. Arshad, Common …xed points for maps on topological vector space valued cone metric spaces, Interant. J. Math. Math. Sci., 2009, Article ID 604084 (2009), 8 pp. 12

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[9] S.H. Cho and J.S. Bae, Fixed point theorems for multivalued maps in cone metric spaces, Fixed Point Theory Appl., 87 (2011). [10] L. Huang and X. Zhang, Cone metric spaces and …xed point theorems of contractive mappings, J. Math. Anal. Appl., 332 (2007) 1468–1476. [11] D. Ili´c and V. Pavlovi´c, Common …xed points for maps on cone metric space, J. Math. Anal. Appl., 341(2008), 876–882. [12] Z. Kadelburg and S. Jankovi´c and S. Radenovi´c, A note on the equivalence of some metric and cone metric …xed point results, Appl. Math. Lett. 24 (2011), 370–374. [13] S. Jankovi´c, Z. Kadelburg and S. Radenovi´c, On cone metric spaces, A survey, Nonlinear Anal., 74(2011), 2591–260. [14] M. Khani and M. Pourmahdian, On the metrizability of cone metric spaces, Topology Appl., 158(2011), 190–193. [15] A. Latif and F.Y. Shaddad, Fixed point results for multivalued maps in cone metric spaces, Fixed Point Theory Appl., 2010 (2010), Article ID 941371. [16] S. Radenovic and B.E. Rhoades, Fixed point theorem for two non-self mappings in cone metric spaces, Comp. Math. Appl., 57 (2009), 1701–1707. [17] S. Rezapour and R. Hamlbarani, Some notes on paper “Cone metric spaces and …xed point theorems of contractive mappings", J. Math. Anal. Appl., 345(2008), 719–724. [18] S. Rezapour and R.H. Haghi, Fixed points of multifunctions on cone metric spaces, Numer. Funct. Anal. Optim., 30(2009), 1–8. [19] S. Rezapour, H. Khandani and S.M. Vaezpour, E¢ cacy of cones on topological vector spaces and application to common …xed points of multifunctions, Rend. Circ. Mat. Palermo, 59(2010), 185–197. ´ [20] W. Shatanawi, V. Cojbaš i´c, S. Radenovi´c and A. Al-Rawashdeh, Mizoguchi-Takahashi-type theorems in tvs-cone metric spaces, Fixed Point Theory Appl., 2012, 2012:106. [21] P. Vetro, Common …xed points in cone metric spaces, Rend. Circ. Mat. Palermo, 56(2007), 464–468.

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ON THE TWISTED q-CHANGHEE POLYNOMIALS OF HIGHER ORDER JIN-WOO PARK

Abstract. The q-Changhee polynomials and numbers are introduced by T. Kim et al in [3]. Some interesting properties of those polynomials are derived from umbral calculus (see [4]). In this paper, we consider Witt-type formula for the n-th twisted q-Changhee numbers and polynomials of higher order and derive some new interesting identities and properties of those polynomials and numbers from the Witt-type formula which are related to special polynomials and numbers.

1. Introduction Let p be an odd prime number. Zp , Qp and Cp will denote the ring of p-adic integers, the field of p-adic numbers and the completion of algebraic closure of Qp . The p-adic norm | · |p is normalized by |p|p = p1 . Let C(Zp ) be the space of continuous functions on Zp . For f ∈ C(Zp ), the fermionic p-adic integral on Zp is defined by T.Kim to be N

pX −1 1 I−q (f ) = f (x)dµ−q (x) = lim f (x)(−q)x , (see [6, 7, 9]). N →∞ [pN ]−q Zp x=0

Z

(1.1)

Let f1 (x) = f (x + 1). Then, by (1.1), we get qI−q (f1 ) + I−q (f ) = [2]q f (0), (see [6, 7]).

(1.2)

By (1.2), we easily see that n

q I−q + (−1)

n−1

I−q = [2]q

n−1 X

(−1)n−1−l f (l),

(1.3)

l=0

where fn (x) = f (x + n) and n ≥ 0. It is well known that the twisted q-Euler polynomials are defined by the generating function to be ∞ X [2]q tn xt e = En,ε,q (x) , (see [13]). t 1 + qεe n! n=0

(1.4)

When x = 0, En,ε,q = En,ε,q (0) are called the n−th twisted q-Euler numbers. For ε = 1, En,1,q (x) = En,q (x) are the n-th q-Euler polynomials, and x = 0 , En,1,q (0) = En,q (0) are the n-th q-Euler numbers. 2000 Mathematics Subject Classification. 11S80, 11B68, 05A30. Key words and phrases. Euler numbers, q-Changhee numbers, twisted q-Changhee numbers of higher order. 1

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Indeed, we note that En,1,q (x) = Hn (x| − q), where Hn (x|λ) are the FrobeniusEuler polynomials which are defined by the generating function to be ∞ 1 − λ tx X tn e = Hn (x|λ) , (see [1]). t e −λ n! n=0

Recently, the q-Changhee polynomials are defined by the generating function to be

∞ X [2]q tn x (1 + t) = Ch (x) , (see [10]). n,q 1 + q εt n! n=0

(1.5)

When x = 0, Chn,ε,q = Chn,ε,q (0) are called the q-Changhee numbers, (see [3]). The Stirling number of the first kind is defined by (x)n = x(x − 1) · · · (x − n + 1) =

n X

S1 (n, l)xl , (see [3]).

(1.6)

l=0

The q-Changhee numbers and polynomials are introduced by T. Kim et. al. in [3], and found interesting identities in [5, 8, 11, 12]. In this paper, we consider the twisted q-Changhee numbers and polynomials of order k which are derived from the multivariate fermionic p-adic q-integral of higher order on Zp , and give some relationship between twisted q-Changhee polynomials and numbers of higher-order and special polynomials and numbers. 2. Twisted q-Changhee numbers and polynomials of higher-order For n ∈ N, let Tp be the p-adic locally constant space defined by Tp = ∪ Cpn = lim Cpn , n→∞

n≥1



n

where Cpn = ω|ω p = 1 is the cyclic group of order pn . 1 For ε ∈ Tp , let us take f (x) = (1 + εt)x for |t|p < p− p−1 . Then by (1.2), we get Z ∞ X [2]q tn (1 + εt)x dµ−q (x) = Chn,ε,q (2.1) = qεt + [2]q n! Zp n=0 where Chn,ε,q are called the n-th twisted q-Changhee numbers. From (2.1), we can derive the following equation: Z ∞ X [2]q tn (1 + εt)x+y dµ−q (y) = Chn,ε,q (x) , (1 + εt)x = qεt + [2]q n! Zp n=0

(2.2)

where Chn,ε,q (x) are called the n-th twisted q-Changhee polynomials. Note that Chn,ε,q (0) = Chn,ε,q are n-th twisted q-Changhee numbers. Since  Z Z  ∞ X x+y x+y n (1 + εt) dµ−q (y) = ε dµ−q (y)tn n Zp Z p n=0 (2.3) Z ∞ X tn n = ε (x + y)n dµ−q (y) , n! Zp n=0 by (2.2) and (2.3), we obtained the following theorem.

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ON THE TWISTED q-CHANGHEE POLYNOMIALS OF HIGHER ORDER

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Theorem 2.1. For n ≥ 0, we have Chn,ε,q (x) = εn

Z (x + y)n dµ−q (y). Zp

From (2.1), we note that ∞ X

εn

  n ∞  X x [2]q qε dµ−q (x)tn = = tn . − n qεt + [2] [2] q q Zp n=0

Z

n=0

(2.4)

Thus, by comparing the coefficients on the both sides, we obtain the following theorem. Theorem 2.2. For n ≥ 0, we have n  Z   x q . dµ−q (x) = − [2]q Zp n Replacing t by ∞ X

et −1 ε

in (2.2), we get

En,q (x)

n=0

 n ∞ tn [2]q xt X 1 et − 1 = t e = Chn,ε,q (x) , n! qe − 1 n! ε n=0

where En,q is the n-th q-Euler polynomials and  n X ∞ ∞ X 1 et − 1 1 Chn,ε,q (x) = Chn,ε,q (x) ε−n n! n! ε n! n=0 n=0 ∞ X m X

=

∞ X

tm S2 (m, n) m! m=n −n t

Chn,ε,q (x)S2 (m, n)ε

m=0 n=0

(2.5)

!

m

m!

(2.6)

,

where S2 (m, n) is the Striling number of the second kind. By comparing the coefficients on the both sides of (2.5) and (2.6), we obtain the following theorem. Theorem 2.3. For n ≥ 0, we have n X En,q (x) = Chm,ε,q (x)S2 (n, m)ε−m . m=0

By Theorem 2.1, we easily get Chn,ε,q (x) =εn =εn

Z (x + y)n dµ−q (y) Zp n X l=0

Z S1 (n, l)

(x + y)l dµ−q (y) = εn

Zp

n X

(2.7) S1 (n, l)El,q (x).

l=0

Therefore, by (2.7), we obtain the following theorem. Theorem 2.4. For n ≥ 0, we have Chn,ε,q (x) = εn

n X

S1 (n, l)El,q (x).

l=0

where S1 (n, l) is the Stirling number of the first kind.

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In viewpoint of (2.3), the n−th twisted q-Changhee numbers of the first kind with order k are defined by the generating function to be Z Z n Ch(k) = ε · · · (x1 + · · · + xk )n dµ−q (x1 ) . . . dµ−q (xk ), (2.8) n,ε Zp

Zp

where n is a positive integer. By (2.8), we easily get  Z Z X ∞ ∞  n X x1 + · · · + xk (k) t ··· Chn,ε,q = (εt)n dµ−q (x1 ) · · · dµ−q (xk ) n! n Z Z p p n=0 (2.9) Z Z n=0 = ··· (1 + εt)n dµ−q (x1 ) · · · dµ−q (xk ). Zp

Zp

From (2.1) and (2.9), we have ∞ X

tn = n!

Ch(k) n,ε,q

n=0

and 

[2]q qεt + [2]q

k =

∞ X

X

n=0

l1 +···+lk =n



[2]q qεt + [2]q

k ,

(2.10)

!  n tn . Chl1 ,ε,q · · · Chlk ,ε,q l1 , . . . , l k n!



By simple calculation. we easily see that  k X n   n ∞  [2]q q n k+n−1 t . = n!ε − qεt + [2]q [2]q n! n n=0

(2.11)

(2.12)

Thus, by (2.10) and (2.12), we get  n+k−1 n =(−q)n εn (k + n − 1)n

n n [2]nq Ch(k) n,ε,q =(−q) n!ε

=(−q)n εn



n X

(2.13)

S1 (n, l)(k + n − 1)l .

l=0

Therefore, by (2.10), (2.11) and (2.13), we obtain the following theorem. Theorem 2.5. For n ≥ 0, we have X (k) [2]nq Ckn,ε,q =[2]nq



 n Chi1 ,ε,q · · · Chlk ,ε,q l1 , . . . , l k

l1 +···+lk =n n X =(−q)n εn S1 (n, l)(k

+ n − 1)l .

l=0

From (2.8), we have Z Z (k) n Chn,ε,q =ε ··· (x1 + · · · + xk )n dµ−q (x1 ) · · · dµ−q (xk ) Zp n

=ε

n X l=0

Zp

Z ···

S1 (n, l) Zp

(2.14)

Z

l

(x1 + · · · + xk ) dµ−q (x1 ) · · · dµ−q (xk ). Zp

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ON THE TWISTED q-CHANGHEE POLYNOMIALS OF HIGHER ORDER

5

Now, we observe that  k X Z ∞ n [2]q (x1 +···+xk )t (k) t ··· = e dµ−q (x1 ) · · · dµ−q (xk ) = E , (2.15) n,q qet + 1 n! Zp Zp n=0

Z

(k)

where En,q are the q-Euler numbers of order k. From (2.14) and (2.15), we obtain the following theorem. Theorem 2.6. For n ≥ 0, we have n Ch(k) n,ε,q = ε

n X

(k)

S1 (n, l)El,q .

l=0 t

e −1 ε ,

we get  t n  k X ∞ n e −1 [2]q (k) 1 (k) t = = , Chn,ε,q E n,q n! ε qet + 1 n! n=0 n=0

Replacing t by ∞ X

(2.16)

and ∞ X n=0

Ch(k) n,ε,q

1 n!



et − 1 ε

n =

∞ X

m X

m=0

n=0

! ε

−n

Ch(k) n,ε,q S2 (m, n)

tm . m!

(2.17)

Thus, by (2.16) and (2.17), we obtain the following theorem. Theorem 2.7. For n ≥ 0, we have n X (k) ε−m Ch(k) En,q = m,ε,q S2 (n, m). m=0

Now we define the twisted q-Changhee polynomials of the first kind with order k as follows: Z Z (k) n Chn,ε,q (x) = ε ··· (x1 + · · · + xk + x)n dµ−q (x1 ) · · · dµ−q (xk ), (2.18) Zp

Zp

where n ≥ 0 and k ∈ N. From (2.18), we can derive the generating function of the twisted q-Changhee polynomials as follows: Z Z ∞ X tn Ch(k) (x) (1 + εt)x1 +···+xk +x dµ−q (x1 ) · · · dµ−q (xk ) = · · · n,ε,q n! Zp Zp n=0 (2.19)  k [2]q = (1 + εt)x . qεt + [2]q It is easy to show that  k ∞ X [2]q (1 + εt)x = qεt + [2]q n=0

!   n tn (k) ε (x)m Chn−m,ε,q . n! m m=0 n X

m

(2.20)

By (2.20), we get Ch(k) n,ε,q (x)

n X

  x n! (k) = ε Chn−m,ε,q m (n − m)! m=0   n X x n! n−m = ε Ch(k) m,ε,q . n − m m! m=0 m

428

(2.21)

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From (2.18), we have Z Z (k) n Chn,ε,q (x) =ε ··· (x1 + · · · + xk + x)n dµ−q (x1 ) · · · dµ−q (xk ) Zp n

=ε

=εn

n X l=0 n X

Zp

Z

Z

(x1 + · · · + xk + x)l dµ−q (x1 ) · · · dµ−q (xk )

···

S1 (n, l) Zp

Zp

(k)

S1 (n, l)El,q (x).

l=0

(2.22) Hence, by (2.22), we obtain the following theorem. Theorem 2.8. For n ≥ 0, we have   n n X X x n! (k) n (k) m Ch(k) S1 (n, l)El,q (x). Chn,ε,q (x) = ε m,ε,q = ε n − m m! m=0 l=0

where

(k) El,q

are the q-Euler polynomials of order k.

Now, we consider the twisted q-Changhee polynomials of second kind with order k as follows: Z Z (k) n c Ch (x) = ε · · · (−x1 − · · · − xk + x)n dµ−q (x1 ) · · · dµ−q (x)k . (2.23) n,ε,q Zp

Zp

By (2.23), we have ∞ X

n

t c = Ch n,ε,q (x) n! n=0 (k)

Z

Z

(1 + εt)−x1 −···−xk +x dµ−q (x1 ) · · · dµ−q (xk )

··· Zp

 =

Zp

[2]q εt + [2]q

(2.24)

k

k+x

(1 + εt)

,

where k is positive integer. Hence, (k)

c Ch n,ε,q (x) Z Z n =ε ··· (−x1 − · · · − xk + x)n dµ−q (x1 ) · · · dµ−q (xk ) Zp

=ε

n

=εn

n X l=0 n X

Zp l

Z

Z ···

S1 (n, l)(−1)

Zp

(x1 + · · · + xk − x)l dµ−q (x1 ) · · · dµ−q (xk )

(2.25)

Zp

(k)

S1 (n, l)(−1)l El,q (−x).

l=0

Therefor, by (2.25), we obtain the following theorem. Theorem 2.9. For n ≥ 0, we have (k)

n c Ch n,ε,q (x) = ε

n X

(k)

S1 (n, l)(−1)l El,q (−x).

l=0

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Now, we consider the n-th twisted q-Changhee polynomials of the first kind relate to n-th twisted q-Changhee polynomials of second kind. (k)

c (−1)n Ch n,ε,q (x) n!Z Z

  −x1 − · · · − xk + x =(−1) ε ··· dµ−q (x1 ) · · · dµ−q (xk ) n Zp Zp  Z Z  x1 + · · · + xk − x + n − 1 =εn ··· dµ−q (x1 ) · · · dµ−q (xk ) n Zp Zp Z  Z  ∞  X n−1 x1 + · · · + xk − x n =ε ··· dµ−q (x1 ) · · · dµ−q (xk ) n − m Zp m Zp m=0   Z Z  n  X n − 1 ε−m x1 + · · · + xk − x =εn m!εm ··· dµ−q (x1 ) · · · dµ−q (xk ) m − 1 m! m Zp Zp m=1  n  (k) X n − 1 n−m Chm,ε,q (−x) . = ε m! m−1 m=1 (2.26) n n

By (2.26) and proceeding similar to (2.26), we have the following theorem. Theorem 2.10. For n ≥ 0, we have (k)  n  (k) c X (−1)n Ch n − 1 n−m Chm,ε,q (−x) n,ε,q (x) ε = , m−1 n! m! m=1

and (k)  n  (k) c X (−1)n Chn,ε,q (x) n − 1 n−m Ch m,ε,q (−x) = , ε n! m! m−1 m=1

By (2.25), c n,ε,q (x) Ch =εn

n X

S1 (n, l)(−1)l

=ε

n

l=0

Z

S1 (n, l)

l X

(x1 + · · · + xk − x)l dµ−q (x1 ) · · · dµ−q (xk )

··· Zp

l=0 n X

Z

Zp l+m

(−1)

m=0

  l (k) E xm , m l−m

and thus we obtain the following theorem. Theorem 2.11. For n ≥ 0, we have c n,ε,q (x) = εn Ch

n X l X

l+m

(−1)

l=0 m=0

  l (k) S1 (n, l)El−m xm . m

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 (2012), no.3, 399-406.

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[2] J. Choi, D. S. Kim, T. Kim and Y. H. Kim, Some arithmetic identities on Bernoulli and Euler numbers arising from the p-adic integrals on Zp , Adv. Stud. Contemp. Math. 22 (2012) 239-247. [3] D. Kim, T. Mansour, S. H. Rim and J. J. Seo, A Note on q-Changhee Polynomials and Numbers, Adv.Studies Theor. Phys., Vol. 8, 2014, no. 1, 35-41. [4] T. Kim, D. S. Kim, T. Mansour, S,-H. Rim and M. Schork Umbral calculus and Sheffer sequences of polynomials, J. Math. Phys. 54, 083504 (2013); doi:10.1063/1.4817853. [5] T. Kim, S.-H. Rim, New Changhee q-Euler numbers and polynomials associated with p-adic q-integrals, Comput. Math.Appl. 54 (2007), no. 4, 484-489. [6] T. Kim, On q-analogye of the p-adic log gamma functions and related integral, J. Number Theory, 76 (1999), no. 2, 320-329. [7] T. Kim, q-Volkenborn integration, Russ. J. Math. Phys., 9 (2002), no. 3, 288-299. [8] T. Kim, Non-Archimedean q-integrals assoiciated with multiple Changhee q-Bernoulli polynomials, Russ. J. Math. Phys. 10 (2003), 91-98. [9] T. Kim, p-adic q-integrals associated with the Changhee-Barnes’ q-Bernoulli polynomials, Integral Transforms Spec. Funct. 15 (2004), no. 5, 415-420. [10] T. Kim, An invariant p-adic q-integral on Zp , Applied Mathematics Letters, 21 (2008), no. 2, 105-108. [11] S. H. Lee, W. J. Kim and Y. S. Jang, Higher-order q-Changhee polynomials, to appear. [12] S. H. Rim, J. W. Park, S. S¿ Pyo and J. Kwon, On the twisted Changhee polynomials and numbers, to appear. [13] C. S. Ryoo, A note on the twisted q-Euler numbers and polynomials with weak weight α, Adv. Studies Theor. Phys., 6 (2012), no. 22, 1109-1116. [14] Y. Simsek, T. Kim, I. S. Pyung, Barnes’ type multiple Changhee q-zeta functions, Adv. Stud. Contemp. Math. 10 (2005), no. 2, 121-129. Department of Mathematics Education, Daegu University, Jillyang, Gyeongsan, Gyeongbuk 712-714, Republic of Korea. E-mail address: [email protected]

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SOME SYMMETRY IDENTITIES FOR THE (h, q)-BERNOULLI POLYNOMIALS UNDER THE THIRD DIHEDRAL GROUP D3 ARISING FROM q-VOLKENBORN INTEGRAL ON Zp S.-H. RIM, T. G. KIM, S. H. LEE

Abstract. In this paper, we give some new identities of symmetry for the (h, q)-Bernoulli polynomials arising from q-Volkenborn integral on Zp .

1. Introduction let p be a fixed prime number. Throughout this paper, Zp , Qp and Cp will, respectively, denote the ring of p-adic integers, the field of p-adic rational numbers and the completion of algebraic closure of Qp . Let vp be the normalized exponential valuation of Cp with |p|p = p−vp (p) = 1/p and let q be an indeterminate in Cp with x 1 |1 − q|p < p− p−1 . The q-extension of x is defined by [x]q = 1−q 1−q . Note that limq→1 [x]q = x. Suppose that f is a uniformly differentiable function on Zp . Then the p-adic q-Vollenborn integral is defined by Kim to be Z Iq (f ) = (1)

f (x)dµq (x) = lim

N →∞

Zp

= lim

N →∞

N pX −1

f (x)µq (x + pN Zp )

x=0

1 [pN ]q

N pX −1

f (x)q x .

x=0

As is well known, Carlitz’s q-Bernoulli numbers are defined by ( 1 if n = 1 n β0,q = 1, q(qβ + 1) − βn,q = 0 if n > 1, with the usual convention about replacing βqn by βn,q (see [1,8,10]). The q-Bernoulli polynomials are given by n   X n βn,q (x) = [x]n−l q lx βl,q q l l=0 n   X 1 n l+1 = (−1)l q lx , (see [10]). (1 − q)n l [l + 1]q l=0

In 1999, Kim gave the formula which is given by Z βn,q (x) = [x + y]nq dµq (x), (n ∈ N ∪ {0}, )

(see [1-15]).

Zp 1

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For h ∈ Z, we consider (h, q)-Bernoulli polynomials as follows: Z (h) βn,q (x) = q (h−1)x [x + y]nq dµq (x), (n ∈ Z≥0 ) Zp

(2)

n   X n lx h+1 1 q (−1)l , = n (1 − q) l [h + l]q

(see [8,10]).

l=0

(h)

(h)

When x = 0, βn,q = βn,q (0) are called the (h, q)-Bernoulli numbers. In this paper, we consider the symmetric identities for the (h, q)-Bernoulli polynomials under the third Dihedral group D3 which are derive from p-adic q-Volkenborn integral on Zp . 2. Symmetric identities for the (h, q)-Bernoulli polynomials Let w1 , w2 , w3 be positive integers. Then we observe that (3) Z

q (h−1)w2 w3 y e[w2 w3 y+w1 w2 w3 x+w1 w3 i+w1 w2 j]q t dµqw2 w3 (y)

Zp N

pX −1 1 q (h−1)w2 w3 y e[w2 w3 y+w1 w2 w3 x+w1 w3 i+w1 w2 j]q t q w2 w3 y = lim N →∞ [pN ]q w2 w3 y=0 N

−1 wX 1 −1 pX 1 = lim q hw2 w3 (k+w1 y) e[w2 w3 (k+w1 y)+w1 w2 w3 x+w1 w3 i+w1 w2 j]q t . N →∞ [w1 pN ]q w2 w3 y=0 k=0

By (3), we get wX 2 −1 w 3 −1 X 1 q (w1 w3 i+w1 w2 j)h [w2 w3 ]q i=0 j=0 Z × q (h−1)w2 w3 y e[w2 w3 y+w1 w2 w3 x+w1 w3 i+w1 w2 j]q t dµqw2 w3 (y)

(4)

Zp wX 2 −1 w 3 −1 w 1 −1 X X 1 q h(w1 w3 i+w1 w2 j+w2 w3 k)+hw1 w2 w3 y N →∞ [w1 w2 w3 pN ]q i=0 j=0

= lim

k=0

×e

[w2 w3 (k+w1 y)+w1 w2 w3 x+w1 w3 i+w1 w2 j]q t

.

From (4), we note that the expression is invariant under any permutation of w1 , w2 , w3 in third Dihedral group D3 . Therefore, by (4), we obtain the following theorem. Theorem 2.1. Let w1 , w2 , w3 be positive integers. Then, the following expressions wσ(2) −1 wσ(3) −1 X X 1 q h(wσ(1) wσ(3) i+wσ(1) wσ(2) j) [wσ(2) wσ(3) ]q i=0 j=0 Z × q (h−1)wσ(2) wσ(3) y e[wσ(2) wσ(3) y+wσ(1) wσ(2) wσ(3) x+wσ(1) wσ(3) i+wσ(1) wσ(2) j]q t dµqwσ(2) wσ(3) (y) Zp

are the same for any σ ∈ D3 .

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Now, we note that (5)   w1 w1 [w2 w3 y + w1 w2 w3 x + w1 w3 i + w1 w2 j]q = [w2 w3 ]q y + w1 x + i+ j w2 w3 qw2 w3 Therefore, by (2), Theorem 1 and (5), we obtain the following theorem. Theorem 2.2. For w1 , w2 , w3 ∈ N, the following expressions wσ(2) −1 wσ(3) −1

[wσ(2) wσ(3) ]n−1 q

X

X

i=0

j=0

q h(wσ(1) wσ(3) i+wσ(1) wσ(2) j) (h)

× βn,qwσ(2) wσ(3) wσ(1) x +

wσ(1)
wσ(1) i+ j wσ(2) wσ(3)

are the same for any σ ∈ D3 . It is not difficult to show that (6)   w1 w1 1 − q w1 w3 i+w1 w2 j + q w1 w3 i+w1 w2 j [y + w1 x]qw2 w3 i+ j y + w1 x + = w2 w 3 q w2 w3 1 − q w2 w3 =

[w1 ]q [w3 i + w2 j]qw1 + q w1 w3 i+w1 w2 j [y + w1 x]qw2 w3 [w2 w3 ]q

From (6), we have n Z  w1 w1 i+ j q (h−1)w2 w3 y dµqw2 w3 (y) y + w1 x + w2 w3 qw2 w3 Zp (7) n−k n   X n [w1 ]q k(w1 w3 i+w1 w2 j) (h) = [w3 i + w2 j]n−k βk,qw2 w3 (w1 x). q w1 q [w2 w3 ]q k k=0

Thus, by Theorem 2 and (7), we get (8) [w2 w3 ]n−1 q

wX 2 −1 w 3 −1 X i=0

q h(w1 w3 i+w1 w2 j)

j=0

 n w1 w1 q (h−1)w2 w3 y y + w1 x + i+ j dµqw2 w3 (y) w2 w3 qw2 w3 Zp

Z

 wX 2 −1 w 3 −1 X n k−1 n−k (h) = [w2 w3 ]q [w1 ]q βk,qw2 w3 (w1 x) q (k+h)(w1 w3 i+w1 w2 j) [w3 i + w2 j]n−k q w1 k i=0 j=0 k=0 n   X n (h) (h) = [w2 w3 ]k−1 [w1 ]n−k βk,qw2 w3 (w1 x) Tn,qw1 (w2 , w3 |k), q q k n  X

k=0

where (9)

(h) Tn,q (w1 , w2 |k) =

wX 1 −1 w 2 −1 X i=0

q (k+h)(w2 i+w1 j) [w2 i + w1 j]n−k . q

j=0

As this expression is invariant under the third Dihedral group D3 , we have the following theorem.

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S.-H. RIM, T. G. KIM, S. H. LEE

Theorem 2.3. For n ≥ 0, w1 , w2 , w3 ∈ N, the following expressions n   X n (h) (h) [wσ(2) wσ(3) ]k−1 [wσ(1) ]n−k βk,qwσ(2) wσ(3) (wσ(1) x) Tn,qwσ(1) (wσ(2) , wσ(3) |k) q q k k=0

are all the same for any σ ∈ D3 . ACKNOWLEDGEMENTS. The present Research has been supported by Jangjeon Research Institute for Mathematics and Physics and has been conducted by the Research Grant of Kwangwoon University in 2014.

References [1] J. Choi, T. Kim, Arithmetic properties for the q-Bernoulli numbers and polynomials, Proc. Jangjeon Math. Soc. 15 (2012), no. 2, 137–143. [2] S. Gaboury, R. Tremblay, B. J. Fugere, Some explicit formulas for certain new classes of Bernoulli, Euler and Genocchi polynomials, Proc. Jangjeon Math. Soc. 17 (2014), no. 1, 115–123. [3] D.V. Dolgy, D. S. Kim, T. Kim, J.-J. Seo, Identities of Symmetry for Carlitz q-Bernoulli Polynomials Arising from q-Volkenborn Integrals on Zp under Symmetry Group S3, Advanced Studies in Theoretical Physics 8(2014), no. 17 , 737 - 744 [4] D. S. Kim, T. Kim, q-Bernoulli polynomials and q-umbral calculus, Sci. China Math. 57 (2014), no. 9, 1867–1874. [5] D. S. Kim, N. Lee, J. Na, K. H. Park, Abundant symmetry for higher-order Bernoulli polynomials (II), Proc. Jangjeon Math. Soc. 16 (2013), no. 3 , 359–378 [6] D. S. Kim, D. V. Dolgy, T. Kim, S.-H. Identities involving Bernoulli and Euler polynomials arising from Chebyshev polynomials, Proc. Jangjeon Math. Soc. 15 (2012), no. 4, 361–370. [7] H. M. Kim, D. S. Kim, T. Kim,S. H. Lee, D. V. Dolgy, B. Lee, Identities for the Bernoulli and Euler numbers arising from the p-adic integral on Zp , Proc. Jangjeon Math. Soc. 15 (2012), no. 2, 155–161 [8] T. Kim, S.H. Rim, Generalized Carlitz’s q-Bernoulli numbers in the p-adic number field, Adv. Stud. Contemp. Math. (Pusan) 2 (2000), 9–19. [9] T. Kim, On the weighted q-Bernoulli numbers and polynomials, Adv. Stud. Contemp. Math. 21 (2011), no. 2, 207–215. [10] T. Kim, J. Choi, Y.-H. Kim, On extended Carlitz’s type q-Euler numbers and polynomials, Adv. Stud. Contemp. Math. 20 (2010), no. 4, 499–505. [11] T. Kim, Y.-H. Kim, B. Lee, Note on Carlitz’s type q-Euler numbers and polynomials, Proc. Jangjeon Math. Soc. 13 (2010), no. 2, 149–155. [12] T. Kim, Y.-H. Kim, K.-W. Hwang, On the q-extensions of the Bernoulli and Euler numbers, related identities and Lerch zeta function, Proc. Jangjeon Math. Soc. 12 (2009), no. 1, 77–92. [13] J.-W. Park, S.-H. Rim, J. Seo, J. Kwon, A note on the modified q-Bernoulli polynomials, Proc. Jangjeon Math. Soc. 16 (2013), no. 4, 451–456. [14] S. H. Rim, J. Joung, J.-H. Jin, S.-J. Lee, A note on the weighted Carlitz’s type q-Euler numbers and q-Bernstein polynomials, Proc. Jangjeon Math. Soc. 15 (2012), no. 2, 195–201.

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[15] J.-J. Seo, S.-H. Rim, S.-H. Lee, D. V. Dolgy, T. Kim, q-Bernoulli numbers and polynomials related to p-adic invariant integral on Zp , Proc. Jangjeon Math. Soc. 16 (2013), no. 3, 321–326

Seog-Hoon Rim Department of Mathematics Education, Kyungpook National University, Tagegu 702-701, S. Korea E-mail: [email protected] Tae Gyun Kim Jangjeon Research Institute for Mathematics and Physics, Hapcheon 678-800, S. Korea Department of Mathematics, Kwangwoon University, Seoul 139-701, Republic of Korea E-mail: [email protected] Sang Hun Lee Division of General Education, Kwangwoon University, Seoul 139-701, Republic of Korea E-mail: [email protected]

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SOME IDENTITIES OF BELL POLYNOMIALS ASSOCIATED WITH p-ADIC INTEGRAL ON Zp SEOG-HOON RIM, HONG KYUNG PAK, J.K. KWON, AND TAE GYUN KIM Abstract. In this paper, we investigate some identities of Bell polynomials associated with special polynomials which are derived from p-adic integral on Zp .

1. Introduction Let p be a fixed odd prime number. Throughout this paper, Zp , Qp and Cp will denote the ring of p-adic integers, the field of p-adic rational numbers and the completion of algebraic closure of Qp . Let νp be the normalized exponential valuation 1 − p−1

of Cp with |p|p = p−νp (p) = p1 . Let q be an indeterminate in Cp with |1 − q|p < p x

and let the q-extension of number x is defined as [x]q = 1−q 1−q . The Euler polynomials of order r are defined by the generating function to be ∞  2 r X tn xt (r) , (see [1 − 18]) e = E (x) n et + 1 n! n=0

and the higher-order Bernoulli polynomials of order r are given by ∞  t r X tn xt (r) e = B (x) , (see [9 − 10]). n et − 1 n! n=0

(r) Bn

(r) Bn (0),

(r)

(r)

When x = 0, = En = En (0) are called higher-order Bernoulli numbers and Euler numbers. Let f (x) be a uniformly continuous function on Zp . Then the bosonic p-adic integral on Zp is defined by Z pN −1 1 X (1) f (x)dµ0 (x) = lim N f (x), (see [12]), N →∞ p Zp x=0

and the fermieuic p-adic integral on Zp is given by Z pN −1 1 X (2) f (x)dµ−1 (x) = lim N f (x)(−1)x , N →∞ p Zp

(see [12]).

x=−1

Thus, we have Z

Z f (x + 1)dµ0 (x) −

(3) Zp

f (x)dµ0 (x) = f 0 (0),

Zp

and Z

Z f (x + 1)dµ−1 (x) +

(4) Zp

f (x)dµ−1 (x) = 2f (0). Zp 1

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As is well know, the higher-order Changhee polynomials are given by (5)

∞  2 r X tn , (1 + t)x = Ch(r) (x) n t+2 n!

(see [11 − 15]),

n=0

and the higher-order Daehee polynomials are defined by the generating function to be (6)

 log(1 + t) r t

x

(1 + t) =

∞ X

Dn(r) (x)

n=0

(r)

(r)

(r)

tn , n!

(see [11 − 15]).

(r)

When x = 0, Chn = Chn (0) and Dn = Dn (0) are called the Changhee numbers and the Daehee numbers with order r. Finally, we introduce the Bell polynomials which are given by the generating function to be (et −1)x

(7)

e

=

∞ X

Beln (x)

n=0

tn , n!

(see [4, 14, 16]).

The purpose of this paper is to given some identities of Bell polynomials associated with special polynomials arising from p-adic integral on Zp . 2. Some identities of Bell polynomials From (2), we note that Z

e(e

t −1)(x+y)

dµ0 (y)

Zp

=

(8)

=

∞ Z X

(x + y)n dµ0 (y)

n=0 Zp ∞ X n X

Bk (x)S2 (n, k)

n=0 k=0

(et − 1)n n!  tn n!

,

where S2 (n, k) is the Stirling number of the second kind. On the other hand, Z (9)

(et −1)(x+y)

e

dµ0 (y) =

Zp

∞ Z X

Beln (x + y)dµ0 (y)

n=0 Zp

tn . n!

Thus, by (8) and (9), we get Z (10)

Beln (x + y)dµ0 (y) = Zp

n X

Bk (x)S2 (n, k).

k=0

By the same method as (10), we get Z (11)

Beln (x + y)dµ−1 (y) = Zp

n X

Ek (x)S2 (n, k).

k=0

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Note that Z

Z

(1 + t)(x1 +···+xr +x) dµ0 (x1 ) · · · dµ0 (xr ) =

··· (12)

 log(1 + t) r t

Zp

Zp

=

∞ X

Dn(r) (x)

n=0 t −1

By replacing t by ee Z

Z

Zp

tn . n!

− 1, we get t −1)(x +···+x +x) r 1

e(e

···

(1 + t)x

dµ0 (x1 ) · · · dµ0 (xr )

Zp

∞ ∞  et − 1 r t X t l  X tm  (r) (e − 1) (e −1)x Bel (x) e = B m l l! m! eet −1 − 1 m=0

=

∞ X

=

l=0 ∞ X

(13) =

∞ X (r)

Bl

S2 (k, l)

k=l k X

l=0 ∞  k X t

k!

m=0

tm  Belm (x) m!

∞
tk  X

(r)

Bl S2 (k, l)

k=0 l=0 ∞ X n X

k!

m=0

Belm (x)

tm  m!

n−m X

 tn Belm (x)n! (r) Bl S2 (n − m, l) m!(n − m)! n! n=0 m=0 l=0 ∞ X n   n−m  tn X X (r) n = Belm (x) Bl S2 (n − m, l) . m n!

=

n=0 m=0

l=0

On the other hand, Z

Z

e(e

··· (14)

Zp

=

t −1)(x

1 +···+xr +x)

dµ0 (x1 ) · · · dµ0 (xr )

Zp

∞ Z X

Z Beln (x1 + · · · + xr + x)dµ0 (x1 ) · · · dµ0 (xr )

··· Zp

n=0 Zp

tn . n!

Therefore, we obtain the following theorem. Theorem 1. For n ≥ 0, we have Z

Z ···

Zp

Beln (x1 + · · · + xr + x)dµ0 (x1 ) · · · dµ0 (xr ) Zp

n   n−m X (r) X n = Belm (x) Bl S2 (n − m, l). m m=0

l=0

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From (12), we note that Z Z t ··· e(e −1)(x1 +···+xr +x) dµ0 (x1 ) · · · dµ0 (xr ) Zp

=

(15)

= = =

∞ X

Zp

Dn(r) (x)

n=0 ∞ X m X m=0 k=0 ∞ X m X

∞ ∞ n X X 1  (et −1) (et − 1)m (r) e −1 = Dk (x) S2 (m, k) n! m! k=0

(r)

Dk (x)S2 (m, k)

1 t (e − 1)m m! ∞ X

(r)

Dk (x)S2 (m, k)

S2 (n, m)

n=m

m=0 k=0 ∞ nX n X m X

m=k

tn n!

o tn (r) . Dk (x)S2 (m, k)S2 (n, m) n!

n=0 m=0 k=0

Therefore, by Theorem 1 and (15), we obtain the following theorem. Theorem 2. For n ≥ 0, we have n   n−m X X (r) n Bl S2 (n − m, l) Belm (x) m m=0

l=0

n X m X

=

(r)

Dk (x)S2 (m, k)S2 (n, m).

m=0 k=0

From (7), we note that xt

e

= =

(16)

∞ X

=

m=0 ∞ X

Belm (x) Belm (x)

m=0 ∞ X n X n=0 m=0

m 1  log(1 + t) m! ∞ X

S1 (n, m)

n=m

tm m!

 tn Belm (x)S1 (n, m) , n!

where S1 (n, m) is the Stirling number of the first kind. Therefore, by (16), we obtain the following theorem. Theorem 3. For n ≥ 0, we have n

x =

n X

Belm (x)S1 (n, m).

m=0

It is easy to show that Z (17)

∞

X tn t = Bn . e dµ0 (x) = t e −1 n! Zp xt

n=0

Thus, by (17), we have Z

xn dµ0 (x) = Bn ,

(n ≥ 0).

Zp

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From Theorem 3, we can derive the following equation: Z Z n X n x dµ0 (x) = Belm (x)dµ0 (x), (18) Bn = S1 (n, m) Zp

5

(n ≥ 0).

Zp

m=0

Therefore, by (10) and (18), we obtain the following theorem. Theorem 4. For n ≥ 0, we have Bn =

n X m X

S1 (n, m)S2 (m, k)Bk .

m=0 k=0

It is not difficult to show that Z ext dµ−1 (x) = (19) Zp

∞

X 2 tn = . E n et + 1 n! n=0

Thus, by (19), we get Z

xn dµ−1 (x) = En ,

(20)

(n ≥ 0).

Zp

From Theorem 3 and (20), we have Z Z n X n x dµ−1 (x) = S1 (n, m) (21) En = Zp

Belm (x)dµ−1 (x).

Zp

m=0

Therefore, by (11) and (21), we obtain the following theorem. Theorem 5. For n ≥ 0, we have En =

n X m X

S1 (n, m)S2 (m, k)Ek .

m=0 k=0

Now, we consider the following equation. (x+x1 +···+xr )t

e

(22)

= = =

∞ X m=0 ∞ X

Belm (x1 + · · · + xr + x) Belm (x1 + · · · + xr + x)

m=0 ∞ X n X n=0 m=0

(log(1 + t))m m! ∞ X n=m

S1 (n, m)

tn n!

 tn Belm (x1 + · · · + xr + x)S1 (n, m) . n!

Thus, by (22), we have the following theorem. Theorem 6. For n ≥ 0, we have (x + x1 + · · · + xr )n =

n X

Belm (x1 + · · · + xr + x)S1 (n, m).

m=0

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SEOG-HOON RIM, HONG KYUNG PAK, J.K. KWON, AND TAE GYUN KIM

From (4), we can easily derive the following equation: Z Z  e(x1 +···+xr +x)t dµ−1 (x1 ) · · · dµ−1 (xr ) = ··· Zp

Zp

(23)

=

2 r xt e et + 1

∞ X

En(r) (x)

n=0

tn . n!

Thus, by (23), we get Z Z (x1 + · · · + xr + x)n dµ−1 (x1 ) · · · dµ−1 (xr ) = En(r) (x). ··· (24) Zp

Zp

By (3), we easily get Z Z  e(x1 +···+xr +x)t dµ0 (x1 ) · · · dµ0 (xr ) = ··· Zp

Zp

(25)

=

t r xt e et − 1

∞ X

Bn(r) (x)

n=0

tn . n!

From (25), we have Z Z (26) ··· (x1 + · · · + xr + x)n dµ0 (x1 ) · · · dµ0 (xr ) = Bn(r) (x). Zp

Zp

From Theorem 6, (24) and (26), we have Z Z n X (r) (27) Bn (x) = S1 (n, m) ··· Belm (x + x1 + · · · + xr )dµ0 (x1 ) · · · dµ0 (xr ) Zp

m=0

Zp

and (28) En(r) (x) =

n X

Z Zp

m=0

Now, we observe that Z ∞ Z X ··· n=0 Zp

Z Zp ∞ X

= =

Belm (x + x1 + · · · + xr )dµ−1 (x1 ) · · · dµ−1 (xr ). Zp

Beln (x + x1 + · · · + xr )dµ0 (x1 ) · · · dµ0 (xr )

Zp

Z

e(e

···

= (29)

Z ···

S1 (n, m)

t −1)(x +···+x +x)t r 1

tn n!

dµ0 (x1 ) · · · dµ0 (xr )

Zp (r) Bm (x)

m=0 ∞ X n X n=0 m=0

m 1  t e −1 m!

 tn (r) Bm (x)S2 (n, m) . n!

Thus, by (29), we get Z Z n X (r) (30) ··· Beln (x1 + · · · + xr + x)dµ0 (x1 ) · · · dµ0 (xr ) = Bm (x)S2 (n, m). Zp

Zp

m=0

Therefore, by (27) and (30), we obtain the following theorem.

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7

Theorem 7. For n ≥ 0, we have Bn(r) (x)

=

n X m X

(r)

S1 (n, m)S2 (m, k)Bk (x).

m=0 k=0

By the same method of (29), we get Z ∞ Z X tn Beln (x + x1 + · · · + xr )dµ−1 (x1 ) · · · dµ−1 (xr ) ··· n! Zp n=0 Zp Z Z t = ··· e(e −1)(x1 +···+xr +x)t dµ−1 (x1 ) · · · dµ−1 (xr ) (31) =

Zp Zp ∞ n  X X n=0 m=0

 tn (r) Em (x)S2 (n, m) . n!

From (31), we have Z Z n X (r) (32) ··· Beln (x1 +· · ·+xr +x)dµ−1 (x1 ) · · · dµ−1 (xr ) = Em (x)S2 (n, m). Zp

Zp

m=0

Therefore, by Theorem 6 and (32), we obtain the following theorem. Theorem 8. For n ≥ 0, we have En(r) (x)

=

n X m X

(r)

S1 (n, m)S2 (m, k)Ek (x).

m=0 k=0

From (4), we have Z

Z

(1 + t)(x1 +···+xr +x) dµ−1 (x1 ) · · · dµ−1 (xr )

··· Zp

Zp

(33)

∞  2 r X tn = (1 + t)x = Ch(r) (x) . n 1+t n! n=0

t

By replacing t by e(e −1) − 1, we get Z Z t ··· e(e −1)(x1 +···+xr +x) dµ−1 (x1 ) · · · dµ−1 (xr ) Zp

=

(34)

= = =

Zp

∞ Z X

Z ···

(x1 + · · · + xr + x)dµ−1 (x1 ) · · · dµ−1 (xr )

Zp m=0 Zp ∞ X 1 (r) Em (x) (et − 1)m m! m=0 ∞ ∞ X X tn (r) Em (x) S2 (n, m) n! n=m m=0 ∞ X n  tn X (r) , Em (x)S2 (n, m) n! n=0 m=0

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and

t −1)r

2r e−(e

e(e

t −1)x

= 2r

∞ X

∞

Bell (−r)

tl  X tm  Belm (x) l! m! m=0

l=0 ∞ n X X

Belm (x)Beln−m (−r)n!  tn m!(n − m)! n! n=0 m=0 ∞  n    tn X X n = . 2r Belm (x)Beln−m (−r) n! m

= 2r

(35)

n=0

m=0

Therefore, by (33),(34) and (35), we obtain the following theorem. Theorem 9. For n ≥ 0, we have

n X

(r) Em (x)S2 (n, m)

m=0

n   X n Belm (x)Beln−m (−r). =2 m r

m=0

Now, we observe that

∞ X

Ch(r) m (x)

m=0

= (36) =

∞ X

∞ ∞ m X X 1  (et −1) (et − 1)k (r) e −1 = Chm (x) S2 (k, m) m! k! m=0

k X

k=0 m=0 ∞ X k X

Ch(r) m (x)S2 (k, m) Ch(r) m (x)S2 (k, m)

k=0 m=0

=

∞ X n X k X

k=m

1 t (e − 1)k k! ∞ X

S2 (n, k)

n=k

tn n!

Ch(r) m (x)S2 (k, m)S2 (n, k)

n=0 k=0 m=0

 tn n!

.

Therefore, by (33), (34) and (36), we obtain the following theorem. Theorem 10. For n ≥ 0, we have

n X m=0

(r) Em (x)S2 (n, m) =

n X k X

Ch(r) m (x)S2 (k, m), S2 (n, k).

k=0 m=0

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From (4), we have Z Z ···

t −1)(x +···+x +x) r 1

dµ−1 (x1 ) · · · dµ−1 (xr )

Zp

Zp

=

e(e

9

2



r

et −1

e(e

t −1)x

e +1 ∞ ∞ X t m X tl  (r) (e − 1) = Em Bell (x) m! l! m=0

= (37)

∞ X

l=0

(r) Em

m=0

=

=

=

∞ X k X

∞

∞ X k=m

tl  tk X Bell (x) S2 (k, m) k! l! l=0

(r) Em S2 (k, m)

k=0 m=0 ∞ n X k XnX

∞  tk X

k!

Bell (x)

l=0

(r) Em (x)S2 (k, m)Beln−k (x)

n=0 k=0 m=0 ∞ nX n   X

k X

n=0 k=0

m=0

n k

tl  l!

o tn n! k!(n − k)! n!

(r) Em (x)S2 (k, m)Beln−k (x)

o tn n!

Therefore, by (34) and (37), we obtain the following theorem. Theorem 11. For n ≥ 0, we have n X k=0

(r) Ek (x)S2 (n, k)

=

k n  X X n k=0

k

(r) Em (x)S2 (k, m)Beln−k (x).

m=0

ACKNOWLEDGEMENTS. The present Research has been conducted by the Research Grant of Kwangwoon University in 2015 References [1] S. Araci, X. Kong, M. Acikgoz, E. Sen, A new approach to multivariate q-Euler polynomials using the umbral calculus, J. Integer Seq. 17 (2014), no. 1, Article 14.1.2, 10 pp. [2] G. E. Andrews, The theory of partitions, Reprint of the 1976 original. Cambridge Mathematical Library. Cambridge University Press, Cambridge, 1998. xvi+255 pp. ISBN: 0-521-63766-X. [3] A. Bayad, Modular properties of elliptic Bernoulli and Euler functions, Adv. Stud. Contemp. Math. 20 (2010), no. 3, 389–401. [4] Bell, E. T. ”Exponential Polynomials.” Ann. Math. 35(1934), 258-277. [5] Comtet, L. Advanced Combinatorics: The Art of Finite and Infinite Expansions, Dordrecht, Netherlands: Reidel, 1974. [6] 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 (2014), no. 1, 115–123 [7] H. W. Gould, T. He, Characterization of (c)-Riordan arrays, Gegenbauer-Humbert-type polynomial sequences, and (c)-Bell polynomials, J. Math. Res. Appl. 33 (2013), no. 5, 505–527. [8] D. S. Kim, T. Kim, Higher-order cauchy of the second kind and poly-cauchy of the second kind mixed type polynomials, Ars Combinatoria 115(2014), pp.435-451. [9] D. S. Kim, D.V. Dolgy, T. Kim, S.-H. Rim, Identities involving Bernoulli and Euler polynomials arising from Chebyshev polynomials, Proc. Jangjeon Math. Soc. 15 (2012), no. 4, 361–370 [10] D. S. Kim, T. Kim, C.S. Ryoo, Sheffer sequences for the powers of sheffer pairs under umbral composition, Adv. Stud. Contemp. Math. 23 (2013), no. 2, 275–285. [11] T. Kim, Identities involving Laguerre polynomials derived from umbral calculus. Russ. J. Math. Phys. 21 (2014), no. 1, 36–45.

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[12] T. Kim, D. V. Dolgy, D.S. Kim, S.-H. Rim, A note on the identities of special polynomials, Ars Combin. 113A (2014), 97–106. [13] Q.-M. Luo, F. Qi, Relationships between generalized Bernoulli numbers and polynomials and generalized Euler numbers and polynomials, Adv. Stud. Contemp. Math. 7 (2003), no. 1, 11–18 [14] T. Mansour, M. Shattuck, A recurrence related to the Bell numbers, Integers 12 (2012), no. 3, 373–384. [15] J. Riordan, An Introduction to Combinatorial Analysis, New York: Wiley, 1980. [16] S. Roman, The umbral calculus. Pure and Applied Mathematics, 111. Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], New York, 1984. x+193 pp. ISBN: 0- 12-594380-6 [17] Z. Zhang, H. Yang, Some closed formulas for generalized Bernoulli-Euler numbers and polynomials, Proc. Jangjeon Math. Soc. 11 (2008), no.2, 191–198 [18] Z. Zhang, J. Yang, Notes on some identities related to the partial Bell polynomials, Tamsui Oxf. J. Inf. Math. Sci. 28 (2012), no. 1, 39–48. Department of Mathematics, Kyungpook National University, Taegu 702-701, S. Korea E-mail address: [email protected] Department of Computer Science, Daegu Haany University, Kyungsan 712-715, S. Korea E-mail address: [email protected] Department of Mathematics, Kyungpook National University, Taegu 702-701, S. Korea E-mail address: [email protected] Jangjeon Research Institute for Mathematics and Physics, Hapcheon 678-800, S. Korea, Department of mathematics, Kwangwoon University, Seoul 139-701, S. Korea E-mail address: [email protected]

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ON A PRODUCT-TYPE OPERATOR FROM WEIGHTED BERGMAN-NEVANLINNA SPACES TO WEIGHTED ZYGMUND SPACES ON THE UNIT DISK ZHI JIE JIANG, HONG BIN BAI, AND ZUO AN LI

Abstract. Let D = {z ∈ C : |z| < 1} be the open unit disk, ϕ an analytic self-mapping of D and ψ an analytic function in D. Let D be the differentiation operator and Wϕ,ψ the weighted composition operator. The boundedness and compactness of the product-type operator Wϕ,ψ D from weighted BergmanNevanlinna spaces to weighted Zygmund spaces on D are characterized.

1. Introduction Let C be the complex plane, D = {z ∈ C : |z| < 1} the open unit disk in C, H(D) the class of all holomorphic functions on D, ϕ a holomorphic self-mapping of D and ψ ∈ H(D). Weighted composition operator Wϕ,ψ on H(D) is defined by Wϕ,ψ f (z) = ψ(z) · f (ϕ(z)), z ∈ D. If ψ ≡ 1 the operator is reduced to, so called, composition operator and usually denote by Cϕ . If ϕ(z) = z, it is reduced to, so called, multiplication operator and usually denote by Mψ . Standard problem is to provide function theoretic characterizations when ϕ and ψ induce a bounded or compact weighted composition operator. Weighted composition operators between various spaces of holomorphic functions on different domains have been studied by numerous authors, see, e.g., [1, 2, 8, 9, 11, 13–17, 19, 21, 23, 28, 34, 35, 45, 49, 50, 53] and the references therein. Let D be the differentiation operator on H(D), that is, Df (z) = f 0 (z), z ∈ D. The product-type operator Cϕ D has been studied, for example, in [4, 18, 20, 25, 26, 29, 41, 44, 46]. In [31] Sharma has studied the following operators from BergmanNevanlinna spaces to Bloch-type spaces: Mψ Cϕ Df (z) = ψ(z)f 0 (ϕ(z)), Mψ DCϕ f (z) = ψ(z)ϕ0 (z)f 0 (ϕ(z)), Cϕ Mψ Df (z) = ψ(ϕ(z))f 0 (ϕ(z)), and Cϕ DMψ f (z) = ψ 0 (ϕ(z))f (ϕ(z)) + ψ(ϕ(z))f 0 (ϕ(z)), 2000 Mathematics Subject Classification. Primary 47B38; Secondary 47B33, 47B37. Key words and phrases. Weighted Bergman-Nevanlinna spaces, product-type operators, weighted Zygmund spaces, little weighted Zygmund spaces. 1

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for z ∈ D and f ∈ H(D). These operators on weighted Bergman spaces, were also studied in [51] and [52] by Stevi´c, Sharma and Bhat. If we consider the product-type operator Wϕ,ψ D, then it is clear that Mψ Cϕ D = Wϕ,ψ D, Mψ DCϕ = Wϕ,ψ·ϕ0 D, Cϕ Mψ D = Wϕ,ψ◦ϕ D and Cϕ DMψ = Wϕ,ψ0 ◦ϕ + Wϕ,ψ◦ϕ D. Quite recently, the present author has considered operator Wϕ,ψ D from weighted Bergman spaces to weighted Zygmund spaces in [10]. This paper is devoted to characterizing the boundedness and compactness of operator Wϕ,ψ D from weighted Bergman-Nevanlinna spaces to weighted Zygmund spaces. It can be regarded as a continuation of the investigation of operators from weighted Bergman-Nevanlinna spaces to other spaces (see. e.g., [12] and [30]). Next we introduce the needed spaces and some facts. Let dA(z) = π1 dxdy be the normalized Lebesgue measure on D. For α > −1, let dAα (z) = (α + 1)(1−|z|2 )α dA(z) be the weighted Lebesgue measure on D. The weighted BergmanNevanlinna space Aα log on D consists of all f ∈ H(D) such that Z log(1 + |f (z)|)dAα (z) < ∞. kf kAαlog = D

It is a Fr´echet space with the translation invariant metric d(f, g) = kf − gkAαlog . For some details of this space, see, e.g., [6], [7], [47] and [54]. For β > 0, the weighted-type Aβ consists of all f ∈ H(D) such that sup(1 − |z|2 )β |f (z)| < ∞. z∈D

This space is a non-separable Banach space with the norm defined by kf kAβ = sup(1 − |z|2 )β |f (z)|. z∈D

The closure of the set of polynomials in Aβ is denoted by Aβ,0 , which is a separable Banach space and consists exactly of those functions f in Aβ satisfying the next condition lim − (1 − |z|2 )β |f (z)| = 0. |z|→1

For β > 0, the weighted Bloch space is defined by  Bβ = f ∈ H(D) : sup(1 − |z|2 )β |f 0 (z)| < ∞ . z∈D

Under the norm kf kBβ = |f (0)| + sup(1 − |z|2 )β |f 0 (z)|, z∈D

it is a Banach space. For more detail on the space, see, e.g. [55]. The closure of the set of polynomials in Bβ is called the little weighted Bloch space and is denoted by Bβ,0 . For a good source for such spaces, we refer to [55]. For β > 0, the weighted Zygmund space Zβ consists of all f ∈ H(D) such that sup(1 − |z|2 )β |f 00 (z)| < ∞. z∈D

It is a Banach space with the norm kf kZβ = |f (0)| + |f 0 (0)| + sup(1 − |z|2 )β |f 00 (z)|. z∈D

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The little weighted Zygmund space Zβ,0 consists those functions f in Zβ satisfying lim (1 − |z|2 )β |f 00 (z)| = 0,

|z|→1−

and it is a closed subspace of the weighted Zygmund space. For weighted-type spaces, weighted Bloch spaces and weighted Zygmund spaces on the unit disk, the upper half plane, the unit ball, the unit polydisk and some operators, see, e.g. [5, 11, 16, 22–24, 27, 28, 32, 33, 36–40, 42, 43, 48] and the references therein. Since the weighted Bergman-Nevanlinna space is a Fr´echet space and not a Banach space, it is necessary to introduce several definitions needed in this paper. Let X and Y be topological vector spaces whose topologies are given by translation invariant metrics dX and dY , respectively, and let L : X → Y be a linear operator. It is said that L is metrically bounded if there exists a positive constant K such that dY (Lf, 0) ≤ KdX (f, 0) for all f ∈ X. When X and Y are Banach spaces, the metrical boundedness coincides with the usual definition of bounded operators between Banach spaces. Recall that L : X → Y is metrically compact if it maps bounded sets into relatively compact sets. When X and Y are Banach spaces, the metrical compactness coincides with the usual definition of compact operators between Banach spaces. When X = Aα log and Y is a Banach space, we define kLkAαlog →Y =

kLf kY ,

sup kf kAα ≤1 log

and we often write kLkAαlog →Y by kLk. Throughout this paper, an operator is bounded (respectively, compact), if it is metrically bounded (respectively, metrically compact). Constants are denoted by C, they are positive and may differ from one occurrence to the next. The notation a  b means that there exists a positive constant C such that a/C ≤ b ≤ Ca. 2. The operator Wϕ,ψ D : Aα log → Zβ (Zβ,0 ) Our first lemma characterizes the compactness in terms of sequential convergence. Since the proof is standard, it is omitted (see, e.g., Proposition 3.11 in [3]). Lemma 2.1. Let α > −1, β > 0 and Y ∈ {Zβ , Zβ,0 }. Then the bounded operator Wϕ,ψ D : Aα log → Y is compact if and only if for every bounded sequence (fn )n∈N in Aα such that fn → 0 uniformly on every compact subset of D as n → ∞, it log follows that lim kWϕ,ψ Dfn kY = 0. n→∞

The next result can be found, for example, in [54]. Lemma 2.2. Let α > −1 and n ∈ N0 = N ∪ {0}. Then for all f ∈ Aα log and z ∈ D, there exists a positive constant C independent of f such that (1 − |z|2 )n |f (n) (z)| ≤ exp

Ckf kAαlog (1 − |z|2 )α+2

.

Now we consider the boundedness of operator Wϕ,ψ D : Aα log → Zβ .

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Theorem 2.3. Let α > −1, β > 0, ϕ be an analytic self-map of D and ψ ∈ H(D). Then for all c > 0, the following statements are equivalent: (i) The operator Wϕ,ψ D : Aα log → Zβ is bounded. (ii) The operator Wϕ,ψ D : Aα log → Zβ is compact. (iii) ψ ∈ Zβ , M0 = sup(1 − |z|2 )β |ψ(z)||ϕ0 (z)|2 < ∞, z∈D

M1 = sup(1 − |z|2 )β |ψ(z)ϕ00 (z) + 2ψ 0 (z)ϕ0 (z)| < ∞, z∈D

(1 − |z|2 )β 00 c |ψ (z)| exp = 0, (1 − |ϕ(z)|2 )α+2 ϕ(z)→∂D 1 − |ϕ(z)|2 lim

(1 − |z|2 )β c ψ(z)ϕ00 (z) + 2ψ 0 (z)ϕ0 (z) exp = 0, 2 2 (1 − |ϕ(z)|2 )α+2 ϕ(z)→∂D (1 − |ϕ(z)| ) lim

and (1 − |z|2 )β c |ψ(z)||ϕ0 (z)|2 exp = 0. 2 3 (1 − |ϕ(z)|2 )α+2 ϕ(z)→∂D (1 − |ϕ(z)| ) lim

Proof. Suppose that (i) holds. Take the functions f (z) = z and f (z) = z 2 , respectively. Since the operator Wϕ,ψ D : Aα log → Zβ is bounded, we have sup(1 − |z|2 )β |ψ 00 (z)| ≤ kWϕ,ψ DzkZβ ≤ CkWϕ,ψ Dk

(1)

z∈D

and sup(1 − |z|2 )β ψ 00 (z)ϕ(z) + 2ψ 0 (z)ϕ0 (z) + ψ(z)ϕ00 (z) ≤ CkWϕ,ψ Dk.

(2)

z∈D

Inequality (1) shows that ψ ∈ Zβ . Also by (1) and the boundedness of ϕ, sup(1 − |z|2 )β |ψ 00 (z)||ϕ(z)| < ∞.

(3)

z∈D

Then by (2), (3) and the boundedness of ϕ, M1 = sup(1 − |z|2 )β |ψ(z)ϕ00 (z) + 2ψ 0 (z)ϕ0 (z)| < ∞.

(4)

z∈D

Let the function f (z) = z 3 . Then sup(1 − |z|2 )β ψ 00 (z)ϕ(z)2 +2ψ(z)ϕ0 (z)2 + 4ψ 0 (z)ϕ0 (z)ϕ(z) + 2ψ(z)ϕ00 (z)ϕ(z) z∈D

≤ CkWϕ,ψ Dk.

(5)

By (1), (4) and (5), M0 = sup(1 − |z|2 )β |ψ(z)||ϕ0 (z)|2 ≤ CkWϕ,ψ Dk < ∞.

(6)

z∈D

For w ∈ D, we choose the function f1 (z) = c1

(1 − |ϕ(w)|2 )α+2

+ c2

(1 − |ϕ(w)|2 )α+4

(1 − ϕ(w)z)2(α+2) (1 − ϕ(w)z)2(α+2)+2 (1 − |ϕ(w)|2 )α+5 (1 − |ϕ(w)|2 )α+6 + c3 − (1 − ϕ(w)z)2(α+2)+3 (1 − ϕ(w)z)2(α+2)+4

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where

5

48α3 + 460α2 + 1398α + 1340 , 24α3 + 214α2 + 655α + 682 16α2 + 104α + 164 c3 = , 6α2 + 37α + 62

c2 = −

and c1 = 1 − c2 − c3 . We also choose the function 2α + 7 (1 − |ϕ(w)|2 )α+2 6α + 21 (1 − |ϕ(w)|2 )α+4 g1 (z) = − 2(α+2) 4α + 8 (1 − ϕ(w)z) 4α + 12 (1 − ϕ(w)z)2(α+2)+2 +

(1 − |ϕ(w)|2 )α+5 (1 − ϕ(w)z)2(α+2)+3

.

For the functions f1 and g1 , we have f1 (ϕ(w)) = f100 (ϕ(w)) = f1000 (ϕ(w)) = 0

(7)

g10 (ϕ(w)) = g100 (ϕ(w)) = 0.

(8)

and

Consequently, (7) and (8) make the function f (z) = f1 (z) exp cg1 (z) to satisfy f 00 (ϕ(w)) = f 000 (ϕ(w)) = 0 and f 0 (ϕ(w)) = C

ϕ(w) c exp , 2 α+3 (1 − |ϕ(w)| ) (1 − |ϕ(w)|2 )α+2

where C = 2c2 + 3c3 − 4. By the boundedness of the operator Wϕ,ψ D : Aα log → Zβ , we find |ϕ(w)|(1 − |w|2 )β 00 c exp ≤ C. ψ (w) (1 − |ϕ(w)|2 )α+3 (1 − |ϕ(w)|2 )α+2 Thus (1 − |w|2 )β 00 c exp = 0. ψ (w) (1 − |ϕ(w)|2 )α+2 ϕ(w)→∂D 1 − |ϕ(w)|2 lim

For w ∈ D, we choose the functions f2 (z) =

3α + 8 (1 − |ϕ(w)|2 )α+2 6α + 22 (1 − |ϕ(w)|2 )α+4 − 3α + 10 (1 − ϕ(w)z)2(α+2) 3α + 10 (1 − ϕ(w)z)2(α+2)+2 +

(1 − |ϕ(w)|2 )α+6 6α + 24 (1 − |ϕ(w)|2 )α+5 − , 3α + 10 (1 − ϕ(w)z)2(α+2)+3 (1 − ϕ(w)z)2(α+2)+4

and g2 (z) =

α + 3 (1 − |ϕ(w)|2 )α+2 (1 − |ϕ(w)|2 )α+4 − . α + 2 (1 − ϕ(w)z)2(α+2) (1 − ϕ(w)z)2(α+2)+2

Then f2 (ϕ(w)) = f20 (ϕ(w)) = f2000 (ϕ(w)) = 0

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and g20 (ϕ(w)) = 0. From this and (9), for the function g(z) = f2 (z) exp cg2 (z) we have g 0 (ϕ(w)) = g 000 (ϕ(w)) = 0 and 2 c ϕ(w) 00 g (ϕ(w)) = C exp , 2 α+4 (1 − |ϕ(w)| ) (1 − |ϕ(w)|2 )α+2 where 24α + 120α + 141 C=− . 3α + 10 By the boundedness of Wϕ,ψ D : Aα log → Zβ , kWϕ,ψ DgkZβ ≤ CkWϕ,ψ Dk, and from which we obtain c |ϕ(w)|2 (1 − |w|2 )β ψ(w)ϕ00 (w) + 2ψ 0 (w)ϕ0 (w) exp ≤ C. 2 α+4 (1 − |ϕ(w)| ) (1 − |ϕ(w)|2 )α+2 This shows that (1 − |w|2 )β c lim = 0. ψ(w)ϕ00 (w) + 2ψ 0 (w)ϕ0 (w) exp (1 − |ϕ(w)|2 )α+2 ϕ(w)→∂D (1 − |ϕ(w)|2 )2 For w ∈ D, we choose the functions f3 (z) =

1 (1 − |ϕ(w)|2 )α+2 (1 − |ϕ(w)|2 )α+4 −2 3 (1 − ϕ(w)z)2(α+2) (1 − ϕ(w)z)2(α+2)+2 (1 − |ϕ(w)|2 )α+6 8 (1 − |ϕ(w)|2 )α+5 − + 3 (1 − ϕ(w)z)2(α+2)+3 (1 − ϕ(w)z)2(α+2)+4

and g3 (z) =

(1 − |ϕ(w)|2 )α+2 (1 − ϕ(w)z)2(α+2)

.

From a calculation, we obtain f3 (ϕ(w)) = f30 (ϕ(w)) = f300 (ϕ(w)) = 0.

(10)

Define the function h(z) = f3 (z) exp cg3 (z). Then by (10), h0 (ϕ(w)) = h00 (ϕ(w)) = 0, and by a direct calculation, 3

ϕ(w) c h (ϕ(w)) = C exp , 2 α+5 (1 − |ϕ(w)| ) (1 − |ϕ(w)|2 )α+2 000

where C = −30(α + 2)2 − 8. Since Wϕ,ψ D : Aα log → Zβ is bounded, we have kWϕ,ψ DhkZβ ≤ CkWϕ,ψ Dk, and so (1 − |z|2 )β |(Wϕ,ψ Dh)00 (z)| ≤ CkWϕ,ψ Dk,

(11)

for all z ∈ D. Letting z = w in (11) yields to (1 − |w|2 )β c |ψ(w)||ϕ0 (w)|2 |ϕ(w)|3 exp ≤ CkWϕ,ψ Dk. (1 − |ϕ(w)|2 )α+5 (1 − |ϕ(w)|2 )α+2

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ON AN OPERATOR

7

Thus (1 − |w|2 )β c C(1 − |ϕ(w)|2 )α+2 |ψ(w)||ϕ0 (w)|2 exp ≤ . 2 3 2 α+2 (1 − |ϕ(w)| ) (1 − |ϕ(w)| ) |ϕ(w)|3

(12)

Taking limit as ϕ(w) → ∂D in (12) gives (1 − |w|2 )β c |ψ(w)||ϕ0 (w)|2 exp = 0. (1 − |ϕ(w)|2 )α+2 ϕ(w)→∂D (1 − |ϕ(w)|2 )3 lim

The proof of the implication is finished. α (iii) ⇒ (ii). Let (fn )n∈N be a sequence in Aα log with supn∈N kfn kAlog ≤ M and fn → 0 uniformly on every compact subset of D as n → ∞. We have that for the constant C in Lemma 2.2, for any ε > 0 there exits a constant δ ∈ (0, 1) such that whenever δ < |ϕ(z)| < 1, it follows that (1 − |z|2 )β 00 C |ψ (z)| exp < ε, 1 − |ϕ(z)|2 (1 − |ϕ(z)|2 )α+2 (1 − |z|2 )β C 00 0 0 exp ψ(z)ϕ (z) + 2ψ (z)ϕ (z) < ε, (1 − |ϕ(z)|2 )2 (1 − |ϕ(z)|2 )α+2 and (1 − |z|2 )β C |ψ(z)||ϕ0 (z)|2 exp < ε. (1 − |ϕ(z)|2 )3 (1 − |ϕ(z)|2 )α+2 Then by Lemma 2.2, for a fixed δ ∈ (0, 1) we have kWϕ,ψ Dfn kZβ = (ψ · fn0 ◦ ϕ)(0) + (ψ · fn0 ◦ ϕ)0 (0) + sup(1 − |z|2 )β (ψ(z)fn0 (ϕ(z))00 z∈D 0 0 0 00 = ψ(0) fn (ϕ(0)) + ψ (0)fn (ϕ(0)) + ψ(0)fn (ϕ(0))ϕ(0)   + sup(1 − |z|2 )β ψ 00 (z)fn0 (ϕ(z)) + ψ(z)ϕ00 (z) + 2ψ 0 (z)ϕ0 (z) fn00 (ϕ(z)) + ψ(z)ϕ0 (z)2 fn000 (ϕ(z)) z∈D   ≤ ψ(0) + ψ 0 (0) fn0 (ϕ(0)) + ϕ(0) ψ(0) fn00 (ϕ(0)) + sup(1 − |z|2 )β ψ 00 (z) fn0 (ϕ(z)) z∈D

2 + sup(1 − |z|2 )β ψ(z)ϕ00 (z) + 2ψ 0 (z)ϕ0 (z) fn00 (ϕ(z)) + sup(1 − |z|2 )β ψ(z) ϕ0 (z) fn000 (ϕ(z)) z∈D z∈D  00 0  0 ≤ ψ(0) + ψ (0) fn (ϕ(0)) + ϕ(0) ψ(0) fn (ϕ(0)) + sup (1 − |z|2 )β ψ 00 (z) fn0 (ϕ(z)) + sup (1 − |z|2 )β ψ 00 (z) fn0 (ϕ(z)) |ϕ(z)|≤δ

δ s(h2 ), then h1 > h2 ; if s(h1 ) = s(h2 ), then h1 = h2 . Xia and Xu [11, 12] further gave some hesitant fuzzy aggregation operators as follows: Let hj (j = 1, 2, · · · , n) be a collection of HFEs, ω = (ω1 , ω2 , · · · , ωn )T be the weight vector of hj (j = n ∑ 1, 2, · · · , n) with ωj ∈ [0, 1] and ωj = 1, then j=1

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Hesitant fuzzy Maclaurin symmetric mean operators

3

(1) The hesitant fuzzy weighted averaging (HF W A) operator   n n  ⊕ ∪  ∏ HF W A(h1 , h2 , · · · , hn ) = (ωj hj ) = 1− (1 − γj )ωj   γj ∈hj , i=1,··· ,n

j=1

j=1

Especially, if ω = (1/n, 1/n, · · · , 1/n)T , then the HF W A operator reduces to the hesitant fuzzy averaging (HF A) operator   n  ∪  ∏ HF A(h1 , h2 , · · · , hn ) = 1− (1 − γj )1/n (1)   γj ∈hj , i=1,··· ,n

j=1

(2) The hesitant fuzzy weighted geometric (HF W G) operator HF W G(h1 , h2 , · · · , hn ) =

n ⊗

∪

hj ωj =

γj ∈hj , i=1,··· ,n

j=1

 n ∏ 

γj ωj

j=1

  

Especially, if ω = (1/n, 1/n, · · · , 1/n)T , then the HF W G operator becomes to the hesitant fuzzy geometric (HF G) operator   n  ∪ ∏ HF G(h1 , h2 , · · · , hn ) = γj 1/n (2)   γj ∈hj , i=1,··· ,n

2.2

j=1

Maclaurin symmetric mean

The M SM introduced by Maclaurin [18] is a useful technique characterized by the ability to capture the interrelationship among the multi-input arguments. The definition of M SM is given as follows. Definition 2.3. [18] Let ai (i = 1, 2, · · · , n) be a collection of nonnegative real numbers and r = 1, 2, · · · , n. If 

∑

r ∏

 1≤i1 0: Keywords: Oscillation, Stochastic di¤erential equations, Zeros of solutions, Wiener process, Itô integral AMS Subject Classi…cation: 60H10; 60H25; 34K11 ————————————————————————————————————————————————–

1

Introduction

During the past few decades, stochastic di¤erential equations (SDEs) are becoming increasingly important as models of stochastic phenomena that play a prominent role in a diverse range of application areas, including mathematical modeling in engineering and physics, geophysical sciences, stochastic control, mechanics, environmental processes, mathematical biology, molecular dynamics for chemistry, epidemiology, economic modeling, industrial mathematics and mathematical …nance [1-10]. Indeed, these models can be stochastic for di¤erent reasons. Therefore, numerous studies have been performed to understanding their dynamical behaviors, particularly in relation to problems of the speci…cation of the stochastic processes governing the behaviors of an underlying quantity, as well as fundamental microscopic laws generate stochastic behaviors in the case of coarse-graining and modeling error and so on [11-16]. However, a complete understanding of SDEs theory requires familiarity with advanced probability and stochastic processes, whereas solutions of such models are themself stochastic processes. Further, in particular, second-order di¤erential equations with random coe¢ cients have found wide variety applications in branches of science. Typically, they are mathematical models of objects under the in‡uence of random forces such that the presence of in…nite set of zeros of solutions for these equations indicates that the evolution of investigated objects is oscillatory. Recently, research work about oscillation phenomenon occupies an important place in di¤erential stochastic theory due to the sensitivity of stochastic forces and behaviors. Moreover, the stochastic theory for these equations, as well as the theory of oscillatory solutions of deterministic equations have been studied extensively and are well-developed. Oscillation and nonoscillatition conditions for both linear and nonlinear di¤erential equations, di¤erence equations and delay equations have been investigated in [17-22]. The oscillating properties for solutions of di¤erence equations can be found in the excellent monograph of Agarwal et al. [23]. Besides, the authors in the monograph [24] were devoted to the problem of relationship between oscillation behavior of solutions for di¤erential equations and the corresponding di¤erence equations. On the contrary, the theory of the oscillation of stochastic system is not well-developed. Incidentally, Mao in [25] considered the stochastic equation of the following form _ (t); x • + kx = hW

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_ (t) is a Wiener process which is nowhere di¤erentiable. It was proved that the solution with initial values where W x(0) = 1; x(0) _ = 0 has in…nitely many zeros, all simple, on each half-line [t0 ; 1) for every t0 0. The …rst two moments of the …rst zero were estimated. In contrast, the more general equation of the form _ (t); x • + k(t; x; x) _ = hW was studied in [26]. The author there demonstrated that this equation has in…nitely many zeros with probability 1. Consequently, the explicit upper and lower estimates for the expected values of these zeros were obtained. However, the Itô stochastic equations of the form _ (t))x = 0 x • + (p(t) + q(t)W was considered by method of asymptotic equivalence in [27,28], whereas the oscillation of solutions was analyzed. In the monographs [29,30], the oscillatory properties of solutions for both linear and nonlinear stochastic delay di¤erential equations with multiplicative noise are given. It was shown that such noise induces an oscillation in solutions. Besides, the oscillation of solutions of …rst order nonlinear stochastic di¤erence equations is investigated in [31]. The purpose of this paper is to study an asymptotic behavior, as t ! 1, of solutions of a second order stochastic Itô equation. Meanwhile, we investigate existence of zeros of its solutions with probability 1. In the sequel, unless otherwise speci…ed, we say that a solution is oscillatory if it has in…nitely many zeros with probability 1 on the half-line [0; 1). A solution which is not oscillatory is called nonoscillatory. This paper is organized in …ve sections including the introduction. In the next section, we present some necessary de…nitions and preliminary results that will be used in this work. In the same time, statement of a second order SDEs is introduced. In Section 3, the discussion of a solution for linear case of second-order SDEs is presented, as well as the conditions of nonoscillatory behavior of its solutions for nonlinear case of SDEs are constructed. Finally, the conclusions are drawn in Section 4.

2

Statement of the problem and auxiliary results

The material in this section is basic in some sense. For the reader’s convenience, we present some necessary de…nitions and auxiliary results related to the SDEs theory that will be used in the remainder of this paper. Let us consider a nonlinear second-order stochastic equation of the following form _ (t) = 0; t x • + p(t; x; x) _ + q(t; x; x) _ W

0:

(1)

While the corresponding system of stochastic Itô equations will be written as dx1 = x2 dt; dx2 =

p(t; x1 ; x2 )dt

(2) q(t; x1 ; x2 )dW (t);

where x 2 R1 ; t 0; W (t) is a standard Wiener process de…ned on the probability space ( ; F; P ), fFt ; t 0g is the family of -algebras adapted to W (t); and the functions p(t; x1 ; x2 ) and q(t; x1 ; x2 ) are continuous with respect to x1 ; x2 2 R1 for t 0, as well as satisfy the Lipschitz condition with respect to x1 , x2 together with linear growth condition. Without loss of generality, we assume that p(t; 0; 0) = q(t; 0; 0) = 0. It should be noted that the presence of stochastic in equation (1) causes new di¢ culties in studying the oscillation of solutions. In this regard, we mention here the following remark: Firstly, solutions of equation (1) are random processes, so their zeros are random variables with certain properties. As a consequence, we need to introduce a new de…nition of zero which is di¤erent of the deterministic case (q = 0). Secondly, from the Strook-Varadham support theorem, it follows that solutions of equation (1) can be nonoscillatory on …nite intervals. Therefore, the oscillatory solutions should be considered only on in…nite intervals. Thirdly, since solutions of equation (1) have only …rst derivative, so we can not use a second derivative to apply the concavity property of the solution between two successive zeros. It is well known that this method is used in the deterministic case. 861

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Nevertheless, system (2) is a particular case of general second-order system, it would seem that this simpli…es its investigation, as well it is the system with a degenerate di¤usion that completes its investigation by probability methods. Subsequently, under the above assumptions of equation (1) and the corresponding system (2), we assume that the solution x(t) = (x1 (t); x2 (t)) of system (2) subject to the initial condition x(t0 ) = x0 satisfy all necessary requirements of the existence of a unique solution for t t0 ; whereas x0 is an Ft0 measurable random variable. In addition, the process x(t) will never reach the origin (0,0), for more details see Lemma 2.3 in [32]. In our notation, let x1 (t) = x(t). Throughout this paper, a solution x(t) of equation (1) is called a nontrivial solution if it satisfy the following condition P fx(t) = 0; t > t0 g = 0 On the other hand, for any nontrivial solution x(t) of equation (1), where t can be de…ned as follows 8 < inf ft > t0 j x1 (t) = 0g ; if ft > t0 j x1 (t) = 0g 6= ?; 1 = : 1; otherwise.

t0

0; the random variable

1

(3)

Now, we will introduce the de…nition of zeros of a solution x(t) on the half-line t > 0.

De…nition 2.1 The random variable with probability 1.

1

is called the …rst zero of a solution x(t) on the interval t

t0 ; if

1

1 j x1 (t) = 0g ; if ft > 1 j x1 (t) = 0g 6= ?; = 2 : 1; otherwise.

(4)

Here, the random variable 2 is called the second zero of a solution x(t) on the interval t t0 ; if 2 < 1 with probability 1. Correspondingly, one can de…ne by induction a sequence of zeros f n g of a solution x(t) on the interval t t0 . Particularly, if t0 = 0: Then, we deal with zeros on the half-line t > 0. De…nition 2.2 A nontrivial solution x(t) of equation (1) is called oscillatory on the half-line t > 0; if it has in…nitely many zeros there. Otherwise, it is called nonoscillatory.

3

Main results and behavior solutions of the SDEs

In this section, some de…nitions and results are brie‡y reviewed to establish and generalize the results to the main equation in this work. Meanwhile, we study the behavior of the zeros of solutions for a class of second-order SDEs subject to some initial conditions, as well as we detect the conditions of nonoscillatory behavior of its solutions.

3.1

Linear stochastic Itô equation

Consider the following equation _ (t); x • + x = f (t)W

(5)

subject to the initial conditions x(0) = 1; x(0) _ = 0;

(6)

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where f (t) is a nonrandom function de…ned on t Zt

4

0 such that a stochastic Itô integral

f (s)dW (s)

0

is de…ned for any t > 0. Note that equation (5) is special case of equation (1), so it satis…es all arguments mentioned in the previous part of our work. Further, we give the following theorem regarding to study the behavior of the zeros of solutions for equation (5) with initial conditions (6). Theorem 3.1 Assume that f (t) satis…es the following conditions: 1. f (t) is di¤erentiable function for t s)f (s))0

2. (sin(t

0 for 0

s

0 such that f (0)

t

0;

2:

Then, the solution of equation (5) subject to initial conditions (6) oscillates on the half-line t mathematical expectation 1 of the …rst zero satis…es the estimation E

1

0: Besides, a

1 2t? ( p ? ); t

(7)

where t is the solution of the equation f (0) = cot(t) on [0; 2 ] and

(z) =

(8) 1 2

Zz

u2 2

e

du:

0

Proof. From Itô formula, the representation of the solution of (5) with initial conditions (6) is given by

x(t) = cos(t) +

Zt

f (s) sin(t

s)dW (s):

(9)

0

Which implies that Zt x(t) = cos(t) + sin(t) f (s) cos(s)dW (s) 0

Zt cos(t) f (s) sin(s)dW (s): 0

Accordingly, the process x(t) can be written as x(t) = cos(t) + W1 (p(t)) sin(t) + W2 (q(t)) cos(t); where p(t) =

Zt

2

2

f (s) cos (s)ds; q(t) =

0

Zt

f 2 (s) sin2 (s)ds;

0

and W1 ; W2 are Wiener processes. In contrast, if we consider x(t) at the times tm = (2m + 12 ) fYm g by Ym = x((2m + 21 ) )

x((2(m

for m = 1; 2; 3; :::; and de…ne a sequence 0 1 (2m+ 12 ) Z B C cos2 (s)f 2 (s)dsA : 1) + 12 ) ); whereas x((2m + 12 ) ) = W1 @ 0

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Then, Y0 = W ( 4 ); Y1 = W1 ( 54 )

W1 ( 4 ); :::; is a sequence of random variables with mean zero and variance

(2m+ 12 )

Z

cos2 (s)f 2 (s)ds:

(2(m 1)+ 12 )

Here, it is worth to mention that 1 x((2m + ) ) = Y0 + Y1 + ::: + Ym : 2

(10)

By the familiar theorems on the limits of sums of independent random variables (e.g. the law of the iterated logarithm), it follows that the sequence fx((2m+ 12 ) )g has in…nitely many switches of sign. Since x(t) is continuous on [0; 1), so it has in…nitely many zeros on [0; 1): Therefore, it oscillates on [0; 1): Now, let us prove the estimation (7) for the …rst zero of the oscillation. By applying the integration-by-parts formula to (9), we obtain Zt

x(t) = cos(t)

0

(sin(t

s)f (s)) dW (s)

cos(t) +

Zt

(sin(t

s)f (s))0 ds;

(11)

0

0

for ! 2 ; where W (t)

1 and 0

s

t

2.

From properties of a Wiener process, it follows that P

where

! j max W (t) > t2[0;T ]

(z) =

Zz

p1 2 2

e

u2 2

1

1 = 2 ( p ); 2 T

(12)

du:

0

As a result, from equation (11), we obtain the estimate x(t)

cos(t)

f (0) sin(t) > 0;

(13)

for t 2 [0; t ); where t is solution of equation (8). Hence, from equations (12) and (13), we have ! 1 1 p P cot (f (0)) 2 ; cot 1 (f (0))

for the …rst zero

3.2

1.

(14)

By using Chebyshev’s inequality, it yields that E

1

2t? ( p1t? ). The proof is complete.

Nonlinear stochastic Itô equation

Let P (a; n) = fx 2 Rm j (x is any set of the form \ P (a ; n );

a; n)

0g, where a; n 2 Rm , and ( ; ) is the usual scalar product. Thus, a polyhedron (15)

2I

where I = f1; :::; N g is a …nite subset of N: Now, consider a system of SDEs dx = f (t; x)dt + g(t; x)dW (t)

(16)

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where x 2 Rm ; f : [0; 1) Rm ! Rm ; g = [gij ] : [0; 1] r dimensional Wiener process.

Rm ! Rm

Rr are mappings, and W (t) is an

De…nition 3.1 A set K 2 Rm is said to be stochastically invariant for system (16), if for any x(0) 2 K and every solution x(t) of equation (1), then P fx(t) 2 K; t > 0g = 1. The next theorem states conditions of an invariance of the set (15) for system (16). Theorem 3.2 [33] Let K =

\

P (a ; n ) be a polyhedron in Rm . Suppose that the coe¢ cients f (t; x) and g(t; x)

2I

0; x 2 Rm ; and satisfy the following conditions:

of system (16) are de…ned for t

1. for each T > 0; there exists a constant KtT > 0 such that for all x 2 K and t 2 [0; T ); 2

2

2

kf (t; x)k + kg(t; x)k

KT (1 + jxj );

2. for all T > 0; x 2 K; y 2 K and t 2 [0; T ); kf (t; x)

f (t; y)k + kg(t; x)

g(t; y)k

KT jx

yj ;

3. for each x 2 K; the functions f ( ; x) and g( ; x); de…ned for t

0; are continuous.

Then, the set K is invariant for the system (16) if and only if the following condition holds: (a) for all

2 I and x 2 K such that (x

(f (t; x); n ) where t

a ; n ) = 0, we have

0 and (gj (t; x); n ) = 0;

0; j = 1; r; and gj is the j

th column of the matrix g = [gij ]:

Now, we use the above theorem to …nd the conditions of nonoscillatory behavior of the solutions of equation (1). As well, we state the following theorem: Theorem 3.3 Suppose that the functions p and q in equation (1) satisfy the conditions (1)-(3) of Theorem 3.2. Moreover, if (1) p(t; x1 ; 0)

0; x1 < 0; t

0;

(2) p(t; x1 ; 0)

0; x1 > 0; t

0;

(3) q(t; x1 ; 0) = 0; t

(17)

0; x1 2 R1 :

Then, all solutions of equation (1) with nonrandom initial values such that x(0) > 0; x(0) _ are not oscillate on the half-line [0; 1).

0 or x(0) < 0; x(0) _

0

Proof. We consider any solution of equation (1) with initial values x(0) > 0; x(0) _ 0. It corresponds to the solution (x1 ; x2 ) of system (2) with initial values x1 (0) > 0; x2 (0) 0. Obviously, there exists > 0 such that 0< x1 (0): Let M be a set such that M = f(x1 ; x2 ) j x1 ; x2 0g. It is a polyhedron, if we set a1 = ; n1 = l1 = (1; 0)T ; \ a2 = 0; n2 = l2 = (0; 1)T . Then, M = P (a ; n ); where I = f1; 2g: Consequently, the boundaries of this 2I

polyhedron are lines 1

= f(x1 ; x2 ) j x1 = ; x2

2

= f(x1 ; x2 ) j x1

0g ;

; x2 = 0g : 865

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Therefore, by using Theorem 3.2, the functions f and q have the form f (t; x1 ; x2 ) = (x2 ; p(t; x1 ; x2 ))T ; g(t; x1 ; x2 ) = (0; q(t; x1 ; x2 ))T ; Next, we verify the conditions of Theorem 3.2. On the boundary (f; n1 ) = (f; l1 ) = x2

1,

we have

0; and (g; n1 ) = (g; l1 ) = 0:

From condition (3) of equation (17), we have (f; n2 ) = (f; l2 ) =

p(t; x1 ; 0)

0;

(g; n2 ) = (g; l2 ) =

q(t; x1 ; 0) = 0:

and

on the boundary

2:

Again from Theorem 3.2, it follows that the set M is the invariant set for the solutions of system (2). Thus, the curve (x1 (t); x2 (t)) does not intersect with probability 1 the line x1 = 0: This means that the solutions of equation (1) with initial values x(0) > 0; x(0) _ 0 do not oscillate. It remains to consider the case with initial values x(0) < 0; x(0) _ 0: We only introduce the polyhedron M1 = f(x1 ; x2 ) j x1 ; x2 0g instead the set M: Hence, the proof is complete.

4

Concluding remarks

The use of SDEs is a natural way to model real-world phenomena under stochastic processes. In this paper, we study the qualitative behavior of nonlinear second order stochastic di¤erential equations. Interest focuses on solutions of such equations which are oscillatory. A nontrivial solution is called oscillatory if it has in…nitely many zeros with probability 1 on half-line. Otherwise, it is called nonoscillatory. The su¢ cient conditions for the oscillation and nonoscillation of solutions are obtained.

References [1] L.J.S. Allen, An Introduction to Stochastic Processes With Applications to Biology, Pearson Education Inc., Upper Saddle River, New Jersey, 2003. [2] P. Greenwood, L.F. Gordillo, R. Kuske, Autonomous stochastic resonance produces epidemic oscillations of ‡uctuating size, Proceedings of Prague Stochastics, 2006. [3] J. Golec, S. Sathananthan, Stability analysis of a stochastic logistic model, Mathematical and Computer Modelling, 38 (2003), 585–593. [4] E. Dogan, E.J. Allen, Derivation of Stochastic Partial Di¤erential Equations for Reaction-Di¤usion Processes, Stochastic Analysis and Applications, 29(3) (2011), 424-443. [5] W.D. Sharp, E.J. Allen, Stochastic neutron transport equations for rod and plane geometries, Annals of Nuclear Energy, 27 (2000), 99–116. [6] K.E. Emmert, L.J.S. Allen, Population extinction in deterministic and stochastic discrete-time epidemic models with periodic coe¢ cients with applications to amphibians, Natural Resource Modeling, 19 (2006), 117–164. [7] D.T. Gillespie, Approximate accelerated stochastic simulation of chemically reacting systems, Journal of Chemical Physics, 115 (2001), 1716–1733. 866

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[8] G.N. Milstein, M.V. Tretyakov, Stochastic Numerics for Mathematical Physics, Springer-Verlag, Berlin, 2004. [9] J.G. Hayes, E.J. Allen, Stochastic point-kinetics equations in nuclear reactor dynamics, Annals of Nuclear Energy, 32 (2005), 572–587. [10] E.J. Allen, L.J.S. Allen, A. Arciniega, P. Greenwood, Construction of equivalent stochastic di¤erential equation models, Stochastic Analysis and Applications, 26 (2008), 274–297. [11] B. Øksendal, Stochastic Di¤erential Equations, 6th edition, Springer-Verlag, Berlin, 2002. [12] A.M. Ateiwi, I.V. Komashynska, The existence and unbounded extension of the solutions of the systems with stochastic impulse action, Journal of Mathematics and Statistics, 2(4) (2006), 460-463. [13] A.M. Ateiwi, Dissipativity of the systems of di¤erential equations with stochastic impulse action, International journal of applied mathematics, 21 (2008), 73-81. [14] A.M. Samoilenko, O.M. Stanzhytskyi, A.M. Ateiwi, On invariant tori for a stochastic Itô system, Journal of dynamics and di¤erential equations, 17(4) (2005), 737-758. [15] D.J. Higham, Mean-square and asymptotic stability of the stochastic theta method, SIAM J. Numer. Anal., 38 (2000), 753–769. [16] I.V. Komashynska, Existence and uniqueness of solutions for a class of nonlinear stochastic di¤erential equations, Abstract and Applied Analysis, 2013 (2013), http://dx.doi.org/10.1155/2013/256809, 7p (Article ID 256809). [17] J.A. Appleby, C. Kelly, Oscillation and non-oscillation in solutions of nonlinear stochastic delay di¤erential equations, Elect. Comm. in Probab., 9 (2004), 106-118. [18] I.T. Kiguradze, Asymptotic properties of solutions of nonautonomous ordinary di¤erential equations, Nauka, Moscow, 1990. [19] A. Lomtatidze, N. Partsvania, Oscillation and non-oscillation criteria for two-dimensional systems of …rst order linear ordinary di¤erential equations, Georgian Math. Journ. 6(3) (1999), 285-298. [20] V. A. C¼ au¸s, Minimal quadratic oscillation for cubic splines, Journal of Computational Analysis and Applications, 9 (2007), 85-92. [21] Ercan Tunç¸ Ahmet Ero¼ glu, Oscillation Results for Second Order Half-Linear Nonhomogeneous Di¤erential Equations with Damping, Journal of Computational Analysis and Applications, 15(2) (2013), 255-263. [22] A. M. Bica, V. A. C¼ au¸s, I. Fechete, S. Mure¸san, Application of the Cauchy-Buniakovski-Schwarz’s Inequality to an Optimal Property for Cubic Splines, Journal of Computational Analysis and Applications, 9 (2007), 43-53. [23] R.P. Agarwal, M. Bohner, S.R. Grace, D. O’Regan, Discrete Oscillation Theory. Hindawi Publishing Corporation, New York, USA, 2005. [24] O.V. Karpenko, V.I. Kravets’, O.M. Stanzhytskyi, Oscillation of solutions of the second-order functional difference equations, Ukr. Math. Journal, 65(2) (2013), 249–259. [25] X. Mao, Stochastic Di¤erential Equations and Their Applications, Horwood Publishing, Chichester, UK, 1997. [26] L. Markus, A. Weerasighe, Stochastic oscillators, Journal of di¤erential equations, 71(3) (1998), 288-314. [27] A.M. Samoilenko, O.M. Stanzhitskyi, Qualitative and asymptotic analysis of di¤erential equations with random perturbations, Singapore, New Jersey, London, Hong Kong: Word Scienti…c, 2011. [28] A.N. Stanzhytskii, A.P. Krenevich, I.G. Novak, Asymptotic equivalence of linear stochastic Itô systems and oscillation of solutions of linear second-order equations, Di¤erential Equations, 47(6) (2011), 799-813. 867

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[29] J.A. Appleby, E. Buckwar, Noise induced oscillation in solutions of stochastic delay equations, Dynamic Systems and Applications, 14(2) (2005), 175-197. [30] J. Shao, F. Meng. Oscillations theorems for second -order forced neutral nonlinear di¤erential equations with delayed argument, Int. J. Di¤. Eq. 2010 (2010), http://dx.doi.org/10.1155/2010/181784, 15p (Article ID 181784). [31] J.A. Appleby, A. Rodkina, On the oscillation of solutions of stochastic di¤erence equations with stateindependent perturbations, Int. J. Di¤er Equ. 2(2) (2007), 139-164. [32] R.Z. Khasminskii, Stochastic stability of di¤erential equations, Sijtho¤ and Noordho¤, 1980. [33] A. Milian, Stochastic viability and a comparison theorem, Colloquium mathematicum, Vol. LXVIII, (1995), 297-316.

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Fixed point theorems and T -stability of Picard iteration for generalized Lipschitz mappings in cone metric spaces over Banach algebras Huaping Huang1∗, Shaoyuan Xu2 , Hao Liu1 , Stojan Radenovi´c3 1. School of Mathematics and Statistics, Hubei Normal University, Huangshi, 435002, China 2. Department of Mathematics and Statistics, Hanshan Normal University, Chaozhou, 521041, China 3. Faculty of Mathematics and Information Technology, Dong Thap University, Dong Thap, Viˆet Nam

Abstract: In this paper, we obtain the existence of non-normal solid cone and some fixed point theorems for generalized Lipschitz contractive mappings in cone metric spaces over Banach algebras. Our results greatly generalize the main work by Xu and Radenovi´c (Fixed Point Theory and Applications, 2014, 2014: 102). Moreover, we verify the P property and T -stability of Picard’s iteration. Further, we give an example to illustrate that our works are never a copy of metric results in the literature. MSC: 47H10; 54H25 Keywords: Generalized Lipschitz constant, P property, T stability, Cone metric space over Banach algebra, Solid cone

1

Introduction Since Huang and Zhang [1] introduced the concept of cone metric space, many scholars

have focused on fixed point theorems in such spaces. There are lots of works on fixed point results in the setting of cone metric spaces (see [2-6]). It is said that [1] is well-known as a result of the fact that cone metric spaces generalize metric spaces and expands the famous Banach contraction principle. But recently, it had not yet been a hot topic since some authors appealed to the equivalence of some metric and cone metric fixed point results ∗

Corresponding author: Huaping Huang. E-mail: [email protected]

1

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(see [7-12]). Owing to these reasons, people set out to lose interest in studying fixed point theorems in cone metric spaces. However, the present situation has gone better since, very recently, Liu and Xu [13] introduced the concept of cone metric space over Banach algebra and obtained some fixed point theorems in normal cone metric spaces over Banach algebras. Moreover, they gave an example to illustrate that the non-equivalence of versions of fixed point theorems between cone metric spaces over Banach algebras and (general) metric spaces (in usual sense), which shows that it is essentially necessary to investigate fixed points in cone metric spaces over Banach algebras. Lately, Xu and Radenovi´c [15] delete the normality of cones and greatly generalize the main results of [13]. Throughout this paper, we obtain the existence of non-normal solid cone for generalized Lipschitz mappings in cone metric spaces over Banach algebras. Moreover, we present some fixed point theorems for such mappings in such setting by omitting the assumptions of normalities of cones. Our theorems include the main results of [13] and [15]. Furthermore, we consider the mapping’s P property and T -stability of Picard’s iteration. Our results greatly unite and extend the main work of [13-15] and [17-21]. In addition, we give an example to illustrate our results in cone metric spaces over Banach algebras are never equivalent to the counterpart of metric spaces. Let A be a Banach algebra with a unit e, and θ the zero element of A. A nonempty closed convex subset K of A is called a cone if {θ, e} ⊂ K, K 2 = KK ⊂ K, K∩(−K) = {θ} and λK + µK ⊂ K for all λ, µ ≥ 0. On this basis, we define a partial ordering  with respect to K by x  y if and only if y − x ∈ K. We shall write x ≺ y to indicate that x  y but x 6= y, while x  y will indicate that y − x ∈ intK, where intK stands for the interior of K. If intK 6= ∅, then K is said to be a solid cone. Write k · k as the norm on A. A cone K is called normal if there is a number M > 0 such that for all x, y ∈ A, θ  x  y implies kxk ≤ M kyk. The least positive number satisfying above is called the normal constant of K. In the sequel we always suppose that A is a Banach algebra with a unit e, K is a solid cone in A, and  is a partial ordering with respect to K. Definition 1.1.([13]) Let X be a nonempty set and A a Banach algebra. Suppose that the mapping d : X × X → A satisfies: (i) θ ≺ d(x, y) for all x, y ∈ X with x 6= y and d(x, y) = θ if and only if x = y; (ii) d(x, y) = d(y, x) for all x, y ∈ X; 2

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(iii) d(x, y)  d(x, z) + d(z, y) 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 Banach algebra. Definition 1.2.([15]) Let (X, d) be a cone metric space, x ∈ X and {xn } a sequence in X. Then (i) {xn } converges to x whenever for every c  θ there is a natural number N such that d(xn , x)  c for all n ≥ N . We denote this by lim xn = x or xn → x (n → ∞). n→∞ (ii) {xn } is a Cauchy sequence whenever for each c  θ there is a natural number N such that d(xn , xm )  c for all n, m ≥ N . (iii) (X, d) is a complete cone metric space if every Cauchy sequence is convergent. Definition 1.3. Let (X, d) be a cone metric space, {yn } a sequence in X and T a self-map of X. Let x0 be a point of X, xn+1 = T xn a Picard’s iteration in X. The iteration procedure xn+1 = T xn is said to be T -stable with respect to T if {xn } converges to a fixed point q of T , and for each c  θ, there exists a natural number N such that d(yn+1 , T yn )  c for all n > N , then lim yn = q. n→∞ Remark 1.4. Comparing Definition 2.1 of [18] and Definition 1.3, we find that, the conditions of the former are stronger than the latter. Actually, if lim d(yn+1 , T yn ) = n→∞

θ, then we must have that for each c  θ there exists an integer N > 0 such that d(yn+1 , T yn )  c for all n > N . But the contrary is not true (see [6]). Lemma 1.5.([6]) Let u, v, w ∈ A. If u  v and v  w, then u  w. Lemma 1.6.([6]) Let A be a Banach algebra and {an } a sequence in A. If an → θ (n → ∞), then for any c  θ, there exists N such that for all n > N , one has an  c. Lemma 1.7.([16]) Let A be a Banach algebra with a unit e, x ∈ A, then the limit 1 lim kxn k n exists and the spectral radius ρ(x) satisfies n→∞

1

1

ρ(x) = lim kxn k n = inf kxn k n . n→∞

If ρ(x) < |λ|, then λe − x is invertible in A, moreover, −1

(λe − x)

∞ X xi = , λi+1 i=0

where λ is a complex constant. Lemma 1.8.([16]) Let A be a Banach algebra with a unit e, a, b ∈ A. If a commutes with b, then ρ(a + b) ≤ ρ(a) + ρ(b),

ρ(ab) ≤ ρ(a)ρ(b). 3

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Main results In this section we give some basic but important properties, which will be used con-

stantly in the sequel. Moreover, we introduce a class of contractive mappings with some generalized Lipschitz constants and prove the existence of non-normal solid cone and several fixed point theorems based on them without the assumption of normalities of cones. In addition, we obtain the fixed point periodic property and T -stability of Picard’s iteration. All results greatly generalize the main assertions of [13-15] and [17-21]. Further, we display an example to illustrate the applications. In the end, we give another example to claim that our results in the setting of cone metric spaces over Banach algebras are never equivalent to those in usual metric spaces. Lemma 2.1. Let A be a Banach algebra and k ∈ A. If ρ(k) < 1, then lim kk n k = 0. n→∞

1

1

Proof. Since ρ(k) = lim kk n k n < 1, then there exists α > 0 such that lim kk n k n < n→∞

n→∞

1

α < 1. Letting n be big enough, we obtain kk n k n ≤ α, then kk n k ≤ αn → 0 (n → ∞). Hence kk n k → 0 (n → ∞). Lemma 2.2. Let A be a Banach algebra with a unit e, {xn } a sequence in A. If xn converges to x in A, and for any n ≥ 1, xn commutes with x, then ρ(xn ) → ρ(x) as n → ∞. Proof. Since xn commutes with x, then it follows by Lemma 1.8 that ρ(xn ) ≤ ρ(xn − x) + ρ(x) ⇒ ρ(xn ) − ρ(x) ≤ ρ(xn − x), ρ(x) ≤ ρ(x − xn ) + ρ(xn ) ⇒ ρ(x) − ρ(xn ) ≤ ρ(x − xn ), thus |ρ(xn ) − ρ(x)| ≤ ρ(xn − x) ≤ kxn − xk ⇒ ρ(xn ) → ρ(x)(n → ∞).

Lemma 2.3. Let A be a Banach algebra with a unit e and K be a solid cone in A. Let {an } and {cn } be two sequences in K satisfying the following inequality: an+1  han + cn ,

(2.1)

where h ∈ K and ρ(h) < 1. If for each c  θ, there exists N such that cn  c for all n > N , then an  c (n > N ).

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Proof. By virtue of ρ(h) < 1, it follows by Lemma 1.7, e − h is invertible and (e − h)−1 = ∞ P hi . Moreover, by Lemma 2.1, it establishes khn k → 0 (n → ∞). Assume c  θ be i=0

arbitrary. Then there exists N1 such that for all n > N1 , we have cn 

(e − h)c . 2

(2.2)

Since khn−N1 aN 1 +1 k ≤ khn−N1 kkaN 1 +1 k → 0 (n → ∞), thus there is N2 such that for all n > N2 , it satisfies c hn−N1 aN 1 +1  . (2.3) 2 Put N = max{N1 , N2 }, then for all n > N , both (2.2) and (2.3) are satisfied. Taking advantage of (2.1), we speculate that an+1 − han  cn , han − h2 an−1  hcn−1 , h2 an−1 − h3 an−2  h2 cn−2 , ········· hn−N1 −1 aN 1 +2 − hn−N1 aN 1 +1  hn−N1 −1 cN 1 +1 . Conbine with the above terms, for all n > N , it follows that an+1  hn−N1 aN 1 +1 + cn + hcn−1 + h2 cn−2 + · · · + hn−N1 −1 cN 1 +1 c (e − h)c  + (e + h + h2 + · · · + hn−N1 −1 ) · 2 2 c (e − h)c  + (e − h)−1 · = c. 2 2 Remark 2.4. Lemma 2.3 greatly generalizes Lemma 1 of [17] and Lemma 1.5 of [18]. Virtually, we delete the normality of K. Moreover, our conditions are weaker than them. Indeed, if an → θ (n → ∞), then an  c (n > N ). But the converse is not true (see [6]). Further, if kkk < 1, it is natural that ρ(k) < 1. Yet, the converse is not true. Theorem 2.5. Let (X, d) be a cone metric space over Banach algebra A and K be a solid cone in A. Suppose that the mapping T : X → X satisfies the following contractive condition: d(T x, T y)  k1 d(x, y) + k2 d(x, T x) + k3 d(y, T y) + k4 d(x, T y) + k5 d(y, T x), 5

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for all x, y ∈ X, where ki ∈ K(i = 1, . . . , 5) are generalized Lipschitz constants with ρ(k1 ) + ρ(k2 + k3 + k4 + k5 ) < 1. If k1 commutes with k2 + k3 + k4 + k5 , then there exists a sequence {xn } in X such that it is a Cauchy sequence. Moreover, if {d(xn , yn )} converges to some non-zero element in A for any two different Cauchy sequence {xn } and {yn }, then K is a non-normal cone. Proof. Fix x0 ∈ X and set xn+1 = T xn = T n+1 x0 . Then we have d(xn+1 , xn ) = d(T xn , T xn−1 )  k1 d(xn , xn−1 ) + k2 d(xn , T xn ) + k3 d(xn−1 , T xn−1 ) + k4 d(xn , T xn−1 ) + k5 d(xn−1 , T xn ) = k1 d(xn , xn−1 ) + k2 d(xn , xn+1 ) + k3 d(xn−1 , xn ) + k5 d(xn−1 , xn+1 )  (k1 + k3 + k5 )d(xn , xn−1 ) + (k2 + k5 )d(xn+1 , xn ).

(2.4)

We also have d(xn+1 , xn ) = d(T xn , T xn−1 ) = d(T xn−1 , T xn )  k1 d(xn−1 , xn ) + k2 d(xn−1 , T xn−1 ) + k3 d(xn , T xn ) + k4 d(xn−1 , T xn ) + k5 d(xn , T xn−1 ) = k1 d(xn , xn−1 ) + k2 d(xn−1 , xn ) + k3 d(xn , xn+1 ) + k4 d(xn−1 , xn+1 )  (k1 + k2 + k4 )d(xn , xn−1 ) + (k3 + k4 )d(xn+1 , xn ).

(2.5)

Add up (2.4) and (2.5) yields that 2d(xn+1 , xn )  (2k1 + k2 + k3 + k4 + k5 )d(xn , xn−1 ) + (k2 + k3 + k4 + k5 )d(xn+1 , xn ), which establishes that (2e − k2 − k3 − k4 − k5 )d(xn+1 , xn )  (2k1 + k2 + k3 + k4 + k5 )d(xn , xn−1 ). Put k = k2 + k3 + k4 + k5 , then (2e − k)d(xn+1 , xn )  (2k1 + k)d(xn , xn−1 ).

(2.6)

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Since ρ(k) ≤ ρ(k1 ) + ρ(k) < 1 < 2, then by Lemma 1.7 it follows that 2e − k is invertible. Furthermore, ∞ X ki = . 2i+1 i=0

−1

(2e − k)

By multiplying in both sides of (2.6) by (2e − k)−1 , we arrive at d(xn+1 , xn )  (2e − k)−1 (2k1 + k)d(xn , xn−1 ).

(2.7)

Denote h = (2e − k)−1 (2k1 + k), then by (2.7) we get d(xn+1 , xn )  hd(xn , xn−1 )  · · ·  hn d(x1 , x0 ). By Lemma 1.8 we conclude that  P P n n ki 6 ρ ρ 2i+1 i=0

i=0

ki




2i+1

6

n P i=0

[ρ(k)]i , 2i+1

which implies by Lemma 2.2 that ρ

P ∞ i=0

ki 2i+1



6

∞ P i=0

[ρ(k)]i . 2i+1

Since k1 commutes with k, it follows that P  P ∞ ∞ −1 ki (2e − k) (2k1 + k) = (2k + k) = 2 1 i+1 2 i=0

= 2k1

P ∞ i=0

ki 2i+1



+k

∞ P i=0

ki 2i+1

i=0

P ∞ = (2k1 + k) i=0

ki 2i+1



ki 2i+1



k1 +

∞ P i=0

ki+1 2i+1

= (2k1 + k)(2e − k)−1 ,

that is to say, (2e − k)−1 commutes with 2k1 + k. Then by Lemma 1.8 we gain

ρ(h) = ρ (2e − k)−1 (2k1 + k) ≤ ρ (2e − k)−1 ρ(2k1 + k)  P  P ∞ ∞ [ρ(k)]i ki [2ρ(k ) + ρ(k)] ≤ [2ρ(k1 ) + ρ(k)] ≤ρ 1 i+1 i+1 2 2 i=0

i=0

=

1 [2ρ(k1 ) + ρ(k)] < 1, 2 − ρ(k)

which establishes that e − h is invertible and khm k → 0 (m → ∞). Thus for all n > m, d(xn , xm )  d(xn , xn−1 ) + d(xn−1 , xn−2 ) + · · · + d(xm+1 , xm )  (hn−1 + hn−2 + · · · + hm )d(x1 , x0 ) = (hn−m−1 + hn−m−2 + · · · + h + e)hm d(x1 , x0 ) P  ∞  hi hm d(x1 , x0 ) i=0

= (e − h)−1 hm d(x1 , x0 ). 7

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Owing to k(e − h)−1 hm d(x1 , x0 )k ≤ k(e − h)−1 kkhm kkd(x1 , x0 )k → 0 (m → ∞), we have (e − h)−1 hm d(x1 , x0 ) → θ (m → ∞). So by using Lemma 1.5 and 1.6, we easily see that {xn } is a Cauchy sequence in X. Now we take y0 ∈ X such that y0 6= x0 . Using the same method as the above mentioned, we can show that {yn } is also a Cauchy sequence if yn+1 = T yn = T n+1 y0 . In the following we suppose the contrary, that is, K is normal. We shall prove that {d(xn , yn )} is convergent in (A, k · k) if K is a normal cone with normal constant M . In fact, in view of the completeness of (A, k · k), it will be enough to show that the sequence {d(xn , yn )} is a Cauchy sequence. To this end, let ε > 0 and choose c  θ and kck
N . It is clear that d(xn , yn )  d(xn , xm ) + d(xm , ym ) + d(ym , yn )  d(xm , ym ) + 2c, d(xm , ym )  d(xm , xn ) + d(xn , yn ) + d(yn , ym )  d(xn , yn ) + 2c.

(2.8) (2.9)

It follows immediately from (2.8) and (2.9) that θ  d(xm , ym ) + 2c − d(xn , yn )  d(xn , yn ) + 2c + 2c − d(xn , yn ) = 4c.

(2.10)

By virtue of the normality of K, (2.10) means that kd(xm , ym ) + 2c − d(xn , yn )k ≤ 4M kck. Hence, it ensures us that kd(xm , ym ) − d(xn , yn )k ≤ kd(xm , ym ) + 2c − d(xn , yn )k + k2ck ≤ (4M + 2)kck < ε, which implies that {d(xn , yn )} is Cauchy and hence convergent. Next, put lim d(xn , yn ) = a, it is evident that θ  a. Finally, we claim that a = θ. n→∞

Actually, if there exists n0 ∈ N such that xn0 = yn0 , the claim is clear. Without loss of

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generality, we suppose that xn 6= yn for all n ∈ N. Notice that d(xn+1 , yn+1 ) = d(T xn , T yn )  k1 d(xn , yn ) + k2 d(xn , T xn ) + k3 d(yn , T yn ) + k4 d(xn , T yn ) + k5 d(yn , T xn ) = k1 d(xn , yn ) + k2 d(xn , xn+1 ) + k3 d(yn , yn+1 ) + k4 d(xn , yn+1 ) + k5 d(yn , xn+1 )  (k1 + k4 + k5 )d(xn , yn ) + (k2 + k5 )d(xn , xn+1 ) + (k3 + k4 )d(yn , yn+1 ). Taking the limit as n → ∞, we obtain that a  (k1 + k4 + k5 )a. Set λ = k1 + k4 + k5 , then it follows that a  λa  · · ·  λn a. Because λ  k1 + k leads to λn  (k1 + k)n , moreover, by Lemma 2.1, ρ(k1 + k) ≤ ρ(k1 ) + ρ(k) < 1 leads to (k1 + k)n → θ (n → ∞), we claim that, for each c  θ, there exists n0 (c) such that λn  c such that for all n > n0 (c). Consequently, a = θ, a contradiction. It is clear that if T is a map which has a fixed point u, then u is also a fixed point of T n for each n ∈ N. It is well known that the converse is not true. If a map T satisfies F (T ) = F (T n ) for each n ∈ N, where F (T ) stands for the set of all fixed points of T , then it is said to have a property P (see [19-21]). The following results are generalizations of the corresponding results in metric and cone metric spaces (see [20-21]). It will be deduced also without using normality of the cone. Theorem 2.6. Let (X, d) be a complete cone metric space over Banach algebra A and K be a solid cone in A. Let T : X → X be a mapping such that F (T ) 6= ∅ and that d(T x, T 2 x)  kd(x, T x)

(2.11)

for all x ∈ X, where k ∈ K is a generalized Lipschitz constant with ρ(k) < 1. Then T has a property P . 9

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Proof. We will always assume that n > 1, since the statement for n = 1 is trivial. Let z ∈ F (T n ). By the hypotheses, it is clear that d(z, T z) = d(T T n−1 z, T 2 T n−1 z)  kd(T n−1 z, T n z) = kd(T T n−2 z, T 2 T n−2 z)  k 2 d(T n−2 z, T n−1 z)  · · ·  k n d(z, T z). On account of ρ(k) < 1, it follows by Lemma 2.1 that kk n k → 0 (n → ∞). Thus kk n d(z, T z)k ≤ kk n kkd(z, T z)k → 0 (n → ∞). Hence d(z, T z) = θ, that is., T z = z. Theorem 2.7. Let (X, d) be a complete cone metric space over Banach algebra A and K be a solid cone in A. Suppose that the mapping T : X → X satisfies the following contractive condition: d(T x, T y)  k1 d(x, y) + k2 d(x, T x) + k3 d(y, T y) + k4 d(x, T y) + k5 d(y, T x), for all x, y ∈ X, where ki ∈ K(i = 1, . . . , 5) are generalized Lipschitz constants with ρ(k1 ) + ρ(k2 + k3 + k4 + k5 ) < 1. If k1 commutes with k2 + k3 + k4 + k5 , then T has a unique fixed point in X. Moreover, for arbitrary x ∈ X, iterative sequence {T n x} converges to the fixed point. Further, T has a property P . Proof. By using Theorem 2.5, we obtain {xn } is a Cauchy sequence in X. Since X is complete, there exists x∗ ∈ X such that xn → x∗ as n → ∞. We shall prove x∗ is the fixed point of T . To this end, on the one hand, we have d(T x∗ , x∗ )  d(T x∗ , T xn ) + d(T xn , x∗ )  k1 d(x∗ , xn ) + k2 d(x∗ , T x∗ ) + k3 d(xn , T xn ) + k4 d(x∗ , T xn ) + k5 d(xn , T x∗ ) + d(T xn , x∗ ) = k1 d(x∗ , xn ) + k2 d(x∗ , T x∗ ) + k3 d(xn , xn+1 ) + (e + k4 )d(x∗ , xn+1 ) + k5 d(xn , T x∗ )  (k1 + k3 + k5 )d(xn , x∗ ) + (k2 + k5 )d(T x∗ , x∗ ) + (e + k3 + k4 )d(xn+1 , x∗ ), which implies that (e − k2 − k5 )d(T x∗ , x∗ )  (k1 + k3 + k5 )d(xn , x∗ ) + (e + k3 + k4 )d(xn+1 , x∗ ).

(2.12)

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On the other hand, we obtain d(T x∗ , x∗ )  d(T xn , T x∗ ) + d(T xn , x∗ )  k1 d(xn , x∗ ) + k2 d(xn , T xn ) + k3 d(x∗ , T x∗ ) + k4 d(xn , T x∗ ) + k5 d(x∗ , T xn ) + d(T xn , x∗ ) = k1 d(xn , x∗ ) + k2 d(xn , xn+1 ) + k3 d(x∗ , T x∗ ) + (e + k5 )d(x∗ , xn+1 ) + k4 d(xn , T x∗ )  (k1 + k2 + k4 )d(xn , x∗ ) + (k3 + k4 )d(T x∗ , x∗ ) + (e + k2 + k5 )d(xn+1 , x∗ ), which means that (e − k3 − k4 )d(T x∗ , x∗ )  (k1 + k2 + k4 )d(xn , x∗ ) + (e + k2 + k5 )d(xn+1 , x∗ ).

(2.13)

Combining (2.12) and (2.13) yields that (2e − k2 − k3 − k4 − k5 )d(T x∗ , x∗ )  (2k1 + k2 + k3 + k4 + k5 )d(xn , x∗ ) + (2e + k2 + k3 + k4 + k5 )d(xn+1 , x∗ ). Denote k = k2 + k3 + k4 + k5 , then (2e − k)d(T x∗ , x∗ )  (2k1 + k)d(xn , x∗ ) + (2e + k)d(xn+1 , x∗ ). Consequently, we deduce that d(T x∗ , x∗ )  (2e − k)−1 (2k1 + k)d(xn , x∗ ) + (2e − k)−1 (2e + k)d(xn+1 , x∗ ). In view of xn → x∗ (n → ∞), then for each that for all n > Nm , one has d(xn , x∗ ) 

c . m

c m

 θ (m = 1, 2, . . .), there exists Nm such

Hence we speculate that

d(T x∗ , x∗ )  (2e − k)−1 (2k1 + k)d(xn , x∗ ) + (2e − k)−1 (2e + k)d(xn+1 , x∗ ) c  [(2e − k)−1 (2k1 + k) + (2e − k)−1 (2e + k)] → θ (m → ∞), m which follows that T x∗ = x∗ . 11

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In the following we shall show the fixed point is unique. Actually, if there is another fixed point y ∗ , then d(x∗ , y ∗ )  k1 d(x∗ , y ∗ ) + k2 d(x∗ , T x∗ ) + k3 d(y ∗ , T y ∗ ) + k4 d(x∗ , T y ∗ ) + k5 d(y ∗ , T x∗ ) = (k1 + k4 + k5 )d(x∗ , y ∗ ). Set λ = k1 + k4 + k5 , then it follows that d(x∗ , y ∗ )  λd(x∗ , y ∗ )  · · ·  λn d(x∗ , y ∗ ). By the proof of Theorem 2.5, we claim that, for each c  θ, there exists n0 (c) such that λn  c such that for all n > n0 (c). Consequently, d(x∗ , y ∗ ) = θ, that is, x∗ = y ∗ . Finally, we shall prove that T has a property P . We have to show that the mapping T satisfies the condition (2.11). Indeed, firstly we have d(T x, T 2 x) = d(T x, T T x)  k1 d(x, T x) + k2 d(x, T x) + k3 d(T x, T 2 x) + k4 d(x, T 2 x) + k5 d(T x, T x)  k1 d(x, T x) + k2 d(x, T x) + k3 d(T x, T 2 x) + k4 d(x, T x) + k4 d(T x, T 2 x), that is, (e − k3 − k4 )d(T x, T 2 x)  (k1 + k2 + k4 )d(x, T x).

(2.14)

Also, we have d(T x, T 2 x) = d(T T x, T x)  k1 d(T x, x) + k2 d(T x, T 2 x) + k3 d(x, T x) + k4 d(T x, T x) + k5 d(x, T 2 x)  k1 d(x, T x) + k2 d(T x, T 2 x) + k3 d(x, T x) + k5 d(x, T x) + k5 d(T x, T 2 x), that is, (e − k2 − k5 )d(T x, T 2 x)  (k1 + k3 + k5 )d(x, T x).

(2.15)

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Adding (2.14) and (2.15) we obtain (2e − k)d(T x, T 2 x)  (2k1 + k)d(x, T x). According to the above proof, we demonstrate that d(T x, T 2 x)  hd(x, T x), where h = (2e − k)−1 (2k1 + k) and ρ(h) < 1. Therefore, by Theorem 2.6, T has a property P. Theorem 2.8. Let (X, d) be a complete cone metric space over Banach algebra A and K be a solid cone in A. Suppose that the mapping T : X → X satisfies the following contractive condition: d(T x, T y)  k1 d(x, y) + k2 d(x, T x) + k3 d(y, T y) + k4 d(x, T y) + k5 d(y, T x), for all x, y ∈ X, where ki ∈ K(i = 1, . . . , 5) are generalized Lipschitz constants with ρ(k1 ) + ρ(k2 + k3 + k4 + k5 ) < 1. If k1 commutes with k2 + k3 + k4 + k5 , then Picard’s iteration is T -stable. Proof. By utilizing Theorem 2.7, we obtain that T has a unique fixed point q in X. Assume that {yn } ⊂ X satisfies the following condition: for each c  θ, there exists N such that for all n > N , d(yn+1 , T yn )  c. Firstly we have d(T yn , q) = d(T yn , T q)  k1 d(yn , q) + k2 d(yn , T yn ) + k3 d(q, T q) + k4 d(yn , T q) + k5 d(q, T yn ) = k1 d(yn , q) + k2 d(yn , T yn ) + k4 d(yn , q) + k5 d(q, T yn )  (k1 + k4 )d(yn , q) + k2 [d(yn , q) + d(q, T yn )] + k5 d(q, T yn ) = (k1 + k2 + k4 )d(yn , q) + (k2 + k5 )d(q, T yn ).

(2.16)

Secondly, we arrive at d(T yn , q) = d(q, T yn ) = d(T q, T yn )  k1 d(q, yn ) + k2 d(q, T q) + k3 d(yn , T yn ) + k4 d(q, T yn ) + k5 d(yn , T q) = k1 d(yn , q) + k3 d(yn , T yn ) + k4 d(q, T yn ) + k5 d(yn , q)  (k1 + k5 )d(yn , q) + k3 [d(yn , q) + d(q, T yn )] + k4 d(q, T yn ) = (k1 + k3 + k5 )d(yn , q) + (k3 + k4 )d(q, T yn ).

(2.17)

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Adding up (2.16) and (2.17) yields that 2d(T yn , q)  (2k1 + k2 + k3 + k4 + k5 )d(yn , q) + (k2 + k3 + k4 + k5 )d(q, T yn ), Denote k = k2 + k3 + k4 + k5 , then we get (2e − k)d(T yn , q)  (2k1 + k)d(yn , q). Based on the proof of Theorem 2.5, it is not hard to verify that d(T yn , q)  hd(yn , q), where h = (2e − k)−1 (2k1 + k) and ρ(h) < 1. Setting an = d(yn , q) and cn = d(yn+1 , T yn ), we claim that an+1 = d(yn+1 , q)  d(yn+1 , T yn ) + d(T yn , q)  cn + han . If for each c  θ, there exists N such that for all n > N , cn = d(yn+1 , T yn )  c. Then, making full use of Lemma 2.3, we get an = d(yn , q)  c, which leads to yn → q as n → ∞. That is to say, the Picard’s iteration is T -stable. Corollary 2.9. Let (X, d) be a complete cone metric space over Banach algebra A and K be a solid cone in A. Suppose that the mapping T : X → X satisfies the following contractive condition: d(T x, T y)  kd(x, y), for all x, y ∈ X, where k ∈ K is a generalized Lipschitz constant with ρ(k) < 1. Then T has a unique fixed point in X. Moreover, for any x ∈ X, iterative sequence {T n x} converges to the fixed point. Further, T has a property P and Picard’s iteration is T -stable. Corollary 2.10. Let (X, d) be a complete cone metric space over Banach algebra A and K be a solid cone in A. Suppose that the mapping T : X → X satisfies the following contractive condition: d(T x, T y) 

k [d(x, T x) + d(y, T y)], 2

for all x, y ∈ X, where k ∈ K is a generalized Lipschitz constant with ρ(k) < 1. Then T has a unique fixed point in X. Moreover, for every x ∈ X, iterative sequence {T n x} converges to the fixed point. Further, T has a property P and Picard’s iteration is T -stable. 14

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Corollary 2.11. Let (X, d) be a complete cone metric space over Banach algebra A and K be a solid cone in A. Suppose that the mapping T : X → X satisfies the following contractive condition: d(T x, T y) 

k [d(x, T y) + d(y, T x)], 2

for all x, y ∈ X, where k ∈ K is a generalized Lipschitz constant with ρ(k) < 1. Then T has a unique fixed point in X. Moreover, for each x ∈ X, iterative sequence {T n x} converges to the fixed point. Further, T has a property P and Picard’s iteration is T -stable. Remark 2.12. Throughout the conclusions above, we focus on fixed point theorems in cone metric spaces over Banach algebras instead of the theorems only in cone metric spaces. All the coefficients are vector elements and the multiplications such as kd(x, y) are vector multiplications instead of usual scalar ones, which may bring us more convenience in applications. Remark 2.13. In our results such as Corollary 2.9, we only suppose the spectral radius of k is less than 1, while kkk < 1 is not assumed. Generally speaking, it is meaningful since by Remark 2.4, the condition ρ(k) < 1 is weaker than that kkk < 1. Remark 2.14. Compared with the main results of [13] and [15], our main results in this paper deal not only with the fixed point theorems for generalized Lipschitz mappings, but also with P property and T -stability of Picard’s iteration, all in the setting of cone metric spaces under the condition that the underlying cones are solid without assumption of normality. These results may be more valuable to put into use since the cones discussed are not necessarily normal under ordinary conditions. Therefore, it is an interesting thing to discuss the fixed point results in cone metric spaces over Banach algebras without the assumption that the underlying cones are normal. The following examples show that our main results will be very useful. Example 2.15. Let A = CR1 [0, 1] and define a norm on A by kxk = kxk∞ +kx0 k∞ . Define multiplication in A as just pointwise multiplication. Then A is a real Banach algebra with a unit e = 1 (e(t) = 1 for all t ∈ [0, 1]). The set K = {x ∈ A : x(t) ≥ 0 for all t ∈ [0, 1]} is a cone in A. Moreover, K is a non-normal solid cone (see [6]). Let X = {a, b, c}. Define d : X × X by d(a, b)(t) = d(b, a)(t) = et , d(b, c)(t) = d(c, b)(t) = 2et , d(c, a)(t) = d(a, c)(t) = 3et and d(x, x)(t) = θ for all t ∈ [0, 1] and each x ∈ X. We have that (X, d) is a solid cone metric space over Banach algebra A. Further, let T : X → X be a mapping defined with T a = T b = b, T c = a and let k1 , k2 , k3 , k4 , k5 ∈ K defined with k1 (t) = 31 t + 12 , k2 (t) = k3 (t) = k4 (t) = k5 (t) =

1 25

for all t ∈ [0, 1]. By careful calculations one can get 15

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that T is not a Banach contraction and all the conditions of Theorems 2.7 are fulfilled. The point x = b is the unique fixed point of T . By using Theorem 2.7 and Theorem 2.8, we can also conclude that T has a P property and Picard’s iteration is T -stable. Example 2.16. Let A = R2 and the norm be k(x1 , x2 )k = |x1 | + |x2 |. Define the multiplication by xy = (x1 , x2 )(y1 , y2 ) = (x1 y1 , x1 y2 + x2 y1 ), where x = (x1 , x2 ), y = (y1 , y2 ) ∈ A. Then A is a Banach algebra with a unit e = (1, 0). Taking X = [0, 0.55] × (−∞, +∞), K = {(x1 , x2 ) ∈ A : x1 , x2 ≥ 0} and d(x, y) = (|x1 − y1 |, |x2 − y2 |) for all x = (x1 , x2 ), y = (y1 , y2 ) ∈ X, we claim that (X, d) is a cone metric space over A and K is a normal solid cone with normal constant M = 1. Define a mapping T : X → X as 1  x1 1
T x = T (x1 , x2 ) = cos − |x1 − | , arctan(1 + |x2 |) + ln(x1 + 1) . 2 2 2 By using mean value theorem of differentials, it follows that d(T x, T y) = d(T (x1 , x2 ), T (y1 , y2 ))  1 y1 1 1
x1 − cos − |x1 − | + |y1 − | , = cos 2 2 2 2 2  | arctan(1 + |x2 |) − arctan(1 + |y2 |) + ln(x1 + 1) − ln(y1 + 1)|  x + y x − y 1  1 1 1 1 1  + |x1 − y1 |, |x2 − y2 | + |x1 − y1 | 4 4 2 2 5 
 , 1 |x1 − y1 |, |x2 − y2 | 8 5  = , 1 d(x, y) 8 for all x, y ∈ X. Put k = ( 58 , 1). Simple calculations show that all conditions of Corollary 2.9 are satisfied. Thus by Corollary 2.9, T has a unique fixed point in X. Further, T has a property P and Picard’s iteration is T -stable. The following statement indicates our fixed point results in cone metric space over Banach algebra A are not equivalent to those in metric spaces. In order to end this, put d1 (x, y) =

inf

{u∈P :d(x,y)u}

kuk,

d2 (x, y) = inf{r ∈ R : d(x, y)  re},

where x, y ∈ X and e = (e1 , e2 ) ∈ intK. Then by Theorem 2.2 of [10], d1 and d2 are both equivalent metrics. Hence we need to consider only one of them. Let us refer to the 16

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metric d2 . We shall prove our conclusions are not equivalent to the well-known Banach contraction principle, which means Theorem 2.4 of [8] does not hold in the setting of cone metric spaces over Banach algebras. As a matter of fact, taking x0 = ( 12 , 0), y 0 = (0, 0), e = (1, 12 ), we have  1 o n 1 1 1 3 cos − , ln  r 1, d2 (T x0 , T y 0 ) = inf r ∈ R : 2 4 4 2 2 n1 1 1 3o 3 = max cos − , 2 ln = 2 ln 2 4 4 2 2 1 ≥ = d2 (x0 , y 0 ), 2 which implies that there does not exist λ ∈ [0, 1) such that d2 (T x, T y) ≤ λd2 (x, y) for all x, y ∈ X. Thus it does not satisfy the contractive condition of Banach contraction principle. That is to say, Theorem 2.4 of [8] is unsuitable for cone metric spaces over Banach algebras. Remark 2.17. Since the contractive mapping in Example 2.15 is generalized Lipschitz mapping, we are easy to make a conclusion that Corollary 2.1 in [5] cannot cope with Example 2.15, which infers that the main results in the setting of cone metric spaces over Banach algebras are very meaningful. Remark 2.18. In Example 2.16, we are not hard to see that the main results in this paper are indeed more different from the standard results of cone metric spaces presented in the literature. Also, Example 2.16 shows that cone metric space over Banach algebra do be a real generalization of metric space even if some works with the assumption of normal cone.

Competing interests The authors declare that they have no competing interests.

Authors’ contributions All authors contributed equally and significantly in writing this paper. All authors read and approved the final manuscript. 17

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Acknowlegements The authors would like to express their thanks to the referees for their helpful comments and suggestions. The research is partially supported by Doctoral Initial Foundation of Hanshan Normal University, China (no. QD20110920).

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Int-soft filters of MTL-algebras Young Bae Jun1 , Seok Zun Song2 , Eun Hwan Roh3 and Sun Shin Ahn4,∗ 1

Department of Mathematics Education, Gyeongsang National University, Jinju 660-701, Korea 2 Department of Mathematics, Jeju National University, Jeju 690-756, Korea 3 Department of Mathematics Education, Chinju National University of Education, Jinju 660-756, Korea 4

Department of Mathematics Education, Dongguk University, Seoul 100-715, Korea

Abstract. The notions of (Boolean) int-soft filters in MTL-algebras are introduced, and several properties are investigated. Characterizations of (Boolean) int-soft filters are discussed, and a condition for an int-soft filter to be Boolean is provided. The extension property for a Boolean int-soft filter is constructed, and the least int-soft filter containing a given soft set is established.

1. Introduction The logic MTL, Monoidal t-norm based logic, was introduced by Esteva and Godo in [3]. This logic is very interesting from many points of view. From the logic point of view, it can be regarded as a weak system of Fuzzy Logic. Indeed, it arises from H´ajek’s Basic Logic BL [4] by replacing the axiom ˆ (A → B)) ↔ (A ∧ B) (A∧ by the weaker axiom ˆ (A → B)) → (A ∧ B). (A∧ In connection with the logic MTL, Esteva and Godo [3] introduced a new algebra, called a MTLalgebra, and studied several basic properties. They also introduced the notion of (prime) filters in MTL-algebras. Vetterlein [8] studied MTL-algebras arising from partially ordered groups. Borzooei, Khosravi Shoar and Americ [1] discussed some types of filters in MTL-algebras. Morton and van Alten [6] considered the algebraic semantics of the monoidal t-norm logic(MTL) with unary operations (modalities). In this paper, we introduce the notion of (Boolean) int-soft filters in MTL-algebras, and investigate several properties. We discuss characterizations of (Boolean) int-soft filters, and provide a condition for an int-soft filter to be Boolean. We establish the extension property for a Boolean int-soft filter. We also construct the least int-soft filter containing a given soft set. 2. Preliminaries 0

2010 Mathematics Subject Classification: 06F35; 03G25; 06D72. Keywords: (Boolean) filter; (Boolean) int-soft filter. ∗ The corresponding author. 0 E-mail: [email protected] (Y. B. Jun); [email protected] (S. Z. Song); [email protected] (E. H. Roh); [email protected] (S. S. Ahn) 0

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J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 20, NO.5, 2016, COPYRIGHT 2016 EUDOXUS PRESS, LLC

Young Bae Jun, Seok Zun, Eun Hwan Roh and Sun Shin Ahn

By a residuated lattice we shall mean a lattice L = (L, ≤, ∧, ∨, ⊙, →, 0, 1) containing the least element 0 and the largest element 1, and endowed with two binary operations ⊙ (called product) and → (called residuum) such that • ⊙ is associative, commutative and isotone. • (∀x ∈ L) (x ⊙ 1 = x). • The Galois correspondence holds, that is, (∀x, y, z ∈ L) (x ⊙ y ≤ z ⇐⇒ x ≤ y → z). In a residuated lattice, the following are true (see [7]): x ≤ y ⇐⇒ x → y = 1. 0 → x = 1, 1 → x = x, x → (y → x) = 1. y ≤ (y → x) → x. x → (y → z) = (x ⊙ y) → z = y → (x → z). x → y ≤ (z → x) → (z → y), x → y ≤ (y → z) → (x → z). y( ≤ x ) ⇒ x → z ≤ y → z, z → y ≤ z → x. ∨ ∧ yi → x = (yi → x). (a7) i∈Γ i∈Γ ∨ ∗ We define x = {y ∈ L | x ⊙ y = 0}, equivalently, x∗ = x → 0. Then (a1) (a2) (a3) (a4) (a5) (a6)

(a8) 0∗ = 1, 1∗ = 0, x ≤ x∗∗ , and x∗ = x∗∗∗ . Based on the H´ajek’s results [4], Axioms of MTL and Formulas which are provable in MTL, Esteva and Godo [3] defined the algebras, so called MTL-algebras, corresponding to the MTLlogic in the following way. Definition 2.1. An MTL-algebra is a residuated lattice L = (L, ≤, ∧, ∨, ⊙, →, 0, 1) satisfying the pre-linearity equation: (x → y) ∨ (y → x) = 1. In an MTL-algebra, the following are true: (a9) x → (y ∨ z) = (x → y) ∨ (x → z). (a10) x ⊙ y ≤ x ∧ y. Definition 2.2 ([3]). Let L be an MTL-algebra. A nonempty subset F of L is called a filter of L if it satisfies (b1) (∀x, y ∈ F ) (x ⊙ y ∈ F ). (b2) (∀x ∈ F ) (∀y ∈ L) (x ≤ y ⇒ y ∈ F ). Since ∧ is not definable from ⊙ and → in a MTL-algebra, one could consider that the further condition (b3) (∀x, y ∈ F ) (x ∧ y ∈ F ) 890

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J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 20, NO.5, 2016, COPYRIGHT 2016 EUDOXUS PRESS, LLC

Int-soft filters of MTL-algebras

should be also required for a filter. However the condition (b3) is indeed redundant because it is a consequence of conditions (b1) and (b2). Namely, since x ⊙ y ≤ x ∧ y, if x, y ∈ F then x ⊙ y ∈ F and thus x ∧ y ∈ F as well. Proposition 2.3. A nonempty subset F of an MTL-algebra L is a filter of L if and only if it satisfies: (b4) 1 ∈ F. (b5) (∀x ∈ F ) (∀y ∈ L) (x → y ∈ F ⇒ y ∈ F ). A soft set theory is introduced by Molodtsov [5], and C ¸ aˇgman et al. [2] provided new definitions and various results on soft set theory. Let P(U ) denote the power set of an initial universe set U and A, B, C, · · · ⊆ E where E is a set of parameters. ( ) Definition 2.4 ([2, 5]). A soft set f˜, A over U is defined to be the set of ordered pairs ( ) {( ) } f˜, A := x, f˜(x) : x ∈ E, f˜(x) ∈ P(U ) , where f˜ : E → P(U ) such that f˜(x) = ∅ if x ∈ / A. ( ) For a soft set f˜, L over U, the set ( ) { } iL f˜; γ = x ∈ L | γ ⊆ f˜(x) ( ) is called the γ-inclusive set of f˜, L . 3. Int-soft filters In what follows, we take an L as a set of parameters. ( MTL-algebra ) ˜ Definition 3.1. A soft set f , L over U is called an int-soft filter of L if it satisfies: ( ) (∀x, y ∈ L) f˜(x ⊙ y) ⊇ f˜(x) ∩ f˜(y) . ( ) (∀x, y ∈ L) x ≤ y ⇒ f˜(x) ⊆ f˜(y) .

(3.1) (3.2)

Example 3.2. Let L = [0, 1] and define a product ⊙ and a residuum → on L as follows: { { 1 if x ≤ y, x ∧ y if x + y > 21 , x → y := x ⊙ y := 0 otherwise (0.5 − x) ∨ y if x > y ( ) for all x, y ∈ L. Then L is an MTL-algebra. Let f˜, L be a soft set over U in which { α if x ∈ (0.5, 1], ˜ f (x) := β otherwise, ( ) where α ⊇ β in P(U ). Then it is routine to verify that f˜, L is an int-soft filter of L. 891

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J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 20, NO.5, 2016, COPYRIGHT 2016 EUDOXUS PRESS, LLC

Young Bae Jun, Seok Zun, Eun Hwan Roh and Sun Shin Ahn

We provide characterizations of an int-soft filter. ( ) Theorem 3.3. A soft set f˜, L over U is an int-soft filter of L if and only if it satisfies: (

) ˜ ˜ (∀x ∈ L) f (1) ⊇ f (x) , ( ) (∀x, y ∈ L) f˜(y) ⊇ f˜(x) ∩ f˜(x → y) .

(3.3) (3.4)

(

) ˜ Proof. Assume that f , L is an int-soft filter of L. Since x ≤ 1 for all x ∈ L, it follows from (3.2) that f˜(x) ⊆ f˜(1) for all x ∈ L. Since x ≤ (x → y) → y, we have x ⊙ (x → y) ≤ y for all x, y ∈ L by the Galois correspondence. It follows from (3.2) and (3.1) that f˜(y) ⊇ f˜(x ⊙ (x → y)) ⊇ f˜(x) ∩ f˜(x → y) for all x, y ∈ L. ( ) Conversely, let f˜, L be a soft set over U which satisfy two conditions (3.3) and (3.4). Let x, y ∈ L be such that x ≤ y. Then x → y = 1, and so f˜(y) ⊇ f˜(x) ∩ f˜(x → y) = f˜(x) ∩ f˜(1) = f˜(x), for all x ∈ L. This proves (3.2). Using (a4), we know that x → (y → (x ⊙ y)) = (x ⊙ y) → (x ⊙ y) = 1. Using (3.3) and (3.4), we have f˜(x ⊙ y) ⊇ f˜(y) ∩ f˜(y → (x ⊙ y)) ( ) ˜ ˜ ˜ ⊇ f (y) ∩ f (x) ∩ f (x → (y → (x ⊙ y))) ( ) = f˜(y) ∩ f˜(x) ∩ f˜(1) = f˜(x) ∩ f˜(y) ( ) for all x, y ∈ L. Therefore f˜, L is an int-soft filter of L.

□

(

) ˜ Theorem 3.4. A soft set f , L over U is an int-soft filter of L if and only if it satisfies: (

) ˜ ˜ ˜ (∀a, b, c ∈ L) a ≤ b → c ⇒ f (c) ⊇ f (a) ∩ f (b) .

(3.5)

(

) ˜ Proof. Assume that f , L is an int-soft filter of L. Let a, b, c ∈ L be such that a ≤ b → c. Then f˜(a) ⊆ f˜(b → c) by (3.2), and so f˜(c) ⊇ f˜(b) ∩ f˜(b → c) ⊇ f˜(b) ∩ f˜(a). 892

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J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 20, NO.5, 2016, COPYRIGHT 2016 EUDOXUS PRESS, LLC

Int-soft filters of MTL-algebras

( ) Conversely, let f˜, L be a soft set over U satisfying (3.5). Since x ≤ x → 1 for all x ∈ L, it follows from (3.5) that f˜(1) ⊇ f˜(x) ∩ f˜(x) = f˜(x) for all x ∈ L. Since x → y ≤ x → y for all x, y ∈ L, we have (

)

f˜(y) ⊇ f˜(x) ∩ f˜(x → y)

for all x, y ∈ L. Therefore f˜, L is an int-soft filter of L.

□

( ) Corollary 3.5. A soft set f˜, L over U is an int-soft filter of L if and only if it satisfies the following assertion: f˜(x) ⊇

n ∩

f˜(ak )

(3.6)

k=1

whenever an → (· · · → (a2 → (a1 → x)) · · · ) = 1 for every a1 , a2 , · · · , an ∈ L. Proof. It is by induction.

(

□

)

Theorem 3.6. For a filter F of L and a ∈ L, let f˜, L be a soft set over U defined by { γ1 if x ∈ {z ∈ L | a ∨ z ∈ F }, ˜ f (x) := γ2 otherwise, ( ) for all x ∈ L where γ2 ⊊ γ1 in P(U ). Then f˜, L is an int-soft filter of L. Proof. Since a ∨ 1 ∈ F, we have 1 ∈ {z ∈ L | a ∨ z ∈ F } and so f˜(1) = γ1 ⊇ f˜(x) for all x ∈ L. Now if y ∈ {z ∈ L | a ∨ z ∈ F }, then clearly f˜(y) = γ1 ⊇ f˜(x) ∩ f˜(x → y). Suppose that y ∈ / {z ∈ L | a ∨ z ∈ F }. Then at least one of x and x → y does not belong to {z ∈ L | a ∨ z ∈ F }. Hence f˜(y) = γ2 = f˜(x) ∩ f˜(x → y), ( ) and therefore f˜, L is an int-soft filter of L. □ ( ) Theorem 3.7. A soft set f˜, L over U is an int-soft filter of L if and only if the nonempty ( ) γ-inclusive set iL f˜; γ is a filter of L for all γ ∈ P(U ). (

) ˜ Proof. Assume that the nonempty γ-inclusive set iL f ; γ is a filter of L for all γ ∈ P(U ). For ( ) ( ) ( ) any x ∈ L, let f˜(x) = γ. Then x ∈ iL f˜; γ . Since iL f˜; γ is a filter of L, we have 1 ∈ iL f˜; γ ( ) and so f˜(x) = γ ⊆ f˜(1). For any x, y ∈ L, let f˜(x → y) ∩ f˜(x) = γ. Then x → y ∈ iL f˜; γ 893

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J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 20, NO.5, 2016, COPYRIGHT 2016 EUDOXUS PRESS, LLC

Young Bae Jun, Seok Zun, Eun Hwan Roh and Sun Shin Ahn

( ) ( ) and x ∈ iL f˜; γ . It follows from (b5) that y ∈ iL f˜; γ . Hence f˜(y) ⊇ γ = f˜(x → y) ∩ f˜(x). ( ) ˜ Therefore f , L is an int-soft filter of L by Theorem 3.3. ( ) ( ) Conversely, suppose that f˜, L is an int-soft filter of L. Let γ ∈ P(U ) be such that iL f˜; γ ̸= ( ) ∅. Then there exists a ∈ iL f˜; γ , and so γ ⊆ f˜(a). It follows from (3.3) that γ ⊆ f˜(a) ⊆ f˜(1). ( ) ( ) ( ) Thus 1 ∈ iL f˜; γ . Let x, y ∈ L be such that x → y ∈ iL f˜; γ and x ∈ iL f˜; γ . Then γ ⊆ f˜(x → y) and γ ⊆ f˜(x). It follows from (3.4) that γ ⊆ f˜(x → y) ∩ f˜(x) ⊆ f˜(y), ( ) ( ) that is, y ∈ iL f˜; γ . Thus iL f˜; γ (̸= ∅) is a filter of L by Proposition 2.3. ( ) ˜ Theorem 3.8. If f , L is an int-soft filter of L, then the set

□

Ωa := {x ∈ L | f˜(x) ⊇ f˜(a)} is a filter of L for every a ∈ L. Proof. Since f˜(1) ⊇ f˜(x) for all x ∈ L. we have 1 ∈ Ωa . Let x, y ∈ L be such that x ∈ Ωa and x → y ∈ Ωa . Then f˜(x) ⊇ f˜(a) and f˜(x → y) ⊇ f˜(a). Since f˜ is an int-soft filter of L, it follows from (3.4) that f˜(y) ⊇ f˜(x) ∩ f˜(x → y) ⊇ f˜(a) so that y ∈ Ωa . Hence Ωa is a filter of L. ( ) Theorem 3.9. Let a ∈ L and let f˜, L be a soft set over U. Then ( ) ˜ (1) If Ωa is a filter of L, then f , L satisfies the following implication: (

)

(∀x, y ∈ L) (f˜(a) ⊆ f˜(x → y) ∩ f˜(x) ⇒ f˜(a) ⊆ f˜(y)).

□

(3.7)

(2) If f˜, L satisfies (3.3) and (3.7), then Ωa is a filter of L. Proof. (1) Assume that Ωa is a filter of L. Let x, y ∈ L be such that f˜(a) ⊆ f˜(x → y) ∩ f˜(x). Then x → y ∈ Ωa and x ∈ Ωa . Using (b5), we have y ∈ Ωa and so f˜(y) ⊇ f˜(a). (2) Suppose that f˜ satisfies (3.3) and (3.7). From (3.3) it follows that 1 ∈ Ωa . Let x, y ∈ L be such that x ∈ Ωa and x → y ∈ Ωa . Then f˜(a) ⊆ f˜(x) and f˜(a) ⊆ f˜(x → y), which imply that f˜(a) ⊆ f˜(x) ∩ f˜(x → y). Thus f˜(a) ⊆ f˜(y) by (3.7), and so y ∈ Ωa . Therefore Ωa is a filter of L. □ ( ) Proposition 3.10. Let f˜, L be an int-soft filter of L. Then the following are equivalent: ( ) (1) (∀x, y, z ∈ L) f˜(x → z) ⊇ f˜(x → (y → z)) ∩ f˜(x → y) . 894

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J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 20, NO.5, 2016, COPYRIGHT 2016 EUDOXUS PRESS, LLC

Int-soft filters of MTL-algebras

( ) (2) (∀x, y ∈ L) f˜(x → y) ⊇ f˜(x → (x → y)) . ( ) ˜ ˜ (3) (∀x, y, z ∈ L) f ((x → y) → (x → z)) ⊇ f (x → (y → z)) . ( ) Proof. (1) ⇒ (2). Suppose that f˜, L satisfies the condition (1). Taking z = y and y = x in (1) and using (3.3), we have f˜(x → y) ⊇ f˜(x → (x → y)) ∩ f˜(x → x) = f˜(x → (x → y)) ∩ f˜(1) = f˜(x → (x → y)) for all x, y, z ∈ L. ( ) (2) ⇒ (3). Suppose that f˜, L satisfies the condition (2) and let x, y, z ∈ L. Since x → (y → z) ≤ x → ((x → y) → (x → z)), it follows that f˜((x → y) → (x → z)) = f˜(x → ((x → y) → z)) ⊇ f˜(x → (x → ((x → y) → z))) = f˜(x → ((x → y) → (x → z))) ⊇ f˜(x → (y → z)).

( ) (3) ⇒ (1). If f˜, L satisfies the condition (3), then

f˜(x → y) ⊇ f˜((x → y) → (x → z)) ∩ f˜(x → y) ⊇ f˜(x → (y → z)) ∩ f˜(x → y). This completes the proof.

(

□

)

Theorem 3.11. For a fixed element a ∈ L, let f˜a , L be a soft set over U defined by { γ1 if a ≤ x, f˜a (x) := γ2 otherwise, ( ) where γ1 ⊋ γ2 in P(U ). Then f˜a , L is an int-soft filter of L if and only if it satisfies the following implication: (∀x, y ∈ L) (a ≤ y → x, a ≤ y ⇒ a ≤ x). (3.8) ) Proof. Assume that f˜a , L is an int-soft filter of L and let x, y ∈ L be such that a ≤ y → x and a ≤ y. Then f˜a (y → x) = γ1 = f˜a (y), and thus (

f˜a (x) ⊇ f˜a (y → x) ∩ f˜a (y) = γ1 which implies that f˜a (x) = γ1 and so a ≤ x. 895

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J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 20, NO.5, 2016, COPYRIGHT 2016 EUDOXUS PRESS, LLC

Young Bae Jun, Seok Zun, Eun Hwan Roh and Sun Shin Ahn

( ) ( ) Conversely, suppose that (3.8) is valid. Note that iL f˜a ; γ2 = L and iL f˜a ; γ1 = {x ∈ ( ) ( ) ˜ ˜ L | a ≤ x}. Obviously 1 ∈ iL fa ; γ1 . Let x, y ∈ L be such that x ∈ iL fa ; γ1 and x → ( ) y ∈ iL f˜a ; γ1 . Then a ≤ x and a ≤ x → y, which imply from (3.8) that a ≤ y, that is, ( ) ( ) ( ) y ∈ iL f˜a ; γ1 . Hence iL f˜a ; γ1 is a filter of L. Using Theorem 3.7, we know that f˜a , L is an int-soft filter of L. □ ( ) Definition 3.12. An int-soft filter f˜, L of L is said to be Boolean if it satisfies the following identity ) ( ∗ ˜ ˜ (∀x ∈ L) f (x ∨ x ) = f (1) . (3.9) ( ) Proposition 3.13. Every Boolean int-soft filter f˜, L of L satisfies the following inclusion: ( ) ∗ ˜ ˜ ˜ (∀x, y, z ∈ L) f (x → z) ⊇ f (x → (z → y)) ∩ f (y → z) . (3.10) Proof. Using (a5), we have y → z ≤ (z ∗ → y) → (z ∗ → z) ≤ (x → (z ∗ → y)) → (x → (z ∗ → z)). It follows from (3.2) that f˜(y → z) ⊆ f˜((x → (z ∗ → y)) → (x → (z ∗ → z))) so from (3.4) that f˜(x → (z ∗ → z)) ⊇ f˜(x → (z ∗ → y)) ∩ f˜((x → (z ∗ → y)) → (x → (z ∗ → z))) ⊇ f˜(x → (z ∗ → y)) ∩ f˜(y → z). Since z ∗ ∨ z = ((z ∗ → z) → z) ∧ ((z → z ∗ ) → z ∗ ) ≤ (z ∗ → z) → z, we have f˜((z ∗ → z) → z) ⊇ f˜(z ∗ ∨ z) = f˜(1). Since x → (z ∗ → z) ≤ ((z ∗ → z) → z) → (x → z), it follows from (3.2) that f˜(x → (z ∗ → z)) ⊆ f˜(((z ∗ → z) → z) → (x → z)). Thus f˜(x → z) ⊇ f˜((z ∗ → z) → z) ∩ f˜(((z ∗ → z) → z) → (x → z)) ⊇ f˜(1) ∩ f˜(x → (z ∗ → z)) = f˜(x → (z ∗ → z)) ⊇ f˜(x → (z ∗ → y)) ∩ f˜(y → z). □

This completes the proof. 896

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Int-soft filters of MTL-algebras

We provide a condition for an int-soft filter to be Boolean. ( ) Proposition 3.14. If an int-soft filter f˜, L of L satisfies the following inclusion ( ) (∀x, y ∈ L) f˜(x) ⊇ f˜((x → y) → x) ,

(3.11)

then it is Boolean. Proof. Using (a2), (a4) and (a5), we have 1 = x → ((x∗ → x) → x) ≤ ((x∗ → x) → x)∗ → x∗ ≤ (x∗ → x) → (((x∗ → x) → x)∗ → x) = ((x∗ → x) → x)∗ → ((x∗ → x) → x) = (((x∗ → x) → x) → 0) → ((x∗ → x) → x). It follows from (3.2), (3.3) and (3.11) that f˜((x∗ → x) → x) ⊇ f˜((((x∗ → x) → x) → 0) → ((x∗ → x) → x)) = f˜(1). Using (a7) and (a9), since (x∗ → x) → x ≤ ((x∗ → x) → x) ∨ ((x∗ → x) → x∗ ) = (x∗ → x) → (x ∨ x∗ ) = (1 ∧ (x∗ → x)) → (x ∨ x∗ ) = ((x → x) ∧ (x∗ → x)) → (x ∨ x∗ ) = ((x ∨ x∗ ) → x) → (x ∨ x∗ ), we get f˜(1) = f˜((x∗ → x) → x) ⊆ f˜(((x ∨ x∗ ) → x) → (x ∨ x∗ )) ⊆ f˜(x ∨ x∗ ), ( ) and so f˜(x ∨ x∗ ) = f˜(1). Therefore f˜, L is Boolean. □ ( ) Proposition 3.15. If an int-soft filter f˜, L of L satisfies the condition (3.10), then it satisfies the condition (3.11). Proof. Since (x → y) → x ≤ x∗ → x, it follows from (3.2) that f˜(x) = f˜(1 → x) ⊇ f˜(1 → (x∗ → x∗ )) ∩ f˜(x∗ → x) ⊇ f˜(1) ∩ f˜((x → y) → x) = f˜((x → y) → x). 897

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J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 20, NO.5, 2016, COPYRIGHT 2016 EUDOXUS PRESS, LLC

Young Bae Jun, Seok Zun, Eun Hwan Roh and Sun Shin Ahn

( ) Hence f˜, L satisfies the condition (3.11). □ ( ) ˜ Proposition 3.16. If an int-soft filter f , L of L satisfies (3.11), then it satisfies the following inclusion: ( ) (∀x, y, z ∈ L) f˜(x → z) ⊇ f˜(x → (y → z)) ∩ f˜(x → y) . (3.12) Proof. Since x → (y → z) = y → (x → z) ≤ (x → y) → (x → (x → z)), it follows from (3.2) that f˜(x → (y → z)) ⊆ f˜((x → y) → (x → (x → z))) so from (3.4) that f˜(x → (x → z)) ⊇ f˜(x → y) ∩ f˜((x → y) → (x → (x → z))) ⊇ f˜(x → y) ∩ f˜(x → (y → z)). Since x → (x → z) ≤ x → (((x → z) → z) → z) = ((x → z) → z) → (x → z), we have f˜(x → z) ⊇ f˜(((x → z) → z) → (x → z)) ⊇ f˜(x → (x → z)) ⊇ f˜(x → y) ∩ f˜(x → (y → z)) by using (3.2) and (3.10). This completes the proof. □ ( ) Proposition 3.17. Every Boolean int-soft filter f˜, L of L satisfies the following inclusion: ( ) (∀x, y, z ∈ L) f˜(x → z) ⊇ f˜(x → (y → z)) ∩ f˜(x → y) . Proof. Note that x → (y → z) = y → (x → z) ≤ (x → y) → (x → (x → z)) and x → (x → z) ≤ x → (((x → z) → z) → z) = ((x → z) → z) → (x → z) for all x, y, z ∈ L. It follows from (3.2), (3.4), and Propositions 3.13 and 3.14 that f˜(x → z) ⊇ f˜(((x → z) → z) → (x → z)) ⊇ f˜(x → (x → z)) ⊇ f˜(x → y) ∩ f˜((x → y) → (x → (x → z))) ⊇ f˜(x → y) ∩ f˜(x → (y → z)). □

This completes the proof.

Combining Propositions 3.13, 3.14 and 3.15, we have a characterization of a Boolean int-soft filter. ( ) Theorem 3.18. Let f˜, L be an int-soft filter of L. Then the following assertions are equivalent: 898

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Int-soft filters of MTL-algebras

(

) f˜, L is Boolean. ( ) ˜ (2) f , L satisfies the condition (3.10). ( ) (3) f˜, L satisfies the condition (3.11). (1)

(

) ˜ Proposition 3.19. Every Boolean int-soft filter f , L of L satisfies: (

) ˜ ˜ (∀x, y ∈ L) f (x → y) ⊆ f (((y → x) → x) → y) .

(3.13)

( ) Proof. Let f˜, L be a Boolean int-soft filter of L. Since y ≤ ((y → x) → x) → y, we have (((y → x) → x) → y) → x ≤ y → x

(3.14)

by (a6). Using (a4), (a5), (a6) and (3.14), we get x → y ≤ ((y → x) → x) → ((y → x) → y) = (y → x) → (((y → x) → x) → y) ≤ ((((y → x) → x) → y) → x) → (((y → x) → x) → y) and so f˜(((y → x) → x) → y) ⊇ f˜(((((y → x) → x) → y) → x) → (((y → x) → x) → y)) ⊇ f˜(x → y) □

by Theorem 3.18(3) and (3.2). ( Theorem 3.20. (Extension Property for Boolean int-soft filter) Let

f˜, L

)

and (˜ g , L) be two ( ) int-soft filters of L such that f˜(1) = g˜(1) and f˜(x) ⊆ g˜(x) for all x ∈ L. If f˜, L is Boolean, then so is (˜ g , L) . ( ) Proof. Assume that f˜, L is a Boolean int-soft filter of L. Then f˜(x ∨ x∗ ) = f˜(1) for all x ∈ L. It follows from the hypothesis that g˜(x ∨ x∗ ) ⊇ f˜(x ∨ x∗ ) = f˜(1) = g˜(1).

(3.15)

Combining (3.15) and (3.3), we have g˜(x ∨ x∗ ) = g˜(1) for all x ∈ L. Hence (˜ g , L) is a Boolean int-soft filter of L. □ ( ) For any soft set f˜, L over U, let (˜ g , L) be a soft set over U in which g˜(x) :=

∪

{

n ∩ k=1

} an → (· · · → (a2 → (a1 → x)) · · · ) = 1, f˜(an ) a1 , a2 , · · · , an ∈ L 899

(3.16)

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Young Bae Jun, Seok Zun, Eun Hwan Roh and Sun Shin Ahn

for all x ∈ L. Let a, b, x ∈ L be such that a ≤ b → x. Take a1 , a2 , · · · , an , b1 , b2 , · · · , bm ∈ L such that an → (· · · → (a2 → (a1 → a)) · · · ) = 1, bm → (· · · → (b2 → (b1 → b)) · · · ) = 1, g˜(a) = g˜(b) =

n ∩ k=1 m ∩

f˜(ak ), f˜(bj ).

j=1

Then bm → (· · · → (b1 → (an → (· · · → (a2 → (a1 → x)) · · · ))) · · · ) = 1, and so g˜(x) ⊇ f˜(a1 ) ∩ f˜(a2 ) ∩ · · · ∩ f˜(an ) ∩ f˜(b1 ) ∩ f˜(b2 ) ∩ · · · f˜(bm ) ( n ) (m ) ∩ ∩ = f˜(ak ) ∩ f˜(bj ) k=1

j=1

= g˜(a) ∩ g˜(b). Hence (˜ g , L) is an int-soft filter of L by Theorem ( 3.4. ) Since( x →) x = 1 for all x ∈ L, we have ˜ L be an int-soft filter of L that f˜(x) ⊆ g˜(x) for all x ∈ L. Thus (˜ g , L) contains f˜, L . Let h, ( ) contains f˜, L . Then { n } ∪ ∩ an → (· · · → (a2 → (a1 → x)) · · · ) = 1, g˜(x) = f˜(an ) a1 , a 2 , · · · , a n ∈ L k=1 { n } ∪ ∩ an → (· · · → (a2 → (a1 → x)) · · · ) = 1, ˜ n) ⊆ h(a a1 , a2 , · · · , an ∈ L k=1 ∪ ˜ ˜ ⊆ h(x) = h(x) ( ) ˜ L . by Corollary 3.5, that is, (˜ g , L) is contained in h, We summarize this as follows: ( ) Theorem 3.21. For any soft set f˜, L over U, the soft set (˜ g , L) over U in which } { n ∪ ∩ an → (· · · → (a2 → (a1 → x)) · · · ) = 1, g˜(x) := f˜(an ) a1 , a2 , · · · , an ∈ L k=1 ( ) for all x ∈ L is the least int-soft filter of L that contains f˜, L . 900

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Int-soft filters of MTL-algebras

Conclusion Based on the soft set theory, we have introduced the notion of (Boolean) int-soft filters in MTL-algebras, and have investigated several properties. We have discussed characterizations of (Boolean) int-soft filters, and have provided a condition for an int-soft filter to be Boolean. We have established the extension property for a Boolean int-soft filter. We have also constructed the least int-soft filter containing a given soft set. Future research will focus on applying the notions and contents to other types of filters in related algebraic structures, and on studying it again by using Boolean algebra instead of P(U ). References [1] R. A. Borzooei, S. Khosravi Shoar and R. Americ, Some typesoffiltersinMTL-algebras, Fuzzy Sets and Systems 187 (2012), 92–102. [2] N. C ¸ aˇgman, F. C ¸ itak and S. Engino˘glu, Soft set theory and uni-int decision making, Eur. J. Oper. Res. 207 (2010) 848–855. [3] F. Esteva and L. Godo, Monoidal t-norm based logic: towards a logic for left-continuous t-norms, Fuzzy Sets and Systems 124 (2001), 271–288. [4] P. H´ajek, Metamathematics of Fuzzy Logic, Kluwer Academic Press, Dordrecht, 1998. [5] D. Molodtsov, Soft set theory - First results, Comput. Math. Appl. 37 (1999) 19–31. [6] W. Morton and C. J. van Alten, Modal MTL-algebras, Fuzzy Sets and Systems 222 (2013), 58–77. [7] E. Turunen, BL-algebras of basic fuzzy logic, Mathware & Soft Computing 6 (1999), 49–61. [8] T. Vetterlein, MTL-algebras arisingfrompartiallyorderedgroups, Fuzzy Sets and Systems 161 (2010), 433– 443.

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Convergence Analysis of New Iterative Approximating Schemes with Errors for Total Asymptotically Nonexpansive Mappings in Hyperbolic Spaces Ting-jian Xiong

a

and Heng-you Lan

a, b ∗

a

b

Department of Mathematics, Sichuan University of Science & Engineering, Zigong, Sichuan 643000, PR China Key Laboratory of Higher Education of Sichuan Province for Enterprise Informationalization and Internet of Things, Zigong, Sichuan 643000, PR China

Abstract. The purpose of this paper is to introduce the concept of total asymptotically nonexpansive mappings and to prove some ∆-convergence theorems of iteration processes with errors to approximating a common fixed point for four total asymptotically nonexpansive mappings in hyperbolic spaces. The results presented in the paper extend and improve some recent results announced in the current literature. Key Words and Phrases. New iterative approximations with errors, asymptotically nonexpansive mapping, total asymptotically nonexpansive mapping, common fixed point, convergence analysis. AMS Subject Classification. 47H09, 47H10, 54E70.

1

Introduction and preliminaries

Most of the problems in various disciplines of science are nonlinear in nature, whereas fixed point theory proposed in the setting of normed linear spaces or Banach spaces majorly depends on the linear structure of the underlying spaces. A nonlinear framework for fixed point theory is a metric space embedded with a ’convex structure’. The class of hyperbolic spaces, nonlinear in nature, is a general abstract theoretic setting with rich geometrical structure for metric fixed point theory. The study of hyperbolic spaces has been largely motivated and dominated by questions about hyperbolic groups, one of the main objects of study in geometric group theory. Throughout this paper, we work in the setting of hyperbolic spaces due to Kohlenbach [1], defined below, which is more restrictive than the hyperbolic type introduced in [2] and more general than the concept of hyperbolic space in [3]. A hyperbolic spaces is a metric space (X, d) together with a mapping W : X 2 × [0, 1] → X satisfying (i) d(u, W (x, y, α)) ≤ αd(u, x) + (1 − α)d(u, y); (ii) d(W (x, y, α), W (x, y, β)) = |α − β|d(x, y); (iii) W (x, y, α) = W (y, x, (1 − α)); (iv) d(W (x, z, α), W (y, w, α)) ≤ αd(x, y) + (1 − α)d(z, w) for all u, x, y, z, w ∈ X and α, β ∈ [0, 1] (see also [4]). A nonempty subset K of a hyperbolic space X is convex if W (x, y, α) ∈ K for all x, y ∈ K and α ∈ [0, 1]. The class of hyperbolic spaces contains normed spaces and convex subsets thereof, the Hilbert open unit ball equipped with the hyperbolic metric [5], Hadamard manifolds as well as CAT(0) spaces in the sense of Gromov (see [6]). ∗ The

corresponding author: [email protected] (H.Y. Lan)

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A hyperbolic space is uniformly convex [7, 8] if for any r > 0 and  ∈ (0, 2] there exists δ ∈ (0, 1] such that for all u, x, y ∈ X, we have 1 d(W (x, y, ), u) ≤ (1 − δ)r, 2 provided d(x, u) ≤ r, d(y, u) ≤ r and d(x, y) ≥ r. A map η : (0, +∞) × (0, 2] → (0, 1], which provides such δ = η(r, ) for given r > 0 and  ∈ (0, 2], is known as a modulus of uniform convexity of X. We call η monotone if it decreases with r (for fixed ), i.e., ∀ > 0, ∀r2 ≥ r1 > 0 (η(r2 , ) ≤ η(r1 , )). In the sequel, let (X, d) be a metric space, and let K be a nonempty subset of X. We shall denote the fixed point set of a mapping T by F (T ) = {x ∈ K : T x = x}. A mapping T : K → K is said to be nonexpansive if d(T x, T y) ≤ d(x, y),

∀x, y ∈ K.

A mapping T : K → K is said to be asymptotically nonexpansive if there exists a sequence {kn } ⊂ [0, +∞) with kn → 0 such that d(T n x, T n y) ≤ (1 + kn )d(x, y),

∀n ≥ 1, x, y ∈ K.

A mapping T : K → K is said to be uniformly L-Lipschitzian if there exists a constant L > 0 such that d(T n x, T n y) ≤ Ld(x, y), ∀n ≥ 1, x, y ∈ K. Definition 1.1 A mapping T : K → K is said to be ({µn }, {ξn }, ρ)-total asymptotically nonexpansive if there exist nonnegative sequences {µn }, {ξn } with µn → 0, ξn → 0 and a strictly increasing continuous function ρ : [0, +∞) → [0, +∞) with ρ(0) = 0 such that
d(T n x, T n y) ≤ d(x, y) + µn ρ d(x, y) + ξn , ∀n ≥ 1, x, y ∈ K. Remark 1.1 From the definitions, it is to know that each nonexpansive mapping is an asymptotically nonexpansive mapping with a sequence {kn = 0}, and each asymptotically nonexpansive mapping is a ({µn }, {ξn }, ρ)-total asymptotically nonexpansive mapping with ξn = 0, and ρ(t) = t, t ≥ 0. The existence of fixed points of various nonlinear mappings has relevant applications in many branches of nonlinear analysis and topology. On the other hand, there are certain situations where it is difficult to derive conditions for the existence of fixed points for certain types of nonlinear mappings. It is worth to mention that fixed point theory for nonexpansive mappings, a limit case of a contraction mapping when the Lipschitz constant is allowed to be 1, requires tools far beyond metric fixed point theory. Iteration schemas are the only main tool for analysis of generalized nonexpansive mappings. Fixed point theory has a computational flavor as one can define effective iteration schemas for the computation of fixed points of various nonlinear mappings. The problem of finding a common fixed point of some nonlinear mappings acting on a nonempty convex domain often arises in applied mathematics. On the other hand, Zhao et al. [9] introduced a mixed type iteration for total asymptotically nonexpansive mappings in hyperbolic spaces, and prove some ∆-convergence theorems for the iteration process approximating to a common fixed point; Zhao et al. [10] consider convergence theorems for total asymptotically nonexpansive mappings in hyperbolic spaces. Furthermore, Fukhar-ud-din and Kalsoom [11] extended iterative process with errors to asymptotically nonexpansive mappings in hyperbolic spaces, and obtained some convergence results. Motivated and inspired by the above works, the purpose of this paper is to introduce the concepts of total asymptotically nonexpansive mappings and to prove some ∆-convergence theorems of iteration process with errors for approximating a common fixed point of four total asymptotically nonexpansive mappings in hyperbolic spaces. The results presented in the paper extend and improve some recent results given in [9-25].

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2

Preliminaries

In order to define the concept of ∆-convergence in the general setup of hyperbolic spaces, we first collect some basic concepts. Let {xn } be a bounded sequence in a hyperbolic space X. For x ∈ X, we define a continuous functional r(·, {xn }) : X → [0, +∞) by r(x, {xn }) = lim sup d(x, xn ). n→∞

The asymptotic radius r({xn }) of {xn } is given by r({xn }) = inf{r(x, {xn }) : x ∈ X}. The asymptotic center AK ({xn }) of a bounded sequence {xn } with respect to K ⊂ X is the set AK ({xn }) = {x ∈ X : r(x, {xn }) ≤ r(y, {xn }), ∀y ∈ K}. This is the set of minimizers of the functional r(·, {xn }). If the asymptotic center is taken with respect to X, then it is simply denoted by A({xn }). It is known that uniformly convex Banach spaces and CAT(0) spaces enjoy the property that ’bounded sequences have unique asymptotic centers with respect to closed convex subsets’. The following lemma is due to Leustean [26] and ensures that this property also holds in a complete uniformly convex hyperbolic space. Lemma 2.1 ([26]) Let (X, d, W ) be a complete uniformly convex hyperbolic space with monotone modulus of uniform convexity. Then every bounded sequence {xn } in X has a unique asymptotic center with respect to any nonempty closed convex subset K of X. Recall that a sequence {xn } in X is said to ∆-converge to x ∈ X if x is the unique asymptotic center of {un } for every subsequence {un } of {xn }. In this case, we write ∆-limn→∞ xn = x and call x the ∆-limit of {xn }. Lemma 2.2 ([27]) Let {an }, {bn } and {ωn } be sequences of nonnegative real numbers satisfying an+1 ≤ (1 + ωn )an + bn ,

∀n ≥ 1.

P∞ P∞ If n=1 ωn < +∞ and n=1 bn < +∞, then limn→∞ an exists. If there exists a subsequence {ani } ⊂ {an } such that ani → 0, then limn→∞ an = 0. Lemma 2.3 ([17]) Let (X, d, W ) be a uniformly convex hyperbolic space with monotone modulus of uniform convexity η. Let x ∈ X and {αn } be a sequence in [a, b] for some a, b ∈ (0, 1). If {xn } and {yn } are sequences in X such that lim sup d(xn , x) ≤ c, lim sup d(yn , x) ≤ c, n→∞

n→∞

lim d(W (xn , yn , αn ), x) = c

n→∞

for some c ≥ 0, then limn→∞ d(xn , yn ) = 0. Lemma 2.4 [17] Let K be a nonempty closed convex subset of uniformly convex hyperbolic space and {xn } be a bounded sequence in K such that A({xn }) = {y} and r({xn }) = ζ. If {ym } is another sequence in K such that limm→∞ r(ym , {xn }) = ζ, then limm→∞ ym = y.

3

Main results

In this section, we give our main results. Theorem 3.1 Let K be a nonempty closed convex subset of a complete uniformly convex hyperbolic space X with monotone modulus of uniform convexity η. Let Ti : K → K, i = 1, 2, be a uniformly Li -Lipschitzian and ({µin }, {ξni }, ρi )-total asymptotically nonexpansive mapping with {µin } and {ξni } satisfying limn→∞ µin = 0, limn→∞ ξni = 0 and a strictly increasing continuous function ρi : [0, +∞) → [0, +∞) with ρi (0) = 0, i = 1, 2, let Si : K → K, i = 1, 2, be a uniˆ i -Lipschitzian and ({ˆ formly L µin }, {ξˆni }, ρˆi )-total asymptotically nonexpansive mapping with {ˆ µin } 3

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and {ξˆni } satisfying limn→∞ µ ˆin = 0, limn→∞ ξˆni = 0 and a strictly increasing continuous function T2 ρˆi : [0, +∞) → [0, +∞) with ρˆi (0) = 0, i = 1, 2. Assume that F := i=1 (F (Ti ) ∩ F (Si )) 6= φ, and for arbitrarily chosen x1 ∈ K, a new iterative approximating scheme {xn } with errors is defined as follows: xn+1 = W (S1n xn , W (T1n yn , un , θn1 ), αn ), yn = W (S2n xn , W (T2n xn , vn , θn2 ), βn ),

(3.1)

where {αn }, {βn }, {γn }, {δn }, {ζn }, {λn } are sequences in [0, 1] and {un }, {vn } are bounded ζn γn δn λn = 1−α , θn2 = 1 − 1−β = 1−β . Let {µin }, {ξni }, ρi , {ˆ µin }, sequences in K and θn1 = 1 − 1−α n n n n {ξˆni }, ρˆi , i = 1, 2, {αn }, {βn }, {γn }, {δn }, {ζn } and {λn } satisfy the following conditions: P∞ P∞ P∞ i P∞ ˆi P∞ i ˆin < +∞, n=1 µn < +∞, n=1 µ n=1 ξn < +∞, n=1 ξn < +∞, n=1 γn < +∞, P∞(i) λ < +∞, i = 1, 2; n=1 n (ii) There exist constants a, b ∈ (0, 1) such that {αn } ⊂ [a, b], {βn } ⊂ [a, b], {δn } ⊂ [a, b], {ζn } ⊂ [a, b] and limn→∞ αn = α ∈ [a, b]; (iii) There exist a constant M ∗ > 0 such that ρi (r) ≤ M ∗ r and ρˆi (r) ≤ M ∗ r, r > 0, i = 1, 2; (iv) d(x, y) ≤ d(Si x, y) for all x, y ∈ K and i = 1, 2. Then the iterative sequence {xn } defined by (3.1) ∆-converges to a common fixed point of F := T2 i=1 (F (Ti ) ∩ F (Si )). ˆ i , i = 1, 2}, µn = max{µin , µ Proof. Set L = max{Li , L ˆin , iP= 1, 2}, and ξn =P max{ξni , ξˆni , i = 1, 2}, ∞ ∞ i i ρ = max{ρ , ρˆ , i = 1, 2}. By condition (i), we know that n=1 µn < +∞, n=1 ξn < +∞. The proof of Theorem 3.1 is divided into three steps. Step 1. We first prove that limn→∞ d(xn , p) exists for each p ∈ F. For any given p ∈ F, since Ti and Si , i = 1, 2, are total asymptotically nonexpansive mappings, by condition (iii) and (3.1), we have d(xn+1 , p) = d(W (S1n xn , W (T1n yn , un , θn1 ), αn ), p) ≤ αn d(S1n xn , p) + (1 − αn )d(W (T1n yn , un , θn1 ), p)
≤ αn d(S1n xn , p) + (1 − αn ) θn1 d(T1n yn , p) + (1 − θn1 )d(un , p) ≤ αn d(S1n xn , p) + δn d(T1n yn , p) + γn d(un , p)
≤ αn [d(xn , p) + µn ρ d(xn , p) + ξn ]
+δn [d(yn , p) + µn ρ d(yn , p) + ξn ] + γn d(un , p) ≤ αn [(1 + µn M ∗ )d(xn , p) + ξn ] +δn [(1 + µn M ∗ )d(yn , p) + ξn ] + γn d(un , p) ≤ αn (1 + µn M ∗ )d(xn , p) + δn (1 + µn M ∗ )d(yn , p) +γn d(un , p) + (αn + δn )ξn ,

(3.2)

where d(yn , p)

= d(W (S2n xn , W (T2n xn , vn , θn2 ), βn ), p) ≤ βn d(S2n xn , p) + (1 − βn )d(W (T2n xn , vn , θn2 ), p) ≤ βn d(S2n xn , p) + (1 − βn )[θn2 d(T2n xn , p) + (1 − θn2 )d(vn , p)] ≤ βn d(S2n xn , p) + ζn d(T2n xn , p) + λn d(vn , p)
≤ βn [d(xn , p) + µn ρ d(xn , p) + ξn ]
+ζn [d(xn , p) + µn ρ d(xn , p) + ξn ] + λn d(vn , p) ≤ (βn + ζn )[(1 + µn M ∗ )d(xn , p) + ξn ] + λn d(vn , p) = (βn + ζn )(1 + µn M ∗ )d(xn , p) + λn d(vn , p) + (βn + ζn )ξn ≤ (1 + µn M ∗ )d(xn , p) + λn d(vn , p) + ξn .

(3.3)

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Substituting (3.3) into (3.2) and simplifying it, we have ≤ δn (1 + µn M ∗ )[(1 + µn M ∗ )d(xn , p) + λn d(vn , p) + ξn ] +αn (1 + µn M ∗ )d(xn , p) + γn d(un , p) + (αn + δn )ξn = (αn + δn + αn µn M ∗ + 2δn µn M ∗ + δn µ2n M ∗ 2 )d(xn , p) +γn d(un , p) + λn δn (1 + µn M ∗ )d(vn , p) +[αn + δn + δn (1 + µn M ∗ )]ξn ≤ [1 + µn M ∗ (αn + 2δn + δn µn M ∗ )]d(xn , p) +γn d(un , p) + λn δn (1 + µn M ∗ )d(vn , p) +[1 + δn (1 + µn M ∗ )]ξn = (1 + ωn )d(xn , p) + bn ,

d(xn+1 , p)

(3.4)

∗ ∗ where ωn = µn M ∗ (α Pn∞+ 2δn + δn µn MP),∞ bn = γn d(un , p) + λn δn (1 + µn M )d(vn , p) + [1 + δn (1 + ∗ µn M )]ξn . Since n=1 µn < +∞, n=1 ξn < +∞ and condition (i),(ii), and {un }, {vn } are bounded sequences in K, it follows from Lemma 2.2 that limn→∞ d(xn , p) exists for each p ∈ F. Step 2. We show that

lim d(xn , Ti xn ) = 0,

n→∞

lim d(xn , Si xn ) = 0,

n→∞

i = 1, 2.

(3.5)

For each p ∈ F, from the proof of Step 1, we know that limn→∞ d(xn , p) exists. We may assume that limn→∞ d(xn , p) = c ≥ 0. If c = 0, then the conclusion is trivial. Next, we deal with the case c > 0. From (3.3), we have d(yn , p) ≤ (1 + µn M ∗ )d(xn , p) + λn d(vn , p) + ξn .

(3.6)

Taking lim sup on both sides in (3.6), we have lim sup d(yn , p) ≤ c.

(3.7)

n→∞

In addition, since
d(T1n yn , p) ≤ d(yn , p) + µn ρ d(yn , p) + ξn ≤ (1 + µn M ∗ )d(yn , p) + ξn and d(S1n xn , p) ≤ (1 + µn M ∗ )d(xn , p) + ξn , we have lim sup d(T1n yn , p) ≤ c

(3.8)

lim sup d(S1n xn , p) ≤ c.

(3.9)

n→∞

and n→∞

Also d(W (T1n yn , un , θn1 ), p) ≤ θn1 d(T1n yn , p) + (1 − θn1 )d(un , p) δn γn = d(T1n yn , p) + d(un , p) 1 − αn 1 − αn γn ≤ d(T1n yn , p) + d(un , p). 1−b

(3.10)

P∞ Since n=1 γn < +∞, limn→∞ γn = 0, by boundedness of {un } in K and (3.8), taking lim sup on both sides in (3.10), we have lim sup d(W (T1n yn , un , θn1 ), p) ≤ c.

(3.11)

n→∞

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By limn→∞ d(xn+1 , p) = c, it is easy to prove that lim d(W (S1n xn , W (T1n yn , un , θn1 ), αn ), p) = c.

(3.12)

n→∞

It follows from (3.9), (3.11), (3.12) and Lemma 2.3 that lim d(S1n xn , W (T1n yn , un , θn1 )) = 0.

(3.13)

n→∞

Since d(xn+1 , p)

= d(W (S1n xn , W (T1n yn , un , θn1 ), αn ), p) ≤ d(S1n xn , p) + d(S1n xn , xn+1 ) ≤ d(S1n xn , p) + (1 − αn )d(S1n xn , W (T1n yn , un , θn1 )),

with the help of (3.13), we have lim inf d(S1n xn , p) ≥ c. n→∞

Combined with (3.9), it yields that lim d(S1n xn , p) = c.

(3.14)

n→∞

Since d(xn+1 , p)

≤ αn d(S1n xn , p) + δn d(T1n yn , p) + γn d(un , p)
≤ αn d(S1n xn , p) + δn [d(yn , p) + µn ρ d(yn , p) + ξn ] +γn d(un , p) ≤ αn d(S1n xn , p) + δn (1 + µn M ∗ )d(yn , p) + γn d(un , p) + δn ξn ≤ αn d(S1n xn , p) + (1 − αn )(1 + µn M ∗ )d(yn , p) +γn d(un , p) + (1 − αn )ξn ,

we get d(xn+1 , p) − αn d(S1n xn , p) 1 − αn ≤ (1 + µn M ∗ )d(yn , p) +

γn d(un , p) + ξn . 1 − αn

By condition (ii), (3.12) and (3.14), we have lim inf d(yn , p) ≥ c. n→∞

Combined with (3.7), it yields that lim d(yn , p) = c.

(3.15)

n→∞

By the same method and (3.15), we can also prove that lim d(S2n xn , W (T2n xn , vn , θn2 )) = 0.

(3.16)

n→∞

It follows from virtue of condition (iv), (3.13), and (3.16) that lim d(xn , W (T1n yn , un , θn1 )) ≤ lim d(S1n xn , W (T1n yn , un , θn1 )) = 0,

(3.17)

lim d(xn , W (T2n xn , vn , θn2 )) ≤ lim d(S2n xn , W (T2n xn , vn , θn2 )) = 0.

(3.18)

n→∞

n→∞

and n→∞

n→∞

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From (3.1) and (3.16), we have d(yn , S2n xn )

= d(W (S2n xn , W (T2n xn , vn , θn2 ), βn ), S2n xn ) ≤ (1 − βn )d(S2n xn , W (T2n xn , vn , θn2 )) → 0 (as n → ∞)

(3.19)

and d(xn , yn ) = d(xn , W (T2n xn , vn , θn2 )) + d(S2n xn , W (T2n xn , vn , θn2 )) + d(S2n xn , yn ). It follows from (3.16), (3.18) and (3.19) that lim d(xn , yn ) = 0.

(3.20)

n→∞

This together with (3.17) implies that d(xn , W (T1n xn , un , θn1 )) ≤ d(xn , W (T1n yn , un , θn1 )) + d(W (T1n yn , un , θn1 ), W (T1n xn , un , θn1 )) ≤ d(xn , W (T1n yn , un , θn1 )) + θn1 d(T1n yn , T1n xn ) δn Ld(yn , xn ) → 0 (n → ∞). ≤ d(xn , W (T1n yn , un , θn1 )) + 1 − αn

(3.21)

On the other hand, from (3.13) and (3.20), we have d(S1n xn , W (T1n xn , un , θn1 )) ≤ d(S1n xn , W (T1n yn , un , θn1 )) + d(W (T1n yn , un , θn1 ), W (T1n xn , un , θn1 )) ≤ d(S1n xn , W (T1n yn , un , θn1 )) + θn1 d(T1n yn , T1n xn ) δn Ld(yn , xn ) → 0 (n → ∞). ≤ d(S1n xn , W (T1n yn , un , θn1 )) + 1 − αn

(3.22)

From (3.21) and (3.22), we have that d(S1n xn , xn )

≤ d(S1n xn , W (T1n xn , un , θn1 )) +d(W (T1n xn , un , θn1 ), xn ) → 0

(n → ∞).

(3.23)

In addition, since d(xn+1 , xn )

= d(W (S1n xn , W (T1n yn , un , θn1 ), αn ), xn ) ≤ αn d(S1n xn , xn ) + (1 − αn )d(W (T1n yn , un , θn1 ), xn ),

it follows from (3.17) and (3.23) that lim d(xn+1 , xn ) = 0.

(3.24)

n→∞

Observe that d(xn , T1n xn )

≤ d(xn , W (T1n xn , un , θn1 )) + d(W (T1n xn , un , θn1 ), T1n xn ) ≤ d(xn , W (T1n xn , un , θn1 )) + (1 − θn1 )d(T1n xn , un ) ≤ d(xn , W (T1n xn , un , θn1 )) γn [d(T1n xn , xn ) + d(xn , p) + d(un , p)], + 1 − αn

then d(xn , T1n xn )

≤

1 − αn d(xn , W (T1n xn , un , θn1 )) 1 − αn − γn γn + [d(xn , p) + d(un , p)]. 1 − αn − γn

(3.25)

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By boundedness of {un } in K and condition (i), (ii) and limn→∞ d(xn , p) exists and (3.21), (3.25), we have lim d(xn , T1n xn ) = 0. (3.26) n→∞

Similarly, we can also prove that lim d(xn , T2n xn ) = 0.

(3.27)

n→∞

For all i = 1, 2, now we know ≤ d(xn , xn+1 ) + d(xn+1 , Tin+1 xn+1 ) +d(Tin+1 xn+1 , Tin+1 xn ) + d(Tin+1 xn , Ti xn ) ≤ (1 + L)d(xn , xn+1 ) + d(xn+1 , Tin+1 xn+1 ) + Ld(Tin xn , xn ).

d(xn , Ti xn )

It follows from (3.24), (3.26) and (3.27) that lim d(xn , Ti xn ) = 0,

n→∞

i = 1, 2.

By virtue of condition (iv), i.e., d(S1 xn , , W (T1n xn , un , θn1 )) ≤ d(S1n xn , , W (T1n xn , un , θn1 )), we have d(xn , S1 xn )

≤ d(xn , , W (T1n xn , un , θn1 )) + d(S1 xn , , W (T1n xn , un , θn1 )) ≤ d(xn , , W (T1n xn , un , θn1 )) + d(S1n xn , , W (T1n xn , un , θn1 )),

from (3.21) and (3.22), which implies that lim d(xn , S1 xn ) = 0.

n→∞

By the same method, we can also prove that lim d(xn , S2 xn ) = 0.

n→∞

Step 3. We shall prove that the sequence {xn } ∆-converges to a common fixed point of F := i=1 (F (Ti ) ∩ F (Si )). In fact, for each p ∈ F , limn→∞ d(xn , p) exist. This implies that the sequence {d(xn , p)} is bounded, so is the sequence {xn }. Hence, by virtue of Lemma 2.1, {xn } has a unique asymptotic center AK ({xn }) = {x}. Let {un } be any subsequence of {xn } with AK ({un }) = {u}. It follows from (3.5) that

T2

lim d(un , Ti un ) = 0.

(3.28)

n→∞

Next, we show that u ∈ F (Ti ), for all i = 1, 2. For this, we define a sequence {zni } in K by i zm = Tim u, for all i = 1, 2. So we calculate i d(zm , un ) ≤ d(Tim u, Tim un ) + d(Tim un , Tim−1 un ) + · · · + d(Ti un , un ) m X
≤ d(u, un ) + µm ρ d(u, un ) + ξm + d(Tik un , Tik−1 un ) k=1

≤ (1 + µm M ∗ )d(u, un ) + ξm +

m X

d(Tik un , Tik−1 un ).

(3.29)

k=1

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Since Ti is uniformly L-Lipschitzian, it follows from (3.29) that i d(zm , un ) ≤ (1 + µm M ∗ )d(u, un ) + ξm + mLd(Ti un , un ).

Taking lim sup on both sides of the above estimate and using (3.28), we have i r(zm , {un })

i = lim sup d(zm , un ) n→∞

≤ (1 + µm M ∗ ) lim sup d(u, un ) + ξm n→∞

= (1 + µm M ∗ )r(u, {un }) + ξm , and so i lim sup r(zm , {un }) ≤ r(u, {un }). m→∞

Based on AK ({un }) = {u} and the definition of asymptotic center AK ({un }) of a bounded sequence {un } with respect to K ⊂ X, we have r(u, {un }) ≤ r(y, {un }),

∀y ∈ K.

This implies that i lim inf r(zm , {un }) ≥ r(u, {un }). m→∞

Hence, we have i lim r(zm , {un }) = r(u, {un }).

m→∞

i = u, namely, limm→∞ Tim u = u. As Ti is uniformly It follows from Lemma 2.4 that limm→∞ zm m continuous, so that Ti u = Ti (limm→∞ Ti u) = limm→∞ Tim+1 u = u. That is, u ∈ F (Ti ). Similarly, we also can show that u ∈ F (Si ), for all i = 1, 2. Hence, u is the common fixed point of Ti and Si , for all i = 1, 2. And we want to show x = u, suppose x 6= u, by the uniqueness of asymptotic centers,

lim sup d(un , u)

< lim sup d(un , x)

n→∞

n→∞

≤ lim sup d(xn , x) n→∞

< lim sup d(xn , u) n→∞

= lim sup d(un , u), n→∞

a contradiction. Thus we have x = u. Since {un } is an arbitrary subsequence of {xn }, A({un }) = {x} for all subsequence {un } of {xn }. This proves that {xn } ∆-converges to a common fixed point of T2 F := i=1 (F (Ti ) ∩ F (Si )). This completes the proof. 2 The following theorem can be obtained from Theorem 3.1 immediately. Theorem 3.2 Let K be a nonempty closed convex subset of a complete uniformly convex hyperbolic space X with monotone modulus of uniform convexity η. Let Ti : K → K, i = 1, 2, be a uniformly Li -Lipschitzian and ({µin }, {ξni }, ρi )-total asymptotically nonexpansive mapping with {µin } and {ξni } satisfying limn→∞ µin = 0, limn→∞ ξni = 0 and a strictly increasing continuous function ρi : ˆ i -Lipschitzian [0, +∞) → [0, +∞) with ρi (0) = 0, i = 1, 2, let Si : K → K, i = 1, 2, be a uniformly L and asymptotically nonexpansive mapping with {kni } ⊂ [0, +∞) satisfying limn→∞ kni = 0. Assume T2 that F := i=1 (F (Ti ) ∩ F (Si )) 6= φ, and for arbitrarily chosen x1 ∈ K, {xn } is defined as follows: xn+1 = W (S1n xn , W (T1n yn , un , θn1 ), αn ), yn = W (S2n xn , W (T2n xn , vn , θn2 ), βn ),

(3.30)

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where {αn }, {βn }, {γn }, {δn }, {ζn }, {λn } are sequences in [0, 1] and {un }, {vn } are bounded ζn γn δn λn = 1−α , θn2 = 1 − 1−β = 1−β . Let {µin }, {ξni }, ρi , {kni }, sequences in K and θn1 = 1 − 1−α n n n n i = 1, 2,P{αn }, {βn }, {γn },P{δn }, {ζn } and {λ n } satisfy the following P P∞ conditions: P∞ ∞ ∞ ∞ i i i (i) n=1 µn < +∞, n=1 ξn < +∞, n=1 kn < +∞, n=1 γn < +∞, n=1 λn < +∞, i = 1, 2; (ii) There exist constants a, b ∈ (0, 1) such that {αn } ⊂ [a, b], {βn } ⊂ [a, b], {δn } ⊂ [a, b], {ζn } ⊂ [a, b] and limn→∞ αn = α ∈ [a, b]; (iii) There exist a constant M ∗ > 0 such that ρi (r) ≤ M ∗ r, r > 0, i = 1, 2; (iv) d(x, y) ≤ d(Si x, y) for all x, y ∈ K and i = 1, 2. T2 Then the sequence {xn } defined by (3.30) ∆-converges to a common fixed point of F := i=1 (F (Ti )∩ F (Si )). Proof. Take ρˆi (t) = t, t ≥ 0, ξˆni = 0, µ ˆin = kni , i = 1, 2, in Theorem 3.1. Since all the conditions in Theorem 3.1 are satisfied, it follows from Theorem 3.1 that the sequence {xn } ∆-converges to a T2 common fixed point of F := i=1 (F (Ti ) ∩ F (Si )). This completes the proof of Theorem 3.2. 2 Theorem 3.3 Let K be a nonempty closed convex subset of a complete uniformly convex hyperbolic space X with monotone modulus of uniform convexity η. Let Ti : K → K, i = 1, 2, be a uniformly Li -Lipschitzian and ({µin }, {ξni }, ρi )-total asymptotically nonexpansive mapping with {µin } and {ξni } satisfying limn→∞ µin = 0, limn→∞ ξni = 0 and a strictly increasing continuous function ρi : ˆ i -Lipschitzian [0, +∞) → [0, +∞) with ρi (0) = 0, i = 1, 2, let Si : K → K, i = 1, 2, be a uniformly L i and asymptotically nonexpansive mapping with {kn } ⊂ [0, +∞) satisfying limn→∞ kni = 0. Assume T2 that F := i=1 (F (Ti ) ∩ F (Si )) 6= φ, and for arbitrarily chosen x1 ∈ K, {xn } is defined as follows: xn+1 = W (S1n xn , T1n yn , αn ), yn = W (S2n xn , T2n xn , βn ),

(3.31)

where {αn } and {βn } are sequences in [0, 1]. Let {µin }, {ξni }, ρi , {kni }, i = 1, 2, {αn }, {βn } satisfy the following P∞ conditions:P∞ P∞ (i) n=1 µin < +∞, n=1 ξni < +∞, n=1 kni < +∞; (ii) There exist constants a, b ∈ (0, 1) such that {αn } ⊂ [a, b], {βn } ⊂ [a, b], limn→∞ αn = α ∈ [a, b]; (iii) There exist a constant M ∗ > 0 such that ρi (r) ≤ M ∗ r, r > 0, i = 1, 2; (iv) d(x, y) ≤ d(Si x, y) for all x, y ∈ K and i = 1, 2. T2 Then the sequence {xn } defined by (3.31) ∆-converges to a common fixed point of F := i=1 (F (Ti )∩ F (Si )). Proof. Take ρˆi (t) = t, t ≥ 0, ξˆni = 0, µ ˆin = kni , i = 1, 2 and γn ≡ λn ≡ 0 in Theorem 3.1. Since all the conditions in Theorem 3.1 are satisfied, it follows from Theorem 3.1 that the sequence {xn } T2 ∆-converges to a common fixed point of F := i=1 (F (Ti ) ∩ F (Si )). This completes the proof. 2 Theorem 3.4 Let K be a nonempty closed convex subset of a complete uniformly convex hyperbolic space X with monotone modulus of uniform convexity η. Let Ti : K → K, i = 1, 2, be a uniformly Li -Lipschitzian and ({µin }, {ξni }, ρi )-total asymptotically nonexpansive mapping with {µin } and {ξni } satisfying limn→∞ µin = 0, limn→∞ ξni = 0 and a strictly increasing continuous T2 function ρi : [0, +∞) → [0, +∞) with ρi (0) = 0, i = 1, 2, Suppose that F := i=1 F (Ti ) 6= φ, and for arbitrarily chosen x1 ∈ K, {xn } is defined as follows: xn+1 = W (xn , T1n yn , αn ), yn = W (xn , T2n xn , βn ),

(3.32)

where {αn } and {βn } are sequences in [0, 1]. Let {µin }, {ξni }, ρi , i = 1, 2, {αn }, {βn } satisfy the following conditions: P∞ P∞ (i) n=1 µin < +∞, n=1 ξni < +∞; (ii) There exist constants a, b ∈ (0, 1) such that {αn } ⊂ [a, b], {βn } ⊂ [a, b]; (iii) There exist a constant M ∗ > 0 such that ρi (r) ≤ M ∗ r, r > 0, i = 1, 2. T2 Then the sequence {xn } defined by (3.32) ∆-converges to a common fixed point of F := i=1 F (Ti ).

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Proof. Take γn ≡ λn ≡ 0 and Si = I, i = 1, 2 in Theorem 3.1. Since all the conditions in Theorem 3.1 are satisfied, it follows from Theorem 3.1 that the sequence {xn } ∆-converges to a common fixed T2 point of F := i=1 F (Ti ). 2 Remark 3.1 The results of Theorems 3.3 and 3.4 improve the corresponding results in Theorem 2.1 of [9] and Theorem 7 of [10], respectively.

Acknowledgements This work was partially supported by the Sichuan Province Cultivation Fund Project of Academic and Technical Leaders, and the Open Research Fund of Key Laboratory of Higher Education of Sichuan Province for Enterprise Informationalization and Internet of Things (2013WZJ01).

References [1] U. Kohlenbach, Some logical metatheorems with applications in functional analysis, Trans. Amer. Math. Soc. 357(1) (2004), 89-128. [2] P.K.F. Kuhfittig, Common fixed points of nonexpansive mappings by iteration. Pacific J. Math. 97(1) (1981), 137-139. [3] S. Reich and I. Shafrir, Nonexpansive iterations in hyperbolic spaces, Nonlinear Anal. 15 (1990), 537-558. [4] M. Abbas, M.A. Khamsi and A.R. Khan, Common fixed point and invariant approximation in hyperbolic ordered metric spaces, Fixed Point Theory Appl. 2011, 2011: 25, 14 pp. [5] K. Goebel and S. Reich, Uniform Convexity, Hyperbolic Geometry and Nonexpansive Mappings, Dekker, New York, 1984. [6] N. Bridson and A. Haefliger, Metric Spaces of Non-Positive Curvature, Springer, Berlin, 1999. [7] L. Leustean, A quadratic rate of asymptotic regularity for CAT(0) spaces, J. Math. Anal. Appl. 325 (2007), 386-399. [8] T, Shimizu and W. Takahashi, Fixed points of multivalued mappings in certain convex metric spaces, Topol. Methods. Nonlinear. Anal. 8 (1996), 197-203. [9] L.C. Zhao, S.S. Chang and J.K. Kim, Mixed type iteration for total asymptotically nonexpansive mappings in hyperbolic spaces, Fixed Point Theory Appl. 2013, 2013: 353, 11 pp. [10] L.C. Zhao, S.S. Chang and X.R. Wang, Convergence theorems for total asymptotically nonexpansive mappings in hyperbolic spaces, J. Appl. Math. 2013, Art. ID 689765, 5 pp. [11] H. Fukhar-ud-din and A. Kalsoom, Fixed point approximation of asymptotically nonexpansive mappings in hyperbolic spaces, Fixed Point Theory Appl. 2014, 2014: 64, 15 pp. [12] R.P. Agarwal, D. O’Regan and D.R. Sahu, Iterative construction of fixed points of nearly asymptotically nonexpansive mappings, J. Nonlinear convex Anal. 8(1) (2007), 61-79. [13] S.S. Chang, Y.J. Cho and H.Y. Zhou, Demiclosed principle and weak convergence problems for asymptotically nonexpansive mappings, J. Korean. Math. Soc. 38(6) (2001), 1245-1260. [14] H. Fukhar-ud-din, Strong convergence of an Ishikawa-type algorithm in CAT(0) spaces, Fixed Point Theory Appl. 2013, 2013: 207, 11 pp. [15] H. Fukhar-ud-din and A.R. Khan, ApproximatIng common fixed points of asymptotically nonexpansive maps in uniformly convex Banach spaces, Comput. Math. Appl. 53 (2007), 1349-1360. [16] F. Gu and Q. Fu, Strong convergence theorems for common fixed points of multistep iterations with errors in Banach spaces, J. Inequal. Appl. 2009, Art. ID 819036, 12 pp.

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[17] A.R. Khan, H. Fukhar-ud-din and M.A.A. Kuan, An implicit algorithm for two finite families of nonexpansive maps in hyperbolic spaces, Fixed Point Theory Appl. 2012, 2012: 54, 12 pp. [18] 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. [19] M.O. Osilike and S.C. Aniagbosor, Weak and strong convergence theorems for fixed points of asymptotically nonexpansive mappings, Math. Compt. Model. 32 (2000), 1181-1191. [20] A. Sahin and M. Basarir, On the strong convergence of a modified S-iteration process for asymptotically quasi-nonexpansive mappings in CAT(0) spaces, Fixed Point Theory Appl. 2013, 2013: 12, 10 pp. [21] J. Schu, Weak and strong convergence to fixed points of asymptotically nonexpansive mappings, Bull. Aust. Math. Soc. 43 (1991), 153-159. [22] J. Schu, Iterative construction of fixed points of asymptotically nonexpansive mappings, J. Math. Anal. Appl. 158 (1991), 407-413. [23] H. Fukhar-ud-din and M.A. Khamsi, Approximating common fixed points in hyperbolic spaces, Fixed Point Theory Appl. 2014, 2014: 113, 15 pp. [24] K.K. Tan, H.K. Xu, Fixed point iteration process for asymptotically nonexpansive mappings, Proc. Amer. Math. Soc. 122(3) (1994), 733-739. [25] Y. Yao, Y.C. Liou, New iterative schemes for asymptotically quasi-nonexpansive mappings, J. Inequal. Appl. 2010, Art. ID 934692, 9 pp. [26] L. Leustean, Nonexpansive iteration in uniformly convex W -hyperbolic spaces, Nonlinear analysis and optimization I, Nonlinear analysis, 193–210, Contemp. Math., 513, Amer. Math. Soc., Providence, RI, 2010. [27] K.K. Tan and H.K. Xu, Approximating fixed point of nonexpansive mapping by the Ishikawa iteration process, J. Math. Anal. Appl. 178 (1993), 301-308.

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SOME OSTROWSKI TYPE INTEGRAL INEQUALITIES FOR DOUBLE INTEGRAL ON TIME SCALES WAJEEHA IRSHAD, MUHAMMAD IQBAL BHATTI, AND MUHAMMAD MUDDASSAR*

A BSTRACT. Weighted montgomery identity on time scales for functions of two variables is established. Corresponding discrete and continuous versions of montgomery identities for functions of two variables are obtained. By using the obtained weighted montgomery identity on time scales, an Ostrowski type inequality for double integrals on time scales is pointed out as well.

1. I NTRODUCTION AND P RELIMINARY R ESULTS The Ostrowski type inequality, which was originally presented by Ostrowski in [14], can be used to estimate the absolute deviation of a function from its integral mean. In [6], Bohner and Matthews derived the Montgomery identity on time scales and established the following Ostrowski inequality on time scales, which unifies and extends corresponding discrete [7], continuous [13] and other cases. Theorem 1. Let a, b, s, t ∈ T with a < b f : [a, b] → R be a differentiable function with the property that, M = supa t, t is right-dense if σ(t) = t, t is leftscattered if ρ(t) < t, t is left-dense if ρ(t) = t, t is isolated if it is left- and right-scattered: ρ(t) < t < σ(t)and t is dense if it is both left- and right-dense ρ(t) = t = σ(t) for t ∈ T. Definition 3. The graininess function, µ : T → [0, ∞), is defined to be µ(t) = σ(t) − t The graininess function essentially describes the step size between two consecutive points in T. Oftentimes the differences in results obtained from discrete and continuous calculus stem from the different value of the graininess function evaluated at a given point t. Definition 4. The derivative in time scale calculus, called the delta derivative, determines the rates of forward change over a time scale. For a function f : T → R and t ∈ Tk , the

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Some Ostrowski type Integral Inequalities for Double Integral on Time Scales

3

delta derivative of f at t, f ∆ (t), is defined to be the number, when it exists, where for any given  > 0, there exists a neighborhood U of t such that [f (σ(t)) − f (s)] − f ∆ (t)[σ(t) − s] ≤ |σ(t) − s|

(2.2)

is true for all s ∈ U . Here, Tk = T \ {m} when T has a left-scattered maximum m; otherwise, Tk = T[1]. Since the delta derivative definition involves the forward jump operator, if the time scale has a left scattered maximum m, then one cannot jump past this point. Therefore, this point is removed from the set of points used to determine the delta derivative. However, if the time scale does not contain such a left-scattered maximum, then Tk is equivalent to the time scale. Take T = R, then σ(t) = t, µ(t) = 0,f ∆ = f 0 is the derivative used in standard calculus. If T = Z, σ(t) = t + 1, µ(t) = 1, f ∆ = ∆f is the forward difference operator used in difference equations. Theorem 2 (Properties of the Delta Derivative). f : T → R be a function and t ∈ Tk as defined above. For such a function, the following properties hold: (1) If f is delta differentiable at t, then f is continuous at t. (2) If t is right-scattered and f is continuous at t, then the delta derivative of f , f ∆ , is defined as follows f ∆ (t) =

f (σ(t)) − f (t) µ(t)

(3) If t is right-dense, then the delta derivative at t is as follows (if and only if the limit exists as a finite number) f ∆ (t) = lim

s→t

f (t) − f (s) t−s

(4) If f is delta differentiable at t, then f (σ(t)) = f (t) + µ(t) f ∆ (t) (5) If T = R, then the delta derivative is f 0 (t) from continuous calculus. (6) If T = Z, then the delta derivative is the forward difference, ∆f , from discrete calculus. Definition 5. If F ∆ (t) = f (t) for all t ∈ Tk , then F (t) is said to be anti-derivative of f (t) and f (t) is said to be delta integrable provided that f (t) is rd-continuous. The cauchy Rs integral of f (t) is defined by r f (t)∆(t) = F (s) − F (r).

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4

W. Irshad, M. I. Bhatti and M. Muddassar

Theorem 3. Let f, g be rd-continuous, a, b, c ∈ T and α, β ∈ R, then Rb Rb Rb (1) a (αf (t) + β g(t))4 t = α a f (t)4 t + β a g(t)4t Rb Ra (2) a f (t)∆t = − b f (t)∆t Rc Rb Rc (3) a f (t)∆t = a f (t)∆t + b f (t)∆t Rb Rb (4) a f (t) g ∆ (t) ∆t = f (b) g(b) − f (a) g(a) − a f ∆ (t) g(σ(t))∆t Theorem 4. If f is ∆-integrable on [a, b], then so is |f |, and Z Z b b |f (t)|∆t f (t)∆t ≤ a a Let T1 , T2 be two time scales. Let σi , ρi and ∆i be the forward jump operator, the backward jump operator and the delta differentiation, respectively on Ti , for i = 1, 2. Let a, b ∈ T1 , c, d ∈ T2 , with a < b, c < d. [a, b) and [c, d) are the half-closed bounded intervals in T1 and T2 respectively, and a ”rectangle” in T1 × T2 by R = [a, b) × [c, d) = {(t1 , t2 ) : t1 ∈ [a, b), t2 ∈ [c, d)} Let f be a real valued function on T1 × T2 . This function f is said to be rd-continuous in t2 if a1 ∈ T1 , then function f is real valued function on T1 × T2 , this function f is said to be rd-continuous in t2 if a1 ∈ T1 , then f (a1 , t2 ) is rd-continuous on T2 . CCrd denotes the set of functions f (a1 , t2 ) on T1 × T2 , having the properties: (1) f is rd-continuous in t1 and t2 . (2) If (x1 , x2 ) ∈ T1 × T2 with x1 right dense and x2 right dense, then f is continuous at (x1 , x2 ). Definition 6. Let gk , hk : T2 → R, k ∈ N0 be defined by g0 (t, s) = h0 (t, s) = 1 for all s, t ∈ T and then recursively by t

Z gk+1 (t, s) =

gk (σ(τ ), s) ∆τ ∀s, t ∈ T

(2.3)

hk (σ(τ ), s)∆τ ∀s, t ∈ T

(2.4)

s

Z hk+1 (t, s) =

t

s

3. M AIN R ESULTS Lemma 1 (Weighted Montgomery Identity on Time Scales). Let g : [a, b] → [0, ∞), G : [c, d] → [0, ∞) be rd-continuous and positive and h : [a, b] → R, H : [c, d] → R be invertible and differentiable, such that g(t1 ) = h∆1 (t1 ) on [a, b] and G(t2 ) = H ∆2 (t2 ). Let a, b, s1 , t1 ∈ T1 , c, d, s2 , t2 ∈ T2 with a < b, c < d,

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Some Ostrowski type Integral Inequalities for Double Integral on Time Scales

5

A = h−1





1 − α2 h(a) + α2 h(b) , B = h−1 1 − α2 h(b) + α2 h(a) ,       C = H −1 1 − β2 G(c) + β2 H(d) , D = H −1 1 − β2 G(d) + β2 H(c) and f : [a, b]T1 × [c, d]T2 → R is ∆1 ∆2 differentiable. Then for all s1 ∈ [A, B],s2 ∈ [C, D], 0 ≤ α, β ≤ 1, we have Z bZ

d

∂ 2 f (s1 , s2 ) ∆1 s1 ∆2 s2 ∆1 s1 ∆2 s2 c n   α o α β β = H(d)− H(c) h(b)− 1− h(b)+ h(a) (f (b, c)−f (a, d)+f (b, d)) 2 2 2 2 α  α + h(a) + h(b) {(1 − β)H(c) + (β − 1)H(d)} (f (a, t2 ) + f (b, t2 )) 2 2   β β + H(c) + H(d) {(1 − α)h(b) + (α − 1)h(a)} (f (t1 , c) − f (t1 , d)) 2 2 W (t1 , t2 , s1 , s2 )

a

+ ((1 − α)h(b) + (α − 1)h(a)) ((1 − β)H(d) + (β − 1)H(c)) f (t1 , t2 )  Z b    β β h0 (s1 )f (σ(s1 ), c)∆1 s1 + H(c) − 1− H(c) + H(d) 2 2 a     Z b β β − H(t2 ) − 1− H(c) + H(d) h0 (s1 )f (σ(s1 ), t2 )∆1 s1 2 2 a Z d − H 0 (s2 ) ((1 − α)h(a) + (α − 1)h(b))) f (t1 , σ(s2 ))∆2 s2 c

Z

d

o n  α α h(a) + h(b) f (a, σ(s2 ))∆2 s2 h(a) − 1 − 2 2 c Z d o n  α α h(b) + h(a) f (b, σ(s2 ))∆2 s2 − H 0 (s2 ) h(b) − 1 − 2 2 c (Z ) Z d b 0 0 + H (s2 ) h (s1 )f (σ(s1 ), σ(s2 ))∆1 s1 ∆2 (t2 )

−

H 0 (s2 )

c

a

where

W1 (t1 , t2 , s1 , s2 ) =

              


h(s1 ) − (1 − α2 )h(a) + α2 h(b) ,
h(s1 ) − α2 h(a) + (1 − α2 )h(b) ,   H(s2 ) − (1 − β2 )H(c) + β2 H(d) ,   H(s2 ) − β2 H(c) + (1 − β2 )H(d) ,

s1 ∈ [a, t1 ) s1 ∈ [t1 , b] s2 ∈ [c, t2 ) s2 ∈ [t2 , d]

Proof. Let’s start with Z

b

Z

d

W (t1 , t2 , s1 , s2 ) a

c

918

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W. Irshad, M. I. Bhatti and M. Muddassar

   Z tZ1 t2n  o α α β β h(s1 )− 1− h(a)+ h(b) = H(s2 )− 1− H(c)+ H(d) 2 2 2 2 a c Z t1Z d n   o 2 α α ∂ f (s1 , s2 ) h(s1 ) − 1 − h(a) + h(b) ∆1 s1 ∆2 s2 + ∆1 s1 ∆2 s2 2 2 a t2     2 β β ∂ f (s1 , s2 ) H(s2 ) − 1− H(d) + H(c) ∆1 s1 ∆2 s2 2 2 ∆1 s1 ∆2 s2    Z bZ t2n o  α α β β h(b)+ h(a) h(s1 )− 1− H(s2 )− 1− H(c)+ H(d) + 2 2 2 2 t1 c Z Z b dn  α  o α ∂ 2 f (s1 , s2 ) h(s1 )− 1− h(b)+ h(a) {H(s2 )− ∆1s1 ∆2s2 + ∆1s1 ∆2s2 2 2 t1 t2    2 β β ∂ f (s1 , s2 ) 1− H(d)+ H(c) ∆1 s1 ∆2 s2 . 2 2 ∆1 s1 ∆2 s2 Now    Z t1n o   α β β α h(a)+ h(b) H(t2 )− 1− H(c)+ H(d) h(s1 )− 1− 2 2 2 2 a      Z t2 ∂f (s1 , t2) β ∂f (s1 , c) β ∂f (s1 , σ(s2 )) H(c)+ H(d) − H(c)− 1− − H 0 (s2 ) ∆2s2 ∆1 s1 2 2 ∆1 s1 ∆1 s1 c    Z t1 n o   β α β α H(d) − 1− h(a) + h(b) H(d) + H(c) + h(s1 ) − 1 − 2 2 2 2 a #     Z d ∂f (s1 , t2 ) ∂f (s1 ,d) β β ∂f (s ,σ(s )) 1 2 − H(t2 )− 1− H(d)+ H(c) − H 0 (s2) ∆2s2 ∆1 s1 2 2 ∆1 s1 ∆1s1 t2    Z bn o   β β α α + H(t2 ) − 1− H(c) + H(d) h(s1 ) − 1 − h(b) + h(a) 2 2 2 2 t1      Z t2 ∂f (s1 , t2) β β ∂f (s1 , c) ∂f (s1 , σ(s2 )) 0 − H(c)− 1− H(c)+ H(d) − H (s2 ) ∆2s2 ∆1 s1 2 2 ∆1 s1 ∆1s1 c    Z bn o   α α β β + h(s1 ) − 1 − h(b) + h(a) H(d) − 1− H(d) + H(c) 2 2 2 2 t1      Z t2 ∂f (s1 , t2 ) β β ∂f (s1 , c) ∂f (s1 , σ(s2 )) − H(t2 )− 1− H(d)+ H(c) − H 0 (s2 ) ∆2s2 ∆1s1 2 2 ∆1s1 ∆1s1 c =

and     nh  i β β α α = H(t2 ) − 1− H(c) + H(d) h(t1 )− 1 − h(a) − h(b) f (t1 , t2 ) 2 2 2 2  Z t1 h   i α α 0 − h(a)− 1 − h(a)− h(b) f (a, t2 )− h (s1 )f (σ(s1 ), t2 )∆1s1 2 2 a

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Some Ostrowski type Integral Inequalities for Double Integral on Time Scales

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    nh  i β α β α − H(c) − 1− h(a) − h(b) f (t1 , c) H(c) + H(d) h(t1 ) − 1 − 2 2 2 2  Z t1  i h  α α 0 h(a)− h(b) f (a, c) − h (s1 ) f (σ(s1 ), c) ∆1 s1 − h(a) − 1 − 2 2 a Z t2 nh  i h  α α α h(a) − h(b) f (t1 , σ(s2 )) − h(a) − 1 − − H 0 (s2 ) h(t1 ) − 1 − 2 2 2 c  Z t1 i α h(a) − h(b) f (a, σ(s2 )) − h0 (s1 )f (σ(s1 ), σ(s2 )) ∆1 s1 ∆2 (s2 ) 2 a

  nh    i α β α β h(a) + h(b) f (t1 , d) H(d) + H(c) h(t1 ) − 1 − H(d) − 1 − 2 2 2 2  Z t1  i h  α α 0 h(a) + h(b) f (a, d) − h (s1 )f (σ(s1 ), d) ∆1 s1 − h(a) − 1 − 2 2 a     nh  i α β β α h(s1 ) − 1 − − H(t2 ) − 1− H(d) + H(c) h(a) − h(b) f (t1 , t2 ) 2 2 2 2  Z t1 h  i  α α 0 − h(a) − 1 − h(a) − h(b) f (a, t2 ) − h (s1 )f (σ(s1 ), d) ∆1 s1 2 2 a Z d i h  nh  α α α − H 0 (s2 ) h(t1 ) − 1 − h(a) − h(b) f (t1 , σ(s2 )) − h(a) − 1 − h(a) 2 2 2 t2  Z t1 i α h0 (s1 )f (σ(s1 ), σ(s2 )) ∆1 s1 ∆2 (s2 ) + − h(b) f (a, σ(s2 )) − 2 a

    nh  i β β α α H(t2 ) − 1− H(c) + H(d) h(b) − 1 − h(b) − h(a) f (b, t2 ) 2 2 2 2 ) Z b h  i α α − h(t1 ) − 1 − h(b) − h(a) f (t1 , t2 ) − h0 (s1 )f (σs1 , t2 ) ∆1 s1 2 2 t1     nh  i α α β β h(t1 ) − 1 − h(b) − h(a) f (b, c) − H(c) − 1− H(c) + H(d) 2 2 2 2 ) Z b h  i α α 0 h (s1 )f (σ(s1 ), c) ∆1 s1 − h(t1 ) − 1 − h(b) − h(a) f (t1 , c) − 2 2 t1 Z y n   h  α α α − h0 (t) h(b)− 1 − h(b) + h(a) f (b, σ(t)) − h(t1 )− 1 − h(b) 2 2 2 c ) Z b i α 0 + h(a) f (t1 , σ(s2 ))− h (s1 )f (σ(s1 ),σ(s2 ))∆1 s ∆2 (s2 )+ 2 t1

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W. Irshad, M. I. Bhatti and M. Muddassar

    nh  i α β β α h(b) − h(a) f (b, d) H(d) − 1− H(d) + H(c) h(b) − 1 − 2 2 2 2 ) Z b  i h  α α 0 h(b) − h(a) f (t1 , d) − h (s1 )f (σ(s1 ), σ(S2 )) ∆1 s1 − h(t1 ) − 1 − 2 2 t1     nh  i β α β α − h(y) − 1− h(b) − h(a) f (b, t2 ) H(d) + H(c) h(b) − 1 − 2 2 2 2 ) Z b i h  α α 0 h(b) + h(a) f (t1 , t2 ) − H (s2 )f (σ(s1 ), t2 ) ∆1 s − h(t1 ) − 1 − 2 2 t1 Z d n   h  α α α h(a)+ h(b) f (t1 , σ(s2 ))− h(a)− 1 − h(a) − H 0 (s2 ) h(t1 )− 1 − 2 2 2 t2  Z t1 i α 0 h (s1 )f (σ(s1 ), σ(s2 ))∆1s1 ∆2(s2 ) + h(b) f (a, σ(t))− 2 a  =

   β β α α H(d)− H(c) h(b)− 1− h(b) + h(a) (f (b, c)−f (a, d)+f (b, d)) 2 2 2 2 α  α + h(a) + h(b) ((1 − β) H(c) + (β − 1) H(d)) (f (a, t2 ) + f (b, t2 )) 2 2   β β + H(c) + H(d) ((1 − α) h(b) + (α − 1) h(a)) (f (t1 , c) − f (t1 , d)) 2 2 + ((1 − α) h(b) + (α − 1) h(a)) ((1 − β) H(d) + (β − 1) H(c)) f (t1 , t2 )   Z b   β β H(c) + H(d) h0 (s1 )f (σ(s1 ), c) ∆1 s1 + H(c) − 1− 2 2 a     Z b β β − h(t2 ) − 1− H(c) + H(d) h0 (s1 )f (σ(s1 ), t2 ) ∆1 s1 2 2 a Z d − H 0 (s2 ) ((1 − α) h(a) + (α − 1) h(b)) f (t1 , σ(s2 )) ∆2 s2 Z

c d

+

o n  α α h(a)+ h(b) f (a, σ(s2 ))∆2 s2 h(a)− 1− 2 2 (Z ) Z d b 0 0 + H (s2 ) h (s1 )f (σ(s1 ), σ(s2 )) ∆1 s1 ∆2 (s2 )

H 0 (s2 )

c

c

a

Remark 1. When T = R

Z bZ

d

d2 f (s1 , s2 ) d1 s1 d2 s2 d1 s1 d2 s2 a c  n  o β β α α = H(d)− H(c) h(b)− 1− h(b)+ h(a) (f (b, c)−f (a, d)+f (b, d)) 2 2 2 2 W (t1 , t2 , s1 , s2 )

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Some Ostrowski type Integral Inequalities for Double Integral on Time Scales

α

9

 α h(b) {(1 − β) H(c) + (β − 1) H(d)} (f (a, t2 ) + f (b, t2 )) 2 2   β β H(c) + H(d) {(1 − α)h(b) + (α − 1) h(a)} (f (t1 , c) − f (t1 , d)) + 2 2 +

h(a) +

+ ((1 − α) h(b) + (α − 1) h(a)) ((1 − β) H(d) + (β − 1) H(c)) f (t1 , t2 )     Z b β β + H(c) − 1− h0 (s1 )f (s1 , c)d1 s1 H(c) + H(d) 2 2 a     Z b β β h0 (s1 )f (s1 , t2 )d1 s1 − H(t2 ) − 1− H(c) + H(d) 2 2 a Z d H 0 (s2 ) ((1 − α) h(a) + (α − 1) h(b)) f (t1 , s2 )d2 s2 − c d

Z

n  o α α h(a) − 1 − h(a) + h(b) f (a, s2 )d2 s2 2 2 c ( ) Z b Z d 0 h0 (s1 )f (s1 , s2 )d1 s1 d2 (t2 ) + H (s2 ) H 0 (s2 )

+

a

c

Remark 2. T = Z

d−1 b−1 X X

W (t1 , t2 , s1 , s2 )

s1 =a s2 =c

 n  o β β α α = H(d)− H(c) h(b)− 1− h(b)+ h(a) (f (b, c)−f (a, d)+f (b, d)) 2 2 2 2 α  α + h(a) + h(b) {(1 − β) H(c) + (β − 1) H(d)} (f (a, t2 ) + f (b, t2 )) 2 2   β β + H(c) + H(d) {(1 − α) h(b) + (α − 1) h(a)} (f (t1 , c) − f (t1 , d)) 2 2 + ((1 − α) h(b) + (α − 1) h(a)) ((1 − β) H(d) + (β − 1) H(c)) f (t1 , t2 )     X b−1 β β h0 (s1 )f (t1 + 1, c) + H(c) − 1− H(c) + H(d) 2 2 s =a 1

    X b−1 β β − H(t2 ) − 1− h0 (s1 )f (t1 + 1, t2 ) H(c) + H(d) 2 2 s =a 1

−

d−1 X

H 0 (s2 ) ((1 − α) h(a) + (α − 1) h(b)) f (t1 , t2 + 1)

s2 =c

+

d−1 X

H 0 (s2 )

s2 =c

+

b−1 X d−1 X s1 =a s2 =c

n  o α α h(a) − 1 − h(a) + h(b) f (a, t2 + 1) 2 2 (Z ) b

H 0 (s2 )

h0 (s1 )f (t1 + 1, t2 + 1)

a

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Remark 3. By taking h(s1 ) = s1 , H(s2 ) = s2 , we obtain

Z bZ

d

W (t1 , t2 , s1 , s2 ) a c

∂ 2 f (s1 , s2 ) ∆1 s1 ∆2 s2 = (1−α) (1−β) (b−a)(d−c)f (t1 , t2 ) ∆1 s1 ∆2 s2

  α β +(b − a)(d − c) (1 − β) [f (a, t2 ) + f (b, t2 )] + (1 − α) [f (t1 , c) + f (t1 , d)] 2 2 Z b αβ β + (b − a)(d − c) {f (b, c) + f (a, d) + f (b, d)} − (d − c) {f (σ(s1 ), c) 4 2 a Z d α +f (σ(s1 ), d)} ∆1 s1 − (b − a) {f (a, σ(t)) + f (b, σ(t))} ∆2 t2 2 c Z bZ d f (σ(s1 ), σ(s2 )) ∆1 s1 ∆2 s2 + a

c

Theorem 5. Under the conditions of Lemma, if f ∆1 ∆2 ∈ L2 ((a, b)T1 × (c, d)T2 ), with h(s1 ) = s1 , H(s2 ) = s2 , then we have h α (1−α) (1−β) (b − a)(d − c)f (t1 , t2 ) + (b − a)(d − c) (1−β) [f (a, t2 ) + f (b, t2 )] 2  β αβ + (1−α) {f (t1 , c)+f (t1 , d)} + (b − a)(d − c) {f (b, c) + f (a, d) + f (b, d)} 2 22 Z b Z d β α − (d − c) {f (σ(s1 ), c) + f (σ(s1 ), d)} ∆1 s1 − (b − a) {f (a, σ(t)) 2 2 a c Z bZ d [f (b, d)−f (a, d)−f (b, c)+f (a, c)] +f (b, σ(t))}∆2 t2 + f (σ(s1 ), σ(s2 )) ∆1 s1 ∆2 s2 − (b−a)2 (d−c)2 a c          b−a b−a b−a b−a × h2 t1 , a+α −h2 a, a+α +h2 b, b−α −h2 t1 , b−α 2 2 2 2          d−c d−c d−c d−c × h2 t2 , c+α −h2 c, c+α +h2 d, d−α −h2 t2 , d−α 2 2 2 2 )# "(        2 3 3 b −a b−a b−a b−a b−a ≤ −2 a+α h2 t1 ,a+α −h2 a, a+α − a+α (t1 −a) 3 2 2 2 2 !         2 b−a b−a b−a b−a h2 b, b−α −h2 t1 , b−α − b−α (b−t1 ) −2 b−α 2 2 2 2 "( )#        2 d3 −c3 d−c d−c d−c d−c × −2 c+α h2 t2 , c+α −h2 c, c+α − c+α (t2 −c) 3 2 2 2 2 !         2 d−c d−c d−c d−c −2 d−α h2 d, d−α −h2 t2 , d−α − d−α (d−t2) 2 2 2 2         1 b−a b−a b−a − h2 t1 , a + α −h2 a, a+α + h2 b, b−α (b−a)(d−c) 2 2 2

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       b−a d−c d−c −h2 t1 , b−α h2 t2 , c+α −h2 c, c+α 2 2 2    q   d−c d−c −h2 t2 , d−α T (f ∆1 ∆2 ). + h2 d, d − α 2 2 where b

Z

Z

T (f ) = c

a

d

1 f (s1 , s2 )∆2 s2 ∆1 s1 − (b − a)(d − c) 2

b

Z

!2

d

Z

f (s1 , s2 )∆2 s2 ∆1 s1 c

a

Proof. From the definition of W (t1 , t2 , s1 , s2 ), and taking h(s1 ) = s1 , H(s2 ) = s2 , we obtain Z bZ

d

Z W (t1 , t2 , s1 , s2 )∆2 s2 ∆1 s1 =

a

b

Z W1 (t1 , s1 )∆1 s1

c

a

d

W2 (t2 , s2 )∆2 s2 c

"Z  #     Z b b b−a b−a s1 − a + α ∆1 s1 + s1 − b − α ∆1 s1 2 2 a t1 # "Z      Z d t2 d−c d−c ∆2 s2 + s2 − d − α ∆2 s2 × s2 − c + α 2 2 t2 c          b−a b−a b−a b−a = h2 t1 , a+α −h2 a, a+α +h2 b, b−α −h2 t1 , b−α 2 2 2 2          d−c d−c d−c d−c × h2 t2 , c+α −h2 c, c+α +h2 d, d−α −h2 t2 , d−α 2 2 2 2 and Z bZ

d 2

Z

W (t1 , t2 , s1 , s2 )∆2 s2 ∆1 s1 = a

c

b

W12 (t1 , s1 )∆1 s

a

Z

d

W22 (t2 , s2 )∆2 s2

c

2 2 )    Z b b−a b−a ∆1 s1 + ∆1 s1 = s1 − a + α s1 − b − α 2 2 a t1 (Z  2 2 )   Z d t2 d−c d−c ∆2 s2 + ∆2 s2 × s2 − c + α s2 − d − α 2 2 t2 c "Z (      2 !) t1 b−a b−a b−a 2 = s1 − 2 a + α s1 − a + α − a+α ∆1 s1 2 2 2 a #     2 !)  Z b( b−a b−a b−a 2 s1 − b−α − b−α ∆1 s1 + s1 −2 b−α 2 2 2 t1 "Z (      2 !) t2 d−c d−c d−c 2 × s2 − 2 c + α s2 − c + α − c+α ∆2 s2 2 2 2 c #      2 !) Z d( d−c d−c d−c 2 + s2 −2 d−α s2 − d − α − d−α ∆2 s2 2 2 2 t2 (Z

t1

924

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12

W. Irshad, M. I. Bhatti and M. Muddassar

"Z ( )#      2! 2 t1 2 s1 +s1 σ(s1 )+(σ(s1 )) b−a b−a b−a ≤ −2 a+α s1 − a+α − a+α ∆1s1 3 2 2 2 a "Z ( )#      2! 2 b 2 s +s1 σ(s1 )+(σ(s1 )) b−a b−a b−a −2 b − α s1 − b−α − b−α ∆1s1 3 2 2 2 t1 "Z ( )#      2! 2 t2 2 s2 +s2σ(s2 )+(σ(s2 )) d−c d−c d−c −2 c+α s2 − c+α − c+α ∆2s2 3 2 2 2 c )# "Z (      2! 2 d 2 d−c d−c d−c s2 +s2σ(s2 )+(σ(s2 )) −2 d−α s2 − d−α − d−α ∆2s2 3 2 2 2 y        2 b−a b−a b−a b−a t1 3 −a3 −2 a+α h2 t1 , a+α −h2 a, a+α − a+α (t1 −a) = 3 2 2 2 2        2 b3 −t1 3 b−a b−a b−a b−a (b−t1) + −2 b−α h2 b, b−α −h2 t1 , b−α − b−α 3 2 2 2 2        2 t2 3 − c3 d−c d−c d−c d−c + −2 c+α h2 t2 , c+α −h2 c, c+α − c+α (t2 −c) 3 2 2 2 2       2  d−c d−c d3 −t32 d−c d−c −2 d−α h2 d, d−α −h2 t2 , d−α − d−α (d−t2 ) + 3 2 2 2 2 Furthermore, we have Z bZ d" W (t1 , t2 , s1 , s2 )− a

c

1 (b−a)(d−c)

Z bZ

#

d

W (t1 , t2 , s1 , s2 )∆2s2 ∆1s1 ∆2s2∆1s1 a c

"

(1 ∂ 2 f (s1 , s2 ) − × ∆1 s1 ∆2 s2 (b − a)(d − c)

Z

b

a

Z

d

c

# ∂ 2 f (s1 , s2 ) ∆2 s2 ∆1 s1 ∆1 s1 ∆2 s2

d 1 ∂ 2f (s1 , s2) ∆2s2∆1s1 − W (t1 , t2 , s1 , s2 )∆2s2∆1s1 W (t1 , t2 , s1 , s2 ) ∆1s1∆2 s2 (b−a)(d−c) a c a c Z bZ d 2 ∂ f (s1 , s2 ) × ∆2 s2 ∆1 s1 a c ∆1 s1 ∆2 s2 Z bZ d ∂ 2 f (s1 , s2 ) [f (b, d)−f (a, d)−f (b, c)+f (a, c)] = W (t1 , t2 , s1 , s2 ) ∆2 s2 ∆1 s1 − ∆1 s1 ∆2 s2 (b−a)(d−c) a c Z bZ d × W (t1 , t2 , s1 , s2 )∆2 s2 ∆1 s1

Z bZ

Z bZ

d

a

c

On the other hand Z Z " b

a c

d

# Z bZ d 1 W (t1 , t2 , s1 , s2 )∆2s2∆1s1 ∆2 s2 ∆1 s1 W (t1 , t2 , s1 , s2 )− (b−a)(d−c) a c " # Z bZ d 2 ∂ 2 f (s1 , s2 ) 1 ∂ f (s1 , s2 ) × − ∆2 s2 ∆1 s1 ∆1 s1 ∆2 s2 (b − a)(d − c) a c ∆1 s1 ∆2 s2

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J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 20, NO.5, 2016, COPYRIGHT 2016 EUDOXUS PRESS, LLC

Some Ostrowski type Integral Inequalities for Double Integral on Time Scales

13

Z bZ d

1

W (t1 , t2 , s1 , s2 )∆2 s2 ∆1 s1 ≤ W (t1 , t2 , ...) −

(b − a)(d − c) a c 2

Z bZ d 2

∂ 2 f (., .) ∂ f (s1 , s2 ) 1

× −

∆1 s∆2 t (b − a)(d − c) a c ∆1 s1 ∆2 s2

2

 Z bZ d W 2 (t1 , t2 , s1 , s2 )∆2s2∆1s1 − = a c

 Z  × a

 Z bZ × a c

d

b

Z

d



c

2

1 (b−a)(d−c)

∂ f (s1 , s2 ) ∆1 s1 ∆2 s2

2 −

Z bZ

!221

d

W (t1 , t2 , s1 , s2 )∆2s2∆1s1  a c

1 (b − a)(d − c)

1 W 2 (t1 , t2 , s1 , s2 )∆2s2∆1 s1 − (b−a)(d−c)

Z bZ

d

Z a

b

Z c

d

!2  12 ∂ f (s1 , s2 )  ∆1 s∆2 s2 2

!2 21 W (t1 , t2 , s1 , s2 )∆2s2∆1s1 

a c

×

q T (f∆1∆2 ).

4. ACKNOWLEDGEMENT We are thankful to Prof. Dr. A. D. Raza, Director General Abdus Salam School of Mathematical Sciences (ASSMS), GC University, Lahore-Pakistan for providing us the opportunity to avail the facilities from ASSMS to get this article complete R EFERENCES [1] R. Agarwal, M. Bohner and A. Peterson, Inequalities on time scales: a survey, Math. Inequal. Appl. 4 (2001), no. 4, 53517557. [2] M. Bohner, M. Fan and J. M. Zhang, Periodicity of scalar dynamic equations and applications to population models, J. Math. Anal. Appl. 330 (2007), 1179. [3] M. Bohner and T. Matthews, Ostrowski inequalities on time scales, JIPAM. J. Inequal. Pure Appl. Math. 9 (2008), no. 1, Article 6, 8 pp. [4] M. Bohner and A. Peterson, Dynamic equations on time scales, Birkhauser Boston, Boston, MA, 2001. [5] M. Bohner and A. Peterson, Advances in dynamic equations on time scales, Birkhauser Boston, Boston, MA, 2003. [6] M. Bohner and T. Matthews, Ostrowski inequalities on time scales, JIPAM. J. Inequal. Pure Appl. Math. 9 (2008), no. 1, Article 6, 8 pp. [7] S. S. Dragomir, The discrete version of Ostrowskis inequality in normed linear spaces, JIPAM. J. Inequal. Pure Appl. Math. 3 (2002), no. 1, Article 2, 16 pp. [8] C. Dinu, Ostrowski type inequalities on time scales, An. Univ. Craiova Ser. Mat. Inform. 34 (2007), 431758.

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14

W. Irshad, M. I. Bhatti and M. Muddassar

[9] B. Karpuz and U. M. Ozkan, Generalized Ostrowskis inequality on time scales, JIPAM. J. Inequal. Pure Appl. Math. 9 (2008), no. 4, Article 112, 7 pp. [10] W. J. Liu and Q. A. Ngo, An Ostrowski-Gruss type inequality on time scales, Comput. Math. Appl. 58 (2009), no. 6, 1207171210. [11] W. J. Liu and Q. A. Ngo, A generalization of Ostrowski inequality on time scales for k points, Appl. Math. Comput. 203 (2008), no. 2, 75417760. [12] W. J. Liu, Q. A. Ngo and W. B. Chen, A new generalization of Ostrowski type inequality on time scales, An. Stiint17. Univ. Ovidius17 Constanta Ser. Mat. 17 (2009), no. 2, 10117114. [13] -D. S. Mitrinovic, J. E. Pecaric and A. M. Fink, Inequalities involving functions and their integrals and derivatives, Kluwer Acad. Publ., Dordrecht, 1991. [14] A. Ostrowski, U17 ber die Absolutabweichung einer differentiierbaren Funktion von ihren Integralmittelwert, Comment. Math. Helv. 10 (1938) 22617227. [15] U. M. Ozkan, M. Z. Sarikaya and H. Yildirim, Extensions of certain integral inequalities on time scales, Appl. Math. Lett. 21 (2008), no. 10, 993171000. [16] M. Z. Sarikaya, New weighted Ostrowski and Cebysev type inequalities on time scales, Comput. Math. Appl. 60 (2010), no. 5, 1510171514. [17] M. Z. Sarikaya, N. Aktan and H. Yildirim, On weighted Cebysev-Gruss type inequalities on time scales, J. Math. Inequal. 2 (2008), no. 2, 18517195. [18] A. Tuna, B. I. Yasar and S. Kutukcu, A note on integral inequalities involving the product¡ins¿¡/ins¿ of two functions on time scales, J. Appl. Funct. Anal. 3 (2008), no. 3, 34117346. [19] S. F. Wang, Q. L. Xue, W. J. Liu, Further generalization of Ostrowski-Gruss type inequalities, Adv. Appl. Math. Anal. 3 (2008), no. 1, 171720. E-mail address: [email protected] E-mail address: [email protected] D EPARTMENT

OF

M ATHEMATICS , U NIVERSITY

OF

E NGINEERING

AND

T ECHNOLOGY,, L AHORE , PAK -

ISTAN

E-mail address, corresponding Author: [email protected] D EPARTMENT

OF

M ATHEMATICS , U NIVERSITY

OF

E NGINEERING

AND

T ECHNOLOGY,, TAXILA , PAK -

ISTAN

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J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 20, NO.5, 2016, COPYRIGHT 2016 EUDOXUS PRESS, LLC

The Henstock-Stieltjes integral for fuzzy-number-valued functions on a infinite interval† Ke-feng Duan∗ College of Mathematics and Statistics, Longdong University, Qingyang Gansu, 745000, P.R. China

Abstract: In this paper, the Henstock-Stieltjes integral for fuzzy-number-valued functions on a infinite interval which is an extension of the usual fuzzy Riemann-Stieltjes integral on infinite interval is firstly defined and discussed. Several necessary and sufficient conditions of the integrability for fuzzy-numbervalued 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. Keywords: Fuzzy numbers; Fuzzy Henstock integral; Stieltjes integral AMS subject classifications. 26E50; 28E10. 1 Introduction Recently, in order to complete the theory of fuzzy integrals and to meet the solving need of the fuzzy differential equations [1-3], fuzzy integrals of fuzzy-number-valued functions have been studied by many authors from different points of views, including Nanda [4], Wu et al. [5] and other authors [6-9]. As an extension for Riemann integral and Lebesgue integral, the Stieltjes integral plays an important role in probability theory, stochastic processes, physics, econometrics, biometrics and numerical analysis[10-13] in the Mathematics analysis. In fact, the establishment of the Stieltjes integral was related to the moment of inertia in physics [14]. Until 1909, Riesz presented a general expression for the linear functional of the space of the continuous functions in a finite interval by Stieltjes integral [15]. After Riesz’ work, people find that the Stieltjes integral is a powerful tool in several branches of mathematics. In the fuzzy analysis, in 1968, Zadeh defined the probability measure of a fuzzy event by using the LebesgueStieltjes integral of the membership function [16]. It is well known that the notion of the Stieltjes integral for fuzzy-number-valued functions was originally proposed by Nanda [4] in 1989. In 1998, Wu [17] discussed and defined 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 [18,19] 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, in 2014, the concept of the Henstock-Stieltjes integral for fuzzy-number-valued functions is defined and discussed, and some useful results for this integral are shown [20]. The expectations of fuzzy random variables were investigated by M. L. Puri and D. A. Ralescu in 1986 [21]. It well known that the notion of a fuzzy random variable as a fuzzy-number-valued function R and the expectation E(X) of a fuzzy random variable X was defined by a fuzzy integral E(X) = X or set-valued integral of Xλ [21]. In 2007, the concept of the fuzzy Henstock integral on infinite interval is 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 [7]. In this paper, the HenstockStieltjes integral for fuzzy-number-valued functions on infinite interval which is an extension of the usual fuzzy Riemann-Stieltjes integral on infinite interval is firstly defined and discussed. Several necessary and sufficient conditions of the integrability for fuzzy-number-valued functions are given by means of the †

The work is supported by the Natural Scientific Fund of Qingyang City (zj2014-10) * Tel.:+8618293439829, E-mail: [email protected] 928

Ke-feng Duan 928-937

J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 20, NO.5, 2016, COPYRIGHT 2016 EUDOXUS PRESS, LLC

Ke-feng Duan: The Henstock-Stieltjes integral for fuzzy-number-valued functions on a infinite interval

Henstock-Stieltjes integral of real-valued functions on infinite interval and Henstock integral of fuzzynumber-valued functions on 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 [22]. 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 [22]: − + + 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˜. Lemma 2.1 [22]. If u ˜ ∈ E 1 , then (1) [˜ u]λ is non-empty bounded closed interval for all λ ∈ [0, 1]; (2) [˜ u]λ1 ⊃ [˜ u]λ2 for any 0 6 λ1 6 λ2 6 1; (3) for any {λn } converging increasingly to λ ∈ (0, 1], ∞ \

[˜ u]λn = [˜ u]λ .

n=1

Conversely, if for all λ ∈ [0, 1], there exists Aλ ⊂ R satisfying (1) ∼ (3), then there exists a unique S u]λ ⊂ A0 . u ˜ ∈ E 1 such that [˜ u]λ = Aλ , λ ∈ (0, 1], and [˜ u]0 = λ∈(0,1] [˜ Definition 2.1 [7, 20, 23]. 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. Let α : [a, +∞] → R be an increasing function. A function f : [a, +∞] → R is Henstock-Stieltjes integrable with respect to α on [a, +∞] if there exists a real number I such that for every ε > 0, there is a function δ(x) > 0 on [a, +∞] such that for any δ-fine division T = {[xi−1 , xi ]; ξi }ni=1 , we have n X | f (ξi )[α(xi ) − α(xi−1 )] − I| < ε. i=1

R +∞

As usual, we write (HS) a f (x)dα = I and (f, α) ∈ HS[a, +∞]. 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 . 3 The fuzzy Henstock-Stieltjes integral on infinite interval and its properties In this section we shall give the definition of the Henstock-Stieltjes integral for fuzzy-number-valued functions on a infinite interval. Definition 3.1. 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 929

Ke-feng Duan 928-937

J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 20, NO.5, 2016, COPYRIGHT 2016 EUDOXUS PRESS, LLC

Ke-feng Duan: The Henstock-Stieltjes integral for fuzzy-number-valued functions on a infinite interval

˜ ∈ 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( f˜(ξi )[α(xi ) − α(xi−1 )], 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

Remark 3.1. It is clear, if f˜(x) is a real-valued function then Definition 3.1 implies the definition of (HS) integral on infinite interval introduced by [20]; if α(x) = x, then Definition 3.1 implies the definition of (F H) integral introduced by Gong et al. [7]. Remark 3.2. From the definition of the fuzzy Henstock-Stieltjes integral and the fact that (E 1 , D) is a complete metric space, we can easily obtain the following conclusions. Theorem 3.1. Let α : [a, +∞] → R be an increasing function. A fuzzy-number-valued function f˜ is fuzzy Henstock-Stieltjes integrable with respect to α on [a, +∞] if and only if for every  > 0, there is a 0 0 0 0 function δ(x) > 0 on [a, +∞] such that for any δ-fine division T = {[u, v]; ξ} and T = {[u , v ]; ξ }, we have X X 0 0 0 D( f˜(ξ)[α(v) − α(u)], f˜(ξ )[α(v ) − α(u )]) < ε. T0

T

Theorem 3.2. 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 +∞ fλ+ (x)dα]. [(F HS) f˜(x)dα]λ = [(HS) fλ− (x)dα, (HS) a

0

a

(3) For any b > a, f˜ ∈ F H[a, b], lim

Rb

b→+∞ a

Z lim

b→+∞ a

b

f˜(x)dα as a fuzzy number exists and f˜(x)dα =

Z

+∞

f˜(x)dα.

a

Proof. First, we Rprove that (1) is equivalent to (2). +∞ ˜ then given ε > 0, there exists a positive-valued function δ(x) on (1) implies (2): If a f˜(x)dα = A, [a, +∞] such that for any δ−fine division of [a, +∞] : T = {[xi−1 , xi ]; ξi }, we have X ˜ < ε, D( f˜(ξi )(α(xi ) − α(xi−1 )), A) i

i.e. sup max{|[ λ∈[0,1]

X

X − + f˜(ξi )(α(xi ) − α(xi−1 ))]− f˜(ξi )(α(xi ) − α(xi−1 ))]+ λ − Aλ |, |[ λ − Aλ |} < ε,

i

i

so for any λ ∈ [0, 1] and any δ−fine division T = {[xi−1 , xi ]; ξi }, we have X X − |[ f˜(ξi )(α(xi ) − α(xi−1 ))]− fλ− (ξi )(α(xi ) − α(xi−1 )) − A− λ − Aλ | = | λ | < ε, i

i

930

Ke-feng Duan 928-937

J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 20, NO.5, 2016, COPYRIGHT 2016 EUDOXUS PRESS, LLC

Ke-feng Duan: The Henstock-Stieltjes integral for fuzzy-number-valued functions on a infinite interval

X X + |[ f˜(ξi )(α(xi ) − α(xi−1 ))]+ fλ+ (ξi )(α(xi ) − α(xi−1 )) − A+ λ − Aλ | = | λ | < ε. i

i

Thus, fλ− and fλ+ are Henstock integrable uniformly with respect to λ ∈ [0, 1] on [a, +∞), and Z +∞ Z +∞ − fλ+ (x)dα]. fλ (x)dα, Aλ = [ a

a

Conversely, since fλ− and fλ+ are Henstock integrable uniformly with respect to λ ∈ [0, 1] on [a, +∞), then given ε > 0, there exists a positive-valued function δ(x) on [a, +∞] such that for any δ−fine division T = {[xi−1 , xi ]; ξi }, and for any λ ∈ [0, 1], we have X | fλ− (ξi )(α(xi ) − α(xi−1 )) − A− (0.1) λ | < ε, i

|

X

fλ+ (ξi )(α(xi ) − α(xi−1 )) − A+ λ | < ε.

(0.2)

i + We can prove that the class of closed intervals {[A− λ , Aλ ] : λ ∈ [0, 1]} determines a fuzzy number. In fact, + [A− λ , Aλ ] satisfies the conditions of lemma 2.1. + − + (1) Since fλ− (x) ≤ fλ+ (x), λ ∈ [0, 1], then A− λ ≤ Aλ , i.e. [Aλ , Aλ ] is a closed interval, λ ∈ [0, 1], (2) For any 0 ≤ λ1 ≤ λ2 ≤ 1,

fλ−1 (x) ≤ fλ−2 (x) ≤ fλ+2 (x) ≤ fλ+1 (x). This implies Z

+∞

a

fλ−1 (x)dα

+∞

Z ≤ a

fλ−2 (x)dα

Z

+∞

fλ+2 (x)dα

≤ a

Z

+∞

fλ+1 (x)dα.

≤ a

+ − + That is, [A− λ1 , Aλ1 ] ⊃ [Aλ2 , Aλ2 ]. (3) For any {λn } increasingly converging to λ ∈ (0, 1], ∞ \

[f˜(x)]λn = [f˜(x)]λ ,

n=1

i.e.

∞ \

[fλ−n (x), fλ+n (x)] = [fλ− (x), fλ+ (x)].

n=1

That is lim f − (x) n→∞ λn

= fλ− (x), lim fλ+n (x) = fλ+ (x). n→∞

Note that f0− (x) ≤ fλ−n (x) ≤ f1− (x), f1+ (x) ≤ fλ+n (x) ≤ f0+ (x). This implies 0 ≤ fλ−n (x) − f0− (x) ≤ f1− (x) − f0− (x), 0 ≤ fλ+n (x) − f1+ (x) ≤ f0+ (x) − f1+ (x). By the non-negativeness and Henstock integrability of f1− − f0− , f0+ − f1+ , we know that f1− − f0− , f0+ − f1+ are Lebesgue integrable (refer to [9]). Hence fλ−n (x) − f0− (x), fλ+n (x) − f1+ (x) are Lebesgue integrable, and Z

+∞

lim

n→∞ a

Z lim

n→∞ a

That is Z lim

n→∞ a

+∞

(fλ−n (x)

−

f0− (x))dα

Z

+∞

(fλ+n (x) − f1+ (x))dα =

fλ−n (x)dα

Z = a

+∞

a

Z

(fλ− (x) − f0− (x))dα,

+∞

(fλ+ (x) − f1+ (x))dα.

a

fλ− (x)dα, lim n→∞ 931

+∞

=

Z a

+∞

fλ+n (x)dα

Z = a

+∞

fλ+ (x). Ke-feng Duan 928-937

J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 20, NO.5, 2016, COPYRIGHT 2016 EUDOXUS PRESS, LLC

Ke-feng Duan: The Henstock-Stieltjes integral for fuzzy-number-valued functions on a infinite interval

Thus,

∞ T

+ − + [A− λn , Aλn ] = [Aλ , Aλ ].

n=1

Combining the inequality (1) and (2) we obtain X ˜ < ε, D( (α(xi ) − α(xi−1 ))f˜(ξi ), A) i

i.e. f˜ ∈ F H[a, +∞),

Z

+∞

˜ f˜(x)dα = A.

a

We’ll prove that (i) is equivalent to (iii) as follows. (1) implies (3): Let ε > 0. Suppose f˜ ∈ F H[a, +∞). There exists a positive-valued function δ on [a, +∞] such that Z +∞ X D( f˜(ξi )(α(xi ) − α(xi−1 )), f˜) < ε. a

i

for any δ−fine division of [a, +∞] : T = {[xi−1 , xi ]; ξi }ni=1 . On the other hand, by the Cuachy Rule about f˜ ∈ F H[a, b] (refer to Th 2.3 of [24]), then f˜ ∈ F H[a, b] for any b > a. There is a positive-valued function δ1 on [a, b] such that for any δ1 −fine division of [a, b] : T = {[xi−1 , xi ]; ξi }ni=1 , we have D(

X

f˜(ξi )(α(xi ) − α(xi−1 )),

Z

b

f˜) < ε.

a

i

We may assume that δ1 ≤ δ for any ξ ∈ [a, b]. Then Z

+∞

D( a

f˜,

Z

b

f˜)

a +∞

Z X ˜ ≤ D( f (ξi )(α(xi ) − α(xi−1 )),

f˜) + D(

a

i

X

f˜(ξi )(α(xi ) − α(xi−1 )),

Z

b

f˜) + D(f˜(+∞)µ([b, +∞))

a

i

< ε + ε = 2ε. Hence Z lim

b→+∞ a

b

f˜(x)dα =

+∞

Z

f˜(x)dα.

a

(3) implies (1): Let ε > 0. Choose a sequence a = b0 < b1 < b2 < ...., bk ↑ +∞. Since f˜ ∈ F H[bk−1 , bk ], k = 1, 2, 3, ..., there exist δk such that D(

X

f˜(ξ)(v − u),

Z

bk

f˜) < ε/2k+2 .

bk−1

[bk−1 ,bk ]

for any δk −fine division on [bk−1 , bk ], k = 1, 2, 3, .... Suppose lim b→+∞ Rb ˜ ˜ b > bN which implies D( a f (x)dα, A) < ε/2. Define ( δ(ξ) =

Rb a

˜ Choose N such that f˜(x)dα = A.

δ1 (ξ), ξ ∈ [b0 , b1 ), δk (ξ), ξ ∈ (bk−1 , bk ), k = 1, 2, 3, ..., min(δk (bk ), δk+1 (bk ) ξ = bk , k = 1, 2, 3, ....

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Ke-feng Duan: The Henstock-Stieltjes integral for fuzzy-number-valued functions on a infinite interval

For any δ−fine division P = {[xi−1 , xi ]; ξi } satisfies i = 1, 2, ..., n, O(ξn ) = [b, +∞) and b > bN , we have X ˜ D( f˜(ξi )(α(xi ) − α(xi−1 )), A) i b

Z ≤ D(

Z b X ˜ + D( f˜, A) f˜, f˜(ξi )(α(xi ) − α(xi−1 )))

a

a

i

Z b n−1 X < ε/2 + D( f˜, f˜(ξi )(α(xi ) − α(xi−1 )) + f˜(+∞)µ([b, +∞)) a

≤ ε/2 +

+∞ X

i=1

ε/2k+2 = 2ε

k=1

Hence, f˜ ∈ F H[a, +∞) and Z

b

lim

b→+∞ a

f˜(x)dα =

Z

+∞

f˜(x)dx.

a

The proof is complete. Theorem 3.3. Let α : [a, +∞) → R be an increasing function such that α ∈ C 1 [a, +∞) and f˜ : [a, +∞) → E 1 be a bounded fuzzy-number-valued function. Then f˜ is fuzzy Henstock-Stieltjes 0 integrable with respect to α on [a, +∞) if and only if f˜α is fuzzy Henstock integrable on [a, +∞). Furthermore, we have Z Z +∞

(F HS)

+∞

f˜(x)dα = (F H)

a

0 f˜(x)α (x)dx,

a

where (F H) integral denotes the fuzzy Henstock integral introduced by Wu et al. [5]. Proof. Since f˜ : [a, +∞) → E 1 is bounded on [a, +∞), supx∈[a,+∞) D(f˜(x), ˜0) exists. Continuity 0 of α on [a, b] implies uniform continuity on [a, b] for any b > a. Hence, for each  > 0, there exists η > 0 such that  0 0 |α (x) − α (y)| < 3 supx∈[a,+∞) D(f˜(x), ˜0) · (b − a) for any x, y ∈ [a, b] satisfying |x − y| < η. Choose a positive-valued function δ1 (x) on [a, b] with δ1 (x) < η for all x ∈ [a, b]. Let T = {[xi−1 , xi ]; ξi }ni=1 be a δ1 -fine division on [a, b], then by Lagrange mean value theorem, there exists x ¯i ∈ [xi−1 , xi ] such that 0

α(xi ) − α(xi−1 ) = α (¯ xi )(α(xi ) − α(xi−1 )),

(1 6 i 6 n).

Since |¯ xi − xi | 6 δ1 (xi ) < η for 1 6 i 6 n, we have 0

0

|α (¯ xi ) − α (xi )|
0 such that for any δ2 -fine division T = {[u, v]; ξ} and T = {[u , v ]; ξ } on [a, b], we have X X ε 0 0 0 f˜(ξ )[α(v ) − α(u )]) < . D( f˜(ξ)[α(v) − α(u)], 3 0 T

T

Define δ(x) on [a, b] by δ(x) = min{δ1 (x), δ2 (x)}. Then for any δ-fine division T = {[u, v]; ξ} and 0 0 0 0 T = {[u , v ]; ξ } on [a, b], we have X X 0 0 0 0 0 0 D( f˜(ξ)α (ξ)(v − u), f˜(ξ )α (ξ )(v − u )) T0

T

X X 0 6 D( f˜(ξ)α (ξ)(v − u), f˜(ξ)[α(v) − α(u)]) T

T

X X 0 0 0 + D( f˜(ξ)[α(v) − α(u)], f˜(ξ )[α(v ) − α(u )]) T0

T

X X 0 0 0 0 0 0 0 0 + D( f˜(ξ )[α(v ) − α(u )], f˜(ξ )α (ξ )(v − u )) T0

T0

< . 0

Hence, f˜α is Henstock integrable on [a, b] for any [a, b] by Theorem 2.3 of [5], and by above formula (*), we know that Z b Z b 0 ˜ (F HS) f (x)dα = (F H) f˜(x)α (x)dx. a

a

0 Applied Theorem 3.1, f˜α is Henstock integrable on [a, +∞). 0 Conversely, if f˜α is Henstock integrable on [a, +∞), then by Theorem 2.3 of [5], for each  > 0, there 0 0 0 0 is a function δ3 (x) > 0 such that for any δ3 -fine division T = {[u, v]; ξ} and T = {[u , v ]; ξ }, we have

X X ε 0 0 0 0 0 0 D( f˜(ξ)α (ξ)(v − u), f˜(ξ )α (ξ )(v − u )) < . 3 0 T

T

Define δ(x) on [a, +∞) by δ(x) = min{δ1 (x), δ3 (x)}. Then for any δ-fine division T = {[u, v]; ξ} and 0 0 0 0 T = {[u , v ]; ξ }, we have X X 0 0 0 f˜(ξ )[α(v ) − α(u )]) D( f˜(ξ)[α(v) − α(u)], T0

T

X X 0 6 D( f˜(ξ)[α(v) − α(u)], f˜(ξ)α (ξ)(v − u)) T

+ D(

X

+ D(

X

T 0

f˜(ξ)α (ξ)(v − u),

X

0 0 0 0 0 f˜(ξ )α (ξ )(v − u ))

T0

T

0 0 0 0 0 f˜(ξ )α (ξ )(v − u ),

T0

X

0 0 0 f˜(ξ )[α(v ) − α(u )])

T0

< . 934

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Ke-feng Duan: The Henstock-Stieltjes integral for fuzzy-number-valued functions on a infinite interval

Hence, f˜ is fuzzy Henstock-Stieltjes integrable with respect to αR on [a, +∞). R +∞ 0 +∞ In the following part, we will prove the equation (F HS) a f˜dα = (F H) a f˜α dx. For any division T : a = x0 < x1 < x2 0 and a positive integer n, choose an open set Gn such that Sn ⊂ Gn and µ(Gn ) < nM 2n . Define δ(x) on [a, +∞) by  1, x ∈ [a, +∞)\S, δ(x) = δ(x), such that (x − δ(x), x + δ(x)) ⊂ Gn , x ∈ Sn , n = 1, 2, · · · . For any δ-fine division T = {[xi−1 , xi ]; ξi }, we have Z +∞ X D( f˜(x)dα, ˜0) = D( f˜(ξi )[α(xi ) − α(xi−1 )] + a

ξi ∈S

= D(

6M

X

ξi ∈S ∞ X

X

f˜(ξi )[α(xi ) − α(xi−1 )], ˜0)

ξi ∈[a,+∞)\S

f˜(ξi )[α(xi ) − α(xi−1 )], ˜0) 6

X

D(f˜(ξi )[α(xi ) − α(xi−1 )], ˜0)

ξi ∈S

X

D(f˜(ξi ), ˜0)(α(xi ) − α(xi−1 )) < M

i=1 ξi ∈Si

∞ X i=1

i·

ε iM 2i

= ε. The proof is complete. 0 Remark 3.3. Let α : [a, +∞) → R be an increasing function and α ∈ C 1 [a, +∞), and |α (x)| 6 M . If f˜(x) = g˜(x) a.e. on [a, +∞) and (f˜, α) ∈ F HS[a, +∞), then (˜ g , α) ∈ F HS[a, +∞) and Z +∞ Z +∞ ˜ (F HS) f (x)dα = (F HS) g˜(x)dα. a

a

Using Theorem 3.4, naturally, we have the following conclusion. Theorem 3.5. Let α : [a, +∞) → R be an increasing function. If f˜(x) = ˜0 a.e.s. on [a, +∞) (i.e. ˜ f (x) = R˜0 on [a, +∞) except a α-Lebesgue-Stieltjes zero measure set), then (f˜, α) ∈ F HS[a, +∞) and +∞ (F HS) a f˜(x)dα = ˜0.

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Ke-feng Duan: The Henstock-Stieltjes integral for fuzzy-number-valued functions on a infinite interval

4 Conclusion The aim of this paper is attempt to extend the theory of the fuzzy Henstock-Stieltjes integral on a infinite interval, we firstly define and discuss the Henstock-Stieltjes integral for fuzzy-number-valued functions on a infinite interval. On the other hand, the integrability of the fuzzy Henstock-Stieltjes integral on a infinite interval are also shown and discussed. In the future, we shall consider the continuity and the differentiability of the primitive for the fuzzy Henstock-Stieltjes integral on a infinite interval, the quadrature rules for the fuzzy Henstock-Stieltjes integral on a infinite interval, the convergence theorems for sequences of the fuzzy Henstock-Stieltjes integrable functions on a infinite interval, and so on.

References [1] B. Bede, Note on /numerical solutions of fuzzy differential equations by predictor-corrector method0, Information Sciences 178 (2008) 1917-1922. [2] Z.T. Gong, Y.B. Shao, Global existence and uniqueness of solutions for fuzzy differential equations under dissipative-type conditions, Computers and Mathematics with Applications 56 (2008) 27162723. [3] O. Kaleva, Fuzzy differential equations, Fuzzy Sets and Systems 24 (1987) 301-317. [4] S. Nanda, On fuzzy integrals, Fuzzy Sets and Systems 32 (1989) 95-101. [5] C.X. Wu, Z.T. Gong, On Henstock integral of fuzzy-number-valued functions(I), Fuzzy Sets and Systems 120 (2001) 523-532. [6] B. Bede, S.G. Gal, Quadrature rules for integrals of fuzzy-number-valued functions, Fuzzy Sets and Systems 145 (2004) 359-380. [7] Z.T. Gong, L. Wang, The numerical calculus of expectations of fuzzy random variables, Fuzzy Sets and Systems 158 (2007) 722-738. [8] K. Kwak, W. Pedrycz, Face Recognition: A study in information fusion using fuzzy integral, Pattren Recognition Letters 26 (2005) 719-733. [9] H.C. Wu, The improper fuzzy Riemann integral and its numerical integration, Information Sciences 111 (1998) 109-137. [10] P. Billingsley, Probability Measures, John Wiley and Sons, Inc., New York, 1968. [11] S.S. Dragomir, The unified treatment of trapezoid, Simpson, and Ostrowski type inequality for monotonic mappings and applications, Mathematical and Computer Modelling 31 (2000) 61-70. [12] L. Egghe, Construction of concentration measures for general Lorenz curves using Riemann-Stieltjes integral, Mathematical and Computer Modelling 35 (2002) 1149-1163. ˇ [13] I. Stajner-Papuga, T. Grbi´c, M. DaokovS, Pseudo-Riemann-Stieltjes integral, Information Sciences 179 (2009) 2923-2933. [14] F.A. Medvedev, Development of the concept of integral, Nauka, Moscow, 1974 (in Russian). [15] F. Riesz, B. Sz.-Nagy, Functional Analysis, Ungar, New York, 1955. [16] L.A. Zadeh, Probability measures of fuzzy events, Journal of Mathematical Analysis and Applications 23 (1968) 421-427. [17] H.C. Wu, The fuzzy Riemann-Stieltjes integral. International Journal of Uncertainty, Fuzziness and Knowledge-Based Systems 6 (1998) 51-67. [18] 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. [19] X.K. Ren, The non-additive measure and the fuzzy Riemann-Stieltjes integral, Ph.D dissertation, Harbin Institute of Technology, 2008. [20] Z.T. Gong, L.L. Wang, The Henstock-Stieltjes Integral for Fuzzy-number-valued Functions, Information Sciences 188 (2012) 276-297 936

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[21] M.L. Puri, D.A. Ralescu, Fuzzy random variables, Journal of Mathematics Analysis and Applications 114 (1986) 409-422. [22] C.V. Negoita, D. Ralescu, Applications of Fuzzy Sets to Systems Analysis, Wiley, New York, 1975. [23] P.Y. Lee, Lanzhou Lectures on Henstock Integration, World Scientific, Singapore, 1989. [24] R. Henstock, Theory of Integration, Butterworth, London, 1963.

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ˇ sev-Gr¨uss type inequalities for double integrals New weighted q-Cebyˇ Zhen Liu1 and Wengui Yang2∗ 1 2

Department of Mathematics, Kashi Teacher’s College, Kashi, Xinjiang 844000, China

Department of Public Education, Sanmenxia Polytechnic, Sanmenxia, Henan 472000, China

Abstract: In this paper, we establish the weighted double q-integrals Montgomery identity for functions ˇ of two independent variables, then obtain weighted q-Cebyˇ sev-Gr¨ uss type inequalities for double integrals. Furthermore, weighted q-Ostrowski type inequalities for double integrals are also given. ˇ Keywords: Cebyˇ sev-Gr¨ uss type inequalities; Ostrowski type inequalities; Montgomery identity; double qintegrals 2010 Mathematics Subject Classification: 34A08; 26D10; 26D15.

1

Introduction and preliminaries

ˇ 1882, P.L. Cebyˇ sev [7] prove that, if f 0 , g 0 ∈ L∞ [a, b], then |T (f, g)| ≤

1 (b − a)2 kf 0 k∞ kg 0 k∞ , 12

(1.1)

where for two functions f, g : [a, b] → R, the functional 1 T (f, g) = b−a

Z

b

f (x)g(x)dx − a

1 b−a

Z

!

b

f (x)dx a

1 b−a

Z

b

! g(x)dx ,

(1.2)

a

and k · k∞ denotes the norm in L∞ [a, b] defined as kf k∞ = ess sup |f (t)|. t∈[a,b]

In 1935, G. Gr¨ uss [13] showed that |T (f, g)| ≤

1 (M − m)(N − n), 4

(1.3)

provided m, M, n and N are real numbers satisfying the conditions, m ≤ f (x) ≤ M,

n ≤ g(x) ≤ N,

(1.4)

for all x ∈ [a, b], where T (f, g) is as defined by (1.2). In 1938, Ostrowski [19] proved the following integral inequality: Let f : I → R, where I ⊂ R is an interval, be a mapping that is differentiable in the interior of I (IntI), and let a, b ∈ IntI, a < b. If |f 0 (t)| ≤ M , ∀t ∈ (a, b), then, " # Z b 2 ) 1 1 (x − a+b 2 f (t)dt ≤ + (b − a)M, (1.5) f (x) − b−a a 4 (b − a)2 for all x ∈ [a, b]. During the past few years, many researchers have given considerable attention to the above results and various generalizations, extensions and variants of these inequalities (1.1), (1.3) and (1.5) have appeared in the literature, see [1, 2, 3, 6, 8, 9, 11, 12, 16, 17, 18, 20, 21, 22] and the references cited therein. Find new ∗ Corresponding author. Email:[email protected] (Z. Liu) and [email protected] (W. Yang)

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inequalities in the multidimensional cases still an interesting problem. In [4, 10], the authors proved the double integrals Montgomery identity: f (x, y) =

1 b−a

b

Z

1 d−c

f (t, y)dt + a

d

Z

f (x, s)ds − c

Z bZ d 1 f (t, s)dtds (b − a)(d − c) a c Z bZ d ∂ 2 f (t, s) P (x, t)Q(y, s) + dtds, (1.6) ∂t∂s c a 2

f (t,s) where f : [a, b] × [c, d] → R is differentiable, the derivative ∂ ∂t∂s is integrable on [a, b] × [c, d], and the Peano kernels P (x, t) and Q(y, x) are defined by ( ( t−a s−c , a ≤ t ≤ x, b−a d−c , c ≤ s ≤ y, and Q(y, s) = P (x, t) = t−b s−d b−a , x ≤ t ≤ b d−c , y ≤ s ≤ d.

Furthermore, Guezane-Lakoud and Aissaoui [14] established new extension of the weighted Montgomery ˇ identity (1.6) for functions of two independent variables, then obtained new Cebyˇ sev type inequalities. For the sake of convenience, some definitions and propositions are cited on q-integral as follows. Some details see [5, 15]. In what follows, q is a real number satisfying 0 < q < 1. Definition 1.1 ([5]). For an arbitrary function f (x), the q-differential is defined by (dq f )(x) = f (qx) − f (x). In particular, dq x = (q − 1)x. q-derivative is defined by (Dq f )(x) =

dq f (x) f (qx) − f (x) = , dq x (q − 1)x

(Dq f )(0) = lim (Dq f )(x). x→0

Clearly, if f (x) is differentiable, then limq→1− (Dq f )(x) = (Dq0 f )(x) = f (x)

and

df (x) dx .

And q-derivatives of higher order by

(Dqn f )(x) = Dq (Dqn−1 f )(x),

n ∈ N.

Definition 1.2 ([5]). Suppose 0 < a < b. The definite q-integral is defined as Z

x

f (t)dq t = x(1 − q)

(Iq f )(x) = 0

∞ X

f (xq n )q n ,

x ∈ [0, b].

(1.7)

n=0

and Z

b

Z

Z f (x)dq x −

f (x)dq x = a

b

0

a

f (x)dq x.

(1.8)

0

Similarly as done for derivatives, an operator Iqn can be defined, namely, (Iq0 f )(x) = f (x)

and

(Iqn f )(x) = Iq (Iqn−1 f )(x),

n ∈ N.

The definite q-integral defined above is too general for our purpose of studying inequalities. For example, Rb if f (x) ≥ 0, it is not necessarily true that a f (x)dq x ≥ 0. From now on, we will use a special type of the definite q-integral, which we will call the restricted definite q-integral. Throughout all the paper, we will use the following notations: cj = bq j ,

for j ∈ {0, 1, · · · , n},

a = cn = bq n .

Definition 1.3 ([5]). Let 0 < q < 1, b > 0, and n ∈ Z+ . The restricted q-integral is defined as

Rb bq n

f (x)dq x.

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The following formula readily follows from (1.7) and (1.8): Z b Z b n−1 n−1 X X f (x)dq x = (1 − q)b f (x)dq x = q j f (bq j ) = (1 − q) cj f (cj ). bq n

a

Note that the restricted integral is easy to check that

j=0

Rb a

j=0

f (x)dq x is just a finite sum, so no questions about convergency arise. It Z

b

Dq f dq x = f (b) − f (a). a

Rb Rb Obviously, if f (x) ≥ g(x) on [a, b], then a f (x)dq x ≥ a g(x)dq x. If 0 < k < n, then Z ck Z b Z b f (x)dq x. f (x)dq x = f (x)dq x + a

ck

a

The following is the formula for the q-integration by parts: Z b Z f (x)(Dq g)(x)dq x = [f (x)g(x)]ba − a

b

g(qx)(Dq f )(x)dq x.

a

Cauciman [15] gave q-integral Gr¨ uss’s inequality as follows: Assume that (1.4) holds, then ! ! Z b Z b 1 Z b 1 1 1 f (x)g(x)dq x − f (x)dq x g(x)dq x ≤ (M − m)(N − n). b − a a 4 b−a a b−a a Rb Rt Assume that w : [a, b] → [0, ∞) satisfying a w(x)dq x = 1. Set W (t) = a w(x)dq x for t ∈ [a, b], W (t) = 0 for t < a, and W (t) = 1 for t > b. We give weighted q-integral Peano kernel Pw (x, t) defined by  W (t), a ≤ t ≤ x, Pw (x, t) = W (t) − 1, x ≤ t ≤ b. Then the following weighted q-integral Montgomery identity holds: (see [23]) Z b Z b f (x) = w(t)f (qt)dq t + Pw (x, t)(Dq f )(t)dq t. a

a

In 2011, Yang [23] obtained the following inequalities: Z b |T (w, f, g)| ≤ kDq f kkDq gk w(x)H 2 (qx)dq x, a

and |T (w, f, g)| ≤

1 2

Z

b

w(x)[|g(qx)|kDq f k + |f (qx)|kDq gk]H(qx)dq x, a

where k · k as khk = supt∈[a,b] |h(t)| for h ∈ C[a, b], Z

b

w(x)f (qx)g(qx)dq x −

T (w, f, g) =

! Z

b

Z

w(x)f (qx)dq x

a

a

b

! w(x)g(qx)dq x ,

a

and Z

b

|Pw (x, t)|dq t

H(x) = a

for all x ∈ [a, b]. Motivated by the results mentioned above, by using weighted q-integral Montgomery identity for functions ˇ of two independent variables, we establish some new weighted q-Cebyˇ sev type inequalities for double integrals. Furthermore, weighted q-Ostrowski type inequalities for double integrals are also given. 3 940

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2

ˇ Weighted q-Cebyˇ sev type inequalities for double integrals

Rb Rd Assume that w : [a, b] → R0 = [0, ∞) and u : [c, d] → R0 satisfying a w(x)dq1 x = c u(y)dq2 y = 1, where Rt Rs 0 < q1 , q2 < 1. Set W (t) = a w(x)dq1 x for t ∈ [a, b] and U (s) = a u(y)dq2 y for s ∈ [c, d], so we have W (a) = U (c) = 0 and W (b) = U (b) = 1. We give the following weighted q-integral Peano kernels Pw (x, t) and Qu (y, t) defined by   U (s), c ≤ s ≤ y, W (t), a ≤ t ≤ x, (2.1) and Qu (y, s) = Pw (x, t) = U (s) − 1, y ≤ s ≤ d. W (t) − 1, x ≤ t ≤ b We use the following notations to simplify details of the presentation. Let ∂fdq(t,s) and ∂fdq(t,s) be partial 1t 2s q-derivative on t and s, respectively. For some suitable functions w : [a, b] → R0 , u : [c, d] → R0 and f, g : Ω = [a, b] × [c, d] → R, we set b

Z

d

Z

w(x)u(y)f (q1 x, q2 y)g(q1 x, q2 y)dq1 xdq2 y

T (w, u, f, g) =

c bZ d

a

Z −

Z w(x)u(y)g(q1 x, q2 y)

a

c b

Z

w(t)f (q1 t, q2 y)dq1 t dq1 xdq2 y a

d

Z

−

Z

c

Z

b

!

d

w(x)u(y)g(q1 x, q2 y) a

!

b

u(s)f (q1 x, q2 s)dq2 s dq1 xdq2 y c

Z

+

! Z

d

b

w(x)u(y)f (q1 x, q2 y)dq1 xdq2 y a

!

d

Z

w(x)u(y)g(q1 x, q2 y)dq1 xdq2 y

c

a

c

and define k · k as khk = sup(t,s)∈Ω |h(t, s)| for h ∈ C(Ω, R). Theorem 2.1. Let f : Ω → R, w : [a, b] → R0 and u : [c, d] → R0 satisfying then Z f (x, y) =

b

Z

a

d

Z

b

Z

c

a

w(x)dq1 x =

Rd c

u(y)dq2 y = 1,

d

u(s)f (x, q2 s)dq2 s −

w(t)f (q1 t, y)dq1 t +

Rb

w(t)u(s)f (q1 t, q2 s)dq1 tdq2 s a

Z

c bZ d

+

Pw (x, t)Qu (y, s) a

c

∂f (t, s) dq tdq s, (2.2) dq1 tdq2 s 1 2

for (x, y) ∈ Ω, where the weighted q-integral Peano kernels Pw (x, t) and Qu (s, y) are defined by (2.1). Proof. According to the weighted q-integral Peano kernels Pw (x, t) and Qu (s, y) and the proof of Theorem 1 in [23], we obtain ! Z bZ d Z b Z d ∂f (t, s) ∂f (t, s) Pw (x, t)Qu (y, s) dq tdq s = Pw (x, t) Qu (y, s) dq s dq1 t dq1 tdq2 s 1 2 dq1 tdq2 s 2 a c a c ! Z b Z d ∂f (t, y) ∂f (t, q2 s) = Pw (x, t) − u(s) dq2 s dq1 t dq1 t dq1 t c a ! Z d Z b Z b ∂f (t, q2 s) ∂f (t, y) dq1 t − u(s) Pw (x, t) dq1 t dq2 s = Pw (x, t) dq1 t dq1 t c a a ! Z ! Z b Z b d = f (x, y) − w(t)f (q1 t, y)dq1 t − u(s) f (x, q2 s) − w(t)f (q1 t, q2 s)dq1 t dq2 s a

Z

b

c

Z w(t)f (q1 t, y)dq1 t −

=f (x, y) − a

a

d

Z

b

Z

u(s)f (x, q2 s)dq2 s + c

d

w(t)u(s)f (q1 t, q2 s)dq1 tdq2 s. a

c

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Thus we have Z Z b w(t)f (q1 t, y)dq1 t + f (x, y) =

d

d

Z

w(t)u(s)f (q1 t, q2 s)dq1 tdq2 s c

a

c

a

b

Z u(s)f (t, q2 s)dq2 s −

b

Z

d

Z

Pw (x, t)Qu (y, s)

+ c

a

∂f (t, s) dq tdq s, dq1 tdq2 s 1 2

and this completes the proof. Rb Rd Theorem 2.2. Let f, g : Ω → R, w : [a, b] → R0 and u : [c, d] → R0 satisfying a w(x)dq1 x = c u(y)dq2 y = 1, then



Z Z

∂f (t, s) ∂g(t, s) b d

w(x)u(y)H 2 (q1 x, q2 y)dq1 xdq2 y, (2.3) |T (w, u, f, g)| ≤

dq tdq s dq tdq s c a 1 2 1 2 where b

Z

d

Z

|Pw (x, t)Qu (y, s)|dq1 tdq2 s.

H(x, y) = c

a

Proof. Since the functions f and g satisfy the hypothesis of Theorem 2.1, the following identities hold: Z b Z d Z bZ d f (x, y) = w(t)f (q1 t, y)dq1 t + u(s)f (t, q2 s)dq2 s − w(t)u(s)f (q1 t, q2 s)dq1 tdq2 s a

c

a

c b

Z

d

Z

+

Pw (x, t)Qu (y, s) a

c

∂f (t, s) dq tdq s, (2.4) dq1 tdq2 s 1 2

and b

Z g(x, y) =

Z

d

b

Z

d

u(s)g(x, q2 s)dq2 s −

w(t)g(q1 t, y)dq1 t + a

Z

w(t)u(s)g(q1 t, q2 s)dq1 tdq2 s

c

a

Z

c b

d

Z

+

Pw (x, t)Qu (y, s) a

c

∂g(t, s) dq tdq s. (2.5) dq1 tdq2 s 1 2

Due to the above two inequalities (2.4) and (2.5), we have b

Z

d

Z

f (q1 x, q2 y) =

b

Z

d

Z

u(s)f (q1 x, q2 s)dq2 s −

w(t)f (q1 t, q2 y)dq1 t + a

w(t)u(s)f (q1 t, q2 s)dq1 tdq2 s

c

a

Z

b

Z

c

d

+

Pw (q1 x, t)Qu (q2 y, s) a

c

∂f (t, s) dq tdq s, (2.6) dq1 tdq2 s 1 2

and b

Z

Z

g(q1 x, q2 y) =

d

Z

d

Z

u(s)g(q1 x, q2 s)dq2 s −

w(t)g(q1 t, q2 y)dq1 t + a

b

c

w(t)u(s)g(q1 t, q2 s)dq1 tdq2 s a

Z

b

Z

+

c

d

Pw (q1 x, t)Qu (q2 y, s) a

c

∂g(t, s) dq tdq s. (2.7) dq1 tdq2 s 1 2

Multiplying (2.6) by (2.7), we obtain b

Z f (q1 x, q2 y) − g(q1 x, q2 y) − b

Z

= a

c

d

w(t)u(s)f (q1 t, q2 s)dq1 tdq2 s a

Z w(t)g(q1 t, q2 y)dq1 t −

a

Z

b

!

d

Z

u(s)f (q1 x, q2 s)dq2 s + c

Z

b

Z

w(t)f (q1 t, q2 y)dq1 t − a

×

d

Z

d

c

Z

b

Z

u(s)g(q1 x, q2 s)dq2 s + c

d

! w(t)u(s)g(q1 t, q2 s)dq1 tdq2 s

a

c

! Z Z ! b d ∂f (t, s) ∂g(t, s) dq tdq s Pw (q1 x, t)Qu (q2 y, s) dq tdq s . Pw (q1 x, t)Qu (q2 y, s) dq1 tdq2 s 1 2 dq1 tdq2 s 1 2 c a

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Consequently, b

Z f (q1 x, q2 y)g(q1 x, q2 y) − f (q1 x, q2 y)

w(t)g(q1 t, q2 y)dq1 t − f (q1 x, q2 y) a

Z

w(t)u(s)g(q1 t, q2 s)dq1 tdq2 s − g(q1 x, q2 y)

b

a Z b

b

c

c d

bZ

w(t)u(s)f (q1 t, q2 s)dq1 tdq2 s

c

b

a

Z

b

u(s)g(q1 x, q2 s)dq2 s

u(s)f (q1 x, q2 s)dq2 s

w(t)u(s)g(q1 t, q2 s)dq1 tdq2 s + g(q1 x, q2 y) a

Z

d

Z

b

Z

w(t)u(s)f (q1 t, q2 s)dq1 tdq2 s −

w(t)g(q1 t, q2 y)dq1 t a

c b

d

Z

d

a Z d

c

d

w(t)u(s)f (q1 t, q2 s)dq1 tdq2 s a

c

d

Z bZ d w(t)u(s)f (q1 t, q2 s)dq1 tdq2 s w(t)u(s)g(q1 t, q2 s)dq1 tdq2 s c a c a c ! Z Z ! Z bZ d b d ∂g(t, s) ∂f (t, s) Pw (q1 x, t)Qu (q2 y, s) dq tdq s Pw (q1 x, t)Qu (q2 y, s) dq tdq s (2.8) dq1 tdq2 s 1 2 dq1 tdq2 s 1 2 a c a c

× =

d

Z

u(s)f (q1 x, q2 s)dq2 s Z

Z

a bZ

u(s)f (q1 x, q2 s)dq2 s c

w(t)g(q1 t, q2 y)dq1 t +

c

c

d

Z

c

u(s)f (q1 x, q2 s)dq2 s

−

u(s)g(q1 x, q2 s)dq2 s c

w(t)u(s)g(q1 t, q2 s)dq1 tdq2 s − g(q1 x, q2 y) Z

d

d

Z w(t)f (q1 t, q2 y)dq1 t

d

Z

a

Z

b

a

w(t)f (q1 t, q2 y)dq1 t

+ −

Z w(t)g(q1 t, q2 y)dq1 t +

a Z b

a Z d

Z

b

Z w(t)f (q1 t, q2 y)dq1 t

−

w(t)f (q1 t, q2 y)dq1 t a

c

a

Z

b

Z

+ f (q1 x, q2 y) +

u(s)g(q1 x, q2 s)dq2 s c

d

bZ

d

Z

Z

Z

u(s)g(q1 x, q2 s)dq2 s +

Multiplying both sides of (2.8) by w(x)u(y), then q-integrating the resultant identity over Ω, we get b

Z

" Z

d

Z

T (w, u, f, g) =

b

Z

w(x)u(y) a

c

a

c

d

! ∂f (t, s) dq tdq s Pw (q1 x, t)Qu (q2 y, s) dq1 tdq2 s 1 2 !# Z bZ d ∂g(t, s) × Pw (q1 x, t)Qu (q2 y, s) dq tdq s dq1 xdq2 y. dq1 tdq2 s 1 2 a c

Finally, using the properties of modulus we observe that " Z Z !



Z Z b d

∂f (t, s) ∂g(t, s) b d

|T (w, u, f, g)| ≤ w(x)u(y) |Pw (q1 x, t)Qu (q2 y, s)|dq1 tdq2 s

dq tdq s dq tdq s a c a c 1 2 1 2 !# Z bZ d × |Pw (q1 x, t)Qu (q2 y, s)|dq1 tdq2 s dq1 xdq2 y a

c



Z Z

∂f (t, s) ∂g(t, s) b d



w(x)u(y)H 2 (q1 x, q2 y)dq1 xdq2 y. = dq1 tdq2 s dq1 tdq2 s a c This completes the proof of Theorem 2.2. Theorem 2.3. Let f, g : Ω → R, w : [a, b] → R0 and u : [c, d] → R0 satisfying 1, then |T (w, u, f, g)| ≤

1 2

Z a

b

Z

Rb a

w(x)dq1 x =

Rd c

u(y)dq2 y =

d

w(x)u(y)



 

∂f (t, s)

+ |f (q1 x, q2 y)| ∂g(t, s) H(q1 x, q2 y)dq xdq y, (2.9) × |g(q1 x, q2 y)| 1 2

dq tdq s

dq tdq s 1 2 1 2 c

where H(x, y) is defined in Theorem 2.2. 6 943

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Proof. Multiplying both sides of (2.6) and (2.7) by w(x)u(y)g(q1 x, q2 y) and w(x)u(y)f (q1 x, q2 y), adding the resulting identities and rewriting, we have w(x)u(y)f (q1 x, q2 y)g(q1 x, q2 y) 1 = 2

Z

b

Z c

a

Z

b

Z

− w(x)u(y)g(q1 x, q2 y)

d

Z w(t)u(s)f (q1 t, q2 s)dq1 tdq2 s + w(x)u(y)f (q1 x, q2 y)

Z

a d

a

c

Z

Z

Z

b

Pw (q1 x, t)Qu (q2 y, s) c

a

b

Z

Z

+ w(x)u(y)f (q1 x, q2 y) a

c

d

w(t)g(q1 t, q2 y)dq1 t !

w(t)u(s)g(q1 t, q2 s)dq1 tdq2 s

d

Z

w(x)u(y)g(q1 x, q2 y)

d

b

c

a

c

1 2

b

u(s)g(q1 x, q2 s)dq2 s − w(x)u(y)f (q1 x, q2 y)

+w(x)u(y)f (q1 x, q2 y) +

d

u(s)f (q1 x, q2 s)dq2 s

w(t)f (q1 t, q2 y)dq1 t + w(x)u(y)g(q1 x, q2 y)

w(x)u(y)g(q1 x, q2 y)

∂f (t, s) dq tdq s dq1 tdq2 s 1 2

! ∂g(t, s) Pw (q1 x, t)Qu (q2 y, s) dq tdq s . (2.10) dq1 tdq2 s 1 2

Q-integrating both sides of (2.10) with respect to x from a to b and y from c to d and rewriting we have ! ∂f (t, s) dq tdq s dq1 xdq2 y w(x)u(y)g(q1 x, q2 y) Pw (q1 x, t)Qu (q2 y, s) dq1 tdq2 s 1 2 a c a c ! ! Z bZ d d ∂g(t, s) w(x)u(y)f (q1 x, q2 y) Pw (q1 x, t)Qu (q2 y, s) dq tdq s dq1 xdq2 y . (2.11) dq1 tdq2 s 1 2 c a c Z

1 T (w, uf, g) = 2 Z bZ + a

b

Z

d

Z

b

Z

d

Finally, from (2.11) and using the properties of modulus we observe that ! ∂f (t, s) dq tdq s dq1 xdq2 y w(x)u(y)|g(q1 x, q2 y)| |Pw (q1 x, t)Qu (q2 y, s)| dq1 tdq2 s 1 2 a c a c ! ! Z bZ d Z bZ d ∂g(t, s) + w(x)u(y)|f (q1 x, q2 y)| |Pw (q1 x, t)Qu (q2 y, s)| dq tdq s dq1 xdq2 y dq1 tdq2 s 1 2 a c a c



 Z Z

∂f (t, s) 1 b d

+ |f (q1 x, q2 y)| ∂g(t, s) ≤ w(x)u(y) |g(q1 x, q2 y)|

2 a c dq1 tdq2 s dq1 tdq2 s ! Z Z Z

1 T (w, uf, g) ≤ 2

b

Z

d

b

b

Z

d

d

×

|Pw (q1 x, t)Qu (q2 y, s)dq1 tdq2 s dq1 xdq2 y a

1 = 2

Z

Z a

b

c

Z c

d





∂f (t, s)

∂g(t, s)



H(q1 x, q2 y)dq1 xdq2 y. w(x)u(y) |g(q1 x, q2 y)| + |f (q1 x, q2 y)| dq1 tdq2 s dq1 tdq2 s 

This completes the proof of Theorem 2.3. Remark 2.4. If q1 , q2 → 1− , by Definitions 1.2 and 1.2, the partial q-derivative and double q-integrals are the usual partial derivative and double integrals, so Theorems 2.2 and 2.3 are reduced to Theorems 3 and 4 in [14]. Rb Rd Theorem 2.5. Let f, g : Ω → R, w : [a, b] → R0 and u : [c, d] → R0 satisfying a w(x)dq1 x = c u(y)dq2 y = 1, then

Z Z

∂f (t, s) b d

w(x)u(y)H(q1 x, q2 y)dq1 xdq2 y, (2.12) |T (w, u, f, g)| ≤ kg(q1 x, q2 y)k dq1 tdq2 s a c

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and

Z Z

∂g(t, s) b d

w(x)u(y)H(q1 x, q2 y)dq1 xdq2 y, |T (w, u, f, g)| ≤ kf (q1 x, q2 y)k

dq tdq s c a 1 2

(2.13)

where H(x, y) is defined in Theorem 2.2. Proof. We prove only (2.12), since the proof of (2.13) is similar. The identity (2.6) shows that b

Z

Z

d

Z

Z

d

u(s)f (q1 x, q2 s)dq2 s −

w(t)f (q1 t, q2 y)dq1 t +

f (q1 x, q2 y) =

b

w(t)u(s)f (q1 t, q2 s)dq1 tdq2 s a

c

a

b

Z

Z

c

d

Pw (q1 x, t)Qu (q2 y, s)

+ a

c

∂f (t, s) dq tdq s, (2.14) dq1 tdq2 s 1 2

for (x, y) ∈ Ω. Now, if we multiply (2.14) by w(x)u(y)g(q1 x, q2 y) and q-integrate over (x, y) ∈ Ω, we deduce Z

b

d

Z

w(x)u(y)f (q1 x, q2 y)g(q1 x, q2 y)dq1 xdq2 y c

a

Z

b

d

Z

Z

=

w(x)u(y)g(q1 x, q2 y) a

c

Z

b

!

b

w(t)f (q1 t, q2 y)dq1 t dq1 xdq2 y a

d

Z

+

Z w(x)u(y)g(q1 x, q2 y)

a

Z

u(s)f (q1 x, q2 s)dq2 s dq1 xdq2 y

c b

!

d

c d

Z

−

b

Z

Z

w(x)u(y)g(q1 x, q2 y)dq1 xdq2 y a

Z

w(t)u(s)f (q1 t, q2 s)dq1 tdq2 s

c b

a d

Z

+

Z

b

d

Z

w(x)u(y)g(q1 x, q2 y) a

c

a

d

c

c

! ∂f (t, s) dq tdq s dq1 xdq2 y, Pw (q1 x, t)Qu (q2 y, s) dq1 tdq2 s 1 2

which provides another representation for the functional T (w, u, f, g) namely, b

Z

Z

d

T (w, u, f, g) =

Z

b

Z

d

w(x)u(y)g(q1 x, q2 y) a

c

a

c

! ∂f (t, s) dq tdq s dq1 xdq2 y, (2.15) Pw (q1 x, t)Qu (q2 y, s) dq1 tdq2 s 1 2

From (2.15) and using modules properties, it yields Z

b

Z

|T (w, u, f, g)| ≤

d

Z

b

d

Z

w(x)u(y)|g(q1 x, q2 y)| a

c

a

c

! ∂f (t, s) dq tdq s dq1 xdq2 y |Pw (q1 x, t)Qu (q2 y, s)| dq1 tdq2 s 1 2 ! Z Z

Z Z b d

∂f (t, s) b d

≤kg(q1 x, q2 y)k w(x)u(y) |Pw (q1 x, t)Qu (q2 y, s)|dq1 tdq2 s dq1 xdq2 y dq1 tdq2 s a c a c

Z Z

∂f (t, s) b d

=kg(q1 x, q2 y)k w(x)u(y)H(q1 x, q2 y)dq1 xdq2 y. dq1 tdq2 s a c This completes the proof of Theorem 2.5.

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3

Weighted q-Ostrowski type inequalities for double integrals

For some given functions f, g : Ω → R and w : [a, b] → R0 and u : [c, d] → R0 satisfying Rd u(y)dq2 y = 1, c

Rb a

w(x)dq1 x =

Z b Z d 1 S(w, u, f, g) = f (q1 x, q2 y)g(q1 x, q2 y) − f (q1 x, q2 y) w(t)g(q1 t, q2 y)dq1 t + u(s)g(q1 x, q2 s)dq2 s 2 a c Z d Z b u(s)f (q1 x, q2 s)dq2 s w(t)f (q1 t, q2 y)dq1 t + g(q1 x, q2 y) × f (q1 x, q2 y) + g(q1 x, q2 y) c a ! Z Z Z Z d

b

d

b

w(t)u(s)g(q1 t, q2 s)dq1 tdq2 s − g(q1 x, q2 y)

−f (q1 x, q2 y)

w(t)u(s)f (q1 t, q2 s)dq1 tdq2 s . c

a

c

a

Rb Rd Theorem 3.1. Let f, g : Ω → R, w : [a, b] → R0 and u : [c, d] → R0 satisfying a w(x)dq1 x = c u(y)dq2 y = 1, then



 

∂g(t, s)

1

+ |g(q1 x, q2 y)| ∂f (t, s) H(q1 x, q2 y), |S(w, u, f, g)| ≤ (3.1) |f (q1 x, q2 y)|

2 dq1 tdq2 s dq1 tdq2 s and



 Z b Z d

∂f (t, s)

∂g(t, s)



+ |g(q1 x, q2 y)| |f (q1 x, q2 y)| w(x)u(y)H(q1 x, q2 y)dq1 xdq2 y. dq1 tdq2 s dq1 tdq2 s a c (3.2) where H(x, y) is defined in Theorem 2.2. 1 |T (w, u, f, g)| ≤ 2



Proof. Multiplying both sides of (2.6) and (2.7) by g(q1 x, q2 y) and f (q1 x, q2 y), adding the resulting identities and rewriting we have f (q1 x, q2 y)g(q1 x, q2 y) =

b

Z

1 2

f (q1 x, q2 y)

Z w(t)g(q1 t, q2 y)dq1 t + f (q1 x, q2 y)

a

Z

b

Z

d

−f (q1 x, q2 y)

d

w(t)f (q1 t, q2 y)dq1 t + g(q1 x, q2 y)

u(s)f (q1 x, q2 s)dq2 s

a

Z

Z

c b

Z

w(t)u(s)g(q1 t, q2 s)dq1 tdq2 s − g(q1 x, q2 y) a

c

+

Z

b

Z

f (q1 x, q2 y)

d

Pw (q1 x, t)Qu (q2 y, s) a

c

Z

b

Z

+ g(q1 x, q2 y) a

c

d

!

d

w(t)u(s)f (q1 t, q2 s)dq1 tdq2 s a

1 2

u(s)g(q1 x, q2 s)dq2 s c

b

Z + g(q1 x, q2 y)

d

c

∂g(t, s) dq tdq s dq1 tdq2 s 1 2

! ∂f (t, s) Pw (q1 x, t)Qu (q2 y, s) dq tdq s , (3.3) dq1 tdq2 s 1 2

which implies 1 S(w, u, f, g) = 2

Z

b

Z

f (q1 x, q2 y)

d

∂g(t, s) dq tdq s dq1 tdq2 s 1 2 ! Z bZ d ∂f (t, s) + g(q1 x, q2 y) Pw (q1 x, t)Qu (q2 y, s) dq tdq s . dq1 tdq2 s 1 2 a c

Pw (q1 x, t)Qu (q2 y, s) a

c

We observe |S(w, u, f, g)| ≤

1 2







∂g(t, s)

+ |g(q1 x, q2 y)| ∂f (t, s) H(q1 x, q2 y). |f (q1 x, q2 y)|

dq tdq s

dq tdq s 1 2 1 2 9 946

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Multiplying both sides of (3.3) by w(x)u(y) and q-integrate over (x, y) ∈ Ω, we deduce ! ∂g(t, s) Pw (q1 x, t)Qu (q2 y, s) w(x)u(y)f (q1 x, q2 y) dq tdq s dq1 xdq2 y dq1 tdq2 s 1 2 c c a a ! ! Z bZ d Z bZ d ∂f (t, s) Pw (q1 x, t)Qu (q2 y, s) w(x)u(y)g(q1 x, q2 y) dq tdq s dq1 xdq2 y . + dq1 tdq2 s 1 2 c a c a

1 T (w, u, f, g) = 2

Z

b

Z

d

Z

b

Z

d

We observe 1 |T (w, u, f, g)| ≤ 2





 Z b Z d

∂f (t, s)

∂g(t, s)



w(x)u(y)H(q1 x, q2 y)dq1 xdq2 y. |f (q1 x, q2 y)| + |g(q1 x, q2 y)| dq1 tdq2 s dq1 tdq2 s a c

This completes the proof of Theorem 3.1. Let g(x, y) = 1, we have the following corollary. Rb Rd Corollary 3.2. Let f : Ω → R, w : [a, b] → R0 and u : [c, d] → R0 satisfying a w(x)dq1 x = c u(y)dq2 y = 1, then Z b Z d Z bZ d w(t)f (q1 t, y)dq1 t − u(s)f (x, q2 s)dq2 s + w(t)u(s)f (q1 t, q2 s)dq1 tdq2 s f (x, y) − a c a c

∂f (t, s)

≤ H(x, y)

dq tdq s , 1 2 where H(x, y) is defined in Theorem 2.2, and especially, let w(x) =

1 b−a

and u(y) =

1 d−c ,

we get

Z b Z d Z bZ d w(t)f (q1 t, y)dq1 t − u(s)f (x, q2 s)dq2 s + w(t)u(s)f (q1 t, q2 s)dq1 tdq2 s f (x, y) − a c a c ! !

 2  2

∂f (t, s) a+b c+d 1

(b − a)2 + 4 x − (d − c)2 + 4 y − ≤

dq tdq s . (3.4) 4(q1 + 1)(q2 + 1) 2 2 1 2 Remark 3.3. If q1 , q2 → 1− , the inequality (3.4) are reduced to the main result in [4].

References ˇ sev type inequalities, [1] F. Ahmad, N. S. Barnett, S. S. Dragomir, New weighted Ostrowski and Cebyˇ Nonlinear Analysis, 71, 12 (2009), 1408-1412. ˇ sev [2] F. Ahmad, P. Cerone, S. S. Dragomir and N. A. Mir, On some bounds of Ostrowski and Cebyˇ type, Journal of Mathematical Inequalities, 4, 1 (2010), 53-65. [3] M. Alomari, M. Darus, S. S. Dragomir, and P. Cerone, Ostrowski type inequalities for functions whose derivatives are s-convex in the second sense, Applied Mathematics Letters, 23, 9 (2010), 10711076. [4] N. S. Barnett, S. S. Dragomir, An Ostrowski type inequality for double integrals and applications for cubature formulae, Soochow Journal of Mathematics, 27, 1 (2001), 1-10. [5] H. Cauciman, Integral inequalities in q-calculus, Computers & Mathematics with Applications, 47, 2-3 (2004), 281-300. ˇ sev type inequalities, Journal of [6] K. Boukerrioua, A. Guezane-Lakoud, On generalization of Cebyˇ Inequalities in Pure and Applied Mathematics, 8, 2 (2007), Art. 15. 10 947

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ˇ ˇ [7] P. L. Ceby 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, 2 (1882), 93-98. ˇ sev functional and [8] P. Cerone and S. S. Dragomir, Some new Ostrowski-type bounds for the Cebyˇ applications, Journal of Mathematical Inequalities, 8, 1 (2014), 159-170. [9] S. S. Dragomir and N. S. Barnett, An Ostrowski type inequality for mapping whose second derivatives are bounded and applications, Journal of the Indian Mathematical Society (N.S.), 66, 1-4 (1999), 237-245. [10] S. S. Dragomir, P. Cerone, N. S. Barnett, J. Roumeliotis, An inequality of the Ostrowski type for double integrals and applications for cubature formulae, Tamsui Oxford Journal of Mathematical Sciences, 16, 1 (2000), 1-16. [11] S. S. Dragomir and A. Sofo, An inequality for monotonic functions generalizing Ostrowski and related results, Computers & Mathematics with Applications, 51, 3-4(2006), 497-506. [12] S. S. Dragomir and S. Wang, An inequality of Ostrowski-Gr¨ uss type and its applications to the estimation of error bounds for some special means and for some numerical quadrature rules, Computers Mathematics with Applications, 33 (1997), 15-20. Rb Rb 1 1 ¨ ¨ ss, Uber [13] G. Gru das maximum des absoluten Betrages von b−a f (x)g(x)dx − (b−a) f (x)dx × 2 a a Rb g(x)dx, Mathematische Zeitschrift, 39, 1 (1935), 215-226. a ˇ sev type inequalities for double integrals, Journal of [14] A. Guezane-Lakoud and F. Aissaoui, New Cebyˇ Mathematical Inequalities, 5, 4 (2011), 453-462. [15] V. Kac and P. Cheung, Quantum Calculus, Springer Verlag, 2002. [16] Z. Liu, A sharp general Ostrowski type inequality for double integrals, Tamsui Oxford Journal of Information and Mathematical Sciences, 28, 2 (2012), 217-226. [17] Z. Liu, A variant of Chebyshev inequality with applications, Journal of Mathematical Inequalities, 7, 4 (2013), 551-561. ´, J. E. Pec ˇaric ´ and N. Ujevic, On new estimation of the remainder in generalized Taylor’s [18] M. Matic formula, Mathematical Inequalities & Applications, 3, 2 (1999), 343-361. ¨ [19] A. Ostrowski, Uber die Absolutabweichung einer differentiierbaren Funktion von ihrem Integralmittelwert, Commentarii Mathematici Helvetici, 10, 1 (1937), 226-227. ˇ sev-Gr¨ [20] B. G. Pachpatte, On Cebyˇ uss type inequalities via Peˇcari´cs extension of the montgomery identity, Journal of Inequalities in Pure and Applied Mathematics, 7, 1 (2006), Art. 11. ˇ sev type for double integrals, Demonstratio Mathematica, [21] B. G. Pachpatte, New inequalities of Cebyˇ 11, 1 (2007), 43-50. [22] K. L. Tseng, S. R. Hwang, and S. S. Dragomir, Generalizations of weighted Ostrowski type inequalities for mappings of bounded variation and their applications, Computers & Mathematics with Applications, 55, 8 (2008), 1785-1793. ˇ sev-Gr¨ [23] W. Yang, On weighted q-Cebyˇ uss type inequalities, Computers & Mathematics with Applications, 61, 5 (2011), 1342-1347.

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QUADRATIC ρ-FUNCTIONAL INEQUALITIES IN NORMED SPACES IKAN CHOI, SUNGHOON KIM∗ , GEORGE A. ANASTASSIOU, AND CHOONKIL PARK∗ Abstract. In this paper, we solve the quadratic ρ-functional inequalities kf (x + y) + f (x − y) − 2f (x) − 2f (y)k

      x+y x−y

+ 2f − f (x) − f (y) , ≤ ρ 2f 2 2 where ρ is a number with |ρ| < 1 and



   x+y x−y

+ 2f − f (x) − f (y)

2f 2 2 ≤ kρ(f (x + y) + f (x − y) − 2f (x) − 2f (y))k,

(0.1)

(0.2)

where ρ is a number with |ρ| < 12 . Using the fixed point method, we prove the Hyers-Ulam stability of the quadratic ρ-functional inequalities (0.1) and (0.2) in normed spaces.

1. Introduction and preliminaries The stability problem of functional equations originated from a question of Ulam [29] 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 [14] gave a first affirmative partial answer to the question of Ulam for Banach spaces. Hyers’ Theorem was generalized by Aoki [2] for additive mappings and by Rassias [20] for linear mappings by considering an unbounded Cauchy difference. A generalization of the Rassias theorem was obtained by G˘avruta [11] by replacing the unbounded Cauchy difference by a general control function in the spirit of Rassias’ approach. 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. A Hyers-Ulam stability problem for the quadratic functional equation was proved by Skof [23] for mappings f : X → Y , where X is a normed space and Y is a Banach space. Cholewa [6] noticed that the theorem of Skof is still true if the relevant domain X is replaced by an Abelian group. Czerwik [7] proved the Hyers-Ulam stability of the  x+y = 12 f (x) + 12 f (y) is called the quadratic functional equation. The functional equation f 2 Jensen type quadratic functional equation. The stability problems of several functional equations 2010 Mathematics Subject Classification. Primary 39B52, 47H10, 39B72. Key words and phrases. Hyers-Ulam stability; quadratic ρ-functional inequality; fixed point. ∗ Corresponding authors.

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have been extensively investigated by a number of authors and there are many interesting results concerning this problem (see [1, 3, 16, 17, 21, 22, 25, 26, 27, 28, 30, 31]). In [12], Gil´anyi showed that if f satisfies the functional inequality k2f (x) + 2f (y) − f (xy −1 )k ≤ kf (xy)k

(1.1)

then f satisfies the Jordan-von Neumann functional equation 2f (x) + 2f (y) = f (xy) + f (xy −1 ). See also [10, 24]. Gil´anyi [13] and Fechner [9] proved the Hyers-Ulam stability of the functional inequality (1.1). Park, Cho and Han [18] proved the Hyers-Ulam stability of additive functional inequalities. Lemma 1.1. (Banach fixed-point theorem) Let (S, d) be a complete metric space and let T : S → S be a strictly contractive mapping with Lipschitz constant α < 1. Then for each given element x ∈ S, there exists a positive integer n0 such that (1) d(T n x, T n+1 x) < ∞, ∀n ≥ n0 ; n (2) the sequence {T x} converges to a fixed point y ∗ of T ; (3) y ∗ is the unique fixed point of T in the set Y = {y ∈ S | d(T n0 x, y) < ∞}; 1 (4) d(y, y ∗ ) ≤ 1−α d(y, T y) for all y ∈ Y . Since we defined the metric d as generalized metric in order to use this lemma in the proof of the problem we extend the lemma. Lemma 1.2. ([8]) Let (S, d) be a complete generalized metric space and let J : S → S be a strictly contractive mapping with Lipschitz constant α < 1. Then for each given element x ∈ S, 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 ∈ S | d(J n0 x, y) < ∞}; 1 d(y, Jy) for all y ∈ Y . (4) d(y, y ∗ ) ≤ 1−α In 1996, Isac and Rassias [15] 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 [4, 5, 19]). 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 normed spaces. 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 normed spaces. Throughout this paper, assume that X is a normed space and Y is a Banach space.

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2. Hyers-Ulam stability of the quadratic ρ-functional inequality (0.1): a fixed point approach In this section, assume that |ρ| < 1. We solve the quadratic ρ-functional inequality (0.1) in normed spaces. Lemma 2.1. A mapping f : X → Y satisfies kf (x + y) + f (x − y) − 2f (x) − 2f (y)k

     

x+y x−y

≤ ρ 2f + 2f − f (x) − f (y)

2 2 for all x, y ∈ X if and only if f : X → Y is quadratic.

(2.1)

Proof. Assume that f : X → Y satisfies (2.1). Letting x = y = 0 in (2.1), we get k2f (0)k ≤ |ρ|k2f (0)k. So f (0) = 0. Letting y = x in (2.1), we get kf (2x) − 4f (x)k ≤ 0 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 kf (x + y) + f (x − y) − 2f (x) − 2f (y)k

     

x−y x+y

+ 2f − f (x) − f (y) ≤ ρ 2f

2 2 |ρ| = kf (x + y) + f (x − y) − 2f (x) − 2f (y)k 2 and so f (x + y) + f (x − y) = 2f (x) + 2f (y) for all x, y ∈ X. The converse is obviously true.



We prove the Hyers-Ulam stability of the quadratic ρ-functional inequalty (0.1) in Banach spaces. Theorem 2.2. Let ϕ : X 2 → [0, ∞) be a function such that there exists an α < 1 with a b , 2 2 for all a, b ∈ X. Let f : X → Y be a mapping satisfying 



ϕ(a, b) ≤ 4αϕ

(2.3)

kf (x + y) + f (x − y) − 2f (x) − 2f (y)k

     

x+y x−y

≤ ρ 2f + 2f − f (x) − f (y)

+ ϕ(x, y) 2 2 for all x, y ∈ X. Then there exists a unique quadratic mapping Q : X → Y such that 1 kf (x) − Q(x)k ≤ ϕ(x, x) 4 − 4α for all x ∈ X.

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(2.4)

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Proof. Consider the set S := {h : X → Y } and let d be the generalized metric on S: d(g, h) := inf{µ ∈ R+ : kg − hk ≤ µϕ(x, x), x ∈ S} It is easy to show that (S, d) is complete. Let J be the linear mapping from S to S such that 1 Jg(x) := g(2x) 4 Let g, h ∈ S be given such that d(g, h) = ε. Then from (2.3) and (2.5), we get

(2.5)



1 1 1

≤ εϕ(2x, 2x) ≤ αεϕ(x, x) kJg(x) − Jh(x)k = g(2x) − h(2x)

4 4 4

This means d(Jg, Jh) ≤ αd(g, h). So the function J : S → S is a contractive mapping such that d(Jg, Jh) ≤ αd(g, h) for 0 ≤ α < 1. Letting y = x in (2.4), we get kf (2x) − 4f (x)k ≤ ϕ(x, x) and so

1 kf (x) − Jf (x)k ≤ ϕ(x, x) 4 for all x ∈ X. Thus we get d(f, Jf ) ≤ 14 . By Lemma 1.2, there exists a mapping Q : X → Y satisfying the following: (1) Q is a fixed point of J, i.e., Q (2a) = 4Q(a)

(2.6)

for all a ∈ 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 (2.6) such that there exists a µ ∈ (0, ∞) satisfying kf (a) − Q(a)k ≤ µϕ (a, a) for all a ∈ X; (2) d(J l f, Q) → 0 as l → ∞. This implies the equality 1  l  f 2 a = Q(a) l→∞ 4l lim

for all a ∈ X; (3) d(f, Q) ≤

1 1−α d(f, Jf ),

which implies the inequality d(f, Q) ≤

1 . 4 − 4α

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So kf (a) − Q(a)k ≤

1 ϕ(a, a) 4 − 4α

for all a ∈ X. Then kQ(x + y) + Q(x − y) − 2Q(x) − 2Q(y)k



1 l l l l

= lim l (f (2 (x + y)) + f (2 (x − y)) − 2f (2 x) − 2f (2 y))

l→∞ 4

ρ

l−1 l−1 l l

≤ lim (2f (2 (x + y)) + 2f (2 (x − y)) − f (2 x) − f (2 y))

l l→∞ 4 1 + lim l ϕ(2l x, 2l y) l→∞ 4

     

x+y x−y

= ρ 2Q + 2Q − Q(x) − Q(y)

2 2 for all x, y ∈ X. Hence Q (x + y) + Q(x − y) = 2Q(x) + 2Q(y) for all x, y. So Q : X → Y is quadratic.



Remark 2.3. We could provethe same statement with the same manner in spite of replacing the condition ϕ(a, b) ≤ 4αϕ a2 , 2b into ϕ(a, b) ≤ 14 αϕ (2a, 2b) by defining J such that Jg(x) = 4g( x2 ) instead of Jg(x) = 14 g(2x). It could be also applied to Theorem 3.2. Corollary 2.4. Let r 6= 2 and θ be nonnegative real numbers, and let f : X → Y be a mapping such that kf (x + y) + f (x − y) − 2f (x) − 2f (y)k

     

x+y x−y r r

≤ ρ 2f + 2f − f (x) − f (y)

+ θ(kxk + kyk ) 2 2 for all x, y ∈ X. Then there exists a unique quadratic mapping Q : X → Y such that 2θ kxkr kf (x) − Q(x)k ≤ |4 − 2r | for all x ∈ X.

3. Hyers-Ulam stability of the quadratic ρ-functional inequality (0.2): a fixed point approach In this section, assume that |ρ| < 12 . We solve the quadratic ρ-functional inequality (0.2) in normed spaces. Lemma 3.1. A mapping f : X → Y satisfies



  

2f x + y + 2f x − y − f (x) − f (y)

2 2 ≤ kρ(f (x + y) + f (x − y) − 2f (x) − 2f (y))k

(3.1)

for all x, y ∈ X if and only if f : X → Y is quadratic.

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Proof. Assume that f : X → Y satisfies (3.1). Letting x = y = 0 in (3.1), we get k2f (0)k ≤ |ρ|k2f (0)k. So f (0) = 0. Letting y = 0 in (3.1), we get

 



4f x − f (x) ≤ 0

2

(3.2)

and so f x2 = 14 f (x) for all x ∈ X. It follows from (3.1) and (3.2) that


1 kf (x + y) + f (x − y) − 2f (x) − 2f (y)k 2

   

x+y x−y

+ 2f − f (x) − f (y) = 2f

2 2 ≤ |ρ|kf (x + y) + f (x − y) − 2f (x) − 2f (y)k and so f (x + y) + f (x − y) = 2f (x) + 2f (y) for all x, y ∈ X. The converse is obviously true.



We prove the Hyers-Ulam stability of the quadratic ρ-functional inequalty (0.2) in Banach spaces. Theorem 3.2. Let ϕ : X 2 → [0, ∞) be a function such that there exists an α < 1 with 

ϕ(a, b) ≤ 4αϕ

a b , 2 2



for all a, b ∈ X. Let f : X → Y be a mapping satisfying f (0) = 0 and

   

2f x + y + 2f x − y − f (x) − f (y)

2 2

≤ kρ(f (x + y) + f (x − y) − 2f (x) − 2f (y))k + ϕ(x, y) for all x, y ∈ X. Then there exists a unique quadratic mapping Q : X → Y such that kf (x) − Q(x)k ≤

1 ϕ(x, 0) 4 − 4α

for all x ∈ X. Proof. Consider the set S := {h : X → Y } and let d be the generalized metric on S: d(g, h) := inf{µ ∈ R+ : kg − hk ≤ µϕ(x, 0), x ∈ S} It is easy to show that (S, d) is complete. Let J be the linear mapping from S to S such that 1 Jg(x) := g(2x) 4

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Let Q : X → Y be defined as in the proof of Theorem 2.2. Then

   

2Q x + y + 2Q x − y − Q(x) − Q(y)

2 2

1

l−1 l−1 l l

= lim l (2f (2 (x + y)) + 2f (2 (x − y)) − f (2 x) − f (2 y))

l→∞ 4

ρ

l l l l

≤ lim (f (2 (x + y)) + f (2 (x − y)) − 2f (2 x) − 2f (2 y))

l→∞ 4l 1 + lim l ϕ(2l x, 2l y) l→∞ 4 = kρ(Q(x + y) + Q(x − y) − 2Q(x) − 2Q(y))k for all x, y ∈ X. Hence Q (x + y) + Q(x − y) = 2Q(x) + 2Q(y) for all x, y. So Q : X → Y is quadratic.



Corollary 3.3. Let r 6= 2 and θ be nonnegative real numbers, and let f : X → Y be a mapping satisfying f (0) = 0 and

   

2f x + y + 2f x − y − f (x) − f (y)

2 2 ≤ kρ(f (x + y) + f (x − y) − 2f (x) − 2f (y))k + θ(kxkr + kykr ) for all x, y ∈ X. Then there exists a unique quadratic mapping Q : X → Y such that θ kf (x) − Q(x)k ≤ kxkr |4 − 2r | for all x ∈ X. Acknowledgments I. Choi and S. Kim were supported by the Seoul Science High School R&E/I&D Program in 2014. C. Park was supported by Basic Science Research Program through the National Research Foundation of Korea funded by the Ministry of Education, Science and Technology (NRF-2012R1A1A2004299). References [1] M. Adam, On the stability of some quadratic functional equation, J. Nonlinear Sci. Appl. 4 (2011), 50–59. [2] T. Aoki, On the stability of the linear transformation in Banach spaces, J. Math. Soc. Japan 2 (1950), 64–66. [3] L. C˘ adariu, L. G˘ avruta and P. G˘ avruta, On the stability of an affine functional equation, J. Nonlinear Sci. Appl. 6 (2013), 60–67. [4] 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. [5] 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). [6] P.W. Cholewa, Remarks on the stability of functional equations, Aequationes Math. 27 (1984), 76–86. [7] S. Czerwik, On the stability of the quadratic mapping in normed spaces, Abh. Math. Sem. Univ. Hamburg 62 (1992), 59–64. [8] 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. [9] W. Fechner, Stability of a functional inequalities associated with the Jordan-von Neumann functional equation, Aequationes Math. 71 (2006), 149–161.

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[10] W. Fechner, On some functional inequalities related to the logarithmic mean, Acta Math. Hungar. 128 (2010), 31–45. [11] P. Gˇ avruta, A generalization of the Hyers-Ulam-Rassias stability of approximately additive mappings, J. Math. Anal. Appl. 184 (1994), 431–43. [12] A. Gil´ anyi, Eine zur Parallelogrammgleichung ¨ aquivalente Ungleichung, Aequationes Math. 62 (2001), 303– 309. [13] A. Gil´ anyi, On a problem by K. Nikodem, Math. Inequal. Appl. 5 (2002), 707–710. [14] D. H. Hyers, On the stability of the linear functional equation, Proc. Natl. Acad. Sci. U.S.A. 27 (1941), 222–224. [15] G. Isac and Th.M. Rassias, Stability of ψ-additive mappings: Appications to nonlinear analysis, Internat. J. Math. Math. Sci. 19 (1996), 219–228. [16] S. Jung, Hyers-Ulam-Rassias Stability of Functional Equations in Mathematical Analysis, Hadronic Press lnc., Palm Harbor, Florida, 2001. [17] C. Park, Homomorphisms between Poisson JC ∗ -algebras, Bull. Braz. Math. Soc. 36 (2005), 79–97. [18] C. Park, Y. Cho and M. Han, Functional inequalities associated with Jordan-von Neumann-type additive functional equations, J. Inequal. Appl. 2007 (2007), Article ID 41820, 13 pages. [19] V. Radu, The fixed point alternative and the stability of functional equations, Fixed Point Theory 4 (2003), 91–96. [20] Th. M. Rassias, On the stability of the linear mapping in Banach spaces, Proc. Amer. Math. Soc. 72 (1978), 297–300. [21] Th.M. Rassias (ed.), Functional Equations and Inequalities, Kluwer Academic, Dordrecht, 2000. [22] 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. [23] F. Skof, Propriet` a locali e approssimazione di operatori, Rend. Sem. Mat. Fis. Milano 53 (1983), 113–129. [24] J. R¨ atz, On inequalities associated with the Jordan-von Neumann functional equation, Aequationes Math. 66 (2003), 191–200. [25] S. Shagholi, M. Eshaghi Gordji and M. Bavand Savadkouhi, Stability of ternary quadratic derivations on ternary Banach algebras, J. Comput. Anal. Appl. 13 (2011), 1097–1105. [26] S. Shagholi, M. Eshaghi Gordji and M. Bavand Savadkouhi, Nearly ternaty cubic homomorphisms in ternary Fr´echet algebras, J. Comput. Anal. Appl. 13 (2011), 1106–1114. [27] D. Shin, C. Park and Sh. Farhadabadi, On the superstability of ternary Jordan C ∗ -homomorphisms, J. Comput. Anal. Appl. 16 (2014), 964–973. [28] 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. [29] S. M. Ulam, A Collection of the Mathematical Problems, Interscience Publ. New York, 1960. [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. Ikan Choi, Sunghoon Kim 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 133-791, Korea E-mail address: [email protected]

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Solution of the Ulam stability problem for quartic (a, b)-functional equations Abdullah Alotaibi1 , John Michael Rassias2 and S.A. Mohiuddine1 1

Department of Mathematics, Faculty of Science, King Abdulaziz University, P.O. Box 80203, Jeddah 21589, Saudi Arabia 2 National and Capodistrian University of Athens, Pedagogical Department, Mathematics and Informatics, 4, Agamemnonos Str., Aghia Paraskevi, Attikis 15342, Greece Email: [email protected]; [email protected]; [email protected] Abstract. The “oldest quartic” functional equation was introduced and solved by the author of this paper (see: Glas. Mat. Ser. III 34 (54) (1999), no. 2, 243-252) which is of the form: f (x + 2y) + f (x − 2y) = 4[f (x + y) + f (x − y)] − 6f (x) + 24f (y). Interesting results have been achieved by S.A. Mohiuddine et al., since 2009. In this paper, we are introducing new quartic functional equations, and establish fundamental formulas for the general solution of such functional equations and for “Ulam stability” of pertinent quartic functional inequalities. Keywords and phrases: Quartic functional equations and inequalities; Various normed spaces; Ulam stability. AMS subject classification (2000): 39B.

1. INTRODUCTION In 1940 S. M. Ulam [1] gave a talk before the Mathematics Club of the University of Wisconsin in which he discussed a number of unsolved problems. Among these was the following famous “stability Ulam question”: We are given a group G and a metric group G0 with metric ρ(., .). Given  > 0, does there exist a δ > 0 such that if f : G → G0 satisfies ρ(f(xy), f(x)f(y)) < δ for all x, y in G, then a homomorphism h : G → G0 exists with ρ(f(x), h(x)) <  for all x ∈ G? By now an affirmative answer has been given in several cases, and some interesting variations of the problem have also been investigated. We shall call such an f : G → G0 an approximate homomorphism. In 1941 D. H. Hyers [2] considered the case of approximately additive mappings f : E → E 0 where E and E 0 are Banach spaces and f satisfies the following Hyers’ inequality

f(x + y) − f(x) − f(y) ≤  for all x, y ∈ E. It was shown that the limit

L(x) = lim 2−n f(2n x) n→∞

exists for all x ∈ E and that L : E → E 0 is the unique additive mapping satisfying f(x) − L(x) ≤ . No continuity conditions are required for this result, but if f(tx) is continuous in the real variable t for each fixed x, then L is linear, and if f is continuous at a single point of E then L : E → E 0 is also continuous. In 1982-1994, a generalization of this result was proved by the author J. M. Rassias [3–7], as follows. He introduced the following weaker condition (or weaker inequality or the generalized Cauchy inequality)



f(x + y) − [f(x) + f(y)] ≤ θ x p y q for all x, y in E, controlled by (or involving) a product of different powers of norms, where θ ≥ 0 and real p, q : r = p + q 6= 1, and retained the condition of continuity of f(tx) in t for fixed x. Besides he investigated that it is possible to replace  in the above Hyers’ inequality, by a non-negative real-valued function such that the pertinent series converges and other conditions hold and still obtain stability results. In all the cases investigated in these results, the approach to the existence question was to prove 957

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asymptotic type formulas: L(x) = limn→∞ 2−n f(2n x); L(x) = limn→∞ 2n f(2−n x). Theorem (J. M. Rassias:1982-1994). Let X be a real normed linear space and let Y be a real complete normed linear space. Assume in addition that f : X → Y is an approximately additive mapping for which there exist constants θ ≥ 0 and p, q ∈ R such that r = p + q 6= 1 and f satisfies the “generalized Cauchy inequality”



f(x + y) − [f(x) + f(y)] ≤ θ x p y q for all x, y ∈ X. Then there exists a unique additive mapping L : X → Y satisfying

f(x) − L(x) ≤

r θ

x |2r − 2|

for all x ∈ X. If in addition f : X → Y is a mapping such that the transformation t → f(tx) is continuous in t ∈ R for each fixed x ∈ X, then L is an R-linear mapping. In 1940, Ulam, S. M. [1] proposed the “Ulam stability problem”: When does a linear transformation near an “approximately linear” transformation exist? Since then, many specialists on this “famous Ulam problem”, have investigated interesting functional equations, for instance: D. H. Hyers [2], in 1941; T. ˇ Aoki [8], in 1950; T. M. Rassias [9], in 1978; Z. Gajda [10], in 1991; T. M. Rassias and P. Semrl [11], in 1992; P. Gˇ avruta [12], in 1994; S.-M. Jung [13], in 1998; K. W. Jun and H. M. Kim [14], in 2002; R. P. Agarwal et al. [15], in 2003, and others. Interesting Ulam-Hyers stability results have been established by S. A. Mohiuddine et al. ( [16–19]). The “oldest quartic” functional equation was introduced and solved by the author of this paper, [20], which is of the form: f(x + 2y) + f(x − 2y) = 4[f(x + y) + f(x − y)] − 6f(x) + 24f(y). Since then various quartic equations have been proposed and solved by a number of experts in the area of functional equations and inequalities. For more details on these concepts, one can be referred to [21–30]. For further research in various normed spaces, we are introducing new quartic functional equations, and establish fundamental formulas for the general solution of such functional equations and for “Ulam stability” of pertinent quartic functional inequalities.

2. ON (a, b)-QUARTIC FUNCTIONAL EQUATIONS 1. Stability of General a-Quartic Functional Equation     2 f(ax + y) + f(x + ay) + a(a − 1)2 f(x − y) = 2(a2 − 1)2 f(x) + f(y) + a(a + 1)2 f(x + y)

(1.1)

where a 6= 0, a 6= ±1. Replacing x = y = 0 in (1.1) one gets 4a2 (1 − a2 )f(0) = 0, or f(0) = 0.

(1.2)

Similarly, substituting x = x, y = 0 in (1.1), we obtain f(ax) = a4 f(x) + (a2 − 1)2 f(0) = a4 f(x)

(1.3)

Also assuming f(2x) = 16f(x), replacing x = x, y = x in (1.1), and setting k = a + 1 6= 0, ±1, one obtains 4f(kx) + a(a − 1)2 f(0) = 4(a2 − 1)2 f(x) + a(a + 1)2 f(2x), or

(1.4)

1 f(kx) + a(a − 1)2 f(0) = (a2 − 1)2 f(x) + 4a(a + 1)2 f(x) = k 4 f(x), or 4 1 f(kx) = k 4 f(x) − a(a − 1)2 f(0) = k 4 f(x). 4

958

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(1.5)

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Without assuming f(2x) = 16f(x), replacing x = x, y = −x in (1.1), and setting l = a − 1 6= 0, ±1, one obtains     2 f(lx) + f(−lx) + a(a − 1)2 f(2x) = 2(a2 − 1)2 f(x) + f(−x) + a(a + 1)2 f(0) (1.6)

Placing −x on x in (1.6), and then subtracting the new equation from (1.6), we get a(a − 1)2 [f(2x) − f(−2x)] = 0. Letting x/2 on x, we find that f is an “even function”, such that f(−x) = f(x). Thus from (1.6), we obtain 4f(lx) + a(a − 1)2 f(2x) = 4(a2 − 1)2 f(x) + a(a + 1)2 f(0). (1.7) Assuming f(2x) = 16f(x), we get 1 f(lx) = l4 f(x) + a(a + 1)2 f(0) = l4 f(x). 4

Without assuming f(2x) = 16f(x), subtracting (1.7) from (1.4), we obtain from (1.2) that "   4 # 4 1 k l 2 f(kx) − f(lx) = a(a + 1)f(2x) = − f(2x). 2 2 2 Replacing x → x/2, we get that   4 k k “f x = f(x) 2 2 

  4 l l if and only if f x = f(x)”. 2 2 

(1.8)

Employing the “quartic mean”, we have equivalently that

f k2 x f 2l x “f k (x) =
= f(x) iff f 2l (x) = l
4 = f(x)”. 2 k 4 2

2

Let X be a real normed linear space and let Y be a real complete normed linear space. Assume f : X → Y , satisfying the following general a-quartic functional inequality



  

2 f(ax + y) + f(x + ay) + a(a − 1)2 f(x − y) − 2(a2 − 1)2 f(x) + f(y) − a(a + 1)2 f(x + y) ≤ c (1.9) where a 6= 0, a 6= ±1. Replacing x = y = 0 in (1.9), one gets

f(0) ≤ c/4a2 |a2 − 1|.

(1.10)

Substituting x = x, y = 0 in (1.9), and employing the triangle inequality, we obtain:

Note that

Therefore

and

Therefore we get

and

2 2

f(ax) − a4 f(x) ≤ c2 = c1 = 2a + |a − 1| c. 2 4a2

  2a2 + |a2 − 1| c1 = c =  2a2

3a2−1 2a2 c

if |a| > 1

a2 +1 2a2 c

if |a| < 1; a 6= 0.

2

f(ax) − a4 f(x) ≤ c2 = c1 = 3a − 1 c, if |a| > 1, 2 4a2

2

f(ax) − a4 f(x) ≤ c3 = c1 = a + 1 c, if |a| < 1; a 6= 0. 2 4a2 2

f(x) − a−4 f(ax) ≤ a−4 c2 = 3a − 1 c, if |a| > 1, 4a6

2

f(x) − a4 f(a−1 x) ≤ c3 = a + 1 c, if |a| < 1; a 6= 0. 4a2

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Thus we can easily obtain the following general inequality

In fact,



f(x) − a−4n f(an x) ≤

f(x) − a−4n f(an x)

≤

≤ ≤

3a2 − 1 (1 − a−4n )c, if |a| > 1. 4a2 (a4 − 1)

(1.11)



f(x) − a−4 f(a1 x) + a−4 f(ax) − a−4 f(a2 x) + . . .

· · · + a−4(n−2) f(an−2 x) − a−4 f(an−1 x)

+ a−4(n−1) f(an−1 x) − a−4 f(an x) (1.12)   3a2 − 1 1 + a−4 + a−8 + · · · + a−4(n−2) + a−4(n−1) c 4a6 3a2 − 1 (1 − a−4n )c, if |a| > 1. 4a2 (a4 − 1)

Also,

f(x) − α4n f(α−n x) ≤ ≤ ≤



f(x) − α4 f(α−1 x) + α4 f(α−1 x) − α4 f(α−2 x)

+ · · · + α4(n−1) f(α−(n−1)x) − α4 f(α−n x)   α2 + 1 1 + α4 + · · · + α4(n−1) c 4α2 α2 + 1 1 (1 − α4n )c, if |α| < 1. 4α2 1 − α4

The “altemative” general inequality for |α| < 1; α 6= 0 is similarly established. Note 1. (i) Assume |α| > 1, and denote

Qn : Qn (x) = α−4n f(αn x).

Claim that sequence {Qn }, |α| > 1 is a Cauchy sequence. In fact, if m > n > 0, then



0 ≤ Qn (x) − Qm (x) = α−4n f(αn x) − α−4m f(αm x)

= |α|−4n f(αn x) − α−4(m−n)f(αm−n .αn x)  3α2 − 1  ≤ |α|−4n 2 4 1 − α−4(m−n) c 4α (α − 1)  2 3α − 1  −4n = |α| − |α|−4nα−4(m−n) c 2 4 4α (α − 1) → 0, as n → ∞ (and m → ∞). Thus {Qn }, |α| > 1, is Cauchy sequence. Similarly, if |α| < 1, α 6= 0, one proves that {Qn }, |α| < 1, α 6= 0 is Cauchy sequence, as well. (ii) Claim the quarticness of Q : Q(x) = lim Qn (x) = lim α−4n f(αn x), α > 1. n→∞

n

n→∞

n

In fact, replacing x → α x, y → α y in the α-quartic functional inequality (1.9) and then multiplying by |α|−4n, and taking limit n → ∞, we obtain

0 ≤ 2[Q(αx + y) + Q(x + αy)] + α(α − 1)2 Q(x − y)

−2(α2 − 1)2 [Q(x) + Q(y)] − α(α + 1)2 Q(x + y) ≤

|α|−4nc → 0, n → ∞. 960

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Thus 2[Q(αx + y) + Q(x + αy)] + α(α − 1)2 Q(x − y) = 2(α2 − 1)2 [Q(x) + Q(y)] + α(α + 1)2 Q(x + y) leading to (1.1) and thus the quarticness of Q : |α| > 1. Similarly, we prove that Q is quartic for |α| < 1, α 6= 0. Thus the existence of Q is complete. If Q(x) = lim Qn (x) = lim n→∞

then

 −4n f(an x) if |a| > 1  a

n→∞ 

,

(1.13)

a4n f(a−n x) if |a| < 1; a 6= 0



2a2 + |a2 − 1| 

f(x) − Q(x) ≤ c·  4a2

1 a4 −1

if |a| > 1

1 1−a4

if |a| < 1; a 6= 0.

(1.14)

Note 2. Claim the uniqueness of Q : Q(x) = lim Qn (x) = lim α−4n f(αn x), α > 1. n→∞

n→∞

In fact, if there is another quartic mapping Q0 satisfying (1.14), then

0 ≤ Q(x) − Q0 (x)



≤ Q(x) − f(x) + f(x) − Q0 (x) ,

or

0

≤ = = ≤ = ≤



Q(x) − Q0 (x)

|α|−4n Q(αn x) − Q0 (αn x)

−4n

α Q(αn x) − α−4n Q0 (αn x)

−4n



α Q(αn x) − α−4n f(αn x) + α−4n f(αn x) − α−4nQ0 (αn x) n



o |α|−4n Q(αn x) − f(αn x) + f(αn x) − Q0 (αn x) 2α2 + |α2 − 1| −4n |α| c → 0, n → ∞ 2α2

or Q(x) = Q0 (x), proving uniqueness of Q : |α| > 1. Similarly, one proves uniqueness of Q : |α| < 1, α 6= 0. Theorem 1.1. Let X be a normed space and Y be a Banach space. If f : X → Y is a mapping satisfying (1.9), then there exists a unique quartic . mapping Q : X → Y, satisfying inequality (1.14).



If f(0) = 0, then f(x) − Q(x) ≤ c 2 1 − a4 , for ∀a 6= 0; ±1.

2. Stability of General (a, b)-Quartic Functional Equation

    2 f(ax + by) + f(bx + ay) + ab(a − b)2 f(x − y) = 2(a2 − b2 )2 f(x) + f(y) + ab(a + b)2 f(x + y), (2.1)

where a 6= ±b, a, b 6= 0, ±1, k = a + b 6= 1, l = a − b 6= 1, and a4 + b4 − a2 b2 − 1 6= 0. Replacing x = y = 0 in this equation, one gets 4(a4 + b4 − a2 b2 − 1)f(0) = 0, or f(0) = 0 961

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Similarly substituting x = x, y = 0 in (2.1), we obtain f(ax) + f(bx) = (a4 + b4 )f(x) + (a2 − b2 )2 f(0) = (a4 + b4 )f(x).

(2.3)

From (2.3), we observe that “f(ax) = a4 f(x)

if and only if f(bx) = b4 f(x).”

(2.4)

Let us introduce the following “(a, b)-quartic functional mean” f(a,b) (x) =

f(ax) + f(bx) . a4 + b 4

(2.5)

From (2.3) and (2.5), we find the quartic functional mean equation fa,b) (x) = f(x)

(2.6)

From (2.4)-(2.5)-(2.6), one establishes “f(a,0) (x) =

f(ax) f(bx) = f(x) iff f(0,b)(x) = = f(x).” 4 a b4

Also assuming f(2x) = 16f(x), replacing x = x, y = x in (2.1) and setting k = a + b 6= 0, ±1, one obtains 4f(k) + ab(a − b)2 f(0) = 4(a2 − b2 )2 f(x) + ab(a + b)2 f(2x), or

(2.8)

1 f(k) + ab(a − b)2 f(0) = (a2 − b2 )2 f(x) + 4ab(a + b)2 f(x) = k 4 f(x), 4

or

1 f(k) = k 4 f(x) − ab(a − b)2 f(0) = k 4 f(x). (2.9) 4 Without assuming f(2x) = 16f(x), replacing x = x, y = −x in (2.1) and setting l = a − b 6= 0, ±1 with a 6= ±b, one obtains     2 f(lx) + f(−lx) + ab(a − b)2 f(2x) = 2(a2 − b2 )2 f(x) + f(−x) + ab(a + b)2 f(0) (2.10)  Placing−x on x in (2.10), and then subtracting the new equation from (2.10), we get ab(a − b)2 f(2x) − f(−2x) = 0. Letting x/2 on x, we find that f is an “even function”, such that f(−x) = f(x). Thus from (2.10), we obtain 4f(lx) + ab(a − b)2 f(2x) = 4(a2 − b2 )2 f(x) + ab(a + b)2 f(0). (2.11) Assuming f(2x) = 16f(x), we get 1 f(lx) = l4 f(x) + ab(a + b)2 f(0) = l4 f(x). 4 Without assuming f(2x) = 16f(x), subtracting (2.11) from (2.8), we obtain from (2.2) that "    # 4 4 1 k l f(kx) − f(lx) = ab(a2 + b2 )f(2x) = − f(2x). 2 2 2

Replacing x → x/2, we obtain that   4    4 k k l l f x − f(x) = f x − f(x). 2 2 2 2 

Therefore, we observe that    4 k k “f x = f(x) 2 2

if and only if f 962

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  4 l l x = f(x)”. 2 2

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Employing the “quartic mean”, we have equivalently that

f 2l x f k2 x “f k (x)
= f(x) iff f 2l (x) = l
4 = f(x)”. 2 k 4 2

2

Let X be a real normed linear space and let Y be a real complete normed linear space. Assume f : X → Y , satisfying the following general (a, b)-quartic functional inequality

2[f(ax +by)+f(bx +ay)] +ab(a−b)2 f(x −y)−2(a2 −b2 )2 [f(x)+f(y)] −ab(a+b)2 f(x +y) ≤ c, (2.13) where a, b 6= 0; a, b 6= ±1. Replacing x = y = 0 in (2.13), one gets

kf(0)k ≤ c/4|a4 + b4 − a2 b2 − 1|.

(2.14)

Substituting x = x, y = 0 in (2.13), and employing the triangle inequality and (2.14), we obtain kf(ax) + f(bx) − (a4 + b4 )f(x)k ≤ or kf(ax) + f(bx) − (a4 + b4 )f(x)k ≤ c

c + (a2 − b2 )2 |f(0)|, 2

(a2 − b2 )2 + 2|a4 + b4 − a2 b2 − 1| . 4|a4 + b4 − a2 b2 − 1|

(2.15)

Substituting x = x, y = x in (2.13), and employing the triangle inequality and (2.14), as well as the following hypothesis kf(2x) − 16f(x)k ≤ c1 (≥ 0), (2.16) and denoting k = a + b 6= 0, ±1, we obtain: kf(kx) − k 4 f(x)k ≤ or kf(kx) − k 4 f(x)k ≤ c2 = or

1 4




1 c + ab(a − b)2 |f(0)| + ab(a + b)2 c1 , 4

 ab(a − b)2 + 4|a4 + b4 − a2 b2 − 1| 2 c + ab(a + b) c 1 , 4|a4 + b4 − a2 b2 − 1| kf(x) − k 4 f(kx)k ≤ k −4 c2

(2.17) (2.18)

if |k| > 1. Thus we easily obtain, the following general inequality: kf(x) − k −4n f(k n x)k ≤

1 (1 − k −4n )c2 , if |k| > 1. k4 − 1

(2.19)

In fact,

f(x) − k −4n f(k n x)

≤

≤ =



f(x) − k −4 f(k 1 x) + k −4 f(k 1 x) − k −4 f(k 2 x) + · · ·

· · · + k −4(n−2) f(k n−2 x) − k −4 f(k n−1 x)

+ k −4(n−1) f(k n−1 x) − k 4 f(k n x) (2.20)   −4 −4 −8 −4(n−2) −4(n−1) k 1+k + k + ···+ k +k c2 k −4 1 (1 − k −4n )c2 = 4 (1 − k −4n )c2 , if |k| > 1. −4 1−k k −1

The“altemative” general inequality is similarly established, as follows kf(x) − k 4n f(k −n x)k ≤

1 (1 − k 4n )c2 , if |k| < 1; k 6= 0. 1 − k4

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(2.21)

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In fact,

f(x) − k 4n f(k −n x)



≤ f(x) − k 4 f(k −1 x) + k 4 f(k −1 x) − k 4 f(k −2 x) + . . .

· · · + k 4(n−2) f(k −(n−2) x) − k 4 f(k −(n−1) x)

+ k 4(n−1) f(k −(n−1) x) − k 4 f(k −n x) (2.22) ≤ (1 + k 4 + k 8 + · · · + k 4(n−2) + k 4(n−1))c2 1 ≤ (1 − k 4n )c2 , if |k| < 1; k 6= 0. 1 − k4

If we denote

 −4n f(k n x)  k

Q(x) = lim Qn (x) = lim n→∞

n→∞ 

It follows

|f(x) − Q(x)| ≤ c2 ·

  

k 4n f(k −n x)

if |k| > 1 if |k| < 1; k 6= 0.

1 k 4 −1

if |k| > 1

1 1−k 4

if |k| < 1; k 6= 0.

(2.23)

Note 3. Following Notes 1-2, we establish the existence and uniqueness of the quartic mapping Q. If f(0) = 0, and f(2x) = 16f(x), then kf(x) − Q(x)k ≤

c , 2|1 − k 4 |

for ∀k 6= 0; ±1.

(2.24)

Theorem 2.1. Let X be normed space and Y a Banach space. If f : X → Y is a mapping satisfying (2.13) then there exists a unique quartic mapping Q : X → Y , satisfying inequality (2.23). If f(0) = 0, and f(2x) = 16f(x), then kf(x) − Q(x)k ≤

c , 2|1 − k 4 |

for ∀k = a + b 6= 0; ±1.

3. General Alternative a-Quartic Functional Equation 2f(ax + y) + f(x + ay) + f(ax − y) =

 a 2 (a + 4a + 1)f(x + y) − (a2 − 4a + 1)f(x − y) 2

+(3a4 − 4a2 + 1)f(x) + (a4 − 4a2 + 3)f(y),

(3.1)

where a 6= 0, a 6= ±1. Replacing x = y = 0 in (3.1), one gets 4a2 (a2 − 1)f(0) = 0, or f(0) = 0.

(3.2)

1 f(ax) = a4 f(x) + (a4 − 4a2 + 3)f(0) = a4 f(x). 3

(3.3)

Substituting x = x, y = 0 in (3.1), we obtain

Let X be a real normed linear space and let Y be a real complete normed linear space. Assume f : X → Y , satisfying the following general alternative a-quartic functional inequality

2f(ax + y) + f(x + ay) + f(ax − y) −

 a 2 (a + 4a + 1)f(x + y) − (a2 − 4a + 1)f(x − y) 2 964

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−(3a4 − 4a2 + 1)f(x) − (a4 − 4a2 + 3)f(y) ≤ c,

(3.4)

where a 6= 0, a 6= ±1. Replacing x = y = 0 in (3.4), one gets

4a2 a2 − 1 f(0) ≤ c, or



f(0) ≤

c . a2 |a2 − 1|

(3.5)

Substituting x = x, y = 0 in (3.4) and employing (3.5), we obtain 2 2

f(x) − a−4 f(ax) ≤ 3a + |a − 3| a−4 c = a−4 c1 . 2 3a

Thus

Similarly, we obtain

Thus

If

(3.6)

 1  −4(n+1) c1 . 1 − a a4 − 1



f(x) − a−4n f(an x) ≤

(3.7)

2 2

f(x) − a4 f(a−1 x) ≤ 3a + |a − 3| c = c1 . 3a2



f(x) − a4n f(a−n x) ≤ Q(x) = lim Qn (x) = lim n→∞

 1  1 − a4(n+1) c1 . 4 1−a

 −4n f(an x)  a

n→∞ 

(3.8)

a4n f(a−n x)



3a2 + |a2 − 3| 

f(x) − Q(x) ≤ c·  3a2

if |a| > 1

(3.9)

, then

if |a| < 1; a 6= 0

1 a4 −1

if |a| > 1

1 1−a4

if |a| < 1; a 6= 0.

(3.10)

Note 4. Following Notes 1-2, we establish existence and uniqueness of Q.

If f(0) = 0, then f(x) − Q(x) ≤

c , 2|1−a4|

for ∀a 6= 0; ±1.

Theorem 3.1. Let X be a normed space and Y a Banach space. If f : X → Y is a mapping satisfying (3.1), then there exists a unique quartic . mapping Q : X → Y, satisfying inequality (3.4).

If f(0) = 0, then f(x) − Q(x) ≤ c 2 1 − a4 , for ∀a 6= 0; ±1. OPEN RESEARCH PROBLEMS

OPEN PROBLEM A. Employing both the “Hyers’ direct method” and the “fixed point method”, it is still “open”, the investigation of “generalized Ulam stabilities” and “generalized Ulam superstabilities” of these quartic functional equations in various normed spaces, domains and groups such as in 1. Banach spaces; 2. Banach algebras; C∗-algebras; 3. N -multi-Banach spaces; multi-Banach spaces; 4. Multi-normed spaces; 5. Quasi-Banach spaces; 6. Quasi-β[beta]-normed spaces; 7. Non-Archimedean normed spaces; 8. Fuzzy normed spaces; 965

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9. Quasi fuzzy normed spaces; 10. Non- Archimedean fuzzy normed spaces; 11. Intuitionistic normed spaces; 12. Random normed spaces; and probabilistic normed spaces; 13. Non-Archimedean RN [Random Normed]-spaces; 14. Intuitionistic random normed spaces; 15. Intuitionistic fuzzy normed spaces; 16. intuitionistic fuzzy Banach algebras; 17. Intuitionistic Non-Archimedean fuzzy normed spaces; 18. Menger normed spaces; 19. Menger probabilistic normed spaces; 20. Non-Archimedean Menger normed spaces; 21. Intuitionistic Menger normed spaces; 22. L-non-Archimedean- fuzzy Euclidean normed spaces; 23. F -spaces; Fr´echet spaces; 24. Banach modules; 25. Distributions and Hyperfunctions; as well as, on: 26. Restricted domains; 27. Heisenberg groups. OPEN PROBLEM B. 28. Exploiting the elementary “M. Hosszu’s method”, due to M. Hosszu (see: “On the Fr´echet’s functional equation””, Bull. Inst. Politech. Iasi 10, (1964), 1-2, 27-28), determine the general solution and the Ulam stability of each one of these quartic functional equations. The advantage of this method is that we do not assume any regularity conditions on the unknown function f. See also the paper of the authors Xu et al.: [31]. 29. Employing two “L. Szekelyhidi’s fundamental results”, due to L. Szekelyhidi (see: “Convolution type functional equation on topological abelian groups”, World Scientific, Singapore, 1991), determine the general solution and the Ulam stability of each one of these quartic functional equations in certain types of groups, such as, “commutative groups”. See: [31]. OPEN PROBLEM C. 30. Exploiting the elementary “M. Hosszu’s method”, due to M. Hosszu (see: “On the Fr´echet’s functional equation”, Bull. Inst. Politech. Iasi 10, (1964), 1-2, 27-28 ), determine the general solution and the Ulam stability of each one of these pertinent “Pexider quartic” functional equations “with or without involution” [32]. We do not assume any regularity conditions on the unknown functions. For “Pexider quartic” equations [31]: (1)

f(ax + y) + f(x + ay) = g(x + y) + g¯(x − y) + h(x) + ¯h(y),

(2)

f(ax + by) + f(bx + ay) = g(x + y) + g¯(x − y) + h(x) + ¯h(y),

¯ g, g¯, h, ¯h. See also the papers of the with fixed integers a, b 6= 0, ±1 and unkown functions f, f, author, et al.: ( [31, 33]). 31. Employing two “L. Szekelyhidi’s fundamental results”, due to L. Szekelyhidi (see: “Convolution type functional equation on topological abelian groups”, World Scientific, Singapore, 1991), determine the general solution and the Ulam stability of each one of these pertinent “Pexider quartic” functional equations with or without “involution” in certain types of groups, such as, “commutative groups”. See: [31, 33]. 32. Investigate Ulam-Hyers stabilities of pertinent “quartic derivations” from a Banach algebra into its Banach modules. See: ( [34, 35]). 966

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33. Establish the solution and stabilities of “conditionally” quartic functional equations, for instance: kxk = kyk. 34. Prove stabilities of “orthogonally” quartic functional equations, “in the sense of J. R¨ atz” [36]: “x ⊥ y ⇐⇒ hx, yi = 0”. 35. Work on stabilities of “quartic-like” functional equations, such as: 
 1. 2 f1 (ax + y) + f2 x + ay + σ(y) + a(a − 1)2 g1 (x − y)   = 2(a2 − 1)2 h1 (x) + h2 (y) + a(a + 1)2 g2 (x + y), σ = σ(y) is “involution”: σ(x + y) = σ(x) + σ(y); σ(σ(x)) = x.

2.

2f(ax + y + ρ) + f(x + ay + τ ) + f(ax − y) =

 a 2 (a + 4a + 1)f(x + y) − (a2 − 4a + 1)f(x − y) 2 +(3a4 − 4a2 + 1)f(x) + (a4 − 4a2 + 3)f(y).

Acknowledgement. The authors gratefully acknowledge the financial support from King Abdulaziz University, Jeddah, Saudi Arabia.

References [1] S.M. Ulam, A Collection of Mathematical Problems, Interscience Tracts in Pure and Applied Mathematics, No.8, Interscience. Publ., New York , 1960; “Problems in Modern Mathematics”, Ch. VI, Science Ed., Wiley, 1940. [2] D.H. Hyers, On the stability of the linear functional equation, Proc. Nat. Acad. Sci., USA, 27 (1941) 222-224. [3] J.M. Rassias, On the stability of the Euler-Lagrange functional equation, C. R. Acad. Bulgare Sci. 45 (1992) 17-20. [4] J.M. Rassias, On the stability of the Euler-Lagrange functional equation, Chinese J. Math. 20 (1992) 185-190. [5] J.M. Rassias, Solution of a problem of Ulam, J. Approx. Theory 57 (1989) 268-273. [6] J.M. Rassias, On a new approximation of approximately linear mappings by linear mappings, Discuss, Math. 7 (1985) 193-196. [7] J.M. Rassias, On approximation of approximately linear mappings by linear mappings, J. Funct. Anal. 46 (1982) 126-130. [8] T. Aoki, On the stability of the linear transformation in Banach spaces, J. Math. Soc. Japan, 2 (1950) 64-66. [9] T.M. Rassias, On the stability of the linear mapping in Banach spaces, Proc. Am. Math. Soc. 72 (1978) 297-300. [10] Z. Gajda, On stability of additive mappings, Int. J. Math. Math. Sci. 14 (1991) 431-434. ˇ [11] T.M. Rassias, P. Semrl, On the behaviour of mappings which do not satisfy Hyers-Ulam stability, Proc. Am. Math. Soc. 114 (1992) 989-993. [12] P. Gˇ avruta, A generalization of the Hyers-Ulam-Rassias stability of approximately additive mappings, J. Math. Anal. Appl. 184 (1994) 431-436. [13] S.-M. Jung, On the Hyers-Ulam stability of the functional equations that have the quadratic property, J. Math. Anal. Appl. 222 (1998) 126-137. [14] K.W. Jun, H.M. Kim, The generalized Hyers-Ulam-Rassias stability of a cubic functional equation, J. Math. Anal. Appl. 274 (2002) 867-878. [15] R.P. Agarwal, B. Xu, W. Zhang, Stability of functional equations in single variable, J. Math. Anal. Appl. 288 (2003) 852-869 [16] S.A. Mohiuddine, H. Sevli, Stability of Pexiderized quadratic functional equation in intuitionistic fuzzy normed space, J. Comp. Appl. Math., 235 (2011) 2137-2146. [17] S.A. Mohiuddine, Stability of Jensen functional equation in intuitionistic fuzzy normed space, Chaos, Solitons Fract., 42 (2009) 2989-2996. [18] M. Mursaleen, S.A. Mohiuddine, On stability of a cubic functional equation in intuitionistic fuzzy normed spaces, Chaos, Solitons Fract., 42 (2009) 2997-3005. [19] S.A. Mohiuddine, A. Alotaibi, Fuzzy stability of of a cubic functional equation via fixed point technique, Adv. Difference Equ., 2012 , 2012:48. [20] J.M. Rassias, Solution of the Ulam stability problem for quartic mappings, Glas. Mat. Ser. III 34(54) (1999) 243-252. 967

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[21] S.-M. Jung, Hyers-Ulam-Rassias Stability of Functional Equations in Mathematical Analysis, Hadronic Press, Palm Harbor (2001) [22] M. Mursaleen, K.J. Ansari, Stability results in intuitionistic fuzzy normed spaces for a cubic functional equation, Appl. Math. Inf. Sci. 7(5) (2013) 1685-1692. [23] A. Najati, C. Park, On the stability of an n-dimensional functional equation originating from quadratic forms, Taiwan. J. Math. 12 (2008) 1609-1624. [24] T.M. Rassias, On the stability of functional equations and a problem of Ulam, Acta Appl. Math. 62 (2000) 123-130. [25] C. Park, Y. Cho, M. Han, Functional inequalities associated with Jordan-von Neumann-type additive functional equations. J. Inequal. Appl. 2007, Article ID 41820 (2007). [26] C. Park, J.R. Lee, D.Y. Shin, Functional equations and inequalities in matrix paranormed spaces, J. Inequal. Appl. 2013, 2013:547 [27] Z. Wang, T.M. Rassias, Intuitionistic fuzzy stability of functional equations associated with inner product spaces, Abstr. Appl. Anal. Volume 2011, Article ID 456182, 19 pages. [28] I.-S. Chang, Higher ring derivation and intuitionistic fuzzy stability, Abstr. Appl. Anal. Volume 2012, Article ID 503671, 16 pages. [29] J. Roh, I.-S. Chang, On the intuitionistic fuzzy stability of ring homomorphism and ring derivation, Abstr. Appl. Anal. Volume 2013, Article ID 192845, 8 pages. [30] M. Mirmostafaee, M.S. Moslehian, Fuzzy versions of Hyers-Ulam-Rassias theorem, Fuzzy Sets Syst. 159 (2008) 720-729. [31] T.Z. Xu, J.M. Rassias, W.X. Xu, A generalized mixed Quadratic-Quartic functional equation, Bull. Malays. Math. Sci. Soc. 35(3) (2012) 633-649. [32] J.M. Rassias, H.-M. Kim, Approximate homomorphisms and derivations between C ∗ -ternary algebras. J. Math. Phys. 49 (2008), no. 6, 063507, 10 pp. [33] M.M. Pourpasha, J.M. Rassias, R. Saadati, S.M. Vaezpour, A fixed point approach to the stability of Pexider quadratic functional equation with involution, J. Inequal. Appl. 2010, Art. ID 839639, 18 pp. [34] M.E. Gordji, N. Ghobadipour, Generalized Ulam-Hyers stabilities of quartic derivations on Banach algebras, Proyecciones J. Math., 29 (2010) 209-226. [35] C. Park, J.M. Rassias, Cubic derivations and quartic derivations on Banach modules, in: “Functional Equations, Difference Inequalities and Ulam Stability Notions” (F. U. N.), Editor: J.M. Rassias, 2010, 119-129, ISBN 978-1-60876-461-7, Nova Science Publishers, Inc. [36] J. R¨ atz, On the orthogonal additive mappings, Aequationes Math. 28 (1985) 35-49.

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Existence of positive solutions for summation boundary value problem for a fourth-order difference equations Thanin Sitthiwirattham Department of Mathematics, Faculty of Applied Science, King Mongkut’s University of Technology North Bangkok, Bangkok 10800, Thailand E-mail address: [email protected]

Abstract In this paper, we study the existence of positive solutions to the differencesummation boundary value problem ∆4 u(t − 2) + a(t)f (u) = 0, 2

u(0) = ∆u(0) = ∆ u(0) = 0,

t ∈ {2, 3, ..., T }, u(T + 2) = α

η ∑

u(s),

s=4

where f is continuous, T ≥ 5 is a fixed positive integer, η ∈ {4, 5, ..., T − 1}, 4T (T +1)(T +2) 0 < α < (η−3)(η+2)(η 2 −η+4) . We show the existence of at least one positive solution if f is either superlinear or sublinear by applying Guo–Krasnoselskii fixed point theorem in cones.

Keywords : Positive solution; Boundary value problem; Fixed point theorem; Cone 2010 Mathematics Subject Classification: 39A15, 34B15

1

Introduction

The existence of solutions for boundary value problems of difference equations has received much attention. For example, see [4-15] and the references therein. 1

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Thanin Sitthiwirattham

Liang et al. in [4] considered the fourth-order boundary value problem of the form  ∆4 x(t − 2) + a(t)f (x) = 0, t ∈ {2, 3, ..., T }, x(0) = x(T + 2) = 0, ∆2 x(0) = ∆2 x(T ) = 0, where T > 2. Existence and uniqueness of solutions are obtained by a fixed point theorem. Ma et al. in [5] considered the fourth-order boundary value problem of the form  ∆4 u(t − 2) − λf (t, u(t)) = 0, t ∈ {2, 3, ..., T }, u(1) = u(T + 1) = ∆2 u(0) = ∆2 u(T ) = 0, where λ is a parameter, T > 5. Existence and uniqueness of solutions are obtained by the theory of fixed-point index in cones. In this paper, we consider the existence of positive solutions to the equation ∆4 u(t − 2) + a(t)f (u) = 0,

t ∈ {2, 3, ..., T },

(1.1)

with summation boundary condition 2

u(0) = ∆u(0) = ∆ u(0) = 0,

u(T + 2) = α

η ∑

u(s),

(1.2)

s=4

where f is continuous. The aim of this paper is to give some results for existence of positive solutions to (1.1)-(1.2). Let N be the nonnegative integer, we let Ni,j = {k ∈ N| i ≤ k ≤ j} and Np = N0,p . By the positive solution of (1.1)-(1.2) we mean that a function u(t) : NT +2 → [0, ∞) and satisfies the problem (1.1)-(1.2). Throughout this paper, we suppose the following conditions hold: (H1) T ≥ 5 is a fixed positive integer, η ∈ {4, 5, ..., T − 1}, constant α > 0 such that 4T (T +1)(T +2) 0 < α < (η−3)(η+2)(η 2 −η+4) . (H2) f ∈ C([0, ∞), [0, ∞)), f is either superlinear or sublinear. Set f0 = lim

u→0+

f (u) , u

f (u) . u→∞ u

f∞ = lim

Then f0 = 0 and f∞ = ∞ correspond to the superlinear case, and f0 = ∞ and f∞ = 0 correspond to the sublinear case. (H3) a ∈ C(N2,T , [0, ∞)), a is not identical zero.

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3

The proof of the main theorem is based upon an application of the following Guo-Krasnoselskii’s fixed point theorem in a cone. Theorem 1.1. Let E be a Banach space, and let K ⊂ E be a cone. Assume Ω1 , Ω2 are open subsets of E with 0 ∈ Ω1 , Ω1 ⊂ Ω2 , and let A : K ∩ (Ω2 \ Ω1 ) −→ K be a completely continuous operator such that (i) ∥Au∥ 6 ∥u∥, u ∈ K ∩ ∂Ω1 , and ∥Au∥ > ∥u∥, u ∈ K ∩ ∂Ω2 ; or (ii) ∥Au∥ > ∥u∥, u ∈ K ∩ ∂Ω1 , and ∥Au∥ 6 ∥u∥, u ∈ K ∩ ∂Ω2 . Then A has a fixed point in K ∩ (Ω2 \ Ω1 ).

2

Preliminaries We now state and prove several lemmas before stating our main results.

Lemma 2.1. Suppose that y(t) ∈ C(N2,T ) and y(t) > 0. Then the linear boundary value problem ∆4 u(t − 2) + y(t) = 0,

2

u(0) = ∆u(0) = ∆ u(0) = 0,

t ∈ N2,T ,

u(T + 2) = α

(2.1) η ∑

u(s),

(2.2)

s=4

has a unique solution u(t) =

[ T ] η−2 t3 ∑ α∑ (T − s + 3)3 y(s) − (η − s + 2)4 y(s) 6Λ 4 s=2

−

t−2 1∑

6

s=2

(t − s + 1)3 y(s),

(2.3)

s=2

where Λ := T (T + 1)(T + 2) −

α (η − 3)(η + 2)(η 2 − η + 4). 4

(2.4)

Proof. In fact, if u(t) is a solution of problem (2.1), by the discrete Taylor expansion formula, we have 1∑ (t − s − 1)3 y(s + 2), 6 t−4

u(t) = C1 t3 + C2 t2 + C3 t1 + C4 −

t ∈ NT +2 .

s=0

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Applying the first boundary condition u(0) = ∆u(0) = ∆2 u(0) = 0 in (2.2), we obtain C2 = C3 = C4 = 0. So, 1∑ (t − s − 1)3 y(s + 2), 6 t−4

u(t) = C1 t3 −

(2.5)

s=0

From (2.5) and the second boundary condition in (2.2) implies α

η ∑

u(s) = αC1

s=4

η ∑

= αC1

s−4

s=4 ξ=0

s=4 η−4 ∑

α ∑∑ (s − ξ − 1)3 y(ξ + 2) 6 η

(s)3 −

(s + 4)3 −

s=0

= C1 (T + 2)3 −

η−4 η−s−4 α∑ ∑ (ξ + 3)3 y(s + 2) 6 s=0 ξ=0

1 6

T −2 ∑

(T − s + 1)3 y(s + 2).

s=0

Solving the above equation for a constant C1 , we get η−4 η−s−4 T −2 1 ∑ α ∑ ∑ 3 C1 = (T − s + 1) y(s + 2) − (ξ + 3)3 y(s + 2) 6Λ 6Λ s=0

s=0 ξ=0

where Λ is defined by (2.4) Therefore, (2.1)-(2.2) has a unique solution [ T ] η−2 t3 ∑ α∑ 3 4 u(t) = (T − s + 3) y(s) − (η − s + 2) y(s) 6Λ 4 −

1 6

s=2 t−2 ∑

s=2

(t − s + 1)3 y(s).

s=2

 Lemma 2.2. The function    −Λ(t − s + 1)3 + t3 (T − s + 3)3 −     1 −Λ(t − s + 1)3 + t3 (T − s + 3)3 , G(t, s) = 3 6Λ   t3 (T − s + 3)3 − αt4 (η − s + 2)4 ,     t3 (T − s + 3)3 ,

972

αt3 4 (η

− s + 2)4 ,

s ∈ N2,t−2 ∩ N2,η−2 s ∈ Nη−1,t−2 s ∈ Nt−1,η−2 s ∈ Nt−1,T ∩ Nη−1,T (2.6)

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Existence of positive solutions for summation boundary value problem ...

5

where Λ is defined by (2.4), is the Green’s function of the problem

− ∆4 u(t − 2) = 0,

t ∈ N2,T ,

2

u(0) = ∆u(0) = ∆ u(0) = 0,

u(T + 2) = α

η ∑

u(s).

(2.7)

s=4

Proof. Suppose t < η. The unique solution of problem (2.1)-(2.2) can be written 1∑ (t − s + 1)3 y(s) 6 s=2 [ t−2 ] η−2 T ∑ ∑ t3 ∑ 3 3 3 + (T − s + 3) y(s) + (T − s + 3) y(s) + (T − s + 3) y(s) 6Λ t−2

u(t) = −

s=2

−

=

[ t−2 3 ∑

αt 24Λ

(η − s + 2)4 y(s) +

s=2

]

η−2 ∑

s=η−1

(η − s + 2)4 y(s)

s=t−1

] αt3 (η − s + 2)4 y(s) 4 s=2 ] η−2 [ 1 ∑ 3 αt3 3 4 + (η − s + 2) y(s) t (T − s + 3) − 6Λ 4

1 6Λ

t−2 [ ∑

s=t−1

− Λ(t − s + 1)3 + t3 (T − s + 3)3 −

s=t−1

T 1 ∑ 3 t (T − s + 3)3 y(s) + 6Λ s=η−1

=

T ∑

G(t, s)y(s).

s=2

Suppose t ≥ η. The unique solution of problem (2.1)-(2.2) can be written [ η−2 ] t−2 ∑ 1 ∑ 3 3 u(t) = − (t − s + 1) y(s) + (t − s + 1) y(s) 6 s=2

t3 + 6Λ

[∑ η−2

s=η−1

(T − s + 3) y(s) + 3

s=2

t−2 ∑

(T − s + 3) y(s) +

s=η−1

3

T ∑

] (T − s + 3) y(s) 3

s=t−1

αt3 ∑ − (η − s + 2)4 y(s) 24Λ η−2

s=2

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Thanin Sitthiwirattham ] η−2 [ 1 ∑ αt3 3 3 3 4 = − Λ(t − s + 1) + t (T − s + 3) − (η − s + 2) y(s) 6Λ 4 s=2 ] t−2 [ 1 ∑ 3 3 3 − Λ(t − s + 1) + t (T − s + 3) y(s) + 6Λ s=η−1

T 1 ∑ 3 + t (T − s + 3)3 y(s) 6Λ s=t−1

=

T ∑

G(t, s)y(s).

s=2

Then the unique solution of problem (2.1)-(2.2) can be written as u(t) = T ∑



G(t, s)y(s). The proof is complete.

s=2

We observe that the condition 0 < α < positive on N2,T × N2,T . {

Let M1 = min

{ M2 = max

4T (T +1)(T +2) . (η−3)(η+2)(η 2 −η+4)

G(t, s) : t ∈ N2,T , s ∈ N2,T G(t, t)

implies G(t, s) is

}

G(t, s) : t ∈ NT +2 , s ∈ N2,T G(t, t)

(2.8) } (2.9)

Lemma 2.3. Let (t, s) ∈ N2,T × N2,T . Then we have G(t, s) ≥ M1 G(t, t.)

(2.10)

where 0 < M1 < 1 is a constant given by ( ) { 4(T − η + 7)3 − 4Λ η−5 − αη 4 η−2 4(T − η + 5)3 − α(η + 1)4 , , M1 = min 4(T + 1)3 − 24α 4(T + 1)3 − 24α ( ) 4(T − η − 5)3 − 4Λ T T−3 − αη 4 6 , , 4(T + 1)3 − 24α 4(T − η + 4)3 } 60(η − 1)3 − Λ(T − η + 2)3 6 , (2.11) T 3 (T − η + 4)3 (T − η + 4)3 Proof. In order that 2.10 holds, it is sufficient that M1 satisfies M1 ≤

min

(t,s)∈N2,T ×N2,T

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G(t, s) . G(t, t)

(2.12)

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7

Then we may choose { M1 ≤ min

min

(t,s)∈N2,η−2 ×N2,T

G(t, s) G(t, s) , min G(t, t) (t,s)∈Nη−1,T ×N2,T G(t, t)

} .

(2.13)

since min

(t,s)∈N2,η−2 ×N2,T

{

= min

G(t, s) G(t, t)

min

t∈N2,η−2

t3 (T − t + 3)3 −

s∈N2,t−2

min

s∈Nt−1,η−2

min

s∈Nη−1,T

{ ≥ min

αt3 4 (η 3 αt − t + 3)3 − 4 (η t3 (T − s + 3)3

αt3 4 (η

−

t3 (T − s + 3)3 −

− s + 2)4

t3 (T

− t + 2)4 }

s + 2)4

,

,

αt3 4 4 (η − t + 2) 3 5)3 − αt4 η 4 (T − η + 5)3 − α4 (η − t + 3)4 , , (T − t + 3)3 − α4 (η − t + 2)4 − t + 2)4

t3 (T − t + 3)3 −

−Λ(t − 1)3 + t3 (T − t +

t∈N2,η−2

αt3 4 (η − t + 2)4

−Λ(t − s + 1)3 + t3 (T − s + 3)3 −

t3 (T − t + 3)3 −

αt3 4 (η

} 33 (T − t + 3)3 − α4 (η − t + 2)4 ) ( { −Λ 1 − 3 + (T − η + 7)3 − α (η)4 } (T − η + 5)3 − α4 (η + 1)4 η−2 4 6 ≥ min , , 4 4 4 (T + 1)3 − α44 (T + 1)3 − α44 (T + 1)3 − α44 ( ) { 4(T − η + 7)3 − 4Λ η−5 − αη 4 } η−2 4(T − η + 5)3 − α(η + 1)4 6 = min , , 4(T + 1)3 − 24α 4(T + 1)3 − 24α 4(T + 1)3 − 24α (2.14) Similarly, we get min

(t,s)∈Nη−1,T ×N2,T

{

≥ min

G(t, s) G(t, t)

( ) } 4(T − η − 5)3 − 4Λ T T−3 − αη 4 60(η − 1)3 − Λ(T − η + 2)3 6 , , 4(T − η + 4)3 T 3 (T − η + 4)3 (T − η + 4)3 (2.15)

The (2.11) is immediate from (2.14)-(2.15)



Lemma 2.4. Let (t, s) ∈ NT +2 × N2,T . Then we have G(t, s) ≤ M2 G(t, t.)

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(2.16)

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where M2 ≥ 1 is a constant given by { 24Λ + (η − 2)3 (T + 1)3 4(T − η + 6)3 − 24α 4(T − η + 4)3 , , , M2 = max α(η − 2)3 η 4 4(T − η + 6)3 − αη 4 4(T − η + 5)3 − αη 4 } 4T 3 − 24α(η − 1)3 T 3 (T − η + 4)3 − 6Λ (T − η + 5)3 , , (2.17) 6(η − 1)3 6(η − 1)3 6 Proof. For t = 0, 1, from (2.6) we get G(0, s) = G(0, 0) = 0; G(1, s) = G(1, 1) = 0. Then we may choose M2 = 1. For t ∈ N2,T , if 2.16 holds, it is sufficient that M2 satisfies G(t, s) M2 ≥ max . (2.18) (t,s)∈N2,T ×N2,T G(t, t) Then we may choose { M2 ≥ max

max

(t,s)∈N2,η−2 ×N2,T

G(t, s) G(t, s) , max G(t, t) (t,s)∈Nη−1,T ×N2,T G(t, t)

} .

(2.19)

since max

(t,s)∈N2,η−2 ×N2,T

{

= max

G(t, s) G(t, t)

t∈N2,η−2

t3 (T − t + 3)3 −

s∈N2,t−2

max

s∈Nt−1,η−2

max

s∈Nη−1,T

{ ≤ max

{ ≤ max

αt3 4 (η 3 − t + 3)3 − αt4 (η t3 (T − s + 3)3

αt3 4 (η

−

t3 (T − s + 3)3 −

− s + 2)4

t3 (T

− t + 2)4 }

t3 (T − t + 3)3 −

−Λ33 + t3 (T + 1)3 − t3 (T − t + 3)3 −

t∈N2,η−2

αt3 4 (η − t + 2)4

−Λ(t − s + 1)3 + t3 (T − s + 3)3 −

max

(T − η + 4)3 (T − t + 3)3 − α4 η 4

αt3 4 (η

αt3 4 (η

}

αt3 4 (η

,

,

− t + 2)4

− t + 4)4

− t + 2)4

s + 2)4

4

,

(T − t + 4)3 − α44 , (T − t + 3)3 − α4 (η − t + 2)4

} 24Λ + (η − 2)3 (T + 1)3 4(T − η + 6)3 − 24α 4(T − η + 4)3 , , α(η − 2)3 η 4 4(T − η + 6)3 − αη 4 4(T − η + 5)3 − αη 4 (2.20)

Similarly, we get max

(t,s)∈Nη−1,T ×N2,T

{

≤ max

G(t, s) G(t, t)

4T 3 − 24α(η − 1)3 T 3 (T − η + 4)3 − 6Λ (T − η + 5)3 , , 6(η − 1)3 6(η − 1)3 6

976

} (2.21)

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For t = T + 1, T + 2 from (2.6) we get ( )] 1 [ α G(T + 1, s) < −Λ(T − s + 2)3 + (T + 1)3 (T − s + 3)3 − (η − s + 2)4 6Λ 4 α 3 4 =− (T + 1) (η − s + 2) 24Λ α 0 so that f (u) 6 ϵ1 u, for 0 < u 6 H1 , where ϵ1 > 0 satisfies ϵ1 M2 M

T ∑

a(s) ≤ 1.

(3.4)

s=2

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Thus, if we let Ω1 = {u ∈ E : ∥u∥ < H1 }, then for u ∈ K ∩ ∂Ω1 , we get (Au)(t) ≤M2

T ∑

G(t, t)a(s)f (u(s)) ≤ ϵ1 M2 M

s=2

≤ϵ1 M2 M

T ∑

a(s)u(s)

s=2 T ∑

a(s)∥u∥ ≤ ∥u∥.

s=2

Thus ∥Au∥ ≤ ∥u∥, u ∈ K ∩ ∂Ω1 . b 2 > 0 such that f (u) ≥ ϵ2 u, for u ≥ H b2, Further, since f∞ = ∞, there exists H where ϵ2 > 0 satisfies ϵ2 M 1 σ

T ∑

G(η − 1, η − 1)a(s) ≥ 1.

(3.5)

s=η−1 b

Let H2 = max{2H1 , Hσ2 } and Ω2 = {u ∈ E : ∥u∥ < H2 }. Then u ∈ K ∩ ∂Ω2 implies b2. min u(t) ≥ σ∥u∥ ≥ H t∈Nη−1,T

Applying (2.8) and (3.5), we get (Au)(η − 1) =M1

T ∑

G(η − 1, s)a(s)f (u(s)) ≥ M1

s=2

≥ε2 M1

T ∑

G(η − 1, η − 1)a(s)f (u(s))

s=η−1

T ∑

T ∑

G(η − 1, η − 1)a(s)y(s) ≥ ε2 M1 σ

s=η−1

G(η − 1, η − 1)a(s)∥u∥

s=η−1

≥∥u∥. Hence, ∥Au∥ ≥ ∥u∥, u ∈ K ∩ ∂Ω2 . By the first part of Theorem 1.1, A has a fixed point in K ∩ (Ω2 \ Ω1 ) such that H1 6 ∥u∥ 6 H2 . Sublinear case. f0 = ∞ and f∞ = 0. Since f0 = ∞, choose H3 > 0 such that f (u) > ϵ3 u for 0 < u 6 H3 , where ε3 > 0 satisfies ϵ3 M 1 σ

T ∑

G(η − 1, η − 1)a(s) > 1.

(3.6)

s=η−1

Let Ω3 = {u ∈ E : ∥u∥ < H3 },

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then for u ∈ K ∩ ∂Ω3 , we get (Au)(η − 1) ≥M1

T ∑

G(η − 1, η − 1)a(s)f (u(s)) ≥ ϵ3 M1

s=η−1

≥ε3 M1 σ

T ∑

G(η − 1, η − 1)a(s)y(s)

s=η−1

T ∑

G(η − 1, η − 1)a(s)∥u∥ ≥ ∥u∥.

s=η−1

Thus, ∥Au∥ > ∥u∥, u ∈ K ∩ ∂Ω3 . b 4 > 0 so that f (u) 6 ϵ4 u for u > H b 4 , where Now, since f∞ = 0, there exists H ϵ4 > 0 satisfies T ∑ ϵ4 M2 M a(s) > 1. (3.7) s=η−1

Subcase 1. Suppose f is bounded, f (u) ≤ L for all u ∈ [0, ∞) for some L > 0. T ∑ Let H4 = max{2H3 , LM2 M a(s)}. s=1

Then for u ∈ K and ∥u∥ = H4 , we get (Au)(η) ≤M2

T ∑

G(t, t)a(s)f (u(s)) ≤ LM2 M

s=2

T ∑

a(s)

s=2

≤H4 = ∥u∥ Thus (Au)(t) ≤ ∥u∥. b

Subcase 2. Suppose f is unbounded, there exist H4 > max{2H3 , Hσ4 } such that f (u) ≤ f (H4 ) for all 0 < u ≤ H4 . Then for u ∈ K with ∥u∥ = H4 from (2.9) and (3.7), we have (Au)(t) ≤M2

T ∑

G(t, t)a(s)f (u(s)) ≤ M2 M

s=2

≤ϵ4 M2 M

T ∑

a(s)f (H4 )

s=2 T ∑

a(s)H4 ≤ H4 = ∥u∥.

s=2

Thus in both cases, we may put Ω4 = {u ∈ E : ∥u∥ < H4 }. Then ∥Au∥ 6 ∥u∥, u ∈ K ∩ ∂Ω4 . By the second part of Theorem 1.1, A has a fixed point u in K ∩ (Ω4 \ Ω3 ), such that H3 6 ∥u∥ 6 H4 . This completes the sublinear part of the theorem. Therefore, the problem (1.1)-(1.2) has at least one positive solution. 

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Existence of positive solutions for summation boundary value problem ...

4

13

Some examples In this section, in order to illustrate our result, we consider some examples.

Example 4.1

Consider the BVP ∆4 u(t − 2) + t2 uk = 0,

t ∈ N2,6 ,

(4.1)

2∑ u(s). u(8) = 3 5

2

u(0) = ∆u(0) = ∆ u(0) = 0,

(4.2)

s=4

Set α = 32 , η = 5, T = 6, a(t) = t2 , f (u) = uk . We can show that T (T + 1)(T + 2) −

α (η − 3)(η + 2)(η 2 − η + 4) = 280 > 0. 4

Case I : k ∈ (1, ∞). In this case, f0 = 0, f∞ = ∞ and (i) of theorem 3.1 holds. Then BVP (4.1)-(4.2) has at least one positive solution. Case II : k ∈ (0, 1). In this case, f0 = ∞, f∞ = 0 and (ii) of theorem 3.1 holds. Then BVP (4.1)-(4.2) has at least one positive solution.

Example 4.2

Consider the BVP

∆4 u(t − 2) + et te (

π sin u + 2 cos u ) = 0, u2

t ∈ N2,8 ,

1∑ u(10) = u(s), 3

(4.3)

6

2

u(0) = ∆u(0) = ∆ u(0) = 0,

(4.4)

s=4

Set α = 31 , η = 6, T = 8, a(t) = et te , f (u) = We can show that T (T + 1)(T + 2) −

π sin u+2 cos u . u2

α (η − 3)(η + 2)(η 2 − η + 4) = 596 > 0. 4

Through a simple calculation we can get f0 = ∞, f∞ = 0. Thus, by (ii) of theorem 3.1, we can get BVP (4.3)-(4.4) has at least one positive solution.

Acknowledgement(s) : This research (KMUTNB-GOV-57-07) is supported by King Mongkut’s University of Technology North Bangkok, Thailand.

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References [1] R.P. Agarwal, Focal Boundary Value Problems for Differential and Difference Equations, Kluwer Academic Publishers, Dordrecht, 1998. [2] R.P. Agarwal, D. O’Regan, P.J.Y. Wong, Positive Solutions of Differential, Difference and Integral Equations, Kluwer Academic Publishers, Dordrecht, 1999. [3] M.A. Krasnoselskii, Positive Solutions of Operator Equations, Noordhoof, Gronignen, 1964. [4] R.J. Liang, Y.H. Zhao, J.P. Sun, A new theoremof existance to fourth-order boundary value problem, International Journal of Differential Equations and Applications, 7 (2003) 257-262 [5] R.Ma, Y. Xu, Existence of positive solution for nonlinear fourth-order difference equations, Computers and Mathematics with Applications 59 (2010) 3770-3777. [6] Z.M. He, J.S. Yu, On the existence of positive solutions of fourth-order difference equations, Appl. Math. Comput. 161 (2005) 139-148. [7] B.G. Zhang, L.J. Kong, Y.J. Sun, X.H. Deng, Existence of positive solutions for BVPs of fourth-order difference equations, Appl. Math. Comput. 131 (2002). 583-591. [8] Q.L. Yao, Existence, multiplicity and infinite solvability of positive solutions to a nonlinear fourth-order periodic boundary value problem, Nonlinear Anal. 63 (2005) 237-246. [9] F.Y. Li, Q. Zhang, Z.P. Liang, Existence and multiplicity of solutions of a kind of fourth-order boundary value problem, Nonlinear Anal. 62 (2005). 803-816. [10] X.L. Liu, W.T. Li, Existence and multiplicity of solutions for fourth-order boundary value problems with parameters, J. Math. Anal. Appl. 327 (2007). 362-375. [11] Y. Yang, J.H. Zhang, Existence of solutions for some fourth-order boundary value problems with parameters, Nonlinear Anal. 69 (2008) 136-1375. [12] G.D. Han, Z.B. Xu, Multiple solutions of some nonlinear fourth-order beam equations, Nonlinear Anal. 68 (2008) 3646-3656.

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[13] 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. [14] J.Reunsumrit, T. Sitthiwirattham, Positive solutions of difference-summation boundary value problem for a second-order difference equation, Journal of Computational Analysis and Applications. (In press) [15] T. Sitthiwirattham, J. Tariboon, Positive Solutions to a Generalized Second Order Difference Equation with Summation Boundary Value Problem. Journal of Applied Mathematics. Vol.2012, Article ID 569313, 15 pages.

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Similarity measure between generalized intuitionistic fuzzy sets and its application to pattern recognition Jin Han Park∗ Department of Applied Mathematics, Pukyong National University Busan 608-737, South Korea [email protected] Jongchul Hwang, Juhyung Kim, Byeongmuk Park, Juyoung Park, Jeongwoo Son, Sihun Lee Busanil Science High School, Busan 660-756, South Korea [email protected] (J. Hwang), [email protected] (J. Kim), [email protected] (B. Park), [email protected] (J.Park), [email protected] (J. Son), [email protected] (S. Lee)

Abstract This paper presents new methods for measuring similarity between generalized intuitionistic fuzzy sets (GIFSs) and its application to pattern recognition. Firstly, the geometrical interpretation of GIFSs is carefully reviewed and then the results of the interpretation is utilized to generate new methods for measuring similarity in order to calculate the degree of similarity between GIFSs. Numerical example is given to illustrate the application of the proposed similarity measures. Finally, we also use the proposed similarity measures to characterize the similarity between linguistic variables.

1

Introduction

As a generalization of fuzzy sets, intuitionistic fuzzy sets (IFSs) were presented by Atanassov [1, 2, 3]. Since IFSs can present the degrees of membership and non-membership with a degree of hesitancy, the knowledge and semantic representation become more meaningful and applicable. These IFSs have been widely studied and applied in various areas, such as logic programming [4], decision making [7, 22], pattern recognition [12, 14, 15, 16, 19, 26] and medical diagnosis ∗ Corresponding author: [email protected] (J.H. Park) This work was supported by a Research Grant of Pukyong National University (2014).

1

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[9, 25], and seem to have more popular than fuzzy sets technology. Mondal and Samanta [18] introduced generalized intuitionistic fuzzy sets (GIFSs) as a generalization of IFSs and studied their basic properties. Park et al. [20] proposed a method to calculate the correlation coefficient of GIFSs. There is a little investigation on GIFSs. Similarity assessment plays a fundamental and important role in inference and approximate reasoning in all applications of intuitionistic fuzzy logic [4]. For different purposes different similarity measures should be used. Based on the importance of the problem, the effectiveness and properties of the different similarity measures for IFSs have been compared and examined by many researchers (e.g. Hung and Yang [12], Li and Cheng [14], Li et al. [15], Liang and Shi [16], Mitchell [19], Szmidt and Baldwin [23]). The analysis of similarity is also a fundamental issue while employing GIFSs. Recently, Park et al. [21] proposed and applied similarity measure to compare generalized intuitionistic fuzzy preferences given by individuals (experts) and evaluated an extent of a group agreement. In this paper, we propose new similarity measures based on the geometrical representation for GIFS. The proposed similarity measures depend on the triplet, membership degree, nonmembership degree, and hesitation margin. This paper proves that the proposed similarity measures satisfy the properties of axiomatic definition for similarity measures. Numerical example is given to illustrate the application of the developed similarity measures. Furthermore, we use the proposed similarity measures to characterize the similarity between linguistic variables.

2

Brief introduction of GIFSs

In the following, we firstly recall basic notions and definitions of GIFSs which can be found in [18]. Let X be the universe of discourse. A generalized intuitionistic fuzzy set (GIFS) A in X is an object having the form A = {(x, µA (x), νA(x)) | x ∈ X}

(1)

where µA , νA : X → [0, 1] denote membership function and non-membership function, respectively, of A and satisfy min{µA (x), νA(x)} ≤ 0.5 for all x ∈ X. Let GIFS(X) denote the set of all GIFSs in X. For an IFS A = {(x, µA(x), νA(x)) | x ∈ X}, it is observed that µA (x) + νA (x) ≤ 1 implies min{µA(x), νA(x)} ≤ 0.5 for each x ∈ X. Thus, every IFS is GIFS. For each GIFS A in X, we call φA (x) = 1 − µA (x) − νA(x)

(2)

a generalized intuitionistic fuzzy index (or a hesitation margin) of x in A and it expresses a lack/excess of knowledge of whether x belongs to A or not. (see, [20]). It is obvious that −0.5 ≤ φA(x) ≤ 1 for each x ∈ X. 2

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Having in mind that for each element x belonging to a GIFS A, the values of membership, non-membership and generalized intuitionistic fuzzy index add up to one, i.e. µA (x) + νA (x) + φA(x) = 1

(3)

and that each of the membership and non-membership are from [0, 1] and the generalized intuitionistic fuzzy index is from [−0.5, 1], we can imagine a cuboid (Figure 1) inside which there is a polygon ADBEGF where the above equation is fulfilled. In other words, the polygon ADBEGF represents a surface where coordinates of any element belonging to a GIFS can be represented. Each point belonging to the polygon ADBEGF is described via three coordinates: (µ, ν, φ). Points A and B represent crisp elements. Point A(1, 0, 0) represents elements fully belonging to a GIFS as µ = 1. Point B(0, 1, 0) represents elements fully not belonging to a GIFS as ν = 1. Point D(0, 0, 1) represents element about which we are not able to say if they belong or not belong to a GIFS (generalized intuitionistic fuzzy index φ = 1). Point E(1, 0.5, −0.5) represents element about which we can say to belong to a GIFS (φ = −0.5). Point F (0.5, 1, −0.5) represents element about which we can say to not belong to a GIFS (φ = −0.5). Such an interpretation is intuitively appealing and provides means for the representation of many aspects of imperfect information. Segment AB (where φ = 0) represents elements belonging to classical fuzzy sets (µ + ν = 1). Triangle ADB (where 0 ≤ φ ≤ 1) represents elements belonging to IFSs (0 ≤ µ + ν ≤ 1). Any other combination of the values characterizing a GIFS can be represented inside the triangles AGF and BEG. In other words, each element belonging to a GIFS can be represented as a point (µ, ν, φ) belonging to the polygon ADBEGF (cf. Figure 1).

Figure 1: A geometrical interpretation of a GIFS

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It is worth mentioning that the geometrical interpretation is directly related to the definition of a GIFS, and it does not need any additional assumptions. By employing the above geometrical representation, a GIFS A can be expressed as A = {(µA (x), νA(x), φA(x)) | x ∈ X}.

(4)

Therefore, this representation of a GIFS will be a point of departure for considering the our method in calculating the degree of similarity between GIFSs. For A, B ∈ GIFS(X), Mondal and Samanta [18] defined the notion of containment as follows: A ⊆ B ⇔ µA (x) ≤ µB (x) and νA(x) ≥ νB (x) ∀x ∈ X.

(5)

As above-mentioned, we can not omit the third parameter (hesitancy degree) in the representation of GIFSs and then redefine the notion of containment as follows: A ⊆ B ⇔ µA (x) ≤ µB (x), νA (x) ≥ νB (x) and φA(x) ≥ φB (x) ∀x ∈ X.

(6)

Definition 2.1 Let S : GIFS(X) × GIFS(X) → [0, 1] be a mapping. S(A, B) is said to be the degree of similarity between A ∈ GIFS(X) and B ∈ GIFS(X) if S(A, B) satisfies the properties (SP1)-(SP4): (SP1) 0 ≤ S(A, B) ≤ 1; (SP2) S(A, B) = 1 if and only if A = B; (SP3) S(A, B) = S(B, A); (SP4) S(A, C) ≤ S(A, B) and S(A, C) ≤ S(B, C) if A ⊆ B ⊆ C, A, B, C ∈ GIFS(X). Definition 2.2 Let D : GIFS(X) × GIFS(X) → [0, 1] be a mapping. D(A, B) is called a distance A ∈ GIFS(X) and B ∈ GIFS(X) if D(A, B) satisfies the properties (DP1)-(DP4): (DP1) 0 ≤ D(A, B) ≤ 1; (DP2) D(A, B) = 0 if and only if A = B; (DP3) D(A, B) = D(B, A); (DP4) D(A, B) ≤ D(A, C) and D(B, C) ≤ D(A, C) if A ⊆ B ⊆ C, A, B, C ∈ GIFS(X). Because distance and similarity measures are complementary concepts, similarity measures can be used to define distance measures and vice verses.

3

New similarity measures between GIFSs

In this section, we take into account three parameters describing GIFSs to propose a new similarity measures between GIFSs based on the geometrical representation of GIFSs. 4

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Let A = {(x, µA (x), νA(x)) | x ∈ X} and B = {(x, µB (x), νB (x)) | x ∈ X} be two GIFSs in X = {x1 , x2 , . . . , xn }. We propose a new similarity measure: Sg (A, B) n

=1−

1X n i=1



|µA (xi ) − µB (xi )| + |νA (xi ) − νB (xi )| + |φA(xi ) − φB (xi )| 4

max(|µA (xi) − µB (xi )|, |νA (xi ) − νB (xi )|, |φA(xi ) − φB (xi )|) + 2



,

(7)

where φA (xi ) and φB (xi ) are, respectively, the hesitancy degree of the element xi ∈ X to the sets A and B. Theorem 3.1 Sg (A, B) is the similarity measure between two GIFSs A and B. Proof For the sake of simplicity, IFSs A and B are denoted by A = {(µA (xi ), νA (xi ), φ(xi )) | xi ∈ X} and B = {(µB (xi ), νB (xi ), φB (xi )) | xi ∈ X}, respectively. Obviously, Sg (A, B) satisfies (SP1) and (SP3) of Definition 1. We only need to prove that Sg (A, B) satisfies (SP2) and (SP4). (SP2): From (6), we have Sg (A, B) = 1 ⇔ µA (xi ) = µB (xi ), νA (xi ) = νB (xi ), φA(xi ) = φB (xi ), ∀xi ∈ X ⇔ A = B. (SP4): For any IFS C = {(µC (xi ), νC (xi ), φC (xi )) | xi ∈ X}, if A ⊆ B ⊆ C, then we have n  X Sg (A, C) = 1 −

1 n

i=1

|µA (xi ) − µC (xi )| + |νA (xi ) − νC (xi )| + |φA (xi ) − φC (xi )| 4

max(|µA(xi ) − µC (xi )|, |νA(xi ) − νC (xi )|, |φA (xi ) − φC (xi )|) + 2



.

It is easy to see that |µA (xi ) − µC (xi )| ≥ |µA (xi ) − µB (xi )|, |νA(xi ) − νC (xi )| ≥ |νA(xi ) − νB (xi )|, |φA (xi ) − φC (xi )| ≥ |φA(xi ) − φB (xi )|. So we have |µA(xi ) − µC (xi )| + |νA(xi ) − νC (xi )| + |φA(xi ) − φC (xi )| 4 max(|µA (xi ) − µC (xi )|, |νA(xi ) − νC (xi )|, |φA(xi ) − φC (xi )|) + 2 |µA (xi ) − µB (xi )| + |νA(xi ) − νB (xi )| + |φA(xi ) − φB (xi )| ≥ 4 max(|µA (xi ) − µB (xi )|, |νA(xi ) − νB (xi )|, |φA(xi ) − φB (xi )|) + 2 5

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and thus we get Sg (A, C) ≤ Sg (A, B). By the same reason, we can get Sg (A, C) ≤ Sg (B, C). However, the elements in the universe may have different importance in pattern recognition. We should consider the weight of the elements so that we can obtain more reasonable results in pattern recognition. Assume that the weight of xi in X is wi , where wi ∈ [0, 1] (i = 1, 2, . . . , n) Pn and i=1 wi = 1. The similarity measure between GIFSs A and B can be obtained by the following form: Sgw (A, B) =1−

n X

wi

i=1

+



|µA (xi ) − µB (xi )| + |νA (xi ) − νB (xi )| + |φA(xi ) − φB (xi )| 4

max(|µA (xi ) − µB (xi )|, |νA(xi ) − νB (xi )|, |φA (xi) − φB (xi )|) 2



.

(8)

Likewise, for Sgw (A, B), the following theorem holds. Theorem 3.2 Sgw (A, B) is the similarity measure between two GIFSs A and B. Proof The proof is similar to that of Theorem 3.1. Remark 3.3 Obviously, if wi = 1/n (i = 1, 2, . . ., n), (8) becomes (7). So, (7) is a special case of (8). Now, we propose another new similarity measure between GIFSs A = {(x, µA (x), νA(x))|x ∈ X} and B = {(x, µB (x), νB (x))|x ∈ X} in X = {x1 , x2 , . . . , xn} as follows: Let ϕµAB (i) = |µA (xi )−µB (xi )|/2, ϕνAB (i) = |νA(xi )−νB (xi )|/2, ϕφAB (i) = |φA (xi ) − φB (xi )|/2 and xi ∈ X. Then v u n uX 1 p p t Sd (A, B) = 1 − √ (9) (ϕµAB (i) + ϕνAB (i) + ϕπAB (i))p , p n i=1 where 1 ≤ p < ∞. Theorem 3.4 Sdp (A, B) is the similarity measure between two GIFSs A and B. Proof Obviously, Sdp (A, B) satisfies (SP1) and (SP3). As to (SP2) and (SP4), we give the following proof. (SP2): From (8), we have Sdp (A, B) = 1 ⇔ ϕµAB (i) = 0, ϕνAB (i) = 0, ϕφAB (i) = 0, ∀i = 1, . . . , n ⇔ µA (xi ) = µB (xi ), νA(xi ) = νB (xi ), φA(xi ) = φB (xi ), ∀xi ∈ X ⇔ A = B. 6

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(SP4): Since A ⊆ B ⊆ C, we have µA (xi ) ≤ µB (xi ) ≤ µC (xi ), νA (xi ) ≥ νB (xi ) ≥ νC (xi ) and φA (xi ) ≥ φB (xi ) ≥ φC (xi ) for any xi ∈ X. Then we have ϕµAB (i) + ϕνAB (i) + ϕφAB (i) = |µA (xi ) − µB (xi )|/2 + |νA(xi ) − νB (xi )|/2 + |φA(xi ) − φB (xi )|/2 ≤ |µA (xi ) − µC (xi )|/2 + |νA(xi ) − νC (xi )|/2 + |φA (xi ) − φC (xi )|/2 = ϕµAC (i) + ϕνAC (i) + ϕφAC (i).

So we have v u n uX p t (ϕµAC (i) + ϕνAC (i) + ϕφAC (i))p ≥ i=1

Therefore, Sdp (B, C).

Sdp (A, C)

≤

Sdp (A, B).

v u n uX p t (ϕµ

AB

(i) + ϕνAB (i) + ϕφAB (i))p .

i=1

In the similar way, it is easy to prove Sdp (A, C) ≤

Similar to (8), considering the weight wi of xi ∈ X, the similarity measure of GIFSs A and B can be obtained as following form. v u n uX p p Sdw (A, B) = 1 − t (10) wi (ϕµAB (i) + ϕνAB (i) + ϕφAB (i))p , i=1

where 1 ≤ p < ∞.

p Likewise, for Sdw (A, B), the following theorem holds.

p Theorem 3.5 Sdw (A, B) is the similarity measure between two GIFSs A and B.

Proof The proof is similar to that of Theorem 3.4.

4

An application to pattern recognition problem

Assume that a question related to pattern recognition is given using GIFSs. Assume that there exist m patterns which are represented by GIFSs Ai = {(xi , µA (xi ), νA(xi )) | xi ∈ X} (i = 1, 2, . . ., m), where X = {x1 , x2 , . . . , xn }. Suppose that there be a sample to be recognized which is represented by GIFS B = {(xi , µB (xi ), νB (xi )) | xi ∈ X}. Set S(Ai0 , B) = max {S(Ai , B)}, 1≤i≤n

(11)

where S(Ai , B) is the similarity measure between Ai and B (i = 1, 2, . . . , n) given by (8) or (10). According to the principle of the maximum degree of similarity between GIFSs, it can be decided that the sample B belongs to some pattern Ai0 . 7

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Example 4.1 Assume that there are three patterns denoted with GIFSs in X = {x1 , x2 , x3 }. Three patterns A1 , A2 and A3 are denoted as follows: A1 A2

= =

A3

=

{(x1 , 0.6, 0.3), (x2, 0.8, 0.3), (x3, 0.7, 0.4)}; {(x1 , 0.5, 0.6), (x2, 0.5, 0.4), (x3, 0.7, 0.5)};

{(x1 , 0.6, 0.5), (x2, 0.7, 0.4), (x3, 0.8, 0.4)}.

Assume that a sample B = {(x1 , 0.6, 0.3), (x2, 0.7, 0.5), (x3, 0.7, 0.5)} is given. Given three kinds of mineral fields, each is featured by the content of three minerals and contains one kind of typical hybrid minerals. The three kinds of typical hybrid minerals are represented by GIFSs A1 , A2 and A3 in X, respectively. Given another kind of hybrid mineral B, to which field does this kind of mineral B most probably belong to ? For convenience, assume that the weight wi of xi in X are equal and p = 2. By (8) and (10), we have Sg (A1 , B) = 0.900, Sg (A2 , B) = 0.800, Sg (A3 , B) = 0.866; Sd2 (A1 , B) = 0.971, Sd2 (A2 , B) = 0.755, Sd2 (A3 , B) = 0.859. From this data, the proposed similarity measures Sg and Sdp show the same classification according to the principle of the maximum degree of similarity between GIFSs. That is, the sample B belongs to the pattern A1 . The results of above example indicates the proposed similarity measure to be good in pattern recognition problems. In the following example, the proposed similarity measure is used to characterize the similarity between linguistic variables. Example 4.2 Let F = {(x, µF (x), νF (x)) : x ∈ X} be a GIFS in X. For any positive real number n, We define the GIFS F n as follows: F n = {(x, (µF (x))n , 1 − (1 − νF (x))n ) | x ∈ X}. Using the above operation, we also define the concentration and dilation of F as follows: • concentration: CON(F ) = F 2 ; • dilation: DIL(F ) = F 1/2 . Like the fuzzy sets, CON(F ) and DIL(F ) may be treated as “very (F )” and “more or less (F )”, respectively. In the next, we consider a GIFS F in X = {6, 7, 8, 9, 10} defined by F = {{(6, 0.2, 0.9), (7, 0.4, 0.7), (8, 0.7, 0.4), (9, 0.9, 0.1), (10, 1, 0)}. With taking into account the characterization of linguistic variables, we regard F as “LARGE” in X. Using the operations of concentration and dilation 8

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• F 1/2 may be treated as “More or less LARGE”, • F 2 may be treated as “Very LARGE”, • F 4 may be treated as “Very very LARGE”. The proposed similarity measure is utilized to calculate the degree of similarity between these GIFSs. The results are summarized in Table 1. In Table 1, L., V.L., V.V.L. and M.L.L. denote LARGE, Very LARGE, Very very LARGE and More or less LARGE, respectively. Table 1: The values calculated by the proposed similarity measure Sd1 Sd1 M.L.L. L. V.L. V.V.L.

M.L.L. 1 0.8562 0.7134 0.6022

L. 0.8562 1 0.8540 0.7428

V.L. 0.7134 0.8540 1 0.8848

V.V.L. 0.6022 0.7428 0.8848 1

From the viewpoint of mathematical operations, the similarities between the above GIFSs require the following conditions: S(M.L.L., L.) S(L., M.L.L.) S(V.L., V.V.L.)

> S(M.L.L., V.L.) > S(M.L.L., V.V.L.), > S(L., V.L.) > S(L., V.V.L.), > S(V.L., L.) > S(V.L., M.L.L.),

(12) (13) (14)

S(V.V.L., V.L.)

> S(V.V.L., L.) > S(V.V.L., M.L.L.).

(15)

From Table 1, it can be seen that the proposed similarity measure Sd1 satisfies the requirements (12)-(15). Therefore, the proposed similarity measure Sd1 is reliable in applications with compound linguistic variables.

5

Conclusions

We apply the principle of maximum degree of similarity measures between GIFSs to solve the pattern recognition problem. Based on the geometrical interpretation of GIFSs, we take into account three parameters (membership, nonmembership, hesitation margin) to propose a similarity measure for calculating the degree of similarity between GIFSs. Numerical example is given to illustrate the application of the developed similarity measures. Furthermore, we use the proposed similarity measures to characterize the similarity between linguistic variables.

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References [1] K. Atanassov, Intuitionistic fuzzy sets, Fuzzy Sets and Systems, 20, 87-96 (1986). [2] K. Atanassov, More on intuitionistic fuzzy sets, Fuzzy Sets and Systems, 33, 37-46 (1989). [3] K. Atanassov, New operations defined over the intuitionistic fuzzy sets, Fuzzy Sets and Systems, 61, 1377-142 (1994). [4] K. Atanassov and G. Gargov, Intuitionistic fuzzy logic, CR Acad. Bulg. Soc., 43, 9-12 (1990). [5] H. Bustince and P. Burillo, Vague sets are intuitionistic fuzzy sets, Fuzzy Sets and Systems, 79, 403-405 (1996). [6] S.M. Chen, Similarity measure between vague sets and elements, IEEE Trans. Systems Man Cybernt., 27, 153-158 (1997). [7] S.M. Chen and J.M. Tan, Handling multi-criteria fuzzy decision-making problems based on vague sets, Fuzzy Sets and Systems, 67, 163-172 (1994). [8] S.K. De, R. Biswas and A.R. Roy, Some operations on intuitionistic fuzzy sets, Fuzzy Sets and Systems, 114, 477-484 (2000). [9] S.K. De, R. Biswas and A.R. Roy, An application of intuitionistic fuzzy sets in medical diagnosis, Fuzzy Sets and Systems, 117, 209-213 (2001). [10] W.L. Gau and D.J. Buehere, Vague sets, IEEE Trans. Systems Man Cybernt., 23, 610-614 (1994). [11] D.H. Hong and C. Kim, A note on similarity measures between vague sets and between elements, Inform. Science, 115, 83-96 (1999). [12] W.L. Hung and M.S. Yang, MS (2004) Similarity measures of intuitionistic fuzzy sets based on Hausdorff distance, Pattern Recognition Lett., 25, 16031611 (2004). [13] A. Kaufman, Introduction a la theorie des sous-ensembles flous, Masson et Cie, Editeurs, 1973. [14] D. Li and C. Cheng, New similarity measures of intuitionistic fuzzy fuzzy sets and applications to pattern recognitions, Pattern Recognition Lett., 23, 221-225 (2002). [15] Y. Li, D.L. Olson and Z. Qin, Similarity measures between intuitinistic fuzzy (vague) sets: A comparative analysis, Pattern Recognition Lett., 28, 278-285 (2007). [16] Z. Liang and P. Shi, Similarity measures on intuitionistic fuzzy fuzzy sets, Pattern Recognition Lett., 24, 2687-2693 (2003). 10

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[17] B. Loewer and R. Laddaga, Destroying the consensus, In B. Loewer (Guest Ed.): Special Issue on Consensus, Synthese, 62 (1985), 79-96. [18] T.K. Mondal and S.K. Samanta, Generalized intuitionistic fuzzy sets, J. Fuzzy Math., 10, 839-861 (2002). [19] H.B. Mitchell, On the Dengfeng-Chuitian similarity measure and its application to pattern recognition, Pattern Recognition Lett., 24, 3101-3104 (2003). [20] J.H. Park, Y.B. Park and K.M. Lim, Correlation coefficient of generalized intuitionistic fuzzy sets by statistical method, Honam Math. J., 28, 317-326 (2006). [21] J.H. Park, J.H. Kang, Y.T. Jang, Y.C. Kwun and J.H. Koo, Similarity measure for generalized intuitionistic fuzzy sets and its application to group decision making. in: Proceeding of the International Conference on e-Commerce, e-Administration, e-Society, e-Education, and e-Technology, Macao, China (2010), 2246-2256. [22] E. Szmidt, Applications of intuitionistic fuzzy sets in decision making, (D. Sc. dissertation) Tech. Univ, Sofia 2000. [23] E. Szmidt and J. Baldwin, New similarity measure for intuitionistic fuzzy set theory and mass assignment theory, Notes on IFSs, 9, 60-76 (2003). [24] E. Szmit and J. Kacprzyk, Distances between intuitionistic fuzzy sets, Fuzzy Sets and Systems, 114, 505-518 (2000). [25] E. Szmit and J. Kacprzyk, A new concept of a similarity measure for intuitionistic fuzzy sets and its use in group decision making, LNAI, 3558, 272-282 (2005). [26] I.K. Vlachos and G.D. Sergiadis, Intuitionistic fuzzy information - Applications to pattern recognition, Pattern Recognition Lett., 28, 197-206 (2007). [27] L.A. Zadeh, Fuzzy sets, Inform. and Control, 8, 338-353 (1965).

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TABLE OF CONTENTS, JOURNAL OF COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 20, NO. 5, 2016

Barnes-Type Peters of the First Kind and Poly-Cauchy of the First Kind Mixed-Type Polynomials, Dae San Kim, Taekyun Kim, Takao Komatsu, and Dmitry V. Dolgy, …………803 On the Hyperstablity of a Functional Equation in Commutative Groups, Muaadh Almahalebi, and Choonkil Park,……………………………………………………………………………..826 A Fractional Finite Difference Inclusion, Dumitru Baleanu, Shahram Rezapour, and Saeid Salehi,………………………………………………………………………………………...…834 Some Implicit Properties of the Second Kind Bernoulli Polynomials of Order α, C. S. Ryoo, and J. Y. Kang,……………………………………………………………………………………...843 An Oscillation of the Solution For a Nonlinear Second-Order Stochastic Differential Equation, Iryna Komashynska, Mohammed AL-Smadi, Ali Ateiwi, and Ayed Al e’damat,……………860 Fixed Point Theorems and T-stability of Picard Iteration For Generalized Lipschitz Mappings in Cone Metric Spaces Over Banach Algebras, Huaping Huang, Shaoyuan Xu, Hao Liu, and Stojan Radenovic,……………………………………………………………………………………...869 Int-soft Filters of MTL-Algebras, Young Bae Jun, Seok Zun Song, Eun Hwan Roh, and Sun Shin Ahn,………………………………………………………………………………………889 Convergence Analysis of New Iterative Approximating Schemes with Errors for Total Asymptotically Nonexpansive Mappings in Hyperbolic Spaces, Ting-jian Xiong, and Heng-you Lan,…………………………………………………………………………………………… 902 Some Ostrowski Type Integral Inequalities for Double Integral on Time Scales, Wajeeha Irshad, Muhammad Iqbal Bhatti, and Muhammad Muddassar,……………………………………….914 The Henstock-Stieltjes Integral for Fuzzy-Number-Valued Functions on a Infinite Interval, Ke-feng Duan,…………………………………………………………………………………928 New Weighted q-Cebysev-Gruss Type Inequalities for Double Integrals, Zhen Liu, and Wengui Yang,…………………………………………………………………………………………...938 Quadratic ρ-Functional Inequalities in Normed Spaces, Ikan Choi, Sunghoon Kim, George A. Anastassiou, and Choonkil Park,……………………………………………………………… 949

TABLE OF CONTENTS, JOURNAL OF COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 20, NO. 5, 2016 (continued) Solution of the Ulam Stability Problem for Quartic (a, b)-Functional Equations, Abdullah Alotaibi, John Michael Rassias, and S.A. Mohiuddine,………………………………………957 Existence of Positive Solutions for Summation Boundary Value Problem for a Fourth-Order Difference Equations, Thanin Sitthiwirattham,………………………………………………..969 Similarity Measure Between Generalized Intuitionistic Fuzzy Sets and Its Application to Pattern Recognition, Jin Han Park, Jongchul Hwang, Juhyung Kim, Byeongmuk Park, Juyoung Park, Jeongwoo Son, and Sihun Lee,…………………………………………………………………984

Volume 20, Number 6 ISSN:1521-1398 PRINT,1572-9206 ONLINE

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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. 20, NO.6, 2016, COPYRIGHT 2016 EUDOXUS PRESS, LLC

ON THE STABILITY OF THE GENERALIZED QUADRATIC SET-VALUED FUNCTIONAL EQUATION HAHNG-YUN CHU† AND SEUNG KI YOO∗

Abstract. focus on the n-dimensional quadratic set-valued functional equation Pn weP Pn PnIn this article, n (4 − n)f ( i=1 xi ) ⊕ f ( j=1 θ(i, j)xj ) = 4 i=1 f (xi ), where n ≥ 2 is an integer. We prove i=1 the Hyers-Ulam stability for the set-valued functional equation.

1. Introduction

The stability problem of functional equation concerning group homomorphisms had been first raised by S. M. Ulam [18] in 1940. Let (G1 , ∗) be a group and let (G2 , , d) be a metric group with the metric d(·, ·). Given ε > 0, does there exists a δ > 0 such that if a mapping h : G1 → G2 satisfies the inequality d(h(x ∗ y), h(x)  h(y)) < δ for all x, y ∈ G1 , then there is a homomorphism H : G1 → G2 with d(h(x), H(x)) < ε for all x ∈ G1 ? The first partial solution to Ulam’s question was provided by D. H. Hyers [8] for Banach spaces. Hyers’ theorem was generalized by T. Aoki [1] for additive mapping. Th. M. Rassias [15] generalized the result of Hyers as follows: Let f : X → Y be a mapping between Banach spaces and let 0 ≤ p < 1 be fixed. If f satisfies the inequality kf (x + y) − f (x) − f (y)k ≤ θ(kxkp + kykp )

(1.1)

for some θ ≥ 0 and for all x, y ∈ X, then there exists a unique additive mapping A : X → Y such that kA(x) − f (x)k ≤

2θ p 2−2p kxk

for all x ∈ X. If f (tx) is continuous in t for each fixed x ∈ X, then A is

linear. Thereafter, P. Gˇ avruta [7] provided a generalization of Th. M. Rassias’ theorem, more precisely speaking, in which he replaced the bound ε(kxkp +kykp ) in (1.1) by control functions φ(x, y) with more general types for the existence of a unique linear mapping. The functional equation f (x+y)+f (x−y) =

∗ Corresponding author † The first author’s research has been performed as a subproject of project Research for Applications of Mathematical Principles (No C21501) and supported by the National Institute of Mathematics Sciences(NIMS). ∗ The corresponding author was supported by Basic Science Research Program through the National Research Foundation of Korea(NRF) funded by the Ministry of Education, Science and Technology(NRF-2012R1A1A2009512). 2010 Mathematics Subject Classification. Primary 39B82; 47H04; 47H10; 54C60 Key words and Phrases. Hyers-Ulam stability, generalized quadratic set-valued functional equation. 1

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HAHNG-YUN CHU, SEUNG KI YOO

2f (x) + 2f (y) is called the quadratic functional equation and every solution of the quadratic functional equation is called a quadratic function. The Hyers-Ulam stsbility of quadratic functional equation was proved by F. Skof [17] for a function f : E1 → E2 where E1 is a normed space and E2 is a Banach space. P. W. Cholewa [3] considered Skof’s theorem to a version of abelian groups. Skof’s result was generalized by S. Czerwik [6] who proved the generalized Hyers-Ulam stability of quadratic functional equation in the spirit of Rassias approach. Kang and Chu [10] extended the quadratic functional equation to the generalized form Pn Pn Pn Pn (4 − n)f ( i=1 xi ) + i=1 f ( j=1 θ(i, j)xj ) = 4 i=1 f (xi ) where n ≥ 2 is an integer and the function θ is defined by ( θ(i, j) =

1 if i 6= j −1 if i = j

and also investigated the Hyers-Ulam stability for the generalized quadratic functional equation. In [12], Lu and Park defined the additive set-valued functional equations f (αx + βy) = rf (x) + sf (y) and f (x + y + z) = 2f ( x+y 2 ) + f (z) and proved the Hyers-Ulam stability of the set-valued functional equations. In [14], Park et al. investigated stability problems of the Jensen additive, quadratic, cubic and quartic set-valued functional equation. Kenary et al. [11] proved the stability for various types of the set-valued functional equation using the fixed point alternative. In recent years, Chu and Yoo [5] studied the Hyers-Ulam stability of the n-dimensional additive set-valued functional equation. In [4], they also investigated the Hyers-Ulam stability of the n-dimensional cubic set-valued functional equation. Let CB(Y ) be the set of all closed bounded subsets of Y and CC(Y ) the set of all closed convex subsets of Y . Let CBC(Y ) be the set of all closed bounded convex subsets of Y . For any elements A, B of CC(Y ), we denote A ⊕ B = A + B. If A is convex, then we obtain that (α + β)A = αA + βA for all α, β ∈ R+ . Let f : X → CBC(Y ) be a mapping. The quadratic set-valued functional equation is defined by f (x + y) ⊕ f (x − y) = 2f (x) ⊕ 2f (y) for all x, y ∈ X. Every solution of the quadratic set-valued functional equation is said to be a quadratic set-valued mapping. In this paper, we introduce the generalized n-dimensional quadratic set-valued functional equation (4 − n)f (

n X i=1

xi ) ⊕

n X i=1

f(

n X

θ(i, j)xj ) = 4

j=1

n X

f (xi )

(1.2)

i=1

where n ≥ 2 is an integer and the function θ is defined by ( 1 if i 6= j θ(i, j) = −1 if i = j and investigate the Hyers-Ulam stability of the functional equation. In the set-valued dynamics, every solution of the generalized n-dimensional quadratic set-valued functional equation is called a n-dimensional quadratic set-valued mapping.

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J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 20, NO.6, 2016, COPYRIGHT 2016 EUDOXUS PRESS, LLC

ON THE STABILITY OF THE GENERALIZED QUADRATIC SET-VALUED FUNCTIONAL EQUATION

3

For a subset A ⊂ Y , the distance function d(·, A) is defined by d(x, A) := inf {k x − y k: y ∈ A} for x ∈ Y . For A, B ∈ CB(Y ), the Hausdorff distance h(A, B) is defined by h(A, B) := inf {α ≥ 0| A ⊆ B + αBY , B ⊆ A + αBY }, where BY is the closed unit ball in Y . In [2], it was proved that (CBC(Y ), ⊕, h) is a complete metric semigroup. R˚ adstr¨ om [16] proved that (CBC(Y ), ⊕, h) is isometrically embedded in a Banach space. The following remark is easily proved by using the notion of the Hausdorff distance. Remark 1.1. Let A, A0 , B, B 0 , C ∈ CBC(Y ) and α > 0. Then we have that (1) h(A ⊕ A0 , B ⊕ B 0 ) ≤ h(A, B) + h(A0 , B 0 ); (2) h(αA, αB) = αh(A, B); (3) h(A, B) = h(A ⊕ C, B ⊕ C). This paper is organized as follows. In section 2, we prove that the generalized n-dimensional setvalued mapping is actually general type of the quadratic set-valued mapping. We also investigate Hyers-Ulam stability for the generalized n-dimensional set-valued funational equation. As applications of the stability, we take to change the control function and obtain the different approaches to unique generalized n-dimensional functional equation. In section 3, we also get the Hyers-Ulam staility for the generalized n-dimensional set-valued functional equation by using the fixed point method which is developed by Margolis and Diaz.

2. Stability of the set-valued functional equation

In this section, we mainly deal with the Hyers-Ulam stability for the generalized n-dimensional quadratic set-valued functional equation. We first study for properties of the n-dimensional quadratic set-valued mapping. Next we prove the Hyers-Ulam stabilities for the generalized n-dimensional quadratic set-valued equation. Especially when n is an even numbers, we find the precise control function depending upon the original function and n-dimensional quadratic set-valued mapping. Similarly we also obtain the precise control function in the odd case for the generalized n-dimensional quadratic set-valued functional equation. Proposition 2.1. Suppose that a mapping f : X → CBC(Y ) defined by n n n n X X X X (4 − n)f ( xi ) ⊕ f( θ(i, j)xj ) = 4 f (xi ) i=1

i=1

j=1

(2.1)

i=1

for all x1 , . . . , xn ∈ X. Then f has the following properties: (1) f (0) = {0} (2) f (x) = f (−x) for all x ∈ X

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HAHNG-YUN CHU, SEUNG KI YOO

(3) f is a quadratic set-valued mapping.

Proof. (1) Putting xi = 0 (i = 1, . . . , n) in (2.1), we have f (0) = {0}. (2) Putting x1 = x and xi = 0 (i = 2, . . . , n) in (2.1), we get (4−n)f (x)⊕f (−x)⊕(n−1)f (x) = 4f (x). Thus f (x) = f (−x) for all x ∈ X. (3) Replacing x1 = x, x2 = y and xi = 0 (i = 3, . . . , n), we have (4−n)f (x+y)⊕f (−x+y)⊕f (x−y)⊕ (n − 2)f (x + y) = 4f (x) ⊕ 4f (y) ⊕ (n − 2)f (0). So we conclude that f (x + y) ⊕ f (x − y) = 2f (x) ⊕ 2f (y). This completes the proof.



Next, we prove the stability of the generalized n-dimensional quadratic set-valued functional equation. To extended precisely to the stability theory for the set-valued functional equation, we state the stability according to dimensions of the equation. Theorem 2.2. Let n ≥ 2 be an integer and let φ : X n → [0, ∞) be a function such that ˜ 1 , . . . , xn ) := φ(x

∞ X 1 φ(2i x1 , . . . , 2i xn ) < ∞ i 4 i=0

(2.2)

for all x1 , . . . , xn ∈ X. Suppose that f : X −→ (CBC(Y ), h) is a mapping with f (0) = {0} and n n n n   X X X X h (4 − n)f ( xi ) ⊕ f( θ(i, j)xj ), 4 f (xi ) ≤ φ(x1 , . . . , xn ) (2.3) i=1

i=1

j=1

i=1

for all x1 , . . . , xn ∈ X. Then there exists a unique n-dimensional quadratic set-valued mapping T : X → (CBC(Y ), h) such that h(f (x), T (x)) ≤

1˜ φ(x, x, 0, . . . , 0) 8

(2.4)

for all x ∈ X. Proof. Putting x1 = x2 = x and x3 = · · · = xn = 0 in (2.3), we have h(

f (2x) 1 , f (x)) ≤ φ(x, x, 0, . . . , 0) 4 8

(2.5)

for all x ∈ X. Replacing x by 2x and dividing by 4 in (2.5) h(

f (4x) 1 , f (2x)) ≤ φ(2x, 2x, 0, . . . , 0) 2 4 32

(2.6)

for all x ∈ X. By (2.5) and (2.6), we get h(

1 f (4x) 1 , f (x)) ≤ φ(x, x, 0, . . . , 0) + φ(2x, 2x, 0, . . . , 0) 42 8 4·8

(2.7)

for all x ∈ X. Using the induction on i, we have that r−1

h(

f (2r x) 1X 1 , f (x)) ≤ φ(2i x, 2i x, 0, . . . , 0) 4r 8 i=0 4i

(2.8)

for any positive integer r and for all x ∈ X.

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5

r

Now, we show that the sequence { f (24r x) } converges for all x ∈ X. For any positive integer r and s, we divide inequality (2.8) by 4s and replace x by 2s x. Then we obtain that the following inequality r−1

h(

1 1X 1 f (2r+s x) f (2s x) , )≤ s ≤ φ(2s+i x, 2s+i x, 0, . . . , 0) r+s s 4 4 4 8 i=0 4i

(2.9)

for all x ∈ X. Since the right-hand side of the inequality (2.9) tends to zero as s tends to infinity, r

the sequence { f (24r x) } is a Cauchy sequence in (CBC(Y), h). Therefore, we can define a mapping T : X → (CBC(Y ), h) as T (x) := limr→∞

f (2r x) 4r

for all x ∈ X. It follows from the definition of T and

(2.2) that n n n n   X X X X 1 h (4 − n)T ( xi ) ⊕ T( θ(i, j)xj ), 4 T (xi ) ≤ lim r φ(2r x1 , . . . , 2r xn ) = 0 r→∞ 4 i=1 i=1 j=1 i=1

(2.10)

for all x1 , . . . , xn ∈ X. Hence, we claim that T is an n-dimensional quadratic set-valued mapping. By letting r → ∞ in (2.8), we have the desired inequality (2.4). Now we prove the uniqueness of T . Let T 0 : X → (CBC(Y ), h) be another n-dimensional quadratic set-valued mapping satisfying (2.4). Therefore, we get the following inequality h(T (x), T 0 (x)) =

1 1˜ r 1 h(T (2r x), T 0 (2r x)) ≤ r φ(2 x, 2r x, 0, . . . , 0) r 4 4 8

for all x ∈ X. Hence, letting r → ∞, the right-hand side of above inequality goes to zero, and it follows that T (x) = T 0 (x) for all x ∈ X.



Corollary 2.3. Let n ≥ 2 be an integer, 0 < p < 2 and θ ≥ 0 be real numbers and let X be a real normed space. Suppose that f : X → (CBC(Y ), h) is a mapping satisfying n n n n n   X X X X X h (4 − n)f ( xi ) ⊕ f( θ(i, j)xj ), 4 f (xi ) ≤ θ kxi kp i=1

i=1

j=1

i=1

i=1

for all x1 , . . . , xn ∈ X. Then there exists a unique n-dimensional quadratic set-valued mapping T : X → (CBC(Y ), h) that satisfies h(f (x), T (x)) ≤

22

θ kxkp − 2p

for all x ∈ X. Proof. The result follows Theorem 2.2 by setting φ(x1 , . . . , xn ) = θ

Pn

i=1

kxi kp for all x1 , . . . , xn ∈

X.



Theorem 2.4. Let n ≥ 2 be an integer and let φ : X n → [0, ∞) be a function such that ˜ 1 , . . . , xn ) := φ(x

∞ X i=1

4i φ(

x1 xn ,... , i ) < ∞ i 2 2

1011

(2.11)

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for all x1 , . . . , xn ∈ X. Suppose that f : X −→ (CBC(Y ), h) is a mapping and n n n n   X X X X h (4 − n)f ( xi ) ⊕ f( θ(i, j)xj ), 4 f (xi ) ≤ φ(x1 , . . . , xn ) i=1

i=1

j=1

(2.12)

i=1

for all x1 , . . . , xn ∈ X. Then there exists a unique n-dimensional quadratic set-valued mapping T : X → (CBC(Y ), h) such that h(f (x), T (x)) ≤

1˜ φ(x, x, 0, . . . , 0) 8

(2.13)

for all x ∈ X. Proof. By (2.11) and (2.12), we get f (0) = {0}. Replacing x by

x 2

and multiplying by 4 in (2.5), we

have the following inequality x 1 x x h(f (x), 4f ( )) ≤ φ( , , 0, . . . , 0) 2 2 2 2 for all x ∈ X. The rest of the proof is similar to the proof of Theorem 2.3.



Corollary 2.5. Let n ≥ 2 be an integer, p > 2 and θ ≥ 0 be real numbers and let X be a real normed space. Suppose that f : X → (CBC(Y ), h) is a mapping satisfying n n n n n   X X X X X h (4 − n)f ( xi ) ⊕ f( θ(i, j)xj ), 4 f (xi ) ≤ θ kxi kp i=1

i=1

j=1

i=1

i=1

for all x1 , . . . , xn ∈ X. Then there exists a unique n-dimensional quadratic set-valued mapping T : X → (CBC(Y ), h) that satisfies h(f (x), T (x)) ≤

θ kxkp 2p − 22

for all x ∈ X. Proof. The result follows Theorem 2.4 by setting φ(x1 , . . . , xn ) = θ

Pn

i=1

kxi kp for all x1 , . . . , xn ∈

X.



Let n be an even positive integer. In this case, we can obtain the control function for the Hausdorff distence between the original mapping and n-dimensional quadratic set-valued mapping.

Theorem 2.6. Let n ≥ 2 be even and let φ : X n → [0, ∞) be a function such that ˜ 1 , . . . , xn ) := φ(x

∞ X 1 φ(2i x1 , . . . , 2i xn ) < ∞ i 4 i=0

(2.14)

for all x1 , . . . , xn ∈ X. Suppose that f : X −→ (CBC(Y ), h) is an even mapping with f (0) = {0} and n n n n   X X X X h (4 − n)f ( xi ) ⊕ f( θ(i, j)xj ), 4 f (xi ) ≤ φ(x1 , . . . , xn ) (2.15) i=1

i=1

j=1

i=1

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7

for all x1 , . . . , xn ∈ X. Then there exists a unique n-dimensional quadratic set-valued mapping T : X → (CBC(Y ), h) such that h(f (x), T (x)) ≤

1 ˜ φ(x, −x, x, −x, . . . , x, −x) 4n

(2.16)

for all x ∈ X. Proof. Put xk = (−1)k−1 x (k = 1, . . . , n) in (2.15). Since f is even and the range of f is convex, we have that h(

f (2x) 1 , f (x)) ≤ φ(x, −x, x, −x, . . . , x, −x) 4 4n

(2.17)

for all x ∈ X. The rest of the proof is similar to proof of Theorem 2.2.



Corollary 2.7. Let n ≥ 2 be even, 0 < p < 2 and θ ≥ 0 be real numbers and let X be a real normed space. Suppose that f : X → (CBC(Y ), h) is an even mapping satisfying n n n n n   X X X X X xi ) ⊕ f( θ(i, j)xj ), 4 f (xi ) ≤ θ kxi kp h (4 − n)f ( i=1

i=1

j=1

i=1

i=1

for all x1 , . . . , xn ∈ X. Then there exists a unique n-dimensional quadratic set-valued mapping T : X → (CBC(Y ), h) that satisfies h(f (x), T (x)) ≤

22

θ kxkp − 2p

for all x ∈ X. Proof. The result follows Theorem 2.6 by setting φ(x1 , . . . , xn ) = θ

Pn

i=1

kxi kp for all x1 , . . . , xn ∈

X.



Theorem 2.8. Let n ≥ 2 be even and let φ : X n → [0, ∞) be a function such that ˜ 1 , . . . , xn ) := φ(x

∞ X i=0

4i φ(

x1 xn , . . . , i+1 ) < ∞ i+1 2 2

(2.18)

for all x1 , . . . , xn ∈ X. Suppose that f : X −→ (CBC(Y ), h) is an even mapping with f (0) = {0} and n n n n   X X X X h (4 − n)f ( xi ) ⊕ f( θ(i, j)xj ), 4 f (xi ) ≤ φ(x1 , . . . , xn ) (2.19) i=1

i=1

j=1

i=1

for all x1 , . . . , xn ∈ X. Then there exists a unique n-dimensional quadratic set-valued mapping T : X → (CBC(Y ), h) such that h(f (x), T (x)) ≤

1˜ φ(x, −x, x, −x, . . . , x, −x) n

(2.20)

for all x ∈ X.

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Proof. Replacing x by

x 2

and multiplying by 4 in (2.17), we have the following inequality

x 1 x x x x x x h(f (x), 4f ( )) ≤ φ( , − , , − , . . . , , − ) 2 n 2 2 2 2 2 2 for all x ∈ X. The rest of the proof is similar to the proof of Theorem 2.3.



Corollary 2.9. Let n ≥ 2 be even, p > 2 and θ ≥ 0 be real numbers and let X be a real normed space. Suppose that f : X → (CBC(Y ), h) is an even mapping satisfying n n n n n   X X X X X h (4 − n)f ( xi ) ⊕ f( θ(i, j)xj ), 4 f (xi ) ≤ θ kxi kp i=1

i=1

j=1

i=1

i=1

for all x1 , . . . , xn ∈ X. Then there exists a unique n-dimensional quadratic set-valued mapping T : X → (CBC(Y ), h) that satisfies h(f (x), T (x)) ≤

θ kxkp 2p − 22

for all x ∈ X. Proof. The result follows Theorem 2.8 by setting φ(x1 , . . . , xn ) = θ

Pn

i=1

kxi kp for all x1 , . . . , xn ∈

X.

 As applications for the theorem, we get the Hyers-Ulam stability for the generalized n-dimensional

set-valued functional equation and especially we deal with the odd case for n. Theorem 2.10. Let n ≥ 2 be odd and let φ : X n → [0, ∞) be a function such that ∞ X 1 ˜ 1 , . . . , xn ) := φ(x φ(3i x1 , . . . , 3i xn ) < ∞ i 9 i=0

(2.21)

for all x1 , . . . , xn ∈ X. Suppose that f : X −→ (CBC(Y ), h) is an even mapping with f (0) = {0} and n n n n   X X X X h (4 − n)f ( xi ) ⊕ f( θ(i, j)xj ), 4 f (xi ) ≤ φ(x1 , . . . , xn ) (2.22) i=1

i=1

j=1

i=1

for all x1 , . . . , xn ∈ X. Then there exists a unique n-dimensional quadratic set-valued mapping T : X → (CBC(Y ), h) such that h(f (x), T (x)) ≤

2 ˜ −x, x, −x, . . . , −x, x) φ(x, 9(n − 1)

(2.23)

for all x ∈ X. Proof. Put xk = (−1)k−1 x (k = 1, . . . , n) in (2.22). Since f is even and the range of f is convex, we have that h(

f (3x) 2 , f (x)) ≤ φ(x, −x, x, −x, . . . , −x, x) 9 9(n − 1)

for all x ∈ X. The rest of the proof is similar to proof of Theorem 2.2.

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9

Corollary 2.11. Let n > 2 be odd, 0 < p < 2 and θ ≥ 0 be real numbers and let X be a real normed space. Suppose that f : X → (CBC(Y ), h) is an even mapping satisfying n n n n n   X X X X X h (4 − n)f ( xi ) ⊕ f( θ(i, j)xj ), 4 f (xi ) ≤ θ kxi kp i=1

i=1

j=1

i=1

i=1

for all x1 , . . . , xn ∈ X. Then there exists a unique n-dimensional quadratic set-valued mapping T : X → (CBC(Y ), h) that satisfies h(f (x), T (x)) ≤

2nθ kxkp (n − 1)(32 − 3p )

for all x ∈ X. Proof. The result follows Theorem 2.10 by setting φ(x1 , . . . , xn ) = θ

Pn

i=1

kxi kp for all x1 , . . . , xn ∈

X.



Theorem 2.12. Let n > 2 be odd and let φ : X n → [0, ∞) be a function such that ˜ 1 , . . . , xn ) := φ(x

∞ X i=0

9i φ(

xn x1 , . . . , i+1 ) < ∞ 3i+1 3

(2.24)

for all x1 , . . . , xn ∈ X. Suppose that f : X −→ (CBC(Y ), h) is an even mapping with f (0) = {0} and n n n n   X X X X h (4 − n)f ( xi ) ⊕ f( θ(i, j)xj ), 4 f (xi ) ≤ φ(x1 , . . . , xn ) i=1

i=1

j=1

(2.25)

i=1

for all x1 , . . . , xn ∈ X. Then there exists a unique n-dimensional quadratic set-valued mapping T : X → (CBC(Y ), h) such that h(f (x), T (x)) ≤

2 ˜ φ(x, −x, x, −x, . . . , −x, x) n−1

(2.26)

for all x ∈ X. Proof. Put xk = (−1)k−1 x (k = 1, . . . , n) in (2.25). Since f is even and the range of f is convex, we have that x 2 x x x x x x h(9f ( ), f (x)) ≤ φ( , − , , − , . . . , − , ) 3 n−1 3 3 3 3 3 3 for all x ∈ X. The rest of the proof is similar to proof of Theorem 2.2.



Corollary 2.13. Let n > 2 be odd, p > 2 and θ ≥ 0 be real numbers and let X be a real normed space. Suppose that f : X → (CBC(Y ), h) is an even mapping satisfying n n n n n   X X X X X h (4 − n)f ( xi ) ⊕ f( θ(i, j)xj ), 4 f (xi ) ≤ θ kxi kp i=1

i=1

j=1

i=1

1015

i=1

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for all x1 , . . . , xn ∈ X. Then there exists a unique n-dimensional quadratic set-valued mapping T : X → (CBC(Y ), h) that satisfies h(f (x), T (x)) ≤

2nθ kxkp (n − 1)(3p − 32 )

for all x ∈ X. Proof. The result follows Theorem 2.12 by setting φ(x1 , . . . , xn ) = θ

Pn

i=1

kxi kp for all x1 , . . . , xn ∈

X.



3. Stability of the set-valued functional equation by fixed point method

As using the fixed point method, we get plenty of the results related to the generalized n-dimensional quadratic set-valued functional equation. We first introduce the generalized metric on the given phase space and recall fundamental results for the fixed point theory. Let X be a set. A function d : X ×X → [0, ∞) is the generalized metric on X if d satisfies the following properties: (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. The following theorem is very useful for proving Hyers-Ulam stability which is due to Margolis and Diaz [13]. Theorem 3.1. 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 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) < ∞}; (4) d(y, y ∗ ) ≤

1 1−L d(y, Jy)

for all y ∈ Y.

Using the alternative fixed point theorem, we investigate the stability of the even dimensional quadratic set-valued functional equation.

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Theorem 3.2. Let n ≥ 2 be even. Suppose that an even mapping f : X −→ (CBC(Y ), h) with f (0) = {0} satisfies the inequality n n n n   X X X X h (4 − n)f ( xi ) ⊕ f( θ(i, j)xj ), 4 f (xi ) ≤ φ(x1 , . . . , xn ) i=1

i=1

j=1

(3.1)

i=1

for all x1 , . . . , xn ∈ X, and there exists a constant L with 0 < L < 1 for which the function φ : X n → [0, ∞) satisfies φ(2x, −2x, 2x, −2x, . . . , 2x, −2x) ≤ 4Lφ(x, −x, x, −x, . . . , x, −x)

(3.2)

for all x ∈ X. Then there exists a n-dimensional quadratic set-valued mapping T : X → (CBC(Y ), h) given by T (x) = limk→∞

f (2k x) 4k

such that

h(f (x), T (x)) ≤

1 φ(x, −x, x, −x, . . . , x, −x) 4n(1 − L)

(3.3)

for all x ∈ X.

Proof. Put xk = (−1)k−1 x (k = 1, . . . , n) in (3.1). Since f is even and the range of f is convex, we have that h(

1 f (2x) , f (x)) ≤ φ(x, −x, x, −x, . . . , x, −x) 4 4n

(3.4)

for all x ∈ X. Let S := {g | g : X → CBC(Y ), g(0) = {0}}. We define a generalized metric on S defined by d(g1 , g2 ) := inf{µ ∈ (0, ∞) | h(g1 (x), g2 (x)) ≤ µφ(x, −x, x, −x, . . . , x, −x), x ∈ X}. It is easy to show that (S, d) is complete (see [9]). Now, we define the mapping J : S → S given by Jg(x) = 41 g(2x) for all x ∈ X. For g1 , g2 ∈ S, let d(g1 , g2 ) = µ. Then 1 1 1 h( g1 (2x), g2 (2x)) ≤ µφ(2x, −2x, 2x, −2x . . . , 2x, −2x) 4 4 4 for all x ∈ X. Then by (3.2), we have h(Jg1 (x), Jg2 (x)) ≤ µLφ(x, −x, x, −x, . . . , x, −x) for all x ∈ X. Therefore, we get d(Jg1 , Jg2 ) ≤ Ld(g1 , g2 ) for any g1 , g2 ∈ S. Hence J is a strictly contractive mapping with Lipschitz constant L. It follows from (3.4) that d(Jf, f ) ≤

1 4n .

By Theorem 3.1, the sequence

k

{J f } converges to a fixed point T : X → (CBC(Y ), h) of J in the set {g ∈ S | d(f, g) < ∞} such that {J k f } → 0 as k → ∞. This implies T (x) = limk→∞ d(f, T ) ≤

1 1−L d(Jf, f )

≤

1 4n(1−L) .

f (2k x) 4k

for all x ∈ X. And we also have

This means that the inequality (3.3) holds. By (3.1),

n n n n   X X X X 1 h (4 − n)T ( xi ) ⊕ T( θ(i, j)xj ), 4 T (xi ) ≤ lim k φ(x, −x, x, −x, . . . , x, −x) = 0 k→∞ 4 i=1 i=1 j=1 i=1

Therefore, T is a unique n-dimensional quadratic set-valued mapping as desired.

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Corollary 3.3. Let n ≥ 2 be even, 0 < p < 2 and θ ≥ 0 be real numbers and let n ≥ 2 be even. Suppose that f : X −→ (CBC(Y ), h) is an even mapping satisfying n n n n n   X X X X X h (4 − n)f ( xi ) ⊕ f( θ(i, j)xj ), 4 f (xi ) ≤ θ kxi kp i=1

i=1

j=1

i=1

i=1

for all x1 , . . . , xn ∈ X. Then there exists a unique n-dimensional quadratic set-valued mapping T : X → (CBC(Y ), h) that satisfies h(f (x), T (x)) ≤

θ kxkp 22 − 2p

for all x ∈ X.

Proof. The proof follows from Theorem 3.2 by setting φ(x2 , . . . , xn ) = θ

Pn

i=1

kxi kp for every x1 , . . . , xn ∈

X. Then we can choose L = 2p−2 and we get the desired result.



Theorem 3.4. Let n ≥ 2 be even. Suppose that an even mapping f : X −→ (CBC(Y ), h) with f (0) = {0} satisfies the inequality n n n n   X X X X f (xi ) ≤ φ(x1 , . . . , xn ) θ(i, j)xj ), 4 f( xi ) ⊕ h (4 − n)f ( i=1

i=1

(3.5)

i=1

j=1

for all x1 , . . . , xn ∈ X, and there exists a constant L with 0 < L < 1 for which the function φ : X n → [0, ∞) satisfies x x x x x x L φ( , − , , − , . . . , , − ) ≤ φ(x, −x, x, −x, . . . , x, −x) 2 2 2 2 2 2 4

(3.6)

for all x ∈ X. Then there exists a n-dimensional quadratic set-valued mapping T : X → (CBC(Y ), h) given by T (x) = limk→∞ 4k f ( 2xk ) such that h(f (x), T (x)) ≤

L φ(x, −x, x, −x, . . . , x, −x) 4n(1 − L)

(3.7)

for all x ∈ X.

Proof. Replacing x by

x 2

and multiplying 4 in (3.4), we have

x 1 x x x x x x L h(f (x), 4f ( )) ≤ φ( , − , , − , . . . , , − ) ≤ φ(x, −x, x, −x, . . . , x, −x) 2 n 2 2 2 2 2 2 4n for all x ∈ X. The rest of the proof is similar to proof of Theorem 2.2.



Corollary 3.5. Let n ≥ 2 be even, p > 2 and θ ≥ 0 be real numbers and let n ≥ 2 be even. Suppose that f : X −→ (CBC(Y ), h) is an even mapping satisfying n n n n n   X X X X X h (4 − n)f ( xi ) ⊕ f( θ(i, j)xj ), 4 f (xi ) ≤ θ kxi kp i=1

i=1

j=1

i=1

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13

for all x1 , . . . , xn ∈ X. Then there exists a unique n-dimensional quadratic set-valued mapping T : X → (CBC(Y ), h) that satisfies h(f (x), T (x)) ≤

n(2p

θ kxkp − 22 )

for all x ∈ X.

Proof. The proof follows from Theorem 3.9 by setting φ(x2 , . . . , xn ) = θ

Pn

i=1

kxi kp for every x1 , . . . , xn ∈

X. Then we can choose L = 22−p and we get the desired result.



Finally, we deal with the Hyers-Ulam stability for the odd dimensional quadratic set-valued functional equation. Theorem 3.6. Let n > 2 be odd. Suppose that an even mapping f : X −→ (CBC(Y ), h) with f (0) = {0} satisfies the inequality n n n n   X X X X h (4 − n)f ( xi ) ⊕ f( θ(i, j)xj ), 4 f (xi ) ≤ φ(x1 , . . . , xn ) i=1

i=1

j=1

(3.8)

i=1

for all x1 , . . . , xn ∈ X, and there exists a constant L with 0 < L < 1 for which the function φ : X n → [0, ∞) satisfies φ(3x, −3x, 3x, −3x, . . . , −3x, 3x) ≤ 9Lφ(x, −x, x, −x, . . . , −x, x)

(3.9)

for all x ∈ X. Then there exists a n-dimensional quadratic set-valued mapping T : X → (CBC(Y ), h) given by T (x) = limk→∞

f (3k x) 9k

such that

h(f (x), T (x)) ≤

2 φ(x, −x, x, −x, . . . , −x, x) 9(n − 1)(1 − L)

(3.10)

for all x ∈ X.

Proof. Put xk = (−1)k−1 x (k = 1, . . . , n) in (3.8). Since f is even and the range of f is convex, we have that x 2 x x x x x x h(9f ( ), f (x)) ≤ φ( , − , , − , . . . , − , ) 3 n−1 3 3 3 3 3 3 for all x ∈ X. The rest of the proof is similar to proof of Theorem 2.2. 

Corollary 3.7. Let n > 2 be odd, 0 < p < 2 and θ ≥ 0 be real numbers and let n ≥ 2 be odd. Suppose that f : X −→ (CBC(Y ), h) is an even mapping satisfying n n n n n   X X X X X h (4 − n)f ( xi ) ⊕ f( θ(i, j)xj ), 4 f (xi ) ≤ θ kxi kp i=1

i=1

j=1

i=1

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for all x1 , . . . , xn ∈ X. Then there exists a unique n-dimensional quadratic set-valued mapping T : X → (CBC(Y ), h) that satisfies h(f (x), T (x)) ≤

2θ kxkp (n − 1)(32 − 3p )

for all x ∈ X.

Proof. The proof follows from Theorem 3.4 by setting φ(x2 , . . . , xn ) = θ X. Then we can choose L = 3p−2 and we get the desired result.

Pn

i=1

kxi kp for every x1 , . . . , xn ∈ 

References [1] T. Aoki, On the stability of the linear transformation in the Banach space, J. Math. Soc. Jap. 2 (1950) 64–66. [2] C. Castaing , M. Valadier, Convex analysis and measurable multifunctions, Lec. Notes in Math., Springer, Berlin, 1977. [3] P. W. Cholewa, Remarks on the stability of functional equations, Aequationes Math. 27 (1984) 76–86. [4] H.-Y. Chu, A. Kim, and S.K. Yoo, On the stability of generalized cubic set-valued functional equation, Appl. Math. Lett. 37 (2014) 7–14. [5] H.-Y. Chu and S. K. Yoo, On the stability of an additive set-valued functional equation, J. Chungcheong Math. Soc. 27 (2014) 455–467. [6] S. Czerwik, On the staility of the quadratic mapping in normed spaces, Abh. Math. Sem. Univ. Hamburg 62 (1992) 59–64. [7] P. Gˇ avruta, A generalization of the Hyers-Ulam-Rassias stability of approximately additive mappings, J. Math. Anal. Appl. 184 (1994) 431–436. [8] D.H. Hyers, On the stability of the linear functional equation, Proc. Nat. Acad. Sci. 27 (1941) 222–224. [9] S.-M. Jung and Z.-H. Lee, A fixed point approach to the stability of quadratic functional equation with involution, Fixed Point Theory Appl. Vol. 2008, Article ID 732086, (2008) 11pages. [10] D.S. Kang and H.-Y. Chu, Stability problem of Hyers-Ulam-Rassias for generalized forms of cubic functional equation, Acta Mathematica Sinica, English Series, 24 (3) (2008) 491–502. [11] H.A. Kenary, H. Rezaei, Y. Gheisari and C. Park, On the stability of set-valued functional equations with the fixed point alternative, Fixed Point Theory and Appl. 2012 2012:81, 17pp. [12] G. Lu and C. Park, Hyers-Ulam stability of additive set-valued functional euqtions, Appl. Math. Lett. 24 (2011) 1312–1316. [13] B. Margolis and J.B. Diaz, A fixed point theorem of the alternative for contractions on a generalized complete metric space, Bull. Amer. Math. Soc. 126 (1968) 305–309. [14] C. Park, D. O’Regan and R. Saadati, Stabiltiy of some set-valued functional equations, Appl. Math. Lett. 24 (2011) 1910–1914. [15] Th.M. Rassias, On the stability of the linear mapping in Banach spaces, Proc. Amer. Math. Soc. 72 (1978) 297–300. [16] H. R˚ adstr¨ om, An embedding theorem for spaces of convex sets, Proc. Amer. Math. Soc. 3 (1952) 165–169. [17] F. Skof, Propriet` a locali e approssimazione di operatori, Rend. Semin. Mat. Fis. Milano, 53 (1983) 113–129. [18] S.M. Ulam, A Collection of the Mathematical Problems, Interscience Publ. New York, 1960. Hahng-Yun Chu, Department of Mathematics, Chungnam National University, 99 Daehangno, Yuseonggu, Daejeon 305-764, Korea E-mail address: [email protected] Seung Ki Yoo, Department of Mathematics, Chungnam National University, 99 Daehangno, Yuseong-gu, Daejeon 305-764, Korea E-mail address: [email protected]

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COMMON BEST PROXIMITY POINTS FOR PROXIMALLY COMMUTING MAPPINGS IN NON-ARCHIMEDEAN PM-SPACES GEORGE A. ANASTASSIOU, YEOL JE CHO, REZA SAADATI, AND YOUNG-OH YANG*

Abstract. In this paper, we prove new common best proximity point theorems for proximally commuting mappings in complete non-Archimedean PM-spaces. Our results generalized the recent results of S. Basha [Common best proximity points: global minimization of multi-objective functions, J. Global Optim. 49(2011), 15–21] and C. Mongkolkeha, P. Kumam [Some common best proximity points for proximity commuting mappings, Optim. Lett. 7 (2013), 1825–1836].

1. Introduction Best proximity point theorems provide sufficient conditions that ensure the existence of approximate solutions which are optimal as well. In fact, if there is no solution to the fixed point equation T x = x for a non-self mapping T : A → B, then it is desirable to determine an approximate solution x such that the error Fx,T x (t) is maximum. A classical best approximation theorem was introduced by Fan [13], that is, if A is a nonempty compact convex subset of a Hausdorff locally convex topological vector space B and T : A → B is a continuous mapping, then there exists an element x ∈ A such that d(x, T x) = d(T x, A). Afterward, several authors, including Prolla [22], Reich [23], Sehgal and Singh [32, 33] and others, have derived some extensions of Fan’s theorem in many directions. Other works of the existence of a best proximity point for contractions can be seen in [2, 5, 12, 15]. In 2005, Anthony Eldred, Kirk and Veeramani [6] have obtained best proximity point theorems for relatively nonexpansive mappings. Since then, best proximity point theorems for several types of contractions have been established in [3, 4, 8, 12, 16, 17, 19, 20, 26, 27, 28, 29, 30, 36, 37, 38, 39, 40]. 2. Preliminaries Throughout this paper, the space of all probability distribution functions (briefly, d.f.’s) is denoted by ∆+ = {F : R ∪ {−∞, +∞} −→ [0, 1] : F is left-continuous and non-decreasing on R, F (0) = 0 and F (+∞) = 1} and the subset D+ ⊆ ∆+ is the set D+ = {F ∈ ∆+ : l− F (+∞) = 1}. Here l− f (x) denotes the left limit of the function f at the point x and l− f (x) = limt→x− f (t). The space ∆+ is partially ordered by the usual point-wise ordering of functions, i.e., F ≤ G if and only if F (t) ≤ G(t) for all t in R. The maximal element for ∆+ in this order is the d.f. given by ( 0, if t ≤ 0, ε0 (t) = 1, if t > 0. Definition 2.1. ([31]) A mapping ∗ : [0, 1] × [0, 1] → [0, 1] is a continuous t–norm if ∗ satisfies the following conditions: (a) ∗ is commutative and associative; (b) ∗ is continuous; (c) a ∗ 1 = a for all a ∈ [0, 1]; 2010 Mathematics Subject Classification. Primary 90C26, 90C30; Secondary 47H09, 47H10. Key words and phrases. Common best proximity point; common fixed point; proximally commuting mapping; PMspace; non-Archimedean PM-space. *The corresponding author. 1

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(d) a ∗ b ≤ c ∗ d whenever a ≤ c and c ≤ d, and a, b, c, d ∈ [0, 1]. Two typical examples of continuous t–norm are a ∗ b = ab and a ∗ b = min(a, b). A t–norm ∗ is said to be positive ([31]) if a ∗ b > 0 whenever a, b ∈ (0, 1]. The notation ∗ < ∗0 means that a ∗ b < a ∗0 b for all a, b ∈ (0, 1). Definition 2.2. (1) A Probabilistic Metric space (briefly, PM-space) is a triple (X, F, ∗), where X is a nonempty set, T is a continuous t–norm and F is a mapping from X × X into D+ such that, if Fx,y denotes the value of F at the pair (x, y), the following conditions hold: (PM1) Fx,y (t) = ε0 (t) for all t > 0 if and only if x = y; (PM2) Fx,y (t) = Fy,x (t); (PM3) Fx,z (t + s) ≥ Fx,y (t) ∗ Fy,z (s) for all x, y, z ∈ X and t, s ≥ 0. (2) If, in the above definition, the triangular inequality (PM3) is replaced by (PM4) Fx,z (max{t, s}) ≥ Fx,y (t) ∗ Fy,z (s) for all x, y, z ∈ X and t, s ≥ 0, then the triple (X, F, ∗) is called a non-Archimedean PM-space (briefly, NA-PM-space). It is easy to check that the triangular inequality (PM4) implies (PM3), that is, every NA-PM-space is itself a PM-space. It is easy to show that (PM4) is equivalent to the following condition: (PM5) Fx,z (t) ≥ Fx,y (t) ∗ Fy,z (t) for all x, y, z ∈ X and t ≥ 0. Example 2.3. Let (X, d) be an ordinary metric space and let θ be a nondecreasing and continuous function from (0, ∞) into (0, 1) such that limt→∞ θ(t) = 1. Some examples of these functions are as follows: t θ(t) = , θ(t) = 1 − e−t , θ(t) = e−1/t . t+1 Let a ∗ b ≤ ab for each a, b ∈ [0, 1]. For each t ∈ (0, ∞), define Fx,y (t) = [θ(t)]d(x,y) for all x, y ∈ X. Then (X, F, ∗) is a NA-PM-space ([1]). For more details and examples of these spaces see also [7], [9], [10], [11], [14], [18], [21], [24], [25], [34], [35], [41] and [42]. Definition 2.4. Let (X, F, T ) be a NA-PM-space. (1) A sequence {xn } in X is said to be convergent to a point x ∈ X if, for any  > 0 and λ > 0, there exists positive integer N such that Fxn ,x () > 1 − λ whenever n ≥ N ; (2) A sequence {xn } in X is called a Cauchy sequence if, for any  > 0 and λ > 0, there exists positive integer N such that Fxn+p ,xn () > 1 − λ whenever n ≥ N and p ∈ N ; (3) A PM-space (X, F, ∗) is said to be complete if every Cauchy sequence in X is convergent to a point in X. Definition 2.5. Let (X, F, ∗) be a PM-space. For each p in X and λ > 0, the strong λ-neighborhood of p is the set Np (λ) = {q ∈ X : Fp,q (λ) > 1 − λ} S and the strong neighborhood system for X is the union p∈V Np , where Np = {Np (λ) : λ > 0}.

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The strong neighborhood system for X determines a Hausdorff topology for X. Theorem 2.6. ([31]) If (X, F, ∗) is a PM-space and {pn } and {qn } are sequences such that pn → p and qn → q as n → ∞, then limn→∞ Fpn ,qn (t) = Fp,q (t) for all continuity point t of Fp,q . Let A and B be two nonempty subsets of a PM-space and t > 0. The following notions and notations are used in the sequel. FA,B (t) := sup{Fx,y (t) : x ∈ A, y ∈ B}, A0 := {x ∈ A : Fx,y (t) = FA,B (t) for some y ∈ B}, B0 := {y ∈ B : Fx,y (t) = FA,B (t) for some x ∈ A}. Definition 2.7. A mapping T : X → X is said to be a contraction if there exists a constant k ∈ [0, 1) such that FT x,T y (kt) ≥ Fx,y (t)

(2.1) for all x, y ∈ X and t > 0.

Definition 2.8. A mapping T : X → X is said to be a weak contraction if FT x,T y (φ(t)) ∗ Fx,y (t) ≥ Fx,y (φ(t))

(2.2)

for all x, y ∈ X and t > 0, where ϕ : [0, ∞) → [0, ∞) is a continuous and nondecreasing function such that, for any t > 0, 0 < ϕ(t) < t. Definition 2.9. A point x ∈ A is said to be a best proximity point of a mapping S : A → B if it satisfies the following condition: Fx,Sx (t)) = FA,B (t) for all x, y ∈ X and t > 0. It can be observed that a best proximity reduces to a fixed point if the underlying mapping is a self-mapping. Definition 2.10. Let S : A → B and T : A → B be two mappings. An element x∗ ∈ A is said to be a common best proximity point if it satisfies the following condition: Fx∗ ,Sx∗ (t) = Fx∗ ,T x∗ (t) = FA,B (t) for each t > 0. Observe that a common best proximity point is an element at which the multi-objective functions x → Fx,Sx (t) and x → Fx,T x (t) attain a common global maximum since Fx,Sx (t) ≤ FA,B (t) and Fx,T x (t) ≤ FA,B (t) for all x and t > 0. Definition 2.11. A mapping S : A → B and T : A → B is said to be a proximally commuting if they satisfy the following condition: [Fu,Sx (t) = Fv,T x (t) = FA,B (t)] =⇒ Sv = T u for all u, v, x, ∈ A and t > 0. It is easy to see that the proximal commutativity of self-mappings become commutativity of the mappings. Definition 2.12. Two mappings S : A → B and T : A → B are said to be a proximally swapped if they satisfy the following condition: [Fy,u (t) = Fy,v (t) = FA,B (t),

Su = T v] =⇒ Sv = T u

for all u, v, ∈ A, y ∈ B and t > 0.

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Definition 2.13. A set A is said to be approximatively compact with respect to a set B if every sequence {xn } in A satisfies the condition that Fy,xn (t) → Fy,A (t) for some y ∈ B and for each t > 0 has a convergent subsequence. Observe that every set is approximatively compact with respect to itself. Also, every compact set is approximatively compact with respect to any set. Moreover, A0 and B0 are nonempty set if A is compact and B is approximatively compact with respect to A. 3. Main result Now, we give our main results in this paper. Theorem 3.1. Let A and B be nonempty closed subsets of a complete NA-PM-space (X, F, ∗) in which the t–norm ∗ is positive and ∗ < min such that A is approximatively compact with respect to B. Also, assume that A0 and B0 are nonempty. Let S : A → B, T : A → B be the non-self mappings satisfying the following conditions: (a) for each x and y are elements in A and t > 0, FSx,Sy (φ(t)) ∗ FT x,T y (t) ≥ FT x,T y (φ(t)), where ϕ : [0, ∞) → [0, ∞) is a continuous and nondecreasing function such that, for any t > 0, 0 < ϕ(t) < t; (b) T is continuous; (c) S and T commute proximally; (d) S and T can be swapped proximally; (e) S(A0 ) ⊆ B0 and S(A0 ) ⊆ T (A0 ). Then there exists an element x ∈ A such that Fx,T x (t) = FA,B (t) and Fx,Sx (t) = FA,B (t). Moreover, if x∗ is another common best proximity point of the mappings S and T , then it is necessary that Fx,x∗ (t) ≥ FA,B (t) ∗ FA,B (t) for all t > 0. Proof. Let x0 be a fixed element in A0 . In view of the fact that S(A0 ) ⊆ T (A0 ), there exists an element x1 ∈ A0 such that Sx0 = T x1 . Again, since S(A0 ) ⊆ T (A0 ), there exists an element x2 ∈ A0 such that Sx1 = T x2 . By the similar fashion, we can find a sequence {xn } in A0 such that (3.1)

Sxn−1 = T xn

for all n ∈ N. It follows that (3.2)

FSxn ,Sxn+1 (φ(t)) ∗ FT xn ,T xn+1 (t) ≥ FT xn ,T xn+1 (φ(t))

and (3.3)

FSxn ,Sxn+1 (φ(t)) ∗ FSxn−1 ,Sxn (t) ≥ FSxn−1 ,Sxn (φ(t))

for all t > 0. Thus we have (3.4)

FSxn ,Sxn+1 (t) ≥ FSxn−1 ,Sxn (t)

for all t > 0, which means that the sequence {FSxn−1 ,Sxn (t)} is non-decreasing and bounded above. Hence there exists r ≤ 1 such that, for any t > 0, (3.5)

lim FSxn−1 ,Sxn (t) = r.

n→∞

If r < 1, then we have (3.6)

FSxn ,Sxn+1 (φ(t)) ∗ FSxn−1 ,Sxn (t) ≥ FSxn−1 ,Sxn (φ(t))

for all t > 0. Taking n → ∞ in the inequality (3.6), by the continuity of ϕ, we get a ∗ r ≥ a, where a = limn→∞ FSxn−1 ,Sxn (φ(t)), which is a contradiction unless r = 1. Therefore, it follows that (3.7)

lim FSxn−1 ,Sxn (t) = 1.

n→∞

By the property of F , we conclude that FSxn−1 ,Sxn (t) tend to 1 for all t > 0.

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Next, we prove that {Sxn } is a Cauchy sequence. We consider two cases. Case I. Suppose that there exits n ∈ N such that Sxn = Sxn+1 . Then we observe that FSxn+1 ,Sxn+2 (φ(t)) ∗ FT xn+1 ,T xn+2 (t) ≥ FT xn+1 ,T xn+2 (φ(t)) and FSxn+1 ,Sxn+2 (φ(t)) ∗ FSxn ,Sxn+1 (t) ≥ FSxn ,Sxn+1 (φ(t)) for all t > 0. Then we have FSxn+1 ,Sxn+2 (t) = 1 for all t > 0, which implies that Sxn+1 = Sxn+2 and so, for each m > n, we conclude that Sxm = Sxn . Hence {Sxn } is a Cauchy sequence in B. Case II. The successive terms of {Sxn } are different. Suppose that {Sxn } is not a Cauchy sequence. Then there exists ε > 0, t > 0 and the subsequences {Sxmk }, {Sxnk } of {Sxn } with nk > mk ≥ k such that (3.8)

FSxmk ,Sxnk (t) ≤ 1 − ε,

FSxmk ,Sxnk −1 (t) > 1 − ε.

By using (3.8) and the triangular inequality, we have (3.9)

1 − ε ≥ FSxmk ,Sxnk (t) ≥ FSxmk ,Sxnk −1 (t) ∗ FSxnk −1 ,Sxnk (t) ≥ (1 − ε) ∗ FSxnk −1 ,Sxnk (t).

Thus, using (3.9) and (3.7), we have FSxmk ,Sxnk (t) → 1 − ε

(3.10)

as k → ∞. Again, by the triangular inequality, we have (3.11)

FSxmk ,Sxnk (t) ≥ FSxmk ,Sxmk +1 (t) ∗ FSxmk +1 ,Sxnk +1 (t) ∗ FSxnk +1 ,Sxnk (t)

and (3.12)

FSxmk +1 ,Sxnk +1 (t) ≥ FSxmk +1 ,Sxmk (t) ∗ FSxmk ,Sxnk (t) ∗ FSxnk ,Sxnk +1 (t).

From (3.7), (3.10), (3.11) and (3.12), it follows that FSxmk +1 ,Sxnk +1 (t) → 1 − ε

(3.13) as k → ∞. In view of the fact that (3.14)

FSxmk +1 ,Sxnk +1 (φ(t)) ∗ FT xmk +1 ,T xnk +1 (t) ≥ FT xmk +1 ,T xnk +1 (φ(t)),

we have (3.15)

FSxmk +1 ,Sxnk +1 (φ(t)) ∗ FSxmk ,Sxnk (t) ≥ FSxmk ,Sxnk (φ(t)).

Letting k → ∞ in the inequality (3.15), we obtain a ∗ (1 − ε) ≥ a, where a = FSxmk +1 ,Sxnk +1 (φ(t)), which is a contradiction by the property of ϕ. Then we deduce that {Sxn } is a Cauchy sequence in B. Since B is a closed subset a complete NA-PM-space X, there exists y ∈ B such that Sxn → y as n → ∞. Consequently, it follows that the sequence {T xn } also converges to y. From S(A0 ) ⊆ B0 , there exists an element un ∈ A such that (3.16)

FSxn ,un (t) = FA,B (t)

for all n ∈ N and t > 0. Thus it follows from (3.1) and (3.16) that (3.17)

FT xn ,un−1 (t) = FSxn−1 ,un−1 (t) = FA,B (t)

for all n ∈ N and t > 0. By (3.16), (3.17) and the fact that the mappings S and T are proximally commuting, we obtain (3.18)

T un = Sun−1

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for all n ∈ N. Moreover, we have (3.19)

Fy,A (t) ≥ ≥ = ≥

Fy,un (t) Fy,Sxn (t) ∗ FSxn ,un (t) Fy,Sxn (t) ∗ FA,B (t) Fy,Sxn (t) ∗ Fy,A (t),

for all t > 0. Therefore, it follows that, for all t > 0, Fy,un (t) → Fy,A (t) as n → ∞. Since A is approximatively compact with respect to B, there exists a subsequence {unk } of the sequence {un } such that {unk } converges to some element u ∈ A. Further, since Fy,unk −1 (t) → Fy,A (t) for all t > 0 and A is approximatively compact with respect to B, there exists a subsequence {unkj −1 } of the sequence {unk −1 } such that {unkj −1 } converges to some element v ∈ A. By the continuity of the mappings S and T , we have (3.20)

T u = lim T unkj = lim Sunkj −1 = Sv j→∞

k→∞

and (3.21)

Fy,u (t)

=

Fy,v (t)

=

lim FSxnk ,unk (t) = FA,B (t),

k→∞

lim FT xnk

j→∞

j

,unk

j

−1

(t) = FA,B (t).

Since S and T can be swapped proximally, we have (3.22)

T v = Su.

Next, we prove that Su = Sv. Suppose that Su 6= Sv. Then, by (3.20), (3.21), (3.22) and the property of ϕ, we have FSu,Sv (φ(t)) ∗ FT u,T v (t) ≥ FT u,T v (φ(t)) and so FSu,Sv (φ(t)) ∗ FSu,Sv (t) ≥ FSu,Sv (φ(t)) for all t > 0, which is a contradiction. Thus Su = Sv and also T u = Su. Since S(A0 ) is contained in B0 , there exists an element x ∈ A such that Fx,T u (t) = FA,B (t) and Fx,Su (t) = FA,B (t). Since S and T are proximally commuting, we have Sx = T x. Consequently, we have (3.23)

FSu,Sx (φ(t)) ∗ FT u,T x (t) ≥ FT u,T x (φ(t))

and so (3.24)

FSu,Sx (φ(t)) ∗ FSu,Sx (t) ≥ FSu,Sx (φ(t))

for all t > 0, which is impossible if Su 6= Sx. Thus we have Su = Sx and hence T u = T x. It follows that Fx,T x (t) = Fx,T u (t) = FA,B (t) and Fx,Sx (t) = Fx,Su (t) = FA,B (t) for all t > 0. Therefore, x is a common best proximity point of S and T . To prove the uniqueness of the point x, suppose that x∗ is another common best proximity point of the mappings S and T . Then we have Fx∗ ,T x∗ (t) = FA,B (t),

Fx∗ ,Sx∗ (t) = FA,B (t)

for all t > 0. Since S and T are proximally commuting, we get Sx = T x and Sx∗ = T x∗ . Consequently, we have (3.25)

FSx∗ ,Sx (φ(t)) ∗ FT x∗ ,T x (t) ≥ FT x∗ ,T x (φ(t))

and so (3.26)

FSx∗ ,Sx (φ(t)) ∗ FSx∗ ,Sx (t) ≥ FSx∗ ,Sx (φ(t))

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for all t > 0, which is impossible if Sx∗ 6= Sx. Thus we have Sx = Sx∗ . Moreover, it can be concluded that Fx,x∗ (t) ≥ Fx,Sx (t) ∗ FSx,Sx∗ (t) ∗ FSx∗ ,x∗ (t) ≥ FA,B (t) ∗ FA,B (t) for all t > 0. This completes the proof.  Corollary 3.2. Let A be a nonempty closed subset of a complete NA-PM-space (X, F, ∗) in which the t–norm ∗ is positive and ∗ < min such that A is compact. Let S : A → A, T : A → A be the self mappings satisfying the following conditions: (a) for each x and y are elements in A and t > 0, FSx,Sy (φ(t)) ∗ FT x,T y (t) ≥ FT x,T y (φ(t)), where ϕ : [0, ∞) → [0, ∞) is a continuous and nondecreasing function such that, for any t > 0, 0 < ϕ(t) < t; (b) T is continuous; (c) S and T commutative; (e) S(A) ⊆ A and S(A) ⊆ T (A). Then S and T have common fixed point. 4. An example Now, we give an example to illustrate Theorem 3.1. Example 4.1. Consider the complete metric space R2 with Euclidean metric. Define t F(x1 ,x2 ),(y1 ,y2 ) (t) = t + |x1 − y1 | + |x2 − y2 | for all t > 0 and F(x1 ,x2 ),(y1 ,y2 ) (t) = 0 for all t ≤ 0. It is easy to show that (X, F, ·) is a NA-PM-space. Let A = {(x, 1) : 0 ≤ x ≤ 1},

B = {(x, −1) : 0 ≤ x ≤ 1}.

Define two mappings S : A → B, T : A → B as follows:

S(x, 1) = 0, −1 , T (x, 1) = x, −1 , t respectively. It is easy to see that FA,B (t) = t+2 , A0 = A and B0 = B. Further, S and T are continuous and A is approximatively compact with respect to B. First, we show that S and T are satisfy the condition (a) of Theorem 3.1 with ϕ : [0, ∞) → [0, ∞) t defined by ϕ(t) = for all t ∈ [0, ∞). Let (x, 1), (y, 1) ∈ A. Without loss of generality, we can take 2 x > y. Then we have

FS(x,1),S(y,1) (φ(t)) ∗ FT (x,1),T (y,1) (t)

=

1·

t t+|x−y| t

2 ≥ t +|x−y| 2 = FT (x,1),T (y,1) (φ(t))

for all t > 0. Next, we show that S and T are proximally commuting. Let (u, 1), (v, 1), (x, 1) ∈ A be such that t t , F(v,1),T (x,1) (t) = FA,B (t) = t+2 t+2 for all t > 0. It follows that u = 0 and v = x and hence
S(v, 1) = 0, −1 = (u, −1) = T (u, 1). F(u,1),S(x,1) (t) = FA,B (t) =

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Finally, we show that S and T are proximally swapped. If it is true that t F(u,1),(y,−1) (t) = F(v,1),(y,−1) (t) = FA,B (t) = , S(u, 1) = T (v, 1) t+2 for some (u, 1), (v, 1) ∈ A and (y, −1) ∈ B, then we get u = v = 0 and so S(v, 1) = T (u, 1). Therefore, all the hypothesis of Theorem 3.1 are satisfied. Furthermore, (0, 1) ∈ A is a common best proximity point of the mappings S and T since F(0,1),S(0,1) (t) = F(0,1),(0,−1) (t) = F(0,1),T ((0,1) (t) = FA,B (t) for all t > 0. Competing interests The authors declare that they have no competing interests. Author’s contributions All authors carried out the proof. All authors conceived of the study, and participated in its design and coordination. All authors read and approved the final manuscript. Acknowledgement The first author was supported by Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Science, ICT and future Planning (2014R1A2A2A01002100). References [1] I. Altun and D. Mihet, Ordered non-Archimedean fuzzy metric spaces and some fixed point results, Fixed Point Theory Appl. 2010, Article ID 782680, 11 pp. [2] M.A. Al-Thagafi and N. Shahzad, Convergence and existence results for best proximity points, Nonlinear Anal. 70 (2009), 3665–3671. [3] M.A. Al-Thagafi and N. Shahzad, Best proximity pairs and equilibrium pairs for Kakutani multimaps, Nonlinear Anal. 70 (2009), 1209–1216. [4] M.A. Al-Thagafi and N. Shahzad, Best proximity sets and equilibrium pairs for a finite family of multi-maps, Fixed Point Theory Appl. 2008, Article ID 457069, 10 pp. [5] A. Anthony Eldred and P. Veeramani, Existence and Convergence of best proximity points, J. Math. Anal. Appl. 323 (2006), 1001–1006. [6] A. Anthony Eldred, W.A. Kirk and P. Veeramani, Proximinal normal structure and relatively nonexpanisve mappings, Studia Math. 171(2005), 283–293. [7] S. Chauhan and B.D. Pant, Fixed point thms for compatible and subsequentially continuous mappings in Menger spaces, J. Nonlinear Sci. Appl. 7 (2014), 78–89. [8] Y.J. Cho, A. Gupta, E. Karapinar, P. Kumam and W. Sintunavarat, Tripled best proximity point theorems in metric spaces, Math. Inequal. Appl. 16(2013), 1197–1216. [9] Y.J. Cho, R. Saadati and J. Vahidi, Approximation of homomorphisms and derivations on non-Archimedean Lie C ∗ -algebras via fixed point method, Discrete Dyn. Nat. Soc. 2012, Article ID 373904, 9 pp. [10] Y.J. Cho and R. Saadati, Lattictic non-Archimedean random stability of ACQ functional equation, Advanc. Differ. Equat. 2011, 2011:31, 12 pp. [11] Y.J. Cho, C. Park and R. Saadati, Functional inequalities in non-Archimedean Banach spaces, Appl. Math. Lett. 23 (2010), 1238–1242. [12] C. Di Bari, T. Suzuki and C. Vetro, Best proximity points for cyclic Meir-Keeler contractions, Nonlinear Anal. 69 (2008), 3790–3794. [13] K. Fan, Extensions of two fixed point thms of F.E. Browder, Math. Z. 112 (1969), 234–240. [14] J.I. Kang and R. Saadati, Approximation of homomorphisms and derivations on non-Archimedean random Lie C ∗ algebras via fixed point method, J. Inequal. Appl. 2012, 2012:251, 10 pp. [15] S. Karpagam and S. Agrawal, Best proximity point thms for p-cyclic Meir-Keeler contractions, Fixed Point Theory Appl. 2009, Article ID 197308, 9 pp.

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[16] W.K. Kim, S. Kum and K.H. Lee, On general best proximity pairs and equilibrium pairs in free abstract economies, Nonlinear Anal. 68 (2008), 2216–222. [17] W.A. Kirk, S. Reich and P. Veeramani, Proximinal retracts and best proximity pair theorms, Numer. Funct. Anal. Optim. 24 (2003), 851–862. [18] D. Mihet¸, Common coupled fixed point thms for contractive mappings in fuzzy metric spaces, J. Nonlinear Sci. Appl. 6 (2013), 35–40. [19] C. Mongkolkeha and P. Kumam, Best proximity point Thms for generalized cyclic contractions in ordered metric spaces, J. Optim. Theory Appl. 155 (2012), 215–226. [20] C. Mongkolkeha and P. Kumam, Some common best proximity points for proximity commuting mappings. Optim. Lett. 7 (2013), 1825–1836. [21] C. Park, M. Eshaghi Gordji and A. Najati, Generalized Hyers-Ulam stability of an AQCQ-functional equation in non-Archimedean Banach spaces, J. Nonlinear Sci. Appl. 3 (2010), 272–281. [22] J.B. Prolla, Fixed point thms for set-valued mappings and existence of best approximations, Numer. Funct. Anal. Optim. 5 (1982-1983) 449–455. [23] S. Reich, Approximate selections, best approximations, fixed points and invariant sets, J. Math. Anal. Appl. 62 (1978), 104–113. [24] R. Saadati, S.M. Vaezpour and Z. Sadeghi, On the stability of Pexider functional equation in non-archimedean spaces, J. Inequal. Appl. 2011, 2011:17, 11 pp. [25] G. Sadeghi, R. Saadati, M. Janfada and J.M. Rassias, Stability of Euler-Lagrange quadratic functional equations in non-Archimedean normed spaces, Hacet. J. Math. Stat. 40 (2011), 571–579. [26] S. Sadiq Basha and P. Veeramani, Best approximations and best proximity pairs, Acta. Sci. Math. (Szeged), 63 (1997), 289–300. [27] S. Sadiq Basha and P. Veeramani, Best proximity pair thms for multifunctions with open fibres, J. Approx. Theory 103 (2000), 119–129. [28] S. Sadiq Basha, P. Veeramani and D.V. Pai, Best proximity pair theorems, Indian J. Pure Appl. Math. 32 (2001), 1237–1246. [29] S. Sadiq Basha, Best proximity point thms generalizing the contraction principle, Nonlinear Anal. 74 (2011), 5844– 5850. [30] S. Sadiq Basha, Common best proximity points: global minimization of multi-objective functions, J. Global Optim. 54 (2012), 367–373. [31] B. Schweizer and A. Sklar, Probabilistic Metric Spaces, Elsevier North Holand, New York, 1983. [32] V.M. Sehgal and S.P. Singh, A generalization to multifunctions of Fan’s best approximation theorem, Proc. Amer. Math. Soc. 102 (1988) 534–537. [33] V.M. Sehgal and S.P. Singh, A theorem on best approximations, Numer. Funct. Anal. Optim. 10 (1989) 181-184. [34] S. Shakeri, A contraction thm in Menger probabilistic metric spaces, J. Nonlinear Sci. Appl. 1 (2008), 189–193. [35] S. Shakeri, A note on the “A contraction thm in Menger probabilistic metric spaces”, J. Nonlinear Sci. Appl. 2 (2009), 25–26. [36] P.S. Srinivasan, Best proximity pair thms, Acta Sci. Math. (Szeged), 67 (2001), 421–429. [37] K. Wlodarczyk, R. Plebaniak and A. Banach, Best proximity points for cyclic and noncyclic set-valued relatively quasi-asymptotic contractions in uniform spaces, Nonlinear Anal. 70 (2009), 3332–3341. [38] K. Wlodarczyk, R. Plebaniak and A. Banach, Erratum to: Best proximity points for cyclic and noncyclic set-valued relatively quasi-asymptotic contractions in uniform spaces, Nonlinear Anal. 71 (2009), 3583–3586. [39] K. Wlodarczyk, R. Plebaniak and C. Obczynski, Convergence theorems, best approximation and best proximity for set-valued dynamic systems of relatively quasi-asymptotic contractions in cone uniform spaces, Nonlinear Anal. 72 (2010), 794–805. [40] W. Sintunavarat and P. Kumam, Coupled best proximity point theorem in metric spaces, Fixed Point Theory Appl. 2012, 2012:93. [41] J. Vahidi, C. Park and R. Saadati, A functional equation related to inner product spaces in non-Archimedean Lrandom normed spaces, J. Inequal. Appl. 2012, 2012:168, 16 pp. [42] C. Zaharia, On the probabilistic stability of the monomial functional equation. J. Nonlinear Sci. Appl. 6 (2013), 51–59. [43] S.S. Zhang, R. Saadati and G. Sadeghi, Solution and stability of mixed type functional equation in non-Archimedean random normed spaces, Appl. Math. Mech. (English Ed.) 32 (2011), 663–676.

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10

G.A. ANASTASSIOU, Y.J. CHO, R. SAADATI, AND Y.O. YANG

George A. Anastassiou, Department of Mathematical Sciences, University of Memphis, Memphis, TN 38152, U.S.A. E-mail address: [email protected] Yeol Je Cho, Department of Mathematics Education and the RINS, Gyeongsang National University, Jinju 660-701, Korea, and Department of Mathematics, Faculty of Science, King Abdulaziz University, Jeddah, Saudi Arabia E-mail address: [email protected] Reza Saadati Department of Mathematics, Iran University of Science and Technology, Tehran, Iran E-mail address: [email protected] Young-Oh Yang, Department of Mathematics, Jeju National University, Jeju 690-756, Korea E-mail address: [email protected]

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The Value Distribution of Some Difference Polynomials of Meromorphic Functions ∗ Jin Tu1†, Hong-Yan Xu2 and Hong Zhang 1

3

College of Mathematics and Information Science, Jiangxi Normal University, Nanchang, Jiangxi, 330022, China

2

Department of Informatics and Engineering, Jingdezhen Ceramic Institute, Jingdezhen, Jiangxi, 333403, China

3

School of the Tourism and Urban Management, Jiangxi University of Finance and Economics, Nanchang, Jiangxi, 330032, China

Abstract The purpose of this paper of some differ∏mis to investigate the nvalue distribution n ∏m ence polynomials G∏ f (z + c ) − af (z) , G (z) = f (z) 1 (z) = j 2 j=1 j=1 f (z + cj ) and G3 (z) = f (z)n m j=1 (f (z + cj ) − f (z)), where f (z) is a meromorphic function and a ∈ C \ {0} and cj , j = 1, 2, . . . , m are complex constants. Key words: meromorphic function; difference polynomial; zeros. Mathematical Subject Classification (2010): 30D35, 39A10.

1

Introduction and main results

This purpose of this paper is to study some properties of value distribution of some complex difference polynomials of meromorphic functions. The fundamental theorems and the standard notations of the Nevanlinna value distribution theory of meromorphic functions will be used(see [7, 15]). In addition, for meromorphic function f , we will use S(r, f ) to denote any quantity satisfying S(r, f ) = o(T (r,∫f )) for all r outside a possible exceptional set E of finite logarithmic measure limr→∞ [1,r)∩E dt t < ∞. We use ρ(f ), λ(f ) and λ( f1 ) to denote the order, the exponent of convergence of zeros and the exponent of convergence of poles of f (z) respectively. ∗ This project is supported by the National Natural Science Foundation of China (11301233, 11261024, 61202313), the Natural Science Foundation of Jiangxi Province in China (20132BAB211001, 20132BAB211002, 20122BAB211005) and the Foundation of Education Bureau of Jiangxi Province in China (GJJ14271,GJJ14272). † Corresponding author.

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Many people were interested in the value distribution of different expressions of meromorphic function and obtained lots of valuable theorems. In 1959, Hayman [8] studied value distribution of meromorphic function and its derivatives, and obtained the following famous theorems. Theorem 1.1 [8]. Let f (z) be a transcendental entire function. Then (i) for n ≥ 3 and a ̸= 0, Ψ(z) = f ′ (z) − af (z)n assumes all finite values infinitely often. (ii) for n ≥ 2, Φ(z) = f ′ (z)f (z)n assumes all finite values except possibly zero infinitely often. However, Mues [12] proved that the conclusion of Theorem 1.1 is not true for n = 3 by providing a counter example and proved that f ′ (z) − af (z)4 has infinitely many zeros. Recently, the topic of difference product in the complex plane C has attracted many researchers, a number of papers have focused on value distribution of differences and differences operator analogues of Nevanlinna theory (including [2, 4, 5, 6, 11]). In 2007, Laine and Yang [9] proved the following result, which is regarded as a difference counterpart of Theorem 1.1. Theorem 1.2 [9]. Let f (z) be a transcendental entire function of finite order, and c be a non-zero complex constant. Then for n ≥ 2, Φ1 (z) = f (z + c)f (z)n assumes every non-zero value a ∈ C infinitely often. It is well known that ∆f (z) = f (z + c) − f (z), where c ∈ C \ {0} is a constant satisfying f (z + c) − f (z) ̸≡ 0, which can be considered as the difference counterpart of f ′ (z). Similarly, ∆f (z) − af (z)n can be considered as the difference counterpart f ′ (z) − af (z)n , where a ∈ C \ {0}. In 2011, Chen [1] considered the difference counterpart of Theorem 1.1 and obtained the following theorems. Theorem 1.3 [1]. Let f (z) be a transcendental entire function of finite order, and let a, c ∈ C \ {0} be constants, with c such that f (z + c) ̸≡ f (z). Set Ψn (z) = ∆f (z) − af (z)n and n ≥ 3 is an integer. Then Ψn (z) assumes all finite values infinitely often, and for every b ∈ C one has λ(Ψn (z) − b) = ρ(f ). Theorem 1.4 [1]. Let f (z) be a transcendental entire function of finite order with a Borel exceptional value 0, and let a, c ∈ C \ {0} be constants, with c such that f (z + c) ̸≡ f (z). Then Ψ2 (z) assumes all finite values infinitely often, and for every b ∈ C one has λ(Ψ2 (z) − b) = ρ(f ). Theorem 1.5 [1]. Let f (z) be a transcendental entire function of finite order with a finite nonzero Borel exceptional value d, and let a, c ∈ C \ {0} be constants, with c such that f (z + c) ̸≡ f (z). Then for every b ∈ C with b ̸= −ad2 , Ψ2 (z) assumes the value b infinitely often, and λ(Ψ2 (z) − b) = ρ(f ). In 2013, Zheng and Chen [16] further investigated the value distribution of some difference polynomial of entire function and obtained the following theorem.

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Theorem 1.6 [16]. Let f (z) be a transcendental entire function of finite order with a finite nonzero Borel exceptional value d, and let a ∈ C \ {0}, c1 , c2 , . . . , cm be complex constants satisfying that at least ∏mone of them is non-zero. Then for 1 ≤ m < n and every b(̸= dm − adn ) ∈ C, G1 (z) = j=1 f (z + cj ) − af (z)n assumes the value b infinitely often and λ(G1 (z) − b) = ρ(f ). Thus, it is natural to ask: On What condition can Theorem 1.6 still hold when f (z) is a transcendental meromorphic function? The main purpose of this article is to study the above questions and obtain the following theorem. Theorem 1.7 Let f (z) be a transcendental meromorphic function of finite order with two Borel exceptional values d, ∞, and let a ∈ C \ {0}, c1 , c2 , . . . , cm be complex constants satisfying that at least one ∏mof them is non-zero. Then for 1 ≤ m < n and every b(̸= dm − adn ) ∈ C, G1 (z) = j=1 f (z + cj ) − af (z)n assumes the value b infinitely often and λ(G1 (z) − b) = ρ(f ). In addition, we further study the value distribution of some difference polynomials of meromorphic function of more general form G2 (z) = f (z)n

m ∏

f (z + cj ), G3 (z) = f (z)n

j=1

m ∏

[f (z + cj ) − f (z)]

j=1

and obtain the following results: Theorem 1.8 Let f (z) be a transcendental meromorphic function of finite order with two Borel exceptional values d, ∞, and let c1 , c2 , . . . , cm be nonzero complex constants. Then for n ≥ 1, G2 (z) assumes every value b(̸= dn+m ) ∈ C infinitely often and λ(G2 (z)− b) = ρ(f ). Corollary 1.1 Let f (z) be a transcendental meromorphic function of finite order with two Borel exceptional values 0, ∞, and let c1 , c2 , . . . , cm be nonzero complex constants. Then for n ≥ 1, G2 (z) assumes every nonzero value b ∈ C infinitely often and λ(G2 (z) − b) = ρ(f ). Example 1.1 Let f (z) = eez −2 +2 , it is easy to see that 0, ∞ are not Borel exceptional values. Let n = 1, m = 2, c1 = πi, c2 = −πi and b = 1, then we have G2 (z) = f (z)f (z + c1 )f (z + c2 ) − 1 = ez4−2 has no zeros. Let n = 2, m = 2, c1 = πi, c2 = −πi and b = 1, then we have G2 (z) = f (z)3 f (z + c1 )f (z + c2 ) − 1 = ez−4 +1 has no zeros. Hence, this shows that the condition in Corollary 1.1 is sharp in a sense. z

Theorem 1.9 Let f (z) be a transcendental meromorphic function of finite order with two Borel exceptional values d, ∞, and let c1 , c2 , . . . , cm be nonzero complex constants and f (z + cj ) ̸≡ f (z) for j = 1, 2, . . . , m. Then for n, m ≥ 1 are two integers, G3 (z) assumes every value b ∈ C \ {0} infinitely often and λ(G3 (z) − b) = ρ(f ). Remark 1.1 When b = 0, the conclusion may not hold. For example, let f (z) = ez , G3 (z) = f (z)n [f (z + πi) − f (z)]. Then G3 (z) = −2e(n+1)z has no zeros.

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Corollary 1.2 Let f (z) be a transcendental entire function of finite order with a Borel exceptional values d, and let c1 , c2 , . . . , cm be nonzero complex constants and f (z + cj ) ̸≡ f (z) for j = 1, 2, . . . , m. Then for n, m ≥ 1 are two integers, G3 (z) assumes every value b ∈ C infinitely often and λ(G3 (z) − b) = ρ(f ). Remark 1.2 It is easily to see that Theorem 1.9 is an improvement of the result in [10, Theorem 1.4], where they consider the case of m = 1 and the value b can replaced by a small function α(z). In fact, our results also can allow the value b to be a polynomial, even be a meromorphic function α(z) ̸≡ 0 satisfying ρ(α) < ρ(f ). Example 1.2 Let f (z) = ez + 2z, c = 2πi, α(z) = 4cz, n = 1 and m = 1. Then we know that f (z) has no Borel exceptional value, and we have G3 (z) = f (z)∆f (z) − 4cz = 2cez , which has no zeros. Hence, the condition on f (z) having a Borel exceptional value is necessary in Corollary 1.2. The following result of this paper is the value distribution of differential and difference polynomial of entire function. Theorem 1.10 Let f (z) be a transcendental entire function of finite order, and a, c1 , . . . , cm be nonzero complex constants. Then for any positive integers n ≥ 2m + 3, Ψ(z) = ∏m f (k) (z) j=1 f (z + cj ) − af (z)n assumes all finite values b ∈ C infinitely often. Regarding Theorem 1.2, we pose the following question. Question 1.1 What can be said if the condition n ≥ 2m + 3 in Theorem 1.10 is replaced with 1 ≤ n ≤ 2m + 2?

2

Some Lemmas

The following lemma is important in the fields of factorization and uniqueness theory of meromorphic functions which is given by Gross [3]. In 2010, Xu and Yi [13] made a small changed form as follows. Lemma 2.1 [13]. Suppose that fj (z)(j = 1, 2, . . . , n + 1) are meromorphic functions and gj (j 1, 2 . . . , n) are entire functions satisfying the following conditions. ∑= n (i) j=1 fj (z)egj (z) ≡ fn+1 . (ii) If 1 ≤ j ≤ n + 1, 1 ≤ k ≤ n, the order of fj is less than the order of egk (z) . If n ≥ 2, 1 ≤ j ≤ n + 1, 1 ≤ h < k ≤ n, and the order of fj (z) is less than the order of egh −gk . Then fj (z) ≡ 0(j = 1, 2, . . . , n + 1). Lemma 2.2 [15]. Let f be a nonconstant meromorphic function and P (f ) = a0 + a1 f + a2 f 2 + · · · + an f n , where a0 , a1 , a2 , · · · , an are constants and an ̸= 0. Then T (r, P (f )) = nT (r, f ) + S(r, f ).

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Lemma 2.3 [2, Theorem 2.1]. Let f (z) be a meromorphic function of finite order ρ and let c be a fixed nonzero complex number, then for each ε > 0, we have ( ) ( ) f (z) f (z + c) m r, + m r, = O(rρ−1+ε ) = S(r, f ). f (z) f (z + c) Lemma 2.4 [2, Corollary 2.5]. Let f (z) be a meromorphic function with order ρ = ρ(f ), ρ < +∞, and let η be a fixed nonzero complex number, then for each ε > 0, we have T (r, f (z + η)) = T (r, f ) + O(rρ−1+ε ) + O(log r). Lemma 2.5 [2, Theorem 2.2]. Let f be a meromorphic function with exponent of convergence of poles λ( f1 ) = λ < +∞, c ̸= 0 be fixed, then for each ε > 0, N (r, f (z + η)) = N (r, f ) + O(rλ−1+ε ) + O(log r). Lemma 2.6 [14]. If f (z) is a transcendental meromorphic function with exponent of convergence of poles λ( f1 ) = λ < +∞, c ̸= 0. Then, for each ε > 0, one has ( λ

1 f (z + c)

)

( =λ

1 f (z)

)

( = λ,

λ

1 ∆f

) ≤ λ.

Lemma 2.7 Let f (z) be a transcendental meromorphic function with exponent of convergence of poles λ( f1 ) = λ < ρ(f ) = ρ < +∞, and let c1 , c2 , . . . , cm be nonzero complex constants, and n, m ≥ 1 be integers. Then ρ(G2 ) = ρ(f ). Proof: We firstly prove that ρ(G2 ) ≤ ρ(f ). We can rewrite G2 (z) as the form G2 (z) = f (z)n+m

m ∏ f (z + cj ) . f (z) j=1

(1)

For each ε(0 < ε < ρ − λ), it follows by Lemma 2.3 and (1) that m(r, G2 ) ≤ (n + m)m(r, f ) +

m ∑ j=1

m(r,

f (z + cj ) ) + O(1) f (z)

(2)

= (n + m)m(r, f ) + O(rρ−1+ε ). By Lemma 2.5, we have N (r, G2 ) ≤ nN (r, f ) +

m ∑

N (r, f (z + cj ))

(3)

j=1

≤ (n + m)N (r, f ) + O(rλ−1+ε ) + O(log r). Since λ < ρ, it follows from (2) and (3) that T (r, G2 ) ≤ (n + m)T (r, f ) + O(rρ−1+ε ) + O(log r).

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So, we can get that ρ(G2 ) ≤ ρ(f ) easily. Next, we prove that ρ(G2 ) ≥ ρ(f ). From Lemma 2.3 and (1), we have (n + m)m(r, f ) = m(r, f n+m ) ≤ m(r, G2 ) +

m ∑

m(r,

j=1

f (z) ) + O(1) f (z + cj )

(5)

= m(r, G2 ) + O(rρ−1+ε ). Since λ(1/f ) = λ < ρ, for any given ε > 0 there exists r0 > 0 such that for all r > r0 we have N (r, f ) ≤ rλ+ε . (6) Thus, it follows from (5) and (6) that T (r, f ) ≤

1 m(r, G2 ) + O(rρ−1+ε ) + O(rλ+ε ), r > r0 . n+m

(7)

Since λ < ρ and 0 < ε < ρ − λ, it follows from (7) that ρ(G2 ) ≥ ρ(f ). Hence, the proof of Lemma 2.7 is proved. 2 By using the same argument as in Lemma 2.7, we can prove the following lemma easily. Lemma 2.8 Let f (z) be a transcendental meromorphic function with exponent of convergence of poles max{λ(f ), λ( f1 )} = λ < ρ(f ) = ρ < +∞, and let c1 , c2 , . . . , cm be nonzero complex constants such that f (z + cj ) ̸≡ f (z)(j = 1, 2, . . . , m), and n, m ≥ 1 be integers. Then ρ(G3 ) = ρ(f ). Lemma 2.9 [15, page 37]. Let f (z) be a nonconstant meromorphic function in the complex plane and l be a positive integer. Then T (r, f (l) (z)) ≤ (l + 1)T (r, f ) + S(r, f ), N (r, f (l) (z)) = N (r, f ) + lN (r, f ).

3 3.1

Proofs of Theorems 1.7, 1.8 and 1.9 The Proof of Theorem 1.7

We first prove ρ(G1 ) = ρ(f ). By Lemma 2.2 and Lemma 2.4, we have ρ(G1 ) ≤ ρ(f ). On the other hand, it follows from Lemma 2.4 that   m ∏ nT (r, f ) = T (r, af n ) + O(1) = T r, f (z + cj ) − G1 (z) + O(1) j=1

≤

m ∑

T (r, f (z + cj )) + T (r, G1 (z)) + O(1)

j=1

= mT (r, f ) + T (r, G1 (z)) + O(rρ−1+ε ) + O(log r), that is, (n − m)T (r, f ) ≤ T (r, G1 (z)) + O(rρ−1+ε ) + O(log r).

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Since 1 ≤ m < n, it follows from (8) that ρ(f ) ≤ ρ(G1 ). Hence, we can prove that ρ(G1 ) = ρ(f ). Since f (z) has two Borel exceptional values d, ∞, then f (z), f (z + cj ) can be written as the form f (z) = d +

g(z) g(z + cj ) exp{αz k }, f (z + cj ) = d + hj (z) exp{αz k }, p(z) p(z + cj )

(9)

where α ̸= 0 is a constant, k(≥ 1) is an integer satisfying ρ(f ) = k, and g(z), hj (z) = k−1 k eαkcj z +···+αcj are entire functions such that g(z)hj (z) ̸≡ 0, ρ(g) < k, ρ(hj ) ≤ k − 1, j = 1, 2 . . . , m, and p(z) is the canonical product formed with the poles of f (z) satisfying ρ(p) = λ(p) = λ( f1 ) < ρ(f ). Set H(z) = g(z) p(z) ̸≡ 0, then we can see that ρ(H) < ρ(f ). Now we prove that λ(G1 − b) = ρ(f ). Suppose that λ(G1 − b) < ρ(f ). Since ρ(G1 ) = ρ(f ) = ρ(G1 − b), then λ(G1 − b) < ρ(G1 − b) = ρ(f ) = k and G1 (z) − b can be rewritten as the form G1 (z) − b =

g1∗ (z) exp{βz k } = H1∗ (z) exp{βz k }, p∗1 (z)

(10)

where β(̸= 0) is a constant, g1∗ (z)(̸≡ 0) is an entire function satisfying ρ(g1∗ ) < k. Thus, 1 1 by Lemma 2.6, we have ρ(p∗1 ) = λ(p∗1 ) ≤ max{λ( f (z) ), λ( f (z+c ), j = 1, 2, . . . , m} = j) 1 ∗ λ( f ) < ρ(f ) = k. So, we have ρ(H1 ) < ρ(f ) = k. Thus, from (9), (10) and the definition of G1 (z), we have   m ∑ ∏ k k H(z + ci )H(z + cj )hi (z)hj (z) e2αz H(z + cj )hj (z)emαz + · · · + dm−2  j=1

 + dm−1  (

m ∑

1≤i a}) > 0}. µ

Further, m is a σ-sup-decomposable measure [17]. More, Mesiar and Pap(see [17]) have showed that any σ-sup-decomposable measure generated as essential supremum of a continuous density can be obtained as a limit of pseudo-additive measures with respect to generted pseudo-addition. In this paper we will consider the semiring ([a, b], ⊕, ⊙) for three (with completely different behavior) cases, namely Case 1(a), 2 and 3(a). Observe that the Case 1(b) and 3(b) are linked to Case 1(a) and 3(a) by duality [21]. First, if the pseudo-operations are defined by a monotone and continuous surjective function g : [a, b] → [0, ∞] (i.e., Case 2), then the pseudo-integral for a measurable function f : X → [a, b] is given by ) ( ∫ ∫⊕ (g ◦ f ) d (g ◦ m) , f ⊙ dm = g −1 X

X

where the integral applied on the right side is the standard Lebesgue integral. In a special case, when X = [c, d], A = B(X) and m = g −1 ◦ µ, then the pseudo-integral reduces on the g-integral ( ) ∫d ∫⊕ −1 f dx = g g(f (x))dx . c

[c,d]

Second, if the semiring is of the form ([a, b], sup, ⊙) (i.e., Case 1(a) and Case 3(a)), then we shall consider complete sup-measure (shortly sup-measure) m only and A = 2X , i.e., for any family {Ei }i∈I of measurable sets, ∪ m( Ei ) = sup m(Ei ). i∈I

i∈I

If X is countable (especially, if X is finite) then any σ-sup-decomposable measure m is complete and, moreover, m(E) = sup ψ(x), where ψ : X → [a, b] is a density function given by ψ(x) = m({x}). Then x∈E

the pseudo-integral for a function f : X → [a, b] is given by ∫⊕ X

f ⊙ dm = sup (f (x) ⊙ ψ(x)), x∈X

where function ψ defines σ-sup-decomposable measure m.

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3

Main results

Theorem 3.1 A σ-⊕-decomposable measure m : A → [a, b] is monotone, if ⊕ satisfies one of the following conditions: (1) x ⊕ y = sup(x, y) on the interval [a, b]; (2) ⊕ has a strictly monotone and continuous surjective generator g. Proof. If E, F ∈ A and E ⊂ F , then F − E ∈ A . Since F = E ∪ (F − E), we get m(F ) = m(F − E) ⊕ m(E). If ⊕ satisfies Condition (1), then we have m(F ) = sup{m(F − E), m(E)} ≥ m(E), i.e., sup{m(E), m(F )} = m(F ). Hence, by x ≼ y if and only if sup(x, y) = y for all x, y ∈ [a, b], we have m(E) ≼ m(F ). If ⊕ satisfies Condition (2), then we have g(m(F )) = g(m(F − E)) + g(m(E)). Since g(x) ≥ 0 for all x ∈ [a, b], we have g(m(F )) ≥ g(m(E)). Hence, by x ≼ y if and only if g(x) ≤ g(y) for all x, y ∈ [a, b], we have m(E) ≼ m(F ). 2 It is easy to see that if F is a σ-⊕-decomposable measure zero set with respect to m, where m is a σ-⊕-decomposable measure, then E is a σ-⊕-decomposable measure zero set with respect to m, for all E ⊂ F. Theorem 3.2 Let (X, A ) be a measurable space. If m : A → [a, b] is a σ-⊕-decomposable measure and {En } ⊂ A (X) is an increasing sequence for which lim En ∈ A (X), then ( m

)

n→∞

lim En = lim m (En ).

n→∞

n→∞

Proof. We might write E0 = ∅. Since {En } is an increasing sequence, {Ei − Ei−1 } is a sequence of pairwise disjoint sets from A , then En =

n ∪

n→∞

i=1

Hence, we have

( m (En ) = m

) n (Ei − Ei−1 ) = ⊕ m (Ei − Ei−1 )

n ∪

and m ∞

) lim En = m

(∞ ∪

n→∞

(Ei − Ei−1 ).

i=1

i=1

i=1

(

∞ ∪

(Ei − Ei−1 ) and lim En =

) ∞ (Ei − Ei−1 ) = ⊕ m (Ei − Ei−1 ). i=1

i=1

n

By ⊕ xi = lim ⊕ xi for all {xi } ⊂ [a, b], we have i=1

n→∞ i=1

∞

n

⊕ m (Ei − Ei−1 ) = lim ⊕ m (Ei − Ei−1 ) .

( Consequently, we get m

i=1

n→∞ i=1

)

lim En = lim m (En ). 2

n→∞

n→∞

Theorem 3.3 Let (X, A ) be a measurable space and m : A → [a, b] be a σ-⊕-decomposable measure, where ⊕ has a strictly increasing (or decreasing) and continuous surjective generator g. If {En } ⊂ A (X) is a decreasing sequence, and there exists at least one l ∈ N such that m(El ) ≺ b (or m(El ) ≺ a). Then 5

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( m

) lim En = lim m (En ).

n→∞

n→∞

Proof. Suppose m(El ) ≺ b for some l ∈ N. For the case m(El ) ≺ a, we can prove it by a similar proof. By Theorem 3.1 and {En } is a decreasing sequence, we have m(En ) ≼ m(El ) ≺ b for all n ≥ l, and therefore

( m

) lim En ≺ b.

n→∞

By the monotonicity of g, we get

( ( )) g(m(El )) < +∞ and g m lim En < +∞. n→∞

Since {En } ia a decreasing sequence, {El − En } is an increasing sequence. By Theorem 3.2, we obtain ( ) ( ) m El − lim En = m lim (El − En ) = lim m (El − En ), n→∞

i.e.,

n→∞

n→∞

( ( )) ( ) g m El − lim En = g lim m (El − En ) . n→∞

n→∞

By the continuity of g and g ◦ m is a σ-additive measure, we have ( ( )) ( ( )) g (m (El )) − g m lim En = g m El − lim En n→∞ n→∞ ( ) = g lim m (El − En ) n→∞

lim g (m (El − En ))

=

n→∞

= g (m (El )) − lim g (m (En )) n→∞ ( ) = g (m (El )) − g lim m (En ) . n→∞

Since g(m(El )) < +∞, we have

( ( )) ( ) g m lim En = g lim m (En ) . n→∞

(

By the strictly monotonicity of g, we get m

)

n→∞

lim En = lim m (En ). 2

n→∞

n→∞

Lemma 3.1 [17] Let m be a σ-sup-decomposable measure which is defined by m(E) = ess sup{ψ(x)|x ∈ E}, µ

on ([0, ∞], B), where ψ : [0, ∞] → [0, ∞] is a continuous density. Then for any generator g there exist a family {mλ } of σ-⊕λ -decomposable measures on ([0, ∞], B), where ⊕λ is generated by g λ (the function g on the power λ), λ ∈ (0, ∞), such that lim mλ = m. λ→∞

Theorem 3.4 Let ([0, ∞], sup, ⊙) be a semiring with ⊙ generated by a strictly increasing and continuous surjective generator g. Let m be the same as in Lemma 3.1. If {En } ⊂ B([0, ∞]) is a decreasing sequence, m(En ) ≺ b for at least one n ∈ N, then ( ) m lim En = lim m (En ). n→∞

n→∞

Proof. Since m is the same one in Lemma 3.1, for the generator g there exist a family {mλ } of σ-⊕λ decomposable measures on ([0, ∞], B), where ⊕λ is generated by g λ , λ ∈ (0, ∞), such that lim mλ = m.

λ→∞

Let l ∈ N such that m(El ) ≺ b. Thus we have 6

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mλ (El ) ≺ b, λ ∈ (0, ∞). Since f (x) = xµ (µ ̸= 0) is strictly increasing, whenever µ > 0, g λ and g are comonotone functions. Hence, by Theorem 3.3, we have ( ) mλ lim En = lim mλ (En ), λ ∈ (0, ∞). n→∞

Consequently, we obtain that lim mλ

( i.e., m

)

λ→∞

(

n→∞

) lim En = lim lim mλ (En ) = lim lim mλ (En ),

n→∞

λ→∞n→∞

n→∞λ→∞

lim En = lim m (En ). 2

n→∞

n→∞

Theorem 3.5 Let (X, ⊕, ⊙) be a semiring with generated pseudo-operations by a strictly monotone and continuous surjective generator g, and let m : A (X) → [a, b] be a σ-⊕-decomposable measure. If f : X → [a, b] is a measurable function with respect to m and E is a σ-⊕-decomposable measure zero set on A (X), then ∫⊕

f ⊙ dm = 0.

E

Proof. If the pseudo-operations are generated by a strictly monotone and continuous surjective generator g, i.e., x⊕y = g −1 (g(x)+g(y)) and x⊙y = g −1 (g(x)·g(y)) for every x, y ∈ [a, b]. Then the pseudo-integral for a measurable function f : X → [a, b] is given by ) ( ∫ ∫⊕ g ◦ f dµ , f ⊙ dm = g −1 E

E

for every measurable set E, where µ = g◦m is a σ-additive measure. Since E is a set of σ-⊕-decomposable measure zero on A (X) and g(0) = 0,∫ we have that E is a measure zero set with respect to µ. Hence, by Theorem C of § 25 of [11], we have g ◦ f dµ = 0. Consequently, by the strictly monotonicity of g, we obtain that

∫⊕

E

f ⊙ dm = g −1 (0) = 0. 2

E

Example 3.1 Let X = R and let the pseudo-addition be represented by a strictly increasing and continuous generator surjective function g : [0, +∞] → [0, +∞], which is defined by g(x) = x for all x ∈ [0, +∞]. It is easy to see that x ⊕ y = x + y and x ⊙ y = x · y for all x, y ∈ [0, +∞]. Hence, we have 0 = 0, 1 = 1. We define a set function m on B(X) by m(E) = µ(E) for all E ∈ B(X), where µ is a Lebesgue measure. It is obvious that m satisfies (1) and (2) of Definition 2.3. Consequently, the set function m is a σ-⊕-decomposable measure. Let E be the set of rational numbers. Since µ(E) = 0, thus m(E) = 0, i.e., E is a set of σ-⊕-decomposable measure zero on B(X). Hence, for any measurable function f we have ∫⊕

f ⊙ dm = 0.

E

Theorem 3.6 Let (X, sup, ⊙) be a semiring with ⊙ generated by a strictly increasing and continuous surjective generator g, and let m : A (X) → [a, b] be a σ-sup-decomposable measure which is defined by m(E) = sup ψ(x), where ψ : X → [a, b] is a density function given by ψ(x) = m({x}). If f : X → [a, b] x∈E

is a measurable function with respect to m and E is a set of σ-sup-decomposable measure zero on A (X), then sup ∫

f ⊙ dm = 0.

E

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Proof. Let m : A → [a, b] be a σ-sup-decomposable measure which is defined by m(E) = sup ψ(x), x∈E

where ψ : X → [a, b] is a density function given by ψ(x) = m({x}). Then the pseudo-integral for a measurable function f : X → [a, b] is given by sup ∫

f ⊙ dm = sup (f (x) ⊙ ψ(x)), x∈E

E

for every measurable set E, where function ψ defines σ-sup-decomposable measure m. Since f (x) ≼ sup f (x) and ψ(x) ≼ sup ψ(x) = m(E) for all x ∈ E, by the monotonicity of g, we have x∈E

x∈E

(

)

g (f (x)) ≤ g sup f (x)

and g (ψ (x)) ≤ g (m (E)),

x∈E

for all x ∈ E. Hence, by g(x) ≥ 0 for all x ∈ [a, b], we have ( ) g (f (x)) · g (ψ (x)) ≤ g sup f (x) · g (m (E)), x∈E

which implies that

( ( ( ) )) ( −1 ) −1 g g (g (f (x)) · g (ψ (x))) ≤ g g g sup f (x) · g (m (E)) , x∈E

for all x ∈ E. Since ⊙ is generated by a generator g, i.e., y ⊙ z = g −1 (g(y)g(z)) for all y, z ∈ [a, b], we get that ( ) g (f (x) ⊙ ψ(x)) ≤ g sup f (x) ⊙ m(E) , x∈E

for all x ∈ E. Hence, we obtain that f (x) ⊙ ψ(x) ≼ sup f (x) ⊙ m(E) x∈E

for all x ∈ E, which implies that sup (f (x) ⊙ ψ(x)) ≼ sup f (x) ⊙ m(E). x∈E

x∈E

Since E is a set of σ-sup-decomposable measure zero on A (X), we have sup f (x) ⊙ m(E) = 0. Conx∈E

sequently, we have sup (f (x) ⊙ ψ(x)) ≼ 0. By the monotonicity of g and g(y) ≥ 0 for y ∈ [a, b], ( x∈E ) ( ) we have 0 ≤ g sup (f (x) ⊙ ψ(x)) ≤ g(0) = 0, i.e., g sup (f (x) ⊙ ψ(x)) = 0, which implies that sup ∫

x∈E

x∈E

f ⊙ dm = 0. 2

E

Theorem 3.7 Let (X, sup, inf) be a semiring, and let m be the same as in Theorem 3.6. If f : X → [a, b] is a measurable function with respect to m and E is a set of σ-sup-decomposable measure zero on A (X), then sup ∫

f ⊙ dm = 0.

E

Proof. Let m be the same as in Theorem 3.6. Then the pseudo-integral for a measurable function f : X → [a, b] is given by sup ∫ E

f ⊙ dm = sup (f (x) ⊙ ψ(x)), x∈E

for every measurable set E, where function ψ defines σ-sup-decomposable measure m. Since f (x) ≼ sup f (x) and ψ(x) ≼ sup ψ(x) = m(E) for all x ∈ E, and y ⊙ z = inf(y, z) for all y, z ∈ [a, b], we have x∈E

x∈E

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{ f (x) ⊙ ψ(x) = inf{f (x), ψ(x)} ≼ inf

} = sup f (x) ⊙ m(E),

sup f (x), m(E) x∈E

x∈E

for all x ∈ E, which implies that sup (f (x) ⊙ ψ(x)) ≼ sup f (x) ⊙ m(E). x∈E

x∈E

Since E is a set of σ-sup-decomposable measure zero on A (X), we have sup f (x) ⊙ m(E) = 0. Hence, we x∈E

have sup (f (x) ⊙ ψ(x)) ≼ 0. By y ≼ z if and only if sup(y, z) = z and y ⊕ z = sup(y, z) for all y, z ∈ [a, b], x∈E

we obtain that sup (f (x) ⊙ ψ(x)) ⊕ 0 = 0. x∈E

By (4) of Definition 2.1, we have sup (f (x) ⊙ ψ(x)) = 0, which implies that

sup ∫

x∈E

f ⊙ dm = 0. 2

E

Theorem 3.8 Let (X, ⊕, ⊙) be a semiring with generated pseudo-operations by a strictly monotone and continuous surjective generator g, and let m : A (X) → [a, b] be a σ-⊕-decomposable measure. If f : X → [a, b] is a measurable function and 0 ≺ f a.e. on a measurable set E with respect to m, and if ∫⊕

f ⊙ dm = 0,

E

then m(E) = 0. Proof. If the pseudo-operations are generated by a strictly monotone and continuous surjective generator g, i.e., y ⊕z = g −1 (g(y)+g(z)) and y ⊙z = g −1 (g(y)·g(z)) for every y, z ∈ [a, b]. Then the pseudo-integral for a measurable function f : X → [a, b] is given by ) ( ∫ ∫⊕ g ◦ f dµ , f ⊙ dm = g −1 E

E

for every measurable set E, where µ = g ◦ m is a σ-additive measure. Let E = E1 ∪E2 and E1 ∩E2 = ∅, where E2 = {x ∈ E |0 ≺ f (x) }. By y ≼ z if and only if g(y) ≤ g(z) for all y, z ∈ [a, b], the strictly monotonicity of g and g(0) = 0, we obtain that g(f (x)) > 0 for all x ∈ E2 . Since 0 ≺ f a.e. on a measurable set E with respect to m, we have m(E1 ) = 0. Hence, by Theorem 3.5, ∫⊕ we have f ⊙ dm = 0, which implies that E1

∫⊕

f ⊙ dm =

E

If

∫⊕

∫⊕

∫⊕

f ⊙ dm ⊕

E1

E2

∫⊕

f ⊙ dm.

E2

f ⊙ dm = 0, then we get that

E

∫⊕

( f ⊙ dm = g

−1

E2

i.e.,

f ⊙ dm =

∫

∫

) g ◦ f dµ

= 0,

E2

g ◦ f dµ = g(0) = 0. By Theorem D of § 25 of [11], we have µ(E2 ) = 0. Hence, we obtain that

E2

m(E2 ) = g −1 (µ(E)) = 0. Consequently, m(E) = m(E1 ) ⊕ m(E2 ) = 0. 2 Theorem 3.9 Let (X, sup, ⊙) be a semiring with ⊙ generated by a strictly increasing and continuous surjective generator g, and let m be the same as in Theorem 3.6. If f : X → [a, b] is a measurable function and 0 ≺ f a.e. on a measurable set E with respect to m, and if sup ∫

f ⊙ dm = 0,

E

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Proof. Let m be the same as in Theorem 3.6. Then the pseudo-integral for a measurable function f : X → [a, b] is given by sup ∫

f ⊙ dm = sup (f (x) ⊙ ψ(x)), x∈E

E

for every measurable set E, where function ψ defines σ-sup-decomposable measure m. Let E = E1 ∪ E2 and E1 ∩ E2 = ∅, where E2 = {x ∈ E |0 ≺ f (x) }. If 0 ≺ f a.e. on a measurable set E with respect to m, then m(E1 ) = 0, which implies that m(E) = sup{m(E1 ), m(E2 )} = m(E2 ). ( sup ) ( ) sup ∫ ∫ If f ⊙ dm = 0, then by g(0) = 0, we have g f ⊙ dm = g sup (f (x) ⊙ ψ(x)) = 0. By the E

x∈E

E

strictly monotonicity of g, we have

(

)

g(f (x) ⊙ ψ(x)) ≤ g sup (f (x) ⊙ ψ(x))

= 0,

x∈E

for all x ∈ E. Since g(y) ≥ 0 for all y ∈ [a, b], we have g(f (x) ⊙ ψ(x)) = 0, for all x ∈ E. If ⊙ is generated by a generator g, i.e., y ⊙ z = g −1 (g(y)g(z)) for all y, z ∈ [a, b], then we get that g(f (x)) · g(ψ(x)) = g(f (x) ⊙ ψ(x)) = 0. Since 0 ≺ f (x) for all x ∈ E2 , we get g(f (x)) > 0 for all x ∈ E2 . Thus, we have g(ψ(x)) = 0, i.e., ψ(x) = 0 for all x ∈ E2 ⊂ E. Consequently, we obtain taht m(E2 ) = sup ψ(x) = 0, which implies that x∈E2

m(E) = 0. 2

Theorem 3.10 Let (X, sup, inf) be a semiring, and let m be the same as in Theorem 3.6. If f : X → [a, b] is a measurable function and 0 ≺ f a.e. on a measurable set E with respect to m, and if sup ∫

f ⊙ dm = 0,

E

then m(E) = 0. Proof. Let m be the same as in Theorem 3.6. Then the pseudo-integral for a measurable function f : X → [a, b] is given by sup ∫

f ⊙ dm = sup (f (x) ⊙ ψ(x)), x∈E

E

for every measurable set E, where function ψ defines σ-sup-decomposable measure m. Let E = E1 ∪ E2 and E1 ∩ E2 = ∅, where E2 = {x ∈ E |0 ≺ f (x) }. If 0 ≺ f a.e. on a measurable set sup ∫ E with respect to m, then m(E1 ) = 0. Hence, by Theorem 3.7, we have f ⊙ dm = 0, which implies E1

that sup ∫

f ⊙ dm =

E

If

sup ∫

sup ∫ E1

f ⊙ dm = 0, then we get that

E

sup ∫

sup ∫

f ⊙ dm ⊕

f ⊙ dm =

E2

sup ∫

f ⊙ dm.

E2

f ⊙ dm = sup (f (x) ⊙ ψ(x)) = 0. Hence, we obtain that x∈E2

E2

f (x) ⊙ ψ(x) ≼ sup (f (x) ⊙ ψ(x)) = 0 = f (x) ⊙ 0, x∈E2

for all x ∈ E2 . By 0 ≺ f (x) for all x ∈ E2 and (2) of Definition 2.2, we have ψ(x) ≼ 0 for all x ∈ E2 , which implies that m(E2 ) = sup ψ(x) ≼ 0. Since f (x) ≼ sup f (x) and ψ(x) ≼ sup ψ(x) = m(E2 ) for x∈E2

x∈E2

x∈E2

all x ∈ E2 , and y ⊙ z = inf(y, z) for all y, z ∈ [a, b], we have { } f (x) ⊙ ψ(x) = inf{f (x), ψ(x)} ≼ inf sup f (x), m(E2 ) = sup f (x) ⊙ m(E2 ), x∈E2

x∈E2

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for all x ∈ E2 , which implies that sup f (x) ⊙ 0 = 0 = sup (f (x) ⊙ ψ(x)) ≼ sup f (x) ⊙ m(E2 ). x∈E2

x∈E2

x∈E2

By 0 ≺ f (x) for all x ∈ E2 and (2) of Definition 2.2, we have 0 ≼ m(E2 ). Consequently, we obtain that m(E2 ) = 0, which implies that m(E) = m(E1 ) ⊕ m(E2 ) = 0. 2 Theorem 3.11 Let (X, ⊕, ⊙) be a semiring with generated pseudo-operations by a strictly monotone and continuous surjective generator g, and let m be a σ-⊕-decomposable measure on A (X) and f be a ∫⊕ measurable function with respect to m on X. Then f ⊙ dm = 0 if and only if f = 0 a.e. for every E

measurable set E. Proof. If the pseudo-operations are generated by a strictly monotone and continuous surjective generator g, i.e., we have y ⊕ z = g −1 (g(y) + g(z)) and x ⊙ y = g −1 (g(y) · g(z)) for every y, z ∈ [a, b]. Then the pseudo-integral for a measurable function f : X → [a, b] is given by ( ) ∫⊕ ∫ f ⊙ dm = g −1 g ◦ f dµ E

E

for every measurable set E, where µ = g ◦ m is a σ-additive measure. ∫⊕ ∫ Suppose f ⊙ dm = 0 for every measurable set E. Since g(0) = 0, we have g ◦ f dµ = g(0) = 0 for E

E

every measurable set E. By Theorem E of § 25 of [11], we have g ◦ f = 0 a.e. for every measurable set E, which implies that f = 0 a.e. for every measurable set E. Suppose f = 0 a.e. for every measurable set E. Let E1 ∪ E2 = E and E1 ∩ E2 = ∅, where ∫⊕ E1 = {x ∈ E|f (x) = 0}. Then, we have m(E2 ) = 0. By Theorem 3.5, we have f ⊙ dm = 0. Hence, E2

we get that ∫⊕

f ⊙ dm =

E

∫⊕

f ⊙ dm ⊕

E1

∫⊕ E2

f ⊙ dm =

∫⊕

f ⊙ dm.

E1

Since f (x) = 0 for all x ∈ E1 , we have g(f (x)) = 0 for all x ∈ E1 . Hence, we obtain that i.e.,

∫⊕

f ⊙ dm = g −1 (0) = 0. Consequently,

E1

∫⊕

∫

g ◦ f dµ = 0,

E1

f ⊙ dm = g −1 (0) = 0. 2

E

Theorem 3.12 Let (X, sup, ⊙) be a semiring with ⊙ generated by a strictly increasing and continuous surjective generator g, and let m be the same as in Theorem 3.6 and f be a measurable function with sup ∫ respect to m on X. Then f ⊙ dm = 0 if and only if f = 0 a.e. for every measurable set E. E

Proof. Let m be the same as in Theorem 3.6. Then the pseudo-integral for a measurable function f : X → [a, b] is given by sup ∫

f ⊙ dm = sup (f (x) ⊙ ψ(x)),

E

x∈E

for every measurable set E, where function ψ defines σ-sup-decomposable measure m. sup ∫ Suppose f ⊙ dm = 0 for every measurable set E. By g(0) = 0, we have E

g

( sup ∫

) ( ) f ⊙ dm = g sup (f (x) ⊙ ψ(x)) = 0. x∈E

E

By the strictly monotonicity of g, we have

11

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(

)

g(f (x) ⊙ ψ(x)) ≤ g sup (f (x) ⊙ ψ(x))

= 0,

x∈E

for all x ∈ E. Since g(y) ≥ 0 for all y ∈ [a, b], we have g(f (x) ⊙ ψ(x)) = 0, for all x ∈ E. Since ⊙ is generated by a generator g, i.e., y ⊙ z = g −1 (g(y)g(z)) for all y, z ∈ [a, b], we get that g(f (x)) · g(ψ(x)) = g(f (x) ⊙ ψ(x)) = 0, for all x ∈ E. Let E = E1 ∪ E2 and E1 ∩ E2 = ∅, where E1 = {x ∈ E|f (x) = 0}. Then, we have f (x) ̸= 0 for all x ∈ E2 , which implies that g(f (x)) ̸= 0 for all x ∈ E2 . Hence, we have g(ψ(x)) = 0, i.e., ψ(x) = 0 for all x ∈ E2 ⊂ E, which implies that m(E2 ) = sup ψ(x) = 0. Hence, f = 0 a.e. for every measurable x∈E2

set E. Suppose f = 0 a.e. for every measurable set E. Let E1 ∪ E2 = E and E1 ∩ E2 = ∅, where sup ∫ E1 = {x ∈ E|f (x) = 0}, then m(E2 ) = 0. By Theorem 3.6, we have f ⊙ dm = 0. Hence, we get that E2 sup ∫

f ⊙ dm =

E

sup ∫

sup ∫

f ⊙ dm ⊕

E1

f ⊙ dm =

E2

sup ∫

f ⊙ dm.

E1

Since f (x) = 0 for all x ∈ E1 , we have f (x) ⊙ ψ(x) = 0 for all x ∈ E1 . Hence, we obtain that sup sup ∫ ∫ f ⊙ dm = sup (f (x) ⊙ ψ(x)) = 0, which implies that f ⊙ dm = 0. 2 E1

x∈E1

E

Theorem 3.13 Let (X, sup, inf) be a semiring, and let m be the same as in Theorem 3.6 and f be a sup ∫ measurable function with respect to m and 0 ≼ f . Then f ⊙ dm = 0 if and only if f = 0 a.e. for E

every measurable set E. Proof. Let m be the same as in Theorem 3.6. Then the pseudo-integral for a measurable function f : X → [a, b] is given by sup ∫

f ⊙ dm = sup (f (x) ⊙ ψ(x)), x∈E

E

for every measurable set E, where function ψ defines σ-sup-decomposable measure m. For every measurable set E, let E = E1 ∪ E2 and E1 ∩ E2 = ∅, where E1 = {x ∈ E|f (x) = 0}. sup ∫ f ⊙ dm = 0 for every measurable set E. By the proof of Theorem 3.10, we have m(E2 ) = Suppose E

0. Hence, f = 0 a.e. for every measurable set E. Suppose f = 0 a.e. for every measurable set E, then m(E2 ) = 0. By Theorem 3.7, we have sup ∫ f ⊙ dm = 0. Hence, we get that E2 sup ∫ E

f ⊙ dm =

sup ∫

sup ∫

f ⊙ dm ⊕

f ⊙ dm =

f ⊙ dm.

E1

E2

E1

sup ∫

Since f (x) = 0 for all x ∈ E1 , we have f (x) ⊙ ψ(x) = 0 for all x ∈ E1 . Hence, we obtain that sup sup ∫ ∫ f ⊙ dm = sup (f (x) ⊙ ψ(x)) = 0, which implies that f ⊙ dm = 0. 2 E1

x∈E1

E

Theorem 3.14 Let (X, ⊕, ⊙) be a semiring with generated pseudo-operations by a strictly monotone and continuous surjective generator g : [a, b] → [0, +∞], and let m : A (X) → [a, b] be a σ-⊕-decomposable measure. For a measurable function f : X → [a, b] with respect to m, define the set function ν : X → [a, b] by ∫⊕

ν(E) =

f ⊙ dm,

E

for any measurable set E. Then ν is a σ-⊕-decomposable measure and absolutely ⊕-continuous with respect to m. 12

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Proof. If the pseudo-operations are generated by a strictly monotone and continuous surjective generator g : [a, b] → [0, +∞], i.e., y ⊕ z = g −1 (g(y) + g(z)) and y ⊙ z = g −1 (g(y) · g(z)) for every y, z ∈ [a, b]. Then the pseudo-integral for a measurable function f : X → [a, b] is given by ( ) ∫⊕ ∫ ν(E) = f ⊙ dm = g −1 g ◦ f dµ , E

E

for every measurable set E, where µ = g ◦ m is a σ-additive measure. Hence, ν is a set function of A to [a,b]. (1) By ∅ is σ-⊕-decomposable measure zero set and Theorem 3.5, we have ν(∅) =

∫⊕ ∅

f ⊙ dm = 0;

(2) For any sequence {Ei }i∈N of pairwise disjoint sets from A , we have   (∞ ) ∫ ⊕ ∫ ∪ ν Ei = f ⊙ dm = g −1  ∞ g ◦ f dµ ∞ ∪ ∪ Ei

i=1

i=1



= g −1 

Ei

∞ ∫ ∑ i=1 E

=

∞

∫⊕

⊕

i=1 Ei



i=1



g ◦ f dµ = g −1 

∞ ∑ i=1

i

 ⊕  ∫ g  f ⊙ dm Ei

∞

f ⊙ dm = ⊕ ν(Ei ). i=1

Consequently, ν is a σ-⊕-decomposable measure. By Theorem 3.5, we obtain that ν is absolutely ⊕continuous with respect to m. 2 Theorem 3.15 Let (X, ⊕, ⊙) be a semiring with generated pseudo-operations by a strictly monotone and continuous surjective generator g, and let m : A (X) → [a, b] be a totally σ-g-finite-decomposable measure. If the σ-g-finite-decomposable measure ν is absolutely ⊕-continuous with respect to m, then there exists a measurable function f on X and g(f (x)) < +∞ for all x ∈ X, such that ∫⊕

ν (E) =

f ⊙ dm,

E

for every measurable set E. The function f is unique in the sense that if also ν (E) =

∫⊕

h ⊙ dm, then

E

f = h a.e. with respect to m. Proof. If the pseudo-operations are generated by a strictly monotone and continuous surjective generator g, i.e., y ⊕ z = g −1 (g(y) + g(z)) and x ⊙ y = g −1 (g(y) · g(z)) for every y, z ∈ [a, b]. Then the pseudo-integral for a measurable function f : X → [a, b] is given by ( ) ∫⊕ ∫ f ⊙ dm = g −1 g ◦ f dµ E

E

for every measurable set E, where µ = g ◦ m is a σ-additive measure. Since m is a totally σ-g-finite-decomposable measure, we have µ is a totally σ-finite measure. If the σ-g-finite-decomposable measure ν is absolutely continuous with respect to m, then g ◦ ν is σ-finite measure and absolutely ⊕-continuous with respect to µ. Hence, by Theorem B of § 31 of [11], there exists a finite valued measurable function p with respect to µ on X such that ∫ g(ν (E)) = pdµ, E

for every measurable set E. By the strictly monotonicity of g, we have p(x) = (g(g −1 (p(x)))) for all x ∈ X. Let f = g −1 ◦ p, then 13

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∫

g(ν (E)) =

pdµ =

E

∫

g ◦ f dµ,

E

for all measurable set E, which implies that ν (E) = g

−1

( ∫

) ⊕ ∫ g ◦ f dµ = f ⊙ dm,

E

for every measurable set E. If ν (E) =

∫⊕

E

h ⊙ dm, then g(ν(E)) =

E

g ◦ hdµ. Hence g ◦ f = g ◦ h a.e. with

E

respect to µ, i.e., f = h a.e. with respect to m. 2

4

∫

Conclusions

In this paper, we mainly discussed two classes of σ-⊕-decomposable measures and the corresponding pseudo-integrals: one is based on the generated pseudo-addition (g-case, see [16, 22]) and the other is based on the idempotent pseudo-operation (sup and inf, see [23, 39]). We got several properties as monotonicity, continuous from above and continuous from below for σ-⊕-decomposable measures. In particular, we obtained the correlation between the measure zero sets with respect to a σ-⊕-decomposable measure and the corresponding pseudo-integrals on them. As an application of the main results, we generalized the classical Radon-Nikodym theorem, which has been extensively studied and discussed [5, 7, 10, 12, 43], to the decomposable measure theory based on pseudo-integrals. We also hope that our results in this paper may lead to significant, new and innovative results in other related fields.

5

Acknowledgements

This work was supported by The National Natural Science Foundations of China (Grant no.11201512 and 61472056), The Natural Science Foundation Project of CQ CSTC (cstc2012jjA00001) and The Graduate Teaching Reform Research Program of Chongqing Municipal Education Commission (No.YJG143010).

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COMPOSITION OPERATOR ON ZYGMUND-ORLICZ SPACE NING XU DEPARTMENT OF MATHEMATICS, TIANJIN UNIVERSITY, TIANJIN 300072 P.R. CHINA. SCHOOL OF SCIENCE, HUAIHAI INSTITUTE OF TECHNOLOGY, LIANYUNGANG, JIANGSU 222005, P.R. CHINA. [email protected]; [email protected] ZE-HUA ZHOU∗ DEPARTMENT OF MATHEMATICS, TIANJIN UNIVERSITY, TIANJIN 300072, P.R. CHINA. [email protected]; [email protected]

Abstract. In this paper, we use Young’s function to define Zygmund-Orlicz space. We study boundedness and compactness of composition operator on Zygmund-Orlicz space.

1. Introduction Let D be the unit disk of the complex plane C and H(D) be the space of all holomorphic function on D. Let µ be a bounded, continuous and positive function denfined on D. A function f ∈ H(D) belongs to µ−Zygmund space , denoted as f ∈ Z µ , if kf kµ := sup µ(z)|f 00 (z)| < ∞. z∈D 2

Clearly, if µ(z) = 1 − |z| , the space Z µ is just the Zygmund space, which is denoted by Z, while when µ(z) = (1 − |z|2 )α with α > 0, the space Z µ becomes the α-Zygmund space which is denoted by Z α . It is readily seen that Z µ is a Banach space with the norm kf kZ µ = |f (0)| + |f 0 (0)| + kf kµ . For some more information of µ-Zygmund space on the unit disk see [3], while for composition and integral-type operators between them on the unit disk see for example [4, 6, 7, 8, 9]. Let A1 , A2 be two linear subspaces of H(D). If φ is a holomorphic self-map of D, such that f ◦ φ belongs to A2 for all f ∈ A1 , then φ induces a linear operator Cφ : A1 → A2 defined as Cφ f = f ◦ φ, called the composition operator with symbol φ. This type of operator appears in studies on isometries of various function spaces. Composition operator has been studied by numerous authors on many subspaces of H(D) and in paticular on Zygmund spaces and µ-Zygmund spaces. In [5], Julio C. Ramos Fern´andez characterized boundedness and compactness of composition operators on Bloch-Orlicz spaces denoted by B ϕ , where ϕ is Young’s function. More precisely, let ϕ : [0, ∞) → [0, ∞) be a strictly increasing convex function such that t = lim ϕ(t) ϕ(0) = 0 and lim ϕ(t) t = 0, t→∞

t→0+

B ϕ = {f ∈ H(D) : sup (1 − |z|2 )ϕ(λ|f 0 (z)|) < ∞} z∈D

for some λ > 0 depending on f . It is easy to see that B ϕ is a Banach space with the norm kf kBϕ = |f (0)| + kf kϕ , The work was supported in part by the National Natural Science Foundation of China (Grant Nos. 11371276; 11301373; 11201331). ∗ Corresponding author. 2010 Mathematics Subject Classification. Primary: 47B38; Secondary: 32H02, 30H05, 30H30, 47B33, 45P05 Key words and phrases. composition operator, Zygmund-Orlicz space, Young’s function.

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Xu and Zhou: Composition operator on Zygmund-Orlicz space 0

where kf kϕ = inf{k > 0 : Sϕ ( fk ) ≤ 1} is a Minkowski’s functional and Sϕ (f ) = supz∈D (1 − |z|2 )ϕ(|f (z)|). This paper is organized as follows: In section 2, we use Young’s function to define the Zygmund-Orlicz space, as a generalization of Zygmund space. The spaces is defined in a similar way as Korenblem-Orlicz space and Bloch-Orlicz space in [1, 2, 5]. We study some of its properties and show that the Zygmund-Orlicz space is isometrically equal to certain µ-Zygmund space for a very special weight µ. In section 3, we characterize boundedness and compactness of composition operator on Zygmund-Orlicz space. Throughout the rest of this paper, C will denote a finite positive constant, and it may differ from one occurrence to the other. 2. Zygmund-Orlicz spaces In this section, we define the Zygmund-Orlicz space Z ϕ using Young’s function. More precisely, Z ϕ is the class of all analytic functions f in D such that sup (1 − |z|2 )ϕ(λ|f 00 (z)|) < ∞ z∈D

for some λ > 0 depending on f . The set Z ϕ is an F-space which we call Zygmund-Orlicz space associated to the function ϕ. We can observe that when ϕ(t) = t with t ≥ 0, we get back the Zygmund spaces Z. It is not hard to see that kf kϕ = inf{k > 0 : Sϕ (

f 00 ) ≤ 1} k

define a seminorm for Z ϕ , where Sϕ (f ) = supz∈D (1 − |z|2 )ϕ(|f (z)|). In fact, it can be show that Z ϕ is a Banach space with the norm kf kZ ϕ = |f (0)| + |f 0 (0)| + kf kϕ .

(2.1)

Also, we can observe that for any f ∈ Z ϕ \ {0}, the following relation Sϕ (

f 00 f 00 ) ≤ Sϕ ( )≤1 kf kZ ϕ kf kϕ

(2.2)

holds. The inequality above allow us to obtain that |f 00 (z)| ≤ ϕ−1 (

1 )kf kϕ 1 − |z|2

for all f ∈ Z ϕ and for all z ∈ D. Furthermore, we have Z |z| Z 1 0 0 00 |f (z)| ≤ |f (0)| + | |f (ζ)||dζ| ≤ (1 + ϕ−1 ( 0

0

1 )dt)kf kZ ϕ . 1 − |z|2 t2

(2.3)

Lemma 1. The Zygmund-Orlicz space is isometrically equal to µ-Zygmund space, where µ(z) =

1 ϕ−1 (

1 1−|z|2 )

with z ∈ D. Thus, for any f ∈ Z ϕ , we have kf kϕ = kf kµ = sup µ(z)|f 00 (z)|. z∈D ϕ

Proof. From (2.2), for any f ∈ Z \ {0} and any z ∈ D, we have (1 − |z|2 )ϕ(

|f 00 (z)| )≤1 kf kϕ

which implies that µ(z)|f 00 (z)| ≤ kf kϕ for all z ∈ D. Thus Z ϕ ⊂ Z µ and kf kµ ≤ kf kϕ . Conversely, if f ∈ Z µ , then µ(z)|f 00 (z)| ≤ kf kµ , for all z ∈ D. From here, we have 00 Sϕ ( kff kϕ ) ≤ 1. Thus, f ∈ Z ϕ and kf kϕ ≤ kf kµ .  The following result will be very useful in the next section and it is a version of Lemma 6 in [5]. for completeness, we include an outline of its proof. 1059

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Xu and Zhou: Composition operator on Zygmund-Orlicz space Lemma 2. Let a ∈ D fixed. There exists a holomorphic function fa ∈ H(D), such that ϕ(|fa0 (z)|) =

1 − |a|2 |1 − a ¯z|2

for all z ∈ D. 2

1−|a| Proof. For z ∈ D, we set u(z) = ϕ−1 ( |1−¯ az|2 ), then u is a real and continuously differentiable function. Therefore, its partial derivatives exist and are continuous throughout D. It is clear 1 0 iv(z) that function u satisfies u(z) ≥ ϕ−1 ( |1−|a| 2 ) > 0 for all z ∈ D. Now we set fa (z) = u(z)e 0 where v is a real function defined on D. Then, in order for fa to be an analytic function on D, its real parts U (z) = u(z) cos v(z) and its imaginary parts V (z) = u(z) sin v(z) must satisfy the Cauchy-Riemann equations. We get the relation uvx = −uy and uvy = ux . We can choose a C 1 real function v defined on D such that fa0 is an analytic function on D Rz 0 1−|a|2 satisfying ϕ(|fa0 (z)|) = |1−¯ az|2 . Of course, fa (z) = 0 fa (ζ)dζ + fa (0) is an analytic function on D, too.  Rz Remark. It is clear that for any a ∈ D, the function ga (z) = 0 fa (s)ds with z ∈ D and fa is the function found in Lemma 2.2, belongs to the space Z ϕ .

S(ga00 ) = sup(1 − |z|2 ) z∈D

where σa (z) = for all a ∈ D.

a−z 1−¯ az

1 − |a|2 = sup(1 − |σa (z)|2 ) = 1 |1 − a ¯z|2 z∈D

(2.4)

denote the automorphism of the disk D. From (2.4), we get kga kϕ = 1

Lemma 3. The composition opreator Cφ is compact on Z ϕ if and only if given a bounded sequence {fn } in Z ϕ such that fn → 0 uniformly on compact subsets of D, then kCφ fn kϕ → 0 as n → ∞. 3. Main results Theorem 1. Let φ be a holomorphic self-map of D. Then Cφ : Z ϕ → Z ϕ is bounded if and only if µ(z) |φ0 (z)|2 < ∞ and sup z∈D µ(φ(z))   Z 1 1 sup µ(z)|φ00 (z)| 1 + ϕ−1 ( )dt < ∞. 1 − |φ(z)t|2 z∈D 0 Proof. Suppose that L1 = sup z∈D

µ(z) |φ0 (z)|2 < ∞, µ(φ(z))

 Z L2 = sup µ(z)|φ00 (z)| 1 + z∈D

0

1

ϕ−1 (

 1 ) dt < ∞. 1 − |φ(z)t|2

(3.5)

Then for all f ∈ Z ϕ \ {0}, We have the following estimate  (f ◦ φ)00 (z)   |f 00 (φ(z))φ02 (z) + f 0 (φ(z))φ00 (z)|  Sϕ = sup(1 − |z|2 )ϕ (L1 + L2 )kf kZ ϕ (L1 + L2 )kf kZ ϕ z∈D  |f 00 (φ(z))||φ0 (z)|2 + |f 0 (φ(z))||φ00 (z)|  ≤ sup(1 − |z|2 )ϕ (L1 + L2 )kf kZ ϕ z∈D R 0 2 1 00  |φ (z)| kf kϕ + (1 + 1 ϕ−1 ( ϕ  µ(φ(z)) 1−|φ(z)t|2 )dt)|φ (z)|kf kZ 0 2 ≤ sup(1 − |z| )ϕ (L1 + L2 )kf kZ ϕ z∈D  1 (L + L )kf k ϕ  1 2 Z ≤ sup(1 − |z|2 )ϕ µ(z) (L1 + L2 )kf kZ ϕ z∈D  1  = sup(1 − |z|2 )ϕ =1 µ(z) z∈D 1060

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Xu and Zhou: Composition operator on Zygmund-Orlicz space where we have used the relations (2.1) (2.2) (2.3) and Lemma 2.1. From here, we can conclude that kCφ f kϕ ≤ (L1 + L2 )kf kZ ϕ . Moreover from (3.5), take z = 0, we have Z 1   1 )dt < ∞, ϕ−1 (1)|φ00 (0)| 1 + ϕ−1 ( 1 − |φ(z)t|2 0 and therefore Z 1+

1

ϕ−1 (

0

1 )dt < ∞. 1 − |φ(0)t|2

(3.6)

Hence with (2.1) (2.3) |f (φ(0))| + |f 0 (φ(0))φ0 (0)|

Z ≤

|φ(0)|

|f (0)| +

|f 0 (ξ)||dξ| + |f 0 (0)| +

0

|φ(0)|

|f 00 (ζ)||dζ|

0 |φ(0)|

Z ≤

Z

|f 00 (ζ)||dζ|)kf kZ ϕ

kf kZ ϕ + 2(1 + 0 1

Z

ϕ−1 (

= kf kZ ϕ + 2(1 + 0

1 )dt)kf kZ ϕ . 1 − |φ(0)t|2

(3.7)

Combined with (3.6) (3.7), we have kCφ f kZ ϕ ≤ Ckf kZ ϕ for all f ∈ Z ϕ , and Cφ : Z ϕ → Z ϕ is bounded. Conversely, suppose that there exists a constant C > 0 such that kf ◦ φkϕ ≤ Ckf kϕ ϕ

for all f ∈ Z . By taking the function f (z) = z ∈ Z ϕ , we obtain sup(1 − |z|2 )ϕ( z∈D

|φ00 (z)| |(Cφ f )00 (z)| ) = sup(1 − |z|2 )ϕ( ) < ∞. k k z∈D

That is sup µ(z)|φ00 (z)| < ∞.

(3.8)

z∈D

Rz For a ∈ D, set ga (z) = 0 fa (s)ds and from the remark , we see that ga ∈ Z ϕ and kga kϕ = 1. With (2.2), we have that 1 ≥ Sϕ (

(ga ◦ φ)00 (z) |f 0 (φ(z))φ02 (z) + fa (φ(z))φ00 (z)| ) = sup(1 − |z|2 )ϕ( a ). Ckga kϕ C z∈D

Hence µ(z)|fa0 (φ(z))||φ0 (z)|2 − µ(z)|fa (φ(z))||φ00 (z)| ≤ µ(z)|fa0 (φ(z))φ02 (z) + fa (φ(z))φ00 (z)| ≤ C, or µ(z)|fa0 (φ(z))||φ0 (z)|2 ≤ C + µ(z)|fa (φ(z))||φ00 (z)|. For ga ∈ Z ϕ , we have ga0 ∈ B ϕ . That is to say fa ∈ B ϕ and from Lemma 2.1 we get fa ∈ Bµ , 1 where µ(z) = −1 . Thus, from [11], we have 1 ϕ ( 1−|z| 2) Z |z| 1 |fa (z)| ≤ C(1 + dt)kfa kBµ . (3.9) µ(t) 0 Hence sup µ(z)|fa0 (φ(z))||φ0 (z)|2 ≤ C + Cµ(z)(1 + z∈D

Z

|φ(z)|

0

1 dt)|φ00 (z)|kfa kBµ . µ(t)

(3.10)

It is obvious that supz∈D µ(z)|fa0 (φ(z))||φ0 (z)|2 < ∞ when |φ(z)| ≤ √12 . Now for √12 < |φ(z)| < 1, fix a ∈ D, we set R a¯ζ Z z Z a¯ζ 1 ( 0 h(t)dt)2 2 ha (z) = (1 − |a| ) ( h(t)dt − R |a|2 )dζ, 2 0 0 h(t)dt 1061

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Xu and Zhou: Composition operator on Zygmund-Orlicz space where h(t) =

1 µ(t) .

Then

R a¯z 1 ( 0 h(t)dt)2 ), R 2 |a|2 h(t)dt 0 0 R a¯z a ¯ h(¯ a z) h(t)dt 00 2 0 ha (z) = (1 − |a| )(¯ ah(¯ az) − ). R |a|2 h(t)dt 0 2 R |a|2 It is easy to see that h0a (a) = 1−|a| h(t)dt and h00a (a) = 0. From above, we can see that 2 0 ϕ ha ∈ Z , and kha kϕ ≤ C. Therefore, kCφ ha kϕ ≤ C. Hence, we have Z h0a (z) = (1 − |a|2 )(

sup(1 − |z|2 )ϕ( z∈D

=

sup(1 − |z|2 )ϕ( z∈D

a ¯z

h(t)dt −

|(Cφ ha )00 (z)| ) C |h00a (φ(z))||φ0 (z)|2 + |h0a (φ(z))||φ00 (z)| ) < ∞. C

From above, put a = φ(z), we obtain sup µ(z)|φ00 (z)|

|φ(z)|2

Z

z∈D

1 dt ≤ C. µ(t)

0

Combined with the boundedness of φ, (3.8) and the following inequality of [12] Z |φ(z)|2 Z |φ(z)| Z |φ(z)|2 1 1 1 dt ≤ dt ≤ C + C dt, µ(t) µ(t) µ(t) 0 0 0

(3.11)

we have sup µ(z)|φ00 (z)||φ(z)|

1

Z

z∈D

ϕ−1 (

0

Z

≤ sup µ(z)|φ00 (z)| z∈D

|φ(z)|

0

≤ sup µ(z)|φ00 (z)|(C + C z∈D

1 )dt 1 − |φ(z)t|2

1 dt µ(t) Z |φ(z)|2 0

1 dt) < ∞ µ(t)

(3.12)

Hence with (3.10), we have sup µ(z)|fa0 (φ(z))||φ0 (z)|2 = sup

z∈D

z∈D

sup µ(z)|φ00 (z)|(1 +

Z

z∈D

1

ϕ−1 (

0

µ(z) |φ0 (z)|2 ≤ C, µ(φ(z))

1 )dt) ≤ C. 1 − |φ(z)t|2

This completes the proof.



Theorem 2. Let φ be a holomorphic self-map of D. Then Cφ : Z ϕ → Z ϕ is compact if and only if Cφ is bounded and lim

|φ(z)|→1

lim

|φ(z)|→1

00

µ(z) |φ0 (z)|2 = 0, µ(φ(z)) Z

µ(z)|φ (z)|(1 + 0

1

ϕ−1 (

1 )dt) = 0. 1 − |φ(z)t|2

(3.13)

(3.14)

Proof. Suppose first that Cφ is bounded and (3.5) holds. Let {fn } be a bounded sequence in Z ϕ converging to 0 uniformly on compact subsets of D. Then, by Lemma 2.3, it is sufficient to show thatkCφ fn kϕ → 0 as n → ∞. From (3.13) and (3.14), for ε1 , ε2 > 0, we can find an r ∈ (0, 1) such that Z 1 µ(z) 1 |φ0 (z)|2 < ε1 and µ(z)|φ00 (z)|(1 + ϕ−1 ( ) < ε2 , 2 µ(φ(z)) 1 − |φ(z)t| 0 1062

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Xu and Zhou: Composition operator on Zygmund-Orlicz space whenever r < |φ(z)| < 1. From here, we have that µ(z)|(Cφ fn )00 (z)|

≤ µ(z)|fn00 (φ(z)||φ0 (z)|2 + |fn0 (φ(z))||φ00 (z)|) µ(z) ≤ |φ0 (z)|2 kfn kϕ µ(φ(z)) Z 1 1 )dtkfn kZ ϕ + µ(z)|φ00 (z)|(1 + ϕ−1 ( 1 − |φ(z)t|2 0 < kfn kϕ ε1 + kfn kZ ϕ ε2 .

(3.15)

On the other hand, since {fn } converges to 0 uniformly on compact subsets of D, sup |fn00 (φ(z))| → |φ(r)|≤r

0, and

sup |fn0 (φ(z))| → 0 as n → ∞. From the boundedness of Cφ , set f (z) = z ∈ Z ϕ

|φ(r)|≤r

and f (z) = z 2 ∈ Z ϕ , we have M1 = sup µ(z)|φ0 (z)|2 < ∞,

M2 = sup µ(z)|φ00 (z)| < ∞.

z∈D

(3.16)

z∈D

Hence, we have sup µ(z)|(Cφ fn )00 (z)| |φ(r)|≤r

sup µ(z)|φ0 (z)|2 |fn00 (φ(z))| + sup µ(z)|φ00 (z)||fn0 (φ(z))|

≤

|φ(r)|≤r

|φ(r)|≤r

≤ M1 sup |fn00 (φ(z))| + M2 sup |fn0 (φ(z))| → 0 |φ(r)|≤r

|φ(r)|≤r

With (3.15), we obtain that Cφ is a compact operator on Z ϕ . Suppose that Cφ : Z ϕ → Z ϕ is compact. It is clear that Cφ : Z ϕ → Z ϕ is bounded. Let {zn } be a sequence in D such that |wn | = |φ(zn )| → 1 as n → ∞. Set Z z gn (z) = (fwn (s) − fwn (0))ds 0 ϕ

then gn ∈ Z and kgn kϕ = 1. Furthermore, gn → 0 uniformly on compact subsets of D as n → ∞. Hence 0 ←

kCφ gn kµ ≥ µ(zn )|gn00 (φ(zn ))φ02 (zn ) + gn0 (φ(zn ))φ00 (zn )|

≥

0 µ(zn )|fφ(z (φ(zn ))||φ0 (zn )|2 n)

−

µ(zn )|fφ(zn ) (φ(zn )) − fφ(zn ) (0)||φ00 (zn )|.

(3.17)

Since Cφ is compact operator on Z ϕ , set Z z  (ln(1 − w¯n ζ))2 1 − ln(1 − w ¯ ζ) dζ. swn (z) = n ln(1 − |wn |2 ) 0 2 ln(1 − |wn |2 ) It is easy to see that s0wn (wn ) = − 21 and s00wn (wn ) = 0. So swn ∈ Z ϕ and swn → 0 uniformly on compact subsets of D as n → ∞. For ε3 > 0, we have ε3 > kCφ swn kµ

=

1 sup µ(zn )|φ00 (zn )|, 2 z∈D

which means lim µ(zn )|φ00 (zn )| = 0.

(3.18)

n→∞

Set 2

z

Z

hn (z) = (1 − |wn | )

Z (

0

0

1063

w¯n ζ

R w¯n ζ 1 ( 0 h(t)dt)2 h(t)dt − R |w |2 )dζ, n 2 h(t)dt 0

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Xu and Zhou: Composition operator on Zygmund-Orlicz space then {hn } ⊂ Z ϕ and {hn } is a sequence converging to 0 uniformly on compact subsets of D. Furthermore, for ε4 > 0, we have  |(C h )00 (z )|  φ n n sup (1 − |zn |2 )ϕ k zn ∈D  |h00 (φ(z ))φ02 (z ) + h0 (φ(z ))φ00 (z )|  n n n n n n = sup (1 − |zn |2 )ϕ k zn ∈D 2 R 2  1−|φ(zn )| |φ00 (zn )| |φ(zn )| 1 dt  2 µ(t) 0 = sup (1 − |zn |2 )ϕ < ε4 . k zn ∈D That is to say, for ε4 > 0, ∃N , such that

√1 2

< |φ(zn )| < 1 whenever n > N , we have

1 − |φ(zn )|2 µ(zn )|φ00 (zn )| 2

Z

|φ(zn )|2

0

1 dt < ε4 . µ(t)

(3.19)

Hence, with (3.18) (3.19), we obtain that Z |φ(zn )| Z |φ(zn )|2 1 1 µ(zn )|φ00 (zn )| dt ≤ µ(zn )|φ00 (zn )|(C1 + C2 dt) µ(t) µ(t) 0 0 Z |φ(zn )|2 1 − |φ(zn )|2 1 00 ≤ Cε3 + C µ(zn )|φ (zn )| dt 2 µ(t) 0 ≤ Cε3 + Cε4 . (3.20) Therefore, with the boundedness of φ and (3.18) (3.20), we have Z 1 1 lim µ(z)|φ00 (z)|(1 + ϕ−1 ( )dt) = 0 2 1 − |φ(z)t| |φ(z)|→1 0 and (3.14) hold. Moreover, with (3.17) we have that µ(zn )|fwn (wn ) − fwn (0)||φ00 (zn )| ≤ µ(zn )|φ00 (zn )|(1 +

Z

|wn |

0

1 dt)kfwn − fwn (0)kBµ µ(t)

≤ (Cε3 + Cε4 )kfwn − fwn (0)kBµ .

(3.21)

From (3.17) (3.21), we obtain that µ(zn ) |φ0 (zn )|2 µ(φ(zn ))

= µ(zn )|fw0 n (wn )||φ0 (zn )|2 ≤ ≤

kCφ gn kµ + µ(zn )|fwn (wn ) − fwn (0)||φ00 (zn )| kCφ gn kµ + (Cε3 + Cε4 )kfwn − fwn (0)kBµ .

This implies that µ(z) |φ0 (z)|2 = 0, µ(φ(z)) and (3.13) holds. This completes the proof. lim

|φ(z)|→1



References [1] Ren´ e Erlin Castillo, Julio C. Ramos Fern´ andez, Estimating the norm of conformal maps in KorenblumOrlicz spaces, Rend. Circ. Mat. Palermo 60(2011), 385-393. [2] Ren´ e Erlin Castillo, Julio C. Ramos Fern´ andez,, Miguel Salazar, Bounded superposition operators between Bloch-Orlicz and α-Bloch spaces, Appl. Math. Comput. 218 (2011), 3441-3450. [3] B.Choe, H. Koo, W.Smith, Composition operators on small spaces, Integr. Equat. Oper. Th. 56 (2006), 357-380. [4] J. Dai, Composition operators on Zygmund spaces of the unit ball, J. Math. Anal. Appl. 394 (2012), 696-705. [5] Julio C. Ramos Fern´ andez, Composition operators on Bloch-Orlicz type spaces, Appl. Math. Comput. 217 (2010),3392-3402. [6] S. Li, S. Stevi´ c, Weighted composition operators from Zygmund spaces into Bloch spaces, Appl. Math. Comput. 206(2) (2008), 825-831. [7] S. Li, S. Stevi´ c, Products of Votarra type operator and composition operator from H ∞ and Bloch spaces to Zygmund spaces, J. Math. Anal. Appl. 345 (2008), 40-52.

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Xu and Zhou: Composition operator on Zygmund-Orlicz space [8] S. Li, S. Stevi´ c, On an integral-type operators from ω-Bloch spaces to µ-Zygmund spaces, Appl. Math. Comput. 215 (2010), 4385-4391. [9] S. Li, 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, Generalized composition operators from logarithmic Bloch spaces to mixed-norm spaces, Util. Math. 77 (2008) 167-172. [11] N. Xu, Extended Ces` aro Operators on µ Bloch Spaces in Cn , J. Math. Research and Exposition, 29(5)(2009),913-922. [12] X.J.Zhang, Weighted composition operator on µ-Bloch spaces in Cn , Science China Mathematics, 35(6) (2005), 601-619. [13] X.L.Zhu, Ingegral-type operators from iterated logarithmic Blochspaces to Zygmund-type spaces, Appl. Math. Comput. 215(2009)1170-1175.

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Multiple positive solutions for m-point boundary value problems with one-dimensional p-Laplacian systems and sign changing nonlinearity ∗ Hanying Feng† , Jian Liu Department of Mathematics, Shijiazhuang Mechanical Engineering College Shijiazhuang 050003, Hebei, P. R. China

Abstract: In this paper, we consider the multipoint boundary value problem for the one-dimensional p-Laplacian system    (φp (u0 ))0 + q1 (t)f (t, u, v) = 0, t ∈ (0, 1), 1   (φp2 (v 0 ))0 + q2 (t)g(t, u, v) = 0, t ∈ (0, 1),  m−2 X    u(0) = ai u(ξi ), u0 (1) = βu0 (0),   i=1

m−2 X     v(0) = ai v(ξi ), v 0 (1) = βv 0 (0),  i=1 pi −2

where φpi (s) = |s| 1 and ai ∈ [0, 1),

m−2 P i=1

s, pi > 1, i = 1, 2, ξi ∈ (0, 1) with 0 < ξ1 < ξ2 < · · · < ξm−2 < ai < 1, β ∈ (0, 1). By using the fixed point index theorem on

cones, we study the existence of positive solutions for the m-point boundary value problem with sign changing nonlinear term. Some sufficient conditions for the existence of multiple positive solutions are obtained. Finally, an example is also included to illustrate the importance of the main result obtained. Keywords: Multipoint boundary value problem, Fixed point theorem, Cone, Positive solution, One-dimensional p-Laplacian. 2010 MR Subject Classification: 34B10, 34B15, 34B18

1

Introduction

In this paper, we study the existence of multiple positive solutions to the boundary value problem (BVP for short) for the one-dimensional p-Laplacian system ½ (φp1 (u0 ))0 + q1 (t)f (t, u, v) = 0, t ∈ (0, 1), (1.1) (φp2 (v 0 ))0 + q2 (t)g(t, u, v) = 0, t ∈ (0, 1), ∗ †

Supported by NNSF of China (11271106) and HEBNSF of China (A2012506010). Corresponding author. E-mail address: [email protected] (H.Feng).

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 m−2 X    ai u(ξi ), u0 (1) = βu0 (0), u(0) =   i=1

m−2 X     v(0) = ai v(ξi ), v 0 (1) = βv 0 (0), 

(1.2)

i=1

|s|pi −2 s,

where φpi (s) = pi > 1, i = 1, 2, ξi ∈ (0, 1) with 0 < ξ1 < ξ2 < · · · < ξm−2 < 1. Multipoint boundary value problems of ordinary differential equations arise in a variety of areas of applied mathematics and physics. For example, the vibrations of a guy wire of a uniform cross-section and composed of N parts of different densities can be set up as a multipoint boundary value problem (see [10]). The study of multipoint boundary value problems for linear second-order ordinary differential equations was initiated by Il’in and Moiseev [2]. Since then there has been much current attention focused on the study of nonlinear multipoint boundary value problems, see ([1, 3, 4, 5, 6, 8, 9, 11, 12, 13, 14, 15]). Karakostas [4] proved the existence of positive solutions for the two-point boundary value problem x00 (t) − sign(1 − α)q(t)f (x, x0 )x0 = 0, t ∈ (0, 1), with one of the following sets of boundary conditions: x(0) = 0, x0 (1) = αx0 (0), or x(1) = 0, x0 (1) = αx0 (0), where α > 0, α 6= 1. By using indices of convergence of the nonlinearities at 0 and at +∞, they provided a priori upper and lower bounds for the slope of the solutions. Ma [11] proved the existence of positive solutions for the multipoint boundary value problem x00 (t) − q(t)f (x, x0 )x0 = 0, t ∈ (0, 1), x(0) =

n−2 X

bi x(ξi ), x0 (1) = αx0 (0),

i=1

where ξi ∈ (0, 1) with 0 < ξ1 < ξ2 < · · · < ξm−2 < 1, bi ∈ [0, 1), α > 1. They provided sufficient conditions for the existence of multiple positive solutions to the above BVP by applying the fixed point theorem in cones. Recently, Ji [3] investigated the following m-point boundary value problem (φp (u0 ))0 + q(t)f (t, u) = 0, t ∈ (0, 1), u(0) =

m−2 X

αi u(ξi ), u(1) =

i=1

m−2 X

βi u(ξi ).

i=1

They obtained sufficient conditions that guarantee the existence of positive solutions by using fixed point theorems on cones. Motivated by these results, our purpose of this paper is to show the existence of multiple positive solutions to multipoint BVP (1.1), (1.2). To date no paper has appeared in the literature which discusses the multipoint boundary value problem for one-dimensional pLaplacian systems when nonlinearity in the differential equation may change sign. This paper attempts to fill this gap in the literature. The interesting point of this paper is the nonlinear terms f and g may change sign. For convenience, we list the following assumptions: 2 1067

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(H1 ) ai ∈ [0, 1) satisfies

m−2 P i=1

ai < 1, β ∈ (0, 1);

(H2 ) f, g ∈ C([0, 1] × [0, +∞) × [0, +∞), (−∞, +∞)) ; (H3 ) q1 , q2 ∈ L1 [0, 1] are nonnegative on (0, 1) and q1 , q2 are not identically zero on any Z 1 Z 1 subinterval of (0, 1). Furthermore, q1 , q2 satisfy 0 < q1 (t)dt < +∞, 0 < q2 (t)dt < 0

+∞.

2

0

Preliminaries

For the convenience of readers, we provide some background material from the theory of cones in Banach spaces. We also state in this section the fixed point index theorem on cones. Definition 2.1. Let E be a real Banach space over R. A nonempty closed set K ⊂ E is said to be a cone provide that (i) au + bv ∈ K for all u, v ∈ K and all a ≥ 0, b ≥ 0, and (ii) u, −u ∈ K implies u = 0. Every cone K ⊂ E induces an ordering in E given by x ≤ y if and only if y − x ∈ K. Definition 2.2. The map α is said to be a nonnegative continuous concave functional on a cone K of a real Banach space E provided that α : K → [0, ∞) is continuous and α(tx + (1 − t)y) ≥ tα(x) + (1 − t)α(y) for all x, y ∈ K and 0 ≤ t ≤ 1. Similarly, we say the map γ is a nonnegative continuous convex functional on a cone K of a real Banach space E provided that γ : P → [0, ∞) is continuous and γ(tx + (1 − t)y) ≤ tγ(x) + (1 − t)γ(y) for all x, y ∈ K and 0 ≤ t ≤ 1. To prove our results, the following fixed point theorem in cones is fundamental. Theorem 2.1. ([7]) Let K be a cone in a real Banach space E. Let D be an open bounded subset of E with Dk = D ∩ K 6= ∅. Assume that A : Dk → K is completely continuous such that A 6= Ax for x ∈ Dk K. Then the following results hold: (1) If kAxk ≤ kxk, x ∈ ∂Dk , then ik (A, Dk ) = 1. (2) If there exists e ∈ K\{0} such that x 6= Ax + λe for all x ∈ ∂Dk and all λ > 0, then ik (A, Dk ) = 0. (3) Let U be open in X such that U ⊂ Dk . If ik (A, Dk ) = 1 and ik (A, Uk ) = 0, then A has a fixed point in Dk \U k . The same result holds if ik (A, Dk ) = 0 and ik (A, Uk ) = 1.

3

Related lemmas

−1 In this paper, we denote C + [0, 1] = {x ∈ C[0, 1] : x(t) ≥ 0, t ∈ [0, 1]}. φ−1 p1 , φp2 are, respectively, the inverse function to φp1 , φp2 . Let E = C[0, 1] × C[0, 1], define norm k(u, v)k = kuk + kvk, where kuk = max |u(t)|, t∈[0,1]

kvk = max |v(t)|, then E is a Banach space. t∈[0,1]

Define the cone K ⊂ E by K = {(u, v) ∈ E | u(t), v(t) ≥ 0, u and v are concave and nondecreasing on [0, 1]} . Lemma 3.1. Assume that (H1 ) − (H3 ) hold. Then, For any x, y ∈ C + [0, 1], the problem ½ (φp1 (u0 ))0 + q1 (t)f (t, x, y) = 0, t ∈ (0, 1), (3.1) (φp2 (v 0 ))0 + q2 (t)g(t, x, y) = 0, t ∈ (0, 1),

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 m−2 X    ai u(ξi ), u0 (1) = βu0 (0), u(0) =   i=1

(3.2)

m−2 X     v(0) = ai v(ξi ), v 0 (1) = βv 0 (0),  i=1

has the unique solution (u, v) as µZ 1 Z t −1 u(t) = φp1 q1 (τ )f (τ, x(τ ), y(τ ))dτ + 0

s

+ 1−

1 m−2 X

m−2 X

ai

Z ai

i=1

µZ

ξi

φ−1 p1

0

s

φp1 (β) 1 − φp1 (β)

Z 0

¶

1

q1 (τ )f (τ, x(τ ), y(τ ))dτ

ds

1

q1 (τ )f (τ, x(τ ), y(τ ))dτ

i=1

¶ Z 1 φp1 (β) (3.3) q1 (τ )f (τ, x(τ ), y(τ ))dτ ds, + 1 − φp1 (β) 0 µZ 1 ¶ Z 1 Z t φp2 (β) −1 v(t) = q2 (τ )g(τ, x(τ ), y(τ ))dτ ds φp2 q2 (τ )g(τ, x(τ ), y(τ ))dτ + 1 − φp2 (β) 0 0 s µZ 1 m−2 X Z ξi 1 −1 φp2 q2 (τ )g(τ, x(τ ), y(τ ))dτ ai + m−2 X 0 s i=1 1− ai i=1

+

φp2 (β) 1 − φp2 (β)

Z

¶

1

0

q2 (τ )g(τ, x(τ ), y(τ ))dτ

(3.4)

ds.

Proof. For any x, y ∈ C + [0, 1], suppose (u, v) is a solution of BVP (3.1), (3.2). By integration of (3.1), it follows µ that ¶ Z t 0 −1 0 u (t) = φp1 φp1 (u (0)) − q1 (τ )f (τ, x(τ ), y(τ ))dτ , 0 µ ¶ Z s Z t −1 0 u(t) = u(0) + φp1 φp1 (u (0)) − q1 (τ )f (τ, x(τ ), y(τ ))dτ ds. 0

0

Using the boundary condition (3.2), we can easily have µZ 1 ¶ Z 1 Z t φp1 (β) −1 q1 (τ )f (τ, x(τ ), y(τ ))dτ ds φp1 q1 (τ )f (τ, x(τ ), y(τ ))dτ + u(t) = 1 − φp1 (β) 0 s 0 µZ 1 m−2 X Z ξi 1 −1 + ai φp1 q1 (τ )f (τ, x(τ ), y(τ ))dτ m−2 X 0 s i=1 1− ai i=1

φp1 (β) + 1 − φp1 (β)

Z

¶

1

q1 (τ )f (τ, x(τ ), y(τ ))dτ

0

In a similar way, we can prove µZ 1 Z t −1 φp2 q2 (τ )g(τ, x(τ ), y(τ ))dτ + v(t) = 0

s

+ 1−

1 m−2 X

m−2 X

ai

i=1

φp2 (β) + 1 − φp2 (β)

i=1

Z 0

Z ai

0

ξi

µZ φ−1 p2

s

ds.

φp2 (β) 1 − φp2 (β)

Z 0

¶

1

q2 (τ )g(τ, x(τ ), y(τ ))dτ

ds

1

q2 (τ )g(τ, x(τ ), y(τ ))dτ ¶

1

q2 (τ )g(τ, x(τ ), y(τ ))dτ

ds.

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Lemma 3.2. Assume that (H1 ) − (H3 ) hold. If f (t, x, y), g(t, x, y) > 0, for x, y ∈ C + [0, 1], t ∈ [0, 1], for the unique solution (u, v) of BVP (3.1), (3.2), then u(t) and v(t) are concave, and u(t), v(t) ≥ 0, u0 (t), v 0 (t) ≥ 0, t ∈ [0, 1]. Proof. From the fact that (φp (u0 ))0 (t) = −q(t)f (t, x(t), y(t)) ≤ 0, we have φp (u0 (t)) is nonincreasing. It follows that u0 (t) is also nonincreasing. Thus, we know that the graph of u(t) is concave down on (0, 1). Then the concavity of u together with boundary u0 (1) = βu0 (0) implies that u0 (t) ≥ 0 for t ∈ [0, 1]. Similarly, we can prove the graph of v(t) is concave down on (0,1) and v 0 (t) ≥ 0 for t ∈ [0, 1]. From u0 (t) ≥ 0, we know that u(ξi ) ≥ u(0), for i = 1, 2, . . . , m − 2. This implies m−2 m−2 P P u(0) = ai u(ξi ) ≥ ai u(0). By 1 −

m−2 P i=1

i=1

i=1

ai > 0, it is obvious that u(0) ≥ 0. Hence u(1) ≥ u(0) ≥ 0. So from the concavity

of u, we know that u(t) ≥ 0, t ∈ [0, 1]. In a similar way, we can know v(t) ≥ 0, t ∈ [0, 1]. Lemma 3.3. If (u, v) ∈ K, η ∈ (0, 1), then u(t) ≥ ηkuk, v(t) ≥ ηkvk, t ∈ [η, 1]. Proof. For u ∈ K, we know u(t) and v(t) are nonnegative, nondecreasing and concave on [0, 1], then u(t) ≥ tu(1) ≥ ηu(1) = ηkuk, v(t) ≥ tv(1) ≥ ηv(1) = ηkvk, t ∈ [η, 1]. We define ϕ(t) = θt, θ ∈ (0, 1), m−2 X ai ξi L=1+

i=1 m−2 X

,

ai  i=1 ¶ µ ¶  µ Z 1 Z 1 Z 1 Z 1   1 1 η   φ−1 q1 (τ )dτ ds η q2 (τ )dτ ds  φ−1   p2 p1 1 − φp1 (β) s 1 − φp2 (β) s η η ¶ , ¶ µ µ Z 1 Z 1 γ1 = min ,   1 1 −1 −1     q1 (τ )dτ q2 (τ )dτ Lφp2   Lφp1 1 − φ (β) 1 − φp2 (β) 0 p1 0 γ = ηγ1 , Kρ = {(u, v) ∈ K : k(u, v)k < ρ}, Kρ∗ = {(u, v) ∈ K : ρϕ(t) < u(t) + v(t) < ρ}, Ωρ = {(u, v) ∈ K : min (u(t) + v(t)) < γρ} 1−

η≤t≤1

= {(u, v) ∈ K : γk(u, v)k ≤ min (u(t) + v(t)) < γρ}. η≤t≤1

Lemma 3.4. ([7]) Ωρ has the following properties: (a) Ωρ is open relative to K. (b) Kγρ ⊂ Ωρ ⊂ Kρ . (c) u ∈ ∂Ωρ if and only if min (u(t) + v(t)) = γρ. η≤t≤1

(d) If u ∈ ∂Ωρ , then γρ ≤ u(t) ≤ ρ, for t ∈ [η, 1].

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Now for convenience we introduce the following notations. Let ½ ¾ ½ ¾ f (t, u, v) g(t, u, v) ρ ρ : u + v ∈ [γρ, ρ] , gγρ = min : u + v ∈ [γρ, ρ] , fγρ = min t∈[η,1] φp1 (ρ) t∈[η,1] φp2 (ρ) ½ ¾ ½ ¾ f (t, u, v) g(t, u, v) ρ ρ fρϕ(t) = max : u + v ∈ [ρϕ(t), ρ] , gρϕ(t) : u + v ∈ [ρϕ(t), ρ] , = max t∈[0,1] φp1 (ρ) t∈[0,1] φp2 (ρ) f (t, u, v) g(t, u, v) f∞ = lim inf min , g∞ = lim inf min , (u,v)→∞ (u,v)→∞ t∈[η,1] φp1 (u + v) t∈[η,1] φp2 (u + v) f (t, u, v) g(t, u, v) , g ∞ = lim sup max , sup max (u,v)→∞ (u,v)→∞ t∈[0,1] φp1 (u + v) t∈[0,1] φp2 (u + v) µ ¶ µ ¶ Z 1 Z 1 1 1 1 −1 −1 q1 (τ )dτ , q2 (τ )dτ , = 2Lφp1 = 2Lφp2 1 − φp1 (β) 0 m2 1 − φp2 (β) 0 µ ¶ µ ¶ Z 1 Z 1 Z 1 Z 1 1 1 1 −1 −1 φp1 q1 (τ )dτ ds, φp2 q2 (τ )dτ ds, = 2η = 2η 1 − φp1 (β) s M2 1 − φp2 (β) s η η

f∞ = 1 m1 1 M1

lim

where (u, v) → ∞ ⇔ kuk + kvk → ∞. Remark 2.1. By (H3 ), it is easy to see that 0 < m1 , m2 , M1 , M2 < ∞, and M1 γ = M1 ηγ1 ≤ ηm1 < m1 , M2 γ = M2 ηγ1 ≤ ηm2 < m2 .

4

Existence of positive solutions

We now give our results on the existence of multiple positive solutions of BVP (1.1), (1.2). Theorem 4.1. Assume (H1 ) − (H3 ) hold. In addition, the following condition (H4 ) holds: (H4 ) There exist ρ1 , ρ2 , ρ3 ∈ (0, ∞), with ρ1 < γρ2 < ρ2 < ρ3 such that (1) f (t, u, v), g(t, u, v) > 0, t ∈ [0, 1], u + v ∈ [ρ1 ϕ(t), ∞); ρ2 ρ2 (2) fρρ11ϕ(t) < φp1 (m1 ), gρρ11ϕ(t) < φp2 (m2 ), fγρ 2 ≥ φp1 (M1 γ), gγρ2 ≥ φp2 (M2 γ), fρρ33ϕ(t) ≤ φp1 (m1 ), gρρ33ϕ(t) ≤ φp2 (m2 ). Then BVP (1.1), (1.2) has at least three positive solutions in K. Proof. We assume that (H4 ) holds. Denote ½ f (t, u, v), u + v ≥ ρ1 ϕ(t), ∗ f (t, u, v) = f (t, u, ρ1 ϕ(t) − u), 0 ≤ u + v < ρ1 ϕ(t). ½ g(t, u, v), u + v ≥ ρ1 ϕ(t), g ∗ (t, u, v) = g(t, u, ρ1 ϕ(t) − u), 0 ≤ u + v < ρ1 ϕ(t). We can see that f ∗ (t, u, v), g ∗ (t, u, v) ∈ C([0, 1] × [0, +∞) × [0, +∞), (0, +∞)). Define the following integral equation systems: µZ 1 Z t −1 q1 (τ )f ∗ (τ, u(τ ), v(τ ))dτ φp1 A(u, v)(t) = s 0 ¶ Z 1 φp1 (β) + q1 (τ )f ∗ (τ, u(τ ), v(τ ))dτ ds 1 − φp1 (β) 0 + 1−

1 m−2 X

m−2 X

ai

Z ai

i=1

i=1

φp1 (β) + 1 − φp1 (β)

Z 0

1

0

ξi

µZ φ−1 p1

1

s

q1 (τ )f ∗ (τ, u(τ ), v(τ )) dτ

¶ ∗

q1 (τ )f (τ, u(τ ), v(τ ))dτ

ds,

(4.1)

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Z B(u, v)(t) =

t

µZ φ−1 p2

1

q2 (τ )g ∗ (τ, u(τ ), v(τ ))dτ 0 s ¶ Z 1 φp2 (β) ∗ q2 (τ )g (τ, u(τ ), v(τ ))dτ ds + 1 − φp2 (β) 0 µZ 1 m−2 X Z ξi 1 −1 q2 (τ )g ∗ (τ, u(τ ), v(τ )) dτ + φp2 ai m−2 X s 0 i=1 1− ai i=1

φp2 (β) + 1 − φp2 (β)

Z 0

¶

1

∗

q2 (τ )g (τ, u(τ ), v(τ ))dτ

ds.

(4.2)

Define operator F (u, v)(t) = (A(u, v)(t), B(u, v)(t)). According to the definition of F and Lemma 3.2, it is easy to show that F (K) ⊂ K. By similar arguments in [5, 12], F : K → K is completely continuous. Now we consider the following modified problem of (1.1) and (1.2): ½ (φp1 (u0 ))0 + q1 (t)f ∗ (t, u, v) = 0, t ∈ (0, 1), (4.3) (φp2 (v 0 ))0 + q2 (t)g ∗ (t, u, v) = 0, t ∈ (0, 1),  m−2 X    u(0) = ai u(ξi ), u0 (1) = βu0 (0),   i=1

m−2 X     v(0) = ai v(ξi ), v 0 (1) = βv 0 (0), 

(4.4)

i=1

From the condition (H4 ), we have ∗ρ2 ∗ρ2 1 1 fρ∗ρ1 ϕ(t) < φp1 (m1 ), gρ∗ρ1 ϕ(t) < φp2 (m2 ), fγρ 2 ≥ φp1 (M1 γ), gγρ2 ≥ φp2 (M2 γ), 3 3 fρ∗ρ3 ϕ(t) ≤ φp1 (m1 ), gρ∗ρ3 ϕ(t) ≤ φp2 (m2 ). Firstly, we show that ik (F, Kρ∗1 ) = 1. 1 1 In fact, by (4.1), (4.2), fρ∗ρ1 ϕ(t) < φp1 (m1 ) and gρ∗ρ1 ϕ(t) < φp2 (m2 ), for (u, v) ∈ ∂Kρ∗1 , we have kA(u, v)(t)k = max |A(u, v)(t)| = A(u, v)(1) 0≤t≤1 µZ 1 Z 1 −1 = φp1 q1 (τ )f ∗ (τ, u(τ ), v(τ ))dτ 0

s

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φp1 (β) + 1 − φp1 (β) + 1−

1 m−2 X

ai

Z

¶

1

∗

q1 (τ )f (τ, u(τ ), v(τ ))dτ

0 m−2 X i=1

Z ai

0

ξi

µZ φ−1 p1

s

1

ds

q1 (τ )f ∗ (τ, u(τ ), v(τ ))dτ

i=1

¶ Z 1 φp1 (β) ∗ q1 (τ )f (τ, u(τ ), v(τ ))dτ ds + 1 − φp1 (β) 0 µZ 1 Z 1 −1 ≤ φp1 q1 (τ )f ∗ (τ, u(τ ), v(τ ))dτ 0 0 ¶ Z 1 φp1 (β) ∗ q1 (τ )f (τ, u(τ ), v(τ ))dτ ds + 1 − φp1 (β) 0 µZ 1 m−2 X Z ξi 1 −1 q1 (τ )f ∗ (τ, u(τ ), v(τ ))dτ φ a + i p1 m−2 X 0 0 i=1 1− ai i=1

¶ Z 1 φp1 (β) ∗ q1 (τ )f (τ, u(τ ), v(τ ))dτ ds + 1 − φp1 (β) 0 µ ¶ Z 1 1 −1 ∗ =Lφp1 q1 (τ )f (τ, u(τ ), v(τ ))dτ 1 − φp1 (β) 0 ¶ µ Z 1 ρ1 ρ1 k(u, v)k −1 φp1 (ρ1 )φp1 (m1 ) q1 (τ )dτ = m1 = = ,

+ . This implies that min (u(t) + v(t)) = γρ2 > γρ2 + λ0 , which is a η≤t≤1 2 2 contradiction. Hence, by Theorem 2.1, it follows that ik (F, Ωρ2 ) = 0. Finally, similar to the proof of ik (F, Kρ∗1 ) = 1, we can show that ik (F, Kρ∗3 ) = 1. We can get the BVP (4.3), (4.4) has at least three positive solutions (u1 , v1 ), (u2 , v2 ) and (u3 , v3 ) such that (u1 , v1 ) ∈ Kρ∗1 , (u1 , v1 ) ∈ Ωρ2 \Kρ∗1 , (u3 , v3 ) ∈ Kρ∗3 \Ωρ2 . As a result, the BVP (4.3), (4.4) has at least three positive solutions (u1 , v1 ), (u2 , v2 ) and (u3 , v3 ) such that u1 + v1 , u2 + v2 , u3 + v3 ∈ [ρ1 ϕ(t), ∞), and f ∗ (t, u, v) = f (t, u, v), g ∗ (t, u, v) = g(t, u, v), u + v ≥ ρ1 ϕ(t), which mean (u1 , v1 ), (u2 , v2 ) and (u3 , v3 ) are also solutions of BVP (1.1), (1.2). Similarly, we can obtain the following conclusions. Theorem 4.2. Assume (H1 ) − (H3 ) hold. In addition, the following condition (H5 ) holds: (H5 ) There exist ρ1 , ρ2 , ρ3 ∈ (0, ∞), with ρ1 < ρ2 < γρ3 such that (1) f (t, u, v), g(t, u, v) > 0, t ∈ [0, 1], u + v ∈ [min{γρ1 , ρ2 ϕ(t)}, ∞); ρ1 ρ1 ρ2 ρ2 (2) fγρ 1 ≥ φp1 (M1 γ), gγρ1 ≥ φp2 (M2 γ), fρ2 ϕ(t) < φp1 (m1 ), gρ2 ϕ(t) < φp2 (m2 ), ρ3 ρ3 fγρ 3 ≥ φp1 (M1 γ), gγρ3 ≥ φp2 (M2 γ). Then BVP (1.1) and (1.2) has at least two positive solutions in K. Theorem 4.3. Assume (H1 ) − (H3 ) hold. In addition, the following condition (H6 ) holds: (H6 ) There exist ρ1 , ρ2 ∈ (0, ∞), with ρ1 < γρ2 such that (1) f (t, u, v), g(t, u, v) > 0, t ∈ [0, 1], u + v ∈ [ρ1 ϕ(t), ∞); ρ2 ρ2 (2) fρρ11ϕ(t) < φp1 (m1 ), gρρ11ϕ(t) < φp2 (m2 ), fγρ 2 ≥ φp1 (M1 γ), gγρ2 ≥ φp2 (M2 γ), 0 ≤ f ∞ < φp1 (m1 ), 0 ≤ g ∞ < φp2 (m2 ). Then BVP (1.1) and (1.2) has at least three positive solutions in K. Proof. We show that (H6 ) implies (H4 ). Let k ∈ (f ∞ , φp1 (m1 )). Then there exists r > ρ2 , such that max f (t, u, v) ≤ kφp1 (u + v) for u + v ∈ [r, ∞) since 0 ≤ f ∞ < φp1 (m1 ). Let ¶ ¾ ½ t∈[0,1] ¾ ½ µ α −1 , ρ2 . α = max f (t, u, v) : ρ1 ϕ(t) ≤ u + v ≤ r and ρ3 > max φp1 φp1 (m) − k t∈[0,1] Then we have max f (t, u, v) ≤ kφp1 (u+v)+α ≤ kφp1 (ρ3 )+α < φp1 (m1 )φp1 (ρ3 ), for u+v ∈ [ρ1 ϕ(t), ρ3 ). t∈[0,1]

This implies that fρρ33ϕ(t) < φp1 (m1 ). Similarly, 0 ≤ g ∞ < φp2 (m2 ) implies that gρρ33ϕ(t) < φp2 (m2 ). Hence, (H4 ) holds, by Theorem 4.1, BVP (1.1), (1.2) has at least three positive solutions in K. Theorem 4.4. Assume (H1 ) − (H3 ) hold. In addition, the following condition (H7 ) holds: (H7 ) There exist ρ1 , ρ2 ∈ (0, ∞), with ρ1 < ρ2 such that (1) f (t, u, v), g(t, u, v) > 0, t ∈ [0, 1], u + v ∈ [min{γρ1 , ρ2 ϕ(t)}, ∞); ρ2 ρ2 ρ1 ρ1 (2) fγρ 1 ≥ φp1 (M1 γ), gγρ1 ≥ φp2 (M2 γ), fρ2 ϕ(t) < φp1 (m1 ), gρ2 ϕ(t) < φp2 (m2 ), φp1 (M1 ) < f∞ ≤ ∞, φp2 (M2 ) < g∞ ≤ ∞. Then BVP (1.1) and (1.2) has at least two positive solutions in K. Proof. We show that (H7 ) implies (H5 ). Since φp1 (M1 ) < f∞ ≤ ∞, then there exists ρ2 ρ3 > , such that γ min f (t, u, v) ≥ φp1 (u+v)φp1 (M1 ) ≥ φp1 (γρ3 )φp1 (M1 ) = φp1 (ρ3 )φp1 (M1 γ), u+v ∈ [γρ3 , ρ3 ).

t∈[η,1]

ρ3 ρ3 This implies that fγρ 3 ≥ φp1 (M1 γ). Similarly, φp2 (M2 ) < g∞ ≤ ∞ implies that gγρ3 ≥ φp2 (M2 γ). Hence, (H5 ) holds, by Theorem 4.2, BVP (1.1), (1.2) has at least two positive solutions in K. By the arguments similar to that of Theorem 3.1 and Theorem 3.2, we obtain the following

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results. Theorem 4.5. Assume (H1 ) − (H3 ) hold. In addition, the following condition (H8 ) holds: (H8 ) There exist ρ1 , ρ2 ∈ (0, ∞), with ρ1 < γρ2 such that (1) f (t, u, v), g(t, u, v) > 0, t ∈ [0, 1], u + v ∈ [ρ1 ϕ(t), ∞); ρ2 ρ2 (2) fρρ11ϕ(t) ≤ φp1 (m1 ), gρρ11ϕ(t) ≤ φp2 (m2 ), fγρ 2 ≥ φp1 (M1 γ), gγρ2 ≥ φp2 (M2 γ). Then BVP (1.1), (1.2) has at least one positive solutions in K. Theorem 4.6. Assume (H1 ) − (H3 ) hold. In addition, the following condition (H9 ) holds: (H9 ) There exist ρ1 , ρ2 ∈ (0, ∞), with ρ1 < ρ2 such that (1) f (t, u, v), g(t, u, v) > 0, t ∈ [0, 1], u + v ∈ [min{γρ1 , ρ2 ϕ(t)}, ∞); ρ1 ρ1 ρ2 ρ2 (2) fγρ 1 ≥ φp1 (M1 γ), gγρ1 ≥ φp2 (M2 γ), fρ2 ϕ(t) ≤ φp1 (m1 ), gρ2 ϕ(t) ≤ φp2 (m2 ). Then BVP (1.1) and (1.2) has at least one positive solutions in K.

5

Example

Now we present an example to illustrate the main result. Example 5.1. Consider the following BVP ½ (|u0 (t)|u0 (t))0 + q1 (t)f (t, u, v) = 0, t ∈ (0, 1), (|v 0 (t)|v 0 (t))0 + q2 (t)g(t, u, v) = 0, t ∈ (0, 1),    u(0) = 1 u( 1 ) + 1 u( 2 ), u0 (1) = 1 u0 (0), 2 3 4 3 2 1 1 1 2 1 0  0  v(0) = v( ) + v( ), v (1) = v (0), 2 3 4 3 2 where    1 (1 + t) 12 (u + v − t )21 + 1 , 0 ≤ u + v ≤ 2, 80 4 1035 f (t, u, v) = 1 1 t 1   (1 + t) 2 (2 − )21 + 35 , u + v > 2, 80 4 10    1 (1 + t) 14 (u + v − t )19 + 1 , 0 ≤ u + v ≤ 2, 40 4 1040 g(t, u, v) = 1 1 1 t 19   (1 + t) 4 (2 − ) + 40 , u + v > 2, 40 4 10

(5.1)

(5.2)

q1 (t) = q2 (t) = 1.

2 1 1 1 Obviously, p1 = p2 = 3, β = 1, ξ1 = , ξ2 = , a1 = , a2 = . Choose ρ1 = 1, ρ2 = 3 3 2 4 √ √ 448 3 1 1 t 3 3 , ρ3 = 1200, η = , θ = , then ϕ(t) = . We note γ = , m1 = m2 = 4 2 2 224 √9 3 3 , M1 = M2 = 4. Consequently, f (t, u, v) satisfies 28 t (1) f (t, u, v), g(t, u, v) > 0, t ∈ [0, 1], u + v ∈ [ , ∞); 2 (2) fρρ11ϕ(t) ≤ 0.018 < φp1 (m1 ) ≈ 0.034, gρρ11ϕ(t) ≤ 0.03 < φp2 (m2 ) ≈ 0.034, ρ2 ρ2 fγρ 2 ≥ 0.239 > φp1 (M1 γ) ≈ 0.009, gγρ2 ≥ 0.147 > φp2 (M2 γ) ≈ 0.009, fρρ33ϕ(t) ≤ 0.026 < φp1 (m1 ) ≈ 0.034, gρρ33ϕ(t) ≤ 0.011 < φp2 (m2 ) ≈ 0.034. Thus with Theorem (4.1), BVP (5.1), (5.2) has at least three positive solutions in K.

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References [1] Y. Guo, Y. Ji, X. Liu, Multiple positive solutions for some multipoint boundary value problems with p-Laplacian, J. Comput. Appl. Math. 216 (2008) 144-156. [2] V. A. Il’in, E. I. Moiseev, Nonlocal boundary value problem of the second kind for a Sturm-Liouville operator, Diff. Eq. 23 (1987) 979-987. [3] D. Ji, M. Feng, W. Ge, Multiple positive solutions for multipoint boundary value problems with sign changing nonlinearity, Appl. Math. Comput. 196 (2008) 511-520. [4] G. L. Karakostas, P. Ch. Tsamatos, Positive solutions of a boundary value problem for second order ordinary differential equations, Electron. J. of Diff. Equ. 49 (2000) 1-9. [5] H. L¨ u, D. O’Regan, C. Zhong, Multiple positive solutions for the one-dimension singular p-Laplacian, Appl. Math. Comput. 46 (2002) 407-422. [6] X. Liu, D. Xia, A. Tang, Note on multiple positive solutions to non-homogenous multipoint BVPS for second order p-Laplacian equations, Ann. of Diff. Eqs. 26 (2010) 30-44. [7] K. Lan. Multiple positive solutions of semilinear differential equations with singularities, J. London Math. Soc. 63 (2001) 690-704. [8] X. Liu, D. Xia, A. Tang, Note on multiple positive solutions to non-homogenous multipoint BVPS for second order p-Laplacian equations, Ann. of Diff. Eqs. 26 (2010) 30-44. [9] D. Ma, Z. Du, W. Ge, Existence and iteration of monotone positive solutions for multipoint boundary value problem with p-Laplacian operator, Comput. Math. Appl. 50 (2005) 729-739. [10] M. Moshinsky, Sobre los problemas de condiciones a la frontiera en una dimension de caracteristicas discontinuas, Bol. Soc. Mat. Mexicana. 7 (1950) 10-25. [11] R. Ma, D. Cao, Positive solutions to an m-point boundary value problem, Appl. Math. J. Chinese Univ. Ser. B. 17 (2002) 24-30. [12] J. Zhao, W. Wang, W. Ge, Three symmetric positive solutions of multipoint boundary value problem with one-dimensional p-Laplacian, Acta Math. Sinica. 52 (2009) 259-268. [13] M. Zong, W. Cai, Three-point boundary value problem for p-Laplacian differential equation at resonance, Ann. of Diff. Eqs. 25 (2009) 249-252. [14] X. Zhao, L. Zhao, W. Ge, Existence of at least three positive solutions to multipoint boundary value problem with p-Laplacian operator, Ann. of Diff. Eqs. 25 (2009) 223-227. [15] Y. Zhou, Y. Cao, Triple positive solutions of the multipoint boundary value problem for second-Order Differential Equations, J. of Math. Research. Exposition. 30 (2010) 475-486.

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J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 20, NO.6, 2016, COPYRIGHT 2016 EUDOXUS PRESS, LLC

An S-partially contractive mapping with a control function φ K. Abodayeh1 , Department of Mathematics and Physical Sciences, Prince Sultan University P. O. Box 66833, Riyadh 11586, Saudi Arabia

Abstract. In this article, we introduce a φ-contraction principle in a partial S-metric space, we show the existence of a fixed point for a self mapping in a partial S-metric space. Also, we show that we have uniqueness only under some specific conditions. Keywords. Partial S-metric space, Banach contraction principle, Fixed point.

1

Introduction and Preliminaries

Finding a fixed point for a self mapping on different types of metric spaces has been one the main topics of research in pure mathematics. it starts with the Banach contraction principle which was introduced by Banach in the early nineties. Since the Banach contraction was introduced, many results were found in fixed point theory field in different type of metric spaces, such as [13], [14], [15],[16],[22], [23], [24],[25], [4], [5], [6], [8],[9], [10], [11], [12], [19], [20], [21],[26],[27],[28]. An S- metric space was introduced in [2]. Definition 1. [2] Let X be a nonempty set. An S-metric space on X is a function S : X 3 → [0, ∞) that satisfies the following conditions, for all x, y, z, a ∈ X : • S(x, y, z) ≥ 0, • S(x, y, z) = 0 if and only if x = y = z, • S(x, y, z) ≤ S(x, x, a) + S(y, y, a) + S(z, z, a). The pair (X, S) is called an S-metric space. In this article, we are interested in partial S-metric space which was introduced in [1]. We recall some definitions of partial metric spaces and state some of their 1

Corresponding Author E-Mail Address: [email protected]

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Abodayeh 1078-1087

J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 20, NO.6, 2016, COPYRIGHT 2016 EUDOXUS PRESS, LLC

2 properties. Definition 2. [1] Let X be a nonempty set. A partial S-metric space on X is a function Sp : X 3 → [0, ∞) that satisfies the following conditions, for all x, y, z, t ∈ X : (P1) x = y if and only if Sp (x, x, x) = Sp (y, y, y) = Sp (x, x, y) (P2) Sp (x, y, z) ≤ Sp (x, x, t) + Sp (y, y, t) + Sp (z, z, t) − Sp (t, t, t) (P3) Sp (x, x, x) ≤ Sp (x, y, z) (P4) Sp (x, x, y) = Sp (y, y, x). The pair (X, Sp ) is called a partial S-metric space. We recall some definitions of partial S-metric spaces and state some of their properties. Definition 3. A sequence {xn }∞ n=0 of elements in X is called Cauchy if the limit limn,m→∞ Sp (xn , xn , xm ) exists and finite. The partial S-metric space (X, Sp ) is called complete if for each Cauchy sequence {xn }∞ n=0 there exists z ∈ X such that Sp (z, z, z) = lim Sp (z, z, xn ) = lim Sp (xn , xn , xm ). n

n,m

Also, (X, Sp ) is a complete partial S-metric space if and only if (X, Sps ) is a complete S-metric space. A sequence {xn }n in a partial S-metric space (X, Sp ) is called 0-Cauchy if limn,m→∞ Sp (xn , xn , xm ) = 0. We say that (X, Sp ) is 0-complete if every 0-Cauchy in X converges to a point x ∈ X such that Sp (x, x, x) = 0. Example 1. (see [1]) Let X = Q ∩ [0, ∞) with the partial metric p(x, y, z) = max{x, y, z}. Then (X, Sp ) is a 0-complete partial metric space which is not complete. Definition 4. Let (X, Sp ) be a complete partial S-metric space. Set ρp = inf{Sp (x, y, z) : x, y, z ∈ X} and define the set Xp = {x ∈ X : Sp (x, x, x) = ρp }. The following Lemma summarizes the relation between certain comparison functions that usually act as control functions in the studied contractive typed mappings in fixed point theory. For such a summary and fixed point theory for φ− contractive mappings, see [18].

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Abodayeh 1078-1087

J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 20, NO.6, 2016, COPYRIGHT 2016 EUDOXUS PRESS, LLC

3 Lemma 1. Let φ : R+ → R+ be a function and relative to the function φ consider the following conditions: • (i) φ is monotone increasing. • (ii) φ(t) < t for all t > 0. • (iii) φ(0) = 0. • (iv) φ is right uppersemicontinuous. • (v) φ is right continuous. • (vi) limn→∞ φn (t) = 0 for all t ≥ 0. Then the following are valid: • (1) The conditions (i) and (ii) imply (iii). • (2) The conditions (ii) and (v) imply (iii). • (3) The conditions (i) and (vi) imply (ii). • (4) The conditions (i) and (iv) imply imply (vi). • (5) If φ satisfies (i) then (iv) ⇔ (v).

2

Main Results

Now, we prove our main result. Theorem 1. Let (X, Sp ) be a complete partial S-metric space. Suppose T : X → X is a given self mapping satisfying: Sp (T x, T x, T y) ≤ max{φ(Sp (x, x, y)), Sp (x, x, x), Sp (y, y, y)},

(1)

where φ is defined as in Lemma 1. Then: (1) the set Xp is nonempty; (2) there is a unique u ∈ Xp such that T u = u;

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Abodayeh 1078-1087

J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 20, NO.6, 2016, COPYRIGHT 2016 EUDOXUS PRESS, LLC

4 Proof. For any x ∈ X, we have Sp (T x, T x, T x) ≤ Sp (x, x, x) and hence the sequence {Sp (T n x, T n x, T n x)}n≥0 is a nonincreasing sequence. Now Define Mx := 2[f −1 (Sp (x, x, T x)) + Sp (x, x, x)], where f (t) = t − φ(t). Notice that f (0) = 0 (and hence f −1 (0) = 0) and f (t) < t for t > 0 and hence f −1 (t) > t for t > 0. Now we prove by induction that Sp (T n x, T n x, x) ≤ Mx , ∀n ≥ 0.

(2)

Notice that the inequality (2) is true for n = 0, 1 since: Sp (x, x, x) ≤ Mx and Sp (T x, T x, x) ≤ f −1 (Sp (T x, T x, x)) ≤ Mx . Suppose that (2) is true for each n ≤ n0 − 1 for some positive integer n0 ≥ 2. Then we have Sp (T n0 x, T n0 x, x) ≤ 2Sp (T n0 x, T n0 x, T x) + Sp (T x, T x, x) ≤ 2 max{φ(Sp (T n0 −1 x, T n0 −1 x, x)), Sp (T n0 −1 x, T n0 −1 x, T n0 −1 x), Sp (x, x, x)} + Sp (T x, T x, x) ≤ 2 max{φ(Sp (T n0 −1 x, T n0 −1 x, x))), Sp (x, x, x)} + Sp (T x, T x, x)

Therefore, we have two cases. Case 1: Sp (T n0 x, T n0 x, x) ≤ φ(Sp (T n0 −1 x, T n0 −1 x, T x)) + Sp (T x, T x, x) ≤ 2[φ(f −1 (Sp (T x, T x, x)) + Sp (x, x, x))] + Sp (T x, T x, x) = 2[f −1 (Sp (T x, T x, x)) + Sp (x, x, x) − f (f −1 (Sp (T x, T x, x)) +Sp (x, x, x))] + Sp (T x, T x, x) ≤ Mx − 2f (f −1 (Sp (T x, T x, x)) + Sp (x, x, x)) + Sp (T x, T x, x) = Mx − Sp (T x, T x, x) − Sp (x, x, x) ≤ Mx . Case 2: Sp (T n0 x, T n0 x, x) ≤ Sp (x, x, x) + Sp (T x, T x, x) ≤ Sp (x, x, x) + f −1 (Sp (T x, T x, x)) = Mx . Hence, we obtain (2). Next we prove that the sequence {Sp (T n x, T n x, T n x)}n≥0 is Cauchy. Equivalently, we show that lim Sp (T n x, T n x, T m x) = rx ,

n,m→∞

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(3)

Abodayeh 1078-1087

J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 20, NO.6, 2016, COPYRIGHT 2016 EUDOXUS PRESS, LLC

5 where rx := inf n Sp (T n x, T n x, T n x). Its clear that rx ≤ Sp (T n x, T n x, T n x) ≤ Sp (T n x, T n x, T m x) for all n, m. Also, given any  > 0, there exists n0 ∈ N such that Sp (T n0 x, T n0 x, T n0 x) < rx +  and φn0 (2Mx ) < rx + . Therefore, for any m, n > 2n0 we have rx ≤ Sp (T n x, T n x, T m x) ≤ max{φ(Sp (T n−1 x, T n−1 x, T m−1 x)), Sp (T n−1 x, T n−1 x, T n−1 x), Sp (T m−1 x, T m−1 x, T m−1 x)} ≤ max{φ2 (Sp (T n−2 x, T n−2 x, T m−2 x)), Sp (T n−2 x, T n−2 x, T n−2 x), Sp (T m−2 x, T m−2 x, T m−2 x)} ≤ max{φn0 (Sp (T n−n0 x, T n−n0 x, T m−n0 x)), Sp (T n−n0 x, T n−n0 x, T n−n0 x), Sp (T m−n0 x, T m−n0 x, T m−n0 x)} ≤ max{φn0 (Sp (T n−n0 x, T n−n0 x, x) + Sp (T m−n0 x, T m−n0 x, x)), Sp (T n−n0 x, T n−n0 x, T n−n0 x), Sp (T m−n0 x, T m−n0 x, T m−n0 x)} < max{φn0 (2Mx ), rx + , rx + } < rx + . Hence, we obtain (3). Since (X, Sp ) is a complete partial S-metric space, there exists z ∈ X such that Sp (z, z, z) = rx . Next, we show that Sp (z, z, z) = p(T z, T z, z). Next, we show that Sp (z, z, z) = Sp (z, z, T z) = Sp (T z, T z, z). For each natural number n we have Sp (z, z, T z) ≤ 2Sp (z, z, zn ) − Sp (zn , zn , zn ) + Sp (T z, T z, zn ). From the contraction condition of our theorem, we deduce that there exists a subsequence of natural numbers {nl } such that Sp (T z, T z, znl ) ≤ φ(Sp (z, z, znl −1 )), for l ≥ 1, or Sp (T z, T z, znl ) ≤ Sp (z, z, z) for l ≥ 1, or Sp (T z, z, znl ) ≤ Sp (znl −1 , znl −1 , znl −1 ), for l ≥ 1, in all of these three cases, if we take the limit as l goes toward ∞ we get Sp (z, z, T z) ≤ Sp (z, z, z). But, we know by the property (iv) of the partial S-metric space that Sp (z, z, z) ≤ Sp (z, z, T z). Therefore, Sp (z, z, z) = Sp (z, z, T z).

(4)

Now we show that Xp (see Definition 4) is nonempty. For each k ∈ N choose xk ∈ X with Sp (xk , xk , xk ) < ρp + 1/k, where xk = T k x. First, we prove that lim Sp (zn , zn , zm ) = ρp .

(5)

m,n→∞

Given  > 0, take n0 := [f −1 (3/)] + 1. If k > n0 , then ρp ≤ Sp (T zk , T zk , T zk ) ≤ Sp (zk , zk , zk ) = rxk ≤ Sp (xk , xk , xk ) < ρp + 1/k < ρp + 1/n0 < ρp + 1/f −1 (3/).

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J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 20, NO.6, 2016, COPYRIGHT 2016 EUDOXUS PRESS, LLC

6 Set Uk := Sp (zk , zk , zk ) − Sp (T zk , T zk , T zk ). Then Uk < 1/f −1 (3/) for k > n0 . Thus, if m, n > n0 then by (4) and the fact that f (and hence f −1 ) is increasing, we have Sp (zn , zn , zm ) ≤ Sp (zn , zn , T zn ) + Sp (T zn , T zn , T zm ) + Sp (T zm , T zm , zm ) −Sp (T zn , T zn , T zn ) − Sp (T zm , T zm , T zm ) = Un + Um + Sp (T zn , T zn , T zm ) < 2/f −1 (3/) + max{φ(Sp (zn , zn , zm )), Sp (zn , zn , zn ), Sp (zm , zm , zm )}
≤ max{f −1 2/f −1 (3/) , 3/f −1 (3/) + ρp } ≤ max{f −1 (2/3)) , ρp + } ≤ ρp +  + f −1 (2/3). Therefore, if we let  → 0 we get (5). Since (X, Sp ) is a complete partial metric space, there exists u ∈ X such that Sp (u, u, u) = limm,n→∞ Sp (zn , zn , zm ) = ρp . Consequently, u ∈ Xp and hence Xp is nonempty. Now choose an arbitrary x ∈ Xp . Then ρp ≤ Sp (T z, T z, T z) ≤ Sp (T z, T z, z) = Sp (z, z, z) = rx = ρp , which, using P2, implies that T z = z. To prove uniqueness of the fixed point we suppose that u, v ∈ Xp are both fixed points of T. Then ρp ≤ Sp (u, u, v) = Sp (T u, T u, T v) ≤ max{φ(Sp (u, u, v)), Sp (u, u, u), Sp (v, v, v)} ≤ max{φ(p(u, v)), ρp }. Case 1: ρp ≤ Sp (u, u, v) ≤ ρp ⇒ Sp (u, u, v) = ρp = Sp (u, u, u) = Sp (v, v, v) ⇒ u = v. Case 2: ⇒ ⇒ ⇒ ⇒ ⇒

Sp (u, u, v) ≤ φ(Sp (u, u, v)) Sp (u, u, v) − φ(Sp (u, u, v)) ≤ 0 f (Sp (u, u, v)) ≤ 0 f (Sp (u, u, v)) = 0 Sp (u, u, v) = 0 u = v.

Thus, the fixed point is unique. Note that the above theorem does not guarantee uniqueness of the fixed point in X. However, if (1) is replaced by the condition below, we can show uniqueness in X. In the next result, we change our contraction condition so that we obtain uniqueness of the fixed point in the whole space X.

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Abodayeh 1078-1087

J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 20, NO.6, 2016, COPYRIGHT 2016 EUDOXUS PRESS, LLC

7 Theorem 2. Let (X, Sp ) be a complete partial S-metric space. Suppose T : X → X is a given self mapping satisfying:   Sp (x, x, x) + Sp (y, y, y) Sp (T x, T x, T y) ≤ max φ(Sp (x, x, y)), , (6) 2 where φ : [0, ∞) → [0, ∞) is as in Theorem 1. Then there is a unique point z ∈ X such that T z = z. Furthermore, z ∈ Xp . Proof. Using Theorem 1 we only need to prove uniqueness. Suppose there exists u, v ∈ X such that T u = u and T v = v. Now   Sp (u, u, u) + Sp (v, v, v) . Sp (u, u, v) = Sp (T u, T u, T v) ≤ max φ(Sp (u, u, v)), 2 Case 1:

⇒ ⇒ ⇒ ⇒ ⇒

Sp (u, u, v) ≤ φ(Sp (u, u, v)) Sp (u, u, v) − φ(Sp (u, u, v)) ≤ 0 f (Sp (u, u, v)) ≤ 0 f (Sp (u, u, v)) = 0 Sp (u, u, v) = 0 u = v.

Case 2: Sp (u, u, u) + Sp (v, v, v) 2 2Sp (u, u, v) − Sp (u, u, u) − Sp (v, v, v) ≤ 0 2Sp (u, u, v) − Sp (u, u, u) − Sp (v, v, v) = 0 u = v. Sp (u, u, v) ≤

⇒ ⇒ ⇒

As a consequence of Theorem2, we obtain the following Corollary. Corollary 1. Let (X, Sp ) be a 0-complete partial S-metric space. Suppose T : X → X is a given self mapping satisfying: Sp (T x, T x, T y) ≤ φ(Sp (x, x, y)),

(7)

where φ : [0, ∞) → [0, ∞) is an increasing function such that f (t) = t − φ(t) is increasing with f −1 is right continuous at 0. Also assume limn→∞ φn (t) = 0 for all t ≥ 0 (and hence φ(0) = 0, φ(t) < t for t > 0 ). Then there is a unique z ∈ X such that T z = z. Also Sp (z, z, z) = 0.

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Abodayeh 1078-1087

J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 20, NO.6, 2016, COPYRIGHT 2016 EUDOXUS PRESS, LLC

8 Example 2. Let X = [0, 1] ∪ [3, 4]. Define Sp : X 3 → [0, ∞), T : X → X and φ : [0, ∞) → [0, ∞) as follows: Sp (x, y, z) = max{x, y, z} ( x , x ∈ [0, 1] 2 T (x) = 7 , x ∈ [3, 4] 5 t φ(t) = 1+t The above definitions satisfy the hypothesis of Theorem 2. In particular, we make the following observations: • (X, p) is a complete partial metric space. • We can easily prove by induction that φn (t) = 0.

t 1+nt

which implies that limn→∞ φn (t) =

• T satisfies condition (6): 1) If {x, y, z} ∩ [3, 4] 6= ∅ then 7 Sp (T x, T y, T z) = max{T x, T y, T z} = 5   Sp (x, x, x) + Sp (y, y, y) ≤ max φ(Sp (x, y, z)), 2 2) If {x, y, z} ⊂ [0, 1] then nx y z o Sp (T x, T y, T z) = max{T x, T y, T z} = max , , 2 2 2   Sp (x, x, x) + Sp (y, y, y) ≤ max φ(Sp (x, y)), . 2 By Theorem 2, there is a unique fixed point which is z = 0.

References [1] N. Mlaiki, “A contraction principle in partial S-metric spaces,” Universal Journal of Mathematics and Mathematical Sciences, 5 (2) (2014) 109-119. [2] S. Sedghi, N. Shobe and A. Aliouche, “A generalization of fixed point theorems in S-metric spaces,” Mat. Vesnik 64 (2012), 258-266.

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9 [3] S. G. Matthews, Partial metric topology, in Proceedings of the 11th Summer Conference on General Topology and Applications, vol. 728, pp. 183197, The New York Academy of Sciences, Gorham, Me, USA, August 1995. [4] T. Abdeljawad, E. Karapinar and K. Ta¸s, Existence and uniqueness of a common fixed point on partial metric spaces, Appl. Math. Lett. 24 (11) (2011), 1900–1904. [5] T. Abdeljawad, E. Karapinar and K. Ta¸s, A generalized contraction principle with control functions on partial metric spaces, 63 (3) (2012), 716-719 . [6] T. Abdeljawad, Fixed points for generalized weakly contractive mappings in partial metric spaces, Math. Comput. Modelling 54 (11-12) (2011), 2923–2927. [7] S. G. Matthews, Partial metric topology. Research Report 212. Dept. of Computer Science. University of Warwick, 1992. [8] S. Oltra and O. Valero, Banach’s fixed point theorem for partial metric spaces, Rend. Istit. Mat. Univ. Trieste 36 (1–2) (2004), 17–26. [9] O. Valero, On Banach fixed point theorems for partial metric spaces, Appl. Gen. Topol. 6 (2) (2005), 229–240. [10] I. Altun, F. Sola and H. Simsek, Generalized contractions on partial metric spaces, Topology and Its Applications 157 (18) (2010), 2778–2785. [11] I. Altun and A. Erduran, Fixed Point Theorems for Monotone Mappings on Partial Metric Spaces, Fixed Point Theory Appl., vol. 2011, Article ID 508730, 10 pages, 2011. doi:10.1155/2011/508730. [12] W. Shatanawi, B. Samet and M. Abbas, Coupled fixed point theorems for mixed monotone mappings in ordered partial metric spaces, Math. Comput. Modelling, doi: 10.1016/j.mcm.2011.08.042. [13] M. S. Khan, M. Sweleh and S. Sessa, Fixed point theorems by alternating distance between the points, Bull. Aust. Math. Soc. 30 (1) (1984), 1–9. [14] B. E. Rhoades, Some theorems on weakly contractive maps, Nonlinear Anal. 47 (4) (2001), 2683–2693. [15] P. N. Dutta and B. S. Choudhury, A generalization of contraction principle in metric spaces, Fixed Point Theory Appl., Artcile ID 406368, 8 pages, vol 2008. [16] D. W. Boyd and S. W. Wong, On nonlinear contractions, Proc. Amer. Math. Soc. 20 (1969),458–464.

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10 [17] D. Ili´c, V. Pavlovi´c and V. Rako˘cevi´c, Some new extensions of Banach’s contraction principle to partial metric spaces, Appl. Math. Lett. 24 (2011), 1326– 1330. [18] I. A. Rus, Generalizeed Contractions and Applications, Cluj University Press, Cluj-Napoca, (2001). [19] S. Romaguera, Fixed point theorems for generalized contractions on partial metric spaces, Topology Appl. 159 (2012), 194-199. [20] M. Abbas, T. Nazir and S. Romaguera, Fixed point results for for generalized cyclic contraction mappings in partial metric spaces, Revista de la Real Academia de Ciencias Exactas, in press, doi: 10.1007/s13398-011-0051-5. [21] I. Altun, A. Erduran, Fixed point theorems for monotone mappings on partial metric spaces, Fixed Point Theory and Applications, Vol 2011, Atricle ID 508730, 10 pages (2011), doi: 10.1155/2011/508730. ´ c, Semi-continuous mappings and fixed point theorems in quasi[22] Lj. B. Ciri´ metric spaces, Publ. Mah. (Debrecen) 54 (1999), 251-261. ´ c Ravi Agarwal B. Samet, Mixed Monotone Generalized Contrac[23] Lj. B. Ciri´ tions in Partially Ordered Probabilistic Metric Spaces, Fixed Point Theory and Applications 2011, 2011:56. [24] S.Sadiq Basha, Naseer Shahzad, R. Jeyaraj, Best proximity point theorems for reckoning optimal approximate solutions, Fixed Point Theory and Applications 2012, 2012:202 (12 November 2012) [25] J. H. Asl, Sh. Rezapour, Naseer Shahzad, On fixed points of α−ψ -contractive multifunctions, Fixed Point Theory and Applications 2012, 2012:212 (26 November 2012). [26] A. G. B. Ahmad, Z. M. Fadail, H. K. Nashine, Z. Kadelburg and S. Radenovi´c, Some new common fixed point results through generalized altering distances on partial metric spaces, Fixed Point Theory and Applications 2012, 2012:120. ´ Raji´c, and S. Radenovi´c, Nonlinear Con[27] A. G. B. Ahmad, Z. M. Fadail, V.C. tractions in 0-Complete Partial Metric Spaces, Abstract and Applied Analysis Volume 2012, Article ID 451239, 12 pages, 2012. [28] N. Shobkolaei, S. Sedghi, J. R. Roshan, I. Altun, Common fixed point of mappings satisfying almost generalized (S,T)-contractive condition in partially ordered partial metric spaces, Appl. Math. Comput. 219 (2012) 443452.

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Approximation by complex q-Gamma operators in compact disks Qing-Bo Caia,b,∗, Cuihua Lia and Xiao-Ming Zengc a

School of Information Science and Engineering, Xiamen University, Xiamen 361005, China b

School of Mathematics and Computer Science, Quanzhou Normal University, Quanzhou 362000, China c

Department of Mathematics, Xiamen University, Xiamen 361005, China E-mail: [email protected]; [email protected]; [email protected]

Abstract. In this paper, the order of simultaneous approximation and Voronovskaya type theorems with quantitative estimate for complex q-Gamma operators attached to analytic functions in compact disks are obtained. 2000 Mathematics Subject Classification: 41A10, 41A25, 41A36. Key words and phrases: quantitative estimate, Voronovskaya type theorem, qGamma operators.

1

Introduction

In recent years, an intensive research has been conducted on polynomials and operators in compact disks, such as [1], [3]-[8]. For a real function of real variable f : [0, ∞) → R, it is well known that the Gamma R∞ 1 n−1 e−t/x dt, x ∈ [0, ∞). In 2005, Zeng operators are given by Gn (f ; x) = xn Γ(n) 0 f (t/n)t [9] obtained the approximation properties of Gn defined above, supposed f satisfies exponential growth condition. he studied the approximation properties to the locally bounded functions and the absolutely continuous functions and obtained some good properties in real disks. In this paper, we introduce complex q-Gamma operators as follows 1 Gn,q (f ; z) = n z Γq (n)

Z

µ

∞/A

f 0

t [n]q

¶

µ t

n−1

Eq

qt − z

¶ dq t.

(1)

We give a suitable exponential growth condition in a parabolic domain for f (z). Let DR ={z ∈ C; |z| < R} be with 1 < R < ∞ and suppose that f : [R, +∞) ∪ DR → C is P k continuous in [R, +∞) ∪ DR , analytic in DR , i.e. f (z) = ∞ k=0 ck z , for all z ∈ DR , and ∗

Corresponding author.

1

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Q. -B. CAI, C. Li and X. -M. Zeng that there exist M, C, B > 0 and A ∈

¢ Ak , for , 1 , with the property |ck | ≤ M q k(k−1)/2 [k] R q!

¡1

all k = 0, 1, ..., which implies |f (z)| ≤ M Eq (A|z|) for all z ∈ DR and |f (x)| ≤ CeBx , for all x ∈ [R, +∞). We recall some concepts of q-calculus. All of the results can be found in [7]. 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 = q = 1. k, Also q-factorial and q-binomial coefficients are defined as follows: ( [k]q [k − 1]q ...[1]q , k = 1, 2, ...; [k]q ! = 1, k = 0, and

"

# n k

= q

[n]q ! , [k]q ![n − k]q !

(n ≥ k ≥ 0).

The q-improper integrals are defined as Z

∞/A

0

µ n¶ n ∞ X q q f (x)dq x = (1 − q) , A > 0, f A A −∞

provided the sums converge absolutely. The q−analogs eq (x) and Eq (x) of the exponential function are given as eq (x) =

∞ X xk 1 1 = , |x| < , |q| < 1, [k]q ! (1 − (1 − q)x)∞ 1 − q q k=0

Eq (x) =

∞ X

q k(k−1)/2

k=0

xk = (1 + (1 − q)x)∞ q , |q| < 1, [k]q !

Q∞

j where (1−x)∞ q = j=0 (1−q x). It is easily observed that eq (x)Eq (−x) = eq (−x)Eq (x) = 1. The q-Gamma integral is defined as

Z Γq (t) =

0

∞/A

xt−1 Eq (−qx)dq x, t > 0,

(2)

which satisfies the following functional equations: Γq (t + 1) = [t]q Γq (t), Γq (1) = 1.

2

Auxiliary Results

In the sequel, we suppose that ek (t) = tk , k = 0, 1, 2, .... In order to obtain the main results, we need the following lemmas:

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Lemma 2.1. For n ∈ N and z ∈ C, we have the following identities: Gn,q (ek ; z) =

[n + k − 1]q ! ek (z), [n − 1]q ![n]kq

(3)

Gn,q (ek ; z) =

[n + k − 1]q z Gn (ek−1 ; z). [n]q

(4)

Proof. From (1) and (2), we have Gn,q (ek ; z) = = =

¶ µ ¶ t k n−1 qt dq t t Eq − [n]q z 0 µ ¶ µ ¶ Z ∞/A µ ¶n+k−1 t qt t zk − E dq q z z z [n]kq Γq (n) 0 1 n z Γq (n)

Z

∞/A µ

Γq (n + k)z k [n + k − 1]q ! = ek (z), [n]kq [n − 1]q ! [n − 1]q ![n]kq

so we proved (3), and (4) is easily obtained according to (3). P k Lemma 2.2. If f is analytic in DR , f (z) = ∞ k=0 ck z , for all z ∈ DR , then for all n ∈ N and 1 ≤ r ≤ R, we have ∞ X Gn,q (f ; z) = ck · Gn,q (ek ; z). (5) k=0

Proof. By Lemma 2.1, we obtain that Gn,q (ek ; z) is a polynomial of degree ≤ k, k = 0, 1, 2, ... for all z ∈ C. From the hypothesis on f in section 1, it follows that Gn,q (f ; z) is analytic in DR (see [2], pp. 1171-1172 and p. 1178). Therefore, it is easy to obtain Lemma 2.2.

3

Main Results

We start with the following quantitative estimates of the convergence for complex qGamma operators attached to an analytic function in a disk of radius R > 1 and center 0. Theorem 3.1. Let DR ={z ∈ C; |z| < R} be with 1 < R < ∞ and suppose that f : S P k [R, +∞) DR → C is continuous in [R, +∞)∪DR , analytic in DR , i.e. f (z) = ∞ k=0 ck z , for all z ∈ DR , and f (z) satisfies exponential-type growth condition in the statement of section 1. (i) Let 1 ≤ r < A1 be arbitrary fixed. For all |z| ≤ r, n ≥ 2 (n ∈ N), we have |Gn,q (f ; z) − f (z)| ≤

Lq,r,A , [n]q

where Lq,r,A = M r2 A2 Cq,r,A , Cq,r,A is a constant depends only on q, r, A. (ii) For the simultaneous approximation by complex q-Gamma operators, we have: if 1 ≤

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r ≤ r1
r and center 0, since for any |z| ≤ r and v ∈ γ, we have |v − z| ≥ r1 − r, by Cauchy’s formulas it follows that for all |z| ≤ r and n ∈ N, n ≥ 2, we have ¯Z ¯ p! ¯¯ Gn,q (f ; v) − f (v) ¯¯ (p) dv |G(p) (f ; z) − f (z)| = n,q ¯ 2π ¯ γ (v − z)p+1 Lq,r1 ,A p! Lq,r1 ,A 2πr1 p!r1 ≤ = , p+1 [n]q 2π (r1 − r) [n]q (r1 − r)p+1 which proves (ii) and Theorem 3.1. Next, we will give Voronovskaya type result in compact disks, for complex q-Gamma operators attached to an analytic function in DR , R > 1 and center 0.

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Theorem 3.2. Suppose that f : DR ∪[R, ∞) → C is continuous and bounded in DR ∪[R, ∞) and analytic in DR . Let 1 ≤ r < A1 be arbitrary fixed, n ≥ 2 (n ∈ N), then we have the following Voronovskaya type result ¯ ¯ 2 00 ¯ ¯ ¯Gn,q (f ; z) − f (z) − z f (z) ¯ ≤ Jq,r,A , ¯ [2]q [n]q ¯ [n]2q P 2 k where Jq,r,A = [2]12 ∞ k=3 [k − 2]q [k − 1]q (rA) < ∞. q

q n [k] [k−1] e (z)

q q k Proof. Denoting Eq,k,n (z) = Gn,q (ek ; z)−ek (z)− , since Eq,0,n (z) = Eq,1,n (z) = [2]q [n]q Eq,2,n (z) = 0, then we have ¯ X ¯ ∞ 2 f 00 (z) ¯ ¯ z ¯≤ ¯Gn,q (f ; z) − f (z) − |ck | · |Eq,k,n (z)|, ¯ [2]q [n]q ¯

k=3

so, it remains to estimate Eq,k,n (z) for k ≥ 3. By Lemma 2.1 and simple calculation, we have

¡ ¢ q n [k − 1]q [k − 2]q [n + k − 1]q − q 2 [n]q k [n + k − 1]q z Eq,k−1,n (z) + z , Eq,k,n (z) = [n]q [2]q [n]2q

this implies, for all |z| ≤ r, k ≥ 3, n ∈ N, |Eq,k,n (z)| ≤

q n [k − 1]2q [k − 2]q k [n + k − 1]q r r , |Eq,k−1,n (z)| + [n]q [2]q [n]2q

taking in the last inequality, k = 3, 4, ..., and reasoning by recurrence, we obtain |Eq,k,n (z)| ≤

[n + k − 1]q [n + k − 2]q [n + 3]q q n · [2]2q k ... r [n]q [n]q [n]q [2]q [n]2q +

[n + k − 1]q [n + k − 2]q [n + 4]q q n [2]q · [3]2q k ... r [n]q [n]q [n]q [2]q [n]2q

[n + k − 1]q q n [k − 3]q [k − 2]2q k q n [k − 2]q [k − 1]2q k +... + r + r [n]q [2]q [n]2q [2]q [n]2q ≤

k [n + k − 1]q [n + k − 2]q [n + 3]q q n rk X ... [j − 2]q [j − 1]2q [n]q [n]q [n]q [2]q [n]2q j=3

≤

2]2q [k

− 1]2q q n rk [n + k − 1]q [n + k − 2]q [n + 3]q [k − ... , [n]q [n]q [n]q [2]q [n]2q

by the hypothesis on ck , we have ¯ ¯ X ∞ 2 00 ¯ ¯ ¯Gn,q (f ; z) − f (z) − z f (z) ¯ ≤ |ck | · |Eq,k,n (z)| ¯ [2]q [n]q ¯ k=3

≤ ≤

1 [2]q [n]2q 1

∞ X [n + k − 1]q [n + k − 2]q k=3 ∞ X

[2]2q [n]2q k=3

[n]q [k − 1]q [n]q [k − 2]q

...

[n + 3]q [k − 2]2q [k − 1]2q (rA)k [3]q [n]q [2]q [k]q

[k − 2]2q [k − 1]q (rA)k ,

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APPROXIMATION BY COMPLEX q-GAMMA OPERATORS IN COMPACT DISKS P∞ k+1 for all |z| ≤ r and n ≥ 2 (n ∈ N). Since the series and its q-derivative k=3 u P∞ k compact disk included in k=3 [k + 1]q u are uniformly and absolutely convergent in any P 2 k the open unit disk, therefore, for 1 ≤ r < A1 , we have [2]21[n]2 ∞ k=3 [k − 2]q [k − 1]q (rA) < q q ∞. Denoting with ||Pk ||r = max{|Pk (z)| : |z| ≤ r}, where Pk (z) is a complex polynomial of degree ≤ k. Now we will give the exact order of approximation by complex q-Gamma operators. Theorem 3.3. In the hypothesis of Theorem 3.1, if f is not a polynomial of degree ≤ 1 in the case (i), we have ||Gn,q (f ) − f ||r ≥

1 Ur (f ), n ∈ N, [n]q

where the constant Ur (f ) depends only on f and r. Proof. Applying the norm || · ||r to the identity ½ 2 · µ ¶¸¾ z 00 1 z2 1 2 00 f (z) + [n]q Gn,q (f ; z) − f (z) − f (z) , Gn,q (f ; z) − f (z) = [n]q [2]q [n]q [2]q [n]q we get 1 ||Gn,q (f ) − f ||r ≥ [n]q

¯¯ ¯¯ ¸¾ ¯¯ ½¯¯ · ¯¯ ¯¯ e2 00 ¯¯ ¯¯ 1 e2 00 ¯¯ 2 ¯¯ ¯ ¯ ¯¯ f − f . [n] G (f ) − f − n,q q ¯¯ [2]q ¯¯ ¯¯ [n]q [2]q [n]q ¯¯r r

e2 00 f ||r > 0. Indeed, Since f is not a polynomial of degree ≤ 1 in DR , it follows that || [2] q

supposing the contrary it follows that z 2 f 00 (z) = 0 for all z ∈ DR , therefore we get f 00 (z) = 0 for all z ∈ DR , by the uniqueness of analytic functions we get f 00 (z) = 0 for all z ∈ DR , that is f is a linear function in DR , which is in contradiction with the hypothesis. Now, by Theorem 3.2, we have ¯ ¯ ∞ 2 00 X ¯ ¯ 1 ¯Gn,q (f ; z) − f (z) − z f (z) ¯ ≤ [k − 2]2q [k − 1]q (rA)k . ¯ [2]q [n]q ¯ [2]2 [n]2q k=3

Therefore, there exists an index n0 (depending only on f and r) such that for all n ≥ n0 , we have ¯¯ ¯¯ ¯¯ ¸ ¯¯ · ¯¯ ¯¯ ¯¯ e2 00 ¯¯ 1 e 1 2 2 00 00 ¯¯ ¯¯ ¯¯ ¯¯ ¯¯ [2]q f ¯¯ − [n]q [n]q ¯¯Gn,q (f ) − f − [2]q [n]q f ¯¯ ≥ [2]2 ||e2 f ||r , q r r which implies ||Gn,q (f ) − f ||r ≥

1 ||e2 f 00 ||r , [2]2q

for all n ≥ n0 . For 1 ≤ n ≤ n0 − 1, we have ||Gn,q (f ) − f ||r ≥

1 1 ([n]q ||Gn,q (f ) − f ||r ) = Vr,n (f ) > 0, [n]q [n]q

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Therefore, finally we obtain ||Gn,q (f ) − f ||r ≥

1 Ur (f ), [n]q

o n for all n, with Ur (f ) = min Vr,1 (f ), Vr,2 (f ), ..., Vr,n0 (f ), [2]12 ||e2 f 00 ||r . q

Combining Theorem 3.1 with Theorem 3.3, we immediately get the following result: Corollary 3.4. In the hypothesis of Theorem 3.1 and Theorem 3.3, we have ||Gn,q (f ) − f ||r ∼

1 , n ∈ N. [n]q

Theorem 3.5. In the hypothesis of Theorem 3.1, if 1 ≤ r ≤ r1 < A1 are arbitrary fixed and f is not a polynomial of degree ≤ p − 1, then for all |z| ≤ r and n, p ∈ N (n ≥ 2), we have 1 (p) ||G(p) ||r ∼ . n,q (f ) − f [n]q Proof. Taking into account the upper estimate in case (ii) of Theorem 3.1, it remains to prove the lower estimate only. Denoting by Γ the circle of radius r1 > r and center 0, since for any |z| ≤ r and v ∈ Γ, we have |v − z| ≥ r1 − r, by Cauchy’s formulas it follows that for all |z| ≤ r and n ∈ N, we get Z Gn,q (f ; v) − f (v) p! (p) G(p) (f ; z) − f (z) = dv, (6) n,q 2πi Γ (v − z)p+1 as we have the identity · ¶¸¾ µ z 2 00 1 z2 2 00 f (z) + [n]q Gn,q (f ; z) − f (z) − f (z) , [2]q [n]q [2]q [n]q (7) applying (7) to (6), we have 1 Gn,q (f ; z) − f (z) = [n]q

½

(p) G(p) (z) n,q (f ; z) − f i  h  Z Z [n]2 Gn,q (f ; v) − f (v) − v2 f 00 (v)   2 00 q [2]q [n]q 1 p! v f (v) 1 p! = dv + dv  [n]q  2πi Γ [2]q (v − z)p+1 [n]q 2πi Γ (v − z)p+1 h i ¯¯  µ ¶ ¯¯ Z [n]2 Gn,q (f ; v) − f (v) − v2 f 00 (v)  q [2]q [n]q 1 p! 1 ¯¯¯¯ e2 f 00 (p) ¯¯¯¯ dv , = ¯¯ ¯¯ + ¯¯  [n]q ¯¯ [2]q [n]q 2πi Γ (v − z)p+1 r

applying the norm || · ||r to the above identity, we have (p) ||G(p) ||r n,q (f ) − f h ¯¯ ¯¯ ¯¯ Z [n]2 G (f ; v) − f (v) − µ ¶(p) ¯¯¯¯  ¯ ¯ 00 n,q q 1 1 ¯¯¯¯ p! ¯¯ ¯¯ e2 f ≥ ¯¯ ¯¯ − ¯ ¯ ¯¯ [n]q ¯¯ [2]q [n]q ¯¯ 2π Γ (v − z)p+1 r

1095

i

v2 00 [2]q [n]q f (v)

¯¯  ¯¯  ¯¯ dv ¯¯¯¯ , ¯¯  r

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APPROXIMATION BY COMPLEX q-GAMMA OPERATORS IN COMPACT DISKS

by using Theorem 3.2, we have h ¯¯ ¯¯ Z [n]2 G (f ; v) − f (v) − n,q ¯¯ p! q ¯¯ ¯¯ 2π (v − z)p+1 Γ ¯¯

i

v2 00 [2]q [n]q f (v)

¯¯ ¯¯ ¯¯ Jq,r1 ,A p!r1 p! 2πr1 dv ¯¯¯¯ ≤ J = , p+1 q,r1 ,A p+1 2π (r − r) (r 1 1 − r) ¯¯ r

¯¯ ³ ¯¯ ¯¯ e2 f 00 ´(p) ¯¯ ¯ ¯¯ > 0, reasoning exactly as in the proof of ¯ by the hypothesis on f , we have ¯¯ [2]q ¯¯ r Theorem 3.3, we immediately get the desired result.

Acknowledgement This work is supported by the Educational Office of Fujian Province of China (Grant No. JA13269), the Startup Project of Doctor Scientific Research of Quanzhou Normal University, Fujian Provincial Key Laboratory of Data Intensive Computing and Key Laboratory of Intelligent Computing and Information Processing, Fujian Province University.

References [1] G. A. Anastassiou, S. G. Gal, Approximation by complex Bernstein-Schurer and KantorovichSchurer polynomials in compact disks, Comput. Math. Appl., 58 (2009), 734-743. [2] F. G. Dressel, J. J. Gergen, W. H. Purcell, Convergence of extended Bernstein polynomials in the complex plane, Pacific J. Math., 13(4) (1963), 1171-1180. [3] S. G. Gal, Approximation and geometric properties of complex Favard-Sz´ asz-Mirakjan operators in compact disks, Atti. Accad. Naz. Lincei Cl. Sci. Fis. Mat. Natur. Rend. Comput. Math. Appl., 56 (2008), 1121-1127. [4] S. G. Gal, Approximation by complex genuine Durrmeyer type polynomials in compact disks, Appl. Math. Comput., 217 (2010), 1913-1920. [5] S. G. Gal, V. Gupta, Approximation by complex Beta operators of first kind in strips of compact disks, Mediterr. J. Math. (2011), doi: 10. 1007/s00009-011-0164-2. [6] V. Gupta, Approximation properties by Bernstein-Durrmeyer type operators, Complex Anal. Oper. Theory (2011), doi: 10. 1007/s11785-011-0167-9. [7] V. G. Kac, P. Cheung, Quantum Calculus, Universitext, Springer-Verlag, New York, 2002. [8] N. I. Mahmudov, Approximation properties of complex q-Sz´ asz-Mirakjan operators in compact disks, Comput. Math. Appl., 60 (2010), 1784-1791. [9] X. M. Zeng, Approximation properties of Gamma operators, J. Math. Anal. Appl., 311 (2005), 389-401.

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Value sharing of meromorphic functions of differential polynomials of finite order Xiao-Bin Zhanga∗ and

Jun-Feng Xub

a

b

College of Science, Civil Aviation University of China, Tianjin 300300, China Department of Mathematics, Wuyi University, Jiangmen, Guangdong 529020, P.R. China

Abstract In this paper, we shall study the uniqueness problems on meromorphic functions of differential polynomials of finite order sharing a value. Our results improve or generalize many previous results on value sharing of meromorphic functions, such as Fang and Hua, Yang and Hua, Lin and Yi, Zhang, Xu, et al.

MSC 2010: 30D35, 30D30. Keywords and phrases: uniqueness, meromorphic function, value sharing.

1

Introduction and main results

Let C denote the complex plane and f (z) be a non-constant meromorphic function on C. We assume the reader is familiar with the standard notion used in the Nevanlinna value distribution theory such as T (r, f ), m(r, f ), N (r, f ) (see [7, 13, 14]), and S(r, f ) denotes any quantity that satisfies the condition S(r, f ) = o(T (r, f )) as r → ∞ outside of a possible exceptional set of finite linear measure. A meromorphic function a(z) is called a small function with respect to f (z), provided that T (r, a) = S(r, f ). ∪ Let f (z) and g(z) be two non-constant meromorphic functions. Let a ∈ C {∞}, we say that f (z), g(z) share a CM (counting multiplicities) if f (z) − a, g(z) − a have the same zeros with the same multiplicities and we say that f (z), g(z) share a IM (ignoring multiplicities) if we do not consider the multiplicities. Nk (r, f ) denotes the truncated counting function bounded by k. Define the order of f as σ(f ) = lim sup r−→∞

log+ T (r, f ) , log r

The following well known theorem in value distribution theory was posed by Hayman and settled by several authors almost at the same time [2, 4]. ∗

Correspoding author: E-mail: [email protected](X.B. Zhang); [email protected](J.F. Xu)

1

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Theorem A: Let f (z) be a transcendental meromorphic function, n ≥ 1 a positive integer. Then f n f ′ = 1 has infinitely many solutions. Fang and Hua [5], Yang and Hua [12] got a unicity theorem respectively corresponding to Theorem A. Theorem B: Let f and g be two non-constant entire (meromorphic) functions, n ≥ 6(n ≥ 11) be a positive integer. If f n (z)f ′ (z) and g n (z)g ′ (z) share 1 CM, then either f (z) = c1 ecz , g(z) = c2 e−cz , where c1 , c2 and c are three constants satisfying 4(c1 c2 )n+1 c2 = −1, or f (z) ≡ tg(z) for a constant t such that tn+1 = 1. Note that f n (z)f ′ (z) = and proved

1 n+1 (z))′ , n+1 (f

Fang [6] considered the case of kth derivative

Theorem C: Let f and g be two non-constant entire functions, and let n, k be two positive integers with n > 2k + 4. If (f n (z))(k) and (g n (z))(k) share 1 CM, then either f (z) = c1 ecz , g(z) = c2 e−cz , where c1 , c2 and c are three constants satisfying (−1)k (c1 c2 )n (nc)2k = 1, or f (z) ≡ tg(z) for a constant t such that tn = 1. Theorem D: Let f and g be two non-constant entire functions, and let n, k be two positive integers with n > 2k + 8. If (f n (z)(f (z) − 1))(k) and (g n (z)(g(z) − 1))(k) share 1 CM, then f (z) ≡ g(z). For more results on this field, see [8, 9, 17]. Corresponding to Theorems C and D, It is natural to ask the following question. Question 1.1. Does Theorem C or D holds if f and g are meromorphic functions? Remark 1.1. Question 1.1 seems to have been solved by Bhoosnurmath and Dyavanal [3], but their proofs contain some gaps that were pointed out by Zhang [15, Annex remarks], Xu et al [10, Remark 2], respectively. The gaps have not been filled as far as we know. Here we give a partial answer to Problem 1.1. Theorem 1.1. Let f (z) and g(z) be two non-constant meromorphic functions with σ(f ) < +∞. Let n, k be two positive integers with n > max{3k + 8, 2(σ(f ) − 1)k}. If [f n (z)](k) and [g n (z)](k) share 1 CM, then one of the following two conclusions holds: (1) f (z) ≡ tg(z) for a constant t such that tn = 1; (2) f = c3 edz , g = c4 e−dz , where c3 , c4 and d are constants such that (−1)k (c3 c4 )n (nd)2k = 1. Remark 1.2. Theorem 1.1 affirmatively answered Problems 1.1. Namely, Theorem C holds for the case of meromorphic functions of finite order, provided that n is sufficiently large. But unfortunately, Theorems D fails if f (z) and g(z) are meromorphic functions without the condition Θ(∞, f ) > 2/n, even if f and g share ∞ CM. We give the following counterexample. Example 1.1. Let f (z) =

h(z)(1 − hn (z)) , 1 − hn+1 (z)

g(z) =

1 − hn (z) , 1 − hn+1 (z)

(1.1)

2

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where n is a positive integer and h(z) is a non-constant meromorphic function. We deduce from (1.1) that f n (f − 1) = g n (g − 1), thus f and g satisfy the conditions of Theorem D, but f ̸≡ g. Note that T (r, f ) = T (r, gh) = nT (r, h) + S(r, f ). By the second fundamental theorem, we deduce N (r, f ) =

n ∑ j=1

N(

1 ) ≥ (n − 2)T (r, h) + S(r, f ), h − aj

where aj (̸= 1) (j = 1, 2, · · · , n) are distinct roots of the algebraic equation hn+1 = 1. Therefore, Θ(∞, f ) = 1 − lim sup r→∞

When n > 3k + 8, then

n 2k

N (r, f ) ≤ 2/n. T (r, f )

+ 1 > 52 , so from Theorem 1.1 we have

Corollary 1.1. Let f (z) and g(z) be two non-constant meromorphic functions with σ(f ) < 3. Let n, k be two positive integers with n > 3k + 8. If [f n (z)](k) and [g n (z)](k) share 1 CM, then one of the following two conclusions holds: (1) f (z) ≡ tg(z) for a constant t such that tn = 1; (2) f = c3 edz , g = c4 e−dz , where c3 , c4 and d are constants such that (−1)k (c3 c4 )n (nd)2k = 1. Consider IM sharing value and we have Theorem 1.2. Let f (z) and g(z) be two non-constant meromorphic functions with σ(f ) < +∞. Let n, k be two positive integers with n > max{9k + 14, 2(σ(f ) − 1)k}. If [f n (z)](k) and [g n (z)](k) share 1 IM, then one of the following two conclusions holds: (1) f (z) ≡ tg(z) for a constant t such that tn = 1; (2) f = c3 edz , g = c4 e−dz , where c3 , c4 and d are constants such that (−1)k (c3 c4 )n (nd)2k = 1. Corollary 1.2. Let f (z) and g(z) be two non-constant meromorphic functions with σ(f ) < 6. Let n, k be two positive integers with n > 9k + 14. If [f n (z)](k) and [g n (z)](k) share 1 IM, then one of the following two conclusions holds: (1) f (z) ≡ tg(z) for a constant t such that tn = 1; (2) f = c3 edz , g = c4 e−dz , where c3 , c4 and d are constants such that (−1)k (c3 c4 )n (nd)2k = 1.

2

Preliminary lemmas and a main proposition

Lemma 2.1. [11] Let f (z) be a non-constant meromorphic function and let a0 (z), a1 (z), · · · , an (z)(̸≡ 0) be small functions of f . Then T (r, an f n + an−1 f n−1 + · · · + a0 ) = nT (r, f ) + S(r, f ). 3

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Lemma 2.2. [16] Let f (z) be a non-constant meromorphic function, s, k be two positive integers. Then Ns (r,

1 f (k)

1 ) ≤ T (r, f (k) ) − T (r, f ) + Ns+k (r, ) + S(r, f ), f

Ns (r,

1 1 ) ≤ kN (r, f ) + Ns+k (r, ) + S(r, f ). f f (k)

Lemma 2.3. Let f (z) be a non-constant meromorphic function of finite order, and let k be a positive integer. Suppose that f (k) ̸≡ 0, then N (r,

1 1 ) ≤ N (r, ) + kN (r, f ) + O(log r). f f (k) ′

Proof. Note that f is of finite order, by Lemma 1.4 in [14, P. 21], we have m(r,

f′ ) = O(log r). f

(k)

Now we prove m(r, f f ) = O(log r) by mathematical induction. Suppose that the conclusion is true for the case of k = m, if k = m + 1, we have f (m+1) f (m) ′ f (m) f ′ =( ) + . f f f f Then we get m(r,

f (m+1) f (m) ′ f (m) f′ ) ≤ m(r, ( ) ) + m(r, ) + m(r, ) + O(1) f f f f (m)

( f f )′ f (m) = m(r, (m) ) + O(log r) f f f

≤ m(r,

(

f (m) f f (m) f

)′

) + m(r,

f (m) ) + O(log r) f

= O(log r). Moreover, we have 1 1 1 f (k) m(r, ) ≤ m(r, (k) ) + m(r, ) = m(r, (k) ) + O(log r). f f f f Hence 1 1 T (r, f ) − N (r, ) ≤ T (r, f (k) ) − N (r, (k) ) + O(log r). f f

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That is N (r,

1 1 ) ≤ T (r, f (k) ) − T (r, f ) + N (r, ) + O(log r) f f (k) 1 = m(r, f (k) ) + N (r, f (k) ) − T (r, f ) + N (r, ) + O(log r) f f (k) 1 ≤ m(r, f ) + m(r, ) + N (r, f ) + kN (r, f ) − T (r, f ) + N (r, ) + O(log r) f f 1 = N (r, ) + kN (r, f ) + O(log r). f

This completes the proof of Lemma 2.3. Lemma 2.4. [12] Let f (z) and g(z) be two non-constant meromorphic functions and n, k be two positive integers, a be a finite nonzero constant. If f and g share a CM, then one of the following cases holds: (i) T (r, f ) ≤ N2 (r, 1/f ) + N2 (r, 1/g) + N2 (r, f ) + N2 (r, g) + S(r, f ) + S(r, g), the same inequality holding for T (r, g); (ii) f g ≡ a2 ; (iii) f ≡ g. Lemma 2.5. Let f (z) and g(z) be non-constant meromorphic functions, n, k be two positive integers with n > k + 2, a be a finite nonzero constant. If [f n ](k) and [g n ](k) share a IM. Then T (r, f ) = O(T (r, g)), T (r, g) = O(T (r, f )) and σ(f ) = σ(g). Proof. Let F = f n . By the second fundamental theorem for small functions, we have T (r, F (k) ) ≤ N (r, f ) + N (r,

1 F (k)

) + N (r,

1 F (k)

−a

) + S(r, F ).

(2.1)

By (2.1) and Lemma 2.1 and Lemma 2.2 with s = 1 applied to F , we have 1 1 ) + N (r, (k) ) + N (r, f ) + S(r, F ) F F −a 1 1 ≤ (k + 1)N (r, ) + N (r, n (k) ) + N (r, f ) + S(r.f ) f [f ] − a 1 ≤ (k + 2)T (r, f ) + N (r, n (k) ) + S(r, f ). [g ] − a nT (r, f ) ≤ Nk+1 (r,

Namely, (n − k − 2)T (r, f ) ≤ N (r,

1 [g n ](k)

−a

) + S(r, f )

≤ n(k + 1)T (r, g) + S(r, f ). Since n > k + 2, we have T (r, f ) = O(T (r, g)). Similarly we have T (r, g) = O(T (r, f )). Thus σ(f ) = σ(g). This completes the proof of Lemma 2.5. By the arguments similar to the proof of Lemma 2.5, we get the following proposition. 5

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Proposition 2.1. Let f be a transcendental meromorphic function, n, k be two positive integers with n > k + 2, a(z)(̸≡ 0, ∞) be a small function of f . Then [f n ](k) − a(z) has infinitely many zeros. Lemma 2.6. [10] Let f and g be two non-constant meromorphic functions, k, n > 2k + 1 be two positive integers. If [f n ](k) = [g n ](k) , then f = tg for a constant t such that tn = 1. Lemma 2.7. Let f, g be two nonconstant meromorphic functions with σ(f ) < +∞, n, k be two positive integers with n > max{3k + 8, 2(σ(f ) − 1)k}. If [f n ](k) [g n ](k) = 1, then f = c3 edz , g = c4 e−dz , where c3 , c4 and d are constants such that (−1)k (c3 c4 )n (nd)2k = 1. Proof. Note that n > k + 2, [f n ](k) and [g n ](k) share 1 IM. Then by Lemma 2.5 we get σ(f ) = σ(g) < +∞. First, we prove f ̸= 0, g ̸= 0.

(2.2)

Suppose that z0 is a zero of f with multiplicity s, then z0 is a pole of g, say multiplicity t, and z0 is a zero of [f n ](k) with multiplicity ns − k, a pole of [g n ](k) with multiplicity nt + k, thus we have ns − k = nt + k, namley n(s − t) = 2k.

(2.3)

Note that n > 3k + 8 and we get a contradiction from (2.3). Thus f has no zero. Similarly, g has no zero. Thus (2.2) holds. Next we prove N (r, f ) = O(log r),

N (r, g) = O(log r).

(2.4)

Rewrite [f n ](k) [g n ](k) = 1 as [f n ](k) =

1 [g n ](k)

.

(2.5)

We deduce from (2.5) that N (r, [f n ](k) ) = N (r,

1 [g n ](k)

).

(2.6)

As N (r, [f n ](k) ) = nN (r, f )+kN (r, f ), this together with (2.2), (2.6) and Lemma 2.3 implies that nN (r, f ) + kN (r, f ) ≤ kN (r, g) + O(log r).

(2.7)

nN (r, g) + kN (r, g) ≤ kN (r, f ) + O(log r).

(2.8)

Similarly we get

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Combining (2.7) and (2.8) yields N (r, f ) + N (r, g) = O(log r).

(2.9)

Thus we obtain (2.4), which means that both f and g have at most finitely many poles. Let f=

eh(z) , p(z)

g=

eh1 (z) , q(z)

(2.10)

where p(z) and q(z) are polynomials with deg(p(z)) = p, deg(q(z)) = q, h(z) and h1 (z) are nonconstant entire functions. By Corollary 1 in [14, P. 65], h(z) and h1 (z) are polynomials with deg(h(z)) = deg(h1 (z)) = h = σ(f ). Then fn =

enh(z) , pn (z)

gn =

enh1 (z) . q n (z)

(2.11)

Let H(z) = nh(z), P (z) = pn (z), H1 (z) = nh1 (z), Q(z) = q n (z). By mathematical induction we get that [f n ](k) =

eH(z) Pk (z) eH1 (z) Qk (z) n (k) , [g ] = , P k+1 (z) Qk+1 (z)

(2.12)

where Pk (z) and Qk (z) are two polynomials with deg(Pk (z)) = k(h−1+np) and deg(Qk (z)) = k(h − 1 + nq). By [f n ](k) [g n ](k) = 1, we have h(z) + h1 (z) ≡ C,

(2.13)

where C is a constant. Furthermore, we get deg(Pk (z)) + deg(Qk (z)) = deg(P k+1 (z)Qk+1 (z)),

(2.14)

which implies that 2k(h − 1) = n(p + q).

(2.15)

N (r, f ) + N (r, g) ̸= 0,

(2.16)

By (2.4), if

then p + q ≥ 1, we deduce from (2.15) that n ≤ 2k(h − 1) = 2k(σ(f ) − 1),

(2.17)

which contradicts the assumption. Therefore N (r, f ) + N (r, g) = 0,

(2.18)

namely both f and g are entire functions and p = q = 0. From (2.15) we obtain that h = 1. Thus h(z) = dz + l3 , h1 (z) = −dz + l4 . Rewrite f and g as f = c3 edz ,

g = c4 e−dz ,

where c3 , c4 and d are nonzero constants. We deduce that (−1)k (c3 c4 )n (nd)2k = 1. This completes the proof of Lemma 2.7.

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Lemma 2.8. [1] Let f (z) and g(z) be two non-constant meromorphic functions and n, k be two positive integers, a be a finite nonzero constant. If f and g share a IM, then one of the following cases holds: (i)T (r, f ) ≤ N2 (r, 1/f )+N2 (r, 1/g)+N2 (r, f )+N2 (r, g)+2N (r, 1/f )+N (r, 1/g)+2N (r, f )+ N (r, g) + S(r, f ) + S(r, g), the similar inequality holding for T (r, g); (ii) f g ≡ a2 ; (iii) f ≡ g.

3

Proof of Theorem 1.1

Let F = [f n ](k) , G = [g n ](k) , F ∗ = f n , G∗ = g n , then F and G share 1 CM. Thus by Lemma 2.5, one of the following cases holds: (i) T (r, F ) ≤ N2 (r, 1/F ) + N2 (r, 1/G) + N2 (r, F ) + N2 (r, G) + S(r, F ) + S(r, G), the same inequality holding for T (r, G); (ii) F G ≡ 1; (iii) F ≡ G. Case (i). By Lemma 2.1 and Lemma 2.2 with s = 2, we obtain T (r, F ∗ ) ≤ Nk+2 (r, 1/F ∗ ) + Nk+2 (r, 1/G∗ ) + (k + 2)N (r, g) + 2N (r, f ) +S(r, f ) + S(r, g) ≤ (k + 2)N (r, 1/f ) + (k + 2)N (r, 1/g) + (k + 2)N (r, g) + 2N (r, f ) +S(r, f ) + S(r, g) ≤ (2k + 4)T (r, g) + (k + 4)T (r, f ) + S(r, f ) + S(r, g), namely nT (r, f ) ≤ (2k + 4)T (r, g) + (k + 4)T (r, f ) + S(r, f ) + S(r, g).

(3.1)

Similarly we have nT (r, g) ≤ (2k + 4)T (r, f ) + (k + 4)T (r, g) + S(r, f ) + S(r, g).

(3.2)

From (3.1) and (3.2) we deduce that (n − 3k − 8)(T (r, f ) + T (r, g) ≤ S(r, f ) + S(r, g),

(3.3)

which is a contradiction since n > 3k + 8. Case (ii). We have [f n ](k) [g n ](k) = 1. By Lemma 2.7 we get the conclusion (2) of Theorem 1.1. Case (iii). We have [f n ](k) ≡ [g n ](k) . By Lemma 2.6 we get the conclusion (1) of Theorem 1.1. This completes the proof of Theorem 1.1.

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4

Proof of Theorem 1.2

Let F = [f n ](k) , G = [g n ](k) , F ∗ = f n , G∗ = g n , then F and G share 1 IM. Thus by Lemma 2.8, one of the following cases holds: (i) T (r, F ) ≤ N2 (r, 1/F ) + N2 (r, 1/G) + N2 (r, F ) + N2 (r, G) + 2N (r, 1/F ) + N (r, 1/G) + 2N (r, F ) + N (r, G) + S(r, F ) + S(r, G), the same inequality holding for T (r, G); (ii) F G ≡ 1; (iii) F ≡ G. Case (i). By Lemma 2.1 and Lemma 2.2 with s = 1, 2, we obtain T (r, F ∗ ) ≤ Nk+2 (r, 1/F ∗ ) + Nk+2 (r, 1/G∗ ) + (k + 2)N (r, g) + 2N (r, f ) 2(Nk+1 (r, 1/F ∗ ) + kN (r, f )) + Nk+1 (r, 1/G∗ ) + kN (r, g) +2N (r, f ) + N (r, g) + S(r, f ) + S(r, g) ≤ (3k + 4)N (r, 1/f ) + (2k + 3)N (r, 1/g) + (2k + 4)N (r, f ) +(2k + 3)N (r, g) + S(r, f ) + S(r, g) ≤ (5k + 8)T (r, f ) + (4k + 6)T (r, g) + S(r, f ) + S(r, g), namely nT (r, f ) ≤ (5k + 8)T (r, f ) + (4k + 6)T (r, g) + S(r, f ) + S(r, g).

(4.1)

Similarly we have nT (r, g) ≤ (5k + 8)T (r, g) + (4k + 6)T (r, f ) + S(r, f ) + S(r, g).

(4.2)

From (4.1) and (4.2) we deduce that (n − 9k − 14)(T (r, f ) + T (r, g) ≤ S(r, f ) + S(r, g),

(4.3)

which is a contradiction since n > 9k + 14. Case (ii). We have [f n ](k) [g n ](k) = 1. By Lemma 2.7 we get the conclusion (2) of Theorem 1.2. Case (iii). We have [f n ](k) ≡ [g n ](k) . By Lemma 2.6 we get the conclusion (1) of Theorem 1.2. This completes the proof of Theorem 1.2.

Acknowledgements This research was supported by the National Natural Science Foundation of China (Grant No. 11171184), the Tian Yuan Special Fund of the National Natural Science Foundation of China (Grant No. 11426215) and Training plan for the Outstanding Young Teachers in Higher Education of Guangdong (No. Yq 2013159).

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References [1] A. Banerjee, Meromorphic functions sharing one value, Int. J. Math. Math. Sci. 22 (2005) 3587–3598. [2] W. Bergweiler, A. Eremenko, On the singularities of the inverse to a meromorphic function of finite order, Rev. Mat. Iberoamericana. 11 (1995), 355–373. [3] S.S. Bhoosnurmath, R.S. Dyavanal, Uniqueness and value-sharing of meromorphic functions, Comput. Math. Appl. 53 (2007), 1191–1205. [4] H.H. Chen, M.L. Fang, On the value distribution of f n f ′ , Sci. China Ser. A. 38 (1995), 789–798. [5] M.L. Fang, X.H. Hua, Entire functions that share one value, J. Nanjing Univ. Math. Biquarterly 13 (1) (1996), 44–48. [6] M.L. Fang, Uniqueness and value-sharing of entire functions, Comput. Math. Appl. 44 (2002), 828–831. [7] W.K. Hayman, Meromorphic Functions, Clarendon Press, Oxford, 1964. [8] W.C. Lin, H.X. Yi, Uniqueness theorems for meromorphic function concerning fixedpoints, Complex Var. Theory Appl. 49 (11) (2004) 793–806. [9] W.C. Lin, H.X. Yi, Uniqueness theorems for meromorphic function, Indian J. Pure Appl. Math. 35 (2004) 121–132. [10] J.F. Xu, F. L¨ u, H.X. Yi, Fixed-points and uniqueness of meromorphic functions, Comput. Math. Appl. 59 (2010), 9–17. [11] C.C. Yang, On deficiencies of differential polynomials II, Math. Z. 125 (1972), 107–112. [12] C.C. Yang, X.H. Hua, Uniqueness and value-sharing of meromorphic functions, Ann. Acad. Sci. Fenn. Math. 22 (2) (1997), 395–406. [13] L. Yang, Value Distribution Theory, Springer-Verlag, Berlin, 1993. [14] C.C. Yang, H.X. Yi, Uniqueness Theory of Meromorphic Functions, Kluwer Acad. Publ. Dordrecht, 2003. [15] J.L. Zhang, Uniqueness theorems for entire functions concerning fixed-points, Comput. Math. Appl. 56 (2008), 3079–3087. [16] J.L. Zhang, L.Z. Yang, Some results related to a conjecture of R. Br¨ uck, J. Inequal. Pure Appl. Math. 8 (1) (2007) Art. 18. [17] X.Y. Zhang and W.C. Lin, Uniqueness and value-sharing of entire functions, J. Math. Anal. Appl. 343 (2008) 938–950.

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Regularized optimization method for determining the space-dependent source in a parabolic equation without iterationI Zewen Wanga,∗, Wen Zhanga , Bin Wub a b

College of Science, East China Institute of Technology, Nanchang, Jiangxi, 330013, P. R. China School of Mathematics and Statistics, Nanjing University of Information Science and Technology, Nanjing, 210044, P. R. China

Abstract In this paper, we consider an inverse problem of identifying a space-dependent source in the parabolic equation which is a classical ill-posed problem. The inverse source problem is formulated into a regularized optimization problem. Then, a non-iterative algorithm based on a sequence well-posed direct problems solved by the finite element method is proposed for solving the optimization problem. In order to obtain a reasonable regularization solution, we utilize the damped Morozov discrepancy principle together with the linear model function method for choosing regularization parameters. Numerical results for one- and two-dimensional examples show that the proposed method is efficient and robust with respect to data noise, especially for reconstructing the discontinuous source functions. Furthermore, the proposed method is successfully used to solve a real example of identifying the magnitude of groundwater pollution source. Keywords: inverse source problem, parabolic equation, optimization, finite element method, discrepancy principle.

1. Introduction Inverse source identification problems arise in many branches of applied science and engineering science, which aim to determine the unknown source from some measurable information related to the source. For example, Identification of a pollution source intensity from some given measurements of the pollutant concentrations is crucial to environmental safeguard in watersheds [1]. In this paper, we consider the inverse problem for determining the unknown space-dependent source in a parabolic equation from a final measurement. As we all know, this inverse source problem is ill-posed since small errors inherently presented in the practical measurement can induce enormous and highly oscillatory errors in I

This work is partially supported by National Natural Science Foundation of China (11161002, 11201238), Young Scientists Training Project of Jiangxi Province (20122BCB23024), Natural Science Foundation of Jiangxi Province (20142BAB201008), Ground Project of Science and Technology of Jiangxi Universities (KJLD14051). ∗ Corresponding author. Email addresses: [email protected]; [email protected] (Zewen Wang), [email protected] (Wen Zhang), [email protected] (Bin Wu)

Preprint submitted to Journal of Computational Analysis and Applications 1107

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reconstructing the unknown heat source. The inverse problem of determining an unknown space-dependent source in the parabolic equation has been considered in a few theoretical papers concerned with existence and uniqueness of the solution[2, 3]. Recently, many authors are interested in numerical reconstruction of the space-dependent source in parabolic equations [4, 5, 7, 6, 8, 9, 10, 11, 12]. In [4], the authors transferred the inverse heat source problems to the problems of numerical differentiation for obtaining stable solutions. An effective meshless numerical method and a finite difference approximate method were proposed in [7] and [6], respectively. In [9], a regularization method based on the quasi-reversibility method together with the error estimate was proposed for identifying an unknown space-dependent source in one dimensional standard heat equation. In [10] and [11], two iterative methods were proposed for finding the spacewise dependent source: one is an iterative algorithm based on a sequence of well-posed direct problems; the other is a variational conjugate gradient-type iterative algorithm which also need to solve a sequence of well-posed direct problems at each iteration. The paper [12] is devoted to identify an unknown heat source depending simultaneously on both space and time variables that is transformed into an optimization problem. The aim of this paper is to construct a regularized optimization method, which is a non-iterative method. We firstly formulate the inverse problem of determining the spacewise dependent source into a regularized optimization problem. Then, the optimization problem is reduced to a system of linear algebraic equations based on a sequence well-posed direct problems solved by the finite element method. This paper is organized as follows. In section 2, the source identification problem is formulated and some properties of the solution of direct problem are given. In section 3, a regularized optimization method is proposed for solve the source identification problem. In section 4, implementations of the regularized optimization method are presented. In section 5, numerical results for one- and two-dimensional examples are given to illustrate the efficiency and stability of the proposed method with respect to data noise. Finally, some conclusions are drawn. 2. Mathematical formulation of the source identification problem Let Ω be a bounded domain possessing piecewise-smooth boundaries in the Euclidean space Rn , n ≥ 1. x = (x1 , x2 , · · · , xn ) denotes an arbitrary point in Ω, and ∂Ω is used for the boundary of the domain Ω. Let us denote by QT a cylinder Ω × (0, T ) consisting of all points (x, t) ∈ Rn+1 with x ∈ Ω and t ∈ (0, T ). 2.1. functional spaces The space L2 (Ω) is a Banach space consisting of all square integrable functions on the domain Ω with the norm Z 1/2 kuk2,Ω = |u(x)|2 dx Ω

and the scalar product

Z (u, v) =

u(x)v(x)dx. Ω

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The Sobolev spaces W2l (Ω), where l is a positive integer, consists of all functions from L2 (Ω) having all generalized derivatives of the first l orders that are square integrable over Ω. The norm of the space W2l (Ω) is defined by 1/2  l X X (l) kDxα uk22,Ω  , kuk2,Ω =  k=0 |α|=k

where α = (α1 , α2 , · · · , αn ) is a multi-index, and |α| = α1 + α2 + · · · + αn , Dxα u ≡

∂ |α| u . ∂xα1 1 ∂xα2 2 · · · ∂xαnn

0

The space W2l (Ω) is a subspace of W2l (Ω) in which the set all functions in Ω that are infinite differentiable and have compact support is dense. The Sobolev space W2l1 ,l2 (QT ) with positive integers li ≥ 0, i = 1, 2 is defined as a Banach space of all functions u belonging to the space L2 (QT ) along with their weak x-derivatives of the first l1 orders and t-derivatives of the first l2 orders. The norm of the space W2l1 ,l2 (QT ) is defined by    1/2 Z l1 X l2 X X (l1 ,l2 )  kuk2,Q = |Dxα u|2 + |Dtk u|2  dxdt . T QT

k=0 |α|=k

k=1

l1 ,l2 The space W2,0 (QT ) is a subspace of W2l1 ,l2 (QT ) in which the set of all smooth functions in QT that vanish on the lateral ∂Ω × [0, T ] is dense.

2.2. The source identification problem The source identification problem considered in this paper is stated as follows: find the function u(x, t) and the unknown source function f (x) which satisfy the following parabolic equation and boundary conditions   ut (x, t) = (Lu)(x, t) + f (x), (x, t) ∈ Ω × (0, T ), (2.1) u(x, 0) = 0, x ∈ Ω,   u(x, t) = 0, (x, t) ∈ ∂Ω × [0, T ], and the final over-specified measurement u(x, T ) = g(x), x ∈ Ω, where

(2.2)

  X n n X ∂ ∂u ∂u Lu ≡ aij (x) + bi (x) + c(x)u. ∂xj ∂xi ∂xi i,j=1

i=1

Moreover, the operator L is supposed to be uniformly elliptic, which means that aij (x) = aji (x) and n n n X X X 2 0 0 in RN , N ≥ 3, where µ ∈ R+ , f ∈ H −1 (RN ), f ≥ 0 and f 6≡ 0 in RN . A well-known result for the homoneneous case is that all positive regular solution of ∗ −∆u = u2 −1 in RN are given by ω :=

!(N −2)/2 p  N (N − 2) 2 + |x|2 ∗

with  > 0. Every ω is a minimizer for the embedding D1,2 (RN ) ,→ L2 (RN ). Namely, the Sobolev constant R |∇u|2 dx RN S := inf
∗ R 2∗ dx 2/2 06=u∈D1,2 (RN ) |u| N R is achived by ω and (1, 1)

∗

||∇ω ||22 = ||ω ||22∗ = S N/2 (cf.[2, 6]).

For convenience, we omit “RN ” and “dx” in integration and, throughout this paper, we will use the letter C > 0 to denote the natural various contents independent of u. Our attempt to show multiplicity of positive solutions for problem (Pµ ) relies on the Ekeland’s variational principle in [13] and the Mountain Pass Theorem in [5]. Since our problem (Pµ ) posesses the critical nonlinearity and the embedding

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Existence and Bifurcation of Positive Global Solutions

3

∗

H 1 (RN ) ,→ L2 (RN ) is not compact, in taking the opportunity of variational structure of problem, the (P S) condition is no longer valid and so the Mountain Pass Theorem in [1] could not be applied directly. However, we can use the Mountain Pass Theorem without the (P S) condition in [5] to get some (P S)c sequence of the variational functional for the second solution with c > 0. In the last decade, the existence and properties of solutions of the problem: ( − ∆u + u = g(x, u), u > 0 in RN , (P0 ) u ∈ H 1 (RN ), N ≥ 2 has been stuied by Struss[24], Lions[22, 23], Ding and Ni[12], Cao[7], Zhu[25] and other authors for the case where g(x, 0) = 0 on RN and g(x, t) has a subcritical superlinear growth. On the other hand, the nonhomogeneous problem with 1 < p < 2∗ − 1: ( − ∆u + u = |u|p−2 u + µf, u > 0 in RN , (P ) u ∈ H 1 (RN ), N ≥ 2, where µ ∈ R+ , f ≥ 0, f ∈ L2 (RN ) was studied in [26,11,14,15]. In the critical case p = 2∗ , the problem is much more difficult than the subcritical case. As we mentioned, the Palais-Smale condition does not hold at some critical levels and the effect of the nonhomogeneous term f to the multiple existence of solutions is delicate. The multiplicity of the solutions of (Pµ ) not only depends on the norm of f but also the decay rate of f . In [10], it has shown that if 2 < N < 6 and |x|N −2 f is bounded, then there exists µ∗ > 0 such that problem (Pµ ) possesses at least two positive solutions with µ ∈]0, µ∗ [. In case that N ≥ 6, there exist µ∗∗ , µ∗ > 0 with µ∗ < µ∗∗ such that for each µ ∈]µ∗∗ , µ∗ [, problem (Pµ ) possesses two positive solutions and for µ ∈]0, µ∗ [ problem (Pµ ) has a unique positive solution. In [11], the authors also gave similar multiplicity results for subcritical caseas. For critical case, In [18], Hirano and Kim consider the multiplicity of solutions of (Pµ ) with −∆+I replaced by ∗ ∗ −∆ + αI, α > 0. They assume that p = 2∗ , 3 ≤ N ≤ 5, f ∈ L2 /(2 −1) (RN ) ∩ L∞ (RN ) with f ≥ 0 and f 6≡ 0, and |x|N −2 f is bounded. It was shown that there exist µ∗ and a function α : (0, µ∗ ) → R+ such that for each α ∈ (0, α(µ)), problem (Pµ ) posesses at least three solutions; the third solution is sign-changing if we assume that there exist exactly two positive solutions. we also refer [21] for critical case. In [19], the effact of the shape of the multiplicity of (P ) was investigated when −∆ + I replaced by −∆,  > 0. In this paper, we do not assume the decay rate on f but assume only uniform boundedness of f which is independent of solution u and x ∈ RN . We study also bifurcation phenomenon and get a bifurcation point of (Pµ ). There seems to have some progress on existence result in elliptic equations. We also refer a multiplicity result on parabolic equations for subcritical case in [20, 16] and elliptic with Neumann boundary condition in [17]. We now state our main results: Proposition 2.3. Assume f ∈ H −1 (RN ), f ≥ 0, f 6≡ 0 in RN and ||µf ||∗ ≤ CN∗ , then problem (Pµ ) has at least one positive solution uµ such that (2.1)

¯R0 }, Iµ (uµ ) := c1 = inf{Iµ : u ∈ B

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¯R0 = {u ∈ H 1 (RN ) : ||u|| ≤ R0 } and C ∗ = where B N

4 N +2

1 2


(N −2)/4 N N +2




S N/4 .

Theorem 3.6. Assume f ∈ H −1 (RN ), f ≥ 0 f 6≡ 0 in RN and satisfies ||µf ||∗ ≤ CN∗ . Then there exists a positive constant µ∗ > 0 such that (Pµ ) possesses at least two positive solutions for 0 < µ < µ∗ , a unique solution for µ = µ∗ and no positive solution if µ > µ∗ . By Uµ , we denote the second solution of (Pµ ). Theorem 4.5. (i) The set {Uµ } is bounded uniformly in H 1 (RN ), (ii) (µ∗ , uµ∗ ) is a bifurcation point. 2. Existence of minimal positive solutions The operator −∆ + I has the maximum principle in H 1 (RN ).

Lemma 2.1.

Proof. Let h ≥ 0 and −∆u + u = h. Suppose that u− 6≡ 0, where uR+ (x) = R max{u(x), 0} and u− (x) = min{u(x), 0}. then 0 < |∇u− |2 + |u− |2 ) = hu− dx which leads a contradiction. This completes the proof. In order to get the existence of positive solutions of (Pµ ), we consider the energy functional Iµ of the problem (Pµ ) defined by 1 Iµ (u) := 2

Z

1 (|∇u| + |u| ) − ∗ 2 2

2

Z

Z

+ 2∗

(u ) − µ

f u, for u ∈ H 1 (RN ).

First, we study the existence of a local mininum for energy functional Iµ and its properities. We denote   (N −2)/4 1 4 N ∗ S N/4 . (2, 1) CN := 2 N +2 N +2 Lemma 2.2. Assume f ∈ H −1 (RN ), f (x) ≥ 0, f (x) 6≡ 0 and ||µf ||∗ ≤ CN∗ , then there exits a positive constant R0 > 0 such that Iµ (u) ≥ 0 for any u ∈ ∂BR0 = {u ∈ H 1 (RN ) : ||u|| = R0 }. Proof.

We consider the function h(t) : [0, +∞) → RN defined by 1 1 ∗ ∗ h(t) = t − ∗ S −2 /2 t2 −1 . 2 2

Note that h(0) = 0, 2∗ − 1 > 1 and h(t) → −∞ as t → ∞. We can show easly there a unique t0 > 0 achieving the maxinum of h(t) at t0 . Since h0 (t0 ) =

1 2∗ − 1 −2∗ /2 2∗ −2 − S t0 = 0, 2 2∗

we have  t0 =

2∗ 2(2∗ − 1)

1/(2∗ −2)

1151

∗ /2(2∗ −2)

S2

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Existence and Bifurcation of Positive Global Solutions

5

Hence, we have (2, 2)

1 h(t0 ) = 2



4 N +2



N N +2

(N −2)/4

S N/4 .

Taking R0 = t0 , for all u ∈ ∂BR0 , Z Z Z 1 1 2 2 + 2∗ Iµ (u) = (|∇u| + |u| ) − ∗ (u ) − µ f u 2 2 1 1 ∗ ∗ (2, 3) ≥ ||u||2 − ∗ S −2 /2 ||u||2 − ||µf ||∗ ||u|| 2 2 = t0 [h(t0 ) − ||µf ||∗ ] From (2, 2) and (2, 3), we have Iµ (u)|∂BR0 ≥ 0. This completes the proof. Proposition 2.3. Assume f ∈ H −1 (RN ), f (x) ≥ 0, f (x) 6≡ 0 in RN and ||µf ||∗ ≤ CN∗ , then problem (Pµ ) has at least one positive solution uµ such that (2.4)

¯R0 }, Iµ (uµ ) := c1 = inf{Iµ : u ∈ B

¯R0 = {u ∈ H 1 (RN ) : ||u|| ≤ R0 }. where B Proof. By Sobolev inequality, the generalized H¨older and Young’s inequality with  > 0, there exists C > 0, we have Z Z Z 1 1 2 2 + 2∗ Iµ (u) = (|∇u| + |u| ) − ∗ (u ) − µ f u 2 2 1 1 ∗ ∗ ≥ ||u||2 − ∗ S −2 /2 ||u||22∗ − ||µf ||∗ ||u|| 2  2 1 1 ∗ ∗ ≥ −  ||u||2 − ∗ S −2 /2 ||u||2 − C ||µf ||2∗ . 2 2 Taking  < 12 , then, for R0 = t0 as in Lemma 2,2, we can find a CR0 > 0 small enough such that (2.5)

Iµ (u)|∂BR0 ≥ CR0 for ||µf ||∗ ≤ CN∗ .

¯R0 and B ¯R0 is a Since there exists a C˜R0 > 0 such that |Iµ (u)| ≤ C˜R0 for all u ∈ B ¯R0 , by complete metric space with respect to the metric d(u, v) = ||u − v||, u, v ∈ B using the Ekeland’s variational principle, from (2.5), we can prove that there exists a ¯R0 and uµ ∈ B ¯R0 such that sequence {un } ⊂ B (2.6)

Iµ (un ) → c1 ,

(2.7)

Iµ0 (un ) → 0,

(2.8)

un → uµ weakly in H 1 (RN ), un → uµ a.e. in RN , ∇un → ∇uµ a.e. in RN

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and


∗ ∗ ∗ ∗ un 2 −1 → uµ 2 −1 weakly in L2 (RN ) as n → ∞. Therefore, uµ is a weak solution of (Pµ ). Hence,

0  (2.9) Iµ (uµ ), ϕ = 0 ∀ϕ ∈ H 1 (RN ). Moreover, by Lemma 2.1, uµ is positive on RN , where Iµ0 is the Frechlet ˙ derivative of Iµ . Next, we are going to prove (2.4). In fact, by the definition of c1 , we know that ¯R0 , that is, Iµ (uµ ) ≥ c1 since uµ ∈ B Z Z Z 1 1 2 2 2∗ (|∇uµ | + |uµ | ) − ∗ |uµ | − µ f uµ ≥ c1 (2.10) Iµ (uµ ) = 2 2 By (2.9) and (2.10), we have Z  Z   1 1 1 2 2 (2.11) − (|∇uµ | + |uµ | ) − 1 − ∗ µ f uµ ≥ c1 2 2∗ 2 On the other hand, by (2.6) - (2.8) and Fatou’s lemma, we get  Z Z 1 1 1 2 2 c1 = lim inf − (|∇un | + |un | ) − lim sup(1 − ∗ )µ f un n 2 2∗ 2 n   Z  Z (2.12) 1 1 1 − ∗ (|∇uµ |2 + |uµ |2 ) − 1 − ∗ µ f uµ . ≥ 2 2 2 Thus, (2.10) and (2.12) imply (2.4) holds. This completes the proof. Remark.

(i) c1 < 0, (ii) c1 is bounded below, (iii) ||uµ || = o(1) as µ → 0+ .

Indeed: (i) For t > 0 and ϕ > 0, we have Z Z Z ∗ Z t2 t2 t2 2 2 2 2∗ Iµ (tϕ) = (|∇ϕ| + |ϕ| ) − ∗ |ϕ| − tµ f ϕ ≤ ||ϕ|| − tµ f ϕ. 2 2 2 By taking t > 0 sufficiently small, we can see c1 < 0. (ii) By (2.9) with ϕ = uµ , and c1 = Iµ (uµ ), we have  Z   Z 1 1 1 2 2 c1 = − (|∇uµ | + |uµ | ) − 1 − ∗ µ f uµ 2 2∗ 2     1 1 1 ≥ − ∗ ||uµ ||2 − 1 − ∗ ||µf ||∗ ||uµ || (2.13) 2 2 2  ∗  2 1 (2 − 1) ≥− ∗ ||µf ||2∗ ∗ 22 2 −2 by Young’s inequality. (iii) Since c1 < 0, from (2.13), we see that ||uµ || → 0 as µ → 0+ . Hence, ||uµ || = o(1) as µ → 0+ . We also have that ||uµ || is uniformly bounded with respect to µ. We will restate results relating to this remark in Proposition 3.4 more precisely. Proposition 2.4. of (Pµ ).

Problem (Pµ ) possesses at least one minimal positive solution

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Existence and Bifurcation of Positive Global Solutions

Proof.

7

Let N be the Nehari manifold   Z Z Z 1 N 2 2 2∗ N := u ∈ H (R ) : |∇u| + |u| = |u| + µf u \ {0} .

Note that ||µf ||∗  1 for µ small enough and for each u ∈ H 1 (RN ) \ {0} , there exists a unique tu > 0 such that Z Z Z 2 2 2 2∗ 2∗ tu |∇u| + |u| − tu |u| − tu µf u = 0 and Iµ (tu u) > 0. Then  N = tu u : u ∈ H 1 (RN ) \ {0} and  N ∼ = S N −1 = u ∈ H 1 (RN ) : ||u|| = 1 . Hence, H 1 (RN ) = H1 ∪ H2 ∪ N , H1 ∩ H2 = φ and 0 ∈ H1 , where

 H1 = tu : u ∈ H 1 (RN ) \ {0} , t ∈ [0, tu [  H2 = tu : u ∈ H 1 (RN ) \ {0} , t > tu .

This implies that for t > 0 with t < tu , tu ∈ H1 . Here, we need to switch our view point, by associating with v a mapping v : [0, ∞[→ H 1 (RN ) defined by (v(t))x = v(x, t), x ∈ RN , t ∈ [0, ∞[. In other words, we consider v not as a function of x and t together, but rather as a mapping v of t into the space H 1 (RN ) of a function of x. We have, for any v0 ∈ H1 , the solution v of the initial value problem:  ∗  dv − ∆v + v = v 2 −1 + µf (x), dt  v(0) = v , 0

converges to uµ as t → ∞, Indeed, in the proof of Proposition 2.3, we know that Iµ (v(t)) is decreasing and limt→∞ Iµ (v(t)) = Iµ (uµ ), where Iµ (uµ ) is the local minimum. Since Z t d Iµ (v(t)) − Iµ (v(s)) = Iµ (v(t))dt s dt  Z t d = v, ∇Iµ (v(t)) dt dt s

Z s

d 2

=−

dt v dt, t

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d 2

v = 0. Thus, v 0 → 0 a.e. in RN as t → ∞ and hence, we have, lim s,t→∞ dt

0  Iµ (v), ϕ → 0, ∀ϕ ∈ C ∞ (RN ). Therefore, we have v → uµ as t → ∞, since Iµ (v(t)) is decreasing and converges to the local minimum Iµ (uµ ). Now, let v0 = tu, where t ∈]0, 1[ and u is a positive solution. Then u ∈ N and v0 ∈ H1 . Since v0 ≤ u and the solution v converges uµ as t → ∞, by the order preserving principle, uµ ≤ u . This completes the proof. Proposition 2.5. Suppose that f ∈ H −1 (RN ), f ≥ 0, f 6≡ 0 in RN and ∗ ||µf ||∗ ≤ CN . Then there exist µ ˜≥µ ¯ > 0 such that (Pµ ) possesses a positive solution for 0 < µ ≤ µ ¯ and no positive solution for µ > µ ¯. Proof. By Proposition 2.3, (Pµ ) has a positive solution if µ ≤ CN∗ /||f ||∗ . Suppose (Pµ ) has a positive solution for some µ = µ ¯. We show that (Pµ ) has a positive solution for any 0 < µ ≤ µ ¯. For fixed 0 < µ < µ ¯, using the Lax-Milgram Theorem, we construct a positive sequence {un } as following; Let −∆u1 + u1 = µf and ∗

−1 −∆un + un = u2n−1 + µf for n ≥ 2.

(2.14)

Then, by the maximum principle, we have 0 < un < un+1 < · · · < u¯ for n ≥ 1. And ||u1 || ≤ µ||f ||∗ and ||u1 ||2∗ ≤ S −1/2 ||u1 || ≤ S −1/2 µ||f ||∗ . Multiplying (2.14) by un , we ∗ ∗ have ||un || ≤ S −2 /2 ||¯ u||2 −1 + µ||f ||∗ . Therefore, there exists u in H 1 (RN ) such that un → u weakly in H 1 (RN ) as n → ∞, un → u a.e. in RN as n → ∞, ∇un → ∇u a.e. in RN , ∗

∗ −1

u2n −1 → u2

∗

weakly in (L2 (RN ))∗ as n → ∞.

Thus, u is a positive solution of (Pµ ). Next, let u be a positive solution of (Pµ ). Then, for any  > 0, multiplying (Pµ ) ∗ by ω2 , we have (2.15)

∗

∗

−∆uω2 + uω2 = u2

∗ −1

∗

∗

ω2 + µf (x)ω2 .

Since 2∗ > 2, for any M > 0, there exists a constant C > 0 such that ∗ −1

u2

≥ M u − C ∀u > 0.

Hence, we have, from (2.15), Z Z Z 2∗ 2∗ − ∆uω + uω ≥

∗ ∗
(M u − C)ω2 + µf (x)ω2 ) .

By Green’s formular, we have Z

∗ ∆uω2

Z =

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∗

u∆ω2 .

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Existence and Bifurcation of Positive Global Solutions

9

Thus, Z (2.16)

µ

∗ f (x)ω2

Z ≤C

∗ ω2

Z  ∗ ∆ω2 ∗ 1 − M − 2∗ ω2 u. + w

Since   ∗ ∆w2 N ∆( + |x|2 )−N N +2 2 2 −2 |x| −  = = 2N (N + 1)( + |x| ) ω2∗ ( + |x|2 )−N N +1 N +1   N +2 2 N 2 −2 = 2N (N + 1)( + 0 ) 0 −  N +1 N +1 = −2N 2 −1 , we get, from (2.16), Z Z Z
∗ 2∗ 2∗ 2 −1 µ f (x)ω ≤ C ω + 2N  + 1 − M ω2 u. If we choose M = 2N 2 −1 + 1, then, by (1.1), we have R ∗ C ω2 CS N/2 R = . µ≤ R f (x)ω2∗ f (x)ω2∗ Hence, there exists µ ¯ > 0 such that (2.17)

R ∗ C w2 CS N/2 R R µ ¯≤µ ˜ + inf = inf . >0 >0 f (x)ω2∗ f (x)ω2∗

Therefore, if µ > µ ¯, then (Pµ ) has no solution and this completes the proof. 3. Multiplicity of positive solutions From now on, we assume that f ∈ H −1 (RN ), f ≥ 0, f 6≡ 0 in RN and f satisfies ||µf ||∗ ≤ CN∗ . We set µ∗ := sup{µ ∈ R+ : (Pµ ) has at least one positive solution in H 1 (RN )}. Then, by Proposition 2.5, we have 0 < µ ¯ ≤ µ∗ < ∞. Remark. The minimal solution uµ of (Pµ ) is increasing with respect to µ. Indeed, suppose µ∗ > ν > µ. Since ∗ −1

−∆uν + uν − u2ν

− µf (x) = (ν − µ)f ≥ 0,

uν > 0 is a supersolution of (Pµ ). Since f (x) ≥ 0 and f (x) 6≡ 0, u ≡ 0 is a subsolution of (Pµ ) for any µ > 0. By the standard barrier method, we can obtain a solution uµ of (Pµ ) such that 0 ≤ uµ ≤ uν on RN . We note that 0 is not a solution of (Pµ ), ν > µ and uµ is a minimal solution of (Pµ ) because uµ also can be derived by an iteration scheme with initial value u(0) = 0. Therefore, by the maximal principle, 0 < uµ < uν on RN which completes the proof.

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Now, consider the corresponding eigenvalue problem: ( ∗ −∆ϕ + ϕ = λ(µ)(2∗ − 1)u2µ −2 ϕ, (3.1)µ ϕ in H 1 (RN ). Let λ1 be the first eigenvalue of (3.1)µ ;i.e., Z Z
∗ 1 N ∗ 2 2 λ1 = λ1 (µ) := inf{ |∇ϕ| + |ϕ| : ϕ ∈ H (R ), (2 − 1) u2µ −2 ϕ2 dx = 1}. Then, 0 < λ1 < ∞ and we can achieve the minimum by some function ϕ1 = ϕ1 (µ) ∈ H 1 (RN ) and ϕ1 > 0 in RN if µ ∈]0, µ∗ [ (cf. [27]). We summarize basic properties for λ1 (µ) : Lemma 3.1. (i) For µ ∈]0, µ∗ [, λ1 (µ) > 1, (ii) If 0 < µ < ν ≤ µ∗ , then λ1 (ν) < λ1 (µ), (iii) λ1 (µ) → +∞ as µ → 0+ . Proof. (i) For given 0 < µ < ν ≤ µ∗ , every solution uν of (Pµ ) with ν ∈ (µ, µ∗ ) is a supersolution of (Pµ ). By Taylor expansion, we have ∗ −1

−∆(uν − uµ ) + u(uν − uµ ) = u2ν

∗

− u2µ −1 + (ν − µ)f ∗

> (2∗ − 1)u2µ −2 (uν − uµ ) and moreover, we get Z Z Z ∇(uν − uµ )∇ϕ1 + (uν − uµ )ϕ1 =

∗ u2ν −1

∗ u2µ −1




Z

− ϕ1 + (ν − µ)f ϕ1 Z ∗ > (2∗ − 1) u2µ −2 (uν − uµ )ϕ1 .

Therefore, from (3.1)µ , we have Z Z Z ∗ ∗ ∇(uν − uµ )∇ϕ1 + (uν − uµ )ϕ1 = λ1 (µ)(2 − 1) u2µ −2 (uν − uµ )ϕ1 , which implies λ1 (µ) > 1. (ii) Since, for 0 < µ < ν ≤ µ∗ , uµ < uν and Z Z Z ∗ 2∗ −2 λ1 (µ)(2 − 1) uµ ϕ1 (µ)ϕ1 (ν) = ∇ϕ1 (µ)∇ϕ1 (ν) + ϕ1 (µ)ϕ1 (ν) Z ∗ ∗ = λ1 (ν)(2 − 1) u2ν −2 ϕ1 (ν)ϕ1 (µ), we have λ1 (µ) > λ1 (ν). (iii) First, we show that ||uµ || → 0 as µ → 0+ . Multiplying (Pµ ) by uµ , we have, Z Z Z
2 2 2∗ |∇uµ | + |uµ | = uµ + µf uµ and hence, for  > 0, we have, by Young’s inequality with ,   1  µ2 2 1− ||u || ≤ − ||f ||2∗ for  > 0. µ λ1 (2∗ − 1) 2 2

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11

Thus, for  > 0 small, we have ||uµ ||2 ≤ C µ2 for some constant C > 0, and hence, ||uµ || = o(1) as µ → 0+ . Next, Multiplying (3.1)µ by ϕ1 (µ), we have, by H¨ older’s inequality, that Z Z
∗ ∗ 2 2 |∇ϕ1 | + |ϕ1 | = λ1 · (2 − 1) u2µ −2 ϕ21 2/2∗ (2∗ −2)/2∗ Z Z 2∗ ∗ 2∗ ϕ1 ≤ λ1 · (2 − 1) uµ Z

∗

∗ u2µ

≤ λ1 · (2 − 1)

(2∗ −2)/2∗ Z

∗ −2)/2

≤ λ1 · (2∗ − 1)S −(2 ∗

∗ −2

and thus, S (2 −2)/2 ≤ λ1 ·(2∗ −1)||uµ ||2 completes the proof.

2

2



|∇ϕ1 | + |ϕ1 | ∗ −2

||uµ ||2

||ϕ1 ||2

. Therefore, we have the desired result. This

Lemma 3.2. Let uµ be a positive solution of (1.3)µ for which λ1 (µ) > 1. Then, for any g ∈ H 1 (RN ), the problem: (3.2)

∗

−∆w + w = (2∗ − 1)u2µ −2 w + g(x),

w ∈ H 1 (RN )

has a solution. Proof.

Consider the functional defined by Z Z Z
1 ∗ 1 2∗ −2 2 2 2 |∇w| + |w| − (2 − 1) uµ w − gw, J(w) = 2 2

w ∈ H 1 (RN ).

From H¨ older’s inequality and Young’s inequality, we have, for any  > 0,   1  1 1 J(w) ≥ − ||w||2 − ||w||2 − ||g||2∗ 2 2λ1 (µ) 2 2   1 1  1 = − − ||w||2 − ||g||2∗ 2 2λ1 (µ) 2 2 and hence, for small  > 0, there exist C1, > 0 and C2, > 0 such that J(w) ≥ C1, ||w||2 − C2, ||g||2∗ .

(3.3)

Let {wn } ⊂ H 1 (RN ) be the minimizing sequence of J. From (3.3), we have {wn } is bounded in H 1 (RN ). Hence, passing subsequence, we may have that there exists w ∈ H 1 (RN ) such that wn → w weakly in H 1 (RN ) as n → ∞, wn → w a.e. in RN as n → ∞ Here, we also note that ∇wn → ∇w a.e. in RN as n → ∞. And

∗

∗ −1

u2n −1 → u˜2

∗

weakly in (L2 (RN ))∗ as n → ∞.

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By Fatou’s Lemma ||w||2 ≤ lim inf ||wn ||2 . n→∞ R ∗ The weak convergence and the fact that u2µ −2 wn2 < ∞ for n ≥ 1 imply Z Z Z Z ∗ 2∗ −2 lim gwn = gw, lim uµ wn = u2µ −2 w n→∞

n→∞

and hence, J(w) ≤ lim J(wn ) = d. n→∞

Then, J(w) = d and w is a minimizer of J. Therefore, w is a critical point of J and w is a solution of (3.2). This completes the proof. Proposition 3.3. For µ = µ∗ , the problem (Pµ ) has a positive solution uµ∗ ∗ and λ1 (µ ) = 1. Moreover, the solution uµ∗ is unique in H 1 (RN ). Proof.

For µ ∈]0, µ∗ [, multiplying (Pµ ) by uµ , we have, by (3.1)µ , Z Z Z
2 2 2∗ |∇uµ | + |uµ | = uµ + µ f uµ Z 1 (|∇uµ |2 + |uµ |2 ) + µ∗ ||f ||∗ ||uµ || ≤ λ1 (µ)(2∗ − 1)   1 µ∗ µ∗ 2 = + ||uµ || + ||f ||2∗ . ∗ λ1 (µ)(2 − 1) 2 2

By taking  > 0 small enough, there exists an constant C > 0 such that ||uµ || ≤ C for all µ ∈]0, µ∗ [. Then, there exists uµ∗ in H 1 (RN ) such that uµ monotonically increasing to uµ∗ as µ → µ∗ and uµ → uµ∗ weakly in H 1 (RN ) as µ → µ∗ . Hence, uµ∗ is a positive solution of (Pµ ) with µ = µ∗ . We note that λ1 (µ) is a continuous function of µ ∈]0, µ∗ ]. Define F : R1 × H 1 (RN ) → H −1 (RN ) by F (µ, u) := ∆u − u + (u+ )2

∗ −1

+ µf (x).

Since uµ → uµ∗ weakly as µ → µ∗ , from Lemma 3.1, λ(µ∗ ) ≥ 1. If λ1 (µ∗ ) > 1, ∗ then Fu (µ∗ , uµ∗ )ϕ = ∆ϕ − ϕ + (2∗ − 1)u2µ∗−2 ϕ = 0 has no nontrivial solution. From Lemma 3.2, F (µ∗ , uµ∗ ) is an isomorphism of R1 × H 1 (RN ) onto H −1 (RN ), and by the implicitly function theorem to F, we find a neighborhood ]µ∗ − δ, µ∗ + δ[ of µ∗ such that (Pµ ) possesses a positive solution if µ ∈]µ∗ − δ, µ∗ + δ[, which contradicts the definition of µ∗ . Therefore, λ1 (µ∗ ) = 1. Suppose vµ∗ is a positive solution of (Pµ∗ ). Then vµ∗ ≥ uµ∗ since uµ∗ is minimal. Let w = vµ∗ − uµ∗ . Then, since λ1 (µ∗ ) = 1, we have ∗

−∆w + w ≥ (2∗ − 1)u2µ∗−2 w. Since ϕ1 = ϕ1 (µ∗ ) is the eigenfunction of the problem (3, 1)µ∗ , we have, Z Z Z Z ∗ ∗ 2∗ −2 ∗ (2 − 1) uµ∗ ϕ1 w = ∇w∇ϕ1 + wϕ1 ≥ (2 − 1) u2µ∗−1 wϕ1 and hence, w ≡ 0. This completes the proof.

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Proposition 3.4. The minimal solution uµ of (Pµ ) increasing continuously ∗ ∗ to uµ as µ → µ and uniformly bounded in H 1 (RN ) for all µ ∈]0, µ∗ ]. Moreover, ||uµ || ≤ O(µ2 ) as µ → 0+ . Proof.

It suffices to find the uniform bound of uµ . Multiplying (Pµ ) by uµ , we have Z Z Z
2∗ 2 2 |∇uµ | + |uµ | = uµ + µf uµ

and hence, for  > 0, we have   1  µ2 2 1− − ||u || ≤ ||f ||2∗ for  > 0. µ λ1 (2∗ − 1) 2 2 Therefore, for  > 0 small, we have ||uµ || ≤ C µ for some constant C > 0. Next, fix τ ∈]0, µ∗ ]. If µ increases to τ, then uµ is increasing up to uτ and uµ → uτ in H 1 (RN ). If it is not the case, then, by multiplying uµ on (Pµ ) again, we have

∗  ||uµ ||2 ≤ uτ2 −1 , uµ + τ hf, uµ i and so

∗ −1)/2

||uµ || ≤ CS −(2

∗ −1

||uτ ||2

+ τ ||f ||∗

for some C > 0. Hence, there exists a sequence {uµj } in H 1 (RN ) conversing weakly to a solution u˜ of (Pτ ) but u˜ 6= uτ . Since {uµj } coverge to u˜ strongly in local L1 sense, by the maximum principle, we have uµj ≤ u˜ < uτ which leads a contradiction to the minimality of uτ . This completes the proof. Remark. From Proposition 3.4 , we have that λ(µ) is a continuously decreasing function from [0, µ∗ ] onto [1, ∞[ and ||uµ || = o(1) as µ → 0+ . Next, we are going to find the second solution. In order to get another positive solution of (Pµ ), we consider the following problem: ( ∗ ∗ −∆v+v = (v + uµ )2 −1 − u2µ −1 in RN , (3.4)µ v ∈ H 1 (RN ), v > 0 in RN and the corresponding variational functional: Z Z

1 1 ∗ ∗ ∗ 2 2 |∇v| + |v| − ∗ (v + + uµ )2 − u2µ − 2∗ u2µ −1 v + Jµ (v) := 2 2 for v ∈ H 1 (RN ). Clearly, we can have another positive solution Uµ = uµ +vµ if we show the problem (3.4)µ possesses a positive solution for µ ∈]0, µ∗ [. We look for a critical point of Jµ which is a weak solution of (3.4)µ by employing standard argument of the Mountain Pass method without the (P S) condition. In the proof of the existance second solution, we make use of some arguments in [9, 10, 11]. Theorem 3.5. all µ ∈]0, µ∗ [.

The problem (Pµ ) possesses at least two positive solutions for

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Proof. (i) Let v ∈ H 1 (RN ) \ {0}, Then, for  > 0, by Young’s inequality, Jµ (v) = ≥

≥ ≥ ≥

Z Z v+ Z

1 ∗ ∗ 2 2 (uµ + s)2 −1 − u2µ −1 dsdx |∇v| + |v| dx − 2 0  Z
1 1 1− |∇v|2 + |v|2 dx 2 λ1 Z Z v+
∗ ∗ ∗ (uµ + s)2 −1 − u2µ −1 − (2∗ − 1)u2µ −2 s dsdx − 0  Z Z Z v+

1 1 ∗ ∗ 2 2 u2µ −2 s + C s2 −1 dsdx 1− |∇v| + |v| dx − 2 λ1 0   Z Z

2∗ 1 1  C ∗ 2 2 2 −2 + 1− ||v|| − uµ v v + dx dx − ∗ 2 λ1 2 2    1 C 1 ∗ ∗ − 1− ||v||2 − ∗ S −2 /2 ||v||2 ∗ 2 λ1 2(2 − 1)λ1 2

for some constant C > 0. Hence, for sufficiently small  > 0, there exist ρ > 0, δ > 0 such that Jµ (v)|∂ B˜ρ ≥ δ > 0, ˜ρ = {u ∈ H 1 (RN )| ||u|| ≤ ρ}. where B (ii) Let v ∈ H 1 (RN ), v ≥ 0 and v 6≡ 0, then, for t > 0, we have Z Z

t2 1 ∗ ∗ ∗ 2 2 Jµ (tv) = |∇v| + |v| dx − ∗ (uµ + tv)2 − u2µ − 2∗ u2µ −1 tv dx 2 2 ∗ Z 2 Z
t t2 ∗ 2 2 |v|2 dx |∇v| + |v| dx − ∗ ≤ 2 2 2 2∗ t t ∗ ≤ ||v||2 − ∗ ||v||22∗ 2 2 Hence, we deduce Jµ (tv) → −∞ as t → ∞. Therefore, for each 0 6≡ v ∈ H 1 (RN ) with v ≥ 0, there exists a constant tv > 0 such that Jµ (tv v) ≤ 0 for t ≥ tv . R R R 2∗ Let K1 (v) := 21 (|∇v|2 + v 2 ) − 21∗ (v + ) − µ f v. Because uµ is the critical point of K1 (u), we can prove that, for v ∈ H 1 (RN ), Jµ (v) = Kµ (v) − Kµ (0) = Kµ (v) − K1 (uµ ).

(3, 5) where 1 Kµ (v) := 2

Z

2

|∇(v + uµ )| + (v + uµ


2

1 − ∗ 2

Z

+

2∗

(v + uµ ) − µ

Z f (x)(v + uµ ).

(iii) From (ii), there exist small t1 > 0 such that, for 0 < t < t1 , Jµ (tω )
t1 such that Jµ (tω ) ≤ 0 for all t ≥ t2 . For t1 ≤ t ≤ t2 , Z Z

t2 1 ∗ ∗ ∗ 2 2 Jµ (tω ) = |∇ω| + |ω | dx − ∗ (uµ + tω )2 − u2µ − 2∗ u2µ −1 tω dx 2 2 2 2∗ t t ∗ < ||ω ||2 − ∗ ||ω ||22∗ 2  2 2 2∗ t t 1 = − ∗ S N/2 ≤ S N/2 . 2 2 N (iv) Let Γ := {γ ∈ C([0, 1], H 1 ); γ(0) = 0, γ(1) = t2 ω } and cµ = inf γ∈Γ maxs∈[0,1] Jµ (γ(s)). Then, we have 1 N/2 S . N We now applying the Mountain Pass Theorem without Palais-Smale condition in [5] to get a sequence {vn } ⊂ H 1 (RN ) such that 0 < α ≤ cµ ≤ supt≥0 Jµ (tω )
0. This completes the proof. Consequently, we have:

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Existence and Bifurcation of Positive Global Solutions

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Theorem 3.6. Assume f ∈ H −1 (RN ), f ≥ 0, f 6≡ 0 in RN and ||µf ||∗ ≤ CN∗ . Then there exists a positive constant µ∗ > 0 such that (Pµ ) possesses at least two positive solutions for 0 < µ < µ∗ , a unique solution for µ = µ∗ and no positive solution if µ > µ∗ . 4. Bifurcation In order to study bifurcation phenomenon, we consider following eigenvalue problem: ( ∗ −∆φ + φ = η(µ)(2∗ − 1)Uµ2 −2 φ, (4.1)µ φ in H 1 (RN ). Let η1 be the first eigenvalue of (4.1)µ ;i.e., Z Z ∗ 2 2 1 N η1 = η1 (µ) inf{ |∇φ| + |φ| ; φ ∈ H (R ), (2∗ − 1)Uµ2 −2 φ2 = 1} and φ1 > 0 be the corresponding eigenfunction. In the proof of the following lemma, we make use of arguments in [3]. Lemma 4.1. Let Uµ be a second positive solution of (Pµ ) obtained in Theorem 3.5. Then η1 (µ) < 1 for 0 < µ < µ∗ . Proof. Suppose contrary that η1 (µ) ≥ 1, Let ψ = Uµ − uµ > 0. Then φ1 and ψ satisfies (4.2)

∗ −2

∆φ1 − φ1 + (2∗ − 1)Uµ2

∗

φ1 ≤ 0 and ∆ψ − ψ + (2∗ − 1)µ2 −2 ψ ≥ 0,

respectively. Set σ = ψ/φ1 ;i.e., ψ = σφ1 . Then, by (4.2), σ∇(φ21 ∇σ) = ψ∇ψ −

(4.3)

ψ ∇φ1 ≥ 0. φ1

Let ζ be a C ∞ function on R+ with 0 ≤ ζ(t) ≤ 1,  1 for 0 ≤ t ≤ 1, ζ(t) := 0 for t ≥ 2.   in RN . Multiplying (4.3) by ζR2 and intergrating over For R > 0, set ζR (t) = ζ |x| R RN , we have by Green’ theorem, Z

ζR2 φ21 |∇σ|2

Z 2 ≤ 2 φ1 ζR σ∇σ · ∇ζR Z 1/2 Z 1/2 2 2 2 2 2 2 ≤2 ζR φ1 |∇σ| φ1 σ |∇ζR | R η1 (µ) >

∗ Uµ2 −1 φ1

Z =µ

f φ1 .

1 . 2∗ −1

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Wan Se Kim

(ii) As µ → 0+ , 1 ||Uµ ||2 S N/2 + o(1) 1 < η (µ) ≤ ≤ → ∗ . 1 ∗ 2 ∗ ∗ ∗ N/2 2 −1 (2 − 1) ||Uµ ||2∗ (2 − 1) (S + o(1)) 2 −1 Thus, η1 (µ) → 1/(2∗ − 1) as µ → 0+ . (iii) follows from (i) of Lemma 3.1, Proposition 3.3, Lemma 4.1 and (i) of Lemma 4.2. This completes the proof. In order to show the existence of a bifurcation point, we make use of Theorem 3.2 in [8]. Now, we have: Theorem 4.5. (i) The set {Uµ } is bounded uniformly in H 1 (RN ), (ii) (µ∗ , uµ∗ ) is a bifurcation point. Proof. (i) It follows immediately from the proof of Lemma 4.2. (ii) For this, define F : R × H 1 (RN ) → H −1 (RN ) by F (µ, u) := ∆u − u + (u+ )2

∗ −1

+ µf (x).

It is easy to see that F (µ, u) is differentiable at solution point (µ, u) for ]0, µ∗ [ and ∗

Fu (µ, uµ )w = ∆w − w + (2∗ − 1)u2µ −2 w is an ismorphism of R × H(RN ) onto H −1 (RN ). Then, by the Implicit Function Theorem, the solution of F (µ, u) near (µ, uµ ) are given by a single continuous cuver and uµ → 0 in H 1 (RN ) as µ → 0. We now are going to prove that (µ∗ , uµ∗ ) is a bifurcation point of F. Since Fu (µ∗ , uµ∗ )φ = 0, φ ∈ H 1 (RN ) has a solution φ1 > 0 in RN , N (Fu (µ∗ , uµ∗ )) = span{φ1 } is one dimensional and codimR (Fu (µ∗ , uµ∗ )) = 1 by the Fredholm alternative. Suppose there exists v ∈ H 1 (RN ) satisfying ∗

∆v − v + (2∗ − 1) u2µ∗−2 v = −f (x). Then Z 0=

∗

∇v · ∇φ1 + vφ1 − (2 −

∗ 1) u2µ∗−2 vφ1




Z =

f φ1 ,

which is impossible because 0 6≡ f ≥ 0. Hence, Fu (µ∗ , uµ∗ ) 6∈ R (Fu (µ∗ , uµ∗ )) . Thus, by Theorem 3.2 in [8], (µ∗ , uµ∗ ) is the bifurcation point near which, the solution of (pµ ) form a curve (µ∗ + τ (s), uµ∗ + sφ1 + z(s)) with s near s = 0 and τ (0) = τ 0 (0) = 0, z(0) = z 0 (0) = 0. Finally, we will show that τ 00 (0) < 0 which implies that the bifurcation curve only turns to the left in the µu−plane. For this, differentiate (Pµ ) in s, we have (4.6)

∗ −2

∆us − us + (2∗ − 1) u2

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Existence and Bifurcation of Positive Global Solutions

21

where us = φ1 + z 0 (s). Multiplying Fu (µ∗ , uµ∗ ) φ1 = 0 by us and (4,6) by φ1 , integrating and substracting, we have Z 0 τ (s) f φ1 Z   ∗ ∗ ∗ u2µ∗−2 − (uµ∗ + sφ1 + z(s))2 −2 (φ1 + z 0 (s))φ1 = (2 − 1)   Z z(s) 2∗ −3 ∗ ∗ φ1 + = −s(2 − 1)(2 − 2) (uµ∗ + θ(sφ1 + z(s))) (φ1 + z 0 (s)) φ1 s for some θ(s) ∈ (0, 1). Therefore, Z  Z Z ∗ τ 0 (s) 00 ∗ ∗ τ (0) f φ1 = lims→0 f φ1 = − (2 − 1) (2 − 2) (uµ∗ )2 −3 φ31 s and τ 00 (0) < 0. This completes proof.

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Wan Se Kim

ACKNOWLEDGEMENT This research was supported by Basic Science Research Program through the National Research Foundation of Korea(NRF) funded by the Ministry of Education(2013057580). This work was done while the author was visiting the Utah State University. He want to thank Professor Zhi-Qiang Wang and all the faculty and staff of the Mathematics dapartment.

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Existence and Bifurcation of Positive Global Solutions

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References [1] A. Ambrosetti and P. H. Rabinowitz, Dual variational methods in critical point theory and applications, J. Funt. Anal. 14, 349-381 (1973). [2] A. Ambrosetti and M. Struwe, 373-379 (1986).

∗ −2

A note on the problem −∆u = λu + u|u|2

,

Manuscripta Mathematica, 54,

[3] H, Berestycky, A. Caffarelli and L. Nerenberg, Further qualitative propertives for elliptic equations in unbounded domains, Ann. Scuola Norm. Sup. Pisa Cl. Sci. 25(4), 99-94 (1997). [4] H. Brezis and E. Lieb, A relation between pointwise convergence of functionals and convergence of functions, Proc. Amer. Soc. 88, 486-490 (1983). [5] H. Brezis and L. Nirenberg, Positive solutions of nonlinear elliptic equations involving critical Sobolev exponent, Comm. Pure Appl. Math. 36, 437-427 (1983). [6] L. Caffarelly, G. Gidas and J. Spruck, Asymptotic symmetry and local behavior of semilinear elliptic equations with critical Sobolev growth, Comm. Pure Appl. Math. 42, 271-1191 (1986). [7] D.-M. Cao, Positive solutions and bifurcation from the essential spectrum of a semilinear elliptic equations on RN , Nonlinear Anal. T.M.A. 15, 1048-1052 (1990), [8] M.G. Crandal and P.R. Rabinowitz, Bifurcation, perturbation of simple eigenvalues and linearized stability, Arch. Rational Mech Anal. 52, 161-180 (1973). [9] Y. Deng, Existence and nodal character of the solutions in RN for semilinear elliptic equation involving critical Sololev exponent, Acta Math. Sinica, 9(4), 385-402 (1989), [10] Y. Deng and Y. Li, Existence and bifurcation of positive solutions for a semilinear elliptic equation with critical exponent, J. Diff. Equa. 130, 179-200 (1996). [11] Y. Deng and Y, Li, Existence of multiple positive solutions for a semilinear elliptic equation, Adv. Differential Equations, 2, 361-382 (1997). [12] W.-Y. Ding and W.-M. Ni, On the existence of positive solutions for a semilinear elliptic equation, Archs Ration Mech. Analysis. 91), 283-307 (1986). [13] L. Ekeland, Convex minimization problem, Bull. Amer. Math. Soc. (NS)1, 443-474 (1976). [14] N. Hirano, Existence of entire positive solutions for nonhomogeneous elliptic equations, Nonlinear Analysis. T.M.A. 29(8), 889-901 (1997). [15] N. Hirano, Multiple existence of solutions for a nonhomogeneous elliptic problems on RN , J. Math. Anal. Appl. 336, 506-522 (2007), [16] N. Hirano and W. S. Kim, Multiple existence of periodic solutions for a nonlinear parabolic problem with singular nonlinearities, Nonlinear Analysis. T. M. A. 54, 445-456 (2003). [17] N. Hirano and W. S. Kim, Multiple existence of solutions for a semilinear elliptic problem with Neumann boundary condition, J. Math. Anal. Appl. 314, 210-218 (2006). [18] N. Hirano and W. S. Kim, Multiple existence of solutions for a nonhomogeneous elliptic problem with critical exponent on RN , J. Diff. Equa. 249, 1799-1816 (2010). [19] N. Hirano and W. S. Kim, Multiple existence of solutions for a nonhomogeneous elliptic problem on RN , Nonlinear Analysis. T.M.A. 74, 4369-4378 (2011). [20] W. S. Kim, Multiple existence of periodic solutions for semilinear parabolic equations with large source, Huston J. Math. 30(1), 283-295 (2004). [21] W. S. Kim, Multiple existence of positive global solutions for parametrized nonhomogeneous elliptic equations involving critical exponents, East Asian Math. J. 30(3), 335-353 (2014). [22] P.L. Lions, The concentration-compactness principle in the calculus of variations, the locally compact case. part 1, Ann. Inst. H. Poincar´ e Analyse non Lin´ eaire, 1(2)), 109-145 (1984). [23] P.L. Lions, The concentration-compactness principle in the calculus of variations, the locally compact case. part 2, Ann. Inst. H. Poincar´ e Analyse non Lin´ eaire, 1(4), 223-283 (1984). [24] W.A. Strauss, Existence of solatary waves in higher dimensions, Comm. Math. Phys. 55, 149-162 (1977). [25] X.-P. Zhu, Multiple entire solutions of a semilinear elliptic equation, Nonlinear analysis T.M.A. 12(11), 1297-1316 (1988). [26] X.-P. Zhu, A perturbation result on positive entire solutions of a semilinear elliptic equation, J. Diff. Equa. 92, 163-178 (1991). [27] X.-P. Zhu, and H.-S. Zhu, Existence of multiple positive solutions of inhomogeneous semilinear elliptic problems in unbounded domain, Proc. Roy. Soc. Edinburgh , 115 A, 301-318 (1990).

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Some inequalities on meromorphic function and its derivative on its Borel direction ∗ Hong Yan Xua† and Cai Feng Yib a

Department of Informatics and Engineering, Jingdezhen Ceramic Institute, Jingdezhen, Jiangxi, 333403, China

b

Institute of Mathematics and informatics, Jiangxi Normal University, Nanchang, Jiangxi, 330022, China

Abstract In view of Nevanlinna theory in the angular domain, we establish some inequalities of meromorphic function concerning its derivation in its Borel direction. By applying these inequalities, we also investigate exceptional values of meromorphic functions with infinite order in the Borel direction. Key words: Infinite order; Borel direction; Exceptional value. Mathematical Subject Classification (2010): 30D30 30D35.

1

Introduction and main results

It is assumed that the reader is familiar with the basic results and the standard notations of the Nevanlinna theory of meromorphic functions (see [7, 17, 20]). We denote by C the open complex S b plane, by C(= C {∞}) the extended complex plane, and by Ω(⊂ C) an angular domain. In addition, the order of meromorphic function f is defined by ρ(f ) = lim sup r→∞

log T (r, f ) , log r

and the exponent of convergence of distinct a-points of f is defined by ρ(a, f ) = lim sup r→∞

log+ N (r, a, f ) . log r

Let f be a meromorophic function of order ρ(0 < ρ < ∞), then we say that a is an exceptional value in the sense of Borel (evB for short) for f for the distinct zeros if ρ(a, f ) < ρ. It is well known that the singular direction of meromorphic function is an interesting topic in the field of complex analysis, such as, Julia direction, Borel direction, T direction, Hayman direction, and so on (see [1, 3, 4, 8, 10, 12, 13, 14]). Moreover, we know that every one singular direction is always responding to exceptional value, such as, the Julia’s direction relating with Picard exceptional value and the Borel’s direction relating with Borel exceptional value, and so ∗ This

work was supported by the NSFC(11561033,11301233, 61202313), the Natural Science Foundation of Jiangxi Province in China(20132BAB211001, 20151BAB201008) and the Foundation of Education Department of Jiangxi (GJJ14644) of China. † Corresponding author

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on. 2011, Peng and Sun [11] gave some examples on T direction which is a singular direction relating with T exceptional value. In the discussion of the topic of singular direction, we find that the characteristics of meromorphic functions in the angular domain played an important role(see [6, 18, 19, 24, 25]). So, we firstly introduce the characteristics of meromorphic functions in the angular domain as follows [5, 25]. For a meromorphic function f on the angular domain Ω(α, β) = {z : α ≤ arg z ≤ β} and 0 < β − α ≤ 2π. Define Z tω dt ω r 1 ( ω − 2ω ){log+ |f (teiα )| + log+ |f (teiβ )|} , Aα,β (r, f ) = π 1 t r t Z β 2ω log+ |f (reiθ )| sin ω(θ − α)dθ, Bα,β (r, f ) = ω πr α X 1 |bµ |ω Cα,β (r, f ) = 2 ( − 2ω ) sin ω(θµ − α), ω |bµ | r 1 k and also zero of f (k) in Ω(θ − ε, θ + ε), then z0 is not zero of f (k) − b in Ω(θ − ε, θ + ε) as b 6= 0. Thus, we have p X

Cθ−ε,θ+ε (r, aj , f ) +

j=1

q X

Cθ−ε,θ+ε (r, bl , f (k) ) − Cθ−ε,θ+ε (r, 0, f (k+1) )

l=1

≤

p X

Cθ−ε,θ+ε (r, aj , f | ≤ k + 1) +

j=1 p X

q X

C θ−ε,θ+ε (r, bl , f (k) ),

(14)

l=1

Cθ−ε,θ+ε (r, aj , f ) − Cθ−ε,θ+ε (r, 0, f (k) ) ≤

j=1

p X

Cθ−ε,θ+ε (r, aj , f | ≤ k).

j=1

Substituting (14) to (13), we get pqSθ−ε,θ+ε (r, f ) ≤ C θ−ε,θ+ε (r, ∞, f ) + (q − 1)

p X

Cθ−ε,θ+ε (r, aj , f | ≤ k)

j=1

+

p X

Cθ−ε,θ+ε (r, aj , f | ≤ k + 1) +

j=1

q X

C θ−ε,θ+ε (r, bl , f (k) )

l=1

+ Qθ−ε,θ+ε (r, f ).

(15)

For any positive integer k, we have Cθ−ε,θ+ε (r, aj , f | ≤ k) ≤ kC θ−ε,θ+ε (r, aj , f )   k mj C θ−ε,θ+ε (r, aj , f | ≤ mj ) + Cθ−ε,θ+ε (r, aj , f ) ≤ mj + 1   k ≤ mj C(r, aj , f | ≤ mj ) + Sθ−ε,θ+ε (r, f ) + O(1), mj + 1

(16)

and i 1 h nl C θ−ε,θ+ε (r, bj , f (k) | ≤ nl ) + Sθ−ε,θ+ε (r, f (k) ) + O(1) nl + 1  1  C θ−ε,θ+ε (r, ∞, f ) ≤ sC θ−ε,θ+ε (r, ∞, f | ≤ s) + Sθ−ε,θ+ε (r, f ) , (17) s+1

C θ−ε,θ+ε (r, bl , f (k) ) ≤

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and since Sθ−ε,θ+ε (r, f (k) ) ≤ Sθ−ε,θ+ε (r, f ) + kC θ−ε,θ+ε (r, ∞, f ) + Qθ−ε,θ+ε (r, f ), then it follows from (15)-(17) that pqSθ−ε,θ+ε (r, f ) ≤ (q − 1)

p X j=1

  k mj C θ−ε,θ+ε (r, aj , f | ≤ mj ) + Sθ−ε,θ+ε (r, f ) mj + 1

p X  k+1  + mj C θ−ε,θ+ε (r, aj , f | ≤ mj ) + Sθ−ε,θ+ε (r, f ) m + 1 j j=1

+

q X l=1

i 1 h nl C θ−ε,θ+ε (r, bl , f (k) | ≤ nl ) + Sθ−ε,θ+ε (r, f (k) ) nl + 1

+ C θ−ε,θ+ε (r, ∞, f ) + Qθ−ε,θ+ε (r, f ) ≤ (q − 1)

+

+

p X kmj C θ−ε,θ+ε (r, aj , f | ≤ mj ) mj + 1 j=1

p X mj (k + 1) j=1 q X l=1

mj + 1

C θ−ε,θ+ε (r, aj , f | ≤ mj )

nl C θ−ε,θ+ε (r, bl , f (k) | ≤ nl ) nl + 1 q X

k )C θ−ε,θ+ε (r, ∞, f ) nl + 1 l=1   p q X X kq + 1 1  + + Sθ−ε,θ+ε (r, f ) + Qθ−ε,θ+ε (r, f ) m + 1 n +1 j l j=1 j=1 + (1 +

≤ (kq + 1)

p X kmj C θ−ε,θ+ε (r, aj , f | ≤ mj ) m j +1 j=1

q X

nl C θ−ε,θ+ε (r, bl , f (k) | ≤ nl ) nl + 1 l=1 ! q X k s C θ−ε,θ+ε (r, ∞, f | ≤ s) + 1+ nl + 1 s + 1

+

l=1

  p q q X X X kq + 1 1 1  k + + + (1 + ) m + 1 n + 1 n + 1 s + 1 j l l j=1 j=1 l=1

× Sθ−ε,θ+ε (r, f ) + Qθ−ε,θ+ε (r, f ).

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Since mj , nl , k, p, q and s are positive integers, it follows from the above inequality that   p q q X X X kq + 1 1 k 1  Sθ−ε,θ+ε (r, f ) pqSθ−ε,θ+ε (r, f ) ≤  + + (1 + ) mj + 1 j=1 nl + 1 nl + 1 s + 1 j=1 l=1

+ (kq + 1)

p X

C θ−ε,θ+ε (r, aj , f | ≤ mj )

j=1 q X

nl C θ−ε,θ+ε (r, bl , f (k) | ≤ nl ) nl + 1 l=1 ! q X s k C θ−ε,θ+ε (r, ∞, f | ≤ s) + 1+ nl + 1 s + 1 +

l=1

+ Qθ−ε,θ+ε (r, f ).

(18)

Thus, from (18), we can prove (2) easily. Therefore, this completes the proof of Theorem 1.2. 2

4

The proof of Theorem 1.3

Proof: Since f is a meromorphic function of infinite order ρ(r) and arg z = θ(0 ≤ θ < 2π) is one Borel direction of ρ(r) order of meromorphic function f , by Lemma 2.6, we can get for any ε(0 < ε < π) log Sθ−ε,θ+ε (r, f ) = 1. (19) lim sup ρ(r) log r r→∞ Since ∞ is an evBB for f for distinct poles of order ≤ s, and aj (j = 1, 2, . . . , p) are evBB for f for distinct zeros of order ≤ mj , and bl (6= 0)(l = 1, 2, . . . , q) are evBB for f (k) for distinct zeros of order ≤ nl , from Definition 1.3 and (19), we have that there exists a number η(0 < η < 1) such that for sufficiently large r, C(r, ∞, f | ≤ s) ≤ (U (r))η ,

(20) η

C θ−ε,θ+ε (r, aj , f | ≤ mj ) < (U (r)) , j = 1, 2, . . . , p,

(21)

C θ−ε,θ+ε (r, bl , f (k) | ≤ nl ) < (U (r))η , l = 1, 2, . . . , q.

(22)

Set Λ :=

p q q X X kq + 1 X 1 k 1 + + (1 + ) . m + 1 n + 1 n + 1 s + 1 j l j=1 j=1 l l=1

From Theorem 1.2, we have (pq − Λ)Sθ−ε,θ+ε (r, f ) ≤ (kq + 1)

p X

C θ−ε,θ+ε (r, aj , f | ≤ mj )

j=1 q X

nl C θ−ε,θ+ε (r, bl , f (k) | ≤ nl ) nl + 1 l=1 ! q X s k C θ−ε,θ+ε (r, ∞, f | ≤ s) + 1+ nl + 1 s + 1 +

l=1

+ Qθ−ε,θ+ε (r, f ).

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(23)

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From (20)-(23), for sufficiently large r, it follows that (pq − Λ)Sθ−ε,θ+ε (r, f ) ≤ O((U (r))η ) + Qθ−ε,θ+ε (r, f ).

(24)

Since η < 1, from (19) and (24) for sufficiently large r, we can get pq − Λ ≤ 0, that is, ! p q q X X kq + 1 X 1 1 1 + + 1+k ≥ pq. mj + 1 nl + 1 s + 1 nl + 1 j=1 l=1

l=1

Thus, this completes the proof of Theorem 1.3. 2

5

Remarks

From the procedure of proofs of Theorems 1.1 and 1.2, we find that the conclusions of Theorems 1.1 and 1.2 can still hold for transcendental meromorphic function f with finite order ρ(0 < ρ < ∞) on the whole complex plane. Thus, it is a natural question to ask: Does the conclusion of Theorem 1.3 still holds when f is a transcendental meromorphic function with finite order ρ(0 < ρ < ∞) on the whole complex plane? In fact, we can not give a positive answer to the above question. Now, we will give a simple procedure to prove this assertion as follows. Firstly, similar to Definitions 1.3 and 1.4, we can get some definitions of exception values of meromorphic function with finite order in the Borel direction, if ρkθ (a, f ) < ρ, ρθ (a, f ) < ρ, ρkθ (a, f (l) ) < ρ, when log Sθ−ε,θ+ε (r, f ) is replaced by log r. Thus, from the definition of Borel direction, (20)-(23) can be replaced by 0

C(r, ∞, f | ≤ s) ≤ rη ,

(25) 0

C θ−ε,θ+ε (r, aj , f | ≤ mj ) < rη , j = 1, 2, . . . , p, C θ−ε,θ+ε (r, bl , f

(k)

(26)

η0

| ≤ nl ) < r , l = 1, 2, . . . , q.

(27)

where η 0 < ρ and r is sufficiently large, and (24) can be replaced by  0 (pq − Λ) Sθ−ε,θ+ε (r, f ) ≤ O rη + Qθ−ε,θ+ε (r, f ).

(28)

However, by Lemmas 2.1-2.5, we can not be sure to derive a contradiction from (28). Therefore, Theorem 1.3 may not be true when f is of finite order.

References [1] M. L. Cargwright, On the directions of Borel of functions which are regular and of finite order in an angle, Proc. London Math. Soc. s2-38 (1) (1935): 503-541. [2] C. T. Chuang, Singular direction of meromorpic functions, Science Press, Beijing, 1982. [3] C. T. Chuang, On Borel directions of meromorphic functions of infinite order (II), Bulletin of the Hong Kong Mathematical Society, 2(2) (1999): 305-323. [4] D. Drasin and A. Weitsman, On the Julia Directions and Borel Directions of Entire Functions, Proc. London Math. Soc. s3-32(2) (1976): 199-212.

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[5] A. A. Gol’dberg, Nevanlinna’s lemma on the logarithmic derivative of a meromorphic function, Mathematical Notes 17(4) (1975): 310-312. [6] A. A. Goldberg and I. V. Ostrovskii, The distribution of values of meromorphic function, Nauka, Moscow, 1970 (in Russian). [7] W. K. Hayman, Meromorphic functions, Oxford Univ. Press, London, 1964. [8] C. N. Linden, On a conjecture of Valiron concerning sets of indirect Borel points, J. London Math. Soc. s1-41 (1) (1966): 304-312. [9] J. R. Long and P. C. Wu, Borel directions and uniqueness of meromorphic functions, Chinese Ann. Math. 33A(3) (2012): 261-266. [10] 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). [11] Y. H. Peng and D. C. Sun, Examples on exceptional values of meromorphic functions, Acta Mathematica Scientia, 31(4) (2011): 1327-1336. [12] M. Tsuji, On Borel’s directions of meromorphic functions of finite order, I, Tohoku Math. J. 2(2) (1950): 97-112. [13] S. J. Wu, Further results on Borel removable sets of entire functions, Ann. Acad. Sci. Fenn. Ser. A. I Math., 19 (1994): 67-81. [14] S. J. Wu, On the distribution of Borel directions of entire function, Chinese Ann. Math. 14A(4) (1993): 400-406. [15] H. Y. Xu, Z. J. Wu and J. Tu, Some inequalities and applications on Borel direction and exceptional values of meromorphic functions, Journal of Inequalities and Applications 2014 (2014): Art. 53, 12 pages. [16] L. Yang and C.C. Yang, Angular distribution of f f 0 , Sci. China Ser. A 37(3)(1994): 284-294. [17] L. Yang, Value distribution theory and its application, Springer/ Science Press, Berlin/ Beijing, 1993/ 1982. [18] L. Yang, Borel directions of meromorphic functions in an angular domain, Sci. in China Ser. A. S1 (1979): 149-164. [19] L. Yang and G.H. Zhuang, The distribution of Borel directions of entire functions, Sci. in China Ser. A. 3 (1976): 157-168. [20] H. X. Yi and C. C. Yang, Uniqueness theory of meromorphic functions, Kluwer Academic Publishers, Dordrecht, 2003; Chinese original: Science Press, Beijing, 1995. [21] K. Y. Zhang, H. Y. Xu and H. X. Yi, Borel directions and uniqueness of meromorphic functions, Abstract and Applied Analysis 2013 (2013): Art. 793810, 8 pages. [22] Q. C. Zhang, Meromorphic functions sharing values in an angular domain, J Math. Anal. Appl. 349(1) (2009): 100-112. [23] Q. C. Zhang, Borel’s directions and shared values, Acta Mathematica Scientia 33B (2) (2013: 471-483. [24] J. H. Zheng, On uniqueness of meromorphic functions with shared values in some angular domains, Canad J. Math. 47(2004): 152-160. [25] J. H. Zheng, Value distribution of meromorphic functions, Springer and Tsinghua University Press, 2010.

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TABLE OF CONTENTS, JOURNAL OF COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 20, NO. 6, 2016

On the Stability of the Generalized Quadratic Set-Valued Functional Equation, Hahng-Yun Chu, and Seung Ki Yoo,…………………………………………………………………………1007 Common Best Proximity Points For Proximally Commuting Mappings In Non-Archimedean PM-Spaces, George A. Anastassiou, Yeol Je Cho, Reza Saadati, and Young-Oh Yang,…1021 The Value Distribution of Some Difference Polynomials of Meromorphic Functions, Jin Tu, Hong-Yan Xu, and Hong Zhang,……………………………………………………………1031 On Properties of Decomposable Measures and Pseudo-Integrals, Dong Qiu, Chongxia Lu, and Nanxiang Yu,………………………………………………………………………………..1043 Composition Operator on Zygmund-Orlicz Space, Ning Xu, and Ze-Hua Zhou,………….1058 Multiple Positive Solutions For m-Point Boundary Value Problems With One-Dimensional p-Laplacian Systems and Sign Changing Nonlinearity, Hanying Feng, and Jian Liu,………1066 An S-Partially Contractive Mapping with a Control Function ϕ, K. Abodayeh,……………1078 Approximation by Complex q-Gamma Operators in Compact Disks, Qing-Bo Cai, Cuihua Li, and Xiao-Ming Zeng,…………………………………………………………………………1088 Value Sharing of Meromorphic Functions of Differential Polynomials of Finite Order, Xiao-Bin Zhang, and Jun-Feng Xu,…………………………………………………………………….1097 Regularized Optimization Method For Determining the Space-Dependent Source in a Parabolic Equation Without Iteration, Zewen Wang, Wen Zhang, and Bin Wu,………………………1107 Knowledge Reduction in Knowledge Bases and its Algorithm, Ningxin Xie,………………1127 Belief Reduction in IVF Decision Information Systems and its Algorithm, Sheng Luo,……1138 Existence and Bifurcation of Positive Global Solutions for Parameterized Nonhomogeneous Elliptic Equations Involving Critical Exponents, Wan Se Kim,………………………………1148 Some Inequalities on Meromorphic Function and its Derivative On Its Borel Direction, Hong Yan Xu, and Cai Feng Yi,……………………………………………………………………..1171

Volume 20, Number 7 ISSN:1521-1398 PRINT,1572-9206 ONLINE

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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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Second order duality for multiobjective optimization problems Meraj Ali Khan, Falleh R. Al-Solamy Abstract In this paper, we first introduce a new class of generalized convex functions, called second order (F, α, ρ, d)-V-convex functions and then discuss appropriate duality results for second order Mangasarian type, Mond-Weir type and general MondWeir type multiobjective duals. Keywords: Multiobjective optimization; Second order generalized convex functions; Weak efficiency; Duality Mathematics Subject Classification 2000: 90C29, 90C30, 90C46, 49N15 1. Introduction Optimization theory is one of the most lively and exciting branch in modern mathematics, in which the importance of convexity is well known. But the notion of convexity does no longer suffice for many mathematical models used in decision sciences, economics, management sciences, stochastics, applied mathematics and engineering. Therefore, various generalizations of convex functions have been provided for the validity of results to larger classes of optimization problems. The generalization of convex functions was originally proposed by Hanson [7], which were named as invex functions by Craven [4], and η-convex functions by Kaul and Kaur [10]. In [9], Jeyakumar and Mond introduced V -invexity and its generalization for vector functions. More specifically, Preda [16] introduced the concept of (F, ρ)-convexity, an extension of F-convexity defined by Hanson and Mond [8] and ρ-convexity given by Vial [17]. Recently, Agarwal et al. [1] introduced a new class of generalized Vtype I functions for a multiobjective problem and discussed sufficiency and duality results. Second order duality was first introduced by Mangasarian [11] for a scalar programming problem. Mond [13] reproved second order duality results of Mangasarian [11] under simpler assumptions, and showed that the second order dual has computational advantages over the first order dual. Zhang and Mond [19] extended the class of (F, ρ)-convex functions to second order (F, ρ)-convex functions and discussed duality results for Mangasarian type, Mond-Weir type and general Mond-Weir type multiobjective duals. Aghezzaf [2] introduced new classes of generalized second order (F, ρ)-convexity for vector-valued functions and established various duality results for mixed type vector dual. In [6], Hachimi and Aghezzaf proposed a new class of generalized second order type I vector-valued functions for multiobjective programming problem and obtained mixed type duality theorems. Ahmad and Husain [3]

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defined a class of generalized second order (F, α, ρ, d)-convex functions and established duality results for Mond-Weir type multiobjective dual. Gulati and Agarwal [5] established Huard type converse duality theorems for second-order scalar and multiobjective dual problems showing certain inconsistencies in the earlier work of Yang et al. [18] and Mond and Zhang [15]. Being inspired by the excellent work of Mond and Zhang [15], Zhang and Mond [19] and Ahmad and Husain [3], we introduce the concept of second order (F, α, ρ, d)V-convex function and its generalizations, which includes most of the introduced classes of generalized convex functions. To characterize the introduced definitions, an example of second order (F, α, ρ, d)-V-convex function is given. Weak, strong and strict converse duality theorems are proved for second order Mangasarian type, Mond-Weir type and general Mond-Weir type multiobjective duals. These results extend the results appeared in [3, 15, 16, 19]. 2. Notations and preliminaries The following conventions for vectors in Rn will be followed: x ≧ y ⇔ xi ≧ yi , i = 1, 2, . . . , n; x ≥ y ⇔ x ≧ y, and there exists at least one i such that xi > yi ; x > y ⇔ xi > yi , i = 1, 2, . . . , n. The index sets are K = {1, 2, . . . , k} and M = {1, 2, . . . , m}. Consider the following nonlinear multiobjective programming problem: (P) Minimize f (x) = [f1 (x), f2 (x), . . . , fk (x)] subject to x ∈ S = {x ∈ X : g(x) ≦ 0}, where X ⊆ Rn is a nonempty open set and the functions f = (f1 , f2 , . . . , fk ) : X → Rk and g = (g1 , g2 , . . . , gm ) : X → Rm are twice differentiable at x¯ ∈ X. Definition 1. A point x¯ ∈ S is said to be a weakly efficient solution of (P), if there exists no other x ∈ S such that f (x) < f (¯ x). The following definitions are due to Mond and Zhang [15]: Definition 2. Function f : X → Rk is said to be second order V -invex at x¯ ∈ X, if there exist functions η : X ×X → Rn and αi : X ×X → R+ \ {0}, i ∈ K such that 1 x)p ≧ αi (x, x¯)[∇fi (¯ x) + ∇2 fi (¯ x)p]η(x, x¯) fi (x) − fi (¯ x) + pT ∇2 fi (¯ 2 holds for all p ∈ Rn and for all x ∈ X.

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Definition 3. Function f : X → Rk is said to be second order V -quasiinvex at x¯ ∈ X, if there exist functions η : X × X → Rn and γi : X × X → R+ \ {0}, i ∈ K such that k


γi (x, x¯)fi (x) ≦

i=1

k


i=1

⇒

k


γi (x, x¯)[fi (¯ x) − 12 pT ∇2 fi (¯ x)p]

x)p]η(x, x¯) ≦ 0 [∇fi (¯ x) + ∇2 fi (¯

i=1

holds for all p ∈ Rn and for all x ∈ X. Definition 4. Function f : X → Rk is said to be second order V -pseudoinvex at x¯ ∈ X, if there exist functions η : X × X → Rn and βi : X × X → R+ \ {0}, i ∈ K such that k


i=1

x)p]η(x, x¯) ≧ 0 ⇒ [∇fi (¯ x)+∇2 fi (¯

k


βi (x, x¯)fi (x) ≧

i=1

k


i=1

holds for all p ∈ Rn and for all x ∈ X.

βi (x, x¯)[fi (¯ x)− 21 pT ∇2 fi (¯ x)p]

Definition 5. A functional F : X × X × Rn → R is said to be sublinear in its third argument, if for any x, x¯ ∈ X, (i) F (x, x¯; a1 + a2 ) ≦ F (x, x¯; a1 ) + F (x, x¯; a2 ) ∀ a1 , a2 ∈ Rn , (ii) F (x, x¯; αa) = αF (x, x¯; a) ∀ α ∈ R, α ≧ 0 and ∀ a ∈ Rn . The following definitions of second order (F, ρ)-convexity and its generalization were introduced by Zhang and Mond [19]. Let F be a functional sublinear in its third argument, φ : X → R be twice differentiable at x¯ ∈ X, d : X × X → R be a metric and ρ ∈ R. Definition 6. Function φ : X → R is said to be second order (F, ρ)-convex at x¯ ∈ X, if 1 φ(x) − φ(¯ x) + pT ∇2 φ(¯ x)p ≧ F (x, x¯; ∇φ(¯ x) + ∇2 φ(¯ x)p) + ρd(x, x¯) 2 holds for all p ∈ Rn and for all x ∈ X. Definition 7. Function φ : X → R is said to be second order (F, ρ)-quasiconvex at x¯ ∈ X, if 1 x)p ⇒ F (x, x¯; ∇φ(¯ φ(x) ≦ φ(¯ x) − pT ∇2 φ(¯ x) + ∇2 φ(¯ x)p) ≦ −ρd(x, x¯) 2 holds for all p ∈ Rn and for all x ∈ X.

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Definition 8. Function φ : X → R is said to be second order (F, ρ)-pseudoconvex at x¯ ∈ X, if 1 F (x, x¯; ∇φ(¯ x) + ∇2 φ(¯ x)p) ≧ −ρd(x, x¯) ⇒ φ(x) ≧ φ(¯ x) − pT ∇2 φ(¯ x)p 2 holds for all p ∈ Rn and for all x ∈ X. Finally, in view of Definitions 2-8 and the concept of generalized second order (F, α, ρ, d)-convex functions [3], we propose our definitions of second order (F, α, ρ, d)-V-convex function and its generalizations as follows: Definition 9. Function f : X → Rk is said to be (strictly) second order (F, α, ρ, d)V-convex at x¯ ∈ X, if there exist functions αi : X × X → R+ \ {0}, i ∈ K, d : X × X → R and ρ = (ρ1 , ρ2 , . . . , ρk ) ∈ Rk such that 1 x)p (>) ≧ F (x, x¯; αi (x, x¯)(∇fi (¯ fi (x) − fi (¯ x) + pT ∇2 fi (¯ x) + ∇2 fi (¯ x)p)) + ρi d2 (x, x¯) 2 holds for all p ∈ Rn and for all x ∈ X. Remark 1. (i) For k = 1 and αi (x, x¯) = 1, the above definition becomes that of (strictly) second order (F, ρ)-convex function introduced by Zhang and Mond [19]. (ii) If ρi = 0, i ∈ K and F (x, x¯; a) = aT η(x, x¯) for a certain mapping η : X × X → Rn , the inequality reduces to that of (strictly) second order V -invex function introduced by Mond and Zhang [15]. (iii) If αi (x, x¯) = α(x, x¯), i ∈ K, then we get the definition of (strictly) second order (F, α, ρ, d)-convex function given by Ahmad and Husain [3]. Following example includes earlierly studied classes as special cases of second order (F, α, ρ, d)-V-convex function. Example 1. Consider the function f = (f1 , f2 , f3 ) : X → R3 , where X = R such that f1 (x) = (x + 2)2 ,

f2 (x) = 2 − x2 ,

f3 (x) = −x2 − 2x.

The feasible region is S = {x ∈ X : x ≧ 2}. a (x2 + x¯2 − 4); α1 (x, x¯) = 2; α2 (x, x¯) = 4; α3 (x, x¯) = 12; Let F (x, x¯; a) = 12 ρ1 = −1; ρ2 = 1; ρ3 = −1; d(x, x¯) = |x − x¯ + 2|; p = 2; x¯ = 2.

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It can be seen that f = (f1 , f2 , f3 ) is second order (F, α, ρ, d)-V-convex for all x ∈ X, f1 (x) − f1 (¯ x) + 12 pT ∇2 f1 (¯ x)p = x2 + 4x − 8 ≧ F (x, x¯; α1 (x, x¯)(∇f1 (¯ x) + ∇2 f1 (¯ x)p)) + ρ1 d2 (x, x¯) = x2 ,

(I)

x)p = − x2 f2 (x) − f2 (¯ x) + 21 pT ∇2 f2 (¯ 5 ≧ F (x, x¯; α2 (x, x¯)(∇f2 (¯ x) + ∇2 f2 (¯ x)p)) + ρ2 d2 (x, x¯) = − x2 , 3 1 T 2 2 f3 (x) − f3 (¯ x) + 2 p ∇ f3 (¯ x)p = −x − 2x + 4 ≧ F (x, x¯; α3 (x, x¯)(∇f3 (¯ x) + ∇2 f3 (¯ x)p)) + ρ3 d2 (x, x¯) = −11x2 .

(II)

(III)

The above inequalities show that f = (f1 , f2 , f3 ) is second order (F, α, ρ, d)-V-convex for all p ∈ R at x¯. If α1 (x, x¯) = α2 (x, x¯) = α3 (x, x¯) = 2, then Inequality (II) does not hold. If α1 (x, x¯) = α2 (x, x¯) = α3 (x, x¯) = 4, then Inequality (I) is not satisfied. Similarly, if α1 (x, x¯) = α2 (x, x¯) = α3 (x, x¯) = 12, then Inequality (I) is not satisfied. Hence, f = (f1 , f2 , f3 ) is not second order (F, α, ρ, d)-convex [3] for all p ∈ R at x¯. Let α1 (x, x¯) = α2 (x, x¯) = α3 (x, x¯) = 1. Then Inequality (II) does not hold. Therefore, f = (f1 , f2 , f3 ) is not second order (F, ρ)-convex [19] for all p ∈ R at x¯. Let ρ1 = ρ2 = ρ3 = 0. Then Inequalities (I) and (II) are not satisfied. Hence f = (f1 , f2 , f3 ) is not second order V-invex [15] for all p ∈ R at x¯. Definition 10. Function f : X → Rk is said to be (strictly) second order (F, α ˜ , ρ˜, d)V-quasiconvex at x¯ ∈ X, if there exist functions α ˜ i : X × X → R+ \ {0}, i ∈ K, d : X × X → R and ρ˜ ∈ R such that k 

α ˜ i (x, x¯)fi (x) ≦

i=1

k  i=1

k

 1 α ˜ i (x, x¯)fi (¯ x) − pT ∇2 α ˜ i (x, x¯)fi (¯ x)p 2 i=1

k  ⇒ F (x, x¯; (∇fi (¯ x) + ∇2 fi (¯ x)p)) + ρ˜d2 (x, x¯) () ≧

i=1

k


i=1

α ¯ i (x, x¯)fi (¯ x) − 12 pT ∇2

holds for all p ∈ Rn and for all x ∈ X.

k


α ¯ i (x, x¯)fi (¯ x)p

i=1

Remark 2. By using the sublinearity of F , one can see from the above definitions that a second order (F, α, ρ, d)-V-convex function is both second order (F, α ¯ , ρ¯, d)k
1 1 , i ∈ K and ρ¯ = ρi ) and second order V-pseudoconvex (with α ¯i = αi ¯) i=1 αi (x, x k
1 1 (F, α ˜ , ρ˜, d)-V-quasiconvex (with α ˜ i = , i ∈ K and ρ˜ = ρi ). Obviously, αi ¯) i=1 αi (x, x the converse is not necessarily true. Following Kuhn-Tucker theorem will be needed in the sequel: Proposition 1 [12]. Let x¯ be a weakly efficient solution of (P) at which the KuhnTucker constraint qualification is satisfied. Then there exist λ ∈ Rk and u ∈ Rm such that m k   ∇uj gj (¯ x) = 0, ∇λi fi (¯ x) + j=1

i=1

m 

uj gj (¯ x) = 0,

j=1

λ ≧ 0,

k 

λi = 1, u ≧ 0.

i=1

3. Mangasarian type duality

In this section, we consider the following second order Mangasarian type dual for (P) and discuss duality results.  (SD) Maximize f1 (y) + uT g(y) − 21 pT ∇2 (f1 (y) + uT g(y))p,

. . . , fk (y) + uT g(y) − 21 pT ∇2 (fk (y) + uT g(y))p

subject to k m   2 (∇λi fi (y) + ∇ λi fi (y)p) + (∇uj gj (y) + ∇2 uj gj (y)p) = 0, i=1

 (1)

j=1

k 

λ ≧ 0,

(2)

λi = 1,

(3)

i=1

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u ≧ 0.

(4)

Let Q be the set of all feasible solutions of (SD). Theorem 1 (Weak duality). Suppose that for all x ∈ S and (y, u, λ, p) ∈ Q, (i) f is second order (F, α, ρ, d)-V-convex at y, and g is second order (F, α ˆ , ρˆ, d)V-convex at y; (ii) (iii) Then

k


λi = 1 and α ˆ j (x, y) = 1, j ∈ M ; and i=1 αi (x, y) m

 λi ρ i + uj ρˆj ≧ 0. i=1 αi (x, y) j=1 k


m

m 

 1 uj gj (y))p, i ∈ K. fi (x) < fi (y) + uj gj (y) − pT ∇2 (fi (y) + 2 j=1 j=1

(5)

Proof. Suppose contrary to the result that (5) cannot hold, i.e., m

m 

 1 uj gj (y))p, i ∈ K, fi (x) < fi (y) + uj gj (y) − pT ∇2 (fi (y) + 2 j=1 j=1 which on using (2), (3), αi (x, y) > 0, i ∈ K and hypothesis (ii) becomes  k  k k m m   λi fi (x) λi fi (y)  1 T 2  λi fi (y)  < + uj gj (y) − p ∇ + uj gj (y) p, α (x, y) α (x, y) 2 α (x, y) i i i i=1 i=1 j=1 i=1 j=1

or

k m k  1 λi fi (x)  λi fi (y)  − − uj gj (y)+ pT ∇2 αi (x, y) i=1 αi (x, y) j=1 2 i=1

 k  m  λi fi (y)  + uj gj (y) p < 0. αi (x, y) j=1 i=1 (6)

According to hypothesis (i), it follows that 1 fi (x) − fi (y) + pT ∇2 fi (y)p ≧ F (x, y; αi (x, y)(∇fi (y) + ∇2 fi (y)p)) + ρi d2 (x, y) 2 and 1 gj (x) − gj (y) + pT ∇2 gj (y)p ≧ F (x, y; α ˆ j (x, y)(∇gj (y) + ∇2 gj (y)p)) + ρˆj d2 (x, y). 2 λi ≧ 0, i ∈ K and second by uj ≧ 0, αi (x, y) with α ˆ j (x, y) = 1, j ∈ M, then summing over i and j respectively, and on using the sublinearity of F , we have On multiplying the first inequality by

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k m k m   λi fi (x)  λi fi (y)  + uj gj (x) − − uj gj (y) α (x, y) α (x, y) i i i=1 j=1 i=1 j=1

 k  m 1 T 2  λi fi (y)  + uj gj (y) p + p ∇ 2 αi (x, y) j=1 i=1     k m   ≧ F x, y; (∇λi fi (y) + ∇2 λi fi (y)p) +F x, y; (∇uj gj (y) + ∇2 uj gj (y)p) j=1

i=1

+

k  λi ρi d2 (x, y) i=1

αi (x, y)

The relations (1), (7) and the sublinearity of F yield

+

m 

uj ρˆj d2 (x, y). (7)

j=1

m k m k   λi fi (x)  λi fi (y)  + uj gj (x) − − uj gj (y) αi (x, y) j=1 αi (x, y) j=1 i=1 i=1

 k  m   1 λi fi (y) + pT ∇2 + uj gj (y) p 2 αi (x, y) j=1 i=1   k m   λi ρ i + uj ρˆj d2 (x, y), ≧ α (x, y) i j=1 i=1

which by virtue of hypothesis (iii) gives

m k m k   λi fi (y)  λi fi (x)  + uj gj (x) − − uj gj (y) α (x, y) α (x, y) i i j=1 i=1 j=1 i=1

1 + pT ∇2 2 By u ≧ 0 and g(x) ≦ 0, it follows that k k m  λi fi (x)  λi fi (y)  1 − − uj gj (y)+ pT ∇2 αi (x, y) i=1 αi (x, y) j=1 2 i=1

 k  m  λi fi (y)  + uj gj (y) p ≧ 0. αi (x, y) j=1 i=1  k  m  λi fi (y)  + uj gj (y) p ≧ 0, α (x, y) i i=1 j=1

a contradiction to (6). This completes the proof.

2

Theorem 2 (Strong duality). Let x¯ be a weakly efficient solution of (P) at which the ¯ ∈ Rk , u¯ ∈ Rm Kuhn-Tucker constraint qualification is satisfied. Then there exist λ ¯ p¯ = 0) ∈ Q and the corresponding objective values of and p¯ ∈ Rn such that (¯ x, u¯, λ, 8 1202

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(P) and (SD) are equal. If, in addition, the hypotheses of weak duality (Theorem ¯ p¯ = 0) is a weakly efficient solution of (SD). 1) hold, then (¯ x, u¯, λ, Proof. Since x¯ is a weakly efficient solution of (P) at which the Kuhn-Tucker ¯ ∈ Rk and constraint qualification is satisfied, from Proposition 1, there exist λ u¯ ∈ Rm such that m k   ¯ i fi (¯ ∇¯ uj gj (¯ x) = 0, ∇λ x) + j=1

i=1

m 

u¯j gj (¯ x) = 0,

j=1

¯ ≧ 0, λ

k 

¯ i = 1, u¯ ≧ 0. λ

i=1

¯ p¯ = 0) ∈ Q and the corresponding objective values of (P) and Therefore, (¯ x, u¯, λ, ¯ p¯ = 0) thus follows from weak duality (SD) are equal. Weak efficiency of (¯ x, u¯, λ, (Theorem 1). 2 ¯ p¯) ∈ Q such that Theorem 3 (Strict converse duality). Let x¯ ∈ S and (¯ y , u¯, λ,  k  m k k m  


1 ¯ i fi (¯ ¯ i fi (¯ ¯ i fi (¯ (i) λ x) ≦ λ y) + u¯j gj (¯ y ) − p¯T ∇2 u¯j gj (¯ y ) p¯; λ y) + 2 i=1 i=1 j=1 i=1 j=1 (ii) f is strictly second order (F, α, ρ, d)-V-convex at y¯ with αi (¯ x, y¯) = 1, i ∈ K and g is second order (F, α ˆ , ρˆ, d)-V-convex at y¯ with α ˆ j (¯ x, y¯) = 1, j ∈ M ; and (iii)

k m
¯ i ρi +
u¯j ρˆj ≧ 0. λ

i=1

j=1

Then x¯ = y¯.

Proof. We assume that x¯ = y¯, and exhibit a contradiction. Using (2)-(4), hypothesis (ii), and the sublinearity of F , we obtain   k k k k     1 ¯ i fi (¯ ¯ i fi (¯ ¯ i fi (¯ ¯ i fi (¯ ¯ i fi (¯ λ x)− λ y )+ p¯T ∇2 λ y )¯ p > F x¯, y¯; (∇λ y ) + ∇2 λ y )¯ p) 2 i=1 i=1 i=1 i=1 +

k 

¯ i ρi d2 (¯ λ x, y¯)

i=1

and m 

m 

m

 1 u¯j gj (¯ y )+ p¯T ∇2 u¯j gj (¯ y )¯ p≧F u¯j gj (¯ x)− 2 j=1 j=1 j=1



m  x¯, y¯; (∇¯ uj gj (¯ y ) + ∇2 u¯j gj (¯ y )¯ p) j=1



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+

m 

u¯j ρˆj d2 (¯ x, y¯).

j=1

Adding these inequalities, we get k  i=1

m k m    1 ¯ ¯ u¯j gj (¯ x)− λi fi (¯ y )− u¯j gj (¯ y )+ p¯T ∇2 λi fi (¯ x)+ 2 j=1 i=1 j=1

>F



k  ¯ i fi (¯ ¯ i fi (¯ p) x¯, y¯; (∇λ y ) + ∇2 λ y )¯ i=1



+F



 k  i=1

k 

¯ i ρi d2 (¯ λ x, y¯) +

m 

j=1

+

k  i=1



u¯j ρˆj d2 (¯ x, y¯)

j=1

k m   ¯ i fi (¯ ¯ i fi (¯ (∇λ y ) + ∇2 λ x¯, y¯; y )¯ y )¯ p) + (∇¯ uj gj (¯ y ) + ∇2 u¯j gj (¯ p) i=1

u¯j gj (¯ y ) p¯

j=1

j=1

i=1

≧F



m  x¯, y¯; (∇¯ uj gj (¯ y ) + ∇2 u¯j gj (¯ p) y )¯

+ 

¯ i fi (¯ λ y) +

m 



m  2 ¯ λi ρi d (¯ x, y¯)+ u¯j ρˆj d2 (¯ x, y¯) (by the sublinearity of F ), j=1

which on using (1) and F (¯ x, y¯; 0) = 0 gives k  i=1

 k 

¯ i fi (¯ λ y) +

u¯j ρˆj

d2 (¯ x, y¯).

m k m    1 ¯ ¯ u¯j gj (¯ x)− λi fi (¯ y )− u¯j gj (¯ y )+ p¯T ∇2 λi fi (¯ x)+ 2 j=1 i=1 j=1

>



k 

¯ i ρi + λ

i=1

m  j=1

i=1



m 



u¯j gj (¯ y ) p¯

j=1

This inequality along with hypothesis (iii), u¯ ≧ 0 and g(¯ x) ≦ 0 yields  k  k k m m      1 ¯ i fi (¯ ¯ i fi (¯ ¯ i fi (¯ λ x) − λ y) − u¯j gj (¯ y ) + p¯T ∇2 λ y) + u¯j gj (¯ y ) p¯ > 0, 2 i=1 i=1 j=1 i=1 j=1 a contradiction to hypothesis (i). Hence, x¯ = y¯.

2

4. Mond-Weir type duality In this section, we present the following Mond-Weir [13] type dual associated to (P): (MD) Maximize (f1 (y) − 12 pT ∇2 f1 (y)p, . . . , fk (y) − 12 pT ∇2 fk (y)p)

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subject to k m   2 (∇λi fi (y) + ∇ λi fi (y)p) + (∇uj gj (y) + ∇2 uj gj (y)p) = 0, i=1

(8)

j=1

1 uj gj (y) − pT ∇2 uj gj (y)p ≧ 0, j ∈ M, 2 λ ≧ 0, k 

(9) (10)

λi = 1,

(11)

u ≧ 0.

(12)

i=1

Let U be the set of all feasible solutions of (MD). In this section and in Section 5, f λ denotes the vector (λ1 f1 , λ2 f2 , . . . , λk fk ) and g u denotes the vector (u1 g1 , u2 g2 , . . . , um gm ). Theorem 4 (Weak duality). Suppose that for all x ∈ S and (y, u, λ, p) ∈ U, (i) f λ is second order (F, α ¯ , ρ¯, d)-V-pseudoconvex at y, and g u is second order (F, α ˜ , ρ˜, d)-V-quasiconvex at y; and (ii) ρ¯ + ρ˜ ≧ 0. Then

1 fi (x) < fi (y) − pT ∇2 fi (y)p, i ∈ K. 2 Proof. Since x ∈ S and (y, u, λ, p) ∈ U, we have

(13)

1 uj gj (x) ≦ 0 ≦ uj gj (y) − pT ∇2 uj gj (y)p, j ∈ M. 2 ˜ j (x, y) > 0, j ∈ M , we get As α m 

m 

m

 1 α ˜ j (x, y)uj gj (x) ≦ α ˜ j (x, y)uj gj (y) − pT ∇2 α ˜ j (x, y)uj gj (y)p. 2 j=1 j=1 j=1

Using second order (F, α ˜ , ρ˜, d)-V-quasiconvexity of g u at y, we obtain   m  F x, y; (∇uj gj (y) + ∇2 uj gj (y)p) + ρ˜d2 (x, y) ≦ 0.

(14)

j=1

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The equation (8) along with the sublinearity of F gives     k m   F x, y; (∇λi fi (y) + ∇2 λi fi (y)p) + F x, y; (∇uj gj (y) + ∇2 uj gj (y)p) i=1

≧F



j=1

k m   2 (∇uj gj (y) + ∇2 uj gj (y)p) x, y; (∇λi fi (y) + ∇ λi fi (y)p) + i=1

j=1



= 0. (15)

Inequalities (14), (15) and hypothesis (ii) imply   k  F x, y; (∇λi fi (y) + ∇2 λi fi (y)p) + ρ¯d2 (x, y) ≧ 0, i=1

which by second order (F, α ¯ , ρ¯, d)-V-pseudoconvexity of f λ at y yields k 

α ¯ i (x, y)λi fi (x) ≧

i=1

k  i=1

k

 1 α ¯ i (x, y)λi fi (y) − pT ∇2 α ¯ i (x, y)λi fi (y)p. 2 i=1

(16)

Now suppose contrary to (13), i.e., 1 fi (x) < fi (y) − pT ∇2 fi (y)p, i ∈ K. 2 Using λ ≧ 0,

k


¯ i (x, y) > 0, i ∈ K, we get λi = 1, and α

i=1 k 

α ¯ i (x, y)λi fi (x)
0, j ∈ M , it follows that m


α ˜ j (¯ x, y¯)¯ uj gj (¯ x) ≦

j=1

m


j=1

α ˜ j (¯ x, y¯)¯ uj gj (¯ y ) − 12 p¯T ∇2

m


α ˜ j (¯ x, y¯)¯ uj gj (¯ y )¯ p.

j=1

On using second order (F, α ˜ , ρ˜, d)-V-quasiconvexity of g u¯ at y¯, we get F (¯ x, y¯;

m


(∇¯ uj gj (¯ y ) + ∇2 u¯j gj (¯ y )¯ p)) + ρ˜d2 (¯ x, y¯) ≦ 0.

(17)

j=1

Now from (8), (17), hypothesis (iii) and the sublinearity of F , we obtain F (¯ x, y¯;

k


¯ i fi (¯ ¯ i fi (¯ (∇λ y ) + ∇2 λ y )¯ p)) + ρ¯d2 (¯ x, y¯) ≧ 0.

i=1 ¯

The strict second order (F, α ¯ , ρ¯, d)-V-pseudoconvexity of f λ at y¯ yields k


i=1

k k

¯ i fi (¯ ¯ i fi (¯ ¯ i fi (¯ x) > α ¯ i (¯ x, y¯)λ y ) − 12 p¯T ∇2 α ¯ i (¯ x, y¯)λ y )¯ p. α ¯ i (¯ x, y¯)λ i=1

i=1

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Since α ¯ i (¯ x, y¯) = δ(¯ x, y¯), i ∈ K, we have k  i=1

¯ i fi (¯ λ x) >

k  i=1

k

 1 ¯ i fi (¯ ¯ i fi (¯ λ y )¯ λ y ) − p¯T ∇2 p, 2 i=1

a contradiction to hypothesis (i). Hence, x¯ = y¯.

2

5. General Mond-Weir type duality For (P), we present the following second order general Mond-Weir type dual: 

uj gj (y) − 12 pT ∇2 (f1 (y) + uj gj (y))p, (GD) Maximize f1 (y) + j∈J◦ j∈J◦ 

. . . , fk (y) + uj gj (y) − 12 pT ∇2 (fk (y) + uj gj (y))p j∈J◦

j∈J◦

subject to

k m   (∇λi fi (y) + ∇2 λi fi (y)p) + (∇uj gj (y) + ∇2 uj gj (y)p) = 0, i=1

(18)

j=1

1 uj gj (y) − pT ∇2 uj gj (y)p ≧ 0, j ∈ Jβ , β = 1, 2, . . . , r, 2 λ ≧ 0, k 

(19) (20)

λi = 1,

(21)

u ≧ 0,

(22)

i=1

where Jβ ⊆ M, β = 0, 1, 2, . . . , r with

r

Jβ = M and Jβ ∩ Jγ = ∅, if β = γ.

β=0

Remark 3. Let Jβ = ∅. Then the dual (GD) reduces to Mangasarian type dual considered in Section 3. If J◦ = ∅, then (GD) becomes Mond-Weir type dual discussed in Section 4. Let Y be the set of all feasible solutions of (GD). Theorem 8 (Weak duality). Suppose that for all x ∈ S and (y, u, λ, p) ∈ Y , (i) (λi fi +uJ◦ gJ◦ )i∈K is second order (F, α ¯ , ρ¯, d)-V-pseudoconvex at y, and (uj gj )j∈Jβ , β = 1, 2, . . . , r is second order (F, α ˜ , ρ˜, d)-V-quasiconvex at y; and (ii) ρ¯ +

r


ρ˜β ≧ 0.

β=1

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Then 

fi (x) < fi (y) +

j∈J◦

 1 uj gj (y) − pT ∇2 {fi (y) + uj gj (y)}p, i ∈ K. 2 j∈J

(23)

◦

Proof. Since x ∈ S and (y, u, λ, p) ∈ Y , we have 1 uj gj (x) ≦ 0 ≦ uj gj (y) − pT ∇2 uj gj (y)p, j ∈ Jβ , β = 1, 2, . . . , r. 2 As α ˜ j (x, y) > 0, j ∈ Jβ , we get


α ˜ j (x, y)uj gj (x) ≦

j∈Jβ




j∈Jβ

α ˜ j (x, y)uj gj (y)− 12 pT ∇2




α ˜ j (x, y)uj gj (y)p, β = 1, 2, . . . , r.

j∈Jβ

The second order (F, α ˜ , ρ˜, d)-V-quasiconvexity of (uj gj )j∈Jβ , β = 1, 2, . . . , r at y implies  
F x, y; (∇uj gj (y) + ∇2 uj gj (y)p) + ρ˜β d2 (x, y) ≦ 0, β = 1, 2, . . . , r. (24) j∈Jβ

Inequality (24) along with (18), hypothesis (ii) and the sublinearity of F yields   k

F x, y; (∇λi fi (y) + ∇2 λi fi (y)p) + (∇uj gj (y) + ∇2 uj gj (y)p) i=1

j∈J◦

+ ρ¯d2 (x, y) ≧ 0. ¯ , ρ¯, d)-V-pseudoconvexity of (λi fi + uJ◦ gJ◦ )i∈K at y, we On using second order (F, α obtain     k k



α ¯ i (x, y) λi fi (x) + uj gj (x) ≧ α ¯ i (x, y) λi fi (y) + uj gj (y) i=1 i=1 j∈J◦ j∈J◦   k

− 12 pT ∇2 α ¯ i (x, y)(λi fi (y) + uj gj (y)) p. (25) i=1

j∈J◦

Now, suppose contrary to the result that (23) cannot hold, then by u ≧ 0 and g(x) ≦ 0, it follows that fi (x) +




uj gj (x) < fi (y) +

j∈J◦




j∈J◦

uj gj (y) − 12 pT ∇2 {fi (y) +




uj gj (y)}p, i ∈ K.

j∈J◦

Using (20), (21), α ¯ i (x, y) > 0, i ∈ K and summing over i, we get k


i=1

α ¯ i (x, y)(λi fi (x)+




j∈J◦

uj gj (x))
0, j ∈ Jβ , it follows that


α ˜ j (¯ x, y¯)¯ uj gj (¯ x) ≦

j∈Jβ




j∈Jβ

α ˜ j (¯ x, y¯)¯ uj gj (¯ y )− 12 p¯T ∇2




α ˜ j (¯ x, y¯)¯ uj gj (¯ y )¯ p, β = 1, 2, . . . , r.

j∈Jβ

The second order (F, α ˜ , ρ˜, d)-V-quasiconvexity of (¯ uj gj )j∈Jβ , β = 1, 2, . . . , r at y¯ gives  
(∇¯ uj gj (¯ y ) + ∇2 u¯j gj (¯ y )¯ p) + ρ˜β d2 (¯ x, y¯) ≦ 0. (26) F x¯, y¯; j∈Jβ

The inequality (26) along with (18), hypothesis (iii) and the sublinearity of F yields   k

¯ i fi (¯ ¯ i fi (¯ y )¯ p) + (∇¯ uj gj (¯ y ) + ∇2 u¯j gj (¯ y )¯ p) F x¯, y¯; (∇λ y ) + ∇2 λ i=1

j∈J◦

+ ρ¯d2 (¯ x, y¯) ≧ 0.

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¯ i fi + u¯T gJ◦ )i∈K at On using strict second order (F, α ¯ , ρ¯, d)-V-pseudoconvexity of (λ J◦ y¯, we obtain     k k



¯ i fi (¯ ¯ i fi (¯ α ¯ i (¯ x, y¯) λ x) + u¯j gj (¯ x) > α ¯ i (¯ x, y¯) λ y) + u¯j gj (¯ y) i=1

i=1

j∈J◦

− 12 p¯T ∇2



j∈J◦

k


¯ i fi (¯ α ¯ i (¯ x, y¯)(λ y) +

i=1




j∈J◦



u¯j gj (¯ y )) p¯,

which by the feasibility of x¯ for (P) gives   k k


¯ i fi (¯ ¯ i fi (¯ x) > α ¯ i (¯ x, y¯) λ y) + u¯j gj (¯ y) α ¯ i (¯ x, y¯)λ i=1 i=1 j∈J◦   k

¯ i fi (¯ − 12 p¯T ∇2 α ¯ i (¯ x, y¯)(λ y) + u¯j gj (¯ y )) p¯. i=1

j∈J◦

Since α ¯ i (¯ x, y¯) = δ(¯ x, y¯), i ∈ K, we obtain

k k


¯ i fi (¯ ¯ i fi (¯ u¯j gj (¯ y ) − 12 p¯T ∇2 λ x) > λ y) +

i=1

i=1

j∈J◦

a contradiction to hypothesis (i). Hence, x¯ = y¯.



 k

¯ i fi (¯ λ y) + u¯j gj (¯ y ) p¯,

i=1

j∈J◦

2

References [1] R. P. Agarwal, I. Ahmad, Z. Husain, A. Jayswal, Optimality and duality in nonsmooth multiobjective optimization involving V- type I invex functions, Journal of Inequalities and Applications, Vol. 2010, Article ID 898626, 14 pages. [2] B. Aghezzaf, Second order mixed type duality in multiobjective programming problems, Journal of Mathematical Analysis and Applications, 285 (2003) 97106. [3] I. Ahmad, Z. Husain, Second order (F, α, ρ, d)-convexity and duality in multiobjective programming, Information Sciences, 176 (2006) 3094-3103. [4] B.D. Craven, Invex functions and constrained local minima, Bulletin of the Australian Mathematical Society, 24 (1981) 357-366. [5] T.R. Gulati, D. Agarwal, On Huard type second-order converse duality in nonlinear programming, Applied Mathematics Letters, 20 (2007) 1057-1063. [6] M. Hachimi, B. Aghezzaf, Second order duality in multiobjective programming involving generalized type-I functions, Numerical Functional Analysis and Optimization, 25 (2005) 725-736. 17 1211

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[7] M.A. Hanson, On sufficiency of the Kuhn-Tucker conditions, Journal of Mathematical Analysis and Applications , 80 (1981) 545-550. [8] M.A. Hanson, B. Mond, Further generalizations of convexity in mathematical programming, Journal of Information and Optimization Sciences, 3 (1982) 2532. [9] V. Jeyakumar, B. Mond, On generalized convex mathematical programming, Journal of the Australian Mathematical Society, Series B, 34 (1992) 43-53. [10] R.N. Kaul, S. Kaur, Optimality criteria in nonlinear programming involving nonconvex functions, Journal of Mathematical Analysis and Applications, 105 (1985) 104-112. [11] O.L. Mangasarian, Second and higher order duality in nonlinear programming, Journal of Mathematical Analysis and Applications, 51 (1975) 607-620. [12] K.M. Miettinen, Nonlinear Multiobjective Optimization, Kluwer Academic, Boston, MA 1999. [13] B. Mond, Second order duality for nonlinear programs, Opsearch, 11 (1974) 90-99. [14] B. Mond, T. Weir, Generalized concavity and duality, In: Generalized Concavity in Optimization and Economics, S. Schaible, W.T. Ziemba (Eds.), Academic Press, New York (1981) 263-279. [15] B. Mond, J. Zhang, Duality for multiobjective programming involving second order V -invex functions, In: Proceedings of the Optimization Miniconference, B.M. Glower, V. Jeyakumar (Eds.), University of New South Wales, Sydney, Australia (1995) 89-100. [16] V. Preda, On efficiency and duality for multiobjective programs, Journal of Mathematical Analysis and Applications, 166 (1992) 365-377. [17] J.P. Vial, Strong and weak convexity of sets and functions, Mathematics of Operations Research, 8 (1983) 231-259. [18] X.M. Yang, X.Q. Yang, K.L. Teo, Huard type second-order converse duality for nonlinear programming, Applied Mathematics Letters, 18 (2005) 205-208. [19] J. Zhang, B. Mond, Second order duality for multiobjective nonlinear programming involving generalized convexity, In: Proceedings of the Optimization Miniconference III, B.M. Glower, B.D. Craven, D. Ralph (Eds.), University of Ballarat (1997) 79-95.

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Meraj Ali Khan Department of Mathematics, University of Tabuk, Tabuk Kingdom of Saudi Arabia E-mail:[email protected] Falleh R. Al Solamy Department of Mathematics King Abdulaziz University P.O. Box 80015, Jeddah 21589, Kingdom of Saudi Arabia E-mail:[email protected]

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ON A FIFTH-ORDER DIFFERENCE EQUATION ´ ∗ , JOSEF DIBL´IK, BRATISLAV IRICANIN, ˇ ˇ SMARDA ˇ STEVO STEVIC AND ZDENEK

Abstract. We investigate the following difference equation xn−3 xn−4 xn−5 xn = , n ∈ N0 , xn−1 xn−2 (an + bn xn−3 xn−4 xn−5 ) where (an )n∈N0 and (bn )n∈N0 are two real sequences and the initial values x−5 , . . . , x−1 are real numbers. The case when the sequences (an )n∈N0 and (bn )n∈N0 are constant is thoroughly studied. Our results considerably extend some results in the recent literature.

1. Introduction There has been a great recent interest in nonlinear difference equations and systems of difference equations (see, for example, [1]-[6], [8]-[14], [18]-[43] and the references therein), and, among them, some renewed interest in the difference equations and systems which can be solved in closed form (see, for example, [1]-[4], [6], [8], [19], [22], [23], [26], [27], [29]-[37], [39]-[43] and the related references therein). For some classical methods for solving difference equations and systems see, for example, [7], [16] and [17]. Many of the papers in the theory deal with difference equations and systems which can be regarded as perturbations of solvable ones (see, for example, [25] and [38]), so that their solutions are frequently compared with the solutions of the solvable ones, or are connected with some other solvable equations as it is the case in [21], [25] and [38]. This fact also shows the importance of solvable difference equations and systems. Among the papers in the area there are some which present formulas of some particular difference equations and/or systems of difference equations which are almost always proved by induction, but do not give any theoretical explanation related to the presented formulas and how the equations/systems can be solved. Paper [22] by S. Stevi´c, in which a natural explanation is given for the formula presented in [8], motivated numerous authors to re-attract their interest in difference equations which can be solved in closed form. Various other explanations and extensions of some results in the literature can be also found in papers [26], [36] and [41]. It is said that the difference equation xn = f (xn−1 , . . . , xn−k ),

n ∈ N0 ,

where k ∈ N, is solvable in closed form if every solution can be written in terms of the initial values x−k , . . . , x−1 and index n only. 2000 Mathematics Subject Classification. Primary 39A20. Key words and phrases. Difference equation, equation solved in closed form, asymptotic behavior. ∗ Corresponding author. 1

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2

´ JOSEF DIBL´IK, BRATISLAV IRICANIN, ˇ ˇ SMARDA ˇ STEVO STEVIC, AND ZDENEK

Paper [43] is one of the papers of above mentioned type. Namely, formulas for solutions of the next four difference equations xn−2 xn−3 xn−4 xn+1 = , n ∈ N0 , (1) xn xn−1 (1 + xn−2 xn−3 xn−4 ) xn−2 xn−3 xn−4 xn+1 = , n ∈ N0 , (2) xn xn−1 (1 − xn−2 xn−3 xn−4 ) xn−2 xn−3 xn−4 xn+1 = , n ∈ N0 , (3) xn xn−1 (−1 + xn−2 xn−3 xn−4 ) xn−2 xn−3 xn−4 xn+1 = , n ∈ N0 , (4) xn xn−1 (−1 − xn−2 xn−3 xn−4 ) are presented in [43] and for some of them are given sketches of the inductive proofs, but there are no theoretical explanations for the formulas. A natural problem is to extend the results in [43] and give theoretical explanations for formulas presented therein. Here, we will study the next difference equation xn−3 xn−4 xn−5 , n ∈ N0 , (5) xn = xn−1 xn−2 (an + bn xn−3 xn−4 xn−5 ) where (an )n∈N0 and (bn )n∈N0 are real sequences and the initial values x−5 , . . . , x−1 are real numbers, which is a natural extension of equations (1)-(4) (we shifted the indices backward for one, since the equation in this form as well as the result might look clearer). To deal with equation (5) we essentially use the idea in [22], later exploited in numerous papers, where a suitable change of variables is used so that the equation therein is transformed into a solvable difference equation (see, for example, [1], [2], [4], [19], [27], [29]-[31], [33]-[37], [39]-[42]). Solution (xn )n≥−s , of the difference equation xn = f (xn−1 , . . . , xn−s ),

n ∈ N0 ,

(6)

where f : R → R, s ∈ N, is called eventually periodic with period p, if there is an n1 ≥ −s such that xn+p = xn , for n ≥ n1 . It is called periodic with period p, if n1 = −s. For some results in this area see, e.g. [5, 9, 10, 11, 12, 13, 14, 15, 18, 20, 24, 28] and the references therein. Throughout the paper we use the following standard conventions s

l ∑

aj = 0,

when k > l,

j=k

and

k−1 ∏

bj = 1,

j=k

where k and l are integers. 2. Formulas for well-defined solutions of equation (5) Assume that (xn )n≥−5 is a solution of equation (5). If x−5 = 0 or x−4 = 0 or x−3 = 0 and x−2 ̸= 0 ̸= x−1 , then from (5) we see that x0 is or not defined (if a0 = 0) or x0 = 0, and consequently x1 is not defined. If x−2 = 0 or x−1 = 0, then

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ON A FIFTH-ORDER DIFFERENCE EQUATION

3

from (5) we see that x0 is not defined. This means that if one of the initial values x−j , j ∈ {1, . . . , 5} is equal to zero, then such a solution is not defined. Now assume that xn0 = 0 for some n0 ∈ N0 , that xi are defined when −5 ≤ i ≤ n0 , and that n0 is the smallest index for which a member of the solution is equal to zero. Then, from (5) we see that xn0 −3 = 0 or xn0 −4 = 0 or xn0 −5 = 0, which along with (5) would imply respectively that xn0 −2 if n0 ≥ 2 is not defined, or xn0 −3 if n0 ≥ 3 is not defined, or xn0 −4 if n0 ≥ 4 is not defined, which would be a contradiction with the fact that xn are defined for −5 ≤ n ≤ n0 . If n0 = 2, then we have that x−1 = 0 or x−2 = 0 or x−3 = 0, if n0 = 1, then we have that x−2 = 0 or x−3 = 0 or x−4 = 0, while if n0 = 0, then we have that x−3 = 0 or x−4 = 0 or x−5 = 0. So, in these three cases we have that at least one of the initial values is equal to zero, and consequently by previous considerations such solutions are not defined. If n0 = 3, then from (5) we have that x0 = 0 or x−1 = 0 or x−2 = 0. If x−1 = 0 or x−2 = 0, then x0 is not defined, while the case x0 = 0 has been previously considered. If n0 = 4, then from (5) we have that x1 = 0 or x0 = 0 or x−1 = 0. If x−1 = 0, then x0 is not defined, while the cases x1 = 0 or x0 = 0 have been previously considered. Thus, according to all above mentioned such solutions are not defined. Hence of some interest are solutions for which x−j ̸= 0,

j ∈ {1, . . . , 5},

since for them it must be xn ̸= 0,

n ≥ −5.

(7)

Now assume that (xn )n≥−5 is a well-defined solution of equation (5). By previous considerations we have that (7) holds, so that for every well-defined solution we can use the following change of variables 1 yn = , n ≥ −3, (8) xn xn−1 xn−2 which transforms equation (5) into the following linear third-order difference equation yn = an yn−3 + bn ,

n ∈ N0 .

(9)

Now note that every integer n ≥ −3 can be written in the following form n = 3m + i, for some m ≥ −1 and i ∈ {0, 1, 2}. Hence, equation (9) can be written in the next form y3m+i = a3m+i y3(m−1)+i + b3m+i ,

m ∈ N0 ,

(10)

where i ∈ {0, 1, 2}. This means that the sequences (y3m+i )m≥−1 , i ∈ {0, 1, 2}, are solutions of the following three linear first order difference equations zm = a3m+i zm−1 + b3m+i ,

m ∈ N0 ,

(11)

i ∈ {0, 1, 2}. The linear first order difference equation is solved in closed form and by using well-known formula for its solution we have that m m m ∏ ∑ ∏ y3m+i = yi−3 a3j+i + b3l+i a3j+i , m ∈ N0 , (12) j=0

l=0

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i ∈ {0, 1, 2}. This formula can be easily obtained, for example, if we multiply the next equalities

by

y3l+i = a3l+i y3(l−1)+i + b3l+i ,

∏m

a3j+i , l = 0, 1, . . . , m, and sum up such obtained equalities ([17]). Now we find formulas for well-defined solutions of equation (5). From (8) with n = 3m + i, we have that j=l+1

x3m+i =

1

y3m+i x3m+i−1 x3m+i−2 y3m+i−1 x3m+i−3 = x3(m−1)+i . = y3m+i x3m+i−1 x3m+i−2 x3m+i−3 y3m+i

(13)

By repeating use of (13) we obtain x3m = x−3

m ∏ y3s−1 , y3s s=0

m ≥ −1,

(14)

and x3m+i = xi−6

m ∏ y3s+i−1 , y3s+i s=−1

m ≥ −1,

(15)

for i ∈ {1, 2}. Using formula (12) in (14) and (15) we obtain formulas for general solution of equation (5) x3m

m ∏ y3(s−1)+2 = x−3 y3s s=0 ∑s−1 ∏s−1 ∏s−1 m ∏ y−1 j=0 a3j+2 + l=0 b3l+2 j=l+1 a3j+2 ∏s ∑s ∏s = x−3 y−3 j=0 a3j + l=0 b3l j=l+1 a3j s=0 ∑s−1 ∏s−1 ∏s−1 m ∏ (x−1 x−2 x−3 )−1 j=0 a3j+2 + l=0 b3l+2 j=l+1 a3j+2 ∏s ∏s ∑s , = x−3 (x−3 x−4 x−5 )−1 j=0 a3j + l=0 b3l j=l+1 a3j s=0

(16)

m ≥ −1, and m ∏ y3s+i−1 y3s+i s=−1 ∏ ∑ ∏ m ∏ yi−4 sj=0 a3j+i−1 + sl=0 b3l+i−1 sj=l+1 a3j+i−1 ∏s ∑s ∏s = xi−6 yi−3 j=0 a3j+i + l=0 b3l+i j=l+1 a3j+i s=−1 ∏s ∑s ∏s m ∏ (xi−4 xi−5 xi−6 )−1 j=0 a3j+i−1 + l=0 b3l+i−1 j=l+1 a3j+i−1 ∏s ∑s ∏s = xi−6 , (xi−3 xi−4 xi−5 )−1 j=0 a3j+i + l=0 b3l+i j=l+1 a3j+i s=−1

x3m+i = xi−6

(17) for m ≥ −1 and for i ∈ {1, 2}.

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3. Case when sequences an and bn are constant In this section we consider the case when the sequences (an )n∈N0 and (bn )n∈N0 are constant, that is, when an = a,

n ∈ N0 .

bn = b,

In this case equation (5) becomes xn−3 xn−4 xn−5 xn = , xn−1 xn−2 (a + bxn−3 xn−4 xn−5 )

n ∈ N0 .

(18)

By using formulas (16) and (17) in this case we obtain

x3m = =

∑s−1 ∏s−1 b j=l+1 a j=0 a + ∏s ∑l=0 ∏s x−3 s −1 (x−3 x−4 x−5 ) j=0 a + l=0 b j=l+1 a s=0 ∑ m s−1 ∏ (x−1 x−2 x−3 )−1 as + b as−l−1 l=0 ∑s x−3 , −1 s+1 (x−3 x−4 x−5 ) a + b l=0 as−l s=0 m ∏ (x−1 x−2 x−3 )−1

∏s−1

(19)

m ≥ −1, and x3m+i = =

∏s ∑s l=0 b j=0 a + ∏s ∑s ∏j=l+1 xi−6 s −1 (x x x ) a + b i−3 i−4 i−5 j=0 l=0 j=l+1 s=−1 ∑ m ∏ (xi−4 xi−5 xi−6 )−1 as+1 + b s as−l ∑sl=0 , xi−6 (xi−3 xi−4 xi−5 )−1 as+1 + b l=0 as−l s=−1 m ∏ (xi−4 xi−5 xi−6 )−1

∏s

a a (20)

for m ≥ −1 and for i ∈ {1, 2}. We have now two cases. 3.1. Case a ̸= 1. In this case formulas (19) and (20) become ∑s−1 m ∏ (x−1 x−2 x−3 )−1 as + b l=0 as−l−1 ∑s x3m = x−3 (x−3 x−4 x−5 )−1 as+1 + b l=0 as−l s=0 = x−3

m ∏

(x−1 x−2 x−3 )−1 as (1 − a) + b(1 − as ) (x−3 x−4 x−5 )−1 as+1 (1 − a) + b(1 − as+1 ) s=0

m ∏ as ((1 − a)(x−1 x−2 x−3 )−1 − b) + b = x−3 , as+1 ((1 − a)(x−3 x−4 x−5 )−1 − b) + b s=0

(21)

m ≥ −1, and x3m+i = xi−6

∑s m ∏ (xi−4 xi−5 xi−6 )−1 as+1 + b l=0 as−l ∑s (xi−3 xi−4 xi−5 )−1 as+1 + b l=0 as−l s=−1

= xi−6

m ∏ (xi−4 xi−5 xi−6 )−1 as+1 (1 − a) + b(1 − as+1 ) (xi−3 xi−4 xi−5 )−1 as+1 (1 − a) + b(1 − as+1 ) s=−1

= xi−6

m ∏ as+1 ((1 − a)(xi−4 xi−5 xi−6 )−1 − b) + b , as+1 ((1 − a)(xi−3 xi−4 xi−5 )−1 − b) + b s=−1

(22)

for m ≥ −1 and for i ∈ {1, 2}.

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´ JOSEF DIBL´IK, BRATISLAV IRICANIN, ˇ ˇ SMARDA ˇ STEVO STEVIC, AND ZDENEK

3.2. Case a = 1. In this case formulas (19) and (20) become m ∏ (x−1 x−2 x−3 )−1 + bs x3m = x−3 , (x−3 x−4 x−5 )−1 + b(s + 1) s=0

(23)

m ≥ −1, and x3m+i = xi−6

m ∏ (xi−4 xi−5 xi−6 )−1 + b(s + 1) , (xi−3 xi−4 xi−5 )−1 + b(s + 1) s=−1

(24)

for m ≥ −1 and for i ∈ {1, 2}. 3.3. Asymptotic behavior of solutions of equation (18). Here we study the asymptotic behavior of well-defined solutions of equation (18). Prior to stating and proving our results we introduce some quantities which will be used in the statements of the results. Set (1 − a)(x−1 x−2 x−3 )−1 − b , L0 = a((1 − a)(x−3 x−4 x−5 )−1 − b) (1 − a)(xi−4 xi−5 xi−6 )−1 − b Li = , (1 − a)(xi−3 xi−4 xi−5 )−1 − b where i = 1, 2, and set 1 1 K0 = − − 1, x−1 x−2 x−3 b x−3 x−4 x−5 b ( ) 1 1 1 Ki = − , b xi−4 xi−5 xi−6 xi−3 xi−4 xi−5 where i = 1, 2. Our first result considers the case |a| > 1, b ̸= 0. Theorem 1. Assume that |a| > 1, b ̸= 0, and (xn )n≥−5 is a well-defined solution of equation (18). Then the following statements are true. (a) If |L0 | < 1, then x3m → 0 as m → +∞. (b) If |L0 | > 1, then |x3m | → +∞ as m → +∞. (c) If L0 = 1, then the sequence (x3m )m≥−1 is constant. (d) If L0 = −1, then the sequences (x6m )m∈N0 and (x6m+3 )m≥−1 are convergent. (e) If |Li | < 1, for some i ∈ {1, 2}, then x3m+i → 0 as m → +∞. (f ) If |Li | > 1, for some i ∈ {1, 2}, then |x3m+i | → +∞ as m → +∞. (g) If Li = 1, for some i ∈ {1, 2}, then the sequence (x3m+i )m≥−2 is constant. (h) If Li = −1, for some i ∈ {1, 2}, then the sequences (x6m+i )m≥−1 and (x6m+3+i )m≥−1 are convergent. Proof. (a), (b) Let ps =

as ((1 − a)(x−1 x−2 x−3 )−1 − b) + b . as+1 ((1 − a)(x−3 x−4 x−5 )−1 − b) + b

(25)

Then we have lim ps = lim

s→+∞

s→+∞

((1 − a)(x−1 x−2 x−3 )−1 − b)/a + (1 − a)(x−3 x−4 x−5 )−1 − b +

1219

b as+1 b

= L0 .

(26)

as+1

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ON A FIFTH-ORDER DIFFERENCE EQUATION

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From (21) and (26), these two statements easily follow. (c) In this case, we have that ps =

((1 − a)(x−1 x−2 x−3 )−1 − b)/a + (1 − a)(x−3 x−4 x−5 )−1 − b +

b as+1 b

= 1,

s ∈ N0 .

From (21) and (27) the result easily follows. (d) Since L0 = −1, and by using the asymptotic relation 1 = 1 − x + O(x2 ), 1+x when x is close to the origin, we have that ps = −

(1 − a)(x−3 x−4 x−5 )−1 − b −

(27)

as+1

(28)

b as+1 b as+1

(1 − a)(x−3 x−4 x−5 )−1 − b + ) ( b = − 1 − s+1 a ((1 − a)(x−3 x−4 x−5 )−1 − b) ( ( 1 )) b × 1 − s+1 + O a ((1 − a)(x−3 x−4 x−5 )−1 − b) a2s ( ) (1) =− 1+O s , a

(29)

for large enough s. From (21), (29), the assumption |a| > 1, and by a known criterion for the convergence of products the result easily follows. (e), (f ) Let qsi =

as+1 ((1 − a)(xi−4 xi−5 xi−6 )−1 − b) + b , as+1 ((1 − a)(xi−3 xi−4 xi−5 )−1 − b) + b

i = 1, 2.

(30)

Then we have lim qsi = lim

s→+∞

s→+∞

(1 − a)(xi−4 xi−5 xi−6 )−1 − b + (1 − a)(xi−3 xi−4 xi−5 )−1 − b +

b as+1 b as+1

= Li ,

i = 1, 2.

(31)

From (22) and (31), these two statements easily follow. (g) In this case we have that qsi =

(1 − a)(xi−4 xi−5 xi−6 )−1 − b + (1 − a)(xi−3 xi−4 xi−5 )−1 − b +

b as+1 b as+1

= 1,

s ∈ N0 .

(32)

From (22) and (32) the result easily follows. (h) Since Li = −1 and by using (28), we have that qsi = −

(1 − a)(xi−3 xi−4 xi−5 )−1 − b −

b as+1 b as+1

(1 − a)(xi−3 xi−4 xi−5 )−1 − b + ( ) b = − 1 − s+1 a ((1 − a)(xi−3 xi−4 xi−5 )−1 − b) ( ( 1 )) b × 1 − s+1 + O a ((1 − a)(xi−3 xi−4 xi−5 )−1 − b) a2s ( ) (1) =− 1+O s , a

(33)

for large enough s. From (22), (33), the assumption |a| > 1, and by a known criterion for the convergence of products the result easily follows. 

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Now we consider the case |a| < 1, b ̸= 0. Theorem 2. Assume that |a| < 1, b = ̸ 0, and (xn )n≥−5 is a well-defined solution of equation (18). Then the sequences (x3m+i )m≥−1 , i ∈ {0, 1, 2}, are convergent. Proof. From (25) and (28), we have ps =

1 + as ((1 − a)(x−1 x−2 x−3 )−1 − b)/b 1 + as+1 ((1 − a)(x−3 x−4 x−5 )−1 − b)/b

=(1 + as ((1 − a)(x−1 x−2 x−3 )−1 − b)/b) × (1 − as+1 ((1 − a)(x−3 x−4 x−5 )−1 − b)/b + O(a2s )) =1 + O(as ),

(34)

for large enough s, while from (30) and by using (28), we have qsi =

1 + as+1 ((1 − a)(xi−4 xi−5 xi−6 )−1 − b)/b 1 + as+1 ((1 − a)(xi−3 xi−4 xi−5 )−1 − b)/b

=(1 + as+1 ((1 − a)(xi−4 xi−5 xi−6 )−1 − b)/b) × (1 − as+1 ((1 − a)(xi−3 xi−4 xi−5 )−1 − b)/b + O(a2s )) =1 + O(as ),

(35)

for large enough s and i ∈ {1, 2}. From (21), (22), (34), (35), the assumption |a| < 1, and by a known criterion for the convergence of products we have that the sequences (x3m+i )m≥−1 , i ∈ {0, 1, 2}, are convergent.  Now we consider the case a = 1, b ̸= 0. Theorem 3. Assume that a = 1, b ̸= 0, and (xn )n≥−5 is a well-defined solution of equation (18). Then the following statements are true. (a) If a = 1 and K0 < 0, then x3m → 0 as m → +∞. (b) If a = 1 and K0 > 0, then |x3m | → +∞ as m → +∞. (c) If a = 1 and K0 = 0, then the sequence (x3m )m≥−1 is constant. (d) If a = 1 and Ki < 0, for some i ∈ {1, 2}, then x3m+i → 0 as m → +∞. (e) If a = 1 and Ki > 0, for some i ∈ {1, 2}, then |x3m+i | → +∞ as m → +∞. (f ) If a = 1 and Ki = 0, for some i ∈ {1, 2}, then the sequence (x3m+i )m≥−1 is constant. Proof. (a), (b) Let rs =

(x−1 x−2 x−3 )−1 + bs . (x−3 x−4 x−5 )−1 + b + bs

(36)

From (36) and by using (28) we have that 1 bx−1 x−2 x−3 s (bx−3 x−4 x−5 )−1 +1 s

1+ rs =

1+ ( = 1+

1

bx−1 x−2 x−3 s (1) K0 =1 + +O 2 , s s

)(

(1) (bx−3 x−4 x−5 )−1 + 1 1− +O 2 s s

)

(37)

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for large enough s. From (23) and (37), and known criteria for the convergence of products, these two statements easily follow. (c) Note that in this case rs = 1, s ∈ N0 , from which the result follows. (d), (e) Let rsi =

(xi−4 xi−5 xi−6 )−1 + b + bs , (xi−3 xi−4 xi−5 )−1 + b + bs

i = 1, 2.

(38)

From (38) and by using (28) we have that rsi

=

1+

(bxi−4 xi−5 xi−6 )−1 +1 s (bxi−3 xi−4 xi−5 )−1 +1 s (bxi−4 xi−5 xi−6 )−1

1+ ( = 1+

+1

)(

(1) (bxi−3 xi−4 xi−5 )−1 + 1 1− +O 2 s s

)

s (1) Ki +O 2 , (39) =1 + s s for large enough s and i ∈ {1, 2}. From (24) and (39), and known criteria for the convergence of products, these two statements easily follow. (f ) Note that in this case rsi = 1, i = 1, 2, s ∈ N0 , from which the result follows.  If a = −1 and b ̸= 0, then from (21) and (22), we have that x3m = x−3

m ∏ (−1)s (2(x−1 x−2 x−3 )−1 − b) + b , (−1)s+1 (2(x−3 x−4 x−5 )−1 − b) + b s=0

m ≥ −1,

(40)

and x3m+i = xi−6

m ∏ (−1)s+1 (2(xi−4 xi−5 xi−6 )−1 − b) + b , (−1)s+1 (2(xi−3 xi−4 xi−5 )−1 − b) + b s=−1

m ≥ −1,

(41)

for some i ∈ {1, 2}. Hence, by using formulas (40) and (41) we have that ( )m (x−1 x−2 x−3 )−1 (x−1 x−2 x−3 )−1 (b − (x−1 x−2 x−3 )−1 ) x6m = x−3 , b − (x−3 x−4 x−5 )−1 (x−3 x−4 x−5 )−1 (b − (x−3 x−4 x−5 )−1 ) (42) ( ) m+1 (x−1 x−2 x−3 )−1 (b − (x−1 x−2 x−3 )−1 ) x6m+3 = x−3 , (43) (x−3 x−4 x−5 )−1 (b − (x−3 x−4 x−5 )−1 ) )m+1 ( (xi−4 xi−5 xi−6 )−1 (b − (xi−4 xi−5 xi−6 )−1 ) , (44) x6m+i = xi−6 (xi−3 xi−4 xi−5 )−1 (b − (xi−3 xi−4 xi−5 )−1 ) ( )m+1 (xi−4 xi−5 xi−6 )−1 (b − (xi−4 xi−5 xi−6 )−1 ) x6m+3+i = xi−3 , (45) (xi−3 xi−4 xi−5 )−1 (b − (xi−3 xi−4 xi−5 )−1 ) for i ∈ {1, 2}. From (42)-(45) it is not difficult to describe the asymptotic behavior of welldefined solutions of equation (18) for the case a = −1, in terms of the quantities N0 :=

(x−1 x−2 x−3 )−1 (b − (x−1 x−2 x−3 )−1 ) (x−3 x−4 x−5 )−1 (b − (x−3 x−4 x−5 )−1 )

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and Ni :=

(xi−4 xi−5 xi−6 )−1 (b − (xi−4 xi−5 xi−6 )−1 ) , (xi−3 xi−4 xi−5 )−1 (b − (xi−3 xi−4 xi−5 )−1 )

i = 1, 2.

Namely, it is easy to see that the following result holds. Theorem 4. Assume that a = −1, b ̸= 0, and (xn )n≥−5 is a well-defined solution of equation (18). Then the following statements are true. (a) If |Ni | < 1 for some i ∈ {0, 1, 2}, then x6m+3j+i → 0, j = 0, 1, as m → +∞. (b) If |Ni | > 1 for some i ∈ {0, 1, 2}, then |x6m+3j+i | → +∞, j = 0, 1, as m → +∞. (c) If Ni = 1 for some i ∈ {0, 1, 2}, then the sequences (x6m+3j+i )m≥−1 , j = 0, 1, are constant. (d) If Ni = −1 for some i ∈ {0, 1, 2}, then the sequences (x6m+3j+i )m≥−1 , j = 0, 1, are two-periodic.

Now we consider the case a ̸= 0, b = 0. In this case equation (18) becomes xn =

xn−3 xn−4 xn−5 , xn−1 xn−2 a

n ∈ N0 ,

(46)

and formulas (21)-(24) also hold. Hence for a ∈ R \ {0}, we have ( x3m = x−3

x−4 x−5 ax−1 x−2

)m+1 ,

(47)

m ≥ −1, and ( x3m+i = xi−3 for m ≥ −1 and for i ∈ {1, 2}. Let x−4 x−5 L3 := ax−1 x−2

and

xi−3 xi−6

)m+1

L3+i :=

,

xi−3 , xi−6

(48)

i ∈ {1, 2}.

By using formulas (47) and (48) it is easy to see that the following result holds. We omit the proof. Theorem 5. Assume that a ̸= 0, b = 0, and (xn )n≥−5 is a well-defined solution of equation (18). Then the following statements are true. (a) (b) (c) (d)

If |L3+i | < 1, for some i ∈ {0, 1, 2}, then x3m+i → 0 as m → +∞. If |L3+i | > 1, for some i ∈ {0, 1, 2}, then |x3m+i | → ∞ as m → +∞. If L3+i = 1, for some i ∈ {0, 1, 2}, then the sequence (x3m+i )m≥−2 is constant. If L3+i = −1, for some i ∈ {0, 1, 2}, then the sequence (x3m+i )m≥−2 is twoperiodic.

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4. Domain of undefinable solutions of equation (5) In Section 2 we proved that solutions of equation (5) for which is x−j = 0 for some j ∈ {1, 2, 3, 4, 5} are not defined. The set of all such initial values is characterized here. Definition 1. ([32]) Consider the difference equation xn = f (xn−1 , . . . , xn−s , n),

n ∈ N0 ,

(49)

where s ∈ N, and x−i ∈ R, i = 1, s. The string of numbers x−s , . . . , x−1 , x0 , . . . , xn0 where n0 ≥ −1, is called an undefined solution of equation (49) if xj = f (xj−1 , . . . , xj−s , j) for 0 ≤ j < n0 + 1, and xn0 +1 is not defined number, that is, the quantity f (xn0 , . . . , xn0 −s+1 , n0 + 1) is not defined. The set of all initial values x−s , . . . , x−1 which generate undefined solutions of equation (49) is called domain of undefinable solutions of the equation. The next result characterizes the domain of undefinable solutions of equation (5) for the case an ̸= 0, bn ̸= 0, n ∈ N0 . Theorem 6. Assume that an ̸= 0, bn ̸= 0, n ∈ N0 . Then the domain of undefinable solutions of equation (5) is the following set U=

} j−1 2 { m ∪ ∪ ∑ b3j+i ∏ 1 1 (x−5 , . . . , x−1 ) ∈ R5 : xi−3 xi−4 xi−5 = , when cm := − ̸= 0 cm a a j=0 3j+i l=0 3l+i m∈N i=0 0

} 5 { ∪∪ (x−5 , . . . , x−1 ) ∈ R5 : x−j = 0 .

(50)

j=1

Proof. As we have already mentioned the set } 5 { ∪ 5 (x−5 , . . . , x−1 ) ∈ R : x−j = 0 , j=1

belongs to the domain of undefinable solutions of equation (5). Now we will consider the case when x−j ̸= 0, j = 1, 5 (i.e. xn ̸= 0 for every n ≥ −5). Such a solution (xn )n≥−5 is not defined if an xn−3 xn−4 xn−5 = − (51) bn for some n ∈ N0 . Since the change of variables (8) implies that equation (5) is transformed to the equations in (10), this along with (51) implies that the solution is not defined if b3m+i (52) y3(m−1)+i = − a3m+i for some m ∈ N0 and i ∈ {0, 1, 2}. Set f3m+i (t) := a3m+i t + b3m+i , m ∈ N0 , i ∈ {0, 1, 2}. −1 Then f3m+i (t) = (t − b3m+i )/a3m+i , m ∈ N0 , i ∈ {0, 1, 2}, so that −1 f3m+i (0) = −

b3m+i , a3m+i

m ∈ N0 , i ∈ {0, 1, 2}.

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´ JOSEF DIBL´IK, BRATISLAV IRICANIN, ˇ ˇ SMARDA ˇ STEVO STEVIC, AND ZDENEK

Now write equations in (10) as y3m+i = f3m+i (y3(m−1)+i ),

m ∈ N0 ,

for i ∈ {0, 1, 2}. Then, we have y3m+i = f3m+i ◦ f3(m−1)+i ◦ · · · ◦ fi (yi−3 ),

m ∈ N0 , i ∈ {0, 1, 2}.

(54)

From (53) and (54) we have that (52) holds for some m ∈ N0 , i ∈ {0, 1, 2}, if and only if −1 (0), yi−3 = fi−1 ◦ · · · ◦ f3m+i that is, j−1 m ∑ b3j+i ∏ 1 yi−3 = − , a a3l+i j=0 3j+i l=0

for some m ∈ N0 and i ∈ {0, 1, 2}, which along with the relations 1 yi−3 = , i ∈ {0, 1, 2}, xi−3 xi−4 xi−5 implies that the first union in (50) belongs to the domain of undefinable solutions too, as desired.  Acknowledgements The second author is supported by the project No. LO1408 “AdMaS UPAdvanced Materials, Structures and Technologies” (supported by Ministry of Education, Youth and Sports of the Czech Republic under the “National Sustainability Programme I”). The fourth author is supported by Project no. FEKT-S-14-2200 of Faculty of Electrical Engineering and Communication, Brno University of Technology, Czech Republic. This paper is also supported by the Serbian Ministry of Science projects III 41025, III 44006 and OI 171007. References [1] M. Aloqeili, Dynamics of a kth order rational difference equation, Appl. Math. Comput. 181 (2006), 1328-1335. [2] M. Aloqeili, Dynamics of a rational difference equation, Appl. Math. Comput. 176 (2006), 768-774. [3] A. Andruch-Sobilo and M. Migda, Further properties of the rational recursive sequence xn+1 = axn−1 /(b + cxn xn−1 ), Opuscula Math. 26 (3) (2006), 387-394. [4] I. Bajo and E. Liz, Global behaviour of a second-order nonlinear difference equation, J. Differ. Equations Appl. 17 (10) (2011), 1471-1486. [5] L. Berg and S. Stevi´ c, Periodicity of some classes of holomorphic difference equations, J. Difference Equ. Appl. 12 (8) (2006), 827-835. [6] L. Berg and S. Stevi´ c, On some systems of difference equations, Appl. Math. Comput. 218 (2011), 1713-1718. [7] L. Brand, A sequence defined by a difference equation, Amer. Math. Monthly 62 (7) (1955), 489-492. [8] C. Cinar, On the positive solutions of difference equation, Appl. Math. Comput. 150 (1) (2004), 21-24. [9] B. Iriˇ canin and S. Stevi´ c, Some systems of nonlinear difference equations of higher order with periodic solutions, Dyn. Contin. Discrete Impuls. Syst. Ser. A Math. Anal. 13 a (3-4) (2006), 499-506. [10] B. Iriˇ canin and S. Stevi´ c, Eventually constant solutions of a rational difference equation, Appl. Math. Comput. 215 (2009), 854-856.

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[11] G. L. Karakostas, Asymptotic 2-periodic difference equations with diagonally self-invertible responces, J. Differ. Equations Appl. 6 (2000), 329-335. [12] C. M. Kent and W. Kosmala, On the nature of solutions of the difference equation xn+1 = xn xn−3 − 1, Int. J. Nonlinear Anal. Appl. 2 (2) (2011), 24-43. [13] C. M. Kent, M. Kustesky M, A. Q. Nguyen and B. V. Nguyen, Eventually periodic solutions of xn+1 = max{An /xn , Bn /xn−1 } when the parameters are two cycles, Dyn. Contin. Discrete Impuls. Syst. Ser. A Math. Anal. 10 (1-3) (2003), 33-49. [14] W. Kosmala, A period 5 difference equation, Int. J. Nonlinear Anal. Appl. 2 (1) (2011), 82-84. [15] R. Kurshan and B. Gopinath, Recursively generated periodic sequences, Canad. J. Math. 24 (6) (1974), 1356-1371. [16] H. Levy and F. Lessman, Finite Difference Equations, Dover Publications, Inc., New York, 1992. [17] D. S. Mitrinovi´ c and J. D. Keˇ cki´ c, Methods for Calculating Finite Sums, Nauˇ cna Knjiga, Beograd, 1984 (in Serbian). [18] G. Papaschinopoulos, C. J. Schinas and G. Stefanidou, On a k-order system of Lyness-type difference equations, Adv. Differ. Equat. Volume 2007, Article ID 31272, (2007), 13 pages. [19] G. Papaschinopoulos and G. Stefanidou, Asymptotic behavior of the solutions of a class of rational difference equations, Inter. J. Difference Equations 5 (2) (2010), 233-249. [20] S. Stevi´ c, On the recursive sequence xn+1 = αn + (xn−1 /xn ) II, Dyn. Contin. Discrete Impuls. Syst. 10a (6) (2003), 911-916. ∏2(k+1) ∏ [21] S. Stevi´ c, On the recursive sequence xn+1 = A/ ki=0 xn−i + 1/ j=k+2 xn−j , Taiwanese J. Math. 7 (2) (2003), 249-259. [22] S. Stevi´ c, More on a rational recurrence relation, Appl. Math. E-Notes 4 (2004), 80-85. [23] S. Stevi´ c, A short proof of the Cushing-Henson conjecture, Discrete Dyn. Nat. Soc. Vol. 2006, Article ID 37264, (2006), 5 pages. [24] S. Stevi´ c, Periodicity of max difference equations, Util. Math. 83 (2010), 69-71. [25] S. Stevi´ c, On a nonlinear generalized max-type difference equation, J. Math. Anal. Appl. 376 (2011), 317-328. [26] S. Stevi´ c, On a system of difference equations, Appl. Math. Comput. 218 (2011), 3372-3378. [27] S. Stevi´ c, On the difference equation xn = xn−2 /(bn + cn xn−1 xn−2 ), Appl. Math. Comput. 218 (2011), 4507-4513. [28] S. Stevi´ c, Periodicity of a class of nonautonomous max-type difference equations, Appl. Math. Comput. 217 (2011), 9562-9566. [29] S. Stevi´ c, On a third-order system of difference equations, Appl. Math. Comput. 218 (2012), 7649-7654. [30] S. Stevi´ c, On some solvable systems of difference equations, Appl. Math. Comput. 218 (2012), 5010-5018. [31] S. Stevi´ c, On the difference equation xn = xn−k /(b + cxn−1 · · · xn−k ), Appl. Math. Comput. 218 (2012), 6291-6296. [32] S. Stevi´ c, Domains of undefinable solutions of some equations and systems of difference equations, Appl. Math. Comput. 219 (2013), 11206-11213. [33] S. Stevi´ c, On a solvable system of difference equations of kth order, Appl. Math. Comput. 219 (2013), 7765-7771. [34] S. Stevi´ c, On a system of difference equations of odd order solvable in closed form, Appl. Math. Comput. 219 (2013) 8222-8230. [35] S. Stevi´ c, On a system of difference equations which can be solved in closed form, Appl. Math. Comput. 219 (2013), 9223-9228. [36] S. Stevi´ c, Representation of solutions of bilinear difference equations in terms of generalized Fibonacci sequences, Electron. J. Qual. Theory Differ. Equ. Vol. 2014, Article no. 67, (2014), 15 pages. [37] S. Stevi´ c, M. A. Alghamdi, A. Alotaibi and N. Shahzad, On a higher-order system of difference equations, Electron. J. Qual. Theory Differ. Equ. Article No. 47, (2013), 18 pages. [38] S. Stevi´ c M. A. Alghamdi, A. Alotaibi and N. Shahzad, Boundedness character of a max-type system of difference equations of second order, Electron. J. Qual. Theory Differ. Equ. Article No. 45, (2014), 12 pages. [39] S. Stevi´ c, M. A. Alghamdi, D. A. Maturi and N. Shahzad, On a class of solvable difference equations, Abstr. Appl. Anal. Vol. 2013, Article ID 157943, (2013), 7 pages.

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´ JOSEF DIBL´IK, BRATISLAV IRICANIN, ˇ ˇ SMARDA ˇ STEVO STEVIC, AND ZDENEK

ˇ [40] S. Stevi´ c, J. Dibl´ık, B. Iriˇ canin and Z. Smarda, On a third-order system of difference equations with variable coefficients, Abstr. Appl. Anal. vol. 2012, Article ID 508523, (2012), 22 pages. ˇ [41] S. Stevi´ c, J. Dibl´ık, B. Iriˇ canin and Z. Smarda, On some solvable difference equations and systems of difference equations, Abstr. Appl. Anal. Vol. 2012, Article ID 541761, (2012), 11 pages. ˇ [42] S. Stevi´ c, J. Dibl´ık, B. Iriˇ canin and Z. Smarda, On a solvable system of rational difference equations, J. Difference Equ. Appl. 20 (5-6) (2014), 811-825. [43] Y. Yazlik, On the solutions and behavior of rational difference equations, J. Comput. Appl. Anal. 17 (3) (2014), 584-594. ´, Mathematical Institute of the Serbian Academy of Sciences, Knez Stevo Stevic Mihailova 36/III, 11000 Beograd, Serbia King Abdulaziz University, Department of Mathematics, P.O. Box 80203, Jeddah 21589, Saudi Arabia E-mail address: [email protected] Josef Dibl´ık, Department of Mathematics and Descriptive Geometry, Faculty of Civil Engineering, 60200, Brno University of Technology, Brno, Czech Republic E-mail address: [email protected], [email protected] ˇanin, Faculty of Electrical Engineering, Belgrade University, BuleBratislav Iric var Kralja Aleksandra 73, 11000 Beograd, Serbia E-mail address: [email protected] ˇ ˇk Smarda, Zdene Department of Mathematics, Faculty of Electrical Engineering and Communication, 61600, Brno University of Technology, Brno, Czech Republic E-mail address: [email protected]

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MODIFIED THREE-STEP ITERATIVE SCHEMES FOR ASYMPTOTICALLY NONEXPANSIVE MAPPINGS IN UNIFORMLY CONVEX METRIC SPACES SHIN MIN KANG1 , ARIF RAFIQ2 , FAISAL ALI3 AND YOUNG CHEL KWUN4,∗

1

Department of Mathematics and RINS, Gyeongsang National University, Jinju 660-701, Korea e-mail: [email protected] 2

3

Department of Mathematics, Lahore Leads University, Lahore 54810, Pakistan e-mail: [email protected]

Centre for Advanced Studies in Pure and Applied Mathematics, Bahauddin Zakariya University, Multan, Pakistan e-mail: [email protected] 4

Department of Mathematics, Dong-A University, Busan 604-714, Korea e-mail: [email protected]

Abstract. We prove the existence of the common fixed point for three asymptotically nonexpensive mappings defined on a A-uniformly convex metric space. A three-step iterative scheme is constructed which converges to the common fixed point. We also generalize the results of several authors. 2010 Mathematics Subject Classification: 47H10, 47J25. Key words and phrases: Iterative schemes, asymptotically nonexpansive mappings, A-uniformly convex metric spaces.

1. Introduction It is well known that the parallelogram law distinguishes the Hilbert spaces from the general Banach spaces. Recently many authors have introduced the idea for solving problems in Banach spaces by establishing identities and inequalities analogous to the parallelogram law (see for example [9, 20]). In 1965, the Banach contraction principle was extended to nonexpansive mappings by Browder [3], Goehde [7] and Kirk [10]. In [10], Kirk proved that ∗

Corresponding author. 1

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there exists a k-Lipschitzian map which has no fixed point. Goebel and Kirk [6] introduced the notion of asymptotically nonexpansive mappings and obtained a generalization of the results obtained in [3, 7, 10]. Afterwards Takahahashi ´ c [20] introduced the notion of convexity in metric spaces. Subsequently, Ciri´ [5], Guay et al. [8], Shimizu and Takahashi [15] and many other authors have studied fixed point theorems on convex metric spaces. Shimizu and Takahashi [16] introduced the concept of uniform convexity in convex metric spaces and studied its properties. Definition 1.1. ([18]) Let (X, d) be a metric space. A mapping W : X × X × [0, 1] → X is said to be a convex structure on X if for each (x, y, λ) ∈ X × X × [0, 1] and u ∈ X, d(u, W (x, y, λ)) ≤ λd(u, x) + (1 − λ)d(u, y). The metric space X together with W is called a convex metric space. Definition 1.2. Let X be a convex metric space. A nonempty subset A of X is said to be convex if W (x, y, λ) ∈ A whenever (x, y, λ) ∈ A × A × [0, 1]. Takahashi [18] has shown that open spheres B(x, r) = {y ∈ X : d(x, y) < r} and closed spheres B(x, r) = {y ∈ X : d(x, y) ≤ r} are convex. All normed spaces and their convex subsets are convex metric spaces. But there are many examples of convex metric spaces which are not embedded in any normed space (see Takahashi [18]). Recently, Beg [1] introduced and studied the notion of 2-uniformly convex metric spaces. Definition 1.3. ([1]) A convex metric space X is said to have property (B) if it satisfies d(W (x, a, α), W (y, a, α)) = αd(x, y). Taking x = a, property (B) implies αd(x, y) = W (y, a, α). Definition 1.4. ([1]) A convex complete metric space X is said to be uniformly convex if for all x, y, a ∈ X, [d (a, W (x, y, 1/2))]2   
1 d(x, y) ≤ [d(a, x)]2 + [d(a, y)]2 , 1−δ 2 max {d(a, x), d(a, y)}

where the function δ is a strictly increasing function on the set of strictly positive numbers and δ(0) = 0.

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Remark 1.5. ([1]) Uniformly convex Banach spaces are uniformly convex metric spaces. Definition 1.6. ([1]) A uniformly convex metric space X is said to be 2uniformly convex if there exists a constant c > 0 such that δ() ≥ c2. Definition 1.7. (1) Let A be a nonempty subset of a metric space X. A mapping T : A → A is said to be asymptotically nonexpansive [6] if there P exists a sequence {kn } ⊂ [1, ∞) with ∞ n=1 (kn − 1) < +∞ such that d(T n x, T n y) ≤ kn d(x, y) for all x, y ∈ A, n ≥ 1.

(2) T is said to be uniformly L-Lipschitzian with a Lipschitzian constant L ≥ 1, i.e., there exists a constant L ≥ 1 such that d(T n x, T ny) ≤ Ld(x, y) for all x, y ∈ A, n ≥ 1. This is a class of mapping introduced by Goebel and Kirk [6], where it is shown that if A is a nonempty bounded closed convex subset of a uniformly convex Banach space and T : A → A is asymptotically nonexpansive, then T has a fixed point and, moreover, the set F (T ) of fixed points of T is closed and convex. Remark 1.8. As an application of the Lagrange mean value theorem, we can see that tq − 1 ≤ qtq (t − 1) P∞ for t ≥ 1 and q > 1. This together with n=1 (kn − 1) < +∞ implies that P∞ q n=1 (kn − 1) < +∞. Theorem 1.9. ([16, Theorem 1]) If (X, d) is uniformly convex complete metric space then every decreasing sequence of nonempty closed bounded convex subsets of X has nonempty intersection.

Definition 1.10. Let (X, d) be a metric space and Y be a topological space. A mapping T : X → X is said to be completely continuous if the image of each bounded set in X is contained in a compact subset of Y. In [1, 2], Beg proved the following remarkable results. Theorem 1.11. Let (X, d) be a uniformly convex metric space having property (B). Then X is 2-unformly convex if and only if there exists a number c > 0 such that 2

2

2

2

2 [d (a, W (x, y, 1/2))] + c [d(x, y)] ≤ [d(a, x)] + [d(a, y)]

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for all a, x, y ∈ X. Theorem 1.12. Let A be a nonempty closed bounded convex subset of a uniformly convex complete metric space (X, d) and T : A → A be an asymptotically nonexpansive mapping. Then T has a fixed point. Theorem 1.13. Let (X, d) be a convex metric space and A be a nonempty convex subset of X. Let L > 0 and T : A → A be uniformly L-Lipschitzian. For x1 ∈ A. Define yn = W (T n xn , xn , 1/2),

xn+1 = W (T nyn , xn , 1/2)

and set cn = d(T n xn , xn ) for all n ∈ N. Then d(xn , T xn ) ≤ cn + cn−1 (L + 3L2 + 2L3 ) for all n ∈ N. Theorem 1.14. Let (X, d) be a 2-uniformly convex metric space having property (B), A be a nonempty closed bounded convex subset of X and T : A → A be P 2 asymptotically nonexpansive with sequence {kn } ∈ [1, +∞)N and ∞ n=1 (kn − 1) < +∞. Let x1 ∈ A and xn+1 = W (T n xn , xn , 1/2) for all n ∈ N. Then limn→∞ d(xn , T xn ) = 0. Theorem 1.15. Let (X, d) be 2-uniformly convex metric space having property (B), A be a nonempty closed bounded convex subset of X and T : A → A be completely continuous asymptotically nonexpansive mapping with sequence P 2 {kn } ∈ [1, +∞)N and ∞ n=1 (kn − 1) < +∞. Let x1 ∈ A and xn+1 = W (T nxn , xn , /1 2) for all n ∈ N. Then {xn } converges to some fixed point of T. In [12], Rafiq introduced the notion of A-uniformly convex metric space defined as follows: Definition 1.16. A convex complete metric space X is said to be A-uniformly convex if for all x, y, a ∈ X,    d(x, y) 2 [d (a, W (x, y, λ))] ≤ max{λ, 1 − λ} 1 − δ max {d(a, x), d(a, y)} (1.2)
2 2 × [d(a, x)] + [d(a, y)] ,

where the function δ is a strictly increasing function on the set of strictly positive numbers and δ(0) = 0.

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Remark 1.17. 1. The inequality (1.2) can be easily proved in a Hilbert space H using the well known identity [9] kλx + (1 − λ)yk2 = λ kxk2 + (1 − λ) kyk2 − λ(1 − λ) kx − yk2 for all x, y ∈ H. 2. For λ = 1/2 in (1.2), we get the inequality (1.1). 3. Uniformly convex Banach spaces are A-uniformly convex metric spaces. Definition 1.18. ([12]) The A-uniformly convex metric space X is said to be (2, A)-uniformly convex if there exists a constant c > 0 such that δ() ≥ c2. The purpose of this paper is to generalize the results of [2, 4, 6, 11, 13, 14, 17, 19, 21] and construct a three-step iterative scheme, convergent to the common fixed point, for three asymptotically nonexpansive mappings defined on a A-uniformly convex metric space. 2. Main Results In the sequel, we will need the following results. The following lemma is now well known. Lemma 2.1. Let {an } and {bn } be sequences of non-negative real numbers P such that an+1 ≤ (1 + bn ) an for all n ≥ 1 and ∞ n=1 bn < ∞. Then limn→∞ an exists. Theorem 2.2. ([2, 12]) Let A be a nonempty closed convex subset of a uniformly convex complete metric space (X, d) and T : A → A be an asymptotically nonexpansive mapping. Then the set F (T ) of fixed points of T is closed and convex. Theorem 2.3. ([12]) Let (X, d) be a A-uniformly convex metric space. Then X is (2, A)-uniformly convex if and only if there exists a number c > 0 such that for all a, x, y in X and λ ∈ [0, 1] , [d (a, W (x, y, λ))]2   ≤ max{λ, 1 − λ} [d(a, x)]2 + [d(a, y)]2 − c [d(x, y)]2 .

(2.1)

Theorem 2.4. Let (X, d) be a (2, A)-uniformly convex metric space, A be a nonempty closed convex subset of X and T, S, H : A → A be asympP∞ totically nonexpansive with sequence {kn } ∈ [1, +∞)N and n=1 (kn − 1) < n n +∞. Let x1 ∈ A and xn+1 = W (T yn , xn , αn ), yn = W (S zn , xn , βn), zn =

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W (H n xn , xn , βn) for all n ∈ N, where {αn }, {βn } and {γn } are the real sequences in [0, 1] satisfying αn , βn , γn ∈ [δ, 1 − δ] for some δ ∈ (0, 1). Then limn→∞ d(T n yn , xn ) = 0 = limn→∞ d(S n zn , xn ) = limn→∞ d(H n xn , xn ). Proof. Since T, S and H are asymptotically nonexpensive, so each mapping possesses a fixed point p ∈ A by Theorem 1.12. Let p ∈ F (T ) ∩ F (S) ∩ F (H). Claim. {xn } is bounded. For this claim, we compute as follows: d (p, xn+1 ) = d (p, W (T n yn , xn , αn )) ≤ αn d(p, T n yn ) + (1 − αn ) d(p, xn ) = αn d(T n p, T n yn ) + (1 − αn ) d(p, xn )

(2.2)

≤ αn kn d(p, yn ) + (1 − αn ) d(p, xn ), d (p, yn ) = d (p, W (S n zn , xn , βn )) ≤ βn d(p, S n zn ) + (1 − βn ) d(p, xn ) = βn d(S n p, S n zn ) + (1 − βn ) d(p, xn )

(2.3)

≤ βn kn d(p, zn ) + (1 − βn ) d(p, xn ), d(p, zn ) = d (p, W (H n xn , xn , βn )) ≤ βn d(p, H n xn ) + (1 − βn ) d(p, xn ) = βn kn d(H n p, H n xn ) + (1 − βn ) d(p, xn ) ≤ βn kn d(p, xn ) + (1 − βn ) d(p, xn )

(2.4)

= [1 + (kn − 1)βn ]d(p, xn ) ≤ kn d(p, xn ). Substituting (2.4) in (2.3) gives 
 d (p, yn ) ≤ 1 + kn2 − 1 βn d(p, xn )

(2.5)

≤ kn2 d(p, xn ).

From (2.5) in (2.2), we get 
 d (p, xn+1 ) ≤ 1 + kn3 − 1 αn d(p, xn ) 
 ≤ 1 + kn3 − 1 d(p, xn ),

which implies that limn→∞ d(p, xn ) exists and {xn } is bounded.

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Let M=supn≥0 d(p, xn ). Now with the help of (2.1), we have [d (p, xn+1 )]2 = [d(p, W (T n yn , xn , αn ))]2   ≤ max{αn , 1 − αn } [d (p, T n yn )]2 + [d(p, xn )]2 − c [d (T n yn , xn )]2

≤ [d (p, T n yn )]2 + [d(p, xn )]2 − max{αn , 1 − αn }c [d (T n yn , xn )]2

(2.6)

= [d (T np, T n yn )]2 + [d(p, xn )]2 − max{αn , 1 − αn }c [d (T n yn , xn )]2 ≤ kn2 [d (p, yn )]2 + [d(p, xn )]2 − max{αn , 1 − αn }c [d (T n yn , xn )]2 , 2

[d (p, yn )]

2

= [d(p, W (S n zn , xn , βn ))]  2 2 2 ≤ max{βn , 1 − βn } [d (p, S n zn )] + [d(p, xn )] − c [d (S n zn , xn )] 2

2

2

≤ [d (p, S n zn )] + [d(p, xn )] − max{βn , 1 − βn }c [d (S n zn , xn )] 2

2

= [d (S n p, S n zn )] + [d(p, xn )] − max{βn, 1 − βn }c [d (S n zn , xn )] 2

(2.7)

2

2

≤ kn2 [d (p, zn )]2 + [d(p, xn )] − max{βn , 1 − βn }c [d (S n zn , xn )] , [d (p, zn )]2 = [d(p, W (H n xn , xn , γn ))]2   ≤ max{γn , 1 − γn } [d (p, H n xn )]2 + [d(p, xn )]2 − c [d (H n xn , xn )]2

≤ [d (p, H n xn )]2 + [d(p, xn )]2 − max{γn , 1 − γn }c [d (H n xn , xn )]2

(2.8)

= [d (H n p, H n xn )]2 + [d(p, xn )]2 − max{γn , 1 − γn }c [d (H n xn , xn )]2 ≤ kn2 [d (p, xn )]2 + [d(p, xn )]2 − max{γn , 1 − γn }c [d (H n xn , xn )]2 = (1 + kn2 ) [d(p, xn )]2 − max{γn , 1 − γn }c [d (H n xn , xn )]2 . Substituting (2.8) in (2.7), we get [d (p, yn )]2 ≤ (1 + kn2 + kn4 ) [d(p, xn )]2 − kn2 max{γn , 1 − γn }c [d (H n xn , xn )]2

(2.9)

− max{βn , 1 − βn }c [d (S n zn , xn )]2 ,

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and by (2.6), we obtain [d (p, xn+1 )]2 ≤ (1 + kn2 + kn4 + kn6 ) [d(p, xn )]2 − max{αn , 1 − αn }c [d (T n yn , xn )]2 − kn2 max{βn , 1 − βn }c [d (S n zn , xn )]2 − kn4 max{γn , 1 − γn }c [d (H n xn , xn )]2

(2.10)

≤ [1 + (kn7 − 1)] [d(p, xn )]2 − max{αn , 1 − αn }c [d (T n yn , xn )]2 − kn2 max{βn , 1 − βn }c [d (S n zn , xn )]2 − kn4 max{γn , 1 − γn }c [d (H n xn , xn )]2 . With the help of condition αn , βn , γn ∈ [δ, 1 − δ] for some δ ∈ (0, 1), it can be easily seen that max{αn , 1 − αn },

max{βn, 1 − βn },

max{γn , 1 − γn } ≥ δ.

(2.11)

Using (2.11) in (2.10) and by kn ≥ 1, we obtain [d (p, xn+1 )]2 ≤ [1 + (kn7 − 1)] [d(p, xn )]2 which implies that


− δc [d (T nyn , xn )]2 + [d (S n zn , xn )]2 + [d (H n xn , xn )]2 ,

δc [d (T n yn , xn )]2 + [d (S n zn , xn )]2 + [d (H n xn , xn )]2 ≤ [d(p, xn )]2 − [d (p, xn+1 )]2 + M 2 (kn7 − 1).




Thus δc

m X 

d(T j yj , xj )

j=1

≤ Hence

M 2

m 2 X j=1

2


kj7 − 1 +

m X 

+

d(H j xj , xj )

j=1

m X j=1

∞ X 

m X  2 + d(S j zj , xj ) j=1

!


[d(p, xj )]2 − [d(p, xj+1 )]2 . 2

< +∞,

2

< +∞,

d(T j yj , xj )

j=1

∞ X 

2

d(S j zj , xj )

j=1

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and

∞ X 

d(H j xj , xj )

j=1

It implies that

2

< +∞.

lim d(T n yn , xn ) = 0,

n→∞

lim d(S n zn , xn ) = 0,

n→∞

and lim d(H n xn , xn ) = 0.

n→∞

This completes the proof.



Theorem 2.5. Let (X, d) be a (2, A)-uniformly convex metric space, A be a nonempty closed convex subset of X and T, S, H : A → A be asymptotically P nonexpansive with sequence {kn } ∈ [1, +∞)N and n=1 ∞ (kn − 1) < +∞. Further let H be completely continuous and T and S are continuous. Let x1 ∈ A and xn+1 = W (T n yn , xn , αn ), yn = W (S n zn , xn , βn ), zn = W (H n xn , xn , βn) for all n ∈ N, where {αn }, {βn } and {γn } are the real sequences in [0, 1] satisfying αn , βn , γn ∈ [δ, 1 − δ] for some δ ∈ (0, 1). Then the sequences {xn } , {yn } and {zn } converge to the common fixed point of T, S and H. Proof. Consider d(xn+1 , H n xn+1 ) ≤ d(xn+1 , xn ) + d(xn , H n xn ) + d(H n xn , H n xn+1 ) ≤ d(xn+1 , xn ) + d(xn , H n xn ) + kn d(xn , xn+1 ) = (1 + kn )d(xn+1 , xn ) + d(xn , H n xn ) = (1 + kn )d(W (T n yn , xn , 1/2), xn ) + d(xn , H n xn ) = (1 + kn ) αn d(T n yn , xn ) + d(xn , H n xn ) → 0 as n → ∞. Thus d(xn+1 , Hxn+1 ) ≤ d(xn+1 , H n+1 xn+1 ) + d(H n+1 xn+1 , Hxn+1 ) ≤ d(xn+1 , H n+1 xn+1 ) + k1 d(xn+1 , H n xn+1 ) → 0 as n → ∞, which implies that lim d(xn , Hxn ) = 0.

n→∞

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Since H is completely continuous and {xn } is bounded, there exists a subsequence {xnk } of {xn } such that {Hxnk } converges. Therefore from limn→∞ d(xn , Hxn ) = 0, {xnk } converges. Let limn→∞ xnk = p. It follows from the continuity of H and limn→∞ d(xn , Hxn ) = 0 that p = Hp. We know that limn→∞ d (p, xn ) exists. But limn→∞ d (p, xnk ) = 0. This implies limn→∞ d (p, xn ) = 0, i.e., limn→∞ xn = p. Since d(xn , zn ) = d(xn , W (H n xn , xn , γn )) = γn d(xn , H n xn ) → 0 as n → ∞, and d(xn , yn ) = d(xn , W (S n zn , xn , βn)) = βn d(xn , S n zn ) → 0 as n → ∞, so limn→∞ zn = p = limn→∞ yn . The following estimates hold: d(xn , xn+1 ) = d(xn , W (T nyn , xn , αn )) = αn d(xn , T n yn ) → 0 as n → ∞, d(yn , yn−1 ) ≤ d(yn , xn ) + d(xn , xn−1 ) + d(xn−1 , yn−1 ) → 0 as n → ∞, d(zn , zn−1 ) ≤ d(zn , xn ) + d(xn , xn−1 ) + d(xn−1 , zn−1 ) → 0 as n → ∞, d(zn , S n zn ) ≤ d(zn , xn ) + d(xn , S n zn ) → 0 as n → ∞, d(yn , T nyn ) ≤ d(yn , xn ) + d(xn , T n yn ) → 0 as n → ∞, d(zn , S n−1 zn ) ≤ d(zn , zn−1 ) + d(zn−1 , S n−1 zn−1 ) + d(S n−1 zn−1 , S n−1 zn ) ≤ d(zn , zn−1 ) + d(zn−1 , S n−1 zn−1 ) + kn−1 d(zn−1 , zn ) = (1 + kn−1 )d(zn , zn−1 ) + d(zn−1 , S

n−1

(2.12)

zn−1 )

→ 0 as n → ∞,

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d(zn , Szn ) ≤ d(zn , S n zn ) + d(S n zn , Szn ) ≤ d(zn , S n zn ) + k1 d(S n−1 zn , zn )

(2.13)

→ 0 as n → ∞, d(yn , T n−1 yn ) ≤ d(yn , yn−1 ) + d(yn−1 , T n−1 yn−1 ) + d(T n−1 yn−1 , T n−1 yn ) ≤ d(yn , yn−1 ) + d(yn−1 , T n−1 yn−1 ) + kn−1 d(yn , yn−1 )

(2.14)

= (1 + kn−1 )d(yn , yn−1 ) + d(yn−1 , T n−1 yn−1 ) → 0 as n → ∞, d(yn , T yn) ≤ d(yn , T n yn ) + d(T n yn , T yn) ≤ d(yn , T n yn ) + k1 d(T n−1 yn , yn )

(2.15)

→ 0 as n → ∞. Now according to the continuity of T and S and by using (2.15) and (2.13), we obtain T p = p = Sp. Thus p is the common fixed point of T, S and H. This completes the proof.  Acknowledgment This study was supported by research funds from Dong-A University. References [1] I. Beg, Inequalities in metric spaces with applications, Topol. Methods Nonlinear Anal., 17 (2001), 183–190. [2] I. Beg, An iteration scheme for asymptotically nonexpansive mappings on uniformly convex metric spaces, Nonlinear Anal. Forum, 6 (2001), 27–34. [3] F. E. Browder, Nonexpansive nonlinear operators in Banach space, Proc. Nat. Acad. Sci. USA, 54 (1965), 1041–1044. [4] Y. J. Cho, H. Zhou and G. Guo, Weak and strong covergence theorems for three-step iterations with errors for asymptotically nonexpansive mappings, Comput Math. Appl., 47 (2004), 707–717. ´ c, On some discontinuous fixed point theorems in convex metric spaces, [5] L. Ciri´ Czechoslovak Math. J., 43 (1993), 319–326. [6] K. Goebel and W. A. Kirk, A fixed point theorem for asymptotically nonexpansive mappings, Proc. Amer. Math. Soc., 35 (1972), 171–174. [7] D. Goehde, Zum Prinzip der kontractiven abbildung, Math. Nachr., 30 (1995), 251–258.

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[8] M. D. Guay, K. L. Singh and J. H. M. Whitfield, Fixed point theorems for nonexpansivemappings in convex metric spaces, Nonlinear Analysis and Applications (S. P. Singh and J. H. Burry, eds.), pp. 179-189, Lectures Notes in Pure and Applied Mathematics, 80, Marcel Dekker Inc., New York, 1982. [9] S. Ishikawa, Fixed points by a new iteration method, Proc. Amer. Math. Soc., 44 (1974), 147–150. [10] W. A. Kirk, A fixed point theorem for mappings which do not increase distances, Amer. Math. Monthly, 72 (1965), 1004–1006. [11] Z. Opial, Weak convergence of the sequence of successive approximations for nonexpansive mappings, Bull. Amer. Math. Soc., 73 (1967), 591–597. [12] A. Rafiq, Ishikawa iteration scheme for asymptotically nonexpansive mappings on uniformly convex metric spaces, Nonlinear Anal. Forum, 12 (2007), 17–27. [13] J. Schu, Weak and strong convergence to fixed points of anymptotically nonexpansive mappings, Bull. Austral. Math. Soc., 43 (1991), 153–159. [14] H. F. Senter and W. G. Dotson, Approximating fixed points of nonexpansive mappings, Proc. Amer. Math. Soc., 44 (1974), 375–380. [15] T. Shimizu and W. Takahashi, Fixed point theorems in certain convex metric spaces, Math. Japon., 37 (1992), 855–859. [16] T. Shimizu and W. Takahashi, Fixed points of multivalued mappings in certain convex metric spaces, Topol. Methods Nonlinear Anal., 8 (1996), 197–203. [17] S. Suantai, Weak and strong convergence criteria of Noor iteration for asymptotically nonexpansive mappings, J. Math. Anal. Appl., 311 (2005), 506–517. [18] W. Takahashi, A convexity in metric spaces and nonexpansive mapping I, Kodai Math. Sem. Rep., 22 (1970), 142–149. [19] K. K. Tan and H. K. Xu, Approximating fixed points of nonexpansive mappings by the Ishikawa iteration process, J. Math. Anal. Appl., 178 (1993), 301–308. [20] H. K. Xu, Inequalities in Banach spaces with applications, Nonlinear Anal., 16 (1991), 1127–1138. [21] B. L. Xu and M. A. Noor, Fixed point iterations for asymptotically nonexpansive mappings in Banach spaces, J. Math. Anal. Appl., 267 (2002), 444–453.

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ON IDENTITIES BETWEEN SUMS OF EULER NUMBERS AND GENOCCHI NUMBERS OF HIGHER ORDER LEE-CHAE JANG AND BYUNG MOON KIM

General Education Institute, Konkuk University, Chungju 138-701, Korea E-mail : [email protected] Department of Mechanical System Engineering, Dongguk University, Gyeongju 780-714, Korea E-mail : [email protected]

Abstract. In this paper we consider differential equations which are closely related to the generating functions of Euler numbers. By using the same method of Kim’s calculation in Kim [24,25], we derive identities involving Euler numbers arising from differential equations. In particular, we derive some new identities between the sums of Euler numbers and Genocchi numbers of higher order.

1. Introduction We consider the Euler numbers defined by the generating function as follows(see [4,8,17,2125]): ∞

E(t) =

∑ 2 tk = E k et + 1 k!

(1)

k=0

and the Genocchi numbers defined by the generating function as follows(see [2,3,5,7,9-16,18,20,2630,32,33,35,36]): ∞

G(t) =

∑ 2t tk = Gk . t e +1 k!

(2)

k=0

Kim(2012) derived some new identities between the sums of products of Frobenius-Euler polynomials and Frobenius-Euler polynomials of higher order (see[1,6,31,34,37,38]). In this paper we derive differential equations which are closely related to the generating function of Euler numbers. By using these differential equations, we derive some identities between the sums of products of Euler numbers and Euler numbers of higher order. In particular, we obtain some identities between the sums of Euler numbers and Genocchi numbers of higher order.

1991 Mathematics Subject Classification. 11B68, 11S40. Key words and phrases. Euler numbers, Genocchi numbers, sums of Euler numbers, differential equations. 1

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LEE-CHAE JANG AND BYUNG MOON KIM

2. Computation of sums of the products of Euler numbers In this section we assume that F = F (t) =

et

1 and F N = F × · · · × F | {z } +1

(3)

N −times

for N ∈ N. Thus, by (3), we get F (t)(1) =

dF (t) dt

−et + 1)2 −1 − e−t 1 = + t (et + 1)2 (e + 1)2 = −F + F 2 . =

(et

(4)

Let us consider the derivative of (4) with respect to t as follows: 2F F (1) = F (1) + F (2) .

(5)

2!F 3 − 2F 2 = F (1) + F (2) .

(6)

Thus, by (5) and (3), we get

From (6), we note that 2!F 3

where F (2) =

d2 F dt2

= 2F 2 + F (1) + F (2) = 2(F + F (1) ) + F (1) + F (2) = 2F + 3F (1) + F (2) ,

(7)

. Thus, by the derivative of (5) with respect to t, we get 2!3F 2 F (1) = 2F (1) + 3F (2) + F (3) .

(8)

2!3F 2 (−F + F 2 ) = 2F (1) + 3F (2) + F (3)

(9)

F (1) = −F + F 2 .

(10)

By (8), we see that

and

By (9), we see that 3!(−1)3 F 4

6F 3 + 2F (1) + 3F (2) + F (3) 3 1 = 6(F + F (1) + F (2) ) + 2F (1) + 3F (2) + F (3) 2 2 = 6F + 11F (1) + 6F (2) + F (3) . =

(11)

Continuing this process, we get (N − 1)!F N =

N −1 ∑

aK (N )F (k)

(12)

k=0 k

where F (k) = ddtFk and N ∈ N. Now we try to find the coefficients ak (N ) in (12). By (12), we differentiate the both sides of (12) as follows: (N − 1)!N F N −1 F (1)

=

N −1 ∑

ak (N )F (k+1)

k=0

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3 N ∑

=

ak−1 (N )F (k) .

(13)

k=1

By (7), we get (N − 1)!N F N −1 F (1)

N !F N (−F + F 2 ) N !(−F N + F N +1 ) −N !F N + N !F N +1 .

= = =

(14)

By (13) and (14), we get N !F N +1

= N !F N +

N ∑

ak−1 (N )F (k)

k=1

= N (N − 1)!F N + = N

N −1 ∑

N ∑

ak−1 (N )F (k)

k=1

ak (N )F (k) +

k=0

N ∑

ak−1 (N )F (k) .

(15)

k=1

In (12), replacing N by N + 1, we have N !F N +1 =

N ∑

ak (N + 1)F (k) .

(16)

k=0

By (15) and (16), we get N ∑

ak (N + 1)F (k) = N

k=0

N −1 ∑

ak (N )F (k) +

k=0

N ∑

ak−1 (N )F (k) .

(17)

k=1

By comparing coefficients on the both sides of (17), we have the followings: a0 (N + 1) = N a0 (N ) and aN (N + 1) = aN −1 (N ).

(18)

For 1 ≤ k ≤ n − 1, we have ak (N + 1) = N ak (N ) = ak−1 (N )

(19)

where ak (N ) = 0 for k ≥ N or k < 0. From (19), we note that a0 (N + 1)

= N a0 (N ) = N (N − 1)a0 (N − 1) = · · · = N (N − 1) · · · 2a0 (2).

(20)

By comparing coefficients on the both sides of (15) with N = 1, F +F

(1)

=

2

1!F =

1 ∑

ak (2)F (k)

k=0

= a0 (2)F + a1 (2)F (1) .

(21)

Thus, by (21), we get a0 (2) = 1 and a1 (2) = 1. Finally, we derive the values of ak (N ) in (12) from (19). Let us consider the following two variable function with variables t, s: ∑ ∑ tN sk , where |t| < 1. g(t, s) = ak (N ) N!

(22)

(23)

N ≥1 0≤k≤N −1

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LEE-CHAE JANG AND BYUNG MOON KIM

Then, Kim [25] derived the followings: ∑ ∑ N! g(t, s) = (k + 1)! N ≥1 0≤k≤N −1

∑ l1 +···+lk+1 =N

tN k 1 s . l1 · · · lk+1 N 1

(24)

By (23) and (24), we get ak (N ) =

N! (k + 1)!

∑ l1 +···+lk+1 =N

1 . l1 · · · lk+1

(25)

Therefore, by (12) and (25), we obtain the following theorem. Theorem 2.1. For n ∈ N, let us consider the following differential equation with respect to t: N −1 ∑ ∑ 1 1 F N (t) = N F (k) (t) (26) (k + 1)! l1 · · · lk+1 k=0

where F (k) (t) =

k

d F (t) dtk

l1 +···+lk+1 =N

and F N (t) = F (t) × · · · × F (t). Them F (t) = | {z }

1 et +1

is a solution of

N −times

(26). We assume that

( |

(N )

where En

2 et + 1

(

) ··· {z

2 et + 1

N −times

) = }

∞ ∑

En(N )

n=0

tn n!

(27)

are called the n-th Euler numbers of order N . By (3) and (27), we get ( ( ) ) 1 1 F N (t) = · · · et + 1 et + 1 | {z } ( ( N −times ) ) 2 1 2 · = · · 2N et + 1 et + 1 | {z } N −times

=

∞ 1 ∑ (N ) tn E 2N n=0 n n!

and F (t)

= =

( ) 1 2 2 N et + 1 ∞ 1 ∑ tl El . 2 l!

(28)

(29)

l=0

From (29), we note that F (t(k) ) = =

dk F (t) dtk ∞ 1∑ tl El+k . 2 l!

(30)

l=0

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Therefore, by (26), (29), and (30), we obtain the following theorem. Theorem 2.2. For N ∈ N, n ∈ Z+ , we have En(N ) = 2N N

N −1 ∑ k=0

1 (k + 1)!

∑ l1 +···+lk+1 =N

En+k . l1 · · · lk+1

(31)

From (28), we can derive the following equation: ( ( ) ) ∞ n ∑ 2 2 (N ) t En = ··· t n! et + 1 e +1 n=0 {z } | ( ∞ N −times ) ( ∞ ) ∑ ∑ t l1 t lN = ··· El1 ElN l1 lN l1 =0 lN =0 | {z } N −times ( ) ∞ ∑ ∑ El1 · · · ElN n! tn = l1 ! · · · lN ! n! n=0 l1 +···+lN =n ( ) ( ) ∞ ∑ ∑ n tn = El1 · · · ElN . l1 · · · lN n! n=0

(32)

l1 +···+lN =n

Therefore, by (31) and (32), we obtain the following corollary. Corollary 2.3. For N ∈ N, n ∈ Z+ , we have ( ) N −1 ∑ ∑ n 1 El1 · · · ElN = 2N N l1 · · · lN (k + 1)! l1 +···+lN =n

k=0

∑ l1 +···+lk+1 =N

En+k . l1 · · · lk+1

(33)

3. Identities between sums of Euler numbers and Genocchi numbers of higher order In this section we assume that ( ) ( ) ∑ ∞ n 2t 2t )t GN = GN (t) t ··· t = G(N n e +1 e +1 n! | {z } n=0

(34)

N −times

where

(N ) Gn

are called the n-th Genocchi numbers of order N . We note that ( ) dk 2t |t=0 = Gk dtk et + 1

(35)

for k ∈ N. By (35), we obtain the following equation: ( ) dk 2t Gk = |t=0 dtk et + 1

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LEE-CHAE JANG AND BYUNG MOON KIM

=

k ( ) ∑ k l=0

l

(2t)l F (k−l)

= F (k) + 2kF (k−1) .

(36)

By (30) and (36), we get ∞ ∑

G =

k=0 ∞ ∑

Gk

tk k! ∞

∑ tk tk +2 kF (k−1) k! k! k=0 k=0 ∞ ∑ ∞ ∑ ∞ ∞ l k ∑ ∑ t t tl tk = El+k +2 kEl+k−1 l! k! l! k! k=0 [ l=0 k=0 l=0 ] ∞ ∞ ∑ ∑ tl tk . = (El+k + 2kEl+k−1 ) l! k! =

F (k)

k=0

(37)

l=0

Therefore, by (37), we obtain the following theorem which is the identities between the sums of Euler numbers and Genocchi numbers. Theorem 3.1. For k ∈ Z+ , we have Gk =

∞ ∑

(El+k + 2kEl+k−1 )

l=0

tl . l!

(38)

From (38), we easily see that (N )

Gk

( =

F (k) + 2kF (k−1)

)(N )

= F (N +k) + 2kF (N +k−1) .

(39)

By (39), we get GN

=

∞ ∑

k (N ) t

Gk

k=0 ∞ ( ∑

k!

) tk F (N +k) + 2kF (N +k−1) k! k=0 ( ) (∞ ) ∞ ∞ ∞ l k ∑ ∑ ∑ ∑ t t tl tk = EN +l+k +2 EN +l+k−1 l! k! l! k! k=0 ( l=0 k=0 l=0 ) ∞ ∞ l k ∑ ∑ t t = EN +l+k + 2kEN +l+k−1 . l! k! =

k=0

(40)

l=0

Therefore, by (40), we obtain the following theorem which is the identities between the sums of Euler numbers and Genocchi numbers of higher order. Theorem 3.2. For N ∈ N, k ∈ Z+ , we have (N )

Gk

=

∞ ∑

(El+k + 2kEl+k−1 )

l=0

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7

Acknowledgement: This paper was supported by Konkuk University in 2014.

References [1] M. Acikgoz, S. Araci, K. Orucoglu, E. Sen, Discontinuous two-point boundary value problems with eigenparameter in the boundary conditions, Proc. Jangjeon Math. 16(2) (2013) 259-275. [2] S. Araci, Novel identities involving Genocchi numbers and polynomials arising from applications of umbral calculus, Appl. Math. Comput. 233 (2014) 599-607. [3] S. Araci, M. Acikgoz, F. Qi, H. Jolany, A note on the modified q-Genocchi numbers and polynomials with weight (α, β), Fasc. Math. 51 (2013) 21-32. [4] S. Araci, M. Acikgoz, A note on the Frobenius-Euler numbers and polynomials associated with Bernstein polynomials, Adv. Stud. Contemp. Math. 22(3) (2012) 399-406. [5] A. Bigeni, Combinatorial study of Dellac configurations and q-extended normalized median Genocchi numbers, Electron. J. Combin. 21(2) (2014) paper 2.32, 27pp. [6] A. Bayad, S. Gaboury, Generalized Dirichlet L-function of arbitrary order with applications, Adv. Stud. Contemp. Math. 23(4) (2013) 607-619. [7] L. Carbone, A. M. Mercurio, F. Palladino, N. Palladino, The correspondence of Brioschi and Genocchi, (Italian) Rend. Accad. Sci. Fis. Mat. Napoli 73(4) (2006) 263-366. [8] I. N. Cancul, Y. Simsek, A note on interpolation functions of the Frobenius-Euler numbers, in: Application of Mathematics in: Technical and Natural Sciences, in: AIP Conf. Proc., Amer. Inst. Phys. 1301 (2010) 59-67. [9] D. Dumont, J. Zeng, Further results on the Euler and Genocchi numbers , Aequationes Math. 47(1) (1994) 31-42. [10] S. Gaboury, R. Tremblay, B.-J. Fugure, Some explicit formulas for certain new classes of Bernoulli, Euler and Genocchi polynomials, Proc. Jangjeon Math. Soc. 17(1) (2014) 115-123. [11] G.-N. Han, J. Zeng, On a q-sequence that generalizes the median Genocchi numbers, Ann. Sci. Math. Qub ebec 23(1) (1999) 63-72. [12] A. F. Horadam, Generation of Genocchi polynomials of first order by recurrence relations, Fibonacci Quart 30(3) (1992) 239-243. [13] L. C. Jang, A study on the distribution of twisted q-Genocchi polynomials, Adv. Stud. Contemp. Math. 18 (2009) 181-189. [14] L. C. Jang , On Multiple generalized w-Genocchi polynomials and their applications ,Mathematical Problems in Engineering, 2010 (2010), Article ID 316870, 8pages. [15] L. C. Jang, T. Kim , q−Genocchi numbers and polynomials associated with fermionic p-adic invariant integrals on Zp ,Absract and Applied Analysis, 2008 (2008), Article ID 232187, 8pages. [16] D. Kang, J.-H. Jeong, B. J. Lee, S.-H. Rim, S. H. Choi, Some identities of higher order Genocchi polynomials arising from higher order Genocchi basis, J. Comput. Anal. Appl. 17(1) (2014) 141-146. [17] T. Kim, New approach to q-Euler polynomials to q-Euler polynomials of higher order , Russ. J. Math. Phys. 17(2) (2010), 201-207. [18] T. Kim, New approach to q-Euler, Genocchi numbers and their interpolation functions , Adv. Stud. Contemp. Math., 18(2) (2009), 105-112. [19] T. Kim, Some identities on the q-Euler polynomials of higher-order and q- Stirling numbers by the fermionic p-adic intergral on Zp , Russ. J. Math. Phys. , 16 (2009), 484-491. [20] T. Kim, On the q-extension of Euler and Genocchi-numbers , J. Math. Anal. Appl. 326 (2007) 1472-1481. [21] T. Kim, B. Lee, Some identities of the Frobenius-Euler polynomials,, Abstract and Applied Analysis, 2009 (2009),Article ID 639439, 7pages. [22] 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 (2009) 484-491. [23] T. Kim, New approach to q-Euler polynomial of higher order, Russ. J. Math. Phys. 17(2) (2010) 218-225. [24] T. Kim, Identities involving Frobenius-Euler polynomials arising from non-linear differential equations, J. Number Theory 132 (2012) 2854-2865. [25] T. Kim, Corrigendum to Identities involving Frobenius-Euler polynomials arising from non-linear differential equations [J. Number Theory 132 (12) (2012) 2854?2865], J. Number Theory 133(2) (2013) 622-625. [26] G. Kreweras, An additive generation for the Genocchi numbers and two of its enumerative meanings, Bull. Inst. Combin. Appl. 20 (1997) 99-103. [27] Q.-M. Luo, The multiplication formulas for the Apostol-Genocchi polynomials, Util. Math. 89 (2012) 179-191.

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[28] G.D. Liu, R. X. Li, Sums of products of Euler-Bernoulli-Genocchi numbers, (Chinese) J. Math. Res. Exposition 22(3) (2002) 469-475. (Chinese) J. Math. Res. Exposition 21 (2001), no. 3, 455–458 [29] M. X. Liu, Z. Z. Zhang, A class of computational formulas involving the multiple sum on Genocchi numbers and the Riemann zeta function, (Chinese) J. Math. Res. Exposition 21(3) (2001) 455-458. [30] Z. R. Li, Y. H. Li, A new class of summation formulae involving the Genocchi number and Riemann zeta function, J. Shandong Univ. Nat. Sci. 42(4) (2007) 5pp. [31] E. Luciano, The treatise of Genocchi and Peano (1884) in the light of unpublished documents, (Italian) Boll. Stor. Sci. Mat. 27(2) (2007) 219-264. [32] H. Ozden, p-adic distribution of the unification of the Bernoulli, Euler and Genocchi polynomials, Appl. Math. Comput. 218(3) (2011) 970-973. [33] K. H. Park, Y.-H. Kim, On some arithmetical properties of the Genocchi numbers and polynomials, Adv. Difference Equ. Art. ID 195049 (2008) 14pp. [34] E. Sen, S. Araci, Computation of eigenvalues and fundamental solutions of a fourth-order boundary value problem, Proc. Jangjeon Math. Soc. 15(4) (2012) 445-464. [35] S.-H. Rim, S.J. Lee, E. J. Moon, J. H. Jin, On the q-Genocchi numbers and polynomials associated with q-zeta function, Proc. Jangjeon Math. Soc. 12(3) (2009) 261-267. [36] S.-H. Rim, J.-H. Jeong, S.-J. Lee, E.-J. Moon, J.-J. Jin, On the symmetric properties for the generalized twisted Genocchi polynomials, Ars Combin. 105 (2012) 267-272. [37] C. S Ryoo, Some identities of the twisted q-Euler numbers and polynomials associated with q-Bernstain polynomials, Proc. Jangjeon Math. Soc. 14 (2011) 239-348. ,, 153 (1996), no. 1-3, 319–333 [38] J. Zeng, Sur quelques proprib etb es de symb etrie des nombres de Genocchi [On some symmetry properties of Genocchi numbers], Discrete Math. 154(1-3) (1996) 319-333.

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An algorithm for multi-attribute decision making based on soft rough sets ∗ Guangji Yu† March 3, 2015 Abstract: Based on soft rough sets, some new concepts such as soft decision systems, soft relative positive regions, relative reduction in soft decision systems and conditional significance relative to decision partition soft sets are proposed. The multi-attribute decision rule in soft decision systems is presented. An algorithm of multi-attribute decision making based on soft rough sets is given. Keywords: Soft rough set; Partition soft set; Soft decision system; Relative reduction; Decision making; Decision rule

1

Introduction

In 1999, Molodtsov [9] initiated soft set theory as a new mathematical tool for dealing with uncertainties which classical mathematical tools cannot handle. Recently, there has been a rapid growth of interest in soft set theory. Many efforts have been devoted to further generalizations and extensions of soft sets. Recently there has been a rapid growth of interest in soft set theory and its applications. Many efforts have been devoted to further generalizations and extensions of soft sets. Maji et al. [11] defined fuzzy soft sets, combining soft sets with fuzzy sets. Maji et al. [12] reported a detailed theoretical study on soft sets, with emphasis on the algebraic operations. Jiang et al. [7] extended soft sets with description logics. Aktas et al. [1] initiated the notion of soft groups, extending fuzzy groups. Feng et al. [2, 5] investigated the relationships among soft sets, rough sets and fuzzy sets. Applications of soft set theory in decision making problems was initiated in [10]. To address fuzzy soft set based decision making problems, Roy et al. [14] presented a novel method of object recognition from an imprecise multi-observer data. Using level soft sets, Feng et al. [3] proposed an adjustable approach to fuzzy soft set based decision making. This approach was further investigated in [4, 8]. Although soft sets have been applied by several authors to the study of ∗ This

work is supported by Quantitative Economics Key Laboratory Program of Guangxi University of Finance and Economics (2014SYS11) and Guangxi University Science and Technology Research Project. † Corresponding Author, School of Information and Statistics, Guangxi University of Finance and Economics, Nanning, Guangxi 530003, P.R.China. [email protected]

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decision making under uncertainty, it seems that soft set based group decision making has not been discussed yet in the literature. Thus the present study can be seen as a first attempt toward the possible application of soft rough approximations in multi-attribute decision making problems under uncertainty. The purpose of this paper is to give a method for multi-attribute decision making applying soft rough sets.

2

Preliminaries

Throughout this paper, U denotes an initial universe, E denotes the set of all possible attributes or parameters, 2U denotes the family of all subsets of U and | ¦ | is the cardinality of a set. We only consider the case where both U and E are nonempty finite sets.

2.1

Soft sets

Definition 2.1 ([9]). Let A ⊆ E. A pair (f, A) is called a soft set over U , if f is a mapping defined by f : A → 2U . In other words, a soft set over U is a parameterized family of subsets of the U . For e ∈ A, f (e) may be considered as the set of e-approximate elements of (f, A). Definition 2.2 ([12]). Let (f, A) and (g, B) be two soft sets over U . (1) (f, A) is called a soft subset of (g, B), if A ⊆ D and f (e) = g(e) for each e (g, B). e ∈ A. We denote it by (f, A) ⊂ (2) (f, A) and (g, B) are called soft equal, if A = B and f (e) = g(e) for each e ∈ A. We denote it by (f, A) = (g, B). e (g, B) and (f, A) ⊃ e (g, B). Obviously, (f, A) = (g, B) if and only if (f, A) ⊂ Definition 2.3 ([12]). Let (f, A), (g, B) and (h, C) be soft sets over U . (1) (h, C) is called the intersection of (f, A) and (g, B), if C = A ∩ B and e (g, B). h(e) = f (e) ∩ g(e) for each e ∈ C. We denote (h, C) by (f, A)∩ (2) (h, C) is called the union of (f, A) and (g, B), if C = A ∪ B and  e ∈ A − B,  f (e), f (e) ∪ g(e), e ∈ A ∩ B, h(e) =  g(e), e ∈ B − A. e (g, B). We denote (h, C) by (f, A) ∪ Definition 2.4 ([12]). Let (f, A) and (g, B) be two soft sets over U . (f, A) AND (g, B) denoted by (f, A) ∧ (g, B) is defined by (f, A) ∧ (g, B) = (h, A × B), where h(a, b) = f (a) ∩ g(b) for each (a, b) ∈ A × B. Definition 2.5 ([5]). A soft set (f, A) over U is called a partition soft set if {f (e)| e ∈ A} forms a partition of U . 2

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Definition 2.6 ([6]). Let A ⊆ E. Let (f, A) be soft sets over U . (f, A) is called U a bijective S soft set, if f is a mapping f : A → 2 such that (1) e∈A f (e) = U . (2) For ei , ej ∈ A and ei 6= ej , f (ei ) ∩ f (ej ) = ∅. In other words, suppose B = {f (ei )| ei ∈ A, ≤ 1 ≤ i ≤ n} ⊆ 2U . From Definition 2.7, the mapping f : A → 2U can be transformed to the mapping f : A → B, which is a bijective function, namely, for every X ∈ B, there is exactly one attribute e ∈ A such that f (e) = X and no unmapped element remains in both A and B. Proposition 2.7. Let (f, A) be a bijective soft set U and let (g, B) be a null e (g, B) is a bijective soft set. soft set over U . (h, C) = (f, A) ∪ Proposition 2.8. Let (f, A) and (g, B) be two bijective soft sets over U . Then (h, A × B) = (f, A) ∧ (g, B) is also a bijective soft set.

2.2

Soft rough sets

Definition 2.9 ([5]). Let (f, A) be a soft set over U and X ⊆ U . Then the pair P = (U, (f, A)) is called a soft approximation space. Based on the soft approximation space P , we define the following two operations aprP X = {x ∈ U | ∃ e ∈ A s.t. x ∈ f (e) ⊆ X}, aprP X = {x ∈ U | ∃ e ∈ A s.t. x ∈ f (e), f (e) ∩ X 6= ∅}. aprP (X) and aprP (X) are called the soft P -lower approximation and the soft P -upper approximation of X, respectively. In general, we refer to the pair (aprP (X), aprP (X)) as the soft rough set of X with respect to P . Moreover, the sets P osP (X) = aprP (X), N egP (X) = U − aprP (X), BndP (X) = aprP (X) − aprP (X) are called the soft P -positive region, the soft P -negative region and the soft P boundary region of X, respectively. X is said to be soft a soft P -definable set if aprP (X) = aprP (X); otherwise, X is called a soft P -rough set. From the analogy with Pawlak rough sets, we also have the following interpretation of above concepts. (1) x ∈ P osP (X) = aprP (X) means that x surely belongs to X with respect to P ; (2) x ∈ aprP (X) means that x possibly belongs to X with respect to P ; (3) x ∈ N egP (X) means that x surely does not belong to X with respect to P. Clearly, aprP (X) and aprP (X) can be expressed equivalently as: aprP (X) = ∪e∈A {f (e)| f (e) ⊆ X}, aprP (X) = ∪e∈A {f (e)| f (e) ∩ X 6= ∅}.

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3

Soft decision systems

It is worth nothing that information systems and soft sets are closely related. Given a soft set (f, A) over U . (f, A) could induce an information system in a natural way. In the section, we mainly discuss soft decision systems. Let (fi , Ci )(i = 1, 2, · · · , n) be bijective soft sets over U where Ci ∩ Cj = ∅ for i 6= j. Denote n

e i=1 (fi , Ci ), (ϕ, K) = (f, C) = ∪

n ^

(fi , Ci ).

i=1

where C =

n S i=1

3.1

Ci and K = C1 × C2 × . . . × Cn .

Soft relative positive regions

Definition 3.1. Let (f, A) and (g, B) be two soft sets over U . Then the soft positive region of (f, A) to (g, B) is defined as follows [ [ {x ∈ U | ∃ e ∈ A s.t. x ∈ f (e) ⊆ g(b)}, P os(f,A) (g, B) = aprP g(b) = b∈B

b∈B

where P = (U, (f, A)). Example 3.2. Let U = {x1 , x2 , x3 , x4 , x5 , x6 , x7 } be a common universe, A = {e1 , e2 , e3 , e4 } and B = {b1 , b2 } be two attribute sets. Suppose that (f, A) and (g, B) are two soft sets over U . The mapping of (f, A) is given below: f (e1 ) = {x1 , x2 }, f (e2 ) = {x4 , x5 , x6 }, f (e3 ) = {x3 , x7 }. The mapping of (g, B) is given below: g(b1 ) = {x1 , x2 , x3 }, g(b2 ) = {x4 , x5 , x6 , x7 }. Then apr(U,(f,A)) g(b1 ) = {x1 , x2 }, apr(U,(f,A)) g(b2 ) = {x4 , x5 , x6 }. So P os(f,A) (g, B) = {x1 , x2 , x4 , x5 , x6 }. Definition 3.3. Let (fi , Ci )(i = 1, 2, · · · , n) be bijective soft sets over U where Ci ∩ Cj = ∅ for i 6= j. Let (g, D) be a partition soft set over U where C ∩ D = ∅. Then the triple (U, (f, C), (g, D)) is called a soft decision system, (f, C) is called the condition bijective soft set and (g, D) is called the decision partition soft set. Accordingly, in a soft decision system (U, (f, C), (g, D)), we have [ [ P os(ϕ,K) (g, D) = aprP g(d) = {x ∈ U | ∃ e ∈ K s.t. x ∈ ϕ(e) ⊆ g(d)}, d∈D

d∈D

where K = C1 × C2 × . . . × Cn and P = (U, (ϕ, K)). We call it soft relative positive regions of soft decision systems. For a given soft decision system, we always consider P os(ϕ,K) (g, D) 6= ∅.

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Example 3.4. Suppose that U = {x1 , x2 , x3 , x4 , x5 , x6 } is a common universe, 3 S which is a set of six shops. C = Ci denotes the attribute set where C1 stands i=1

for the empowerment of sales personnel, C2 stands for the perceived quality of goods, and C3 stands for the high traffic location, respectively. The value sets of these attributes are C1 = {high, medium, low}, C2 = {good, average} and C3 = {no, yes}, respectively. And D = {profit,loss} describes shop profit or loss. Suppose that the six shops are characterized by the condition bijective soft e 3i=1 (fi , Ci ), and the management benefit of shop is characterized by the set ∪ decision partition soft set (g, D). The mapping of each bijective soft set over U is defined as follows: f1 (high) = {x1 , x6 }, f1 (medium) = {x2 , x3 , x5 }, f1 (low) = {x4 }, f2 (good) = {x1 , x2 , x3 }, f2 (average) = {x4 , x5 , x6 }, f3 (no) = {x1 , x2 , x3 , x4 }, f3 (yes) = {x5 , x6 }. The mapping of the decision partition soft set over U is defined as follows: g(profit) = {x1 , x3 , x6 }, g(loss) = {x2 , x4 , x5 }. Then we can view each bijective soft set (fi , Ci ) as a collection of approximations as follows: (f1 , C1 ) = {high = {x1 , x6 }, medium = {x2 , x3 , x5 }, low = {x4 }}, (f2 , C2 ) = {good = {x1 , x2 , x3 }, average = {x4 , x5 , x6 }}, (f3 , C3 ) = {no = {x1 , x2 , x3 , x4 }, yes = {x5 , x6 }}. Similarly, (g, D) = {profit = {x1 , x3 , x6 }, loss = {x2 , x4 , x5 }}. Denote 3 V e 3i=1 (fi , Ci ), (ϕ, K) = (fi , Ci ), (f, C) = ∪ where C =

3 S i=1

i=1

Ci and K = C1 × C2 × C3 .

Let ei ∈ K, then e1 =high and good and no, e2 =medium and good and no, e3 =low and average and no, e4 =medium and average and yes, e5 =high and average and yes. ϕ(e1 ) = {x1 }, ϕ(e2 ) = {x2 , x3 }, ϕ(e3 ) = {x4 }, ϕ(e4 ) = {x5 }, ϕ(e5 ) = {x6 }. Besides, we have the tabular form of (ϕ, K) given in Table 2. Table 1: Tabular representation of (ϕ, K) x1 x2 x3 x4 x5 x6

e1 1 0 0 0 0 0

e2 0 1 1 0 0 0

e3 0 0 0 1 0 0

e4 0 0 0 0 1 0

e5 0 0 0 0 0 1

So (U, (f, C), (g, D)) is a soft decision system on how to choose profitable shops. Thus apr(U,(f,C)) g(prof it) = {x1 , x6 }, apr(U,(f,C)) g(loss) = {x4 , x5 }. 5

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Therefore P os(ϕ,K) (g, D) = {x1 , x4 , x5 , x6 }.

3.2

Relative reduction in soft decision systems

Definition 3.5. Let (U, (f, C), (g, D)) be a soft decision system and let 1 ≤ j ≤ n. Then (1) (fj , Cj ) is called a soft dispensable set of (f, C) relative to (g, D), if n V P os(ϕ,K) (g, D) = P os(ψ,Q) (g, D), where (ψ, Q) = (fi , Ci ). Otherwise, i=1,i6=j

(fj , Cj ) is called a soft indispensable set of (f, C) relative to (g, D). (2) (f, C) is called a soft independent set relative to (g, D), if every soft bijective set (fi , Ci ) of (f, C) is a soft indispensable set relative to (g, D). Otherwise, (f, C) is called a soft dependent set relative to (g, D). (3) The unit set of all the soft indispensable set of (f, C) relative to (g, D) is called the core of (f, C) relative to (g, D), denoted by core((f, C), (g, D)). Definition 3.6. Let (U, (f, C), (g, D)) be a soft decision system. Let k = 1, 2, . . . , m and 1 ≤ jk ≤ n, denote m V 0 0 em (f 0 , C 0 ) = ∪ (fjk , Cjk ). k=1 (fjk , Cjk ) and (ϕ , K ) = k=1

(f 0 , C 0 ) is called a relative reduction in (U, (f, C), (g, D)), if (1) P os(ϕ,K) (g, D) = P os(ϕ0 ,K 0 ) (g, D), (2) (f 0 , C 0 ) is a soft independent set relative to (g, D). Example 3.7. In Example 3.4, denote (ϕ1 , K1 ) = (f1 , C1 ) ∧ (f2 , C2 ), (ϕ2 , K2 ) = (f1 , C1 ) ∧ (f3 , C3 ), (ϕ3 , K3 ) = (f2 , C2 ) ∧ (f3 , C3 ). We have P os(ϕ1 ,K1 ) (g, D) = P os(ϕ2 ,K2 ) (g, D) = P os(ϕ,K) (g, D) = {x1 , x4 , x5 , x6 }, P os(ϕ3 ,K3 ) (g, D) = {x4 }. But P os(f1 ,C1 ) (g, D) = {x1 , x4 , x6 } 6= P os(ϕ,K) (g, D), P os(f3 ,C3 ) (g, D) = ∅ 6= P os(ϕ,K) (g, D). So (f1 , C1 ) ∪ (f2 , C2 ) and (f1 , C1 ) ∪ (f3 , C3 ) are both relative reductions in (U, (f, C), (g, D)).

3.3

Dependent degree of decision partition soft sets

Definition 3.8. Let (f, A) and (g, B) be two soft sets over U . (f, A) is said to depend on (g, B) to a degree k (0 ≤ k ≤ 1), denoted (f, A) ⇒k (g, B), if k = γ((f, A), (g, B)) =

| P os(f,A) (g, B) | . |U |

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Accordingly, in a soft decision system (U, (f, C), (g, D)), we have k = γ((ϕ, K), (g, D)) =

| P os(ϕ,K) (g, D) | . |U |

We call it the dependent degree of decision partition soft sets upon condition bijective soft sets. It characters a degree of condition bijective soft sets in classifying decision partition soft sets. Obviously, we have 0 ≤ k ≤ 1. If k = 1, then (g, D) is completely dependent on (f, C). If k = 0, then (g, D) is completely independent on (f, C). Example 3.9. In example 3.4, the dependent degree of the decision partition e 3i=1 (fi , Ci ) : soft set (g, D) upon the condition bijective soft set (f, C) = ∪ k = γ(

3 ^

(fi , Ci ), (g, D)) =

i=1

|{x1 , x4 , x5 , x6 }| 4 2 = = |U | 6 3

Proposition 3.10. Let (U, (f, C), (g, D)) be a soft decision system. Let m, n ∈ N and m < n. Then γ(

m ^

(fi , Ci ), (g, D)) ≤ γ(

i=1

n ^

(fi , Ci ), (g, D)).

i=1

Proof. Since we have S

|

γ((ϕ, K), (g, D)) = =

γ((ϕ0 , K 0 ), (g, D)) = = where K =

n V i=1

Ci , K 0 =

m V i=1

aprP g(d) | | P os(ϕ,K) (g, D) | d∈D = |U | |U | S | {x ∈ U | ∃ e ∈ K s.t. x ∈ ϕ(e) ⊆ g(d)} | d∈D

,

|U | S

aprP 0 g(d) | (g, D) | d∈D = |U | |U | S 0 | {x ∈ U | ∃ e ∈ K s.t. x ∈ ϕ0 (e) ⊆ g(d)} | | P os

|

(ϕ0 ,K 0 )

d∈D

,

|U | Ci , P = (U, (f, C)) and P 0 = (U, (f 0 , C 0 )).

By Definition 2.6, for any (c1 , c2 , · · · , cn ) ∈ C1 × C2 × · · · × Cn , we have ϕ(c1 , c2 , · · · , cn ) = f1 (c1 ) ∩ f2 (c2 ) ∩ · · · ∩ fm (cm ) ∩ · · · ∩ fn (cn ). Moreover, for any (c1 , c2 , · · · , cm ) ∈ C1 × C2 × · · · × Cm , we also have ϕ0 (c1 , c2 , · · · , cm ) = f1 (c1 ) ∩ f2 (c2 ) ∩ · · · ∩ fm (cm ). 7

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For m, n ∈ N and m < n, aprP 0 g(d) ⊆ aprP g(d). So [ [ aprP 0 g(d) ⊆ aprP g(d). d∈D

Hence γ(

m ^

d∈D

(fi , Ci ), (g, D))} ≤ γ(

i=1

n ^

(fi , Ci ), (g, D)).

i=1

In other words, condition bijective soft sets can explain the most detailed classification of decision partition soft sets. And deleting some condition bijective soft sets can lose some information about decision partition soft sets. Thus, more information (more condition bijective soft sets) can result in bigger dependent degree of decision partition soft sets.

3.4

Conditional significance relative to decision partition soft sets

Definition 3.11. Let (U, (f, C), (g, D)) be a soft decision system and 1 ≤ j ≤ n. The conditional significance of (fj , Cj ) in (f, C) relative to (g, D) is denoted and defined as follows s((fj , Cj ), (f, C), (g, D)) = γ(

n ^

(fi , Ci ), (g, D)) − γ(

i=1

n ^

(fi , Ci ), (g, D)).

i=1,i6=j

This definition indicates the decrease of the dependent degree of decision partition soft sets when deleting one bijective soft set (fj , Cj ) from (f, C). The following results are easily obtained from the above definitions. Proposition 3.12. Let (U, (f, C), (g, D)) be a soft decision system and 1 ≤ j ≤ n. (1) 0 ≤ s((fj , Cj ), (f, C), (g, D)) ≤ 1, (2)(fj , Cj ) is a soft indispensable set of (f, C) to (g, D) if and only if s((fj , Cj ), (f, C), (g, D)) > 0, e {(fj , Cj ) | s((fj , Cj ), (f, C), (g, D)) > 0, j = (3) core((f, C), (g, D)) = ∪ 1, 2, · · · , n}. Theorem 3.13. Let (U, (f, C), (g, D)) be a soft decision system. Let k = 1, 2, . . . , m and 1 ≤ jk ≤ n, denote m V 0 0 em (f 0 , C 0 ) = ∪ (fjk , Cjk ), k=1 (fjk , Cjk ) and (ϕ , K ) = 0

where C =

m S jk =1

k=1

0

Cjk and K = Cj1 × Cj2 × . . . × Cjm .

If γ((ϕ0 , K 0 ), (g, D)) = γ((ϕ, K), (g, D)) and s((fj , Cj ), (f 0 , C 0 ), (g, D)) > 0, then (f 0 , C 0 ) is a relative reduction of (U, (f, C), (g, D)). 8

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3.5

The multi-attribute decision rule in soft decision systems

Definition 3.14. Let (U, (f, C), (g, D)) be a soft decision system. Let e ∈ K = C1 × C2 × . . . × Cn , d ∈ D. The soft rough membership function of ϕ(e) relative to g(d) is denoted and defined as follows ξ(ϕ(e), g(d)) =

|ϕ(e) ∩ g(d)| . |ϕ(e)|

Definition 3.15. Let (U, (f, C), (g, D)) be soft decision system. Let k = 1, 2, . . . , m and 1 ≤ jk ≤ n, denote m V 0 0 em (f 0 , C 0 ) = ∪ (fjk , Cjk ), k=1 (fjk , Cjk ) and (ϕ , K ) = where C =

m S jk =1

k=1

0

Cjk and K = Cj1 × Cj2 × . . . × Cjm . Let (f 0 , C 0 ) be a relative

reduction of (U, (f, C), (g, D)). We call If e, then d (ξ(ϕ0 (e), g(d))) the multi-attribute decision rule by induced (f 0 , C 0 ) in (U, (f, C), (g, D)), where e ∈ K 0 , d ∈ D and ξ(ϕ0 (e), g(d)) denotes the soft rough membership function of ϕ0 (e) relative to g(d), which expresses the support degree of rules.

4

An algorithm for multi-attribute decision making based on soft rough sets

Based on above definitions and results, we will give an algorithm for the multi-attribute decision rule. Algorithms: Step 1. Construct a soft decision system (U, (f, C), (g, D)). n V Step 2. Calculate the dependent degree of (g, D) upon (fi , Ci ) (j = i=1,i6=j

0, 1, 2, . . . , n). Step 3. Calculate each conditional significance of (fj , Cj ) in (f, C) relative to (g, D) by Definition 3.11. Step 4. Find core((f, C), (g, D)) by Proposition 3.12. Step 5. Find relative reductions in (U, (f, C), (g, D)) by Theorem 3.13. (1) If γ(core((f, C), (g, D)), (g, D)) = γ((f, C), (g, D)), then core((f, C), (g, D)) is a relative reduction in (U, (f, C), (g, D)). In this case, the process stops. Otherwise, it continues (2). (2) Denote m V (fjk , Cjk ), where k = 1, 2, . . . , m and 1 ≤ jk ≤ n. core((f, C), (g, D)) = k=1

9

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(a) Calculate the conditional significance of each bijective soft set (fi , Ci ) em (i 6= jk ) about ∪ k=1 (fjk , Cjk ) relative to (g, D) by Definition 3.11. (b) Select (fi , Ci ) with maximal conditional significance one by one. If there are many soft sets with the same maximal significant, we choose the attribute e (fi , Ci ) is a relative set containing the most elements. So core((f, C), (g, D))∪ reduction in (U, (f, C), (g, D)). Step 6. Obtain decision rules by relative reductions in the soft decision system (U, (f, C), (g, D)).(Fig.1) An algorithm

5

Conclusions

This method is based on cases of library history data analysis, then we can find the useful information. The multi-attribute decision rule and the support degree of rules provides scientific objective basis. This method reduces the search domain and hence does a more efficient retrieval than the existing methods. Therefore, the new evaluation method can help users to decide the component adapter scheme and reduce pressure and subjectivity in the component reuse process adapter decision-making.

References [1] H.Aktas, N.Ca˘ g man, Soft sets and soft groups, Information Sciences, 177(2007), 2726-2735. 10

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[2] F.Feng, C.Li, B.Davvaz, M.Irfan Ali, Soft sets combined with fuzzy sets and rough sets: a tentative approach, Soft Computing, 14(2010), 899-911. [3] F.Feng, Y.B.Jun, X.Liu, L.Li, An adjustable approach to fuzzy soft set based decision making, Computers and Mathematics with Applications, 234(2010), 10-20. [4] F.Feng, Y.Li, V.Leoreanu-Fotea, Application of level soft sets in decision making based on interval-valued fuzzy soft sets, Computers and Mathematics with Applications, 60(2010), 1756-1767. [5] F.Feng, X.Liu, V.Leoreanu-Fotea, Y.B.Jun, Soft sets and soft rough sets, Information Sciences, 181(2011), 1125-1137. [6] K.Gong, Z.Xiao, X.Zhang, The bijective soft set with its operations, Computers and Mathematics with Applications, 60(2010), 2270-2278 [7] Y.Jiang, Y.Tang, Q.Chen, J.Wang, S.Tang, Extending soft sets with description logics, Computers and Mathematics with Applications, 59(2010), 2087-2096. [8] Y.Jiang, Y.Tang, Q.Chen, An adjustable approach to intuitionistic fuzzy soft sets based decision making, Applied Mathematical Modelling, 35(2011), 824-836. [9] D.Molodtsov, Soft set theory-First result, Computers and Mathematics with Applications, 37(1999), 19-31. [10] P.K.Maji, A.R.Roy, An application of soft sets in a decision making problem, Computers and Mathematics with Applications, 44(2002), 1077-1083. [11] P.K.Maji, R.Biswas, A.R.Roy, Fuzzy soft sets, Journal of Fuzzy Mathematics, 9(2001), 589-602. [12] P.K.Maji, R.Biswas, A.R.Roy, Soft set theory, Computers and Mathematics with Applications, 45(2003), 555-562. [13] Z.Pawlak, Rough sets, International Journal of Computing and Information Sciences, 11(1982), 341-356. [14] A.R.Roy, P.K.Maji, A fuzzy soft set theoretic approach to decision making problems, Computers and Mathematics with Applications, 203(2007), 412418.

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Fixed point results for modular ultrametric spaces Cihangir Alaca Celal Bayar University Faculty of Science and Arts Department of Mathematics Muradiye Campus 45140 Manisa, Turkey E-mail: [email protected] Meltem Erden Ege Celal Bayar University Institute of Natural and Applied Sciences Department of Mathematics Muradiye Campus 45140 Manisa, Turkey E-mail: [email protected] Choonkil Park∗ Hanyang University Research Institute for Natural Sciences Seoul 133-791, Republic of Korea E-mail: [email protected]

Abstract In this study, we define the notion of modular ultrametric space. We present a fixed point theorem in modular spherically complete ultrametric space, and prove coincidence point theorem for three self maps in a modular spherically complete ultrametric space.

1

Introduction

Fixed point theory is a developing field of mathematics with various applications to engineering, applied mathematics, some disciplines of sciences, etc. Fixed point theorems play a key role in this theory. Under certain conditions, we get some results related to a self map on any set, which allows one or more fixed points by means of them. Ultrametric space is a kind of metric space but it has the strong triangle inequality, i.e., d(x, y) < max{d(x, z), d(z, y)}. 2010 Mathematics Subject Classification: Primary 46A80, 47H10, 54E35. Key words and phrases: Modular ultrametric space; coincidence point; fixed point. ∗ Corresponding author: Choonkil Park (email: [email protected]) 1259

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C. Alaca, M.E. Ege, C. Park This metric is also known non-Archimedean metric. The notion of ultrametric is utilized outside mathematics. For example, ultrametric distances are tools of taxonomy and phylogenetic tree. The notion of ultrametric space was introduced by Van Rooij [32]. Gajic [13] proved a fixed point theorem for a class of generalized contractive mapping on ultrametric space. Rao et al. [30] introduced two coincidence point theorems for three and four self maps in a spherically complete ultrametric space. Some fixed point results on ultrametric spaces were given by Kirk and Shahzad [17]. There are also some studies in [10, 21]. Modular space was appeared by Nakano [24] in 1950. Many authors [19, 20, 25, 26, 27, 28, 29] gave some remarks on modular spaces. The concept of a modular metric space more general than a metric space was presented by Chistyakov [6]. He also developed the theory of modular metric spaces in [7, 8]. Chaipunya et al. [5] showed the existence of fixed point and uniqueness of quasi-contractive mappings in modular metric spaces. Azadifar et al. [4] proved the existence and uniqueness of a common fixed point of compatible mappings of integral type in modular metric spaces. Hussain and Salimi [14] investigated the existence of fixed points of generalized α-admissible modular contractive mappings in modular metric spaces. Kilinc and Alaca [15] defined (ε, k)-uniformly locally contractive mappings and η-chainable concept and proved a fixed point theorem for these concepts in complete modular metric spaces. Many studies were done in [1, 2, 3, 9, 11, 12, 16, 18, 22, 31, 33, 34]. In this paper, we first introduce the notion of modular ultrametric space. We give some fixed point theorems in a modular spherically complete ultrametric space.

2

Preliminaries

Definition 2.1. [27]. A modular on a real linear space X is a functional ρ : X −→ [0, ∞] satisfying the following statements: (A1) ρ(0) = 0; (A2) If x ∈ X and ρ(αx) = 0 for all positive real numbers α, then x = 0; (A3) ρ(−x) = ρ(x) for all x ∈ X; (A4) ρ(αx + βy) ≤ ρ(x) + ρ(y) for all α, β ≥ 0 with α + β = 1 and x, y ∈ X. Let X be a nonempty set and λ ∈ (0, ∞). We indicate that the function ω : (0, ∞) × X × X → [0, ∞] is denoted by ωλ (x, y) = ω(λ, x, y) for all λ > 0 and x, y ∈ X. Definition 2.2. [7]. Let X be a nonempty set. The function ω : (0, ∞) × X × X → [0, ∞] is called a metric modular on X if, for all x, y, z ∈ X, the following conditions hold: (i) ωλ (x, y) = 0 for all λ > 0 ⇔ x = y; (ii) ωλ (x, y) = ωλ (y, x) for all λ > 0; (iii) ωλ+µ (x, y) ≤ ωλ (x, z) + ωµ (z, y) for all λ, µ > 0. 1260

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Modular ultrametric spaces Let’s recall the definitions of two sets Xω and Xω∗ [7]: Xω ≡ Xω (x0 ) = {x ∈ X : ωλ (x, x0 ) → 0 as λ → ∞} and Xω∗ ≡ Xω∗ (x0 ) = {x ∈ X : ∃λ = λ(x) > 0 such that ωλ (x, x0 ) < ∞}. Definition 2.3. [32]. Let (X, d) be a metric space. If the metric d satisfies the strong triangle inequality d(x, y) ≤ max{d(x, z), d(z, y)} for all x, y, z ∈ X, then it is called ultrametric on X. The pair (X, d) is said to be ultrametric space.

3

Modular ultrametric spaces

In this section, we first give some new definitions. Definition 3.1. Let (X, ω) be a modular metric space. If ω satisfies the strong triangle inequality ωλ (x, y) ≤ max{ωλ (x, z), ωλ (z, y)} for all x, y, z ∈ X, then it is called a modular ultrametric on X. Definition 3.2. Let (X, ω) be a modular ultrametric space. For r > 0 and x ∈ Xω , we define the open sphere Bω (x, r) and the closed sphere Bω [x, r] with centre x and radius r as follows: Bω (x, r) ={y ∈ Xω : ωλ (x, y) < r} Bω [x, r] ={y ∈ Xω : ωλ (x, y) ≤ r}. Definition 3.3. The modular ultrametric space Xω∗ is called a modular spherically complete ultrametric space if every nest of balls has a nonempty intersection. Theorem 3.4. Let Xω∗ be a modular spherically complete ultrametric space. Assume that there exists an element x = x(λ) ∈ Xω∗ such that ωλ (x, T x) < ∞. If T : Xω∗ → Xω∗ is a map such that for every x, y ∈ Xω∗ , x 6= y, (3.1)

ωλ (T x, T y) < max{ωλ (x, T x), ωλ (x, y), ωλ (y, T y)},

then T has a unique fixed point. Proof. Let Ba = Bω [a, ωλ (a, T a)] be the closed sphere centered at a with the radius ωλ (a, T a) and let A be the collection of these spheres for all a ∈ Xω∗ . It is clear that the relation Ba ≤ Bb ⇔ Bb ⊆ Ba is a partial order on A. Now we pay attention to a totally ordered subfamily A1 of A. Since Xω∗ is modular spherically complete, we have \ Ba = B 6= ∅. Ba ∈A1

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C. Alaca, M.E. Ege, C. Park Let b ∈ B, ba ∈ A1 and x ∈ Bb . Then we get the following: ωλ (x, b) ≤ ωλ (b, T b) ≤ max{ωλ (b, a), ωλ (a, T a), ωλ (T a, T b)} (3.2)

= max{ωλ (a, T a), ωλ (T a, T b)}.

The case ωλ (T a, T b) ≤ ωλ (a, T a) implies that ωλ (x, b) ≤ ωλ (a, T a). In case of ωλ (T a, T b) > ωλ (a, T a), it follows from (3.2) that ωλ (x, b) ≤ ωλ (b, T b) ≤ωλ (T a, T b) < max{ωλ (a, T a), ωλ (a, b), ωλ (b, T b)} = max{ωλ (a, T a), ωλ (b, T b)}. By ωλ (b, T b) ≤ ωλ (a, T a), we have ωλ (x, b) ≤ ωλ (a, T a) and ωλ (b, T b) > ωλ (a, T a) shows that ωλ (b, T b) < ωλ (b, T b), which is a contradiction. So ωλ (x, b) ≤ ωλ (a, T a) for x ∈ Bb . Since we have ωλ (x, a) ≤ ωλ (a, T a), x ∈ Ba and Bb ⊆ Ba for any Ba ∈ A1 . Thus Bb is the upper bound for the family A. From Zorn’s Lemma, we conclude that A has a maximal element Bz where z ∈ Xω∗ . Now we prove z = T z. Suppose that z 6= T z. The inequality (3.1) implies that ωλ (T z, T (T z)) < ωλ (z, T z). If y ∈ BT z , then ωλ (y, T z) ≤ ωλ (T z, T (T z)) < ωλ (z, T z). In this case, we get ωλ (y, z) ≤ max{ωλ (y, T z), ωλ (T z, z)} = ωλ (T z, z), i.e., y ∈ Bz and BT z ⊆ Bz . Moreover, z ∈ / BT z since ωλ (z, T z) > ωλ (T z, T (T z)). As a consequence, we have BT z ( Bz but it contradicts to the maximality of Bz . Hence we have z = T z. It only remains to show the uniqueness. For this purpose, we take u as a different fixed point. For u 6= z, we have ωλ (z, u) = ωλ (T z, T u) < max{ωλ (T z, z), ωλ (z, u), ωλ (u, T u)} = ωλ (z, u) which is a contradiction. This completes the proof. Theorem 3.5. Let Xω∗ be a modular ultrametric space, and let f, S, T : Xω∗ → Xω∗ be maps satisfying (1) f (Xω∗ ) is modular spherically complete, 1262

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Modular ultrametric spaces (2) ωλ (Sx, T y) < max{ωλ (f x, f y), ωλ (f x, Sx), ωλ (f y, T y)} for x, y ∈ Xω∗ , x 6= y, (3) f S = Sf, f T = T f, ST = T S, (4) S(Xω∗ ) ⊆ f (Xω∗ ), T (Xω∗ ) ⊆ f (Xω∗ ). Then either f w = Sw or f w = T w for some w ∈ Xω∗ . Proof. For a ∈ Xω∗ , let Ba = [f a, max{ωλ (f a, Sa), ωλ (f a, T a)}] denote the closed sphere centered at f a with the radius max{ωλ (f a, Sa), ωλ (f a, T a)}. Let A be the collection of all the spheres for all a ∈ f (Xω∗ ). We state that the relation Ba ≤ Bb iff Bb ⊆ Ba is a partial order on A. For a totally ordered subfamily A1 of A, since f (Xω∗ ) is modular spherically complete, we have \ Ba = B 6= ∅. Ba ∈A1

Let f b ∈ B where b ∈

f (Xω∗ )

and Ba ∈ A1 . Then we have f b ∈ Ba and so

ωλ (f b, f a) ≤ max{ωλ (f a, Sa), ωλ (f a, T a)}.

(3.3)

If a = b, then Ba = Bb . Assume that a 6= b and x ∈ Bb . It follows from the condition (2) and (3.3) that ωλ (x, f b) ≤ max{ωλ (f b, Sb), ωλ (f b, T b)} ≤ max{ωλ (f b, f a), ωλ (f a, T a), ωλ (T a, Sb), ωλ (f b, f a), ωλ (f a, Sa), ωλ (Sa, T b)} < max{ωλ (f b, f a), ωλ (f a, T a), ωλ (f a, Sa), max{ωλ (f b, f a), ωλ (f b, Sb), ωλ (f a, T a)}, max{ωλ (f a, f b), ωλ (f a, Sa), ωλ (f b, T b)}} = max{ωλ (f a, Sa), ωλ (f a, T a)}. Thus (3.4)

ωλ (x, f b) < max{ωλ (f a, Sa), ωλ (f a, T a)}.

From (3.3) and (3.4), we get ωλ (x, f a) ≤ max{ωλ (x, f b), ωλ (f b, f a)} ≤ max{ωλ (f a, Sa), ωλ (f a, T a)}. Therefore, x ∈ Ba . We have also Bb ⊆ Ba for any Ba ∈ A1 and Bb is an upper bound in A for the family A1 . By Zorn’s Lemma, there is a maximal element, denoted by Bz , in A, where z ∈ F (Xω∗ ). There exists an element w ∈ Xω∗ such that z = f w. Suppose f w 6= Sw and f w 6= T w. Since f S = Sf , ωλ (Sf w, T Sw) < max{ωλ (f 2 w, f Sw), ωλ (f 2 w, Sf w), ωλ (f Sw, T Sw)} = ωλ (f 2 w, f Sw).

(3.5) Since f T = T f ,

ωλ (ST w, T f w) < max{ωλ (f T w, f 2 w), ωλ (f T w, ST w), ωλ (f 2 w, T f w)} (3.6)

= ωλ (f 2 w, f T w) 1263

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C. Alaca, M.E. Ege, C. Park Since ST = T S, it follows from (3.5) and (3.6) that ωλ (Sf w, S 2 w) ≤ max{ωλ (Sf w, T Sw), ωλ (T Sw, T f w), ωλ (T f w, S 2 w)} < max{ωλ (f 2 w, f Sw), ωλ (f 2 w, f T w), max{ωλ (f Sw, f 2 w), ωλ (f Sw, S 2 w), ωλ (f 2 w, T f w)}}

(3.7)

= max{ωλ (f 2 w, f Sw), ωλ (f 2 w, f T w)}. From (3.5) and (3.7), we have max{ωλ (Sf w, T Sw), ωλ (Sf w, S 2 w)}

(3.8)

< max{ωλ (f 2 w, f Sw), ωλ (f 2 w, f T w)}. By (3.5) and (3.6), ωλ (T f w, T 2 w) ≤ max{ωλ (T f w, T Sw), ωλ (T Sw, Sf w), ωλ (Sf w, T 2 w)} < max{ωλ (f 2 w, f T w), ωλ (f 2 w, f Sw), max{ωλ (f 2 w, f T w), ωλ (f 2 w, Sf w), ωλ (f T w, T 2 w)}}

(3.9)

= max{ωλ (f 2 w, f T w), ωλ (f 2 w, f Sw)}. From (3.6) and (3.9), we have max{ωλ (ST w, T f w), ωλ (T f w, T 2 w)} (3.10)

< max{ωλ (f 2 w, f T w), ωλ (f 2 w, f Sw)}.

If max{ωλ (f 2 w, f T w), ωλ (f 2 w, f Sw)} = ωλ (f 2 w, f Sw), then from (3.8), we have max{ωλ (Sf w, T Sw), ωλ (Sf w, S 2 w)} < ωλ (f 2 w, f Sw) which gives f 2 w ∈ / BSw . Hence f z ∈ / BSw . But f z ∈ Bz . Hence Bz * BSw . It is a contradiction to the maximality of Bz in A, since Sw ∈ S(Xω∗ ) ⊆ f (Xω∗ ). If max{ωλ (f 2 w, f T w), ωλ (f 2 w, f Sw)} = ωλ (f 2 w, f T w), then from (3.10), max{ωλ (ST w, T f w), ωλ (T f w, T 2 w)} < ωλ (f 2 w, f T w) which gives f 2 w ∈ / BT w . Hence f z ∈ / BT w . Since f z ∈ Bz , we get Bz * BT w . It contradicts to the maximality of Bz in A, since T w ∈ T (Xω∗ ) ⊆ f (Xω∗ ). As a result, either f w = Sw or f w = T w. Proposition 3.6. Let Xω∗ be a modular spherically complete ultrametric space and let f, T : Xω∗ → Xω∗ be maps satisfying T (Xω∗ ) ⊆ f (Xω∗ ) and (3.11)

ωλ (T x, T y) < max{ωλ (f x, f y), ωλ (f x, T x), ωλ (f y, T y)}

for all x, y ∈ Xω∗ , with x 6= y. Then there exists z ∈ Xω∗ such that f z = T z. Moreover, if f and T are coincidentally commuting at z, then z is a unique common fixed point of f and T . 1264

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Modular ultrametric spaces Proof. Let Ba = [f a, ωλ (f a, T a)] represent the closed sphere centered at f a with radius ωλ (f a, T a) and let A be the collection of these spheres for all a ∈ X. By the same reasoning as in Theorem 3.5, we conclude that A has a maximal element Bz for an element z ∈ Xω∗ . Let’s assume f z 6= T z. Since T z ∈ T (X) ⊆ f (X), there exists w ∈ Xω∗ such that T z = f w. It is clear that w 6= z. From (3.11), we have ωλ (f w, T w) =ωλ (T z, T w) < max{ωλ (f z, f w), ωλ (f z, T z), ωλ (f w, T w)} =ωλ (f z, f w). Thus f z ∈ / Bw and Bz * Bw . This contradicts to the maximality of Bz . So f z = T z. On the other hand, we suppose that f and T are coincidentally commuting at z. Then f 2 z = f (f z) = f T z = T f z = T (T z) = T 2 z. Suppose f z 6= z. By (3.11), we conclude that ωλ (T f z, T z) < max{ωλ (f 2 z, f z), ωλ (f 2 z, T f z), ωλ (f z, T z)} =ωλ (T f z, T z), which is a contradiction. Thus z = f z = T z. Now we show the uniqueness. Let u be a different fixed point. For u 6= z, we have ωλ (z, u) = ωλ (T z, T u) < max{ωλ (f z, f u), ωλ (f z, T z), ωλ (f u, T u)} = ωλ (z, u), which is a contradiction. As a consequence, we have the required result.

References [1] A.A.N. Abdou, On asymptotic pointwise contractions in modular metric spaces, Abstr. Appl. Anal. 2013, Art. ID 501631 (2013). [2] B. Azadifar, M. Maramaei, Gh. Sadeghi, Common fixed point theorems in modular G-metric spaces, J. Nonlinear Anal. Appl. (in press). [3] B. Azadifar, M. Maramaei, Gh. Sadeghi, On the modular G-metric spaces and fixed point theorems, J. Nonlinear Sci. Appl. 6, 293-304 (2013). [4] B. Azadifar, Gh. Sadeghi, R. Saadati, C. Park, Integral type contractions in modular metric spaces, J. Inequal. Appl. 2013, 2013:483 (2013). [5] P. Chaipunya, Y.J. Cho, P. Kumam, Geraghty-type theorems in modular metric spaces with an application to partial differential equation, Adv. Difference Equ. 2012, 2012:83 (2012). [6] V.V. Chistyakov, Modular metric spaces generated by F -modulars, Folia Math. 15, 3–24 (2008). 1265

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C. Alaca, M.E. Ege, C. Park [7] V.V. Chistyakov, Modular metric spaces I. Basic conceps, Nonlinear Anal. 72, 1–14 (2010). [8] V.V. Chistyakov, Fixed points of modular contractive maps, Dokl. Math. 86, 515–518 (2012). [9] Y.J. Cho, R. Saadati, G. Sadeghi, Quasi-contractive mappings in modular metric spaces, J. Appl. Math. 2012, Art. ID 907951 (2012). [10] S. Priess-Crampe, P. Ribenboim, The common point theorem for ultrametric spaces, Geom. Dedicata 72, 105–110 (1998). [11] H. Dehghan, M.E. Gordji, A. Ebadian, Comment on ‘Fixed point theorems for contraction mappings in modular metric spaces’, Fixed Point Theory Appl. 2012, 2012:144 (2012). [12] M.E. Ege, C. Alaca, On Banach fixed point theorem in modular b-metric spaces (preprint). [13] L. Gajic, On ultrametric space, Novi Sad J. Math. 31, 69–71 (2001). [14] N. Hussain, P. Salimi, Implicit contractive mappings in modular metric and fuzzy metric spaces, Scientific World J. (in press). [15] E. Kilinc, C. Alaca, A fixed point theorem in modular metric spaces, Adv. Fixed Point Theory 4, 199–206 (2014). [16] E. Kilinc, C. Alaca, Fixed point results for commuting mappings in modular metric spaces, J. Appl. Funct. Anal. (in press). [17] W.A. Kirk, N. Shahzad, Some fixed point results in ultrametric spaces, Topology Appl. 159, 3327–3334 (2012). [18] P. Kumam, Fixed point theorems for nonexpansive mapping in modular spaces, Arch Math. (Brno) 40, 345–353 (2004). [19] W.A.J. Luxemburg, Banach function spaces, Thesis, Delft, Inst. of Techn. Assen., The Netherlands (1955). [20] S. Mazur, W. Orlicz, On some classes of linear spaces, Studia Math. 17, 97–119 (1958). [21] S.N. Mishra, R. Pant, Generalization of some fixed point theorems in ultrametric spaces, Adv. Fixed Point Theory 4, 41–47 (2014). [22] C. Mongkolkeha, W. Sintunavarat, P. Kumam, Fixed point theorems for contraction mappings in modular metric spaces, Fixed Point Theory Appl.2011, 2011:93 (2011). 1266

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Modular ultrametric spaces [23] A.F. Monna, Remarques sur les metriques non-archimediennes I, II, Indagationes Math. 53, 470–481, 625–637 (1950). [24] H. Nakano, Modulared Semi-Ordered Linear Space, Maruzen Co., Ltd., Tokyo (1950). [25] J. Musielak, W. Orlicz, On modular spaces, Studia Math. 18, 49–65 (1959). [26] J. Musielak and W. Orlicz, Some remarks on modular spaces, Bull. Acad. Polon. Sci. Ser. Sci. Math. Astr. Phys. 7, 661–668 (1959). [27] W. Orlicz, A note on modular spaces I, Bull. Acad. Polon. Sci. Ser. Sci. Math. Astronom. Phys. 9, 157–162 (1961). [28] W. Orlicz, Collected Papers: Part II, Polish Scientific Publishers, Warsaw, 851–1688 (1988). [29] L. Maligranda, Orlicz spaces and interpolation: Seminars in Mathematics 5, Universidade Estadual de Campinas, Departamento de Matem´atica, Campinas (1989). [30] K.P.R. Rao, G.N.V. Kishore, T.R. Rao, Some coincidence point theorems in ultra metric spaces, Int. J. Math. Anal. (Ruse) 1, 897–902 (2007). [31] W. Takahashi, N.-C. Wong, J.-C. Yao, Fixed point theorems for new generalized hybrid mappings in Hilbert spaces and applications, Taiwanese J. Math. 17, 1597–1611 (2013). [32] A.C.M. Van Rooij, Non-Archimedean Functional Analysis, Marcel Dekker, New York (1978). [33] L.-J. Zhu, Y.-C. Liou, Y. Yao, C.-C. Chyu, Algorithmic and analytic approaches to the split feasibility problems and fixed point problems, Taiwanese J. Math. 17, 1839–1853 (2013). [34] L.-J. Zhu, M. Ren, Y.-C. Liou, Y. Yao, From equilibrium problems and fixed point problems to minimization problems, Taiwanese J. Math. 18, 1041–1061 (2014).

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On the backward difference scheme for a class of SIRS epidemic models with nonlinear incidence Zhidong Teng, Ying Wang, Mehbuba Rehim College of Mathematics and Systems Science, Xinjiang University Urumqi 830046, Xinjiang, People’s Republic of China E-mail: [email protected], [email protected]

Abstract. In this paper, we construct a backward difference scheme for a class of SIRS epidemic models with nonlinear incidence rate βf (S)g(I) and vaccination in susceptible. The dynamical properties of the scheme are investigated. By using the inductive method and the linearization method of difference equations, the positivity and the boundedness of solutions, the existence and local stability of equilibria are obtained. By constructing new discrete type Lyapunov functions, under the conditions which functions f (S) and g(I) satisfy assumptions (H1 ) − (H3 ), the global stability of the equilibria is obtained. That is, the disease-free equilibrium is globally asymptotically stable if basic reproduction number R0 ≤ 1, and the endemic equilibrium is globally asymptotically stable if R0 > 1. Keywords: discrete SIRS epidemic model; backward difference scheme; nonlinear incidence; local and global stability; discrete Lyapunov function.

1. Introduction As we well known, for some practical purposes, especially the numerical computing, it is often necessary to discretize the continuous-time model to a corresponding discrete difference scheme, that is discrete dynamical model. In recent years, aim at the continuous-time SIR and SIRS epidemic models, the various discrete dynamical models are constructed, and the dynamical properties of these models are studied in many articles, for example, see [1-22] and the reference 1 1268

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therein. Many important results have been established. These results focus on: the computation of the basic reproduction number, the local and global stability of the disease-free equilibrium and endemic equilibrium, the permanence, persistence and extinction of the disease, the bifurcation and chaos phenomena, etc. Particularly, we see that, in [1,2], the authors studied a class of discrete SIRS epidemic models with time delays and bilinear incidence derived from corresponding continuous models by applying the nonstandard finite difference scheme (See [2326]), and the sufficient conditions on the global asymptotic stability of the diseasefree equilibrium and the permanence of the disease are established. In [3], the authors studied a discrete SIRS epidemic model with bilinear incidence derived from corresponding continuous model by applying the backward difference scheme, and the sufficient and necessary conditions on the global asymptotic stability of the disease-free equilibrium and endemic equilibrium are established. In [4], the authors discussed a class of discrete SIRS epidemic models with general nonlinear incidence derived from corresponding continuous model by applying the forward difference scheme, and the sufficient conditions for the existence and local stability of the disease-free equilibrium and endemic equilibrium are obtained. In [5], the authors discussed a class of discrete SIRS epidemic models with standard incidence discretized from corresponding continuous model by applying the forward difference scheme, and the sufficient condition for the global stability of the endemic equilibrium is established. However, from above articles we easily see that the studies on the backward difference scheme for SIRS epidemic models with nonlinear incidence are few. In this paper, we construct a backward difference scheme for a class of continuous-time SIRS epidemic models with nonlinear incidence βf (S)g(I) and vaccination in susceptible. We will study the dynamical properties, especially the global asymptotic stability of the disease-free equilibrium and endemic equilibrium for this discrete model. Firstly, the basic properties of the model, such as, the positivity and the boundedness of solutions, the existence and local stability of equilibria are discussed by using the inductive method and the linearization method of difference equations. Further, by constructing new discrete type Lyapunov functions which is different from those given in [3] and using the theory of stability of difference equations, we will establish the global asymptotic stability of equilibria under the assumptions (H1 ) − (H3 ) (see

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Section 2). That is, when assumptions (H1 )−(H3 ) hold, the disease-free equilibrium is globally asymptotically stable if and only if basic reproduction number R0 ≤ 1, and the endemic equilibrium is globally asymptotically stable if and only if R0 > 1. The organization of this paper is as follows. In the second section we firstly introduce a backward difference scheme, that is discrete dynamical model, for SIRS epidemic models with nonlinear incidence, and further give some basic assumptions. In the third section the results on the positivity and boundedness of solutions, the existence and local stability of equilibria for the model are stated and proved. In the fourth section we will state and prove the theorems on the global asymptotic stability of the disease-free equilibrium and endemic equilibrium for the model. Lastly, in the fifth section we will give a conclusion.

2. Model description We consider the following continuous SIRS epidemic model with nonlinear incidence and vaccination in susceptible dS = A − βf (S)g(I) − d1 S − ηS + δR, dt dI (1) = βf (S)g(I) − d2 I − γI, dt dR = ηS + γI − d3 R − δR, dt where S(t), I(t) and R(t) denote the numbers of susceptible, infected and recovered individuals at time t, respectively. A is the recruitment rate of the total population, d1 , d2 , and d3 represent the death rate of susceptible, infected and recovered individuals, respectively. Particularly, death rate d2 includes the natural death rate and the disease-related death rate of the infected individuals. δ is the rate at which recovered individuals lose immunity and return to the susceptible class. γ is the natural recovery rate of the infective individuals, β is the proportionality constant. f (S) and g(I) are continuous functions defined on [0, ∞). The transmission of the infection is governed by a nonlinear incidence rate βf (S)g(I). In this paper, we always assume that δ is nonnegative constant, and A, d1 , d2 , d3 , β, γ are positive constants. Now, we use the backward difference scheme to discretize model (1). Let h > 0 be the time step size. Since dS(t) S(t + h) − S(t) = lim , h→0 dt h

dI(t) I(t + h) − I(t) = lim , h→0 dt h 3 1270

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R(t + h) − R(t) dR(t) = lim . h→0 dt h lim (A − βf (S(t + h))g(I(t + h)) − (d1 + η)S(t + h) + δR(t + h))

h→0

= A − βf (S(t))g(I(t)) − (d1 + η)S(t) + δR(t), lim (βf (S(t + h))g(I(t + h)) − (d2 + γ)I(t + h))

h→0

= βf (S(t))g(I(t)) − (d2 + γ)I(t) and

lim (ηS(t + h) + γI(t + h) − (d3 + δ)R(t + h))

h→0

= ηS(t) + γI(t) − (d3 + δ)R(t), we can assume from model (1) for any h > 0 S(t + h) − S(t) = A − βf (S(t + h))g(I(t + h)) h −(d1 + η)S(t + h) + δR(t + h), I(t + h) − I(t) = βf (S(t + h))g(I(t + h)) − (d2 + γ)I(t + h), h R(t + h) − R(t) = ηS(t + h) + γI(t + h) − (d3 + δ)R(t + h). h

(2)

Denote t = n, t + h = n + 1, S(t) = Sn , I(t) = In , R(t) = Rn , S(t + h) = Sn+1 , I(t + h) = In+1 and R(t + h) = Rn+1 , then from (2) we further obtain the following discrete SIRS epidemic model with nonlinear incidence and vaccination in susceptible Sn+1 − Sn = h[A − βf (Sn+1 )g(In+1 ) − (d1 + η)Sn+1 + δRn+1 ], In+1 − In = h[βf (Sn+1 )g(In+1 ) − (d2 + γ)In+1 ],

(3)

Rn+1 − Rn = h[ηSn+1 + γIn+1 − (d3 + δ)Rn+1 ]. In this paper, our main aim namely is to investigate the dynamical properties of model (3). The initial condition for model (3) is given in the following form S0 > 0, I0 > 0, R0 ≥ 0.

(4)

For model (3) we firstly introduce the following assumption. (H1 ) Functions f (S) and g(I) are continuously differentiable and monotone increasing on R, f (0) = g(0) = 0,

I g(I)

is monotone increasing on (0, +∞) and

g ′ (0) > 0. Remark 1. It is obvious that assumption (H1 ) is basic for model (3). In fact, when f (S) = S p (where 0 < p ≤ 1) or f (S) =

S 1+αS

and g(I) =

I , 1+ωI

then

assumption (H1 ) naturally holds. 4 1271

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Remark 2. If function g(I) satisfies that second order derivative g ′′ (I) exists and g ′′ (I) ≤ 0 for all I ∈ [0, ∞), then we can easily prove that

I g(I)

is monotone

increasing on I ∈ (0, +∞). Define a function F (u, v) as follows. For any u, v ∈ R, if u ̸= v then F (u, v) = f (u)−f (v) , u−v

and if u = v then F (u, u) = f ′ (u). In order to obtain the global asymp-

totic stability of equilibria of model (3), we need to further introduce the following assumption. (H2 ) (d1 + d3 )γ − (d2 + d3 )η > 0. (H3 ) There are positive constants K1 and K3 such that 4d1 [d3 + k4 (d3 + δ)] > (d1 + d3 − k4 η − K1 δF (u, v))2 and 4K1 (d1 + η)d2 F (u, v) > (d1 + d2 − K3 βF (u, v))2 for any 0 < u, v ≤

A d

with u ̸= v, where k4 =

d2 +d3 γ

and d = min{d1 , d2 , d3 }.

Remark 3. When f (S) ≡ S, we have F (u, v) ≡ 1. Choosing positive constants K1 =

d1 +d3 −k4 η δ

and K2 =

d1 +d2 , β

Remark 4. When f (S) = such that F (u, v) =

1 . (1+ωξ)2

then assumption (H3 ) naturally holds. S , 1+ωS

we have that there is a ξ = ξ(u, v) ∈ (0, Ad )

Obviously,

1 1 ≤ F (u, v) = ≤1 (1 + ωξ)2 (1 + ω Ad )2 for all u, v ∈ (0, Ad ] with u ̸= v. Choose positive constants 2(d1 + d3 − k4 η)(1 + ω Ad )2 K1 = , δ(1 + (1 + ω Ad )2 )

2(d1 + d2 )(1 + ω Ad )2 K3 = , β(1 + (1 + ω Ad )2 )

then we can easily obtain that 2(1 + ω Ad )2 2 (d1 + d3 − k4 η − K1 δF (u, v)) ≤ (1 − ) (d1 + d3 − k4 η)2 , A 2 1 + (1 + ω d ) 2

2(1 + ω Ad )2 2 (d1 + d2 − K3 βF (u, v)) ≤ (1 − ) (d1 + d2 )2 1 + (1 + ω Ad )2 2

and 4K1 (d1 + η)d2 F (u, v) ≥ 8(d1 + η)d2

d1 + d3 − k4 η δ(1 + (1 + ω Ad )2 )

for all u, v ∈ (0, Ad ] with u ̸= v. 5 1272

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Therefore, for f (S) =

S , 1+ωS

if we assume that the following conditions hold

4d1 (d3 + k4 (d3 + δ)) > (1 − and

2(1 + ω Ad )2 2 ) (d1 + d3 − k4 η)2 1 + (1 + ω Ad )2

2(1 + ω Ad )2 2 d1 + d3 − k4 η 8(d1 + η)d2 > (1 − ) (d1 + d2 )2 , A 2 A 2 δ(1 + (1 + ω d ) ) 1 + (1 + ω d )

(5)

(6)

then it can be easily proved that assumption (H3 ) holds. Particularly, when ω = 0, that is f (S) ≡ S, we easily see that conditions (5) and (6) naturally hold.

3. Basic properties Firstly, on the existence of positive solutions with initial condition (4) and the boundedness of all solutions of model (3), we have the following results. Theorem 1. Suppose that (H1 ) holds. Then model (3) has a unique positive solution (Sn , In , Rn ) for all n ≥ 0 with initial condition (4), and lim sup(Sn + In + Rn ) ≤ n→∞

A , d

where d = min{d1 , d2 , d3 }. Proof: When n = 0, from model (3) we have (1 + h(d1 + η))S1 = S0 + h[A + δR1 − βf (S1 )g(I1 )], (7)

(1 + h(d2 + γ))I1 = I0 + hβf (S1 )g(I1 ), (1 + h(d3 + δ))R1 = R0 + h[ηS1 + γI1 ]. Solving S1 from (7), we obtain 1 [S0 + h(A + δR1 ) + I0 − (1 + h(d2 + γ))I1 ] 1 + h(d1 + η) δR0 1 [S0 + I0 + h(A + = ) 1 + h(d1 + η) 1 + h(d3 + δ) h2 γδ h2 δη −(1 + h(d2 + γ) − )I1 + S1 ]. 1 + h(d3 + δ) 1 + h(d3 + δ)

S1 =

Therefore, (7) is equivalent to 1 δR0 [S0 + I0 + h(A + 1 + h(d1 + η) 1 + h(d3 + δ) h2 γδ )I1 ], −(1 + h(d2 + γ) − 1 + h(d3 + δ) I0 + hβf (S1 )g(I1 ) R0 + hγI1 I1 = , R1 = , 1 + h(d2 + γ) 1 + h(d3 + δ)

S1 = a−1

(8)

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where a=1−

h2 δη . (1 + h(d1 + η))(1 + h(d3 + δ))

Obviously, a > 0, 1 + h(d2 + γ) − I¯1 =

h2 γδ 1+h(d3 +δ)

> 0. Let

S0 + I0 + h(A + 1 + h(d2 + γ) −

δR0 ) 1+h(d3 +δ) , h2 γδ 1+h(d3 +δ)

then from (8) we have S1 > 0 when 0 < I1 < I¯1 , S1 < 0 when I1 > I¯1 , and S1 = 0 when I1 = I¯1 . Let Ψ∗ (I1 ) , with

I1 1 I0 − + hβf (S1 )) ( g(I1 ) 1 + h(d2 + γ) g(I1 )

1 δR0 ) [S0 + I0 + h(A + 1 + h(d1 + η) 1 + h(d3 + δ) h2 γδ −(1 + h(d2 + γ) − )I1 ]. 1 + h(d3 + δ)

S1 = a−1

Then, from (8) we have Ψ∗ (I1 ) = 0. Under assumption (H1 ), we obtain that Ψ∗ (I1 ) is monotonically increasing for I1 > 0 and limI1 →0 Ψ∗ (I1 ) = −∞. On the other hand, when I1 = I¯1 we have f (S1 ) = f (0) = 0 and hence, I¯1 I0 1 − g(I¯1 ) 1 + h(d2 + γ) g(I¯1 ) 1 ¯ I0 = ( I − ) > 0. 1 1 + h(d2 + γ) g(I¯1 )

Ψ∗ (I¯1 ) =

Therefore, Ψ∗ (I1 ) = 0 has a unique positive solution y ∗ ∈ (0, I¯1 ). That is, y∗ =

1 (I0 − hβf (S1 )g(y ∗ )). 1 + h(d2 + γ)

Now, we show that y ∗ is the unique solution of Ψ∗ (I1 ) = 0 on (0, ∞). Otherwise, there is a y ′ ∈ [I¯1 , ∞) such that Ψ∗ (y ′ ) = 0. Since y ′ ≥ I¯1 , we have that S1 ≤ 0 when I1 = y ′ . From (H1 ), we have f (S) ≤ 0 for any S ≤ 0. Hence, from Ψ∗ (y ′ ) = 0 I0 I0 we further have y ′ ≤ . On the other hand, since I¯1 > , we obtain 1+h(d2 +γ)

′

y >

I0 , 1+h(d2 +γ)

1+h(d2 +γ)

which leads to a contradiction.

Therefore, we have I1 = y ∗ > 0. Again from (8), we further also have S1 > 0 and R1 > 0. This shows that from (7) we can obtain a unique positive solution (S1 , I1 , R1 ).

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When n = 1, a similarly argument as in above, we can obtain a unique positive solution (S2 , I2 , R2 ) satisfying model (3) at n = 1. By using the induction, we finally obtain a unique positive solution (Sn , In , Rn ) for all n > 0 satisfying model (3). Let Nn = Sn + In + Rn , then from model (3) we have Nn = Nn−1 + h(A − d1 Sn − d2 In − d3 Rn ) ≤ Nn−1 + h(A − dNn ). Hence, Nn ≤

hA + Nn−1 . 1 + hd

By using iteration method, we obtain hA N0 hA hA + + ··· + + 2 n 1 + hd (1 + hd) (1 + hd) (1 + hd)n A 1 N0 = [1 − ]+ . n d (1 + hd) (1 + hd)n

Nn ≤

Therefore, it holds that lim supn→+∞ Nn ≤

A . d

This completes the proof.

The basic reproduction number for model (3) can be defined by R0 = where S 0 =

A(d3 +δ) . d1 (d3 +δ)+d3 η

βf (S 0 )g ′ (0) , d2 + γ

On the existence of equilibria of model (3), we have the

following result. Theorem 2. Suppose that (H1 ) holds. (1). If R0 ≤ 1, then model (3) has only a unique disease-free equilibrium E 0 = (S 0 , 0, R0 ), where S 0 is given in the above and R0 =

ηA . d1 (d3 +δ)+d3 η

(2). If R0 > 1, then model (3) has a unique endemic equilibrium E ∗ = (S ∗ , I ∗ , R∗ ), except for disease-free equilibrium E 0 . Proof: We know that any equilibrium E = (S, I, R) of model (3) satisfies the following equation A − βf (S)g(I) − (d1 + η)S + δR = 0, βf (S)g(I) − (d2 + γ)I = 0,

(9)

ηS + γI − (d3 + δ)R = 0. Firstly, when I = 0, we have A − (d1 + η)S + δR = 0,

ηS − (d3 + δ)R = 0.

From this, we directly obtain disease-free equilibrium E 0 = (S 0 , 0, R0 ). 8 1275

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Second, when I > 0, from equation (9), we obtain R=

ηS + γI , δ + d3

S = S0 −

d2 (δ + d3 ) + d3 γ I. d1 (δ + d3 ) + d3 η

Substituting S into the second equation of (9) and we have βf (S 0 −

d2 (δ + d3 ) + d3 γ I)g(I) − (d2 + γ)I = 0. d1 (δ + d3 ) + d3 η

Let H(I) = βf (S 0 −

d2 (δ + d3 ) + d3 γ g(I) I) − (d2 + γ). d1 (δ + d3 ) + d3 η I

By assumption (H1 ), H(I) is strictly monotone decreasing on (0, +∞) and satisfies lim+ H(I) = βf (S 0 )g ′ (0) − (d2 + γ) = (d2 + γ)(R0 − 1)

I→0

¯ = −(d2 + γ) < 0, where I¯ = and we also have H(I)

S 0 (d1 (d3 +δ)+d3 η) . d2 (d3 +δ)+d3 γ

When R0 ≤ 1, we have limI→0+ H(I) ≤ 0. Consequently, there is not any I ∗ > 0 such that H(I ∗ ) = 0. Therefore, model (3) only has a unique disease-free equilibrium E0 . When R0 > 1, we have limI→0+ H(I) > 0. Therefore, there exists a unique ¯ such that H(I ∗ ) = 0. Furthermore, we have S ∗ = S 0 − d2 (δ+d3 )+d3 γ I ∗ > 0 I ∗ ∈ (0, I) d1 (δ+d3 )+d3 η ηS ∗ +γI ∗ δ+d3 ∗ ∗ ∗

and R∗ =

> 0. This implies that model (3) has a unique endemic equilibrium

E ∗ = (S , I , R ). This completes the proof. Further, on the local stability of equilibria of model (3), we have the following result. Theorem 3. Suppose that (H1 ) holds. (1). When R0 < 1, then disease-free equilibrium E 0 of model (3) is locally asymptotically stable. (2). When R0 > 1, then disease-free equilibrium E 0 of model (3) is unstable, and endemic equilibrium E ∗ is locally asymptotically stable. Proof: Calculating the linearization system of model (3) at equilibrium E 0 , we have

un+1 − un = h[βf (S 0 )g ′ (0)vn+1 − (d1 + η)un+1 + δwn+1 ], vn+1 − vn = h[βf (S 0 )g ′ (0)vn+1 − (d2 + γ)vn+1 ],

(10)

wn+1 − wn = h[ηun+1 + γvn+1 − (d3 + δ)wn+1 ]. From the second equation of system (10), we have vn+1 =

vn . 1 + h[d2 + γ − βf (S 0 )g ′ (0)]

(11)

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When R0 < 1, we obtain 0
1. Hence, norms of two eigenvalues of A−1 are less than one. Since limn→∞ vn = 0, from (12) we can obtain limn→∞ un = 0 and limn→∞ wn = 0. This shows that equilibrium E 0 of model (3) is locally stable. 1 When R0 > 1, since | 1+h[d2 +γ−βf | > 1, from (11) we obtain limn→∞ vn = (S 0 )g ′ (0)]

∞. Therefore, E 0 is unstable. Calculating the linearization system of model (3) at equilibrium E ∗ , we have un+1 = un − h[βf ′ (S ∗ )g(I ∗ )un+1 − βf (S ∗ )g ′ (I ∗ )vn+1 −(d1 + η)un+1 + δwn+1 ], vn+1 = vn + h[βf ′ (S ∗ )g(I ∗ )un+1 + βf (S ∗ )g ′ (I ∗ )vn+1 − (d2 + γ)vn+1 ],

(13)

wn+1 = wn + h[ηun+1 + γvn+1 − (d3 + δ)wn+1 ]. Let



 1 + h[βf ′ (S ∗ )g(I ∗ ) + d1 + η] hβf (S ∗ )g ′ (I ∗ ) −hδ   A= −hβf ′ (S ∗ )g(I ∗ ) 1 − h[βf (S ∗ )g ′ (I ∗ ) − d2 − γ] 0  −hη −hγ 1 + h[d3 + δ] and Xn = (un , vn , wn )T , then equation (13) can be rewrote into Xn+1 = A−1 Xn .

(14)

It is clear that if all eigenvalues λ of matrix −A satisfy |λ| > 1, then all eigenvalues σ of matrix A−1 will satisfy |σ| < 1. The characteristic equation of −A is |λE +A| = 0, where E is the unit matrix. Let r =

λ+1 , h

then we easily obtain

|λE + A| = r3 + ar2 + br + c, 10 1277

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where a = d1 + η + βf ′ (S ∗ )g(I ∗ ) + d3 + δ + d2 + γ − βf (S ∗ )g ′ (I ∗ ), b = d1 (d3 + δ) + d3 η + (d3 + δ)βf ′ (S ∗ )g(I ∗ ) + (d2 + γ)βf ′ (S ∗ )g(I ∗ ) (d3 + δ + d1 + η)(d2 + γ − βf (S ∗ )g ′ (I ∗ )) and c = (d1 (d3 + δ) + d3 η)(d2 + γ − βf (S ∗ )g ′ (I ∗ )) + (d2 (d3 + δ) + d3 γ)βf ′ (S ∗ )g(I ∗ ). g(I) I

From assumption (H1 ), we easily obtain

− g ′ (I) ≥ 0 for all I > 0. Since

βf (S ∗ )g(I ∗ ) − (d2 + γ)I ∗ = 0, we obtain d2 + γ − βf (S ∗ )g ′ (I ∗ ) ≥ 0. Hence, we have a > 0, b > 0 and c > 0. By calculating, we further obtain ab − c = (d1 + η + d3 + δ)[(d2 + γ − βf (S ∗ )g ′ (I ∗ ))2 +(d3 + δ)(d1 + βf ′ (S ∗ )g(I ∗ )) + d3 η] +(d2 + γ − βf (S ∗ )g ′ (I ∗ ))[(d3 + δ + d1 + η)2 +βf ′ (S ∗ )g(I ∗ )(2(d3 + δ) + d2 + γ + d1 + η)] +βf ′ (S ∗ )g(I ∗ )[(d1 + η)(d2 + γ) + d1 (d3 + δ) + γδ] +(βf ′ (S ∗ )g(I ∗ ))2 (d2 + γ + d3 + δ) > 0. Therefore, by the Routh-Hurwitz criterion all roots of equation r3 + ar2 + br + c = 0 have the negative real parts. Since λ = hr − 1, we further obtain that all eigenvalues λ of matrix −A satisfy |λ| > 1. Therefore, the zero solution X = 0 of equation (14) is asymptotically stable. This shows that equilibrium E ∗ is locally asymptotically stable. This completes the proof. Remark 5. From Theorems 3 we directly see that assumptions (H2 ) and (H3 ) only are used to obtain the global asymptotic stability of equilibria of model (3). Remark 6. From the results obtained in this section, we easily see that the backward difference scheme, that is discrete dynamical model (3), for a class of SIRS epidemic models (1) with nonlinear incidence is provided for us with excellent properties in the local stability of equilibria and the permanence of disease. These properties nearly are same to corresponding continuous-time model (1).

4. The global stability 11 1278

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Now, we study the stability of equilibria of model (3). Firstly, on the global stability of disease-free equilibrium E 0 , we have the following result: Theorem 4. Suppose that (H1 ) − (H3 ) hold. If R0 ≤ 1, then for any time step size h > 0 disease-free equilibrium E 0 of model (3) is globally asymptotically stable. Proof: Model (3) can be rewritten as the following form Sn+1 − Sn = h[−(d1 + η)(Sn+1 − S 0 ) − βg(In+1 )(f (Sn+1 ) − f (S 0 )) +δ(Rn+1 − R0 ) − βf (S 0 )g(In+1 )], In+1 − In = h[βg(In+1 )(f (Sn+1 ) − f (S 0 ))

(15)

−(d2 + γ)In+1 + βf (S 0 )g(In+1 )], Rn+1 − Rn = h[η(Sn+1 − S 0 ) + γIn+1 − (d3 + δ)(Rn+1 − R0 )]. We consider the following Lyapunov function 1 Wn = (Sn − S 0 + In + Rn − R0 )2 + k1 2 k4 +(k2 + k3 )In + (Rn − R0 )2 , 2

∫

Sn

(f (τ ) − f (S 0 ))dτ

S0

where ki (i = 1, 2, 3, 4) are positive constants which will be determined in the following. Calculating difference of Wn along solutions of equation (15), by assumption (H1 ) we have

∫

Wn+1 − Wn = k1

Sn+1

(f (τ ) − f (S 0 ))dτ + (k2 + k3 )(In+1 − In )

Sn

k4 + [(Rn+1 − R0 )2 − (Rn − R0 )2 ] 2 1 + [(Sn+1 − S 0 + In+1 + Rn+1 − R0 )2 2 −(Sn − S 0 + In + Rn − R0 )2 ] = k1 (Sn+1 − Sn )(f (Sn+1 ) − f (S 0 )) + (k2 + k3 )(In+1 − In ) k4 [(Rn+1 − Rn )(Rn − Rn+1 + 2(Rn+1 − R0 )] 2 1 + [(Sn+1 − Sn + In+1 − In + Rn+1 − Rn ) 2 ×(Sn − Sn+1 + 2(Sn+1 − S 0 ) + In − In+1 +

+2In+1 + Rn − Rn+1 + 2(Rn+1 − R0 )] ≤ k1 (Sn+1 − Sn )(f (Sn+1 ) − f (S 0 )) + (k2 + k3 )(In+1 − In ) +k4 (Rn+1 − Rn )(Rn+1 − R0 ) + (Sn+1 − S 0 + In+1 + Rn+1 − R0 ) ×(Sn+1 − Sn + In+1 − In + Rn+1 − Rn ) = k1 h[−(d1 + η)(Sn+1 − S 0 ) − βg(In+1 )(f (Sn+1 ) − f (S 0 )) 12 1279

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+δ(Rn+1 − R0 ) − βf (S 0 )g(In+1 )](f (Sn+1 ) − f (S 0 )) +k2 h[βg(In+1 )(f (Sn+1 ) − f (S 0 )) − (d2 + γ)In+1 +βf (S 0 )g(In+1 )] + k3 h(βf (Sn+1 )g(In+1 ) − (d2 + γ)In+1 ) +k4 h[η(Sn+1 − S 0 ) + γIn+1 − (d3 + δ)(Rn+1 − R0 )] ×(Rn+1 − R0 ) + h[−d1 (Sn+1 − S 0 ) − d2 In+1 −d3 (Rn+1 − R0 )] × (Sn+1 − S 0 + In+1 + Rn+1 − R0 ) Since R0 =

βf (S 0 )g ′ (0) d2 +γ

≤ 1, we have βf (S 0 )g ′ (0) ≤ d2 + γ. Under assumption (H1 ),

we have g(In+1 ) g(I) ≤ lim+ = g ′ (0). I→0 In+1 I Choosing constants k2 = k1 f (S 0 ) and k4 =

d2 +d3 , γ

we further have

Wn+1 − Wn ≤ −k1 hβg(In+1 )(f (Sn+1 ) − f (S 0 ))2 − d1 h(Sn+1 − S 0 )2 2 −d2 hIn+1 − k4 h(d3 + δ)(Rn+1 − R0 )2

−d3 h(Rn+1 − R0 )2 − h(d1 + d2 )(Sn+1 − S 0 )In+1 −h(d1 + d3 − k4 η)(Sn+1 − S 0 )(Rn+1 − R0 ) −(d1 + η)k1 h[f (Sn+1 ) − f (S 0 )](Sn+1 − S 0 ) +k1 hδ(f (Sn+1 ) − f (S 0 ))(Rn+1 − R0 ) g(In+1 ) − g ′ (0)] In+1 g(In+1 ) +k3 hβIn+1 [f (Sn+1 )( − g ′ (0)) + g ′ (0)(f (Sn+1 ) − f (S 0 ))] In+1 2 ≤ −d1 h(Sn+1 − S 0 )2 − d2 hIn+1 − (k4 (d3 + δ) + d3 )h(Rn+1 − R0 )2 +k2 hβf (S 0 )In+1 [

+k1 hδ(f (Sn+1 ) − f (S 0 ))(Rn+1 − R0 ) −(d1 + η)k1 h[f (Sn+1 ) − f (S 0 )](Sn+1 − S 0 ) −(d1 + d2 )h(Sn+1 − S 0 )In+1 + k3 hβg ′ (0)In+1 (f (Sn+1 ) − f (S 0 )) −(d1 + d3 − k4 η)h(Sn+1 − S 0 )(Rn+1 − R0 ) 2 − (k4 (d3 + δ) + d3 )h(Rn+1 − R0 )2 = −d1 h(Sn+1 − S 0 )2 − d2 hIn+1

−(d1 + d3 − k4 η)h(Sn+1 − S 0 )(Rn+1 − R0 ) −(d1 + d2 )h(Sn+1 − S 0 )In+1 f (Sn+1 ) − f (S 0 ) ](Sn+1 − S 0 )2 Sn+1 − S 0 f (Sn+1 ) − f (S 0 ) +k1 hδ( )(Sn+1 − S 0 )(Rn+1 − R0 ) Sn+1 − S 0

−(d1 + η)k1 h[

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f (Sn+1 ) − f (S 0 ) ) Sn+1 − S 0 = −h[(Sn+1 − S 0 , In+1 )P (Sn+1 − S 0 , In+1 )T +k3 hβg ′ (0)In+1 (Sn+1 − S 0 )(

+(Sn+1 − S 0 , Rn+1 − R0 )Q(Sn+1 − S 0 , Rn+1 − R0 )T ], where ( P =

k1 (d1 + η)F (Sn+1 , S 0 ) p12 p12

)

( ,

Q=

d2 ,

d1

q12

)

q12 [d3 + k4 (d3 + δ)]

with 1 p12 = (d1 + d2 − k3 βg ′ (0)F (Sn+1 , S 0 )), 2 1 q12 = (d1 + d3 − k4 η − k1 δF (Sn+1 , S 0 )). 2 K3 , g ′ (0)

Further, we choose k1 = K1 and k3 =

then assumption (H3 ) implies that

matrices P and Q are positive definite. This implies that Wn+1 − Wn < 0 for all (Sn , In , Rn ) ̸= (S 0 , 0, R0 ). By the Lyapunov’s theorems on the global asymptotical stability for difference equations [28], we obtain that disease-free equilibrium E 0 is globally asymptotically stable. This completes the proof. Remark 7. In articles [1-3,5], the authors studied the global properties of solutions for the various discrete difference scheme, such as the nonstandard finite difference scheme, backward difference scheme and forward difference scheme, for continuous-time SIRS epidemic models. The condition that the death rate (d1 ) of susceptible is less than or equal to the death rate (d2 ) of infected and the death rate (d3 ) of recovered, that is, d1 ≤ min{d2 , d3 } is required. Therefore, the global asymptotic stability of the disease-free equilibrium can be established only when the basic reproduction number R0 ≤ 1, except for some basic assumptions, for example, such as assumption (H1 ) for model (3). However, in this paper we do not require the condition d1 ≤ min{d2 , d3 } for model (3). Therefore, in order to obtain the global stability of the disease-free equilibrium of model (3), a new Lyapunov function is constructed and the assumption (H3 ) is introduced. On the global stability of the endemic equilibrium E ∗ , we have the following result. 14 1281

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Theorem 5. Suppose that (H1 ) − (H3 ) hold. If R0 > 1, then for any time step size h > 0 endemic equilibrium E ∗ of model (3) is globally asymptotically stable. Proof: The model (3) can be rewritten as the following form: Sn+1 − Sn = h[−βg(In+1 )[f (Sn+1 ) − f (S ∗ )] − (d1 + η)(Sn+1 − S ∗ ) +δ(Rn+1 − R∗ ) − βf (S ∗ )(g(In+1 ) − g(I ∗ ))], In+1 − In = h[βg(In+1 )(f (Sn+1 ) − f (S ∗ )) − (d2 + γ)(In+1 − I ∗ )

(16)

+βf (S ∗ )(g(In+1 ) − g(I ∗ ))], Rn+1 − Rn = h[η(Sn+1 − S ∗ ) + γ(In+1 − I ∗ ) − (d3 + δ)(Rn+1 − R∗ )]. Since (d2 + γ)I ∗ = βf (S ∗ )g(I ∗ ), we also have In+1 − In = h[βf (Sn+1 )g(In+1 ) − (d2 + γ)In+1 ] g(I ∗ ) g(In+1 ) − f (S ∗ ) ∗ ] In+1 I g(In+1 ) g(I ∗ ) g(I ∗ ) = βhIn+1 [f (Sn+1 )( − ∗ ) + ∗ (f (Sn+1 ) − f (S ∗ ))] In+1 I I

= βhIn+1 [f (Sn+1 )

and

Sn+1 − Sn + In+1 − In + Rn+1 − Rn = −d1 (Sn+1 − S ∗ ) − d2 (In+1 − I ∗ ) − d3 (Rn+1 − R∗ ).

(17)

(18)

We consider the following Lyapunov function

∫ Sn 1 ∗ ∗ ∗ 2 Vn = (Sn − S + In − I + Rn − R ) + k1 (f (τ ) − f (S ∗ ))dτ 2 ∫ S∗ In In k4 g(τ ) − g(I ∗ ) dτ + k3 (In − I ∗ − I ∗ ln ∗ ) + (Rn − R∗ )2 , +k2 g(τ ) I 2 I∗

where ki (i = 1, 2, 3, 4) are positive constants which will be determined in the following. Calculating difference of Vn along equation (16), then by (17) and (18) we have

∫ Vn+1 − Vn = k1 [

Sn+1

∫

∗

(f (τ ) − f (S ))dτ ] + k2

Sn

In+1

In

g(τ ) − g(I ∗ ) dτ g(τ )

In+1 k4 +k3 (In+1 − In − I ln ) + [(Rn+1 − R∗ )2 − (Rn − R∗ )2 ] In 2 1 ∗ ∗ + [(Sn+1 − S + In+1 − I + Rn+1 − R∗ )2 2 −(Sn − S ∗ + In − I ∗ + Rn − R∗ )2 ] ∗

≤ k1 (Sn+1 − Sn )(f (Sn+1 ) − f (S ∗ )) +k2 (In+1 − In )(

g(In+1 ) − g(I ∗ ) In+1 − I ∗ ) + k3 (In+1 − In ) g(In+1 ) In+1

k4 (Rn+1 − Rn )(Rn − Rn+1 + 2(Rn+1 − R∗ )) 2 1 + (Sn+1 − Sn + In+1 − In + Rn+1 − Rn ) 2 +

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×(Sn − Sn+1 + 2(Sn+1 − S ∗ ) + In − In+1 +2(In+1 − I ∗ ) + Rn − Rn+1 + 2(Rn+1 − R∗ )) ≤ k1 (Sn+1 − Sn )(f (Sn+1 ) − f (S ∗ )) + k4 (Rn+1 − Rn )(Rn+1 − R∗ ) g(In+1 ) − g(I ∗ ) In+1 − I ∗ ) + k3 (In+1 − In ) g(In+1 ) In+1 +(Sn+1 − Sn + In+1 − In + Rn+1 − Rn ) +k2 (In+1 − In )(

×(Sn+1 − S ∗ + In+1 − I ∗ + Rn+1 − R∗ ) = k1 h[−βg(In+1 )[f (Sn+1 ) − f (S ∗ )] − (d1 + η)(Sn+1 − S ∗ ) + δ(Rn+1 − R∗ ) −βf (S ∗ )(g(In+1 ) − g(I ∗ ))](f (Sn+1 ) − f (S ∗ )) +k2 h[βg(In+1 )(f (Sn+1 ) − f (S ∗ )) − (d2 + γ)(In+1 − I ∗ ) g(In+1 ) − g(I ∗ ) ) g(In+1 ) g(In+1 ) g(I ∗ ) g(I ∗ ) +k3 h[βf (Sn+1 )( − ∗ ) + β ∗ (f (Sn+1 ) − f (S ∗ ))](In+1 − I ∗ ) In+1 I I +k4 h[η(Sn+1 − S ∗ ) + γ(In+1 − I ∗ ) − (d3 + δ)(Rn+1 − R∗ )](Rn+1 − R∗ )

+βf (S ∗ )(g(In+1 ) − g(I ∗ ))](

−h(d1 (Sn+1 − S ∗ ) + d2 (In+1 − I ∗ ) + d3 (Rn+1 − R∗ )) ×(Sn+1 − S ∗ + In+1 − I ∗ + Rn+1 − R∗ ) Choosing constants k2 = k1 f (S ∗ ) and k4 =

d2 +d3 , γ

we further have

Vn+1 − Vn ≤ −d1 h(Sn+1 − S ∗ )2 − d2 h(In+1 − I ∗ )2 −h[k4 (d3 + δ) + d3 ](Rn+1 − R∗ )2 − (d1 + d2 )h(Sn+1 − S ∗ )(In+1 − I ∗ ) −(d1 + d3 − k4 η)h(Rn+1 − R∗ )(Sn+1 − S ∗ ) −k1 (d1 + η)h(Sn+1 − S ∗ )(f (Sn+1 ) − f (S ∗ )) +k1 hδ(Rn+1 − R∗ )(f (Sn+1 ) − f (S ∗ )) g(In+1 ) − g(I ∗ ) [βf (S ∗ )(g(In+1 )) − g(I ∗ )) − (d2 + γ)(In+1 − I ∗ )] g(In+1 ) g(In+1 ) g(I ∗ ) +k3 hβf (Sn+1 )(In+1 − I ∗ )( − ∗ ) In+1 I g(I ∗ ) +k3 hβ ∗ (In+1 − I ∗ )(f (Sn+1 ) − f (S ∗ )). I

+k2 h

∗

From assumption (H1 ) and d2 + γ = βf (S ∗ ) g(II ∗ ) , we have g(In+1 ) − g(I ∗ ) [βf (S ∗ )(g(In+1 ) − g(I ∗ )) − (d2 + γ)(In+1 − I ∗ )] k2 g(In+1 ) g(In+1 ) − g(I ∗ ) = k2 [βf (S ∗ )g(In+1 ) − (d2 + γ)In+1 ] g(In+1 ) k2 βf (S ∗ )In+1 g(In+1 ) g(I ∗ ) = (g(In+1 ) − g(I ∗ ))[ − ∗ ]≤0 g(In+1 ) In+1 I 16 1283

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and k3 βf (Sn+1 )(In+1 − I ∗ )(

g(In+1 ) g(I ∗ ) − ∗ ) ≤ 0. In+1 I

Hence, Vn+1 − Vn ≤ −d1 h(Sn+1 − S ∗ )2 − d2 h(In+1 − I ∗ )2 −h[k4 (d3 + δ) + d3 ](Rn+1 − R∗ )2 − (d1 + d2 )h(Sn+1 − S ∗ )(In+1 − I ∗ ) −(d1 + d3 − k4 η)h(Rn+1 − R∗ )(Sn+1 − S ∗ ) −k1 (d1 + η)h(Sn+1 − S ∗ )(f (Sn+1 ) − f (S ∗ )) +k1 hδ(Rn+1 − R∗ )(f (Sn+1 ) − f (S ∗ )) g(I ∗ ) (In+1 − I ∗ )(f (Sn+1 ) − f (S ∗ )) I∗ = −d1 h(Sn+1 − S ∗ )2 − d2 h(In+1 − I ∗ )2 +k3 hβ

−h[k4 (d3 + δ) + d3 ](Rn+1 − R∗ )2 − (d1 + d2 )h(Sn+1 − S ∗ )(In+1 − I ∗ ) −(d1 + d3 − k4 η)h(Rn+1 − R∗ )(Sn+1 − S ∗ ) f (Sn+1 ) − f (S ∗ ) Sn+1 − S ∗ f (Sn+1 ) − f (S ∗ ) +k1 hδ(Rn+1 − R∗ )(Sn+1 − S ∗ ) Sn+1 − S ∗ ∗ f (Sn+1 ) − f (S ∗ ) g(I ) +k3 hβ ∗ (In+1 − I ∗ )(Sn+1 − S ∗ ) I Sn+1 − S ∗ ∗ ∗ ∗ = −h[(Sn+1 − S , In+1 − I )P (Sn+1 − S , In+1 − I ∗ )T −k1 (d1 + η)h(Sn+1 − S ∗ )2

+(Sn+1 − S ∗ , Rn+1 − R∗ )Q(Sn+1 − S ∗ , Rn+1 − R∗ )T ], where P =

(

k1 (d1 + η)F (Sn+1 , S ∗ ) p12 p12

)

( ,

Q=

d2

d1

q12

)

q12 [d3 + k4 (d3 + δ)]

with

1 g(I ∗ ) p12 = (d1 + d2 − k3 β ∗ F (Sn+1 , S ∗ )), 2 I 1 q12 = (d1 + d3 − k4 η − k1 δF (Sn+1 , S ∗ )). 2 ∗ Further, we choose k1 = K1 and k3 = K3 g(II ∗ ) , then assumption (H3 ) implies

that matrices P and Q are positive definite. This implies that Vn+1 − Vn < 0 for all (Sn , In , Rn ) ̸= (S ∗ , I ∗ , R∗ ). By the Lyapunov’s theorems on the globally asymptotical stability for difference equations [28], we directly obtained that the endemic equilibrium E ∗ is globally asymptotically stable. This completes the proof. 17 1284

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Remark 8. From the above discussion we immediately see that constant R0 is the basic reproduction number of model (3) and it can completely determine the global asymptotic stability of model (3). Remark 9. From the above discussions we easily see that assumption (H2 ) only is used to ensure the positivity of constant K1 . When η = 0, that is, there is not vaccination in susceptible, then assumption (H2 ) naturally holds. As consequences of Theorems 4 and 5, combining Remarks 3 and 4 we have the following corollaries. Corollary 1. Assume that in model (3) f (S) ≡ S and (H1 ) and (H2 ) hold. (1). If R0 ≤ 1, then disease-free equilibrium E 0 of model (3) is globally asymptotically stable. (2). If R0 > 1, then endemic equilibrium E ∗ of model (3) is globally asymptotically stable. Corollary 2. Assume that in model (3) f (S) ≡

S , 1+ωS

(H1 ) and (H2 ), and

conditions (5) and (6) hold. (1). If R0 ≤ 1, then disease-free equilibrium E 0 of model (3) is globally asymptotically stable. (2). If R0 > 1, then endemic equilibrium E ∗ of model (3) is globally asymptotically stable. Remark 10. In [3], the following backward difference scheme for SIRS epidemic model with the bilinear incidence is studied Sn+1 = Sn + B − µ1 Sn+1 − βSn+1 In+1 + δRn+1 In+1 = In + βSn+1 In+1 − (µ2 + γ)In+1

(19)

Rn+1 = Rn + γIn+1 − (µ3 + δ)Rn+1 . The condition µ1 ≤ min{µ2 , µ3 } is required. By constructing the discrete Lyapunov 0

∗

functions UδE and UδE (see the proof of Theorem 2.1 in [3]), the authors established that if the basic reproduction number R0 ≤ 1, then disease-free equilibrium E 0 of model (19) is globally asymptotically stable, and if R0 > 1, then endemic equilibrium E ∗ of model (19) is globally asymptotically stable. 0

∗

By computing, we easily see that the Lyapunov functions UδE and UδE are not applicable for model (3). Therefore, in this paper we construct a class of new Lyapunov functions to study the global asymptotic stability of model (3).

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Furthermore, we also see that above Corollary 1 is an extension of the main results given in [3] in the nonlinear incidence case. Remark 11. Analyzing the conditions and results given in Corollary 1 and Theorems 3 given in Section 3, we can propose an important and interesting open problem for general model (3): whether only when assumption (H1 ) holds, we can obtain that the disease-free equilibrium is globally asymptotically stable if and only if R0 ≤ 1, and the endemic equilibrium is globally asymptotically stable if and only if R0 > 1.

5. Conclusion In [4], the dynamical properties of the forward difference scheme for a class of SIRS epidemic models with general nonlinear incidence are investigated. It is shown that when step size h is small enough the disease-free equilibrium and endemic equiloibrium are local asymptotically stable, and along step size h increase, the scheme will occur the bifurcation phenomena. In this paper, the dynamical properties of the backward difference scheme for a class of SIRS epidemic models with nonlinear incidence βf (S)g(I) are investigated. From the main results obtained in this paper, we see that the backward difference scheme, that is discrete dynamical model (3), is provided for us with excellent dynamical properties for any step size h in the local and global stability of equilibria. These properties nearly are same to corresponding continuous-time model (1). Furthermore, we also see that the results on the global asymptotic stability of the endemic equilibrium for the backward difference scheme for SIRS epidemic model with bilinear incidence obtained in [3] are directly extended. By constructing new discrete Lyapunov functions we established the sufficient and necessary conditions on the global asymptotic stability of the disease-free equilibrium and endemic equilibrium for a class of discrete SIRS epidemic models with general nonlinear incidence βf (S)g(I), vaccination in susceptible and different death rates d1 , d2 and d3 . That is, under assumptions (H1 ) − (H3 ), the disease-free equilibrium is globally asymptotically stable if and only if basic reproduction number R0 ≤ 1, and the endemic equilibrium is globally asymptotically stable if and only if R0 > 1. However, we also see that assumption (H3 ) is very strong. For the local stability of the disease-free equilibrium and endemic equilibrium for model (3), the assump19 1286

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tion (H3 ) is not required. Therefore, an interesting and important open problem is whether the assumption (H3 ) can be weakened in the studies of the global stability of equilibria of model (3). On the other hand, we know that there is the nonstandard difference scheme to discretize continuous-time model (1) with nonlinear incidence. For the the nonstandard difference scheme of model (1) whether we also can establish the same results, like in this paper, still is an interesting open problem.

Acknowledgements This work is supported by the National Natural Science Foundation of China (Grant Nos. 11271312, 11261058).

References [1] M. Sekiguchi, E. Ishiwata, Global dynamics of a discretized SIRS epidemic model with time delay, J. Math. Anal. Appl., 371, 195-202 (2010). [2] M. Sekiguchi, Permanence of a discrete SIRS epidemic model with time delays, Appl. Math. Letters, 23, 1280-1285 (2010). [3] Y. Enatsu, Y. Muroya, A simple discrete-time analogue preserving the global stability of a continuous SIRS epidemic model, Inter. J. Biomath., 6, 1350001-1 (2013). [4] Z. Hu, Z. Teng, H. Jiang, Stability analysis in a class of discrete SIRS epidemic models, Nonlinear Anal.: RWA, 13, 2017-2033 (2012). [5] L. Wang, Z. Teng, H. Jiang, Global attractivity of a discrete SIRS epidemic model with standard incidence rate, Math. Meth. Appl. Sci., 36, 601-619 (2013). [6] E.Y. Rodin, Discrete model of an epidemic, Math. Comput. Modelling, 12, 121-128 (1989). [7] L.J.S. Allen, Some discrete-time SI, SIR, and SIS epidemic models, Math. Biosci., 124, 83-105 (1994).

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[8] L.J.S. Allen, A.M. Burgin, Comparison of deterministic and stochastic SIS and SIR models in discrete time, Math. Biosci., 163, 1-33(2000). [9] R. Willox, B. Grammaticos, A.S. Carstea, A. Ramani, Epidemic dynamics: discrete-time and cellular automaton models, Physica A, 328, 13-22 (2003). [10] A. Ramani, A.S. Carstea, R. Willox, B. Grammaticos, Oscillating epidemics: a discrete-time model, Physica A, 333, 278-292 (2004). [11] J. Satsuma, R. Willox, A. Ramani, B. Grammaticos, A.S. Carstea, Extending the SIR epidemic model, Physica A, 336, 369-375 (2004). [12] A. D’Innocenzo, F. Paladini, L. Renna, A numerical investigation of discrete oscillating epidemic models, Physica A, 364, 497-512 (2006). [13] M.K. Oli, M. Venkataraman, P.A. Klein, L.D. Wendland, M.B. Brown, Population dynamics of infectious disease: A discrete time model, Ecol Modelling, 198, 183-194 (2006). [14] G. Izzo, A. Vecchio, A discrete time version for models of population dynamics in the presence of an infection, J. Comput. Appl. Math., 210, 210-221 (2007). [15] L.J.S. Allen, P. van den Driessche, The basic reproduction number in some discrete-time epidemic models, J. Diff. Equat. Appl., 14, 1127-1147 (2008). [16] Y. Enatsu, Y. Nakata, Y. Muroya, Globa stability for a class of discrete SIR epidemic models, Math. Biosci. Eng., 7, 347-361 (2010). [17] M. Sekiguchi, E. Ishiwata, Dynamics of a discretized SIR epidemic model with pulse vaccination and time delay, J. Comput. Appl. Math., 236, 997-1008 (2011). [18] Y. Muroya, Y. Nakata, G. Izzo, A. Vecchio, Permanence and global stability of a class of discrete epidemic models, Nonlinear Anal.: RWA, 12, 2105-2117 (2011). [19] Y. Enatsu, Y. Nakata, Y. Muroya, G. Izzo, A. Vecchio, Global dynamics of difference equations for SIR epidemic models with a class of nonlinear incidence rates, J. Diff. Equat. Appl., 18, 1163-1181 (2012). 21 1288

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[20] X. Ma, Y. Zhou, H. Cao, Global stability of the endemic equilibrium of a discrete SIR epidemic model, Adv. Diff. Equat., 2013, 42 (2013). [21] P.L. Salceanu, Robust uniform persistence in discrete and continuous nonautonomous systems, J. Math. Anal. Appl., 398, 487-500 (2013). [22] R.E. Mickens, A SIR-model with square-root dynamics: An NSFD scheme, J. Diff. Equ. Appl., 16, 209-216 (2010). [23] R.E. Mickens, Nonstandard Finite Difference Model of Differential Equations, World Scientific, Singapore, 1994. [24] R.E. Mickens, Application of Nonstandard Finite Difference Schemes, World Scientific, Singapore, 2000. [25] R.E. Mickens, Nonstandard finite difference schemes for differential equations, J. Diff. Equ. Appl., 8, 823-847 (2002). [26] R.E. Mickens, Numerical integration of population models satisfyng conservation laws: NSFD methods, J. Biol. Dyn., 1, 427-436 (2007). [27] X. Zhao, Dynamical Systems in Population Biology, Springer-Verlag, New York, 2003. [28] J.P. LaSalle. The Stability of Dynamical Systems, Society for Industrial and Applied Mathematics, Philadelphia, 1976.

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Bounds for the largest eigenvalue of nonnegative tensors Jun He∗, School of mathematics and computer science, Zunyi Normal College, Zunyi, Guizhou, 563002, P.R. China

Abstract In this paper, we establish some eigenvalue properties of nonnegative tensors. We derive new bounds for the largest eigenvalue (Z-eigenvalue, H-eigenvalue, and B-eigenvalue) of nonnegative tensors. Numerical examples show the efficiency of these bounds. Key words: Nonnegative tensor; Spectral radius; Eigenvalue; Bound AMSC (2010): 15A18; 15A69; 65F15; 65F10

1

Introduction

Eigenvalue problems of higher order tensors have become an important topic of study in a new applied mathematics branch, numerical multilinear algebra, and they have a wide range of practical applications [2, 5, 6, 7, 8, 9, 10, 17]. First, we recall some definitions on tensors. Let R be the real field. An m-th order n dimensional square tensor A consists of nm entries in R, which is defined as follows: A = (Ai1 i2 ···im ), Ai1 i2 ···im ∈ R, 1 ≤ i1 , i2 , · · · im ≤ n. A is called nonnegative if Ai1 i2 ···im ≥ 0. To an n-vector x, real or complex, we define the n-vector:  n   X  Axm−1 =  aii2 ···im xi2 · · · xin  , i2 ,··· ,im =1

x ∗

[m−1]

=

1≤i≤n

(xim−1 )1≤i≤n .

E-mail: [email protected]

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In this paper, we continue this research on the eigenvalue problems for tensors. In section 2, bounds for the largest Z-eigenvalue are obtained, and proved to be tighter than that in Corollary 4.5 in [16]. In section 3, bounds for the largest H-eigenvalue are given. Moreover, the upper bound for the largest B-eigenvalue is presented in section 4.

2

Notation and preliminaries.

The following two definitions were first introduced and studied by Qi and Lim [4, 11]. Definition 2.1. Let A be an m-order and n-dimensional tensor. A pair (λ, x) ∈ C × (Cn \ {0}) is called an eigenvalue-eigenvector (or simply eigenpair) of A if they satisfy the equation Axm−1 = λx[m−1] . We call (λ, x) an H-eigenpair if they are both real. Definition 2.2. Let A be an m-order and n-dimensional tensor. A pair (λ, x) ∈ C × (Cn \ {0}) is called an E-eigenvalue and E-eigenvector (or simply E-eigenpair) of A if they satisfy the equation     Axm−1 = λx, (1)    xT x = 1. We call (λ, x) an Z-eigenpair if they are both real. Recently, Chang et al. [1, 2] generalized the notion of eigenvalues of higher order tensors to tensor pairs (or tensor pencils). Definition 2.3. Let A, B be two m-order and n-dimensional tensors. A pair (λ, x) ∈ C × (Cn \ {0}) is called an B-eigenvalue and B-eigenvector of A relative to B if they satisfy the equation Axm−1 = λBx[m−1] . The following definition for irreducibility has been introduced in [1, 11]. Definition 2.4. The tensor A is called reducible if there exists a nonempty proper index subset J ⊂ {1, 2, · · · , n} such that ai1 ,i2 ,··· ,im = 0, ∀i1 ∈ J, ∀i2 , · · · , im < J. If A is not reducible, then we call A to be irreducible. In this paper, let N = {1, 2, . . . , n}, we define the ith row sum of A as Ri (A) = n P aii2 ···im , and denote the largest and the smallest row sums of A by

i2 ,··· ,im =1

Rmax (A) = max Ri (A), Rmin (A) = min Ri (A). i=1,··· ,n

i=1,··· ,n

Furthermore, a real tensor of order m dimension n is called the unit tensor, if its entries are δi1 ...im for i1 , . . . , im ∈ N, where ( 1, if i1 = . . . = im δi1 ...im = 0, otherwise. 2

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And we define ri (A) =

X δii2 ...im =0

3

X

aii2 ...im , rij (A) =

aii2 ...im = ri (A) − ai j... j .

δii2 ...im =0, δ ji2 ...im =0

Bounds for the largest Z-eigenvalue.

First, we list some results about the largest Z-eigenvalue of tensors. Definition 3.1. Let A be an m-order and n-dimensional tensor. We define σ(A) the Z-spectrum of A by the set of all Z-eigenvalues of A. Assume σ(A) , ∅, then the Z-spectral radius of A is denoted by ρ(A) = max{|λ| : λ ∈ σ(A)}. In [3], Chang, Pearson, and Zhang gave the following bounds for the Z-eigenvalues of an m-order n-dimensional tensor A. Lemma 3.2. (Proposition 3.3 in [3]) Let A be an m-order and n-dimensional tensor. Then n X √ a . (2) ρ(A) ≤ n max i∈N

ii2 ...im

i2 ,...,im =1

For the positively homogeneous operators, Song and Qi [16] studied the relationship between the Gelfand formula and the spectral radius as well as the upper bound of the spectral radius. From Corollary 4.5 in [16], we can get the following Lemma: Lemma 3.3. (Corollary 4.5 in [16]) Let A be an m-order and n-dimensional tensor. Then n X a . ρ(A) ≤ max (3) ii2 ...im

i∈N

i2 ,...,im =1

Obviously, the bound in (3) is better than the bound in (2). Here, we give another proof of Lemma 3.3, which is very simple. Proof. Suppose that λ is an Z-eigenvalue of A with eigenvector x. Assume that |xi | = max |x j |. j∈N

Consider the i-th equation of (1). We have λxi =

n X

aii2 ···im xi2 . . . xim .

i2 ,··· ,im =1

By |xi | ≤

√

m−1

|xi | ≤ 1, we can get n X

n X xi2 xim aii2 ...im . |λ| ≤ |aii2 ···im | m−1 √ . . . m−1 √ ≤ |xi | |xi | i2 ,...,im =1 i2 ,··· ,im =1 3

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Thus, we complete the proof. 2 Note that λ and x may be non-real here. A tensor A is called weakly symmetric if the associated homogeneous polynomial Axm satisfies ∇Axm = mAxm−1 . This concept was first introduced and used by Chang, Pearson and Zhang [3] for studying the properties of Z-eigenvalue of nonnegative tensors and presented the following Perron-Frobenius Theorem for the Z-eigenvalue of nonnegative tensors, which was later reproved as Lemma 4.7 by Song and Qi in [16], using a different technique. Lemma 3.4. Suppose that m-order n-dimensional tensor A is weakly symmetric, nonnegative and irreducible. Then ρ(A) is a positive Z-eigenvalue with a positive Z-eigenvector. Based on the above Lemma, we give the main result of this section. Theorem 3.5. Suppose that m-order n-dimensional tensor A is weakly symmetric, nonnegative and irreducible. Then 1 1 {ai...i + a j... j + rij (A) + ∆i,2 j (A)}, i, j∈N, j,i 2

ρ(A) ≤ max where

∆i, j (A) = (ai...i − a j... j + rij (A))2 + 4ai j... j r j (A). Proof. Let x = (x1 , . . . , xn )T be an Z-eigenvector of A corresponding to ρ(A), that is, Axm−1 = ρ(A)x,

(4)

Let xt ≥ x s ≥ max{xk , k = 1, . . . , n, k , t, s}. Obviously, by Lemma 3.4, we have xt > 0, x s > 0. From Corollary 4.10 in [3], we have ρ(A) − ai...i ≥ 0, i = 1, . . . , n. Consider the equation of (1), by xtm−1 ≤ xt , xm−1 ≤ x s , we can get s X + at...t (xtm−1 − xt ) (ρ(A) − at...t )xt = ati2 ...im xi2 . . . xim + ats...s xm−1 s δti2 ...im =0, δ si2 ...im =0

≤

X

ati2 ...im xtm−1 + ats...s xm−1 s

δti2 ...im =0, δ si2 ...im =0

≤ rts (A)xt + ats...s x s ,

(5)

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equivalently, (ρ(A) − at...t − rts (A))xt ≤ ats...s x s . Moreover, from equality (1), we similarly get X (ρ(A) − a s...s )x s = a si2 ...im xi2 . . . xim + a s...s (xm−1 − xs) s δ si2 ...im =0

≤ r s (A)xtm−1 + a s...s (xm−1 − xs) s ≤ r s (A)xt .

(6)

Multiplying equation (5) and (6), we get (ρ(A) − at...t − rts (A))(ρ(A) − a s...s ) ≤ ats...s r s (A). Then, solving for ρ(A), 1 1 1 1 ρ(A) ≤ {ai...i + a j... j + rij (A) + ∆i,2 j (A)} ≤ max {ai...i + a j... j + rij (A) + ∆i,2 j (A)}. i, j∈N, j,i 2 2

Thus, we complete the proof. 2 From Theorem 3.5 in [12], we know that Xn 1 1 j 2 aii2 ...im , max {ai...i + a j... j + ri (A) + ∆i, j (A)} ≤ max i2 ,...,im =1 i, j∈N, j,i 2 i∈N that is to say, our new bound in Theorem 3.5 is always better than the bound in Lemma 3.3. We now show the efficiency of the upper bound in Theorem 3.5 by the following example which was introduced in [3]. Example 3.1. Consider the tensor A = (ai jkl ) of order 4 dimension 2 with entries defined as follows: 1 1 a1111 = , a2222 = 3, and ai jkl = elsewhere. 2 3 By Lemma 3.2, we have ρ(A) ≤ 10.6666. By Lemma 3.3, we have ρ(A) ≤ 5.3333. By Theorem 3.5, we have ρ(A) ≤ 5.1667. In fact, ρ(A) = 3.1092. Hence, the bound in Theorem 3.5 is tight and sharper.

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4

Bounds for the largest H-eigenvalue

In this section, we give the lower bound and the upper bound for the largest Heigenvalue of an m-order n-dimensional nonnegative tensor A. Definition 4.1. Let A be an m-order and n-dimensional tensor. We define the H-spectrum of A, denoted H(A) to be the set of all H-eigenvalues of A. Assume H(A) , 0, then the H-spectral radius of A, denoted µ(A), is defined as µ(A) = max{|λ| : λ ∈ H(A)}. First, we introduce some results for H-eigenvalue of nonnegative tensors [1, 13, 14], which are generalized from nonnegative matrices. Theorem 4.2. If A is irreducible and nonnegative, then there exists a number µ(A) > 0 and a vector x0 > 0, such that Ax0m−1 = µ(A)x0[m−1] . Moreover, if λ is an eigenvalue of A, then |λ| ≤ µ(A). Lemma 4.3. (Lemma 5.2 in [13]) Let A be an m-order and n-dimensional nonnegative tensor. Then Rmin (A) ≤ µ(A) ≤ Rmax (A). (7) According to some eigenvalue inclusion theorems, Li, Li and Kong [12] obtained the following upper bound for the spectral radius of a nonnegative tensor, which is sharper than the upper bound in Lemma 4.3. Lemma 4.4. (Theorem 3.3 in [12]) Suppose that m-order n-dimensional tensor A is nonnegative. Then  1 1 ai...i + a j... j + rij (A) + ∆i,2 j (A) , µ(A) ≤ max i, j∈N, j,i 2 where

 2 ∆i, j (A) = ai...i − a j... j + rij (A) + 4ai j... j r j (A).

In the following Theorem, we give new bounds for the spectral radius of a nonnegative tensor. Theorem 4.5. Suppose that m-order n-dimensional tensor A is nonnegative. Then   1 1 1 1 j j ai...i + a j... j + ri (A) + ∆i,2 j (A) ≤ µ(A) ≤ max min ai...i + a j... j + ri (A) + ∆i,2 j (A) , j∈N, j,i 2 i∈N j∈N, j,i 2

min max i∈N

where

 2 ∆i, j (A) = ai...i − a j... j + rij (A) + 4ai j... j r j (A).

Proof. First, we assume that tensor A is strictly positive and let x be the unique positive eigenvector corresponding to µ(A), i.e. Axm−1 = µ(A)x[m−1] .

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Assume 0 < xt = max xi , then, for any s , t, we can get i∈N

X

(µ(A) − at...t )xtm−1 − ats...s xm−1 = s

ati2 ...im xi2 . . . xim ,

δti2 ...im =0, δ si2 ...im =0

X

(µ(A) − a s...s )xm−1 − a st...t xtm−1 = s

a si2 ...im xi2 . . . xim .

δti2 ...im =0, δ si2 ...im =0

Solving for xt we obtain X

((µ(A) − a s...s )(µ(A) − at...t ) − a st...t ats...s ) xtm−1 = (µ(A) − a s...s )

ati2 ...im xi2 . . . xim

δti2 ...im =0, δ si2 ...im =0

X

+ ats...s

a si2 ...im xi2 . . . xim .

(8)

δti2 ...im =0, δ si2 ...im =0

Recalling that 0 < xt = max xi , we have i∈N

(µ(A) − a s...s )(µ(A) − at...t ) − a st...t ats...s = (µ(A) − a s...s )

X

ati2 ...im

δti2 ...im =0, δ si2 ...im =0

+ ats...s

X δti2 ...im =0, δ si2 ...im =0

a si2 ...im

xi xi2 ... m xs xs

xi xi2 ... m xs xs

≤ (µ(A) − a s...s )rts (A) + ats...s rts (A).

(9)

Therefore

 1 1 2 at...t + a s...s + rts (A) + ∆t,s (A) . 2 This must be true for every s , t, then, we get  1 1 µ(A) ≤ min at...t + a j... j + rtj (A) + ∆t,2 j (A) . j∈N, j,t 2 µ(A) ≤

And this could be true for any t ∈ N, that is  1 1 µ(A) ≤ max min ai...i + a j... j + rij (A) + ∆i,2 j (A) . i∈N j∈N, j,i 2 Similarly, assume 0 < xT = min xi , we can get i∈N

 1 1 ai...i + a j... j + rij (A) + ∆i,2 j (A) . j∈N, j,i 2

µ(A) ≥ min max i∈N

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If the tensor A is not strictly positive, we denote by D = (di1 ...im ) the m-order ndimensional tensor with di1 ...im = 1, for all i1 ∈ N, . . . , im ∈ N. Hence, A + tD is strictly positive for any chosen positive real number t, and then letting t → 0, the result follows by continuity. 2 From the proof of the Theorem 3.5 in [12], we can get the following result:     1 1 j j 1 1 2 2 max min 2 ai...i + a j... j + ri (A) + ∆i, j (A) ≤ max 2 ai...i + a j... j + ri (A) + ∆i, j (A) i∈N j∈N, j,i

i, j∈N, j,i

≤ R, where

 2 ∆i, j (A) = ai...i − a j... j + rij (A) + 4ai j... j r j (A).

We now compare the lower bound in Theorems 4.5 with that in Lemma 4.3. Theorem 4.6. Suppose that m-order n-dimensional tensor A is nonnegative. Then  1 1 ai...i + a j... j + rij (A) + ∆i,2 j (A) , Rmin (A) ≤ min max i∈N j∈N, j,i 2 where

 2 ∆i, j (A) = ai...i − a j... j + rij (A) + 4ai j... j r j (A).

Proof. First, we assume that tensor A is strictly positive. Equivalently, we will prove that, if  1 1 ai...i + a j... j + rij (A) + ∆i,2 j (A) . µ(A) ≥ min max i∈N j∈N, j,i 2 Then, we can get µ(A) ≥ Rmin (A). If µ(A) satisfies the lower bound in the Theorem 4.5 and the matrix is positive, similar to the proof of Theorem 4.5, if assume 0 < xT = min xi , for any s , T , we can get i∈N

(µ(A) − a s...s )(µ(A) − aT ...T ) ≥ (µ(A) − a s...s )rTs (A) + aT s...s r s (A). If we assumed that µ(A) ≤ R s (A), then we have that µ(A) − a s...s ≤ r s (A). So, (µ(A) − a s...s )(µ(A) − aT ...T − rTs (A) ≥ aT s...s r s (A) ≥ aT s...s (µ(A) − a s...s ), that is (µ(A) − a s...s )(µ(A) − aT ...T − rTs (A) − aT s...s ) ≥ 0. From Lemma 3.2 in [12], we know µ(A) − a s...s ≥ 0, then, we obtain µ(A) − aT ...T − rTs (A) − aT s...s ≥ 0, 8

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that is µ(A) ≥ RT (A) ≥ Rmin (A). If the tensor A is not strictly positive, we denote by D = (di1 ...im ) the m-order ndimensional tensor with di1 ...im = 1, for all i1 ∈ N, . . . , im ∈ N. Hence, A + tD is strictly positive for any chosen positive real number t, and then letting t → 0, the result follows by continuity. 2 We now show the efficiency of the bounds in Theorem 4.5 by the following example. Example 4.1. Consider the tensor A = (ai jkl ) of order 4 dimension 3 with entries defined as follows: a1111 = 1, a1222 = 1, a1333 = 1, a2111 = 2, a2222 = 2, a2333 = 2, a3111 = 3, a3222 = 3, a3333 = 3, and ai jkl = 0 elsewhere. By Lemma 4.3, we have 3 ≤ µ(A) ≤ 9. By Lemma 4.4, we have µ(A) ≤ 8. By Theorem 4.5, we have 5 ≤ µ(A) ≤ 7. In fact, µ(A) = 6. Hence, the bound in Theorem 4.5 is tight and sharper.

5

Bounds for the largest B-eigenvalue

In this section, we focus our attention on the largest B-eigenvalue of a m-order n-dimensional nonnegative tensor A relative to B. Definition 5.1. Let A, B be two m-order and n-dimensional tensors. We define the B-spectrum of A relative to B, denoted T (A) to be the set of all B-eigenvalues of A relative to B. Assume T (A) , 0, then the B-spectral radius of A, denoted ν(A), is defined as ν(A) = max{|λ| : λ ∈ T (A)}. For an m-order n-dimensional tensor A, let  [ 1 ] FA = Axm−1 m−1 , Song and Qi [15] proved the Perron-Frobenius property for nonnegative tensor pairs (A, B) without the requirement of the tensor inversion. Lemma 5.2. (Corollary 4.2 in [15]) Let A, B be two weakly irreducible and nonnegative tensors with order m and dimension n and FA FB = FB FA . If ∃x > 0 such 9

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that Bxm−1 ≥ x[m−1] , then A has a unique positive B-eigenvalue with a corresponding positive B-eigenvector. Based on the above Lemma, we give the main results of this section. Theorem 5.3. Under the conditions of Lemma 5.2 and bi...i > 0 for all i ∈ N. Then ν(A) ≤ max i∈N

Ri (A) . bi...i

Proof. Let x be the unique positive eigenvector corresponding to ν(A), i.e. Axm−1 = ν(A)Bxm−1 . Assume 0 < xt = max xi , then, from the i-th equation of the above equation, we can i∈N get    X  X ati2 ...im xi2 . . . xim + at...t xtm−1 = ν(A)  bti2 ...im xi2 . . . xim + bt...t xtm−1  , δti2 ...im =0

δti2 ...im =0

Dividing both sides by xtm−1 and rearranging yields     X X xim xi xi xi2   + bt...t  − at...t = ati2 ...im 2 . . . m ≤ rt (A). ν(A)  bti2 ...im . . . xt xt xt xt δ =0 δ =0 ti2 ...im

ti2 ...im

Hence, Rt (A)

ν(A) ≤

P δti2 ...im =0

x bti2 ...im xi2t

...

x im xt

+ bt...t

≤

Rt (A) . bt...t

Thus, we complete the proof. 2 If ∃x > 0 such that Bxm−1 ≥ x[m−1] and suppose that m-order n-dimensional tensor B is nonnegative and diagonal, we can get bi...i ≥ 1, for all i ∈ N. Similar to the proof of Theorem 4.5, we can get some new bounds for ν(A), including the upper bound and the lower bound. Theorem 5.4. Under the conditions of Lemma 5.2 and suppose that m-order ndimensional tensor B is nonnegative and diagonal. Then j

min max

i∈N j∈N, j,i

1

ai...i b j... j +a j... j bi...i +b j... j ri (A)+∆i,2j (A) 2bi...i b j... j

≤ ν(A) j

≤ max min

i∈N j∈N, j,i

1

ai...i b j... j +a j... j bi...i +b j... j ri (A)+∆i,2j (A) 2bi...i b j... j

,

10

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where  2 ∆i, j (A) = ai...i b j... j − a j... j bi...i + b j... j rij (A) + 4ai j... j bi...i b j... j r j (A). If bi...i = 1 for all i ∈ N, then, the results in Theorem 5.4 reduce to the result in Theorem 4.5. Acknowledgements. This research is supported by 973 Program (2013CB329404), NSFC (61170309), Chinese Universities Specialized Research Fund for the Doctoral Program (20110185110020). The first author is supported by the Fundamental Research Funds for Central Universities.

References [1] K. C. Chang, K. Pearson, T. Zhang, Perron-Frobenius theorem for nonnegative tensors, Comm. Math. Sci., 6 (2008), 507-520. [2] K. C. Chang, K. Pearson, T. Zhang, On eigenvalue problems of real symmetric tensors, J. Math. Anal. Appl., 350 (2009), 416-422. [3] K. C. Chang, K. Pearson, T. Zhang, Some variational principles for Z-eigenvalues of nonnegative tensors, Linear Algebra Appl. 438 (2013) 4166-4182. [4] L. Qi, Eigenvalues of a real supersymmetric tensor, J. Symbolic Comput., 40 (2005), 1302-1324. [5] L. Qi, Eigenvalues and invariants of tensor, J. Math. Anal. Appl., 325 (2007), 1363-1377. [6] L. Qi, Symmetric nonnegative tensors and copositive tensors, Linear Algebra Appl. 439 (1) (2013) 228-238. [7] Y. Liu, G. Zhou, N. F. Ibrahim, An always convergent algorithm for the largest eigenvalue of an irreducible nonnegative tensor, J. Comput. Appl. Math., 235 (2010), 286-292. [8] Michael K. Ng, L. Qi, G. Zhou, Finding the largest eigenvalue of a non-negative tensor, SIAM J. Matrix Anal. Appl. 31 (2009),1090-1099 [9] G. Zhou, L. Caccetta, L. Qi, Convergence of an algorithm for the largest singular value of a nonnegative rectangular tensor, Linear Algebra Appl. 438 (2013) 959-968. [10] L. Zhang, L. Qi, G. Zhou, M-tensors and the positive definiteness of a multivariate form, preprint, arXiv:1202.6431, 2012. [11] L. H. Lim, Singular values and eigenvalues of tensors: A variational approach, in Proceedings of the IEEE International Workshop on Computational Advances in MultiSensor Adaptive Processing (CAMSAP 05), Vol. 1, IEEE Computer Society Press, Piscataway, NJ, 2005, 129-132.

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[12] C. Li, Y. Li, X. Kong, New eigenvalue inclusion sets for tensors, Numerical Linear Algebra Appl. 21 (2014) 39-50. [13] Y. Yang, Q. Yang, Further results for Perron-Frobenius Theorem for nonnegative tensors, SIAM. J. Matrix Anal. Appl. 31 (2010), 2517-2530. [14] Y. Yang, Q. Yang, Further results for Perron-Frobenius Theorem for nonnegative tensors II, SIAM. J. Matrix Anal. Appl., 32 (2011)1236-1250 [15] Y. Song, L. Qi, The existence and uniqueness of eigenvalues for monotone homogeneous mapping pairs, Nonlinear Analysis 75 (2012), 5283-5293. [16] Y. Song, L. Qi, Spectral properties of positively homogeneous operators induced by higher order tensors, SIAM. J. Matrix Anal. Appl. 34(2013), 1581-1595. [17] J. He, T. Huang, Upper bound for the largest Z-eigenvalue of positive tensors, Applied Mathematics Letters 38(2014), 110-114. [18] A. Berman, R. J. Plemmons, Nonnegative Matrices in the Mathematical Sciences, Classics in Appl. Math. 9, SIAM, Philadelphia, 1994.

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A note on fractional neutral integro-differential inclusions with state-dependent delay in Banach spaces Selvaraj Suganya∗,

Dumitru Baleanu,† Mani Mallika Arjunan‡

Abstract We have applied different fixed point theorems to examine the existence results for fractional neutral integro-differential inclusions (FNIDI) with state-dependent delay (SDD) in Banach spaces. We tend to conjointly discuss the cases once the multivalued nonlinear term takes convex values further as nonconvex values. An example is offered to demonstrate the obtained results. Keywords: Fractional order differential equations, state-dependent delay, multivalued map, fixed point theorem, Banach spaces, semigroup theory. 2010 Mathematics Subject Classification: 26A33, 34A08, 35R12, 34A60, 34G20, 34K05, 45J05.

1

Introduction The aim of the manuscript is to investigate the existence of mild solutions for neutral integro-differential

inclusions of fractional order as given below  d x(t) − G (t, x%(t,xt ) ) ∈ dt

Z 0

t

 (t − s)α−2  A x(s) − G (s, x%(s,xs ) ) ds Γ(α − 1) + F (t, x%(t,xt ) ),

a.e. t ∈ I = [0, b],

x0 = ς ∈ B,

(1.1) (1.2)

such that 1 < α < 2 and A : D(A ) ⊂ X → X denotes a linear densely defined operator of sectorial type on a complex Banach space (X, | · |), the convolution integral within the equation is understood because the Riemann-Liouville fractional integral (see [4]) and F : I × B → P(X) represents a multivalued map (P(X) is the family of nonempty subsets of X), G : I × B → X, and % : I × B → (−∞, b] are apposite functions, and B is a theoretical phase space axioms outlined in Preliminaries. We recall that for any continuous function x defined on (−∞, b] and for any t ≥ 0, we designate by xt the part of B defined by xt (θ) = x(t + θ) for θ ≤ 0. Here xt (·) speaks to the historical backdrop of the state from every θ ∈ (−∞, 0] likely the current time t. Fractional differential equations have picked up hefty grandness as a final result of their exertion in numerous field of science and engineering. In the latest years, there has been a major growth in differential systems ∗ Department

of Mathematics, C. B. M. College, Kovaipudur, Coimbatore - 641 042, Tamil Nadu, India.

E. Mail: selvarajsug-

[email protected] † Corresponding Author: Department of Mathematics and Computer Sciences, Faculty of Arts and Sciences, Cankaya University, 06530 Ankara, Turkey and Institute of Space Sciences, Magurele-Bucharest, Romania, E. Mail: [email protected] ‡ Department of Mathematics, C. B. M. College, Kovaipudur, Coimbatore - 641 042, Tamil Nadu, India. E. Mail: [email protected]

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comprising fractional derivatives, e.g. the monographs of Abbas et al. [5], Baleanu et al. [6], Podlubny [7], Diethelm [8], Kilbas et al. [9], Tarasov [10] and Anastassiou [11], and the papers [12, 13, 14, 15, 16, 17, 18, 19], and the references cited therein. As it is known, a delay differential equation (DDE) may be a special sort of functional differential equation (FDE). FDEs with SDD seem often in applications as models of equations and for this intention the report of this kind of equations received nice care in latest years. For points of interest, we recommend the reader to check the papers by by Abada et al. [20], Ait Dads et al. [21], Anguraj et al. [22], Benchohra et al. [23], Cuevas et al. [24], Hernandez et al. [25, 26], Mallika Arjunan et al. [27] and Yan et al. [28]. In the situation where F is either a single or a multivalued map, the problem (1.1)-(1.2) with G = 0 was investigated on a compact interval in Agarwal et al. [29], Benchohra et al. [30, 31]. On unbounded interval when F is a single map, the problem (1.1)-(1.2) with G = 0 was discussed by Benchohra et al. [32]. According to the knowledge of the authors , there is no work on the existence results for FNIDI with SDD in Banach spaces, which is communicated in the structure (1.1)-(1.2). Roused by this thought, in this paper, we concentrate on this problem, which is common generalizations of the idea of mild solution for fractional neutral equations well known in the theory of integer order systems. This manuscript has the following structure. In section 2, we show some preliminaries and lemmas to be utilized to demonstrate our primary results. In section 3, we show two results for the problem (1.1)-(1.2) when the right-hand side is convex valued. The principal result is focused on a fixed point theorem of BohnenblustKarlin [1], and the second one on the nonlinear alternative of Leray-Schauder type [2]. The final existence result is given for a nonconvex valued right-hand side by utilizing a fixed point theorem for contraction multivalued maps thanks to Covitz and Nadler [3]. An application is presented in Section 4.

2

Preliminaries Let C(I , X) be the Banach space of all continuous functions from I into X with the norm kxk∞ = sup{|x(t)| : t ∈ I }. Let B(X) signifies the Banach space of all bounded linear operators from X into X. A measurable function x : I → X is Bochner integrable if and only if |x| is Lebesgue integrable. ( For extra

insights about Bochner integral, see Yosida [33]). Let L1 (I , X) signify the Banach space of all continuous functions x : I → X which are Bochner integrable and have norm Z kxkL1 =

b

|x(t)|dt for all x ∈ L1 (I , X).

0

We expect that the phase space (B, k · kB ) is a semi-normed linear space of functions mapping (−∞, 0] into X, and fulfilling the subsequent elementary adages as a result of Hale and Kato ( see more details in [34, 35]). (P1 ) If x : (−∞, b] → X, b > 0, is continuous on I and x0 ∈ B, then for every t ∈ I the accompanying conditions hold: (i) xt is in B; (ii) |x(t)| ≤ Hkxt kB ;

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(iii) kxt kB ≤ D1 (t) sup{|x(s)| : 0 ≤ s ≤ t} + D2 (t)kx0 kB , where H > 0 is a constant and D1 (·) : [0, +∞) → [0, +∞) is continuous, D2 (·) : [0, +∞) → [0, +∞) is locally bounded, and D1 , D2 are independent of x(·). (P2 ) For the function x(·) in (P1 ), xt is a B-valued continuous function on I . (P3 ) The space B is complete. Designate D1∗ = sup{D1 (t) : t ∈ I } and D2∗ = sup{D2 (t) : t ∈ I }. Now, we briefly review some known results from the solution operator. The Laplace transformation of a function f ∈ L1loc (R+ , X) is defined by ∞

Z

e−λt f (t)dt,

L(f )(λ) = fb(λ) =

Re(λ) > ω,

0

if the integral is definitely convergent for Re(λ) > ω. We mention the subsequent definition [4]. Definition 2.1. Let A : D(A ) ⊂ X → X be a closed and linear operator on a Banach space X. We call A is the generator of a solution if there exist ω ∈ R and a strongly continuous function Sα : R+ → B(X) such that {λα : and λ

α−1

(λ − A ) α

−1

Re (λ) > ω} ⊂ ρ(A ), Z

x=

∞

e−λt Sα (t)xdt,

Re λ > ω,

x ∈ X.

0

In this case, Sα (t) is called operator function created by A . The idea of a solution operator, as characterized above, is nearly identified with the ones of a resolvent family [37]. Having in mind the uniqueness of the Laplace transform, in the fringe case α = 1, the family Sα (t) relates to a strongly continuous semigroup (see Pazy [38]), while in the case α = 2 a solution operator relates to the idea of a cosine family; see [39]. The subsequent result is an immediate outcome of [40, Proposition 3.1 and Lemma 2.2]. Proposition 2.1. Let Sα (t) be a solution operator on X with generator A . Then, we have (a) Sα (t) is strongly continuous for t ≥ 0 and S(0) = I; (b) Sα (t)D(A ) ⊂ D(A ) and A Sα (t)x = Sα (t)A x for all x ∈ D(A ), t ≥ 0; (c) For every x ∈ D(A ) and t ≥ 0, Z Sα (t)x = x + 0

Z (d) Let x ∈ D(A ). Then 0

t

t

(t − s)α−1 A Sα (s)xds. Γ(α)

(t − s)α−1 Sα (s)xds ∈ D(A ) and Γ(α) Sα (t)x = x + A

Z

t

0

(t − s)α−1 Sα (s)xds. Γ(α)

Definition 2.2. A solution operator {Sα (t)}t>0 is called uniformly continuous if lim kSα (t) − Sα (s)kB(X) = 0.

t→s

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Before we finish this section, we review some known results from multivalued analysis that we will apply in the spin-off. We recall that P(X) = {Y ⊂ X : Y 6= ∅}, Pcl (X) = {Y ∈ P(X) : Y closed},

Pb (X) = {Y ∈ P(X) : Y bounded },

Pcp (X) = {Y ∈ P(X) : Y compact}, Pcp,c (X) = {Y ∈ P(X) : Y compact and convex}. Remark 2.1. For extra points of interest on this, we suggest the reader to [13]. Definition 2.3. The multivalued map F : I × B → P(X) is said to be Carath´eodory if (i) t 7→ F (t, u) is measurable for each u ∈ B; (ii) u 7→ F (t, u) is upper semicontinuous for almost all t ∈ I . Let SF ,x be a set characterized by SF ,x = {v ∈ L1 (I , X) : v(t) ∈ F (t, x%(t,xt ) ) a.e. t ∈ I }. Definition 2.4. A multivalued operator Υ : X → Pcl (X) is called: (a) Λ-Lipschitz if there exists Λ > 0 such that Hd (Υ(x), Υ(x)) ≤ Λ d(x, x) for all x, x ∈ X; (b) a contraction if it is Λ-Lipschitz with Λ < 1. Presently, we express the accompanying lemmas which are important to make our primary result. Lemma 2.1 ([41]). Let X be a Banach space. Let F : I × B → Pcp,c (X) be an L1 -Carath´eodory multivalued map and let Ψ be a linear continuous mapping from L1 (I , X) to C(I , X), then the operator Ψ ◦ SF : C(I , X) → Pcp,c (C(I , X)), x

7→ (Ψ ◦ SF )(x) := Ψ(SF ,x )

has a closed graph operator in C(I , X) × C(I , X). Lemma 2.2 (Bohnenblust-Karlin’s [1]). Let X be a Banach space and D ∈ Pcl,c (X). Suppose that the operator G : D → Pcl,c (D) is upper semicontinuous and the set G(D) is relatively compact in X. Then G has a fixed point in D. Lemma 2.3 (Covitz and Nadler [3]). Let (X, d) be a complete metric space. If Υ : X → Pcl (X) is a contraction, then Fix Υ 6= ∅. For more details on multivalued maps see the books of Graef et al. [42] and Castaing et al. [43].

3

Existence results We demonstrate below the existence of solutions for the problem (1.1)-(1.2). To start with, we delineate the

mild solution for the problem (1.1)-(1.2).

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Definition 3.1. We affirm that the function x : (−∞, b] → X is a mild solution of (1.1)-(1.2) if x(t) = ς(t) for all t ≤ 0, the constraint of x(·) to the interval [0, b] is continuous and there exists v(·) ∈ L1 (I , X), such that v(t) ∈ F (t, x%(t,xt ) ) a.e. t ∈ [0, b], and x fulfills the consecutive integral equation:   x(t) = Sα (t) ς(0) − G (0, ς(0)) + G (t, x%(t,xt ) ) +

Z

t

Sα (t − s)v(s)ds for each t ∈ I .

(3.1)

0

Let us set R(%− ) = {%(s, ς) : (s, ς) ∈ I × B, %(s, ς) ≤ 0}. We generally expect that % : I × B → (−∞, b] is continuous. Moreover, we suppose: (Hς ) The function t → ςt is continuous from R(%− ) into B and there exists a continuous and bounded function Lς : R(%− ) → (0, ∞) such that for every t ∈ R(%− ).

kςt kB ≤ Lς (t)kςkB

Lemma 3.4. [26, Lemma 3.1] If x : (−∞, b] → X is a function such that x0 = ς, then kxs kB ≤ (D2∗ + Lς )kςkB + D1∗ sup{|x(θ)| : θ ∈ [0, max{0, s}]}, where Lς =

s ∈ R(%− ) ∪ I ,

sup Lς (t). t∈R(%− )

3.1

Existence results: The convex case In this section, we are dealing with the existence results for the structure (1.1)-(1.2). We expect that F is

a compact and convex valued multivalued map and we apply Lemma 2.2 to build our first result. Thus, we have: (H1) The solution operator Sα (t)t∈I is compact for t > 0, and there is M > 0 such that kSα (t)kB(X) ≤ M,

for each t ∈ I .

(H2) The multivalued map F : I × B → Pcp,c (X) is Carath´eodory. (H3) There exists a continuous function k : I → R+ such that |F (t, u) − F (t, v)| ≤ k(t)ku − vkB , and k ∗ = sup

Z

t ∈ I , u, v, ∈ B,

t

k(s)ds < ∞.

t∈I

0

(H4) The function t → F (t, 0) = F0 ∈ L1 (I , R+ ) with F ∗ = kF0 kL1 . (H5) The function G (t, ·) is continuous on I and there exists a constant KG > 0 such that |G (t, u) − G (t, v)| ≤ KG ku − vkB ,

for each u, v ∈ B,

and G ∗ = sup |G (t, 0)| < ∞. t∈I

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(H6) For each t ∈ I and any bounded set V ⊂ B, the set {F (t, u), G (t, u) : u ∈ B} is relatively compact in X. (H7) For any bounded set V ⊂ B, the function {t → G (t, x%(t,xt ) ) : x ∈ V} is equicontinuous on I . Theorem 3.1. Assume that (H1)-(H7) and (Hς ) hold. Then, the problem (1.1)-(1.2) has a mild solution on (−∞, b] provided that h

i D1∗ (KG + M k ∗ ) < 1.

(3.2)

Proof. We will transmute the structure (1.1)-(1.2) into a fixed point problem. We conceive the set V = {x : (−∞, b] → X : x|I

is continuous and x0 ∈ B},

where x|I is the constraint of x to the real compact interval on I . Recognize the multivalued operator Υ : V → P(V) defined by Υ(h) = {h ∈ V} with   ς(t), t ≤ 0; Z t h(t) =    Sα (t) ς(0) − G (0, ς(0)) + G (t, x%(t,xt ) ) + Sα (t − s)v(s)ds,

t ∈ I,

0

where v ∈ SF ,x%(s,xs ) . For ς ∈ B, we express the function y(·) : (−∞, b] → X by  ς(t), t ≤ 0; y(t) = S (t)ς(0), t ∈ I , α

then y0 = ς. For every function z ∈ V with z0 = 0, we designate by z the function clear by  0, t ≤ 0; z(t) = z(t), t ∈ I . If x(·) fulfills (3.1), we are able to decompose it as x(t) = z(t) + y(t), t ∈ I , which suggests xt = zt + yt , for each t ∈ I and also the function z(·) fulfills z(t) = G (t, zρ(t,zt +yt ) + yρ(t,zt +yt ) ) − Sα (t)G (0, ς(0)) +

t

Z

Sα (t − s)v(s)ds,

t ∈ I,

0

where v(s) ∈ SF ,z%(s,zs +ys ) +y%(s,zs +ys ) . Let Vb0 = {z ∈ V : z(0) = 0 ∈ B}. For some z ∈ Vb0 , we have kzkb = sup kz(t)k + kz0 kB = sup kz(t)k. t∈I

t∈I

Thus Vb0 is a Banach space with the norm k·kb . We delimit the operator Υ : Vb0 → P(Vb0 ) by Υ(z) = {h ∈ Vb0 } with h(t) = G (t, zρ(t,zt +yt ) + yρ(t,zt +yt ) ) − Sα (t)G (0, ς(0)) +

Z

t

Sα (t − s)v(s)ds,

t ∈ J,

0

where v(s) ∈ SF ,z%(s,zs +ys ) +y%(s,zs +ys ) . We recall that the operator Υ has a fixed point if and only if Υ has a fixed point. Thus, let us demonstrate that Υ has a fixed point. Let Br = {z ∈ Vb0 : z(0) = 0, kzkb ≤ r}, where r is any fixed real number. It is perfect that Br is a closed, convex, bounded set in Vb0 . 6 1307

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Remark 3.1. By hypotheses (H3)-(H5) we obtain: (i) t

Z

|F (s, z%(s,zs +ys ) + y%(s,zs +ys ) ) − F (s, 0) + F (s, 0)|ds Z t Z t k(s)kz%(s,zs +ys ) + y%(s,zs +ys ) kB ds + M |F (s, 0)|ds. ≤M

M

0

(3.3)

0

0

Since kz%(s,zs +ys ) + y%(s,zs +ys ) kB ≤ kz%(s,zs +ys ) kB + ky%(s,zs +ys ) kB ≤ D1∗ |z(s)| + (D2∗ + Lς )kz0 kB + D1∗ |y(s)| + (D2∗ + Lς )kςkB ≤ D1∗ |z(s)| + (D2∗ + Lς + D1∗ M H)kςkB ≤ D1∗ |z(s)| + C1 , where C1 = (D2∗ + Lς + D1∗ M H)kςkB . Then (3.3) becomes t

Z

|F (s, z%(s,zs +ys ) + y%(s,zs +ys ) ) − F (s, 0) + F (s, 0)|ds Z t Z t ≤M k(s)kz%(s,zs +ys ) + y%(s,zs +ys ) kB ds + M |F (s, 0)|ds 0 0 Z t   ≤M k(s) D1∗ |z(s)| + C1 ds + M F ∗

M

0

0

≤ M D1∗ rk ∗ + M C1 k ∗ + M F ∗ . (ii) |G (t, zρ(t,zt +yt ) + yρ(t,zt +yt ) ) − G (t, 0) + G (t, 0)| ≤ KG kzρ(t,zt +yt ) + yρ(t,zt +yt ) kB + G ∗ ≤ KG D1∗ r + KG C1 + G ∗ . (iii) Z 0

η1

kSα (η2 − s) − Sα (η1 − s)kB(X) |F (s, z%(s,zs +ys ) + y%(s,zs +ys ) ) − F (s, 0)|ds Z η1 + kSα (η2 − s) − Sα (η1 − s)kB(X) |F (s, 0)|ds Z0 η2 + kSα (η2 − s)kB(X) |F (s, z%(s,zs +ys ) + y%(s,zs +ys ) ) − F (s, 0)|ds η1 η2

Z

kSα (η2 − s)kB(X) |F (s, 0)|ds

+ η1

≤ D1∗ r

Z

η1

kSα (η2 − s) − Sα (η1 − s)kB(X) k(s)ds Z0 η1

+ C1 kSα (η2 − s) − Sα (η1 − s)kB(X) k(s)ds Z η1 0 + kSα (η2 − s) − Sα (η1 − s)kB(X) |F (s, 0)|ds 0

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+

Z

D1∗ r

η2

kSα (η2 − s)kB(X) k(s)ds

Z

η1 η2

kSα (η2 − s)kB(X) k(s)ds

+ C1 η1

Z

η2

kSα (η2 − s)kB(X) |F (s, 0)|ds.

+ η1

Presently, we might demonstrate that Υ fulfills all the assumptions of Bohnenblust-Karlin’s theorem. For better comprehensibility, we break the verification into succession of steps. Step 1: Υ(z) is convex for each z ∈ Vb0 . In fact, if h1 and h2 have a place with Υ(z), then there exists v1 , v2 ∈ SF ,z%(s,zs +ys ) +y%(s,zs +ys ) such that, for t ∈ I , we have hi (t) = G (t, z%(t,zt +yt ) + y%(t,zt +yt ) ) − Sα (t)G (0, ς(0)) +

Z

t

Sα (t − s)vi (s)ds,

i = 1, 2.

0

Let 0 ≤ d ≤ 1. Then, for every t ∈ I , we have (λh1 + (1 − λ)h2 )(t) = G (t, z%(t,zt +yt ) + y%(t,zt +yt ) ) − Sα (t)G (0, ς(0)) +

Z

t

  Sα (t − s) λv1 (s) + (1 − λ)v2 (s) ds.

0

Since F has convex values, SF ,z%(s,zs +ys ) +y%(s,zs +ys ) is convex, we see that (λh1 + (1 − λ)h2 ) ∈ Υ(z). Step 2: Υ(Br ) ⊆ Br for some r > 0. We assert that there exists a positive number r such that Υ(br ) ⊆ Br . On the off chance that it is not true, then for every positive number r, there exists a function zr ∈ Br and h ∈ Υ(zr ) such that |h(t)| > r for some t ∈ I . Then from Remark 3.1, we have r < |h(t)| ≤ |G (t, z%(t,zt +yt ) + y%(t,zt +yt ) ) − G (t, 0) + G (t, 0)| + kSα (t)kB(X) |G (0, ς(0))| Z t + kSα (t − s)kB(X) |F (s, z%(s,zs +ys ) + y%(s,zs +ys ) ) − F (s, 0) + F (s, 0)|ds 0

≤ KG D1∗ r + KG C1 + G ∗ + M |G (0, ς(0))| + M k ∗ D1∗ r + M C1 k ∗ + M F ∗ ≤ KG D1∗ r + M k ∗ D1∗ r + C2 , where C2 = KG C1 + G ∗ + M |G (0, ς(0))| + M C1 k ∗ + M F ∗ is independent of r. Dividing both sides by r and taking the lower limit, we have h

i D1∗ (KG + M k ∗ ) ≥ 1.

This contradicts to (3.2). Hence for some positive number r, Υ(Br ) ⊆ Br . Step 3: Υ(Br ) is relatively compact. We know that Br is bounded and Υ(Br ) ⊆ Br , it is clear that Υ(Br ) is bounded. It remains to show that Υ(Br ) is equicontinuous. Let η1 , η2 ∈ I with η1 < η2 and z ∈ Υ(Br ). Then from the remark 3.1 (iii), we have |h(η2 ) − h(η1 )| ≤ |G (η2 , z%(η2 ,zη2 +yη2 ) + y%(η2 ,zη2 +yη2 ) ) − G (η1 , z%(η1 ,zη1 +yη1 ) + y%(η1 ,zη1 +yη1 ) )| + kSα (η2 ) − Sα (η1 )kB(X) |G (0, ς(0))| Z η1 + kSα (η2 − s) − Sα (η1 − s)kB(X) |F (s, z%(s,zs +ys ) + y%(s,zs +ys ) )|ds Z0 η2 + kSα (η2 − s)kB(X) |F (s, z%(s,zs +ys ) + y%(s,zs +ys ) )|ds η1

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≤ |G (η2 , z%(η2 ,zη2 +yη2 ) + y%(η2 ,zη2 +yη2 ) ) − G (η1 , z%(η1 ,zη1 +yη1 ) + y%(η1 ,zη1 +yη1 ) )| + kSα (η2 ) − Sα (η1 )kB(X) |G (0, ς(0))| Z η1 + D1∗ r kSα (η2 − s) − Sα (η1 − s)kB(X) k(s)ds 0 Z η1 + C1 kSα (η2 − s) − Sα (η1 − s)kB(X) k(s)ds Z η1 0 kSα (η2 − s) − Sα (η1 − s)kB(X) |F (s, 0)|ds + 0 Z η2 + D1∗ r kSα (η2 − s)kB(X) k(s)ds Z

η1 η2

kSα (η2 − s)kB(X) k(s)ds

+ C1 η1

Z

η2

kSα (η2 − s)kB(X) |F (s, 0)|ds.

+ η1

At the point when η2 → η1 , the right-hand side of the overhead inequality has a tendency to zero, subsequent to by (H7) and Sα (t) is uniformly continuous, this demonstrates the equicontinuity. As a result of Steps 1-3, together with the Arzela-Ascoli’s theorem, we conclude that the operator Υ is completely continuous. Step 4: Υ has a closed graph. Suppose that z n → z ∗ , hn ∈ Υ(z n ) with hn → h∗ . We claim that h∗ ∈ Υ(z ∗ ). In fact, assumption hn ∈ Υ(z n ) n suggests that there exists vn ∈ SF ,z%(s,z n +y s

hn (t) =

n G (t, z%(t,z n t +yt )

s)

+y%(s,zsn +ys )

such that, for every t ∈ I , t

Z + y%(t,ztn +yt ) ) − Sα (t)G (0, ς(0)) +

∗ We must demonstrate that there exists v∗ ∈ SF ,z%(s,z ∗ +y s

s)

+y%(s,zs∗ +ys )

Sα (t − s)vn (s)ds. 0

such that, for each t ∈ I ,

∗ h∗ (t) = G (t, z%(t,z + y%(t,zt∗ +yt ) ) − Sα (t)G (0, ς(0)) + ∗ t +yt )

Z

t

Sα (t − s)v∗ (s)ds. 0

Set Z t n Θn (t) = hn (t) − G (t, z%(t,z Sα (t − s)vn (s)ds, n +y ) + y%(t,z n +yt ) ) − Sα (t)G (0, ς(0)) + t t t 0 Z t ∗ Θ∗ (t) = h∗ (t) − G(t, z%(t,z Sα (t − s)v∗ (s)ds. ∗ +y ) + y%(t,z ∗ +yt ) ) − Sα (t)G (0, ς(0)) + t t t 0

We have, for every t ∈ I , kΘn (t) − Θ∗ (t)k → 0 as n → ∞. Recognize the linear continuous operator Ψ : L1 (I , X) → Vb0 defined by Z t Ψ(v)(t) = Sα (t − s)v(s)ds. 0

From Lemma 2.1 and the definition of Ψ, it follows that Ψ ◦ SF is a closed graph operator, and for every t ∈ I , n Θn (t) ∈ Ψ(SF ,z%(s,z n +y s

s)

+y%(s,zsn +ys ) ).

∗ Since z n → z ∗ and Ψ ◦ SF is a closed graph operator, then there exists v∗ ∈ SF ,z%(s,z ∗ +y s

for each t ∈ I , ∗ h∗ (t) − G (t, z%(t,z + y%(t,zt∗ +yt ) ) + Sα (t)G (0, ς(0)) = ∗ t +yt )

Z

s)

+y%(s,zs∗ +ys )

such that,

t

Sα (t − s)v∗ (s)ds. 0

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Hence h∗ ∈ Υ(z ∗ ). As a result of Lemma 2.2, we find that Υ has a fixed point z on the interval (−∞, b]. Along these lines, x = z + y is a fixed point of the operator Υ which is the mild solution of the structure (1.1)-(1.2). Our next result is focused on the Leray-Schauder’s alternative fixed point theorem [2]. In order to utilize this theorem, we require the subsequent further hypothesis: (H3∗ ) There exists a function ϑ ∈ L1 (I , R+ ) and a continuous non-decreasing function  : R+ → (0, ∞) such that

|F (t, u)| ≤ ϑ(t)(kukB ) for a.e. t ∈ I If µ = 1 − D1∗ KG > 0 and

where C = C1 +

D1∗ µ

h

D1∗ M µ

Z

b

Z

∞

ϑ(s)ds < C

0

i

M |G (0, ς(0))| + KG C1 + G ∗ .

and each u ∈ B.

ds , (s)

Theorem 3.2. Assume that (H1), (H2), (H3∗ ) and (H5)-(H8) are fulfilled. Then, the problem (1.1)-(1.2) has a mild solution on (−∞, b]. Proof. Let z be solutions of the inclusion z ∈ λΥ(z), for any λ ∈ (0, 1), then there exists v ∈ SF,z%(s,zs +ys ) +y%(s,zs +ys ) such that |z(t)| ≤ kSα (t)kB(X) |G (0, ς(0))| + |G (t, z%(t,zt +yt ) + y%(t,zt +yt ) ) − G (t, 0) + G (t, 0)| Z t + kSα (t − s)kB(X) |F (s, z%(s,zs +ys ) + y%(s,zs +ys ) )|ds 0

≤ M |G (0, ς(0))| + KG kz%(t,zt +yt ) + y%(t,zt +yt ) kB + G ∗ Z t +M ϑ(s)(kz%(s,zs +ys ) + y%(s,zs +ys ) kB )ds. 0

From the remark 3.1, we have Z t |z(t)| ≤ M |G (0, ς(0))| + KG C1 + KG D1∗ |z(t)| + G ∗ + M ϑ(s)(D1∗ |z(s)| + C1 )ds 0 i MZ t 1h ∗ ≤ ϑ(s)(D1∗ |z(s)| + C1 )ds. M |G (0, ς(0))| + KG C1 + G + µ µ 0 Then D1∗ |z(t)| + C1 ≤ C1 +

i D ∗M D1∗ h M |G (0, ς(0))| + KG C1 + G ∗ + 1 µ µ

Z

t

ϑ(s)(D1∗ |z(s)| + C1 )ds.

0

We conceive the function β characterized by β(t) = sup{D1∗ |z(s)| + C1 : 0 ≤ s ≤ b},

t ∈ I.

Let t∗ ∈ [0, t] be such that β(t) = D1∗ |z(t∗ )| + C1 kςkB . By the aforementioned inequality, we sustain i D ∗M Z t D1∗ h ∗ β(t) ≤ C1 + M |G (0, ς(0))| + KG C1 + G + 1 ϑ(s)(β(s))ds. µ µ 0 Let us occupy the right-hand side of the overhead inequality as v(t), for all t ∈ I . Then, we sustain β(t) ≤ v(t),

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From the meaning of v, we obtain v(0) = C1 + and v 0 (t) =

i D1∗ h M |G (0, ς(0))| + KG C1 + G ∗ = C µ D1∗ M ϑ(t)(β(t)), µ

a.e. t ∈ I .

Applying the non-decreasing character of , we conclude v 0 (t) ≤ and hence Z

v(t)

v(0)=C

D1∗ M ϑ(t)(v(t)), µ

ds D ∗M ≤ 1 (s) µ

Z

t

0

a.e. t ∈ I ,

D ∗M ϑ(s)ds ≤ 1 µ

Z

b

Z

∞

ϑ(s)ds < C

0

ds . (s)

In this manner, for each t ∈ I , there exists a constant Λ∗ such that v(t) ≤ Λ∗ and henceforth β(t) ≤ Λ∗ . Since kzkB ≤ β(t), we have kzkB ≤ Λ∗ . Set U = {z ∈ Vb0 : kzk∞ < Λ∗ + 1}. From Theorem 3.1, we realize that the operator Υ : U → Υ(z) is completely continuous. Besides, from the decision of U, there is no z ∈ ∂U such that z = λΥ(z), for λ ∈ (0, 1). As an outcome of the nonlinear alternative of Leray-Schauder type [2], we conclude that Υ has a fixed point z in U, then the structure (1.1)-(1.2) has at least one mild solution on (−∞, b].

3.2

Existence results: Nonconvex case The next step is to demonstrate the existence results for the structure (1.1)-(1.2). Our result is focused

around the Lemma 2.3. Theorem 3.3. Assume that the subsequent hypotheses hold: (H8) F : I × B → Pcp (X) has the assets that F (·, u) : I → Pcp (X) is measurable, for each u ∈ B. (H9) There exists ℘ ∈ L1 (I , R+ ) such that Hd (F (t, u), F (t, u)) ≤ ℘(t)ku − ukB ,

for every u, u ∈ B,

and d(0, F (t, 0)) ≤ ℘(t)

a.e. t ∈ I .

(H10) There exists a positive constant L∗ > 0 such that |G (t, u) − G (t, u)| ≤ L∗ ku − ukB ,

a.e t ∈ I

and for all u.u ∈ B.

Then the problem (1.1)-(1.2) has at least one mild solution on (−∞, b]. Remark 3.2. For every z ∈ Vb0 , the set SF ,z is nonempty, since, by (H8), F has a measurable choice [43, Theorem III.6]. Proof. Let Υ : Vb0 → P(Vb0 ), where Υ is defined in Theorem 3.1 be solutions of the problem (1.1)-(1.2). Presently, we might demonstrate that the operator Υ fulfills all the states of Lemma 2.3. For our comfort, we split up the proof into two steps: 11 1312

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Step 1: Υ(z) ∈ Pcl (Vb0 ) for all z ∈ Vb0 . In fact, let (z n )n≥0 ∈ Υ(z) be such that z n → z˜ ∈ Vb0 . Then z˜ ∈ Vb0 and there exists vn ∈ SF ,z%(s,zs +ys ) +y%(s,zs +ys ) such that, for every t ∈ I , z (t) = G (t, z%(t,zt +yt ) + y%(t,zt +yt ) ) − Sα (t)G (0, ς(0)) + n

t

Z

Sα (t − s)vn (s)ds. 0

Utilizing the way that F has compact values and from (H9), we may go to a subsequence if important to get that vn converges to v in L1 (I , X) and consequently v ∈ SF ,z%(s,zs +ys ) +y%(s,zs +ys ) . Then, for every t ∈ I , Z t z n (t) → z˜ = G (t, z%(t,zt +yt ) + y%(t,zt +yt ) ) − Sα (t)G (0, ς(0)) + Sα (t − s)v(s)ds. 0

So z˜ ∈ Υ(z). Step 2: There exists Λ < 1 such that Hd (F (z), F (z)) ≤ Λkz − zk∞

for all z, z ∈ Vb0 .

Let z, z ∈ Vb0 and h ∈ Υ(z). Then there exists v(t) ∈ F (t, z%(t,zt +yt ) + y%(t,zt +yt ) ) such that, for every t ∈ I , Z t h(t) = G (t, z%(t,zt +yt ) + y%(t,zt +yt ) ) − Sα (t)G (0, ς(0)) + S(t − s)v(s)ds. 0

From (H9), it takes after that   Hd F (t, z%(t,zt +yt ) + y%(t,zt +yt ) ), F (t, z %(t,zt +yt ) + y%(t,zt +yt ) ) ≤ ℘(t)kz%(t,zt +yt ) − z %(t,zt +yt ) kB . Therefore, there is w ∈ F (t, z %(t,zt +yt ) + y%(t,zt +yt ) ) such that t ∈ I.

|v(t) − w| ≤ ℘(t)kz%(t,zt +yt ) − z %(t,zt +yt ) kB , Recognize U : I → P(X) specified by

U (t) = {w ∈ X : |v(t) − w| ≤ ℘(t)kz%(t,zt +yt ) − z %(t,zt +yt ) kB }. T Since the multivalued operator V (t) = U (t) F (t, z %(t,zt +yt ) + y%(t,zt +yt ) ) is measurable [43, Proposition III.4], there exists a function v(t), which is measurable choice v. Along these lines, v(t) ∈ F (t, z %(t,zt +yt ) + y%(t,zt +yt ) ), and utilizing phase space axioms, for every t ∈ J, we obtain |v(t) − v(t)| ≤ ℘(t)kz%(t,zt +yt ) − z %(t,zt +yt ) kB ≤ ℘(t)D1∗ |z(t) − z(t)|. For every t ∈ I , give us a chance to characterize h(t) = G (t, z %(t,zt +yt ) + y%(t,zt +yt ) ) − Sα (t)G (0, ς(0)) +

Z

t

S(t − s)v(s)ds. 0

Then, for every t ∈ I , |h(t) − h(t)| ≤ |G (t, z%(t,zt +yt ) + y%(t,zt +yt ) ) − G (t, z %(t,zt +yt ) + y%(t,zt +yt ) )| Z t + kSα (t − s)kB(X) |v(s) − v(s)|ds 0 Z t ∗ ≤ L∗ kz%(t,zt +yt ) − z %(t,zt +yt ) kB + M D1 ℘(s)|z(s) − z(s)|ds 0 Z t ≤ L∗ D1∗ |z(t) − z(t)| + ℘(s)|z(s) − z(s)|ds 0

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h ih i Z th ih i ≤ L∗ D1∗ eτ L(t) e−τ L(t) |z(t) − z(t)| + ℘(s)eτ L(s) e−τ L(s) |z(s) − z(s)| ds 0

Z t h τ L(s) i0 e ds ≤ L∗ D1∗ eτ L(t) kz − zkV + kz − zkV τ 0 h 1i ≤ L∗ D1∗ + eτ L(t) kz − zkV , τ Z t where τ > 0, L(t) = ℘(s)ds, ℘(t) = M D1∗ ℘(t), and k · kV is the Bielecki-type norm on Vb0 defined by 0

kzkV = sup{e−τ L(t) kz(t)k : t ∈ I }. Thus, we obtain

h 1i kh − hkV ≤ L∗ D1∗ + kz − zkV . τ By exchanging the parts of z and z, we have h 1i Hd (Υ(z), Υ(z)) ≤ L∗ D1∗ + kz − zkV . τ   1 Settling τ > 0 and for L∗ D1∗ + τ < 1, implies Υ is a contraction, and by Lemma 2.3, it has a fixed point z, which represents a mild solution (1.1)-(1.2).

4

Application We consider the FNIDI with SDD, namely: h i Z 0 (t − s)α−2  ∂ 2 i ∂h − r u(s, ξ) − g(s, u(s − σ(u(s, 0)), ξ)) ds u(t, ξ) − g(t, u(t − σ(u(t, 0)), ξ)) ∈ ∂t Γ(α − 1) ∂ξ 2 t h i + f1 (t, u(t − σ(u(t, 0)), ξ)), f2 (t, u(t − σ(u(t, 0)), ξ)) , 0 ≤ t ≤ b, 0 ≤ ξ ≤ π, (4.1) u(t, 0) = u(t, π) = 0, u(θ, ξ) = u0 (θ, ξ),

t ∈ I,

θ ∈ (−∞, 0],

(4.2) ξ ∈ J = [0, π],

(4.3)

 ∂2  − r , r > 0 stands for the operator with respect to the ∂ξ 2 special variable ξ, f1 , f2 : I × B → R are measurable in t and continuous in x, and g : I × B → R are where 1 < α < 2, (u0 , σ) ∈ C(R, [0, ∞)), Lξ =

appropriate functions. We expect that for every t ∈ I , f1 (t, ·) is lower semicontinuous , and assume that for each t ∈ I , f2 (t, ·) is upper semicontinuous. Consider X = L2 ([0, π], R) and the operator A : Lξ : D(A ) ⊂ X → X with domain D(A ) = {u ∈ X : u00 ∈ X,

u(0) = u(π) = 0}.

A is densely defined in X and is sectorial. As a result A represents a generator of a solution operator on X. For the phase space, we pick B = Cγ = {ς ∈ C((−∞, 0] : X) : lim eγθ ς(θ) θ→−∞

exists in X} invested with the

norm |ς| =

eγθ |ς(θ)|.

sup −∞ 0 with a ≥ 0 are defined by

Jaα+ f (x) =

x

1 Γ(α)

Z

1 Γ(α)

Z

(x − t)α−1 f (t)dt,

a 1. If p |f 0 | (p−1) is preinvex on K, then for every a, b ∈ K with η (b, a) 6= 0 the following inequality holds: 1 f (a) + f (a + η (b, a)) − 2 η (b, a)

  p−1 p p p (p−1) (p−1)   0 0 η (b, a) |f (a)| + |f (b)| f (x)dx ≤ .  2 2 (1 + p)1/p 

a+η(b,a) Z

a

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4

S. Q, M. IQBAL, AND MUHAMMAD MUDDASSAR

The aim of this paper is to establish left Hermite−Hadamard type inequalities for Riemann−Liouville fractional integral using the identity obtained for fractional integrals. 2. Main Results In order to obtain our results, we modified [15, Lemma 2.1] as following: Lemma 2. Let A ⊆ R be an open invex subset with respect to η : A × A → R and a, b ∈ A with a < a + η (b, a) .Suppose that f : A → R be a differentiable function.If f 0 is preinvex function on A and f 0 ∈ L[a, a + η (b, a)], then the following identity for Riemann−Liouville fractional integrals holds:  f

2a + η(b, a) 2

 −

4 i Γ(α + 1) h α η(b, a) X α J Ik , (5) + f (a + η(b, a)) + Ja+η(b,a)− f (a) = a 2η α (b, a) 2 k=1

where 1/2

Z I1 =

I2 =

0

(−tα )f 0 (b + tη(a, b))dt,

0 1

Z

1/2

Z

tα f 0 (a + tη(b, a))dt,

I3 =

(tα − 1)f 0 (a + tη(b, a))dt, I4 =

1

Z

1/2

(1 − tα )f 0 (b + tη(a, b))dt.

1/2

Proof. Integrating by parts Z 1/2 I1 = tα f 0 (a + tη(b, a))dt 0

1/2 Z 1/2 α tα f (a + tη(b, a))dt tα−1 f (a + tη(b, a))dt = − η(b, a) η(b, a) 0 0   Z 1/2 2a + η(b, a) 2−α α = f − tα−1 f (a + tη(b, a))dt. η(b, a) 2 η(b, a) 0 Analogously: I2 =

2−α f η(b, a)



2a + η(b, a) 2

 −

α η(b, a)

1/2

Z

tα−1 f (b + tη(a, b))dt

0

and Z

1

I3 =

(tα − 1)f 0 (a + tη(b, a))dt

1/2

1 Z 1 (tα − 1)f ((a + tη(b, a)) α − tα−1 f (a + tη(b, a))dt η(b, a) η(b, a) 1/2 1/2   Z 1 −α 1−2 2a + η(b, a) α = f − tα−1 f (a + tη(b, a))dt. η(b, a) 2 η(b, a) 1/2

=

Analogously: I4 =

1 − 2−α f η(b, a)



2a + η(b, a) 2

 −

1321

α η(b, a)

Z

1

tα−1 f (b + tη(a, b))dt

1/2

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GENERALIZATION OF INEQUALITIES ANALOGOUS TO H−H INEQUALITY

5

Adding above equalities, we get    Z 1 Z 1 2 2a + η(b, a) α tα−1 f (a + tη(b, a))dt+ tα−1 f ((b + tη(a, b))dt f − η(b, a) 2 η(b, a) 0 0 = I1 + I2 + I3 + I4 . Now making substitution u = (a + tη(b, a)), we have Z 1 Z a+η(b,a) 1 tα−1 f (a + tη(b, a))dt = α (u − a)α−1 f (u)du η (b, a) 0 a Γ(α) α = α J − f (a), η (b, a) b likewise Z

1/2

tα−1 f (b + tη(a, b)) =

0

Γ(α) α J + f (a + η(b, a)), η α (b, a) a

which completes our proof. New upper bound for the left-hand side of (2) for convex functions is proposed in the following theorem. Theorem 7. Let A ⊆ R be an open invex subset with respect to η : A × A → R and a, b ∈ A with a < a + η (b, a) .Such that f 0 ∈ L[a, a + η (b, a)].Suppose that f : A → R be a differentiable function. If |f 0 | is preinvex function on A then the following inequality for fractional integrals holds for 0 < α ≤ 1 :     α f 2a + η(b, a) − Γ(α + 1) J α+ f (a + η(b, a))+Ja+η(b,a)− f (a) 2 2η α (b, a) a η(b, a) ( |f 0 (a)| + |f 0 (b)| ) (6) ≤ α+1 2 (α + 1) Proof. By using the properties of modulus on Lemma 2, we have   4 h i η(b, a) X f 2a + η(b, a) − Γ(α + 1) J α+ f (a + η(b, a)) + J α ≤ |Ik |. f (a) − a+η(b,a) 2 2η α (b, a) a 2 k=1

0

Now, using preinvexity of |f |, we have Z 1/2 Z α 0 |I1 | ≤ t |f (a + tη(b, a))|dt ≤ 0

tα |f 0 (1 − t)a + tb|dt

0

≤ |f 0 (a)|

Z

1/2

tα (1 − t)dt + |f 0 (b)|

0

=

1/2

Z

1/2

tα+1 dt

0

α+3 1 |f 0 (a)| + α+2 |f 0 (b)|. 2α+2 (α + 1)(α + 2) 2 (α + 2)

Analogously: |I2 | ≤

α+3 1 |f 0 (b)| + α+2 |f 0 (a)|. 2α+2 (α + 1)(α + 2) 2 (α + 2)

By using preinvexity on |f 0 | and fact that for α ∈ (0, 1] and ∀ t1 , t2 ∈ [0, 1], |t1 α − t2 α | ≤ |t1 − t2 |α ,

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6

S. Q, M. IQBAL, AND MUHAMMAD MUDDASSAR

|I3 | ≤ |f 0 (a)|

Z

≤ |f 0 (a)|

Z

1

(1 − tα )(1 − t) dt + |f 0 (b)| (1 − t)α+1 dt + |f 0 (b)|

Z

1/2

1 2α+2 (α

1

(1 − tα )t dt

1/2

1/2 1

=

Z

+ 2)

1

(t − tα+1 ) dt

1/2

|f 0 (a)| +

α+3 |f 0 (b)|, + 1)(α + 2)

2α+2 (α

similarly |I4 | ≤

1 2α+2 (α

+ 2)

|f 0 (b)| +

α+3 |f 0 (a)|, + 1)(α + 2)

2α+2 (α

which completes the proof. Corollary 1. If we take η(b, a) = b − a in Theorem 7, then inequality (6) becomes inequality as   (b − a) α ≤ f a + b − Γ(α + 1) [J α+ f (b)+Jb− f (a)] ( |f 0 (a)|+|f 0 (b)| ). (7) a α α+1 2 2(b − a) 2 (α + 1)

Remark 2. If we take α = 1, in Corollary 1 then inequality (7) becomes inequality as obtained in [15, T heorem 2.2] . The corresponding version for powers of the absolute value of the derivative is incorporated in the following theorem. Theorem 8. Let A ⊆ R be an open invex subset with respect to η : A × A → R and a, b ∈ A with a < a + η (b, a) .Such that f 0 ∈ L[a, a + η (b, a)].Suppose that p f : A → R be a differentiable function. If |f 0 | p−1 is preinvex function on A for p some fixed p ≥ 1 with q = p−1 , then the following inequality for fractional integrals holds for 0 < α ≤ 1 :   h i f 2a + η(b, a) − Γ(α + 1) J α+ f (a + η(b, a)) + J α − f (a) a a+η(b,a) 2 2η α (b, a) "   0 1/q # 0 q 0 q 1/q q η(b, a) |f (b)| +3|f (a)| |f (a)| + 3|f 0 (b)|q ≤ α+1 + . (8) 4 4 2 (αp + 1)1/p

Proof. From Lemma 2 and using H¨older inequality with properties of modulus, we have   4 h i η(b, a) X f 2a + η(b, a) − Γ(α + 1) J α+ f (a + η(b, a)) + J α ≤ |Ik |. f (a) − a+η(b,a) 2 2(b − a)α a 2 k=1

0 q

By using the convexity of |f | , we have !1/p Z !1/q Z 1/2 1/2 q αp 0 |I1 | ≤ t dt |f (a + tη(b, a))| dt 0

 ≤  =

0

1 2αp+1 (αp + 1)

1/p

1 2αp+1 (αp + 1)

1/p 

0

q

Z

|f (a)|

1/2 0

Z

(1 − t)dt + |f (b)| 0

|f 0 (b)|q +3|f 0 (a)|q 8

1323

q

!1/q

1/2

t dt 0

1/q ,

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GENERALIZATION OF INEQUALITIES ANALOGOUS TO H−H INEQUALITY

7

similarly  |I2 | ≤

1 2αp+1 (αp + 1)

1/p 

3|f 0 (b)|q +|f 0 (a)|q 8

1/q ,

now !1/p

1

Z

α p

Z

!1/q

1

(1 − t ) dt

|I3 | ≤

q

0

|f (tb + (1 − t)a)| dt

1/2

.

1/2

Let α ∈ (0, 1] and ∀ t1 , t2 ∈ [0, 1], |t1 α − t2 α | ≤ |t1 − t2 |α , therefore Z

1 α p

Z

1

(1 − t)αp dt =

(1 − t ) dt ≤ 1/2

1/2

1 2αp+1 (αp + 1)

Hence 1 2αp+1 (αp + 1)

1/p 

1 |I4 | ≤ αp+1 2 (αp + 1) which completes the proof.

1/p 

 |I3 | ≤

|f 0 (b)|q +3|f 0 (a)|q 8

1/q ,

and 

3|f 0 (b)|q + |f 0 (a)|q 8

1/q ,

Corollary 2. .If we take η(b, a) = b − a in Theorem 8, then inequality (8) becomes inequality (2.1) of [16, Theorem 2.3]   f a + b − Γ(α + 1) [J α+ f (b) + J α− f (a)] a b α 2 2(b − a) " 1/q  0 1/q # (b − a) |f (a)|q + 3|f 0 (b)|q |f 0 (b)|q +3|f 0 (a)|q ≤ α+1 + . (9) 4 4 2 (αp + 1)1/p

Remark 3. If we take α = 1, in Corollary 2 then inequality (9) becomes inequality (2.1) of [15, T heorem 2.3] . Another similar result may be extended in the following theorem. Theorem 9. Let A ⊆ R be an open invex subset with respect to η : A × A → R and a, b ∈ A with a < a + η (b, a) .Such that f 0 ∈ L[a, a + η (b, a)].Suppose that p f : A → R be a differentiable function. If |f 0 | p−1 is preinvex function on A for p some fixed p > 1 with q = p−1 , then the following inequality for fractional integrals holds for α > 0:   η(b, a) f 2a + η(b, a) − Γ(α + 1) [J α+ f (b) + J α− f (a)] ≤ × a b 2 2η α (b, a) 2α+1 (α + 1) # " q q 1/q  q q 1/q (α + 3) |f 0 (b)| +(α + 1) |f 0 (a)| (α + 3) |f 0 (a)| +(α + 1) |f 0 (b)| + . 2(α + 2) 2(α + 2)

(10)

Proof. Using the well-known power-mean integral inequality for q > 1 we have !1−1/q Z !1/q Z 1/2

1/2

0

q

tα |f 0 (a + tη(b, a))| dt

tα dt

|I1 | ≤

0

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8

S. Q, M. IQBAL, AND MUHAMMAD MUDDASSAR q

By preinvexity of |f 0|  1−1/q  1/q 1 α+3 1 q q 0 0 |I1 | ≤ |f (a)| + |f (b)| 2α+1 (α + 1) 2α+2 (α + 1)(α + 2) 2α+2 (α + 2)   q q 1/q 1 (α + 3) |f 0 (a)| + (α + 1) |f 0 (b)| = α+1 2 (α + 1) 2(α + 2) Analogously:  q q 1/q 1 (α + 3) |f 0 (b)| + (α + 1) |f 0 (a)| |I2 | ≤ α+1 2 (α + 1) 2(α + 2)  q q 1/q 1 (α + 3) |f 0 (b)| + (α + 1) |f 0 (a)| |I3 | ≤ α+1 2 (α + 1) 2(α + 2) and  q q 1/q 1 (α + 3) |f 0 (a)| + (α + 1) |f 0 (b)| |I4 | ≤ α+1 2 (α + 1) 2(α + 2) Combining all the obtained inequalities, we get desired inequality. Which completes the proof. Corollary 3. Let A ⊆ R be an open invex subset with respect to η : A × A → R and a, b ∈ A with a < a + η (b, a) .Such that f 0 ∈ L[a, a + η (b, a)].Suppose that q f : A → R be a differentiable function. If |f 0 | is preinvex function on A for some fixed q > 1 then the following inequality for fractional integrals holds for     Z a+η(b,a) 2a + η(b, a) η(b, a) 1 + 21/q 1 f (x)dx−f [|f 0 (a)|+|f 0 (b)|], (11) ≤ η(b, a) a 2 8 31/p

Proof. If we take α = 1 in Theorem 9, then inequality (10) becomes as:   1 Z a+η(b,a) 2a + η(b, a) f (x)dx − f η(b, a) a 2 "   1/q # 1/q η(b, a) 2|f 0 (a)|q + |f 0 (b)|q |f 0 (a)|q + 2|f 0 (b)|q ≤ + , 8 3 3 which can be made equivalent to (11) by using the fact: n n n X X X ai r + bi r , (ai + bi )r ≤ i=1

i=1

i=1

for 0 ≤ r < 1, a1 , a2 , ..., an ≥ 0 and b1 , b2 , ..., bn ≥ 0. Remark 4. Inequality (11) is an improvement of obtained inequality as in [16, T heorem 2.1]. 3. Applications to special means In what follows we give certain generalizations of some notions for a positive valued function of a positive variable. Definition 3. A function M : R+ → R+ ,is called a Mean function if it has the following properties: (1) Homogeneity : M (ax, ay) = aM (x, y) , for all a > 0, (2) Symmetry : M (x, y) = M (y, x) , (3) Reflexivity : M (x, x) = x,

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GENERALIZATION OF INEQUALITIES ANALOGOUS TO H−H INEQUALITY

9

(4) Monotonicity : If x 6 x and y 6 y 0 then M (x, y) = M (x0 , y 0 ) , (5) Internality : min {x, y} 6 M (x, y) 6max{x, y} . We consider some mens for arbitrary positive real numbers a and b (see for instance [7]). The arithmetic mean a+b A(a, b) = , a, b ∈ R 2 The geometric mean √ G(a, b) = ab , a, b ∈ R The harmonic mean H(a, b) =

2ab , a+b

a, b ∈ R\{0}

The power mean  P (a, b) =

ar + br 2

 r1 ,

r>1

.The identric mean I(a, b) =

  

1 e



a 1  b−a

b

b aa

if a = b if a 6= b

,

a, b > 0

The logarithmic mean  L(a, b) = Generalized logarithmic mean  a    n1 Ln (a, b) = n+1 n+1 b −a  (n+1)(b−a)

a b−a lnb−lna

if a = b if a = 6 b

,

if a = b if a 6= b

, n ∈ Z\{−1, 0}; a, b > 0

Now, using the results of Section Main Results, some new inequalities are derived for the above means.It is well known that Lp is monotonic nondecreasing over p ∈ R,with L−1 := L and L0 := I. In particular, we have the following inequality H 6 G 6 L 6 I 6 A. Now letting a and b be positive real numbers such that a < b.Consider the function M := M (a, b) : [a, a + η (b, a)] × [a, a + η (b, a)] → R+ ,which is one of the above mentioned means, therefore one can obtain variant inequalities for these means as follows: Setting η (b, a) = M (b − a) in (6) , (8) , (10) , one can obtain the following interesting inequalities involving means:     α f 2a + M (b, a) − Γ(α + 1) J α+ f (a + M (b, a))+Ja+M (b,a)− f (a) 2 2M α (b, a) a M (b, a) ( |f 0 (a)| + |f 0 (b)| ). (12) ≤ α+1 2 (α + 1)

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10

S. Q, M. IQBAL, AND MUHAMMAD MUDDASSAR

  h i f 2a + M (b, a) − Γ(α + 1) J α+ f (a + M (b, a)) + J α f (a) − a+M (b,a) 2 2M α (b, a) a " #    1/q 1/q |f 0 (a)|q + 3|f 0 (b)|q M (b, a) |f 0 (b)|q +3|f 0 (a)|q ≤ α+1 + . 4 4 2 (αp + 1)1/p

(13)

  h i f 2a + M (b, a) − Γ(α + 1) J α+ f (a + M (b, a)) + J α ≤ f (a) − a+M (b,a) 2 2M α (b, a) a " 1    1# M (b, a) (α+ 3)|f 0 (a)|q +(α+ 1)|f 0 (b)|q q (α+ 3)|f 0 (b)|q +(α+ 1)|f 0 (a)|q q + . (14) 2α+1 (α+1) 2(α+ 2) 2(α + 2) Proof. Letting M = A, G, H, P, I, L, LP in (12) , (13) , and (14) , we can get the required inequalities, and the details are left to interested reader. 4. Acknowledgments The authors are grateful to Dr S. M. Junaid Zaidi, Rector, COMSATS Institute of Information Technology, Pakistan for providing excellent research facilities. The author S. Qaisar was partially supported by the Higher Education Commission of Pakistan [grant number No. 21-52/SRGP/R&D/HEC /2014. References [1] T. Antczak, Mean value in invexity analysis, Nonlinear Analysis 60 (2005) 1471-1484. [2] M. Aslam Noor, On Hadamard integral inequalities invoving two log-preinvex functions, J. Inequal. Pure Appl. Math., 8 (2007), No. 3, 1-6, Article 75. [3] M. Aslam Noor, Hermite-Hadamard integral inequalities for log-preinvex functions, J. Math. Anal. Approx. Theory, 2 (2007), 126-131. [4] M. Aslam Noor, Some new classes of nonconvex functions, Nonl. Funct. Anal. Appl.,11(2006),165-171. [5] A. Barani, A.G. Ghazanfari, S.S. Dragomir, Hermite-Hadamard inequality through prequsiinvex functions, RGMIA Research Report Collection, 14(2011), Article 48, 7 pp. [6] A. Barani, A.G. Ghazanfari, S.S. Dragomir, Hermite-Hadamard inequality for functions whose derivatives absolute values are preinvex, RGMIA Research Report Collection, 14(2011), Article 64, 11 pp. [7] P. S. Bullen, Hand book of means and their inequalities, Kluwer Academic Publishers, Dordrecht, 2003. [8] Z. Dahmani, On Minkowski and Hermite-Hadamard integral inequalities via fractional integration, Ann. Funct. Anal. 1(1) (2010), 51-58. [9] Z. Dahmani, L. Tabharit, S. Taf, New generalizations of Gruss inequality usin Riemann– Liouville fractional integrals, Bull. Math. Anal. Appl. 2 (3) (2010)93–99. [10] Z. Dahmani, L. Tabharit, S. Taf, Some fractional integral inequalities, Nonl. Sci. Lett. A, 1(2) (2010), 155-160. [11] 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) 91-95. [12] S.S. Dragomir and C.E.M. Pearce, Selected Topics on Hermite-Hadamard Inequalities and Applications, RGMIA Monographs,Victoria University, 2000. [13] R. Gorenflo, F. Mainardi, Fractional calculus: integral and differential equations of fractional order, Springer Verlag, Wien (1997), 223-276. [14] M.A. Hanson, On sufficiency of the Kuhn-Tucker conditions, J. Math. Anal. Appl. 80 (1981) 545-550. [15] U.S. Kirmaci, Inequalities for differentiable mappings and applicatios to special means of real numbers to midpoint formula, Appl. Math. Comp., 147 (2004), 137–146.

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[16] U. S. Kirmaci and M. E. Ozdemir, On some inequalities for differentiable mappings and applications to special means of real numbers and to midpoint formula, Appl. Math. Comp., 153 (2004), 361-368. [17] S. Miller and B. Ross, An introduction to the Fractional Calculus and Fractional Differential Equations, John Wiley & Sons, USA, 1993, 2. [18] S.R.Mohan and S.K. Neogy, On invex sets and preinvex functions, J. Math. Anal. Appl. 189 (1995), 901-908. [19] C.E.M. Pearce and J. Pecaric ´c, Inequalities for diffrentiable mapping with application to special means and quadrature formula. Appl. Math. Lett., 13 (2000), 51-55. [20] R. Pini, Invexity and generalized convexity, Optimization 22 (1991) 513-525. [21] J. Pecaric, F. Proschan and Y.L. Tong, Convex functions, partial ordering and statistical applications, Academic Press, New York, 1991. [22] I. Podlubni, Fractional Differential Equations, Academic Press, San Diego, 1999. [23] M. Z. Sarikaya, H. Bozkurt and N. Alp, On Hermite-Hadamard type integral inequalities for preinvex and log-preinvex functions, arXiv:1203.4759v1. [24] M.Z. Sarikaya, E. Set, H. Yaldiz and N. Basak, Hermite-Hadamard’s inequalities for fractional integrals and related fractional inequalities, Mathematical and Computer Modelling, 57(9-10), 2403−2407, (2013). [25] T. Weir, and B. Mond, Preinvex functions in multiple bjective optimization, Journal of Mathematical Analysis and Applications, 136 (1998) 29-38. [26] X.M. Yang, X.Q. Yang and K.L. Teo, Characterizations and applications of prequasiinvex functions, properties of preinvex functions, J. Optim. Theo. Appl. 110 (2001) 645-668. E-mail address: [email protected] Department of Mathematics, COMSATS Institute of Information Technology,Sahiwal, Pakistan. E-mail address: [email protected] University of Engineering and Technology, Lahore, Pakistan E-mail address: [email protected] University of Engineering and Technology, Taxila, Pakistan.

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The Borel direction and uniqueness of meromorphic function

∗

Hong Yan Xua† and Hua Wangb a

Department of Informatics and Engineering, Jingdezhen Ceramic Institute, Jingdezhen, Jiangxi, 333403, China

b

Department of Informatics and Engineering, Jingdezhen Ceramic Institute, Jingdezhen, Jiangxi, 333403, China

Abstract The purpose of this paper is to investigate the relationship between Borel directions and shared-set of meromorphic functions and obtain some results of meromorphic functions sharing one finite set in an angular domain containing a Borel line. Key words: Meromorphic function; Borel direction; Uniqueness. Mathematical Subject Classification (2010): 30D30.

1

Introduction and main results

S b We use C to denote the open complex plane, C(= C {∞}) to denote the extended complex plane, and Ω(⊂ C) to denote an angular domain. We assume that the readers are familiar with the standard notations and fundamental results of Nevanlinna value distribution theory of meromorphic functions (see [7, 16]). b and Ω := {z : α ≤ arg z ≤ β} ⊆ C. Define Let S be a set of distinct elements in C [ E(S, Ω, f ) = {z ∈ Ω|fa (z) = 0, counting multiplicities}, a∈S

E(S, Ω, f ) =

[

{z ∈ Ω|fa (z) = 0, ignoring multiplicities},

a∈S

where fa (z) = f (z) − a if a ∈ C and f∞ (z) = 1/f (z). Let f and g be two non-constant meromorphic functions in C. We say f and g share the set S CM (counting multiplicities) in Ω if E(S, Ω, f ) = E(S, Ω, g); we say f and g share the set S IM b we say f and g (ignoring multiplicities) in Ω if E(S, Ω, f ) = E(S, Ω, g). If S = {a}, where a ∈ C, share the value a CM in Ω if E(S, Ω, f ) = E(S, Ω, g), and we say f and g share the value a IM in Ω if E(S, Ω, f ) = E(S, Ω, g). If Ω = C, we give the simple notation as before, E(S, f ), E(S, f ) and so on(see [17]). In 1926, R. Nevanlinna (see [11]) proved his famous five-value and four-value theorems. After this very work, many investigations studied the uniqueness of meromorphic functions with shared values in the whole complex plane (see [17]). Around 2003, Zheng [18, 19] was the first to study the uniqueness of meromorphic functions sharing five values and four values in some angular domain under some condition. ∗ The first author was supported by the NSF of China(11561033, 11301233, 61202313), the Natural Science Foundation of Jiangxi Province in China (20132BAB211001, 20151BAB201008), and the Foundation of Education Department of Jiangxi (GJJ14644) of China. † Corresponding author

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Theorem 1.1 ([18, Theorem 1.1]). Let f (z) and g(z) be both transcendental meromorphic funcb and an integer p ≥ 0, tions and let f (z) be of the finite lower order µ and such that for some a ∈ C (p) δ = δ(a, f ) > 0. For q pair of real numbers αj , βj satisfying −π ≤ α1 < β1 ≤ α2 < β2 ≤ · · · ≤ αq < βq ≤ π, and

q X

4 (αj+1 − βj ) < arcsin σ j=1

r

δ , 2

(1)

π π , · · · , βq −α }, assume that f (z) and g(z) have five where σ = max{ω, µ} and ω = max{ β1 −α 1 q Sq distinct IM shared values in Ω = j=1 {z : αj ≤ arg z ≤ βj }. If ω < λ(f ), then f (z) ≡ g(z).

After Zheng’s work, there were many interesting results about the uniqueness with shared values in the angular domain, see [1, 9, 14, 15, 20]. In 2006, Lin, Mori and Tohge [9] dealt with the uniqueness problem on meromorphic functions sharing three finite sets in an angular domain and obtained the following theorems. Theorem 1.2 (see [9, Thereom 1]). Let S1 = {∞}, S2 = {ω|ω n−1 (ω + a) − b = 0}, S3 = {0}, where n(≥ 4) is an integer, and a, b are two nonzero constants, such that the algebraic equation ω n−1 (ω + a) − b = 0 has no multiple roots. Assume that f is a meromorphic function of lower b and δ := δ(ι, f ) > 0 for some ι ∈ C\{0, b order µ(f ) ∈ (1/2, ∞) in C −a}. Then, for each σ < ∞ with µ(f ) ≤ σ ≤ λ(f ), there exists an angular domain Ω = Ω(α, β) := {z : α < arg z < β} with 0 ≤ α < β ≤ 2π and ( r ) δ 4 π , 2π − arcsin (2) β − α > max σ σ 2 such that if the conditions E(S3 , f ) = E(S3 , g) and E(Sj , Ω, f ) = E(Sj , Ω, g) (j = 1, 2) hold for a meromorphic function g of finite order or, more generally, with the growth satisfying either log T (r, g) = O(log T (r, f )) or lim

r→∞

log log T (r, g) = 0, r 6∈ E1 , min{log r, log T (r, f )}

(3)

where E1 is a set of finite linear measure, then f ≡ g. It is well known that Borel direction is an important singular direction for meromorphic function in the fields of complex analysis, and Borel directions played an important role in the topic of angular distribution(see [8, 12, 13]). Valiron [16] proved that every meromorphic function of finite order ρ > 0 has at least one Borel direction of order ρ, where the order of meromorphic function T (r,f ) f is defined by ρ = ρ(f ) = lim sup loglog . r r→∞

In 2012, Long and Wu[10] was the first to investigate the problem concerning Borel direction and shared value of meromorphic functions and obtained the following theorems. Theorem 1.3 (see [10, Theorem 1.1]) Let f be meromorphic function of infinite order ρ(r), g ∈ M (ρ(r)), arg z = θ(0 ≤ θ < 2π) be one Borel direction of ρ(r) order of meromorphic function f , b = 1, 2, 3, 4, 5) be five distinct complex numbers. If f and g share ai (i = 1, 2, 3, 4, 5) IM in ai ∈ C(i the angular domain Ω(θ − ε, θ + ε) for any ε(0 < ε < π), then f ≡ g. Definition 1.1 [2]. Let f be a meromorphic function of infinite order, ρ(r) is a real function satisfying the following conditions: (i) ρ(r) is continuous, non-decreasing for r ≥ r0 and ρ(r) → ∞ as r → ∞; (ii) r log U (R) = 1, R = r + , lim r→∞ log U (r) log U (r)

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where U (r) = rρ(r) (r ≥ r0 ); (iii) lim sup r→∞

log T (r, f ) = 1. log U (r)

Then ρ(r) is called infinite order of meromorphic function f . This definition is given by Xiong Qinglai[2]. Let ρ(r) be infinite order of meromorphic function f , we will denote by M (ρ(r)) the set of T (r,g) meromorphic function g satisfying 0 < lim supr→∞ log ρ(r) log r ≤ 1, that is, M (ρ(r)) :=

  log T (r, g) g : 0 < lim sup ≤1 . ρ(r) log r r→∞

Let α < β, β − α < 2π, r > 0, and Ω(α, β, r) := {z : α ≤ arg z ≤ β, 0 < |z| ≤ r}. The definition of Borel direction of meromorphic functions f of infinite order ρ(r) is defined as follows. Definition 1.2 [2]. Let f be meromorphic functions of infinite order ρ(r), if for any ε(0 < ε < π), the equality log n(Ω(θ − ε, θ + ε, r), f = a) = 1, lim sup ρ(r) log r r→∞ b at most except two exception, where n(Ω(θ −ε, θ +ε, r), f = a) holds for any complex number a ∈ C, is the counting function of zero of the function f − a in the angular domain Ω(θ − ε, θ + ε), counting multiplicities. Then the ray arg z = θ is called a Borel direction of ρ(r) order of meromorphic function f . Remark 1.1 Chuang [2] proved that every meromorphic function f with infinite order ρ(r) has as least one Borel direction of infinite order ρ(r). In this paper, we will investigate the uniqueness problem of meromorphic functions sharing one finite set in an angular domain containing a Borel line. We will mainly consider the following finite set S = {w ∈ A : P1 (w) = 0}, where P1 (w) =

(n − 1)(n − 2) n n(n − 1) n−2 w − n(n − 2)wn−1 + w − c, 2 2

c is a complex number satisfying c 6= 0, 1. Theorem 1.4 Let f be meromorphic function of infinite order ρ(r), g ∈ M (ρ(r)), arg z = θ(0 ≤ θ < 2π) be one Borel direction of ρ(r) order of meromorphic function f , if E(S, Ω(θ −ε, θ +ε), f ) = E(S, Ω(θ − ε, θ + ε), g) and n is an integer ≥ 11, then f ≡ g. A set S is called a unique range set for meromorphic functions on the Borel direction arg z = θ, if for any two nonconstant meromorphic functions f and g the condition E(S, Ω(θ − ε, θ + ε), f ) = E(S, Ω(θ − ε, θ + ε), g) implies f ≡ g. We denote by ]S the cardinality of a set S. Thus, from Theorem 1.4, we can get the following corollary Corollary 1.1 There exists one finite set S with ]S = 11, such that any two meromorphic functions f and g on the Borel direction, which f (z) is meromorphic function of infinite order ρ(r), g ∈ M (ρ(r)), arg z = θ(0 ≤ θ < 2π) be one Borel direction of ρ(r) order of meromorphic function f , and E(S, Ω(θ − ε, θ + ε), f ) = E(S, Ω(θ − ε, θ + ε), g).

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2

Some Lemmas

We first introduce the basic notations and definitions of meromorphic functions in an angular domain as follows(see [7, 18, 19]). Let f be a meromorphic function on the angular domain Ω(α, β) = {z : α ≤ arg z ≤ β} and 0 < β − α ≤ 2π. Define Z tω ω r 1 dt ( ω − 2ω ){log+ |f (teiα )| + log+ |f (teiβ )|} , Aα,β (r, f ) = π 1 t r t Z β 2ω log+ |f (reiθ )| sin ω(θ − α)dθ, Bα,β (r, f ) = ω πr α X 1 |bµ |ω Cα,β (r, f ) = 2 ( − 2ω ) sin ω(θµ − α), ω |bµ | r 1 α} ∈ A for all α ∈ (−∞, ∞). In section 2, we give Choquet integrals of non-negative, continuous and decreasing functions on R− with respect to fuzzy measure according to the ideas of [8]. In section 3, we investigate some properties and examples of derivatives of those functions with respect to distorted 2010 Mathematics Subject Classification : 28E10, 46A55. Key words and phrases : fuzzy measures, Choquet integral. Correspondence should be addressed to Jongsung Choi, [email protected]. 1

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Lebesgue measures. Finally, we show that existence and non-existence of derivatives depend on fuzzy measures.

2. Choquet integral of g ∈ M− In this paper, we assume that X = R− . Let A be the smallest σ-algebra of subsets of X. Then (X, A, ν) is called a fuzzy measure space. As in [8], Choquet integral of g with respect to a fuzzy measure ν on a set A is defined by Z (2.1)

(C)

Z

A

∞

ν ({t|g(t) ≥ α} ∩ A) dα.

g(t)dν = 0

Let M− be the set of measurable, non-negative, continuous and decreasing functions such that g : R− → R+ . Definition 2.1. ([8]) Let m : R+ → R+ be a continuous and increasing function m(0) = 0. A fuzzy measure νm , a distorted Lebesgue measure, is defined by (2.2)

νm (·) = m(λ(·)),

where λ([a, b]) = b − a for all [a, b] ⊂ R− . From Definition 2.1 and (2.1), we have the following theorem([8]). Theorem 1. We assume that ν([t, t]) = 0 for all t ∈ R− . Let g ∈ M− , then Choquet integral of g with respect to ν on [t, 0] is represented as ∞

Z

0

Z

ν 0 ([t, τ ])g(τ )dτ.

ν({t|g(t) ≥ α} ∩ [t, 0])dα =

(2.3) 0

t

In particular, for ν = νm , Z

∞

Z ν({t|g(t) ≥ α} ∩ [t, 0])dα =

(2.4) 0

0

m0 (τ − t)g(τ )dτ.

t

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Proof. Let α = g(τ ), dα = g 0 (τ )dτ and g −1 (α) = τ . By the definition of Choquet integral, we have that Z Z ∞ ν({t|g(t) ≥ α} ∩ [t, 0])dα (C) g(τ )dν(τ ) = 0

[t,0]

Z

g(0)

Z

g(t)

ν({t|g(t) ≥ α} ∩ [t, 0])dα +

= 0

ν({t|g(t) ≥ α} ∩ [t, 0])dα g(0)

Z

g(t)

= ν([t, 0])g(0) +

ν([t, g −1 (α)])dα

g(0) t

Z

ν([t, τ ])g 0 (τ )dτ 0 h it Z t ν 0 ([t, τ ])g(τ )dτ = ν([t, 0])g(0) + ν([t, τ ])g(τ ) − 0 0 Z 0 ν 0 ([t, τ ])g(τ )dτ = ν([t, 0])g(0) + ν([t, t])g(t) − ν([t, 0])g(0) + t Z 0 = ν 0 ([t, τ ])g(τ )dτ.

= ν([t, 0])g(0) +

t

For ν = νm , we obtain Z Z (C) g(τ )dν(τ ) = [t,0]

0

Z

0

ν ([t, τ ])g(τ )dτ =

t

0

m0 (τ − t)g(τ )dτ.

t

 From Theorem 1, we have the following corollary. Corollary 2. Let g(t) = k be a constant function for all t ∈ R− and k ∈ R+ . Then Z ∞ ν({t|g(t) ≥ α} ∩ [t, 0])dα = kν([t, 0]). 0

In particular, for ν = νm , Z ∞ ν({t|g(t) ≥ α} ∩ [t, 0])dα = km(−t). 0

Proof. From (2.1) we have that Z Z ∞ (C) g(τ )dν(τ ) = ν({t|g(t) = k ≥ α} ∩ [t, 0])dα [t,0]

0

Z =

k

ν([t, 0])dα = kν([t, 0]). 0

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For ν = νm , we obtain that Z g(τ )dν(τ ) = kν([t, 0]) = km(−t). (C) [t,0]

 Definition 2.2. Let f be a continuous and decreasing function with f (0) = 0. The derivative of f with respect to a fuzzy measure νm is defined as the inverse operation of Choquet integral based on (2.4) by (2.5)

df (t) = Dm (f ) = g(t), dνm (t)

if g(t) is found to be an element of M− . From (2.5), let us consider a class of f ’s for a given m(t) such that Z 0 n o − m0 (τ − t)g(τ )dτ, g ∈ M− . (2.6) Tm (M ) = f |f (t) = t

3. Derivatives with respect to distorted Lebesgue measures In this section, we discuss some properties of derivatives of continuous and decreasing functions with respect to distorted Lebesgue measures. From the conditions of g(t) in (2.5), we obtain the following theorem. Theorem 3. Dm (f ) is linear for f ∈ Tm (M− ) and non-negative constants. Proof. Let f1 (t), f2 (t) ∈ Tm (M− ) and k ∈ R+ . From the condition of f1 (t) and f2 (t), we have that Z 0 m0 (τ − t)Dm (f1 (τ ))dτ (3.1) f1 (t) = t

and Z (3.2)

f2 (t) =

0

m0 (τ − t)Dm (f2 (τ ))dτ.

t

Adding (3.1) and (3.2), we obtain that Z 0 n o f1 (t) + f2 (t) = m0 (τ − t) Dm (f1 (τ )) + Dm (f2 (τ )) dτ. t

By the definitions of M− and Tm (M− ), we know that Dm (f1 (t)) + Dm (f2 (t)) ∈ M−

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From (3.1), we see that Z kf1 (t) =

0

m0 (τ − t)kDm (f1 (τ ))dτ.

t −

Since kDm (f1 (t)) ∈ M , we have kf1 (t) ∈ Tm (M− ).



From Definition 2.2 and Theorem 3, we have the following theorem. Theorem 4. For t ∈ R− , we have the followings: d (1) m(−t) = 1, dνm  Z  0 d n−1 m(τ − t)(−τ ) dτ = (−t)n , n = 1, 2, . . . (2) n dνm  t  Z 0 d aτ (3) m(τ − t)e dτ = eat , a ≤ 0, m(−t) − a dνm  t  Z 0 d m(τ − t) (4) dτ = ln(1 − t). dνm 1−τ t Proof. (1) From (2.4), we have that Z 0 m0 (τ − t)dτ = m(−t). t −

n

(2) By (−t) ∈ M , n = 1, 2, . . ., we obtain that Z 0 m0 (τ − t)(−τ )n dτ t Z 0 h i0 n = m(τ − t)(−τ ) +n m(τ − t)(−τ )n−1 dτ t t Z 0 =n m(τ − t)(−τ )n−1 dτ. t

(3) Similarly, by eat ∈ M− for all a ≤ 0, we have that Z 0 Z 0 h i0 0 aτ aτ m (τ − t)e dτ = m(τ − t)e −a m(τ − t)eaτ dτ t t t Z 0 = m(−t) − a m(τ − t)eaτ dτ. t −

(4) Since ln(1 − t) ∈ M , we have Z 0 Z 0 m(τ − t) 0 0 m (τ − t) ln(1 − τ )dτ = [m(τ − t) ln(1 − τ )]t + dτ 1−τ t t Z 0 m(τ − t) = dτ. 1−τ t

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Thus (4) is proved.



By Definition 2.1 and (2.3), we have the following remark. Remark 1. We assume that g(t) is even function. Let τ − t = p, dτ = dp, and 0 ≤ p ≤ −t. Then, we have Z 0 Z 0 Z 0 g(t)dν = ν ([t, τ ])g(τ )dτ = ν 0 ([t, τ ])g(−τ )dτ (C) [t,0]

t

Z

t −t

Z

0

ν (p)g(−p − t)dp =

= Z0 α =

α

ν 0 (τ )g(α − τ )dτ

0

ν 0 (α − τ )g(τ )dτ.

0

In particular, for ν = νm , Z Z (C) g(t)dν = − [t,0]

α

m0 (α − τ )g(τ )dτ.

0

From Remark 1, the Choquet integral (2.3) on R− is considered as a convolution. That is, under the assumption of even function, we can apply the Laplace transformation for calculations of the Choquet integrals. As you see (2.5) and (2.6) of Definition 2.2., to find g(t) is same with the solvability of a Volterra integral equation of the first kind for given f (t) and a fuzzy measure νm ([4]). In fact, fuzzy measures play important roles in the existence of derivatives. Now we give a theorem to explain the relation with the existence of derivatives and fuzzy measures. Theorem 5. (The dependence on fuzzy measures) Let t ∈ R− . (1) If νm ([t, 0]) = m(−t) = e−t − 1, then −t ∈ / Tm (M− ), that is, 6 ∃Dm (−t). (2) If νm ([t, 0]) = m(−t) = −t, then −t ∈ Tm (M− ), that is, ∃Dm (−t). Proof. Suppose that −t ∈ Tm (M− ). From (2.6) of Definition 2.2., we know that Z 0 Z 0 0 (3.3) −t = m (τ − t)x1 (τ )dτ = e(τ −t) x1 (τ )dτ, t

t −

where Dm (−t) = x1 (t) ∈ M . But differentiating (3.3), we obtain Z 0 −t −1 = −e eτ x1 (τ )dτ − x1 (t) = t − x1 (t). t

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From the definition of M− , we obtain x1 (t) = t + 1 ∈ / M− . (1) is proved by this contradiction. To prove (2), it is sufficient to find x2 (t) ∈ M− such that Z 0 Z 0 0 m (τ − t)x2 (τ )dτ = x2 (τ )dτ. (3.4) −t = t

t

Differentiating (3.4), we have x2 (t) = 1 ∈ M− .  Acknowledgments The authors would like to express thanks to referees for careful reading this paper. The present Research has been conducted by the Research Grant of Kwangwoon University in 2014. References [1] J. Choi, On properties of derivatives with respect to fuzzy measures, J. Chungcheong Math. Soc., vol. 27(2014), No. 3, pp. 469–474. [2] G. Choquet, Th´eorie de capacit´es, Ann. Inst. Fourier, 5(1955), pp. 131–295. [3] S. Graf, A Radon-Nikodym theorem for capacities, J. Reine Angew. Math., 1980(1980), pp. 192–214. [4] G. Gripenberg, On Volterra equations of the first kind, Integral Equation Oper. Theory, 3(1980), pp. 473–488. [5] U. H˝ ohle, Integration with respect to fuzzy measures, in: Proceedings of the IFAC Symposium on Theory and Application of Digital Control, New Delhi, 1982, pp. 35–37. [6] T. Murofushi and M. Sugeno, A theory of fuzzy measures; representations, the Choquet integral, and null sets, J. Math. Anal. and Appl., vol. 159(1991), pp. 532–547. [7] M. Sugeno, Theory of fuzzy integrals and its applications, Doctoral dissertation, Tokyo Institute of Technology, 1974. [8] M. Sugeno, A note on derivatives of functions with respect to fuzzy measures, Fuzzy Sets and Systems, 222(2013), pp. 1–17. Hyun-Mee Kim, Mathematics Education Major, Graduate School of Education, Kookmin University, Seoul 136-702, Republic of Korea, Young-Hee Kim, Division of General Education-Mathematics, Kwangwoon University, Seoul 139-701, Republic of Korea, Jongsung Choi, Division of General Education-Mathematics, Kwangwoon University, Seoul 139-701, Republic of Korea,

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Some new inequalities for the gamma function Xiaodong Cao

Abstract In this paper, we present some new inequalities for the gamma function. The main tools are the multiple-correction method developed in [6, 7] and a generalized Mortici’s lemma.

1

Introduction

Duo to its importance in mathematics, the problem of finding new and sharp inequalities for the gamma function and, in particular for large values of x Z ∞ (1.1) Γ(x) := tx−1 e−t dt, x > 0, 0

has attracted the attention of many researchers (see [2, 3, 8, 9, 12, 14, 15, 16, 17, 18] and the references therein). Let’s recall some of the classical results. Maybe one of the most well-known formula for approximation the gamma function is the Stirling’s formula  x x √ Γ(x + 1) ∼ 2πx (1.2) , x → +∞. e See, e.g. [1, p. 253]. The following two formulas give slightly better estimates than Stirling’s formula, !x+ 1 2 √ x + 12 (1.3) , (Burnside [5], 1917 ), Γ(x + 1) ∼ 2π e r √  x x 1 Γ(x + 1) ∼ 2π (1.4) x + , (Gosper [10], 1978). e 6 Address: Department of Mathematics and Physics, Beijing Institute of Petro-Chemical Technology, Beijing, 102617, PR China. E-mail: [email protected] Tel/Fax:(+86)010-81292176 MSC: 33B15;41A20;41A25 Key words and phrases: Gamma function, Rate of convergence, Continued fraction, Multiple-correction This work is supported by the National Natural Science Foundation of China (Grant No.11171344) and the Natural Science Foundation of Beijing (Grant No.1112010).

1

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Ramanujian [22] proposed the claim (without proof) for the gamma function (1.5)

1  √  x x θx 6 3 2 Γ(x + 1) = π 8x + 4x + x + , e 30

3 where θx → 1 as x → +∞ and 10 < θx < 1. This open problem was solved by Karatsuba[13]. Thus (1.5) provides a more accurate estimate for the gamma function (see Sec. 2 below). In this paper, we will continue the previous works [6, 7], and introduce a class of new approximations to improve these inequalities. Throughout the paper, the notation Ψ(k; x) denotes a polynomial of degree k in x with all coefficients non-negative, which may be different at each occurrence. Let (an )n≥1 and (bn )n≥0 be two sequences of real numbers with an 6= 0 for all n ∈ N . The generalized continued fraction

τ = b0 +

a1

= b0 +

a2

b1 +

b2 +

..

∞ a1 a2 an · · · = b0 + K n=0 bn b1 + b2 +

.

is defined as the limit of the nth approximant n ak An = b0 + K k=1 bk Bn

as n tends to infinity. See [2, p.105].

2

A generalized Mortici’s lemma

Mortici [14] established a very useful tool for measuring the rate of convergence, which says that a sequence (xn )n≥1 converging to zero is the fastest possible when the difference (xn − xn+1 )n≥1 is the fastest possible. Since then, Mortici’s lemma has been effectively applied in many paper such as [6, 7, 17, 18]. The following lemma is a generalization of Mortici’s lemma. Lemma 1. If limx→+∞ f (x) = 0, and there exists the limit lim xλ (f (x) − f (x + 1)) = l ∈ R,

(2.1)

x→+∞

with λ > 1, then there exists the limit lim xλ−1 f (x) =

(2.2)

x→+∞

l . λ−1

Proof. It is not very difficult to prove that for x > 2 (2.3)

1 = (λ − 1)xλ−1

Z x

+∞

∞

dt X 1 ≤ ≤ λ t (x + j)λ j=0

Z

+∞

x−1

dt 1 = . λ t (λ − 1)(x − 1)λ−1

2

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For ε > 0, we assume that l − ε ≤ xλ (f (x) − f (x + 1)) ≤ l + ε for every real number x greater than or equal to the rank X0 > 0. By adding the inequalities of the form 1 1 (2.4) (l − ε) λ ≤ f (x) − f (x + 1) ≤ (l + ε) λ , x x we get (2.5)

(l − ε)

m−1 X j=0

m−1

X 1 1 ≤ f (x) − f (x + m) ≤ (l + ε) λ (x + j) (x + j)λ j=0

for every x ≥ X0 and m ≥ 1. By taking the limit as m → ∞, then multiplying by xλ−1 , we obtain ∞ ∞ X X 1 1 λ−1 λ−1 (l − ε)xλ−1 (2.6) ≤ x f (x) ≤ (l + ε)x . λ (x + j) (x + j)λ j=0

j=0

It follows from (2.3) that l−ε l+ε xλ−1 ≤ xλ−1 f (x) ≤ . λ−1 λ − 1 (x − 1)λ−1

(2.7)

Now by taking the limit as x → +∞, this completes the proof of the lemma at once. An example Let’s consider the Ramanujan’s asymptotic formula (1.5). Let the error term E(x) be defined by the following relation 1  √  x x 1 6 3 2 (2.8) Γ(x + 1) = π (1 + E(x)) . 8x + 4x + x + e 30 It follows readily from the recurrence formula Γ(x + 1) = xΓ(x) that   1 (2.9) ln (1 + E(x)) − ln (1 + E(x + 1)) = − 1 + x ln 1 + x 3 1 8(x + 1) + 4(x + 1)2 + (x + 1) + + ln 1 6 8x3 + 4x2 + x + 30

1 30

.

By using the Mathematica software, we expand the right-hand function in the above formula as a power series in terms of 1/x: 11 1 ln (1 + E(x)) − ln (1 + E(x + 1)) = (2.10) + O( 6 ). 2880x5 x Thus, by Lemma 1 we have 11 (2.11) lim x4 ln (1 + E(x)) = . x→+∞ 11520 Noting that limu→0

ln(1+u) u

= 1, one get finally

11 . 11520 Remark 1. Just as Motici’s lemma, Lemma 1 also provides a method for finding the limit of a function as x tends to infinity. (2.12)

lim x4 E(x) =

x→+∞

3

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Gosper-type inequalities

In this section, we use an example to illustrate the idea of this paper. To this end, we introduce some class of correction function (MCk (x))k≥0 such that the relative error function Ek (x) has the fastest possible rate of convergence, which are defined by the relations r √  x x 1 Γ(x + 1) = 2π x + + MCk (x) · exp(Ek (x)). (3.1) e 6 If limx→+∞ xµ f (x) = l 6= 0 with constant µ > 0, we say that the function f (x) is order x−µ , and write the exponent of convergence µ = µ(f (x)). Clearly if µ(Ek (x)) = µk , we have the following asymptotic formula r √  x x
1 Γ(x + 1) = 2π x + + MCk (x) · 1 + O(x−µk ) , x → +∞. (3.2) e 6 Let us briefly review a so-called multiple-correction method presented in our previous paper [6, 7]. Actually, the multiple-correction method is a recursive algorithm, and one of its advantages is that by repeating correction process we always can accelerate the convergence, i.e. the sequence (µ(Ek (x)))k≥0 is a strictly increasing. The key step is to find a suitable structure of MCk (x). In general, the correction function MCk (x) is a finite generalized continued fraction (see [7] or (3.8) below) or a hyper-power series (see [6] or (4.7) below) in x. It is not difficult to see that (3.1) is equivalent to (3.3)

ln Γ(x + 1) =

1 1 ln(2π) + x (ln x − 1) + ln (x + MCk (x)) + Ek (x). 2 2

By the recurrence formula Γ(x + 1) = xΓ(x), we have for x > 0 

(3.4)

1 Ek (x) − Ek (x + 1) = −1 + x ln 1 + x

 +

1 (x + 1) + 16 + MCk (x + 1) ln . 2 x + 16 + MCk (x)

κ0 Now by taking the initial-correction function MC0 (x) = x+λ and using Mathematica software, 0 we expand Ek (x) − Ek (x + 1) into a power series in terms of 1/x:

(3.5) E0 (x) − E0 (x + 1) =

1 − 72 + κ0 17 − 945κ0 − 810κ0 λ0 + + x3 540x4   −641 + 33120κ0 − 12960κ20 + 43200κ0 λ0 + 25920κ0 λ20 1 +O . 5 12960x x6

The fastest possible function E0 (x) − E0 (x + 1) is obtained when the first two coefficients in the 1 above formula vanish. In this case, we find κ0 = 72 , λ0 = 31 90 , and   5929 1 E0 (x) − E0 (x + 1) = (3.6) +O . 1166400x5 x6

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By Lemma 1, we can check that 5929 . 4665600

lim x4 E0 (x) =

(3.7)

x→+∞

We continue the above correction process to successively determine the correction function MCk (x) until some k ∗ you want. On one hand, to find the related coefficients, we often use an appropriate symbolic computations software because it’s huge of computations. On the other hand, the exact expressions at each occurrence also need lot of space. Hence in this paper we omit many related details. For interesting readers, see our previous paper [6, 7]. In fact, we can prove that for 0 ≤ k ≤ 3 k

(3.8)

MCk (x) =

K j=0

κj , x + λj

where 1 , 72 5929 , κ1 = 32400 76899172249 , κ2 = 248039857296 786873417270631211749921 κ3 = , 851541507731717527392144

31 , 90 481937 λ1 = , 3735270 7745462509019287 λ2 = , 19149278075101482 2098335745817751685364201067279071 λ3 = . 30311088872486921466334781589254970

κ0 =

λ0 =

By Lemma 1 again, we get for some constant Ck 6= 0 (3.9)

lim x2k+4 Ek (x) = Ck ,

x→+∞

(k = 0, 1, 2, 3),

i.e. µ(Ek (x)) = 2k + 4 for k = 0, 1, 2, 3. Thus we obtain more accurate approximation formulas: r   √  x x 1 Γ(x + 1) = 2π (3.10) x + + MCk (x) · 1 + O(x−(2k+4) ) , x → +∞. e 6 Pr (m) It should be noted that if we rewrite MCk (x) in the form of Q , where P, Q are polynomials s (m) with r = k and s = k + 1, theoretically at least, for a large x the above formula may reduce or eliminate numerically computations compared with the previous results, see e.g. [9, 12]. This is the main advantage of the multiple-correction method. The following theorem tells us how to obtain sharp inequalities.

Theorem 1. Let MCk (x) be defined as (3.8). Let x ≥ 1, then we have for k = 0, 2, r √  x x 1 (3.11) Γ(x + 1) > 2π x + + MCk (x), e 6 and for k = 1, 3, (3.12)

Γ(x + 1)
0 (or Ek (x) < 0), it suffices to show that the equality fk (x) > 0 (or fk (x) < 0) holds under the condition limx→+∞ Ek (x) = 0. By the Stirling’s formula (1.2), we can show that the condition limx→+∞ Ek (x) = 0 always holds. In what follows, we will apply this condition many times. By using Mathematica software, we may prove that for x ≥ 1 Ψ1 (8; x) > 0, x(1 + + + + 552x + 1080x2 )2 (1709 + 2712x + 1080x2 )2 Ψ2 (13; x)(x − 1) + 1463 · · · 9447 < 0, f100 (x) = − 2 x(1 + x) )(1359251 + 2829648x + 5976432x2 )2 Ψ3 (16; x) Ψ4 (20; x) f200 (x) = > 0, x(1 + x)2 Ψ5 (28; x) Ψ6 (25; x)(x − 1) + 17135 · · · 66999 f300 (x) = < 0. x(1 + x)2 Ψ7 (36; x)

f000 (x) =

x)2 (31

90x)2 (121

90x)2 (77

We only give the proof of inequalities in case k = 3, other may be proved similarly. In this case, we see that for x ≥ 1 the inequality (3.12) is equivalent to E3 (x) < 0. As limx→+∞ E3 (x) = 0, it suffices to prove that f3 (x) < 0 for x ≥ 1. Since f30 (x) is strictly decreasing, but limx→+∞ f30 (x) = 0, so f30 (x) > 0. Thus f3 (x) is strictly increasing with limx→+∞ f3 (x) = 0, so f3 (x) < 0. This completes the proof of Theorem 1. By the multiple-correction method, we also find another kind of inequalities. Theorem 2. Let the k-th correction function MCk (x) be defined by MC0 (x) = MCk (x) =

(x +

κ0 23 2 90 )

+ λ0

,

k κj κ0 K , 23 2 (x + 90 ) + λ0 + j=1 x + λj

(k ≥ 1),

where 1 4007 , λ0 = , 144 21600 4394 130311599 κ1 = , λ1 = , 637875 15575040 7894414898425 265702682899837009577 κ2 = , λ2 = − , 119793516544 34427631789478287360 1897560849252106177858465792 30320380455616293004898928163131563244811979 κ3 = , λ3 = . 77174813342532578267347147395 6134364315672065325746652708240298034227200 κ0 = −

Then we have (3.13)

Γ(x + 1)
0, x ≥ 13, x(1 + x)2 (1 + 6x)2 (7 + 6x)2 Ψ2 (16; x) Ψ3 (20; x)(x − 1) + 13798 · · · 89479 g100 (x) = − < 0, x ≥ 1, x(1 + x)2 (1 + 6x)2 (7 + 6x)2 Ψ4 (24; x) Ψ5 (26; x)(x − 6) + 97250 · · · 34321 > 0, x ≥ 6, g200 (x) = x(1 + x)2 (1 + 6x)2 (7 + 6x)2 Ψ6 (32; x) Ψ7 (32; x)(x − 1) + 836559 · · · 37479 g300 (x) = − < 0, x ≥ 1. x(1 + x)2 (1 + 6x)2 (7 + 6x)2 Ψ8 (40; x) g000 (x) =

Lastly, just as the proof of Theorem1, Theorem 2 follows from the above inequalities readily.

4

Ramanujan-type inequalities

Theorem 3. Let the k-th correction function MCk (x) be defined as k

(4.1)

MCk (x) =

K j=0

aj , x + bj

7

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where 79 11 , b0 = , 240 154 459733 1455925 a1 = , b1 = − , 711480 70798882 10259108965771635091 49600874140433 , b2 = , a2 = 101450127018720 19545564575317443762 169085305336152527131511003963 6141448535908002711219920016488834171 a3 = , b3 = − . 101221579151797375403194730976 203275987838924050801436670299517447102 Let x ≥ 1, then for k = 0, 2,  1 6 √  x x 1 3 2 (4.2) 8x + 4x + x + + MCk (x) , Γ(x + 1) < π e 30 a0 = −

and for k = 1, 3,  1 6 √  x x 1 3 2 8x + 4x + x + Γ(x + 1) > π + MCk (x) . e 30

(4.3)

Proof. We define the relative error function Ek (x) by the relation 1  6 √  x x 1 3 2 + MCk (x) (4.4) Γ(x + 1) = π 8x + 4x + x + exp(Ek (x)). e 30 Thus (4.5)

  1 Ek (x) − Ek (x + 1) = − 1 + x ln 1 + x 1 3 + MCk (x + 1) 1 8(x + 1) + 4(x + 1)2 + (x + 1) + 30 + ln . 1 3 2 6 8x + 4x + x + 30 + MCk (x)

By using Mathematica software and Lemma 1, we can check (4.6)

µ(Ek (x)) = 2k + 6,

(k = 0, 1, 2, 3).

We let Uk (x) = Ek (x) − Ek (x + 1). By making use of Mathematica software again, we can prove Ψ1 (13; x)(x − 1) + 416838558509297754261614731715717 < 0, 3x(1 + x)2 (79 + 154x)2 (233 + 154x)2 (Ψ21 (3; x)(x − 1) + 363565)2 Ψ222 (4; x) Ψ3 (19; x)(x − 1) + 85653 · · · 25001 U100 (x) = > 0, x(1 + x)2 Ψ4 (28; x) Ψ5 (25; x)(x − 1) + 32968 · · · 13479 U200 (x) = − < 0, x(1 + x)2 Ψ6 (36; x) Ψ7 (31; x)(x − 1) + 17145 · · · 57723 > 0. U300 (x) = 3x(1 + x)2 Ψ8 (44; x) U000 (x) =

Similar to the proof of Theorem 1, we can get the desired assertions from the above inequalities.

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Theorem 4. Let the first-correction function MC∗1 (x) be defined by MC∗1 (x) =

(4.7)

κ0 κ1 + 3 , 2 x + λ0 x + λ10 x + λ11 x + λ12

where 11 , 240 459733 κ1 = , 15523200 717183502490887 , λ11 = 520777318696096

κ0 = −

79 , 154 71181889 λ10 = , 70798882 1118629052995381153799 = . 1958878792277282473920 λ0 =

λ12

Then for x ≥ 1, the following inequality holds true  1 6 √  x x 1 3 2 ∗ 8x + 4x + x + + MC1 (x) . Γ(x + 1) < π e 30

(4.8)

Proof. Let the first-correction error function E1∗ (x) be defined by (4.9)

1  6 √  x x 1 ∗ 3 2 Γ(x + 1) = π exp(E1∗ (x)). + MC1 (x) 8x + 4x + x + e 30

Hence (4.10)

E1∗ (x)

−

E1∗ (x

  1 + 1) = − 1 + x ln 1 + x 1 3 + MC∗1 (x + 1) 1 8(x + 1) + 4(x + 1)2 + (x + 1) + 30 . + ln 1 6 8x3 + 4x2 + x + 30 + MC∗1 (x)

By using Mathematica software and Lemma 1, we have µ(E1∗ (x)) = 10.

(4.11)

Now we let V (x) = E1∗ (x) − E1∗ (x + 1). By using Mathematica again, we have (4.12)

V100 (x) = −

Ψ1 (33; x)(x − 1) + 96057 · · · 27429 < 0. 3x (Ψ2 (3; x)) Ψ3 (12; x) (Ψ4 (6; x)(x − 1) + 2169 · · · 3461)2 Ψ5 (14; x) 2

By the same approach as the proof of Theorem 1, the inequality (4.8) follows from the (4.12). Remark 2. It is an interesting question whether our method may be used to obtain some sharp bounds for the ratio of the gamma functions, see e.g. [11, 19, 20, 21].

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References [1] M. Abramowitz, I. A. Stegun(Editors), Handbook of Mathematical Functions With Formulas, Graphs, and Mathematical Tables, Applied Mathematics Series 55, ninth printing, National Bureau of Standards, Washington D.C., 1972. [2] H. Alzer, On Ramanujan’s double inequality for the gamma function, Bull. London Math. Soc. 35 (2003), no. 5, 601–607. [3] N. Batir, Very accurate approximations for the factorial function. J. Math. Inequal. 4(3)(2010), 335–344. [4] B.C. Berndt, Ramanujan’s Notebooks, Part II, Springer-Verlag, 1989. [5] W. Burnside, A rapidly convergent series for log N!. Messenger Math. 46(1917), 157–159. [6] X.D. Cao, H.M. Xu and X. You, Multiple-correction and faster approximation, J. Number Theory 149(2015),327–350. Availabe at: http://dx.doi.org/10.1016/j.jnt.2014.10.016. [7] X.D. Cao, Multiple-Correction and Continued Fraction J. Math. Anal. Appl. 424(2015)1425–1446. Availabe at: http://dx.doi.org/10.1016/j.jmaa.2014.12.014.

Approximation, Availabe at

[8] C.-P. Chen and L. Lin, Remarks on asymptotic expansions for the gamma function. Appl. Math. Lett. 25(2012), 2322–2326. [9] Chao-Ping Chen and Jing-Yun Liu, Inequalities and asymptotic expansions for the gamma function, Journal of Number Theory 149(2015),313–326. [10] R.W. Gosper, Decision procedure for indefinite hypergeometric summation. Proc. Natl. Acad. Sci. USA 75(1978), 40–42. [11] S. Guo, J. Xu and F. Qi, Some exact constants for the approximation of the quantity in the Wallis’ formula, Journal of Inequalities and Applications 2013, 2013:67, 7 pp. [12] M.D. Hirschhorn and M.B. Villarino, A refinement of Ramanujans factorial approximation, Ramanujan J. 34(2014),73–81. [13] E.A. Karatsuba, On the asymptotic representation of the Euler gamma function by Ramanujan, J. Comput. Appl. Math. 135 (2001), no. 2, 225–240. [14] C. Mortici, New approximations of the gamma function in terms of the digamma function, Applied Mathematics Letters, 23 (2010) 97–100. [15] C. Mortici, On Ramanujan’s large argument formula for the gamma function, Ramanujan J. 26 (2011), no. 2, 185-192. [16] C. Mortici, Ramanujan’s estimate for the gamma function via monotonicity arguments, Ramanujan J. 25 (2011), no. 2, 149–154. 10

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[17] C. Mortici, A new fast asymptotic series for the gamma function, Ramanujan J. DOI 10.1007/s11139-041-9589-0. [18] C. Mortici, Sharp bounds for gamma function in terms of xx−1 , Applied Mathematics and Computation, 249(2015),278–285. [19] Feng Qi, Bounds for the Ratio of Two Gamma Functions, Journal of Inequalities and Applications, Volume 2010, Article ID 493058, 84 pp. [20] Feng Qi, Integral representations and complete monotonicity related to the remainder of Burnside’s formula for the gamma function, Journal of Computational and Applied Mathematics, 268 (2014), 155–167. [21] Feng Qi and Qiu-Ming 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 pp. [22] S. Ramanujan, The Lost Notebook and Other Unpublished Papers. Narosa, Springer, New Delhi, Berlin (1988). Intr. by G.E. Andrews. Xiaodong Cao Department of Mathematics and Physics, Beijing Institute of Petro-Chemical Technology, Beijing, 102617, P. R. China E-mail: [email protected]

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Mathematical analysis of humoral immunity viral infection model with Hill type infection rate M. A. Obaid Department of Mathematics, Faculty of Science, King Abdulaziz University, P.O. Box 80203, Jeddah 21589, Saudi Arabia. Email: [email protected]

Abstract In this paper, we propose and analyze a viral infection model with humoral immunity. The incidence rate is given by Hill type infection rate. We have derived two threshold parameters, R0 and R1 which completely determined the global properties of the model. By constructing suitable Lyapunov functions and applying LaSalle’s invariance principle we have established the global asymptotic stability of all steady states of the model. We have proven that, if R0 ≤ 1, then the infection-free steady state is globally asymptotically stable (GAS), if R1 ≤ 1 < R0 , then the chronic-infection steady state without humoral immune response is GAS, and if R1 > 1, then the chronic-infection steady state with humoral immune response is GAS. Keywords: Virus infection; Global stability; Immune response; Lyapunov function; Hill type infection rate.

1

Introduction

In recent years, considerable attention has been paid to study the dynamical behaviors of viruses such as human immunodeficiency virus (HIV) (see e.g. [1]-[11]), hepatitis B virus (HBV) [12]-[14], hepatitis C virus (HCV) [15]-[17], human T cell leukemia (HTLV) [18] and dengue virus [19], etc. There are many benifits from mathematical models of viral infection include: (i) it provide important quantitative insights into viral dynamics in vivo, (ii) it can improve diagnosis and treatment strategies which yield to raise hopes of patients with viruses, (iii) it can be used to estimate key parameter values that control the infection process. Nowak and Bangham [2] proposed the basic viral infection model which contains three variables x, y and v representing the populations of the uninfected target cells, infected cells and free virus particles, respectively. In [20]-[26], the basic model has been modified to take into consideration the humoral immune response. The basic model of viral infection with humoral immune response has

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been introduced by Murase et. al. [20] and Shifi Wang [26] as: x˙ = λ − dx − βxv,

(1)

y˙ = βxv − ay,

(2)

v˙ = ky − cv − rzv,

(3)

z˙ = gzv − µz,

(4)

where z denotes the population of the B cells. Parameters λ, k and g represent, respectively, the rate at which new healthy cells are generated from the source within the body, the generation rate constant of free viruses produced from the infected cells and the proliferation rate constant of B cells. Parameters d, a, c and µ are the natural death rate constants of the uninfected cells, infected cells, free virus particles and B cells, respectively. Parameter β is the infection rate constant and r is the removal rate constant of the virus due to humoral immune response. All the parameters given in model (1)-(4) are positive. In model (1)-(4), the incidence rate is supposed to be bilinear, βxv, which is based on the law of mass action. In reality, bilinear incidence rate is not accurate to describe the viral dynamics during the full course of infection. In [27], the incidence rate is given by Hill type infection rate. However, the humoral immune response has been neglected. Our aim in this paper is to propose a viral infection model with humoral immune response and investigate its global stability analysis. The incidence rate is given by Hill type infection rate. Using Lyapunov functions, we prove that the global stability of the model is determined by two threshold parameters, the basic infection reproduction number R0 and the humoral immune response activation number R1 . We have proven that, if R0 ≤ 1, then the uninfected steady state is globally asymptotically stable (GAS), if R1 ≤ 1 < R0 , then the infected steady state without humoral immune response is GAS, and if R1 > 1, then the infected steady state with humoral immune response is GAS.

2

The model

In this section, we propose a viral infection model with humoral immune response. The incidence rate is given by a Hill type infection rate. x˙ = λ − dx −

βxn v , γ n + xn

(5)

βxn v − ay, γ n + xn

(6)

v˙ = ky − cv − rvz,

(7)

z˙ = gvz − µz,

(8)

y˙ =

where γ and n are positive constants. Next, we study the properties of the solutions of the model.

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2.1

Positive invariance

We note that model (5)-(8) is biologically acceptable in the sense that no population goes negative. It is straightforward to check the positive invariance of the non-negative orthant R4≥0 by model (5)-(8). In the following, we show the boundednes of the solution of model (5)-(8).



Proposition 1. There exist positive numbers Li , i = 1, 2, 3 such that the compact set Ω = (x, y, v, z) ∈ R4≥0 : 0 ≤ x, y ≤ L1 , 0 ≤ v ≤ L2 , 0 ≤ z ≤ L3 is positively invariant. Proof. Let X1 (t) = x(t) + y(t), then X˙ 1 = λ − dx − ay ≤ λ − s1 X1 ,

λ . Since x(t) > 0 and y(t) ≥ 0, s1 then 0 ≤ x(t), y(t) ≤ L1 if 0 ≤ x(0) + y(0) ≤ L1 . On the other hand, let X2 (t) = v(t) + gr z(t), then where s1 = min{d, a}. Hence X1 (t) ≤ L1 , if X1 (0) ≤ L1 , where L1 =

rµ r X˙ 2 = ky − cv − z ≤ kL1 − s2 (v + z) = kL1 − s2 X2 , g g kL1 . Since v(t) ≥ 0 and s2 z(t) ≥ 0, then 0 ≤ v(t) ≤ L2 and 0 ≤ z(t) ≤ L3 if 0 ≤ v(0) + gr z(0) ≤ L2 , where L3 = gLr 2 .

where s2 = min{c, µ}. Hence X2 (t) ≤ L2 , if X2 (0) ≤ L2 , where L2 =

2.2

Steady states

In this subsection, we calculate the steady states of model (5)-(8) and derive two thresholds parameters. The steady states of model (5)-(8) satisfy the following equations: βxn v = 0, γ n + xn βxn v − ay = 0, n γ + xn

λ − dx −

(9) (10)

ky − cv − rvz = 0,

(11)

(gv − µ)z = 0.

(12)

Equation (12) has two possible solutions, z = 0 or v = µ/g. If z = 0, then from Eqs. (10)-(11) we obtain kβxn v − cv = 0. a(γ n + xn )

(13)

Equation (13) has two possibilities, v = 0 or v 6= 0. If v = 0, then y = 0 and x =

λ d

which leads to

the uninfected steady state E0 = (x0 , 0, 0, 0), where x0 = λd . If v 6= 0, then from Eqs. (9) and (13) we obtain k βxn v k(λ − dx) = n n ac γ + x ac ac ⇒ x = x0 − v. dk

(14)

v=

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Then, Eq. (13) becomes


n ac v v dk
n ac x0 − dk v

kβ x0 − aγ n + a

− cv = 0.

Let us define a function Ψ1 as Ψ1 (v) =


n ac v dk v
n ac x0 − dk v

kβ x0 − aγ n + a

It is clear that Ψ1 (0) = 0, and when v = v =

x0 dk ac

− cv = 0.

> 0, then Ψ1 (v) = −cv < 0. Since Ψ1 (v) is

continuous for all v ≥ 0, then we have Ψ01 (0) Therefore, if Ψ01 (0) > 0 i.e.

n

k βx0 ac γ n +xn 0

 =c

 k βxn0 −1 . ac γ n + xn0

> 1, then there exist a v1 ∈ (0, v) such that Ψ1 (v1 ) = 0. From

Eq. (11) we obtain y1 = kc v1 > 0 and from Eq. (9) we define a function Ψ2 as: Ψ2 (x) = λ − dx −

βxn v1 = 0. γ n + xn

βxn

0 We have Ψ2 (0) = λ > 0 and Ψ2 (x0 ) = − γ n +x n v1 < 0. Since f (x) = 0

xn γ n +xn

is a strictly increasing

function of x, then Ψ2 is a strictly decreasing function of x, then there exist a unique x1 ∈ (0, x0 ) such that Ψ2 (x1 ) = 0. It means that, an infected steady state without humoral immune response E1 = (x1 , y1 , v1 , 0) exists when

n k βx0 ac γ n +xn 0

> 1. Now we are ready to define the basic infection reproduction

number as: k βxn0 . ac γ n + xn0 µ The other possibility of Eq. (12) z 6= 0 leads to v2 = . Inserting v2 in Eq. (9) we define a function g Ψ3 as: βxn v2 Ψ3 (x) = λ − dx − n = 0. γ + xn R0 =

βxn v

0 2 Note that Ψ3 is a strictly decreasing function of x. Clearly, Ψ3 (0) = λ > 0 and Ψ3 (x0 ) = − γ n +x n < 0. 0

Thus, there exists a unique x2 ∈ (0, x0 ) such that Ψ3 (x2 ) = 0. It follows from Eq. (11) that,   c k βxn2 βxn2 v2 , z2 = −1 . y2 = a(γ n + xn2 ) r ac γ n + xn2 Thus y2 > 0, and if

n k βx2 v2 acv2 γ n +xn 2

> 1, then z2 > 0 when . Now we define the humoral immune response

activation number as: R1 =

k βxn2 , ac γ n + xn2

Hence, z2 can be rewritten as z2 = rc (R1 − 1). It follows that, there exists an infected steady state with humoral immune response E2 (x2 , y2 , v2 , z2 ) when R1 > 1. Since x1 < x0 , then R1 =

k βxn2 k βxn0 < = R0 . ac γ n + xn2 ac γ n + xn0

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From above we have the following result. Lemma 1 (i) if R0 ≤ 1, then there exists only one positive equilibrium E0 , (ii) if R1 ≤ 1 < R0 , then there exist two positive steady states E0 and E1 , and (iii) if R1 > 1, then there exist three positive steady states E0 , E1 and E2 .

3

Global stability analysis

In this section, we establish the global stability of the three steady states of system (5)-(8) employing the direct Lyapunov method and LaSalle’s invariance principle. Theorem 1. If R0 ≤ 1, then E0 is GAS. Proof. Define a Lyapunov functional W0 as follows: Zx W0 = x − x0 − x0

Calculating

xn0 (γ n + sn ) a ar ds + y + v + z. n n n s (γ + x0 ) k kg

dW0 dt

along the trajectories of (5)-(8) as:    βxn v dW0 xn0 (γ n + xn ) βxn v λ − dx − = 1− n n + dt x (γ + xn0 ) γ n + xn γ n + xn a ar + (ky − cv − rvz) + (gvz − µz) k kg    xn0 (γ n + xn ) x βxn0 v ac =λ 1− n n 1 − + − v− n n n x (γ + x0 ) x0 γ + x0 k     x xn0 (γ n + xn ) ac k βxn0 1− =λ 1− n n + x (γ + xn0 ) x0 k ac (γ n + xn0 ) λγ n (xn − xn0 ) (x0 − x) ac arµ = + (R0 − 1)v − z. n n n x x0 (γ + x0 ) k kg

We have (xn − xn0 ) (x0 − x) ≤ 0 for all x, n > 0. Then if R0 ≤ 1 then

− ay (16) arµ z kg  arµ −1 v− z kg (17)

dW0 dt

≤ 0 for all x, v, z > 0. Thus, n o 0 the solutions of system (5)-(8) converge to Ω, the largest invariant subset of dW = 0 [28]. Clearly, dt it follows from Eq. (17) that

dW0 dt

= 0 if and only if x = x0 , 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 can see from Eq. (7) that 0 = v˙ = ky, and thus y = 0. Hence

dW0 dt

= 0 if and only if x = x0 , y = 0, v = 0 and z = 0. From

LaSalle’s invariance principle, E0 is GAS. Theorem 2. If R1 ≤ 1 < R0 , then E1 is GAS. Proof. We construct the following Lyapunov functional Zx W1 = x − x1 − x1

xn1 (γ n + sn ) ds + y1 H sn (γ n + xn1 )

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y y1



a + v1 H k



v v1

 +

ar z. kg

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The time derivative of W1 along the trajectories of (5)-(8) is given by       βxn v y1 βxn v xn1 (γ n + xn ) dW1 λ − dx − n + 1− − ay = 1− n n dt x (γ + xn1 ) γ + xn y γ n + xn a v1  ar + 1− (ky − cv − rvz) + (gvz − µz) . k v kg Applying λ = dx1 +

βxn 1 v1 γ n +xn 1

(18)

and collecting terms of Eq. (18) we get

    βxn1 v1 xn1 (γ n + xn ) xn1 (γ n + xn ) (dx1 − dx) + n 1− n n 1− n n x (γ + xn1 ) γ + xn1 x (γ + xn1 ) βxn v y1 yv1 ac ar arµ − n + ay1 − a + v1 + v1 z − z. γ + xn y v k k kg

dW1 = dt

Using the equilibrium conditions for E1 ,

βxn 1 v1 γ n +xn 1

= ay1 , cv1 = ky1 , we obtain

     x xn1 (γ n + xn ) xn1 (γ n + xn ) βxn1 v1 dW1 1− 1− n n + n = dx1 1 − n n dt x (γ + xn1 ) x1 γ + xn1 x (γ + xn1 )   βxn1 v1 xn (γ n + xn1 )vy1 µ βxn1 v1 βxn1 v1 yv1 βxn1 v1 ar − n v1 − z + − n + + γ + xn1 xn1 (γ n + xn )v1 y γ n + xn1 γ + xn1 y1 v γ n + xn1 k g   dγ n (xn − xn1 ) (x1 − x) ar µ = + v1 − z n n n x (γ + x1 ) k g   xn1 (γ n + xn ) xn (γ n + xn1 )vy1 yv1 βxn1 v1 3− n n − − . + n γ + xn1 x (γ + xn1 ) xn1 (γ n + xn )v1 y y1 v

(19)

Clearly, the first term of Eq. (19) is less than or equal zero. Because the geometrical mean is less than or equal to the arithmetical mean, then the third and fourth terms of Eq. (19) are less than or equal zero. Now we show that if R1 ≤ 1 then v1 ≤

µ r

= v2 . This can be achieved if we show that

sgn (x2 − x1 ) = sgn (v1 − v2 ) = sgn (R1 − 1) . We have 

xn2 xn1 − n γ n + x2 γ n + xn1

 (x2 − x1 ) > 0,

(20)

Suppose that, sgn (x2 − x1 ) = sgn (v2 − v1 ). Using the conditions of the steady states E1 and E2 we have βxn2 v2 βxn1 v1 − γ n + xn2 γ n + xn1 βxn v2 βxn v1 βxn v1 βxn v1 = n 2 n− n 2 n+ n 2 n− n 1 n γ + x2 γ + x2 γ + x2 γ + x1   n n βx2 x2 xn1 = n (v2 − v1 ) + βv1 − n , γ + xn2 γ n + xn2 γ + xn1

(λ − dx2 ) − (λ − dx1 ) =

and from inequality (20) we get sgn (x1 − x2 ) = sgn (x2 − x1 ), which leads to contradiction. Thus, sgn (x2 − x1 ) = sgn (v1 − v2 ) . Using the steady state conditions for E1 we have R1 − 1 =

k ac



βxn2 βxn1 − n γ n + x2 γ n + xn1

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n

k βx1 ac γ n +xn 1

= 1, then

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From inequality (20) we get: sgn (R1 − 1) = sgn (v1 − v2 ) . It follows that, if R1 ≤ 1 then v1 ≤

µ r

= v2 . Therefore, if R1 ≤ 1 then

dW1 dt

≤ 0 for all x, y, v, z > 0,

where the equality occurs at the equilibrium E1 . LaSalle’s invariance principle implies the global stability of E1 . Theorem 3. If R1 > 1,then E2 is GAS. Proof. We construct the following Lyapunov functional Zx W2 = x − x2 − x2

xn2 (γ n + sn ) ds + y2 H sn (γ n + xn2 )



y y2



a + v2 H k



v v2



ar + z2 H kg



z z2

 .

We calculate the time derivative of W2 along the trajectories of (5)-(8) as:       βxn v y2 βxn v dW2 xn2 (γ n + xn ) λ − dx − n + 1− − ay = 1− n n dt x (γ + xn2 ) γ + xn y γ n + xn a v2  ar  z2  + 1− (ky − cv − rvz) + 1− (gvz − µz) . k v kg z Applying λ = dx2 +

βxn 2 v2 γ n +xn 2

(21)

and collecting terms of Eq. (21) we get

    xn2 (γ n + xn ) βxn2 v2 xn2 (γ n + xn ) 1− n n (dx2 − dx) + n 1− n n x (γ + xn2 ) γ + xn2 x (γ + xn2 ) n n βx v ac yv2 βx v y2 + n 2 n− n + ay2 − v − a n γ + x2 γ +x y k v ac ar arµ ar arµ + v2 + v2 z − z − z2 v + z2 . k k kg k kg

dW2 = dt

Using the equilibrium conditions for E2 βxn2 v2 = ay2 , ky2 = cv2 + rv2 z2 , µ = gv2 , γ n + xn2 we get      xn2 (γ n + xn ) x βxn2 v2 xn2 (γ n + xn ) dW2 = dx2 1 − n n 1− + n 1− n n dt x (γ + xn2 ) x2 γ + xn2 x (γ + xn2 ) βxn v2 xn (γ n + xn2 )vy2 βxn2 v2 βxn2 v2 yv2 βxn2 v2 − n 2 n n n + − + γ + x2 x2 (γ + xn )v2 y γ n + xn2 γ n + xn2 y2 v γ n + xn2   dγ n (xn − xn2 ) (x2 − x) βxn2 v2 xn2 (γ n + xn ) xn (γ n + xn2 )vy2 yv2 = + n 3− n n − − . xn (γ n + xn2 ) γ + xn2 x (γ + xn2 ) xn2 (γ n + xn )v2 y y2 v

(22)

Thus, if R1 > 1 then x2 , y2 , v2 and z2 > 0. Clearly, we get that the first and second terms of Eq. (22) are less than or equal zero. Since the arithmetical mean is greater than or equal to the geometrical mean, then

dW2 dt

≤ 0. It can be seen that,

dW2 dt

= 0 if and only if x = x2 , y = y2 and v = v2 . From Eq.

(7), if v = v2 and y = y2 , then v˙ = 0 and 0 = ky2 − cv2 − rv2 z, which yields z = z2 and hence equal to zero at E2 . LaSalle’s invariance principle implies global stability of E2 .

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dW2 dt

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Conclusion

In this paper, we have proposed and analyzed a viral infection model with humoral immune response. The model is a four dimensional that describe the interaction between the uninfected target cells, infected cells, free virus particles and B cells. The incidence rate has been represented by Hill type infection rate. We have derived two threshold parameters, the basic reproduction number R0 and the humoral immune response number R1 which completely determined the basic and global properties of the viral infection model. Using Lyapunov method and applying LaSalle’s invariance principle we have proven that if R0 ≤ 1, then the uninfected steady state is GAS, if R1 ≤ 1 < R0 , then the infected steady state without humoral immune response is GAS, and if R1 > 1, then the infected steady state with humoral immune response is GAS.

5

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. A. Nowak and R. M. May, “Virus dynamics: Mathematical Principles of Immunology and Virology,” Oxford Uni., Oxford, 2000. [2] M. A. Nowak and C. R. M. Bangham, Population dynamics of immune responses to persistent viruses, Science, 272 (1996), 74-79. [3] A. S. Perelson and P. W. Nelson, Mathematical analysis of HIV-1 dynamics in vivo, SIAM Rev., 41 (1999), 3-44. [4] A.M. Elaiw, and S.A. Azoz, Global properties of a class of HIV infection models with BeddingtonDeAngelis functional response, Math. Method Appl. Sci., 36 (2013), 383-394. [5] L. Wang, M.Y. Li, Mathematical analysis of the global dynamics of a model for HIV infection of CD4 + T cells, Math. Biosc., 200 (2006), 44-57. [6] Y. Zhao, D. T. Dimitrov, H. Liu and Y. Kuang, Mathematical insights in evaluating state dependent effectiveness of HIV prevention interventions, Bull. Math. Biol., 75 (2013), 649-675. [7] K. Hattaf and N. Yousfi, Global stability of a virus dynamics model with cure rate and absorption, Journal of the Egyptian Mathematical Society, (In press).

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[8] D. S. Callaway and A. S. Perelson, HIV-1 infection and low steady state viral loads, Bull. Math. Biol., 64 (2002), 29-64. [9] 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. [10] A. M. Elaiw, Global properties of a class of virus infection models with multitarget cells, Nonlinear Dynam., 69 (2012) 423-435. [11] A. M. Elaiw, Global properties of a class of HIV models, Nonlinear Anal. Real World Appl., 11 (2010), 2253–2263. [12] S. Eikenberry, S. Hews, J. D. Nagy and Y. Kuang, The dynamics of a delay model of HBV infection with logistic hepatocyte growth, Math. Biosc. Eng., 6 (2009), 283-299. [13] S. A. Gourley, Y. Kuang and J. D. Nagy, Dynamics of a delay differential equation model of hepatitis B virus infection, J. Biol. Dyn., 2 (2008), 140-153. [14] J. Li, K. Wang, Y. Yang, Dynamical behaviors of an HBV infection model with logistic hepatocyte growth, Math. Comput. Modelling, 54 (2011), 704-711. [15] R. Qesmi, J. Wu, J. Wu and J.M. Heffernan, Influence of backward bifurcation in a model of hepatitis B and C viruses, Math. Biosci., 224 (2010) 118–125. [16] R. Qesmi, S. ElSaadany, J.M. Heffernan and J. Wu, A hepatitis B and C virus model with age since infection that exhibit backward bifurcation, SIAM J. Appl. Math., 71 (4) (2011) 1509–1530. [17] 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. [18] M. Y. Li and H. Shu, Global dynamics of a mathematical model for HTLV-I infection of CD4+ T cells with delayed CTL response, Nonlinear Anal. Real World Appl., 13 (2012), 1080-1092. [19] P. Tanvi, G. Gujarati, and G. Ambika, Virus antibody dynamics in primary and secondary dengue infections, J. Math. Biol., (In press). [20] A. Murase, T. Sasaki and T. Kajiwara, Stability analysis of pathogen-immune interaction dynamics, J. Math. Biol., 51 (2005), 247-267. [21] W. Dominik, R. M. May and M. A. Nowak, T he role of antigen-independent persistence of memory cytotoxic T lymphocytes, Int. Immunol. 12 (4) (2000), 467-477.

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[22] M. A . Obaid, Global dynamics of a viral infection model with exposed state and antibodies, Journal of Computational and Theoretical Nanoscience, (in press). [23] 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. [24] A. M. Elaiw, A. Alhejelan, Global dynamics of virus infection model with humoral immune response and distributed delays. Journal of Computational Analysis and Applications, 17 (2014), 515-523. [25] T. Wang, Z. Hu, F. Liao and Wanbiao, Global stability analysis for delayed virus infection model with general incidence rate and humoral immunity, Math. Comput. Simulation, 89 (2013), 13-22. [26] S. Wang and D. Zou, Global stability of in host viral models with humoral immunity and intracellular delays, J. Appl. Math. Mod., 36 (2012), 1313-1322. [27] N. Bairagi, D. Adak, Global analysis of HIV-1 dynamics with Hill type infection rate and intracellular delay, Appl. Math. Model., 38 (2014), 5047-5066. [28] J.K. Hale, and S. Verduyn Lunel, Introduction to functional differential equations, SpringerVerlag, New York, 1993.

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A Parameterized Series Representation for Ap´ ery’s Constant ζ(3)

HORST ALZER

a

b

a

and

JONATHAN SONDOW

b

Morsbacher Str. 10, 51545 Waldbr¨ol, Germany email: [email protected]

209 West 97th Street, New York, NY 10025, USA email: [email protected]

Abstract. We prove that if λ ≤ 1/2, then n   ∞ X X n 1 (−λ)n−k δk ζ(3) = n+1 k (1 − λ) n=1

k=1

with Hk 1 δk = 2 − k k



 π2 (2) − Hk , 6

(2)

where Hk and Hk denote the harmonic numbers and the generalized harmonic numbers of order 2, respectively. Keywords. Ap´ery’s constant, series representation, harmonic numbers. 2010 Mathematics Subject Classification. 11M35

1

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1. Introduction The famous Riemann zeta function is defined for all complex numbers s with 1 by the Dirichlet series (1)

ζ(s) =

∞ X 1 1 1 = 1 + s + s + ··· . ns 2 3

n=1

In this note we are concerned with the special case s = 3, that is, with ζ(3) =

∞ X 1 = 1.20205 . . . . n3

n=1

This number is known in the literature as Ap´ery’s constant. It is named after the Greek-French mathematician Roger Ap´ery (1916–1994), who proved in 1979 that ζ(3) is irrational; see [4]. A central role in his proof is played by the elegant series representation ∞

ζ(3) =

5 X (−1)n−1
. 2 n3 2n n n=0

Ap´ery’s constant has been the subject of much attention. in the P It appears 2 + r 2 )−2 , 2n(n solution of finding sharp bounds for the Mathieu series ∞ n=1 it has applications in physics and it also occurs in the solution of probability problems; see [1], [8] and [10, A002117]. Euler, Ramanujan and numerous other researchers provided various integral and series representations for ζ(3) and related constants. We refer to Srivastava’s survey paper [12] and the references therein; see also [2]. As is well-known, Euler proved that the numbers ζ(2n) (n = 1, 2, 3, ...) are irrational. Thus, it is natural to ask whether the values ζ(2n + 1) (n = 2, 3, 4, ...) are also irrational. This is a classical open problem. Recent progress on this subject was made by Rivoal [9], who established that infinitely many of the numbers ζ(2n+1) are irrational, and Zudilin [14], who proved that at least one of the numbers ζ(5), ζ(7), ζ(9), ζ(11) is irrational. It is the aim of this note to present a new singly-parameterized series representation for ζ(3) in terms of the classical harmonic numbers Hk =

k X 1 j=1

j

=1+

1 1 1 + + ··· + 2 3 k

(k = 1, 2, ...).

and the generalized harmonic numbers of order 2 (2) Hk

k X 1 1 1 1 = = 1 + 2 + 2 + ··· + 2 2 j 2 3 k

(k = 1, 2, ...).

j=1

Our method of proof, which can be used to obtain series representations for other mathematical constants as well, is explained in detail in [3]. A key

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role is played by the remarkable integral representation Z 1 1 log t log(1 − t) (2) ζ(3) = dt, 2 0 t (1 − t) which was published by Janous [7] in 2006. 2. Main Result The following series representation for Ap´ery’s constant is valid. Theorem. Let λ be a real number with λ ≤ 1/2. Then, n   ∞ X X n 1 (−λ)n−k δk (3) ζ(3) = n+1 (1 − λ) k n=1

k=1

with Hk 1 δk = 2 − k k

(4)



 π2 (2) . − Hk 6

Proof. Let λ ≤ 1/2 and 0 < t < 1. Then, t−λ < 1. 1−λ Expanding in a geometric series, we obtain −1
j≥1

X 1 1 = ζ(3), = i2 j i3 i≥1

as claimed. If we combine this with (6) and reverse the order of summation, we get ζ(3) =

∞ X n=1

δn =

∞ X ∞ X ν=1 n=1

∞

(2)

X ζ(2) − Hν 1 = . ν(n + ν)2 ν ν=1

Together with (7), this proves Euler’s famous relation [5] ∞ X Hn i=1

= 2ζ(3).

n2

Example 2. The case λ = −1 yields ∞ n   X 1 X n δk . 2ζ(3) = k 2n n=1

k=1

This may be compared to the series ∞ n   X 3 1 X n (−1)k ζ(3) = 2 2n k (k + 1)3 n=0

k=0

which in turn is the case s = 3 of a global series for ζ(s) due to Hasse [6] and rediscovered in [11]. Example 3. The cases λ = 1/2, −1/2, 1/4 give   ∞ n X X n k n ζ(3) = (−1) (−1) 2k+1 δk , k n=1 k=1 ∞ n X 1 X n 3 ζ(3) = 2k δk , 2 3n k n=1 k=1 ∞ n   X 1 X 3 n ζ(3) = (−4)k δk , 4 (−3)n k n=1

k=1

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respectively. 4. Concluding Remarks We conclude the paper with three remarks. Remark 1. If we multiply both sides of (3) by (1 − λ)a+1 (a ∈ R) and differentiate with respect to λ, then we find that ∞ n   X X
1 n (a + 1) ζ(3) = (−λ)n−k−1 n − aλ + (λ − 1)k δk . n+1 (1 − λ) k n=1

k=1

Applying this with a = 1, λ = −1/4 and (3) with λ = −1/4 yields ∞ n   X 5 1 X n k ζ(3) = 4 (4n − 5k) δk . 4 5n k n=1

k=1

Remark 2. Using the asymptotic formulas Hk ∼ log k

(2)

and ζ(2) − Hk ∼

1 k

(k → ∞)

we obtain

log k (k → ∞). k2 For k = 1, 2, . . . , 10, we have the values δk ∼

δk = 0.35506 . . . , 0.17753 . . . , 0.10909 . . . , 0.07487 . . . , 0.05506 . . . , 0.04246 . . . , 0.03389 . . . , 0.02777 . . . , 0.02324 . . . , 0.01977 . . . . Remark 3. Applying the series representation (6) and [13, Theorem 11d] we conclude that the sequence {δk }∞ k=0 is not only decreasing and convex but even completely monotonic, that is, (−1)n ∆n δk ≥ 0

for k, n = 0, 1, 2, ...,

where ∆ denotes the forward difference operator defined by ∆ 0 δk = δ k ,

∆n δk = ∆n−1 δk+1 − ∆n−1 δk

(k = 0, 1, 2, ...; n = 1, 2, ...).

References [1] H. Alzer, J. L. Brenner, O. G. Ruehr, On Mathieu’s inequality, J. Math. Anal. Appl. 218 (1998), 607–610. [2] H. Alzer, D. Karayannakis, H. M. Srivastava, Series representations for some mathematical constants, J. Math. Anal. Appl. 320 (2006), 145–162. [3] H. Alzer, S. Koumandos, Series and product representations for some mathematical constants, Period. Math. Hung. 58 (2009), 71-82. [4] R. Ap´ery, Irrationalit´e de ζ(2) et ζ(3), Ast´erisque 61 (1979), 11–13. [5] L. Euler, Meditationes circa singulare serierum genus, Novi Comm. Acad. Sci. Petropolitanae 20 (1775), 140–186; reprinted in Opera Omnia, ser. I, vol. 15, B. G. Teubner, Berlin, 1927, pp. 217–267.

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[6] H. Hasse, Ein Summierungsverfahren f¨ ur die Riemannsche ζ-Reihe, Math. Zeit. 32 (1930), 458–464. [7] W. Janous, Around Ap´ery’s constant, J. Inequal. Pure Appl. Math. 7(1) (2006), article 35, 8 pages. [8] C. Nash, D. J. O’Connor, Determinants of Laplacians, the Ray-Singer torsion on lens spaces and the Riemann zeta function, J. Math. Phys. 36 (1995), 1462–1505. [9] T. Rivoal, La fonction zˆeta de Riemann prend une infinit´e de valeurs irrationnelles aux entiers impairs, Compt. Rend. Acad. Sci. 331 (2000), 267–270. [10] N. J. A. Sloane, The On-line Encyclopedia of Integer Sequences, http://oeis.org, 2010. [11] J. Sondow, Analytic continuation of Riemann’s zeta function and values at negative integers via Euler’s transformation of series, Proc. Amer. Math. Soc. 120 (1994), 421–424. [12] H. M. Srivastava, Some families of rapidly convergent series representations for the zeta functions, Taiwanese J. Math. 4 (2000), 560–598. [13] D. V. Widder, The Laplace Transform, Princeton Univ. Press, Princeton, NJ, 1941. [14] W. Zudilin, One of the numbers ζ(5), ζ(7), ζ(9), ζ(11) is irrational, Russ. Math. Surv. 56 (2001), 774–776.

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TABLE OF CONTENTS, JOURNAL OF COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 20, NO. 7, 2016

Second Order Duality for Multiobjective Optimization Problems, Meraj Ali Khan, and Falleh R. Al-Solamy,……………………………………………………………………………………1195 On A Fifth-Order Difference Equation, Stevo Stevic, Josef Diblik, Bratislav Iricanin, and Zdenek Smarda,……………………………………………………………………………………….1214 Modified Three-Step Iterative Schemes for Asymptotically Nonexpansive Mappings In Uniformly Convex Metric Spaces, Shin Min Kang, Arif Rafiq, Faisal Ali, and Young Chel Kwun,…………………………………………………………………………………………1228 On Identities between Sums of Euler Numbers and Genocchi Numbers of Higher Order, Lee-Chae Jang, and Byung Moon Kim,………………………………………………………1240 An Algorithm for Multi-Attribute Decision Making Based On Soft Rough Sets, Guangji Yu,1248 Fixed Point Results for Modular Ultrametric Spaces, Cihangir Alaca, Meltem Erden Ege, and Choonkil Park,………………………………………………………………………………..1259 On the Backward Difference Scheme for a Class of SIRS Epidemic Models With Nonlinear Incidence, Zhidong Teng, Ying Wang, and Mehbuba Rehim,………………………………1268 Bounds for the Largest Eigenvalue of Nonnegative Tensors, Jun He,………………………1290 A Note On Fractional Neutral Integro-Differential Inclusions With State-Dependent Delay In Banach Spaces, Selvaraj Suganya, Dumitru Baleanu, and Mani Mallika Arjunan,…………1302 New Hermite-Hadamard's Inequalities for Preinvex Functions via Fractional Integrals, Shahid Qaisar, Muhammad Iqbal, and Muhammad Muddassar,……………………………………1318 The Borel Direction and Uniqueness of Meromorphic Function, Hong Yan Xu, Hua Wang,1329 Pseudo-Valuations on BCH-Algebras with Respect To Subalgebras and (Closed) Ideals, Young Bae Jun, and Sun Shin Ahn,…………………………………………………………………1341 Derivatives of Decreasing Functions with Respect to Fuzzy Measures, H.M. Kim, Y.H. Kim, and J. Choi,………………………………………………………………………………………..1352 Some New Inequalities for the Gamma Function, Xiaodong Cao,……………………….….1359

TABLE OF CONTENTS, JOURNAL OF COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 20, NO. 7, 2016 (continued) Mathematical Analysis of Humoral Immunity Viral Infection Model with Hill Type Infection Rate, M. A. Obaid,……………………………………………………………………………1370 A Parameterized Series Representation for Apery's Constant ζ(3), Horst Alzer, and Jonathan Sondow,……………………………………………………………………………………….1380