JOURNAL OF COMPUTATIONAL ANALYSIS AND APPLICATIONS VOLUME 29, 2021

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Volume 29, Number 1 ISSN:1521-1398 PRINT,1572-9206 ONLINE

January 2021

Journal of Computational Analysis and Applications EUDOXUS PRESS, LLC

Journal of Computational Analysis and Applications ISSNno.’s:1521-1398 PRINT,1572-9206 ONLINE SCOPE OF THE JOURNAL An international publication of Eudoxus Press, LLC (six 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,[email protected], St.Martin Univ.,Olympia,WA,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 $800, Electronic OPEN ACCESS. Individual:Print $400. For any other part of the world add $160 more(handling and postages) to the above prices for Print. No credit card payments. Copyright©2021 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.

Turkey, [email protected] 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

Dumitru Baleanu Department of Mathematics and Computer Sciences, Cankaya University, Faculty of Art and Sciences, 06530 Balgat, Ankara,

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

Duisburg, D-47048 Germany 011-49-203-379-3542 e-mail: [email protected] Approximation Theory, Computer Aided Geometric Design 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

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.

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

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

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

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

Margareta Heilmann Faculty of Mathematics and Natural Sciences, University of Wuppertal Gaußstraße 20 D-42119 Wuppertal, Germany, [email protected] Approximation Theory (Positive Linear Operators)

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

Xing-Biao Hu Institute of Computational Mathematics AMSS, Chinese Academy of Sciences

H. H. Gonska Department of Mathematics University of Duisburg

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Beijing, 100190, CHINA [email protected] Computational Mathematics

Approximation Theory, Splines, Wavelets, Neural Networks 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 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,

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

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Probability

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] Approximation Theory, Banach spaces, Classical Analysis

Roberto Triggiani Department of Mathematical Sciences University of Memphis Memphis, TN 38152 PDE, Control Theory, Functional Analysis, [email protected] Juan J. Trujillo University of La Laguna Departamento de Analisis Matematico C/Astr.Fco.Sanchez s/n 38271. LaLaguna. Tenerife. SPAIN

T. E. Simos Department of Computer Science and Technology Faculty of Sciences and Technology University of Peloponnese GR-221 00 Tripolis, Greece

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Tel/Fax 34-922-318209 [email protected] Fractional: Differential EquationsOperators-Fourier Transforms, Special functions, Approximations, and Applications

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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

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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

Xiao-Jun Yang State Key Laboratory for Geomechanics and Deep Underground Engineering, China University of Mining and Technology, Xuzhou 221116, China Local Fractional Calculus and Applications, Fractional Calculus and Applications, General Fractional Calculus and Applications, Variable-order Calculus and Applications, Viscoelasticity and Computational methods for Mathematical [email protected]

Jessada Tariboon Department of Mathematics King Mongut’s University of Technology N. Bangkok 1518 Pracharat 1 Rd., Wongsawang, Bangsue, Bangkok, Thailand 10800 [email protected], Time scales Differential/Difference Equations, Fractional Differential Equations

Richard A. Zalik Department of Mathematics Auburn University Auburn University, AL 36849-5310 USA. Tel 334-844-6557 office 678-642-8703 home Fax 334-844-6555 [email protected]

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J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 29, NO.1, 2021, COPYRIGHT 2021 EUDOXUS PRESS, LLC

A new techniques applied to Volterra–Fredholm integral equations with discontinuous kernel M. E. Nasr 1,2

1

and M. A. Abdel-Aty

2

Department of Mathematics, Faculty of Science, Benha University, Benha 13518, Egypt 1

Department of Mathematics, Collage of Science and Arts–Gurayat, Jouf University, Kingdom of Saudi Arabia

Abstract The purpose of this paper is to establish the general solution of a Volterra–Fredholm integral equation with discontinuous kernel in a Banach space. Banach’s fixed point theorem is used to prove the existence and uniqueness of the solution. By using separation of variables method, the problem is reduced to a Volterra integral equations of the second kind with continuous kernel. Normality and continuity of the integral operator are also discussed. Mathematics Subject Classification(2010): 45L05; 46B45; 65R20. Key–Words: Banach space, Volterra–Fredholm integral equation, Separation of variables method.

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Introduction It is well-known that the integral equations govern many mathematical models of various

phenomena in physics, economy, biology, engineering, even in mathematics and other fields of science. The illustrative examples of such models can be found in the literature, (see, e.g., [5, 6, 9, 11, 12, 14, 18, 20]). Many problems of mathematical physics, applied mathematics, and engineering are reduced to Volterra–Fredholm integral equations, see [1, 2]. Analytical solutions of integral equations, either do not exist or it’s hard to compute. Eventual an exact solution is computable, the required calculations may be tedious, or the resulting solution may be difficult to interpret. Due to this, it is required to obtain an efficient numerical solution. There are numerous studies in literature concerning the numerical solution of integral equations such as [4, 8, 10, 13, 16, 17, 21]. In this present paper, the existence and uniqueness solution of the Eq. (1) are discussed and proved in the space L2 (Ω) × C[0, T ], 0 ≤ T < 1. Moreover, the normality and continuity of the 11

Nasr 11-24

J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 29, NO.1, 2021, COPYRIGHT 2021 EUDOXUS PRESS, LLC

integral operator are obtained. A numerical method is used to translate the Volterra–Fredholm integral equation (1) to a Volterra integral equations of the second kind with continuous kernel, The outline of the paper is as follows: Sect. 1 is the introduction; In Sect. 2, the existence of a unique solution of the Volterra–Fredholm integral equation is discussed and proved using Picard’s method and Banach’s fixed point method. Sect. 3, include the general solution of the Volterra–Fredholm integral equation by applying the method of separation of variables. A brief conclusion is presented in Sect. 4. Consider the following linear Volterra–Fredholm integral equation: Z t Z tZ F (t, τ )ψ(x, τ )dτ = g(x, t), Φ(t, τ )k(|x − y|)ψ(y, τ )dydτ − λ µψ(x, t) − λ 0

(1)

0

Ω

(x = x¯(x1 , x1 , . . . , xn ),

y = y¯(y1 , y1 , . . . , yn )),

where µ is a constant, defined the kind of integral equation, λ is constant, may be complex and has many physical meaning. The function ψ(x, t) is unknown in the Banach space L2 (Ω) × C[0, T ], 0 ≤ T < 1, where Ω is the domain of integration with respect to position and the time t ∈ [0, T ] and it called the potential function of the Volterra–Fredholm integral equation. The kernels of time Φ(t, τ ), F (t, τ ) are continuous in C[0, T ] and the known function g(x, t) is continuous in the space L2 (Ω) × C[0, T ], 0 ≤ t ≤ T. In addition the kernel of position k(|x − y|) is discontinuous function.

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The existence of a unique solution of the Volterra– Fredholm integral equation In this paper, for discussing the existence and uniqueness of the solution of Eq. (1), we

assume the following conditions: (i) The kernel of position k(|x − y|) ∈ L2 ([Ω] × [Ω]), x, y ∈ [Ω] satisfies the discontinuity condition: Z Z

 12

2

k (|x − y|)dxdy Ω

= k∗,

k ∗ is constant.

Ω

(ii) The kernels of time Φ(t, τ ), F (t, τ ) ∈ C[0, T ] and satisfies |Φ(t, τ )| ≤ M1 , |F (t, τ )| ≤ M2 , s.t M1 , M2 are constants, ∀t, τ ∈ [0, T ]. (iii) The given function g(x, t) with its partial derivatives with respect to the position and time is continuous in the space L2 (Ω) × C[0, T ], 0 ≤ τ ≤ T < 1 and its norm is defined as,  12 Z t Z 2 kg(x, t)k = max g (x, τ )dx dτ = N, N is a constant. 0≤t≤T

0

Ω

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Nasr 11-24

J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 29, NO.1, 2021, COPYRIGHT 2021 EUDOXUS PRESS, LLC

Theorem 1. If the conditions (i)–(iii) are satisfied, then Eq. (1) has a unique solution ψ(x, t) in the Banach space L2 (Ω) × C[0, T ], 0 ≤ T < 1, under the condition, |λ|
0, one has ρ(xn − xm ) <  for sufficiently large m, n ∈ N, (ii) {xn } is called ρ-convergent to a point x ∈ Xρ if ρ(xn − x) → 0 as n → ∞, and (iii) a subset K of Xρ is called ρ-complete if each ρ-Cauchy sequence is ρ-convergent to a point in K. A modular space Xρ is said to satisfy the 42 -condition if there exists k ≥ 2 such that Xρ (2x) ≤ kXρ (x) for all x ∈ X. Example 1.2. ([9], [11], [12]) A convex function ζ defined on the interval [0, ∞), nondecreasing and continuous, such that ζ(0) = 0, ζ(α) > 0 for α > 0, ζ(α) → ∞ as α → ∞, is called an Orlicz function. Let (Ω, Σ, µ) be a measure space and L0 (µ) the set of all measurable real valued (or complex valued) R functions on Ω. Deine the Orlicz modular ρζ on L0 (µ) by the formula ρζ (f ) = Ω ζ(|f |)dµ. The associated modular space with respect to this modular is called an Orlicz space, and will be denoted by (Lζ , Ω, µ) or briefly Lζ . In other words, Lζ = {f ∈ L0 (µ) | ρζ (λf ) < ∞ for some λ > 0}. It is known that the Orlicz space Lζ is ρζ -complete. Moreover, (Lζ , k · kρζ ) is a Banach space, where the Luxemburg norm k · kρζ is defined as follows Z  |f |  n o kf kρζ = inf λ > 0 ζ dµ ≤ 1 . λ Ω Further, if µ is the Lebesgue measure on R and ζ(t) = et − 1, then ρζ does not satisfy the 42 -condition.

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CHANG IL KIM 34-41

J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 29, NO.1, 2021, COPYRIGHT 2021 EUDOXUS PRESS, LLC

QUADRATIC FUNCTIONAL INEQUALITY IN MODULAR SPACES AND ITS STABILITY 3

For a modular space Xρ , a nonempty subset C of Xρ , and a mapping T : C −→ C, the orbit of T at x ∈ C is the set O(x) = {x, T x, T 2 x, · · ·}. If δρ (x) = sup{ρ(u − v) | u, v ∈ O(x)} < ∞, then one says that T has a bounded orbit at x. Lemma 1.3. [5] Let Xρ be a modular space whose induced modular is lower semicontinuous and let C ⊆ Xρ be a ρ-complete subset. If T : C −→ C is a ρcontraction, that is, there is a constant L ∈ [0, 1) such that ρ(T x − T y) ≤ Lρ(x − y), ∀x, y ∈ C and T has a bounded orbit at a point x0 ∈ C, then the sequence {T n x0 } is ρconvergent to a point w ∈ C. For any modular ρ on X and any linear space V , we define a set M M := {g : V −→ Xρ | g(0) = 0} and the generalized function ρe on M by for each g ∈ M, ρe(g) := inf{c > 0 | ρ(g(x)) ≤ cψ(x, 0), ∀x ∈ V }, where ψ : V 2 −→ [0, ∞) is a mapping. The proof of the following lemma is similar to the proof of Lemma 10 in [17]. Lemma 1.4. Let V be a linear space, Xρ a ρ-complete modular space where ρ is convex lower semi-continuous and f : V −→ Xρ a mapping with f (0) = 0. Let ψ : V 2 −→ [0, ∞) be a mapping such that (1.2)

ψ(ax, ax) ≤ a2 Lψ(x, x)

for all x, y ∈ V and some a and L with a ≥ 2 and 0 ≤ L < 1. Then we have the following : (1) M is a linear space, (2) ρe is a convex modular, and (3) Mρe = M and Mρe is ρe-complete, and (4) ρe is lower semi-continuous. 2. Solutions of (1.1) In this section, we consider solutions of (1.1). For any f : V −→ Xρ , let Af (x, y) = k[f (ax + by) + f (ax − by) − 2a2 f (x) − 2b2 f (y)] and Bf (x, y) = f (x + y) + f (x − y) − 2f (x) − 2f (y). Lemma 2.1. Let ρ be a convex modular on X and f : V −→ Xρ an even mapping with f (0) = 0. Suppose that ka2 ≥ 1 and b2 > a2 . Then f is a quadratic mapping if and only if f is a solution of (1.1).

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CHANG IL KIM 34-41

J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 29, NO.1, 2021, COPYRIGHT 2021 EUDOXUS PRESS, LLC

4

CHANGIL KIM AND GILJUN HAN

Proof. Since k 6= 0 and f is even, we have f (ax) = a2 f (x), f (bx) = b2 f (x)

(2.1)

for all x ∈ V . Putting y = ay in (1.1), by (2.1), we have (2.2)

ρ(f (x + ay) + f (x − ay) − 2f (x) − 2a2 f (y)) ≥ ρ(ka2 [f (x + by) + f (x − by) − 2f (x) − 2b2 f (y)])

for all x, y ∈ V and letting y = (2.3)

y a

in (2.2), by (2.1), we have

ρ(Bf (x, y)) ≥ ρ(ka2 [f (x + py) + f (x − py) − 2f (x) − 2p2 f (y)])

for all x, y ∈ V ,where p = ab . Since ρ is convex and ka2 ≥ 1, by (2.3), (2.4)

ρ(Bf (x, y)) ≥ ka2 ρ(f (x + py) + f (x − py) − 2f (x) − 2p2 f (y))

for all x, y ∈ V . Letting x = py in (2.3), by (2.1), we have (2.5)

ρ(f (px + y) + f (px − y) − 2p2 f (x) − 2f (y)) ≥ kb2 ρ(f (x + y) + f (x − y) − 2f (x) − 2f (y))

for all x, y ∈ V , because ρ is convex and b2 > a2 . Interchanging x and y in (2.5), we have (2.6)

ρ(f (x + py) + f (x − py) − 2f (x) − 2p2 f (y)) ≥ kb2 ρ(Bf (x, y))

for all x, y ∈ V . By (M4), (2.4), and (2.6), we have (2.7)

ρ(f (x + py) + f (x − py) − 2f (x) − 2p2 f (y)) ≥ k 2 a2 b2 ρ(f (x + py) + f (x − py) − 2f (x) − 2p2 f (y))

for all x, y ∈ V . Since k 2 a2 b2 > 1, by (2.7) and (M1), we get f (x + py) + f (x − py) − 2f (x) − 2p2 f (y) = 0 for all x, y ∈ V and hence f is a quadratic mapping. The converse is trivial.



Theorem 2.2. Let ρ be a convex modular on X and f : V −→ Xρ a mapping with f (0) = 0. Suppose that ka2 ≥ 2 and b2 > a2 . Then f is a quadratic mapping if and only if f is a solution of (1.1). Proof. By (1.1), we have 1 1 ρ(Af (x, y)) + ρ(Af (−x, −y)) 2 2 1 1 (2.8) ≤ ρ(Bf (x, y)) + ρ(Bf (−x, −y)) 2 2 1 1 ≤ ρ(2Bfo (x, y)) + ρ(2Bfe (x, y)) 2 2 for all x, y ∈ V and similarly, we have ρ(Afo (x, y)) ≤

1 1 ρ(2Bfo (x, y)) + ρ(2Bfe (x, y)) 2 2 for all x, y ∈ V . Letting x = 0 in (2.8), by (M4), we have

(2.9)

(2.10)

ρ(Afe (x, y)) ≤

1 kb2 ρ(4fo (y)) ≥ ρ(2kb2 fo (y)) ≥ ρ(4fo (y)) 2 2

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QUADRATIC FUNCTIONAL INEQUALITY IN MODULAR SPACES AND ITS STABILITY 5

for all y ∈ V , because ρ is convex and kb2 > 2. Since kb2 > 1, by (2.10) and (M1), we have fo (y) = 0 for all y ∈ V and hence by (2.9), we have ρ(Afe (x, y)) ≤ ρ(2Bfe (x, y))

(2.11)

2

for all x, y ∈ V . Since ka ≥ 2 and b2 > a2 , by Lemma 2.1 and (2.11), 2fe is a quadratic mapping and since fo (x) = 0 for all x ∈ X, f is a quadratic mapping.  For k = 1 in Theorem 2.2, we have the following corollary: Corollary 2.3. Let ρ be a convex modular on X and f : V −→ Xρ a mapping with f (0) = 0. Suppose that b2 > a2 ≥ 2. The f is quadratic if and only if ρ(Bf (x, y)) ≥ ρ(f (ax + by) + f (ax − by) − 2a2 f (x) − 2b2 f (y)) for all x, y ∈ V . Corollary 2.4. Let ρ be a convex modular on X and f : V −→ Xρ a mapping with f (0) = 0. Suppose that ka2 ≥ 2 and b2 > a2 . Then the following are equivalent (1) f is quadratic, (2) f satisfies (1.1), and (3) f satisfies the following ρ(rBf (x, y)) ≥ ρ(rAf (x, y)) for all x, y ∈ V and all real number r. 3. The generalized Hyers-Ulam stability for (1.1) in modular spaces Throughout this section, we assume that every modular is convex and lower semicontinuous. In this section, we will prove the generalized Hyers-Ulam stability for (1.1). Lemma 3.1. Let ρ be a convex modular on X and t a real number with 2 ≤ t. Then 1 1  1 1 ρ x + y ≤ ρ(x) + ρ(y) t t t t for all x, y ∈ X. Proof. Since ρ is a convex modular on X, we have  1 1  1 1  1  1 1 ρ y ≤ ρ(x) + ρ(y) ρ x + y ≤ ρ(x) + 1 − t t t t t−1 t t for all x, y ∈ X, because 2 ≤ t.



Theorem 3.2. Let ρ be a modular on X, V a linear space, Xρ a ρ-complete modular space and f : V −→ Xρ a mapping with f (0) = 0. Suppose that a ≥ 2, k ≥ a2 , and b2 > a2 . Let φ : V 2 −→ [0, ∞) be a mapping such that (3.1)

φ(ax, ay) ≤ a2 Lφ(x, y)

for all x, y ∈ V and some L with 0 < L < 1 and (3.2)

ρ(rAf (x, y)) ≤ ρ(rBf (x, y)) + |r|φ(x, y)

for all x, y ∈ V and all real number r. Then there exists a unique quadratic mapping Q : V −→ Xρ such that   1 1 (3.3) ρ Q(x) − 2 f (x) ≤ 4 φ(x, 0) a ka (1 − L) for all x ∈ V .

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Proof. By Lemma 1.4, ρe is a lower semi-continuous convex modular on Mρe, Mρe = 1 M, and Mρe is ρe-complete. Define T : Mρe −→ Mρe by T g(x) = 2 g(ax) for all a g ∈ Mρe and all x ∈ V . Let g, h ∈ Mρe. Suppose that ρe(g − h) ≤ c for some nonnegative real number c. Then by (3.1), we have 1 ρ(T g(x) − T h(x)) ≤ 2 ρ(g(ax) − h(ax)) ≤ Lcφ(x, 0) a for all x ∈ V and so ρe(T g − T h) ≤ Le ρ(g − h). Hence T is a ρe-contraction. Since 2k > 1, by (3.2), for r = 1, we get   1 1 (3.4) ρ f (ax) − a2 f (x) ≤ ρ(2kf (ax) − 2ka2 f (x)) ≤ φ(x, 0) 2k 2k for all x ∈ X. Since a ≥ 2, by (3.4),  1 1 1 φ(x, 0) (3.5) ρ(T f (x)−f (x)) = ρ 2 f (ax)−f (x) ≤ 2 ρ(f (ax)−a2 f (x)) ≤ a a 2ka2 for all x ∈ X. Now, we claim that T has a bounded orbit at a12 f . By Lemma 3.1 and (3.5), for any nonnegative integer n, we obtain  1   1 1  1 1 1 ρ T n f (x) − f (x) ≤ ρ T n f (x) − 2 f (ax) + ρ 2 f (ax) − f (x) a a a a a a  1  1 n−1 1 1 ≤ 2ρ T f (ax) − f (ax) + φ(x, 0) a a a 2ka3 for all x ∈ V and by (3.1), we have n−1 1  1 1 X i 1 (3.6) ρ T n f (x) − f (x) ≤ L φ(x, 0) ≤ φ(x, 0) 3 3 a a 2ka i=0 2ka (1 − L) for all x ∈ V and all n ∈ N. By Lemma 3.1 and (3.6), we get (3.7)  1  1 1 1 1 φ(x, 0) ρ 2 T n f (x) − 2 T m f (x) = ρ 2 T n f (x) − 2 T m f (x) ≤ 4 a a a a ka (1 − L) for all x ∈ V and all nonnegative integers n, m. Hence T has a bounded orbit at 1 a2 f . By Lemma 1.3, there is a Q ∈ Mρe such that {T n a12 f } is ρe-convergent to Q. Since ρe is lower semi-continuous, we get   1  1  0 ≤ ρe(T Q − Q) ≤ lim inf ρe T Q − T n+1 2 f ≤ lim inf Le ρ Q − Tn 2f = 0 n→∞ n→∞ a a and hence Q is a fixed point of T in Mρe. Since a ≥ 2, there is a a natural number t with k < at−6 and 2kb2 < at−3 and hence we have i 1h 1 ρ t AQ (x, y) − 2n+2 Af (an x, an y) a a   1 2k  1 k  ≤ t ρ Q(ax + by) − 2n+2 f (an+1 x + an by) + t−2 ρ Q(x) − 2n+2 f (an x) a a a a  2kb2   k  1 1 n+1 n + t ρ Q(ax − by) − 2n+2 f (a x − a by) + t ρ Q(y) − 2n+2 f (an y) a a a a for all x, y ∈ V and all n ∈ N. Letting n → ∞ in the above inequality, we get 1h i 1 (3.8) lim ρ t AQ (x, y) − 2n+2 Af (an x, an y) = 0 n→∞ a a

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QUADRATIC FUNCTIONAL INEQUALITY IN MODULAR SPACES AND ITS STABILITY 7 1 for all x, y ∈ V , because { a2n+2 f } is ρe-convergent to Q. Similarly, we have i 1h 1 (3.9) lim ρ t BQ (x, y) − 2n+2 Bf (an x, an y) = 0 n→∞ a a 2 for all x, y ∈ V . Since a ≤ k, by (3.2), we have   1 A (x, y) ρ Q kat+1 i 1   1 1  1 h 1 n n n n A (x, y) − A (a A (a ≤ ρ x, a y) + ρ x, a y) Q f f a kat a2n+2 a a2n+t+4 i 1   1 1 1h 1 ≤ 3 ρ t AQ (x, y) − 2n+2 Af (an x, an y) + ρ 2n+t+4 Bf (an x, bn y) a a a a a 1 + 2n+t+5 φ(an x, an y) a i  1 1h 1 1  1 ≤ 3 ρ t AQ (x, y) − 2n+2 Af (an x, an y) + 2 ρ t+1 BQ (x, y) a a a a a i n L 1 1h 1 + 3 ρ t 2n+2 Bf (an x, an y) − BQ (x, y) + t+5 φ(x, y) a a a a for all x, y ∈ V and all n ∈ N. Letting n → ∞ in the last inequality, by (3.8) and (3.9), we get  1   1  ρ A (x, y) ≤ ρ B (x, y) Q Q kat+1 at+1 for all x, y ∈ V . By Corollary 2.3, Q is a quadratic mapping. Moreover, since ρ is lower semi-continuous, by (3.7), we have (3.3). 

Corollary 3.3. Let X and Y be normed spaces and , θ, and p real numbers with  ≥ 0, θ ≥ 0, and 0 < p < 1. Suppose that a ≥ 2, k ≥ a2 , and b2 > a2 . Let f : X −→ Y be a mapping such that f (0) = 0 and kAf (x, y)k ≤ kBf (x, y)k +  + θ(kxk2p + kyk2p + kxkp kykp ) for all x, y ∈ X. Then there is a quadratic mapping Q : X −→ Y such that 1 kQ(x) − f (x)k ≤ ( + θkxk2p ) k(a2 − a2p ) for all x ∈ X. Proof. Let ρ(z) = kzk for all y ∈ Y and φ(x, y) =  + θ(kxk2p + kyk2p + kxkp kykp ) for all x, y ∈ V . Then ρ is a convex modular on a normed space Y , Y = Yρ , and φ(ax, ay) ≤ a2p φ(x, y) for all x, y ∈ V . By Theorem 3.2, we have the results.  Using Example 1.1, we get the following example. Example 3.4. Let θ, and p be real numbers with θ ≥ 0 and 0 < p < 1. Suppose that a ≥ 2, k ≥ a2 , and b2 > a2 . Let ζ be an Orlicz function and Lζ the Orlicz space. Let f : V −→ Lζ be a mapping such that f (0) = 0 and Z Z ζ(rAf (x, y))dµ ≤ ζ(rBf (x, y))dµ + |r|θ(kxk2p + kyk2p + kxkp kykp ) Ω

Ω

for all x, y ∈ X and all real number r. Then there is a quadratic mapping Q : X −→ Y such that Z   1 θ ζ Q(x) − 2 f (x) dµ ≤ 2 2 kxk2p 2p ) a ka (a − a Ω for all x ∈ X.

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References [1] I. Amemiya, On the representation of complemented modular lattices, J. Math. Soc. Japan. 9(1957), 263-279. [2] T. Aoki, On the stability of the linear transformation in Banach spaces, J. Math. Soc. Japan 2(1950), 64-66. [3] P. Gˇ avruta, A generalization of the Hyer-Ulam-Rassias stability of approximately additive mappings, J. Math. Anal. Appl. 184(1994), 431-436. [4] D. H. Hyers, On the stability of the linear functional equation, Proc. Nat. Acad. Sci. 27(1941), 222-224. [5] M. A. Khamsi, Quasicontraction mappings in modular spaces without 42 -condition, Fixed Point Theory and Applications, 2008(2008), 1-6. [6] S. Koshi and T. Shimogaki, On F-norms of quasi-modular spaces, J. Fac. Sci., Hokkaido Univ., Ser. 1 15(1961), 202-218. [7] W. A. Luxemburg, Banach function spaces. Ph.D. Thesis, Technische Hogeschool te Delft, Netherlands, 1955. [8] B. Mazur, Modular curves and the Eisenstein ideal. Publ. Math. IHS 47(1977), 33-186. [9] J. Musielak, Orlicz Spaces and Modular Spaces. Lecture Notes in Mathematics, SpringerVerlag, Berlin, 1983. [10] J. Musielak and W. Orlicz, On modular spaces, Studia Mathematica, 18(1959), 49-65. [11] H. Nakano, Modular semi-ordered spaces, Tokyo, Japan, 1950. [12] W. Orlicz, Collected Papers, vols. I, II. PWN, Warszawa 1988. [13] Th. M. Rassias, On the stability of the linear mapping in Banach sapces, Proc. Amer. Math. Sco. 72(1978), 297-300. [14] G. Sadeghi, A fixed point approach to stability of functional equations in modular spaces, Bulletin of the Malaysian Mathematical Sciences Society, 37(2014), 333-344. [15] Ph. Turpin, Fubini inequalities and bounded multiplier property in generalized modular spaces, Comment. Math. 1(1978), 331-353. [16] S. M. Ulam, Problems in Modern Mathematics, Wiley, New York, 1964. [17] K. Wongkum, P. Chaipunya, and P. Kumam, On the generalized Ulam-Hyers-Rassias stability of quadratic mappings in modular spaces without 42 -conditions, 2015(2015), 1-6. [18] S. Yamamuro, On conjugate spaces of Nakano spaces, Trans. Am. Math. Soc. 90(1959), 291-311. Department of Mathematics Education, Dankook University, 152, Jukjeon-ro, Sujigu, Yongin-si, Gyeonggi-do, 16890, Korea E-mail address: [email protected] Department of Mathematics Education, Dankook University, 152, Jukjeon-ro, Sujigu, Yongin-si, Gyeonggi-do, 16890, Korea E-mail address: [email protected]

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Complex Multivariate Taylor’s formula George A. Anastassiou Department of Mathematical Sciences University of Memphis Memphis, TN 38152, U.S.A. [email protected] Abstract We derive here a Taylor’s formula with integral remainder in the several complex variables and we estimate its remainder.

2010 Mathematics Subject Classi…cation : 32A05, 32A10, 32A99. Key Words and phrases: Taylor’s formula in several complex variables, remainder estimation.

1

Main Results

We need the following vector Taylor’s formula: Theorem 1 (Shilov, [3], pp. 93-94) Let n 2 N and f 2 C n ([a; b] ; X), where [a; b] R and (X; k k) is a Banach space. Then f (b) = f (a) +

n X1

i

(b

a) i!

i=1

f

(i)

(a) +

1 (n

1)!

Z

b

(b

n 1

x)

f (n) (x) dx:

(1)

a

The remainder here is the Riemann X-valued integral (de…ned similar to numerical one) given by Qn

1

=

1 (n

1)!

Z

b

(b

n 1

x)

f (n) (x) dx;

(2)

a

with the property: kQn

1k

max

a x b

f (n) (x)

(b

n

a) : n!

(3)

The derivatives above are de…ned similar to the numerical ones. We make Remark 2 Here Q is an open convex subset of Ck , k 2; z := (z1 ; :::; zk ), x0 := (x01 ; :::; x0k ) 2 Q. Let f : Q ! C be a coordinate-wise holomorphic 1

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function. Then, by the famous Hartog’s fundamental theorem ([1], [2]) f is jointly holomorphic and jointly continuous on Q. Let n 2 N. Each nth order complex partial derivative is denoted by f := @@x f , where := ( 1 ; :::; k ), k P i 2 Z+ , i = 1; :::; k and j j := i = n: i=1

Consider

gz (t) := f (x0 + t (z

x0 )) ,

0

t

1:

(4)

Clearly it holds that x0 + t (z x0 ) 2 Q and gz (t) 2 C, 8 t 2 [0; 1]. Then we derive 2 !j 3 k X @ (zi x0i ) gz(j) (t) = 4 f 5 (x01 + t (z1 x01 ) ; :::; x0k + t (zk @x i i=1

x0k )) ; (5)

for all j = 0; 1; :::; n: Notice here that any mixed partials commute. We remind that (C; j j) is a Banach space. By Shilov’s Theorem 1, about the Taylor’s formula for Banach space valued functions, we obtain Theorem 3 It holds n X1

f (z1 ; :::; zk ) = gz (1) =

j=0

where 1

Rn (z; 0) =

(n

1)!

Z

(j)

gz (0) + Rn (z; 0) ; j!

1

(1

n 1

)

(6)

gz(n) ( ) d ;

(7)

0

and notice that gz (0) = f (x0 ) : We make Remark 4 Notice that (by (7)) we get max gz(n) ( )

jRn (z; 0)j

0

1

1 : n!

(8)

We also have for j = 0; 1; :::; n:

gz(j)

(0) = :=(

X

1 ;:::;

0

k );

i=1;:::;k; j j:=

+

j 2Z

k P

i=1

i =j

B j! B B k @Q i=1

1 i!

k C Y C (zi C A i=1

x0i )

i

!

f (x0 ) :

(9)

2

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Furthermore it holds

gz(n)

( )= :=(

0

X

1 ;:::;

k );

i=1;:::;k; j j:=

0

k P

+

j 2Z

i =n

i=1

B n! B B k @Q

1

k C Y C (zi C A i=1

i!

i=1

i

x0i )

!

f (x0 + (z

x0 )) ;

(10)

1: Another version of (6) is f (z1 ; :::; zk ) = gz (1) =

n (j) X gz (0)

where Rn (z; 0) =

1 (n

1)!

Z

1

n 1

(1

+ Rn (z; 0) ;

j!

j=0

)

gz(n) ( )

gz(n) (0) d :

(11)

(12)

0

Identities (6) and (11) are the multivariate complex Taylor’s formula with integral remainders. We give Example 5 Let n = k = 2. Then gz (t) = f (x01 + t (z1

x01 ) ; x02 + t (z2

x02 )) ; t 2 [0; 1] ;

and gz0 (t) = (z1

x01 )

@f (x0 + t (z @x1

x0 )) + (x2

@f (x0 + t (z @x1

x0 ))

x02 )

@f (x0 + t (z @x2

x0 )) : (13)

In addition, gz00 (t) = (z1

x01 )

0

+(x2

@f (x0 + t (z @x2

x02 )

= (z1

x01 ) (z1

x01 )

@f 2 ( ) + (z2 @x21

(z2

x02 ) (z1

x01 )

@f 2 ( ) + (z2 @x1 @x2

x02 )

@f 2 ( ) + @x2 @x1

x02 )

@f 2 ( ) : @x22

x02 )

@f 2 ( )+ @x2 @x1

x0 ))

0

(14)

Hence, gz00 (t) = (z1 (z1

2

x01 )

x01 ) (z2

@f 2 ( ) + (z1 @x21 x02 )

x01 ) (z2

@f 2 ( ) + (z2 @x1 @x2

2

x02 )

@f 2 ( ); @x22

(15)

where := x0 + t (z x0 ). Notice that gz (t) ; gz0 (t) ; gz00 (t) 2 C: 3

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We make Remark 6 We de…ne Z

kf kp;zx0 :=

1

jf (x0 + (z

0

where zx0 denotes the segment zx0 We also de…ne

1 p

p

x0 ))j d

, p

1;

(16)

Q.

kf k1;zx0 := max jf (x0 + (z 2[0;1]

x0 ))j :

(17)

By (10) we obtain X

gz(n) ( ) :=(

0

1 ;:::;

k );

i=1;:::;k; j j:=

8

+

k P

j 2Z

i =n

i=1

B n! B B k @Q

1 i!

i=1

k C Y C jzi C A i=1

p;zx0

:=(

X

1 ;:::;

+

k );

i=1;:::;k; j j:=

k P

j 2Z

i =n

i=1

k X i=1

where

i

jf (x0 + (z

p

B n! B B k @Q

1 i!

i=1

k C Y C jzi C A i=1

!n

jzi

x0i j

1, it holds

x0i j

i

!

kf kp;zx0 (19)

kf kp;zx0 ;

kf kp;zx0 := max kf kp;zx0 ;

(20)

j j=n

for all 1 p That is

x0 ))j ;

(18)

2 [0; 1] : Therefore, by norm properties for 1 0

gz(n)

x0i j

!

1: gz(n)

p;zx0

kz

n

x0 kl1

kf kp;zx0 ;

(21)

for all 1 p 1: Therefore by (8) we obtain jRn (z; 0)j

kz

x0 kl1

n

n!

kf k1;zx0

:

(22)

Next, we put things together and we further estimate Rn (z; 0) : 4

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Theorem 7 Here p; q > 1 :

1 p

+

jRn (z; 0)j

kz

x0 kl1

1 q

= 1. It holds

min

n

8 > > > > > > > > < > > > > > > > > :

min

kgz(n) k1;zx0 n!

;

kgz(n) k1;zx0

;

(n 1)!

kgz(n) kp;zx0

9 > > > > > > > > = 1

(n 1)!(q(n 1)+1) q

8 > > > > > > > < > > > > > > > :

kf k1;zx

;

kf k1;zx

;

0

n!

0

(n 1)!

kf kp;zx

0

> > > > > > > > ;

(23)

(24) 1

:

(n 1)!(q(n 1)+1) q

Proof. Based on (7), Hölder’s inequality and (21).

References [1] C. Caratheodory, Theory of Functions of a complex variable, Volume Two, Chelsea publishing Company, New York, 1954. [2] S.G. Krantz, Function theory of several complex variables, second edition, AMS Chelsea publishing Providence, Rhode Island, 2001. [3] G.E. Shilov, Elementary Functional Analysis, Dover Publications Inc., New York, 1996.

5

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J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 29, NO.1, 2021, COPYRIGHT 2021 EUDOXUS PRESS, LLC

On the Barnes-type multiple twisted q-Euler zeta function of the second kind C. S. Ryoo Department of Mathematics, Hannam University, Daejeon 34430, Korea

Abstract : In this paper we introduce the Barnes-type multiple twisted q-Euler numbers and polynomials of the second kind, by using fermionic p-adic invariant integral on Zp . Using these numbers and polynomials, we construct the Barnes-type multiple twisted q-Euler zeta function of the second kind. Finally, we obtain the relations between these numbers and polynomials and Barnes-type multiple twisted q -Euler zeta function. Key words : p-adic invariant integral on Zp , Euler numbers and polynomials of the second kind, q-Euler numbers and polynomials of the second kind, Barnes-type multiple twisted q-Euler numbers and polynomials of the second kind, Barnes-type multiple twisted q -Euler zeta function. 2000 Mathematics Subject Classification : 11B68, 11S40, 11S80. 1. Introduction Recently, Bernoulli numbers, Bernoulli polynomials, q-Bernoulli numbers, q-Bernoulli polynomials, the second kind Bernoulli number, the second kind Bernoulli polynomials, Euler numbers of the second kind , Euler polynomials of the second kind, Genocchi numbers, Genocchi polynomials, tangent numbers, tangent polynomials, and Bell polynomials were studied by many authors(see [1, 2, 3, 4, 9]). Euler numbers and polynomials possess many interesting properties and arising in many areas of mathematics and physics. In [5], by using Euler numbers Ej and polynomials Ej (x) of the second kind, we investigated the alternating sums of powers of consecutive odd integers. Also we carried out computer experiments for doing demonstrate a remarkably regular structure of the complex roots of the second kind Euler polynomials En (x)(see [6]). The outline of this paper is as follows. We introduce the Barnes-type multiple twisted q-Euler numbers and polynomials of the second kind, by using fermionic p-adic invariant integral on Zp . In Section 2, we construct the Barnes-type multiple twisted q-Euler zeta function of the second kind. Finally, we obtain the relations between these numbers and polynomials and Barnes-type multiple twisted q -Euler zeta function. Throughout this paper we use the following notations. By Zp we denote the ring of p-adic rational integers, Qp denotes the field of rational numbers, N denotes the set of natural numbers, C denotes the complex number field, and Cp denotes the completion of algebraic closure of Qp . Let νp be the normalized exponential valuation of Cp with |p|p = p−νp (p) = p−1 . For g ∈ U D(Zp ) = {g|g : Zp → Cp is uniformly differentiable function}, the fermionic p-adic invariant integral on Zp of the function g ∈ U D(Zp ) is defined by ∫ I−1 (g) =

Zp

g(x)dµ−1 (x) = lim

N →∞

47

N p∑ −1

g(x)(−1)x , see [1, 2, 4].

(1.1)

x=0

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From (1.1), we note that

∫

∫ Zp

g(x + 1)dµ−1 (x) +

Zp

g(x)dµ−1 (x) = 2g(0).

(1.2)

First, we introduced the second kind Euler numbers En . The second kind Euler numbers En are defined by the generating function: ∞ ∑ 2et tn = E . n e2t + 1 n=0 n!

(1.3)

We introduce the second kind Euler polynomials En (x) as follows: ∞ 2et xt ∑ tn e = En (x) . 2t e +1 n! n=0

(1.4)

In [5, 6], we studied the second kind Euler numbers En and polynomials En (x) and investigate their properties. 2. Barnes-type multiple twisted q-Euler numbers and polynomials of the second kind In this section, we assume that w1 , . . . , wk ∈ Zp and a1 , . . . , ak ∈ Z. Let Tp = ∪N ≥1 CpN = N limN →∞ CpN , where CpN = {ω|ω p = 1} is the cyclic group of order pN . For ω ∈ Tp , we denote by ϕω : Zp → Cp the locally constant function x 7−→ ω x . We construct the Barnes-type multiple twisted q-Euler polynomials of the second kind, En,ω,q (w1 , . . . , wk ; a1 , . . . , ak | x). For k ∈ N, we define Barnes-type multiple twisted q-Euler polynomials of the second kind as follows: ∫ |

∫ Zp

··· {z

Zp

ω x1 +···+xk q a1 x1 +···+ak xk e(x+2w1 x1 +···+2wk xk +k)t dµ−1 (x1 ) · · · dµ−1 (xk )

}

k−times

2k ekt ext (ωq a1 e2w1 t + 1)(ωq a2 e2w2 t + 1) · · · (ωq ak e2wk t + 1) ∞ ∑ tn = En,ω,q (w1 , . . . , wk ; a1 , . . . , ak | x) . n! n=0

=

(2.1)

In the special case, x = 0, En,ω,q (w1 , . . . , wk ; a1 , . . . , ak | 0) = En,ω,q (w1 , . . . , wk ; a1 , . . . , ak ) are called the n-th Barnes-type multiple twisted q-Euler numbers of the second kind. By (2.1) and Taylor expansion of e(x+2w1 x1 +···+2wk xk +k)t , we have the following theorem. Theorem 1. For positive integers n and k, we have En,ω,q (w1 , . . . , wk ; a1 , . . . , ak | x) ∫ ∫ = ··· ω x1 +···+xk q a1 x1 +···+ak xk (x + 2w1 x1 + · · · + 2wk xk + k)n dµ−1 (x1 ) · · · dµ−1 (xk ). |

Zp

Zp

{z } k−times

In the case when x = 0 in (2.1), we have the following corollary. Corollary 2. For positive integers n, we have En,ω,q (w1 , . . . , wk ; a1 , . . . , ak ) ∫ ∫ ∑k ∑k = ··· ω i=1 xi q i=1 ai xi (2w1 x1 + · · · + 2wk xk + k)n dµ−1 (x1 ) · · · dµ−1 (xk ). |

Zp

(2.2)

Zp

{z } k−times

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By Theorem 1 and (2.2), we obtain En,ω,q (w1 , . . . , wk ; a1 , . . . , ak | x) =

n ( ) ∑ n

l

l=0

where

(n) k

El,ω,q (w1 , . . . , wk ; a1 , . . . , ak )xn−l ,

(2.3)

is a binomial coefficient.

We define distribution relation of Barnes-type multiple twisted q-Euler polynomials of the second kind as follows: For m ∈ N with m ≡ 1( mod 2), we obtain ∞ ∑

En,ω,q (w1 , . . . , wk ; a1 , . . . , ak | x)

n=0

=

tn n!

2k ekmt (ω m q a1 m e2w1 mt + 1)(ω m q a2 m e2w2 mt + 1) · · · (ω m q ak m e2wk mt + 1)   x + 2w1 l1 + · · · + 2wk lk + k − mk  (mt) m−1 ∑ ∑k ∑k m l1 +···+lk li ai li i=1 i=1 × (−1) ω q e . l1 ,...,lk =0

From the above equation, we obtain ∞ ∑

En,ω,q (w1 , . . . , wk ; a1 , . . . , ak | x)

n=0

=

∞ ∑

m−1 ∑

mn

n=0

(−1)l1 +···+lk ω

tn n!

∑k

i=1 li

q

∑k i=1

ai li

l1 ,...,lk =0

(

× En,ωm ,qm

x + 2w1 l1 + · · · + 2wk lk + k − mk w1 , . . . , wk ; a1 , . . . , ak | m

)

tn . n!

tn in the above equation, we arrive at the following theorem. n! Theorem 3 (Distribution relation). For m ∈ N with m ≡ 1( mod 2), we have

By comparing coefficients of

En,ω,q (w1 , . . . , wk ; a1 , . . . , ak | x) m−1 ∑

= mn

(−1)l1 +···+lk ω

l1 ,...,lk =0

× En,ωm ,qm

∑k

i=1 li

q

∑k i=1

ai li

( ) x + 2w1 l1 + · · · + 2wk lk + k − mk w1 , . . . , wk ; a1 , . . . , ak | . m

From (2.1), we derive ∫ ∫ ··· ω x1 +···+xk q a1 x1 +···+ak xk e(x+2w1 x1 +···+2wk xk +k)t dµ−1 (x1 ) · · · dµ−1 (xk ) |

Zp

Zp

{z } k−times ∞ ∑ = 2k

(2.4) (−1)m1 +···+mk ω

∑k i=1

mi

q

∑k i=1

ai mi (x+2w1 m1 +···+2wk mk +k)t

e

.

m1 ,...mk =0

From (2.2) and (2.4), we note that En,ω,q (w1 , . . . , wk ; a1 , . . . , ak | x) = 2k

∞ ∑

(−1)m1 +···+mr q

∑k i=1

ai mi

(x + 2w1 m1 + · · · + 2wk mk + k)n .

(2.5)

m1 ,...mr =0

By using binomial expansion and (2.1), we have the following addition theorem.

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Theorem 4(Addition theorem). Barnes-type multiple twisted q-Euler polynomials of the second kind En,ω,q (w1 , . . . , wk ; a1 , . . . , ak | x) satisfies the following relation: En,ω,q (w1 , . . . , wk ; a1 , . . . , ak | x + y) =

n ( ) ∑ n El,ω,q (w1 , . . . , wk ; a1 , . . . , ak | x)y n−l . l l=0

3. Barnes-type multiple twisted q-Euler zeta function of the second kind In this section, we assume that q ∈ C with |q| < 1 and the parameters w1 , . . . , wk are positive. dl Let ω be the pN -th root of unity. By applying derivative operator, l |t=0 to the generating function dt of Barnes-type multiple twisted q-Euler polynomials of the second kind, En,ω,q (w1 , . . . , wk ; a1 , . . . , ak | x), we define Barnes-type multiple twisted q-Euler zeta function of the second kind. This function interpolates the Barnes-type multiple twisted q-Euler polynomials of the second kind at negative integers. By (2.1), we obtain 2k ekt ext (ωq a1 e2w1 t + 1) · · · (ωq ak e2wk t + 1) ∞ ∑ tn = En,ω,q (w1 , . . . , wk ; a1 , . . . , ak | x) . n! n=0

Fω,q (w1 , . . . , wk ; a1 , . . . , ak | x, t) =

(3.1)

Hence, by (3.1), we obtain ∞ ∑

En,ω,q (w1 , . . . , wk ; a1 , . . . , ak | x)

n=0 ∞ ∑

= 2k

(−1)m1 +···+mk ω

∑k i=1

tn n!

mi

q

∑k i=1

ai mi (x+2w1 m1 +···+2wk mk +k)t

e

.

m1 ,...mr =0

By applying derivative operator,

dl |t=0 to the above equation, we have dtl

En,ω,q (w1 , . . . , wk ; a1 , . . . , ak | x) = 2k

∞ ∑

(−1)m1 +···+mk ω

∑k i=1

mi

q

∑k i=1

a i mi

(x + 2w1 m1 + · · · + 2wk mk + k)n .

(3.2)

m1 ,...mk =0

By (3.2), we define the Barnes-type multiple twisted q-Euler zeta function of the second kind ζω,q (w1 , . . . , wk ; a1 , . . . , ak | s, x) as follows: Definition 5. For s, x ∈ C with Re(x) > 0, a1 , . . . , ak ∈ C, we define ζω,q (w1 , . . . , wk ; a1 , . . . , ak | s, x) = 2

k

∞ ∑ m1 ,...,mk

∑k

∑k

(−1)m1 +···+mk ω i=1 mi q i=1 ai mi , (x + 2w1 m1 + · · · + 2wk mk + k)s =0

(3.3)

For s = −l in (3.3) and using (3.2), we arrive at the following theorem. Theorem 6. For positive integer l, we have ζω,q (w1 , . . . , wk ; a1 , . . . , ak | −l, x) = El (w1 , . . . , wk ; a1 , . . . , ak | x). By (2.6), we define multiple twisted q-Euler zeta function of the second kind ζω,q (w1 , . . . , wk ; a1 , . . . , ak | s) as follows:

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Definition 7. For s ∈ C, we define ζω,q (w1 , . . . , wk ; a1 , . . . , ak | s) = 2

k

∞ ∑ m1 ,...,mk

∑k

∑k

(−1)m1 +···+mk ω i=1 mi q i=1 ai mi , (2w1 m1 + · · · + 2wk mk + k)s =0

(3.4)

For s = −l in (3.4) and using (2.5), we arrive at the following theorem. Theorem 8. For positive integer l, we have ζω,q (w1 , . . . , wk ; a1 , . . . , ak | −l) = El (w1 , . . . , wk ; a1 , . . . , ak ). Acknowledgment This work was supported by 2020 Hannam University Research Fund.

REFERENCES 1. Kim, T.(2008). Euler numbers and polynomials associated with zeta function, Abstract and Applied Analysis, Art. ID 581582. 2. Kim, T.(2008) Symmetry p-adic invariant integral on Zp for Bernoulli and Euler polynomials, Journal of Difference Equations and Applications, v.12, pp. 1267-1277. 3. Liu, G.(2006). Congruences for higher-order Euler numbers, Proc. Japan Acad., v.82 A, pp. 30-33. 4. Ryoo, C.S., Kim, T., Jang, L. C.(2007). Some relationships between the analogs of Euler numbers and polynomials, Journal of Inequalities and Applications, v.2007, ID 86052, pp. 122. 5. Ryoo, C.S.(2011). On the alternating sums of powers of consecutive odd integers, Journal of Computational Analysis and Applications, v.13, pp. 1019-1024. 6. Ryoo, C.S.(2010). Calculating zeros of the second kind Euler polynomials, Journal of Computational Analysis and Applications, v.12, pp. 828-833. 7. Ryoo, C.S.(2014). Note on the second kind Barnes’ type multiple q-Euler polynomials, Journal of Computational Analysis and Applications, v.16, pp. 246-250. 8. Ryoo, C.S.(2015). On the second kind Barnes-type multiple twisted zeta function and twisted Euler polynomials, Journal of Computational Analysis and Applications, v.18, pp. 423-429. 9. Ryoo, C.S.(2020). Symmetric identities for the second kind q-Bernoulli polynomials, Journal of Computational Analysis and Applications, v.28, pp. 654-659. 10. Ryoo, C.S.(2020). On the second kind twisted q-Euler numbers and polynomials of higher order, Journal of Computational Analysis and Applications, v.28, pp. 679-684. 11. Ryoo, C.S.(2020). Symmetric identities for Dirichlet-type multiple twisted (h, q)-l-function and higher-order generalized twisted (h, q)-Euler polynomials, Journal of Computational Analysis and Applications, v.28, pp. 537-542.

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Some Approximation Results of Kantorovich type operators Prashantkumar Patel April 26, 2019 In this manuscript, we investigate a variant of the operators define by Lupa¸s. We compute the rate of convergence for different class of functions. In section 3, the weighted approximation results are established. At the end, stated the problems for further research. Keyword: Positive linear operators; Rate Convergence; Weighted approximation 2000 Mathematics Subject Classification: primary 41A25, 41A30, 41A36.

1

Introduction

In [1], Lupa¸s proposed to study the following sequence of linear and positive operators Pn[0] (f, x) = 2−nx

  ∞ X k (nx)k f , 2k k! n

x ≥ 0,

f : [0, ∞) → R,

(1.1)

k=0

where (nx)0 = 1 and (nx)k = nx(nx + 1)(nx + 2) . . . (nx + k − 1), k[ ≥ 1. [0] Ea and We can consider that Pn , n ≥ 1, are defined on E where E = a>0

Ea is the subspace of all real valued continuous functions f on [0, ∞) such as sup(exp(−ax)|f (x)|) < ∞. The space Ea is endowed with the norm kf ka = x≥0

sup(exp(−ax)|f (x)|) with respect to which it becomes a Banach space. x≥0

In recent year, Patel and Mishra [2] generalized Jain operators type variant of the Lupa¸s operators defined as Pn[β] (f, x) =

∞ X (nx + kβ)k k=0

2k k!

2−(nx+kβ) f

  k , n

x ≥ 0, f : [0, ∞) → R, (1.2)

where (nx + kβ)0 = 1, (nx + kβ)1 = nx and (nx + kβ)k = nx(nx + kβ + 1)(nx + kβ + 2) . . . (nx + kβ + k − 1), k ≥ 2. [0] We mention that β = 0, the operators Pn reduce to Lupa¸s operators (1.1). In 1

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[2], the authors have used following Lagrange’s formula to define the operators (1.2):    k ∞ X z 1 dk−1 k 0 ((f (z) )φ (z) . φ(z) = φ(0) + k! dz k−1 f (z) z=0

(1.3)

k=1

But, if we use following Lagrange’s formula then the generalization of the operators (1.1) is written better way: 

z df (z) φ(z) 1 − f (z) dz

−1

   k ∞ X z 1 dk k ((f (z) )φ(z) = . k! dz k f (z) z=0 k=0

By choosing φ(z) = (1 − z)−α and f (z) = (1 − z)β , we have  −1 (1 − z)−α 1 − zβ(1 − z)−1  k ∞ X z 1 . (α + kβ)(α + kβ + 1) . . . (α + kβ + k − 1) = k! (1 − z)−β k=0

Taking z = 21 , we get 1 = (1 − β)

∞ X 1 (α + βk)k 2−(α+βk) . k 2 k!

k=0

Now, we may define the operators as Pn[β] (f, x)

=

∞ X k=0

  k pβ (k, nx)f n

(1.4)

∞ X 1 (nx + βk)k 2−(nx+βk) . where (nx + βk)0 = 1 k 2 k! k=0 and (nx + βk)k = (nx + βk)(nx + βk + 2) . . . (nx + βk + k − 1), k ≥ 1 and β+1 0≤ 0, Peetre’s K-functional is define by K2 (f, δ) =

{kf − gk + δkg 00 k},

inf

2 [0,∞) g∈CB

2 where CB [0, ∞) = {g ∈ CB [0, ∞) : g 0 , g 00 ∈ CB [0, ∞)}. By DeVore and Lorentz [4, P.177, Theorem 2.4], there exists an absolute constant C > 0 such that  √  (2.9) K2 (f, δ) ≤ Cω2 f, δ ,

where the second order modulus of smoothness of g ∈ CB [0, ∞) is defined as ω2 (g; δ) = sup sup |g(x) − 2g(x + h) + g(x + 2h)|,

δ > 0,

0 0.

0 0 one has    1 1 [β] 0 0 |Kn (f, x) − f (x)| ≤ , bn kf ||C[0,a] + cn ω1 f ; √ 2n(1 − β)2 n where p bn = 2anβ + (1 + β)2 and cn = 2 n2 a2 β 2 + 2na (1 + 2β) + 1 + 35β  p 1 + (1 − β)−2 na2 β 2 + 2a (1 + 2β) + (1 + 35β)n−1 . Proof: We can write f (x) − f (t) = (x − t)f 0 (x) + (x − t)(f 0 (ξ) − f 0 (x)), where ξ = ξ(t, x) is a point of the interval determinate by x and t. If we Z (k+1)/n multiply both members of this inequality by npβ (k, nx) dt and sum k/n

over k , there follows |Kn[β] (f, x) − f (x)|

≤

|f 0 (x)|Ωn,1 (x) Z ∞ X +n pβ (k, nx)

(k+1)/n

|x − t| · |f 0 (ξ) − f 0 (t)|dt

k/n

k=0

2xnβ + (1 + β)2 ≤ 2n(1 − β)2 Z ∞ X +n pβ (k, nx) 



max |f 0 (x)|

(2.11)

x∈[0,a] (k+1)/n

|x − t|(1 + δ −1 |t − x|)ω1 (f 0 ; δ)dt.

k/n

k=0

8

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According to Cauchy’s inequality, we have

n

∞ X

Z

(k+1)/n

|x − t|dt

pβ (k, nx) k/n

k=0

∞ √ X ≤ n pβ (k, nx)

(Z

("

∞ X

2

(x − t) dt k/n

k=0

√ ≤ n

)1/2

(k+1)/n

#" pβ (k, nx)

k=0

∞ X

Z

#)1/2

(k+1)/n 2

(x − t) dt

pβ (k, nx)

.

k/n

k=0

Hence, n

∞ X k=0

Z

(k+1)/n

|x − t|dt ≤

pβ (k, nx)

q

Ωn,2 (x).

(2.12)

k/n

Using inequalities (2.12) in (2.11), we write   2 [β] |Kn (f, x) − f (x)| ≤ 2anβ+(1+β) kf 0 kC[0,a] 2n(1−β)2   p p + Ωn,2 (x) 1 + δ −1 Ωn,2 (x) ω1 (f 0 ; δ)dt.

(2.13)

p q n2 a2 β 2 + 2na (1 + 2β) + 1 + 35β 1 , Inserting δ = √ and using Ωn,2 (x) ≤ n(1 − β)2 n x ∈ [0, a] , the proof of our theorem is complete. Theorem 4 Let f ∈ CB [0, ∞). Then for all x ∈ [0, ∞) there exists a constant A > 0 such that   x (1 + β)2 [β] |Kn (f, x) − f (x)| ≤ Aω2 (f, ξn (x)) + ω1 f, + , 1−β 2n(1 − β)2 where ξn (x) =

3n2 x2 β 2 +6nx(1+2β)+1+35β 3n2 (1−β)4

+



2xnβ+(1+β)2 2n(1−β)2

2

.

Proof: Consider the following operator   x (1 + β)2 [β] [β] ˆ Kn (f, x) = Kn (f, x) − f + + f (x). 1−β 2n(1 − β)2

(2.14)

ˆ n[β] and Lemma 1, we have By the definition of the operators K ˆ n[β] (t − x, x) = 0. K 2 Let g ∈ CB [0, ∞) and x ∈ [0, ∞). By Taylor’s formula of g, we get Z t 0 g(t) − g(x) = (t − x)g (x) + (t − u)g 00 (u)du, t ∈ [0, ∞). x

9

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One may write ˆ n[β] (g, x) − g(x) K

Z t  ˆ n[β] (t − x, x) + K ˆ n[β] (t − u)g 00 (u)du, x = g 0 (x)K x Z t  ˆ n[β] = K (t − u)g 00 (u)du, x x  Z t [β] (t − u)g 00 (u)du, x = Kn x

Z −

(1+β)2 x 1−β + 2n(1−β)2

x



 x (1 + β)2 − u du. + 1−β 2n(1 − β)2

Now, using the following inequalities Z t 00 ≤ (t − x)2 kg 00 k (t − u)g (u)du

(2.15)

x

and Z

(1+β)2 x 1−β + 2n(1−β)2

x



  2 x x (1 + β)2 (1 + β)2 ≤ + − u du + kg 00 k, 1−β 2n(1 − β)2 1−β 2n(1 − β)2

we reach to ( ˆ n[β] (g, x) − g(x)| |K

Kn[β] ((t

≤

(1 + β)2 x + − x) , x) + 1−β 2n(1 − β)2 2



2 )

3n2 x2 β 2 + 6nx (1 + 2β) + 1 + 35β 3n2 (1 − β)4  2 ) 2xnβ + (1 + β)2 kg 00 k. + 2n(1 − β)2

kg 00 k

 ≤

(2.16)

ˆ n[β] and Kn[β] , we have By means of the definitions of the operators K [β] ˆ n (g, x) − g(x) ˆ n[β] (f − g, x)| + |(f − g)(x)| + K |Kn[β] (f, x) − f (x)| ≤ |K   x (1 + β)2 − f (x) + f + 1−β 2n(1 − β)2 and ˆ n[β] (f, x) ≤ |Kn[β] (f, x)| + 2kf k ≤ kf kKn[β] (1, x) + 2kf k = 3kf k. K Thus, we may conclude that |Kn[β] (f, x) − f (x)|

ˆ n[β] (g, x) − g(x)| ≤ 4kf − gk + |K   x (1 + β)2 . + f + − f (x) 1−β 2n(1 − β)2 10

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In the light of inequality (2.16), one gets |Kn[β] (f, x) − f (x)|

≤ 4kf − gk  2 2 2 3n x β + 6nx (1 + 2β) + 1 + 35β + 3n2 (1 − β)4  2 ) 2xnβ + (1 + β)2 + kg 00 k 2n(1 − β)2   x (1 + β)2 +ω1 f, . + 1−β 2n(1 − β)2

Therefore taking the infimum over all g ∈ CB [0, ∞) on the right-hand side of the last inequality and considering (2.9), we find that   x (1 + β)2 [β] |Kn (f, x) − f (x)| ≤ 4K2 (f, ξn (x)) + ω1 f, + 1−β 2n(1 − β)2   x (1 + β)2 ≤ 4Cω2 (f, ξn (x)) + ω1 f, + 1−β 2n(1 − β)2   (1 + β)2 x ≤ Aω2 (f, ξn (x)) + ω1 f, + , 1−β 2n(1 − β)2 which completes the proof. Theorem 5 Let 0 < γ ≤ 1, β ∈ [0, 1) and f ∈ CB [0, ∞). Then if f ∈ LipM (γ), that is, the inequality |f (t) − f (x)| ≤ M |t − x|γ , x, t ∈ [0, ∞) holds, then for each x ∈ [0, ∞), we have γ |Kn[β] (f, x) − f (x)| ≤ dn2 (x), where dn (x) =

3n2 x2 β 2 + 6nx (1 + 2β) + 1 + 35β and M > 0 is a constant. 3n2 (1 − β)4

Proof: Let f ∈ CB [0, ∞) ∩ LipM (γ). By the linearity and monotonicity of [β] the operators Kn , we get |Kn[β] (f, x) − f (x)|

≤ Kn[β] (|f (t) − f (x)|, x) ≤ M Kn[β] (|t − x|γ , x) Z (k+1)/n ∞ X = Mn pβ (k, nx) |t − x|γ dt. k=0

k/n

11

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Now, applying the H¨ older inequality two times successively with p =

2 , q = γ

2 , we obtain 2−γ |Kn[β] (f, x)

− f (x)|

≤ M

∞ X

Z

! γ2

(k+1)/n γ

|t − x| dt

pβ (k, nx) n k/n

k=0 γ

≤ M (Ωn,2 (x)) 2 γ  2 2 2 3n x β + 6nx (1 + 2β) + 1 + 35β 2 . ≤ M 3n2 (1 − β)4 This completes the proof.

3

Weighted approximation properties [β]

Now, we introduce convergence properties of the operators Kn via the weighted Korovkin type theorem given by Gadzhiev in [5, 6]. For this purpose, we recall some definitions and notations. Let ρ(x) = 1+ x2 and Bρ [0, ∞) be the space of all functions having the property |f (x)| ≤ Mf ρ(x), where x ∈ [0, ∞) and Mf is a positive constant depending only on f . The set Bρ [0, ∞) is equipped with the norm kf kρ =

|f (x)| . 2 1 x∈[0,∞) + x sup

Cρ [0, ∞) denotes the space of all continuous functions belonging to Bρ [0, ∞). By Cρ0 [0, ∞), we denote the subspace of all functions f ∈ Cρ [0, ∞) for which lim

x→∞

|f (x)| < ∞. ρ(x)

Theorem 6 ([5, 6]) Let {An} be a sequence of positive linear operators acting from Cρ [0, ∞) to Bρ [0, ∞) and satisfying the conditions limn→∞ kAn (tv ; x) − xv kρ = 0, v = 0, 1, 2. Then for any function f ∈ Cρ0 [0, ∞), lim kAn (f ; ·) − f (·)kρ = 0.

n→∞

Note that, a sequence of linear positive operators An acts from Cρ [0, ∞) to Bρ [0, ∞) if and only if kAn (ρ; x)k ≤ Mρ , where Mρ is positive constant. This fact also given in [5, 6]. 12

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[β ]

Theorem 7 Let {Kn n } be the sequence of linear positive operators defined by (1.5) and βn ∈ [0, 1) with βn → 0 as n → ∞. Then for each f ∈ Cρ0 [0, ∞), we have lim kKn[β] (f ; x) − f (x)kρ = 0. n→∞

Proof: Using Lemma 1, we may write [β ]

|Kn n (ρ, x)| 1 + x2 x∈[0,∞) sup

≤

3 + 2βn + βn2 1 + 2 (1 − βn ) n(1 − βn )3 +




1 + 20βn + 12βn2 + 2βn3 + βn4 + 1. 3n2 (1 − βn )4

Since lim βn = 0, , there exists a positive constant M ∗ such that n→∞


3 + 2βn + βn2 1 1 + 20βn + 12βn2 + 2βn3 + βn4 + + ≤ M∗ 2 3 (1 − βn ) n(1 − βn ) 3n2 (1 − βn )4 for each n. Hence, we get kKn[βn ] (ρ, x)kρ ≤ 1 + M ∗ , [β ]

which shows that {Kn n } is a sequence of positive linear operators acting from Cρ [0, ∞) to Bρ [0, ∞). In order to complete the proof, it is enough to prove that the conditions of Theorem 6 lim kKn[βn ] (tv ; x) − xv kρ = 0, v = 0, 1, 2 n→∞

are satisfied. It is clear that lim kKn[βn ] (1; x) − 1kρ = 0

n→∞

By Lemma 1, we have kKn[βn ] (t; x) − xkρ

  1 x (1 + βn )2 1 sup −1 + 1 − βn 1 + x2 2n(1 − βn )2 1 + x2 x∈∞ βn (1 + βn )2 ≤ . + 1 − βn 2n(1 − βn )2 =

Thus taking into consideration the conditions βn → 0 as n → ∞, we can conclude that lim kKn[βn ] (t; x) − xkρ = 0

n→∞

13

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Similarly, one gets kKn[βn ] (t2 ; x) − x2 kρ
  2 2 3 + 2β + β 1 x x n n −1 + = sup (1 − βn )2 1 + x2 n(1 − βn )3 1 + x2 x∈∞ 1 + 20βn + 12βn2 + 2βn3 + βn4 1 + 3n2 (1 − βn )4 1 + x2
  3 + 2βn + βn2 1 1 + 20βn + 12βn2 + 2βn3 + βn4 ≤ sup −1 + + (1 − βn )2 n(1 − βn )3 3n2 (1 − βn )4 x∈∞
2 2 2 3 4 2βn − βn 3 + 2βn + βn 1 + 20βn + 12βn + 2βn + βn ≤ sup + + 2 3 n(1 − βn ) 3n2 (1 − βn )4 x∈∞ (1 − βn ) which leads to lim kKn[βn ] (t2 ; x) − x2 kρ = 0 with βn → 0.

n→∞

Thus the proof is completed. [β] Now, we compute the order of approximation of the operators Kn in terms of the weighted modulus of continuity Ω2 (f, δ) (see[7]) defined by Ω2 (f, δ) =

sup x≥0,0 0. For each function f : (0, ∞) → F of exponential order. Transform of f so that Z∞ g(u) = f (t) e−itu dt.

Let g denote the Fourier

−∞

Then, at points of continuity of f , we have Z∞

1 f (x) = 2π

g(u) e−ixu du,

−∞

this is called the inverse Fourier transforms. The Fourier transform of f is denoted by F(ξ). We also introduce a notion, the convolution of two functions. Definition 2.1. (Convolution). Given two functions f and g, both Lebesgue integrable on (−∞, +∞). Let S denote the set of x for which the Lebesgue integral Z∞ f (t) g(x − t) dt

h(x) = −∞

exists. This integral defines a function h on S called the convolution of f and g. We also write h = f ∗ g to denote this function. Theorem 2.2. The Fourier transform of the convolution of f (x) and g(x) is the product of the Fourier transform of f (x) and g(x). That is, F{f (x) ∗ g(x)} = F{f (x)} F{g(x)} = F (s) G(s)

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or F

 ∞ Z 

−∞

  f (t) g(x − t) dt = F (s) G(s), 

where F (s) and G(s) are the Fourier transforms of f (x) and g(x), respectively. Definition 2.3. [41] The Mittag-Leffler function of one parameter is denoted by Eα (z) and defined as ∞ X 1 Eα (z) = zk Γ(αk + 1) k=0

where z, α ∈ C and Re(α) > 0. If we put α = 1, then the above equation becomes E1 (z) =

∞ X k=0

∞

X zk 1 zk = = ez . Γ(k + 1) k k=0

Definition 2.4. [41] The generalization of Eα (z) is defined as a function Eα,β (z) =

∞ X k=0

1 zk Γ(αk + β)

where z, α, β ∈ C, Re(α) > 0 and Re(β) > 0. Now, we give the definition of Mittag-Leffler-Hyers-Ulam stability and Mittag-LefflerHyers-Ulam-Rassias stability of the differential equations (1.1), (1.2), (1.3) and (1.4). Definition 2.5. The linear differential equation (1.1) is said to have the Mittag-LefflerHyers-Ulam stability, if there exists a constant K > 0 with the following property: For every  > 0, let x(t) be a continuously differentiable function satisfies the inequality |x0 (t) + l x(t)| ≤ Eα (tα ), where Eα is a Mittag-Leffler function, then there exists some y : (0, ∞) → F satisfies the differential equation (1.1) such that |x(t) − y(t)| ≤ KEα (tα ), for any t > 0. We call such K as the Mittag-Leffler-Hyers-Ulam stability constant for the differential equation (1.1). Definition 2.6. The linear differential equation (1.2) is said to have the Mittag-LefflerHyers-Ulam stability, if there exists a constant K > 0 with the following property: For every  > 0, let x(t) be a continuously differentiable function satisfies the inequality |x0 (t) + l x(t) − r(t)| ≤ Eα (tα ), where Eα is a Mittag-Leffler function, then there exists some y : (0, ∞) → F satisfies the differential equation (1.2) such that |x(t) − y(t)| ≤ KEα (tα ), for any t > 0. We call such K as the Mittag-Leffler-Hyers-Ulam stability constant for the differential equation (1.2). Definition 2.7. The linear differential equation (1.3) is said to have the Mittag-LefflerHyers-Ulam stability, if there exists a constant K > 0 with the following property: For every  > 0, let x(t) be a twice continuously differentiable function satisfying |x00 (t) + l x0 (t) + m x(t)| ≤ Eα (tα ),

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where Eα is a Mittag-Leffler function, then there exists some y : (0, ∞) → F satisfies the differential equation (1.3) such that |x(t) − y(t)| ≤ KEα (tα ), for any t > 0. We call such K as the Mittag-Leffler-Hyers-Ulam stability constant for the differential equation (1.3). Definition 2.8. The linear differential equation (1.4) is said to have the Mittag-LefflerHyers-Ulam stability, if there exists a constant K > 0 with the following property: For every  > 0, let x(t) be a twice continuously differentiable function satisfying |x00 (t) + l x0 (t) + m x(t) − r(t)| ≤ Eα (tα ), where Eα is a Mittag-Leffler function, then there exists some y : (0, ∞) → F satisfies the differential equation (1.4) such that |x(t) − y(t)| ≤ KEα (tα ), for any t > 0. We call such K as the Mittag-Leffler-Hyers-Ulam stability constant for the differential equation (1.4). Definition 2.9. We say that the homogeneous linear differential equation (1.1) has the Mittag-Leffler-Hyers-Ulam-Rassias stability, if there exists a constant K > 0 with the following property: For every  > 0, let x(t) be a continuously differentiable function, if there exists φ : (0, ∞) → (0, ∞) satisfies the inequality |x0 (t) + l x(t)| ≤ φ(t)Eα (tα ), where Eα is a Mittag-Leffler function, then there exists some y : (0, ∞) → F satisfies the differential equation (1.1) such that |x(t) − y(t)| ≤ Kφ(t)Eα (tα ), for any t > 0. We call such K as the Mittag-Leffler-Hyers-Ulam-Rassias stability constant for the equation (1.1). Definition 2.10. We say that the non-homogeneous linear differential equation (1.2) has the Mittag-Leffler-Hyers-Ulam-Rassias stability, if there exists a constant K > 0 with the following property: For every  > 0, let x(t) be a continuously differentiable function, if there exists φ : (0, ∞) → (0, ∞) satisfies the inequality |x0 (t) + l x(t) − r(t)| ≤ φ(t)Eα (tα ), where Eα is a Mittag-Leffler function, then there exists some y : (0, ∞) → F satisfies the differential equation (1.2) such that |x(t) − y(t)| ≤ Kφ(t)Eα (tα ), for any t > 0. We call such K as the Mittag-Leffler-Hyers-Ulam-Rassias stability constant for the equation (1.2). Definition 2.11. We say that the homogeneous linear differential equation (1.3) has the Mittag-Leffler-Hyers-Ulam-Rassias stability, if there exists a constant K > 0 with the following property: For every  > 0, let x(t) be a twice continuously differentiable function, if there exists φ : (0, ∞) → (0, ∞) satisfies the inequality |x00 (t) + l x0 (t) + m x(t)| ≤ φ(t)Eα (tα ), where Eα is a Mittag-Leffler function, then there exists some y : (0, ∞) → F satisfies the differential equation (1.3) such that |x(t) − y(t)| ≤ Kφ(t)Eα (tα ), for any t > 0. We call such K as the Mittag-Leffler-Hyers-Ulam-Rassias stability constant for the equation (1.3). Definition 2.12. We say that the non-homogeneous linear differential equation (1.4) has the Mittag-Leffler-Hyers-Ulam-Rassias stability, if there exists a constant K > 0 with the following property: For every  > 0, let x(t) be a twice continuously differentiable function, if there exists φ : (0, ∞) → (0, ∞) satisfies the inequality |x00 (t) + l x0 (t) + m x(t) − r(t)| ≤ φ(t)Eα (tα ),

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where Eα is a Mittag-Leffler function, then there exists some y : (0, ∞) → F satisfies the differential equation (1.4) such that |x(t) − y(t)| ≤ Kφ(t)Eα (tα ), for any t > 0. We call such K as the Mittag-Leffler-Hyers-Ulam-Rassias stability constant for the equation (1.4). 3. Mittag-Leffler-Hyers-Ulam Stability In the following theorems, we prove the Mittag-Leffler-Hyers-Ulam stability of the homogeneous and non-homogeneous linear differential equations (1.1), (1.2), (1.3) and (1.4). Firstly, we prove the Mittag-Leffler-Hyers-Ulam stability of first order homogeneous differential equation (1.1). Theorem 3.1. The differential equation (1.1) has Mittag-Leffler-Hyers-Ulam stability. Proof. Let l be a constant in F. For every  > 0, there exists a positive constant K such that x : (0, ∞) → F be a continuously differentiable function satisfies the inequality |x0 (t) + l x(t)| ≤ Eα (tα )

(3.1)

for all t > 0. We will prove that, there exists a solution y : (0, ∞) → F satisfying the differential equation y 0 (t) + l y(t) = 0 such that |x(t) − y(t)| ≤ KEα (tα ) for any t > 0. Let us define a function p : (0, ∞) → F such that p(t) =: x0 (t) + l x(t) for each t > 0. In view of (3.1), we have |p(t)| ≤ Eα (tα ). Now, taking Fourier transform to p(t), we have F{p(t)} = F{x0 (t) + l x(t)} P (ξ) = F{x0 (t)} + l F{x(t)} = −iξX(ξ) + l X(ξ) = (l − iξ)X(ξ) X(ξ) =

P (ξ) . (l − iξ)

Thus F{x(t)} = X(ξ) = Taking Q(ξ) =

P (ξ) (l + iξ) . l2 − ξ 2

(3.2)

1 , then we have (l − iξ) 1 F{q(t)} = (l − iξ)

⇒

q(t) = F

−1



1 (l − iξ)

 .

Now, we set y(t) = e−lt and taking Fourier transform on both sides, we get Z∞ F{y(t)} = Y (ξ) =

e −∞

−lt

ist

e

Z0 dt =

−lt

e −∞

e

ist

Z∞ dt +

e−lt eist dt = 0.

(3.3)

0

Now, F{y 0 (t) + l y(t)} = F{y 0 (t)} + l F{y(t)} = −iξY (ξ) + l Y (ξ) = (l − iξ)Y (ξ).

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Then by using (3.3), we have F{y 0 (t) + l y(t)} = 0, since F is one-to-one operator, thus y 0 (t) + l y(t) = 0, Hence y(t) is a solution of the differential equation (1.1). Then by using (3.2) and (3.3) we can obtain P (ξ) (l + iξ) = P (ξ) Q(ξ) = F{p(t)} F{q(t)} l2 − ξ 2 F{x(t) − y(t)} = F{p(t) ∗ q(t)}.

F{x(t)} − F{y(t)} = X(ξ) − Y (ξ) = ⇒

Since the operator F is one-to-one and linear, which gives x(t) − y(t) = p(t) ∗ q(t). Taking modulus on both sides, we have ∞ ∞ Z Z |x(t) − y(t)| = |p(t) ∗ q(t)| = p(t) q(t − s) ds ≤ |p(t)| q(t − s) ds ≤ KEα (tα ). −∞

−∞

R∞ Where K = q(t − s) ds exists for each value of t. Then by virtue of Definition 2.5 the −∞ homogeneous linear differential equation (1.1) has the Mittag-Leffler-Hyers-Ulam stability.  Now, we are going prove the Mittag-Leffler-Hyers-Ulam stability of the non-homogeneous linear differential equation (1.2) using Fourier transform method. Theorem 3.2. The differential equation (1.2) has Mittag-Leffler-Hyers-Ulam stability. Proof. Let l be a constant in F. For every  > 0, there exists a positive constant K such that x : (0, ∞) → F be a continuously differentiable function satisfies the inequality |x0 (t) + l x(t) − r(t)| ≤ Eα (tα )

(3.4)

for all t > 0. We have to show that there exists a solution y : (0, ∞) → F satisfying the nonhomogeneous differential equation y 0 (t) + l y(t) = r(t) such that |x(t) − y(t)| ≤ KEα (tα ), for any t > 0. Let us define a function p : (0, ∞) → F such that p(t) =: x0 (t) + l x(t) − r(t) for each t > 0. In view of (3.4), we have |p(t)| ≤ Eα (tα ). Now, taking Fourier transform to p(t), we have F{p(t)} = F{x0 (t) + l x(t) − r(t)} P (ξ) = F{x0 (t)} + l F{x(t)} − F{r(t)} = −iξX(ξ) + l X(ξ) − R(ξ) = (l − iξ)X(ξ) − R(ξ) X(ξ) =

P (ξ) + R(ξ) . (l − iξ)

Thus F{x(t)} = X(ξ) =

{P (ξ) + R(ξ)} (l + iξ) . l2 − ξ 2

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Let us choose Q(ξ) as

1 , then we have (l − iξ)

1 F{q(t)} = (l − iξ)

⇒

q(t) = F

−1



1 (l − iξ)

 .

Now, we set y(t) = e−lt + (r(t) ∗ q(t)) and taking Fourier transform on both sides, we get Z∞ F{y(t)} = Y (ξ) =

e−lt eist dt +

R(ξ) R(ξ) = (l − iξ) (l − iξ)

(3.6)

−∞

Now, F{y 0 (t) + l y(t)} = −iξY (ξ) + l Y (ξ) = R(ξ). Then by using (3.6), we have F{y 0 (t) + l y(t)} = F {r(t)}, since F is one-to-one operator, thus y 0 (t) + l y(t) = r(t), Hence y(t) is a solution of the differential equation (1.2). Then by using (3.5) and (3.6) we have {P (ξ) + R(ξ)} (l + iξ) R(ξ) − 2 2 l −ξ (l − iξ) = P (ξ) Q(ξ) = F{p(t)} F{q(t)}

F{x(t)} − F {y(t)} = X(ξ) − Y (ξ) =

⇒

F{x(t) − y(t)} = F{p(t) ∗ q(t)}

Since the operator F is one-to-one and linear, which gives x(t) − y(t) = p(t) ∗ q(t). Taking modulus on both sides, we have ∞ ∞ Z Z |x(t) − y(t)| = |p(t) ∗ q(t)| = p(t) q(t − s) ds ≤ |p(t)| q(t − s) ds ≤ KEα (tα ). −∞ −∞ R∞ Where K = q(t − s) ds , the integral exists for each value of t. Hence, by the virtue of −∞ Definition 2.6 the non-homogeneous differential equation (1.2) has the Mittag-Leffler-HyersUlam stability.  Now, we prove the Mittag-Leffler-Hyers-Ulam stability of the homogeneous and nonhomogeneous second order linear differential equations (1.3) and (1.4). Theorem 3.3. The differential equation (1.3) has Mittag-Leffler-Hyers-Ulam stability. Proof. Let l, m be constants in F such that there exist µ, ν ∈ F with µν = m, µ+ν = −l and µ 6= ν. For every  > 0, there exists a positive constant K such that x : (0, ∞) → F be a twice continuously differentiable function satisfying the inequality |x00 (t) + l x0 (t) + m x(t)| ≤ Eα (tα )

(3.7)

for all t > 0. We will show that there exists a solution y : (0, ∞) → F satisfying the homogeneous differential equation y 00 (t) + l y 0 (t) + m y(t) = 0 such that |x(t) − y(t)| ≤ KEα (tα ),

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for any t > 0. Let us define a function p : (0, ∞) → F such that p(t) =: x00 (t) + l x0 (t) + m x(t) for each t > 0. In view of (3.7), we have |p(t)| ≤ Eα (tα ). Now, taking Fourier transform to p(t), we have F{p(t)} = F{x00 (t) + l x0 (t) + m x(t)} P (ξ) = F{x00 (t)} + l F{x0 (t)} + m F{x(t)} = (ξ 2 − iξl + m) X(ξ) X(ξ) =

ξ2

P (ξ) . − iξl + m

Since l, m are constants in F such that there exist µ, ν ∈ F with µ + ν = −l, µν = m and µ 6= ν, we have (ξ 2 − iξl + m) = (iξ − µ) (iξ − ν). Thus F{x(t)} = X(ξ) = Let Q(ξ) =

P (ξ) . (iξ − µ) (iξ − ν)

(3.8)

1 , then we have (iξ − µ) (iξ − ν) 1 F{q(t)} = (iξ − µ) (iξ − ν)

Now, setting y(t) as

⇒

q(t) = F

−1



1 (iξ − µ) (iξ − ν)

µe−µt − νe−νt and taking Fourier transform, we obtain µ−ν Z∞ −µt µe − νe−νt ist F{y(t)} = Y (ξ) = e dt = 0. µ−ν

 .

(3.9)

−∞

Now, F{y 00 (t) + l y 0 (t) + m y(t)} = (ξ 2 − iξl + m) Y (ξ). Then by using (3.9), we have F{y 00 (t) + l y 0 (t) + m y(t)} = 0. Since F is one-to-one operator, then y 00 (t) + l y 0 (t) + m y(t) = 0, Hence y(t) is a solution of the differential equation (1.3). Then by using (3.8) and (3.9) we can obtain P (ξ) = P (ξ) Q(ξ) = F{p(t)} F{q(t)} − iξl + m F{x(t) − y(t)} = F{p(t) ∗ q(t)}

F{x(t)} − F{y(t)} = X(ξ) − Y (ξ) = ⇒

ξ2

Since the operator F is one-to-one and linear, which gives x(t) − y(t) = p(t) ∗ q(t). Taking modulus on both sides, we have ∞ ∞ Z Z q(t − s) ds ≤ KEα (tα ). |x(t) − y(t)| = |p(t) ∗ q(t)| = p(t) q(t − s) ds ≤ |p(t)| −∞ −∞ R∞ Where K = q(t − s) ds , the integral exists for each value of t. Then by virtue of −∞ Definition 2.7 the homogeneous linear differential equation (1.3) has the Mittag-Leffler-HyersUlam stability. 

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Theorem 3.4. The differential equation (1.4) has Mittag-Leffler-Hyers-Ulam stability. Proof. Let l, m be constants in F such that there exist µ, ν ∈ F with µν = m, µ+ν = −l and µ 6= ν. For every  > 0, there exists a positive constant K such that x : (0, ∞) → F is a twice continuously differentiable function satisfying the inequality |x00 (t) + l x0 (t) + m x(t) − r(t)| ≤ Eα (tα )

(3.10)

for all t > 0. We have to prove that there exists a solution y : (0, ∞) → F satisfying the non-homogeneous differential equation y 00 (t) + l y 0 (t) + m y(t) = r(t) such that |x(t) − y(t)| ≤ KEα (tα ), for any t > 0. Assume that x(t) is a continuously differentiable function satisfying the inequality (3.10). Let us define a function p : (0, ∞) → F such that p(t) =: x00 (t) + l x0 (t) + m x(t) − r(t) for each t > 0. In view of (3.10), we have |p(t)| ≤ Eα (tα ). Now, taking Fourier transform to p(t), we have F{p(t)} = F{x00 (t) + l x0 (t) + m x(t) − r(t)} P (ξ) = F{x00 (t)} + l F{x0 (t)} + m F{x(t)} − F{r(t)} = (ξ 2 − iξl + m) X(ξ) − R(ξ) X(ξ) =

P (ξ) + R(ξ) . ξ 2 − iξl + m

Since l, m are constants in F such that there exist µ, ν ∈ F with µ + ν = −l, µν = m and µ 6= ν, we have (ξ 2 − iξl + m) = (iξ − µ) (iξ − ν). Thus F{x(t)} = X(ξ) =

P (ξ) + R(ξ) . (iξ − µ) (iξ − ν)

Q(ξ) = F{q(t)} =

1 , (iξ − µ) (iξ − ν)

(3.11)

Taking

setting µe−µt − νe−νt + (r(t) ∗ q(t)) µ−ν and taking Fourier transform on both sides, we get y(t) =

Z∞ F{y(t)} = Y (ξ) =

µe−µt − νe−νt ist R(ξ) R(ξ) e dt + = . (3.12) µ−ν (iξ − µ) (iξ − ν) (iξ − µ) (iξ − ν)

−∞

Now, F{y 00 (t) + l y 0 (t) + m y(t)} = F{y 00 (t)} + l F{y 0 (t)} + m F{y(t)} = (ξ 2 − iξl + m) Y (ξ) = R(ξ). Then by using (3.12), we have F{y 00 (t) + l y 0 (t) + m y(t)} = F{r(t)}, since F is one-to-one operator, thus y 00 (t) + l y 0 (t) + m y(t) = r(t), Hence y(t) is a solution of the differential

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equation (1.4). Then by using (3.11) and (3.12) we can obtain P (ξ) + R(ξ) R(ξ) − (iξ − µ) (iξ − ν) (iξ − µ) (iξ − ν) = P (ξ) Q(ξ) = F{p(t)} F{q(t)}

F{x(t)} − F{y(t)} = X(ξ) − Y (ξ) =

⇒

F{x(t) − y(t)} = F{p(t) ∗ q(t)}

Since the operator F is one-to-one and linear, which gives x(t) − y(t) = p(t) ∗ q(t). Taking modulus on both sides, we have ∞ ∞ Z Z |x(t) − y(t)| = |p(t) ∗ q(t)| = p(t) q(t − s) ds ≤ |p(t)| q(t − s) ds ≤ KEα (tα ). −∞ −∞ R∞ Where K = q(t − s) ds , the integral exists for each value of t. Then by virtue of −∞ Definition 2.8 the non-homogeneous linear differential equation (1.4) has the Mittag-LefflerHyers-Ulam stability.  4. Mittag-Leffler-Hyers-Ulam-Rassias Stability In the following theorems, we are going to investigate the Mittag-Leffler-Hyers-UlamRassias stability of the differential equations (1.1), (1.2), (1.3) and (1.4). Theorem 4.1. The differential equation (1.1) has Mittag-Leffler-Hyers-Ulam-Rassias stability. Proof. Let l be a constant in F. For every  > 0, there exists a positive constant K such that x : (0, ∞) → F be a continuously differentiable function and φ : (0, ∞) → (0, ∞) be an integrable function satisfies |x0 (t) + l x(t)| ≤ φ(t)Eα (tα )

(4.1)

for all t > 0. We will prove that, there exists a solution y : (0, ∞) → F which satisfies the differential equation y 0 (t) + l y(t) = 0 such that |x(t) − y(t)| ≤ Kφ(t)Eα (tα ) for any t > 0. Let us define a function p : (0, ∞) → F such that p(t) =: x0 (t) + l x(t) for each t > 0. In view of (4.1), we have |p(t)| ≤ φ(t)Eα (tα ). Now, taking Fourier transform to p(t), we have P (ξ) (l + iξ) F{x(t)} = X(ξ) = . (4.2) l2 − ξ 2   1 1 −1 Choosing Q(ξ) = , then we have q(t) = F . Now, we set y(t) = e−lt and (l − iξ) (l − iξ) taking Fourier transform on both sides, we get Z∞ F{y(t)} = Y (ξ) = e−lt eist dt = 0. (4.3) −∞

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Hence F{y 0 (t) + l y(t)} = −iξY (ξ) + l Y (ξ) = (l − iξ)Y (ξ) Then by using (4.3), we have F{y 0 (t) + l y(t)} = 0, since F is one-to-one operator, thus y 0 (t) + l y(t) = 0, Hence y(t) is a solution of the differential equation (1.1). Then by using (4.2) and (4.3) we can obtain F{x(t) − y(t)} = F{p(t) ∗ q(t)} Since the operator F is one-to-one and linear, which gives x(t) − y(t) = p(t) ∗ q(t). Taking modulus on both sides, we have ∞ ∞ Z Z p(t) q(t − s) ds ≤ |p(t)| q(t − s) ds ≤ Kφ(t)Eα (tα ). |x(t) − y(t)| = |p(t) ∗ q(t)| = −∞ −∞ R∞ Where K = q(t − s) ds , the integral exists for each value of t and φ(t) is an integrable −∞ function. Then by virtue of Definition 2.9 the differential equation (1.1) has the MittagLeffler-Hyers-Ulam-Rassias stability.  Now, we prove the Mittag-Leffler-Hyers-Ulam-Rassias stability of the non-homogeneous linear differential equation (1.2) with the help of Fourier Transforms. Theorem 4.2. The differential equation (1.2) has Mittag-Leffler-Hyers-Ulam-Rassias stability. Proof. Let l be a constant in F. For every  > 0, there exists a positive constant K such that x : (0, ∞) → F is a continuously differentiable function and φ : (0, ∞) → (0, ∞) an integrable function satisfying |x0 (t) + l x(t) − r(t)| ≤ φ(t)Eα (tα )

(4.4)

for all t > 0. We will now prove that, there exist a solution y : (0, ∞) → F, which satisfies the differential equation y 0 (t) + l y(t) = r(t) such that |x(t) − y(t)| ≤ Kφ(t)Eα (tα ), for any t > 0. Let us define a function p : (0, ∞) → F such that p(t) =: x0 (t) + l x(t) − r(t) for each t > 0. In view of (4.4), we have |p(t)| ≤ φ(t)Eα (tα ). Now, taking Fourier transform to p(t), we have F{x(t)} = X(ξ) = Now, let us take Q(ξ) as

{P (ξ) + R(ξ)} (l + iξ) . l2 − ξ 2

(4.5)

1 ; then we have (l − iξ)

1 F{q(t)} = (l − iξ)

⇒

q(t) = F

79

−1



1 (l − iξ)

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We set y(t) = e−lt + (r(t) ∗ q(t)) and taking Fourier transform on both sides, we get Z∞ R(ξ) R(ξ) F{y(t)} = Y (ξ) = e−lt eist dt + = (l − iξ) (l − iξ)

13

(4.6)

−∞

Now, F{y 0 (t) + l y(t)} = F{y 0 (t)} + l F{y(t)} = −iξY (ξ) + l Y (ξ) = R(ξ) Then by using (4.6), we have F{y 0 (t) + l y(t)} = F {r(t)}, since F is one-to-one operator, thus y 0 (t) + l y(t) = r(t). Hence y(t) is a solution of the differential equation (1.2). Then by using (4.5) and (4.6) we can obtain F{x(t) − y(t)} = F{p(t) ∗ q(t)}. Since the operator F is one-to-one and linear, it gives x(t)−y(t) = p(t)∗q(t). Taking modulus on both sides, we have ∞ ∞ Z Z |x(t) − y(t)| = |p(t) ∗ q(t)| = p(t) q(t − s) ds ≤ |p(t)| q(t − s) ds ≤ K φ(t)Eα (tα ). −∞ −∞ R∞ If K = q(t − s) ds the integral exists for each value of t and φ(t) is an integrable function. −∞ Hence by the virtue of Definition 2.10 the differential equation (1.2) has the Mittag-LefflerHyers-Ulam-Rassias stability.  Now, we are going to establish the Mittag-Leffler-Hyers-Ulam-Rassias stability of the second order homogeneous differential equation (1.3). Theorem 4.3. The second order linear differential equation (1.3) has Mittag-LefflerHyers-Ulam-Rassias stability. Proof. Let l, m are constants in F such that there exist µ, ν ∈ F with µν = m, µ+ν = −l and µ 6= ν. For every  > 0, there exists a positive constant K such that x : (0, ∞) → F is a twice continuously differentiable function and φ : (0, ∞) → (0, ∞) an integrable function satisfying the inequality |x00 (t) + l x0 (t) + m x(t)| ≤ φ(t)Eα (tα )

(4.7)

for all t > 0. We will now prove that there exists a solution y : (0, ∞) → F satisfying the homogeneous differential equation (1.3) such that |x(t) − y(t)| ≤ Kφ(t)Eα (tα ), for any t > 0. Let us define a function p : (0, ∞) → F such that p(t) =: x00 (t) + l x0 (t) + m x(t) for each t > 0. In view of (4.7), we have |p(t)| ≤ φ(t)Eα (tα ). Now, taking Fourier transform to p(t), we have P (ξ) = F{x00 (t)} + l F{x0 (t)} + m F{x(t)} = (ξ 2 − iξl + m) X(ξ) X(ξ) =

ξ2

P (ξ) . − iξl + m

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Since l, m be constants in F such that there exist µ, ν ∈ F with µ + ν = −l, µν = m and µ 6= ν, we have (ξ 2 − iξl + m) = (iξ − µ) (iξ − ν). Thus F{x(t)} = X(ξ) =

P (ξ) . (iξ − µ) (iξ − ν)

(4.8)

1 1 , then we have F{q(t)} = and we define (iξ − µ) (iξ − ν) (iξ − µ) (iξ − ν) µe−µt − νe−νt a function y(t) = and taking Fourier transform on both sides, we get µ−ν Choosing Q(ξ) as

Z∞ F{y(t)} = Y (ξ) =

µe−µt − νe−νt ist e dt = 0. µ−ν

(4.9)

−∞

Now, F{y 00 (t) + l y 0 (t) + m y(t)} = (ξ 2 − iξl + m) Y (ξ). Then by using (4.9), we have F{y 00 (t)+l y 0 (t)+m y(t)} = 0, since F is one-to-one operator, thus y 00 (t)+l y 0 (t)+m y(t) = 0, Hence y(t) is a solution of the differential equation (1.3). Then by using (4.8) and (4.9) we can obtain P (ξ) − iξl + m = P (ξ) Q(ξ) = F{p(t)} F{q(t)}

F{x(t)} − F{y(t)} = X(ξ) − Y (ξ) =

⇒

ξ2

F{x(t) − y(t)} = F{p(t) ∗ q(t)}

Since the operator F is one-to-one and linear, which gives x(t) − y(t) = p(t) ∗ q(t). Taking modulus on both sides, we have ∞ Z |x(t) − y(t)| = |p(t) ∗ q(t)| = p(t) q(t − s) ds −∞ ∞ Z ≤ |p(t)| q(t − s) ds ≤ Kφ(t)Eα (tα ). −∞ R∞ q(t − s) ds exists for each value of t and φ(t) is an integrable function. Where K = −∞ Then by the virtue of Definition 2.11 the homogeneous linear differential equation (1.3) has the Mittag-Leffler-Hyers-Ulam-Rassias stability.  Finally, we are going to investigate the Mittag-Leffler-Hyers-Ulam-Rassias stability of the second order non-homogeneous differential equation (1.4). Theorem 4.4. The second order linear differential equation (1.4) has the Mittag-LefflerHyers-Ulam-Rassias stability. Proof. Let l, m be constants in F such that there exist µ, ν ∈ F with µν = m, µ+ν = −l and µ 6= ν. For every  > 0, there exists a positive constant K such that x : (0, ∞) → F is

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a twice continuously differentiable function and φ : (0, ∞) → (0, ∞) an integrable function satisfying the inequality |x00 (t) + l x0 (t) + m x(t) − r(t)| ≤ φ(t)Eα (tα )

(4.10)

for all t > 0. We have to prove that there exists a solution y : (0, ∞) → F satisfying the non-homogeneous differential equation (1.4) such that |x(t) − y(t)| ≤ Kφ(t)Eα (tα ), for any t > 0. Let us define a function p : (0, ∞) → F such that p(t) =: x00 (t) + l x0 (t) + m x(t) − r(t) for each t > 0. In view of (4.10), we have |p(t)| ≤ φ(t)Eα (tα ). Now, taking the Fourier transform to p(t), we have P (ξ) = F{x00 (t)} + l F{x0 (t)} + m F{x(t)} − F{r(t)} = (ξ 2 − iξl + m) X(ξ) − R(ξ) X(ξ) =

P (ξ) + R(ξ) . ξ 2 − iξl + m

Since l, m be constants in F such that there exist µ, ν ∈ F with µ + ν = −l, µν = m and µ 6= ν, we have (ξ 2 − iξl + m) = (iξ − µ) (iξ − ν). Thus F{x(t)} = X(ξ) = Assuming Q(ξ) = F{q(t)} =

P (ξ) + R(ξ) . (iξ − µ) (iξ − ν)

(4.11)

1 and defining a function (iξ − µ) (iξ − ν)

y(t) =

µe−µt − νe−νt + (r(t) ∗ q(t)) µ−ν

and also taking Fourier transform on both sides, we get Z∞ F{y(t)} = Y (ξ) =

µe−µt − νe−νt ist R(ξ) R(ξ) e dt + = . (4.12) µ−ν (iξ − µ) (iξ − ν) (iξ − µ) (iξ − ν)

−∞

Now, F{y 00 (t) + l y 0 (t) + m y(t)} = (ξ 2 − iξl + m) Y (ξ) = R(ξ). Then by using (4.12), we have F{y 00 (t) + l y 0 (t) + m y(t)} = F{r(t)}, since F is one-to-one operator; thus y 00 (t) + l y 0 (t) + m y(t) = r(t). Hence y(t) is a solution of the differential equation (1.4). Then by using (4.11) and (4.12) we can obtain P (ξ) + R(ξ) R(ξ) − (iξ − µ) (iξ − ν) (iξ − µ) (iξ − ν) = P (ξ) Q(ξ) = F{p(t)} F{q(t)}

F{x(t)} − F{y(t)} =

⇒

F{x(t) − y(t)} = F{p(t) ∗ q(t)}

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Since the operator F is one-to-one and linear, which gives x(t) − y(t) = p(t) ∗ q(t). Taking modulus on both sides, we have |x(t) − y(t)| = |p(t) ∗ q(t)| ∞ Z = p(t) q(t − s) ds −∞ ∞ Z ≤ |p(t)| q(t − s) ds ≤ Kφ(t)Eα (tα ). −∞ R∞ Where K = q(t − s) ds , the integral exists for each value of t. Then by the virtue of −∞ Definition 2.12 the non-homogeneous linear differential equation (1.4) has the Mittag-LefflerHyers-Ulam-Rassias stability.  Conclusion: We have proved the Mittag-Leffler-Hyers-Ulam stability and Mittag-LefflerHyers-Ulam-Rassias stability of the linear differential equations of first order and second order with constant co-efficients using the Fourier Transforms method. That is, we established the sufficient criteria for Mittag-Leffler-Hyers-Ulam stability and Mittag-Leffler-Hyers-UlamRassias stability of the linear differential equation of first order and second order with constant co-efficients using Fourier Transforms method. Additionally, this paper also provides another method to study the Mittag-Leffler-Hyers-Ulam stability of differential equations. Also, this paper shows that the Fourier Transform method is more convenient to study the Mittag-Leffler-Hyers-Ulam stability and Mittag-Leffler-Hyers-Ulam-Rassias stability of the linear differential equation with constant co-efficients. References [1] M.R. Abdollahpoura, R. Aghayaria, M.Th. Rassias, Hyers-Ulam stability of associated Laguerre Differential equations in a subclass of analytic functions, Journal of Mathematical Analysis and Applications, 437 (2016), 605-612. [2] Abbas Najati, Jung Rye Lee, Choonkil Park, and Themistocles M. Rassias, On the stability of a Cauchy type functional equation, Demonstr. Math. 51 (2018) 323-331. [3] Alsina, C., Ger, R.: On Some inequalities and stability results related to the exponential function. Journal of Inequalities Appl. 2, 373–380 (1998) [4] Aoki, T.: On the stability of the linear transformation in Banach Spaces. J. Math. Soc. Japan, 2, 64–66 (1950) [5] Alqifiary, Q.H., Jung, S.M.: Laplace Transform And Generalized Hyers-Ulam stability of Differential equations. Elec., J. Diff., Equations, 2014 (80), 1–11 (2014) [6] Bourgin, D.G.: Classes of transformations and bordering transformations. Bull. Amer. Math. Soc. 57, 223–237 (1951) [7] I. Fakunle, P.O. Arawomo, Hyers-Ulam stability of certain class of Nonlinear second order differential equations, Global Journal of Pure and Applied Mathematics, 14 (8) (2018) 1029-1039. [8] P. Gavruta, S. M. Jung and Y. Li, Hyers - Ulam Stability for Second order linear differential equations with boundary conditions, Elec. J. of Diff. Equations, Vol. 2011 (2011), No. 80, pp. 1-5. [9] Hyers, D.H.: On the Stability of a Linear functional equation. Proc. Natl. Acad. Sci. USA, 27, 222–224 (1941)

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[10] Iz-iddine EL-Fassi, Choonkil Park, and Gwang Hui Kim, Stability and hyperstability of a quadratic functional equation and a characterization of inner product spaces, Demonstr. Math. 51 (2018) 295-303. [11] Jung, S.M.: Hyers-Ulam stability of linear differential equation of first order. Appl. Math. Lett. 17, 1135–1140 (2004) [12] Jung, S.M.: Hyers-Ulam stability of linear differential equations of first order (III). J. Math. Anal. Appl. 311, 139–146 (2005) [13] Jung, S.M.: Hyers-Ulam stability of linear differential equations of first order (II). Appl. Math. Lett. 19, 854–858 (2006) [14] Jung, S.M.: Hyers-Ulam stability of a system of first order linear differential equations with constant coefficients. J. Math. Anal. Appl. 320, 549–561 (2006) [15] S. M. Jung, Approximate solution of a Linear Differential Equation of Third Order, Bull. of the Malaysian Math. Sciences Soc. (2) 35 (4), 2012, 1063-1073. [16] Y. Li, Y. Shen, Hyers-Ulam stability of linear differential equations of second order, Appl. Math. Lett. 23 (2010), 306-309. [17] Miura, T.: On the Hyers-Ulam stability of a differentiable map. Sci. Math. Japan, 55, 17–24 (2002) [18] Miura, T., Jung, S.M., Takahasi, S.E.: Hyers-Ulam-Rassias stability of the Banach space valued linear differential equation y 0 = λy. J. Korean Math. Soc. 41, 995–1005 (2004) [19] Miura, T., Miyajima, S., Takahasi, S.E.: A Characterization of Hyers-Ulam Stability of first order linear differential operators. J. Math. Anal. Appl. 286, 136–146 (2003) [20] Miura, T., Takahasi, S.E., Choda, H.: On the Hyers-Ulam stability of real continuous function valued differentiable map. Tokyo J. Math. 24, 467–476 (2001) [21] Nazarianpoor, M., Rassias, J.M., Sadeghi, GH.: Solution and stability of Quattuorvigintic Functional equation in Intuitionistic Fuzzy Normed spaces. Iranian Journal of Fuzzy system, 15 (4), 13–30 (2018) [22] R. Murali and A. Ponmana Selvan, On the Generalized Hyers-Ulam Stability of Linear Ordinary Differential Equations of Higher Order, International Journal of Pure and Applied Mathematics, 117 (12) (2017) 317-326. [23] R. Murali and A. Ponmana Selvan, Hyers-Ulam-Rassias Stability for the Linear Ordinary Differential Equation of Third order, Kragujevac Journal of Mathematics, 42 (4) (2018) 579-590. [24] R. Murali and A. Ponmana Selvan, Hyers-Ulam Stability of Linear Differential Equation, Computational Intelligence, Cyber Security and Computational Models, Models and Techniques for Intelligent Systems and Automation, Communications in Computer and Information Science, 844. Springer, Singapore (2018), 183-192. [25] Obloza, M.: Hyers stability of the linear differential equation. Rockznik Nauk-Dydakt. Prace Math. 13, 259–270 (1993) [26] Obloza, M.: Connection between Hyers and Lyapunov stability of the ordinary differential equations. Rockznik Nauk-Dydakt. Prace Math. 14, 141–146 (1997) [27] M. N. Qarawani, Hyers-Ulam stability of Linear and Nonlinear differential equation of second order, Int. Journal of Applied Mathematical Research, 1 (4), 2012, 422-432. [28] M. N. Qarawani, Hyers-Ulam stability of a Generalized second order Nonlinear Differential equation, Applied Mathematics, 2012, 3, pp. 1857-1861. [29] Rassias, J.M.: On approximation of approximately linear mappings by linear mappings. J. Funct., Anal. 46, 126–130 (1982) [30] Eshaghi Gordji, M., Javadian, A., Rassias, J.M.: Stability of systems of bi-quadratic and additive-cubic functional equations in Frechet’s spaces. Functional Analysis, Approximation and Computation, 4 (1), 85–93 (2012) [31] Rassias, J.M., Murali, R., Rassias, M.J., Antony Raj, A.: General Solution, stability and Non-stability of Quattuorvigintic functional equation in Multi-Banach spaces. Int. J. Math. And Appl. 5, 181–194 (2017) [32] Ravi, K., Rassias, J.M., Senthil Kumar, B.V.: Ulam-Hyers stability of undecic functional equation in quasi-beta-normed spaces fixed point method. Tbilisi Mathematical Science, 9 (2), 83–103 (2016) [33] Ravi, K., Rassias, J.M., Pinelas, S., Suresh, S.: General solution and stability of Quattuordecic functional equation in quasi-beta-normed spaces. Advances in pure mathematics, 6, 921–941 (2016) doi: 10.4236/apm.2016.612070

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[34] Xu, T.Z., Rassias, J.M., Xu, W.X.: A fixed point approach to the stability of a general mixed additivecubic functional equation in quasi fuzzy normed spaces. International Journal of the Physical Sciences, 6(2), 313–324 (2011) [35] Ravi, K., Rassias, J.M., Senthil Kumar, B.V.: Generalized Ulam-Hyers stability of the Harmonic Mean functional equation in two variables. International Journal of Analysis and Applications, 1 (1), 1–17 (2013) [36] Rassias, Th.M.: On the stability of the linear mappings in Banach Spaces. Proc. Amer. Math. Soc. 72, 297–300 (1978) [37] Rus, I.A.: Ulam Stabilities of Ordinary Differential Equations in a Banach Space. Carpathian J. Math. 26 (1), 103–107 (2010) [38] K. Ravi, R. Murali and A. Ponmana Selvan, Hyers - Ulam stability of nth order linear differential equation with initial and boundary condition, Asian Journal of Mathematics and Computer Research, 11 (3), 2016, 201-207. [39] Takahasi, S.E., Miura, T., Miyajima, S.: On the Hyers-Ulam stability of the Banach space-valued differential equation y 0 = αy. Bulletin Korean Math. Soc. 39, 309-315 (2002) [40] Ulam, S.M.: Problem in Modern Mathematics, Chapter IV, Science Editors, Willey, New York (1960) [41] Vida Kalvandi, Nasrin Eghbali and John Micheal Rassias, Mittag-Leffler-Hyers-Ulam stability of fractional differential equations of second order, J. Math. Extension, 13 (1) (2019) 1 - 15. [42] Wang, G., Zhou, M., Sun, L.: Hyers-Ulam stability of linear differential equations of first order. Appl. Math. Lett. 21, 1024–1028 (2008) [43] J. Xue, Hyers - Ulam stability of linear differential equations of second order with constant coefficient, Italian Journal of Pure and Applied Mathematics, No. 32, (2014) 419-424. Pedagogical Department E.E., Section of Mathematics and Informatics, National and Capodistrain University of Athens, Athens 15342, Greece. E-mail address: [email protected], [email protected] PG and Research Department of Mathematics, Sacred Heart College (Autonomous), Tirupattur - 635 601, Vellore Dt., Tamil Nadu, India E-mail address: [email protected] PG and Research Department of Mathematics, Sacred Heart College (Autonomous), Tirupattur - 635 601, Vellore Dt., Tamil Nadu, India E-mail address: [email protected]

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On Some Systems of Three Nonlinear Difference Equations E. M. Elsayed1,2 and Hanan S. Gafel1,3 1 King AbdulAziz University, Faculty of Science, Mathematics Department, P. O. Box 80203, Jeddah 21589, Saudi Arabia. 2 Mathematics Department, Faculty of Science, Mansoura University, Mansoura 35516, Egypt. 3 Mathematics Department, Faculty of Science, Taif University-K.S.A. E-mail: [email protected], [email protected]. Abstract We consider in this paper, the solution of the following systems of difference equation: xn+1 =

xn−2 yn−2 zn−2 , yn+1 = , zn+1 = ±1 + xn−2 yn−1 zn ±1 + yn−2 zn−1 xn ±1 + zn−2 xn−1 yn

where the initial conditions x−2 , x−1 , x0 , y−2 , y−1 , y0 , z−2 , z−1 , z0 are arbitrary non zero real numbers.

Keywords: difference equations, recursive sequences, periodic solutions, system of difference equations, stability. Mathematics Subject Classification: 39A10. ––––––––––––––––––––––

1

Introduction

Difference equations related to differential equations as discrete mathematics related to continuous mathematics. Most of these models are described by nonlinear delay difference equations; see, for example, [9], [10]. The subject of the qualitative study of the nonlinear delay population models is very extensive, and the current research work tends to center around the relevant global dynamics of the considered systems of difference equations such as oscillation, boundedness of solutions, persistence, global 1

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stability of positive steady sates, permanence, and global existence of periodic solutions. See [13], [17], [19]-[22], [26], [28], [29] and the references therein. In particular, Agarwal and Elsayed [1] deal with the global stability, periodicity character and gave the solution form of some special cases of the recursive sequence xn+1 = axn +

bxn xn−3 . cxn−2 + dxn−3

Camouzis et al. [5] studied the global character of solutions of the difference equation xn+1 =

δxn−2 + xn−3 . A + xn−3

Clark and Kulenovic [7] investigated the global asymptotic stability of the system xn+1 =

xn , a + cyn

yn+1 =

yn . b + dxn

In [9], Din studied the boundedness character, steady-states, local asymptotic stability of equilibrium points, and global behavior of the unique positive equilibrium point of a discrete predator-prey model given by xn+1 =

αxn − βxn yn , 1 + γxn

yn+1 =

δxn yn . xn + ηyn

Elsayed et al. [23] discussed the global convergence and periodicity of solutions of the recursive sequence xn+1 = axn +

b + cxn−1 . d + exn−1

Elsayed and El-Metwally [24] discussed the periodic nature and the form of the solutions of the nonlinear difference equations systems xn+1 =

xn yn−2 , yn−1 (±1 ± xn yn−2 )

yn+1 =

yn xn−2 . xn−1 (±1 ± yn xn−2 )

Gelisken and Kara [25] studied some behavior of solutions of some systems of rational difference equations of higher order and they showed that every solution is periodic with a period depends on the order. In [27] Kurbanli discussed a three-dimensional system of rational difference equations xn−1 yn−1 xn xn+1 = , yn+1 = , zn+1 = . xn−1 yn − 1 yn−1 xn − 1 zn−1 yn

Touafek et al. [33] studied the sufficient conditions for the global asymptotic stability of the following systems of rational difference equations: xn+1 =

xn−3 , ±1 ± xn−3 yn−1

yn+1 =

yn−3 . ±1 ± yn−3 xn−1

2

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with a real number’s initial conditions. Our goal in this paper is to investigate the form of the solutions of the system of three difference equations xn+1 =

xn−2 yn−2 zn−2 , yn+1 = , zn+1 = , (1) ±1 + xn−2 yn−1 zn ±1 + yn−2 zn−1 xn ±1 + zn−2 xn−1 yn

where the initial conditions x−2 , x−1 , x0 , y−2 , y−1 , y0 , z−2 , z−1 , z0 are arbitrary real numbers. Moreover, we obtain some numerical simulation to the equation are given to illustrate our results.

2

The System xn+1 =

xn−2 1+xn−2 yn−1 zn ,

yn+1 =

yn−2 1+yn−2 zn−1 xn ,

zn+1 =

zn−2 1+zn−2 xn−1 yn

In this section, we study the solution of the following system of difference equations. xn+1 =

xn−2 , 1 + xn−2 yn−1 zn

yn+1 =

yn−2 zn−2 , zn+1 = , 1 + yn−2 zn−1 xn 1 + zn−2 xn−1 yn

(2)

where n ∈ N0 and the initial conditions are arbitrary real numbers. The following theorem is devoted to the form of the solutions of system (1). Theorem 1. Suppose that {xn , yn , zn } are solutions of the system (1). Then for n = 0, 1, 2, ..., we have the following formulas n−1 Q (1 + (3i + 1)x−1 y0 z−2 ) (1 + (3i)x−2 y−1 z0 ) , x3n−1 = x−1 , i=0 (1 + (3i + 1)x−2 y−1 z0 ) i=0 (1 + (3i + 2)x−1 y0 z−2 ) n−1 Q (1 + (3i + 2)x0 y−2 z−1 ) , = x0 i=0 (1 + (3i + 3)x0 y−2 z−1 )

x3n−2 = x−2 x3n

n−1 Q (1 + (3i + 1)x−2 y−1 z0 ) (1 + (3i)x0 y−2 z−1 ) , y3n−1 = y−1 , i=0 (1 + (3i + 1)x0 y−2 z−1 ) i=0 (1 + (3i + 2)x−2 y−1 z0 ) n−1 Q (1 + (3i + 2)x−1 y0 z−2 ) , = y0 i=0 (1 + (3i + 3)x−1 y0 z−2 )

y3n−2 = y−2 y3n

n−1 Q

n−1 Q (1 + (3i + 1)x0 y−2 z−1 ) (1 + (3i)x−1 y0 z−2 ) , z3n−1 = z−1 , i=0 (1 + (3i + 1)x−1 y0 z−2 ) i=0 (1 + (3i + 2)x0 y−2 z−1 ) n−1 Q (1 + (3i + 2)x−2 y−1 z0 ) , = z0 i=0 (1 + (3i + 3)x−2 y−1 z0 )

z3n−2 = z−2 z3n

n−1 Q

n−1 Q

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Proof. For n = 0 the result holds. Suppose that the result holds for n − 1. n−2 Q (1 + (3i + 1)x−1 y0 z−2 ) (1 + (3i)x−2 y−1 z0 ) , x3n−4 = x−1 , i=0 (1 + (3i + 1)x−2 y−1 z0 ) i=0 (1 + (3i + 2)x−1 y0 z−2 ) n−2 Q (1 + (3i + 2)x0 y−2 z−1 ) , = x0 i=0 (1 + (3i + 3)x0 y−2 z−1 )

x3n−5 = x−2 x3n−3

n−2 Q (1 + (3i + 1)x−2 y−1 z0 ) (1 + (3i)x0 y−2 z−1 ) , y3n−4 = y−1 , i=0 (1 + (3i + 1)x0 y−2 z−1 ) i=0 (1 + (3i + 2)x−2 y−1 z0 ) n−2 Q (1 + (3i + 2)x−1 y0 z−2 ) , = y0 i=0 (1 + (3i + 3)x−1 y0 z−2 )

y3n−5 = y−2 y3n−3

n−2 Q

n−2 Q (1 + (3i + 1)x0 y−2 z−1 ) (1 + (3i)x−1 y0 z−2 ) , z3n−4 = z−1 , i=0 (1 + (3i + 1)x−1 y0 z−2 ) i=0 (1 + (3i + 2)x0 y−2 z−1 ) n−2 Q (1 + (3i + 2)x−2 y−1 z0 ) . = z0 i=0 (1 + (3i + 3)x−2 y−1 z0 )

z3n−5 = z−2 z3n−3

n−2 Q

n−2 Q

It follows from Eq.(1) that x3n−2 =

x3n−5 1 + x3n−5 y3n−4 z3n−3 x−2

= 1 + (x−2

n−2 Q i=0

1 + x−2 y−1 z0 x−2

=

i=0

1 + x−2 y−1 z0 = x−2

n−2 Q i=0

n−2 Q

x−2 n−2 Q i=0

n−2 Q

(1+(3i)x−2 y−1 z0 ) (1+(3i+1)x−2 y−1 z0 )

i=0 n−2 Q

(1+(3i)x−2 y−1 z0 ) )(y−1 (1+(3i+1)x−2 y−1 z0 )

= n−2 Q

n−2 Q

i=0

i=0

(1+(3i+1)x−2 y−1 z0 ) )(z0 (1+(3i+2)x−2 y−1 z0 )

(1+(3i)x−2 y−1 z0 ) (1+(3i+1)x−2 y−1 z0 )

n−2 Q i=0

(1+(3i+2)x−2 y−1 z0 ) ) (1+(3i+3)x−2 y−1 z0 )

(1+(3i)x−2 y−1 z0 ) −2 y−1 z0 ) (( (1+(3i+1)x )( (1+(3i+1)x )( (1+(3i+2)x−2 y−1 z0 ) )) (1+(3i+2)x−2 y−1 z0 ) (1+(3i+3)x−2 y−1 z0 ) −2 y−1 z0 )

(1+(3i)x−2 y−1 z0 ) (1+(3i+1)x−2 y−1 z0 ) n−2 Q i=0

(1+(3i)x−2 y−1 z0 ) ( (1+(3i+3)x ) −2 y−1 z0 )

(1 + (3i)x−2 y−1 z0 ) 1 x−2 y−1 z0 (1 + (3i + 1)x−2 y−1 z0 ) 1 + ( (1+(3n−3)x−2 y−1 z0 ) )

(1 + (3n − 3)x−2 y−1 z0 ) (1 + (3i)x−2 y−1 z0 ) ( ) i=0 (1 + (3i + 1)x−2 y−1 z0 ) (1 + (3n − 3)x−2 y−1 z0 ) + x−2 y−1 z0 n−2 Q (1 + (3i)x−2 y−1 z0 ) (1 + (3n − 3)x−2 y−1 z0 ) ( ). = x−2 i=0 (1 + (3i + 1)x−2 y−1 z0 ) (1 + (3n − 2)x−2 y−1 z0 ) = x−2

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Then, we see that x3n−2 = x−2 Also, we see from Eq.(1) that y3n−2 =

n−1 Q i=0

(1 + (3i)x−2 y−1 z0 ) . (1 + (3i + 1)x−2 y−1 z0 )

y3n−5 1 + y3n−5 z3n−4 x3n−3 y−2

= 1 + (y−2 y−2

=

n−2 Q

i=0 n−2 Q i=0

(1+(3i)x0 y−2 z−1 ) (1+(3i+1)x0 y−2 z−1 )

= y−2

i=0

n−2 Q

n−2 Q i=0

(1+(3i)x0 y−2 z−1 ) (1+(3i+1)x0 y−2 z−1 )

i=0 n−2 Q

(1+(3i)x0 y−2 z−1 ) )(z−1 (1+(3i+1)x0 y−2 z−1 )

1 + x0 y−2 z−1 n−2 Q

n−2 Q

i=0

(1+(3i+1)x0 y−2 z−1 ) )(x0 (1+(3i+2)x0 y−2 z−1 )

n−2 Q i=0

(1+(3i+2)x0 y−2 z−1 ) ) (1+(3i+3)x0 y−2 z−1 )

(1+(3i)x0 y−2 z−1 ) (1+(3i+3)x0 y−2 z−1 )

(1 + (3i)x0 y−2 z−1 ) ( (1 + (3i + 1)x0 y−2 z−1 ) 1 +

1

) x0 y−2 z−1 1+(3n−3)x0 y−2 z−1 −

1 + (3n − 3)x0 y−2 z−1 (1 + (3i)x0 y−2 z−1 ) ( ) i=0 (1 + (3i + 1)x0 y−2 z−1 ) 1 + (3n − 3)x0 y−2 z−1 + x0 y−2 z−1 n−2 Q (1 + (3i)x0 y−2 z−1 ) 1 + (3n − 3)x0 y−2 z−1 ( = y−2 ). i=0 (1 + (3i + 1)x0 y−2 z−1 ) 1 + (3n − 2)x0 y−2 z−1 = y−2

Then, we see that y3n−2 = y−2 Finally, we see that

n−1 Q i=0

(1 + (3i)x0 y−2 z−1 ) (1 + (3i + 1)x0 y−2 z−1 )

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z3n−2 =

z3n−5 1 + z3n−5 x3n−4 y3n−3 z−2

= 1 + (z−2 z−2

=

n−2 Q

i=0 n−2 Q i=0

(1+(3i)x−1 y0 z−2 ) (1+(3i+1)x−1 y0 z−2 )

= z−2 = z−2 Then

i=0

n−2 Q i=0

n−2 Q i=0

i=0

(1+(3i+1)x−1 y0 z−2 ) )(y0 (1+(3i+2)x−1 y0 z−2 )

n−2 Q i=0

(1+(3i+2)x−1 y0 z−2 ) ) (1+(3i+3)x−1 y0 z−2 )

(1+(3i)x−1 y0 z−2 ) (1+(3i+3)x−1 y0 z−2

(1 + (3i)x−1 y0 z−2 ) ( (1 + (3i + 1)x−1 y0 z−2 ) 1 +

1 x−1 y0 z−2 1+(3n−3)x−1 y0 z−2

)

(1 + (3i)x−1 y0 z−2 ) 1 + (3n − 3)x−1 y0 z−2 ( ). (1 + (3i + 1)x−1 y0 z−2 ) 1 + (3n − 2)x−1 y0 z−2 z3n−2 = z−2

This completes the proof.

3

(1+(3i)x−1 y0 z−2 ) (1+(3i+1)x−1 y0 z−2 )

i=0 n−2 Q

(1+(3i)x−1 y0 z−2 ) )(x−1 (1+(3i+1)x−1 y0 z−2 )

1 + x−1 y0 z−2 n−2 Q

n−2 Q

n−1 Q i=0

(1 + (3i)x−1 y0 z−2 ) . (1 + (3i + 1)x−1 y0 z−2 )

The System xn+1 =

xn−2 1+xn−2 yn−1 zn , yn+1

=

yn−2 −1+yn−2 zn−1 xn , zn+1

=

zn−2 −1+zn−2 xn−1 yn

In this section, we obtain the form of the solutions of the system of three difference equations xn−2 yn−2 zn−2 xn+1 = , yn+1 = , zn+1 = , (3) 1 + xn−2 yn−1 zn −1 + yn−2 zn−1 xn −1 + zn−2 xn−1 yn

where n ∈ N0 and the initial conditions are arbitrary nonzero real numbers. Theorem 2. Suppose that {xn , yn , zn } are solutions of the system (2). Then for n = 0, 1, 2, ..., we have the following formulas x3n−2 =

x−2 x−1 (x−1 y0 z−2 − 1) x0 , x3n = , x3n−1 = , 1 + nx−2 y−1 z0 (n + 1)x−1 y0 z−2 − 1 1 + nx0 y−2 z−1

(−1)n+1 y−2 (1 + (n − 1)x0 y−2 z−1 ) , y3n−1 = (−1)n y−1 (1 + nx−2 y−1 z0 ), x0 y−2 z−1 − 1 n (−1) y0 ((n + 1)x−1 y0 z−2 − 1) , = x−1 y0 z−2 − 1

y3n−2 = y3n

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(−1)n+1 z−2 (−1)n+1 z−1 (x0 y−2 z−1 − 1) (−1)n z0 , z3n−1 = , z3n = . nx−1 y0 z−2 − 1 (n − 1)x0 y−2 z−1 + 1 1 + nx−2 y−1 z0

z3n−2 =

Proof. For n = 0 the result holds. Suppose that the result holds for n − 1. x3n−5 =

x−2 x−1 (x−1 y0 z−2 − 1) x0 , x3n−3 = , x3n−4 = , 1 + (n − 1)x−2 y−1 z0 nx−1 y0 z−2 − 1 1 + (n − 1)x0 y−2 z−1

(−1)n y−2 (1 + (n − 2)x0 y−2 z−1 ) , y3n−4 = (−1)n−1 y−1 (1 + (n − 1)x−2 y−1 z0 ), x0 y−2 z−1 − 1 (−1)n−1 y0 (nx−1 y0 z−2 − 1) , = x−1 y0 z−2 − 1

y3n−5 = y3n−3

z3n−5 =

(−1)n z−2 (−1)n z−1 (x0 y−2 z−1 − 1) (−1)n−1 z0 , z3n−4 = , z3n−3 = , (n − 1)x−1 y0 z−2 − 1 (n − 2)x0 y−2 z−1 + 1 1 + (n − 1)x−2 y−1 z0

from system (2) we can prove as follow x3n−2 =

x3n−5 1 + x3n−5 y3n−4 z3n−3 x−2 1+(n−1)x−2 y−1 z0

=

n−1

(−1) z0 x−2 1 + ( 1+(n−1)x )((−1)n−1 y−1 (1 + (n − 1)x−2 y−1 z0 ))( 1+(n−1)x ) −2 y−1 z0 −2 y−1 z0 x−2 x−2 = = 1 + (n − 1)x−2 y−1 z0 + x−2 y−1 z0 1 + nx−2 y−1 z0

Also, we get y3n−1 = = =

y3n−4 −1 + y3n−4 z3n−3 x3n−2 (−1)n−1 y−1 (1 + (n − 1)x−2 y−1 z0 ) n−1

(−1) z0 −2 −1 + ((−1)n−1 y−1 (1 + (n − 1)x−2 y−1 z0 ))( 1+(n−1)x )( 1+nxx−2 ) y−1 z0 −2 y−1 z0

(−1)n y−1 (1 + (n − 1)x−2 y−1 z0 )(1 + nx−2 y−1 z0 ) = (−1)n y−1 (1 + nx−2 y−1 z0 ) 1 + (n − 1)x−2 y−1 z0

z3n = = =

z3n−3 −1 + z3n−3 x3n−2 y3n−1 −1 +

(−1)n−1 z0 1+(n−1)x−2 y−1 z0 (−1)n−1 z0 −2 ( 1+(n−1)x )( 1+nxx−2 )((−1)n y−1 (1 y−1 z0 −2 y−1 z0 (−1)n z0 (−1)n z0

1 + (n − 1)x−2 y−1 z0 + x−2 y−1 z0

=

+ nx−2 y−1 z0 ))

1 + nx−2 y−1 z0

.

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4

The System xn+1 =

xn−2 −1+xn−2 yn−1 zn ,

yn+1 =

yn−2 1+yn−2 zn−1 xn ,

zn+1 =

zn−2 −1+zn−2 xn−1 yn

In this section, we study the solution of the following system of difference equations xn−2 yn−2 zn−2 , yn+1 = , zn+1 = , (4) xn+1 = −1 + xn−2 yn−1 zn 1 + yn−2 zn−1 xn −1 + zn−2 xn−1 yn where n ∈ N0 and the initial conditions are arbitrary nonzero real numbers. Theorem 3. Suppose that {xn , yn , zn } are solutions of the system (3). Then for n = 0, 1, 2, ..., we have the following formulas x3n−2 =

x−2 (−1)n+1 x−1 (x−1 y0 z−2 − 1) (−1)n x0 , x3n−1 = , x3n = , nx−2 y−1 z0 − 1 (n − 1)x−1 y0 z−2 + 1 1 + nx0 y−2 z−1

y3n−2 =

y−2 y−1 (x−2 y−1 z0 − 1) y0 , y3n−1 = , y3n = , nx0 y−2 z−1 + 1 (n + 1)x−2 y−1 z0 − 1 nx−1 y0 z−2 + 1

(−1)n+1 z−2 ((n − 1)x−1 y0 z−2 + 1) , z3n−1 = (−1)n z−1 (nx0 y−2 z−1 + 1), x−1 y0 z−2 − 1 (−1)n z0 ((n + 1)x−2 y−1 z0 − 1) . z3n = x−2 y−1 z0 − 1 Proof. For n = 0 the result holds. Suppose that the result holds for n − 1 z3n−2 =

x3n−5 = y3n−5 =

x−2 (−1)n x−1 (x−1 y0 z−2 − 1) (−1)n−1 x0 , x3n−4 = , x3n−3 = , (n − 1)x−2 y−1 z0 − 1 (n − 2)x−1 y0 z−2 + 1 1 + (n − 1)x0 y−2 z−1

y−2 y−1 (x−2 y−1 z0 − 1) y0 , y3n−4 = , y3n−3 = , (n − 1)x0 y−2 z−1 + 1 nx−2 y−1 z0 − 1 (n − 1)x−1 y0 z−2 + 1

(−1)n z−2 ((n − 2)x−1 y0 z−2 + 1) , z3n−4 = (−1)n−1 z−1 ((n − 1)x0 y−2 z−1 + 1), x−1 y0 z−2 − 1 (−1)n−1 z0 (nx−2 y−1 z0 − 1) , z3n−3 = x−2 y−1 z0 − 1 from system (3) we can prove as follow x3n−4 x3n−1 = −1 + x3n−4 y3n−3 z3n−2

z3n−5 =

= = =

(−1)n x−1 (x−1 y0 z−2 −1) (n−2)x−1 y0 z−2 +1 n+1 (−1)n x−1 (x−1 y0 z−2 −1) ((n−1)x−1 y0 z−2 +1) −1 + ( (n−2)x−1 y0 z−2 +1 )( (n−1)x−1y0y0 z−2 +1 )( (−1) z−2 ) x−1 y0 z−2 −1 n (−1) x−1 (x−1 y0 z−2 − 1) −((n − 2)x−1 y0 z−2 + 1) + ((−1)n x−1 )((−1)n+1 y0 z−2 ) (−1)n+1 x−1 (x−1 y0 z−2 − 1)

(n − 1)x−1 y0 z−2 + 1

.

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Also, we get y3n−3 1 + y3n−3 z3n−2 x3n−1

y3n = =

1+

y0 (n−1)x−1 y0 z−2 +1 n+1 ((n−1)x−1 y0 z−2 +1) (−1)n+1 x−1 (x−1 y0 z−2 −1) ( (n−1)x−1y0y0 z−2 +1 )( (−1) z−2 )( (n−1)x−1 y0 z−2 +1 ) x−1 y0 z−2 −1

y0 (n − 1)x−1 y0 z−2 + 1 + y0 ((−1)n+1 z−2 )((−1)n+1 x−1 ) y0 = nx−1 y0 z−2 + 1

=

z3n−2 = =

= =

z3n−5 −1 + z3n−5 x3n−4 y3n−3

(−1)n z−2 ((n−2)x−1 y0 z−2 +1) x−1 y0 z−2 −1 (−1)n z−2 ((n−2)x−1 y0 z−2 +1) (−1)n x−1 (x−1 y0 z−2 −1) −1 + ( )( (n−2)x−1 y0 z−2 +1 )( (n−1)x−1y0y0 z−2 +1 ) x−1 y0 z−2 −1 (−1)n z−2 ((n−2)x−1 y0 z−2 +1) x−1 y0 z−2 −1 −((n−2)x−1 y0 z−2 +1) (n−1)x−1 y0 z−2 +1 (−1)n+1 z−2 ((n − 1)x−1 y0 z−2 + 1)

.

x−1 y0 z−2 − 1

This completes the proof.

5

The System xn+1 =

xn−2 −1+xn−2 yn−1 zn ,

yn+1 =

yn−2 −1+yn−2 zn−1 xn ,

zn+1 =

zn−2 1+zn−2 xn−1 yn

In this section, we investigate the solution of the following system of difference equations xn−2 yn−2 zn−2 xn+1 = , yn+1 = , zn+1 = , (5) −1 + xn−2 yn−1 zn −1 + yn−2 zn−1 xn 1 + zn−2 xn−1 yn where the initial conditions n ∈ N0 are arbitrary non zero real numbers. The following theorem is devoted to the form of the solutions of system (4). Theorem 4. Suppose that {xn , yn , zn } are solutions of the system (4). Then for n = 0, 1, 2, ..., we have the following formulas (−1)n+1 x−2 ((n − 1)x−2 y−1 z0 + 1) , x3n−1 = (−1)n x−1 (nx−1 y0 z−2 + 1), x−2 y−1 z0 − 1 n (−1) x0 ((n + 1)x0 y−2 z−1 − 1) , = x0 y−2 z−1 − 1

x3n−2 = x3n

9

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(−1)n+1 y−2 (−1)n+1 y−1 (x−2 y−1 z0 − 1) (−1)n y0 , y3n−1 = , y3n = , nx0 y−2 z−1 − 1 (n − 1)x−2 y−1 z0 + 1 nx−1 y0 z−2 + 1 z−2 z−1 (x0 y−2 z−1 − 1) z0 z3n−2 = , z3n−1 = , z3n = . nx−1 y0 z−2 + 1 (n + 1)x0 y−2 z−1 − 1 nx−2 y−1 z0 + 1 Proof. For n = 0 the result holds. Suppose that the result holds for n − 1 y3n−2 =

(−1)n x−2 ((n − 2)x−2 y−1 z0 + 1) , x3n−4 = (−1)n−1 x−1 ((n − 1)x−1 y0 z−2 + 1), x−2 y−1 z0 − 1 (−1)n−1 x0 (x0 y−2 z−1 − 1) , = x0 y−2 z−1 − 1

x3n−5 = x3n−3

(−1)n y−2 (−1)n y−1 (x−2 y−1 z0 − 1) (−1)n−1 y0 , y3n−4 = , y3n−3 = , (n − 1)x0 y−2 z−1 − 1 (n − 2)x−2 y−1 z0 + 1 (n − 1)x−1 y0 z−2 + 1 z−2 z−1 (x0 y−2 z−1 − 1) z0 z3n−5 = , z3n−4 = , z3n−3 = , (n − 1)x−1 y0 z−2 + 1 nx0 y−2 z−1 − 1 (n − 1)x−2 y−1 z0 + 1 from system (4) we can prove as follow x3n−3 x3n = −1 + x3n−3 y3n−2 z3n−1 y3n−5 =

=

=

−1 +

(−1)n−1 x0 (nx0 y−2 z−1 −1) x0 y−2 z−1 −1 (−1)n−1 x0 (nx0 y−2 z−1 −1) (−1)n+1 y−2 z−1 (x0 y−2 z−1 −1) ( )( nx )( (n+1)x ) x0 y−2 z−1 −1 0 y−2 z−1 −1 0 y−2 z−1 −1

(−1)n−1 x0 (nx0 y−2 z−1 −1) x0 y−2 z−1 −1 x0 y−2 z−1 −((n+1)x0 y−2 z−1 −1) ((n+1)x0 y−2 z−1 −1)

=

(−1)n x0 ((n + 1)x0 y−2 z−1 − 1) x0 y−2 z−1 − 1

Also, we get y3n−1 = =

=

z3n−2 = = =

y3n−4 −1 + y3n−4 z3n−3 x3n−2 −1 +

(−1)n y−1 (x−2 y−1 z0 −1) (n−2)x−2 y−1 z0 +1 n+1 (−1)n y−1 (x−2 y−1 z0 −1) ((n−1)x−2 y−1 z0 +1) ( (n−2)x−2 y−1 z0 +1 )( (n−1)x−2z0y−1 z0 +1 )( (−1) x−2 ) x−2 y−1 z0 −1

(−1)n+1 y−1 (x−2 y−1 z0 −1) (n−2)x−2 y−1 z0 +1 (n−2)x−2 y−1 z0 +1+x−2 y−1 z0 (n−2)x−2 y−1 z0 +1

=

(−1)n+1 y−1 (x−2 y−1 z0 − 1) (n − 1)x−2 y−1 z0 + 1

z3n−5 1 + z3n−5 x3n−4 y3n−3 z−2 (n−1)x−1 y0 z−2 +1

n−1

(−1) y0 z−2 1 + ( (n−1)x−1 )((−1)n−1 x−1 ((n − 1)x−1 y0 z−2 + 1))( (n−1)x ) y0 z−2 +1 −1 y0 z−2 +1 z−2 (n−1)x−1 y0 z−2 +1 (n−1)x−1 y0 z−2 +1+x−1 y0 z−2 (n−1)x−1 y0 z−2 +1

=

z−2 nx−1 y0 z−2 + 1 10

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This completes the proof. The following cases can be proved similarly.

6

On The System xn+1 =

xn−2 −1+xn−2 yn−1 zn ,

yn+1 =

yn−2 −1+yn−2 zn−1 xn ,

zn+1 =

zn−2 −1+zn−2 xn−1 yn

In this section we study the solution of the following system of difference equations xn+1 =

xn−2 yn−2 zn−2 , yn+1 = , zn+1 = , (6) −1 + xn−2 yn−1 zn −1 + yn−2 zn−1 xn −1 + zn−2 xn−1 yn

where the initial conditions n ∈ N0 are arbitrary non zero real numbers. Theorem 5. Let {xn , yn , zn }+∞ n=−2 be solutions of system (5). Then +∞ +∞ 1- {xn }n=−2 , {yn }n=−2 and {zn }+∞ n=−2 and are periodic with period six i.e., xn+6 = xn , yn+6 = yn ,

zn+6 = zn .

2- We have the following form x6n−2 = x−2 , x6n−1 = x−1 , x6n = x0 , x−2 x0 , x6n+2 = x−1 (x−1 y0 z−2 − 1), x6n+3 = , x−2 y−1 z0 − 1 x0 y−2 z−1 − 1 y6n−2 = y−2 , y6n−1 = y−1 , y6n = y0 , y−2 y0 , y6n+2 = y−1 (x−2 y−1 z0 − 1), y6n+3 = , y6n+1 = x0 y−2 z−1 − 1 x−1 y0 z−2 − 1 z6n−2 = z−2 , z6n−1 = z−1 , z6n = z0 , z−2 z0 , z6n+2 = z−1 (x0 y−2 z−1 − 1), z6n+3 = , z6n+1 = x−1 y0 z−2 − 1 x−2 y−1 z0 − 1

x6n+1 =

Or equivalently

¾ x−2 x0 , x−1 (x−1 y0 z−2 − 1), , x−2 y−1 z0 − 1 x0 y−2 z−1 − 1 ¾ ½ y−2 y0 +∞ , y−1 (x−2 y−1 z0 − 1), . {yn }n=−2 = y−2 , y−1 , y0 , x0 y−2 z−1 − 1 x−1 y0 z−2 − 1 ¾ ½ z−2 z0 +∞ {zn }n=−2 = z−2 , z−1 , z0 , , z−1 (x0 y−2 z−1 − 1), . x−1 y0 z−2 − 1 x−2 y−1 z0 − 1

{xn }+∞ n=−2

½ = x−2 , x−1 , x0 ,

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7

On The System xn+1 =

xn−2 −1−xn−2 yn−1 zn ,

yn+1 =

yn−2 −1−yn−2 zn−1 xn ,

zn+1 =

zn−2 −1−zn−2 xn−1 yn

In this section we study the solution of the following system of difference equations yn−2 zn−2 , zn+1 = , −1 − yn−2 zn−1 xn −1 − zn−2 xn−1 yn (7) where the initial conditions n ∈ N0 are arbitrary non zero real numbers. Theorem 6. Let {xn , yn , zn }+∞ n=−2 be solutions of system (6). Then +∞ +∞ 1- {xn }n=−2 , {yn }n=−2 and {zn }+∞ n=−2 and are periodic with period six i.e., xn+1 =

xn−2 , −1 − xn−2 yn−1 zn

yn+1 =

xn+6 = xn , yn+6 = yn ,

zn+6 = zn .

2- We have the following form x6n−2 = x−2 , x6n−1 = x−1 , x6n = x0 , x−2 x0 , x6n+2 = −x−1 (x−1 y0 z−2 + 1), x6n+3 = − , x−2 y−1 z0 + 1 x0 y−2 z−1 + 1 y6n−2 = y−2 , y6n−1 = y−1 , y6n = y0 , y−2 y0 , y6n+2 = −y−1 (x−2 y−1 z0 + 1), y6n+3 = − , y6n+1 = − x0 y−2 z−1 + 1 x−1 y0 z−2 + 1 z6n−2 = z−2 , z6n−1 = z−1 , z6n = z0 , z−2 z0 , z6n+2 = −z−1 (x0 y−2 z−1 + 1), z6n+3 = − , z6n+1 = − x−1 y0 z−2 + 1 x−2 y−1 z0 + 1

x6n+1 = −

Or equivalently ½ +∞ {xn }n=−2 = x−2 , x−1 , x0 , − {yn }+∞ n=−2 {zn }+∞ n=−2

½ = y−2 , y−1 , y0 , − ½ = z−2 , z−1 , z0 , −

x−2 x0 , −x−1 (x−1 y0 z−2 + 1), − x−2 y−1 z0 + 1 x0 y−2 z−1 + 1

y−2 y0 , −y−1 (x−2 y−1 z0 + 1), − x0 y−2 z−1 + 1 x−1 y0 z−2 + 1 z−2 z0 , −z−1 (x0 y−2 z−1 + 1), − x−1 y0 z−2 + 1 x−2 y−1 z0 + 1

¾

¾

¾

,

.

.

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8

The System xn+1 =

xn−2 1−xn−2 yn−1 zn ,

yn+1 =

yn−2 1−yn−2 zn−1 xn ,

zn+1 =

zn−2 1−zn−2 xn−1 yn

In this section, we study the solution of the following system of difference equations. xn+1 =

xn−2 , 1 − xn−2 yn−1 zn

yn+1 =

yn−2 zn−2 , zn+1 = 1 − yn−2 zn−1 xn 1 − zn−2 xn−1 yn

(8)

where n ∈ N0 and the initial conditions are arbitrary nonzero real numbers. The following theorem is devoted to the form of the solutions of system (7). Theorem 7. Suppose that {xn , yn , zn } are solutions of the system (7). Then for n = 0, 1, 2, ..., we have the following formulas n−1 Q (−1 + (3i + 1)x−1 y0 z−2 ) (−1 + (3i)x−2 y−1 z0 ) , x3n−1 = x−1 , i=0 (−1 + (3i + 1)x−2 y−1 z0 ) i=0 (−1 + (3i + 2)x−1 y0 z−2 ) n−1 Q (−1 + (3i + 2)x0 y−2 z−1 ) , = x0 i=0 (−1 + (3i + 3)x0 y−2 z−1 )

x3n−2 = −x−2 x3n

n−1 Q (−1 + (3i + 1)x−2 y−1 z0 ) (−1 + (3i)x0 y−2 z−1 ) , y3n−1 = y−1 , i=0 (−1 + (3i + 1)x0 y−2 z−1 ) i=0 (−1 + (3i + 2)x−2 y−1 z0 ) n−1 Q (−1 + (3i + 2)x−1 y0 z−2 ) , = y0 i=0 (−1 + (3i + 3)x−1 y0 z−2 )

y3n−2 = −y−2 y3n

n−1 Q

n−1 Q (−1 + (3i + 1)x0 y−2 z−1 ) (−1 + (3i)x−1 y0 z−2 ) , z3n−1 = z−1 , i=0 (−1 + (3i + 1)x−1 y0 z−2 ) i=0 (−1 + (3i + 2)x0 y−2 z−1 ) n−1 Q (−1 + (3i + 2)x−2 y−1 z0 ) , = z0 i=0 (−1 + (3i + 3)x−2 y−1 z0 )

z3n−2 = −z−2 z3n

n−1 Q

n−1 Q

Proof. For n = 0 the result holds. Suppose that the result holds for n − 1.

n−2 Q (−1 + (3i + 1)x−1 y0 z−2 ) (−1 + (3i)x−2 y−1 z0 ) , x3n−4 = x−1 , i=0 (−1 + (3i + 1)x−2 y−1 z0 ) i=0 (−1 + (3i + 2)x−1 y0 z−2 ) n−2 Q (−1 + (3i + 2)x0 y−2 z−1 ) , = x0 i=0 (−1 + (3i + 3)x0 y−2 z−1 )

x3n−5 = −x−2 x3n−3

n−2 Q (−1 + (3i + 1)x−2 y−1 z0 ) (−1 + (3i)x0 y−2 z−1 ) , y3n−4 = y−1 , i=0 (−1 + (3i + 1)x0 y−2 z−1 ) i=0 (−1 + (3i + 2)x−2 y−1 z0 ) n−2 Q (−1 + (3i + 2)x−1 y0 z−2 ) , = y0 i=0 (−1 + (3i + 3)x−1 y0 z−2 )

y3n−5 = −y−2 y3n−3

n−2 Q

n−2 Q

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n−2 Q (−1 + (3i + 1)x0 y−2 z−1 ) (−1 + (3i)x−1 y0 z−2 ) , z3n−4 = z−1 , i=0 (−1 + (3i + 1)x−1 y0 z−2 ) i=0 (−1 + (3i + 2)x0 y−2 z−1 ) n−2 Q (−1 + (3i + 2)x−2 y−1 z0 ) , = z0 i=0 (−1 + (3i + 3)x−2 y−1 z0 )

z3n−5 = −z−2 z3n−3

n−2 Q

It follows from Eq.(7) that x3n−2 =

x3n−5 1 + x3n−5 y3n−4 z3n−3 Q

n−2

−x−2

=

Q

n−2

1+(−x−2

−x−2

= 1 − x−2 y−1 z0 =

i=0

1 − x−2 y−1 z0 = −x−2

i=0

n−2 Q

n−2 Q i=0

n−2 Q

−x−2

n−2 Q

i=0 n−2

(−1+(3i)x−2 y−1 z0 ) )(y−1 (−1+(3i+1)x−2 y−1 z0 )

i=0

(−1+(3i)x−2 y−1 z0 ) (−1+(3i+1)x−2 y−1 z0 )

Q

i=0

n−2 Q i=0

Q

n−2 (−1+(3i+1)x−2 y−1 z0 ) )(z0 (−1+(3i+2)x−2 y−1 z0 )

(−1+(3i)x−2 y−1 z0 ) (−1+(3i+1)x−2 y−1 z0 )

i=0

(−1+(3i+2)x−2 y−1 z0 ) ) (−1+(3i+3)x−2 y−1 z0 )

(−1+(3i)x−2 y−1 z0 ) −2 y−1 z0 ) (( (−1+(3i+1)x )( (−1+(3i+1)x )( (−1+(3i+2)x−2 y−1 z0 ) )) (−1+(3i+2)x−2 y−1 z0 ) (−1+(3i+3)x−2 y−1 z0 ) −2 y−1 z0 )

(−1+(3i)x−2 y−1 z0 ) (−1+(3i+1)x−2 y−1 z0 )

n−2 Q i=0

(−1+(3i)x−2 y−1 z0 ) ( (−1+(3i+3)x ) −2 y−1 z0 )

(−1 + (3i)x−2 y−1 z0 ) 1 x−2 y−1 z0 (−1 + (3i + 1)x−2 y−1 z0 ) 1 + ( (−1+(3n−3)x−2 y−1 z0 ) )

(−1 + (3n − 3)x−2 y−1 z0 ) (−1 + (3i)x−2 y−1 z0 ) ( ) i=0 (−1 + (3i + 1)x−2 y−1 z0 ) (−1 + (3n − 3)x−2 y−1 z0 ) + x−2 y−1 z0 n−2 Q (−1 + (3i)x−2 y−1 z0 ) (−1 + (3n − 3)x−2 y−1 z0 ) ( ) = −x−2 i=0 (−1 + (3i + 1)x−2 y−1 z0 ) (−1 + (3n − 2)x−2 y−1 z0 ) = −x−2

Then, we see that x3n−2 = −x−2

n−1 Q i=0

(−1 + (3i)x−2 y−1 z0 ) (−1 + (3i + 1)x−2 y−1 z0 )

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Also, we see from Eq.(1) that y3n−2 =

y3n−5 1 + y3n−5 z3n−4 x3n−3 Q

n−2

−y−2

= 1+(−y−2

=

Q

n−2 i=0

i=0

1 − x0 y−2 z−1 = −y−2

n−2 Q i=0

n−2 Q

i=0 n−2

(−1+(3i)x0 y−2 z−1 ) )(z−1 (−1+(3i+1)x0 y−2 z−1 )

n−2 Q

−y−2

(−1+(3i)x0 y−2 z−1 ) (−1+(3i+1)x0 y−2 z−1 ) n−2 Q i=0

(−1+(3i)x0 y−2 z−1 ) (−1+(3i+1)x0 y−2 z−1 )

Q

i=0

Q

n−2 (−1+(3i+1)x0 y−2 z−1 ) )(x0 (−1+(3i+2)x0 y−2 z−1 )

i=0

(−1+(3i+2)x0 y−2 z−1 ) ) (−1+(3i+3)x0 y−2 z−1 )

(−1+(3i)x0 y−2 z−1 ) (−1+(3i+3)x0 y−2 z−1 )

(−1 + (3i)x0 y−2 z−1 ) ( (−1 + (3i + 1)x0 y−2 z−1 ) 1 +

1

) x0 y−2 z−1 −1+(3n−3)x0 y−2 z−1 −

−1 + (3n − 3)x0 y−2 z−1 (−1 + (3i)x0 y−2 z−1 ) ( ) i=0 (−1 + (3i + 1)x0 y−2 z−1 ) −1 + (3n − 3)x0 y−2 z−1 + x0 y−2 z−1 n−2 Q (−1 + (3i)x0 y−2 z−1 ) −1 + (3n − 3)x0 y−2 z−1 ( = −y−2 ) i=0 (−1 + (3i + 1)x0 y−2 z−1 ) −1 + (3n − 2)x0 y−2 z−1

= −y−2

Then, we see that

y3n−2 = −y−2 Finally, we see that z3n−2 =

n−1 Q i=0

(−1 + (3i)x0 y−2 z−1 ) (−1 + (3i + 1)x0 y−2 z−1 )

z3n−5 1 + z3n−5 x3n−4 y3n−3 Q

n−2

=

=

(−1+(3i)x−1 y0 z−2 ) (−1+(3i+1)x−1 y0 z−2 ) i=0 n−2 n−2 n−2 (−1+(3i)x−1 y0 z−2 ) (−1+(3i+1)x−1 y0 z−2 ) (−1+(3i+2)x−1 y0 z−2 ) 1+(−z−2 )(x−1 )(y0 ) (−1+(3i+1)x−1 y0 z−2 ) (−1+(3i+2)x−1 y0 z−2 ) (−1+(3i+3)x−1 y0 z−2 ) i=0 i=0 i=0

−z−2

Q

−z−2

n−2 Q i=0

1 − x−1 y0 z−2 = −z−2 = −z−2

n−2 Q i=0

n−2 Q i=0

(−1+(3i)x−1 y0 z−2 ) (−1+(3i+1)x−1 y0 z−2 ) n−2 Q i=0

Q

Q

(−1+(3i)x−1 y0 z−2 ) (−1+(3i+3)x−1 y0 z−2

(−1 + (3i)x−1 y0 z−2 ) ( (−1 + (3i + 1)x−1 y0 z−2 ) 1 +

1

) x−1 y0 z−2 −1+(3n−3)x−1 y0 z−2 −

(−1 + (3i)x−1 y0 z−2 ) −1 + (3n − 3)x−1 y0 z−2 ( ) (−1 + (3i + 1)x−1 y0 z−2 ) −1 + (3n − 2)x−1 y0 z−2 15

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Then, z3n−2 = −z−2 This completes the proof

8.1

n−1 Q i=0

(−1 + (3i)x−1 y0 z−2 ) (−1 + (3i + 1)x−1 y0 z−2 )

Numerical Examples

For confirming the results of this section, we consider the following numerical example which represent solutions to the previous systems. Example 1. We consider interesting numerical example for the difference equations system (1) with the initial conditions x−2 = 13, x−1 = 0.4, x0 = 3, y−2 = 0.5, y−1 = 7, y0 = 3.7, z−2 = 0.9, z−1 = 17 and z0 = 0.72. (See Fig. 1). plot of xn+1=xn−2/(1+xn−2yn−1zn),yn+1=yn−2/(1+xnyn−2zn−1);zn+1=zn−2/(1+xn−1ynzn−2); 18 xn 16

yn zn

14

x(n),y(n),Z(n)

12 10 8 6 4 2 0

0

5

10

15 n

20

25

30

Figure 1. Example 2. We put the initial conditions for system (2) as follows: x−2 = 1.3, x−1 = −0.4, x0 = 0.3, y−2 = 0.5, y−1 = 0.1, y0 = −0.7, z−2 = −0.9, z−1 = 0.7 and z0 = 0.2. (See Fig. 2).

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plot of x

=x

n+1

/(1+x

n−2

y

z ),y

n−2 n−1 n

=y

n+1

/(−1+x y

n−2

z

);z

n n−2 n−1

=z

n+1

/(−1+x

n−2

y z

);

n−1 n n−2

2.5 x 2

y z

1.5

n n n

x(n),y(n),Z(n)

1 0.5 0 −0.5 −1 −1.5 −2 −2.5

0

5

10

15

20 n

25

30

35

40

Figure 2. Example 3. For the difference equations system (3) where the initial conditions x−2 = 1.3, x−1 = 0.4, x0 = 0.3, y−2 = 0.25, y−1 = 0.1, y0 = 0.7, z−2 = 0.9, z−1 = 0.7 and z0 = 0.2. (See Fig. 3). plot of x

=x

n+1

/(−1+x

n−2

y

z ),y

n−2 n−1 n

=y

n+1

/(1+x y

n−2

z

);z

n n−2 n−1

=z

n+1

/(−1+x

n−2

y z

);

n−1 n n−2

5 x 4

y z

3

n n n

x(n),y(n),Z(n)

2 1 0 −1 −2 −3 −4 −5

0

5

10

15

20 n

25

30

35

40

Figure 3. 17

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Example 4. We assume x−2 = 1.3, x−1 = 0.4, x0 = 0.3, y−2 = 0.25, y−1 = 0.1, y0 = 0.7, z−2 = 0.9, z−1 = 0.7 and z0 = 0.2 for system (4) see Fig. 4. plot of xn+1=xn−2/(−1+xn−2yn−1zn),yn+1=yn−2/(−1+xnyn−2zn−1);zn+1=zn−2/(1+xn−1ynzn−2); 7 x 6

y

n n

zn

5

x(n),y(n),Z(n)

4 3 2 1 0 −1 −2

0

5

10

15

20

25 n

30

35

40

45

50

Figure 4. Example 5. See Fig. 5, if we take system (5) with x−2 = 3, x−1 = −0.4, x0 = 2, y−2 = −0.5, y−1 = 0.9, y0 = 0.7, z−2 = 0.19, z−1 = −0.4 and z0 = 0.1.

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plot of x

=x

n+1

/(−1+x

n−2

y

z ),y

n−2 n−1 n

=y

n+1

/(−1+x y

n−2

z

);z

n n−2 n−1

=z

n+1

/(−1+x

n−2

y z

);

n−1 n n−2

3 xn

2

yn z

1

n

x(n),y(n),Z(n)

0 −1 −2 −3 −4 −5 −6 −7

0

5

10

15 n

20

25

30

Figure 5. Example 6. See Fig. 6, if we consider system (6) with x−2 = −9, x−1 = 0.4, x0 = −2, y−2 = 0.2, y−1 = 0.7, y0 = 1.8, z−2 = 9, z−1 = −0.4 and z0 = −2. plot of x

=x

n+1

/(−1−x

n−2

y

z ),y

n−2 n−1 n

=y

n+1

/(−1−x y

n−2

z

);z

n n−2 n−1

=z

n+1

/(−1−x

n−2

y z

);

n−1 n n−2

10 xn

8

y z

6

n n

x(n),y(n),Z(n)

4 2 0 −2 −4 −6 −8 −10

0

2

4

6

8

10 n

12

14

16

18

20

Figure 6. 19

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Example 7. We take the difference equations system (7) with the initial conditions x−2 = 9, x−1 = 4, x0 = 2, y−2 = 3, y−1 = 7, y0 = 18, z−2 = 11, z−1 = −4 and z0 = 5. (See Fig. 7). plot of x

=x

n+1

/(1−x

n−2

y

z ),y

n−2 n−1 n

=y

n+1

/(1−x y

n−2

z

);z

n n−2 n−1

=z

n+1

/(1−x

n−2

y z

);

n−1 n n−2

20 xn y z

x(n),y(n),Z(n)

15

n n

10

5

0

−5

0

5

10

15

20 n

25

30

35

40

Figure 7.

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. [2] A. M. Ahmed and A.M. Youssef, A solution form of a class of higherorder rational difference equations, Journal of the Egyptian Mathematical Society, 21 (3) (2013), 248-253. [3] M. Avotina, On three second-order rational difference equations with period-two solutions, International Journal of Difference Equations, 9 (1) (2014), 23—35. [4] N. Battaloglu, C. Cinar and I. Yalçınkaya, The dynamics of the difference equation, ARS Combinatoria, 97 (2010), 281-288. δxn−2 + xn−3 [5] E. Camouzis, and E. Chatterjee, On the dynamics of xn+1 = , A + xn−3 Journal of Mathematical Analysis and Applications, 331 (2007), 230-239. 20

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[6] C. Cinar, I. Yalcinkaya and R. Karatas, On the positive solutions of pyn the difference equation system xn+1 = ymn , yn+1 = xn−1 , Journal of yn−1 Institute of Mathematics & Computer Sciences, 18 (2005), 135-136. [7] D. Clark, and M. R. S. Kulenovic, A coupled system of rational difference equations, Computers & Mathematics with Applications, 43 (2002) 849-867. [8] X. Deng, X. Liu, Y. Zhang, Periodic and subharmonic solutions for a 2nth-order difference equation involving p-Laplacian, Indagationes Mathematicae New Series, 24 (3) (2013), 613-625. [9] Q. Din, Qualitative nature of a discrete predator-prey system, Contemporary Methods in Mathematical Physics and Gravitation, l (1) (2015), 27-42. [10] Q. Din, On a system of rational difference equation, Demonstratio Mathematica, Vol. XLVII (2) (2014), 324-335. [11] E. M. Elabbasy, H. El-Metwally and E. M. Elsayed, Global behavior of the solutions of difference equation, Advances in Difference Equations, 2011, (2011): 28. [12] E. M. Elabbasy, H. El-Metwally and E. M. Elsayed, Some properties and expressions of solutions for a class of nonlinear difference equation, Utilitas Mathematica, 87 (2012), 93-110. [13] M. M. El-Dessoky, and E. M. Elsayed, On the solutions and periodic nature of some systems of rational difference equations, Journal of Computational Analysis and Applications, 18 (2) (2015), 206-218. [14] E. M. Elsayed, On the global attractivity and the solution of recursive sequence, Studia Scientiarum Mathematicarum Hungarica, 47 (3) (2010), 401—418. [15] E. M. Elsayed, Solution and attractivity for a rational recursive sequence, Discrete Dynamics in Nature and Society, Volume 2011, Article ID 982309, 17 pages. [16] E. M. Elsayed, On the solution of some difference equations, European Journal of Pure and Applied Mathematics, 4 (3) (2011), 287-303. [17] E. M. Elsayed, Solutions of rational difference system of order two, Mathematical and Computer Modelling, 55 (2012), 378–384. [18] E. M. Elsayed, Behavior and expression of the solutions of some rational difference equations, Journal of Computational Analysis and Applications, 15 (1) (2013), 73-81.

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[19] E. M. Elsayed, On the solutions and periodic nature of some systems of difference equations, International Journal of Biomathematics, 7 (6) (2014), 1450067, (26 pages). [20] E. M. Elsayed, Solution for systems of difference equations of rational form of order two, Computational and Applied Mathematics, 33 (3) (2014), 751-765. [21] E. M. Elsayed and C. Cinar, On the solutions of some systems of Difference equations, Utilitas Mathematica, 93 (2014), 279-289. [22] E. M. Elsayed, On a system of two nonlinear difference equations of order two, Proceedings of the Jangjeon Mathematical Society, 18 (3) (2015), 353-368. [23] E. M. Elsayed, M. M. El-Dessoky and Asim Asiri, Dynamics and behavior of a second order rational difference equation, Journal of Computational Analysis and Applications, 16 (4) (2014), 794-807. [24] E. M. Elsayed and H. A. El-Metwally, On the solutions of some nonlinear systems of difference equations, Advances in Difference Equations 2013, 2013:16, Published: 7 June 2013. [25] A. Gelisken and M. Kara, Some General Systems of Rational Difference Equations, Journal of Difference Equations, Volume 2015, Article ID 396757, 7 pages. [26] A. S. Kurbanli, C. Cinar and I. Yalçınkaya, On the behavior of positive solutions of the system of rational difference equations, Mathematical and Computer Modelling, 53 (2011), 1261-1267. [27] A. Kurbanli, C. Cinar and M. Erdo˘ gan, On the behavior of solutions xn−1 , yn+1 = of the system of rational difference equations xn+1 = xn−1 yn − 1 xn yn−1 , zn+1 = , Applied Mathematics, 2 (2011), 1031-1038. yn−1 xn − 1 zn−1 yn [28] H. Ma and H. Feng, On Positive Solutions for the Rational Difference Equation Systems, International Scholarly Research Notices, Volume 2014, Article ID 857480, 4 pages. [29] G. Papaschinopoulos and C. J. Schinas, On the dynamics of two exponential type systems of difference equations, Computers & Mathematics with Applications, 64 (7) (2012), 2326-2334. [30] H. Sedaghat, Nonlinear Difference Equations, Theory with Applications to Social Science Models, Kluwer Academic Publishers, Dordrect, 2003. [31] S. Stevi´c, Domains of undefinable solutions of some equations and systems of difference equations, Applied Mathematics and Computation, 22

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219 (2013), 11206-11213. [32] S. Stevic, M. A. Alghamdi, A. Alotaibi, and E. M. Elsayed, Solvable product—type system of difference equations of second order, Electronic Journal of Differential Equations, Vol. 2015 (2015) (169), 1—20. [33] N. Touafek and E. M. Elsayed, On the solutions of systems of rational difference equations, Mathematical and Computer Modelling, 55 (2012), 1987–1997.

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APPROXIMATION OF SOLUTIONS OF THE INHOMOGENEOUS GAUSS DIFFERENTIAL EQUATIONS BY HYPERGEOMETRIC FUNCTION S. OSTADBASHI, M. SOLEIMANINIA, R. JAHANARA AND CHOONKIL PARK∗ Abstract. In this paper, we solve the inhomogeneous Gauss differential equation and apply this result to estimate the error bound occurring when an analytic function is approximated by an appropriate hypergeometric function.

1. Introduction More than a half century ago, Ulam [22] posed the famous Ulam stability problem which was partially solved by Hyers [7] in the framework of Banach spaces. The Hyers’ theorem was generalized by Aoki [4] for additive mappings. In 1978, Rassias [14] extended the theorem of Hyers by considering the unbounded Cauchy difference inequality kf (x + y) − f (x) − f (y)k 6 ε(kxkp + kykp ),

(ε ≥ 0, p ∈ [0, 1)).

Since then, the stability problems of various functional equations have been studied by many authors (see [1, 6, 8, 9, 13, 15, 17, 18, 19, 20]). Alsina and Ger [3] were the first authors who investigated the Hyers-Ulam stability of differential equations. They proved that if a differentiable function f : I → R is a solution of the differential inequality |y 0 (t) − y(t)| ≤ , where I is an open subinterval of R, then there exists a solution f0 : I → R of the differential equation y 0 (t) = y(t) such that |f (t)−f0 (t)| ≤ 3 for any t ∈ I. From then on, many research papers about the Hyers-Ulam stability of differential equations have appeared in the literature, see [2, 5, 10, 11, 12, 21, 23] for instance. The form of the homogeneous Gauss differential equation has the form x(1 − x)y 00 + [r − (1 + s + t)x]y 0 − sty = 0.

(1.1)

It is easy to see that y1 = 1 +

st (st)(s + 1)(t + 1) 2 (st)(s + 1)(s + 2)(t + 1)(t + 2) 5 x+ x + x + ··· 1!r 2!r(r + 1) 3!r(r + 1)(r + 2)

2010 Mathematics Subject Classification. Primary 39B82, 35B35. Key words and phrases. Gauss differential equation; analytic function; hypergeometric function; approximation. ∗ Corresponding author: Choonkil Park (email: [email protected], fax: +82-2-2281-0019).

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and (s − r + 1)(t − r + 1) x 1!(2 − r) (s − r + 1)(s − r + 2)(t − r + 1)(t − r + 2) 2 + x 2!(2 − r)(3 − r) (s − r + 1)(s − r + 2)(s − r + 3)(t − r + 1)(t − r + 2)(t − r + 3) 5 + x + ...] 3!(2 − r)(3 − r)(4 − r)

y2 =x1−r [1 +

are a fundamental set of solutions of equation (1.1) (if r 6= 1). The series y1 known the hypergeometric function is convergent for |x| < 1 and is represented by y1 = F (s, t, r, x). Note that y2 = x1−r F (s − r + 1, t − r + 1, 2 − r, x) is of the same type. Thus the general solution is yc = c1 y1 + c2 y2 = c1 F (s, t, r, x) + c2 x1−r F (s − r + 1, t − r + 1, 2 − r, x). 2. Inhomogeneous Gauss differential equation In this section, we consider the solution of inhomogeneous Gauss differential equation of the form x(1 − x)y 00 + [r − (1 + s + t)x]y 0 − sty =

+∞ X

am xm ,

(2.1)

m=0

where the coefficients an ’s of the power series are given such that the radius of convergence is positive. Theorem 2.1. Assume that the radius of convergence of the power series R0 > 0 and ck R1 = lim | | > 0. k→∞ ck+1

P+∞

m=0 am x

m

is

(2.2)

Let ρ be a positive number defined by ρ = min{1, R0 , R1 }. Then every solution y : (−ρ, ρ) → C of differential equation (2.1) can be expressed by y(x) = yc (x) +

+∞ X

cm xm ,

(2.3)

m=1

where c1 = 1r a0 and cm =

am−1 m(m − 1 + r) +

i+1 i X Y Y 1 m−1 1 am−i−1 (m + s − j)(m + t − j) m! i=1 m − j + r j=1 j=1

(2.4)

for any m ∈ {2, 3, · · · }.

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Proof. We will show that each function y : (−ρ, ρ) → C defined by (2.3) is a solution of the inhomogeneous Gauss differential equation (2.1), where yc is a solution of homogeneous Gauss differential equation (1.1). For this purpose, it is only necessary to show that yp (x) = P∞ m satisfies differential equation (2.1). Therefore, letting y (x) = P∞ m in p m=1 cm x m=1 cm x differential equation (2.1), we obtain +∞ X

m

m(m + 1)cm+1 x + r

m=1

+∞ X

m

(m + 1)cm+1 x −

m=0

+∞ X

m(m − 1)cm xm

m=2

− (1 + s + t)

+∞ X

m

mcm x − st

m=1

+∞ X

cm x

m

=

m=1

+∞ X

am xm .

m=0

Hence rc1 +

+∞ X

[(m + 1)(m + r)cm+1 − (m + s)(m + t)cm ]xm =

m=1

+∞ X

am xm .

m=0

Therefore, we get c1 = 1r a0 and cm+1 =

1 (m + s)(m + t) am + cm , (m + 1)(m + r) (m + 1)(m + r)

(m = 1, 2, ...).

By some manipulations, we obtain cm =

am−1 m(m − 1 + r) +

i+1 i Y X Y 1 1 m−1 am−i−1 (m + s − j)(m + t − j) m! i=1 m − j + r j=1 j=1

(2.5)

for any m ∈ {2, 3, ...}. The condition (2.2) implies that the radius of convergence of yp (x) = c xm is R1 . By using the ratio test, we can easily show that the radius of convergence m=1 m of yc is 1. Thus y is certainly defined on (−ρ, ρ). 

P+∞

Corollary 2.2. Assume that the assumptions of Theorem 2.1 hold. Then there exists C > 0 such that +∞ X

cm xm ≤

m=1

+

+∞ X

am−1 xm m(m − 1 + r) m=1

+∞ X +∞ X

i Y Cam−2 −st (1 − )xm+i−1 . 2 (m + i − 1) (m + i − j − 1)(m + i − j + s + t − 1) i=1 m=2 j=0

Proof. Since there exists a constant C > 0 with i Y 1 i+1 1 C Y 1 ≤ 2 m! j=1 m − j + r m j=0 (m − j)(m − j + s + t)

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for any m = 2, 3, ... and for any i = 1, 2, ..., it follows from (2.5) that +∞ X

cm x

m

m=1

+

≤

= c1 x +

+∞ X

cm x

m=2

m

+∞ X 1 am−1 = a0 x + xm r m(m − 1 + r) m=1

i+1 i X Y Y 1 1 m−1 am−i−1 (m + s − j)(m + t − j)xm m! m − j + r m=2 i=1 j=1 j=0 +∞ X

+∞ i X m−1 X Cam−i−1 Y am−1 (m + s − j)(m + t − j) m xm + x 2 m(m − 1 + r) m (m − j)(m − j + s + t) m=1 m=2 i=1 j=0 +∞ X

+∞ X

+∞ i X m−1 X Cam−i−1 Y am−1 −st m = x + (1 − )xm 2 m(m − 1 + r) m (m − j)(m − j + s + t) m=1 m=2 i=1 j=0

=

+∞ X

+∞ X +∞ X am−1 xm + Ami xm m(m − 1 + r) m=1 i=1 m=i+1

=

+∞ X +∞ X am−1 xm + Am+i−1i xm+i−1 , m(m − 1 + r) m=1 i=1 m=2 +∞ X

where we define Ami :=

i −st Cam−i−1 Y (1 − ) 2 m (m − j)(m − j + s + t) j=0

for all i = 1, 2, · · · and m = 2, 3, · · · .



3. Approximation property of hypergeometric function In this section, we investigate an approximation property of hypergeometric functions. More precisely, we will prove that if an analytic function satisfies the condition (2.2), then it can be approximated by a hypergeometric function. Suppose that y is a given function expressed as a power series of the form y(x) =

∞ X

bm xm ,

(3.1)

m=0

whose radius of convergence is R0 > 0. Then we obtain x(1 − x)y 00 +[r − (1 + s + t)x]y 0 − sty = =

∞ X m=0 ∞ X

[(m + 1)(m + r)bm+1 − (m + s)(m + t)bm ]xm

(3.2)

am xm ,

m=0

where we define am := (m + 1)(m + r)bm+1 − (m + s)(m + t)bm

(3.3)

for all m ∈ {0, 1, 2, 3, · · · }.

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Lemma 3.1. If the am ’s, the bm ’s and the cm ’s are as defined in (3.3), (3.1) and (2.4), then cm = bm −

m m−1 Y b0 Y 1 (m + s − j)(m + t − j) m! j=1 m − j + r j=1

(3.4)

for all m ∈ {0, 1, 2, 3, ...}. Proof. The proof is clear by induction on m. For m = 1 and by (3.3) we have c1 =

1 st a0 = (rb1 − stb0 ) = b1 − b0 . r r r

(3.5)

Assume now that formula (3.3) is true for some m. It follows from (2.4), (3.3) and (3.4) that am (m + s)(m + t) + cm (m + 1)(m + r) (m + 1)(m + r)   1 (m + 1)(m + r)bm+1 − (m + s)(m + t)bm = (m + 1)(m + r)

cm+1 =

+

m m−1  Y (m + s)(m + t)  b0 Y 1 bm − (m + s − j)(m + t − j) (m + 1)(m + r) m! j=1 m − j + r j=1

= bm+1 −

m+1 m Y Y b0 1 (m + 1 + s − j)(m + 1 + t − j), (m + 1)! j=1 m + 1 − j + r j=1

as desired.



Theorem 3.2. Let R and R0 be positive constants with R < R0 . Assume that y : (−R, R) → C is a function of the form (3.1) whose radius of convergence is R1 . Also, bm ’s and cm ’s are given by (3.3) and (3.4), respectively. If R < min{1, R0 , R1 }, then there exist a hypergeox metric function yh : (−R, R) → C and a constant d > 0 such that |y(x) − yh (x)| ≤ d 1−x for all x ∈ (−R, R). Proof. We assume that y can be represented by a power series (3.1) whose radius of convergence is R < R0 . So x(1 − x)

+∞ X

m(m − 1)bm xm−2 + [r − (1 + s + t)x]

+∞ X

mbm xm−1 − st

m=1

m=2

+∞ X

mbm xm

m=0

is also a power series whose radius of convergence is R0 , more precisely, in view of (3.2) and (3.3), we have x(1 − x)

+∞ X

m−2

m(m − 1)bm x

+ [r − (1 + s + t)x]

m=0

− st

+∞ X

m=0 +∞ X

mbm x

m=0

113

mbm xm−1 m

=

+∞ X

am xm

m=0

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m for all x ∈ (−R, R). Since the power series +∞ on its interm=0 am x is absolutely convergent P+∞ val of convergence, which includes the interval [−R, R] and the power series m=0 |am xm | is continuous on [−R, R]. So there exists a constant d1 > 0 with

P

n X

|am xm | ≤ d1

m=0

for all integers n ≥ 0 and for any x ∈ (−R, R). On the other hand, since +∞ X k=1

|

stπ 2 −st |≤ =: d2 , (m − k − 1)(m − k − 1 + t + s) 6

(m = 2, 3, ...),

we have |

+∞ Y

(1 −

k=1

−st | ≤ d2 , (m − k − 1)(m − k − 1 + t + s)

(m = 2, 3, · · · )

(see [16, Theorem 6.6.2]). Hence, substituting i − j for k in the above infinite product, there exists a constant d3 with |

i Y j=0

(1 −

−st | ≤ d3 (m − i − j − 1)(m − i − j − 1 + t + s)

for all i = 1, 2, · · · and m = 2, 3, · · · . Therefore, it follows Lemma 2.2 that |

∞ X

cm xm | ≤ d1 d3

m=0

x 1−x

(3.6)

for all x ∈ (−R0 , R0 ). This completes the proof of our theorem.



Corollary 3.3. Assume that R and R0 are positive constants with R < R0 . Let y : (R, R0 ) → C be a function which can be represented by a power series of the form (3.1) whose radius of convergence is R0 . Moreover, assume that there exists a positive number R1 satisfying the condition (2.2) with bm ’s and cm ’s given in (3.1) and Lemma 3.1. If R < min{1, R0 , R1 } then there exists a hypergeometric function yh : (−R, R) → C such that |y(x)−yh (x)| = O(x) as x → 0. Example 3.4. Now, we will introduce an example concerning the hypergeometric function 1 for differential equation (2.1) with st = 16 . Given a constant R with 0 < R < 1 and assume that a function y : (−R, R) −→ C can be expressed as a power series of the form (3.1), where bm = {0,1

, 4m

m=0 m≥1.

It is easy to see that the radius of convergence of the above power series is R1 = 4. Since b0 = 0 it follows from Lemma 3.1 that cm = bm for each m ∈ {0, 1, 2, 3, ...}. Moreover, there exists a positive constant R1 such that the condition (2.2) is satisfied R1 = lim | k→+∞

ck ck+1

| = lim |

114

k→+∞

bk bk+1

| = 4.

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Now we assume r = s = t = 14 . Then we get +∞ X

|am xm | ≤

m=0

+∞ X 4(m + 1 )2 − (m + 1)(m + 1 ) 1 15 4 4 + |x| + |x|m m+2 16 64 4 m=2

≤

X 3m(m + 1 ) 1 15 +∞ 4 + + 16 64 m=2 4m+2

≤

X 1 1 15 +∞ 1 15 1 27 + + ≤ + + = m+2 16 64 m=2 2 16 64 8 64

for all x ∈ (−R, R). Since R < min{1, R0 , R1 } = 1, we can conclude from (3.6) that there exists a solution function yh : (−R, R) → C of the Gauss differential equation (2.1) with 27 x r = s = t = 41 satisfying |y(x) − yh (x)| ≤ 64 1−x for all x ∈ (−R, R). Acknowledgments This work was supported by Basic Science Research Program through the National Research Foundation of Korea funded by the Ministry of Education, Science and Technology (NRF-2017R1D1A1B04032937). References [1] M. Adam and S. Czerwik, On the stability of the quadratic functional equation in topological spaces, Banach J. Math. Anal. 1 (2007), 245–251. [2] Z. Ali, P. Kumam, K. Shah and A. Zada, Investigation of Ulam stability results of a coupled system of nonlinear implicit fractional differential equations, Math. 7 (2019), Art. No. 341. [3] C. Alsina and R. Ger, On some inequalities and stability results related to the exponential function, J. Inequal. Appl. 2 (1998), 373–380. [4] T. Aoki, On the stability of linear trasformation in Banach spaces, J. Math. Soc. Japan 2 (1950), 64–66. [5] E. Bicer and C. Tunc, On the Hyers-Ulam stability of certain partial differential equations of second order, Nonlinear Dyn. Syst. Theory 17 (2017), 150–157. [6] M. Eshaghi Gordji , C. Park and M. B. Savadkouhi, The stability of a quartic type functional equation with the fixed point alterbative, Fixed Point Theory 11 (2010), 265–272. [7] D. H. Hyers, On the stability of the linear functional equation, Proc. Nat. Acad. Sci. USA 27 (1941), 222–224. [8] D. H. Hyers and Th. M. Rassias, Approximate homomorphisms, Aequationes Math. 44 (1992), 125–153. [9] S. Jung, Hyers-Ulam-Rassias Stability of Functional Equations in Mathematical Analysis, Hadronic Press, Palm Harbor, Fla, USA, 2001. [10] S. Jung, Legendre’s differential equation and its Hyers-Ulam stability, Abst. Appl. Anal. 2007 (2007), Article ID 56419. [11] B. Kim and S. Jung, Bessel’s differential equation and its Hyers-Ulam stability, J. Inequal. Appl. 2007 (2007), Article ID 21640. [12] K. Liu, M. Feˇckan, D. O’Regan and J. Wang, Hyers–Ulam stability and existence of solutions for differential equations with Caputo-Fabrizio fractional derivative, Math. 7 (2019), Art. No. 333. [13] S. Ostadbashi and M. Soleimaninia, On Pexider difference for a Pexider cubic functional equation, Math. Reports 18(68) (2016), 151–162. [14] Th. M. Rassias, On the stability of the linear mapping in Banach spaces, Proc. Am. Math. Soc. 72 (1978), 297–300. [15] J. M. Rassias, On the Ulam stability of mixed type mappings on restricted domains, J. Math. Anal. Appl. 276 (2002), 747–762. [16] M. Reed, Fundamental Ideas of Analysis, John Wiley and Sons, New York, 1998.

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[17] S. Shagholi, M. Bavand Savadkouhi and M. Eshaghi Gordji, Nearly ternary cubic homomorphism in ternary Fr´echet algebras, J. Comput. Anal. Appl. 13 (2011), 1106–1114. [18] S. Shagholi, M. Eshaghi Gordji and M. Bavand Savadkouhi, Stability of ternary quadratic derivation on ternary Banach algebras, J. Comput. Anal. Appl. 13 (2011), 1097–1105. [19] D. Shin, C. Park and Sh. Farhadabadi, On the superstability of ternary Jordan C ∗ -homomorphisms, J. Comput. Anal. Appl. 16 (2014), 964–973. [20] 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. [21] C. Tun¸c, A study of the stability and boundness of the solutions of nonlinear differential equations of the fifth order, Indian J. Pure Appl. Math. 33 (2002), 519–529. [22] S. M. Ulam, Problems in Modern Mathematics, John Wiley and Sons, New York, NY, USA, Science edition, 1964. [23] E. Zadrzynska and W. M. Zajaczkowski, On stability of solutions to equations describing incompressible heat- coducting motions under Navier’s boundary conditions, Acta Appl. Math. 152 (2017), 147–170. S. Ostadbashi, M. Soleimaninia, R. Jahanara Department of Mathematics, Urmia University, Urmia, Iran E-mail address: [email protected]; [email protected]; [email protected] Choonkil Park Research Institute for Natural Sciences, Hanyang University, Seoul 04763, Korea E-mail address: [email protected]

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ON TOPOLOGICAL ROUGH GROUPS 3 ¨ NOF ALHARBI1 , HASSEN AYDI2 , CENAP OZEL AND CHOONKIL PARK4∗

1

Department of Mathematics, King Abdulaziz University, P.O. Box 80203, Jeddah 21589, Saudi Arabia Department of Mathematics, College of Education of Jubail, Imam Abdulrahman Bin Faisal University, P.O. 12020, Industrial Jubail 31961, Saudi Arabia 3 Department of Mathematics, King Abdulaziz University, P.O. Box 80203, Jeddah 21589, Saudi Arabia 4 Research Institute of Natural Sciences, Hanyang University, Seoul 04763, Korea

2

Abstract. In this paper, we give an introduction for rough groups and rough homomorphisms. Then we present some properties related to topological rough subgroups and rough subsets. Finally we construct the product of topological rough groups and give an illustrated example.

1. Introduction In [2], Bagirmaz et al. introduced the concept of topological rough groups. They extended the notion of a topological group to include algebraic structures of rough groups. In addition, they presented some examples and properties. The main purpose of this paper is to introduce some basic definitions and results about topological rough groups and topological rough subgroups. We also introduce the Cartesian product of topological rough groups. This paper is as follows: Section 2 gives basic results and definitions on rough groups and rough homomorphisms. In Section 3, following results and definitions of [2], we give some more interesting and nice results about topological rough groups. Finally, in Section 4 we prove that the product of topological rough groups is a topological rough group. Further, an example is provided. This paper has been produced from the PhD thesis of Alharbi registered in King Abdulaziz University. 2. Rough groups and rough homomorphisms First, we give the definition of rough groups introduced by Biswas and Nanda in 1994 [3]. 2010 Mathematics Subject Classification. Primary: 22A05, 54A05. Secondary: 03E25. Key words and phrases. rough group; topological rough group; topological rough subgroup; product of topological rough groups. ∗ Corresponding author: Choonkil Park (email: [email protected], fax: +82-2-2281-0019, office: +822-2220-0892). [email protected] , [email protected] , [email protected] , [email protected] , [email protected] .

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Let (U, R) be an approximation space. For a subset X ⊆ U , X = {[x]R : [x]R ∩ X 6= ∅} and X = {[x]R : [x]R ⊆ X}. Suppose that ∗ is a binary operation defined on U . We will use xy instead of x ∗ y for each composition of elements x, y ∈ U as well as for composition of subsets XY , where X, Y ⊆ U . Definition 2.1. [2] Let G = (G, G) be a rough set in the approximation space (U, R). Then G = (G, G) is called a rough group if the following conditions are satisfied: (1) for all x, y ∈ G, xy ∈ G (closed); (2) for all x, y, z ∈ G, (xy)z = x(yz) (associative law); (3) for all x ∈ G, there exists e ∈ G such that xe = ex = x (e is the rough identity element); (4) for all x ∈ G, there exists y ∈ G such that xy = yx = e (y is the rough inverse element of x. It is denoted as x−1 ). Definition 2.2. [2] A nonempty rough subset H = (H, H) of a rough group G = (G, G) is called a rough subgroup if it is a rough group itself. A rough set G = (G, G) is a trivial rough subgroup of itself. Also the rough set e = (e, e) is a trivial rough subgroup of the rough group G if e ∈ G. Theorem 2.1. [2] A rough subset H is a rough subgroup of the rough group G if the two conditions are satisfied: (1) for all x, y ∈ H, xy ∈ H; (2) for all y ∈ H, y −1 ∈ H. Also, a rough normal subgroup can be defined. Let N be a rough subgroup of the rough group G. Then N is called a rough normal subgroup of G if for all x ∈ G, xN = N x. 0

Definition 2.3. [4] Let (U1 , R1 ) and (U2 , R2 ) be two approximation spaces and ∗, ∗ be two binary operations on U1 and U2 , respectively. Suppose that G1 ⊆ U1 , G2 ⊆ U2 are rough 0 groups. If the mapping ϕ : G1 → G2 satisfies ϕ(x ∗ y) = ϕ(x) ∗ ϕ(y) for all x, y ∈ G1 , then ϕ is called a rough homomorphism. Definition 2.4. [4] A rough homomorphism ϕ from a rough group G1 to a rough group G2 is called: (1) a rough epimorphism (or surjective) if ϕ : G1 → G2 is onto. (2) a rough embedding (or monomorphism) if ϕ : G1 → G2 is one-to -one. (3) a rough isomorphism if ϕ : G1 → G2 is both onto and one-to-one.

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3. Topological rough groups We study a topological rough group, which has an ordinary topology on a rough group, i.e., a topology τ on G induced a subspace topology τG on G. Suppose that (U, R) is an approximation space with a binary operation ∗ on U . Let G be a rough group in U . Definition 3.1. [2] A topological rough group is a rough group G with a topology τG on G satisfying the following conditions: (1) the product mapping f : G × G → G defined by f (x, y) = xy is continuous with respect to a product topology on G × G and the topology τ on G induced by τG ; (2) the inverse mapping ι : G → G defined by ι(x) = x−1 is continuous with respect to the topology τ on G induced by τG . Elements in the topological rough group G are elements in the original rough set G with ignoring elements in approximations. Example 3.1. Let U = {0, 1, 2} be any group with 3 elements. Let U/R = {{0, 2}, {1}} be a classification of equivalent relation. Let G = {1, 2}. Then G = {1} and G = {0, 1, 2} = U . A topology on G is τG = {ϕ, G, {1}, {2}, {1, 2}} and the relative topology is τ = {ϕ, G, {1}, {2}}. The conditions in Definition 3.1 are satisfied and hence G is a topological rough group. Example 3.2. Let U = R and U/R = {{x : x > 0}, {x : x < 0}} be a partition of R. Consider G = R∗ = R − 0. Then G is a rough group with addition. It is also a topological rough group with the standard topology on R. Example 3.3. Consider U = S4 the set of all permutations of four objects. Let (∗) be the multiplication operation of permutations. Let U/R = {E1 , E2 , E3 , E4 } be a classification of U, where E1 = {1, (12), (13), (14), (23), (24), (34)}, E2 = {(123), (132), (142), (124), (134), (143), (234), (243)}, E3 = {(1234), (1243), (1342), (1324), (1423), (1432)}, E4 = {(12)(34), (13)(24), (14)(23)}. Let G = {(12), (123), (132)}. Then G = E1 ∪ E2 . Clearly, G is a rough group. Consider a topology on G as τG = {ϕ, G, {(12)}, {1, (123), (132)}, {1, (12), (123), (132)}}. Then the relative topology on G is τ = {ϕ, G, {(12)}, {(123), (132)}}. The conditions in Definition 3.1 are satisfied and hence G is a topological rough group. Proposition 3.1. [2] Let G be a topological rough group and fix a ∈ G. Then

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(1) the mapping La : G → G defined by La (x) = ax, is one-to-one and continuous for all x ∈ G. (2) the mapping Ra : G → G defined by Ra (x) = xa, is one-to-one and continuous for all x ∈ G. (3) the inverse mapping ι : G → G is a homeomorphism for all x ∈ G. Proposition 3.2. [2] Let G be a topological rough group. Then G = G−1 . Proposition 3.3. [2] Let G be a topological rough group and V ⊆ G. Then V is open (resp. closed) if and only if V −1 is open (resp. closed). Proposition 3.4. [2] Let G be a topological rough group and W be an open set in G with e ∈ W . Then there exists an open set V with e ∈ V such that V = V −1 and V V ⊆ W . Proposition 3.5. [2] Let G be a rough group. If G = G, then G is a topological group. Definition 3.2. Let G be a topological rough group. Then a subset U of G is called rough symmetric if U = U −1 . From the definition of rough subgroups, we obtain the following result. Corollary 3.1. Every rough subgroup of a topological rough group is rough symmetric. Theorem 3.1. Let G be a topological rough group. Then the closure of any rough symmetric subset A of G is again rough symmetric. Proof. Since the inverse mapping ι : G → G is a homeomorphism, cl(A) = (cl(A))−1 .



Theorem 3.2. Let G be a topological rough group and H be a rough subgroup. Then cl(H) is a rough group in G. (1) Identity element: H ⊆ cl(H) implies that H ⊆ cl(H) and so e ∈ cl(H). Since cl(H) ⊆ G, we have ex = xe = x for all x ∈ cl(H). (2) Inverse element: cl(H)−1 ⊆ cl(H −1 ) = cl(H). (3) Closed under product: Let x, y ∈ cl(H). Then xy ∈ G, which implies that there exists an open set U ∈ G such that xy ∈ U. We will prove that U ∧ H 6= ϕ. Consider the multiplication mapping µ : G × G → G. This implies that there exist open sets W, V of G such that x ∈ W, y ∈ V, W ∧ H 6= ϕ, V ∧ H 6= ϕ. Since the 0 0 topology on G is a relative topology on G, there exist open sets W , V of G such 0 0 0 0 that W ⊆ W , V ⊆ V . Hence W ∧ H 6= ϕ, V ∧ H 6= ϕ. Then µ(W × V ) ∧ H 6= ϕ, but we have µ(W × V ) ⊆ U , which implies H ∧ U 6= ϕ. So xy ∈ cl(H) ⊆ cl(H). This implies that cl(H) is a rough group of G. Thus cl(H) is a rough group in G. 

Proof.

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Definition 3.3. Let (X, τ ) be a topological rough space of approximation space (U, R), and let B ⊆ τ be a base for τ . For x ∈ X, the family Bx = {O ∈ B : x ∈ O} ⊆ B is called a base at x. Theorem 3.3. Let G be a topological rough space with G group. For g ∈ G, the base at g is equal to Bg = {gO : O ∈ Be }, where e is the identity element of a rough group G. 4. Product of topological rough groups Let (U, R1 ) and (V, R2 ) be approximation spaces with binary operations ∗1 and ∗2 , 0 0 respectively. Consider the Cartesian product of U and V : let x, x ∈ U and y, y ∈ V . 0 0 0 0 0 0 Then (x, y), (x , y ) ∈ U × V . Define ∗ as (x, y) ∗ (x , y ) = (x ∗1 x , y ∗2 y ). Then ∗ is a binary operation on U × V . In [1], Alharbi et al. proved that the product of equivalence relations is also an equivalence relation on U × V . Theorem 4.1. [1] Let G1 ⊆ U and G2 ⊆ V be two rough groups. Then the Cartesian product G1 × G2 is a rough group. The following conditions are satisfied: 0

0

0

0

0

0

(1) For all (x, y), (x , y ) ∈ G1 × G2 , (x1 , y1 ) ∗ (x2 , y2 ) = (x1 ∗1 x2 , y1 ∗2 y2 ) ∈ G1 × G2 . (2) Associative law is satisfied over all elements in G1 × G2 . 0 0 (3) There exists an identity element (e, e ) ∈ G1 × G2 such that ∀(x, x ) ∈ G1 × 0 0 0 0 0 G2 , (x, x ) × (e, e ) = (e, e ) × (x, x ) = (ex, e0 x ) = (x.x0 ). 0 0 (4) For all (x, x ) ∈ G1 × G2 , there exists an element (y, y ) ∈ G1 × G2 such that 0 0 0 0 0 (x, x ) ∗ (y, y ) = (y, y ) ∗ (x, x ) = (e, e ). Example 4.1. Consider Example 3.1 where U = {0, 1, 2} and U/R = {{0, 2}, {1}}. Then the Cartesian product U × U is as follows: U × U = {(0, 0), (0, 2), (0, 1), (2, 0), (2, 2), (2, 1), (1, 0), (1, 2), (1, 1)}. Then the new classification is {{0, 0), (0, 2), (2, 0), (2, 2)}, {(0, 1), (2, 1)}, {(1, 0), (1, 2)}, {(1, 1)}}. Consider the rough group G = {1, 2}. Then the Cartesian product G × G is G × G = {(2, 2), (2, 1), (1, 2), (1, 1)}, where G × G = G × G = U × U. From the definition of a rough group, we have that

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(1) the multiplication of elements in G × G is closed under G × G, i.e. (2, 2)(2, 2) = (1, 1), (2, 2)(2, 1) = (1, 0), (2, 2)(1, 1) = (0, 0), (2, 2)(1, 2) = (0, 1), (2, 1)(2, 1) = (1, 2), (2, 1)(1, 1) = (0, 2), (2, 1)(1, 2) = (0, 0), (1, 1)(1, 1) = (2, 2), (1, 1)(1, 2) = (2, 0); 0 0 (2) there exists (0, 0) ∈ G×G such that for every (g, g ) ∈ G×G, we have (0, 0)(g, g ) = 0 (g, g ); (3) for every element of G × G, there exists an inverse element in G × G, where (1, 1)−1 = (2, 2) ∈ G × G, (2, 1)−1 = (1, 2) ∈ G × G; (4) the associative law is satisfied. Hence G × G is a rough group. From Example 3.1, we have τG = {ϕ, G, {1}, {2}, {1, 2}} as a topology on G. Then τG × τG is the product topology of G × G. Also we have τ = {ϕ, G, {1}, {2}} as a relative topology on G. So τ × τ is a topology on G × G induced by τG × τG . Consider the multiplication mapping µ : (G × G) × (G × G) → G × G. This mapping is continuous with respect to topology τ × τ and the product topology on (G × G) × (G × G). Also, we can show that the inverse mapping ι : G × G → G × G is continuous. Hence G × G is a topological rough group. Acknowledgement The authors wish to thank the Deanship for Scientific Research (DSR) at King Abdulaziz University for financially funding this project under grant no. KEP-PhD-2-130-39. 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 (NRF2017R1D1A1B04032937). References ¨ [1] N. Alharbi, H. Aydi, C. Ozel, Rough spaces on rough sets (preprint). [2] N. Bagirmaz, I. Icen, A.F. Ozcan, Topological rough groups, Topol. Algebra Appl. 4 (2016), 31–38. [3] R. Biswas, S. Nanda, Rough groups and rough subgroups, Bull. Polish Acad. Sci. Math. 42 (1994), 251–254. [4] C.A. Neelima, P. Isaac, Rough anti-homomorphism on a rough group, Global J. Math. Sci. Theory Pract. 6, (2014), no. 2, 79–85.

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ON THE FARTHEST POINT PROBLEM IN BANACH SPACES A. YOUSEF1 , R. KHALIL2 AND B. MUTABAGANI3 Abstract. A long standing conjecture in theory of Banach spaces is:" Every uniquely remotal set in a Banach is a singleton". This is known as the farthest point Conjecture. In an attempt to solve this problem, we give our contribution toward solving it, in the positive direction, by proving that every such subset E in the sequence space `1 is a singleton.

1. Introduction Let X be a normed space, and E be a closed and bounded subset of X. We de…ne the real valued function D(:; E) : X ! R by D(x; E) = supfkx

ek : e 2 Eg;

the farthest distance function. We say that E is remotal if for every x 2 X, there exists e 2 E such that D(x; E) = kx ek. In this case, we denote the set fe 2 E : D(x; E) = kx ekg by F (x; E). It is clear that F (:; E) : X ! E is a multi-valued function. However, if F (:; E) : X ! E is a single-valued function, then E is called uniquely remotal. In such case, we denote F (x; E) by F (x); if no confusion arises. The study of remotal and uniquely remotal sets has attracted many mathematicians in the last decades, due to its connection with the geometry of Banach spaces. We refer the reader to [1], [3], [5], [6] and [8] for samples of these studies. However, uniquely remotal sets are of special interest. In fact, one of the most interesting and hitherto unsolved problems in the theory of farthest points, known as the the farthest point problem, which is stated as: If every point of a normed space X admits a unique farthest point in a given bounded subset E, then must E be a singleton ?. This problem gained its importance when Klee [4] proved that: singletoness of uniquely remotal sets is equivalent to convexity of Chybechev sets in Hilbert spaces (which is an open problem too, in the theory of nearest points). Since then, a considerable work has been done to answer this question, and many partial results have been obtained toward solving this problem. We refer the reader to [1], [3], [6] and [8] for some related work on uniquely remotal sets. 1991 Mathematics Subject Classi…cation. Primary 46B20; Secondary 41A50; 41A65. Key words and phrases. Uniquely remotal, Singleton, Banach space, farthest point problem. 1

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Centers of sets have played a major role in the study of uniquely remotal sets, see [1], [2] and [3]. Recall that a center c of a subset E of a normed space X is an element c 2 X such that D(c; E) = inf D(x; E): x2X

Whether a set has a center or not is another question. However, in inner product spaces, any closed bounded set does have a center [1]. In [7] it was proved that if E is a uniquely remotal subset of a normed space, admitting a center c, and if F , restricted to the line segment [c; F (c)] is continuous at c, then E is a singleton. Then recently, a generalization has been obtained in [9], where the authors proved the singletoness of uniquely remotal sets if the farthest point mapping F restricted to [c; F (c)] is partially continuous at c. Furthermore, a generalization of Klee’s result in [4], "If a compact subset E, with a center c, is uniquely remotal in a normed space X, then E must be a singleton", was also obtained in [9]. In this article, we prove that every uniquely remotal subset of thePsequence space `1 (R) is a singleton. Recall that `1 (R) = fx = (xn ) : xn 2 R and 1 n=1 jxn j < 1g. 2. Preliminaries In this section, we prove the following propositions that play a key role in the proof of the main result. Throughout the rest of the paper, F will denote the farthest distance singlevalued function associated with a uniquely remotal set E. Proposition 2.1. Let E be a uniquely rematal subset of a Banach space X. Let (xn ) be a sequence in X such that (xn ) converges to x 2 X. If F (xn ) = y for all n, where y 2 E, then F (x) = y. Proof. Suppose that F (x) 6= y. Since E is uniquely remotal, then there exists w 2 E such that F (x) = w. Further, there exists > 0 such that jjx wjj > jjx yjj + : Also, there exists n0 2 N such that jjxn xjj < 2 for all n n0 . Therefore, for m n0 jjxm

wjj

jjx > jjx

wjj

jjxm

yjj +

> jjxm

yjj +

> jjxm

yjj:

xjj

2 2

jjxm

xjj

This contradicts that y = F (xm ). Hence, we must have F (x) = y.

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ON THE FARTHEST POINT PROBLEM IN BANACH SPACES

3

Proposition 2.2. Let K be a compact subset of a Banach space X and E be uniquely remotal in X. Then there exist x 2 K and e 2 E such that D(E; K) = supfjjy

jj : y 2 K;

2 Eg = jje

xjj:

Proof. From the de…nition of D(E; K), there exist two sequences (en ) and (xn ) in E and K respectively such that D(E; K) = lim jjen n!1

xn jj:

Since K is compact, then there exists a subsequence (xnk ) of (xn ) such that (xnk ) converges to x in K. So, D(E; K) = lim jjenk k!1

xnk jj:

The de…nition of D(E; K) implies that D(E; K) jje0 x 2 K. Therefore, lim jjenk xnk jj jjx F (x)jj: 0

x0 jj for all e0 2 E and

k!1

But Thus

jjenk

xnk jj

jjenk

xjj + jjx

lim jjxnk

k!1

xnk jj

ynk jj

jjx

jjxnk

xjj + jjx

F (x)jj:

F (x)jj:

Since x 2 K and F (x) 2 E, it follows that D(E; K) = jjx the proof.

F (x)jj, which ends

3. Main Results Let E be a uniquely remotal subset of a Banach space X. Let x0 be an element in X and e0 2 E be the unique farthest point from x0 , i.e F (x0 ) = e0 . Consider the closed ball B[x0 ; jjx0 e0 jj] = B[x0 ; D(x0 ; E)]: Then clearly e0 lies on the boundary of B[x0 ; D(x0 ; E)]. Let J = fB[y; jjy follows:

e0 jj : F (y) = e0 g; and de…ne the relation "

" on J as

B1 B2 if B2 B1 : It is easy to see that the relation " " is a partial order. Now, we claim the following. Theorem 3.1. J has a maximal element. Proof. Let T be any chain in J. Consider the net fjjy e0 jj : 2 Ig. Notice that if B 1 B 2 then jjy 2 e0 jj jjy 1 e0 jj. Let r = inf jjy e0 jj. Then it is 2I

easy to see that if the in…mum is attained at some 0 , then B 0 [y 0 ; jjy 0 e0 jj] is an upper bound for T . If the in…mum is not attained then there exists a sequence

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(Bn ) in T such that lim jjyn n!1

e0 jj = inf jjy

e0 jj = r.

2I

We claim that (yn ) has a convergent subsequence. If not, then there exists > 0 such that jjyn ym jj > for all n; m. Clearly we can assume that < r. Since lim jjyn n!1

e0 jj = r, then there exists n0 2 N such that jjyn

e0 jj < r + 2

for all n n0 . But jjyn0 yn0 +1 jj > , so Bn0 Bn0 +1 . Farther, r jjyn0 e0 jj and jjyn0 +1 e0 jj < r+ 2 : Without loss of generality, we can assume, for simplicity, that yn0 = 0. Then the element v = (1 + jjyn r+1 jj yn0 +1 ) 2 Bn0 +1 . 0

Now, jjv 0jj = jjvjj = jjyn0 +1 jj + r > r + . Thus, v 62 Bn0 which contradicts the fact that Bn0 +1 Bn0 . Hence, there is a subsequence (ynk ) that converges to some element, say y. By assumption F (ynk ) = e0 for all nk , which implies by Proposition 2.1 that F (y) = e0 . Thus, B[y; jjy e0 jj] 2 J. It su¢ ces now to show that B[y; jjy e0 jj] B for all 2 I. If this is not true then there exists w 2 B[y; jjy e0 jj] such that w 62 Bm1 for some m1 . Since (Bn ) is a chain, then w 62 Bnk for all nk > m1 . Furthermore, jjw ynk jj > r + 0 for some 0 > 0 and all nk > m1 . But jjw ynk jj jjy F (y)jj = jjy

jjynk yjj + jjy wjj, where jjynk yjj ! 0 and jjy wjj < rjj. It follows that lim inf jjw ynk jj r, which contradicts nk

the fact that jjw ynk jj > r + 0 . This means that B[y; jjy e0 jj] is an upper bound for the chain T . Hence, By Zorn’s lemma J has a maximal element. Now we are ready to prove the main result of this paper. Theorem 3.2. Every uniquely remotal set in `1 (R) is a singleton. Proof. Let E be a uniquely remotal set in `1 , and let e^ be the unique farthest point in E from 0, i.e. F (0) = e^. By Theorem 3.1, J = fB[y; jjy e^jj] : F (y) = e^g has a maximal element say B[^ v ; jj^ v e^jj]. Without loss of generality, we may assume that v^ = 0 and jj^ ejj = 1 so that the maximal element is the unit ball of `1 . Let e^ = (b1 ; b2 ; b3 ; : : : ). Since jj^ ejj = 1 then with no loss of generality we can assume that b1 6= 0. Further, assume b1 > 0. So, b1 > m10 for some m0 2 N. Let 1 = (1; 0; 0; : : : ) and consider the sequence ( n1 ) in `1 , where n > m0 . Then F ( n1 ) 6= e^ for all n > m0 , since if F ( n1 ) = e^ for some n > m0 , then for 1 w 2 B[ n1 ; jj n1 e^jj], we have jjwjj jj n1 jj jjw jj jj n1 e^jj. But b1 > n1 , n so jj n1 e^jj = jj^ ejj n1 = jj^ ejj jj n1 jj. Thus, jjwjj jj^ ejj = 1 and accordingly w 2 B[0; 1], which contradicts the maximality of B[0; 1]. Hence, F ( n1 ) 6= e^ for all n > m0 .

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ON THE FARTHEST POINT PROBLEM IN BANACH SPACES

5

Let F ( n1 ) = zn = (cn1 ; cn2 ; cn3 ; : : : ). Then we must have cn1 < n1 for all n > m0 . 1 Otherwise, we obtain that jjzn jj = jjzn jj jj n1 jj 1 n1 = jj^ e n1 jj, which n contradicts the fact that F ( n1 ) = zn . Now, since n1 ! 0, then jjzn jj ! 1. Further, the sequence cn1 converges to , where 0. Consider the set P = fb1 1 g. Then, clearly D(^ e; P ) = P1 n n D(zn ; P ) = jjzn b1 1 jj = jc1 b1 j + j=2 jcj j. Therefore, lim D(zn ; P ) = (b1 + j j) + lim

n!1

n!1

= b1 + j j + (1 = 1 + b1

1 X j=2

P1

j=2

jbj j < 1. Also,

jcnj j

j j)

Since D(P; E) D(P; zn ) for all n, we get that D(P; E) 1 + b1 . On the other hand, D(P; E) = sup jjb1 1 ejj b1 + 1, since jjejj 1 for every e 2 E. Thus, e2E

D(P; E) = 1 + b1 : By Proposition 2.2, D(P; E) = jjb1 1 + b1

1

e0 jj for some e0 2 E. So,

b1 + jje0 jj

1 + b1 ;

which implies that jje0 jj = 1. Therefore, e0 is another farthest point in E from 0, i.e. F (0) = fe0 ; e^g, which contradicts the unique remotality of E. Hence, E must be a singleton. References [1] A. Astaneh, On Uniquely Remotal Subsets of Hilbert Spaces, Indian Journal Of

Pure and Applied Mathematics. 14(10) (1983) 1311–1317. [2] A. Astaneh, On Singletoness of Uniquely Remotal Sets, Indian Journal Of Pure and

Applied Mathematics 17(9)(1986) 1137–1139. [3] M. Baronti, A note on remotal sets in Banach spaces, Publications de L’institute

mathematique 53(67) (1993) 95–98. [4] Klee, V., Convexity of Chebychev sets, Math. Ann. 142 (1961) 292–304. [5] M. Martin, T.S.S.R.K Rao, On Remotality for Convex Sets in Banach Spaces, Jour-

nal of Approximation Theory. 162(2)(2010) 392–396. [6] T. D. Narang, On singletoness of uniquely remotal sets , Periodica Mathematika

Hungarica 21(1990) 17–19. [7] A. Niknam, On Uniquely Remotal Sets, Indian Journal Of Pure and Applied Math-

ematics 15(10) (1984) 1079–1083. [8] M. Sababheh, R. Khalil, A study of Uniquely Remotal Sets, Journal of Computa-

tional Analysis and Applications 13(7)(2010) 1233–1239. [9] M. Sababheh, A. Yousef and R. Khalil, Uniquely Remotal Sets in Banach Spaces,

Filomat 31:9 (2017), 2773— 2777. 1

Department of Mathematics, The University of Jordan , Al Jubaiha, Amman 11942, Jordan. E-mail address: [email protected]

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6

A. YOUSEF, R. KHALIL AND B. MUTABAGANI 2

Department of Mathematics, The University of Jordan , Al Jubaiha, Amman 11942, Jordan. E-mail address: [email protected] 3

Department of Mathematics, The University of Jordan , Al Jubaiha, Amman 11942, Jordan. E-mail address: [email protected]

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On the stability of 3-Lie homomorphisms and 3-Lie derivations Vahid Keshavarz1 , Sedigheh Jahedi1∗ , Shaghayegh Aslani2 , Jung Rye Lee3∗ and Choonkil Park4 1

Department of Mathematics, Shiraz University of Technology, P. O. Box 71555-313, Shiraz, Iran 2

Department of Mathematics, Bonab University, P. O. Box 55517-61167, Bonab, Iran 3 4

Department of Mathematics, Daejin University, Kyunggi 11159, Korea

Research Institute for Natural Sciences, Hanyang University, Seoul 04763, Korea

e-mail: [email protected], [email protected], [email protected], [email protected], [email protected] Abstract. In this paper, we prove the Hyers-Ulam stability of 3-Lie homomorphisms in 3-Lie algebras for Cauchy-Jensen functional equation. We also prove the Hyers-Ulam stability of 3-Lie derivations on 3-Lie algebras for Cauchy-Jensen functional equation.

1. Introduction and preliminaries The stability problem of functional equations had been first raised by Ulam [21]. In 1941, Hyers [10] gave a first affirmative answer to the question of Ulam for Banach spaces. The generalizations of this result have been published by Aoki [2] for (0 < p < 1), Rassias [19] for (p < 0) and Gajda [8] for (p > 1) for additive mappings and linear mappings by a general control function θ(kxkp + kykp ), respectively. In 1994, Gˇavruta [9] generalized these theorems for approximate additive mappings controlled by the unbounded Cauchy difference with regular conditions, i.e., who replaced θ(kxkp + kykp ) by a general control function ϕ(x, y). Several stability problems for various functional equations have been investigated in [1, 4, 6, 7, 12, 14, 15, 16, 17, 18, 20]. A Lie algebra is a Banach algebra endowed with the Lie product [x, y] :=

(xy − yx) . 2

Similarly, a 3-Lie algebra is a Banach algebra endowed with the product h i [x, y]z − z[x, y] [x, y], z := . 2 Let A and B be two 3-Lie algebras. A C-linear mapping H : A → B is called a 3-Lie homomorphism if H([[x, y], z]) = [[H(x), H(y)], H(z)] 0∗

Corresponding authors. Keywords: Jensen functional equation, 3-Lie algebra, 3-Lie homomorphisms, 3-Lie derivation, HyersUlam stability. 0 2010 Mathematics Subject Classification. Primary 39B52; 39B82; 22D25; 17A40. 0

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for all x, y, z ∈ A. A C-linear mapping D : A → A is called a 3-Lie derivation if   D [[x, y], z] = [[D(x), y], z] + [[x, D(y)], z] + [[x, y, ], D(z)] for all x, y, z ∈ A (see [22]). Throughout this paper, we suppose that A and B are two 3-Lie algebras. For convenience, we use the following abbreviation for a given mapping f : A → B µx + µz µy + µz µx + µy + µz) + f ( + µy) + f ( + µx) Dµ f (x, y, z) := f ( 2 2 2 − 2µf (x) − 2µf (y) − 2µf (z) for all µ ∈ T1 := {λ ∈ C : |λ| = 1} and all x, y, z ∈ A. Throughout this paper, assume that A is a 3-Lie algebra with norm k · k and that B is a 3-Lie algebra with norm k · k. 2. Stability of 3-Lie homomorphisms in 3-Lie algebras We need the following lemmas which have been given in for proving the main results. Lemma 2.1. ([11]) Let X be a uniquely 2-divisible abelian group and Y be linear space. A mapping f : X → Y satisfies x+y x+z y+z + z) + f ( + y) + f ( + x) = 2[f (x) + f (y) + f (z)] 2 2 2 for all x, y, z ∈ X if and only if f : X → Y is additive. f(

(2.1)

Lemma 2.2. Let X and Y be linear spaces and let f : X → Y be a mapping such that Dµ f (x, y, z) = 0

(2.2)

for all µ ∈ T1 and all x, y, z ∈ A. Then the mapping f : X → Y is C-linear. Proof. By Lemma 2.2, f is additive. Letting y = z = 0 in (2.1), we get 2f µ x2




= µf (x) and so

1

f (µx) = µf (x) for all x ∈ X and all µ ∈ T . By the same reasoning as in the proof of [13, Theorem 2.1], the mapping f : X → Y is C-linear.



In the following, we investigate the Hyers-Ulam stability of (2.1). Theorem 2.3. Let ϕ : A3 → [0, ∞) be a function such that ϕ(x, e y, z) :=

∞ X 1 ϕ(2n x, 2n y, 2n z) < ∞ n 2 n=0

(2.3)

for all x, y, z ∈ A. Suppose that f : A → B is a mapping satisfying kDµ f (x, y, z)k ≤ ϕ(x, y, z),

(2.4)

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

(2.5)

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for all µ ∈ T1 and all x, y, z ∈ A. Then there exists a unique 3-Lie homomorphism H : A → B such that 1 kf (x) − H(x)k ≤ ϕ(x, e x, x) (2.6) 6 for all x ∈ A. Proof. Letting µ = 1 and x = y = z in (2.4), we get k3f (2x) − 6f (x)k ≤ ϕ(x, x, x)

(2.7)

for all x ∈ A. If we replace x by 2n x in (2.7) and divide both sides by 3 · 2n+1 . then we get 1 1 f (2n x)k ≤ ϕ(2n x, 2n x, 2n x) n 2 3 · 2n+1 for all x ∈ A and all nonnegative integers n. Hence k

k

1

2n+1

f (2n+1 x) −

n

1

2

f (2n+1 x) − n+1

X 1 1 1 f (2m x)k =k [ k+1 f (2k+1 x) − k f (2k x)]k m 2 2 2 k=m n X

≤ ≤

k

k=m n X

1 6

1 1 f (2k+1 x) − k f (2k x)k 2k+1 2

k=m

(2.8)

1 ϕ(2k x, 2k x, 2k x) 2k

for all x ∈ A and all nonnegative integers n ≥ m ≥ 0. It follows from (2.3) and (2.8) that the sequence { 21n f (2n x)} is a Cauchy sequence in B for all x ∈ A. Since B is complete, the sequence { 21n f (2n x)} converges for all x ∈ A. Thus one can define the mapping H : A → B by 1 H(x) := lim n f (2n x) n→∞ 2 for all x ∈ A. Moreover, letting m = 0 and passing the limit n → ∞ in (2.8), we get (2.6). It follows from (2.3) that 1 kDµ f (2n x, 2n y, 2n z)k 2n 1 ≤ lim n ϕ(2n x, 2n y, 2n z) = 0 n→∞ 2 for all x, y, z ∈ A and all µ ∈ T1 . So Dµ H (x, y, z) = 0 for all µ ∈ T1 and all x, y, z ∈ A. By Lemma 2.2, kDµ H(x, y, z)k = lim

n→∞

the mapping H : A → B is C-linear. It follows from (2.5) that kH([[x, y], z]) − [[H(x), H(y)], H(z)]k 1 = lim n kf ([[2n x, 2n y], 2n z]) − [[f (2n x), f (2n y)], f (2n z)]k n→∞ 8 1 1 ≤ lim n ϕ(2n x, 2n y, 2n z) ≤ lim n ϕ(2n x, 2n y, 2n z) = 0 n→∞ 2 n→∞ 8 for all x, y, z ∈ A. Thus H([[x, y], z]) = [[H(x), H(y)], H(z)] for all x, y, z ∈ A.

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Therefore, the mapping H : A → B is a 3-Lie homomorphism.



Corollary 2.4. Let ε, θ, p1 , p2 , p3 , q1 , q2 , q3 be positive real numbers such that p1 , p2 , p3 < 1 and q1 , q2 , q3 < 3. Suppose that f : A → B is a mapping such that kDµ f (x, y, z)k ≤ θ(kxkp1 + kykp2 + kzkp3 ),

(2.9)

kf ([[x, y], z]) − [[f (y), f (z)], f (x)]k ≤ ε(kxkq1 + kykq2 + kzkq3 )

(2.10)

for all µ ∈ T1 and all x, y, z ∈ A. Then there exists a unique 3-Lie homomorphism H : A → B such that kf (x) − H(x)k ≤

θ 1 1 1 kxkp1 + kxkp2 + kxkp3 } { p p 1 2 3 2−2 2−2 2 − 2p3

for all x ∈ A. Theorem 2.5. Let Φ : A3 → [0, ∞) be a function such that ∞ X n=1

8n ψ(

x y z , , ) 1 and q1 , q2 , q3 > 3. Suppose that f : A → B is a mapping satisfying (2.9) and (2.10). Then there exists a unique 3-Lie homomorphism H : A → B such that kf (x) − H(x)k ≤

1 1 1 θ { kxkp1 + p2 kxkp2 + p3 kxkp3 } 3 2p1 − 2 2 −2 2 −2

for all x ∈ A.

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3. Stability of 3-Lie derivations on 3-Lie algebras In this section, we prove the Hyers-Ulam stability of 3-Lie derivations on 3-Lie algebras for the functional equation Dµ f (x, y, z) = 0. Theorem 3.1. Let ϕ : A3 → [0, ∞) be a function satisfying (2.3). Suppose that f : A → A is a mapping satisfying kDµ f (x, y, z)k ≤ ϕ(x, y, z), kf ([[x, y], z]) − [[f (x), y], z] − [[x, f (y)], z] − [[x, y], f (z)]k ≤ ϕ(x, y, z)

(3.1)

for all µ ∈ T1 and all x, y, z ∈ A. Then there exists a unique 3-Lie derivation D : A → A such that kf (x) − D(x)k ≤

1 ϕ(x, e x, x) 6

(3.2)

for all x ∈ A, where ϕ e is given in Theorem 2.3. Proof. By the proof of Theorem 2.3, there exists a unique C-linear mapping D : A → A satisfying (3.2) and D(x) := lim

n→∞

1 f (2n x) 2n

for all x ∈ A. It follows from (3.1) that kD([[x, y], z]) − [[D(x), y], z] − [[x, D(y)], z] − [[x, y], D(z)]k 1 kf ([[2n x, 2n y], 2n z]) − [[f (2n x), 2n y], 2n z] − [[2n x, f (2n y)], 2n z] − [[2n x, 2n x], f (2n z)]k 8n 1 ≤ lim n ϕ(2n x, 2n y, 2n z) = 0 n→∞ 8

= lim

n→∞

for all x, y, z ∈ A. So D([[x, y], z]) = [[D(x), y], z] + [[x, G(y)], z] + [[x, y], D(z)] for all x, y, z ∈ A. Therefore, the mapping D : A → A is a 3-Lie derivation.



Corollary 3.2. Let ε, θ, p1 , p2 , p3 , q1 , q2 , q3 be positive real numbers such that p1 , p2 , p3 < 1 and q1 , q2 , q3 < 3. Suppose that f : A → A is a mapping such that kDµ f (x, y, z)k ≤ θ(kxkp1 + kykp2 + kzkp3 ),

(3.3)

kf ([[x, y], z]) − [[f (x), y], z] − [[x, f (y)], z] − [[x, y], f (z)]k ≤ ε(kxkq1 + kykq2 + kzkq3 )

(3.4)

for all µ ∈ T1 and all x, y, z ∈ A. Then there exists a unique 3-Lie derivation D : A → A such that kf (x) − D(x)k ≤

θ 1 1 1 { kxkp1 + kxkp2 + kxkp3 } 3 2 − 2p1 2 − 2p2 2 − 2p3

for all x ∈ A.

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Theorem 3.3. Let ψ : A3 → [0, ∞) be a function satisfying (2.11). Suppose that f : A → A is a mapping satisfying kDµ f (x, y, z)k ≤ ψ(x, y, z), kf ([[x, y], z]) − [[f (x), y], z] − [[x, f (y)], z] − [[x, y], f (z)]k ≤ ψ(x, y, z) for all µ ∈ T1 and all x, y, z ∈ A. Then there exists a unique 3-Lie derivation D : A → A such that 1e kf (x) − D(x)k ≤ ψ(x, x, x) (3.5) 6 for all x ∈ A, where ψe is given in Theorem 2.5. Proof. By the proof of Theorem 2.3, there exists a unique C-linear mapping D : A → A satisfying (3.5) and D(x) := lim 2n f ( n→∞

x ) 2n

for all x ∈ A. The rest of proof is similar to the proof Theorem 3.1.



Corollary 3.4. Let ε, θ, p1 , p2 , p3 , q1 , q2 and q3 be non-negative real numbers such that p1 , p2 , p3 > 1 and q1 , q2 , q3 > 3. Suppose that f : A → B is a mapping satisfying (3.3) and (3.4). Then there exists a unique 3-Lie derivation D : A → A such that θ 1 1 1 kf (x) − H(x)k ≤ { p1 kxkp1 + p2 kxkp2 + p3 kxkp3 } 3 2 −2 2 −2 2 −2 for all x ∈ A. Acknowledgments This work was supported by Daejin University. References [1] M. Almahalebi, A. Charifi, C. Park and S. Kabbaj, Hyerstability results for a generalized radical cubic functional equation related to additive mapping in non-Archimedean Banach spaces, J. Fixed Point Theory Appl. 20 (2018), 2018:40. [2] T. Aoki, On the stability of the linear transformation in Banach spaces, J. Math. Soc. Japan 2 (1950), 64–66. [3] M. Eshaghi Gordji, S. Bazeghi, C. Park and S. Jang, Ternary Jordan ring derivations on Banach ternary algebras: A fixed point approach, J. Comput. Anal. Appl. 21 (2016), 829–834. [4] M. Eshaghi Gordji, V. Keshavarz, J. Lee and D. Shin, Stability of ternary m-derivations on ternary Banach algebras, J. Comput. Anal. Appl. 21 (2016), 640–644. [5] M. Eshaghi Gordji, V. Keshavarz, J. Lee, D. Shin and C. Park, Approximate ternary Jordan ring homomorphisms in ternary Banach algebras, J. Comput. Anal. Appl. 22 (2017), 402–408. [6] M. Eshaghi Gordji, V. Keshavarz, C. Park and S. Jang, Ulam-Hyers stability of 3-Jordan homomorphisms in C ∗ -ternary algebras, J. Comput. Anal. Appl. 22 (2017), 573–578. [7] M. Eshaghi Gordji, V. Keshavarz, C. Park and J. Lee, Approximate ternary Jordan bi-derivations on Banach Lie triple systems, J. Comput. Anal. Appl. 21 (2017), 45–51.

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[8] Z. Gajda, On stability of additive mappings, Int. J. Math. Math. Sci. 14 (1991), 431–434. [9] P. Gˇavruta, A generalization of the Hyers-Ulam-Rassias stability of approximately additive mappings, J. Math. Anal. Appl. 184 (1994), 431-436. [10] D. H. Hyers, On the stability of the linear functional equation, Proc. Nat. Acad. Sci. U. S. A. 27 (1941), 222–224. [11] A. Najati and A. Ranjbari, Stability of homomorphisms for a 3D CauchyJensen type functional equation on C ∗ -ternary algebras, J. Math. Anal. Appl. 341 (2008), 62-79. [12] L. Naranjani, M. Hassani and M. Mirzavaziri, Local higher derivations on C ∗ -algebras are higher derivations, Int. J. Nonlinear Anal. Appl. 9 (2018), 111–115. [13] C. Park, Homomorphisms between Poisson JC ∗ -algebras, Bull. Braz. Math. Soc. 36 (2005), 79–97. [14] C. Park, C ∗ -ternary biderivations and C ∗ -ternary bihomomorphisms, Math. 6 (2018), Art. 30. [15] C. Park, Bi-additive s-functional inequalities and quasi-∗-multipliers on Banach algebras, Math. 6 (2018), Art. 31. [16] C. Park, J. Lee and D. Shin, Stability of J ∗ -derivations, Int. J. Geom. Methods Mod. Phys. 9 (2012), Art. ID 1220009, 10 pages. [17] H. Piri, S. Aslani, V. Keshavarz, Th. M. Rassias, C. Park and Y. Park, Approximate ternary quadratic 3-derivations on ternary Banach algebras and C ∗ -ternary rings, J. Comput. Anal. Appl. 24 (2018), 1280–1291. [18] M. Raghebi Moghadam, Th. M. Rassias, V. Keshavarz, C. Park and Y. Park, Jordan homomorphisms in C ∗ -ternary algebras and JB ∗ -triples, J. Comput. Anal. Appl. 24 (2018), 416–424. [19] Th. M. Rassias, On the stability of the linear mapping in Banach spaces, Proc. Am. Math. Soc. 72 (1978), 297–300. [20] R. F. Rostami, Lie ternary (σ, τ, xi)-derivations on Banach ternary algebras, Int. J. Nonlinear Anal. Appl. 9 (2018), 41–53. [21] S. M. Ulam, Problems in Modern Mathematics, Chapter VI, Science ed., Wiley, New York, 1940. [22] H. Yuan and L. Chen, Lie n superderivations and generalized Lie n superderivations of superalgebras, Open Math. 16 (2018), 196–209.

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NEUTROSOPHIC EXTENDED TRIPLET GROUPS AND HOMOMORPHISMS IN C ∗ -ALGEBRAS JUNG RYE LEE, CHOONKIL PARK∗ , AND XIAOHONG ZHANG Abstract. C ¸ elik, Shalla and Olgun [2] defined neutro-homomorphisms in neutrosophic extended triplet groups and Zhang et al. [8] investigated neutro-homomorphisms in neutrosophic extended triplet groups. In this note, we apply the results on neutro-homomorphisms in neutrosophic extended triplet groups to investigate C ∗ -algebra homomorphisms in unital C ∗ -algebras.

1. Introduction and preliminaries As an extension of fuzzy sets and intuitionistic fuzzy sets, Smarandache [4] proposed the new concept of neutrosophic sets. Definition 1.1. ([5, 6]) Let N be a nonempty set together with a binary operation ∗. Then N is called a neutrosophic extended triplet set if, for any a ∈ N , there exist a neutral of a (denoted by neut(a)) and an opposite of a (denoted by anti(a)) such that neut(a) ∈ N , anti(a) ∈ N and a ∗ neut(a) = neut(a) ∗ a = a, a ∗ anti(a) = anti(a) ∗ a = neut(a). The triplet (a, neut(a), anti(a)) is called a neutrosophic extended triplet. Note that, for a neutrosophic triplet set (N, ∗) and a ∈ N , neut(a) and anti(a) may not be unique. Definition 1.2. ([5, 6]) Let (N, ∗) be a neutrosophic extended triplet set. Then N is called a neutrosophic extended triplet group (NETG) if the following conditions hold: (1) (N, ∗) is well-defined, i.e., for any a, b ∈ N , one has a ∗ b ∈ N ; (2) (N, ∗) is associative, i.e., (a ∗ b) ∗ c = a ∗ (b ∗ c) for all a, b, c ∈ N . N is called a commutative neutrosophic extended triplet group if, for all a, b ∈ N , a ∗ b = b ∗ a. Let A be a unital C ∗ -algebra with multiplication operation •, unit e and unitary group U (A) := {u ∈ A | u∗ • u = u • u∗ = e}. Then u • v ∈ U (A) and (u • v) • w = u • (v • w) for all u, v, w ∈ U (A) (see [3]). So (U (A), •) is an NETG. 2010 Mathematics Subject Classification. Primary 46L05, 03E72, 94D05. Key words and phrases. neutro-homomorphism; neutrosophic extended triplet group; homomorphism in unital C ∗ -algebra; perfect neutrosophic extended triplet group. ∗ Corresponding author: C. Park (email: [email protected], fax: +82-2-2211-0019). 136

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Proposition 1.3. ([7]) Let (N, ∗) be an NETG. Then (1) neut(a) is unique for each a ∈ N ; (2) neut(a) ∗ neut(a)= neut(a) for each a ∈ N . Note that u • e = e • u = u for any u ∈ (U (A), •). By Proposition 1.3, neut(u) = e for each u ∈ U (A). Definition 1.4. ([7]) Let (N, ∗) be an NETG. Then N is called a weak commutative neutrosophic extended triplet group (briefly, WCNETG) if a ∗ neut(b) = neut(b) ∗ a for all a, b ∈ N . Since neut(v) = e for all v ∈ U (A), u • neut(v) = neut(v) • u for all u, v ∈ U (A). So (U (A), •) is a WCNETG.

2. Neutrosophic extended triplet groups and C ∗ -algebra homomorphisms in unital C ∗ -algebras Definition 2.1. ([8]) Let (N, ∗) be a WCNETG. Then N is called a perfect NETG if anti(neut(a))= neut(a) for all a ∈ N . Since anti(e) = e and neut(u) = e for all u ∈ U (A), anti(neut(u)) = neut(u) = e for all u ∈ U (A). Thus (U (A), •) is a perfect NETG. Definition 2.2. ([1, 2]) Let (N1 , ∗) and (N2 , ∗) be neutrosophic extended triplet groups. A mapping f : N1 → N2 is called a neutro-homomorphism if f (x ∗ y) = f (x) ∗ f (y) for all x, y ∈ N1 . From now on, assume that A is a unital C ∗ -algebra with multiplication operation •, unit e and unitary group U (A) and that B is a unital C ∗ -algebra with multiplication operation • and unitary group U (B). Definition 2.3. Let (U (A), •) and (U (B), •) be unitary groups of unital C ∗ -algebras A and B, respectively. A mapping h : U (A) → U (B) is called a neutro-∗-homomorphism if h(u • v) = h(u) • h(v), h(u∗ ) = h(u)∗ for all u, v ∈ U (A). Theorem 2.4. Let A and B be unital C ∗ -algebras. Let H : A → B be a C-linear mapping and let h : (U (A), •) → (U (B), •) be a neutro-∗-homomorphism. If H|U (A) = h, then H : A → B is a C ∗ -algebra homomorphism. 137

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Proof. Since H : A → B is C-linear and x, yP ∈ A are finite linear combinations of unitary P n elements (see [3]), i.e., x = m λ u , y = j=1 j j i=1 µi vi (λj , µi ∈ C, uj , vi ∈ U (A)), H(x • y) = H((

m X

n m X n X X λj uj ) • ( µi vi )) = H( λj µi (uj • vi ))

j=1

=

m X n X

i=1

j=1 i=1 m X n X

λj µi H(uj • vi ) =

j=1 i=1

=

λj µi h(uj • vi )

j=1 i=1

m X n X

λj µi h(uj ) • h(vi ) =

j=1 i=1

m X n X

λj µi H(uj ) • H(vi )

j=1 i=1

m n X X = H( λj uj ) • H( µi vi ) = H(x) • H(y) j=1

i=1

for all x, y ∈ A. Since H : A → B is C-linear Pm and each x ∈ A is a finite linear combination of unitary elements (see [3]), i.e., x = j=1 λj uj (λj ∈ C, uj ∈ U (A)), ∗

H(x ) = H((

m X

∗

λj uj ) ) = H(

j=1

=

m X j=1

m X j=1

m X

λj H(uj )∗ = (

λj u∗j )

=

m X

λj H(u∗j )

j=1

λj H(uj ))∗ = H(

j=1

=

m X

λj h(u∗j )

j=1 m X

=

m X

λj h(uj )∗

j=1

λj uj )∗ = H(x)∗

j=1

for all x ∈ A. Thus the C-linear mapping H : A → B is a C ∗ -algebra homomorphism.  3. Conclusions In this note, we have studied unitary groups of unital C ∗ -algbras as neutrosophic extended triplet groups and have extended neutro-homomorphisms in neutrosophic extended triplet groups to neutro-∗-homomorphisms in unitary groups of unital C ∗ -algebras. We have obtained C ∗ -algebra homomorphisms in unital C ∗ -algebras by using neutro-∗homomorphisms in unitary groups of unital C ∗ -algebras. Acknowledgments This work was supported by Daejin University. Competing interests The authors declare that they have no competing interests. References [1] M. Bal, M.M. Shalla, N. Olgun, Neutrosophic triplet cosets and quotient groups, Symmetry 10 (2018), 10:126. doi:10.3390/sym10040126. [2] M. C ¸ elik, M.M. Shalla, N. Olgun, Fundamental homomorphism theorems for neutrosophic extended triplet groups, Symmetry 10 (2018), 10:321. doi:10.3390/sym10080321. 138

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[3] R.V. Kadison, J.R. Ringrose, Fundamentals of the Theory of Operator Algebras: Elementary Theory, Academic Press, New York, 1983. [4] F. Smarandache, Neutrosophic set–A generialization of the intuituionistics fuzzy sets, Int. J. Pure Appl. Math. 3 (2005), 287–297. [5] F. Smarandache, Neutrosophic Perspectives: Triplets, Duplets, Multisets, Hybrid Operators, Modal Logic, Hedge Algebras and Applications, Pons Publishing House, Brussels, Belgium, 2017. [6] F. Smarandache, M. Ali, Neutrosophic triplet group, Neural Comput. Appl. 29 (2018), 595–601. [7] X. Zhang, Q.Q. Hu, F. Smarandache, X.G. An, On neutrosophic triplet groups: Basic properties, NT-subgroups and some notes, Symmetry 10 (2018), 10:289. doi:10.3390/sym10070289. [8] X. Zhang, X. Mao, F. Smarandache, C. Park, On homomorphism theorem for perfect neutrosophic extended triplet groups, Information 9 (2018), 9:237. doi:10.3390/info9090237. Jung Rye Lee Department of Mathematics, Daejin University, Kyunggi 11159, Korea E-mail address: [email protected] Choonkil Park Research Institute for Natural Sciences, Hanyang University, Seoul 04763, Korea E-mail address: [email protected] Xiaohong Zhang Department of Mathematics, School of Arts and Sciences, Shaanxi University of Science and Technology, Xi’an, P. R. China; Department of Mathematics, College of Arts and Sciences, Shanghai Maritime University, Shanghai, P.R. China E-mail address: [email protected]

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Orthogonal stability of a quadratic functional inequality: a fixed point approach Shahrokh Farhadabadi1∗ and Choonkil Park2 1

Computer Engineering Department, Komar University of Science and Technology, Sulaymaniyah 46001, Kurdistan Region, Iraq 2 Research Institute for Natural Sciences Hanyang University, Seoul 133-791, Korea e-mail: [email protected]; [email protected]

Abstract. Let f : X → Y be a mapping from an orthogonality space (X , ⊥) into a real Banach space (Y, k · k). Using fixed point method, we prove the Hyers-Ulam stability of the orthogonally quadratic functional inequality

 x − y − z  y − x − z 

x + y + z

f + f + f

2 2 2

z − x − y

(0.1) (z) ≤ +f − f (x) − f (y)

f

2 for all x, y, z ∈ X with x⊥y, x⊥z and y⊥z. Keywords: Hyers-Ulam stability; quadratic functional equation; fixed point method; quadratic functional inequality; orthogonality space.

1. Introduction and preliminaries Studying functional equations by focusing on their approximate and exact solutions conduces to one of the most substantial significant study brunches in functional equations, what we call “the theory of stability of functional equations”. This theory specifically analyzes relationships between approximate and exact solutions of functional equations. Actually a functional equation is considered to be stable if one can find an exact solution for any approximate solution of that certain functional equation. Another related and close term in this area is superstability, which has a similar nature and concept to the stability problem. As a matter of fact, superstability for a given functional equation occurs when any approximate solution is an exact solution too. In such this situation the functional equation is called superstable. In 1940, the most preliminary form of stability problems was proposed by Ulam [58]. He gave a talk and asked the following: “when and under what conditions does an exact solution of a functional equation near an approximately solution of that exist?” In 1941, this question that today is considered as the source of the stability theory, was formulated and solved by Hyers [26] for the Cauchy’s functional equation in Banach spaces. Then the result of Hyers was generalized by Aoki [1] for additive mappings and by Rassias [47] for linear mappings by considering the unbounded Cauchy difference kf (x + y) − f (x) − f (y)k ≤ ε(kxkp + kykp ), (ε > 0

∗

2010 Mathematics Subject Classification: 39B55, 39B52, 47H10. Corresponding author: S. Farhadabadi (email: [email protected])

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0, p ∈ [0, 1)). In 1994, G˘ avrut¸a [23] provided a further generalization of Rassias’ theorem in which he replaced the unbounded Cauchy difference by the general control function ϕ(x, y) for the existence of a unique linear mapping. The first author treating the stability of the quadratic functional equation f (x + y) + f (x − y) = 2f (x) + 2f (y) was Skof [55] by proving that if f is a mapping from a normed

space X into a Banach space Y satisfying f (x + y) + f (x − y) − 2f (x) − 2f (y) ≤ ε, for some

ε > 0, then there is a unique quadratic mapping g : X → Y such that f (x) − g(x) ≤ 2ε . Cholewa [13] extended the Skof’s theorem by replacing X by an abelian group G. The Skof’s result was later generalized by Czerwik [14] in the spirit of Ulam-Hyers-Rassias. For more epochal information and various aspects about the stability of functional equations theory, we refer the reader to the monographs ([6, 11, 12, 15, 16, 20, 27, 30, 41, 42, 43, 46], [48]–[51], [54]), which also include many interesting results concerning the stability of different functional equations in many various spaces. 1

Assume that (X , h·, ·i) is a real inner product space with the usual Hilbert norm k · k = h·, ·i 2 . Moreover, consider the orthogonal Cauchy functional equation f (x + y) = f (x) + f (y),

x⊥y

in which ⊥ is an abstract orthogonality relation. By the Pythagorean theorem, f : X → R defined by f (x) = kxk2 = hx, xi is a solution of the conditional equation. Of course, this function does not satisfy the additivity equation everywhere. Thus orthogonally Cauchy functional equation is not equivalent to the classic Cauchy equation on the whole inner product space (X , h·, ·i). Pinsker [44] characterized orthogonally additive functionals on an inner product space when the orthogonality is the ordinary one in such spaces. Sundaresan [56] generalized this result to arbitrary Banach spaces equipped with the Birkhoff-James orthogonality. The orthogonal Cauchy functional equation was first investigated by Gudder and Strawther [25]. They defined ⊥ by a system consisting of five axioms and described the general semi-continuous real-valued solution of conditional Cauchy functional equation. In 1985, R¨ atz [52] introduced his new definition of orthogonality by using more restrictive axioms than of Gudder and Strawther. Furthermore, he investigated the structure of orthogonally additive mappings. R¨ atz and Szab´o [53] investigated the problem in a rather more general framework. We now recall the concept of orthogonality space in the sense of R¨atz [52], and then proceed it to prove our results for the orthogonally functional inequality (0.1). Definition 1.1. Suppose X is a real vector space with dim X ≥ 2 and ⊥ is a binary relation on X with the following properties: (O1 ) totality of ⊥ for zero: x⊥0, 0⊥x for all x ∈ X ; (O2 ) independence: if x, y ∈ X − 0, x⊥y, then x, y are linearly independent; (O3 ) homogeneity: if x, y ∈ X , x⊥y, then αx⊥βy for all α, β ∈ R; (O4 ) the Thalesian property: if P is a 2-dimensional subspace of X , x ∈ P and λ ∈ R+ , which is the set of nonnegative real numbers, then there exists y0 ∈ P such that x⊥y0 and x + y0 ⊥λx − y0 . The pair (X , ⊥) is called an orthogonality space and it becomes an orthogonality normed space when the orthogonality space equipped with a normed structure.

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3

Some interesting examples are (i) The trivial orthogonality on a vector space X defined by (O1 ), and for non-zero elements x, y ∈ X , x⊥y if and only if x, y are linearly independent. (ii) The ordinary orthogonality on an inner product space (X , h·, ·i) given by x⊥y if and only if hx, yi = 0. (iii) The Birkhoff-James orthogonality on a normed space (X , k · k) defined by x⊥y if and only if



x + λy ≥ x for all λ ∈ R. The relation ⊥ is called symmetric if x⊥y implies that y⊥x for all x, y ∈ X . Clearly examples (i) and (ii) are symmetric but example (iii) is not. It is remarkable to note, however, that a real normed space of dimension greater than 2 is an inner product space if and only if the Birkhoff-James orthogonality is symmetric. There are several orthogonality notions on a real normed space such as Birkhoff-James, Boussouis, Singer, Carlsson, unitary-Boussouis, Roberts, Phythagorean, isosceles and Diminnie (see [3]–[5], [10, 18, 29]). Ger and Sikorska [24] investigated the orthogonal stability of the Cauchy functional equation f (x + y) = f (x) + f (y), namely, they showed that if f is a mapping from an orthogonality space X into a real Banach space Y and kf (x + y) − f (x) − f (y)k ≤ ε, for all x, y ∈ X with x⊥y and some ε > 0, then there exists exactly one orthogonally additive mapping g : X → Y such that kf (x) − g(x)k ≤

16 3 ε,

for

all x ∈ X . Consider the classic quadratic functional equation f (x + y) + f (x − y) = 2f (x) + 2f (y) on the real inner product space (X , h·, ·i). Then the important parallelogram identity kx + yk2 + kx − yk2 = 2kxk2 + kyk2 which holds entirely in a square norm on an inner product space, shows that f : X → R defined by f (x) = kxk2 = hx, xi, is a solution for the quadratic functional equation on the whole inner product space X , (particularly in where x⊥y). The orthogonally quadratic functional equation f (x + y) + f (x − y) = 2f (x) + 2f (y),

x⊥y

was first investigated by Vajzovi´c [59] when X is a Hilbert space, Y is the scalar field, f is continuous and ⊥ means the Hilbert space orthogonality. Later, Drljevi´c [19], Fochi [22], Moslehian [34, 35] and Szab´o [57] generalized the Vajzovi´c’s results. See also [36, 37, 40]. The following quadratic 3-variables functional equation f

x + y + z  2

+f

x − y − z 

+f

y − x − z 

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

2

+f

z − x − y 2 (1.1)

has been introduced and solved by S. Farhadabadi, J. Lee and C. Park on vector spaces in [21]. It has been also shown that the functional equation (1.1) is equivalent to the classic quadratic functional

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equation in vector spaces. In any inner product space (X , h·, ·i), it is easy to verify that       x+y+z x+y+z x−y−z x−y−z y−x−z y−x−z , + , + , 2 2 2 2 2 2  

   z−x−y z−x−y + , = x, x + y, y + z, z 2 2 for all x, y, z ∈ X . For this obvious reason, similar to the classic quadratic functional equation, the mapping f (x) = hx, xi can also be a solution for the 3-variables equation (1.1) on the whole inner product space X (particularly, for the case x⊥y, y⊥z and x⊥z). Fixed point theory has a basic role in applications of many considerable branches in mathematics specially in stability problems. In 1996, Isac and Rassias [28] were the first to provide applications of stability theory of functional equations for the proof of new fixed point theorems with applications. In view of the fact that, we will use methods related to fixed point theory, we give briefly some useful information, a definition and a fundamental result in fixed point theory. Definition 1.2. 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 1.3. ([7, 17]) Let (X , d) be a complete generalized metric space and let J : X → X be a strictly contractive mapping with Lipschitz constant α < 1. Then for each given element x ∈ X , either d(J n x, J n+1 x) = ∞ for all nonnegative integers n or there exists a positive integer n0 such that (1) d(J n x, J n+1 x) < ∞,

∀n ≥ n0 ;

(2) the sequence {J x} converges to a fixed point y ∗ of J ; n

(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−α d(y, J y)

for all y ∈ Y.

In 2003, C˘adariu and Radu [7, 8, 45] exerted the above definition and fixed point theorem to prove some stability problems for the Jensen and Cauchy functional equations. During the last decade, by applying fixed point methods, stability problems of several functional equations have been extensively investigated by a number of authors (see [2, 8, 9, 31, 33, 38, 39, 45]). Throughout this paper, (X , ⊥) is an orthogonality space and (Y, k · k) is a real Banach space.

2. Solution and Hyers-Ulam stability of the functional inequality (0.1) In this section, we first solve the orthogonally quadratic functional inequality (0.1) by proving an orthogonal superstability proposition, and then we prove its Hyers-Ulam stability in orthogonality spaces.

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Definition 2.1. A mapping f : X → Y is called an (exact) orthogonally quadratic mapping if f (x + y) + f (x − y) = f (x) + f (y)

(2.1)

for all x, y ∈ X , with x⊥y. And it is called an approximate orthogonally quadratic mapping if

 x − y − z  y − x − z 

x + y + z

f + f + f

2 2 2

z − x − y

(z) ≤ +f − f (x) − f (y)

f

2

(2.2)

for all x, y, z ∈ X with x⊥y, y⊥z and x⊥z. Proposition 2.2. Each approximate orthogonally quadratic mapping in the form of (2.2) is also an (exact) orthogonally quadratic mapping satisfying (2.1). Proof. Assume that f : X → Y is an approximate orthogonally quadratic mapping satisfying (2.2). Since 0⊥0, letting x = y = z = 0 in (2.2), we have



2f (0) ≤ f (0) = 0 and so f (0) = 0. Since (x + y)⊥0 for all x, y ∈ X , replacing x, y and z by x + y, 0 and 0 in (2.2), respectively, we conclude that

x + y

 −x − y 

+ 2f − f (x + y) ≤ f (0) = 0,

2f 2 2 which implies f

x + y

2 for all x, y ∈ X (particularly, with x⊥y).

+f

 −x − y  2

=

1 f (x + y) 2

Replacing y by −y in the above equality, we get y − x 1 x − y f +f = f (x − y) 2 2 2 for all x, y ∈ X (particularly, with x⊥y).

(2.3)

(2.4)

Since x⊥0 for all x ∈ X , letting z = 0 in (2.2), we obtain

x + y x − y y − x  −x − y 

+f +f +f

f 2 2 2 2



−f (x) − f (y) ≤ f (0) = 0 and so f

x + y

2 for all x, y ∈ X with x⊥y.

+f

 −x − y  2

+f

x − y 2

+f

y − x 2

= f (x) + f (y)

(2.5)

It follows from (2.3), (2.4) and (2.5) that 1 1 f (x + y) + f (x − y) = f (x) + f (y) 2 2 for all x, y ∈ X with x⊥y, which is the equation (2.1). Hence f : X → Y is an (exact) orthogonally quadratic mapping.



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Theorem 2.3. Let ϕ : X 3 → [0, ∞) be a function such that ϕ(0, 0, 0) = 0 and there exists an α < 1 with x y z  , , 2 2 2 for all x, y, z ∈ X , with x⊥y, y⊥z and x⊥z. Let f : X → Y be an even mapping satisfying

 x − y − z  y − x − z 

x + y + z

f +f +f

2 2 2



z − x − y



(z) +f ≤ − f (x) − f (y)

+ ϕ(x, y, z)

f

2 ϕ(x, y, z) ≤ 4αϕ

(2.6)

(2.7)

for all x, y, z ∈ X , with x⊥y, y⊥z and x⊥z. Then there exists a unique orthogonally quadratic mapping Q : X → Y such that

f (x) − Q(x) ≤

α ϕ(x, 0, 0) 1−α

(2.8)

for all x ∈ X .  Proof. Consider the set S := h : X → Y and introduce the generalized metric on S: n o

d(g, h) = inf µ ∈ R+ : g(x) − h(x) ≤ µϕ(x, 0, 0), ∀x ∈ X , where, as usual, inf ∅ = +∞. It is easy to show that (S, d) is complete (see [32]). Now we consider the linear mapping J : S → S such that J g(x) :=

1 g(2x) 4

for all g ∈ S and all x ∈ X . Since 0⊥0, letting x = y = z = 0 in (2.7), we have



2 f (0) ≤ f (0) + ϕ(0, 0, 0). So f (0) = 0.

Since x⊥0 for all x ∈ X , letting y = z = 0 in (2.7), we get 4f

x 2




− f (x) ≤ ϕ(x, 0, 0) for all

x ∈ X . Dividing both sides by 4, putting 2x instead of x and then using (2.6), we obtain

1

1

f (2x) − f (x) ≤ ϕ(2x, 0, 0) ≤ αϕ(x, 0, 0) 4 4 for all x ∈ X , which clearly yields d(J f, f ) ≤ α. (2.9)

Let g, h ∈ S be given such that d(g, h) = ε. Then g(x) − h(x) ≤ εϕ(x, 0, 0) for all x ∈ X . Hence the definition of J g and (2.6), result that





J g(x) − J h(x) = 1 g(2x) − 1 h(2x) ≤ 1 εϕ(2x, 0, 0) ≤ αεϕ(x, 0, 0)

4

4 4 for all x ∈ X , which implies that d(J g, J h) ≤ αε = αd(g, h) for all g, h ∈ S. Thus J is a strictly contractive mapping with Lipschitz constant α < 1. According to Theorem 1.3, there exists a mapping Q : X → Y satisfying the following:

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(1) Q is a fixed point of J , i.e., J Q = Q, and so 1 Q(2x) = Q(x) 4 for all x ∈ X . The mapping Q is a unique fixed point of J in the set  M = g ∈ S : d(g, f ) < ∞ .

(2.10)

This signifies that Q is a unique mapping satisfying (2.10) such that there exists a µ ∈ (0, ∞) satisfying

f (x) − Q(x) ≤ µϕ(x, 0, 0) for all x ∈ X ; (2) d(J n f, Q) → 0 as n → ∞. So, we conclude that lim

n→∞

1 f (2n x) = Q(x) 4n

(2.11)

for all x ∈ X ; (3) d(f, Q) ≤

1 1−α d(f, J f ),

which gives by (2.9) the inequality d(f, Q) ≤

α . 1−α

This proves that the inequality (2.8) holds. To end the proof we show that Q is an orthogonally quadratic mapping. By (2.11), (2.7), (2.6) and the fact that α < 1,

x + y + z  x − y − z  y − x − z  z − x − y

+Q +Q +Q

Q 2 2 2 2

−Q(x) − Q(y)

1

= lim n f 2n−1 (x + y + z) + f 2n−1 (x − y − z) n→∞ 4





+f 2n−1 (y − z − x) + f 2n−1 (z − x − y) − f 2n x − f 2n y



1 1

≤ lim n f 2n z + lim n ϕ 2n x, 2n y, 2n z n→∞ 4 n→∞ 4

≤ Q(z) + lim αn ϕ(x, y, z)

n→∞ = Q(z) for all x, y, z ∈ X , with x⊥y, x⊥z and y⊥z. And, now applying Proposition 2.2, we obatin that Q is an orthogonally quadratic mapping and the proof is complete.



Theorem 2.4. Let ϕ : X 3 → [0, ∞) be a function such that ϕ(0, 0, 0) = 0 and there exists an α < 1 with
α ϕ 2x, 2y, 2z 4 for all x, y, z ∈ X , with x⊥y, y⊥z and x⊥z. Let f : X → Y be an even mapping satisfying (2.7). Then ϕ(x, y, z) ≤

there exists a unique orthogonally quadratic mapping Q : X → Y such that

f (x) − Q(x) ≤ 1 ϕ(x, 0, 0) 1−α for all x ∈ X .

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Proof. Let (S, d) be the generalized metric space defined in the proof of Theorem 2.3. Now we consider the linear mapping J : S → S such that x J g(x) := 4g 2 for all x ∈ X . Similar to the proof of Theorem 2.3, from (2.7) one can get

x

− f (x) ≤ ϕ(x, 0, 0)

4f 2 for all x ∈ X , which means d(f, J f ) ≤ 1. We can also show that J is a strictly contractive mapping with Lipschitz constant α < 1. So by applying Theorem 1.3 again, we have 1 1 d(f, J f ) ≤ 1−α 1−α which implies that the inequality (2.12) holds. d(f, Q) ≤

The rest of the proof is similar to the proof of the previous theorem.



Corollary 2.5. Let X be a normed orthogonality space. Let δ be a nonnegative real number and p 6= 2 be a positive real number. Let f : X → Y be an even mapping satisfying

x + y + z 

x − y − z  y − x − z  z − x − y

+f +f +f − f (x) − f (y)

f 2 2 2 2

p p p
≤ f (z) + δ kxk + kyk + kzk for all x, y, z ∈ X , with x⊥y, y⊥z and x⊥z. Then there exists a unique orthogonally quadratic mapping Q : X → Y such that

p

f (x) − Q(x) ≤ 2 δkxkp 2p − 4

for all x ∈ X . p

p

Proof. Define ϕ(x, y, z) := δ kxk + kyk + kzk

p


for all x, y, z ∈ X .

First assume that 0 < p < 2. Take α := 2p−2 . Since p < 2, obviously α < 1. Hence there exists an α < 1 such that p p p
ϕ(x, y, z) = δ kxk + kyk + kzk p p p
= 4α2−p δ kxk + kyk + kzk  x p y p z p 





= 4αδ + + 2 2 2 x y z  = 4αϕ , , 2 2 2 for all x, y, z ∈ X (particularly, with x⊥y, y⊥z and x⊥z). The recent term allows to use Theorem 2.3. So by applying Theorem 2.3, it follows from (2.8) that p

f (x) − Q(x) ≤ 2 δkxkp 4 − 2p for all x ∈ X . For the case p > 2, taking α := 22−p , and then applying Theorem 2.4, we similarly obtain the desired result.



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[53] J. R¨atz and Gy. Szab´ o, On orthogonally additive mappings IV, Aequationes Math. 38 (1989), 73–85. [54] K. Ravi, E. Thandapani and B. V. Senthil Kumar, Solution and stability of a reciprocal type functional equation in several variables, J. Nonlinear Sci. Appl. 7 (2014), 18–27. [55] F. Skof, Propriet` a locali e approssimazione di operatori, Rend. Sem. Mat. Fis. Milano. 53 (1983), 113–129. [56] K. Sundaresan, Orthogonality and nonlinear functionals on Banach spaces, Proc. Amer. Math. Soc. 34 (1972), 187–190. [57] Gy. Szab´ o, Sesquilinear-orthogonally quadratic mappings, Aequationes Math. 40 (1990), 190–200. [58] S.M. Ulam, Problems in Modern Mathematics, Wiley, New York, 1960. ¨ [59] F. Vajzovi´c, Uber das Funktional H mit der Eigenschaft: (x, y) = 0 ⇒ H(x + y) + H(x − y) = 2H(x) + 2H(y), Glasnik Mat. Ser. III 2 (22) (1967), 73–81.

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INTEGRAL INEQUALITIES FOR ASYMMETRIZED SYNCHRONOUS FUNCTIONS S. S. DRAGOMIR 1;2 Abstract. In this paper we establish some integral inequalities for the product of asymmetrized synchronous/asynchronous functions. Some examples for integrals of monotonic functions, including power, logarithmic and sin functions are also provided.

1. Introduction For a function f : [a; b] ! C we consider the symmetrical transform of f on the interval [a; b] ; denoted by f[a;b] or simply f , when the interval [a; b] is implicit, as de…ned by 1 (1.1) f (t) := [f (t) + f (a + b t)] ; t 2 [a; b] : 2 The anti-symmetrical transform of f on the interval [a; b] is denoted by f~[a;b] ; or simply f~ and is de…ned by 1 f~ (t) := [f (t) 2

f (a + b

t)] ; t 2 [a; b] :

It is obvious that for any function f we have f + f~ = f: If f is convex on [a; b] ; then for any t1 ; t2 2 [a; b] and ; 0 with + = 1 we have 1 f ( t1 + t2 ) = [f ( t1 + t2 ) + f (a + b t1 t2 )] 2 1 = [f ( t1 + t2 ) + f ( (a + b t1 ) + (a + b t2 ))] 2 1 [ f (t1 ) + f (t2 ) + f (a + b t1 ) + f (a + b t2 )] 2 1 1 = [f (t1 ) + f (a + b t1 )] + [f (t2 ) + f (a + b t2 )] 2 2 = f (t1 ) + f (t2 ) ; which shows that f is convex on [a; b] : Consider the real numbers a < b and de…ne the function f0 : [a; b] ! R, f0 (t) = t3 : We have [6] i 3 1h3 3 1 3 2 3 f0 (t) := t + (a + b t) = (a + b) t2 (a + b) t + (a + b) 2 2 2 2 1991 Mathematics Subject Classi…cation. 26D15; 25D10. µ Key words and phrases. Monotonic functions, Synchronous functions, Cebyšev’ s inequality, Integral inequalities. 1

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S. S. DRAGOM IR 1;2

2

for any t 2 R.

Since the second derivative f0

00

(t) = 3 (a + b) ; t 2 R, then f0 is strictly convex

> 0 and strictly concave on [a; b] if a+b on [a; b] if 2 < 0: Therefore if a < 0 < b a+b with 2 > 0; then we can conclude that f0 is not convex on [a; b] while f0 is convex on [a; b] : We can introduce the following concept of convexity [6], see also [9] for an equivalent de…nition. a+b 2

De…nition 1. We say that the function f : [a; b] ! R is symmetrized convex (concave) on the interval [a; b] if the symmetrical transform f is convex (concave) on [a; b] : Now, if we denote by Con [a; b] the closed convex cone of convex functions de…ned on [a; b] and by SCon [a; b] the closed convex cone of symmetrized convex functions, then from the above remarks we can conclude that (1.2)

Con [a; b]

SCon [a; b] :

Also, if [c; d] [a; b] and f 2 SCon [a; b] ; then this does not imply in general that f 2 SCon [c; d] : We have the following result [6], [9] : Theorem 1. Assume that f : [a; b] ! R is symmetrized convex and integrable on the interval [a; b] : Then we have the Hermite-Hadamard inequalities Z b 1 f (a) + f (b) a+b : f (t) dt (1.3) f 2 b a a 2 We also have [6]:

Theorem 2. Assume that f : [a; b] ! R is symmetrized convex on the interval [a; b] : Then for any x 2 [a; b] we have the bounds (1.4)

f

a+b 2

f (x)

f (a) + f (b) : 2

For a monograph on Hermite-Hadamard type inequalities see [8]. In a similar way, we can introduce the following concept as well: De…nition 2. We say that the function f : [a; b] ! R is asymmetrized monotonic nondecreasing (nonincreasing) on the interval [a; b] if the anti-symmetrical transform f~ is monotonic nondecreasing (nonincreasing) on the interval [a; b] : If f is monotonic nondecreasing on [a; b] ; then for any t1 ; t2 2 [a; b] we have 1 1 f~ (t2 ) f~ (t1 ) = [f (t2 ) f (a + b t2 )] [f (t1 ) f (a + b t1 )] 2 2 1 1 = [f (t2 ) f (t1 )] + [f (a + b t1 ) f (a + b t2 )] 2 2 0; which shows that f : [a; b] ! R is asymmetrized monotonic nondecreasing on the interval [a; b] : Consider the real numbers a < b and de…ne the function f0 : [a; b] ! R, f0 (t) = t2 : We have i 1h2 1 2 2 f~0 (t) := t (a + b t) = (a + b) t (a + b) 2 2

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INTEGRAL INEQUALITIES FOR ASYM M ETRIZED SYNCHRONOUS FUNCTIONS

and f~0

0

3

(t) = a + b; therefore f : [a; b] ! R is asymmetrized monotonic nonde-

creasing (nonincreasing) on the interval [a; b] provided a+b 2 > 0 (< 0) : So, if we take > 0; then f is asymmetrized monotonic nondecreasing on [a; b] a < 0 < b with a+b 2 but not monotonic nondecreasing on [a; b] : If we denote by M% [a; b] the closed convex cone of monotonic nondecreasing functions de…ned on [a; b] and by AM% [a; b] the closed convex cone of asymmetrized monotonic nondecreasing functions, then from the above remarks we can conclude that AM% [a; b] :

M% [a; b]

(1.5)

Also, if [c; d] [a; b] and f 2 AM% [a; b] ; then this does not imply in general that f 2 AM% [c; d] : We recall that the pair of functions (f; g) de…ned on [a; b] are called synchronous (asynchronous) on [a; b] if (1.6)

(f (t)

f (s)) (g (t)

g (s))

( )0

for any t; s 2 [a; b] : It is clear that if both functions (f; g) are monotonic nondecreasing (nonincreasing) on [a; b] then they are synchronous on [a; b] : There are also functions that change monotonicity on [a; b] ; but as a pair they are still synchronous. For instance if a < 0 < b and f; g : [a; b] ! R, f (t) = t2 and g (t) = t4 ; then (f (t)

f (s)) (g (t)

g (s)) = t2

s2

t4

s4 = t2

2

s2

t2 + s2

0

for any t; s 2 [a; b] ; which show that (f; g) is synchronous. De…nition 3. We say that the pair of functions (f; g) de…ned on [a; b] is called asymmetrized synchronous (asynchronous) on [a; b] if the pair of transforms f~; g~ is synchronous (asynchronous) on [a; b] ; namely f~ (t)

(1.7)

f~ (s) (~ g (t)

g~ (s))

( )0

for any t; s 2 [a; b] : It is clear that if f; g are asymmetrized monotonic nondecreasing (nonincreasing) on [a; b] then they are asymmetrized synchronous on [a; b] : One of the most important results for synchronous (asynchronous) and integrable µ functions f; g on [a; b] is the well-known Cebyš ev’s inequality: (1.8)

1 b

a

Z

a

b

f (t) g (t) dt

( )

1 b

a

Z

a

b

f (t) dt

1 b

a

Z

b

g (t) dt:

a

µ For integral inequalities of Cebyš ev’s type, see [1]-[5], [7], [10]-[18] and the references therein. Motivated by the above results, we establish in this paper some inequalities for asymmetrized synchronous (asynchronous) functions on [a; b] : Some examples for power, logarithm and sin functions are provided as well.

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4

2. Main Results We have the following result: Theorem 3. Assume that f; g are asymmetrized synchronous (asynchronous) and integrable functions on [a; b]. Then Z b (2.1) f~ (t) g (t) dt ( ) 0: a

Proof. We consider only the case of symmetrized synchronous and integrable functions. µ 1. By the Cebyš ev’s inequality (1.8) for f~; g~ we get Z b Z b Z b 1 1 1 f~ (t) g~ (t) dt f~ (t) dt g~ (t) dt: (2.2) b a a b a a b a a We have

Z

b

"Z

1 f~ (t) dt = 2

a

Z

b

f (t) dt

a

#

b

f (a + b

t) dt = 0

a

since, by the change of variable s = a + b t; t 2 [a; b] ; Z b Z b f (a + b t) dt = f (s) ds: a

a

Also,

(2.3)

Z

a

b

1 f~ (t) g~ (t) = 4 =

1 4 1 4

b

[f (t)

f (a + b

t)] [g (t)

g (a + b

t)] dt

a

Z

b

[f (t) g (t) + f (a + b

a Z b

[f (t) g (a + b

"Z

"Z

t) g (a + b

t) + f (a + b

a

1 = 4 1 4

Z

b

f (t) g (t) dt +

a

Z

t)] dt

t) g (t)] dt

b

f (a + b

t) g (a + b

t) dt

a

b

f (t) g (a + b

t) dt +

a

Z

b

f (a + b

t) g (t) dt

a

Z b 1 f (t) g (t) dt = 2 a Z b = f~ (t) g (t) dt

Z

b

f (a + b

t) g (t) dt

a

!

#

#

a

since, by the change of variable s = a + b t; t 2 [a; b] ; we have Z b Z b f (a + b t) g (a + b t) dt = f (t) g (t) dt a

and

Z

a

a

b

f (t) g (a + b

t) dt =

Z

b

f (a + b

t) g (t) dt:

a

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5

By (2.2) we then get the desired result (2.1). 2. An alternative proof is as follows. Since f~; g~ are synchronous, then a+b f~ 2

f~ (t)

g~ (t)

g~

a+b 2

0

for any t 2 [a; b] ; which is equivalent to (2.4) f~ (t) g~ (t) 0 for any t 2 [a; b] ; or to

[f (t) f (a + b t)] [g (t) g (a + b t)] 0 for any t 2 [a; b] : This is a property of interest for asymmetrized synchronous functions. If we integrate the inequality (2.4) and use the identity (2.3) we get the desired result (2.1). Remark 1. The inequality (2.1) can be written in an equivalent form as Z b Z b f (t) g (t) dt f (a + b t) g (t) dt; a

or as

a

Z

b

f (t) g (t) dt

a

Z

b

f (t) g (t) dt:

a

Theorem 4. If both f; g are asymmetrized monotonic nondecreasing (nonincreasing) and integrable functions on [a; b] ; then Z b 1 1 jf (b) f (a)j jg (b) g (a)j f~ (t) g (t) dt 0; (2.5) 4 b a a and ( ) Z b Z b 1 1 1 (2.6) min jf (b) f (a)j jg (t)j dt; jg (b) g (a)j jf (t)j dt 2 b a a b a a Z b 1 f~ (t) g (t) dt 0: b a a Proof. Assume that both f; g are asymmetrized monotonic nondecreasing and integrable functions on [a; b] ; then they are asymmetrized synchronous and by (2.1) we get the second inequality in (2.5). We also have f~ (a) f~ (t) f~ (b) for any t 2 [a; b] ; namely 1 1 1 [f (b) f (a)] [f (t) f (a + b t)] [f (b) f (a)] ; 2 2 2 for any t 2 [a; b] ; which implies that 12 [f (b) f (a)] 0 and (2.7) for any t 2 [a; b] : Similarly, we have (2.8)

1 jf (t) 2 1 2

[g (b)

1 jg (t) 2

f (a + b

t)j

g (a)]

0 and

g (a + b

t)j

155

1 [f (b) 2

f (a)]

1 [g (b) 2

g (a)]

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for any t 2 [a; b] : If we multiply (2.7) and (2.8), then we get 1 (2.9) [f (t) f (a + b t)] [g (t) g (a + b t)] 4 1 = j[f (t) f (a + b t)] [g (t) g (a + b t)]j 4 1 [f (b) f (a)] [g (b) g (a)] 4 for any t 2 [a; b] : Since Z b Z 1 b f~ (t) g (t) dt = [f (t) f (a + b t)] [g (t) g (a + b t)] dt 0 4 a a 1 [f (b) f (a)] [g (b) g (a)] (b a) ; 4 where for the last inequality we used (2.9), hence we get the …rst inequality in (2.5). Also, we observe that Z b Z b Z b 1 ~ ~ f (t) g (t) dt 0 [f (b) f (a)] f (t) g (t) dt = jg (t)j dt 2 a a a and since Z b Z b f~ (t) g (t) dt = f (t) g~ (t) dt; a

then also

Z

a

b

1 [g (b) f (t) g~ (t) dt 2 a and the inequality (2.6) is also proved.

g (a)]

Z

a

b

jf (t)j dt

Remark 2. If the functions f; g : [a; b] ! R are either both of them nonincreasing or nondecreasing on [a; b] ; then they are integrable and we have the inequalities (2.5) and (2.6). We have the following re…nement of the inequality in (2.1). Theorem 5. Assume that f; g are asymmetrized synchronous and integrable functions on [a; b]. Then Z b 1 (2.10) f~ (t) g (t) dt b a a Z b Z b Z b 1 1 1 ~ ~ f (t) j~ g (t)j dt f (t) dt j~ g (t)j dt 0: b a a b a a b a a Proof. By the continuity property of modulus, we have h i h i f~ (t) f~ (s) [~ g (t) g~ (s)] = f~ (t) f~ (s) [~ g (t) = f~ (t)

f~ (t) f~ (t)

=

156

f~ (s) j~ g (t)

g~ (s)] g~ (s)j

f~ (s) j~ g (t) f~ (s)

(~ g (t)

g~ (s)j g~ (s))

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7

for any t; s 2 [a; b] : 2 Taking the double integral mean on [a; b] and using the properties of the integral versus the modulus, we have Z bZ bh i 1 (2.11) f~ (t) f~ (s) [~ g (t) g~ (s)] dtds 2 (b a) a a Z bZ b 1 g (t)j j~ g (s)j) dtds : f~ (s) (j~ f~ (t) 2 (b a) a a Since, by Korkine’s identity we have Z bZ bh i 1 ~ (t) f~ (s) [~ f g (t) g~ (s)] dtds 2 (b a) a a " # Z b Z b Z b 1 1 1 ~ ~ =2 f (t) g~ (t) dt f (t) dt g~ (t) dt b a a b a a b a a Z b 2 f~ (t) g~ (t) dt = b a a and Z

1 2

(b =2

a) "

b

a

b

f~ (t)

f~ (s)

(j~ g (t)j

j~ g (s)j) dtds

a

1

b

Z

a

Z

b

f~ (t) j~ g (t)j dt

a

1 b

a

Z

b

a

1 f~ (t) dt b a

Z

b

a

#

j~ g (t)j dt ;

hence by (2.11) we have 1 b

a

Z

b

a

1 b

a

f~ (t) g~ (t) dt Z

b

a

f~ (t) j~ g (t)j dt

1 b

a

Z

b

a

1 f~ (t) dt b a

Z

b

a

j~ g (t)j dt :

By using the identity (2.3) we get the desired result (2.10). Remark 3. We remark that, if f~; g are synchronous, then by a similar argument to the one above for g $ g~ we have Z b 1 (2.12) f~ (t) g (t) dt b a a Z b Z b Z b 1 1 1 f~ (t) jg (t)j dt f~ (t) dt jg (t)j dt 0: b a a b a a b a a Also, since 1 b

a

Z

a

b

f~ (t) g (t) dt =

1 b

157

a

Z

b

f (t) g~ (t) dt;

a

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then if we assume that (f; g~) are synchronous we also have Z b 1 (2.13) f~ (t) g (t) dt b a a Z b Z b Z b 1 1 1 jf (t)j j~ g (t)j dt jf (t)j dt j~ g (t)j dt b a a b a a b a a

0:

Now, if f and g have the same monotonicity, then f~; g~ ; f~; g ; (f; g~) are synchronous and we have Z b o n 1 0; (2.14) f~ (t) g (t) dt max C f~; g~ ; C f~; g ; jC (f; g~)j b a a where

C (h; `) :=

1 b

a

Z

a

b

1

jh (t) ` (t)j dt

b

a

Z

a

b

jh (t)j dt

1 b

a

Z

a

b

j` (t)j dt

provided h and ` are integrable on [a; b] : We say that the function h : [a; b] ! R is H-r-Hölder continuous with the constant H > 0 and power r 2 (0; 1] if (2.15)

jh (t)

h (s)j

H jt

r

sj

for any t; s 2 [a; b] : If r = 1 we call that h is L-Lipschitzian when H = L > 0: Theorem 6. Assume that f; g are asymmetrized synchronous with f is H1 -r1 Hölder continuous and g is H2 -r2 -Hölder continuous on [a; b] : Then Z b 1 1 r +r (2.16) H1 H2 (b a) 1 2 f~ (t) g (t) dt 0: 4 (r1 + r2 + 1) b a a If particular, if f is L1 -Lipschitzian and g is L2 -Lipschitzian, then Z b 1 1 2 L1 L2 (b a) f~ (t) g (t) dt 0: (2.17) 12 b a a Proof. From (2.3) we have Z Z b 1 b ~ [f (t) f (a + b t)] [g (t) g (a + b t)] dt 0 f (t) g (t) dt = 4 a a Z b 1 = j[f (t) f (a + b t)] [g (t) g (a + b t)]j dt 4 a Z b Z b r +r 1 2r1 +r2 a+b 1 2 r +r H1 H2 j2t a bj 1 2 dt = H1 H 2 t dt 4 4 2 a a Z b r +r b a r1 +r2 +1 2 a+b 1 2 2 = 2 r 1 r2 H1 H2 t dt = 2 r1 r2 H1 H2 2 a+b 2 2 2 r1 + r2 + 1 2 =

1 H1 H2 (b 4 (r1 + r2 + 1)

r1 +r2 +1

a)

;

which is equivalent to the desired result (2.16).

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9

3. Some Examples Consider the identity function ` : [a; b] ! R de…ned by ` (t) = t: If g is monotonic nondecreasing, then by (2.5) and (2.14) we have Z b 1 a+b 1 t (3.1) (b a) [g (b) g (a)] g (t) dt 4 b a a 2 max fjC1;` (g)j ; jC2;` (g)j ; jC3;` (g)jg 0; where

b

a

b

a

and

Z

1

b

t

a

1

C2;` (g) :=

C3;` (g) :=

Z

1

C1;` (g) :=

Z

b

t

a

a+b 2

g~ (t) dt

a+b 2

g (t) dt

b

1

jt~ g (t)j dt

Z

1 4 1 4

b

jtj dt

Z

b

j~ g (t)j dt;

a

Z

b

jg (t)j dt

a

Z

1

b

j~ g (t)j dt: b a a b a a b a a If g is monotonic nondecreasing and L-Lipschitzian on [a; b] ; then by (2.17) we get Z b 1 1 a+b 2 L (b a) g (t) dt ( 0) : (3.2) t 12 b a a 2

Consider the power function f : [a; b] (0; 1) ! R, f (t) = tp with p > 0: If g is monotonic nondecreasing, then by (2.5) and (2.14) we get Z b p p 1 p 1 t (a + b t) p (3.3) (b a ) [g (b) g (a)] g (t) dt 4 b a a 2 max fjC1;p (g)j ; jC2;p (g)j ; jC3;p (g)jg 0; where

Z

1

C1;p (g) :=

b

a Z

1 b

a 1

C2;p (g) :=

b 1 b

and

Z

a b

tp tp

a

Z

a Z

a

b

a b

a

b

tp tp

(a + b 2 (a + b 2 (a + b 2 (a + b 2

p

t)

j~ g (t)j dt

p

t)

dt

1 b

a

Z

b

j~ g (t)j dt;

a

p

t)

jg (t)j dt

p

t)

dt

1 b

a

Z

a

b

jg (t)j dt

Z b bp+1 ap+1 1 C3;p (g) := t j~ g (t)j dt j~ g (t)j dt: (p + 1) (b a) b a a a If g is monotonic nondecreasing and L-Lipschitzian on [a; b] ; then by (2.17) we get 8 p 1 if p 1 < b p 2 (3.4) L (b a) : p 1 12 a if p 2 (0; 1) Z b p p 1 t (a + b t) g (t) dt ( 0) : b a a 2 b

p

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Consider the function f : [a; b] (0; 1) ! R, f = ln : If g is monotonic nondecreasing, then by (2.5) and (2.14) we have Z b t 1 b 1 ln (3.5) ln [g (b) g (a)] g (t) dt 4 a 2 (b a) a a+b t max fjC1;ln (g)j ; jC2;ln (g)j ; jC3;ln (g)jg 0; where b

a Z

1 b C2;ln (g) :=

a

a b

1=2

t a+b

ln

1=2

t a+b

ln

a

b 1

Z

a Z

a

b

a b

a

Z

1 b

a

a

Z

1 b

a

b

j~ g (t)j dt;

a

b

jln tj dt

1 b

a

Z

a

b

j~ g (t)j dt

1=2

jg (t)j dt

t 1=2

t a+b

ln

dt

t

t a+b

ln

j~ g (t)j dt

t

jln tj j~ g (t)j dt

1

b

b

b

a

and C1;ln (g) :=

a Z

1 b

Z

1

C1;ln (g) :=

dt

t

1 b

a

Z

a

b

jg (t)j dt:

If g is monotonic nondecreasing and L-Lipschitzian on [a; b] ; then by (2.17) we get Z b 1 1 t 2 (3.6) L (b a) ln g (t) dt ( 0) : 6a b a a a+b t 2; 2

Consider the function f : [a; b] nondecreasing, then by (2.5) we have (3.7)

1 sin 2

b

a 2

[g (b)

g (a)]

1 b

a

! R, f = sin : If g is monotonic Z

b

a

sin t

a+b 2

g (t) dt

0:

If g is monotonic nondecreasing and L-Lipschitzian on [a; b] ; then by (2.17) we get 8 a < b 0; < cos b if 2 1 2 max fcos a; cos bg if a 0: 4h The function K is decreasing on (0; 1) and increasing on [1; 1) ; K (h) 1 for any h > 0 and K (h) = K h1 for any h > 0: In the recent paper [1] we have obtained the following additive and multiplicative reverse of Young’s inequality (1.3)

(1.4)

K (h) :=

0

(1

)a + b

a1

b

(1

) (a

b) (ln a

ln b)

1991 Mathematics Subject Classi…cation. 26D15; 26D10, 47A63, 47A30, 15A60. Key words and phrases. Young’s Inequality, Convex functions, Arithmetic mean-Geometric mean inequality, Heinz means. 1

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and h i a )a + b exp 4 (1 ) K 1 ; b b for any a; b > 0 and 2 [0; 1] ; where K is Kantorovich’s constant. The operator version of (1.4) is as follows [1]:

(1.5)

1

(1

a1

Theorem 1. Let A; B be two positive operators. For positive real numbers m; m0 ; M0 0 M; M 0 , put h := M 2 [0; 1] : m ; h := m0 and let (i) If 0 < mI A m0 I < M 0 I B M I; then (1.6)

0

Ar B

A] B

(1

) (h

1) ln hA

and, in particular (1.7) (ii) If 0 < mI (1.8)

0

ArB

A]B

B

m0 I < M 0 I

0

Ar B

A

A] B

1 (h 1) ln hA: 4 M I; then (1

h

)

1 h

ln hA

and, in particular (1.9)

0

ArB

A]B

1h 1 ln hA: 4 h

The operator version of (1.5) is [1]: Theorem 2. For two positive operators A; B and positive real numbers m; m0 ; M; M 0 satisfying either of the following conditions (i) 0 < mI A m0 I < M 0 I B M I; (ii) 0 < mI B m0 I < M 0 I A M I; we have (1.10)

Ar B

exp [4 (1

) (K (h)

1)] A] B

and, in particular (1.11)

ArB

exp [K (h)

1] A]B:

For other recent results on geometric operator mean inequalities, see [2]-[12], [14] and [16]-[17]. We recall that Specht’s ratio is de…ned by [15] 8 1 hh 1 > > < e ln h h 1 1 if h 2 (0; 1) [ (1; 1) ; (1.12) S (h) := > > : 1 if h = 1:

It is well known that limh!1 S (h) = 1; S (h) = S h1 > 1 for h > 0; h 6= 1. The function is decreasing on (0; 1) and increasing on (1; 1) : In the recent paper [6] we obtained amongst other the following result for the Heinz operator mean of A; B that are positive invertible operators that satisfy the condition mA B M A for some constants M > m > 0; (1.13)

! (m; M ) A]B

H (A; B)

163

(m; M ) A]B;

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FURTHER INEQUALITIES FOR HEINZ OPERATOR M EAN

where

and

8 S mj2 1j if M < 1; > > > > < (m; M ) := max S mj2 1j ; S M j2 > > > > : S M j2 1j if 1 < m; 8 > S Mj > > > > > < 1 if m ! (m; M ) := > > > > > > : S mj

1 2

j

1j

if m

M;

if M < 1;

1

M;

j

if 1 < m:

1 2

1

3

Motivated by the above results we establish in this paper some new additive and multiplicative reverse inequalities for the Heinz operator mean. 2. Additive Reverse Inequalities for Heinz Mean We have the following generalization of Theorem 1: Theorem 3. Assume that A; B are positive invertible operators and the constants M > m > 0 are such that (2.1)

mA

Then for any

(2.3)

M A:

2 [0; 1] we have

(2.2) where

B

(0

) Ar B

A] B

(1

)

(m; M ) A

8 (m 1) ln m if M < 1; > > > > < max f(m 1) ln m; (M 1) ln M g if m (m; M ) := > > > > : (M 1) ln M if 1 < m:

1

M;

In particular, we have (2.4)

(0

) ArB

A]B

1 4

(m; M ) A:

Proof. We consider the function D : (0; 1) ! [0; 1) de…ned by D (x) = (x 1) ln x: We have that D0 (x) = ln x + 1 x1 and D00 (x) = x+1 x2 for x 2 (0; 1) : This shows that the function is convex on (0; 1) ; monotonic decreasing on (0; 1) and monotonic increasing on [1; 1) with the minimum 0 realized in x = 1: From the inequality (1.4) we have (0 for any x > 0; (2.5)

) (1

)+ x

x

(1

) D (x)

2 [0; 1] and hence (0

) (1

)I + X

X

(1

) max D (x) m x M

for the positive operator X that satis…es the condition 0 < mI 0 < m < M and 2 [0; 1] :

164

X

M I for

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If the condition (2.1) holds true, then by multiplying in both sides with A 1=2 we get mI A 1=2 BA 1=2 M I and by taking X = A 1=2 BA 1=2 in (2.5) we get (2.6)

(1

1=2

)I + A

1=2

BA

1=2

A

BA

1=2

1=2

Now, if we multiply (2.6) in both sides with A (2.7)

(0

) (1 (1

)A + B

A1=2 A

(1

) max D (x) m x M

we get

1=2

1=2

BA

A1=2

) max D (x) A m x M

for any 2 [0; 1] : Finally, since

8 (m 1) ln m if M < 1; > > > > < max f(m 1) ln m; (M max D (x) = m x M > > > > : (M 1) ln M if 1 < m;

1) ln M g if m

1

M;

then by (2.7) we get the desired result (2.2).

Corollary 1. With the assumptions of Theorem 3 we have (2.8)

(0

) ArB

H (A; B)

Proof. From (2.2) we have by replacing (2.9)

(0

) Ar1

B

(1

)

with 1

A]1

B

(m; M ) A: that

(1

)

(m; M ) A:

Adding (2.2) with (2.9) and dividing by 2 we get (2.8). Corollary 2. Let A; B be two positive operators. For positive real numbers m; M0 0 2 [0; 1] : m0 ; M; M 0 , put h := M m ; h := m0 and let 0 (i) If 0 < mI A m I < M 0 I B M I; then (2.10)

(0

(ii) If 0 < mI

B

(2.11)

(0

) ArB

H (A; B)

m0 I < M 0 I ) ArB

A

(1

) (h

1) ln hA:

M I; then

H (A; B)

(1

)

h

1 h

ln hA:

Proof. If the condition (i) is valid, then we have M M0 I = h0 I X hI = I; 0 m m which, by (2.8) gives the desired result (2.10). If the condition (ii) is valid, then we have 1 1 0< I X I < I; h h0 which, by (2.8) gives I
0: If we take in (2.13) c = a1 (2.14)

a1

p

b + a b1 2

p

c+d 2

=

1 (c 4

cd

b and d = a b1 1 1 a 4

ab

1 2

d) (ln c

ln d)

then we get a b1

b

1) ln x (see the

ln a1

ln a b1

b

for any a; b > 0 and 2 [0; 1] : This inequality is of interest in itself. Now, if we take in (2.14) a = 1 and b = x; then we get (2.15)

0

p x + x1 x 2 2 1 = x x1 4 1 = 1 D x2 1 4x

1 x 4

x1

ln x =

1 4x1

ln x1

ln x x2

1

1 ln x2

1

for any x > 0 and 2 [0; 1] : Now, if x 2 [m; M ] (0; 1), then by (2.15) we get the upper bound (0

)

x + x1 2

p

1 4m1

x

max D x2

1

x2[m;M ]

:

Using the continuous functional calculus, we then have (2.16)

(0

)

X + X1 2

1 4m1

X 1=2

max D x2

1

x2[m;M ]

If the condition (2.1) holds true, then by multiplying in both sides with A 1=2 we get mI A 1=2 BA 1=2 M I and by taking X = A 1=2 BA 1=2 in (2.16) we get (2.17)

0

A 1 4m1

1=2

BA

1=2

1=2

+ A 2

max D x2

BA

1=2 1

A

1=2

BA

1=2

1=2

1

x2[m;M ]

for any 2 [0; 1] : Now, if we multiply (2.17) in both sides with A1=2 we get the desired result (2.12). Corollary 3. Let A; B be as in Corollary 2.

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(i) If 0 < mI (2.18)

m0 I < M 0 I

A

(0

) H (A; B) A]B 8 2 1 > < h 1 1

> :

4 (h0 ) (ii) If 0 < mI (2.19)

B

(0

1 ln h2

2

1

if

1

1 ln (h0 )

A

M I; then

(h0 )

m0 I < M 0 I

B

M I; then 1 2; 1

2

2

1

) H (A; B) A]B 8 > h 2 +1 1 ln h 2 +1 if 1 1 < h > 4 : (h0 ) 2 +1 1 ln (h0 ) 2

2 0; 21 :

if

1 2; 1

2 +1

;

;

2 0; 21 :

if

Proof. If the condition (i) is valid, then we have I
0 are such that the condition (2.1) is valid. Then for any 2 [0; 1] we have (3.1)

Ar B

A] B exp [4 (1

) (z (m; M )

1)]

where

8 K (m) if M < 1; > > > > < max fK (m) ; K (M )g if m z (m; M ) := > > > > : K (M ) if 1 < m; In particular, we have (3.2)

ArB

A]B exp [z (m; M )

1

M;

1] :

Proof. From the inequality (1.5) we have for a = 1 and b = x that (3.3)

(1

)+ x

x exp 4 (1

1 x

) K

= x exp [4 (1

1

) (K (x)

1)]

for any x > 0 and hence (3.4)

(1

)I + X

X

max exp [4 (1

m x M

= X exp 4 (1

)

) (K (x)

1)]

max K (x)

1

m x M

for any operator X with the property that 0 < mI X M I and for any 2 [0; 1] : If the condition (2.1) holds true, then by multiplying in both sides with A 1=2 we get mI A 1=2 BA 1=2 M I and by taking X = A 1=2 BA 1=2 in (3.4) we get (3.5)

(1

)I + A A

1=2

= A

1=2

1=2

BA

1=2

BA

1=2

BA

1=2

max exp [4 (1

m x M

exp 4 (1

)

) (K (x)

1)]

max K (x)

1

m x M

for any 2 [0; 1] : Now, if we multiply (3.5) in both sides with A1=2 we get (3.6)

for any

(1

) A + BA A1=2 A

1=2

= A1=2 A

1=2

BA

1=2

BA

1=2

A1=2 max exp [4 (1 m x M

A1=2 exp 4 (1

)

) (K (x)

1)]

max K (x)

1

m x M

2 [0; 1] :

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Since

8 K (m) if M < 1; > > > > < max fK (m) ; K (M )g if m max K (x) = m x M > > > > : K (M ) if 1 < m;

1

M;

then by (3.6) we get the desired result (3.1).

Corollary 4. With the assumptions of Theorem 5 we have (3.7)

ArB

exp [4 (1

) (z (m; M )

1)] H (A; B) :

Corollary 5. For two positive operators A; B and positive real numbers m; m0 ; M; M 0 satisfying either of the following conditions: (i) 0 < mI A m0 I < M 0 I B M I; (ii) 0 < mI B m0 I < M 0 I A M I; we have (3.8)

ArB

exp [4 (1

) (K (h)

1)] H (A; B) :

We also have: Theorem 6. Assume that A; B are positive invertible operators and the constants M > m > 0 are such that the condition (2.1) is valid. Then for any 2 [0; 1] we have (3.9) where

(3.10)

H (A; B)

exp [

(m; M )

8 K mj2 1j if M < 1; > > > > < (m; M ) := max K mj2 1j ; K M j2 > > > > : K M j2 1j if 1 < m:

Proof. From the inequality (1.5) we have for c+d

p2 cd

(3.11) for any c; d > 0: If we take in (3.11) c = a1 1

(3.12)

1] A]B

a

b +a b 2

exp K

=

c d

exp K

if m

1

M;

1 2

1

b and d = a b1 1

1j

then we get a b

1 2

1

p

ab

for any a; b > 0 for any 2 [0; 1]: This is an inequality of interest in itself. If we take in (2.19) a = x and b = 1; then we get x1

(3.13) for any x > 0: Now, if x 2 [m; M ] 1

(3.14)

x

+x 2

exp K x1

2

1

(0; 1) then by (2.20) we have p +x x exp max K x1 2 x2[m;M ]

169

p

2

x;

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FURTHER INEQUALITIES FOR HEINZ OPERATOR M EAN

for any x 2 [m; M ] : If 2 0; 12 ; then

max K x1

2

x2[m;M ]

If

2

1 2; 1

, then

max K x1

2

x2[m;M ]

8 K m1 2 if M < 1; > > > > < max K m1 2 ; K M 1 = > > > > : K M 1 2 if 1 < m:

2

max K x2 1 8 K m2 1 if M < 1; > > > > < max K m2 1 ; K M 2 = > > > > : K M 2 1 if 1 < m: =

if m

9

1

M;

x2[m;M ]

1

if m

1

M;

Therefore, by (3.14) we have x1

(3.15)

+x 2

exp [ (m; M )

p 1] x

for any x 2 [m; M ] (0; 1) and for any 2 [0; 1]: If X is an operator with mI X M I; then by (3.15) we have X1

+X 2

exp [ (m; M )

1] X 1=2 :

If the condition (2.1) holds true, then by multiplying in both sides with A 1=2 we get mI A 1=2 BA 1=2 M I and by taking X = A 1=2 BA 1=2 in (3.15) we get 1 2

(3.16)

A

1=2

BA

1=2

exp [ (m; M )

1

+ A 1] A

1=2

1=2

BA

BA 1=2

1=2 1=2

:

Now, if we multiply (3.16) in both sides with A1=2 we get the desired result (3.9). Finally, we have Corollary 6. For two positive operators A; B and positive real numbers m; m0 ; M; M 0 satisfying either of the following conditions: (i) 0 < mI A m0 I < M 0 I B M I; (ii) 0 < mI B m0 I < M 0 I A M I; M0 0 we have for h = M m and h = m0 that h i (3.17) H (A; B) exp K hj2 1j 1 A]B; where

2 [0; 1]:

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10

S. S. DRAGOM IR 1;2

References [1] S. S. Dragomir, Some new reverses of Young’s operator inequality, Preprint RGMIA Res. Rep. Coll. 18 (2015), Art. 130. [http://rgmia.org/papers/v18/v18a130.pdf]. [2] S. S. Dragomir, On new re…nements and reverses of Young’s operator inequality, Preprint RGMIA Res. Rep. Coll. 18 (2015), Art. 135. [http://rgmia.org/papers/v18/v18a135.pdf]. [3] S. S. Dragomir, Some inequalities for operator weighted geometric mean, Preprint RGMIA Res. Rep. Coll. 18 (2015), Art. 139. [http://rgmia.org/papers/v18/v18a139.pdf ]. [4] S. S. Dragomir, Re…nements and reverses of Hölder-McCarthy operator inequality, Preprint RGMIA Res. Rep. Coll. 18 (2015), Art. 143. [http://rgmia.org/papers/v18/v18a143.pdf]. [5] S. S. Dragomir, Some reverses and a re…nement of Hölder operator inequality, Preprint RGMIA Res. Rep. Coll. 18 (2015), Art. 147. [http://rgmia.org/papers/v18/v18a147.pdf]. [6] S. S. Dragomir, Some inequalities for Heinz operator mean, Preprint RGMIA Res. Rep. Coll. 18 (2015), Art. 163. [http://rgmia.org/papers/v18/v18a163.pdf]. [7] S. Furuichi, On re…ned Young inequalities and reverse inequalities, J. Math. Inequal. 5 (2011), 21-31. [8] S. Furuichi, Re…ned Young inequalities with Specht’s ratio, J. Egyptian Math. Soc. 20 (2012), 46–49. [9] F. Kittaneh and Y. Manasrah, Improved Young and Heinz inequalities for matrix, J. Math. Anal. Appl. 361 (2010), 262-269. [10] F. Kittaneh and Y. Manasrah, Reverse Young and Heinz inequalities for matrices, Lin. Multilin. Alg., 59 (2011), 1031-1037. [11] F. Kittaneh, M. Krni´c, N. Lovriµcevi´c and J. Peµcari´c, Improved arithmetic-geometric and Heinz means inequalities for Hilbert space operators, Publ. Math. Debrecen, 2012, 80(3-4), 465–478. [12] M. Krni´c and J. Peµcari´c, Improved Heinz inequalities via the Jensen functional, Cent. Eur. J. Math. 11 (9) 2013,1698-1710. [13] F. Kubo and T. Ando, Means of positive operators, Math. Ann. 264 (1980), 205–224. [14] W. Liao, J. Wu and J. Zhao, New versions of reverse Young and Heinz mean inequalities with the Kantorovich constant, Taiwanese J. Math. 19 (2015), No. 2, pp. 467-479. [15] W. Specht, Zer Theorie der elementaren Mittel, Math. Z. 74 (1960), pp. 91-98. [16] M. Tominaga, Specht’s ratio in the Young inequality, Sci. Math. Japon., 55 (2002), 583-588.H. [17] G. Zuo, G. Shi and M. Fujii, Re…ned Young inequality with Kantorovich constant, J. Math. Inequal., 5 (2011), 551-556. 1 Mathematics, College of Engineering & Science, Victoria University, PO Box 14428, Melbourne City, MC 8001, Australia. E-mail address : [email protected] URL: http://rgmia.org/dragomir 2 School of Computer Science & Applied Mathematics, University of the Witwatersrand, Private Bag 3, Johannesburg 2050, South Africa

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Global Dynamics of Monotone Second Order Difference Equation S. Kalabuˇsi´c† , M. R. S Kulenovi´c‡1 and M. Mehulji´c§ †

Department of Mathematics University of Sarajevo, Sarajevo, Bosnia and Herzegovina ‡ Department of Mathematics University of Rhode Island, Kingston, Rhode Island 02881-0816, USA §

Division of Mathematics Faculty of Mechanical Engineering, University of Sarajevo, Bosnia and Herzegovina

Abstract. We investigate the global character of the difference equation of the form xn+1 = f (xn , xn−1 ),

n = 0, 1, . . .

with several period-two solutions, where f is decreasing in the first variable and is increasing in the second variable. We show that the boundaries of the basins of attractions of different locally asymptotically stable equilibrium solutions or period-two solutions are in fact the global stable manifolds of neighboring saddle or non-hyperbolic equilibrium solutions or period-two solutions. We illustrate our results with the complete study of global dynamics of a certain rational difference equation with quadratic terms. Keywords. asymptotic stability, attractivity, bifurcation, difference equation, global, local stability, period two; AMS 2000 Mathematics Subject Classification: 37B25, 37D10, 39A20, 39A30.

1

Introduction and Preliminaries

Let I be some interval of real numbers and let f ∈ C 1 [I × I, I] be such that f (I × I) ⊆ K where K ⊆ I is a compact set. Consider the difference equation xn+1 = f (xn , xn−1 ),

n = 0, 1, . . .

(1)

where f is a continuous and decreasing in the first variable and increasing in the second variable. The following result gives a general information about global behavior of solutions of Equation (1). Theorem 1 ([4]) Let I ⊆ R and let f ∈ C[I × I, I] be a function which is non-decreasing in first and non-increasing in second ∞ argument. Then for every solution of Equation (1) the subsequences {x2n }∞ n=0 and {x2n+1 }n=−1 of even and odd terms of the solution are eventually monotonic. The consequence of Theorem 1 is that every bounded solution of (1) converges to either an equilibrium or periodtwo solution or to the singular point on the boundary. Consequently, most important question becomes determining the basins of attraction of these solutions as well as the unbounded solutions. The answer to this question follows from an application of the theory of monotone maps in the plane which will be presented in Preliminaries. In [1, 2, 3] authors consider difference equation (1) with several equilibrium solutions as well as the period-two solutions and determine the basins of attraction of different equilibrium solutions and the period-two solutions. In this paper we consider Equation (1) which has up to two equilibrium solutions and up to two minimal period-two solutions which are in South-East ordering. More precisely, we will give sufficient conditions for the precise description of the basins of attraction of different equilibrium solutions and period-two solutions. The results can be immediately extended to the case of any number of the equilibrium solutions and the period-two solutions by replicating our main results. This paper is organized as follows. In the rest of this section we will recall several basic results on competitive systems in the plane from [7, 15, 16, 17] which are included for completeness of presentation. Our main results about some global dynamics scenarios for monotone systems in the plane and their application to global dynamics of 1

Corresponding author, e-mail: [email protected]

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Equation (1) are given in section 2. As an application of the results from section 2 in section 3 the global dynamics of difference equation γxn−1 xn+1 = , n = 0, 1, . . . (2) Ax2n + Bxn xn−1 + Cxn−1 with all non-negative parameters and initial conditions is presented. All global dynamic scenarios for Equation (1) will be illustrated in the case of Equation (2), which global dynamics can be shortly described as the sequence of exchange of stability bifurcations between an equilibrium and one or two period-two solutions. We now give some basic notions about monotone maps in the plane. Definition 2 Let R be a subset of R2 with nonempty interior, and let T : R → R be a map (i.e., a continuous function). Set T (x, y) = (f (x, y), g(x, y)). The map T is competitive if f (x, y) is non-decreasing in x and nonincreasing in y, and g(x, y) is non-increasing in x and non-decreasing in y. If both f and g are nondecreasing in x and y, we say that T is cooperative. If T is competitive (cooperative), the associated system of difference equations  xn+1 = f (xn , yn ) , n = 0, 1, . . . , (x−1 , x0 ) ∈ R (3) yn+1 = g(xn , yn ) is said to be competitive (cooperative). The map T and associated difference equations system are said to be strongly competitive (strongly cooperative) if the adjectives non-decreasing and non-increasing are replaced by increasing and decreasing. Consider a partial ordering  on R2 . Two points x, y ∈ R2 are said to be related if x  y or y  x. Also, a strict inequality between points may be defined as x ≺ y if x  y and x 6= y. A stronger inequality may be defined as x = (x1 , x2 )  y = (y1 , y2 ) if x  y with x1 6= y1 and x2 6= y2 . The map T is monotone if x  y implies T (x)  T (y) for all x, y ∈ R, and it is strongly monotone on R if x ≺ y implies that T (x)  T (y) for all x, y ∈ R. The map is strictly monotone on R if x ≺ y implies that T (x) ≺ T (y) for all x, y ∈ R. Clearly, being related is invariant under iteration of a strongly monotone map. Throughout this paper we shall use the North-East ordering (NE) for which the positive cone is the first quadrant, i.e. this partial ordering is defined by (x1 , y1 ) ne (x2 , y2 ) if x1 ≤ x2 and y1 ≤ y2 and the South-East (SE) ordering defined as (x1 , y1 ) se (x2 , y2 ) if x1 ≤ x2 and y1 ≥ y2 . Now we can show that a map T on a nonempty set R ⊂ R2 which is monotone with respect to the North-East ordering is cooperative and a map monotone with respect to the South-East ordering is competitive. For x ∈ R2 , define Q` (x) for ` = 1, . . . , 4 to be the usual four quadrants based at x = (x1 , x2 ) and numbered in a counterclockwise direction, for example, Q1 (x) = {y = (y1 , y2 ) ∈ R2 : x1 ≤ y1 , x2 ≤ y2 }. Basin of attraction of a fixed point (¯ x, y¯) of a map T , denoted as B((¯ x, y¯)), is defined as the set of all initial points (x0 , y0 ) for which the sequence of iterates T n ((x0 , y0 )) converges to (¯ x, y¯). Similarly, we define a basin of attraction of a periodic point of period p. The fixed point A(x, y) of the map T is said to be non-hyperbolic point of stable type if one of the roots of characteristic equation evaluated in A is 1 or −1 and the second root is in (−1, 1). The next four results, from [16, 17], are useful for determining basins of attraction of fixed points of competitive maps. Related results have been obtained by H. L. Smith in [7, 19] and in [18]. Theorem 3 Let T be a competitive map on a rectangular region R ⊂ R2 . Let x ∈ R be a fixed point of T such that ∆ := R ∩ int (Q1 (x) ∪ Q3 (x)) is nonempty (i.e., x is not the NW or SE vertex of R), and T is strongly competitive on ∆. Suppose that the following statements are true. a. The map T has a C 1 extension to a neighborhood of x. b. The Jacobian JT (x) of T at x has real eigenvalues λ, µ such that 0 < |λ| < µ, where |λ| < 1, and the eigenspace E λ associated with λ is not a coordinate axis. Then there exists a curve C ⊂ R through x that is invariant and a subset of the basin of attraction of x, such that C is tangential to the eigenspace E λ at x, and C is the graph of a strictly increasing continuous function of the first coordinate on an interval. Any endpoints of C in the interior of R are either fixed points or minimal period-two points. In the latter case, the set of endpoints of C is a minimal period-two orbit of T . Theorem 4 For the curve C of Theorem 3 to have endpoints in ∂R, it is sufficient that at least one of the following conditions is satisfied. i. The map T has no fixed points nor periodic points of minimal period two in ∆. ii. The map T has no fixed points in ∆, det JT (x) > 0, and T (x) = x has no solutions x ∈ ∆. iii. The map T has no points of minimal period-two in ∆, det JT (x) < 0, and T (x) = x has no solutions x ∈ ∆.

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For maps that are strongly competitive near the fixed point, hypothesis b. of Theorem 3 reduces just to |λ| < 1. This follows from a change of variables [19] that allows the Perron-Frobenius Theorem to be applied. Also, one can show that in such case no associated eigenvector is aligned with a coordinate axis. The next result is useful for determining basins of attraction of fixed points of competitive maps. Theorem 5 Assume the hypotheses of Theorem 3, and let C be the curve whose existence is guaranteed by Theorem 3. If the endpoints of C belong to ∂R, then C separates R into two connected components, namely W− := {x ∈ R \ C : ∃y ∈ C with x se y},

W+ := {x ∈ R \ C : ∃y ∈ C with y se x} ,

(4)

such that the following statements are true. (i) W− is invariant, and dist(T n (x), Q2 (x)) → 0 as n → ∞ for every x ∈ W− . (ii) W+ is invariant, and dist(T n (x), Q4 (x)) → 0 as n → ∞ for every x ∈ W+ . (B) If, in addition to the hypotheses of part (A), x is an interior point of R and T is C 2 and strongly competitive in a neighborhood of x, then T has no periodic points in the boundary of Q1 (x) ∪ Q3 (x) except for x, and the following statements are true. (iii) For every x ∈ W− there exists n0 ∈ N such that T n (x) ∈ int Q2 (x) for n ≥ n0 . (iv) For every x ∈ W+ there exists n0 ∈ N such that T n (x) ∈ int Q4 (x) for n ≥ n0 . If T is a map on a set R and if x is a fixed point of T , the stable set W s (x) of x is the set {x ∈ R : T n (x) → x} and unstable set W u (x) of x is the set



x ∈ R : there exists {xn }0n=−∞ ⊂ R s.t. T (xn ) = xn+1 , x0 = x, and

 lim xn = x

n→−∞

When T is non-invertible, the set W s (x) may not be connected and made up of infinitely many curves, or W u (x) may not be a manifold. The following result gives a description of the stable and unstable sets of a saddle point of a competitive map. If the map is a diffeomorphism on R, the sets W s (x) and W u (x) are the stable and unstable manifolds of x. Theorem 6 In addition to the hypotheses of part (B) of Theorem 5, suppose that µ > 1 and that the eigenspace E µ associated with µ is not a coordinate axis. If the curve C of Theorem 3 has endpoints in ∂R, then C is the stable set W s (x) of x, and the unstable set W u (x) of x is a curve in R that is tangential to E µ at x and such that it is the graph of a strictly decreasing function of the first coordinate on an interval. Any endpoints of W u (x) in R are fixed points of T . Remark 7 We say that f (u, v) is strongly decreasing in the first argument and strongly increasing in the second argument if it is differentiable and has first partial derivative D1 f negative and first partial derivative D2 f positive in a considered set. The connection between the theory of monotone maps and the asymptotic behavior of Equation (1) follows from the fact that if f is strongly decreasing in the first argument and strongly increasing in the second argument, then the second iterate of a map associated to Equation (1) is a strictly competitive map on I × I, see [16]. Set xn−1 = un and xn = vn in Equation (1) to obtain the equivalent system un+1 = vn , vn+1 = f (vn , un )

n = 0, 1, . . . .

Let T (u, v) = (v, f (v, u)). The second iterate T 2 is given by T 2 (u, v) = (f (v, u), f (f (v, u), v)) and it is strictly competitive on I × I, see [16]. Remark 8 The characteristic equation of Equation (1) at an equilibrium solution (¯ x, x ¯): λ2 − D1 f (¯ x, x ¯)λ − D2 f (¯ x, x ¯) = 0,

(5)

has two real roots λ, µ which satisfy λ < 0 < µ, and |λ| < µ, whenever f is strictly decreasing in first and increasing in second variable. Thus the applicability of Theorems 3-6 depends on the nonexistence of minimal period-two solution.

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2

Main Results

In this section we present some global dynamics scenarios which feasibility will be illustrated in Section 3. Theorem 9 Consider the competitive map T generated by the system (3) on a rectangular region R with nonempty interior. Suppose T has no minimal period-two solutions in R, is strongly competitive on int R, is C 2 in a neighborhood of any fixed point and b. of Theorem 3 holds. (a) Assume that T has a saddle fixed points E1 , E3 and locally asymptotically stable fixed point E2 , such that E1 se E2 se E3 , and E0 , which is South-west corner of the region R is either repeller or singular point. Furthermore assume that E1 se E0 se E3 and that the ray through E0 and E1 (resp. E0 and E2 ) is stable manifold of E1 (resp. E2 ). If T has no period-two solutions then every solution which starts in the interior of the region bounded by the global stable manifolds W s (E1 ) and W s (E3 ) converges to E2 . (b) Assume that T has locally asymptotically stable fixed points E1 , E3 and a saddle fixed point E2 , such that E1 se E2 se E3 , and E0 , which is South-west corner of the region R is either repeller or singular point. Furthermore assume that E1 se E0 se E3 and that the ray through E0 and E1 (resp. E0 and E3 ) is attracted to E1 (resp. E3 ). If T has no period-two solutions then every solution which starts below (resp. above) the stable manifold W s (E2 ) converges to E1 (resp. E3 ). (c) Assume that T has exactly five fixed points E1 , . . . , E5 , E1 se E2 se E3 se E4 se E5 where E1 , E3 , E5 are saddle points, and E2 , E4 are locally asymptotically stable points. Assume that E0 , which is South-west corner of the region R, is either repeller or singular point such that E1 se E0 se E5 and that the ray through E0 and E1 (resp. E0 and E5 ) is part of the basin of attraction of E1 (resp. E5 ). If T has no period-two solutions then every solution which starts in the interior of the region bounded by the global stable manifolds W s (E1 ) and W s (E3 ) converges to E2 while every solution which starts in the interior of the region bounded by the global stable manifolds W s (E3 ) and W s (E5 ) converges to E4 . (d) Assume that T has exactly five fixed points E1 , . . . , E5 , E1 se E2 se E3 se E4 se E5 where E1 , E3 , E5 are locally asymptotically stable points, and E2 , E4 are saddle points. Assume that E0 , which is South-west corner of the region R, is either repeller or singular point such that E1 se E0 se E5 and that the ray through E0 and E1 (resp. E0 and E5 ) is part of the basin of attraction of E1 (resp. E5 ). If T has no period-two solutions then every solution which starts below (resp. above) the stable manifold W s (E4 ) (resp. W s (E2 )) converges to E5 (resp. E1 ). Every solution which starts between the stable manifolds W s (E2 ) and W s (E4 ) converges to E3 . Proof. (a) The existence of the global stable and unstable manifolds of the saddle point equilibria E1 and E3 is guaranteed by Theorems 3 - 6. In view of uniqueness of these manifolds we have that W s (E1 ) has end points in E0 and (0, ∞) while W s (E3 ) has end points in E0 and (∞, 0). Furthermore W u (E1 ) and W u (E3 ) have end points in E2 . Now, by Corollary 2 in [16] every solution which starts in the interior of the ordered interval [[E1 , E2 ]] is attracted to E2 and similarly every solution which starts in the interior of the ordered interval [[E2 , E3 ]] is attracted to E2 . Furthermore, for every (x0 , y0 ) ∈ [[E1 , E3 ]] \ ([[E1 , E2 ]] ∪ [[E2 , E3 ]] ∪ {E0 }) one can find the points (xl , yl ) ∈ [[E1 , E2 ]] and (xu , yu ) ∈ [[E1 , E2 ]] such that (xl , yl ) se (x0 , y0 ) se (xu , yu ) and so T n ((xl , yl )) se T n ((x0 , y0 )) se T n ((xu , yu )), n ≥ 1, which implies that T n ((x0 , y0 )) → E2 . Finally, for every (x0 , y0 ) ∈ R \ ([[E1 , E3 ]] ∪ {E0 })) one can find the points (xL , yL ) ∈ W u (E1 ), (xU , yU ) ∈ W u (E3 ) such that (xL , yL ) se (x0 , y0 ) se (xU , yU ) which implies that T n ((x0 , y0 )) will eventually enter [[E1 , E3 ]] and so it will converge to E2 . (b) The existence of the stable and unstable manifolds of the saddle point equilibrium E2 is guaranteed by Theorems 3-6. The endpoints of the unstable manifold are E1 and E3 . First one can assume that the initial point (x0 , y0 ) ∈ [[E1 , E2 ]] \ {E0 }. In view of Corollary 2 in [16] the interior of [[E1 , E2 ]] is subset of the basin of attraction of E1 . If the initial point (x0 , y0 ) ∈ / [[E1 , E2 ]] but it is betweenW s (E1 ) and the ray through E0 and E1 then one can find te points (xl , yl ) the ray through E0 and E1 and (xu , yu ) ∈ W s (E1 ) such that (xl , yl ) se (x0 , y0 ) se (xu , yu ) and so T n ((xl , yl )) se T n ((x0 , y0 )) se T n ((xu , yu )), n ≥ 1, which means T n ((x0 , y0 )) will eventually enter [[E1 , E2 ]] and so T n ((x0 , y0 )) → E2 . The proof when the initial point (x0 , y0 ) is below W s (E2 ) is similar. (c) The proof is similar to the one in case (a) and will be ommitted. This dynamic scenario is a replication of dynamic scenario in (a).

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(d) The proof is similar to the one in case (b) and will be ommitted. This dynamic scenario is exactly replication of dynamic scenario in (b). In the case of Equation (1) we have the following results which are direct application of Theorem 9. Theorem 10 Consider Equation (1) and assume that f is decreasing in first and increasing in the second variable on the set (a, b)2 , where a is either the repeller or a singular point of f , such that f is C 2 in a neighborhood of any fixed point. (a) Assume that Equation (1) has locally asymptotically stable equilibrium solutions x ¯ > a and the unique saddle point minimal period-two solution {P1 , Q1 }, P1 se (a, a) se Q1 . Assume that the stable manifold of P1 (resp. Q1 ) is the line through (a, a) and P1 (resp. the line through (a, a) and Q1 ). Then the equilibrium x ¯ is globally asymptotically stable for all x−1 , x0 > a. (b) Assume that Equation (1) has the saddle equilibrium solution x ¯ > a and the unique locally asymptotically stable minimal period-two solution {P1 , Q1 }, P1 se (a, a) se Q1 . Assume that the stable manifold of P1 (resp. Q1 ) is the line through (a, a) and P1 (resp. the line through (a, a) and Q1 ). Then the period-two solution {P1 , Q1 } attracts all initial points off the global stable manifold W s (E(¯ x, x ¯)). (c) Assume that Equation (1) has a saddle equilibrium solution x ¯ > a. Assume that Equation (1) has two minimal period-two solutions {P1 , Q1 } and {P2 , Q2 } such that P1 se P2 se E(¯ x, x ¯) se Q2 se Q1 , where {P2 , Q2 } is locally asymptotically stable and {P1 , Q1 } is a saddle point and assume that the global stable manifold of P1 (resp. Q1 ) is the line through (a, a) and P1 (resp. the line through (a, a) and Q1 ). Then every solution which starts off the union of global stable manifolds W s (E(¯ x, x ¯)) ∪ W s (P1 ) ∪ W s (Q1 ) converges to the period-two solution {P2 , Q2 }. (d) Assume that Equation (1) has locally asymptotically stable equilibrium solution x ¯ > a. Asume that Equation (1) has two minimal period-two solutions {P1 , Q1 } and {P2 , Q2 } such that P1 se P2 se E(¯ x, x ¯) se Q2 se Q1 , where {P1 , Q1 } is locally asymptotically stable and {P2 , Q2 } is a saddle point. If the line through (a, a) and P1 (resp. the line through (a, a) and Q1 ) is a part of the basin of attraction of {P1 , Q1 } then every solution which starts between the stable manifolds W s (P2 ) and W s (Q2 ) converges to x ¯ while every solution which starts below W s (Q2 ) (resp. above W s (P2 )) converges to the period-two solution {P1 , Q1 }. Proof. (a) In view of Remark 7 the second iterate T 2 of the map T associated with Equation (1) is strictly competitive. Applying Theorem 9 part (a) to T 2 , where we set E1 = P1 , E2 = (¯ x, x ¯), E3 = Q1 we complete the proof. (b) The proof follows from Theorem 9 part (b) applied to T 2 , where we set E1 = P1 , E2 = (¯ x, x ¯), E3 = Q1 and observation that locally asymptotically stable fixed point (resp. saddle point) for T has the same character for T 2. (c) The proof is similar to the proof in case (a) and will be ommitted. (d) The proof follows from Theorem 9 part (d) applied to T 2 , where we set E1 = P1 , E2 = P2 , E3 = (¯ x, x ¯), E4 = Q2 , E5 = Q1 and the observation that locally asymptotically stable fixed point (resp. saddle point) for T has the same character for T 2 .

Remark 11 The term ”saddle point” in formulation of statements of Theorems 9 and 10 can be replaced by the term ”non-hyperbolic point of stable type”. Results related to Theorem 9 were obtained in [1, 2] and the results related to Theorem 10 were obtained in [6, 9, 10]. Furthermore Cases (b) and (c) of Theorem 9 can be extended to the case when we have any odd number of the equilibrium points which alternate its stability between two types: locally asymptotically stable and saddle points or non-hyperbolic equilibrium points of the stable type. The transition from Case (a) to Case (b) and from Case (c) to Case (d) in Theorem 9 is an exchange of stability bifurcation, while in the case of Theorem 10 these two bifurcations are two global period doubling bifurcations.

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3

Case study: Equation xn+1 =

γxn−1 Ax2n +Bxn xn−1 +Cxn−1

We investigate global behavior of Equation (2), where the parameters γ, A, B, C are positive numbers and the initial conditions x−1 , x0 are arbitrary nonnegative numbers such that x−1 + x0 > 0. Equation (2) is a special case of equations αx2n + βxn xn−1 + γxn−1 xn+1 = , n = 0, 1, . . . (6) Ax2n + Bxn xn−1 + Cxn−1 and Ax2n + Bxn xn−1 + Cx2n−1 + Dxn + Exn−1 + F xn+1 = , n = 0, 1, . . . (7) ax2n + bxn xn−1 + cx2n−1 + dxn + exn−1 + f The comprehensive linearized stability analysis of Equation (6) was given in [9] and some special cases were considered in [10]. Some special cases of Equation (7) have been considered in the series of papers [5, 6, 11, 12, 19]. Describing the global dynamics of Equation (7) is a formidable task as this equation contains as a special cases many equations with complicated dynamics, such as the linear fractional difference equation xn+1 =

Dxn + Exn−1 + F , dxn + exn−1 + f

n = 0, 1, ....

(8)

Equation (2) has 0 as a singular point and the first quadrant as the region R.

3.1

Local stability analysis

By using the substitution yn =

C x γ n

Equation (2) is reduced to the equation

xn+1 = 2

xn−1 , n = 0, 1, ... A0 x2n + B 0 xn xn−1 + xn−1

(9)

2

γ γ 0 0 0 where A0 = C 2 A and B = C 2 B. In the sequel we consider Equation (9) where A and B will be replaced with A and B respectively. First, we notice that under the conditions on parameters all solutions of Equation (9) are in interval (0, 1] and that 0 is a singular point. Equation (9) has the unique positive equilibrium x ¯ given by √ −1+ 1+4(A+B) x ¯= . (10) 2(A+B)

The partial derivatives associated to Equation (9) at the equilibrium x ¯ are 4(2A+B) −y(2Ax+By) Ax2 4A fy0 = (Ax2 +Bxy+y) = . fx0 = (Ax √ √ 2 +Bxy+y)2 = − 2, 2 (1+ 1+4A+4B ) (1+ 1+4A+4B )2 x ¯ x ¯ Characteristic equation associated to Equation (9) at the equilibrium is λ2 +

4(2A+B) √ 2λ 1+4A+4B )

(1+

−

4A √ 2 1+4A+4B )

(1+

= 0.

By applying the linearized stability Theorem, see [13], we obtain the following result. Theorem 12 The unique positive equilibrium solution x ¯ of Equation (9) is: i) locally asymptotically stable when B + 3A > 4A2 ; ii) a saddle point when B + 3A < 4A2 ; iii) a non-hyperbolic point of stable type (with eigenvalues λ1 = −1 and λ2 =

1 4A

< 1) when B + 3A = 4A2 .

In the next lemma we prove the existence of period two solutions of Equation (9). Lemma 13 Equation (9) has the minimal period-two solution {(0, 1), (1, 0)} and the minimal period-two solution {P (φ, ψ), Q(ψ, φ)}, where √ √ A− (A−B)(A(−3+4A)−B−B A+ (A−B)(A(−3+4A)−B−B φ= and ψ = (11) 2A(A−B) 2A(A−B) if and only if 3 < A < 1 and B + 3A < 4A2 4

or

A > 1 and B + 3A > 4A2 .

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Proof. A minimal period-two solution is a positive solution of the following system  x + (B − A)y − 1 = 0 −Axy + y = 0.

(12)

where φ + ψ = x and φψ = y. The system (12) has two solutions x = 1, y = 0 and x=

1 , A

A−1 . A(B − A)

y=

For second solution we have that x, y, x2 − 4y > 0 if and only if 3 < A < 1 and B + 3A < 4A2 4

or

A > 1 and B + 3A > 4A2 .

Now, φ and ψ are solution of equation t2 −

1 A−1 t− = 0, A A(B − A)

and the proof is complete. The following theorem describes the local stability nature of the period-two solutions. Theorem 14 Consider Equation (9). i) The minimal period two solution {(0, 1), (1, 0)} is: a) locally asymptotically stable when A > 1; b) a saddle point when A < 1; c) a non-hyperbolic point of the stable type when A = 1. ii) The minimal period two solution {P (φ, ψ), Q(ψ, φ)}, given by (11) is: a) locally asymptotically stable when

3 4

< A < 1 and B + 3A < 4A2 ;

b) a saddle point when A > 1 and B + 3A > 4A2 . iii) If A = B = 1 the minimal period two solution {φ, 1 − φ} (0 < φ < 1) is non-hyperbolic. Proof. In order to prove this theorem, we associate the second iterate map to Equation (9). We have     u g(u, v) T2 = v h(u, v) where g(u, v) =

u , Av 2 + Buv + u

h(u, v) =

v v+

Au2 (Av2 +Buv+u)2

The Jacobian of the map T 2 has the following form    0 φ gu (φ, ψ) JT 2 = ψ h0u (φ, ψ) where gu0 =

(Av 2

Av 2 , + Buv + u)2 3

gv0 = −

gv0 (φ, ψ) h0v (φ, ψ)

+

Buv Av 2 +Buv+u

.



u(Bu + 2Av) , (Av 2 + Buv + u)2

2

3

(u+Buv+Av )(Buv(1+Bv)+A(2u+Bv )) h0u = − (A2 vAv 5 +u2 v(1+Bv)(1+B+Bv)+Au(u+v 3 (2+B+2Bv))2 ,

h0v =

u(u+Buv+Av 2 )(B 2 u2 v 2 (1+Bv)+A2 v 2 (5u+2Bv 3 )+Au(u+3Buv+Bv 3 (2+3Bv 3 ))) . (A2 v 5 +u2 v(1+Bv)(1+B+Bv)+Au(u+v 3 (2+B+2Bv))2

Set S = gu0 (φ, ψ) + h0v (φ, ψ) and D = gu0 (φ, ψ)h0v (φ, ψ) − gv0 (φ, ψ)h0u (φ, ψ). After some lengthy calculation one can see that:

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i) for the minimal period-two solution {(0, 1), (1, 0)} we have S=

1 and D = 0 A

and applying the linearized stability Theorem [13] we obtain that the minimal period-two solution {(0, 1), (1, 0)} of Equation (9) is: a) locally asymptotically stable when A > 1; b) a saddle point when A < 1; c) a non-hyperbolic point of the stable type when A = 1. ii) For the positive minimal period two solution {P (φ, ψ), Q(ψ, φ)} we have S=

6A4 +A(B−2)B−B 2 −3A3 (3+2B)+A2 (4+B(6+B)) , A2 (A−B)2

D=

(A−1)2 . (A−B)2

Applying the linearized stability Theorem [13] we obtain that the minimal period-two solution {P (φ, ψ), Q(ψ, φ)} of Equation (9) is: a) locally asymptotically stable when

3 4

< A < 1 and B + 3A < 4A2 ;

b) a saddle point when A > 1 and B + 3A > 4A2 . iii) If A = B = 1 then S = 1 + φ2 (1 − φ)2 ,

D = φ2 (1 − φ)2

from which the proof follows.

3.2

Global results and basins of attraction

In this section we present global dynamics results for Equation (9). Theorem 15 If B + 3A > 4A2 and 0 < A < 1 then Equation (9) has a unique equilibrium solution E(x, x) given by (10) which is locally asymptotically stable and the minimal period-two solution {P (0, 1), Q(1, 0)} which is a saddle point. Furthermore, the global stable manifold of the period-two solution {P, Q} is given by W s ({P, Q}) = W s (P ) ∪ W s (Q) where W s (P ) and W s (Q) are the coordinate axes. The basin of attraction B(E) = {(x, y) : x ≥ 0, y ≥ 0}. More precisely i) If (u0 , v0 ) ∈ W s (P ) then the subsequence of even-indexed terms {(u2n , v2n )} is attracted to P , and the subsequence of odd-indexed terms {(u2n+1 , v2n+1 )} is attracted to Q. ii) If (u0 , v0 ) ∈ W s (Q) then the subsequence of even-indexed terms {(u2n , v2n )} is attracted to Q, and the subsequence of odd-indexed terms {(u2n+1 , v2n+1 )} is attracted to P . iii) If (u0 , v0 ) ∈ R(W s (P ) ∪ W s (Q)) (the region between W s (P ) and W s (Q)) then the sequence {(un , vn )} is attracted to E(x, x). See Figure 1 for visual illustration. Proof. The proof is direct application of Theorem 10 part (a). Theorem 16 If B + 3A > 4A2 and A = 1 then Equation (9) has a unique equilibrium solution E(x, x) which is locally asymptotically stable and the minimal period-two solution, {P (0, 1), Q(1, 0)} which is a non-hyperbolic point of stable type. Furthermore, the global stable manifold of the period-two solution {P, Q} is given by W s ({P, Q}) = W s (P ) ∪ W s (Q) where W s (P ) and W s (Q) are the coordinate axes. The global dynamics is given in Theorem 15. Proof. In view of Remark 11 the proof is direct application of Theorem 10 part (a).

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Figure 1: Visual illustration of Theorem 15. Theorem 17 If B + 3A > 4A2 and A > 1 then Equation (9) has a unique equilibrium solution E(x, x) which is locally asymptotically stable and two minimal period-two solutions {P1 (0, 1), Q1 (1, 0)} which is locally asymptotically stable and {P2 (φ, ψ), Q2 (ψ, φ)} given by (11), which is a saddle point. Furthermore, the global stable manifold of the period-two solution {P2 , Q2 } is given by W s ({P2 , Q2 }) = W s (P2 ) ∪ W s (Q2 ) where W s (P2 ) and W s (Q2 ) are continuous increasing curves, that divide the first quadrant into two connected components, namely W1+ := {x ∈ R \ W s (P2 ) : ∃y ∈ W s (P2 ) with y se x} and W1− := {x ∈ R \ W s (P2 ) : ∃y ∈ W s (P2 ) with x se y} W2+ := {x ∈ R \ W s (Q2 ) : ∃y ∈ W s (Q2 ) with y se x} and W2− = {x ∈ R \ W s (Q2 ) : ∃y ∈ W s (Q2 ) with x se y} respectively such that the following statements are true. i) If (u0 , v0 ) ∈ W s (P2 ) then the subsequence of even-indexed terms {(u2n , v2n )} is attracted to P2 and the subsequence of odd-indexed terms {(u2n+1 , v2n+1 )} is attracted to Q2 . ii) If (u0 , v0 ) ∈ W s (Q2 ) then the subsequence of even-indexed terms {(u2n , v2n )} is attracted to Q2 and the subsequence of odd-indexed terms {(u2n+1 , v2n+1 )} is attracted to P2 . iii) If (u0 , v0 ) ∈ W1− (the region above W s (P2 )) then the subsequence of even-indexed terms {(u2n , v2n )} is attracted to P1 and the subsequence of odd-indexed terms {(u2n+1 , v2n+1 )} tends to Q1 . iv) If (u0 , v0 ) ∈ W2+ (the region below W s (Q2 )) then the subsequence of even-indexed terms {(u2n , v2n )} tends to Q1 and the subsequence of odd-indexed terms {(u2n+1 , v2n+1 )} tends to P1 . v) If (u0 , v0 ) ∈ W1+ ∩ W2− (the region between W s (P2 ) and W s (Q2 )) then the sequence {(un , vn )} is attracted to E(x, x). Shortly the basin of attraction of E is the region between W s (P2 ) and W s (Q2 ) while the rest of the first quadrant without W s (P2 ) ∪ W s (Q2 ) ∪ (0, 0) is the basin of attraction of {P1 , Q1 }. See Figure 2 for visual illustration. Proof. The proof is direct application of Theorem 10 part (d).

Theorem 18 If B + 3A < 4A2 and 34 < A < 1 then Equation (9) has a unique equilibrium solution E(x, x) which is a saddle point and minimal period-two solution {P1 (0, 1), Q1 (1, 0)} which is a saddle point and {P2 (φ, ψ), Q2 (ψ, φ)}, given by (11) which is locally asymptotically stable. Furthermore, there exists a set CE which is an invariant subset of the basin of attraction of E. The set CE is a graph of a strictly increasing continues function of the first variable

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Figure 2: Visual illustration of Theorem 17. on (0, ∞) interval and separates R(0, 0), where R = (0, ∞) × (0, ∞) into two connected and invariant components W − (x, x) and W + (x, x). The global stable manifold of the period-two solution {P1 , Q1 } is given by W s ({P1 , Q1 }) = W s (P1 ) ∪ W s (Q1 ) where W s (P1 ) and W s (Q1 ) are continuous nondecreasing curves which represent the coordinate axes. The basin of attraction of {P2 , Q2 } is the first quadrant without W s (P1 ) ∪ W s (Q1 ) ∪ (0, 0) ∪ CE . More precisely i) Every initial point (u0 , v0 ) in CE is attracted to E. ii) If (u0 , v0 ) ∈ W s (P1 ) then the subsequence of even-indexed terms {(u2n , v2n )} is attracted to P1 and the subsequence of odd-indexed terms {(u2n+1 , v2n+1 )} is attracted to Q1 . iii) If (u0 , v0 ) ∈ W s (Q1 ) then the subsequence of even-indexed terms {(u2n , v2n )} is attracted to Q1 and the subsequence of odd-indexed terms {(u2n+1 , v2n+1 )} is attracted to P1 . iv) If (u0 , v0 ) ∈ W − (x, x) (the region between CE and W s (P1 )) then the subsequence of even-indexed terms {(u2n , v2n )} is attracted to P2 and the subsequence of odd-indexed terms {(u2n+1 , v2n+1 )} tends to Q2 . v) If (u0 , v0 ) ∈ W + (x, x) (the region between CE and W s (Q1 )) then the subsequence of even-indexed terms {(u2n , v2n )} tends to Q2 and the subsequence of odd-indexed terms {(u2n+1 , v2n+1 )} tends to P2 . See Figure 3 for visual illustration. Proof. Theorem 12 implies that there exists a unique equilibrium solution E(x, x) which is a saddle point and Theorem 14 implies that minimal period-two solution {P1 (0, 1), Q1 (1, 0)} is a saddle point and {P2 (φ, ψ), Q2 (ψ, φ)} is locally asymptotically stable. Now the proof is direct application of Theorem 10 part (c).

Figure 3: Visual illustration of Theorem 18. Theorem 19 If B + 3A < 4A2 and A = 1 then Equation (9) has a unique equilibrium solution E(x, x), which is a saddle point and the minimal period-two solution {P1 (0, 1), Q1 (1, 0)} which is a non-hyperbolic point of stable type.

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Furthermore, the global stable manifold W s (E) is continuous increasing curve which divides first quadrant and the global stable manifold of the period-two solution {P1 , Q1 } is given by W s ({P1 , Q1 }) = W s (P1 )∪W s (Q1 ) where W s (P1 ) and W s (Q1 ) are the coordinate axes. The basin of attraction B({P1 , Q1 }) = {(x, y) : x ≥ 0, y ≥ 0}(W s (E)∪(0, 0))}. More precisely i) Every initial point (u0 , v0 ) in W s (E) is attracted to E. i) If (u0 , v0 ) ∈ W + (E) (the region below W s (E)) then the subsequence of even-indexed terms {(u2n , v2n )} is attracted to Q1 and the subsequence of odd-indexed terms {(u2n+1 , v2n+1 )} is attracted to P1 . iii) If (u0 , v0 ) ∈ W − (E) (the region above W s (E)) then the subsequence of even-indexed terms {(u2n , v2n )} is attracted to P1 and the subsequence of odd-indexed terms {(u2n+1 , v2n+1 )} is attracted to Q1 . See Figure 4 for visual illustration. Proof. From Theorem 12 Equation (9) has a unique equilibrium point E(x, x) which is a saddle point. Theorem 14 implies that the period-two solution {P, Q} is a non-hyperbolic point. In view of Remark 11 the proof is direct application of Theorem 10 part (b).

Figure 4: Visual illustration of Theorem 19. Theorem 20 If B + 3A < 4A2 and A > 1 then Equation (9) has a unique equilibrium solution E(x, x) which is a saddle point and the minimal period-two solution {P (0, 1), Q(1, 0)} which is locally asymptotically stable. The global behavior is the same as in Theorem 19. Proof. The proof is direct application of Theorem 10 part (b). Theorem 21 Assume that B + 3A = 4A2 . a) If 43 < A < 1 then Equation (9) has a unique equilibrium point E(x, x) which is a non-hyperbolic point of stable type and the minimal period-two solution {P (0, 1), Q(1, 0)} which is a saddle point. Then every initial point (u0 , v0 ) in R is attracted to E. b) If A > 1 then Equation (9) has a unique equilibrium solution E(x, x) which is a non-hyperbolic point of the stable type and the minimal period-two solution {P (0, 1), Q(1, 0)} which is locally asymptotically stable. The global behavior is the same as in Theorem 19. c) If A = 1 then Equation (9) has a unique equilibrium solution E(x, x) and infinitely many minimal period-two solution {P (φ, 1 − φ), Q(1 − φ, φ)} (0 < φ < 1) which are a non-hyperbolic points of stable type. i) There exists a continuous increasing curve CE which is a subset of the basin of attraction of E ii) For every minimal period-two solution {P (φ, 1 − φ), Q(1 − φ, φ)} (0 < φ < 1) there exists the global stable manifold given by W s ({P, Q}) = W s (P ) ∪ W s (Q) where W s (P ) and W s (Q) are continuous increasing curves. If (u0 , v0 ) ∈ W s (P ) then the subsequence of even-indexed terms {(u2n , v2n )} tends to P and the subsequence of odd-indexed terms {(u2n+1 , v2n+1 )} tends to Q. If (u0 , v0 ) ∈ W s (Q) then the subsequence of even-indexed terms {(u2n , v2n )} tends to Q and the subsequence of odd-indexed terms {(u2n+1 , v2n+1 )} tends to P. The union of these stable manifolds and CE foliates the first quadrant without the singular point (0, 0).

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See Figure 5 for visual illustration. Proof. 1 1 a) From Theorem 12 Equation (9) has a unique equilibrium point E(x, x) = ( 2A , 2A ) which is non-hyperbolic of stable type. From Theorem 14 Equation (9) has a unique minimal period-two solution {P1 (0, 1), Q1 (1, 0)} which is a saddle point. In view of Remark 11 the proof is direct application of Theorem 10 part (a). 1 1 b) From Theorem 12 Equation (9) has a unique equilibrium point E(x, x) = ( 2A , 2A ), which is non-hyperbolic of stable type. From Theorem 14 Equation (9) has a unique minimal period-two solution {P1 (0, 1), Q1 (1, 0)} which is locally asymptotically stable point. In view of Remark 11 the proof is direct application of Theorem 10 part (b). 1 1 c) From Theorem 12 Equation (9) has a unique equilibrium point E(x, x) = ( 2A , 2A ) which is non-hyperbolic. All conditions of Theorem 5 are satisfied, which yields the existence a continuous increasing curve CE which is a subset of the basin of attraction of E. The proof of the statement ii) follows from Theorems 3, 5, 14 and Theorem 5 in [8].

Remark 22 The global dynamics of Equation (9) can be described in the language of bifurcation theory as follows: when B + 3A 6= 4A2 , then the period-doubling bifurcation happens when A is passing through the value 1 in such a way that for A > 1 new interior period-two solution emerges and exchange stability with already existing period-two solution on the boundary. Another bifurcation happens when B + 3A < 4A2 in which case the stability of the unique equilibrium changes from local attractor to the saddle point. Finally, there is a bifurcation at another critical value B + 3A = 4A2 when A is passing through the critical value 1, which is one of exchange stability between the unique equilibrium and unique period-two solution, with specific dynamics at A = 1, when there is an infinite number of period-two solutions which basins of attraction filled up the first quadrant without the origin. See [16] for similar results.

Figure 5: Visual illustration of Theorem 21.

References [1] E. Bertrand and M. R. S. Kulenovi´c, Global Dynamic Scenarios for Competitive Maps in the Plane, Adv. Difference Equ., Volume 2018 (2018):307, 28 p. [2] A. Bilgin, M. R. S. Kulenovi´c and E. Pilav, Basins of Attraction of Period-Two Solutions of Monotone Difference Equations, Adv. Difference Equa., Volume 2016 (2016), 25p.

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[3] A. Brett and M. R. S. Kulenovi´c, Basins of Attraction of Equilibrium Points of Monotone Difference Equations, Sarajevo J. Math., 5(2009), 211-233 . [4] E. Camouzis and G. Ladas. When does local asymptotic stability imply global attractivity in rational equations? J. Difference Equ. Appl., 12 (2006), 863–885. [5] M. Dehghan, C. M. Kent, R. Mazrooei-Sebdani, N. L. Ortiz and H. Sedaghat, Monotone and oscillatory solutions of a rational difference equation containing quadratic terms, J. Difference Equ. Appl., 14 (2008), 1045–1058. [6] M. Gari´c Demirovi´c, M. R. S. Kulenovi´c and M. Nurkanovi´c, Basins of Attraction of Equilibrium Points of Second Order Difference Equations, Appl. Math. Letters, 25(2012), 2110–2115. [7] M. Hirsch and H. L. Smith, Monotone Maps: A Review, J. Difference Equ. Appl. 11(2005), 379–398. [8] S. Kalabuˇsi´c, M. R. S. Kulenovi´c and E. Pilav, Global Dynamics of Anti-Competitive Systems in the Plane, Dyn. Contin. Discrete Impuls. Syst. Ser. A Math. Anal., 20(2013), 477-505. [9] S. Kalabuˇsi´c, M. R. S. Kulenovi´c and M. Mehulji´c, Global Period-doubling Bifurcation of Quadratic Fractional Second Order Difference Equation, Discrete Dyn. Nat. Soc., (2014), Art. ID 920410, 13p. [10] S. Kalabuˇsi´c, M. R. S. Kulenovi´c and M. Mehulji´c, Global Dynamics and Bifurcations of Two Quadratic Fractional Second Order Difference Equations, J. Comp. Anal. Appl., 20(2016), 132–143. [11] C. M. Kent and H. Sedaghat, Global attractivity in a quadratic-linear rational difference equation with delay. J. Difference Equ. Appl. 15 (2009), 913–925. [12] C. M. Kent and H. Sedaghat, Global attractivity in a rational delay difference equation with quadratic terms, J. Difference Equ. Appl., 17 (2011), 457–466. [13] M. R. S. Kulenovi´c and G. Ladas, Dynamics of Second Order Rational Difference Equations with Open Problems and Conjectures, Chapman and Hall/CRC, Boca Raton, London, 2001. [14] M. R. S. Kulenovi´c and O. Merino, Discrete Dynamical Systems and Difference Equations with Mathematica, Chapman and Hall/CRC, Boca Raton, London, 2002. [15] M. R. S. Kulenovi´c and O. Merino, Competitive-Exclusion versus Competitive-Coexistence for Systems in the Plane, Discrete Cont. Dyn. Syst.Ser. B 6(2006), 1141–1156. [16] M. R. S. Kulenovi´c and O. Merino, Global Bifurcation for Competitive Systems in the Plane, Discrete Contin. Dyn. Syst. B, 12 (2009), 133–149. [17] M. R. S. Kulenovi´c and O. Merino, Invariant manifolds for competitive discrete systems in the plane. Internat. J. Bifur. Chaos Appl. Sci. Engrg. 20 (2010), 2471-2486. [18] M. R. S. Kulenovi´c and O. Merino, Invariant Curves for Planar Competitive and Cooperative Maps, J. Difference Equ. Appl., 24(2018), 898–915. [19] H. L. Smith, Planar Competitive and Cooperative Difference Equations,J. Difference Equ. Appl. 3(1998), 335357.

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Global Dynamics of Generalized Second-Order Beverton–Holt Equations of Linear and Quadratic Type E. Bertrand

†

and M. R. S Kulenovi´c‡1

†

Department of Mathematics Sacred Heart University, Fairfield, CT 06825, USA ‡

Department of Mathematics University of Rhode Island, Kingston, Rhode Island 02881-0816, USA Abstract. We investigate second-order generalized Beverton–Holt difference equations of the form xn+1 =

af (xn , xn−1 ) , 1 + f (xn , xn−1 )

n = 0, 1, . . . ,

where f is a function nondecreasing in both arguments, the parameter a is a positive constant, and the initial conditions x−1 and x0 are arbitrary nonnegative numbers in the domain of f . We will discuss several interesting examples of such equations and present some general theory. In particular, we will investigate the local and global dynamics in the event f is a certain type of linear or quadratic polynomial, and we explore the existence problem of period-two solutions. Keywords. attractivity, difference equation, invariant sets, periodic solutions, stable set .

AMS 2010 Mathematics Subject Classification: 39A20, 39A28, 39A30, 92D25

1

Introduction and Preliminaries

Consider the following second-order difference equation: xn+1 =

af (xn , xn−1 ) , 1 + f (xn , xn−1 )

n = 0, 1, . . . .

(1)

Here f is a continuous function nondecreasing in both arguments, the parameter a is a positive real number, and the initial conditions x−1 and x0 are arbitrary nonnegative numbers in the domain of f . Equation (1) is a generalization of the first-order Beverton–Holt equation xn+1 =

axn , 1 + xn

n = 0, 1, . . . ,

(2)

where a > 0 and x0 ≥ 0. The global dynamics of Equation (2) may be summarized as follows, see [9, 15]:  0 if a ≤ 1 lim xn = (3) n→∞ a − 1 if a > 1 and x0 > 0. Many variations of Equation (2) have been studied. German biochemist Leonor Michaelis and Canadian physician Maud Menten used the model in their study of enzyme kinetics in 1913; see [20]. Additionally, Jacques Monod, a French biochemist, happened upon the model empirically in his study of microorganism growth around 1942; see [20]. It was not until 1957 that fisheries scientists Ray Beverton and Sidney Holt used the model in their study of population dynamics, see [1, 9]. The so-called Monod differential equation [20] is given by rS 1 dN · = , (4) N dt a+S where N (t) is the concentration of bacteria at time t, dN dt is the growth rate of the bacteria, S(t) is the concentration of the nutrient, r is the maximum growth rate of the bacteria, and a is a half-saturation 1

Corresponding author, e-mail: [email protected]

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constant (when S = a, the right-hand side of Equation (4) equals r/2). Based on experimental data, the following system of two differential equations for the nutrient S and bacteria N , as presented in [20], is given by dS 1 rS dN rS =− N , =N , (5) dt γ a+S dt a+S where the constant γ is called the growth yield. Both Equation (4) and System (5) contain the function f (x) = rx/(a + x) known as the Monod function, Michaelis-Menten function, Beverton–Holt function, or Holling function of the first kind; see [1, 5, 9, 11]. One possible two-generation population model based on Equation (2), xn+1 =

a1 xn a2 xn−1 + , 1 + xn 1 + xn−1

n = 0, 1, . . . ,

(6)

where ai > 0 for i = 1, 2 and x−1 , x0 ≥ 0, was considered in [18]. The global dynamics of Equation (6) may be summarized as follows:  0 if a1 + a2 ≤ 1 lim xn = n→∞ a1 + a2 − 1 if a1 + a2 > 1 and x0 + x−1 > 0. This result was extended in [5] to the case of a k-generation population model based on Equation (2) of the form k−1 X ai xn−i xn+1 = , n = 0, 1, . . . , (7) 1 + xn−i i=0

where ai ≥ 0 for i = 0, 1, . . . , k − 1,

k−1 P

ai > 0, and x1−k , . . . , x0 ≥ 0. It was shown that the global dynam-

i=0

ics of Equation (7) may be given precisely by (3), where a =

k−1 P

ai and we consider all initial conditions

i=0

positive. The simplest model of Beverton–Holt type which exhibits two coexisting attractors and the Allee effect is the sigmoid Beverton–Holt (or second-type Holling) difference equation xn+1 =

ax2n , 1 + x2n

n = 0, 1, . . . ,

(8)

where a > 0 and x0 ≥ 0. The dynamics of Equation (8) may be concisely summarized as follows:   0 if a < 2 or (a ≥ 2 and x0 < x− ) x− if a ≥ 2 and x0 = x− lim xn = n→∞  x+ if a ≥ 2 and x0 > x− ,

(9)

where x− and x+ are the two positive equilibria when a ≥ 2; see [1, 5]. One possible two-generation population model based on Equation (8), xn+1 =

a2 x2n−1 a1 x2n + , 1 + x2n 1 + x2n−1

n = 0, 1, . . . ,

(10)

where ai > 0 for i = 1, 2 and x−1 , x0 ≥ 0, was considered in [4]. However, the summary of the global dynamics of Equation (10) is not an immediate extension of the global dynamics of Equation (8) as given in

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(9); see [4]. Equation (10) can have up to three equilibrium solutions and up to three period-two solutions. In the case when Equation (10) has three equilibrium solutions and three period-two solutions, the zero equilibrium, the larger positive equilibrium, and one period-two solution are attractors with substantial basins of attraction, which together with the remaining equilibrium and the global stable manifolds of the saddle-point period-two solutions exhaust the first quadrant of initial conditions. This behavior happens when the coefficient a2 is in some sense dominant to a1 ; see [4]. Such behavior is typical for other models in population dynamics such as xn+1 =

a2 x2n−1 a1 xn + , 1 + xn 1 + x2n−1

and xn+1 = a1 xn +

a2 x2n−1 , 1 + x2n−1

n = 0, 1, . . .

n = 0, 1, . . . ,

which were also investigated in [4]. In the case of a k-generation population model based on the sigmoid Beverton–Holt difference equation with k > 2, one can expect to have attractive period-k solutions as well as chaos. The first model of the form given in Equation (1), where f is a linear function in both variables (that is, f (u, v) = cu + dv for c, d, u, v ≥ 0) was considered in [19] to describe the global dynamics in part of the parametric space. Here we will extend the results from [19] to the whole parametric space. In this paper we will then restrict ourselves to the case when f (u, v) is a quadratic polynomial, which will give similar global dynamics to that presented for Equation (10). The corresponding dynamic scenarios will be essentially the same for any polynomial function of the type f (u, v) = cuk +dum where c, d ≥ 0 and m, k are positive integers. Higher values of m and k may only create additional equilibria and period-two solutions but should replicate the global dynamics seen in the quadratic case presented in this paper. The global dynamics of some higher-order transcendental-type generalized Beverton-Holt equation was considered in [3]. Let the function F : [0, ∞)2 → [0, a) be defined as follows: F (u, v) =

af (u, v) . 1 + f (u, v)

(11)

Then Equation (1) becomes xn+1 = F (xn , xn−1 ) for all n = 0, 1, . . . , where F (u, v) is nondecreasing in both of its arguments. The following theorem from [2] immediately applies to Equation (1). Theorem 1 Let I be a set of real numbers and F : I × I → I be a function which is nondecreasing in the first variable and nondecreasing in the second variable. Then, for every solution {xn }∞ n=−1 of the equation xn+1 = F (xn , xn−1 ) ,

x−1 , x0 ∈ I,

n = 0, 1, . . . ,

(12)

∞ the subsequences {x2n }∞ n=0 and {x2n−1 }n=0 of even and odd terms of the solution are eventually monotonic.

The consequence of Theorem 1 is that every bounded solution of Equation (12) converges to either an equilibrium, a period-two solution, or to a singular point on the boundary. It should be noticed that Theorem 1 is specific for second-order difference equations and does not extend to difference equations of order higher than two. Furthermore, the powerful theory of monotone maps in the plane [16, 17] can be applied to Equation (1) to determine the boundaries of the basins of attraction of the equilibrium

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solutions and period-two solutions. Finally, when f (u, v) is a polynomial function, all computation needed to determine the local stability of all equilibrium solutions and period-two solutions is reduced to the theory of counting the number of zeros of polynomials in a given interval, as given in [12]. This theory will give more precise results than the global attractivity and global asymptotic stability results in [7, 8]. However, in the case of difference equations of the form xn+1 =

ag(xn , xn−1 , . . . , xn+1−k ) , 1 + g(xn , xn−1 , . . . , xn+1−k )

n = 0, 1, . . . ,

k ≥ 1,

where a > 0 and g is nondecreasing in all its arguments, Theorem 1 does not apply for k > 2, but the results from [7, 8, 13] can give global dynamics in some regions of the parametric space. The following theorem from [10] is often useful in determining the global attractivity of a unique positive equilibrium. Theorem 2 Let I ⊆ [0, ∞) be some open interval and assume that F ∈ C[I × I, (0, ∞)] satisfies the following conditions: (i) F (x, y) is nondecreasing in each of its arguments; (ii) Equation (12) has a unique positive equilibrium point x ∈ I and the function F (x, x) satisfies the negative feedback condition: (x − x)(F (x, x) − x) < 0 for every x ∈ I\{x}. Then every positive solution of Equation (12) with initial conditions in I converges to x. The following result from [4] will be used to describe the global dynamics of Equation (1). Theorem 3 Assume that difference equation (12) has three equilibrium points U1 ≤ x ¯0 < x ¯SW < x ¯N E where the equilibrium points x ¯0 and x ¯N E are locally asymptotically stable. Further, assume that there exists a minimal period-two solution {Φ1 , Ψ1 } which is a saddle point such that (Φ1 , Ψ1 ) ∈ int(Q2 (ESW )). In this case there exist four continuous curves W s (Φ1 , Ψ1 ), W s (Ψ1 , Φ1 ), W u (Φ1 , Ψ1 ), W u (Ψ1 , Φ1 ), where W s (Φ1 , Ψ1 ), W s (Ψ1 , Φ1 ) are passing through the point ESW , and are graphs of decreasing functions. The curves W u (Φ1 , Ψ1 ), W u (Ψ1 , Φ1 ) are the graphs of increasing functions and are starting at E0 . Every solution which starts below W s (Φ1 , Ψ1 ) ∪ W s (Ψ1 , Φ1 ) in the North-east ordering converges to E0 and every solution which starts above W s (Φ1 , Ψ1 ) ∪ W s (Ψ1 , Φ1 ) in the North-east ordering converges to EN E , i.e. W s (Φ1 , Ψ1 ) = C1+ = C2+ and W s (Ψ1 , Φ1 ) = C1− = C2− . This paper is organized as follows. The next section deals with the local stability of equilibrium solutions and period-two solutions of the general second-order difference equation (12), where F (u, v) is nondecreasing in both of its arguments. In view of the results for monotone maps in [16, 17] and their applications to second-order difference equations in [4, 5], the local dynamics of the equilibrium solutions and period-two solutions will determine the global dynamics in hyperbolic cases and some nonhyperbolic cases as well. The third section will provide some examples of global dynamic scenarios of Equation (1) when the function f (u, v) is either linear in both variables or linear in one variable and quadratic in the other variable. The obtained results will be interesting from a modeling point of view as they show that the appearance of period-two solutions with substantial basins of attraction (sets which contain open subsets) is controlled by the coefficient of the xn−1 term that is affected by the size of the grandparents’ population. The same phenomenon was observed in the case of Equation (10).

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2

Local Stability

In this section we provide general conditions to determine the local stability of equilibrium solutions and period-two solutions. It is clear that xn ≤ a for all n ≥ 1. In light of Theorem 1, since all solutions are bounded, if there are no singular points on the boundary of the domain of F , it immediately follows that all solutions to Equation (1) converge to an equilibrium or a period-two solution. An equilibrium x of Equation (1) satisfies x(1 + f (x, x)) = af (x, x).

(13)

Clearly x0 = 0 is an equilibrium point if and only if (0, 0) is in the domain of f and f (0, 0) = 0. The linearized equation of Equation (1) about an equilibrium x is zn+1 = Fu (x, x)zn + Fv (x, x)zn−1 , n = 0, 1, . . . . Since f is a nondecreasing function, it follows that Fu (x, x) ≥ 0, Fv (x, x) ≥ 0. Therefore, if λ(x) = Fu (x, x) + Fv (x, x) =

a(fu (x, x) + fv (x, x)) , (1 + f (x, x))2

then in view of Corollary 2 of [13] we may conclude that   locally asymptotically stable nonhyperbolic x is  unstable

(14)

if λ(x) < 1 if λ(x) = 1 if λ(x) > 1.

Further, Theorem 2.13 of [15] implies that if x is unstable, then  if δ(x) > 1  a repeller nonhyperbolic if δ(x) = 1 x is  a saddle point if δ(x) < 1, where δ(x) = Fv (x, x) − Fu (x, x) =

a(fv (x, x) − fu (x, x)) . (1 + f (x, x))2

(15)

Let (φ, ψ) be a period-two solution of Equation (1). The Jacobian matrix of the corresponding map T = G2 , where G(u, v) = (v, F (v, u)) and F is given by Equation (11), is given in Theorem 12 of [6]. The linearized equation evaluated at (φ, ψ) is λ2 − T rJT (φ, ψ)λ + DetJT (φ, ψ) = 0, where T rJT (φ, ψ) = D2 F (ψ, φ) + D1 F (F (ψ, φ), ψ) · D1 F (ψ, φ) + D2 F (F (ψ, φ), ψ) and DetJT (φ, ψ) = D2 F (F (ψ, φ), ψ) · D2 F (ψ, φ).

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3

Examples

In this section we present four examples of different forms of Equation (1) where the transition function f (u, v) is linear or quadratic polynomial in its variables which effects the global dynamics.

3.1

Linear-Linear: f (u, v) = cu + dv

We consider the difference equation xn+1 =

a(cxn + dxn−1 ) , 1 + cxn + dxn−1

n = 0, 1, . . . ,

(16)

where c ≥ 0 and d > 0. If d = 0, then Equation (16) becomes Equation (2) after a reduction of parameters. By Equation (13) we know that x0 = 0 is always a fixed point and x+ = a(c+d)−1 is a unique positive fixed c+d point for a(c + d) > 1. Since λ(x0 ) = a(c + d), we have that   locally asymptotically stable nonhyperbolic x0 is  unstable

if a(c + d) < 1 if a(c + d) = 1 if a(c + d) > 1.

Further, notice that λ(x+ ) = 

a(c + d) 1 c and (1 + cφ + dψ)(1 + dψ + dφ) = a(d − c). Now ψ+φ=

a ((c + d)(ψ + φ) + 2(cφ + dψ)(cψ + dφ)) , a(d − c)

or equivalently, 2c(ψ + φ) + 2(cφ + dψ)(cψ + dφ) = 0.

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Since ψ + φ > 0, it must be the case that c = 0, and then 2d2 ψφ = 0 so that one of either φ or ψ equals adψ zero. Without loss of generality assume φ = 0. But then ψ = 1+dψ , and hence ψ = ad−1 = x+ . Thus the d only non-equilibrium solution of System (17) is the period-two solution {x+ , 0, x+ , 0, . . .}, which exists for ad > 1 and c = 0. Now we formulate our main result about the global dynamics of Equation (16). Theorem 4 Consider Equation (16). (a) If a(c + d) ≤ 1, then x0 = 0 is a global attractor of all solutions.  ad−1 (b) If c = 0 and ad > 1, then there exists a period-two solution ad−1 d , 0, d , 0, . . . . x+ is a global attractor of all solutions with positive initial conditions. Any solution with exactly one initial condition equal to zero will converge to the period-two solution. (c) If c > 0 and a(c + d) > 1, x+ is a global attractor of all nonzero solutions. Proof. (a) If a(c + d) ≤ 1, then x0 = 0 is the only equilibrium, and no period-two solutions exist. By Theorem 1 all solutions must converge to zero. (b) Suppose c = 0 and ad > 1, and consider I = (0, ∞). Notice that F (x, x) =

adx ≷ x ⇐⇒ x+ ≷ x, 1 + dx

and therefore by Theorem 2 we have that all solutions with initial conditions in I converge to x+ . Now suppose one initial condition is zero, so without loss of generality assume x−1 = 0 and x0 > 0. Then x1 = 0 and adx0 ad − 1 x2 = = x+ ≷ x0 . ≷ x0 ⇐⇒ 1 + dx0 d Further, one can show x2 ≶ x+ ⇐⇒ x0 ≶ x+ . By induction, lim x2k = x+ and x2k−1 = 0 for all k→∞

k = 0, 1, . . .. Thus all solutions with exactly one initial condition equal to zero will converge to the period-two solution {x+ , 0, x+ , 0, . . .}. (c) When c > 0 and a(c + d) > 1, x+ is locally asymptotically stable while x0 is unstable. As in the proof of (b) we can employ Theorem 2 to show that all solutions with positive initial conditions must converge to x+ . Since c > 0 and d > 0, if x0 + x−1 > 0, then x1 = F (x0 , x−1 ) > 0 (and also x2 > 0), so the solution eventually has consecutive positive terms and must converge to x+ . 2

3.2

Translated Linear-Linear: f (u, v) = cu + dv + k

We briefly consider the difference equation xn+1 =

a(cxn + dxn−1 + k) , 1 + cxn + dxn−1 + k

191

n = 0, 1, . . . ,

(18)

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where c ≥ 0, d ≥ 0, c + d > 0, and k > 0. We notice in this example f (0, 0) = k > 0, so the origin cannot be an equilibrium. More specifically, an equilibrium of Equation (18) must satisfy (c + d)x2 + (k + 1 − a(c + d)) x − ak = 0 Since c + d > 0 and ak > 0 by Descartes’ Rule of Signs it must be the case that there exists a unique positive equilibrium x+ . Theorem 5 Consider Equation (18) such that c + d > 0 and k > 0. The unique positive equilibrium x+ is a global attractor. 2

Proof. The result follows from a straightforward application of Theorem 1.4.8 of [14].

3.3

Quadratic-Linear: f (u, v) = cu2 + dv

We consider the difference equation xn+1 =

a(cx2n + dxn−1 ) , 1 + cx2n + dxn−1

n = 0, 1, . . . .

(19)

Remark 1 For the analysis that follows, we will consider Equation (19) with c > 0 and d > 0. Notice that when c = 0 Equation (19) is a special case of Equation (16), and the global dynamics for this case is discussed in Theorem 4. When d = 0 Equation (19) is essentially Equation (8), the dynamics of which may be seen in (9). An equilibrium solution of Equation (19) satisfies cx3 + dx2 + x = acx2 + adx so that all nonzero equilibria satisfy cx2 + (d − ac)x + (1 − ad) = 0,

(20)

whence we easily deduce the possible solutions x± =

ac − d ±

p

(d − ac)2 + 4c(ad − 1) , 2c

which are real if and only if R = (d − ac)2 + 4c(ad − 1) ≥ 0. Notice that R ≥ 0 ⇐⇒ d2 − 2acd + a2 c2 + 4acd − 4c ≥ 0 ⇐⇒ (ac + d)2 ≥ 4c. Here we have that λ(x) =

(21)

a(2cx + d) . (1 + cx2 + dx)2

Theorem 6 Equation (19) always has the zero equilibrium x0 = 0, and   locally asymptotically stable if ad < 1 nonhyperbolic if ad = 1 x0 is  a repeller if ad > 1. 2

Proof. The proof follows from the fact that λ(x0 ) = δ(x0 ) = ad. The next result gives the local stability of positive equilibrium solutions.

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Theorem 7 Assume c > 0 and d > 0. (1) Suppose either (a) d ≥ ac and 1 ≥ ad, or (b) d < ac, 1 > ad, and R < 0. Then Equation (19) has no positive equilibria. (2) Suppose either (a) 1 < ad, or (b) d < ac and 1 = ad. Then Equation (19) has the positive equilibrium solution x+ , and it is locally asymptotically stable. (3) Suppose d < ac, 1 > ad, and R = 0. Then Equation (19) has the positive equilibrium solution x± , and it is nonhyperbolic of stable type (that is one characteristic value is λ1 = ±1 and the other |λ1 | < 1). (4) Suppose d < ac, 1 > ad, and R > 0. Then Equation (19) has two positive equilibria, x+ and x− ; x+ is locally asymptotically stable, and x− is a saddle point. Proof. The existence of positive equilibria follows from Descartes’ Rule of Signs. Using Equation (14), notice that λ(x) =

a(2cx + d) a(2cx + d) 2cx + d 1 cx = = = + . (a(cx + d))2 a(cx + d)2 a(cx + d) a(cx + d)2 (1 + cx2 + dx)2

Further, for the parametric values for which x+ exists, cx+ a(cx+ + d) − 1 ≤ a(cx+ + d)2 a(cx+ + d) ⇐⇒ cx+ ≤ (cx+ + d) (a(cx+ + d) − 1) = (cx+ + d)(cx2+ + dx+ )

λ(x+ ) ≤ 1 ⇐⇒

⇐⇒ c ≤ (cx+ + d)2 ⇐⇒ 4c ≤ (2cx+ + 2d)2 = (ac + d +

√

R)2 ,

which is true by (21). Thus if R > 0, x+ is locally asymptotically stable, and if R = 0, x± is nonhyperbolic. In the latter case the characteristic equation of the linearization of Equation (19) about x± , y 2 = Fu (x± , x± )y + Fv (x± , x± ), reduces to acy 2 − (ac − d)y − d = 0, which has characteristic values y1 = 1 d and y2 = − ac , where −1 < y2 < 0 since ac > d. Thus in this case x± is nonhyperbolic of stable type. When x− exists, then √ λ(x− ) > 1 ⇐⇒ 4c > (ac + d − R)2 √ ⇐⇒ 4c + (ac + d) R > (ac + d)2 √ ⇐⇒ (ac + d) R > (ac + d)2 − 4c = R ⇐⇒ (ac + d)2 > R = (ac + d)2 − 4c, which is true since c > 0. To show more specifically that x− is a saddle point when R > 0, we must show that δ(x− ) < 1, where δ is defined by Equation (15). Notice  √  4 2d − ac + R a(d − 2cx− ) a(d − 2cx− ) 4(d − 2cx− ) √ δ(x− ) = = = = , 2 2 2 2 (a(cx− + d)) a(2cx− + 2d) (1 + cx− + dx− ) a(ac + d − R)2

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and so we have that   √  √ 2 δ(x− ) < 1 ⇐⇒ 4 2d − ac + R < a ac + d − R √ ⇐⇒ (2 + a(ac + d)) R < a(ac + d)2 − 4d. The right-hand side of the latter inequality is positive since a(ac + d)2 − 4d > 4ac − 4d = 4(ac − d) > 0 by assumption. But then

2 δ(x− ) < 1 ⇐⇒ (2 + a(ac + d))2 (ac + d)2 − 4c < a(ac + d)2 − 4d ⇐⇒ 3a3 c2 d + 6a2 cd2 + 3ad3 − 3a2 c2 − 2acd − 3d2 − 4c < 0
⇐⇒ (ad − 1) 3d2 + 3a2 c2 + 2c(3ad + 2) < 0, which is automatically true since the latter factor is strictly positive and ad < 1. Thus indeed x− is a saddle point when it exists for R > 0. 2 Theorem 8 There exist no minimal period-two solutions to Equation (19) if c, d > 0. Proof. Suppose there exist φ, ψ > 0 with φ 6= ψ such that  a(cφ2 + dψ) af (φ, ψ)   =  ψ =   1 + f (φ, ψ) 1 + cφ2 + dψ   af (ψ, φ)    φ = 1 + f (ψ, φ)

=

.

(22)

a(cψ 2

+ dφ) 1 + cψ 2 + dφ

From System (22) we notice that ψ−φ=

a(ψ − φ)(d − c(ψ + φ)) , (1 + cφ2 + dψ)(1 + cψ 2 + dφ)

whence it immediately follows that (1 + cφ2 + dψ)(1 + cψ 2 + dφ) = a(d − c(ψ + φ)). But then ψ+φ=

2(cφ2 + dψ)(cψ 2 + dφ) + c(ψ 2 + φ2 ) + d(ψ + φ) . d − c(ψ + φ)

Thus we have that necessarily   2a2 (cφ2 + dψ)(cψ 2 + dφ) c(ψ 2 + φ2 ) + d(ψ + φ) 2φψ = = a (ψ + φ) − >0 a(d − c(ψ + φ)) d − c(ψ + φ) since both ψ, φ > 0. But this implies that (ψ + φ)(d − c(ψ + φ)) > c(ψ 2 + φ2 ) + d(ψ + φ) ⇐⇒ d(ψ + φ) − c(ψ + φ)2 > c(ψ 2 + φ2 ) + d(ψ + φ) ⇐⇒ 0 > c(ψ 2 + φ2 ) + c(ψ + φ)2 , a clear contradiction since c > 0.

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Now suppose there exists a period-two solution {φ, ψ, φ, ψ, . . .} with φ 6= ψ but φψ = 0. Suppose without loss of generality that φ = 0. Now  adψ   ψ = af (0, ψ) =   1 + f (0, ψ) 1 + dψ  ,  2  af (ψ, 0) acψ   =  0 = 1 + f (ψ, 0) 1 + cψ 2 which immediately leads to the contradiction ψ = φ = 0 for c > 0. Thus Equation (19) has no minimal period-two solutions. 2 The next result describes the global dynamics of Equation (19). Theorem 9 Consider Equation (19) under the condition c > 0 and d > 0. (1) Suppose either (a) d ≥ ac and 1 ≥ ad, or (b) d < ac, 1 > ad, and R < 0. Then x0 is a global attractor of all solutions. (2) Suppose either (a) 1 < ad, or (b) d < ac and 1 = ad. Then x+ is a global attractor of all nonzero solutions. (3) Suppose d < ac, 1 > ad, and R = 0. Then Equation (19) has the equilibria x0 = 0, which is locally asymptotically stable, and x± , which is nonhyperbolic of stable type. There exists a continuous curve C passing through E = (x± , x± ) such that C is the graph of a decreasing function. The set of initial conditions Q1 = {(x−1 , x0 ) : x−1 ≥ 0, x0 ≥ 0} is the union of two disjoint basins of attraction, namely Q1 = B(E0 ) ∪ B(E), where E0 = (x0 , x0 ), B(E0 ) = {(x−1 , x0 ) : (x−1 , x0 ) ≺ne (x, y) for some (x, y) ∈ C}, and B(E) = {(x−1 , x0 ) : (x, y) ≺ne (x−1 , x0 ) for some (x, y) ∈ C} ∪ C. (4) Suppose d < ac, 1 > ad, and R > 0. Then Equation (19) has the equilibria x0 = 0, which is locally asymptotically stable, x− , which is a saddle point, and x+ , which is locally asymptotically stable. There exist two continuous curves W s (E− ) and W u (E− ), both passing through E− = (x− , x− ), such that W s (E− ) is the graph of a decreasing function and W u (E− ) is the graph of an increasing function. The set of initial conditions Q1 = {(x−1 , x0 ) : x−1 ≥ 0, x0 ≥ 0} is the union of three disjoint basins of attraction, namely Q1 = B(E0 ) ∪ B(E− ) ∪ B(E+ ), where E0 = (x0 , x0 ), E+ = (x+ , x+ ), B(E− ) = W s (E− ), B(E0 ) = {(x−1 , x0 ) : (x−1 , x0 ) ≺ne (x, y) for some (x, y) ∈ W s (E− )}, and B(E+ ) = {(x−1 , x0 ) : (x, y) ≺ne (x−1 , x0 ) for some (x, y) ∈ W s (E− )} Proof. (1) The proof in this case follows from Theorems 1, 7, and 8 along with the fact that x0 = 0 is the sole equilibrium of Equation (19).

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(2) The proof used to show that all solutions with positive initial conditions converge to x+ follows from an application of Theorem 2 (as used above in the proof of Theorem 4). Notice that x1 = F (x0 , x−1 ) > 0 if either x0 > 0 or x−1 > 0 (and similar for x2 ), so I = (0, ∞) is an attracting and invariant interval. Thus all nonzero solutions must converge to x+ . (3) The proof follows from an application of Theorems 1-4 of [17] applied to the cooperative second iterate of the map corresponding to Equation (19). The proof is completely analogous to the proof of Theorem 5 in [4], so we omit the details. 2

(4) The proof follows from an immediate application of Theorem 5 in [4].

3.4

Linear-Quadratic: f (u, v) = cu + dv 2

We consider the difference equation xn+1 =

a(cxn + dx2n−1 ) , 1 + cxn + dx2n−1

n = 0, 1, . . . .

(23)

Remark 2 For the analysis that follows, we will consider Equation (23) with c > 0 and d > 0. Notice that when d = 0 Equation (23) reduces to Equation (2), a special case of Equation (16). When c = 0 Equation (23) is essentially Equation (8) with delay. An equilibrium of (23) satisfies dx3 + cx2 + x = acx + adx2 so that all nonzero equilibria satisfy dx2 + (c − ad)x + (1 − ac) = 0,

(24)

whence we easily deduce the possible solutions x± =

ad − c ±

p

(c − ad)2 + 4d(ac − 1) , 2d

which are real if and only if R = (c − ad)2 + 4d(ac − 1) ≥ 0. Notice that R ≥ 0 ⇐⇒ c2 − 2acd + a2 d2 + 4acd − 4d ≥ 0 ⇐⇒ (ad + c)2 ≥ 4d. Here we have that λ(x) =

(25)

a(c + 2dx) . (1 + cx + dx2 )2

Theorem 10 Equation (23) always has the zero equilibrium x0 = 0, and   locally asymptotically stable if ac < 1 nonhyperbolic if ac = 1 x0 is  unstable if ac > 1. 2

Proof. The proof follows from the fact that λ(x0 ) = ac. Theorem 11 Consider Equation (23) and assume c > 0 and d > 0.

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(1) Suppose either (a) c ≥ ad and 1 ≥ ac, or (b) c < ad, 1 > ac, and R < 0. Then Equation (23) has no positive equilibria. (2) Suppose either (a) 1 < ac, or (b) c < ad and 1 = ac. Then Equation (23) has the positive equilibrium solution x+ , and it is locally asymptotically stable. (3) Suppose c < ad, 1 > ac, and R = 0. Then Equation (23) has the positive equilibrium solution x± , and it is nonhyperbolic of stable type. (4) Suppose c < ad, 1 > ac, and R > 0. Then Equation (23) has two positive equilibria, x+ and x− ; x+ is locally asymptotically stable, and x− is unstable. Let K = a2 d2 + 14acd − 3c2 − 3a3 cd2 − 6a2 c2 d − 3ac3 − 4d. (i) If K < 0, then x− is a saddle point. (ii) If K > 0, then x− is a repeller. (iii) If K = 0, then x− is nonhyperbolic of unstable type (that is one characteristic value is λ1 = ±1 and the other |λ1 | > 1). Proof. Much of the analysis is similar to the considerations in the proof of Theorem 7. Notice that λ(x) =

a(c + 2dx) a(c + 2dx) c + 2dx 1 dx = = = + . 2 2 2 2 (a(c + dx)) a(c + dx) a(c + dx) a(c + dx)2 (1 + cx + dx )

For the parametric values for which x+ exists, dx+ a(c + dx+ ) − 1 ≤ 2 a(c + dx+ ) a(c + dx+ ) ⇐⇒ dx+ ≤ (c + dx+ ) (a(c + dx+ ) − 1) = (c + dx+ )(cx+ + dx2+ )

λ(x+ ) ≤ 1 ⇐⇒

⇐⇒ d ≤ (c + dx+ )2 ⇐⇒ 4d ≤ (2c + 2dx+ )2 = (ad + c +

√

R)2 ,

which is true by (25). Thus if R > 0, x+ is locally asymptotically stable, and if R = 0, x± is nonhyperbolic. In the latter case the characteristic equation of the linearization of Equation (23) about x± , y 2 = Fu (x± , x± )y + Fv (x± , x± ), reduces to ady 2 − cy + c − ad = 0, which has characteristic values y1 = 1 and y2 = c−ad ad , where −1 < y2 < 0 since ad > c. Thus in this case x± is nonhyperbolic of stable type. When x− exists, √ λ(x− ) > 1 ⇐⇒ 4d > (ad + c − R)2 √ ⇐⇒ 4d + (ad + c) R > (ad + c)2 √ ⇐⇒ (ad + c) R > (ad + c)2 − 4d = R ⇐⇒ (ad + c)2 > R = (ad + c)2 − 4d which is true since d > 0. To more specifically classify x− , we must calculate δ(x− ). Notice  √  4 ad − 2c − R a(2dx− − c) a(2dx− − c) 4(2dx− − c) √ δ(x− ) = = , = = 2 2 2 (a(c + dx− )) a(2c + 2dx− ) (1 + cx− + dx− )2 a(ad + c − R)2

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and so we have that   √  √ 2 δ(x− ) ≷ 1 ⇐⇒ 4 ad − 2c − R ≷ a ad + c − R √ ⇐⇒ (a(ad + c) − 2) R ≷ a(ad + c)2 − 4ad + 4c = aR + 4c. Notice that R > 0 automatically implies a(ad + c) > 2, as 0 < (ad + c)2 − 4d < a2 d2 + 2acd + a2 d2 − 4d = 2d (a(ad + c) − 2) since c < ad. Therefore we may square both sides to obtain δ(x− ) ≷ 1 ⇐⇒ (a(ad + c) − 2)2 R ≷ (aR + 4c)2
⇐⇒ R a2 (ad + c)2 − 4a(ad + c) + 4 ≷ a2 R2 + 8acR + 16c2
⇐⇒ R a2 R − 4ac + 4 ≷ a2 R2 + 8acR + 16c2 ⇐⇒ R(1 − 3ac) − 4c2 ≷ 0 ⇐⇒ a2 d2 + 14acd − 3c2 − 3a3 cd2 − 6a2 c2 d − 3ac3 − 4d ≷ 0. Thus if K = a2 d2 + 14acd − 3c2 − 3a3 cd2 − 6a2 c2 d − 3ac3 − 4d,

(26)

K < 0 implies x− is a saddle point and K > 0 implies it is a repeller. If K = 0, x− is nonhyperbolic, and we expect in such case to be nonhyperbolic of unstable type. Indeed one can show that in the event K = 0, the characteristic equation of the linearization of Equation (23) about x− , y 2 = Fu (x− , x− )y + Fv (x− , x− ), has roots y1 = −1 and y2 = Fu (x− , x− ) + 1 > 1, which immediately shows the desired result. 2 The investigation of the existence of periodic solutions of Equation (23) is an interesting one that involves a thorough analysis of potential parametric cases. This analysis will reveal the potential for the existence of several nonzero periodic solutions. The juxtaposition of Equation (19) with Equation (23) illustrates an interesting phenomenon in which, loosely speaking, the dominance of the delay term xn−1 contributes to the possibility of periodic solutions arising. A minimal period-two solution {φ, ψ, φ, ψ, . . .} with φ, ψ > 0 and φ 6= ψ must satisfy  a(cφ + dψ 2 ) af (φ, ψ)   = ψ =    1 + f (φ, ψ) 1 + cφ + dψ 2 .  2)  af (ψ, φ) a(cψ + dφ    φ = = 1 + f (ψ, φ) 1 + cψ + dφ2

(27)

Eliminating either ψ or φ from System (27) we obtain

dφ2 + (c − ad)φ + (1 − ac) h(φ) = 0, or dψ 2 + (c − ad)ψ + (1 − ac) h(ψ) = 0, where h(x) = −d3 x6 + d2 (c + 2ad)x5 − d(c2 + 2d + 3acd + a2 d2 )x4 + d(c + 3ac2 + 2ad + 3a2 cd)x3 2

3

2 2

3

2

2

(28)

2 2

− (c + ac + d + 2acd + 3a c d + a cd )x + ac(1 + ac)(2c + ad)x − a c (1 + ac). Since dx2 + (c − ad)x + (1 − ac) 6= 0 for any x that is not a solution of the equilibrium equation (24), minimal period-two solutions must be the solutions of the equation h(x) = 0.

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Theorem 12 Any real solutions of Equation (29) are positive numbers for c, d > 0, and there exist up to three minimal period-two solutions of Equation (23). Furthermore, let K be as defined in Equation (26), and define the following expressions: J = 4a5 cd4 − 8a4 c2 d3 + 12a3 c3 d2 − 24a3 cd3 − 8a2 c4 d + 28a2 c2 d2 − a2 d3 + 4ac5 + 4ac3 d + 32acd2 + 4c4 + 8c2 d + 4d2 ∆1 = 6d6
∆2 = d10 8a2 d2 − 16acd − 7c2 − 24d ∆3 = −2d12 8a5 cd5 + 13a4 c2 d4 + 10a3 c3 d3 − 44a3 cd4 + 4a2 c4 d2 − 34a2 c2 d3 − 4a2 d4 − 19ac5 d
+14ac3 d2 + 44acd3 + 6c6 + 7c4 d + 5c2 d2 + 16d3

∆4 = c2 d13 −16a9 cd8 − 12a8 c2 d7 + 24a7 c3 d6 + 152a7 cd7 − 68a6 c4 d5 + 80a6 c2 d6 + 8a6 d7 + 48a5 c5 d4 −164a5 c3 d5 − 464a5 cd6 − 60a4 c6 d3 + 20a4 c4 d4 − 180a4 c2 d5 − 64a4 d6 + 56a3 c7 d2 − 332a3 c5 d3 +388a3 c3 d4 + 488a3 cd5 − 48a2 c8 d + 272a2 c6 d2 + 255a2 c4 d3 + 152a2 c2 d4 + 136a2 d5 + 24ac9
+8ac7 d + 124ac5 d2 + 180ac3 d3 − 152acd4 + 24c8 + 68c6 d + 32c4 d2 − 44c2 d3 − 32d4 ∆5 = 2c4 d13 J 3a8 c2 d6 + 2a7 cd6 − 18a6 c2 d5 − a6 d6 + 6a5 c5 d3 + 10a5 c3 d4 − 8a5 cd5 − 10a4 c4 d3 +44a4 c2 d4 + 6a4 d5 + 54a3 c5 d2 − 25a3 c3 d3 − 6a3 cd4 + 3a2 c8 − 8a2 c6 d + 35a2 c4 d2 − 39a2 c2 d3
−9a2 d4 + 6ac7 + 2ac5 d + 4ac3 d2 + 14acd3 + 3c6 + 10c4 d + 11c2 d2 + 4d3 ∆6 = a2 c6 d14 (ac + 1)KJ 2 . (1) If ∆i > 0 for all 2 ≤ i ≤ 6 then Equation (29) has six real roots. Consequently, Equation (23) has three minimal period-two solutions. (2) If ∆j ≤ 0 for some 2 ≤ j ≤ 5 and ∆i > 0 for i 6= j, then Equation (29) has two distinct real roots and two pairs of conjugate imaginary roots. Consequently, Equation (23) has one minimal period-two solution. (3) If ∆i ≤ 0, ∆i+1 ≥ 0 (such that at least one of these is strict) for some 2 ≤ i ≤ 4, and if ∆6 < 0, then Equation (29) has three pairs of conjugate imaginary roots. Consequently, Equation (23) has no minimal period-two solutions.

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Proof. The proof of the first statement follows from Descartes’ Rule of Signs. Let disc(h) denote the 12 × 12 discrimination matrix as defined in [12]:  a6 a5 a4 a3 a2 a1 a0 0 0 0 0 0  0 6a 5a 4a 3a 2a a1 0 0 0 0 0 6 5 4 3 2   0 a a a a a a a 0 0 0 0 6 5 4 3 2 1 0   0 0 6a 5a 4a 3a 2a a 0 0 0 0 6 5 4 3 2 1   0 0 a6 a5 a4 a3 a2 a1 a0 0 0 0   0 0 6a6 5a5 4a4 3a3 2a2 a1 0 0 0  0 disc(h) =  0 0 a6 a5 a4 a3 a2 a1 a0 0 0  0   0 0 0 0 6a6 5a5 4a4 3a3 2a2 a1 0 0   0 0 0 0 a6 a5 a4 a3 a2 a1 a0 0   0 0 0 0 0 6a6 5a5 4a4 3a3 2a2 a1 0   0 0 0 0 0 a6 a5 a4 a3 a2 a1 a0 0 0 0 0 0 0 6a6 5a5 4a4 3a3 2a2 a1

           .         

Here ak equals the coefficient of the degree-k term of h as defined in Equation (28); that is, a6 = −d3 , a5 = d2 (c + 2ad), a4 = −d(c2 + 2d + 3acd + a2 d2 ), a3 = d(c + 3ac2 + 2ad + 3a2 cd), a2 = −(c2 + ac3 + d + 2acd + 3a2 c2 d + a3 cd2 ), a1 = ac(1 + ac)(2c + ad), and a0 = −a2 c2 (1 + ac). Let ∆k denote the determinant of the submatrix of disc(h) formed by its first 2k rows and 2k columns for k = 1, 2, . . . , 6. Then the values of ∆k are listed above, and the veracity of the statements above may now be verified by employing Theorem 1 of [12]. Notice that ∆1 > 0 for all d > 0. 2 Remark 3 The parametric conditions discussed above do not exhaust all of the parametric space but cover a substantial region of parameters for which Equation (23) possesses hyperbolic dynamics. We will use the sufficient conditions provided in Theorems 10, 11, and 12 to obtain some global dynamic scenarios discussed in [4]. We will not investigate the dynamics of Equation (23) when it has one or no positive fixed point since in such cases the dynamics should be similar to the dynamics of Equation (19) discussed in Theorem 9. The following theorem relies on results from [4] and summarizes potential hyperbolic dynamic scenarios for Equation (23) in the event it possesses three fixed points and zero, one, or three pairs of hyperbolic period-two points. In particular, Theorem 3 is applicable to case (ii) of the following result. See also the statement and proof of Theorem 11 in [4]. Theorem 13 Consider Equation (23) and assume 0 < c < ad, ac < 1 such that R > 0. (i) If ∆i > 0 for all 2 ≤ i ≤ 6 then Equation (23) has three equilibria x0 < x− < x+ , where x0 and x+ are locally asymptotically stable and x− is a repeller, and three minimal period-two solutions {φ1 , ψ1 }, {φ2 , ψ2 }, and {φ3 , ψ3 }. Here (φ1 , ψ1 ) ≺ne (φ2 , ψ2 ) ≺ne (φ3 , ψ3 ), {φ1 , ψ1 } and {φ3 , ψ3 } are saddle points, and {φ2 , ψ2 } is locally asymptotically stable. The global behavior of Equation (23) is described by Theorem 8 of [4]. In this case there exist four continuous curves W s (φ1 , ψ1 ), W s (ψ1 , φ1 ), W s (φ3 , ψ3 ), W s (ψ3 , φ3 ) that have endpoints at E− = (x− , x− ) and are graphs of decreasing functions. Every solution which starts below W s (φ1 , ψ1 ) ∪ W s (ψ1 , φ1 ) in the northeast ordering converges to E0 = (x0 , x0 ) and every solution which starts above W s (φ3 , ψ3 )∪W s (ψ3 , φ3 ) in the northeast ordering converges to E+ = (x+ , x+ ). Every solution which starts above W s (φ1 , ψ1 ) ∪ W s (ψ1 , φ1 ) and below W s (φ3 , ψ3 ) ∪ W s (ψ3 , φ3 ) in the northeast ordering converges to {φ2 , ψ2 }. For example, this happens 389 for a = 1, c = 2176 , and d = 249 64 .

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(ii) If ∆j ≤ 0 for some 2 ≤ j ≤ 5 and ∆i > 0 for i 6= j, then Equation (23) has three equilibria x0 < x− < x+ , where x0 and x+ are locally asymptotically stable and x− is a repeller, and one period-two solution {φ1 , ψ1 }, which is a saddle point. The global behavior of Eq. (23) is described by Theorem 7 of [4]. In this case there exist four continuous curves W s (φ1 , ψ1 ), W s (ψ1 , φ1 ), W u (φ1 , ψ1 ), W u (ψ1 , φ1 ), where W s (φ1 , ψ1 ), W s (ψ1 , φ1 ) have endpoints at E− = (x− , x− ) and are graphs of decreasing functions. The curves W u (φ1 , ψ1 ), W u (ψ1 , φ1 ) are graphs of increasing functions and start at E0 = (x0 , x0 ). Every solution which starts below W s (φ1 , ψ1 ) ∪ W s (ψ1 , φ1 ) in the northeast ordering converges to E0 and every solution which starts above W s (φ1 , ψ1 ) ∪ W s (ψ1 , φ1 ) in the northeast ordering converges to E+ = (x+ , x+ ) . For example, this happens for a = 1, c = 51 , and d = 237 64 . (iii) If ∆i ≤ 0 and ∆i+1 ≥ 0 (such that at least one of these is strict) for some 2 ≤ i ≤ 4, and if ∆6 < 0, then Eq. (23) has three equilibria x0 < x− < x+ , where x0 and x+ are locally asymptotically stable and x− is a saddle point, and no period-two solution. The global behavior of Equation (23) is 493 described by Theorem 5 of [4] or Theorem 9 case (4). For example, this happens for a = 1, c = 1024 , 157 and d = 48 . Equation (23) exhibits global dynamics similar to that of Equation (10), which was investigated in [4]. Therefore, we pose the following conjecture. Conjecture 1 There exists a topological conjugation between the maps in Equations (10) and (23).

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TABLE OF CONTENTS, JOURNAL OF COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 29, NO. 1, 2021

A New Techniques Applied to Volterra-Fredholm Integral Equations with Discontinuous Kernel, M. E. Nasr and M. A. Abdel-Aty,……………………………………………………………11 A Comparative Study of Three Forms Of Entropy On Trade Values Between Korea and Four Countries, Jacob Wood and Lee-Chae Jang,…………………………………………………25 Quadratic Functional Inequality in Modular Spaces and Its Stability, Chang Il Kim and Giljun Han,……………………………………………………………………………………….….34 Complex Multivariate Taylor’s Formula, George A. Anastassiou,…………………………..42 On the Barnes-Type Multiple Twisted q-Euler Zeta Function of the Second Kind, C. S. Ryoo,47 Some Approximation Results of Kantorovich Type Operators, Prashantkumar Patel,………52 Mittag-Leffler-Hyers-Ulam Stability of Linear Differential Equations using Fourier Transforms, J.M. Rassias, R. Murali, and A. Ponmana Selvan, ……………………………………………68 On Some Systems of Three Nonlinear Difference Equations, E. M. Elsayed and Hanan S. Gafel,…………………………………………………………………………………………86 Approximation of Solutions of the Inhomogeneous Gauss Differential Equations by Hypergeometric Function, S. Ostadbashi, M. Soleimaninia, R. Jahanara, and Choonkil Park,109 On Topological Rough Groups, Nof Alharbi, Hassen Aydi, Cenap Ozel, and Choonkil Park,117 On the Farthest Point Problem In Banach Spaces, A. Yousef, R. Khalil, and B. Mutabagani,123 On the Stability of 3-Lie Homomorphisms and 3-Lie Derivations, Vahid Keshavarz, Sedigheh Jahedi, Shaghayegh Aslani, Jung Rye Lee, and Choonkil Park,…………………………….129 Neutrosophic Extended Triplet Groups and Homomorphisms in C*-Algebras, Jung Rye Lee, Choonkil Park, and Xiaohong Zhang,……………………………………………………….136 Orthogonal Stability of a Quadratic Functional Inequality: a Fixed Point Approach, Shahrokh Farhadabadi and Choonkil Park,…………………………………………………………… 140 Integral Inequalities for Asymmetrized Synchronous Functions, S. S. Dragomir,………….151 Further Inequalities for Heinz Operator Mean, S. S. Dragomir,……………………………162

TABLE OF CONTENTS, JOURNAL OF COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 29, NO. 1, 2021 (continues) Global Dynamics of Monotone Second Order Difference Equation, S. Kalabušić, M. R. S Kulenović, and M. Mehuljić,…………………………………………………………………172 Global Dynamics of Generalized Second-Order Beverton-Holt Equations of Linear and Quadratic Type, E. Bertrand and M. R. S Kulenović,…………………………………………185

Volume 29, Number 2 ISSN:1521-1398 PRINT,1572-9206 ONLINE

March 2021

Journal of Computational Analysis and Applications EUDOXUS PRESS, LLC

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J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 29, NO.2, 2021, COPYRIGHT 2021 EUDOXUS PRESS, LLC

A study of a coupled system of nonlinear second-order ordinary differential equations with nonlocal integral multi-strip boundary conditions on an arbitrary domain Bashir Ahmada,1 , Ahmed Alsaedia , Mona Alsulamia,b and Sotiris K. Ntouyasa,c

a

Nonlinear Analysis and Applied Mathematics (NAAM)-Research Group, Department of Mathematics, Faculty of Science, King Abdulaziz University, P.O. Box 80203, Jeddah 21589, Saudi Arabia. E-mail addresses: bashirahmad− [email protected], [email protected]. b

Department of Mathematics, Faculty of Science, University of Jeddah, P.O. Box 80327, Jeddah 21589, Saudi Arabia E-mail address: [email protected]

c

Department of Mathematics, University of Ioannina, 451 10 Ioannina, Greece E-mail address: [email protected]

Abstract

In this paper, we study a nonlinear system of second order ordinary differential equations with nonlocal integral multi-strip coupled boundary conditions. Leray-Schauder alternative criterion, Schauder fixed point theorem and Banach contraction mapping principle are employed to obtain the desired results. Examples are constructed for the illustration of the obtained results. We emphasize that our results are new and enhance the literature on boundary value problems of coupled systems of ordinary differential equations. Several new results appear as special cases of our work.

Keywords: System of ordinary differential equations; integral boundary condition; multi-strip; existence; fixed point. MSC 2000: 34A34, 34B10, 34B15.

1

Corresponding author

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AHMAD 215-235

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2

B. Ahmad, A. Alsaedi, M. Alsulami and S.K. Ntouyas

1

Introduction

This paper is concerned with the following coupled system of nonlinear second-order ordinary differential equations:  00 u (t) = f (t, u(t), v(t)), t ∈ [a, b], (1.1) v 00 (t) = g(t, u(t), v(t)), t ∈ [a, b], supplemented with the nonlocal integral multi-strip coupled boundary conditions of the form:  Z Z ηj Z b Z ηj m m b X X   0  u(s)ds = u (s)ds = γj v(s)ds + λ1 , ρj v 0 (s)ds + λ2 ,   a

    

Z

b

v(s)ds = a

j=1 m X j=1

a

ξj

Z

ηj

σj

Z u(s)ds + λ3 ,

ξj

b 0

v (s)ds = a

j=1 m X j=1

ξj

Z

ηj

δj

(1.2)

0

u (s)ds + λ4 , ξj

where f, g : [a, b] × R × R → R are given continuous functions, a < ξ1 < η1 < ξ2 < η2 < · · · < ξm < ηm < b, and γj , ρj , σj and δj ∈ R+ (j = 1, 2, . . . , m), λi ∈ R (i = 1, 2, 3, 4). Mathematical modeling of several real world phenomena lead to the occurrence of nonlinear boundary value problems of differential equations. During the past few decades, the topic of boundary value problems has evolved as an important and interesting area of investigation in view of its extensive applications in diverse disciplines such as fluid mechanics, mathematical physics, etc. For application details, we refer the reader to the text [1], while some recent works on boundary value problems of ordinary differential equations can be found in the papers ([2]-[5]). Much of the literature on boundary value problems involve classical boundary conditions. However, these conditions cannot cater the complexities of the physical and chemical processes occurring within the domain. In order to cope with this situation, the concept of nonlocal boundary conditions was introduced. Such conditions relate the boundary values of the unknown function to its values at some interior positions of the domain. For a detailed account of nonlocal nonlinear boundary value problems, for instance, see ([6]-[16]) and the references cited therein. Computational fluid dynamics (CFD) technique are directly concerned with the boundary data [1]. However, the assumption of circular cross-section in the fluid flow problems is not justifiable in many situations. The concept of integral boundary conditions played a key role in resolving this issue as such conditions can be applied to arbitrary shaped structures. Integral boundary conditions are also found to be quite useful in the study of thermal and hydrodynamic problems. In fact, one can find numerous applications of integral boundary conditions in the fields like chemical engineering, thermoelasticity, underground water flow, population dynamics, etc. ([17]-[20]). For some recent results on boundary value problems integral boundary conditions, we refer the reader to a series of articles ([21]-[32]) and the references cited therein.

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A study of a coupled system of nonlinear ordinary differential equations

3

Motivated by the importance of nonlocal and integral boundary conditions, we introduce a new kind of coupled integral boundary conditions (1.2) and solve a nonlinear coupled system of second-order ordinary differential equations (1.1) equipped with these conditions. Our main results rely on Leray-Schauder alternative and Banach contraction mapping principle. The rest of the paper is organized as follows. In Section 2, we present an auxiliary lemma. The main results for the problem (1.1) and (1.2) are discussed in Section 3. We also construct examples illustrating the obtained results. The paper concludes with some interesting observations.

2

An auxiliary lemma

The following lemma plays a key role in defining the solution for the problem (1.1) − (1.2). Lemma 2.1 For f1 , g1 ∈ C([a, b], R), the solution of the linear system of differential equations u00 (t) = f1 (t), t ∈ [a, b], v 00 (t) = g1 (t), t ∈ [a, b],

(2.1)

subject to the boundary conditions (1.2) is equivalent to the system of integral equations Z t u(t) = (t − s)f1 (s)ds a Z i 1 n b h1 A1 (b − a)(b − s) + L1 + (b − a)A2 (t − a) (b − s)f1 (s)ds − A3 2 a Z bh m m i X X 1 + A1 (b − s) γj (ηj − ξj ) + L2 + A2 (t − a) ρj (ηj − ξj ) 2 a j=1 j=1 Z Z m o n h η s j 1 X ×(b − s)g1 (s)ds + γj A1 (b − a)(s − p) + ρj L1 (2.2) A3 j=1 ξj a m Z ηj Z s h m i X X +ρj (b − a)A2 (t − a) g1 (p)dpds + σj A 1 γj (ηj − ξj )(s − p) j=1

+δj L2 + δj A2 (t − a)

m X

ξj

a

j=1

i o ρj (ηj − ξj ) f1 (p)dpds + Ω1 (t),

j=1

Z v(t) = a

t

  Z b h A1 (b − a)2 − A2 n 1 P (b − s) (t − s)g1 (s)ds − A3 2 m a j=1 γj (ηj − ξj )

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B. Ahmad, A. Alsaedi, M. Alsulami and S.K. Ntouyas

+L3 + A2 (t − a)

m X

i δj (ηj − ξj ) (b − s)f1 (s)ds

j=1

Z bh i o 1 A1 (b − a)(b − s) + L4 + A2 (b − a)(t − a) (b − s)g1 (s)ds + 2 a   2 Z Z m A1 (b − a) − A2 1 n X ηj s h + (s − p) Pm + ρj L3 A3 j=1 ξj a j=1 (ηj − ξj ) +δj A2 (t − a)

m X

(2.3)

i ρj (ηj − ξj ) g1 (p)dpds

j=1

+

m Z ηj X j=1

ξj

Z sh i o σj A1 (b − a)(s − p) + δj L4 + δj A2 (b − a)(t − a) f1 (p)dpds a

+Ω2 (t), where 2

A1 = (b − a) − A2 = (b − a)2 −

m X j=1 m X

ρj γj

j=1

L1

m  X j=1 m  X

 δj (ηj − ξj )2 ,

(2.4)

 σj (ηj − ξj )2 ,

A3 = A1 A2 6= 0,

j=1

m m   (η − a)2 (ξ − a)2  X X j j − σj + δj = (b − a) γj (ηj − ξj ) 2 2 j=1 j=1 j=1 m X

m

m

 (b − a)4 (b − a)2  X  X − γj δj (ηj − ξj )2 , − 2 2 j=1 j=1 L2

(2.6)

m m  X  (η − a)2 (ξ − a)2  X   j j − ρj σj (ηj − ξj )2 + (b − a)2 = γj 2 2 j=1 j=1 j=1 m X

m (b − a)3 X − (ηj − ξj )(ρj + γj ), 2 j=1

L3

(2.5)

(2.7)

m   X (ηj − a)2 (ξj − a)2  = − (b − a)2 (σj + δj ) − A2 δj 2 2 j=1

+

m i X (b − a)3 h A2 − (b − a)2 Pm − δj (ηj − ξj ) , 2 j=1 γj (ηj − ξj ) j=1 m

(2.8)

m

X  (ηj − a)2 (ξj − a)2 h X (b − a) − σj ρj (ηj − ξj )2 L4 = Pm (η − ξ ) 2 2 j j=1 j=1 j j=1

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2

+(b − a) − A2

i

5

m  (b − a)4 (b − a)2 X  − + Pm ρj A2 − (b − a)2 , 2 2 j=1 γj j=1

(2.9)

m h i X 1 n A1 (b − a)λ1 + L1 + A2 (b − a)(t − a) λ2 + A1 γj (ηj − ξj )λ3 Ω1 (t) = A3 j=1

h

+ L2 + A2 (t − a)

m X

i o ρj (ηj − ξj ) λ4 ,

(2.10)

j=1

  2 m n h i A (b − a) − A X 1 2 1 P Ω2 (t) = λ1 + L3 + A2 (t − a) δj (ηj − ξj ) λ2 m A3 j=1 γj (ηj − ξj ) j=1 h i o + A1 (b − a)λ3 + L4 + A2 (b − a)(t − a) λ4 . Proof. Integrating the linear system (2.1) twice from a to t, we get Z t u(t) = c1 + c2 (t − a) + (t − s)f1 (s)ds,

(2.11)

(2.12)

a

Z

t

(t − s)g1 (s)ds,

v(t) = c3 + c4 (t − a) +

(2.13)

a

where c1 , c2 , c3 and c4 are arbitrary real constants. Using the boundary conditions (1.2) in (2.12) and (2.13), together with notations (2.4), we obtain m m  (η − a)2 (ξ − a)2  X X (b − a)2 j j c2 − γj (ηj − ξj )c3 − γj − c4 2 2 2 j=1 j=1 Z ηj Z s m X (b − s)2 f1 (s)ds + γj (s − p)g1 (p)dpds + λ1 , 2 ξ a j j=1

(b − a)c1 + Z

b

=− a

(2.14) (b−a)c2 −

m X

Z ρj (ηj −ξj )c4 = −

(b−s)f1 (s)ds+ a

j=1 m X

b

m X

Z

ηj

s

Z

ρj

j=1

g1 (p)dpds+λ2 , (2.15) ξj

a

m X

 (η − a)2 (ξ − a)2  (b − a)2 j j − σj (ηj − ξj )c1 − σj − c2 + (b − a)c3 + c4 2 2 2 j=1 j=1 Z b Z ηj Z s m X (b − s)2 =− g1 (s)ds + σj (s − p)f1 (p)dpds + λ3 , 2 a ξj a j=1 (2.16) −

m X j=1

Z δj (ηj − ξj )c2 + (b − a)c4 = −

b

(b − s)g1 (s)ds + a

m X j=1

Z

ηj

Z

δj

s

f1 (p)dpds + λ4 . ξj

a

(2.17)

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Solving the equations (2.15) and (2.17) for c2 and c4 , we find that c2

Z b m   X 1 h = (b − s) (b − a)f1 (s) + − ρj (ηj − ξj )g1 (s) ds A1 a j=1 Z ηj Z s  m m  X X + ρj (b − a)g1 (p) + δj (ηj − ξj )f1 (p) dpds j=1

ξj

a

j=1

+(b − a)λ2 +

m X

i ρj (ηj − ξj )λ4 ,

(2.18)

j=1

c4

Z b m X  1 h − (b − s) = δj (ηj − ξj )f1 (s) + (b − a)g1 (s) ds A1 a j=1 Z Z m m   η s j X X + δj ρj (ηj − ξj )g1 (p) + (b − a)f1 (p) dpds j=1

ξj

a

+

j=1 m X

i δj (ηj − ξj )λ2 + (b − a)λ4 .

(2.19)

j=1

Using (2.18) and (2.19) in (2.14) and (2.16) and then solving the resulting equations for c1 and c3 , we obtain Z bh i 1 n 1 − (b − a)A1 (b − s) + L1 (b − s)f1 (s)ds c1 = A3 2 a Z bh m i X 1 A1 (b − s) γj (ηj − ξj ) + L2 (b − s)g1 (s)ds − 2 a j=1 m Z ηj Z s h i X A1 γj (b − a)(s − p) + ρj L1 g1 (p)dpds + +

j=1 ξj m Z ηj X j=1

ξj

a

Z sh X m i A1 γj σj (ηj − ξj )(s − p) + δj L2 f1 (p)dpds a

j=1

+A1 (b − a)λ1 + L1 λ2 + A1

m X

o γj (ηj − ξj )λ3 + L2 λ4 ,

j=1

c3

  Z b h A1 (b − a)2 − A2 n i 1 P = − (b − s) + L3 (−s)f1 (s)ds A3 2 m a j=1 γj (ηj − ξj ) Z bh i 1 − A1 (b − a)(b − s) + L4 (b − s)g1 (s)ds 2 a

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7

  Z s h A1 (b − a)2 − A2 X m i Pm + γj (s − p) + ρj L3 g1 (p)dpds a j=1 γj (ηj − ξj ) j=1 j=1 ξj Z Z m i ηj sh X + σj A1 (b − a)(s − p) + δj L4 f1 (p)dpds m Z ηj X

ξj

j=1

a



 o A1 (b − a)2 − A2 λ1 + L3 λ2 + A1 (b − a)λ3 + L4 λ4 . + Pm j=1 γj (ηj − ξj ) Inserting the values of c1 , c2 , c3 and c4 in (2.12) and (2.13), we get the solutions (2.2) and (2.3). The converse follows by direct computation. This completes the proof. 2

3

Main results

Let us introduce the space X = {u(t)|u(t) ∈ C([a, b])} equipped with norm kuk = sup{|u(t)|, t ∈ [a, b]}. Obviously (X , k · k) is a Banach space and consequently, the product space (X × X , ku, vk) is a Banach space with norm k(u, v)k = kuk + kvk for (u, v) ∈ X × X . By Lemma 2.1, we define an operator T : X × X → X × X as T (u, v)(t) := (T1 (u, v)(t), T2 (u, v)(t)), where t

Z bh 1 n 1 (t − s)f (s, u(s), v(s))ds + T1 (u, v)(t) = − A1 (b − a)(b − s) A3 2 a a i +L1 + (b − a)A2 (t − a) (b − s)f (s, u(s), v(s))ds Z bh m m i X X 1 − A1 (b − s) γj (ηj − ξj ) + L2 + A2 (t − a) ρj (ηj − ξj ) 2 a j=1 j=1 Z Z m X ηj s h ×(b − s)g(s, u(s), v(s))ds + γj A1 (b − a)(s − p) (3.1) Z

j=1

ξj

a

i +ρj L1 + ρj (b − a)A2 (t − a) g(p, u(p), v(p))dpds m Z ηj Z s h m X X + σ j A1 γj (ηj − ξj )(s − p) + δj L2 j=1

ξj

a

+δj A2 (t − a)

j=1 m X

i o ρj (ηj − ξj ) f (p, u(p), v(p))dpds + Ω1 (t),

j=1

Z T2 (u, v)(t) = a

t

  Z b h A1 (b − a)2 − A2 n 1 P − (b − s) (t − s)g(s, u(s), v(s))ds + A3 2 m a j=1 γj (ηj − ξj )

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+L3 + A2 (t − a)

m X

i δj (ηj − ξj ) (b − s)f (s, u(s), v(s))ds

j=1

Z bh i 1 A1 (b − a)(b − s) + L4 + A2 (b − a)(t − a) (b − s) − 2 a   2 Z Z m A1 (b − a) − A2 X ηj s h ×g(s, u(s), v(s))ds + (s − p) Pm a j=1 (ηj − ξj ) j=1 ξj +ρj L3 + δj A2 (t − a)

m X

(3.2)

i ρj (ηj − ξj ) g(p, u(p), v(p))dpds

j=1

+

m Z ηj X j=1

ξj

Z sh

i σj A1 (b − a)(s − p) + δj L4 + δj A2 (b − a)(t − a)

a

o

×f (p, u(p), v(p))dpds + Ω2 (t). In order to prove our main results, we need the following assumptions. (H1 ) There exist real constants mi , ni ≥ 0, (i = 1, 2) and m0 > 0, n0 > 0 such that ∀u, v ∈ R, we have |f (t, u, v)| ≤ m0 + m1 |u| + m2 |v|, |g(t, u, v)| ≤ n0 + n1 |u| + n2 |v|. (H2 ) There exist nonnegative functions α(t), β(t) ∈ L(0, 1) and u, v ∈ R, such that |f (t, u, v)| ≤ α(t) + 1 |u|p1 + 2 |v|p2 , 1 , 2 > 0, 0 < p1 , p2 < 1, |g(t, u, v)| ≤ β(t) + d1 |u|l1 + d2 |v|l2 , d1 , d2 > 0, 0 < l1 , l2 < 1. (H3 ) There exist `1 and `2 such that for all t ∈ [a, b] and ui , vi ∈ R, i = 1, 2, we have |f (t, u1 , v1 ) − f (t, u2 , v2 )| ≤ `1 (|u1 − u2 | + |v1 − v2 |), |g(t, u1 , v1 ) − g(t, u2 , v2 )| ≤ `2 (|u1 − u2 | + |v1 − v2 |). For the sake of convenience in the forthcoming analysis, we set q1

1 n (b − a)4 (b − a)2 (b − a)4 (b − a)2 + |A1 | + |L1 | + |A2 | = 2 |A3 | 6 2 2 m m X  X  (η − a)3 (ξ − a)3  j j +|A1 | γj σj (ηj − ξj ) − 3! 3! j=1 j=1 +

m X

 (η − a)2 (ξ − a)2  j j δj |L2 | − 2 2 j=1

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+|A2 |(b − a)

m X

ρj

j=1

q¯1

9

m  X

 (η − a)2 (ξ − a)2 o j j δj (ηj − ξj ) − , 2 2 j=1

(3.3)

m m (b − a)3 X (b − a)3 X 1 n (b − a)2 |A1 | + |A2 | = γj (ηj − ξj ) + |L2 | ρj (ηj − ξj ) |A3 | 6 2 2 j=1 j=1 m  (η − a)2 (ξ − a)2   (η − a)3 (ξ − a)3  X j j j j − + ρj |L1 | − +|A1 |(b − a) γj 3! 3! 2 2 j=1 j=1 m X

q2

m X

 (η − a)2 (ξ − a)2 o j j − , 2 2 j=1   2 n 1 A1 (b − a) − A2 (b − a)3 (b − a)2 (b − a)3 = + |L3 | + |A2 | Pm |A3 | 6 2 2 j=1 γj (ηj − ξj ) m m X X  (ηj − a)3 (ξj − a)3  − × δj (ηj − ξj ) + |A1 |(b − a) σj 3! 3! j=1 j=1 +|A2 |(b − a)

+

2

ρj

(3.4)

m X

 (η − a)2 (ξ − a)2  j j δj |L4 | − 2 2 j=1

m  (η − a)2 (ξ − a)2 o X j j +|A2 |(b − a) − , δj 2 2 j=1 (b − a)2 1 n (b − a)4 (b − a)2 (b − a)4 = + |A1 | + |L4 | + |A2 | 2 |A3 | 6 2 2   m  A1 (b − a)2 − A2 X (ηj − a)3 (ξj − a)3  P − + m 3! 3! j=1 (ηj − ξj ) j=1 2

q¯2

(3.5)

m X

 (η − a)2 (ξ − a)2  j j + ρj |L3 | − 2 2 j=1 m m X  X  (η − a)2 (ξ − a)2 o j j +|A2 |(b − a) δj ρj (ηj − ξj )) − , 2 2 j=1 j=1

(3.6)

¯ 1 = sup |Ω1 (t)|, λ ¯ 2 = sup |Ω2 (t)|. λ t∈[a,b]

(3.7)

t∈[a,b]

Moreover, we set Q1 = q1 + q2 ,

Q2 = q¯1 + q¯2 ,

¯=λ ¯1 + λ ¯2, λ

(3.8)

¯ i (i=1,2) are given in the equations (3.3) − (3.7) and where qi , q¯i and λ Q0 = min{1 − (Q1 m1 + Q2 n1 ), 1 − (Q1 m2 + Q2 n2 )}, mi , ni ≥ 0 (i = 1, 2).

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(3.9)

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3.1

B. Ahmad, A. Alsaedi, M. Alsulami and S.K. Ntouyas

Existence of solutions

In this subsection, we discuss the existence of solutions for the problem (1.1)-(1.2) by using standard fixed poit theorems. Lemma 3.1 (Leray-Schauder alternative [33]). Let T : K → K be a completely continuous operator (i.e., a map that restricted to any bounded set in K is compact). Let ω(T ) = {x ∈ K : x = ϕT (x) for some 0 < ϕ < 1}. Then either the set ω(T ) is unbounded, or T has at least one fixed point. Theorem 3.2 Assume that condition (H1 ) holds. In addition it is assumed that Q1 m1 + Q2 n1 < 1 and Q1 m2 + Q2 n2 < 1,

(3.10)

where Q1 and Q2 are given by (3.8). Then there exist at least one solution for problem (1.1) − (1.2) on [a, b] Proof. First of all, we show that the operator T : X × X → X × X is completely continuous. Notice that the operator T is continuous as the functions f and g are continuous. Let Υ ⊂ X × X be bounded. Then there exist positive constants κf and κg such that |f (t, u(t), v(t))| ≤ κf , |g(t, u(t), v(t))| ≤ κg , ∀(u, v) ∈ Υ. Then, for any (u, v) ∈ Υ, we can obtain Z Z t 1 n b h1 A1 (b − a)(b − s) |T1 (u, v)(t)| = sup (t − s)f (s, u(s), v(s))ds − A3 2 t∈[a,b] a a i +L1 + (b − a)A2 (t − a) (b − s)f (s, u(s), v(s))ds Z bh m m i X X 1 + A1 (b − s) γj (ηj − ξj ) + L2 + A2 (t − a) ρj (ηj − ξj ) 2 a j=1 j=1 Z m Z o 1 n X ηj s h γj A1 (b − a)(s − p) ×(b − s)g(s, u(s), v(s))ds + A3 j=1 ξj a i +ρj L1 + ρj (b − a)A2 (t − a) g(p, u(p), v(p))dpds m Z ηj Z s h m X X + σ j A1 γj (ηj − ξj )(s − p) + δj L2 j=1

ξj

a

+δj A2 (t − a)

j=1 m X

i o ρj (ηj − ξj ) f (p, u(p), v(p))dpds + Ω1 (t)

j=1

n (b − a)2

1 h (b − a)4 (b − a)2 (b − a)4 ≤ κf + |A1 | + |L1 | + |A2 | 2 |A3 | 6 2 2 m m X  X  (η − a)3 (ξ − a)3  j j +|A1 | γj σj (ηj − ξj ) − 3! 3! j=1 j=1

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+

11

m X

 (η − a)2 (ξ − a)2  j j δj |L2 | − 2 2 j=1

m m X  X  (η − a)2 (ξ − a)2 io j j − +|A2 |(b − a) ρj δj (ηj − ξj ) 2 2 j=1 j=1 m n 1 h (b − a)3 X (b − a)2 +κg |A1 | γj (ηj − ξj ) + |L2 | |A3 | 6 2 j=1 m (b − a)3 X +|A2 | ρj (ηj − ξj ) 2 j=1

+|A1 |(b − a)

m X

γj

j=1

+

 (η − a)3 (ξ − a)3  j j − 3! 3!

m X

 (η − a)2 (ξ − a)2  j j − ρj |L1 | 2 2 j=1 2

+|A2 |(b − a)

m X

ρj

j=1

 (η − a)2 (ξ − a)2 io j j ¯1 − +λ 2 2

¯1, ≤ κf q1 + κg q¯1 + λ which implies that ¯1. kT1 (u, v)k ≤ κf q1 + κg q¯1 + λ Similarly, it can be found that ¯2. kT2 (u, v)k ≤ κf q2 + κg q¯2 + λ ¯ (Q1 , Q2 and λ ¯ are given by Consequently, we get kT (u, v)(t)k ≤ κf Q1 + κg Q2 + λ (3.8)), which implies that the operator T is uniformly bounded. Next, we show that T is equicontinuous. For t1 , t2 ∈ [a, b] with t1 < t2 , we have |T1 (u, v)(t2 ) − T1 (u, v)(t1 )| Z Z t1 h i ≤ κf (t2 − s) − (t1 − s) ds + a

t2

t1

(t2 − s)ds

Z m m X  Z ηj Z s X i (t2 − t1 ) n h b δj ρj (ηj − ξj )dpds κf (b − a)(b − s)ds + + |A1 | a ξj a j=1 j=1 Z Z Z m m h bX i ηj s X +κg ρj (ηj − ξj )(b − s)ds + ρj (b − a)dpds a

j=1

+(b − a)λ2 +

j=1 m X

ρj (ηj − ξj )λ4

ξj

a

o

j=1

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B. Ahmad, A. Alsaedi, M. Alsulami and S.K. Ntouyas h (t2 − t1 )2 i (t2 − t1 ) n h (b − a)3 ≤ κf (t2 − t1 )(t1 − a) + + κf 2 |A1 | 2 m m  X  X  (η − a)2 (ξ − a)2 i j j − + ρj δj (ηj − ξj ) 2 2 j=1 j=1 +κg

m hX

m  (η − a)2 (ξ − a)2 i X (b − a)2 j j ρj (ηj − ξj ) + (b − a) − ρj 2 2 2 j=1 j=1

+(b − a)λ2 +

m X

o ρj (ηj − ξj )λ4 → 0 independent of u and v as (t2 − t1 ) → 0.

j=1

Similarly, one can obtain |T2 (u, v)(t2 ) − T2 (u, v)(t1 )| m h (t2 − t1 )2 i (t2 − t1 ) n h X (b − a)3 + κf δj (ηj − ξj ) ≤ κg (t2 − t1 )(t1 − a) + 2 |A1 | 6 j=1 m  (η − a)2 (ξ − a)2 i X j j +(b − a) δj − 2 2 j=1

+κg +

h (b − a)3

m X

2

m m X  X  (η − a)2 (ξ − a)2 i j j + − δj ρj (ηj − ξj ) 2 2 j=1 j=1

o

δj (ηj − ξj )λ2 + (b − a)λ4 → 0 independent of u and v as (t2 − t1 ) → 0.

j=1

Finally, we will verify that the set ω = {(u, v) ∈ X × X |(u, v) = ϕT (u, v), 0 < ϕ < 1} is bounded. Let (u, v) ∈ ω. Then (u, v) = ϕT (u, v) and for any t ∈ [a, b], we have u(t) = ϕT1 (u, v)(t), v(t) = ϕT2 (u, v)(t). Then ¯1 |u(t)| ≤ q1 (m0 + m1 kuk + m2 kvk) + q¯1 (n0 + n1 kuk + n2 kvk) + λ ¯1, = q1 m0 + q¯1 n0 + (q1 m1 + q¯1 n1 )kuk + (q1 m2 + q¯1 n2 )kvk + λ and ¯2 |v(t)| ≤ q2 (m0 + m1 kuk + m2 kvk) + q¯2 (n0 + n1 kuk + n2 kvk) + λ ¯2. = q2 m0 + q¯2 n0 + (q2 m1 + q¯2 n1 )kuk + (q2 m2 + q¯2 n2 )kvk + λ Hence, we have kuk + kvk ≤ (q1 + q2 )m0 + (¯ q1 + q¯2 )n0 + [(q1 + q2 )m1 + (¯ q1 + q¯2 )n1 ]kuk

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¯1 + λ ¯2, +[(q1 + q2 )m2 + (¯ q1 + q¯2 )n2 ]kvk + λ which, in view of (3.9) and (3.10), yields k(u, v)k ≤

¯ Q1 m0 + Q2 n0 + λ , Q0

for any t ∈ [a, b], which proves that the set ω is bounded. Hence, by Lemma 3.1, the operator T has at least one fixed point. Therefore, the problem (1.1) − (1.2) has at least one solution on [a, b]. This completes the proof. 2 Next, we apply Schauder fixed point theorem to prove the existence of solutions for the problem (1.1)-(1.2) by imposing the the sub-growth condition on the nonlinear functions involved in the problem. Theorem 3.3 Assume that (H2 ) holds. Then, there exist at least one solution on [a, b] for the problem (1.1) − (1.2). Proof. Define a set Y in the Banach space X × X by Y = {(u, v) ∈ X × X : k(u, v)k ≤ y}, where 1

1

1

1

¯ 7Q1 α(t), 7Q2 β(t), (7Q1 1 ) 1−p1 , (7Q1 2 ) 1−p2 , (7Q2 d1 ) 1−l1 , (7Q2 d2 ) 1−l1 }. y > max{7λ, In order to show that T : Y → Y. We have Z Z t 1 n b h1 |T1 (u, v)(t)| = sup (t − s)f (s, u(s), v(s))ds − A1 (b − a)(b − s) A3 2 t∈[a,b] a a i +L1 + (b − a)A2 (t − a) (b − s)f (s, u(s), v(s))ds Z bh m m i X X 1 + A1 (b − s) γj (ηj − ξj ) + L2 + A2 (t − a) ρj (ηj − ξj ) 2 a j=1 j=1 Z Z m o 1 n X ηj s h ×(b − s)g(s, u(s), v(s))ds + γj A1 (b − a)(s − p) A3 j=1 ξj a i +ρj L1 + ρj (b − a)A2 (t − a) g(p, u(p), v(p))dpds m Z ηj Z s h m X X + σ j A1 γj (ηj − ξj )(s − p) + δj L2 j=1

ξj

a

+δj A2 (t − a)

j=1 m X

i o ρj (ηj − ξj ) f (p, u(p), v(p))dpds + Ω1 (t)

j=1

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B. Ahmad, A. Alsaedi, M. Alsulami and S.K. Ntouyas     ¯1, ≤ α(t) + 1 |u|p1 + 2 |v|p2 q1 + β(t) + d1 |u|l1 + d2 |v|l2 q¯1 + λ

which implies that 

p2

p1

kT1 (u, v)k ≤ α(t) + 1 |u| + 2 |v|





l2

l1

q1 + β(t) + d1 |u| + d2 |v|



¯1. q¯1 + λ

Analogously, we have     ¯2. kT2 (u, v)k ≤ α(t) + 1 |u|p1 + 2 |v|p2 q2 + β(t) + d1 |u|l1 + d2 |v|l2 q¯2 + λ In consequence, 

p1

p2

kT (u, v)k ≤ α(t) + 1 |u| + 2 |v|





l1

l2

Q1 + β(t) + d1 |u| + d2 |v|



¯ ≤ y, Q2 + λ

¯ are given by (3.8). Therefore, we conclude that T : Y → Y, where where Q1 , Q2 and λ T1 (u, v)(t) and T2 (u, v)(t) are continuous on [a, b]. Now we prove that T is completely continuous operator by fixing that G = max |f (t, u(t), v(t))|, H = max |g(t, u(t), v(t))|. t∈[a,b]

t∈[a,b]

Letting t, τ ∈ [a, b] with a < t < τ < b and (u, v) ∈ Y, we get |T1 (u, v)(τ ) − T1 (u, v)(t)| h (τ − t)2 i (τ − t) n h (b − a)3 ≤ G (τ − t)(t − a) + + G 2 |A1 | 2 m m X  X  (η − a)2 (ξ − a)2 i j j + ρj δj (ηj − ξj ) − 2 2 j=1 j=1 m  (η − a)2 (ξ − a)2 i X (b − a)2 j j +H ρj (ηj − ξj ) + (b − a) ρj − 2 2 2 j=1 j=1 m hX

+(b − a)λ2 +

m X

o

ρj (ηj − ξj )λ4 → 0 as (τ − t) → 0.

j=1

In a similar manner, one can obtain |T2 (u, v)(τ ) − T2 (u, v)(t)| m h (τ − t)2 i (τ − t) n h X (b − a)3 ≤ H (τ − t)(t − a) + + G δj (ηj − ξj ) 2 |A1 | 6 j=1 +(b − a)

m  (η − a)2 (ξ − a)2 i X j j δj − 2 2 j=1

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+H +

h (b − a)3

m X

2

15

m m X  X  (η − a)2 (ξ − a)2 i j j + δj ρj (ηj − ξj ) − 2 2 j=1 j=1

o δj (ηj − ξj )λ2 + (b − a)λ4 → 0 as (τ − t) → 0.

j=1

Thus the operator T Y ⊂ Y is equicontinuous and uniformaly bounded set. Hence T is a completely continuous operator. So, by Schauder fixed point theorem, there exist a solution to the problem (1.1) − (1.2). 2

3.2

Uniqueness of solutions

Here we establish the uniqueness of solutions for the problem (1.1) − (1.2) by means of Banach’s contraction mapping principle. Theorem 3.4 Assume that (H3 ) holds and that Q1 `1 + Q2 `2 < 1,

(3.11)

where Q1 and Q2 are given by (3.8). Then the problem (1.1)−(1.2) has a unique solution on [a, b]. Proof. Define supt∈[a,b] |f (t, 0, 0)| = N1 , supt∈[a,b] |g(t, 0, 0)| = N2 and r≥

¯ Q1 N1 + Q2 N2 + λ . 1 − (Q1 `1 + Q2 `2 )

Then we show that T Br ⊂ Br , where Br = {(u, v) ∈ X × X : k(u, v)k ≤ r}. For any (u, v) ∈ Br , t ∈ [a, b], we find that |f (s, u(s), v(s))| = |f (s, u(s), v(s)) − f (s, 0, 0) + f (s, 0, 0)| ≤ |f (s, u(s), v(s)) − f (s, 0, 0)| + |f (s, 0, 0)| ≤ `1 (kuk + kvk) + N1 ≤ `1 k(u, v)k + N1 ≤ `1 r + N1 , and |g(s, u(s), v(s))| = |g(s, u(s), v(s)) − g(s, 0, 0) + g(s, 0, 0)| ≤ |g(s, u(s), v(s)) − g(s, 0, 0)| + |g(s, 0, 0)| ≤ `2 (kuk + kvk) + N2 ≤ `2 k(u, v)k + N2 ≤ `2 r + N2 . Then, for (u, v) ∈ Br , we obtain |T1 (u, v)(t)| ≤

Z bh Z t 1 n 1 sup (t − s)f (s, u(s), v(s))ds + − A1 (b − a)(b − s) A3 2 t∈[a,b] a a

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B. Ahmad, A. Alsaedi, M. Alsulami and S.K. Ntouyas i +L1 + (b − a)A2 (t − a) (b − s)f (s, u(s), v(s))ds Z bh m m i X X 1 − A1 (b − s) γj (ηj − ξj ) + L2 + A2 (t − a) ρj (ηj − ξj ) 2 a j=1 j=1 m Z ηj Z s h X ×(b − s)g(s, u(s), v(s))ds + γj A1 (b − a)(s − p) j=1

ξj

a

i +ρj L1 + ρj (b − a)A2 (t − a) g(p, u(p), v(p))dpds m Z ηj Z s h m X X + σ j A1 γj (ηj − ξj )(s − p) + δj L2 j=1

ξj

a

+δj A2 (t − a)

j=1 m X

i o ρj (ηj − ξj ) f (p, u(p), v(p))dpds + Ω1 (t)

j=1

1 n (b − a)4 (b − a)2 |A1 | + |L1 | 2 |A3 | 6 2 m  (η − a)2 (ξ − a)2  (b − a)4 X j j + δj |L2 | − +|A2 | 2 2 2 j=1 ≤ [`1 r + N1 ] ×

n (b − a)2

+

m m X  X  (η − a)3 (ξ − a)3  j j − +|A1 | γj σj (ηj − ξj ) 3! 3! j=1 j=1 m m X  X  (η − a)2 (ξ − a)2 o j j +|A2 |(b − a) ρj δj (ηj − ξj ) − 2 2 j=1 j=1 m n 1 n (b − a)3 X (b − a)2 +[`2 r + N2 ] × |A1 | γj (ηj − ξj ) + |L2 | |A3 | 6 2 j=1 m (b − a)3 X +|A2 | ρj (ηj − ξj ) + |A1 |(b − a) 2 j=1 m  (η − a)2 (ξ − a)2   (η − a)3 (ξ − a)3  X j j j j − + ρj |L1 | − × γj 3! 3! 2 2 j=1 j=1 m X

2

+|A2 |(b − a)

m X j=1

ρj

 (η − a)2 (ξ − a)2 o j j ¯1 − +λ 2 2

¯1. ≤ q1 (`1 r + N1 ) + q¯1 (`2 r + N2 ) + λ Hence ¯1. kT1 (u, v)k ≤ q1 (`1 r + N1 ) + q¯1 (`2 r + N2 ) + λ Likewise, we find that ¯2. kT2 (u, v)k ≤ q2 (`1 r + N1 ) + q¯2 (`2 r + N2 ) + λ

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From the above estimates, it follows that that kT (u, v)k ≤ r. Next we show that the operator T is a contraction. For (u1 , v1 ), (u2 , v2 ) ∈ X × X , we have

|T1 (u1 , v1 )(t) − T1 (u2 , v2 )(t)| nZ t ≤ sup (t − s) f (s, u1 (s), v1 (s)) − f (s, u2 (s), v2 (s)) ds t∈[a,b]

a

Z i 1 n b h1 + |A1 |(b − a)(b − s) + L1 + (b − a)|A2 |(t − a) (b − s) |A | a 2 3 × f (s, u1 (s), v1 (s)) − f (s, u2 (s), v2 (s)) ds Z bh m m i X X 1 |A1 |(b − s) γj (ηj − ξj ) + L2 + |A2 |(t − a) ρj (ηj − ξj ) + 2 a j=1 j=1 o ×(b − s) g(s, u1 (s), v1 (s)) − g(s, u2 (s), v2 (s)) ds Z m Z i 1 n X ηj s h + γj |A1 |(b − a)(s − p) + ρj L1 + ρj (b − a)|A2 |(t − a) |A3 | j=1 ξj a × g(p, u1 (p), v1 (p)) − g(p, u2 (p), v2 (p)) dpds m m m Z ηj Z s h i X X X σj |A1 | γj (ηj − ξj )(s − p) + δj L2 + δj |A2 |(t − a) ρj (ηj − ξj ) + j=1

ξj

a

j=1

j=1

o × f (p, u1 (p), v1 (p)) − f (p, u2 (p), v2 (p)) dpds n (b − a)2 1 h (b − a)4 (b − a)2 ≤ `1 (|u1 − u2 | + |v1 − v2 |) × + |A1 | + |L1 | 2 |A3 | 6 2 m m    4 3 X X (ηj − a) (ξj − a)3  (b − a) + |A1 | γj σj (ηj − ξj ) − +|A2 | 2 3! 3! j=1 j=1 +

m X

 (η − a)2 (ξ − a)2  j j δj |L2 | − 2 2 j=1

m m X  X  (η − a)2 (ξ − a)2 io j j +|A2 |(b − a) ρj δj (ηj − ξj ) − 2 2 j=1 j=1

+`2 (|u1 − u2 | + |v1 − v2 |) × +|A2 |

m n 1 h (b − a)3 X (b − a)2 |A1 | γj (ηj − ξj ) + |L2 | |A3 | 6 2 j=1

m m  (η − a)3 (ξ − a)3  X (b − a)3 X j j ρj (ηj − ξj ) + |A1 |(b − a) γj − 2 3! 3! j=1 j=1

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B. Ahmad, A. Alsaedi, M. Alsulami and S.K. Ntouyas m X

m  (η − a)2 (ξ − a)2   (η − a)2 (ξ − a)2 io X j j j j 2 + ρj |L1 | − + |A2 |(b − a) − ρj 2 2 2 2 j=1 j=1

≤ (`1 q1 + `2 q¯1 )(|u1 − u2 | + |v1 − v2 |), which yields kT1 (u1 , v1 ) − T1 (u2 , v2 )k ≤ (`1 q1 + `2 q¯1 )(|u1 − u2 | + |v1 − v2 |). Similarly, kT2 (u1 , v1 ) − T2 (u2 , v2 )k ≤ (`1 q2 + `2 q¯2 )(|u1 − u2 | + |v1 − v2 |). So, it follows from the above inequalities that kT (u1 , v1 ) − T (u2 , v2 )k ≤ (Q1 `1 + Q2 `2 )(ku1 − u2 k + kv1 − v2 k), where Q1 and Q2 are given by (3.8). By the given assumption (3.11), it follows that the operator T is a contraction. Thus, by Banach’s contraction mapping principle, we deduce that the operator T has a fixed point, which corresponds to a unique solution of the problem (1.1)-(1.2) on [a, b]. 2 Example 3.5 Consider the following second order system of ordinary differential equations   1  |u|   u00 (t) = + v(t) + e−t , t ∈ [2, 3], 10 + t2 1 + |u| (3.12)    v 00 (t) = √ 1 u(t) + tan−1 v(t) + cos (t − 2), t ∈ [2, 3], 3 32 + t2 subject to the boundary conditions  Z Z 3 Z ηj Z ηj 3 3 3 X X   0  v 0 (s)ds + 1, γj v(s)ds + 2, u (s)ds = ρj   2 u(s)ds = ξj 2 ξj j=1 j=1 Z Z Z Z 3 3 3 ηj 3 ηj X X  3 1  0  u(s)ds + v(s)ds = σ u0 (s)ds + , , v (s)ds = δ  j j  2 2 2 2 ξj ξj j=1 j=1

(3.13)

where a = 2, b = 3, m = 3, λ1 = 2, λ2 = 1, λ3 = 3/2, λ4 = 1/2, γ1 = 2/5, γ2 = 21/40, γ3 = 13/20, ρ1 = 1/3, ρ2 = 1/2, ρ3 = 2/3, σ1 = 3/7, σ2 = 5/7, σ3 = 1, δ1 = 3/8, δ2 = 5/8, δ3 = 7/8, ξ1 = 15/7, η1 = 16/7, ξ2 = 17/7, η2 = 18/7, ξ3 = 19/7, η3 = 20/7. Using the given data, we find that `1 = 71 , `2 = 19 , A1 ≈ 0.827806 6= 0, A2 ≈ 0.793367 6= 0, A3 ≈ 0.656754, |L1 | = 0.03337, |L2 | ≈ 0.225389, |L3 | ≈ 0.027121, |L4 | ≈ 0.185097, q1 ≈ 1.963984, q2 ≈ 1.422591, q¯1 ≈ 1.290164 and q¯2 ≈ 1.851349. Also Q1 `1 + Q2 `2 ≈ 0.832853 < 1 (Q1 and Q2 are given by (3.8)). Thus, all the conditions of Theorem 3.4 are satisfied. Hence it follows by the conclusion of Theorem 3.4 that the problem (3.12) − (3.13) has a unique solution on [2, 3].

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4

19

Conclusions

The salient features of this work includes (i) considering a coupled system of nonlinear ordinary differential equations on an arbitrary domain (ii) a new kind of integral multistrip coupled boundary conditions. The results obtained for the given problem are new and significantly contribute to the existing literature on the topic. As a special case, our results correspond to the uncoupled integral boundary conditions of the form: Z

b

Z

Z

0

a

a

b

Z v(s)ds = λ3 ,

u (s)ds = λ2 ;

u(s)ds = λ1 , a

b

b

v 0 (s)ds = λ4 ,

a

if we take all γj = 0, ρj = 0, σj = 0, δj = 0 (j = 1, . . . , m) in the results of this paper.

References [1] L. Zheng, X. Zhang, Modeling and analysis of modern fluid problems. Mathematics in Science and Engineering, Elsevier/Academic Press, London, 2017. [2] F.T. Akyildiz, H. Bellout, K. Vajravelu, R.A. Van Gorder, Existence results for third order nonlinear boundary value problems arising in nano boundary layer fluid flows over stretching surfaces, Nonlinear Anal. Real World Appl. 12 (2011), 2919-2930. [3] A. Abid, P. Eloe, Positive solutions of boundary value problems for ordinary differential equations with dependence on higher order derivatives, Comm. Appl. Nonlinear Anal. 24 (2017), 101-108. [4] M. Ashordia, On boundary value problems for systems of nonlinear generalized ordinary differential equations, Czechoslovak Math. J. 67 (2017), 579-608. [5] R. Liu, Z. Wu, Well-posedness of a class of two-point boundary value problems associated with ordinary differential equations, Adv. Difference Equ. (2018), 2018:54. [6] M. Greguˇs, F. Neumann, F. M. Arscott, Three-point boundary value problems in differential equations, Proc. London Math. Soc. 3 (1964), 459-470. [7] V.A. Il’in, E.I. Moiseev, Nonlocal boundary-value problem of the first kind for a Sturm-Liouville operator in its differential and finite difference aspects, Differential Equations 23 (1987), 803-810. [8] C.P. Gupta, Solvability of a three-point nonlinear boundary value problem for a second order ordinary differential equations, J. Math. Anal. Appl. 168 (1998), 540-551.

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[9] M.R. Grossinho, F.M. Minhos, Existence result for some third order separated boundary value problems, Nonlinear Anal. 47 (2001), 2407-2418. [10] P.W. Eloe, B. Ahmad, Positive solutions of a nonlinear nth order boundary value problem with nonlocal conditions, Appl. Math. Lett. 18 (2005), 521-527. [11] S. Clark, J. Henderson, Uniqueness implies existence and uniqueness criterion for non local boundary value problems for third-order differential equations, Proc. Amer. Math. Soc. 134 (2006), 3363-3372. [12] J.R.L. Webb, G. Infante, Positive solutions of nonlocal boundary value problems: A unified approach, J. London Math. Soc. 74 (2006), 673-693. [13] J.R. Graef, J.R.L. Webb, Third order boundary value problems with nonlocal boundary conditions, Nonlinear Anal. 71 (2009), 1542-1551. [14] L. Wang, M. Pei, W. Ge, Existence and approximation of solutions for nonlinear second-order four-point boundary value problems, Math. Comput. Modelling 50 (2009), 1348-1359. [15] Y. Sun, L. Liu, J. Zhang, R.P. Agarwal, Positive solutions of singular threepoint boundary value problems for second-order differential equations, J. Comput. Appl. Math. 230 (2009), 738-750. [16] M. Feng, X. Zhang, W. Ge, Existence theorems for a second order nonlinear differential equation with nonlocal boundary conditions and their applications, J. Appl. Math. Comput. 33 (2010), 137-153. [17] J.R. Cannon, The solution of the heat equation subject to the specification of energy, Quart. Appl. Math. 21 (1963), 155-160. [18] N.I. Ionkin, The solution of a certain boundary value problem of the theory of heat conduction with a nonclassical boundary condition (Russian), Diff. Uravn. 13 (1977), 294-304. [19] C. Taylor, T. Hughes, C. Zarins, Finite element modeling of blood flow in arteries, Comput. Methods Appl. Mech. Engrg. 158 (1998), 155-196. [20] F. Nicoud, T. Schfonfeld, Integral boundary conditions for unsteady biomedical CFD applications, Int. J. Numer. Meth. Fluids 40 (2002), 457-465. [21] S.K. Ntouyas, Nonlocal Initial and Boundary Value Problems: A survey, Handbook on Differential Equations: Ordinary Differential Equations, Edited by A. Canada, P. Drabek and A. Fonda, Elsevier Science B. V., 2005, 459-555. [22] Z. Yang, Positive solutions of a second order integral boundary value problem, J. Math. Anal. Appl. 321 (2006), 751-765.

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[23] J.R.L. Webb, G. Infante, Positive solutions of nonlocal boundary value problems involving integral conditions, NoDEA, Nonlinear Differ. Equ. Appl. 15 (2008), 45-67. [24] B. Ahmad, A. Alsaedi, B.S. Alghamdi, Analytic approximation of solutions of the forced Duffing equation with integral boundary conditions. Nonlinear Anal. Real World Appl. 2008, 9, 1727–1740. [25] A. Boucherif, Second-order boundary value problems with integral boundary conditions, Nonlinear Anal. TMA. 70 (2009), 364-371. [26] B. Ahmad, S.K. Ntouyas, H.H. Alsulami, Existence results for n-th order multipoint integral boundary-value problems of differential inclusions, Electron. J. Differential Equations 2013, No. 203, 13 pp. [27] Y. Li, H. Zhang, Positive solutions for a nonlinear higher order differential system with coupled integral boundary conditions, J. Appl. Math. 2014, Art. ID 901094, 7 pp. [28] B. Ahmad, A. Alsaedi, A. Assolami, Relationship between lower and higher order anti-periodic boundary value problems and existence results, J. Comput. Anal. Appl. 16 (2014), 210-219. [29] J. Henderson, Smoothness of solutions with respect to multi-strip integral boundary conditions for nth order ordinary differential equations, Nonlinear Anal. Model. Control 19 (2014), 396-412. [30] I.Y. Karaca, F.T. Fen, Positive solutions of nth-order boundary value problems with integral boundary conditions, Math. Model. Anal. 20 (2015), 188-204. [31] B. Ahmad, A. Alsaedi, N. Al-Malki, On higher-order nonlinear boundary value problems with nonlocal multipoint integral boundary conditions, Lith. Math. J. 56 (2016), 143-163. [32] M. Boukrouche, D.A. Tarzia, A family of singular ordinary differential equations of the third order with an integral boundary condition, Bound. Value Probl. (2018), 2018:32. [33] A. Granas, J. Dugundji, Fixed Point Theory, Springer-Verlag, New York, 2005.

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Explicit identities involving truncated exponential polynomials and phenomenon of scattering of their zeros C. S. RYOO Department of Mathematics, Hannam University, Daejeon 34430, Korea

Abstract : In this paper, we study differential equations arising from the generating functions of truncated exponential polynomials. We give explicit identities for the truncated polynomials. Using numerical investigation, we observe the behavior of complex roots of the truncated polynomials en (x). By means of numerical experiments, we demonstrate a remarkably regular structure of the complex roots of the truncated polynomials en (x). Key words : Differential equations, complex roots, truncated polynomials. AMS Mathematics Subject Classification : 05A19, 11B83, 34A30, 65L99. 1. Introduction Recently, many mathematicians have studied in the area of the Bernoulli numbers and polynomials, Euler numbers and polynomials, tangent numbers and polynomials, Genocchi numbers and polynomials, Laguerre polynomials, and Hermite polynomials. These numbers and polynomials possess many interesting properties and arising in many areas of mathematics, physics, and applied engineering(see [1-14]). By using software, many mathematicians can explore concepts much more easily than in the past. The ability to create and manipulate figures on the computer screen enables mathematicians to quickly visualize and produce many problems, examine properties of the figures, look for patterns, and make conjectures. This capability is especially exciting because these steps are essential for most mathematicians to truly understand even basic concept. Numerical experiments of Euler polynomials, Bernoulli polynomials, tangent polynomials, Genocchi polynomials, Laguerre polynomials, and Hermite polynomials have been the subject of extensive study in recent year and much progress have been made both mathematically and computationally. Using computer, a realistic study for the zeros of truncated polynomials en (x) is very interesting. The main purpose of this paper is to observe an interesting phenomenon of ‘scattering’ of the zeros of the truncated polynomials en (x) in complex plane. Throughout this paper, we always make use of the following notations: N = {1, 2, 3, · · · } denotes the set of natural numbers, N0 = {0, 1, 2, 3, · · · } denotes the set of nonnegative integers, Z denotes the set of integers, R denotes the set of real numbers, C denotes the set of complex numbers. We first give the definitions of the truncated exponential polynomials. It should be mentioned that the definition of truncated exponential polynomials en (x) can be found in [1, 3]. The truncated exponential polynomials en (x) are defined by means of the generating function:

(

1 1−t

) ext =

∞ ∑

en (x)tn ,

|t| < 1.

(1.1)

n=0

We recall that G. Dattoli and M. Migliorati(see [3]) studied some properties of truncated exponential polynomials en (x). The truncated exponential polynomials en (x) satisfy the following relations d en (x) = en−1 (x), dx ( )) ( x d en+1 (x) = 1 + 1− en (x). n+1 dx

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(k)

The Miller-Lee polynomials Gn (x)(see [1]), are defined by means of the following generating function ( )k+1 ∞ ∑ 1 n ext = G(k) (1.2) n (x)t . 1−t n=0 Differential equations arising from the generating functions of special polynomials are studied by many authors in order to give explicit identities for special polynomials. In this paper, we study linear differential equations arising from the generating functions of truncated exponential polynomials en (x). We give explicit identities for truncated exponential polynomials en (x). 2. Differential equations associated with truncated exponential polynomials In this section, we study linear differential equations arising from the generating functions of truncated exponential polynomials. Let ( ) 1 ext . (2.1) F = F (t, x) = 1−t Then, by (2.1), we get F

and

( F (2) =

(1)

d dt

( ) 1 d d ext = F (t, x) = dt dt 1 − t ( )2 ( ) 1 1 = ext + x ext 1−t 1−t ( ) 1 = + x F (t, x), 1−t

)2 F (t, x)

( )2 ) 1 1 xt = e F (t, x) + + x F (t, x)F (1) 1−t 1−t (( )2 ( )) 2 2x 2 = + +x F (t, x). 1−t 1−t (

(2.2)

(2.3)

Continuing this process, we can guess that ( F (N ) = ( =

d dt

)N

N ∑

F (t, x) ) −i

ai (N, x)(1 − t)

(2.4) F (t, x),

(N = 0, 1, 2, . . .).

i=0

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Taking the derivative with respect to t in (2.4), we obtain dF (N ) F (N +1) = dt ) (N ) (N ∑ ∑ = ai (N, x)(1 − t)−i F (1) (t, x) iai (N, x)(1 − t)−i−1 F (t, x) + i=0

=

(N ∑

i=0

) iai (N, x)(1 − t)−i−1

i=0

+

(N ∑

F (t, x) ) −i

ai (N, x)(1 − t)

(

(2.5)

) (1 − t)−1 + x F (t, x)

i=0

(N ) (N ) ∑ ∑ −i−1 −i = (i + 1)ai (N, x)(1 − t) F (t, x) + xai (N, x)(1 − t) F (t, x) i=0

=

(N ∑

) xai (N, x)(1 − t)−i

F (t, x) +

(N +1 ∑

i=0

i=0

)

iai−1 (N, x)(1 − t)−i

F (t, x).

i=1

On the other hand, by replacing N by N + 1 in (2.4), we get ) (N +1 ∑ −i (N +1) ai (N + 1, x)(1 − t) F (t, x). F =

(2.6)

i=0

By (2.5) and (2.6), we have (N ) (N +1 ) ∑ ∑ −i −i xai (N, x)(1 − t) F (t, x) + iai−1 (N, x)(1 − t) F (t, x) i=0

=

(N +1 ∑

i=1

) ai (N + 1, x)(1 − t)−i

(2.7)

F (t, x)..

i=0

Comparing the coefficients on both sides of (2.7), we obtain a0 (N + 1, x) = xa0 (N, x),

(2.8)

aN +1 (N + 1, x) = (N + 1)aN (N, x), and ai (N + 1, x) = xai (N, x) + iai−1 (N, x), (1 ≤ i ≤ N ).

(2.9)

In addition, by (2.2) and (2.4), we get F = F (0) = a0 (0, x)F (t, x) = F (t, x).

(2.10)

a0 (0, x) = 1.

(2.11)

Thus, by (2.10), we obtain

It is not difficult to show that (1 − t)−1 F (t, x) + xF (t, x) =

1 ∑

ai (1, x)(1 − t)−i F (t, x)

(2.12)

i=0

= a0 (1, x)F (t, x) + a1 (1, x)(1 − t)−1 F (t, x). Thus, by (2.12), we also get a0 (1, x) = x,

238

a1 (1, x) = 1.

(2.13)

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From (2.8), we note that a0 (N + 1, x) = xa0 (N, x) = x2 a0 (N − 1, x) = · · · = xN +1 , and aN +1 (N + 1, x) = (N + 1)aN (N, x) = · · · = (N + 1)!.

(2.14)

For i = 1, 2, 3 in (2.9), we get a1 (N + 1, x) =

N ∑

xk a0 (N − k, x),

k=0 N −1 ∑

a2 (N + 1, x) = 2

xk a1 (N − k, x), and

k=0 N −2 ∑

a3 (N + 1, x) = 3

xk a2 (N − k, x).

k=0

Continuing this process, we can deduce that, for 1 ≤ i ≤ N, ai (N + 1, x) = i

N∑ −i+1

xk ai−1 (N − k, x).

(2.15)

k=0

Now, we give explicit expressions for ai (N + 1, x). By (2.14) and (2.15), we get N ∑

a1 (N + 1, x) =

xk1 a0 (N − k1 , x) = xN (N + 1),

k1 =0 N −1 ∑

a2 (N + 1, x) = 2

xk1 a1 (N − k1 , x) = 2!

k1 =0

and

xN −1 (N − k1 ),

k1 =0 N −2 ∑

a3 (N + 1, x) = 3

N −1 ∑

xk2 a2 (N − k2 , x)

k2 =0

= 3!

N −2 N −k 2 −2 ∑ ∑ k2 =0

xN −k2 −2 (N − k2 − k1 − 1).

k1 =0

Continuing this process, we have ai (N + 1, x) = i!

N∑ −i+1 N −k∑ i−1 −i+1 ki−1 =0

N −ki−1 −···−k2 −i+1

∑

···

ki−2 =0

xN −ki −···−k2 −i+1

k1 =0

(2.16)

× (N − ki−1 − ki−2 − · · · − k2 − k1 − i + 2). Note that, here the matrix ai (j, x)0≤i,j≤N +1 is given by  1 x  0 1!  0 0   0 0 . . . . . . 0

0

x2 2x

x3 ·

··· ···

2! 0 .. .

· 3! .. .

··· ··· .. .

0

0

···

 xN +1  (N + 1)xN    ·   ·   ..   . (N + 1)!

Therefore, by (2.16), we obtain the following theorem.

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Theorem 1. For N = 0, 1, 2, . . . , the functional equation (N ( )i ) ∑ 1 (N ) F = ai (N, x) F 1−t i=0 (

has a solution F = F (t, x) =

1 1−t

) ext ,

where a0 (N, x) = xN , aN (N, x) = N !, N −i N −k i−1 −i ∑ ∑

ai (N, x) = i!

ki−1 =0

N −ki−1 −···−k2 −i

∑

···

ki−2 =0

xN −ki−1 −···−k2 −i

k1 =0

× (N − ki−1 − ki−2 − · · · − k2 − k1 − i + 1), (1 ≤ i ≤ N ). From (1.1), we note that ( F (N ) =

d dt

)N F (t, x) =

∞ ∑ (k + N )! k=0

k!

ek+N (x)tk .

(2.17)

From Theorem 1, (1.2), and (2.17), we can derive the following equation: ∞ ∑ (k + N )! k=0

k!

)i ) 1 ek+N (x)t = ai (N, x) F 1−t i=0 ( )i+1 N ∑ 1 = ext ai (N, x) 1 − t i=0 ) (∞ N ∑ ∑ (i) k = ai (N, x) Gk (x)t k

(N ∑

(

i=0

(N ∞ ∑ ∑

=

k=0

k=0 (i) ai (N, x)Gk (x)

(2.18)

) tk .

i=0

By comparing the coefficients on both sides of (2.18), we obtain the following theorem. Theorem 2. For k = 0, 1, . . . , and N = 0, 1, 2, . . . , we have ∑ k! (i) ai (N, x)Gk (x), (k + N )! i=0 N

ek+N (x) =

(2.19)

where a0 (N, x) = xN , aN (N, x) = N !, ai (N, x) = i!

N −i N −k i−1 −i ∑ ∑ ki−1 =0

N −ki−1 −···−k2 −i

···

ki−2 =0

∑

xN −ki−1 −···−k2 −i

k1 =0

× (N − ki−1 − ki−2 − · · · − k2 − k1 − i + 1), (1 ≤ i ≤ N ). Let us take k = 0 in (2.19). Then, we have the following corollary.

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Corollary 3. For N = 0, 1, 2, . . . , we have 1 ∑ (i) ai (N, x)G0 (x). N ! i=0 N

eN (x) = For N = 1, 2, . . . , the functional equation ( F

(N )

=

N ∑

( ai (N, x)

i=0

(

has a solution F = F (t, x) =

1 1−t

1 1−t

)i ) F

) ext .

Here is a plot of the surface for this solution.

FHt,xL 0.75

0.5

0.25

4 2 0

-0.5

x

0

-0.25

-2

0 t

t

FHt,xL

10 7.5 5 2.5 0

0.5

-4

-0.5

-0.75 -4

-2

0 x

2

4

Figure 1: The surface for the solution F (t, x)

In Figure 1(left), we plot of the surface for this solution. In Figure 1(right), we shows a higherresolution density plot of the solution. 3. Zeros of the truncated exponential polynomials This section aims to demonstrate the benefit of using numerical investigation to support theoretical prediction and to discover new interesting pattern of the zeros of the truncated exponential polynomials en (x). By using computer, the truncated exponential polynomials en (x) can be determined explicitly. We display the shapes of the truncated exponential polynomials en (x) and investigate the zeros of the truncated exponential polynomials en (x). We investigate the beautiful zeros of the truncated exponential polynomials en (x) by using a computer. We plot the zeros of the en (x) for n = 20, 30, 40, 50 and x ∈ C(Figure 2). In Figure 2(top-left), we choose n = 20. In Figure 2(top-right), we choose n = 30. In Figure 2(bottom-left), we choose n = 40. In Figure 2(bottom-right), we choose n = 50. Stacks of zeros of en (x) for 1 ≤ n ≤ 40, forming a 3D structure are presented(Figure 3). In Figure 3(top-left), we plot stacks of zeros of en (x) for 1 ≤ n ≤ 40. In Figure 3(top-right), we draw x and y axes but no z axis in three dimensions. In Figure 3(bottom-left), we draw y and z axes but no x axis in three dimensions. In Figure 3(bottom-right), we draw x and z axes but no y axis in three dimensions. Our numerical results for approximate solutions of real zeros of the truncated exponential polynomials en (x) are displayed(Tables 1, 2).

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Im(x)

30

30

20

20

10

10

0

Im(x)

0

-10

-10

-20

-20

-30

-20

-30 0

20

40

-20

0

Re(x)

Im(x)

30

20

20

10

10

0

Im(x)

-10

-20

-20

-20

-30 0

40

20

40

0

-10

-30

20 Re(x)

30

20

40

-20

0

Re(x)

Re(x)

Figure 2: Zeros of en (x)

Table 1. Numbers of real and complex zeros of en (x) degree n

real zeros

complex zeros

1

1

0

2

0

2

3

1

2

4

0

4

5

1

4

6

0

6

7

1

6

8

0

8

9

1

8

10

0

10

11

1

10

12

0

12

13

1

12

14

0

14

How many zeros does en (x) have? We are not able to decide if en (x) has n distinct solutions(see Table 1, Table 2). We would also like to know the number of complex zeros Cen (x) of en (x), Im(x) ̸= 0. Since n is the degree of the polynomial en (x), the number of real zeros Ren (x) lying on the real line Im(x) = 0 is then Ren (x) = n − Cen (x) , where Cen (x) denotes complex zeros. See Table 1 for tabulated values of Ren (x) and Cen (x) .

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ImHwL 10 0

10

-10 40

ImHwLL 0

30

s

20

-10

10 0 -10

-10

0

0

10

20

10

ReHwL

ReHwL

20 40

40

30

30

20 s

s 20 10

10

0 -10 10

0

-10

0

10

20

0

ReHwL

ImHwL

Figure 3: Stacks of zeros of en (x) for 1 ≤ n ≤ 40

Conjecture 5. Prove that en (x) = 0 has n distinct solutions. Using computers, many more values of n have been checked. It still remains unknown if the conjecture fails or holds for any value n. Since n is the degree of the polynomial en (x), the number of real zeros Ren (x) lying on the real plane Im(x) = 0 is then Ren (x) = n − Cen (x) , where Cen (x) denotes complex zeros. See Table 1 for tabulated values of Ren (x) and Cen (x) . Conjecture 6. Prove that the numbers of complex zeros Cen (x) of en (x), Im(x) ̸= 0 is [n] Cen (x) = 2 , 2 where [

] denotes taking the integer part.

Conjecture 7. For n ∈ N0 , if n ≡ 1 (mod 2), then Ren (x) = 1, if n ≡ 0 (mod 2), then Ren (x) = 0. The plot of real zeros of the truncated exponential polynomials en (x) for 1 ≤ n ≤ 50 structure are presented(Figure 4). It is expected that en (x), x ∈ C, has Im(x) = 0 reflection symmetry analytic complex functions (see Figure 2, Figure 3, Figure 4). For a ∈ R, we expect that en (x), x ∈ C, has not Re(x) = a reflection symmetry analytic complex functions. We observe a remarkable regular structure of the complex roots of the truncated exponential polynomials en (x). We also hope to verify a remarkable regular structure of the complex roots of the truncated exponential polynomials en (x)(Table 1). Next, we calculated an approximate solution satisfying en (x) = 0, x ∈ C. The

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40

n 20

0

-5

-10

0

ReHxL

Figure 4: Real zeros of en (x) for 1 ≤ n ≤ 50

results are given in Table 2. Table 2. Approximate solutions of en (x) = 0, x ∈ C degree n

x

1

−1.0000

2 3 4

5

−1.0000 − 1.0000i, −1.5961,

−0.7020 − 1.8073i,

−0.7020 + 1.8073i

−1.7294 − 0.8890i,

−1.7294 + 0.8890i

−0.2706 − 2.5048i,

−0.2706 + 2.5048i

−2.1806,

−1.6495 − 1.6939i,

0.2398 − 3.1283i, 6

−1.0000 + 1.0000i

−2.3618 − 0.8384i, −1.4418 + 2.4345i,

−1.6495 + 1.6939i

0.2398 + 3.1283i

−2.3618 + 0.8384i, 0.8036 − 3.6977i,

−1.4418 − 2.4345i 0.8036 + 3.6977i

Acknowledgement: This work was supported by the National Research Foundation of Korea(NRF) grant funded by the Korea government(MEST) (No. 2017R1A2B4006092).

REFERENCES 1. Andrews, L.C.(1985). Special Functions for Engineers and Applied Mathematicians, Macmillan Publishing Company, New York. 2. Andrews, G. E., Askey, R., Roy, R.(1999). Special Functions. Cambridge, England: Cambridge University Press. 3. Dattoli, G., Migliorati, M.(2006). The truncated exponential polynomials, the associated Hermite forms and applications, International Journal of Mathematics and Mathematical Sciences, v.2006, Article ID 98175, pp. 1-10.

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4. Erdelyi, A., Magnus, W., Oberhettinger, F., Tricomi, F.G.(1981). Higher Transcendental Functions, v.3. New York: Krieger. 5. Kim, T.(2008). Euler numbers and polynomials associated with zeta function, Abstract and Applied Analysis, Art. ID 581582. 6. Liu, G.(2006). Congruences for higher-order Euler numbers, Proc. Japan Acad., v.82 A, pp. 30-33. 7. Ryoo, C.S., Kim, T., Jang, L.C.(2007). Some relationships between the analogs of Euler numbers and polynomials, Journal of Inequalities and Applications, v.2007, ID 86052, pp. 1-22. 8. Ryoo, C.S.(2014). Note on the second kind Barnes’ type multiple q-Euler polynomials, Journal of Computational Analysis and Applications, v.16, pp. 246-250. 9. Ryoo, C.S.(2015). On the second kind Barnes-type multiple twisted zeta function and twisted Euler polynomials, Journal of Computational Analysis and Applications, v.18, pp. 423-429. 10. Ryoo, C.S.(2017). On the (p, q)-analogue of Euler zeta function, J. Appl. Math. & Informatics v. 35, pp. 303-311. 11. Ryoo, C.S.(2019). Some symmetric identities for (p, q)-Euler zeta function, J. Computational Analysis and Applications v. 27, pp. 361-366. 12. Ryoo, C.S.(2020). Symmetric identities for the second kind q-Bernoulli polynomials, Journal of Computational Analysis and Applications, v.28, pp. 654-659. 13. Ryoo, C.S.(2020). On the second kind twisted q-Euler numbers and polynomials of higher order, Journal of Computational Analysis and Applications, v.28, pp. 679-684. 14. Ryoo, C.S.(2020). Symmetric identities for Dirichlet-type multiple twisted (h, q)-l-function and higher-order generalized twisted (h, q)-Euler polynomials, Journal of Computational Analysis and Applications, v.28, pp. 537-542.

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On generalized degenerate twisted (h, q)-tangent numbers and polynomials C. S. RYOO Department of Mathematics, Hannam University, Daejeon 34430, Korea

Abstract : We introduced the generalized twisted (h, q)-tangent numbers and polynomials. In this paper, our goal is to give generating functions of the generalized degenerate twisted (h, q)-tangent numbers and polynomials. We also obtain some explicit formulas for generalized degenerate twisted (h, q)-tangent numbers and polynomials. Key words : Generalized tangent numbers and polynomials, degenerate generalized twisted (h, q)tangent numbers and polynomials. AMS Mathematics Subject Classification : 11B68, 11S40, 11S80. 1. Introduction Many mathematicians have studied in the area of the Bernoulli numbers and polynomials, Euler numbers and polynomials, Genocchi numbers and polynomials, tangent numbers and polynomials(see [1-16]). In [2], L. Carlitz introduced the degenerate Bernoulli polynomials. Recently, Feng Qi et al.[3] studied the partially degenerate Bernoull polynomials of the first kind in p-adic field. In this paper, we obtain some interesting properties for generalized degenerate tangent numbers and polynomials. Throughout this paper we use the following notations. Let p be a fixed odd prime number. By Zp we denote the ring of p-adic rational integers, Q denotes the field of rational numbers, Qp denotes the field of p-adic rational numbers, C denotes the complex number field, and Cp denotes the completion of algebraic closure of Qp , N denotes the set of natural numbers and Z+ = N ∪ {0}. Let r be a positive integer, and let ζ be rth root of 1. Let χ be Dirichlet’s character with conductor d ∈ N with d ≡ 1(mod 2). Then the generalized twisted (h, q)-tangent numbers (h)

associated with associated with χ, Tn,χ,q,ζ , are defined by the following generating function 2

∑d−1

∞ ∑ χ(a)(−1)a ζ a q ha e2at tn (h) = Tn,χ,q,ζ . d hd 2dt ζ q e +1 n! n=0

a=0

(1.1) (h)

We now consider the generalized twisted (h, q)-tangent polynomials associated with χ, Tn,χ,q,ζ (x), are also defined by ) ( ∑ ∞ d−1 ∑ 2 a=0 χ(a)(−1)a ζ a q ha e2at tn (h) xt e = Tn,χ,q,ζ (x) . (1.2) d hd 2dt ζ q e +1 n! n=0 When χ = χ0 , above (1.1) and (1.2) will become the corresponding definitions of the twisted (h, q)(h) (h) tangent numbers Tn,q,w and polynomials Tn,q,w (x). If q → 1, above (1.1) and (1.2) will become the corresponding definitions of the generalized twisted tangent numbers Tn,χ,w and polynomials Tn,χ,w (x). We recall that the classical Stirling numbers of the first kind S1 (n, k) and S2 (n, k) are defined by the relations(see [7]) (x)n =

n ∑

S1 (n, k)xk and xn =

k=0

n ∑

S2 (n, k)(x)k ,

k=0

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respectively. Here (x)n = x(x − 1) · · · (x − n + 1) denotes the falling factorial polynomial of order n. The numbers S2 (n, m) also admit a representation in terms of a generating function ∞ ∑

(et − 1)m tn = . n! m!

(1.3)

tn (log(1 + t))m = . n! m!

(1.3)

S2 (n, m)

n=m

We also have

∞ ∑

S1 (n, m)

n=m

The generalized falling factorial (x|λ)n with increment λ is defined by (x|λ)n =

n−1 ∏

(x − λk)

(1.5)

k=0

for positive integer n, with the convention (x|λ)0 = 1. We also need the binomial theorem: for a variable x, ∞ ∑ tn (1 + λt)x/λ = (1.6) (x|λ)n . n! n=0 2. On the generalized degenerate twisted (h, q)-tangent polynomials In this section, we define the generalized degenerate twisted (h, q)-tangent numbers and polynomials, and we obtain explicit formulas for them. Let χ be Dirichlet’s character with conductor d ∈ N with d ≡ 1(mod 2), and let ζ be rth root of 1. For h ∈ Z, the generalized degenerate (h)

twisted (h, q)-tangent polynomials associated with associated with χ, Tn,χ,q,ζ (x|λ), are defined by the following generating function ∑d−1 ∞ ∑ 2 a=0 (−1)a χ(a)ζ a q ha (1 + λt)2a/λ tn (h) x/λ (1 + λt) = (2.1) Tn,χ,q,ζ (x|λ) d dh 2/λ n! ζ q (1 + λt) +1 n=0 and their values at x = 0 are called the generalized degenerate twisted (h, q)-tangent numbers and (h)

denoted Tn,χ,q,ζ (λ). From (2.1) and (1.2), we note that ∞ ∑ n=0

(h)

lim Tn,χ,q,ζ (x|λ)

λ→0

∑d−1 2 a=0 (−1)a χ(a)ζ a q ha (1 + λt)2a/λ tn = lim (1 + λt)x/λ n! λ→0 ζ d q dh (1 + λt)2/λ + 1 ( ∑ ) d−1 2 a=0 χ(a)(−1)a ζ a q ha e2at = ext ζ d q hd e2dt + 1 =

∞ ∑

(h)

Tn,χ,q,ζ (x)

n=0

tn . n!

Thus, we get (h)

(h)

lim Tn,χ,q,ζ (x|λ) = Tn,χ,q,ζ (x), (n ≥ 0).

λ→0

From (2.1) and (1.6), we have ∞ ∑ n=0

(h)

Tn,χ,q,ζ (x|λ)

∑d−1 2 a=0 (−1)a χ(a)ζ a q ha (1 + λt)2a/λ tn = (1 + λt)x/λ n! ζ d q dh (1 + λt)2/λ + 1 ( ∞ )( ∞ ) ∑ ∑ (h) tl tm = (x|λ)l Tn,χ,q,ζ (λ) m! l! m=0 l=0 ( n ( ) ) ∞ ∑ ∑ n (h) tn Tl,χ,q,ζ (λ)(x|λ)n−l . = l n! n=0

(2.2)

l=0

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tm in the above equation, we have the following theorem: m! Theorem 1. For n ≥ 0, we have

By comparing coefficients of

n ( ) ∑ n (h) T (λ)(x|λ)n−l . l l,χ,q,ζ

(h)

Tn,χ,q,ζ (x|λ) =

l=0

For χ = χ0 , we have ∞ ∑

(h)

Tn,χ,q,ζ (x|λ)

n=0

tn 2 = h (1 + λt)x/λ n! ζq (1 + λt)2/λ + 1 =

∞ ∑

t (h) Tn,q,ζ (x|λ)

m!

m=0

(2.3)

m

.

Theorem 2. For n ≥ 0 and χ = χ0 , we have (h)

(h)

Tn,χ,q,ζ (x|λ) = Tn,q,ζ (x|λ).

For d ∈ N with d ≡ 1(mod2), we have ∞ ∑

(h)

Tn,χ,q,ζ (x|λ)

n=0

2 tn = n!

∑d−1

a a ha 2a/λ a=0 (−1) χ(a)ζ q (1 + λt) (1 ζ d q dh (1 + λt)2d/λ + 1

+ λt)x/λ

∑ 2 x/λ (1 + λt) (−1)l χ(l)(1 + λt)2l/λ ζq h (1 + λt)2d/λ + 1 l=0 ( d−1 ( )) n ∞ ∑ ∑ 2l + x λ t (h) = dn (−1)l χ(l)Tn,qd ,ζ d . d d n! n=0 d−1

=

(2.4)

l=0

m

t in the above equation, we have the following theorem: m! Theorem 3. Let χ be Dirichlet’s character with conductor d ∈ N with d ≡ 1(mod 2). Then

By comparing coefficients of

we have

( ) d−1 ∑ 2l + x λ (h) l (1) =d (−1) χ(l)Tn,qd ,ζ d , d d l=0 ( ) d−1 ∑ 2l + x (h) (h) l n (2) Tn,χ,q,ζ (λ) = d (−1) χ(l)Tn,qd ,ζ d . d (h) Tn,χ,q,ζ (x|λ)

n

l=0

For m ∈ Z+ , we obtain we can derive the following relation: ∞ ∑

(h)

ζ d q hd Tm,χ,q,ζ (2d|λ)

m=0

=2

∞ ∑ tm tm (h) + Tm,χ,q,ζ (2d|λ) m! m=0 m!

d−1 ∑ (−1)l χ(l)ζ l q hl (1 + λt)2l/λ l=0

=

∞ ∑

(

m=0

By comparing of the coefficients

d−1 ∑ 2 (−1)n−1−l χ(l)ζ l q hl (2l|λ)m l=0 m

t m!

(2.5) )

tm . m!

on the both sides of (2.5), we have the following theorem.

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Theorem 4. For m ∈ Z+ , we have (h)

(h)

ζ d q hd Tm,χ,q,ζ (2d|λ) + Tm,χ,q,ζ (λ) = 2

d−1 ∑ (−1)l χ(l)ζ l q hl (2l|λ)m . l=0

From (2.1), we have ∞ ∑

(h) Tn,χ,q,ζ (x

n=0

2 tn + y|λ) = n! =

2

∑d−1

a a ha 2a/λ a=0 (−1) χ(a)ζ q (1 + λt) (1 ζ d q dh (1 + λt)2d/λ + 1

+ λt)(x+y)/λ

∑d−1

(

a a ha (2a+x)/λ a=0 (−1) χ(a)ζ q (1 + λt) (1 d dh 2d/λ ζ q (1 + λt) +1

) tn = (y|λ)n n! n! n=0 n=0 ) ( n ( ) ∞ ∑ ∑ n (h) tn = . Tl,χ,q,ζ (x|λ)(y|λ)n−l n! l n=0 ∞ ∑

tn (h) Tn,χ,q,ζ (x|λ)

)(

+ λt)y/λ (2.6)

∞ ∑

l=0

Therefore, by (2.6), we have the following theorem. Theorem 5. For n ∈ Z+ , we have (h)

Tm,χ,q,ζ (x + y|λ) =

n ( ) ∑ n (h) T (x|λ)(y|λ)n−k . k kχ,q,ζ

k=0 (h)

From Theorem 5, we note that Tn,χ,q,ζ (x|λ) is a Sheffer sequence. By replacing t by 2

eλt − 1 in (2.1), we obtain λ

∑d−1

( λt )n ∞ χ(a)(−1)a ζ a q ha e2at xt ∑ (h) e −1 1 e = (x|λ) T n,χ,q,ζ ζ d q hd e2dt + 1 λ n! n=0

a=0

∞ ∑

∞ ∑

tm m! m=n n=0 ) ( ∞ m ∑ ∑ tm (h) = . Tn,χ,q,ζ (x|λ)λm−n S2 (m, n) m! m=0 n=0 =

(h) Tn,χ,q,ζ (x|λ)λ−n

S2 (m, n)λm

(2.7)

Thus, by (2.7) and (1.2), we have the following theorem. Theorem 6. For n ∈ Z+ , we have (h)

Tm,χ,q,ζ (x) =

m ∑

(h)

λm−n Tn,χ,q,ζ (x|λ)S2 (m, n).

n=0

By replacing t by log(1 + λt)1/λ in (1.2), we have ∑d−1 ∞ )n 1 ( ∑ 2 a=0 (−1)a χ(a)ζ a q ha (1 + λt)(2a+x)/λ (h) Tn,χ,q,ζ (x) log(1 + λt)1/λ = n! ζ d q hd (1 + λt)2d/λ + 1 n=0 ∞ ∑

=

m=0

and ∞ ∑ n=0

(h) Tn,χ,q,ζ (x)

(

tm (h) Tm,χ,q,ζ (x|λ) , m!

(m ) ∞ )n 1 ∑ ∑ (h) tm = Tn,χ,q,ζ (x)λm−n S1 (m, n) . log(1 + λt)1/λ n! m=0 n=0 m!

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(2.8)

(2.9)

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Thus, by (2.8) and (2.9), we have the following theorem. Theorem 8. For n ∈ Z+ , we have (h)

Tm,χ,q,ζ (x|λ) =

m ∑

(h)

Tn,χ,q,ζ (x)λm−n S1 (m, n).

n=0

Acknowledgment Acknowledgement: This work was supported by the National Research Foundation of Korea(NRF) grant funded by the Korea government(MEST) (No. 2017R1A2B4006092).

REFERENCES 1. Carlitz, L.(1979). Degenerate Stirling, Bernoulli and Eulerian numbers, Utilitas Math., v.15, pp. 51-88. 2. Carlitz, L.(1956). A degenerate Staudt-Clausen theorem, Arch. Math. (Basel), v.7, pp. 28-33. 3. Qi, F., Dolgy, D.V., Kim, T., Ryoo, C. S.(2015). On the partially degenerate Bernoulli polynomials of the first kind, Global Journal of Pure and Applied Mathematics, v.11, pp. 2407-2412. 4. Kim, T.(2015). Barnes’ type multiple degenerate Bernoulli and Euler polynomials, Appl. Math. Comput. v.258, pp. 556-564. 5. Ozden, H., Cangul, I.N., Simsek, Y.(2009). Remarks on q-Bernoulli numbers associated with Daehee numbers, Adv. Stud. Contemp. Math., v.18, pp. 41-48. 6. Ryoo, C.S.(2013). A Note on the tangent numbers and polynomials, Adv. Studies Theor. Phys., v.7, pp. 447 - 454. 7. Young, P.T.(2008).

Degenerate Bernoulli polynomials, generalized factorial sums, and their

applications, Journal of Number Theory, v.128, pp. 738-758. 8. Kim, T.(2008). Euler numbers and polynomials associated with zeta function, Abstract and Applied Analysis, Art. ID 581582. 9. Liu, G.(2006). Congruences for higher-order Euler numbers, Proc. Japan Acad., v.82 A, pp. 30-33. 10. Ryoo, C.S., Kim, T., Jang, L.C.(2007). Some relationships between the analogs of Euler numbers and polynomials, Journal of Inequalities and Applications, v.2007, ID 86052, pp. 1-22. 11. Ryoo, C.S.(2014). Note on the second kind Barnes’ type multiple q-Euler polynomials, Journal of Computational Analysis and Applications, v.16, pp. 246-250. 12. Ryoo, C.S.(2015). On the second kind Barnes-type multiple twisted zeta function and twisted Euler polynomials, Journal of Computational Analysis and Applications, v.18, pp. 423-429. 13. Ryoo, C.S.(2019). Some symmetric identities for (p, q)-Euler zeta function, J. Computational Analysis and Applications v.27, pp. 361-366.

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14. Ryoo, C.S.(2020). Symmetric identities for the second kind q-Bernoulli polynomials, Journal of Computational Analysis and Applications, v.28, pp. 654-659. 15. Ryoo, C.S.(2020). On the second kind twisted q-Euler numbers and polynomials of higher order, Journal of Computational Analysis and Applications, v.28, pp. 679-684. 16. Ryoo, C.S.(2020). Symmetric identities for Dirichlet-type multiple twisted (h, q)-l-function and higher-order generalized twisted (h, q)-Euler polynomials, Journal of Computational Analysis and Applications, v.28, pp. 537-542.

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New Oscillation Criteria of First Order Neutral Delay Di¤erence Equations of Emden–Fowler Type S. H. Saker1 ; and M. A. Arahet2 1

Department of Mathematics, Faculty of Science, Mansoura University, Mansoura 35516, Egypt, E-mail: [email protected]. 2

Department of Mathematics, Faculty of Science and Arts, Amran University, Yamen E-mail: [email protected].

Abstract In this paper, we will establish some new su¢ cient condition for oscillation of solutions of a certain class of …rst-order neutral delay di¤erence equations of the form (xn pn xn 1 ) + qn xn = 0; where is a quotient of odd positive integers. We will consider the sublinear and super linear cases. The results will be obtained by using the oscillation theorems of second order delay di¤erence equations. 2010 Mathematics Subject Classi…cation: 34C10, 34K11, 34B05. Keywords and phrases: Neutral di¤erence equation, oscillation, Riccati technique.

1

Introduction

In recent decades there has been much research activity concerning oscillation and nonoscillation of …rst and second order delay and neutral delay di¤erence equations, we refer the reader to the papers [1, 2, 3, 4, 5, 6, 7, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23] and the references cited therein. In the following, we recall some results of …rst order neutral delay di¤erence equations of sublinear and super linear types that motivate the contents of this paper. Xiaoyan Lin in [12] studied the oscillatory behavior of solutions of the neutral di¤erence equations with nonlinear neutral term of the form (1.1)

xn

pn xn

+ qn xn

= 0; for n 2 Nn0 ;

where and are quotient of odd positive integers, and are nonnegative integers and fpn g and fqn g are two sequences of nonnegative real numbers. The authors obtained necessary and su¢ cient conditions for existence of oscillatory solutions and studied the two cases when 0 < < 1 and when > 1: As usual, a nontrivial solution xn of (1.1) is called nonoscillatory if it eventually positive or eventually negative, otherwise it is called oscillatory and is the forward di¤erence operator de…ned by xn = xn+1 xn and Ni = fi + 1; i + 2; :::g: Lalli [11] established several su¢ cient conditions for oscillation of the equation (1.2)

(xn + pxn

k)

+ qn f (xn

) = Fn ; n

n0 ;

1

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where = 1, p is a nonnegative real number, k 2 N = f1; 2; :::g, is a sequence of nonnegative integers with limn!1 n = 1; and fFn g, fqn g are sequences of real numbers and f is a real valued function satisfying xf (x) > 0 for x 6= 0. El-Morshedy et al. [6] considered the equation (1.3)

g (xn + pn x

n

) + f (n; x n ) = 0;

where 0 pn < p < 1, n and n are sequences of integers such that limn!1 n = limn!1 = 1 and n+1 > n for all n n0 . They established several su¢ cient conditions for oscillation when the function f satis…es the condition f (n; x) h (x)

qn ; x 6= 0 and n

n0 ;

where qn 0 for n n0 , h 2 C (R,R) and xh(x) > 0 for all x 6= 0. Recently Murugesan and Suganthi [13] discussed the oscillatory behavior of all solutions of the …rst order nonlinear neutral delay di¤erence equation [

(rn (an xn

p n xn

))] + qn xn

= 0;

where rn and an are sequences of positive real numbers pn and qn are sequences of nonnegative real numbers, and are positive integers. Following this trend in this paper, we will consider the …rst order neutral delay di¤erence equation (1.4)

(xn

pn xn

1)

+ qn xn

= 0; for n 2 Nn0 ;

Our aim in this paper is to establish some new su¢ cient conditions for oscillation of (1.4) by using a new technique when 0 < pn p 1 and we will consider the sublinear and the super linear cases: The new technique depends on the application of an invariant substitution which transforms the …rst nonlinear neutral di¤erence equation to a second nonlinear di¤erence equation. This allows us to obtain several su¢ cient conditions for oscillation of (1.4) by employing the oscillation conditions of second order delay di¤erence equations by using the Riccati technique.

2

Main results

In this section, we prove the main results but before we do this, we apply an invariant substitution which transforms the …rst order neutral equation to a non-neutral second order di¤erence equations. This substitution is given by (2.1)

yn+1 = xn

n Y 1 ; p i=1 i

n Y

where

pi = O (n) ;

i=1

This gives us that (2.2)

xn = yn+1

n Y

pi ;

xn

i=1

1

= yn

n Y1

pi ;

and xn

i=1

= yn

+1

nY

pi :

i=1

From (2.2), we have (2.3)

xn

p n xn

1

=

yn

n Y

pi :

i=1

2

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Substituting (2.3) into (1.4), we obtain ! n nY Y (2.4) yn pi + qn pi yn i=1

Setting dn = (2.5)

Yn

i=1

+1

= 0:

i=1

pi ; and Qn = qn dn

then (2.4) becomes

(dn yn ) + Qn yn

(

1)

= 0; n 2 N0 :

In this section, we intend to use the Riccati transformation technique for obtaining several new oscillation criteria for (1.4). First we state some fundamental lemmas for second order di¤erence equations that will be used in the proofs of the main results (see [15]). Lemma 2.1 Assume that pn is a real sequence with 0 < pn Furthermore assume that

p < 1 for all n 2 N:

1 X 1 = 1: d n=1 n

(2.6)

Let y be a positive solution of (2.5): Then (I ): y(n) 0; y(n) n y(n) for n N , (II ): y is nondecreasing, while y(n)=n is nonincreasing for n

N:

Lemma 2.2 Assume that pn is a real sequence with 0 < pn p < 1 for all n 2 N: Furthermore assume that (2.6) holds. If yn be a nonoscillatory solution of (2.5) such that yn 0; yn 0, then limn!1 yn = 0 and hence (2.7)

lim

n!1

xn = 0; dn

where xn is a solution of (1.4). Throughout this paper, we will assume that the real sequences pn ; qn are nonnegative, is a quotient of odd positive integers, is a nonnegative integer. Now, we state and prove the su¢ cient conditions which ensure that each solution of equation (1.4) is oscillatory or satis…es (2.7). We start with the case when 0 < 1: Theorem 2.3 Assume that (H1 ) holds and exists a positive sequence n such that, " n X di +1 1 (2.8) lim sup Q i i n!1

where dn = 0< 1:

Yn

i=1

dn

(i + 2

)

1

(

i

i=n0

pi and Qn = qn dn

0: Furthermore, assume that there 2 i)

#

= 1;

: Then every solution of (1.4) oscillates for all

Proof. Assume to the contrary that xn be a nonoscillatory solution of (1.4) such that xn 1 , xn , xn > 0 for all large n n1 > n0 su¢ ciently large. We shall consider only this case, since the substitution yn = xn transforms equation (1.4) into an equation of the same form. From (2.1) we see that yn is a positive solution of (2.5) such that yn > 0 and yn +1 > 0 for n > n1 > n0 su¢ ciently large. From equation (2.5), we have (2.9)

(dn yn ) =

Qn yn

+1

0; n

n1 ;

3

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and then dn yn is an eventually nonincreasing sequence. We …rst show that dn yn 0 for n n0 : In fact, if there exists an integer n1 n0 such that dn1 yn1 = c < 0 then (2.9) implies that dn yn c for n n1 that is yn c=dn ; and hence (2.10)

yn

n X1

yn1 + c

i=n1

1 ! di

1; as n ! 1;

which contradicts the fact that yn > 0 for n n0 then dn yn we can prove that 2 yn > 0 for n n1 : Therefore we have (2.11)

yn > 0;

yn

0; and

+1

dn+1

2

yn

0: Also since

0; for n

dn

0,

n1 :

From (2.9) and (2.11) (2.12)

dn

+1

yn

(yn+1 ) and yn

yn

+1

:

De…ning the sequence un by the Riccati substitution (2.13)

un =

n

dn yn ; yn +1

for n > n1 :

This implies that un > 0; and n

un = dn+1 yn+1

yn

+

n

+1

(dn yn ) : yn +1

Hence (2.14)

"

un = dn+1 yn+1

yn

n

yn

yn

n

+1 +1 yn

#

+1

+2

+

(dn yn ) : yn +1

n

From this, (2.5) and (2.14) we see that (2.15)

n

un

dn+1 yn+1 n yn yn +2 yn +1

un+1

n+1

+1

n Qn :

From (2.5) and (2.14), we have (2.16)

un

n Qn

n

+

dn+1 yn+1

un+1

n+1

yn

n

yn2

+1

:

+2

By using the inequality (see [8]), (2.17)

x

y

x

1

(x

y) ; for all x 6= y > 0 where 0
0 for all large n n1 > n0 su¢ ciently large. We shall consider only this case, since the substitution yn = xn transforms equation (1.4) into an equation of the same form. As in the proof of Theorem 2.3, we have by (2.6) that (2.26)

yn > 0;

yn

0;

(dn ( yn ))

0; n

n1 :

De…ne the sequence un by (2.27)

un :=

n

dn yn : yn

Then un > 0; and (2.28)

un = dn+1 yn+1

n

yn

+

n

(dn yn ) : yn

In view of (2.5), (2.28), we have (2.29)

un

n qn

+

n

un+1

n+1

n dn+1

yn+1 yn yn+1 yn

:

From (2.26), we see that (2.30)

dn

yn

dn+1 yn+1 ; and yn+1

yn

:

6

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Substituting (2.30) into (2.29), we have (2.31)

un

n qn

n

+

n dn+1

un+1

yn+1 yn

:

2

yn+1

n+1

Now, by using the inequality y > 21

x

(x

y) ; for all x > y > 0 and

yn

> 21

> 1;

we …nd that (2.32)

yn

= yn+1

(yn+1

) = 21

yn

( yn

) :

Substituting (2.32) into (2.31), we have (2.33)

un

n qn

n

+

un+1

21

un+1

21

yn+1 ( yn

n dn+1

yn+1

n+1

) 2

:

From (2.30) and (2.33), we obtain +1

(2.34)

un

n

n qn +

n+1

(dn+1 ) n (dn )

+1

( yn+1 )

2

yn+1

:

Hence, +1

(2.35)

un

n qn

n

+

n+1

2

21

(dn+1 ) (dn )

un+1

n

( yn+1 ) 2

yn+1

( yn+1 )

1

:

From the de…nition of un , we get that (2.36)

un

n

n qn +

21

un+1

n+1

n 2

n+1

(dn+1 ) (dn )

1

u2n+1

1:

( yn+1 )

Since fdn ( yn )g is a positive and nonincreasing sequence, there exists a n2 ciently large such that dn ( yn ) 1=M for some positive constant M and n hence by (2.26), we have 1 1 : 1 > (M dn+1 ) ( yn+1 )

n1 su¢ n1 , and

Substituting the last inequality into (2.36), we obtain (2.37)

un

n

n qn +

1

M 2

un+1

n+1

2

n

(dn+1 )

2

2

1 (dn

n+1

)

u2n+1 ;

so that un

n qn

2q 4
1 is a constant. From Theorem 2.6 we have the following result. Corollary 2.7 Assume that all the assumptions of Theorem 2.6 hold, except the condition (2.25) is replaced by # " n X (ds ) ((s + 1) s )2 (2.39) lim sup s qs = 1: 1 2 2 n!1 23 (M ) (ds+1 ) s s=n0 Then, every solution of (1.4) oscillates for all

1.

As a variant of the Riccati transformation technique used above, we will derive some oscillation criterion which can be considered as a discrete analogy of the Philos condition for oscillation of second order di¤erential equation by introducing the following class of sequences that will be used in this chapter and later: Let $0 = f(m; n) : m > n

n0 g; $ = f(m; n) : m

The double sequence Hm;n 2 if: (I): H(m; m) = 0 on $; (II): H(m; n) > 0 on $0 ; (III): 2 Hm;n = Hm;n+1 Hm;n sequence hm;n such that hm;n =

0 for m

n

n

n0 g:

0; and there exists a double

H p2 m;n ; for m > n Hm;n

0:

Theorem 2.8 Assume that (2.6) hold. Let f n g1 n=1 be a positive sequence and Hm;n 2 : If " # m 1 2 p 1 X n (2.40) lim sup Hm;n n qn Bn hm;n Hm;n = 1; m!1 Hm;0 n=0 n+1 where Bn :=

(dn 23

M

2 n+1 2 1 (d n+1 )

Then every solution of (1.4) oscillates for all

)

:

2 n

1.

8

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Proof. We proceed as in the proof of Theorem 2.6, we may assume that (1.4) has a nonoscillatory solution xn such that xn > 0: As in the proof of Theorem 2.6 we get that (2.26) holds. De…ne fun g by (2.27) as before, then we have un > 0 and there is some M > 0 such that (2.37) holds. For the sake of convenience, let us set n

=

21

(M )

1

(dn+1 ) (dn )

2

2

n

n

n

:

Then, we have from (2.37) that (2.41)

n qn

un +

un+1

n+1

2 2 un+1 :

n+1

Therefore, we get (2.42)

m X1

Hm;n

m X1

n qn

n=n1

Hm;n un +

n=n1

m X1

n

Hm;n

un+1

n+1

n=n1

m X1

Hm;n

n=n1

2 n un+1 2: n+1

The rest of the proof is similar to the proof of [15, Theorem 2.3.6]. As an immediate consequence of Theorem 2.8, we get the following: Corollary 2.9 Assume that all the assumptions of Theorem 2.8 hold, except that the condition (2.40) is replaced by lim sup

m!1

lim sup

m!1

1 Hm;0

m X1

n=n0

(dn (M )

1

1 Hm;0 )

m X1

Hm;n

n qn

n=n0 2 n+1 2 2

(dn+1 )

= 1; n

hm;n

n+1

n

Then every solution of (1.4) oscillates for all

1.

p Hm;n

2

< 1:

By choosing the sequence Hm;n in appropriate manners, we can derive several oscillation criteria for (1.4). For instance, let us consider the double sequence fHm;n g de…ned by 9 Hm;n = (m n) ; 1; m n 0; > = (2.43) Hm;n = log m+1 ; 1; m n 0; n+1 > ; Hm;n = (m n)( ) > 2; m n 0; where (m

n)(

)

= (m

2 (m

n)(

)

n)(m

n + 1):::(m

= (m

n

1)(

)

n+

(m

1); and

n)(

)

=

(m

n)(

1)

:

Then Hm;m = 0 for m 0 and Hm;n > 0 and 2 Hm;n 0 for m > n 0: Hence we have the following result which gives new su¢ cient conditions for the oscillation of (1.4) of Kamenev type. Corollary 2.10 Assume that all the assumptions of Theorem 2.8 hold, except that the condition (2.40) is replaced by " # m 1 2 1 X n+1 2 (2.44) lim sup (m n) n qn Vm;n = 1; m!1 m n=0 4 n 9

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where Vm;n :=

(m

n)

2

n

2

n+1

Then every solution of (1.4) oscillates for all

q

(m

n)

:

1.

Corollary 2.11 Assume that all the assumptions of Theorem 2.8 hold, except that the condition (2.40) is replaced by " # 2 m 2 X1 1 m+1 n+1 2 (2.45) lim sup log Rm;n = 1; n qn m!1 (log(m + 1)) n=0 n+1 4 n where

0

Rm;n = @

2

log

n+1

m+1 n+1

2

n n+1

Then, every solution of (1.4) oscillates for all

s

log

m+1 n+1

1.

1

A:

Corollary 2.12 Assume that all the assumptions of Theorem 2.8 hold, except that the condition (2.40) is replaced by " # m 1 2 1 X n+1 2 ( ) (m n) Un = 1; (2.46) lim sup ( ) n qn m!1 m 4 n n=0 where

2

Un :=

n

m

n+

1

Then, every solution of (1.4) oscillates for all

:

n+1

1.

In the following theorem, we consider the case when 0 < Theorem 2.13 Assume that (2.6) holds and 1 X

(2.47)

dn

n

0: If

qn = 1:

dn

n=n0

< 1:

Then every solution of (1.4) oscillates for all 0 < < 1: Proof. Proceeding as in Theorem 2.6, we assume that (1.4) has a nonoscillatory solution, say xn > 0 and xn > 0 for all n n0 . From the proof of Theorem 2.6 we know that yn > 0; then yn is nondecreasing sequence. Since dn 0 we obtain that 2 yn 0 and then yn is a nonincreasing for all n n1 n0 . Then, we have yn (n n1 ) yn n n2 2n1 + 1: Then which implies that yn 2 yn for n (2.48)

yn

n 2

n

yn

yn+1 ; f or n

2

N = n2 + :

From (2.5) and (2.48) by dividing by zn+1 = (dn yn+1 ) > 0 and summing from 2N to k, we obtain (2.49)

k X

n=2N

n 2dn

qn

k X

n=2N

(zn ) ; (zn+1 )

k

2N:

10

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Since y

z

y

1

(y

z) for

< 1 and y > z > 0;

we see that zn1

1 = zn+1

zn1

(1

)(z(n + 1))

z(n):

Substituting in (2.49), we see that k X

n=2N

n 2dn

k X

qn

zn1

n=2N


0 as n ! 1; and hence there exists n1 n0 > 0 such that yn b : Therefore from (2.52) we have (dn yn )

qn b :

The rest of the proof is similar to the proof of [15, Theorem 2.3.7] and hence is omitted. By choosing f n g1 n=1 in appropriate manners, we may obtain di¤erent oscillation criteria. For instance, let n = n for n 0 and > 1: Then we have the following oscillation conditions of all solutions of (1.4) when (2.50) holds. 11

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Corollary 2.15 Assume that all assumptions of Theorem 2.14 hold, except that the condition (2.25) is replaced by (2.39). Then, every solution of (1.4) oscillates or limn!1 xn =dn = 0 for all 1. Theorem 2.16 Assume that (2.50) and (2.51) hold: Furthermore, assume that there exists a double sequence Hm;n 2 such that (2.40) holds. Then every solution of (1.4) oscillates or limn!1 xn =dn = 0 for all 1. Indeed, suppose that fxn g is an eventually positive solution of (1.4). Then as seen in the proof of Theorem 2.3, either f xn g is eventually positive or is eventually negative. In the case when f yn g is eventually positive, we may follow the proof of Theorem 2.8 and obtain a contradiction. If f yn g is eventually negative, then we may follow the proof of Theorem 2.14 to show that fyn g converges to zero. By choosing Hm;n in appropriate manners, we can derive several oscillation criteria for (2.5) when (2.50) holds. For instance, let us consider the double sequence Hm;n de…ned again by (2.43). Hence we have the following results. Corollary 2.17 Assume that all the assumptions of Theorem 2.16 hold, except that the condition (2.40) is replaced by (2.44). Then, every solution of (1.4) oscillates or limn!1 xn =dn = 0 for all 1. Corollary 2.18 Assume that all the assumptions of Theorem 2.16 hold, except that the condition (2.40) is replaced by( 2.45) or (2.46). Then, every solution of (1.4) oscillates or limn!1 xn =dn = 0 for all 1: Theorem 2.19 Assume that (2.50) and (2.47) hold. Let f n g1 n=1 such that (2.51) holds. Then every solution of (1.4) oscillates or limn!1 xn =dn = 0 for all 0 < < 1: Indeed, suppose that fxn g is an eventually positive solution of (1.4). Then as seen in the proof of Theorem 2.6, either f yn g is eventually positive or is eventually negative. In the case when f yn g is eventually positive , we may follow the proof of Theorem 2.13 and obtain a contradiction. If f yn g is eventually negative, then we may follow the proof of Theorem 2.14 to show that fxn =dn g converges to zero. From Theorem 2.14 if n = 1; we see that the Riccati inequality associated with the equation (1.4) is given by (2.53)

un +

n qn

+

1 2 u an n+1

0;

where (2.54)

An =

2

1

(dn

(M )

1

(dn+1 )

) 2

2

> 0;

for every positive constant M > 0: Using the inequality (2.53) and proceeding as in the proof [15, Theorem 2.3.8], we can prove the following Hille and Nehari type results. Theorem 2.20 Assume that (H1 ) holds and lim inf n!1

or lim inf n!1

n An

1 X

dn

0. Furthermore, assume that

q(s) >

n+1

1 ; 4

1 n 1 n X 1 X s2 qs + lim inf qs > 4: n!1 n An n+1 An N

Then every solution of (1.4) is oscillatory.

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References [1] R. P. Agarwal, M. M. S. Manuel and E. Thandapani, Oscillatory and nonoscillatory behavior of second order neutral delay di¤erence equations, Math. Comput. Modelling 24 (1996), 5–11. [2] R. P. Agarwal, M. M. S. Manuel, and E. Thandapani, Oscillatory and nonoscillatory behavior of second order neutral delay di¤erence equations II, Appl. Math. Lett. 10 (1997), 103–109. [3] A. Bezubik, M. Migda, M. Nockowska and E. Schmeidel, Trichotomy of nonoscillatory solutions to second-order neutral di¤erence equation with quasi-di¤erence, Adv. Di¤. Eqns. (2015). [4] M. Budericenic, Oscillation of a second order neutral di¤erence equation, Bull Cl. Sci. Math. Nat. Sci. Math. 22 (1994), 1–8. [5] M. P. Chen, B. S. Lalli and J. S. Yu, Oscillation in neutral delay di¤erence equations with variable coe¢ cients, Comp. Math. Applic. 29 (3), 5-12, (1995). [6] H. A. El-Morshedy and S. R. Grace, Oscillation of some nonlinear di¤erence equations, J. Math. Anal. Appl. 281 (2003) 10–21. [7] I. Gyori and G. Ladas, Oscillation Theory of Delay Di¤ erential Equations with Applications, Clarendon Press, Oxford, (1991). [8] G. H. Hardy, J. E. Littlewood and G. Polya, Inequalities, 2nd Ed. Cambridge Univ. Press 1952. [9] H. J. Li and C. C. Yeh, Oscillation criteria for second order neutral delay di¤erence equations, Comp. Math. Appl. 36 (1998), 123–132. [10] B. S. Lalli, B. G. Zhang and J. Z. Li, On the oscillation of solutions and existence of positive solutions of neutral di¤erence equations, J. Math. Anal. Appl. 158 (1991), 213-233. [11] B. S. Lalli, Oscillation theorems for neutral di¤erence equations, Comp. Math. Appl. 28, (1994) 191-202. [12] X. Lin, Oscillation of solutions of neutral di¤erence equations with a nonlinear neutral term, Comp. Math. Appl. 52 (2006) 439-448. [13] A. Murugesan, and R. Suganthi, Oscillation criteria for …rst order nonlinear neutral delay di¤erence equations with variable coe¢ cients, Inter. J. Math. Stat. Inven. 4 (2016) 35-40. [14] S. H. Saker, New oscillation criteria for second order nonlinear neutral delay di¤erence equations, Appl. Math. Comput. 142 (2003), 99–111. [15] S. H. Saker, Oscillation Theory of Delay Di¤ erential and Di¤ erence Equations Second and Third Orders, Verlag Dr. Muller (2010). [16] S. H. Saker, and S. S. Cheng, Kamenev type oscillation criteria for nonlinear di¤erence equations, Czechoslovak Math. J. 54 (2004), 955-967. [17] Y. G. Sun and S. H. Saker, Oscillation for second order nonlinear neutral delay di¤erence equations, Appl. Math. Comput. 163 (2005), 909–918. 13

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[18] A. Sternal and B. Szmanda, Asymptotic and oscillatory behavior of certain di¤erence equations, Le Mat. 51 (1996), 77–86. [19] Y. Shoukaku, On the oscillation of solutions of …rst-order di¤erence equations with delay, Comm. Math. Analysis 20 (2017), 62-67. [20] X. H. Tang and Y. J. Liu, Oscillations for nonlinear delay di¤erence equations, Tamkang J. Math. 32 (2001), 275–280. [21] X. H. Tang, Necessary and su¢ cient conditions of oscillation for a class of neutral di¤erence equations with variable coe¢ cients (in Chinese), J. Hunan University 26 (6) (1996), 20-26. [22] E. Thandapani and R. Arul, Oscillation properties of second order nonlinear neutral delay di¤erence equations, Indian J. Pure Appl. Math. 28 (1997), 1567–1571. [23] G. Zhang and Y. Gao, Oscillation Theory for Di¤ erence Equations, Publishing House of Higher Education, Beijing, (2001).

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RICCATI TECHNIQUE AND OSCILLATION OF SECOND ORDER NONLINEAR NEUTRAL DELAY DYNAMIC EQUATIONS S. H. SAKER 1 ; AND A. K. SETHI2 Abstract. In this paper, by using the Riccati technique which reduces the higher order dynamic equations to a Riccati dynamic inequality, we will establish some new su¢ cient conditions for oscillation of the second order nonlinear neutral dynamic equation (r(t)((x(t) + p(t)x( (t))) ) ) on time scales where ,

+ q(t)x ( (t)) + v(t)x ( (t)) = 0;

are quotient of odd positive integers.

Mathematics Sub ject Classi…cation(2010): 34C10, 34K11, 39A21, 34A40, 34N05. Keywords: Oscillation, nonoscillation, neutral, delay dynamic equations, time scales, neutral delay equations

1. Introduction The theory of time scales has been introduced by Stefan Hilger in [14] in 1988 in his Ph.D thesis in order to unify continuous and discrete analysis. In the last decades the subject is going fast and simultaneously extending to the other areas of research and many researchers have contributed on di¤erent aspects of this new theory, see the survey paper by Agarwal et al. [1] and the references cited therein. In the last few years, there has been an increasing interest in obtaining su¢ cient conditions for the oscillation or nonoscillation of solutions of di¤erent classes of dynamic equations on a time scale T which may be an arbitrary closed subset of real numbers R, and as special cases contains the continuous and the discrete results as well, we refer the reader to papers ([3],[6], [7], [21]) and the references cited therein. Following this trend, in this paper, we are concerned with oscillation of a certain class of nonlinear neutral delay dynamic equations of the form (1.1)

(r(t)((x(t) + p(t)x( (t))) ) ) + q(t)x ( (t)) + v(t)x ( (t)) = 0; for t 2 [t0 ; 1)T ,

where ; ; are quotient of odd positive integers, r 2 Crd ([t0 ; 1)T ; (0; 1)) and p; q 2 Crd ([t0 ; 1)T ; R+ ) with 0 p(t) < 1, q(t); v(t) 0 and , , 2 Crd ([t0 ; 1)T ; R+ ) and (t) t; (t) t; (t) t with limt!1 (t) = 1 = lim (t) = 1 = limt!1 (t). By a t!1

1 solution of (1.1), we mean a nontrivial real-valued function x(t) 2 Crd ([Tx ; 1); R), Tx t0 1 which has the properties that r(z ) ) 2 Crd ([Tx ; 1); R) such that x(t) satis…es (1.1) for all [Tx ; 1)T .

We mention here that the neutral delay di¤erential equations appear in modelling of the networks containing lossless transmission lines (as in high-speed computers where the lossless transmission lines are used to interconnect switching circuits), in the study of vibrating masses attached to an elastic bar, as the Euler equation in some variational problems, theory of automatic control and in neuromechanical systems in which inertia plays an important role, we refer the reader to the papers by Boe and Chang [4], Brayton and Willoughby [8] and to the books by Driver [9], Hale [13] and Popov [16] and reference cited therein. For more details of time scale analysis we refer the reader to the two books by Bohner and Peterson [5], [6] which summarize and organize much of the time scale calculus. Throughout the paper, we will denote the time scale by the symbol T. For example, the real numbers R, 1

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2

the integers Z and the natural numbers N are time scales. For t 2 T, we de…ne the forward jump operator : T ! T by (t) := inffs 2 T : s > tg. A time-scale T equipped with the order topology is metrizable and is a K space; i.e. it is a union of at most countably many compact sets. The metric on T which generates the order topology is given by d(r; s) := j (r; s)j ; where (:) = (:; ) for a …xed 2 T is de…ned as follows: The mapping : T ! R+ = [0; 1) such that (t) := (t) t is called graininess. When T = R, then (t) = t and (t) 0 for all t 2 T. If T = N, then (t) = t+1 and (t) 1 for all t 2 T. The backward jump operator : T ! T is de…ned by (t) := supfs 2 T : s < tg: The mapping : : T ! R+ (t) is called the backward graininess. If 0 such that (t) = t (t) > t, we say that t is right-scattered , while if (t) < t, we say that t is left-scattered. Also, if t < sup T and (t) = t, then t is called right-dense, and if t > inf T and (t) = t, then t is called left-dense. A function f : T ! R is called right-dense continuous (rd continuous) if it is continuous at right-dense points in T and its left-sided limits exist (…nite) at left-dense points in T. For a function f : T ! R, we de…ne the derivative f as follows: Let t 2 T. If there exists a number 2 R such that for all " > 0 there exists a neighborhood U of t with jf ( (t))

f (s)

( (t)

s)j

"j (t)

sj;

for all s 2 U , then f is said to be di¤erentiable at t, and we call t and denote it by f (t). For example, if T = R, then 0

f (t) = f (t) = lim

f (t +

t!0

t) t

f (t)

the delta derivative of f at

, for all t 2 T:

If T = N, then f (t) = f (t + 1) f (t) for all t 2 T. For a function f : T ! R (the range R of f may be actually replaced by any Banach space) the (delta) derivative is de…ned by f (t) =

f ( (t)) (t)

f (t) ; t

if f is continuous at t and t is right–scattered. If t is not right–scattered then the derivative is de…ned by f ( (t)) f (s) f (t) f (s) f (t) = lim = lim ; s!t t!1 t s t s provided this limit exists. A function f : [a; b] ! R is said to be right–dense continuous (rd continuous) if it is right continuous at each right–dense point and there exists a …nite left limit at all left–dense points, and f is said to be di¤erentiable if its derivative exists. The space of rd continuous functions is denoted by Cr (T, R). A useful formula is f =f+ f ;

wheref := f

:

A time scale T is said to be regular if the following two conditions are satis…ed simultaneously: (a): For all t 2 T, ( (t)) = t; (b): For all t 2 T, ( (t)) = t: Remark 1.1. If T is a regular time scale, then both operators and are invertible with and 1 = .

1

=

The following formulae give the product and quotient rules for the derivative of the product f g and the quotient f =g (where gg 6= 0) of two di¤erentiable function f and g: Assume f ; g : T ! R are delta di¤erentiable at t 2 T, then (1.2)

(f g)

(1.3)

f g

= f g+f g =

= fg + f g ;

f g fg : gg

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3

The chain rule formula that we will use in this paper is (1.4)

Z1

(x (t)) =

[hx + (1

1

h)x]

dhx (t);

0

which is a simple consequence of Keller’s chain rule [5, Theorem 1.90]. Note that when T = R, we have Z b Z b 0 (t) = t; (t) = 0; f (t) = f (t); f (t) t = f (t)dt: a

a

When T = Z, we have

(t) = t + 1;

(t) = 1; f (t) =

Z

f (t);

b

f (t) t =

a

f (t) =

f (t))

;

Z

b

b

f (t) t =

a

f (t):

t=a

When T =hZ, h > 0; we have (t) = t + h; (t) = h; (f (t + h) h f (t) = h

b 1 X

a

h

h X

f (a + kh)h:

k=0

When T = ft : t = q k , k 2 N0 , q > 1g, we have (t) = qt; (t) = (q 1)t; Z 1 1 X (f (q t) f (t)) f (t) = q f (t) = ; f (t) t = f (q k ) (q k ): (q 1) t t0 k=0 p 2 2 2 When T = N0 = ft : t 2 Ng; we have (t) = ( t + 1) and p p p (t) = 1 + 2 t; f (t) = 0 f (t) = (f (( t + 1)2 ) f (t))=1 + 2 t: When T =PTn = ftn : n 2 Ng where (tn g is the harmonic numbers that are de…ned by t0 = 0 n and tn = k=1 k1 ; n 2 N0 ; we have 1 ; f (t) = n+1 p p When T2 =f n : n 2 Ng; we have (t) = t2 + 1; (tn ) = tn+1 ;

(tn ) =

p (t) = t2 + 1

t; f (t) =

1 f (tn )

p (f ( t2 + 1) p 2 f (t) = t2 + 1

f (t)) : t

p (f ( 3 t3 + 1) p 3 f (t) = 3 3 t +1

f (t)) : t

p p When T3 =f 3 n : n 2 Ng; we have (t) = 3 t3 + 1 and p 3 (t) = t3 + 1

t; f (t) =

= (n + 1)f (tn ):

Now, we pass to the antiderivative and the integration on time scales for detla di¤erentiable functions. For a; b 2 T; and a delta di¤erentiable function f; the Cauchy integral of f is de…ned by Z b f (t) t = f (b) f (a): a

An integration by parts formula reads Z b b (1.5) f (t)g (t) t = f (t)g(t)ja

Z

a

and in…nite integrals are de…ned as Z 1

f (t) t = lim

b!1

a

268

b

f (t)g (t) t;

a

Z

b

f (t) t:

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4

It is well known that rd continuous functions possess antiderivative. If f is rd continuous and F = f , then Z (t) f (s) s = F ( (t)) F (t) = (t)F (t) = (t)f (t): t

Note that the integration formula on a discrete time scale is de…ned by Z b X f (t) t = f (t) (t): a

t2(a;b)

We say that a solution x of (1.1) has a generalized zero at t if x (t) = 0 and has a generalized zero in (t; (t)) in case x (t) x (t) < 0 and (t) > 0. To investigate the oscillation properties of (1.1) it is proper to use the notions such as conjugacy and disconjugacy of the equation (1.1). Equation (1.1) is disconjugate on the interval [t0 ; b]T , if there is no nontrivial solution of (1.1) with two (or more) generalized zeros in [t0 ; b]T . Equation (1.1) is said to be nonoscillatory on [t0 ; 1]T if there exists c 2 [t0 ; 1]T such that this equation is disconjugate on [c; d]T for every d > c. In the opposite case (1.1) is said to be oscillatory on [t0 ; 1]T . A solution x (t) of (1.1) is said to be oscillatory if it is neither eventually positive nor eventually negative, otherwise it is oscillatory. We say that (1.1) is right disfocal (left disfocal) on [a; b]T if the solutions of (1.1) such that x (a) = 0 (x (b) = 0) have no generalized zeros in [a; b]T . In recent two decades some authors have been studied the oscillation of the second order nonlinear neutral delay dynamic equations on time scales and established several su¢ cient conditions for oscillation of some di¤erent types of equations by employing the Riccati transformation technique. For example, Saker [18] has studied the oscillation of second order neutral delay dynamic equations of Emden-fowler type of the form [a(t)(y(t) + r(t)y( (t))] + p(t)jy( (t))j signy( (t))) = 0; on time scale T, where, > 1, a(t), p(t), r(t) and (t) are real-valued function de…ned on T. Also Saker [19] studied the oscillation of the superlinear and sublinear neutral delay dynamic equations of the form [a(t)([y(t) + p(t)y( (t)))] ) ] + q(t)y ( (t))) = 0; on time scale, where > 0 is a quotient of odd positive integers. The main results has been obtained under the conditions (t) : T ! T, (t) : T ! T, (t) t, (t) t for all t 2 T and R1 1 1 t = 1, a (t) t and 0 p(t) < 1. lim (t) = lim (t) = 1, t0 a(t) t!1

t!1

Thandapani et. al [24] studied the oscillation of second order nonlinear neutral dynamic equations on time scale of the form (r(t)((y(t) + p(t)y(t

)) ) ) + q(t)y (t

) = 0; t 2 T;

where T is a time scales. They obtained their results under the conditions 1 and > 0 are quotients of odd positive integers, ; are …xed nonnegative constants such that the delay function (t) = t < t and (t) = t < t satisfying : T ! T and : T ! T for all t 2 T, q(t) and (t) real valued rd-continuous functions de…ned on T, p(t) is a positive and rd-continuous function T such that 0 p(t) < 1. Sun et al. [22] studied the oscillation of a second order quasiliniear neutral delay dynamic equation on time scales of the form (r(t)((x(t) + p(t)x( (t))) ) ) + q1 (t)x ( 1 (t)) + q2 (t)x ( 2 (t)) = 0; on time scale T, where ; ; are quotients of odd positive integers, r, p, q1 , q2 are rd-continuous function on T and r; q1 ; q2 are positive, 1 < p0 p(t) < 1, p0 > 0, the delay functions t for t 2 T and i (t) ! 1 as t ! 1, for i = 0; 1; 2 and there i : T ! T satisfying i (t) exists a function : T ! T which satisfying (t) 1 (t), (t) 2 (t), (t) ! 1 as t ! 1.

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5

Gao et al. [12] established some oscillation theorems for second order neutral functional dynamic equations on time scale of the form (r(t)((x(t) + p(t)x( (t))) ) ) + q1 (t)x ( (t)) + q2 (t)x ( (t)) = 0; where ; ; are ratios of odd positive integers by using the comparison theorems for oscillation. Sethi [26] considered the second order sublinear neutral delay dynamic equations of the form (r(t)((x(t) + p(t)x( (t))) ) ) + q(t)x ( (t)) + v(t)x ( (t)) = 0; under the assumptions: R1 1 1 (H0 ) 0 t = +1; r(t) 1 R1 1 (H1 ): 0 t < 1; r(t) where 0 < 1 is a quotient of odd positive integers, q; v ! [0; 1) and p; q; v : T ! T are rd-continuous functions and ; ; : T ! T are positive rd-continuous functions such that limt!1 (t) = 1 = lim (t) = 1 = limt!1 (t) and obtained some su¢ cient conditions for t!1 oscillation. Our aim in this paper is to establish some new su¢ cient conditions for oscillation of the equation (1.1) by employing the Riccati technique and some basic lemmas studied the behavior of nonoscillatory solutions. Our motivation of the present work has come under two ways. First is due to the work in [17] and [22] and second is due to the work in [10]. 2. Main Results In this section, we establish some su¢ cient conditions for oscillation of all solutions of (1.1) under the hypothesis (H0 ). Throughout the paper, we use the notation (2.1)

z(t) = x(t) + p(t)x( (t)):

1 Lemma 2.1. [2] Assume that (H0 ) holds and r(t) 2 Crd ([(a; 1); R+ ) such that r (t) 0. Let x(t) be an eventually positive real valued function such that (r(t)(x (t)) ) 0, for t t1 > t0 . Then x (t) > 0 and x (t) < 0 for t t1 > t0 .

Lemma 2.2. [2] Assume that the assumptions of Lemma 2.1 holds and let (t) be a positive continuous function such that (t) t and lim (t) = 1. Then there exists tl > t1 such that t!1

for each l 2 (0; 1)

x( (t)) x( (t))

Proof. Indeed, for t

t1

u( (t))

u( (t)) =

which implies that

Z

u(t1 )) =

(t) : (t)

(t)

u (s) s

( (t)

(t)))u ( (t);

(t)

u( (t)) u( (t)) On the other hand, it follows that u( (t))

l

1 + ( (t) Z

(t)))

u ( (t) : u( (t))

(t)

u (s) s

(u(t)

t1 )u ( (t)):

t1

That is for each l 2 (0; 1), there exists a tl > t1 such that l( (t))

u( (t)) ; t u ( (t))

tl :

Consequently, u( (t)) u( (t))

1 + ( (t)

(t)))

u ( (t)) u( (t))

(t) : l (t)

The proof is complete.

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6

In the following, for simplicity, we denote Z 1 l (s) a1 (t) := [q(s)(1 p( (s))] (s) t and

"

A1 (t; K1 ) := a1 (t) + K1

Z

1

t

s+

Z

1

[v(s)(1

l (s) (s)

p( (s)))]

t

1

1 r(s)

(a1 (s))1+

1

s

#1

s;

, for t 2 [t0 ; 1)T ;

where K1 > 0 is an arbitrary constant. Theorem 2.1. Assume that (H0 ) holds and let 0 p(t) (t) (t) and (t) 1 for t 2 [t0 ; 1)T . If (H1 ): lim sup a1 (t) < 1, Rt!1 1 1 1 (H2 ): t0 ( r(s) ) A1 (s; K1 ) s = 1. Then every solution of (1.1) oscillates on [t0 ; 1)T .

a < 1, r (t) > 0 and


0 for t t0 . Hence there exists t 2 [t0 ; 1)T such that x(t) > 0; x( (t)) > 0; x( (t)) > 0 and x( (t)) > 0 for t t1 . Using (2.1), we see that (1.1) becomes (2.2)

(r(t)(z (t)) ) =

q(t)x ( (t))

v(t)x ( (t))

0; for t

t2 :

So r(t)(z (t)) is nonincreasing on [t1 ; 1)T , that is, either z (t) > 0 or z (t) < 0. By Lemma 2.1, it follows that z (t) > 0 for t t2 . Hence there exists t3 > t2 such that z(t)

p(t)z( (t))

= x(t) + p(t)x( (t)) p(t)x( (t)) p(t)p( (t))p( ( (t))) = x(t) p(t)p( (t))p( ( (t))) x(t);

which implies that x(t)

(1

p(t))z(t); for t 2 [t3 ; 1)T :

Therefore (1.1) can be written as (r(t)(z (t)) ) + q(t)(1 where


(t) and due to (2.3) and (2.4), we have w (t)

q(1

p )

v(1

p )

(z ) (z )

w (z ) ; for t 2 [t3 ; 1)T ; z

Now, by using the chain rule [6], we get that Z 1 (z (t)) = [(1 h)z(t) + hz( (t))] ( 0 (z(t))] 1 z (t); > 1; (z( (t)))] 1 z (t); 0
1 (z (t)) z(t) ; (z( (t))) 1 z (t) z (t); f or 0 < z (t) Using the fact that t

1:

(t), we have (z ) z

z ; z

> 0 on [t3 ; 1)T :

v(1

p )

Therefore (2.4) yields that (2.5)

w 1

Now, since r z

q(1

p )

(z ) (z )

is nonincreasing on [t3 ; 1)T , then for t

(2.6)

z

1

r

z ; t z

w

1

(w ) (z ) ; t

t3 :

(t), we have that t3 :

Substituting (2.6) into (2.5), we get w

q(1

p )

(z ) (z )

v(1

p )

(z ) (z )

r

1

1

(w )1+ (z )

1; t

t3 :

Since z(t) is nondecreasing on [t3 ; 1)T , then there exists t4 > t3 and C > 0 such that (z( (t)))

1

1

(z(t))

C;

for t

t4 :

By using Lemma 2.2, it follows from the last inequality that w (t) Cr

1

q(1

1

(t)(w (t))1+ ; t

p( (t)))

l (t) (t)

v(1

l (t) (t)

p( (t)))

tl > t4 :

Integrating the above inequality from t to u (t < u) for t, u 2 [t4 ; 1)T , we obtain w(t)

w(u) w(t) Z u q(1 p ) t

that is,

w(t)

l (t) (t)

+ v(1

a1 (t) + K1

Z

1

r

1

p )

l (t) (t)

(s)w( (s))1+

+ Cr

1

s; t

1

(t)(w (t))1+

1

s;

t1 ;

t

where K1 = C . Indeed, w(t) > a1 (t) implies that Z 1 1 1 w(t) a1 (t) + K1 r (s)(a1 ( (s)))1+

s = A1 (t; K1 ):

t

Since t

(t) we see

r(z )

(r(z ) ) ;

which implies that r(z ) (z )

(r(z ) ) =w (z )

(A1 (t; k1 )) ;

that is, 1

where

(z ) z r (A1 (t; k1 )); t 2 [t5 ; 1]T ; = ( ) > 1. Using the chain rule, we have Z 1 (z 1 (t)) = (1 ) [(1 h)z(t) + hz( (t))] dhz (t) 0

(1

)(z( (t)))

272

z (t);

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8

that is, (z 1

( (t))) 1

z( (t))

z ( (t)):

Hence (z 1 (t)) 1 and then due to (2.6), we see that

(z( (t)))

z (t);

1 (z 1 (t)) r (t)(A1 (t; k1 )); t 2 [t5 ; 1)T : 1 Integrating above inequality from t5 to t, we get Z t 1 1 r(s) A1 (s; K1 ) s < 1;

t5

which is a contradiction to (H2 ). The proof is complete. Theorem 2.2. Let 0 p(t) p(t) 1, r (t) 0 for t 2 [t0 ; 1)T and and assume that (H0 ), and (H1 ) hold. Furthermore assume that Rt 1 (H3 ): lim sup t0 r (s)A1 (s; K1 ) s > 1.

=

= , (t)

(t)

t!1

Then every solution of (1.1) oscillates.

Proof. Proceeding as in the proof of Theorem 2.1, we have w(t)

A1 (t; K1 ) f or t 2 [t4 ; 1)T :

1

Using the fact that r z

is nonincreasing on [t4 ; 1)T , we get Z t Z t 1 z(t) = z(t4 ) + z (s) s = z(t4 ) + r (s) r(s) t4

1

z (s)

s

t4

1

r (t)z (t)r

1

(s) s;

that is, Z

1

r(t) z (t) z(t)

(2.7)

t

r(s)

1

1

s

Consequently, 1

A1 (t; K1 ) implies that

r(t) z 0 (t) z(t

1

w (t) = Z

t

r

; t

t4 ;

t4

1

Z

(s) s A1 (t; K1 )

t

r

1

1

(s) s

;

t2

1;

t4

which contradicts (H3 ). Hence the theorem is proved. Theorem 2.3. Let 0 p(t) p(t) 1, r (t) 0 for t 2 [t0 ; 1)T and > > , (t) (t) and assume that (H0 ) and (H2 ) hold: Furthermore assume that 1 Rt R1 1 1 ( ) 1 1+ 1 (H4 ): lim sup(a1 (t)) r (s) s a1 (t) + K1 t (a1 (s)) s = 1: r(s) t0 t!1

Then every solution of (1.1) oscillates.

Proof. Proceeding as in the proof of Theorem 2.1, we obtain (2.2) and (2.3) and hence w(t) > a1 (t), for t 2 [t4 ; 1). Consequently, it follows from (2.3) that 1

r z

1

> z a1 ; for t

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We have (rz ) ) 1

r z

0 implies that there exists a constant C > 0 and t5 > t4 such that

C, for t

(2.8)

1

1

t5 , that is C

r z

> z a1 and hence 1

C a1 (t)

z(t)

; for t 2 [t5 ; 1)T ;

which implies that (2.9)

(z )

(

)

C

(

)

(a1 )

(

)

for t 2 [t5 ; 1)T :

Due to (2.5), (2.6) and using Lemma 2.2, we have that w (t)

q(1

l (t) (t)

p( (t)))

Cr

1

v(1

1

(t)(w (t))1+ (z (t))

(

l (t) (t)

p( (t))) )

:

Integrating the last inequality as in the proof of Theorem 2.1 and using (2.8), we obtain for t t1 t5 that Z 1 1 1 (2.10) w(t) a1 (t) + K3 r (s)(a1 (s))1+ s; f or t 2 [tl ; 1)T ; t

where K1 = C (2.11)

(

)

. Substitute (2.9) into (2.3), it is easy to verify that 1 Z 1 ( ) r (t)z 1 1 (t) h (z(t) a1 (t) + K1 r (s)(a1 (s))1+ z(t) t

Using (2.7) and (2.9) in (2.11), we can …nd Z t ( ) C a1 (t) r +K1

Z

t2

1

r

1

(s)(a1 (s))1+

t

Therefore, for t (a1 (t))

(

1

1

i1 s ; for t

t1 we have Z t Z h ) 1 r (s) s a1 (t) + K1 t2

h

1

(s) s

1

r

1

2

s

i1

:

a1 (t)

[t1 ; 1)T :

(s)(a1 (s))1+

1

t

which contradicts (H4 ). This completes the proof of theorem. Theorem 2.4. Let 0 p(t) 1, r (t) 0 for t 2 [t0 ; 1)T and (H0 ), (H2 ) and (H3 ) hold. Then every solution of (1.1) oscillates.




> , (t)

(t). If


1: t!1

Then every solution of (1.1) oscillates.

Proof. The proof of the theorem follows from Theorem 2.2 and Theorem 2.8. Hence the details are omitted. Theorem 2.10. Let 1 p(t) a < 1, r (t) 0 ( (t)) = > > , (t) (t). If (H0 ), (H2 ), (H5 ) (H7 ) and Rt R1 1 ( ) 1 (H8 ): lim sup(a1 (t)) r (s) s a1 (t) + K3 t r(s) t0 t!1

( (t)), 1

( (t)) =

( (t)),

1

1+ 1

(a1 (s))

s

= 1:

Then every solution of (1.1) oscillates.

Acknowledgement The second author is supported by Rajiv Gandhi National Fellowship (UGC), New Delhi, India, through the letter No. F1-17.1/2013-14/RGNF-2013-14-SC-ORI-42425, dated Feb. 6, 2014. References [1] R. P. Agarwal, M. Bohner, D. O’Regan, A. Peterson; Dynamic equations on time scales: a survey, J. Math. Anal. Appl. Math. V0l.141, N0.1-2 (2002), 1–26. [2] R. P. Agarwal, D. O’Regan, S. H. Saker; Oscillation criteria for second order nonlinear delay dynamic equations, J. Math. Anal. Appl., 300(2004), 203–217. [3] R. P. Agarwal, D. O’Regan, S .H. Saker; Oscillation criteria for nonlinear perturbed dynamic equations of second order on time scales, J. Appl. Math. Compu., 20(2006), 133–147. [4] E. Boe and H. C. Chang, Dynamics of delayed systems under feedback control, Chem. Engng. Sci. Vol.44 (1989), 1281-1294. [5] M. Bohner, A. Peterson; Dynamic equations on time scales: An introduction with Applications , Birkhauser, Boston.,(2001). [6] M. Bohner, A. Peterson; Advance in dynamic equations on time scales, Birkhauser, Boston.,(2001). [7] M. Bohner, S. H. Saker; Oscillation of second order nonlinear dynamic equations on time scales, Rocky. Mount. J. Math., 34 (2004), 1239-1254. [8] R. K. Brayton and R. A. Willoughby, On the numerical integration of a symmetric system of di¤ erencedi¤ erential equations of Neutral type, J. Math. Anal. Appl. 18 (1976), 182-189. [9] D. R. Driver, A mixed neutral systems, Nonlinear Anal. 8 (1984), 155-158. [10] L. H. Erbe, T. S. Hassan, A. Peterson; Oscillation criteria for sublinear half linear delay dynamic equations, Int. J. Di¤erence Equ., 3(2008), 227–245. [11] L. H. Erbe, T. S. Hassan, A. Peterson; Oscillation criteria for half linear delay dynamic equations, Nolinear. Dyns. Systems theory., may 27(2008), 1–9. [12] C. Gao, T. Li, S. Tang, E. Thandapani; Osillation theorems for second order neutral functional dynamic equations on time scale, Elec. J. Di¤. Equn., 2011(2011), 1–9. [13] J. K. Hale, Theory of Functional Di¤ erential Equations, Springer-Verlag, New York, (1977). [14] S. Hilger; Analysis on measure chains-a uni…ed approach to continuous and discrete calculus, Results in mathematics, Vol-18, N0,1-2 .,(1990), 18–56. [15] T. S. Hassan; Oscillation criteria for half linear dynamic equations, J. Math. Anal.Appl ., 345(2008), 176–185. [16] E. P. Popov, Automatic Regulation and Control, Nauka, Moscow (in Russian), (1966). [17] S. H. Saker; Oscillation of second order nonlinear neutral delay dynamic equations on time scales, J. Comp. Appl. Math., 2(2006), 123–141. [18] S. H. Saker; Oscillation second order neutral delay dynamic equations of Emden-Fow ler type, Dynamic Systems and Appl ., 15(2006), 629–644. [19] S. H. Saker; Oscillation of superlinear and sublinear neutral delay dynamic equations, Communication in Applied. Annal ., 12(2008), 173–188. [20] S. H. Saker, D. O’Regan; New oscillation criteria for second order neutral functional dynamic equations via the general Riccati substitution, Commun. Sci. Numer. Simulate ., 16(2011), 423–434. [21] Q. Yang, Z. Xu; Oscillation criteria for second order quasilinear neutral delay di¤ ertial equations on time scales, Coput. Math. Appl., 62(2011),3682–3691.

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[22] Y. Sun, Z. Han, T.Li and G. Zhang ; Oscillation criteria for second order quasilinear neutral delay dynamic equations on time scales, Hinawi Pub. Coporation. Advance in Di¤ernce Eun (2010),1–14. [23] A. K. Tripathy; Some oscillation results of second order nonlinear dynamic equations of neutral type, Nonlinear Analysis, 71(2009), e1727–e1735. [24] E. Thandapani, V. Piramanantham; Oscillation criteria for second order nonlinear neutral dynamic equations on time scales, Tamkang. J. Mathematics. Equs., 43(2012), 109–122. [25] A. K. Tripathy; Riccati transformation and sublinear oscillation for second order neutral delay dynamic equations, J.Appl.Math and Informatics, 30(2012), 1005–1021. [26] A. K. Sethi; Oscillation of second order sublinear neutral delay dynamic equations via riccati transformation, J. Appl. Math and Informatics, Volume 36(2018), No 3-4, 213–229. 1 Department

of Mathematics, Faculty of Science, Mansoura University, Mansoura 35516, Egypt,, Sambalpur University, Sambalpur-

E-mail:[email protected], 2 Department of Mathematics, 768019, India,, E-mail: [email protected].

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Semilocal Convergence of a Newton-Secant Solver for Equations with a Decomposition of Operator Ioannis K. Argyros1 , Stepan Shakhno2 , Halyna Yarmola2 1 Department of Mathematics, Cameron University, Lawton, USA, OK 73505; [email protected], 2 Faculty of Applied Mathematics and Informatics, Ivan Franko National University of Lviv, Lviv, Ukraine, 79000; [email protected], [email protected] May 4, 2019 Abstract. We provide the semilocal convergence analysis of the NewtonSecant solver with a decomposition of a nonlinear operator under classical Lipschitz conditions for the first order Fr´echet derivative and divided differences. We have weakened the sufficient convergence criteria, and obtained tighter error estimates. We give numerical experiments that confirm theoretical results. The same technique without additional conditions can be used to extend the applicability of other iterative solvers using inverses of linear operators. The novelty of the paper is that the improved results are obtained using parameters which are special cases of the ones in earlier works. Therefore, no additional information is needed to establish these advantages. Keywords: Newton-Secant solver; semilocal convergence analysis; Fr´echet derivative; divided differences; decomposition of nonlinear operator AMS Classification: 45B05, 47J05, 65J15, 65J22

1

Introduction

One of the important problems in Computational Mathematics including Mathematical Biology, Chemistry, Economic, Physics, Engineering and other disciplines is finding solutions of nonlinear equations and systems of nonlinear equations [1-14]. For most of these problems, to find the exact solution is difficult or impossible. Therefore, the development and research of numerical methods for solving nonlinear problems is an urgent task.

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Newton-Secant Solver...

A popular solver for dealing with nonlinear equations is Newton’s [2, 3, 4]. But it is not applicable, if functions are nondifferentiable. In this case, we can apply solvers with divided differences [1, 2, 3, 7, 8, 10, 11]. If it is possible to decompose into differentiable and nondifferentiable parts, it is advisable to use combined methods [2, 3, 5, 6, 12, 13, 14]. Consider a nonlinear equation F (x) + G(x) = 0,

(1)

where the operators F and G are defined on a open convex set D of a Banach space E1 with values in a Banach space E2 , F is a Fr´echet differentiable operator, G is a continuous operator for which differentiability is not assumed. It is necessary to find an approximate solution x∗ ∈ D that satisfies equation (1). In this paper, we consider the Newton-Secant solver xn+1 = xn − [F 0 (xn ) + G(xn−1 , xn )]−1 (F (xn ) + G(xn )), n = 0, 1, ....

(2)

This iterative process √ was proposed in [6] and studied in [2, 3, 13], and the 1+ 5 was established. It is shown that (2) converges faster convergence order 2 than the Secant solver. In this paper, we study solver (2) under the classical Lipschitz conditions for first-order Fr´echet derivative and divided differences. Our technique allows to get the weaker convergence criteria, and tighter error estimates. This way, we extended the applicability of the results obtained in [13].

2

Convergence Analysis

Let L(E1 , E2 ) be a space of linear bounded operators from E1 into E2 . Set ¯ τ ) denote its closure. S(x, τ ) = {y ∈ E1 : ky − xk < τ } and let S(x, Define quadratic polynomial ϕ by ϕ(t) = α1 t2 + α2 t + α3 and parameters r, and r1 by r=

1 − (q0 + q¯0 )a , p0 + q0 + 2¯ p0 + q¯0 + ¯q 0 r1 =

1 − q¯0 a , 2¯ p0 + q¯0 + ¯q 0

where α1 = p0 + q0 + 2¯ p0 + q¯0 + ¯q 0 , α2 = −[1 − (q0 + q¯0 )a + (2¯ p0 + q¯0 + ¯q 0 )c] and α3 = (1 − q¯0 a)c,

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Argyros, Shakhno, Yarmola

where p0 , p¯0 , q0 , q¯0 , ¯q 0 , a and c are nonnegative numbers.
Suppose that (q0 + q¯0 )a < 1 and ϕ 12 r ≤ 0. Then, it is simple algebra to show, function ϕ has a unique root r¯0 ∈ (0, 2r ], and r ≤ r1 , γ¯ =

p0 r¯0 + q0 (¯ r0 + a) ∈ [0, 1) 1 − q¯0 a − (2¯ p0 + q¯0 + ¯q 0 )¯ r0

and r¯0 ≥

c . 1 − γ¯

Set D0 = D ∩ S(x0 , r1 ). Definition 2.1. We call an operator that acts from E1 into E2 and is denoted by G(x, y) a first-order divided difference for the operator G by fixed points x and y (x 6= y), if the equality G(x, y)(x − y) = G(x) − G(y) is satisfied. Theorem 2.2. Suppose that: 1) F and G are nonlinear operators on an open convex set D of a Banach space E1 into a Banach space E2 ; 2) F is a Fr´echet-differentiable operator, and let G is a continuous operator; 3) G(·, ·) is the first-order divided differences of the operator G defined on the set D; 4) the linear operator A0 = F 0 (x0 ) + G(x−1 , x0 ), where x−1 , x0 ∈ D, is invertible; 5) the following conditions are satisfied for all x, y, ∈ D 0 0 kA−1 p0 kx0 − xk, 0 (F (x0 ) − F (x))k ≤ 2¯

(3)

kA−1 ¯0 kx−1 − xk, 0 (G(x−1 , x0 ) − G(x, x0 ))k ≤ q

(4)

kA−1 0 (G(x, x0 )

− G(x, y))k ≤ ¯q 0 kx0 − yk,

(5)

and for all x, y, u ∈ D0 0 0 kA−1 0 (F (x) − F (y))k ≤ 2p0 kx − yk,

(6)

kA−1 0 (G(x, y) − G(u, y))k ≤ q0 kx − uk;

(7)

6) a, c are nonnegative numbers such that kx0 − x−1 k ≤ a, kA−1 0 (F (x0 ) + G(x0 ))k ≤ c, c > a, (q0 + q¯0 )a < 1,

281

ϕ

1
r ≤ 0; 2

(8) (9)

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Newton-Secant Solver...

¯ 0 , r¯0 ) ⊂ D. 7) S(x Then, the solver (2) is well-defined and the sequence generated by it converges to the solution x∗ of equation (1), so that for each n ∈ {−1, 0, 1, 2, ...}, the following inequalities are satisfied kxn − xn+1 k ≤ tn − tn+1 ,

(10)

kxn − x∗ k ≤ tn − t¯∗ ,

(11)

where sequence {tn }n≥−1 defined by the formulas t−1 = r¯0 + a, t0 = r¯0 , t1 = r¯0 − c, tn+1 − tn+2 = γ¯n (tn − tn+1 ), n ≥ 0, p˜0 (tn − tn+1 ) + q˜0 (tn−1 − tn+1 ) , 0 ≤ γ¯n < γ¯ 1 − q¯0 a − 2¯ p0 (t0 − tn+1 ) − q¯0 (t0 − tn ) − ¯q 0 (t0 − tn+1 ) (12) is decreasing, nonnegative, and converges to t¯∗ , so that r¯0 − c/(1 − γ¯ ) ≤ t¯∗ < t0 , where   p¯0 , n = 0 q¯0 , n = 0 p˜0 = , q˜0 = p0 , n > 0 q0 , n > 0. γ¯n =

Proof. We use mathematical induction to show that, for each k ≥ 0 the following inequalities are satisfied tk+1 ≥ tk+2 ≥ r¯0 −

1 − γ¯ k+2 c c ≥ r¯0 − ≥ 0, 1 − γ¯ 1 − γ¯

(13)

tk+1 − tk+2 ≤ γ¯ (tk − tk+1 ).

(14)

Setting k = 0 in (12), we get t1 − t2 =

p˜0 (t0 − t1 ) + q˜0 (t−1 − t1 ) (t0 − t1 ) ≤ γ¯ (t0 − t1 ), 1 − q¯0 a − 2¯ p0 (t0 − t1 ) − ¯q 0 (t0 − t1 )

t0 ≥ t1 , t1 ≥ t2 ≥ t1 −¯ γ (t0 −t1 ) ≥ r¯0 −(1+¯ γ )c = r¯0 −

(1 − γ¯ 2 )c c ≥ r¯0 − ≥ 0. 1 − γ¯ 1 − γ¯

Suppose that (13) and (14) are true for k = 0, 1, ..., n − 1. Then, for k = n, we obtain
p˜0 (tn − tn+1 ) + q˜0 (tn−1 − tn+1 ) (tn − tn+1 ) tn+1 − tn+2 = 1 − q¯0 a − 2¯ p0 (t0 − tn+1 ) − q¯0 (t0 − tn ) − ¯q 0 (t0 − tn+1 ) ≤

p˜0 tn + q˜0 tn−1 (tn − tn+1 ) ≤ γ¯ (tn − tn+1 ), 1 − q¯0 a − 2¯ p0 t0 − q¯0 t0 − ¯q 0 t0

tn+1 ≥ tn+2 ≥ tn+1 − γ¯ (tn − tn+1 ) ≥ r¯0 −

282

1 − γ¯ n+2 c c ≥ r¯0 − ≥ 0. 1 − γ¯ 1 − γ¯

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Thus, {tn }n≥0 is a decreasing nonnegative sequence, and converges to t¯∗ ≥ 0. Let us prove that the method (2) is well-defined, and for each n ≥ 0 the inequality (10) is satisfied. Since t−1 − t0 = a, t0 − t1 = c and conditions (8) are fulfilled then x1 ∈ S(x0 , r¯0 ) and (10) is satisfied for n ∈ {−1, 0}. Let conditions (8) be satisfied for k = 0, 1, ..., n. Let us prove that the method (2) is well-defined for k = n + 1. Denote An = F 0 (xn ) + G(xn−1 , xn ). Using the Lipschitz conditions (3) – (5), we have −1 −1 0 0 kI − A−1 0 An+1 k = kA0 (A0 − An+1 )k ≤ kA0 (F (x0 ) − F (xn+1 ))k

+kA−1 0 (G(x−1 , x0 ) − G(xn , x0 ) + G(xn , x0 ) − G(xn , xn+1 ))k ≤ 2¯ p0 kx0 − xn+1 k + q¯0 (kx−1 − x0 k + kx0 − xn k) + ¯q 0 kx0 − xn+1 k ≤ 2¯ p0 kx0 − xn+1 k + q¯0 a + q¯0 kx0 − xn k + ¯q 0 kx0 − xn+1 k ≤ q¯0 a + 2¯ p0 (t0 − tn+1 ) + q¯0 (t0 − tn ) + ¯q 0 (t0 − tn+1 ) ≤ q¯0 a + 2¯ p0 r¯0 + q¯0 r¯0 + ¯q 0 r¯0 < 1. According to the Banach lemma on inverse operators [2] An+1 is invertible, and kA−1 ¯0 a − 2¯ p0 kx0 − xn+1 k − q¯0 kx0 − xn k + ¯q 0 kx0 − xn+1 k)−1 . n+1 A0 k ≤ (1 − q By the definition of the divided difference and conditions (6), (7), we obtain kA−1 0 (F (xn+1 ) + G(xn+1 ))k = kA−1 0 (F (xn+1 ) + G(xn+1 ) − F (xn ) − G(xn ) − An (xn − xn+1 ))k R1 0 0 ≤ kA−1 0 ( 0 {F (xn+1 + t(xn − xn+1 )) − F (xn )}dt)kkxn − xn+1 k +kA−1 0 (G(xn+1 , xn ) − G(xn−1 , xn ))kkxn − xn+1 k ≤ (˜ p0 kxn − xn+1 k + q˜0 (kxn − xn+1 k + kxn−1 − xn k))kxn − xn+1 k. In view of condition (10), we have kxn+1 − xn+2 k = kA−1 n+1 (F (xn+1 ) + G(xn+1 ))k −1 ≤ kA−1 n+1 A0 kkA0 (F (xn+1 ) + G(xn+1 ))k

p˜0 kxn − xn+1 k + q˜0 (kxn − xn+1 k + kxn−1 − xn k) kxn − xn+1 k 1 − q¯0 a − 2¯ p0 kx0 − xn+1 k − q¯0 kx0 − xn+1 k + ¯q 0 kx0 − xn k
p˜0 (tn − tn+1 ) + q˜0 (tn−1 − tn+1 ) (tn − tn+1 ) ≤ = tn+1 − tn+2 . 1 − q¯0 a − 2¯ p0 (t0 − tn+1 ) − q¯0 (t0 − tn ) − ¯q 0 (t0 − tn+1 )

≤

Thus, the method (2) is well-defined for each n ≥ 0 . Hence it follows that kxn − xk k ≤ tn − tk , −1 ≤ n ≤ k.

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Newton-Secant Solver...

Therefore, the sequence {xn }n≥0 is fundamental, so it converges to some ¯ 0 , r¯0 ). Inequality (11) is obtained from (15) for k → ∞. Let us x∗ ∈ S(x show that x∗ solves the equation F (x) + G(x) = 0. Indeed, we get in turn that A−1 ˜0 kxn − xn+1 k 0 (F (xn+1 ) + G(xn+1 )) ≤ p
+˜ q0 (kxn − xn+1 k + kxn−1 − xn k) kxn − xn+1 k → 0, n → ∞. u t

Hence, F (x∗ ) + G(x∗ ) = 0. √ 1+ 5 Remark 2.3. The order of convergence of method (2) is equal to . 2 Proof. In view of tn − tn+1 ≤ γ¯ (tn−1 − tn ), and (12), we obtain
p˜0 (tn − tn+1 ) + q˜0 (tn − tn+1 + tn−1 − tn ) (tn − tn+1 ) tn+1 − tn+2 = 1 − q¯0 a − 2¯ p0 (t0 − tn+1 ) − q¯0 (t0 − tn ) − ¯q 0 (t0 − tn+1 ) ≤

p˜0 γ¯ (tn−1 − tn ) + q˜0 (1 + γ¯ )(tn−1 − tn ) (tn − tn+1 ) 1 − q¯0 a − 2¯ p0 (t0 − tn+1 ) − q¯0 (t0 − tn ) − ¯q 0 (t0 − tn+1 )
p¯0 γ¯ + q¯0 (1 + γ¯ ) (tn − tn+1 )(tn−1 − tn ) = 1 − q¯0 a − 2¯ p0 (t0 − tn+1 ) − q¯0 (t0 − tn ) − ¯q 0 (t0 − tn+1 ) ≤

p˜0 γ¯ + q˜0 (1 + γ¯ ) (tn − tn+1 )(tn−1 − tn ). 1 − q¯0 a − 2¯ p0 t0 − q¯0 t0 − ¯q 0 t0

Denote C¯ =

p¯0 γ¯ + q¯0 (1 + γ¯ ) . Clearly, 1 − q¯0 a − 2¯ p0 t0 − q¯0 t0 − ¯q 0 t0 ¯ n−1 − t¯∗ )(tn − t¯∗ ). tn+1 − tn+2 ≤ C(t

(16)

Since, for each k > 2, the estimate is satisfied tn+k−1 − tn+k ≤ γ¯ k−2 (tn+1 − tn+2 ), we get tn+1 − tn+k = tn+1 − tn+2 + tn+2 − tn+3 + . . . + tn+k−1 − tn+k ≤ (1 + γ¯ + . . . + γ¯ k−2 )(tn+1 − tn+2 ) =

1 1 − γ¯ k−1 (tn+1 − tn+2 ) ≤ (tn+1 − tn+2 ). 1 − γ¯ 1 − γ¯

In view of (16), for k → ∞, we have tn+1 − t¯∗ ≤

C¯ (tn−1 − t¯∗ )(tn − t¯∗ ) 1 − γ¯

Hence, it √ follows that the order of convergence of the sequence {tn }n≥0 is 1+ 5 equal to , and, according (11), the sequence {xn }n≥0 converges with the 2 same order. u t

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Remark 2.4. (a) The following conditions were used for each x, y, u, v ∈ D in [13] 0 0 kA−1 (17) 0 (F (y) − F (x))k ≤ 2P0 ky − xk, kA−1 0 (G(x, y) − G(u, v))k ≤ Q0 (kx − uk + ky − vk),

(18)

c , Q0 a + 2P0 r0 + 2Q0 r0 < 1, 1−γ P0 r0 + Q0 (r0 + a) γ= , 0 ≤ γ < 1. 1 − Q0 a − 2P0 r0 − 2Q0 r0

(19)

r0 ≥

But, then we have p¯0 q¯0 ¯q 0

≤ P0 , ≤ Q0 , ≤ Q0 ,

since D0 ⊆ D, (3) and (4), (5), (7) are weaker than (17) and (18) respectively for r¯0 ≤ r0 . Notice that sufficient convergence criteria (9) imply (19) but not necessarily vice versa, unless if p¯0 = P0 , q¯0 = ¯q 0 = Q0 and r¯0 = r0 . A simple inductive argument shows that γ¯n ≤ γn ,

(20)

tn − tn+1 ≤ sn − sn+1 ,

(21)

where s−1 = r0 + a, s0 = r0 , s1 = r0 − c, sn+1 − sn+2 = γn (sn − sn+1 ), n ≥ 0, γn =

P0 (sn − sn+1 ) + Q0 (sn−1 − sn+1 ) , 0 ≤ γn ≤ γ. 1 − Q0 a − 2P0 (s0 − sn+1 ) − Q0 (2s0 − sn − sn+1 )

Notice that the corresponding quadratic polynomial ϕ1 to ϕ is defined similarly by ϕ1 (t) = b1 t2 + b2 t + b3 where b1 = 3P0 + 3Q0 , b2 = −[1 − 2Q0 a + (2P0 + 2Q0 )c] and b3 = (1 − Q0 a)c. We have by these definitions that α1 < b1 , α2 < b2 , but α3 > b3 .

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Newton-Secant Solver...

Therefore, we cannot tell, if r0 < r¯0 or r¯0 < r0 or r0 = r¯0 . But, we have γ ≤ γ¯ ⇒ r0 ≤ r¯0 , sn ≤ tn ,

(22)

s∗ ≤ t¯∗ = lim tn n→∞

and

γ¯ ≤ γ ⇒ r¯0 ≤ r0 ⇒ C¯ ≤ C, tn ≤ sn ,

(23)

t¯∗ ≤ s∗ = lim sn , n→∞

α2 (solving It is simple algebra to show that ϕ(r) ≥ 0, and for rmin = − 2α1 r r1 ϕ0 (t) = 0), rmin ≥ , rmin ≤ . Hence, one may replace the second inequation 2 2 in (9) by ϕ(λr) ≤ 0 for some λ ∈ (0, 12 ] to obtain a better information about the location of r¯0 , if λ 6= 12 , especially in the case when we do not actually need to compute r¯0 . (b) The Lipschitz parameters p¯0 , q¯0 , ¯q 0 can become even smaller, if we define the set D1 = D ∩ S(x1 , r1 − c) for r1 > c to replace D0 in Theorem 2.2., since D1 ⊆ D0 .

3

Numerical experiments

Let us define function F + G : R → R, where F (x) = ex−0.5 + x3 − 1.3, G(x) = 0.2x|x2 − 2|. The exact solution of F (x) + G(x) = 0 is x∗ = 0.5. Let D = (0, 1). Then F 0 (x) = ex−0.5 + 3x2 , G(x, y) =

0.2x(2 − x2 ) − 0.2y(2 − y 2 ) = 0.2(1 − x2 − xy − y 2 ). x−y

A0 = ex0 −0.5 + 3x20 + 0.2(1 − x2−1 − x−1 x0 − x20 ), 0 0 |A−1 0 (F (x) − F (y))| ≤

|A−1 0 (G(x, y) − G(u, v))| =

e0.5 + 3|x + y| |x − y|, |A0 |

0.2 |(u + x + y)(u − x) + (v + y + u)(v − y)|. |A0 |

Let x0 = 0.57, x−1 = 0.571. Then, we have a = 0.001, c ≈ 0.0660157, p¯0 ≈ 1.4118406, q¯0 ≈ 0.1901483, ¯q 0 ≈ 0.2282491, r1 ≈ 0.3083854, D0 = D ∩ S(x0 , r1 ) = (0.2616146, 0.8783854),

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p0 ≈ 1.5362481, q0 ≈ 0.2340358, P0 ≈ 1.6982621, r ≈ 0.1994221, ϕ( 21 r) ≈ −0.0051722 < 0. So, p¯0 < P0 ,

Q0 ≈ 0.2664386, and q¯0 < Q0 , ¯q 0 < Q0 .

By solving inequalities ϕ(t) ≤ 0 and ϕ1 (t) ≤ 0, we get (1)

(2)

(1)

(2)

t ∈ [0.0824903, 0.1596319] ⇒ r¯0 ≈ 0.0824903, r¯0 ≈ 0.1596319, t ∈ [0.0924062, 0.1211750] ⇒ r0 ≈ 0.0924062, r0 ≈ 0.1211750. (1)

(1)

Then r¯0 = r¯0 ≈ 0.0824903, r0 = r0 ≈ 0.0924062, and S(x0 , r¯0 ) = (0.4875097, 0.6524903), γ¯ ≈ 0.1997151 < 1, C¯ ≈ 0.8023108, S(x0 , r0 ) = (0.4775938, 0.6624062), γ ≈ 0.2855916 < 1, C ≈ 1.2998717. In Table 1, there are results that confirm estimates (10), (11) and (21). Table 2 shows that sequences {tn } and {sn } converge to t¯∗ ≈ 0.0073550 and s∗ ≈ 0.0144209, respectively, and confirms (20) and (23). Table 1: Obtained results for ε = 10−7 n 1 2 3 4

|xn−1 − xn | 0.0660157 0.0040123 0.0000281 1.761e-08

tn−1 − tn 0.0660157 0.0087609 0.0003573 0.0000040

sn−1 − sn 0.0660157 0.0113203 0.0006452 0.0000040

|xn − x∗ | 0.0039843 0.0000281 1.761e-08 7.438e-14

tn − t¯∗ 0.0091195 0.0003586 0.0000013 1.440e-10

sn − s∗ 0.0119695 0.0006492 0.0000040 1.033e-09

Table 2: Obtained results for ε = 10−7 n -1 0 1 2 3 4 5

4

tn 0.0834903 0.0824903 0.0164746 0.0077136 0.0077136 0.0073550 0.0073550

sn 0.0934062 0.0924062 0.0263904 0.0150701 0.0144249 0.0144209 0.0144209

γ¯n−2

γn−2

0.1327096 0.0407873 0.0035475 0.0001136

0.1714793 0.0569927 0.0061771 0.0002592

Conclusions

We investigated the semilocal convergence of Newton-Secant solver under classical center and restricted Lipschitz conditions. This technique weakens the

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Newton-Secant Solver...

10

sufficient convergence criteria without adding more conditions and uses constants that are specializations of earlier ones. Moreover, tighter estimate errors are obtained. The theoretical results are confirmed by numerical experiments. Our technique can be used to extend the applicability of other iterative methods using inverses of linear operators [1-14] along the same lines.

References [1] S. Amat, On the local convergence of Secant-type methods, Intern. J. Comput. Math., 81, 1153-1161 (2004). ´ [2] I.K. Argyros, A.A. Magre˜ n´an, A Contemporary Study of Iterative Methods, Elsevier (Academic Press), New York, 2018. ´ [3] I.K. Argyros, A.A. Magre˜ n´an, Iterative Methods and Their Dynamics with Applications: A Contemporary Study, CRC Press, 2017. [4] I.K. Argyros, S. Hilout, On an improved convergence analysis of Newtons method, Applied Mathematics and Computation, 25, 372-386 (2013). [5] I.K. Argyros, S.M. Shakhno, H.P. Yarmola, Two-Step Solver for Nonlinear Equations, Symmetry, 11(2):128 (2019). [6] E. C˜ atinac, On some iterative methods for solving nonlinear equations, Rev. Anal. Numer., Theorie Approximation, 23(I), 47-53 (1994). [7] M.A. Hernandez, M.J. Rubio, The Secant method and divided differences H¨ older continuous, Applied Mathematics and Computation, 124(2), 139-149 (2001). [8] V.A. Kurchatov, On one method of linear interpolation for solving functional equations, Dokl. AN SSSR. Ser. Mathematics. Physics., 198(3), 524526 (1971) (in Russian). [9] F.-A. Potra, V. Pt´ak, Nondiscrete induction and iterative processes. Research Notes in Mathematics, 103, Pitman Advanced Publishing Program, Boston, MA, USA, 1984. [10] S.M. Shakhno, On the difference method with quadratic convergence for solving nonlinear operator equations, Matematychni Studii, 26, 105–110 (2006) (in Ukrainian). [11] S.M. Shakhno, Application of nonlinear majorants for investigation of the secant method for solving nonlinear equations, Matematychni Studii, 22, 79-86 (2004) (in Ukrainian). [12] S.M. Shakhno, Convergence of the two-step combined method and uniqueness of the solution of nonlinear operator equations, J. Comp. App. Math., 261, 378-386 (2014).

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[13] S.M. Shakhno, I.V. Melnyk, H.P. Yarmola, Analysis of the Convergence of a Combined Method for the Solution of Nonlinear Equations, J. Math. Sci., 201, 32-43 (2014). [14] S.M. Shakhno, H.P. Yarmola, Two-point method for solving nonlinear equations with nondifferentiable operator, Matematychni Studii, 36, 213-220 (2011) (in Ukrainian).

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Global behavior of a nonlinear higher-order rational di¤erence equation A. M. Ahmed1 Mathematics Department, College of Science, Jouf University, Sakaka (2014), Kingdom of Saudi Arabia E-mail: [email protected] & [email protected] Abstract In this paper, we investigate the global behavior of the di¤erence equation xn 1 xn+1 = ; n = 0; 1; 2; ::: k k P p 1 Q + xn 2mi xn 2mj i=1

j=1

with positive parameters and non-negative initial conditions.

Keywords: Recursive sequences; Global asymptotic stability; Oscillation; Period two solutions; Semicycles. Mathematics Subject Classi…cation: 39A10.

————————————————— 1 On leave from: Department of Mathematics, Faculty of Science, Al-Azhar University , Nasr City (11884), Cairo, Egypt.

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1. INTRODUCTION Di¤erence equations appear as natural descriptions of observed evolution phenomena because most measurements of time evolving variables are discrete and as such these equations are in their own right important mathematical models. More importantly, di¤erence equations also appear in the study of discretization methods for di¤erential equations. Several results in the theory of di¤erence equations have been obtained as more or less natural discrete analogues of corresponding results of di¤erential equations. The study of these equations is quite challenging and rewarding and is still in its infancy. We believe that the nonlinear rational di¤erence equations are of paramount importance in their own right, and furthermore, that results about such equations o¤er prototypes for the development of the basic theory of the global behavior of nonlinear di¤erence equations. Recently there has been a lot of interest in studying the global attractivity, boundedness character, periodicity and the solution form of nonlinear di¤erence equations. El-Owaidy et al [1] investigated the global asymptotic behavior and the periodic character of the solutions of the di¤erence equation xn+1 =

xn 1 + xpn

;

n = 0; 1; 2; :::

2

where the parameters ; ; and p are non-negative real numbers. Other related results on rational di¤erence equations can be found in refs. [2-15]. In this paper, we investigate the global asymptotic behavior and the periodic character of the solutions of the di¤erence equation xn

xn+1 = +

k P

i=1

1

xpn 12mi

k Q

; xn

n = 0; 1; 2; :::

(1.1)

2mj

j=1

where the parameters ; ; and p are positive real numbers, k 2 f1; 2; :::g; fmi gki=1 be positive integers such that mi > mi 1 ; i = 2; :::k and the initial conditions x 2mk ; x 2mk +1 ; :::; x0 are non-negative real numbers. The results in this work are consistent with the results in [1] when k = 1 and m1 = 1: 2

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The results in this work are consistent with the results in [3] when k = 2; m1 = 1 and m2 = 2: We need the following de…nitions. De…nition 1. Let I be an interval of real numbers and let f : I k+1 ! I be a continuously di¤erentiable function. Consider the di¤erence equation (1.2)

xn+1 = f (xn ; xn 1 ; :::; xn k ); n = 0; 1; :::; with x k ; x k+1 ; :::; x0 2 I: Let x be the equilibrium point of Eq.(1.2). linearized equation of Eq.(1.2) about the equilibrium point x is yn+1 = c1 yn + c2 yn

1

+ ::: + ck+1 yn

where @f c1 = @x (x; x; :::; x) , c2 = @x@f (x; x; :::; x); :::; ck+1 = n n 1 The characteristic equation of Eq.(1.3) is k+1 X

k+1

ci

k i+1

(1.3)

k

@f @xn

The

k

(x; x; :::; x):

(1.4)

= 0:

i=1

(i) The equilibrium point x of Eq.(1.2) is locally stable if for every there exists > 0 such that for all x k ; x k+1 ; :::; x 1 ,x0 2 I with jx

k

xj + jx

k+1

xj + ::: + jx0

> 0;

xj < ;

we have jxn

xj
0; such that for all x k ; x k+1 ; :::; x 1 , x0 2 I with jx k xj + jx k+1 xj + ::: + jx0 xj < ; we have lim xn = x:

n!1

3

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(iii) The equilibrium point x of Eq.(1.2) is global attractor if for all x k ; x x0 2 I; we have lim xn = x:

k+1 ; :::; x 1 ,

n!1

(iv) The equilibrium point x of Eq.(1.2) is globally asymptotically stable if x is locally stable, and x is also a global attractor of Eq.(1.2). (v) The equilibrium point x of Eq.(1.2) is unstable if x is not locally stable. De…nition 2. A positive semicycle of fxn g1 n= k of Eq.(1.2) consists of a ‘string’ of terms fxl ; xl+1 ; :::; xm g ; all greater than or equal to x; with l k and m < 1 and such that either l = k or l > k and xl 1 < x and either m = 1 or m < 1 and xm+1 < x: A negative semicycle of fxn g1 n= k of Eq.(1.2) consists of a ‘string’ of terms fxl ; xl+1 ; :::; xm g ; all less than x; with l k and m < 1 and such that either l = k or l > k and xl 1 x and either m = 1 or m < 1 and xm+1 x: De…nition 3. A solution fxn g1 n= exists N k such that either xn

x 8n

k

N

of Eq.(1.2) is called nonoscillatory if there or xn < x 8n

N ;

and it is called oscillatory if it is not nonoscillatory. (a) A sequence fxn g1 n=

k

is said to be periodic with period p if

xn+p = xn for all n

k:

(1.5)

(b) A sequence fxn g1 n= k is said to be periodic with prime period p if it is periodic with period p and p is the least positive integer for which (1.5) holds. We need the following theorem. Theorem 1.1. (i) If all roots of Eq.(1.4) have absolute value less than one, then the equilibrium point x of Eq.(1.2) is locally asymptotically stable. (ii) If at least one of the roots of Eq.(1.4) has absolute value greater than one, then x is unstable. The equilibrium point x of Eq.(1.2) is called a saddle point if Eq.(1.4) has roots both inside and outside the unit disk. 4

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2. Main results In this section, we investigate the dynamics of Eq.(1.1) under the assumptions that all parameters in the equation are positive and the initial conditions are non-negative. 1 p+k 1

The change of variables xn = yn reduces Eq.(1.1) to the di¤erence equation ryn 1 yn+1 = ; n = 0; 1; 2; ::: (2.1) k k P p 1 Q 1+ yn 2mi yn 2mj i=1

j=1

where r = > 0: Note that y1 = 0 is always an equilibrium point of Eq.(2.1). When r > 1;

Eq.(2.1) also possesses the unique positive equilibrium y2 =

r 1 k

1 k+p 1

:

Theorem 2.1. The following statements are true (i) If r < 1; then the equilibrium point y1 = 0 of Eq.(2.1) is locally asymptotically stable. (ii) If r > 1; then the equilibrium point y1 = 0 of Eq.(2.1) is a saddle point. (iii) When r > 1; then the positive equilibrium point y2 = Eq.(2.1) is unstable.

r 1 k

1 k+p 1

of

Proof: The linearized equation of Eq.(2.1) about the equilibrium point y1 = 0 is zn+1 = rzn 1 ;

n = 0; 1; 2; :::

so, the characteristic equation of Eq.(2.1) about the equilibrium point y1 = 0 is 2mk +1

r

2mk 1

= 0;

and hence, the proof of (i) and (ii) follows from Theorem A. For (iii), we assume that r > 1; then the linearized equation of Eq.(2.1) about 1

the equilibrium point y2 = r k 1 k+p 1 has the form 1) 1) zn+1 = zn 1 (r 1)(p+k zn 2m1 (r 1)(p+k zn 2m2 ::: rk rk 0; 1; 2; :::

(r 1)(p+k 1) zn 2mk ; rk

n=

5

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so, the characteristic equation of Eq.(2.1) about the equilibrium point y2 = r 1 k

1 k+p 1

is

f( )=

2mk +1

2mk 1

+

(r

1)(p + k rk

k 1) P

2mk 2mi

= 0;

i=1

It is clear that f ( ) has a root in the interval ( 1; 1); and so, y2 = is an unstable equilibrium point.

r 1 k

1 k+p 1

This completes the proof. Theorem 2.2. Assume that r < 1; then the equilibrium point y1 = 0 of Eq.(2.1) is globally asymptotically stable. Proof: We know by Theorem 2.1 that the equilibrium point y1 = 0 of Eq.(2.1) is locally asymptotically stable. So, let fyn g1 n= 2mk be a solution of Eq.(2.1). It su¢ ces to show that limn!1 yn = 0: Since 0

ryn

yn+1 = 1+

k P

i=1

1

ynp 12mi

k Q

ryn yn

1

< yn 1 :

2mj

j=1

So, the even terms of this solution decrease to a limit (say L1 terms decrease to a limit (say L2 0), which implies that L1 =

rL1 1+

1 kLk+p 2

and

L2 =

0), and the odd

rL2 1 + kLk+p 1

1

:

1 If L1 6= 0 ) Lk+p = r k 1 < 0; which is a contradiction, so L1 = 0; which implies 2 that L2 = 0: So, limn!1 yn = 0; which the proof is complete.

Theorem 2.3. Assume that r = 1; then Eq.(2.1) possesses the prime period two solution :::; ; 0; ; 0; ::: (2.2) with > 0: Furthermore, every solution of Eq.(2.1) converges to a period two solution (2.2) with 0: 6

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Proof: Let :::; ; ; ; ; ::: be period two solutions of Eq.(2.1). Then =

r 1+k

k+p 1

; and

r 1+k

=

k+p 1

;

so, k

=

(r

1)(

)

k+p 2

k+p 2

0;

If k + p > 2, then we have r 1 0: If r < 1; then this implies that < 0 or < 0; which is impossible, so r = 1: If k + p < 2, then we have r 1 0: If r > 1; then we have either = = 0; which is impossible or = = 1 r 1 k+p 1 ; which is impossible, so r = 1: k If k + p = 2, then we have(r 1)( ) = 0; which implies that r = 1: To complete the proof, assume that r = 1 and let fyn g1 n= 2k be a solution of Eq.(2.1), then ! k k P Q yn 1 ynp 12mi yn 2mj yn+1

yn

1

i=1

=

1+

k P

i=1

j=1

ynp 12mi

k Q

0;

yn

n = 0; 1; 2; :::

2mj

j=1

So, the even terms of this solution decrease to a limit (say odd terms decrease to a limit (say 0). Thus, = which implies that k

1+k

k+p 1

k+p 1

and

=

k+p 1

= 0 and k

1+k

k+p 1

0), and the

;

= 0: Then the proof is complete.

Theorem 2.4. Assume that r > 1; and let fyn g1 n= such that

2mk

be a solution of Eq.(2.1)

y

2mk ; y 2mk +2 ; :::; y0

y2 and y

2mk +1 ; y 2mk +3 ; :::; y 1

< y2 ;

(2.3)

y

2mk ; y 2mk +2 ; :::; y0

< y2 and y

2mk +1 ; y 2mk +3 ; :::; y 1

y2 :

(2.4)

or Then fyn g1 n=

2mk oscillates

about y2 =

r 1 k

1 k+p 1

with a semicycle of length one.

7

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Proof: Assume that (2.3) holds. (The case where (2.4) holds is similar and will be omitted.) Then, ry

y1 = 1+

k P

i=1

and

1

1 y p 2m i

k Q

< y

1+

k P

i=1

1 y p 2m i +1

1

= y2

2mj

j=1

ry0

y2 =

ry2 1 + ky2 k+p

k Q

> y

ry2 1 + ky2 k+p

= y2

1

2mj +1

j=1

and then the proof follows by induction. Theorem 2.5. Assume that r > 1; then Eq.(2.1) possesses an unbounded solution. Proof: From Theorem 2.4, we can assume without loss of generality that the solution fyn g1 n= 2k of Eq.(2.1) is such that y2n

1

< y2 =

r

1

1 k+p 1

and

k

Then

ry2n

y2n+1 = 1+

k P

i=1

and y2n+2 = 1+

k P

i=1

p 1 y2n 2mi

y2n > y2 = 1 k Q

lim y2n = 1

1 k+p 1

;

k

1

= y2n

for n

mk +1:

1

2mj

j=1

p 1 y2n 2mi +1

n!1

y2n

1

ry2n 1 1 + ky2 k+p


y2n

ry2n 1 + ky2 k+p

1

= y2n

2mj +1

j=1

and

lim y2n+1 = 0:

n!1

Then, the proof is complete. Acknowledgement: The author would like to express his gratitude to the anonymous referees of Journal of Computational Analysis and Applications for their interesting remarks.

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References [1] H. M. El-Owaidy, A. M. Ahmed and A. M. Youssef, The dynamics of the xn 1 recursive sequence xn+1 = ; Appl. Math. Lett., vol. 18, no. 9, pp. + xpn 2 1013–1018, 2005. [2] A. M. Ahmed, On the dynamics of a higher order rational di¤erence equation, Discrete Dynamics in Nature and Society, vol. 2011, Article ID 419789, 8 pages, doi:10.1155/2011/419789. [3] A. M. Ahmed and Ibrahim M. Ahmed, On the dynamics of a rational di¤erence equation, J. Pure and Appl. Math. Advances and Appliications 18 (1) (2017), 25-35. [4] C. Cinar: On the positive solutions of the di¤erence equation xn+1 = xn 1 ; Appl. Math. Comp. 150, 21-24(2004). 1 + xn xn 1 xn 1 ; Appl. Math. [5] C. Cinar: On the di¤erence equation xn+1 = 1 + xn xn 1 Comp.158, 813-816(2004). [6] C. Cinar, On the positive solutions of the di¤erence equation xn+1 = axn 1 ; Appl. Math. Comp., 156 (2004) 587-590. 1 + bxn xn 1 [7] C. Cinar,R. Karatas ,I. Yalcinkaya: On solutions of the di¤erence equaxn 3 tion xn+1 = ; Mathematica Bohemica. 132 (3), 2571 + xn xn 1 xn 2 xn 3 261(2007). [8] E. M. Elabbasy, H. El-Metwally and E. M. Elsayed, Global attractivity and periodic character of a fractional di¤erence equation of order three, Yokohama Math. J., 53 (2007), 89-100. [9] E. M. Elabbasy, H. El-Metwally and E. M. Elsayed, On the di¤erence equaxn k ; J. Conc. Appl. Math., 5(2) (2007), 101-113. tions xn+1 = Qk + i=0 xn i

[10] E. M. Elabbasy, H. El-Metwally and E. M. Elsayed, Qualitative behavior of higher order di¤erence equation, Soochow Journal of Mathematics, 33 (4) (2007), 861-873. 9

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[11] E. M. Elabbasy, H. El-Metwally and E. M. Elsayed, On the Di¤erence Equa1 xn 1 +:::+ak xn k tion xn+1 = ab00xxnn+a , Mathematica Bohemica, 133 (2) (2008), +b1 xn 1 +:::+bk xn k 133-147. [12] E. M. Elabbasy and E. M. Elsayed, On the Global Attractivity of Di¤erence Equation of Higher Order, Carpathian Journal of Mathematics, 24(2) (2008), 45-53. [13] M. Emre Erdogan, Cengiz Cinar, I. Yalç¬nkaya, On the dynamics of the recursive sequence xn+1 = + x2 xnxn4 +1 xn 2 x2 ; Comp. & Math. Appl. Math. n 2 n 4 61, 2011, 533–537. [14] V. L. Kocic and G. Ladas, Global Behavior of Nonlinear Di¤erence Equations of Higher Order with Applications, Kluwer Academic Publishers, Dordrecht, 1993. [15] M. R. S. Kulenovic and G. Ladas, Dynamics of Second Order Rational Difference Equations with Open Problems and Conjectures, Chapman & Hall / CRC Press, 2001.

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Weighted composition operator acting between some classes of analytic function spaces A. El-Sayed Ahmed 1,2 1 Taif University, Faculty of Science, Math. Dept. Taif, Saudi Arabia 2 Sohag University, Faculty of Science, Math. Dept. Egypt e-mail: [email protected] and Aydah Al-Ahmadi3 3 Jouf University Colleague of Sciences and arts at Al-Qurayyat- Mathematics Department, Jouf- Saudi Arabia

Abstract In this paper, we define some general classes of weighted analytic function spaces in the unit disc. For the new classes, we investigate boundedness and compactness of the weighted composition operator uCφ under some mild conditions on the weighted functions of the classes.

1

Introduction

Let H(D) denote the class of analytic functions in the unit disk D. As usual, two quantities Lf and Mf , both depending on analytic function f on the unit disk D, are said to be equivalent, and written in the form Lf ≈ Mf , if there exists a positive constant C such that 1 Mf ≤ Lf ≤ C Mf . C The notation A . B means that there exists a positive constant C1 such that A ≤ C1 B. For 0 < α < ∞. The weighted type space Hα∞ is the space of all f ∈ H(D) such that kf kHα∞ = sup(1 − |z|2 )α |f (z)| < ∞. z∈D

∞ ∞ ∞ and Hα, 0 denotes the closed subspace of Hα such that f ∈ Hα satisfies

(1 − |z|2 )α |f (z)| → 0 as |z| → 1. az 1 1−¯ az Let the Green’s function g(z, a) = ln | 1−¯ obius transfora−z | = ln |ϕa (z)| , where ϕa (z) = a−z stands for M¨ mation. The following classes of weighted function spaces are defined in [7]:

Definition 1.1 Let K : [0, ∞) → [0, ∞) be a nondecreasing function and let f be an analytic function in D then f ∈ NK if Z 2 kf kNK = sup |f (z)|2 K(g(z, a))dA(z) < ∞, a∈D

D

AMS 2010 classification: 30H30, 30C45, 46E15. Key words and phrases: analytic classes, weighted composition operators.

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2 where dA(z) defines the normalized area measure on D, so that A(D) ≡ 1. Now, if Z lim |f (z)|2 K(g(z, a))dA(z) = 0, |a|→1

D

then f is said to belong to the class NK,0 . Clearly, if K(t) = tp , then NK = Np (see [19]) , since g(z, a) ≈ (1 − |ϕa (z)|2 ). For K(t) = 1 it gives the Bergman space A2 (see [17]). It is easy to check that k · kNK is a complete semi-norm on NK and it is M¨obius invariant in the sense that kf ◦ ϕa kNK = kf kNK , a ∈ D, whenever f ∈ NK and ϕa ∈ Aut(D) is the group of all M¨obius maps of D. If NK consists of just the constant functions, we say that it is trivial. We assume from now that all K : [0, ∞) → [0, ∞) to appear in this paper is right-continuous and nondecreasing function such that the integral Z

Z

1/e

∞

K(log(1/ρ))ρ dρ = 0

K(t)e−2t dt < ∞.

1

From a change of variables we see that the coordinate function z belongs to NK space if and only if Z ¡ ¢ (1 − |a|2 )2 K log(1/|z|) d A(z) < ∞. sup 4 |1 − a ¯ z| a∈D D Simplifying the above integral in polar coordinates, we conclude that NK space is nontrivial if and only if Z 1 ¡ ¢ (1 − t)2 K log(1/r) rdr < ∞. sup (1) 2 )3 (1 − tr t∈(0,1) 0 An important tool in the study of NK space is the auxiliary function φK defined by K(st) , 0 < s < ∞. 0 0, also suppose that condition (4) is satisfied and ∞ α ∈ (0, ∞). Then uCφ : NK, ω → Hα, ω is bounded if and only if sup z∈D

|u(z)|(1 − |z|2 )α < ∞. (1 − |φ(z)|2 )ω(1 − |z|2 ))

(5)

Proof: First assume that (5) holds. Then ¯ ¯ ¯ (1 − |z|2 )α ¯ ¯ ¯ = sup |u(z)|f (φ(z))¯ ω(1 − |z|2 ) ¯ z∈D

∞ kuCφ f kHα, ω

.

sup z∈D

|u(z)|(1 − |z|2 )α (1 − |φa (z)|2 ) sup |f (φ(z))| 2 2 (1 − |φa (z)| )ω(1 − |z| ) z∈D ω(1 − |φ(z)|2 )

.

∞ kf kH1, sup ω

≤

λkf kNK, ω ,

z∈D

|u(z)|(1 − |z|2 )α (1 − |φa (z)|2 )ω(1 − |φa (z)|2 )

where λ is a positive constant. ∞ Conversely, assume that uCφ : NK, ω → Hα, ω is bounded, then ∞ kuCφ f kHα, . kf kNK, ω . ω

Fix a z0 ∈ D, and let hw be the test function in Lemma 2.4 with w = φ(z0 ). Then 1 & khw kNK, ω

≥ ≥ =

∞ λ1 kuCφ hw kHα, ω

|u(z0 )|(1 − |w|2 ) (1 − |z0 |2 )α |1 − wφa (z0 )|2 )ω(1 − |z0 |2 ) |u(z0 )|(1 − |z0 |2 )α , (1 − |φa (z0 |2 ))ω(1 − |z0 |2 )

where λ1 is a positive constant. The proof of Theorem 3.1 is therefore established. Theorem 3.2 Let u ∈ H(D), suppose that ω : (0, 1] −→ [0, ∞), K : [0, ∞) → [0, ∞) be nondecreasing right continuous functions with ω(kt) = kω(t) , k > 0, also suppose that condition (4) is satisfied and ∞ α ∈ (0, ∞). Then the weighted composition operator uCφ : Hα, ω → NK, ω is bounded if and only if Z sup a∈D

D

|u(z)|2 ω 2 (1 − |φ(z)|2 ) K(g(z, a))dA(z) < ∞. (1 − |φ(z)|2 )2α (ω 2 (1 − |z|2 ))

(6)

Proof: First we assume that condition (6) holds and let Z |u(z)|2 ω 2 (1 − |φ(z)|2 ) sup K(g(z, a))dA(z) < C, 2 2α 2 2 a∈D D (1 − |φ(z)| ) (ω (1 − |z| )) 305

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7 ∞ where C is a positive constant. If f ∈ Hα, ω , then for all a ∈ D we have

Z kuCφ f kNK, ω

|u(z)|2 |f (φ(z)|2

= sup a∈D

D

Z

K(g(z, a) dA(z) ω 2 (1 − |z|2 )

|u(z)|2 (1 − |φ(z)|2 )2α |f (φ(z)|2 ω 2 (1 − |φ(z)|2 )K(g(z, a)) . dA(z) 2 2 ω 2 (1 − |φ(z)|2 ) (1 − |φ(z)|2 )2α a∈D D ω (1 − |z| ) Z |u(z)|2 ω 2 (1 − |φ(z)|2 ) ≤ kf k2Hα, sup K(g(z, a))dA(z) ∞ 2 2α ω 2 (1 − |z|2 ) ω a∈D D (1 − |φ(z)| ) = sup

≤ Ckf k2Hα, ∞ . ω ∞ Conversely, assume that uCφ : Hα, ω → NK, ω is bounded, then

kuCφ f k2NK, ω . kf k2Hα, ∞ . ω fixing a point z0 ∈ D , with w = φ(z0 ) then we set that ¢ ¡ ω 1 − wφ(z0 ) fw (z) = , (1 − wz)α ∞ . 1. Then, it is easy to check that kfw kHα, ω

Z kuCφ fw k2NK, ω

D

|u(z0 )|2 ω 2 (1 − |φ(z0 )|2 )K(g(z0 , a)) dA(z0 ) ¢2α ¡ 1 − φ(z0 )φ(z0 ) ω 2 (1 − |z0 |2 )

D

|u(z0 )|2 ω 2 (1 − |φ(z0 )|2 )K(g(z0 , a)) dA(z0 ) ¡ ¢2α 1 − |φ(z0 )|2 ω 2 (1 − |z0 |2 )

= sup a∈D

=

Z

sup a∈D

.

kfw k2Hα, ∞ . ω

Theorem 3.3 Let u ∈ H(D), suppose that ω : (0, 1] −→ [0, ∞), K : [0, ∞) → [0, ∞) be nondecreasing right continuous functions with ω(kt) = kω(t) , k > 0, also suppose that condition (4) is satisfied and ∞ α ∈ (0, ∞). Then, the operator uCφ : NK, ω → Hα, ω is compact if and only if lim

sup

r→1 |φ(z)|>r

|u(z)|(1 − |z|2 )α = 0. (1 − |φ(z)|2 )ω(1 − |z|2 )

(7)

∞ Proof: First assume that uCφ : NK, ω → Hα, ω is compact and suppose that there exists ε0 > 0 a sequence (zn ) ⊂ D such that

|u(zn )|(1 − |zn |2 )α ≥ ε0 (1 − |φ(zn )|2 )ω(1 − |zn |2 )

whenever |φ(zn )| > 1 −

1 . n

Clearly, we can assume that wn = φ(zn ) −→ w0 ∈ ∂D as n → ∞. 2

(1 − |wn | ) be the test function in Lemma 2.4. Then hwn → hw0 with respect to the compact (1 − wn z)2 open topology. Define fn = hwn − hw0 . Then kfn kNK, ω ≤ 1 (see Lemma 2.4) and fn → 0 uniformly on ∞ compact subsets of D. Thus, ufn ◦ φ → 0 in Hα, ω by assumption. But, for n big enough, we obtain Let hwn =

∞ kuCφ fn kHα, ω

¯ ¯ (1 − |zn |2 )α ≥ |u(zn )|¯hwn (φ(zn )) − hw0 (φ(zn ))¯ ω(1 − |zn |2 ) ¯ ¯ 2 α ¯ (1 − |wn |2 )(1 − |w0 |2 ) ¯¯ |u(zn )|(1 − |zn | ) ¯ ¯ ¯ 1− ≥ ¯, ¯1 − w0 wn ¯ (1 − |φ(zn )|2 )ω(1 − |zn |2 ) ¯ {z } | | {z } ≥ ε0

=1

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8 which is a contradiction. Conversely, assume that for all ε > 0 there exists r ∈ (0, 1) such that |u(z)|(1 − |z|2 )α < ε whenever |φ(z)| > r. (1 − |φ(z)|2 )ω(1 − |z|2 ) Let (fn )n be a bounded sequence in NK, ω norm which converges to zero on compact subsets of D. Clearly, we may assume that |φ(z)| > r. Then ∞ kuCφ fn kHα, ω

(1 − |z|2 )α ω(1 − |z|2 ) z∈D ¯ ¯ |u(z)|(1 − |z|2 )α ¯fn (φ(z))¯(1 − |φ(z)|2 ). sup 2 )ω(1 − |z|2 ) (1 − |φ(z)| z∈D

= sup |u(z)||fn (φ(z))| =

It is not hard to show that ∞ ||.||H1, . ||.||NK,ω . ω

Thus, we obtain that ∞ ∞ kuCφ fn kHα, ≤ ε kfn kH1, ≤ ε kfn kNK, ω ≤ ε. ω ω

It follows that uCφ is a compact operator. This completes the proof of the theorem. Remark 3.1 It is still an open problem to extend the results of this paper in Clifford analysis, for several studies of function spaces in Clifford analysis, we refer to [1, 2, 3, 4, 5, 6] and others. Remark 3.2 It is still an open problem to study properties for differences of weighted composition oper∞ ators between NK, ω and Hα, ω classes. For more information of studying differences of weighted composition operators, we refer to [14, 22, 23, 26] and others.

References [1] A. El-Sayed Ahmed, On weighted α-Besov spaces and α-Bloch spaces of quaternion-valued functions, Numer. Func. Anal. Optim.29(2008), 1064-1081. [2] A. El-Sayed Ahmed, Lacunary series in quaternion B p, q - spaces, Complex Var. Elliptic Equ, 54(7)(2009), 705-723. [3] A. El-Sayed Ahmed, Lacunary series in weighted hyperholomorphic B p; q (G) spaces, Numer. Funct. Anal. Optim, 32(1)(2011), 41-58. [4] A. El-Sayed Ahmed, Hyperholomorphic Q-classes, Math. Comput. Modelling, 55(2012) 1428-1435. [5] A. El-Sayed Ahmed and A. Ahmadi, On weighted Bloch spaces of quaternion-valued functions, AIP Conference Proceedings, 1389(2011), 272-275. [6] A. El-Sayed Ahmed, K. Gu¨rlebeck, L. F.Res´ ndis and L.M. Tovar, Characterizations for the Bloch space by B p; q spaces in Clifford analysis, Complex. var. Elliptic. Equ., l51(2)(2006),119- 136. [7] A. El-Sayed Ahmed and M. A. Bakhit, Holomorphic NK and Bergman-type spaces, Birkhuser Series on Oper Theo Adva.Appl. (2009), Birkhuser Verlag Publisher Basel-Switzerland, 195 (2009), 121138. [8] F. Colonna, Weighted composition operators between and BMOA, Bull. Korean Math. Soc. 50(1)(2013), 185-200. [9] C. C. Cowen and B. D. Maccluer, Composition Operators on Spaces of Analytic Functions, Studies in Advanced Mathematics. CRC Press, Boca Raton, FL, 1995. 307

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9 [10] K. M. Dyakonov, Weighted Bloch spaces, H p ,and BMOA, J. Lond. Math. Soc. 65(2)(2002), 411-417. [11] M. Ess´en, H. Wulan, and J. Xiao, Several Function- theoretic aspects characterizations of M¨obius invariant QK -spaces, J. Funct. Anal. 230 (2006), 78-115. [12] O. Hyv¨arinen and M. Lindstr¨om, Estimates of essential norms of weighted composition operators between Bloch-type spaces, J. Math. Anal. Appl. 393(1)(2012), 38-44. [13] A. S. Kucik, Weighted composition operators on spaces of analytic functions on the complex halfplane, Complex Anal. Oper. Theory 12(8)(2018), 1817-1833. [14] Y. Liang, New characterizations for differences of weighted differentiation composition operators from a Bloch-type space to a weighted-type space, Period. Math. Hung. 77(1)(2018), 119-138. [15] H. Li and T. Ma, Generalized composition operators from Bµ spaces to QK,ω (p, q) spaces, Abstr. Appl. Anal. Volume 2014, Special Issue (2014), Article ID 897389, 6 pages. [16] J. S. Manhas and R. Zhao, New estimates of essential norms of weighted composition operators between Bloch type spaces, J Math. Anal. Appl. 389(1)(2012), 32-47. [17] V. G. Miller and T. L. Miller, The Ces´aro operator on the Bergman space, Arch. Math. 78 (2002), 409-416. [18] A. Montes-Rodriguez, Weighted composition operators on weighted Banach spaces of analytic functions, J. Lond. Math. Soc. 61(3)(2000), 872-884. [19] N. Palmberg, Composition operators acting on Np - spaces, Bull. Belg. Math. Soc. Simon Stevin, 14 (2007), 545 - 554. [20] J .B. Garnett, Bounded analytic functions, Academic Press, New York, (1981). [21] R. A. Rashwan, A. El-Sayed Ahmed and A. Kamal, Some characterizations of weighted holomorphic Bloch space, Eur. J. Pure Appl. Math, 2 (2009), 250-267. [22] X. Song and Z. Zhou, Differences of weighted composition operators from Bloch space to H 1 on the unit ball, J. Math. Anal. Appl. 401(1)(2013), 447-457. [23] M. H. Shaabani, Fredholmness of multiplication of a weighted composition operator with its adjoint on H 2 and A2α , J. Inequal. Appl. (2018): 23. [24] D. Thompson, Normaloid weighted composition operators on H 2 , J. Math. Anal. Appl. 467(2)(2018), 1153-1162. [25] P. T. Tien and L. H. Khoi, Weighted composition operators between different Fock spaces, Potential Anal. 50(2)(2019), 171-195. [26] P. T. Tien and L. H. Khoi, Differences of weighted composition operators between the Fock spaces, Monatsh. Math. 188(1)(2019), 183-193. [27] S. Ueki, Weighted composition operators on the Fock space, Proc. Amer. Math. Soc. 135 (2007), 1405-1410. [28] X. Zhang and J. Xiao, Weighted composition operator between two analytic function spaces, Adv. Math. 35(4) (2006), 477-486. [29] L. Zhang and H. Zeng, Weighted differentiation composition operators from weighted Bergman space to nth weighted space on the unit disk, J. Ineq. Appl. (2011) :65.

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1

HERMITE-HADAMARD TYPE INEQUALITIES FOR THE ABK-FRACTIONAL INTEGRALS ARTION KASHURI

Abstract. The author introduced the new fractional integral operator called ABK-fractional integral and proved four identities for this type. By applying the established identities, some integral inequalities connected with the right hand side of the Hermite-Hadamard type inequalities for the ABK-fractional integrals are given. Various special cases have been identified. The ideas of this paper may stimulate further research in the field of integral inequalities.

1. Introduction The class of convex functions is well known in the literature and is usually defined in the following way: Definition 1.1. Let I be an interval in R. A function f : I −→ R, is said to be convex on I if the inequality f (λe1 + (1 − λ)e2 ) ≤ λf (e1 ) + (1 − λ)f (e2 ) (1.1) holds for all e1 , e2 ∈ I and λ ∈ [0, 1]. Also, we say that f is concave, if the inequality in (1.1) holds in the reverse direction. The following inequality, named Hermite-Hadamard inequality, is one of the most famous inequalities in the literature for convex functions. Theorem 1.2. Let f : I ⊆ R −→ R be a convex function and e1 , e2 ∈ I with e1 < e2 . Then the following inequality holds:   Z e2 e1 + e2 1 f (e1 ) + f (e2 ) f ≤ . (1.2) f (x)dx ≤ 2 e 2 − e 1 e1 2 This inequality (1.2) is also known as trapezium inequality. The trapezium inequality has remained an area of great interest due to its wide applications in the field of mathematical analysis. Authors of recent decades have studied (1.2) in the premises of newly invented definitions due to motivation of convex function. Interested readers see the references [2],[4]-[20],[22]-[27]. In [8], Dragomir and Agarwal proved the following results connected with the right part of (1.2). Lemma 1.3. Let f : I ◦ ⊆ R → R be a differentiable mapping on I ◦ , e1 , e2 ∈ I ◦ with e1 < e2 . If f 0 ∈ L[e1 , e2 ], then the following equality holds: Z e2 Z f (e1 ) + f (e2 ) 1 (e2 − e1 ) 1 − f (x)dx = (1 − 2t)f 0 (te1 + (1 − t)e2 )dt. (1.3) 2 e2 − e1 e1 2 0 1

2010 Mathematics Subject Classification: Primary: 26A09; Secondary: 26A33, 26D10, 26D15, 33E20. Key words and phrases. Hermite-Hadamard inequality, H¨ older inequality, power mean inequality, Katugampola fractional integral, AB-fractional integrals. 1

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2

A. KASHURI

Theorem 1.4. Let f : I ◦ ⊆ R → R be a differentiable mapping on I ◦ , e1 , e2 ∈ I ◦ with e1 < e2 . If |f 0 | is convex on [e1 , e2 ], then the following inequality holds: Z e2 f (e1 ) + f (e2 ) (e2 − e1 ) 1 f (x)dx ≤ − (|f 0 (e1 )| + |f 0 (e2 )|) . (1.4) 2 e2 − e1 8 e1

Now, let us recall the following definitions. Definition 1.5. Xcp (e1 , e2 ) (c ∈ R), 1 ≤ p ≤ ∞ denotes the space of all complex-valued Lebesgue measurable functions f for which kf kXcp < ∞, where the norm k · kXcp is defined by Z e2 1 p dt p c p t f (t) kf kXc = (1 ≤ p < ∞) t e1 and for p = ∞ kf kXc∞ = ess sup tc f (t) . e1 ≤t≤e2

Recently, in [12], Katugampola introduced a new fractional integral operator which generalizes the Riemann-Liouville and Hadamard fractional integrals as follows: Definition 1.6. Let [e1 , e2 ] ⊂ R be a finite interval. Then, the left and right side Katugampola fractional integrals of order α (> 0) of f ∈ Xcp (e1 , e2 ) are defined by Z tρ−1 ρ1−α x ρ α f (t)dt, x > e1 (1.5) Ie+ f (x) = 1 Γ(α) e1 (xρ − tρ )1−α and

ρ1−α = Γ(α) where ρ > 0, if the integrals exist. ρ α Ie− f (x) 2

Z

e2

x

tρ−1 f (t)dt, x < e2 , (tρ − xρ )1−α

(1.6)

In [3], Atangana and Baleanu produced two new fractional derivatives based on the Caputo and the Riemann-Liouville definitions of fractional order derivatives. They declared that their fractional derivative has a fractional integral as the antiderivative of their operators. The Atangana-Baleanu (AB) fractional order derivative is known to possess nonsingularity as well as nonlocality of the kernel, which adopts the generalized Mittag-Leffler function, see [15],[21]. Definition 1.7. The fractional AB-integral of the function f ∈ H ∗ (e1 , e2 ) is given by Z t 1−ν ν ν−1 AB ν I f (t) = f (t) + (t − u) f (u)du, t > e1 , e1 t B (ν) B (ν) Γ (ν) e1

(1.7)

where e1 < e2 , 0 < ν < 1 and B (ν) > 0 satisfies the property B (0) = B (1) = 1. Similarly, we give the definition of the (1.7) opposite side is given by Z e2 1−ν ν ν−1 AB ν I f (t) = f (t) + (u − t) f (u)du, t < e2 . e2 t B (ν) B (ν) Γ (ν) t Here, Γ(ν) is the Gamma function. Since the normalization function B (ν) > 0 is positive, it immediately follows that the fractional AB-integral of a positive function is positive. It should be noted that, when the order ν → 1, we recover the classical integral. Also, the initial function is recovered whenever the fractional order ν → 0. Motivated by the above literatures, the main objective of this paper is to establish some new estimates for the right hand side of Hermite-Hadamard type integral inequalities for new fractional integral operator called the ABK-fractional integral operator. Various special cases will be identified. The ideas of this paper may stimulate further research in the field of integral inequalities.

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3

2. Hermite-Hadamard inequalities for ABK-fractional integrals Now, we are in position to introduce the left and right side ABK-fractional integrals as follows. Definition 2.1. Let [e1 , e2 ] ⊂ R be a finite interval. Then, the left and right side ABKfractional integrals of order ν ∈ (0, 1) of f ∈ Xcp (e1 , e2 ) are defined by Z t 1−ν uρ−1 ρ1−ν ν ABK ρ ν I f (t) = f (u)du, t > e1 ≥ 0 (2.1) f (t) + + t ρ e1 B (ν) B(ν)Γ(ν) e1 (t − uρ )1−ν and ABK ρ ν It f (t) e− 2

1−ν ρ1−ν ν = f (t) + B (ν) B(ν)Γ(ν)

Z t

e2

uρ−1 f (u)du, t < e2 , (uρ − tρ )1−ν

(2.2)

where ρ > 0 and B (ν) > 0 satisfies the property B (0) = B (1) = 1. Remark 2.2. Since the normalization function B (ν) > 0 is positive, it immediately follows that the fractional ABK-integral of a positive function is positive. It should be noted that, when the ρ → 1, we recover the AB-fractional integral. Also, using the same idea as in [12], the ABKfractional integral operators are well-defined on Xcp (e1 , e2 ) . Finally, using the same idea as in [1], the interested reader can find new nonlocal fractional derivative of it with Mittag-Leffler nonsingular kernel, several formulae and many applications. Let represent Hermite-Hadamard’s inequalities in the ABK-fractional integral forms as follows: Theorem 2.3. Let ν ∈ (0, 1) and ρ > 0. Let f : [eρ1 , eρ2 ] → R be a function with 0 ≤ e1 < e2 and f ∈ Xcp (eρ1 , eρ2 ) . If f is a convex function on [eρ1 , eρ2 ], then the following inequalities for the ABK-fractional integrals hold:  ρ  ν 2 (eρ2 − eρ1 ) e1 + eρ2 1−ν f + [f (eρ1 ) + f (eρ2 )] B (ν) Γ (ν + 1) ρ2−ν 2 B (ν) i h ρ ρ ABK ρ ν ABK ρ ν ρ f (e ) ρ f (e ) + I (2.3) I ≤ − + 1 2 e e e2 e 1 2  1ρ  ρ ν (e2 − e1 ) + ρ(1 − ν)Γ(ν) ≤ [f (eρ1 ) + f (eρ2 )] . ρB (ν) Γ(ν) Proof. Let t ∈ [0, 1]. Consider xρ , y ρ ∈ [eρ1 , eρ2 ], defined by xρ = tρ eρ1 + (1 − tρ )eρ2 , y ρ = (1 − tρ )eρ1 + tρ eρ2 . Since f is a convex function on [eρ1 , eρ2 ], we have   ρ f (xρ ) + f (y ρ ) x + yρ . ≤ f 2 2 Then, we get  2f

eρ1 + eρ2 2



≤ f (tρ eρ1 + (1 − tρ )eρ2 ) + f ((1 − tρ )eρ1 + tρ eρ2 ) .

(2.4)

ν Multiplying both sides of (2.4) by B(ν)Γ(ν) tρν−1 , then integrating the resulting inequality with respect to t over [0, 1], we obtain  ρ  2 e1 + eρ2 f ρB (ν) Γ (ν) 2 Z 1 Z 1 ν ν ≤ tρν−1 f (tρ eρ1 + (1 − tρ )eρ2 ) dt + tρν−1 f ((1 − tρ )eρ1 + tρ eρ2 ) dt B (ν) Γ (ν) 0 B (ν) Γ (ν) 0 ν−1 Z e2  ρ ν e 2 − xρ xρ−1 = f (xρ ) ρ dx ρ ρ B (ν) Γ (ν) e1 e2 − e1 e2 − eρ1

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+

ν B (ν) Γ (ν)

Z

e2



e1

y ρ − eρ1 eρ2 − eρ1

ν−1

y ρ−1 dy − eρ1

f (y ρ )

eρ2

Therefore, it follows that  ρ  ν 2 (eρ2 − eρ1 ) e1 + eρ2 1−ν f + [f (eρ1 ) + f (eρ2 )] B (ν) Γ (ν + 1) ρ2−ν 2 B (ν) i h ABK ρ ν ABK ρ ν ρ ρ ρ f (e ) + ρ f (e ) I ≤ I − + 2 1 e e e e 1

2

1

2

and the left hand side inequality of (2.3) is proved. For the proof of the right hand side inequality of (2.3) we first note that if f is a convex function, then f (tρ eρ1 + (1 − tρ )eρ2 ) ≤ tρ f (eρ1 ) + (1 − tρ ) f (eρ2 ) and f ((1 − tρ )eρ1 + tρ eρ2 ) ≤ (1 − tρ ) f (eρ1 ) + tρ f (eρ2 ). By adding these inequalities, we have f (tρ eρ1 + (1 − tρ )eρ2 ) + f ((1 − tρ )eρ1 + tρ eρ2 ) ≤ f (eρ1 ) + f (eρ2 ).

(2.5)

ν ρν−1 B(ν)Γ(ν) t

and integrating the resulting inequality Then multiplying both sides of (2.5) by with respest to t over [0, 1], we obtain Z 1 Z 1 ν ν ρν−1 ρ ρ ρ ρ t f (t e1 + (1 − t )e2 ) dt + tρν−1 f ((1 − tρ )eρ1 + tρ eρ2 ) dt B (ν) Γ (ν) 0 B (ν) Γ (ν) 0 Z 1 ν [f (eρ1 ) + f (eρ2 )] ≤ tρν−1 dt B (ν) Γ (ν) 0 i.e. h

ABK ρ ν Ieρ f (eρ2 ) e+ 2 1

+

i

ABK ρ ν Ieρ f (eρ1 ) e− 1 2

 ≤

(eρ2 − eρ1 )ν + ρ(1 − ν)Γ(ν) ρB (ν) Γ(ν)



[f (eρ1 ) + f (eρ2 )] .

The proof of this theorem is complete.



Corollary 2.4. If we take ρ → 1 in Theorem 2.3, then the following Hermite-Hadamard’s inequalities for the AB-fractional integrals hold:   ν e1 + e2 1−ν 2 (e2 − e1 ) f + [f (e1 ) + f (e2 )] B (ν) Γ (ν + 1) 2 B (ν)  AB ν  AB ν ≤ (2.6) e Ie f (e2 ) + e2 Ie1 f (e1 )   1 2 ν (e2 − e1 ) + (1 − ν)Γ(ν) [f (e1 ) + f (e2 )] . ≤ B (ν) Γ(ν) Remark 2.5. If in Corollary 2.4, we let ν → 1, then the inequalities (2.6) become the inequalities (1.2). 3. The ABK-fractional inequalities for convex functions For establishing some new results regarding the right side of Hermite-Hadamard type inequalities for the ABK-fractional integrals we need to prove the following four lemmas. Lemma 3.1. Let ν ∈ (0, 1) and ρ > 0 and f : [eρ1 , eρ2 ] → R be a differentiable mapping on (eρ1 , eρ2 ) with 0 ≤ e1 < e2 . Then the following equality for the ABK-fractional integrals exist:   ν h i (eρ2 − eρ1 ) 1−ν ρ ν ρ ABK ρ ν ρ ρ f (e ) + ρ f (e ) + [f (eρ1 ) + f (eρ2 )] − ABK I I + − 2 1 e e ν e1 e2 2 1 ρ B (ν) Γ (ν) B (ν)

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(eρ2 − eρ1 ) ρν B (ν) Γ (ν)

=

Z

1

ν

[(1 − tρ ) − tρν ] tρ−1 f 0 (tρ eρ1 + (1 − tρ )eρ2 ) dt.

5

(3.1)

0

Proof. Integrating by parts, we get Z 1 ν I1 = (1 − tρ ) tρ−1 f 0 (tρ eρ1 + (1 − tρ ) eρ2 ) dt 0

= =

1 Z 1 ν ν (1 − tρ ) ν−1 ρ−1 ρ ρ ρ ρ (1 − tρ ) f (t e1 + (1 − t ) e2 ) − ρ t f (tρ eρ1 + (1 − tρ ) eρ2 ) dt ρ ρ(eρ1 − eρ2 ) e − e 1 2 0 0 Z 1 ν f (eρ2 ) ν−1 ρ−1 (1 − tρ ) t f (tρ eρ1 + (1 − tρ ) eρ2 ) dt. − ρ(eρ2 − eρ1 ) eρ1 − eρ2 0

Similarly, Z I2

=

1

tρ(ν+1)−1 f 0 (tρ eρ1 + (1 − tρ ) eρ2 ) dt

0

1 Z 1 tρ(ν+1)−1 ν ρ ρ ρ ρ f (t e + (1 − t ) e ) − tρ(ν+1) f (tρ eρ1 + (1 − tρ ) eρ2 ) dt ρ ρ 1 2 ρ(eρ1 − eρ2 ) e − e 1 2 0 0 Z 1 ν f (eρ1 ) = − ρ − tρ(ν+1) f (tρ eρ1 + (1 − tρ ) eρ2 ) dt. ρ(e2 − eρ1 ) eρ1 − eρ2 0

=

ν+1

(eρ2 − eρ1 ) , using definition of the ABK-fractional ρν B (ν) Γ (ν) integrals and subtracting them, we get the result.  Thus, by multiplying I1 and I2 with

Remark 3.2. If in Lemma 3.1, we let ρ → 1, then we get the following equality for the ABfractional integrals:   ν   (e2 − e1 ) 1−ν ν AB ν + [f (e1 ) + f (e2 )] − AB e1 Ie2 f (e2 ) + e2 Ie1 f (e1 ) B (ν) Γ (ν) B (ν) ν+1 Z 1 (e2 − e1 ) ν = [(1 − t) − tν ] f 0 (te1 + (1 − t)e2 ) dt. (3.2) B (ν) Γ (ν) 0 Remark 3.3. If in Lemma 3.1, we let ρ, ν → 1, then we obtain the equality (1.3). Lemma 3.4. Let ν ∈ (0, 1) and ρ > 0 and f : [eρ1 , eρ2 ] → R be a differentiable mapping on (eρ1 , eρ2 ) with 0 ≤ e1 < e2 . Then the following equality for the ABK-fractional integrals exist:   ν h i 1−ν (eρ2 − eρ1 ) ρ ρ ABK ρ ν ρ ABK ρ ν ρ ρ f (e ) + ρ f (e ) + [f (e ) + f (e )] − I I + − 1 2 2 1 e e e1 e2 2 1 ρν B (ν) Γ (ν) B (ν) ρ ρ ν+1 Z 1   (e2 − e1 ) = tρ(ν+1)−1 f 0 ((1 − tρ )eρ1 + tρ eρ2 ) − f 0 (tρ eρ1 + (1 − tρ )eρ2 ) dt. (3.3) ν−1 ρ B (ν) Γ (ν) 0 Proof. The proof is similarly as Lemma 3.1, so we omit it.



Remark 3.5. If in Lemma 3.4, we let ρ → 1, then we get the following equality for the ABfractional integrals:   ν   (e2 − e1 ) 1−ν ν AB ν + [f (e1 ) + f (e2 )] − AB e1 Ie2 f (e2 ) + e2 Ie1 f (e1 ) B (ν) Γ (ν) B (ν) ν+1 Z 1   (e2 − e1 ) = tν f 0 ((1 − t)e1 + te2 ) − f 0 (te1 + (1 − t)e2 ) dt. (3.4) B (ν) Γ (ν) 0

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Lemma 3.6. Let ν ∈ (0, 1) and ρ > 0 and f : [eρ1 , eρ2 ] → R be a twice differentiable mapping on (eρ1 , eρ2 ) with 0 ≤ e1 < e2 . Then the following equality for the ABK-fractional integrals exist:   ν h i (eρ2 − eρ1 ) 1−ν ABK ρ ν ABK ρ ν ρ ρ ρ ρ ρ f (e ) + ρ f (e ) )] − I I ) + f (e + [f (e + − 2 2 1 1 e2 e1 e1 e2 ρν B (ν) Γ (ν) B (ν) ( Z ν+2 1  ν (eρ2 − eρ1 ) = 1 − tρ(ν+1) tρ−1 f 00 ((1 − tρ )eρ1 + tρ eρ2 ) dt × ρν−1 B (ν) Γ (ν + 2) 0 ) Z 1 ρ(ν+2)−1 00 ρ ρ ρ ρ − t f (t e1 + (1 − t )e2 ) dt . 0

Proof. By using twice integration by parts the proof is similarly as Lemma 3.1, so we omit it.  Remark 3.7. If in Lemma 3.6, we let ρ → 1, then we get the following equality for the ABfractional integrals:   ν   (e2 − e1 ) 1−ν ν AB ν + [f (e1 ) + f (e2 )] − AB e1 Ie2 f (e2 ) + e2 Ie1 f (e1 ) B (ν) Γ (ν) B (ν) ν+2

ν (e2 − e1 ) B (ν) Γ (ν + 2) (Z Z 1  × 1 − tν+1 f 00 ((1 − t)e1 + te2 ) dt −

=

0

)

1

t

ν+1 00

f (te1 + (1 − t)e2 ) dt .

0

Lemma 3.8. Let ν ∈ (0, 1) and ρ > 0 and f : [eρ1 , eρ2 ] → R be a twice differentiable mapping on (eρ1 , eρ2 ) with 0 ≤ e1 < e2 . Then the following equality for the ABK-fractional integrals exist:   ν h i 1−ν (eρ2 − eρ1 ) ρ ν ρ ν + [f (eρ1 ) + f (eρ2 )] − ABK Ieρ f (eρ2 ) + ABK Ieρ f (eρ1 ) + − ν e e 2 1 ρ B (ν) Γ (ν) B (ν) 1 2 ρ ρ ν+2 Z 1   ν (e2 − e1 ) 1 − (1 − tρ )ν+1 − tρ(ν+1) tρ−1 f 00 (tρ eρ1 + (1 − tρ )eρ2 ) dt. (3.5) = ν ρ B (ν) Γ (ν + 2) 0 Proof. By using twice integration by parts and Lemma 3.1, we get the desired result.



Remark 3.9. If in Lemma 3.8, we let ρ → 1, then we get the following equality for the ABfractional integrals:   ν   1−ν (e2 − e1 ) ν AB ν + [f (e1 ) + f (e2 )] − AB e1 Ie2 f (e2 ) + e2 Ie1 f (e1 ) B (ν) Γ (ν) B (ν) ν+2 Z 1   ν (e2 − e1 ) = 1 − (1 − t)ν+1 − tν+1 f 00 (te1 + (1 − t)e2 ) dt. (3.6) B (ν) Γ (ν + 2) 0 Using Lemmas 3.1, 3.4, 3.6 and 3.8, we can obtain the following the ABK-fractional integral inequalities. Theorem 3.10. Let ν ∈ (0, 1) and ρ > 0 and f : [eρ1 , eρ2 ] → R be a differentiable mapping q on (eρ1 , eρ2 ) with 0 ≤ e1 < e2 . If |f 0 | is convex on [eρ1 , eρ2 ] for q > 1 and p1 + 1q = 1, then the following inequality for the ABK-fractional integrals holds:   i h (eρ2 − eρ1 )ν 1−ν ρ ABK ρ ν ρ ABK ρ ν ρ ρ ρ ρ ) + f (e )] − I f (e ) + I f (e ) + [f (e 2 1 1 2 e1 e2 ρν B (ν) Γ (ν) e− e+ B (ν) 2 1 s ν+1 f 0 (eρ ) q + f 0 (eρ ) q p q ν (eρ2 − eρ1 ) 1 2 p ≤ , (3.7) × D(p, ρ, ν) 1 2 ρν+ q B (ν) Γ (ν)

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where Z D(p, ρ, ν) :=

1 2

Z i h ρ pν pρν ρ−1 t dt + (1 − t ) − t

1

h

7

i tpρν − (1 − tρ )pν tρ−1 dt

1 2

0

(



2 1 = 1− 1− ρ ρ(pν + 1) 2

pν+1 −

1 2ρ(pν+1)

) .

q

Proof. Using Lemma 3.1, convexity of |f 0 | , H¨older inequality and properties of the modulus, we have   h i (eρ2 − eρ1 )ν 1−ν ρ ρ ABK ρ ν ρ ABK ρ ν ρ Ieρ f (e2 ) + e− Ieρ f (e1 ) ρν B (ν) Γ (ν) + B (ν) [f (e1 ) + f (e2 )] − e+ 2 1 1 2 ν+1

ν (eρ2 − eρ1 ) ρν B (ν) Γ (ν)  p1 Z Z 1 p ρ ν ρν ρ−1 × (1 − t ) − t t dt

≤

0

q  q1 ρ ρ ρ ρ f (t e1 + (1 − t )e2 ) dt

1

t

ρ−1 0

0

! p1 Z 21 h Z 1h ν+1 i i ν (eρ2 − eρ1 ) ≤ (1 − tρ )pν − tpρν tρ−1 dt + tpρν − (1 − tρ )pν tρ−1 dt 1 ρν B (ν) Γ (ν) 0 2 Z 1  q1 q q
× tρ−1 tρ f 0 (eρ1 ) + (1 − tρ ) f 00 (eρ2 ) dt 0 s ν+1 f 0 (eρ ) q + f 0 (eρ ) q p q ν (eρ2 − eρ1 ) 2 1 p × D(p, ρ, ν) = . 1 2 ρν+ q B (ν) Γ (ν) The proof of this theorem is complete.



Corollary 3.11. With the notations in Theorem 3.10, if we take |f 0 | ≤ K, the following inequality for the ABK-fractional integrals holds:   h i (eρ2 − eρ1 )ν 1−ν ρ ρ ρ ρ ABK ρ ν ABK ρ ν Ieρ f (e2 ) + e− Ieρ f (e1 ) ρν B (ν) Γ (ν) + B (ν) [f (e1 ) + f (e2 )] − e+ 2 1 1 2 ≤

νK (eρ2 − eρ1 ) ρ

ν+ q1

ν+1

×

p p D(p, ρ, ν).

(3.8)

B (ν) Γ (ν)

Corollary 3.12. With the notations in Theorem 3.10, if we take ρ → 1, the following inequality for the AB-fractional integrals holds:   (e2 − e1 )ν  AB ν  1−ν AB ν B (ν) Γ (ν) + B (ν) [f (e1 ) + f (e2 )] − e1 Ie2 f (e2 ) + e2 Ie1 f (e1 ) s ν+1 f 0 (e1 ) q + f 0 (e2 ) q p q ν (e2 − e1 ) p ≤ × D(p, 1, ν) . (3.9) B (ν) Γ (ν) 2 Theorem 3.13. Let ν ∈ (0, 1) and ρ > 0 and f : [eρ1 , eρ2 ] → R be a differentiable mapping on q (eρ1 , eρ2 ) with 0 ≤ e1 < e2 . If |f 0 | is convex on [eρ1 , eρ2 ] for q ≥ 1, then the following inequality for the ABK-fractional integrals holds:   h i (eρ2 − eρ1 )ν 1−ν ρ ρ ABK ρ ν ρ ABK ρ ν ρ ρ ρ + [f (e ) + f (e )] − I f (e ) + I f (e ) 1 2 2 1 e2 e1 ρν B (ν) Γ (ν) e+ e− B (ν) 1 2 ν+1

≤

ν (eρ2 − eρ1 ) 1− 1 [D(1, ρ, ν)] q ρν B (ν) Γ (ν)

315

(3.10)

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 q q E(ρ, ν) f 0 (eρ1 ) + (F (ρ, ν) − E(ρ, ν)) f 0 (eρ2 )

×

q q + G(ρ, ν) f 0 (eρ1 ) + (F (ρ, ν) − G(ρ, ν)) f 0 (eρ2 )

 q1 ,

where Z

1 2

E(ρ, ν) := 0

h

"  #  i 1 1 1 β ; 2, ν + 1 − ρ(ν+2) ; (1 − tρ )ν − tρν t2ρ−1 dt = ρ 2ρ 2 (ν + 2) 1 2

Z F (ρ, ν) :=

h

1

Z i ρ−1 (1 − t ) − t t dt = ρ ν

ρν

h

i tρν − (1 − tρ )ν tρ−1 dt

1 2

0

" # ν+1  1 1 1 = − ρ(ν+1) ; 1− 1− ρ ρ(ν + 1) 2 2 1

Z G(ρ, ν) :=

1 2

h

t

ρν

" #   1 i 1 1 − 2ρ(ν+2) 1 2ρ−1 − (1 − t ) t dt = +β ; 2, ν + 1 − β(2, ν + 1) , ρ ν+2 2ρ ρ ν

where β(· ; ·, ·), β(·, ·) are respectively the incomplete and complete beta functions and D(1, ρ, ν) is defined as in Theorem 3.10 for value p = 1. q

Proof. Using Lemma 3.1, convexity of |f 0 | , the well-known power mean inequality and properties of the modulus, we have   h i (eρ2 − eρ1 )ν 1−ν ρ ρ ρ ρ ABK ρ ν ABK ρ ν ρ ρ + [f (e ) + f (e I f (e I f (e )] − ) + ) 1 2 2 1 e2 e1 ρν B (ν) Γ (ν) e+ e− B (ν) 1 2 1− q1 ν+1 Z 1 ν (eρ2 − eρ1 ) ρ ν ρν ρ−1 ≤ (1 − t ) − t t dt ρν B (ν) Γ (ν) 0 Z 1 q  q1 ρ ν ρν ρ−1 0 ρ ρ ρ ρ × (1 − t ) − t t f (t e1 + (1 − t )e2 ) dt 0

ν+1

ν (eρ2 − eρ1 ) 1− 1 [D(1, ρ, ν)] q ν ρ B (ν) Γ (ν) (Z 1 i 2 h q q
× (1 − tρ )ν − tρν tρ−1 tρ f 0 (eρ1 ) + (1 − tρ ) f 0 (eρ2 ) dt

≤

0

Z

1

+ 1 2

) q1 h i
ρν ρ ν ρ−1 ρ 0 ρ q ρ 0 ρ q t − (1 − t ) t t f (e1 ) + (1 − t ) f (e2 ) dt ν+1

ν (eρ2 − eρ1 ) 1− 1 [D(1, ρ, ν)] q ρν B (ν) Γ (ν)  q q × E(ρ, ν) f 0 (eρ1 ) + (F (ρ, ν) − E(ρ, ν)) f 0 (eρ2 )

=

q q + G(ρ, ν) f 0 (eρ1 ) + (F (ρ, ν) − G(ρ, ν)) f 0 (eρ2 ) The proof of this theorem is complete.

 q1 . 

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9

Corollary 3.14. With the notations in Theorem 3.13, if we take |f 0 | ≤ K, the following inequality for the ABK-fractional integrals holds:   h i (eρ2 − eρ1 )ν 1−ν ρ ρ ABK ρ ν ρ ABK ρ ν ρ Ieρ f (e2 ) + e− Ieρ f (e1 ) ρν B (ν) Γ (ν) + B (ν) [f (e1 ) + f (e2 )] − e+ 2 1 1 2 ν+1

≤

νK (eρ2 − eρ1 ) ρν B (ν) Γ (ν)

[D(1, ρ, ν)] .

(3.11)

Corollary 3.15. With the notations in Theorem 3.13, if we take ρ → 1, the following inequality for the AB-fractional integrals holds:   (e2 − e1 )ν  AB ν  1−ν AB ν B (ν) Γ (ν) + B (ν) [f (e1 ) + f (e2 )] − e1 Ie2 f (e2 ) + e2 Ie1 f (e1 ) ν+1

ν (e2 − e1 ) 1− 1 [D(1, 1, ν)] q B (ν) Γ (ν)  q q × E(1, ν) f 0 (e1 ) + (F (1, ν) − E(1, ν)) f 0 (e2 )

≤

q q + G(1, ν) f 0 (e1 ) + (F (1, ν) − G(1, ν)) f 0 (e2 )

(3.12)

 q1 .

(3.13)

Theorem 3.16. Let ν ∈ (0, 1) and ρ > 0 and f : [eρ1 , eρ2 ] → R be a differentiable mapping q on (eρ1 , eρ2 ) with 0 ≤ e1 < e2 . If |f 0 | is convex on [eρ1 , eρ2 ] for q > 1 and p1 + 1q = 1, then the following inequality for the ABK-fractional integrals holds:   h i (eρ2 − eρ1 )ν 1−ν ρ ρ ABK ρ ν ρ ABK ρ ν ρ ρ ρ + [f (e ) + f (e )] − I f (e ) + I f (e ) 1 2 2 1 e2 e1 ρν B (ν) Γ (ν) e+ e− B (ν) 1 2 ν+1

1 1 (eρ2 − eρ1 ) √ × p ≤ ν−1 q p ρ B (ν) Γ (ν) ρ +1 p(ρ(ν + 1) − 1) + 1 (q ) q ρ q ρ q ρ q ρ q q q 0 0 0 0 × |f (e1 )| + ρ|f (e2 )| + ρ|f (e1 )| + |f (e2 )| .

(3.14)

q

Proof. Using Lemma 3.4, convexity of |f 0 | , H¨older inequality and properties of the modulus, we have   h i (eρ2 − eρ1 )ν 1−ν ρ ρ ABK ρ ν ρ ABK ρ ν ρ ρ ρ I I + [f (e ) + f (e )] − f (e ) + f (e ) 1 2 2 1 e2 e1 ρν B (ν) Γ (ν) e+ e− B (ν) 1 2 Z 1  p1 ν+1 (eρ2 − eρ1 ) p(ρ(ν+1)−1) ≤ × t dt ρν−1 B (ν) Γ (ν) 0 ( Z q  q1 Z 1 0 ρ ρ ρ ρ × + f (t e1 + (1 − t )e2 ) dt 0

1

0

) q  q1 0 ρ ρ ρ ρ f ((1 − t )e1 + t e2 ) dt

Z 1  p1 ν+1 (eρ2 − eρ1 ) p(ρ(ν+1)−1) × t dt ≤ ρν−1 B (ν) Γ (ν) 0 ( Z  q1 1
ρ 0 ρ q ρ 0 ρ q × t f (e1 ) + (1 − t ) f (e2 ) dt 0

Z +

1

 q1 )
q q ρ ρ (1 − tρ ) f 0 (e1 ) + tρ f 0 (e2 ) dt

0

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(eρ2 − eρ1 ) 1 1 √ × p q p ρν−1 B (ν) Γ (ν) ρ +1 p(ρ(ν + 1) − 1) + 1 ) (q q ρ q ρ q ρ q ρ q q q 0 0 0 0 × |f (e1 )| + ρ|f (e2 )| + ρ|f (e1 )| + |f (e2 )| . =

The proof of this theorem is complete.



Corollary 3.17. With the notations in Theorem 3.16, if we take |f 0 | ≤ K, the following inequality for the ABK-fractional integrals holds:   h i (eρ2 − eρ1 )ν 1−ν ρ ρ ABK ρ ν ρ ABK ρ ν ρ Ieρ f (e2 ) + e− Ieρ f (e1 ) ρν B (ν) Γ (ν) + B (ν) [f (e1 ) + f (e2 )] − e+ 2 1 1 2 ν+1

≤

2K (eρ2 − eρ1 ) 1 . × p p ρν−1 B (ν) Γ (ν) p(ρ(ν + 1) − 1) + 1

(3.15)

Corollary 3.18. With the notations in Theorem 3.16, if we take ρ → 1, the following inequality for the AB-fractional integrals holds:   (e2 − e1 )ν  AB ν  1−ν AB ν B (ν) Γ (ν) + B (ν) [f (e1 ) + f (e2 )] − e1 Ie2 f (e2 ) + e2 Ie1 f (e1 ) r ν+1 0 q 0 q 2 (e2 − e1 ) q |f (e1 )| + |f (e2 )| × . (3.16) ≤ √ p 2 pν + 1B (ν) Γ (ν) Theorem 3.19. Let ν ∈ (0, 1) and ρ > 0 and f : [eρ1 , eρ2 ] → R be a differentiable mapping on q (eρ1 , eρ2 ) with 0 ≤ e1 < e2 . If |f 0 | is convex on [eρ1 , eρ2 ] for q ≥ 1, then the following inequality for the ABK-fractional integrals holds:   h i (eρ2 − eρ1 )ν 1−ν ρ ρ ρ ρ ABK ρ ν ABK ρ ν ρ ρ + [f (e ) + f (e I f (e I f (e )] − ) + ) 1 2 2 1 e2 e1 ρν B (ν) Γ (ν) e+ e− B (ν) 1 2 ν+1

ν (eρ2 − eρ1 ) √ ρν q ν + 2B (ν) Γ (ν + 2) (q ) q ρ q ρ q ρ q ρ q q q 0 0 0 0 × |f (e1 )| + (ν + 1)|f (e2 )| + (ν + 1)|f (e1 )| + |f (e2 )| . ≤

(3.17)

q

Proof. Using Lemma 3.4, convexity of |f 0 | , the well-known power mean inequality and properties of the modulus, we have   h i (eρ2 − eρ1 )ν 1−ν ABK ρ ν ρ ρ ABK ρ ν ρ ρ Ieρ f (e2 ) + e− Ieρ f (e1 ) ρν B (ν) Γ (ν) + B (ν) [f (e1 ) + f (e2 )] − e+ 2 1 1 2 Z 1 1− q1 ν+1 (eρ2 − eρ1 ) ρ(ν+1)−1 ≤ × t dt ρν−1 B (ν) Γ (ν) 0 ( Z q  q1 1 ρ(ν+1)−1 0 ρ ρ ρ ρ × t f (t e1 + (1 − t )e2 ) dt 0

Z + 0

1

) q  q1 ρ ρ tρ(ν+1)−1 f 0 ((1 − tρ )e1 + tρ e2 ) dt ν+1

≤

(eρ2 − eρ1 ) × ν−1 ρ B (ν) Γ (ν)

Z

1

t

ρ(ν+1)−1

1− q1 dt

0

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( Z

1

×

t

ρ(ν+1)−1

ρ 0



t f

q (eρ1 )



11

 q1 0 ρ q
+ (1 − t ) f (e2 ) dt ρ

0 1

Z

t

+

ρ(ν+1)−1

 q1 )
0 ρ q ρ 0 ρ q (1 − t ) f (e1 ) + t f (e2 ) dt ρ

0 ν+1

ν (eρ2 − eρ1 ) √ ρν q ν + 2B (ν) Γ (ν + 2) ) (q q ρ ρ ρ ρ q q × |f 0 (e1 )|q + (ν + 1)|f 0 (e2 )|q + (ν + 1)|f 0 (e1 )|q + |f 0 (e2 )|q . =

The proof of this theorem is complete.



Corollary 3.20. With the notations in Theorem 3.19, if we take |f 0 | ≤ K, the following inequality for the ABK-fractional integrals holds:   i h (eρ2 − eρ1 )ν 1−ν ρ ABK ρ ν ρ ρ ρ ABK ρ ν Ieρ f (e1 ) Ieρ f (e2 ) + e− ρν B (ν) Γ (ν) + B (ν) [f (e1 ) + f (e2 )] − e+ 2 1 2 1 ν+1

≤

2νK (eρ2 − eρ1 ) . ρν B (ν) Γ (ν + 2)

(3.18)

Corollary 3.21. With the notations in Theorem 3.19, if we take ρ → 1, the following inequality for the AB-fractional integrals holds:   (e2 − e1 )ν  AB ν  1−ν AB ν B (ν) Γ (ν) + B (ν) [f (e1 ) + f (e2 )] − e1 Ie2 f (e2 ) + e2 Ie1 f (e1 ) ν+1

ν (e2 − e1 ) ≤ √ q ν + 2B (ν) Γ (ν + 2)

(3.19)

) p p q q × |f 0 (e1 )|q + (ν + 1)|f 0 (e2 )|q + (ν + 1)|f 0 (e1 )|q + |f 0 (e2 )|q . (

Theorem 3.22. Let ν ∈ (0, 1) and ρ > 0 and f : [eρ1 , eρ2 ] → R be a twice differentiable mapping q on (eρ1 , eρ2 ) with 0 ≤ e1 < e2 . If |f 00 | is convex on [eρ1 , eρ2 ] for q > 1 and p1 + 1q = 1, then the following inequality for the ABK-fractional integrals holds:   h i (eρ2 − eρ1 )ν 1−ν ρ ρ ρ ρ ABK ρ ν ABK ρ ν ρ ρ + [f (e ) + f (e I f (e I f (e )] − ) + ) 1 2 2 1 e2 e1 ρν B (ν) Γ (ν) e+ e− B (ν) 1 2 ( s r ν+2 1 p p(ν + 1) q |f 00 (eρ1 )|q + |f 00 (eρ2 )|q ν (eρ2 − eρ1 ) × (3.20) ≤ ν−1 ρ B (ν) Γ (ν + 2) ρ p(ν + 1) + 1 2 s ) 00 ρ q 00 ρ q 1 q |f (e1 )| + ρ|f (e2 )| +p . p ρ+1 p(ρ(ν + 2) − 1) + 1 q

Proof. Using Lemma 3.6, convexity of |f 00 | , H¨older inequality and properties of the modulus, we have   h i (eρ2 − eρ1 )ν 1−ν ABK ρ ν ρ ρ ABK ρ ν ρ ρ ρ ρ Ie f (e2 ) + e− Ie f (e1 ) ρν B (ν) Γ (ν) + B (ν) [f (e1 ) + f (e2 )] − e+ 2 1 1 2 ν+2

≤

ν (eρ2 − eρ1 ) ν−1 ρ B (ν) Γ (ν + 2)

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( Z

1

× 0

 p1 Z p ρ(ν+1) ρ−1 t dt 1 − t

t

+

t

ρ−1 00

0

1

Z

q  q1 ρ ρ ρ ρ f ((1 − t )e1 + t e2 ) dt

1

p(ρ(ν+2)−1)

 p1 Z dt

1

0

0

) q  q1 00 ρ ρ ρ ρ f (t e1 + (1 − t )e2 ) dt

ν+2

ν (eρ2 − eρ1 ) ρν−1 B (ν) Γ (ν + 2) ( Z  p1 Z p 1 ρ(ν+1) ρ−1 × t dt 1 − t ≤

0

t

+

t

ρ−1

 q1 00 ρ q
00 ρ q ρ (1 − t ) f (e1 ) + t f (e2 ) dt ρ

0

1

Z

1

p(ρ(ν+2)−1)

 p1 Z dt

0

1

 q1 )
ρ q ρ 00 ρ q t f (e1 ) + (1 − t ) f (e2 ) dt ρ 00

0

( s r 1 p p(ν + 1) q |f 00 (eρ1 )|q + |f 00 (eρ2 )|q ρ p(ν + 1) + 1 2 s ) 00 ρ q 00 ρ q 1 q |f (e1 )| + ρ|f (e2 )| +p . p ρ+1 p(ρ(ν + 2) − 1) + 1 ν+2

ν (eρ − eρ1 ) = ν−1 2 × ρ B (ν) Γ (ν + 2)

The proof of this theorem is complete.



Corollary 3.23. With the notations in Theorem 3.22, if we take |f 00 | ≤ K, the following inequality for the ABK-fractional integrals holds:   h i (eρ2 − eρ1 )ν 1−ν ρ ρ ρ ρ ABK ρ ν ABK ρ ν ρ ρ + [f (e ) + f (e I f (e I f (e )] − ) + ) 1 2 2 1 e2 e1 ρν B (ν) Γ (ν) e+ e− B (ν) 1 2 ( s ) ν+2 1 p p(ν + 1) 1 νK (eρ2 − eρ1 ) × + p . (3.21) ≤ ν−1 p ρ B (ν) Γ (ν + 2) ρ p(ν + 1) + 1 p(ρ(ν + 2) − 1) + 1 Corollary 3.24. With the notations in Theorem 3.22, if we take ρ → 1, the following inequality for the AB-fractional integrals holds:   (e2 − e1 )ν  AB ν  1−ν AB ν B (ν) Γ (ν) + B (ν) [f (e1 ) + f (e2 )] − e1 Ie2 f (e2 ) + e2 Ie1 f (e1 ) hp i r p ν+2 p(ν + 1) + 1 q |f 00 (e )|q + |f 00 (e )|q ν (e2 − e1 ) 1 2 × p (3.22) ≤ p B (ν) Γ (ν + 2) 2 p(ν + 1) + 1 Theorem 3.25. Let ν ∈ (0, 1) and ρ > 0 and f : [eρ1 , eρ2 ] → R be a twice differentiable mapping q on (eρ1 , eρ2 ) with 0 ≤ e1 < e2 . If |f 00 | is convex on [eρ1 , eρ2 ] for q ≥ 1, then the following inequality for the ABK-fractional integrals holds:   i h (eρ2 − eρ1 )ν 1−ν ABK ρ ν ρ ABK ρ ν ρ ρ ρ ρ ρ Ie f (e2 ) + e− Ie f (e1 ) ρν B (ν) Γ (ν) + B (ν) [f (e1 ) + f (e2 )] − e+ 2 1 2 1 ν+2

≤ ( ×

ν+1 ρ(ν + 2)

1− q1 s q

ν (eρ2 − eρ1 ) ρν−1 B (ν) Γ (ν + 2)

(3.23)

(ν + 1)(ν + 4) 00 ρ q (ν + 1) 00 ρ q f (e1 ) + f (e2 ) 2ρ(ν + 2)(ν + 3) 2ρ(ν + 3)

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 +

1 ρ(ν + 2)

1− q1 s q

13

) 1 1 f 00 (eρ ) q + f 00 (eρ ) q . 1 2 ρ(ν + 3) ρ(ν + 2)(ν + 3) q

Proof. Using Lemma 3.6, convexity of |f 00 | , the well-known power mean inequality and properties of the modulus, we have   h i (eρ2 − eρ1 )ν 1−ν ρ ρ ABK ρ ν ρ ABK ρ ν ρ ρ ρ + [f (e ) + f (e )] − I f (e ) + I f (e ) 1 2 2 1 e2 e1 ρν B (ν) Γ (ν) e+ e− B (ν) 1 2 ν+2

ν (eρ2 − eρ1 ) ρν−1 B (ν) Γ (ν + 2) ( Z 1− q1 Z 1h i ρ(ν+1) ρ−1 1−t t dt ×

≤

0

Z

1

+

1

h

ρ(ν+1)

1−t

0

1− q1 Z ρ(ν+2)−1 t dt

1

0

0

i

q  q1 ρ ρ ρ ρ f ((1 − t )e1 + t e2 ) dt

ρ−1 00

t

) q  q1 ρ ρ tρ(ν+2)−1 f 00 (tρ e1 + (1 − tρ )e2 ) dt

1− q1 1h i ν (eρ2 − ρ(ν+1) ρ−1 × 1 − t t dt ρν−1 B (ν) Γ (ν + 2) 0 Z 1 h  q1 i
ρ(ν+1) ρ−1 ρ 00 ρ q ρ 00 ρ q × 1−t t (1 − t ) f (e1 ) + t f (e2 ) dt ( Z

ν+2 eρ1 )

≤

0

Z

1

+

t

ρ(ν+2)−1

 p1 Z dt

0

1

t

ρ(ν+2)−1

ρ 00



t f

q (eρ1 )



 q1 ) 00 ρ q
+ (1 − t ) f (e2 ) dt ρ

0 ν+2

ν (eρ2 − eρ1 ) ρν−1 B (ν) Γ (ν + 2) ( 1− q1 s ν+1 (ν + 1)(ν + 4) 00 ρ q (ν + 1) 00 ρ q q × f (e1 ) + f (e2 ) ρ(ν + 2) 2ρ(ν + 2)(ν + 3) 2ρ(ν + 3) )  1− q1 s q q 1 1 1 ρ ρ q f 00 (e ) + f 00 (e ) . + 1 2 ρ(ν + 2) ρ(ν + 3) ρ(ν + 2)(ν + 3)

=

The proof of this theorem is complete.



Corollary 3.26. With the notations in Theorem 3.25, if we take |f 00 | ≤ K, the following inequality for the ABK-fractional integrals holds:   h i (eρ2 − eρ1 )ν 1−ν ABK ρ ν ρ ABK ρ ν ρ ρ ρ ρ ρ ) + f (e )] − I f (e ) + I f (e ) + [f (e 1 2 2 1 e2 e1 ρν B (ν) Γ (ν) e+ e− B (ν) 1 2 ν+2

≤

νK (eρ2 − eρ1 ) . ρν B (ν) Γ (ν + 2)

(3.24)

Corollary 3.27. With the notations in Theorem 3.25, if we take ρ → 1, the following inequality for the AB-fractional integrals holds:   (e2 − e1 )ν  AB ν  1−ν AB ν B (ν) Γ (ν) + B (ν) [f (e1 ) + f (e2 )] − e1 Ie2 f (e2 ) + e2 Ie1 f (e1 ) ν+2

≤

ν (e2 − e1 ) B (ν) Γ (ν + 2)

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×

+

s

q q (ν + 4) f 00 (e1 ) + (ν + 2) f 00 (e2 ) 2(ν + 3) ) q q q 1 q 00 00 √ (ν + 2) f (e1 ) + f (e2 ) . (ν + 2) q ν + 3

(

ν+1 ν+2

q

Theorem 3.28. Let ν ∈ (0, 1) and ρ > 0 and f : [eρ1 , eρ2 ] → R be a twice differentiable mapping q on (eρ1 , eρ2 ) with 0 ≤ e1 < e2 . If |f 00 | is convex on [eρ1 , eρ2 ] for q > 1 and p1 + 1q = 1, then the following inequality for the ABK-fractional integrals holds:   h i (eρ2 − eρ1 )ν 1−ν ρ ρ ABK ρ ν ρ ABK ρ ν ρ Ieρ f (e2 ) + e− Ieρ f (e1 ) ρν B (ν) Γ (ν) + B (ν) [f (e1 ) + f (e2 )] − e+ 2 1 1 2 s s ν+2 f 00 (eρ ) q + f 00 (eρ ) q q ν (eρ2 − eρ1 ) 2 1 p p(ν + 1) − 1 ≤ . (3.25) ρν+1 B (ν) Γ (ν + 2) p(ν + 1) + 1 2 q

Proof. Using Lemma 3.8, convexity of |f 00 | , H¨older inequality and properties of the modulus, we have   h i (eρ2 − eρ1 )ν 1−ν ρ ρ ρ ρ ABK ρ ν ABK ρ ν Ieρ f (e2 ) + e− Ieρ f (e1 ) ρν B (ν) Γ (ν) + B (ν) [f (e1 ) + f (e2 )] − e+ 2 1 1 2 ν+2

ν (eρ2 − eρ1 ) ρν B (ν) Γ (ν + 2)  p1 Z Z 1 p ρ ν+1 ρ(ν+1) ρ−1 × −t 1 − (1 − t ) t dt

≤

0

≤

=

0

1

t

q  q1 ρ ρ ρ ρ f (t e1 + (1 − t )e2 ) dt

ρ−1 00

Z 1  q1
p(ν + 1) − 1 ρ−1 ρ 00 ρ q ρ 00 ρ q × t t f (e1 ) + (1 − t ) f (e2 ) dt 1 0 ρν+ p B (ν) Γ (ν + 2) p(ν + 1) + 1 s s ν+2 f 00 (eρ ) q + f 00 (eρ ) q q p(ν + 1) − 1 ν (eρ2 − eρ1 ) 2 1 p . ρν+1 B (ν) Γ (ν + 2) p(ν + 1) + 1 2 ν (eρ2 − eρ1 )

ν+2

s p

The proof of this theorem is complete.



Corollary 3.29. With the notations in Theorem 3.28, if we take |f 00 | ≤ K, the following inequality for the ABK-fractional integrals holds:   i h (eρ2 − eρ1 )ν 1−ν ρ ABK ρ ν ρ ρ ρ ABK ρ ν Ieρ f (e2 ) + e− Ieρ f (e1 ) ρν B (ν) Γ (ν) + B (ν) [f (e1 ) + f (e2 )] − e+ 1 2 2 1 s ν+2 νK (eρ2 − eρ1 ) p p(ν + 1) − 1 ≤ . (3.26) ν+1 ρ B (ν) Γ (ν + 2) p(ν + 1) + 1 Corollary 3.30. With the notations in Theorem 3.28, if we take ρ → 1, the following inequality for the AB-fractional integrals holds:   (e2 − e1 )ν  AB ν  1−ν AB ν + [f (e ) + f (e )] − I f (e ) + I f (e ) 1 2 2 1 e1 e2 e2 e1 B (ν) Γ (ν) B (ν) s s ν+2 f 00 (e1 ) q + f 00 (e2 ) q q ν (e2 − e1 ) p(ν + 1) − 1 p ≤ . (3.27) B (ν) Γ (ν + 2) p(ν + 1) + 1 2 Theorem 3.31. Let ν ∈ (0, 1) and ρ > 0 and f : [eρ1 , eρ2 ] → R be a twice differentiable mapping q on (eρ1 , eρ2 ) with 0 ≤ e1 < e2 . If |f 00 | is convex on [eρ1 , eρ2 ] for q ≥ 1, then the following inequality

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for the ABK-fractional integrals holds:   h (eρ2 − eρ1 )ν 1−ν ρ ρ ABK ρ ν + [f (e ) + f (e )] − Ieρ f (eρ2 ) + 1 2 ρν B (ν) Γ (ν) e+ 2 B (ν) 1 1   ν+2 1− q ν (eρ2 − eρ1 ) ν ≤ ρν B (ν) Γ (ν + 2) ρ(ν + 2) s   q q ν × q C(ρ, ν) f 00 (eρ1 ) + − C(ρ, ν) f 00 (eρ2 ) , ρ(ν + 2)

ABK ρ ν Ieρ f (eρ1 ) e− 1 2

15

i (3.28)

where C(ρ, ν) :=

1 ρ



 ν+1 − β(2, ν + 2) . 2(ν + 3) q

Proof. Using Lemma 3.8, convexity of |f 00 | , the well-known power mean inequality and properties of the modulus, we have   h i (eρ2 − eρ1 )ν 1−ν ρ ρ ρ ρ ABK ρ ν ABK ρ ν Ieρ f (e2 ) + e− Ieρ f (e1 ) ρν B (ν) Γ (ν) + B (ν) [f (e1 ) + f (e2 )] − e+ 2 1 1 2 ν+2

ν (eρ2 − eρ1 ) ρν B (ν) Γ (ν + 2) Z 1 h 1− q1 i ρ ν+1 ρ(ν+1) ρ−1 × 1 − (1 − t ) −t t dt

≤

0

Z × 0

1

q  q1 h i ρ ν+1 ρ(ν+1) ρ−1 00 ρ ρ ρ ρ 1 − (1 − t ) −t t f (t e1 + (1 − t )e2 ) dt

1− q1 ν+2  ν (eρ2 − eρ1 ) ν ρν B (ν) Γ (ν + 2) ρ(ν + 2) Z 1 h  q1 i q q
× 1 − (1 − tρ )ν+1 − tρ(ν+1) tρ−1 tρ f 00 (eρ1 ) + (1 − tρ ) f 00 (eρ2 ) dt 0 1− q1 s   ν+2  ν ν (eρ2 − eρ1 ) ν ρ q q 00 f 00 (eρ ) q . = − C(ρ, ν) C(ρ, ν) f (e ) + 2 1 ρν B (ν) Γ (ν + 2) ρ(ν + 2) ρ(ν + 2) ≤

The proof of this theorem is complete.



Corollary 3.32. With the notations in Theorem 3.31, if we take |f 00 | ≤ K, the following inequality for the ABK-fractional integrals holds:   h i (eρ2 − eρ1 )ν 1−ν ABK ρ ν ρ ρ ABK ρ ν ρ ρ Ieρ f (e2 ) + e− Ieρ f (e1 ) ρν B (ν) Γ (ν) + B (ν) [f (e1 ) + f (e2 )] − e+ 2 1 1 2 ν+2

≤

ν 2 K (eρ2 − eρ1 ) . ρν+1 B (ν) Γ (ν + 3)

(3.29)

Corollary 3.33. With the notations in Theorem 3.31, if we take ρ → 1, the following inequality for the AB-fractional integrals holds:   (e2 − e1 )ν  AB ν  1−ν AB ν B (ν) Γ (ν) + B (ν) [f (e1 ) + f (e2 )] − e1 Ie2 f (e2 ) + e2 Ie1 f (e1 ) 1− q1 ν+2  ν ν (e2 − e1 ) ≤ B (ν) Γ (ν + 2) ν + 2

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s ×

q

q C(1, ν) f 00 (e1 ) +



 q ν − C(1, ν) f 00 (e2 ) . ν+2

Theorem 3.34. Let ν ∈ (0, 1) and ρ > 0. Let f and g be real valued, nonnegative and convex functions on [eρ1 , eρ2 ], where 0 ≤ e1 < e2 . Then the following inequality for the ABK-fractional integrals holds: i h ρ ABK ρ ν ρ ABK ρ ν ρ ρ ρ f (e )g(e ) ρ f (e )g(e ) + I I − + 1 1 2 2 e1 e2 e2 e1 !
ρ ν ν 2 ν ν + ν + 2 (e2 − eρ1 ) 1−ν 2ν 2 (eρ2 − eρ1 ) ≤ + M (eρ1 , eρ2 ) + N (eρ1 , eρ2 ), (3.30) B (ν) ρB (ν) Γ (ν + 3) B (ν) Γ (ν + 3) where M (eρ1 , eρ2 ) = f (eρ1 )g(eρ1 ) + f (eρ2 )g(eρ2 ) and N (eρ1 , eρ2 ) = f (eρ1 )g(eρ2 ) + f (eρ2 )g(eρ1 ). Proof. Since f and g are convex on [eρ1 , eρ2 ], then f (tρ eρ1 + (1 − tρ )eρ2 ) ≤ tρ f (eρ1 ) + (1 − tρ )f (eρ2 )

(3.31)

g(tρ eρ1 + (1 − tρ )eρ2 ) ≤ tρ g(eρ1 ) + (1 − tρ )g(eρ2 ). From (3.31) and (3.32), we get

(3.32)

and

f (tρ eρ1 + (1 − tρ )eρ2 )g(tρ eρ1 + (1 − tρ )eρ2 ) ≤ t2ρ f (eρ1 )g(eρ1 ) + (1 − tρ )2 f (eρ2 )g(eρ2 ) + tρ (1 − tρ )[f (eρ1 )g(eρ2 ) + f (eρ2 )g(eρ1 )]. Similarly, f ((1 − tρ )eρ1 + tρ eρ2 )g((1 − tρ )eρ1 + tρ eρ2 ) ≤ (1 − tρ )2 f (eρ1 )g(eρ1 ) + t2ρ f (eρ2 )g(eρ2 ) + tρ (1 − tρ )[f (eρ1 )g(eρ2 ) + f (eρ2 )g(eρ1 )]. By adding the above two inequalities, it follows that f (tρ eρ1 + (1 − tρ )eρ2 )g(tρ eρ1 + (1 − tρ )eρ2 ) + f ((1 − tρ )eρ1 + tρ eρ2 )g((1 − tρ )eρ1 + tρ eρ2 ) ≤ (2t2ρ − 2tρ + 1)[f (eρ1 )g(eρ1 ) + f (eρ2 )g(eρ2 )] + 2tρ (1 − tρ )[f (eρ1 )g(eρ2 ) + f (eρ2 )g(eρ1 )]. ν tρν−1 and integrating the resulting inMultiplying both sides of above inequality by B(ν)Γ(ν) equality with respest to t over [0, 1], we obtain Z 1 ν tρν−1 f (tρ eρ1 + (1 − tρ )eρ2 )g(tρ eρ1 + (1 − tρ )eρ2 )dt B (ν) Γ (ν) 0 Z 1 ν + tρν−1 f ((1 − tρ )eρ1 + tρ eρ2 )g((1 − tρ )eρ1 + tρ eρ2 )dt B (ν) Γ (ν) 0 Z 1 ν ≤ tρν−1 (2t2ρ − 2tρ + 1)[f (eρ1 )g(eρ1 ) + f (eρ2 )g(eρ2 )]dt B (ν) Γ (ν) 0 Z 1 ν tρν−1 2tρ (1 − tρ )[f (eρ1 )g(eρ2 ) + f (eρ2 )g(eρ1 )]dt + B (ν) Γ (ν) 0 Z Z νM (eρ1 , eρ2 ) 1 ρν−1 2ρ 2νN (eρ1 , eρ2 ) 1 ρν−1 ρ = t (2t − 2tρ + 1)dt + t t (1 − tρ )dt B (ν) Γ (ν) 0 B (ν) Γ (ν) 0 ν(ν 2 + ν + 2) 2ν 2 = M (eρ1 , eρ2 ) + N (eρ1 , eρ2 ). ρB (ν) Γ (ν + 3) B (ν) Γ (ν + 3) By the change of variables and with simple integral calculations, we get the desired result. 

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HERMITE-HADAMARD TYPE INEQUALITIES FOR THE ABK-FRACTIONAL INTEGRALS

17

Corollary 3.35. With the notations in Theorem 3.34, if we choose f = g, the following inequality for the ABK-fractional integrals holds: h i ABK ρ ν 2 ρ ABK ρ ν 2 ρ ρ f (e ) + ρ f (e ) I I + − 2 1 e2 e1 e1 e2 !
ρ ρ ν ν 2 ν ν + ν + 2 (e2 − e1 ) 1−ν 2ν 2 (eρ2 − eρ1 ) ≤ M1 (eρ1 , eρ2 ) + + N1 (eρ1 , eρ2 ),(3.33) B (ν) ρB (ν) Γ (ν + 3) B (ν) Γ (ν + 3) where M1 (eρ1 , eρ2 ) = f 2 (eρ1 ) + f 2 (eρ2 ),

N1 (eρ1 , eρ2 ) = 2f (eρ1 )f (eρ2 ).

Corollary 3.36. With the notations in Theorem 3.34, if we take ρ → 1, the following inequality for the AB-fractional integrals holds:  AB ν  AB ν e1 Ie2 f (e2 )g(e2 ) + e2 Ie1 f (e1 )g(e1 ) !
ν ν ν ν 2 + ν + 2 (e2 − e1 ) 2ν 2 (e2 − e1 ) 1−ν + M (e1 , e2 ) + N (e1 , e2 ). (3.34) ≤ B (ν) B (ν) Γ (ν + 3) B (ν) Γ (ν + 3) Remark 3.37. With the notations in our theorems given in Section 3, if we take ρ, ν → 1, then we get some classical integral inequalities.

Acknowledgements The author would like to thank the referee for valuable comments and suggestions. References [1] Abdeljawad, T. and Baleanu, D., Integration by parts and its applications of a new nonlocal fractional derivative with Mittag-Leffler nonsingular kernel, arXiv:1607.00262v1 [math.CA], (2016). [2] Aslani, S.M., Delavar, M.R. and Vaezpour, S.M., Inequalities of Fej´ er type related to generalized convex functions with applications, Int. J. Anal. Appl., 16(1) (2018), 38–49. [3] Atangana, A. and Baleanu, D., Discrete fractional differences with nonsingular discrete Mittag-Leffler kernels, Adv. Differ. Equ., 2016(232) (2016). [4] Chen, F.X. and Wu, S.H., Several complementary inequalities to inequalities of Hermite-Hadamard type for s-convex functions, J. Nonlinear Sci. Appl., 9(2) (2016), 705–716. [5] Chu, Y.M., Khan, M.A., Khan, T.U. and Ali, T., Generalizations of Hermite-Hadamard type inequalities for M T -convex functions, J. Nonlinear Sci. Appl., 9(5) (2016), 4305–4316. [6] Delavar, M.R. and Dragomir, S.S., On η-convexity, Math. Inequal. Appl., 20 (2017), 203–216. [7] Delavar, M.R. and De La Sen, M. Some generalizations of Hermite-Hadamard type inequalities, SpringerPlus, 5(1661) (2016). [8] Dragomir, S.S. and Agarwal, R.P., Two inequalities for differentiable mappings and applications to special means of real numbers and trapezoidal formula, Appl. Math. Lett., 11(5) (1998), 91–95. [9] Hristov, J., Response functions in linear viscoelastic constitutive equations and related fractional operators, Math. Model. Nat. Phenom., 14(3) (2019), 1–34. [10] Jleli, M. and Samet, B., On Hermite-Hadamard type inequalities via fractional integral of a function with respect to another function, J. Nonlinear Sci. Appl., 9 (2016), 1252–1260. [11] Kashuri, A. and Liko, R., Some new Hermite-Hadamard type inequalities and their applications, Stud. Sci. Math. Hung., 56(1) (2019), 103–142. [12] Katugampola, U.N., New approach to a generalized fractional integral, Appl. Math. Comput., 218 (2011), 860–865. [13] Kermausuor, S., Nwaeze, E.R. and Tameru, A.M., New integral inequalities via the Katugampola fractional integrals for functions whose second derivatives are strongly η-convex, Mathematics, 7(183) (2019), 1–14. [14] Khan, M.A., Chu, Y.M., Kashuri, A., Liko, R. and Ali, G., New Hermite-Hadamard inequalities for conformable fractional integrals, J. Funct. Spaces, (2018), Article ID 6928130, pp. 9. [15] Kumar, D., Singh, J. and Baleanu, D., Analysis of regularized long-wave equation associated with a new fractional operator with Mittag-Leffler type kernel, Phys. A, Stat. Mech. Appl. 492 (2018), 155–167. [16] Liu, W., Wen, W. and Park, J., Hermite-Hadamard type inequalities for M T -convex functions via classical integrals and fractional integrals, J. Nonlinear Sci. Appl., 9 (2016), 766–777.

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A. KASHURI

[17] Luo, C., Du, T.S., Khan, M.A., Kashuri, A. and Shen, Y., Some k-fractional integrals inequalities through generalized λφm -M T -preinvexity, J. Comput. Anal. Appl., 27(4) (2019), 690–705. [18] Mihai, M.V., Some Hermite-Hadamard type inequalities via Riemann-Liouville fractional calculus, Tamkang J. Math, 44(4) (2013), 411–416. [19] Omotoyinbo, O. and Mogbodemu, A., Some new Hermite-Hadamard integral inequalities for convex functions, Int. J. Sci. Innovation Tech., 1(1) (2014), 1–12. ¨ [20] Ozdemir, M.E., Dragomir, S.S. and Yildiz, C., The Hadamard’s inequality for convex function via fractional integrals, Acta Mathematica Scientia, 33(5) (2013), 153–164. [21] Owolabi, K.M., Modelling and simulation of a dynamical system with the Atangana-Baleanu fractional derivative, Eur. Phys. J. Plus 133(1) (2018), pp. 15. [22] Sarikaya, M.Z. and Yaldiz, H., On weighted Montogomery identities for Riemann-Liouville fractional integrals, Konuralp J. Math., 1(1) (2013), 48–53. [23] Set, E., Noor, M.A., Awan, M.U. and G¨ ozpinar, A., Generalized Hermite-Hadamard type inequalities involving fractional integral operators, J. Inequal. Appl., 169 (2017), 1–10. [24] Wang, H., Du, T.S. and Zhang, Y., k-fractional integral trapezium-like inequalities through (h, m)-convex and (α, m)-convex mappings, J. Inequal. Appl., 2017(311) (2017), pp. 20. [25] Xi, B.Y and Qi, F., Some integral inequalities of Hermite-Hadamard type for convex functions with applications to means, J. Funct. Spaces Appl., 2012 (2012), Article ID 980438, pp. 14. [26] Zhang, X.M., Chu, Y.M. and Zhang, X.H., The Hermite-Hadamard type inequality of GA-convex functions and its applications, J. Inequal. Appl., (2010), Article ID 507560, pp. 11. [27] Zhang, Y., Du, T.S., Wang, H., Shen, Y.J. and Kashuri, A., Extensions of different type parameterized inequalities for generalized (m, h)-preinvex mappings via k-fractional integrals, J. Inequal. Appl., 2018(49) (2018), pp. 30. Artion Kashuri Department of Mathematics, Faculty of Technical Science, University Ismail Qemali, Vlora, Albania E-mail address: [email protected]

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A unified convergence analysis for single step-type methods for non-smooth operators S. Amat∗

I. Argyros†

S. Busquier‡ M.A. Hern´andez-Ver´on§ Eulalia Mart´ınez¶ May 7, 2019

Abstract This paper is devoted to the approximation of solutions for nonlinear equations by using iterative methods. We present a unified convergence analysis for some Newton-type methods. We consider both semilocal and local analysis. In the first one, the hypotheses are imposed on the initial guess and in the second on the solution. The results can be applied for smooth and non-smooth operators. In the numerical section we study two applications, first one, it is devoted to a nonlinear integral equation of Hammerstein type and in second one, we approximate the solution of a nonlinear PDE related to image denoising.

1

Introduction

There are several situations in which the modeling of a problem leads us to calculate a solution of an equation F (x) = 0. (1) This equation can represent a differential equation, ordinary or partial, an integral equation, an integro-differential equation or a simple system of equations. In general, mathematical methods that obtain exact solutions of (1) are not known, so that iterative methods are usually used to solve (1) [9, 10, 1, 2, 3, 4, 5, 7, 12]. For a greater generality, ∗

Departamento de Matem´ atica Aplicada y Estad´ıstica. Universidad Polit´ecnica de Cartagena (Spain). e-mail:[email protected] † Departament of Mathematics Sciences. Cameron University (USA). e-mail:[email protected] ‡ Departamento de Matem´ atica Aplicada y Estad´ıstica. Universidad Polit´ecnica de Cartagena (Spain). e-mail:[email protected] § Departamento de Matem´ aticas y Computaci´on. Universidad de La Rioja (Spain). email:[email protected] ¶ Instituto Universitario de Matem´ atica Multidisciplinar. Universitat Polit`ecnica de Val`encia (Spain). e-mail:[email protected]

1

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in this study, we consider F : D ⊂ X → Y, where X, Y are Banach spaces and D is a nonempty, open and convex set. And we pay attention to F is continuous and Fr´echet non-differentiable. In this case, to approximate a solution of (1), iterative methods using divided differences are usually applied instead of using derivatives [12]-[11]. It is common to approximate derivatives by divided differences for obtaining derivative free iterative schemes. So, given an operator G : D ⊂ X → Y, let us denote by L(X, Y ) the space of bounded linear operators from X into Y , an operator [x, y; G] ∈ L(X, Y ) is called a first order divided difference for the operator G on the points x and y (x 6= y) in D if [x, y; G](x − y) = G(x) − G(y).

(2)

Steffensen’s method [13] is the most used iterative method using divided differences in the algorithm, which is ( x0 given in D, (3) xn+1 = xn − [xn , xn + F (xn ); F ]−1 F (xn ), n ≥ 0. As we can see in [14], Steffensen’s method has a problem of accessibility that can be solved by using a procedure of decomposition ([15]) for operator F , the Fr´echet differentiable part and the non-differentiable part. So, we consider F (x) = F1 (x) + F2 (x)

(4)

where F1 , F2 : D ⊂ X → Y , F1 is Fr´echet differentiable and F2 is continuous and Fr´echet non-differentiable. Thus, in [14], we consider the method of Newton-Steffensen, given by the following algorithm ( x0 given in D, (5) xn+1 = xn − (F10 (xn ) + [xn , xn + F (xn ); F2 ])−1 (F1 (xn ) + F2 (xn )), n ≥ 0, with X = Y , which improves significantly the accessibility of method (3) and has quadratic convergence. By using this procedure of decomposition for operator F , we see that we can also consider the application of iterative methods that use derivatives when F is non-differentiable. So, for example, we can consider the well-known Newton’s method, which algorithm is ( x0 given in D, (6) xn+1 = xn − [F 0 (xn )]−1 F (xn ), n ≥ 0, Obviously, Newton’s method is not applicable, under form (6), when F is not Fr´echet differentiable. However, if we consider decomposition of F given in (4), we can use the following algorithm ( x0 given in D, (7) xn+1 = xn − [F10 (xn )]−1 (F1 (xn ) + F2 (xn )), n ≥ 0, 2

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which is known as method of Zincenko [17]. The main aim of this paper consists of defining one-point iterative methods of Newtontype, as we can see previously, to obtain a general study for the convergence, local and semilocal, for these type of iterative methods. Moreover, in view of the last two considerations, with these one point iterative methods we can to improve the accessibility of one-point iterative methods that use divided differences and, in addition, to extend the application of iterative methods that use derivatives when F is Fr´echet non-differentiable. For this aim, we consider the one-point iterative methods of Newton-type given by the following algorithm ( x0 given in D, (8) xn+1 = xn − L−1 n ≥ 0, n (F1 (xn ) + F2 (xn )), where Ln := L(xn ) with L(.) : D → L(X, Y ). Clearly, method (8) can be used to solve equations containing a nondifferentiable term. There are a lot of iterative methods that can be written as algorithm (8), in addition to modifications of Steffensen and Newton given in (5) and (7), where L(x) = F10 (x) + [x, x + F2 (x); F2 ] and L(x) = F10 (x), respectively. At the same time, we can also consider two interesting cases. Firstly, the generalized Steffensen methods [6], that are very used in the approximation of solutions of non-differentiable operators equations and the algorithm is ( x0 given in D, xn+1 = xn − [xn − aF (xn ), xn + bF (xn ); F ]−1 F (xn ),

n ≥ 0.

Then, it is clear that we can define the generalized Newton-Steffensen method from 8) with L(x) = F10 (x) + [x − aF2 (x), x + bF2 (x); F2 ], so we have the final iterative function given as: ( x0 given in D, (9) xn+1 = xn − (F10 (xn ) + [xn − aF2 (xn ), xn + bF2 (xn ); F2 ])−1 F (xn ), n ≥ 0. where a, b ∈ R. In the same way as Newton’s method, from Stirling method [16], ( x0 given in D, xn+1 = xn − [F10 (xn − F (xn ))]−1 F (xn ),

n ≥ 0,

(10)

we can define a modification of Newton-type, that can be applied to Fr´echet non-differentiable operators. For this, just consider (8) with L(x) = F10 (x − F (x)). In both cases, we choose X = Y . Obviously, we can include a lot of iterative methods in (8) if F is Fr´echet differentiable. So, in this paper, we study the convergence of algorithm (8). We analyze the semilocal and local convergences, so that we have a study of convergence of a lot of iterative methods that are usually used and can be written by algorithm (8). 3

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Section 2 is devoted to the theoretical analysis about local and semilocal convergence for a very general single step Newton-like methods. In Section 3 we make a comparison for the behavior of some of these methods by solving a non-differentiable problem. In Section 4, we consider an application related to image denoising. Finally, in Section 5 we give some conclusions.

2

Convergence Analysis for single step Newton-like methods

In this section, we present both semilocal and local convergence analysis. In the first one, the hypotheses are imposed on the initial guess and in the second on the solution. The results can be applied for smooth and non-smooth operators.

2.1

Local Convergence Analysis

In this section, we first present the local followed by the semilocal convergence of method (8). Let v0 : [0, +∞) → [0, +∞) be a nondecreasing continuous function with v0 (0) = 0. Suppose that the equation v0 (t) = 1 (11) has at least one positive root r0 . Let also v : [0, r0 ) → [0, +∞) be a nondecreasing v(t) − 1. continuous function. Define function v¯ on the interval [0, r0 ) by v¯(t) = 1−v 0 (t) Suppose equation v¯(t) = 0 (12) has at least one positive root. Denote by r the smallest such root. It follows that for each t ∈ [0, r) 0 ≤ v0 (t) < 1 (13) and 0 ≤ v¯(t) < 1.

(14)

The local convergence analysis of method (8) uses the conditions (A): • (a1 ) There exist a solution x∗ ∈ D of equation (4), and B ∈ L(X, Y ) such that B −1 ∈ L(Y, X). • (a2 ) Condition (11) holds and for each x ∈ D kB −1 (L(x) − B)k ≤ v0 (kx − x∗ k), where v0 is defined previously and r0 is given in (11). Set D0 = D ∩ U¯ (x∗, r0 ).

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• (a3 ) For L : D0 → L(X, Y ), any solution y of equation (4) and each x ∈ D0 kB −1 (F1 (x) + F2 (x) − L(x)(x − y))k ≤ v(kx − yk)kx − yk, where v is defined previously. • (a4 ) U¯ (x∗ , r) ⊂ D, where r is given in (12). • (a5 ) There exist r∗ ≥ r such that ξ :=

v(r∗ ) ∈ [0, 1). 1 − v0 (r)

Set D1 = D ∩ U¯ (x∗, r∗ ).

Remark 1

• Condition (a3 ) can be replaced by the stronger: for each x, y, z ∈ D0 kB −1 (F1 (x) + F2 (x) − L(x)(x − y))k ≤ v1 (kx − yk)kx − yk,

where function v1 is as v. But for each t ≥ 0 v(t) ≤ v1 (t). • Linear operator B does not necessarily depend on the solution x∗ . It is used to determine the invertibility of linear operator L(·) appearing in the method. The invertibility of B can be assured by an additional condition of the form ||I − B|| < 1 0 or some other way. A possible choice for B is B = B(x∗ ) or B = F1 (x∗ ). • It follows from the definition of r0 and r that r0 ≥ r. We can present the local convergence analysis of method (8) based on the aforementioned conditions (A).

Theorem 2 Suppose that the conditions (A) hold. Then, sequence xk generated by method (8) for x0 ∈ U (x∗ , r) − x∗ is well defined in U (x∗ , r), remains in U (x∗ , r) and converges to x∗ . Moreover, the following estimates hold. kxk+1 − x∗ k ≤

v(kxk − x∗ k) kxk − x∗ k ≤ kxk − x∗ k < r. 1 − v0 (kxk − x∗ k)

(15)

The vector x∗ is the only solution of equation (4) in D1 , where D1 is given in (a5). Proof We base the proof on k and mathematical induction. Let x ∈ U (x∗ , r). Using (8), (a1) and (a2), we have in turn that kB −1 (L(x) − B)k ≤ v0 (kx − x∗ k) ≤ v0 (r) < 1.

(16)

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It follows by (16) and the Banach lemma on invertible operators [] that L(x)−1 ∈ L(Y, X) and 1 . (17) kL(x)−1 Bk ≤ 1 − v0 (kx − x∗ k) In particular, estimate (17) holds for x = x0 , so x1 is well defined by method (8) for k = 0. We also get by method (8) (for k = 0), (a1), (a3), (14) and (17) (for k = 0) that kx1 − x∗ k = kx0 − x∗ − L(x0 )−1 (F1 (x0 ) + F2 (x0 ))k = k[−L(x0 )−1 B][B −1 (F1 (x0 ) + F2 (x0 ) − L(x0 )(x0 − x∗ ))]k ≤ kL(x0 )−1 BkkB −1 (F1 (x0 ) + F2 (x0 ) − L(x0 )(x0 − x∗ ))k v(kx0 − x∗ k) ≤ kx0 − x∗ k ≤ kx0 − x∗ k < r, ∗ 1 − v0 (kx0 − x k)

(18)

which shows estimate (15) for k = 0, and x1 ∈ U (x∗ , r). Simply, replace x0 , x1 by xi , xi+1 in the preceding estimates to complete the induction for estimate (15). Then, in view of the estimate kxi+1 − x∗ k ≤ ξkxi − x∗ k < r, where ξ=

(19)

v(kx0 − x∗ k) ∈ [0, 1), 1 − v0 (kx0 − x∗ k)

we deduce that limi→+∞ xi = x∗ and xi+1 ∈ U (x∗ , r). Moreover, to show the uniqueness part, let y ∗ ∈ D1 with F1 (y ∗ ) + F2 (y ∗ ) = 0. Using (a3), (a5) and estimate (18), we obtain in turn that kxi+1 − y ∗ k ≤ kL(xi )−1 BkkB −1 (F1 (xi ) + F2 (xi ) − L(xi )(xi − y ∗ ))k v(kxi − y ∗ k) kxi − y ∗ k ≤ 1 − v0 (kxi − x∗ k) ≤ ξkxi − y ∗ k < ξ i+1 kx0 − y ∗ k,

(20)

which shows limi→+∞ xi = y ∗ . But, we showed limi→+∞ xi = x∗ . Hence, we conclude that x∗ = y ∗ . 

2.2

Semilocal Convergence Analysis

As in the local case it is convenient to define some functions and parameters for the semilocal analysis. Let w0 : [0, +∞) → [0, +∞) be a continuous and nondecreasing function. Suppose that equation w0 (t) = 1. (21) 6

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has at least one positive root. Denote by ρ0 the smallest such root. Let also w : [0, ρ0 ) × [0, ρ0 ) → [0, +∞) be a nondecreasing continuous function. Moreover, for η ≥ 0, define parameters C1 and C2 by C1 = C2 =

w(η, 0) , 1 − w0 (η) η w( 1−C , η) 1 η ) 1 − w0 ( 1−C 1

and function C : [0, ρ0 ) → [0, +∞) by C(t) = (

w(t,t) . 1−w0 (t)

Suppose that equation

C1 C2 + C1 + 1)η − t = 0 1 − C(t)

(22)

has as least one positive root. Denote by ρ the smallest such root. Next, we show the semilocal convergence analysis of method (8) in an analogous way, under the conditions (H): • (h1) There exists x0 ∈ D and B ∈ L(X, Y ) such that B −1 ∈ L(Y, X). • (h2) Condition (21) holds and for each x ∈ D kB −1 (L(x) − B)k ≤ w0 (kx − x0 k), where w0 is as defined previously, and ρ0 is given in (21). T Set D2 = D U¯ (x0 , ρ0 ). • (h3) For L(·) : D2 → L(X, Y ), and each x, y ∈ D2

kB −1 (F1 (y) − F1 (x) + F2 (y) − F2 (x) − L(x)(y − x))k ≤ w(ky − x0 k, kx − x0 k)ky − xk, where w is as defined previously. • (h4) U¯ (x0 , ρ) ⊆ D and condition (22) holds for ρ, where kx1 − x0 k ≤ η. • (h5) There exists ρ∗ ≥ ρ such that ξ0 := Set D2 = D

w(ρ, ρ∗ ) ∈ [0, 1). 1 − w0 (ρ)

T¯ ∗ ∗ U (x , ρ ).

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Then, as in the local case but using the (H) instead of the (A) conditions, we have in turn the estimates: kx2 − x1 k ≤ kx2 − x0 k ≤ = < kx3 − x2 k ≤ ≤

w(kx1 − x0 k, kx0 − x0 k) = C1 kx1 − x0 k, 1 − w0 (kx1 − x0 k) kx2 − x1 k + kx1 − x0 k ≤ (1 + C1 )kx1 − x0 k 1 − C12 kx1 − x0 k 1 − C1 kx1 − x0 k η < ρ, 1 − C1 w(kx2 − x0 k, kx1 − x0 k) kx2 − x1 k 1 − w0 (kx2 − x0 k) η , η) w( 1−C 1 kx2 − x1 k = C2 kx2 − x1 k η 1 − w0 ( 1−C ) 1

kx3 − x0 k ≤ kx3 − x2 k + kx2 − x1 k + kx1 − x0 k ≤ C2 kx2 − x1 k + C1 kx1 − x0 k + kx1 − x0 k ≤ (C2 C1 + C1 + 1)kx1 − x0 k, w(kx3 − x0 k, kx2 − x0 k) kx3 − x2 k kx4 − x3 k ≤ 1 − w0 (kx3 − x0 k) ≤ C(ρ)kx3 − x2 k ≤ C(ρ)C2 kx2 − x1 k ≤ C(ρ)C2 C1 kx1 − x0 k, ... kxi+1 − xi k ≤ C(ρ)kxi − xi−1 k ≤ C(ρ)i−2 kx3 − x2 k kxi+1 − x0 k ≤ kxi+1 − xi k + ... + kx4 − x3 k + kx3 − x0 k ≤ C(ρ)kxi − xi−1 k + ... + C(ρ)kx3 − x2 k +(C2 C1 + C1 + 1)kx1 − x0 k ≤ C(ρ)i−2 kx3 − x2 k + ... + C(ρ)kx3 − x2 k +(C2 C1 + C1 + 1)kx1 − x0 k 1 − C(ρ)i−1 ≤ ( C2 C1 + C1 + 1)kx1 − x0 k 1 − C(ρ) C1 C2 < ( + C1 + 1)η ≤ ρ, 1 − C(ρ)

(23)

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kxi+j − xi k ≤ kxi+j − xi+j−1 k + kxi+j−1 − xi+j−2 k + ... + kxi+1 − xi k ≤ (C(ρ)i+j−3 + ... + C(ρ)i−2 )kx3 − x2 k 1 − C(ρ)j−1 ≤ C(ρ)i−2 kx3 − x2 k 1 − C(ρ) 1 − C(ρ)j−1 C2 C1 kx1 − x0 k ≤ C(ρ)i−2 1 − C(ρ) 1 − C(ρ)j−1 ≤ C(ρ)i−2 C2 C1 η. 1 − C(ρ)

(24)

It follows from (23) that xi ∈ U (x0 , ρ) and from (24) that sequence xi is complete in X and as such it converges to some x∗ ∈ U¯ (x0 , ρ). By letting i → +∞ in the estimate kB −1 (F1 (xi ) + F2 (xi ))k = kB −1 (F1 (xi ) + F2 (xi ) − F1 (xi−1 ) − F2 (xi−1 ) − Bi−1 (xi − xi−1 ))k ≤

w(kxi − x0 k, kxi−1 − x0 k)kkxi − xi−1 k w(ρ, ρ) ≤ kxi − xi−1 k, 1 − w0 (kxi − x0 k) 1 − w0 (ρ)

we obtain F1 (x∗ ) + F2 (x∗ ) = 0. The uniqueness part is omitted as identical to the one in the local convergence case. Hence, we arrived at the semilocal convergence result for method (8).

Theorem 3 Suppose that the conditions (H) hold. Then, sequence xk generated by method (8) for x0 ∈ D is well defined in U (x0 , ρ) remains in U (x0 , ρ) and converges x∗ ∈ U¯ (x0 , ρ) to a solution of equation (4). Moreover, the vector x∗ is the only solution of equation (4) in D3 , where D3 is defined previously. The same comments introduced in the previous remark are valid. We emphasize the theoretical importance of this theorem because it presents a unified studied of the local and semilocal convergence of a big variety of Newton-Type methods and Steffensen type methods, so the study is applicable to differentiable an non differentiable equations.

3

Numerical Experiments

In this section, we consider a nonlinear integral equation of Hammerstein type, which can be used to describe applied problems in the fields of electro-magnetics, fluid dynamics, in the kinetic theory of gases and, in general, in the reformulation of boundary value problems. These equations are of the form: Z b x(s) = f (s) − K(s, t)Φ(x(t))dt, a ≤ s ≤ b, (25) a

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where x(s), f (s) ∈ C[a, b], with −∞ < a < b < ∞, and Φ is a polynomial function. One of the most used techniques to solve this kind of equations consists of expressing them as a nonlinear operator in a Banach space and solving the following operator equation: Z b F (x)(s) = x(s) − f (s) + K(s, t)Φ(x(t))dt = 0, (26) a

where F : D ⊆ C[a, b] → C[a, b] with D a non-empty open convex subset of C[a, b] with the max-norm kνk = maxs∈[a,b] |ν(s)|. We consider (25), where K is the Green function in [a, b] × [a, b], and then use a discretization process to transform equation (26) into a finite dimensional problem by approximating the integral by an adequate quadrature formula Z b p X wi q(ti ), q(t) dt ' a

i=1

where the nodes ti and the weights wi are known. If we denote the approximations of x(ti ) and f (ti ) by xi and fi , respectively, with i = 1, 2, . . . , p, then equation (26) is equivalent to the following system of nonlinear equations: p X xi = f i + aij Φ(xj ), j = 1, 2, . . . , p, (27) j=1

where ( aij = wj K(ti , tj ) =

(b−ti )(tj −a) , b−a (b−tj )(ti −a) wj , b−a

wj

j ≤ i, j > i.

Now, system (27) can be written as F : ∆ ⊆ Rp −→ Rp ,

F(x) ≡ x − f − A z = 0,

(28)

where x = (x1 , x2 , . . . , xp )T ,

f = (f1 , f2 , . . . , fp )T ,

A = (aij )pi,j=1 ,

z = (Φ(x1 ), Φ(x2 ), . . . , Φ(xp ))T . After that, we choose a = 0, b = 1, K(s, t) as the Green function in [0, 1] × [0, 1] and Φ(x(t)) = x(t)3 + |x(t)| in (25). Then, the system of nonlinear equations given in (28) is of the form F(x) = x − f − A (vx + wx ) = 0, F : Rp −→ Rp , (29) where vx = (x31 , x32 , . . . , x3p )T ,

wx = (|x1 |, |x2 |, . . . , |xp |)T .

It is obvious that the function F defined in (29) is nonlinear and non-differentiable. So, we consider F(x) = F1 (x) + F2 (x) where: F1 (x) = x − f − Avx

and F2 (x) = −Awx . 10

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As in Rp we can consider divided difference of first order that do not need that the function F is differentiable (see [16]), we use the divided difference of first order given by [u, v; G] = ([u, v; G]ij )pi,j=1 ∈ L(Rp , Rp ), where [u, v; G]ij =

1 (Gi (u1 , . . . , uj , vj+1 , . . . , vp ) − Gi (u1 , . . . , uj−1 , vj , . . . , vp )) , uj − vj

(30)

if uj 6= vj , in other case [u, v; G]ij = 0, for u = (u1 , u2 , . . . , up )T and v = (v1 , v2 , . . . , vp )T . Now, to compare the behavior of different methods we consider the case f = 0 in (29). Obviously, for this problem, x∗ = 0 is a solution of F(x) = 0. Then, the system of nonlinear equations given in (29) is of the form F(x) = x − A z,

zj = x3j + |xj |, j = 1, . . . , p.

(31)

The numerical results are obtained by using MATLAB 2018 and working with variable precision arithmetic with 100 digits. In Table 1 we can see the results obtained by using the methods mentioned in our study. First of all we take nodes and weights of Trapezoidal rule with n = 10 subintervals for approximatting the integral and starting guess x0 (t) = 1/2 ∀t ∈ [0, 1]. We compare the distance between consecutive iterates of the first 7 iterations of each method. In the case of the Newton-Steffensen General method (9), the parameters involved are a = 0.5 and b = 1.5.

1 2 3 4 5 6 7

Stirling (10) Zincenko (7) Steffensen (3) New-Steff. (5) New-Steff. Gen.(9) 1.5887 1.1637 7.4375 2.9044 2.9044 6.0578e − 01 3.0210e − 01 2.7350e − 01 1.3867 1.3867 4.7941e − 01 1.2065e − 01 1.8235e − 02 3.2041e − 01 1.2942e − 01 4.1942e − 01 4.9511e − 02 5.5411e − 05 2.8725e − 04 2.8725e − 04 3.5456e − 01 2.0403e − 02 2.8134e − 09 1.3552e − 12 1.3552e − 12 1.9024e − 01 8.4133e − 03 3.0173e − 18 3.1538e − 37 3.3246e − 37 2.9676e − 02 3.4697e − 03 3.9490e − 36 1.7796e − 111 2.1782e − 111 Table 1: Results with different methods in the first iterations.

In Table 2 we work with same conditions, we obtain the iterations that each method needs to satisfy the stopping criterion ||xk+1 − xk || ≤ 10−40 . It should be noted that the first two methods never meet the required tolerance because they are not convergent and, therefore, the methods end when the required iterations are completed (in this case 15 iterations at most). Second, we observe a good approximation to the order of convergence of each method p in case the method converges. In the last two rows of Table 2 we compare 11

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Stirling (10) Zincenko (7) Steffensen (3) New-Steff. (5) New-Steff. Gen.(9) k 15 15 8 7 7 p 1.0000 1.0000 1.9994 3.0142 3.0148 ||xk−1 − xk || 2.3258e − 04 2.9041e − 06 6.9382e − 72 1.7796e − 111 2.1782e − 111 ||F (xk )|| 9.5985e − 05 1.1977e − 06 1.2745e − 107 7.8863e − 219 6.8587e − 219 Table 2: Numerical results for comparing the proposed methods.

the difference between the last iterates of each method and we also see the norm of the function evaluted in the last iteration. Now, we also want to use the Gauss-Legendre quadrature to approximate the integral of equation (25). Moreover, by using the Newton-Steffensen method we compare two different possibilities for implementing the divided differences given in (30), that is, in Tables 1 and 2 we obtain the divided difference like [xn , xn + F1 (xn ) + F2 (xn ), F2 ] but we want to compare with [xn , xn + F2 (xn ), F2 ]. The results in Table 3 show that the use of first form used for obtaining the divided differences gives better residual errors, which was expected because F1 (xn ) + F2 (xn ) tends to zero quicker than F2 (xn ). Even in some different example the value F2 (xn ) could not tend to zero, in this case only first form of obtaining the divided differences considered would work. In Table 3 we have also included the computational time, as can be observed in the last row, notice that the use of Gauss-Legedre quadrature needs much more time than the trapezoidal rule although in some cases reaches better accuracy. ||xn − xn−1 || Iterations T rapezoidal rule n [x, x + F1 + F2 , F2 ] [x, x + F2 , F2 ] 1 2.9044 2.9044 2 1.3867 1.3867 3 3.2041e − 01 1.2942e − 01 4 2.8725e − 04 2.8725e − 04 5 1.3552e − 12 1.3552e − 12 6 3.1538e − 37 3.3489e − 28 7 1.7796e − 111 1.3651e − 43

Gauss − Legendre [x, x + F1 + F2 , F2 ] [x, x + F2 , F2 ] 2.7204 2.7204 1.1355 1.1355 6.6978e − 02 6.6978e − 02 3.4608e − 05 3.4608e − 05 2.1448e − 15 2.1448e − 15 1.124e − 45 1.124e − 45 8.0773e − 137 7.8571e − 137

Table 3: Results with Trapezoidal rule and Gauss-Legendre method by using different form of obtaining the divided differences.

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T rapezoidal rule n [x, x + F1 + F2 , F2 ] [x, x + F2 , F2 ] k 7 8 p 3.0142 unstable ||xk−1 − xk || 1.7796e − 111 4.3463e − 59 ||F (xk )|| 7.8863e − 219 1.0160e − 74 time 17.796129 20.6134

Gauss − Legendre [x, x + F1 + F2 , F2 ] [x, x + F2 , F2 ] 7 7 3.0099 3.0103 8.0772e − 137 7.8571e − 137 1.3057e − 243 1.5367e − 138 282.5403 309.3090

Table 4: Numerical results and computational time for comparing the proposed methods.

4

Approximating the solution of a nonlinear PDE related to image denoising

In some steps of the manipulation of an image, some random noise is usually introduced. This noise makes the later steps of processing the image difficult and inaccurate. In many applications like astrophysics, astronomy or meteorology we have to manipulate images contaminated by noise. The image processing becomes difficult and inaccurate. For these reasons, usually some image denoising strategies are developed. In this paper, we center our attention in the PDE framework. Let f : Ω → R be a signal or image which we would like to denoise. The usual PDE frameworks start with constrained optimization problems like Minimize in u : R(u) subject to ku − f k2L2 (Ω) = |Ω|σ 2 . where n = u − f denotes the noise. If there is no good estimate of the variance of the noise, then we may consider the unconstrained optimization problem. Different linear regularization functionals R(u) can be consider, the most used is k∇ukL2 . This type of functionals introduce diffusion near the edges of the images, this is their main limitation. The TV norm does not penalize discontinuities in u, and thus allows us to improve the approximation near the edges. Z |∇u(x)|dx. Ω

For the linear model its Euler–Lagrange equation, with Neumann’s boundary conditions for u, is − 4u + λ(u − f ) = 0, (32)

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which comes from the corresponding unconstrained problem with the norm k∇uk2L2 (Ω) and where the positive parameter λ determines the relative importance of the smoothness of u and the quality of the approximation to the given signal f .. For the TV- model we have ∇u ) + λ(u − f ) = 0. (33) |∇u| p In practice, the term |∇u| is replaced by |∇u|2 + , but even after this regularization, Newton’s method does not work satisfactorily in the sense that its domain of convergence is very small. This is especially true if the regularizing parameter  is small. On the other hand, while the singularity and nondifferentiability of the term w = ∇u/|∇u| is the source of numerical problems, w itself is usually smooth because it is in fact the unit vector normal to the level sets of u. The numerical difficulties arise only because we linearize it the wrong way. Thus we should introduce a new variable w; namely −∇·(

∇u , w=p |∇u|2 and replace (33) by the equivalent system of nonlinear PDEs: −∇ · w + λ(u − f ) = 0, p w |∇u|2 − ∇u = 0. Without the inclusion of the above regularization parameter , this system is nonlinear and nondifferentiable .

4.1

Discretization and numerical implementation

We present a comparison between the nonlinear model and the linear model using a simple finite difference discretization procedure. For a regular mesh of size h = 1/m, m ∈ N (xi = i · h, i = 0, . . . , m), if in each iteration k we approximate the divergence and the gradient operators (these operators are the same in 1D) by vi − vi−1 ∇ · v(xi ) = ∇v(xi ) ≈ , h we obtain a nonlinear system for the unknowns wi and ui . That is, wi − wi−1 − − λ(ui − fi ) = 0, h r ui − ui−1 2 ui − ui−1 wi · ( ) − = 0, h h

w1 = wm = 0, u0 = f0 , um = fm ,

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for i = 1, . . . , m − 1. We then consider the nonlinear and nondifferentiable operator F2i−1 (u, w, λh ) = wi − wi−1 + λh (ui − fi ) = 0, p F2i (u, w, λh ) = wi (ui − ui−1 )2 + − (ui − ui−1 ) = 0,

1 ≤ i ≤ m − 1,

with λh = h λ, w0 = wm = 0, u0 = f0 and um = fm . For the discretization of the linear model we can consider the system −

ui+1 − 2ui + ui−1 − λ(ui − fi ) = 0, h2

u0 = f0 , um = fm ,

for i = 1, . . . , m − 1.

1.5

1.4

1.2

1 1 0.8

0.6 0.5 0.4

0.2 0 0

−0.2

−0.5

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

−0.4

1

Figure 1: Original signal with a jump singularity.

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Figure 2: Solid lines = nonlinear model, starred lines = linear model and + lines = signal with noise. Noise level = 0.3, λ = 10.

In Figure 2, the solid lines are the function reconstructed by the nonlinear model approximated by the linearization based on a dual variable, solving the nonlinear system of equations by Steffensen’s method 3 and the starred lines are given by the standard linear model, solving the associated linear system of equations by Gauss’s method. The line with ‘+’ is the noisy signal. The linear model introduces too much diffusion, giving a continuous function.

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5

Conclusions

We have to point out the generalization of this study in which we have analyzed the local and semilocal convergence for Newton type methods and Steffensen like methods, so we can consider Newton-Steffensen’s methods. The main idea it is to apply these kind of study to non-differentiable equations by taking in to account the advantages of consider the decomposition of the nonlinear equation into a sum of the differentiable part and the one non-differentiable.

Acknowledgements Research of the first and third authors supported in part by Programa de Apoyo a la investigaci´on de la fundaci´on S´eneca-Agencia de Ciencia y Tecnolog´ıa de la Regi´on de Murcia 19374/PI/14 and by MTM2015-64382-P. Research of the fourth and fifth authors supported in part by the project MTM2014-52016-C2-1-2-P of the Spanish Ministry of Science and Innovation and by the project of Generalitat Valenciana Prometeo/2016/089.

References [1] Amat, S.; Busquier, S.; Guti´errez, J. M., On the local convergence of secant-type methods. Int. J. Comput. Math. 81 (2004), no. 9, 1153-1161. [2] Amat, S.; Busquier, S., Convergence and numerical analysis of a family of two-step Steffensen’s methods. Comput. Math. Appl. 49 (2005), no. 1, 13-22. [3] Amat, S.; Busquier, S., On a Steffensen’s type method and its behavior for semismooth equations. Appl. Math. Comput. 177 (2006), no. 2, 819-823. [4] Amat, S.; Busquier, S., A two-step Steffensen’s method under modified convergence conditions. J. Math. Anal. Appl. 324 (2006), no. 2, 1084-1092. [5] Amat, S.; Busquier, S.; Guti´errez, J. M., An adaptive version of a fourth-order iterative method for quadratic equations. J. Comput. Appl. Math. 191 (2006), no. 2, 259-268. [6] Amat, S.; Ezquerro J.A.; Hern´andez M.A, On a Steffensen-like method for solving nonlinear equations, Calcolo (2016) 53, 171-188. [7] Alarc´on, V.; Amat, S.; Busquier, S.; L´opez, D.J., A Steffensen’s type method in Banach spaces with applications on boundary-value problems. J. Comput. Appl. Math. 216 (2008), no. 1, 243-250. [8] Argyros I.K., A new convergence theorem for Steffensen’s method on Banach spaces and applications, South west J. Pure Appl. Math. 1 (1997) 23-29. 16

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[9] Argyros I.K.; Magre˜ na´n A.A., Iterative Methods and Their Dynamics with Applications: A Contemporary Study, CRC Press, 2017. [10] Argyros I.K.; Magre˜ n´an A.A., A Contemporary Study of Iterative Methods, Academic Press, 2018. [11] Chen K.W., Generalization of Steffensen’s method for operator equations in Banach spaces, Comment. Math. Univ. Carolin. 5 (1964), no. 2, 47-77. [12] Ezquerro J.A.; Hern´andez M.A.; Romero N.; Velasco A.I., On Steffensen’s method on Banach spaces,J. Comput. Appl. Math. 249 (2013) 9-23. [13] Grau-S´anchez, M.; Noguera, M.; Amat, S., On the approximation of derivatives using divided difference operators preserving the local convergence order of iterative methods. J. Comput. Appl. Math. 237 (2013), no. 1, 363-372. [14] Hern´andez, M.A.; Mart´ınez, E., Improving the accessibility of Steffensen’s method by decomposition of operators, J.Comput. Appl. Math. 330 (2018) 536-552. [15] Hern´andez, M.A.; Rubio, M. J., On a Newton-Kurchatov-type Iterative Process, Numer. Funct. Anal. 37 (2016), no. 1, 65-79. [16] Ostrowski A.M., Solution of Equations in Euclidean and Banach Space, Academic Press, NewYork, (1973). [17] Zincenko A.I., Some approximate methods of solving equations with nondifferentiable operators, Dopovidi Akad Nauk, (1963) 156-161.

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On the Localization of Factored Fourier Series ¨ Hikmet Seyhan OZARSLAN Department of Mathematics Erciyes University 38039 Kayseri, TURKEY [email protected]

Abstract In the present paper, a theorem concerning local property of |A, pn |k summability of ¯ , pn |k summability of factored Fourier series, which generalizes a result dealing with |N factored Fourier series, has been obtained. Also, some results have been given. 2010 AMS Mathematics Subject Classification : 26D15, 40D15, 40F05, 40G99, 42A24. Keywords and Phrases :Absolute matrix summability, Fourier series, H¨ older inequality, Infinite series, Local property, Minkowski inequality, Summability factors.

1 Let

Introduction P

an be an infinite series with its partial sums (sn ) and (pn ) be a sequence of positive

numbers such that Pn =

n X

pv → ∞ as

n → ∞,

(P−i = p−i = 0,

i ≥ 1) .

v=0

Let A = (anv ) be a normal matrix, i.e., a lower triangular matrix of nonzero diagonal entries. Then A defines the sequence-to-sequence transformation, mapping the sequence s = (sn ) to As = (An (s)), where An (s) =

n X

anv sv ,

n = 0, 1, ...

v=0

1

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The series

P

an is said to be summable |A, pn |k , k ≥ 1, if (see [21])  ∞  X Pn k−1 n=1

If we take anv = If we take anv =

pv Pn , pv Pn

|An (s) − An−1 (s)|k < ∞.

pn

¯ , pn |k summability (see [2]). then |A, pn |k summability reduces to |N and pn = 1 for all values of n (resp. anv =

pv Pn

and k = 1), |A, pn |k

¯ , pn |) summability. Also, if summability reduces to |C, 1|k summability (see [11]) (resp. |N we take pn = 1 for all values of n, then |A, pn |k summability reduces to |A|k summability (see [22]). Furthermore, if we take anv =

pv Pn ,

then |A|k summability reduces to |R, pn |k

summability (see [4]). A sequence (λn ) is said to be convex if ∆2 λn ≥ 0 for every positive integer n, where ∆2 λn = ∆(∆λn ) and ∆λn = λn − λn+1 (see [24]). Let f (t) be a periodic function with period 2π, and integrable (L) over (−π, π). Without any loss of generality we may assume that the constant term in the Fourier series of f (t) is zero, so that Z

π

f (t)dt = 0 −π

and f (t) ∼

∞ X

(an cosnt + bn sinnt) =

∞ X

Cn (t),

n=1

n=1

where (an ) and (bn ) denote the Fourier coefficients. It is well known that the convergence of the Fourier series at t = x is a local property of the generating function f (i.e. it depends only on the behaviour of f in an arbitrarily small neighbourhood of x), and hence the summability of the Fourier series at t = x by any regular linear summability method is also a local property of the generating function f (see [23]).

2

Known Results

There are many different applications of Fourier series. Some of them can be find in [1], [5]-[10], [12]-[20]. Furthermore, Bor [3] has proved the following theorem. 2

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Theorem 1 Let k ≥ 1 and (pn ) be a sequence such that Pn = O(npn ),

(1)

Pn ∆pn = O(pn pn+1 ).

(2)

¯ , pn | of the series Then the summability |N k

P

Cn (t)λn Pn npn

at a point can be

ensured by local property, where (λn ) is a convex sequence such that

P

n−1 λn is

convergent.

3

Main Result

The purpose of this paper is to generalize Theorem 1 by using the definition of |A, pn |k summability. Now, let us introduce some further notations. Let A = (anv ) be a normal matrix, we associate two lower semimatrices A¯ = (¯ anv ) and Aˆ = (ˆ anv ) as follows: a ¯nv =

n X

ani ,

n, v = 0, 1, ...

(3)

i=v

a ˆ00 = a ¯00 = a00 ,

a ˆnv = a ¯nv − a ¯n−1,v ,

n = 1, 2, ...

(4)

and it is well known that An (s) =

n X

anv sv =

n X

a ¯nv av

(5)

v=0

v=0

and ¯ n (s) = ∆A

n X

a ˆnv av .

(6)

v=0

Now, we will prove the following theorem. Theorem 2 Let k ≥ 1 and A = (anv ) be a positive normal matrix such that an0 = 1, n = 0, 1, ...,

(7)

an−1,v ≥ anv , f or n ≥ v + 1,

(8)

3

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pn =O , Pn 

ann



(9)

|ˆ an,v+1 | = O (v |∆v a ˆnv |) ,

(10)

where ∆v (ˆ anv ) = a ˆnv − a ˆn,v+1 . Let the sequence (pn ) be such that the conditions (1) and (2) of Theorem 1 are satisfied. Then the summability |A, pn |k of the series

P

Cn (t)λn Pn npn

at

a point can be ensured by local property, where (λn ) is as in Theorem 1. Here, if we take anv =

pv Pn ,

then we get Theorem 1.

We should give the following lemmas for the proof of Theorem 2. Lemma 3 ([13]) If the sequence (pn ) is such that the conditions (1) and (2) of Theorem 1 are satisfied, then 

∆

Pn npn



1 . n

 

=O

(11)

Lemma 4 ([10]) If (λn ) is a convex sequence such that is non-negative and decreasing, and n∆λn → 0

as

P

n−1 λn is convergent, then (λn )

n → ∞.

Lemma 5 Let k ≥ 1 and let the sequence (pn ) be such that the conditions (1) and (2) of Theorem 1 are satisfied. If (sn ) is bounded and the conditions (7)-(10) are satisfied, then the series ∞ X an λ n Pn n=1

(12)

npn

is summable |A, pn |k , where (λn ) is as in Theorem 1. Remark 6 Since (λn ) is a convex sequence, therefore (λn )k is also convex sequence and X1

n

(λn )k < ∞.

(13)

4

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4

Proof of Lemma 5

Let (Mn ) denotes the A-transform of the series n X

¯ n = ∆M

P

a ˆnv

v=1

an λn Pn npn .

Then, we have

av λv Pv vpv

by (5) and (6).

Now, we get n−1 X

¯ n = ∆M

v=1 n−1 X

=



∆v 

∆v

v=1

a ˆnv λv Pv vpv

X v

a ˆnv λv Pv vpv



ar +

r=1

sv +

n a ˆnn Pn λn X av npn v=1

ann Pn λn sn npn

n−1 n−1 X Pv λv ∆v (ˆ Xa ann Pn λn anv ) ˆn,v+1 ∆λv Pv sn + sv + sv npn vpv vpv v=1 v=1

=

n−1 X

Pv a ˆn,v+1 λv+1 ∆ + vpv v=1 



sv

= Mn,1 + Mn,2 + Mn,3 + Mn,4 by applying Abel’s transformation. For the proof of Lemma 5, it is sufficient to show that  ∞  X Pn k−1 n=1

pn

|Mn,r |k < ∞,

f or

r = 1, 2, 3, 4.

First, we have  m  X Pn k−1 n=1

pn

k

|Mn,1 |

=

 m  X Pn k−1 ann Pn λn k sn np p

n=1

= O(1) = O(1)

n

m X n=1 m X

n



Pn pn

k−1 

pn Pn

k

1 (λn )k = O(1) n n=1

1 nk as



Pn pn

k

(λn )k |sn |k

m → ∞,

by (9), (1) and (13).

5

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From H¨older’s inequality, we have m+1 X n=2

Pn pn

k−1

k k−1 n−1 anv ) X Pv λv ∆v (ˆ = sv vpv n=2 v=1 ( m+1 X  Pn k−1 n−1 X  Pv  m+1 X

k

|Mn,2 |

≤

pn

n=2 m+1 X

≤

Pn pn

n=2

Pn pn

|∆v (ˆ anv )| (λv )|sv |

vpv

v=1

k−1 (n−1 X v=1

Pv vpv

)k

k

k

k

|∆v (ˆ anv )|(λv ) |sv |

) (n−1 X

)k−1

|∆v (ˆ anv )|

v=1

By (4) and (3), we have that ∆v (ˆ anv ) = a ˆnv − a ˆn,v+1 = a ¯nv − a ¯n−1,v − a ¯n,v+1 + a ¯n−1,v+1 = anv − an−1,v .

(14)

Thus using (8), (3) and (7)

n−1 X

|∆v (ˆ anv )| =

v=1

n−1 X

(an−1,v − anv ) ≤ ann .

(15)

v=1

Hence, we get m+1 X n=2

Pn pn

k−1

k

|Mn,2 |

= O(1) = O(1)

m+1 X n=2 m  X v=1

Pn pn

Pv pv

k−1

k

ak−1 nn

(n−1   X Pv k 1 v=1 m+1 X

pv

vk

)

|∆v (ˆ anv )|(λv )

k

1 (λv )k |∆v (ˆ anv )| . vk n=v+1

Here, from (14) and (8), we obtain m+1 X

|∆v (ˆ anv )| =

n=v+1

m+1 X

(an−1,v − anv ) ≤ avv .

n=v+1

Then, m+1 X n=2

Pn pn

k−1

k

|Mn,2 |

= O(1)

 m  X Pv k 1 v=1

pv

vk

(λv )k avv

6

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 m  X Pv k−1 1

= O(1)

v=1 m X

= O(1)

pv

vk

v k−1

1 (λv )k vk

v=1 m X

(λv )k

1 (λv )k = O(1) v v=1

= O(1)

m → ∞,

as

by (9), (1) and (13). Now, by (1) and H¨ older’s inequality, we have m+1 X n=2

Pn pn

k−1

|Mn,3 |k =

k k−1 n−1 ˆn,v+1 ∆λv Pv X a sv vpv v=1 ( m+1 X  Pn k−1 n−1 X

m+1 X n=2

= O(1)

Pn pn

= O(1)

|ˆ an,v+1 |∆λv |sv |

pn

n=2 m+1 X n=2

)k

v=1

Pn pn

k−1 (n−1 X

|ˆ an,v+1 |∆λv |sv |k

) (n−1 X

)k−1

|ˆ an,v+1 |∆λv

v=1

v=1

Now, (4), (3), (7) and (8) imply that a ˆn,v+1 = a ¯n,v+1 − a ¯n−1,v+1 = =

n X

n−1 X

ani − v X

i=0

i=0

ani −

= 1−

v X

ani −

ani − 1 +

i=0

=

an−1,i

i=v+1

i=v+1 n X

n−1 X

an−1,i +

i=0 v X

v X

an−1,i

i=0

an−1,i

i=0

v X

(an−1,i − ani ) ≥ 0

(16)

i=0

and from this, using (4), (3) and (8), we have |ˆ an,v+1 | = a ¯n,v+1 − a ¯n−1,v+1 =

n X

n−1 X

ani −

i=v+1

an−1,i

i=v+1 n−1 X

= ann +

(ani − an−1,i )

i=v+1

≤ ann . 7

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Hence, we get m+1 X n=2

Pn pn

k−1

k

|Mn,3 |

= O(1) = O(1)

m+1 X n=2 m+1 X n=2 m X

= O(1)

Pn pn

k−1

Pn pn

k−1

v=1

ak−1 nn

n−1 X

|ˆ an,v+1 |∆λv

v=1 (n−1 X

(n−1 X

)k−1

∆λv

v=1

)

|ˆ an,v+1 |∆λv

v=1 m+1 X

∆λv

ak−1 nn

|ˆ an,v+1 |.

n=v+1

Now, by (16), (3) and (7), we find m+1 X

|ˆ an,v+1 | ≤ 1.

(17)

n=v+1

Thus, m+1 X n=2

Pn pn

k−1

|Mn,3 |k = O(1)

m X

∆λv = O(1)

as

m → ∞,

v=1

by Lemma 4. Since ∆ m+1 X n=2



Pn pn

Pv vpv



=O

k−1

  1 v

k

|Mn,4 |

by Lemma 3 and also by using (10), we have that =

k−1 n−1   k Pv X a ˆn,v+1 λv+1 ∆ sv vp v v=1 ( )k m+1 X1 X  Pn k−1 n−1

m+1 X n=2

= O(1)

Pn pn

n=2

= O(1)

m+1 X n=2

pn Pn pn

v=1

k−1 n−1 X v=1

v

|ˆ an,v+1 |(λv+1 )|sv |

1 |ˆ an,v+1 |(λv+1 )k |sv |k v

(n−1 X

)k−1

|∆v (ˆ anv )|

v=1

From (15) and (9), m+1 X n=2

Pn pn

k−1

k

|Mn,4 |

= O(1)

m+1 X n=2

= O(1)

m X 1

v v=1

Pn pn

k−1

(λv+1 )k

ak−1 nn

n−1 X v=1

m+1 X

1 |ˆ an,v+1 |(λv+1 )k v

|ˆ an,v+1 |.

n=v+1

Again using (17), m+1 X n=2

Pn pn

k−1

k

|Mn,4 |

= O(1)

m X 1

v v=1

(λv+1 )k = O(1)

as

m → ∞,

by (13). Hence the proof of Lemma 5 is completed. 8

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5

Proof of Theorem 2

The convergence of the Fourier series at t = x is a local property of f (i.e., it depends only on the behaviour of f in an arbitrarily small neighbourhood of x), and hence the summability of the Fourier series at t = x by any regular linear summability method is also a local property of f . Since the behaviour of the Fourier series, as far as convergence is concerned, for a particular value of x depends on the behaviour of the function in the immediate neighbourhood of this point only, hence the truth of Theorem 2 is a consequence of Lemma 5.

6

Conclusions

For anv =

pv Pn

and pn = 1 for all values of n, then we get a result concerning |C, 1|k

summability factors of Fourier series. If we take anv =

pv Pn

and k = 1, then we get a result

¯ , pn | summability factors of Fourier series (see [13]). concerning |N

References [1] S. N. Bhatt, An aspect of local property of | R, logn, 1 | summability of the factored Fourier series, Proc. Nat. Inst. Sci. India Part A, 26 (1960), 69–73. [2] H. Bor, On two summability methods, Math. Proc. Cambridge Philos Soc., 97(1) (1985), 147–149. ¯ , pn |k summability of factored Fourier series, Bull. Inst. [3] H. Bor, Local property of | N Math. Acad. Sinica, 17(2) (1989), 165–170. [4] H. Bor, On the relative strength of two absolute summability methods, Proc. Amer. Math. Soc., 113(4) (1991), 1009-1012. [5] H. Bor, Some new results on absolute Riesz summability of infinite series and Fourier series, Positivity, 20(3) (2016), 599-605. 9

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[6] H. Bor, On absolute weighted mean summability of infinite series and Fourier series, Filomat, 30(10) (2016), 2803-2807. [7] H. Bor, Absolute weighted arithmetic mean summability factors of infinite series and trigonometric Fourier series, Filomat, 31(15) (2017), 4963-4968. [8] H. Bor, An application of quasi-monotone sequences to infinite series and Fourier series, Anal. Math. Phys., 8(1) (2018), 77-83. [9] H. Bor, On absolute summability of factored infinite series and trigonometric Fourier series, Results Math., 73(3) (2018), Art. 116, 9 pp. [10] H. C. Chow, On the summability factors of Fourier series, J. London Math. Soc., 16 (1941), 215-220. [11] T. M. Flett, On an extension of absolute summability and some theorems of Littlewood and Paley, Proc. London Math. Soc., 7 (1957), 113-141. [12] K. Matsumoto, Local property of the summability |R, λn, 1|, Tˆohoku Math. J. (2), 8 (1956), 114–124. ¯ , pn | summability of Fourier series, Bull. Inst. Math. [13] K. N. Mishra, Multipliers for | N Acad. Sinica, 14 (1986), 431–438. [14] R. Mohanty, On the summability | R, logω, 1 | of a Fourier Series, J. London Math. Soc., 25 (1950), 67–72. ¨ ¯ , pα | summability factors, Soochow J. Math., 27(1) [15] H. S. Ozarslan, A note on |N n k (2001), 45-51. ¨ ¨ gd¨ [16] H. S. Ozarslan and H. N. O˘ uk, Generalizations of two theorems on absolute summability methods, Aust. J. Math. Anal. Appl., 1 (2004), Article 13 , 7 pp. ¨ ¯ , pn |k summability factors, Int. J. Pure Appl. Math., [17] H. S. Ozarslan, A note on |N 13(4) (2004), 485-490. 10

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¨ [18] H. S. Ozarslan, Local properties of factored Fourier series, Int. J. Comp. Appl. Math., 1 (2006), 93-96. ¨ [19] H. S. Ozarslan, On the local properties of factored Fourier series, Proc. Jangjeon Math. Soc., 9(2) (2006), 103-108. ¯ , pn ; δ|k summability of factored Fourier [20] H. Seyhan, On the local property of ϕ − |N series, Bull. Inst. Math. Acad. Sinica, 25(4) (1997), 311–316. [21] W. T. Sulaiman, Inclusion theorems for absolute matrix summability methods of an infinite series. IV, Indian J. Pure Appl. Math., 34(11) (2003), 1547–1557. [22] N. Tanovi˘ c-Miller, On strong summability, Glas. Mat. Ser. III, 14(34) (1979), 87–97. [23] E. C. Titchmarsh, Theory of Functions, Second Edition, Oxford University Press, London, 1939. [24] A. Zygmund, Trigonometric Series, Instytut Matematyczny Polskiej Akademi Nauk, Warsaw, 1935.

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Analysis of Solutions of Some Discrete Systems of Rational Difference Equations M. B. Almatrafi. Department of Mathematics, Faculty of Science, Taibah University, P.O. Box 30002, Saudi Arabia. E-mails: [email protected] Abstract The major objective of this article is to determine and formulate the analytical solutions of the following systems of rational recursive equations: xn+1 =

xn−1 yn−3 , yn−1 (±1 ∓ xn−1 yn−3 )

yn+1 =

yn−1 xn−3 , xn−1 (∓1 ± yn−1 xn−3 )

n = 0, 1, ...,

where the initial conditions x−3 , x−2 , x−1 , x0 , y−3 , y−2 , y−1 and y0 are required to be arbitrary non-zero real numbers. We also introduce some graphs describing these exact solutions under a suitable choice of some initial conditions.

Keywords: difference equations, system of recursive equations, periodicity, local stability, global stability. Mathematics Subject Classification: 39A10.

1

Introduction

The global interest in exploring the qualitative behaviours of discrete systems of recursive equations has been recently emerged due to the significance of difference equations in modelling a considerable number of discrete phenomena. More specifically, recursive equations are utilized in describing some real life problems that originate in genetics in biology, queuing problems, enegineering, physics, etc. Some experts put effort to analyse dynamical systems of difference equations. Take, for instance, the following ones. Almatrafi et al. [1] studied the local stability, global attractivity, periodicity and solutions for a special case for the difference equation bxn−1 xn+1 = axn−1 − . cxn−1 − dxn−3 Clark and Kulenovic [6] investigated the global attractivity of the system xn+1 =

xn , a + cyn

yn+1 =

355

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The author in [8] explored the equilibrium points and the stability of a discrete Lotka-Volterra model shown as follows: xn+1 =

αxn − βxn yn , 1 + γxn

yn+1 =

δyn + xn yn . 1 + ηyn

The positive solutions of the system un+1 =

αun−1 , q β + γvnp vn−2

vn+1 =

α1 vn−1 . 1 β1 + γ1 upn1 uqn−2

˝ were obtained in [14] by G˝ um˝ u¸s and Ocalan. Moreover, Kurbanli et al. [18] solved the dynamical systems of recursive equations given by xn+1 =

yn−1 xn xn−1 , yn+1 = , zn+1 = . yn xn−1 − 1 xn yn−1 − 1 yn zn−1

In [19] Mansour et al. presented the analytical solutions of the system xn+1 =

xn−1 , α − xn−1 yn

yn+1 =

yn−1 . β + γyn−1 xn

Finally, the author in [23] demonstrated the dynamics of the system xn+1 =

xn−2 , B + yn yn−1 yn−2

yn+1 =

yn−2 . A + xn xn−1 xn−2

To attain more information on the qualitative behaviours of dynamical difference equations, one can refer to refs [1–5, 7, 9–13, 15–17, 20–22] In this paper, the rational solutions of the following discrete systems of difference equations will be discovered and given in four different theorems: xn+1 =

xn−1 yn−3 , yn−1 (±1 ∓ xn−1 yn−3 )

yn+1 =

yn−1 xn−3 , xn−1 (∓1 ± yn−1 xn−3 )

n = 0, 1, ...,

where the initial values are as described previously.

2 2.1

Main Results First System xn+1 =

xn−1 yn−3 yn−1 (1−xn−1 yn−3 ) ,

yn+1 =

yn−1 xn−3 xn−1 (1−yn−1 xn−3 )

This subsection concentrates on obtaining the solutions of a dynamical system of fourth order difference equations given by the form: xn+1 =

xn−1 yn−3 yn−1 xn−3 , yn+1 = , n = 0, 1, ... , yn−1 (1 − xn−1 yn−3 ) xn−1 (1 − yn−1 xn−3 )

(1)

where the initial values are as shown previously. The following fundamental theorem presents the solutions of system (1).

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Theorem 1 Assume that {xn , yn } is a solution to system (1) and let x−3 = α, x−2 = β, x−1 = γ, x0 = δ, y−3 = , y−2 = η, y−1 = µ and y0 = ω.Then, for n = 0, 1, ... we have n−1

n−1

γ n n Π [(2i) αµ − 1]

δ n η n Π [(2i) βω − 1]

i=0 n−1

x4n−3 =

,

αn−1 µn Π [(2i + 1) γ − 1]

i=0 n−1

δ n+1 η n Π [(2i + 1) βω − 1]

γ n+1 n Π [(2i + 1) αµ − 1] x4n−1 = α n µn

,

β n−1 ω n Π [(2i + 1) δη − 1]

i=0 n−1

i=0 n−1

i=0 n−1

x4n−2 =

,

i=0 n−1

x4n = β nωn

Π [(2i + 2) γ − 1]

i=0

.

Π [(2i + 2) δη − 1]

i=0

And

n−1

n−1

αn µn Π [(2i) γ − 1] i=0 n−1

y4n−3 = γ n n−1

y4n−1 =

β n ω n Π [(2i) δη − 1] ,

δ n η n−1

Π [(2i + 1) αµ − 1]

i=0 n−1 αn µn+1 Π [(2i + 1) γ − 1] i=0 , n−1 n n γ  Π [(2i + 2) αµ − 1] i=0

i=0 n−1

y4n−2 =

,

Π [(2i + 1) βω − 1]

i=0 n−1

β n ω n+1 Π [(2i + 1) δη − 1] i=0 n−1

y4n = δnηn

.

Π [(2i + 2) βω − 1]

i=0

Proof. For n = 0, our results hold. Next, let n > 1 and suppose that the relations hold for n − 1. That is n−2

n−2

δ n−1 η n−1 Π [(2i) βω − 1]

γ n−1 n−1 Π [(2i) αµ − 1] i=0 n−2

x4n−7 = αn−2 µn−1

,

β n−2 ω n−1

Π [(2i + 1) γ − 1]

i=0 n−2

x4n−5 =

,

Π [(2i + 1) δη − 1]

i=0 n−2

γ n n−1 Π [(2i + 1) αµ − 1]

i=0 n−2 αn−1 µn−1 Π [(2i i=0

i=0 n−2

x4n−6 =

δ n η n−1 Π [(2i + 1) βω − 1]

,

x4n−4 =

+ 2) γ − 1]

i=0 n−2 β n−1 ω n−1 Π [(2i i=0

. + 2) δη − 1]

And n−2

n−2

αn−1 µn−1 Π [(2i) γ − 1] y4n−7 =

β n−1 ω n−1 Π [(2i) δη − 1]

i=0 n−2

,

y4n−6 =

γ n−1 n−2 Π [(2i + 1) αµ − 1]

i=0 n−2

αn−1 µn Π [(2i + 1) γ − 1]

y4n−5 =

,

δ n−1 η n−2 Π [(2i + 1) βω − 1]

i=0 n−2

i=0 n−2 γ n−1 n−1 Π [(2i i=0

i=0 n−2

β n−1 ω n Π [(2i + 1) δη − 1]

, + 2) αµ − 1]

357

y4n =

i=0 n−2 δ n−1 η n−1 Π [(2i i=0

. + 2) βω − 1]

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Now, it can be obviously observed from system (1) that x4n−3 =

x4n−5 y4n−7 y4n−5 (1 − x4n−5 y4n−7 ) n−2

n−2

γ n n−1 Π [(2i+1)αµ−1]

i=0 n−2 n−1 n−1 α µ Π [(2i+2)γ−1] i=0

=

n−2

i=0 n−2

Π [(2i+1)αµ−1]

i=0

n−2

γ n n−1 Π [(2i+1)αµ−1]

1−

i=0 n−2

γ n−1 n−1

γ n−1 n−2

n−2

"

αn−1 µn Π [(2i+1)γ−1]

αn−1 µn−1 Π [(2i)γ−1]

i=0 n−2

αn−1 µn−1

Π [(2i+2)αµ−1]

i=0

Π [(2i+2)γ−1]

i=0

#

αn−1 µn−1 Π [(2i)γ−1] γ n−1 n−2

i=0 n−2

Π [(2i+1)αµ−1]

i=0

n−2

γ Π [(2i)γ−1]

i=0 n−2

Π [(2i+2)γ−1]

i=0

=

n−2

1−

i=0 n−2

γ n−1 n−1

n−2

"

αn−1 µn Π [(2i+1)γ−1] Π [(2i+2)αµ−1]

γ Π [(2i)γ−1]

#

i=0 n−2

Π [(2i+2)γ−1]

i=0

i=0

n−2

n−2

γ n n Π [(2i) γ − 1] Π [(2i + 2) αµ − 1] i=0 i=0   = n−2 n−2 n−2 n−1 n α µ Π [(2i + 1) γ − 1] Π [(2i + 2) γ − 1] − γ Π [(2i) γ − 1] i=0

i=0

i=0

n−2

n−1

γ n n Π [(2i + 2) αµ − 1]

=

i=0 − n−1 αn−1 µn Π [(2i i=0

γ n n Π [(2i) αµ − 1] i=0 n−1

= + 1) γ − 1]

αn−1 µn

.

Π [(2i + 1) γ − 1]

i=0

Now, system (1) gives us that y4n−3 =

y4n−5 x4n−7 x4n−5 [1 − y4n−5 x4n−7 ] n−2

n−2

αn−1 µn Π [(2i+1)γ−1]

=

i=0 n−2 n−1 n−1 γ  Π [(2i+2)αµ−1] i=0 n−2

i=0 n−2

Π [(2i+1)γ−1]

i=0

αn−1 µn Π [(2i+1)γ−1]

1−

i=0 n−2

αn−1 µn−1

αn−2 µn−1

n−2

"

γ n n−1 Π [(2i+1)αµ−1]

γ n−1 n−1 Π [(2i)αµ−1]

i=0 n−2

γ n−1 n−1

Π [(2i+2)γ−1]

i=0

Π [(2i+2)αµ−1]

i=0

n−2

γ n−1 n−1 Π [(2i)αµ−1] αn−2 µn−1

#

i=0 n−2

Π [(2i+1)γ−1]

i=0

n−2

αµ Π [(2i)αµ−1]

i=0 n−2

Π [(2i+2)αµ−1]

=

i=0 n−2

1−

i=0 n−2

αn−1 µn−1

n−2

"

γ n n−1 Π [(2i+1)αµ−1] Π [(2i+2)γ−1]

i=0

αµ Π [(2i)αµ−1]

#

i=0 n−2

Π [(2i+2)αµ−1]

i=0

n−2

n−2

αn µn Π [(2i) αµ − 1] Π [(2i + 2) γ − 1] i=0 i=0   = n−2 n−2 n−2 n n−1 γ  Π [(2i + 1) αµ − 1] Π [(2i + 2) αµ − 1] − αµ Π [(2i) αµ − 1] i=0

i=0

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n−2

n−1

αn µn Π [(2i + 2) γ − 1]

=−

i=0 n−1 γ n n−1 Π [(2i i=0

αn µn Π [(2i) γ − 1] i=0 n−1

= γ n n−1

+ 1) αµ − 1]

.

Π [(2i + 1) αµ − 1]

i=0

Hence, the rest of the results can be similarly proved.

2.2

Second System xn+1 =

xn−1 yn−3 yn−1 (−1+xn−1 yn−3 ) ,

yn+1 =

yn−1 xn−3 xn−1 (−1+yn−1 xn−3 )

Our leading duty in this subsection is to determine the solutions of the following discrete systems: xn−1 yn−3 yn−1 xn−3 xn+1 = , yn+1 = . (2) yn−1 (−1 + xn−1 yn−3 ) xn−1 (−1 + yn−1 xn−3 ) The initial values of this system are arbitrary real numbers. Theorem 2 Suppose that {xn , yn } is a solution to system (2) and assume that x−3 = α, x−2 = β, x−1 = γ, x0 = δ, y−3 = , y−2 = η, y−1 = µ and y0 = ω.Then, for n = 0, 1, ... we have γ n n , αn−1 µn (γ − 1)n γ n+1 n (αµ − 1)n = , α n µn

δnηn , β n−1 ω n (δη − 1)n δ n+1 η n (βω − 1)n = . β nωn

x4n−3 =

x4n−2 =

x4n−1

x4n

And

α n µn , γ n n−1 (αµ − 1)n αn µn+1 (γ − 1)n = , γ n n

β nωn , δ n η n−1 (βω − 1)n β n ω n+1 (δη − 1)n = . δnηn

y4n−3 =

y4n−2 =

y4n−1

y4n

Proof. It is obvious that all solutions are satisfied for n = 0. Next, we suppose that n > 1 and assume that the solutions hold for n − 1. That is δ n−1 η n−1 γ n−1 n−1 , x = , 4n−6 αn−2 µn−1 (γ − 1)n−1 β n−2 ω n−1 (δη − 1)n−1 γ n n−1 (αµ − 1)n−1 δ n η n−1 (βω − 1)n−1 = , x4n−4 = . αn−1 µn−1 β n−1 ω n−1

x4n−7 = x4n−5 And

αn−1 µn−1 β n−1 ω n−1 , y = , 4n−6 γ n−1 n−2 (αµ − 1)n−1 δ n−1 η n−2 (βω − 1)n−1 αn−1 µn (γ − 1)n−1 β n−1 ω n (δη − 1)n−1 = , y = . 4n−4 γ n−1 n−1 δ n−1 η n−1

y4n−7 = y4n−5

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We now turn to illustrate the first result. System (2) leads to

x4n−3 = = =

x4n−5 y4n−7 y4n−5 (−1 + x4n−5 y4n−7 ) γ n n−1 (αµ−1)n−1 αn−1 µn−1 αn−1 µn−1 γ n−1 n−2 (αµ−1)n−1

h

αn−1 µn (γ−1)n−1 −1 γ n−1 n−1 n n

αn−1 µn

+

γ n n−1 (αµ−1)n−1 αn−1 µn−1 αn−1 µn−1 γ n−1 n−2 (αµ−1)n−1 n n

i

γ  γ  = n−1 n . n−1 α µ (γ − 1)n (γ − 1) [−1 + γ]

Similarly, it is easy to see from system (2) that

y4n−3 = = =

y4n−5 x4n−7 x4n−5 (−1 + y4n−5 x4n−7 ) αn−1 µn (γ−1)n−1 γ n−1 n−1 γ n−1 n−1 αn−2 µn−1 (γ−1)n−1

h

γ n n−1 (αµ−1)n−1 −1 αn−1 µn−1 n n

γ n n−1

+

αn−1 µn (γ−1)n−1 γ n−1 n−1 γ n−1 n−1 αn−2 µn−1 (γ−1)n−1 n n

i

α µ α µ = n n−1 . n−1 γ  (αµ − 1)n (αµ − 1) [−1 + αµ]

The remaining solutions of system (2) can be clearly justified in a similar technique. Thus, the proof is complete.

2.3

Third System xn+1 =

xn−1 yn−3 yn−1 (1−xn−1 yn−3 ) ,

yn−1 xn−3 xn−1 (−1+yn−1 xn−3 )

yn+1 =

The central point of this subsection is to resolve a system of fourth order rational recursive equations given by the form: xn+1 =

xn−1 yn−3 , yn−1 (1 − xn−1 yn−3 )

yn+1 =

yn−1 xn−3 , xn−1 (−1 + yn−1 xn−3 )

(3)

where the initial values are as described previously. Theorem 3 Let {xn , yn } be a solution to system (3) and suppose that x−3 = α, x−2 = β, x−1 = γ, x0 = δ, y−3 = , y−2 = η, y−1 = µ and y0 = ω. Then, for n = 0, 1, ... we have (−1)n γ n n

x4n−3 =

,

n−1

αn−1 µn

Π [(2i + 1) γ − 1]

x4n−1 =

 (αµ − 1)n

n−1

,

x4n =

αn µn Π [(2i + 2) γ − 1]

,

n−1

β n−1 ω n

i=0 n n+1 n

(−1) γ

(−1)n δ n η n

x4n−2 = (−1)n δ

Π [(2i + 1) δη − 1]

i=0 n+1 n

η (βω − 1)n

n−1

.

β n ω n Π [(2i + 2) δη − 1]

i=0

i=0

And

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n−1

n−1

(−1)n αn µn Π [(2i) γ − 1] i=0

y4n−3 =

(−1)n β n ω n Π [(2i) δη − 1] ,

γ n n−1 (αµ − 1)n

i=0

y4n−2 =

n−1

n−1

(−1)n αn µn+1 Π [(2i + 1) γ − 1] i=0 γ n n

y4n−1 =

,

δ n η n−1 (βω − 1)n

(−1)n β n ω n+1 Π [(2i + 1) δη − 1] ,

i=0 δnηn

y4n =

.

Proof. The results are true for n = 0. Next, we suppose that n > 1 and assume that the relations hold for n − 1. That is (−1)n−1 γ n−1 n−1

x4n−7 =

n−2

αn−2 µn−1 (−1)

γ 

(αµ − 1)n−1

n−2

β n−2 ω n−1 ,

n−2

αn−1 µn−1

(−1)n−1 δ n−1 η n−1

x4n−6 =

Π [(2i + 1) γ − 1]

i=0 n−1 n n−1

x4n−5 =

,

(−1)

x4n−4 =

Π [(2i + 1) δη − 1]

i=0 n−1 n n−1

δ η

Π [(2i + 2) γ − 1]

(βω − 1)n−1

n−2

β n−1 ω n−1

i=0

,

.

Π [(2i + 2) δη − 1]

i=0

And n−2

n−2

(−1)n−1 β n−1 ω n−1 Π [(2i) δη − 1]

(−1)n−1 αn−1 µn−1 Π [(2i) γ − 1] y4n−7 =

i=0

γ n−1 n−2

(αµ − 1)

,

n−1

i=0

y4n−6 =

δ n−1 η n−2

n−2 i=0

(−1)n−1 β n−1 ω n Π [(2i + 1) δη − 1] ,

γ n−1 n−1

,

n−2

(−1)n−1 αn−1 µn Π [(2i + 1) γ − 1] y4n−5 =

(βω − 1)n−1 i=0

y4n−4 =

.

δ n−1 η n−1

Now, we establish the proofs of two relations. Firstly, system (3) gives us that x4n−3 =

x4n−5 y4n−7 y4n−5 (1 − x4n−5 y4n−7 ) n−2

(−1)n−1 γ n n−1 (αµ−1)n−1 αn−1 µn−1

=

(−1)n−1 αn−1 µn−1 Π [(2i)γ−1]

n−2

Π [(2i+2)γ−1]

i=0

γ n−1 n−2 (αµ−1)n−1

i=0

n−2

(−1)n−1 αn−1 µn Π [(2i+1)γ−1]

n−2

" 1−

i=0

γ n−1 n−1

(−1)n−1 γ n n−1 (αµ−1)n−1 αn−1 µn−1

n−2

Π [(2i+2)γ−1]

(−1)n−1 αn−1 µn−1 Π [(2i)γ−1]

#

i=0

γ n−1 n−2 (αµ−1)n−1

i=0

n−2

γ Π [(2i)γ−1]

i=0 n−2

Π [(2i+2)γ−1]

=

i=0 n−2

(−1)n−1 αn−1 µn Π [(2i+1)γ−1] i=0

γ n−1 n−1

n−2

" 1−

γ Π [(2i)γ−1]

#

i=0 n−2

Π [(2i+2)γ−1]

i=0

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n−2

(−1)−n+1 γ n n Π [(2i) γ − 1] i=0   = n−2 n−2 n−2 n−1 n α µ Π [(2i + 1) γ − 1] Π [(2i + 2) γ − 1] − γ Π [(2i) γ − 1] i=0

i=0

−n+1

=

− (−1)

i=0

n

n n

γ 

(−1) γ 

=

n−1

n n

.

n−1

αn−1 µn Π [(2i + 1) γ − 1]

αn−1 µn Π [(2i + 1) γ − 1]

i=0

i=0

Next, it can be noticed from system (3) that y4n−5 x4n−7 y4n−3 = x4n−5 (−1 + y4n−5 x4n−7 ) n−2

(−1)n−1 αn−1 µn Π [(2i+1)γ−1]

(−1)n−1 γ n−1 n−1

i=0

γ n−1 n−1

n−2

αn−2 µn−1 Π [(2i+1)γ−1] i=0

= (−1)n−1 γ n n−1 (αµ−1)n−1 αn−1 µn−1

"

n−2

−1 +

(−1)n−1 αn−1 µn

Π [(2i+1)γ−1]

Π [(2i+2)γ−1]

n−2 i=0

n−1

(−1)−n+1 αn µn Π [(2i + 2) γ − 1] i=0

n−1

(αµ − 1)

#

αn−2 µn−1 Π [(2i+1)γ−1]

i=0

γ n n−1

(−1)n−1 γ n−1 n−1

i=0

γ n−1 n−1

n−2

=

n−2

− (−1)n−1 αn µn Π [(2i) γ − 1] i=0

=

[−1 + αµ]

γ n n−1

(αµ − 1)n

n−1

(−1)n αn µn Π [(2i) γ − 1] =

i=0

γ n n−1

.

(αµ − 1)n

The proofs of the remaining relations can be likewise achieved. Therefore, they are omitted.

2.4

Fourth System xn+1 =

xn−1 yn−3 yn−1 (−1+xn−1 yn−3 ) ,

yn+1 =

yn−1 xn−3 xn−1 (1−yn−1 xn−3 )

Our fundamental task in this subsection is to develop fractional solutions to the system of recursive equations given by the form: yn−1 xn−3 xn−1 yn−3 , yn+1 = , (4) xn+1 = yn−1 (−1 + xn−1 yn−3 ) xn−1 (1 − yn−1 xn−3 ) where the initial conditions are required to be non-zero real numbers. Theorem 4 Assume that {xn , yn } is a solution to system (4) and suppose that x−3 = α, x−2 = β, x−1 = γ, x0 = δ, y−3 = , y−2 = η, y−1 = µ and y0 = ω. Then, for n = 0, 1, ... we have n−1

n−1

(−1)n γ n n Π [(2i) αµ − 1] x4n−3 =

i=0

αn−1 µn (γ − 1)n

(−1)n δ n η n Π [(2i) βω − 1] ,

x4n−2 =

n−1

i=0 α n µn

,

n−1

(−1)n γ n+1 n Π [(2i + 1) αµ − 1] x4n−1 =

i=0

β n−1 ω n (δη − 1)n

(−1)n δ n+1 η n Π [(2i + 1) βω − 1] ,

x4n =

i=0 β nωn

.

And

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(−1)n αn µn

y4n−3 =

n−1

γ n n−1

(−1) α µ

(γ − 1)n

y4n =

(−1)n β n ω

Π [(2i + 2) αµ − 1]

Π [(2i + 1) βω − 1]

i=0 n+1

(δη − 1)n

n−1

δnηn

i=0

,

n−1

δ n η n−1

,

n−1

γ n n

(−1)n β n ω n

y4n−2 =

Π [(2i + 1) αµ − 1]

i=0 n n n+1

y4n−1 =

,

.

Π [(2i + 2) βω − 1]

i=0

Proof. The relations hold for n = 0. Next, we let n > 1 and assume that the formulas hold for n − 1. That is n−2

n−2

(−1)n−1 γ n−1 n−1 Π [(2i) αµ − 1] i=0

x4n−7 =

,

n−1

αn−2 µn−1

(−1)n−1 δ n−1 η n−1 Π [(2i) βω − 1]

(γ − 1)

i=0

x4n−6 =

β n−2 ω n−1

n−2

n−2

(−1)n−1 δ n η n−1 Π [(2i + 1) βω − 1]

(−1)n−1 γ n n−1 Π [(2i + 1) αµ − 1] i=0

x4n−5 =

,

αn−1 µn−1

,

(δη − 1)n−1 i=0

x4n−4 =

.

β n−1 ω n−1

And (−1)n−1 αn−1 µn−1

y4n−7 =

,

n−2

y4n−6 =

γ n−1 n−2 Π [(2i + 1) αµ − 1] i=0 n−1 n

n−1

y4n−5 =

(−1)

n−1

,

n−2

γ n−1 n−1

n−2

,

δ n−1 η n−2 Π [(2i + 1) βω − 1]

n−1

µ (γ − 1)

α

(−1)n−1 β n−1 ω n−1

(−1)

y4n−4 =

Π [(2i + 2) αµ − 1]

β

ω n (δη − 1)n−1

n−2

δ n−1 η n−1

i=0

i=0 n−1

.

Π [(2i + 2) βω − 1]

i=0

We now turn to verify the proof of two relations. It can be obviously seen from system (4) that x4n−3 =

x4n−5 y4n−7 y4n−5 (−1 + x4n−5 y4n−7 ) n−2

(−1)n−1 γ n n−1 Π [(2i+1)αµ−1]

(−1)n−1 αn−1 µn−1

i=0

αn−1 µn−1

n−2

γ n−1 n−2 Π [(2i+1)αµ−1] i=0

= (−1)n−1 αn−1 µn (γ−1)n−1

"

n−2

−1 +

(−1)n−1 γ n n−1

i=0

i=0

(γ − 1)

i=0

#

n−2

γ n−1 n−2 Π [(2i+1)αµ−1] n−1

(−1)−n+1 γ n n Π [(2i + 2) αµ − 1] n−1

(−1)n−1 αn−1 µn−1 i=0

n−2

αn−1 µn

Π [(2i+1)αµ−1]

αn−1 µn−1

γ n−1 n−1 Π [(2i+2)αµ−1]

=

n−2

[−1 + γ]

− (−1)n−1 γ n n Π [(2i) αµ − 1] =

i=0

αn−1 µn (γ − 1)n

n−1

(−1)n γ n n Π [(2i) αµ − 1] =

i=0 n−1 α µn (γ

− 1)n

.

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Further, it can be attained from system (4) that y4n−3 =

y4n−5 x4n−7 x4n−5 (1 − y4n−5 x4n−7 ) n−2

(−1)n−1 αn−1 µn (γ−1)n−1 γ n−1 n−1

=

(−1)n−1 γ n−1 n−1 Π [(2i)αµ−1] i=0

αn−2 µn−1 (γ−1)n−1

n−2

Π [(2i+2)αµ−1]

i=0

n−2

(−1)n−1 γ n n−1 Π [(2i+1)αµ−1]

n−2

" 1−

i=0

αn−1 µn−1

(−1)n−1 αn−1 µn (γ−1)n−1 γ n−1 n−1

n−2

Π [(2i+2)αµ−1]

(−1)n−1 γ n−1 n−1 Π [(2i)αµ−1]

#

i=0

αn−2 µn−1 (γ−1)n−1

i=0

n−2

αµ Π [(2i)αµ−1]

i=0 n−2

Π [(2i+2)αµ−1]

i=0

=

n−2

(−1)n−1 γ n n−1 Π [(2i+1)αµ−1]

n−2

"

i=0

αn−1 µn−1

1−

αµ Π [(2i)αµ−1]

#

i=0 n−2

Π [(2i+2)αµ−1]

i=0

n−2

(−1)−n+1 αn µn Π [(2i) αµ − 1] i=0   = n−2 n−2 n−2 n n−1 γ  Π [(2i + 1) αµ − 1] Π [(2i + 2) αµ − 1] − αµ Π [(2i) αµ − 1] i=0

i=0

n

(−1) α µ

=

n−1

γ n n−1

i=0

n n

.

Π [(2i + 1) αµ − 1]

i=0

Other results can be proved in a similar way. Thus, the remaining proofs are omitted.

2.5

Numerical Examples

This subsection aims to present graphical confirmations to the whole solutions obtained in the previous subsections. Here, we plot the solutions (by using MATLAB software) under specific selections of some initial conditions. Example 1. This example shows the paths of the solutions of system (1). The initial conditions of this example are given as follows: x−3 = 3, x−2 = 1, x−1 = 5, x0 = 2, y−3 = 1, y−2 = 3, y−1 = 5 and y0 = 5. See Figure 1.

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Plot of The First System 5 x(n) y(n)

4

x(n),y(n)

3 2 1 0 −1 −2 0

10

20

30

40

50

60

70

Figure 1: The behaviour of the solution of system (1). Example 2. In Figure 2, we illustrate the behaviour of the solution of system (2) under the following selection of initial conditions: x−3 = 3.4, x−2 = 0.7, x−1 = 2, x0 = 3, y−3 = 1.5, y−2 = 1.5, y−1 = 0.5 and y0 = 1.22. Plot of The Second System 200 x(n) y(n) 150

x(n),y(n)

100

50

0

−50

−100 0

10

20

30

40

50

n

Figure 2: The behaviour of the solution of system (2). Example 3. Figure 3 illustrates the curves of the solutions of system (3) when we assume that x−3 = 0.7, x−2 = 2.1, x−1 = 1, x0 = 0.5, y−3 = 0.1, y−2 = 0.2, y−1 = 2.2 and y0 = 0.5.

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Plot of The Third System 3000 x(n) y(n) 2000

x(n),y(n)

1000

0

−1000

−2000

−3000 0

10

20

30

40

50

60

70

Figure 3: The behaviour of the solution of system (3). Example 4. The solutions of system (4) are depicted in Figure 4 under the following initial data: x−3 = 0.2, x−2 = 1, x−1 = 0.3, x0 = 0.2, y−3 = 3, y−2 = 1, y−1 = 2 and y0 = 0.3. Plot of The Fourth System

16

2

x 10

x(n) y(n)

1.5 1

x(n),y(n)

0.5 0 −0.5 −1 −1.5 −2 −2.5 0

10

20

30

40

50

60

70

n

Figure 4: The behaviour of the solution of system (4).

References [1] M. B. Almatrafi, E. M. Elsayed and Faris Alzahrani, Investigating Some Properties of a Fourth Order Difference Equation, J. Computational Analysis and Applications, 28

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(2) (2020), 243-253. [2] M. B. Almatrafi, E. M. Elsayed and Faris Alzahrani, Qualitative Behavior of Two Rational Difference Equations, Fundamental Journal of Mathematics and Applications, 1 (2) (2018) 194-204. [3] M. B. Almatrafi, E. M. Elsayed, Solutions And Formulae For Some Systems Of Difference Equations, MathLAB Journal, 1 (3) (2018), 356-369. [4] M. B. Almatrafi, E. M. Elsayed and Faris Alzahrani, Qualitative Behavior of a Quadratic Second-Order Rational Difference Equation, International Journal of Advances in Mathematics, 2019 (1) (2019), 1-14. [5] E. O. Alzahrani, M. M. El-Dessoky, E. M. Elsayed and Y. Kuang, Solutions and Properties of Some Degenerate Systems of Difference Equations, J. Comp. Anal. Appl., 18 (2), (2015), 321-333. [6] D. Clark and M. R. S. Kulenovic, A Coupled System of Rational Difference Equations, Comput. Math. Appl., 43 (2002), 849–867. [7] S. E. Das and M.Bayram, On a System of Rational Difference Equations, World Applied Sciences Journal 10 (11) (2010), 1306-1312. [8] Q. Din, Dynamics of a Discrete Lotka-Volterra Model, Advances in Difference Equations, 2013, 2013:95 doi:10.1186/1687-1847-2013-95. [9] Q. Din, On a System of Rational Difference Equation, Demonstratio Mathematica, Vol. XLVII (2) (2014), 324-335. [10] Q. Din, On a System of Fourth-Order Rational Difference Equations, Acta Universitatis Apulensis, 39 (2014), 137-150. [11] M. M. El-Dessoky, A. Khaliq, A. Asiri and A. Abbas, Global Attractivity and Periodic Nature of a Higher order Difference Equation, Journal of Computational Analysis & Applications, 26 (1) (2019), 294-304. [12] E. M. Elsayed and Marwa M. Alzubaidi, The Form of the Solutions of System of Rational Difference Equation, Journal of Mathematical Sciences and Modelling, 1(3) (2018), 181191. [13] A. Gelisken and M. Kara, Some General Systems of Rational Difference Equations, Journal of Difference Equations, vol. 2015 (2015), Article ID 396757, 7 pages. ˝ Ocalan, ˝ [14] M. G˝ um˝ u¸s and O. The Qualitative Analysis of a Rational System of Difference Equations, Journal of Fractional Calculus and Applications, 9 (2) (2018), 113-126. [15] A. Gurbanlyyev, On a System of Difference Equations, European Journal of Mathematics and Computer Science, 3 (1) (2016), 1-14.

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[16] Y. Halim, Global Character of Systems of Rational Difference Equations, Electronic Journal of Mathematical Analysis and Applications, 3 (1) (2015), 204-214. [17] A. Q. Khan, Q. Din, M. N. Qureshi and T. F. Ibrahim, Global Behavior of an AntiCompetitive System of Fourth-Order Rational Difference Equations, Computational Ecology and Software, 4 (1) (2014), 35-46. [18] A. S. Kurbanli, C. C ¸ inar, M. E. Erdo˘gan, On The Behavior of Solutions of The System of Rational Difference Equations xn+1 = xn−1 /yn xn−1 −1, yn+1 = yn−1 /xn yn−1 −1, zn+1 = xn /yn zn−1 , Applied Mathematics, 2 (2011), 1031-1038. [19] M. Mansour, M. M. El-Dessoky and E. M. Elsayed, On The Solution of Rational Systems of Difference Equations, Journal of Computational Analysis and Applications, 15 (5) (2013), 967-976. [20] R. Memarbashi, Sufficient Conditions for The Exponential Stability of Nonautonomous Difference Equations, Appl. Math. Letter, 21 (2008), 232–235. [21] O. Ocalan, Global Dynamics of a Non-autonomous Rational Difference Equation, Journal of Applied Mathematics and Informatics, 32 (5-6) (2014), 843-848. ¨ [22] O. Ozkan and A. S. Kurbanli, On a System of Difference Equations, Discrete Dynamics in Nature and Society, Volume 2013, Article ID 970316, 7 pages. [23] Q. Zhang, L. Yang and J. Liu, Dynamics of a System of Rational Third Order Difference Equation, Ad. Differ. Equ., doi:10.1186/1687-1847-2012-136, 8 pages.

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The ELECTRE multi-attribute group decision making method based on interval-valued intuitionistic fuzzy sets Cheng-Fu Yang∗ School of Mathematics and Statistics of Hexi University, Zhangye Gansu,734000, P. R. China

Abstract In this paper, based on the ELECTRE method and new ranking for the interval-valued intuitionistic fuzzy set (IVIFS), the IVIF ELECTRE method to solve multi-attribute group decision-making problems with interval-valued intuitionistic fuzzy input data is proposed, it is extending the intuitionistic fuzzy set (IF) ELECTRE method. This method firstly use AHP (Analytic hierarchy process) to find the weights of attribute, and use new ranking method for IVIFS and similarity measure between IVIFS to determine the weights of decision makers (DMs), then give the concordance set, midrange concordance set, weak concordance set and cosponging discordance set, midrange discordance set, weak discordance set. From this, the concordance matrix, discordance index, concordance dominance matrix and discordance dominance matrix are proposed. Finally, the ranking order of all the alternatives Ai (i = 1, 2, . . . , n) and the best alternative are obtained. A numerical example is taken to illustrate the feasibility and practicability of the proposed method. Keywords: Interval-valued intuitionistic fuzzy sets; ELECTRE method; Multi-attribute group decision making

1

Introduction

Since the multi-attribute decision making (MADM) was introduced in 1960‘ s, it has been a hot topic in decision making and systems engineering, and been proven as a useful tool due to its broad applications in a number of practical problems. But in some real-life situations, a decision maker (DM) may not be able to accurately express his/her preferences for alternatives due to that (1) the DM may not possess a precise or sufficient level of knowledge of the problem; (2) the DM is unable to discriminate explicitly the degree to which one alternative are better than others. In order to handle inexact and imprecise data, in 1965 Zadeh [38] introduced fuzzy set (FS) theory. In 1983 Atanassov [1,2] generalized FS to intuitionistic fuzzy set (IFS) by using two characteristic functions to express the degree of membership and the degree of non-membership of elements of the universal set. Since IFS tackled the drawback of the single membership value in FS theory, IFS has been widely applied to the multi-attribute decision making (MADM) [4,7,8,1014,20,22,23,28] and multi-attribute group decision making (MAGDM) [18,19,21]. In 1989 Atanassov and Gargov [3] further generalized the IFS in the spirit of the ordinary intervalvalued fuzzy set (IVFS) and defined the concept of interval-valued intuitionistic fuzzy set (IVIFS), which enhances greatly the representation ability of uncertainty than IFS. Similar to the IFS, IVIFSs were also ∗ Corresponding author Address: School of Mathematics and Statistics of Hexi University, Zhangye, Gansu,734000, P. R. China. Tel.:+86 0936 8280868; Fax:+86 0936 8282000. E-mail: [email protected] (C.F.Yang).

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used in the problems of MADM [6,15-17,28,32] and MAGDM [29,31,33,34]. In these researches, some are extension of classic decision making methods in IVIFS environment. For example, Li [15] developed the closeness coefficient-based nonlinear-programming method for interval-valued intuitionistic fuzzy MADM with incomplete preference information, Li [16] proposed the TOPSIS-based nonlinear-programming methodology for MADM with IVIFSs, Li [17] proposed the linear-programming method for MADM with IVIFSs. These decision methods under interval-valued intuitionistic fuzzy environments also generalize the classic decision making methods, such as TOPSIS and LINMAP. In [32], Wang et al. proposed a expect to apply ELECTRE and PROMETHEE motheds to MADM and MAGDM with IVIFS. In this paper, based on the new ranking method of interval in [27] and similarity measure of IVIFSs in [35, 37], the IF ELECTRE [30] method is applied to MAGDM with IVIFS, and obtain IVIFS ELECTRE method for solving MAGDM problems under IVIF environments. This paper is organized as follows. Section 2 briefly reviews the analytic hierarchy process (AHP). Section 3 and Section 4 introduce the new ranking method of intervals and similarity measure between IVIFSs, respectively. Section 5 formulates an MAGDM problem in which the evaluation of alternatives in each attribute is expressed by IVIF sets, and also develops an extended ELECTRE method. Section 6 demonstrates the feasibility and applicability of the proposed method by applying it to the MAGDM problem of the air-condition. Finally, Section 7 presents the conclusions.

2

Analytic hierarchy process (AHP)

AHP was introduced for the first time in 1980 by Thomas L. Saaty [24]. For years, AHP has been used in various fields such as social sciences, health planning and management. Many researchers have preferred to use AHP to find the weights of attribute [25,26]. Due to the fact that attribute weights in the decisionmaking problems are various, it is not correct to assign all of them as equalled [5]. To solve the problem of indicating the weights, some methods like AHP, eigenvector, entropy analysis, and weighted least square methods were used. For the calculation of attribute weight in AHP the following steps are used: (i) Arrange the attribute in n × n square matrix form as rows and columns. (ii) Using pairwise comparisons, the relative importance of one attribute over another can be expressed as follow: If two attribute have equal importance in pairwise comparison enter 1; if one of them is moderately more important than the other enter 3 and for the other enter 1/3; if one of them is strongly more important enter 5 and for the other enter 1/5; if one of them is very strongly more important enter 7 and for the other enter 1/7, and if one of them is extremely important enter 9 and for the other enter 1/9. 2, 4, 6 and 8 can be entered as intermediate values. Thus, pairwise comparison matrix is obtained as a result of the pairwise comparisons. Note that all elements in the comparison matrix are positive, in other words ai j > 0 (i, j = 1, 2, . . . , n). (a) To find the maximum eigenvalue λ of the comparison matrix. CI (b) Calculate consistency index CI = λ−n n−1 and consistency ratio CR = RI , where RI is the random consistency index given by Saaty.(Table 1) (c) If CR ≥ 0.1, then adjusts elements ai j (i, j = 1, 2, . . . , n) of the comparison matrix, (a) and (b) choices are done iteratively until CR < 0.1. (d) Compute eigenvector of the maximum eigenvalue of the comparison matrix. (e) Normalized eigenvector. n RI

1 0

2 0

3 0.58

Table1:Random consistency index RI. 4 5 6 7 8 0.90 1.12 1.24 1.32 1.41

9 1.45

10 1.49

11 1.51

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3

Ranking method for intervals

Let x = [a, b] ⊆ [0, 1] and y = [c, d] ⊆ [0, 1] be two intervals. Since the location relations between x = [a, b] and y = [c, d] include the following six cases, Wan and Dong [27] calculated the occurrence probability for the fuzzy(or random) event x ≥ y, denoted by P(x ≥ y), under different cases. Case1: a < b ≤ c < d, P(x ≥ y) = 0.

(1)

Case2: a ≤ c < b < d or a < c < b ≤ d, P(x ≥ y) =

(b − c)2 . 2(b − a)(d − c)

(2)

Case3: a ≤ c < d < b or a < c < d ≤ b or a ≤ c < d ≤ b, P(x ≥ y) =

2b − d − c . 2(b − a)

(3)

Case4: c ≤ a < b < d or c < a < b ≤ d, P(x ≥ y) =

b + a − 2c . 2(d − c)

(4)

Case5: c ≤ a < d < b or c < a < d ≤ b, P(x ≥ y) =

2bd + 2ac − 2bc − a2 − d2 . 2(b − a)(d − c)

(5)

Case6: c ≤ d ≤ a < b or c < a < b ≤ d, P(x ≥ y) = 1.

(6)

In order to rank intervals a˜ i = [ai , bi ] (i = 1, 2, . . . , n), Wang and Dong [27] construct the matrix of possibility degree as P = (Pi j )n×n , where Pi j = P(˜ai ≥ a˜ j ) (i = 1, 2, . . . , n; j = 1, 2, . . . , n). Then, the ranking vector ω = (ω1 , ω2 , ..., ωn )T is derived as follows: n X n ωi = ( Pi j + − 1)/(n(n − 1)) (i = 1, 2, · · · , n). 2 j=1

(7)

The larger the value of ωi , the bigger the corresponding intervals a˜ i = [ai , bi ]. In other words, for the two intervals a˜ i = [ai , bi ] and a˜ j = [a j , b j ], if ωi ≥ ω j , then [ai , bi ] ≥ [a j , b j ].

4

Similarity measure between IVIFSs Definition 1.[3] An IVIFS A in the universe set of discourse X is defined as A = {hx, µA (x), νA (x)i |x ∈ X } ,

where µA (x) ⊆ [0, 1] and νA (x) ⊆ [0, 1] denote respectively the membership degree interval and the nonmembership degree interval of x to A,with the condition: 3

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supµA (x)+ supνA (x) ≤ 1, ∀x ∈ X. Since IVIFS is composed of two ordered interval pairs, Xu [31,32] called them interval-valued intuitionistic fuzzy numbers(IVIFNs) and simply denoted by G = ([a, b], [c, d]), where [a, b] ⊆ [0, 1], [c, d] ⊆ [0, 1] and b + d ≤ 1. Definition 2.[37] Let Gi = ([ai , bi ], [ci , di ]) (i = 1, 2) be two IVIFNs, the normalized Hamming distance between G1 and G2 can be defined as: d(G1 , G2 ) =

1 (|a1 − a2 | + |b1 − b2 | + |c1 − c2 | + |d1 − d2 | + π01 − π02 + π001 − π002 ), 4

(8)

where πGi = [π0i , π00i ] = [1 − bi − di , 1 − ai − ci ] (i = 1, 2) is called the degree of indeterminacy or called the degree of hesitancy of the IVIFN Gi . Definition 3.[35, 37] Let Gi = ([ai , bi ], [ci , di ]) (i = 1, 2) be two IVIFNs, then   i f G1 = G2 = Gc2 ,   1, c d(G ,G ) (9) s(G1 , G2 ) =  1 2   d(G1 ,G2 )+d(G c , otherwise 1 ,G ) 2

is called the degree of similarity between G1 and G2 , where Gc2 = ([c2 , d2 ], [a2 , b2 ]) is denoted as the complement of G2 . Definition 4.[37] Let A and B be two IVIFSs in X, then s(A, B) =

n n d(G Aj , (G Bj )c ) 1X 1X s(G Aj , G Bj ) = n j=1 n j=1 d(G Aj , G Bj ) + d(G Aj , (G Bj )c )

(10)

is called the degree of similarity between A and B , where G Aj and G Bj are j-th IVIFNs of A and B , respectively. Definition 5.[6, 27] Let Gi (i = 1, 2, . . . , n) be a collection of the IVIFNs, where Gi = ([ai , bi ], [ci , di ]). If n P

Yω (G1 , G2 , · · · , Gn ) =

ω jG j

j=1 n P

,

(11)

ωj

j=1

where ω = (ω1 , ω2 , ..., ωn )T is the weight vector, then the function Yω is called the weighted average operator for the IVIFNs. Particularly, if ω j ( j = 1, 2, . . . , n) are crisp values, then the weighted average operator Yω is calculated as follows:  P  P  n n n n n P P P ω jG j  ω j a j ω j b j   ω j c j ω j d j   j=1   j=1  j=1 j=1 j=1  ,   . Yω (G1 , G2 , · · · , Gn ) = n =  n , n , (12) n n P P P  P   P  ωj ωj ωj   ωj ω j   j=1

5 5.1

j=1

j=1

j=1

j=1

MAGDM problems and ELECTRE method with IVIFSs Problems description for MAGDM with IVIFSs

Assume that there are m alternatives {A1 , A2 , . . . , Am } and k experts {p1 , p2 , . . . , pk }, each alternative Ai has n attributes {a1 , a2 , . . . , an }. For each alternative Ai , each expert gives evaluation on different attribute. 4

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The multi-attribute group decision making (MAGDM) is choose the best one from all alternatives according to these evaluations. Assume that GtMi j = [ati j , bti j ] and GtNi j = [cti j , dit j ] are respectively the membership degree and non-membership degree of alternative Ai ∈ A on an attribute a j given by DM pt to the fuzzy concept ”excellent”. In other words, the evaluation of Ai on a j given by pt is an IVIFN as follows: Gti j = (GtMi j , GtNi j ),

(13)

where [ati j , bti j ] ⊆ [0, 1], [cti j , dit j ] ⊆ [0, 1] and bti j + dit j ≤ 1 (1 ≤ i ≤ m, 1 ≤ j ≤ n, 1 ≤ t ≤ k).

5.2

Determination of the weights of DMs

Since the different DMs play different roles during the process of decision making, thus the importance of DMs should be taken into consideration. The weight vector of DMs is denoted by z = (z1 , z2 , . . . , zk )T . In the following, an approach determined the weights of DMs is given. Suppose that the evaluation of alternative Ai given by DM pt on each attribute are respectively the IVIFNs Gti1 , Gti2 , ..., Gtin . By Eq.(12), the individual overall attribute value of Ai given by pt is obtained as follows: Eit = ([ati , bti ], [cti , dit ]) = Yω (Gti1 , Gti2 , · · · Gtin ),

(14)

where ω = (ω1 , ω2 , ..., ωn )T is the weight vector of attributes. Let E t = (E1t , E2t , . . . , Emt ) and E u = (E1u , E2u , . . . , Emu ) are evaluation vectors of all alternatives given by DMs pt and pu , respectively. Using Eqs.(8-10), the similarity degree stu between E t and E u is obtained, and the similarity matrix S is constructed as follows: S = (stu )k×k .

(15)

Obviously, S is a non-negative symmetric matrix, by the Perron-Frobenius theorem [12], there exists the maximum module eigenvalue λ > 0, and the corresponding eigenvector x = (x1 , x2 , . . . , xk )T satisfies that xt > 0 (t = 1, 2, . . . , k) and λx = S x. Let z = λx = S x, then each component of z is the weight of corresponding expert. The normalized vector z, the weight zt (t = 1, 2, . . . , k) of DM pt is obtained as follows: zt =

5.3

xt (t = 1, 2, · · · , k). (x1 + x2 + · · · + xk )

(16)

ELECTRE methods based on IVIFS

Based on the idea of ELECTRE method, a new approach, named as IVIF ELECTRE, is formulated to solve a MCDM problem under interval-valued intuitionistic fuzzy environment. For each pair of alternatives k and l (k, l = 1, 2, . . . , m and k , l), each attribute in the different alternatives can be divided into two distinct subsets. The concordance set Ekl of Ak and Al is composed of all attribute for which Ak is preferred to Al . In other words, Ekl = { j|xk j ≥ xl j }, where J = { j| j = 1, 2, . . . , n}, xk j and xl j denoted the evaluation of DM in the jth attribute to alternative Ak and Al , respectively. The complementary subset, which is the discordance set, is Fkl = { j|xk j < xl j }. In the proposed IVIF ELECTRE method, we can classify different types of concordance and discordance sets using the concepts of score function, accuracy function and hesitation degree, and use concordance and discordance sets to construct concordance and discordance matrices, respectively. The decision makers can choose the best alternative using the concepts of positive and negative ideal points.

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Xu [31] and Xu and Chen [36] defined the score function S (G) and accuracy function H(G) for an IVIFN G=([a,b],[c,d]) as follows: S (G) = 21 (a + b − c − d),

(17)

H(G) = 12 (a + b + c + d).

(18)

Here, we define the hesitation degree for an IVIFN G=([a,b],[c,d]) as follows: π(G) = 1 − 21 (a + b + c + d).

(19)

From (18) and (19), easy to see that a higher accuracy degree H(G) correlates with a lower hesitancy degree π(G). Considering the better alternative has the higher score degree or higher accuracy degree in cases where alternatives have the same score degree. A higher score degree refers to a larger membership degree or smaller non-membership degree, and a higher accuracy degree refers to a smaller hesitation degree. Based on this, using the above three functions to compare IVIF values of different alternatives. The concordance set can be classified as concordance set, midrange concordance set and weak concordance set. Similarly, The discordance sets can also be classified as the discordance set, midrange discordance set, and weak discordance set. Next, the concordance set, midrange concordance set, weak concordance set, discordance set, midrange discordance set, weak discordance set are defined respectively as follows. Let Gk j = ([ak j , bk j ], [ck j , dk j ]) and Gl j = ([al j , bl j ], [cl j , dl j ]) denote the jth attribute value of alternative Ak and Al , respectively. The concordance set Ckl is composed of all attribute for which Ak is preferred to Al ,i.e., Ckl = { j|[ak j , bk j ] ≥ [al j , bl j ], [ck j , dk j ] < [cl j , dl j ] and [π0k j , π00k j ] < [π0l j , π00l j ]},

(20)

where J = { j| j = 1, 2, . . . , n}. The midrange concordance set Ckl0 is defined as Ckl0 = { j|[ak j , bk j ] ≥ [al j , bl j ], [ck j , dk j ] < [cl j , dl j ] and [π0k j , π00k j ] ≥ [π0l j , π00l j ]}.

(21)

The major difference between (20) and (21) is the hesitancy degree; the hesitancy degree at the kth alternative with respect to the jth attribute is higher than the lth alternative with respect to the jth attribute in the midrange concordance set. Thus, Eq. (20) is more concordant than (21). The weak concordance set Ckl00 is defined as Ckl00 = { j|[ak j , bk j ] ≥ [al j , bl j ] and [ck j , dk j ] ≥ [cl j , dl j ]}.

(22)

The degree of non-membership at the kth alternative with respect to the jth attribute is higher than the lth alternative with respect to the jth attribute in the weak concordance set; thus, Eq. (21) is more concordant than (22). The discordance set is composed of all attribute for which Ak is not preferred to Al . The discordance set Dkl is formulated as follows: Dkl = { j|[ak j , bk j ] < [al j , bl j ], [ck j , dk j ] ≥ [cl j , dl j ] and [π0k j , π00k j ] ≥ [π0l j , π00l j ]},

(23)

The midrange discordance set D0kl is defined as D0kl = { j|[ak j , bk j ] < [al j , bl j ], [ck j , dk j ] ≥ [cl j , dl j ] and [π0k j , π00k j ] < [π0l j , π00l j ]}.

(24)

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The weak discordance set D00kl is defined as D00kl = { j|[ak j , bk j ] < [al j , bl j ] and [ck j , dk j ] < [cl j , dl j ]}.

(25)

The IVIF ELECTRE method is an integrated IVIFS and ELECTRE method. The relative value of the concordance set of the IVIF ELECTRE method is measured through the concordance index. The concordance index ekl between Ak and Al is defined as: ekl = min∗ {wC ∗ × d(Gk j , Gl j )},

(26)

j∈C

where d(Gk j , Gl j ) is defined in (8), denoted the distance between jth attribute of alternatives Ak and Al , and wC ∗ is equal to wC , wC 0 or wC 00 , which denoted the weight of the concordance, midrange concordance, and weak concordance sets, respectively. The concordance matrix E is defined as follows:   e12 · · · ··· e1m   −   e − e23 ··· e2m  21   ··· − ··· · · ·  , (27) E =  · · ·   e  · · · · · · − e (m−1)m   (m−1)1  em1 em2 · · · em(m−1) − where the maximum value of ekl is denoted by e∗ , which is the positive ideal point, and a higher value of ekl indicates that Ak is preferred to Al . the discordance index is defined as follows: hkl = max∗ {wD∗ × d(Gk j , Gl j )},

(28)

j∈D

where d(Gk j , Gl j ) is defined in (8), denoted the distance between jth attribute of alternatives Ak and Al , and wD∗ is equal to wD , wD0 or wD00 , which denoted the weight of the discordance, midrange discordance, and weak discordance sets, respectively. The discordance matrix H is defined as follows:   − h12 · · · ··· h1m     h21 − h23 ··· h2m    ··· − ··· · · ·  , H =  · · · (29)   h − h(m−1)m   (m−1)1 · · · · · · hm1 hm2 · · · hm(m−1) − where the maximum value of hkl is denoted by h∗ , which is the negative ideal point, and a higher value of Hkl indicates that Ak is less favorable than Al . The concordance dominance matrix calculation process is based on the concept that the chosen alternative should have the shortest distance from the positive ideal solution, thus, the concordance dominance matrix K is defined as follows:   k12 · · · ··· k1m   −   k21 − k23 ··· k2m    ··· − ··· · · ·  , K =  · · · (30)   k − k(m−1)m   (m−1)1 · · · · · · km1 km2 · · · km(m−1) − 7

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where kkl = e∗ − ekl , which refers to the separation of each alternative from the positive ideal solution. A higher value of kkl indicates that Ak is less favorable than Al . The discordance dominance matrix calculation process is based on the concept that the chosen alternative should have the farthest distance from the negative ideal solution, thus, the discordance dominance matrix L is defined as follows:   l12 · · · ··· l1m   −   l − l23 ··· l2m   21  ··· − ··· · · ·  , (31) L =  · · ·   l  · · · · · · − l (m−1)1 (m−1)m   lm1 lm2 · · · lm(m−1) − where lkl = h∗ − hkl , which refers to the separation of each alternative from the negative ideal solution. A higher value of lkl indicates that Ak is preferred to Al . In the aggregate dominance matrix determining process, the distance of each alternative to both positive and negative ideal points can be calculated to determine the ranking order of all alternatives. The aggregate dominance matrix R is defined as follows:   r12 · · · ··· r1m   −   r − r23 ··· r2m  21   ··· − ··· · · ·  , (32) R =  · · ·   r  · · · · · · − r (m−1)1 (m−1)m   rm1 rm2 · · · rm(m−1) − where rkl =

lkl , kkl + lkl

rkl refers to the relative closeness to the ideal solution, with a range from 0 to 1. A higher value of rkl indicates that the alternative Ak is simultaneously closer to the positive ideal point and farther from the negative ideal point than the alternative Al , thus, it is a better alternative. m P 1 Let T k = m−1 rkl , k = 1, 2, · · · , m, (33) l=1,l,k

and T k is the final value of evaluation. All alternatives can be ranked according to T k . The best alternative T ∗ , which is simultaneously the shortest distance to the positive ideal point and the farthest distance from the negative ideal point, can be generated and defined as follows: T ∗ = max {T k },

(34)

1≤k≤m

where A∗ is the best alternative.

5.4

Group decision making method

In the following we shall utilize the AHP and interval-valued intuitionistic fuzzy weighted average operator Y ( i.e. Eq. (12)) to propose a new MAGDM method with IVIFN information. The detailed steps are summarized as follows: Step 1. DMs use IVIFSs to represent the evaluation information in the each attribute of alternatives; Step 2. Use AHP to calculate the weight of attribute; Step 3. Calculate the individual overall attribute value of each alternative by Eq.(14); Step 4. Obtain the similarity matrix of the DMs according to Eq.(10); 8

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Step 5. Derive the weight value of each DM from Eq.(16); Step 6. Using the weight of DM to integrate the same attribute value of different DMs of each alternative in terms of Eq.(14); Step 7. By the possibility degree ranking method for intervals in Section 3, calculate the ranking vector of the membership degree interval, the non-membership degree interval and the hesitancy degree interval of between the difference alternatives on each attribute, respectively. Step 8. Obtain the concordance, midrange concordance, weak concordance, discordance, midrange discordance and weak discordance set according to Eqs.(20)-(25), respectively; Step 9. Compute the concordance matrix, discordance matrix, concordance dominance matrix, discordance dominance matrix and aggregate dominance matrix in terms of Eqs.(26)-(32); Step 10. Obtain the ranking order of all alternatives and the best alternative according to Eqs.(33)-(34).

6

Numerical example

In this section, we use the air-condition system selection problem given by [27] to verify the feasibility of the proposed method. The problem is described as follows: Suppose there exist three air-condition systems {A1 , A2 , A3 }, four attributes a1 (economical), a2 (function), a3 (being operative) and a4 (longevity) are taken into consideration in the selection problem. Three experts (DMs) {p1 , p2 , p3 } participate in the decision making. The membership degrees and non-membership degrees for the alternative Ai on the attribute a j given by expert pt were listed in Tables 2 − 4. Attribute a1 a2 a3 a4 Attribute a1 a2 a3 a4 Attribute a1 a2 a3 a4

Table 2: IVIFNs given by the expert p1 . A1 A2 A3 ([0.4, 0.8], [0.0, 0.1]) ([0.5, 0.7],[0.1, 0.2]) ([0.5, 0.7],[0.2, 0.3]) ([0.3, 0.6], [0.0, 0.2]) ([0.3, 0.5],[0.2, 0.4]) ([0.6, 0.8],[0.1, 0.2]) ([0.2, 0.7], [0.2, 0.3]) ([0.4, 0.7],[0.0, 0.2]) ([0.4, 0.7],[0.1, 0.2]) ([0.3, 0.4], [0.4, 0.5]) ([0.1, 0.2],[0.7, 0.8]) ([0.6, 0.8],[0.0, 0.2]) Table 3: IVIFNs given by the expert p2 . A1 A2 A3 ([0.5, 0.9], [0.0, 0.1]) ([0.7, 0.8], [0.1, 0.2]) ([0.5, 0.6], [0.1, 0.4]) ([0.4, 0.5], [0.3, 0.5]) ([0.5, 0.6], [0.2, 0.3]) ([0.6, 0.7], [0.1, 0.2]) ([0.5, 0.8], [0.0, 0.1]) ([0.5, 0.8], [0.0, 0.2]) ([0.4, 0.8], [0.1, 0.2]) ([0.4, 0.7], [0.1, 0.2]) ([0.5, 0.6], [0.3, 0.4]) ([0.2, 0.6], [0.2, 0.3]) Table 4: IVIFNs given by the expert p3 . A1 A2 A3 ([0.3, 0.9], [0.0, 0.1]) ([0.3, 0.8], [0.1, 0.2]) ([0.2, 0.6], [0.1, 0.2]) ([0.2, 0.5], [0.1, 0.4]) ([0.5, 0.6], [0.1, 0.3]) ([0.2, 0.6], [0.2, 0.3]) ([0.4, 0.7], [0.1, 0.2]) ([0.2, 0.8], [0.0, 0.2]) ([0.3, 0.6], [0.1, 0.3]) ([0.3, 0.6], [0.3, 0.4]) ([0.3, 0.5], [0.2, 0.3]) ([0.4, 0.7], [0.1, 0.2])

In the following, we will illustrate the decision making process. (1) Calculation of weights of attributes In order to find the weights of attributes, A commission, which is organized by sampling method, determined the importance of attribute by using AHP. A 4 × 4 size matrix is formed because 4 attribute are considered in this study. All the diagonal elements of the matrix will be 1, the elements of symmetrical position with respect to the diagonal are reciprocal, in other words, if ai j is ith row and jth column element of matrix, then its symmetrical position is filled using a ji = 1/ai j formula.

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The comparison matrix W is obtained as follows:   1  1 W =  2  3 4

2 1 3 6

1 3 1 3

1 3

1 4 1 6 1 3

1

    .  

By computing the eigenvalues and the eigenvectors of W, we obtained that the maximum eigenvalue of W was 4.0875, the corresponding eigenvector was ω = (0.1905, 0.1230, 0.4046, 0.8849)T , consistency index CI=0.0292 and consistency ratio CR = 0.0324 < 0.1. Normalized eigenvectors, the four attributes weights are obtained as follows: ω1 = 0.1213, ω2 = 0.0765, ω3 = 0.2517, ω4 = 0.5505. (2) Calculate the individual overall attribute value of each alternative By Eq.(14), the individual overall attribute value of each alternative can be obtained as in Table 5. Eit p1 p2 p3

Table 5: The individual overall attribute values of the alternatives for weight vector of attributes. A1 A2 A3 ([0.2870,0.5393],[0.2705,0.3782]) ([0.2393,0.4095],[0.4128,0.5456]) ([0.5375,0.7627],[0.0571,0.2121]) ([0.4373,0.7341],[0.0780,0.1857]) ([0.5242,0.6746],[0.1926,0.3178]) ([0.3173,0.6580],[0.1551,0.2793]) ([0.3175,0.6539],[0.1980,0.3133]) ([0.2901,0.6196],[0.1299,0.2627]) ([0.3353,0.6551],[0.1077,0.2328])

(3) Calculation of the similarity matrix and the weight vector of DMs The similarity matrix for the DMs is constructed by Eq.(10) as follows:    1 0.5415 0.6059    1 0.7577  . S =  0.5415   0.6059 0.7577 1 Because the maximum eigenvalue of S is 2.2746, the corresponding eigenvector is x = (0.5373, 0.5878, 0.6048)T , the expert0 s weights are obtained from Eq.(16) as follows: z1 = 0.3106, z2 = 0.3398, z3 = 0.3496. (4) Integrate the attribute value of different DMs By Eq.(14), the attribute value of different DMs are respectively integrated as in Table 6.

a1 a2 a3 a4

Table 6: The attribute value of different DMs in the different alternatives and different attributes. A1 A2 A3 ([0.3990,0.8689],[0,0.1]) ([0.4980,0.7689],[0.1,0.2]) ([0.3951,0.6311],[0.1311,0.2990]) ([0.2990,0.5311],[0.1369,0.3719]) ([0.4379,0.5689],[0.1650,0.3311]) ([0.4602,0.6961],[0.1350,0.2350]) ([0.3719,0.7340],[0.0971,0.1971]) ([0.3641,0.7689],[0,0.2]) ([0.3650,0.6990],[0.1,0.2350]) ([0.3340,0.5719],[0.2631,0.3631]) ([0.3058,0.4408],[0.3893,0.4893]) ([0.3942,0.6971],[0.1029,0.2340])

(5) Calculate the ranking vector The ranking vector of the membership degree interval, the non-membership degree interval and the hesitancy degree interval of between the difference alternatives on each attribute is calculated by Eqs.(1-7), respectively, as in Table 7.

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Table 7: The attribute value of different DMs in the different alternatives and different attributes. membership degree interval non-membership degree interval hesitancy degree interval A1 A2 A1 A2 A1 A2 0.5006 0.4994 0.25 0.75 0.5873 0.4127 A1 A3 A1 A3 A1 A3 a1 0.6286 0.3714 0.25 0.75 0.5388 0.4612 A2 A3 A2 A3 A2 A3 0.6808 0.3192 0.3207 0.6793 0.4341 0.5659 A1 A2 A1 A2 A1 A2 0.3214 0.6786 0.5135 0.4865 0.5878 0.4122 A1 A3 A1 A3 A1 A3 a2 0.27295 0.72705 0.6477 0.3523 0.59905 0.40095 A2 A3 A2 A3 A2 A3 0.34565 0.65435 0.6764 0.3236 0.5173 0.4827 A1 A2 A1 A2 A1 A2 0.48325 0.51675 0.6177 0.3823 0.4723 0.5277 A1 A3 A1 A3 A1 A3 a3 0.52875 0.47125 0.4246 0.5754 0.4995 0.5005 A2 A3 A2 A3 A2 A3 0.54255 0.45745 0.3426 0.6574 0.5273 0.4727 A1 A2 A1 A2 A1 A2 0.66115 0.33885 0.25 0.75 0.56895 0.43105 A1 A3 A1 A3 A1 A3 a4 0.35955 0.64045 0.75 0.25 0.44015 0.55985 A2 A3 A2 A3 A2 A3 0.2633 0.7367 0.75 0.25 0.365 0.635

(6) Determine the concordance, midrange concordance, weak concordance, discordance, midrange discordance and weak discordance set Applying Eqs.(20-25) and Table 7, the concordance, midrange concordance, weak concordance, discordance, midrange discordance and weak discordance set is calculated, respectively, as follows:        − − 3   − 1, 4 1   − − −        C =  2 − 1  , C 0 =  3 − 3  , C 00 =  − − −  ,       4 −   2 2 −   4 − − −   − 2 2   − 3 4   − − −        D =  − − 2  , D0 =  1, 4 − 4  , D00 =  − − −  .       3 1 − 1 3 − − − − For example, c13 = {3}, which is in the 1st (horizontal) row and the 3rd (vertical) column of the concordance set, is ”3.” c12 = {−}, which is in the 1st row and 2nd column of the concordance set, is ”empty,” and so forth. (7) Compute the concordance matrix, discordance matrix, concordance dominance matrix, discordance dominance matrix and aggregate dominance matrix We give the relative weights as: [ωC , ωC 0 , ωC 00 , ωD , ωD0 , ωD00 ] = [1, 32 , 31 , 1, 23 , 13 ]. By Eqs.(26)-(32), the concordance matrix, discordance matrix, concordance dominance matrix, discordance dominance matrix and aggregate dominance matrix are obtained, respectively, as follows:       − 0.08575 0.02235  − 0.1039 0.16309      − 0.05697  , H =  0.09967 − 0.18088  , E =  0.04759     0.09643 0.07862 − 0.12298 0.1204 −       − 0.01068 0.07408  − 0.07698 0.01779      . − 0.03946  , L =  0.08121 − 0 K =  0.04884     0 0.01781 − 0.0579 0.06048 − 11

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   − 0.8782 0.1936    − 0 R =  0.6246   1 0.7725 − (8) Compute the ranking order of all alternatives and obtain the best alternative Applying Eq.(33),    0.5359    T =  0.3123    0.88625 The optimal ranking order of alternatives is given by A3  A1  A2 . The best alternative is A3 . The ranking order given by [27] is identical. The best air-condition system is A3 . This example shows the effectiveness of the ranking method proposed in this paper.

7

Conclusion

Regarding the MAGDM problem, the IVIF theory provides a useful and convenient way to reflect the ambiguous nature of subjective judgments and assessments. In this paper, firstly, using the normalized Hamming distance between IVIFS to construct similarity matrix and obtain the wights of DMs. Then, using possibility degree of IVIF to calculate the ranking vector. Based on this, the concordance and discordance sets, concordance and discordance matrices etc. are obtained. Finally, by computing the ranking order of all alternatives, decision makers can choose the best alternative, the example verify the correctness of the method.

References [1] K.T. Atanassov, Intuitionistic fuzzy sets, Seventh Scientific Session of ITKR, Sofia, June (Dep. in CINTI, Nd 1697/84), 1983. [2] K.T. Atanassov, Intuitionistic fuzzy sets, Fuzzy Sets Syst. 20(1) (1986) 87-96. [3] K.T. Atanassov, G. Gargov, Interval-valued intuitionistic fuzzy sets, Fuzzy Sets and Systems 31(3) (1989) 343-349. [4] G. Beliakov, H. Bustinc, D.P. Goswami, U.K. Mukherjee, N.R. Pal, On averaging operators for Atanassov0 s intuitionistic fuzzy sets, Information Sciences 181(6) (2011) 1116-1124. [5] M.F. Chen, G.H. Tzeng, Combing grey relation and TOPSIS concepts for selectingan expatriate host country, Math. Comput. Model. 40 (2004) 1473-1490. [6] S.M. Chen, L.W. Lee, H.C. Liu, S.W. Yang, Multiattribute decision making based on interval-valued intuitionistic fuzzy values, Expert Systems with Applications 39 (2012) 10343-10351. [7] T.Y. Chen, C.H. Li, Objective weights with intuitionistic fuzzy entropy measures and computational experiment analysis, Applied Soft Computing 11(2011) 5411-5423. [8] Z.P. Chen, W. Yang, A new multiple criteria decision making method based on intuitionistic fuzzy information, Expert Systems with Applications 39(4)(2012) 4328-4334. [9] Z.X. Jiang, G.L. Shi, Matrix Theory and Application, Beijing University of Aeronautics and Astronautics Pressing, Beijing, 1998. [10] D.F. Li, Some measures of dissimilarity in intuitionistic fuzzy structures, Journal of Computer and System Sciences 68(1) (2004) 115-122.

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[11] D.F. Li, Multiattribute decision making models and methods using intuitionistic fuzzy sets, Journal of Computer and System Sciences 70(1)(2005) 73-85. [12] D.F. Li, The GOWA operator based approach to multiattribute decision making using intuitionistic fuzzy sets, Mathematical and Computer Modelling 53(5-6) (2011) 1182-1196. [13] D.F. Li, Multiattribute decision making method based on generalized OWA operators with intuitionistic fuzzy sets, Expert Systems with Applications 37(12) (2010) 8673-8678. [14] D.F. Li, Extension of the LINMAP for multiattribute decision making under Atanassovs intuitionistic fuzzy environment, Fuzzy Optimization and Decision Making 7(1) (2008) 17-34. [15] D.F. Li, Closeness coefficient based nonlinear programming method for interval-valued intuitionistic fuzzy multiattribute decision making with incomplete preference information, Applied Soft Computing 11(4) (2011)34023418. [16] D.F. Li, TOPSIS-based nonlinear-programming methodology for multiattribute decision making with intervalvalued intuitionistic fuzzy sets, IEEE Transactions on Fuzzy Systems 18(2) (2010) 299-311. [17] D.F. Li, Linear programming method for MADM with in terval-valued intuitionistic fuzzy sets, Expert Systems with Applications 37(8) (2010)5939-5945. [18] D.F. Li, G.H. Chen, Z.G. Huang, Linear programming method for multiattribute group decision makingusing IF sets, Information Sciences 180(9) (2010)1591-1609. [19] D.F. Li, L.L. Wang, G.H. Chen, Group decision making methodology based on the Atanassov0 s intuitionistic fuzzy set generalized OWA operator, International Journal of Uncertainty, Fuzziness and Knowledge-Based Systems 18 (6) (2010) 801-817. [20] D.F. Li, Y.C. Wang, Mathematical programming approach to multiattribute decision making under intuitionistic fuzzy environments, International Journal of Uncertainty, Fuzziness and Knowledge-Based Systems 16(4) (2008) 557-577. [21] D.F. Li, Y.C. Wang, S. Liu, F. Shan, Fractional programming methodology for multi-attribute group decision making using IFS, Applied Soft Computing 9(1) (2009) 219-225. [22] L. Lin, X.H. Yuan, Z.Q. Xia, Multicriteria fuzzy decision-making methods based on intuitionistic fuzzy sets, Journal of Computer and System Sciences 73(1) (2007) 84-88. [23] H.W. Liu, G.J. Wang, Multicriteria decision making methods based on intuitionistic fuzzy sets, European Journal of Operational Research 179(1) (2007) 220-233. [24] T.L. Saaty, The Analytic Hierarchy Process: Planning, Priority Setting, McGrawHill, New York, 1980. [25] Y. Shimizu, T. Jindo, A fuzzy logic analysis method for evaluating human sen-sitivities, Int. J. Ind. Ergon. 15 (1995) 39-47. [26] S.H. Tsaur, T.Y. Chang, C.H. Yen, The evaluation of airline service quality by fuzzy MCDM, Tour. Manage. 23 (2002) 107-115. [27] S.p. Wan, J.y. Dong, A possibility degree method for interval-valued intuitionistic fuzzy multi-attribute group decision making, Journal of Computer and System Sciences 80 (2014) 237-256. [28] W.Z. Wang, Comments on ”Multicriteria fuzzy decision making method based on an ovel accuracy function under interval-valued intuitionistic fuzzy environment” by Jun Ye, Expert Systems with Applications 38 (2011) 1318613187. [29] G.W. Wei, Some induced geometric aggregation operators with intuitionistic fuzzy information and their application to group decision making, Applied Soft Computing 10 (2010) 423-431. [30] M.C. Wu, T.Y. Chen, The ELECTRE multicriteria analysis approach based on Atanassovs intuitionistic fuzzy sets, Expert Systems with Applications 38(2011) 12318-12327. [31] Z.S. Xu, Methods for aggregating interval-valued intuitionistic fuzzy information and their application to decision making, Control and Decision 22(2) (2007) 215-219.

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[32] Z.S. Xu, Models for multipleattribute decision making with intuitionistic fuzzy information, International Journal of Uncertainty, Fuzziness and Knowledge-Based Systems 15 (2007) 285-297. [33] Z.S. Xu, On similarity measures of interval-valued intuitionistic fuzzy sets and their application to patternre cognitions, Journalof Southeast University 23(2007) 139-143. [34] Z.S. Xu, Choquet integrals of weighted intuitionistic fuzzy information, Information Sciences 180(2010) 726-736. [35] Z.S. Xu, Erratum to: Intuitionistic and interval-valued intutionistic fuzzy preference relations and their measures of similarity for the evaluation of agreement within a group and Some similarity measures of intuitionistic fuzzy sets and their applications to multiple attribute decision making, Fuzzy Optim Decis Making (2012) 11:351-352 [36] Z.S. Xu, J. Chen, An approach to group decision making based on interval-valued intuitionistic judgment matrices, System Engineering Theory and Practice27(4) (2007) 126-133. [37] Z.S. Xu, R. R. Yager, Intuitionistic and interval-valued intutionistic fuzzy preference relations and their measures of similarity for the evaluation of agreement within a group, Fuzzy Optim Decis Making (2009) 8:123-139. [38] L.A. Zadeh, Fuzzy Sets, Inform. Control 8 (1965) 338-353.

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The interior and closure of fuzzy topologies induced by the generalized fuzzy approximation spaces Cheng-Fu Yang∗ School of Mathematics and Statistics of Hexi University, Zhangye Gansu,734000, P. R. China

Abstract With respect to the Alexandrov fuzzy topologies induced by the generalized fuzzy approximation spaces, Wang defined interior of fuzzy set. In this paper, we give the closure of fuzzy set and discuss some properties of the interior and closure. Keywords: Alexandrov fuzzy topology; the generalized fuzzy approximation spaces; interior; closure; properties

1

Introduction

In his classical paper [36], Zadeh introduced the notation of fuzzy sets and fuzzy set operation. Subsequently, Chang [2] applied some basic concepts from general topology to fuzzy sets and developed a theory of fuzzy topological spaces. Pu etc.[18] defined a fuzzy point which took a crisp singleton, equivalently, an ordinary point, as a special case and gave the concepts of interior and closure operator w.r.t. fuzzy topology. Later, Lai and Zhang [11] modified the second axiom in Chang’s definition of fuzzy topology to define an Alexandrov fuzzy topology. The concept of Rough sets were introduced by Z. Pawlak [19] in 1982 as an powerful mathematical tool for uncertain data while modeling the problems in many fields [17,20,27]. Because the rough sets defined with equivalence relations limited the application of it. Thus many authors changed the equivalence relations into different binary relations to expand the application of it [23,35,37,38]. In recent years, the rough sets has been combined with some mathematical theories such as algebra and topology [1,5,6,8,10, 14, 16, 21, 25, 26, 28, 29]. With respect to different binary relations, the topological properties of rough sets were further investigated in [7,14,33,34]. In 1990, Dubois and Prade [3] combining fuzzy sets and rough sets proposed rough fuzzy sets and fuzzy rough sets. Afterward Morsi and Yakout [15] investigated fuzzy rough sets defined with left-continuous tnorms and R-implicators with respect to fuzzy similarity relations. Radzikowska and Kerre [24] defined a broad family of fuzzy rough sets based on t-norms and fuzzy implicators, which are called generalized fuzzy rough sets here. In recent years, the topological properties of fuzzy rough sets were further studied in many literatures [4,9,12,13,22]. Recently, with respect to the lower fuzzy rough approximation operator determined by a fuzzy implicator, Wang [30] studied various fuzzy topologies induced by different fuzzy relations and proved that I-lower fuzzy rough approximation operators were the interior operator w.r.t. some Alexandrov fuzzy topology. ∗ Corresponding author Address: School of Mathematics and Statistics of Hexi University, Zhangye, Gansu,734000, P. R. China. E-mail: [email protected]

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In this paper, we give closure operator w.r.t. some Alexandrov fuzzy topology given by Wang in [30]. Combined with the definition of Wang’s interior, discuss some properties of the interior and closure of fuzzy set.

2

Preliminary Definition 2.1.[36] A fuzzy set A in X is a set of ordered pairs: A = {(x, A(x)) : x ∈ X}

where A(x) : X → [0, 1] is a mapping and A(x) states the grade of belongness of x in A. The family of all fuzzy sets in X is denoted by F (X). Let α ∈ [0, 1], then a fuzzy set A ∈ F (X) is a constant, while A(x) = α for all x ∈ X, denoted as αX . Deflnition 2.2.[36] Let A, B be two fuzzy sets of F (X) (1) A is contained in B if and only if A(x) ≤ B(x) for every x ∈ X. (2) The union of A and B is a fuzzy set C, denoted by A ∪ B = C, whose membership function C(x) = A(x) ∨ B(x) for every x ∈ X. (3) The intersection of A and B is a fuzzy set C, denoted by A ∩ B = C, whose membership function C(x) = A(x) ∧ B(x) for every x ∈ X. (4) The complement of A is a fuzzy set, denoted by Ac , whose membership function Ac (x) = 1 − A(x) for every x ∈ X. A binary operation T : [0, 1] × [0, 1] → [0, 1] (resp. S : [0, 1] × [0, 1] → [0, 1]) is called a t-norm (resp. t-conorm) on [0, 1] if it is commutative, associative, increasing in each argument and has a unit element 1 (resp. 0). A mapping I : [0, 1] × [0, 1] → [0, 1] is called a fuzzy implicator on [0, 1] if it satisfies the boundary conditions according to the Boolean implicator, and is decreasing in the first argument and increasing in the second argument. Definition 2.3.[30] A fuzzy implicator I is said to satisfy (1) the left neutrality property ((NP), for short), if I(1, b) = b for all b ∈ [0, 1]; (2) the confinement principle ((CP), for short), if I(a, b) = 1 ⇔ a ≤ b, for all a, b ∈ [0, 1]; (3) the regular property ((RP), for short), if NI is an involutive negation, where NI is defined as NI (a) = I(a, 0) for all a ∈ [0, 1]. Definition 2.4. [11] A subset τ ⊆ F (X) is called an Alexandrov fuzzy topology if it satisfies: (1) αX ∈ τ for all α ∈ [0, 1], (2) ∩i∈Λ Ai ∈ τ for all {Ai }i∈Λ ⊆ τ, (3) ∪i∈Λ Ai ∈ τ for all {Ai }i∈Λ ⊆ τ. Every member of τ is called a τ-open fuzzy set. A fuzzy set is τ-closed if and only if its complement is τ-open. In the sequel, when no confusion is likely to arise, we shall call a τ-open (τ-closed) fuzzy set simply an open (closed) set. Definition 2.5. [18,31]. Let τ ⊆ F (X) be a fuzzy topology. Then the interior of A ∈ F (X) w.r.t. fuzzy topology τ denoted as Ao is defined as follows: Ao = ∪{B ∈ τ|B ⊆ A}. 2

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The operator Ao is called an interior operator w.r.t. fuzzy topology τ. According to definition of the fuzzy topology, obviously Ao is an open set. Definition 2.6. [18]. Let τ ⊆ F (X) be a fuzzy topology. Then the closure of A ∈ F (X) w.r.t. fuzzy topology τ denoted as A is defined as follows: A = ∩{B|B ⊇ A, Bc ∈ τ} The operator A is called a closure operator w.r.t. fuzzy topology τ. According to De Morgan’s Law and definition of the fuzzy topology, A is a closed set.

3

Fuzzy topologies induced by the generalized fuzzy approximation spaces

A fuzzy set R ∈ F (X × Y) is called a fuzzy relation from X to Y. If X = Y, then R is a fuzzy relation on X. For every fuzzy relation R on X, a fuzzy relation R−1 is defined as R−1 (x, y) = R(y, x) for all x, y ∈ X. Let R be a fuzzy relation from X to Y . Then the triple (X, Y, R) is called a fuzzy approximation space. When X = Y and R is a fuzzy relation on X, we also call (X, R) a fuzzy approximation space. Definition 3.1.[30]. Let R be a fuzzy relation on X. Then R is said to be (1) reflexive if R(x, x) = 1 for all x ∈ X; (2) symmetric if R(x, y) = R(y, x) for all x, y ∈ X; (3) T -transitive if T (R(x, y), R(y, z)) ≤ R(x, z) for all x, y, z ∈ X. If T = ∧, then T -transitive is said to be transitive for short. A fuzzy relation R is called a fuzzy tolerance if it is reflexive and symmetric, and a fuzzy T -preorder if it is reflexive and T -transitive. Similarly, a fuzzy relation R is called a fuzzy preorder if it is reflexive and transitive. Definition 3.2.[24,30,32]. Let (X, Y, R) be a fuzzy approximation space. Then the following mappings R, R : F (Y) → F (X) are defined as follows: for all A ∈ F (Y) and x ∈ X, R(A)(x) = ∧ I(R(x, y), A(y)) and R(A)(x) = ∨ T (R(x, y), A(y)). y∈Y

y∈Y

The mappings R and R are called I−lower and T −upper fuzzy rough approximation operators, respectively. The pair (R(A), R(A)) is called a generalized fuzzy rough set of A w.r.t. (X, Y, R). Also known as generalized fuzzy approximation spaces. Let R be a fuzzy relation on X. Then a fuzzy set A ∈ F (X) is said to be (1) I-lower definable w.r.t. fuzzy relation R if R(A) = A; the family of all I − lower definable fuzzy sets w.r.t. R is denoted as DI (R). (2) T -upper definable w.r.t. fuzzy relation R if R(A) = A; the family of all T − upper definable fuzzy sets w.r.t. R is denoted as DT (R). Proposition 3.3.[30]. Let (X, R) be a fuzzy approximation space and R be reflexive. Then (1)DI (R) is an Alexandrov fuzzy topology, if I satisfies (NP). (2)DT (R) is an Alexandrov fuzzy topology. Let (X, R) be a fuzzy approximation space. In [30] Wang defined RI (R) = {R(A)|A ∈ F (X)} and RT (R) = {R(A)|A ∈ F (X)}. 3

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To discuss the properties of generalized fuzzy rough sets, Radzikowska and Kerre [19] introduced the following auxiliary conditions: for a fuzzy implicator I and a t-norm T , (C1) I(a, I(b, c)) = I(T (a, b), c) for all a, b, c ∈ [0, 1], (C2) I(a, I(b, c)) ≥ I(T (a, b), c) for all a, b, c ∈ [0, 1], (C3) I(a, I(b, c)) ≤ I(T (a, b), c) for all a, b, c ∈ [0, 1]. If (C1) (resp. (C2), (C3)) holds for I and T , then we say that I satisfies (C1) (resp. (C2), (C3)) for T . Proposition 3.4.[30]. Let (X, R) be a fuzzy approximation space and R be a fuzzy T -preorder. Then (1) RI (R) is an Alexandrov fuzzy topology and RI (R) = DI (R), if I satisfies (NP) and (C2) for T . (2) RT (R) is an Alexandrov fuzzy topology and RT (R) = DT (R). The above DI (R), DT (R), RI (R) and RT (R) are called fuzzy topologies induced by the generalized fuzzy approximation spaces.

4

The interior and closure of fuzzy set

Proposition 4.1.[30]. Let R be a fuzzy T -preorder on X, and I satisfy (NP) and (C2) for T . Then R is the interior operator w.r.t. Alexandrov fuzzy topology DI (R). Proposition 4.2. Let R be a fuzzy T -preorder on X, and I satisfy (NP) and (C2) for T . Then A is an open set w.r.t. Alexandrov fuzzy topology DI (R) iff R(A) = Ao = A. Proof. Suppose A is an open set w.r.t. Alexandrov fuzzy topology DI (R), again A ⊆ A, due to definition of Ao , A ⊆ Ao . On the other hand, ∀x ∈ X, R(A)(x) = ∧ I(R(x, y), A(y)) ≤ I(R(x, x), A(x)) = I(1, A(x)) = A(x). y∈X

This means R(A) = A ⊆ A. Thus R(A) = Ao = A. Conversely, suppose R(A) = Ao = A, Ao is an open set, thus A is an open set. o

Proposition 4.3. Let R be a fuzzy T -preorder on X, and I satisfy (NP) and (C2) for T . Then for any A ∈ F(X) , [R(Ac )]c is the closure operator w.r.t. Alexandrov fuzzy topology DI (R). Proof. For any A ∈ F(X) , since R(Ac ) is an open set, thus (R(Ac ))c is a closed set. Again ∀x ∈ X, R(Ac )(x) = ∧ I(R(x, y), Ac (y))≤ I(R(x, x), Ac (x)) = I(1, Ac (x)) = Ac (x), y∈X

this means (R(Ac ))c ⊇ A. On the other hand, for any A ⊆ B ∈ F(X) and Bc ∈ DI (R). By Proposition 4.2, R(Bc ) = Bc , and ∀x ∈ X, R(Ac )(x) = ∧ I(R(x, y), Ac (y)) ≥ ∧ I(R(x, x), Bc (x)) = R(Bc )(x). y∈X

y∈X

We obtain R(Ac ) ⊇ R(Bc ) = Bc . This means (R(Ac ))c ⊆ B. By Definition of the closure, for any A ∈ F(X) , [R(Ac )]c is the closure operator w.r.t. Alexandrov fuzzy topology DI (R) i.e. [R(Ac )]c = A. Proposition 4.4. Let R be a fuzzy T -preorder on X, and I satisfy (NP) and (C2) for T . Then A is a closed set w.r.t. Alexandrov fuzzy topology DI (R) iff (R(Ac ))c = A = A. Proof. Suppose A is a closed set w.r.t. Alexandrov fuzzy topology DI (R), then Ac is an open set. Therefore R(Ac ) = Ac , and then A = (R(Ac ))c = A. 4

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Conversely, suppose A = (R(Ac ))c = A, A is a closed set, thus A is a closed set. Proposition 4.5. Let R be a fuzzy T -preorder on X, I satisfy (NP) and (C2) for T . Then for any A, B ∈ F(X) the following formula hold w.r.t. Alexandrov fuzzy topology DI (R). (1) A ⊆ A; (2) A = A; (3) If A ⊆ B, then A ⊆ B; (4) A ∪ B = A ∪ B. Proof. (1) For all x ∈ X, A(x) = (R(Ac ))c (x) = 1 − R(Ac )(x) = 1 − ∧ I(R(x, y), Ac (y)) y∈X

≥ 1 − I (R(x, x), Ac (x))= 1 − I(1, Ac (x))= 1 − Ac (x) = A(x), thus A ⊆ A. (2) Since A is a closed set, By Proposition 4.4, A = A. (3) By A ⊆ B, we obtain Ac ⊇ Bc . According to Definition 3.2, obviously R(Ac ) ⊇ R(Bc ), and then A = (R(Ac ))c ⊆ (R(Bc ))c = B. (4) Since A ⊆ A ∪ B, B ⊆ A ∪ B, by (2) A ⊆ A ∪ B and B ⊆ A ∪ B. Thus A ∪ B ⊆ A ∪ B. On the other hand, by (1) A ⊆ A, B ⊆ B. Thus A ∪ B ⊆ A ∪ B. And then A ∪ B ⊆ A ∪ B. Again A ∪ B is a closed set, according to Proposition 4.4 A ∪ B = A ∪ B. Thus A ∪ B ⊆ A ∪ B. Thereby A ∪ B = A ∪ B. Proposition 4.6. Let R be a fuzzy T -preorder on X, I satisfy (NP) and (C2) for T . Then for any A ∈ F(X) , the following formula hold w.r.t. Alexandrov fuzzy topology DI (R). (1) A = [(Ac )o ]c ; (2) Ao = [Ac ]c ; (3) [A]c = [Ac ]o ; (4) Ac = [Ao ]c . Proof. (1) By Proposition 4.2, (Ac )o = R(Ac ), thus [(Ac )o ]c = [R(Ac )]c = A. (2),(3),(4) can be proven in a similar way as for item (1). Proposition 4.7. Let R be a fuzzy T -preorder on X, I satisfy (NP) and (C2) for T . Then for any A, B ∈ F(X) and A ⊆ B, the following holds w.r.t. Alexandrov fuzzy topology DI (R). (1) Ao ⊆ Bo ; (2) Aoo = Ao ; (3) (A ∩ B)o = Ao ∩ Bo . Proof. (1) ∀x ∈ X, R(A)(x) = ∧ I(R(x, y), A(y)) ≤ ∧ I(R(x, y), B(y)) = R(B)(x). Thus Ao ⊆ Bo . y∈X

y∈X

(2) Since Ao is a open set, by Proposition 4.2, Aoo = Ao . (3) By Proposition 4.6 (2) and Proposition 4.5 (4), (A ∩ B)o = ((A ∩ B)c )c = (Ac ∪ Bc )c = (Ac ∪ Bc )c = (Ac )c ∩ (Bc )c = Ao ∩ Bo .

References [1] A. A. Allam, M. Y. Bakeir, E. A. Abo-Tabl, Some Methods for Generating Topologies by Relations, Bull. Malays. Math. Sci. Soc. (2) 31(1) (2008), 35-45. [2] C. L. Chang, Fuzzy Topological Spaces, J. Math. Anal. Appl 24(1968)182-190.

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[3] D. Dubois, H. Prade, Rough fuzzy sets and fuzzy rough sets, Int. J. Gen. Syst. 17 (1990) 191-209. [4] J. Hao, Q. Li, The relationship between L-fuzzy rough set and L-topology, Fuzzy Sets Syst. 178 (2011) 74-83. [5] J. Jarvinen, Properties of Rough Approximations, J. Adv. Comput. Intel. and Intelligent Inform. Vol.9 No.5 (2005) 502-505. [6] J. Jarvinen, On the Structure of Rough Approximations, Fund. Inform.53 (2002) 135-153. [7] J. Kortelainen, On relationship between modified sets, topological spaces and rough sets, Fuzzy Sets Syst. 61 (1994) 91-95. [8] M. Kondo, On the structure of generalized rough sets, Inform. Sci. 176 (2006) 589-600. [9] Y. C. Kim, Relationships between Alexandrov (fuzzy) topologies and upper approximation operators, J. Math. Comput. Sci. 4 (2014) 558-573. [10] E. F. Lashin et al, Rough set theory for topological spaces, Int. J. Aprrox. Reason. 40 (2005) 35-43. [11] H. Lai, D. Zhang, Fuzzy preorder and fuzzy topology, Fuzzy Sets Syst. 157 (2006) 1865-1885. [12] Z. Li, R. Cui, T -similarity of fuzzy relations and related algebraic structures, Fuzzy Sets Syst. 275 (2015) 130143. [13] Z. Li, R. Cui, Similarity of fuzzy relations based on fuzzy topologies induced by fuzzy rough approximation operators, Inf. Sci. 305 (2015) 219-233. [14] Z. Li, T. Xie, Q. Li, Topological structure of generalized rough sets, Comput. Math. with Appl. 63 (2012) 10661071. [15] N. N. Morsi, M.M. Yakout, Axiomatics for fuzzy rough sets, Fuzzy Sets Syst. 100 (1998) 327-342. ¨ [16] A. F. Ozcan, N. Ba˘grmaz, H. Tas¸bozan, i. ic¸en, Topologies and Approximation Operators Induced by Binary Relations, IECMSA-2013, Sarajevo, Bosnia and Herzegovina, August 2013. [17] L. Polkowski and A. Skowron, Eds., Rough Sets and Current Trends in Computing, vol. 1424, Springer,Berlin, Germany, 1998. [18] P. m. Pu and Y. m. Liu, Fuzy topology I, J. Math. Anal. Appl. 76(1980)571-599. [19] Z. Pawlak, Rough sets, Int. J. Comput. Inf. Sci. 11 (1982) 341-356. 337-369. [20] Z. Pawlak, Rough sets and intelligent data analysis, Inform. Sci. 147 (2002) 1-12. [21] Z. Pei, D. Pei , L. Zheng, Topology vs generalized rough sets, Int. J. Aprrox. Reason. 52 (2011) 231-239. [22] K. Qin, Z. Pei, On the topological properties of fuzzy rough sets, Fuzzy Sets Syst. 151 (2005) 601-613. [23] K. Qin, J. Yang, Z. Pei, Generalized rough sets based on reflexive and transitive relations, Inf. Sci. 178 (2008) 4138-4141. [24] A. M. Radzikowska, E.E. Kerre, A comparative study of fuzzy rough sets, Fuzzy Sets Syst. 126 (2002) 137-155. [25] A. Skowron, On the topology in information systems, Bull. Polish Acad. Sci. Math. 36 (1988) 477-480. [26] A. S. Salama, M.M.E. Abd El-Monsef, New topological approach of rough set generalizations, Internat. J. Computer Math. Vol. 8, No.7, (2011) 1347-1357. [27] M. L. Thivagar, C. Richard, N. R. Paul, Mathematical Innovations of a Modern Topology in Medical Events, Internat. J. Inform. Sci. 2(4) (2012) 33-36. [28] M. Vlach, Algebraic and Topological Aspects of Rough Set Theory, Fourth International Workshop on Computational Intelligence Applications, IEEE SCM Hiroshima Chapter, Hiroshima University, Japan, December, 2008. [29] A. Wiweger, On topological rough sets, Bull. Polish Acad. Sci. Math. 37 (1988) 51-62. [30] C. Y. Wang, Topological characterizations of generalized fuzzy rough sets, Fuzzy Sets and Systems 312 (2017) 109-125.

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[31] W. Z. Wu, On some mathematical structures of T -fuzzy rough set algebras in infinite universes of discourse, Fundam. Inform. 108 (2011)337-369. [32] W. Z. Wu, Y. Leung, J.S. Mi, On characterizations of (J , J)-fuzzy rough approximation operators, Fuzzy Sets Syst. 154 (2005) 76-102. [33] H. Yu, W. Zhan, On the topological properties of generalized rough sets, Inf. Sci. 263 (2014) 141-152. [34] L. Y. Yang, L.S. Xu, Topological properties of generalized approximation spaces, Inf. Sci. 181 (2011) 3570-3580. [35] Y. Y. Yao, Constructive and algebraic methods of the theory of rough sets, Inf. Sci. 109 (1998) 21-47. [36] L. A. Zadeh, Fuzzy Sets, Inform. Control 8 (1965) 338-353. [37] W. Zhu, Generalized rough sets based on relations, Inf. Sci. 177 (2007) 4997-5011. [38] W. Zhu, Relationship between generalized rough sets based on binary relation and covering, Inf. Sci. 179 (2009) 210-225.

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Weighted Lim’s Geometric Mean of Positive Invertible Operators on a Hilbert Space Arnon Ploymukda1 , Pattrawut Chansangiam1∗ 1

Department of Mathematics, Faculty of Science, King Mongkut’s Institute of Technology Ladkrabang, Bangkok 10520, Thailand.

Abstract We generalize the weighted Lim’s geometric mean of positive definite matrices to positive invertible operators on a Hilbert space. This mean is defined via a certain bijection map and parametrized over Hermitian unitary operators. We derive an explicit formula of the weighted Lim’s geometric mean in terms of weighted metric/spectral geometric means. This kind of operator mean turns out to be a symmetric Lim-P´ alfia weighted mean and satisfies the idempotency, the permutation invariance, the joint homogeneity, the self-duality, and the unitary invariance. Moreover, we obtain relations between weighted Lim geometric means and Tracy-Singh products via operator identities.

Keywords: positive invertible operator, metric geometric mean, spectral geometric mean, Lim’s geometric mean, Tracy-Singh product Mathematics Subject Classifications 2010: 47A64, 47A80.

1

Introduction

Recall that the Riccati equation for positive definite matrices A and B: XA−1 X = B

(1)

 1  12 1 1 1 X = A]B := A 2 A− 2 BA− 2 A 2 ,

(2)

has a unique positive solution

known as the metric geometric mean of A and B. This kind of mean was introduced by Ando [2] as the maximum (with respect to the L¨ owner partial ∗ Corresponding

author. Email: [email protected]

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Weighted Lim’s Geometric Mean of Operators

order) of positive semidefinite matrices X satisfying   A X > 0. X B The above two definitions of the metric geometric mean are equivalent. See a nice discussion about the Riccati equation and the metric geometric mean of matrices in [4]. Fiedler and Pt´ ak [3] modified the notion of the metric geometric mean to the spectral geometric mean: 1

1

A♦B = (A−1 ]B) 2 A(A−1 ]B) 2 .

(3)

One of the most important properties of the spectral geometric mean is the positive similarity between (A♦B)2 and AB. This shows that the eigenvalues of A♦B coincide with the positive square roots of the eigenvalues of AB. Lee and Lim [5] introduced the notion of metric geometric means and spectral geometric means on symmetric cones of positive definite matrices and developed various properties of these means. Lim [6] provided a new geometric mean of positive definite matrices varying over Hermitian unitary matrices, including the metric geometric mean as a special case. The new mean has an explicit formula in terms of metric and spectral geometric means. He established basic properties of this mean including the idempotency, joint homogeneity, permutation invariance, non-expansiveness, self-duality, and a determinantal identity. He also gave this new geometric mean for the weighted case. Lim and P´alfia [7] presented a unified framework for weighted inductive means on the cone of positive definite matrices. The metric geometric mean, spectral geometric mean, and Lim geometric mean [6] are basic examples of the two-variable weighted mean. In this paper, we extend the notion of weighted Lim’s geometric mean [6] to the case of Hilbert-space operators via a certain bijection map (see Section 2). This operator mean is parametrized over Hermitian unitary operators. An explicit formula of the weighted Lim’s geometric mean is given in term of weighted metric geometric means and spectral geometric means. This kind of operator mean serves the idempotency, the permutation invariance, the joint homogeneity, the self-duality, and the unitary invariance. Moreover, we establish certain operator identities involving Lim weighted geometric means and Tracy-Singh products (see Section 3). Our results include certain literature results involving weighted metric geometric means.

2

Lim’s geometric mean of operators

In this section, we discuss the notion of Lim’s geometric mean of positive invertible operators on any complex Hilbert space. Throughout, let H be a complex Hilbert space. Denoted by B(H) the Banach space of bounded linear operators on H. The set of all positive invertible operators on H is denoted by P.

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A. Ploymukda, P. Chansangiam

First of all, we recall the notions of metric/spectral geometric means of operators. Recall that for any t ∈ [0, 1], the t-weighted metric geometric mean of A, B ∈ P is defined by  1 t 1 1 1 A]t B = A 2 A− 2 BA− 2 A 2

(4)

For briefly, we write A]B for A]1/2 B. The spectral geometric mean of A, B ∈ P is defined by 1

1

A♦B = (A−1 ]B) 2 A(A−1 ]B) 2 .

(5)

We list some basic properties of metric and spectral geometric means. Lemma 1 (e.g. [1, 3, 4]). Let A, B ∈ P and t ∈ [0, 1]. Then (i) A]t A = A, (ii) (αA)]t (βB) = α1−t β t (A]t B), (iii) A]t B = B]1−t A, (iv) (A]t B)−1 = A−1 ]t B −1 , (v) (Riccati Lemma) A]B is the unique positive invertible solution of XA−1 X = B, (vi) (T ∗ AT )]t (T ∗ BT ) = T ∗ (A]t B)T for any invertible operator T ∈ B(H), (vii) (T ∗ AT )♦(T ∗ BT ) = T ∗ (A♦B)T for any unitary operator T ∈ B(H). For a Hermitian unitary operator U ∈ B(H), we set P+ U := {X ∈ P : U XU = X},

−1 P− } U := {X ∈ P : U XU = X

Lemma 2. Let U ∈ B(H) be a Hermitian unitary operator. Then the map − ΦU : P+ U × PU → P,

1

1

(A, B) 7→ A 2 BA 2

(6)

is bijective with the inverse map given by X 7→ (X](U XU ), X♦(U X −1 U )).

(7)

Proof. The proof is quite similar to [6, Theorem 2.6]. Let A1 , A2 ∈ P+ U and 1

1

1

1

2 2 2 2 B1 , B2 ∈ P− U such that ΦU (A1 , B1 ) = ΦU (A2 , B2 ), i.e. A1 B1 A1 = A2 B2 A2 . + Since Ai ∈ PU , we have

−1 U A−1 = A−1 i U = (U Ai U ) i , 1

1

1

U Ai2 U = (U Ai U ) 2 = Ai2 .

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Weighted Lim’s Geometric Mean of Operators

1

+ 2 and thus A−1 i , Ai ∈ PU for i = 1, 2. It follows that

B1−1 = U B1 U   1 1 1 −1 − = U A1 2 A22 B2 A22 A1 2 U 1 1


−1
−1
= U A1 2 U U A22 U U B2 U U A22 U U A1 2 U −1

1

1

−1

= A1 2 A22 B2−1 A22 A1 2  1 1  1 − −1 1 − 1 −1 21 − 12 A2 A1 = A1 2 A22 A2 2 A12 B1 A12 A2 2 −1 −1
−1
−1 = A1 2 A2 A1 2 B1−1 A1 2 A2 A1 2 , −1

− 21

i.e. A1 2 A2 A1

is a solution of XB1−1 X = B1−1 . Since B1−1 ]B1 = I is the −1

−1

unique solution of XB1−1 X = B1−1 (Lemma 1 (v)), we conclude A1 2 A2 A1 2 = I. This implies that A1 = A2 and then B1 = B2 . Hence, ΦU is injective. Next, 1 1 let X ∈ P. and consider A = X](U XU ) and B = X♦(U X −1 U ) = A− 2 XA− 2 . Consider
U AU = U X](U XU ) U = (U XU )](U 2 XU 2 ) = (U XU )]X = X](U XU ) = A and 1 1
U BU = U A− 2 XA− 2 U = 1



1 1 U A− 2 U (U XU ) U A− 2 U

1

= A 2 X −1 A 2 = B −1 . − + This implies that A ∈ P+ U and B ∈ PU . We have that there exist A ∈ PU and 1 1 2 2 B ∈ P− U such that ΦU (A, B) = A BA = X. Thus, ΦU is surjective. Therefore ΦU is bijective.

By the bijectivity of ΦU , we can define the t-weighted Lim geometric mean of operators as follows: Definition 3. Let U ∈ B(H) be a Hermitian unitary operator and t ∈ [0, 1]. Let X = ΦU (A1 , B1 ), Y = ΦU (A2 , B2 ) ∈ P. The t-weighted Lim geometric mean of X and Y is defined by GU (t; X, Y ) = ΦU (A1 ]t A2 , B1 ]t B2 ).

(8)

We denote GU (X, Y ) = GU (1/2; X, Y ) the Lim geometric mean. The next theorem gives an explicit formula of GU (X, Y ). Theorem 4. Let U be a Hermitian unitary operator and t ∈ [0, 1]. Let X, Y ∈ P. We have 1

1

GU (t; X, Y ) = (A1 ]t A2 ) 2 (B1 ]t B2 )(A1 ]t A2 ) 2 , where A1 = X](U XU ), A2 = Y ](U Y U ), B1 = X♦(U X In particular, GI (X, Y ) = X]t Y .

393

−1

(9)

U ) and B2 = Y ♦(U Y −1 U ).

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− Proof. Since fU is surjective, there exist A1 , A2 ∈ P+ U and B1 , B2 ∈ PU such that X = ΦU (A1 , B1 ) and Y = ΦU (A2 , B2 ). By using the inverse map (7), we have −1 (A1 , B1 ) = Φ−1 U )) U (X) = (X](U XU ), X♦(U X −1 (A2 , B2 ) = Φ−1 U )). U (Y ) = (Y ](U Y U ), Y ♦(U Y − For the case U = I, we have P+ I = P and PI = {I}. It follows that B1 = B2 = I. By Lemma 1, we have A1 = X]X = X and A2 = Y ]Y = Y . Hence, 1

1

GI (t; X, Y ) = (X]t Y ) 2 (I]t I)(X]t Y ) 2 = X]t Y. Now, we give the definition of the Lim-P´alfia weighted mean [7] in the case of operators. Definition 5. The (two-variable) Lim-P´ alfia weighted mean of positive invertible operators is the map M : [0, 1] × P × P → P satisfying (i) M(0, X, Y ) = X, (ii) M(1, X, Y ) = Y , (iii) (Idempotency) M(t, X, X) = X for all t ∈ [0, 1]. We say that M is symmetric if (iv) (Permutation invariancy) M(t, X, Y ) = M(1 − t, Y, X) for all t ∈ [0, 1]. Theorem 6. The t-weighted Lim geometric mean of operators is a symmetric Lim-P´ alfia weighted mean. Proof. Let U ∈ B(H) be a Hermitian unitary operator and t ∈ [0, 1]. Let X, Y ∈ P. Write X = ΦU (A1 , B1 ) and Y = ΦU (A2 , B2 ). We have by Lemma 1 that GU (0; X, Y ) = ΦU (A1 ]0 A2 , B1 ]0 B2 ) = ΦU (A1 , B1 ) = X, GU (1; X, Y ) = ΦU (A1 ]1 A2 , B1 ]1 B2 ) = ΦU (A2 , B2 ) = Y, GU (t; X, X) = ΦU (A1 ]t A1 , B1 ]t B1 ) = ΦU (A1 , B1 ) = X. This implies that GU is a Lim-P´ alfia weighted mean. Using Lemma 1 again, we get GU (t; X, Y ) = ΦU (A1 ]t A2 , B1 ]t B2 ) = ΦU (A2 ]1−t A1 , B2 ]1−t B1 ) = GU (1 − t; Y, X). Thus, GU is symmetric. Corollary 7. The t-weighted metric geometric mean of operators is a symmetric Lim-P´ alfia weighted mean.

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Weighted Lim’s Geometric Mean of Operators

Theorem 8. Let U ∈ B(H) be a Hermitian unitary operator and t ∈ [0, 1]. Let X = ΦU (A1 , B1 ) and Y = ΦU (A2 , B2 ). We have 1−t t t (i) GU (t; X, I) = ΦU (A1−t 1 , B1 ) and GU (t; I, Y ) = ΦU (A2 , B2 ),

(ii) (Joint Homogeneity) GU (t; αX, βY ) = α1−t β t GU (t; X, Y ) for any α, β > 0, (iii) (Self-duality) GU (t; X, Y )−1 = GU (t; X −1 , Y −1 ), (iv) (Unitary invariance) GU (t; T ∗ XT, T ∗ Y T ) = T ∗ GU (t; X, Y )T where T ∈ B(H) is a unitary operator such that U T = T U , (v) GU (t; U XU, U Y U ) = U GU (t; X, Y )U , (vi) GU (X, X −1 ) = I. Proof. The first assertion is immediate from Definition 3. For the joint homogeneity, note that 1

1

αX = αΦU (A1 , B1 ) = α A12 B1 A12




1

1

= A12 (αB1 )A12 = ΦU (A1 , αB1 ).

Similarly, βY = ΦU (A2 , βB2 ). Using Lemma 1, we obtain GU (t; αX, βY ) = ΦU (A1 ]t A2 , (αB1 )]t (βB2 )) = ΦU (A1 ]t A2 , α1−t β t (B1 ]t B2 )) = α1−t β t ΦU (A1 ]t A2 , B1 ]t B2 ) = α1−t β t GU (t; X, Y ). For the self-duality, consider  1 −1 1 −1 −1 −1 = A1 2 B1−1 A1 2 = ΦU (A−1 X −1 = ΦU (A1 , B1 )−1 = A12 B1 A12 1 , B1 ). −1 Similarly, Y −1 = ΦU (A−1 2 , B2 ). Applying Lemma 1, we get

GU (t; X, Y )−1 = ΦU (A1 ]t A2 , B1 ]t B2 )−1 = ΦU ((A1 ]t A2 )−1 , (B1 ]t B2 )−1 ) −1 −1 −1 −1 = ΦU (A−1 , Y −1 ) 1 ]t A2 , B1 ]t B2 ) = GU (t; X

Now, let us prove the assertion (iv). Since T is unitary, we have by Lemma 1 that (T ∗ XT )][U (T ∗ XT )U ] = (T ∗ XT )][T ∗ (U XU )T ] = T ∗ [X](U XU )]T = T ∗ A1 T, (T ∗ XT )♦[U (T ∗ XT )−1 U ] = (T ∗ XT )♦[U T X −1 T ∗ U ] = (T ∗ XT )♦[T ∗ (U X −1 U )T ] = T ∗ [X♦(U X −1 U )]T = T ∗ B1 T.

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Similarly, (T ∗ Y T )][U (T ∗ Y T )U ] = T ∗ A2 T

and

(T ∗ Y T )♦[U (T ∗ Y T )−1 U ] = T ∗ B2 T

Then T ∗ XT = ΦU (T ∗ A1 T, T ∗ B1 T ) and T ∗ Y T = ΦU (T ∗ A2 T, T ∗ B2 T ). Thus GU (t; T ∗ XT, T ∗ Y T ) 1

1

= [(T ∗ A1 T )]t (T ∗ A2 T )] 2 [(T ∗ B1 T )]t (T ∗ B2 T )][(T ∗ A1 T )]t (T ∗ A2 T )] 2 1

1

= [T ∗ (A1 ]t A2 )T ] 2 [T ∗ (B1 ]t B2 )T ][T ∗ (A1 ]t A2 )T ] 2 1

1

= [T ∗ (A1 ]t A2 ) 2 T ][T ∗ (B1 ]t B2 )T ][T ∗ (A1 ]t A2 ) 2 T ] 1

1

= T ∗ (A1 ]t A2 ) 2 (B1 ]t B2 )(A1 ]t A2 ) 2 T = T ∗ GU (t; X, Y )T. Setting T = U , we get the result in the assertion (v). For the last assertion, −1 since X −1 = ΦU (A−1 1 , B1 ), we have −1 GU (X, X −1 ) = ΦU (A1 ]A−1 1 , B1 ]B1 ) = ΦU (I, I) = I.

3

Weighted Lim geometric means and TracySingh products

In this section, we investigate relations between Weighted Lim geometric means and Tracy-Singh products of operators. Let us recalling the notion of TracySingh product.

3.1

Preliminaries on the Tracy-Singh product of operators

The projection theorem for Hilbert space allows us to decompose H =

n M

Hi

(10)

i=1

where all Hi are Hilbert spaces. For each i = 1, . . . , n, let Pi be the natural projection map from H onto Hi . Each operator A ∈ B(H) can be uniquely determined by an operator matrix n,n

A = [Aij ]i,j=1 where Aij : Hj → Hi is defined by Aij = Pi APj∗ for each i, j = 1 . . . , n. Recall that the tensor product of A, B ∈ B(H) is the operator A ⊗ B ∈ B(H ⊗ H) such that for all x, y ∈ H, (A ⊗ B)(x ⊗ y) = (Ax) ⊗ (By).

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Weighted Lim’s Geometric Mean of Operators

n,n

n,n

Definition 9. Let A = [Aij ]i,j=1 and B = [Bij ]i,j=1 be operators in B(H). The Tracy-Singh product of A and B is defined to be   (11) A  B = [Aij ⊗ Bkl ]kl ij which is a bounded linear operator from

Ln,n

i,j=1

Hi ⊗ Hj into itself.

Lemma 10 ([9, 10, 11]). Let A, B, C, D ∈ B(H). (i) (A  B)(C  D) = (AC)  (BD). (ii) If A, B ∈ P, then A  B ∈ P. (iii) If A, B ∈ P, then (A  B)α = Aα  B α for any α ∈ R. (iv) If A and B are Hermitian, then A  B is also. (v) If A and B are unitary, then A  B is also. Lemma 11 ([8]). Let A, B, C, D ∈ P and t ∈ [0, 1]. Then (A  B)]t (C  D) = (A]t C)  (B]t D). For each i = 1, . . . , k, let Hi be a Hilbert space and decompose Hi =

ni M

Hi,r

r=1 1

where all Hi,r are Hilbert spaces. We set i=1 Ai = A1 . For k ∈ N − {1} and Ai ∈ B(Hi ) (i = 1, . . . , k), we use the notation k

A

i

= ((A1  A2 )  · · ·  Ak−1 )  Ak .

i=1

3.2

The compatibility between weighted Lim geometric means and Tracy-Singh products

The following theorem provides an operator identity involving t-weighted Lim geometric means and Tracy-Singh products. Theorem 12. Let U, V be Hermitian unitary operators, X1 , X2 , Y1 , Y2 ∈ P and t ∈ [0, 1]. GU (t; X1 , Y1 )  GV (t; X2 , Y2 ) = GU V (t; X1  X2 , Y1  Y2 ).

(12)

Proof. Write X1 = ΦU (A1 , B1 ),

Y1 = ΦU (C1 , D1 ),

397

X2 = ΦV (A2 , B2 ),

Y2 = ΦV (C2 , D2 ),

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A. Ploymukda, P. Chansangiam

− + − where A1 , C1 ∈ P+ U , B1 , D1 ∈ PU , A2 , C2 ∈ PV , B2 , D2 ∈ PV . Since U and V are Hermitian unitary operators, we have by Lemma 10 that U  V is also a Hermitian unitary operator. Thus GU V (t; X1  X2 , Y1  Y2 ) is well-defined. By Lemma 10, we get

(U  V )(A1  A2 )(U  V ) = (U A1 U )  (V A2 V ) = A1  A2 and (U  V )(B1  B2 )(U  V ) = (U B1 U )  (V B2 V ) = B1−1  B2−1 = (B1  B2 )−1 . − + Thus A1 A2 ∈ P+ U V and B1 B2 ∈ PU V . Similarly, we have C1 C2 ∈ PU V − and D1  D2 ∈ PU V . Using Lemma 10, we get

X1  X2 = ΦU (A1 , B1 )  ΦV (A2 , B2 ) 1
1 1
1 = A12 B1 A12  A22 B2 A22 1

1

1

1

= (A12  A22 )(B1  B − 2)(A12  A22 ) 1

1

= (A1  A2 ) 2 (B1  B2 )(A1  A2 ) 2 = ΦU V (A1  A2 , B1  B2 ). Similarly, Y1  Y2 = ΦU V (C1  C2 , D1  D2 ). Then
GU V (t; X1  X2 , Y1  Y2 ) = ΦU V (A1  A2 )]t (C1  C2 ), (B1  B2 )]t (D1  D2 ) . We have by applying Lemmas 10 and 11 that GU (t;X1 , Y1 )  GV (t; X2 , Y2 ) = ΦU (A1 ]t C1 , B1 ]t D1 )  ΦV (A1 ]t C2 , B2 ]t D2 )   1 1 1 1 = (A1 ]t C1 ) 2 (B1 ]t D1 )(A1 ]t C1 ) 2  (A2 ]t C2 ) 2 (B2 ]t D2 )(A2 ]t C2 ) 2   1 1  1 1 = (A1 ]t C1 ) 2  (A2 ]t C2 ) 2 (B1 ]t D1 )  (B2 ]t D2 ) (A1 ]t C1 ) 2  (A2 ]t C2 ) 2  1   1 = (A1 ]t C1 )  (A2 ]t C2 ) 2 (B1 ]t D1 )  (B2 ]t D2 ) (A1 ]t C1 )  (A2 ]t C2 ) 2
= ΦU V (A1 ]t C1 )  (A2 ]t C2 ), (B1 ]t D1 )  (B2 ]t D2 )
= ΦU V (A1  A2 )]t (C1  C2 ), (B1  B2 )]t (D1  D2 ) = GU V (t; X1  X2 , Y1  Y2 ). Corollary 13. Let k ∈ N and t ∈ [0, 1]. For each 1 6 i 6 k, let Ui ∈ B(H) be a Hermitian unitary operator and Xi , Yi ∈ P. Then k

 i=1

 GUi (t; Xi , Yi ) = GU t;

k

k

  Xi ,

i=1

 Yi ,

(13)

i=1

k

where U = i=1 Ui .

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Weighted Lim’s Geometric Mean of Operators

Proof. Since Ui is a Hermitian unitary operator for all i = 1, . . . , k, we have by k Lemma 10 that i=1 Ui is also. Using the positivity of the Tracy-Singh product, k k we get i=1 Xi , i=1 Yi ∈ P. Hence, the right hand side of (13) is well-defined. We reach the result by applying Thoerem 12 and induction on k. From Corollary 13, setting Ui = I for all i = 1, . . . , k, we have k



k

(X ] Y ) =  i t i

i=1

i=1

  Xi ]t

k



 Yi .

i=1

This equality were proved already in [8, Corollary 1].

Acknowledgement The first author would like to thank the Royal Golden Jubilee Ph.D. Scholarship, grant no. PHD60K0225, from Thailand Research Fund.

References [1] T. Ando, Concavity of certain maps on positive definite matrices and applications to Hadamard products, Linear Algebra Appl. 26, 203–241 (1979), DOI: 10.1016/0024-3795(79)90179-4. [2] T. Ando, Topics on Operator Inequalities. Hokkaido Univ., Sapporo (1978). [3] M. Fiedler and V. Pt´ak, A new positive definite geometric mean of two positive definite matrices, Linear Algebra Appl. 251, 1–20 (1997), DOI: 10.1016/0024-3795(95)00540-4. [4] J.D. Lawson, and Y. Lim, The geometric mean, matrices, metrics, and more, Amer. Math. Monthly 108, 797–812 (2001), DOI: 10.2307/2695553. [5] H. Lee and Y. Lim, Metric and spectral geometric means on symmetric cones, Kyungpook Math. J. 47(1), 133–150 (2007). [6] Y. Lim, Factorizations and geometric means of positive definite matrices, Linear Algebra Appl. 437(9), 2159–2172 (2012), DOI: 10.1016/j.laa.2012.05.039. [7] Y. Lim and M. P´alfia, Weighted inductive means, Linear Algebra Appl. 45, 59–83 (2014), DOI: 10.1016/j.laa.2014.04.002. [8] A. Ploymukda and P. Chansangiam, Geometric means and Tracy-Singh products for positive operators, Communications in Mathematics and Applications 9(4), 475–488 (2018), DOI: 10.26713/cma.v9i4.547.

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[9] A. Ploymukda, P. Chansangiam and W. Lewkeeratiyutkul, Algebraic and order properties of Tracy-Singh products for operator matrices, J. Comput. Anal. Appl. 24(4), 656–664 (2018). [10] A. Ploymukda, P. Chansangiam and W. Lewkeeratiyutkul, Analytic properties of Tracy-Singh products for operator matrices, J. Comput. Anal. Appl. 24(4), 665–674 (2018). [11] A. Ploymukda, P. Chansangiam and W. Lewkeeratiyutkul, Tracy-Singh products and classes of operators, J. Comput. Anal. Appl. 26(8), 1401– 1413 (2019).

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TABLE OF CONTENTS, JOURNAL OF COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 29, NO. 2, 2021

A Study of a Coupled System of Nonlinear Second-Order Ordinary Differential Equations with Nonlocal Integral Multi-Strip Boundary Conditions on an Arbitrary Domain, Bashir Ahmad, Ahmed Alsaedi, Mona Alsulami, and Sotiris K Ntouyas,…………………………………215 Explicit Identities Involving Truncated Exponential Polynomials and Phenomenon of Scattering of Their Zeros, C. S. RYOO,………………………………………………………………236 On Generalized Degenerate Twisted (h, q)-Tangent Numbers and Polynomials, C. S. RYOO,246 New Oscillation Criteria of First Order Neutral Delay Difference Equations of Emden-Fowler Type, S. H. Saker and M. A. Arahet,………………………………………………………….252 Riccati Technique and Oscillation of Second Order Nonlinear Neutral Delay Dynamic Equations, S. H. Saker and A. K. Sethi,………………………………………………………266 Semilocal Convergence of a Newton-Secant Solver for Equations with a Decomposition of Operator, Ioannis K. Argyros, Stepan Shakhno, and Halyna Yarmola,…………………….279 Global Behavior of a Nonlinear Higher-Order Rational Difference Equation, A. M. Ahmed,290 Weighted Composition Operator Acting Between Some Classes of Analytic Function Spaces, A. El-Sayed Ahmed and Aydah Al-Ahmadi,…………………………………………………300 Hermite-Hadamard Type Inequalities for the ABK-Fractional Integrals, Artion Kashuri,……309 A Unified Convergence Analysis for Single Step-Type Methods for Non-Smooth Operators, S. Amat, I. Argyros, S. Busquier, M.A. Hernandez-Veron, and Eulalia Martinez,……………327 On the Localization of Factored Fourier Series, Hikmet Seyhan Ozarslan,………………….344 Analysis of Solutions of Some Discrete Systems of Rational Difference Equations, M. B. Almatrafi,…………………………………………………………………………………….355 The ELECTRE Multi-Attribute Group Decision Making Method Based on Interval-Valued Intuitionistic Fuzzy Sets, Cheng-Fu Yang, …………………………………………………369 The Interior and Closure of Fuzzy Topologies Induced by the Generalized Fuzzy Approximation Spaces, Cheng-Fu Yang,……………………………………………………………………383

TABLE OF CONTENTS, JOURNAL OF COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 29, NO. 2, 2021 (continues)

Weighted Lim's Geometric Mean of Positive Invertible Operators on a Hilbert Space, Arnon Ploymukda and Pattrawut Chansangiam, ……………………………………………………390

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J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 29, NO.3, 2021, COPYRIGHT 2021 EUDOXUS PRESS, LLC

A Numerical Technique for Solving Fuzzy Fractional Optimal Control Problems† Altyeb Mohammeda,b , Zeng-Tai Gonga,∗ , Mawia Osmana a College of Mathematics and Statistics, Northwest Normal University, Lanzhou 730070, China b Faculty of Mathematical Science, University of Khartoum, Khartoum, Sudan

Abstract In this paper, the fuzzy fractional optimal control problem with both fixed and free final state conditions has been considered. Our problem is defined in the sense of Riemann-Liouville fractional derivative based on Hukuhara difference, and the dynamic constraint is described by a fractional differential equation of order less than 1. Using fuzzy variational approach, a necessary conditions of our problem has been derived. A numerical technique based on Gr¨ unwald-Letnikov definition of fractional derivative and the relation between right Riemann-Liouville fractional derivative and right Caputo fractional derivative is proposed. Finally, some numerical examples are given to illustrate our main results. Keywords: Fuzzy fractional calculus;Gr¨ unwald-Letnikov fractional derivative;Fuzzy fractional optimal control problem;Fixed final state problem;Free final state problem;Fuzzy variational approach;Necessary conditions. 1. Introduction Optimal control is the standard method for solving dynamic optimization problems, which deal with finding a control law for a given system such that a certain optimality criterion is achieved. It’s playing an increasingly important role in modern system design, and considered to be a powerful mathematical tool that can be used to make decisions in real life. On the other hand, accurate modeling of some real problems in scientific fields and engineering, sometimes lead to a set of fractional differential and integral equations. Fractional optimal control problem is an optimal control problem whose dynamic system is described by fractional differential equations. We can define the fractional optimal control problem in sense of different definitions of fractional derivative, for example Riemann-Liouville fractional derivative, Caputo fractional derivative and so on. Due to, uncertainty in the input, output and manner of many dynamical systems, meanwhile, fuzziness is a way to express an uncertain phenomena in real world. Thus, importing fuzziness in the optimal control theory, give a better display of the problems with control parameters in real world such as physical models and dynamical systems. In the last decade, fuzzy fractional optimal control problems have attracted a great deal of attention and the interest in the filed of fuzzy fractional optimal control problems has increased. In [1], Fard and Soolaki, prove the necessary optimality conditions of pontryagin type for a class of fuzzy fractional optimal control problems with the fuzzy fractional derivative described in the Caputo sense. In [2], Fard and Salehi studied the constrained and unconstrained fuzzy fractional variational problems containing the Caputo-type fractional derivatives using the approach of the generalized differentiability. In [3], Karimyar and Fakharzadeh introduced the solution of fuzzy fractional optimal control problems by using Mittag-Leffler function. In this paper, we will study a fixed and free final state fuzzy fractional optimal control problems with the fuzzy fractional derivative described in Riemann-Liouville type in sense of Hukuhara difference. †

This work is supported by National Natural Science Foundation of China(61763044). Corresponding Author:Zeng-Tai Gong. Tel.: +869317971430. E-mail addresses: [email protected] email: [email protected] ∗

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Altyeb Mohammed, Zeng-Tai Gong and Mawia Osman: A Numerical Technique for Solving Fuzzy Fractional ...

Then, we derive the necessary conditions of that problems based on fuzzy variational approach. A numerical algorithm is proposed to solve the necessary conditions to find the optimal fuzzy control and optimal fuzzy state as a solutions of our problems. The definitions of a strong and weak solutions of our problems are given, to guarantee the optimal solutions are a fuzzy functions. This paper is organized as follows. In Section 2 we introduce and generalize some basic concepts and notations that are key to our discussion. In Section 3 we present basic elements of fuzzy fractional calculus and fuzzy calculus of variations. In Section 4 we establish our main results, Theorem(4.1), that provides the necessary conditions of fuzzy fractional optimal control problems with both fixed and free final state conditions. In Section 5 we propose a numerical technique to solve the necessary conditions. Finally, we discuss the applicability of the main theorem and the numerical algorithm through an examples. 2. Definitions and preliminaries Here, we start with basic definitions and lemmas needed in the other sections for a better understanding of this work. The details of this concepts are clearly found in [7, 9, 10, 11, 12, 17]. Definition 2.1 A fuzzy set A˜ : R → [0, 1] is called a fuzzy number if A˜ is normal, convex fuzzy set, ˜ upper semi-continuous and suppA = {x ∈ R|A(x) > 0} is compact, where M denotes the closure of M . 1 In the rest of this paper we use E to denote the fuzzy number space. Where it is α−level set a ˜[α] = {x ∈ R : a ˜(x) ≥ α} = [al (α), ar (α)], ∀α ∈ (0, 1], and 0−level set a ˜[0] is defined as {x ∈ R|˜ a(x) > 0}. Obviously, the α-level set a ˜[α] = [al (α), ar (α)] is bounded closed l r interval in R for all α ∈ [0, 1], where a (α) and a (α) denote the left-hand and right-hand end points of a ˜[α], respectively. a ˜ is a crisp number with value k if its membership function is defined by, { 1 ,x = k a ˜(x) = 0 , x ̸= k Thus,

{ ˜0(x) =

1 ,x = 0 0 , x ̸= 0.

Let u ˜, v˜ ∈ E 1 , k ∈ R, we can define the addition and scalar multiplication by using α-level set respectively as (˜ a + ˜b)[α] = a ˜[α] + ˜b[α], (k˜ a)[α] = k˜ a[α], where a ˜[α] + ˜b[α] means the usual addition of two intervals of R, and k˜ a[α] means the usual product between a scalar and interval of R. Furthermore, the opposite of the fuzzy number a ˜ is −˜ a, i.e., −˜ a(x) = a ˜(−x), it means, −˜ a[α] = [−ar (α), −al (α)]. The binary operation ”.” in R can be extended to the binary operation ”⊙” of two fuzzy numbers by using the extension principle. Let a ˜ and ˜b be fuzzy numbers, then (˜ a ⊙ ˜b)(z) = sup min{˜ a(x), ˜b(x)}. x·y=z

Using α-level set the product (˜ a ⊙ ˜b) is defined by [ (˜ a ⊙ ˜b)[α] = min{al (α)bl (α), al (α)br (α), ar (α)bl (α), ar (α)br (α)}, ] max{al (α)bl (α), al (α)br (α), ar (α)bl (α), ar (α)br (α)} . The metric structure is given by the Hausdorff distance D : E 1 × E 1 × R → R+ ∪ {0}, D(˜ a, ˜b) = sup max{| al (α) − bl (α) |, | ar (α) − br (α) |}. α∈[0,1]

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Altyeb Mohammed, Zeng-Tai Gong and Mawia Osman: A Numerical Technique for Solving Fuzzy Fractional ...

A special class of fuzzy numbers is the class of triangular fuzzy numbers. For a1 < a2 < a3 and a1 , a2 , a3 ∈ R, the triangular fuzzy number a ˜ is generally denoted by a ˜ = (a1 , a2 , a3 ) is determined by a1 , a2 , a3 such that al (α) = a1 + (a2 − a1 )α and ar (α) = a3 − (a3 − a2 )α, when α = 0 then a ˜[0] = [a1 , a3 ] and when α = 1 then a ˜[1] = [a2 , a2 ] = a2 . We know that, we can identify a fuzzy number a ˜ ∈ E 1 by the left and right hand functions of its α−level set, the following lemma introduce the properties of this functions. Lemma 2.1 Suppose that al : [0, 1] → R and ar : [0, 1] → R satisfy the conditions: C1: al is bounded increasing function, C2: ar is bounded decreasing function, C3: al (1) ≤ ar (1), C4: lim al (α) = al (k) and lim ar (α) = ar (k), for all 0 < k ≤ 1, α→k−

α→k−

C5: lim al (α) = al (0) and lim ar (α) = ar (0). α→0+

α→0+

Then a ˜ : R → [0, 1] defined by a ˜(x) = sup{α|al (α) ≤ x ≤ ar (α)} is a fuzzy number with a ˜[α] = l r l r [a (α), a (α)]. Moreover, if a ˜ : R → [0, 1] is a fuzzy number with a ˜[α] = [a (α), a (α)], then the functions al (α) and ar (α) satisfy conditions C1- C5. Definition 2.2 (H-difference). Let a ˜, ˜b ∈ E 1 , where a ˜[α] = [al (α), ar (α)] and ˜b[α] = [bl (α), br (α)] for all α ∈ [0, 1], the H-difference is defined by a ˜ ⊖ ˜b = c˜

⇐⇒

a ˜ = ˜b + c˜.

Obviously, a ˜⊖a ˜ = ˜0, and the α-level set of H-difference is (˜ a ⊖ ˜b)[α] = [al (α) − bl (α), ar (α) − br (α)], ∀α ∈ [0, 1]. Definition 2.3 (Partial ordering). Let a ˜, ˜b ∈ E 1 , we write a ˜ ≼ ˜b, if al (α) ≤ bl (α) and ar (α) ≤ br (α) for all α ∈ [0, 1]. We also write a ˜ ≺ ˜b, if a ˜ ≼ ˜b and there exists α0 ∈ [0, 1] such that al (α0 ) < bl (α0 ) or r r a (α0 ) < b (α0 ). Furthermore, a ˜ = ˜b, if a ˜ ≼ ˜b and a ˜ ≽ ˜b. In other words, a ˜ = ˜b, if a ˜[α] = ˜b[α] for all α ∈ [0, 1]. In the sequel, we say that a ˜, ˜b ∈ E 1 are comparable if either a ˜ ≼ ˜b or a ˜ ≽ ˜b, and non-comparable otherwise. From now we consider S as a subset of R. Definition 2.4 (Fuzzy valued function). The function f˜ : S → E 1 is called a fuzzy-valued function if f˜(t) is assign a fuzzy number for any e ∈ S. We also denote f˜(t)[α] = [f l (t, α), f r (t, α)], where f l (t, α) = (f˜(t))l (α) = min{f˜(t)[α]} and f r (t, α) = (f˜(t))r (α) = max{f˜(t)[α]}. Therefore any fuzzyvalued function f˜ may be understood by f l (t, α) and f r (t, α) being respectively a bounded increasing function of α and a bounded decreasing function of α for α ∈ [0, 1]. And also it holds f l (t, α) ≤ f r (t, α) for any α ∈ [0, 1]. Definition 2.5 (Continuity of a fuzzy valued function). We say that f˜ : S → E 1 is continuous at t ∈ S, if both f l (t, α) and f r (t, α) are continuous functions at t ∈ S for all α ∈ [0, 1]. If f˜(t) is continuous in the metric D, then its definite integral exists and defined by   b ∫b ∫ ∫b f˜(t)[α]dt =  f l (t, α)dt, f r (t, α)dt . a

a

a

Definition 2.6 (Distance measure between fuzzy valued functions). Suppose that f˜, g˜ : S → E 1 are two fuzzy functions. We define the distance measure between f˜ and g˜ by DE 1 (f˜(x), g˜(x)) = sup H(f˜(x)[α], g˜(x)[α]) 0≤α≤1

= max{ sup

d(z, g˜(x)[α]),

z∈f˜(x)[α]

sup

d(f˜(x)[α], y)}, ∀x ∈ S.

y∈˜ g (x)[α]

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Altyeb Mohammed, Zeng-Tai Gong and Mawia Osman: A Numerical Technique for Solving Fuzzy Fractional ...

Where H is the Hausdorff metric on the family of all nonempty compact subsets of R, and d(a, B) = inf d(a, b). b∈B

Moreover, we can define ∥ f˜(x) ∥2E 1 = DE 1 (f˜(x), f˜(x)), ∀x ∈ S, for any f˜ : S → E 1 . 3. Elements of fuzzy fractional calculus and fuzzy calculus of variations Several definitions of a fractional derivative have been studied, such as Riemann-Liouville, Gr¨ unwaldLetnikov, Caputo and so on. In this paper, we deal with the problems defined by Riemann-Liouville fractional derivative. In this section, we first introduce the definition of fuzzy Riemann-Liouville integrals and derivatives in sense of Hukuhara difference. Definition 3.1(see [6]) Let f˜(x) be continuous and Lebesgue integrable fuzzy valued function in [a, b] ∈ R and 0 < β ≤ 1, then the fuzzy Riemann-Liouville integral of f˜(x) of order β is defined by ∫ x 1 β ˜ f˜(t)(x − t)β−1 dt, a Ix f (x) = Γ(β) a where Γ(β) is the Gamma function and x > a. Theorem 3.1(see [6]) Let f˜(x) be continuous and Lebesgue integrable fuzzy valued function in [a, b] ∈ R. The fuzzy Riemann-Liouville integral of f˜(x) can be expressed as follows [ ] β ˜ β l β r I f (x) [α] = I f (x, α), I f (x, α) , 0 ≤ α ≤ 1, a x a x a x where β l a Ix f (x, α) = β r a Ix f (x, α)

=

∫ x 1 f l (t, α)(x − t)β−1 dt, Γ(β) a ∫ x 1 f r (t, α)(x − t)β−1 dt. Γ(β) a

In the next definition, we define the fuzzy Riemann-Liouville fractional derivative of order 0 < β < 1 of a fuzzy valued function f˜(x). Definition 3.2(see [6]) Let f˜(x) be continuous and Lebesgue integrable fuzzy valued function in [a, b] ∈ ∫ x f˜(t)dt ˜ R. x0 ∈ (a, b) and then: G(x) = 1 β . We say that f is Riemann-Liouville H-differentiable Γ(1−β)

a (x−t)

of order 0 < β < 1 at x0 , if there exist an element a Dxβ f˜(x0 ) ∈ E 1 such that for h > 0 sufficiently small (1) a Dxβ f˜(x0 ) = lim

h→0+

G(x0 +h)⊖G(x0 ) h

= lim

G(x0 )⊖G(x0 −h) , h

G(x0 )⊖G(x0 +h) −h

= lim

G(x0 −h)⊖G(x0 ) , −h

G(x0 +h)⊖G(x0 ) h

= lim

G(x0 −h)⊖G(x0 ) , −h

G(x0 )⊖G(x0 +h) −h

= lim

G(x0 )⊖G(x0 −h) . h

h→0+

or (2) a Dxβ f˜(x0 ) = lim

h→0+

h→0+

or (3) a Dxβ f˜(x0 ) = lim

h→0+

h→0+

or (4) a Dxβ f˜(x0 ) = lim

h→0+

h→0+

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J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 29, NO.3, 2021, COPYRIGHT 2021 EUDOXUS PRESS, LLC

Altyeb Mohammed, Zeng-Tai Gong and Mawia Osman: A Numerical Technique for Solving Fuzzy Fractional ...

For sake of simplicity, we say that the fuzzy valued function f˜(x) is Riemann-Liouville [(i)−β]−differentiable if it is differentiable as in the Definition(3.2) case(i), i = 1, 2, 3, 4 respectively. Theorem 3.2(see [6]) Let f˜(x) be continuous and Lebesgue integrable fuzzy valued function in [a, b] ∈ R and f˜(x)[α] = [f l (x, α), f r (x, α)],then for α ∈ [0, 1], x ∈ (a, b) and β ∈ (0, 1) (i) Let us consider f˜ is Riemann-Liouville [(1) − β]−differentiable fuzzy-valued function, then: [ ] β ˜ β l β r D f (x )[α] = D f (x , α), D f (x , α) . 0 0 0 a x a x a x (ii) Let us consider f˜ is Riemann-Liouville [(2) − β]−differentiable fuzzy-valued function, then: [ ] β ˜ β r β l D f (x )[α] = D f (x , α), D f (x , α) . 0 0 0 a x a x a x Where

[ β l a Dx f (x0 , α)

= [

β r a Dx f (x0 , α) =

1 d Γ(1 − β) dx 1 d Γ(1 − β) dx

∫ ∫

x a x a

] f l (t, α)dt , (x − t)β x=x0 ] f r (t, α)dt . (x − t)β x=x0

Theorem 3.3(see [6]) Let f˜(x) be continuous and Lebesgue integrable fuzzy valued function in [a, b] is a Riemann-Liouville H-differentiable of order 0 < β < 1 on each point x ∈ (a, b) in the sense of Definition(3.2) case(3) or case(4), then a Dxβ f˜(x) ∈ R for all x ∈ (a, b). Now we state some elements of fuzzy calculus of variations. Definition 3.3(Fuzzy increment[10]). Suppose that x ˜(.) and x ˜(.) + δ x ˜(.) are fuzzy functions for which ˜ denoted by ∆J, ˜ is the fuzzy functional J˜ is defined. The increment of J, ˜ x + δx ˜ ∆J˜ := J(˜ ˜) ⊖ J(x),

(3.1)

Where δ x ˜(.) is the variation of x ˜(.). ˜ x, δ x Because the increment ∆J˜ depends on the fuzzy functions x ˜ and δ x ˜, we denote ∆J˜ by ∆J(˜ ˜). Definition 3.4(Differentiability of a fuzzy functional[10, 15]). Suppose that ∆J˜ can be written as ˜ x, δ x ˜ x, δ x ∆J(˜ ˜) := δ J(˜ ˜) + ˜j(˜ x, δ x ˜)· ∥ δ x ˜ ∥E 1 ,

(3.2)

Where δ J˜ is linear in δ x ˜. We say that J˜ is differentiable with respect to x ˜ if for any ϵ > 0 , DE 1 (˜j(˜ x, δ x ˜), 0) < ϵ, as ∥ δ x ˜(.) ∥E 1 → 0. ˜ 0 , t1 ] represent the class of all fuzzy continuous functions on [t0 , t1 ]. From now C[t ˜ 0 , t1 ], has a fuzzy Definition 3.5(Fuzzy relative minimum[10]) A fuzzy functional J˜ with domain C[t ∗ ∗ relative minimizer x ˜ =x ˜ (t), if ˜ x) ≽ J(˜ ˜ x∗ ), J(˜ (3.3) ˜ 0 , t1 ]. for all fuzzy functions x ˜ ∈ C[t It is clear that the inequality (3.3) holds iff J l (˜ x, α) ≥ J l (˜ x∗ , α), and J r (˜ x, α) ≥ J r (˜ x∗ , α),

(3.4)

˜ 0 , t1 ]. for all α ∈ [0, 1] and all x ˜ ∈ C[t The following theorem is the fundamental theorem of the calculus of variations in fuzzy environment. ˜ 0 , t1 ] be two fuzzy functions of t ∈ [t0 , t1 ], and J(˜ ˜ x) differentiable fuzzy Theorem 3.4 Let x ˜, δ x ˜ ∈ C[t ∗ ˜ ˜ functional of x ˜. If x ˜ is a fuzzy minimizer of J, then the variation of J regardless of any boundary conditions must vanish on x ˜∗ , that is, ˜ x∗ , δ x δ J(˜ ˜) = 0, (3.5)

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˜ 0 , t1 ]. for all admissible δ x ˜ having the property x ˜ + δx ˜ ∈ C[t It is obviously that the equality (3.5) holds if and only if δJ l (˜ x∗ (t)[α], δ x ˜(t)[α], t, α) = 0,

(3.6)

δJ r (˜ x∗ (t)[α], δ x ˜(t)[α], t, α) = 0,

(3.7)

for all α ∈ [0, 1], t ∈ [t0 , t1 ] and all admissible δ x ˜ where, δx ˜(t)[α] = [δxl (t, α), δxr (t, α)]. Proof. See [10] 4. Fuzzy fractional optimal control problem In this section, we first define fuzzy fractional optimal control problem with fixed and free final state conditions, and then we derive necessary conditions for optimality by applying fuzzy variational approaches to our problem. We define fuzzy fractional optimal control problem as: ∫t1 f˜(˜ x(t), u ˜(t), t)dt,

˜ x(t1 ), t1 ) + ˜ u) = ϕ(˜ min J(˜ u ˜

t0

subject to:

(4.1)

β ˜ t0 Dt x

= g˜(˜ x(t), u ˜(t), t) x ˜(t0 ) = x ˜0 .

For fixed final state problem we have additional condition x ˜(t1 ) = x ˜1 . Where f˜, g˜ : E 1 × E 1 × R → E 1 are assumed to be continuous first and second partial derivatives on t ∈ I = [t0 , t1 ] ⊆ R with respect to all their arguments and Riemann integrable, the fuzzy state x ˜(t) and the fuzzy control u ˜(t) are functions of t ∈ I, and the fuzzy state function x ˜(t) is Riemann-Liouville [(1) − β]−differentiable fuzzy-valued function and satisfies appropriate boundary conditions, and β ∈ (0, 1). Definition 4.1 We say that an admissible fuzzy curve (˜ x∗ , u ˜∗ ) is solution of (4.1), if for all admissible fuzzy curve (˜ x, u ˜) of (4.1), ˜ x∗ , u ˜ x, u J(˜ ˜∗ ) ≼ J(˜ ˜). Note that, we consider an admissible fuzzy control u ˜ is not bounded. Remark 4.1 If we choose β = 1, problem (4.1) is reduced to classical fuzzy optimal control problem. Definition 4.2(Fuzzy Hamiltonian Function). We define fuzzy Hamiltonian function as, ˜ ˜ g (˜ ˜ x(t), u H(˜ ˜(t), λ(t), t) = f˜(˜ x(t), u ˜(t), t) + λ(t)˜ x(t), u ˜(t), t).

(4.2)

˜ ˜ x(t), u H(˜ ˜(t), λ(t), t)[α] = [H l (xl , ul , λl , t, α), H r (xr , ur , λr , t, α)].

(4.3)

It means that, for any α ∈ [0, 1], and where H l (xl , ul , λl , t, α) and H r (xr , ur , λr , t, α) are classical Hamiltonian functions. ˜ ˜ Remark 4.2 In the following theorem, we assume that J l (˜ x(t), u ˜(t), λ(t), t) (or J r (˜ x(t), u ˜(t), λ(t), t)) l l l r r is stated in terms containing only x (t, α), u (t, α) and λ (t, α) (or only x (t, α), u (t, α) and λr (t, α)) in order to simplify the result presentations.

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4.1 Derivation of Necessary Conditions Now we are in the position to state a fundamental result of this work in the following theorem. Theorem 4.1(Necessary Conditions) Assume that x ˜∗ (t) be an admissible fuzzy state and u ˜∗ (t) be an ∗ admissible fuzzy control. Then the necessary conditions for u ˜ to be an optimal control for (4.1) and for all α ∈ [0, 1], t ∈ [t0 , t1 ] are: β ∗l t0 Dt x (t, α)

=

β ∗ t0 Dt x (t, α) = r

∂H l ∗l l l (x (t, α), u∗ (t, α), λ∗ (t, α), t, α), ∂λl

(4.4)

∂H r ∗r r r (x (t, α), u∗ (t, α), λ∗ (t, α), t, α), r ∂λ

(4.5)

∂H l ∗l l l (x (t, α), u∗ (t, α), λ∗ (t, α), t, α), l ∂x ∂H r ∗r r r C β ∗r (x (t, α), u∗ (t, α), λ∗ (t, α), t, α), t Dt1 λ (t, α) = ∂xr C β ∗l t Dt1 λ (t, α)

=

(4.6) (4.7)

∂H l ∗l l l (x (t, α), u∗ (t, α), λ∗ (t, α), t, α) = 0, ∂ul ∂H r ∗r r r (x (t, α), u∗ (t, α), λ∗ (t, α), t, α) = 0. r ∂u

(4.8) (4.9)

∂ϕl λ (t1 , α) = , ∂xl t=t1 ∂ϕr r λ (t1 , α) = . ∂xr t=t1

with

l

(4.10) (4.11)

for free final state problems. Proof. First we adopt fuzzy lagrange multiplier to form an augmented functional incorporating the constraints, then we modify the performance index as, ] ∫t1 [ ( ) ˜ d ϕ β ˜ g˜(˜ J˜a (˜ u) = f˜(˜ x(t), u ˜(t), t) + +λ x(t), u ˜(t), t) ⊖t0 Dt x ˜ dt, dt

(4.12)

t0

It means that, t ∫1 [ [ ] ( )] dϕl β l l l r r l l l l l l l  Ja (u , α), Ja (u , α) = + λ (t, α) g (x , u , t, α) − t0 Dt x f (x , u , t, α) + dt, dt t0  ∫t1 [ ( )] r dϕ + λr g r (xr , ur , t, α) − t0 Dtβ xr dt . f r (xr , ur , t, α) + dt t0

In the remaining of the proof we will ignore the similar arguments and only we consider the left hand of all functions of its α-level set. Jal (ul , α)

] ∫t1 [ dϕl β l l l l l l l l l dt. = f (x (t), u (t), t, α) + λ (t, α)g (x (t), u (t), t, α) − λ (t, α)t0 Dt x (t, α) + dt t0

(4.13)

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Using the definition of fuzzy Hamiltonian function, then we can rewrite equation (4.13) as, ∫t1 [ Jal (ul , α)

] dϕl β l l H (x (t), u (t), λ (t), t, α) + − λ (t, α)t0 Dt x (t, α) . dt l

=

l

l

l

(4.14)

t0

Taking variation of equation (4.14), we obtain ∫t1 δJal (ul , α)

=

∂H l l ∂H l l ∂H l l ∂ϕl l δx + δu + δλ + δx − δλl t0 Dtβ xl − λl δ t0 Dtβ xl , ∂xl ∂ul ∂λl ∂xl

(4.15)

t0

where δxl , δλl and δul are the variations of xl , λl and ul respectively. Using the formula for fractional integration by parts, integrate the last term on the RHS of (4.15), then we obtain ∫t1 ( δJal (ul , α)

= t0

( ) ) ) ( l ∂H l C β l ∂H l ∂H l l ∂ϕ β l l l l δxl (t1 ). −t Dt1 λ δx + δu + − t0 Dt x δλ dt + −λ l l l l ∂x ∂u ∂λ ∂x t=t1 (4.16)

C Dβ t t1

where represent the classical right Caputo fractional derivative. l ∗ u is an extremal if the variation of Jal is zero, that is, for all α ∈ [0, 1] we require ∫t1 ( t0

) ( ) ( l ) ∂H l C β l ∂H l l ∂H l ∂ϕ β l l l l −t Dt1 λ δx + δu + − t0 Dt x δλ dt + −λ δxl (t1 ) = 0. l l l l ∂x ∂u ∂λ ∂x t=t1

(4.17)

It is convenient to choose the coefficients of δxl , δul , and δλl in (4.17) to be zero. This leads to β ∗ t0 Dt x (t, α) =

∂H l ∗l l l (x (t, α), u∗ (t, α), λ∗ (t, α), t, α), l ∂λ

(4.18)

C β ∗l t Dt1 λ (t, α)

∂H l ∗l l l (x (t, α), u∗ (t, α), λ∗ (t, α), t, α), l ∂x

(4.19)

l

=

∂H l ∗l l l (x (t, α), u∗ (t, α), λ∗ (t, α), t, α) = 0, l ∂u Finally, we have

(

) ∂ϕl l δxl (t1 ) = 0, −λ l ∂x t=t1

(4.20)

(4.21)

1. For the fixed final state problem δxl (t1 ) = 0, 2. For the free final state problem

(

(4.22)

) ∂ϕl l − λ = 0. ∂xl t=t1

(4.23)

Equations (4.18)−(4.20) represents the necessary conditions for u∗ to be an optimal with the condition (4.22) for the fixed final state problem and (4.23) for the free final state problem. r By following the same steps(using the right hand of all functions of its α-level set ) for δJar (u∗ , α) = 0, for all α ∈ [0, 1] and t ∈ [0, 1], we will obtain l

β ∗l t0 Dt x (t, α)

=

∂H r ∗r r r (x (t, α), u∗ (t, α), λ∗ (t, α), t, α), r ∂λ 420

(4.24)

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C β ∗r t Dt1 λ (t, α)

=

∂H l ∗r r l (x (t, α), u∗ (t, α), λ∗ (t, α), t, α), ∂xr

(4.25)

∂H r ∗r r r (x (t, α), u∗ (t, α), λ∗ (t, α), t, α) = 0. r ∂u

(4.26)

Equations (4.24) − (4.26) represents the necessary conditions u∗ ) to be an extremal with the con( for r l ditions δxr (t1 ) = 0 for the fixed final state problem and ∂ϕ = 0 for the free final state ∂xr − λ r

t=t1

problem. The above equations form a set of necessary conditions that the left and right hand functions of its α−level set of the fuzzy optimal control u ˜∗ and fuzzy optimal state x ˜∗ must satisfy. [ l ] 2 r ∗ ∗ ∗ ∗ ∗ We know that, u ˜ (t) and x ˜ (t) are a fuzzy numbers with u ˜ (t)[α] = u (t, α), u (t, α) and [ l ] r l r l r x ˜∗ (t)[α] = x∗ (t, α), x∗ (t, α) if u∗ (t, α), u∗ (t, α), x∗ (t, α) and x∗ (t, α) satisfy are related properties in C1-C5 of Lemma(2.1). In the following definition, based on the conditions C1 and C2 of Lemma(2.1), we introduce the definition of strong and weak solutions of our problem. Definition 4.3(Strong and Weak Solutions). ∗

∗

1. (Strong Solution). We say that u ˜∗ (t)[α] and x ˜∗ (t)[α] are strong solutions of (4.1) if ul (t, α), ur (t, α) ∗ ∗ l r ,x (t, α) and x (t, α) obtained from (4.4) − (4.11) satisfy the conditions C1-C2 of Lemma(2.1), for all t ∈ [t0 , t1 ] and α ∈ [0, 1]. ∗

∗

2. (Weak Solution). We say that u ˜∗ (t)[α] and x ˜∗ (t)[α] are weak solutions of (4.1) if ul (t, α), ur (t, α) ∗ ∗ l r ,x (t, α) and x (t, α) obtained from (4.4) − (4.11) do not satisfy the conditions C1-C2 of Lemma(2.1), then we define u ˜∗ (t)[α] and x ˜∗ (t)[α] as: u ˜∗ (t)[α] =  r∗ l∗ r∗ l∗ r∗  [2u (t, 1) − u (t, α), u (t, α)], if u , u are decreasing functions of α, ∗ ∗ ∗ ∗ ∗ [ul (t, α), 2ul (t, 1) − ur (t, α)], if ul , ur are increasing functions of α,  [ur∗ (t, α), ul∗ (t, α)], if ul∗ is decreasing and ur∗ is increasing of α and, x ˜∗ (t)[α] =  r∗ l∗ r∗ l∗ r∗  [2x (t, 1) − x (t, α), x (t, α)], if x , x are decreasing functions of α, ∗ ∗ ∗ ∗ ∗ [xl (t, α), 2xl (t, 1) − xr (t, α)], if xl , xr are increasing functions of α,  [xr∗ (t, α), xl∗ (t, α)], if xl∗ is decreasing and xr∗ is increasing of α for all t ∈ [t0 , t1 ] and α ∈ [0, 1]. Now, we consider fixed and free final state problems with a quadratic performance index. 4.2 Fixed Final State Problem We can define fuzzy fractional optimal control problem with fixed final state as ˜ u) = 1 min J(˜ u ˜ 2

∫t1

[

] q(t)˜ x2 + r(t)˜ u2 dt,

t0

subject to:

(4.27)

β ˜ 0 Dt x

= a(t)˜ x + b(t)˜ u, x ˜(t0 ) = x ˜0 , x ˜(t1 ) = x ˜1 .

where q(t) ≥ 0 and r(t) > 0.

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Theorem(4.1), give the necessary conditions for u∗ to be an optimal as l

β l t0 Dt x

= a(t)xl + b(t)ul ,

(4.28)

C β l t Dt1 λ

= q(t)xl + a(t)λl ,

(4.29)

r(t)ul + b(t)λl = 0.

(4.30)

Equations (4.28) and (4.30) gives β l t0 Dt x

= a(t)xl − r−1 (t)b2 (t)λl .

(4.31)

We will obtain xl (t, α) and ul (t, α) by solving Equations (4.29) − (4.31) with the boundary conditions xl (t0 ) = xl0 and xl (t1 ) = xl1 . r Similarly Theorem(4.1), give the necessary conditions for u∗ to be an optimal as β r t0 Dt x

= a(t)xr + b(t)ur ,

(4.32)

C β r t Dt1 λ

= q(t)xr + a(t)λr ,

(4.33)

r(t)ur + b(t)λr = 0.

(4.34)

Equations (4.32) and (4.34) gives β r t0 Dt x

= a(t)xr − r−1 (t)b2 (t)λr .

(4.35)

We will obtain xr (t, α) and ur (t, α) by solving Equations (4.33) − (4.35) with the boundary conditions xr (t0 ) = xr0 and xr (t1 ) = xr1 . 4.3 Free Final State Problem We can define fuzzy fractional optimal control problem with free final state as ˜ x(t1 ), t1 ) + 1 ˜ u) = ϕ(˜ min J(˜ u ˜ 2

∫t1 t0

subject to:

[ ] q(t)˜ x2 + r(t)˜ u2 dt, (4.36)

β ˜ t0 Dt x

= a(t)˜ x + b(t)˜ u, x ˜(t0 ) = x ˜0 .

where q(t) ≥ 0 and r(t) > 0. Following the same steps, we will obtain xl (t, α) and ul (t, α) by solving Equations (4.29) − (4.31) with respect to the conditions ( l ) ∂ϕ l l l . (4.37) x (t0 ) = x0 and λ (t1 , α) = ∂xl t=t1 Also we will obtain xr (t, α) and ur (t, α) by solving Equations (4.33) − (4.35) with respect to the conditions ( r ) ∂ϕ r r r . (4.38) x (t0 ) = x0 and λ (t1 , α) = ∂xr t=t1 In the next section we propose an algorithm used to find the solution of both cases numerically, the details of this algorithm in [4, 5].

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5. Numerical technique Considering the both cases of fixed and free final state problems defined above, in order to find the solution of our problems, we use the Gr¨ unwald-Letnikov(GL-for short) approximation of the left Riemann-Liouville fractional derivative and using the relation between right Riemann-Liouville fractional derivative and right Caputo fractional derivative and then use GL-approximation, we can approximate (4.31) and (4.29) as m ∑ (β) h−β wj xlm−j = a(mh)xlm − r−1 (mh)b2 (mh)λlm , (5.1) j=0

for m = 1, 2, ..., N , and m ∑

h−β wj λlm+j = q(mh)xlm + a(mh)λlm + (β)

j=0

λlN (t1 − mh)−β , γ(1 − β)

(5.2)

for m = N − 1, N − 2, ..., 0, respectively. Where N is the number of equal divisions of the interval [0, t1 ], t1 the nodes are labeled as 0, 1, ..., N . The size of each division is given as h = N , and tj = jh represent the time at node j. The coefficients are defined as ( ) β j β wj = (−1) . (5.3) j Where xli and λli represent the numerical approximations of xl (t, α) and λl (t, α) at node i. Similarly, we can approximate (4.35) and (4.33) as m ∑

h−β wj xrm−j = a(mh)xrm − r−1 (mh)b2 (mh)λrm , (β)

(5.4)

j=0

for m = 1, 2, ..., N , and m ∑

h−β wj λrm+j = q(mh)xrm + a(mh)λlm + (β)

j=0

λrN (t1 − mh)−β , γ(1 − β)

(5.5)

for m = N − 1, N − 2, ..., 0, respectively. Also xri and λri represent the numerical approximations of xr (t, α) and λr (t, α) at node i. In general, Equations (5.1) and (5.2) or Equations (5.4) and (5.5) give a set of 2N equations in terms of 2N variables, i.e., Ax = b, it means that, we can use any linear equation solver to find the solution. ˜ the vector x is constructed as Regardless the left and right bounds of the fuzzy numbers x ˜ and λ, follows • For fixed final state problem x = [x1 x2 ... xN −1 λ0 λ1 ... λN ]T . • For free final state problem x = [x1 x2 ... xN λ0 λ1 ... λN −1 ]T . In the next section, we will give four examples can serve to illustrate our main results.

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6. Numerical examples Example 6.1 Find the fuzzy control that minimize ˜ u(t)) = 1 J(˜ 2

∫1

[ 2 ] x ˜ +u ˜2 dt

0

subject to: β ˜ 0 Dt x

= t˜ x+u ˜, x ˜(0) = (0, 1, 2),

x ˜(1) = (−2, −1, 1).

Solution.We have, q(t) = r(t) = b(t) = t1 = 1, and a(t) = t, Then for the left bound of state and control Theorem(4.1) gives, β l 0 Dt x

= txl − λl ,

(6.1)

C β l t D1 λ

= xl + tλl ,

(6.2)

ul + λl = 0.

(6.3)

and the boundary conditions xl (0, α) = α, xl (1, α) = −2 + α. For the right bound of state and control, Theorem(4.1) gives, β r 0 Dt x

= txr − λr ,

(6.4)

C β r t D1 λ

= xr + tλr ,

(6.5)

ur + λr = 0.

(6.6)

and the boundary conditions xr (0, α) = 2 − α, xr (1, α) = 1 − 2α. Now, we use the numerical method to solve the above equations with the related boundary conditions, then we obtain the following results. ∗ Figure(1(a)) show that the state x ˜∗ (t) as a function of α, we observe that xl (t, α) is an increasing ∗ ∗ ∗ ∗ ∗ function of α, xr (t, α) is a decreasing function of α and xl (t, 1) = xr (t, 1), thus, xl (t, α) and xr (t, α) satisfy the conditions of Lemma(2.1). ∗ Figure(1(b)) show that the control u ˜∗ (t) as a function of α, we find that ul (t, α) is an increasing ∗ ∗ ∗ ∗ function of α, ur (t, α) is a decreasing function of α and xl (t, 1) = xr (t, 1), it means that ul (t, α) and ∗ ur (t, α) satisfy the conditions of Lemma(2.1), furthermore, x ˜∗ (t) and u ˜∗ (t) represent a strong fuzzy solution of this problem. Example 6.2 Find the fuzzy control that minimize ˜ u(t)) = 1 J(˜ 2

∫2 u ˜2 dt 1

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subject to: β ˜ 0 Dt x

= (2t − 1)˜ x ⊖ sin(t)˜ u, x ˜(1) = (0, 1, 2), x ˜(2) = (−2, −1, 1).

Solution.We have, q(t) = 0, r(t) = t0 = 1, b(t) = − sin(t), and a(t) = (2t − 1), then for the left bound of the state and control, Theorem(4.1) gives, β l 1 Dt x

= (2t − 1)xl − sin2 (t)λl ,

C β l t D2 λ

= (2t − 1)λl ,

ul − sin(t)λl = 0.

(6.7) (6.8) (6.9)

and the boundary conditions xl (0, α) = α, xl (1, α) = −2 + α. For the right bound of state and control Theorem(4.1) gives, β r 1 Dt x

= (2t − 1)xr − sin2 (t)λr ,

C β r t D2 λ

= (2t − 1)λr ,

u − sin(t)λ = 0. r

r

(6.10) (6.11) (6.12)

and the boundary conditions xr (0, α) = 2 − α, xr (1, α) = 1 − 2α. Now, we use the numerical method to solve the above equations with the related boundary conditions, then we obtain the following results. ∗ Figure(2(a)) show that the state x ˜∗ (t) as a function of α, we observe that xl (t, α) is an increasing ∗ ∗ ∗ ∗ ∗ function of α, xr (t, α) is a decreasing function of α and xl (t, 1) = xr (t, 1), thus, xl (t, α) and xr (t, α) satisfy the conditions of Lemma(2.1). ∗ Figure(2(b)) show that the control u ˜∗ (t) as a function of α, we find that ul (t, α) is a decreasing ∗ ∗ ∗ ∗ function of α, ur (t, α) is an increasing function of α and xl (t, 1) = xr (t, 1), it means that ul (t, α) ∗ and ur (t, α) do not satisfy the conditions C1-C2 of Lemma(2.1), then we use the definition(4.3) of weak solution, we find that [ ∗ ] ∗ u ˜∗ (t)[α] = ur (t, α), ul (t, α) . Furthermore, x ˜∗ (t) and u ˜∗ (t) represent a weak fuzzy solution of this problem. Example 6.3 Find the fuzzy control that minimize ˜ u(t)) = 1 J(˜ 2

∫1

[ 2 ] x ˜ +u ˜2 dt

0

subject to: β ˜ 0 Dt x

= −(0, 1, 3)˜ x+u ˜, x ˜(0) = (1, 1, 1), x ˜(1) = (0, 0, 0).

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Solution.We know that, [

] [ ] β l β r l l r r D x , D x = −(3 − 2α)x + u , −αx + u , 0 t 0 t

then we have, q(t) = r(t) = b(t) = x0 = t1 = 1, a(t) = −(3 − 2α) and a(t) = −α for the left and right derivatives respectively, then for the left bound of the state and control Theorem(4.1) gives, β l 0 Dt x

= −(3 − 2α)xl − λl ,

C β l t D1 λ

(6.13)

= xl − (3 − 2α)λl ,

(6.14)

ul + λl = 0.

(6.15)

and the boundary conditions xl (0, α) = 1, xl (1, α) = 0. For the right bound of the state and control Theorem(4.1) gives, β r 1 Dt x

= −αxr − λr ,

C β r t D2 λ r

= xr − αλr ,

u + λr = 0.

(6.16) (6.17) (6.18)

and the boundary conditions xr (0, α) = 1, xr (1, α) = 0. Now, we use the numerical method to solve the above equations with the related boundary conditions, then we obtain the following results. ∗ Figure(3(a)) show that the state x ˜∗ (t) as a function of α, we observe that xl (t, α) is an increasing ∗ ∗ ∗ ∗ ∗ function of α, xr (t, α) is a decreasing function of α and xl (t, 1) = xr (t, 1), thus, xl (t, α) and xr (t, α) satisfy the conditions of Lemma(2.1). ∗ Figure(3(b)) show that the control u ˜∗ (t) as a function of α, we find that ul (t, α) is a decreasing ∗ ∗ ∗ ∗ function of α, ur (t, α) is an increasing function of α and xl (t, 1) = xr (t, 1), it means that ul (t, α) ∗ and ur (t, α) do not satisfy the conditions C1-C2 of Lemma(2.1), then we use the definition(4.3) of weak solution, we find that [ ∗ ] ∗ u ˜∗ (t)[α] = ur (t, α), ul (t, α) . Furthermore, x ˜∗ (t) and u ˜∗ (t) represent a weak fuzzy solution of this problem. Example 6.4 Find the fuzzy control that minimize 1 ˜ u(t)) = 1 x ˜2 (1) + J(˜ 2 2

∫1

[

] x ˜2 + u ˜2 dt

0

subject to: β ˜ 0 Dt x

= −(0, 1, 3)˜ x+u ˜, x ˜(0) = (1, 1, 1).

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Altyeb Mohammed, Zeng-Tai Gong and Mawia Osman: A Numerical Technique for Solving Fuzzy Fractional ...

Solution.We have, q(t) = r(t) = b(t) = x0 = t1 = 1, a(t) = −(3 − 2α) and a(t) = −α for the left and right derivatives respectively, then Theorem(4.1) gives, β l t0 Dt x

= −(3 − 2α)xl − λl ,

C β l t Dt1 λ

(6.19)

= xl − (3 − 2α)λl ,

(6.20)

ul + λl = 0.

(6.21)

and the boundary conditions xl (0, α) = 1, λl (0, α) = xl (1, α). For the right bound of the state and control Theorem(4.1) gives, β r 1 Dt x

= −αxr − λr ,

C β r t D2 λ

= xr − αλr ,

ur + λr = 0.

(6.22) (6.23) (6.24)

and the boundary conditions xr (0, α) = 1, λr (0, α) = xr (1, α). Now, we use the numerical method to solve the above equations with the related boundary conditions, then we obtain the following results. ∗ Figure(4(a)) show that the state x ˜∗ (t) as a function of α, we observe that xl (t, α) is an increasing ∗ ∗ ∗ ∗ ∗ function of α, xr (t, α) is a decreasing function of α and xl (t, 1) = xr (t, 1), thus, xl (t, α) and xr (t, α) satisfy the conditions of Lemma(2.1). ∗ Figure(4(b)) show that the control u ˜∗ (t) as a function of α, we find that ul (t, α) is a decreasing ∗ ∗ ∗ ∗ function of α, ur (t, α) is an increasing function of α and xl (t, 1) = xr (t, 1), it means that ul (t, α) ∗ and ur (t, α) do not satisfy the conditions C1-C2 of Lemma(2.1), then we use the definition(4.3) of weak solution, we find that [ ∗ ] ∗ u ˜∗ (t)[α] = ur (t, α), ul (t, α) . Furthermore, x ˜∗ (t) and u ˜∗ (t) represent a weak fuzzy solution of this problem. 7. Conclusion In this paper, the necessary conditions of fuzzy fractional optimal control problem with both fixed and free final state conditions at the final time has been derived using fuzzy variational approach. Our problems is defined in the sense of Riemann-Liouville fractional derivative based on Hukuhara difference. A numerical technique is proposed based on Gr¨ unwald-Letnikov definition of fractional derivative. The concepts of strong and weak solutions of our problems are given. lastly, four examples are provided to show the effectiveness of Theorem(4.1) and the numerical algorithm.

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(a)

(b)

Figure 1: Example(6.1) (a) the state at t = 0.1, β = 0.77 (b) the control at t = 0.1, β = 0.77.

(a)

(b)

Figure 2: Example(6.2) (a) the state at t = 0.1, β = 0.77 (b) the control at t = 0.1, β = 0.77.

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Altyeb Mohammed, Zeng-Tai Gong and Mawia Osman: A Numerical Technique for Solving Fuzzy Fractional ...

(a)

(b)

Figure 3: Example(6.3) (a) the state at t = 0.1, β = 0.77 (b) the control at t = 0.1, β = 0.77.

(a)

(b)

Figure 4: Example(6.4) (a) the state at t = 0.1, β = 0.77 (b) the control t = 0.1, β = 0.77.

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References [1] O.S. Fard, J. Soolaki, and D.F.M. Torres, A necessary condition of pontryagin type for fuzzy fractional optimal control problems. Discrete Contin. Dyn. Syst. Ser. S, in press. [2] O.S. Fard and M. Salehi, A survey on fuzzy fractional variational problems, J. Comput. Appl. Math. 271 (2014)71-82. [3] M.J. Karimyar, A.J. Fakharzadeh, Fractional Fuzzy Optimal Control Problem Governed by Fuzzy System, International Journal of Research and Allied Sciences, (SI) (2018)151-158. [4] R.K. Biswas, S. Sen, Fractional optimal control problems with specified final time, J. Comput. Nonlinear Dyn. 6 (2011)1-6. [5] T.M. Atanackovi´c, S. Konjik, and S. Pilipovi´c, Variational problems with fractional derivatives: Euler Lagrange equations, J. Phys. A:Math. Theor., 41 (2008)095201. [6] S. Abbasbandy, T. Allahviranloo, M. R.B. Shahryari, and S. Salahshour, Fuzzy local fractional differential equations, International Journal of Industrial Mathematics, 4(3) (2012)231-246. [7] J.J. Buckley, T. Feuring, Introduction to fuzzy partial differential equations, Fuzzy Sets and Systems. 105 (1999)241-248 [8] S.S.L. Chang, L.A. Zadeh, On fuzzy mappings and control, IEEE Trans. on SMC. 1 (1972)30-34. [9] D. Dubois, H. Prade, Operation on fuzzy numbers, Internat. J. Systems Sci. 9 (1978)613-626. [10] B. Farhadinia, Pontryagin’s Minimum Principle for Fuzzy Optimal Control Problems, Iranian. J. fuzzy system. 11 (2014)27-43. [11] B. Farhadinia, Necessary optimality conditions for fuzzy variational problems, Information Science. 181 (2011)1348-1357. [12] R. Goetschel, W. Voxman, Elementary fuzzy calculus, Fuzzy Sets and Systems. 18 (1986)31-43. [13] M.I. Kamien, N.L Schwarz, Dynamic optimization: the calculus of variations and optimal control, North Holland Amsterdam(1991) [14] S. Lenhart, J.T. Workman, Optimal control appLied to biological models, Chapman and Hall/CRC(2007). [15] C.C. Remsing, Lecture notes in linear control, Phodes University South Africa(2006). [16] J. Soolaki, O.S. Fard, A necessary condition of Pontryagin type for fuzzy control systems, Computational and Applied Mathematics. 37 (2018)1263-1278. [17] L.A. Zadeh, Fuzzy sets, Informations and Control. 8 (1965)338-353.

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Differential Transform Method for Solving Fuzzy Fractional Wave Equation† Mawia Osman 1 , Zeng-Tai Gong1,∗ , Altyeb Mohammed 1,2 1 College of Mathematics and Statistics, Northwest Normal University, Lanzhou 730070, China 2 Faculty of Mathematical Science, University of Khartoum, Khartoum, Sudan

Abstract: In this letter, the differential transform method (DTM) is applied to solve fuzzy fractional wave equation. The elemental properties of this method are investigated based on the two-dimensional differential transform method (DTM), generalized Taylor’s formula and fuzzy Coputo’s derivative. The proposed method is also illustrated by using some examples. The results reveal that DTM is a highly effective scheme for obtaining analytical solutions of the fuzzy fractional wave equation. Mathematics Subject Classification. 65L05, 26E50 Keyword: Fuzzy numbers; Fuzzy fractional wave equation; Differential transform method; Fuzzy Caputo’s derivative; Generalized Taylor formula.

1

Introduction

In 1965, the fuzzy sets were introduced for the first time by Zadeh in [28]. hundreds of examples have been supplied where the nature of uncertainty in the behavior of given system processes are fuzzy rather than stochastic nature. In the last few years, many authors have interested in the study of the theoretical framework of fuzzy initial value problems. Chang and Zadeh in [6] have introduced the concept of fuzzy derivative. Kandel and Byatt in [12] have initially presented the concept of the fuzzy differential equation. Bede and Gal in [4] have studied the concept of strongly generalized differentiable of fuzzy valued functions, which enlarged the class of differentiable fuzzy valued functions. In 1695, the fractional calculus was first studied. The subject of fractional calculus has gained importance during the past three decades due mainly to its demonstrated applications in different area of physics and engineering in [16]. Fuzzy fractional differential equations (FFDE) play an important role in modelling of science and engineering problems. Padmapriya and Kaliyappan in [22] established analytical and numerical methods to solve fuzzy fractional differential equations. the concept of differential of fuzzy function with two variables and fuzzy wave equations studied in [26]. In the last years many authors have developed and introduced some variant methods for solving fuzzy wave equation. Kermani in [15] used finite difference method to solve the fuzzy wave equation numerically. Also, Martin and Radek in [25] used f-transforms to solve the fuzzy wave equation. Zhou in [29] has presented the concept of the differential transform method (DTM), this method constructs an analytical solution inform of a polynomial, which is different from the tradition higher order Taylor formula method. Recently some researchers used differential transform method (DTM) to solve fuzzy fractional differential equations and fuzzy differential equations in [9, 23, 1, 19, 20]. This paper is structured as follows. In Section 2, we call some definitions on fuzzy numbers, fuzzy functions and fuzzy Caputo’s derivative. In Section 3, The generalization of Taylor’s formula is presented. In Section 4, the generalized two-dimensional differential transform method (DTM) for † ∗

This work is supported by National Natural Science Foundation of China (61763044). Corresponding Author: Zeng-Tai Gong. Tel.: +869317971430. E-mail addresses: email: [email protected]

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the solution of the fuzzy wave equation with space and time-fractional derivatives are developed and derived. Examples are shown in Section 5. Finely, conclusion is given in section 6.

2

Basic concepts

The results about fuzzy numbers space E 1 , we recall that E 1 = {˜ u : R → [0, 1] : u fies (1)(4) below } (refer to [6])

satis-

1. u ˜ is normal, i.e., there exists x0 ∈ R such that u ˜(x0 ) = 1; 2. u ˜ is convex, i.e., for all and λ ∈ [0, 1], x, y ∈ R, u ˜(λx + (1 − λ)y) ≥ min{˜ u(x), u ˜(y)}, holds; 3. u ˜ is upper semicontinuous, i.e., for any x0 ∈ R, u ˜(x0 ) ≥ lim u e(x); x−→x± 0

4. supp u ˜ = {x ∈ R|˜ u(x) > 0} is the support of u ˜, and its closure cl (supp u ˜) is compact. For 0 < r ≤ 1, denote [˜ u]r = {x : u ˜(x) ≥ r}. Then from (1)-(4), follows that the r-level set [˜ u]r is a closed and bounded interval for all r ∈ [0, 1]. For u ˜, v˜ ∈ E 1 , k ∈ R, the addition and scalar multiplication are defined using the equations [˜ u + v˜]r = [˜ u]r + [˜ v ]r , [k˜ u]r = k[˜ u]r , respectively. Define D : E 1 × E 1 → R+ ∪ {0} using the equation D(˜ u, v˜) = sup d([˜ u]r [˜ v ]r ), r∈[0,1]

where d is Hausdorff metric space as d([˜ u]r , [˜ v ]r ) = inf{ε : [˜ u]r ⊂ N ([˜ v ]r , ε), [˜ v ]r ⊂ N ([˜ u]r , ε)} = max{|ur − v r |, |ur − v r |}, where N ([˜ u]r , ε), N ([˜ v ]r , ε) is the ε-neighborhood of [˜ u]r , [˜ v ]r , respectively, and ur , v r , ur , v r are endpoints of [˜ u]r , [˜ v ]r , respectively. By using the results of [13], we see that • (E 1 , D) is complete metric space, • D(˜ u + w, ˜ v˜ + w) ˜ = D(˜ u, v˜) for all u ˜, v˜, w ˜ ∈ E1, • D(k˜ u, k˜ v ) = |k|D(˜ u, v˜). In addition, we can introduce a partial order in E 1 by u ˜ ≤ v˜ if and only if [˜ u]r ≤ [˜ v ]r , r ∈ [0, 1] if and only if ur ≤ v r , ur ≤ v r , r ∈ [0, 1]. For applications of the partial order on E 1 (refer to [27]). As the fuzzy number is resolved by using the interval u ˜r = [ur , ur ], see [8] defined another statements, parametrically, of fuzzy numbers as in following. Definition 2.1.[31, 32] For arbitrary fuzzy numbers u ˜, v˜ ∈ E 1 , u ˜ = [ur , ur ], v˜ = [v r , v r ], the quantity D(˜ u, v˜) = supr∈[0,1] max{|ur − v r |, |ur − v r |} is the distance between u ˜ and v and also the following properties hold:

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• (E 1 , D) is a complete metric space, • D(˜ u ⊕ w, ˜ v˜ ⊕ w) ˜ = D(˜ u, v˜), ∀˜ u, v˜, w ˜ ∈ E1, • D(˜ u ⊕ v˜, w ˜ ⊕ e˜) ≤ D(˜ u, w) ˜ + D(˜ v , e˜), ∀˜ u, v˜, w, ˜ e˜ ∈ E 1 , • D(˜ u ⊕ v˜, ˜0) ≤ D(˜ u, ˜0) + D(˜ v , ˜0), ∀˜ u, v˜ ∈ E 1 , • D(k ⊙ u ˜, k ⊙ v˜) = |k|D(˜ u, v˜), ∀˜ u, v˜ ∈ E 1 , k ∈ R, • D(k1 ⊙ u ˜ , k2 ⊙ u ˜) = |k1 − k2 |D(˜ u, ˜0), ∀˜ u ∈ E 1 , k1 , k2 ∈ R, with k1 · k2 ≥ 0. Let us recall the definition of the Hukuhara difference (H-difference) in [33]. Suppose that u ˜, v˜ ∈ E 1 . The Hukuhara H-difference has been presented as a set w ˜ for which u ˜ ⊖gH v˜ = w ˜⇔u ˜ = v˜ ⊕ w. ˜ The H-difference is unique, but it does not always exist (a necessary condition for u ˜ ⊖gH v˜ to exist is that u ˜ contains a translate {c} ⊕ v˜ of v˜). A generalization of the Hukuhara difference aims to overcome this situation. Definition 2.2.[33, 31] The generalized Hukuhara difference between two fuzzy numbers u ˜, v˜ ∈ E 1 is defined as following: { (i) u ˜ = v˜ ⊕ w, ˜ u ˜ ⊖gH v˜ = w ˜⇔ (2.1) or (ii) v˜ = u ˜ ⊕ (−w). ˜ In terms of the r−levels, we get [˜ u ⊖gH v˜] = [min{ur − v r , ur − v r }, max{ur − v r , ur − v r }] and if ˜ ⊖ v˜ = u ˜ ⊖gH v˜; the conditions for existence of w ˜=u ˜ ⊖gH v˜ ∈ E 1 are the H-difference exists, then u { Case (i) { Case (ii)

wr = ur − v r and wr = ur − v r , ∀r ∈ [0, 1], with wr increasing, wr decreasing, wr ≤ wr .

(2.2)

wr = ur − v r and wr = ur − v r , ∀r ∈ [0, 1], with wr increasing, wr decreasing, wr ≤ wr .

(2.3)

It is easy to show that (i) and (ii) are both valid if and only if w ˜ is a crisp number. In the case, it is possible that the gH-difference of two fuzzy numbers does not exist. To address this shortcoming, a new difference between fuzzy numbers was introduced in [33]. Lemma 2.1.[10, 24] A fuzzy number u ˜ in parametric form is a pair [ur , ur ] of function ur and ur for any r ∈ [0, 1], which satisfies the following requirements. • ur is a bounded non-decreasing left continuous function in (0,1]; • ur is a bounded non-increasing left continuous function in (0,1]; • ur ≤ ur . Some the author of the classified fuzzy numbers into several types of fuzzy membership function. To the deepest of our study, triangular fuzzy membership function or also often referred to as triangular fuzzy numbers are the most widely used membership function. In order to describe the fuzzy numbers and real numbers clearly, in convenience, the fuzzy numbers and fuzzy-valued functions in the whole paper are added with a tilde sign at the top, while the real-value function and interval-value functions are written directly. A fuzzy valued function f˜ of two variables is a rule that assigns to each ordered pair of real numbers, (x, t), in a set D, a unique fuzzy numbers denoted by f˜(x, t). The set D is the domain of f˜ and its range is the set of values taken by f , i.e., {f˜(x, t)|(x, t) ∈ D}. The parametric representation of the fuzzy valued function f : D → E 1 is expressed by f (x, t)(r) = [f (x, t)(r), f (x, t)(r)], for all (x, t) ∈ D and r ∈ [0, 1]. Suppose f : D → E 1 be a fuzzy valued function of two variable. Then, we say that the fuzzy limit of f (x, t) as (x, t) approaches to (a, b) is L ∈ E 1 , and we write lim(x,t)→(a,b) f (x, t) = L if for every

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number ε > 0, there is a corresponding number δ > 0 such that if (x, t) ∈ D, ∥ (x, t) − (a, b) ∥< δ ⇒ D(f (x, t), L) < ε, where ∥ · ∥ denotes the Euclidean norm in Rn (ref. to [3]) A fuzzy valued function f : D → E 1 is said to be fuzzy continuous at (x0 , t0 ) ∈ D if lim(x,t)→(x0 ,t0 ) f (x, t) = f (x0 , t0 ). We say that f is fuzzy continuous on D if f is fuzzy continuous at every point (x0 , t0 ) in D (ref. to [3, 30]). Definition 2.3.[11] Suppose that u ˜(x, t) : D → E 1 and (x0 , t) ∈ D. We say that u ˜ is strongly generalized differentiable on (x0 , t) if there exists an element ∂∂xu˜ |(x0 ,t) ∈ E 1 such that i. for all h > 0 sufficiently small, ∃˜ u(x0 + h, t) ⊖gH u ˜(x0 , t), u ˜(x0 , t) ⊖gH u ˜(x0 − h, t) and the limits (in the metric D) (x0 , t) ⊖gH u ˜(x0 − h, t) u ˜(x0 + h, t) ⊖gH u ˜(x0 , t) ∂u ˜ = lim = = | , h→0+ h→0+ h h ∂x (x0 ,t) lim

or ii. for all h > 0 sufficiently small, ∃gH u ˜(x0 , t) ⊖gH u ˜(x0 + h, t), u ˜(x0 − h, t) ⊖gH u ˜(x0 , t) and the limits u ˜(x0 , t) ⊖gH u ˜(x0 + h, t) u ˜(x0 − h, t) ⊖gH u ˜(x0 , t) ∂u ˜ = lim = | , h→0+ h→0+ −h −h ∂x (x0 ,t) lim

or iii. for all h > 0 sufficiently small, ∃˜ u(x0 + h, t) ⊖gH u ˜(x0 , t), u ˜(x0 − h, t) ⊖gH u ˜(x0 , t) and the limits u ˜(x0 + h, t) ⊖gH u ˜(x0 , t) u ˜(x0 − h, t) ⊖gH u ˜(x0 , t) ∂u ˜ = lim = | , h→0+ h→0+ h −h ∂x (x0 ,t) lim

or iv. for all h > 0 sufficiently small, ∃˜ u(x0 , t) ⊖gH u ˜(x0 + h, t), u ˜(x0 , t) ⊖gH u ˜(x0 − h, t) and the limits lim

h→0+

u ˜(x0 , t) ⊖gH u ˜(x0 + h, t) u ˜(x0 , t) ⊖gH u ˜(x0 − h, t) ∂u ˜ = lim = | . h→0+ −h h ∂x (x0 ,t)

Definition 2.4.[4] Suppose that u ˜(x, t) : D → E 1 and (x0 , t) ∈ D. We define the n th-order derivative of u ˜ as follows: we say that u ˜ is strongly generalized differentiable of the n th-order at (x0 , t) s if there exists an element ∂∂ xu˜s |(x0 ,t) ∈ E 1 , ∀s = 1, 2, · · ·, n such that i. for all h > 0 sufficiently small, ∃˜ u(s−1) (x0 +h, t)⊖gH u ˜(s−1) (x0 , t), u ˜(s−1) (x0 , t)⊖gH u ˜(s−1) (x0 −h, t) and the limits (in the metric D) u ˜(s−1) (x0 , t) ⊖gH u ˜(s−1) (x0 − h, t) u ˜(s−1) (x0 + h, t) ⊖gH u ˜(s−1) (x0 , t) ∂su ˜ = lim = | , h→0+ h→0+ h h ∂ xs (x0 ,t) lim

or ii. for all h > 0 sufficiently small, ∃˜ u(s−1) (x0 , t)⊖gH u ˜(s−1) (x0 +h, t), u ˜(s−1) (x0 −h, t)⊖gH u ˜(s−1) (x0 , t) and the limits u ˜(s−1) (x0 , t) ⊖gH u ˜(s−1) (x0 + h, t) u ˜(s−1) (x0 − h, t) ⊖gH u ˜(s−1) (x0 , t) ∂su ˜ = lim = | , h→0+ h→0+ −h −h ∂ xs (x0 ,t) lim

or

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iii. for all h > 0 sufficiently small, ∃˜ u(s−1) (x0 +h, t)⊖gH u ˜(s−1) (x0 , t), u ˜(s−1) (x0 −h, t)⊖gH u ˜(s−1) (x0 , t) and the limits u ˜(s−1) (x0 + h, t) ⊖gH u ˜(s−1) (x0 , t) u ˜(s−1) (x0 − h, t) ⊖gH u ˜(s−1) (x0 , t) ∂su ˜ = lim = | , h→0+ h→0+ h −h ∂ xs (x0 ,t) lim

or iv. for all h > 0 sufficiently small, ∃˜ u(s−1) (x0 , t)⊖gH u ˜(s−1) (x0 +h, t), u ˜(s−1) (x0 , t)⊖gH u ˜(s−1) (x0 −h, t) and the limits u ˜(s−1) (x0 , t) ⊖gH u ˜(s−1) (x0 + h, t) u ˜(s−1) (x0 , t) ⊖gH u ˜(s−1) (x0 − h, t) ∂su ˜ = lim = | . h→0+ h→0+ −h h ∂ xs (x0 ,t) lim

2.1

Fuzzy Coputo’s derivative

We denote C F [a, b] as a space of all fuzzy valued functions which are continuous on [a, b], and the space of all Kaleva integrable fuzzy-valued functions on the bounded interval [a, b] ⊂ R by K F [a, b], we denote the space of fuzzy value functions f˜(x) which have continuous H-derivative up to order n − 1 on [a, b] such that f˜(n−1) (x) ∈ AC F ([a, b]) by AC (n)F ([a, b]), where AC F ([a, b]) denote the set of all fuzzy-valued functions which are absolutely continuous (ref. to [13, 9]). Definition 2.5.[2] Suppose f˜(x) ∈ C F [a, b]∩K F [a, b], the fuzzy Riemann Liouville integral of fuzzy valued function f˜ is defined as following: α ˜ α α (Ia+ f )(x, r) = [(Ia+ f )(x, r), (Ia+ f )(x, r)],

where 0 ≤ r ≤ 1 α (Ia+ f )(x, r)

1 = Γ(α)

α f )(x, r) = (Ia+

1 Γ(α)

∫

x

f (t)(r)dt , 0 ≤ r ≤ 1, (x − t)1−α

x

f (t)(r)dt , 0 ≤ r ≤ 1. (x − t)1−α

a

∫

a

Suppose f˜(x) ∈ C F ((0, a]) ∩ K F (0, a), be a given function such that f˜(t, r) = [f (t, r), f (t, r)] for α f˜(t; r) the fuzzy fractional Riemann-Liouville derivative of all t ∈ (0, a] and 0 ≤ r ≤ 1. We define D∗a order 0 < α < 1 of f˜ in the parametric from, [ ∫ t ] ∫ 1 d d t α ˜ −α −α D∗a f (t; r) = (t − s) f (s, r)ds, (t − s) f (s, r)ds , Γ(1 − α) dt 0 dt 0 α f˜(t) ∈ E 1 . In fact, provided that equation defines a fuzzy number D∗a α ˜ α α D∗a f (t, r) = [D∗a f (t, r), D∗a f (t, r)]. α f˜(t) = Obviously, D∗a

3

d 1−α ˜ f (t) dt I

f or t ∈ (0, a].

Generalized Taylor’s formula

In this section, we present the generalized Taylor’s formula that involves Caputo fractional derivative. α )j f (x) ∈ C(a, b] for j = 0, 1, ·····, n+1, where 0 < α ≤ 1, that we get Theorem 3.1.[21] Let that (D∗a

f (x) =

n α )n+1 f )(ζ) ∑ (x − a)iα ((D∗a α i ((D∗a ) f )(a+) + (x − a)(n+1)α , Γ(iα + 1) Γ((n + 1)α + 1)

(3.4)

i=0

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α is the Caputo fractional derivative of order α, where (D α )j = with a ≤ ζ ≤ x, ∀x ∈ (a, b] and D∗a ∗a α α α D∗a D∗a · · · ·D∗a . In case of α = 1, the generalized Taylor’s formula (3.4) reduces to the classical Taylor’s formula. α )j f (x) ∈ C(a, b] for j = 0, 1, · · · · ·, N + 1, where 0 < α ≤ 1. If Theorem 3.2.[17] Let that (D∗a x ∈ [a, b], then N ∑ (x − a)iα α i f (x) ≃ ((D∗a ) f )(a+). (3.5) Γ(iα + 1) i=0

α (x) has the from Furthermore, there is a value ζ with a ≤ ζ ≤ x so that the error term RN α RN (x) =

α )N +1 f )(ζ) ((D∗a (x − a)(N +1)α . Γ((N + 1)α + 1)

(3.6)

α (x) increases when we choose large N and decreases as value of x moves away The accuracy of RN from the center a. Hence, we must choose N large enough so that the error does not exceed a specified bound. In the following theorem, we find precise condition under which the exponents hold for arbitrary fractional operators. ∗ Theorem 3.3.[18] Let xλ g(x), where λ∗ > −1 and g(x) has the generalized power ∑∞that f (x) =nα series expansion g(x) = n=0 an (x − a) with radius of convergence R > 0, where 0 < α ≤ 1. Then γ β γ+β D∗a D∗a f (x) = D∗a f (x)

(3.7)

for all x ∈ (0, R) if one of the following conditions is satisfied: 1. β < λ∗ + 1, and γ arbitrary, 2. β ≥ λ∗ + 1, γ arbitrary,, and aj = 0 for j = 0, 1, · · · · ·, m − 1, where m − 1 < β ≤ m.

4

Differential transform method and fuzzy fractional wave equation

4.1

Generalized two-dimensional differential transform method

In this section, we will derive the generalized two-dimensional differential transform method (DTM) that we get developed for the solution of the wave equation with space and time-fractional derivatives. The proposed method is based on Taylor’s formula. Consider a function of two variables u(x, t), and Let that it can be represented as a product of two single variable functions, u(x, t) = f (x)g(t). Based on the properties of generalized two dimensional differential transform method, function u(x, t) can be represented as. u(x, t) =

∞ ∑ j=0

Fα (j) · (x − x0 )

jα

∞ ∑

Gβ (h) · (t − t0 )

hβ

=

∞ ∑ ∞ ∑

Uα,β (j, h)(x − x0 )jα (t − t0 )hβ ,

(4.8)

j=0 h=0

h=0

where 0 < α, β ≤ 1, Uα,β (j, h) = Fα (j)Gβ (h) is called the spectrum of u(x, t). If function u(x, t) is analytical and differentiated continuously with respect to time t∗ in the domain of interest, then we define the generalized two-dimensional differential transform method (DTM) of the function u(x, t) as follows: 1 Uα,β (j, h) = [(Dxα0 )j (Dtβ0 )h u(x, t)](x0 ,t0 ) , (4.9) Γ(αj + 1)Γ(βh + 1) where (Dxα0 )j = Dxα0 ·Dxα0 ·····Dxα0 . In this work, the lowercase u(x, t) represents the original function while the uppercase Uα,β (j, h) stands for the transformed function. The generalized differential transform method (DTM) inverse of Uα,β (j, h) is defined as follows u(x, t) =

∞ ∑ ∞ ∑

Uα,β (j, h) · (x − x0 )jα (t − t0 )hβ

(4.10)

j=0 h=0

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In case of α = 1 and β = 1. then generalized two-dimensional differential transform (DTM) (4.9) reduces to the classical two-dimensional DTM [5]. From equation (4.9) and (4.10), some basic properties of the generalized two-dimensional differential transform (DTM) are introduced below (ref. to [17]). Theorem 4.1 If u(x, t) = v(x, t) ± w(x, t), then Uα,β (j, h) = Vα,β (j, h) ± Wα,β (j, h). Theorem 4.2 If u(x, t) = cv(x, t), then Uα,β (j, h) = cVα,β (j, h). Theorem 4.3 If u(x, t) = v(x, t)w(x, t), then Uα,β (j, h) =

j ∑ h ∑

Vα,β (r, h − s)Wα,β (j − r, s).

(4.11)

r=0 s=0

Theorem 4.4 If u(x, t) = Dxα0 v(x, t) and 0 < α ≤ 1, then we get Uα,β (j, h) =

Γ(α(j + 1) + 1) Vα,β (j + 1, h). Γ(αj + 1)

(4.12)

Theorem 4.5 If u(x, t) = Dxα0 Dtβ0 v(x, t) and 0 < α, β ≤ 1, then we get Uα,β (j, h) =

Γ(α(j + 1) + 1)Γ(β(h + 1) + 1) Vα,β (j + 1, h + 1). Γ(αj + 1)Γ(βh + 1)

(4.13)

Theorem 4.6 If u(x, t) = (x − x0 )nα (t − t0 )mα , then Uα,β (j, h) = δ(j − n)(h − m). Theorem 4.7 If u(x, t) = Dxγ0 v(x, t), m − 1 < γ ≤ m and v(x, t) = f (x)g(t), where f (x) satisfies the conditions in Theorem 3.3, then Uα,β (j, h) =

Γ(αj + γ + 1) Uα,β (j + γ/α, h). Γ(αj + 1)

(4.14)

Theorem 4.8 If u(x, t) = Dxγ0 Dtη0 v(x, t), where m−1 < γ ≤ m, n−1 < η ≤ n and v(x, t) = f (x)g(t), where the functions f (x) and g(x) satisfy the conditions given in Theorem 3.3, then Uα,β (j, h) =

4.2

Γ(αj + γ + 1) Γ(βh + η + 1) Uα,β (j + γ/α, h + η/β). Γ(αj + 1) Γ(βh + 1)

(4.15)

Fuzzy fractional wave equation

Consider the fuzzy fractional wave equation with the indicated initial conditions and boundary conditions. ∂αu ˜ ∂2u ˜ 2 = c ⊙ , 0 < α ≤ 2, 0 < x < L, t > 0, (4.16) α ∂t ∂x2 subject to the boundary conditions u ˜(0, t) = 0, and u ˜(L, t) = 0,

(4.17)

u ˜(x, 0) = f˜(x), and u ˜t (x, 0) = g˜(x).

(4.18)

and initial conditions.

We note that the case (i) of Definition 2.3 is coincident with the Hukuhara derivative [14]. We say that a function is (i) differentiable if it is differentiable as in (i) of Definition 2.3, a function is (ii)

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differentiable if it is differentiable as in (ii) of Definition 2.3. In this paper we consider the two cases (i) and (ii). In Ref. [4] the authors consider four cases: the case (i) in [14] is coincident with (i); the case (iii) of Definition 2.1 is equivalent to (ii); in the other cases, the derivative is trivial because it is reduced to crisp element. For details see Theorem 7 in [4]. Thus, we only consider the cases (i) and (ii). Lemma 4.2. [7]. Let u ˜(x, t) : D → E 1 . Then the following statements hold. (i) If u ˜(x, t) is (i)-partial differentiable for x (i.e. u ˜ is partial differentiable for x under the meaning of Definition 2.1 (i), similarly to t), then [ ] [ ] ∂u ˜ ∂u(x, t)(r) ∂ u ¯(x, t)(r) = , ; (4.19) ∂x r ∂x ∂x (ii) If u ˜(x, t) is (ii)-partial differentiable for x (i.e. u ˜ is partial differentiable for x under the meaning of Definition 2.1 (ii), similarly to t), then [ ] [ ] ∂u ¯(x, t)(r) ∂u(x, t)(r) ∂u ˜ = , . (4.20) ∂x r ∂x ∂x Remark 4.1. For u ˜(x, t) : D → E 1 , the following results hold. [ 2 ] [ 2 ] ∂ u ˜ ∂ u(x, t)(r) ∂ 2 u ¯(x, t)(r) = , , ∂x2 r ∂x2 ∂x2

(4.21)

2

in cases for that (i, i), (ii, ii)- ∂∂xu˜2 exist; [

∂2u ˜ ∂x2

]

[

r

] ∂2u ¯(x, t)(r) ∂ 2 u(x, t)(r) = , . ∂x2 ∂x2

(4.22)

2

in cases for that (i, ii), (ii, i)- ∂∂t2u˜ exist. n Remark 4.2. In this paper, we only consider that the cases of (i − ii)n - ∂∂tnue such that [ n ] [ n ] ∂ u ˜ ∂ u(x, t)(r) ∂ n u ¯(x, t)(r) = , , ∂xn r ∂xn ∂xn

(4.23)

where (i − ii)n - ∂∂tnue stands for n time derivative in the cases (i) or (ii). n

5

Examples Example 5.1. Consider the following fuzzy fractional wave equation

(A) ∂2u ˜ ∂2u ˜ = 4 ⊙ 2 ∂t ∂x2 subject to the boundary conditions

0 ≤ x ≤ 1, 0 < t,

u ˜(0, t) = u ˜(1, t) = 0,

0 < t,

(5.24)

(5.25)

and initial conditions u ˜(x, 0) = f˜(x) = k˜n ⊙ sin(πx), ∂u ˜(x, 0) = g˜(x) = 0, ∂t

438

0 ≤ x ≤ 1, 0 ≤ x ≤ 1.

(5.26)

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where k˜n ∈ E 1 , n=1,2,3,... fuzzy number is defined by  s ∈ [0, 1],   s, ˜ 2−s s ∈ (1, 2], k(s) =   0 s∈ / [0, 2],

(5.27)

and [k˜n ](r) = rn , [k˜n ](r) = (2 − r)n . The parametric form of (5.24) is ∂2u ∂2u = 4 ∂t2 ∂x2 2 ∂ u ∂2u = 4 , ∂t2 ∂x2

0 ≤ x ≤ 1, 0 ≤ x ≤ 1,

0 < t,

(5.28)

0 < t,

(5.29)

for r ∈ [0, 1], and where u stands for u(x, t)(r), similar to u. Taking the differential transform of equations (5.28) and (5.29), we get (j + 2)(j + 1)U (i, j + 2)(r) = 4(i + 2)(i + 1)U (i + 2, j)(r),

(5.30)

(j + 2)(j + 1)U (i, j + 2)(r) = 4(i + 2)(i + 1)U (i + 2, j)(r).

(5.31)

From the initial given by equation (5.26), we get u(x, 0)(r) =

∞ ∑

i

U (i, 0)(r)x = k(r) sin(πx) = r

n

i=0

u(x, 0)(r) =

∞ ∑

∞ ∑

(i−1) 2

i=1,3,....

U (i, 0)(r)x = k(r) sin(πx) = (2 − r) i

n

i=0

(−1) i! ∞ ∑

i=1,3,....

The corresponding spectra can be obtained as follows,    0, U (i, 0)(r) = (−1) (i−1) 2   rn π i , i!    0, U (i, 0)(r) = (−1) (i−1) 2   (2 − r)n π i , i!

π i xi , (i−1) 2

(−1) i!

(5.32)

π i xi .

(5.33)

for i is even, (5.34) for i is odd for i is even, (5.35) for i is odd

and from equation (5.26) it can be obtained that, ∞

∂u(x, 0)(r) ∑ = U (i, 1)(r)xi = 0, ∂t

(5.36)

∂u(x, 0)(r) = ∂t

(5.37)

i=0 ∞ ∑

U (i, 1)(r)xi = 0.

i=0

Hence, u(i, 1)(r) = 0, u(i, 1)(r) = 0.

439

(5.38) (5.39)

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Substituting equations (5.34) -(5.39) to equations (5.30) and (5.31), all spectra can be found as,  for i is even or j is odd   0, U (i, j)(r) = 2j (−1) (i+j−1) (5.40) 2   rn π i+j , for i is odd or j is even i!j!  for i is even or j is odd   0, (i+j−1)

U (i, j)(r) =

2j (−1) 2   (2 − r)n π i+j , for i is odd or j is even i!j! So, the closed from of the solution can be easily written as ∞ ∑ ∞ ∑ (i+j−1) 2j (−1) 2 π i+j xi tj j!i! i=0 j=0 i=0 j=0    ∞ ∞ ∑ ∑ (i−1) j 1 1 (−1) 2 (πx)i   (−1) 2 (2πt)j  = rn  i! j!

u(x, t)(r) = k n

∞ ∑ ∞ ∑

(5.41)

U (i, j)(r)xi tj = rn

i=1,3,...

j=0,2,...

n

= r sin(πx) cos(2πt), ∞ ∞ n∑∑

(5.42)

∞ ∑ ∞ ∑ (i+j−1) 2j u(x, t)(r) = k U (i, j)(r)x t = (2 − r) (−1) 2 π i+j xi tj j!i! i=0 j=0 i=0 j=0    ∞ ∞ ∑ ∑ (i−1) j 1 1 = (2 − r)n  (−1) 2 (πx)i   (−1) 2 (2πt)j  i! j! i j

n

i=1,3,...

j=0,2,...

= (2 − r)n sin(πx) cos(2πt).

(5.43)

(B) Consider the following fuzzy fractional wave equation (5.24) with the boundary conditions: u ˜(0, t) = u ˜(1, t) = 0,

0 < t,

(5.44)

and initial conditions u ˜(x, 0) = f˜(x) = k˜n ⊕ sin(πx), ∂u ˜(x, 0) = g˜(x) = 0, ∂t By following the same steps, we will find that the solution. can be easily written as ∞ ∑ ∞ ∞ ∑ ∞ ∑ ∑ n i j n U (i, j)(r)x t = r + u(x, t)(r) = k +

0 ≤ x ≤ 1, 0 ≤ x ≤ 1.

(5.45)

So, the closed from of the solution

(i+j−1) 2j (−1) 2 π i+j xi tj j!i! i=0 j=0 i=0 j=0    ∞ ∞ ∑ ∑ (i−1) j 1 1 = rn +  (−1) 2 (πx)i   (−1) 2 (2πt)j  i! j!

i=1,3,...

j=0,2,...

n

= r + (sin(πx) cos(2πt)), (5.46) ∞ ∞ ∞ ∞ j ∑∑ ∑∑ 2 (i+j−1) n u(x, t)(r) = k + U (i, j)(r)xi tj = (2 − r)n + (−1) 2 π i+j xi tj j!i! i=0 j=0 i=0 j=0    ∞ ∞ ∑ ∑ (i−1) j 1 1 = (2 − r)n +  (−1) 2 (πx)i   (−1) 2 (2πt)j  i! j! i=1,3,...

j=0,2,...

= (2 − r) + (sin(πx) cos(2πt)). n

440

(5.47)

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(C) Consider the following fuzzy fractional wave equation (5.24) with the boundary conditions: u ˜(0, t) = u ˜(1, t) = 0,

0 < t,

(5.48)

and initial conditions u ˜(x, 0) = f˜(x) = k˜n ⊖gH sin(πx), ∂u ˜(x, 0) = g˜(x) = 0, ∂t

0 ≤ x ≤ 1, 0 ≤ x ≤ 1.

(5.49)

where k˜n ∈ E 1 , n=1,2,3,... , fuzzy number is defined by  s ∈ [0.5, 1],   2(s − 0.5), ˜ 2(1.5 − s), s ∈ (1, 1.5], k(s) =   0 s∈ / [0.5, 1.5],

(5.50)

and{k˜n }(r) = (0.5 + 0.5r)n , {k˜n }(r) = (1.5 − 0.5r)n . By following the same steps, we will find that the solution. So, the closed from of the solution can be easily written as ∞ ∞ ∑ ∑ (i+j−1) 2j U (i, j)(r)x t = (0.5 + 0.5r) − (−1) 2 π i+j xi tj u(x, t)(r) = k − j!i! i=0 j=0 i=0 j=0    ∞ ∞ ∑ ∑ (i−1) j 1 1 = (0.5 + 0.5r)n −  (−1) 2 (πx)i   (−1) 2 (2πt)j  i! j! n

∞ ∞ ∑ ∑

i j

n

i=1,3,...

j=0,2,...

= (0.5 + 0.5r) − (sin(πx) cos(2πt)) , (5.51) ∞ ∞ ∞ ∑ ∞ ∑ j ∑ ∑ (i+j−1) 2 n u(x, t)(r) = k − U (i, j)(r)xi tj = (1.5 − 0.5r)n − (−1) 2 π i+j xi tj j!i! i=0 j=0 i=0 j=0    ∞ ∞ ∑ ∑ (i−1) j 1 1 = (1.5 − 0.5r)n −  (−1) 2 (πx)i   (−1) 2 (2πt)j  i! j! n

i=1,3,...

j=0,2,...

= (1.5 − 0.5r) − (sin(πx) cos(2πt)) . n

(5.52)

Example 5.2. Consider the following fuzzy time-fractional wave equation. (A) ∂ 1.5 u ˜ ∂2u ˜ = , 1.5 ∂t ∂x2

t > 0,

(5.53)

subject to the initial conditions u ˜(x, 0) = f˜(x) = k˜n ⊙ sin(x),

∂u ˜(x, 0) = g˜(x) = k˜n ⊙ (− sin(x)). ∂t

where k˜n ∈ E 1 , n=1,2,3,..., fuzzy number is defined by  s ∈ [0.5, 1],   2(s − 0.5), ˜ 2(1.5 − s), s ∈ (1, 1.5], k(s) =   0 s∈ / [0.5, 1.5],

441

(5.54)

(5.55)

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and{k˜n }(r) = (0.5 + 0.5r)n , {k˜n }(r) = (1.5 − 0.5r)n . The parametric form of (5.53) is ∂ 1.5 u ∂2u = , ∂t1.5 ∂x2 ∂ 1.5 u ∂2u = , ∂t1.5 ∂x2

t > 0,

(5.56)

t > 0.

(5.57)

for r ∈ [0, 1], and where u stands for u(x, t)(r), similar to u. Let the solution u(x, t) = f (x)g(t) where the function g(t) satisfies the conditions given in Theorem 3.3. Then selecting α = 0.5, β = 1 and applying the generalized two-dimensional differential transform method (DTM) to both sides of equations (5.56) and (5.57) by Theorem 4.7, equations (5.56) and (5.57) Transforms to U 0.5,1 (j, h + 3)(r) = U 0.5,1 (j, h + 3)(r) =

(j + 1)(j + 2)Γ( h2 + 1) Γ( h2 + 25 ) (j + 1)(j + 2)Γ( h2 + 1) Γ( h2 + 25 )

U 0.5,1 (j + 2, h)(r),

(5.58)

U 0.5,1 (j + 2, h)(r).

(5.59)

The generalized two-dimensional differential transform of the initial conditions (5.54) are given by U 0.5,1 (j, 0)(r) = (0.5 + 0.5r)n

1 πj sin( ), j! 2

U 0.5,1 (j, 1)(r) = 0,

(5.60) (5.61)

−1 πj sin( ), j! 2 1 πj U 0.5,1 (j, 0)(r) = (1.5 − 0.5r)n sin( ), j! 2 U 0.5,1 (j, 1)(r) = 0, −1 πj sin( ). U 0.5,1 (j, 2)(r) = (1.5 − 0.5r)n j! 2

U 0.5,1 (j, 2)(r) = (0.5 + 0.5r)n

(5.62) (5.63) (5.64) (5.65)

Utilizing the recurrence relation (5.58), (5.59) and the transformed initial conditions (5.60) -(5.65), the first few components of U0.5,1 (j, h) can be calculated. So, the solution u(x, t) of equations (5.56) and (5.57) is obtained ( ) 1 3 1 5 1 3 n t + ..... x u(x, t)(r) = (0.5 + 0.5r) 1 − t − 5 t 2 + 7 t 2 + Γ(4) Γ( 2 ) Γ( 2 ) ( ) 3 5 1 1 1 1 1 + (0.5 + 0.5r)n − + t + t2 − t2 − t3 + .... x3 3! 3! 3!Γ(4) 3!Γ( 52 ) 3!Γ( 72 ) ) ( 3 5 1 1 1 1 1 + (0.5 + 0.5r)n − t− t2 + t2 + t3 − ... x5 5! 5! 5!Γ(4) 5!Γ( 52 ) 5!Γ( 72 )   3j 3j ∞ ∞ +1 j j ∑ ∑ (−1) t 2 (−1) t 2 u(x, t)(r) = (0.5 + 0.5r)n  sin(x) − sin(x) , 3j 3j Γ( + 1) Γ( + 2) 2 2 j=0 j=0 ( ) 3 3 = (0.5 + 0.5r)n E 3 ,1 (−t 2 ) sin(x) − tE 3 ,2 (−t 2 ) sin(x) , (5.66) 2

2

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( u(x, t)(r) = (1.5 − 0.5r)n (

) 1 3 1 5 1 3 1 − t − 5 t2 + 7 t2 + t + ..... · x Γ(4) Γ( 2 ) Γ( 2 )

) 3 5 1 1 1 1 1 + (1.5 − 0.5r)n − + t + t2 − t2 − t3 + .... · x3 3! 3! 3!Γ(4) 3!Γ( 52 ) 3!Γ( 72 ) ( ) 3 5 1 1 1 1 1 3 n − t− t2 + t2 + t − ... · x5 + (1.5 − 0.5r) 5! 5! 5!Γ(4) 5!Γ( 52 ) 5!Γ( 72 )   3j 3j ∞ ∞ jt 2 j t 2 +1 ∑ ∑ (−1) (−1) u(x, t)(r) = (1.5 − 0.5r)n  sin(x) − sin(x) , 3j 3j Γ( Γ( + 1) + 2) 2 2 j=0 j=0 ( ) 3 3 = (1.5 − 0.5r)n E 3 ,1 (−t 2 ) sin(x) − tE 3 ,2 (−t 2 ) sin(x) . (5.67) 2

2

Which is the exact solution of the fuzzy time fractional wave equations (5.56) and (5.57) where Eα,β (z) is the two parameters mittag-Leffer function defined by Eα,β (z) = k˜n ⊙

∞ ∑ n=0

zn . Γ(αn + β)

(5.68)

(B) Consider the following fuzzy time-fractional wave equation (5.53) with the initial conditions: u ˜(x, 0) = f˜(x) = k˜n ⊕ sin(x),

∂u ˜(x, 0) = g˜(x) = k˜n ⊕ (− sin(x)). ∂t

(5.69)

By following the same steps, we will find that the solution. Utilizing the recurrence relation (5.58), (5.59) and the transformed initial conditions (5.60) -(5.65), the first few components of U0.5,1 (j, h) can be calculated. So, the solution u(x, t) of equations (5.56) and (5.57) is obtained ( ) 1 3 1 5 1 3 n u(x, t)(r) = (0.5 + 0.5r) + 1 − t − 5 t 2 + 7 t 2 + t + ..... x Γ(4) Γ( 2 ) Γ( 2 ) ( ) 3 5 1 1 1 1 1 3 3 n 2 2 + (0.5 + 0.5r) + − + t + 5 t − 7 t − 3!Γ(4) t + .... x 3! 3! 3!Γ( 2 ) 3!Γ( 2 ) ( ) 3 5 1 1 1 1 1 + (0.5 + 0.5r)n + − t− t2 + t2 + t3 − ... x5 5! 5! 5!Γ(4) 5!Γ( 25 ) 5!Γ( 72 )   3j 3j ∞ ∞ +1 j j ∑ ∑ (−1) t 2 (−1) t 2 u(x, t)(r) = (0.5 + 0.5r)n +  sin(x) − sin(x) , 3j 3j Γ( + 1) Γ( + 2) 2 2 j=0 j=0 ( ) 3 3 = (0.5 + 0.5r)n + E 3 ,1 (−t 2 ) sin(x) − tE 3 ,2 (−t 2 ) sin(x) , (5.70) 2

2

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( u(x, t)(r) = (1.5 − 0.5r)n + (

) 1 3 1 5 1 3 1 − t − 5 t2 + 7 t2 + t + ..... · x Γ(4) Γ( 2 ) Γ( 2 )

) 3 5 1 1 1 1 1 + (1.5 − 0.5r)n + − + t + t2 − t2 − t3 + .... · x3 3! 3! 3!Γ(4) 3!Γ( 52 ) 3!Γ( 72 ) ( ) 3 5 1 1 1 1 1 3 n − t− t2 + t2 + t − ... · x5 + (1.5 − 0.5r) + 5! 5! 5!Γ(4) 5!Γ( 52 ) 5!Γ( 27 )   3j 3j ∞ ∞ jt 2 j t 2 +1 ∑ ∑ (−1) (−1) u(x, t)(r) = (1.5 − 0.5r)n +  sin(x) − sin(x) , 3j 3j Γ( Γ( + 1) + 2) 2 2 j=0 j=0 ( ) 3 3 = (1.5 − 0.5r)n + E 3 ,1 (−t 2 ) sin(x) − tE 3 ,2 (−t 2 ) sin(x) . (5.71) 2

2

Which is the exact solution of the fuzzy time fractional wave equations (5.56) and (5.57) where Eα,β (z) is the two parameters mittag-Leffer function defined by Eα,β (z) = k˜n ⊕

∞ ∑ n=0

zn . Γ(αn + β)

(5.72)

(C) Consider the following fuzzy time fractional wave equation (5.53) with initial conditions: u ˜(x, 0) = f˜(x) = k˜n ⊖gH sin(x),

∂u ˜(x, 0) = g˜(x) = k˜n ⊖gH (− sin(x)). ∂t

(5.73)

By following the same steps, we will find that the solution. Utilizing the recurrence relation (5.58), (5.59) and the transformed initial conditions (5.60) -(5.65), the first few components of U0.5,1 (j, h) can be calculated. So, the solution u(x, t) of equations (5.56) and (5.57) is obtained ( ) 1 3 1 5 1 3 n u(x, t)(r) = (0.5 + 0.5r) − 1 − t − 5 t 2 + 7 t 2 + t + ..... x Γ(4) Γ( 2 ) Γ( 2 ) ( ) 3 5 1 1 1 1 1 3 3 n 2 2 + (0.5 + 0.5r) − − + t + 5 t − 7 t − 3!Γ(4) t + .... x 3! 3! 3!Γ( 2 ) 3!Γ( 2 ) ( ) 3 5 1 1 1 1 1 + (0.5 + 0.5r)n − − t− t2 + t2 + t3 − ... x5 5! 5! 5!Γ(4) 5!Γ( 25 ) 5!Γ( 72 )   3j 3j ∞ ∞ +1 j j ∑ ∑ (−1) t 2 (−1) t 2 u(x, t)(r) = (0.5 + 0.5r)n −  sin(x) − sin(x) , 3j 3j Γ( + 1) Γ( + 2) 2 2 j=0 j=0 ( ) 3 3 = (0.5 + 0.5r)n − E 3 ,1 (−t 2 ) sin(x) − tE 3 ,2 (−t 2 ) sin(x) , (5.74) 2

2

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( u(x, t)(r) = (1.5 − 0.5r)n − (

) 1 3 1 5 1 3 1 − t − 5 t2 + 7 t2 + t + ..... · x Γ(4) Γ( 2 ) Γ( 2 )

) 3 5 1 1 1 1 1 + (1.5 − 0.5r)n − − + t + t2 − t2 − t3 + .... · x3 3! 3! 3!Γ(4) 3!Γ( 52 ) 3!Γ( 72 ) ( ) 3 5 1 1 1 1 1 3 n − t− t2 + t2 + t − ... · x5 + (1.5 − 0.5r) − 5! 5! 5!Γ(4) 5!Γ( 52 ) 5!Γ( 27 )   3j 3j ∞ ∞ jt 2 j t 2 +1 ∑ ∑ (−1) (−1) u(x, t)(r) = (1.5 − 0.5r)n −  sin(x) − sin(x) , 3j 3j Γ( Γ( + 1) + 2) 2 2 j=0 j=0 ( ) 3 3 = (1.5 − 0.5r)n − E 3 ,1 (−t 2 ) sin(x) − tE 3 ,2 (−t 2 ) sin(x) . (5.75) 2

2

Which is the exact solution of the fuzzy time fractional wave equations (5.56) and (5.57) where Eα,β (z) is the two parameters mittag-Leffer function defined by Eα,β (z) = k˜n ⊖gH

∞ ∑ n=0

zn . Γ(αn + β)

(5.76)

Example 5.3. Consider the following fuzzy linear space time fractional wave equation (A) ∂ 1.5 u ˜ ˜ 1 2 ∂ 1.25 u x ⊙ 1.25 = 1.5 ∂t 2 ∂x subject to the initial conditions u ˜(x, 0) = f˜(x) = k˜n ⊙

∞ ∑

x > 0,

t > 0,

(5.77)

∞

n

an x ,

n=0

∑ ∂u ˜(x, 0) = g˜(x) = k˜n ⊙ bn xn . ∂t

(5.78)

n=0

where k˜n ∈ E 1 , n=1,2,3,... fuzzy number is defined by  s ∈ [0, 1],   s, ˜ 2−s s ∈ (1, 2], k(s) =   0 s∈ / [0, 2],

(5.79)

and [k˜n ](r) = rn , [k˜n ](r) = (2 − r)n . The parametric form of (5.77) is ∂ 1.5 u 1 2 ∂ 1.25 u = x ∂t1.5 2 ∂x1.25 ∂ 1.5 u 1 2 ∂ 1.25 u = x ∂t1.5 2 ∂x1.25

x > 0,

t>0

(5.80)

x > 0,

t>0

(5.81)

for r ∈ [0, 1], and where u stands for u(x, t)(r), similar to u. Let the solution u(x, t) can be represented as a product of single-valued functions, u(x, t) = f (x)g(t) where the functions f (x) and g(t) satisfy the conditions given in Theorem 3.3. Selecting α = 0.5, β = 0.25 and applying the generalized two-dimensional differential transform to both

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sides of equations (5.80) and (5.81), the fuzzy linear space-time fractional wave equations (5.80) and (5.81) transform to   1 Γ(h/2 + 1)Γ(j/4 + 7/4) U j≥2 (j + 3, h)(r), 2 Γ(h/2 + 5/2)Γ(j/4 + 2/4) 1/2,1/4 U 1/2,1/4 (j, h + 3)(r) = (5.82)  0, j < 2.   1 Γ(h/2 + 1)Γ(j/4 + 7/4) U (j + 3, h)(r), j≥2 2 Γ(h/2 + 5/2)Γ(j/4 + 2/4) 1/2,1/4 U 1/2,1/4 (j, h + 3)(r) = (5.83)  0, j < 2. The generalized two-dimensional transforms of the initial conditions (5.78) are given by U 1/2,1/4 (j, 0)(r) = rn aj ,

(5.84)

U 1/2,1/4 (j, 1)(r) = 0,

(5.85)

n

U 1/2,1/4 (j, 2)(r) = r bj ,

(5.86)

U 1/2,1/4 (j, 0)(r) = (2 − r) aj ,

(5.87)

U 1/2,1/4 (j, 1)(r) = 0,

(5.88)

n

U 1/2,1/4 (j, 2)(r) = (2 − r) bj . n

(5.89)

Utilizing the recurrence relation (5.82), (5.83) and the transformed initial conditions (5.84) -(5.89), the first few components of U1/2,1/4 (j, h) are calculated. So, from equation (4.8), the approximate solution of the fuzzy linear space-time-fractional wave equations (5.80) and (5.81) can be derived as ( ) Γ(7/4) Γ(7/4) n 3/2 5/2 u(x, t)(r) = r a0 + b0 t + a3 t + b3 t Γ(5/2)Γ(2/4) Γ(7/2)Γ(2/4) ( ) Γ(7/4) Γ(7/4) n 3/2 5/2 + r a1 + b1 t + a4 t + b4 t · x1/4 Γ(5/2)Γ(2/4) Γ(7/2)Γ(2/4) ) ( Γ(7/4) Γ(7/4) 3/2 5/2 n a5 t + b5 t · x2/4 + r a2 + b2 t + Γ(5/2)Γ(2/4) Γ(7/2)Γ(2/4) ( ) Γ(7/4) Γ(7/4) n 3/2 5/2 + r a3 + b3 t + a6 t + b6 t · x3/4 Γ(5/2)Γ(2/4) Γ(7/2)Γ(2/4) ( ) Γ(7/4) Γ(7/4) 3/2 5/2 n a7 t + b7 t · x + · · ·, (5.90) + r a4 + b4 t + Γ(5/2)Γ(2/4) Γ(7/2)Γ(2/4) ) ( Γ(7/4) Γ(7/4) 3/2 5/2 u(x, t)(r) = (2 − r) a0 + b0 t + a3 t + b3 t Γ(5/2)Γ(2/4) Γ(7/2)Γ(2/4) ( ) Γ(7/4) Γ(7/4) 3/2 5/2 n a4 t + b4 t · x1/4 + (2 − r) a1 + b1 t + Γ(5/2)Γ(2/4) Γ(7/2)Γ(2/4) ( ) Γ(7/4) Γ(7/4) n 3/2 5/2 + (2 − r) a2 + b2 t + a5 t + b5 t · x2/4 Γ(5/2)Γ(2/4) Γ(7/2)Γ(2/4) ( ) Γ(7/4) Γ(7/4) 3/2 5/2 n a6 t + b6 t · x3/4 + (2 − r) a3 + b3 t + Γ(5/2)Γ(2/4) Γ(7/2)Γ(2/4) ( ) Γ(7/4) Γ(7/4) n 3/2 5/2 + (2 − r) a4 + b4 t + a7 t + b7 t · x + · · ·. Γ(5/2)Γ(2/4) Γ(7/2)Γ(2/4) n

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(5.91)

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(B) Consider the following fuzzy linear-space-time-fractional wave equation (5.77) with the initial conditions: u ˜(x, 0) = f˜(x) = k˜n ⊕

∞ ∑

∞

∑ ∂u ˜(x, 0) = g˜(x) = k˜n ⊕ bn xn . ∂t

n

an x ,

n=0

(5.92)

n=0

By following the same steps, we will find that the solution. Utilizing the recurrence relation (5.82), (5.83) and the transformed initial conditions (5.84) -(5.89), the first few components of U1/2,1/4 (j, h) are calculated. So, from equation (4.8), the approximate solution of the fuzzy linear space-time-fractional wave equations (5.80) and (5.81) can be derived as ( ) Γ(7/4) Γ(7/4) a3 t3/2 + b3 t5/2 u(x, t)(r) = rn + a0 + b0 t + Γ(5/2)Γ(2/4) Γ(7/2)Γ(2/4) ( ) Γ(7/4) Γ(7/4) n 3/2 5/2 + r + a1 + b1 t + a4 t + b4 t · x1/4 Γ(5/2)Γ(2/4) Γ(7/2)Γ(2/4) ( ) Γ(7/4) Γ(7/4) n 3/2 5/2 + r + a2 + b2 t + a5 t + b5 t · x2/4 Γ(5/2)Γ(2/4) Γ(7/2)Γ(2/4) ) ( Γ(7/4) Γ(7/4) 3/2 5/2 n a6 t + b6 t · x3/4 + r + a3 + b3 t + Γ(5/2)Γ(2/4) Γ(7/2)Γ(2/4) ( ) Γ(7/4) Γ(7/4) n 3/2 5/2 + r + a4 + b4 t + (5.93) a7 t + b7 t · x + · · ·, Γ(5/2)Γ(2/4) Γ(7/2)Γ(2/4) ( ) Γ(7/4) Γ(7/4) 3/2 5/2 u(x, t)(r) = (2 − r) + a0 + b0 t + a3 t + b3 t Γ(5/2)Γ(2/4) Γ(7/2)Γ(2/4) ) ( Γ(7/4) Γ(7/4) a4 t3/2 + b4 t5/2 · x1/4 + (2 − r)n + a1 + b1 t + Γ(5/2)Γ(2/4) Γ(7/2)Γ(2/4) ( ) Γ(7/4) Γ(7/4) n 3/2 5/2 + (2 − r) + a2 + b2 t + a5 t + b5 t · x2/4 Γ(5/2)Γ(2/4) Γ(7/2)Γ(2/4) ( ) Γ(7/4) Γ(7/4) n 3/2 5/2 + (2 − r) + a3 + b3 t + a6 t + b6 t · x3/4 Γ(5/2)Γ(2/4) Γ(7/2)Γ(2/4) ) ( Γ(7/4) Γ(7/4) n 3/2 5/2 + (2 − r) + a4 + b4 t + a7 t + b7 t · x + · · ·. (5.94) Γ(5/2)Γ(2/4) Γ(7/2)Γ(2/4) n

(C) Consider the following fuzzy linear space-time-fractional wave equation (5.77) with the initial conditions: u ˜(x, 0) = f˜(x) = k˜n ⊖gH

∞ ∑ n=0

an xn ,

∞ ∑ ∂u ˜(x, 0) = g˜(x) = k˜n ⊖gH bn xn . ∂t

(5.95)

n=0

By following the same steps, we will find that the solution. Utilizing the recurrence relation (5.82), (5.83) and the transformed initial conditions (5.84) -(5.89), the first few components of U1/2,1/4 (j, h) are calculated. So, from equation (4.8), the approximate solution of the fuzzy linear

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space-time-fractional wave equations (5.80) and (5.81) can be derived as ( ) Γ(7/4) Γ(7/4) n 3/2 5/2 u(x, t)(r) = r − a0 + b0 t + a3 t + b3 t Γ(5/2)Γ(2/4) Γ(7/2)Γ(2/4) ( ) Γ(7/4) Γ(7/4) n 3/2 5/2 + r − a1 + b1 t + a4 t + b4 t · x1/4 Γ(5/2)Γ(2/4) Γ(7/2)Γ(2/4) ( ) Γ(7/4) Γ(7/4) n 3/2 5/2 + r − a2 + b2 t + a5 t + b5 t · x2/4 Γ(5/2)Γ(2/4) Γ(7/2)Γ(2/4) ( ) Γ(7/4) Γ(7/4) n 3/2 5/2 + r − a3 + b3 t + a6 t + b6 t · x3/4 Γ(5/2)Γ(2/4) Γ(7/2)Γ(2/4) ( ) Γ(7/4) Γ(7/4) n 3/2 5/2 + r − a4 + b4 t + a7 t + b7 t · x + · · ·, Γ(5/2)Γ(2/4) Γ(7/2)Γ(2/4)

(5.96)

( ) Γ(7/4) Γ(7/4) 3/2 5/2 u(x, t)(r) = (2 − r) − a0 + b0 t + a3 t + b3 t Γ(5/2)Γ(2/4) Γ(7/2)Γ(2/4) ( ) Γ(7/4) Γ(7/4) n 3/2 5/2 · x1/4 + (2 − r) − a1 + b1 t + a4 t + b4 t Γ(5/2)Γ(2/4) Γ(7/2)Γ(2/4) ( ) Γ(7/4) Γ(7/4) n 3/2 5/2 + (2 − r) − a2 + b2 t + a5 t + b5 t · x2/4 Γ(5/2)Γ(2/4) Γ(7/2)Γ(2/4) ( ) Γ(7/4) Γ(7/4) n 3/2 5/2 + (2 − r) − a3 + b3 t + a6 t + b6 t · x3/4 Γ(5/2)Γ(2/4) Γ(7/2)Γ(2/4) ) ( Γ(7/4) Γ(7/4) 3/2 5/2 n a7 t + b7 t · x + · · ·. (5.97) + (2 − r) − a4 + b4 t + Γ(5/2)Γ(2/4) Γ(7/2)Γ(2/4) n

6

Conclusions

In this paper, the differential transform method (DTM) has been successfully applied for solving fuzzy fractional wave equation. The proposed method is also illustrated by three examples. The new method is investigated based on the two-dimensional differential transform method, generalized Taylor’s formula and fuzzy Caputo,s derivative. The results reveal that DTM is a highly effective scheme for obtaining analytical solutions of the fuzzy fractional wave equation.

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Figure 1: Example (5.1), Case (A), t = 0.000001, x = 0.1, n = 1.

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Figure 2: Example (5.1), Case (B), t = 0.03, x = 0.1, n = 2.

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Figure 3: Example (5.1), Case (C), t = 0.0001, x = 0.001, n = 3.

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References [1] T. Allahviranloo, N.A. Kiani, N. Motamedi, Solving fuzzy differential equations by differential transform method, Information Sciences 170 (2009) 956-966. [2] A. Ahmadian, M. Suleiman, S. Sahahshour, D. Baleanu, A Jacobi operational matrix for solving a fuzzy linear fractional differential equation, Advances in Difference Equations (2013) 1-29. [3] T. Allahviranlooa, Z. Gouyandeha, A. Armanda, A. Hasanoglub, On fuzzy solutions for heat equation based on generalized Hukuhara differentiability, Fuzzy Sets and Systems 265 (2015) 1-23. [4] B. Bede, S.G. Gal, Generalizations of the differentiability fuzzy-number-valued functions with applications to fuzzy differential equations, Fuzzy Sets and Systems 151 (2005) 581-599. [5] N. Bildik, A. Konuralp, F. Bek, S. Kucukarslan, Solution of different type of the partial differential equation by differential transform method and adomian’s decomposition method, Appl. Math. Comput. 172 (2006) 551-567. [6] S.L. Chang, L.A. Zadeh, On fuzzy mapping and control, lEEE Trans, Systems Man Cybernet 2 (1972) 30-34. [7] Y.C. Cano, H.R. Flores, On new solutions of fuzzy differential equations, Chaos Soliton Fract. 38 (2008) 112-119. [8] M. Friedman, M. Ma, A. kandel, Numerical solutions of fuzzy differential and integral equations, fuzzy modeling and dynamics, Fuzzy Sets and Systems 106 (1999) 35-48. [9] B. Ghazanfari, P. Ebrahimi, Differential transformation method for solving fuzzy fractional heat equations, International Journal of Mathematical Moballing and Computations 5 (2015) 81-89. [10] Z.T. Gong, H. Yang, lll-Posed fuzzy initial-boundary value problems based on generalized differentiability and regularization, Fuzzy Sets and Systems 295 (2016) 99-113. [11] Z.T. Gong, Y.D. Hao, Fuzzy Laplace transform based on the Henstock integral and its applications in discontinuous fuzzy systems, Fuzzy Sets and Systems 283 (2018) 1-28. [12] A. Kandel, W.J. Byatt, Fuzzy differential equations, in: Proceedings of International Conference on Cybernetics and Society, Tokyo, 1978. [13] O. Kaleva, fuzzy differential equations, Fuzzy Sets and Systems 24 (1987) 301-317. [14] O. Kaleva, A note on fuzzy differential equations, Nonlinear Anal. 64 (2006) 895-900. [15] M.A. Kermani, Numerical method for solving fuzzy wave equation, American Institute of Physics conference Proceeding 1558 (2013) 2444-2447. [16] S. Momani, Analytical approximate solution for fractional heat-like and wave-like equations with variable coefficients using the decomposition method, Applied Mathematics and Computation 165 (2005) 459-472. [17] S. Momani, Z. Odidat, V.S. Erturk, Generalized differential transform method for solving a spaceand time-fractional diffusion-wave equation, Physics Letters A 370 (2007) 379-387. [18] S. Momani, Z. Odibat, A novel method for nonlinear fractional partial differential equations: Combination of DTM and generalized Taylor’s formula, Journal of Computational and Applied Mathematics 220 (2008) 85-95. [19] O.H. Mohammed, F.S. Fadhel, F.A.A. Khaleq, differential transform method for solving fuzzy fractional initial value problems, Journal of Basrah Researches 37 (2011) 158-170.

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[20] O.H. Mohammed, S.A. Ahmed, Solving fuzzy fractional boundary value problems using fractional differential transform method, Journal of Al-Nahrain University 16 (2013) 225-232. [21] Z.M. Odibat, N.T. Shawagfeh, Generalized Taylor’s formula, Applied Mathematics and Computation 186 (2007) 286-293. [22] V. PadmaPriya, M. Kaliyappan, A Review of fuzzy fractional differential equations, International Journal of Pure and Applied Mathematics 113 (2017) 203-216. [23] A. Rivaz, O.S. Fard, T.A. Bidgoli, Solving fuzzy fractional differential equations by generalized differential transform method, SeMA Springer (2015) 1-22. [24] N.A.A. Rahman, M.Z. Ahmad, Solving fuzzy fractional differential equation using fuzzy sumudu transform, Journal of Nonlinear Sciences and Applications 10 (2017) 2620-2632. [25] M. Stepnicka, R. Valasek, Fuzzy Transforms and Their Application on Wave Equation, Journal of Electrical Engineering (2004) 1-7. [26] W.Y. Shi, A.B. Ji, X.D. Dai, Differential of fuzzy function with two variables and fuzzy wave equations, Fourth International Conference on Fuzzy Systems and Knowledge Discovery (2007) 24-27. [27] W.C. Xin, M. Ming, Embedding problem of fuzzy number space: Part I∗ , Fuzzy Sets and Systems 44 (1991) 33-38. [28] L.A. Zaden, Fuzzy sets, lnformation and Control 8 (1965) 338-353. [29] J.K. Zhou, Differential Transformation and Its Applications for Electrical Circuits (in Chinese), Huazhong University Press, Wuhan, China, 1986. [30] G.Q. Zhang, fuzzy continuous function and its properties, Fuzzy Sets and Systems 43 (1991) 159171. [31] H. Yang, Z. Gong, I11-Posedness for fuzzy Fredholm integral equations of the first kind and regularization methods, Fuzzy Sets and Systems 358 (2019) 132-149. [32] C. Wu, Z. Gong, On Henstock integral of fuzzy-number-valued functions (1), Fuzzy Sets Systems 120 (2001) 523-532. [33] B. Bede, L. Stefanini, Generalized differentiability of fuzzy-valued functions, Fuzzy Setes and Systems 230 (2013) 119-141.

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Some Generalized k-Fractional Integral Inequalities for Quasi-Convex Functions Ghulam Farid1 , Chahn Yong Jung2,∗ , Sami Ullah3, Waqas Nazeer4,∗, Muhammad Waseem5 and Shin Min Kang6,∗

1

2

COMSATS University Islamabad, Attock Campus, Attock 43600, Pakistan e-mail: [email protected]; [email protected]

Department of Business Administration, Gyeongsang National University, Jinju 52828, Korea e-mail: [email protected] 3

4

5

Department of Mathematics, Air University, Islamabad 44000, Pakistan e-mail: [email protected]

Division of Science and Technology, University of Education, Lahore 54000, Pakistan e-mail: [email protected]

Department of Mathematics, COMSATS University Islamabad, Vehari Campus, Vehari 61100, Pakistan e-mail: [email protected] 6

Department of Mathematics, Gyeongsang National University, Jinju 52828, Korea e-mail: [email protected] Abstract

Fractional integral operators generalize the concept of definite integration. Therefore these operators play a vital role in the advancement of subjects of sciences and engineering. The aim of this study is to establish the bounds of a generalized fractional integral operator via quasi-convex functions. These bounds behave as a formula in unified form, and estimations of almost all fractional integrals defined in last two decades can be obtained at once by choosing convenient parameters. Moreover, several related fractional integral inequalities are identified. 2010 Mathematics Subject Classification: 26A51, 26A33, 26D15 Key words and phrases: convex function, quasi-convex function, fractional integral operators, bounds

1

Introduction

A function f : I → R is said to be convex if the following inequality holds: f (ta + (1 − t)b) ≤ tf (a) + (1 − t)f (b) ∗

(1.1)

Corresponding authors

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for all a, b ∈ I and t ∈ [0, 1]. If inequality (1.1) is reversed, then the function f will be the concave on [a, b]. Convex functions are very useful in mathematical analysis. A lot of integral inequalities have been established due to convex functions in literature (for details see, [2–6, 10, 11, 18–20]. Quasi-convexity is also class of convex functions which is defined as follows: Definition 1.1. ([10]) A function f : I → R is said to be quasi-convex if the following inequality holds: f (ta + (1 − t)b) ≤ max{f (a), f (b)} (1.2) for all a, b ∈ I and t ∈ [0, 1]. Example 1.2. ([11, p. 83]) The function f : [−2, 2] → R, given by  1 x ∈ [−2, −1] f (x) = x2 x ∈ (−1, 2] is not a convex function on [−2, 2] but it is quasi-convex function on [−2, 2]. It is noted that class of quasi-convex functions contain the class of finite convex functions defined on finite closed intervals. For some recent citations and utilizations of quasiconvex functions one can see [2, 10, 11, 20] and references therein. Fractional integral operators play an important role in generalizing the mathematical inequalities. In recent years, authors have proved various interesting mathematical inequalities due to different fractional integral operators, for example see [3–8 11, 15, 20]. The upcoming definitions and remark provide a detailed information of recent and classical fractional integral operators. Definition 1.3. Let f ∈ L1 [a, b] with 0 ≤ a < b. Then Riemann-Liouville fractional integral operators of order µ > 0 are defined by Z x 1 µ Ia+ f (x) = (1.3) (x − t)µ−1 f (t)dt, x > a Γ(µ) a and µ

Ib− f (x) =

1 Γ(µ)

Z

b

(t − x)µ−1 f (t)dt,

(1.4)

x < b,

x

where Γ(µ) is the Gamma function defined by Γ(µ) =

R∞ 0

tµ−1 e−t dt.

Definition 1.4. ([16]) Let f ∈ L1 [a, b] with 0 ≤ a < b. Then Riemann-Liouville kfractional integral operators of order µ, k > 0 are defined by Z x µ 1 µ k Ia+ f (x) = (1.5) (x − t) k −1 f (t)dt, x > a kΓk (µ) a and µ k Ib− f (x)

1 = kΓk (µ)

Z

b

µ

(t − x) k −1 f (t)dt,

(1.6)

x < b,

x

where Γk (µ) is the k-Gamma function defined as Γk (µ) =

455

R∞ 0

k

tµ−1 e− tk dt.

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Definition 1.5. ([14]) Let f ∈ L1 [a, b] with 0 ≤ a < b. Also let g be an increasing and positive function on (a, b], having a continuous derivative g 0 on (a, b). The left-sided and right-sided fractional integrals of a function f with respect to another function g on [a, b] of order µ > 0, are defined by Z x 1 µ I + f (x) = (1.7) (g(x) − g(t))µ−1 g 0 (t)f (t)dt, x > a g a Γ(µ) a and µ g Ib− f (x)

=

1 Γ(µ)

b

Z

(g(t) − g(x))µ−1g 0 (t)f (t)dt,

x < b.

(1.8)

x

Definition 1.6. ([15]) Let f ∈ L1 [a, b] with 0 ≤ a < b. Also let g be an increasing and positive function on (a, b], having a continuous derivative g 0 on (a, b). The left-sided and right-sided fractional integrals of a function f with respect to another function g on [a, b] of order µ, k > 0 are defined by Z x µ 1 µ k I f (x) = (1.9) (g(x) − g(t)) k −1 g 0 (t)f (t)dt, x > a g a+ kΓk (µ) a and µ k g Ib− f (x)

=

1 kΓk (µ)

Z

b

µ

(g(t) − g(x)) k −1 g 0 (t)f (t)dt,

x < b.

(1.10)

x

These are compact formulas which give almost all fractional integrals by choosing suitable formations of function g. In this context the following remark is important: Remark 1.7. Fractional integrals elaborated in (1.9) and (1.10) particularly produce several known fractional integrals corresponding to different settings of k and g. (i) For k = 1 (1.9) and (1.10) fractional integrals coincide with (1.7) and (1.8). (ii) For taking g as identity function (1.9) and (1.10) fractional integrals coincide with (1.5) and (1.6). (iii) For k = 1, along with g as identity function (1.9) and (1.10) fractional integrals coincide with (1.3) and (1.4). ρ (iv) For k = 1 and g(x) = xρ , ρ > 0, (1.9) and (1.10) produce Katugampola fractional integrals defined by Chen et al. in [1]. τ +s (v) For k = 1 and g(x) = xτ +s , (1.9) and (1.10) produce generalized conformable fractional integrals defined by Khan et al. in [13]. s s (vi) If we take g(x) = (x−a) , s > 0 in (1.9) and g(x) = − (b−x) s s , s > 0 in (1.10), then conformable (k, s)-fractional integrals are achieved as defined by Sidra et al. in [9]. 1+s (vii) If we take g(x) = x1+s , then conformable fractional integrals are achieved as defined by Sarikaya et al. in [17]. (x−a)s (b−x)s (viii) If we take g(x) = s , s > 0 in (1.9) and g(x) = − s , s > 0 in (1.10) with k = 1, then conformable fractional integrals are achieved as defined by Jarad et al. in [12]. The rest of paper is organized as follows: In Section 2, the bounds of sum of left-sided and right-sided generalized fractional integrals via quasi-convex function are established. First result provides an upper bound for generalized fractional integrals, and some particular cases are elaborated. Then bounds

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along with particular cases, in modulus form have been presented. Furthermore, Hadamarad type bounds are formulated. In Section 3, applications of results of Section 2 are given. Moreover concluding remarks are included at the end. In the next sections the notation Mab (f ) = max{f (a), f (b)} has been used frequently.

2

Main Results

Firstly, the following theorem set the formula for upper bounds of fractional integrals via quasi-convex functions in a unified form. Theorem 2.1. Let f, g : [a, b] −→ R be two functions such that g be differentiable and f ∈ L[a, b] with a < b. Also let f be positive, quasi-convex and g be strictly increasing function with g 0 ∈ L[a, b]. Then for x ∈ [a, b] and µ, ν ≥ k, the following inequality holds: µ

µ k g Ia+ f (x)

+νg

Ibk− f (x)

ν

(g(x) − g(a)) k x (g(b) − g(x)) k b ≤ Ma (f ) + Mx (f ). kΓk (µ) kΓk (ν)

(2.1)

Proof. As f is quasi-convex, therefore for t ∈ [a, x], f (t) ≤ Max (f ). Under assumptions on function g, for all x ∈ [a, b], t ∈ [a, x] and µ ≥ k, the following inequality holds: µ

µ

g 0 (t)(g(x) − g(t)) k −1 ≤ g 0 (t)(g(x) − g(a)) k −1 .

(2.2)

From aforementioned two inequalities, the following integral inequality is yielded: Z x Z x µ µ −1 0 −1 x k k (g(x) − g(t)) f (t)g (t)dt ≤ (g(x) − g(a)) Ma (f ) g 0 (t)dt. a

(2.3)

a

By using (1.9) of Definition 1.6, the following bound of fractional integral defined in (1.9) is obtained: µ

µ k g Ia+ f (x)

≤

(g(x) − g(a)) k x Ma (f ). kΓk (µ)

(2.4)

Again from quasi-convexity of f , for t ∈ [x, b], f (t) ≤ Mxb (f ). Also for x ∈ [a, b], t ∈ [x, b] and ν ≥ k, the following inequality holds: ν

ν

g 0 (t)(g(t) − g(x)) k −1 ≤ g 0 (t)(g(b) − g(x)) k −1 .

(2.5)

From aforementioned two inequalities, the following integral inequality is yielded: Z b Z b ν ν −1 0 −1 b k k (g(t) − g(x)) f (t)g (t)dt ≤ (g(b) − g(x)) Mx (f ) g 0 (t)dt. x

(2.6)

x

By using (1.10) of Definition 1.6, the following bound of fractional integral defined in (1.10) is obtained: ν

ν k g Ib− f (x)

(g(b) − g(x)) k b ≤ Mx (f ). kΓk (ν)

(2.7)

From (2.4) and (2.7), the bound of sum of left-sided and right-sided fractional integrals is achieved.

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Special cases of Theorem 2.1, are discussed in the following corollaries. Corollary 2.2. If we take µ = ν in (2.1), then we get the following fractional integral inequality: µ k g Ia+ f (x)

+µg Ibk− f (x) ≤

  µ µ 1 (g(x) − g(a)) k Max (f ) + (g(b) − g(x)) k Mxb (f ) . (2.8) kΓk (µ)

Corollary 2.3. If we take k = 1 in (2.1), then we get the following generalized (RL) fractional integral inequality: µ ν g Ia+ f (x) +g

Ib− f (x) ≤

(g(x) − g(a))µ x (g(b) − g(x))ν b Ma (f ) + Mx (f ). Γ(µ) Γ(ν)

(2.9)

Corollary 2.4. If we take g(x) = x in (2.1), then we get the following (RL) k-fractional integral inequality: µ

µ k Ia+ f (x)

ν

+

Ibk− f (x)

ν

(x − a) k x (b − x) k b ≤ Ma (f ) + M (f ). kΓk (µ) kΓk (ν) x

(2.10)

Corollary 2.5. If we take g(x) = x and k = 1 in (2.1), then we get the following (RL) fractional integral inequality: µ

Ia+ f (x) +ν Ib− f (x) ≤

(x − a)µ x (b − x)ν b Ma (f ) + Mx (f ). Γ(µ) Γ(ν)

(2.11)

Corollary 2.6. Under the assumptions of above theorem if f is increasing on [a, b], then from (2.1), we get the following fractional integral inequality: µ

µ k g Ia+ f (x)

+νg

Ibk− f (x)

ν

(g(b) − g(x)) k (g(x) − g(a)) k ≤ f (x) + f (b). kΓk (µ) kΓk (ν)

(2.12)

Corollary 2.7. Under the assumptions of above theorem if f is decreasing on [a, b], then from (2.1), we get the following fractional integral inequality: µ

µ k ν g Ia+ f (x) +g

Ibk− f (x)

ν

(g(x) − g(a)) k (g(b) − g(x)) k ≤ f (a) + f (x). kΓk (µ) kΓk (ν)

(2.13)

Next theorem provides the bound of generalized fractional integrals in modulus form. Theorem 2.8. Let f, g : [a, b] −→ R be two differentiable functions with a < b. Also let |f 0 | be quasi-convex and g be strictly increasing with g 0 ∈ L[a, b]. Then for x ∈ [a, b] and µ, ν, k > 0, the following inequality holds: ! µ ν (g(x) − g(a)) k (g(b) − g(x)) k µ k ν k f (a) + f (b) (2.14) g Ia+ f (x) +g Ib− f (x) − Γk (µ + k) Γk (ν + k) µ

ν

(g(x) − g(a)) k (x − a) x 0 (g(b) − g(x)) k (b − x) b 0 ≤ Ma (|f |) + Mx (|f |). Γk (µ + k) Γk (ν + k)

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Proof. As |f 0 | is quasi-convex, therefore for t ∈ [a, x], we have |f 0 (t)| ≤ Max (|f 0 |).

(2.15)

f 0 (t) ≤ Max (|f 0 |).

(2.16)

From (2.15), we have Under assumptions of the function g, the following inequality holds: µ

µ

(g(x) − g(t)) k ≤ (g(x) − g(a)) k

(2.17)

for all x ∈ [a, b], t ∈ [a, x] and µ, k > 0. From (2.16) and (2.17), we have Z x Z µ µ (g(x) − g(t)) k f 0 (t)dt ≤ (g(x) − g(a)) k Max (|f 0 |) a

x

dt,

(2.18)

a

the left hand side calculate as follows: Z x µ (g(x) − g(t)) k f 0 (t)dt a

= f (t)(g(x) − g(t))

µ k

|xa

µ + k

µ k

Z

x

µ

(g(x) − g(t)) k −1 f (t)g 0 (t)dt

a

= −f (a)(g(x) − g(a)) + Γk (µ + k)µg Iak+ f (x). Using above calculation in (2.18), we get the following inequality: µ

µ k g Ia+ f (x) −

µ

(g(x) − g(a)) k (g(x) − g(a)) k (x − a) x 0 f (a) ≤ Ma (|f |). Γk (µ + k) Γk (µ + k)

(2.19)

Also from (2.15), we can write f 0 (t) ≥ −Max (|f 0 |).

(2.20)

Following the same procedure as we did for (2.16), we also have µ

µ

(g(x) − g(a)) k (g(x) − g(a)) k (x − a) x 0 f (a) −µg Iak+ f (x) ≤ Ma (|f |). Γk (µ + k) Γk (µ + k) From (2.19) and (2.21), we get the following modulus inequality: µ (g(x) − g(a)) kµ (x − a) (g(x) − g(a)) k µ k I f (x) − f (a) Max (|f 0 |). g a+ ≤ Γk (µ + k) Γk (µ + k)

(2.21)

(2.22)

Again by using quasi-convexity of |f 0 |, for t ∈ [x, b], we have |f 0 (t)| ≤ Mxb (|f 0 |).

(2.23)

Now for x ∈ [a, b], t ∈ [x, b] and ν, k > 0, the following inequality holds: ν

ν

(g(t) − g(x)) k ≤ (g(b) − g(x)) k .

459

(2.24)

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By adopting the same way as we have done for (2.16), (2.17) and (2.20) one can get from (2.23) and (2.24) the following modulus inequality: ν (g(b) − g(x)) kν (b − x) (g(b) − g(x)) k ν k f (b) ≤ Mxb (|f 0 |). (2.25) g Ib− f (x) − Γk (ν + k) Γk (ν + k)

From (2.22) and (2.25) via triangular inequality, we get the modulus inequality in (2.14), which is required. Special cases of Theorem 2.8, are discussed in the following corollaries.

Corollary 2.9. If we take µ = ν in (2.14), then we get the following fractional integral inequality:   µ k µ µ 1 I + f (x) +µ I k− f (x) − k f (a) + (g(b) − g(x)) k f (b) (g(x) − g(a)) g b g a (2.26) Γk (µ + k)   µ µ 1 ≤ (g(x) − g(a)) k (x − a)Max (|f 0 |) + (g(b) − g(x)) k (b − x)Mxb (|f 0 |) . Γk (µ + k)

Corollary 2.10. If we take k = 1 in (2.14), then we get the following generalized (RL) fractional integral inequality:   µ ν µ Ia+ f (x) +ν Ib− f (x) − (g(x) − g(a)) f (a) + (g(b) − g(x)) f (b) (2.27) g g Γ(µ + 1) Γ(ν + 1) (g(x) − g(a))µ(x − a) x 0 (g(b) − g(x))ν (b − x) b 0 ≤ Ma (|f |) + Mx (|f |). Γ(µ + 1) Γ(ν + 1)

Corollary 2.11. If we take g(x) = x in (2.14), then we get the following (RL) k-fractional integral inequality: ! µ ν k k (x − a) (b − x) µ k f (a) + f (b) (2.28) Ia+ f (x) +ν Ibk− f (x) − Γk (µ + k) Γk (ν + k) µ

ν

(x − a) k +1 x 0 (b − x) k +1 b 0 ≤ Ma (|f |) + M (|f |). Γk (µ + k) Γk (ν + k) x

Corollary 2.12. If we take g(x) = x and k = 1 in (2.14), then we get the following (RL) fractional integral inequality:   µ ν µ Ia+ f (x) +ν Ib− f (x) − (x − a) f (a) + (b − x) f (b) (2.29) Γ(µ + 1) Γ(ν + 1) (x − a)µ+1 x 0 (b − x)ν+1 b 0 ≤ Ma (|f |) + M (|f |). Γ(µ + 1) Γ(ν + 1) x Corollary 2.13. Under the assumptions of above theorem if |f 0 | is increasing on [a, b], then from (2.14), we get the following fractional integral inequality: ! µ ν (g(x) − g(a)) k (g(b) − g(x)) k µ k ν k f (a) + f (b) (2.30) g Ia+ f (x) +g Ib− f (x) − Γk (µ + k) Γk (ν + k) µ

ν

(g(x) − g(a)) k (x − a) 0 (g(b) − g(x)) k (b − x) 0 ≤ |f (x)| + |f (b)|. Γk (µ + k) Γk (ν + k)

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Corollary 2.14. Under the assumptions of above theorem if |f 0 | is decreasing on [a, b], then from (2.14), we get the following fractional integral inequality: ! µ ν (g(b) − g(x)) k (g(x) − g(a)) k µ k ν k f (a) + f (b) (2.31) g Ia+ f (x) +g Ib− f (x) − Γk (µ + k) Γk (ν + k) µ

ν

(g(x) − g(a)) k (x − a) 0 (g(b) − g(x)) k (b − x) 0 ≤ |f (a)| + |f (x)|. Γk (µ + k) Γk (ν + k)

We need the following lemma in the proof of next result. Lemma 2.15. Let f : [0, ∞) → R be a quasi-convex function. If f (x) = f (a + b − x), then for x ∈ [a, b], the following inequality holds:   a+b f ≤ f (x). (2.32) 2 Proof. We have a+b 1 = 2 2



   x−a b−x 1 x−a b−x b+ a + a+ b . b−a b−a 2 b−a b−a

As f is quasi-convex, therefore for x ∈ [a, b], we have   a+b f ≤ max {f (x), f (a + b − x)}. 2

(2.33)

(2.34)

Using given condition f (x) = f (a + b − x) in (2.34), then inequality in (2.32) is established. Theorem 2.16. Let f, g : [a, b] −→ R be two functions such that g be differentiable and f ∈ L[a, b] with a < b. Also let f be positive, quasi-convex, f (x) = f (a + b − x) and g be strictly increasing with g 0 ∈ L[a, b]. Then for x ∈ [a, b] and µ, ν, k > 0, the following inequalities hold: #  µ " ν a+b (g(b) − g(a)) k +1 (g(b) − g(a)) k +1 f + (2.35) 2 Γk (ν + 2k) Γk (µ + 2k) ≤

ν+k k Ib− f (a) g

+µ+k Iak+ f (b) g

" # µ ν 1 (g(b) − g(a)) k +1 (g(b) − g(a)) k +1 ≤ + Mab (f ). k Γk (ν + k) Γk (µ + k) Proof. As f is quasi-convex, therefore for x ∈ [a, b], we have f (x) ≤ Mab (f ).

(2.36)

Under assumptions of the function g, the following inequality holds: ν

ν

g 0 (x)(g(x) − g(a)) k ≤ g 0 (x)(g(b) − g(a)) k

(2.37)

for all x ∈ [a, b] and ν, k > 0.

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From (2.36) and (2.37), we have Z b Z b ν ν (g(x) − g(a)) k f (x)g 0(x)dx ≤ (g(b) − g(a)) k Mab (f ) g 0 (x)dx. a

a

By using (1.10) of Definition 1.6, we get ν

ν+k k Ib− f (a) g

≤

(g(b) − g(a)) k +1 b Ma (f ). kΓk (ν + k)

(2.38)

Now for x ∈ [a, b] and µ, k > 0, the following inequality inequality holds: µ

µ

g 0 (x)(g(b) − g(x)) k ≤ g 0 (x)(g(b) − g(a)) k .

(2.39)

From (2.36) and (2.39), we have Z b Z b µ µ (g(b) − g(x)) k f (x)g 0(x)dx ≤ (g(b) − g(a)) k Mab (f ) g 0 (x)dx. a

a

By using (1.9) of Definition 1.6, we get µ

µ+k k Ia+ f (b) g

≤

(g(b) − g(a)) k +1 b Ma (f ). kΓk (µ + k)

Adding (2.38) and (2.40), we get the following inequality " # µ ν +1 +1 k k 1 (g(b) − g(a)) (g(b) − g(a)) ν+k k Ib− f (a) +µ+k Iak+ f (b) ≤ + Mab (f ). g g k Γk (ν + k) Γk (µ + k)

(2.40)

(2.41)

ν

Now on the other hand multiplying (2.32) with (g(x) − g(a)) k g 0 (x), then integrating over [a, b], we have  Z b Z b ν ν a+b 0 f (g(x) − g(a)) k g 0 (x)f (x)dx. (2.42) (g(x) − g(a)) k g (x)dx ≤ 2 a a By using (1.10) of Definition 1.6, we get   ν k(g(b) − g(a)) k +1 a+b f ≤ kΓk (ν + k)ν+k Ibk− f (a). g ν +k 2

(2.43)

µ

Similarly, multiplying (2.32) with (g(b) − g(x)) k g 0 (x), then integrating over [a, b], we have µ   k(g(b) − g(a)) k +1 a+b f ≤ kΓk (µ + k)µ+k Iak+ f (b). (2.44) g µ+k 2 Adding (2.43) and (2.44), we get the following inequality #  µ " ν a+b (g(b) − g(a)) k +1 (g(b) − g(a)) k +1 f Iak+ f (b). (2.45) + ≤ν+k Ibk− f (a) +µ+k g g 2 Γk (ν + 2k) Γk (µ + 2k) From (2.41) and (2.45), we get the inequalities in (2.35), which is required.

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Special cases of Theorem 2.16, are discussed in the following corollaries. Corollary 2.17. If we take µ = ν in (2.35), then we get the following fractional integral inequality: # "  µ (g(b) − g(a)) k +1 a+b 2f Iak+ f (b) ≤ µ+k Ibk− f (a) +µ+k g g 2 Γk (µ + 2k) " # µ 2 (g(b) − g(a)) k +1 ≤ Mab (f ). k Γk (µ + k) Corollary 2.18. If we take k = 1 in (2.35), then we get the following generalized (RL) fractional integral inequality:    a+b (g(b) − g(a))ν+1 (g(b) − g(a))µ+1 f + (2.46) 2 Γ(ν + 2) Γ(µ + 2) ≤

≤



ν+1 µ+1 g Ib− f (a) +g (g(b) − g(a))ν+1

Γ(ν + 1)

Ia+ f (b)

 (g(b) − g(a))µ+1 + Mab (f ). Γ(µ + 1)

Corollary 2.19. If we take g(x) = x in (2.35), then we get the following (RL) k-fractional integral inequality: #  " µ ν a+b (b − a) k +1 (b − a) k +1 f + (2.47) 2 Γk (ν + 2k) Γk (µ + 2k) ≤

ν+k k Ib− f (a) +µ+k

Iak+ f (b)

" # µ ν 1 (b − a) k +1 (b − a) k +1 ≤ + Mab (f ). k Γk (ν + k) Γk (µ + k) Corollary 2.20. If we take g(x) = x and k = 1 in (2.35), then we get the following (RL) fractional integral inequality:    a+b (b − a)ν+1 (b − a)µ+1 f + (2.48) 2 Γ(ν + 2) Γ(µ + 2) ν+1

Ib− f (a) +µ+1 Ia+ f (b)  (b − a)ν+1 (b − a)µ+1 ≤ + Mab (f ). Γ(ν + 1) Γ(µ + 1) ≤



Corollary 2.21. Under the assumptions of above theorem if f is increasing on [a, b], then from (2.35), we get the following fractional integral inequality: #  µ " ν a+b (g(b) − g(a)) k +1 (g(b) − g(a)) k +1 f + (2.49) 2 Γk (ν + 2k) Γk (µ + 2k) ≤

ν+k k Ib− f (a) g

+µ+k Iak+ f (b) g

" # µ ν 1 (g(b) − g(a)) k +1 (g(b) − g(a)) k +1 ≤ + f (b). k Γk (ν + k) Γk (µ + k)

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Corollary 2.22. Under the assumptions of above theorem if f is decreasing on [a, b], then from (2.35), we get the following fractional integral inequality: #  µ " ν (g(b) − g(a)) k +1 a+b (g(b) − g(a)) k +1 + (2.50) f 2 Γk (ν + 2k) Γk (µ + 2k) ≤

ν+k k Ib− f (a) g

+µ+k Iak+ f (b) g

# µ ν 1 (g(b) − g(a)) k +1 (g(b) − g(a)) k +1 ≤ + f (a). k Γk (ν + k) Γk (µ + k) "

3

Applications

In this section we give applications of the results proved in the previous section. First we apply Theorem 2.1 and get the following result. Theorem 3.1. Under the assumptions of Theorem 2.1, we have the following fractional integral inequality: ! µ ν 1 (g(b) − g(a)) k (g(b) − g(a)) k µ k ν k + Mab (f ). (3.1) g Ia+ f (b) +g Ib− f (a) ≤ k Γk (µ) Γk (ν) Proof. If we put x = a in (2.1), then we have ν

ν k g Ib− f (a)

((g(b) − g(a)) k b ≤ Ma (f ). kΓk (ν)

(3.2)

If we put x = b in (2.1), then we have µ

µ k g Ia+ f (b)

(g(b) − g(a)) k b ≤ Ma (f ). kΓk (µ)

(3.3)

Adding inequalities (3.2) and (3.3), we get (3.1). Special cases of Theorem 3.1, are discussed in the following corollaries. Corollary 3.2. If we take µ = ν in (3.1), then we have the following fractional integral inequality: µ

µ k µ k g Ia+ f (b) +g Ib− f (a)

2(g(b) − g(a)) k b ≤ Ma (f ). kΓk (µ)

(3.4)

Corollary 3.3. ([2]) If we take µ = k = 1 and g(x) = x in (3.4), then we get the following inequality: Z b 1 f (t)dt ≤ Mab (f ). (3.5) b−a a Next we apply Theorem 2.8 to obtain required results.

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Theorem 3.4. Under the assumptions of Theorem 2.8, we have the following fractional integral inequality: ! µ ν (g(b) − g(a)) k (g(b) − g(a)) k µ k ν k f (a) + f (b) (3.6) g Ia+ f (b) +g Ib− f (a) − Γk (µ + k) Γk (ν + k) ! µ ν (g(b) − g(a)) k (g(b) − g(a)) k ≤ + (b − a)Mab (|f 0 |). Γk (µ + k) Γk (ν + k) Proof. If we put x = a in (2.14), then we have ν (g(b) − g(a)) kν (b − a) (g(b) − g(a)) k ν k f (b) ≤ Mab (|f 0 |). g Ib− f (a) − Γk (ν + k) Γk (ν + k)

If we put x = b in (2.14), then we have µ (g(b) − g(a)) kµ (b − a) (g(b) − g(a)) k µ k f (a) ≤ Mab (|f 0 |). g Ia+ f (b) − Γk (µ + k) Γk (µ + k)

(3.7)

(3.8)

Adding inequalities (3.7) and (3.8), we get (3.6).

Special cases of Theorem 3.4, are discussed in the following corollaries. Corollary 3.5. If we take µ = ν in (3.6), then we have the following fractional integral inequality: µ (g(b) − g(a)) k µ k µ k (3.9) (f (a) + f (b)) g Ia+ f (b) +g Ib− f (a) − Γk (µ + k) µ

2(g(b) − g(a)) k (b − a) b 0 ≤ Ma (|f |). Γk (µ + k)

Corollary 3.6. If we take µ = k = 1 and g(x) = x in (3.9), then we get the following inequality: Z b 1 f (a) + f (b) b 0 f (t)dt − (3.10) b − a ≤ (b − a)Ma (|f |). 2 a By applying Theorem 2.16 similar relations can be established we leave it for the reader.

4

Concluding Remarks

The aim of this study is to explore bounds of fractional integrals in a compact form by using the concept of quasi-convexity. The authors are succeeded in the formulation of bounds of generalized fractional integrals (1.9) and (1.10). Theorem 2.1 provides upper bounds, Theorem 2.8 gives bounds in modulus form while Theorem 2.16 formulates bounds of Hadamard type. Section 3 consists of the applications of these bounds. Also some particular case of all these results are shown. Remark 1.7 includes all possible fractional integrals associated with generalized fractional integrals (1.9) and (1.10). The readers can obtain bounds for desired fractional integrals by putting the corresponding function g from Remark 1.7.

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Acknowledgement This research work is supported by Higher Education Commission of Pakistan under NRPU 2016, Project No. 5421.

References [1] H. Chen and U. N. Katugampola, Hermite-Hadamard and Hermite-Hadamard-Fej´er type inequalities for generalized fractional integrals, J. Math. Anal. Appl., 446 (2017), 1274–1291. [2] S. S. Dragomir and C. E. M. Pearce, Quasi-convex functions and Hadamard’s inequality, Bull. Austral. Math. Soc., 57 (1998), 377–385. [3] G. Farid, Some Riemann-Liouville fractional integral inequalities for convex functions, J. Anal., (2018), doi.org/10.1007/s41478-0079-4. [4] G. Farid, K. A. Khan, N. Latif, A. U. Rehman and S. Mehmood, General fractional integral inequalities for convex and m-convex functions via an extended generalized Mittag-Leffler function, J. Inequal. Appl., 2018 (2018), Paper No. 243, 12 pages. [5] G. Farid, W. Nazeer, M. S. Saleem, S. Mehmood and S. M. Kang, Bounds of RiemannLiouville fractional integrals in general form via convex functions and their applications, Math., 6 (2018), Article ID 248, 10 pages. [6] G. Farid, A. U. Rehman and S. Mehmood, Hadamard and Fej´er-Hadamard type integral inequalities for harmonically convex functions via an extended generalized Mittag-Leffler function, J. Math. Comput. Sci., 8 (2018), 630–643. [7] G. Farid, A. U. Rehman and M. Zahra, On Hadamard inequalities for k-fractional integrals, Nonlinear Funct. Anal. Appl., 21 (2016), 463–478. [8] G. Farid, A. U. Rehman and M. Zahra, On Hadamard inequalities for relative convex function via fractional integrals, Nonlinear Anal. Forum, 21 (2016), 77–86. [9] S. Habib, S. Mubeen, M. N. Naeem, Chebyshev type integral inequalities for generalized k-fractional conformable integrals, J. Inequal. Spec. Funct., 9 (2018), 53–65. [10] R. Hussain, A. Ali, A. Latif and G. Gulshan, Some k-fractional associates of HermiteHadamard’s inequality for quasi-convex functions and applications to special means, Fract. Differ. Calc., 7 (2017), 301–309. [11] D. A. Ion, Some estimates on the Hermite-Hadamard inequality through quasi-convex functions, Annals of University of Craiova, Math. Sci. Ser., 34 (2007), 82–87. [12] F. Jarad, E. Ugurlu, T. Abdeljawad and D. Baleanu, On a new class of fractional operators, Adv. Difference Equ., 2017 (2017), Paper No. 247, 16 pages. [13] T. U. Khan and M. A. Khan, Generalized conformable fractional operators, J. Comput. Appl. Math., 346 (2019), 378–389.

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[14] A. A. Kilbas, H. M. Srivastava and J. J. Trujillo, Theory and applications of fractional differential equations, North-Holland Mathematics Studies, 204, Elsevier, New YorkLondon, 2006. [15] Y. C. Kwun, G. Farid, N. Latif, W. Nazeer and S. M. Kang, Generalized RiemannLiouville k-fractional integrals associated with Ostrowski type inequalities and error bounds of Hadamard Inequalities, IEEE Access, 6 (2018), 64946–64953. [16] S. Mubeen and G. M. Habibullah, k-fractional integrals and application, Int. J. Contemp. Math. Sci., 7 (2012), 89–94. [17] M. Z. Sarikaya, M. Dahmani, M. E. Kiris and F. Ahmad, (k, s)-Riemann-Liouville fractional integral and applications, Hacet. J. Math. Stat., 45 (2016), 77–89. ¨ [18] E. Set, M. Sardari, M. E. Ozdemir and J. Rooin, On generalizations of the Hadamard inequality for (α, m)-convex functions, RGMIA, Res. Rep. Coll., 12 (2009), Article ID 4, 10 pages. [19] W. Sun and Q. Liu, New Hermite-Hadamard type inequalities for (α, m)-convex functions and applications to special means, J. Math. Inequal., 11 (2017), 383–397. [20] S. Ullah, G. Farid, K. A. Khan, A. Waheed and S. Mehmood, Generalized fractional inequalities for quasi-convex functions, Adv. Difference Equ., 2019 (2019), Paper No. 15, 16 pages.

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Asymptotically Almost Automorphic Mild Solutions for Second Order Nonautonomous Semilinear Evolution Equations Mouffak Benchohraa,b , Gaston M. N’Gu´ er´ ekatac and Noreddine d Rezoug a

Laboratory of Mathematics, Djillali Liabes, University of Sidi Bel-Abbes, PO Box 89, Sidi Bel-Abbes 22000, Algeria. E-mail: [email protected] b Department

of Mathematics, College of Science, King Saud University, P.O. Box 2455, Riyadh 11451, Saudi Arabia

c Department

d

of Mathematics, Morgan State University, 1700 E. Cold Spring Lane, Baltimore M.D. 21252, USA Gaston.N’[email protected]

Department of Mathematics, University of Relizane, Relizane, 48000, Algeria. E-mail: [email protected] Abstract The aim of this paper is to study the existence of asymptotically almost automorphic mild solution to some classes of second order semilinear evolution equation via the techniques of measure of noncompactness. The investigation is based on a new fixed point result which is a generalization of the well known Darbo’s fixed point theorem. Finally examples are given to illustrate the analytical findings.

Key words: Asymptotically almost automorphic, second order nonautonomous differential equations, mild solution, evolution system, Kuratowski measures of noncompactness, fixed point. AMS Subject Classification : 34G20

1

Introduction

This work is mainly concerned with the existence of asymptotically almost automorphic mild solution for second differential equations. More

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M. Benchohra, G.M. N’Gu´er´ekata and N. Rezoug

precisely, we will consider the following problem y 00 (t) − A(t)y(t) = f (t, y(t)), t ∈ R+ := [0, +∞),

(1)

y(0) = y0 , y 0 (0) = y1 ,

(2)

where {A(t)}t∈R+ is a family of linear closed operators from E into E that generate an evolution system of linear bounded operators {U(t, s)}(t,s)∈R+ ×R+ for 0 ≤ s ≤ t < +∞, f : R+ ×E → E is a Carath´eodory function, and (E, |·|) is a real Banach space. Evolution equations arise in many areas of applied mathematics [2, 37]. This type of equations has received much attention in recent years [1]. There are many results concerning the second-order differential equations, see for example [8, 11, 12, 20, 28, 35]. In recent years there has been an increasing interest in studying the abstract non-autonomous second order initial value problem y 00 (t) − A(t)y(t) = f (t, y(t)), t ∈ [0, T ], (3) y(0) = y0 , y 0 (0) = y1 .

(4)

The reader is referred to [10, 19, 22, 36] and the references therein. In the above mentioned works, the existence of solutions to the problem (3)-(4) is related to the existence of an evolution operator U(t; s) for the homogeneous equation y 00 (t) = A(t)y(t), for t ≥ 0. For this purpose there are many techniques to show the existence of U(t, s) which has been developed by Kozak [25]. On the other hand, since Bochner [13] introduced the concept of almost automorphy, the automorphic functions have been applied to many areas including ordinary as well as partial differential equations, abstract differential equations, functional differential equations, integral equations, etc.; see [16, 21, 18, 27, 7]. We also refer the reader to the monographs by N’Gu´er´ekata [30, 31] for the basic theory of almost automorphic functions and applications. The concept of asymptotically almost automorphy was introduced by N’Gu´er´ekata [29]. Since then, these functions have generated lot of developments and applications, see [39, 14, 24, 17] and the references therein. In the previous works, people have established the existence of asymptotically almost automorphic mild solution of differential equations under the conditions that f satisfies or not the Lipschitz condition.

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Asymptotically almost automorphic mild solutions

3

In this paper we use the technique of measures of noncompactness. It is well known that this method provides an excellent tool for obtaining existence of solutions of nonlinear differential equation. This technique works fruitfully for both integral and differential equations. More details are found in Aissani and Benchohra [3], Akhmerov et al. [4], Al´ vares [5], Bana´s and Goebel [9], Olszowy and W¸edrychowicz [33], Olszowy [34], and the references therein. Inspired by the above works,, in this work, using the properties of the analytic semigroups, Kuratowski measure of noncompactness, fixed point theorem, we obtain an existence result without assuming that the nonlinearity f satisfies a Lipschitz type condition. This work is organized of as follows. In Section 2, we recall some fundamental properties of asymptotically almost automorphic and facts about evolution systems. Section 3 is devoted to establishing some criteria the existence of asymptotically almost automorphic mild solutions to the problem (1)-(2). Furthermore, appropriate examples are provided in section 4 to show the feasibility of our results.

2

Preliminaries and basic results

In this section we recall certain definitions and lemmas to be used subsequently in this paper. Throughout this paper, we denote by E a Banach space with the norm | · |. Let BC(R+ , E) be the Banach space of all bounded and continuous functions y mapping R+ into E endowed with the usual supremum norm kyk∞ = sup |y(t)|. t∈R+

In what follows, let {A(t), t ∈ R+ } be a family of closed linear operators on the Banach space E with domain D(A(t)) which is dense in E and independent of t. In this work the existence of solution the problem (1)-(2) is related to the existence of an evolution operator U(t, s) for the following homogeneous problem y 00 (t) = A(t)y(t) t ∈ R+ . (5)

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M. Benchohra, G.M. N’Gu´er´ekata and N. Rezoug

This concept of evolution operator has been developed by Kozak [25] and recently used by Henr´ıquez et al. [22]. Definition 2.1 A family U of bounded operators U(t, s) : E → E, (t, s) ∈ ∆ := {(t, s) ∈ R+ × R+ : s ≤ t}, is called an evolution operator of the equation (5) if de following conditions hold: (e1 ) For any x ∈ E the map (t, s) 7−→ U(t, s)x is continuously differentiable and (a) for each t ∈ R, U(t, t)x = 0, ∀x ∈ E, ∂ (b) for all (t, s) ∈ ∆ and for any x ∈ E, U(t, s)x|t=s = x and ∂t ∂ U(t, s)x|t=s = −x. ∂s (e2 ) For all (t, s) ∈ ∆, if x ∈ D(A(t)), then (t, s) 7−→ U(t, s)x is of class C 2 and

∂ U(t, s)x ∈ D(A(t)), the map ∂s

∂2 U(t, s)x = A(t)U(t, s)x, ∂t2 ∂2 (b) U(t, s)x = U(t, s)A(s)x, ∂s2 ∂2 (c) U(t, s)x|t=s = 0. ∂s∂t

(a)

(e3 ) For all (t, s) ∈ ∆, then ∂3 U(t, s)x and ∂s2 ∂t

∂3 ∂ U(t, s)x ∈ D(A(t)), there exist 2 U(t, s)x, ∂s ∂t ∂s

∂ ∂3 U(t, s)x = A(t) (t)U(t, s)x. 2 ∂t ∂s ∂s ∂ Moreover, the map (t, s) 7−→ A(t) (t)U(t, s)x is continuous, ∂s 3 ∂ ∂ U(t, s)x = U(t, s)A(s)x. (b) ∂s2 ∂t ∂t (a)

Throughout this paper, we will use the following definition of the concept of Kuratowski measure of noncompactness [9].

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Asymptotically almost automorphic mild solutions

5

Definition 2.2 The Kuratowski measure of noncompactness α is defined by α(D) = inf{r > 0 : D has a finite cover by sets of diameter ≤ r}, for a bounded set D in any Banach space E. Let us recall the basic properties of Kuratowski measure of noncompactness. Lemma 2.3 [9] Let E be a Banach space and C, D ⊂ E be bounded, then the following properties hold: (i1 ) α(D) = 0 if only if D is relatively compact, (i2 ) α(D) = α(D) ; D the closure of D, (i3 ) α(C) ≤ α(D) when C ⊂ D, (i4 ) α(C +D) ≤ α(C)+α(D) where C +D = {x | x = y + z; y ∈ C; z ∈ D}, (i5 ) α(aD) = |a|α(D) for any a ∈ R, (i6 ) α(ConvD) = α(D), where ConvD is the convex hull of D, (i7 ) µ(C ∪ D) = max(α(C), α(D)), (i8 ) α(C ∪ {x}) = α(C) for any x ∈ E. Denote by ω T (y, ε) the modulus of continuity of y on the interval [0, T ] i.e. ω T (y, ε) = sup {|y(t) − y(s)| ; t, s ∈ [0, T ], |t − s| ≤ ε} . Moreover, let us put  ω T (D, ε) = sup ω T (y, ε); y ∈ D , ω0T (D) = lim ω T (D, ε). ε→0

Lemma 2.4 [15] Let E be a Banach space, D ⊂ E be bounded. Then there exists a countable set D0 ⊂ D, such that α(D) ≤ 2α(D0 ).

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M. Benchohra, G.M. N’Gu´er´ekata and N. Rezoug

+ Lemma 2.5 [23] Let D = {yn }+∞ n=0 ⊂ C(R , E) be a bounded and countable set. Then α(D(t)) is Lebesgue integrable on R+ , and Z t ∞ Z t α yn (s))ds α(D(s))ds, t ∈ R+ . ≤2 0

n=0

0

Now, we recall some basic definitions and results on almost automorphic functions and asymptotically almost automorphic functions (for more details, see [13, 31, 38]). Definition 2.6 A continuous function f : R → E is said to be almost automorphic if for every sequence of real numbers {τn0 }, there exists a subsequence {τn } such that g(t) = lim f (t + τn ) n→∞

is well defined for each t ∈ R and lim g(t − τn ) = f (t)

n→∞

for each t ∈ R.

Denote by AA(R, E) the set of all such functions. Lemma 2.7 [30] AA(R, E) is a Banach space with the supremum norm kf k∞ = sup |f (t)|. t∈R

Definition 2.8 A continuous function f : R × E → E is said to be almost automorphic in t ∈ R for each y ∈ E if for every sequence of real numbers {τn0 }, there exists a subsequence {τn } such that lim f (t + τn , y) = g(t, y)

n→∞

is well defined for each t ∈ R and lim g(t − τn , y) = f (t, y)

n→∞

for each t ∈ R and each y ∈ E. The collection of those functions is denoted by AA(R × E, E). Example 2.9 [40] The function f : R × E → E given by   1 √ f (t, y) = sin cos y 2 + cos t + cos 2t is almost automorphic in t ∈ R for each y ∈ E, where E = L2 ([0, 1]).

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The space of all continuous functions h : R+ → E such that lim h(t) = 0 t→∞

is denoted by C0 (R+ , E). Moreover, we denote C0 (R+ × E, E); the space of all continuous functions from R × E to E satisfying lim h(t, y) = 0 in t and t→∞

uniformly in y ∈ E.

Remark 2.10 Note that if ν(t) ∈ C0 (R+ , E), then Z t e−(t−s) ν(s)ds ∈ C0 (R+ , E). 0

Definition 2.11 A continuous function f : R+ → E is said to be asymptotically almost automorphic if it can be decomposed as f (t) = g(t) + h(t), where g(t) ∈ AA(R, E), h(t) ∈ C0 (R+ , E). Denote by AAA(R+ , E) the set of all such functions. Example 2.12 The function f : R → R defined by   1 √ f (t) = sin + e−t 2 + cos t + cos 2t is an asymptotically almost automorphic function with   1 √ g(t) = sin ∈ AA(R, R), h(t) = e−t ∈ C0 (R+ , R). 2 + cos t + cos 2t Lemma 2.13 [31],[32]. AAA(R+ , E) is also a Banach space with the norm kf k∞ = sup |f (t)|. t∈R+

Definition 2.14 A continuous function f : R+ × E → E is said to be asymptotically almost automorphic if it can be decomposed as f (t, y) = g(t, y) + h(t, y), where g(t, y) ∈ AA(R × E, E),

h(t, y) ∈ C0 (R+ × E, E).

Denote by AAA(R+ × E, E) the set of all such functions.

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Example 2.15 The function f : R+ × E → E given by   1 √ cos y + e−t |y| f (t, x) = sin 2 + cos t + cos 2t is asymptotically almost automorphic in t ∈ R+ for each y ∈ E, where E = L2 ([0, 1]).  g(t, y) = sin

 1 √ cos y ∈ AA(R × E, E), 2 + cos t + cos 2t h(t, y) = e−t |y| ∈ C0 (R+ × E, E).

. Lemma 2.16 [26] f : R × E → E is almost automorphic, and assume that f (t, ·) is uniformly continuous on each bounded subset K ⊂ E uniformly for t ∈ R, that is for any ε > 0, there exists % > 0 such that y, z ∈ K and |y(t) − z(t)| < % imply that |f (t, y) − f (t, z)| < ε for all t ∈ R. Let ϕ : R → E be almost automorphic. Then the function F : R → E defined by F (t) = f (t, ϕ(t)) is almost automorphic. Theorem 2.17 [6] Let Ω be a nonempty, bounded, closed and convex subset of a Banach space E, and let Γ : Ω → Ω be a continuous operator satisfying the inequality α(Γ(D)) ≤ Ψ(α(D)) for any nonempty subset D of Ω, where Ψ : R+ → R+ is a nondecreasing function such that lim Ψn (t) = 0 for each t ≥ 0.

n→+∞

Then Γ has at least one fixed point in the set Ω.

3

Main results

Definition 3.1 A function y ∈ BC(R+ , E) is said to be a mild solution to the problem (1)-(2) if y satisfies the integral equation Z t ∂ y(t) = − U(t, 0)y0 + U(t, 0)y1 + U(t, s)f (s, y(s))ds. ∂s 0

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For the proof of our main theorem, we need the following hypotheses: (H1 ) (a) There exists a constant M ≥ 1 and δ > 0, such that kU(t, s)kB(E) ≤ M e−δ(t−s)

for any (t, s) ∈ ∆

and for any sequence of real numbers {τn0 }, we can extract a subsequence {τn } and for any ε > 0, there exists N ∈ N such that kU (t + τn , s + τn ) − U (t, s)kB(E) ≤ εe−δ(t−s) , kU (t − τn , s − τn ) − U (t, s)kB(E) ≤ εe−δ(t−s) for each t, s ∈ R. for all n > N, for each t, s ∈ R, t ≥ s. f ≥ 0 and δ > 0, such that: (H2 ) There exist a constant M

∂

fe−δ(t−s) , (t, s) ∈ ∆.

U(t, s) ≤M

∂s

B(E) (H3 ) The function f : R+ × E → E is Carath´eodory and asymptotically almost automorphic i.e., f (t, y) = g(t, y) + h(t, y) with h(t, y) ∈ C0 (R+ × E, E),

g(t, y) ∈ AA(R × E, E),

and g(t, y) is uniformly continuous on any bounded subset K ⊂ E uniformly for t ∈ R. Moreover, (a) There exist p ∈ Lq (R, R+ ), q ∈ [1, ∞) and a continuous nondecreasing function ψ : [0, ∞) → (0, ∞) such that for all t ∈ R+ and y ∈ E, |g(t, y)| ≤ p(t)ψ(|y|)

and

lim inf

|y|→+∞

ψ(|y|) = ρ1 . |y|

(b) There exist a function β(t) ∈ C0 (R, R+ ) and a nondecreasing function Φ : R+ → R+ such that for all t ∈ R+ and y ∈ E with |y| ≤ R, |h(t, y)| ≤ β(t)φ(|y|)

and

476

lim inf

R→+∞

φ(R) = ρ2 . R

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(H4 ) There exist a locally integrable function η : R → R+ and a continuous nondecreasing function ϕ : R+ → R+ such that for any nonempty bounded set D ⊂ E we have : α(f (t, D)) ≤ η(t)ϕ(α(D)) for a.e t ∈ R+ . Additionally we assume that lim (ψ + φ)n (t) = 0 for a.e t ∈ R+ . Let n→+∞

β(t) be the function involved in the assumption (H3 ), then Z

t

e−(t−s) β(s)ds ∈ C0 (R+ , R+ ).

0

Put Z ρ = sup t∈R+

t

e−(t−s) β(s)ds.

0

We need the following technical lemma. Lemma 3.2 Assume that (H1 ) hold. If ϕ(t) ∈ AA(R, E), then t

Z

U (t, s)ϕ(s)ds, t ∈ R,

Λ(t) := −∞

belongs to AA(R, E). Proof. From (H1 ) it is clear that Λ(t) is well-defined and continuous on R. Since ϕ(t) ∈ AA(R, E), it follows that for every sequence of real numbers {τn0 }, we can extract a subsequence {τn } such that (c1 ) lim ϕ(t + τn ) − ϕ(t) e = 0 for each t ∈ R and, n→∞

(c2 ) lim ϕ(t e − τn ) − ϕ(t) = 0 for each t ∈ R. n→∞

Notes that ϕ e is also bounded on R, and measurable. Define Z

t

e = Λ(t) −∞

U(t, s)ϕ(s)ds, e

477

t ∈ R.

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e is well-defined. For t ∈ R, Since ϕ e is measurable, Λ For t ∈ R, we have e Λy)(t + τn ) − (Λy)(t) Z t+τn Z t U(t, s)ϕ(s)ds e U(t + τn , s)ϕ(s)ds − = −∞ Z−∞ Z t t = U(t + τn , s + τn )ϕ(s + τn )ds − U(t, s)ϕ(s)ds e −∞ −∞ Z t kU(t + τn , s + τn )kB(E) |ϕ(s + τn ) − ϕ(s))| e ds ≤ Z −∞ t + kU(t + τn , s + τn ) − U(t, s))kB(E) ϕ(s)ds e Z−∞ t M e−δ(t−s) |ϕ(s + τn ) − ϕ(s))| e ds ≤ −∞ Z t + εe−δ(t−s) |ϕ(s)| e ds −∞ Z t ≤M e−δ(t−s) ds sup |ϕ(s + τn ) − ϕ(s))| e s∈R Z t −∞ +ε e−δ(t−s) ds sup |ϕ(s)| e s∈R

−∞

M ε ≤ sup |ϕ(s + τn ) − ϕ(s))| e + sup |ϕ(s)| e . δ s∈R δ s∈R Using (c1 ), we obtain that for n → ∞, e Λ(t + τn ) → Λ(t). Analogously, one can prove that, e − τn ) → Λ(t) for each t ∈ R as n → ∞. Λ(t This we show that Λ ∈ AA(R, E). Theorem 3.3 Assume that the hypotheses (H1 ) − (H4 ) are satisfied. If M ρ1 kpkLq + M δ −1 ρρ2 < 1,

(6)

and M max(4kηkL1 , kpkLq δ

−1+ 1q

) < 1,

(7)

then the problem (1)-(2) has a asymptotically almost automorphic mild solution.

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Proof. Consider the operator N : AAA(R+ , E) → AAA(R+ , E) defined by ∂ U(t, 0)y0 + U(t, 0)y1 + ∂s

(N y)(t) = −

Z

t

U(t, s)f (s, y(s))ds,

(8)

0

where y ∈ AAA(R+ , E) with y = γ + ζ, γ is the principal term and ζ the corrective term of y. We need to prove that N is weel- defined, that is N (AAA(R+ , E)) ⊂ AAA(R+ , E). Let σ(t) = −

∂ U(t, 0)y0 + U(t, 0)y1 , ∂s

then ∂ |σ(t)| = | − ∂s U(t, 0)y0 + U(t, 0)y1 | ∂ ≤ | ∂s U(t, 0)y0 | + |U(t, 0)y1 | fe−δt |y0 | + M e−δt |y1 |. ≤M

Since δ > 0, we get lim |(σ(t)| = 0. that is t→+∞

σ ∈ C0 (R+ , E).

(9)

By assumption f = g + h where g is the principal term and h the corrective term. So we can write f (t, y(t)) = g(t, γ(t)) + f (t, y(t)) − f (t, γ(t)) + h(t, γ(t)) = g(t, γ(t)) + H(t, y(t)),

(10)

In view of (10), we have Z

t

U(t, s)f (s, y(s))ds

W (t) = 0

Z

t

t

Z U(t, s)g(s, γ(s))ds +

= 0

Z

U(t, s)H(s, y(s))ds 0

t

Z

0

U(t, s)g(s, γ(s))ds −

= −∞ Z t

U(t, s)g(s, γ(s))ds −∞

U(t, s)H(s, y(s))ds

+ 0

= (I1 y)(t) + (I2 y)(t),

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where t

Z

U(t, s)g(s, γ(s))ds,

(I1 y)(t) = −∞

t

Z

U(t, s)H(s, y(s))ds

(I2 y)(t) = 0 0

Z

U(t, s)g(s, y(s))ds

− −∞

= (J1 y)(t) + (J2 y)(t), where t

Z

U(t, s)H(s, y(s))ds,

(J1 y)(t) = 0

Z

t

U(t, s)g(s, γ(s))ds.

(J2 y)(t) = −∞

Using (H3 ) and Lemma 2.16 , we deduce that s → g(s, γ(s)) is in AA(R, E). Thus, by Lemma 3.2 we obtain (I1 y)(t) ∈ AA(R, E).

(11)

Let’s prove that J1 ∈ C0 (R+ , E), J2 ∈ C0 (R+ , E). Ideed by definition H ∈ C0 (R+ , E), that means given ε > 0, there exists T > 0 such that if t ≥ T, we have |H(t, y)| ≤ ε. Therefore if t ≥ T, we get Z

t

Z

t

kU(t, s)kB(E) |H(s, y(s))|ds ≤ M ε

e−δ(t−s) ds

T

T

≤

M ε, δ

then |(J1 y)(t)| ≤

M ε δ

if t ≥ T.

So, J1 ∈ C0 (R+ , E).

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Next, let us show that J2 ∈ C0 (R+ , E). Z 0 |(J2 y)(t)| ≤ kU(t, s)kB(E) |g(s, y(s))|ds −∞ T

Z ≤ M sup |g(t, y(t))| t∈R

+ M kgk∞

e−δ(t−s) ds

0

e−δ(t δ

→ 0 as → ∞.

So, J2 ∈ C0 (R+ , E).

(13)

Finaly combining (9),(11), (12) and (13) proves our claim that N ∈ AAA(R+ , E). Next, we will prove that the operator N satisfies all the assumptions of Theorem 2.17. We will break the proof into several steps. Let  BR = y ∈ AAA(R+ , E) : kyk∞ ≤ R , where R be any positive constant. Then BR is a bounded, closed and convex subset of AAA(R+ , E). Step 1: N (y) ∈ BR for any y ∈ BR . In fact, if we assume that the assertion is false, then R < |(N y)(t)|. This yields that Z

t

R < |(N y)(t)| ≤

kU(t, s)kB(E) |g(s, y(s)|ds

0

Z

t

kU(t, s)kB(E) |h(s, y(s)|ds

+ 0

Z

t

≤

kU(t, s)kB(E) p(s)ψ(|y(s)|)ds

0

Z

t

kU(t, s)kB(E) β(s)φ(|y(s)|)ds Z t ≤ M ψ(R) e−δ(t−s) p(s)ds 0 Z t + M φ(R) e−δ(t−s) β(s)ds. +

0

0

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For t ≥ 0, it follows from the H¨older inequality that R < |(N y)(t)| ≤ M ψ(R)kpkLq + M ρ2 φ(R). Dividing both sides by R and taking the lim inf as R → +∞, we have M ρ1 kpkLq + M δ −1 ρρ2 > 1, which contradicts (6). Hence, the operator N transforms the set BR into itself. Step 2. N is continuous. Let (yn )n∈N be a sequence in BR such that yn → y in BR . Case 1. If t ∈ [0, T ]; T > 0, then, we have Z t |f (s, yn (s)) − f (s, y(s))| ds. |(N yn )(t) − (N y)(t)| ≤ M 0

Since the functions f is Carath´eodory, the Lebesgue dominated convergence theorem implies that kN yn − N yk∞ → 0

as n → +∞.

Case 2. Since the functions f is Carath´eodory, we can see that |f (s, yn (s)) − f (s, y(s))| ≤

δε M

for t ≥ T.

If t ∈ (T, ∞), T > 0, then (14) and the hypotheses give us that Z t |N yn (t) − N y(t)| ≤ kU(t, s)kB(E) f (s, yn (s)) − f (s, y(s)) ds 0 Z δε t −δ(t−s) e ds ≤M M 0 M δε ≤ δ M ≤ ε.

(14)

(15)

Then the inequality (15) reduces to kN (yn ) − N (y)k∞ → 0

as n → ∞.

Now, we conclude that N is continuous from BR to BR . Step 3: N (BR ) is equicontinuous.

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Let t1 , t2 ∈ [0, T ] with t2 > t1 and y ∈ BR . Then, we have |(N 1 y)(t2 ) − (N1 y(t1 )| Z t1 = (U(t2 , s) − U(t1 , s))g(s, y(s)) Z 0t2 + U(t2 , s)g(s, y(s))ds Zt1 t1 + (U(t2 , s) − U(t1 , s))h(s, y(s)) Z 0t2 + U(t2 , s)h(s, y(s))ds Zt1t1 ≤ kU(t2 , s) − U(t1 , s)kB(E) p(s)ψ(|y(s)|)ds 0Z t2 +M e−δ(t−s) p(s)ψ(|y(s)|)ds. Z t1t1 + kU(t2 , s) − U(t1 , s)kB(E) β(s)φ(|y(s)|)ds 0Z t2 +M e−δ(t−s) β(s)φ(|y(s)|)ds. t1

It follows from the H¨older inequality that |(N1 y)(t2 ) − (N1 y(t1 )| Z t1 ≤ kU(t2 , s) − U(t1 , s)kB(E) p(s)ψ(|y(s)|)ds 0  qδ 1− 1q qδ M kpkLq ψ(R) − q−1 (t−t2 ) − q−1 (t−t2 ) e − e + 1− 1 Z t1 δ q + kU(t2 , s) − U(t1 , s)kB(E) β(s)φ(|y(s)|)ds 0

M φ(R) sup β(t) +

t∈R

δ

(e−δ(t−t2 ) − e−δ(t−t1 ) ).

The right-hand side of the above inequality tends to zero as t2 − t1 → 0, which implies that N (BR ) is equicontinuous. Consider the measure of noncompacteness µ(B) defined on the family of bounded subsets of the space AAA(R+ , E) (see [33]) by

µ(B) = ω0T (B) + sup α(B(t)) + lim sup{|y(t)| : t ≥ T, y ∈ E}. T →+∞

t∈J

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−1+ 1

q )(ϕ + ψ)(µ(B)) for all B ⊂ Step 4: µ(N (B)) ≤ M max(4kηkL1 , kpkLq δ BR . For all B ⊂ BR , N (B) is bounded. Hence, by Lemma 2.4, there exists a countable set B1 = {y}∞ n=1 ⊂ B, such that

(N (B)) ≤ 2α(N (B1 )).

(16)

Using the properties of α, Lemma 2.4, Lemma 2.5 and assumptions (H1 ) and (H4 ), we get ∞ 

t

Z α(N B1 (t)) ≤ α

U(t, s)f (s, yn (s))ds 0 t

Z

{α (f (s, yn (s))ds))}∞ n=0 ds

≤ 2M 0 t

Z

n=0

≤ 2M 0

η(s)ϕ ({(α(yn (s))}∞ n=0 ))) ds

t

Z ≤ 2M

η(s)ϕ(α(B(s)))ds. 0

Form inequality (16), it follows that Z α(N B(t)) ≤ 4M

t

η(s)ϕ(α(B(s)))ds, 0

then α(N (B(t)) ≤ 4M kηkL1 ϕ( sup α(B(t))). t∈R+

Since

sup α(B(t)) ≤ sup α(B(t)) + lim sup{|y(t)| : t ≥ T, y ∈ E}), t∈R+

t∈R+

t→+∞

then α(N (B(t)) ≤ 4M kηkL1 ϕ( sup α(B(t)) + lim sup{|y(t)| : t ≥ T, y ∈ E}).(17) t∈R+

t→+∞

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On the other hand, we have fe−δt |y1 | + M e−δt |y0 | |(N y)(t)| ≤ M Z t e−δ(t−s) p(s)ψ(|γ(s)|)ds + |(I2 y)(t)| + M −∞ T

Z + M

e−δ(t−s) p(s)ψ(|γ(s)|)ds.

−∞ t

Z

e−δ(t−s) p(s)ψ(|γ(s)|)ds + |I2 (t)|.

+ M

T −δt

fe ≤ M Z + M

|y1 | + M e−δt |y0 |

T

e−δ(t−s) p(s)dsψ(sup |γ(s)|) s∈R

−∞ t

Z + M

e−δ(t−s) p(s)dsψ(sup{|γ(t)| : t ≥ T, y ∈ E})

T

+ sup{|(I2 y)(t)| : t ≥ T, y ∈ E}).

Next, applying the H¨older inequality we derive fe−δt |y1 | + M e−δt |y0 | |(N y)(t)| ≤ M M kpkLq −δ(t−T ) + e ψ(kyk∞ ). 1− 1 δ q M kpkLq − qδ t 1− 1 (1 − e q−1 ) q ψ(sup{|y(t)| : t ≥ T, y ∈ E}) + 1 1− δ q + sup{|(I2 y)(t)| : t ≥ T, y ∈ E}).

Then fe−δt |y1 | + M e−δt |y0 | |(N y)(t)| ≤ M M kpkLq −δT + e ψ(kyk∞ ). 1− 1 δ q M kpkLq + ψ(sup{|y(t)| : t ≥ T, y ∈ E}) 1− 1 δ q + sup{|(I2 y)(t)| : t ≥ T, y ∈ E}).

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Since δ ≥ 0, I2 ∈ C0 (R+ , E) and lim sup{|y(t)| : t ≥ T, y ∈ E} ≤ sup α(B(t))+ lim sup{|y(t)| : t ≥ T, y ∈ E},

T →+∞

T →+∞

t∈R

then lim sup{|(N y)(t) : t ≥ T, y ∈ E})

T →+∞ M kpkLq

≤

δ

1− 1 q

ψ(sup α(B(t)) + lim sup{|y(t)| : t ≥ T, y ∈ E}).

(18)

T →+∞

t∈J

Further, combining (17) and (18), we get sup α((N B)(t)) + lim sup{|(N y)(t) : t ≥ T, y ∈ E}) T →+∞

t∈J

≤ 4M kηkL1 ϕ(sup α(B(t)) + lim sup{|y(t)| : t ≥ T, y ∈ E}) T →+∞

t∈J

(19)

L + M kpk ψ(sup α(B(t)) + lim sup{|y(t)| : t ≥ T, y ∈ E} 1− 1 q

δ

q

T →+∞

t∈J

L ≤ M max(4kηkL1 , kpk )(ϕ + ψ)(sup α(B(t)) + lim sup{|y(t)| : t ≥ T, y ∈ E}). 1− 1 q

δ

q

T →+∞

t∈J

From Step 3 and inequality (19), we conclude that   kpkLq (ϕ + ψ)(µ(B)). µ(N (B)) ≤ M max 4kηkL1 , 1− 1 δ q It follows from Lemma 2.17 that N has at least one fixed point y ∈ BR , which is just a asymptotically almost automorphic mild solution of problem (1)-(2) on R+ .

4

An Example

Consider the second order differential equation of the form;                           

 ∂ ∂2 1 ∂2 √ z(t, τ ) = z(t, τ ) + 2 sin z(t, τ ) ∂t2 ∂τ 2 2 + cos t + cos 2t ∂t   sin2 t 1 √ + √ sin (|z(t, τ )| + ln (1 + |z(t, τ )|)) 2 + cos t + cos 2t 12 1 + t2 sin2 t sin πz(t, τ ) + √ , t ∈ R+ , τ ∈ [0, π], 15 1 + t2 (1 + |z(t, τ )|) z(t, 0) = z(t, π) = 0, t ∈ R+ , ∂ z(0, τ ) = ψ(τ ), τ ∈ [0, π]. ∂t (20)

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Let E = L2 ([0, π], R+ ) be the space of 2-integrable functions from [0, π] into R+ , and let H 2 ([0, π], R+ ) be the Sobolev space of functions x : [0, π] → R+ , such that x00 ∈ L2 ([0, π], R+ ). We consider the operator A1 z(τ ) = z 00 (τ ) with domain D(A1 ) = H 2 (R+ , C), which is the infinitesimal generator of strongly continuous cosine function C(t) on E. Moreover, A1 has discrete spectrum, the spectrum of A1 consists of eigenvalues n2 for n ∈ Z, with associated eigenvector 1 ωn (ξ) = √ einξ , n ∈ Z, 2π the set {ωn ∈ Z} is an orthonormal basis of E. In particular, A1 x = −

∞ X

n2 hx, wn iwn for x ∈ D(A).

n=1

The cosine function C(t) is given by C(t)x =

∞ X

cos(nt)hx, wn iwn for x ∈ D(A), t ∈ R+ ,

n=1

form a cosine function on H, with associated sine function S(t)x =

∞ X sin(nt) n=1

n

hx, wn iwn for x ∈ D(A), t ∈ R+ .

From [35], for all x ∈ H 2 ([0, π], R+ ), t ∈ R+ , kC(t)kB(E) ≤ e−t and kS(t)kB(E) ≤ e−t . Now, we define an operator A(t) : D(A) ⊂ H → H by  D(A(t)) = D(A) A(t) = A1 + b(t, τ ).   1 √ where b(t, τ ) = 2 sin 2 + cos t + cos 2t Note that A(t) generates an evolutionary process U(t, s) of the form Rt

U(t, s) = S(t − s)e s b(t,s)ds   1 √ Since b(t, τ ) = 2 sin ≤ 2, we have 2 + cos t + cos 2t U(t, s) = S(t − s)e−2(t−s)

487

(21)

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Asymptotically almost automorphic mild solutions

21

and kUkB(E) ≤ kSkB(E) e−2(t−s) ≤ e−3(t−s) We conclude that U(t, s) is a evolutionary process exponentially stable with M = 1 and δ = 3. It follows from the estimate (21) that U(t, s) : E → E is well defined and satisfies the conditions of Definition 2.1. Hence conditions (H1 ) and (H2 ) are satisfied. Now, let z(t)(τ ) = w(t)(τ ), t ≥ 0, τ ∈ [0, π], g(t, z)(τ ) =

  sin2 t 1 √ √ sin (|z(t, τ )| + ln (1 + |z(t, τ )|)), 2 + cos t + cos 2t 12 1 + t2

h(t, z)(τ ) =

sin2 t sin πz(t, τ ) √ . 15 1 + t2 (1 + |z(t, τ )|)

Then it is easy to verify that g : R × E × E is continuous and g ∈ AA(R × E; E). We can estimate for the functions g: g(t, z)(τ ) ≤

sin2 t √ (|z(t, τ )| + ln (1 + |z(t, τ )|)). 12 1 + t2

Hence conditions (H3 )(a) is satisfied with sin2 t p(t) = √ , 3 1 + t2

1 ψ(t) = (t + ln(1 + t)). 4

1 Then it is easy to verify that p ∈ L2 (R) and ρ1 = . 4 On the other hand, it is clear that h : R+ × E × E is continuous and h ∈ C0 (R+ × E; E). We can also estimate for the functions h: π |z(t, τ )|. h(t, z)(τ ) ≤ √ 15 1 + t2

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M. Benchohra, G.M. N’Gu´er´ekata and N. Rezoug

Hence conditions (H3 )(b) is satisfied with β(t) =

π , 15 1 + t2 √

φ(R) = R.

π Then it is easy to verify that β ∈ C0 (R+ , R), ρ2 = 1 and ρ ≤ . 15 Furthermore: f (t; z) = g(t; z) + h(t; z) ∈ AA(R+ × E; E).

We can also estimate for the functions f : 2 sin2 t f (t, z)(τ ) ≤ √ |z(t, τ )|. 1 + t2

(22)

By (22), for every t ∈ J, and B ∈ D ⊂ E, we have α(f (t, D) ≤

sin2 t √ α(D), 12 1 + t2

Hence conditions (H4) is satisfied with 1 η(t) = √ , 6 1 + t2

ϕ(t) =

sin2 t . 2

Moreover, we have (ψ + ϕ)(t) =

sin2 t 1 + (t + ln(1 + t)) ≤ t. 2 4

We conclude that (see Lemma 2.1. [6]) lim (ψ + φ)n (t) = 0 for a.e t ∈ R+ .

n→+∞

Consequently, can be written in the abstract form (1)-(2) with A(t) and f as defined above. Thus, Theorem 3.3 yields that equation (20) has a asymptotically almost automorphic mild solution.

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[10] C. J. K. Batty, R. Chill, S. Srivastava, Maximal regularity for second order non-autonomous Cauchy problems, Studia Math. 189 (2008) 205223. [11] M. Benchohra, J. Henderson and N. Rezoug, Global existence results for second order evolution equations, Comm. Appl. Nonlinear Anal. 23 (2016), 57-67.

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[12] M. Benchohra and S. K. Ntouyas, Existence of mild solutions on noncompact intervals to second order initial value problems for a class of differential inclusions with nonlocal conditions, Comput. Math. Appl. 39 (2000), 11-18. [13] S. Bochner, A new approach in almost-periodicity. Proc. Nat. Acad. Sci. USA, 48 (1962), 2039-2043. [14] J. Cao, Z. Huang, G. M. N’Gu´er´ekata, Existence of asymptotically almost automorphic mild solutions for nonautonomous semilinear evolution equations. Electron. J. Differential Equations 2018, Paper No. 37, 16 pp. [15] P. Chen, Y. Li, Monotone iterative technique for a class of semilinear evolution equations with nonlocal conditions, Results Math. 63 (2013), 731-744. [16] T. Dianaga, G. M. N’Gu´er´ekata, Almost automorphic solutions to some classes of partial evolution equations, Appl. Math. Lett. 20 (2007), 462466. [17] H. S. Ding, T. J. Xiao, J. Liang. Asymptotically almost automorphic solutions for some integrodifferential equations with nonlocal initial conditions. J. Math. Anal. Appl. 338 (2008), no. 1, 141–151. [18] K. Ezzinbi, and G. M. N’Gu´er´ekata, Almost automorphic solutions for some partial functional differential equations. J. Math. Anal. Appl. 328 (1) (2007), 344-358. [19] F. Faraci, A. Iannizzotto, A multiplicity theorem for a perturbed second-order non-autonomous system, Proc. Edinb. Math. Soc. 49 (2006), 267-275. [20] H. O. Fattorini, Second Order Linear Differential Equations in Banach Spaces, North-Holland Mathematics Studies, Vol. 108, North-Holland, Amsterdam, 1985. [21] J. A. Goldstein, G.M. N’Gu´er´ekata, Almost automorphic solutions of semilinear evolution equations, Proc. Amer. Math. Soc. 133 (2005), 2401-2408.

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[22] H. Henr´ıquez, V. Poblete, J. Pozo, Mild solutions of non-autonomous second order problems with nonlocal initial conditions. J. Math. Anal. Appl. 412 (2014), no. 2, 1064-1083. [23] H.P. Heinz, On the behaviour of measure of noncompactness with respect to differentiation and integration of rector-valued functions, Nonlinear Anal. 7 (1983), 1351-1371. [24] V. Kavitha, S. Abbas, R. Murugesu, Asymptotically almost automorphic solutions of fractional order neutral integro-differential equations. Bull. Malays. Math. Sci. Soc. 39 (2016), no. 3, 1075–1088 [25] M. Kozak, A fundamental solution of a second-order differential equation in a Banach space, Univ. Iagel. Acta Math. 32 (1995), 275-289. [26] J. Liang, J. Zhang, T. Xiao. Composition of pseudo almost automorphic and asymptotically almost automorphic functions. J. Math. Anal. Appl. 340 (2008), no. 2, 1493-1499. [27] I. Mishra, D. Bahuguna, S. Abbas, Existence of almost automorphic solutions of neutral functional differential equation, Nonlinear Dyn. Syst. Theory 11 no. 2 (2011) 165-172. [28] H. M¨onch, Boundry value problems for nonlinear ordinary differential equations of second order in Banach spaces, Nonlinear Anal. 4(5) (1980) 985-999. [29] G. M. N’Gu´er´ekata, Sur les solutions presque automorphes d’´equations diff´erentielles abstraites, Ann. Sci. Math. Qu´ebec, 1(1981), 69-79. [30] G. M. N’Gu´er´ekata, Almost Automorphic and Almost Periodic Functions in Abstract Spaces, Kluwer Academic, New York, London, Moscow, 2001. [31] G. M. N’Gu´er´ekata, Topics in Almost Automorphy, Springer, New York, Boston, Dordrecht, London, Moscow, 2005. [32] G. M. N’Gu´er´ekata, Spectral Theory for Bounded Functions and Applications to Evolution Equations. Mathematics Research Developments. Nova Science Publishers, Inc., New York, 2017. [33] L. Olszowy, S. W¸edrychowicz. On the existence and asymptotic behaviour of solutions of an evolution equation and an application to the Feynman-Kac theorem. Nonlinear Anal. 74 (2011), no. 17, 6758-6769.

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[34] L. Olszowy. Existence of mild solutions for semilinear nonlocal cauchy problems in separable Banach spaces. Z Anal. Anwend. 32 (2) (2013), 215-232. [35] C. C. Travis and G. F. Webb, Second order differential equations in Banach spaces, in: Nonlinear Equations in Abstract Spaces, Proc. Internat. Sympos. (Univ. Texas, Arlington, TX, 1977), Academic Press, New-York, 1978, 331-361. [36] T. Winiarska, Evolution equations of second order with operator dependent on t, Selected Problems of Math. Cracow Univ. Tech. 6 (1995) 299-314. [37] J. Wu, Theory and Application of Partial Functional Differential Equations, Springer-Verlag, New York, 1996. [38] T. J. Xiao, X. X. Zhu, J. Liang. Pseudo-almost automorphic mild solutions to nonautonomous differential equations and applications. Nonlinear Anal. 70 (2009), no. 11, 4079-4085. [39] X. J. Zheng, C. Z. Ye, H. S. Ding, Asymptotically almost automorphic solutions to nonautonomous semilinear evolution equations. Afr. Diaspora J. Math. 12 (2011), no. 2, 104–112 [40] Z. Zhao, Y. Chang, J. Nieto, Almost automorphic and pseudo-almost automorphic mild solutions to an abstract differential equation in Banach spaces. Nonlinear Anal. Theory Methods Appl. 72 (2010), 18861894.

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Applying hybrid coupled fixed point theory to the nonlinear hybrid system of second order differential equations TAMER NABILa,b1 a

b

Department of Mathematics, Faculty of Science, King Khaled University, Abha 9004, Saudi Arabia

Suez Canal University, Faculty of Computers and Informatics, Department of Basic Science, Ismailia, Egypt Abstract In this work, we apply hybrid fixed point theory method to prove the existence of solution of systems of second order ordinary nonlinear hybrid differential equations with periodic boundaries.

Key words and phrases. Second order differential equation; hybrid systems; boundary value problems; coupled fixed point.

AMS Mathematics subject Classification. 34A12, 34A38.

1

Introduction

Let X 6= φ, and F : X × X → X is a mapping. A point (x, y) ∈ X × X is said to a coupled fixed point of F in X × X if F (x, y) = x and F (y, x) = y. The notation of coupled fixed point was introduced in 2006 by Baskar and Lshmikantham [1]. It will known that the existence of the fixed point play an important role for showing the existence of solutions of nonlinear integral [2, 3] , differential equations [4, 5] and iterative process [6]. In 1964, Krasnoselskii [7] initiated the idea of study the hybrid fixed point theory for the function which can be written as the sum of two other functions.In 2013, Dhage [2] obtained hybrid fixed point theorems for the operator which can be written as the sum of two other operators using Krasnoselskii fixed point theorem techniques and developed a Krasnoselskii fixed point technique helpful to analyze the existence of solution of nonlinear Volterra fractional integral equations under some conditions. Recently, in 2015, Dhage and Dhage [8] proved the existence of solutions of the boundary value problems of second order ordinary nonlinear differential equations using the hybrid fixed point theorem which was obtained by themselves [9] . 1 t [email protected] −

1

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More recently, in 2017, Yang et al. [10] introduced the notation of hybrid coupled fixed point theorems and applied this ideas to prove the existence of system of fractional differential equations of order α, 0 < α < 1. In this paper, we apply the hybrid fixed point theorem to study and prove the existence of solution of a certain system of boundary -value problems with periodic boundaries( for short BVPPB) of second order ordinary nonlinear hybrid differential equations. To this end, the remainder of the article is organized as follows. Section 2, is given some preliminaries and basic definitions. Section3, is established the existence results of coupled nonlinear system of second order differential equations.

2

Preliminaries

Throughout this paper, let E be a nonempty set and (E, ≤, k.k) be a partially ordered normed linear space. If Q : E → E is a mapping. Then Q is said to be monotone nondecreasing if a ≤ b the Q(a) ≤ Q(b) , for all a, b ∈ E. Two elements a, b ∈ E are said to be comparable if a ≤ b or b ≤ a. If C is nonempty subset of E,C is said to be chain if each two elements a, b ∈ C are comparable. Definition 1 [10] Let Q : E → E be a mapping. Q is called partially compact if for each chain C subset of E, Q(C) is reltively compact subset of E. Definition 2 [2, 10] Let Q : E → E be a mapping. Given an element a ∈ E. Define orbit Γ(a; Q) as: Γ(a; Q) = {a, Qa, Q2 a, Q3 a, ....., Qn a, ....}. If for any sequence {an } ∈ Γ(a; Q) such that: an → a∗ as n → ∞ then Qan → Qa∗ , for each a ∈ E, then Q is called Γ− orbitally continuous in E. Furthermore, (E, ≤, k.k) is said to be Γ− orbitally complete if each sequence {an } ∈ Γ(a; Q) converges to an element a∗ ∈ E. Definition 3 [2, 10] A mapping φ : E → E is said to be D− function if it is upper semi continuous and monotone nondecreasing such that: φ(0) = 0.

2

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Definition 4 [2, 10] A mapping Υ : E → E is called partially nonlinear D− contraction in E , if for each comparable elements a, b ∈ E, there exist a D− function φ : 0. The periodic boundary value problem of second-order nonlinear differential equation can be written as: d2 x(t) = f (t, x(t) + h(t, x(t)), dt2

x(0) = x(b), x0 (0) = x0 (b)

(1)

for all t ∈ I, where f, h : I × < → < are continuous function. The solution of the differential equation (1) is the function x ∈ C 2 (I, µ(A ∩ Fβ ) f or any β > α}. It is easy to see that Z _ − f dµ = (α ∧ µ(A ∩ Fα )). A

α∈Γ

Remark 2.8. A binary operator T on [0, 1] is called a t-seminorm[16] if it satisfies the above condition (T1 ) and (T2 ). Notice that if T is a t-seminorm, for any x, y ∈ [0, 1], we have T (x, y) ≤ T (x, 1) = x and T (x, y) ≤ T (1, y) = y, and consequently, T (x, y) ≤ TM (x, y). ´ By using the concept of t-seminorm, Garc´ıa and Alvarez [16] proposed the following family of fuzzy integral. Definition 2.9. Let T be a t-seminorm. Then the seminormed Sugeno’s fuzzy integral of a function f ∈ F+ over A ∈ Σ with respect to T and the fuzzy measure µ is defined by Z _ f dµ = T (α, µ(A ∩ Fα )). T,A

α∈[0,1]

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Notice that the Sugeno integral of f ∈ F+ over A ∈ Σ is the seminormed Sugeno’s fuzzy integral of f over A ∈ Σ with respect to the t-seminorm TM . ´ Proposition 2.10. (Garc´ıa and Alvarez [16])Let (X, Σ, µ) be a monotone measure space and T be a t-seminorm. Then, 1: For any A ∈ Σ and f, g ∈ F+ with f ≤ g, we have Z Z f dµ ≤ T,A

gdµ.

T,A

2: For A, B ∈ Σ with A ⊂ B and any f ∈ F+ , Z Z f dµ ≤ T,A

f dµ.

T,B

Definition 2.11. [7] Let I ⊂ R − {0} is a real interval. A function f : I → R is said to be harmonically convex on I if the inequality  f

ab ta + (1 − t)b

 ≤ tf (b) + (1 − t)f (a)

(2.1)

holds, for all a, b ∈ I and t ∈ [0, 1]. If the inequality (2.1) is reversed, then f is said to be harmonically concave. We note that for t = 21 , we have the definition of Jensen type of harmonic convex functions, that is   f (a) + f (b) 2ab ≤ , ∀a, b ∈ I. f a+b 2 Proposition 2.12. [7] Let I ⊂ R − {0} be a real interval and f : I → R is function, then: 1: if I ⊂ (0, +∞) and f is convex and nondecreasing, then f is harmonically convex. 2: if I ⊂ (0, +∞) and f is harmonically convex and nonincreasing, then f is convex. 3: if I ⊂ (−∞, 0) and f is harmonically convex and nondecreasing, then f is convex. 4: if I ⊂ (−∞, 0) and f is convex and nonincreasing, then f is harmonically convex. → R defined by g(t) = f ( 1t ),   then f is harmonically convex on [a, b] if and only if g is convex in the usual sense on 1b , a1 . Proposition 2.13. [4] If [a, b] ⊂ I ⊆ (0, ∞) and we consider the function g :

1

1 b, a



Proposition 2.14. [6] A function f : (0, ∞) → R is harmonically convex if and only if xf (x) is convex. Theorem 2.15. Let f : [a, b] ⊆ (0, ∞) → [0, +∞) be a convex function with f (a) 6= f (b).Then    Z b _ α(b − a) + af (b) − bf (a) − f dµ ≤ α ∧ µ [a, b] ∩ x ≥ f (b) − f (a) a α∈Γ

where Γ = [f (a), f (b)) for f (b) > f (a) and Γ = [f (b), f (a)) for f (a) > f (b). Proof. As f is convex function, for x ∈ [a, b] we have,   x−a x−a x−a x−a f (x) = f (1 − )a + b ≤ (1 − )f (a) + f (b) b−a b−a b−a b−a

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the Hermite-Hadamard inequality for the Sugeno integral based on harmonically convex functions

5

and so by (3) of Proposition 2.3  Z b Z b Z b x−a x−a − f dµ ≤ − (1 − )f (a) + f (b) dµ = − g(x)dµ. b−a b−a a a a In order to calculate the integral in the right hard part of the last inequality, we consider the distribution function F (α) given by    x−a b−x f (a) + f (b) ≥ α . F (α) = µ([a, b] ∩ {g ≥ α}) = µ [a, b] ∩ b−a b−a If f (a) < f (b), then      α(b − a) + af (b) − bf (a) α(b − a) + af (b) − bf (a) F (α) = µ [a, b] ∩ x ≥ =µ [ , b] . f (b) − f (a) f (b) − f (a) Thus Γ = [f (a), f (b)) and we only consider α ∈ [f (a), f (b)). If f (a) > f (b), then      α(b − a) + af (b) − bf (a) α(b − a) + af (b) − bf (a) = µ [a, ] . F (α) = µ [a, b] ∩ x ≤ f (b) − f (a) f (b) − f (a) Thus Γ = [f (b), f (a)) and only need α ∈ [f (b), f (a)). This completes the proof.



Remark 2.16. In the case f (a) = f (b) in Theorem 2.15, we have g(x) = f (x) and so Z b Z b Z b − f dµ ≤ − gdµ = − f (a)dµ = f (a) ∧ µ([a, b]). a

a

a

Corollary 2.17. Let f : [a, b] ⊆ (0, ∞) → (0, ∞) be a convex function and Σ be the Borel field and µ be the Lebesgue measure on X = R, then  W   α(b−a)+af (b)−bf (a)  α ∧ (b − ) , f (a) < f (b)  α∈[f (a),f (b)) f (b)−f (a)       Z b  f (a) ∧ (b − a) , f (a) = f (b) − f dµ ≤  a     W  α(b−a)+af (b)−bf (a)  − a) , f (a) > f (b) α ∧ (  α∈[f (b),f (a)) f (b)−f (a)    So

Z b − f dµ ≤ a

                

(b−a)f (b) f (b)−f (a)+(b−a)

∧ (b − a) , f (a) < f (b)

f (a) ∧ (b − a) (b−a)f (a) f (a)−f (b)+(b−a)

, f (a) = f (b) ∧ (b − a) , f (a) > f (b).

Proof. In the case where f (a) < f (b), we have   _ α(b − a) + af (b) − bf (a) (b − a)f (b) α ∧ (b − ) = . f (b) − f (a) f (b) − f (a) + (b − a) α∈[f (a),f (b))

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In fact, α =

(b−a)f (b) f (b)−f (a)+(b−a)

is as the solution of the equation F (α) = α, where F is the distribution function. Rb So taking into account (1) of Proposition 2.3 (−a f dµ ≤ µ([a, b]) = b − a) and Remark 2.4 we have Z b (b − a)f (b) ∧ (b − a). − f dµ ≤ f (b) − f (a) + (b − a) a Proofs the other cases is analogous.



Note that Corollary 2.17 is the same as the Sadarangani Theorem [3].

3. Main Results Let I ⊂ R − {0} be a harmonically convex function and a, b ∈ I with a < b and f ∈ L([a, b]). The following inequalities  Z b ab f (x) f (a) + f (b) 2ab ≤ . (3.1) dx ≤ a+b b − a a x2 2 holds. This double inequality is known in the literature as Hermite-Hadamard integral inequality for harmoni

f

cally convex functions. Unfortunately, as we will see in the following example, in general, the Hermite-Hadamard inequality is not valid in the fuzzy context. Example 3.1. Let µ be the usual Lebesgue measure on R and the function f (x) =

3 2 7x

on X = [ 12 , 1].

Obviously, this function is convex and nondecreasing as a result f is harmonically convex function on [ 12 , 1]. With the above inequality we have Z 1 Z 1 3 1 3 f (x) 3 − dx = − dx = ∧ µ([ , 1] = ' 0.42. 2 1 1 x 7 7 2 7 2 2 on the other hand,

f ( 12 )+f (1) 2

=

15 56

' 0.26.

This proves that the right-hand side of inequality (3.1) is not satisfied for the Sugeno integrals. The aim of this work is to show a the Hermite-Hadamard type inequality for the Sugeno integral in the case where f is a harmonically convex function. Lemma 3.2. Let f : [a, b] ⊆ (0, ∞) → (0, ∞) be a harmonically convex function which is not concave, then  W   α(b−a)+af (b)−bf (a)  , b] , f (a) < f (b) α ∧ µ[  f (b)−f (a)   α∈[f (a),f (b))     Z b  f (a) ∧ µ([a, b]) , f (a) = f (b) − f dµ ≤  a     W  α(b−a)+af (b)−bf (a)  ] , f (a) > f (b).  α∈[f (b),f (a)) α ∧ µ[a, f (b)−f (a)    Proof. Since f : [a, b] ⊆ (0, ∞) → (0, ∞) is harmonically convex function on the interval [a, b], then by Proposition 2.13 the function g : [ 1b , a1 ] → R, g(s) = f ( 1s ) is convex on [ 1b , a1 ]. Obviously for any x ∈ [a, b], f (x) = g( x1 ),

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and therefor applying Theorem 2.15 to g, we have

 W   1 α(b−a)+ag( 1b )−bg( a )  α ∧ µ[ , b] , g( a1 ) < g( 1b ) 1 1  1 1 ),g( )) α∈[g(  g( b )−g( a ) a b      Z b Z b  1 , g( a1 ) = g( 1b ) g( a1 ) ∧ µ([a, b]) − f (x)dµ = − g( )dµ ≤  x a a     1 W  α(b−a)+ag( 1b )−bg( a )  α ∧ µ[a, ] , g( a1 ) > g( 1b ) 1 1  1 α∈[g( b ),g( a ))  g( 1b )−g( a )  

 W   α(b−a)+af (b)−bf (a)  α ∧ µ[ , b]  α∈[f (a),f (b)) f (b)−f (a)        f (a) ∧ µ([a, b]) =   W     α(b−a)+af (b)−bf (a)  ]  α∈[f (b),f (a)) α ∧ µ[a, f (b)−f (a)   

, f (a) < f (b) , f (a) = f (b) , f (a) > f (b).



Corollary 3.3. Let f : [a, b] ⊆ (0, ∞) → (0, ∞) be a harmonically convex function which is not concave, Σ be the Borel field and µ be the Lebesgue measure on X = R, then

Z b − f dµ ≤ a

                

(b−a)f (b) f (b)−f (a)+b−a

∧ (b − a) , f (a) < f (b)

f (a) ∧ (b − a) (b−a)f (a) f (a)−f (b)+b−a

, f (a) = f (b) ∧ (b − a) , f (a) > f (b).

Remark 3.4. If [a, b] ⊆ (0, ∞) and f is harmonically convex and nonincreasing, then taking into account (2) of Proposition 2.12 the function f is convex and hance the upper bound for the Sugeno integral of f mentioned in article ”Hermite-Hadamard inequality for fuzzy integral”, were written by K. sadarangani is established.

Remark 3.5. If [a, b] ⊆ (−∞, 0) and f is harmonically convex and nondecreasing, then taking into account (3) of Proposition 2.12 the function f is convex and hance the upper bound for the Sugeno integral of f is established.

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Example 3.6. Let µ be a Lebesgue measure and consider function f (x) = e− x on [ 13 , 34 ]. Obviously, this function is non-negative and harmonically convex but neither convex, nor concave. we have,  Z 34 o n 1 _ 1 3 α ∧ µ [ , ] ∩ e− x ≥ α − f dµ = 1 3 4 3 α≥0     _ 1 3 1 = α ∧ µ [ , ] ∩ − ≥ ln α 3 4 x α≥0    _ 1 3 = α ∧ µ [ , ] ∩ {−1 ≥ xlnα} 3 4 α≥0     _ 1 3 −1 = α∧µ [ , ]∩ x≥ . 3 4 lnα α≥0

As result with the solution of the equation 3 1 + =α lnα 4 R3 we conclude that α ' 0/175. Then −14 f dµ ' 0/175. On the other hand, since f ( 34 ) =

3

1

and f ( 13 ) =

4

e3

Z 34 − f dµ ≤ 1 3

f ( 34 )

1 e3 .

By Corollary 3.3, we have

f ( 34 )( 34 − 13 ) 3 1 ∧( − ) 4 3 − f ( 31 ) + ( 34 − 13 )

' 0/234 ∧

5 = 0/234 ∧ 0/416 = 0/234 12

that is a logical inequality. Example 3.7. The function f (x) = x − ln(x + 1) is nondecreasing and harmonic convex function on [ 21 , 1]. f (1) = 1 − ln 2 and f ( 12 ) =

1 2

− ln( 23 ). As f (1) > f ( 12 ), Corollary 3.3 gives us, Z 1 − f dµ ≤ 1 2

(1 − 21 )f (1) f (1) − f ( 12 ) +

1 2

1 1 1 ∧ ( ) ' 0.718 ∧ = . 2 2 2

Thus, we find an upper bound for the Sugeno integral of this function on [ 12 , 1]. 2

Example 3.8. The function f (x) = ex

+x

is nondecreasing and harmonic convex function on [1, 2] and f (1) = e2

and f (2) = e5 . As follows we find an upper bound for the Sugeno integral of this function, Z 2 2 e5 − ex +x dµ ≤ 5 ∧ (1) ' 1.0449 ∧ 1 = 1. e − e2 + 1 1 Remark 3.9. f (x) = log(x) is a harmonically convex function but not convex, that is why in the Corollary 3.3, does not apply because it is concave. For concave function, we use the Sadarangani paper.

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Corollary 3.10. Let f : [a, b] ⊆ (0, ∞) → R be a harmonically convex function which is not concave and g : R → R is a linear function, then f ◦ g is harmonically convex[10] and so,  W   α(b−a)+af (g(b))−bf (g(a))  , b] , f (g(a)) < f (g(b)) α ∧ µ[  α∈[f (g(a)),f (g(b))) f (g(b))−f (g(a))     Z b  f (g(a)) ∧ µ([a, b]) , f (g(a)) = f (g(b)) − (f ◦ g)dµ ≤  a        α(b−a)+af (g(b))−bf (g(a))  W α ∧ µ[a, ] , f (g(a)) > f (g(b)). α∈[f (g(b)),f (g(a))) af (g(b))−bf (g(a)) Corollary 3.11. Let f : [a, b] ⊆ (0, ∞) → R be a harmonically convex function which is not concave and g : R → R is a linear function, Σ be the Borel field and µ be the Lebesgue measure on X = R, then f ◦ g is harmonic convex function[10] and so,  (b−a)f (g(b))   f (g(b))−f (g(a))+b−a ∧ (b − a) , f (g(a)) < f (g(b))    Z b  f (g(a)) ∧ (b − a) , f (g(a)) = f (g(b)) − (f ◦ g)dµ ≤  a     (b−a)f (g(a))  f (g(a))−f (g(b))+b−a ∧ (b − a) , f (g(a)) > f (g(b)). Remark 3.12. In the case g be harmonic convex function and f be relative convex function, we know that f ◦ g is harmonically convex function [11]. Thus similar results of Corollary 3.10 and Corollary 3.11 hold. Corollary 3.13. Let f : [a, b] ⊆ (0, ∞) → (0, ∞) be a harmonically convex function which is not concave function, Σ be the Borel field and µ be the Lebesgue measure on X = R, then  (b−a)2 f (b)  , f (a) < f (b)  f (b)−f (a)+b−a      Z  (b − a)f (a) , f (a) = f (b) f dµ ≤  TP ,[a,b]   (b−a)2 f (a)   , f (a) > f (b).    f (a)−f (b)+b−a Proof. For harmonically convex function f : [a, b] ⊆ (0, ∞) → (0, ∞) with f (a) 6= f (b) according to Proposition 2.10 and Corollary 3.3 with t-norm Tp , we have  (b−a)f (b)   f (b)−f (a)+b−a .(b − a) , f (a) < f (b)      Z  f (a).(b − a) , f (a) = f (b) f dµ ≤  TP ,[a,b]   (b−a)f (a)    f (a)−f (b)+b−a .(b − a) , f (a) > f (b)  

=

                

(b−a)2 f (b) f (b)−f (a)+b−a

, f (a) < f (b)

(b − a)f (a)

, f (a) = f (b)

(b−a)2 f (a) f (a)−f (b)+b−a

, f (a) > f (b).

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 Example 3.14. Let µ be the Lebesgue measure on R. Consider the function f (x) =

1 x2

on X = [1, 3].

Obviously, this function is harmonically convex and positive on X = [1, 3]. As f (1) = 1 and f (3) = 19 , using Corollary 3.13, we can get the following estimate: Z (3 − 1)2 f (1) 18 1 dµ ≤ = . 2 f (1) − f (3) + (3 − 1) 13 TP ,[1,3] x Now, let’s introduce the most important theorem of this article. With the help of it, an upper bound in the framework of the Sugeno integral for Hermite-Hadamard inequality of harmonically convex functions can be established. Theorem 3.15. Let f : [a, b] ⊆ (0, ∞) → (0, ∞) be a harmonically convex function which is not concave, then  W   α(b−a)+af (b)−bf (a)  α ∧ µ[ , b] , f (a) < f (b)  α∈[f (a),f (b)) f (b)−f (a)       Z b Z b  f (x) f (a) ∧ µ([a, b]) , f (a) = f (b) − m0 2 dµ ≤ − f dµ ≤  x a a     W  α(b−a)+af (b)−bf (a)  α ∧ µ[a, ] , f (a) > f (b)  α∈[f (b),f (a)) f (b)−f (a)    where m0 = min{a2 , b2 }. Proof. Let f be a harmonically convex function which is not concave and m0 = min{a2 , b2 }. By Proposition 2.5 we have, Z b Z b f (x) − m0 2 dµ = − µ([a, b] ∩ Fα )dm x a a where m is the Lebesgue measure and Fα = {x ∈ X : m0

(3.2)

f (x) ≥ α}. x2

Obviously, 

  x2 [a, b] ∩ f (x) ≥ α ⊆ ([a, b] ∩ {f (x) ≥ α}) . m0

By monotonicity µ, we deduce    x2 µ [a, b] ∩ f (x) ≥ α ≤ µ ([a, b] ∩ {f (x) ≥ α}) . m0 Now, by Proposition 2.3 and Proposition 2.5, we obtain  Z b  Z b Z b x2 − µ [a, b] ∩ {f ≥ α} dm ≤ − µ ([a, b] ∩ {f ≥ α}) dm = − f dµ. m0 a a a

(3.3)

Combining ( 3.2 , 3.3), we have Z b Z b f (x) − m0 2 dµ ≤ − f dµ. x a a The last inequality follows from Lemma 3.2.

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Corollary 3.16. If f : [a, b] ⊆ (0, ∞) → (0, ∞) be a harmonically convex function which is not concave then,  W   α(b−a)+abf (b)−baf (a)  , b] , af (a) < bf (b) α ∧ µ[  α∈[af (a),bf (b)) bf (b)−af (a)       Z b  af (a) ∧ µ([a, b]) , af (a) = bf (b) − xf (x)dµ ≤  a     W  α(b−a)+abf (b)−baf (a)  α ∧ µ[a, ] , af (a) > bf (b).  α∈[bf (b),af (a)) bf (b)−af (a)    Proof. f is harmonically convex function.Therefore, according to the Proposition 2.14 xf (x) is convex. Finally, the proof is complete by using Theorem 2.15.



Corollary 3.17. If f : [a, b] ⊆ (0, ∞) → (0, ∞) be a harmonically convex function which is not concave, Σ be Borel field and µ be a Lebesgue measure on X = R, then  (b−a)bf (b)  ∧ (b − a) , af (a) < bf (b)   bf (b)−af (a)+b−a     Z b  af (a) ∧ (b − a) , af (a) = bf (b) − xf (x)dµ ≤  a   (b−a)af (a)   ∧ (b − a) , af (a) > bf (b).    af (a)−bf (b)+b−a Example 3.18. Let µ be the usual Lebesgue measure on X and the function f (x) =

3 2 5x

on X = [1, 2].

Obviously, this function is convex and nondecreasing. So by (1) of Proposition 2.12 f is harmonically convex on [1, 2]. With use the Corollary 3.17 we have Z 2 (2 − 1)2f (2) ∧ (2 − 1) ' 0.923. − xf (x)dx ≤ 2f (2) − f (1) + (2 − 1) 1 R2 On the other hand, −1 xf (x)dx ' 0.87. This show that the Corollary 3.17 is valid.

4. Conclusion In this paper, we have researched the Hermite-Hadamard inequality for the Sugeno integral based on harmonically convex functions. For further investigations we propose to consider the Hermite-Hadamard inequality for the Choquet integral, and also for some other non-additive integrals. In the future research, we will continue to explore other integral inequalities for non-additive measures and integrals based on harmonically convex function. References [1] S. Abbaszadeh, A. Ebadian, M. Jaddi, H¨older type integral inequalities with different pseudo-operations, Asian-European Journal of Mathematics, To apear. [2] S. Abbaszadeh, M. Eshaghi, M. de la Sen, The Sugeno fuzzy integral of log-convex functions, J. Inequal.Appl. (2015) 2015: 362. [3] J. Caballero, K. Sadarangani, Hermite-Hadamard inequality for fuzzy integrals, Appl. Math. Comput. 215(2009) 2134-2138.

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[4] S. Dragomir, Inequalities of Jensen type for HA-convex functions, RGMIA Mono-graphs, Victoria University, 2015. [5] S. S. Dragomir, Inequalities of Hermite-Hadamard Type for HA-Convex Functions, Moroccan J. Pure and Appl. Anal(MJPAA), 3(1), 2017, 3(1), 2017, 83-101. [6] S. S .Dragomir. New inequalities of Hermit-Hadamard type for HA-convex function,J. Numer. Anal. Approx. Theory, vol. 47 (2018) no. 1, pp. 26-41. ˙ scan, Hermite-Hadamard type inequalities for harmonically convex functions, Hacettepe Journal of [7] I. I¸ Mathematics and Statistics, 43 (6) (2014), 935-942. [8] M. Jaddi, A. Ebadian, M. de la Sen, S. Abbaszadeh, An equivalent condition to the Jensen inequality for the generalized Sugeno integral, Journal of Inequalities and Applications 2017, n o. 285, 11 pp. [9] E.P. Klement, R. Mesiar and E. Pap, Triangular norms, Trends in Logic, Kluwer Academic Publishers, Dordrecht, 2000. [10] M. A. Noor, K. I. Noor and S. Iftikhar, Some Characterizations of Harmonic Convex Functions, International Journal of Analysis and Applications, Volume 15, Number 2 (2017), 179-187. [11] M. A. Noor, K. I. Noor, M. U. Awan, Some Characterizations of Harmonically log-Convex Functions, Proc. Jangjeon Math. Soc., 17(1), 51-61, (2014). [12] D. Ralescu and G. Adams, G., The fuzzy integral, J. Math. Anal. Appl. 75 (1980) no. 2, 562-570. [13] H. Rom´an-Flores, A. Flores-Franuliˇc, Y. Chalco-Cano, The fuzzy integral for monotone functions, Appl. Math. Comput. 185 (2007) 492-498. [14] H. Rom´an-Flores, Y. Chalco-Cano, H-Continuity of fuzzy measures and set defuzzification, Fuzzy Sets Syst. 157 (2006) 230-242. [15] H. Rom´an-Flores, A. Flores-Franuliˇc, R. Bassanezi, M. Rojas-Medar, On the level-continuity of fuzzy integrals, Fuzzy Sets Syst. 80 (1996) 339-344. ´ [16] F. Su´arez Garc´ıa and P, Gil Alvarez, Two families of fuzzy integrals, Fuzzy Sets and Systems. 18 (1986) no. 1, 67-81. [17] M. Sugeno, Theory of Fuzzy Integrals and its Applications, Ph.D. Dissertation, Tokyo Institute of Technology, 1974. [18] Z. Wang, George J. Klir, Generalized Measure Theory. Springer, New York (2009). [19] Z. Wang, G. Klir, Fuzzy Measure Theory, Plenum Press, New York, 1992.

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On the Order, Type and Zeros of Meromorphic Functions and Analytic Functions of [p, q]-Order in the Unit Disc Jin Tu1∗ , Ke Qi Hu1 , Hong Zhang2 1. College of Mathematics and Information Science, Jiangxi Normal University, Nanchang, 330022, China 2. School of the Tourism and Urban Management, Jiangxi University of Finance and Economics, Nanchang 330032, China Corresponding author: [email protected]

Abstract In this paper, the authors investigate the [p, q]-order and [p, q]-type of f1 + f2 , f1 f2 , f1 /f2 , where f1 , f2 are meromorphic functions or analytic functions with the same [p, q]-order and different [p, q]-type in the unit disc, and the authors also study the [p, q]-order and [p, q]-type of f and its derivative. At the end, the authors investigate the relationship between two different [p, q]-convergence exponents of f . The obtained results are the improvements and supplements to many previous results. Key words:meromorphic function; analytic function; unit disc; [p, q]-order; [p, q]-type AMS Subject Classification(2000): 30D35, 30D15

1.

Notations and Results

We use C to denote the complex plane and ∆ = {z : |z| < 1} to denote the unit disc. By a meromorphic function f , we mean a meromorphic function in the complex plane or a meromorphic function in the unit disc. We shall assume that readers are familiar with the fundamental results and the standard notations of the Nevanlinna value distribution theory in the complex plane or in the unit disc (see [4, 10, 14 − 17, 19, 20]). Firstly for r ∈ (0, +∞), we define exp1 r = er and expi+1 r = exp(expi r), i ∈ N, for all r sufficiently large in (0, +∞), we define log1 r = log r and logi+1 r = log (logi r) , i ∈ N, we also denote exp0 r = r = log0 r and exp−1 r = log1 r. Moreover, R dt . Throughout this paper, we denote the logarithmic measure of a set E ⊂ [0, 1) by ml E = E 1−t we use p, q to denote positive integers satisfying 1 ≤ q ≤ p. Secondly, we recall some notations about meromorphic functions and analytic functions. Definition 1.1 (see [4, 17, 19, 20]). The order σ(f ) and lower order µ(f ) of a meromorphic function f in the complex plane are respectively defined by σ(f ) = lim

r→∞

log T (r, f ) , log r

µ(f ) = lim

r→∞

log T (r, f ) , log r

1

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J. Tu, K.Q. Hu, H. Zhang

where T (r, f ) is the characteristic function of a meromorphic function f in the complex plane or in the unit disc. Definition 1.2 (see [4, 19, 20]). Let f be a meromorphic function in the complex plane or an entire function satisfying 0 < σ(f ) < ∞, then the type of f is respectively defined by τ (f ) = lim

r→∞

T (r, f ) , rσ(f )

τM (f ) = lim

r→∞

log M (r, f ) . rσ(f )

Definition 1.3 (see [8, 9, 11, 13]). The [p, q]-order of a meromorphic function f in the complex plane is defined by logp T (r, f ) . r→∞ logq r

σ[p,q] (f ) = lim

If f is a transcendental entire function, the [p, q]-order of f is defined by (see [11, 13]) logp T (r, f ) logp+1 M (r, f ) = lim . r→∞ r→∞ logq r logq r

σ[p,q] (f ) = lim

If f is a polynomial, then σ[p,q] (f ) = 0 for any p ≥ q ≥ 1. From Definition 1.3, if q = 1, we denote σ[1,1] = σ1 (f ) = σ(f ), and σ[p,1] = σp (f ). Similar with Definition 1.2, we can also give the definitions of τp (f ) and τM,p (f ) when p > 1. In order to keep accordance with Definition 1.1, we give Definition 1.3 by making a small change to the original definition of entire functions of [p, q]-order (see [8, 9]). Definition 1.4 (see [3, 7]). The iterated p-order of a meromorphic function f in ∆ is defined by logp T (r, f ) r→1− − log(1 − r)

σp (f ) = lim

(p ∈ N).

For an analytic function f in ∆, we also define σM,p (f ) = lim

r→1−

logp+1 M (r, f ) . − log(1 − r)

Remark 1.1. If p = 1, then we denote σ1 (f ) = σ(f ) and σM,1 (f ) = σM (f ), and we have σ(f ) ≤ σM (f ) ≤ σ(f ) + 1 (see [6, 12, 16, 17]) and σM,p (f ) = σp (f ) (p ≥ 2) (see [3, 7]). Definition 1.5 (see [2]). Let f be a meromorphic function in ∆, then the [p, q]-order and lower [p, q]-order of f are respectively defined by logp T (r, f )  , 1 r→1− log q 1−r

σ[p,q] (f ) = lim

µ[p,q] (f ) = lim

r→1−

545

logp T (r, f )  . 1 logq 1−r

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The Order, Type and Zeros of Meromorphic Functions and Analytic Functions of [p, q]-Order in the Unit Disc 3

Definition 1.6 (see [2]). Let f be an analytic function in ∆, then the [p, q]-order and lower [p, q]-order about maximum modulus of f are respectively defined by σM,[p,q] (f ) = lim

r→1−

logp+1 M (r, f )   , 1 logq 1−r

µM,[p,q] (f ) = lim

r→1−

logp+1 M (r, f )   . 1 logq 1−r

Definition 1.7 (see [2]). The [p, q]-type of a meromorphic function f of [p, q]-order in ∆ with 0 < σ[p,q] (f ) = σ1 < ∞ is defined by logp−1 T (r, f ) iσ1 .  τ[p,q] (f ) = lim h 1 r→1− log q−1 1−r For an analytic function f in ∆, and the [p, q]-type about maximum modulus of f of [p, q]-order with 0 < σM,[p,q] (f ) = σ2 < ∞ is defined by logp M (r, f )  iσ2 . τM,[p,q] (f ) = lim h 1 r→1− log q−1 1−r Definition 1.8 The lower [p, q]-type of a meromorphic function f of lower [p, q]-order in ∆ with 0 < µ[p,q] (f ) = µ1 < ∞ is defined by logp−1 T (r, f ) iµ1 .  τ [p,q] (f ) = lim h r→∞ 1 logq−1 1−r Similarly for an analytic function f in ∆, and the lower [p, q]-type about maximum modulus of f of lower [p, q]-order with 0 < µM,[p,q] (f ) = µ2 < ∞ is defined by logp M (r, f )  iµ2 . τ M,[p,q] (f ) = lim h r→∞ 1 logq−1 1−r Remark 1.2. From Definitions 1.7 and 1.8, it is easy to see that τ[p,q] (f ) ≤ τM,[p,q] (f ) and τ [p,q] (f ) ≤ τ M,[p,q] (f ).   1 to denote the unintegrated counting Definition 1.9. For any a ∈ C ∪ {∞}, we use n r, f −a function for the sequence of a-point of a meromorphic function f in ∆. Then the [p, q]-exponents   1 is defined by of convergence of a-point of f about n r, f −a   1 logp n r, f −a   . λn[p,q] (f, a) = lim 1 r→1− log q 1−r

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  1 Definition 1.10. Let N r, f −a be the integrated counting function for the sequence of a-point of a meromorphic function f in ∆. Then the [p, q]-exponents of convergence of a-point of f about   1 N r, f −a is defined by   1 logp N r, f −a   . λN [p,q] (f, a) = lim− 1 r→1 logq 1−r Remark 1.3. Similar with Definitions 1.9 and 1.10, we can of the [p, q] alsogive thedefinitions  n 1 1 exponents of convergence of distinct a-point of f about n r, f −a and N r, f −a , i.e., λ[p,q] (f, a) N

and λ[p,q] (f, a). The order and type are two important indicators in revealing the growth of the entire functions or meromorphic functions, many authors have investigated the growth of entire functions or meromorphic functions in the complex plane or in the unit disc (e.g., see[4, 8 − 10, 14 − 20]) since the first half of the twentieth century. In the following, we list some classic results in the complex plane. Theorem A(see [4, 10, 19, 20]). If f1 and f2 are meromorphic functions of finite order with σ(f1 ) = σ3 and σ(f2 ) = σ4 , then σ(f1 + f2 ) ≤ max{σ3 , σ4 }, σ(f1 f2 ) ≤ max{σ3 , σ4 }, σ(f1 /f2 ) ≤ max{σ3 , σ4 }; if σ3 < σ4 , then σ(f1 + f2 ) = σ(f1 f2 ) = σ(f1 /f2 ) = σ4 . Theorem B (see [20]). If f1 and f2 are meromorphic functions of finite order, then µ(f1 + f2 ) ≤ min{max{σ(f1 ), µ(f2 )}, max{µ(f1 ), σ(f2 )}}, µ(f1 f2 ) ≤ min{max{σ(f1 ), µ(f2 )}, max{µ(f1 ), σ(f2 )}}. Furthermore, if σ(f1 ) < µ(f2 ), then µ(f1 + f2 ) = µ(f1 f2 ) = µ(f2 ); or if σ(f2 ) < µ(f1 ), then µ(f1 + f2 ) = µ(f1 f2 ) = µ(f1 ). Theorem C (see [10]). If f1 and f2 are entire functions of finite order satisfying σ(f1 ) = σ(f2 ) = σ5 , then the following two statements hold: (i) If τM (f1 ) = 0 and 0 < τM (f2 ) < ∞, then σ(f1 f2 ) = σ5 , τM (f1 f2 ) = τM (f2 ). (ii) If τM (f1 ) < ∞ and τM (f2 ) = ∞, then σ(f1 f2 ) = σ5 , τM (f1 f2 ) = ∞. Theorem D (see [18]). Let f1 (z) and f2 (z) be entire functions satisfying 0 < σp (f1 ) = σp (f2 ) = σ6 < ∞, 0 ≤ τM,p (f1 ) < τM,p (f2 ) ≤ ∞. Then the following statements hold: (i) If p ≥ 1, then σp (f1 + f2 ) = σ6 , τM,p (f1 + f2 ) = τM,p (f2 ); (ii) If p > 1, then σp (f1 f2 ) = σ6 , τM,p (f1 f2 ) = τM,p (f2 ). Theorem E (see [18]). Let p ≥ 1, f (z) be an entire function or a meromorphic function in the complex plane satisfying 0 < σp (f ) < ∞. If p ≥ 1, then σp (f ) = σp (f 0 ), τM,p (f 0 ) = τM,p (f ); if p > 1, then σp (f ) = σp (f 0 ), τp (f 0 ) = τp (f ). From Theorems A-E, we can easily obtain the following similar propositions in the unit disc.

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The Order, Type and Zeros of Meromorphic Functions and Analytic Functions of [p, q]-Order in the Unit Disc 5

Propositions (i) If f1 and f2 are meromorphic functions satisfying σ[p,q] (f1 ) = σ6 and σ[p,q] (f2 ) = σ7 in ∆, then σ[p,q] (f1 ± f2 ) ≤ max{σ6 , σ7 }, σ[p,q] (f1 f2 ) ≤ max{σ6 , σ7 } and σ[p,q] (f1 /f2 ) ≤ max{σ6 , σ7 }. (ii) If σ6 6= σ7 in Proposition (i), then σ[p,q] (f1 ± f2 ) = σ[p,q] (f1 f2 ) = σ[p,q] (f1 /f2 ) = max{σ6 , σ7 }. (iii) If f1 and f2 are meromorphic functions in ∆, then µ[p,q] (f1 + f2 ) ≤ max{σ[p,q] (f1 ), µ[p,q] (f2 )} or µ[p,q] (f1 +f2 ) ≤ max{µ[p,q] (f1 ), σ[p,q] (f2 )} and µ[p,q] (f1 f2 ) ≤ max{σ[p,q] (f1 ), µ[p,q] (f2 )} or µ[p,q] (f1 f2 ) ≤ max{µ[p,q] (f1 ), σ[p,q] (f2 )}. (iv) If f1 and f2 are meromorphic functions in ∆ satisfying σ[p,q] (f1 ) < µ[p,q] (f2 ) ≤ ∞, then µ[p,q] (f1 + f2 ) = µ[p,q] (f1 f2 ) = µ[p,q] (f1 /f2 ) = µ[p,q] (f2 ). (v) If f1 and f2 are analytic functions in ∆ satisfying σM,[p,q] (f1 ) = σ8 and σM,[p,q] (f2 ) = σ9 , then σM,[p,q] (f1 ± f2 ) ≤ max{σ8 , σ9 } and σM,[p,q] (f1 f2 ) ≤ max{σ8 , σ9 }. If σ8 6= σ9 , then σM,[p,q] (f1 ± f2 ) = max{σ8 , σ9 }. (vi) If f1 and f2 are analytic functions in ∆, then max{µM,[p,q] (f1 ± f2 ), µM,[p,q] (f1 f2 )} ≤ max{σM,[p,q] (f1 ), µM,[p,q] (f2 )} or max{µM,[p,q] (f1 ±f2 ), µM,[p,q] (f1 f2 )} ≤ max{µM,[p,q] (f1 ), σM,[p,q] (f2 )}. (vii) If f1 and f2 are analytic functions
of [p, q]-order in ∆, for any r ∈ [0, 1), by the inequal4 T 1+r ity T (r, f ) ≤ log+ M (r, f ) ≤ 1−r 2 , f (see [4, 17]), we easily obtain that if p = q ≥ 2 and σ[p,q] (f ) > 1, or p > q ≥ 1, then σ[p,q] (f ) = σM,[p,q] (f ) and τ[p,q] (f ) = τM,[p,q] (f ). Similarly, we have µ[p,q] (f ) = µM,[p,q] (f ) and τ [p,q] (f ) = τ M,[p,q] (f ) if p = q ≥ 2 and µ[p,q] (f ) > 1, or p > q ≥ 1. Combining Theorems D and E, a natural question is: Can we get the similar results with Theorems D, E for meromorphic functions or analytic functions of [p, q] order in ∆? In fact, we obtain the following results: Theorem 1.1. Let f1 and f2 be meromorphic functions in ∆ satisfying 0 < σ[p,q] (f1 ) = σ[p,q] (f2 ) = σ10 < ∞ and 0 ≤ τ1 = τ[p,q] (f1 ) < τ[p,q] (f2 ) = τ2 ≤ ∞. Then σ[p,q] (f1 + f2 ) = σ[p,q] (f1 f2 ) = σ[p,q] (f1 /f2 ) = σ10 , and the following two statements hold: (i) If p > 1 and p ≥ q ≥ 1, then τ[p,q] (f1 + f2 ) = τ[p,q] (f1 f2 ) = τ[p,q] (f1 /f2 ) = τ[p,q] (f2 ). (ii) If p = q = 1, then τ2 − τ1 ≤ max{τ (f1 + f2 ), τ (f1 f2 ), τ (f1 /f2 )} ≤ τ2 + τ1 . Theorem 1.2. Let f1 and f2 be meromorphic functions in ∆ satisfying 0 < σ[p,q] (f1 ) = µ[p,q] (f2 ) < ∞ and 0 ≤ τ[p,q] (f1 ) < τ [p,q] (f2 ) ≤ ∞, then µ[p,q] (f1 +f2 ) = µ[p,q] (f1 f2 ) = µ[p,q] (f1 /f2 ) = µ[p,q] (f2 ). And if p > 1 and p ≥ q ≥ 1, we have τ [p,q] (f1 + f2 ) = τ [p,q] (f1 f2 ) = τ [p,q] (f1 /f2 ) = τ [p,q] (f2 ). In the following, when f1 and f2 are analytic functions of [p, q]-order in the unit disc we have the similar results.

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Theorem 1.3. Let f1 and f2 be analytic functions in ∆ satisfying 0 < σM,[p,q] (f1 ) = σM,[p,q] (f2 ) = σ11 < ∞ and 0 ≤ τM,[p,q] (f1 ) < τM,[p,q] (f2 ) ≤ ∞, then σM,[p,q] (f1 +f2 ) = σ11 and τM,[p,q] (f1 +f2 ) = τM,[p,q] (f2 ). Remark 1.4. By Proposition (vii), we know that Theorem 1.3 is of the same with Theorem 1.1 for p > q ≥ 1 and p = q ≥ 2, σ[p,q] (f ) > 1. For the case p = q = 1, the result of Theorem 1.3 is better than that of Theorem 1.1. Corollary 1.1. Let f1 and f2 be analytic functions in ∆ satisfying 0 < σM,[p,q] (f1 ) = µM,[p,q] (f2 ) < ∞ and 0 ≤ τM,[p,q] (f1 ) < τ M,[p,q] (f2 ) ≤ ∞, then µM,[p,q] (f1 + f2 ) = µM,[p,q] (f2 ) and τ M,[p,q] (f1 + f2 ) = τ M,[p,q] (f2 ). Theorem 1.4. Let f be an analytic function of [p, q]-order in ∆, then σM,[p,q] (f ) = σM,[p,q] (f 0 ), µM,[p,q] (f ) = µM,[p,q] (f 0 ). If 0 < σM,[p,q] (f ) < ∞ or 0 < µM,[p,q] (f ) < ∞, then τM,[p,q] (f ) = τM,[p,q] (f 0 ), τ M,[p,q] (f ) = τ M,[p,q] (f 0 ). Theorem 1.5. Let f be a meromorphic function of [p, q]-order in ∆, then (i) If p ≥ q ≥ 2 and p > q = 1, then σ[p,q] (f ) = σ[p,q] (f 0 ), µ[p,q] (f ) = µ[p,q] (f 0 ) and τ[p,q] (f ) = τ[p,q] (f 0 ), τ [p,q] (f ) = τ [p,q] (f 0 ) for 0 < σ[p,q] (f ) < ∞ or 0 < µ[p,q] (f ) < ∞. (ii) If p = q = 1, then σ(f ) = σ(f 0 ), µ(f ) = µ(f 0 ) and τ[1,1] (f 0 ) ≤ 2τ[1,1] (f ), τ [1,1] (f 0 ) ≤ 2τ [1,1] (f ). Theorem 1.6. Let f be a meromorphic function of [p, q]-order in ∆, a ∈ C ∪ {∞}. Then the following statements hold: n (i) If p > q ≥ 1, then λN [p,q] (f, a) = λ[p,q] (f, a). (ii) If p = q = 1, then λN (f, a) ≤ λn (f, a) ≤ λN (f, a) + 1n(see [12]). o n N (iii) If p = q ≥ 2, then λN [p,p] (f, a) ≤ λ[p,p] (f, a) ≤ max λ[p,p] (f, a), 1 . Furthermore, we have n N N N n λN [p,p] (f, a) = λ[p,p] (f, a) if λ[p,p] (f, a) ≥ 1, and if λ[p,p] (f, a) < 1 then λ[p,p] (f, a) ≤ λ[p,p] (f, a) ≤ 1. n

N

Remark 1.4. The conclusions of Theorem 1.6 also hold between λ[p,q] (f, a) and λ[p,q] (f, a).

2.

Preliminary Lemmas

Lemma 2.1 (see [4, 19, 20]). Let f1 , f2 , · · ·, fm (z) be meromorphic functions in ∆, where m ≥ 2 is a positive integer. Then (i) T (r, f1 f2 · · · fm ) ≤

m P

T (r, fi ),

i=1

(ii) T (r, f1 + f2 + · · · + fm ) ≤

m P

T (r, fi ) + log m.

i=1

Lemma 2.2 (see [6, 17]). Let f be a meromorphic function in ∆, and let k ≥ 1 be an integer.

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The Order, Type and Zeros of Meromorphic Functions and Analytic Functions of [p, q]-Order in the Unit Disc 7

Then

! f (k) m r, = S(r, f ), f n  o R 1 where S(r, f ) = O log+ T (r, f ) + log 1−r , possibly outside a set E1 ⊂ [0, 1) with E1 ∞.

dt 1−t


0 and r → 1− , we have σ10     1 T (r, f1 ) ≤ expp−1 (τ1 + ε) logq−1 , (3.1) 1−r    σ10  1 T (r, f2 ) ≤ expp−1 (τ2 + ε) logq−1 . (3.2) 1−r By using (3.1)-(3.2) and Lemma 2.1, we have T (r, f1 + f2 ) ≤ T (r, f1 ) + T (r, f2 ) + log 2 σ10     σ10     1 1 + expp−1 (τ2 + ε) logq−1 + log 2 ≤ expp−1 (τ1 + ε) logq−1 1−r 1−r    σ10  1 ≤ 2 expp−1 (τ2 + ε) logq−1 . 1−r Hence σ[p,q] (f1 + f2 ) ≤ σ10 . In addition, if p = q = 1, we can get τ[p,q] (f1 + f2 ) ≤ τ1 + τ2 , if p > 1, then τ[p,q] (f1 + f2 ) ≤ τ2 for any p ≥ q ≥ 1. On the other hand, for any given ε > 0, there exists a − sequence {rn }∞ n=1 → 1 satisfying    σ10  1 T (rn , f1 ) ≤ expp−1 (τ1 + ε) logq−1 , (3.3) 1 − rn

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T (rn , f2 ) ≥ expp−1 (τ2 − ε) logq−1



1 1 − rn

σ10  .

(3.4)

By (3.3)-(3.4) and Lemma 2.1, we obtain T (rn , f1 + f2 ) ≥ T (rn , f2 ) − T (rn , f1 ) − log 2       σ10  σ10  1 1 ≥ expp−1 (τ2 − ε) logq−1 − expp−1 (τ1 + ε) logq−1 − log 2. 1 − rn 1 − rn (3.5) By (3.5) we have σ[p,q] (f1 + f2 ) ≥ σ10 . Furthermore, if p = q = 1, then τ[p,q] (f1 + f2 ) ≥ τ2 − τ1 and τ[p,q] (f1 + f2 ) ≥ τ2 for p > 1 and p ≥ q ≥ 1. Therefore, we have σ[p,q] (f1 + f2 ) = σ10 , and if p > 1, then τ[p,q] (f1 + f2 ) = τ[p,q] (f2 ), if p = q = 1, then τ2 − τ1 ≤ τ[p,q] (f1 + f2 ) ≤ τ2 + τ1 . Since T (r, f1 f2 ) ≤ T (r, f1 ) + T (r, f2 ),   T (r, f1 f2 ) ≥ T (r, f2 ) − T (r, f1 ) − O(1) and T r, f12 = T (r, f2 ) + O(1), by the above proof, σ[p,q] (f1 f2 ) = σ[p,q] (f1 /f2 ) = σ10 , τ[p,q] (f1 f2 ) = τ[p,q] (f1 /f2 ) = τ[p,q] (f2 ) for p > 1 and τ2 − τ1 ≤ max{τ (f1 f2 ), τ (f1 /f2 )} ≤ τ2 + τ1 if p = q = 1 also can hold. Moreover, Theorem 1.1 also holds for τ[p,q] (f2 ) = τ2 = ∞. Proof of Theorem 1.2. Without loss of generality, we suppose that 0 ≤ τ3 = τ[p,q] (f1 ) < τ [p,q] (f2 ) = τ4 < ∞. Assume that σ[p,q] (f1 ) = µ[p,q] (f2 ) = µ3 , and by Definition 1.8, it is easy to − see that for any given ε > 0, there exists a sequence {rn }∞ n=1 → 1 satisfying    µ3  1 T (rn , f1 ) < expp−1 (τ3 + ε) logq−1 , (3.6) 1 − rn µ3     1 . (3.7) T (rn , f2 ) < expp−1 (τ4 + ε) logq−1 1 − rn By (3.6)-(3.7) and Lemma 2.1, we have

 ≤ expp−1

T (rn , f1 + f2 ) ≤ T (rn , f1 ) + T (rn , f2 ) + log 2  µ3     µ3  1 1 + expp−1 (τ4 + ε) logq−1 + log 2 (τ3 + ε) logq−1 1 − rn 1 − rn    µ3  1 ≤ 2 expp−1 (τ4 + ε) logq−1 . 1 − rn 

Hence µ[p,q] (f1 + f2 ) ≤ µ3 . In addition, if p > 1, then τ [p,q] (f1 + f2 ) ≤ τ4 for p ≥ q ≥ 1. On the other hand, for any given ε > 0 and r → 1− , we have    µ3  1 T (r, f1 ) ≤ expp−1 (τ3 + ε) logq−1 , (3.8) 1−r    µ3  1 T (r, f2 ) ≥ expp−1 (τ4 − ε) logq−1 . (3.9) 1−r

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The Order, Type and Zeros of Meromorphic Functions and Analytic Functions of [p, q]-Order in the Unit Disc 9

By (3.8)-(3.9) and Lemma 2.1, we obtain T (r, f1 + f2 ) ≥ T (r, f2 ) − T (r, f1 ) − log 2       µ3  µ3  1 1 −expp−1 (τ3 + ε) logq−1 −log 2. (3.10) ≥ expp−1 (τ4 − ε) logq−1 1−r 1−r By (3.10) we have µ[p,q] (f1 + f2 ) ≥ µ3 and τ [p,q] (f1 + f2 ) ≥ τ4 for p > 1 and p ≥ q ≥ 1. Thus we have µ[p,q] (f1 + f2 ) = µ(f2 ) and if p > 1 and p ≥ q ≥ 1, then τ [p,q] (f1 + f2 ) = τ [p,q] (f2 ). Since T (r, f1 f2 ) ≤ T (r, f1 ) + T (r, f2 ), T (r, f1 f2 ) ≥ T (r, f2 ) − T (r, f1 ) − O(1) and   T r, f12 = T (r, f2 ) + O(1), by the above proof, µ[p,q] (f1 f2 ) = µ[p,q] (f1 /f2 ) = µ[p,q] (f2 ) and τ [p,q] (f1 f2 ) = τ [p,q] (f1 /f2 ) = τ [p,q] (f2 ) also hold if p > 1 and p ≥ q ≥ 1. The conclusions of Theorem 1.2 also hold for τ3 = τ[p,q] (f1 ) < τ [p,q] (f2 ) = τ4 = ∞. Proof of Theorem 1.3. Set 0 ≤ τ5 = τM,[p,q] (f1 ) < τM,[p,q] (f2 ) = τ6 < ∞, by Definition 1.7, − for any given ε (0 < 2ε < τ6 − τ5 ), there exists a sequence {rn }∞ n=1 → 1 satisfying σ11     1 , (3.11) M (rn , f1 ) ≤ expp (τ5 + ε) logq−1 1 − rn    σ11  1 M (rn , f2 ) > expp (τ6 − ε) logq−1 . (3.12) 1 − rn We can choose a sequence {zn }∞ n=1 satisfying |zn | = rn (n = 1, 2, · · ·) and |f2 (zn )| = M (rn , f2 ), by (3.11)-(3.12) we have M (rn , f1 + f2 ) ≥ |f1 (zn ) + f2 (zn )| ≥ |f2 (zn )| − |f1 (zn )| ≥ M (rn , f2 ) − M (rn , f1 ) σ11  σ11        1 1 ≥ expp (τ6 − ε) logq−1 − expp (τ5 + ε) logq−1 1 − rn 1 − rn    σ11  1 1 ≥ expp (τ6 − ε) logq−1 (rn → 1− ). 2 1 − rn Hence σM,[p,q] (f1 + f2 ) ≥ σ11 and τM,[p,q] (f1 + f2 ) ≥ τ6 . On the other hand, we have M (r, f1 + f2 ) ≤ M (r, f1 ) + M (r, f2 ) σ11  σ11        1 1 + expp (τ6 + ε) logq−1 ≤ expp (τ5 + ε) logq−1 1−r 1−r    σ11  1 ≤ 2 expp (τ6 + ε) logq−1 , 1−r therefore σM,[p,q] (f1 + f2 ) ≤ σ11 and τM,[p,q] (f1 + f2 ) ≤ τ6 . Thus we can get σM,[p,q] (f1 + f2 ) = σ11 and τM,[p,q] (f1 + f2 ) = τM,[p,q] (f2 ). Moreover, Theorem 1.3 also holds for τM,[p,q] (f1 ) < τM,[p,q] (f2 ) = τ6 = ∞.

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Proof of Theorem 1.4. Since f is an analytic function in the unit disc, from the formula Z z f 0 (ζ)dζ (|z| = r < 1) f (z) = f (0) + 0

where the integral route is a line from 0 to z in the unit disc. We obtain that Z z f 0 (ζ)dζ| ≤ |f (0)| + rM (r, f 0 ) ≤ |f (0)| + M (r, f 0 ), M (r, f ) ≤ |f (0)| + | 0

i.e. M (r, f 0 ) ≥ M (r, f ) − |f (0)|.

(3.13)

By (3.13), we have σM,[p,q] (f 0 ) ≥ σM,[p,q] (f ), µM,[p,q] (f 0 ) ≥ µM,[p,q] (f ). On the other hand, in the circle |z| = r ∈ (0, 1), we take a point z0 satisfying |f 0 (z0 )| = M (r, f 0 ). By the Cauchy inequality Z 1 f (ζ) f 0 (z0 ) = dζ, 2πi C (ζ − z0 )2 where C = {ζ : |ζ − z0 | = s(r) − r} and s(r) = 1 − d(1 − r), d ∈ (0, 1). We deduce that 1 M (r, f ) = |f (z0 )| ≤ 2π 0

0

Z 0

2π

f (ζ) M (s(r), f ) (ζ − z0 )2 (s(r) − r) dθ ≤ s(r) − r ,

i.e. M (s(r), f ) . (1 − d)(1 − r)

M (r, f 0 ) ≤

(3.14)

By (3.14), then σM,[p,q] (f 0 ) ≤ σM,[p,q] (f ), µM,[p,q] (f 0 ) ≤ µM,[p,q] (f ). Hence σM,[p,q] (f ) = σM,[p,q] (f 0 ), µM,[p,q] (f 0 ) = µM,[p,q] (f ).

(3.15)

If 0 < σM,[p,q] (f ) < ∞ and by (3.13), (3.15), we can get τM,[p,q] (f 0 ) ≥ τM,[p,q] (f ). Then by (3.14)-(3.15), if p ≥ q = 1 we can obtain   i h  1   0 logp (1−r)(1−d) log M (s(r), f )  logp M (r, f ) p ≤ max ,  σM,[p,1] (f 0 )  σM,[p,1] (f )  σM,[p,1] (f )   1 1 1   1−r

1−r

1−r

  h i   1    logp (1−r)(1−d) logp M (s(r), f ) 1 σM,[p,1] (f )  ≤ max  , σM,[p,1] (f ) , h iσM,[p,1] (f ) · d   1 1   1−r

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The Order, Type and Zeros of Meromorphic Functions and Analytic Functions of [p, q]-Order in the Unit Disc11

let d → 1, therefore τM,[p,q] (f 0 ) ≤ τM,[p,q] (f ), then τM,[p,q] (f ) = τM,[p,q] (f 0 ). If p ≥ q ≥ 2, then   h i   1   0 log p (1−r)(1−d) logp M (r, f ) logp M (s(r), f ) ≤ max , h iσM,[p,q] (f 0 ) h iσM,[p,q] (f ) h iσM,[p,q] (f )  ,     1 1 1   logq−1 1−r logq−1 1−r logq−1 1−r thus we have τM,[p,q] (f 0 ) ≤ τM,[p,q] (f ) and τM,[p,q] (f ) = τM,[p,q] (f 0 ). If 0 < µM,[p,q] (f ) < ∞, we can similarly obtain τ M,[p,q] (f ) = τ M,[p,q] (f 0 ). Proof of Theorem 1.5. By Lemma 2.2, we have  f0 + 2N (r, f ) T (r, f ) = m(r, f ) + N (r, f ) ≤ m(r, f ) + m r, f     f0 1 ≤ 2T (r, f ) + m r, ≤ (2 + ε)T (r, f ) + O log (r 6∈ E1 ). (3.16) f 1−r 0

0



0

From (3.16) and Lemma 2.3, we have σ[p,q] (f 0 ) ≤ σ[p,q] (f ), µ[p,q] (f 0 ) ≤ µ[p,q] (f ) for p ≥ q ≥ 1, τ[p,q] (f 0 ) ≤ τ[p,q] (f ), τ [p,q] (f 0 ) ≤ τ [p,q] (f ), for p > 1 and τ[1,1] (f 0 ) ≤ 2τ[1,1] (f ), τ [1,1] (f 0 ) ≤ 2τ [1,1] (f ) for p = q = 1. On the other hand, set R = s(r) = 1 − d(1 − r), d ∈ (0, 1) in Lemma 2.4, we have   1 3 T (r, f ) < 2 + log T (s(r), f 0 ). (3.17) π (1 − d)(1 − r) By (3.17) and by the similar proof in Theorem 1.4, we have σ[p,q] (f ) ≤ σ[p,q] (f 0 ), µ[p,q] (f ) ≤ µ[p,q] (f 0 ) for p ≥ q ≥ 1, τ[p,q] (f ) ≤ τ[p,q] (f 0 ), τ [p,q] (f ) ≤ τ [p,q] (f 0 ) for p ≥ q ≥ 2 and τ[p,q] (f ) ≤ ( d1 )σ[p,q] (f ) τ[p,q] (f 0 ) for p > q = 1, letting d → 1, therefore the following statements hold: If p ≥ q ≥ 2 and p > q = 1, then σ[p,q] (f ) = σ[p,q] (f 0 ), µ[p,q] (f ) = µ[p,q] (f 0 ) and τ[p,q] (f ) = τ[p,q] (f 0 ) for 0 < σ[p,q] (f ) < ∞, τ [p,q] (f ) = τ [p,q] (f 0 ) for 0 < µ[p,q] (f ) < ∞. If p = q = 1, then σ(f ) = σ(f 0 ), µ(f ) = µ(f 0 ) and τ[1,1] (f 0 ) ≤ 2τ[1,1] (f ), τ [1,1] (f 0 ) ≤ 2τ [1,1] (f ). Without loss of generality, assume that f (a) 6= 0, by       Z r n t, 1 − n 0, 1 f −a f −a 1 N r, = dt (0 < r < 1), f −a t 0

Proof of Theorem 1.6.

we have  n r,

1 f −a



1 ≤ log 1 +

Z 1−r 2r




r

r+ 1−r 2

  1 n t, f −a t

1 dt ≤ log 1 +

 1−r 2r


N

1+r 1 , 2 f −a

1−r − where 0 < r < 1, log(1 + 1−r 2r ) ∼ 2r , r → 1 . By (3.18), we have        1 1 2r   logp n r, f −a logp N 1+r , log p 2 f −a 1−r   ≤ max lim     . lim , lim 1 1 1 r→1−  r→1− log r→1− log logq 1−r q 1−r q 1−r

554

 , (3.18)

(3.19)

Jin Tu 544-556

J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 29, NO.3, 2021, COPYRIGHT 2021 EUDOXUS PRESS, LLC

12

J. Tu, K.Q. Hu, H. Zhang

By (3.19), we can obtain (i) if p > q ≥ 1, then λn[p,q] (f, a) ≤ λN [p,q] (f, a); n N (ii) if p = q = 1, then λ (f, a) ≤ λ (f, a)n+ 1; o (iii) if p = q ≥ 2, then λn[p,p] (f, a) ≤ max λN (f, a), 1 . [p,p] On the other hand, by     Z r n t, 1       f −a 1 1 r 1 N r, = ≤ n r, log + O(1), dt + N r0 , f −a t f − a f − a r 0 r0

(3.20)

where 0 < r0 < r < 1. By (3.20), we can get n (i) if p > q ≥ 1, then λN [p,q] (f, a) ≤ λ[p,q] (f, a); (ii) if p = q = 1, then λN (f, a) ≤ λn (f, a); n (iii) if p = q ≥ 2, then λN [p,p] (f, a) ≤ λ[p,p] (f, a). Therefore, the conclusions of Theorem 1.6 hold. Author Contributions: Conceptualization and providing original idea, J. Tu; Formal analysis and manuscript editing, K.Q.Hu and H. Zhang. Funding: This project is supported by the National Natural Science Foundation of China(11561031, 11861005, 41472130), the Natural Science Foundation of China of Jiangxi Province in China(20161BAB201020). Conflicts of Interest: The authors declare that there are no conflicts of interest regarding the publication of this paper.

References [1] S. Bank, General theorem concerning the growth of solutions of first-order algebraic differential equations, Composition Math., no. 25, pp. 61-70, 1972. [2] B. Bela¨ıdi, Growth of solutions to linear equations with analytic coefficients of [p,q]-order in the unit disc, Electron. J. Diff. Equ., vol.156 no.2, pp. 25-38, 2011. [3] T. B. Cao and H. X. Yi, The growth of solutions of linear equations with coeffients of iterated order in the unit disc, J. Math. Anal. Appl., vol.319 pp. 278-294, 2006. [4] W. Hayman, Meromorphic Functions, Clarendon Press, Oxford, 1964. [5] W. Hayman, On the characteristic of functions meromorphic in the unit disk and of their integrals, Acta Math., vol.112, pp.181-214, 1964. [6] J. Heittokangas, On complex differential equations in the unit disc, Ann. Acad. Sci. Fenn. Math. Diss., vol.122, pp.1-54, 2000. [7] J. Heittokangas, R. Korhonen, and J. R¨ atty¨ a, Fast growing solutions of linear differential equations in the unit disc, Results Math., vol.49, pp.265-278, 2006.

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The Order, Type and Zeros of Meromorphic Functions and Analytic Functions of [p, q]-Order in the Unit Disc13

[8] O. P. Juneja, G. P. Kapoor, and S. K. Bajpai, On the (p,q)-order and lower (p,q)-order of an entire function, J. Reine Angew. Math., vol.282, pp.53-67, 1976. [9] O. P. Juneja, G. P. Kapoor, and S. K. Bajpai, On the (p,q)-type and lower (p,q)-type of an entire function, J. Reine Angew. Math., vol.290, pp.180-190, 1977. [10] B. Ja. Levin, Distribution of Zeros of Entire Functions, revised edition Transl. Math. Monographs Vol. 5, Amer. Math. Soc. Providence, 1980. [11] L. M. Li and T.B. Cao, Solutions for linear differentialequations with meromorphic coefficients of (p,q)-order in the plane, Electron. J. Diff. Equ.,vol.2012, no.195, pp.1-15, 2012. [12] C. Linden, The representation of regular functions, J. London Math. Soc., vol.39, pp.19-30, 1964. [13] J. Liu, J. Tu, and L. Z. Shi, Linear differential equations with coefficients of (p,q)-order in the complex plane, J. Math. Anal. Appl, vol.372 pp.55-67, 2010. [14] R. Nevanlinna, Le th´eor`eme de Picard-Borel et la theorie des functions meromorphes, Gauthier-Villans Pairs, 1929. [15] D. Shea, L.R. Sons, Value distribution theory for meromorphic functions of slow growth in the disc, Houston J. Math., vol.12, no.2, pp.249-266, 1986. [16] L. R. Sons, Unbouned functions in the unit disc, Internet. J. Math. Math. Sci., vol.6, no.2, pp.201-242, 1983. [17] M. Tsuji, Potential Theory in Modern Function Theory, Chelsea, New York, 1975, reprint of the 1959 edition. [18] J. Tu, Y. Zeng, H. Y. Xu, The Order and Type of Meromorphic Functions and Entire Functions of Finite Iterated Order, J. Computational Analysis And Applications, vol.21, (5), pp.994-1003, 2016. [19] L. Yang, Value Distribution Theory and Its New Research, Science Press, Beijing, 1982 (in Chinese). [20] H. X. Yi and C. C. Yang, The Uniqueness Theory of Meromorphic Function, Science Press, Beijing, 1995 (in Chinese).

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Reachable sets for semilinear integrodifferential control systems Hyun-Hee Roh1 and Jin-Mun Jeong2,∗ 1,2 Department

of Applied Mathematics, Pukyong National University Busan 48513, South Korea

E-mail: [email protected], ∗ [email protected](Corresponding author)

Abstract In this paper, we consider a control system for semilinear integrodifferential equations in Hilbert spaces with Lipschitz continuous nonlinear term. Our method is to find the equivalence of approximate controllability for the given semilinear system and the linear system excluded the nonlinear term, which is based on results on regularity for the mild solution. Finally, we give a simple example to which our main result can be applied. Keywords: approximate controllability, semilinear control system, lipschtiz continuity, approximate controllability, reachable set AMS Classification Primary 35B37; Secondary 93C20

1

Introduction

Let H and V be real Hilbert spaces such that V is a dense subspace in H. In this paper, we are concerned with the control results for the following retarded semilinear control system in Hilbert space H: ( Rt 0 x (t) = Ax(t) + g(t, x(t), 0 k(t, s, x(s))ds)) + Bu(t), t > 0, (1.1) x(0) = x0 , 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(2019R1F1A1048077)

1

557

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where t > 0, B is a bounded linear controller, and u(t) is an appropriate control functions. Let A be the operator associated with a bounded sesquilinear form defined in V × V satisfying G˚ arding inequality. Then it is well known that S(t) generated by A is an analytic semigroup in both H and V ∗ , where V ∗ is the dual space of V , and so the system (1.1) may be considered as an system in both H and V ∗ . g is is a nonlinear mapping as detailed in Section 2. Whether the reachable set associated with control space in dense subset of H. This is called an approximate controllability problem. As for linear evolution systems in general Banach, there are many papers and monographs, see [1, 2], Triggiani [3], Curtain and Zwart [4] and references and therein. The controllability for nonlinear control systems has been studied by many authors, for example, control of nonlinear infinite dimensional systems in [5], controllability for parabolic equations with uniformly bounded nonlinear terms in [6], local controllability of neutral functional differential systems in [7]. Recently, the approximate controllability for semilinear control systems can be founded in [8, 9, 10], their results give sufficient condition on strict assumptions on the control action operator B. Similar considerations of semilinear systems have been dealt with in many references [11, 12, 13, 14]. We investigate the equivalence of approximate controllability for (1.1) such that excluded the nonlinear term and the controller. The solution mapping from the initial space to the solution space is Lipschitz continuous in [0, T ]. We no longer require the strict range condition on B, and the uniform boundedness in [6] but instead we need the regularity and a variation of solutions of the given equations. For the basis of our study we construct the fundamental solution and establish variations of constant formula of solutions for the linear systems, see [15, 16]. Based on L2 -regularity properties of semilinear integrodifferential equations in Hilbert space and the regularity of solutions discussed in Section 2. We will obtain the relations between the reachable set of the semilinear system and that of its corresponding linear system in Section 3. Finally, a simple example to which our main result can be applied is given.

2

Regularity for retarded semilinear equations

If H is identified with its dual space we may write V ⊂ H ⊂ V ∗ densely and the corresponding injections are continuous. The norms on V , H and V ∗ will be denoted by || · ||, | · | and || · ||∗ , respectively. The duality pairing between the element v1 of V ∗ and the element v2 of V is denoted by (v1 , v2 ), which is the ordinary inner product in H if v1 , v2 ∈ H.

558

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For l ∈ V ∗ we denote (l, v) by the value l(v) of l at v ∈ V . The norm of l as element of V ∗ is given by |(l, v)| . ||l||∗ = sup v∈V ||v|| Therefore, we assume that V has a stronger topology than H and, for brevity, we may regard that ||u||∗ ≤ |u| ≤ ||u||, ∀u ∈ V. (2.1) Let a(·, ·) be a bounded sesquilinear form defined in V × V and satisfying G˚ arding’s inequality Re a(u, u) ≥ ω1 ||u||2 − ω2 |u|2 , (2.2) where ω1 > 0 and ω2 is a real number. Let A be the operator associated with this sesquilinear form: (Au, v) = −a(u, v), u, v ∈ V. (2.3) Then A is a bounded linear operator from V to V ∗ by the Lax-Milgram Theorem. The realization of A in H which is the restriction of A to D(A) = {u ∈ V : Au ∈ H} is also denoted by A. It is well known that A generates an analytic semigroup in both of H and V ∗ (see [17]). From the following inequalities ω1 ||u||2 ≤ Re a(u, u) + ω2 |u|2 ≤ |Au| |u| + ω2 |u|2 ≤ max{1, ω2 }||u||D(A) |u|, where ||u||D(A) = (|Au|2 + |u|2 )1/2 is the graph norm of D(A), it follows that there exists a constant C > 0 such that 1/2

||u|| ≤ C||u||D(A) |u|1/2 .

(2.4)

Thus we have the following sequence D(A) ⊂ V ⊂ H ⊂ V ∗ ⊂ D(A)∗ ,

(2.5)

where each space is dense in the next one, which is continuous injection. Lemma 2.1. With the notations (2.1), (2.4), and (2.5), we have (V, V ∗ )1/2,2 = H, (D(A), H)1/2,2 = V, where (V, V ∗ )1/2,2 denotes the real interpolation space between V and V ∗ (Section 1.3.3 of [18]).

559

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Assumption (K). Let k : R+ × [ − h, 0] × V → H be a nonlinear mapping satisfying the following: (K1) For any x ∈ V the mapping k(·, ·, x) is measurable; (K2) There exist positive constants K0 , K1 such that |k(t, s, x) − k(t, s, y)| ≤ K1 ||x − y||, |k(t, s, 0)| ≤ K0 for all (t, s) ∈ R+ × [−h, 0] and x, y ∈ V . Assumption (G). Let g : R+ × V × H → H be a nonlinear mapping satisfying the following: (G1) For any x ∈ V , y ∈ H the mapping g(·, x, y) is measurable; (G2) There exist positive constants L0 , L1 , L2 such that |g(t, x, y) − g(t, xˆ, yˆ)| ≤ L1 ||x − xˆ|| + L2 |y − yˆ|, |g(t, 0, 0)| ≤ L0 for all t ∈ R+ , x, xˆ ∈ V , and y, yˆ ∈ H. For x ∈ L2 (−h, T ; V ), T > 0 we set Z G(t, x) = g(t, x(t),

t

k(t, s, x(s))ds). 0

The above operator g is the semilinear case of the nonlinear part of quasilinear equations considered by Yong and Pan [19]. The mild solution of (1.1) is represented by Z t  x(t) = S(t)x0 + G(s, x(s)0 + Bu(s) ds, t ≥ 0. 0

Lemma 2.2. Let x ∈ L2 (0, T ; V ), T > 0. Then G(·, x) ∈ L2 (0, T ; H) and √ ||G(·, x)||L2 (0,T ;H) ≤ (L0 + K0 L2 ) T + (L1 + L2 K1 T )||x||L2 (0,T ;V ) . Moreover if x1 , x2 ∈ L2 (0, T ; V ), then ||G(·, x1 ) − G(·, x2 )||L2 (0,T ;H) ≤ (L1 + L2 K1 T )||x1 − x2 ||L2 (0,T ;V ) .

560

(2.6)

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Proof. Hence, from (K2), (G2) and the above inequality it is easily seen that ||G(·, x)||L2 (0,T ;H) ≤ ||G(·, 0)|| + ||G(·, x) − G(·, 0)|| Z · √ k(·, s, x(s))ds||L2 (0,T ;H) ≤ L0 T + L1 ||x||L2 (0,T ;V ) + L2 || 0 √ √ ≤ L0 T + L1 ||x||L2 (0,T ;V ) + L2 K1 T ||x||L2 (0,T ;V ) + K0 L2 T √ ≤ (L0 + K0 L2 ) T + (L1 + L2 K1 T )||x||L2 (0,T ;V ) Similarly, we can prove (2.6). In view of Lemma 2.2, we can apply the regularity results of Theorem 3.1 of [10] to (1.1), and so we obtain the following results. Proposition 2.1. 1) Let x0 ∈ H and k ∈ L2 (0, T ; V ∗ ), T > 0. Then there exists a unique solution x of (2.7) belonging to L2 (0, T ; V ) ∩ W 1,2 (0, T ; V ∗ ) ⊂ C([0, T ]; H) and satisfying ||x||L2 (0,T ;V )∩W 1,2 (0,T ;V ∗ ) ≤ C1 (|x0 | + ||k||L2 (0,T ;V ∗ ) ),

(2.7)

where C1 is a constant depending on T . 2) If x0 ∈ H and k ∈ L2 (0, T ; V ∗ ), then the mapping H × L2 (0, T ; V ∗ ) 3 (x0 , k) 7→ x ∈ L2 (0, T ; V ) ∩ W 1,2 (0, T ; V ∗ ) is Lipschitz continuous. Here, we note that by using interpolation theory, we have that for z ∈ L2 (0, T ; V )∩ W 1,2 (0, T ; V ∗ ), there exists a constant C2 > 0 such that ||z||C([0,T ];H) ≤ C2 ||z||L2 (0,T ;V )∩W 1,2 (0,T ;V ∗ ) .

3

(2.8)

Approximately reachable sets

Let U be a Banach space and the controller operator B is bounded linear operator from another Banach space U to X.

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Let S(t) be an analytic semigroup generation by A. Then we may assume that there exists a positive constant C0 such that ||S(t)|| ≤ C0 ,

||AS(t)|| ≤ C0 /t(t > 0).

(3.1)

The solution x(t) = x(t; x0 , G, u) of initial value problem (1,1) is the following form: Z t S(t − s){G(t, x(s)) + Bu(s)}ds, t > 0, x(t; x0 , G, u) = S(t)x0 + 0

For T > 0, x0 ∈ H and u ∈ L2 (0, T ; U ) we define reachable sets as follows. LT (x0 ) = {x(T ; x0 , 0, u) : u ∈ L2 (0, T ; U )}, RT (x0 ) = {x(T ; x0 , G, u) : u ∈ L2 (0, T ; U )}, [ [ L(x0 ) = LT (x0 ), R(x0 ) = RT (x0 ). T >0

T >0

Definition 3.1. (1) System (1.1) is said to be H-approximately controllable for initial value x0 (resp. in time T ) if R(x0 ) = H ( resp. RT (x0 ) = H). (2) The linear system corresponding (1.1) is said to be H-approximately controllable for initial value x0 (resp. in time T ) if L(x0 ) = H ( resp. LT (x0 ) = H). Remark 3.1. Since A generate an analytic semigroup, the following (1)-(4) are equivalent for the linear system (see [2, Theorem 3.10]). (1) L(x0 ) = H

∀x0 ∈ H.

(2) L(0) = H. (3) LT (x0 ) = H

∀x0 ∈ H.

(4) LT (0) = H. Theorem 3.1. For any T > 0 we have RT (0) ⊂ LT (0). Proof. Let z0 ∈ / LT (0). Since LT (0) is a balanced closed convex subspace, we have αz0 ∈ / LT (0) for every α ∈ R, and inf{||z0 − z|| : z ∈ LT (0)} = d. By the formula (2.7) we have ||x(·; 0, G, u)||L2 (0,T ;V ) ≤ C1 ||B||||u||L2 (0,T ;U ) ,

562

(3.2)

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where C1 is the constant in Proposition 2.1. For every u ∈ L2 (0, T ; U ), we choose a constant α > 0 such that √ (3.3) C0 {(L0 + K0 L2 ) T + (L1 + L2 K1 T )C1 ||B||||u||L2 (0,T ;U ) } < αd. Hence form (3.2), (3.3) and by using H¨older inequality, it follows that |x(T ; 0, G, u) − αz0 | Z T Z T ≥| S(T − s)Bu(s)ds − αz0 | − | S(T − s)G(s, x(s))ds| 0 0 √ ≥ αd − C0 {(L0 + K0 L2 ) T + (L1 + L2 K1 T )||x||L2 (0,T ;V ) } √ ≥ αd − C0 {(L0 + K0 L2 ) T + (L1 + L2 K1 T )C1 ||B||||u||L2 (0,T ;U ) } > 0. Thus, we have αz0 ∈ / RT (0). Lemma 3.1. Suppose that k ∈ L2 (0, T ; H) and x(t) = t ≤ T . Then there exists a constant C3 such that

and

Rt 0

S(t − s)k(s)ds for 0 ≤

||x||L2 (0,T ;D(A)) ≤ C1 ||k||L2 (0,T ;H) , ||x||L2 (0,T ;H) ≤ C3 T ||k||L2 (0,T ;H) ,

(3.4) (3.5)

√ ||x||L2 (0,T ;V ) ≤ C3 T ||k||L2 (0,T ;H) .

(3.6)

Proof. The assertion (3.4) is immediately obtained by (2.7). Since RT Rt RT Rt ||x||2L2 (0,T ;H) = 0 | 0 S(t − s)k(s)ds|2 dt ≤ C0 0 ( 0 |k(s)|ds)2 dt RT Rt 2 RT ≤ C0 0 t 0 |k(s)|2 dsdt ≤ C0 T2 0 |k(s)|2 ds it follows that ||x||L2 (0,T ;H) ≤ T

p

C0 /2||k||L2 (0,T ;H) .

From (2.4), (3.4), and (3.5) it holds that p ||x||L2 (0,T ;V ) ≤ C C1 T (M/2)1/4 ||k||L2 (0,T ;H) . So, if we take a constant C3 > 0 such that p p C3 = max{ C0 /2, C C1 (C0 /2)1/4 }, the proof is complete.

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Theorem 3.2. Under Assumptions (K) and (G), for any x0 ∈ H we have LT (x0 ) ⊂ RT (x0 ). . Proof. Let u ∈ L2 (0, T ; U ) be arbitrary fixed. Then by (2.7) we have ||xu ||L2 (0,T ;V ) ≤ C1 (|x0 | + ||B||||u||L2 (0,T ;U ) ), where xu is the solution of (1.1) corresponding to the control u. For any  > 0, we can choose a constant δ > 0 satisfying √  −1 min{ δ, δ} < min 7C3 (L1 + L2 K1 T )) , (3.7) √ −1   C3 (L0 + K0 L2 T ) ,  −1  C3 (L1 + L2 K1 T )(C1 C2 ||xu ||L2 (0,T ;V )∩W 1,2 (0,T ;V ∗ ) + ) ,  −1  C3 (C0 ||xu ||L2 (0,T ;V )∩W 1,2 (0,T ;V ∗ ) + )(L1 + L2 K1 T ) , √  2 −1  /6.  (C3 (L0 + K0 L2 ) T + )(L1 + L2 K1 T ) Set x1 := x(T − δ; x0 , G, u) = S(T − δ)x0 + Z T −δ Z + S(T − δ − s)G(s, xu (s))ds + 0

T −δ

S(T − δ − s)Bu(s)ds,

0

where xu (t) = x(t; x0 , G, u) for 0 < t ≤ T . Consider the following problem: ( 0 y (t) = Ay(t) + Bu(t), δ < t ≤ T, y(T − δ) = x1 , y(s) = 0 − h ≤ s ≤ 0.

(3.8)

The solution of (3.8) with respect to the control w ∈ L2 (T − δ, T ; U ) is denoted by Z

T

S(T − s)Bw(s)ds

yw (T ) = S(δ)x1 +

(3.9)

T −δ

Z

T −δ

= S(T )x0 + S(δ) S(T − δ − s)G(s, xu (s))ds 0 Z T −δ Z T + S(δ) S(T − δ − s)Bu(s)ds + S(T − s)Bw(s)ds. T −δ

0

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Then since z ∈ LT (x0 ), and LT (x0 ) = L(0) is independent of the time T and initial data x0 (see Remark 2.1), there exists w1 ∈ L2 (T − δ, T ; U ) such that  |yw1 (t) − z| < , 6 T −δ≤t≤T

(3.10)

sup

and hence, by (3.9), Z t  S(T − s)Bw1 (s)ds| ≤ C0 ||xu ||L2 (0,T −δ;V ) + , | 6 T −δ

t − δ ≤ t ≤ T.

(3.11)

Now, we set ( v(s) =

u w1 (s)

if 0 ≤ s ≤ T − δ, if T − δ < s < T.

Then v ∈ L2 (0, T ; U ). Observing that Z t xv (t; G, v) = S(t)x0 + S(t − τ ){G(τ, xv (τ )) + Bv(τ )}dτ, 0

from (3.9) and (3.10) we obtain that |x(T ; x0 , G, v) − z| ≤ |yw1 (T ) − z| + |x(T ; x0 , G, v) − yw1 (T )| (3.12) ≤ |yw1 (T ) − z| Z Z T −δ T + S(T − s)G(s, xv (s))ds − S(δ) S(T − δ − s)G(s, xu (s))ds 0 0 Z T −δ Z T S(T − δ − s)Bu(s)ds + S(T − s)Bv(s)ds − S(δ) 0 0 Z T S(T − s)Bw1 (s)ds − T −δ Z  T S(T − s)G(s, xw1 (s))ds ≤ + 6 T −δ  ≤ + II. 6 Here, we remind that the xw1 is represented by xw1 (t) =S(t)x(T − δ; x0 , G, u) Z t Z + S(T − s)G(s, xw1 (s))ds + T −δ

t

S(T − s)Bw1 (s))ds

T −δ

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for T − δ < t ≤ T . Here, by (2.7) we have ||S(·)x(T − δ; x0 , G, u)||L2 (0,T ;V ) ≤ C1 |x(T − δ; x0 , G, u)| ≤ C1 C2 ||xu ||L2 (0,T ;V )∩W 1,2 (0,T ;V ∗ ) . Put

Z

(3.13)

t

S(t − s)G(s, xw1 (s))ds,

p(t) =

T − δ < t ≤ T,

T −δ

and Z

T

S(t − s)Bw1 (s)ds T − δ < t ≤ T.

q(t) := t−δ

Then with aid of (3.6) of Lemma 3.1 and Lemma 2.2, we have √ ||p||L2 (T −δ,T ;V ) ≤ C3 δ||G(·, xw1 )||L2 (T −δ,T ;V ) √ √ ≤ C3 δ{(L0 + K0 L2 ) T + (L1 + L2 K1 T )||xw1 ||L2 (T −δ,T ;V ) }, and by (3.11),

√

 δ(C0 ||xu ||L2 (0,T −δ;V ) + ). 6 √ Since C3 δ(L1 + L2 K1 T )) < 1 by virtue of (3.7), by (3.13)-(3.15), we get ||q||L2 (T −δ,T ;V ) ≤

(3.14)

(3.15)

||xw1 ||L2 (T −δ,T ;V ) ≤{C1 C2 ||xu ||L2 (0,T ;V )∩W 1,2 (0,T ;V ∗ ) (3.16) √  + δ(C0 ||xu ||L2 (0,T −δ;V ) + ) 6 √ √ + C3 δT (L0 + K0 L2 )}{1 − C3 δ(L1 + L2 K1 T ))}−1 . Hence, with aid of (3.6), (3.7), (3.16), and by using the H¨older inequality, we have Z T II = S(T − s)G(s, xw1 (s))ds (3.17) T −δ √ ≤ C3 δT {(L0 + K0 L2 ) + (L1 + L2 K1 T )||xw1 ||L2 (T −δ,T ;V ) } √ √  ≤ C3 δT (L0 + K0 L2 ) + C3 δ(L1 + L2 K1 T ) C1 C2 ||xu ||L2 (0,T ;V )∩W 1,2 (0,T ;V ∗ ) √  + δ(C0 ||xu ||L2 (0,T −δ;V ) + ) 6 √ √  −1 5 + C3 δT (L0 + K0 L2 ) 1 − C3 δ(L1 + L2 K1 T )) < . 6 Therefore, by (3.12) and (3.17), we have ||x(T ; x0 , G, v) − z||H < , that is, z ∈ RT (x0 ) and the proof is complete.

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Remark 3.2. Noting that H([0, T ]; U ) is dense in L2 (0, T ; U ), we can obtain the same results of Theorem 3.2 corresponding to (1.1) with control space H([0, T ]; U ) = {w : [0, T ] → U : |w(t) − w(s)| ≤ H0 |t − s|θ , 0 < θ < 1, H0 > 0} instead of L2 (0, T ; U ) From Theorems 3.1-2, we obtain the following control results of (1.1). Corollary 3.1. Under Assumptions (K) and (G), for T > 0 we have LT (x0 ) = H ⇐⇒ RT (x0 ) = H. Therefore, the approximate controllability of linear system (1.1) with g = 0 is equivalent to the condition for the approximate controllability of the nonlinear system (1.1). Acknowledgement 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(2015R1D1A1A09059030).

References [1] H.O. Fattorini, On complete controllability of linear systems, J. Differential Equations 3 (1967), 391–402. [2] H.O. Fattorini, Boundary control systems, SIAM J. Control Optimization 6 (1968), 349–402. [3] R. Triggiani, Existence of rank conditions for controllability and obserbavility to Banach spaces and unbounded operators, SIAM J. Control Optim. 14 (1976), 313–338. [4] R. F. Curtain and H. Zwart,An Introduction to Infinite Dimensional Linear Systems Theory, Springer-Velag, New-York, 1995. [5] V. Barbu,Analysis and Control of Nonlinear Infinite Dimensional Systems, Academic Press Limited, 1993. [6] M. Yamamoto and J. Y. Park, Controllability for parabolic equations with uniformly bounded nonlinear terms, J. optim. Theory Appl., 66(1990), 515-532.

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[7] X. Fu,Controllability of neutral functional differential systems in abstract space, Appl. Math. Comput. 141 (2003), 281-296. [8] H. X. Zhou, Approximate controllability for a class of semilinear abstract equations, SIAM J. Control Optim., 21(1983). [9] K. Naito, Controllability of semilinear control systems dominated by the linear part, SIAM J. Control Optim., 25(1987), 715-722. [10] J. M. Jeong, Y. C. Kwun and J. Y. Park, Approximate controllability for semilinear retarded functional differential equations, J. Dynamics and Control Systems, 5 (1999), no. 3, 329-346. [11] N. I. Mahmudov, Approximate controllability of semilinear deterministic and stochastic evolution equations in abstract spaces, SIAM J. Control Optim. 42(2006), 175-181. [12] L. G´orniewicz, S. K. Ntouyas and D. O’Reran, Controllability of semilinear differential equations and inclusions via semigroup theory in Banach spaces, Rep. Math. Phys. 56(2005), 437-470. [13] N. Sukavanam and Nutan Kumar Tomar, Approximate controllability of semilinear delay control system, Nonlinear Func.Anal.Appl. 12(2007), 53-59. [14] L. Wang, Approximate controllability and approximate null controllability of semilinear systems, Commun. Pure and Applied Analysis 5(2006), 953-962. [15] J. M. Jeong, Stabilizability of retarded functional differential equation in Hilbert space, Osaka J. Math. 28(1991), 347–365. [16] H. Tanabe, Fundamental solutions for linear retarded functional differential equations in Banach space, Funkcial. Ekvac., 35(1992), 149-177. [17] H. Tanabe, Equations of Evolution, Pitman-London, 1979. [18] H. Triebel, Interpolation Theory, Function Spaces, Differential Operators, North-Holland, 1978. [19] J. Yong and L. Pan, Quasi-linear parabolic partial differential equations with delays in the highest order partial derivatives, J. Austral. Math. Soc. 54 (1993), 174–203. [20] S. Nakagiri, Structural properties of functional differential equations in Banach spaces, Osaka J. Math. 25 (1988), 353–398.

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[21] J. M. Jeong, Y. C. Kwun and J. Y. Park, Approximate controllability for semilinear retarded functional differential equations, J. Dynamics and Control Systems, 5 (1999), no. 3, 329–346.

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On modified degenerate poly-tangent numbers and polynomials C. S. RYOO Department of Mathematics, Hannam University, Daejeon 34430, Korea

Abstract : In this paper we introduce the modified degenerate degenerate poly-tangent polynomials and numbers. We also give some properties, explicit formulas, several identities, a connection with modified degenerate poly-tangent numbers and polynomials, and some integral formulas. Finally, we investigate the zeros of the modified degenerate poly-tangent polynomials by using computer. Key words : Tangent numbers and polynomials, degenerate poly-tangent numbers and polynomials, Cauchy numbers, Stirling numbers, modified degenerate poly-tangent polynomials. AMS Mathematics Subject Classification : 11B68, 11S40, 11S80. 1. Introduction Many mathematicians have studied in the area of the Bernoulli numbers and polynomials, Euler numbers and polynomials, Genocchi numbers and polynomials, tangent numbers and polynomials, poly-Bernoulli numbers and polynomials, poly-Euler numbers and polynomials(see [1-11]). In this paper, we define modified degenerate poly-tangent polynomials and numbers and study some properties of the modified degenerate poly-tangent polynomials and numbers. Throughout this paper, we always make use of the following notations: N denotes the set of natural numbers and Z+ = N ∪ {0}. Carlitz [1] has defined the degenerate Stirling numbers of the first kind and second kind, S1 (n, k, λ) and S2 (n, k, λ) by means of (

(

1 − (1 − t)λ λ

)k = k!

∞ ∑

S1 (n, k, λ)

n=k

tn , n!

(1.1)

∞ )k ∑ tn (1 + λt)1/λ − 1 = k! S2 (n, k, λ) . n!

(1.2)

n=k

Howard [12] has defined the degenerate weighted Stirling numbers of the first kind and second kind, S1 (n, k, x, λ) and S2 (n, k, x, λ) by means of ( (1 − t)λ−x

1 − (1 − t)λ λ

)k = k!

∞ ∑

S1 (n, k, x, λ)

n=k

tn , n!

∞ ( )k ∑ tn (1 + λt)x/λ (1 + λt)1/λ − 1 = k! S2 (n, k, x, λ) . n!

(1.3)

(1.4)

n=k

The generalized falling factorial (x|λ)n with increment λ is defined by (x|λ)n =

n−1 ∏

(x − λk).

k=0

The generalized raising factorial < x|λ >n with increment λ is defined by < x|λ >n =

n−1 ∏

(x + λk).

k=0

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for positive integer n, with the convention (x|λ)0 = 1. We also need the binomial theorem: for a variable x, (1 + λt)x/λ =

∞ ∑

(x|λ)n

n=0

The degenerate poly-Bernoulli numbers generating function

(k) Bn (λ)

tn . n!

were introduced by Kaneko [5] by using the following

∞ ∑ Lik (1 − e−t ) tn (k) , = B (λ) n n! 1 − (1 + λt)−1/λ n=0

where Lik (t) =

(k ∈ Z),

(1.5)

∞ ∑ tn nk n=1

(1.6)

is the kth polylogarithm function. (k) The degenerate poly-Euler polynomials En (x, λ) are defined by generating function ∞ ∑ Lik (1 − e−t ) tn x/λ (k) (1 + λt) = , E (x, λ) n n! (1 + λt)1/λ + 1 n=0

(k ∈ Z).

(1.7)

The familiar degenerate tangent polynomials Tn (x, λ) are defined by the generating function([7]):

(

2 (1 + λt)2/λ + 1

) (1 + λt)x/λ =

∞ ∑

Tn (x, λ)

n=0

tn , n!

(|2t| < π).

(1.8)

When x = 0, Tn (0, λ) = Tn (λ) are called the degenerate tangent numbers. The degenerate tangent (r)

polynomials Tn (x, λ) of order r are defined by (

2 (1 + λt)2/λ + 1

)r (1 + λt)x/λ =

∞ ∑

T(r) n (x, λ)

n=0

tn , n!

(|2t| < π).

(1.9)

It is clear that r = 1 we recover the degenerate tangent polynomials Tn (x, λ). (r) The degenerate Bernoulli polynomials Bn (x, λ) of order r are defined by the following generating function (

t (1 + λt)1/λ − 1

)r (1 + λt)x/λ =

∞ ∑

B(r) n (x, λ)

n=0

tn , n!

(|t| < 2π).

(1.10)

(r)

The degenerate Frobenius-Euler polynomials of order r, denoted by Hn (u, x, λ), are defined as (

1−u (1 + λt)1/λ − u

)r (1 + λt)x/λ =

∞ ∑

H(r) n (u, x, λ)

n=0

tn . n!

(1.11)

The values at x = 0 are called degenerate Frobenius-Euler numbers of order r; when r = 1, the polynomials or numbers are called ordinary degenerate Frobenius-Euler polynomials or numbers. (k)

The degenerate poly-tangent polynomials Tn (x, λ) are defined by the generating function: ∞ ∑ tn 2Lik (1 − e−t ) x/λ (1 + λt) = Tn(k) (x, λ) , 2/λ n! (1 + λt) +1 n=0 (k)

(k ∈ Z).

(1.12)

(k)

When x = 0, Tn (0, λ) = Tn (x, λ) are called the degenerate poly-tangent numbers. Many kinds of of generalizations of these polynomials and numbers have been presented in the literature(see [1-12]). In the following section, we introduce the modified degenerate poly-tangent polynomials and numbers. After that we will investigate some their properties. We also give some relationships

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both between these polynomials and modified degenerate poly-tangent polynomials and between these polynomials and cauchy numbers. Finally, we investigate the zeros of the modified degenerate poly-tangent polynomials by using computer.

2. Modified degenerate poly-tangent polynomials In this section, we define modified degenerate poly-tangent numbers and polynomials and provide some of their relevant properties. (k) The modified degenerate poly-tangent polynomials Tn (x, λ) are defined by the generating ( ) ∞ ∑ 2Lik 1 − (1 + λt)−1/λ tn x/λ (k) , (1 + λt) = T (x, λ) n n! (1 + λt)2/λ + 1 n=0

function:

(k)

(k ∈ Z).

(2.1)

(k)

When x = 0, Tn (0, λ) = Tn (x, λ) are called the degenerate poly-tangent numbers. Upon setting k = 1 in (2.1), we have Tn(1) (x, λ) =

n ( ) ∑ n n−1 λ S1 (l, 1)Tn−l (x, λ) for n ≥ 1. l l=0

By (2.1), we get ∞ ∑ n=0

t Tn(k) (x, λ)

(

n

n!

=

( )) 2Lik 1 − (1 + λt)−1/λ (1 + λt)x/λ (1 + λt)2/λ + 1

∞ ∑

∞

tn ∑ tn (x|λ)n n! n=0 n! n=0 ( ) ( ) ∞ n ∑ ∑ n tn (k) = Tl (λ)(x|λ)n−l . l n! n=0 =

Tn(k) (λ)

(2.2)

l=0

By comparing the coefficients on both sides of (2.2), we have the following theorem. Theorem 2.1. For n ∈ Z+ , we have Tn(k) (x, λ) =

n ( ) ∑ n (k) T (λ)(x|λ)n−l . l l l=0

(k)

The following elementary properties of the degenerate poly-tangent numbers Tn (λ) and polynomials

(k) Tn (x, λ)

are readily derived form (2.1). We, therefore, choose to omit details involved.

Theorem 2.2. For k ∈ Z, we have n ( ) ∑ n (k) T (x, λ)(y|λ)n−l . l l l=0 ( ) n ∑ (k) (k) l n (2) Tn (2 − x, λ) = (−1) T (2, λ) < x|λ) >l . l n−l

(1) Tn(k) (x + y, λ) =

l=0

Theorem 2.3 For any positive integer n, we have (1)

(2)

n ( ) ∑ n (k) = T (x, λ)((m − 1)x|λ)n−l . l l l=0 n−1 ∑ (n) (k) Tn(k) (x + 1, λ) − Tn(k) (x, λ) = T (x, λ)(1|λ)n−l . l l

Tn(k) (mx, λ)

(2.3)

l=0

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From (1.6), (1.8), and (2.1), we get ( )) Lik 1 − (1 + λt)−1/λ = 2 (1 + λt)x/λ 2/λ + 1 n! (1 + λt) n=0 )l+1 ∞ ( ∑ 1 − (1 + λt)−1/λ 2(1 + λt)x/λ = (l + 1)k (1 + λt)2/λ + 1 l=0 ) ∞ l+1 ( −i/λ x/λ ∑ ∑ 1 (1 + λt) l+1 i 2(1 + λt) = (−1) (l + 1)k i=0 i (1 + λt)2/λ + 1 l=0   ) ∞ ∞ n ( ) l+1 ( ∑ ∑ ∑ ∑ 1 n tn l+1  = (−1)i Tj (x, λ)(−1)n−j < i|λ >(n−j)  k i j (l + 1) i=0 n! n=0 j=0 l=0   ( ) ( ) ∞ ∞ ∑ l+1 ∑ n ∑ ∑ l+1 tn 1 n+i−j n  (−i) = Tj (x, λ) < i|λ >(n−j)  . k (l + 1) i n! j n=0 i=0 j=0 ∞ ∑

(

tn Tn(k) (x, λ)

(2.4)

l=0

By comparing the coefficients on both sides of (2.4), we have the following theorem. Theorem 2.4 For n ∈ Z+ , we have Tn(k) (x, λ)

=

( )( ) ∞ ∑ l+1 ∑ n ∑ (−i)n+i−j l + 1 n l=0 i=0 j=0

(l + 1)k

i

j

Tj (x, λ) < i|λ >(n−j)

( ) ∞ ∑ l+1 ∑ (−i)i l + 1 = Tn (x − i, λ). (l + 1)k i i=0 l=0

By (2.1), we note that )l+1 ∞ ∞ ( ∑ ∑ 1 − (1 + λt)−1/λ l 2l/λ =2 (−1) (1 + λt) (1 + λt)x/λ k n! (l + 1) n=0 l=0 l=0 ) ∞ ∑ l ( −1/λ i+1 ∑ 1 − (1 + λt) =2 (−1)l−i (1 + λt)(2l−2i)/λ (1 + λt)x/λ k (i + 1) l=0 i=0 ( ) ∞ l ∑ i+1 ∑∑ 2(−1)l+j−i i+1 j −j/λ = (1 + λt)(2l−2i+x)/λ (1 + λt) k (i + 1) l=0 i=0 j=0   ( )( n ) ∞ ∞ ∑ l ∑ i+1 ∑ n n ∑ ∑ 2(−1)l+j−i i+1 (2l − 2i + x|λ) < j|λ > m (n−m) j m  t . = (i + 1)k n! n=0 i=0 j=0 m=0 ∞ ∑

t Tn(k) (x, λ)

n

l=0

Comparing the coefficients on both sides, we have the following theorem. Theorem 2.5 For n ∈ Z+ , we have Tn(k) (x, λ)

=

∞ ∑ l ∑ i+1 ∑ n ∑ 2(−1)l+j−i l=0 i=0 j=0 m=0

=

∞ ∑ l ∑ i+1 ∑ 2(−1)l+j−i l=0 i=0 j=0

(i+1)( n ) j

m

(2l − 2i + x|λ)m < j|λ >(n−m) (i + 1)k

(i+1) j

(2l − 2i − j + x|λ)m

(i + 1)k

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3. Some identities involving degenerate poly-tangent numbers and polynomials In this section, we give several combinatorics identities involving degenerate poly-tangent numbers and polynomials in terms of degenerate Stirling numbers, generalized falling factorial functions, generalized raising factorial functions, Beta functions, degenerate Bernoulli polynomials of higher order, and degenerate Frobenius-Euler functions of higher order. By (2.1) and by using Cauchy product, we get ∞ ∑

Tn(k) (x, λ)

n=0

tn n!

( )) ( )−x 2Lik 1 − (1 + λt)−1/λ = 1 − (1 − (1 + λt)−1/λ ) 2/λ (1 + λt) +1 ( ) ∞ ( ) 2Lik 1 − (1 + λt)−1/λ ∑ x + l − 1 = (1 − (1 + λt)−1/λ )l l (1 + λt)2/λ + 1 l=0 ( ) ( ) ∞ 1/λ ∑ ((1 + λt) − 1)l 2Lik 1 − (1 + λt)−1/λ −l/λ (1 + λt) = < x >l l! (1 + λt)2/λ + 1 l=0 (

∞ ∑

(3.1)

∞

∞ ∑

tn tn ∑ (k) = < x >l Tn (−l, λ) S2 (n, l, λ) n! n=0 n! n=0 l=0 ( ) ∞ ∞ ∑ n ( ) ∑ ∑ n tn (k) = S2 (i, l, λ)Tn−i (−l, λ) < x >l , i n! n=0 l=0 i=l

where < x >l = x(x + 1) · · · (x + l − 1)(l ≥ 1) with < x >0 = 1. By comparing the coefficients on both sides of (3.1), we have the following theorem. Theorem 3.1 For n ∈ Z+ , we have Tn(k) (x, λ) =

∞ ∑ n ( ) ∑ n (k) S2 (i, l, λ)Tn−i (−l, λ) < x >l . i l=0 i=l

By (2.1) and by using Cauchy product, we get ∞ ∑

Tn(k) (x, λ)

n=0

(

tn n!

( )) ( )−x 2Lik 1 − (1 + λt)−1/λ −1/λ = 1 − (1 − (1 + λt) ) (1 + λt)2/λ + 1 ( ) ∞ ( ) 2Lik 1 − (1 + λt)−1/λ ∑ x + l − 1 = (1 − (1 + λt)−1/λ )l l (1 + λt)2/λ + 1 l=0 ( ( )) ∞ −l/λ ∑ (1 + λt) ((1 + λt)1/λ − 1)l 2Lik 1 − (1 + λt)−1/λ = < x >l l! (1 + λt)2/λ + 1 l=0 =

=

∞ ∑

< x >l

∞ ∑

n=0 l=0 (∞ n ( ∞ ∑ ∑∑ n=0

l=0 i=l

S2 (n, l, −l, λ)

(3.2)

∞ tn tn ∑ (k) Tn (λ) n! n=0 n!

) ) n tn (k) , S2 (i, l, −l, λ)Tn−i (λ) < x >l n! i

where < x >l = x(x + 1) · · · (x + l − 1)(l ≥ 1) with < x >0 = 1. By comparing the coefficients on both sides of (3.2), we have the following theorem.

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Theorem 3.2 For n ∈ Z+ , we have Tn(k) (x, λ)

∞ ∑ n ( ) ∑ n (k) = S2 (i, l, −l, λ)Tn−i (λ) < x >l . i l=0 i=l

By (2.1) and by using Cauchy product, we get ( ( )) ( ∞ )x ∑ 2Lik 1 − (1 + λt)−1/λ tn (k) 1/λ Tn (x, λ) = ((1 + λt) − 1) + 1 n! (1 + λt)2/λ + 1 n=0 ( ) ∞ ( ) )l 2Lik 1 − (1 + λt)−1/λ ∑ x ( 1/λ = (1 + λt) − 1 l (1 + λt)2/λ + 1 l=0 ( ( )l ( )) ∞ ∑ (1 + λt)1/λ − 1 2Lik 1 − (1 + λt)−1/λ = (x)l l! (1 + λt)2/λ + 1 l=0

(3.3)

∞ ∞ ∞ ∑ ∑ tn ∑ (k) tn (x)l S2 (n, l, λ) T n! n=0 n n! n=0 l=0 (∞ n ( ) ) ∞ ∑ ∑∑ n tn (k) (x)l S2 (i, l, λ)Tn−i = . i n! n=0

=

l=0 i=l

By comparing the coefficients on both sides of (3.3), we have the following theorem. Theorem 3.3 For n ∈ Z+ , we have Tn(k) (x, λ)

∞ ∑ n ( ) ∑ n (k) = (x)l S2 (i, l, λ)Tn−i . i l=0 i=l

By (1.2), (1.10), (2.1), and by using Cauchy product, we get ∞ ∑

Tn(k) (x, λ)

n=0

(

tn n!

( )) 2Lik 1 − (1 + λt)−1/λ = (1 + λt)x/λ (1 + λt)2/λ + 1 )r ( ∞ ∑ ((1 + λt)1/λ − 1)r r! tn t x/λ (k) = (1 + λt) (λ) T n r! tr (1 + λt)1/λ − 1 n! n=0 (∞ )( ∞ ) ∑ tn ((1 + λt)1/λ − 1)r ∑ (r) tn r! = Bn (x, λ) Tn(k) (λ) r! n! n! tr n=0 n=0 ( n ( ) ) ) ∞ n−l ( ∑ ∑ n ∑ n−l tn (r) (k) l = Bi (x, λ)Tn−l−i (λ) . (l+r) S2 (l + r, r, λ) i n! r n=0 i=0 l=0

By comparing the coefficients on both sides, we have the following theorem. Theorem 3.4 For n ∈ Z+ and r ∈ N, we have Tn(k) (x, λ)

=

n ∑ n−l ∑ l=0 i=0

(n)(n−l) (k)

(r)

(l+ri) S2 (l + r, r)Tn−l−i Bi (x, λ).

l

r

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By (1.2), (1.11), (2.1), and by using Cauchy product, we get ∞ ∑

Tn(k) (x, λ)

n=0

tn n!

( ) 2Lik 1 − (1 + λt)−1/λ = (1 + λt)x/λ (1 + λt)2/λ + 1 ( ) ( )r −1/λ 1−u ((1 + λt)1/λ − u)r x/λ 2Lik 1 − (1 + λt) = (1 + λt) (1 − u)r (1 + λt)1/λ − u (1 + λt)2/λ + 1 ( ) ( ) ∞ r ∑ 2Lik 1 − (1 + λt)−1/λ tn ∑ r 1 (r) r−i i/λ = Hn (u, x, λ) (1 + λt) (−u) n! i=0 i (1 − u)r (1 + λt)2/λ + 1 n=0 r ( ) ∞ ∞ ∑ ∑ 1 r tn ∑ (k) tn r−i (r) = (−u) H (u, x, λ) Tn (i, λ) n r (1 − u) i=0 i n! n=0 n! n=0 ) ( ( ) ( ) r n ∞ ∑ ∑ ∑ r n tn 1 (k) (r) r−i . (−u) H (u, x, λ)T (i, λ) = l n−l (1 − u)r i=0 i n! l n=0 l=0

By comparing the coefficients on both sides, we have the following theorem. Theorem 3.5 For n ∈ Z+ and r ∈ N, we have r ∑ n ( )( ) ∑ 1 r n (r) (k) Tn(k) (x, λ) = (−u)r−i Hl (u, x, λ)Tn−l (i, λ). (1 − u)r i=0 i l l=0

By (1.2), (1.11), (2.1), and by using Cauchy product, we get ∞ ∑

Tn(k) (x, λ)

n=0

tn n!

( ) 2Lik 1 − (1 + λt)−1/λ (1 + λt)1/λ + 1 = (1 + λt)x/λ 2/λ (1 + λt) +1 (1 + λt)1/λ + 1 ( ) ( ) 2Lik 1 − (1 + λt)−1/λ (1 + λt)1/λ 1 x/λ = (1 + λt) + (1 + λt)1/λ + 1 (1 + λt)2/λ + 1 (1 + λt)2/λ + 1 (∞ )( ∞ ) ∑ ∑1 tn tn = En(k) (x, λ) (Tn (1, λ) + Tn (λ)) n! 2 n! n=0 n=0 ( ) ( ) ∞ n ∑ 1∑ n tn (k) = (Tn (1, λ) + Tn (λ)) En−l (x, λ) . 2 n! l n=0 l=0

By comparing the coefficients on both sides, we have the following theorem. Theorem 3.6 For n ∈ Z+ and r ∈ N, we have n ( ) 1∑ n (k) Tn(k) (x, λ) = (Tn (1, λ) + Tn (λ)) En−l (x, λ). 2 l l=0

By (1.2), (1.11), (2.1), and by using Cauchy product, we get ( ) ∞ −1/λ ∑ 2Lik 1 − (1 + λt)−1/λ tn (k) x/λ 1 − (1 + λt) Tn (x, λ) = (1 + λt) n! (1 + λt)2/λ + 1 1 − (1 + λt)−1/λ n=0 ( ) ( ) Lik 1 − (1 + λt)−1/λ 2(1 + λt)x/λ 2(1 + λt)(x−1)/λ − = 1 − (1 + λt)−1/λ (1 + λt)2/λ + 1 (1 + λt)2/λ + 1 )( ∞ ) (∞ ∑ ∑ tn tn (k) (Tn (x, λ) − Tn (x − 1, λ)) = Bn (λ) n! n! n=0 n=0 ( n ( ) ) ∞ ∑ ∑ n tn (k) = (Tn (x, λ) − Tn (x − 1, λ)) Bn−l (x, λ) . l n! n=0 l=0

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By comparing the coefficients on both sides, we have the following theorem. Theorem 3.7 For n ∈ Z+ and r ∈ N, we have Tn(k) (x, λ)

=

n ( ) ∑ n l=0

l

(k)

(Tn (x, λ) − Tn (x − 1, λ)) Bn−l (λ).

By Theorem 3.6 and Theorem 3.7, we have the following corollary. Corollary 3.8 For n ∈ Z+ and r ∈ N, we have n ( ) ∑ n l=0

=2

l

(k)

(Tn (1, λ) + Tn (λ)) En−l (x, λ)

n ( ) ∑ n l=0

l

(k)

(Tn (x, λ) − Tn (x − 1, λ)) Bn−l (λ).

3. Distribution of zeros of the degenerate poly-tangent polynomials This section aims to demonstrate the benefit of using numerical investigation to support theoretical prediction and to discover new interesting pattern of the zeros of the degenerate poly-tangent (k)

(k)

polynomials Tn (x, λ). The degenerate poly-tangent polynomials Tn (x, λ) can be determined explicitly. A few of them are (k)

T0 (x, λ) = 0, (k)

T1 (x, λ) = 1, (k)

T2 (x, λ) = −3 + 21−k − λ + 2x (k)

T3 (x, λ) = 4 − 3 · 22−k + 2 · 31−k + 9λ − 3 · 21−k λ + 2λ2 − 9x + 3 · 21−k x − 6λx + 3x2 , (k)

T4 (x, λ) = 3 + 33−2k + 7 · 21−k + 3 · 23−k − 8 · 31−k − 4 · 32−k − 24λ + 3 · 23−k λ + 3 · 24−k λ − 4 · 32−k λ − 33λ2 + 11 · 21−k λ2 − 6λ3 + 16x − 3 · 24−k x + 8 · 31−k x + 54λx − 3 · 22−k λx − 3 · 23−k λx + 22λ2 x − 18x2 + 3 · 22−k x2 − 18λx2 + 4x3 . (k)

We investigate the beautiful zeros of thedegenerate poly-tangent polynomials Tn (x, λ) by (k) using a computer. We plot the zeros of the poly-tangent polynomials Tn (x, λ) for n = 30, k = −5, −1, 1, 5, λ = 1/2, and x ∈ C(Figure 1). In Figure 1(top-left), we choose n = 30 and k = −5. In Figure 1(top-right), we choose n = 30 and k = −1. In Figure 1(bottom-left), we choose n = 30 and (k) k = 1. In Figure 1(bottom-right), we choose n = 30 and k = 5. Stacks of zeros of Tn (x, λ) for 1 ≤ n ≤ 30 from a 3-D structure are presented(Figure 2). In Figure 2(left), we choose k = −5. In Figure 2(middle), we choose k = 1. In Figure 2(right), we choose k = 5. Our numerical results for (k) approximate solutions of real zeros of Tn (x, λ), λ = 1/2 are displayed(Tables 1, 2).

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Im(x)

15

15

10

10

5

5

0

Im(x)

0

-5

-5

-10

-10

-15 -800

-600

-400

-200

-15 0

-10

0

Re(x)

Im(x)

15

10

10

5

5

0

Im(x)

-5

-10

-10

-15

-10

0

20

10

20

0

-5

-15

10 Re(x)

15

10

20

-10

Re(x)

0 Re(x)

(k)

Figure 1: Zeros of Tn (x, λ)

(k)

Table 1. Numbers of real and complex zeros of Tn (x, λ) degree n

real

k = −10 complex zeros

2

1

0

1

0

1

0

3

2

0

2

0

2

0

4

3

0

3

0

3

0

5

4

0

4

0

4

0

6

5

0

5

0

5

0

7

6

0

2

4

2

4

8

5

2

3

4

3

4

9

6

2

4

4

4

4

10

5

4

5

4

5

4

11

6

4

6

4

6

4

12

7

4

7

4

5

6

real

k=1 complex zeros

real

k = 10 complex zeros

(k)

The plot of real zeros of Tn (x, λ) for 1 ≤ n ≤ 30 structure are presented(Figure 3). In Figure 3(left), we choose k = −5 and λ = 1/2. In Figure 3(middle),we choose k = 1 and λ = 1/2. In Figure 3(right), we choose k = 5 and λ = 1/2. We observe a remarkable regular structure of the complex roots of the degenerate poly-tangent

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(k)

Figure 2: Stacks of zeros of Tn (x, λ) for 1 ≤ n ≤ 30

(k)

Figure 3: Real zeros of Tn (x, λ) for 1 ≤ n ≤ 30

(k)

polynomials Tn (x, λ). We also hope to verify a remarkable regular structure of the complex roots (k) of the degenerate poly-tangent polynomials Tn (x, λ)(Table 1). (k)

Next, we calculated an approximate solution satisfying poly-tangent polynomials Tn (x, λ) = 0 for x ∈ R. The results are given in Table 2 and Table 3. (k)

Table 2. Approximate solutions of Tn (x, λ) = 0, λ = 1/2, k = −5 degree n

x

2

30.250 −53.896,

3

−77.421,

4

−100.91,

5 6 7

−124.39, −147.85,

−16.655,

−8.8591,

−11.489,

−14.080,

−2.9699

−3.9628,

−4.7720,

−5.4611,

579

−6.1044

−1.6365

−2.3421,

−3.0181,

−0.66874

−1.0879,

0.076439

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(k)

Table 3. Approximate solutions of Tn (x, λ) = 0, λ = 1/2, k = 5 degree n

x

2

1.7188

3

0.95682,

4 5 6

0.44597, 0.13979, 0.090663,

2.2234,

1.4750,

0.71964,

7

2.9807 3.9869

3.4758,

2.7246,

1.9752,

4.7844

4.7571,

5.3017

3.9751

By numerical computations, we will make a series of the following conjectures: (k)

Conjecture 4.1. Prove that Tn (x, λ), x ∈ C, has Im(x, λ) = 0 reflection symmetry analytic (k) complex functions. However, Tn (x, λ), k ̸= 1, has not Re(x, λ) = a reflection symmetry for a ∈ R. Using computers, many more values of n have been checked. It still remains unknown if the (k)

conjecture fails or holds for any value n(see Figures 1, 2, 3). We are able to decide if Tn (x, λ)) = 0 has n − 1 distinct solutions(see Tables 1, 2, 3). (k)

Conjecture 4.2. Prove that Tn (x, λ)) = 0 has n − 1 distinct solutions. (k)

Since n−1 is the degree of the polynomial Tn (x, λ), the number of real zeros RT (k) (x,λ) lying on n

the real plane Im(x, λ) = 0 is then RT (k) (x,λ) = n − 1 − CT (k) (x,λ) , where CT (k) (x,λ) denotes complex n n n zeros. See Table 1 for tabulated values of RT (k) (x,λ) and CT (k) (x,λ) . The author has no doubt that n

n

investigations along these lines will lead to a new approach employing numerical method in the (k) research field of the degenerate poly-tangent polynomials Tn (x, λ) which appear in mathematics and physics. Acknowledgement: This work was supported by the National Research Foundation of Korea(NRF) grant funded by the Korea government(MEST) (No. 2017R1A2B4006092).

REFERENCES 1. G.E. Andrews, R. Askey, R. Roy,(1999). Special Functions, v.71, Combridge Press, Cambridge, UK. 2. R. Ayoub,(1974). Euler and zeta function, Amer. Math. Monthly, v.81, pp. 1067-1086. 3. L. Comtet,(1974). Advances Combinatorics, Riedel, Dordrecht. 4. D. Kim, T. Kim,(2015). Some identities involving Genocchi polynomials and numbers, ARS Combinatoria, v.121, pp. 403-412 5. M. Kaneko,(1997). Poly-Bernoulli numbers, J. Th´eor. Nombres Bordeaux, v.9, pp. 199-206 6. N. I. Mahmudov,(2012). q-analogue of the Bernoulli and Genocchi polynomials and the SrivastavaPint´ er addition theorems, Discrete Dynamicss in Nature and Society v.2012, ID 169348, 8 pages.

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7. C. S. Ryoo,(2013). A note on the tangent numbers and polynomials, Adv. Studies Theor. Phys. v.7, pp. 447 - 454. 8. C. S. Ryoo,(2016). Modified degenerate tangent numbers and polynomials, Global Journal of Pure and Applied Mathematics, v. 12, pp. 1567-1574. 9. C.S. Ryoo, R.P. Agarwal,(2017). Some identities involving q-poly-tangent numbers and polynomials and distribution of their zeros, Advances in Difference Equations, v.213, DOI 10.1186/s13662-017-1275-2. 10. H. Shin, J. Zeng,(2010). The q-tangent and q-secant numbers via continued fractions, European J. Combin. v.31, pp. 1689-1705 11. P. T. Young,(2008). Degenerate Bernoulli polynomials, generalized factorial sums, and their applications, Journal of Number Theorey., v.128, pp. 738-758 12. F. T. Howard,(1985). Degenerate weighted stirling numbers, Discrete Mathematics, v.57, pp. 45-58

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On the Carlitz’s type twisted (p, q)-Euler polynomials and twisted (p, q)-Euler zeta function C. S. RYOO Department of Mathematics, Hannam University, Daejeon 34430, Korea

Abstract : In this paper we construct Carlitz’s type twisted (p, q)-Euler zeta function. In order to define Carlitz’s type twisted (p, q)-Euler zeta function, we introduce the Carlitz’s type twisted (p, q)-Euler numbers and polynomials by generalizing the Euler numbers and polynomials, Carlitz’s type q-Euler numbers and polynomials. We also give some interesting properties, explicit formulas, a connection with Carlitz’s type twisted (p, q)-Euler numbers and polynomials. Finally, we investigate the zeros of the Carlitz’s type twisted (p, q)-Euler polynomials by using computer. Key words : Euler numbers and polynomials, q-Euler numbers and polynomials, (h, q)-Euler numbers and polynomials, Carlitz’s type twisted (p, q)-Euler numbers and polynomials, (p, q)-Euler zeta function, twisted (p, q)-Euler zeta function. AMS Mathematics Subject Classification : 11B68, 11S40, 11S80. 1. Introduction Many mathematicians have studied in the area of the Bernoulli numbers and polynomials, Euler numbers and polynomials, Genocchi numbers and polynomials, tangent numbers and polynomials(see [1-10]). In this paper, we define Carlitz’s type twisted (p, q)-Euler numbers and polynomials and study some properties of the Carlitz’s type twisted (p, q)-Euler numbers and polynomials. Throughout this paper, we always make use of the following notations: N denotes the set of natural numbers, Z+ = N ∪ {0} denotes the set of nonnegative integers, Z− 0 = {0, −1, −2, −3, . . .} denotes the set of nonpositive integers, Z denotes the set of integers, R denotes the set of real numbers, and C denotes the set of complex numbers. We remember that the classical Euler numbers En and Euler polynomials En (x) are defined by the following generating functions(see [1, 2, 3, 4, 5]) ∞ ∑ 2 tn = E , n et + 1 n=0 n!

and

(

2 et + 1

) ext =

∞ ∑

En (x)

n=0

(|t| < π).

tn , n!

(1.1)

(|t| < π).

(1.2)

respectively. The (p, q)-number is defined as [n]p,q =

pn − q n = pn−1 + pn−2 q + pn−3 q 2 + · · · + p2 q n−3 + pq n−2 + q n−1 . p−q

It is clear that (p, q)-number contains symmetric property, and this number is q-number when p = 1. In particular, we can see limq→1 [n]p,q = n with p = 1. By using (p, q)-number, we define the (p, q)-analogue of Euler polynomials and numbers, which generalized the previously known numbers and polynomials, including the Carlitz’s type q-Euler

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numbers and polynomials. We begin by recalling here the Carlitz’s type q-Euler numbers and polynomials(see 1, 2, 3, 4, 5]). Definition 1. The Carlitz’s type q-Euler polynomials En,q (x) are defined by means of the generating function Fq (t, x) =

∞ ∑ n=0

En,q (x)

∞ ∑ tn = [2]q (−1)m q m e[m+x]q t . n! m=0

(1.3)

and their values at x = 0 are called the Carlitz’s type q-Euler numbers and denoted En,q . Many kinds of of generalizations of these polynomials and numbers have been presented in the literature(see [1-10]). Based on this idea, we generalize the Carlitz’s type q-Euler number En,q and q-Euler polynomials En,q (x). It follows that we define the following (p, q)-analogues of the the Carlitz’s type q-Euler number En,q and q-Euler polynomials En,q (x) (see [6, 7, 9, 10]). Definition 2. For 0 < q < p ≤ 1, the Carlitz’s type (p, q)-Euler numbers En,p,q and polynomials En,p,q (x) are defined by means of the generating functions Fp,q (t) =

∞ ∑

En,p,q (x)

∞ ∑ tn = [2]q (−1)m q m e[m]p,q t . n! m=0

(1.4)

En,p,q (x)

∞ ∑ tn = [2]q (−1)m q m e[m+x]p,q t , n! m=0

(1.5)

n=0

and Fp,q (t, x) =

∞ ∑ n=0

respectively. In the following section, we define Carlitz’s type twisted (p, q)-Euler zeta function. We introduce the Carlitz’s type twisted (p, q)-Euler polynomials and numbers. After that we will investigate some their properties. Finally, we investigate the zeros of the Carlitz’s type twisted (p, q)-Euler polynomials by using computer. 2. Twisted (p, q)-Euler numbers and polynomials In this section, we define twisted (p, q)-Euler numbers and polynomials and provide some of their relevant properties. Let r be a positive integer, and let ω be rth root of 1. Definition 2. For 0 < q < p ≤ 1, the Carlitz’s type twisted (p, q)-Euler numbers En,p,q,ω and polynomials En,p,q,ω (x) are defined by means of the generating functions Fp,q,ω (t) =

∞ ∑ n=0

and

En,p,q,ω (x)

∞ ∑ tn = [2]q (−1)m q m ω m e[m]p,q t . n! m=0

∞ ∑

∞ ∑ tn (−1)m q m ω m e[m+x]p,q t , Fp,q,ω (t, x) = En,p,q,ω (x) = [2]q n! n=0 m=0

(2.1)

(2.2)

respectively. Setting p = 1 in (2.1) and (2.2), we can obtain the corresponding definitions for the Carlitz’s type twisted q-Euler number En,q,ω and q-Euler polynomials En,q,ω (x) respectively. Obviously, if we put ω = 1, then we have En,p,q,ω (x) = En,p,q (x),

En,p,q,ω = En,p,q .

Putting p = 1, we have lim En,p,q,ω (x) = En,ω (x),

q→1

583

lim En,p,q,ω = En,ω .

q→1

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By using above equation (2.1), we have ∞ ∑

En,p,q,ω

n=0

∞ ∑ tn = [2]q (−1)m q m ω m e[m]p,q t n! m=0 ( ) )n ∑ ( n ( ) ∞ ∑ n 1 1 tn l (−1) . = [2]q l+1 n−l p−q l 1 + ωq p n! n=0

(2.3)

l=0

By comparing the coefficients

tn n!

in the above equation, we have the following theorem.

Theorem 3. For n ∈ Z+ , we have ( En,p,q,ω = [2]q

1 p−q

)n ∑ n ( ) n 1 (−1)l . l 1 + ωq l+1 pn−l l=0

If we put p = 1 in the above theorem we obtain ( En,p,q,ω = [2]q

1 1−q

)n ∑ n ( ) n 1 . (−1)l 1 + ωq l+1 l l=0

By (2.2), we obtain ( En,p,q,ω (x) = [2]q

1 p−q

)n ∑ n ( ) n 1 (−1)l q xl p(n−l)x . l 1 + ωq l+1 pn−l

(2.4)

l=0

By using (2.2) and (2.4), we obtain ) ( ( )n ∑ n ( ) ∞ ∞ ∑ ∑ n 1 1 tn tn l xl (n−l)x (−1) q p [2]q En,p,q,ω (x) = l+1 n−l l n! n=0 p−q 1 + ωq p n! n=0 l=0

= [2]q

∞ ∑

(2.5)

(−1)m q m ω m e[m+x]p,q t .

m=0

Since [x + y]p,q = py [x]p,q + q x [y]p,q , we see that En,p,q,ω (x) = [2]q

( )l l ( ) n ( ) ∑ ∑ n l 1 1 xl k q [x]n−l (−1) . p,q l k p − q 1 + ωq k+1 pn−k

(2.6)

k=0

l=0

(h)

Next, we introduce Carlitz’s type twisted (h, p, q)-Euler polynomials En,p,q,ω (x). (h)

Definition 4. The Carlitz’s type twisted (h, p, q)-Euler polynomials En,p,q,ω (x) are defined by (h) En,p,q (x) = [2]q

∞ ∑

(−1)m q m phm ω m [m + x]np,q .

(2.7)

m=0

By using (2.7) and (p, q)-number, we have the following theorem. Theorem 5. For n ∈ Z+ , we have ( (h) En,p,q,ω (x) = [2]q

1 p−q

)n ∑ n ( ) n 1 . (−1)l q xl p(n−l)x l+1 l 1 + ωq pn−l+h l=0

By (2.6) and Theorem 2.4, we have En,p,q,ω (x) =

n ( ) ∑ n l=0

584

l

(n−l)

xl [x]n−l p,q q El,p,q,ω

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The following elementary properties of the (p, q)-analogue of Euler numbers En,p,q,ω and polynomials En,p,q,ω (x) are readily derived form (2.1) and (2.2). We, therefore, choose to omit details involved. Theorem 6. (Distribution relation) For any positive integer m(=odd), we have ) ( m−1 ∑ [2]q a+x n a a a En,p,q,ω (x) = [m]p,q , n ∈ Z+ . (−1) q ω En,pm ,qm ,ωm [2]qm m a=0 Theorem 7. (Property of complement) For n ∈ Z+ , we have En,p−1 ,q−1 ,ω−1 (1 − x) = (−1)n ωpn q n En,p,q,ζ (x). Theorem 8. For n ∈ Z+ , we have { ωqEn,p,q,ω (1) + En,p,q,ω =

[2]q , 0,

if n = 0, if n = ̸ 0.

By (2.1) and (2.2), we get − [2]q

∞ ∞ n−1 ∑ ∑ ∑ (−1)l+n q l+n ω l+n e[l+n]p,q t + [2]q (−1)l q l ω l e[l]p,q t = [2]q (−1)l q l ω l e[l]p,q t . l=0

l=0

(2.8)

l=0

Hence we have ( ) ∞ ∞ n−1 ∑ ∑ ∑ tm tm tm l l l m q ω Em,p,q,ω (n) + Em,p,q,ω = [2]q (−1) q ω [l]p,q . m! m=0 m! m=0 m! m=0

n+1 n

(−1)

n

∞ ∑

(2.9)

l=0

By comparing the coefficients

tm m!

on both sides of (2.9), we have the following theorem.

Theorem 9. For n ∈ Z+ , we have n−1 ∑

(−1)l q l ω l [l]m p,q =

l=0

(−1)n+1 q n ω n Em,p,q,ω (n) + Em,p,q,ω . [2]q

We investigate the zeros of the twisted (p, q)-Euler polynomials En,p,q,ω (x) by using a computer. We plot the zeros of the twisted (p, q)-Euler polynomials En,p,q,ω (x) for x ∈ C(Figure 1). In Figure 2πi

1(top-left), we choose n = 20, p = 1/2, q = 1/10 and ω = e 2 . In Figure 1(top-right), we choose 2πi n = 40, p = 1/2, q = 1/10 and ω = e 2 . In Figure 1(bottom-left), we choose n = 20, p = 1/2, q = 1/10 and ω = e

2πi 4

. In Figure 1(bottom-right), we choose n = 40, p = 1/2, q = 1/10 and ω = e

2πi 4

.

3. Twisted (p, q)-Euler zeta function By using twisted (p, q)-Euler numbers and polynomials, (p, q)-Euler zeta function and Hurwitz (p, q)-Euler zeta function is defined. These functions interpolate the twisted (p, q)-Euler numbers En,p,q,ω , and polynomials En,p,q,ω (x), respectively. From (2.1), we note that ∞ ∑ dk = [2] (−1)n q m ω m [m]kp,q F (t) q p,q,ω dtk t=0 m=0 = Ek,p,q,ω , (k ∈ N). By using the above equation, we are now ready to define twisted (p, q)-Euler zeta function.

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Im(x)

0.4

0.4

0.2

0.2

0.0

0.0

Im(x)

-0.2

-0.2

-0.4

-0.4

-0.4

-0.2

0.0

0.2

-0.4

0.4

-0.2

Re(x)

Im(x)

0.4

0.4

0.2

0.2

0.0

-0.2

-0.2

-0.4

-0.4

-0.2

0.2

0.4

0.2

0.4

0.0

Im(x)

-0.4

0.0 Re(x)

0.0

0.2

-0.4

0.4

-0.2

0.0 Re(x)

Re(x)

Figure 1: Zeros of En,p,q,ω (x)

Definition 10. Let s ∈ C with Re(s) > 0. ζp,q,ω (s) = [2]q

∞ ∑ (−1)n q n ω n . [n]sp,q n=1

(3.1)

Note that ζp,q,ω (s) is a meromorphic function on C. Note that, if p = 1, q → 1, then ζp,q,ω (s) = ζE (s) which is the Euler zeta functions(see [4]). Relation between ζp,q,ω (s) and Ek,p,q,ω is given by the following theorem. Theorem 11. For k ∈ N, we have ζp,q,ω (−k) = Ek,p,q,ω . Observe that ζp,q,ω (s) function interpolates Ek,p,q,ω numbers at non-negative integers. By using (2.2), we note that ∞ ∑ dk F (t, x) = [2] (−1)m q m ω m [m + x]kp,q p,q,ω q dtk t=0 m=0 and

(

d dt

)k ( ∑ ∞

tn En,p,q (x) n! n=0

)

= Ek,p,q (x), for k ∈ N.

(3.2)

(3.3)

t=0

By (3.2) and (3.3), we are now ready to define the Hurwitz (p, q)-Euler zeta function.

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Definition 12. Let s ∈ C with Re(s) > 0 and x ∈ / Z− 0. ζp,q,ω (s, x) = [2]q

∞ ∑ (−1)n q n ω n . [n + x]sp,q n=0

(3.4)

Note that ζp,q,ω (s, x) is a meromorphic function on C. Obverse that, if p = 1 and q → 1, then ζp,q,ω (s, x) = ζE (s, x) which is the Hurwitz Euler zeta functions(see [1, 3, 6]). Relation between ζp,q,ω (s, x) and Ek,p,q,ω (x) is given by the following theorem. Theorem 13. For k ∈ N, we have ζp,q,ω (−k, x) = Ek,p,q,ω (x). Observe that ζp,q,ω (−k, x) function interpolates Ek,p,q,ω (x) numbers at non-negative integers.

Acknowledgment Acknowledgement: This work was supported by 2020 Hannam University Research Fund.

REFERENCES 1. Kim, T.(2008). Euler numbers and polynomials associated with zeta function, Abstract and Applied Analysis, Art. ID 581582. 2. Liu, G.(2006). Congruences for higher-order Euler numbers, Proc. Japan Acad., v.82 A, pp. 30-33. 3. Ryoo, C.S., Kim, T., Jang, L.C.(2007). Some relationships between the analogs of Euler numbers and polynomials, Journal of Inequalities and Applications, v.2007, ID 86052, pp. 1-22. 4. Ryoo, C.S.(2014). Note on the second kind Barnes’ type multiple q-Euler polynomials, Journal of Computational Analysis and Applications, v.16, pp. 246-250. 5. Ryoo, C.S.(2015). On the second kind Barnes-type multiple twisted zeta function and twisted Euler polynomials, Journal of Computational Analysis and Applications, v.18, pp. 423-429. 6. Ryoo, C.S.(2017). On the (p, q)-analogue of Euler zeta function, J. Appl. Math. & Informatics v. 35, pp. 303-311. 7. Ryoo, C.S.(2019). Some symmetric identities for (p, q)-Euler zeta function, J. Computational Analysis and Applications v. 27, pp. 361-366. 8. Ryoo, C.S.(2020). Symmetric identities for the second kind q-Bernoulli polynomials, Journal of Computational Analysis and Applications, v.28, pp. 654-659. 9. Ryoo, C.S.(2020). On the second kind twisted q-Euler numbers and polynomials of higher order, Journal of Computational Analysis and Applications, v.28, pp. 679-684. 10. Ryoo, C.S.(2020). Symmetric identities for Dirichlet-type multiple twisted (h, q)-l-function and higher-order generalized twisted (h, q)-Euler polynomials, Journal of Computational Analysis and Applications, v.28, pp. 537-542.

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STABILITY OF SET-VALUED PEXIDER FUNCTIONAL EQUATIONS ZIYING LU, GANG LU∗ , YUANFENG JIN∗ , DONG YUN SHIN∗ , AND CHOONKIL PARK Abstract. In this paper, we investigate a set-valued solution of the following Pexider functional equation F (ax + by) = αG(x) + βH(y) with three unknown functions F , G and H, where a, b, α, β are positive real scalars.

1. Introduction and preliminaries Assume that Y is a topological vector space satisfying the T0 separation axiom. For real numbers s, t and sets A, B ⊂ Y we put sA + tB := {y ∈ Y ; y = sa + tb, a ∈ A, b ∈ B}. Suppose that the space 2Y of all subsets of Y is endowed with the Hausdorff topology (see [4]). A set-valued function F : X → 2Y is said to be additive if it satisfies the Cauchy functional equation F (x1 + x2 ) = F (x1 ) + F (x2 ), x1 , x2 ∈ X. The family of all closed and convex subsets of Y will be denoted by CC(Y ), and the sets of all real, rational and positive integer numbers are denoted by R, Q, N, respectively. Lemma 1.1. [1] Let λ and µ be real numbers. If A and B are nonempty subsets of a real vector space X, then λ(A + B) = λA + λB, (λ + µ)A ⊆ λA + µB. Moreover, if A is a convex set and λ, µ ≥ 0, then we have (λ + µ)A = λA + µA. Lemma 1.2. [3] Let A, B be subsets of Y and assume that B is closed and convex. If there exists a bounded and nonempty set C ⊂ Y such that A + C ⊂ B + C, then A ⊂ B. Lemma 1.3. If (An )n∈N and (Bn )n∈N are decreasing sequences of compact subsets of Y , then T T T n∈N (An + Bn ) = n∈N An + n∈N Bn . T Lemma 1.4. If (An )n∈N is a decreasing sequence of compact subsets of Y , then An → n∈N An . Lemma 1.5. If A is a bounded subset of Y and (sn )n∈N is a real sequence converging to an s ∈ R, then sn A → sA. Lemma 1.6. If An → A and Bn → B, then An + Bn → A + B. 2010 Mathematics Subject Classification. Primary 54C60, 39B52, 47H04, 49J54. Key words and phrases. Hyers-Ulam stability, additive set-valued functional equation, closed and convex subset, cone. ∗ Corresponding authors.

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Lemma 1.7. If An → A and An → B, then clA = clB. Lemma 1.3–1.7 are rather known and can be easily verified. The proofs of them can be found in [1, 2].

2. Set-valued solution of the Pexider functional equation In the section, we give the solution of the Pexider functional equation. Theorem 2.1. Assume that (X, +) is a vector space and Y is a T0 topological vector space. If set-valued functions F : X → CC(Y ), G : X → CC(Y ) and H : X → CC(Y ) satisfy the functional equation F (ax + by) = αG(x) + βH(y)

(2.1)

for all x, y ∈ X, where a, b, α and β are positive real numbers, then there exist an additive set-valued function F0 : X → CC(Y ) and sets A, B ∈ CC(Y ) such that F (x) = αF0 (x) + αA + βB,

G(x) = F0 (ax) + A

and

H(x) = F0 (bx) + B

for all x ∈ X. Proof. First, assume that 0 ∈ G(0) and 0 ∈ H(0). Then, for all x, y ∈ X, we have  x x y  y F (x + y) = F a + b = αG + βH a b a   b x y + βH(0) + αG(0) + βH ⊂ αG a b = F (x) + F (y). Letting x = y in the above equation, we get F (2x) ⊂ 2F (x), which implies that the sequence T (2−n F (2n x))n∈N is decreasing. Put F0 (x) := n∈N 2−n F (2n x), x ∈ X. It is clear that F0 (x) ∈ CC(Y ) for all x ∈ X. Similarly, we get   ax  αG(2x) + βH(0) = F (2ax) = F ax + b b    ax  ax = αG(x) + βH ⊂ αG(x) + αG(0) + βH b b = αG(x) + F (ax) = αG(x) + αG(x) + βH(0) = 2αG(x) + βH(0). In view of Lemma 1.2, we obtain that G(2x) ⊂ 2G(x), and consequently the sequence (2−n G(2n x))n∈N is decreasing. Applying Lemma1.3 and this equality F (a2n x) = αG(2n x) + βH(0), n ∈ N, we obtain \ \ \ F0 (ax) = 2−n F (a2n x) = α 2−n G(2n x) + β 2−n H(0). n∈N

n∈N

n∈N

T But n∈N 2−n H(0) = {0}, since the set H(0) is bounded. Therefore F0 (ax) = α n∈N 2−n G(2n x) for all x ∈ X. In an analogous way we show that the sequence (2−n H(2n x))n∈N is decreasing T

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T

−n H(2n x)

for all x ∈ X. Hence,using once more Lemma 1.3, we get   n   n  \ \ 2 x1 2 x2 −n n n −n αG + βH F0 (x1 + x2 ) = 2 F (2 x1 + 2 x2 ) = 2 a b n∈N n∈N     \ \ 2n x2 2n x1 + 2−n βH = 2−n αG a b

and F0 (bx) = β

n∈N 2

n∈N

n∈N

= F0 (x1 ) + F0 (x2 ), x1 , x2 ∈ X, which means that the set-valued function F0 is additive. Now observe that F (nbx) + (n − 1)βH(0) = F (bx) + (n − 1)βH(x)

(2.2)

for all x ∈ X and n ∈ N. Indeed, for n = 1 the equality is trivial. Assume that it holds for a natural number k. Then, in virtue of (2.1), we obtain   kbx F ((k + 1)bx) + kβH(0) = αG + βH(x) + kβH(0) = F (kbx) + βH(x) + (kβ − β)H(0) a = F (bx) + (k − 1)βH(x) + βH(x) = F (x) + kβH(x). which proves that (2.2) holds for n = k + 1. Thus, by induction, it holds for all n ∈ N. In particular, we have x F (2n x) + (2n − 1)H(0) = F (x) + (2n − 1)H , b and so x −n n −n −n −n 2 F (2 x) + (1 − 2 )H(0) = 2 F (x) + (1 − 2 )H b T −n n −n n for all x ∈ X. By Lemma 1.4, 2 F (2 x) → n∈N 2 F (2 x) = F0 (x).
−n )H x → On
the other hand, by Lemma 1.5, 1−2−n H(0) → H(0), 2−n F (x) → {0} and (1−2 b

H xb . Thus, using Lemmas 1.6 and 1.7, we get cl[F0 (x) + H(0)] = clH xb , whence H xb =
F0 (x) + H(0) for all x ∈ X. Similarly, we can obtain G xa = F0 (x) + G(0), x ∈ X. Let A := G(0) and B := H(0). Then G(x) = F0 (ax) + A and H(x) = F0 (bx) + B for all x ∈ X. Moreover F (x) = αF0 (x) + αA + βB, x ∈ X. This finishes our proof in the case that 0 ∈ G(0) and 0 ∈ H(0). In the opposite case, fix arbitrarily points a ∈ G(0) and b ∈ H(0), and consider the set-valued functions F1 , G − 1, H1 : X → CC(Y ) defined by F1 (x) := F (x) − αa − βb, G1 (x) := G(x) − a and H1 := H(x) − b, x ∈ X. These set-valued functions satisfy the equation (2.1) and moreover 0 ∈ G1 (0) and 0 ∈ H1 (0). Therefore, by what we have discussed previously, we can get the same result. This completes the proof.  In [2], Nikodem proved that a set-valued function F0 : [0, ∞) → CC(Y ), where Y is a locally convex Hausdorff space, is additive if and only if there exists an additive function f : [0, ∞) → Y and a set K ∈ CC(Y ) such that F0 (x) = f (x) + xK, x ∈ [0, ∞). Thus we can get the following. Theorem 2.2. Let Y be a locally convex Hausdorff space. The set-valued functions F : [0, ∞) → CC(Y ), G : [0, ∞) → CC(Y ) and H : [0, ∞) → CC(Y ) satisfy the functional equation (2.1) if and only if there exist an additive function f : [0, ∞) → Y and sets K, A, B ∈ CC(Y ) such that F (x) = αf (x) + αKx + αA + βB, G(x) = f (ax) + akx + A

590

and

H(x) = f (bx) + bkx + B

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for all x ∈ [0, ∞). Funding This work was supported by National Natural Science Foundation of China (No.11761074), the projection of the Department of Science and Technology of JiLin Province (No. JJKH20170453KJ) and the Education Department of Jilin Province (No. 20170101052JC). Authors’ contributions All authors conceived of the study, participated in its design and coordination, drafted the manuscript, participated in the sequence alignment, and read and approved the final manuscript.

References [1] [2] [3] [4]

K. Nikodem, On Jensen’s functional equation for set-valued functions, Radovi Mat. 3 (1987), 23–33. K. Nikodem, Set-valued solutions of the Pexider functional equation, Funkcialay Ekvacioj 31 (1988), 227–231. H. Radstr¨ ˙ om, An embedding theorem for space of convex sets, Proc. Am. Math. Soc. 3 (1952),165–169. H. Radstr¨ ˙ om, One-parameter semigroups of subsets of a real linear space, Ark. Mat. 4 (1960), 87–97.

ZiYing Lu Department of Mathematics, School of Science, ShenYang University of Technology, Shenyang 110870, P.R. China E-mail address: [email protected] Gang Lu Department of Mathematics, School of Science, ShenYang University of Technology, Shenyang 110870, P.R. China E-mail address: [email protected] Yuanfeng Jin Department of Mathematics, Yanbian University, Yanji 133001, P.R. China E-mail address: [email protected] Dong Yun Shin Department of Mathematics, University of Seoul, Seoul 02504, Korea E-mail address: [email protected] Choonkil Park Research Institute for Natural Sciences, Hanyang University, Seoul 04763, Korea E-mail address: [email protected]

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On symmetries and solutions of certain sixth order difference equations

D. Nyirenda1, and M. Folly-Gbetoula1,

1

School of Mathematics, University of the Witwatersrand, Johannesburg, South Africa. Abstract We use the Lie group analysis method to investigate the invariance properties and the solutions of xn+1 =

xn−5 xn−3 . xn−1 (an + bn xn−5 xn−3 )

We show that this equation has a two-dimensional Lie algebra and that its solutions can be presented in a unified manner. Besides presenting solutions of the recursive sequence above where an and bn are sequences of real numbers, some specific cases are emphasized.

Key words: Difference equation; symmetry; reduction; group invariant solutions, periodicity MSC 2010: 39A05, 39A23, 70G65

1

Introduction

Difference equations are important in mathematical modelling, especially where discrete time evolving variables are concerned. They also occur when studying discretization methods for differential equations. Countless results in the subject of difference equations have been recorded. For rational difference equations of order greater than 1, the study can be quite challenging at the same time rewarding. Rewarding in the sense that such a study lays ground for the theory of global properties of difference equations (not necessarily rational) of higher order. In [4], the author developed an effective symmetry based algorithm to deal with the obtention of solutions of difference equations of any order. However, the calculation one deals with in this application to difference equations of order greater than one can become cumbersome but with great recompense often times. The method consists of finding a group of transformations that maps solutions onto themselves. Symmetry method is a valuable tool and it has been used to solve several difference equations [1–3, 7, 8]. In this paper, our objective is to obtain the symmetry operators of xn+1 =

xn−5 xn−3 xn−1 (an + bn xn−5 xn−3 )

(1)

1

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where an and bn are real sequences and to find its solutions by way of symmetries. Without loss of generality, we equivalently study the forward difference equation un+6 =

un un+2 . un+4 (An + Bn un un+2 )

(2)

We refer the interested reader to [4, 9] for a deeper knowledge of Lie analysis.

2

Definitions and Notation

In this section, we briefly present some definitions and notation (largely from Hydon in [4]) indispensable for the understanding of Lie symmetry analysis of difference equations. Definition 2.1 Let G be a local group of transformations acting on a manifold M . A subset S ⊂ M is called G-invariant, and G is called symmetry group of S, if whenever x ∈ S, and g ∈ G is such that g · x is defined, then g · x ∈ S. Definition 2.2 Let G be a connected group of transformations acting on a manifold M . A smooth real-valued function V : M → R is an invariant function for G if and only if X(V) = 0 for all x ∈ M, and every infinitesimal generator X of G. Definition 2.3 A parameterized set of point transformations, Γε : x 7→ x ˆ(x; ε),

(3)

where x = xi , i = 1, . . . , p are continuous variables, is a one-parameter local Lie group of transformations if the following conditions are satisfied: 1. Γ0 is the identity map if x ˆ = x when ε = 0 2. Γa Γb = Γa+b for every a and b sufficiently close to 0 3. Each xˆi can be represented as a Taylor series (in a neighborhood of ε = 0 that is determined by x), and therefore xˆi (x : ε) = xi + εξi (x) + O(ε2 ), i = 1, ..., p.

(4)

Assuming that the sixth-order difference equation has the form un+6 =Ψ(n, un , . . . , un+5 ),

n∈D

(5)

for some smooth function Ω and a regular domain D ⊂ Z. To deduce the symmetry group of (5), we search for a one parameter Lie group of point transformations Γε : (n, un ) 7→ (n, un + εQ(n, un )), (6) 2

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in which ε is the parameter and Q a continuous function, referred to as characteristic. Let X =Q(n, un )

∂ ∂ ∂ + Q(n + 1, un+1 ) + · · · + Q(n + 5, un+5 ) ∂un ∂un+1 ∂un+5

(7)

be the corresponding ‘prolonged’ infinitesimal generator and S : n 7→ n + 1 the shift operator. The linearized symmetry condition is given by S 6 Q − XΨ = 0.

(8a)

Upon knowledge of the characteristic Q, it is important to introduce the canonical coordinate Z dun , (9) Sn = Q(n, un ) a useful tool which allows one to obtain the invariant V.

3

Main results

As earlier emphasized, our equation under study is un+6 = Ψ =

un un+2 . un+4 (An + Bn un un+2 )

(10)

Appliying the criterion of invariance (8) to (10), we get Q(n + 6, un+6 ) + − −

un un+2 Q (n + 4, un+4 ) u2n+4 (An + Bn un un+2 )

An un

2 Q (n

un+4 ((An + Bn un un+2 ) An un+2

+ 2, un+2 )

2 Q (n, un )

un+4 (An + Bn un un+2 )

= 0.

(11)

In order to eliminate un+3 , we invoke implicit differentiation with respect to un (regarding un+4 as a function of un , un+2 and un+3 ) via the operator L=

∂ Ψun ∂ − . ∂un Ψun+4 ∂un+4

With some simplification, one gets (An + Bn un un+2 ) Q (n + 4, un+4 ) un+4 + Bn un Q(n + 2, un+2 ) − (An + Bn un un+2 ) Q0 (n, un )   An + 2Bn un+2 + Q (n, un ) = 0. (12) un

(An + Bn un un+2 ) Q0 (n + 4, un+4 ) −

3

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Note that the symbol 0 stands for the derivative with respect to the continuous variable. After twice differentiating (12) with respect to un , keeping un+2 and un+4 fixed, we are led to the equation − Bn un un+2 Q000 (n, un ) − An Q000 (n, un ) + +

2An 0 An 00 Q (n, un ) − Q (n, un ) un un 2

2An Q (n, un ) = 0 un 3

(13)

Note that the characteristic in (13) is not a function of un+2 and so we split (13) up with respect to un+2 to get the system 1 : Q000 (n, un ) −

2 2 1 00 Q (n, un ) + 2 Q0 (n, un ) − 3 Q (n, un ) = 0 (14a) un un un

un+2 : Q000 (n, un ) = 0.

(14b)

We find that the solution to (14) is Q (n, un ) = αn un 2 + βn un

(15)

for some arbitrary functions αn and βn that depend on n. Substituting (16) and its first, second and third shifts in (11), and then replacing the expression of un+3 given in (10) in the resulting equation yields Bn un 2 un+2 2 un+4 2 αn+4 + Bn un 2 un+2 2 un+4 (βn+4 + βn+3 ) − An un 2 un+2 un+4 αn − An un un+2 2 un+4 αn+2 + An un un+2 un+4 2 αn+4 + un 2 un+2 2 αn+2 − An (βn + βn+2 − βn+4 − βn+3 ) = 0.

(16)

Equating all coefficients of all powers of shifts of un to zero and simplifying the resulting system, we get its reduced form αn = 0, βn + βn+2 = 0.

(17) (18)

The two independent solutions of the linear second-order difference equation above are given by βn = β n and βn = β¯n , (19) ¯ where β = exp{iπ/2} and β = − exp{iπ/2} is its complex conjugate. The characteristic functions are given by Q1 (n, un ) = β n un and Q2 (n, un ) = β¯n un , (20) and so the Lie algebra of (10) is generated by X1 =β n un

∂ ∂ ∂ + β n+2 un+2 + β n+4 un+4 ∂un ∂un+2 ∂un+4

(21)

X2 =β¯n un

∂ ∂ ∂ + β¯n+2 un+2 + β¯n+4 un+4 . ∂un ∂un+2 ∂un+4

(22)

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Using the canonical coordinate Z Z dun dun 1 Sn = = = n ln |un | Q1 (n, un ) β n un β

(23)

and (17), we derive the invariant function V˜n as follows: V˜n = Sn β n + Sn+2 β n+2 .

(24)

X1 (V˜n ) = β n + β n+2 = 0

(25)

X2 (V˜n ) = β¯n + β¯n+2 = 0.

(26)

Actually,

and

For the sake of convenience, we use |Vn | = exp{−V˜n }

(27)

instead/ In other words, Vn = ±1/(un un+2 ). Using (10) and (27), one can prove that Vn+4 = An Vn ± Bn . (28) Utilizing the plus sign, the solution of (28) can be written as ! n−1 ! n−1 n−1 Y X Y V4n+j =Vj A4k1 +j + B4l+j A4k2 +j , k1 =0

l=0

(29)

k2 =l+1

where j = 0, 1, 2, 3. From here, obtaining the solution of (10) is straightforward. We first employ (23) to get |un | = exp (βn Sn ) .

(30)

Secondly, we employ (24) to obtain n−1 n−1 1 X n ¯k1 ˜ 1 X ¯n k2 ˜ |un | = exp β c1 + β c2 − β β Vk1 − β β Vk2 2 2 n

¯n

k1 =0

! .

(31)

k2 =0

Lastly, we use (27) to get ! n−1 n−1 X X 1 1 |un | = exp β n c1 + β¯n c2 + β n β¯k1 ln |Vk1 | + β¯n β k2 ln |Vk2 | , 2 2 k1 =0

k2 =0

(32a) in which Vk is given in (28) with γ(n, k) = β n β¯k . Note that the constants c1 and c2 satisfy c1 + c2 = ln |u0 |

and

β(c1 − c2 ) = ln |u1 |.

(32b)

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Note. Equations in (32) give the solutions of (2) in a unified manner. On a further note, γ(n, k) = β n β¯k satisfies ¯ γ(0, 2) = −1, γ(1, 0) = β, γ(1, 2) = −β, γ(1, 3) = −1, γ(0, 1) = β, γ(n, n) = 1, γ(n + 2, k) = −γ(n, k), γ(n, k + 2) = −γ(n, k), γ(4n, k) = γ(0, k), γ(n, 4k) = γ(n, 0).

(33)

From un given in (32a) and equation (33), observe that |u4n+j | = exp Hj +

4n+j−1 X

! Re(γ(0, k1 )) ln |Vk1 |

(34)

k1 =0

in which Hj = β j c1 + β¯j c2 . For j = 0, we have, |u4n | = exp(H0 + ln |V0 | − ln |V2 | + . . . + ln |V4n−4 | − ln |V4n−2 |) n−1 Y V4s = exp(H0 ) V4s+2 . s=0

(35)

It can be shown that there is no need for the absolute values via the utilization of the fact that 1 Vi = . (36) ui ui+2 In order to deduce exp(H0 ), we set n = 0 in (35) and note that |u0 | = exp(H0 ). Thus n−1 Y V4s u4n = u0 . V s=0 4s+2 Similarly, replacing n with 4n + j for j = 0, 1, 2, 3, we obtain U4n+j = uj

n−1 Y s=0

V4s+j . V4s+j+2

(37)

Nevertheless, from (28), using the plus sign we are led to ! n−1 ! n−1 n−1 Y X Y V4n+j =Vj A4k1 +j + B4l+j A4k2 +j , k1 =0

l=0

(38)

k2 =l+1

6

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1 u0 u2 .

for j = 0, 1, 2, 3, where V0 = U4n = u0

n−1 Y s=0

= u0

V4s V4s+2 !

s−1 Q

V0

n−1 Y s=0

Thus, using (37) with j = 0, we get

+

A4k1 !

V2

A4k1 +2

s−1 P

+

k1 =0

= u0

n−1 Y s=0

u4 u0

+ u0 u2

! A4k1 +2

+ u2 u4

=

n−1 Y s=0

A4k1

+ u0 u2

k1 =0 s−1 Q

A4k1 +2

+ u2 u4

k1 =0

!

s−1 Q

B4l+2

A4k2 +2

k2 =l+1 s−1 P

s−1 P

s−1 Q

B4l

l=0

!

A4k2

k2 =l+1

s−1 P

!

!

s−1 Q

B4l

l=0 s−1 Q

u1−n un4 0

s−1 P l=0

k1 =0

A4k2 +2

k2 =l+1

k1 =0 s−1 Q

!

s−1 Q

B4l+2

! A4k1

A4k2

k2 =l+1

l=0

s−1 Q

!

s−1 Q

B4l

l=0

k1 =0 s−1 Q

s−1 P

! A4k2

k2 =l+1 s−1 Q

B4l+2

l=0

! A4k2 +2

k2 =l+1

For j = 1, we have U4n+1 = u1

n−1 Y s=0

V4s+1 V4s+3 !

s−1 Q

=

u1−n un5 1

n−1 Y

k1 =0

s=0

s−1 Q

A4k1 +1

+ u1 u3 !

A4k1 +3

+ u3 u5

k1 =0

s−1 P

!

s−1 Q

B4l+1

A4k2 +1

l=0

k2 =l+1

s−1 P

s−1 Q

B4l+3

l=0

! A4k2 +3

k2 =l+1

For j = 2, we have U4n+2 = u2

n−1 Y s=0

V4s+2 V4s+4 s−1 Q

=

un0 u−n 4 u2

n−1 Y s=0

! A4k1 +2

+ u2 u4

k1 =0

s−1 P

B4l+2

l=0 s Q

! A4k1

k1 =0

+ u0 u2

s P l=0

!

s−1 Q

A4k2 +2

k2 =l+1

B4l

s Q

! A4k2

k2 =l+1

7

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For j = 3, we get U4n+3 = u3

n−1 Y s=0

V4s+3 V4s+5 !

s−1 Q

=

un1 u−n 5 u3

A4k1 +3

n−1 Y

k1 =0

s=0

s Q

+ u3 u5 !

A4k1 +1

+ u1 u3

k1 =0

s−1 P

s−1 Q

B4l+3

l=0

k2 =l+1

s P

s Q

B4l+1

l=0

! A4k2 +3 ! A4k2 +1

k2 =l+1

Hence, our solution in terms of xn (n > 0) is given by ! ! s−1 s−1 s−1 Q P Q b4l a4k1 + x−5 x−3 a4k2 n−1 Y k1 =0 l=0 k2 =l+1 1−n n ! !, x4n−5 = x−5 x−1 s−1 s−1 s−1 Q P Q s=0 a4k1 +2 + x−3 x−1 b4l+2 a4k2 +2 k1 =0

!

s−1 Q

x4n−4 =

n x1−n −4 x0

l=0

a4k1 +1

n−1 Y

k1 =0

s=0

s−1 Q

+ x−4 x−2 !

a4k1 +3

+ x−2 x0

x4n−3 =

s−1 P

s Q

+ x−3 x−1

s−1 P

a4k2 +3

b4l+2

+ x−5 x−3

!

s−1 Q

a4k2 +2

l=0

k2 =l+1

Ps

s Q

! a4k1

! , (40)

k2 =l+1

k1 =0

s=0

a4k2 +1

s−1 Q

b4l+3

! a4k1 +2

! (39)

s−1 Q k2 =l+1

l=0

s−1 Q n−1 Y

b4l+1

l=0

k1 =0

xn−5 x−n −1 x−3

s−1 P

k2 =l+1

l=0

k1 =0

b4l

! a4k2

k2 =l+1

(41) and !

s−1 Q

x4n−2 =

xn−4 x−n 0 x−2

n−1 Y s=0

a4k1 +3

k1 =0 s Q

+ x−2 x0

s−1 P

b4l+3

l=0

! a4k1 +1

k1 =0

+ x−4 x−2

s P

s−1 Q

! a4k2 +3

k2 =l+1

b4l+1

l=0

s Q

! a4k2 +1

k2 =l+1

(42) In the following sections, we specifically look at some special cases.

4

The case an , bn are 1-periodic

Let an = a and bn = b, where a and b are non-zero constants.

8

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4.1

Case: a 6= 1

We have x4n−5 =

n x1−n −5 x−1

n x4n−4 = x1−n −4 x0

x4n−3 = xn−5 x−n −1 x−3

as + bx−5 x−3 1−a 1−a

s=0

as + bx−3 x−1 1−a 1−a

s

,

s

n−1 Y

as + bx−4 x−2 1−a 1−a

s=0

as + bx−2 x0 1−a 1−a

s

, s

n−1 Y

as + bx−3 x−1 1−a 1−a

s+1

s=0

x4n−2 = xn−4 x−n 0 x−2

s

n−1 Y

as+1 + bx−5 x−3 1−a 1−a

,

s

n−1 Y

as + bx−2 x0 1−a 1−a

s+1

s=0

as+1 + bx−4 x−2 1−a 1−a

as long as any of the denominators does not vanish. Case: a = −1 We have

x4n−5 =

  b n−1 2 c −1+bx−5 x−3  1−n n  x x ,  −1 −5 −1+bx−3 x−1 

if n is odd;

   n−1   x1−n xn −1+bx−5 x−3 b 2 c+1 , if n is even; −1 −1+bx−3 x−1 −5

x4n−4 =

  b n−1 2 c  1−n n −1+bx−4 x−2  x x ,  0 −4 −1+bx x −2 0 

if n is even;

   n−1   x1−n xn −1+bx−4 x−2 b 2 c+1 , if n is odd; 0 −4 −1+bx−2 x0

x4n−3 =

 n −n  b n−1 2 c+1 x−5 x−1 x−3 −1+bx−3 x−1   ,   −1+bx−5 x−3 −1+bx−5 x−3

if n is odd;

  b n−1   2 c+1 −1+bx−3 x−1 xn x−n x , if n is even; −5 −1 −3 −1+bx−5 x−3 and

x4n−2 =

 n −n  b n−1 2 c+1 x−4 x0 x−2 −1+bx−2 x0   ,  −1+bx−4 x−2 −1+bx−4 x−2 

if n is odd;

  b n−1   2 c+1 −1+bx−2 x0 xn x−n x , if n is even; −2 −1+bx−4 x−2 −4 0 9

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where bx−i x2−i 6= 1 for i = 2, 3, 4, 5.

4.2

Case: a = 1

The solution is given by n x4n−5 = x1−n −5 x−1

n−1 Y s=0

n−1 Y 1 + bx−4 x−2 s 1 + bx−5 x−3 s n , x4n−4 = x1−n , −4 x0 1 + bx−3 x−1 s 1 + bx−2 x0 s s=0

x4n−3 = xn−5 x−n −1 x−3

n−1 Y

1 + bx−3 x−1 s , 1 + bx−5 x−3 (s + 1)

s=0

x4n−2 = xn−4 x−n 0 x−2

n−1 Y

1 + bx−2 x0 s . 1 + bx−4 x−2 (s + 1)

s=0

5

The case an , bn are 2-periodic

∞ In this case, we have {an }∞ n=0 = a0 , a1 , a0 , a1 , . . ., and similarly {bn }n=0 = b0 , b1 , b0 , b1 , . . . where a0 6= a1 , and b0 = 6 b1 . Then the solution is given by

x4n−5 =

1−n n x−5 x−1

as0 + b0 x−5 x−3

n−1 Y

as0

s=0

+ b0 x−3 x−1

s−1 P l=0 s−1 P

al0 , al0

l=0

x4n−4 =

n x1−n −4 x0

n−1 Y

as1 + b1 x−4 x−2 as1 + b1 x−2 x0

s=0

s−1 P

al1

l=0 s−1 P

,

al1

l=0

x4n−3 = xn−5 x−n −1 x−3

as0 + b0 x−3 x−1

n−1 Y

s+1 s=0 a0

s−1 P

al0

l=0 s P

+ b0 x−5 x−3

al0

l=0

and x4n−2 =

xn−4 x−n 0 x−2

as1 + b1 x−2 x0

n−1 Y s=0

s−1 P

al1

l=0 s P

as+1 + b1 x−4 x−2 1

al1

l=0

as long as any of the denominators does not vanish.

10

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6

The case an , bn are 4-periodic

We assume that {an } = a0 , a1 , a2 , a3 , a0 , a1 , a2 , a3 , . . . and {bn } = b0 , b1 , b2 , b3 , b0 , b1 , b2 , b3 , · · · . The solution is given by

1−n n x4n−5 = x−5 x−1

as0 + b0 x−5 x−3

n−1 Y

as2 + b2 x−3 x−1

s=0

s−1 P l=0 s−1 P

al0 ,

(43)

,

(44)

al2

l=0

x4n−4 =

n x1−n −4 x0

n−1 Y

as1 + b1 x−4 x−2 as3

s=0

+ b3 x−2 x0

s−1 P

al1

l=0 s−1 P

al3

l=0

x4n−3 = xn−5 x−n −1 x−3

as2 + b2 x−3 x−1

n−1 Y

s+1 s=0 a0

s−1 P

al2

l=0 s P

+ b0 x−5 x−3

(45) al0

l=0

and x4n−2 =

xn−4 x−n 0 x−2

as3 + b3 x−2 x0

n−1 Y s=0

s−1 P

al3

l=0 s P

as+1 + b1 x−4 x−2 1

(46) al1

l=0

as long as any of the denominators does not vanish.

7

Conclusion

In this paper, non-trivial symmetries for difference equations of the form (1) were found. Consequently, the results were used to find formulas for the solutions of the equations. Specific cases of the solutions were also discussed.

References [1] Mensah Folly-Gbetoula, Invariance analysis of a four-dimensional system of fourth-order difference equations with variable coefficients, J. Computational Analysis and Applications, 28:6 (2020), 949–961. [2] M. Folly-Gbetoula and A.H. Kara, Symmetries, conservation laws, and integrability of difference equations, Advances in Difference Equations, 2014, 2014. [3] M. Folly-Gbetoula (2017), Symmetry, reductions and exact solutions of the difference equation un+2 = (aun )/(1 + bun un+1 ),Journal of Difference Equations and Applications, 23:6 (2017). 11

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[4] P. E. Hydon, Difference Equations by Differential Equation Methods, Cambridge University Press (2014). [5] N. Joshi and P. Vassiliou, The existence of Lie Symmetries for First-Order Analytic Discrete Dynamical Systems, Journal of Mathematical Analysis and Applications 195 (1995), 872-887 (1995) [6] D. Levi, L. Vinet and P. Winternitz, Lie group formalism for difference equations, J. Phys. A: Math. Gen. 30, 633-649 (1997). [7] N. Mnguni, D. Nyirenda and M. Folly-Gbetoula, On solutions of some fifth-order difference equations, Far East Journal of Mathematical Sciences, 102:12 (2017) 3053-3065. [8] D. Nyirenda and M. Folly-Gbetoula, Invariance analysis and exact solutions of some sixth-order difference equations, J. Nonlinear Sci. Appl., 10 (2017) 6262-6273. [9] P. J. Olver, Applications of Lie Groups to Differential Equations, Second Edition, Springer, New York (1993). [10] G. R. W. Quispel and R. Sahadevan, Lie symmetries and the integration of difference equations, Physics Letters A, 184 (1993) 64-70.

12

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TABLE OF CONTENTS, JOURNAL OF COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 29, NO. 3, 2021

A Numerical Technique for Solving Fuzzy Fractional Optimal Control Problems, Altyeb Mohammed, Zeng-Tai Gong, and Mawia Osman,…………………………………………413 Differential Transform Method for Solving Fuzzy Fractional Wave Equation, Mawia Osman, Zeng-Tai Gong, and Altyeb Mohammed,……………………………………………………431 Some Generalized k-Fractional Integral Inequalities for Quasi-Convex Functions, Ghulam Farid, Chahn Yong Jung, Sami Ullah, Waqas Nazeer, Muhammad Waseem, and Shin Min Kang,.454 Asymptotically Almost Automorphic Mild Solutions for Second Order Nonautonomous Semilinear Evolution Equations, Mouffak Benchohra, Gaston M. N'Guérékata, and Noreddine Rezoug,………………………………………………………………………………………468 Applying Hybrid Coupled Fixed Point Theory to the Nonlinear Hybrid System of Second Order Differential Equations, Tamer Nabil,……………………………………………………….494 Co-Ordinated Convex Functions of Three Variables and Some Analogous Inequalities with Applications, Rashida Hussain, Asghar Ali, Asia Latif, and Ghazala Gulshan,……………505 Studies on the Higher Order Difference Equation, M. M. El-Dessoky,……………………518 Hermite-Hadamard Inequality for Sugeno Integral Based on Harmonically Convex Functions, Ali Ebadian and Maryam Oraki,……………………………………………………………532 On the Order, Type and Zeros of Meromorphic Functions and Analytic Functions of [p, q]-Order in the Unit Disc, Jin Tu, Ke Qi Hu, and Hong Zhang,………………………………………544 Reachable Sets for Semilinear Integrodifferential Control Systems, Hyun-Hee Roh and Jin-Mun Jeong,…………………………………………………………………………………………557 On Modified Degenerate Poly-Tangent Numbers and Polynomials, C. S. RYOO,…………570 On the Carlitz's Type Twisted (p, q)-Euler Polynomials and Twisted (p, q)-Euler Zeta Function, C. S. RYOO,…………………………………………………………………………………582 Stability of Set-Valued Pexider Functional Equations, Ziying Lu, Gang Lu, Yuanfeng Jin, Dong Yun Shin, and Choonkil Park,………………………………………………………………588

TABLE OF CONTENTS, JOURNAL OF COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 29, NO. 3, 2021 (continues)

On Symmetries and Solutions of Certain Sixth Order Difference Equations, D. Nyirenda and M. Folly-Gbetoula,………………………………………………………………………………592

Volume 29, Number 4 ISSN:1521-1398 PRINT,1572-9206 ONLINE

August 2021

Journal of Computational Analysis and Applications EUDOXUS PRESS, LLC

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Exact Solitary Wave Solutions for Wick-type Stochastic (2+1)-dimensional Coupled KdV equations Hossam A. Ghany1 and M. Zakarya2 1 Department of Mathematics, Helwan University, Cairo (11282), Egypt. [email protected] 2 Department of Mathematics, Faculty of Science, Al-Azhar University, Assiut (71524), Egypt. mohammed [email protected] Abstract Variable coefficients and Wick-type stochastic (2+1)-dimensional coupled KdV equations are investigated. By using the F-expansion method , Hermite transform and white noise theory, the white noise functional solutions for Wick-type stochastic (2+1)dimensional coupled KdV equations are obtained. The exact travelling wave solutions are expressed in terms of Jacobi elliptic (JEF), trigonometric and hyperbolic functions.

Keywords: KdV equations; F-expansion method; Hermite transform; Wick product. PACS No. : 05.40. ± a, 02.30.Jr.

1

Introduction

In this paper, we shall explore exact solutions for the following variable coefficients (2+1)dimensional coupled KdV equations.   ut + φ1 (t)uvx + φ2 (t)vux + φ3 (t)uxxx = 0, (1.1)  ux + vy = 0,

where (t, x) ∈ R+ × R and φ1 (t) , φ2 (t) and φ3 (t) are bounded measurable or integrable functions on R+ . Random wave is an important subject of stochastic partial differential equations (PDEs). Many authors have studied this subject. Wadati first introduced and studied the stochastic KdV equations and gave the diffusion of soliton of the KdV equation under Gaussian noise in [30, 32] and others [3, 4, 5, 25] also researched stochastic KdV-type 1

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equations. Xie first introduced Wick-type stochastic KdV equations on white noise space and showed the auto- Backlund transformation and the exact white noise functional solutions in [37]. Furthermore, Xie [38, 39, 40, 41], Ghany et al. [11, 12, 13, 15, 16, 17, 18, 19, 20] researched some Wick-type stochastic wave equations using white noise analysis. In this paper we use F-expansion method for finding new periodic wave solutions of nonlinear evolution equations in mathematical physics, and we obtain some new periodic wave solutions for (2+1)-dimensional coupled KdV equations. This method is more powerful and will be used in further works to establish more entirely new solutions for other kinds of nonlinear partial differential equations arising in mathematical physics. The effort in finding exact solutions to nonlinear equations is important for the understanding of most nonlinear physical phenomena. For instance, the nonlinear wave phenomena observed in fluid dynamics, plasma, and optical fibers[24]. Many effective methods have been presented, such as tanh-function method [34, 42, 8], variational iteration method [6, 7], exp-function method [22, 23, 36, 43, 44] , homotopy perturbation method [10, 29, 35], homotopy analysis method [1], tanh-coth method [33, 34, 31], Jacobi elliptic function expansion method [27, 28, 9, 26] and F-expansion method [45, 46, 47, 48]. The main objective of this paper is using the F-expansion method to construct white noise functional solutions for wick-type stochastic (2+1)-dimensional coupled KdV equations via hermite transform, wick-type product and white noise analysis. If equation (1.1) is considered in a random environment, we can get stochastic (2+1)-dimensional coupled KdV equations. In order to give the exact solutions of stochastic (2+1)-dimensional coupled KdV equations, we only consider this problem in white noise environment. We shall study the following Wick-type stochastic (2+1)-dimensional coupled KdV equations.  Ut + Φ1 (t)  U  Vx + Φ2 (t)  V  Ux + Φ3 (t)  Uxxx = 0, (1.2) Ux + Vy = 0, where “  ” is the Wick product on the Kondratiev distribution space (S)−1 which was defined in [21] and Φ1 (t), Φ2 (t) and Φ3 (t) are (S)−1 -valued functions.

2

Description of the F-expansion Method

In order to at the same time obtain more periodic wave solutions expressed by various Jacobi elliptic functions to nonlinear wave equations, we introduce an F-expansion method which can be thought of as a succinctly over-all generalization of Jacobi elliptic function expansion. We briefly show what is F-expansion method and how to use it to obtain various periodic wave solutions to nonlinear wave equations. Suppose a nonlinear wave equation for u(t, x) is given by θ1 (u, ut , ux , uy , uxx , uxxx , ...) = 0,

(2.1)

2

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where u = u(t, x) is an unknown function, θ1 is a polynomial in u and its various partial derivatives in which the highest order derivatives and nonlinear terms are involved. In the following we give the main steps of a deformation F-expansion method. Step 1. Look for traveling wave solution of Eq.(2.1) by taking Z t u(t, x, y) = u(ξ) , ξ(t, x, y) = kx + ly + μ ω(τ )dτ + c, (2.2) 0

Hence, under the transformation (2.2). Eq.(2.1) can be transformed into the following ordinary differential equation (ODE) as following θ2 (u, μωu0 , ku0 , lu0 , k 2 u00 , k 3 u000 , ...) = 0,

(2.3)

Step 2. Suppose that u(ξ) can be expressed by a finite power series of F (ξ) of the form u(t, x, y) = u(ξ) =

N X

ai F i (ξ),

(2.4)

i=1

where a0 , a1 , ..., aN are constants to be determined later, while F 0 (ξ) in(2.4) satisfy [F 0 (ξ)]2 = P F 4 (ξ) + QF 2 (ξ) + R,

(2.5)

and hence holds for F (ξ)  0 00 F F = 2P F 3 F 0 + QF F 0 ,    00 F = 2P F 3 + QF, F 000 = 6P F 2 F 0 + QF 0 ,    ...

(2.6)

where P, Q, and R are constants. Step 3. The positive integer N can be determined by considering the homogeneous balance between the highest derivative term and the nonlinear terms appearing in (2.3). Therefore, we can get the value of N in (2.4). Step 4. Substituting (2.4) into (2.3) with the condition (2.5), we obtain polynomial in F i (ξ)[F 0 (ξ)]j , (i = 0 ± 1, ±2, ..., j = 0, 1) . Setting each coefficient of this polynomial to be zero yields a set of algebraic equations for a0 , a1 , ..., aN , μ and ω . Step 5. Solving the algebraic equations with the aid of Maple we have a0 , a1 , ..., aN , μ and ω can be expressed by (P, Q, R) . Substituting these results into F-expansion (2.4), then a general form of traveling wave solution of Eq. (2.1) can be obtained. Step 6. Since the general solutions of (2.4) have been well known for us Choose properly ( P, Q and R .) in ODE (2.5) such that the corresponding solution F (ξ) of it is one of Jacobi elliptic functions. (See Appendices A, B and C .)[45, 46, 47]

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3

New Exact Wave Solutions of Eq. (1.2)

Taking the Hermite transform, white noise theory, and F-expansion method to explore new exact wave solutions for Eq.(1.2). Applying Hermite transform to Eq.(1.2), we get the deterministic equation.  et (t, x, y, z) + Φ f1 (t, z)U f2 (t, z)Ve (t, x, y, z)U ex (t, x, y, z) e (t, x, y, z)Vex (t, x, y, z) + Φ  U    f exxx (t, x, y, z) = 0, +Φ3 (t, z)U (3.1)     U ex (t, x, y, z) + Vey (t, x, y, z) = 0,

where z = (z1 , z2 , ...) ∈ (CN ) is a vector parameter. To look for the travelling wave sof1 (t, z) := φ1 (t, z) , Φ f2 (t, z) := φ2 (t, z) , lution of Eq.(3.1), we make the transformations Φ e (t, x, y, z) := u(t, x, y, z) = u(ξ(t, x, y, z)) and Ve (t, x, y, z) := v(t, x, y, z) = f3 (t, z) := φ3 (t, z) , U Φ v(ξ(t, x, y, z)) with Z t ξ(t, x, y, z) = kx + ly + μ ω(τ, z)dτ + c, 0

where k, μ and c are arbitrary constants which satisfy kμ 6= 0 , ω(τ, z) is a nonzero function of the indicated variables to be determined later. Hence, Eq.(3.1) can be transformed into the following (ODE).  0 0 0 000 μωu + kφ1 uv + kφ2 vu + k 3 φ3 u = 0, (3.2) 0 0 ku + lv = 0,

where the prime denote to the differential with respect to ξ . In view of F-expansion method, the solution of Eq. (3.1), can be expressed in the form.  i u(t, x, y, z) = u(ξ) = ΣN i=1 ai F (ξ), (3.3) M i v(t, x, y, z) = v(ξ) = Σi=1 bi F (ξ), where ai and bi are constants to be determined later. considering homogeneous balance between the highest order nonlinear terms and the highest order partial derivative of u in (3.2), then we can obtain N = M = 2 so (3.3) can be rewritten as following  u(t, x, y, z) = a0 + a1 F (ξ) + a2 F 2 (ξ), (3.4) v(t, x, y, z) = b0 + b1 F (ξ) + b2 F 2 (ξ), where a0 , a1 , a2 , b0 , b1 and b2 are constants to be determined later. Substituting (3.4) with the conditions (2.5),(2.6) into (3.2) and collecting all terms with the same power of

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F i (ξ)[F 0 (ξ)]j , (i = 0 ± 1, ±2, ..., j = 0, 1) . as following  0 [μωa1 + ka0 b1 φ1 + ka1 b0 φ2 + k 3 a1 φ3 Q]F   0    +[2μωa2 + 2ka0 b2 φ1 + ka1 b1 φ1 + 2ka2 b0 φ2 + ka1 b1 φ2 + 8k 3 a2 φ3 Q]F F 0 +k[2a1 b2 φ1 + a2 b1 φ1 + 2a2 b1 φ2 + a1 b1 φ2 + 6k 2 a1 φ3 P ]F 2 F 0   +2ka2 [b2 φ1 + b2 φ2 + 12k 2 φ3 P ]F 2 F = 0,    0 0 (ka1 + lb1 )F + 2[ka2 + lb2 ]F F = 0.

(3.5)

Setting each coefficients of F i (ξ)[F 0 (ξ)]j to be zero, we get a system of algebraic equations which can be expressed by.  μωa1 + ka0 b1 φ1 + ka1 b0 φ2 + k 3 a1 φ3 Q = 0,         2μωa2 + 2ka0 b2 φ1 + ka1 b1 φ1 + 2ka2 b0 φ2 + ka1 b1 φ2 + 8k 3 a2 φ3 Q = 0,          k[2a1 b2 φ1 + a2 b1 φ1 + 2a2 b1 φ2 + a1 b1 φ2 + 6k 2 a1 φ3 P ] = 0, (3.6)  2  2ka2 [b2 φ1 + b2 φ2 + 12k φ3 P ] = 0,         ka1 + lb1 = 0,        2[ka2 + lb2 ] = 0.

with solving by Maple to get the following coefficients  a2 = b2 = 0, a0 , b0 = arbitrary constant,     a1 = 6lkφ3 (t,z)P , φ2 (t,z)

(3.7)

2

(t,z)P b1 = − 6k φφ23(t,z) ,     k2 a0 φ1 (t,z)−lk[b0 φ2 (t,z)+k2 φ3 (t,z)Q] . ω= lμ

Substituting by coefficient (3.7) into (3.4) yields general form solutions of Eq. (1.2). u(t, x, y, z) = a0 +

6lkφ3 (t, z)P F (ξ), φ2 (t, z)

(3.8)

v(t, x, y, z) = b0 −

6k 2 φ3 (t, z)P F (ξ), φ2 (t, z)

(3.9)

with ξ(t, x, y, z) = kx + ly +

Z

0

t

k 2 a0 φ1 (τ, z) − lk[b0 φ2 (τ, z) + k 2 φ3 (τ, z)Q] dτ. l

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From Appendix A, we give the special cases as following. Case I: If we take P = 14 , Q =

m2 −2 2

and R =

m2 4

, we have F (ξ) → ns(ξ) ± ds(ξ) ,

  3lkφ3 (t, z) ns (ξ1 (t, x, y, z)) ± ds (ξ1 (t, x, y, z)) , u1 (t, x, y, z) = a0 + 2φ2 (t, z)

(3.10)

  3k 2 φ3 (t, z) v1 (t, x, y, z) = b0 − ns (ξ1 (t, x, y, z)) ± ds (ξ1 (t, x, y, z)) , 2φ2 (t, z)

(3.11)

with ξ1 (t, x, y, z) = kx + ly +

Z t 0

2k 2 a0 φ1 (τ, z) − lk[2b0 φ2 (τ, z) + k 2 φ3 (τ, z)(m2 − 2)] 2l



dτ.

In the limit case when m → o , we have ns(ξ)±ds(ξ) → 2 csc(ξ) , thus (3.10),(3.11) become. u2 (t, x, y, z) = a0 +

3lkφ3 (t, z) csc (ξ2 (t, x, y, z)), φ2 (t, z)

(3.12)

v2 (t, x, y, z) = b0 −

3k 2 φ3 (t, z) csc (ξ2 (t, x, y, z)), φ2 (t, z)

(3.13)

with ξ2 (t, x, y, z) = kx + ly +

Z t 0

k 2 a0 φ1 (τ, z) − lk[b0 φ2 (τ, z) − k 2 φ3 (τ, z)] l



dτ.

In the limit case when m → 1 we have ns(ξ) ± ds(ξ → coth(ξ) ± (ξ) , thus (3.10).(3.11) become.   3lkφ3 (t, z) u3 (t, x, y, z) = a0 + coth ξ3 (t, x, y, z) ± (ξ3 (t, x, y, z)) , (3.14) 2φ2 (t, z) 3k 2 φ3 (t, z) v3 (t, x, y, z) = b0 − 2φ2 (t, z)





[coth ξ3 (t, x, y, z) ± (ξ3 (t, x, y, z)) ,

(3.15)

with ξ3 (t, x, y, z) = kx + ly +

Z t 0

2k 2 a0 φ1 (τ, z) − lk[2b0 φ2 (τ, z) − k 2 φ3 (τ, z)] 2l



dτ.

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Case II: If we take P = 1, Q = −(1 + m2 ) and R = m2 , then F (ξ) → ns(ξ) , 6lkφ3 (t, z) ns (ξ4 (t, x, y, z)), φ2 (t, z)

(3.16)

6k 2 φ3 (t, z) v4 (t, x, y, z) = b0 − ns (ξ4 (t, x, y, z)), φ2 (t, z)

(3.17)

u4 (t, x, y, z) = a0 +

with ξ4 (t, x, y, z) = kx + ly +

Z t 0

2k 2 a0 φ1 (τ, z) − lk[2b0 φ2 (τ, z) + k 2 φ3 (τ, z)(m2 − 2)] l



dτ.

In the limit case when m → o we have ns(ξ) ± ds(ξ) → csc(ξ) , thus (3.10),(3.11) become. u5 (t, x, y, z) = a0 +

6lkφ3 (t, z) csc (ξ2 (t, x, y, z)), φ2 (t, z)

(3.18)

v5 (t, x, y, z) = b0 −

6k 2 φ3 (t, z) csc (ξ2 (t, x, y, z)). φ2 (t, z)

(3.19)

In the limit case when m → 1 we have ns(ξ) → coth(ξ) , thus (3.10).(3.11) become. u6 (t, x, y, z) = a0 +

6lkφ3 (t, z) coth (ξ5 (t, x, y, z)), 2φ2 (t, z)

(3.20)

v6 (t, x, y, z) = b0 −

6k 2 φ3 (t, z) coth (ξ5 (t, x, y, z)), 2φ2 (t, z)

(3.21)

with ξ5 (t, x, y, z) = kx + ly +

Z t 0

k 2 a0 φ1 (τ, z) − lk[b0 φ2 (τ, z) − 2k 2 φ3 (τ, z)] l



dτ.

Case III: If we take P = 1, Q = (2 − m2 ) and R = 1 − m2 , then F (ξ) → cs(ξ) , u7 (t, x, y, z) = a0 +

6lkφ3 (t, z) cs (ξ6 (t, x, y, z)), φ2 (t, z)

(3.22)

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v7 (t, x, y, z) = b0 −

6k 2 φ3 (t, z) cs (ξ6 (t, x, y, z)), φ2 (t, z)

(3.23)

with ξ6 (t, x, y, z) = kx + ly +

Z t 0

k 2 a0 φ1 (τ, z) − lk[2b0 φ2 (τ, z) + k 2 φ3 (τ, z)(2 − m2 )] l



dτ.

In the limit case when m → o we have cs(ξ) → cot(ξ) , thus (3.10),(3.11) become. 6lkφ3 (t, z) cot (ξ7 (t, x, y, z)), φ2 (t, z)

(3.24)

6k 2 φ3 (t, z) cot (ξ7 (t, x, y, z)), v8 (t, x, y, z) = b0 − φ2 (t, z)

(3.25)

u8 (t, x, y, z) = a0 +

ξ7 (t, x, y, z) = kx + ly +

Z t 0

k 2 a0 φ1 (τ, z) − lk[b0 φ2 (τ, z) + 2k 2 φ3 (τ, z)] l



dτ.

In the limit case when m → 1 we have cs(ξ) → (ξ) , thus (3.10).(3.11) become. u9 (t, x, y, z) = a0 +

6lkφ3 (t, z) (ξ8 (t, x, y, z)), φ2 (t, z)

(3.26)

v9 (t, x, y, z) = b0 −

6k 2 φ3 (t, z) (ξ8 (t, x, y, z)), φ2 (t, z)

(3.27)

with ξ8 (t, x, y, z) = kx + ly +

Z t 0

k 2 a0 φ1 (τ, z) − lk[b0 φ2 (τ, z) + k 2 φ3 (τ, z)] l



dτ.

Obviously, there are another solutions for Eq.(1.2). These solutions come from setting different values for the coefficients P, Q and R . (see Appendix A, B and C.)[46, 47]. The above mentioned cases are just to clarify how far our technique is applicable.

4

White Noise Functional Solutions of Eq.(1.2)

In this section, we employ the results of the Section 3 by using Hermite transform to obtain exact white noise functional solutions for Wick-type stochastic (2+1)-dimensional coupled KdV equations (1.2). The properties of exponential and trigonometric functions yield that there exists a bounded open set G ⊂ R+ × R2 , ρ < ∞, λ > 0 such that the so8

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lution u(t, x, y, z) of Eq. (3.1) and all its partial derivatives which are involved in Eq. (3.1) are uniformly bounded for (t, x, y, z) ∈ G × Kρ (λ) , continuous with respect to (t, x, y) ∈ G for all z ∈ Kρ (λ) and analytic with respect to z ∈ Kρ (λ) , for all (t, x, y) ∈ G . From e (t, x, y)(z) Theorem 4.1.1 in [21], there exists U (t, x, y, z) ∈ (S)−1 such that u(t, x, y, z) = U for all (t, x, y, z) ∈ G × Kρ (λ) and U (t, x, y) solves Eq.(1.2) in (S)−1 . Hence, by applying the inverse Hermite transform to the results of Section 3, we get exact white noise functional solutions of Eq. (1.2) as follows. • White noise functional solutions of JEF type:   3lkΦ3 (t)   U1 (t, x, y) = a0 +  ns (Ξ1 (t, x, y)) ± ds (Ξ1 (t, x, y)) , 2Φ2 (t)

(4.1)

  3k 2 Φ3 (t)   V1 (t, x, y) = b0 −  ns (Ξ1 (t, x, y)) ± ds (Ξ1 (t, x, y)) , 2Φ2 (t)

(4.2)

U2 (t, x, y) = a0 +

6lkΦ3 (t)  ns (Ξ2 (t, x, y)), Φ2 (t)

(4.3)

V2 (t, x, y) = b0 −

6k 2 Φ3 (t)  ns (Ξ2 (t, x, y)), Φ2 (t)

(4.4)

U3 (t, x, y) = a0 +

6lkΦ3 (t)  cs (Ξ3 (t, x, y)), Φ2 (t)

(4.5)

V3 (t, x, y) = b0 −

6k 2 Φ3 (t)  cs (Ξ3 (t, x, y)), Φ2 (t)

(4.6)

with Ξ1 (t, x, y) = kx + ly +

Z t

2k 2 a0 Φ1 (τ ) − lk[2b0 Φ2 (τ ) + k 2 φ3 (τ )(m2 − 2)] 2l



dτ,

Z t

2k 2 a0 Φ1 (τ ) − lk[2b0 Φ2 (τ ) + k 2 Φ3 (τ )(m2 − 2)] l



dτ,

0

Ξ2 (t, x, y) = kx + ly +

0

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Ξ3 (t, x, y) = kx + ly +

Z t 0

k 2 a0 Φ1 (τ ) − lk[2b0 Φ2 (τ ) + k 2 Φ3 (τ )(2 − m2 )] l



dτ.

• White noise functional solutions of trigonometric type: U4 (t, x, y) = a0 +

3lkΦ3 (t)  csc (Ξ4 (t, x, y)), Φ2 (t)

(4.7)

V4 (t, x, y) = b0 −

3k 2 Φ3 (t)  csc (Ξ4 (t, x, y)), Φ2 (t)

(4.8)

U5 (t, x, y) = a0 +

6lkΦ3 (t)  csc (Ξ4 (t, x, y)), Φ2 (t)

(4.9)

V5 (t, x, y) = b0 −

6k 2 Φ3 (t)  csc (Ξ4 (t, x, y)), Φ2 (t)

(4.10)

U6 (t, x, y) = a0 +

6lkΦ3 (t)  cot (Ξ5 (t, x, y)), Φ2 (t)

(4.11)

V6 (t, x, y) = b0 −

6k 2 Φ3 (t)  cot (Ξ5 (t, x, y)), Φ2 (t)

(4.12)

with Ξ4 (t, x, y) = kx + ly +

Z t 0

Ξ5 (t, x, y) = kx + ly +

Z t 0

k 2 a0 Φ1 (τ ) − lk[b0 Φ2 (τ ) − k 2 Φ3 (τ )] l



dτ,



dτ.

k 2 a0 Φ1 (τ ) − lk[b0 Φ2 (τ ) + 2k 2 Φ3 (τ )] l

• White noise functional solutions of hyperbolic type:   3lkΦ3 (t)    coth (Ξ6 (t, x, y)) ± (Ξ6 (t, x, y)) , U7 (t, x, y) = a0 + 2Φ2 (t)

(4.13)

  3k 2 Φ3 (t)   V7 (t, x, y) = b0 −  coth (Ξ6 (t, x, y)) ± (Ξ6 (t, x, y)) , 2Φ2 (t)

(4.14)

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U8 (t, x, y) = a0 +

6lkΦ3 (t)  coth (Ξ7 (t, x, y)), 2Φ2 (t)

(4.15)

V8 (t, x, y) = b0 −

6k 2 Φ3 (t)  coth (Ξ7 (t, x, y)), 2Φ2 (t)

(4.16)

U9 (t, x, y) = a0 +

6lkΦ3 (t)   (Ξ8 (t, x, y)), Φ2 (t)

(4.17)

V9 (t, x, y) = b0 −

6k 2 Φ3 (t)   (Ξ8 (t, x, y)), Φ2 (t)

(4.18)

with Ξ6 (t, x, y) = kx + ly +

Z t 0

Ξ7 (t, x, y) = kx + ly +

Z t 0

Ξ8 (t, x, y) = kx + ly +

dτ,



dτ,



dτ.

k 2 a0 Φ1 (τ ) − lk[b0 Φ2 (τ ) − 2k 2 Φ3 (τ )] l

Z t 0



2k 2 a0 Φ1 (τ ) − lk[2b0 Φ2 (τ ) − k 2 Φ3 (τ )] 2l

k 2 a0 Φ1 (τ ) − lk[b0 Φ2 (τ ) + k 2 Φ3 (τ )] l

We observe that, for different forms of Φ1 , Φ2 and Φ3 , we can get different exact white noise functional solutions of Eq. (1.2) from Eqs. (4.1)-(4.18).

5

Example

It is well known that Wick version of function is usually difficult to evaluate. So, in this section, we give non-Wick version of solutions of Eq. (1.2). Let Wt = B˙ t be the Gaussian ft (z) = white noise, motion. We have the Hermite transform W R t where Bt is the Brownian P 2 ∞ t t2   i=1 zi 0 ηi (s)ds [21]. Since exp (Bt ) = exp(Bt − 2 ), we have cot (Bt ) = cot(Bt − 2 ) , 2 2 2 csc (Bt ) = csc(Bt − t2 ) , coth (Bt ) = coth(Bt − t2 ) and  (Bt ) = (Bt − t2 ). Suppose that. Φ1 (t) = ψ1 Φ3 (t), Φ2 (t) = ψ2 Φ3 (t) and Φ3 (t) = Γ(t) + ψ3 Wt where ψ1 , ψ2 and ψ3 are arbitrary constants and Γ(t) is integrable or bounded measurable function on R+ . Therefore, for Φ1 (t)Φ2 (t)Φ3 (t) 6= 0 . thus exact white noise functional solutions of Eq. (1.2)

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are as follows. U10 (t, x, y) = a0 +

3lk csc (Ω1 (t, x, y)), ψ2

(5.1)

V10 (t, x, y) = b0 −

3k 2 csc (Ω1 (t, x, y)), ψ2

(5.2)

6lk csc Ω1 (t, x, y), ψ2

(5.3)

V11 (t, x, y) = b0 −

6k 2 csc (Ω1 (t, x, y)), ψ2

(5.4)

U12 (t, x, y) = a0 +

6lk cot (Ω2 (t, x, y)), ψ2

(5.5)

V12 (t, x, y) = b0 −

6k 2 cot (Ω2 (t, x, y)), ψ2

(5.6)

U11 (t, x, y) = a0 +

with Z

k 2 a0 ψ1 − lk[b0 ψ2 − k 2 ]
Ω1 (t, x, y) = kx + ly + l

t 0

Z

and

k 2 a0 ψ1 − lk[b0 ψ2 + 2k 2 ]
Ω2 (t, x, y) = kx + ly + l 3lk U13 (t, x, y) = a0 + 2ψ2



3k 2 V13 (t, x, y) = b0 − 2ψ2



 t2 Γ(τ )dτ + ψ3 [Bt − ] , 2

t 0

 t2 Γ(τ )dτ + ψ3 [Bt − ] , 2 

(5.7)



(5.8)

coth (Ω3 (t, x, y)) ± (Ω3 (t, x, y)) ,

coth (Ω2 (t, x, y)) ± (Ω3 (t, x, y)) ,

U14 (t, x, y) = a0 +

6lk coth (Ω4 (t, x, y)), 2ψ2

(5.9)

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6k 2 coth (Ω4 (t, x, y)), 2ψ2

(5.10)

U15 (t, x, y) = a0 +

6lk (Ω5 (t, x, y)), ψ2

(5.11)

V15 (t, x, y) = b0 −

6k 2 (Ω5 (t, x, y)), ψ2

(5.12)

V14 (t, x, y) = b0 −

with Z

2k 2 a0 ψ1 − lk[2b0 ψ2 − k 2 ]
Ω3 (t, x, y) = kx + ly + 2l

Z

k 2 a0 ψ1 − lk[b0 ψ2 − 2k 2 ]
Ω4 (t, x, y) = kx + ly + l

Z

k 2 a0 ψ1 − lk[b0 ψ2 + k 2 ]
Ω5 (t, x, y) = kx + ly + l

6

t 0

t 0

t 0

 t2 Γ(τ )dτ + ψ3 [Bt − ] , 2

 t2 Γ(τ )dτ + ψ3 [Bt − ] , 2

 t2 Γ(τ )dτ + ψ3 [Bt − ] . 2

Conclusion

We have discussed the solutions of (SPDEs) driven by Gaussian white noise. There is a unitary mapping between the Gaussian white noise space and the Poisson white noise space. This connection was given by Benth and Gjerde [2]. By the aid of this connection, we can derive some stochastic exact soliton solutionsfor our problem. In this paper, using Hermite transformation, white noise theory and F-expansion method, we study the white noise functional solutions of the Wick-type stochastic (2+1)-dimensional coupled KdV equations. This paper shows that the F-expansion method is sufficient to solve many stochastic nonlinear equations in mathematical physics. The method which we have proposed in this paper is standard, direct and computerized method, which allows us to do complicated and tedious algebraic calculation. It is shown that the algorithm can be also applied to other nonlinear (PDEs) in mathematical physics such as modified Hirota-Satsuma coupled KdV, KdVBurgers, modified KdV Burgers, Sawada-Kotera, Zhiber-Shabat equations and BenjaminBona-Mahony equations. Since the equation (1.2) has other solutions if select other values of P, Q and R (see Appendices A, B, C), and there are many other of exact solutions for wick-type stochastic (2+1)-dimensional coupled KdV equations.

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Appendix A. The ODE and Jacobi Elliptic Functions Relation between values of ( P , Q , R ) and corresponding F (ξ) in ODE. (F 0 )2 (ξ) = P F 4 (ξ) + QF 2 (ξ) + R, P m2 −m2 −1 1 1 − m2

m2 − 1 1 − m2

−m2 (1 − m2 )

Q −1 − m2

R 1

2m2 − 1

1 − m2

−1 − m2

m2

2 − m2

2m2 − 1 2 − m2 2 − m2

2m2 − 1

F (ξ) snξ, cdξ =

cnξ dnξ

cnξ

m2 − 1

dnξ ξ nsξ = sn1 ξ , dcξ = dn cnξ ncξ = cn1 ξ

−m2 −1

ndξ =

1

scξ =

1

sdξ =

1

dnξ snξ cnξ snξ dnξ cnξ snξ dnξ snξ cnξ

1

2 − m2

1 − m2

csξ =

1

2m2 − 1

−m2 (1 − m2 )

dsξ =

m2 4

m2 −2 2

m2 4

1 4

1−2m2 2

1 4

m2 −1 4

m2 +1 2

m2 −1 4

dnξ 1±msnξ

1−m2 4

m2 +1 2

1−m2 4

−1 4

m2 +1 2

1 4 1 4

m2 +1

−(1−m2 )2 4 (1−m2 )2 4 m2 4

ξ ncξ ± iscξ 1±cn snξ

m4 4

m2 −2 2

2 m2 −2 2

snξ , √ 1±dnξ 1−m2 ±dnξ

1 4

snξ ± icnξ, nsξ ± csξ,

dnξ √ , m snξ i 1−m2 snξ±cnξ 1±dnξ cnξ √ , snξ , 1−m2 snξ±dnξ 1±cnξ

mcnξ ± dnξ snξ cnξ±dnξ nsξ ± dsξ

Appendix B. the jacobi elliptic functions degenerate into trigonometric functions when m → 0. snξ → sin ξ, cnξ → cos ξ, dnξ → 1, scξ → tan ξ, sdξ → sin ξ, cdξ → cos ξ, nsξ → csc ξ, ncξ → sec ξ, ndξ → 1, csξ → cot ξ, dsξ → csc ξ, dcξ → sec ξ.

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Appendix C. the jacobi elliptic functions degenerate into hyperbolic functions when m → 1. snξ → tan ξ, cnξ → ξ, dnξ → ξ, scξ → sinh ξ, sdξ → sinh ξ, cdξ → 1, nsξ → coth ξ, ncξ → cosh ξ, ndξ → cosh, csξ → ξ, dsξ → ξ, dcξ → 1.

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[20] H. A. Ghany and A. Hyder. Chin. Phys. B, 23 (2014): 0605031-7. [21] H. Holden, B. Ø sendal, J. U b ø e and T. Zhang. Stochastic partial differential equations, Bihk¨auser: Basel, (1996). [22] J .H. He and X. H. Wu. Chaos Solitons and Fractals, 30 (2006):700-708. [23] J. H. He and M. A. Abdou. Chaos Solitons and Fractals, 34 (2007): 1421-1429. [24] R. S. Johnson, J. Fluid Mech.,42 (1970), 49-60. [25] V.V. Konotop, L. Vazquez. Nonlinear Random Waves, World Scientific, Singapore, (1994). [26] S. K. Liu, Z. T. Fu, S. D. Liu, Q. Zhao. Phys. Lett. A, 289 (2001): 69-74. [27] J. Liu, L. Yang, K. Yang Chaos, Solitons and Fractals, 20 (2004): 1157-1164. [28] E. J. Parkes, B. R. Duffy, P. C. Abbott Phys. Lett. A, 295 (2002): 280-286. [29] F. Shakeri and M. Dehghan. Math. Comput. Model, 48 (2008): 486-498. [30] M. Wadati. J. Phys. Soc. Jpn., 52 (1983): 2642-2648. [31] A. M. Wazwaz. Chaos Solitons Fractals, 188 (2007): 1930-1940. [32] M. Wadati and Y. Akutsu. J. Phys. Soc. Jpn., 53 (1984): 3342-3350. [33] A. M. Wazwaz. Appl. Math. Comput, 177 (2006): 755-760. [34] A. M. Wazwaz. Appl. Math. Comput, 169 (2005): 321-338. [35] X. Ma, L. Wei, and Z. Guo. J. Sound Vibration, 314 (2008): 217-227. [36] X. H. Wu and J.H. He. Chaos Solitons and Fractals, 38 (2008): 903-910. [37] Y. C. Xie . Phys. Lett. A, 310 (2003): 161-167. [38] Y. C. Xie. Solitons and Fractals, 21 (2004): 473-480. [39] Y. C. Xie. Solitons and Fractals, 20 (2004): 337-342. [40] Y. C. Xie. J. Phys. A: Math. Gen, 37 (2004): 5229-5236. [41] Y. C. Xie. Phys. Lett. A, 310 (2003): 161-167. [42] S. Zhang, T. C. Xia. Communications in Theoretical Physics, 46 (2006): 985-990. [43] S. D. Zhu. Int J Non Sci Numer Simul, 8 (2007): 461-464. 16

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[44] S. D. Zhu. Int J Non Sci Numer Simul, 8 (2007): 465-468. [45] Yubin Zhou. Mingliang Wang. Yueming Wang. Physics Letters A, 308 (2003): 31-36. [46] Sheng Zhang. Tiecheng Xia. Applied Mathematics and Computation, 189 (2007): 836843. [47] Sheng Zhang. TieCheng Xia. Applied Mathematics and Computation, 183 (2006): 11901200. [48] Sheng Zhang. Tiecheng Xia. Communications in Nonlinear Science and Numerical Simulation, 13 (2008): 1294-1301.

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Exact Solutions for Stochastic Fractional Zhiber-Shabat Equations Hossam A. Ghany1 and Ashraf Fathallah 2 1 Department Mathematics, Helwan University, Cairo, Egypt [email protected] 2 Department of Mathematics, Misr International University, Cairo 11341, Egypt. [email protected]

Abstract This paper is devoted to give exact solutions of the variable coefficient fractional Zhiber -Shabat equation with space-time-fractional derivatives. Moreover, by using the Hermite transform and the homogeneous balance principle, the white noise functional solutions for the Wick-type stochastic fractional Zhiber-Shabat equation are explicitly shown. Detailed computations and implemented examples are explicitly provided. Keywords: Fractional Zhiber-Shabat equations; White noise; Stochastic; Hermite transform. MSC: 60H30; 60H15; 35R60

1

Introduction

The main task of this paper is to explore exact solutions for the following fractional Zhiber-Shabat equation with variable coefficients: ∂xα1 ∂tα2 u + p(t)eu + q(t)e−u + r(t)e−2u = 0

(1.1)

where ∂xα1 , ∂tα2 (0 < α1 , α2 < 0) are the modified Riemann-Liouville fractional derivatives defined by Jumarie [6] and q(t), p(t) and r(t) are bounded measurable or integrable functions on R+ . Random waves is an important subject of random fractional partial differential equations. Recently, both mathematicians and physicists have devoted considerable effort to the study of explicit solutions to nonlinear integer-order differential equation. In the past decades, an important progress has been made in the research of the exact solutions of nonlinear partial differential equations (PDEs). To seek various exact solutions of multifarious physical models described by nonlinear

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PDEs, various methods have been proposed. There are many authers studied this subject. Wadati first introduced and studied the stochastic KdV equation and gave the diffusion of soliton of the KdV equation under Gaussian noise in ([10]-[12]). Xie firstly researched Wick-type stochastic KdV equation on white noise space and showed the auto-Bachlund transformation and the exact white noise functional solutions in [14], furthermore, Chen and Xie ([1]-[3]) and Xie ([15]-[17]) researched some Wick-type stochastic wave equations using white noise analysis method. Recently, Uˇgurlu and Kaya[9] gave the tanh function method, Wazzan [13] showed the modified tanh-coth method, these methods have been applied to derive nonlinear transformations and exact solutions of nonlinear PDEs in mathematical physics. If Eqn.(1.1) is considered in random environment, we can get random fractional Zhiber-Shabat equation with space-fractional derivatives. In order to give the exact solutions of random fractional Zhiber-Shabat equation with space-fractional derivatives, we only consider this problem in white noise environment. Wick-type stochastic generalized fractional Zhiber-Shabat equations with space-fractional derivatives is the perturbation of Eqn.(1.1) by random force W (t)  R (U, Uxt ) , which represented by: ∂xα1 ∂tα2 U + P (t)  eU + Q(t)  e(−U ) + R(t)e(−2U ) = W (t)  R (U, Uxα1 tα2 )

(1.2)

.

where W (t) is Gaussian white noise, i.e., W (t) = B (t) and B(t) is a Brownian motion, R(u, uxα1 tα2 ) ∂ α1 +α2 u = −β1 ∂xα1 ∂tα2 u − β2 eu − β3 e−u − β4 e−2u is a functional of u, ∂xα1 ∂tα2 u := ∂x α1 ∂xα2 = uxα1 xα2  for some constants β1 , ..., β4 and R is the Wick version of the functional R. ”  ” is the Wick product on the Kondratiev distribution space (S)−1 and P (t), Q(t) and R(t) are white noise functionals. Eqn.(1.2) can be seen as the perturbation of the coefficients p(t), q(t) and r(t) of Eqn.(1.1) by white noise functionals.

This paper is devoted to give white noise functional solution for Wick-type stochastic generalized fractional Zhiber-Shabat equations with space-fractional derivatives. Moreover, the Hermite transform and the homogenous balance principle are employed to find the exact solution for stochastic fractional Zhiber-Shabat equation with variable coefficient. Finally, implemented examples are explicitly shown.

2

Preliminaries

There are different definitions for fractional derivatives, for more details (see [5, 6]). In our paper we use the modified Riemann-Liouville derivative defined by Jumarie [6]:  Rx 1 −α−1 [f (y) − f (0)]dy,   α < 0,  Γ(1 − α) 0 (x − y)   1 d Rx −α [f (y) − f (0)]dy, 0 < α < 1, Dxα f (x) = (2.1) 0 (x − y)   Γ(1 − α) dx    (α−n) (n) f (x) , n ≤ α < n + 1, n ∈ N which has merits over the original one, for example, the α-order derivative of a constant is zero. Some properties of the modified Riemann-Liouville derivative were summarized in [5] , three useful

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formulas of them are  Γ(1 + β)   Dxα xβ = xβ−α , β > 0,   Γ(1 + β − α)  Dxα (u(x)v(x)) = u(x)Dxα v(x) + v(x)Dxα u(x),  α    df α du  α Dx [f (u(x)] = Dx u(x) = Duα f (u). du dx

(2.2)

Now, we outline the main idea of the modified fractional sub-equation method. Many authors considered nonlinear FPDE, say, in two variables F (u, ux , ut , Dxα u, Dtα u, ...) = 0,

0 0 and n > 0 such that u(x, t, z), uxt (x, t, z) are uniformally bounded for all (t, x, z) ∈ S × Km (n) , continuous with respect to (t, x) ∈ S for all z ∈ Km (n) and analytic with respect to z ∈ Km (n) for all (t, x) ∈ S . Using Theorem 2.1 of Xie [17], there exists a stochastic process U (t, x) such that the Hermite transformation of U (t, x) is u(t, x, z) for all S × Km (n) , and U (t, x) is the solution of (1.2). This implies that U (t, x) is the inverse Hermite transformation of u(t, x, z) . Hence, for Λ1 Λ2 Λ3 6= 0 the white noise functional solutions of Eqn.(1.2) as follows: s μxα1 νtα2  −1 Λ3 (t) 3{exp ( Γ(1+α1 ) + Γ(1+α2 ) ) − 1}  p U1 (x, t) = ln {±i + − Λ2 (t) Λ2 (t) ± i Λ2 (t)Λ3 (t) μxα1 νtα2 2μν (4.1) {exp ( + ) − 1}−2 } Λ2 (t) Γ(1 + α1 ) Γ(1 + α2 ) s

U2 (x, t) = ln {±i

U3 (x, t) = ln {−

U4 (x, t) = ln {−

Λ3 (t) μxα1 2μν νtα2 − {exp ( + ) − 1}−2 } Λ2 (t) Λ2 (t) Γ(1 + α1 ) Γ(1 + α2 )

(4.2)

νtα2 νtα2 μxα1 μxα1 μν {coth ( + ) ± csch ( + )}−2 } (4.3) Γ(1 + α1 ) Γ(1 + α2 ) Γ(1 + α1 ) Γ(1 + α2 ) 2Λ2 (t)

μν μxα1 μxα1 νtα2 νtα2 {tanh ( + ) ± isech ( + )}−2 }(4.4) 2Λ2 (t) Γ(1 + α1 ) Γ(1 + α2 ) Γ(1 + α1 ) Γ(1 + α2 )

U5 (x, t) = ln {−

U6 (x, t) = ln {−

μxα1 μxα1 νtα2 νtα2 μν {coth ( + ) ± csch ( + )}2 } (4.5) Γ(1 + α1 ) Γ(1 + α2 ) Γ(1 + α1 ) Γ(1 + α2 ) 2Λ2 (t)

μν μxα1 μxα1 νtα2 νtα2 {tanh ( + ) ± isech ( + )}2 } (4.6) 2Λ2 (t) Γ(1 + α1 ) Γ(1 + α2 ) Γ(1 + α1 ) Γ(1 + α2 )

We observe that for different form of Λ2 (t) and Λ3 (t) , we can get different solutions of (1.2) from (3.1)-(3.6).

Example and Concluding Remarks

5 .

noise, where Bt is Brown motion. We have the Hermite transLet Bt be the Gaussian Rwhite P∞ . t e form B (t, z) = k=1 zk 0 ηk (s)ds . Science exp (Bt ) = exp(Bt − t2 /2) , we have tanh (Bt ) = tanh(Bt − t2 /2) , coth (Bt ) = cot(Bt − t2 /2) , sech (Bt ) = sech(Bt − t2 /2) and csch (Bt ) = . csch(Bt − t2 /2) . Suppose Λ3 (t) = αΛ2 (t) and Λ2 (t) = λ2 (t) + βBt , where α, β are arbitrary

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constants and λ2 (t) is integrable or bounded measurable function on R+ . The white noise functional solutions of (1.2) are as follows: If Λ1 (t)Λ2 (t)Λ3 (t) 6= 0 √

α1

μx 3{exp( Γ(1+α + 1)

ν(t−βBt +0.5βt2 )α2 ) Γ(1+α2 )

√

− 1}−1

− Λ2 (t)(1 + ±i α) μxα1 ν(t − βBt + 0.5βt2 )α2 2μν {exp( + ) − 1}−2 } Γ(1 + α1 ) Γ(1 + α2 ) Λ2 (t)

U7 (x, t) = ln{±i α +

(5.1)

√ μxα1 ν(t − βBt + 0.5βt2 )α2 2μν U8 (x, t) = ln{±i α − {exp( + ) − 1}−2 } Λ2 (t) Γ(1 + α1 ) Γ(1 + α2 )

U9 (x, t) = ln{−

U10 (x, t) = ln{−

μxα1 ν(t − βBt + 0.5βt2 )α2 μν {coth( + )± Γ(1 + α1 ) Γ(1 + α2 ) 2Λ2 (t) ν(t − βBt + 0.5βt2 )α2 −2 μxα1 + )} } csch( Γ(1 + α1 ) Γ(1 + α2 ) μxα1 ν(t − βBt + 0.5βt2 )α2 μν {tanh( + )±i Γ(1 + α1 ) Γ(1 + α2 ) 2Λ2 (t) ν(t − βBt + 0.5βt2 )α2 −2 μxα1 + )} } sech( Γ(1 + α1 ) Γ(1 + α2 )

U11 (x, t) = ln{−

U12 (x, t) = ln{−

μxα1 ν(t − βBt + 0.5βt2 )α2 μν {coth( + )± Γ(1 + α1 ) Γ(1 + α2 ) 2Λ2 (t) ν(t − βBt + 0.5βt2 )α2 2 μxα1 + )} } csch( Γ(1 + α1 ) Γ(1 + α2 )

μxα1 ν(t − βBt + 0.5βt2 )α2 μν {tanh( + )±i Γ(1 + α1 ) Γ(1 + α2 ) 2Λ2 (t) ν(t − βBt + 0.5βt2 )α2 2 μxα1 + )} } sech( Γ(1 + α1 ) Γ(1 + α2 )

(5.2)

(5.3)

(5.4)

(5.5)

(5.6)

Finally, we remark that for α1 = α2 = 0, p(t) =1 and q(t) = r(t) = 0 , Eqn.(1.1) reduces to the Liouville equation. For α1 = α2 = 0, r(t) =0 and q(t) = p(t) = 1 , Eqn.(1.1) reduces to the Sinh-Gordon equation. For α1 = α2 = 0, p(t) = r(t) =1 and q(t) = 0 , Eqn.(1.1) reduces to the the well known Dodd-Bullough-Mikhailov equation. Moreover, for α1 = α2 = 0, p(t) =0 , q(t) = −1 and r(t) = 1 , gives Tzitzeica-Dodd-Bullough equation. Hence, our results in this work can be considered as a continuation of our results in our previous papers [4,5], this work gives directly exact solutions for wick-type stochastic form to each one of the above equations. Also, we remark that, since the Riccati equation has other solution if select other values of c1 , c2 and c3 , there are many other exact solutions of variable coefficient and wick-type stochastic Zhiber-Shabat equations

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References [1] Chen. B, Xie. Y.C., 2005, Exact solutions for wick-type stochastic coupled KadomtsevPetviashili equations, J. Phys. A, 38, 815-22. [2] Chen. B, Xie. Y.C., 2005, Exact solutions for generalized stochastic Wick-type KdV-mKdV equations, Chaos Solitons Fractals, 23,281-7. [3] Chen. B, Xie. Y.C., 2007, Periodic-like solutions of variable coefficient and Wicktype stochastic NLS equations, J. Comput. Appl. Math., 203,249-63. [4] Ghany H. A., 2011, Exact Solutions for Stochastic Generalized Hirota-Satsuma Coupled KdV Equations, Chin. J. Phys., 49,926-940. [5] Ghany H. A., Mohammed S. A., 2012, White Noise Functional Solutions for the Wick-type Stochastic Fractional KdV-Burgers-Kuramoto Equations, Chin. J. Phys., 50,619627. [6] Jumarie G.,2006, Modifed Riemann-Liouville derivative and fractional Taylor series of non differentiable functions further results, Comp. Math. Appl., 51, 1367-1376. [7] Malfliet W., 1992, Solitary wave solutions of nonlinear wave equations, Am. J. Phys, 60(7),6504. [8] Podlubny. I., 1999, Fractional Differential Equations, Academic Press, San Diego. [9] Uˇgurlu. Y., Kaya. D., 2007, Analytic method for solitary solutions of some partial differential equations, Phys. Lett. A, 370,251-9. [10] Wadati. M., 1983, Stochastic Korteweg de Vries equation, J. Phys. Soc. Jpn, 52,2642-8. [11] Wadati. M., 1990, Deformation of solitons in random media, J. Phys. Soc. Jpn, 59,4201-3. [12] Wadati. M., Akutsu. Y., 1984, Stochastic Korteweg de Vries equation, J. Phys. Soc. Jpn, 53,3342-50. [13] Wazzan. L., A modified tanh-coth method for solving the KdV and the KdV-Burgers equations, to appear on Commu. Nonlinear Sci. Numer. Simul. [14] Xie. Y. C., 2003, Exact solutions for stochastic KdV equations, Phys. Lett. A, 310,161-7. [15] Xie. Y. C., 2004, Positonic solutions for Wick-type stochastic KdV equations, Chaos Solitons Fractals, 20,337-42.

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[16] Xie. Y. C., 2004, An auto-Backlund transformation and exact solutions for wick type stochastic generalized KdV equations, J. Phys. A: Math. Gen., 37,5229-36. [17] Xie. Y. C., 2004, Exact solutions of the Wick-type stochastic Kadomtsev-Peviashvili equations, Chaos Solitons Fractals, 21,473-80. [18] Zhang. J.L., Ren. D.F. and Wang. M.L., 2003, The periodic wave solutions for the generalized Nizhnik-Novikov-Veselov equation, Chin. Phys, 12(8), 825-830. [19] Zhang. J.L, Zong. Q.A., Liu. D. and Gao. Q., 2010, A generalized exp-function method for fractional Riccati differential equations, Communication Fractional Calculus, 1, 48-56.

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Invariance, solutions, periodicity and asymptotic behavior of a class of fourth order difference equations Mensah Folly-Gbetoula

∗

School of Mathematics, University of the Witwatersrand, 2050, Johannesburg, South Africa. Abstract We construct Lie symmetry generators of some fourth order difference equations. We use these generators to derive similarity variables that make it possible to obtain exact solutions. In some cases, we study periodicity and asymptotic behavior of the solutions.

2010 Mathematics Subject Classification: 39A11, 39A05. Key words: Difference equation; symmetry; reduction; group invariant solutions

1

Introduction

Several years back, Sophus Lie studied the invariance property of equations under a group of transformations. The approach used was later known as Lie symmetry method. This method has been used to solve differential equations, and recently it has been applied to difference equations. Although Maeda studied difference equations via Lie symmetry analysis in twentieth century [9, 10], it is Hydon who really rekindled the interest for solving difference equations via symmetry. For Hydon’s work, refer to [8]. Most often, difference equations arise as a result of discretizing differential equations, especially in phenomena that depend on time. There are many ways in which a differential equation can be discretized (see [4]). Difference equations have numerous applications. For example, biological systems, population dynamics, economics, physics (see [1, 2]). Although difference equations appear simple, finding their solutions can be incredibly difficult. The symmetry approach to finding solutions of difference equations is recent and the reader can refer to [8] and some recent articles [5–7, 11, 12] for further knowledge on this method. In this paper, we consider the system of difference equations xn xn+1 (1) xn+4 = xn+3 (an + bn xn xn+1 ) where (an )n∈N0 and (bn )n∈N0 are non-zero sequences of real numbers. For equation (1), we derive all Lie point symmetries and give formulas for solutions in closed form. We also discuss periodicity and asymptotic behavior of solutions in some cases. ∗ [email protected]

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1.1

Preliminaries

In this section, we give a background on symmetry methods for difference equations. Our definitions and notation come from [3, 8, 13]. Consider the difference equations xn+4 =Ω(xn , xn+1 , xn+3 ),

(2)

where n denotes the independent variable; xn the dependent variable. In this case un+i denotes the ‘i-th shift’ of un . Consider the group of transformations (n, xn ) 7→ (n, x ˜n = xn + εQ1 (n, xn ) + O(ε2 )),

(3)

where Q is the characteristic of the group of point transformations. Let X = Q(n, xn )

∂ ∂xn

(4)

be the corresponding infinitesimal generator. The group of transformations (3) is a symmetry group if and only if Q(n + 4, Ω) − X (Ω) = 0,

(5)

whenever (2) holds. Here, X = Q(n, xn )

∂ ∂ ∂ + Q(n, xn+1 ) + Q(n + 3, xn+3 ) ∂xn ∂xn+1 ∂xn+3

denotes the prolongation of X to all shifts of xn appearing in the right hand sides of equations in (2). Equation (5), known as the linearized symmetry condition, can be solved for Q by applying the appropriate differential operators. The characteristic, together with the canonical coordinate Z dxn s= , (6) Q(n, xn ) are necessary in the reduction of order of (2). The following definition can be used to check if a given function is invariant under a given group of transformations. Definition 1 [13] Let G be a connected group of transformations acting on a manifold M . A smooth real-valued function ζ : M → R is an invariant function for G if and only if X(ζ) = 0

for all

x ∈ M.

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2

Main results

2.1

Symmetry and difference invariant

To obtain the the criterion which gives the Lie point symmetries of (1), we force (5) on xn+4 =

xn xn+1 . xn+3 (an + bn xn xn+1 )

(7)

This leads to xn xn+1 (an + bn xn xn+1 )Q(n + 3, xn+3 ) xn+3 2 (an + bn xn xn+1 )2 an [xn Q(n + 1, xn+1 ) + xn+1 Q(n, xn )] = 0. − xn+3 (an + bn xn xn+1 )2

Q(n + 4, xn+4 ) +

(8)

The methodology of solving these functional equations is given as follows: • Firstly, apply the differential operator This leads (after simplification) to

∂ ∂xn

+

xn+1 ∂ xn ∂xn+1

xn+1 Q0 (n + 1, xn+1 ) − xn+1 Q0 (n, xn ) − Q(n + 1, xn+1 ) +

on equation (8). an Q(n, xn ) = 0. xn

• Secondly, differentiate with respect to xn , separate by powers of xn+1 and solve the resulting system of over determining equations for Q. This gives Q(n, xn ) = α(n)xn + β(n) for some functions α and β of n. • Lastly, substitute the latter in (8) to eliminate any dependency among the arbitrary functions that appear in Q. This leads to the constraints α(n) + α(n + 1) = 0 and

β(n) = 0.

(9)

We have omitted the details in the computation. The constraints in (9) are readily solved (α(n) = (−1)n ) and we have Q = (−1)n xn .

(10)

Consequently, Equation (1) admits a one dimensional Lie algebra: X = (−1)n xn

∂ . ∂xn

The canonical coordinate is given by Z dxn sn = = (−1)n ln |xn | (−1)n xn

(11)

(12)

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and the difference invariant which is inspired by the form of the final constraints (9) is given by un = (−1)n sn + (−1)n+1 sn+1 .

(13)

It is not difficult to verify, using Definition 1 together with (11), that (13) is indeed invariant under the group of transformations of (1). For simplicity, we prefer using the compatible variable |un | = exp(−un )

(14)

which is also invariant. This gives a convenient choice of the change variables which does not require lucky guesses. With this variable un , it follows that un+3 = an un + bn

(15)

whose solution is given by u3n+j = uj

n−1 Y

! a3k1 +j

+

k1 =0

n−1 X

b3l+j

l=0

!

n−1 Y

a3k2 +j

,

j = 0, 1, 2.

(16)

k2 =l+1

To obtain the solutions of (1), we go up the hierarchy created by the changes of variables. By evaluating (13) as a telescoping series, we have (−1)n sn = (−1)n−1

n−1 X

(−1)k1 uk1 + (−1)n s0

(= ln |xn | from

(12)), (17)

k1 =0

i.e. ( n−1

xn = exp (−1)

n−1 X

) k1

n

(−1) uk1 + (−1) s0

,

(18)

k1 =0

= exp{

n−1 X

(−1)n+k1 ln uk1 + ln x0 },

(19)

k1 =0

where all the uk1 ’s are obtained using (16). Note. Equation (19) gives the closed form solution of (1) in a unified manner. Looking at the form of ul in (16), we rephrase (19) as follows: x6n+j

(6n+j−1 ) X 6n+j+k1 = exp (−1) ln uk1 + ln x0 ,

(20)

k1 =0

=xj

n−1 Y i=0

2 Y u6i+j+2r u r=0 6i+j+2r+1

! ,

(21)

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j = 0, 1, . . . , 5. More clearly, x6n =x0

n−1 Y i=0

x6n+1 =x1

n−1 Y i=0

x6n+2 =x2

n−1 Y i=0

x6n+3 =x3

n−1 Y i=0

x6n+4 =x4

n−1 Y i=0

x6n+5 =x5

n−1 Y i=0

u3(2i) u3(2i)+2 u3(2i+1)+1 , u3(2i)+1 u3(2i+1) u3(2i+1)+2

(22a)

u3(2i)+1 u3(2i+1) u3(2i+1)+2 , u3(2i)+2 u3(2i+1)+1 u3(2i+2)

(22b)

u3(2i)+2 u3(2i+1)+1 u3(2i+2) , u3(2i+1) u3(2i+1)+2 u3(2i+2)+1

(22c)

u3(2i+1) u3(2i+1)+2 u3(2i+2)+1 , u3(2i+1)+1 u3(2i+2) u3(2i+2)+2

(22d)

u3(2i+1)+1 u3(2i+2) u3(2i+2)+2 , u3(2i+1)+2 u3(2i+2)+1 u3(2i+3)

(22e)

u3(2i+1)+2 u3(2i+2)+1 u3(2i+3) . u3(2i+2) u3(2i+2)+2 u3(2i+3)+1

(22f)

We then substitute the expressions given in (16) in (22) to get

x6n =x0

i=0

2i Q

u1

2i−1 Q

2i−1 P

l1 =0

j=0 l2 =j+1

u0

n−1 Y

a3l1 +

2i Q

2i−1 P

2i−1 Q

l=0

j=0

l2 =j+1

a3l+1 +

2i Q

b3j+2

2i Q

j=0

u0

2i Q

l2 =j+1

b3j+1

a3l2 +1

u0

2i−1 P

2i−1 Q

2i Q

l=0

j=0

l2 =j+1

l=0

a3l+2 +

2i+1 Q

b3j

2i Q

a3l +

a3l+1 +

2i P

2i Q

a3l2 j=0 l2 =j+1

2i P j=0

b3j

b3j+1

2i Q

a3l2 +1 l2 =j+1

2i Q

a3l2 +2 l2 =j+1

2i+1 P

a3l +

a3l2 +2 u1

b3j+2

b3j+2

j=0 2i+1 Q

akl2 j=0 l2 =j+1

l=0

2i−1 Q

l=0

2i Q

(23a)

j=0

2i P

b3j

,

l1 =0

a3l+2 +

2i P

l2 =j+1

a3l1 +1 +

u2

a3l +

akl2 +2 l2 =j+1

akl2 +2

2i−1 Q

l=0

l=0

2i−1 Q

b3j+2

2i Q

2i−1 P

u0

2i−1 P

a3l+2 +

l=0

2i−1 Q

u1

2i−1 Q

u2

a3l2 +1

akl2 +1 l2 =j+1

j=0

i=0

u2

2i P

a3l+2 +

b3j+1

b3j+1

j=0

l=0

x6n+1 =x1

2i P

a3l+1 +

n−1 Y

a3l2

2i−1 Q

u1

l=0

u2

2i−1 Q

b3j

,

(23b)

a3l2

j=0 l2 =j+1

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n−1 Y

x6n+2 =x2

2i−1 Q

2i−1 P

2i−1 Q

2i Q

l1 =0

j=0

l2 =j+1

l=0

u2

a3l1 +2 +

i=0

u0

2i Q

2i+1 Q l=0

2i+1 Q

u1

x6n+3 =x3

a3l2 j=0 l2 =j+1 2i+1 Q

j=0

l2 =j+1

i=0

b3j+1

2i Q

u0 2i Q

2i+1 P

l=0

j=0

2i+1 Q

n−1 Y

x6n+4 =x4

i=0

u1

2i+1 Q

j=0

l2 =j+1

a3l+2 + 2i+1 P

l=0

j=0

a3l+2 +

b3j+2

2i+1 Q

u0

2i+1 P

a3l +

l=0

2i Q

a3l2 +2 l2 =j+1

j=0

l=0

a3l2 +1

2i P

2i+1 Q

b3j

a3l2 j=0 l2 =j+1

(23d)

,

2i P

2i Q

b3j+1

2i+1 Q

a3l2 +1

u0

l2 =j+1

b3j+2

2i Q

2i+1 P

a3l +

l=0

a3l2 j=0 l2 =j+1

2i+1 Q

2i+1 P

l=0

j=0

a3l2 +2 u1

l2 =j+1

2i+1 Q

b3j

a3l+1 +

2i+1 Q

b3j+1

akl2 +1 l2 =j+1

a3l2 +2 l2 =j+1

2i+2 Q

b3j

(23e)

,

a3l2

j=0 l2 =j+1

u2

2i Q

a3l1 +2 +

2i P

b3j+2

j=0

l1 =0 2i+1 Q

i=0

u0

2i+2 Q l=0

2i+1 Q

b3j

a3l2

2i+1 P

l=0

j=0

a3l+1 +

2i+1 Q

2i+1 P

l=0

j=0

u2

2i+1 Q

b3j+1

a3l2 +1 l2 =j+1 2i+1 Q

b3j+2

a3l+2 +

a3l2 +2 l2 =j+1

2i+2 Q

a3l2 j=0 l2 =j+1

b3j

2i+2 Q

2i+2 P

2i+2 Q

l=0

j=0

l2 =j+1

a3l+1 +

2i+1 Q

a3l2 +2 u1

j=0 l2 =j+1

2i+2 P

a3l +

2i Q

l2 =j+1

2i+1 P

a3l +

l=0

u1

2i Q

2i Q

2i+1 Q

2i+2 P

a3l +

u0

u2

b3j+2

l=0

x6n+5 =x5

2i P

j=0

a3l+2 +

n−1 Y

a3l2

a3l2 +2

j=0

2i+2 Q

(23c)

l2 =j+1

b3j+2

2i+1 Q

u0

b3j+1

2i+1 P

l=0

u2

2i Q

a3l2 +1 l2 =j+1

a3l1 +1 +

2i Q

j=0

2i+1 Q

l1 =0

u2

2i Q

a3l2 +2 l2 =j+1

b3j+2

b3j+1

a3l+2 +

l=0

2i P

a3l+2 +

,

b3j

j=0

a3l+1 +

2i Q

2i P

2i P

a3l+1 +

2i+1 Q

u2

2i Q

a3l2 +1 l2 =j+1

j=0

l=0

j=0 l2 =j+1

l=0

u1

u2

2i Q

a3l2 +1

a3l1 +

l1 =0

u1

a3l2

b3j+1

2i+1 Q

2i+1 P

l=0

2i Q

2i P

a3l+1 +

b3j

a3l+1 +

n−1 Y

b3j

j=0 l2 =j+1

2i+1 P

a3l +

2i P

a3l +

l=0

u0

a3l2 +2 u1

b3j+2

b3j+1

.

(23f)

a3l2 +1

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We rewrite (23) in terms of initial conditions only as follows:

x6n =x0

i=0

2i Q

2i−1 Q

2i−1 P

l1 =0

j=0 l2 =j+1

a3l1 + x0 x1

n−1 Y

j=0

l2 =j+1

l=0

2i P 2i P

b3j+2

2i Q

2i−1 Q

2i Q

j=0

l2 =j+1

l=0

b3j+1

2i Q

l=0

j=0

l2 =j+1

l=0

b3j+2

b3j+2

2i+1 P

a3l + x0 x1

2i−1 Q

a n−1 Y l1 =0 3l1 +2 2i Q

i=0

2i+1 P

2i Q

n−1 Y

2i Q

2i−1 Q

2i Q

j=0

l2 =j+1

l=0

2i Q

2i Q

b3j+2

2i+1 Q

j=0

l2 =j+1

b3j+1

a3l1 + x0 x1

2i+1 P

l=0

j=0

2i P

2i Q

b3j+2

2i Q

a3l2 +2 l2 =j+1

j=0

(24c) 2i Q

a3l2

b3j+1

a3l+2 + x2 x3 2i+1 Q

a3l2 +1

2i P

b3j+2

2i+1 P

a3l + x0 x1

l=0

2i Q

a3l2 +2 l2 =j+1

j=0

l=0

l2 =j+1

2i+1 Q

b3j

a3l2 j=0 l2 =j+1

2i+1 Q

a3l2 +1 l2 =j+1

j=0

l2 =j+1

a n−1 Y l1 =0 3l1 +1

2i P

a3l+2 + x2 x3

2i Q

a3l2 +1 l2 =j+1

j=0

l=0

j=0 l2 =j+1 2i 2i P Q

2i+1 Q

2i Q

b3j+1

,

b3j

2i+1 P

a3l+2 + x2 x3

b3j+2

+ x1 x2

a3l+2 + x2 x3

2i P

b3j+1

2i P

2i+1 P

l=0

j=0

a3l+2 + x2 x3

2i Q

2i+1 Q

2i Q

a3l2 +2

l2 =j+1

2i+1 P

a3l + x0 x1

a3l2 +1

l=0

l2 =j+1

b3j+2

j=0

2i+1 Q

(24d)

,

a3l2 +2

j=0

l=0

2i+2 Q

a3l2

2i P

a3l+1 + x1 x2

b3j+1

a3l+1 + x1 x2

l=0

a3l2 +1 l2 =j+1

j=0

a3l2 +1

j=0

2i+1 Q

2i Q

b3j

2i+1 P

l=0

i=0

2i P

a3l2 +2

a3l2 j=0 l2 =j+1

a3l+1 + x1 x2

2i+1 Q

2i Q

2i+1 Q

l1 =0

i=0

b3j+1

b3j

a3l+1 + x1 x2

l=0

2i Q

a3l2 j=0 l2 =j+1

2i P

a3l+1 + x1 x2

b3j

(24b)

j=0 l2 =j+1

l=0

2i P

,

2i−1 P

+ x2 x3

a3l + x0 x1

a3l + x0 x1

b3j a3l2 j=0 l2 =j+1

a3l + x0 x1

2i+1 Q

akl2 j=0 l2 =j+1

2i Q

2i+1 Q

l=0

2i+1 Q

a3l2 +2

a3l2 +2 l2 =j+1

j=0

2i+1 Q

b3j

a3l2 j=0 l2 =j+1

2i+1 Q

2i+1 P

l=0

j=0

a3l+1 + x1 x2

2i+1 Q

b3j+1

akl2 +1 l2 =j+1

2i+1 Q

b3j+2

a3l2 +2 l2 =j+1

2i+2 P

a3l + x0 x1

l=0

a3l2 +1

2i−1 Q

l=0

2i Q

(24a)

2i−1 P

2i+1 Q

b3j

,

2i−1 P

a3l+2 + x2 x3

l=0

2i P

l2 =j+1

+ x1 x2

2i P

a3l + x0 x1

akl2 +2 l2 =j+1

akl2 +2

2i−1 Q

a3l+2 + x2 x3

2i−1 Q

b3j+2

2i Q

akl2 +1 l2 =j+1

j=0

a n−1 Y l1 =0 3l1 +1

a3l2 +1

b3j+1

b3j+1

j=0

i=0

x6n+4 =x4

j=0

l=0

2i−1 Q

x6n+3 =x3

l=0 2i Q

l=0

x6n+2 =x2

2i−1 P

a3l+2 + x2 x3

2i−1 Q

a3l+2 + x2 x3

2i Q

2i−1 Q

2i−1 P

a3l+1 + x1 x2

l=0

x6n+1 =x1

a3l2

2i−1 Q

a3l+1 + x1 x2

2i Q

2i−1 Q

b3j

2i+2 Q

b3j

,

(24e)

a3l2

j=0 l2 =j+1

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2i Q

x6n+5 =x5

a n−1 Y l1 =0 3l1 +2

+ x2 x3

b3j+2

j=0

2i+1 Q

i=0

2i P

2i+1 Q

b3j

a3l2

j=0 l2 =j+1

l=0 2i+2 Q

2i+2 P

a3l + x0 x1

l=0

2i+1 P

l=0

j=0

a3l+1 + x1 x2

2i+1 Q

b3j+1

2i+1 Q

2i+1 P

l=0

j=0

a3l+2 + x2 x3

a3l2 +1 l2 =j+1 2i+1 Q

b3j+2

a3l2 +2 l2 =j+1

2i+2 Q

a3l2 j=0 l2 =j+1

b3j

2i+2 Q

2i+2 P

2i+2 Q

l=0

j=0

l2 =j+1

a3l+1 + x1 x2

2i+1 Q

a3l2 +2

l2 =j+1

2i+1 P

a3l + x0 x1

2i Q

b3j+1

(24f)

,

a3l2 +1

where x4 = x0 x1 /(x3 (a0 + b0 x0 x1 )) and x5 = x2 x3 (a0 + b0 x0 x1 )/(x0 (a1 + b1 x1 x2 )). In the following subsections, we study some special cases.

2.2

The case where (an ) and (bn ) are 3 periodic sequences

Let an = {a0 , a1 , a2 , a0 , a1 , a2 , . . . } and bn = {b0 , b1 , b2 , b0 , b1 , b2 , . . . }. Equations in (23) reduce to

x6n =x0

n−1 Y i=0

x6n+1 =x1

n−1 Y i=0

x6n+2 =x2

n−1 Y i=0

2i−1 P

2i−1 P

2i P

j=0

j=0

j=0

aj2 a2i+1 + b1 x1 x2 1

aj0 a2i 2 + b2 x2 x3

a2i 0 + b0 x0 x1

2i−1 P

2i P

+ b0 x0 x1 aj1 a2i+1 0

a2i 1 + b1 x1 x2

j=0

j=0

2i−1 P

2i P

j=0

j=0

aj1 a2i+1 + b0 x0 x1 0

a2i 1 + b1 x1 x2

2i−1 P

2i P

j=0

j=0

+ b1 x1 x2 aj2 a2i+1 1

a2i 2 + b2 x2 x3

x6n+3 =x3

i=0

x6n+4 =x4

n−1 Y i=0

2i P

+ b2 x2 x3 aj0 a2i+1 2

a2i+1 + b0 x0 x1 0

2i P

u2 a2i+1 + b2 2

a2i+1 + b1 x1 x2 1

2i P

x6n+5 =x5

i=0

a2i+1 + b2 x2 x3 2

,

aj0

j=0

2i P

2i P

2i+1 P

aj1 a2i+2 + b0 x0 x1 0

aj1

j=0 2i+1 P

aj1

a2i+2 + b1 x1 x2 1

j=0

2i+1 P

aj2

j=0

j=0

j=0

2i P

2i+1 P

2i+1 P

j=0

j=0

2i P

2i P

aj0 a2i+2 + b2 x2 x3 2

aj2

2i+1 P

2i+2 P

j=0

j=0

aj2 a2i+2 + b1 x1 x2 1

aj1 a2i+3 + b0 x0 x1 0

2i+2 P

j=0

j=0

j=0

aj1 a2i+3 + b0 x0 x1 0

aj0

2i+1 P

2i+1 P

2i+2 P

j=0

j=0

j=0

aj0 a2i+2 + b2 x2 x3 2

a2i+2 + b0 x0 x1 0

,

aj0

2i+1 P

+ b1 x1 x2 aj2 a2i+2 1

,

2i+1 P

aj0 a2i+2 + b2 x2 x3 2

aj1 a2i+2 + b0 x0 x1 0

,

2i+1 P

aj2 a2i+2 + b1 x1 x2 1

aj2

aj1 a2i+2 + b0 x0 x1 0

aj0

j=0

j=0

j=0

a2i+1 + b2 x2 x3 2

aj2

2i+1 P

j=0

aj0

j=0

a2i+1 + b1 x1 x2 1

2i P

aj1 a2i+2 + b0 x0 x1 0

j=0

a2i+1 + b0 x0 x1 0

, aj2

j=0

j=0

aj2 a2i+1 + b1 x1 x2 1

j=0

n−1 Y

aj0 a2i+1 + b2 x2 x3 2

2i P

a2i 2 + b2 x2 x3

2i P j=0

2i−1 P

j=0

n−1 Y

aj0 a2i+1 + b2 x2 x3 2

aj1

aj2 a2i+3 + b1 x1 x2 1

.

aj1

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2.3

The case where (an ) and (bn ) are real constants

Let an = a and bn = b. Equations in (23) give rise to

x6n =x0

n−1 Y

2i P

j=0

j=0

j=0

2i P

j=0

j=0

aj a2i+1 + bx0 x1

j=0

j=0

aj a2i+1 + bx0 x1

2i P

x6n+3 =x3

j=0

j=0

x6n+4 =x4

x6n+5 =x5

n−1 Y i=0

2.3.1

aj

j=0

j=0

2i P

2i+1 P

2i P

aj a2i+1 + bx2 x3

a2i+1 + b0 x0 x1

aj a2i+1 + b2 x2 x3

j=0 2i P

a2i+1 + bx1 x2

2i P

a2i+1 + bx2 x3

aj

aj a2i+2 + bx1 x2

aj

j=0

j=0

2i+1 P

2i+1 P

aj a2i+2 + bx2 x3

j=0

j=0

j=0

j=0

aj

aj a2i+2 + bx2 x3

2i+1 P

aj a2i+2 + bx1 x2

2i+2 P

aj a2i+3 + bx0 x1

j=0

j=0

2i P

2i+1 P

2i+2 P

j=0

j=0

aj a2i+3 + bx0 x1

2i+1 P

2i+2 P

j=0

j=0

j=0

aj a2i+2 + bx2 x3

(25e)

.

(25f)

aj

2i+1 P

a2i+2 + bx0 x1

,

aj

j=0

aj a2i+2 + bx1 x2

, (25d)

aj

2i+1 P

aj a2i+2 + bx0 x1

(25c)

2i+1 P

aj a2i+2 + bx1 x2

2i+1 P

j=0

,

aj

j=0

2i P

aj a2i+2 + bx0 x1

j=0 2i P

2i+1 P

aj a2i+2 + bx0 x1

j=0

2i P

(25b)

,

2i+1 P

aj a2i+2 + bx0 x1

j=0

a2i+1 + bx2 x3

i=0

aj

j=0

j=0

aj a2i+1 + b1 x1 x2

a2i + bx2 x3

j=0

n−1 Y

2i P

2i P

a2i+1 + b1 x1 x2

i=0

(25a)

, aj

2i−1 P

j=0

n−1 Y

2i P

aj a2i+1 + bx2 x3

2i−1 P

aj a2i+1 + bx1 x2

aj

j=0

2i P

a2i+1 + bx0 x1

i=0

aj a2i+1 + bx2 x3

2i−1 P

a2i + b2 x2 x3

n−1 Y

aj a2i+1 + bx1 x2

2i−1 P

a2i + bx1 x2

n−1 Y i=0

x6n+2 =x2

2i−1 P

aj a2i + bx2 x3

a2i + bx1 x2

i=0

x6n+1 =x1

2i−1 P

a2i + bx0 x1

aj a2i+3 + bx1 x2

aj

The case where a = 1

Equations in (25) simplify to x6n =x0

n−1 Y i=0

x6n+1 =x1

1 + (2i + 1)bx1 x2 1 + 2ibx0 x1 1 + 2ibx2 x3 , 1 + 2ibx1 x2 1 + (2i + 1)bx0 x1 1 + (2i + 1)bx2 x3

n−1 Y

1 + 2ibx1 x2 1 + (2i + 1)bx0 x1 1 + (2i + 1)bx2 x3 , 1 + 2ibx2 x3 1 + (2i + 1)bx1 x2 1 + (2i + 2)bx0 x1

(26b)

1 + (2i + 1)bx1 x2 1 + (2i + 2)bx0 x1 1 + 2ibx2 x3 , 1 + (2i + 1)bx0 x1 1 + (2i + 1)bx2 x3 1 + (2i + 2)bx1 x2

(26c)

i=0

x6n+2 =x2

n−1 Y i=0

(26a)

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x6n+3 =x3

n−1 Y i=0

x6n+4 =x4

n−1 Y i=0

x6n+5 =x5

n−1 Y i=0

2.3.2

1 + (2i + 1)bx0 x1 1 + (2i + 1)bx2 x3 1 + (2i + 2)bx1 x2 , 1 + (2i + 1)bx1 x2 1 + (2i + 2)bx0 x1 1 + (2i + 2)bx2 x3

(26d)

1 + (2i + 1)bx1 x2 1 + (2i + 2)bx0 x1 1 + (2i + 2)bx2 x3 , 1 + (2i + 1)bx2 x3 1 + (2i + 2)bx1 x2 1 + (2i + 3)bx0 x1

(26e)

1 + (2i + 1)bx2 x3 1 + (2i + 2)bx1 x2 1 + (2i + 3)bx0 x1 . 1 + (2i + 2)bx0 x1 1 + (2i + 2)bx2 x3 1 + (2i + 3)bx1 x2

(26f)

The case where a = −1

Let an = −1 and bn = b. Equations in (23) result in x6n =

x0 (x1 x2 b − 1)n x1 (x0 x1 b − 1)n (x2 x3 b − 1)n , x6n+1 = , n n (x0 x1 b − 1) (x2 x3 b − 1) (x1 x2 b − 1)n

x6n+2 =

x6n+4 =

2.4

x3 (x0 x1 b − 1)n (x2 x3 b − 1)n x2 (x1 x2 b − 1)n , , x = 6n+3 (x0 x1 b − 1)n (x2 x3 b − 1)n (x1 x2 b − 1)n

x0 x1 (x1 x2 b − 1)n x2 x3 (x0 x1 b − 1)n+1 (x2 x3 b − 1)n , x6n+5 = . n+1 n x3 (x0 x1 b − 1) (x2 x3 b − 1) x0 (x1 x2 b − 1)n+1

Existence of six periodic solutions

From (26), if a = 1 and b = 0, then the solution of (1) is periodic with period six as long as u0 6= x2 or x1 6= x3 . It should also be noted that the solutions are periodic with period two when x0 = x2 and x1 = x3 . The graphs below are cases where the solutions are six periodic.

Figure 2: a = 1, b = 0, x0 = 0.7, x1 = −0.2, x2 = 0.33, x3 = −0.8.

Figure 1: a = 1, b = 0, x0 = 0.1, x1 = 0.2, x2 = 0.3, x3 = 0.44.

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2.5

Existence of 12-periodic solutions

Using (27), we have that if a = −1 and b = 0, then the solution of (1) is periodic with period twelve. The graphs below are cases where the solutions are twelve periodic.

Figure 3: a = −1, b = 0, x0 = 2.2, x1 = 1.1, x2 = 0.9, x3 = 0.3.

3

Figure 4: a = −1, b = 0, x0 = 0.2, x1 = 1.1, x2 = −0.9, x3 = 0.3.

Asymptotic behavior of the solutions for constant coefficients

Theorem 1 Let {xn }n∈N be the solution to the sequence in (1) where an = 1 for all n ≥ 0 and bn = b 6= 0. Then lim xn = 0.

n→∞

Proof 1 Using (26), we have that x6n =x0

n−1 Y i=0

1 + (2i + 1)bx1 x2 1 + 2ibx0 x1 1 + 2ibx2 x3 1 + 2ibx1 x2 1 + (2i + 1)bx0 x1 1 + (2i + 1)bx2 x3

n−1 Y

1 + (2i + 1)bx1 x2 1 + 2ibx0 x1 1 + 2ibx2 x3 1 + (2i + 1)bx 1 + (2i)bx1 x2 0 x1 1 + (2i + 1)bx2 x3 i=0    −1   n−1 −1 Y bx0 x1 bx2 x3 bx1 x2 =x0 1+ 1+ 1+ 1 + 2ibx0 x1 1 + 2ibx2 x3 1 + 2ibx1 x2 i=0 =x0

We know that 1 + 2ixk xk+1 → ∞ as i → ∞. Hence, there is a sufficiently large integer t such that for i ≥ t, we have 1 + 2ixk xk+1 ∼ 2ixk xk+1 .

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Thus x6n =x0 Γ(t)

n−1 Y

 1+

i=t+1

1 2i

"

n−1 Y

−1 

1 2i

−1 

−1



1+



1 =x0 Γ(t) exp ln 1 + 2i i=t+1

1+

1 2i

1 + ln 1 + 2i



−1



1 + ln 1 + 2i

# ,

where Γ(t) =

t  Y

1+

i=0

bx0 x1 1 + 2ibx0 x1

−1  1+

bx2 x3 1 + 2ibx2 x3

−1  1+

 bx1 x2 . 1 + 2ibx1 x2

Utilizing the expansion ln(1 + x) = x + O(x2 ), (1 + x)−1 = 1 − x + O(x2 ), for x → 0, we obtain n−1 Y

   1 1 exp − + O 2 2i i i=t+1 " n−1  # n−1    Y X 1 1 exp O 2 . =x0 Γ(t) exp − 2i i i=t+1 i=t+1

x6n =x0 Γ(t)

Therefore, lim x6n = 0

as

n→∞

n → ∞.

Similarly, lim x6n+j = 0

n→∞

as

n → ∞,

for j = 1, 2, 3, 4, 5.

References [1] R.P. Agarwal, Difference Equations and Inequalities, Dekker, New York (1992). [2] L. Berezansky and E. Braverman, On impulsive BevertonHolt difference equations and their applications, J. Difference Equ. Appl. 10:9 (2004), 851– 868. [3] G. Bluman and S. Anco, Symmetry and Integration Methods for Differential Equations, Springer, New York (2002). [4] V. A. Dorodnitsyn, R. Kozlov and P. Winternitz,Lie group classiffcation of second-order ordinary difference equations, J. Math. Phys. 41 (2000), 480–504. [5] M. Folly-Gbetoula and A.H. Kara, Symmetries, conservation laws, and integrability of difference equations, Advances in Difference Equations, 2014:224 (2014). 12

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[6] Folly-Gbetoula M, Ndlovu L, Kara A H and A Love , Symmetries, Associated First Integrals, and Double Reduction of Difference Equations, Abstract and Applied Analysis 2014, Article ID 490165, (2014) 6 pages. [7] M. Folly-Gbetoula and D. Nyirenda, On some sixth-order rational recursive sequences, Journal of computational analysis and applications, 27:6 (2019) 1057–1069. [8] P. E. Hydon, Difference Equations by Differential Equation Methods, Cambridge University Press, Cambridge, (2014). [9] S. Maeda, Canonical structure and symmetries for discrete systems, Math. Japonica 25 (1980), 405–420. [10] S. Maeda, The similarity method for difference equations, IMA J. Appl. Math.38 (1987), 129–134. [11] N. Mnguni, M. Foly-Gbetoula, Invariance Analysis of a Third Order Diference Equation with Variable Coefficients, Dynamics of Continuous, Discrete and Impulsive Systems 25 (2018), 63-73. [12] M. Mnguni, D. Nyirenda and M. Folly-Gbetoula, On solutions of some fifth-order difference equations, Far East Journal of Mathematical Sciences, 102:12 (2017) 3053-3065. [13] P. J. Olver, Applications of Lie Groups to Differential Equations, Second Edition, Springer, New York (1993).

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GENERALIZED ZWEIER I-CONVERGENT SEQUENCE SPACES OF FUZZY NUMBERS KAVITA SAINI AND KULDIP RAJ

Abstract. In the present paper we introduce Zweier ideal convergent sequences spaces of fuzzy numbers by using lacunary sequence, infinite matrix and generalized difference matrix operator Api . We study some topological and algebraic properties of these sequence spaces. Some inclusion relations related to these spaces are also establish.

1. Introduction and Preliminaries Initially the idea of I-convergence was introduced by Kostyrko et al.[10]. Gurdal [7] studied the ideal convergence sequences in 2-normed spaces. Later on, it was further studied by Savas [21], Savas and Hazarika [8], Tripathy and Dutta [25], Tripathy and Hazarika [26], Raj et al.[17]. Let X be a non-empty set, then a family of sets I ⊂ 2X is called an ideal iff for each X1 , X2 ∈ I, we have X1 ∪X2 ∈ I and for each X1 ∈ I and each X2 ⊂ X1 , we have X2 ∈ I. A non-empty family of sets U ⊂ 2X is a filter on X iff φ ∈ / U, for each X1 , X2 ∈ U, we have X1 ∩ X2 ∈ U and each X1 ∈ U and each X1 ⊂ X2 , we have X2 ∈ U. An ideal I is said to be non-trivial ideal if I = 6 φ and X ∈ / I. Clearly, I ⊂ 2X is a non-trivial ideal iff U = U (I) = {X − X1 : X1 ∈ I} is a filter on X. A non-trivial ideal I ⊂ 2X is said to be admissible iff {x : x ∈ X} ⊂ I. A non-trivial ideal is called maximal if there cannot exists any non-trivial ideal J 6= I containing I as a subset. A sequence x = (xk ) of points in R is said to be I-convergent to a real number x0 if {k ∈ N : |xk − x0 | ≥ } ∈ I, for every  > 0 (see [10]). We denote it by I − lim xk = x0 . Kızmaz [9] introduced the notion of difference sequence spaces and studied l∞ (∆), c(∆) and c0 (∆). Further this notion generalized by Et and C ¸ olak [5] by introducing the spaces l∞ (∆i ), c(∆i ) and c0 (∆i ). The new type of generalization of the difference sequence spaces was introduced by Tripathy and Esi [27] who studied the spaces l∞ (∆iv ), c(∆iv ) and c0 (∆iv ). Let i, v be non-negative integers, then for Z = l∞ , c, c0 we have sequence spaces Z(∆iv ) = {x = (xk ) ∈ w : (∆iv xk ) ∈ Z}, i−1 0 where ∆iv x = (∆iv xk ) = (∆i−1 v xk − ∆v xk+1 ) and ∆v xk = xk for all k ∈ N, which is equivalent to the following binomial representation   i X i ∆iv xk = (−1)n xk+vn . n n=0

Ba¸sar and Atlay [2] introduced and studied the generalized difference matrix A(m, n) = 2010 Mathematics Subject Classification. 40A05, 40A30. Key words and phrases. Musielak-Orlicz function, ideal convergence, generalized difference matrix operator, fuzzy real number. 1

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(ars (m, n)) which is a generalization of ∆1(1) -difference operator as follows:   m, (s = r); n, (s = r − 1); ars (m, n) =  0, 0 ≤ s ≤ r − 1 or s > r. for all r, s ∈ N and m, n ∈ R − {0}. Ba¸sarir and Kayik¸ci [3] introduced the generalized difference matrix Ap of order p and the binomial representation of this operator is p   X p p A (xk ) = mp−v nv xk−v , v v=0 where m, n ∈ R − {0} and r ∈ N. Recently, Ba¸sarir et al.[4] studied the following generalized difference sequence spaces Z(Api ) = {x = (xk ) ∈ w : (Api xk ) ∈ Z}, for Z = l∞ , c¯, c¯0 , where c¯, c¯0 are the sets of statistically convergent and statistically null convergent respectively and the binomial representation of operator Api is as follows: p   X p Api (xk ) = mp−v nv xk−iv . v v=0 S¸eng¨on¨ ul [22] defined the sequence y = (yk ) which is frequently used as the Z−transformation of the sequence x = (xk ) that is, yk = βxk + (1 − β)xk−1 , where x−1 = 0, k 6= 0, 1 < k < ∞ and Z   β, 1 − β, zik =  0,

denotes the matrix Z = (zik ) defined by (i = k); (i − 1 = k)(i, k ∈ N); otherwise.

S¸eng¨on¨ ul [22] introduced the Zweier sequence spaces Z and Z0 as follows: Z = {x = (xk ) ∈ w : Z(x) ∈ c} and Z0 = {x = (xk ) ∈ w : Z(x) ∈ c0 }. An Orlicz function M : [0, ∞) → [0, ∞) is convex, continuous and non-decreasing function which also satisfy M (0) = 0, M (x) > 0 for x > 0 and M (x) → ∞ as x → ∞. Lindenstrauss and Tzafriri [11] used the idea of Orlicz function to define the following sequence space: `M

    ∞ X |xk | < ∞, for some ρ > 0 , = x∈ω: M ρ k=1

which is called as an Orlicz sequence space. An Orlicz function is said to satisfy ∆2 −condition if for a constant R, M (Qx) ≤ RQM (x) for all values of x ≥ 0 and for Q > 1. A sequence M = (Mk ) of Orlicz functions is called as Musielak-Orlicz function.To know more about sequence spaces see ([1], [15], [16], [24], [18], [19] and [28]) and references therein. An increasing non-negative integer sequence θ = (kr ) with k0 = 0 and kr − kr−1 → ∞ as r → ∞ is known as lacunary sequence. The intervals determined by θ will be denoted kr by Ir = (kr−1 , kr ]. We write hr = kr − kr−1 and qr denotes the ratio kr−1 . The space of

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lacunary strongly convergent sequence was defined by Freedman et al. [6] as follows: o n 1 X |xk − L| = 0, for some L . Nθ = x = (xk ) : lim r→∞ hr k∈Ir

The space Nθ is a BK− space with the norm   1 X kxk = sup |xk | . hr k∈Ir

Let λ = (λnk ) be an infinite matrix of real or complex numbers λnk , where n, k ∈ N. Then a ∞ X matrix transformation of x = (xk ) is denoted as λx and λx = λn (x)) if λn (x) = λnk xk k=1

converges for each n ∈ N. The concept of fuzzy numbers and arithmetic operations with these numbers were first introduced and investigated by Zadeh [29] in 1965. Subsequently many authors have discussed various aspects of the theory and applications of fuzzy sets such as fuzzy topological spaces, similarity relations and fuzzy orderings, fuzzy measures of fuzzy events and fuzzy mathematical programming. The theory of sequences of fuzzy numbers was first studied by Matloka [12]. He studied some of their properties and showed that every convergent sequences of fuzzy numbers is bounded. Later on Nanda [13] introduced sequences of fuzzy numbers and studied that the set of all convergent sequences of fuzzy numbers forms a complete metric space. Further, the theory of sequences of fuzzy numbers have been discussed by Savas and Mursaleen [20], Tripathy and Nanda [23], Hazarika and Savas [8] and many more. Let B denote the set of all closed bounded intervals U = [u1 , u2 ] on the real line R. For U, V ∈ B, we define U ≤ V iff u1 ≤ v1 and u2 ≤ v2 and we define d(U, V ) = max{|u1 − v1 |, |u2 , v2 |}. It is well known that d defines a metric on B and (B, d) is a complete metric space (see [14]). A fuzzy number is a function U : R → [0, 1], which satisfy the following conditions: (i) U is normal i.e there exits an x0 such that U (x0 ) = 1, (ii) U is convex i.e for x, y ∈ R and 0 ≤ τ ≤ 1, U (τ x + (1 − τ )y) ≥ min{U (x), U (y)}, (iii) U is upper semi-continuous, (iv) the closure of the set supp(U ) is compact, where supp(U ) = {x ∈ R : U (x) > 0} and it is denoted by [U ]0 . The set of all fuzzy numbers are denoted by RF . Let [U ]0 = x ∈ R : u(x) > 0 and the r-level set is [U ]r = {x ∈ R : u(x) ≥ r}, (0 ≤ r ≤ 1). The set [U ]r is a closed and bounded interval of R. For any U, V ∈ RF and λ ∈ R, it is positive to define uniquely the sum U ⊕ V and the product U V as follows: [U ⊕ V ]r = [U ]r + [V ]r and [λ U ]r = λ[U ]r . (r)

(r)

(r)

Now, denote the interval [U ]r by [u1 , u2 ], where u1 r ∈ [0, 1]. Now, define dˆ : RF × RF → R by

(r)

≤ u2

(r)

(r)

and u1 , u2

∈ R, for

ˆ V ) = sup d([U ]r , [V ]r ). d(U, r∈[0,1]

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Definition 1.1. A sequence x = (xk ) of fuzzy numbers is said to be convergent to a fuzzy number x0 if for every  > 0 there exist a positive integer n0 such that ˆ k , x0 ) < , for k > n0 . d(x Definition 1.2. A sequence x = (xk ) of fuzzy numbers is said to be I- convergent to a fuzzy number x0 if for every  > 0 such that ˆ k , x0 ) ≥ } ∈ I. {k ∈ N : d(x Throughout the article, we denote Zweier fuzzy number sequence Z(x) by x0 for x ∈ ω F . Let I be an admissible ideal of N, M = (Mk ) be a Musielak-Orlicz function, q = (qk ) be a bounded sequence of positive real numbers, λ = (λnk ) be an infinite matrix, θ be a lacunary sequence and ω F is the set of all sequences of fuzzy real numbers. In the present paper we define lacunary Zweier I−convergent, lacunary Zweier I−null and lacunary Zweier I−bounded sequence spaces of fuzzy numbers as follows: Z I(F ) [AP i , θ, λ, M, q] = qk   ˆ p 0   d(Ai xk , x0 ) 1 X F λnk Mk ≥ ∈I x = (xk ) ∈ ω : n ∈ N : lim r→∞ hr ρ k∈Ir  for some ρ > 0 and x0 ∈ RF , I(F )

Z0

[AP i , θ, λ, M, q] = qk     ˆ p 0 1 X d(Ai xk , ¯0) F ≥ ∈I λnk Mk x = (xk ) ∈ ω : n ∈ N : lim r→∞ hr ρ k∈Ir  for some ρ > 0

and I(F )

Z∞ [AP i , θ, λ, M, q] =   qk   ˆ p 0 1 X d(Ai xk , ¯0) x = (xk ) ∈ ω F : ∃K > 0 s.t. n ∈ N : λnk Mk ≥K ∈I hr ρ k∈Ir  for some ρ > 0 , where, ¯ 0(t) =



1, if t = 0; 0, otherwise.

If 0 < qk ≤ sup qk = D, C = max(1, 2D−1 ). Then (1.1)

|ck + dk |qk ≤ C(|ck |qk + |dk |qk ),

for all ck , dk ∈ R and for all k ∈ N. The main purpose of this paper is to study some classes of lacunary Zweier sequences of fuzzy numbers defined by means of generalized difference matrix operator, Musielak-Orlicz function and infinite matrix. We shall make an effort to study some interesting algebraic and topological properties of concerning sequence spaces. Also, we examine some interrelations between these sequence spaces.

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2. Main Results Theorem 2.1. Let M = (Mk ) be a Musielak-Orlicz function, q = (qk ) be a bounded sequence of positive real numbers and θ be a lacunary sequence. Then the sequence spaces I(F ) I(F ) Z I(F ) [AP [AP [AP i , θ, λ, M, q], Z0 i , θ, λ, M, q] and Z∞ i , θ, λ, M, q] are closed under addition and scalar multiplication. I(F )

Proof. Consider x = (xk ), y = (yk ) ∈ Z0 [AP i , θ, λ, M, q] and α, β are scalars. Then there exist positive numbers ρ1 > 0 and ρ2 > 0 such that  ˆ p 0 qk   d(Ai xk , x0 )  1 X λnk Mk ≥ ∈I n ∈ N : lim r→∞ hr ρ1 2 k∈Ir

and

 ˆ p 0  qk  d(Ai yk , y0 ) 1 X  λnk Mk n ∈ N : lim ≥ ∈ I. r→∞ hr ρ2 2 k∈Ir

Api

Since is linear and by using the continuity of Musielak-Orlicz function M, we have the following inequality:  ˆ p qk d(Ai (α(x0k ) + β(yk0 ))) 1 X λnk Mk lim r→∞ hr |α|ρ1 + |β|ρ2 k∈Ir

 qk ˆ p 0 |α| 1 X d(Ai xk , x0 ) λnk Mk ≤ D lim r→∞ hr |α|ρ1 + |β|ρ2 ρ1 k∈Ir   ˆ p y 0 , y0 ) qk 1 X |β| d(A i k + D lim λnk Mk r→∞ hr |α|ρ1 + |β|ρ2 ρ2 k∈Ir qk  ˆ p 0 1 X d(Ai xk , x0 ) λnk Mk ≤ DK lim r→∞ hr ρ1 k∈Ir   ˆ p y 0 , y0 ) qk 1 X d(A i k + DK lim λnk Mk , r→∞ hr ρ2 k∈Ir   |β|ρ2 1 , . where K = max 1, |α|ρ|α|ρ 1 +|β|ρ2 |α|ρ1 +|β|ρ2 Thus, we have   ˆ p qk  1 X d(Ai (α(x0k ) + β(yk0 ))) λnk Mk n ∈ N : lim ≥ r→∞ hr |α|ρ1 + |β|ρ2 k∈Ir

  ˆ p 0 qk  d(Ai xk , x0 )  1 X λnk Mk ≥ ⊆ n ∈ N : DK lim r→∞ hr ρ1 2 k∈Ir   ˆ p 0 qk  1 X d(Ai yk , y0 )  ∪ n ∈ N : DK lim λnk Mk ≥ . r→∞ hr ρ2 2 k∈Ir

Since the sets on right hand side of above relation belong to I. Thus, the sequence space I(F ) Z0 [AP i , θ, λ, M, q] is closed under addition and scalar multiplication . Similarly, we can prove others.  Theorem 2.2. Let M = (Mk ) be a Musielak-Orlicz function, q = (qk ) and v = (vk ) be two bounded sequences of positive real numbers with 0 < qk ≤ vk for each k and ( vqkk ) be bounded. Then

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I(F )

(i) Z0 [AP [AP i , θ, λ, M, v], ⊆ Z0 i , θ, λ, M, q], I(F ) P I(F ) (ii) Z [Ai , θ, λ, M, v], ⊆ Z [AP i , θ, λ, M, q], I(F ) I(F ) P (iii)Z∞ [AP , θ, λ, M, v], ⊆ Z [A ∞ i i , θ, λ, M, q]. Proof. The proof of the theorem is straightforward, so we omit it.



Theorem 2.3. Let M = (Mk ) be a Musielak-Orlicz function and q = (qk ) be a bounded seI(F ) I(F ) I(F ) [AP quence of positive numbers. Then Z0 [AP i , θ, λ, M, q] ⊂ Z∞ i , θ, λ, M, q], ⊆ Z P [Ai , θ, λ, M, q]. Proof. We know that the first inclusion is obvious. Next, we show that Z I(F ) [AP i , θ, λ, M, q] ⊂ I(F ) p P I(F ) Z∞ [Ai , θ, λ, M, q]. Let (xk ) ∈ Z [Ai , θ, λ, M, q]. Then we have 1 hr

X k∈Ir

 ˆ p 0 qk d(Ai xk , ¯ 0) λnk Mk ρ qk  ˆ p 0 d(Ai xk , x0 ) C X λnk Mk hr ρ k∈Ir   ˆ 0 , ¯0) qk C X d(x + λnk Mk hr ρ k∈Ir qk  ˆ p 0 C X d(Ai xk , x0 ) λnk Mk ≤ hr ρ k∈Ir D     ˆ d(x0 , ¯0) , + C max 1, sup λnk Mk ρ ≤

I(F )

where sup qk = D and C = max(1, 2D−1 ). Therefore, (xk ) ∈ Z∞ completes the proof of the theorem.

[Api , θ, λ, M, q]. This 

Theorem 2.4. Let M = (Mk ) and M0 = (Mk0 ) be two Musielak-Orlicz functions. Then the folowing inclusions holds: T I(F ) I(F ) I(F ) (i) Z0 [Api , θ, λ, M, q] Z0 [Api , θ, λ, M0 , q] ⊂ Z0 [Api , θ, λ, M + M0 , q], T I(F ) p p I(F ) 0 (ii) Z [Ai , θ, λ, M, q] Z [A , θ, λ, M , q] ⊂ Z I(F ) [Api , θ, λ, M + M0 , q], T I(F ) ip I(F ) I(F ) p (iii) Z∞ [Ai , θ, λ, M, q] Z∞ [Ai , θ, λ, M0 , q] ⊂ Z∞ [Api , θ, λ, M + M0 , q]. I(F )

Proof. (xk ) ∈ Z0 [Api , θ, λ, M, q]  Suppose  qk ˆ p x0 ,¯ d(A i k 0) λnk (Mk + Mk0 ) ρ

T

I(F )

Z0

[Api , θ, λ, M0 , q]. Then, we have

  ˆ p 0   ˆ p 0 qk qk d(Ai xk , ¯0) d(Ai xk , ¯0) + C λnk Mk0 , ≤ C λnk Mk ρ ρ which consequently implies that  ˆ p 0 qk X d(Ai xk , ¯ 0) 0 1 λ (M + M ) nk k k hr ρ k∈Ir  ˆ p 0 qk C X d(Ai xk , ¯0) ≤ λnk Mk hr ρ k∈Ir   ˆ p 0 ¯ qk C X 0 d(Ai xk , 0) + λnk Mk . hr ρ k∈Ir

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This implies (xk ) ∈ Z0 way.

7

[Api , θ, λ, M + M0 , q]. We can prove the other cases in the same 

Theorem 2.5. Let M = (Mk ) and M0 = (Mk0 ) be two Musielak-Orlicz functions. Then the folowing inclusion holds: I(F )

I(F )

[Api , θ, λ, M0 , q] ⊆ Z0

[Api , θ, λ, M.M0 , q].  X λnk max{h0 , D Proof. For given  > 0 and choose 0 such that sup 0 } < . Choose Z0

n

k∈Ir I(F )

0 < ϕ < 1 such that Mk (t) < 0 , for all k ∈ N. Let x = (xk ) ∈ Z0 for some ρ > 0, we have

B1 =

[Api , θ, λ, M0 , q]. Then

  ˆ p 0 qk  1 X d(Ai xk , ¯0) n ∈ N : lim λnk Mk0 ≥ ϕD ∈ I. r→∞ hr ρ k∈Ir

If n ∈ / B1 , then we have qk  ˆ p 0 ¯ 1 X 0 d(Ai xk , 0) λnk Mk < ϕD . lim r→∞ hr ρ k∈Ir

This implies  ˆ p 0 qk d(Ai xk , ¯0) Mk0 < ϕD for all k ∈ N. ρ Hence, Mk0

q ˆ p 0 d(Ai xk , ¯0) k < ϕ for all k ∈ N. ρ

Therefore,  Mk

Mk0

qk ˆ p 0 d(Ai xk , ¯0) < 0 for all k ∈ N. ρ

Thus, we get   ˆ p 0 qk X  ¯ 1 X 0 d(Ai xk , 0) λnk Mk Mk lim < sup λnk max{h0 , D 0 } < . r→∞ hr ρ n k∈Ir

k∈Ir

Now, we have qk   ˆ p 0 1 X d(Ai xk , ¯0) λnk Mk Mk0 < . r→∞ hr ρ lim

k∈Ir

This implies    ˆ p 0 qk  1 X d(Ai xk , ¯0) n ∈ N : lim λnk Mk Mk0 ≥  ⊂ B1 ∈ I. r→∞ hr ρ k∈Ir

This completes the proof.



Theorem 2.6. If lim qk > 0 and x = (xk ) →

x0 (Z I(F ) [Api , θ, λ, M, q]), I(F )

then x0 is unique.

[Api , θ, λ, M, q])

Proof. Let lim qk = u0 . Consider that (xk ) → x0 (Z and (xk ) → p I(F ) y0 (Z [Ai , θ, λ, M, q]). So, there exist ρ1 , ρ2 > 0, such that   ˆ p 0 qk  1 X d(Ai xk , x0 )  (2.1) X1 = n ∈ N : lim λnk Mk ≥ ∈I r→∞ hr ρ1 2 k∈Ir

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and (2.2)

X2 =

  ˆ p 0 qk  1 X d(Ai xk , y0 )  n ∈ N : lim λnk Mk ∈ I. ≥ r→∞ hr ρ2 2 k∈Ir

Define ρ = max{2ρ1 , 2ρ2 }. Then we have  ˆ qk X d(x0 , y0 ) λnk Mk ρ

k∈Ir

≤ D

X k∈Ir

 ˆ p 0 qk  ˆ p 0 qk X d(Ai xk , x0 ) d(Ai xk , y0 ) λnk Mk λnk Mk +D . ρ ρ k∈Ir

Then from (2.1) and (2.2), we have  ˆ qk   X d(x0 , y0 ) λnk Mk ≥ n∈N: ρ k∈Ir

⊆ ∪

  qk  ˆ p 0 X  d(Ai xk , x0 ) n∈N:D λnk Mk ≥ ρ1 2 k∈Ir      X ˆ p x0 , y0 ) qk  d(A i k n∈N:D λnk Mk ≥ ρ2 2 k∈Ir

⊆ X1 ∪ X2 ∈ I. Also,  ˆ qk  ˆ u0 d(x0 , y0 ) d(x0 , y0 ) Mk → Mk as k → ∞. ρ ρ Then, we have qk   ˆ u0  ˆ d(x0 , y0 ) d(x0 , y0 ) = Mk = 0. Mk k→∞ ρ ρ lim

Thus, x0 = y0 .



Theorem 2.7. Let M = (Mk ) be a Musielak-Orlicz function and q = (qk ) be a bounded sequence of positive real numbers, I(F ) I(F ) (a) If 0 < inf qk ≤ qk ≤ 1 for all k, then Z0 [Api , θ, λ, M, q] ⊆ Z0 [Api , θ, λ, M] and p p Z I(F ) [Ai , θ, λ, M, q] ⊆ Z I(F ) [Ai , θ, λ, M]. I(F ) I(F ) (b) If 1 ≤ qk ≤ sup qk = D < ∞ for all k, then Z0 [Api , θ, λ, M] ⊆ Z0 [Api , θ, λ, M, q] p p I(F ) I(F ) and Z [Ai , θ, λ, M] ⊆ Z [Ai , θ, λ, M, q]. Proof. (a) Suppose (xk ) ∈ Z I(F ) [Api , θ, λ, M, q]. Since 0 < inf qk ≤ qk ≤ 1, then we have  ˆ p 0  1 X d(Ai xk , x0 ) lim λnk Mk r→∞ hr ρ k∈Ir

 ˆ p 0 qk 1 X d(Ai xk , x0 ) λnk Mk . ≤ lim r→∞ hr ρ k∈Ir

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Thus,   ˆ p 0   1 X d(Ai xk , x0 ) n ∈ N : lim λnk Mk ≥ r→∞ hr ρ k∈Ir

⊆

 ˆ p 0  qk  1 X d(Ai xk , x0 ) λnk Mk n ∈ N : lim ≥  ∈ I. r→∞ hr ρ k∈Ir

The other part can be proved in the same way. (ii) Suppose (xk ) ∈ Z I(F ) [Api , θ, λ, M]. Since 1 ≤ qk ≤ sup qk = D < ∞. Then for each 0 <  < 1, there exists a positive integer m0 such that  ˆ p 0  d(Ai xk , x0 ) 1 X λnk Mk ≤  < 1, r→∞ hr ρ lim

k∈Ir

for all n ≥ m0 . This implies qk  ˆ p 0 d(Ai xk , x0 ) 1 X λnk Mk lim r→∞ hr ρ k∈Ir

≤

 ˆ p 0  1 X d(Ai xk , x0 ) λnk Mk . r→∞ hr ρ lim

k∈Ir

Thus,   ˆ p 0 qk  1 X d(Ai xk , x0 ) λnk Mk n ∈ N : lim ≥ r→∞ hr ρ k∈Ir

    ˆ p 0 1 X d(Ai xk , x0 ) ⊆ n ∈ N : lim λnk Mk ≥  ∈ I. r→∞ hr ρ k∈Ir

The other part can be proved in the same way.



References [1] R. Anand, C. Sharma and K. Raj, Seminormed double sequence spaces of four dimensional matrix and Musielak-Orlicz function, J. Inequal. Appl., (2018), 2018:285. [2] E. Ba¸sar and B. Altay, On the space of sequences of p-bounded variation and related matrix mappings, Ukrainian Math. J., 55 (2003), 136-147. [3] M. Ba¸sarir and M. Kayik¸ci, On the generalized B m -Riesz difference sequence spaces and β-property, J. Inequal. Appl., 2009 (2009), Article ID 385029, 18 pages. [4] M. Ba¸sarir, S. Kayikci and E. E. Kara, Some generalized difference statistically convergent sequence spaces in 2-normed space, J. Inequal. Appl., 37 (2013), 2013: 177. [5] M. Et and R. C ¸ olak, On some generalized difference sequence spaces and related matrix transformations, Hokkaido Math. J., 26 (1997), 483-492. [6] A. R. Freedman, J. J. Sember and M. Raphael, Some Cesaro-type summability spaces, Proc. London Math. Soc., 37 (1978), 508-520. [7] M. Gurdal, On ideal convergent sequences in 2-normed spaces, Thai J. Math., 4 (2006), 85-91. [8] B. Hazarika and E. Savas, Some I-convergent lambda-summable difference sequences spaces of fuzzy real numbers defined by a sequence of Orlicz functions, Math. Comput. Modelling, 54 (2011), 29862998. [9] H. Kızmaz, On certain sequence spaces, Canad. Math. Bull., 24 (1981), 169-176. [10] P. Kostyrko, T. Salat and W. Wilczynski, I-convergence, Real Anal. Exchange, 26 (2000-2001), 669686. [11] J. Lindenstrauss and L. Tzafriri, An Orlicz sequence spaces, Israel J. Math., 10 (1971), 379-390. [12] M. Matloka, Sequences of fuzzy numbers, J. Math. Anal. Appl., 28 (1986), 28-37. [13] S. Nanda., On sequences of fuzzy number, Fuzzy Sets Syst., 33 (1989), 123-126.

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[14] M. L. Puri and D. A. Ralescu, Differentials of fuzzy functions, J. Math. Anal. Appl., 91 (1983), 552-558. [15] K. Raj, A. Choudhary and C. Sharma, Almost strongly Orlicz double sequence spaces of regular matrices and their applications to statistical convergence, Asian-Eur. J. Math., 11 1850073 (2018), doi.org/10.1142/S1793557118500730. [16] K. Raj, C. Sharma and A. Choudhary, Applications of Tauberian theorem in Orlicz spaces of double difference sequences of fuzzy numbers, J. Intell. Fuzzy Systems, 35 (2018), 2513-2524. [17] K. Raj, A. Abzhapborav and A. Khassymkan, Some generalized difference sequences of ideal convergence and Orlicz functions, J. Comput. Anal. Appl., 22 (2017), 52-63. [18] K. Raj, and R. Anand, Double difference spaces of almost null and almost convergent sequences for Orlicz function, J. Comput. Anal. Appl., 24 (2018), 773-783. [19] A. Choudhary and K. Raj, Applications of double difference fractional order operators to originate some spaces of sequences, J. Comput. Anal. Appl., 28 (2020), 94-103. [20] E. Savas and M. Mursaleen, On statistically convergent double sequence of fuzzy numbers, Inform. Sci., 162 (2004), 183-192. [21] E. Savas, A sequence spaces in 2-normed space defined by ideal convergence and an Orlicz function, Abst. Appl. Anal., (2011), Article ID 741382, 1-8. [22] M. S ¸ eng¨ on¨ ul, On The Zweier Sequence Space, Demonstratio Math., 40 (2007), 181-196. [23] B. C. Tripathy and S. Nanda, Absolute value of fuzzy real number and fuzzy sequence spaces, Jour. Fuzzy Math., 8 (2000), 883-892 [24] B. C. Tripathy, S. Debnath and S. Saha, On some difference sequence spaces of interval numbers, Proyecciones, 37 (2018), 603-612. [25] B. C. Tripathy and A. J. Dutta, On I-acceleration convergence of sequences of fuzzy numbers, Math. Modell. Analysis, 17 (2012), 549-557. [26] B. C. Tripathy and B. Hazarika, Some I-convergent sequence spaces defined by Orlicz functions, Acta. Math. Appl. Sinica, 27 (2011), 149-154. [27] B. C. Tripathy and A. Esi, A new type of difference sequences spaces, Int. J. Sci. and Tech., 1 (2006), 11-14. [28] B. C. Tripathy and R. Goswami, Statistically convergent multiple sequences in probalistic normed spaces, Politehn. Univ. Bucharest Sci. Bull. Ser. A Appl. Math. Phys., 78 (2016), 83-94. [29] L. A. Zadeh, Fuzzy sets, Inform. and Control, 8 (1965), 338-353. School of Mathematics, Shri Mata Vaishno Devi University, Katra-182320, J & K (India) E-mail address: [email protected] E-mail address: [email protected]

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Some convergence results using K ∗ iteration process in CAT (0) spaces Kifayat Ullah, Dong Yun Shin, Choonkil Park and Bakhat Ayaz Khan Abstract. In this paper, some strong and ∆-convergence results for Suzuki generalized nonexpansive mappings in the setting of complete CAT (0) spaces are proved. We are using newly introduced K ∗ iteration process for approximation of fixed point. We also give an example to show the efficiency of the K ∗ iteration process. Our results are extension, improvement and generalization of many well known results in the literature of fixed point theory in CAT (0) spaces. Mathematics Subject Classification (2010). Primary 47H09, 47H10. Keywords. Suzuki generalized nonexpansive mapping; CAT (0) space; K ∗ iterative process; ∆-convergence; strong convergence.

1. Introduction It is well-known that several mathematics problems are naturally formulated as fixed point problem T x = x, where T is some suitable mapping, may be nonlinear. For example, for given functions ζ : [a, b] ⊆ R → R and ξ : [a, b] × [a, b] × R → R, the solution of following nonlinear integral equation Zb x(c) = ζ(c) +

ξ(c, r, x(r))dr, a

where x ∈ C[a, b] (the set of all continuous real-valued functions defined on [a, b] ⊆ R), is equivalently to fixed point problems for the following mapping T : C[a, b] → C[a, b] defined by 0∗ Corresponding

author: Dong Yun Shin (email: [email protected]). This work was supported by Basic Science Research Program through the National Research Foundation of Korea funded by the Ministry of Education, Science and Technology (NRF-2017R1D1A1B04032937). We would like to thank Prof. Balwant Singh Thakur for technical assistance.

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Zb (T x)(c) = ζ(c) +

ξ(c, r, x(r))dr a

for all x ∈ C[a, b]. The well-known Banach contraction theorem uses the Picard iteration process for approximation of fixed point. Many iterative processes have been developed to approximate fixed points of contraction type of mapping in CAT (0) type spaces of ground spaces. Some of the other well-known iterative processes are those of Mann [17], Ishikawa [10], Noor [8], Abbas [1], Agarwal [2], Phuengrattana and Suantai [19], Karahan and Ozdemir [11], Chugh, Kumar and Kumar [6], Sahu and Petrusel [20], Khan [14], Gursoy and Karakaya [9], Thakur, Thakur and Postolache [22] and so on. See also [13, 23, 25] for more information on CAT (0) spaces and applications. Recently, Ullah and Arshad [24] introduced a new three steps iteration process as the K ∗ iteration process and proved that it is strong and converges fast as compared to all above mentioned iteration processes. They use uniformly convex Banach space as a ground space. Motivated by above, in this paper, first we develop an example of Suzuki generalized nonexpansive mappings is given which is not nonexpansive. We compare the speed of convergence of the K ∗ iteration process with the leading two steps S-iteration process and leading three steps Picard-S-iteration process for Suzuki generalized nonexpansive mappings, and graphic representation is also given. Finally, we prove some strong and ∆-convergence theorems for Suzuki generalized nonexpansive mappings in the setting of CAT (0) spaces.

2. Preliminaries Let (X, d) be a metric space. A geodesic from x to y in X is a mapping c from closed interval [0, l] ⊂ R to X such that c(0) = x, c(l) = y, and 0 0 0 d(c(t), c(t )) = |t − t | for all t, t ∈ [0, l]. In particular, c is an isometry and d(x, y) = l. The image of c is called a geodesic (or metric) segment joining x and y. The space (X, d) is said to be a geodesic space if every two points of X is joined by a geodesic and X is said to be uniquely geodesic if there is exactly one geodesic joining x and y for each x, y ∈ X, which we denote by [x, y], called the segment joining x to y. A geodesic triangle ∆(x1, x2, x3 ) in a geodesic metric space (X, d) consists of three points x1 , x2 , x3 in X (the vertices of ∆) and a geodesic segment between each pair of vertices (the edges of ∆). A comparison triangle for the ¯ 1, x2, x3 ) := ∆(¯ triangle ∆(x1, x2, x3 ) in (X, d) is a triangle ∆(x x1, x ¯2, x ¯3 ) in R2 such that dR2 (¯ xi , x ¯j ) = d(xi , xj ) for i, j ∈ {1, 2, 3}. A geodesic space is said be a CAT (0) space if all geodesic triangles of appropriate size satisfy the following comparison axiom. 669

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¯ be a comparison CAT (0): Let ∆ be a geodesic triangle in X and ∆ triangle for ∆. Then ∆ is said to satisfy the CAT (0) inequality if for x, y ∈ ¯ ∆ and all comparison points x ¯, y¯ ∈ ∆, d(x, y) ≤ dE 2 (¯ x, y¯). If x, y1, y2 are points in CAT (0) space and if y0 is the midpoint of the segment [y1 , y2 ], then the CAT (0) inequality implies 1 1 1 d(x, y1 )2 + d(x, y2 )2 − d(y1 , y2 )2 . (CN) 2 2 4 This is the (CN ) inequality of Burhat and Tits [5]. We recall the following result from Dhompongsa and Panyanak [8]. d(x, y0 )2 ≤

Lemma 2.1. ([8]) For x, y ∈ X and α ∈ [0, 1], there exists a unique point z ∈ [x, y] such that d(x, z) = αd(x, y) and d(y, z) = (1 − α)d(x, y).

(2.1)

The notation ((1 − α)x ⊕ αy) is used for the unique point z satisfying (2.1). CAT (0) space may be regarded as a metric version of Hilbert space. For example, in CAT (0) space we have the following extended version of parallelogram law: d(z, αx ⊕ (1 − α)y)2 = αd(x, z)2 + (1 − α)d(z, y)2 − α(1 − α)d(x, y)2 (2.2) for any α ∈ [0, 1], x, y ∈ X. If α = 12 , then the inequality (2.2) becomes the (CN ) inequality. In fact, a geodesic space is a CAT (0) space if and only if it satisfies the (CN ) inequality (cf. [5]). Complete CAT (0) spaces are often called Hadmard spaces. For more on these spaces, please refer to [3, 4]. Lemma 2.2. ([14, Lemma 2.4]) For x, y, z ∈ X and α ∈ [0, 1], we have d(z, αx ⊕ (1 − α)y) ≤ αd(z, x) + (1 − α)d(z, y). Let C be a nonempty closed convex subset of a CAT (0) space X let {xn } be a bounded sequence in X. For x ∈ X, we set r(x, {xn }) = lim sup d(xn , x). n→∞

The asymptotic radius of {xn } relative to C is given by r(C, {xn }) = inf{r(x, {xn }) : x ∈ C} and the asymptotic center of {xn } relative to C is the set A(C, {xn }) = {x ∈ C : r(x, {xn }) = r(C, {xn })}. It is well known that, in a complete CAT (0) space, A(C, {xn }) consists of exactly one point. We now recall the definition of ∆-convergence in CAT (0) space. 670

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Definition 2.3. A sequence {xn } in a CAT (0) space X is said to be ∆convergent to x ∈ X if x is the unique asymptotic center of {ux } for every subsequence {ux } of {xn }. In this case, we write ∆-limn xn = x and call x the ∆-lim of {xn }. Recall that a bounded sequence {xn } in X is said to be regular if r({xn }) = r{ux } for every subsequence {ux } of {xn }. Since in a CAT (0) space every regular sequence ∆-converges, we see that every bounded sequence in X has a ∆-convergent subsequence. A CAT (0) space X is said to satisfy the Opial0 s property [17] if for each sequence {xn } in X, ∆-converges to x ∈ X, we have lim sup d(xn , x) < lim sup d(xn , y) n→∞

n→∞

for all y ∈ X such that y 6= x. Definition 2.4. A point p is called a fixed point of a mapping T if T (p) = p and F (T ) represents the set of all fixed points of the mapping T. Definition 2.5. Let C be a nonempty subset of a CAT (0) space X. (i) A mapping T : C → C is called a contraction if there exists α ∈ (0, 1) such that d(T x, T y) ≤ αd(x, y) for all x, y ∈ C. (ii) A mapping T : C → C is called nonexpansive if d(T x, T y) ≤ d(x, y) for all x, y ∈ C. (iii) A mapping is a quasi-nonexpansive if for all x ∈ C and p ∈ F (T ), we have d(T x, p) ≤ d(x, p). In 2008, Suzuki [21] introduced the concept of generalized nonexpansive mappings which is a condition on mappings called condition (C). A mapping T : C → C is said to satisfy condition (C) if for all x, y ∈ C, we have 1 d(x, T x) ≤ d(x, y) implies d(T x, T y) ≤ d(x, y). 2 Suzuki [21] showed that the mapping satisfying condition (C) is weaker than nonexpansiveness. The mapping satisfying condition (C) is called a Suzuki generalized nonexpansive mapping. Suzuki [21] obtained fixed point theorems and convergence theorems for Suzuki generalized nonexpansive mapping. In 2011, Phuengrattana [18] proved convergence theorems for Suzuki generalized nonexpansive mappings using the Ishikawa iteration in uniformly convex Banach spaces and CAT (0) spaces. Recently, fixed point theorems for Suzuki generalized nonexpansive mapping have been studied by a number of authors, see, e.g., [22] and references therein. 671

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The following are some basic properties of Suzuki generalized nonexpansive mappings whose proofs in the setup of CAT (0) spaces follow the same lines as those of [12, Propostions 11, 14, 19] and therefore we omit them. Proposition 2.6. Let C be a nonempty subset of a CAT (0) space X and T : C → C be any mapping. (i) [21, Proposition 1] If T is nonexpansive, then T is a Suzuki generalized nonexpansive mapping. (ii) [21, Proposition 2] If T is a Suzuki generalized nonexpansive mapping and has a fixed point, then T is a quasi-nonexpansive mapping. (iii) [21, Lemma 7] If T is a Suzuki generalized nonexpansive mapping, then d(x, T y) ≤ 3d(T x, x) + d(x, y) for all x, y ∈ C. Lemma 2.7. [21, Theorem 5] Let C be a weakly compact convex subset of a CAT (0) space X. Let T be a mapping on C. Assume that T is a Suzuki generalized nonexpansive mapping. Then T has a fixed point. Lemma 2.8. [16, Lemma 2.9] Suppose that X is a complete CAT (0) space and x ∈ X. If {tn } is a sequence in [b, c] for some b, c ∈ (0, 1) and {xn }, {yn } are sequences in X such that for some r ≥ 0, we have lim sup d(xn , x) ≤ r,

n→∞

lim sup d(yn , x) ≤ r,

n→∞

lim sup d(tn xn + (1 − tn )yn , x)

n→∞

= r,

then lim d(xn , yn ) = 0.

n→∞

Lemma 2.9. [7, Proposition 2.1] If C is a closed comvex subset of a complete CAT (0) space X and if {xn } is a bounded sequence in C, then the asymptotic center of {xn } is in C. Lemma 2.10. [15] Every bounded sequence in a complete CAT (0) space always has a ∆-convergent subsequence. Lemma 2.11. [15, Proposition 3.7] Let C is a closed comvex subset of a complete CAT (0) space X and T : C → X be a Suzuki generalized nonexpansive mapping. Then the conditions {xn } ∆-converges to x and d(T xn , xn ) → 0 imply x ∈ C and T x = x. The following is an example of Suzuki generalized nonexpansive mapping which is not nonexpansive. Example 1. Define a mapping T : [0, 1] → [0, 1] by
  1 − x if x ∈ 0, 61 Tx = x+5 1 6 if x ∈ 6 , 1 . 672

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We need to prove that T is a Suzuki generalized nonexpansive but not nonexpansive. 15 and y = 61 , then we have If x = 96 d(T x, T y)

= |T x − T y| 15 31 = 1 − − 96 36 5 = 288 1 > 96 = d(x, y).

Hence T is not a nonexpansive mapping. To verify that T is a Suzuki generalized nonexpansive mapping, consider the following cases: 
 1 Case I: Let x ∈ 0, 61 . Then 12 d( x, T x) = 1−2x ∈ 31 , 12 . For 2 2 d(x, T x) ≤  1−2x 1 1 d(x, y), we have 2 ≤ y − x, i.e., 2 ≤ y and hence y ∈ 2 , 1 . We have y + 5 y + 6x − 1 1 < d(T x, T y) = − (1 − x) = 6 6 6 and

1 1 2 d(x, y) = |x − y| > − = . 6 2 6 1 Hence 2 d(x, T x) ≤ d(x, y)  1 =⇒  d(T x, T 1y) ≤ d(x, y). 1 x+5 Case II: Let x ∈ , 1 . Then 2 d( x, T x) = 2 6 − x = 5−5x ∈ 6 12  25  ≤ |y − x| , which gives two 0, 72 . For 12 d(x, T x) ≤ d(x, y), we have 5−5x 12 possibilities:  37  5−5x 5+7x (a) Let x < y. Then ≤ y − x =⇒ y ≥ =⇒ y ∈ 12 12 72 , 1 ⊂ 1  6 , 1 . So x + 5 y + 5 1 = d(x, y) ≤ d(x, y). d(T x, T y) = − 6 6 6

Hence 21 d(x, T x) ≤ d(x, y) =⇒ d(T x, T y) ≤ d(x, y). 5−5x − y =⇒ y ≤ x − 5−5x = 17x−5 =⇒ 12 12 (b)13 Let  x > y. Then 12 ≤ x17x−5  5 y ∈  − 72, 1 . Since y ∈ [0, 1], y ≤ 12 =⇒ x ∈ 12 , 1 . So the case is 5 x ∈ 12 , 1 and y ∈ [0, 1] .   1  5 Now the case  5 that  x ∈ 12, 1 1
and y ∈ 6 , 1 is the same case as that of (a). So let x ∈ 12 , 1 and y ∈ 0, 6 . Then x + 5 − (1 − y) d(T x, T y) = 6 x + 6y − 1 . = 6  5 1 
For convenience, first we consider x ∈ 12 , 2 and y ∈ 0, 61 . Then d(T x, T y) ≤ 1 3 12 and d(x, y) ≥ 12 . Hence d(T x, T y) ≤ d(x, y). 673

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Table 1. Some values produced by S, Picard-S and K ∗ IP

x0 x1 x2 x3 x4 x5 x6 x7 x8 x9 x10

K∗ Picard-S 0.9 0.9 0.99809713998382 0.99722222222222 0.99997729192914 0.99993300629392 0.99999985210113 0.99999849779947 0.99999999971662 0.99999996779523 1 0.99999999933035 1 0.99999999998638 1 0.99999999999973 1 0.99999999999999 1 1 1 1

S 0.9 0.98333333333333 0.99758822658104 0.99967552468466 0.99995826261755 0.99999479283092 0.99999936458953 0.99999992375668 0.99999999097156 0.99999999894221 0.99999999987715

Table 2. Some values produced by S, Picard-S and K ∗ IP

x0 x1 x2 x3 x4 x5 x6 x7 x8 x9 x10

K∗ Picard-S 0.5 0.5 0.99048569991909 0.99722222222222 0.99988645964572 0.99993300629392 0.99999926050565 0.99999926050566 0.99999999858311 0.99999996779523 1 0.99999999933035 1 0.99999999998638 1 0.99999999999973 1 0.99999999999999 1 1 1 1

S 0.5 0.98333333333333 0.99758822658104 0.99967552468466 0.99995826261755 0.99999479283092 0.99999936458953 0.99999992375668 0.99999999097156 0.99999999894221 0.99999999987715

  
Next consider x ∈ 12 , 1 and y ∈ 0, 61 . Then d(T x, T y) ≤ d(x, y) ≥ 62 . Hence d(T x, T y) ≤ d(x, y). So

1 6

and

1 d(x, T x) ≤ d(x, y) =⇒ d(T x, T y) ≤ d(x, y). 2 Hence T is a Suzuki generalized nonexpansive mapping. In order to show the efficiency of K ∗ iteration process, we use Example 1 with x0 = 0.9, x0 = 0.5 and get the above Tables 1 and 2. Graphic representation is given in Figure 1. Let n ≥ 0 and {αn } and {βn } be real sequences in [0, 1]. Ullah and Arshad [24] introduced a new iteration process known as the K ∗ iteration process  x0 ∈ C    zn = (1 − βn )xn + βn T xn yn = T ((1 − αn )zn + αn T zn )    xn+1 = T yn . 674

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x0 =0.5 1.00

0.998

0.99

0.996

0.98

0.994

0.97 xn+1

xn+1

x0 =0.9 1.000

0.992

0.96

0.990

0.95

0.988

0.94 0.93

0.986 0.90

0.92

0.94

0.96

0.98

1.00

0.5

0.6

xn

0.7

0.8

0.9

xn

Figure 1. Convergence of iterative sequences generated by K ∗ (red line), Picard-S (blue line) and S (green line) iteration process to the fixed point 1 of the mapping T defined in Example 1.

They also proved that the K ∗ iteration process is faster than the PicardS iteration and S-iteration processes with the help of a numerical example.

3. Convergence results for Suzuki generalized nonexpansive mappings In this section, we prove some strong and ∆-convergence theorems of a sequence generated by a K ∗ iteration process for Suzuki generalized nonexpansive mappings in the setting of CAT (0) space. The K ∗ iteration process in the language of CAT (0) space is given by x0 ∈ C zn = (1 − βn )xn ⊕ βn T xn . yn = T ((1 − αn )zn ⊕ αn T zn ) xn+1 = T yn

(3.1)

Lemma 3.1. Let C be a nonempty closed convex subset of a CAT (0) space X and T : C → C be a Suzuki generalized nonexpansive mapping with F (T ) 6= ∅. For arbitrarily chosen x0 ∈ C, let the sequence {xn } be generated by (3.1). Then lim d(xn , p) exists for any p ∈ F (T ). n→∞

Proof. Let p ∈ F (T ) and z ∈ C. Since T is a Suzuki generalized nonexpansive mapping, 1 d(p, T p) = 0 ≤ d(p, z) implies that d(T p, T z) ≤ d(p, z). 2 By Proposition 2.6 (ii), we have 675

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d(zn , p)

= ≤ ≤ =

d(((1 − βn )xn ⊕ βn T xn ), p) (1 − βn )d(xn , p) + βn d(T xn , p) (1 − βn )d(xn , p) + βn d(xn , p) d(xn , p).

(3.2)

= ≤ ≤ ≤ ≤ =

d((T (1 − αn )zn ⊕ αn T zn ), p) d(((1 − αn )zn ⊕ αn T zn ), p) (1 − αn )d(zn , p) + αn d(T zn , p) (1 − αn )d(xn , p) + αn d(zn , p) (1 − αn )d(xn , p) + αn d(xn , p) d(xn , p).

(3.3)

Using (3.2), we get d(yn , p)

Similarly by using (3.3), we have d(xn+1 , p)

= d(T yn , p) ≤ d(yn , p) ≤ d(xn , p).

This implies that {d(xn , p)} is bounded and nonincreasing for all p ∈ F (T ). Hence lim d(xn , p) exists, as required.  n→∞

Theorem 3.2. Let C be a nonempty closed convex subset of a CAT (0) space X and T : C → C be a Suzuki generalized nonexpansive mapping. For arbitrary chosen x0 ∈ C, let the sequence {xn } be generated by (3.1) for all n ≥ 1, where {αn } and {βn } are sequences of real numbers in [a, b] for some a, b with 0 < a ≤ b < 1. Then F (T ) 6= ∅ if and only if {xn } is bounded and lim d(T xn , xn ) = 0. n→∞

Proof. Suppose F (T ) 6= ∅ and let p ∈ F (T ). Then, by Theorem 3.2, lim d(xn , p) n→∞

exists and {xn } is bounded. Put lim d(xn , p) = r.

n→∞

(3.4)

From (3.2) and (3.4), we have lim supd(zn , p) ≤ lim supd(xn , p) = r. n→∞

(3.5)

n→∞

By Proposition 2.6 (ii) we have lim supd(yn , p) ≤ lim supd(xn , p) = r. n→∞

(3.6)

n→∞

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On the other hand, by using (3.2), we have d(xn+1 , p)

= ≤ = ≤ ≤ ≤ =

d(T yn , p) d(yn , p) d((T (1 − αn )zn ⊕ αn T zn ), p) d(((1 − αn )zn ⊕ αn T zn ), p) (1 − αn )d(zn , p) + αn d(T zn , p) (1 − αn )d(xn , p) + αn d(zn , p) d(xn , p) − αn d(xn , p) + αn d(zn , p).

This implies that d(xn+1 , p) − d(xn , p) ≤ d(zn , p) − d(xn , p). αn So d(xn+1 , p) − d(xn , p) ≤

d(xn+1 , p) − d(xn , p) ≤ d(zn , p) − d(xn , p), αn

which implies that d(xn+1 , p) ≤ d(zn , p). Therefore, r ≤ lim inf d(zn , p).

(3.7)

n→∞

By (3.5) and (3.7), we get r

= = =

lim d(zn , p)

n→∞

lim d(((1 − βn )xn + βn T xn ), p)

n→∞

lim d(βn (T xn , p) + (1 − βn )(xn , p)).

(3.8)

n→∞

From (3.4), (3.6), (3.8) and Lemma 2.8, we have that lim d(T xn , xn ) = n→∞

0. Conversely, suppose that {xn } is bounded and lim d(T xn , xn ) = 0. Let n→∞

p ∈ A(C, {xn }). By Proposition 2.6 (iii), we have r(T p, {xn })

=

lim supd(xn , T p) n→∞

≤ lim sup(3d(T xn , xn ) + d(xn , p)) n→∞

≤ lim supd(xn , p) n→∞

= r(p, {xn }). This implies that T p ∈ A(C, {xn }). Since X is uniformly convex, A(C, {xn }) is a singleton and hence we have T p = p. So F (T ) 6= ∅.  Now we are in the position to prove ∆-convergence theorem. 677

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Theorem 3.3. Let C be a nonempty closed convex subset of a complete CAT (0) space X and T : C → C be a Suzuki generalized nonexpansive mapping with F (T ) 6= ∅. Let {tn } and {sn } be sequences in [0, 1] such that {tn } ∈ [a, b] and {sn } ∈ [0, b] or {tn } ∈ [a, 1] and {sn } ∈ [a, b for some a, b with 0 < a ≤ b < 1. For an arbitrary element x1 ∈ C, {xn } ∆-converges to a fixed point of T . Proof. Since F (T ) 6= ∅, by Theorem 3.3, we have S that {xn } is bounded and lim d(T xn , xn ) = 0. We now let ww {xn } := A({un }) where the union is n→∞

taken over all subsequences {un } of {xn }. We claim that ww {xn } ⊂ F (T ). Let u ∈ ww {xn }. Then there exists a subsequence {un } of {xn } such that A({un }) = {u}. By Lemmas 2.9 and 2.10, there exists a subsequence {vn } of {un } such that ∆-limn {vn } = v ∈ C. Since lim d(vn , T vn ) = 0, v ∈ F (T ) n→∞ by Lemma 2.11. We claim that u = v. Suppose not. Since T is a Suzuki generalized nonexpansive mapping and v ∈ F (T ), limn d(xn , v) exists by Theorem 3.2. Then by uniqueness of asymptotic centers, lim supd(vn , v)
0 such that d(x, T x) ≥ f (d(x, F (T ))) for all x ∈ C, where d(x, F (T )) = inf p∈F (T ) d(x, p). Now we prove the strong convergence theorem using condition (I). Theorem 3.5. Let C be a nonempty closed convex subset of a CAT (0) space X and T : C → C be a Suzuki generalized nonexpansive mapping. For arbitrary chosen x0 ∈ C, let the sequence {xn } be generated by (3.1) for all n ≥ 1, where {αn } and {βn } are sequences of real numbers in [a, b] for some a, b with 0 < a ≤ b < 1 such that F (T ) 6= ∅. If T satisfies condition (I), then {xn } converges strongly to a fixed point of T . Proof. By Lemma 3.1, we see that lim d(xn , p) exists for all p ∈ F (T ) and n→∞

so lim d(xn , F (T )) exists. Assume that lim d(xn , p) = r for some r ≥ 0. If n→∞ n→∞ r = 0, then the result follows. Suppose r > 0. Then from the hypothesis and condition (I), f (d(xn , F (T ))) ≤ d(T xn , xn ).

(3.9)

Since F (T ) 6= ∅, by Theorem 3.3, we have lim d(T xn , xn ) = 0. So (3.9) n→∞ implies that lim f (d(xn , F (T ))) = 0.

(3.10)

n→∞

Since f is a nondecreasing function, from (3.10), we have lim d(xn , F (T )) = n→∞

0. Thus we have a subsequence {xnk } of {xn } and a sequence {yk }, yk ∈ F (T ), such that 1 d(xnk , yk ) < k for all k ∈ N. 2 So using (3.4), we get d(xnk+1 , yk ) ≤ d(xnk , yk ) < 679

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Hence d(yk+1 , yk ) ≤ d(yk+1 , xk+1 ) + d(xk+1 , yk ) 1 1 + k ≤ k+1 2 2 1 < → 0, as k → ∞. 2k−1 This shows that {yk } is a Cauchy sequence in F (T ) and so it converges to a point p. Since F (T ) is closed, p ∈ F (T ) and then {xnk } converges strongly to p. Since lim d(xn , p) exists, we have that xn → p ∈ F (T ). Hence n→∞ the proof is complete. 

References [1] M. Abbas and T. Nazir, A new faster iteration process applied to constrained minimization and feasibility problems, Mat. Vesn. 66 (2014), 223–234. [2] 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 (2007), 61–79. [3] M. Bridson and A. Heaflinger, Metric Space of Non-positive Curvature, Springer-Verlag, Berlin, 1999. [4] D. Burago, Y. Burago and S. Inavo, A Course in Metric Geometry, in: Graduate Studies in Mathematics, Vol. 33, Americal Mathematical Society, Providence, 2001. [5] F. Burhat and J. Tits, Groups reductifs sur un curps local, Inst. Hautes Etudes Sci. Publ. Math. 41 (1972), 5–251. [6] R. Chugh, V. Kumar and S. Kumar, Strong convergence of a new three step iterative scheme in Banach spaces, Am. J. Comput. Math. 2 (2012), 345–357. [7] S. Dhompongsa, W. A. Kirk and B. Panyanak, Nonexpansive set-valued mappings in metric and Banach spaces, J. Nonlinear Convex Anal. 8 (2007), 35–45. [8] S. Dhompongsa and B. Panyanak, On ∆-convergence theorem in CAT (0) spaces, Comput. Math. Appl. 56 (2008), 2572–2579. [9] F. Gursoy and V. Karakaya, A Picard-S hybrid type iteration method for solving a differential equation with retarded argument, arXiv:1403.2546v2 (2014). [10] S. Ishikawa, Fixed points by a new iteration method, Proc. Am. Math. Soc. 44 (1974), 147–150. [11] I. Karahan and M. Ozdemir, A general iterative method for approximation of fixed points and their applications, Adv. Fixed Point Theory 3 (2013), 520–526. [12] E. Karapinar and K. Tas, Generalized (C)-conditions and related fixed point theorems, Comput. Math. Appl. 61 (2011), 3370–3380. [13] H. Khatibzadeh and S. Ranjbar, A variational inequality in complete CAT (0) spaces, J. Fixed Point Theory Appl. 17 (2015), 557–574. [14] S.H. Khan, A Picard-Mann hybrid iterative process, Fixed Point Theory Appl. 2013, Article No. 69 (2013). [15] W. A. Kirk and B. Panyanak, A concept of convergence in geodesic spacces, Nonlinear Anal.–TMA 68 (2008), 3689–3696. 680

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[16] W. Lawaong and B. Panyanak, Approximating fixed points of nonexpansive nonself mappings in CAT (0) spaces, Fixed Point Theory Appl. 2010, Art. ID 367274 (2010). [17] W.R. Mann, Mean value methods in iteration, Proc. Am. Math. Soc. 4 (1953), 506–510. [18] W. Phuengrattana, Approximating fixed points of Suzuki-generalized nonexpansive mappings, Nonlinear Anal. Hybrid Syst. 5 (2011), 583–590. [19] W. Phuengrattana and S. Suantai, On the rate of convergence of Mann, Ishikawa, Noor and SP-iterations for continuous functions on an arbitrary interval, J. Comput. Appl. Math. 235 (2011), 3006–3014. [20] D. R. Sahu and A. Petrusel, Strong convergence of iterative methods by strictly pseudocontractive mappings in Banach spaces, Nonlinear Anal.–TMA 74 (2011), 6012–6023. [21] T. Suzuki, Fixed point theorems and convergence theorems for some generalized nonexpansive mappings, J. Math. Anal. Appl. 340 (2008), 1088–1095. [22] B.S Thakur, D. Thakur and M. Postolache, A new iterative scheme for numerical reckoning fixed points of Suzuki’s generalized nonexpansive mappings, Appl. Math. Comput. 275 (2016), 147–155. [23] G. C. Ugwunnadi, A. R. Khan and M. Abbas, A hybrid proximal point algorithm for finding minimizers and fixed points in CAT (0) spaces, J. Fixed Point Theory Appl. 20 (2018), 20:82. [24] K. Ullah and M. Arshad, New three step iteration process and fixed point approximation in Banach spaces, preprint. [25] Z. Yang and Y.J. Pu, Generalized Browder-type fixed point theorem with strongly geodesic convexity on Hadamard manifolds with applications, Indian J. Pure Appl. Math. 43 (2012), 129–144. Kifayat Ullah Department of Mathematics, University of Science and Technology, Township Campus, 44000, Bannu, Pakistan e-mail: [email protected] Dong Yun Shin Department of Mathematics, University of Seoul, Seoul 02504, Korea e-mail: [email protected] Choonkil Park Research Institute for Natural Sciences, Hanyang University, Seoul 04763, Korea e-mail: [email protected] Bakhat Ayaz Khan Department of Mathematics, University of Science and Technology, Township Campus, 44000, Bannu, Pakistan e-mail: [email protected]

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Nonlinear Discrete Inequalities Method for the Ulam Stability of First Order Nonlinear Difference Equations

3

R.Dhanasekaran1 , E.Thandapani2 and J.M.Rassias3 1 Department of Mathematics, Vel Tech Rangarajan Dr.Sagunthala, R and D Institute of Science and Technology, Chennai - 600 062, India. e-mail: [email protected] 2 Ramanujan Institute for Advanced Study in Mathematics, University of Madras, Chennai - 600 005, India. e-mail: [email protected] Pedagogical Department E.E.,Section of Mathematics and Informatics, National and Capodistrian University of Athens, Athens 15342,GREECE. e-mail:[email protected] Abstract

In this paper, first we derive some nonlinear discrete inequalities, and then as an application, we study the Ulam stability of the first order nonlinear difference equation ∆y(n) = f (n, y(n)), n ≥ n0 , where f is a given function. The obtained result on Ulam stability is new to the literature in the sense that our approach does not require the explicit form of solutions of the investigated equations. 2010 Mathematics Subject Classification: 39A30,39B82 Keywords and Phrases: Ulam stability, discrete inequality, nonlinear difference equation. 1. Introduction In the passed years, the Ulam stability of functional equations received a great attention.In general, we say that an equation is stable in the sense of Ulam if for

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every approximate solution of that equation there exists an exact solution of the equation near it. For more details on Ulam stability, one can refer to [13]. The problem of the Ulam stability of difference equations is related to the notion of the perturbation of discrete dynamical systems. In [2–5, 7–9, 11, 12, 14, 17], the authors studied Ulam stability of linear difference equations and in [16], the authors obtained some results on Ulam stability for some second order linear difference equations. In all these papers, the authors studied the Ulam stability of first and second order linear difference equations and it seems that no results dealing with Ulam stability for the nonlinear difference equations are available in the literature. Therefore the purpose of this paper is to study that Ulam stability of the following first order nonlinear difference equation ∆y(n) = f (n, y(n)), n ≥ N,

(1.1)

where f ∈ C(N, R) and N denotes the set of all non-negative integers, without using the explicit form of the solutions. Next, we present the definition of the Ulam stability for difference equations. Definition 1.1. The equation (1.1) is called stable in Ulam sense if there exists a constant L ≥ 0 such that for every  > 0 and every {y(n)} in R satisfying |∆y(n) − f (n, y(n))| ≤ , n ≥ 0

(1.2)

there exists a sequence {x(n)} in R with the properties ∆x(n) = f (n, x(n)), n ≥ 0

(1.3)

|y(n) − x(n)| ≤ L, n ≥ 0.

(1.4)

and

A sequence {y(n)} which satisfies (1.2) for some  > 0 is called an approximate solution of the nonlinear difference equation (1.1), and we reformulate the above definition as: the equation (1.1) is called Ulam stable if for every approximate solution of it there exists an exact solutions close to it. If in Definition 1.1, the

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number  is replaced by a sequence of positive numbers {(n)} and L from (1.4)by a sequence of positive numbers {η(n)} the equation (1.1) is called genelized stable in the Ulam sense. In this paper first we derive some nonlinear discrete inequalities, and as an application we investigate the Ulam stability of equations (1.1). 2. Nonlinear Discrete Inequalities In this section, we present some nonlinear discrete inequalities which provide us a powerful tool for investigating the Ulam stability of a nonlinear first order difference equations. We begin with the following results which can be found in: [[6], Theorem 41, pp.39]. Lemma 2.1. If a > 0 and 0 < α ≤ 1, then aα ≤ αa + (1 − α) and the equality holds if α = 1. Theorem 2.2. Let {u(n)}, {f (n)}, {g(n)} and {h(n)} be nonnegative real sequences defined for all n ∈ N, and u(n) ≤ f (n) + g(n)

n−1 X

h(s)uα (s),

(2.1)

s=0

where 0 < α ≤ 1. Then u(n) ≤ f (n) + g(n)

n−1 X

h(s)(αf (s) + (1 − α)) exp

s=0

n−1 X

! αf (t)g(t) .

(2.2)

t=s+1

Proof. Defining a sequence R(n) by R(n) =

n−1 X

h(s)uα (s),

s=0

then R(0) = 0 and u(n) ≤ f (n) + g(n)R(n). Now using Lemma 2.1, one can obtain ∆R(n) = h(n)uα (n) ≤ h(n)(f (n) + g(n)R(n))α ≤ (αh(n)f (n) + (a − α)h(n)) + αh(n)g(n)R(n)

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or R(n + 1) − (1 + αh(n)g(n))R(n) ≤ h(n)(αf (n) + (1 − α)). Multiplying (2.3) by

Qn

s=0 (1

(2.3)

+ αh(s)g(s))−1 , we have !

n Y ∆ R(n) (1 + αh(s)g(s))−1

n Y ≤ h(n)(αf (n) + (1 − α)) (1 + αh(s)g(s))−1 .

s=0

s=0

Summing up the last inequality from 0 to n − 1, we obtain R(n) ≤

n−1 X

h(s)(αf (s) + (1 − α))

s=0

≤

n−1 X

n−1 Y

(1 + αh(t)g(t))

t=s+1 n−1 X

h(s)(αf (s) + (1 − α)) exp

s=0

! αh(t)g(t) .

(2.4)

t=s+1

Using (2.4) in u(n) ≤ f (n) + g(n)R(n), we have the desired inequality (2.2). This completes the proof.



Corollary 2.3. Let u(n) and p(n) be non-negative real sequences defined for all n ∈ N such that u(n) ≤ c +

n−1 X

p(s)uα (s)

(2.5)

s=0

where c ≥ 0 and 0 < α ≤ 1. Then !   n−1 X cα + (1 − α) u(n) ≤ exp αp(s) . α s=0

(2.6)

Proof. Let f (n) = c ≥ 0, g(n) = 1 and h(n) = p(n) in (2.2), we have u(n) ≤ c +

n−1 X

p(s)(αc + 1 − α)

s=0

n−1 Y

(1 + αp(t))

t=s+1

n−1 n−1 Y (αc + (1 − α)) X = c+ αp(s) (1 + αp(t)) α s=0 t=s+1 ! n−1 Y (αc + (1 − α)) (1 + αp(s)) − 1 = c+ α s=0 !   n−1 X αc + (1 − α) ≤ αp(s) . exp α s=0

The proof is now complete.



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Theorem 2.4. Let u(n), p(n) and h(n) be non-negative real sequences for all n ∈ N and u(n) ≤ c +

n−1 X

p(s)u(s) +

s=0

n−1 X

h(s)uα (s),

(2.7)

s=0

where c ≥ 0 and 0 < α ≤ 1. Then u(n) ≤

c + (1 − α)

n−1 X

! h(s) exp

s=0

n−1 X

! (p(s) + αh(s)) .

(2.8)

s=0

Proof. Let R(n) be the right hand side of (2.7). Then R(0) = c and u(n) ≤ R(n) and ∆R(n) = p(n)u(n) + h(n)uα (n) ≤ p(n)R(n) + h(n)Rα (n) ≤ p(n)R(n) + h(n)(αR(n) + (1 − α)) = (p(n) + αh(n))R(n) + (1 − α)h(n)

(2.9)

where we have used Lemma 2.1. Now from (2.9), we have R(n + 1) − (1 + p(n) + αh(n))R(n) ≤ (1 − α)h(n). Arguing as in the proof of Theorem 2.2, one can easily obtain the desired result and hence the details are omitted.



Remark 2.1. (a) If α = 1 in Theorem 2.2, then it reduced to the well-known Pachpatte inequality [10], in 2002. For 0 < α < 1 the estimate (2.2) of Theorem 2.2 is new to the literature. (b) If α = 1 and g(n) ≡ 1, then Theorem 2.2 reduced to a well-known result due to Sugiyama [15], in 1969. Remark 2.2. If α = 1 in Corollary 2.3, then it reduced to the discrete analogue of the well-known Gronwall-Bellman inequality [1]. Remark 2.3. The result obtained in Theorem 2.4 is different from that one by Willet and Wong [18] for the case 0 < α < 1.

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3. Ulam Stability As an application of the discrete inequalities established in Section 2, we investigate the Ulam stability of equation (1.1). Theorem 3.1. Let p(n) be a positive real sequence for all n ∈ N such that |f (n, u) − f (n, v)| ≤ p(n)|u − v|α

(3.1)

where 0 < α ≤ 1, and ∞ X

p(n) < ∞.

(3.2)

n=0

If for a positive real sequence φ(n) such that

P∞

n=0

φ(n) < ∞, and

|∆y(n) − f (n, y(n))| ≤ φ(n)

(3.3)

then there exists a real sequence x(n) and a constant k > 0 satisfying ∆x(n) = f (n, x(n))

(3.4)

such that |y(n) − x(n)| ≤ k; that is, equation (1.1) has the Ulam stability. Proof. From the inequality (3.3), we have y(n) ≤ y(0) +

n−1 X

f (s, y(s)) +

s=0

n−1 X

φ(s)

(3.5)

s=0

and from the equation (3.4), we obtain x(n) = x(0) +

n−1 X

f (s, x(s)).

(3.6)

s=0

Combining (3.5) and (3.6) yields |y(n) − x(n)| ≤ |y(0) − x(0)| +

n−1 X

|f (s, y(s)) − f (s, x(s))| +

s=0

n−1 X

φ(s).

s=0

Using the condition (3.1) in the above inequality, we have |y(n) − x(n)| ≤ M1 +

n−1 X

p(s)|y(s) − x(s)|α + M2

(3.7)

s=0

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where M1 = |y(0) − x(0)| and

P∞

n=0

φ(n) ≤ M2 by hypothesis. Now applying

Corollary 2.3 in (3.7), we obtain ! n−1 X ((M1 + M2 )α + (1 − α)) exp αp(s) . (3.8) |y(n) − x(n)| ≤ α s=0 P It follows from (3.2) that there is a constant M3 > 0 such that ∞ n=0 p(n) ≤ M3 , and using this in (3.8), one obtains |y(n) − x(n)| ≤ k where k =

((M1 +M2 )α+(1−α)) α

exp(αM3 ). This completes the proof.



4. Conclusion In this paper, first we have obtained some new nonlinear discrete inequalities and then as an application we investigate the Ulam stability of a nonlinear first order difference equation. In this approach, we do not need to require the explicit form of the solution of the studied equation, where as in [3,4,7-9,11,12,14] the authors used the explicit form of the solutions to prove their established results.

References [1] R.P.Agarwal, Difference Equations and Inequalities, Second Edition, Marcel Dekker, New York, 2000. [2] A.R.Baias, F.Blaga and D.Popa, Best Ulam constant for a linear difference equations, Carpathian J. Math., 35(2019), 13-21. [3] J.Brzdek, D.Popa and B.Xu, The Hyers-Ulam stabiility of nonlinear recurrences, J. Math. Anal. Appl., 335(2007), 443-449. [4] J.Brzdek, D.Popa and B.Xu, Remarks on stability of linear recurrence of higher order, Appl. Math.Lett., 23(2010), 1459-1463. [5] J.Brzdek, D.Popa and B.Xu, On nonstability of the linear recurrence of order one, J. Math. Anal. Appl., 367(2010), 146-153.

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[6] G.H.Hardy, J.E.Littlewood and G.Polya, Inequalities, Cambridge Univ. Press, Cambridge, 1934. [7] S.M.Jung, Hyers-Ulam stability of the first order matrix differene equations, Adv. Diff. Equ., 2015(2015), Art. ID.2015:170, 13 pages. [8] S.M.Jung and Y.W.Nam, On the Hyers-Ulam stability of the first order difference equation, J. Funct. Spaces, Vol.2016, Art. ID:6078298, 6 pages. [9] M.Onitsuka, Hyers-Ulam stability of first order nonhomogeneous difference equations with a constant stepsize, Appl. Math. Comput., 330(2018), 143-151. [10] B.G.Pachpatte, Inequalities For Finite Difference Equations, Marcel Dekker, New York, 2002. [11] D.Popa, Hyers-Ulam stability of the linear recurrence with constant coefficients, Adv. Diff. Equ., 2(2005), 101-107. [12] D.Popa, Hyers-Ulam-Rassias stability of a linear recurrence, J. Math. Anal. Appl., 309(2005), 591-597. [13] J.M.Rassias, E.Thandapani, K.Ravi and B.V.Senthil kumar,Book: Functional Equations and Inequalities: Solutions and Stability Results,Series on Concrete and Applicable Mathematics:Vol.21, World Sci. Pub.Comp., Singapore, 2017,Pages 396. [14] Y.Shen and Y.Li, The Z-transform method for the Ulam stability of linear difference equations with constant coefficients, Adv. Diff. Equ., (2018), 2018:396, 15 pages. [15] S.Sugiyama, On the stability of difference equations, Bull.Sci.Engr.Research Lab., Waseda Univ., 45(1969), 140-144. [16] A.K.Tripathy, Hyers-Ulam stability of second order linear difference equation, Inter. J. Diff. Equ. Appl., 16(2017), 53-65. [17] A.K.Tripathy and P.Senapati, Hyers-Ulam stability of first order linear difference operators on Banach spaces, J. Adv. Math., 14(2018), 7475-7485. [18] D.Willett and J.S.W.Wong, On the discrete analogues of some generalizations

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of Gronwall’s inequality, Monatsh. Math., 69(1965), 362-367.

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Algebras and Smarandache Types Jung Mi Ko1 and Sun Shin Ahn2,∗ 1

Department of Mathematics, Gangneung-Wonju National University, Gangneung 25457, Korea 2

Department of Mathematics Education, Dongguk University, Seoul 04620, Korea

Abstract. In this paper we introduce the notion of a BQ-algebra and show that is is equivalent to an abelian group. For deep investigations of several algebraic structures, we introduce the notions of a Smarandache V algebra-type U -algebra and a Smarandache V -algebra-trans-type U -algebra, and apply the notions to several algebras.

1. Introduction W. B. Vasantha Kandasamy ([8]) studied the concept of Smarandache groupoids, ideals of groupoids, Smarandache Bol groupoids and strong Bol groupoids, and obtained many interesting results about them. Smarandace semigroups are very important for the study of congruences, and it was studied by R. Padilla ([18]). It will be very interesting to study the Smarandache structure in general algebraic structures. Kim et al. ([11]) defined the concept of a Smarndache d-algebra and investigated some related properties of it. Seo et al. ([19]) introduced the concept of a Smarndache fuzzy BCI-algebra and investigated some related properties of it. Neggers et al. ([17]) defined the notion of a B-algebra and investigated some related properties of it. Some properties of B-algebra are studied in ([3, 12, 13]). In this paper, we introduce the notion of a BQ-algebras and show that it is equivalent to an abelian group. Moreover, we introduce the notions of a Smarandache V -algebra-type U -algebra and a Smarandache V -algebra-trans-type U -algebra, and apply the notions to several algebras. 2. Preliminaries A B-algebra ([17]) is a non-empty set X with a selected point 0 and a binary operation “∗” satisfying the following axioms: (i) x ∗ x = 0, (ii) x ∗ 0 = x, (iii) (x ∗ y) ∗ z = x ∗ (z ∗ (0 ∗ y)) for any x, y, z ∈ X. A B-algebra (X, ∗, 0) is said to be 0-commutative ([2]) if x ∗ (0 ∗ y) = y ∗ (0 ∗ x) for any x, y ∈ X. Let (X, ∗, 0) be a B-algebra and let g ∈ X. We define g [0] := 0, g [1] := g [0] ∗ (0 ∗ g) = 0 ∗ (0 ∗ g) = g and g [n] := g [n−1] ∗ (0 ∗ g) where n ≥ 1.

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2010 Mathematics Subject Classification: 06F35; 03G25. Keywords: Smarandache algebra, point algebra, p-derived algebra. The corresponding author. Tel: +82 2 2260 3410, Fax: +82 2 2266 3409 0 E-mail: [email protected] (J. M. Ko); [email protected] (S. S. Ahn) 0

∗

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J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 29, NO.4, 2021, COPYRIGHT 2021 EUDOXUS PRESS, LLC

Jung Mi Ko and Sun Shin Ahn

Theorem 2.1. Let (X, ∗, 0) be a B-algebra and let g ∈ X. Then { [m−n] g if m ≥ n, [m] [n] g ∗g = [n−m] 0∗g otherwise.

Theorem 2.2. ([10]) Every 0-commutative B-algebra is a BCI-algebra. Theorem 2.3. ([10]) The following are equivalent: (i) X is an abelian group, (ii) X is a p-semisimple BCI-algebra, (iii) X is a 0-commutative B-algebra. Let (X, ∗, 0) be a B-algebra. Given x, y ∈ X, we define x ∗⟨1⟩ y := x ∗ y, x ∗⟨2⟩ y := (x ∗ y) ∗ y, x ∗⟨n⟩ y := (x ∗⟨n−1⟩ y) ∗ y where n ≥ 3. For general references for BCK/BCI-algebras, we refer to [5, 6, 14].

3. Several algebras Let (X, ∗) be a groupoid (or a binary system, an algebra), i.e., X is a set and “∗” is a binary operation on X. If we take an element p in X which plays an important role in (X, ∗), then we say that p is a selected point and we write it by (X, ∗, p). Such an algebra (X, ∗, p) is said to be a pointed algebra. Example 3.1. Let (X, ∗) be a group with identity e. The identity element e plays an important role in (X, ∗) and hence we may write it by (X, ∗, e) and e becomes a selected point in (X, ∗). We regard all algebras below as pointed algebras without loss of generality. For simplicity’s sake, we shall write p = 0, not intending 0 to have the usual meaning. Thus, in Example 3.1, (X, ∗, e) becomes (X, ∗, 0) unless it is important to distinguish the algebra (X, ∗, 0) which contains the subalgebra (not necessary a subgroup) (Y, ∗) with its selected point p to produce (Y, ∗, p). Example 3.2. Consider X := {a, b, c, d} with the following table: ∗ a b c d

a a a a a

b a a b b

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c a a c c

d a b c d

Jung Mi Ko 691-699

J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 29, NO.4, 2021, COPYRIGHT 2021 EUDOXUS PRESS, LLC

Algebras and Smarandache Types

Then (X, ∗, d) is an pointed algebra and the selected point d is the right identity. Consider Y := {a, c} and Z := {a, d} with the following tables: ∗ a c a a a c a c

∗ a d a a a d a d

Then (Y, ∗, c) is a pointed algebra with a selected point c is the right identity, and (Z, ∗, d) is also a pointed algebra with a special point d is the left identity. Definition 3.3. Let (X, ∗, p) be a pointed algebra. Define a binary operation “•” on X by x • y := x ∗ (p ∗ y) for any x, y ∈ X. Then the algebra (X, •, p) is called a p-derived algebra from (X, ∗, p). Example 3.4. (i) Let (X, ∗, e) be a group with identity e. If (X, •, e) is an e-derived algebra of (X, ∗, e), then (X, •) = (X, ∗), since e is the identity, we have x • y = x ∗ (e ∗ y) = x ∗ y for all x, y ∈ X. (ii) Let (X, ∗, p) be a left-zero-semigroup with a selected point p. If (X, •, p) is a p-derived algebra of (X, ∗, p), then (X, ∗) = (X, •). Let X be a d-algebra and x ∈ X. Define x ∗ X := {x ∗ a|a ∈ X}. X is said to be edge ([16]) if for any x ∈ X, x ∗ X = {x, 0}. Lemma 3.5. ([16]) Let X be an edge d-algebra. Then (i) x ∗ 0 = x for all x ∈ X. (ii) (x ∗ (x ∗ y)) ∗ y = 0 for all x, y ∈ X. Example 3.6. (i) Let (X, ∗, 0) be an edge d-algebra. If (X, •, 0) is an e-derived algebra of (X, ∗, 0), then (X, •) is a left-zero-semigroup. (ii) Let (X, ∗, 0) be a BCK-algebra. If (X, •, 0) is an e-derived algebra of (X, ∗, 0), then (X, •) is a left-zero-semigroup. In fact, x • y = x ∗ (0 ∗ y) = x ∗ 0 = x for all x, y ∈ X. In terms of list of axioms to be used to describe the various algebra types we note the following section of axioms: (1) (2) (3) (4) (5) (6) (7) (8)

x ∗ x = 0 for all x ∈ X. x ∗ 0 = x for all x ∈ X. 0 ∗ x = x for all x ∈ X. x ∗ y = y ∗ x for all x, y ∈ X. x ∗ y = y ∗ x = 0 ⇔ x = y for all x, y ∈ X. x ∗ y = y ∗ x = 0 ⇒ x = y for all x, y ∈ X. x ∗ y = y ∗ x ⇒ x = y for all x, y ∈ X. 0 ∗ x = 0 for all x ∈ X.

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Jung Mi Ko 691-699

J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 29, NO.4, 2021, COPYRIGHT 2021 EUDOXUS PRESS, LLC

Jung Mi Ko and Sun Shin Ahn

(9) (10) (11) (12) (13) (14) (15) (16) (17) (18)

(x ∗ y) ∗ z = (x ∗ z) ∗ y for all x, y, z ∈ X. (x ∗ y) ∗ z = x ∗ (z ∗ y) for all x, y, z ∈ X. (x ∗ y) ∗ z = x ∗ (z ∗ (0 ∗ y)) for all x, y, z ∈ X. (x ∗ y) ∗ z = (x ∗ z) ∗ (y ∗ z) for all x, y, z ∈ X. (x ∗ y) ∗ (0 ∗ y) = x for all x, y ∈ X. x ∗ (y ∗ z) = (x ∗ y) ∗ z for all x, y, z ∈ X. (x ∗ (x ∗ y)) ∗ y = 0 for all x, y ∈ X. ((x ∗ y) ∗ (x ∗ z)) ∗ (x ∗ y) = 0 for all x, y, z ∈ X. for any x ∈ X, there exists y ∈ X with x ∗ y = 0. for any x ∈ X, there exists y ∈ X with y ∗ x = 0.

An algebra (X, ∗, 0) is called a group if it satisfies (2),(3), (14), (17), and (18). An algebra (X, ∗) is called a semigroup if it satisfies (14). An algebra (X, ∗, 0) is called a semigroup with identity if it satisfies (2),(3), and (14). An algebra (X, ∗, 0) is called a B-algebra ([17]) if it satisfies (1),(2), and (11). An algebra (X, ∗, 0) is called a BG-algebra ([9]) if it satisfies (1),(2), and (13). An algebra (X, ∗, 0) is called a BH-algebra ([7]) if it satisfies (1), (2), and (6). An algebra (X, ∗, 0) is called a Q-algebra ([15]) if it satisfies (1), (2), and (9). An algebra (X, ∗, 0) is called a d-algebra ([16]) if it satisfies (1), (5), and (8). An algebra (X, ∗, 0) is called a BCK-algebra ([14]) if it satisfies (1), (5), (8), (15), and (16). An algebra (X, ∗, 0) is called a gBCK-algebra ([4]) if it satisfies (1), (2), (9) and (12). An algebra (X, ∗, 0) is called an abelian group if it satisfies (2), (3), (4), (14), (17), and (18). An algebra (X, ∗, 0) is called a commutative semigroup if it satisfies (4) and (14). 4. BQ-algebras In this section, we introduce the notion of a BQ-algebra and we show that it is equivalent to an abelian group. An algebra (X, ∗, 0) is said to be a BQ-algebra if it satisfies the conditions (1), (2), (9) and (11). Theorem 4.1. Let (X, ∗, 0) be a BQ-algebra. If we define x • y := x ∗ (0 ∗ y) for any x, y ∈ X, then (X, •, 0) is an abelian group. Proof. Since (X, ∗, 0) is a BQ-algebra, it is both a B-algebra and a Q-algebra. It was proved that if (X, ∗, 0) is a B-algebra, then (X, •, 0) is a group ([1]). By (9) we obtain (x ∗ (0 ∗ y)) ∗ (0 ∗ z) = (x∗(0∗z))∗(0∗y) for any x, y, z ∈ X. It follows that (x•y)•z = (x•z)•y for any x, y, z ∈ X. If we take x := 0, then (0•y)•z = (0•z)•y. Since (X, ∗, 0) is a B-algebra, we have 0•y = 0∗(0∗y) = y for any y ∈ X. Hence we obtain y • z = z • y for any y, z ∈ X. This proves that (X, •, 0) is an abelian group. □ Theorem 4.2. Let (X, •, 0) be an abelian group. If we define x ∗ y := x • y −1 for any x, y ∈ X, then (X, ∗, 0) is a BQ-algebra.

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J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 29, NO.4, 2021, COPYRIGHT 2021 EUDOXUS PRESS, LLC

Algebras and Smarandache Types

Proof. (1) For any x ∈ X, we have x∗x = x•x−1 = 0. (2) For any x ∈ X, x∗0 = x•0−1 = x•0 = x. (9) Given x, y, z ∈ X, since (X, •, 0) is a group, we obtain (x ∗ y) ∗ z = (x • y −1 ) • z −1 = (x • z −1 ) • y −1 = (x ∗ z) ∗ y. (11) Given x, y, z ∈ X, since (X, •, 0) is a group, we have x ∗ (z ∗ (0 ∗ y)) = x • [z • [(y −1 )−1 • 0−1 ]]−1 = x • [z • (y • 0)]−1 = x • (z • y)−1 = x • (y −1 • z −1 ). Similarly, we prove that (x ∗ y) ∗ z = (x • y −1 ) • z −1 . Since (X, •, 0) is a group, we obtain (x ∗ y) ∗ z = x ∗ (z ∗ (0 ∗ y)). Hence (X, ∗, 0) is a BQ-algebra. □ By Theorems 4.1 and 4.2, we conclude that the class of all BQ-algebras is equivalent to the class of all abelian groups. The interesting fact to note is that we are able to take advantage of the relationship x • y = x ∗ (0 ∗ y) to understand better what the meaning of the class of BQ-algebra is. Other such questions around in this setting as well as others. E.q., what class of B-algebras corresponds to the class of solvable groups ? Can it be considered to be of the form: B“V ”-algebras corresponds to solvable groups where “V ”-algebras is some nicely identifiable class, as the same as the class for BQ-algebras ? 5. Smarandache types Let (X, ∗) be an U -algebra. Then (X, ∗) is said to be a Smarandache V -algebra-type U -algebra if there exists Y ⊆ X such that (Y, ∗) is a non-trivial subalgebra of (X, ∗) and |Y | ≥ 2, and (Y, ∗) is a V -algebra. For example, a B-algebra (X, ∗, 0) is said to be a Smarandache Q-algebra-type B-algebra it it contains a non-trivial sub-B-algebra (Y, ∗, 0) of (X, ∗, 0) and |Y | ≥ 2, and (Y, ∗, 0) is a Q-algebra. Similarly, a Q-algebra (X, ∗, 0) is called a Smarandache group-type Q-algebra if it contains a non-trivial sub-Q-algebra (Y, ∗, 0) of (X, ∗, 0), and (Y, ∗, 0) is a group where |Y | ≥ 2. Theorem 5.1. There is no Smarandache d-algebra-type commutative groupoid. Proof. Assume that there is a Smarandache d-algebra-type commutative groupoid (X, ∗, 0). Then there exists Y ⊆ X such that (Y, ∗, 0) is a non-trivial subgroupoid of a commutative groupoid (X, ∗, 0), |Y | ≥ 2 and (Y, ∗, 0) is a d-algebra. It follows that 0 ∗ y = 0 for all y ∈ Y . Since (X, ∗, 0)

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Jung Mi Ko 691-699

J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 29, NO.4, 2021, COPYRIGHT 2021 EUDOXUS PRESS, LLC

Jung Mi Ko and Sun Shin Ahn

is a commutative groupoid and Y ⊆ X, we obtain 0 ∗ y = y ∗ 0 = 0 for all y ∈ Y . Since (X, ∗, 0) is a d-algebra and Y ⊆ X, we obtain y = 0, i.e., |Y | = 1, a contradiction. □ Theorem 5.2. There is no Smarandache semigroup-type d-algebra. Proof. Assume that there is a Smarandache semigroup-type d-algebra (X, ∗, 0). Then there exists Y ⊆ X such that (Y, ∗, 0) is a non-trivial subalgebra of a d-algebra (X, ∗, 0), |Y | ≥ 2 and (Y, ∗, 0) is a semigroup. It follows that 0 ∗ (y ∗ 0) = 0 for any y ∈ Y , since Y ⊂ X and (X, ∗, 0) is a d-algebra. Hence y ∗ 0 = y ∗ (0 ∗ (y ∗ 0)) = (y ∗ 0) ∗ (y ∗ 0) = 0. Since Y ⊆ X and (X, ∗, 0) is a d-algebra, we obtain 0 ∗ y = 0 for all y ∈ Y . By (6), we have y = 0, i.e., |Y | = 1, a contradiction. □ Theorem 5.3. A Smarandache group-type B-algebra is equal to a Smarandache Boolean-grouptype B-algebra. Proof. Since every Boolean group is a group, it is enough to show that every Smarandache grouptype B-algebra is a Smarandache Boolean-group-type B-algebra. Assume (X, ∗, 0) is a Smarandache group-type B-algebra. Then there exists Y ⊆ X such that |Y | ≥ 2, (Y, ∗, 0) is a non-trivial subalgebra of a B-algebra and (Y, ∗, 0) is a group. For any y ∈ Y , since Y ⊆ X and (X, ∗, 0) is a B-algebra, we obtain y ∗ y = 0. Since (Y, ∗) is a group, the order of y is 2 in the group (Y, ∗) for any y ̸= 0 in Y and hence (Y, ∗, 0) is a Boolean group, proving the theorem. □ Corollary 5.4. A Smarandache group-type Q-algebra is equal to a Smarandache Boolean-grouptype Q-algebra. Proof. Every Q-algebra has also the condition (1), and the proof is similar to the proof of Theorem 5.3. □ Theorem 5.5. Every Smarandache B-algebra-type group is a Smarandache Boolean-group-type group. Proof. Let (X, ∗, 0) be a Smarandache B-algebra-type group. Then there exists Y ⊆ X such that |Y | ≥ 2, (Y, ∗, 0) is a non-trivial subgroup of a group (X, ∗, 0) and (Y, ∗, 0) is a B-algebra. It follows that y ∗ y = 0 for all y ∈ Y . Since Y ⊆ X and (X, ∗, 0) is a group, we obtain y = y −1 in the group. Hence x ∗ y −1 = x ∗ y ∈ Y , which shows that (Y, ∗) is a subgroup of (X, ∗) and the order of y is 2. Thus (Y, ∗) is a Boolean group. This proves that (X, ∗, 0) is a Smarandache Boolean-group-type group. □

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Jung Mi Ko 691-699

J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 29, NO.4, 2021, COPYRIGHT 2021 EUDOXUS PRESS, LLC

Algebras and Smarandache Types

Theorem 5.6. Let (X, ∗, 0) be a Smarandache L-algebra-type M -algebra. If every L-algebra is an N -algebra, then (X, ∗, 0) is a Smarandache N -algebra-type M -algebra. □

Proof. It is easy and omit the proof.

Theorem 5.7. Let (X, ∗, 0) be a Smarandache 0-commutative-B-algebra-type M -algebra. Then (X, ∗, 0) is a Smarandache BCI-algebra-type M -algebra, where M -algebra is any algebra. □

Proof. By applying Theorems 2.2 and 5.6, we prove the theorem. Theorem 5.8. Let (X, ∗, 0) be an M -algebra. Then the following are equivalent: (i) X is a Smarandache abelian-group-type M -algebra (ii) X is a Smarandache p-semisimple BCI-algebra-type M -algebra, (iii) X is a Smarandache 0-commutative B-algebra-type M -algebra.

□

Proof. It follows immediately from Theorems 2.3 and 5.6.

Proposition 5.9. If (X, ∗, 0) is a Smarandache Q-algebra-type group, then it is a Smarandache Boolean-group-type group. Proof. Let (X, ∗, 0) be a Smarandache Q-algebra-type group. Then there exists Y ⊆ X such that |Y | ≥ 2, (Y, ∗, 0) is a non-trivial subgroup of a group (X, ∗, 0) and (Y, ∗, 0) is a Q-algebra. Since Y is a Q-algebra, we have y ∗ y = 0 for any y ∈ Y . This means the order of y is 2 in the group (Y, ∗), i.e., y = y −1 , which shows that (Y, ∗, 0) is a Boolean-group. Hence (X, ∗, 0) is a Smarandache Boolean-group-type group. □ Theorem 5.10. Any non-trivial d-algebra cannot be a Smarandache group-type d-algebra. Proof. Assume there exists a Smarandache group-type d-algebra (X, ∗, 0). Then there exists Y ⊆ X such that (Y, ∗, 0) is a non-trivial sub-d-algebra of (X, ∗, 0) and (Y, ∗, 0) is a group where |Y | ≥ 2. Since (Y, ∗, 0) is a group and (X, ∗, 0) is a d-algebra, we have y = 0 ∗ y = 0 for all y ∈ Y . It follows that |Y | = 1, a contradiction. □ Theorem 5.11. Any non-trivial group cannot be a Smarandache d-algebra-type group. Proof. Assume that there exists a Smarandache d-algebra-type group (X, ∗, 0). Then there exists Y ⊆ X such that (Y, ∗, 0) is a non-trivial subgroup of a group (X, ∗, 0), and (Y, ∗, 0) is a d-algebra and |Y | ≥ 2. Then 0 ∗ x = 0 for all x ∈ Y . Since (Y, ∗, 0) is a group, we obtain x = 0 for all x ∈ Y , proving that |Y | = 1, a contradiction. □ Theorem 5.12. Any non-trivial gBCK-algebra cannot be a Smarandache group-type gBCKalgebra. Proof. Let (X, ∗, 0) be a Smarandache group-type gBCK-algebra. Then there exists Y ⊆ X such that (Y, ∗, 0) is a non-trivial sub-gBCK-algebra of (X, ∗, 0), and (Y, ∗, 0) is a group and |Y | ≥ 2. Since Y ⊆ X and (X, ∗, 0) is a gBCK-algebra, we obtain y ∗ y = 0 for all y ∈ Y . It follows from

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Jung Mi Ko 691-699

J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 29, NO.4, 2021, COPYRIGHT 2021 EUDOXUS PRESS, LLC

Jung Mi Ko and Sun Shin Ahn

(Y, ∗, 0) is a group that the order of y is 2, i.e., (Y, ∗, 0) is a Boolean group. Now, since (Y, ∗, 0) is a gBCK-algebra, we have (x ∗ y) ∗ z = (x ∗ z) ∗ (y ∗ z) for all x, y, z ∈ X. It follows that (x ∗ x) ∗ x = (x ∗ x) ∗ (x ∗ x) for all x ∈ X. Since (X, ∗, 0) is a group, we obtain x = 0 for all x ∈ X, proving that |X| = 1, a contradiction. □ Corollary 5.13. Any non-trivial group cannot be a Smarandache gBCK-algebra-type group. □

Proof. The proof is similar to Theorem 5.12, and we omit it.

Definition 5.14. Let (X, ∗, p) be an L-algebra and let (Y, ∗, p) be both a sub-L-algebra of (X, ∗, p) and an M -algebra. (X, ∗, p) is said to be a Smarandache N -algebra-trans-type L-algebra if (Y, ∗, p) is isomorphic with an N -algebra (Y, ⊙, q). (X, ∗, p)

(Y, ∗, p)

∼ =

/ (Y, ⊙, q)

where L−, M −, N − algebras are arbitrary algebras. Theorem 5.15. If (X, ∗, 0) is a Smarandache B-algebra-type Q-algebra, then it is a Smarandache abelian-group-trans-type Q-algebra. Proof. Let (X, ∗, 0) be a Smarandache B-algebra-type Q-algebra. Then there exists Y ⊆ X such that (Y, ∗, 0) is a non-trivial sub-Q-algebra of a Q-algebra (X, ∗, 0), |Y | ≥ 2 and (Y, ∗, 0) is a B-algebra. Define x • y := x ∗ (0 ∗ y) for any x, y ∈ Y . Then (Y, •, 0) is an abelian group. In fact, since Y is both a Q-algebra and B-algebra, (Y, ∗, 0) is a BQ-algebra. By Theorem 4.1, (Y, •, 0) is an abelian group. By Theorems 4.1 and 4.2, (Y, ∗, 0) ∼ = (Y, •, 0). This shows that (X, ∗, 0) is a Smarandache abelian-group-trans-type Q-algebra. □ Corollary 5.16. If (X, ∗, 0) is a Smaradache Q-algebra-type B-algebra, then it is a Smarandache abelian-group-trans-type B-algebra. □

Proof. It is similar to Theorem 5.15. Proposition 5.17. Every B-algebra is a Smarandache BQ-algebra-trans-type B-algebra.

Proof. Let (X, ∗, 0) be a B-algebra. Define x • y := x ∗ (0 ∗ y) for all x, y ∈ X. Then (X, •, 0) is a group. Let x ∈ X such that x ̸= 0. Let ⟨x⟩ be a cyclic group generated by x. Then ⟨x⟩ is a non-trivial abelian subgroup of (X, •, 0). If we let Yx := {x ∗⟨n⟩ (0 ∗ x) | n ∈ Z}, then Yx ∼ = ⟨x⟩. By Theorems 4.1 and 4.2, Yx is a non-trivial BQ-algebra. This shows that X is a Smarandache BQ-algebra-trans-type B-algebra. □

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J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 29, NO.4, 2021, COPYRIGHT 2021 EUDOXUS PRESS, LLC

Algebras and Smarandache Types

6. Conclusion We introduced the notion of a BQ-algebra and proved that it is equivalent to an abelian group. For detailed investigations among several algebraic structures, we introduced the notions of a Smarandache V -type U -algebra and a Smarandache V -trans-type U -algebra, and applied this notions to several algebras. For further investigations, we will apply the notions of a hyper structure theory and several fuzzy related algebras to the notions of a Smarandache V -type U algebra and a Smarandache V -trans-type U -algebra. References [1] P. J. Allen, J. Neggers and H. S. Kim, B-algebras and groups, Sci. Math. Japo. 59(2004), 23-29. [2] J. R. Cho and H. S. Kim, On B-algebras and quasigroups, Quasigroups and Related Systems 8(2001), 1-6. [3] J. S. Han and S. S. Ahn, Quotient B-algebras induced by an int-soft normal subalgebras, J. Comput. Anal. Appl. 26(2019), no. 5, 791-801. [4] S. M. Hong, Y. B. Jun and M. A. Ozturk, Generalizations of BCK-algebras, Sci. Math. Japo. 58(2003), 603-611. [5] Y. Huang, BCI-algebras, Science Press, Beijing, 2006. [6] A. Iorgulescu, Algebras of logic as BCK-algebras, Editura ASE, Bucharest, 2008. [7] Y. B. Jun, E. H. Roh and H. S. Kim, On BH-algebras, Sci. Mathematicae 1(1998), 347-354. [8] W. B. V. Kandasamy, Smarandache groupoids, http://www.gallwp.unm.edu/∼smarandache/Groupoids.pdf. [9] C. B. Kim and H. S. Kim, On BG-algebras, Demonstratio Math. 41(2008), 497-505. [10] H. S. Kim and H. G. Park, On 0-commutative B-algebras, Sci. Math. Japo. 62(2005), 31-36. [11] Y. H. Kim, Y. H. Kim and S. S. Ahn, Smarandache d-algebras, Honam Math. J. 40(2018), no.3, 539-548. [12] J. M. Ko and S. S. Ahn, On fuzzy B-algebras over t-norm, J. Comput. Anal. Appl. 19(2015), no. 6, 975-983. [13] J. M. Ko and S. S. Ahn, Hesitant fuzzy normal subalgebras in B-algebras, J. Comput. Anal. Appl. 26(2019), no. 6, 1084-1094. [14] J. Meng and Y. B. Jun, BCK-algebras, Kyungmoon Sa, Seoul, 1994. [15] J. Neggers, S. S. Ahn and H. S. Kim, On Q-algebras, Int. J. Math. & Math. Sci. 27(2001), 749-757. [16] J. Neggers, and H. S. Kim, On d-algebras, Math. Slovaca 49(1999), 19-26. [17] J. Neggers and H. S. Kim, On B-algebras, Mate. Vesnik 54(2002), 21-29. [18] R. Padilla, Smarandache algebric structures, Bull. Pure Appl. Sci., 1998, 17E, 119-121. [19] Y. J. Seo and S. S. Ahn, Smarndache fuzzy BCI-algebras, J. Comput. Anal. Appl. 24(2018), no. 4, 619-627.

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Jung Mi Ko 691-699

J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 29, NO.4, 2021, COPYRIGHT 2021 EUDOXUS PRESS, LLC

Nonlinear differential equations associated with degenerate (h, q)-tangent numbers Cheon Seoung Ryoo Department of Mathematics, Hannam University, Daejeon 34430, Korea

Abstract : In this paper, we study nonlinear differential equations arising from the generating functions of degenerate (h, q)-tangent numbers. We give explicit identities for the degenerate (h, q)tangent numbers. Key words : Nonlinear differential equations, (h, q)-tangent numbers and polynomials, degenerate tangent numbers, degenerate (h, q)-tangent numbers, higher-order degenerate tangent numbers. AMS Mathematics Subject Classification : 05A19, 11B83, 34A30, 65L99. 1. Introduction Recently, many mathematicians have studied in the area of the degenerate Euler numbers and polynomials, degenerate Bernoulli numbers and polynomials, degenerate Genocchi numbers and polynomials, and degenerate tangent numbers and polynomials(see [1, 2, 3, 4, 5, 6, 7]). In [1], L. Carlitz introduced the degenerate Bernoulli polynomials. Recently, Feng Qi et al.[2] studied the partially degenerate Bernoull polynomials of the first kind in p-adic field. The degenerate (h, q)(h) tangent numbers Tn,q (λ) are defined by the generating function: ∞ ∑

(h) Tn,q (λ)

n=0

tn 2 = h . n! q (1 + λt)2/λ + 1

(1.1)

(k,h)

The degenerate (h, q)-tangent numbers of higher order, Tn,λ,q are defined by means of the following generating function ( )k ∑ ∞ 2 tn (k,h) = (1.2) Tn,q (λ) . h 2/λ n! q (1 + λt) +1 n=0 We recall that the classical Stirling numbers of the first kind S1 (n, k) and S2 (n, k) are defined by the relations(see [7]) n n ∑ ∑ (x)n = S1 (n, k)xk and xn = S2 (n, k)(x)k , k=0

k=0

respectively. Here (x)n = x(x − 1) · · · (x − n + 1) denotes the falling factorial polynomial of order n. We also have ∞ ∑ n=m

S2 (n, m)

∞ ∑ tn (et − 1)m tn (log(1 + t))m = and S1 (n, m) = . n! m! n! m! n=m

(1.3)

The generalized falling factorial (x|λ)n with increment λ is defined by (x|λ)n =

n−1 ∏

(x − λk)

(1.4)

k=0

for positive integer n, with the convention (x|λ)0 = 1. We also need the binomial theorem: for a variable x, (1 + λt)

x/λ

=

∞ ∑ n=0

700

(x|λ)n

tn . n!

(1.5)

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Many mathematicians have studied in the area of the linear and nonlinear differential equations arising from the generating functions of special numbers and polynomials in order to give explicit identities for special polynomials. In this paper, we study nonlinear differential equations arising from the generating functions of degenerate (h, q)-tangent numbers . We give explicit identities for the degenerate (h, q)-tangent numbers . 2. Nonlinear differential equations associated with degenerate (h, q)-tangent numbers In this section, we study nonlinear differential equations arising from the generating functions of degenerate twisted (h, q)-tangent numbers. Let F = F (t, λ, q, h) =

∞ ∑ 2 tn (h) = . T (λ) n,q n! q h (1 + λt)2/λ + 1 n=0

(2.1)

Then, by (2.1), we have F

(1)

( ) ∂ 2 ∂ = F (t, λ, q, h) = ∂t ∂t q h (1 + λt)2/λ + 1 ( ) ( )2 −4 2 1 1 = + 1 + λt q h (1 + λt)2/λ + 1 1 + λt q h (1 + λt)2/λ + 1 −2F + F 2 = . 1 + λt

(2.2)

By (2.2), we have F 2 = 2F + (1 + λt)F (1) .

(2.3)

Taking the derivative with respect to t in (2.3), we obtain 2F F (1) = 2F (1) + λF (1) + (1 + λt)F (2)

(2.4)

= (λ + 2)F (1) + (1 + λt)F (2) . From (2.2), (2.3), and (2.4), we have 2F 3 = 4F + (1 + λ)(1 + λt)F (1) + (1 + λt)2 F (2) . Continuing this process, we can guess that N !F N +1 =

N ∑

ai (N, λ, q, h)(1 + λt)i F (i) ,

(2.5)

(N = 0, 1, 2, . . .),

i=0

( where F (i) =

∂ ∂t

)i F (t, λ, q, h). Differentiating (2.5) with respect to t, we have

(N + 1)!F N F (1) =

N ∑

iλai (N, λ, q, h)(1 + λt)i−1 F (i) +

i=0

( (N + 1)!F F

(1)

= (N + 1)!F

ai (N, λ, q, h)(1 + λt)i F (i+1)

(2.6)

i=0

and N

N ∑

N

−2F + F 2 1 + λt

701

)

( = (N + 1)!

F N +2 − 2F N +1 1 + λt

) .

(2.7)

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J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 29, NO.4, 2021, COPYRIGHT 2021 EUDOXUS PRESS, LLC

By (2.5), (2.6), and (2.7), we have (N + 1)!F N +2 = 2(N + 1)!F N +1 +

N ∑

λiai (N, λ, q, h)(1 + λt)i F (i) +

i=0

N ∑

ai (N, λ)(1 + λt)i+1 F (i+1)

i=0

= 2(N + 1)

N ∑

ai (N, λ, q, h)(1 + λt)i F (i)

(2.8)

i=0

+

N ∑

λiai (N, λ, q, h)(1 + λt)i F (i) +

i=0

=

N ∑

N ∑

ai (N, λ, q, h)(1 + λt)i+1 F (i+1)

i=0

(2(N + 1) + λi) ai (N, λ, q, h)(1 + λt)i F (i) +

i=0

N +1 ∑

ai−1 (N, λ, q, h)(1 + λt)i F (i) .

i=1

Now replacing N by N + 1 in (2.5), we find (N + 1)!F N +2 =

N +1 ∑

ai (N + 1, λ, q, h)(1 + λt)i F (i) .

(2.9)

i=0

By (2.8) and (2.9), we have N +1 ∑

ai (N + 1, λ, q, h)(1 + λt)i F (i) =

i=0

N ∑

(2(N + 1) + λi) ai (N, λ, q, h)(1 + λt)i F (i)

i=0

+

N +1 ∑

i

ai−1 (N, λ, q, h)(1 + λt) F

(i)

(2.10)

.

i=1

Comparing the coefficients on both sides of (2.10), we obtain 2(N + 1)a0 (N, λ, q, h) = a0 (N + 1, λ, q, h),

(2.11)

aN +1 (N + 1, λ, q, h) = aN (N, λ, q, h), and ai (N + 1, λ, q, h) = (2(N + 1) + λi) ai (N, λ, q, h) + ai−1 (N, λ, q, h), (1 ≤ i ≤ N ).

(2.12)

In addition, by (2.5), we have F = a0 (0, λ, q, h)F,

(2.13)

a0 (0, λ, q, h) = 1.

(2.14)

which gives

It is not difficult to show that F 2 = a0 (1, λ, q, h)F + a1 (1, λ, q, h)(1 + λt)F (1) = 2F + (1 + λt)F (1) .

(2.15)

Thus, by (2.15), we also find a0 (1, λ, q, h) = 2,

a1 (1, λ, q, h) = 1.

(2.16)

From (2.11), we note that a0 (N + 1, λ, q, h) = 2(N + 1)a0 (N, λ, q, h) = 4(N + 1)N a0 (N − 1, λ, q, h) = · · · = 2N +1 (N + 1)!,

702

(2.17)

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J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 29, NO.4, 2021, COPYRIGHT 2021 EUDOXUS PRESS, LLC

and aN +1 (N + 1, λ, q, h) = aN (N, λ, q, h) = · · · = 1.

(2.18)

For i = 1, 2, 3 in (2.11), then we find that ) ( λ a0 (N − k, λ, q, h), 2k N + 1 + 2 k k=0 ( ) N −1 ∑ λ k 2 N + 1 + × 2 a1 (N − k, λ, q, h), a2 (N + 1, λ, q, h) = 2 k k=0 ( ) N −2 ∑ λ a3 (N + 1, λ, q, h) = 2k N + 1 + × 3 a2 (N − k, λ, q, h). 2 k

a1 (N + 1, λ, q, h) =

N ∑

k=0

Continuing this process, we can deduce that, for 1 ≤ i ≤ N, ai (N + 1, λ, q, h) =

N∑ −i+1 k=0

( ) λ 2k N + 1 + × i ai−1 (N − k, λ, q, h). 2 k

(2.19)

Note that, here the matrix ai (j, λ, q, h)0≤i,j≤N +1 is given by 

1  0  0   0 . . . 0

2 1 0

2!22 · 1

3!23 · ·

··· ··· ···

0 .. . 0

0 .. . 0

1 .. . 0

··· .. .

 (N + 1)!2N +1  ·    ·   ·   ..   .

···

1

Now, we give explicit expressions for ai (N + 1, λ, q, h). By (2.17), (2.18), and (2.19), we have a1 (N + 1, λ, q, h) =

N ∑ k1 =0

=

N ∑ k1 =0

a2 (N + 1, λ, q, h) =

N −1 ∑

2k2

k2 =0

=

N −1 N −k 2 −1 ∑ ∑ k2 =0

k1 =0

2

λ N +1+ 2

) a0 (N − k1 , λ, q, h) k1

( ) λ 2 (N − k1 )! N + 1 + , 2 k1 N

( ) λ N +1+ ×2 a1 (N − k2 , λ, q, h) 2 k2

( ) ( ) λ λ 2N −1 (N − k2 − k1 − 1)! N + 1 + × 2 , N − k2 + 2 2 k1 k2

and a3 (N + 1, λ, q, h) =

N −2 ∑

2k3

k3 =0

=

( k1

) ( λ N +1+ ×3 a2 (N − k3 , λ, q, h) 2 k3

) ( λ 2N −2 (N − k3 − k2 − k1 − 2)! N + 1 + × 3 2 k3 k3 =0 k2 =0 k1 =0 ) ( ) ( λ λ × · · · × N − k3 + × 2 N − k3 − k2 − 1 + 2 2 k1 k2 N −2 N −k 3 −2 N −k∑ 3 −k2 −2 ∑ ∑

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J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 29, NO.4, 2021, COPYRIGHT 2021 EUDOXUS PRESS, LLC

Continuing this process, we have ai (N + 1, λ, q, h) =

N∑ −i+1 N −k i −i+1 ∑ ki =0

N −ki−1 −···−k2 −i+1

∑

···

ki−1 =0

2N −i+1

k1 =0

× (N − ki − ki−1 − · · · − k2 − k1 − i + 1)! ( ) ( ) λ λ × N +1+ ×i N − ki + × (i − 1) 2 2 ki ki−1 ) ( λ × N − ki − ki−1 − 1 + × (i − 2) 2 ki−2 ( ) λ × N − ki − ki−1 − ki−2 − 2 + × (i − 3) ··· 2 ki−3 ) ( λ . × N − ki − ki−1 − ki−2 − · · · − k2 − i + 2 + 2 k1

(2.20)

Therefore, by (2.20), we obtain the following theorem. Theorem 1. For N = 0, 1, 2, . . . , the nonlinear functional equation N !F N +1 =

N ∑

ai (N, λ, q, h)(1 + λt)i F (i)

i=0

has a solution F = F (t, λ, q, h) =

2 , q h (1 + λt)2/λ + 1

where a0 (N, λ, q, h) = 2N N !, aN (N, λ, q, h) = 1, ai (N, λ, q, h) =

N −i N −k ∑ ∑i −i

···

ki =0 ki−1 =0

N −ki −···−k ∑ 2 −i

(2q h − x)N −i

k1 =0

× (N − ki − ki−1 − · · · − k2 − k1 − i)! ( ) ( ) λ λ × N + ×i N − ki − 1 + × (i − 1) 2 2 ki ki−1 ( ) λ × N − ki − ki−1 − 2 + × (i − 2) 2 ki−2 ( ) λ × N − ki − ki−1 − −ki−2 − 3 + × (i − 3) ··· 2 ki−3 ( ) λ × N − ki − ki−1 − ki−2 − · · · − k2 − i + 1 + . 2 k1 From (1.1) and (1.2), we note that ( N !F

N +1

= N!

2 q h (1 + λt)2/λ + 1

)N +1 = N!

∞ ∑

(N +1,h) Tn,q (λ)

n=0

tn . n!

(2.21)

From (2.5), we note that F (i) =

∞ ∑ ( ∂ )i tl (h) F (t, λ, q, h) = Ti+l,q (λ) . ∂t l!

(2.22)

l=0

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J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 29, NO.4, 2021, COPYRIGHT 2021 EUDOXUS PRESS, LLC

From Theorem 1, (1.5), (2.21), and (2.22), we can derive the following equation: N !F N

∞ ∑ n=0

∑ tn = ai (N, λ, q, h)(1 + λt)i F (i) n! i=0 N

(N +1,h) Tn,q (λ)

=

N ∑

ai (N, λ, q, h)

i=0

∞ ∑

∞

(i)k λk

k=0

tk ∑ (h) tl Ti+l,q (λ) k! l!

(2.23)

l=0

) (N n ( ) ∞ ∑ ∑∑ n tn k (h) . = ai (N, λ, q, h)(i)k λ Tn−k+i,q (λ) n! k n=0 i=0 k=0

By comparing the coefficients on both sides of (2.23), we obtain the following theorem. Theorem 2. For k, N = 0, 1, 2, . . . , we have (N +1,h) N !Tn,q (λ) =

N ∑ n ( ) ∑ n i=0 k=0

k

(h)

ai (N, λ, q, h)(i)k λk Tn−k+i,q (λ),

where a0 (N, λ) = N !2N , ai (N, λ) =

aN (N, λ) = 1,

N −i N −k ∑ ∑i −i

···

N −ki −···−k ∑ 2 −i

ki =0 ki−1 =0

2N −i

k1 =0

× (N − ki − ki−1 − · · · − k2 − k1 − i)! ) ( λ ··· × N − ki − ki−1 − −ki−2 − 3 + × (i − 3) 2 ki−3 ( ) λ × N − ki − ki−1 − ki−2 − · · · − k2 − i + 1 + . 2 k1 Acknowledgement: This work was supported by the National Research Foundation of Korea(NRF) grant funded by the Korea government(MEST) (No. 2017R1A2B4006092). REFERENCES 1. Carlitz, L.(1979). Degenerate Stirling, Bernoulli and Eulerian numbers, Utilitas Math. v.15, pp. 51-88. 2. Qi, F., Dolgy, D.V., Kim, T., Ryoo, C.S.(2015). On the partially degenerate Bernoulli polynomials of the first kind, Global Journal of Pure and Applied Mathematics, v.11, pp. 2407-2412. 3. Ryoo, C.S.(2015). Notes on degenerate tangent polynomials, Global Journal of Pure and Applied Mathematics v.11, pp. 3631-3637. 4. Ryoo, C.S.(2015). Note on degenerate tangent polynomials of higher order, Global Journal of Pure and Applied Mathematics v.11, pp. 4547-4554. 5. Ryoo, C.S.(2014). A numerical investigation on the zeros of the tangent polynomials, J. App. Math. & Informatics, v.32, pp. 315-322. 6. Ryoo, C.S.(2011). On the alternating sums of powers of consecutive odd integers, Journal of Computational Analysis and Applications, v.13, pp. 1019-1024. 7. Young, P.T.(2008) Degenerate Bernoulli polynomials, generalized factorial sums, and their applications, Journal of Number Theory, v. 128, pp. 738-758

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On the symmetries of the second kind (h, q)-Bernoulli polynomials C. S. RYOO Department of Mathematics, Hannam University, Daejeon 34430, Korea Abstract : In this paper, by applying the symmetry of the fermionic p-adic q-integral on Zp , we give recurrence identities the second kind (h, q)-Bernoulli polynomials and the sums of powers of consecutive (h, q)-odd integers. Key words : Bernoulli numbers and polynomials, the second kind Bernoulli numbers and polynomials, the second kind q-Bernoulli numbers and polynomials, the second kind (h, q)-Bernoulli numbers and polynomials. AMS Mathematics Subject Classification : 11B68, 11S40, 11S80. 1. Introduction Bernoulli numbers, Bernoulli polynomials, q-Bernoulli numbers, q-Bernoulli polynomials, the second kind Bernoulli number and the second kind Bernoulli polynomials were studied by many authors(see [1-8]). Bernoulli numbers and polynomials posses many interesting properties and arising in many areas of mathematics and physics. In [5], by using the second kind Bernoulli numbers Bj and polynomials Bj (x), we investigated the q-analogue of sums of powers of consecutive odd integers(see [6]). Let k be a positive integer. Then we obtain Ok (n − 1) =

n−1 ∑

(2i + 1)k−1 =

i=0

Bk (2n) − Bk . 2k (h)

(h)

In [4], we introduced the second kind (h, q)-Bernoulli numbers Bn,q and polynomials Bn,q (x). By using computer, we observed an interesting phenomenon of ‘scattering’ of the zeros of the second kind (h) (h, q)-Bernoulli polynomials Bn,q (x) in complex plane. Also we carried out computer experiments for doing demonstrate a remarkably regular structure of the complex roots of the second kind (h) (h, q)-Bernoulli polynomials Bn,q (x). In this paper, we give recurrence identities the second kind (h, q)-Bernoulli polynomials and the sums of powers of consecutive (h, q)-odd integers. Throughout this paper, we always make use of the following notations: N = {1, 2, 3, · · · } denotes the set of natural numbers, Z denotes the set of integers, R denotes the set of real numbers, C denotes the set of complex numbers, Zp denotes the ring of p-adic rational integers, Qp denotes the field of p-adic rational numbers, and Cp denotes the completion of algebraic closure of Qp . Let νp be the normalized exponential valuation of Cp with |p|p = p−νp (p) = p−1 . When one talks of q-extension, q is considered in many ways such as an indeterminate, a complex number q ∈ C, or p-adic number q ∈ Cp . If q ∈ C one normally assume that |q| < 1. If q ∈ Cp , we normally assume that |q − 1|p < p− p−1 so that q x = exp(x log q) for |x|p ≤ 1. For 1

g ∈ U D(Zp ) = {g|g : Zp → Cp is uniformly differentiable function}, the p-adic q-integral was defined by [2, 5] ∫

p −1 1 ∑ g(x)q x . Iq (g) = g(x)dµq (x) = lim N →∞ [pN ] Zp x=0 N

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The bosonic integral was considered from a physical point of view to the bosonic limit q → 1, as follows: ∫ pN −1 1 ∑ I1 (g) = lim Iq (g) = g(x)dµ1 (x) = lim N (1.1). g(x) (see [2]). q→1 N →∞ p Zp x=0 By (1.1), we easily see that I1 (g1 ) = I1 (g) + g ′ (0),

(1.2)

dg(x) where g1 (x) = g(x + 1) and g ′ (0) = . dx x=0 First, we introduce the second kind Bernoulli numbers Bn and polynomials Bn (x). The second kind Bernoulli numbers Bn and polynomials Bn (x) are defined by means of the following generating functions (see [3]):

∞ ∑ 2tet tn = B . n e2t − 1 n=0 n!

and

(

2tet e2t − 1

) ext =

∞ ∑

Bn (x)

n=0

tn n!

respectively. (h)

The second kind (h, q)-Bernoulli polynomials, Bn,q (x) are defined by means of the generating function: ( ) ∞ ∑ (h log q + 2t)et tn xt (h) e = (1.3) Bn,q (x) . h 2t q e −1 n! n=0 (h)

The second kind (h, q)-Bernoulli numbers En,q are defined by means of the generating function: ∞ n ∑ (h log q + 2t)et (h) t = . Bn,q h 2t q e −1 n! n=0

(1.4)

In (1.2), if we take g(x) = q hx e(2x+1)t , then we have ∫ (h log q + 2t)et . q hx e(2x+1)t dµ1 (x) = q h e2t − 1 Zp

(1.5)

for |t| ≤ p− p−1 , h ∈ Z. In (1.2), if we take g(x) = e2nxt , then we also have ∫ 2nt e2nxt dµ1 (x) = 2nt . e −1 Zp 1

(1.6)

for |t| ≤ p− p−1 . It will be more convenient to write (1.2) as the equivalent bosonic integral form ∫ ∫ g(x + 1)dµ1 (x) = g(x)dµ1 (x) + g ′ (0), (see [2]). (1.7) 1

Zp

Zp

For n ∈ N, we also derive the following bosonic integral form by (1.7), ∫

∫ Zp

g(x + n)dµ1 (x) =

Zp

g(x)dµ1 (x) +

n−1 ∑

g ′ (k), where g ′ (k) =

k=0 (h)

dg(x) . dx x=k

(1.8) (h)

In [4], we introduced the second kind (h, q)-Bernoulli numbers Bn,q and polynomials Bn,q (x) and investigate their properties. The following elementary properties of the second kind (h, q)(h)

(h)

Bernoulli numbers Bn,q and polynomials Bn,q (x) are readily derived form (1.1), (1.2), (1.3) and (1.4). We, therefore, choose to omit details involved.

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Theorem 1. For h ∈ Z, q ∈ Cp with |1 − q|p < p− p−1 , we have ∫ (h) Bn,q = q hx (2x + 1)n dµ1 (x), 1

Zp

∫

(h) Bn,q (x)

= Zp

q hy (x + 2y + 1)n dµ1 (y).

Theorem 2. For any positive integer n, we have n ( ) ∑ n (h) (h) Bn,q (x) = Bk,q xn−k . k k=0

Theorem 3. For any positive integer m, we obtain (h) Bn,q (x)

=m

n−1

m−1 ∑

( q

hi

(h) Bn,qm

i=0

2i + x + 1 − m m

) for n ≥ 0.

2. On the symmetries of the second kind (h, q)-Bernoulli polynomials In this section, we assume that q ∈ Cp and h ∈ Z. We investigate interesting properties of symmetry p-adic invariant integral on Zp for the second kind (h, q)-Bernoulli polynomials. W also obtain recurrence identities the second kind (h, q)-Bernoulli polynomials. By (1.7), we obtain 1 h log q + 2t

(∫ Zp

n =

)

∫ q hx q hn e(2x+2n+1)t dµ1 (x) −

∫

∫

Zp

q hx e(2x+1)t dµ1 (x)

Zp

q hnx e2ntx dµ1 (x)

Zp

q hx e(2x+1)t dµ1 (x) (2.1)

By (1.8), we obtain 1 h log q + 2t

(∫ q

q

e

Zp

=

)

∫ hx hn (2x+2n+1)t

(n−1 ∞ ∑ ∑ k=0

dµ1 (x) − )

q hi (2i + 1)k

i=0

q

hx (2x+1)t

e

Zp

dµ1 (x) (2.2)

tk . k!

For each integer k ≥ 0, let (h)

Ok,q (n) = 1k + q h 3k + q 2h 5k + q 3h 7k + · · · + q nh (2n + 1)k . (h)

The above sum Ok,q (n) is called the sums of powers of consecutive (h, q)-odd integers. From the above and (2.2), we obtain (∫ ) ∫ 1 tk q hx q hn e(2x+2n+1)t dµ1 (x) − q hx e(2x+1)t dµ1 (x) h log q + 2t k! Zp Zp =

∞ ∑ k=0

(h) Ok,q (n

tk − 1) . k!

Thus, we have ( ∫ ∫ ∞ ∑ q hx (2x + 2n + 1)k dµ1 (x) − q hn k=0

Zp

Zp

) q

hx

k

(2x + 1) dµ1 (x)

(2.3)

∞

∑ tk tk (h) = (h log q+2t)Ok,q (n−1) k! k! k=0

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By comparing coefficients

tk in the above equation, we have k! (h)

(h log q + 2t)Ok,q (n − 1) ∫ ∫ hx k = q (2x + 2n + 1) dµ1 (x) − Zp

Zp

q hx (2x + 1)k dµ1 (x)

.

By using the above equation we arrive at the following theorem: Theorem 4. Let k be a positive integer. Then we obtain (h)

(h)

(h) (h) q hn Bn,q (2n) − Bn,q = h log qOk,q (n − 1) + 2kOk−1,q (n − 1).

(2.4)

Remark 5. For the alternating sums of powers of consecutive integers, we have ( ) n−1 ∑ (h) (h) lim h log qOk,q (n − 1) + 2kOk−1,q (n − 1) = (2i + 1)k−1

q→1

i=0

=

Bk (2n) − Bk , for k ∈ N. 2k

By using (2.1) and (2.3), we arrive at the following theorem: Theorem 6. Let n be positive integer. Then we have ∫ ∞ ( ) tm ∑ n Zp q hx e(2x+1)t dµ1 (x) (h) ∫ = O (n − 1) . m,q m! q hnx e2ntx dµ1 (x) Zp m=0

(2.5)

Let w1 and w2 be positive integers. By using (1.5) and (1.6), we have ∫ ∫ q h(w1 x1 +w2 x2 ) e(w1 (2x1 +1)+w2 (2x2 +1)+w1 w2 x)t dµ1 (x1 )dµ1 (x2 ) Zp Zp ∫ q hw1 w2 x e2w1 w2 xt dµ1 (x) Zp

(2.6)

(h log q + 2t)ew1 t ew2 t ew1 w2 xt (q hw1 w2 e2w1 w2 t − 1) = (q hw1 e2w1 t − 1)(q hw2 e2w2 t − 1) By using (2.4) and (2.6), after calculations, we obtain ) ( )( ∫ ∫ w1 Zp q hw2 x2 e(2x2 +1)(w2 t) dµ1 (x2 ) 1 hw1 x1 (w1 (2x1 +1)+w1 w2 x)t ∫ S= q e dµ1 (x1 ) w1 Zp q hw1 w2 x e2w1 w2 tx dµ1 (x) Zp ( ) ( ) ∞ ∞ m m ∑ 1 ∑ (h) t t (h) = B w1 (w2 x)w1m Om,qw2 (w1 − 1)w2m . w1 m=0 m,q m! m! m=0 By using Cauchy product in the above, we have   ∞ m ( ) ∑ ∑ m tm (h) (h) j−1 m−j  S= Bj,qw1 (w2 x)w1 Om−j,qw2 (w1 − 1)w2  m! j m=0 j=0

(2.7)

(2.8)

By using the symmetry in (2.7), we have ( )( ∫ ) ∫ w2 Zp q hw1 x1 e(2x1 +1)(w1 t) dµ1 (x1 ) 1 hw2 x2 (w2 (2x2 +1)+w1 w2 x)t ∫ S= q e dµ1 (x2 ) w2 Zp q hw1 w2 x e2w1 w2 tx dµ1 (x) Zp ( ) ( ) ∞ ∞ m m ∑ 1 ∑ (h) (h) mt mt = B w2 (w1 x)w2 Om,qw1 (w2 − 1)w1 . w2 m=0 m,q m! m! m=0

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Thus we have

 m ( ) ∑ m tm (h) (h)  S= Bj,qw2 (w1 x)w2j−1 Om−j,qw1 (w2 − 1)w1m−j  j m! m=0 j=0 ∞ ∑



(2.9)

tm in the both sides of (2.8) and (2.9), we arrive at the following theorem: m! Theorem 7. Let w1 and w2 be positive integers. Then we obtain m ( ) ∑ m (h) (h) Bj,qw1 (w2 x)w1j−1 Om−j,qw2 (w1 − 1)w2m−j j j=0 m ( ) ∑ m (h) (h) = Bj,qw2 (w1 x)w2j−1 Om−j,qw1 (w2 − 1)w1m−j , j j=0

By comparing coefficients

(h)

(h)

where Bk,q (x) and Om,q (k) denote the second kind (h, q)-Bernoulli polynomials and the sums of powers of consecutive (h, q)-odd integers, respectively. By using Theorem 2, we have the following corollary: Corollary 8. Let w1 and w2 be positive integers. Then we have j ( )( ) m ∑ ∑ m j (h) (h) w1m−k w2j−1 xj−k Bk,qw2 Om−j,qw1 (w2 − 1) j k j=0 k=0

=

j ( m ∑ ∑ j=0 k=0

)( ) m j (h) (h) wj−1 w2m−k xj−k Bk,qw1 Om−j,qw2 (w1 − 1), j k 1

By using (2.6), we have ( )( ∫ ) ∫ w1 Zp q hw2 x2 e(2x2 +1)(w2 t) dµ1 (x2 ) 1 w1 w2 xt ∫ S= e q hw1 x1 e(2x1 +1)w1 t dµ1 (x1 ) w1 q hw1 w2 x e2w1 w2 tx dµ1 (x) Zp Zp   ( ) w −1 ∫ 1 ∑ 1 w1 w2 xt = e q hw1 x1 e(2x1 +1)w1 t dµ1 (x1 )  q w2 hj e(2j+1)(w2 t)  w1 Zp j=0 =

w∑ 1 −1

q

w2 hj

q

=



w∑ 1 −1



n=0

hw1 x1

2x1 +1+w2 x+(2j+1)

e

w2 w1

)

( (h)

q w2 hj Bn,qw1

j=0

w2 w2 x + (2j + 1) w1

(2.10)

(w1 t)

dµ1 (x1 )

Zp

j=0 ∞ ∑

(

∫

)

 w1n−1 

tn . n!

By using the symmetry property in (2.10), we also have )( ∫ ) ( ∫ w2 Zp q hw1 x1 e(2x1 +1)(w1 t) dµ1 (x1 ) 1 w1 w2 xt ∫ e q hw2 x2 e(2x2 +1)w2 t dµ1 (x2 ) S= w2 q hw1 w2 x e2w1 w2 tx dµ1 (x) Zp Zp   ( ) w −1 ∫ 2 ∑ 1 w1 w2 xt = e q hw2 x2 e(2x2 +1)w2 t dµ1 (x2 )  q w1 hj e(2j+1)(w1 t)  w2 Zp j=0 =

w∑ 2 −1

q

w1 hj

=

n=0

q

hw2 x2

2x2 +1+w1 x+(2j+1)

e

w1 w2

)

Zp

j=0 ∞ ∑

(

∫



w∑ 2 −1



j=0

( (h)

q w1 hj Bn,qw2

w1 w1 x + (2j + 1) w2

710

)

(2.11)

(w2 t)

dµ1 (x2 )  w2n−1 

tn . n!

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tn in the both sides of (2.10) and (2.11), we have the following theorem. n! Theorem 9. Let w1 and w2 be positive integers. Then we obtain ( ) w∑ 1 −1 w2 (h) w1n−1 q w2 hj Bn,qw1 w2 x + (2j + 1) w 1 j=0 (2.12) ( ) w∑ 2 −1 w1 n−1 w1 hj (h) = q Bn,qw2 w1 x + (2j + 1) w2 . w2 j=0

By comparing coefficients

Observe that if h = 1, then (2.12) reduces to Theorem 5 in [9](see [5, 9]). Substituting w1 = 1 into (2.12), we arrive at the following corollary. Corollary 10. Let w2 be positive integer. Then we obtain ( ) w∑ 2 −1 x − w2 + 2j + 1 (h) (h) . Bn,q (x) = w2n−1 q hj Bn,qw2 w2 j=0

(2.13)

The Corollary 10 is shown to yield the known distribution relation of the second kind (h, q)Bernoulli polynomials(see Theorem 3). Note that if q → 1, then (2.13) reduces to distribution relation of the second kind Bernoulli polynomials(see [8]). Corollary 11. Let w2 be positive integer. Then we have ) ( w∑ 2 −1 x − w2 + 2j + 1 Bn (x) = w2n−1 . Bn w2 j=0 Acknowledgement: This work was supported by 2020 Hannam University Research Fund. REFERENCES 1. Adelberg, A.(1992). On the degrees of irreducible factors of higher order Bernoulli polynomials, Acta Arith., v.62, pp. 329-342. 2. Kim, T.(2002). q-Volkenborn integration, Russ. J. Math. phys., v.9, pp. 288-299. 3. Ryoo, C.S.(2011). Distribution of the roots of the second kind Bernoulli polynomials, Journal of Computational Analysis and Applications, v.13, pp. 971-976. 4. Ryoo, C.S.(2012). Zeros of the second kind (h, q)-Bernoulli polynomials, Applied Mathematical Sciences, v.6, pp. 5869-5875. 5. Ryoo, C.S.(2020). Symmetric identities for the second kind q-Bernoulli polynomials, Journal of Computational Analysis and Applications, v.28, pp. 654-659. 6. Ryoo, C.S.(2011). On the alternating sums of powers of consecutive odd integers, Journal of Computational Analysis and Applications, v.13, pp. 1019-1024. 7. Ryoo, C.S.(2015). Symmetric identities for Carlitzs twisted q-Bernoulli numbers and polynomials associated with p-adic invariant integral on Zp , Global Journal of Pure and Applied Mathematics, v.11, pp. 2413-2417. 8. Ryoo, C.S.(2010). A note on the second kind Bernoulli polynomials, Far East J. Math. Sci., v.42, pp. 109-115. 9. Ryoo, C.S.(2011). A note on q-extension of the second kind Bernoulli numbers and polynomials, Far East J. Math. Sci., v.58, pp. 75-82.

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SOME NEW FUZZY BEST PROXIMITY POINT THEOREMS IN NON-ARCHIMEDEAN FUZZY METRIC SPACES ¨ ¨ MUZEYYEN SANGURLU SEZEN 1 , HUSEYIN IS¸IK

2,†

Abstract. In this paper, we define fuzzy weak P-property. Then we prove a fuzzy best proximity point theorems for γ-contractions with condition fuzzy weak P-property. Later, we give definition of fuzzy isometric distance between two functions in non-Archimedean fuzzy metric spaces. Also, we introduce γ-proximal contraction type-1 and type-2 contraction respectively via functions preserving fuzzy isometric distance and providing fuzzy isometry. Then, we obtain some fuzzy best proximity results for γ-proximal contractions types in non-Archimedean fuzzy metric spaces. Finally, we present some examples to illustrate the validity of the definitions and results obtained in the paper.

1. Introduction and Preliminaries The Banach contraction principle found by Banach has an important resonance in mathematics as well as in other fields [1]. Later, the subject of fixed point theory attracted the attention of many aouthors and caused this subject to be discussed in different areas of mathematics and different topological spaces. Then, authors intensively introduced many works regarding the fixed point theory. On the other hand, the concept of fuzzy metric space was introduced in different ways by some authors (see [2, 7]). Importantly, Gregori and Sapena [5] introduced the notion of fuzzy contractive mapping and gave some fixed point theorems for complete fuzzy metric spaces in the sense of George and Veeramani, and also for Kramosil and Michalek’s fuzzy metric spaces which are complete in Grabiec’s sense. At the same time, there are presented by many authors by expanding the Banach’s result in the literature (see [9–11, 14, 16, 20, 21]). In this work, we prove some fuzzy best proximity point results for mappings providing γ-proximal contractions. Then, we give some examples are supplied in order to support the useability of our results. Also, we show that our main results are more general than known results in the existing literature. 2010 Mathematics Subject Classification. 47H10,54H25. Key words and phrases. γ-proximal contraction, fuzzy best proximity point, non-Archimedean fuzzy metric space. † Corresponding author. 1

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Definition 1. [12] A binary operation ∗ : [0, 1] × [0, 1] → [0, 1] is called a continuous triangular norm (in short, continuous t−norm) if it satisfies the following conditions: (TN-1) ∗ is commutative and associative, (TN-2) ∗ is continuous, (TN-3) ∗(a, 1) = a for every a ∈ [0, 1], (TN-4) ∗(a, b) ≤ ∗(c, d) whenever a ≤ c, b ≤ d and a, b, c, d ∈ [0, 1]. An arbitrary t−norm ∗ can be extended (by associativity) in a unique way to an nary operator taking for (x1 , x2 , ..., xn ) ∈ [0, 1]n , n ∈ N , the value ∗(x1 , x2 , ..., xn ) is defined, n−1 0 n in [4], by ∗I=1 xi , xn ) = ∗(x1 , x2 , ..., xn ). ˙ xi = 1, ∗I=1 ˙ xi = ∗(∗I=1 ˙ Definition 2. [3] A fuzzy metric space is an ordered triple (X, M, ∗) such that X is a nonempty set, ∗ is a continuous t-norm and M is a fuzzy set on X 2 × (0, ∞), satisfying the following conditions, for all x, y, z ∈ X, s, t > 0 : (FM-1) M (x, y, t) > 0, (FM-2) M (x, y, t) = 1 iff x = y, (FM-3) M (x, y, t) = M (y, x, t), (FM-4) M (x, z, t + s) ≥ M (x, y, t) ∗ M (y, z, s), (FM-5) M (x, y, ·) : (0, ∞) → [0, 1] is continuous. If, in the above definition, the triangular inequality (FM-4) is replaced by (NA) M (x, z, max{t, s}) ≥ M (x, y, t) ∗ M (y, z, s) for all x, y, z ∈ X, s, t > 0, or equivalently, M (x, z, t) ≥ M (x, y, t) ∗ M (y, z, t) then the triple (X, M, ∗) is called a non-Archimedean fuzzy metric space [6]. Definition 3. Let (X, M, ∗) be a fuzzy metric space (or non-Archimedean fuzzy metric space). Then (i) A sequence {xn } in X is said to converge to x in X, denoted by xn → x, if and only if lim M (xn , x, t) = 1 for all t > 0, i.e. for each r ∈ (0, 1) and t > 0, there n→∞

exists n0 ∈ N such that M (xn , x, t) > 1 − r for all n ≥ n0 [7, 13]. (ii) A sequence {xn } is a M-Cauchy sequence if and only if for all ε ∈ (0, 1) and t > 0, there exists n0 ∈ N such that M (xn , xm , t) ≥ 1 − ε for all m > n ≥ n0 [3, 13]. A sequence {xn } is a G-Cauchy sequence if and only if lim M (xn , xn+p , t) = 1 for n→∞

any p > 0 and t > 0 [4, 5, 15]. (iii) The fuzzy metric space (X, M, ∗) is called M-complete (G-complete) if every MCauchy (G-Cauchy)sequence is convergent.

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Definition 4. [18,19] Let A, B be a non-empty subset of a non-Archimedean fuzzy metric space (X, M, ∗). The mapping g : A → A is said to be a fuzzy isometric if M (gx1 , gx2 , t) = M (x1 , x2 , t) for all x1 , x2 ∈ A. Definition 5. [17] For t > 0, a non-empty subset A of a fuzzy metric space (X, M, ∗) is said to be t-approximatively compact if for each x in X and each sequence yn in A with M (yn , x, t) −→ M (A, x, t), there exists a subsequence ynk of yn converging to an element y0 in A. Definition 6. [22] Let γ : [0, 1) → R be a strictly increasing, continuous mapping and for each sequence {an }n∈N of positive numbers lim an = 1 if and only if lim γ(an ) = +∞. n→∞ n→∞ Let Γ is the family of all γ functions. A mapping T : X → X is said to be a γ-contraction if there exists a δ ∈ (0, 1) such that M (T x, T y, t) < 1 ⇒ γ(M (T x, T y, t)) ≥ γ(M (x, y, t)) + δ

(1.1)

for all x, y ∈ X and γ ∈ Γ. 2. Main Results In this section, we present some definitions and deduce some best proximity point results in non-Archimedean fuzzy metric spaces. Let A0 (t) and B0 (t) two nonempty subsets of a fuzzy metric space (X, M, ∗). We will use the following notations: M (A, B, t) = sup {M (x, y, t) : x ∈ A, y ∈ B} ; A0 (t) = {x ∈ A : M (x, y, t) = M (A, B, t) for some y ∈ B} ; B0 (t) = {y ∈ B : M (x, y, t) = M (A, B, t) for some x ∈ A} . Now, let us state our main results. Definition 7. Let (A, B) be a pair of nonempty subsets of a non-Archimedean fuzzy metric space X with A0 6= 0. Then the pair (A, B) is said to have the fuzzy weak P-property if and ony if  M (x1 , y1 , t) = M (A, B, t) =⇒ M (x1 , x2 , t) ≥ M (y1 , y2 , t) M (x2 , y2 , t) = M (A, B, t) where x1 , x2 ∈ A0 and y1 , y2 ∈ B0 .

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Example 8. Let X = R×R and M : X × X × (0, ∞) → (0, 1] be the non-Archimedean fuzzy metric given by M (x, y, t) =

t t + d(x, y)

for all t > 0, where d : X × X → [0, ∞) is the standart metric d(x, y) = |x − y| for all x ∈ X. Let A = {(0, 0)}, B = {(1, 0), (−1, 0)}. Then here, d(A, B) = 1 and M (A, B, t) = t . Let us consider t+1 M (u1 , x1 , t) = M (A, B, t) M (u2 , x2 , t) = M (A, B, t). Herefrom, we have (u1 , x1 ) = ((0, 0), (1, 0)) and (u2 , x2 ) = ((0, 0), (−1, 0)) M (u1 , u2 , t) = M ((0, 0), (0, 0), t) = 1 >

t = M (x1 , x2 , t). t+2

Then it is easy to see that (A, B) is said to have the fuzzy weak P-property. Definition 9. Let A, B be a nonempty subset of a non-Archimedean fuzzy metric space (X, M, ∗). Given T : A → B and a fuzzy isometry g : A → A, the mapping T is said to preserve fuzzy isometric distance with respect to g if M (T gx1 , T gx2 , t) = M (T x1 , T x2 , t) for all x1 , x2 ∈ A. Example 10. Let X = R× [0, 1] and M : X ×X ×(0, ∞) → (0, 1] be the non-Archimedean fuzzy metric given by M (x, y, t) =

t t + d(x, y)

for all t > 0, where d : X × X → [0, ∞) is the standart metric d(x, y) = |x − y| for all x ∈ X. Let A = {(x, 0) : for all x ∈ R}. Define g : A → A by g(x, 0) = (−x, 0). Then t M (x, y, t) = t+d(x,y) = M (gx, gy, t), where x = (x1 , 0) and y = (y1 , 0) ∈ A. Therefore, g is a fuzzy isometry. Theorem 11. Let A and B be two nonempty, closed subsets of a non-Archimedean fuzzy metric space (X, M, ∗) such that A0 (t) is nonempty. Let T : A → B be γ-contraction such that T (A0 (t)) ⊆ B0 (t). Suppose that the pair (A, B) has the fuzzy P-property. Then, there exists a unique x∗ in A such that M (x∗ , T x∗ , t) = M (A, B, t).

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Proof. Let we choose an element x0 in A0 (t). Since T (A0 (t)) ⊆ B0 (t), we can find x1 ∈ A0 (t) such that M (x1 , T x0 , t) = M (A, B, t). Further, since T (A0 (t)) ⊆ B0 (t), it follows that there is an element x2 in A0 (t) such that M (x2 , T x1 , t) = M (A, B, t). Recursively, we obtain a sequence {xn } in A0 (t) satisfying M (xn+1 , T xn , t) = M (A, B, t),

for all n ∈ N.

(2.1)

(A, B) satisfies the fuzzy weak P-property, therefore from (2.1) we obtain M (xn , xn+1 , t) ≥ M (T xn−1 , T xn , t),

for all n ∈ N.

(2.2)

Now we will prove that the sequence {xn } is convergent in A0 (t). If there exists n0 ∈ N such that M (T xn0 −1 , T xn0 , t) = 1, then by (2.2) we get M (xn0 , xn0 +1 , t) = 1 which implies xn0 = xn0 +1 . Therefore, we get T xn0 = T xn0 +1 =⇒ M (T xn0 , T xn0 +1 , t) = 1.

(2.3)

From (2.2) and (2.3), we have that M (xn0 +2 , xn0 +1 , t) ≥ M (T xn0 +1 , T xn0 , t) = 1 =⇒ xn0 +2 = xn0 +1 . Therefore, xn = xn0 , for all n ≥ n0 and {xn } is convergent in A0 (t). Also, we obtain M (xn0 , T xn0 , t) = M (xn0 +1 , T xn0 , t) = M (A, B, t). This shows that xn0 is a fuzzy best proximity point of T and the proof is completed. Due to this reason, we suppose that M (T xn−1 , T xn , t) 6= 1, for all n ∈ N. In view of (1.1) and by (2.2), we get γ(M (xn , xn+1 , t)) ≥ γ(M (xn−1 , xn , t)) + δ ≥ γ(M (xn−2 , xn−1 , t)) + 2δ ... ≥ γ(M (x0 , x1 , t)) + nδ.

(2.4)

Letting n → ∞ , from (2.4) we get lim γ(M (xn , T xn+1 , t)) = +∞.

n→∞

Then, we have lim M (xn , xn+1 , t) = 1.

(2.5)

n→∞

Now, we want to show that {xn } is a Cauchy sequence. Suppose to the contrary, that {xn } is not a Cauchy sequence. Then there are ε ∈ (0, 1) and t0 > 0 such that for all

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k ∈ N there exist n(k), m(k) ∈ N with n(k) > m(k) > k and M (xn(k) , xm(k) , t0 ) ≤ 1 − ε .

(2.6)

Assume that m(k) is the least integer exceeding n(k) satisfying the inequality (2.6). Then, we have M (xm(k)−1 , xn(k) , t0 ) > 1 − ε and so, for all k ∈ N, we get 1−ε

≥ M (xn(k) , xm(k) , t0 ) ≥ M (xm(k)−1 , xm(k) , t0 ) ∗ M (xm(k)−1 , xn(k) , t0 ) ≥ M (xm(k)−1 , xm(k) , t0 ) ∗ (1 − ε ).

(2.7)

By taking k → ∞ in (2.7) and using (2.5), we obtain lim M (xn(k) , xm(k) , t0 ) = 1 − ε.

(2.8)

k→∞

From (FM-4), we get M (xm(k)+1 , xn(k)+1 , t0 ) ≥ M (xm(k)+1 , xm(k) , t0 ) ∗M (xm(k) , xn(k) , t0 ) ∗M (xn(k)1 , xn(k)+1 , t0 ). (2.9) Taking the limit as k → ∞ in (2.9), we obtain lim M (xn(k)+1 , xm(k)+1 , t0 ) = 1 − ε .

k→∞

(2.10)

By applying the inequality (1.1) with x = xm(k) and y = xn(k) γ(M (xn(k)+1 , xm(k)+1 , t)) ≥ γ(M (xn(k) , xm(k) , t)) + δ.

(2.11)

Taking the limit as k → ∞ in (2.11), applying (1.1), from (2.8), (2.10) and continuitiy of γ, we obtain γ(1 − ε) ≥ γ(1 − ε) + δ

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which is a contradiction. Thus {xn } is a Cauchy sequence in X. Since A0 (t) is a closed subset of the complete non-Archimedean fuzzy metric space (X, M, ∗), there exists x∗ ∈ A0 (t) such that lim xn = x∗ .

n→∞

Since T is continuous, we obtain T xn → T x∗ . Also, from continuity of the fuzzy metric function M , we have M (xn+1 , T xn , t) = M (x∗ , T x∗ , t). From (2.1), M (x∗ , T x∗ , t) = M (A, B, t). So, we prove that x∗ is a fuzzy best proximity point of T. The uniqueness of the best proximity point of T. From the condition that T is γ-contraction, we get x1 , x2 ∈ A such that x1 6= x2 and M (x1 , T x1 , t) = M (x2 , T x2 , t) = M (A, B, t). Then by the fuzzy weak P-property of (A, B), we have M (x1 , x2 , t) ≥ M (T x1 , T x2 , t). Also x1 6= x2 =⇒ M (x1 , x2 , t) 6= 1. Hence, γ(M (x1 , x2 , t)) ≥ γ(M (T x1 , T x2 , t)) ≥ γ(M (x1 , x2 , t)) + δ > γ(M (x1 , x2 , t)) which is a contradiction. Therefore the fuzzy best proximity point is unique.



Corollary 12. Let (X, M, ∗) be a non-Archimedean fuzzy metric space and A0 (t) a nonempty closed subsets of X. Let T : A → A be a γ-contraction. Then, there exists a unique x∗ in A. Example 13. Let X = [0, 1]×R and M : X ×X ×(0, ∞) → (0, 1] be the non-Archimedean fuzzy metric given by as in Example 10. Let A = {(0, x) : for all x ∈ R}, B = {(1, y) : t for all y ∈ R}. Then here A0 (t) = A, B0 (t) = B, d(A, B) = 1 and M (A, B, t) = t+1 . 1 Let γ : [0, 1) → R such that γ = 1−x for all x ∈ X . Now, define T : A → B by T (0, x) = (1, x6 ). Then, we get T (A0 (t)) = B0 (t) . Let us consider M (u1 , T x1 , t) = M (A, B, t) M (u2 , T x2 , t) = M (A, B, t).

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Herefrom, we have (u1 , x1 ) = ((0, − z61 ), (0, −z1 )) or (u2 , x2 ) = ((0, − z62 ), (0, −z2 )). Then from (1.1), we obtain, t γ(M (u1 , u2 , t)) = γ(M ((0, −z6 1 ), (0, −z6 2 ), t)) = γ( ) |z1 −z2 | t+ 6 1 t 1 > = γ( ) = t t t + |z1 − z2 | 1 − |z1 −z2 | 1 − t+|z1 −z2 | t+

6

= γ(M (x1 , x2 , t)). That is, γ(M (u1 , u2 , t)) > γ(M (x1 , x2 , t)). Therefore, there exixts a δ ∈ (0, 1) such that γ(M (u1 , u2 , t)) ≥ γ(M (x1 , x2 , t)) + δ Then it is easy to see that T is a γ-contraction and (0, 0) is a unique fuzzy best proximity point of T. Definition 14. ( γ-proximal contraction of Type-1) Let A and B be two nonempty subsets of a non-Archimedean fuzzy metric space (X, M, ∗) such that A0 (t) is nonempty. Suppose that a mapping T : A → B is said to be a γ-proximal contraction if there exists a δ ∈ (0, 1) for all u1 , u2 , x1 , x2 ∈ X such that   M (u1 , T x1 , t) = M (A, B, t) M (u2 , T x2 , t) = M (A, B, t) =⇒ γ(M (u1 , u2 , t)) ≥ γ(M (x1 , x2 , t)) + δ. (2.12)  M (u1 , u2 , t), M (x1 , x2 , t) < 1 Definition 15. (γ-proximal contraction of Type-2) Let A and B be two nonempty subsets of a non-Archimedean fuzzy metric space (X, M, ∗) such that A0 (t) is nonempty. Suppose that a mapping T : A → B is said to be a γ-proximal contraction if there exists a δ ∈ (0, 1) for all u1 , u2 , x1 , x2 ∈ X such that  M (u1 , T x1 , t) = M (A, B, t)  =⇒ γ(M (T u1 , T u2 , t)) ≥ γ(M (T x1 , T x2 , t))+δ. M (u2 , T x2 , t) = M (A, B, t)  M (T u1 , T u2 , t), M (T x1 , T x2 , t) < 1 (2.13) Theorem 16. Let A and B be two nonempty, closed subsets of a non-Archimedean fuzzy metric space (X, M, ∗) such that A0 (t) is nonempty. Suppose that T : A → B and g : A → A satisfy the following conditions:

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(i) (ii) (iii) (iv)

9

T (A0 (t)) ⊆ B0 (t), T : A → B is a continuous γ-proximal contraction of type-1, g is a fuzzy isometry, A0 (t) ⊆ g(A0 (t)).

Then, there exists a unique element x in A such that M (gx, T x, t) = M (A, B, t). Proof. Let we choose an element x0 in A0 (t). Since T (A0 (t)) ⊆ B0 (t) and A0 (t) ⊆ g(A0 (t)), we can find x1 ∈ A0 (t) such that M (gx1 , T x0 , t) = M (A, B, t). Further, since T x1 ∈ T (A0 (t)) ⊆ B0 (t) and and A0 (t) ⊆ g(A0 (t)), it follows that there is an element x2 in A0 (t) such that M (gx2 , T x1 , t) = M (A, B, t). Recursively, we obtain a sequence {xn } in A0 (t) satisfying M (gxn+1 , T xn , t) = M (A, B, t),

for all n ∈ N.

(2.14)

Now we will prove that the sequence {xn } is convergent in A0 (t). If there exists n0 ∈ N such that M (gxn0 , T xn0 +1 , t) = 1, then it is clear that sequence {xn } is convergent. Hence, let M (gxn0 , gxn0 +1 , t) 6= 1, for all n ∈ N. From T is a γ-proximal contraction of type-1 and (2.14), we have γ(M (gxn , gxn+1 , t)) ≥ γ(M (xn−1 , xn , t)) + δ ⇒ γ(M (xn , xn+1 , t)) ≥ γ(M (xn−1 , xn , t)) + δ ... ≥ γ(M (x0 , x1 , t)) + nδ.

(2.15)

Letting n → ∞ , from (2.15) we get lim γ(M (xn , T xn+1 , t)) = +∞.

n→∞

Then, if we similarly continue as the process in the proof of Theorem 11, we have {xn } is a Cauchy sequence. Since is a closed subset of the complete non-Archimedean fuzzy metric space (X, M, ∗), there exists x ∈ A0 (t) such that lim xn = x. n→∞ Since T ,g and M are continuous, passing to the limit n → ∞, we have M (gx, T x, t) = M (A, B, t). Let x∗ be in A0 (t) such that M (gx∗ , T x∗ , t) = M (A, B, t). Now, we will show that x = x∗ . Suppose to the contrary, let x 6= x∗ . Therefore, M (x, x∗ , t) 6= 1. Since T is a γ-proximal contraction of type-1 and g is an isometry, we have γ(M (x, x∗ , t)) = γ(M (gx, gx∗ , t)) ≥ γ(M (x, x∗ , t)) + δ > γ(M (x, x∗ , t))

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which is a contradiction. Hence, x = x∗ . Therefore, the proof of Theorem 16 is completed.  If we take g is the identity mapping, we obtain the following result. Corollary 17. Let A and B be two nonempty, closed subsets of a non-Archimedean fuzzy metric space (X, M, ∗) such that A0 (t) is nonempty. Assume that A is approximatively compact with respect to B. Also, suppose that T : A → B satisfy the following conditions: (i) T (A0 (t)) ⊆ B0 (t), (ii) T : A → B is a continuous γ-proximal contraction of type-1, Then, T has a unique fuzzy best proximity point in A. Example 18. Let X = R×[−2, 2] and M : X × X × (0, ∞) → (0, 1] be the nonArchimedean fuzzy metric given by M (x, y, t) =

t t + d(x, y)

for all t > 0, where d : X × X → [0, ∞) is the standart metric d(x, y) = |x − y| for all x ∈ X. Let A = {(x, −2) : for all x ∈ R}, B = {(y, 2) : for all y ∈ R}. Then here t . Let γ : [0, 1) → R such that A0 (t) = A, B0 (t) = B, d(A, B) = 4 and M (A, B, t) = t+4 1 γ = 1−x2 for all x ∈ X . Now, define T : A → B and g : A → A by x T (x, −2) = ( , 2) and g(x, −2) = (−x, −2) 2 Clearly, g is fuzzy isometry. Then, we have, we get T (A0 (t)) = B0 (t) and A0 (t) = g(A0 (t)). Let us consider M (gu1 , T x, t) = M (A, B, t) M (gu2 , T x, t) = M (A, B, t). Herefrom, we have (u1 , x1 ) = ((− z21 , −2), (z1 , −2)) or (u2 , x2 ) = ((− z22 , −2), (z2 , −2)). We claim that T is a γ-proximal contraction type-1. Now, putting u1 = (− z21 , −2), x1 = (z1 , −2), u2 = (− z22 , −2) and x2 = (z2 , −2) in (2.12), we have γ(M (gu1 , gu2 , t)) = γ(M (( z21 , −2), ( z22 , −2), t) = γ( =

1 1−(

t t+

|z1 −z2 | 2

)2

>

t t+

1 1−

( t+|z1t−z2 | )2

|z1 −z2 | 2

= γ(

) t

t + |z1 − z2 |

)

= γ(M (x1 , x2 , t)).

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That is, we have γ(M (u1 , u2 , t)) > γ(M (x1 , x2 , t)). Therefore, there exixts a δ ∈ (0, 1) such that γ(M (u1 , u2 , t) ≥ γ(M (x1 , x2 , t)) + δ. Then it is easy to see that T is a γ-proximal contraction type-1. It now follows from Theorem 16 that (0, −2) is a unique fuzzy best proximity point of T. Theorem 19. Let A and B be two nonempty, closed subsets of a non-Archimedean fuzzy metric space (X, M, ∗) such that A0 (t) is nonempty. Assume that A is approximatively compact with respect to B. Also, suppose that T : A → B and g : A → Asatisfy the following conditions: (i) (ii) (iii) (iv) (v)

T (A0 (t)) ⊆ B0 (t), T : A → B is a continuous γ-proximal contraction of type-2, g is a fuzzy isometry, A0 (t) ⊆ g(A0 (t)), T preserves fuzzy isometric distance with respect to g.

Then, there exists an element x in A such that M (gx, T x, t) = M (A, B, t). Moreover, if x∗ is another element of A such that M (gx∗ , T x∗ , t) = M (A, B, t). Proof. Let we choose an element T x0 in T (A0 (t)). Since T x0 ∈ T (A0 (t)) ⊆ B0 (t) and A0 (t) ⊆ g(A0 (t)), we can find x1 ∈ A0 (t) such that M (gx1 , T x0 , t) = M (A, B, t). Further, since T (A0 (t)) ⊆ B0 (t) and and A0 (t) ⊆ g(A0 (t)), it follows that there is an element x2 in A0 (t) such that M (gx2 , T x1 , t) = M (A, B, t). Recursively, we obtain a sequence {xn } in A0 (t) satisfying M (gxn+1 , T xn , t) = M (A, B, t),

for all n ∈ N.

(2.16)

Now we will prove that the sequence {T xn } is convergent in B. If there exists n0 ∈ N such that M (T gxn0 , T gxn0 +1 , t) = 1, then it is clear that sequence {T xn } is convergent. Hence, let M (T gxn0 , T gxn0 +1 , t) 6= 1, for all n ∈ N. From T is a γ-proximal contraction of type-2, T preserves fuzzy isometric distance with respect to g and (2.16), we have γ(M (T gxn , T gxn+1 , t)) ≥ γ(M (T xn−1 , T xn , t)) + δ ⇒ γ(M (T xn , T xn+1 , t)) ≥ γ(M (T xn−1 , T xn , t)) + δ ... ≥ γ(M (T x0 , T x1 , t)) + nδ.

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Letting n → ∞ , from (2.17) we get lim γ(M (T xn , T xn+1 , t)) = +∞.

n→∞

Then, if we similarly continue as the process in the proof of Theorem 11, we have {T xn } is a Cauchy sequence in B. Since B is a closed subset of the complete non-Archimedean fuzzy metric space (X, M, ∗), there exists y ∈ B such that lim T xn = y. From the triangular inequality, we obtain n→∞

M (y, A, t) ≥ M (y, gxn , t) ≥ M (y, T xn−1 , t) ∗ M (T xn−1 , gxn , t) = M (y, T xn−1 , t) ∗ M (A, B, t) ≥ M (y, T xn−1 , t) ∗ M (y, A, t).

(2.18)

Passing to the limit as n → ∞ in (2.18), we have lim M (y, gxn , t) = M (y, A, t).

n→∞

Since A0 (t) is approximatively compact with respect to B, there exists a subsequence {gxnk } of {gxn } such that converges to some z in A0 (t). Therefore, we have M (z, y, t) = lim M (gxnk , T gxnk −1 , t) = M (y, A, t). k→∞

Hence, it implies that z ∈ A0 (t). Since A0 (t) ⊆ g(A0 (t)), there exists x ∈ A0 (t) such that z = gx. Taking to the limit as lim gxnk = gx and g is a fuzzy isometry, we obtain k→∞

lim xnk = x.

k→∞

Since T is continuous and {T xn } is convergent to y, we have lim T xnk = T x = y.

k→∞

Hence, it follows that M (gx, T x, t) = lim M (gxnk , T gxnk , t) = M (A, B, t). k→∞

∗

Let x be in A0 (t) such that M (gx∗ , T x∗ , t) = M (A, B, t). Now, we will show that T x = T x∗ . Suppose to the contrary, let T x 6= T x∗ . Therefore, M (x, T x∗ , t) 6= 1. Since T is a γ-proximal contraction of type-2 and T preserves fuzzy isometric distance with respect to g, we have γ(M (T x, T x∗ , t)) = γ(M (T gx, T gx∗ , t)) ≥ γ(M (x, x∗ , t)) + δ > γ(M (x, x∗ , t)) which is a contradiction. Hence, T x = T x∗ . Therefore, the proof of Theorem 19 is completed. 

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If we take g is the identity mapping, we obtain the following result. Corollary 20. Let A and B be two nonempty, closed subsets of a non-Archimedean fuzzy metric space (X, M, ∗) such that A0 (t) is nonempty. Assume that A is approximatively compact with respect to B. Also, suppose that T : A → B satisfy the following conditions: (i) T (A0 (t)) ⊆ B0 (t), (ii) T : A → B is a continuous γ-proximal contraction of type-2, Then, T has a unique fuzzy best proximity point in A. Moreover, if x∗ is another fuzzy best proximity point T , then T x = T x∗ . Example 21. Let X = [0, 1]×R and M : X ×X ×(0, ∞) → (0, 1] be the non-Archimedean fuzzy metric given by t M (x, y, t) = t + d(x, y) for all t > 0, where d : X × X → [0, ∞) is the standart metric d(x, y) = |x − y| for all x ∈ X. Let A = {(0, x) : for all x ∈ R}, B = {(1, y) : for all y ∈ R}. Then here t A0 (t) = A, B0 (t) = B, d(A, B) = 1 and M (A, B, t) = t+1 . Let γ : [0, 1) → R such that 1 γ = √1−x for all x ∈ X . Now, define T : A → B and g : A → A by x T (0, x) = (1, ) and g(0, x) = (0, −x) 3 Clearly, g is a fuzzy isometry. Then, we have, we get T (A0 (t)) = B0 (t) and A0 (t) = g(A0 (t)). Let us consider M (gu1 , T x1 , t) = M (A, B, t) M (gu2 , T x2 , t) = M (A, B, t). . Clearly, T is preserve isometric distance with respect to g. That is M (T gx1 , T gx2 , t) = M (T x1 , T x2 , t). We claim that T is a γ−proximal contraction type-2. Now, putting u1 = (0, − z31 ), x1 = (0, z1 ), u2 = (0, − z32 , ) and x2 = (0, z2 ) in (2.13), we have t γ(M (T gu1 , T gu2 , t) = γ(M (1, z91 ), (1, z92 ), t) = γ( ) |z1 −z2 | t+ 9 1 1 t = q >q = γ( ) |z1 −z2 | t t t+ 3 1 − |z1 −z2 | 1 − |z1 −z2 | t+

9

t+

3

= γ(M (T x1 , T x2 , t)). Since, T preserves isometric distance with respect to g, we have γ(M (T u1 , T u2 , t)) > γ(M (T x1 , T x2 , t)).

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Therefore, there exixts a δ ∈ (0, 1) such that γ(M (T u1 , T u2 , t)) ≥ γ(M (T x1 , T x2 , t)) + δ. Then it is easy to see that T is a γ-proximal contraction type-2. It now follows from Theorem 19 that (0, 0) is a unique fuzzy best proximity point of T.

References [1] S. Banach, Sur les op´erations dans les ensembles abstraits et leurs applications aux ´equations int´egrales, Fund. Math., 3 (1922), 133-181. [2] Z. Deng, Fuzzy pseudometric spaces, J. Math. Anal. Appl., 86 (1922), 74-95. [3] A. George and P. Veeramani, On some results in fuzzy metric spaces, Fuzzy Sets and Systems, 64 (1994), 395-399. [4] M. Grabiec, Fixed points in fuzzy metric spaces, Fuzzy Sets and Systems 27 (1988), 385-389. [5] V. Gregori and A. Sapena, Sn fixed point theorems in fuzzy metric spaces, Fuzzy Sets and Systems 125 (2001), 245-253. [6] V. Istr˘ a¸tescu, An Introduction to Theory of Probabilistic Metric Spaces, with Applications, Ed, Tehnic˘a, Bucure¸sti, in Romanian, (1974). [7] O. Kramosil, and J. Michalek, Fuzzy metric and statistical metric spaces, Kybernetika, 11 (1975), 336-344. [8] D. Mihet, Fuzzy ψ-contractive mappings in non-Archimedean fuzzy metric spaces, Fuzzy Sets and Systems, 159 (2008), 739 -744. [9] D. Mihet, A class of contractions in fuzzy metric spaces, Fuzzy Sets Syst., 161 (2010), 1131-1137. [10] P. Salimi, C. Vetro and P. Vetro, Some new fixed point results in non-Archimedean fuzzy metric spaces, Nonlinear Analysis: Modelling and Control, 18 2013, 3, 344-358. [11] M. Sangurlu and D. Turkoglu, Fixed point theorems for (ψ ◦ϕ)-contractions in a fuzzy metric spaces, J. Nonlinear Sci. Appl., 8 (2015), 687-694. [12] B. Schweizer and A. Sklar, Statistical metric spaces, Pacific Journal of Mathematics, 10 (1960) 385-389. [13] B. Schweizer, and A. Sklar, Probabilistic Metric Spaces, North-Holland, Amsterdam, 1983. [14] D. Turkoglu and M. Sangurlu, Fixed point theorems for fuzzy (ψ)-contractive mappings in fuzzy metric spaces, Journal of Intelligent and Fuzzy Systems, 26(2014), 1, 137-142. [15] R. Vasuki and P. Veeramani, Fixed point theorems and Cauchy sequences in fuzzy metric spaces, Fuzzy Sets and Systems 135 (2003), 3, 409-413. [16] C. Vetro, Fixed points in weak non-Archimedean fuzzy metric spaces, Fuzzy Sets and Systems, 162 (2011), 84-90. [17] Z. Razaa, N. Saleem, M. Abbas, Optimal coincidence points of proximal quasi-contraction mappings in non-Archimedean fuzzy metric spaces, Journal of Nonlinear Science and Appl., 9 (2016), 37873801. [18] M. Abbas, N. Saleem, M. De la Sen, Optimal coincidence point results in partially ordered nonArchimedean fuzzy metric spaces, Fixed Point Theory Appl., 2016 (2016), 44 pages.

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[19] N. Saleem, M. Abbas, Z. Raza, Optimal coincidence best approximation solution in non-Archimedean Fuzzy Metric Spaces, Iranian J. Fuzzy Sys., 13 (2016), 113-124. [20] C. Vetro, P. Salimi, Best proximity point results in non-Archimedean fuzzy metric spaces, Fuzzy Information and Engineering, 5 (2013), 417-4291. [21] V. S. Raj, A Best proximity point theorem for weakly contractive non-self mappings, Nonlinear Anal., 74 (2011), 449-455. [22] M. Sangurlu Sezen, Fixed Point Theorems for New Type Contractive Mappings, Journal of Function Spaces, 2019 (2019), Article ID 2153563, 6 pages. 1

Department of Mathematics, Faculty of Science and Arts, University of Giresun, ¨ re, Giresun, Turkey. Gu E-mail address: [email protected] 2

Department of Mathematics, Faculty of Science and Arts, Mus¸ Alparslan University, Mus¸ 49250, Turkey. E-mail address: [email protected]

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Exact solutions of conformable fractional Harry Dym equation Asma ALHabees

Department of Mathematics, Faculty of Science, The University of Jordan, Amman 11942, Jordan *Corresponding author e-mail address: [email protected]

Abstract: The aim of this paper is to find exact solutions for the conformable fractional Harry Dym Equation. In this work we deal with three different forms of conformable fractional Harry Dym Equation and for each form a suitable wave variable substitution is found. Each substitution transform its corresponding problem to an ordinary differential equation, What is more, the resulted ordinary differential equations in the three cases are the same. General solutions are obtained by applying the direct integration method on the resulted ordinary differential equation. These obtained solutions are found for some particular choices for the constants values. The behavior of every solution is discussed and illustrated in graphs. The tedious integrals and difficult computations associated with calculations in this paper are performed and simplified by using Mathematica 9.0. Keywords: Conformable fractional derivative, Harry Dym Equation, Conformable Harry Dym Equation, Exact solutions.

1. Introduction Recently, differential equations with fractional derivatives attracted the interest of many researchers; since such equations describe effectively many phenomena in applied sciences such as physics, biology, technology, and engineering [3, 7, 14]. Harry Dym equation (HD) was so named related to the name of its discoverer Harry Dym in his unpublished paper 1973-1974, although it appeared to first time in Kruskal and Moser [9]. HD equation represents a system which gathers non-linearity and dispersion, also it is a completely integrable nonlinear evolution equation which obeys an infinite number of conservation laws, but it does not have the Painleve property. More properties for HD equation discussed in details can be found in the reference [4]. Moreover HD equation can be connected to the Korteweg-ge Vries equation which has many applications in hydrodynamics [4, 15]. Many efforts have been done to find exact and approximate solutions for both HD equation and fractional HD equation like algebraic geometric solution of the HD equation[13], solitions solutions of the (2+1) dimensional HD equation via Darboux transformation [2], explicit solutions for HD equation [1], exact solution of the HD equation [12], an efficient approach for fractional HD equation by using sumudu transform [10], symmetries and exact solutions of the time fractional HD equation with Rieman-Liouville derivative [5], and a fractional model of HD equation and its approximate solution [11]. Fractional derivatives have many definitions [14] but the most used of these definitions are RiemannLiouville derivative and Caputo derivative. They were defined as follows: (i) Riemann - Liouville Definition. For 𝛼𝛼 ∈ [n-1, n), the α derivative of 𝑓𝑓 is:

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J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 29, NO.4, 2021, COPYRIGHT 2021 EUDOXUS PRESS, LLC

𝐷𝐷𝑎𝑎𝛼𝛼 𝑓𝑓(𝑡𝑡) =

1 𝑑𝑑 𝑛𝑛 𝑡𝑡 𝑓𝑓(𝑥𝑥) 𝑑𝑑𝑑𝑑 ∫ Γ(n−α) 𝑑𝑑𝑡𝑡 𝑛𝑛 𝑎𝑎 (𝑡𝑡−𝑥𝑥)𝛼𝛼 −𝑛𝑛 +1

(ii) Caputo Definition. For 𝛼𝛼 ∈ [n-1, n), the α derivative of 𝑓𝑓 is: 𝐷𝐷𝑎𝑎𝛼𝛼 𝑓𝑓(𝑡𝑡) =

𝑡𝑡 1 𝑓𝑓 (𝑛𝑛) (𝑥𝑥) � 𝑑𝑑𝑑𝑑 Γ(𝑛𝑛 − 𝛼𝛼) 𝑎𝑎 (𝑡𝑡 − 𝑥𝑥)𝛼𝛼−𝑛𝑛+1

Recently, a new definition called conformable fractional derivative was introduced by authors in [6], Since then the interest of it keeps growing and many equations were solved using such definition [8]. In this paper we intend to find exact solutions for fractional HD equation in the sense of this definition rather than Rieman-Liouville definition or Caputo definition. The rest of the paper is organized as follows: Basics of conformable fractional derivative are stated in section 2, in section 3 solutions for conformable fractional HD equation are found, in section 4 some examples are discussed.

2. Basic results on conformable fractional derivatives. Now, Let us summarize the basic properties of the conformable fractional derivative definition. Definition [6]: Given a function𝑓𝑓: [0, ∞) → ℝ. And 𝑡𝑡 > 0, 𝛼𝛼 ∈ (0, 1], then the conformable fractional derivative of order α is defined as 𝑇𝑇𝛼𝛼 (𝑓𝑓)(𝑡𝑡) = lim𝜖𝜖→0

𝑓𝑓�𝑡𝑡+𝜖𝜖𝑡𝑡 1−𝛼𝛼 �−𝑓𝑓(𝑡𝑡)

,

𝜖𝜖

𝑇𝑇𝛼𝛼 is called the conformable fractional derivative of 𝑓𝑓 of order 𝛼𝛼 . Let 𝑓𝑓 𝛼𝛼 (𝑡𝑡) stands for 𝑇𝑇𝛼𝛼 (𝑓𝑓)(𝑡𝑡) =

𝑑𝑑 𝛼𝛼 𝑓𝑓 𝑑𝑑𝑡𝑡 𝛼𝛼

.

If 𝑓𝑓 is α-differentiable in some(0, 𝑏𝑏), 𝑏𝑏 > 0, and lim𝑡𝑡→0+ 𝑓𝑓 𝛼𝛼 (𝑡𝑡) exists, then by definition:

𝑓𝑓 𝛼𝛼 (0) = lim𝑡𝑡→0+ 𝑓𝑓 𝛼𝛼 (𝑡𝑡)

Theorem 1 [6]: Let 𝛼𝛼 ∈ (0, 1] and 𝑓𝑓, 𝑔𝑔 be α-differentiable at a point 𝑡𝑡 > 0. Then 1. 𝑇𝑇𝛼𝛼 (𝑎𝑎𝑎𝑎 + 𝑏𝑏𝑏𝑏) = 𝑎𝑎 𝑇𝑇𝛼𝛼 (𝑓𝑓) + 𝑏𝑏 𝑇𝑇𝛼𝛼 (𝑔𝑔), 𝑓𝑓𝑓𝑓𝑓𝑓 𝑎𝑎𝑎𝑎𝑎𝑎 𝑎𝑎, 𝑏𝑏 ∈ ℝ. 2. 𝑇𝑇𝛼𝛼 (𝑡𝑡 𝑝𝑝 ) = 𝑝𝑝𝑡𝑡 𝑝𝑝−𝛼𝛼 𝑓𝑓𝑓𝑓𝑓𝑓 𝑎𝑎𝑎𝑎𝑎𝑎 𝑝𝑝 ∈ ℝ.

3. 𝑇𝑇𝛼𝛼 (𝜆𝜆) = 0 𝑓𝑓𝑓𝑓𝑓𝑓 𝑎𝑎𝑎𝑎𝑎𝑎 𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐 𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓 𝑓𝑓(𝑡𝑡) = 𝜆𝜆. 4. 𝑇𝑇𝛼𝛼 (𝑓𝑓𝑓𝑓) = 𝑓𝑓 𝑇𝑇𝛼𝛼 (𝑔𝑔) + 𝑔𝑔 𝑇𝑇𝛼𝛼 (𝑓𝑓). 𝑓𝑓 𝑔𝑔

5. 𝑇𝑇𝛼𝛼 � � =

𝑔𝑔 𝑇𝑇𝛼𝛼 (𝑓𝑓)−𝑓𝑓 𝑇𝑇𝛼𝛼 (𝑔𝑔) 𝑔𝑔 2

.

6. If, in addition, 𝑓𝑓 is differentiable, then 𝑇𝑇𝛼𝛼 (𝑓𝑓)(𝑡𝑡) = 𝑡𝑡1−𝛼𝛼

728

𝑑𝑑𝑑𝑑 𝑑𝑑𝑑𝑑

.

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Theorem 2 [8]: let 𝑓𝑓 be an α-differentiable function in conformable sense and differentiable and suppose that 𝑔𝑔 is also differentiable and defined in the range of 𝑓𝑓. Then 𝑇𝑇𝛼𝛼 (𝑓𝑓𝑓𝑓𝑓𝑓) (𝑡𝑡) = 𝑡𝑡1−𝛼𝛼 𝑔𝑔′ (𝑡𝑡)𝑓𝑓 ′ �𝑔𝑔(𝑡𝑡)�.

More properties, definitions and theorems as Roll’s Theorem and Mean Value Theorem for conformable fractional derivative are expressed in the work [6],

3. Fractional Harry Dym Equation. The classical HD equation is: 𝑢𝑢𝑡𝑡 = 𝑢𝑢3 𝑢𝑢𝑥𝑥𝑥𝑥𝑥𝑥

(∗)

Where 𝑢𝑢(𝑥𝑥, 𝑡𝑡) is a function of two real variables 𝑥𝑥 𝑎𝑎𝑎𝑎𝑎𝑎 𝑡𝑡. Let us write:

𝑢𝑢𝑡𝑡𝛼𝛼 = 𝑇𝑇𝑡𝑡𝛼𝛼 𝑢𝑢 =

𝜕𝜕 𝛼𝛼 𝑢𝑢 , 𝜕𝜕𝑡𝑡 𝛼𝛼

𝑢𝑢𝑥𝑥𝛼𝛼 = 𝑇𝑇𝑥𝑥𝛼𝛼 𝑢𝑢 =

Now we will solve three fractional forms of(∗): (i) (ii) (iii)

𝑢𝑢𝑡𝑡𝛼𝛼 = 𝑢𝑢3 𝑢𝑢𝑥𝑥𝑥𝑥𝑥𝑥 .

𝑢𝑢𝑡𝑡 =

𝑢𝑢𝑡𝑡𝛼𝛼 =

Where 𝛼𝛼 ∈ (0, 1].

(3𝛼𝛼) 𝑢𝑢3 𝑢𝑢𝑥𝑥 . (3𝛼𝛼) 𝑢𝑢3 𝑢𝑢𝑥𝑥 .

𝜕𝜕 𝛼𝛼 𝑢𝑢 , 𝜕𝜕𝑥𝑥 𝛼𝛼

(3𝛼𝛼)

𝑢𝑢𝑥𝑥

(3𝛼𝛼)

= 𝑇𝑇𝑥𝑥

𝑢𝑢 = 𝑇𝑇𝑥𝑥𝛼𝛼 𝑇𝑇𝑥𝑥𝛼𝛼 𝑇𝑇𝑥𝑥𝛼𝛼 𝑢𝑢 .

(1)

(2)

(3)

Using suitable wave variable substitution in each form will transform the equation to an ordinary differential equation as follows: 𝑐𝑐 𝛼𝛼

1. For form (i) let the wave variable substitution 𝜂𝜂 = 𝑥𝑥 + 𝑡𝑡 𝛼𝛼 and 𝑢𝑢(𝑥𝑥, 𝑡𝑡) = 𝑣𝑣( 𝜂𝜂) . So one can write

𝑢𝑢 = 𝑣𝑣 ∘ 𝜂𝜂 , now apply Theorem 2 to find 𝑢𝑢𝑡𝑡𝛼𝛼 . You will get that 𝑢𝑢𝑡𝑡𝛼𝛼 = 𝑡𝑡1−𝛼𝛼 𝜂𝜂 ′ (𝑡𝑡)𝑣𝑣 ′ �𝜂𝜂(𝑡𝑡)� = 𝑐𝑐𝑣𝑣 ′ , 𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎 𝑢𝑢3 = 𝑣𝑣 3 𝑎𝑎𝑎𝑎𝑎𝑎 𝑢𝑢𝑥𝑥𝑥𝑥𝑥𝑥 = 𝑣𝑣 ′′′ . Hence equation (1) is transformed to: 𝑐𝑐𝑣𝑣 ′ = 𝑣𝑣 3 𝑣𝑣 ′′′

1 𝛼𝛼

(4)

2. For form (ii) let the wave variable substitution 𝜂𝜂 = 𝑥𝑥 𝛼𝛼 + 𝑐𝑐𝑐𝑐 and 𝑢𝑢(𝑥𝑥, 𝑡𝑡) = 𝑣𝑣( 𝜂𝜂) = 𝑣𝑣 ∘ 𝜂𝜂. so (3𝛼𝛼)

𝑢𝑢𝑡𝑡𝛼𝛼 = 𝑐𝑐𝑣𝑣 ′ , 𝑢𝑢3 = 𝑣𝑣 3 𝑎𝑎𝑎𝑎𝑎𝑎 𝑢𝑢𝑥𝑥

= 𝑣𝑣 ′′′ . Then equation (2) is transformed to: 𝑐𝑐𝑣𝑣 ′ = 𝑣𝑣 3 𝑣𝑣 ′′′

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(4)

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1 𝛼𝛼

𝑐𝑐 𝛼𝛼

3. For form (iii) let the wave variable substitution 𝜂𝜂 = 𝑥𝑥 𝛼𝛼 + 𝑡𝑡 𝛼𝛼 and 𝑢𝑢(𝑥𝑥, 𝑡𝑡) = 𝑣𝑣( 𝜂𝜂) . so 𝑢𝑢𝑡𝑡𝛼𝛼 = (3𝛼𝛼)

𝑐𝑐𝑣𝑣 ′ , 𝑢𝑢3 = 𝑣𝑣 3 𝑎𝑎𝑎𝑎𝑎𝑎 𝑢𝑢𝑥𝑥

= 𝑣𝑣 ′′′ . Then equation (3) is transformed to: 𝑐𝑐𝑣𝑣 ′ = 𝑣𝑣 3 𝑣𝑣 ′′′

(4)

Now to solve the resulted ordinary differential equation (4), rewrite it as: 𝑐𝑐 ′ 𝑣𝑣 ′′′ + � 2 � = 0 2𝑣𝑣

(5)

Integrate (5) with respect to η, gets

𝑐𝑐 𝑐𝑐1 = 2 2 2𝑣𝑣

(6)

𝑐𝑐 + 𝑐𝑐1 𝑣𝑣 + 𝑐𝑐 2 𝑣𝑣

(7)

𝑣𝑣 ′′ +

Multiply (6) by 𝑣𝑣′ then integrate with respect to 𝜂𝜂 yields (𝑣𝑣 ′ )2 =

Using the separation of variables changes (7) to

𝑣𝑣 𝑑𝑑𝑑𝑑 = ±� 𝑑𝑑𝑑𝑑 𝑐𝑐₁𝑣𝑣² + 𝑐𝑐₂𝑣𝑣 + 𝑐𝑐

(8)

Integrate both sides of (8) using Mathematica 9.0 you will obtain 𝑣𝑣 𝜂𝜂 = ± � � 𝑑𝑑𝑑𝑑 + 𝑐𝑐3 𝑐𝑐₁𝑣𝑣² + 𝑐𝑐₂𝑣𝑣 + 𝑐𝑐

(9)

𝐴𝐴𝐴𝐴𝐴𝐴𝐴𝐴 [𝐸𝐸𝐸𝐸𝐸𝐸𝐸𝐸𝐸𝐸𝐸𝐸𝐸𝐸𝐸𝐸 𝐸𝐸(𝑖𝑖 𝑠𝑠𝑠𝑠𝑠𝑠ℎ−1 (𝐺𝐺), 𝐾𝐾) − 𝐸𝐸𝐸𝐸𝐸𝐸𝐸𝐸𝐸𝐸𝐸𝐸𝐸𝐸𝐸𝐸 𝐹𝐹(𝑖𝑖 𝑠𝑠𝑠𝑠𝑠𝑠ℎ−1 (𝐺𝐺), 𝐾𝐾)] + 𝑐𝑐3 𝑐𝑐₁𝐺𝐺

𝜂𝜂 = ±𝑖𝑖

𝑣𝑣 𝑐𝑐₁𝑣𝑣² +𝑐𝑐₂𝑣𝑣+𝑐𝑐

Where: A = �

, 𝐵𝐵 = −𝑐𝑐₂ + �−4𝑐𝑐𝑐𝑐₁ + 𝑐𝑐2 2 ,

730

𝐶𝐶 = �1 +

(10)

2𝑐𝑐₁𝑣𝑣

𝑐𝑐₂−�−4𝑐𝑐𝑐𝑐₁+𝑐𝑐₂²

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𝐷𝐷 = �1 +

2𝑐𝑐₁𝑣𝑣

𝑐𝑐2 +�−4𝑐𝑐𝑐𝑐₁+𝑐𝑐₂²

2𝑐𝑐₁𝑣𝑣

, 𝐺𝐺 = �

𝑐𝑐2 +�−4𝑐𝑐𝑐𝑐₁+𝑐𝑐₂²

and 𝐾𝐾 =

𝑐𝑐₂+�−4𝑐𝑐𝑐𝑐₁+𝑐𝑐₂² 𝑐𝑐₂−�−4𝑐𝑐𝑐𝑐₁+𝑐𝑐₂²

.

𝐸𝐸𝐸𝐸𝐸𝐸𝐸𝐸𝐸𝐸𝐸𝐸𝐸𝐸𝐸𝐸 𝐹𝐹 and 𝐸𝐸𝐸𝐸𝐸𝐸𝐸𝐸𝐸𝐸𝐸𝐸𝐸𝐸𝐸𝐸 𝐸𝐸 are elliptic integrals of the first and second kind respectively.

For some particular choices to the constants c, c₁ and c₂ in equation (9) one can get simpler solutions as

follows:

•

Let 𝑐𝑐₁ = 𝑐𝑐₂ = 0 , then 𝜂𝜂 = ± 3

2 3

𝑣𝑣 �

2

𝑣𝑣 = (𝑐𝑐₃ ± √𝑐𝑐 𝜂𝜂)3 2 •

𝑣𝑣 𝑐𝑐

+ 𝑐𝑐₃ , hence

(11) �𝑐𝑐𝑐𝑐+𝑐𝑐2 𝑣𝑣 2 𝑐𝑐2

Let 𝑐𝑐₁ = 0, 𝑐𝑐₂ ≠ 0, then 𝜂𝜂 = ± �

−

𝑐𝑐

3

𝑐𝑐2 2

log( 2𝑐𝑐2 √𝑣𝑣 + 2�𝑣𝑣𝑐𝑐₂² + 𝑐𝑐2 𝑐𝑐 )� + 𝑐𝑐₃

Other suggested constants are: 1. Let c2 = 2√cc1 .

2. Let c2 = −2√cc1 .

You can easily using Mathematica 9.0 to perform the integration of equation (9) to get formula of 𝜂𝜂

after you determine the suggested constants, however the difficulty that faces is how to get 𝑣𝑣 with respect to 𝜂𝜂 explicitly , except the formula in (11), this what was discussed in [12], Hence it seems that

formula (11) is the only explicit solution for equations (1), (2) and (3). So results can be summarized as follows: 3

2

𝑐𝑐

∙ The solution of equation (1) is 𝑢𝑢(𝑥𝑥, 𝑡𝑡) = (c3 ± √𝑐𝑐 � 𝑥𝑥 + 𝑡𝑡 𝛼𝛼 �)3 . 2 𝛼𝛼 3

1

2

∙ The solution of equation (2) is 𝑢𝑢(𝑥𝑥, 𝑡𝑡) = (c3 ± √𝑐𝑐 ( 𝑥𝑥 𝛼𝛼 + 𝑐𝑐𝑐𝑐) )3 . 2 𝛼𝛼 3

1

𝑐𝑐

2

∙ The solution of equation (3) is 𝑢𝑢(𝑥𝑥, 𝑡𝑡) = (c3 ± √𝑐𝑐 ( 𝑥𝑥 𝛼𝛼 + 𝑡𝑡 𝛼𝛼 ) )3 . 2 𝛼𝛼 𝛼𝛼

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Remarks: 1. The same ordinary differential equation is obtained from the three different forms of conformable fractional Harry Dym- Equation after using special wave variable for each form. 2. A function could be α-differentiable at a point but not differentiable, illustrating example was discussed in [6].

4. Examples. 3

𝑐𝑐

2

Example 1: Let 𝛼𝛼 = 0.7 , for the graph of equation (1) solution 𝑢𝑢(𝑥𝑥, 𝑡𝑡) = (c3 + √𝑐𝑐 � 𝑥𝑥 + 𝑡𝑡 𝛼𝛼 �)3 with 2 𝛼𝛼 respect to 𝑥𝑥 and t, with 𝑐𝑐₃ = 4 and 𝑐𝑐 = 1 see Figure 1.

3

1

2

Fig. 1 The graph of 𝑢𝑢(𝑥𝑥, 𝑡𝑡) = (4 + 2 � 𝑥𝑥 + 𝛼𝛼 𝑡𝑡 𝛼𝛼 �)3 at 𝛼𝛼 = 0.7 for example 1

3

𝑐𝑐

2

Example 2: The graph of equation (1) solution 𝑢𝑢(𝑥𝑥, 𝑡𝑡) = (c3 + √𝑐𝑐 � 𝑥𝑥 + 𝑡𝑡 𝛼𝛼 �)3 versus 𝑥𝑥 at 𝑡𝑡 = 1 , 2 𝛼𝛼 𝑐𝑐₃ = 4 𝑎𝑎𝑎𝑎𝑎𝑎 𝑐𝑐 = 1 for different values of 𝛼𝛼 is in Figure 2.

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3

1

2

Fig. 2 The graph of 𝑢𝑢(𝑥𝑥, 𝑡𝑡) = (4 + 2 � 𝑥𝑥 + 𝛼𝛼 �)3 versus 𝑥𝑥 at 𝑡𝑡 = 1 𝑎𝑎𝑎𝑎 𝛼𝛼 = 1, 0.9 𝑎𝑎𝑎𝑎𝑎𝑎 0.7 for example 2

3

1

2

Example 3: Let 𝛼𝛼 = 0.9 , for the graph of equation (2) solution 𝑢𝑢(𝑥𝑥, 𝑡𝑡) = (c3 + √𝑐𝑐 ( 𝑥𝑥 𝛼𝛼 + 𝑐𝑐𝑐𝑐) )3 2 𝛼𝛼 with respect to 𝑥𝑥 and t, with 𝑐𝑐₃ = 4 𝑎𝑎𝑎𝑎𝑎𝑎 𝑐𝑐 = 1 see Figure 3.

3

1

2

Fig. 3 The graph of 𝑢𝑢(𝑥𝑥, 𝑡𝑡) = (4 + 2 � 𝛼𝛼 𝑥𝑥𝛼𝛼 + 𝑡𝑡�)3 at 𝛼𝛼 = 0.9 for example 3

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Example 4: The graph of equation (2) solution 𝑢𝑢(𝑥𝑥, 𝑡𝑡) = �4 +

4 𝑎𝑎𝑎𝑎𝑎𝑎 𝑐𝑐 = 1 for different values of α is in Figure 4.

3

3 2

1 � 𝛼𝛼

𝛼𝛼

2 3

𝑥𝑥 + 𝑡𝑡�� versus x at 𝑡𝑡 = 0, 𝑐𝑐₃ =

2

1

Fig. 4 The graph of 𝑢𝑢(𝑥𝑥, 𝑡𝑡) = (4 + 2 �𝛼𝛼 𝑥𝑥𝛼𝛼 + 𝑡𝑡�)3 versus 𝑥𝑥 at t = 0 𝑎𝑎𝑎𝑎 𝛼𝛼 = 1, 0.9 𝑎𝑎𝑎𝑎𝑎𝑎 0.7 for example 4

3

1

𝑐𝑐

2

Example 5: Let 𝛼𝛼 = 0.9 , for the graph of equation (3) solution 𝑢𝑢(𝑥𝑥, 𝑡𝑡) = (c3 + √𝑐𝑐 ( 𝑥𝑥 𝛼𝛼 + 𝑡𝑡 𝛼𝛼 ) )3 2 𝛼𝛼 𝛼𝛼 with respect to 𝑥𝑥 and t, with 𝑐𝑐₃ = 4 𝑎𝑎𝑎𝑎𝑎𝑎 𝑐𝑐 = 1 see Figure 5.

3

1

1

2

Fig. 5 The graph of 𝑢𝑢(𝑥𝑥, 𝑡𝑡) = (4 + 2 � 𝛼𝛼 𝑥𝑥𝛼𝛼 + 𝛼𝛼 𝑡𝑡𝛼𝛼 �)3 at 𝛼𝛼 = 0.9

for example 5

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3

1

𝑐𝑐

2

Example 6: The graph of equation (3) solution 𝑢𝑢(𝑥𝑥, 𝑡𝑡) = (c3 + √𝑐𝑐 ( 𝑥𝑥 𝛼𝛼 + 𝑡𝑡 𝛼𝛼 ) )3 versus 𝑥𝑥 at 𝑡𝑡 = 1 2 𝛼𝛼 𝛼𝛼

, 𝑐𝑐₃ = 4 𝑎𝑎𝑎𝑎𝑎𝑎 𝑐𝑐 = 1 for different values of 𝛼𝛼 is in Figure 6.

3 1

2

1

Fig. 6 The graph of 𝑢𝑢(𝑥𝑥, 𝑡𝑡) = (4 + 2 �𝛼𝛼 𝑥𝑥𝛼𝛼 + 𝛼𝛼 𝑡𝑡𝛼𝛼 �)3 versus 𝑥𝑥 at t = 1 𝑎𝑎𝑎𝑎 𝛼𝛼 = 1, 0.9 𝑎𝑎𝑎𝑎𝑎𝑎 0.7 for example 6

References [1] B. Fuchssteinert , T. Schulzet, S. Carllot, Explicit solutions for Harry Dym equation, J Phys A: Math Gen.25(1992) 223-30 [2] A.A. Halim, Solition solutions of the (2+1) dimentional Harry Dym equation via Darboux transformation, Chaos Solitions Fractals 36 ( 2008) 646-53. [3] J.H. He, Some applications of non linear fractional differential equations and their approximations, Bull. Sci. Technol. 15 (2) (1999) 86--90. [4] W.Hereman, P.P. Banerjee, M.R. Chatterjee, On the nonlocal equations and nonlocal charges associated with the Harry-Dym hierarchy Korteweg-de Vries equation, J. Phys. A: Math. 22 (1989) 241-252. [5] Q. Huang , R. Zhdanov, Symmetries and exact solutions of the time fractional Harry- Dym equation with Rieman-Liouville derivative, Physica A. 409 (2014) 110-118. [6] R. Khalil, M. Al horani, A. Yousef , M. Sababheh, Anew definition of fractional derivative, Journal of Computational Applied Mathematics, 264 (2014) 65-70. [7] A.A. Kilbas, H.M. Srivastava, J.J Trujillo,Theory and Applications of Fractional Differential

Equations. North-Holland Math. Stud. 204 (2006). [8] A. Korkmaz, Exact Solutions to Some Conformable Time Fractional Equations in Benjamin-Bona-Mohany Family. https://arxiv.org/abs/1611.07086, 2007 (accessed 3 December 2007).

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[9] M.D.Kruskal , J.Moser, Dynamical system: Theory and Applications (Lecture Notes in Physics 38),

Springer, Berlin (1975). [10] D. Kumar , J. Singh, A. Kılıçman, An Efficient Approach for Fractional Harry Dym Equation by Using Sumudu Transform, Abstract and Applied Analysis. Article ID 608943(2013) 8 pages . [11] S. Kumar, M. P. Tripathi , O. P. Singh, A fractional model of Harry Dym equation and its approximate solution, Ain Shams Engineering Journa l .4 (2013),111-115. [12] R. Mokhtari, Exact solutions of the Harry- Dym equation, Commun. Theor. Phys. 55 (2).(2011) 204-208.

[13] D.P. Novikov, Alalgebraic geometric solution of the Harry Dym equation, Siberian Math J. 40 (1) (1999). 136-140. [14] I. Podlubny, Fractional differential equations. An introduction to fractional derivatives fractional differential equations some methods of their Solution and some of their applications, Academic Press, San Diego, 1999. [15] G.L. Vasconcelos, L.P. Kadanoff, Stationary solutions for the Saffman-Taylor problem with surface tention, . Phys Re A. 44 (10). (1991) 6490-6495.

. .

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Some Properties of the q-Exponential Functions Mahmoud J. S. Belaghi Bah¸ce¸sehir University, Istanbul, Turkey [email protected] Abstract. This paper aims to investigate some striking properties of the q-exponential functions more profoundly. To achieve this, at first, the Gauss q-binomial formula is generalized and based on the formula, important properties of the q-exponential functions are established. Keywords. q-Exponential function, q-Binomial formula. Mathematics Subject Classification. 11B65, 05A30.

1

Introduction t

The q-analogue of any real number t is defined as [t]q = 1−q 1−q and the q-factorial, denoted by [n]q !, is defined [1, 2] as ( 1 if n = 0, [n]q ! = (1) [n]q × [n − 1]q × · · · × [1]q if n = 1, 2, . . . . The q-analogue of (a + x)n , denoted by (a + x)nq , is defined [3] as ( 1 n = 0, n (a + x)q = Qn−1 m (a + q x) n = 1, 2, . . . . m=0

(2)

It is also defined for any complex number α as (a + x)α q =

(a + x)∞ q , (a + q α x)∞ q

(3)

Qn m α where (a + x)∞ q := limn→∞ m=0 (a + q x), and the principal value of q is considered, 0 < q < 1. n Yet, the q-Maclaurin series expansion of (a + x)q is n   X k n n an−k xk q (2) (4) (a + x)q = k q k=0


n

[n]q ! [k]q ![n−k]q !

are called q-binomial coefficients. Expression (4) is called Gauss q-binomial where k q = formula (see [3], p.
15). In the q-binomial coefficients, if |q| < 1 and n tends to infinity (see [3], p. 30) we 1 obtain limn→∞ nk q = (1−q) k . More details about the identities involving q-binomial coefficients can be q

found in reference [4]. One can also recall definitions of the q-functions [2, 5, 6] as follows: exq

∞ X 1 1 n = = x , (1 − (1 − q)x)∞ [n] q! q n=0

Eqx = (1 + (1 − q)x)∞ q =

|x| < 1,

∞ X

1 n (n2 ) x q , [n]q ! n=0

x ∈ C.

(5) (6)

It can be seen that exq Eq−x = 1 and exq−1 = Eqx . The product of the two functions are investigated in a more detailed way in [6, 7, 8]. The contribution of the corresponding references can be summarized in the following theorem: Theorem 1. For all x, y ∈ C the following equation holds exq Eqy =

∞ X

1 q (x + y)nq = e(x+y) q [n] ! q n=0

(7)

where (x + y)nq is defined in (4). In the light of aforementioned preliminaries, this paper aims at studying about the q-exponential functions more closely. At first, the Gauss q-binomial formula is generalized and based on the formula, some properties of the q-exponential functions are established.

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2

q-Exponential Functions

First, let us generalize the q-binomial formula given in (4). The generalization of the q-binomial can then be carried out as follows. Theorem 2. For any x, y, z ∈ C and positive integer n, the following identity holds: n   X n (x + y)nq = (x − z)kq (z + y)n−k . q k q

(8)

k=0

Proof. The induction is used to prove the theorem. Equation (8) is valid for n = 1. Assuming that (8) holds for any n and we show that it holds for n + 1. Then
(x + y)n+1 = (x + y)nq q k (z + q n−k y) + (x − q k z) q n   n   X X n n = q k (x − z)kq (z + y)qn+1−k + (x − z)k+1 (z + y)n−k q q k q k q k=0 k=0 n   X n n+1 n+1 = (z + y)q + (x − z)q + q k (x − z)kq (z + y)n+1−k q k q k=1  n  X n + (x − z)kq (z + y)n+1−k q k−1 q k=1 n+1 X n + 1  = (x − z)kq (z + y)n+1−k . q k q k=0

Thus, the proof is complete. It is realized that the identity in Theorem 2 can be re-written as (x +

y)nq

=

n   X n

k

k=0

(x − z)n−k (z + y)kq . q

(9)

q

Its proof can be readily derived form the proof of Theorem 2. Theorem 2 and its re-expression (9) allow one to conclude the striking identities given as follows: • For y = 0 and z = 1, the q-Taylor expansion of xn about x = 1, (see [3], p. 23) becomes xn =

n   X n

k

k=0

(x − 1)kq . q

• For x = 1, y = −ab and z = a, the following identity (see [2], p. 25 ) is obtained (1 − ab)nq =

n   X n

k

k=0

an−k (1 − a)kq (1 − b)n−k . q

q

• For y = −x, the identity n   X n k=0

k

(x − z)kq (z − x)n−k = 0. q

q

is found. • For the case of z = 0 in (9), the q-binomial formula in (4) is reached. • For x = 1, y = −ab and z = b in (9); the identity (see [2], p. 25 ) (1 −

ab)nq

=

n   X n k=0

k

bk (1 − a)kq (1 − b)n−k q q

is stated.

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Theorem 3. For x, y, z ∈ C, the following equations hold ∞

(x + y)q

∞

(z + y)q and

(10)

k ∞ (z+y)q X 1 (z + y)q 1 (1−q)x = e = . q [k]q ! (1 − q)k xk

(11)

k=0

∞

(x + y)q

∞

(x − z)q

k ∞ (x−z)q X 1 (x − z)q 1 (1−q)z = e , q [k]q ! (1 − q)k z k

=

k=0

Proof. As n → ∞ in equation (8), it is arrived at ∞

(x + y)q = lim

n→∞

= lim

n→∞

=

n   X n k=0 n  X k=0

k n k

k

n−k

(x − z)q (z + y)q

q n

 (x − q

k z)q

(z + y)q

k

(z + yq n−k )q

∞ ∞ X 1 1 k (z + y)q . (x − z) q [k]q ! (1 − q)k zk

k=0

∞

Dividing both sides of the last equation by (z + y)q gives ∞

(x + y)q

∞

(z + y)q

=

k ∞ X 1 (x − z)q 1 . [k]q ! (1 − q)k z k

k=0

(x−z)q

By using Theorem 1, the right hand side of the previous equation can be re-written as eq(1−q)z which completes the proof of equation (10). In a similar manner, the latter can be proven. Example 1. If we take x = 1 and y = −az in equation (11), we will get (see [2], p. 8 ) ∞

(1 − az)q (1 −

∞ z)q

=

k k ∞ ∞ X X (1 − a)q k 1 (z − az)q = z = k [k]q ! (1 − q)k (1 − q)q

k=0

1 φ0

(a; −; q, z) .

k=0

The function on the right hand side of the above equation is called basic hypergeometric series and more details about it can be found in [2]. Now we concentrate about the q-exponential functions. At first, product of the q-exponential functions is given in the next theorem and then some properties of the q-exponential functions are derived. Remark 1. For |x| < 1 and |q| < 1, the following identity holds ∞ X (1 − y)∞ (x − y)kq q = . (1 − x)∞ (1 − q)kq q

(12)

k=0

Theorem 4. For x, y, z ∈ C, the following identity holds eq(x+y)q = eq(x−z)q eq(z+y)q .

(13)

Proof. The identity (7) is taken to expand the q-exponential functions on the right hand side of (13), and thus ∞

∞ X


X 1
1 (x − z)nq (z + y)nq [n]q ! [n]q ! n=0 n=0   ∞ n X 1 X n = (x − z)kq (z + y)qn−k [n] ! k q q n=0

q q e(x−z) e(z+y) = q q

k=0

=

∞ X

1 (x + y)nq = eq(x+y)q . [n] ! q n=0

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Corollary 1. For x, z, ∈ C, the following identity holds eq−(x+z)q =

1 (z+x)q eq

.

Proof. By taking y := −x and z := −z in Theorem 4, the requirement can be easily carried out. Theorem 5. For x ∈ C and m, n ∈ Z, the following identity m−1 Q ((j+1)−j)q x   eq if m > n  j=n qx e(m−n) = q n−1 Q (j−(j+1))q x    eq if m < n j=m

holds. Proof. First, consider the case of m > n. The theorem is proven by induction. For the basis step, m = n + 1, the theorem is valid. Take the case m = k, k > n. Then it needs to be proven that it holds for the case m = k + 1. By using identity (13) and the induction, it can be reached qx q x (k−n)q x qx e((k+1)−n) = e((k+1)−k) eq = e((k+1)−k) q q q

k−1 Y

eq((j+1)−j)q x =

j=n

k Y

qx e((j+1)−j) q

j=n

which completes the proof of the first part. For the case of m < n, Corollary 1 is used. Then the result of the first part is applied to get qx e(m−n) = q

1 (n−m)q (x)

eq

1

= Qn−1

((j+1)−j)q x

j=m eq

=

n−1 Y

qx e(j−(j+1)) q

j=m

which completes the proof. Corollary 2. For x ∈ C, and positive integers m and n, the following identities hold: emx = q

m−1 Y

eq((j+1)−j)q x ,

(14)

j=0

Eq−nx =

n−1 Y

qx e(j−(j+1)) q

(15)

j=0

Proof. Consideration of (7) with n = 0 and m any positive integer in Theorem 5 leads to the complete proof of the first identity. Replacing m and n values between each other in the first identity gives the proof of the second one. Now then, the n-th q-derivative of the q-exponential functions is found in the next theorem. Theorem 6. For α, β, x ∈ C and positive integer n, qx Dqn e(α+β) = (α + β)nq e(α+q q q

n

β)q x

.

(16)

Proof. We use the induction to prove the theorem. For the case of n = 1, we need to get the q-derivative (α+β)q x of eq . So we use equation (7) and then take the q-derivative to obtain qx Dq e(α+β) = Dq q

∞ ∞ X  X 1 1 qx (α + β)kq xk = (α + β) (α + qβ)kq xk = (α + β) e(α+qβ) . q [k]q ! [k]q ! k=0

k=0

Assuming that (16) holds for a given k and to prove that it holds for k + 1, we need to obtain the (α+β)q x q-derivative of Dqk eq . Hence     k k+1 β)q x β)q x qx qx Dqk+1 e(α+β) = Dq Dqk e(α+β) = (α + β)kq Dq e(α+q = (α + β)k+1 e(α+q . q q q q q Thus the proof is complete.

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Theorem 7. For |x| < 1, |q| < 1 and any arbitrary α, the following identity holds e(1−q q

α

)q x

=

1 (1 − (1 − q)x)α q

(17)

Proof. To prove the theorem, we use equations (3), (5), (6) and (7). Then we have e(1−q q

α

)q x

= exq Eq−q

α

x

=

1 1 (1 − (1 − q)q α x)∞ q = ∞ (1 − (1 − q)x)q (1 − (1 − q)x)α q

which completes the proof. (q α −1)q x

Remark 2. Equation (17) can be rewritten as eq

= (1 − (1 − q)x)α q.

In the next example, we show that the q-binomial theorem (see: [1] P. 247 or [9] P. 488) can be proven shortly by using Theorem 1. Example 2. For |x| < 1 and |q| < 1, ∞ X (1 − a)kq

∞ X (1 − a)kq

k=0

k=0

xk = (1 − q)kq

[k]q !

(1−a)q x (1 − ax)∞ ( x ) ( −ax ) x
k q = eq (1−q) = eq 1−q Eq 1−q = . 1−q (1 − x)∞ q

Note that to reach this result; (7) in the second and third equations, and (5) and (6) in the last equation have been considered.

3

Conclusions and Recommendation

Some striking properties of the q-exponential functions have been analyzed in detail. In doing so, the Gauss q-binomial identity has generalized and based on it, remarkable properties of the q-exponential have been established. For further studies, similar discussion can be carried out for q-trigonometric functions.

4

Acknowledgment

I am thankful to Dr. M. Sari of Yildiz Technical University for patiently helping, advising me and also spending his valuable time to revise the paper.

References [1] Ernst, T., A Comprehensive Treatment of Q-calculus, Springer, 2012. [2] Gasper, G., Rahman, M., Basic hypergeometric series, Vol. 96 Cambridge university press, 2004. [3] Kac, V., Cheung, P., Quantum Calculus, Springer, 2002. [4] Gould, H. W., The q-Series Generalization of a Formula of Sparre Andersen. Mathematica Scandinavica, Vol. 9, pp. 90–94, 1961. [5] Exton, H., q-Hypergeometric Functions and ApplicationsEllis, Horwood, Chichester, 1983. [6] Jackson, F. H., A basic-sine and cosine with symbolical solutions of certain differential equations. Proceedings of the Edinburgh Mathematical Society, Vol. 22, pp. 28–39, 1904. [7] Hahn, W., Beitr¨ age zur Theorie der Heineschen Reihen. Die 24 Integrale der hypergeometrischen q-Differenzengleichung. Das q-Analogon der LaplaceTransformation, Mathematische Nachrichten, Vol. 2, no. 6, pp. 340–379, 1949. [8] Jackson, F. H., On basic double hypergeometric functions, The Quarterly Journal of Mathematics, Vol. 1, pp. 69–82, 1942. [9] Andrews, G. E., Askey, R., Roy, R., Special functions, Vol. 71, Cambridge university press, 1999.

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BCI-implicative ideals of BCI-algebras using neutrosophic quadruple structure Young Bae Jun1 , Seok-Zun Song2,∗ and and G. Muhiuddin 1

3

Department of Mathematics Education, Gyeongsang National University, Jinju 52828, Korea 2 Department of Mathematics, Jeju National University, Jeju 63243, Korea 3

Department of Mathematics, University of Tabuk, Tabuk 71491, Saudi Arabia

Abstract. Neutrosophic quadruple structure is used to study BCI-implicative ideal in BCI-algebra. The conceot of neutrosophic quadruple BCI-implicative ideal based on nonempty subsets in BCI-algebra is introduced, and their related properties are investigated. Relationship between neutrosophic quadruple ideal, neutrosophic quadruple BCI-implicative ideal, neutrosophic quadruple BCI-positive implicative ideal and neutrosophic quadruple BCIcommutative ideal are consulted. Conditions for the neutrosophic quadruple set to be neutrosophic quadruple BCI-implicative ideal are provided. A characterization of a neutrosophic quadruple BCI-implicative ideal is displayed, and the extension property of neutrosophic quadruple BCI-implicative ideal is established.

1. Introduction In [14], Smarandche has introduced the neutrosophic quadruple numbers for the first time. Using the notion of Smarandache’s neutrosophic quadruple numbers, Akinleye et al. [2] presented the notion of neutrosophic quadruple algebraic structures. In particular, they studied neutrosophic quadruple rings. Agboola et al. [1] studied neutrosophic quadruple algebraic hyperstructures, in particular, they developed neutrosophic quadruple semihypergroups, neutrosophic quadruple canonical hypergroups and neutrosophic quadruple hyperrings. Using BCK/BCI-algebras, Jun et al. [7] have established neutrosophic quadruple BCK/BCI-algebra, and have studied neutrosophic quadruple (positive implicative) ideal in neutrosophic quadruple BCK-algebra and neutrosophic quadruple closed ideal in neutrosophic quadruple BCI-algebra. Muhiuddin et al. [13] have studied neutrosophic quadruple q-ideal and (regular) neutrosophic quadruple ideal in neutrosophic quadruple BCI-algebra. Muhiuddin et al. [12] also have studied implicative neutrosophic quadruple ideal in neutrosophic quadruple BCK-algebra. In this article, we study BCI-implicative ideal in BCI-algebra using neutrosophic quadruple structure. We define neutrosophic quadruple BCI-implicative ideal based on nonempty subsets in BCI-algebra, and investigate their related properties. We consult relationship between neutrosophic quadruple ideal, neutrosophic quadruple BCI-implicative ideal, neutrosophic quadruple BCI-positive implicative ideal and neutrosophic quadruple BCIcommutative ideal. We provide conditions for the neutrosophic quadruple set to be neutrosophic quadruple BCI-implicative ideal. We discuss a characterization of an neutrosophic quadruple BCI-implicative ideal, and establish the extension property of neutrosophic quadruple BCI-implicative ideal. 0

2010 Mathematics Subject Classification: 06F35, 03G25, 08A72. Keywords: neutrosophic quadruple BCK/BCI-algebra; neutrosophic quadruple BCI-implicative ideal; neutrosophic quadruple BCI-positive implicative ideal; neutrosophic quadruple BCI-commutative ideal. ∗ The corresponding author. 0 E-mail : [email protected] (Y. B. Jun); [email protected] (S. Z. Song); [email protected] (G. Muhiuddin) 0

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Young Bae Jun, Seok-Zun Song and G. Muhiuddin 2. Preliminaries A BCK/BCI-algebra, which is an important class of logical algebras, is introduced by K. Is´eki (see [4, 5]) and it is being studied by many researchers. A BCI-algebra is a set X with a binary operation “·” and a special element “0” that satisfies the following conditions: (I) (∀x, y, z ∈ X) (((x · y) · (x · z)) · (z · y) = 0), (II) (∀x, y ∈ X) ((x · (x · y)) · y = 0), (III) (∀x ∈ X) (x · x = 0), (IV) (∀x, y ∈ X) (x · y = 0, y · x = 0 ⇒ x = y). If a BCI-algebra X satisfies the following identity: (V) (∀x ∈ X) (0 · x = 0), then X is called a BCK-algebra. Any BCK/BCI-algebra X satisfies the following conditions: (∀x ∈ X) (x · 0 = x) ,

(2.1)

(∀x, y, z ∈ X) (x ≤ y ⇒ x · z ≤ y · z, z · y ≤ z · x) ,

(2.2)

(∀x, y, z ∈ X) ((x · y) · z = (x · z) · y) ,

(2.3)

(∀x, y, z ∈ X) ((x · z) · (y · z) ≤ x · y)

(2.4)

where x ≤ y if and only if x · y = 0. Any BCI-algebra X satisfies the following conditions (see [3]): (∀x, y ∈ X)(x · (x · (x · y)) = x · y),

(2.5)

(∀x, y ∈ X)(0 · (x · y) = (0 · x) · (0 · y)),

(2.6)

(∀x, y ∈ X)(0 · (0 · (x · y)) = (0 · y) · (0 · x)).

(2.7)

An element a in a BCI-algebra X is said to be minimal (see [3]) if the following assertion is valid. (∀x ∈ X)(x ≤ a ⇒ x = a).

(2.8)

Note that the zero element 0 in a BCI-algebra X is minimal (see [3]). A nonempty subset S of a BCK/BCI-algebra X is called a subalgebra of X if x · y ∈ S for all x, y ∈ S. A subset G of a BCK/BCI-algebra X is called an ideal of X if it satisfies: 0 ∈ G,

(2.9)

(∀x ∈ X) (∀y ∈ G) (x · y ∈ G ⇒ x ∈ G) .

(2.10)

A subset G of a BCI-algebra X is called • a closed ideal of X (see [3]) if it is an ideal of X which satisfies: (∀x ∈ X)(x ∈ G ⇒ 0 · x ∈ G),

(2.11)

• a BCI-positive implicative ideal of X (see [8, 9]) if it satisfies (2.9) and (∀x, y, z ∈ X) (((x · z) · z) · (y · z) ∈ G, y ∈ G ⇒ x · z ∈ G) , 743

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BCI-implicative ideals of BCI-algebras using neutrosophic quadruple structure • a BCI-commutative ideal of X (see [10]) if it satisfies (2.9) and (x · y) · z ∈ G, z ∈ G ⇒ x · ((y · (y · x)) · (0 · (0 · (x · y)))) ∈ G

(2.13)

for all x, y, z ∈ X, • a BCI-implicative ideal of X (see [8]) if it satisfies (2.9) and (((x · y) · y) · (0 · y)) · z ∈ G, z ∈ G ⇒ x · ((y · (y · x)) · (0 · (0 · (x · y)))) ∈ G

(2.14)

for all x, y, z ∈ X. Note that every BCI-implicative ideal is an ideal, but the converse is not true (see [8]). Lemma 2.1 ([8]). A subset K of X is a BCI-implicative ideal of a BCI-algebra X if and only if it is an ideal of X that satisfies the following condition. ((x · y) · y) · (0 · y) ∈ K ⇒ x · ((y · (y · x)) · (0 · (0 · (x · y)))) ∈ K

(2.15)

for all x, y ∈ X. Lemma 2.2 ([10]). An ideal K of X is a BCI-commutative ideal of X if and only if it satisfies: x · y ∈ K ⇒ x · ((y · (y · x)) · (0 · (0 · (x · y)))) ∈ K

(2.16)

for all x, y, z ∈ X. Lemma 2.3 ([9]). An ideal K of X is a BCI-positive implicative ideal of X if and only if it satisfies: ((x · y) · y) · (0 · y) ∈ K ⇒ x · y ∈ K

(2.17)

for all x, y, z ∈ X. We refer the reader to the books [3, 11] for further information regarding BCK/BCI-algebras, and to the site “http://fs.gallup.unm.edu/neutrosophy.htm” for further information regarding neutrosophic set theory. We consider neutrosophic quadruple numbers based on a set instead of real or complex numbers. Let X be a set. A neutrosophic quadruple X-number is an ordered quadruple (a, xT, yI, zF ) where a, x, y, z ∈ X and T, I, F have their usual neutrosophic logic meanings (see [7]). The set of all neutrosophic quadruple X-numbers is denoted by Nq (X), that is, Nq (X) := {(a, xT, yI, zF ) | a, x, y, z ∈ X}, and it is called the neutrosophic quadruple set based on X. If X is a BCK/BCI-algebra, a neutrosophic quadruple X-number is called a neutrosophic quadruple BCK/BCI-number and we say that Nq (X) is the neutrosophic quadruple BCK/BCI-set. Let X be a BCK/BCI-algebra. We define a binary operation (a, xT, yI, zF )

on Nq (X) by

(b, uT, vI, wF ) = (a · b, (x · u)T, (y · v)I, (z · w)F )

for all (a, xT, yI, zF ), (b, uT, vI, wF ) ∈ Nq (X). Given a1 , a2 , a3 , a4 ∈ X, the neutrosophic quadruple BCK/BCInumber (a1 , a2 T, a3 I, a4 F ) is denoted by a ˜, that is, a ˜ = (a1 , a2 T, a3 I, a4 F ), 744

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Young Bae Jun, Seok-Zun Song and G. Muhiuddin and the zero neutrosophic quadruple BCK/BCI-number (0, 0T, 0I, 0F ) is denoted by ˜0, that is, ˜0 = (0, 0T, 0I, 0F ). Then (Nq (X); , ˜ 0) is a BCK/BCI-algebra (see [7]), which is called neutrosophic quadruple BCK/BCI-algebra, and it is simply denoted by Nq (X). We define an order relation “” and the equality “=” on Nq (X) as follows: x ˜  y˜ ⇔ xi ≤ yi for i = 1, 2, 3, 4, x ˜ = y˜ ⇔ xi = yi for i = 1, 2, 3, 4 for all x ˜, y˜ ∈ Nq (X). It is easy to verify that “” is an equivalence relation on Nq (X). Let X be a BCK/BCI-algebra. Given nonempty subsets K and J of X, consider the set Nq (K, J) := {(a, xT, yI, zF ) ∈ Nq (X) | a, x ∈ K & y, z ∈ J}, which is called the neutrosophic quadruple set based on K and J. The set Nq (K, K) is denoted by Nq (K), and it is called the neutrosophic quadruple set based on K. 3. Neutrosophic quadruple BCI-implicative ideals In what follows, let X denote a BCI-algebra unless otherwise specified. Definition 3.1. Let K and J be nonempty subsets of X. Then the neutrosophic quadruple set based on K and J is called a neutrosophic quadruple BCI-implicative ideal (briefly, NQ-BCI-implicative ideal) over (X, K, J) if it is a BCI-implicative ideal of Nq (X). If K = J, then we say that it is an NQ-BCI-implicative ideal over (X, K). Example 3.2. Consider a BCI-algebra X = {0, 1, 2, 3, 4, 5} with the binary operation ·, which is given in Table 1.

Table 1. Cayley table for the binary operation “·” · 0 1 2 3 4 5

0 0 1 2 3 4 5

1 0 0 2 3 3 3

2 0 1 0 3 4 5

3 3 3 3 0 1 1

4 3 3 3 0 0 1

5 3 3 3 0 0 0

Then the neutrosophic quadruple BCI-algebra Nq (X) has 64 elements. If we take K = {0, 1, 2}, then the neutrosophic quadruple set based on K has 81-elements, that is, Nq (K) = {˜0, ζ˜i | i = 1, 2, · · · , 80}, and it is an NQ-BCI-implicative ideal over (X, K) where 0˜ = (0, 0T, 0I, 0F ), ζ˜1 = (0, 0T, 0I, 1F ), ζ˜2 = (0, 0T, 0I, 2F ), ζ˜3 = (0, 0T, 1I, 0F ), ζ˜6 = (0, 0T, 2I, 0F ), ζ˜9 = (0, 1T, 0I, 0F ),

ζ˜4 = (0, 0T, 1I, 1F ), ζ˜5 = (0, 0T, 1I, 2F ), ζ˜7 = (0, 0T, 2I, 1F ), ζ˜8 = (0, 0T, 2I, 2F ), ζ˜10 = (0, 1T, 0I, 1F ), ζ˜11 = (0, 1T, 0I, 2F ), 745

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BCI-implicative ideals of BCI-algebras using neutrosophic quadruple structure ζ˜12 = (0, 1T, 1I, 0F ), ζ˜15 = (0, 1T, 2I, 0F ), ζ˜18 = (0, 2T, 0I, 0F ),

ζ˜13 = (0, 1T, 1I, 1F ), ζ˜16 = (0, 1T, 2I, 1F ), ζ˜19 = (0, 2T, 0I, 1F ),

ζ˜14 = (0, 1T, 1I, 2F ), ζ˜17 = (0, 1T, 2I, 2F ), ζ˜20 = (0, 2T, 0I, 2F ),

ζ˜21 = (0, 2T, 1I, 0F ), ζ˜24 = (0, 2T, 2I, 0F ), ζ˜27 = (1, 0T, 0I, 0F ),

ζ˜22 = (0, 2T, 1I, 1F ), ζ˜25 = (0, 2T, 2I, 1F ), ζ˜28 = (1, 0T, 0I, 1F ),

ζ˜23 = (0, 2T, 1I, 2F ), ζ˜26 = (0, 2T, 2I, 2F ), ζ˜29 = (1, 0T, 0I, 2F ),

ζ˜30 = (1, 0T, 1I, 0F ), ζ˜33 = (1, 0T, 2I, 0F ), ζ˜36 = (1, 1T, 0I, 0F ),

ζ˜31 = (1, 0T, 1I, 1F ), ζ˜34 = (1, 0T, 2I, 1F ), ζ˜37 = (1, 1T, 0I, 1F ),

ζ˜32 = (1, 0T, 1I, 2F ), ζ˜35 = (1, 0T, 2I, 2F ), ζ˜38 = (1, 1T, 0I, 2F ),

ζ˜39 = (1, 1T, 1I, 0F ), ζ˜42 = (1, 1T, 2I, 0F ), ζ˜45 = (1, 2T, 0I, 0F ),

ζ˜40 = (1, 1T, 1I, 1F ), ζ˜43 = (1, 1T, 2I, 1F ), ζ˜46 = (1, 2T, 0I, 1F ),

ζ˜41 = (1, 1T, 1I, 2F ), ζ˜44 = (1, 1T, 2I, 2F ), ζ˜47 = (1, 2T, 0I, 2F ),

ζ˜48 = (1, 2T, 1I, 0F ), ζ˜51 = (1, 2T, 2I, 0F ), ζ˜54 = (2, 0T, 0I, 0F ),

ζ˜49 = (1, 2T, 1I, 1F ), ζ˜52 = (1, 2T, 2I, 1F ), ζ˜55 = (2, 0T, 0I, 1F ),

ζ˜50 = (1, 2T, 1I, 2F ), ζ˜53 = (1, 2T, 2I, 2F ), ζ˜56 = (2, 0T, 0I, 2F ),

ζ˜57 = (2, 0T, 1I, 0F ), ζ˜60 = (2, 0T, 2I, 0F ), ζ˜63 = (2, 1T, 0I, 0F ),

ζ˜58 = (2, 0T, 1I, 1F ), ζ˜61 = (2, 0T, 2I, 1F ), ζ˜64 = (2, 1T, 0I, 1F ),

ζ˜59 = (2, 0T, 1I, 2F ), ζ˜62 = (2, 0T, 2I, 2F ), ζ˜65 = (2, 1T, 0I, 2F ),

ζ˜66 = (2, 1T, 1I, 0F ), ζ˜69 = (2, 1T, 2I, 0F ), ζ˜72 = (2, 2T, 0I, 0F ),

ζ˜67 = (2, 1T, 1I, 1F ), ζ˜70 = (2, 1T, 2I, 1F ), ζ˜73 = (2, 2T, 0I, 1F ),

ζ˜68 = (2, 1T, 1I, 2F ), ζ˜71 = (2, 1T, 2I, 2F ), ζ˜74 = (2, 2T, 0I, 2F ),

ζ˜75 = (2, 2T, 1I, 0F ), ζ˜78 = (2, 2T, 2I, 0F ),

ζ˜76 = (2, 2T, 1I, 1F ), ζ˜79 = (2, 2T, 2I, 1F ),

ζ˜77 = (2, 2T, 1I, 2F ), ζ˜80 = (2, 2T, 2I, 2F ).

Theorem 3.3. Every NQ-BCI-implicative ideal is a neutrosophic quadruple ideal. Proof. It is straightforward since every BCI-implicative ideal is an ideal in BCI-algebras.



The converse of Theorem 3.3 is not true in general as seen in the following example. Example 3.4. Let X = {0, 1, 2, 3, 4} be a set with the binary operation ·, which is given in Table 2.

Table 2. Cayley table for the binary operation “·” · 0 1 2 3 4

0 0 1 2 3 4

1 0 0 2 3 4

2 0 0 0 2 4 746

3 0 0 0 0 4

4 4 4 4 4 0 Young Bae Jun 742-757

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Young Bae Jun, Seok-Zun Song and G. Muhiuddin Then X is a BCI-algebra (see [8]), and the neutrosophic quadruple BCI-algebra Nq (X) has 625 elements. If we take K = {0, 1}, then the neutrosophic quadruple set based on K has 16-elements, that is, Nq (K) = {˜0, ζ˜i | i = 1, 2, · · · , 15}, and it is a neutrosophic quadruple ideal over (X, K) where ˜0 = (0, 0T, 0I, 0F ), ζ˜1 = (0, 0T, 0I, 1F ), ζ˜2 = (0, 0T, 1I, 0F ), ζ˜3 = (0, 0T, 1I, 1F ), ζ˜4 = (0, 1T, 0I, 0F ), ζ˜5 = (0, 1T, 0I, 1F ), ζ˜6 = (0, 1T, 1I, 0F ), ζ˜7 = (0, 1T, 1I, 1F ), ζ˜8 = (1, 0T, 0I, 0F ), ζ˜9 = (1, 0T, 0I, 1F ), ζ˜10 = (1, 0T, 1I, 0F ), ζ˜11 = (1, 0T, 1I, 1F ), ζ˜12 = (1, 1T, 0I, 0F ), ζ˜13 = (1, 1T, 0I, 1F ), ζ˜14 = (1, 1T, 1I, 0F ), ζ˜15 = (1, 1T, 1I, 1F ). If we take x ˜ = (2, 2T, 2I, 2F ) and y˜ = (3, 3T, 3I, 3F ) in Nq (X), then (((˜ y

x ˜)

(˜0

x ˜)

= ((((3, 3T, 3I, 3F ) ((0, 0T, 0I, 0F )

˜0

x ˜))

(2, 2T, 2I, 2F ))

(2, 2T, 2I, 2F )))

(2, 2T, 2I, 2F ))

(0, 0T, 0I, 0F )

= (0, 0T, 0I, 0F ) ∈ Nq (K). But y˜

((˜ x

(˜ x

(˜ 0

y˜))

= (3, 3T, 3I, 3F ) ((0, 0T, 0I, 0F )

(˜0

(˜ y

x ˜))))

(((2, 2T, 2I, 2F ) ((0, 0T, 0I, 0F )

((2, 2T, 2I, 2F ) ((3, 3T, 3I, 3F )

(3, 3T, 3I, 3F ))) (2, 2T, 2I, 2F )))))

= (2, 2T, 2I, 2F ) ∈ / Nq (K). Hence Nq (K) is not a BCI-implicative ideal of Nq (X), and so it is not an NQ-BCI-implicative ideal over (X, K). Lemma 3.5 ([7]). If K and J are ideals of X, then the neutrosophic quadruple set based on K and J is a neutrosophic quadruple ideal over (X, K, J). Theorem 3.6. The neutrosophic quadruple set based on BCI-implicative ideals K and J of X is an NQ-BCIimplicative ideal over (X, K, J). Proof. Let K and J be BCI-implicative ideals of X. Since 0 ∈ K ∩ J, we get ˜0 ∈ Nq (K, J). Let x ˜ = (x1 , x2 T, x3 I, x4 F ), y˜ = (y1 , y2 T, y3 I, y4 F ) and z˜ = (z1 , z2 T, z3 I, z4 F ) be elements of Nq (X) such that (((˜ x

y˜)

y˜)

(˜0

y˜)) 747

z˜ ∈ Nq (K, J) Young Bae Jun 742-757

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BCI-implicative ideals of BCI-algebras using neutrosophic quadruple structure and z˜ ∈ Nq (K, J). Then z˜ = (z1 , z2 T, z3 I, z4 F ) ∈ Nq (K, J) and

(((˜ x

y˜)

y˜)

(˜ 0

y˜))

= ((((x1 , x2 T, x3 I, x4 F ) ((0, 0T, 0I, 0F )

z˜ (y1 , y2 T, y3 I, y4 F ))

(y1 , y2 T, y3 I, y4 F )))

(y1 , y2 T, y3 I, y4 F ))

(z1 , z2 T, z3 I, z4 F )

= (((((x1 · y1 ) · y1 ) · (0 · y1 )) · z1 ), ((((x2 · y2 ) · y2 ) · (0 · y2 )) · z2 )T, ((((x3 · y3 ) · y3 ) · (0 · y3 )) · z3 )I, ((((x4 · y4 ) · y4 ) · (0 · y4 )) · z4 )F ) ∈ Nq (K, J).

Hence zi ∈ K and (((xi · yi ) · yi ) · (0 · yi )) · zi ∈ K for i = 1, 2; and zj ∈ J and (((xj · yj ) · yj ) · (0 · yj )) · zj ∈ K for j = 3, 4. Since K and J are BCI-implicative ideals of X, it follows that xi · ((yi · (yi · xi )) · (0 · (0 · (xi · yi )))) ∈ K and xj · ((yj · (yj · xj )) · (0 · (0 · (xj · yj )))) ∈ J for i = 1, 2 and j = 3, 4. Thus

x ˜

((˜ y

(˜ y

x ˜)) · (˜0

(˜0

(˜ x

y˜))))

= (x1 , x2 T, x3 I, x4 F ) · (((y1 , y2 T, y3 I, y4 F ) · ((y1 , y2 T, y3 I, y4 F )· (x1 , x2 T, x3 I, x4 F ))) · ((0, 0T, 0I, 0F ) · ((0, 0T, 0I, 0F )· ((x1 , x2 T, x3 I, x4 F ) · (y1 , y2 T, y3 I, y4 F ))))) = (x1 · ((y1 · (y1 · x1 )) · (0 · (0 · (x1 · y1 )))), (x2 · ((y2 · (y2 · x2 )) · (0 · (0 · (x2 · y2 )))))T, (x3 · ((y3 · (y3 · x3 )) · (0 · (0 · (x3 · y3 )))))I, (x4 · ((y4 · (y4 · x4 )) · (0 · (0 · (x4 · y4 )))))F ) ∈ Nq (K, J).

Hence Nq (K, J) is a BCI-implicative ideal of Nq (X), and therefore the neutrosophic quadruple set based on K and J is an NQ-BCI-implicative ideal over (X, K, J).



Corollary 3.7. The neutrosophic quadruple set based on a BCI-implicative ideal K of X is an NQ-BCI-implicative ideal over (X, K).

Proposition 3.8. Every neutrosophic quadruple set based on BCI-implicative ideals K and J of X satisfies the following condition. ((˜ x y˜) y˜) (˜ 0 y˜) ∈ Nq (K, J) ⇒x ˜ ((˜ y (˜ y x ˜)) (˜0 (˜0 (˜ x

y˜)))) ∈ Nq (K, J).

(3.1)

for all x ˜, y˜, z˜ ∈ Nq (X). 748

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Young Bae Jun, Seok-Zun Song and G. Muhiuddin Proof. Let ((˜ x

y˜)

y˜)

(˜ 0

y˜) ∈ Nq (K, J) for all x ˜, y˜, z˜ ∈ Nq (X). Then

((((x1 · y1 ) · y1 ) · (0 · y1 )) · 0, ((((x2 · y2 ) · y2 ) · (0 · y2 )) · 0)T, ((((x3 · y3 ) · y3 ) · (0 · y3 )) · 0)I, ((((x4 · y4 ) · y4 ) · (0 · y4 )) · 0)F ) = (((x1 · y1 ) · y1 ) · (0 · y1 ), (((x2 · y2 ) · y2 ) · (0 · y2 ))T, (((x3 · y3 ) · y3 ) · (0 · y3 ))I, (((x4 · y4 ) · y4 ) · (0 · y4 ))F ) = (((x1 , x2 T, x3 I, x4 F )

(y1 , y2 T, y3 I, y4 F ))

((0, 0T, 0I, 0F )

(y1 , y2 T, y3 I, y4 F ))

= ((˜ x

(˜ 0

y˜)

y˜)

(y1 , y2 T, y3 I, y4 F ))

y˜) ∈ Nq (K, J),

and so (((xi · yi ) · yi ) · (0 · yi )) · 0 ∈ K for i = 1, 2 and (((xj · yj ) · yj ) · (0 · yj )) · 0 ∈ J for j = 3, 4. Since 0 ∈ K ∩ J, and since K and J are BCI-implicative ideals of X, it follows that xi · ((yi · (yi · xi )) · (0 · (0 · (xi · yi )))) ∈ K for i = 1, 2, and xj · ((yj · (yj · xj )) · (0 · (0 · (xj · yj )))) ∈ J for j = 3, 4. Hence we have x ˜

((˜ y

(˜ y

x ˜))

(˜0

(˜0

(˜ x

y˜))))

= (x1 · ((y1 · (y1 · x1 )) · (0 · (0 · (x1 · y1 )))), (x2 · ((y2 · (y2 · x2 )) · (0 · (0 · (x2 · y2 )))))T, (x3 · ((y3 · (y3 · x3 )) · (0 · (0 · (x3 · y3 )))))I, (x4 · ((y4 · (y4 · x4 )) · (0 · (0 · (x4 · y4 )))))F ) ∈ Nq (K, J). This completes the proof.



We provide conditions for a neutrosophic quadruple set to be an NQ-BCI-implicative ideal. Theorem 3.9. Let K and J be ideals of X such that ((x · y) · y) · (0 · y) ∈ K (resp., J) ⇒ x · ((y · (y · x)) · (0 · (0 · (x · y)))) ∈ K (resp., J)

(3.2)

for all x, y ∈ X. Then the neutrosophic quadruple set based on K and J is an NQ-BCI-implicative ideal over (X, K, J). Proof. Assume that (((x · y) · y) · (0 · y)) · z ∈ K (resp., J) for all x, y ∈ X and z ∈ K (resp., J). Then ((x · y) · y) · (0 · y) ∈ K (resp., J) since K and J are ideals of X. It follows from the condition (3.2) that x · ((y · (y · x)) · (0 · (0 · (x · y)))) ∈ K (resp., J). Hence K and J are BCI-implicative ideals of X, and therefore the neutrosophic quadruple set based on K and J is an NQ-BCI-implicative ideal over (X, K, J) by Theorem 3.6.  Corollary 3.10. Let K be an ideal of X such that ((x · y) · y) · (0 · y) ∈ K ⇒ x · ((y · (y · x)) · (0 · (0 · (x · y)))) ∈ K

(3.3)

for all x, y ∈ X. Then the neutrosophic quadruple set based on K is an NQ-BCI-implicative ideal over (X, K). 749

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BCI-implicative ideals of BCI-algebras using neutrosophic quadruple structure Theorem 3.11. Let K and J be ideals of X such that 0 · x ∈ K (resp., J),

(3.4)

((x · y) · y) · (0 · y) ∈ K (resp., J) ⇒ x · (y · (y · x)) ∈ K (resp., J)

(3.5)

for all x, y ∈ X. Then the neutrosophic quadruple set based on K and J is an NQ-BCI-implicative ideal over (X, K, J). Proof. Assume that ((x · y) · y) · (0 · y) ∈ K (resp., J) for all x, y ∈ X. Then x · (y · (y · x)) ∈ K (resp., J) by (3.5). Using (I), (II), (2.3), (2.5), (2.6) and (3.4), we have (x · ((y · (y · x)) · (0 · (0 · (x · y))))) · (x · (y · (y · x))) ≤ (y · (y · x)) · ((y · (y · x)) · (0 · (0 · (x · y)))) ≤ 0 · (0 · (x · y)) = 0 · ((0 · x) · (0 · y)) = 0 · ((((0 · y) · x) · (0 · y)) · (0 · y)) = 0 · ((((0 · (0 · (0 · y))) · x) · (0 · y)) · (0 · y)) = 0 · ((((0 · x) · (0 · y)) · (0 · y)) · (0 · (0 · y))) = 0 · (((0 · (x · y)) · (0 · y)) · (0 · (0 · y))) = 0 · (0 · (((x · y) · y) · (0 · y))) ∈ K (resp., J). It follows that x · ((y · (y · x)) · (0 · (0 · (x · y)))) ∈ K (resp., J). Hence K and J are BCI-implicative ideals of X by Lemma 2.1. Therefore the neutrosophic quadruple set based on K and J is an NQ-BCI-implicative ideal over (X, K, J) by Theorem 3.6.



Corollary 3.12. Let K be an ideal of X such that 0 · x ∈ K,

(3.6)

((x · y) · y) · (0 · y) ∈ K ⇒ x · (y · (y · x)) ∈ K

(3.7)

for all x, y ∈ X. Then the neutrosophic quadruple set based on K is an NQ-BCI-implicative ideal over (X, K). Theorem 3.13. Let X be a BCI-algebra satisfying the conditions: (∀x, y ∈ X)(x · y = ((x · y) · y) · (0 · y)),

(3.8)

(∀x, y ∈ X)(x · (x · y) = y · (y · (x · (x · y)))).

(3.9)

If K and J are closed ideals of X, then the neutrosophic quadruple set based on K and J is an NQ-BCI-implicative ideal over (X, K, J). 750

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Young Bae Jun, Seok-Zun Song and G. Muhiuddin Proof. Let K and J be closed ideals of X. Assume that ((x·y)·y)·(0·y) ∈ K (resp., J). Then 0·(((x·y)·y)·(0·y)) ∈ K (resp., J). Using the conditions (3.8), (3.9), (2.3), (2.5), (I) and (III), we have (x · (y · (y · x))) · (((x · y) · y) · (0 · y)) = (x · (y · (y · x))) · (x · y) = (x · (x · y)) · (y · (y · x)) = (y · (y · (x · (x · y)))) · (y · (y · x)) = (y · (y · (y · x))) · (y · (x · (x · y))) (3.10)

= (y · x) · (y · (x · (x · y))) ≤ (x · (x · y)) · x = 0 · (x · y) = 0 · (((x · y) · y) · (0 · y)) ∈ K (resp., J). It follows that x · (y · (y · x)) ∈ K (resp., J), and so that x · ((y · (y · x)) · (0 · (0 · (x · y)))) ∈ K (resp., J)

in the proof of Theorem 3.18. Thus K and J are BCI-implicative ideals of X by Lemma 2.1, and therefore the neutrosophic quadruple set based on K and J is an NQ-BCI-implicative ideal over (X, K, J) by Theorem 3.6.  Corollary 3.14. Let X be a BCI-algebra satisfying the conditions (3.8) and (3.9). If K is a closed ideal of X, then the neutrosophic quadruple set based on K is an NQ-BCI-implicative ideal over (X, K). Corollary 3.15. Let X be a BCI-algebra satisfying the condition: (∀x, y ∈ X)((x · (x · y)) · (y · x) = y · (y · x)).

(3.11)

If K and J are closed ideals of X, then the neutrosophic quadruple set based on K and J is an NQ-BCI-implicative ideal over (X, K, J). Proof. If X satisfies the condition (3.11), then it satisfies two conditions (3.8) and (3.9) (see [?, ?]). Hence the result is induced from Theorem 3.13.



Corollary 3.16. Let X be a BCI-algebra satisfying the condition (3.11). If K is a closed ideal of X, then the neutrosophic quadruple set based on K is an NQ-BCI-implicative ideal over (X, K). Theorem 3.17. Let X be a BCI-algebra satisfying the condition (3.9) and (∀x, y ∈ X)((x · (y · x)) · (0 · (y · x)) = x).

(3.12)

If K and J are closed ideals of X, then the neutrosophic quadruple set based on K and J is an NQ-BCI-implicative ideal over (X, K, J). Proof. Let K and J be closed ideals of X. The conditions (3.12) and (III) lead to the following fact. (z · y) · (((z · y) · (z · (z · y))) · (0 · (z · (z · y)))) = 0. 751

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BCI-implicative ideals of BCI-algebras using neutrosophic quadruple structure It follows from (2.1), (I), (2.2), (2.3) and (III) that (z · y) · (((z · y) · y) · (0 · y)) = ((z · y) · (((z · y) · y) · (0 · y))) · 0 = ((z · y) · (((z · y) · y) · (0 · y))) · ((z · y) · (((z · y) · (z · (z · y)))· (0 · (z · (z · y))))) ≤ (((z · y) · (z · (z · y))) · (0 · (z · (z · y)))) · (((z · y) · y) · (0 · y)) ≤ (((z · y) · y) · (0 · (z · (z · y)))) · (((z · y) · y) · (0 · y))

(3.14)

≤ (0 · y) · (0 · (z · (z · y))) ≤ (z · (z · y)) · y = (z · y) · (z · y) = 0. Hence (z · y) · (((z · y) · y) · (0 · y)) = 0 since 0 is a minimal element of X, that is, z · y ≤ ((z · y) · y) · (0 · y).

(3.15)

On the other hand, we get (((z · y) · y) · (0 · y)) · (z · y) = (((z · y) · y) · (z · y)) · (0 · y) = (((z · y) · (z · y)) · y) · (0 · y) = (0 · y) · (0 · y) = 0 by (2.3) and (III), that is, ((z · y) · y) · (0 · y) ≤ z · y.

(3.16)

Conditions (3.15) and (3.16) induce z · y = ((z · y) · y) · (0 · y). Therefore the neutrosophic quadruple set based on K and J is an NQ-BCI-implicative ideal over (X, K, J) by Theorem 3.13.



We now consider extension property of NQ-BCI-implicative ideal. Theorem 3.18. For any nonempty subsets K and J of X, let A and B be closed ideals of X such that K ⊆ A and J ⊆ B. If K and J are BCI-implicative ideals of X, then the neutrosophic quadruple set based on A and B is an NQ-BCI-implicative ideal over (X, A, B), which is larger than the NQ-BCI-implicative ideal over (X, K, J). Proof. Assume that K and J are BCI-implicative ideals of X. It is clear that Nq (K, J) ⊆ Nq (A, B). Let ((x · y) · y) · (0 · y) ∈ A (resp., B) for all x, y ∈ X. Then 0 · (((x · y) · y) · (0 · y)) ∈ A (resp., B) since A and B are closed ideals of X. Using (2.3) and (III) induce (((x · (((x · y) · y) · (0 · y))) · y) · y) · (0 · y) = (((x · y) · y) · (0 · y)) · (((x · y) · y) · (0 · y))

(3.17)

= 0 ∈ K (resp., J), which implies from Lemma 2.1 that (x · (((x · y) · y) · (0 · y))) · ((y · (y · (x · (((x · y) · y) · (0 · y)))))· (0 · (0 · ((x · (((x · y) · y) · (0 · y))) · y))))

(3.18)

∈ K ⊆ A (resp., J ⊆ B). 752

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Young Bae Jun, Seok-Zun Song and G. Muhiuddin Since 0 · (((x · y) · y) · (0 · y)) = ((0 · (x · y)) · (0 · y)) · (0 · (0 · y)) = (((0 · x) · (0 · y)) · (0 · y)) · (0 · (0 · y)) = (((0 · (0 · (0 · y))) · x) · (0 · y)) · (0 · y) = (((0 · y) · x) · (0 · y)) · (0 · y)

(3.19)

= (0 · x) · (0 · y) = 0 · (x · y) by (2.6), (2.3), (2.5) and (III), we have 0 · (0 · ((x · (((x · y) · y) · (0 · y))) · y)) = 0 · (0 · ((x · y) · (((x · y) · y) · (0 · y)))) = 0 · ((0 · (x · y)) · (0 · (((x · y) · y) · (0 · y))))

(3.20)

= 0 · ((0 · (x · y)) · (0 · (x · y))) = 0. Combining (3.18) and (3.20) implies that (x · (y · (y · (x · (((x · y) · y) · (0 · y)))))) · (((x · y) · y) · (0 · y)) = (x · (((x · y) · y) · (0 · y))) · (y · (y · (x · (((x · y) · y) · (0 · y)))))

(3.21)

∈ A (resp., B). Since A and B are ideals of X, it follows that x · (y · (y · (x · (((x · y) · y) · (0 · y))))) ∈ A (resp., B).

(3.22)

On the other hand, we have (x · (y · (y · x))) · (x · (y · (y · (x · (((x · y) · y) · (0 · y)))))) ≤ (y · (y · (x · (((x · y) · y) · (0 · y))))) · (y · (y · x)) ≤ (y · x) · (y · (x · (((x · y) · y) · (0 · y)))) ≤ (x · (((x · y) · y) · (0 · y))) · x

(3.23)

= 0 · (((x · y) · y) · (0 · y)) ∈ A (resp., B). By (3.22) and (3.23), we get x · (y · (y · x)) ∈ A (resp., B). Using (3.19), (I), (II) we get (x · ((y · (y · x)) · (0 · (0 · (x · y))))) · (x · (y · (y · x))) ≤ (y · (y · x)) · ((y · (y · x)) · (0 · (0 · (x · y)))) (3.24)

≤ 0 · (0 · (x · y)) = 0 · (0 · (((x · y) · y) · (0 · y))) ∈ A (resp., B).

It follows that x · ((y · (y · x)) · (0 · (0 · (x · y)))) ∈ A (resp., B). Hence A and B are BCI-implicative ideals of X by Lemma 2.1. Therefore the neutrosophic quadruple set based on A and B is an NQ-BCI-implicative ideal over (X, A, B) by Theorem 3.6.

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BCI-implicative ideals of BCI-algebras using neutrosophic quadruple structure Corollary 3.19. For any nonempty subset K of X, let A be a closed ideal of X such that K ⊆ A. If K is a BCI-implicative ideals of X, then the neutrosophic quadruple set based on A is an NQ-BCI-implicative ideal over (X, A), which is larger than the NQ-BCI-implicative ideal over (X, K). 4. Relations between NQ-BCI-commutative ideal, NQ-BCI-positive implicative ideal and NQ-BCI-implicative ideal Theorem 4.1. For any nonempty subsets K and J of X, every NQ-BCI-implicative ideal over (X, K, J) is an NQ-BCI-commutative ideal over (X, K, J). Proof. Let K and J be nonempty subsets of X such that the neutrosophic quadruple set based on K and J is an NQ-BCI-implicative ideal over (X, K, J). Let x, y, z ∈ X be such that z ∈ K (resp., J) and (((x·y)·y)·(0·y))·z ∈ K (resp., J). Then (z, zT, zI, zF ) ∈ Nq (K, J) and ((((x, xT, xI, xF ) ((0, 0T, 0I, 0F )

(y, yT, yI, yF )) (y, yT, yI, yF )))

(y, yT, yI, yF )) (z, zT, zI, zF )

= ((((x · y) · y) · (0 · y)) · z, ((((x · y) · y) · (0 · y)) · z)T, ((((x · y) · y) · (0 · y)) · z)I, ((((x · y) · y) · (0 · y)) · z)F ) ∈ Nq (K, J) Since Nq (K, J) is a BCI-implicative ideal of Nq (X), it follows that (x · ((y · (y · x)) · (0 · (0 · (x · y)))), (x · ((y · (y · x)) · (0 · (0 · (x · y)))))T, (x · ((y · (y · x)) · (0 · (0 · (x · y)))))I, (x · ((y · (y · x)) · (0 · (0 · (x · y)))))F ) = (x, xT, xI, xF ) ((0, 0T, 0I, 0F )

(((y, yT, yI, yF ) ((0, 0T, 0I, 0F )

((y, yT, yI, yF )

((x, xT, xI, xF )

(x, xT, xI, xF ))) (y, yT, yI, yF )))))

∈ Nq (K, J). Hence x · ((y · (y · x)) · (0 · (0 · (x · y)))) ∈ K (resp., J), and so K and J are BCI-implicative ideals of X. Thus K and J are ideals of X. Assume that x · y ∈ K (resp., J) for all x, y ∈ X. Then (((x · y) · y) · (0 · y)) · (x · y) = (0 · y) · (0 · y) = 0 ∈ K (resp., J) by using (2.3) and (III), which implies that ((x · y) · y) · (0 · y) ∈ K (resp., J). Hence (((x · y) · y) · (0 · y)) · 0 ∈ K (resp., J) and 0 ∈ K (resp., J). Since K (resp., J) is a BCI-implicative ideal of X, it follows that x · ((y · (y · x)) · (0 · (0 · (x · y)))) ∈ K (resp., J). Therefore K (resp., J) is a BCI-commutative ideal of X by Lemma 2.2, and consequently the neutrosophic quadruple set based on K and J is an NQ-BCI-commutative ideal over (X, K, J).



The converse of Theorem 4.1 is not true in general. In fact, Nq (K) in Example 3.4 is not a BCI-implicative ideal of Nq (X). But it is routine to verify that Nq (K) is a BCI-commutative ideal of Nq (X). 754

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Young Bae Jun, Seok-Zun Song and G. Muhiuddin Lemma 4.2 ([6]). If K and J are BCI-positive implicative ideals of X, then the neutrosophic quadruple set based on K and J is an NQ-BCI-positive implicative ideal over (X, K, J). Theorem 4.3. For any nonempty subsets K and J of X, every NQ-BCI-implicative ideal over (X, K, J) is an NQ-BCI-positive implicative ideal over (X, K, J). Proof. Let K and J be nonempty subsets of X such that Nq (K, J) is a BCI-implicative ideal of Nq (X). Then K and J are ideals of X (see the proof of Theorem 4.1). Let x, y ∈ X be such that ((x · y) · y) · (0 · y) ∈ K (resp., J). Then x · ((y · (y · x)) · (0 · (0 · (x · y)))) ∈ K (resp., J) by Lemma 2.1. Note that (x · y) · (x · ((y · (y · x)) · (0 · (0 · (x · y))))) ≤ ((y · (y · x)) · (0 · (0 · (x · y)))) · y = (0 · (y · x)) · (0 · (0 · (x · y))) = (0 · (x · y)) · (y · x) = ((0 · x) · (0 · y)) · (y · x) = (0 · (0 · x)) · x = 0 ∈ K (resp., J). It follows that x · y ∈ K (resp., J). Hence K and J are BCI-positive implicative ideals of X by Lemma 2.3, and therefore Nq (K, J) is a BCI-positive implicative ideal of Nq (X) by Lemma 4.2.



In the following example, we can see that the converse of Theorem 4.3 is not true in general. Example 4.4. Let X = {0, 1, 2, 3, 4} be a set with the binary operation “·”, which is given in Table 3.

Table 3. Cayley table for the binary operation “·” · 0 1 2 3 4

0 0 1 2 3 4

1 0 0 2 3 4

2 0 1 0 3 4

3 0 0 0 0 4

4 4 4 4 4 0

Then X is a BCI-algebra (see [8]), and the neutrosophic quadruple BCI-algebra Nq (X) has 625 elements. If we take K = {0, 2}, then the neutrosophic quadruple set based on K has 16-elements, that is, Nq (K) = {˜0, ρ˜i | i = 1, 2, · · · , 15}, where ˜0 = (0, 0T, 0I, 0F ), ρ˜1 = (0, 0T, 0I, 2F ), ρ˜2 = (0, 0T, 2I, 0F ), ρ˜3 = (0, 0T, 2I, 1F ), ρ˜4 = (0, 2T, 0I, 0F ), ρ˜5 = (0, 2T, 0I, 2F ), 755

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BCI-implicative ideals of BCI-algebras using neutrosophic quadruple structure ρ˜6 = (0, 2T, 2I, 0F ), ρ˜7 = (0, 2T, 2I, 2F ), ρ˜8 = (2, 0T, 0I, 0F ), ρ˜9 = (2, 0T, 0I, 2F ), ρ˜10 = (2, 0T, 2I, 0F ), ρ˜11 = (2, 0T, 2I, 2F ), ρ˜12 = (2, 2T, 0I, 0F ), ρ˜13 = (2, 2T, 0I, 2F ), ρ˜14 = (2, 2T, 2I, 0F ), ρ˜15 = (2, 2T, 2I, 2F ). It is routine to verify that Nq (K) is an NQ-BCI-positive implicative ideal over (X, K). (1, 1T, 1I, 1F ) and α ˜ 3 = (3, 3T, 3I, 3F ) in Nq (X), then ˜0 ∈ Nq (K) and (((α ˜1

α ˜3)

α ˜3)

(˜0

(˜0

(˜ α1

α ˜ 3 ))

If we take α ˜1 =

˜0 = ˜0 ∈ Nq (K).

But, α ˜1

((˜ α3

(˜ α3

α ˜ 1 ))

(˜ 0

α ˜ 3 )))) = α ˜1

(˜0

˜0) = α ˜1 ∈ / Nq (K).

Hence Nq (K) is not an NQ-BCI-implicative ideal over (X, K). We display a characterization of an NQ-BCI-implicative ideal. Theorem 4.5. For any nonempty subsets K and J of X, the neutrosophic quadruple set based on K and J is both an NQ-BCI-commutative ideal and an NQ-BCI-positive implicative ideal over (X, K, J) if and only if it is an NQ-BCI-implicative ideal over (X, K, J). Proof. For the sufficiency, see Theorems 4.1 and 4.3. Let Nq (K, J) be both an NQ-BCI-commutative ideal and an NQ-BCI-positive implicative ideal over (X, K, J). Then K and J are both a BCI-commutative ideal and a BCI-positive implicative ideal of X. Assume that ((x · y) · y) · (0 · y) ∈ K (resp., J) for all x, y ∈ X. Then x · y ∈ K (resp., J) by Lemma 2.3, and so x · ((y · (y · x)) · (0 · (0 · (x · y)))) ∈ K(resp., J) by Lemma 2.2. It follows from Lemma 2.1 that K and J are BCI-implicative ideals of X. Therefore the neutrosophic quadruple set based on K and J is an NQ-implicative ideal over (X, K, J) by Theorem 3.6.



Corollary 4.6. For any nonempty subset K of X, the neutrosophic quadruple set based on K is both an NQ-BCIcommutative ideal and an NQ-BCI-positive implicative ideal over (X, K) if and only if it is an NQ-BCI-implicative ideal over (X, K). 5. Conclusions Smarandache introduced the notion of neutrosophic quadruple numbers by considering an entry (i.e., a number, an idea, an object, etc.) which is represented by a known part (a) and an unknown part (bT, cI, dF ) where a, b, c and d are real or complex numbers and T , I, F have their usual neutrosophic logic meanings. Jun et al. made up neutrosophic quadruple BCK/BCI-algebras and (positive) implicative neutrosophic quadruple BCK-algebras using neutrosophic quadruple numbers based on BCK/BCI-algebras (instead of real or complex numbers). In this article, we have studied BCI-implicative ideal in BCI-algebra using neutrosophic quadruple structure. We have introduced neutrosophic quadruple BCI-implicative ideal based on nonempty subsets in BCIalgebra, and have investigated their related properties. We have consulted relationship between neutrosophic quadruple ideal, neutrosophic quadruple BCI-implicative ideal, neutrosophic quadruple BCI-positive implicative ideal and neutrosophic quadruple BCI-commutative ideal. We have provided conditions for the neutrosophic 756

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Young Bae Jun, Seok-Zun Song and G. Muhiuddin quadruple set to be neutrosophic quadruple BCI-implicative ideal. We have discussed a characterization of an NQ-BCI-implicative ideal, and have established the extension property of neutrosophic quadruple BCI-implicative ideal. Based on the contents and ideas of this manuscript, we will study neutrosophic quadruple structure for various algebraic sub-structures in the future. Acknowledgement

The second author, Seok-Zun Song, was supported by Basic Science Research Pro-

gram through the National Research Foundation of Korea (NRF) funded by the Ministry of Education (No. 2016R1D1A1B02006812). The last author, G. Muhiuddin, is partially supported by the research grant S-00641439, Deanship of Scientific Research, University of Tabuk, Tabuk-71491, Saudi Arabia.

References [1] A.A.A. Agboola, B. Davvaz and F. Smarandache, Neutrosophic quadruple algebraic hyperstructures, Ann Fuzzy Math. Inform. 14 (2017), no. 1, 29–42. [2] S.A. Akinleye, F. Smarandache and A.A.A. Agboola, On neutrosophic quadruple algebraic structures, Neutrosophic Sets and Systems 12 (2016), 122–126. [3] Y. Huang, BCI-algebra, Science Press, Beijing, 2006. [4] K. Is´eki, On BCI-algebras, Math. Seminar Notes 8 (1980), 125–130. [5] K. Is´eki and S. Tanaka, An introduction to the theory of BCK-algebras, Math. Japon. 23 (1978), 1–26. [6] Y.B. Jun, S.Z. Song and S.J. Kim, Neutrosophic quadruple BCI-positive implicative ideals, Mathematics 2019, 7, 385; doi:10.3390/math7050385 [7] Y.B. Jun, S.Z. Song, F. Smarandache and H. Bordbar, Neutrosophic quadruple BCK/BCI-algebras, Axioms 2018, 7, 41; doi:10.3390/axioms7020041 [8] Y.L. Liu, Y. Xu, and J. Meng, BCI-implicative ideals of BCI-algebras, Inform. Sci. 177 (2007), 4987–4996. [9] Y.L. Liu and X.H. Zhang, Characterization of weakly positive implicative BCI-algebras, J. Hanzhong Teachers College (Natural) (1) (1994), 4–8. [10] J. Meng, An ideal characterization of commutative BCI-algebras, Pusan Kyongnam Math. J. 9 (1993), no. 1, 1–6. [11] J. Meng and Y. B. Jun, BCK-algebras, Kyungmoonsa Co. Seoul, Korea 1994. [12] G. Muhiuddin, A.N. Al-Kenani, E.H. Roh and Y.B. Jun, Implicative neutrosophic quadruple BCK-algebras and ideals, Symmetry 2019, 11, 277; doi:10.3390/sym11020277. [13] G. Muhiuddin, F. Smarandache and Y.B. Jun, Neutrosophic quadruple ideals in neutrosophic quadruple BCI-algebras, Neutrosophic Sets and Systems 25 (2019), 161–173. [14] F. Smarandache, Neutrosophic quadruple numbers, refined neutrosophic quadruple numbers, absorbance law, and the multiplication of neutrosophic quadruple numbers, Neutrosophic Sets and Systems, 10 (2015), 96–98.

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Isolation numbers of matrices over nonbinary Boolean semiring LeRoy B. Beasley1 , Madad Khan2 and Seok-Zun Song3,∗ 1

Department of Mathematics and Statistics, Utah State University, Logan, UT84322-3900, USA Department of Mathematics, COMSATS University Islamabad, Abbottabad Campus, Pakistan

2

3

Department of Mathematics, Jeju National University, Jeju 63243, Korea

Abstract. Let Bk be the nonbinary Boolean semiring and A be a m × n Boolean matrix over Bk . The Boolean rank of a Boolean matrix A is the smallest k such that A can be factored as an m × k Boolean matrix times a k × n Boolean matrix. The isolation number of A is the maximum number of nonzero entries in A such that no two are in any row or any column, and no two are in a 2 × 2 submatrix of all nonzero entries. We have that the isolation number of A is a lower bound on the Boolean rank of A. We also compare the isolation number with the binary Boolean rank of the support of A, and determine the equal cases of them.

1. Introduction There are many papers on the study of rank of matrices over several semirings containing binary Boolean algebra, fuzzy semiring, semiring of nonegative integers, and so on ([2], [3], [6], and [7]). But there are few papers on isolation numbers of matrices. Gregory et al ([7]) introduced set of isolated entries and compared binary Boolean rank with biclique covering number. Recently Beasley ([2]) introduced isolation number of Boolean matrix and compare it with binary Boolean rank. In this paper, we investigate the possible isolation number of Boolean matrix and compare it with Boolean rank of Boolean matrix and the binary Boolean rank of the support of the Boolean matrix. 2. Preliminaries Definition 2.1. A semiring S consists of a set S with two binary operations, addition and multiplication, such that: · S is an Abelian monoid under addition (the identity is denoted by 0); · S is a monoid under multiplication (the identity is denoted by 1, 1 6= 0); · multiplication is distributive over addition on both sides; · s0 = 0s = 0 for all s ∈ S. Definition 2.2. A semiring S is called antinegative if the zero element is the only element with an additive inverse.

0

2010 Mathematics Subject Classification: 15A23; 15A03; 15B15. Keywords: Boolean rank; nonbinary Boolean semiring; binary Boolean algebra; isolation number. ∗ The corresponding author. 0 E-mail : [email protected] (L. B. Beasley); [email protected] (S. Z. Song); [email protected] (M. Khan) 0

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LeRoy B. Beasley, Madad Khan and Seok-Zun Song Definition 2.3. A semiring S is called a Boolean semiring if S is equivalent to a set of subsets of a given set X, the sum of two subsets is their union, and the product is their intersection. The zero element 0 is the empty set and the identity element 1 is the whole set X. Let Sk = {a1 , a2 , · · · , ak } be a set of k-elements, P(Sk ) be the set of all subsets of Sk . Then P(Sk ) is the Boolean semiring of all subsets of Sk with operations in above definition. Let Bk be a Boolean semiring of subsets of Sk = {a1 , a2 , · · · , ak }, that is a subset of P(Sk ). It is straightforward to see that a Boolean semiring Bk is a commutative and antinegative semiring. Moreover, all of its elements, except 0 and 1, are zero-divisors. If Bk consists of only 0 (the empty subset) and 1 (the whole set Sk ) then it is called a binary Boolean semiring, which is denoted as B1 . If Bk is not a binary Boolean semiring then it is called a nonbinary Boolean semiring. Throughout the paper, we assume that m ≤ n and Bk denotes a nonbinary Boolean semiring, which contains at least 3 elements. Let Mm,n (Bk ) denote the set of m × n matrices with entries from a Boolean semiring Bk . Let Mn (Bk ) = Mm,n (Bk ) if m = n, let Im denote the m × m identity matrix, Om,n denote the zero matrix in Mm,n (Bk ), Jm,n denote the matrix of all ones in Mm,n (Bk ). The subscripts are usually omitted if the order is obvious, and we write I, O, J. Definition 2.4. The matrix A ∈ Mm,n (Bk ) is said to be of Boolean rank r if there exist matrices B ∈ Mm,r (Bk ) and C ∈ Mr,n (Bk ) such that A = BC and r is the smallest positive integer such that such a factorization exists. We denote b(A) = r. By definition, the unique matrix with Boolean rank equal to 0 is the zero matrix O. Now let Mm,n (B1 ) denote the set of all m × n binary Boolean matrices with entries in B1 . The binary Boolean rank of A ∈ Mm,n (B1 ) is the Boolean rank over B1 and denoted b1 (A). Definition 2.5. For two (binary) Boolean matrices A and B, A dominates B if ai,j = 0 implies bi,j = 0. Given a matrix X ∈ Mm,n (Bk ), we let x(j) denote the j th column of X and x(i) denote the ith row. Now if b(A) = r and A = BC is a factorization of A ∈ Mm,n (Bk ), then A = b(1) c(1) + b(2) c(2) + · · · + b(r) c(r) . Since each of the terms b(i) c(i) is a Boolean rank one matrix, the Boolean rank of A is also the minimum number of Boolean rank one matrices whose sum is A. The binary Boolean rank has many applications in combinatorics, especially graph theory, for example, if A ∈ Mm,n (B1 ) is the adjacency matrix of the bipartite graph G with bipartition (X, Y ), the binary Boolean rank of A is the minimum number of bicliques that cover the edges of G, called the biclique covering number. Definition 2.6. Given a matrix A ∈ Mm,n (Bk ), a set of isolated entries ([7]) is a set of entries, usually written as E = {ai,j } such that ai,j 6= 0, no two entries in E are in the same row, no two entries in E are in the same column, and, if ai,j , ak,l ∈ E then, ai,l = 0 or ak,j = 0. That is, isolated entries are independent entries and any   ai,j ai,l two isolated entries ai,j and ak,l do not lie in a submatrix of A of the form with all entries nonzero. ak,j ak,l The isolation number of A, ι(A), is the maximum size of a set of isolated entries. Note that ι(A) = 0 if and only if A = O. 759

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Isolation numbers of matrices over nonbinary Boolean semiring Example 2.7. Let σ ∈ Bk be neither 0 nor 1 and     A=  

1 σ 1 0 0

1 1 0 σ 0

σ 0 0 0 1

0 1 0 1 σ

0 0 σ 1 1

      

be a Boolean matrix over Bk and E1 is the set of σ 0 s which are located at the positions {a1,3 , a2,1 , a3,5 , a4,2 , a5,4 } of A. The entries σ 0 s of A are isolated entries and hence ι(A) = 5. But the entries of A in the position in E2 = {a1,1 , a2,2 , a3,5 , a4,4 , a5,3 } are not isolated. Suppose that A ∈ Mm,n (Bk ) and b(A) = r. Then there are r Boolean rank one matrices Ai such that A = A1 + A2 + · · · + Ar .

(2.1) 



N O with all nonzero O O entries in N , it is easily seen that the matrix consisting of two isolated entries of A cannot be dominated by any Because each Boolean rank one matrix can be permuted to a matrix of the form

one Ai among the Boolean rank one summand of A in (2.1). Thus i(A) ≤ b(A).

(2.2)

Many functions, sets and relations concerning matrices do not depend upon the magnitude or nature of the individual entries of a matrix, but rather only on whether the entry is zero or nonzero. These combinatorially significant matrices have become increasingly important in recent years. Of primary interest is the binary Boolean rank. Finding the binary Boolean rank of a (0, 1)-matrix is an NP-Complete problem ([8]), and consequently finding bounds on the binary Boolean rank of a matrix is of interest to those researchers that would use binary Boolean rank in their work. If the (0, 1)-matrix is the reduced adjacency matrix of a bipartite graph, the isolation number of the matrix is the maximum size of a non-competitive matching in the bipartite graph. This is related to the study of such combinatorial problems as the patient hospital problem, the stable marriage problem, etc. An additional reason for studying the isolation number is that it is a lower bound on the Boolean rank of a Boolean matrix over Bk . While finding the isolation number as well as finding the Boolean rank of a Boolean matrix is an NP-Complete problem ([1]), for some matrices finding the isolation number can be easier than finding the Boolean rank especially if the matrix is sparse: Example 2.8. Let σ ∈ Bk and         F =      

1 1 1 1 1 1 σ 0 1

1 1 1 1 1 1 1 σ 0

1 1 1 1 1 1 0 1 σ

σ 1 0 0 0 0 0 0 0

0 σ 0 0 0 0 0 0 0

1 1 σ 0 0 0 0 0 0

1 1 1 0 0 0 0 0 0

1 1 1 0 0 0 0 0 0

1 1 1 0 0 0 0 0 0

              

be a Boolean matrix in M9 (Bk ). 760

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LeRoy B. Beasley, Madad Khan and Seok-Zun Song Then we can easily see b(F ) 5 6 from first 3 rows and columns, however to find that Boolean rank is not 5, requires much calculation if the isolation number is not considered. However, the isolation number is easily seen to be 6, both computationally and visually, the σ’s in this matrix represent a set of isolated entries. Thus we have b(F ) = 6 by (2.2). Note that if any of the 1’s in F are replaced by zeros, the resulting matrix still has Boolean rank 6 as well as isolation number 6. Terms not specifically defined here can be found in Brualdi and Ryser [5] for matrix terms, or Bondy and Murty [4] for graph theoretic terms. For our use in the next section, we define the support matrix of a Boolean matrix. If A ∈ Mm,n (Bk ), then the support of A is the binary Boolean matrix A = (ai,j ) ∈ Mm,n (B1 ) such that ai,j = 1 if ai,j 6= 0 and ai,j = 0 if ai,j = 0. 3. Comparisons between isolation numbers and Boolean ranks over Mm,n (Bk ) In this section, we compare the isolation number with Boolean rank of a Boolean matrix, and also we compare the isolation number with binary Boolean rank of the support of a Boolean matrix. Lemma 3.1. For A, B ∈ Mm,n (Bk ), b(A + B) ≤ b(A) + b(B). And for A, B ∈ Mm,n (B1 ), b1 (A + B) ≤ b1 (A) + b1 (B). Proof. It follows from the definition of Boolean rank and equation (2.1). Lemma 3.2. For A, B ∈ Mm,n (Bk ), A + B = A + B in Mm,n (B1 ). Proof. It follows from the facts that Bk is an antinegative semiring and 1 + 1 = 1 in B1 .

Lemma 3.3. For A ∈ Mm,n (Bk ), b1 (A) ≤ b(A). Proof. If b(A) = r, then A has a Boolean rank one factorization such that A = b(1) c(1) + b(2) c(2) + · · · + b(r) c(r) with B = [b(1) b(2) · · · b(r) ] ∈ Mm,k (Bk ) and C = [c(1) c(2) · · · c(k) ]t ∈ Mk,n (Bk ) from (2.1). Therefore b1 (A) = b1 (b(1) c(1) + b(2) c(2) + · · · + b(r) c(r) ) = b1 (b(1) c(1) + b(2) c(2) + · · · + b(r) c(r) ) ≤ r, from Lemma 3.2. Hence b1 (A) ≤ b(A). We may have strict inequality in Lemma 3.3 as we see in the following example.   1 {x} Example 3.4. Let S3 = {x, y, z} and B3 = {0, {x}, {x, y}, 1} with 1 = {x, y, z}. Consider X = {x, y} {x, y}     1 1 1 {x} and Y = in M2 (B3 ). Then b(X) = 2 but b1 (X) = b1 ( ) = 1. Hence b1 (X) < b(X). But {x, y} {x} 1 1     1 b(Y ) = b1 (Y ) = 1 since Y = 1 {x} over B3 . {x, y}

Lemma 3.5. For A = [ai,j ] ∈ Mm,n (Bk ), ι(A) = ι(A). 761

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Isolation numbers of matrices over nonbinary Boolean semiring Proof. If ai,j and ak,l are any isolated entries in A, then i 6= k and j 6= l, and that ai,l = 0 or ak,j = 0. Hence ai,j and ak,l are isolated entries in A, so we have ι(A) ≤ ι(A). Conversely, if ai,j and ak,l are any isolated entries in A, then ai,j 6= 0 and ak,l 6= 0 and that ai,l = ai,l = 0 or ak,j = ak,j = 0. Hence ai,j and ak,l are isolated entries in A, so we have ι(A) ≤ ι(A).

Theorem 3.6. If A ∈ Mm,n (Bk ), then ι(A) = 1 if and only if b1 (A) = 1. Proof. Let A ∈ Mm,n (Bk ). If b1 (A) = 1 then A 6= O so that ι(A) 6= 0 and since ι(A) = ι(A) ≤ b1 (A) by (2.2), we have ι(A) = 1. Conversely, suppose on the contrary that there exists a matrix A = [ai,j ] ∈ Mm,n (Bk ) such that ι(A) = 1, b1 (A) > 1. Then, there exists two non-equal and nonzero rows of A, say ith and jth. Hence, without loss of generality, there exists a k such that ai,k = 1 and aj,k = 0. Then, ai,k and any unit entry in jth row of B constitute a set of two isolated entries. Thus, ι(A) = ι(A) > 1, a contradiction. It follows that the subset of Mm,n (Bk ) of matrices with isolation number 1 is the same as the set of matrices whose support has Boolean rank 1. For A = A1 + A2 + · · · + Ar with b(A) = r, let Ri denote the indices of the nonzero rows of Ai and Cj denote the indices of the nonzero columns of Aj , i, j = 1. · · · , k. Let ri = |Ri |, the number of nonzero rows of Ai and cj = |Cj |, the number of nonzero columns of Aj . Lemma 3.7. Let A ∈ Mm,n (Bk ). Then if b(A) ≥ b1 (A) = 2 then ι(A) = 2, and if ι(A) = 2 then b1 (A) 6= 3. Proof. If b1 (A) = 2, then ι(A) > 1 by Theorem 3.6. Since ι(A) = ι(A) ≤ b1 (A) from Lemma 3.5 and (2.2), we have that ι(A) = ι(A) = 2. Now, suppose that ι(A) = 2 and that b1 (A) = 3. Then, we have a factorization of A as A = C × D with C ∈ Mm,3 (B1 ) and D ∈ M3,n (B1 ). Then, the three rows of D generate all the rows of A. Since b1 (A) = 3, D cannot have binary Boolean rank 2 or less. Thus, we have b1 (D) = 3. Therefore, we have a factorization of D as D = E × F with E ∈ M3,3 (B1 ) and F ∈ M3,n (B1 ). Then, the three column of E generate all the columns of D and b1 (E) = 3. Therefore, it is sufficient to consider 3 × 3 matrices of binary Boolean rank 3. However, there are only 10 following 3 × 3 matrices of binary Boolean rank     1 0 0 1 0 0 B1 =  0 1 0  , B2 =  0 1 0  , 0 0 1 0 1 1     1 0 0 1 0 0 B5 =  1 1 0  , B6 =  1 1 0  , 0 0 1 0 1 1   1 0 1 B9 =  1 1 0  , 0 0 1

3 up to permutations:     1 0 0 1 0 0 B3 =  0 1 0  , B4 =  0 1 0  , 1 0 1 1 1 1     1 0 0 1 0 0 B7 =  1 1 0  , B8 =  1 1 0  , 1 0 1 1 1 1   1 0 1 B10 =  1 1 0  . 0 1 1

Since B5 can be permuted to B2 and B7 can be permuted to B4 , and B9 can be permuted to B6 with transposing. Therefore, there are only seven non-equivalent 3 × 3 matrices of binary Boolean rank 3. However, these matrices 762

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LeRoy B. Beasley, Madad Khan and Seok-Zun Song have three isolation entries on the main diagonal. Thus, we have a contradiction to the conditions that ι(B) = 2 and rB1 (B) = 3. Thus, if ι(A) = 2 then b1 (A) 6= 3. Theorem 3.8. Let A ∈ Mm,n (Bk ). Then, ι(A) = 2 if and only if b1 (A) = 2. Proof. From Lemma 3.7, we have the sufficiency. So we only need show the necessity. Suppose there exists A ∈ Mm,n (Bk ) with ι(A) = ι(A) = 2 and b1 (A) > 2. By Lemma 3.7, b1 (A) 6= 3, and hence b1 (A) ≥ 4. Thus we choose A such that if b1 (A) > b1 (C) > 2 then ι(C) > 2. Suppose that A = A1 + A2 + · · · + Ar for r = b1 (A) where each Ai is binary Boolean rank 1, i.e., r is the minimum r such that b1 (A) = r and ι(A) = 2. Suppose that A1 has the fewest number of nonzero rows of the Ai ’s. As in the proof of the above lemma 3.7, permute the rows of A so that A1 has nonzero rows 1, 2, · · · , r1 . For j = 1, · · · , r1 , let Bj be the matrix whose first j rows are the first j rows of A and whose last m − j rows are all zero. Let Cj be the matrix whose first j rows are all zero and whose last m − j rows are the last m − j rows of A. Then A = Bj + Cj . Further any set of isolated entries of Cj is a set of isolated entries for A. Now, from b1 (A) ≤ b1 (Bj ) + b1 (Cj ), and the fact that b1 (Cj ) = b1 (Cj−1 ) or b1 (Cj ) = b1 (Cj−1 ) − 1, there is some t such that b1 (Ct ) = b1 (A) − 1. Since b1 (Ct ) < r by the choice of A, for this t, we have that ι(Ct ) > 2 since b1 (Ct ) ≥ 3. That is, ι(A) = ι(A) > 2, which is impossible since ι(A) = 2. Therefore b1 (A) = 2. Now, as we can see in the following example, there is a Boolean matrix A ∈ Mm,n (Bk ) such that ι(A) = 3 and b1 (A) is relative large, depending on m and n. Example 3.9. For n ≥ 3, let Dn = J \ I ∈ Mn (B1 ). Then, it is easily shown that ι(Dn ) = 3 while b1 (Dn ) = r    h where r = min h : n ≤ , see [6](Corollary 2). So, ι(D20 ) = 3 while b1 (D20 ) = 6. h 2

Definition 3.10. A tournament matrix [T ] ∈ Mn (Bk ) is the adjacency matrix of a directed graph called a tournament, T . It is characterized by [T ]◦[T ]t = O and [T ]+[T ]t = J −I, where ◦ denotes entrywise multiplication of two matrices. Now, for each r = 1, 2, · · · , min{m, n}, can we characterize the matrices in Mm,n (Bk ) for which ι(A) = b1 (A) ? Of course it is done if r = 1 or r = 2 in the above theorems, but only in those cases. For r = m we can also find a characterization: Theorem 3.11. Let 1 ≤ m ≤ n and A ∈ Mm,n (Bk ). Then, ι(A) = b1 (A) = m if and only if there exist permutation matrices P ∈ Mm (B1 ) and Q ∈ Mn (B1 ) such that P AQ = [B|C] where B = Im + T ∈ Mm (B1 ) where T ∈ Mm (B1 ) is dominated by a tournament matrix. (There are no restrictions on C.) Proof. Suppose that ι(A) = m. Then we permute A by permutation matrices P and Q so that the set of isolated entries are in the (d, d) positions, d = 1, · · · , m. That is, if X = P AQ then I = {x1,1 , x2,2 , · · · , xm,m } is the set of isolated entries in X. Therefore X = [B|C], with bi,i = xi,i = 1 and bi,j · bj,i = 0 for every i and j 6= i from the definition of the isolated entries. Thus, B = Im + T where T is an m square matrix which is dominated by a tournament matrix. Thus, P AQ = [B|C] where B = Im + T and clearly there are no conditions on C. 763

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Isolation numbers of matrices over nonbinary Boolean semiring Conversely, if P AQ = [B|C] and B = Im + T where T is an m square matrix which is dominated by a tournament matrix, then the diagonal entries of B form a set of isolated entries for P AQ and hence A has a set of m isolated entries. Thus ι(A) = b1 (A) = m. Corollary 3.12. Let 1 ≤ m ≤ n and A ∈ Mm,n (Bk ). If there exist permutation matrices P ∈ Mm (B1 ) and Q ∈ Mn (B1 ) such that P AQ = [B|C] where B ∈ Mm (Bk ) is a diagonal matrix or a triangular matrix with nonzero diagonal entries, then ι(A) = b1 (A) = m.

4. Conclusions In this paper, we investigated the nonbinary Boolean rank of a matrix A and the rank of its support for the given isolation number k over nonbinary Boolean semirings. Thus, we proved that the isolation number of A is the same as the Boolean rank of the support of it if the isolation numbers are 1 and 2. If the isolation number were greater than 2, then we showed by example that binary Boolean rank of the support of the given nonbinary Boolean matrix may be strictly greater than the isolation number of the matrix. In addition, in some special cases involving tournament matrices, we obtained that the isolation number of the given matrix and the Boolean rank of its support of the nonbinary Boolean matrix are the same. Acknowledgement

The third author, Seok-Zun Song, was supported by Basic Science Research Pro-

gram through the National Research Foundation of Korea (NRF) funded by the Ministry of Education (No. 2016R1D1A1B02006812).

References [1] K. Akiko, Complexity of the sex-equal stable marriage problem (English summary), Japan J. Indust. Appl. Math., 10(1993), 1-19. [2] L. B. Beasley, Isolation number versus Boolean rank, Linear Algebra Appl., 436(2012), 3469-3474. [3] L. B. Beasley and N. J. Pullman, Nonnegative rank-preserving operators, Linear Algebra Appl., 65(1985), 207-223. [4] J. A. Bondy and U. S. R. Murty, Graph Theory, Graduate texts in Mathematics 244, Springer, New York, 2008. [5] R. Brualdi and H. Ryser, Combinatorial Matrix Theory, Cambridge University Press, New York, 1991. [6] D. de Caen, D.A. Gregory,and N. J. Pullman, The Boolean rank of zero-one matrices, Proceedings of the Third Caribbean Conference on Combinatorics and Computing (Bridgetown), 169-173, Univ. West Indies, Cave Hill Campus, Barbados, 1981 [7] D. Gregory, N. J. Pullman, K. F. Jones and J. R. Lundgren, Biclique coverings of regular bigraphs and minimum semiring ranks of regular matrices. J. Combin. Theory Ser. B, 51(1991), 73-89. [8] G. Markowsky, Ordering D-classes and computing the Schein rank is hard, Semigroup Forum, 44(1992), 373-375.

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ORTHOGONALLY EULER-LAGRANGE TYPE CUBIC FUNCTIONAL EQUATIONS IN ORTHOGONALITY NORMED SPACES CHANG IL KIM AND GILJUN HAN∗

Abstract. In this paper, we investigate the orthogonally Euler-Lagrange type cubic functional equation f (ax + by) + f (ax − by) − ab2 [f (x + y) + f (x − y)] − 2a(a2 − b2 )f (x) + c[f (x + 2y) − 3f (x + y) + 3f (x) − f (x − y) − 6f (y)] = 0, x⊥y for fixed non-zero rational numbers a, b and a fixed non-zero real number c with a2 6= b2 and a 6= ±1 and prove the generalized Hyers-Ulam stability for it by using the fixed point method,

1. Introduction Assume that X is a real inner product space and f : X −→ R is a solution of the orthogonally Cauchy functional equation f (x + y) = f (x) + f (y), < x, y >= 0. By the Pythagorean theorem, f (x) = kxk2 is a solution of the conditional equation. Of course, this function does not satisfy the additivity equation everywhere. Thus, orthogonal Cauchy equation is not equivalent to the classic Cauchy equation on the whole inner product space. The orthogonally Cauchy functional equation f (x + y) = f (x) + f (y), x⊥y in which ⊥ is an abstract orthogonality relation, was first investigated by Gudder and Strawther [5]. R¨ atz [16] introduced a new definition of orthogonality by using more restrictive axioms than of Gudder and Strawther. Moreover, he investigated the structure of orthogonally additive mappings. R¨ atz and Szab´o [17] investigated the problem in a rather more general framework. Definition 1.1. [17] Let X be a real vector space with dim X ≥ 2 and ⊥ a binary relation on X with the following properties: (O1) totality for zero: x⊥0 and 0⊥x for all x ∈ X; (O2) independence: if x, y ∈ X − {0}, x⊥y, then x, y are linearly independent; (O3) homogeneity: if x, y ∈ X, x⊥y, then αx⊥βy for all α, β ∈ R; (O4) the Thalesian property: if P is a 2-dimensional subspace of X, x ∈ P and a non-negative real number k, then there exists an y ∈ P such that x⊥y and x + y⊥kx − y. The pair (X, ⊥) is called an orthogonality space. By an orthogonality normed space, we mean an orthogonality space having a normed structure. 2010 Mathematics Subject Classification. 39B55, 47H10, 39B52, 46H25. Key words and phrases. Hyers-Ulam stability, fixed point theorem, orthogonally cubic functional equation, orthogonality space. * Corresponding author. 1

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Remark 1.2. (i) The trivial orthogonality on a vector space X defined by (O1) and for non-zero elements x, y ∈ X, x⊥y if and only if x, y are linearly independent. (ii) The ordinary orthogonality on an inner product space (X, < ·, · >) given by x⊥y if and only if < x, y >= 0. (iii) The Birkhoff-James orthogonality on a normed space (X, k · k) defined by x⊥y if and only if kx + kyk ≥ kxk for all k ∈ R. The relation ⊥ is called symmetric if x⊥y implies that y⊥x for all x, y ∈ X. Then clearly examples (i) and (ii) are symmetric but example (iii) is not. However, that a real normed space of dimension greater than 2 is an inner product space if and only if the Birkhoff-James orthogonality is symmetric. In 1940, S. M. Ulam proposed the following stability problem (cf. [19]): “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 the next year, Hyers [6] gave a partial solution of Ulam, s problem for the case of approximate additive mappings. In 1978, Rassias [14] extended the theorem of Hyers by considering the unbounded Cauchy difference. The result of Rassias has provided a lot of influence in the development of what we now call the generalized Hyers-Ulam stability or Hyers-Ulam stability of functional equations. Ger and Sikorska [4] investigated the orthogonal stability of the Cauchy functional equation (1.1)

f (x + y) = f (x) + f (y), x⊥y

and Vajzovi´c [20] investigated the orthogonally additive-quadratic equation (1.2)

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

when X is a Hilbert space, Y is a scalar field, f is continuous and ⊥ means the Hilbert space orthogonality. Later, many mathematicians have investigated the orthogonal stability of functional equations ([3], [9], [10], [11], [12], [13], and [18]). In 2001, Rassias [15] introduced the following cubic functional equation (1.3)

f (x + 2y) − 3f (x + y) + 3f (x) − f (x − y) − 6f (y) = 0

and every solution of the cubic functional equation is called a cubic mapping and Jun, Kim, and Chang [8] introduced the Euler-Lagrange cubic functional equation. In this paper, we consider the following orthogonally Euler-Lagrange type cubic functional equation (1.4)

f (ax + by) + f (ax − by) − ab2 [f (x + y) + f (x − y)] − 2a(a2 − b2 )f (x) + c[f (x + 2y) − 3f (x + y) + 3f (x) − f (x − y) − 6f (y)] = 0, x⊥y.

for fixed non-zero rational numbers a, b and a fixed non-zero real numbers c with a2 6= b2 and a 6= ±1 and prove the generalized Hyers-Ulam stability for it. Every solution of (1.4) is called an orthogonally Euler-Lagrange type cubic mapping. Throughtout this paper, (X, ⊥) is an orthogonality normed space with the norm k · kX and (Y, k · k) is a Banach space.

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ORTHOGONALLY EULER-LAGRANGE TYPE CUBIC FUNCTIONAL EQUATIONS...

3

2. Solutions of (1.4) In this section, we investigate solutiuons of (1.4). We will show that a mapping f satisfying (1.4) is an orthogonally cubic mapping. Theorem 2.1. Let f : X −→ Y be a mapping with f (0) = 0. If f satisfies (1.4) and c 6= 0, then f is an orthogonally cubic mapping. Proof. Suppose that f satisfies (1.4). Setting y = 0 in (1.4), we have f (ax) = a3 f (x)

(2.1)

for all x ∈ X and setting x = 0 and y = x in (1.4), we have (2.2)

f (bx) + f (−bx) = (ab2 + 9c)f (x) + (ab2 + c)f (−x) − cf (2x)

for all x ∈ X. Replacing x by −x in (2.2), we have (2.3)

f (bx) + f (−bx) = (ab2 + 9c)f (−x) + (ab2 + c)f (x) − cf (−2x)

for all x ∈ X. Since c 6= 0, by (2.2) and (2.3), we have f (2x) − f (−2x) = 8[f (x) − f (−x)]

(2.4)

for all x ∈ X. Relpacing y by ay in (1.4), by (2.2), we have (2.5)

a3 [f (x + by) + f (x − by)] − (ab2 + 3c)f (x + ay) − (ab2 + c)f (x − ay) + cf (x + 2ay) − (2a3 − 2ab2 − 3c)f (x) − 6cf (ay) = 0

for all x, y ∈ X with x⊥y and letting y = (2.6)

y b

in (2.5), we have

a3 [f (x + y) + f (x − y)] − (ab2 + 3c)f (x + py) − (ab2 + c)f (x − py) + cf (x + 2py) − (2a3 − 2ab2 − 3c)f (x) − 6cf (py) = 0

for all x, y ∈ X with x⊥y, where p = ab . Letting y = −y in (2.6), we have (2.7)

a3 [f (x − y) + f (x + y)] − (ab2 + 3c)f (x − py) − (ab2 + c)f (x + py) + cf (x − 2py) − (2a3 − 2ab2 − 3c)f (x) − 6cf (−py) = 0

for all x, y ∈ X with x⊥y. By (2.6) and (2.7), we have (2.8)

c[f (x + 2py) − f (x − 2py)] − 2c[f (x + py) − f (x − py)] − 6c[f (py) − f (−py)] = 0

for all x, y ∈ X with x⊥y. Letting y = p1 y in (2.8), we have (2.9)

[f (x + 2y) − f (x − 2y)] − 2[f (x + y) − f (x − y)] − 6[f (y) − f (−y)] = 0

for all x, y ∈ X with x⊥y. Let fo (x) = (2.10)

f (x)−f (−x) . 2

Then fo satisfies (2.9). Letting x = 0 in (2.9), we have fo (2y) = 8fo (y)

for all y ∈ X. Letting x = 2x in (2.9), by (2.10), we have (2.11)

4[fo (x + y) − fo (x − y)] = fo (2x + y) − fo (2x − y) + 6fo (y)

for all x, y ∈ X with x⊥y. Interchanging x and y in (2.11), we have (2.12)

4[fo (x + y) + fo (x − y)] = fo (x + 2y) + fo (x − 2y) + 6fo (x)

for all x, y ∈ X with x⊥y. By (2.9) and (2.12), we have fo (x + 2y) − 3fo (x + y) + 3fo (x) − fo (x − y) − 6fo (y) = 0

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for all x, y ∈ X with x⊥y and hence f0 is an orthogonally cubic mapping. Let fe (x) = (2.13)

f (x)+f (−x) . 2

Then fe satisfies (2.9) and so we have

fe (x + 2y) − fe (x − 2y) − 2[fe (x + y) − fe (x − y)] = 0

for all x, y ∈ X with x⊥y. Letting y = x in (2.13), we have fe (3x) = 2fe (2x) + fe (x) for all x ∈ X and letting y = 2x in (2.13), we have fe (4x) = 2fe (3x) − 2fe (x) for all x ∈ X. Hence we have fe (4x) = 4fe (2x) for all x ∈ X and so fe (2x) = 4fe (x), fe (3x) = 9fe (x), fe (4x) = 16fe (x) for all x ∈ X. By induction on n, we have fe (nx) = n2 fe (x) for all x ∈ X and all n ∈ N and hence fe (rx) = r2 fe (x) for all x ∈ X and all rational number r. By (2.1), since a is a non-zero rational number with a 6= 1, f (x) = 0 for all x ∈ X. Hence f = fo + fe = fo is an orthogonally cubic mapping.  3. The Generalized Hyers-Ulam stability for (1.4) In this section, we prove the generalized Hyers-Ulam stability for the orthogonally cubic functional equation (1.4) by using the fixed point method. In 1996, Isac and Rassias [7] were the first to provide applications of stability theory of functional equations for the proof of new fixed point theorems with applications. Theorem 3.1. [1], [2] Let (X, d) be a complete generalized metric space and let J : X −→ X be a strictly contractive mapping with some Lipschitz constant L with 0 < L < 1. Then for each given element x ∈ X, either d(J n x, J n+1 x) = ∞ for all nonnegative integer 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) < ∞} and 1 d(y, Jy) for all y ∈ Y . (4) d(y, y ∗ ) ≤ 1−L For any mapping f : X −→ Y , we define the difference operator Df : X 2 −→ Y by Df (x, y) = f (ax + by) + f (ax − by) − ab2 [f (x + y) + f (x − y)] − 2a(a2 − b2 )f (x) + c[f (x + 2y) − 3f (x + y) + 3f (x) − f (x − y) − 6f (y)] for all x, y ∈ X.

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Theorem 3.2. Assume that φ : X 2 −→ [0, ∞) is a function such that φ(x, y) ≤

(3.1)

L φ(ax, ay) |a|3

for all x, y ∈ X and some real number L with 0 < L < 1. Let f : X −→ Y be a mapping such that f (0) = 0 and kDf (x, y)k ≤ φ(x, y)

(3.2)

for all x, y ∈ X with x⊥y. Then there exists a unique orthogonally cubic mapping F : X −→ Y such that kF (x) − f (x)k ≤

(3.3)

L 2|a|3 (1

− L)

φ(x, 0)

for all x ∈ X. Proof. Consider the set S = {g | g : X −→ Y } and define the generalized metric d on S by d(g, h) = inf{c ∈ [0, ∞) | kg(x) − h(x)k ≤ c φ(x, 0), ∀x ∈ X}. Then (S, d) is a complete metric space([9]). Define a mapping T : S −→ S by T g(x) = a3 g( xa ) for all x ∈ X and all g ∈ S. Let g, h ∈ S and d(g, h) ≤ c for some c ∈ [0, ∞). Then by (3.1), we have

x  x 

kT g(x) − T h(x)k = |a|3 g −h

≤ cLφ(x, 0) a a for all x ∈ X. Hence we have d(T g, T h) ≤ Ld(g, h) for all g, h ∈ S and so T is a strictly contractive mapping. Putting y = 0 in (3.2), we get k2f (ax) − 2a3 f (x)k ≤ φ(x, 0) for all x ∈ X and hence

 x  L

φ(x, 0)

f (x) − a3 f

≤ a 2|a|3 L for all x ∈ X and hence d(f, T f ) ≤ 2|a| 3 < ∞. By Theorem 3.1, there exists a mapping F : X −→ Y which is a fixed point of T such that d(T n f, F ) → 0 as n → ∞ and L kF (x) − f (x)k ≤ φ(x, 0) 2|a|3 (1 − L)

for all x ∈ X. Replacing x, y by axn , ayn in (3.2), respectively, and multiplying (3.2) by |a|3n , by (O3), we have

 x y 

3n

a Df n , n ≤ Ln φ(x, y) a a for all x, y ∈ X with x⊥y and all n ∈ N. Letting n → ∞ in the last inequality, we get DF (x, y) = 0 for all x, y ∈ X with x⊥y and by Theorem 2.1, F is an orthogonally cubic mapping.

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Now, we will show the uniqueness of F . Let G : X −→ Y be another orthogonally cubic mapping with (3.3). Since F and G are fixed points of T , by (3.3), we get kG(x) − F (x)k = kT n G(x) − T n F (x)k ≤ kT n G(x) − T n f (x)k + kT n F (x) − T n f (x)k ≤

Ln+1 φ(x, 0) |a|3 (1 − L)

for all x ∈ V and for all n ∈ N. Since 0 < L < 1, letting n → ∞ in the above inequality, we have F = G.  Related with Theorem 3.2, we can also have the following theorem. And the proof is similar to that of Theorem 3.2. Theorem 3.3. Assume that φ : X 2 −→ [0, ∞) is a function such that φ(ax, ay) ≤ |a|3 Lφ(x, y)

(3.4)

for all x, y ∈ X and some real number L with 0 < L < 1. Let f : X −→ Y be a mapping such that satisfying (3.2). Then there exists a unique orthogonally cubic mapping F : X −→ Y such that 1 φ(x, 0) kF (x) − f (x)k ≤ (3.5) 2|a|3 (1 − L) for all x ∈ X. Proof. Consider the set S = {g | g : X −→ Y } and define the generalized metric d on S by d(g, h) = inf{c ∈ [0, ∞) | kg(x) − h(x)k ≤ c φ(x, 0), ∀x ∈ X}. Then (S, d) is a complete metric space([9]). Define a mapping T : S −→ S by T g(x) = a13 g(ax) for all x ∈ X and all g ∈ S. Let g, h ∈ S and d(g, h) ≤ c for some c ∈ [0, ∞). Then by (3.4), we have kT g(x) − T h(x)k =

1 kg(ax) − h(ax)k ≤ cLφ(x, 0) |a|3

for all x ∈ X. Hence we have d(T g, T h) ≤ Ld(g, h) for all g, h ∈ S and so T is a strictly contractive mapping. Putting y = 0 in (3.2), we get k2f (ax) − 2a3 f (x)k ≤ φ(x, 0) for all x ∈ X and hence

1 1

φ(x, 0)

f (x) − 3 f (ax) ≤ a 2|a|3 1 for all x ∈ X and hence d(f, T f ) ≤ 2|a| 3 < ∞. By Theorem 3.1, there exists a mapping F : X −→ Y which is a fixed point of T such that d(T n f, F ) → 0 as n → ∞ and 1 kF (x) − f (x)k ≤ φ(x, 0) 3 2|a| (1 − L) for all x ∈ X. Replacing x, y by an x, an y in (3.2), respectively, and multiplying (3.2) by |a|−3n , by (O3), we have

−3n

Df (an x, an y) ≤ Ln φ(x, y)

a

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for all x, y ∈ X with x⊥y and all n ∈ N. Letting n → ∞ in the last inequality, we get DF (x, y) = 0 for all x, y ∈ X with x⊥y and by Theorem 2.1, F is an orthogonally cubic mapping. Now, we will show the uniqueness of F . Let G : X −→ Y be another orthogonally cubic mapping with (3.3). Since F and G are fixed points of T , by (3.3), we get kG(x) − F (x)k = kT n G(x) − T n F (x)k ≤ kT n G(x) − T n f (x)k + kT n F (x) − T n f (x)k Ln ≤ 3 φ(x, 0) |a| (1 − L) for all x ∈ V and for all n ∈ N. Since 0 < L < 1, letting n → ∞ in the above inequality, we have F = G.  As an example of φ(x, y) in Theorem 3.2 and Theorem 3.3, we can take φ(x, y) = 2p (kxkpX kxkpX + kxk2p X + kykX ) for some positive real numbers  and p. Then we can formulate the following corollary : Corollary 3.4. Let (X, ⊥) be an orthogonality normed space with the norm k · kX and (Y, k · k) a Banach space. Let f : X −→ Y be a mapping such that (3.6)

2p kDf (x, y)k ≤ (kxkpX kxkpX + kxk2p X + kykX )

for all x, y ∈ X with x⊥y and a fixed positive number p with p 6= exists a unique orthogonally cubic mapping F : X −→ Y such that 1 kxk2p kF (x) − f (x)k ≤ 2p 2 |a| − |a|3

3 2.

Then there

for all x ∈ X. By Theorem 2.1, if c = − 13 ab2 , then we have the following orthogonally EulerLagrange type cubic functional equation : 2 1 f (ax + by) + f (ax − by) − ab2 f (x − y) − ab2 f (x + 2y) 3 3 − a(2a2 − b2 )f (x) + 2ab2 f (y) = 0 for all x, y ∈ X with x⊥y. By Corollary 3.6, we have the following exmaple. Example 3.5. Let (X, ⊥) be an orthogonality normed space with the norm k · kX and (Y, k · k) a Banach space. Let f : X −→ Y be a mapping such that 1 2 kf (ax + by) + f (ax − by) − ab2 f (x − y) − ab2 f (x + 2y) 3 3 p p 2p 2 2 2 − a(2a − b )f (x) + 2ab f (y)k ≤ (kxkX kxkX + kxk2p X + kykX ) for all x, y ∈ X with x⊥y and a fixed positive number p with p 6= 23 . Then there exists a unique orthogonally cubic mapping F : X −→ Y such that 1 kxk2p kF (x) − f (x)k ≤ 2p 2 |a| − |a|3 for all x ∈ X.

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It should be remarked that if a functional inequality can be deformed into the type of (3.2), then a solution of the original functional equation is cubic. In the following theorems, we give a simple example. Theorem 3.6. Let φ : X 2 −→ [0, ∞) be a function such that 1 Lφ(2x, 2y) 8 for all x, y ∈ X, some real number L with 0 < L < 1 and f : X −→ Y a mapping such that f (0) = 0 and φ(x, y) ≤

(3.7)

(3.8)

kf (2x + y) + f (2x − y) − 2f (x + y) − 2f (x − y) − 12f (x)k ≤ φ(x, y)

for all x, y ∈ X with x⊥y. Then there exists a unique orthogonally cubic mapping F : X −→ Y such that L kF (x) − f (x)k ≤ [3φ(x, 0) + 8φ(0, x)] 16(1 − L) for all x ∈ X. Proof. Letting x = 0 in (3.8), we have kf (y) + f (−y)k ≤ φ(0, y)

(3.9)

for all y ∈ X and letting y = 0 in (3.8), we have 1 φ(x, 0) 2 for all y ∈ X. Letting y = 2y in (3.8), by (3.10), we have kf (2x) − 8f (x)k ≤

(3.10)

k8f (x + y) + 8f (x − y) − 2f (x + 2y) − 2f (x − 2y) − 12f (x)k 1 1 ≤ φ(x + y, 0) + φ(x − y, 0) + φ(x, 2y) 2 2 for all x, y ∈ X with x⊥y. Interchang x and y in (3.8), by (3.9), we get (3.11)

(3.12)

kf (x + 2y) − f (x − 2y) − 2f (x + y) + 2f (x − y) − 12f (y)k ≤ φ(y, x) + φ(0, x − 2y) + 2φ(0, x − y)

for all x, y ∈ X with x⊥y. Putting a = 2, b = 1, and c = −4 in Df (x, y), by (3.8), (3.11), and (3.12), we have kDf (x, y)k ≤ ψ(x, y) for all x, y ∈ X, where 1 1 ψ(x, y) = φ(x, y) + 2φ(y, x) + φ(x + y, 0) + φ(x − y, 0) + φ(x, 2y) 2 2 + 2φ(0, x − 2y) + 4φ(0, x − y) Since ψ satisfies (3.1), by Theorem 3.2, we get the result.



Similar to Theorem 3.6, we have the following theorem : Theorem 3.7. Let φ : X 2 −→ [0, ∞) be a function such that (3.13)

φ(2x, 2y) ≤ 8Lφ(2x, 2y)

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for all x, y ∈ X, some real number L with 0 < L < 1 and f : X −→ Y a mapping satisfying f (0) = 0 (3.8). Then there exists a unique orthogonally cubic mapping F : X −→ Y such that 1 kF (x) − f (x)k ≤ [3φ(x, 0) + 8φ(0, x)] 16(1 − L) for all x ∈ X. By Theorem 3.6 and Theorem 3.7, we have the following corollary : Corollary 3.8. Let f : X −→ Y be a mapping such that f (0) = 0 and kf (2x+y)+f (2x−y)−2f (x+y)−2f (x−y)−12f (x)k ≤ kxkp kykp +kxk2p +kyk2p . for all x, y ∈ X and a fixed positive real number p with p 6= 23 . Then there exists a unique orthogonally cubic mapping F : X −→ Y such that 11 kF (x) − f (x)k ≤ kxk2p 2|8 − 22p | for all x ∈ X. References [1] L. Cˇ adariu and V. Radu, Fixed points and the stability of Jensens functional equation, J Inequal Pure Appl. Math. 4(2003), 1-7. [2] J. B. 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. [3] M. Fochi, Functional equations in A-orthogonal vectors, Aequationes Math. 38(1989), 28-40. [4] R. Ger and J. Sikorska, Stability of the orthogonal additivity, Bull. Polish. Acad. Sci. Math. 43(1995), 143-151. [5] S. Gudder and D. Strawther, Orthogonally additive and orthogonally increasing functions on vector spaces, Pacific. J. Math. 58(1975), 427-436. [6] D. H. Hyers, On the stability of the linear functional equation, Proc. Natl. Acad. Sci. 27(1941), 222-224. [7] G. Isac and Th. M. Rassias, Stability of ψ-additive mappings: applications to nonlinear analysis, Intern. J. Math. Math. Sci. 19(1996), 219-228. [8] K. Jun, H. Kim and I. Chang, On the Hyers-Ulam stability of an Euler-Lagrange type cubic functional equation, J. Comput. Anal. Appl., 7 (2005) 21-33 . [9] D. Mihe and V. Radu, On the stability of the additive Cauchy functional equation in random normed spaces, J. Math. Anal. Appl. 343(2008), 567-572. [10] M. S. Moslehian, On the orthogonal stability of the Pexiderized quadratic equation, J. Differ. Equat. Appl. 11(2005), 999-1004. [11] M. S. Moslehian, On the stability of the orthogonal Pexiderized Cauchy equation, J. Math. Anal. Appl. 318(2006), 211-223. [12] M. S. Moslehian and Th. M. Rassias, Orthogonal stability of additive type equations, Aequationes Math. 73(2007), 249-259. [13] C. Park, Orthogonal Stability of an Additive-Quadratic Functional Equation, Fixed Point Theory and Applications 2011(2011), 1-11. [14] Th. M. Rassias, On the stability of the linear mapping in Banach sapces, Proc. Amer. Math. Sco. 72(1978), 297-300. [15] J. M. Rassias, Solution of the Ulam stability problem for cubic mappings, Glasnik Matematiˇ cki, 36(2001), 63-72. [16] J. R¨ atz, On orthogonally additive mappings, Aequationes Math. 28(1985), 35-49 . [17] J. R¨ atz and G. Y. Szab´ o, On orthogonally additive mappings IV, Aequationes Math. 38(1989), 73-85. [18] G. Y. Szab´ o, Sesquilinear-orthogonally quadratic mappings, Aequationes Math. 40(1990), 190-200. [19] S. M. Ulam, Problems in Modern Mathematics, Wiley, New York, 1960, Chapter VI.

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[20] F. Vajzovi´ c, ber das Funktional H mit der Eigenschaft: (x, y) = 0 ⇒ H(x + y) + H(x − y) = 2H(x) + 2H(y), Glasnik Mat. Ser III. 2(1967), 73-81. Department of Mathematics Education, Dankook University, 152, Jukjeon-ro, Sujigu, Yongin-si, Gyeonggi-do, 16890, Korea E-mail address: [email protected] Department of Mathematics Education, Dankook University, 152, Jukjeon-ro, Sujigu, Yongin-si, Gyeonggi-do, 16890, Korea E-mail address: [email protected]

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CERTAIN SUBCLASS OF HARMONIC MULTIVALENT FUNCTIONS DEFINED BY DERIVATIVE OPERATOR ˘ ADRIANA CATAS ¸ 1∗ , ROXANA S ¸ ENDRUT ¸ IU2 AND LOREDANA-FLORENTINA IAMBOR3

Abstract. In the present paper, we investigate new properties of a new subclass of multivalent harmonic functions in the open unit disc U = {z ∈ C : |z| < 1}, under certain conditions involving a new generalized differential operator. Furthermore, a representation theorem, an integral property and convolution conditions for the subclass denoted by f H (p, m, δ, α, λ, l) are also obtained. Finally, we will give an application AL of neighborhood.

Keywords: differential operator, harmonic function, extreme points, convolution, neighborhood. 2000 Mathematical Subject Classification: 30C45.

1. Introduction A continuous complex-valued function f = u + iv defined in a simply connected complex domain D is said to be harmonic in D if both u and v are real harmonic in D. In any simple connected domain we can write f = h + g¯, where h and g are analytic in D. A necessary and sufficient condition for f to be univalent and sense preserving in D is that |h0 (z)| > |g 0 (z)|, z ∈ D. (See also Clunie and Sheil-Small [5] for more details.) Denote by SH (p, n), (p, n ∈ N = {1, 2, . . .}) the class of functions f = h + g¯ that are harmonic multivalent and sense-preserving in the unit disc U = {z ∈ C : |z| < 1}. Then for f = h + g¯ ∈ SH (p, n) we may express the analytic functions h and g as

(1.1)

h(z) = z p +

∞ X

ak z k , g(z) =

k=p+n

∞ X

bk z k , |bp+n−1 | < 1.

k=p+n−1 1

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A. C˘ ata¸s, R. S ¸ endrut¸iu, L.F. Iambor

Let S˜H (p, n, m), (p, n ∈ N, m ∈ N0 ∪ {0}) denote the family of functions fm = h + g¯m that are harmonic in D with the normalization (1.2) ∞ ∞ X X p k m h(z) = z − |ak |z , gm (z) = (−1) |bk |z k , |bp+n−1 | < 1. k=p+n

k=p+n−1

Definition 1.1. [4] Let H(U ) denote the class of analytic functions in the open unit disc U = {z ∈ C : |z| < 1} and let A(p) be the subclass of the functions belonging to H(U ) of the form ∞ X

p

h(z) = z +

ak z k .

k=p+n

For m ∈ N0 , λ ≥ 0, δ ∈ N0 , l ≥ 0 we define the generalized differential operator m (p, l) on A(p) by the following infinite series Iλ,δ (1.3)

m (p, l)h(z) = (p + l)m z p + Iλ,δ

∞ X

[p + λ(k − p) + l]m C(δ, k)ak z k ,

k=p+n

where  (1.4)

C(δ, k) =

k+δ−1 δ

 =

Γ(k + δ) . Γ(k)Γ(δ + 1)

Remark 1.2. When λ = 1, p = 1, l = 0, δ = 0 we get S˘al˘agean differential operator [13]; p = 1, m = 0 gives Ruscheweyh operator [12]; p = 1, l = 0, δ = 0 implies Al-Oboudi differential operator of order m (see [1]); λ = 1, p = 1, l = 0 operator (1.3) reduces to Al-Shaqsi and Darus differential operator [2] and when p = 1, l = 0 we reobtain the operator introduced by Darus and Ibrahim in [6]. Definition 1.3. [4] Let f ∈ SH (p, n), p ∈ N. Using the operator (1.3) for f = h + g¯ given by (1.1) we define the differential operator of f as (1.5)

m m m (p, l)g(z) Iλ,δ (p, l)f (z) = Iλ,δ (p, l)h(z) + (−1)m Iλ,δ

where (1.6)

m Iλ,δ (p, l)h(z)

m p

= (p + l) z +

∞ X

[p + λ(k − p) + l]m C(δ, k)ak z k

k=p+n

and (1.7)

m Iλ,δ (p, l)g(z) =

∞ X

[p + λ(k − p) + l]m C(δ, k)bk z k .

k=p+n−1

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Certain subclass of harmonic multivalent functions defined by derivative operator

3

Remark 1.4. When λ = 1, l = 0, δ = 0 the operator (1.5) reduces to the operator introduced earlier in [8] by Jahangiri et al. Definition 1.5. [4] A function f ∈ SH (p, n) is said to be in the class ALH (p, m, δ, α, λ, l) if ) ( m+1 Iλ,δ (p, l)f (z) 1 ≥ α, 0 ≤ α < 1, Re (1.8) m (p, l)f (z) p+l Iλ,δ m f is defined by (1.5), for m ∈ N . where Iλ,δ 0 Finally, we define the subclass

f H (p, m, δ, α, λ, l) ≡ ALH (p, m, δ, α, λ, l) ∩ S˜H (p, n, m). AL

(1.9)

Remark 1.6. The class ALH (p, m, δ, α, λ, l) includes a variety of well-known subclasses of SH (p, n). For example, letting n = 1 we get ALH (1, 1, 0, α, 1, 0) ≡ HK(α) in [7], for n = 1, ALH (1, m − 1, 0, α, 1, 0) ≡ SH (t, u, α) in [14], ALH (p, n + p, 0, α, 1, 0) ≡ SHp (n, α) in [11] and n = 1, ALH (1, m, δ, α, 1, 0) ≡ MH (m, δ, α) in [3]. Theorem 1.7. [4] Let fm = h + g¯m be given by (1.2). Then fm ∈ f H (p, m, δ, α, λ, l) if and only if AL ∞ X [(p + l)(1 − α) + λ(k − p)]dp,k (m, λ, l)C(δ, k) |ak |+ (p + l)m+1 (1 − α)

(1.10)

k=p+n ∞ X

+

k=p+n−1

[(p + l)(1 + α) + λ(k − p)]dp,k (m, λ, l)C(δ, k) |bk | ≤ 1, (p + l)m+1 (1 − α)

where λn ≥ α(p + l), 0 ≤ α < 1, m ∈ N0 , λ ≥ 0 and dp,k (m, λ, l) = [p + λ(k − p) + l]m .

(1.11)

Remark 1.8. The harmonic function ∞ X (p + l)m+1 (1 − α) (1.12) f (z) = z p + xk z k + [(p + l)(1 − α) + λ(k − p)]dp,k (m, λ, l)C(δ, k) k=p+n

∞ X

(p + l)m+1 (1 − α) yk z k , [(p + l)(1 + α) + λ(k − p)]dp,k (m, λ, l)C(δ, k) k=p+n−1 P P∞ where k=p+n |xk | + ∞ k=p+n−1 |yk | = 1, 0 ≤ α < 1, m ∈ N0 , λn ≥ α(p + l), λ ≥ 0 and dp,k (m, λ, l) is given in (1.11), show that the coefficient bound expressed by (1.10) is sharp. +

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A. C˘ ata¸s, R. S ¸ endrut¸iu, L.F. Iambor

2. Convex combination and extreme points f H (p, m, δ, α, λ, l) is closed under In this section, we show that the class AL convex combination of its members. For i = 1, 2, 3, ..., let the functions fmi (z) be p

fmi (z) = z −

(2.1)

∞ X

k

|ak,i |z + (−1)

k=p+n

∞ X

m

|bk,i |¯ zk .

k=p+n−1

f H (p, m, δ, α, λ, l) is closed under convex combiTheorem 2.1. The class AL nation. f H (p, m, δ, α, λ, l), where the functions Proof. For i = 1, 2, 3, ..., let fmi (z) ∈ AL fmi (z) are defined by (2.1). Then by (1.10) we have ∞ X [(p + l)(1 − α) + λ(k − p)]dp,k (m, λ, l)C(δ, k) |ak,i |+ (p + l)m+1 (1 − α)

(2.2)

k=p+n ∞ X

+

k=p+n−1 ∞ X

For

[(p + l)(1 + α) + λ(k − p)]dp,k (m, λ, l)C(δ, k) |bk,i | ≤ 1. (p + l)m+1 (1 − α)

ti = 1, 0 ≤ ti ≤ 1, the convex combination of fmi may be written as

i=1 ∞ X

ti fmi (z) = z p −

i=1

∞ X

∞ X

k=p+n

i=1

! ti |ak,i | z k + (−1)m

∞ X

∞ X

k=p+n−1

i=1

! ti |bk,i | z¯k .

Then by (2.2) one obtains ∞ X [(p + l)(1 − α) + λ(k − p)]dp,k (m, λ, l)C(δ, k) · (p + l)m+1 (1 − α)

k=p+n

∞ X

! ti |ak,i | +

i=1

∞ X

! ∞ X [(p + l)(1 + α) + λ(k − p)]dp,k (m, λ, l)C(δ, k) + · ti |bk,i | = (p + l)m+1 (1 − α) i=1 k=p+n−1  ∞ ∞  X X [(p + l)(1 − α) + λ(k − p)]dp,k (m, λ, l)C(δ, k) ti · |ak,i |+  (p + l)m+1 (1 − α) i=1

+

∞ X k=p+n−1

k=p+n

 ∞  X [(p + l)(1 + α) + λ(k − p)]dp,k (m, λ, l)C(δ, k) |b | ≤ ti = 1, k,i  (p + l)m+1 (1 − α) i=1

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Certain subclass of harmonic multivalent functions defined by derivative operator ∞ X

and therefore

f H (p, m, δ, α, λ, l). ti fmi (z) ∈ AL

5



i=1

Further, we will determine a representation theorem for functions in f ALH (p, m, δ, α, λ, l) from which we also establish the extreme points of closed f H (p, m, δ, α, λ, l) denoted by clcoAL f H (p, m, δ, α, λ, l). convex hulls of AL f H (p, m, δ, α, λ, l) Theorem 2.2. Let fm (z) given by (1.2). Then fm (z) ∈ AL if and only if (2.3)

fm (z) = Xp hp (z) +

∞ X k=p+n

where hp (z) = (2.4)

∞ X

Xk hk (z) +

Yk gmk (z),

k=p+n−1

zp

hk (z) = z p −

(p + l)m+1 (1 − α) zk , [(p + l)(1 − α) + λ(k − p)]dp,k (m, λ, l)C(δ, k) k = p + n, p + n + 1, ...,

and (2.5) gmk (z) = z p + (−1)m

(p + l)m+1 (1 − α) z¯k , [(p + l)(1 + α) + λ(k − p)]dp,k (m, λ, l)C(δ, k)

k = p + n − 1, p + n, ..., P∞ P with Xk ≥ 0, Yk ≥ 0, Xp = 1 − ∞ k=p+n−1 Yk . k=p+n Xk − f H (p, m, δ, α, λ, l) are {hk } and {gm }. In particular, the extreme points of AL k Proof. For the functions fm of the form (2.3), we have fm (z) = Xp hp (z) +

∞ X

Xk hk (z) +

k=p+n ∞ X

= zp −

k=p+n

Yk gmk (z) =

k=p+n−1

(p + l)m+1 (1 − α) Xk z k + [(p + l)(1 − α) + λ(k − p)]dp,k (m, λ, l)C(δ, k)

∞ X

+(−1)m

∞ X

k=p+n−1

(p + l)m+1 (1 − α) Yk z¯k . [(p + l)(1 + α) + λ(k − p)]dp,k (m, λ, l)C(δ, k)

Consequently, ∞ X [(p + l)(1 − α) + λ(k − p)]dp,k (m, λ, l)C(δ, k) ak + (p + l)m+1 (1 − α)

k=p+n

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A. C˘ ata¸s, R. S ¸ endrut¸iu, L.F. Iambor ∞ X

+

k=p+n−1

[(p + l)(1 + α) + λ(k − p)]dp,k (m, λ, l)C(δ, k) bk = (p + l)m+1 (1 − α) ∞ X

=

Xk +

k=p+n

where ak =

∞ X

Yk = 1 − Xp ≤ 1,

k=p+n−1

(p + l)m+1 (1 − α) Xk [(p + l)(1 − α) + λ(k − p)]dp,k (m, λ, l)C(δ, k)

(p + l)m+1 (1 − α) Yk [(p + l)(1 + α) + λ(k − p)]dp,k (m, λ, l)C(δ, k) f H (p, m, δ, α, λ, l). and therefore fm ∈ clcoAL f H (p, m, δ, α, λ, l). Conversely, suppose that fm ∈ clcoAL bk =

Setting (2.6)

Xk =

[(p + l)(1 − α) + λ(k − p)]dp,k (m, λ, l)C(δ, k) |ak |, (p + l)m+1 (1 − α)

k = p + n, p + n + 1, ..., [(p + l)(1 + α) + λ(k − p)]dp,k (m, λ, l)C(δ, k) Yk = |bk | (p + l)m+1 (1 − α) k = p + n − 1, p + n, ..., P P∞ and Xp = 1 − k=p+n Xk − ∞ k=p+n−1 Yk . We note by Theorem 1.7 that 0 ≤ Yk ≤ 1, 0 ≤ Xk ≤ 1, and Xp ≥ 0. We obtain the required representation since fm can be written as ∞ X

fm (z) = z p −

k=p+n p

=z −

∞ X k=p+n

+(−1)m

k=p+n−1

k=p+n−1

(p + l)m+1 (1 − α)Yk z¯k = [(p + l)(1 + α) + λ(k − p)]dp,k (m, λ, l)C(δ, k)

∞ X

(z p − hk (z))Xk +

k=p+n

=

∞ X k=p+n

|bk |¯ zk =

(p + l)m+1 (1 − α)Xk zk + [(p + l)(1 − α) + λ(k − p)]dp,k (m, λ, l)C(δ, k)

∞ X

= zp −

∞ X

|ak |z k + (−1)m

hk (z)Xk +

∞ X

(gmk (z) − z p )Yk =

k=p+n−1 ∞ X

 gmk (z)Yk + z p 1 −

k=p+n−1

∞ X

k=p+n

780

Xk −

∞ X

 Yk  =

k=p+n−1

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Certain subclass of harmonic multivalent functions defined by derivative operator ∞ X

= Xp hp (z) +

∞ X

Xk hk (z) +

k=p+n

7

Yk gmk (z),

k=p+n−1

as required.



3. Integral property and convolution conditions In this section we will examine the closure properties of the class f ALH (p, m, δ, α, λ, l) under the generalized Bernardi-Libera-Livingston integral operator and also convolution properties of the same class. Now, for f = h + g¯ given by (1.1) we define the modified generalized Bernardi-Libera-Livingston integral operator of f as Lc (f (z)) = Lc (h(z)) + Lc (g(z)),

(3.1) where

c+p Lc (h(z)) = zc

Z

c > −p,

z

tc−1 h(t)dt

0

and

Z c + p z c−1 t g(t)dt. zc 0 Putting g = 0 in (3.1), we get the definition of the generalized BernardiLibera-Livingston integral operator on analytic functions, (see [9], [10]). Lc (g(z)) =

f H (p, m, δ, α, λ, l). Then Lc (f ) belongs to the class Theorem 3.1. Let f ∈ AL f H (p, m, δ, α, λ, l). AL Proof. From the representation of Lc (f ), it follows that Z c + p z c−1 Lc (f (z)) = t (h(t) + g¯m (t))dt = zc 0     Z z Z ∞ ∞ X c + p  z c−1  p X k m c−1  = t t − |a |t dt + (−1) t  k zc 0 0 =z −

∞ X

k

|Ak |z + (−1)

m

∞ X

k=p+n

k=p+n−1

Ak =

c+p c+p ak , Bk = bk . c+k c+k

where



 |bk |tk  dt =

k=p+n−1

k=p+n

p



|Bk |¯ zk ,

Further, one obtains ∞ X [(p + l)(1 − α) + λ(k − p)]dp,k (m, λ, l)C(δ, k) c + p · |ak |+ (p + l)m+1 (1 − α) c+k k=p+n

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A. C˘ ata¸s, R. S ¸ endrut¸iu, L.F. Iambor ∞ X

+

k=p+n−1 ∞ X

+

[(p + l)(1 + α) + λ(k − p)]dp,k (m, λ, l)C(δ, k) c + p · |bk | ≤ (p + l)m+1 (1 − α) c+k

k=p+n ∞ X

[(p + l)(1 − α) + λ(k − p)]dp,k (m, λ, l)C(δ, k) |ak |+ (p + l)m+1 (1 − α)

k=p+n−1

[(p + l)(1 + α) + λ(k − p)]dp,k (m, λ, l)C(δ, k) |bk | ≤ 1. (p + l)m+1 (1 − α)

f H (p, m, δ, α, λ, l), by Theorem 1.7 we have Lc (f ) ∈ Since f ∈ AL f H (p, m, δ, α, λ, l). AL  For the harmonic functions ∞ X |ak |z k + (−1)m (3.2) f1 (z) = z p −

∞ X

|bk |¯ z k , |bp+n−1 | < 1,

k=p+n−1

k=p+n

and f2 (z) = z p −

(3.3)

∞ X

|Ak |z k + (−1)m

k=p+n

∞ X

|Bk |¯ z k , |Bp+n−1 | < 1,

k=p+n−1

we define the convolution of f1 and f2 as ∞ X p (f1 ∗ f2 )(z) = f1 (z) ∗ f2 (z) = z − |ak Ak |z k + (−1)m k=p+n

∞ X

|bk Bk |¯ zk .

k=p+n−1

In the following theorem, we examine the convolution properties of the class f ALH (p, m, δ, α, λ, l). f H (p, m, δ, α, λ, l) and f2 ∈ Theorem 3.2. For 0 ≤ β ≤ α < 1 let f1 ∈ AL f H (p, m, δ, β, λ, l). Then f1 ∗f2 ∈ AL f H (p, m, δ, α, λ, l) ⊂ AL f H (p, m, δ, β, λ, l). AL f H (p, m, δ, α, λ, l) and f2 ∈ AL f H (p, m, δ, β, λ, l). Obviously, Proof. Let f1 ∈ AL the coefficients of f1 and f2 must satisfy similar conditions to the inequality (1.10). Therefore, for the coefficients of f1 ∗ f2 we can write ∞ X [(p + l)(1 − β) + λ(k − p)]dp,k (m, λ, l)C(δ, k) |ak Ak |+ (p + l)m+1 (1 − β)

k=p+n

+

∞ X k=p+n−1 ∞ X k=p+n

[(p + l)(1 + β) + λ(k − p)]dp,k (m, λ, l)C(δ, k) |bk Bk | ≤ (p + l)m+1 (1 − β) [(p + l)(1 − β) + λ(k − p)]dp,k (m, λ, l)C(δ, k) |ak |+ (p + l)m+1 (1 − β)

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Certain subclass of harmonic multivalent functions defined by derivative operator ∞ X

+

k=p+n−1 ∞ X

+

k=p+n ∞ X

9

[(p + l)(1 + β) + λ(k − p)]dp,k (m, λ, l)C(δ, k) |bk | ≤ (p + l)m+1 (1 − β)

[(p + l)(1 − α) + λ(k − p)]dp,k (m, λ, l)C(δ, k) |ak |+ (p + l)m+1 (1 − α)

k=p+n−1

[(p + l)(1 + α) + λ(k − p)]dp,k (m, λ, l)C(δ, k) |bk | ≤ 1, (p + l)m+1 (1 − α)+

f H (p, m, δ, α, λ, l). In view of Theorem 1.7, it follows that because f1 ∈ AL f f H (p, m, δ, β, λ, l). f1 ∗ f2 ∈ ALH (p, m, δ, α, λ, l) ⊂ AL  4. An application of neighborhood Let us define a generalized (n, η)-neighborhood of a function f given in (1.2) to be the set n Nn,η (f ) = Fm (z) ∈ S˜H (p, n, m) : ∞ X [(p + l)(1 − α) + λ(k − p)]dp,k (m, λ, l)C(δ, k) |ak − Ak |+ (p + l)m+1 (1 − α)

k=p+n

 

∞ X

[(p + l)(1 + α) + λ(k − p)]dp,k (m, λ, l)C(δ, k) |bk − Bk | ≤ η  (p + l)m+1 (1 − α) k=p+n−1 P∞ P k m zk . where Fm (z) = z p − ∞ k=p+n−1 |Bk |¯ k=p+n |Ak |z + (−1) +

Theorem 4.1. Let fm = h + g¯m be given by (1.2). If the functions fm satisfy the conditions  ∞ X [(p + l)(1 − α) + λ(k − p)]dp,k (m, λ, l)C(δ, k) k· (4.1) |ak |+ (p + l)m+1 (1 − α) k=p+n  [(p + l)(1 + α) + λ(k − p)]dp,k (m, λ, l)C(δ, k) α + |b | ≤ 1 − Up,δ (m, λ, l) k (p + l)m+1 (1 − α) and
p+n−α−1 α 1 − Up,δ (m, λ, l) , (4.2) η≤ p+n−α λn ≥ α(p + l), where α Up,δ (m, λ, l) =

[(p + l)(1 + α) + λ(n − 1)]dp,p+n−1 (m, λ, l)C(δ, p + n − 1) |bp+n−1 | (p + l)m+1 (1 − α)

f H (p, m, δ, α, λ, l). then Nn,η (f ) ⊂ AL

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A. C˘ ata¸s, R. S ¸ endrut¸iu, L.F. Iambor

Proof. Let fm satisfy (4.1) and Fm ∈ Nn,η (f ). We have ∞ X [(p + l)(1 − α) + λ(k − p)]dp,k (m, λ, l)C(δ, k) |Ak |+ (p + l)m+1 (1 − α)

k=p+n ∞ X

[(p + l)(1 + α) + λ(k − p)]dp,k (m, λ, l)C(δ, k) |Bk | ≤ (p + l)m+1 (1 − α) k=p+n−1 ∞  X [(p + l)(1 − α) + λ(k − p)]dp,k (m, λ, l)C(δ, k) ≤η+ |ak |+ (p + l)m+1 (1 − α) k=p+n  [(p + l)(1 + α) + λ(k − p)]dp,k (m, λ, l)C(δ, k) α |b | + Up,δ (m, λ, l) ≤ k (p + l)m+1 (1 − α)  ∞ X [(p + l)(1 − α) + λ(k − p)]dp,k (m, λ, l)C(δ, k) 1 k· η+ |ak |+ p+n−α (p + l)m+1 (1 − α) k=p+n  [(p + l)(1 + α) + λ(k − p)]dp,k (m, λ, l)C(δ, k) α (m, λ, l) ≤ |b | + Up,δ k (p + l)m+1 (1 − α) 1 α α ≤η+ (m, λ, l) ≤ 1. (m, λ, l)) + Up,δ (1 − Up,δ p+n−α   α (m, λ, l) Hence, for η ≤ p+n−α−1 1 − U we deduce that fm ∈ p,δ p+n−α f H (p, m, δ, α, λ, l). AL  +

References [1] F. M. Al-Oboudi, On univalent functions defined by a generalized S˘ al˘ agean operator, Inter. J. of Math. and Mathematical Sci., 27(2004), 1429-1436. [2] K. Al-Shaqsi, M. Darus, An operator defined by convolution involving polylogarithms functions, Journal of Math. and Statistics, 4(1)(2008), 46-50. [3] K. Al-Shaqsi, M. Darus, On Harmonic Functions Defined by Derivative Operator, Journal of Inequalities and Applications, vol. 2008, Article ID 263413, doi: 10.1155/2008/263413. [4] A. C˘ ata¸s, R. S ¸ endrut¸iu On harmonic multivalent functions defined by a new derivative operator, Journal of Computational Analysis and Applications, Volume: 28, Issue: 5, Pages: 775-780, 2020. [5] J. Clunie and T. Sheil-Small, Harmonic Univalent Functions, Ann. Acad. Sci. Fenn, Ser. A I. Math. 9(1984), 3-25. [6] M. Darus, R. W. Ibrahim, On new classes of univalent harmonic functions defined by generalized differential operator, Acta Universitatis Apulensis, 18(2009), 61-69. [7] J. M. Jahangiri, Coefficient bounds and univalence criteria for harmonic functions with negative coefficients, Ann. Univ. Mariae Curie-Sklowdowska Sect. A, 52(1998), 57-66. [8] J. M. Jahangiri, G. Murugusundaramoorthy and K. Vijaya S˘ al˘ agean type harmonic univalent functions South. J. Pure Appl. Math., 2(2002), 77-82.

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11

[9] R. J. Libera, Some classes of regular univalent functions, Proc. Am. Math. Soc. 63(1965), 755-758. [10] A. E. Livingston, On the radius of univalence of certain analytic functions, Proc. Am. Math. Soc. 17(1966), 352-357. [11] Om P. Ahuja and J. M. Jahangiri, Multivalent harmonic starlike functions with missing coefficients, Math. Sci. Res. J., 7(9)(2003), 347-352. [12] St. Ruscheweyh, New criteria for univalent functions, Proc. Amer. Math. Soc., 49(1975), 109-115. [13] Gr. S ¸ t. S˘ al˘ agean, Subclasses of univalent functions, Lecture Notes in Math., Springer Verlag, Berlin, Heidelberg and New York, 1013(1983), 362-372. [14] Sibel Yalcin, A new class of S˘ al˘ agean-type harmonic univalent functions Appl. Math. Letters, 18(2005), 191-198. 1

Department of Mathematics and Computer Sciences, University of Oradea, ˘t Str. Universita ¸ ii, No.1, 410087 Oradea, Romania ∗ Corresponding author: [email protected] 2

Faculty of Environmental Protection, University of Oradea, Str. B-dul Gen. Magheru, No.26, 410048 Oradea, Romania E-mail address: [email protected] 3

Department of Mathematics and Computer Sciences, University of Oradea, ˘t Str. Universita ¸ ii, No.1, 410087 Oradea, Romania E-mail addresses: [email protected]

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ARGUMENT ESTIMATES FOR CERTAIN ANALYTIC FUNCTIONS N. E. CHO, M. K. AOUF, AND A. O. MOSTAFA Abstract. TThe purpose of the present paper is to investigate some argument properties for certain analytic functions in the open unit disk. The main results presented in here generalize some previous those concerning starlike function of reciprocal of order beta and strongly starlike functions.

1. Introduction Let A be the class of analytic functions f (z) of the form ∞ X an z n (z ∈ U = {z : z ∈ C , |z| < 1}). f (z) = z +

(1.1)

n=2

A function f (z) ∈ A is said to be in the class C(α) of convex functions of order α if and only if   zf 00 (z) Re 1 + 0 > α (0 ≤ α < 1) (1.2) f (z) and is said to be in the class S∗ (α) of starlike functions of order α if and only if  0  zf (z) > α (0 ≤ α < 1). (1.3) Re f (z) We note that C(0) = C and S∗ (0) = S∗ , where C and S∗ are, respectively, the well-known classes of convex and starlike functions. The classical result of Marx [5] and Strahh¨acker [8] asserts that a convex function is starlike of order 1/2, that is,   0   zf (z) 1 zf 00 (z) > 0 (z ∈ U) =⇒ Re > (z ∈ U). (1.4) Re 1 + 0 f (z) f (z) 2 If f (z) ∈ S∗ satisfies the condition   f (z) Re > β (0 ≤ β < 1; z ∈ U), zf 0 (z)

(1.5)

2010 Mathematics Subject Classification. 30C45. Key words and phrases. univalent functions, starlike function of reciprocal of order β, strongly starlike functions, convex functions. 1

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N. E. CHO, M. K. AOUF, AND A. O. MOSTAFA

then f (z) is said to be starlike of reciprocal of order β ( see Nunokawa et al. [4] ). In [7] Sakaguchi proved that: If f (z) ∈ A and g(z) ∈ S∗ , then    0  f (z) f (z) > 0 (z ∈ U) =⇒ Re > 0 (z ∈ U). (1.6) Re g 0 (z) g(z) In [6] Pommerenke generalized Sakaguchi’s result as follows. If f (z) ∈ A and g(z) ∈ C and 0 arg f (z) ≤ π α (0 < α ≤ 1; z ∈ U), g 0 (z) 2

(1.7)

then arg f (z2 ) − f (z1 ) ≤ π α (|z1 | < 1, |z2 | < 1). g(z2 ) − g(z1 ) 2

(1.8)

Recently, Nunokawa et al. [4] generalized Pommerenke’s result as follows. If f (z) ∈ A and g(z) ∈ C, then g(z) is starlike of reciprocal of order β and 0 π f (z) arg ≤ α + tan−1 αβ (z ∈ U; 0 < α ≤ 1; 0 ≤ β < 1), g 0 (z) 2 1+α

(1.9)

then π f (z) arg ≤ α (z ∈ U). g(z) 2

(1.10)

Also Kanas et al. [1] generalized Sakaguchi’s result as follows. If f (z) ∈ A and g(z) ∈ S∗ , then ( 1−α  0 α )   f (z) f (z) f (z) > 0 (z ∈ U; 0 ≤ α ≤ 1) =⇒ Re > α (z ∈ U), Re g(z) g 0 (z) g(z) (1.11) where the powers in (1.11) are meant as the principal values. Also Kanas et al. [1] defined the class H(α) as follows.     f (z) f 0 (z) ∗ H(α) = f (z) ∈ A, g(z) ∈ S : Re (1 − α) +α 0 > 0 (0 ≤ α ≤ 1) . g(z) g (z) (1.12) In the present paper, we extend some results obtained by Kanas et al. [1], Liu [2], Nunokawa et al. [4], Pommerenke [6] and Sakaguchi [7] by using Nunowawa’s lemma [3].

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2. Main results To derive our results, we need the following lemma due to Nunokawa [3]. Lemma 2.1. [3] Let a function p(z) with p(0) = 1 and p(z) 6= 0 be analytic in U. If there exists a point z0 ∈ U such that π |arg p(z)| < α (|z| < |z0 | , α > 0), 2 then π z0 p0 (z0 ) = ikα and |arg p(z0 )| = α, p(z0 ) 2 where   1 1 π k≥ a+ ≥ 1 when arg p(z0 ) = α 2 a 2 and   1 1 π k≤− a+ ≤ −1 when arg p(z0 ) = − α, 2 a 2 where 1 p(z0 ) α = ±ia(a > 0). Theorem 2.2. Let f (z) ∈ A, g(z) ∈ C and g(z) is starlike of reciprocal of order β. Suppose that   0 arg (1 − λ) f (z) + λ f (z) − γ < π ρ (0 ≤ λ ≤ 1; 0 ≤ γ < 1; z ∈ U), (2.1) 2 g(z) g 0 (z) where   2 αβλ −1 ρ = α + tan (0 < α ≤ 1; 0 ≤ γ < 1). (2.2) π 1 + αλ Then we have   π f (z) arg < α (z ∈ U). − γ (2.3) 2 g(z) Proof. Let   1 f (z) p(z) = −γ . 1 − γ g(z) Then p(z) is analytic in U, p(0) = 1 and p(z) 6= 0. It follows from (2.4) that   f 0 (z) zp0 (z) g(z) = γ + (1 − γ)p(z) 1 + . g 0 (z) p(z) zg 0 (z)

(2.4)

(2.5)

Also, from (2.4) and (2.5), we have   f (z) f 0 (z) zp0 (z) g(z) (1 − λ) +λ 0 − γ = (1 − γ)p(z) 1 + λ . g(z) g (z) p(z) zg 0 (z)

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If there exists a point z0 ∈ U such that |arg p(z)|
0). Since g(z) ∈ C, from Marx-Strohh¨acker’s theorem [5,8], we have  0  zg (z) Re > 1/2 (z ∈ U), g(z) so that g(z) ∈ S∗ (1/2). Putting

zg 0 (z) g(z)

= u + iv, where u > 1/2. Then

2 1 − u − iv 2 1 − 2u + u2 + v 2 g(z) < 1. zg 0 (z) − 1 = u + iv = u2 + v 2 Therefore, g(z) < 1 (z ∈ U), − 1 zg 0 (z)

(2.7)

  Im g(z) < 1 (z ∈ U), zg 0 (z)

(2.8)

which implies that

and from the assumption of the theorem, we have   g(z) Re > β (0 ≤ β < 1; z ∈ U). zg 0 (z)

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For the case |arg p(z0 )| = π2 α, from (2.5), (2.6) and (2.8), we have   f (z0 ) f 0 (z0 ) arg (1 − λ) +λ 0 −γ g(z0 ) g (z0 )    g(z0 ) z0 p0 (z0 ) = arg p(z0 ) + arg 1 + λ p(z0 ) z0 g 0 (z0 )    π g(z0 ) g(z0 ) = α + arg 1 + iαkλ Re 0 + iIm 0 2 z0 g (z0 ) z0 g (z0 )     π g(z0 ) g(z0 ) = α + arg 1 − αkλ Im 0 + ikαλRe 0 2 z0 g (z0 ) z0 g (z0 )    αkλRe g(z0 0 )  π z0 g (z0 ) −1 α + tan =  1 + αkλ Im g(z0 )  2 0 z0 g (z0 )     αkλβ αλβ π π −1 −1 α + tan ≥ ≥ α + tan . 2 1 + αkλ 2 1 + αλ This contradicts the assumption of the theorem, then π |arg p(z)| < α (z ∈ U). 2 For the case |arg p(z0 )| = − π2 α, applying the same method above, we have a contradiction. This completes the proof of Theorem 2.2.

Remark. Putting λ = 1 in Theorem 1, we get the result obtained by Liu [2, Theorem 2.1]. Also, from Theorem 1, we have the results obtained by Kanas [1], Nunokawa [4] and Sakaguchi [7]. Theorem 2.3. Let f (z) ∈ A, g(z) ∈ C and g(z) is starlike of reciprocal of order β (0 ≤ β < 1). Suppose that  µ  0 γ f (z) π arg f (z) < ρ (z ∈ U), (2.10) g(z) g 0 (z) 2 where 2γ tan−1 ρ = (µ + γ)α + π



αβ 1+α

 (z ∈ U),

(2.11)

µ and γ are fixed positive real numbers with 0 < µ + γ ≤ 1 and 0 < α ≤ 1. Then π f (z) arg < α (z ∈ U). (2.12) g(z) 2

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Proof. Let us define the function p(z) by (2.4). It follows from (2.4) and (2.5) that µ  0 γ γ   f (z) f (z) zp0 (z) g(z) µ+γ = (p(z)) 1+ g(z) g 0 (z) p(z) zg 0 (z) and  arg

f (z) g(z)

µ 

f 0 (z) g 0 (z)

γ = µ arg

f (z) f 0 (z) + γ arg 0 g(z) g (z)

  zp0 (z) g(z) . = (µ + γ) arg p(z) + γ arg 1 + p(z) zg 0 (z) (2.13) Suppose that there exists a point z0 ∈ U such that π π |arg p(z)| < α (|z| < |z0 |) and |arg p(z0 )| = α (0 < α ≤ 1). 2 2 Then, using Lemma 1, we have z0 p0 (z0 ) = ikβ. p(z0 ) For the case arg p(z) = π2 α, from (2.7), (2.8) and (2.13), we have µ  0 γ    f (z0 ) z0 p0 (z0 ) g(z0 ) f (z0 ) = (µ + γ) arg p(z0 ) + γ arg 1 + arg g(z0 ) g 0 (z0 ) p(z0 ) z0 g 0 (z0 )    π g(z0 ) g(z0 ) = (µ + γ) α + γ arg 1 + iαk Re 0 + iIm 0 2 z0 g (z0 ) z0 g (z0 )     g(z0 ) π g(z0 ) = (µ + γ) α + γ arg 1 − αk Im 0 + iαkRe 0 2 z0 g (z0 ) z0 g (z0 )    αkRe g(z0 0 )  π −1 z0 g (z0 ) = (µ + γ) α + γ tan  1 + αk Im g(z0 )  2 z0 g 0 (z0 )   αβk π ≥ (µ + γ) α + γ tan−1 2 1 + αk   π αβ −1 ≥ (µ + γ) α + γ tan . 2 1+α This contradicts the assumption of the theorem, then we have π |arg p(z)| < α (z ∈ U). 2 π For the case |arg p(z0 )| = − 2 α, applying the same method above, we have a contradiction. This completes the proof of Theorem 2.3.

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Putting µ = 1 − γ (γ > 0) in Theorem 2, we obtain the following corollary. Corollary 1. Let f (z) ∈ A, g(z) ∈ C and g(z) is starlike of reciprocal of order β (0 < β ≤ 1). Suppose that 1−γ  0 γ  f (z) π f (z) arg < ρ (γ > 0; z ∈ U), g(z) g 0 (z) 2 2γ ρ=α+ tan−1 π



αβ 1+α

 (0 < α ≤ 1).

Then arg f (z) < π α (z ∈ U). g(z) 2

Remark. Putting γ = 1 in Corollary 1, we have the result obtained by Nunokawa et al. [ [4], Theorem 2.3].

Acknowledgement The second author was supported by the Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education, Science and Technology (No. 2019R1I1A3A01050861).

References [1] S. Kanas, A. Lecko and A. Moleda, Certain generalization of the Sakaguchi lemma, Zeszyty Nauk. Pol. Rzes. Folia Sci. Univ. Tech. Resov., 38(1987), 35-41. [2] Jin-Lin Liu, Some argument inequalities for certain analytic functions, Math. Solvaca, 62(2012), 25-28. [3] M. Nunokawa, On the order of strongly convex functions, Proc. Japan Acad. Ser. A, 69(1993), no. 7, 234-237. [4] M. Nunokawa, S. Owa, J. Nishiwaki, K. Kuroni and T. Hayanni, Differential subordination and argument property, Comput. Math. Appl., 56(2008), 2733-2736. [5] A. Marx, Untersuchungen u ¨ber schlichte Abildung, Math. Ann., 107(1932-1933), 40-67. [6] Ch. Pommerenke, On close-to-convex analytic functions, Trans. Amer. Math. Soc., 114(1965), no. 1, 176-186. [7] K. Sakaguchi, On a certain univalent mapping, J. Math. Soc. Japan, 11(1959), 72-75. [8] E. Strohh¨ acher, Beitrage zur theorie der schlichten funktionen, Math. Z., 37(1933), 356-380.

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(N. E. Cho) Department of Applied Mathematics, Pukyong National University, Busan 48513, Korea Email address: [email protected] (M. K. Aouf) Department of Mathematics Faculty of Science, Mansoura University, Mansoura 35516, Egypt Email address: [email protected] (A. O. Mostafa) Department of Mathematics Faculty of Science, Mansoura University, Mansoura 35516, Egypt Email address: [email protected]

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Some properties of the second kind degenerate q-Euler polynomials associated with the p-adic integral on Zp C. S. RYOO Department of Mathematics, Hannam University, Daejeon 34430, Korea

Abstract : In this paper, we introduce the second kind degenerate q-Euler numbers and polynomials associated with the p-adic integral on Zp . We also obtain some explicit formulas for the second kind degenerate q-Euler numbers and polynomials. Key words : Euler numbers and polynomials, the second kind Euler numbers and polynomials, the second kind degenerate Euler numbers and polynomials, the second kind degenerate q-Euler numbers and polynomials, p-adic integral on Zp . AMS Mathematics Subject Classification : 11B68, 11S40, 11S80. 1. Introduction Throughout this paper we use the following notations. By Zp we denote the ring of p-adic rational integers, Qp denotes the field of rational numbers, N denotes the set of natural numbers, C denotes the complex number field, Cp denotes the completion of algebraic closure of Qp , N denotes the set of natural numbers and Z+ = N∪{0}, and C denotes the set of complex numbers. Let p be a fixed odd prime number. Let νp be the normalized exponential valuation of Cp with |p|p = p−νp (p) = p−1 . When one talks of q-extension, q is considered in many ways such as an indeterminate, a complex number q ∈ C, or p-adic number q ∈ Cp . If q ∈ C one normally assumes that |q| < 1. If q ∈ Cp , we normally assume that |q − 1|p < p− p−1 so that q x = exp(x log q) for |x|p ≤ 1. We say that f is uniformly differentiable function at a point a ∈ Zp and denote this property by g ∈ U D(Zp ), if the difference quotients 1

Fg (x, y) =

g(x) − g(y) x−y

have a limit l = g ′ (a) as (x, y) → (a, a). For g ∈ U D(Zp ), the fermionic p-adic invariant integral on Zp is defined by ∫ ∑ I−1 (g) = g(x)dµ−1 (x) = lim g(x)(−1)x , (see [3]). (1) N →∞

Zp

0≤x 1; a ∈ / Z− 0 ), and lq (s) =

∞ ∑ (−1)n q n , [n]sq n=1

(Re(s) > 1).

Inspired by their work, the (p, q)-extension of the twisted q-L-function can be defined as follow: Let ζ be rth root of 1 and ζ ̸= 1. For s, x ∈ C with Re(x) > 0, the twisted (p, q)-L-function Lp,q,ζ (s, x) is define by ∞ ∑ (−1)m ζ m . Lp,q,ζ (s, x) = [2]q [m + x]sp,q m=0

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2. Twisted (p, q)-Euler numbers and polynomials In this section, we define twisted (p, q)-Euler numbers and polynomials and provide some of their relevant properties. Let r be a positive integer, and let ζ be rth root of 1. Definition 1. For 0 < q < p ≤ 1, the Carlitz’s type twisted (p, q)-Euler numbers En,p,q,ζ and polynomials En,p,q,ζ (x) are defined by means of the generating functions Gp,q,ζ (t) =

∞ ∑

En,p,q,ζ

n=0

and Gp,q,ζ (t, x) =

∞ ∑

∞ ∑ tn = [2]q (−1)m ζ m e[m]p,q t . n! m=0

En,p,q,ζ (x)

n=0

(2.1)

∞ ∑ tn = [2]q (−1)m ζ m e[m+x]p,q t , n! m=0

(2.2)

respectively. Setting p = 1 in (2.1) and (2.2), we can obtain the corresponding definitions for the Carlitz’s type twisted q-Euler number En,q,ζ and q-Euler polynomials En,q,ζ (x), respectively. By (2.1), we get ( ) ( )n ∑ ∞ n ( ) ∞ ∑ ∑ tn 1 n 1 tn m m [m]p,q t l = En,p,q,ζ = [2]q [2]q (−1) . (−1) ζ e l n−l n! p−q l 1 + ζq p n! n=0 n=0 m=0 ∞ ∑

l=0

By comparing the coefficients

tn n!

in the above equation, we have the following theorem.

Theorem 2. For n ∈ Z+ , we have ( )n ∑ n ( ) 1 n 1 En,p,q,ζ = [2]q (−1)l . l p−q 1 + ζpn−l q l l=0

By (2.2), we obtain ( En,p,q,ζ (x) = [2]q

1 p−q

)n ∑ n ( ) n 1 (−1)l q xl p(n−l)x . l 1 + ζpn−l q l

(2.3)

l=0

(h)

Next, we introduce Carlitz’s type twisted (h, p, q)-Euler polynomials En,p,q,ζ (x). (h)

Definition 3. The Carlitz’s type twisted (h, p, q)-Euler polynomials En,p,q,ζ (x) are defined by (h)

En,p,q,ζ (x) = [2]q

∞ ∑

(−1)m phm ζ m [m + x]np,q .

(2.4)

m=0 (h)

(h)

(h)

When x = 0, En,p,q,ζ = En,p,q,ζ (0) are called the twisted (h, p, q)-Euler numbers En,p,q,ζ . By using (2.4) and (p, q)-number, we have the following theorem. Theorem 4. For n ∈ Z+ , we have )n ∑ ( n ( ) n 1 1 (h) . (−1)l q xl p(n−l)x En,p,q,ζ (x) = [2]q p−q l 1 + ζpn−l+h q l l=0

By (2.4) and Theorem 2, we have n ( ) ∑ n (n−l)x (l) q En−l,p,q,ζ [x]lp,q l l=0 n ( ) ∑ n xl (n−l)y (l) En,p,q,ζ (x + y) = p q En−l,p,q,ζ (x)[y]lp,q . l

En,p,q,ζ (x) =

(2.5)

l=0

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By (2.1) and (2.2), we get − [2]q

∞ ∞ n−1 ∑ ∑ ∑ (−1)l+n ζ l+n e[l+n]p,q t + [2]q (−1)l ζ l e[l]p,q t = [2]q (−1)l ζ l e[l]p,q t . l=0

l=0

l=0

Hence we have (−1)

) ( ∞ ∞ n−1 ∑ ∑ ∑ tm tm tm l l m (−1) ζ [l]p,q + Em,p,q,ζ = [2]q . ζ Em,p,q,ζ (n) m! m=0 m! m=0 m! m=0

n+1 n

∞ ∑

(2.6)

l=0

By comparing the coefficients

tm m!

on both sides of (2.6), we have the following theorem.

Theorem 5. For m ∈ Z+ , we have n−1 ∑

(−1)l ζ l [l]m p,q =

l=0

(−1)n+1 ζ n Em,p,q,ζ (n) + Em,p,q,ζ . [2]q

3. Twisted (p, q)-l-function and twisted (p, q)-L-function By using twisted (p, q)-Euler numbers and polynomials, twisted (p, q)-L-function is defined. These functions interpolate the twisted (p, q)-Euler numbers En,p,q,ζ , and polynomials En,p,q,ζ (x), respectively. From (2.1), we note that ∞ ∑ dk G (t) = [2] (−1)n ζ m [m]kp,q = Ek,p,q,ζ , (k ∈ N). p,q,ζ q dtk t=0 m=0 By using the above equation, we are now ready to define twisted (p, q)-l-function. Definition 6. Let s ∈ C with Re(s) > 0. ∞ ∑ (−1)n ζ n . [n]sp,q n=1

lp,q,ζ (s) = [2]q

(3.1)

Relation between lp,q,ζ (s) and Ek,p,q,ζ is given by the following theorem. Theorem 7. For k ∈ N, we have lp,q,ζ (−k) = Ek,p,q,ζ . By using (2.2), we note that ∞ ∑ dk G (t, x) = [2] (−1)m ζ m [m + x]kp,q p,q,ζ q dtk t=0 m=0 and

(

d dt

)k ( ∑ ∞

tn En,p,q,ζ (x) n! n=0

)

= Ek,p,q,ζ (x), for k ∈ N.

(3.2)

(3.3)

t=0

By (3.2) and (3.3), we are now ready to define the twisted (p, q)-L-function. Definition 8. Let s ∈ C with Re(s) > 0 and x ∈ / Z− 0. Lp,q,ζ (s, x) = [2]q

∞ ∑ (−1)n ζ n . [n + x]sp,q n=0

(3.4)

Note that Lp,q,ζ (s, x) is a meromorphic function on C. Relation between Lp,q,ζ (s, x) and Ek,p,q,ζ (x) is given by the following theorem. Theorem 9. For k ∈ N, we have Lp,q,ζ (−k, x) = Ek,p,q,ζ (x). Observe that Lp,q,ζ (−k, x) function interpolates Ek,p,q,ζ (x) numbers at non-negative integers.

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3. Some symmetric identities for twisted (p, q)-L-function Let w1 , w2 ∈ N with w1 ≡ 1 (mod 2), w2 ≡ 1 (mod 2). For n ∈ Z+ , we obtain certain symmetric identities for twisted (p, q)-L-function. Theorem 10. Let w1 , w2 ∈ N with w1 ≡ 1 (mod 2), w2 ≡ 1 (mod 2). Then we obtain [w2 ]sp,q [2]qw2

w∑ 1 −1

( (−1)j ζ w2 j Lpw1 ,qw1 ,ζ w1

s, w2 x +

j=0

=

[w1 ]sp,q [2]qw1

w∑ 2 −1

j w1 j

(−1) ζ

Lpw2 ,qw2 ,ζ w2

j=0

w2 j w1

)

Proof. Note that [xy]q = [x]qy [y]q for any x, y ∈ C. In (3.4), by substitute w2 x + and replace q, p, and ζ by q w1 , pw1 and ζ w1 , respectively, we derive next result 1 Lpw1 ,qw1 ,ζ w1 [2]qw1

( ) ∑ ∞ w2 [ s, w2 x + j = w1 m=0

=

∞ ∑ m=0

[

w2 j for x in w1

(−1)m ζ w1 m ]s w2 m + w2 x + j w1 pw1 ,qw1 (−1)m ζ w1 m ]s w1 m + w1 w2 x + w2 j w1 pw1 ,q w1

= [w1 ]sp,q

∞ w∑ 2 −1 ∑ m=0 i=0

=

(4.1)

) ( w1 j . s, w1 x + w2

[w1 ]sp,q

∞ w∑ 2 −1 ∑ m=0 i=0

(4.2)

(−1)w2 m+i ζ w1 (w2 m+i) [w1 (w2 m + i) + w1 w2 x + w2 j]sp,q

(−1)m (−1)i ζ w1 w2 m ζ w1 i . [w1 w2 (x + m) + w1 i + w2 j]sp,q

Thus, from (4.2), we can derive the following equation. ( ) 1 −1 [w2 ]sp,q w∑ w2 (−1)j ζ w2 j Lpw1 ,qw1 ,ζ w1 s, w2 x + j [2]qw1 j=0 w1 =

[w1 ]sp,q [w2 ]sp,q

∞ w∑ 2 −1 w 1 −1 ∑ ∑ (−1)j+i+m ζ w1 w2 m ζ w1 i ζ w2 j [w1 w2 (x + m) + w1 i + w2 j]sp,q m=0 i=0 j=0

(4.3)

By using the same method as (4.3), we have ) ( 2 −1 [w1 ]sp,q w∑ w1 (−1)j ζ w1 j Lpw2 ,qw2 ,ζ w2 s, w1 x + j [2]qw2 j=0 w2 =

[w1 ]sp,q [w2 ]sp,q

∞ w∑ 2 −1 w 1 −1 ∑ ∑ (−1)j+i+m ζ w1 w2 m ζ w2 i ζ w1 j [w1 w2 (x + m) + w1 j + w2 i]sp,q m=0 j=0 i=0

(4.4)

Therefore, by (4.3) and (4.4), we have the following theorem.  Taking w2 = 1 in Theorem 10, we obtain the following corollary. Corollary 11. Let w1 ∈ N with w1 ≡ 1 (mod 2). For n ∈ Z+ , we obtain Lp,q,ζ (s, w1 x) =

( ) w∑ 1 −1 j [2]q j j w w w (−1) ζ L s, x + . 1 1 1 p ,q ,ζ [2]qw1 [w1 ]sp,q j=0 w1

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Let us take s = −n in Theorem 10. For n ∈ Z+ , we obtain certain symmetry identities for twisted (p, q)-Euler polynomials. Theorem 12. Let w1 , w2 ∈ N with w1 ≡ 1 (mod 2), w2 ≡ 1 (mod 2). For n ∈ Z+ , we obtain [w1 ]np,q [2]qw2

w∑ 1 −1

j w2 j

(−1) ζ

En,pw1 ,qw1 ,ζ w1

j=0

= [w2 ]np,q [2]qw1

w∑ 2 −1

( ) w2 w2 x + j w1

(−1)j ζ w1 j En,pw2 ,qw2 ,ζ w2

j=0

( ) w1 w1 x + j . w2

Taking w2 = 1 in Theorem 12, we obtain the following distribution relation. Corollary 13. Let w1 ∈ N with w1 ≡ 1 (mod 2). For n ∈ Z+ , we obtain ) ( w∑ 1 −1 [2]q j n j j [w1 ]p,q . En,p,q,ζ (w1 x) = (−1) ζ En,pw1 ,qw1 ,ζ w1 s, x + [2]qw1 w1 j=0 By (2.5), we have w∑ 1 −1

( j w2 j

(−1) ζ

En,pw1 ,qw1 ,ζ w1

j=0

=

w∑ 1 −1

j w2 j

(−1) ζ

j=0

=

w∑ 1 −1

j w2 j

(−1) ζ

j=0

w2 w2 x + j w1

)

]i [ n ( ) ∑ n w2 j(n−i) w1 w2 xi (i) w2 j q p En−i,pw1 ,qw1 ,ζ w1 (w2 x) w1 pw1 ,qw1 i i=0

( )i n ( ) ∑ n w2 j(n−i) w1 w2 xi (i) [w2 ]p,q i q p En−i,pw1 ,qw1 ,ζ w1 (w2 x) [j]pw2 ,qw2 i [w ] 1 p,q i=0

Hence we have the following theorem. Theorem 14. Let w1 , w2 ∈ N with w1 ≡ 1 (mod 2), w2 ≡ 1 (mod 2). For n ∈ Z+ , we obtain w∑ 1 −1

( j w2 j

(−1) ζ

En,pw1 ,qw1 ,ζ w1

j=0

w2 w2 x + j w1

)

w∑ n ( ) 1 −1 ∑ n w1 w2 xi (i) = [w2 ]ip,q [w1 ]−i p E (w x) (−1)j ζ w2 j q w2 (n−i)j [j]ipw2 ,qw2 . w w w 2 p,q n−i,p 1 ,q 1 ,ζ 1 i i=0 j=0

For each integer n ≥ 0, let An,i,p,q,ζ (w) = the alternating twisted (p, q)-power sums.

∑w−1 j=0

(−1)j ζ j q j(n−i) [j]ip,q . The sum An,i,p,q,ζ (w) is called

By Theorem 14, we have [2]qw2 [w1 ]np,q

w∑ 1 −1

j w2 j

(−1) ζ

En,pw1 ,qw1 ,ζ w1

j=0

= [2]qw2

) ( w2 j w2 x + w1

n ( ) ∑ n w1 w2 xi (i) [w2 ]ip,q [w1 ]n−i En−i,pw1 ,qw1 ,ζ w1 (w2 x)An,i,pw2 ,qw2 ,ζ w2 (w1 ) p,q p i i=0

(4.5)

By using the same method as in (4.5), we have [2]qw1 [w2 ]np,q

w∑ 2 −1

j w1 j

(−1) ζ

En,pw2 ,qw2 ,ζ w2

j=0

= [2]qw1

) ( w1 j w1 x + w2

n ( ) ∑ n w1 w2 xi (i) [w1 ]ip,q [w2 ]n−i En−i,pw2 ,qw2 ,ζ w2 (w1 x)An,i,pw1 ,qw1 ,ζ w1 (w2 ) p,q p i i=0

803

(4.6)

RYOO 799-804

J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 29, NO.4, 2021, COPYRIGHT 2021 EUDOXUS PRESS, LLC

Therefore, by (4.5) and (4.6) and Theorem 12, we have the following theorem. Theorem 15. Let w1 , w2 ∈ N with w1 ≡ 1 (mod 2), w2 ≡ 1 (mod 2). For n ∈ Z+ , we obtain n ( ) ∑ n w1 w2 xi (i) [w1 ]ip,q [w2 ]n−i En−i,pw2 ,qw2 ,ζ w2 (w1 x)An,i,pw1 ,qw1 ,ζ w1 (w2 ) p,q p i i=0 n ( ) ∑ n w1 w2 xi (i) = [2]qw2 [w2 ]ip,q [w1 ]n−i En−i,pw1 ,qw1 ,ζ w1 (w2 x)An,i,pw2 ,qw2 ,ζ w2 (w1 ). p,q p i i=0

[2]qw1

By Theorem 15, we obtain the interesting symmetric identity for the twisted (h, p, q)-Euler numbers (h) En,p,q,ζ in complex field. Corollary 16. Let w1 , w2 ∈ N with w1 ≡ 1 (mod 2), w2 ≡ 1 (mod 2). For n ∈ Z+ , we obtain [2]

n ( ) ∑ n q w1 i=0

= [2]qw2

i

(i)

w1 w2 xi [w1 ]ip,q [w2 ]n−i An,i,pw1 ,qw1 ,ζ w1 (w2 )En−i,pw2 ,qw2 ,ζ w2 p,q p

n ( ) ∑ n (i) w1 w2 xi [w2 ]ip,q [w1 ]n−i An,i,pw2 ,qw2 ,ζ w2 (w1 )En−i,pw1 ,qw1 ,ζ w1 . p,q p i i=0

Acknowledgement: This work was supported by 2020 Hannam University Research Fund. REFERENCES 1. Agarwal, R.P.; Kang, J.Y.; Ryoo, C.S.(2018). Some properties of (p, q)-tangent polynomials, Journal of Computational Analysis and Applications, v.24, pp. 1439-1454. 2. Araci, S.; Duran, U.; Acikgoz, M.; Srivastava, H.M.(2016). A certain (p, q)-derivative operator and associated divided differences, Journal of Inequalities and Applications, v. 2016:301, DOI 10.1186/s13660-016-1240-8. 3. Duran, U.; Acikgoz, M.; Araci, S.(2016). On (p, q)-Bernoulli, (p, q)-Euler and (p, q)-Genocchi polynomials, J. Comput. Theor. Nanosci., v.13, pp. 7833-7846. 4. Hwang, K.W.; Ryoo, C.S.(2019). Some symmetric identities for degenerate Carlitz-type (p, q)-Euler numbers and polynomials, Symmetry, v.11, 830; doi:10.3390/sym11060830. 5. Luo, Q.M.; Zhou, Y.(2011). Extension of the Genocchi polynomials and its q-analogue, Utilitas Mathematica, v.85, pp. 281 - 297. 6. Ryoo, C.S.(2017). Some identities on the higher-order twisted q-Euler numbers and polynomials, J. Computational Analysis and Applications, v.22, pp. 825-830. 7. Ryoo, C.S.(2017). On the (p, q)-analogue of Euler zeta function, J. Appl. Math. & Informatics v.35, pp. 303-311. 8. Ryoo, C.S.(2020). Symmetric identities for Dirichlet-type multiple twisted (h, q)-l-function and higher-order generalized twisted (h, q)-Euler polynomials, Journal of Computational Analysis and Applications, v.28, pp. 537-542.

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RYOO 799-804

TABLE OF CONTENTS, JOURNAL OF COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 29, NO. 4, 2021

Exact Solitary Wave Solutions for Wick-type Stochastic (2+1)-dimensional Coupled KdV Equations, Hossam A. Ghany and M. Zakarya,……………………………………………617 Exact Solutions for Stochastic Fractional Zhiber-Shabat Equations, Hossam A. Ghany and Ashraf Fathallah,…………………………………………………………………………..634 Invariance, Solutions, Periodicity and Asymptotic Behavior of a Class of Fourth Order Difference Equations, Mensah Folly-Gbetoula,……………………………………………645 Generalized Zweier I-Convergent Sequence Spaces of Fuzzy Numbers, Kavita Saini and Kuldip Raj,…………………………………………………………………………………………658 Some Convergence Results Using K* Iteration Process in CAT(0) Spaces, Kifayat Ullah, Dong Yun Shin, Choonkil Park, and Bakhat Ayaz Khan,…………………………………………668 Nonlinear Discrete Inequalities Method for the Ulam Stability of First Order Nonlinear Difference Equations, R.Dhanasekaran, E.Thandapani, and J.M.Rassias,………………….682 Algebras and Smarandache Types, Jung Mi Ko and Sun Shin Ahn,……………………….691 Nonlinear Differential Equations Associated with Degenerate (h,q)-Tangent Numbers, Cheon Seoung Ryoo,………………………………………………………………………………700 On the Symmetries of the Second Kind (h,q)-Bernoulli Polynomials, C. S. Ryoo,……….706 Some New Fuzzy Best Proximity Point Theorems in Non-Archimedean Fuzzy Metric Spaces, Muzeyyen Sangurlu Sezen and Huseyin Isik,………………………………………………712 Exact Solutions of Conformable Fractional Harry Dym Equation, Asma ALHabees,…….727 Some Properties of the q-Exponential Functions, Mahmoud J. S. Belaghi,……………….737 BCI-Implicative Ideals of BCI-Algebras Using Neutrosophic Quadruple Structure, Young Bae Jun, Seok-Zun Song, and G. Muhiuddin,…………………………………………………..742 Isolation Numbers of Matrices Over Nonbinary Boolean Semiring, LeRoy B. Beasley, Madad Khan, and Seok-Zun Song,…………………………………………………………………758 Orthogonally Euler-Lagrange Type Cubic Functional Equations in Orthogonality Normed Spaces, Chang Il Kim and Giljun Han,……………………………………………………..765

TABLE OF CONTENTS, JOURNAL OF COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 29, NO. 4, 2021 (continues)

Certain Subclass of Harmonic Multivalent Functions Defined By Derivative Operator, Adriana Cătaș, Roxana Șendruțiu, and Loredana-Florentina Iambor,………………………………775 Argument Estimates for Certain Analytic Functions, N. E. Cho, M. K. Aouf, and A. O. Mostafa,…………………………………………………………………………………….786 Some Properties of the Second Kind Degenerate q-Euler Polynomials Associated with the pAdic Integral On ℤp, C. S. Ryoo,……………………………………………………………794 Some Symmetric Identities for Twisted (p,q)-L-Function. C. S. Ryoo,…………………….799

Volume 29, Number 5 ISSN:1521-1398 PRINT,1572-9206 ONLINE

October 2021

Journal of Computational Analysis and Applications EUDOXUS PRESS, LLC

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J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 29, NO.5, 2021, COPYRIGHT 2021 EUDOXUS PRESS, LLC

Differential equations and inclusions involving mixed fractional derivatives with four-point nonlocal fractional boundary conditions Bashir Ahmada ,

Sotiris K. Ntouyasb,a,1 ,

Ahmed Alsaedia

a

Nonlinear Analysis and Applied Mathematics (NAAM)-Research Group, Department of Mathematics, Faculty of Science, King Abdulaziz University, P.O. Box 80203, Jeddah 21589, Saudi Arabia b

Department of Mathematics, University of Ioannina, 451 10 Ioannina, Greece

E-mail: bashirahmad− [email protected] (B. Ahmad), [email protected] (S.K. Ntouyas), [email protected] (A. Alsaedi) Abstract We study a new class of boundary value problems of mixed fractional differential equations and inclusions involving both left Caputo and right RiemannLiouville fractional derivatives, and nonlocal four-point fractional boundary conditions. We apply the standard tools of the fixed-point theory to obtain the sufficient criteria for the existence and uniqueness of solutions for the problems at hand. Illustrative examples for the obtained results are also presented.

Keywords: Fractional differential equations; fractional differential inclusions; fractional derivative; boundary value problem; existence; fixed point theorems. MSC 2000: 34A08, 34B15, 34A60.

1

Introduction

Fractional calculus deals with the study of fractional order integrals and derivatives and their diverse applications [1, 2, 3]. Riemann-Liouville and Caputo are kinds of fractional derivatives. They all generalize the ordinary integral and differential operators. However, the fractional derivatives have fewer properties than the corresponding classical ones. As a result, it makes these derivatives very useful at describing the anomalous phenomena, see [4, 5, 6] and references cited therein. Some solutions of equations containing left and right fractional derivatives were investigated [7, 8, 9]. The left and the right derivatives found interesting applications in fractional variational principles, fractional control theory as well as in fractional Lagrangian and Hamiltonian dynamics. In [10], the existence of an extremal solution to a nonlinear system with the right-handed Riemann-Liouville fractional derivative 1

Corresponding author 817

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was discussed. In [11, 12], the authors studied the existence of solutions for fractional boundary value problems involving both the left Riemann-Liouville and the right Caputo fractional derivatives. In this paper, we investigate the existence and uniqueness of solutions for a mixed fractional differential equation involving both left Caputo and right Riemann-Liouville types fractional derivatives associated with nonlocal four-point fractional boundary conditions. Precisely, we study the following problems: ( c α β D1− D0+ y(t) = f (t, y(t)), t ∈ J := [0, 1], (1.1) β y(0) = 0, D0+ y(ξ) = 0, y(1) = δy(η), 0 < η < 1, and (

c

β α D1− D0+ y(t) ∈ F (t, y(t)), t ∈ J := [0, 1],

β y(0) = 0, D0+ y(ξ) = 0, y(1) = δy(η), 0 < ξ, η < 1,

(1.2)

β α and D0+ denote the left Caputo fractional derivative of order α ∈ (1, 2] where c D1− and the right Riemann-Liouville fractional derivative of order β ∈ (0, 1] respectively, f : J × R → R is a given function, F : [0, 1] × R → P(R) is a multivalued map, P(R) is the family of all nonempty subsets of R and δ ∈ R is an appropriate constant. Here β we remark that the problem (1.1) with y 0 (0) = 0 in palce of D0+ y(ξ) = 0, was studied recently in [13]. The rest of the paper is organized as follows. In Section 2, we recall some basic definitions of fractional calculus and prove a basic result that plays a key role in the forthcoming analysis. Section 3 contains the existence and uniqueness results for the problem (1.1), which rely on fixed point theorems due to Banach, Krasnoselskii and Leray-Schauder nonlinear alternative. In Section 4, we discuss existence results for the problem (1.2), which rely on nonlineqar alternative for Kakutani maps and Covitz and Nadler fixed point theorem. Finally in Section 5 we study illustrative examples for the obtained results.

2

Preliminaries

In this section, we introduce notations, definitions, and preliminary facts [14] that we need in the sequel. Definition 2.1 We define the left and right Riemann-Liouville fractional integrals of order α > 0 of a function g : (0, ∞) → R as Z t (t − s)α−1 α g(s)ds, (2.1) I0+ g(t) = Γ(α) 0 Z 1 (s − t)α−1 α I1− g(t) = g(s)ds, (2.2) Γ(α) t 818

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provided the right-hand sides are point-wise defined on (0, ∞), where Γ is the Gamma function. Definition 2.2 The left Riemann-Liouville fractional derivative and the right Caputo fractional derivative of order α > 0 of a continuous function g : (0, ∞) → R such that g ∈ C n ((0, ∞), R) are respectively given by α D0+ g(t) = c

dn n−α (I g)(t), dtn 0+

α n−α (n) D1− g(t) = (−1)n I1− g (t),

where n − 1 < α < n. The following lemma, dealing with a linear variant of the problem (1.1), plays an important role in the forthcoming analysis. Lemma 2.3 Let h ∈ C(J, R) and P = [(1 − δη β+1 ) − (β + 1)ξ(1 − δη β )] 6= 0. The function y is a solution of the problem (

c

β α D1− D0+ y(t) = h(t), t ∈ J := [0, 1],

y(0) = 0,

β D0+ y(ξ)

(2.3)

= 0, y(1) = δy(η), 0 < ξ, η < 1,

if and only if [tβ+1 (1 − δη β ) − tβ (1 − δη β+1 )] α I1− h(t)|t=ξ y(t) = + P Γ(β + 1)  [tβ+1 − ξ(β + 1)tβ ]  β α β α δI0+ I1− h(t)|t=η − I0+ I1− h(t)|t=1 , + P β α I0+ I1− h(t)

(2.4)

α where I1− y(s) is defined by (2.2). α Proof. Applying the right fractional integral I1− to both sides of the equation in the problem (2.3), we get β α y(t) = I1− h(t) + c0 + c1 t. (2.5) D0+ β Using the condition D0+ y(ξ) = 0 in (2.5), we obtain α c0 + c1 ξ = −I1− h(t)|t=ξ .

(2.6)

β Next we apply the left fractional integral I0+ to the equation (2.5) to get β α y(t) = I0+ I1− h(t) + c0

tβ tβ+1 + c1 + c2 tβ−1 . Γ(β + 1) Γ(β + 2) 819

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Making use of the conditions y(0) = 0 and y(1) = δy(η) in (2.7) yields c2 = 0 and (1 − δη β ) (1 − δη β+1 ) β α β α c0 + c1 = δI0+ I1− h(t)|t=η − I0+ I1− h(t)|t=1 . Γ(β + 1) Γ(β + 2)

(2.8)

Solving (2.7) and (2.8) for c0 and c1 , we find that i  Γ(β + 2) h (1 − δη β+1 ) α β α β α c0 = − I h(t)|t=ξ + ξ δI0+ I1− h(t)|t=η − I0+ I1− h(t)|t=1 , P Γ(β + 2) 1− i Γ(β + 2) h β α (1 − δη β ) α β α c1 = I1− h(t)|t=1 + δI0+ I1− h(t)|t=η − I0+ I1− h(t)|t=ξ . P Γ(β + 1) Substituting the values of c0 and c1 in (2.6), we get the solution (2.4). By direct computation, we can obtain the converse of this lemma. This completes the proof. 2 Remark 2.4 Let khk = supt∈[0,1] |h(t). Then we have the following estimate: ( ) [tβ + (1 + δη β )|µ2 (t)|] (1 − ξ)α kyk ≤ khk max |µ1 (t)| + , t∈[0,1] Γ(α + 1) Γ(α + 1)Γ(β + 1)

(2.9)

where µ1 (t) =

tβ+1 (1 − δη β ) − tβ (1 − δη β+1 ) , P Γ(β + 1)

µ2 (t) =

tβ+1 − ξ(β + 1)tβ . P

(2.10)

Indeed, we have Z t Z Z 1 (t − s)β−1 1 (u − s)α−1 (s − ξ)α−1 |y(t)| ≤ khk duds + |µ1 (t)|khk ds Γ(β) Γ(α) Γ(α) 0 s ξ " Z Z η (η − s)β−1 1 (u − s)α−1 duds +|µ2 (t)|khk δ Γ(β) Γ(α) s 0 # Z 1 Z (1 − s)β−1 1 (u − s)α−1 + duds Γ(β) Γ(α) 0 s Z t Z 1 (t − s)β−1 (1 − s)α (s − ξ)α−1 = khk ds + |µ1 (t)|khk ds Γ(β) Γ(α + 1) Γ(α) 0 ξ " Z # Z 1 η α β−1 α β−1 (η − s) (1 − s) (1 − s) (1 − s) +|µ2 (t)|khk δ ds + ds Γ(β) Γ(α + 1) Γ(β) Γ(α + 1) 0 0 ( ) α β β (1 − ξ) [t + (1 + δη )|µ2 (t)|] ≤ khk max |µ1 (t)| + , t∈[0,1] Γ(α + 1) Γ(α + 1)Γ(β + 1) where we taken (1 − s)α ≤ 1. For computation convenience, we introduce the notation: n (1 − ξ)α [tβ + (1 + δη β )|µ2 (t)|] o Λ = max |µ1 (t)| + . t∈[0,1] Γ(α + 1) Γ(α + 1)Γ(β + 1) 820

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5

Existence and uniqueness results for the problem (1.1)

Let X = C([0, 1], R) denotes the Banach space of all continuous functions from [0, 1] → R equipped with the norm kyk = sup {|y(t)| : t ∈ [0, 1]}. In view of Lemma 2.3, we transform the problem (1.1) into a fixed point problem as y = Gy, (3.1) where the operator G : X → X is defined by Z t Z 1 (t − s)β−1 α (s − ξ)α−1 I1− f (s, y(s))ds + µ1 (t) f (s, y(s))ds (3.2) Gy(t) = Γ(β) Γ(α) 0 ξ Z 1 h Z η (η − s)β−1 i (1 − s)β−1 α α I1− f (s, y(s))ds − I1− f (s, y(s))ds , +µ2 (t) δ Γ(β) Γ(β) 0 0 where µ1 , µ2 are defined by (2.10). Our first result deals with the existence and uniqueness of solutions for the problem (1.1). Theorem 3.1 Let f : [0, 1] × R → R be a continuous function such that: (H1 ) |f (t, y) − f (t, z)| ≤ L|y − z|, for all t ∈ [0, 1], y, z ∈ R, L > 0. Then the problem (1.1) has a unique solution on [0, 1] if LΛ < 1,

(3.3)

where Λ is defined by (2.11). MΛ to establish that 1 − LΛ GBr ⊂ Br , where Br = {y ∈ X : kyk ≤ r} and G is defined by (3.2). Using the condition (H1 ), we have

Proof. Let us define supt∈[0,1] |f (t, 0)| = M and select r ≥

|f (t, y)| = |f (t, y) − f (t, 0) + f (t, 0)| ≤ |f (t, y) − f (t, 0)| + |f (t, 0)| ≤ Lkyk + M ≤ Lr + M.

(3.4)

Then, for y ∈ Br , by using Remark 2.4, we obtain (Z Z Z 1 t (t − s)β−1 1 (u − s)α−1 (s − ξ)α−1 kGyk ≤ (Lr + M ) duds + |µ1 (t)| ds Γ(β) Γ(α) Γ(α) 0 s ξ " Z Z η (η − s)β−1 1 (u − s)α−1 +|µ2 (t)| δ duds Γ(β) Γ(α) 0 s 821

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B. Ahmad, S. K. Ntouyas, A. Alsaedi #) Z (1 − s)β−1 1 (u − s)α−1 + duds Γ(β) Γ(α) 0 s (Z Z 1 t (s − ξ)α−1 (t − s)β−1 (1 − s)α = (Lr + M ) ds + |µ1 (t)| ds Γ(β) Γ(α + 1) Γ(α) ξ 0 " Z #) Z 1 η (η − s)β−1 (1 − s)α (1 − s)β−1 (1 − s)α +|µ2 (t)| δ ds + ds Γ(β) Γ(α + 1) Γ(β) Γ(α + 1) 0 0 Z

1

≤ (Lr + M )Λ < r. This show that Gy ∈ Br , y ∈ Br . Thus GBr ⊂ Br . Next we show that G is a contraction. For that, let y, z ∈ X . Then, for each t ∈ [0, 1], we have k(Gy) − (Gz)k Z t Z (t − s)β−1 1 (u − s)α−1 ≤ |f (u, y(u)) − f (u, z(u))|duds Γ(β) Γ(α) 0 s Z 1 (s − ξ)α−1 |f (s, y(s)) − f (s, z(s))|ds +|µ1 (t)| Γ(α) ξ " Z Z η (η − s)β−1 1 (u − s)α−1 +|µ2 (t)| δ |f (u, y(u)) − f (u, z(u))|duds Γ(β) Γ(α) 0 s # Z Z 1 (1 − s)β−1 1 (u − s)α−1 |f (u, y(u)) − f (u, z(u))|duds + Γ(β) Γ(α) s 0 ≤ LΛky − zk, which, in view of the given condition LΛ < 1, implies that G is a contraction. In consequence, it follow by the contraction mapping principle that there exists a unique solution for the problem (1.1) on [0, 1]. This completes the proof. 2 Our next existence result for the problem (1.1) relies on Krasnoselskii’s fixed point theorem. Lemma 3.2 (Krasnoselskii’s fixed point theorem) [15]. Let S be a closed, bounded, convex and nonempty subset of a Banach space X. Let Y1 , Y2 be the operators mapping S into X such that (a) Y1 s1 + Y2 s2 ∈ S whenever s1 , s2 ∈ S; (b) Y1 is compact and continuous; (c) Y2 is a contraction mapping. Then there exists s3 ∈ S such that s3 = Y 1 s3 + Y 2 s3 . Theorem 3.3 Let f : [0, 1] × R → R be a continuous function satisfying the condition (H1 ). In addition we assume that: (H2 ) |f (t, y)| ≤ m(t), for all (t, y) ∈ [0, 1] × R and m ∈ C([0, 1], R+ ). 822

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Then there exists at least one solution for the problem (1.1) on [0, 1] provided that ( L sup t∈[0,1]

β

α

t (1 − ξ) + |µ1 (t)| Γ(α + 1)Γ(β + 1) Γ(α + 1)

) < 1.

(3.5)

Proof. Setting supt∈[0,1] |m(t)| = kmk, we fix % ≥ kmkΛ,

(3.6)

where Λ is defined by (2.11), and consider B% = {y ∈ X : kyk ≤ %}. Introduce the operators G1 and G2 on B% as follows: Z (t − s)β−1 1 (u − s)α−1 G1 y(t) = f (u, y(u))duds Γ(β) Γ(α) 0 s Z 1 (s − ξ)α−1 +µ1 (t) f (s, y(s))ds, Γ(α) ξ Z

t

and " Z G2 y(t) = µ2 (t) δ Z + 0

1

η

Z (η − s)β−1 1 (u − s)α−1 f (u, y(u))duds Γ(β) Γ(α) 0 s # Z (1 − s)β−1 1 (u − s)α−1 f (u, y(u))duds . Γ(β) Γ(α) s

Observe that G = G1 + G2 . Now we verify the hypotheses of Krasnoselskii’s fixed point theorem in the following steps. (i) For y, z ∈ B% , we have kG1 y + G2 zk = sup |(G1 y)(t) + (G2 z)(t)| t∈[0,1]

(Z ≤ kmk sup t∈[0,1]

" Z +|µ2 (t)| δ Z + 0

1

0 η

t

(t − s)β−1 Γ(β)

Z s

1

(u − s)α−1 duds + |µ1 (t)| Γ(α)

Z ξ

1

(s − ξ)α−1 ds Γ(α)

(η − s)β−1 1 (u − s)α−1 duds Γ(β) Γ(α) 0 s #) Z (1 − s)β−1 1 (u − s)α−1 duds Γ(β) Γ(α) s Z

≤ kmkΛ ≤ %, where we used (3.6). Thus G1 y + G2 z ∈ B% . 823

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(ii) We show that G1 is a contraction. Indeed, by using the assumption (H1 ) together with (3.5) and the fact that (1 − s)α < 1, (1 < α ≤ 2) we have (Z

t

|G1 y(t) − G1 z(t)| ≤ Lky − zk 0

(t − s)β−1 Γ(β)

Z s

1

(u − s)α−1 duds Γ(α)

)

1

(s − ξ)α−1 ds Γ(α) ξ (Z ) Z 1 t (s − ξ)α−1 (t − s)β−1 ≤ Lky − zk ds + |µ1 (t)| ds Γ(α) ξ 0 Γ(β)Γ(α + 1) ( ) tβ (1 − ξ)α ≤ L sup + |µ1 (t)| ky − zk, Γ(α + 1)Γ(β + 1) Γ(α + 1) t∈[0,1] Z

+|µ1 (t)|

which implies that ( kG1 y − G1 zk ≤ L sup t∈[0,1]

) tβ (1 − ξ)α + |µ1 (t)| ky − zk. Γ(α + 1)Γ(β + 1) Γ(α + 1)

Hence G1 is a contraction by (3.5). (iii) Using the continuity of f, it is easy to show that the operator G2 is continuous. Further, G2 is uniformly bounded on B% as kG2 xk =

sup |(G2 y)(t)| ≤ t∈[0,1]

kmkM2 (δη β + 1) , Γ(α + 1)Γ(β + 1)

M2 = sup |µ2 (t)|. t∈[0,1]

In order to establish that G2 is compact, we define sup(t,y)∈[0,1]×B% |f (t, y)| = f . Thus, for 0 < t1 < t2 < 1, we have " Z Z η (η − s)β−1 1 (u − s)α−1 |(G2 y)(t2 ) − (G2 y)(t1 )| ≤ |µ2 (t2 ) − µ2 (t1 )|f δ duds Γ(β) Γ(α) 0 s # Z 1 Z (1 − s)β−1 1 (u − s)α−1 + duds Γ(β) Γ(α) 0 s δη β + 1 ≤ |µ2 (t2 ) − µ2 (t1 )|f → 0 as t1 → t2 , Γ(α + 1)Γ(β + 1) independent of y. This shows that G2 is relatively compact on B% . As all the conditions of the Arzel´a-Ascoli theorem are satisfied, so G2 is compact on B% . In view of steps (i)-(iii), the conclusion of Krasnoselskii’s fixed point theorem applies and hence there exists at least one solution for the problem (1.1) on [0, 1]. The proof is completed. 2 824

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Remark 3.4 Interchanging the roles of the operators G1 and G2 in the foregoing result, we can obtain a second result by requiring the condition: LM1

δη β + 1 < 1, Γ(α + 1)Γ(β + 1)

M1 = sup |µ1 (t)|, t∈[0,1]

instead of (3.5). The following existence result is based on Leray-Schauder nonlinear alternative. Lemma 3.5 (Nonlinear alternative for single valued maps)[16]. Let E be a Banach space, C a closed, convex subset of E, U an open subset of C and 0 ∈ U. Suppose that F : U → C is a continuous, compact (that is, F (U ) is a relatively compact subset of C) map. Then either (i) F has a fixed point in U , or (ii) there is a u ∈ ∂U (the boundary of U in C) and λ ∈ (0, 1) with u = λF (u). Theorem 3.6 Let f : [0, 1] × R → R be a continuous function. Assume that (H3 ) There exist a function g ∈ C([0, 1], R+ ), and a nondecreasing function ψ : R+ → R+ such that |f (t, y)| ≤ g(t)ψ(kyk), ∀(t, y) ∈ [0, 1] × R. (H4 ) There exists a constant K > 0 such that K > 1. kgkψ(K)Λ Then the problem (1.1) has at least one solution on [0, 1]. Proof. Consider the operator G : X → X defined by (3.2). We show that G maps bounded sets into bounded sets in X = C([0, 1], R). For a positive number r, let Br = {y ∈ C([0, 1], R) : kyk ≤ r} be a bounded set in C([0, 1], R). Then, by using the fact that (1 − s)α−1 ≤ 1 (1 < α ≤ 2) we have (Z Z Z 1 t (t − s)β−1 1 (u − s)α−1 (s − ξ)α−1 |Gy(t)| ≤ kgkψ(r) duds + |µ1 (t)| ds Γ(β) Γ(α) Γ(α) 0 s ξ " Z Z η (η − s)β−1 1 (u − s)α−1 +|µ2 (t)| δ duds Γ(β) Γ(α) 0 s #) Z 1 Z (1 − s)β−1 1 (u − s)α−1 + duds Γ(β) Γ(α) 0 s ≤ kgkψ(r)Λ, 825

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which, on taking the norm for t ∈ [0, 1], yields kGyk ≤ kgkψ(r)Λ. Next we show that G maps bounded sets into equicontinuous sets of C([0, 1], R). Let t1 , t2 ∈ [0, 1] with t1 < t2 and y ∈ Br , where Br is a bounded set of C([0, 1], R). Then, using the fact that (1 − s)α−1 ≤ 1 (1 < α ≤ 2) and the computations for G2 in previous theorem, we obtain |Gy(t2 ) − Gy(t1 )| ( Z Z t2 t1 [(t − s)β−1 − (t − s)β−1 ] β−1 (t − s) 2 2 1 ds + ds ≤ kgkψ(r) 0 Γ(β) Γ(β) t1 ) Z 1 (s − ξ)α−1 δη β + 1 +|µ1 (t2 ) − µ1 (t1 )| ds + |µ2 (t2 ) − µ2 (t1 )| Γ(α) Γ(α + 1)Γ(β + 1) ξ ( 2(t2 − t1 )β + tβ2 − tβ1 (1 − ξ)α ≤ kgkψ(r) + |µ1 (t2 ) − µ1 (t1 )| Γ(β + 1) Γ(α + 1) ) δη β + 1 , +|µ2 (t2 ) − µ2 (t1 )| Γ(α + 1)Γ(β + 1) which tends to zero independently of y ∈ Br as t2 − t1 → 0. As G satisfies the above assumptions, therefore it follows by the Arzel´a-Ascoli theorem that G : C([0, 1], R) → C([0, 1], R) is completely continuous. The result will follow from the Leray-Schauder nonlinear alternative once it is shown that the set of all solutions to the equation y = λGy is bounded for λ ∈ [0, 1]. For that, let y be a solution of y = λGy for λ ∈ [0, 1]. Then, for t ∈ [0, 1], we have (Z Z Z 1 t (s − ξ)α−1 (t − s)β−1 1 (u − s)α−1 |y(t)| = |λGy(t)| ≤ duds + |µ1 (t)| ds Γ(β) Γ(α) Γ(α) s ξ 0 " Z Z η (η − s)β−1 1 (u − s)α−1 duds +|µ2 (t)| δ Γ(β) Γ(α) s 0 #) Z 1 Z (1 − s)β−1 1 (u − s)α−1 + duds |g(t)|ψ(kyk) Γ(β) Γ(α) 0 s ≤ kgkψ(kyk)Λ, which implies that kyk ≤ 1. kgkψ(kyk)Λ In view of (H4 ), there is no solution y such that kyk 6= K. Let us set U = {y ∈ X : kyk < K}. 826

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The operator G : U → X is continuous and completely continuous. From the choice of U , there is no u ∈ ∂U such that u = λG(u) for some λ ∈ (0, 1). Consequently, by the nonlinear alternative of Leray-Schauder type [16, Theorem 5.2], we deduce that G has a fixed point u ∈ U which is a solution of the problem (1.1). This completes the proof. 2

4

Existence results for the problem (1.2)

Before presenting the existence results for the problem (1.2), we outline the necessary concepts on multi-valued maps [17], [18]. For a normed space (X, k · k), let Pcl (X) = {Y ∈ P(X) : Y is closed}, Pb (X) = {Y ∈ P(X) : Y is bounded}, Pcp (X) = {Y ∈ P(X) : Y is compact}, and Pcp,c (X) = {Y ∈ P(X) : Y is compact and convex }. A multi-valued map G : X → P(X) is convex (closed) valued if G(x) is convex (closed) for all x ∈ X. The map G is bounded on bounded sets if G(B) = ∪x∈B G(x) is bounded in X for all B ∈ Pb (X) (i.e. supx∈B {sup{|y| : y ∈ G(x)}} < ∞). G is called upper semi-continuous (u.s.c.) on X if for each x0 ∈ X, the set G(x0 ) is a nonempty closed subset of X, and if for each open set N of X containing G(x0 ), there exists an open neighborhood N0 of x0 such that G(N0 ) ⊆ N. G is said to be completely continuous if G(B) is relatively compact for every B ∈ Pb (X). If the multi-valued map G is completely continuous with nonempty compact values, then G is u.s.c. if and only if G has a closed graph, i.e., xn → x∗ , yn → y∗ , yn ∈ G(xn ) imply y∗ ∈ G(x∗ ). G has a fixed point if there is x ∈ X such that x ∈ G(x). The fixed point set of the multivalued operator G will be denoted by Fix G. A multivalued map G : [0, 1] → Pcl (R) is said to be measurable if for every y ∈ R, the function t 7−→ d(y, G(t)) = inf{|y − z| : z ∈ G(t)} is measurable. For each y ∈ X , define the set of selections of F by SF,y := {v ∈ L1 ([0, 1], R) : v(t) ∈ F (t, y(t)) for a.e. t ∈ [0, 1]}. Definition 4.1 A function y ∈ C([0, 1], R) is said to be a solution of the boundary β value problem (1.2) if y(0) = 0, D0+ y(ξ) = 0, y(1) = δy(η), 0 < ξ, η < 1, and there exists a function v ∈ SF,y such that v(t) ∈ F (t, y(t)) and Z 1 (s − ξ)α−1 (t − s)β−1 α I1− v(s)ds + µ1 (t) v(s)ds y(t) = Γ(β) Γ(α) ξ 0 Z 1 h Z η (η − s)β−1 i (1 − s)β−1 α α +µ2 (t) δ I1− v(s)ds − I1− v(s)ds , t ∈ [0, 1]. Γ(β) Γ(β) 0 0 Z

4.1

t

The upper semicontinuous case

In the case when F has convex values we prove an existence result based on nonlinear alternative of Leray-Schauder type. 827

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Definition 4.2 A multivalued map F : [0, 1] × R → P(R) is said to be Carath´eodory if (i) t 7−→ F (t, y) is measurable for each y ∈ R; (ii) y 7−→ F (t, y) is upper semicontinuous for almost all t ∈ [0, 1]. Further a Carath´eodory function F is called L1 −Carath´eodory if (iii) for each ρ > 0, there exists ϕρ ∈ L1 ([0, 1], R+ ) such that kF (t, y)k = sup{|v| : v ∈ F (t, y)} ≤ ϕρ (t) for all y ∈ R with kyk ≤ ρ and for a.e. t ∈ [0, 1]. We define the graph of G to be the set Gr (G) = {(x, y) ∈ X × Y : y ∈ G(x)} and recall two results for closed graphs and upper-semicontinuity. Lemma 4.3 ([17, Proposition 1.2]) If G : X → Pcl (Y ) is u.s.c., then Gr (G) is a closed subset of X × Y ; i.e., for every sequence {xn }n∈N ⊂ X and {yn }n∈N ⊂ Y , if when n → ∞, xn → x∗ , yn → y∗ and yn ∈ G(xn ), then y∗ ∈ G(x∗ ). Conversely, if G is completely continuous and has a closed graph, then it is upper semi-continuous. Lemma 4.4 ([19]) Let X be a separable Banach space. Let F : [0, 1] × X → Pcp,c (X) be an L1 − Carath´eodory multivalued map and let Θ be a linear continuous mapping from L1 ([0, 1], X) to C([0, 1], X). Then the operator Θ ◦ SF,x : C([0, 1], X) → Pcp,c (C([0, 1], X)), y 7→ (Θ ◦ SF,y )(y) = Θ(SF,y ) is a closed graph operator in C([0, 1], X) × C([0, 1], X). For the forthcoming analysis, we need the following lemma. Lemma 4.5 (Nonlinear alternative for Kakutani maps)[16]. Let E be a Banach space, C a closed convex subset of E, U an open subset of C and 0 ∈ U. Suppose that F : U → Pcp,c (C) is a upper semicontinuous compact map. Then either (i) F has a fixed point in U , or (ii) there is a u ∈ ∂U and λ ∈ (0, 1) with u ∈ λF (u). Theorem 4.6 Assume that: (B1 ) F : [0, 1] × R → P(R) is L1 -Carath´eodory and has nonempty compact and convex values; 828

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(B2 ) there exist a function φ ∈ C([0, 1], R+ ), and a nondecreasing function Ω : R+ → R+ such that kF (t, y)kP := sup{|w| : w ∈ F (t, y)} ≤ φ(t)Ω(kyk) for each (t, y) ∈ [0, 1] × R; (B3 ) there exists a constant M > 0 such that M > 1, kφkΛΩ(M ) where Λ is defined by (2.11). Then the boundary value problem (1.2) has at least one solution on [0, 1]. Proof. Define an operator ΩF : X → P(X ) by ΩF (y) = {h ∈ X : h(t) = N (y)(t)} where Z 1 (s − ξ)α−1 (t − s)β−1 α N (y)(t) = I1− v(s)ds + µ1 (t) v(s)ds Γ(β) Γ(α) 0 ξ Z 1 h Z η (η − s)β−1 i (1 − s)β−1 α α +µ2 (t) δ I1− v(s)ds − I1− v(s)ds . Γ(β) Γ(β) 0 0 Z

t

We will show that ΩF satisfies the assumptions of the nonlinear alternative of LeraySchauder type. The proof consists of several steps. As a first step, we show that ΩF is convex for each y ∈ X . This step is obvious since SF,y is convex (F has convex values), and therefore we omit the proof. In the second step, we show that ΩF maps bounded sets (balls) into bounded sets in X . For a positive number ρ, let Bρ = {y ∈ X : kyk ≤ ρ} be a bounded ball in X . Then, for each h ∈ ΩF (y), y ∈ Bρ , there exists v ∈ SF,y such that Z t Z Z 1 (s − ξ)α−1 (t − s)β−1 1 (u − s)α−1 v(u)duds + µ1 (t) v(s)ds h(t) = Γ(β) Γ(α) Γ(α) s ξ 0 " Z Z η (η − s)β−1 1 (u − s)α−1 +µ2 (t) δ v(u)duds Γ(β) Γ(α) 0 s # Z 1 Z (1 − s)β−1 1 (u − s)α−1 + v(u)duds . Γ(β) Γ(α) 0 s Then, by using the fact that (1 − s)α−1 ≤ 1 (1 < α ≤ 2) we have (Z Z Z 1 t (t − s)β−1 1 (u − s)α−1 (s − ξ)α−1 |h(t)| ≤ kgkΩ(r) duds + |µ1 (t)| ds Γ(β) Γ(α) Γ(α) 0 s ξ 829

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B. Ahmad, S. K. Ntouyas, A. Alsaedi " Z +|µ2 (t)| δ Z + 0

1

η

Z (η − s)β−1 1 (u − s)α−1 duds Γ(β) Γ(α) 0 s #) Z (1 − s)β−1 1 (u − s)α−1 duds Γ(β) Γ(α) s

≤ kφkΩ(r)Λ, which, on taking the norm for t ∈ [0, 1]. yields khk ≤ kφkΩ(r)Λ. Now we show that ΩF maps bounded sets into equicontinuous sets of X . Let t1 , t2 ∈ [0, 1] with t1 < t2 and y ∈ Bρ . For each h ∈ ΩF (y), using the fact that (1 − s)α−1 ≤ 1 (1 < α ≤ 2), we obtain |h(t2 ) − h(t1 )| ( Z Z t2 t1 [(t − s)β−1 − (t − s)β−1 ] β−1 (t2 − s) 2 1 ≤ kφkΩ(r) ds + ds 0 Γ(β)Γ(α + 1) t1 Γ(β)Γ(α + 1) ) Z 1 α−1 β (s − ξ) δη + 1 +|µ1 (t2 ) − µ1 (t1 )| ds + |µ2 (t2 ) − µ2 (t1 )| Γ(α) Γ(α + 1)Γ(β + 1) ξ ( 2(t2 − t1 )β + tβ2 − tβ1 (1 − ξ)α ≤ kφkΩ(r) + |µ1 (t2 ) − µ1 (t1 )| Γ(β + 1)Γ(α + 1) Γ(α + 1) ) δη β + 1 +|µ2 (t2 ) − µ2 (t1 )| , Γ(α + 1)Γ(β + 1) which tends to zero independently of y ∈ Br as t2 − t1 → 0. As ΩF satisfies the above assumptions, therefore it follows by the Arzel´a-Ascoli theorem that ΩF : C([0, 1], R) → C([0, 1], R) is completely continuous. In our next step, we show that ΩF is upper semicontinuous. To this end it is sufficient to show that ΩF has a closed graph, by Lemma 4.3. Let yn → y∗ , hn ∈ ΩF (yn ) and hn → h∗ . Then we need to show that h∗ ∈ ΩF (y∗ ). Associated with hn ∈ ΩF (yn ), there exists vn ∈ SF,yn such that for each t ∈ [0, 1], Z 1 Z t (t − s)β−1 α (s − ξ)α−1 hn (t) = I1− vn (s)ds + µ1 (t) vn (s)ds Γ(β) Γ(α) 0 ξ Z 1 h Z η (η − s)β−1 i (1 − s)β−1 α α +µ2 (t) δ I1− vn (s)ds − I1− vn (s)ds . Γ(β) Γ(β) 0 0 Thus it suffices to show that there exists v∗ ∈ SF,y∗ such that for each t ∈ [0, 1], Z t Z 1 (t − s)β−1 α (s − ξ)α−1 h∗ (t) = I1− v∗ (s)ds + µ1 (t) v∗ (s)ds Γ(β) Γ(α) 0 ξ 830

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η

15

(η − s)β−1 α I1− v∗ (s)ds − Γ(β)

Z 0

1

i (1 − s)β−1 α I1− v∗ (s)ds . Γ(β)

Let us consider the linear operator Θ : L1 ([0, 1], R) → X given by Z t Z 1 (t − s)β−1 α (s − ξ)α−1 v 7→ Θ(v)(t) = I1− v(s)ds + µ1 (t) v(s)ds Γ(β) Γ(α) 0 ξ Z 1 i h Z η (η − s)β−1 (1 − s)β−1 α α I1− v(s)ds − I1− v(s)ds . +µ2 (t) δ Γ(β) Γ(β) 0 0 Observe that khn (t) − h∗ (t)k Z 1

Z t (t − s)β−1 (s − ξ)α−1

α = I1− (vn − v∗ )(s)ds + µ1 (t) (vn − v∗ )(s)ds Γ(β) Γ(α) 0 ξ Z 1 h Z η (η − s)β−1 i (1 − s)β−1 α

α +µ2 (t) δ I1− (vn − v∗ )(s)ds − I1− (vn − v∗ )(s)ds → 0, Γ(β) Γ(β) 0 0

as n → ∞. Thus, it follows by Lemma 4.4 that Θ ◦ SF is a closed graph operator. Further, we have hn (t) ∈ Θ(SF,yn ). Since yn → y∗ , therefore, we have Z t Z 1 (t − s)β−1 α (s − ξ)α−1 h∗ (t) = I1− v∗ (s)ds + µ1 (t) v∗ (s)ds Γ(β) Γ(α) 0 ξ Z 1 h Z η (η − s)β−1 i (1 − s)β−1 α α +µ2 (t) δ I1− v∗ (s)ds − I1− v∗ (s)ds . Γ(β) Γ(β) 0 0 Finally, we show there exists an open set U ⊆ X with y ∈ / θΩF (y) for any θ ∈ (0, 1) and all y ∈ ∂U. Let θ ∈ (0, 1) and y ∈ θΩF (y). Then there exists v ∈ L1 ([0, 1], R) with v ∈ SF,y such that, for t ∈ [0, 1], we can obtain |y(t)| = |θΩF (y)(t)| Z Z 1 Z t (s − ξ)α−1 (t − s)β−1 1 (u − s)α−1 |v(u)|duds + |µ1 (t)| |v(s)|ds ≤ Γ(β) Γ(α) Γ(α) s ξ 0 " Z Z η (η − s)β−1 1 (u − s)α−1 +|µ2 (t)| δ |v(u)|duds Γ(β) Γ(α) 0 s # Z 1 Z (1 − s)β−1 1 (u − s)α−1 + |v(u)|duds Γ(β) Γ(α) 0 s ≤ kφkΩ(kyk)Λ, which implies that kyk ≤ 1. kφkΩ(kyk)Λ 831

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In view of (B3 ), there exists M such that kyk 6= M . Let us set U = {y ∈ X : kyk < M }. Note that the operator ΩF : U → P(X ) is upper semicontinuous and completely continuous. From the choice of U , there is no y ∈ ∂U such that y ∈ θΩF (y) for some θ ∈ (0, 1). Consequently, by the nonlinear alternative of Leray-Schauder type (Lemma 4.5), we deduce that ΩF has a fixed point y ∈ U which is a solution of the problem (1.2). This completes the proof. 2

4.2

The Lipschitz case

We prove in this subsection the existence of solutions for the problem (1.2) with a nonconvex valued right-hand side by applying a fixed point theorem for multivalued maps due to Covitz and Nadler [21]. Let (X, d) be a metric space induced from the normed space (X; k·k). Consider Hd : P(X) × P(X) → R ∪ {∞} defined by Hd (A, B) = max{supa∈A d(a, B), supb∈B d(A, b)}, where d(A, b) = inf a∈A d(a; b) and d(a, B) = inf b∈B d(a; b). Then (Pb,cl (X), Hd ) is a metric space and (Pcl (X), Hd ) is a generalized metric space (see [20]). Definition 4.7 A multivalued operator N : X → Pcl (X) is called (a) γ−Lipschitz if and only if there exists γ > 0 such that Hd (N (x), N (y)) ≤ γd(x, y) for each x, y ∈ X and (b) a contraction if and only if it is γ−Lipschitz with γ < 1. Lemma 4.8 ([21]) Let (X, d) be a complete metric space. If N : X → Pcl (X) is a contraction, then F ixN 6= ∅. Theorem 4.9 Assume that: (A1 ) F : [0, 1] × R → Pcp (R) is such that F (·, y(t)) : [0, 1] → Pcp (R) is measurable for each y ∈ R; (A2 ) Hd (F (t, y), F (t, y¯) ≤ q(t)|y − y¯| for almost all t ∈ [0, 1] and y, y¯ ∈ R with q ∈ C([0, 1], R+ ) and d(0, F (t, 0)) ≤ q(t) for almost all t ∈ [0, 1]. Then the problem (1.2) has at least one solution on [0, 1] if kqkΛ < 1,

(4.1)

where Λ is defined by (2.11). Proof. Consider the operator ΩF : X → P(X ) defined in the beginning of the proof of Theorem 4.6. Observe that the set SF,y is nonempty for each y ∈ X by the assumption (A1 ), so F has a measurable selection (see Theorem III.6 [22]). Now we show that the operator ΩF satisfies the assumptions of Lemma 4.8. To show that ΩF (y) ∈ Pcl (X ) 832

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for each y ∈ X , let {un }n≥0 ∈ ΩF (y) be such that un → u (n → ∞) in X . Then u ∈ C([0, 1], R) and there exists vn ∈ SF,y such that, for each t ∈ [0, 1], Z t Z 1 (t − s)β−1 α (s − ξ)α−1 un (t) = I1− vn (s)ds + µ1 (t) vn (s)ds Γ(β) Γ(α) 0 ξ Z 1 i h Z η (η − s)β−1 (1 − s)β−1 α α I1− vn (s)ds − I1− vn (s)ds . +µ2 (t) δ Γ(β) Γ(β) 0 0 As F has compact values, we pass onto a subsequence (if necessary) to obtain that vn converges to v in L1 ([0, 1], R). Thus, v ∈ SF,y and for each t ∈ [0, 1], we have Z t Z 1 (t − s)β−1 α (s − ξ)α−1 un (t) → u(t) = I1− v(s)ds + µ1 (t) v(s)ds Γ(β) Γ(α) 0 ξ Z 1 h Z η (η − s)β−1 i (1 − s)β−1 α α +µ2 (t) δ I1− v(s)ds − I1− v(s)ds . Γ(β) Γ(β) 0 0 Hence, u ∈ ΩF (y). Next we show that there exists θˆ := kqkΛ < 1 such that ˆ − y¯k for each y, y¯ ∈ X . Hd (ΩF (y), ΩF (¯ y )) ≤ θky Let y, y¯ ∈ X and h1 ∈ ΩF (y). Then there exists v1 (t) ∈ F (t, y(t)) such that, for each t ∈ [0, 1], Z 1 Z t (s − ξ)α−1 (t − s)β−1 α I1− v1 (s)ds + µ1 (t) v1 (s)ds h1 (t) = Γ(β) Γ(α) ξ 0 Z 1 h Z η (η − s)β−1 i (1 − s)β−1 α α +µ2 (t) δ I1− v1 (s)ds − I1− v1 (s)ds . Γ(β) Γ(β) 0 0 By (A2 ), we have Hd (F (t, y), F (t, y¯) ≤ q(t)|y − y¯|. So, there exists w ∈ F (t, y¯) such that |v1 (t) − w| ≤ q(t)|y(t) − y¯(t)|, t ∈ [0, 1]. Define U : [0, 1] → P(R) by U (t) = {w ∈ R : |v1 (t) − w| ≤ q(t)|y(t) − y¯(t)|}. Since the multivalued operator U (t) ∩ F (t, y¯) is measurable (Proposition III.4 [22]), there exists a function v2 (t) which is a measurable selection for U (t) ∩ F (t, y¯). So v2 (t) ∈ F (t, y¯) and for each t ∈ [0, 1], we have |v1 (t) − v2 (t)| ≤ q(t)|y(t) − y¯(t)|. For each t ∈ [0, 1], let us define Z t Z 1 (t − s)β−1 α (s − ξ)α−1 h2 (t) = I1− v2 (s)ds + µ1 (t) v2 (s)ds Γ(β) Γ(α) 0 ξ 833

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B. Ahmad, S. K. Ntouyas, A. Alsaedi h Z +µ2 (t) δ 0

η

(η − s)β−1 α I1− v2 (s)ds − Γ(β)

Z 0

1

i (1 − s)β−1 α I1− v2 (s)ds . Γ(β)

Thus |h1 (t) − h2 (t)| Z 1 Z t (s − ξ)α−1 (t − s)β−1 α I1− |v1 − v2 |(s)ds + |µ1 (t)| |v1 − v2 |(s)ds ≤ Γ(β) Γ(α) ξ 0 Z 1 h Z η (η − s)β−1 i (1 − s)β−1 α α +|µ2 (t)| δ I1− |v1 − v2 |(s)ds + I1− |v1 − v2 |(s)ds Γ(β) Γ(β) 0 0 ≤ kqkΛky − y¯k, which yields kh1 − h2 k ≤ kqkΛky − y¯k. Analogously, interchanging the roles of y and y, we can obtain Hd (ΩF (y), ΩF (¯ y )) ≤ kqkΛky − y¯k. By the condition (4.1), it follows that ΩF is a contraction and hence it has a fixed point y by Lemma 4.8, which is a solution of the problem (1.2). This completes the proof.2

5

Examples

(a) We construct examples for the illustration of the results obtained in Section 3. For that, we consider the following problem:  7/4 3/4  D1− D0+ y(t) = f (t, y(t)), t ∈ J := [0, 1], (5.1)  y(0) = 0, D3/4 y(ξ) = 0, y(1) = (5/2)y(2/3), 0+ Here α = 7/4, β = 3/4, ξ = 1/3, η = 2/3, δ = 5/2, and f (t, y) =

2

√

1 t2 + 81



cos y +

|y|  e−2t + . 1 + |y| t+4

(5.2)

With the given data, it is found that P = [(1 − δη β+1 ) − (β + 1)ξ(1 − δη β )] ≈ 0.262961 6= 0, n tβ (1 − ξ)α o sup + |µ1 (t)| ≈ 1.454491, Γ(α + 1) t∈[0,1] Γ(α + 1)Γ(β + 1) and Λ ≈ 4.503584 (Λ is given by (2.11)). Furthermore, |f (t, y1 ) − f (t, y2 )| ≤ L|y1 − y2 | with L = 1/9 so that LΛ ≈ 0.0.500398 < 1. Clearly the hypothesis of Theorem 3.1 is satisfied and hence the problem (5.1) has a unique solution by the conclusion of Theorem 3.1. 834

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In order to illustrate Theorem 3.3, we notice that (3.5) is satisfied as L

n

tβ (1 − ξ)α o + |µ1 (t)| ≈ 0.161610 < 1, Γ(α + 1)Γ(β + 1) Γ(α + 1)

and

1 e−2t |f (t, y)| ≤ m(t) = √ + . t2 + 81 t + 4 As all the assumptions of Theorem 3.3 hold true, we deduce from the conclusion of Theorem 3.3 that the problem (5.1) has at least one solution on [0, 1]. Now we demonstrate the application of Theorem 3.6 by considering the nonlinear function 1 2 e−t  . (5.3) y + tan−1 y + f (t, y) = √ π 10 t + 36 −t

Clearly |f (t, y)| ≤ g(t)ψ(kyk), where g(t) = √et+36 , ψ(kyk = ( 11 + kyk). By the con10 dition (H4 ), we find that K > 3.310535. Thus all the conditions of Theorem 3.6 are satisfied and consequently, the problem (5.1) with f (t, y) given by (5.3) has has at least one solution on [0, 1]. (b) Here we illustrate the results obtained in Section 4. Let us consider the following fractional differential inclusion involving both left Caputo and right Riemann-Liouville types fractional derivatives equipped with fractional boundary conditions:  7/4 3/4  D1− D0+ y(t) ∈ F (t, y(t)), t ∈ J := [0, 1], (5.4)  y(0) = 0, D3/4 y(ξ) = 0, y(1) = (5/2)y(2/3), 0+ In order to illustrate Theorem 4.6, we take # "  |y(t)| 1  e−t  1 + |y(t)| + , sin y(t) + . F (t, y(t)) = √ 2 9+t 80 t2 + 49 2(1 + |y(t)|)

(5.5)

Clearly |F (t, y(t))| ≤ φ(t)Ω(kyk), where φ(t) = √t21+49 and Ω(kyk) = kyk + 1. Using the condition (B3 ), we find that M > 1.804018. As the hypothesis of Theorem 4.6 is satisfied, the problem (5.4) with F (t, y(t)) given by (5.5) has at least one solution on [0, 1]. Now we illustrate Theorem 4.9 by considering " # 1 sin x(t) 1 F (t, x(t)) = √ , + . (5.6) 50 100 + t2 (6 + t) Obviously q(t) = 1(6+t) with kqk = 1/6 and d(0, F (t, 0)) ≤ q(t) for almost all t ∈ [0, 1]. Moreover, kqkΛ ≈ 0.750597. Thus all the assumptions of Theorem 4.9 hold true and consequently its conclusion applies to the problem (5.4) with F (t, y(t)) given by (5.6). 835

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References [1] S.G. Samko, A.A. Kilbas, O.I. Marichev Fractional Integrals and Derivatives, Theory and Applications, Gordon and Breach, Linghorne, PA 1993. [2] I. Podlubny, Fractional Differential Equations, Academic Press, San Diego, CA 1999. [3] A. Kilbas, M.H. Srivastava, J.J. Trujillo, Theory and Application of Fractional Differential Equations, North Holland Mathematics Studies, vol. 204, 2006. [4] F. Mainardi, Fractional relaxation-oscillation and fractional diffusion-wave phenomena, Chaos, Solitons and Fractals 7 (1996), 1461-1447. [5] A.A. Kilbas, M. Rivero, J.J. Trujillo, Existence and uniqueness theorems for differential equations of fractional order in weighted spaces of continuous functions, Frac. Calc. Appl. Anal. 6 (2003), 363-400. [6] M.F. Silva, J.A.T. Machado, A.M. Lopes, Modelling and simulation of artificial locomotion systems, Robotica 23 (2005), 595-606. [7] T.M. Atanackovic, B. Stankovic, On a differential equation with left and right fractional derivatives, Fract. Calc. Appl. Anal. 10 (2007), 139-150. [8] T. Abdeljawad (Maraaba), D. Baleanu, F. Jarad, Existence and uniqueness theorem for a class of delay differential equations with left and right Caputo fractional derivatives, J. Math. Phys. 49 (2008), 083507. [9] D. Baleanu, J.J. Trujillo, On exact solutions of a class of fractional Euler-Lagrange equations, Nonlinear Dynamics 52 (2008), 331-335. [10] L. Zhang, B. Ahmad, G. Wang, The existence of an extremal solution to a nonlinear system with the right-handed Riemann-Liouville fractional derivative, Appl. Math. Lett. 31 (2014), 1-6. [11] R. Khaldi, A. Guezane-Lakoud, Higher order fractional boundary value problems for mixed type derivatives, J. Nonlinear Funct. Anal. (2017), Article ID 30. [12] A. Guezane Lakoud, R. Khaldi, A. Klcman, Existence of solutions for a mixed fractional boundary value problem, Adv. Difference Equ. (2017) 2017:164. [13] B. Ahmad, S.K. Ntouyas, A. Alsaedi, Existence theory for nonlocal boundary value problems involving mixed fractional derivatives, Nonlinear Anal. Model. Control 24 (2019), 1-21. 836

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BVP for mixed fractional derivatives

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[14] A.A. Kilbas, H.M. Srivastava, J.J. Trujillo, Theory and Applications of Fractional Differential Equations. North-Holland Mathematics Studies, 204. Elsevier Science B.V., Amsterdam, 2006. [15] M.A. Krasnoselskii, Two remarks on the method of successive approximations, Uspekhi Mat. Nauk. 10 (1955), 123-127. [16] A. Granas, J. Dugundji, Fixed Point Theory, Springer-Verlag, New York, 2005. [17] K. Deimling, Multivalued Differential Equations, Walter De Gruyter, Berlin-New York, 1992. [18] Sh. Hu, N. Papageorgiou, Handbook of Multivalued Analysis, Theory I, Kluwer, Dordrecht, 1997. [19] A. Lasota, Z. Opial, An application of the Kakutani-Ky Fan theorem in the theory of ordinary differential equations, Bull. Acad. Polon. Sci. Ser. Sci. Math. Astronom. Phys. 13 (1965), 781-786. [20] M. Kisielewicz, Differential Inclusions and Optimal Control, Kluwer, Dordrecht, The Netherlands, 1991. [21] H. Covitz, S.B. Nadler Jr., Multivalued contraction mappings in generalized metric spaces, Israel J. Math. 8 (1970), 5-11. [22] C. Castaing, M. Valadier, Convex Analysis and Measurable Multifunctions, Lecture Notes in Mathematics 580, Springer-Verlag, Berlin-Heidelberg-New York, 1977.

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L2-primitive process for retarded stochastic neutral functional differential equations in Hilbert spaces Yong Han Kanga , Jin-Mun Jeongb and Seong Ho Choc a Institute

of Liberal Education, Catholic University of Daegu Kyeongsan 680-749, Korea E-mail: [email protected]

b,c Department

of Applied Mathematics, Pukyong National University Busan 608-737, Korea

E-mail:[email protected](CA), [email protected]

Abstract In this paper, we study the existence of solutions and L2 -primitive process for retarded stochastic neutral functional differential equations in Hilbert spaces. We no longer require the Azera-Ascoli theorem to prove the existence of continuous solutions of nonlinear differential systems, but instead we apply the regularity results of general linear differential equations to the case the L2 -primitive process for retarded stochastic neutral functional differential systems with unbounded principal operators, delay terms and local Lipschitz continuity of the nonlinear term. Finally, we give a simple example to which our main result can be applied. Keywords: stochastic neutral differential equations, retarded system, L2 primitive process, analytic semigroup, fractional power AMS Classification 34K50, 93E03, 60H15 This research was supported by Basic Science Research Program through the National Research Foundation of Korea(NRF) funded by the Ministry of Education(NRF-2014R1A6A3A01009038).

1

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2

1

Introduction

In this paper, we study the existence of solutions and L2 -primitive process for the following retarded stochastic neutral functional differential equations in Hilbert spaces: ( R0 d[x(t) + g(t, xt )] = [Ax(t) + −h a1 (s)A1 x(t + s)ds + k(t)]dt + f (t, xt )dW (t), x(0) = φ0 ∈ L2 (Ω, H), x(s) = φ1 (s), s ∈ [−h, 0). (1.1) where t > 0, h > 0, a1 (·) is H¨older continuous, k is a forcing term, W (t) stands for K-valued Brownian motion or Winner process with a finite trace nuclear covariance operator Q, and g, f , are given functions satisfying some assumptions. Moreover, A : D(A) ⊂ H → H is unbounded and A1 ∈ B(H), where B(X, Y ) is the collection of all bounded linear operators from X into Y , and B(X, X) is simply written as B(X). This kind of systems arises in many practical mathematical models, such as, population dynamics, physical, biological and engineering problems, etc. (see [6, 11, 23]). Many authors have studied for the theory of stochastic differential equations in a variety of ways (see [4] [7] and reference therein), impulsive stochastic neutral differential equations [14, 21], approximate controllability of stochastic equations [5, 27, 26]. As for the retarded differential equations, Jeong et al [17, 18], Wang [32], and Sukavanam et al. [28] have discussed the regularity of solutions and controllability of the semilinear retarded systems, and see [8, 15, 16, 24] and references therein for the linear retarded systems. In [10, 12, 13], the authors have discussed the existence of solutions for mild solutions for the neutral differential systems with state-dependence delay. Most studies about the neutral initial value problems governed by retarded semilinear parabolic equation have been devoted to the control problems. Recently, second order neutral impulsive integrodifferential systems have been studied in [2, 25], and Stochastic differential systems with impulsive conditions in [1, 3, 29]. Further, as for impulsive neutral stochastic differential inclusions with nonlocal initial conditions have been studied for the existence results by Lin and Hu [22], and controllability results by [19]. Let (Ω, F, P ) be a complete probability space furnished with complete family of right continuous increasing sub σ-algebras {Ft , t ∈ I} satisfying Ft ⊂ F. An H valued random variables is an F -measurable function x(t) : Ω → H. Usually we suppress the dependence on w ∈ Ω in the stochastic process S = {x(t, w) : Ω → H :

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3 t ∈ [0, T ]}and write x(t) instead of x(t, w) and x(t) : [0, T ] → H in the space of S. Then we have to study on results in connection with solutions of random differential and integral equations in Hilbert spaces. It should be ensured that x(t, w) is a H-valued random variable with finite second moments and L2 -primitive process of (1.1) for all t ∈ T in order to study stationary random function, Brownian motion, Markov process, and etc. But the papers treating the regularity for second moments of the systems and L2 -primitive process for retarded stochastic neutral functional differential equations in Hilbert spaces are not many. In this paper, we propose a different approach of the earlier works used AzeraAscoli theorem to prove the existence of the mild solutions of functional differential systems in the Banach space of all continuous functions. Our approach is that regularity results of general differential equations results of the linear cases of Di Blasio et al. [8] and semilinear cases of [17] remain valid under the above formulation of the stochastic neutral differential system (1.1) even though the system (1.1) contains unbounded principal operators, delay term and local Lipschitz continuity of the nonlinear term. The paper is organized as follows. In Section 2, we construct the strict solution of the semilinear functional differential equations and introduce basic properties. In Section 3, by using properties of the strict solutions in dealt in Section 2, we will obtain the L2 -primitive process of (1.1), and a variation of constant formula of L2 primitive process of (1.1) on the solution space. Finally, we give a simple example to which our main result can be applied.

2

Preliminaries and Lemmas

The inner product and norm in H are denoted by (·, ·) and | · |, respectively. V is another Hilbert space densely and continuously embedded in H. The notations || · || and || · ||∗ denote the norms of V and V ∗ as usual, respectively. For brevity we may regard that ||u||∗ ≤ |u| ≤ ||u||, u ∈ V. (2.1) Let a(·, ·) be a bounded sesquilinear form defined in V × V and satisfying G˚ arding’s inequality Re a(u, u) ≥ c0 ||u||2 − c1 |u|2 , c0 > 0, c1 ≥ 0. (2.2) Let A be the operator associated with the sesquilinear form −a(·, ·): ((c1 − A)u, v) = −a(u, v),

u, v ∈ V.

It follows from (2.2) that for every u ∈ V Re (Au, u) ≥ c0 ||u||2 .

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4 Then A is a bounded linear operator from V to V ∗ according to the Lax-Milgram theorem, and its realization in H which is the restriction of A to D(A) = {u ∈ V ; Au ∈ H} is also denoted by A. Then A generates an analytic semigroup S(t) = etA in both H and V ∗ as in Theorem 3.6.1 of [30]. Moreover, there exists a constant C0 such that 1/2 ||u|| ≤ C0 ||u||D(A) |u|1/2 , (2.3) for every u ∈ D(A), where ||u||D(A) = (|Au|2 + |u|2 )1/2 is the graph norm of D(A). Thus we have the following sequence D(A) ⊂ V ⊂ H ⊂ V ∗ ⊂ D(A)∗ , where each space is dense in the next one and continuous injection. Lemma 2.1. With the notations (2.3), (2.4), we have (V, V ∗ )1/2,2 = H, (D(A), H)1/2,2 = V, where (V, V ∗ )1/2,2 denotes the real interpolation space between V and V ∗ (Section 1.3.3 of [31]). If X is a Banach space and 1 < p < ∞, Lp (0, T ; X) is the collection of all strongly measurable functions from (0, T ) into X the p-th powers of norms are integrable. For the sake of simplicity we assume that the semigroup S(t) generated by A is uniformly bounded, that is, There exists a constant M0 such that ||S(t)||B(H) ≤ M0 ,

||AS(t)||B(H) ≤

M0 . t

(2.4)

The following lemma is from [30, Lemma 3.6.2]. Lemma 2.2. There exists a constant M0 such that the following inequalities hold: ||S(t)||B(H,V ) ≤ t−1/2 M0 , ||S(t)||B(V ∗ ,V ) ≤ t−1 M0 ,

(2.5) (2.6)

||AS(t)||B(H,V ) ≤ t−3/2 M0 .

(2.7)

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5 First, consider the following initial value problem for the abstract linear parabolic equation ( R0 dx(t) = Ax(t) + −h a1 (s)A1 x(t + s)ds + k(t), 0 < t ≤ T, dt (2.8) x(0) = φ0 , x(s) = φ1 (s) s ∈ [−h, 0). By virtue of Theorem 2.1 of [15] or [8], we have the following result on the corresponding linear equation of (2.8). Lemma 2.3. 1) For (φ0 , φ1 ) ∈ V × L2 (−h, 0; D(A)) and k ∈ L2 (0, T ; H), T > 0, there exists a unique solution x of (2.8) belonging to L2 (0, T ; D(A)) ∩ W 1,2 (0, T ; H) ⊂ C([0, T ]; V ) and satisfying ||x||L2 (0,T ;D(A))∩W 1,2 (0,T ;H) ≤ C1 (||φ0 || + ||φ1 ||L2 (−h,0;D(A)) + ||k||L2 (0,T ;H) ),

(2.9)

where C1 is a constant depending on T and ||x||L2 (0,T ;D(A))∩W 1,2 (0,T ;H) = max{||x||L2 (0,T ;D(A)) , ||x||W 1,2 (0,T ;H) } (2) Let (φ0 , φ1 ) ∈ H × L2 (−h, 0; V ) and k ∈ L2 (0, T ; V ∗ ), T > 0. Then there exists a unique solution x of (2.8) belonging to L2 (0, T ; V ) ∩ W 1,2 (0, T ; V ∗ ) ⊂ C([0, T ]; H) and satisfying ||x||L2 (0,T ;V )∩W 1,2 (0,T ;V ∗ ) ≤ C1 (|φ0 | + ||φ1 ||L2 (−h,0;V ) + ||k||L2 (0,T ;V ∗ ) ),

(2.10)

where C1 is a constant depending on T . Let the solution spaces W0 (T ) and W1 (T ) of strong solutions be defined by W0 (T ) = L2 (0, T ; D(A)) ∩ W 1,2 (0, T ; H), W1 (T ) = L2 (0, T ; V ) ∩ W 1,2 (0, T ; V ∗ ). Here, we note that by using interpolation theory, we have W0 (T ) ⊂ C([0, T ]; V ),

W1 (T ) ⊂ C([0, T ]; H).

Thus, there exists a constant c1 > 0 such that ||x||C([0,T ];V ) ≤ c1 ||x||W0 (T ) ,

||x||C([0,T ];H) ≤ c1 ||x||W1 (T ) .

842

(2.11)

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6 Lemma 2.4. Suppose that k ∈ L2 (0, T ; H) and x(t) = t ≤ T . Then there exists a constant C2 such that

and

Rt 0

S(t − s)k(s)ds for 0 ≤

||x||L2 (0,T ;D(A)) ≤ C1 ||k||L2 (0,T ;H) , ||x||L2 (0,T ;H) ≤ C2 T ||k||L2 (0,T ;H) ,

(2.12)

√ ||x||L2 (0,T ;V ) ≤ C2 T ||k||L2 (0,T ;H) .

(2.13)

Proof. The first assertion is immediately obtained by (2.9). Since Z T Z t 2 S(t − s)k(s)ds|2 dt | ||x||L2 (0,T ;H) = 0 0 Z T Z t ≤ M0 ( |k(s)|ds)2 dt 0 0 Z T Z t t |k(s)|2 dsdt ≤ M0 0 0 2 Z T T ≤ M0 |k(s)|2 ds, 2 0 it follows that ||x||L2 (0,T ;H) ≤ T

p

M0 /2||k||L2 (0,T ;H) .

(2.14)

From (2.3), (2.12), and (2.14) it holds that p ||x||L2 (0,T ;V ) ≤ C0 C1 T (M0 /2)1/4 ||k||L2 (0,T ;H) . So, if we take a constant C2 > 0 such that p p C2 = max{ M0 /2, C0 C1 (M0 /2)1/4 }, the proof is complete. In what follows in this section, we assume c1 = 0 in (2.2) without any loss of generality. So we have that 0 ∈ ρ(A) and the closed half plane {λ : Re λ ≥ 0} is contained in the resolvent set of A. In this case, it is possible to define the fractional power Aα for α > 0. The subspace D(Aα ) is dense in H and the expression ||x||α = ||Aα x||,

x ∈ D(Aα )

defines a norm on D(Aα ). It is also well known that Aα is a closed operator with its domain dense and D(Aα ) ⊃ D(Aβ ) for 0 < α < β. Due to the well known fact that A−α is a bounded operator, we can assume that there is a constant C−α > 0 such that ||A−α ||L(H) ≤ C−α , ||A−α ||L(V ∗ ,V ) ≤ C−α . (2.15)

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7 Lemma 2.5. For any T > 0, there exists a positive constant Cα such that the following inequalities hold for all t > 0: ||Aα S(t)||L(H) ≤

Cα , tα

||Aα S(t)||L(H,V ) ≤

Cα 3α/2 t

.

(2.16)

Proof. The relation is from the inequalities (2.6) and (2.7) by properties of fractional power of A and the definition of S(t).

3

Existence of solutions

In this paper (H, | · |) and (K, | · |K ) denote real separable Hilbert spaces. Consider the following retarded semilinear impulsive neutral differential system in Hilbert space H: ( R0 d[x(t) + g(t, xt )] = [Ax(t) + −h a1 (s)A1 x(t + s)ds + k(t)]dt + f (t, xt )dW (t), x(0) = φ0 ∈ L2 (Ω, H), x(s) = φ1 (s), s ∈ [−h, 0). (3.1) Let (Ω, F, P ) be a complete probability space furnished with complete family of right continuous increasing sub σ-algebras {Ft , t ∈ I} satisfying Ft ⊂ F. An H valued random variables is an F -measurable function x(t) : Ω → H and the collection of random variables S = {x(t, w) : Ω → H : t ∈ [0, T ], w ∈ Ω} is a stochastic process. Generally, we just write x(t) instead of x(t, w) and x(t) : [0, T ] → H in the space of S Let {en }∞ n=1 be a complete orthonormal basis of K, and P∞ let√Q ∈ B(K, K) be an operator defined by Qen = λn en with finite Tr(Q) = n=1 λn = λ < ∞ (Tr denotes the trace of the operator), where λn ≥ 0(n = 1, 2, · · · ), and B(K, K) denotes the space of all bounded linea operators from K into K. {W (t) : t ≥ 0} be a cylindrical K-valued Wiener process with a finite trace nuclear covariance operator Q over (Ω, F, P ), which satisfies that W (t) =

∞ p X

λn wi (t)en ,

t ≥ 0,

n=1

where {wi (t)}∞ i=1 be mutually independent one dimensional standard Wiener processes over (Ω, F, P ). Then the above K-valued stochastic process W (t) is called a Q-Wiener process.

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8 We assume that Ft = σ{W (s) : 0 ≤ s ≤ t} is the σ-algebra generated by w and FT = F. Let ψ ∈ B(K, H) and define |ψ|2Q = Tr(ψQψ ∗ ) =

∞ p X | λn ψen |2 . n=1

If |ψ|2Q < ∞, then ψ is called a Q-Hilbert-Schmidt operator. BQ (K, H) stands for the space of all Q-Hilbert-Schmidt operators. The completion BQ (K, H) of B(K, H) with respect to the topology induced by the norm |ψ|Q , where |ψ|2Q = (ψ, ψ) is a Hilbert space with the above norm topology. Let V be a dense subspace of H as mentioned in Section 2. For T > 0 we define Z T 2 ||x(s)||2 ds) < ∞} M (−h, T ; V ) = {x : [−h, T ] → V : E( −h

with norm defined by ||x||

M 2 (0,T ;V

)

Z  = E(

T

||x(s)||2 ds)

1/2

.

−h

The spaces M 2 (−h, 0; V ), M 2 (0, T ; V ), and M 2 (0, T ; V ∗ ) are also defined as the same way and the basic theory of the class of all nonanticipative functions can be founded in [9]. For h > 0, we assume that φ1 : [−h, 0) → V is a given initial value satisfying Z 0 E( ||φ1 (s)||2 ds) < ∞, −h

that is, φ1 ∈ M 2 (−h, 0; V ). In this note, a random variable x(t) : Ω → H will be called an L2 -primitive process if x ∈ M 2 (−h, T ; V ). For each s ∈ [0, T ], we define xs : [−h, 0] → H as xs (r) = x(s + r),

−h ≤ r ≤ 0.

We will set Π = M 2 (−h, 0; V ). Definition 3.1. A stochastic process x : [−h, T ] × Ω → H is called a solution of (3.1) if (i) x(t) is measurable and Ft -adapted for each t ≥ 0.

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9 (ii) x(t) ∈ H has c´adl´ag paths on t ∈ (0, T ) such that Z

0

t

AS(t − s)g(s, xs )ds (3.2) x(t) =S(t)[φ + g(0, x0 )] − g(t, xt ) + 0 Z t Z  0 S(t − s) a1 (τ )A1 x(s + τ )dτ ds + f (s, xs )dW (s)} + 0 −h Z t S(t − s)k(s)ds, + 0 0

x(0) =φ ,

x(s) = φ1 (s),

(iii) x ∈ M 2 (0, T ; V ) i.e., E(

RT 0

s ∈ [−h, 0).

||x(s)||2 ds) < ∞ and C([0, T ]; H).

To establish our results, we introduce the following assumptions on system (3.1). Assumption (A). We assume that a1 (·) is H¨older continuous of order ρ: |a1 (0)| ≤ H1 ,

|a1 (s) − a1 (τ )| ≤ H1 (s − τ )ρ .

Assumption (G). Let g : [0, T ] × Π → H be a nonlinear mapping satisfying the following conditions hold: (i) For any x ∈ Π, the mapping g(·, x) is strongly measurable. (ii) There exist positive constants Lg and β > 2/3 such that E|Aβ g(t, x)|2 ≤ Lg (||x||Π + 1)2 , E|Aβ g(t, x) − Aβ g(t, xˆ)|2 ≤ Lg ||x − xˆ||2Π , for all t ∈ [0, T ], and x, xˆ ∈ Π. Assumption (F). Let f : R×Π → B(K, H) be a nonlinear mapping satisfying the following: (i) For any x ∈ Π, the mapping f (·, x) is strongly measurable. (ii) There exists a function Lf : R+ → R such that E|f (t, x) − f (t, y)|2 ≤ Lf (r)||x − y||2Π ,

t ∈ [0, T ]

hold for ||x||Π ≤ r and ||y||Π ≤ r.

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10 (iii) The inequality E|f (t, x)|2 ≤ Lf (r)(||x||Π + 1)2 holds for every t ∈ [0, T ] and ||x||Π ≤ r. Lemma 3.1. Let x ∈ M 2 (−h, T ; V ). Then the mapping s 7→ xs belongs to C([0, T ]; Π), and for each 0 < t ≤ T ||xt ||Π ≤ ||x||M 2 (−h,t;V ) = ||φ1 ||Π + ||x||M 2 (0,t;V ) , E(||x||2L2 (0,t;V ) ) = ||x||2M 2 (0,t;V ) , √ ||x· ||L2 (0,t;Π) ≤ t||x||M 2 (−h,t;V ) .

(3.3)

Proof. The first paragraph is easy to verify. In fact, it is from the following inequality; Z Z 0 
 t 2 2 ||x(τ )||2 dτ ≤ ||x||2M 2 (−h,t;V ) , t > 0. ||x(t + τ )|| dτ ≤ E ||xt ||Π = E −h

−h

The second paragraph is immediately obtained by definition. From the inequality (3.3), we have Z

t

||xs ||2Π ds

Z

t

 E

=

0

0

Z =

t

 E

Z

0

||x(s + τ )||2 dτ

−h Z s

0

||x(τ )||2 dτ


2


2

ds

ds ≤ t||x||2M 2 (−h,t;V ) ,

s−h

which completes the last paragraph. One of the main useful tools in the proof of existence theorems for nonlinear functional equations is the following Sadvoskii’s fixed point theorem. Lemma 3.2. (Krasnoselski [20]) Suppose that Σ is a closed convex subset of a Banach space X. Assume that K1 and K2 are mappings from Σ into X such that the following conditions are satisfied: (i) (K1 + K2 )(Σ) ⊂ Σ, (ii) K1 is a completely continuous mapping, (iii) K2 is a contraction mapping. Then the operator K1 + K2 has a fixed point in Σ.

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11 From now on, we establish the following results on the solvability of the equation (3.1). Theorem 3.1. Let Assumptions (A), (G) and (F) be satisfied. Assume that (φ0 , φ1 ) ∈ L2 (Ω, H) × Π and k ∈ M 2 (0, T ; V ∗ ) for T > 0. Then, there exists a solution x of the system (3.1) such that x ∈ M 2 (0, T ; V ) ∩ C([0, T ]; H). Moreover, there is a constant C3 independent of the initial data (φ0 , φ1 ) and the forcing term k such that ||x||M 2 (−h,T ;V ) ≤ C3 (1 + E(|φ0 |2 ) + ||φ1 ||Π + ||k||M 2 (0,T ;V ∗ ) ).

(3.4)

Proof. Let √ p 
1/2  r := 2 C1 C−α Lg (||φ1 ||Π + 1) + 3C1 E(|φ0 |2 ) + ||φ1 ||2Π + ||k||2M 2 (0,T1 ;V ∗ ) , and √ p
N := 3C−α Lg ||φ1 ||Π + r + 1 p + (3β − 2)−1/2 (3β)−1/2 C1−β Lg (||φ1 ||Π + r + 1) q + C2 Tr(Q) Lf (r)(||φ1 ||Π + r + 1), where β > 2/3, C1 and C2 is constants in Lemma 2.3 and Lemma 2.4, respectively. Let 1/2 3β/2 T1γ := max{T1 , T1 } and choose 0 < T1 < T such that T1γ N ≤

p √ p r E(|φ0 |2 ) + ||φ1 ||Π + ||k||M 2 (0,T1 ;V ∗ ) , = C1 C−α Lg (||φ1 ||Π + 1) + 3C1 2 (3.5)

and ˆ :=T1γ N

q p p √ 3C−α Lg + (3β − 1)−1/2 (3β)−1/2 C1−β Lg + C2 Tr(Q) Lf (r) < 1. (3.6)

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Yong Han Kang 838-861

J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 29, NO.5, 2021, COPYRIGHT 2021 EUDOXUS PRESS, LLC

12 Let J be the operator on M 2 (0, T1 ; V ) defined by Z t 0 1 (Jx)(t) =S(t)[φ + g(0, φ )] − g(t, xt ) + AS(t − s)g(s, xs )ds 0 Z t Z  0 S(t − s) a1 (τ )A1 x(s + τ )dτ ds + f (s, xs )dW (s) + 0 −h Z t S(t − s)k(s)ds. + 0

It is easily seen that J is continuous from C([0, T1 ]; H) into itself. Let Σ = {x ∈ M 2 (−h, T ; V ) : x(0) = φ0 , and x(s) = φ1 (s)(s ∈ [−h, 0))}. and Σr = {x ∈ Σ : ||x||M 2 (0,T1 ;V ) ≤ r}, which is a bounded closed subset of M 2 (0, T1 ; V ). Now, we give the proof of Theorem 3.1 in the following several steps: Now, in order to show that the operator J has a fixed point in Σr ⊂ M 2 (0, T1 ; V ), we take the following steps. Step 1. J maps Σr into Σr . By (2.10), (2.15) and Assumption (G), and noting x0 = φ1 , we know Z  T1    E kS(t)g(0, x0 )k2 dt = E C12 |g(0, φ1 )|2 (3.7) 0 
2  = E C12 ||A−β ||B(H) |Aβ g(0, φ1 )| ≤ (C1 C−α )2 Lg (||φ1 ||Π + 1)2 . From (2.10) of Lemma 2.3 it follows Z Z t Z  T1  0 2  0

E S(t)φ + S(t − s) a1 (τ )A1 x(s + τ )dτ + k(s) ds dt 0 0 −h  2 0  1 ≤ E C1 {|φ | + ||φ ||L2 (−h,0;V ) + ||k||L2 (0,T1 ;V ∗ ) }2
≤ 3C12 E[|φ0 |2 ] + ||φ1 ||2Π + ||k||2M 2 (0,T1 ;V ∗ ) . By using Assumption (G) and Lemma 3.1, we have Z T1

−β β


2

A A g(t, xt ) 2 dt ||g(·, x· )||M 2 (0,T1 ;V ) = E 0 Z T1

2 2 2 Aβ g(t, xt ) 2 dt ≤ C−α E Lg T1 ||xt ||Π + 1 ≤ C−α 0
2 ≤ 3C−α Lg T1 ||φ1 ||2Π + ||x||M 2 (0,T1 ;V ) + 1

849

(3.8)

(3.9)

Yong Han Kang 838-861

J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 29, NO.5, 2021, COPYRIGHT 2021 EUDOXUS PRESS, LLC

13 Define H1 : M 2 (0, T1 ; V ) → M 2 (0, T1 ; V ) by t

Z

AS(t − s)g(s, xs )ds.

(H1 x)(t) = 0

Then from Lemma 2.5 and Assumption (G) we have ||AS(t − s)g(s, xs )|| = ||A1−β S(t − s)||B(H,V ) |Aβ g(s, xs )| C1−β ≤ |Aβ g(s, xs )|, (t − s)3(1−β)/2 and hence, by using H´older inequality and Assumption (G), ||H1 x||2M 2 (0,T1 ;V ) Z

T1

Z

=E



T1

Z



0

Z

t

2  AS(t − s)g(s, xs )ds dt

(3.10)

0

t


2  C1−β β |A g(s, x )|ds dt s 3(1−β)/2 0 0 (t − s) Z Z t  T1 2  −1 3β−2 ≤E C1−β (3β − 2) t |Aβ g(s, xs )|2 dsdt 0 0 Z T1 −1 2 2 ≤ (3β − 2) C1−β Lg (||xs ||Π + 1) t3β−1 dt ≤E



≤ (3β − = (3β −

0 3β 2 2) (3β) C1−β Lg T1 (||x||M 2 (−h,T1 ;V ) + 1)2 2 2)−1 (3β)−1 C1−β Lg T13β (||φ1 ||Π + ||x||M 2 (0,T1 ;V ) −1

−1

+ 1)2

Let t

Z

S(t − s)f (s, xs )dW (s).

(H2 x)(t) = 0

For (2.13) of Lemma 2.4 it follows ||H2 x||2M 2 (0,T1 ;V )

=E ≤ ≤ ≤ ≤



Z

T1



Z

t

2  S(t − s)f (s, xs )dW (s) dt

0 0 2 E[C2 Tr(Q)2 T1 ||f (s, xs )||2L2 (0,T ;V ∗ ) ] C22 Tr(Q)2 T1 ||f (s, xs )||2M 2 (0,T ;V ∗ ) C22 Tr(Q)2 T1 Lf (r)(||xs ||Π + 1)2 C22 Tr(Q)2 T1 Lf (r)(||φ1 ||Π + ||x||M 2 (0,T1 ;V )

850

(3.11)

+ 1)2

Yong Han Kang 838-861

J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 29, NO.5, 2021, COPYRIGHT 2021 EUDOXUS PRESS, LLC

14 Therefore, from (3.7)-(3.11) it follows that p ||Jx||M 2 (0,T1 ;V ) ≤C1 C−α Lg (||φ1 ||Π + 1) √
1/2 + 3C1 E[|φ0 |2 ] + ||φ1 ||2Π + ||k||2M 2 (0,T1 ;V ∗ ) √ p
+ 3C−α T1 Lg ||φ1 ||Π + ||x||M 2 (0,T1 ;V ) + 1 p 3β/2 + (3β − 2)−1/2 (3β)−1/2 C1−β Lg T1 (||φ1 ||Π + ||x||M 2 (0,T1 ;V ) + 1) q + C2 Tr(Q) T1 Lf (r)(||φ1 ||Π + ||x||M 2 (0,T1 ;V ) + 1) p ≤C1 C−α Lg (||φ1 ||Π + 1) √
1/2 + T1γ N ≤ r, + 3C1 E[|φ0 |2 ] + ||φ1 ||2Π + ||k||2M 2 (0,T1 ;V ∗ ) and so, J maps Σr into Σr . Define mapping K1 + K2 on L2 (0, T1 ; V ) by the formula (Jx)(t) = (K1 x)(t) + (K2 x)(t), Z t Z s (K1 x)(t) = S(t − s) a1 (τ − s)A1 x(τ )dτ ds, 0

0

and Z

0

t

(K2 x)(t) =S(t)[φ + g(0, x0 )] − g(t, xt ) + AS(t − s)g(s, xs )ds 0 Z Z t  0 a1 (τ − s)A1 φ1 (τ )dτ ds + f (s, xs )dW (s)} + S(t − s) s−h 0 Z t + S(t − s)k(s)ds. 0

Step 2. K1 is a completely continuous mapping. We can now employ Lemma 3.2 with Σr . Assume that a sequence {xn } of 2 M (0, T1 ; V ) converges weakly to an element x∞ ∈ M 2 (0, T1 ; V ), i.e., w−limn→∞ xn = x∞ . Then we will show that lim ||K1 xn − K1 x∞ ||M 2 (0,T1 ;V ) = 0,

n→∞

(3.12)

which is equivalent to the completely continuity of K1 since M 2 (0, T1 ; V ) is reflexive. For a fixed t ∈ [0, T1 ], let x∗t (x) = (K1 x)(t) for every x ∈ M 2 (0, T1 ; V ). Then x∗t ∈ M 2 (0, T1 ; V ∗ ) and we have limn→∞ x∗t (xn ) = x∗t (x∞ ) since w−limn→∞ xn = x∞ . Hence, lim (K1 xn )(t) = (K1 x∞ )(t), t ∈ [0, T1 ]. n→∞

851

Yong Han Kang 838-861

J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 29, NO.5, 2021, COPYRIGHT 2021 EUDOXUS PRESS, LLC

15 Set

s

Z

a1 (τ − s)A1 x(τ )dτ.

h(s) = 0

Then by using H´older inequality we obtain the following inequality Z s (a1 (τ − s) − a1 (0))A1 x(τ )dτ (3.13) |h(s)| ≤ 0 Z s a1 (0)A1 x(τ )dτ + 0 Z s 

1/2 √ −1 2ρ+1 1/2 ≤ (2ρ + 1) s + s H1 ||A1 ||B(H) ||x(τ )||2 dτ . 0

Thus, by (2.5) and (3.13) it holds Z

t

2 S(t − s)h(s)ds 0 Z Z t
t  √ 2 1 2 2 −1 (2ρ+1)/2

≤ (H1 ||A1 ||B(H) ) ||x(τ )|| dτ s ds ((2ρ + 1) s + 1/2 (t − s) 0 0  2 2 2 ≤ (H1 ||A1 ||B(H) ) ||x||L2 (0,t;V ) (2ρ + 1)−1 B(1/2, (2ρ + 3)/2)tρ+1 + B(1/2, 3/2)t .

||(K1 x)(t)|| = 2

:= c2 ||x||2L2 (0,t;V ) , where c2 is a constant and B(·, ·) is the Beta function. Here we used B(1/2, (2ρ + 3)/2)t

ρ+1

Z =

t

(t − s)−1/2 s(2ρ+1)/2 ds.

0

And we know sup ||E[(K1 x)(t)]2 || ≤ c2 ||x||2M 2 (0,T1 ;V ) ≤ ∞.

0≤t≤T1

Therefore, by Lebesgue’s dominated convergence theorem it holds Z

2




Z

||(K1 xn )(t)|| dt = E

lim E

n→∞

T1

0

T1


||(K1 x∞ )(t)||2 dt ,

0

i.e., limn→∞ ||K1 xn ||M 2 (0,T1 ;V ) = ||K1 x∞ ||M 2 (0,T1 ;V ) . Since M 2 (0, T1 ; V ) is a reflexive space, it holds (3.12). Step 3. K2 is a contraction mapping.

852

Yong Han Kang 838-861

J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 29, NO.5, 2021, COPYRIGHT 2021 EUDOXUS PRESS, LLC

16 For every x1 and x2 ∈ Σr , we have (K2 x1 )(t) − (K2 x2 )(t) =g(t, x2t ) − g(t, x1t ) Z t
AS(t − s) g(t, x1s ) − g(t, x2s ) ds + Z0 t + S(t − s){f (s, x1 (s)) − f (s, x2 (s))}dW (s). 0

In a similar way to (3.9)-(3.11) and Proposition 2.3, we have p p  3β/2 ||K2 x1 − K2 x2 ||M 2 (0,T1 ;V ) ≤ C−α T1 Lg + (3β − 2)−1/2 (3β)−1/2 C1−β T1 Lg q + C2 T1 Lf (r)Tr(Q) ||x1 − x2 ||M 2 (0,T1 ;V ) ˆ ||x1 − x2 ||M 2 (0,T ;V ) . ≤N 1 So by virtue of the condition (3.6) the contraction mapping principle gives that the solution of (3.1) exists uniquely in M 2 (0, T1 ; V ). This has proved the local existence and uniqueness of the solution of (3.1). Step 4. We drive a priori estimate of the solution. To prove the global existence, we establish a variation of constant formula (3.4) of solution of (3.1). Let x be a solution of (3.1) and φ0 ∈ H. Then we have that from (3.7)-(3.11) it follows that p ||x||M 2 (0,T1 ;V ) ≤C1 C−α Lg (||φ1 ||Π + 1) √
1/2 + 3C1 E[|φ0 |2 ] + ||φ1 ||2Π + ||k||2M 2 (0,T1 ;V ∗ ) √ p
+ 3C−α T1 Lg ||φ1 ||Π + ||x||M 2 (0,T1 ;V ) + 1 p 3β/2 + (3β − 2)−1/2 (3β)−1/2 C1−β Lg T1 (||φ1 ||Π + ||x||M 2 (0,T1 ;V ) + 1) q + C2 Tr(Q)T1 T1 Lf (r)(||φ1 ||Π + ||x||M 2 (0,T1 ;V ) + 1) ˆ ||x||L2 (0,T ;V ) + N ˆ1 , ≤N 1 where p ˆ1 =C1 C−α Lg (||φ1 ||Π + 1) N √
1/2 + 3C1 E[|φ0 |2 ] + ||φ1 ||2Π + ||k||2M 2 (0,T1 ;V ∗ ) √ p
+ 3C−α T1 Lg ||φ1 ||Π + 1 p 3β/2 + (3β − 2)−1/2 (3β)−1/2 C1−β Lg T1 (||φ1 ||Π + 1) q + C2 Tr(Q) T1 Lf (r)(||φ1 ||Π + 1).

853

Yong Han Kang 838-861

J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 29, NO.5, 2021, COPYRIGHT 2021 EUDOXUS PRESS, LLC

17 Taking into account (3.6) there exists a constant C3 such that ˆ )−1 N ˆ1 ||x||L2 (0,T ;V ) ≤(1 − N 1

≤C3 (1 + E(|φ0 |2 ) + ||φ1 ||Π + ||k||M 2 (0,T1 ;V ∗ ) ), which obtain the inequality (3.4). Now we will prove that E[x(T1 )2 ] < ∞ in order that the solution can be extended to the interval [T1 , 2T1 ]. Define a mapping H3 : L2 (0, T1 ; V ) → L2 (0, T1 ; V ) as Z Z t  0 0 S(t − s) a1 (τ )A1 x(s + τ )dτ + k(s) ds (H3 x)(t) = S(t)[φ + g(0, x0 )] + −h

0

The from (2.11) and Lemma 2.3 it follows that E|(H3 x)(T1 )|2 ≤ c1 E||H3 x||2W1 (3.14)  0 2 ≤ 3c1 C1 E |φ + g(0, φ1 )| + ||φ1 ||L2 (−h,0;V ) + ||k||L2 (0,T1 ;V ∗ )  ≤ c1 C1 E|φ0 + g(0, φ1 )|2 + ||φ1 ||2M 2 (−h,0;V ) + ||k||2M 2 (0,T1 ;V ∗ ) := I, and from (2.4) and Assumption (F), Z T1 2 2 E|(H2 x)(T1 )| =E S(T1 − s)f (s, xs )dW (s)

(3.15)

0

≤M02 Tr(Q)2 T1 Lf (r)(||xs ||Π + 1)2 ≤M02 Tr(Q)2 T1 Lf (r)(||φ1 ||Π + ||x||M 2 (0,T1 ;V ) + 1)2 := II. Moreover, by using Assumption (G) we have

2 E|g(T1 , xT1 )|2 ≤ E A−β Aβ g(t, xT1 ) , ≤ C− ALP HA2 Lg ||xT1 ||Π + 1

(3.16)
2

≤ C− ALP HA2 Lg ||φ1 ||Π + ||x||M 2 (0,T1 ;V ) + 1


2

:= III,

and Z

2

T1

AS(T1 − s)g(s, xs )ds|2

E|(H1 x)(T1 )| = E|

(3.17)

0

Z

T1

A1−β W (T1 − s)Aβ g(s, xs )ds|2

= E| 0

Z

T1

2 C1−β β |A (g(s, x )|ds s (t − s)3(1−β)/2 0 Z t  2 2 −1 3β−2 |Aβ (g(s, xs )|2 ds ≤ E C1−β (3β − 2) T1 ≤E

=



2 C1−β (3β

−1

− 2)

0 3β−1 T1 Lg (||x||M 2 (0,T1 ;V )

854

+ ||φ1 ||M 2 (−h,0;V ) + 1)2 := IV.

Yong Han Kang 838-861

J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 29, NO.5, 2021, COPYRIGHT 2021 EUDOXUS PRESS, LLC

18 Thus,by (3.14)-(3.17) we have Z T1 AS(T1 − s)g(s, xs )ds E|x(T1 )| = E (H3 x)(T1 ) − g(T1 , xT1 ) + 0 Z T1 S(T1 − s)f (s, xs )dW (s) + 2

0

≤ I + II + III + IV < ∞. Hence we can solve the equation in [T1 , 2T1 ] with the initial (x(T1 ), xT1 ) and an analogous estimate to (3.4). Since the condition (3.6) is independent of initial values, the solution can be extended to the interval [0, nT1 ] for any natural number n, and so the proof is complete. Remark 3.1. Thanks for Lemma 2.3, we note that the solution of (3.1) with the conditions of Theorem 3.1 satisfies also that Z

T

E(

||x0 (s)||2∗ ds) < ∞.

−h

Here we note that by a simple calculation using the properties of analytic semigroup, it is immediately seen that x ∈ M 2 (−h, T ; H). Now, we obtain that the solution mapping is continuous in the following result, which is useful for the control problem and physical applications of the given equation. Theorem 3.2. Let Assumptions (A), (G) and (F) be satisfied. Assuming that the initial data (φ0 , φ1 ) ∈ L2 (Ω, H) × Π and the forcing term k ∈ M 2 (0, T ; V ∗ ). Then the solution x of the equation (3.1) belongs to x ∈ M 2 (0, T ; V ) and the mapping L2 (Ω, H) × Π × M 2 (0, T ; V ∗ ) 3 (φ0 , φ1 , k) 7→ x ∈ M 2 (0, T ; V )

(3.18)

is continuous. Proof. From Theorem 3.1, it follows that if (φ0 , φ1 , k) ∈ L2 (Ω, H)×Π×M 2 (0, T ; V ∗ ) then x belongs to M 2 (0, T ; V ). Let (φ0i , φ1i , ki ) and xi be the solution of (3.1) with

855

Yong Han Kang 838-861

J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 29, NO.5, 2021, COPYRIGHT 2021 EUDOXUS PRESS, LLC

19 (φ0i , φ1i , ki ) in place of (φ0 , φ1 , k) for i = 1, 2. Let xi (i = 1, 2) ∈ Σr . Then it holds x1 (t) − x2 (t) = S(t)[(φ01 − φ02 ) + (g(0, x10 ) − g(0, x20 ))] Z t 2 1 AS(t − s)(g(s, x1s ) − g(t, x2s ))ds − (g(t, xt ) − g(t, xt )) + 0 Z Z t  0 S(t − s) a1 (τ )A1 (x1 (s + τ ) − x2 (s + τ ))dτ ds + 0 −h Z t S(t − s){((F x1 )(s) − (F x2 )(s)) + (k1 (s) − k2 (s))}ds. + Z0 t + S(t − s)(k1 (s) − k2 (s))ds 0

Hence, by applying the same argument as in the proof of Theorem 3.1, we have ˆ ||x1 − x2 ||L2 (0,T ;V ) + N ˆ2 , ||x1 − x2 ||M 2 (0,T1 ;V ) ≤N 1 where p ˆ2 =C1 C− ALP HA Lg (||φ1 − φ1 ||Π ) N 1 2 √
1/2 + 3C1 E[|φ01 − φ01 |2 ] + ||φ11 − φ12 ||2Π + ||k1 − k2 ||2M 2 (0,T1 ;V ∗ ) √ p + 3C− ALP HA T1 Lg ||φ11 − φ12 ||Π ) p (3β+1)/2 (||φ11 − φ12 ||Π ) + (3β − 2)−1/2 (3β)−1/2 C1−β Lg T1 q + C2 Tr(Q) T1 Lf (r)(||φ11 − φ12 ||Π ), which implies ˆ2 (1 − N ˆ )−1 . ||x||M 2 (0,T1 ;V ) ≤N Therefore, it implies the inequality (3.18).

4

Example

Let H = L2 (0, π), V = H01 (0, π), V ∗ = H −1 (0, π).

856

Yong Han Kang 838-861

J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 29, NO.5, 2021, COPYRIGHT 2021 EUDOXUS PRESS, LLC

20 Consider the following retarded neutral stochastic differential system in Hilbert space H:  R0   d[x(t, y) + g(t, xt (t, y))] = [Ax(t, y) + −h a1 (s)A1 x(t + s, y)ds + k(t, y)]dt +F (t, x(t, y))dW (t), (t, y) ∈ [0, T ] × [0, π],   0 2 x(0, y) = φ (y) ∈ L (Ω, H), x(s, y) = φ1 (s, y), (s, y) ∈ [−h, 0) × [0, π], (3.19) where h > 0, a1 (·) is H¨older continuous, A1 ∈ B(H), and W (t) stands for a standard cylindrical Winner process in H defined on a stochastic basis (Ω, F, P ). Let Z a(u, v) = 0

π

du(y) dv(y) dy. dy dy

Then A = ∂ 2 /∂y 2

with

D(A) = {x ∈ H 2 (0, π) : x(0) = x(π) = 0}.

The eigenvalue and the eigenfunction of A are λn = −n2 and zn (y) = (2/π)1/2 sin ny, respectively. Moreover, (a1) {zn : n ∈ N } is an orthogonal basis of H and S(t)x =

∞ X

2

en t (x, zn )zn ,

∀x ∈ H, t > 0.

n=1

Moreover, there exists a constant M0 such that ||S(t)||B(H) ≤ M0 . (a2) Let 0 < α < 1. Then the fractional power Aα : D(Aα ) ⊂ H → H of A is given by ∞ X α n2α (x, zn )zn , D(Aα ) := {x : Aα x ∈ H}. A x= n=1

In particular, A−1/2 x =

∞ X 1 (x, zn )zn , and ||A−1/2 || = 1. n n=1

The nonlinear mapping f is a real valued function belong to C 2 ([0, ∞)) which satisfies the conditions (f1) f (0) = 0, f (r) ≥ 0 for r > 0,

857

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J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 29, NO.5, 2021, COPYRIGHT 2021 EUDOXUS PRESS, LLC

21 0

(f2) |f (r) ≤ c(r + 1) and |f 00 (r)| ≤ c for r ≥ 0 and c > 0. If we present

0

F (t, x(t, y)) = f (|x(t, y)|2 )x(t, y), Then it is well known that F is a locally Lipschitz continuous mapping from the whole V into H by Sobolev’s imbedding theorem (see [30, Theorem 6.1.6]). As an example of q in the above, we can choose q(r) = µ2 r + η 2 r2 /2 (µ and η is constants). Define g : [0, T ] × Π → H as Z 0 ∞ Z t X n2 t a2 (s)x(t + s)ds, zn )zn , , t > 0. g(t, xt ) = e ( n=1

0

−h

Then it can be checked that Assumption (G) is satisfied. Indeed, for x ∈ Π, we know Z 0 a2 (s)x(t + s)ds, Ag(t, xt ) = (S(t) − I) −h

where I is the identity operator form H to itself and |a2 (s) − a2 (τ )| ≤ H2 (s − τ )κ ,

|a2 (0)| ≤ H2 ,

s, τ ∈ [−h, 0]

for a constant κ > 0. Hence we have Z 2  0 E|Ag(t, xt )| ≤(M0 + 1) (a2 (s) − a2 (0))x(t + s)dτ −h Z 0 2 + a2 (0)x(t + s)dτ −h  ≤(M0 + 1)2 H22 (2κ + 1)−1 h2ρ+1 + h ||xt ||2Π . 2

2

It is immediately seen that Assumption (G) has been satisfied. Thus, all the conditions stated in Theorem 3.1 have been satisfied for the equation (3.19), and so there exists a solution x of the equation (3.19) such that Z T Z T 2 ||x(s)|| ds) < ∞, and E( ||x0 (s)||2∗ ds) < ∞. E( −h

−h

References [1] A. Anguraj and A. Vinodkumar, Existence, uniqueness ans stability of impulsive Stochastic partial neutral functional differential equations with infinite delays, J. Appl. Math. and Informatics 28 (2010), 3–4.

858

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22 [2] G. Arthi and K. Balachandran, Controllability of damped second-order neutral functional differential systems with impulses, Taiwanese J. Math. 16 (2012), 89106. [3] K. Balachandran and R. Sathaya, Controllability of nonlocal Stochastic quasilinear integrodifferential systems, Electron J. Qual. Theory Differ. 50 (2011), 1–16. [4] P. Balasubramaniam, Existence of solutions of functional stochastic differential inclusions, Tamkang J. Math. 33 (2002), 35–43. [5] P. Balasubramaniam, J. Y. Park and P. Muthukumar, Approximate controllability of neutral stochastic functional differential systems with infinite delay, Stoch. Anal. Appl. 28 (2010), 389–400. [6] A. T. Bharucha-Ried, random Integral Equation, Academic Press, Newyork, 1982. [7] R. F. Curtain, Stochastic evolution equations with general white noise disturbance, J. Math. Anal. Appl. 60(1977), 570–595. [8] G. Di Blasio, K. Kunisch and E. Sinestrari, L2 −regularity for parabolic partial integrodifferential equations with delay in the highest-order derivatives, J. Math. Anal. Appl. 102(1984), 38–57. [9] A. Friedman, Stochastic Differential Equations & Applications, Academic Press, INC. 1975. [10] X. Fu,Controllability of neutral functional differential systems in abstract space, Appl. Math. Comput. 141 (2003), 281-296. [11] W. Grecksch and C. Tudor, Stochastic Evolution Equations: A Hilbert space Apprauch, Akademic Verlag, Berlin, 1995. [12] E. Hern´andez and M. Mckibben, On state-dependent delay partial neutral functional differential equations, Appl. Math. Comput. 186(1) (2007). 294–301. [13] E. Hern´andez, M. Mckibben, and H. Henr´rquez, Existence results for partial neutral functional differential equations with state-dependent delay, Math. Comput. Modell. 49 (2009), 1260–1267. [14] L. Hu and Y. Ren, Existence results for impulsive neutral stochastic functional integro-differential equations with infinite dalays, Acta Appl. Math. 111(2010), 303-317.

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23 [15] J. M. Jeong, Stabilizability of retarded functional differential equation in Hilbert space, Osaka J. Math. 28(1991), 347–365. [16] J. M. Jeong, Retarded functional differential equations with L1 -valued controller, Funkcialaj Ekvacioj 36(1993), 71–93. [17] J. M. Jeong, Y. C. Kwun and J. Y. Park, Approximate controllability for semilinear retarded functional differential equations, J. Dynamics and Control Systems, 5 (1999), no. 3, 329–346. [18] J. M. Jeong and H. G. Kim, Controllability for semilinear functional integrodifferential equations, Bull. Korean Math. Soc. 46 (2009), no. 3, 463–475. [19] S. Kathikeyan and K. Balachandran, Controllability of nonlinear stochastic neutral impulsive systems, Nonlinear Anal. 3(2009), 1117–1135. [20] M. A. Krannoselski, Topological Methods in the Theory of Nonlinear Integral Equations, Pergamon Press, New York, 1964. [21] A. Lin and L.hu, Existence results for impulsive neutral stochastic functional integro-differential inclusions with nonlocal conditions, Comput. Math. Appl. 59(2010), 64-73. [22] A. Lin and L. Hu, Existence results for impulsive neutral stochastic differential inclusions with nonlocal initial conditions, Comput. Math. Appl. 59(2010), 64–73. [23] M. Metivier and J. Pellaumail, Stochastic Integration, Academic Press, Newyork, 1980 [24] S. Nakagiri, Structural properties of functional differential equations in Banach spaces, Osaka J. Math. 25(1988), 353–398. [25] B. Radhakrishnan and K. Balachandran, Controllability results for second order neutral impulsive integrodifferential systems, J. Optim. Theory Appl. 151 (2011), 589-612. [26] Y. Ren, L. Hu and R. Sakthivel, Controllability of impulsive neutral stochastic functional differential systems with infinite delay, J. Comput. Anal. Appl. 235(2011), 2603–2614. [27] R. Sakthivel, N. I. Mahmudov and S. G. Lee, Controllability of non-linear impulsive stochastic systems, Int. J. Control, 82(2009), 801–807.

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24 [28] N. Sukavanam and Nutan Kumar Tomar, Approximate controllability of semilinear delay control system, Nonlinear Func.Anal.Appl. 12(2007), 53–59. [29] L. Hu and Y. Ren, Existence for impulsive Stochastic functional integrodifferential equations,Acta Appl. Math. 111(2010), 303–317. [30] H. Tanabe, Equations of Evolution, Pitman-London, 1979. [31] H. Triebel, Interpolation Theory, Function Spaces, Differential Operators, North-Holland, 1978. [32] L. Wang, Approximate controllability for integrodifferential equations and multiple delays, J. Optim. Theory Appl. 143(2009), 185–206.

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FUZZY STABILITY OF CUBIC FUNCTIONAL EQUATIONS WITH EXTRA TERMS CHANG IL KIM AND GILJUN HAN∗

Abstract. In this paper, we consider the generalized Hyers-Ulam stability for the following cubic functional equation f (x + 2y) − 3f (x + y) + 3f (x) − f (x − y) − 6f (y) + Gf (x, y) = 0. with an extra term Gf which is a functional operator of f .

1. Introduction and preliminaries In 1940, Ulam proposed the following stability problem (cf. [20]): “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 an unique homomorphism h : G1 −→ G2 with d(f (x), h(x)) < δ for all x ∈ G1 ?” In the next year, Hyers [8] gave a partial solution of Ulam, s problem for the case of approximate additive mappings. Subsequently, his result was generalized by Aoki ([1]) for additive mappings and by Rassias [18] for linear mappings to consider the stability problem with unbounded Cauchy differences. During the last decades, the stability problem of functional equations have been extensively investigated by a number of mathematicians ([3], [4], [5], [7], and [16]). Katsaras [11] defined a fuzzy norm on a vector space to construct a fuzzy vector topological structure on the space. Later, some mathematicians have defined fuzzy norms on a vector space in different points of view. In particular, Bag and Samanta [2] gave an idea of fuzzy norm in such a manner that the corresponding fuzzy metric is of Kramosil and Michalek type [13]. In this paper, we use the definition of fuzzy normed spaces given in [2],[14], [15]. Definition 1.1. Let X be a real vector space. A function N : X × R −→ [0, 1] is called a fuzzy norm on X if for any x, y ∈ X and any s, t ∈ R, (N1) N (x, t) = 0 for t ≤ 0; (N2) x = 0 if and only if N (x, t) = 1 for all t > 0; t (N3) N (cx, t) = N (x, |c| ) if c 6= 0; (N4) N (x + y, s + t) ≥ min{N (x, s), N (y, t)}; (N5) N (x, ·) is a nondecreasing function of R and limt→∞ N (x, t) = 1; (N6) for any x 6= 0, N (x, ·) is continuous on R. In this case, the pair (X, N ) is called a fuzzy normed space. 2010 Mathematics Subject Classification. 39B52, 46S40. Key words and phrases. fuzzy normed space, cubic functional equation. * Corresponding author. 1

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Let (X, N ) be a fuzzy normed space. (i) A sequence {xn } in X is said to be convergent in (X, N ) if there exists an x ∈ X such that limn→∞ N (xn − x, t) = 1 for all t > 0. In this case, x is called the limit of the sequence {xn } in X and one denotes it by N − limn→∞ xn = x. (ii) A sequence {xn } in X is said to be Cauchy in (X, N ) if for any  > 0 and any t > 0, there exists an m ∈ N such that N (xn+p − xn , t) > 1 −  for all n ≥ m and all positive integer p. It is well known that every convergent sequence in a fuzzy normed space is Cauchy. A fuzzy normed space is said to be complete if each Cauchy sequence in it is convergent and a complete fuzzy normed space is called a fuzzy Banach space. For example, it is well known that for any normed space (X, || · ||), the mapping NX : X × R −→ [0, 1], defined by ( 0, if t ≤ 0 NX (x, t) = t t+||x|| , if t > 0 is a fuzzy norm on X. In 1996, Isac and Rassias [9] were the first to provide applications of stability theory of functional equations for the proof of new fixed point theorems with applications. Theorem 1.2. [6] Let (X, d) be a complete generalized metric space and let J : X −→ X be a strictly contractive mapping with some Lipschitz constant L with 0 < L < 1. Then for each given element x ∈ X, either d(J n x, J n+1 x) = ∞ for all nonnegative integer 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) < ∞} and 1 (4) d(y, y ∗ ) ≤ d(y, Jy) for all y ∈ Y . 1−L In 2001, Rassias [19] introduced the following cubic functional equation (1.1)

f (x + 2y) − 3f (x + y) + 3f (x) − f (x − y) − 6f (y) = 0

and the following cubic functional equations were investigated (1.2)

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

in ([10]). Every solution of a cubic functional equation is called a cubic mapping and Kim and Han [12] investigated the following cubic functional equation f (x + 2y) − 3f (x + y) + 3f (x) − f (x − y) − 6f (y) + k[f (mx + y) + f (mx − y) − m[f (x + y) + f (x − y)] − 2(m3 − m)f (x)] = 0 for some rational number m and some real number k and proved the stability for it in fuzzy normed spaces. In this paper, we investigate the following functional equation which is added a term by Gf to (1.1) f (x + 2y) − 3f (x + y) + 3f (x) − f (x − y) − 6f (y) + Gf (x, y) = 0, where Gf is a functional operator depending on functions f . The definition of Gf is given in section 2 and prove the stability for it in fuzzy normed spaces. Throughout this paper, we assume that X is a linear space, (Y, N ) is a fuzzy Banach space, and (Z, N 0 ) is a fuzzy normed space.

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2. Cubic functional equations with extra terms For given l ∈ N and any i ∈ {1, 2, · · ·, l}, let σi : X × X −→ X be a binary operation such that σi (rx, ry) = rσi (x, y) for all x, y ∈ X and all r ∈ R. It is clear that σi (0, 0) = 0. Also let F : Y l −→ Y be a linear, continuous function. For a map f : X −→ Y , define Gf (x, y) = F (f (σ1 (x, y)), f (σ2 (x, y)), · · ·, f (σl (x, y))). Now consider the functional equation (2.1)

f (x + 2y) − 3f (x + y) + 3f (x) − f (x − y) − 6f (y) + Gf (x, y) = 0

with the functional operator Gf . Theorem 2.1. Suppose that the mapping f : X −→ Y is a solution of (2.1) with f (0) = 0. Then f is cubic if and only if f (2x) = 8f (x) and Gf (y, x) = Gf (y, −x) for all x, y ∈ X. Proof. Suppose that f (2x) = 8f (x) and Gf (y, x) = Gf (y, −x) for all x, y ∈ X. Interchanging x and y in (2.1) , we have (2.2)

f (2x + y) − 3f (x + y) + 3f (y) − f (y − x) − 6f (x) + Gf (y, x) = 0.

for all x, y ∈ X and letting x = −x in (2.2) , we have (2.3)

f (−2x + y) − 3f (−x + y) + 3f (y) − f (x + y) − 6f (−x) + Gf (y, −x) = 0.

for all x, y ∈ X. By (2.2) and (2.3), we have (2.4)

f (2x + y) − f (−2x + y) − 2f (x + y) + 2f (y − x) − 6f (x) + 6f (−x) = 0.

for all x, y ∈ X, because Gf (y, x) = Gf (y, −x). Letting y = x in (2.4), we have f (3x) − 22f (x) + 5f (−x) = 0.

(2.5)

for all x ∈ X and letting y = 2x in (2.4), by (2.5), we have f (4x) − 2f (3x) − 4f (x) + 6(−x) = 16f (x) + 16f (−x) = 0. for all x ∈ X, because f (2x) = 8f (x). Hence f is odd and by (2.2) and (2.3), f satisfies (1.2). Thus f is a cubic mapping. The converse is trivial.  3. The Generalized Hyers-Ulam stability for (2.1) In this section, we prove the generalized Hyers-Ulam stability of (2.1) in fuzzy normed spaces. For any mapping f : X −→ Y , we define the difference operator Df : X 2 −→ Y by Df (x, y) = f (x + 2y) − 3f (x + y) + 3f (x) − f (x − y) − 6f (y) + Gf (x, y) for all x, y ∈ X. Theorem 3.1. Let φ : X 2 −→ Z be a function such that there is a real number L satisfying 0 < L < 1 and (3.1)

N 0 (φ(2x, 2y), t) ≥ N 0 (8Lφ(x, y), t)

for all x, y ∈ X and all t > 0. Let f : X −→ Y be a mapping such that f (0) = 0 and (3.2)

N (Df (x, y), t) ≥ N 0 (φ(x, y), t)

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for all x, y ∈ X and all t > 0 and (3.3) N (f (2x) − 8f (x), t) ≥ min{N 0 (aφ(x, 0), t), N 0 (bφ(0, x), t), N 0 (cφ(x, −x), t)} for all x ∈ X, all t > 0 and some nonnegative real numbers a, b, c. Further, assume that if g satisfies (2.1), then g is a cubic mapping. Then there exists an unique cubic mapping C : X −→ Y such that   1 N f (x) − C(x), t 8(1 − L) (3.4) 0 ≥ min{N (aφ(x, 0), t), N 0 (bφ(0, x), t), N 0 (cφ(x, −x), t)} for all x ∈ X and all t > 0. Proof. Let ψ(x, t) = min{N 0 (aφ(x, 0), t), N 0 (bφ(0, x), t), N 0 (cφ(x, −x), t)}. Consider the set S = {g | g : X −→ Y } and the generalized metric d on S defined by d(g, h) = inf{c ∈ [0, ∞) | N (g(x) − h(x), ct) ≥ ψ(x, t), ∀x ∈ X, ∀t > 0}. Then (S, d) is a complete metric space(see [17]). Define a mapping J : S −→ S by Jg(x) = 2−3 g(2x) for all x ∈ X and all g ∈ S. Let g, h ∈ S and d(g, h) ≤ c for some c ∈ [0, ∞). Then by (3.1), we have N (Jg(x) − Jh(x), cLt) ≥ N (2−3 (g(2x) − h(2x)), cLt) ≥ ψ(x, t) for all x ∈ X and all t > 0. Hence we have d(Jg, Jh) ≤ Ld(g, h) for any g, h ∈ S and so J is a strictly contractive mapping. By (3.3), d(f, Jf ) ≤ 81 < ∞ and by Theorem 1.2, there exists a mapping C : X −→ Y which is a fixed point of J such that d(J n f, C) → 0 as n → ∞. Moreover, C(x) = N − limn→∞ 2−3n f (2n x) for all 1 x ∈ X and d(f, C) ≤ 8(1−L) and hence we have (3.4). Replacing x, y, and t by 2n x, 2n y, and 23n t in (3.2), respectively, we have N (Df (2n x, 2n y), 23n t) ≥ N 0 (φ(2n x, 2n y), 23n t) ≥ N 0 (Ln φ(x, y), t) for all x, y ∈ X and all t > 0. Letting n −→ ∞ in the last inequality, we have C(x + 2y) − 3C(x + y) + 3C(x) − C(x − y) − 6C(y) + GC (x, y) = 0 for all x, y ∈ X and thus C is a cubic mapping. Now, we show the uniqueness of C. Let C0 : X −→ Y be another cubic mapping with (3.4). Then C0 is a fixed ponit of J in S and by (3.4), we get d(Jf, C0 ) ≤ d(Jf, JC) ≤ Ld(f, C0 ) ≤

L 0. Let f : X −→ Y be a mapping satisfying f (0) = 0, (3.2), and (3.3). Further, assume that if g satisfies (2.1), then g is a cubic mapping.

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Then there exists an unique cubic mapping C : X −→ Y such that the inequality   L N f (x) − C(x), t 8(1 − L) (3.6) 0 ≥ min{N (aφ(x, 0), t), N 0 (bφ(0, x), t), N 0 (cφ(x, −x), t)} for all x ∈ X and all t > 0. Proof. Let ψ(x, t) = min{N 0 (aφ(x, 0), t), N 0 (bφ(0, x), t), N 0 (cφ(x, −x), t)}. Consider the set S = {g | g : X −→ Y } and the generalized metric d on S defined by d(g, h) = inf{c ∈ [0, ∞) | N (g(x) − h(x), ct) ≥ ψ(x, t), ∀x ∈ X, ∀t > 0}. Then (S, d) is a complete metric space(see [17]). Define a mapping J : S −→ S by Jg(x) = 8g(2−1 x) for all x ∈ X and all g ∈ S. Let g, h ∈ S and d(g, h) ≤ c for some c ∈ [0, ∞). Then by (3.2) and (3.5), we have N (Jg(x) − Jh(x), cLt) ≥ N (8(g(2−1 x) − h(2−1 x)), cLt) ≥ ψ(x, t) for all x ∈ X and all t > 0. Hence we have d(Jg, Jh) ≤ Ld(g, h) for any g, h ∈ S and so J is a strictly contractive mapping. By (3.3), we get   L  L  (3.7) N f (x) − 8f (2−1 x), t ≥ ψ 2−1 x, t ≥ ψ(x, t) 8 8 for all x ∈ X and all t > 0. Hence d(f, Jf ) ≤ L8 < ∞ and by Theorem 1.2, there exists a mapping C : X −→ Y which is a fixed point of J such that d(J n f, C) → 0 as n → ∞. Moreover, C(x) = N − limn→∞ 23n f (2−n x) for all x ∈ X and d(f, C) ≤ L 8(1−L) and hence we have (3.6). The rest of the proof is similar to that of Theorem 3.1.  Using Theorem 3.1 and Theorem 3.2, we have the following corollaries. Corollary 3.3. Let φ : X 2 −→ Z be a function with (3.1). Let f : X −→ Y be a mapping such that f (0) = 0 and (3.2). Further, assume that if g satisfies (2.1), then g is a cubic mapping and that (3.8)

N (Gf (0, x), t) ≥ min{N 0 (a1 φ(x, 0), t), N 0 (a2 φ(0, x), t), N 0 (a3 φ(x, −x), t)}, N (Gf (x, −x), t) ≥ min{N 0 (b1 φ(x, 0), t), N 0 (b2 φ(0, x), t), N 0 (b3 φ(x, −x), t)}

for all x ∈ X, all t > 0 and for some nonnegative real numbers ai , bi (i = 1, 2, 3). Then there exists an unique cubic mapping C : X −→ Y such that   7 N f (x) − C(x), t 24(1 − L) (3.9) 0 ≥ min{N (c1 φ(x, 0), t), N 0 (c2 φ(0, x), t), N 0 (c3 φ(x, −x), t)} for all x ∈ X and all t > 0, where c1 = max{a1 , b1 }, c2 = max{1, a2 , b2 }, and c3 = max{1, a3 , b3 }. Proof. Setting x = 0 and y = x in (3.2), we have (3.10)

N (f (2x) − 9f (x) − f (−x) + Gf (0, x), t) ≥ N 0 (φ(0, x), t)

for all x ∈ X and all t > 0. Setting y = −x in (3.2), we have (3.11)

N (3f (x) − 5f (−x) − f (2x) + Gf (x, −x), t) ≥ N 0 (φ(x, −x), t)

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for all x ∈ X and all t > 0. Hence by (3.10) and (3.11), we get (3.12)

N (6f (x) + 6f (−x) − Gf (0, x) − Gf (x, −x), 2t) ≥ min{N 0 (φ(0, x), t), N 0 (φ(x, −x), t)}

for all x ∈ X and all t > 0. Thus by (3.8), (3.10), and (3.12), we get  7  N f (2x) − 8f (x), t 3 n 5 5  = min N (f (2x) − 9f (x) − f (−x) + Gf (0, x), t), N Gf (0, x), t , 6 6  1 1 1  1 1 o N f (x) + f (−x) − Gf (0, x) − Gf (x, −x), t , N Gf (x, −x), t 6 6 3 6 6 ≥ min{N 0 (c1 φ(x, 0), t), N 0 (c2 φ(0, x), t), N 0 (c3 φ(x, −x), t)} for all x ∈ X and all t > 0. By Theorem 3.1, there exists an unique cubic mapping C : X −→ Y with (3.9).  Corollary 3.4. Let φ : X 2 −→ Z be a function with (3.5). Let f : X −→ Y be a mapping satisfying f (0) = 0 and (3.2). Further, assume that if g satisfies (2.1), then g is a cubic mapping and that (3.8) hold. Then there exists an unique cubic mapping C : X −→ Y such that the inequality   L t N f (x) − C(x), 8(1 − L) (3.13) 0 ≥ min{N (c1 φ(x, 0), t), N 0 (c2 φ(0, x), t), N 0 (c3 φ(x, −x), t)} holds for all x ∈ X and all t > 0, where c1 = max{a1 , b1 }, c2 = max{1, a2 , b2 }, and c3 = max{1, a3 , b3 }. Proof. By (??), we get   7L  L  N f (x) − 8f (2−1 x), t ≥ ψ 2−1 x, t 24 8 ≥ min{N 0 (c1 φ(x, 0), t), N 0 (c2 φ(0, x), t), N 0 (c3 φ(x, −x), t)} for all x ∈ X and all t > 0. By Theorem 3.2, there exists an unique cubic mapping C : X −→ Y with (3.13).  From now on, we consider the following functional equation f (x + 2y) − 3f (x + y) + 3f (x) − f (x − y) − 6f (y) (3.14) + k[f (2x + y) + f (2x − y) − 2f (x + y) − 2f (x − y) − 12f (x)] = 0 for some positive real number k. Lemma 3.5. [12] A mapping f : X −→ Y satisfies (3.14) if and only if f is a cubic mapping. Using Theorem 2.1, Theorem 3.1, and Theorem 3.2, we have the following example. Example 3.6. Let f : X −→ Y be a mapping such that f (0) = 0 and (3.15) N (f (x + 2y) − 3f (x + y) + 3f (x) − f (x − y) − 6f (y) + k[f (2x + y) + f (2x − y) t − 2f (x + y) − 2f (x − y) − 12f (x)], t) ≥ t + kxk2p + kyk2p + kxkp kykp

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for all x, y ∈ X, all t > 0 and some positive real numbers k, p with p 6= 32 . Then there exists an unique cubic mapping C : X −→ Y such that N (f (x) − C(x), t) ≥

(3.16)

2k|8 − 22p |t 2k|8 − 22p |t + kxk2p

for all x ∈ X. Proof. Let Gf (x, y) = k[f (2x + y) + f (2x − y) − 2f (x + y) − 2f (x − y) − 12f (x)] and φ(x, y) = kxk2p + kyk2p + kxk2p kykp . Then Gf (y, x) = Gf (y, −x) for all x, y ∈ X and f satisfies (3.2). Letting y = 0 in (3.15), we have 1  N (f (2x) − 8f (x), t) ≥ N 0 φ(x, 0), t 2k for all x ∈ X and all t > 0, where ( 0, if t ≤ 0 0 N (r, t) = t , if t>0 t+|r| for all r ∈ R. By Theorem 3.1, and Theorem 3.2, there exists an unique mapping C : X −→ Y with (2.1) and (3.16). Since Gf (y, x) = Gf (y, −x) for all x, y ∈ X , GC (y, x) = GC (y, −x) for all x, y ∈ X and letting y = 0 in DC (x, y) = 0, we have C(2x) = 8C(x) for all x ∈ X. By Theorem 2.1, we have the result.  We can use Corollary 3.3 and Corollary 3.4 to get a classical result in the framework of normed spaces. As an example of φ(x, y) in Corollary 3.3 and Corollary 3.4, we can take φ(x, y) = (kxkp kykp + kxk2p + kyk2p ). Then we can formulate the following example. Example 3.7. Let X be a normed space and Y a Banach space. Suppose that if g satisfies (2.1), then g is a cubic mapping. Let f : X −→ Y be a mapping such that f (0) = 0 and kDf (x, y)k ≤ (kxkp kykp + kxk2p + kyk2p )

(3.17)

for all x, y ∈ X and a fixed positive real numbers p,  with p 6= 23 . Suppose that kGf (0, x)k ≤  max{a1 , a2 , a3 }kxk2p , kGf (x, −x)k ≤  max{b1 , b2 , b3 }kxk2p for all x ∈ X, all t > 0 and for some nonnegative real numbers ai , bi (i = 1, 2, 3). Then there is an unique cubic mapping C : X −→ Y such that kf (x) − C(x)k ≤

7 max{3, a1 , a2 , 3a3 , b1 , b2 , 3b3 }kxk2p 3|8 − 22p |

for all x ∈ X. Proof. Define a fuzzy norm N 0 on R by ( NR (x, t) =

t t+|x| ,

if t > 0

0,

if t ≤ 0

for all x ∈ R and all t > 0. Similary we can define a fuzzy norm NY on Y . Then (Y, NY ) is a fuzzy Banach space. Let φ(x, y) = (kxkp kykp + kxk2p + kyk2p ). Then by definitions NY and N 0 , the following inequality holds : NY (Df (x, y), t) ≥ NR (φ(x, y), t)

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for all x, y ∈ X and all t > 0. By Corollary 3.3 and Corollary 3.4, we have the result.  References [1] T. Aoki, On the stability of the linear transformation in Banach spaces, J. Math. Soc. Japan 2(1950), 64-66. [2] T. Bag and S. K. Samanta, Finite dimensional fuzzy normed linear spaces, J. Fuzzy Math. 11(2003), 687-705. [3] P.W.Cholewa, Remarkes on the stability of functional equations, Aequationes Math. 27(1984), 76-86. [4] K. Ciepli´ nski, Applications of fixed point theorems to the Hyers-Ulam stability of functional equation-A survey, Ann. Funct. Anal. 3(2012), no. 1, 151-164. [5] S. Czerwik, On the stability of the quadratic mapping in normed spaces, Bull. Abh. Math. Sem. Univ. Hamburg 62(1992), 59-64. [6] J. B. 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. [7] P. Gˇ avruta, A generalization of the Hyer-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] G. Isac and Th. M. Rassias, Stability of ψ-additive mappings, Appications to nonlinear analysis, Internat. J. Math. and Math. Sci. 19(1996), 219-228. [10] 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. [11] A. K. Katsaras, Fuzzy topological vector spaces II, Fuzzy Sets Syst 12(1984), 143-154. [12] C.I. Kim and G. J. Han, Fuzzy stability for a class of cubic, Journal of Intelligent and Fuzzy Systems 33(2017), 37793787 functional equations [13] I. Kramosil and J. Mich´ alek, Fuzzy metric and statistical metric spaces, Kybernetica 11(1975), 336-344. [14] A. K. Mirmostafaee and M. S. Moslehian, Fuzzy versions of Hyers-Ulam-Rassias theorem, Fuzzy Sets Syst. 159(2008), 720-729. [15] A. K. Mirmostafaee, M. Mirzavaziri, and M. S. Moslehian, Fuzzy stability of the Jensen functional equation, Fuzzy Sets Syst. 159(2008), 730-738. [16] M. Mirzavaziri and M. S. Moslehian, A fixed point approach to stability of a quadratic equation, Bulletin of the Brazilian Mathematical Society 37(2006), 361-376. [17] M. S. Moslehian and T. H. Rassias, Stability of functional equations in non-Archimedean spaces, Applicable Anal. Discrete Math. 1(2007), 325-334. [18] Th. M. Rassias, On the stability of the linear mapping in Banach sapces, Proc. Amer. Math. Sco. 72(1978), 297-300. [19] J. M. Rassias, Solution of the Ulam stability problem for cubic mappings, Glasnik Matematiˇ cki 36(2001), 63-72. [20] S. M. Ulam, Problems in modern mathematics, Wiley, New York, 1960, Chapter VI. Department of Mathematics Education, Dankook University, 152, Jukjeon-ro, Sujigu, Yongin-si, Gyeonggi-do, 16890, Korea E-mail address: kci206@@hanmail.net Department of Mathematics Education, Dankook University, 152, Jukjeon-ro, Sujigu, Yongin-si, Gyeonggi-do, 16890, Korea E-mail address: [email protected]

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DRYGAS FUNCTIONAL EQUATIONS WITH EXTRA TERMS AND ITS STABILITY YOUNG JU JEON AND CHANG IL KIM∗

Abstract. In this paper, we consider the generalized Hyers-Ulam stability for the following functional equation with an extra term Gf f (x + y) + f (x − y) + Gf (x, y) = 2f (x) + f (y) + f (−y), where Gf is a functional operator of f .

1. Introduction and preliminaries In 1940, Ulam [12] 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 an unique homomorphism h : G1 −→ G2 with d(f (x), h(x)) < δ for all x ∈ G1 ?” In 1941, Hyers [6] 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 p < 1 and for all x, y ∈ X, where f : X −→ Y is a function between Banach spaces. The paper of Rassias [11] has provided a lot of influence in the development of what we call the generalized Hyers-Ulam stability or Hyers-Ulam-Rassias stability of functional equations. A generalization of the Rassias theorem was obtained by Gˇavruta [5] 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. A generalized Hyers-Ulam stability problem for the quadratic functional equation was proved by Skof [10] 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 the quadratic functional equation and Park [9] proved the generalized Hyers-Ulam stability of the quadratic functional eqution in Banach modules over a C ∗ -algebra. 2010 Mathematics Subject Classification. 39B52, 39B82. Key words and phrases. Hyers-Ulam Stability, Banach Space. * Corresponding author. 1

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In this paper, we are interested in what kind of terms can be added to the Drygas functional equation [4] f (x + y) + f (x − y) = 2f (x) + f (y) + f (−y) while the generalized Hyers-Ulam stability still holds for the new functional equation. We denote the added term by Gf (x, y) which can be regarded as a functional operator depending on the variables x, y, and functions f . Then the new functional equation can be written as (1.2)

f (x + y) + f (x − y) + Gf (x, y) = 2f (x) + f (y) + f (−y).

In fact, the functional operator Gf (x, y) was introduced and considered in the cases of additive, quadratic functional equations with somewhat different point of view by the authors([7], [8]). 2. Solutions of 1.2 as additive-quadratic mappings Let X and Y be normed spacese. For given l ∈ N and any i ∈ {1, 2, · · ·, l}, let σi : X × X −→ X be a binary operation such that σi (rx, ry) = rσi (x, y) for all x, y ∈ X and all r ∈ R. It is clear that σi (0, 0) = 0. Also let F : Y l −→ Y be a linear, continuous function. For a map f : X −→ Y , define Gf (x, y) = F (f (σ1 (x, y)), f (σ2 (x, y)), · · ·, f (σl (x, y))). From now on, for any mapping f : X −→ Y , we deonte f (x) − f (−x) f (x) + f (−x) , fe (x) = 2 2 First, we consider the following functional equation fo (x) =

(2.1)

af (x + y) + bf (x − y) − cf (y − x) = (a + b)f (x) − cf (−x) + (a − c)f (y) + bf (−y)

for fixed real numbers a, b, c with a = b − c and a 6= 0. We can easily show the following lemma. Lemma 2.1. Let f : X −→ Y be a mapping. Then f satisfies (2.1) if and only if f is an additive-quadratic mapping. Definition 2.2. The functional operator G is called additive-quadratic if whenever Gh (x, y) = 0 for all x, y ∈ X, h is an additive-quadratic mapping. Lemma 2.3. Let f : X −→ Y be a mapping satisfying (1.2) and G additvequadratic. Then the following are equivalent : (1) f is additive-quadratic, (2) the following equality (2.2)

Gf (x, y) = −Gf (y, x)

holds for all x, y ∈ X, and (3) there exist real numbers b, c such that b 6= c and (2.3)

bGf (x, y) = cGf (y, x)

holds for all x, y ∈ X.

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Proof. (1) (⇒) (2) (⇒) (3) are trivial. (3) (⇒) (1) By (2.2), we have f (0) = 0 and by (1.2), we have Gf (x, y) = 2f (x) + f (y) + f (−y) − f (x + y) − f (x − y), and Gf (y, x) = 2f (y) + f (x) + f (−x) − f (x + y) − f (y − x) for all x, y ∈ X. Hence by (2.3), we have (b+c)f (x+y)+bf (x−y)−cf (y −x) = (2b+c)f (x)+cf (−x)+(b+2c)f (y)+bf (−y) for all x, y ∈ X and by Lemma 2.1, we have that f is additive-quadratic.



3. The generalized Hyers-Ulam stability of (1.2) In this section, we deal with the generalized Hyers-Ulam stability of (1.2). Throughout this paper, assume that G is additive-quadratic and the following inequalities hold kGh (x, x)k ≤ kGh (0, x)k +

t X

|bi |kGh (δi x, 0)k if h : odd,

i=1

(3.1) kGh (x, x)k ≤

r X

|pi |kGh (0, αi x)k +

s X

|ai |kGh (λi x, 0)k if h : even

i=1

i=1

for some r, s, t ∈ N ∪ {0}, some real numbers pi , ai , bi , αi , λi , and δi and for all x ∈ X. Theorem 3.1. Let φ : X 2 −→ [0, ∞) be a function such that ∞ X

(3.2)

2−n φ(2n x, 2n y) < ∞

n=0

for all x, y ∈ X. Let f : X −→ Y be an odd mapping such that (3.3)

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

for all x, y ∈ X. Then there exists an odd mapping A : X −→ X such that A satisfies (1.2) and (3.4) kA(x) − f (x)k ≤

∞ X

t i h X 2−n−1 φ(2n x, 2n x) + φ(0, 2n x) + |bi |φ(2n δi x, 0) .

n=0

i=1

for all x ∈ X. Further, if Gf satisfies (2.2), then A : X −→ X is an unique additive mapping with (3.4). Proof. By (3.3), we have kGf (x, 0)k ≤ φ(x, 0), kGf (0, x)k ≤ φ(0, x) for all x, y ∈ X. Setting y = x in (3.3), we have (3.5)

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

for all x ∈ X. Hence by (3.1) and (3.5), we have (3.6)

t h i X kf (x) − 2−1 f (2x)k ≤ 2−1 φ(x, x) + φ(0, x) + |bi |φ(δi x, 0) i=1

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for all x ∈ X. By (3.6), we have kf (x) − 2−n f (2n x)k ≤

n−1 X

t h i X 2−k−1 φ(2k x, 2k x) + φ(0, 2k x) + |bi |φ(2k δi x, 0) i=1

k=0

for all x ∈ X and all n ∈ N . For m, n ∈ N ∪ {0} with 0 ≤ m < n, k2−m f (2m x) − 2−n f (2n x)k = 2−m kf (2m x) − 2−(n−m) f (2n−m (2m x))k (3.7) ≤

n−1 X

t h i X 2−k−1 φ(2k x, 2k x) + φ(0, 2k x) + |bi |φ(2k δi x, 0) i=1

k=m −n

n

for all x ∈ X. By (3.2) and (3.7), {2 f (2 x)} is a Cauchy sequence in Y and since Y is a Banach space, there exists a mapping A : X −→ Y such that A(x) = limn→∞ 2−n f (2n x) for all x ∈ X. By (3.7), we have (3.4). Replacing x and y by 2n x and 2n y in (3.3), respectively and deviding (3.3) by n 2 , we have k2−n [f (2n (x + y)) + f (2n (x − y)) + Gf (2n x, 2n y) − 2f (2n x)]k ≤ 2−n φ(2n x, 2n y) for all x, y ∈ X and letting n → ∞, we can show that A satisfies (1.2). Since f is odd, A is odd. Suppose that Gf satisfies (2.2). Then clearly, we can show that GA satisfies (2.2) and hence by Lemma 2.3, A is an additive-quadratic mapping. Since A is odd, A is an additive mapping. Now, we show the uniqueness of A. Let E : X −→ Y be an additive mapping with (3.4). Since A and E are additive, kA(x) − E(x)k = kA(2n x) − E(2n x)k ∞ t i h X X ≤ 2−k 2−n φ(2n x, 2n x) + φ(0, 2n x) + |bi |φ(2n δi x, 0) n=0

i=1

for all x ∈ X and all k ∈ N. Hence, letting k → ∞, by (3.2), we have A = E.



Similar to Theorem 3.1, we have the following theorem. Theorem 3.2. Let φ : X 2 −→ [0, ∞) be a function such that ∞ X

(3.8)

2n φ(2−n x, 2−n y) < ∞

n=0

for all x, y ∈ X. Let f : X −→ Y be an odd mapping satisfying (3.3). Then there exists an odd mapping A : X −→ X such that A satisfies (1.2) and (3.9) kA(x)−f (x)k ≤

∞ X

t h i X 2n−1 φ(2−n x, 2−n x)+φ(0, 2−n x)+ |bi |φ(2−n δi x, 0)

n=0

i=1

for all x ∈ X. Further, if Gf satisfies (2.2), then A : X −→ X is an unique additive mapping with (3.9)

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Proof. By (3.3), we have kGf (x, 0)k ≤ φ(x, 0), kGf (0, x)k ≤ φ(0, x) for all x, y ∈ X. Setting y = x = x2 in (3.5), we have

x x x x  x 

(3.10) − 2f ,

f (x) + Gf

≤φ , 2 2 2 2 2 for all x ∈ X. Hence by (3.1), (3.3), and (3.10), we have (3.11)

t

 x  X

≤ φ(x, x) + φ(0, x) + |bi |φ(δi x, 0) (x) − 2f

f

2 i=1

for all x ∈ X. By (3.11), we have kf (x) − 2n f (2−n x)k ≤

n−1 X

t h i X 2k φ(2−k x, 2−k x) + φ(0, 2−k x) + |bi |φ(2−k δi x, 0) i=1

k=0

for all x ∈ X and all n ∈ N . For m, n ∈ N ∪ {0} with 0 ≤ m < n, k2m f (2−m x) − 2n f (2−n x)k = 2m kf (2−m x) − 2(n−m) f (2−(n−m) (2−m x))k (3.12) ≤

n−1 X

t h i X 2k φ(2−k x, 2−k x) + φ(0, 2−k x) + |bi |φ(2−k δi x, 0) i=1

k=m n

−n

for all x ∈ X. By (3.12), {2 f (2 is similar to Theorem 3.1.

x)} is a Cauchy sequence in Y . The rest of proof 

Theorem 3.3. Let φ : X 2 −→ [0, ∞) be a function such that ∞ X

(3.13)

2−2n φ(2n x, 2n y) < ∞

n=0

for all x, y ∈ X. Let f : X −→ Y be an even mapping such that (3.14)

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

for all x, y ∈ X. Then there exists an even mapping Q : X −→ X such that (3.15) ∞ r s h i X X X kQ(x) − f (x)k ≤ 2−2n−2 φ(2n x, 2n x) + |pi |φ(0, 2n ai x) + |ai |φ(2n λi x, 0) n=0

i=1

i=1

for all x ∈ X. Further, if Gf satisfies (2.2), then Q : X −→ Y is an unique quadratic mapping with (3.15) Proof. Setting y = x in (3.14), we have k22 f (x) − f (2x) + Gf (x, x)k ≤ φ(x, x) for all x ∈ X and by (3.14), we have kGf (x, 0)k ≤ φ(x, 0), kGf (0, x)k ≤ φ(0, x)

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for all x ∈ X. Since f is even, letting y = x in (3.14), by (3.1), we have kf (x) − 2−2 f (2x)k h i ≤ 2−2 φ(x, x) + kGf (x, x)k r s h i X X ≤ 2−2 φ(x, x) + |pi |kGf (0, αi x)k + |ai |kGf (λi x, 0)k i=1

i=1

r s h i X X ≤ 2−2 φ(x, x) + |pi |φ(0, αi x) + |ai |φ(λi x, 0) i=1

i=1

for all x ∈ X. Hence we have kf (x) − 2−2n f (2n x)k (3.16)

≤

n−1 X

r s h i X X 2−2k−2 φ(2k x, 2k x) + |pi |φ(0, 2k ai x) + |ai |φ(2k λi x, 0) i=1

k=0

i=1

for all x ∈ X and all n ∈ N . For m, n ∈ N ∪ {0} with 0 ≤ m < n, by (3.16) k2−2m f (2m x) − 2−2n f (2n x)k = 2−2m kf (2m x) − 2−2(n−m) f (2n−m (2m x))k (3.17) ≤

n−1 X

s r i h X X |ai |φ(2k λi x, 0) 2−2k−2 φ(2k x, 2k x) + |pi |φ(0, 2k ai x) + i=1

k=m

i=1

for all x ∈ X. By (3.17), {2−2n f (2n x)} is a Cauchy sequence in Y . The rest of proof is similar to Theorem 3.1.  Theorem 3.4. Let φ : X 2 −→ [0, ∞) be a function such that ∞ X

(3.18)

22n φ(2−n x, 2−n y) < ∞

n=0

for all x, y ∈ X. Let f : X −→ Y be an even mapping satisfying (3.14). Then there exists an even mapping Q : X −→ X such that (3.19) ∞ r s h i X X X kQ(x)−f (x)k ≤ 22n φ(2−n x, 2−n x)+ |pi |φ(0, 2−n ai x)+ |ai |φ(2−n λi x, 0) n=0

i=1

i=1

for all x ∈ X. Further, if Gf satisfies (2.2), then Q : X −→ Y is an unique quadratic mapping with (3.19) Proof. Setting y = x = x2 in (3.14), we have

   x x  x x

2 x

− f (x) + Gf ,

2 f

≤φ , 2 2 2 2 2 for all x ∈ X. By (3.14), we have kGf (x, 0)k ≤ φ(x, 0), kGf (0, x)k ≤ φ(0, x)

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for all x ∈ X and so, we have

   x x   x x 

2 x

− f (x) ≤ φ , + Gf ,

2 f

2 2 2 2 2 s r

 x 

 x x X x 

X

≤φ , + |ai | Gf λi , 0 |pi | Gf 0, αi + 2 2 2 2 i=1 i=1 ≤φ

r s x x X   x  x X + + , |pi |φ 0, αi |ai |φ λi , 0 2 2 2 2 i=1 i=1

for all x ∈ X. Similar to Theorem 3.1, we have the result.



Theorem 3.5. Let φ : X 2 −→ [0, ∞) be a function with (3.2). Let f : X −→ Y be a mapping with (3.3). Then there exists a mapping F : X −→ X such that F satisfies (1.2) and kF (x) − f (x)k ∞ r s h i X X X ≤ 2−2n−2 φ1 (2n x, 2n x) + |pi |φ1 (0, 2n x) + |ai |φ1 (λi 2n x, 0) (3.20)

n=0 ∞ X

+

i=1

h

2−n−1 φ1 (2n x, 2n x) + φ1 (0, 2n x) +

n=0

i=1 t X

i |bi |φ1 (δi 2n x, 0)

i=1

h

i for all x ∈ X, where φ1 (x, y) = 21 φ(x, y) + φ(−x, −y) . Further, if Gf satisfies (2.2), then F : X −→ X is an unique additive-quadratic mapping with (3.20) Proof. By (3.3), we have (3.21)

kfe (x + y) + fe (x − y) + Gfe (x, y) − 2fe (x) − 2fe (y)k ≤ φ1 (x, y)

for all x, y ∈ X. By Theorem 3.3, there exists an even mapping Q : X −→ Y such that Q(x) = limn−→∞ 2−2n fe (2n x) for all x ∈ X, Q(x + y) + Q(x − y) + GQ (x, y) = 2Q(x) + 2Q(y)

(3.22)

for all x, y ∈ X, and (3.23)

kQ(x) − fe (x)k ∞ r s h i X X X 2−2n−2 φ1 (2n x, 2n x) + |pi |φ1 (0, 2n ai x) + |ai |φ1 (2n λi x, 0) ≤ n=0

i=1

i=1

for all x ∈ X. Similarly, there exists an odd mapping A : X −→ Y such that A(x) = limn−→∞ 2−n fo (2n x) for all x ∈ X, (3.24)

A(x + y) + A(x − y) + GA (x, y) − 2A(x) = 0

for all x, y ∈ X, and (3.25) kA(x)−fo (x)k ≤

∞ X

t h i X 2−n−1 φ1 (2n x, 2n x)+φ1 (0, 2n x)+ |bi |φ1 (2n δi x, 0)

n=0

i=1

for all x ∈ X. Let F = Q+A. Since Q is even and A is odd, 2Q(y) = F (y)+F (−y) and by (3.22) and (3.24), F satisfies (1.2). Since kF (x)−f (x)k ≤ kQ(x)−fe (x)k+kA(x)−fo (x)k, by (3.23) and (3.25), we have (3.20).

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Suppose that Gf satisfies (2.2). Then clearly, we can show that GF satisfies (2.2) and hence by Lemma 2.3, F is an additive-quadratic mapping. The proof of the uniqueness of F is similar to Theorem 3.1.  Theorem 3.6. Let φ : X 2 −→ [0, ∞) be a function such that ∞ X 2n φ(2−n x, 2−n y) < ∞ n=0

for all x, y ∈ X. Let f : X −→ Y be a mapping with (3.3). Then there exist a mapping F : X −→ X such that (3.26) kF (x) − f (x)k ∞ r s h i X X X ≤ 22n−2 φ1 (2−n x, 2−n x) + |pi |φ1 (0, 2−n x) + |ai |φ1 (λi 2−n x, 0) n=0 ∞ X

+

i=1

i=1

h

2n−1 φ1 (2−n x, 2−n x) + φ1 (0, 2−n x) +

n=0

t X

i |bi |φ1 (δi 2−n x, 0)

i=1

i φ(x, y) + φ(−x, −y) . Further, if Gf satisfies (2.2), then F : X −→ X is an unique additive-quadratic mapping with (3.26). for all x ∈ X, where φ1 (x, y) =

1 2

h

4. Applicaions In this section, we illustrate how the theorems in section 3 work well for the generalized Hyers-Ulam stability of various additive-quadratic functional equations. As examples of φ(x, y) in Theorem 3.5 and Theorem 3.6, we can take φ(x, y) = (kxkp kykp + kxk2p + kyk2p ). Then we can formulate the following theorem : Theorem 4.1. Assume that all of the conditions in Theorem 3.1 hold and Gf satisfies (2.2). Let p be a real number with 0 < p < 12 , 1 < p. Let f : X −→ Y be a mapping such that (4.1) kf (x+y)+f (x−y)−2f (x)−f (y)−f (−y)+Gf (x, y)k ≤ (kxkp kykp +kxk2p +kxk2p ) for all x, y ∈ X. Then there exists an unique additive-quadratic mapping F : X −→ Y such that ( Ψ1 (x), if 0 < p < 21 kF (x) − f (x)k ≤ Ψ2 (x), if 1 < p for all x ∈ X, where r s t h i  h i  X X X 2p 2p Ψ1 (x) = 3 + |pi | + |ai ||λi |2p kxk + 4 + |b ||δ | kxk2p i i p p 4 − 4 2 − 4 i=1 i=1 i=1 and h

Ψ2 (x) = 3 +

r X i=1

|pi | +

s X i=1

2p

|ai ||λi |

i 4p−1  4p − 4

kxk

2p

t h i 22p−1  X + 4+ |bi ||δi |2p p kxk2p 4 − 2 i=1

Lemma 4.2. Let G be the operator defined by Gf (x, y) = f (2x + y) − f (x + 2y) + f (x − y) − f (y − x) − 3f (x) + 3f (y) for all mapping f : X −→ Y . Then G is additive-quadratic.

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DRYGAS FUNCTIONAL EQUATIONS WITH EXTRA TERMS AND ITS STABILITY

9

Proof. Suppose that Gf (x, y) = 0 for all x, y ∈ X. Then we have f (2x + y) − f (x + 2y) + f (x − y) − f (y − x) − 3f (x) + 3f (y) = 0.

(4.2)

and so we have fe (2x + y) − fe (x + 2y) − 3fe (x) + 3fe (y) = 0

(4.3)

for all x, y ∈ X and letting y = y − x in (4.3), we have fe (x + y) − fe (x − 2y) − 3fe (x) + 3fe (x − y) = 0

(4.4)

for all x, y ∈ X. Letting y = −y in (4.4), we have fe (x − y) − fe (x + 2y) − 3fe (x) + 3fe (x + y) = 0

(4.5)

for all x, y ∈ X. By (4.4) and (4.5), we have fe (x+2y)+fe (x−2y)−2fe (x)−8fe (y)−4[fe (x+y)+fe (x−y)−2fe (x)−2fe (y)] = 0 for all x, y ∈ X and so fe is quadratic. Since fo is an odd mapping, by (4.2), we have fo (2x + y) − fo (x + 2y) + 2fo (x − y) − 3fo (x) + 3fo (y) = 0

(4.6)

for all x, y ∈ X and letting y = −x − y in (4.6), we have fo (x − y) + fo (x + 2y) + 2fo (2x + y) − 3fo (x) − 3fo (x + y) = 0

(4.7)

for all x, y ∈ X. By (4.6) and (4.7), we have fo (2x + y) + fo (x − y) − 2fo (x) + fo (y) − fo (x + y) = 0

(4.8)

for all x, y ∈ X and letting y = −y in (4.10), we have fo (2x − y) + fo (x + y) − 2fo (x) − fo (y) − fo (x − y) = 0

(4.9)

for all x, y ∈ X. By (4.10) and (4.9), we have fo (2x + y) + fo (2x − y) − 4fo (x) = 0

(4.10)

for all x, y ∈ X and hence fo is additive. Thus f is an additive-quadratic mapping.  By Lemma 2.3, Theorem 4.1, and Lemma 4.2, we have the following theorem : Theorem 4.3. Let f : X −→ Y be a mapping such that kf (x + 2y) − f (2x + y) + f (x + y) + f (y − x) + f (x) − 4f (y) − f (−y)k ≤ (kxkp kykp + kxk2p + kxk2p ) for all x, y ∈ X and some a real number p with 0 < p < 21 , 1 < p. Then there exists an unique additive-quadratic mapping F : X −→ Y such that i h 3 4 2p  + if 0 < p < 12 p p  2−4 kxk ,  4−4 kF (x) − f (x)k ≤

h    3×4p−1 + 4p −4

2×4p 4p −2

i kxk2p ,

if 1 < p

for all x ∈ X. Proof. For a mapping h : X −→ Y , let Gh (x, y) = h(2x + y) − h(x + 2y) + h(x − y) − h(y − x) − 3h(x) + 3h(y). By Lemma 4.2, G is additive-quadratic and f, G satisfiy (4.1). Since Gf satisfies (2.2) in Lemma 2.3, by Theorem 4.1, we have the result. 

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YOUNG JU JEON AND CHANGIL KIM

References [1] T. Aoki, On the stability of the linear transformation in Banach spaces, J. Math. Soc. Japan 2(1950), 64-66. [2] P.W.Cholewa, Remarkes on the stability of functional equations, Aequationes Math. 27(1984), 76-86. [3] S. Czerwik, On the stability of the quadratic mapping in normed spaces, Bull. Abh. Math. Sem. Univ. Hamburg 62(1992), 59-64. [4] H. Drygas, Quasi-inner products and their applications, In : Advances in Multivariate Statistical Analysis (ed. A. K. Gupta), Reidel Publ. Co., 1987, 13–30. [5] P. Gˇ avruta, A generalization of the Hyer-Ulam-Rassias stability of approximately additive mappings, J. Math. Anal. Appl. 184(1994), 431-436. [6] D. H. Hyers, On the stability of the linear functional equation, Proc. Nat. Acad. Sci. 27(1941), 222-224. [7] C. I. Kim, G. Han, and J. W. Chang, Stability of the generalized version of Euler-Lagrange type quadratic equation, J. Comput. Anal. Appl. 21(2016), 156-169. [8] C. I. Kim, G. Han, and S. A. Shim, Hyers-Ulam Stability for a Class of Quadratic Functional Equations via a Typical Form, Abs. and Appl. Anal. 2013(2013), 1-8. [9] C. G. Park, On the stability of the quadratic mapping in Banach modules, J. Math. Anal. Appl. 276(2002), 135-144. [10] F. Skof, Propriet´ a locali e approssimazione di operatori, Rend. Sem. Mat. Fis. Milano 53(1983), 113-129. [11] Th. M. Rassias, On the stability of the linear mapping in Banach sapces, Proc. Amer. Math. Sco. 72(1978), 297-300. [12] S. M. Ulam, Problems in modern mathematics, Wiley, New York, 1960, Chapter VI. Department of Mathematics Education, College of Education, ChonBuk National University, 567 Baekje-daero, deokjin-gu, Jeonju-si, Jeollabuk-do 54896 Republic of Korea Email address: [email protected] Department of Mathematics Education, Dankook University, 152, Jukjeon-ro, Sujigu, Yongin-si, Gyeonggi-do, 16890, Korea Email address: [email protected]

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J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 29, NO.5, 2021, COPYRIGHT 2021 EUDOXUS PRESS, LLC

Theoretical and Numerical Discussion for the Mixed Integro–Differential Equations M. E. Nasr 1,2 1

1

and M. A. Abdel-Aty

2

Department of Mathematics, Faculty of Science, Benha University, Benha 13518, Egypt

Department of Mathematics, Collage of Science and Arts -Al Qurayyat, Jouf University, Kingdom of Saudi Arabia

Abstract In this paper, we tend to apply the proposed modified Laplace Adomian decomposition method that is the coupling of Laplace transform and Adomian decomposition method. The modified Laplace Adomian decomposition method is applied to solve the Fredholm–Volterra integro–differential equations of the second kind in the space L2 [a, b]. The nonlinear term will simply be handled with the help of Adomian polynomials. The Laplace decomposition technique is found to be fast and correct. Several examples are tested and also the results of the study are discussed. The obtained results expressly reveal the complete reliability, efficiency, and accuracy of the proposed algorithmic rule for solving the Fredholm–Volterra integro–differential equations and therefore will be extended to other problems of numerous nature. Mathematics Subject Classification: 41A10, 45J05, 65R20. Key-Words: Fredholm-Volterra Integro-Differential Equations; Adomian Decomposition Method; Laplace Transform Method; Laplace Adomian Decomposition Method. 1. Introduction Mathematical modeling of real-life problems usually results in functional equations, such as differential, integral, and integro-differential equations. Many mathematical formulations of physical phenomena reduced to integro-differential equations, like fluid dynamics, biological models, chemical mechanics and contact problems, see [6, 14, 19]. Many problems from physics and engineering and alternative disciplines cause linear and nonlinear integral equations. Now, for the solution of those equations several analytical and numerical methods are introduced, however numerical methods are easier than analytical methods and most of the time numerical methods are used to solve these equations we refer to [1, 2, 18]. Laplace Adomians decomposition method was first introduced by Suheil A. Khuri [16,17] and has been with successfully used to find the solution of differential equations [20]. This method generates a solution in the form of a series whose terms are determined by a recursive relation using the Adomian polynomials. 880

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Most of the nonlinear integro-differential equations don’t have an exact analytic solution, therefore approximation and numerical technique should be used, there are only a number of techniques for the solution of integro-differential equations, since it’s relatively a new subject in arithmetic. The modified laplace decomposition technique has applied for solving some nonlinear ordinary, partial differential equations. Recently, the authors have used many methods for the numerical or the analytical solution of linear and nonlinear Fredholm and Volterra integral and integrodifferential equations of the second kind [8, 9, 11, 12, 21]. In this paper, we consider the Fredholm–Volterra integro–differential equations of the second kind with continuous kernels with respect to position. We applied Laplace transform and Adomian polynomials to solve nonlinear Fredholm–Volterra integro–differential equations. Al– Towaiq and Kasasbeh [7] have applied the modification of Laplace decomposition method to solve linear interval Fredholm integro–differential equations of the form : Z b 0 k(x, t)u(t)dt; u(a) = α. u (x) = f (x) + a

But in this paper, we will study the modification of Laplace Adomian decomposition method to solve the nonlinear interval Fredholm–Volterra integro–differential equation of the form: Z b Z u φ(u + q) = p(u) + λ k(u, v)µ(v, φ(v))dv + λ ψ(u, v)ν(v, φ(v))dv; (q 0, Z u φ(v)ψ(u − v)dv, (φ ∗ ψ)(u) =

(5)

0

the function φ ∗ ψ is called the convolution of φ and ψ. Theorem 1. The convolution theorem L[φ ∗ ψ](u) = L[φ(u)] ∗ L[ψ(u)].

(6)

Lemma 1. Laplace Transform of an Integral: If Φ(s) = L[φ(u)] then Z u  Φ(s) L φ(v)dv = . s 0

(7)

Theorem 2. The Laplace transform L[φ(u)] of the derivatives are defined by L[φ(n) (u)] = sn L[φ(u)] − sn−1 φ(0) − sn−2 φ0 (0) − · · · − φ(n−1) (0).

(8)

2.2 Adomians Decomposition method Consider the general functional equation: φ = p + N1 φ + N2 φ, 882

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where N1 , N2 are a nonlinear operators, p is a known function, and we are seeking the solution φ satisfying (9). We assume that for every p , Eq. (9) has one and only one solution. The Adomians technique consists of approximating the solution of (9) as an infinite series φ=

∞ X

φn ,

(10)

n=0

and decomposing the nonlinear operators N1 , N2 as respectively N1 φ =

∞ X

An ,

N2 φ =

n=0

∞ X

Bn ,

(11)

n=0

where An , Bn are polynomials (called Adomian polynomials)of {φ0 , φ1 , . . . , φn } [4, 5] given by " !# ∞ X 1 dn N1 λ i φi ; n = 0, 1, 2, . . . An = n! dλn i=0 " !#λ=0 ∞ n X 1 d Bn = N2 λ i φi ; n = 0, 1, 2, . . . n! dλn i=0 λ=0 P∞ P P∞ The proofs of the convergence of the series n=0 φn , ∞ n=0 Bn are given in [3, 13]. n=0 An and Substituting (10) and (11) into (9) yields, we get ∞ X

φn = p +

n=0

∞ X

An +

n=0

∞ X

Bn .

n=0

Thus, we can identify φ0 =p, φn+1 =An (φ0 , φ1 , . . . , φn ) + Bn (φ0 , φ1 , . . . , φn );

n = 0, 1, 2, . . .

Thus all components of φ can be calculated once the An , Bn are given. We then define the n-terms approximate to the solution φ by Ψn [φ] =

n−1 X

φi

, with

i=0

lim Ψn [φ] = φ.

n→∞

3. Description of the Method The purpose of this section is to discuss the use of modified Laplace decomposition algorithm for the Fredholm–Volterra integro–differential equation. Applying the Laplace transform (denoted by L) on the both sides of the equation yield (2), we have   Z b  dφ(u) L[φ(u)] + qL =L[p(u)] + λL k(u, v)µ(v, φ(v))dv du a Z u  + λL ψ(u, v)ν(v, φ(v))dv ,

(12)

0

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using the differentiation property of Laplace transform (8) we get Z b  L[φ(u)] + qsL[φ(u)] − qφ(0) =L[p(u)] + λL k(u, v)µ(v, φ(v))dv a  Z u ψ(u, v)ν(v, φ(v))dv . + λL

(13)

0

Thus, the given equation is equivalent to  Z b qφ(0) L[p(u)] λ L[φ(u)] = k(u, v)µ(v, φ(v))dv + + L (1 + qs) (1 + qs) (1 + qs) a Z u  λ + L ψ(u, v)ν(v, φ(v))dv . (1 + qs) 0

(14)

The Adomian decomposition method and the Adomian polynomials can be used to handle (14) and to address the nonlinear terms µ(v, φ(v)), ν(v, φ(v)). We first represent the linear term φ(u) at the left side by an infinite series of components given by φ(u) =

∞ X

φn (u),

(15)

n=0

where the components φn ; n ≥ 0 will be determined recursively. However, the nonlinear terms µ(v, φ(v)), ν(v, φ(v)) at the right side of Eq. (14) will be represented by an infinite series of the Adomian polynomials An , Bn respectively in the form µ(v, φ(v)) =

∞ X

An (v),

ν(v, φ(v)) =

∞ X

Bn (v),

(16)

n=0

n=0

where An , Bn ; n ≥ 0 are defined by " !# ∞ X 1 dn An = µ λi φi ; n! dλn i=0 " !# λ=0 ∞ n X 1 d Bn = ν λi φi ; n! dλn i=0

n = 0, 1, 2, . . . n = 0, 1, 2, . . .

λ=0

where the so-called Adomian polynomials An , Bn can be evaluated for all forms of nonlinearity [22]. In other words, assuming that the nonlinear function is µ(v, φ(v)), ν(v, φ(v)), therefore the Adomian polynomials are given by A0 = µ(φ0 ),

B0 = ν(φ0 ),

A1 = φ1 µ0 (φ0 ),

B1 = φ1 ν 0 (φ0 ),

1 A2 = φ2 µ0 (φ0 ) + φ21 µ00 (φ0 ), 2

1 B2 = φ2 ν 0 (φ0 ) + φ21 ν 00 (φ0 ). 2

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Substituting (15) and (16) into (14), we will get "Z # "∞ # ∞ b X X L[p(u)] λ qφ(0) + + L k(u, v) An (v)dv L φn (u) = (1 + qs) (1 + qs) (1 + qs) a 0 0 "Z # ∞ u X λ + L ψ(u, v) Bn (v)dv . (1 + qs) 0 0

(17)

The Adomian decomposition method presents the recursive relation L[p(u)] λ qφ(0) + + , (1 + qs) (1 + qs) (1 + qs) Z b Z u   λ λ L[φ1 (u)] = L L k(u, v)A0 (v)dv + ψ(u, v)B0 (v)dv , (1 + qs) (1 + qs) a 0   Z b Z u λ λ k(u, v)A1 (v)dv + ψ(u, v)B1 (v)dv . L L L[φ2 (u)] = (1 + qs) (1 + qs) a 0 L[φ0 (u)] =

In general, the recursive relation is given by  Z b λ L[φn+1 (u)] = k(u, v)An (v)dv L (1 + qs) a  Z u λ + ψ(u, v)Bn (v)dv , L (1 + qs) 0

(18) (19) (20)

(21) n = 0, 1, 2, . . .

A necessary condition for Eq. (21) to work is that λ = 0. s→∞ (1 + qs) lim

Applying inverse Laplace transform to Eqs. (18)–(21), so our required recursive relation φ0 (u) = G(u),

(22)

Z b  λ φn+1 (u) =L L k(u, v)An (v)dv (1 + qs)  Za u  λ −1 +L L ψ(u, v)Bn (v)dv , (1 + qs) 0

(23)

and −1



where G(u) may be a function that arises from the source term and also the prescribed initial conditions, the initial solution is very important, the choice of (22) as the initial solution always leads to noise oscillation during the iteration procedure, the modified laplace decomposition method [15] suggests that the operate G(u) defined above in (18) be rotten into two parts: G(u) = G1 (u) + G2 (u). 885

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Instead of iteration procedure (22) and (23), we suggest the following modification φ0 (u) = G1 (u),  Z b λ k(u, v)A0 (v)dv L φ1 (u) = G2 (u) + L (1 + qs) a   Z u λ −1 L +L ψ(u, v)B0 (v)dv , (1 + qs) 0  Z b  λ −1 φn+1 (u) = L L k(u, v)An (v)dv (1 + qs) a  Z u  λ −1 L ψ(u, v)Bn (v)dv , n = 0, 1, 2, . . . +L (1 + qs) 0 −1



We then define the n-terms approximate to the solution φ(u) by Ψn [φ(u)] =

n−1 X

φi (u),

i=0

with

lim Ψn [φ(u)] = φ(u).

n→∞

In this paper, the obtained series solution converges to the exact solution. 3.1 A Test of Convergence In fact, on every interval the inequality kφi+1 k2 < βkφi k2 is required to hold for i = 0, 1, . . . , n, wherever 0 < β < 1 may be a constant and n is that the maximum order of the approximate used in the computation. Of course, this is often only a necessary condition for convergence, as a result of it might be necessary to compute kφi k2 for each i = 0, 1, . . . , n so as to conclude that the series is convergent. 4. Application of the Laplace transform–Adomian decomposition method In this section, the Laplace transform–Adomian decomposition method for solving Fredholm– Volterra integro–differential equation is illustrated in the two examples given below. To show the high accuracy of the solution results from applying the present method to our problem (2) compared with the exact solution, the maximum error is defined as: Rn = kφExact (u) − Ψn [φ(u)]k∞ , where n = 1, 2, . . . represents the number of iterations. Example 1 Consider the nonlinear Fredholm–Volterra integro–differential equation Z Z u 1 1 2 φ(u + 0.2) = p(u) + cos(u)φ (v)dv + φ3 (v)dv, 4 0 0 886

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where 1 (−3 − u3 cos(u)). 12 Using Taylor Expansion after neglecting the second derivative in the equation (24) we get, Z u Z 1 1 dφ(u) 2 = p(u) + cos(u)φ (v)dv + φ3 (v)dv; φ(0) = 0. (25) φ(u) + 0.2 du 4 0 0 p(u) =

The exact solution for this problem is φ(u) = cos(u) − sin(u). First, we apply the Laplace transform to both sides of (25)   Z u Z 1   dφ(u) 1 3 2 φ (v)dv , cos(u)φ (v)dv + L L[φ(u)] + 0.2L = L[p(u)] + L du 4 0 0 Using the property of Laplace transform and the initial conditions, we get   Z u Z 1 1 3 2 L[φ(u)] + 0.2sL[φ(u)] = L[p(u)] + L φ (v)dv , cos(u)φ (v)dv + L 4 0 0

(26)

(27)

or equivalently L[p(u)] 1 L[φ(u)] = + L 1 + 0.2s 4 + 0.8s

Z 0

1



1 cos(u)φ (v)dv + L 1 + 0.2s 2

Z

u



3

φ (v)dv .

(28)

0

Substituting the series assumption for φ(u) and the Adomian polynomials for φ2 (u), φ3 (u) as given above in (15) and (16) respectively into Eq. (28) we obtain "∞ # # "Z ∞ 1 X X L[p(u)] 1 L φn (u) = + L cos(u) An (v)dv 1 + 0.2s 4 + 0.8s 0 n=0 n=0 "Z ∞ # uX 1 + L Bn (v)dv . 1 + 0.2s 0 n=0

(29)

The recursive relation is given below L[p(u)] , 1 + 0.2s Z 1  Z u  1 1 L cos(u)A0 (v)dv + L B0 (v)dv , L[φ1 (u)] = 4 + 0.8s 1 + 0.2s 0 0 Z 1  Z u  1 1 L[φn+1 (u)] = L cos(u)An (v)dv + L Bn (v)dv , 4 + 0.8s 1 + 0.2s 0 0 L[φ0 (u)] =

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(30)

J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 29, NO.5, 2021, COPYRIGHT 2021 EUDOXUS PRESS, LLC

where An , Bn are the Adomian polynomials for the nonlinear terms φ2 (u), φ3 (u) respectively. The Adomian polynomials for µ(v, φ(v)) = φ2 (u), ν(v, φ(v)) = φ3 (u) are given by A0 = φ20 ,

B0 = φ30 ,

A1 = 2φ0 φ1 ,

B1 = 3φ20 φ1 ,

A2 = 2φ0 φ2 + φ21 ,

B2 = 3φ20 φ2 + 3φ0 φ21 , B3 = 3φ20 φ3 + 6φ0 φ1 φ2 + φ31 .

A3 = 2φ0 φ3 + 2φ1 φ2 ,

Taking the inverse Laplace transform of both sides of the first part of (30), and using the recursive relation (30) gives 1 1 φ0 (u) = 1 − u − u2 + u3 + u4 − . . . 2 12 1 2 1 3 1 4 1 5 φ1 (u) = u − u − u + u + . . . 2 3 8 6 1 4 1 5 φ2 (u) = u − u + . . . 12 12

(31)

Thus the series solution is given by n−1 X

   1 3 1 5 1 2 1 4 n = 1, 2, . . . Ψn [φ(u)] = φi (u) = 1 − u + u + . . . − u − u + u + . . . 2! 4! 3! 5! i=0     1 2 1 4 1 3 1 5 φ(u) = lim Ψn [φ(u)] = lim 1 − u + u + ... − u − u + u + ... , n→∞ n→∞ 2! 4! 3! 5! 

that converges to the exact solution φ(u) = cos(u) − sin(u). Example 2 Consider the nonlinear Fredholm–Volterra integro–differential equation Z 1 Z u φ(u + 0.01) = p(u) + φ(v)dv + e−u φ2 (v)dv, 0

(32)

0

where 1 p(u) = 1 − e−u + 0.0100502eu . 4 Using Taylor Expansion after neglecting the second derivative in the equation (32) we get, Z 1 Z u dφ(u) φ(u) + 0.01 = p(u) + φ(v)dv + e−u φ2 (v)dv; φ(0) = 1. (33) du 0 0 The exact solution for this problem is φ(u) = eu . 888

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First, we apply the Laplace transform to both sides of (33)  Z u   Z 1  dφ(u) −u 2 φ(v)dv + L e φ (v)dv , = L[p(u)] + L L[φ(u)] + 0.01L du 0 0 using the property of Laplace transform and the initial conditions, we get  Z u  Z 1 −u 2 φ(v)dv + L e φ (v)dv , L[φ(u)] + 0.01sL[φ(u)] − 0.01 = L[p(u)] + L 0

(34)

(35)

0

or equivalently Z 1  0.01 L[p(u)] 1 L[φ(u)] = + + L φ(v)dv 1 + 0.01s 1 + 0.01s 1 + 0.01s 0  Z u 1 −u 2 e φ (v)dv . L + 1 + 0.01s 0

(36)

Substituting the series assumption for φ(u) and the Adomian polynomials for φ2 (u) as given above in (15) and (16) respectively into above equation, we obtain "∞ # # "Z ∞ 1X X 0.01 L[p(u)] 1 L φn (u) = φn (v)dv + + L 1 + 0.01s 1 + 0.01s 1 + 0.01s 0 n=0 n=0 "Z # ∞ u X 1 −u + e L An (v)dv , 1 + 0.01s 0 n=0

(37)

the recursive relation is given below 0.01 L[p(u)] + , 1 + 0.01s 1 + 0.01s Z 1  Z u  1 1 −u L[φ1 (u)] = L φ0 (v)dv + L e A0 (v)dv , 1 + 0.01s 1 + 0.01s 0 0 Z 1  Z u  1 1 −u L φn (v)dv + L e An (v)dv , L[φn+1 (u)] = 1 + 0.01s 1 + 0.01s 0 0 L[φ0 (u)] =

(38)

where An are the Adomian polynomials for the nonlinear terms φ2 (u). The Adomian polynomials for µ(v, φ(v)) = φ2 (u) is given by A0 = φ20 , A1 = 2φ0 φ1 , A2 = 2φ0 φ2 + φ21 , A3 = 2φ0 φ3 + 2φ1 φ2 ,

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Taking the inverse Laplace transform of both sides of the first part of (38), and using the recursive relation (38) gives 1 1 13 φ0 (u) = 1 + u − u3 − u4 − u5 + . . . 2 2 40 1 2 2 3 5 4 7 5 φ1 (u) = u + u + u + u + ... 2 3 12 120 1 11 φ2 (u) = u4 + u5 + . . . 8 40

(39)

Thus the series solution is given by n−1 X

  1 2 1 3 1 4 1 5 n = 1, 2, . . . Ψn [φ(u)] = φi (u) = 1 + u + u + u + u + u + . . . 2! 3! 4! 5! i=0   1 2 1 3 1 4 1 5 , φ(u) = lim Ψn [φ(u)] = lim 1 + u + u + u + u + u + ... n→∞ n→∞ 2! 3! 4! 5! that converges to the exact solution φ(u) = eu . 5. Conclusions In this work, the Laplace decomposition technique has been successfully applied to finding the approximate solution of the nonlinear Fredholm–Volterra integro–differential equation. The method is extremely powerful and efficient find analytical moreover as numerical solutions for wide classes of nonlinear Fredholm–Volterra integro–differential equations. It provides a lot of realistic series solutions that converge very rapidly in real physical issues. The main advantage of this technique is that the fact that it provides the analytical solution. Some examples are given and therefore the results reveal that the method is extremely effective. some of the nonlinear equations are examined by the modified technique to Illustrate the effectiveness and convenience of this technique, and in all cases, the modified technique performed excellently. The results reveal that the proposed technique is extremely effective and easy. Acknowledgements The authors are very grateful to Prof. Dr. M. A. Abdou, (Dep. of Maths. Faculty of Education, Alexandria University) for their help and suggestions in the process of numerical calculation. We are also very grateful to the reviewers for their constructive suggestions towards upgrading the quality of the manuscript.

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References [1] M. A. Abdou, M. E. Nasr, M. A. Abdel-Aty, A study of normality and continuity for mixed integral equations, J. of Fixed Point Theory Appl., 20(1) (2018). https://doi.org/10.1007/s11784-018-0490-0 [2] M. A. Abdou, M. E. Nasr, M. A. Abdel-Aty, Study of the Normality and Continuity for the Mixed Integral Equations with Phase-Lag Term, Inter. J. of Math. Analysis, 11 (2017), 787–799. https://doi.org/10.12988/ijma.2017.7798 [3] K. Abbaoui, Y. Cherruault, New Ideas for Proving Convergence of Decomposition Methods, Computers and Mathematics with Applications, 29(7) (1995), 103-108. https://doi.org/10.1016/0898-1221(95)00022-Q [4] G. Adomian, Nonlinear Stochastic Systems Theory and Applications to Physics, Kluwer Academic Publishers, Dordrecht, 1989. https://doi.org/10.1007/978-94-009-2569-4 [5] G. Adomian, Solving Frontier Problems of Physics: The Decomposition Method, Kluwer Academic Publishers, Dordrecht, 1994. [6] A. Alawneh, K. Al–Khaled, M. Al-Towaiq, Reliable Algorithms for Solving IntegroDifferential Equations with Applications, International Journal of Computer Mathematics, 87(2010), 1538–1554. https://doi.org/10.1080/00207160802385818 [7] M. Al-Towaiq, A. Kasasbeh, Modified Algorithm for Solving Linear Integro–Differential Equations of the Second Kind, American Journal of Computational Mathematics, 7(2) (2017), 157. [8] L. M. Delves, J. L. Mohamed, Computational Methods for Integral Equations, New York, London, Cambridge, 1988. [9] M. M. EL-Borai, M. A. Abdou, M. M. EL-Kojok, On a discussion of nonlinear integral equation of type Volterra–Fredholm, J. KSIAM, 10(2) (2006), 59–83. [10] Z. Fang, H. Li, Y. Liu, S. He, An expanded mixed covolume element method for integro– differential equation of Sobolev type on triangular grids, Advances in Difference Equations, (1) 2017, 143. [11] M. A. Golberg, Numerical Solution of Integral Equations, Dover, New York, 1990. [12] C. D. Green, Integral Equation Methods, New York, 1969. 891

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[13] S. Guellal, Y. Cherruault, Practical Formula for Calculation of Adomians Polynomials and Application to the Convergence of the Decomposition Method,International Journal Bio– Medical Computing, 36(3) (1994), 223–228. https://doi.org/10.1016/0020-7101(94)90057-4 [14] A. J. Jerri, Introduction to Integral Equations with Applications. John Wiley & Sons, New York, 1999 . [15] Y. Khan, A effective modification of the Laplace decomposition method for nonlinear equations, Int. J. Nonlin. Sci. Num. Simul., 10 (2009), 1373–1376. [16] S. A. Khuri, A Laplace decomposition algorithm applied to a class of nonlinear differential equations, J. Math. Annl.Appl., 4 (2001), 141–155. [17] S. A. Khuri, A new approach to Bratus problem, Appl. Math. Comp., 147, (2004), 31–136. [18] M. E. Nasr, M. A. Abdel-Aty, Analytical discussion for the mixed integral equations, J. of Fixed Point Theory Appl., 20(3) (2018). https://doi.org/10.1007/s11784-018-0589-3 [19] R. Saadati, B. Raftari, H. Abibi, S. M. Vaezpour, S. Shakeri, A comparison between the variational iteration method and trapezoidal rule for linear integro–differential equations, World Applied Sciences Journal, 4(3) (2008), 321–325. [20] M. I. Syam, A. Hamdan, An efficient method for solving Bratu equations, Appl. Math. Comp., 2 (176), (2006), 704–713. [21] A. M. Wazwaz, A comparison study between the modified decomposition method and the traditional methods for solving nonlinear integral equations, Appl. Math. Comput., 181 (2006), 1703–1712 [22] A. M. Wazwaz, A new algorithm for calculating Adomian polynomials for nonlinear operators, Appl. Math. Comput., 111 (2002), 33–51.

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Representation of the Matrix for Conversion between Triangular Bézier Patches and Rectangular Bézier Patches P. Sabancigil, M. Kara and N. I. Mahmudov Department of Mathematics Eastern Mediterranean University Famagusta, T.R. North Cyprus Mersin 10, Turkey Abstract In this paper we studied Bézier surfaces that are very famous techniques and widely used in Computer Aided Geometric Design. Mainly there are two types of Bézier surfaces which are rectangular and triangular Bézier patches. In this paper we will give a representation for the conversion matrix which converts one type to another.

1

Introduction

The theory of Bézier curves has an important role and they are numerically the most stable among all polynomial bases currently used in CAD systems. On the other hand in these days Bézier surfaces are very famous techniques and widely used in Computer Aided Geometric Design [1]-[13]. Mainly there are two types of Bézier surfaces which are rectangular and triangular Bézier patches and they are de…ned in terms of the univariate Bernstein polynomials Bin (s) = ni si (1 s)n i and the bivariate Bernstein polynomial n n Bi;j;k (u; v; w) = i;j;k ui v j wk where u + v + w = 1: A triangular Bézier patch of degree n with control points Ti;j;k is de…ned by X n T (u; v; w) = Ti;j;k Bi;j;k (u; v; w); u; v; w 0; u + v + w = 1: i+j+k=n

and a rectangular Bézier patch of degree n P (s; t) =

n X n X

m with control points Pi;j is represented by Pij Bin (s)Bjn (t)

0

s; t

1; (see [3])

i=0 j=0

Since the two patches have di¤erent geometric properties it is not easy to use both of them in the same CAD system and conversion of one type to another is needed.

2

Construction of the Conversion Matrices

The following theorem gives the conversion of degree n triangular Bézier patch to degenerate rectangular Bézier patch of degree n n: De…nition 1 For all nonnegative integers x the falling factorial is de…ned by (x)n = x(x

1):::(x

n + 1) =

n Y

(x

(k

1))

k=1

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Theorem 2 A degree n triangular Bézier patch T (u; v; w) can be represented as a degenerate Bézier patch of degree n n : n X n X P (s; t) = Pij Bin (s)Bjn (t); 0 s; t 1 i=0 j=0

where the control points Pij are determined by 0 0 1 Pi0 B B Pi1 C B B C B .. C = A1 A2 :::Ai B @ @ . A Pin

1

Ti0 Ti1 .. . Ti;n

C C C; A

i

i = 0; 1; 2; :::; n:

and Ai (i = 0; 1; :::; n) are degree elevation operators in the form 2

6 6 6 6 Ak = 6 6 6 4

1

0

1 n+1 k

n k n+1 k 2 n+1 k

0 .. . 0 0

0 0

::: ::: ::: .. . 0 0

n k 1 n+1 k

.. . 0 0

.. . 0 0

0 0 0 .. .

0 0 0 .. .

n k n+1 k

1 n+1 k

0

1

3 7 7 7 7 7 7 7 5

(n k+2) (n k+1)

Until now no one has studied the generalization of the product A1 A2 :::Ak mentioned in the above theorem and indeed the product of these matrices is not easy to calculate for di¤erent values of n and k: Here we will give the generalization of this product which will make all the computations easier. Theorem 3 The following formula is true h i (k) A1 A2 :::Ak = Ak = ai;j where

i 1 j 1

(k)

ai;j = (k)n = k(k

1):::(k

(k)i

(n+1) (n k+1)

j

(n

(n)i

1

n + 1) =

n Y

k)j

1

(k

(j

;

; 1)) and

j=1

2

6 6 6 6 Ak = 6 6 6 4

1

0

1 n+1 k

n k n+1 k 2 n+1 k

0 .. . 0 0

.. . 0 0

0 0 n k 1 n+1 k

.. . 0 0

::: ::: ::: .. . 0 0

0 0 0 .. .

0 0 0 .. .

n k n+1 k

1 n+1 k

0

1

3 7 7 7 7 7 7 7 5

:

(n k+2) (n k+1)

Proof. For k = 1; A1 = A1 : Suppose it is true for k, that is A1 A2 :::Ak = Ak : We will show that it also true for k + 1; i.e Ak Ak+1 = Ak+1 :

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Let ci;j be the element at the ith row, j th column of the matrix Ak Ak+1 : ci;j =

nX k+1

(k)

(k+1)

ai;m am;j

m=1

=

i 1 m 1

nX k+1

(k)i

m=1

where i = f1; 2; :::; n + 1g and j = f1; 2; :::; n For j = 1 (…rst column) ci;1 =

nX k+1

m

(n

(n)i

1

k)m

1 (k+1) am;j ;

kg :

(k)

(k+1)

ai;m am;1

m=1

=

i 1 m 1

nX k+1

(k)i

m=1

For i = 1 and j = 1 c1;1 =

nX k+1

(k)

m

(n

(n)i

1

k)m

1 (k+1) am;1 :

(k+1)

a1;m am;1

m=1

= =

0 m 1

nX k+1

m=1 (k+1) a1;1

(k)1

m

(n

k)m

(n)0 (k+1)

= 1 = a1;1

1 (k+1) am;1

:

For i = 2 and j = 1; c2;1 =

nX k+1

(k)

(k+1)

a2;m am;1

m=1

=

nX k+1

1 m 1

nX k+1

n m 1

(k)2

m=1

=

m

(n

k)m

(n)1

1 (k+1) am;1

(k + 1)1 k+1 = : n (n)1

For i = n + 1 and j = 1; cn+1;1 =

(k)n+1

m

(n

k)m

(n)n

m=1

1 (k+1) am;1

(k + 1)k(k 1):::(k n + 2) = n(n 1)(n 2):::1 (k + 1)n = : (n)n For j = 2 (second solumn), for i = 1 and j = 2 c1;2 =

nX k+1

(k+1)

(k)

a1;m am;2

m=1

c1;2 = c1;2 =

nX k+1

m=1 (k+1) a1;2

0 m 1

(k)1

m

(n)0 (k+1)

= 0 = a1;2

895

(n

k)0

(k+1)

am;2

:

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J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 29, NO.5, 2021, COPYRIGHT 2021 EUDOXUS PRESS, LLC

For i = 2 and j = 2 c2;2 =

nX k+1

(k)

(k+1)

ai;m am;2

m=1

=

1 m 1

nX k+1

(k)2

=

(n

k)m

(n)1

m=1

n

m

k n

1

(k+1)

= a2;2

1 (k+1) am;2

:

For i = n + 1 and j = 2 cn+1;2 =

nX k+1

(k)

(k+1)

an+1;m am;n+1

m=1

=

nX k+1

n m 1

(k)n+1

m

(n

k)m

1 (k+1) am;2

(n)n

m=1

n(k + 1)k(k 1):::(k n + 3)(n k = n(n 1)(n 2):::1 n(k + 1)n 1 (n k 1)1 (k+1) = = an+1;2 : (n)n For j = n

k (last column), for i = 1 and j = n c1;n

k

=

nX k+1

(k)

1)

k (k+1) k

a1;m am;n

m=1

= = For i = 2 and j = n

0 m 1

nX k+1

m=1 (k+1) a1;n k

(k)1

m

(n

k)m

(n)0

1 (k+1) am;n k

(k+1) k:

= 0 = a1;n

k c2;n

k

=

nX k+1

(k)

(k+1)

a2;m am;j

m=1

=

1 m 1

nX k+1

(k)2

m

(n

k)m

(n)1

m=1

1 (k+1) am;n k

(k+1) k:

= 0 = a2;n For i = n + 1 and j = n

k cn+1;n

k

=

nX k+1

(k)

(k+1) k

an+1;m am;n

m=1

=

nX k+1

n m 1

(k)n+1

(k+1)

= 1 = an+1;n (n k)

h i (k+1) = ai;j

(n

k)m

(n)n

m=1

Hence, Ak Ak+1 = [ci;j ](n+1)

m

1 (k+1) am;n k

k: (k+1)

(n+1) (n k)

896

; where ai;j

=

(ji 11)(k+1)i

j (n

(n)i

k 1)j

1

:

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Remark 4 Sum of the elements in each row of the matrix Ak is equal to 1: Now in the following theorem we consider the inverse process Theorem 5 A rectangular Bézier patch P (s; t) of degree n patch T (u; v; w) of degree n : X n T (u; v; w) = Ti;j;k Bi;j;k (u; v; w);

n can be represented as a Triangular Bézier

u; v; w

0; u + v + w = 1:

i+j+k=n

where the control points Ti;j;k are determined by 0 B B B @

1

Ti0 Ti1 .. . Ti;n

C C C = Bi Bi A

i

1 :::B1

0

1

Pi0 Pi1 .. .

B B B @

C C C A

Pin

i = 0; 1; 2; :::; n:

and Bi (i = 0; 1; :::; n) are degree elevation operators in the form 2 1 t t 0 0 0 6 0 1 t t 0 0 6 6 0 0 1 t t 0 6 Bk = 6 . .. . . . . .. .. .. .. 6 .. . 6 4 0 0 0 0 1 t t 0 0 0 0 0 1 t

0 0 0 .. .

3

7 7 7 7 7 7 7 0 5 t (n

k+1) (n k+2)

Proof. Indeed P (s; t) =

n X n X

Pi;j Bin (s)Bjn (t)

i=0 j=0

=

n X n X

Pi;j Bin (s) tBjn

i=0 j=0

=

=

1 1 (t)

8

n n t}, L∗ (f ; t) := {x ∈ X|f (x) < t}.

L(f ; t) := {x ∈ X|f (x) ≤ t},

3. Energetic subsets In what follows, let X denote a BE-algebra unless otherwise specified. Definition 3.1. A nonempty subset A of a BE-algebra X is said to be S-energetic if it satisfies (S) (∀a, b ∈ X)(a ∗ b ∈ A ⇒ {a, b} ∩ A 6= ∅). Definition 3.2. A nonempty subset A of a BE-algebra X is said to be F -energetic if it satisfies (F ) (∀x, y ∈ X)(y ∈ A ⇒ {x ∗ y, x} ∩ A 6= ∅).

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J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 29, NO.5, 2021, COPYRIGHT 2021 EUDOXUS PRESS, LLC

Permeable values with applications in BE-algebras

Example 3.3. (1) Let X := {1, a, b, c} be a BE-algebra with the following Cayley table: ∗ 1 a b c

1 1 1 1 1

a a 1 1 1

b b a 1 a

c c a a 1

It is easy to show that A := {b, c} is a S-energetic subset of X. But B := {a} is not an Senergetic subset of X since c ∗ b = a ∈ B and {c, b} ∩ B = ∅. It is routine to verify that C := {c} is an S-energetic subset of X. But it is not an F -energetic subset of X, since c ∈ C and {b ∗ c, b} ∩ C = ∅. (2) Let X := {1, a, b, c} be a BE-algebra with the following Cayley table: ∗ 1 a b c

1 1 1 1 1

a a 1 1 a

b b a 1 a

c c a a 1

It is easy to show that A := {a, b} is an F -energetic subset of X. Theorem 3.4. For any nonempty subset A of X, if A is a subalgebra of a BE-algebra X, then X \ A is an S-energetic a subset of X. Proof. Let a, b ∈ X be such that a ∗ b ∈ X \ A. If {a, b} ∩ (X \ A) = ∅, then a, b ∈ A and so a ∗ b ∈ A since A is a subalgebra of X. This is a contradiction. Thus {a, b} ∩ (X \ A) 6= ∅. Therefore X \ A is an S-energetic subset of X.  Theorem 3.5. For any nonempty subset A of X, if A is a filter of a BE-algebra X, then X \ A is an F -energetic a subset of X. Proof. Let x, y ∈ X be such that y ∈ X \ A. If {x ∗ y, x} ∩ X \ A = ∅, then x ∗ y, x ∈ A and so y ∈ A, since A is a filter of X. This is a contradiction. Therefore {x ∗ y, x} ∩ X \ A 6= ∅. Thus X \ A is an F -energetic subset of X.  Theorem 3.6. Let A be a nonempty subset of a BE-algebra X with 1 ∈ / A. If A is F -energetic, then X \ A is a filter of X. Proof. Obviously, 1 ∈ X \ A. Let x, y ∈ X be such that x ∗ y, x ∈ X \ A. Assume that y ∈ A. Then {x ∗ y, x} ∩ A 6= ∅ by (F ). Hence x ∗ y ∈ A or x ∈ A, which is a contradiction. Therefore y ∈ X \ A. This completes the proof.  Theorem 3.7. If f is a fuzzy filter of a BE-algebra X, then the nonempty lower t-level set L(f ; t) is an F -energetic subset of X for all t ∈ [0, 1].

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J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 29, NO.5, 2021, COPYRIGHT 2021 EUDOXUS PRESS, LLC

Jung Mi Ko, Young Hie Kim, and Sun Shin Ahn

Proof. Assume that L(f ; t) 6= ∅ for t ∈ [0, 1] and let x, y ∈ X be such that y ∈ L(f ; t). Then t ≥ f (y) ≥ min{f (x ∗ y), f (x)}. Hence f (x ∗ y) ≤ t or f (x) ≤ t, i.e., x ∗ y ∈ L(f ; t) or x ∈ L(f ; t). Thus {x ∗ y, x} ∩ L(f ; t) 6= ∅. Therefore L(f ; t) is an F -energetic subset of X.  Corollary 3.8. If f is a fuzzy filter of a BE-algebra X, then the nonempty stronger lower t-level set L∗ (f ; t) is an F -energetic subset of X. Since L(f ; t) ∪ U ∗ (f ; t) = X and L(f ; t) ∩ U ∗ (f ; t) = ∅ for all t ∈ [0, 1], we have the following corollary. Corollary 3.9. If f is a fuzzy filter of a BE-algebra X, then U ∗ (f ; t) is empty set or a filter of X for all t ∈ [0, 1]. For any a, b ∈ X, we consider sets Xab := {x ∈ X|a ∗ (b ∗ x) = 1} and Aba := X \ Xab . Obviously, a, b ∈ / Aba , Aba = Aab and 1 ∈ / Aba . In the following example, we know that there exist a, b ∈ X such that Aba may not be F -energetic. Example 3.10. Let X := {1, a, b, c, d, 0} be a BE-algebra [2] with the following Cayley table ∗ 1 a b c d 0

1 1 1 1 1 1 1

a a 1 1 a 1 1

b b a 1 b a 1

c c c c 1 1 1

d d c c a 1 1

0 0 d c b a 1

Then Adc = {0, b} and it is not F -energetic since b ∈ Adc but {a ∗ b, a} ∩ Adc = ∅. We consider conditions for the set Aba to be F -energetic. Theorem 3.11. If X is a self distributive BE-algebra X, then Aba is F -energetic for all a, b ∈ X. Proof. Let y ∈ Aba for any a, b, y ∈ X. Assume that {x ∗ y, x} ∩ Aba = ∅ for any x ∈ X. Then x∗y ∈ / Aba and x ∈ / Aba and so a ∗ (b ∗ (x ∗ y)) = 1 and a ∗ (b ∗ x) = 1. Using (BE3) and the self distributivity of X, we have 1 =a ∗ (b ∗ (x ∗ y)) = a ∗ ((b ∗ x) ∗ (b ∗ y)) =(a ∗ (b ∗ x)) ∗ (a ∗ (b ∗ y)) = 1 ∗ (a ∗ (b ∗ y)) = a ∗ (b ∗ y) and so y ∈ / Aba . This is a contradiction, and therefore {a ∗ b, a} ∩ Aba 6= ∅. Hence Aba is an F -energetic subset of X for all a, b ∈ X. 

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J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 29, NO.5, 2021, COPYRIGHT 2021 EUDOXUS PRESS, LLC

Permeable values with applications in BE-algebras

Definition 3.12. A fuzzy set f in a BE-algebra X is called an anti fuzzy subalgebra of X if f (x ∗ y) ≤ max{f (x), f (y)} for all x, y ∈ X. A fuzzy set f in a BE-algebra X is called an anti fuzzy filter of X if it satisfies (AF1 ) (∀x ∈ X)(f (1) ≤ f (x)); (AF2 ) (∀x, y ∈ X)(f (y) ≤ max{f (x ∗ y), f (x)}). Proposition 3.13. For any anti fuzzy filter of a BE-algebra X, then following are valid. (i) (∀x, y ∈ X)(x ≤ y ⇒ f (y) ≤ f (x)); (ii) (∀x, y, z ∈ X)(f (x ∗ z) ≤ max{f (x ∗ (y ∗ z)), f (y)}); (iii) (∀a, x ∈ X)(f ((a ∗ x) ∗ x) ≤ f (a)). Proof. (i) Let x, y ∈ X be such that x ≤ y. Then x ∗ y = 1. It follows from Definition 3.12 that f (y) ≤ max{f (x ∗ y), f (x)} = max{f (1), f (x)} = f (x). (ii) Using (AF2 ) and (BE4), we have f (x∗z) ≤ max{f (y∗(x∗z)), f (y)} = max{f (x∗(y∗z)), f (y)} for any x, y, z ∈ X. (iii) Taking y := (a ∗ x) ∗ x and x := a in (AF2 ), we have f ((a ∗ x) ∗ x) ≤ max{f (a ∗ ((a ∗ x) ∗ x)), f (a)} = max{f ((a ∗ x) ∗ (a ∗ x)), f (a)} = max{f (1), f (a)} = f (a) for any a, x ∈ X.  Theorem 3.14. Any fuzzy set of a BE-algebra X satisfying (AF1 ) and Proposition 3.13 (ii) is an anti fuzzy filter of X. Proof. Taking x := 1 in Proposition 3.13 (ii) and (BE3), we have f (z) = f (1 ∗ z) ≤ max{f (1 ∗ (y ∗ z)), f (y)} = max{f (y ∗ z), f (y)} for all y, z ∈ X. Hence f is an anti fuzzy filter of X.  Corollary 3.15. For any fuzzy set f of a BE-algebra X, f is an anti fuzzy filter of X if and only if it satisfies (AF1 ) and Proposition 3.13 (ii). Theorem 3.16. Any fuzzy set f of a BE-algebra X is an anti fuzzy filter of X if and only if it satisfies the following conditions: (i) (∀x, y ∈ X)(f (y ∗ x) ≤ f (x)); (ii) (∀x, a, b ∈ X)(f ((a ∗ (b ∗ x)) ∗ x) ≤ max{f (a), f (b)}). Proof. Assume that f is an anti fuzzy filter of X. It follows from Definition 3.12 that f (y ∗ x) ≤ max{f (x ∗ (y ∗ x)), f (x)} = max{f (1), f (x)} = f (x) for all x, y ∈ X. Using Proposition 3.13, we have f ((a ∗ (b ∗ x)) ∗ x) ≤ max{f ((a ∗ (b ∗ x)) ∗ (b ∗ x)), f (b)} ≤ max{f (a), f (b)} for any a, b, x ∈ X. Conversely, let f be a fuzzy set satisfying conditions (i) and (ii). Setting y := x in (i), we have f (x∗x) = f (1) ≤ f (x) for all x ∈ X. Using (ii), we obtain f (y) = f (1∗y) = f ((x∗y)∗(x∗y))∗y) ≤ max{f (x ∗ y), f (y)} for all x, y ∈ X. Hence f is an anti fuzzy filter of X.  Proposition 3.17. For any fuzzy set of a BE-algebra X, then f is an anti fuzzy filter of X if and only if (∗) (∀x, y, z ∈ X)(z ≤ x ∗ y ⇒ f (y) ≤ max{f (x), f (z)}).

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Jung Mi Ko 902-909

J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 29, NO.5, 2021, COPYRIGHT 2021 EUDOXUS PRESS, LLC

Jung Mi Ko, Young Hie Kim, and Sun Shin Ahn

Proof. Assume that f is an anti fuzzy filter of X. Let x, y, z ∈ X be such that z ≤ x ∗ y. By Proposition 3.13, we have f (y) ≤ max{f (x ∗ y), f (x)} ≤ max{f (z), f (x)}. Conversely, suppose that f satisfies (∗). By (BE2), we have x ≤ x ∗ 1 = 1. Using (∗), we have f (1) ≤ f (x) for all x ∈ X. It follows from (BE1) and (BE4) that x ≤ (x ∗ y) ∗ y for all x, y ∈ X. Using (∗), we have f (y) ≤ max{f (x ∗ y), f (x)}. Therefore f is an anti fuzzy filter of X.  4. Permeable values in BE-algebras Definition 4.1. Let f be a fuzzy set in a BE-algebra X. A number t ∈ [0, 1] is called a permeable S-value for f if U (f ; t) 6= ∅ and the following assertion is valid. (4.1) (∀a, b ∈ X)(f (a ∗ b) ≥ t ⇒ max{f (a), f (b)} ≥ t). Example 4.2. Consider a BE-algebra X = {1, a, b, c} as in Example 3.3 (1). Let f be a fuzzy set of X defined by f (1) = 0.2, f (a) = 0.3, and f (b) = f (c) = 0.6. Take t ∈ (0.3, 0.6]. Then U (f ; t) = {b, c}. It is easy to check that t is a permeable S-value for f . Theorem 4.3. Let f be a fuzzy subalgebra of a BE-algebra X. If t ∈ [0, 1] is a permeable S-value for f , then the nonempty upper t-level set U (f ; t) is an S-energetic subset of X. Proof. Let a, b ∈ X be such that a ∗ b ∈ U (f ; t). Then f (a ∗ b) ≥ t and so max{f (a), f (b)} ≥ t. Therefore f (a) ≥ t or f (b) ≥ t, i.e., a ∈ U (f ; t) or b ∈ U (f ; t). Hence {a, b} ∩ U (f ; t) 6= ∅. Thus U (f ; t) is an S-energetic subset of X.  Since U (f ; t) ∪ L∗ (f ; t) = X and U (f ; t) ∩ L∗ (f ; t) = ∅ for all t ∈ [0, 1], we have the following corollary. Corollary 4.4. Let f be a fuzzy subalgebra of a BE-algebra X. If t ∈ [0, 1] is a permeable S-value for f , then L∗ (f ; t) is empty or a subalgebra of X. Definition 4.5. Let f be a fuzzy set in a BE-algebra X. A number t ∈ [0, 1] is called an anti permeable S-value for f if L(f ; t) 6= ∅ and the following assertion is valid. (4.2) (∀a, b ∈ X)(f (a ∗ b) ≤ t ⇒ min{f (a), f (b)} ≤ t). Example 4.6. Consider a BE-algebra X = {1, a, b, c} as in Example 3.3 (1). Let f be a fuzzy set of X defined by f (1) = 0.4, f (a) = f (b) = 0.5, and f (c) = 0.3. Take t ∈ [0.3, 0.4). Then L(f ; t) = {c}. It is easy to check that t is an anti permeable S-value for f . Theorem 4.7. Let f be an anti fuzzy subalgebra of a BE-algebra X. For any anti permeable S-value t ∈ [0, 1] for f , we have L(f ; t) 6= ∅ ⇒ L(f ; t) is an S-energetic subset of X. Proof. Let a, b ∈ X be such that a ∗ b ∈ L(f ; t). Then f (a ∗ b) ≤ t and so min{f (a), f (b)} ≤ t. Thus f (a) ≤ t or f (b) ≤ t, i.e., a ∈ L(f ; t) or b ∈ L(f ; t). Hence {a, b} ∩ L(f ; t) 6= ∅. Therefore L(f ; t) is an S-energetic subset of X. 

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Jung Mi Ko 902-909

J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 29, NO.5, 2021, COPYRIGHT 2021 EUDOXUS PRESS, LLC

Permeable values with applications in BE-algebras

Theorem 4.8. Let f be a fuzzy subalgebra of a BE-algebra X and let t ∈ [0, 1] be such that L(f ; t) 6= ∅. Then t is an anti permeable S-value for f . Proof. Let a, b ∈ X be such that f (a∗b) ≤ t for all t ∈ [0, 1]. Then min{f (a), f (b)} ≤ f (a∗b) ≤ t. Therefore t is an anti permeable S-value for f .  Definition 4.9. Let f be a fuzzy set in a BE-algebra X. A number t ∈ [0, 1] is called a permeable F -value for f if U (f ; t) 6= ∅ and the following assertion is valid. (4.3) (∀x, y ∈ X)(f (y) ≥ t ⇒ max{f (x ∗ y), f (x)} ≥ t). Example 4.10. Consider the BE-algebra X = {1, a, b, c, d, 0} as in Example 3.10. Let f be a fuzzy set in X defined by f (1) = 0.2, f (a) = f (b) = 0.4, and f (c) = f (d) = f (0) = 0.7. If t ∈ (0.4, 0.7], then U (f ; t) = {0, c, d} and it is easy to check that t is a permeable F -value for f . Theorem 4.11. Let f be a fuzzy filter of a BE-algebra X. If t ∈ [0, 1] is a permeable F -value for f , then the nonempty upper t-level set U (f ; t) is an F -energetic subset of X. Proof. Assume that U (f ; t) 6= ∅ for t ∈ [0, 1]. Let y ∈ X be such that y ∈ U (f ; t). Then t ≤ f (y). It follows from (4.3) that t ≤ max{f (x ∗ y), f (x)} for all x ∈ X. Hence f (x ∗ y) ≥ t or f (x) ≥ t, i.e., x ∗ y ∈ U (f ; t) or x ∈ U (f ; t). Hence {x ∗ y, x} ∩ U (f ; t) 6= ∅. Therefore U (f ; t) is an F -energetic subset of X.  Since U (f ; t) ∪ L∗ (f ; t) = X and U (f ; t) ∩ L∗ (f ; t) = ∅ for all t ∈ [0, 1], we have the following corollary. Corollary 4.12. Let f be a fuzzy filter of a BE-algebra X. If t ∈ [0, 1] is a permeable F -value for f , then L∗ (f ; t) is empty or a filter of X. Theorem 4.13. For a fuzzy set f in a BE-algebra X, if there exists a subset K of [0, 1] such that {U (f ; t), L∗ (f ; t)} is a partition of X and L∗ (f ; t) is a filter of X for all t ∈ K, then t is a permeable F -value for f . Proof. Assume that f (y) ≥ t for any y ∈ X. Then y ∈ U (f ; t) and so {x ∗ y, x} ∩ U (f ; t) 6= ∅ for any x ∈ X, since U (f ; t) is an F -energetic subset of X. Hence x ∗ y ∈ U (f ; t) or x ∈ U (f ; t) and so max{f (x ∗ y), f (x)} ≥ t. Therefore t is a permeable F -value for f .  Theorem 4.14. Let f be a fuzzy set in a BE-algebra X with U (f ; t) 6= ∅ for t ∈ [0, 1]. If f is an anti fuzzy filter of X, then t is a permeable F -value for f . Proof. Let y ∈ X be such that f (y) ≥ t. Then t ≤ f (y) ≤ max{f (x ∗ y), f (x)} for all x ∈ X. Hence t is a permeable F -value for f .  Theorem 4.15. Let f be an anti fuzzy filter of a BE-algebra X. Then the following assertion is valid. (∀t ∈ [0, 1])(U (f ; t) 6= ∅ ⇒ U (f ; t) is an F -energetic subset of X).

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Jung Mi Ko 902-909

J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 29, NO.5, 2021, COPYRIGHT 2021 EUDOXUS PRESS, LLC

Jung Mi Ko, Young Hie Kim, and Sun Shin Ahn

Proof. Let y ∈ X be such that y ∈ U (f ; t). Then f (y) ≥ t. By (AF2 ), we have t ≤ f (y) ≤ max{f (x ∗ y), f (x)} for all x ∈ X. Hence f (x ∗ y) ≥ t or f (x) ≥ t, i.e., x ∗ y ∈ U (f ; t) or x ∈ U (f ; t). Therefore {x ∗ y, x} ∩ U (f ; t) 6= ∅. Thus U (f ; t) is an F -energetic subset of X.  Definition 4.16. Let f be a fuzzy set in a BE-algebra X. A number t ∈ [0, 1] is called an anti permeable F -value for f if L(f ; t) 6= ∅ and the following assertion is valid. (4.4) (∀x, y ∈ X)(f (y) ≤ t ⇒ min{f (x ∗ y), f (x)} ≤ t). Theorem 4.17. Let f be a fuzzy set in a BE-algebra X with L(f ; t) 6= ∅ for t ∈ [0, 1]. If f is a fuzzy filter of X, then t is an anti permeable F -value for f . Proof. Let y ∈ X be such that f (y) ≤ t. Then min{f (x ∗ y), f (x)} ≤ f (y) ≤ t for all x ∈ X. Hence t is an anti permeable F -value for f .  Theorem 4.18. Let f be an anti fuzzy filter of a BE-algebra X. If t ∈ [0, 1] is an anti permeable F -value for f , then the lower t-level set L(f ; t) is an F -energetic subset of X. Proof. Let y ∈ X be such that y ∈ L(f ; t). Then f (y) ≤ t. It follows from (4.4) that min{f (x ∗ y), f (x)} ≤ t for all x ∈ X. Hence x ∗ y ∈ L(f ; t) or x ∈ L(f ; t) and so {x ∗ y, x} ∩ L(f ; t) 6= ∅. Therefore L(f ; t) is an F -energetic subset of X.  Corollary 4.19. Let f be an anti fuzzy filter of a BE-algebra X. If t ∈ [0, 1] is an anti permeable F -value for f , then U ∗ (f ; t) is empty or a filter of X. Theorem 4.20. For a fuzzy set f in a BE-algebra X, if there exists a subset K of [0, 1] such that {U ∗ (f ; t), L(f ; t)} is a partition of X and U ∗ (f ; t) is a filter of X for all t ∈ K, then t is an anti permeable F -value for f . Proof. Assume that f (y) ≤ t for any y ∈ X. Then y ∈ L(f ; t) and so {x ∗ y, x} ∩ L(f ; t)} 6= ∅ for all x ∈ X, since L(f ; t) is an F -energetic subset of X. Hence f (x ∗ y) ≤ t or f (x) ≤ t and so min{f (x ∗ y), f (x)} ≤ t. Therefore t is anti permeable F -value for f .  References [1] S. S. Ahn, Y. H. Kim, J. M. Ko, Filters in commutative BE-algerbas, Commun. Korean Math. Soc. 27 (2012), no. 2, 233–242. [2] S. S. Ahn, K. S. So, On ideals and upper sets in BE-algerbas, Sci. Math. Jpn. 68 (2008), no. 2, 279–285. [3] Y. B. Jun, S. S. Ahn, E. H. Roh, Energetic subsets and permeable values with applications in BCK/BCIalgebras, Appl. Math. Sci. 7 (2013), no. 89, 4425–4438. [4] Y. B. Jun, E. H. Roh and S. Z. Song, Commutative energetic subsets of BCK-algebras, Bulletin of the Section of Logic 45 (2016), 53–63. [5] H. S. Kim, Y. H. Kim, On BE-algerbas, Sci. Math. Jpn. 66 (2007), no. 1, 113–116. [6] H. S. Kim, K. J. Lee, Extended upper sets in BE-algerbas, Bull. Malays. Math. Sci. Soc. 34 (2011), no. 3, 511–520. [7] L. A. Zadeh, Fuzzy sets, Information and Control 8 (1965), 338-353.

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Jung Mi Ko 902-909

J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 29, NO.5, 2021, COPYRIGHT 2021 EUDOXUS PRESS, LLC

Rapid gradient penalty schemes and convergence for solving constrained convex optimization problem in Hilbert spaces Natthaphon Artsawanga,1 , Kasamsuk Ungchittrakoola,b,∗ a Department b Research

of Mathematics, Faculty of Science, Naresuan University, Phitsanulok 65000, Thailand Center for Academic Excellence in Nonlinear Analysis and Optimization, Faculty of Science, Naresuan University, Phitsanulok 65000, Thailand

Abstract The purposes of this paper are to establish and study the convergence of a new gradient scheme with penalization terms called rapid gradient penalty algorithm (RGPA) for minimizing a convex differentiable function over the set of minimizers of a convex differentiable constrained function. Under the observation of some appropriate choices for the available properties of the considered functions and scalars, we can generate a suitable algorithm that weakly converges to a minimal solution of the considered constraint minimization problem. Further, we also provide a numerical example to compare the rapid gradient penalty algorithm (RGPA) and the algorithm introduced by Peypouquet [20]. Keywords: Rapid gradient penalty algorithm, penalization, constraint minimization, fenchel conjugate 2010 Mathematics Subject Classification. 65K05, 65K10, 90C25. 1. Introduction Let H be a real Hilbert space with the norm and inner product given by k·k and h·, ·i, respectively. Let f : H → R and g : H → R be convex and (Fr´echet) differentiable functions on the space H and the gradients ∇f and ∇g are Lipschitz continuous operators with constants Lf and Lg , respectively. We consider the following constrained convex optimization problem min

x∈arg min g

f (x).

(1.1)

Throughout the paper, we also assume that the solution set S := arg min{f (x) : x ∈ arg min g} is a nonempty set. Further, without loss of generality, we may assume that min g = 0. Due to the interesting applications of (1.1) in many branches of mathematics and sciences, many researchers have paid attention to solve the problem (1.1) which can be mentioned briefly as follows: In 2010, Attouch and Czarnecki [1] initially presented and studied a numerical algorithm called the multiscale asymptotic gradient (MAG) for solving general constrained convex optimization problem. They proved that every sequence generated by (MAG) converges weakly to a solution of their ∗ Corresponding

author. Tel.:+66 55963250; fax:+66 55963201. Email addresses: [email protected] (Kasamsuk Ungchittrakool), [email protected] Natthaphon Artsawang 1 Supported by The Royal Golden Jubilee Project Grant no. PHD/0158/2557, Thailand.

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J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 29, NO.5, 2021, COPYRIGHT 2021 EUDOXUS PRESS, LLC

considered problem. It seems that their representation is the starting point for the development of numerical algorithms in the context of solving type of constrained convex optimization problem (see, for instance [2–4, 7–10, 18]) and the references therein. Inspired by Attouch and Czarnecki [1], in 2012 Peypouquet [20] proposed and analyzed an algorithm called diagonal gradient scheme (DGS) via gradient method and exterior penalization scheme for constrained minimization of convex functions. He also provided a weak convergence to find a solution of the considered constrained minimization of convex functions. Several applications are provided such as relaxed feasibility, mathematical programming with convex inequality constraints, and Stokes equation and signal reconstruction, etc. In 2013, Shehu et al. [21] studied the problem (1.1) in the case when the constrained set is simple enough and also proposed an algorithm for solving (1.1). In the last two decades, intensive research efforts dedicated to algorithms of inertial type and their convergence behavior can be noticed (see [6, 11, 13–17, 19]). In 2017, Bot et al. [9] considered the problem of minimizing a smooth convex objective function subject to the set of minima of another differentiable convex function. They proposed a new algorithm called gradient-type penalty with inertial effects method (GPIM) for solving the problem (1.1). They also illustrated the usability of their method via a numerical experiment for image classification via support vector machines. In the remaining part of this section, we recall some elements of convex analysis. For a function h : H → R := R ∪ {−∞, +∞} we denote by dom h = {x ∈ H : h(x) < +∞} its effective domain and say that h is proper, if dom h 6= ∅ and h(x) 6= −∞ for all x ∈ H. The Fenchel conjugate of h is h∗ : H → R, which is defined by h∗ (z) = sup {hz, xi − h(x)} for all z ∈ H. x∈H

The subdifferential of h at x ∈ H, with h(x) ∈ R, is the set ∂h(x) := {v ∈ H : h(y) − h(x) ≥ hv, y − xi ∀y ∈ H}. We take by convention ∂h(x) := ∅, if h(x) ∈ {±∞}. The convex and differentiable function T : H → R has a Lipschitz continuous gradient with Lipschitz constant LT > 0, if k∇T (x) − ∇T (y)k ≤ LT kx − yk for all x, y ∈ H. Let C ⊂ H be a nonempty closed convex set. The indicator function is defined as: ( 0 if x ∈ C δC (x) = +∞ otherwise. The support function of C is defined as: σC (x) := supc∈C hx, ci for all x ∈ H. The normal cone C at a point x is ( {x ∈ H : hx, c − xi ≤ 0 for all c ∈ C}, if x ∈ C NC (x) := ∅, otherwise. We denote by Ran(NC ) for the range of NC . Notice that δC∗ = σC . Moreover, it holds that x ∈ NC (x) if and only if σC (x) = hx, xi. Inspired by the research works in this direction, we are interested in the development and improvement of the method for finding solutions of the considered problem, that is, we wish to establish the algorithm called rapid gradient penalty algorithm (RGPA) for solving (1.1) which is generated by a controlling sequence of scalars together with the gradient of objective and feasibility gap functions as follows:   x1 ∈ H; (RGPA) yn = xn − λn ∇f (xn ) − λn βn ∇g(xn );   xn+1 = yn + αn (yn − xn ) for all n ≥ 1,

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J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 29, NO.5, 2021, COPYRIGHT 2021 EUDOXUS PRESS, LLC

where {λn } and {βn } are sequences of positive parameters and {αn } ⊆ (0, 1). For n ≥ 1, we write Ωn := f + βn g, which is also (Fr´echet) differentiable function. Therefore, ∇Ωn is Lipschitz continuous with constant Ln := Lf + βn Lg . In particular, if we setting αn = 0 for all n ≥ 1, the algorithm (RGPA) can be reduced to (DGS) in Peypouquet [20]. In order to support our ideas, we also provide a numerical example to simulate an event for solving problem (1.1). We also compare the time and the iteration between two algorithms including (RGPA) and (DGS). 2. The Hypotheses In this section, we will carry out the main assumptions to prove the convergence results for rapid gradient penalty algorithm (RGPA). In order to prove the convergence results, the following assumptions will be proposed. Assumption A (I) The function f is bounded from below; (II) There exists a positive K > 0 such that βn+1 − βn ≤ Kλn+1 βn+1 ,

Ln 2

−

α2n −1 2λn

1 2λn

≤ −K and

+ (1 + αn )2 K < 0 for all n ≥ 1; ∞ X λn = +∞ and lim inf λn βn > 0; (III) {αn } ∈ l2 \ l1 , n→∞

n=1

(IV) For each p ∈ Ran(Narg min g ), we have

∞ X

     p p λn βn g ∗ − σarg min g < +∞. β β n n n=1

Remark 2.1. The conditions in Assumption A sparsely extend the hypotheses in [20]. The differences are given by the second and third inequality in (II), which here involves a sequence {αn } which controls the inertial terms, and by {αn } ∈ l2 \ l1 . In the following remark, we present some situations where Assumption A is verified. 1 Remark 2.2. Let K > 0, q ∈ (0, 1), δ > 0 and γ ∈ (0, 3L1 g ) be any given. Then we set αn := n+1 ∞ X for all n ≥ 1, which implies that lim αn = 0, αn2 < +∞ and αn ≤ 21 for all n ≥ 1. We also set n→∞

βn := Since βn ≥

n=1

3γ[Lf + 2(K + δ)] γ + γKnq and λn := for all n ≥ 1. 1 − 3γLg βn

3γ[Lf +2(K+δ)] , 1−3γLg

we have for each n ≥ 1 βn (1 − 3γLg ) ≥ 3γ[Lf + 2(K + δ)].

It follows that

1 − βn Lg ≥ Lf + 2(K+ δ) for all n ≥ 1, 3λn

which implies that − (K + δ) ≥

Ln 1 − for all n ≥ 1. 2 6λn

(2.1)

According to (2.1), we obtain that −K ≥

Ln 1 1 − and > 2λn K for all n ≥ 1. 2 2λn 3

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Artsawang 910-921

J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 29, NO.5, 2021, COPYRIGHT 2021 EUDOXUS PRESS, LLC

Let us consider, for each n ≥ 1 − 3 + 9 2λn K −3 + αn2 − 1 + (1 + αn )2 K ≤ 4 4 < 4 2λn 2λn 2λn

3 4

= 0.

On the other hand, βn+1 − βn = γK[(n + 1)q − nq ] ≤ γK = Kλn+1 βn+1 . Hence, we can conclude that Assumption A (II) holds. ∞ ∞ X X 1 Since q ∈ (0, 1), we obtain that = +∞, so λn = +∞. Notice that λn βn = γ for all β n=1 n n=1 n ≥ 1. It follows that lim inf λn βn = lim inf γ > 0. Thus Assumption A (III) holds. n→∞

n→∞

Finally, since g ∗ − σarg min g ≥ 0. If g(x) ≥ k2 dist2 (x, arg min g) where k > 0, then g ∗ (x) − 1 σarg min g (x) ≤ 2k kxk2 for all x ∈ H. Therefore, for each p ∈ Ran(Narg min g ), we obtain that      λn p p − σarg min g ≤ kpk2 . λn βn g ∗ βn βn 2kβn      ∞ ∞ X X λn p p Thus, λn βn g ∗ − σarg min g converges, if converges, which is equivalently β β β n n n=1 n=1 n ∞ X 1 converges. This holds for the above choices of {βn } and {λn } when q ∈ ( 21 , 1). to 2 β n n=1 3. Convergence analysis for convexity In this section, we will prove the convergence of the sequence of {xn } generated by (RGPA) and of the sequence of objective values {f (xn )}. We start the convergence analysis of this section with three technical lemmas. Lemma 3.1. Let x∗ be an arbitrary element in S and set p∗ := −∇f (x∗ ). Then for each n ≥ 1 kxn+1 − x∗ k2 − kxn − x∗ k2 + (1 + αn )λn βn g(xn ) ≤ (1 + αn )2 kxn − yn k2   ∗  ∗  2p 2p + (1 + αn )λn βn g ∗ − σarg min g . βn βn

(3.1)

Proof. Applying to the first-order optimality condition, we have 0 ∈ ∇f (x∗ ) + Narg min g (x∗ ). It follows that p∗ = −∇f (x∗ ) ∈ Narg min g (x∗ ). n − βn ∇g(xn ) = ∇f (xn ). Note that for each n ≥ 1, xnλ−y n By monotonicity of ∇f , we obtain that   xn − yn ∗ ∗ − βn ∇g(xn ) + p , xn − x = h∇f (xn ) − ∇f (x∗ ), xn − x∗ i λn

≥0

, ∀n ≥ 1,

and hence, for each n ≥ 1 2 hxn − yn , xn − x∗ i ≥ 2λn βn h∇g(xn ), xn − x∗ i − 2λn hp∗ , xn − x∗ i .

913

(3.2)

Artsawang 910-921

J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 29, NO.5, 2021, COPYRIGHT 2021 EUDOXUS PRESS, LLC

Since g is convex and differentiable, we have for each n ≥ 1 h∇g(xn ), x∗ − xn i + g(xn ) ≤ g(x∗ ) = 0, whence 2λn βn g(xn ) ≤ 2λn βn h∇g(xn ), xn − x∗ i.

(3.3)

2hxn − yn , xn − x∗ i = kxn − yn k2 + kxn − x∗ k2 − kyn − x∗ k2 .

(3.4)

On the other hand,

Combining (3.2), (3.3) and (3.4), we get that kyn − x∗ k2 ≤ kxn − yn k2 + kxn − x∗ k2 − 2λn βn g(xn ) + 2λn hp∗ , xn − x∗ i.

(3.5)

Since x∗ ∈ S and p∗ ∈ Narg min g (x∗ ), we have σarg min g (p∗ ) = hp∗ , x∗ i. In (3.5), we observe that 2λn hp∗ , xn − x∗ i − λn βn g(xn ) = 2λn hp∗ , xn i − λn βn g(xn ) − 2λn hp∗ , x∗ i  ∗    ∗ 2p 2p = λn βn , xn − g(xn ) − , x∗ βn βn   ∗  ∗  2p 2p ≤ λn βn g ∗ − σarg min g . βn βn

(3.6)

Combining (3.6) and (3.5), we obtain that  ∗    ∗ 2p 2p ∗ − σarg min g . kyn − x k ≤ kxn − yn k + kxn − x k − λn βn g(xn ) + λn βn g βn βn (3.7) ∗ 2

2

∗ 2

On the other hand, we observe that kxn+1 − x∗ k2 = kyn + αn (yn − xn ) − x∗ k2 = k(1 + αn )(yn − x∗ ) + αn (x∗ − xn )k2 = (1 + αn )kyn − x∗ k2 − αn kxn − x∗ k2 + αn (1 + αn )kxn − yn k2 .

(3.8)

By (3.7) and (3.8), we obtain the desired result. Lemma 3.2. For all n ≥ 1, we have α2 − 1 Ωn+1 (xn+1 ) ≤ Ωn (xn ) + (βn+1 − βn )g(xn+1 ) + n kyn − xn k2 2λn   Ln 1 + − kxn+1 − xn k2 . 2 2λn Proof. Since ∇Ω is Ln -Lipschitz continuous and by Descent Lemma (see [5, Theorem 18.15]), we obtain that Ln Ωn (xn+1 ) ≤ Ωn (xn ) + h∇Ωn (xn ), xn+1 − xn i + kxn+1 − xn k2 . 2 n Recall that − ynλ−x = ∇Ωn (xn ). n

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It follows that f (xn+1 ) + βn g(xn+1 ) 

 yn − xn Ln ≤ f (xn ) + βn g(xn ) − , xn+1 − xn + kxn+1 − xn k2 λn 2 1 1 1 Ln = f (xn ) + βn g(xn ) − kyn − xn k2 − kxn+1 − xn k2 + kyn − xn+1 k2 + kxn+1 − xn k2 2λn 2λn 2λn 2   α2 − 1 Ln 1 = f (xn ) + βn g(xn ) + n kyn − xn k2 + − kxn+1 − xn k2 . 2λn 2 2λn Adding βn+1 g(xn+1 ) to both sides, we have f (xn+1 ) + βn+1 g(xn+1 ) ≤ f (xn ) + βn g(xn ) + (βn+1 − βn )g(xn+1 )   αn2 − 1 Ln 1 2 + kyn − xn k + − kxn+1 − xn k2 , 2λn 2 2λn which means that Ωn+1 (xn+1 ) ≤ Ωn (xn ) + (βn+1 − βn )g(xn+1 ) +

  Ln αn2 − 1 1 kyn − xn k2 + − kxn+1 − xn k2 . 2λn 2 2λn

For n ≥ 1 and x∗ ∈ S, we denote by Λn := f (xn ) + (1 − (1 + αn )Kλn )βn g(xn ) + Kkxn − x∗ k2 = Ωn (xn ) − (1 + αn )Kλn βn g(xn ) + Kkxn − x∗ k2 . Lemma 3.3. Let x∗ ∈ S and set p∗ := −∇f (x∗ ). Then there exists θ > 0 such that for each n ≥ 1  ∗    ∗ 2p 2p 2 ∗ − σarg min g . Λn+1 − Λn + θkyn − xn k ≤ (1 + αn )Kλn βn g βn βn Proof. From Lemma 3.2 and Assumption A (II), we obtain that αn2 − 1 kyn − xn k2 2λn α2 − 1 kyn − xn k2 . ≤ (1 + αn+1 )Kλn+1 βn+1 g(xn+1 ) + n 2λn

Ωn+1 (xn+1 ) − Ωn (xn ) ≤ Kλn+1 βn+1 g(xn+1 ) +

(3.9)

On the other hand, multiplying (3.1) by K, we have Kkxn+1 − x∗ k2 − Kkxn − x∗ k2 + (1 + αn )Kλn βn g(xn )   ∗  ∗  2p 2p ∗ ≤ (1 + αn ) Kkxn − yn k + (1 + αn )Kλn βn g − σarg min g . (3.10) βn βn 2

2

Combining (3.9) and (3.10), we have    ∗  ∗   2 2p 2p αn − 1 2 2 ∗ Λn+1 − Λn ≤ + (1 + αn ) K kyn − xn k + (1 + αn )Kλn βn g − σarg min g . 2λn βn βn (3.11) For each n ≥ 1,

α2n −1 2λn

+ (1 + αn )2 K < 0, we have there exists θ > 0 such that αn2 − 1 + (1 + αn )2 K < −θ. 2λn 915

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From (3.11), we have 

2

Λn+1 − Λn + θkyn − xn k ≤ (1 + αn )Kλn βn g

∗



2p∗ βn



 − σarg min g

2p∗ βn

 .

This completes the proof. The next lemma is an important role in convergence analysis (see in [3, Lemma 2] or [12, Lemma 3.1]). Lemma 3.4. Let {γn }, {δn } and {εn } be real sequences. Assume that {γn } is bounded from below, ∞ X {δn } is non-negative and εn < +∞ such that n=1

γn+1 − γn + δn ≤ εn for all n ≥ 1. Then lim γn exists and n→∞

∞ X

δn < +∞.

n=1

Lemma 3.5. Let x∗ ∈ S. Then the following statements hold: (i) The sequence {Λn } is bounded from below and lim Λn exists; n→∞

(ii)

∞ X

kyn − xn k2 < +∞;

n=1

(iii) lim kxn − x∗ k2 exists and n→∞

∞ X

λn βn g(xn ) < +∞;

n=1

(iv) lim Ωn (xn ) exists; n→∞

(v) lim g(xn ) = 0 and every weak cluster point of the sequence {xn } lies in arg min g. n→∞

Proof. We set p∗ := −∇f (x∗ ). (i). From Assumption A (II) implies 1−(1+αn )Kλn ≥ 0. Since f is convex and differentiable, we have for each n ≥ 1 Λn = f (xn ) + (1 − (1 + αn )Kλn )βn g(xn ) + Kkxn − x∗ k2 ≥ f (xn ) + Kkxn − x∗ k2   ∗ √ p ≥ f (x∗ ) + h∇f (x∗ ), xn − x∗ i + Kkxn − x∗ k2 = f (x∗ ) − √ , 2K(xn − x∗ ) + Kkxn − x∗ k2 2K ∗ 2 kp k kp∗ k2 ≥ f (x∗ ) − − Kkxn − x∗ k2 + Kkxn − x∗ k2 = f (x∗ ) − . 4K 4K Therefore, {Λn } is bounded from below. Next, we set γn = Λn , δn = θkyn − xn k2 and   ∗  ∗  2p 2p εn = (1 + αn )Kλn βn g ∗ − σarg min g . βn βn Recall that min g = 0. Thus g ≤ δarg min g . Therefore σarg min g = (δarg min g )∗ ≤ g ∗ and hence, g ∗ − σarg min g ≥ 0. It follows that   ∗  ∗    ∗  ∗  2p 2p 2p 2p εn = (1 + αn )Kλn βn g ∗ − σarg min g ≤ 2Kλn βn g ∗ − σarg min g . βn βn βn βn By using Assumption A (IV) and p∗ ∈ Narg min g (x∗ ), we have

∞ X

εn < +∞. Applying Lemma

n=1

3.3 and Lemma 3.4, we obtain that lim Λn exists. n→∞

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(ii). Follows immediately from Lemmas 3.3 and 3.4. (iii). We set γn = kxn − x∗ k2 , δn = (1 + αn )λn βn g(xn ) and   ∗  ∗  2p 2p 2 2 ∗ εn = (1 + αn ) kyn − xn k + (1 + αn )λn βn g − σarg min g . βn βn From statement (ii), Lemma 3.4 and Lemma 3.1, we get that lim kxn − x∗ k exists and

n→∞

∞ X

λn βn g(xn ) < +∞.

n=1

For (iv) since for each n ≥ 1 Ωn (xn ) = Λn + (1 + αn )Kλn βn g(xn ) − Kkxn − x∗ k2 , by using (i), (iii) and lim αn = 0, we have lim Ωn (xn ) exists. n→∞

n→∞

(v). It follows from Assumption A (III) that lim inf λn βn > 0. According to statement (iii) n→∞

implies lim g(xn ) = 0. Let x be any weak cluster point of the sequence {xn }. Therefore, there exists n→∞

subsequence {xnk } of {xn } converges weakly to x as k → ∞. By the weak lower semicontinuity of g, we get that g(x) ≤ lim inf g(xnk ) ≤ lim g(xn ) = 0, n→∞

k→∞

which means that x ∈ arg min g. This completes the proof. Lemma 3.6. Let x∗ ∈ S. Then ∞ X

λn [Ωn (xn ) − f (x∗ )] < +∞.

n=1

Proof. Since f is differentiable and convex function, we obtain that for each n ≥ 1 f (x∗ ) ≥ f (xn ) + h∇f (xn ), x∗ − xn i. Since g is differentiable, convex function and min g = 0 , we obtain that for each n ≥ 1 0 = g(x∗ ) ≥ g(xn ) + h∇g(xn ), x∗ − xn i, which implies that 0 ≥ βn g(xn ) + hβn ∇g(xn ), x∗ − xn i, for all n ≥ 1. Therefore, we can conclude that ∗



∗

f (x ) ≥ Ωn (xn ) + h∇Ωn (xn ), x − xn i = Ωn (xn ) +

 xn − yn ∗ , x − xn . λn

(3.12)

From (3.12), we obtain that 2λn [Ωn (xn ) − f (x∗ )] ≤ 2hxn − yn , xn − x∗ i = kxn − yn k2 + kxn − x∗ k2 − kyn − x∗ k2 .

(3.13)

On the other hand, for each n ≥ 1 kyn − x∗ k2 = kxn+1 − αn (yn − xn ) − x∗ k2 = kxn+1 − x∗ k2 + αn2 kyn − xn k2 − 2hαn (xn+1 − x∗ ), yn − xn i = kxn+1 − x∗ k2 + αn2 kyn − xn k2 − αn2 kxn+1 − x∗ k2 − kyn − xn k2 + kαn (xn+1 − x∗ ) − (yn − xn )k2 ≥ kxn+1 − x∗ k2 + αn2 kyn − xn k2 − αn2 kxn+1 − x∗ k2 − kyn − xn k2 , which implies that −kyn − x∗ k2 ≤ −kxn+1 − x∗ k2 − αn2 kyn − xn k2 + αn2 kxn+1 − x∗ k2 + kyn − xn k2 .

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Combining (3.13) and (3.14), we have for all n ≥ 1 2λn [Ωn (xn ) − f (x∗ )] ≤ (2 − αn2 )kxn − yn k2 + kxn − x∗ k2 − kxn+1 − x∗ k2 + αn2 kxn+1 − x∗ k2 ≤ 2kxn − yn k2 + kxn − x∗ k2 − kxn+1 − x∗ k2 + αn2 kxn+1 − x∗ k2 . Therefore, according to Lemma 3.5 (iii), we get that the sequence {kxn − x∗ k} is bounded, which means that there exists M > 0 such that kxn − x∗ k ≤ M for all n ≥ 1. By Assumption A (III) and Lemma 3.5, we obtain that 2

∞ X

λn [Ωn (xn ) − f (x∗ )] ≤ 2

n=1

∞ X

kyn − xn k2 + kx1 − x∗ k2 + M 2

n=1

∞ X

αn2 < +∞.

n=1

The following proposition will play an important role in convergence analysis, which is the main result of this paper. Proposition 3.7 ([5, Opial Lemma]). Let H be a real Hilbert space, C ⊆ H be nonempty set and {xn } be any given sequence such that: (i) For every z ∈ C, lim kxn − zk exists; n→∞

(ii) Every weak cluster point of the sequence {xn } lies in C. Then the sequence {xn } converges weakly to a point in C. Let {xn } be define by (RGPA). Then {xn } converges weakly to a point in S. Moreover, the sequence {f (xn )} converges to the optimal objective value of the optimization problem (1.1). Proof. From Lemma 3.5 (iii), lim kxn − x∗ k exists for all x∗ ∈ S. Let x be any weak cluster point n→∞

of {xn }. Then there exists a subsequence {xnk } of {xn } such that {xnk } converges weakly to x as k → ∞. According to Lemma 3.5 (v) implies x ∈ arg min g. It suffices to show that f (x) ≤ f (x) for ∞ X λn = +∞, and by Lemma 3.6 and Lemma 3.5 (iv), we have all x ∈ arg min g. Since n=1

lim Ωn (xn ) − f (x∗ ) ≤ 0 for all x∗ ∈ S.

n→∞

Therefore, f (x) ≤ lim inf f (xnk ) ≤ lim Ωn (xn ) ≤ f (x∗ ), k→∞

n→∞

∀x∗ ∈ S, which implies that x ∈ S.

Applying Opial Lemma, we obtain that {xn } converges weakly to a point in S. The last statement follows immediately from the above. 4. Numerical experiments In this section, we present the convergence of the algorithm proposed (RGPA) in this paper by the minimization problem with linear equality constraints. Firstly, we are given a linear system of the form Ax = b, where A ∈ Rm×n and b ∈ Rm . In addition, we assume that n > m. In this section, the system has many solutions. This leads us to the question of which solution should be considered. As a result, we may consider the following problem, say, the minimization problem with linear equality constraints. 1 kxk2 2 subject to Ax = b,

minimize

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Table 1: Comparison of the convergence of (RGPA) and (DGS) for the parameters K = 0.001 and q ∈ ( 12 , 1).

q 0.6 0.7 0.8 0.9

(RGPA) Time (sec) #(Iters) 2.38 566 2.31 568 2.46 581 44.96 11458

(DGS) Time (sec) #(Iters) 10.23 2221 107.78 25336 384.00 90636 447.11 103487

or , equivalently, minimize

1 kxk2 2

1 subject to x ∈ arg min kA(·) − bk2 . 2 The above problem can be written in the form of the problem (1.1) as 1 kxk2 2 1 subject to g(x) := kA(x) − bk2 . 2 In this setting, we have ∇f (x) = x and notice that ∇f is 1-Lipschitz continuous. Furthermore, we get that ∇g(x) = A> (Ax−b) and notice that ∇g is kAk2 -Lipschitz continuous. All the numerical experiments were performed under MATLAB (R2015b). We begin with the problem by random matrix A in Rm×n , vector x1 ∈ Rn and b ∈ Rm with m = 1000 and n = 4000 generated by using MATLAB command randi, which the entries of A, x1 and b are integer in [-10,10]. Next, we are going to compare a performance of the algorithms (RGPA) and (DGS). The choice of the parameters for the computational experiment is based on Remark 2.2. We chooses 1 1 γ = 4kAk 2 and δ = 1. We consider different choices of the parameters K ∈ (0, 1] and q ∈ ( 2 , 1). We will terminate the algorithms (RGPA) and (DGS) when the errors become small, i.e., minimize f (x) :=

kxn − x∗ k ≤ 10−6 , where x∗ = A> (AA> )−1 b. In Table 1 we present a comparison of the convergence between two algorithms including (RGPA) and (DGS) for the parameters K = 0.001 and different choices for the parameters q ∈ ( 21 , 1). We observe that when q = 0.6 leads to lowest computation time for (RGPA) and (DGS) with 2.38 second and 10.23 second, respectively. Furthermore, we also observe that (DGS) hit a big number of iterations than (RGPA) for all choices of parameter q. In Table 2 we present a comparison of the convergence of (RGPA) and (DGS) for the parameters q = 0.6 and K ∈ (0, 1]. We observe that the number of iterations and computation time for (RGPA) smaller than the number of iterations for (DGS) for each choice of parameters K. Furthermore, (RGPA) needs tiny computation time to reach the optimality tolerance than (DGS) for each choice of parameter K. We observe that our algorithm (RGPA) performs an advantage behavior when comparing with algorithm (DGS) for all different choices of parameters. Note that the number of iterations for (RGPA) smaller than the number of iterations for (DGS). Furthermore, (RGPA) needs tiny computation time to reach optimality tolerance than (DGS) for each different choice of parameters. 5. Conclusions We have presented a new gradient penalty scheme, say, rapid gradient penalty algorithm (RGPA). We provide sufficient conditions to guarantee the convergences of (RGPA) for the considered con919

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Table 2: Comparison of the convergence of (RGPA) and (DGS) for the parameters q = 0.6 and K ∈ (0, 1].

K 0.001 0.005 0.01 0.05 0.1 0.5 1

(RGPA) Time (sec) #(Iters) 2.38 566 2.40 585 6.63 1612 83.22 20480 107.41 26257 79.95 18606 51.46 13414

(DGS) Time (sec) #(Iters) 10.23 2221 171.46 40888 254.93 64469 288.39 65722 212.02 52464 100.33 24419 67.20 16616

strained convex optimization problem (1.1). We also provide a numerical example to compare the performance of the algorithms (RGPA) and (DGS). As a result, we observe that our algorithm (RGPA) performs an advantage behavior when comparing with (DGS) for all different choices of parameters. Acknowledgements The second author would like to thank Naresuan University and The Thailand Research Fund for financial support. Moreover, N. Artsawang is also supported by The Royal Golden Jubilee Program under Grant PHD/0158/2557, Thailand. Disclosure statement The authors declare that there is no conflict of interests regarding the publication of this paper. Funding N. Artsawang was supported by the Thailand Research Fund through the Royal Golden Jubilee PhD Program under Grant PHD/0158/2557, Thailand. References [1] H. Attouch, M-O. Czarnecki, Asymptotic behavior of coupled dynamical systems with multiscale aspects, J. Differ. Equat. 248 (2010), 1315–1344. [2] H. Attouch, M-O. Czarnecki and J. Peypouquet, Coupling forward-backward with penalty schemes and parallel splitting for constrained variational inequalities, SIAM J. Optim. 21 (2011), 1251–1274. [3] H. Attouch, M-O. Czarnecki and J. Peypouquet, Prox-penalization and splitting methods for constrained variational problems, SIAM J. Optim. 21 (2011), 149–173. [4] S. Banert, RI. Bot¸, Backward penalty schemes for monotone inclusion problems, J. Optim. Theory Appl. 166 (2015), 930–948. [5] HH. Bauschke and PL. Combettes, Convex Analysis and Monotone Operator Theory in Hilbert Spaces, CMS Books in Mathematics. New York:Springer 2011. [6] A. Beck and M. Teboulle, A fast iterative shrinkage-thresholding algorithm for linear inverse problems, SIAM J. Imag. Sci. 2 (2009), 183–202. 920

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[7] RI. Bot¸ and ER. Csetnek, Forward-backward and Tseng’s type penalty schemes for monotone inclusion problems, Set-Valued Var. Anal. 22 (2014), 313–331. [8] RI. Bot¸ and ER. Csetnek, Approaching the solving of constrained variational inequalities via penalty term-based dynamical systems, J. Math. Anal. Appl. 435 (2016), 1688–1700. [9] RI. Bot¸, ER. Csetnek and N. Nimana, Gradient-type penalty method with inertial effects for solving constrained convex optimization problems with smooth data, Optim. Lett. 12(1) (2018), 17–33. [10] RI. Bot¸, ER. Csetnek and N. Nimana, An Inertial Proximal-Gradient Penalization Scheme for Constrained Convex Optimization Problems, Vietnam J. Math. 46(1) (2018), 53–71. [11] C. Chen, RH. Chan, S. Ma and J. Yang, Inertial proximal ADMM for linearly constrained separable convex optimization, SIAM J. Imaging Sci. 8(4) (2015), 2239–2267. [12] PL. Combettes, Quasi-Fej´erian analysis of some optimization algorithms, In: Butnariu, D., Censor, Y., Reich, S. (eds.) Inherently Parallel Algorithms for Feasibility and Optimization. pp. 115-152. Elsevier:Amsterdam ;(2001). [13] P-E. Maing´e, Convergence theorems for inertial KM-type algorithms, J. Comput. Appl. Math. 219 (2008), 223–236. [14] P-E. Maing´e and A. Moudafi, Convergence of new inertial proximal methods for DC programming, SIAM J. Optim. 19(1) (2008), 397–413. [15] A. Moudafi and M. Oliny, Convergence of a splitting inertial proximal method for monotone operators, J. Comput. Appl. Math. 155 (2003), 447–454. [16] Y. Nesterov, A method for solving the convex programming problem with convergence rate O(1/k2), Dokl. Akad. Nauk SSSR 269(3) (1984), 543–547. [17] Y. Nesterov, Gradient methods for minimizing composite objective function, Technical report, CORE DISCUSSION PAPER (2007). [18] N. Noun and J. Peypouquet, Forward-backward penalty scheme for constrained convex minimization without inf-compactness, J. Optim. Theory Appl. 158 (2013), 787–795. [19] P. Ochs, Y. Chen, T. Brox and T. Pock, iPiano:Inertial proximal algorithm for non-convex optimization, SIAM J. Imaging Sci. 7(2) (2014), 1388–1419. [20] J. Peypouquet, Coupling the gradient method with a general exterior penalization scheme for convex minimization, J. Optim. Theory Appl. 153 (2012), 123–138. [21] Y. Shehu, OS. Iyiola and CD. Enyi, Iterative approximation of solutions for constrained convex minimization problem, Arab. J. Math. 2(4) (2013), 393–402.

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Approximation by modified Lupa¸s operators based on (p, q)–integers 1

Asif Khan1 , Zaheer Abbas2 , Mohd Qasim2 and M. Mursaleen1,3,4 Department of Mathematics, Aligarh Muslim University, Aligarh–202002, India 2 Department of Mathematical Sciences , Baba Ghulam Shah Badshah University, Rajouri–185234, India 3 Department of Medical Research, China Medical University Hospital, China Medical University (Taiwan), Taichung, Taiwan 4 Department of Computer Science and Information Engineering, Asia University, Taichung, Taiwan [email protected]; [email protected]; [email protected]; [email protected] Abstract

The purpose of this paper is to introduce a new modification of Lupa¸s operators in the frame of post quantum setting and to investigate their approximation properties. First using the relations between q-calculus and post quantum calculus, the post quantum analogue of operators constructed will be linear and positive but will not follow Korovkin’s theorem. Hence a new modification of q-Lupa¸s operators is constructed which will preserve test functions. Based on these modification of operators, approximation properties have been investigated. Further, the rate of convergence of operators by mean of modulus of continuity and functions belonging to the Lipschitz class as well as Peetre’s K-functional are studied. Keywords and phrases: Lupa¸s operators; Post quantum analogue; q analogue; Peetre’s K-functional; Korovkin’s type theorem; Convergence theorems. AMS Subject Classification (2010): 41A10, 41A25, 41A36. 1. Introduction and preliminaries A. Lupa¸s [17] introduced the linear positive operators at the International Dortmund Meeting held in Witten (Germany, March, 1995) as follows:   ∞ X (mu)l al l −mu Lm (f ; u) = (1 − a) f , u ≥ 0, (1.1) l! m l=0

with f : [0, ∞) → R. If we impose Lm (u) = u, we get a = 12 . Thus operators (1.1) becomes   ∞ X (mu)l l Lm (f ; u) = 2−mu f , u ≥ 0, (1.2) l!2l m l=0

where (mu)l is the rising factorial defined as: (mu)0 = 1, (mu)l = mu(mu + 1)(mu + 2) · · · (mu + l − 1), l ≥ 0. The q-analogue of Lupa¸s operators (1.2) is defined in [26] as:   ∞ X ([m]q u)l [l]q p,q −[m]q u Lm (f ; u) = 2 f , u ≥ 0. [l]!2l [m]q

(1.3)

l=0

1

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2

2. Construction of new operators and auxiliary results Let us recall certain notations and definitions of (p, q)-calculus. Let p > 0, q > 0. For each non negative integer l, m, m ≥ l ≥ 0, the (p, q)-integer, (p, q)-binomial are defined, as

[j]p,q = pj−1 + pj−2 q + pj−3 q 2 + ... + pq j−2 + q j−1 =

       

pj −q j p−q ,

when p 6= q 6= 1,

j pj−1 ,

when p = q 6= 1,

       [j]q , j,

when p = 1, when p = q = 1.

where [j]q denotes the q-integers and m = 0, 1, 2, · · · . The formula for (p, q)-binomial expansion is as follows:   m X l(l−1) (m−l)(m−l−1) m m 2 2 am−l bl um−l v l , q (au + bv)p,q := p l p,q l=0

2 2 m−1 (u + v)m u + q m−1 v), p,q = (u + v)(pu + qv)(p u + q v) · · · (p m 2 2 m−1 (1 − u)p,q = (1 − u)(p − qu)(p − q u) · · · (p − q m−1 u),

where (p, q)-binomial coefficients are defined by   [m]p,q ! m = . l p,q [l]p,q ![m − l]p,q ! Details on (p, q)-calculus can be found in [9, 11, 21]. In the case of p = 1, the above notations reduce to q-analogues and one can easily see that [m]p,q = pm−1 [m]q/p . Mursaleen et al. [21] introduced (p, q)-calculus in approximation theory and constructed post quantum analogue of Bernstein operators. On the other hand Khalid and Lobiyal defined the (p, q)- analogue of Lupa¸s Bernstein operators in [12] and have shown its application in computer aided geometric design for construction of Beizer curves and surfaces. For another applications of extra parameters p in the field of approximation on compact disk, one can refer [4]. For related literature, one can refer [1, 2, 9, 3, 13, 14, 18, 19, 20, 22, 23, 25, 24] papers based on q and (p, q) integers in approximation theory and CAGD. Motivated by the above mentioned work, we introduce a new analogue of Lupa¸s operators. The post quantum analogue of (1.3) are as follows: Definition 2.1. Let f : [0, ∞) → R, 0 < q < p ≤ 1 and for any m ∈ N. we define the (p, q)-analogue of Lupa¸s operators as  l−m  ∞ X ([m]p,q u)l p [l]p,q p,q −[m]p,q u Lm (f ; u) = 2 f , u ≥ 0. (2.1) [l]p,q !2l [m]p,q l=0

The operators (2.1) are linear and positive. For p = 1, the operators (2.1) turn out to be q-Lupa¸s operators defined in (1.3). Next, we prove some auxiliary results for (2.1). Lemma 2.2. Let 0 < q < p ≤ 1 and m ∈ N. We have (i) Lp,q m (1; u) = 1, u (ii) Lp,q m (t; u) = pm−1 (2−p)[m]p,q u+1 , 2 (iii) Lp,q m (t ; u) =

u [m]p,q p2m−2 (2−p3 )[m]p,q u+1

2

qu + p2m−4 (2−p + p2m−4 (2−p2 )qu [m]p,q u+2 2 )[m]p,q u+2 [m]

.

p,q

Proof. we have

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(i)

−[m]p,q u Lp,q m (1; u) = 2

∞ X ([m]p,q u)l l=0

[l]p,q !2l

= 1.

(ii)

Lp,q m (t; u)

= = =

2−[m]p,q u 2−[m]p,q u

∞ X ([m]p,q u)l pl−m [l]p,q

[l]p,q !2l

l=0 ∞ X

([m]p,q u)([m]p,q u + 1)l−1 pl−m [l]p,q [l]p,q [l − 1]p,q !2l [m]p,q

l=1 ∞ X

2−[m]p,q u u

l=0

=

2

u

pm−1

([m]p,q u + 1)l l+1−m p [l]p,q !2l+1

∞ X ([m]p,q u + 1)l pl

−[m]p,q u−1

=

[l]p,q !2l

l=0

(pm−1 )(2

[m]p,q

u . − p)([m]p,q u+1)

(iii)

2 Lp,q m (t ; u)

= =

2−[m]p,q u 2−[m]p,q u

∞ X ([m]p,q u)l p2l−2m [l]2p,q l=0 ∞ X

[l]p,q !2l

[m]2p,q

([m]p,q u)([m]p,q u + 1)l−1 p2l−2m [l]2p,q [l]p,q [l − 1]p,q !al [m]2p,q

l=1 ∞ X

([m]p,q u + 1)l p2l+2−2m [l + 1]p,q [l]p,q !2l+1 [m]p,q

=

2−[m]p,q u u

=

2−[m]p,q u−1 X ([m]p,q u + 1)l p2l [l + 1]p,q u p2m−2 [l]p,q !2l [m]p,q

l=0

∞

l=0

=

2

−[m]p,q u−1

p2m−2

u

∞ X ([m]p,q u + 1)l p2l [pl + q[l]p,q ] l=0 ∞ X

[l]p,q !2l

[m]p,q

=

2−[m]p,q u−1 u p2m−2

=

2−[m]p,q u−1 X ([m]p,q u + 1)l p2l q[l]p,q u p2m−2 [l]p,q !2l [m]p,q

=

I1 + I2 (say),

l=0 ∞

([m]p,q u + 1)l p3l [l]p,q !2l [m]p,q

l=0



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we find that I1 and I2 are ∞

I1

= =

I2

=

2−[m]p,q u−1 X ([m]p,q u + 1)l p3l u p2m−2 [l]p,q !2l [m]p,q l=0 u . [m] u+1 2m−2 [m]p,q p (2 − p3 ) p,q ∞ 2−[m]p,q u−1 X ([m]p,q u + 1)l p2l q[l]p,q u p2m−2 [l]p,q !2l [m]p,q l=0

∞

=

2−[m]p,q u−1 ([m]p,q u + 1) X ([m]p,q u + 2)l−1 p2l [l]p,q qu p2m−2 [l]p,q [l − 1]p,q !2l [m]p,q l=0

= = =

2

−[m]p,q u−1

([m]p,q u + 1) qu [m]p,q p2m−2

2−[m]p,q u−2 ([m]p,q u + 1) qu [m]p,q p2m−4 qu2 p2m−4 (2

−

[m] u+2 p2 ) p,q

+

∞ X ([m]p,q u + 2)l−1 p2l l=1 ∞ X l=0

[l − 1]p,q !2l ([m]p,q u + 2)l p2l [l]p,q !2l qu

p2n−4 (2

−

[m] u+2 p2 ) p,q [m]p,q

.

On adding I1 and I2 , we get 2 Lp,q m (t ; u)

=

u [m]p,q p2m−2 (2 −

[m] u+1 p3 ) p,q

+

qu2 p2m−4 (2 −

[m] u+2 p2 ) p,q

+

qu [m]p,q u+2

p2m−4 (2 − p2 )

. [m]p,q

The sequence of (p, q)-Lupa¸s operators constructed in (2.1) however do not preserve the test functions t and t2 . Hence one can not guarantee approximation via these operators. Therefore, we construct the modified (p, q)- Lupa¸s operators as follows: Lemma 2.3. Let 0 < q < p ≤ 1 and m ∈ N. For f : [0, ∞) → R, we define the (p, q)-analogue of Lupa¸s operators as: −[m]p,q u e p,q L m (f ; u) = 2

  ∞ X ([m]p,q u)l [l]p,q f , u ≥ 0. [l]p,q !al [m]p,q

(2.2)

l=0

The operators (2.2) are linear and positive. For p = 1, the operators (2.2) turn out to be q-Lupa¸s operator defined in (1.3). We shall investigate approximation properties of the operators (2.2). We obtain rate of convergence of the operators via modulus of continuities. We also obtain approximation behaviors of the operators for functions belonging to Lipschitz spaces. Lemma 2.4. Let 0 < q < p ≤ 1 and m ∈ N. We have (i) (ii) (iii)

e p,q (1; u) = 1, L m e Lp,q m (t; u) = u, u e p,q (t2 ; u) = L m (2−p)([m]p,q u+1) [m]

p,q

+

qu [m]p,q

+ qu2 .

Proof. We have (i) e p,q (1; u) = 2−[m]p,q u L m

∞ X ([m]p,q u)l l=0

925

[l]p,q !2l

= 1.

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(ii) e p,q L m (t; u)

∞ X ([m]p,q u)l [l]p,q [l]p,q !2l [m]p,q

=

2−[m]p,q u

=

∞ X ([m]p,q u)([m]p,q u + 1)l−1 [l]p,q 2−[m]p,q u [l]p,q [l − 1]p,q !2l [m]p,q

l=0

l=1

=

−[m]p,q u−1

2

u

∞ X ([m]p,q u + 1)l

[l]p,q !2l

l=0

= u. (iii) 2 e p,q L m (t ; u)

∞ X ([m]p,q u)l [l]2p,q [l]p,q !2l [m]2p,q

=

2−[m]p,q u

=

∞ X ([m]p,q u)([m]p,q u + 1)l−1 [l]2p,q 2−[m]p,q u [l]p,q [l − 1]p,q !2l [m]2p,q

l=0

=

l=1 ∞ X

2−[m]p,q u u

([m]p,q u + 1)l [l + 1]p,q [l]p,q !2l+1 [m]p,q

l=0 ∞ X −[m]p,q u−1

u

([m]p,q u + 1)l [pl + q[l]p,q ] [l]p,q !2l [m]p,q

=

2

=

2−[m]p,q u−1 X ([m]p,q u + 1)l l p u [m]p,q [l]p,q !2l

l=0 ∞

l=0 ∞ X

−[m]p,q u−1

+

2

[m]p,q

qu

l=0

([m]p,q u + 1)l [l]p,q [l]p,q !2l

= I1 + I2 (Say). After solving I1 and I2 , we get ∞

I1

= =

I2

=

2−[m]p,q u−1 X ([m]p,q u + 1)l l p u [m]p,q [l]p,q !2l l=0 u . (2 − p)([m]p,q u+1) [m]p,q ∞ 2−[m]p,q u−1 X ([m]p,q u + 1)l [l]p,q qu [m]p,q [l]p,q !2l l=0

−[m]p,q u−1

=

2

[m]p,q

qu

∞ X ([m]p,q u + 1)([m]p,q u + 2)l−1 [l]p,q

[l]p,q [l − 1]p,q !2l

l=1

∞

= =

2−[m]p,q u−2 ([m]p,q u + 1)qu X ([m]p,q u + 2)l [n]p,q [l]p,q !2l l=0 qu + qu2 . [m]p,q

On adding I1 and I2 , we get 2 e p,q L m (t ; u)

=

u qu + + qu2 . [m]p,q (2 − p)([m]p,q u+1) [m]p,q 

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Corollary 2.5. Using Lemma 2.4, we get the following central moments. e p,q L n (t − u; u) = 0 2 e p,q L n ((t − u) ; u) =

u (2−p)([m]p,q u+1) [m]p,q

+

qu [m]p,q

+ qu2 − u2 = δm (u) (say).

Remark 2.6. One can observe that lim [m]p,q =

m→∞

  0, 

p,q ∈ (0, 1),

1 1−q ,

p = 1 and q ∈ (0, 1).

Thus for approximation processes, one need to choose convergent sequences (pm ) and (qm ) such that for each n, 0 < qm < pm ≤ 1 and pm , qm → 1 so that [m]pm ,qm → ∞ as m → ∞. Theorem 2.7. Let f ∈ CB [0, ∞) and qm ∈ (0, 1), pm ∈ (qm , 1] such that qm → 1, pm → 1, as m → ∞. Then for each u ∈ [0, ∞) we have e pmm ,qm (f ; u) = f (u). lim L

n→∞

Proof. By Korovkin’s theorem it is enough to show that pm ,qm m em (t ; u) = um , m = 0, 1, 2. lim L

m→∞

By Lemma 2.4, it is clear that e pmm ,qm (1; u) = 1 lim L

m→∞

e pmm ,qm (1; u) = u lim L

m→∞

and e pmm ,qm (t2 ; u) = lim lim L

m→∞

m→∞



u (2 − pm )([m]pm ,qm u+1) [m]pm ,qm

+

qm u + qm u2 [m]pm ,qm



= u2 . This completes the proof.

 3. Direct results

Let CB [0, ∞) be the space of real-valued continuous and bounded functions f defined on the interval [0, ∞). The norm k · k on the space CB [0, ∞) is given by k f k= sup | f (x) | . 0≤x 0 and W = {s ∈ CB [0, ∞) : s , s ∈ CB [0, ∞)}. Then as in ([4], p. 177, Theorem 2.4), there euists an absolute constant C > 0 such that √ K2 (f, δ) ≤ Cω2 (f, δ). (3.1)

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Second order modulus of smoothness of f ∈ CB [0, ∞) is as follows √ ω2 (f, δ) = sup√ sup | f (u + 2h) − 2f (u + h) + f (u) | . 0 1 e pmm ,qm (f ; u) − f (u) | |L = 0. m→∞ u∈[0,∞) (1 + u2 )α lim

sup

Proof. Let for any fixed u0 > 0, e pmm ,qm (f ; u) − f (u) | |L (1 + u2 )α u∈[0,∞)

≤

sup

sup u≤u0

≤

e pmm ,qm (f ; u) − f (u) | e pmm ,qm (f ; u) − f (u) | |L |L + sup (1 + u2 )α (1 + u2 )α u≥u0

k Lpmm ,qm (f ) − f k[c0 ,u0 ] + k f ku2 sup

u≤u0

| f (u) | + sup . 2 α u≥u0 (1 + u )

e pmm ,qm (1 + t2 ; u) | |L (1 + u2 )α (5.2)

Since, | f (u) |≤ Mf (1 + u2 ) we have, sup u≥u0

| f (u) | Mf Mf ≤ sup ≤ . 2 α 2 α−1 (1 + u ) (1 + u2 )α−1 u≥u0 (1 + u )

Let  > 0, and let us choose u0 large then we have Mf  < (1 + u0 2 )α−1 3

(5.3)

and in view of (2.4), we get, pm ,qm em (1 + t2 ; u) | |L m→∞ (1 + u2 )α

k f ku2 lim

= k f ku2

1 + u2 (1 + u2 )α

k f ku2 (1 + u2 )α−1 k f ku2 ≤ (1 + u0 2 )α−1  ≤ . 3 By using Theorem 5.3, the first term of inequality (5.2) becomes  pm ,qm em kL (f ) − f k[c0 ,u0 ] < , as m → ∞. 3 Hence we get the required proof by combining (5.3)-(5.5) ≤

(5.4)

(5.5)

e pmm ,qm (f ; u) − f (u) | |L < . (1 + u2 )α u∈[0,∞) sup

 Acknowledgment. The third author (M. Qasim) is grateful to Council of Scientific and Industrial Research (CSIR), India, for providing the Senior Research Fellowship under the grant number 09/1172(0001)/2017-EMR-I.

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References [1] T Acar, A. Aral and S. A Mohiuddine, Approximation By Bivariate (p, q)-Bernstein Kantorovich operators, Iran. J. Sci. Technol. Trans. A Sci. 42(2018), 655–662. [2] T. Acar, S.A. Mohiudine and M. Mursaleen, Approximation by (p, q)-Baskakov Durrmeyer Stancu operators, Complex Anal. Oper. Theory. 12(6) (2018), 1453–1468. [3] H. Ben Jebreen, M. Mursaleen and M. Ahasan, On the convergence of Lupa¸s (p, q)Bernstein operators via contraction principle, J. Inequl. Appl. (2019) 2019:34. [4] R. A. DeVore and G. G. Lorentz, Constructive Approximation, Grundlehren Math. Wiss. [Fundamental principales of Mathematical Sciences], Springer-Verlag, Berlin, (1993). [5] A.D. Gadjiev, Theorems of the type of P. P. Korovkins theorems, Math. Zametki 20(5) (1976), 7811–7786 [6] A.D. Gadzhiev, A problem on the convergence of a sequence of positive linear operators on unbounded sets, and theorems that are analogous to P. P. Korovkin’s theorem. Dokl. Akad. Nauk SSSR 218 (1974) (Russian),1001–1004. [7] A.D. Gadzhiev, Weighted approximation of continuous functions by positive linear operators on the whole real axis. Izv. Akad. Nauk Azerbazan. SSR Ser. Fiz.-Tehn. Mat. Nauk (5) (1975) (Russian),4145. [8] A.D. Gadjiev, R. O. Efendiyev and E. Ibikli, On Korovkin type theorem in the space of locally integrable functions, Czechoslovak Math. J. 53(1) (2003), 45–53. [9] M. N. Hounkonnou and J.D.B Kyemba, R(p, q)-calculus: differentiation and integration, SUT J. Math. 49(2) (2013), 145–167. ˙ [10] E. Ibikli and E. A. Gadjieva, The order of approximation of some unbounded functions by the sequence of positive linear operators, Turkish J. Math. 19(3) (1995) 331–337. [11] R. Jagannathan and K. Srinivasa Rao, Two-parameter quantum algebras, twin-basic numbers, and associated generalized hypergeometric series, Proceedings of the International Conference on Number Theory and Mathematical Physics, 20–21 December 2005. [12] Khalid Khan and D.K. Lobiyal, B´ezier curves based on Lupa¸s (p, q)-analogue of Bernstein functions in CAGD, J. Comput. Appl. Math. 317 (2017), 458–477. [13] Khalid Khan, D. K. Lobiyal and Adem Kilicman, A de Casteljau Algorithm for Bernstein type Polynomials based on (p, q)-integers, Appl. Appl. Math. 13(2) (2018). [14] Khalid Khan, D.K. Lobiyal and Adem Kilicman, B´ezier curves and surfaces based on modified Bernstein polynomials, Azerb. J. Math. 9(1) 2019. [15] P. P. Korovkin, Linear Operators and Approximation Theory , Hindustan Publ. Co., Delhi, 1960. [16] B. Lenze, On Lipschitz type maximal functions and their smoothness spaces, Nederl Akad Indag Math. 50 (1988), 53–63. [17] A. Lupa¸s, The approximation by some positive linear operators. In: proceedings of the International Dortmund meeting on Approximation Theory (M.W. Muller et al., eds.), akademie Verlag, Berlin, 1995, 201–229. [18] V.N. Mishra and S. Pandey, On (p, q) Baskakov-Durrmeyer-Stancu Operators, Adv. Appl. Clifford Algebr. 27(2) (2017), 1633-1646. [19] V.N. Mishra and S. Pandey, On Chlodowsky variant of (p, q) Kantorovich-Stancu-Schurer operators, Int. J. of Anal. Appl. 11(1) (2016), 28–39. [20] M. Mursaleen and A. Khan, Generalized q-Bernstein-Schurer operators and some approximation theorems, Jour. Function Spaces Appl. Volume (2013) Article ID 719834, 7 pages. [21] M. Mursaleen, K. J. Ansari and Asif Khan, On (p, q)-analogue of Bernstein Operators, Appl. Math. Comput. 266 (2015), 874-882. [Erratum: Appl. Math. Comput. 278 (2016), 70-71]. ¨ u and O. Do˘ [22] M. Orkc¨ gru, q-Sz´ asz-Mirakyan-Kantorovich type operators preserving some test functions, Appl. Math. Lett. 24 (2011), 1588–1593. [23] G. M. Phillips, Bernstein polynomials based on the q-integers, The Heritage of P. L. Chebyshev, Ann. Numer. Math. 4 (1997), 511–518.

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[24] N. Rao and A. Wafi. (p, q)-Bivariate-Bernstein-Chlowdosky operators, Filomat 32(2) (2018), 369–378. [25] N. Rao and A. Wafi. Bivariate-Schurer-Stancu operators based on (p, q)-integers, Filomat 32(4) (2018), 1251–1258. [26] K.K. Sing, A.R. Gairola and Deepmala, Approximation theorems for q- analogue of a linear operator by A. Lupa¸s, Int. Jour. Anal. Appl. 1(12) (2016), 30–37.

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ADDITIVE-QUADRATIC FUNCTIONAL INEQUALITIES IN FUZZY NORMED SPACES AND STABILITY YOUNG JU JEON AND CHANG IL KIM∗

Abstract. In this paper, we investigate the following functional inequality N (f (x − y) + f (y − z) + f (z − x) − 2[f (x) + f (y) + f (z)] − f (−x) − f (−y) − f (−z), t) ≥ N (f (x + y + z), t) and prove the generalized Hyers-Ulam stability for it in fuzzy Banach spaces.

1. Introduction and preliminaries The concept of a fuzzy norm on a linear space was introduced by Katsaras [11] in 1984. Later, Cheng and Mordeson [3] gave a new definition of a fuzzy norm in such a manner that the corresponding fuzzy metric is of Kramosil and Michalek type [13]. Definition 1.1. Let X be a real vector space. A function N : X × R −→ [0, 1] is called a fuzzy norm on X if for all x, y ∈ X and all c, s, t ∈ R, (N1) N (x, t) = 0 for all t ≤ 0; (N2) x = 0 if and only if N (x, t) = 1 for all t > 0; t (N3) N (cx, t) = N (x, |c| ) if c 6= 0; (N4) N (x + y, s + t) ≥ min{N (x, s), N (y, t)}; (N5) N (x, ·) is a nondecreasing function on R and limt→∞ N (x, t) = 1; (N6) for any x 6= 0, N (x, ·) is continuous on R. In this case, the pair (X, N ) is called a fuzzy normed space. Let (X, N ) be a fuzzy normed space. A sequence {xn } in X is said to be convergent if there exists an x ∈ X such that limn→∞ N (xn − x, t) = 1 for all t > 0. In this case, x is called the limit of the sequence {xn } in (X, N ) and one denotes it by N − limn→∞ xn = x. A sequence {xn } in (X, N ) is said to be Cauchy if for any  > 0, there is an m ∈ N such that for any n ≥ m and any positive integer p, N (xn+p − xn , t) > 1 −  for all t > 0. A fuzzy normed space is said to be complete if each Cauchy sequence in it is convergent and a complete fuzzy normed space is called a fuzzy Banach space. In 1940, Ulam proposed the following stability problem (cf.[21]): “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 an unique homomorphism h : G1 −→ G2 with d(f (x), h(x)) < δ for all x ∈ G1 ?” 2010 Mathematics Subject Classification. 39B62, 39B72, 54A40, 47H10. Key words and phrases. Hyers-Ulam stability, additive-quadratic functional inequality, fuzzy normed space, fixed point theorem. * Corresponding author. 1

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YOUNG JU JEON AND CHANGIL KIM

In the next year, Hyers [10] gave a partial solution of Ulam, s problem for the case of approximate additive mappings. Subsequently, his result was generalized by Aoki [1] for additive mappings, and by Rassias [17] for linear mappings, to consider the stability problem with unbounded Cauchy differences. A generalization of the Rassias’ theorem was obtained by Gˇ avruta [7] by replacing the unbounded Cauchy difference by a general control function in the spirit of the Rassias’ approach. In 2008, for the first time, Mirmostafaee and Moslehian [14], [15] used the definition of a fuzzy norm in [2] to obtain a fuzzy version of stability for the Cauchy functional equation (1.1)

f (x + y) = f (x) + f (y)

and the quadratic functional equation (1.2)

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

Gl´ anyi [8] and R¨ atz [18] showed that if a mapping f : X −→ Y satisfies the following functional inequality (1.3)

k2f (x) + 2f (y) − f (xy −1 )k ≤ kf (xy)k,

then f satisfies the following Jordan-Von Neumann functional equation 2f (x) + 2f (y) − f (xy −1 ) = f (xy). for an abelian group X divisible by 2 into an inner product space Y. Gl´ anyi [9] and Fechner [6] proved the Hyers-Ulam stability of (1.3). The stability problems of several functional equations and inequalities have been extensively investigated by a number of authors and there are many interesting results concerning the stability of various functional equations and inequalities. Now, we consider the following fixed point theorem on generalized metric spaces. Definition 1.2. Let X be a non-empty set. Then a mapping d : X 2 −→ [0, ∞] is called a generalized metric on X if d satisfies the following conditions: (D1) d(x, y) = 0 if and only if x = y, (D2) d(x, y) = d(y, x), and (D3) d(x, y) ≤ d(x, z) + d(z, y). In case, (X, d) is called a generalized metric space. Theorem 1.3. [4] Let (X, d) be a complete generalized metric space and let J : X −→ X a strictly contractive mapping with some Lipschitz constant L with 0 < L < 1. Then for each given element x ∈ X, either d(J n x, J n+1 x) = ∞ for all nonnegative integers n or there exists a positive integer n0 such that (1) d(J n x, J n+1 x) < ∞ for all n ≥ n0 ; (2) the sequence {J n x} converges to a fixed point y ∗ of J ; (3) y ∗ is the unique fixed point of J in the set Y = {y ∈ X | d(J n0 x, y) < ∞} and 1 (4) d(y, y ∗ ) ≤ d(y, Jy) for all y ∈ Y . 1−L The following function equation f : X −→ Y is called the Drygas functional equation : f (x + y) + f (x − y) = 2f (x) + f (y) + f (−y)

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for all x, y ∈ X. The Drygas functional equation has been studied by Szabo [20] and Ebanks, Fˇ aiziev and Sahoo [5]. The solutions of the Drygas functional equation in abelian group are obtained by H. Stetkær in [19]. In this paper, we investigate the following functional inequality which is related with the Drygas type functional equation (1.4)

N (f (x − y) + f (y − z) + f (z − x) − 2[f (x) + f (y) + f (z)] − f (−x) − f (−y) − f (−z), t) ≥ N (f (x + y + z), t)

and prove the generalized Hyers-Ulam stability for it in fuzzy Banach spaces. Throughout this paper, we assume that X is a linear space, (Y, N ) is a fuzzy Banach space, and (R, N 0 ) is a fuzzy normed space. 2. Solutions and the stability for (1.4) In this section, we investigate the functional equation (1.4) and prove the generalized Hyers-Ulam stability for it in fuzzy Banach spaces. For any mapping f : X −→ Y , let f (x) + f (−x) f (x) − f (−x) , fe (x) = . 2 2 In [12], the authors proved the following theorem: fo (x) =

Lemma 2.1. [12] Let f : X −→ Y be a mapping with f (0) = 0. Then f is quadratic if and only if f satisfies the following functional equation f (ax + by) + f (ax − by) − 2a2 f (x) − 2b2 f (y) = k[f (x + y) + f (x − y) − 2f (x) − 2f (y)] for all x, y ∈ X, a fixed nonzero rational number a and fixed real numbers b, k with a2 6= b2 . Using this, we have the following theorem: Theorem 2.2. If a mapping f : X −→ Y saisfies (1.4), then f is an additivequadratic mapping. Proof. Suppose that f satisfies (1.4). Setting x = y = z = 0 in (1.4), by (N3), we have  t N (f (0), t) ≤ N (6f (0), t) = N f (0), 6 for all t > 0 and by (N5), N (f (0), 6t ) ≤ N (f (0), t) for all t > 0. Hence we have N (f (0), t) = N (f (0), 6t) for all t > 0. By induction, we get N (f (0), t) = N (f (0), 6n t) for all t > 0 and all n ∈ N. By (N5), we get N (f (0), t) = lim N (f (0), 6n t) = 1 n→∞

for all t > 0 and hence by (N2), f (0) = 0. Letting z = −x − y in (1.4), we have N (f (x − y) + f (x + 2y) + f (−2x − y) − 2f (x) − 2f (y) − 2f (−x − y) − f (−x) − f (−y) − f (x + y), t) ≥ N (0, t) = 1

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for all x, y ∈ X and all t > 0 and so by (N2), we get f (x − y) + f (x + 2y) + f (−2x − y)

(2.1)

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

for all x, y ∈ X. By (2.1), we have fo (x − y) + fo (x + 2y) − fo (2x + y) = −fo (x + y) + fo (x) + fo (y)

(2.2)

for all x, y ∈ X and interchanging x and y in (2.2), we have −fo (x − y) + fo (2x + y) − fo (x + 2y) = −fo (x + y) + fo (x) + fo (y)

(2.3)

for all x, y ∈ X. By (2.2) and (2.3), we have fo (x + y) = fo (x) + fo (y) for all x, y ∈ X and hence fo is an additive mapping. By (2.1), we have fe (x − y) + fe (2x + y) + fe (x + 2y) = 3fe (x + y) + 3fe (x) + 3fe (y)

(2.4)

for all x, y ∈ X and letting y = −y in (2.4), we have fe (x + y) + fe (2x − y) + fe (x − 2y) = 3fe (x − y) + 3fe (x) + 3fe (y)

(2.5)

for all x, y ∈ X. By (2.4) and (2.5), we have fe (2x + y) + fe (2x − y) + fe (x + 2y) + fe (x − 2y)

(2.6)

= 2fe (x + y) + 2fe (x − y) + 6fe (x) + 6fe (y)

for all x, y ∈ X. Letting y = 0 in (2.6), we get (2.7)

fe (2x) = 4fe (x)

for all x ∈ X and letting y = 2y in (2.6), by (2.7), we have 4fe (x + y) + 4fe (x − y) + fe (x + 4y) + fe (x − 4y)

(2.8)

= 2fe (x + 2y) + 2fe (x − 2y) + 6fe (x) + 24fe (y)

for all x, y ∈ X. By (2.6) and (2.8), we have 2fe (2x + y) + 2fe (2x − y) + fe (x + 4y) + fe (x − 4y) = 18fe (x) + 36fe (y)

(2.9)

for all x, y ∈ X. Letting y = 2y in (2.9), by (2.7), we have fe (x + 8y) + fe (x − 8y) + 8fe (x + y) + 8fe (x − y) = 18fe (x) + 144fe (y) for all x, y ∈ X. By Lemma 2.1, fe is a quadratic mapping. Thus f is an additivequadratic mapping.  Now, we will prove the generalized Hyers-Ulam stability of (1.4) in fuzzy normed spaces. For any mapping f : X −→ Y , let Df (x, y, z) = f (x−y)+f (y−z)+f (z−x)−2[f (x)+f (y)+f (z)]−f (−x)−f (−y)−f (−z). Theorem 2.3. Assume that φ : X 3 −→ [0, ∞) is a function such that  x y z   L  (2.10) N0 φ , , , t ≥ N0 φ(x, y, z), t 2 2 2 4 for all x, y, z ∈ X, t > 0 and some L with 0 < L < 1. Let f : X −→ Y be a mapping such that f (0) = 0 and (2.11)

N (Df (x, y, z), t) ≥ min{N (f (x + y + z), t), N 0 (φ(x, y, z), t)}

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for all x, y, z ∈ X and all t > 0. Then there exists an unique additive-quadratic mapping F : X −→ Y such that   L (2.12) N f (x) − F (x), t ≥ min{N 0 (φ(x, −x, 0), t), N 0 (φ(−x, x, 0), t)} 4(1 − L) for all x ∈ X and all t > 0. Further, we have (2.13)

h 2n (2n + 1)  x  2n (2n − 1)  x i f n + f − n n→∞ 2 2 2 2

F (x) = N − lim

for all x ∈ X. Proof. Consider the set S = {g | g : X −→ Y } and the generalized metric d on S defined by d(g, h) = inf{c ∈ [0, ∞)| N (g(x) − h(x), ct) ≥ min{N 0 (φ(x, −x, 0), t), N 0 (φ(−x, x, 0), t)}, ∀x ∈ X, ∀t > 0}. Then (S, d) is a complete metric space([16]). Define a mapping J : S −→ S by Jg(x) = 3g( x2 ) + g(− x2 ) for all g ∈ S and all x ∈ X. Let g, h ∈ S and d(g, h) ≤ c for some c ∈ [0, ∞). Then by (2.10), we have N (Jg(x) − Jh(x), cLt)  x x  x   x +g − − 3h −h − , cLt = N 3g 2 2 2 2 n  x x 1    x  x 1 o ≥ min N g −h , cLt , N g − −h − , cLt 2 2 4 2 2 4 n  x x  1    x x  1 o 0 0 ≥ min N φ , − , 0 , Lt , N φ − , , 0 , Lt 2 2 4 2 2 4 ≥ min{N 0 (φ(x, −x, 0), t), N 0 (φ(−x, x, 0), t)} for all x ∈ X and all t > 0. Hence we have d(Jg, Jh) ≤ Ld(g, h) for any g, h ∈ S and so J is a strictly contractive mapping. Putting y = −x and z = 0 in (2.11), we get (2.14)

N (f (2x) − 3f (x) − f (−x), t) ≥ N 0 (φ(x, −x, 0), t)

for all x ∈ X, t > 0 and hence   x  x L  L  N f (x) − Jf (x), t = N f (x) − 3f −f − , t 4 2 2 4  x x  L  ≥ N 0 φ , − , 0 , t ≥ min{N 0 (φ(x, −x, 0), t), N 0 (φ(−x, x, 0), t)} 2 2 4 for all x ∈ X, t > 0 and so we have d(f, Jf ) ≤ L4 < ∞. By Theorem 1.3, there exists a mapping F : X −→ Y which is a fixed point of J such that d(J n f, F ) → 0 as n → ∞. By induction, we have J n f (x) =

2n (2n + 1)  x  2n (2n − 1)  x f n + f − n 2 2 2 2

for all x ∈ X and all n ∈ N and hence we have (2.13).

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t Replacing x, y, z, and t by 2xn , 2yn , 2zn , and 22n in (2.11), respectively, by (2.11), we have  x y z  1  N Dfe n , n , n , 2n t 2 2 2 2 n        o (2.15) ≥ min N Df x , y , − z , 1 t , N Df − x , − y , z , 1 t 2n 2n 2n 22n 2n 2n 2n 22n ≥ min{N 0 (Ln φ(x, y, z), t), N 0 (Ln φ(−x, −y, z), t)}

for all x, y, z ∈ X and all n ∈ N. Letting n → ∞ in (2.15), we have DFe (x, y, z) = 0 for all x, y, z ∈ X and by Lemma 2.1, Fe is an quadratic mapping. Similarly, Fo is an additive mapping and thus F is an additive-quadratic mapping. Since d(f, Jf ) ≤ L4 , by Theorem 1.3, we have (2.12). Now, we show the uniqueness of F . Let G be another additive-quadratic mapping with (2.12). Then clearly, G is a fixed point of J and (2.16)

d(Jf, G) = d(Jf, JG) ≤ Ld(f, G) ≤

L2 0 and some L with 0 < L < 1. Let f : X −→ Y be a mapping with f (0) = 0 and (2.11). Then there exists an unique additive-quadratic mapping F : X −→ Y such that   1 (2.18) N f (x) − F (x), t ≥ min{N 0 (φ(x, −x, 0), t), N 0 (φ(−x, x, 0), t)} 2(1 − L) for all x ∈ X and all t > 0. Further, we have i h 2n + 1 2n − 1 n n f (2 x) − f (−2 x) F (x) = lim n→∞ 22n+1 22n+1 for all x ∈ X. Proof. Consider the set S = {g | g : X −→ Y } and the generalized metric d on S defined by d(g, h) = inf{c ∈ [0, ∞)| N (g(x) − h(x), ct) ≥ min{N 0 (φ(x, −x, 0), t), N 0 (φ(−x, x, 0), t)}, ∀x ∈ X, ∀t > 0}. Then (S, d) is a complete metric space([16]). Define a mapping J : S −→ S by Jg(x) = 83 g(2x) − 18 g(−2x) for all g ∈ S and all x ∈ X. Let g, h ∈ S and d(g, h) ≤ c for some c ∈ [0, ∞). Then by (2.10), we have N (Jg(x) − Jh(x), cLt) 3  1 3 1 = N g(2x) − g(−2x) − h(2x) + h(−2x), cLt 8 8 8 8 ≥ min{N (g(2x) − h(2x), 2cLt), N (g(−2x) − h(−2x), 2cLt)} ≥ min{N 0 (φ(2x, −2x, 0), 2Lt), N 0 (φ(−2x, 2x, 0), 2Lt)} ≥ min{N 0 φ(x, −x, 0), t), N 0 (φ(−x, x, 0), t)}

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for all x ∈ X. Hence we have d(Jg, Jh) ≤ Ld(g, h) for any g, h ∈ S and since 0 < L < 1, J is a strictly contractive mapping. By (2.14), we get  t N f (x) − Jf (x), 2 3 1 t = N [f (2x) − 3f (x) − f (−x)] − [f (−2x) − 3f (−x) − f (x)], 8 8 2 ≥ min{N 0 (φ(x, −x, 0), t), N 0 (φ(−x, x, 0), t)} for all x ∈ X and all t > 0. Thus d(f, Jf ) ≤ similar to Theorem 2.3.

1 2

< ∞. The rest of proof the proof is 

As examples of φ(x, y, z) and N 0 (x, t) in Theorem 2.3 and Theorem 2.4, we can take φ(x, y, z) = (kxkp + kykp + kzkp ) and ( t , if t > 0 0 N (x, t) = t+k|x| 0, if t ≤ 0 for all x ∈ R, t > 0, and for some  > 0, where k = 1, 2. Then we can formulate the following corollary: Corollary 2.5. Let X be a normed space and (Y, N ) a fuzzy Banach space. Let f : X −→ Y be a mapping such that o n t N (Df (x, y, z), t) ≥ min N (f (x + y + z), t), p p p t + k(kxk + kyk + kzk ) for all x, y, z ∈ X, t > 0, a fixed real number p with 0 < p < 1 or 2 < p. Then there is an unique additive-quadratic mapping F : X −→ Y such that  (2p − 4)t   , if 2 < p  p   (2 − 4)t + 2kkxkp (2.19) N (f (x) − F (x), t) ≥   (2 − 2p )t    , if 0 < p < 1 (2 − 2p )t + 2kkxkp for all x ∈ X and all t > 0. For any f : X −→ Y , let (2.20)

D1 f (x, y) =f (x − y) + f (x + 2y) + f (−2x − y) − 2f (x) − 2f (y) − 2f (−x − y) − f (−x) − f (−y) − f (x + y)

Using Corollary 2.5, we have the following corollary: Corollary 2.6. Let X be a normed space and (Y, N ) a fuzzy Banach space. Let f : X −→ Y be a mapping such that (2.21)

N (D1 f (x, y), t) ≥

t t + k(kxkp + kykp + kx + ykp )

for all x, y, z ∈ X, t > 0, a fixed real number p with 0 < p < 1 or 2 < p. Then there is an unique additive-quadratic mapping F : X −→ Y with (2.19). We remark that the functional inequality (1.4) is not stable for p = 1 in Corollary 2.6. The following example shows that the inequality (2.21) is not stable for p = 1.

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Example 2.7. Define mappings t, s : R −→ R by   if |x| < 1 x, t(x) = −1, if x ≤ −1   1, if 1 ≤ x, ( x2 , if |x| < 1 s(x) = 1, ortherwise and a mapping f : R −→ R by f (x) =

∞ h X t(2n x) n=0

2n

+

s(2n x) i 4n

We will show that f satisfies the following inequality |D1 f (x, y)| ≤ 112(|x| + |y| + |x + y|)

(2.22)

for all x, y ∈ R and so f satisfies (2.21). But there do not exist an additive-quadratic mapping F : R −→ R and a non-negative constant K such that |F (x) − f (x)| ≤ K|x|

(2.23) for all x ∈ R.

Proof. Note that to (x) = t(x), so (x) = 0, and |fo (x)| ≤ 2 for all x ∈ R. First, suppose that 41 ≤ |x| + |y| + |x + y|. Then |D1 fo (x, y)| ≤ 48(|x| + |y| + |x + y|). Now suppose that 14 > |x| + |y| + |x + y|. Then there is a non-negative integer m such that 1 1 ≤ |x| + |y| + |x + y| < m+2 2m+3 2 and so 1 1 1 2m |x| < , 2m |y| < , 2m |x + y| < . 4 4 4 Hence we have {2m x, 2m y, 2m (x − y), 2m (x + y), 2m (x + 2y), 2m (2x + y)} ⊆ (−1, 1) and so for any n = 0, 1, 2, · · ·, m, |D1 t0 (2n x, 2n y)| = 0, because t(x) = to (x) = x on (−1, 1). Thus ∞ X 1 |D1 fo (x, y)| = D1 to (2n x, 2n y) n 2 n=0 m ∞ X X 1 1 n n n n ≤ D t (2 x, 2 y) + D t (2 x, 2 y) 1 o 1 o 2n 2n n=0 n=m+1

≤

12 ≤ 48(|x| + |y| + |x + y|), 2m+1

because |D1 t0 (2n x, 2n y)| ≤ 6.

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Note that te (x) = 0, se (x) = s(x), and |fe (x)| ≤ 34 for all x ∈ R. First, suppose that 14 ≤ |x| + |y| + |x + y|. Then |D1 fe (x, y)| ≤ 64(|x| + |y| + |x + y|). Now suppose that 14 > |x| + |y| + |x + y|. Then there is a non-negative integer k such that 1 1 ≤ |x| + |y| + |x + y| < 2k+2 22k+4 2 and so 22k |x|
2Kx 2n 2n n=0 n=0 which contradicts to (2.24).



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References [1] T. Aoki, On the stability of the linear transformation in Banach space, J.Math.Soc.Japan 2 (1950), 64-66. [2] T. Bag and S. K. Samanta, Finite dimensional fuzzy normed linear spaces, J. Fuzzy Math. 11(2003), 687-705. [3] S. C. Cheng and J. N. Mordeson, Fuzzy linear operator and fuzzy normed linear spaces, Bull. Cal. Math. Soc. 86(1994), 429–436. [4] J. B. 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. [5] B. R. Ebanks, PL. Kannappan, and P. K. Sahoo, A common generalization of functional equations characterizing normed and quasi-inner product spaces, Canad. Math. Bull. 35(3)(1992), 321-327. [6] W. Fechner, Stability of a functional inequality associated with the Jordan-Von Neumann functional equation, Aequationes Mathematicae 71(2006), 149-161. [7] P. Gˇ avruta, A generalization of the Hyer-Ulam-Rassias stability of approximately additive mappings, J. Math. Anal. Appl. 184(1994), 431-436. [8] A. Gil´ anyi, Eine zur Parallelogrammgleichung a ¨quivalente Ungleichung, Aequationes Mathematicae 62(2001), 303-309. [9] A. Gil´ anyi, On a problem by K. Nikoden, Mathematical Inequalities and Applications 4(2002), 707-710. [10] D. H. Hyers, On the stability of the linear functional equation, Proc. Natl. Acad. Sci. USA 27(1941), 222-224. [11] A. K. Katsaras, Fuzzy topological vector spaces II, Fuzzy Sets Syst 12(1984), 143-154. [12] C. I. Kim, G. Han, and S. A. Shim, Hyers-Ulam Stability for a Class of Quadratic Functional Equations via a Typical Form, Abs. and Appl. Anal. 2013(2013), 1-8. [13] I. Kramosil and J. Mich´ alek, Fuzzy metric and statistical metric spaces, Kybernetica 11(1975), 336-344. [14] A. K. Mirmostafaee and M. S. Moslehian, Fuzzy almost quadratic functions, Results Math. 52(2008), 161–177. [15] A. K. Mirmostafaee and M. S. Moslehian, Fuzzy versions of Hyers-Ulam-Rassias theorem, Fuzzy Sets Syst. 159(2008), 720-729. [16] M. S. Moslehian and Th. M. Rassias, Stability of functional equations in non-Archimedean spaces, Applicable Anal. Discrete Math. 1(2007), 325-334. [17] Th. M. Rassias, On the stability of the linear mapping in Banach spaces, Proc. Amer. Math. Soc. 72(1978), 297-300. [18] J. R¨ atz, On inequalities associated with the Jordan-von Neumann functional equation, Aequationes Mathematicae 66(2003), 191-200. [19] H. Stetkær, Functional equations on abelian groups with involution, II, Aequationes Math. 55(1998), 227-240. [20] G. Y. Szab´ o, Some functional equations related to quadratic functions, Glasnik Math. 38(1983), 107-118. [21] S. M. Ulam, Problems in Modern Mathematics, Wiley, New York, 1960, Chapter VI. Department of Mathematics Education, College of Education, ChonBuk National University, 567, Baekje-daero, deokjin-gu, Jeonju-si, Jeollabuk-do, 54896, Korea Email address: [email protected] Department of Mathematics Education, Dankook University, 152, Jukjeon-ro, Sujigu, Yongin-si, Gyeonggi-do, 16890, Korea Email address: [email protected]

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GENERALIZED ADDITIVE-CUBIC FUCNTIONAL EQUATION AND ITS STABILITY CHANG IL KIM

Abstract. In this paper, we establish some stability results for the following additive-cubic functional equation with an extra term Gf f (2x + y) + f (2x − y) + Gf (x, y) = 2f (x + y) + 2f (x − y) + 2f (2x) − 4f (x). in Banach spaces, where Gf is a functional operator of f . Using these, we give new additive-cubic functional equations and prove their stability.

1. Introduction In 1940, Ulam [12] raised the following question concerning the stability of group homomorphisms: “Under what conditions does there is an additive mapping near an approximately additive mapping between a group and a metric group ? ” In the next year, Hyers [5] gave a partial solution of Ulam, s problem for the case of additive mappings. Hyers ’s result, using unbounded Cauchy different, was generalized for additive mappings in [1] and for a linera mapping in [11]. Some stability results for additive, quardartic and mixed additve-cubic functional equations were investigated ([2], [3], [4], [6], [7], [8], [9], [10]). The generalized Hyers–Ulam stability for the mixed additive-cubic functional equation (1.1)

f (2x + y) + f (2x − y) = 2f (x + y) + 2f (x − y) + 2f (2x) − 4f (x)

in quasi-Banach spaces has been investigated by Najati and Eskandani [8]. Functional equation (1.1) is called an additive-cubic functional equation, since the function f (x) = ax3 + bx is its solution. Every solution of this mixed additive-cubic functional equation is said to be an additive-cubic mapping. In this paper, we are interested in what kind of a term Gf (x, y) can be added to (1.1) while the solution of the new functional equation is also an additive-cubic funtional equation and the generalized Hyers-Ulam stability for it still holds, where Gf (x, y) is a functional operator depending on the variables x, y, and function f . The new functional equation can be written as (1.2) f (2x + y) + f (2x − y) + Gf (x, y) = 2f (x + y) + 2f (x − y) + 2f (2x) − 4f (x). We give some new functional equations in section 3 as examples of our results and prove the generalized Hyers-Ulam stability for these. 2010 Mathematics Subject Classification. 39B62, 39B72. Key words and phrases. Hyers-Ulam stability, additive-cubic functional inequality. * Corresponding author. 1

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2. The generalized Hyers-Ulam stability for (1.2) Let X be a real normed linear space and Y a real Banach space. For given l ∈ N and any i ∈ {1, 2, · · ·, l}, let σi : X × X −→ X be a binary operation such that σi (rx, ry) = rσi (x, y) for all x, y ∈ X and all r ∈ R. Also let F : Y l −→ Y be a linear, continuous function. For a map f : X −→ Y , define Gf (x, y) = F (f (σ1 (x, y)), f (σ2 (x, y)), · · ·, f (σl (x, y))). Throughout this section we always assume that Gf satisfies the following two conditions unless a specific expression for Gf is given. Condition P1 : Suppose that f : X −→ Y is a mapping satisfying f (2x) = 2f (x) and (2.1)

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

for all x, y ∈ X. Then f is an additive mapping. Condition P2 : Suppose that f : X −→ Y is a mapping satisfying f (2x) = 8f (x) and (2.2)

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

for all x, y ∈ X. Then f is a cubic mapping. For any f : X −→ Y , let 1 1 1 4 f (x) − f (2x), fc (x) = − f (x) + f (2x) 3 6 3 6 Now, we prove the following main theorem. fa (x) =

Theorem 2.1. Let Gt be a functional operator satisfying Condition P1 and Condition P2 . Further, suppose that there is a real number λ(λ 6= −1) such that (2.3)

Gt (x, 2x) + 2Gt (x, x) − 2Gt (0, x) = λ[t(4x) − 10t(2x) + 16t(x)]

for all x ∈ X and all mapping t : X −→ Y. Let φ : X 2 −→ [0, ∞) be a function such that ∞ X 2−n φ(2n x, 2n y) < ∞ (2.4) n=0

for all x, y ∈ X. Let f : X −→ Y be a mapping such that f (0) = 0 and kf (2x + y) + f (2x − y) + Gf (x, y)

(2.5)

− 2f (x + y) − 2f (x − y) − 2f (2x) + 4f (x)k ≤ φ(x, y)

for all x, y ∈ X. Then there exists an unique additive-cubic mapping F : X −→ Y such that kFa (x) − fa (x)k (2.6)

≤

∞ X 1 2−n [φ(2n x, 2n+1 x) + 2φ(2n x, 2n x) + 2φ(0, 2n x)] 12|λ + 1| n=0

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and kFc (x) − fc (x)k (2.7)

≤

∞ X 1 2−3n [φ(2n x, 2n+1 x) + 2φ(2n x, 2n x) + 2φ(0, 2n x)] 48|λ + 1| n=0

for all x ∈ X. Proof. By (2.5), we have (2.8)

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

(2.9)

kf (3x) − 4f (2x) + 5f (x) + Gf (x, x)k ≤ φ(x, x),

and (2.10)

kf (4x) − 2f (3x) − 2f (2x) − 2f (−x) + 4f (x) + Gf (x, 2x)k ≤ φ(x, 2x)

for all x ∈ X. By (2.3), (2.8), (2.9), and (2.10), we have (2.11)

k2−1 fa (2x) − fa (x)k ≤

1 [φ(x, 2x) + 2φ(x, x) + 2φ(0, x)] 12|λ + 1|

for all x ∈ X. By (2.11), for m, n ∈ N ∪ {0} with 0 ≤ m < n, we have k2−n fa (2n x) − 2−m fa (2m x)k (2.12)

= 2−m k2−(n−m) fa (2n−m · 2m x) − fa (2m x)k ≤

n−1 X 1 2−k [φ(2k x, 2k+1 x) + 2φ(2k x, 2k x) + 2φ(0, 2k x)] 12|λ + 1| k=m

for all x ∈ X. By (2.12), {2−n fa (2n x)} is a Cauchy sequence in Y and since Y is a Banach space, there exists a mapping A : X −→ Y such that A(x) = lim 2−n fa (2n x) n→∞

for all x ∈ X . Moreover, by (2.12), we have kA(x) − fa (x)k (2.13)

≤

∞ X 1 2−n [φ(2n x, 2n+1 x) + 2φ(2n x, 2n x) + 2φ(0, 2n x)] 12|λ + 1| n=0

for all x ∈ X. By (2.5), we have kfa (2x + y) + fa (2x − y) + Gfa (x, y) − 2fa (x + y) − 2fa (x − y) 1 4 − 2fa (2x) + 4fa (x)k ≤ φ(x, y) + φ(2x, 2y) 3 6 for all x, y ∈ X. Replacing x and y by 2n x and 2n y in (2.14), respectively and deviding (2.5) by 2n , we have (2.14)

k2−n fa (2n (2x + y)) + 2−n fa (2n (2x − y)) + 2−n Gfa (2n x, 2n y) − 2 · 2−n fa (2n (x + y)) − 2 · 2−n fa (2n (x − y)) − 2 · 2−n fa (2n+1 x) 4 1 + 4 · 2−n fa (2n x)k ≤ · 2−n φ(2n x, 2n y) + · 2−n φ(2n+1 x, 2n+1 y) 3 6

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for all x, y ∈ X. Letting n → ∞ in the last inequality, we have A(2x + y) + A(2x − y) + lim 2−n Gfa (2n x, 2n y) n→∞

(2.15)

− 2A(x + y) − 2A(x − y) − 2A(2x) + 4A(x) = 0

for all x, y ∈ X and since F is continuous, lim 2−n Gfa (2n x, 2n y)

n→∞

= lim F (2−n fa (2n σ1 (x, y)), 2−n fa (2n σ2 (x, y)), · · ·, 2−n fa (2n σl (x, y))) n→∞

= GA (x, y) for all x, y ∈ X. Hence by (2.15), we have (2.16) A(2x + y) + A(2x − y) + GA (x, y) = 2A(x + y) + 2A(x − y) + 2A(2x) − 4A(x) for all x, y ∈ X. Relpacing x by 2n x in (2.11) and deviding (2.11) by 2n , we have k2−n−1 fa (2n · 2x) − 2−n fa (2n x)k ≤

2−n [φ(2n x, 2n+1 x) + 2φ(2n x, 2n x) + 2φ(0, 2n x)] 12|λ + 1|

for all x ∈ X and letting n → ∞ in the above inequality, we have (2.17)

A(2x) = 2A(x)

for all x, y ∈ X. By (2.16) and (2.17), A satisfies (2.1). By Condition P1 , A is an additive mapping . By (2.3), (2.8), (2.9), and (2.10), we have (2.18)

k8−1 fc (2x) − fc (x)k ≤

1 [φ(x, 2x) + 2φ(x, x) + 2φ(0, x)] 48|λ + 1|

for all x ∈ X. By (2.18), for m, n ∈ N ∪ {0} with 0 ≤ m < n, we have k2−3n fc (2n x) − 2−3m fc (2m x)k (2.19)

= 2−3m k2−3(n−m) fc (2n−m · 2m x) − fc (2m x)k ≤

n−1 X 1 2−3k [φ(2k x, 2k+1 x) + 2φ(2k x, 2k x) + 2φ(0, 2k x)] 48|λ + 1| k=m

for all x ∈ X. By (2.19), {2−3n fc (2n x)} is a Cauchy sequence in Y and since Y is a Banach space, there exists a mapping C : X −→ Y such that C(x) = lim 2−3n h(2n x) n→∞

for all x ∈ X . Moreover, by (2.19), we have kC(x) − fc (x)k (2.20)

≤

∞ X 1 2−3n [φ(2n x, 2n+1 x) + 2φ(2n x, 2n x) + 2φ(0, 2n x)] 48|λ + 1| n=0

for all x ∈ X. By (2.5), we have (2.21)

kfc (2x + y) + fc (2x − y) + Gfc (x, y) − 2fc (x + y) − 2fc (x − y) 1 1 − 2fc (2x) + 4fc (x)k ≤ φ(x, y) + φ(2x, 2y) 3 6

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for all x, y ∈ X. Replacing x and y by 2n x and 2n y in (2.21), respectively and deviding (2.21) by 23n , we have k2−3n fc (2n (2x + y)) + 2−3n fc (2n (2x − y)) + 2−3n Gfc (2n x, 2n y) − 2 · 2−3n fc (2n (x + y)) − 2 · 2−3n fc (2n (x − y)) − 2 · 2−n fc (2n+1 x) 1 1 + 4 · 2−3n fc (2n x)k ≤ · 2−3n φ(2n x, 2n y) + · 2−3n φ(2n+1 x, 2n+1 y) 3 6 for all x, y ∈ X. Letting n → ∞ in the last inequality, we have C(2x + y) + C(2x − y) + lim 2−3n Gfc (2n x, 2n y) n→∞

(2.22)

− 2C(x + y) − 2C(x − y) − 2C(2x) + 4C(x) = 0

for all x, y ∈ X and since F is continuous, lim 2−3n Gfc (2n x, 2n y)

n→∞

= lim F (2−3n h(2n σ1 (x, y)), 2−3n h(2n σ2 (x, y)), · · ·, 2−3n h(2n σl (x, y))) n→∞

= GC (x, y) for all x, y ∈ X. Hence by (2.22), we have (2.23) C(2x + y) + C(2x − y) + GC (x, y) = 2C(x + y) + 2C(x − y) + 2C(2x) − 4C(x) for all x, y ∈ X. Relpacing x by 2n x in (2.18) and deviding (2.18) by 23n , we have k2−3 · 2−3n fc (2n · 2x) − 2−3n fc (2n x)k ≤

2−3n [φ(2n x, 2n+1 x) + 2φ(2n x, 2n x) + 2φ(0, 2n x)] 48|λ + 1|

for all x ∈ X and letting n → ∞ in the above inequality, we have (2.24)

C(2x) = 8C(x)

for all x, y ∈ X. By (2.23) and (2.24), C satisifes (2.2). By Condition P2 , C is a cubic mapping. Let F = A + C. Then F is an additive-cubic mapping, Fa = A, and Fc = C. By (2.13) and (2.20), we have (2.6) and (2.7). For the uniqueness of F , let H be another additive-cubic mapping with (2.6) and (2.7). Then Fa and Ha are additive mappings and hence kFa (x) − Ha (x)k = 2−k kFa (2k x) − Ha (2k x)k ≤

∞ X 1 2−n [φ(2n x, 2n+1 x) + 2φ(2n x, 2n x) + 2φ(0, 2n x)] 6|λ + 1| n=k

for all x ∈ X. Hence, letting k → ∞ in the above inequality, we have Fa = Ha and similarly, we have Fc = Hc . Thus F = H.  Similarly, we have the following theorem: Theorem 2.2. Let Gt be a functional operator satisfying Condition P1 , Condition P2 , and (2.25)

Gt (x, 0) = −Gt (0, x).

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for all x ∈ X and all mapping t : X −→ Y. Further, suppose that there are real numbers λ, δ(λ 6= −1) such that Gt (x, 2x) + 2Gt (x, x) − 2Gt (0, x)

(2.26)

= λ[t(4x) − 10t(2x) + 16t(x)] + δ[f (x) + f (−x)]

for all x ∈ X and all mapping t : X −→ Y. Let φ : X 2 −→ [0, ∞) be a function with (2.4). Let f : X −→ Y be a mapping with f (0) = 0 and (2.5). Then there exists an unique additive-cubic mapping F : X −→ Y such that ∞ X 1 kFa (x) − fa (x)k ≤ 2−n [φ(2n x, 2n+1 x) 12|λ + 1| n=0 + 2φ(2n x, 2n x) + |δ|φ(2n x, 0) + (2 + |δ|)φ(0, 2n x)] and kFc (x) − fc (x)k ≤

∞ X 1 2−3n [φ(2n x, 2n+1 x) 48|λ + 1| n=0

+ 2φ(2n x, 2n x) + |δ|φ(2n x, 0) + (2 + |δ|)φ(0, 2n x)] for all x ∈ X. Proof. By (2.8) and (2.25), we have kf (x) + f (−x)k ≤ φ(x, 0) + φ(0, x) for all x ∈ X, because kGf (x, 0)k ≤ φ(x, 0) and Gf (x, 0) = −Gf (0, x). Similar to the proof of Theorem 2.1, we have k(1 + λ)[f (4x) − 10f (2x) + 16f (x)]k ≤ φ(x, 2x) + 2φ(x, x) + 2φ(0, x) + |δ|kf (x) + f (−x)k ≤ φ(x, 2x) + 2φ(x, x) + |δ|φ(x, 0) + (2 + |δ|)φ(0, x) for all x ∈ X and so we get 1 [φ(x, 2x) + 2φ(x, x) + |δ|φ(x, 0) + (2 + |δ|)φ(0, x)] 12|λ + 1| for all x ∈ X. The rest of this proof is similar to the proof of Theorem 2.1.  k2−1 fa (2x) − fa (x)k ≤

3. Applications In this section, using Theorem 2.1 and Theorem 2.2, we will prove the generalized Hyers-Ulam stability for some additive-cubic functional equations. First, we consider the following functional equation : (3.1) f (2x + y) + f (2x − y) − f (4x) = 2f (x + y) + 2f (x − y) − 8f (2x) + 12f (x). Theorem 3.1. Let φ : X 2 −→ [0, ∞) be a function with (2.4). Let f : X −→ Y be a mapping such that f (0) = 0 and (3.2)

kf (2x + y) + f (2x − y) − f (4x) − 2f (x + y) − 2f (x − y) + 8f (2x) − 12f (x)k ≤ φ(x, y)

for all x, y ∈ X. Then there exists an unique additive-cubic mapping F : X −→ Y such that ∞ 1 X −n (3.3) kFa (x) − fa (x)k ≤ 2 [φ(2n x, 2n+1 x) + 2φ(2n x, 2n x) + 2φ(0, 2n x)] 24 n=0

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and (3.4) kFc (x) − fc (x)k ≤

∞ 1 X −3n 2 [φ(2n x, 2n+1 x) + 2φ(2n x, 2n x) + 2φ(0, 2n x)] 96 n=0

for all x ∈ X. Proof. Let Gf (x, y) = −f (4x) + 10f (2x) − 16f (x). Then f satisfies (2.5) and Gt (x, 2x) + 2Gt (x, x) − 2Gt (0, x) = −3[t(4x) − 10t(2x) + 16t(x)] for all x ∈ X and all mapping t : X −→ Y. If t : X −→ Y is a mapping with t(2x) = 2t(x) for all x ∈ X and (2.1), then Gt (x, y) = 0 for all x, y ∈ X and so t is an additive mapping. Hence Gt satifies Condition P1 and similarly Gt satifies Condition P2 . By Theroem 2.1, there is an unique additive-cubic mapping F : X −→ Y with (3.3) and (3.4).  Using the above theorem, we have the following corollaries: Corollary 3.2. Let f : X −→ Y be a mapping. Then f satisfies (3.1) if and only if f is an additive-cubic mapping. Ostadbashi and Kazemzadeh [9] investigated the following additive-cubic functinal equation : (3.5)

f (2x + y) + f (2x − y) − f (4x) = 2f (x + y) + 2f (x − y) − 8f (2x) + 10f (x) − 2f (−x).

Corollary 3.3. Let φ : X 2 −→ [0, ∞) be a function with (2.4). Let f : X −→ Y be a mapping such that f (0) = 0 and (3.6)

kf (2x + y) + f (2x − y) − f (4x) − 2f (x + y) − 2f (x − y) + 8f (2x) − 10f (x) + 2f (−x)k ≤ φ(x, y)

for all x, y ∈ X. Then there exists an unique additive-cubic mapping F : X −→ Y such that ∞ 1 X −n 2 [φ1 (2n x, 2n+1 x) + 2φ1 (2n x, 2n x) + 2φ1 (0, 2n x)] (3.7) kFa (x) − fa (x)k ≤ 24 n=0 and (3.8) kFc (x)−fc (x)k ≤

∞ 1 X −3n 2 [φ1 (2n x, 2n+1 x)+2φ1 (2n x, 2n x)+2φ1 (0, 2n x)] 96 n=0

for all x ∈ X, where φ1 (x, y) = φ(x, y) + φ(0, x). Proof. By (3.6), we have kf (x) + f (−x)k ≤ φ(0, x) for all x ∈ X and hence we have kf (2x + y) + f (2x − y) − f (4x) − 2f (x + y) − 2f (x − y) + 8f (2x) − 12f (x)k ≤ φ(x, y) + φ(0, x) = φ1 (x, y) for all x, y ∈ X. By Theorem 3.3, we have the results.

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Finally, we consider the following new functional equation : (3.9)

f (2x + y) + f (2x − y) − 2f (x + y) + 3f (x − y) − 5f (y − x) − 10(x) + 14f (y) − 2f (2y) = 0.

Lemma 3.4. Let Gf be a functional operator such that Gf (x, y) = −Gf (y, x)

(3.10)

for all mapping f : X −→ Y and all x, y ∈ X. Condition P2 hold.

Then Condition P1 and

Proof. Suppose that f : X −→ Y is a mapping with f (2x) = 2f (x) and (2.1). Letting y = 0 in (2.1), we have (3.11)

Gf (x, 0) = 0

for all x ∈ X and by (3.10) and (3.11), we get Gf (x, 0) = −Gf (0, x) = −[f (x) + f (−x)] = 0 for all x ∈ X. Hence (3.12)

f (−x) = −f (x)

for all x ∈ X. Interchaging x and y in (2.1), by (3.12), we have (3.13)

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

for all x, y ∈ X and by (2.1), (3.10), and (3.13), we have (3.14)

f (2x + y) + f (2x − y) + f (x + 2y) − f (x − 2y) = 4f (x + y)

for all x, y ∈ X. Letting y = −y in (3.14), we have (3.15)

f (2x + y) + f (2x − y) + f (x − 2y) − f (x + 2y) = 4f (x − y)

for all x, y ∈ X. By (3.14) and (3.15), we have (3.16)

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

for all x, y ∈ X. Letting x = x + y in (3.16), we get (3.17)

f (x + 2y) + f (x) = f (x + 3y) + f (x − y)

for all x, y ∈ X and letting x = 2x in (3.16), we get (3.18)

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

for all x, y ∈ X. Letting y = x + y in (3.18), we get (3.19)

f (3x + y) + f (x − y) = 2f (2x + y) − 2f (y)

for all x, y ∈ X and interchaging x and y in (3.19), we have (3.20)

f (x + 3y) − f (x − y) = 2f (x + 2y) − 2f (x)

for all x, y ∈ X. By (3.17) and (3.20), we have (3.21)

f (x + 2y) − 3f (x) + 2f (x − y) = 0

for all x, y ∈ X. Letting x = x − y in (3.21), we get (3.22)

f (x + y) − 3f (x − y) + 2f (x − 2y) = 0

for all x, y ∈ X and letting y = −y in (3.22), we get (3.23)

f (x − y) − 3f (x + y) + 2f (x + 2y) = 0

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for all x, y ∈ X. By (3.21) and (3.23), we have f (x + y) + f (x − y) − 2f (x) = 0 for all x, y ∈ X and hence f is an additive mapping. Thus Condition P1 holds. (2) Suppose that f : X −→ Y is a mapping with f (2x) = 8f (x) and (2.2). Similar to (1), we have Gf (x, 0) = −Gf (0, x) = 0, f (−x) = −f (x) for all x, y ∈ X. Interchaging x and y in (2.2), we have (3.24)

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

for all x, y ∈ X and by (2.2), (3.10), and (3.24), we have (3.25) f (2x + y) + f (2x − y) + f (x + 2y) − f (x − 2y) = 4f (x + y) + 12f (x) + 12f (y) for all x, y ∈ X. Letting y = −y in (3.25), we have (3.26) f (2x + y) + f (2x − y) + f (x − 2y) − f (x + 2y) = 4f (x − y) + 12f (x) − 12f (y) for all x, y ∈ X. By (3.25) and (3.26), we have f (2x + y) + f (2x − y) = 2f (x + y) + 2f (x − y) + 12f (x) for all x, y ∈ X and hence f is a cubic mapping. Thus Condition P2 holds.



Using Lemma 3.4, we investigate solutions and the generalized Hyers-Ulam stability for (3.9). Theorem 3.5. Let φ : X 2 −→ [0, ∞) be a function with (2.4). Let f : X −→ Y be a mapping such that f (0) = 0 and (3.27)

kf (2x + y) + f (2x − y) − 2f (x + y) + 3f (x − y) − 5f (y − x) − 10(x) + 14f (y) − 2f (2y)k ≤ φ(x, y)

for all x, y ∈ X. Then there exists an unique additive-cubic mapping F : X −→ Y such that ∞ 1 X −n kFa (x) − fa (x)k ≤ 2 [φ(2n x, 2n+1 x) + 2φ(2n x, 2n x) 12 n=0 (3.28) + 5φ(2n x, 0) + 7φ(0, 2n x)] and (3.29)

kFc (x) − fc (x)k ≤

∞ 1 X −3n 2 [φ(2n x, 2n+1 x) + 2φ(2n x, 2n x) 48 n=0

+ 5φ(2n x, 0) + 7φ(0, 2n x)] for all x ∈ X. Proof. Let Gf (x, y) = 5[f (x − y) − f (y − x)] − 14[f (x) − f (y)] + 2[f (2x) − f (2y)]. Then f satisfies (2.5) and Gt (x, 2x) + 2Gt (x, x) − 2Gt (0, x) = −2[t(4x) − 10t(2x) + 16t(x)] − 5[f (x) + f (−x)] for all x ∈ X and all mapping t : X −→ Y. Since Gf satifies (3.10), by Lemma 3.4, Condition P1 and Condition P2 satisfy. By Theroem 2.2, there is an unique additive-cubic mapping F : X −→ Y with (3.28) and (3.29). 

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Corollary 3.6. Let f : X −→ Y be a mapping. Then f satisfies (3.9) if and only if f is an additive-cubic mapping. References [1] T. Aoki, On the stability of the linear transformation in Banach spaces, J. Math. Soc. Japan 2(1950), 64-66. [2] L. Cˇ adariu and V. Radu, Fixed points and the stability of Jensen’s functional equation, J. Inequal. Pure Appl. Math. 4(2003), 1-7. [3] J. B. 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. [4] P. Gˇ avruta, A generalization of the Hyer-Ulam-Rassias stability of approximately additive mappings, J. Math. Anal. Appl. 184(1994), 431-436. [5] D. H. Hyers, On the stability of the linear functional equation, Proc. Natl. Acad. Sci. 27(1941), 222-224. [6] K. Jun, H. Kim and I. Chang, On the Hyers-Ulam stability of an Euler-Lagrange type cubic functional equation, J. Comput. Anal. Appl., 7 (2005), 21-33 . [7] 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. [8] A. Najati and G. Z. Eskandani, Stability of a mixed additive and cubic functional equation in quasi-Banach spaces, J. Math. Anal. Appl., 342(2008), 1318-1331. [9] S. Ostadbashi and J. Kazemzadeh, Orthogonal stability of mixed type additive and cubic functional equation, Int. J. Nonlinear. Anal. Appl., 6(2015), 35-43 [10] C. Park, Orthogonal Stability of an Additive-Quadratic Functional Equation, Fixed Point Theory and Applications, 2011(2011), 1-11. [11] Th. M. Rassias, On the stability of the linear mapping in Banach sapces, Proc. Amer. Math. Soc., 72(1978), 297-300. [12] S. M. Ulam, Problems in Modern Mathematics, Chapter VI, Wiley, New York, 1960. Department of Mathematics Education, Dankook University, 152, Jukjeon-ro, Sujigu, Yongin-si, Gyeonggi-do, 16890, Korea Email address: [email protected]

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TOEPLITZ DUALS OF FIBONACCI SEQUENCE SPACES KULDIP RAJ, SURUCHI PANDOH AND KAVITA SAINI

Abstract. In this paper we introduce and study some classes of almost strongly convergent difference sequences of Fibonacci numbers defined by a sequence of modulus functions. We also make an effort to study some topological properties and inclusion relations between these classes of sequences. Further, we compute toeplitz duals of theses classes and study matrix transformations on these classes of sequences.

1. Introduction and Preliminaries Let w be the vector space of all real sequences. We shall write c, c0 and l∞ for the sequence spaces of all convergent, null and bounded sequences. Moreover, we write bs and cs for the spaces of all bounded and convergent series, respectively. Also, we use the conventions that e = (1, 1, 1, ...) and e(n) is the sequence whose only non-zero term is 1 in the nth place for each n ∈ N. Let X and Y be two sequence spaces and A = (ank ) be an infinite matrix of real numbers ank , where n, k ∈ N. Then we say that A defines a matrix transformation from X into Y and we denote it by writing A : X → Y if for every sequence x = (xk ) ∈ X, the sequence Ax = {An (x)} and the A-transform of x is in Y , where (1.1)

An (x) =

∞ X

ank xk (n ∈ N).

k=0

By (X, Y ) we denote the class of all matrices A such that A : X → Y . Thus, A ∈ (X, Y ) if and only if the series on the right-hand side of (1.1) converges for each n ∈ N and every x ∈ X, and we have Ax ∈ Y for all x ∈ X. The matrix domain XA of an infinite matrix A in a sequence space X is defined by (1.2)

XA = {x = (xk ) ∈ w : Ax ∈ X}

which is a sequence space. By using the matrix domain of a triangle infinite matrix, so many sequence spaces have recently been defined by several authors, (see [1], [2], [15], [25]). In the literature, the matrix domain X∆ is called the difference sequence space whenever X is a normed or paranormed sequence space, where ∆ denotes the backward difference matrix ∆ = (∆nk ) and ∆0 = (∆0nk ) denotes the forward difference matrix (the transpose of the matrix ∆), which are defined by  (−1)n−k , n − 1 ≤ k ≤ n, ∆nk = 0 , 0 ≤ k < n − 1 or k > n  (−1)n−k , n ≤ k ≤ n + 1, ∆0nk = 0 , 0 ≤ k < n or k > n + 1 2010 Mathematics Subject Classification. 11B39, 46A45, 46B45. Key words and phrases. Fibonacci numbers, difference matrix, modulus function, paranorm space, α−, β−, γ− duals, matrix transformations. 1

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for all k, n ∈ N respectively. The notion of difference sequence spaces was introduced by Kızmaz [16], who defined the sequence spaces X(∆) = {x = (xk ) ∈ w : (xk − xk+1 ) ∈ X} for X = l∞ , c and c0 . The difference space bνp , consisting of all sequences (xk ) such that (xk − xk−1 ) is in the sequence space lp , was studied in the case 0 < p < 1 by Altay and Ba¸sar [3] and in the case 1 ≤ p ≤ ∞ by Ba¸sar and Altay [7] and C ¸ olak et al. [9]. Kiri¸s¸ci and Ba¸sar [15] have been introduced and studied the generalized difference sequence spaces ˆ = {x = (xk ) ∈ w : B(r, s)x ∈ X} X where X denotes any of the spaces l∞ , lp , c and c0 , (1 ≤ p < ∞) and B(r, s)x = (sxk−1 + rxk ) with r, s ∈ R\{0}. Following Kiri¸s¸ci and Ba¸sar [15], S¨onmez [31] have been examined the sequence space X(B) as the set of all sequences whose B(r, s, t)-transforms are in the space X ∈ {l∞ , lp , c, c0 }, where B(r, s, t) denotes the triple band matrix B(r, s, t) = {bnk (r, s, t)} defined by  r, n = k    s, n = k + 1 bnk (r, s, t) = t, n = k + 2    0, otherwise for all k, n ∈ N and r, s, t ∈ R\{0}. Also in ([10-13], [26]) authors studied certain difference sequence spaces. A B-space is a complete normed space. A topological sequence space in which all coordinate functionals πk , πk (x) = xk , are continuous is called a K-space. A BK-space is defined as a K-space which is also a B-space, that is, a BK-space is a Banach space with continuous coordinates. For example, the space lp (1 ≤ p < ∞) is a BK-space with ∞ X  p1 and c0 , c and l∞ are BK-spaces with kxk∞ = sup |xk |. The sequence kxkp = |xk |p k

k=0

space X is said to be solid (see [17, p. 48]) if and only if e = {(vk ) ∈ w : ∃(xk ) ∈ X such that |vk | ≤ |xk | for all k ∈ N} ⊂ X. X A sequence (bn ) in a normed space X is called a Schauder basis for X if for every P x ∈ X there is a unique sequence (αn ) of scalars such that x = n αn bn , i.e., limm kx − m X αn bn k = 0. n=0

The following lemma (known as the Toeplitz Theorem) contains necessary and sufficient condition for regularity of a matrix. Lemma 1.1. (Wilansky, 1984): Matrix A = (ank )∞ n,k=1 is regular if and only if the following three conditions hold: (1) There exists M > 0 such that for every n = 1, 2, ... the following inequality holds: ∞ X

|ank | ≤ M ;

k=1

(2) lim ank = 0 for every k = 1, 2, ... n→∞

(3) lim

n→∞

∞ X

ank = 1.

k=1

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The sequence {fn }∞ n=0 of Fibonacci numbers is given by the linear recurrence relations f0 = f1 = 1 and fn = fn−1 + fn−2 , n ≥ 2. Fibonacci numbers have many interesting properties and applications in arts, sciences and architecture. For example, the ratio sequences of Fibonacci numbers converges to the golden ratio which is important in sciences and arts. Also, in [18] some basic properties of Fibonacci numbers are given as follows: √ 1+ 5 fn+1 = = φ (golden ratio), lim n→∞ fn 2 n X fk = fn+2 − 1 (n ∈ N), k=0

X 1 converges, fk k

fn−1 fn+1 − fn2 = (−1)n+1

(n ≥ 1) (Cassini formula).

2 Substituting for fn+1 in Cassini’s formula yields fn−1 + fn fn−1 − fn2 = (−1)n+1 .

Now, let A = (ank ) be an infinite matrix and list the following conditions: X (1.3) sup ank < ∞ n∈N

(1.4) (1.5) (1.6)

(1.7)

(1.8)

k

lim ank = 0 for each k ∈ N

n→∞

∃αk ∈ C 3 lim ank = αk for each k ∈ N n→∞

lim

n→∞

X

ank = 0

k

∃α ∈ C 3 lim

n→∞

sup

X

ank = α

k

X X ank < ∞

k∈H n

k∈K

where C and H denote the set of all complex numbers and the collection of all finite subsets of N, respectively. Now, we may give the following lemma on the characterization of the matrix transformations between some classical sequence spaces. Lemma 1.2. The following statements hold: (a) A = (ank ) ∈ (c0 , c0 ) if and only if (1.3) and (1.4) hold. (b) A = (ank ) ∈ (c0 , c) if and only if (1.3) and (1.5) hold. (c) A = (ank ) ∈ (c, c0 ) if and only if (1.3), (1.4) and (1.6) hold. (d) A = (ank ) ∈ (c, c) if and only if (1.3), (1.5) and (1.7) hold. (e) A = (ank ) ∈ (c0 , l∞ ) = (c, l∞ ) if and only if condition (1.3) holds. (f ) A = (ank ) ∈ (c0 , l1 ) = (c, l1 ) if and only if condition (1.8) holds. Recently, Kara [19] has defined the sequence spaces lp (Fˆ ) as follows: lp (Fˆ ) = {x ∈ w : Fˆ x ∈ lp }, (1 ≤ p ≤ ∞)

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where Fˆ = (fˆnk ) is the double band matrix defined by the sequence (fn ) of Fibonacci numbers as follows  fn+1   − fn , k = n − 1, fn (k, n ∈ N). fˆnk = , k = n, f   n+1 0 , 0 ≤ k < n − 1 or k > n Also, in [20] Kara et al. have characterized some classes of compact operators on the spaces lp (Fˆ ) and l∞ (Fˆ ), where 1 ≤ p < ∞. The inverse Fˆ −1 = (gnk ) of the Fibonacci matrix Fˆ is given by ( 2 fn+1 fk fk+1 , 0 ≤ k ≤ n, (k, n ∈ N). gnk = 0 , k>n that is,         

1

0

4 1 9 1 25 1 64 1

4 2 9 2 25 2 64 2

0 0 9 6 25 6 64 6

0 0 0 25 15 64 15

0 0 0 0 64 40

0 0 0 0 0 .. .

 ... ...   ...   ...   ...   .. .

.. .. .. .. .. . . . . . ˆ It is obvious that the matrix F is a triangular matrix, that is, fnn 6= 0 and fnk = 0 for k > n (n = 1, 2, 3...). Also, it follows by Lemma 1.1 that the method Fˆ is regular. In [8] Ba¸sarir et al. introduce the Fibonacci difference sequence spaces c0 (Fˆ ) and c(Fˆ ) as the set of all sequences whose Fˆ -transforms are in the spaces c0 and c, respectively, i.e., ( )  f  f n n+1 c0 (Fˆ ) = x = (xn ) ∈ w : lim xn − xn−1 = 0 , n→∞ fn+1 fn and

)   f fn+1 n xn − xn−1 = l . x = (xn ) ∈ w : ∃l ∈ C 3 lim n→∞ fn+1 fn

( c(Fˆ ) =

Define the sequence y = (yn ) by the Fˆ -transform of a sequence x = (xn ), i.e., ( x0 , n=0 ˆ (n ∈ N). (1.9) yn = Fn (x) = fn+1 fn fn+1 xn − fn xn−1 , n ≥ 1 A linear functional L on l∞ is said to be a Banach limit if it has the following properties: (1) L(x) ≥ 0 if n ≥ 0 (i.e. xn ≥ 0 for all n), (2) L(e) = 1, where e = (1, 1, ...), (3) L(Dx) = L(x), where the shift operator D is defined by D(xn ) = {xn+1 } (see [6]). Let B be the set of all Banach limits on l∞ . A sequence x = (xk ) ∈ l∞ is said to be almost convergent if all Banach limits of x = (xk ) coincide. In [22], it was shown that ( ) n 1X xk+s exits, uniformly in s cˆ = x = (xk ) : lim n→∞ n k=1

In ([23], [24]) Maddox defined strongly almost convergent sequences. Recall that a sequence x = (xk ) is strongly almost convergent if there is a number l such that n 1X lim |xk+s − l| = 0, uniformly in s. n→∞ n k=1

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Let X be a linear metric space. A function p : X → R is called paranorm, if (1) (2) (3) (4)

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

A paranorm p for which p(x) = 0 implies x = 0 is called total paranorm and the pair (X, p) is called a total paranormed space. It is well known that the metric of any linear metric space is given by some total paranorm (see [33], Theorem 10.4.2, pp. 183). A modulus function is a function f : [0, ∞) → [0, ∞) such that (1) (2) (3) (4)

f (x) = 0 if and only if x = 0, f (x + y) ≤ f (x) + f (y), for all x, y ≥ 0, f is increasing, f is continuous from the right at 0.

It follows that f must be continuous everywhere on [0, ∞). The modulus function may x , then f (x) is bounded. be bounded or unbounded. For example, if we take f (x) = x+1 p If f (x) = x , 0 < p < 1 then the modulus function f (x) is unbounded. Subsequently, modulus function has been discussed in ([4], [27], [28], [29], [30]) and references therein. Let F = (Fk ) be a sequence of modulus functions, p = (pk ) be any bounded sequence of positive real numbers and u = (uk ) be a sequence of strictly positive real numbers. In this paper we define ( the following sequence spaces: " # ) n f pk X 1 f k k+1 uk Fk xk − xk−1 =0 , c0 (Fˆ , F, u, p) = x = (xk ) ∈ w : lim n→∞ n fk+1 fk k=1 and " # ) n f pk 1X fk+1 k x = (xk ) ∈ w : ∃l ∈ C 3 lim uk Fk xk − xk−1 =l . n→∞ n fk+1 fk

( c(Fˆ , F, u, p) =

k=1

If Fk (x) = x, for all k ∈ N. Then above sequence spaces reduces to c0 (Fˆ , u, p) and c(Fˆ , u, p). By taking pk = 1 and uk = 1, for all k ∈ N, then we get the sequence spaces c0 (Fˆ , F) and c(Fˆ , F). With the notation of (1.2), the sequence spaces c0 (Fˆ , F, u, p) and c(Fˆ , F, u, p) can be redefined as follows: (1.10)

c0 (Fˆ , F, u, p) = {c0 (F, u, p)}Fˆ and c(Fˆ , F, u, p) = {c(F, u, p)}Fˆ .

The following inequality will be used throughout the paper. If 0 ≤ pk ≤ sup pk = H, K = max(1, 2H−1 ) then (1.11)

|ak + bk |pk ≤ K{|ak |pk + |bk |pk }

for all k and ak , bk ∈ C. Also |a|pk ≤ max(1, |a|H ) for all a ∈ C. In this paper, we introduce the sequence spaces c0 (Fˆ , F, u, p) and c(Fˆ , F, u, p). We investigate some topological properties of these new sequence spaces and establish some inclusion relations between these spaces. Also we determine the α−, β− and γ− duals of these spaces and construct the matrix transformation of the spaces (c0 (Fˆ , F, u, p), X) and (c(Fˆ , F, u, p), X), where X denote the spaces l∞ , f, c, f0 , c0 , bs, f s and l1 .

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2. Some topological properties of the spaces c0 (Fˆ , F, u, p) and c(Fˆ , F, u, p) Theorem 2.1. Let F = (Fk ) be a sequence of modulus functions, p = (pk ) be a bounded sequence of positive real numbers and u = (uk ) be a sequence of strictly positive real numbers. Then c0 (Fˆ , F, u, p) and c(Fˆ , F, u, p) are linear spaces over the field R of real numbers. Proof. Let x = (xk ), y = (yk ) ∈ c0 (Fˆ , F, u, p) and λ, µ ∈ C. Then there exist integers Mλ and Nµ such that |λ| ≤ Mλ and |µ| ≤ Nµ . Using inequality (1.11) and definition of modulus " function, we have # n  f   f  pk X f f k k+1 k k+1 1 uk Fk λ xk − xk−1 + µ yk − yk−1 n fk+1 fk fk+1 fk k=1

" # " # n n f pk f pk 1X fk+1 fk+1 1X k k + uk Fk |λ| xk − xk−1 uk Fk |µ| yk − yk−1 ≤ n fk+1 fk n fk+1 fk k=1

k=1

" " # # n n f f pk pk 1X fk+1 fk+1 1X k k ≤K uk Fk Mλ uk Fk Nµ xk − xk−1 yk − yk−1 +K n fk+1 fk n fk+1 fk k=1

k=1

"

n f fk+1 1X k uk Fk xk − xk−1 ≤ KMλH n fk+1 fk k=1

# pk

" # n f pk X f 1 k k+1 uk Fk yk − yk−1 +KNµH n fk+1 fk k=1

→ 0 as n → ∞. Thus λx + µy ∈ c0 (Fˆ , F, u, p). This proves that c0 (Fˆ , F, u, p) is a linear space. Similarly we can prove that c(Fˆ , F, u, p) is a linear space over the real field R.  Theorem 2.2. Let F = (Fk ) be a sequence of modulus functions and p = (pk ) be a bounded sequence of positive real numbers and u = (uk ) be a sequence of strictly positive real numbers. Then c0 (Fˆ , F, u, p) and c(Fˆ , F, u, p) are paranormed space with the paranorm defined by " # !1 n f pk M 1X fk+1 k g(x) = sup uk Fk xk − xk−1 n fk+1 fk k=1

where 0 ≤ pk ≤ sup pk = H, M = max(1, H). Proof. Since the proof is similar for the space c(Fˆ , F, u, p), we consider only the space fk c0 (Fˆ , F, u, p). Clearly g(−x) = g(x), for all x ∈ c0 (Fˆ , F, u, p). It is trivial that fk+1 xk − fk+1 fk xk−1

= 0, for x = 0. Hence we get g(0) = 0. Since pMk ≤ 1, using Minkowski’s inequality, we have " # !1 n  f   f  pk M X f f k k+1 k k+1 1 uk Fk xk − xk−1 + yk − yk−1 n fk+1 fk fk+1 fk k=1 " # !1 n f f pk M X fk+1 fk+1 k k 1 ≤ n uk Fk xk − xk−1 + uk Fk yk − yk−1 fk+1 fk fk+1 fk k=1 " # !1 " # !1 n n f pk M f pk M X fk+1 1X fk+1 k k 1 ≤ n uk Fk xk − xk−1 + uk Fk yk − yk−1 . fk+1 fk n fk+1 fk k=1

k=1

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Now it follows that g(x) is subadditive. Finally to check the continuity of scalar multiplication let us take any real number ρ. By definition of modulus function Fk , we have " # !1 n f pk M 1X fk+1 k g(ρx) = sup uk Fk ρ xk − xk−1 n fk+1 fk k k=1

H M

≤ Cρ g(x). where Cρ is a positive integer such that |ρ| ≤ Cρ . Now, Let ρ → 0 for any fixed x with g(x) = 0. By definition for |ρ| < 1, we have " # n f pk 1X fk+1 k (2.1) uk Fk xk − xk−1 <  for n > N (). n fk+1 fk k=1

Also for 1 ≤ n < N , taking ρ small enough. Since Fk is continuous, we have # " n pk f 1X fk+1 k (2.2) xk − xk−1 < . uk Fk n fk+1 fk k=1

Now from equation (2.1) and (2.2), we have g(ρx) → 0 as ρ → 0. This completes the proof.



Theorem 2.3. Let F = (Fk ) be a sequence of modulus functions, u = (uk ) be a sequence of strictly positive real numbers. If p = (pk ) and q = (qk ) are bounded sequences of positive real numbers with 0 ≤ pk ≤ qk < ∞ for each k, then c0 (Fˆ , F, u, p) ⊆ c(Fˆ , F, u, q). Proof. Let x ∈ c0 (Fˆ , F, u, p). Then " # n f pk 1X fk+1 k uk Fk xk − xk−1 −→ 0 as n → ∞. n fk+1 fk k=1

This implies that "

f fk+1 k uk Fk xk − xk−1 fk+1 fk

# pk ≤ 1,

for sufficiently large values of k. Since Fk is increasing and pk ≤ qk we have " # " # n n f qk f pk 1X fk+1 1X fk+1 k k uk Fk xk − xk−1 ≤ uk Fk xk − xk−1 n fk+1 fk n fk+1 fk k=1

k=1

−→

0 as n → ∞.

Hence x ∈ c(Fˆ , F, u, q). This completes the proof. Theorem 2.4. Let F = (Fk ) be a sequence of modulus functions and % = lim t→∞ Then c0 (Fˆ , F, u, p) ⊆ c0 (Fˆ , u, p).

 Fk (t) > 0. t

Proof. In order to prove that c0 (Fˆ , F, u, p) ⊆ c0 (Fˆ , u, p). Let % > 0. By definition of %, we have Fk (t) ≥ %(t), for all t > 0. Since % > 0, we have t ≤ %1 Fk (t) for all t > 0. Let x = (xk ) ∈ c0 (Fˆ , F, u, p). Thus, we have " # " # n n f pk f pk fk+1 fk+1 1X 1 X k k uk xk − xk−1 ≤ uk Fk xk − xk−1 n fk+1 fk %n fk+1 fk k=1

k=1

which implies that x = (xk ) ∈ c0 (Fˆ , u, p). This completes the proof.

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Theorem 2.5. Let F 0 = (Fk0 ) and F 00 = (Fk00 ) are sequences of modulus functions, then c0 (Fˆ , F 0 , u, p) ∩ c0 (Fˆ , F 00 , u, p) ⊆ c0 (Fˆ , F 0 + F 00 , u, p). Proof. Let x = (xk ) ∈ c0 (Fˆ , F 0 , u, p) ∩ c0 (Fˆ , F 00 , u, p). Therefore " # n pk 1X fk+1 0 fk −→ 0 as n → ∞. uk Fk xk − xk−1 n fk+1 fk k=1

and

" # n pk 1X fk+1 00 fk uk Fk xk − xk−1 −→ 0 as n → ∞. n fk+1 fk k=1

Then we have " # n pk X fk+1 0 00 fk 1 uk (Fk + Fk ) xk − xk−1 n fk+1 fk k=1 ( n " # ) pk 1X fk+1 0 fk ≤ K uk Fk xk − xk−1 n fk+1 fk k=1 ( n " # ) pk fk+1 1X 00 fk uk Fk xk − xk−1 + K n fk+1 fk k=1

−→

0 as n → ∞.

# f pk fk+1 k + xk − xk−1 −→ 0 as n → ∞. Thus fk+1 fk k=1 Therefore x = (xk ) ∈ c0 (Fˆ , F 0 + F 00 , u, p) and this completes the proof. 1 n

n X

"

uk (Fk0

Fk00 )



Theorem 2.6. Let F = (Fk ) and F 0 = (Fk0 ) be two sequences of modulus functions, then c0 (Fˆ , F 0 , u, p) ⊆ c0 (Fˆ , FoF 0 , u, p). Proof. Let x = (xk ) ∈ c0 (Fˆ , F 0 , u, p). Then we have " # n pk 1X fk+1 0 fk lim uk Fk xk − xk−1 = 0. n→∞ n fk+1 fk k=1

Let  > 0 and " choose δ > 0 with 0 < δ #< 1 such that Fk (t) <  for 0 ≤ t ≤ δ. fk Write yk = uk Fk0 fk+1 xk − fk+1 x and consider k−1 fk n

1X 1X 1X [Fk (yk )]pk = [Fk (yk )]pk + [Fk (yk )]pk n n 1 n 2 k=1

where the first summation is over yk ≤ δ and second summation is over yk ≥ δ. Since Fk is continuous, we have 1X (2.3) [Fk (yk )]pk < H n 1 and for yk > δ, we use the fact that yk
δ Fk (yk ) < 2Fk (1)

yk . δ

Hence 1 X  1X [Fk (yk )]pk ≤ max 1, (2Fk (1)δ −1 )H [yk ]pk . n 2 n

(2.4)

k

From equation (2.3) and (2.4), we have c0 (Fˆ , F 0 , u, p) ⊆ c0 (Fˆ , FoF 0 , u, p). This completes the proof.



Theorem 2.7. The sets c0 (Fˆ , F, u, p) and c(Fˆ , F, u, p) are BK-spaces with the norm kxkc0 (Fˆ ,F ,u,p) = kxkc(Fˆ ,F ,u,p) = kFˆ xk∞ . Proof. Since (1.10) holds, c0 and c are the BK-spaces with respect to their natural norms and the matrix Fˆ is a triangle; Theorem 4.3.12 of Wilansky [33, p.63] gives the fact that the spaces c0 (Fˆ , F, u, p) and c(Fˆ , F, u, p) are BK-spaces with the given norms. This completes the proof.  Remark 2.8. One can easily check that the absolute property does not hold on the spaces c0 (Fˆ , F, u, p) and c(Fˆ , F, u, p), that is, kxkc0 (Fˆ ,F ,u,p) 6= k|x|kc0 (Fˆ ,F ,u,p) and kxkc(Fˆ ,F ,u,p) 6= k|x|k ˆ for at least one sequence in the spaces c0 (Fˆ , F, u, p) and c(Fˆ , F, u, p), and c(F ,F ,u,p)

this shows that c0 (Fˆ , F, u, p) and c(Fˆ , F, u, p) are the sequence spaces of non-absolute type, where |x| = (|xk |). Theorem 2.9. The Fibonacci difference sequence spaces c0 (Fˆ , F, u, p) and c(Fˆ , F, u, p) of non-absolute type are linearly isomorphic to the spaces c0 and c respectively, i.e., c0 (Fˆ , F, p, u) ∼ = c0 and c(Fˆ , F, p, u) ∼ = c. Proof. To prove this, we should show the existence of a linear bijection between the spaces c0 (Fˆ , F, u, p) and c0 . Consider the transformation T defined with the notation of (1.9), from c0 (Fˆ , F, u, p) to c0 by x → y = T x. The linearity of T is clear. Further it is trivial that x = 0 whenever T x = 0 and hence T is injective. We assume that y = (yk ) ∈ c0 , for 1 ≤ p ≤ ∞ and defined the sequence x = (xk ) by

xk =

k 2 X fk+1 yj , for all k ∈ N. f f j=0 j j+1

Then we have ( lim

k→∞

# ) " n k k−1 f X pk 2 fk+1 1X fk+1 X fk2 k uk Fk yj − yj = lim yk = 0 k→∞ n fk+1 j=0 fj fj+1 fk j=0 fj fj+1 k=1

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which shows that x ∈ c0 (Fˆ , F, p, u). Additionally, we have for every x ∈ c0 (Fˆ , F, p, u) that " # n 1 X f pk fk+1 k kxkc0 (Fˆ ,F ,p,u) = sup uk Fk xk − xk−1 fk+1 fk k∈N n k=1 " # n k k−1 1 X f X pk 2 fk+1 fk+1 X fk2 k = sup uk Fk yj − yj fk+1 j=0 fj fj+1 fk j=0 fj fj+1 k∈N n k=1

=



sup |yk |pk



k∈N

= kyk∞ < ∞. Consequently, we see from here that T is surjective and norm preserving. Hence, T is a linear bijection which shows that the spaces c0 (Fˆ , F, u, p) and c0 are linearly isomorphic. It is clear here that if the spaces c0 (Fˆ , F, u, p) and c0 are respectively replaced by the spaces c(Fˆ , F, u, p) and c, then we obtain the fact that c(Fˆ , F, p, u) ∼ = c. This concludes the proof.  Now, we give some inclusion relations concerning with the space c0 (Fˆ , F, u, p) and c(Fˆ , F, u, p). Theorem 2.10. The inclusion c0 (Fˆ , F, u, p) ⊂ c(Fˆ , F, u, p) strictly holds. Proof. It is clear that the inclusion c0 (Fˆ , F, u, p) ⊂ c(Fˆ , F, u, p) holds. Further, to show k 2 X fk+1 that this inclusion is strict, consider the sequence x = (xk ) = . Then, we obtain fj2 j=0 (1.9) for all k ∈ N that " # " # n k k−1 2 n f X pk 2 f  pk fk+1 1X fk+1 X fk+1 1X k k+1 uk Fk − uk Fk = n fk+1 j=0 fj2 fk j=0 fj2 n fk k=1 k=1 " # n f  pk X k+1 1 uk Fk → ϕ, as k → ∞. This is to say that Fˆ (x) ∈ c\c0 . which shows that n fk k=1 Thus, the sequence x is in the c(Fˆ , F, u, p) but not in c0 (Fˆ , F, u, p). Hence, the inclusion c0 (Fˆ , F, u, p) ⊂ c(Fˆ , F, u, p) is strict.  Theorem 2.11. The space l∞ does not include the spaces c0 (Fˆ , F, u, p) and c(Fˆ , F, u, p). 2 2 Proof. Let us consider the sequence x = (xk ) = (fk+1 ). Since fk+1 → ∞ as k → ∞ and (0) ˆ ˆ F (x) = e = (1, 0, 0, ...), the sequence x is in the space c0 (F , F, u, p) but is not in the space l∞ . This shows that the space l∞ does not include the space c0 (Fˆ , F, u, p) and the space c(Fˆ , F, u, p), as desired. 

Theorem 2.12. The inclusions c0 ⊂ c0 (Fˆ , F, u, p) and c ⊂ c(Fˆ , F, u, p) strictly holds. Proof. Let X = c0 or c. Since the matrix Fˆ = (fnk ) satisfies the conditions  f X fn+1  1 5 n sup |fnk | = sup + =2+ = , fn 2 2 n∈N n∈N fn+1 k

lim fnk = 0,

n→∞

lim

n→∞

X k

 f fn+1  1 n − = −ϕ n→∞ fn+1 fn ϕ

fnk = lim

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TOEPLITZ DUALS OF FIBONACCI SEQUENCE SPACES

11

we conclude by parts (a) and (c) of Lemma 1.2 that (Fˆ , F, u, p) ∈ (X, X). This leads that (Fˆ , F, u, p)x ∈ X for any x ∈ X. Thus, x ∈ X(Fˆ ,F ,u,p) . This shows that X ⊂ X(Fˆ ,F ,u,p) . 2 Now, let x = (xk ) = (fk+1 ). Then, it is clear that x ∈ X(Fˆ ,F ,u,p) \X. This says that the inclusion X ⊂ X(Fˆ ,F ,u,p) is strict.  Theorem 2.13. The spaces c0 (Fˆ , F, u, p) and c(Fˆ , F, u, p) are not solid. 2 Proof. Consider the sequences r = (rk ) and s = (sk ) defined by rk = fk+1 and sk = k+1 ˆ (−1) for all k ∈ N. Then, it is clear that r ∈ c0 (F , F, u, p) and s ∈ l∞ . Nevertheless 2 rs = {(−1)k+1 fk+1 } is not in the space c0 (Fˆ , F, u, p), since " # n f pk 1X f k k+1 2 uk Fk (−1)k+1 fk+1 (−1)k fk2 − n fk+1 fk k=1 " # pk n
1X k+1 uk Fk 2(−1) fk fk+1 = for all k ∈ N. n k=1

This shows that the multiplication l∞ c0 (Fˆ , F, u, p) of the spaces l∞ and c0 (Fˆ , F, u, p) is not a subset of c0 (Fˆ , F, u, p). Hence, the space c0 (Fˆ , F, u, p) is not solid. It is clear here that if the spaces c0 (Fˆ , F, u, p) is replaced by the space c(Fˆ , F, u, p), then we obtain the fact c(Fˆ , F, u, p) is not solid. This completes the proof.  It is known from Theorem 2.3 of Jarrah and Malkowsky [14] that the domain XT of an infinite matrix T = (tnk ) in a normed sequence space X has a basis if and only if X has a basis, if T is a triangle. As a direct consequence of this fact, we have (−1)

(n)

Corollary 2.14. Define the sequences c(−1) = {ck }k∈N and c(n) = {ck }k∈N for every fixed n ∈ N by ( k 2 X 0 , 0≤k ≤n−1 fk+1 (n) (−1) 2 and ck = ck = fk+1 f f fn fn+1 , k ≥ n j=0 j j+1 Then, the following statements hold: (a) The sequence {c(n) }∞ c (Fˆ , F, u, p) and every sequence x ∈ n=0 is a basis for the space P ˆ0 ˆ c0 (F , F, u, p) has a unique representation x = n Fn (x)c(n) . ˆ (b) The sequence {c(n) }∞ n=−1 is a basis for the space c(F , F, u, p) and every sequence z = P ˆ (zn ) ∈ c(F , F, u, p) has a unique representation z = lc(−1) + n [Fˆn (z) − l]c(n) , where l = lim Fˆn (z). n→∞

3. The α−, β− and γ− duals of the spaces c0 (Fˆ , F, u, p) and c(Fˆ , F, u, p) and some matrix transformations The α−, β− and γ− duals of the sequence space X are respectively defined by X α = {a = (ak ) ∈ w : ax = (ak xk ) ∈ l1 for all x = (xk ) ∈ X}, X β = {a = (ak ) ∈ w : ax = (ak xk ) ∈ cs for all x = (xk ) ∈ X} and X γ = {a = (ak ) ∈ w : ax = (ak xk ) ∈ bs for all x = (xk ) ∈ X} In this section, we determine α−, β− and γ− duals of the sequence spaces c0 (Fˆ , F, u, p) and c(Fˆ , F, u, p), and characterize the classes of infinite matrices from the spaces c0 (Fˆ , F, u, p) and c(Fˆ , F, u, p) to the spaces c0 , c, l∞ , f, f0 , bs, f s, cs and l1 , and from the space f to the

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spaces c0 (Fˆ , F, u, p) and c(Fˆ , F, u, p). The following two lemmas are essential for our results. Lemma 3.1. [8] Let X be any of the spaces c0 or c and a = (an ) ∈ w, and the matrix B = (bnk ) be defined by Bn = an Fˆn−1 , that is ,  an gnk , 0 ≤ k ≤ n, bnk = 0 , k>n for all k, n ∈ N. Then a ∈ XFβˆ if and only if B ∈ (X, l1 ). Lemma 3.2 (5, Theorem 3.1). Let C = (cnk ) be defined via a sequence a = (ak ) ∈ w and the inverse matrix V = (vnk ) of the triangle matrix Z = (znk ) by  Pn j=k aj vjk , 0 ≤ k ≤ n, cnk = 0 , k>n for all k, n ∈ N. Then for any sequence space X, XZγ = {a = (ak ) ∈ w : C ∈ (X, l∞ )}, XZβ = {a = (ak ) ∈ w : C ∈ (X, c)}. Combining Lemmas (1.2), (3.1), and (3.2), we have Corollary 3.3. Consider the sets d1 , d2 , d3 and d4 defined as follows: #pk ( " ) n X f2 X1X n+1 d1 = a = (ak ) ∈ w : sup uk Fk an 0,

r∈N

(27)

we obtain ∞ X m  X

(24k−2 − 1)(4k)(4k − 1)|B4k−2 | ϕ(t) = 1 − 4k (2 − 1)(s + 4(m − k) + 3)(s + 4(m − k) + 4)|B4k | s=0 k=1  X ∞ 2(24k − 1)|B4k |ts+4m+1 ts+4m+1 + . (4k)!(s + 4(m − k) + 2)! (s + 4m + 1)! s=0



For s ≥ 0 and m ≥ k ≥ 1, we have (s + 4(m − k) + 3)(s + 4(m − k) + 4) ≥ (s + 3)(s + 4) ≥ 12 and then ϕ(t) ≥

∞ X m X s=0 k=1 ∞ X

+

2(24k − 1)|B4k | (4k)!(s + 4(m − k) + 2)!

  (24k−2 − 1)(4k)(4k − 1)|B4k−2 | s+4m+1 1− t 12(24k − 1)|B4k |

ts+4m+1 (s + 4m + 1)!

s=0 ∞ m XX

  2(24k − 1)|B4k | (24k−2 − 1)(4k)(4k − 1)|B4k−2 | s+4m+1 ≥ 1− t (4k)!(s + 4(m − k) + 2)! 12(24k − 1)|B4k | s=0 k=2   ∞ ∞ X X ts+4m+1 30|B4 | |B2 | . 1− ts+4m+1 + + (4!)(s + 4m − 2)!) 5|B | (s + 4m + 1)! 4 s=0 s=0 Using inequality (21) with v = 2k − 1 for k ∈ N, we get   ∞ X m ∞ X 2(24k − 1)|B4k | π 2 + 1 s+4m+1 X ts+4m+1 ϕ(t) > 1− t + > 0, (4k)!(s + 4(m − k) + 2)! 12 (s + 4m + 1)! s=0 k=2 s=0 which complete the proof. Lemma 2.4. For a positive integer m, the function 2m−1 X (22k − 1)B2k 1 M (x) = − G(x) + , x kx2k k=1

x>0

(28)

is strictly completely monotonic. Proof. Using the formula (25) and the integral representation of G(x), we have # Z ∞" 2m−1 X (22k − 1)B2k t2k−1 e−xt M (x) = (1 + et ) − (et − 1) dt t k(2k − 1)! 1 + e 0 k=1 Z ∞ e−xt = µ(t) dt, 1 + et 0

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where t

µ(t) = (1 + e )

2m−1 X k=1

2(22k − 1)B2k 2k−1 t − (et − 1). (2k)!

Now 2m−1 X

µ(t) =

k=1 2m−1 X

=

k=1 2m−1 X

=

k=1

2m−1 ∞ ∞ 2(22k − 1)B2k 2k−1 X 2(22k − 1)B2k X tr+2k−1 X tr t + − µ(t) (2k)! (2k)! r! r! r=0 r=1 k=1 2m−1 ∞ ∞ r X ts t 2(22k − 1)B2k 2k−1 X 2(22k − 1)B2k X t + − (2k)! (2k)! (s − 2k + 1)! r=1 r! k=1 s=2k−1 2m−1 4m−2 4m−2 X tr 2(22k − 1)B2k 2k−1 X 2(22k − 1)B2k X ts t + − (2k)! (2k)! (s − 2k + 1)! r! r=1 k=1 s=2k−1

∞ X

+

2m−1 X

s=4m−1 k=1

∞ X 2(22k − 1)B2k ts tr − . (2k)! (s − 2k + 1)! r=4m−1 r!

Rewrite infinite summations from 0 and split finite summations by even and odd power of t, we obtain µ(t) =

2m−1 X

2m−1 2m−1 2m−1 X t2r−1 2(22k − 1)B2k 2k−1 X 2(22k − 1)B2k X t2s−1 t + − (2k)! (2k)! (2s − 2k)! (2r − 1)! r=1 k=1 s=k

k=1 2m−1 X

+

+

2m−1 2m−1 X t2r 2(22k − 1)B2k X t2s − (2k)! (2s − 2k + 1)! (2r)! r=1 s=k

k=1 ∞ 2m−1 X X s=0 k=1

∞

X 2(22k − 1)B2k tr+4m−1 ts+4m−1 − , (2k)! (s + 4m − 2k)! r=0 (r + 4m − 1)!

which can be rewritten as µ(t) =

2m−1 X

2m−1 s 2m−1 2(22s − 1)B2s 2s−1 X 1 X 2(22k − 1)(2s!)B2k 2s−1 X t2s−1 t + t − (2s)! (2s)! k=1 (2k)!(2s − 2k)! (2s − 1)! s=1 s=1

s=1 2m−1 X

+

s 2m−1 X 1 2(22k − 1)(2s + 1)!B2k 2s X t2s t − (2s + 1)! k=1 (2k)!(2s − 2k + 1)! (2s)! s=1

s=1 ∞ 2m−1 X X

∞

X ts+4m−1 ts+4m−1 2(22k − 1)B2k + − (2k)! (s + 4m − 2k)! s=0 (s + 4m − 1)! s=0 k=1 " # s−1 2m−1 X X
2t2s−1 = 2(22s − 1)B2s − s + (22k − 1) 2s B 2k 2k (2s)! s=1 k=1 " # 2m−1 s X X
t2s + −(2s + 1) + 2(22k − 1) 2s+1 B 2k 2k (2s + 1)! s=1 k=1 " # ∞ 2m−1 X X 1 s + 4m − 3 2(22k − 1)B2k + + ts+4m−1 . 2 (s + 4m − 1)! (2k)!(s + 4m − 2k)! s=0 k=2

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Using the identities (12) and (13) with the relation (27), µ(t) satisfies ∞ m−1 X X

 2(24k+2 − 1)B4k+2 2(24k − 1)B4k + ts+4m−1 µ(t) > (4k)!(s + 4(m − k))! (4k + 2)!(s + 4(m − k) − 2)! s=0 k=1 " ∞ m−1   XX (24k − 1)(4k + 1)(4k + 2)|B4k | > 1 − 4k+2 (2 − 1)(s + 4(m − k) − 1)(s + 4(m − k))|B4k+2 | s=0 k=1  2(24k+2 − 1)|B4k+2 |ts+4m−1 . (4k + 2)!(s + 4(m − k) − 2)! For s ≥ 0 and m − k ≥ 1, we have (s + 4(m − k) − 1)(s + 4(m − k)) ≥ (s + 3)(s + 4) ≥ 12 and then µ satisfies µ(t) >

∞ m−1 X X s=0 k=1

π2 + 1 1− 12



2(24k+2 − 1)|B4k+2 |ts+4m−1 > 0, (4k + 2)!(s + 4(m − k) − 2)!

which complete the proof. From the complete monotonicity of the two functions F (x) and M (x) with the asymptotic expansion (9), we get the following double inequality which posed as a conjecture in [21]. Lemma 2.5. The following double inequality holds 2m X (22k − 1)B2k k=1

k

x

−2k

< G(x) − x

−1


0.

(29)

From the positivity of the two functions ϕ(t) and µ(t) in the proofs of Lemmas 2.3 and 2.4, we obtain the following result: Lemma 2.6. The following double inequality holds 2m 2k 2k X 2 (2 − 1)B2k k=1

(2k)!

x

2k−1

≤ tanh(x) ≤

2l−1 X k=1

22k (22k − 1)B2k 2k−1 x , (2k)!

l, m ∈ N ; x ≥ 0

(30)

and the inequality is reversed if x ≤ 0. Equality holds if x = 0. Remark 1. In the case |x| < holds, since

π 2

and l or m = tends to ∞, in the inequality (30) in fact equality

tanh(x) =

∞ X 22k (22k − 1)B2k k=1

(2k)!

x2k−1 ,

|x|
a, a ∈ R with limx→∞ K(x) = 0. Then K(x) > 0, if K(x) > K(x + 1) for all x > a and K(x) < 0, if K(x) < K(x + 1) for all x > a. To present our next result, we can easily prove the following simple modification on Lemma 2.7: Corollary 2.8. Let K be a real-valued function defined on x > a, a ∈ R with limx→∞ K(x) = 0. Then for m ∈ N, K(x) > 0, if K(x) > K(x+m) for all x > a and K(x) < 0, if K(x) < K(x+m) for all x > a. Proof. For m ∈ N, if we have K(x) > K(x + m) and limx→∞ K(x) = 0, then K(x) > K(x + m) > ... > K(x + rm) > ... > lim K(x + rm) = lim K(y) = 0. r→∞

y→∞

The other case is similarly treated. Lemma 2.9. The function q(x) =

1 G(x) −

1 x

− 2x2 ,

x>0

(31)

is strictly increasing. Proof. For x > 0, we have q 0 (x) =

L(x) , [G(x) − x1 ]2

where L(x) = −G0 (x) − 4xG2 (x) + 8G(x) −

(4x + 1) . x2

Now,   L(x + 1) − L(x) = G0 (x) − G0 (x + 1) + 4x G2 (x) − G2 (x + 1) − 4G2 (x + 1) 4x2 + 6x + 1 − 8 [G(x) − G(x + 1)] + 2 x (x + 1)2 and using equation (4) and its derivative, we get L(x + 1) − L(x) = 2G0 (x) − 4G2 (x + 1) +

6x2 + 10x + 3 , L1 (x). x2 (x + 1)2

Consider the difference   L1 (x + 2) − L1 (x) = 2 [G0 (x + 2) − G0 (x)] − 4 G2 (x + 3) − G2 (x + 1) 4 (27 + 135x + 220x2 + 158x3 + 51x4 + 6x5 ) − x2 (x + 1)2 (x + 2)2 (x + 3)2

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and using equation (4) and its derivative, we obtain   4x5 + 34x4 + 98x3 + 99x2 + 3x − 9 16 G(x + 1) − L1 (x + 2) − L1 (x) = (x + 1)(x + 2) 4x2 (x + 1)(x + 2)(x + 3)2 16 , L2 (x). (x + 1)(x + 2) Using equation (4), the function L2 (x) satisfies L2 (x + 2) − L2 (x) = −

2x2 (x

3(7x + 15)(7x + 20) < 0. + 1)(x + 2)2 (x + 3)2 (x + 4)(x + 5)2

From the asymptotic formula (9) and its derivative ∞

G0 (x) ∼ −

X 2(22k − 1)B2k 1 − , x2 k=1 x2k+1

x→∞

(32)

we have lim L(x) = lim L1 (x) = lim L2 (x) = 0.

x→∞

x→∞

x→∞

Hence, using Corollary 2.8, we get that L(x) > 0 for all x > 0 which completes the proof. As a consequence of the monotonicity of the function q(x) with the asymptotic expansion (9), we obtain the following inequality: Lemma 2.10. The following double inequality holds 1 1 1 < G(x) − < 2 , +α x 2x + β

x>0

2x2

(33)

where α = 1 and β = 0 are the best possible constants. Remark 2. The double inequality (33) is a refinement of the double inequality (10). Lemma 2.11. The function U (x) = G(x) −

1 1 − 2 , x 2x + 1

x>0

(34)

is strictly completely monotonic. Proof. Using the formula (25), the integral representation of G(x) and the Laplace transform of sine function, we have Z ∞ U (x) = λ(t)e−xt dt, 0

where

et − 1 1 λ(t) = t − √ sin e +1 2

980



t √ 2

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Since sin z < 1, we get t

λ(t) >

e −1 1 − √ > 0, t e +1 2

t > ln

! √ 2+1 √ ≈ 1.76275 . 2−1

Also, from the generalization of Redheffer-Williams’s inequality [40], [41], [42], [46] π 2 − x2 12 − x2 sin x ≤ ≤ , π 2 + x2 x 12 + x2 and the inequality (30) for m = 4, we obtain λ(t) > q √ 4 3399−120 ≈ 2.58051. 17

0 0 for 0 < t
0

(35)


√ r−2l+2 ( 2x)

x>0

(36)

2. For even positive integer r , we have √ r r2 +1 X r! r!( 2) G(r) (x) > r+1 + (−1)l+1 x (2x2 + 1)r+1 l=1

r+1 2l−1

Also, as a consequence of the proof of Lemma 2.11, we obtain the following inequality: Lemma 2.13. The following double inequality holds √ 1 tanh(x) ≥ √ sin( 2x), 2

x ≥ 0.

(37)

Equality holds iff x = 0.

3

Applications: Some inequalities of Wallis ratio

The Wallis ratio Wm =

1.3.5...(2m − 1) Γ(m + 1/2) =√ , 2.4.6...(2m) π Γ(m + 1)

m∈N

(38)

plays an important role in mathematics especially in special functions, combinatorics, graph theory and many other branches. For further details about its history and applications, we refer to [7], [16], [18], [20], [26]-[29].

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Guo, Xu and Qi [14] deduced the inequality  m  m √ √ 1 C2 1 C1 1− m − 1 < Wm ≤ 1− m − 1, m ≥ 2 (39) m 2m m 2m p with the best possible constants C1 = πe and C2 = 43 . Recently, Qi and Mortici [37] presented the following improvement of the double inequality (39) r

r  m+1/3  m+1/3 1 e 1 e 1 1− 1− < Wm < e 144m3 , πm 2(m + 1/3) πm 2(m + 1/3)

m ∈ N. (40)

Also, Zhang, Xu and Situ [47] presented the inequality 1 √ eπm

1 1 m− 12m m− 12m+16   1 1 1 1+ < Wm ≤ √ 1+ , 2m 2m eπm

m ∈ N.

(41)

Recently, Cristea [8] improved the upper bound of the inequality (41) by 1 Wm ≤ √ eπm

1 1  m− 12m + 1 2− 48m 2880m3 1 1+ , 2m

m∈N

(42)

which is better than the upper bound of the inequality (40).

3.1

New proof of Slavi´ c inequality

Slavi´ c [43] presented the following double inequality ! ! 2l−1 2m −2k −2k X X 1 1 (1 − 2 )B2k Γ(x + 1/2) (1 − 2 )B2k √ exp < √ exp < , 2k−1 k(1 − 2k)x Γ(x + 1) k(1 − 2k)x2k−1 x x k=1 k=1

(43)

where x > 0 and l, m ∈ N . In the following sequel, we will present a new proof of Slavi´ c inequality (43). Consider the two functions ! 2l−1 X (1 − 2−2k )B2k Γ(x + 1/2) √ x exp , l∈N SL (x) = Γ(x + 1) k(2k − 1)x2k−1 k=1 and

2m X Γ(x + 1/2) √ (1 − 2−2k )B2k SU (x) = x exp Γ(x + 1) k(2k − 1)x2k−1 k=1

! ,

m ∈ N.

Using Lemma 2.5, we obtain SL0 (x) 1 = G(2x) − − SL (x) 2x

2l−1 X k=1

982

(1 − 2−2k )B2k kx2k

! < 0,

l∈N

Mahmoud 970-986

J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 29, NO.5, 2021, COPYRIGHT 2021 EUDOXUS PRESS, LLC

and SU0 (x) 1 = G(2x) − − SU (x) 2x

2m X (1 − 2−2k )B2k

! > 0,

kx2k

k=1

m ∈ N.

Then the function SL (x) is decreasing and the function SU (x) is increasing and using the asymptotic expansion of the ratio of two gamma functions [19]   (a − b)(a + b − 1) Γ(x + a) a−b −2 1+ ∼x + O(x ) , a, b ≥ 0 (44) Γ(x + b) 2x as x → ∞, we have lim SL (x) = lim SU (x) = 1.

x→∞

x→∞

Hence we get SL (x) > 1

and

SU (x) < 1,

which complete the proof of Slavi´ c inequality (43). Remark 3. In the case of l = 1, m = 1 and x = m, the inequality (43) will gives −1

−1

1

e 8m e 8m + 192m3 √ < Wm < √ , πm πm

m∈N

(45)

which is better than inequality (40) of Qi and Mortici [37].

3.2

New upper bound of Wn

Consider the function ML (x) =

√ Γ(x + 1/2) √ −1 √ tan−1 (2 2x)− π ] 2 xe 2 2 [ , Γ(x + 1)

x > 0.

Using the inequality (33), we get ML0 (x) 1 1 = G(2x) − − 2 >0 ML (x) 2x 8x + 1 and using the expansion (44), we have limx→∞ ML (x) = 1. Then ML (x) < 1 and we obtain the following result: Lemma 3.1. The following double inequality holds 1 √ tan−1 (2 Γ(x + 1/2) e2 2[ √ < Γ(x + 1) x

√ 2x)− π2 ]

,

x > 0.

(46)

Remark 4. In the case of x = m in the inequality (46), we have √

1 √ tan−1 (2 2m)− π2 ] e2 2[ √ Wm < , πm

m∈N

(47)

which is better than inequality (42) of Cristea [8].

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References [1] M. Abramowitz, I. A. Stegun (Eds.), Handbook of Mathematical Functions with Formulas, Graphs and Mathematical Tables, Dover, New York, 1965. [2] H. Alzer and C. Berg, Some classes of completely monotonic functions, Annales Acad. Sci. Fenn. Math. 27(2), 445-460, 2002. [3] G. E. Andrews, R.Askey and R.Roy, Special Functions, Cambridge Univ. Press, 1999. [4] G. B. Arfken and H. J. Weber, Mathematical Methods For Physicists, 6th edition, Elsevir academic press, 2005. ´ Besenyei, On complete monotonicity of some functions related to means, Math. Inequal. [5] A. Appl. 16, no. 1, 233239, 2013. [6] T. Buri´ c, N. Elezovi´ c, Some completely monotonic functions related to the psi function, Math. Inequal. Appl., 14(3), 679-691, 2011. [7] C.P. Chen, and F. Qi, The best bounds in Wallis’ inequality, Proceedings of the Mathematical Society, 133, no. 2, 397-401, 2004. [8] V. G. Cristea, A direct approach for proving Wallis ratio estimates and an improvement of Zhang-Xu-Situ inequality, Stud. Univ. Babe¸s-Bolyai Math. 60, No. 2, 201-209, 2015. [9] A. Elbert and A. Laforgia, On some properties of the gamma function, Proc. Amer. Math. Soc., 128 (9), 2667-2673, 2000. [10] A. Erd´ elyi et al., Higher Transcendental Functions Vol. I-III, California Institute of Technology - Bateman Manuscript Project, 1953-1955 McGraw-Hill Inc., reprinted by Krieger Inc. 1981. [11] W. Feller, An Introduction to Probability Theory and Its Applications, V. 2, Academic Press, New York, 1966. [12] A. Z. Grinshpan and M. E. H. Ismail, Completely monotonic functions involving the gamma and q−gamma functions, Proc. Amer. Math. Soc., 134, 1153-1160, 2006. [13] B.-N. Guo and F. Qi, A property of logarithmically absolutely monotonic functions and the logarithmically complete monotonicity of a power-exponential function, Politehn. Univ. Bucharest Sci. Bull. Ser. A Appl. Math. Phys. 72, no. 2, 2130, 2010. [14] S. Guo, J.-G. Xu and F. Qi, Some exact constants for the approximation of the quantity in the Wallis formula, J. Inequal. Appl. 2013, 2013:67, 7 pages. [15] F. Hausdorff, Summationsmethoden und Momentfolgen I, Math. Z. 9, 74-109, 1921. [16] M. D. Hirschhorn, , Comments on the paper ”Wallis sequence estimated through the EulerMaclaurin formula: even the Wallis product π could be computed fairly accurately” by Lampret, Aust. Math. Soc. Gazette, 32(2005), 104.

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[17] V. Kac and Pokman Cheung, Quantum Calculus, Springer-Verlag, 2002. [18] D. K. Kazarinoff, On Wallis’ formula, Edinburgh Math. Notes, 40, 19-21, 1956. [19] A. Laforgia and P. Natalini, On the asymptotic expansion of a ratio of gamma functions, J. Math. Anal. Appl. 389, 833-837, 2012. [20] L. Lin, J.-E. Deng and C.-P. Chen, Inequalities and asymptotic expansions associated with the Wallis sequence, J. Inequal. Appl., 251, 2014. [21] M. Mahmoud and R. P. Agarwal, Bounds for Batemans G-function and its applications, Georgian Mathematical Journal, Vol. 23, Issue 4, 579-586, 2016 [22] M. Mahmoud and H. Almuashi, On some inequalities of the Bateman’s G−function, J. Comput. Anal. Appl., Vol. 22, No.4, , 672-683, 2017. [23] M. Mahmoud, A. Talat and H. Moustafa, Some approximations of the Bateman’s G−function, J. Comput. Anal. Appl., Vol. 23, No. 6, 1165-1178, 2017. [24] M. Mahmoud and H. Almuashi, Generalized Bateman’s G−function and its bounds, J. Computational Analysis and Applications, Vol. 24, No. 1, 2018. [25] C. Mortici, A sharp inequality involving the psi function, Acta Universitatis Apulensis, 41-45, 2010. [26] C. Mortici, Sharp inequalities and complete monotonicity for the Wallis ratio, Bull. Belg. Math. Soc. Simon Stevin, 17, 929-936, 2010. [27] C. Mortici, A new method for establishing and proving new bounds for the Wallis ratio, Math. Inequal. Appl., 13, 803-815, 2010. [28] C. Mortici, Completely monotone functions and the Wallis ratio, Appl. Math. Lett., 25, no. 4, 717-722, 2012. [29] C. Mortici, C., Estimating π from the Wallis sequence, Math. Commun., 17, 489-495, 2012. [30] V. Namias, A simple derivation of Stirlings asymptotic series. Amer. Math. Monthly, 93(1), 25-29, 1986. [31] K. Oldham, J. Myland and J. Spanier, An Atlas of Functions, 2nd edition. Springer, 2008. [32] F. Qi and C.-P. Chen, A complete monotonicity property of the gamma function, J. Math. Anal. Appl. 296, 603-607, 2004. [33] F. Qi, D.-W. Niu and B.-N. Guo, Refinements, generalizations, and applications of Jordans inequality and related problems, Journal of Inequalities and Applications 2009 (2009), Article ID 271923, 52 pages. [34] F. Qi, S. Guo, B.-N. Guo, Complete monotonicity of some functions involving polygamma functions, J. Comput. Appl. Math., 233(9), 2149-2160, 2010.

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[35] F. Qi and S.-H. Wang, Complete monotonicity, completely monotonic degree, integral representations, and an inequality related to the exponential, trigamma, and modified Bessel functions, Glob. J. Math. Anal. 2, no. 3, 91-97, 2014. [36] F. Qi, A double inequality for ratios of Bernoulli numbers, RGMIA Research Report Collection 17,Article 103, 4 pages, 2014. [37] F. Qi and C. Mortici, Some best approximation formulas and inequalities for the Wallis ratio, Applied Mathematics and Computation 253, 363-368, 2015. [38] F. Qi and W.-H. Lic, Integral representations and properties of some functions involving the logarithmic function, Filomat 30:7, 1659-1674, 2016. [39] S.-L. Qiu and M. Vuorinen, Some properties of the gamma and psi functions with applications, Math. Comp., 74, no. 250, 723-742, 2004. [40] R. Redheffer, Problem 5642, Amer. Math. Monthly 75, No. 10, 1125, 1968. [41] R. Redheffer, Correction, Amer. Math. Monthly 76, No. 4, 422, 1969. [42] J. Sandor and B. A. Bhayo, On an inequality of Redheffer, Miskolc Mathematical Notes, Vol. 16, No. 1, 475-482, 2015. [43] D. V. Slavi´ c, On inequalities for Γ(x + 1)/Γ(x + 1/2), Univ. Beograd. Publ. Elektrotehn. Fak. Ser. Mat. Fiz. No. 498 – No. 541, (1975), 17–20. [44] H. Van Haeringen, Completely monotonic and related functions, J. Math. Anal. Appl., 204, 389-408, 1996. [45] D. V. Widder, The Laplace Transform, Princeton University Press, Princeton, 1946. [46] J. Williams, Solution of problem 5642, Amer. Math. Monthly, Vol. 10, No. 76, 1153-1154, 1969. [47] X.-M. Zhang, T.-Q. Xu and L.-B. Situ, Geometric convexity of a function involving gamma function and application to inequality theory, J. Inequal. Pure Appl. Math., 8(2007), No. 1, 9 pages. [48] T.-H. Zhao, Z.-H. Yang and Y.-M. Chu, Monotonicity properties of a function involving the Psi function with applications, Journal of Inequalities and Applications (2015) 2015:193.

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On Ramanujan’s asymptotic formula for n! Ahmed Hegazi1 , Mansour Mahmoud2 and Hend Salah3 Mansoura University, Faculty of Science, Mathematics Department, Mansoura 35516, Egypt. 1 [email protected] 2 [email protected] 3 Moon

[email protected]

. . Abstract In this paper, we present the following new asymptotic formula of factorial n r √  n n 6 1 n! ∼ π − U (n), n→∞ 8n3 + 4n2 + n + e 30
−1 9480 919466 1455925 639130140029 where U (n) = 240 depending on Ra11 n + 847 + 65219 n + 5021863 n2 − 92804028240 n3 + ... manujan’s approximation formula for n! and we deduce the following upper bound for i1/6 h √ 1 1 gamma function Γ(x + 1) < π (x/e)x 8x3 + 4x2 + x + 30 + 240x + , x > 0. 9480 11

847

2010 Mathematics Subject Classification: 41A60, 41A25, 33B15. Key Words: Factorial, Ramanujan’s formula, asymptotic formula, best possible constant, rate of convergence,bounds.

1

Introduction.

In many science branches, we need estimations of big factorials. Stirling’s formula  n n √ n! ∼ 2πn , n→∞ e is the most well known and used approximation formula for factorial n, which is satisfactory in many branches such as statistical physics and statistics but we need more precise estimates in many pure mathematics studies. For more details about Stirling’s formula refinements and its related inequalities, we refer to [2], [12], [22]. Other known formula for estimating n! for large values of n is Ramanujan formula: r √  n n 6 1 n! ∼ π 8n3 + 4n2 + n + , e 30

987

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(1)

J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 29, NO.5, 2021, COPYRIGHT 2021 EUDOXUS PRESS, LLC

which is a refinement of Stirling’s formula and was recorded in the book ”The lost notebook and other unpublished papers” as a conjecture of Srinivasa Ramanujan based on some numerical evidence. For more details please refer to [1], [4], [13], [24], [29]. Starting from Ramanujan formula (1), Karatsuba presented the following asymptotic formula [13]  1/6 √ 1 11 79 3539 x 3 2 − + Γ(x + 1) ∼ π (x/e) 8x + 4x + x + + + ... , (2) 30 240x 3360x2 201600x3 R∞ where Γ(x) = 0 e−r rx−1 dr, x > 0 is the ordinary gamma function and n! = Γ(n + 1) for n ∈ N . Mortici [23] improve the Ramanujan formula by establishing the following asymptotic formula:  1/6   √ 1 11 13 1 x 3 2 exp − Γ(x + 1) ∼ π (x/e) 8x + 4x + x + + + + ... , 30 11520x4 3440x5 691200x6 (3) which is faster than formula (2). Dumitrescu and Mortici [9] introduced the following class of approximations: s √ α β 1 + + , α, β, δ ∈ R Γ(x + 1) ∼ 2πx (x/e)x 6 1 + 2(x − δ) 2(x − δ)2 2(x − δ)3

(4)

which is a generalization of the Ramanujan’s formula (1) at δ = 0, α = 1/8 and β = 1/240. More various results involving approximations for the gamma function and the factorial can be found in [7], [8], [15], [16], [25], [26], [30] and the references therein. In sequel, we need the following important Lemma, which is due to Mortici in 2010 and is a very useful tool for constructing asymptotic expansions and measuring the convergence rate of a family of null sequences [19]: Lemma 1.1. If {σm }m∈N is a null sequence and there is s ∈ R and n > 1 such that lim mn (σm − σm+1 ) = s,

(5)

m→∞

then we have lim mn−1 σm =

m→∞

s . n−1

From Lemma (1.1), we can conclude that the convergence rate of the sequence {σm }m∈N will increase with the increasing of the value of n in relation (5). Several approximations, formulas and inequalities have been produced using the technique developed by this Lemma. For more details please refer to [5], [6], [11], [14], [17], [20], [21], [28] and the references therein. In the rest of this paper, we will present a new asymptotic formula of n! depending on Ramanujan’s asymptotic formula (1) and we deduce a new upper bound for the ordinary gamma function related to our new asymptotic formula.

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J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 29, NO.5, 2021, COPYRIGHT 2021 EUDOXUS PRESS, LLC

2

Main results.

In our first step, we will try to find the best possible constants k1 and k2 in the approximation formula r √  n n 6 1 1 n! ∼ π − , n→∞ (6) 8n3 + 4n2 + n + e 30 k1 n + k2 by defining a sequence An satisfies r √  n n 6 1 1 n! = π 8n3 + 4n2 + n + − e An , n ≥ 1. e 30 k1 n + k2 Then

If



1 12k1



  1 25 29 1 5k2 An − An+1 = − + + − 2 5 n 48k1 96k1 2016 n6   −9031k13 + 158200k12 + 100800k1 k2 + 33600k22 1 + + O(n−8 ). 268800k13 n7    5k2 11 29 25 − 2880 6= 0 and − 48k + 6= 0, then the sequence An − An+1 has a rate of − 2 96k1 2016 1 11 − 12k1 2880



1

worse than n−6 . So, we will consider (

that is, k1 =

240 11

and k2 =

9480 . 847

1 11 =0 − 2880 12k1 5k2 25 29 − 48k2 − 96k1 + 2016 1

=0

Now by Lemma (1.1), we obtain the following result:

Lemma 2.1. The sequence √

 1 1 − An = ln n! − ln π − n ln n − n − ln n 8n3 + 4n2 + n + 6 30

1 240 n+ 11

 9480 847

(7)

has a rate of convergence equal to n−6 , where lim n7 (An − An+1 ) =

n→∞

459733 . 124185600

In our second step, we will try to find the best possible constants T1 , T2 and T3 in the approximation formula s √  n n 6 1 1 n! ∼ π 8n3 + 4n2 + n + − 240 , n→∞ (8) T1 T2 T3 9480 e 30 + n + + + 2 3 11 847 n n n by defining a sequence Bn satisfies s √  n n 6 1 n! = π 8n3 + 4n2 + n + − e 30

989

240 n 11

+

9480 847

1 + Tn1 +

T2 n2

+

T3 n3

eBn ,

n ≥ 1.

Hegazi 987-994

J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 29, NO.5, 2021, COPYRIGHT 2021 EUDOXUS PRESS, LLC

Hence Bn − Bn+1 = + − + + − + − − − +

(919466 − 65219T1 ) (45457643T1 − 10043726T2 − 637955952) + 248371200 n7 32784998400 n8 1 (4253517961T12 − 1277759560770T1 + 466430635440T2 9 265066712064000 n 92804028240T3 + 16394247383595) 1 (−5933657555595T12 + 1965125297982T1 T2 10 54427031543808000 n 750735798062481T1 − 361540539736530T2 + 118464342048360T3 8420494064916176) 1 (−277410187898459T13 + 143136026144382810T12 11 301743462878871552000 n 79155247002714960T1 T2 + 12105171835569120T1 T3 10550047712231492850T1 + 6052585917784560T22 + 6180552136457196960T2 2679997511635567200T3 + 101393364617835255540) O(n−12 ).

To obtain the best possible values of the constants T1 , T2 and T3 , we put  65219T1 = 919466  45457643T1 − 10043726T2 = 637955952 ,  4253517961T12 − 1277759560770T1 + 466430635440T2 − 92804028240T3 = −16394247383595 that is, T1 = 919466 , T2 = 65219 following result:

1455925 5021863

and T3 = − 639130140029 . Hence by Lemma (1.1), we get the 92804028240

Lemma 2.2. The sequence √

 1 1 Bn = ln n! − ln π − n ln n − n − ln n 8n3 + 4n2 + n + 6 30  1 − 240 919466 1455925 639130140029 n + 9480 + 65219 + 5021863 − 92804028240 11 847 n n2 n3

(9)

has a rate of convergence equal to n−9 , where lim n10 (Bn − Bn+1 ) =

n→∞

142970656174139 . 108854063087616000

In our third step, we can follow the same technique to get the following result: Lemma 2.3. The sequence Cn defined by r √  n n 6 1 8n3 + 4n2 + n + − V (n) eCn , n! = π e 30 where V (n) =

1 240 n 11

+

9480 847

+

919466 65219 n

+

1455925 5021863 n2

990

−

639130140029 92804028240 n3

+

T4 n4

+

T5 n5

+

T6 n6

,

Hegazi 987-994

J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 29, NO.5, 2021, COPYRIGHT 2021 EUDOXUS PRESS, LLC

converges to zero as n−12 with the best possible constants T4 = and T6 = − 5422052608484409095873 since 396565429333244371200 lim n13 (Cn − Cn+1 ) = −

n→∞

142970656174139 , 42875461046880

T5 =

288878734012247231 22009403337398400

384377015548794481311979 . 19141959578859903385600000

Hence, we get the asymptotic formula r √  n n 6 1 − U (n), 8n3 + 4n2 + n + n! ∼ π e 30

n→∞

(10)

where  U (n) =

240 9480 919466 1455925 639130140029 142970656174139 n+ + + − + 2 3 11 847 65219 n 5021863 n 92804028240 n 42875461046880 n4 −1 288878734012247231 5422052608484409095873 + − + ... . 22009403337398400 n5 396565429333244371200 n6

3

An inequality of Gamma function.

In this section, we will follow a method presented by Elbert and Laforgia in their paper [10] (see also, [3], [27], [32] and its simple modification in [18]): Corollary 3.1. Let T (t) be a real-valued function defined on t > t0 ∈ R with limt→∞ T (t) = 0. Then T (t) > 0, if T (t) > T (t + 1) for all t > t0 and T (t) < 0, if T (t) < T (t + 1) for all t > t0 . Now, Consider the following function   √ 1 1 1 3 2 +x−x ln(x)+ln Γ(x+1)−ln( π), F (x) = − ln 8x + 4x + x + 240x 9480 + 6 30 + 847 11

x>0

which satisfies lim F (x) = 0.

x→∞

  −1 847 1 3 2 F (x) − F (x + 1) = ln 8x + 4x + x + + − x ln(x) + x ln(x + 1) 6 18480x + 9480 30   1 847 31 3 2 ln 8(x + 1) + 4(x + 1) + x + + −1 + 6 18480x + 27960 30 + H(x) The function H(x) satisfies H 00 (x) =

H1 (x) < 0, H2 (x)

991

x>0

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J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 29, NO.5, 2021, COPYRIGHT 2021 EUDOXUS PRESS, LLC

where H1 (x) = − − − − −

1.84724 × 1029 x16 − 2.37023 × 1030 x15 − 1.39723 × 1031 x14 − 5.01631 × 1031 x13 1.22596 × 1032 x12 − 2.15964 × 1032 x11 − 2.83269 × 1032 x10 − 2.81806 × 1032 x9 2.14586 × 1032 x8 − 1.25283 × 1032 x7 − 5.57791 × 1031 x6 − 1.86841 × 1031 x5 4.59618 × 1030 x4 − 7.9786 × 1029 x3 − 9.149 × 1028 x2 − 6.15185 × 1027 x 1.83421 × 1026 < 0

and H2 (x) = 3x(x+1)2 (154x+79)2 (154x+233)2 147840x4 + 149760x3 + 56400x2 + 10096x + 1163 147840x4 + 741120x3 + 1392720x2 + 1163536x + 365259


2


2

.

Then H(x) is strictly concave function satisfies     28855461 1 log −6 0 and hence we get the following inequality Lemma 3.2.  √ 1 x Γ(x + 1) < π (x/e) 8x3 + 4x2 + x + + 30

240x 11

1 +

1/6 9480 847

,

x > 0.

(11)

Remark 1. In 2018, Yang and Tian [31] presented the inequality

Γ(x + 1)
0, the Moreau envelope of g of parameter γ is the convex function ¾ ½ 1 2 γ ky − xk ∀x ∈ H. g(x) = inf g(y) + y∈H 2γ 1 ky−xk2 is proper, strongly convex and lowe semicontinuous, For all x ∈ H, the function y 7→ g(y)+ 2γ γ thus the infimum is attained, i.e. g : H → R. 1 ky − xk2 is called proximal point of g at x and it is The unique minimum of y 7→ g(y) + 2γ denoted by proxg (x). The operator proxg (x) : H → H ¾ ½ 1 2 ky − xk x 7→ arg min g(y) + 2γ y∈H

is well-defined and is said to be the proximity operator of g. When g = iC (the indicator function of the convex set C), one has proxiC (x) = PC (x) for all x ∈ H. We also recall that the subdifferential of g : H → (−∞, ∞] at x ∈ domg is defined as the set of all subgradient of g at x ∂g(x) := {w ∈ H : g(y) − g(x) ≥ hw, y − xi ∀y ∈ H}. The function g is called subdifferentiable at x if ∂g(x) 6= ∅, g is said to be subdifferentiable on a subset C ⊂ H if it is subdifferentiable at each point x ∈ C, and it is said to be subdifferentiable, if it is subdifferentiable at each point x ∈ H, i.e., if dom(∂g) = H. The normal cone of C at x ∈ C is defined by NC (x) := {q ∈ H : hq, y − xi ≤ 0, ∀y ∈ C}. Definition 2.2 ([34, 35]). A bifunction ψ : H × H → R is called: (i) β-strongly monotone on C if there exists β > 0 such that ψ(x, y) + ψ(y, x) ≤ −βkx − yk2

∀x, y ∈ C;

(ii) monotone on C if ψ(x, y) + ψ(y, x) ≤ 0

∀x, y ∈ C;

(iii) pseudomonotone on C if ψ(x, y) ≥ 0 ⇒ ψ(y, x) ≤ 0 ∀x, y ∈ C. (iv) β-strongly pseudomonotone on C if there exists β > 0 such that ψ(x, y) ≥ 0 ⇒ ψ(y, x) ≤ −βkx − yk2

∀x, y ∈ C.

It is easy to see from the aforementioned definitions that the following implications hold, (i) ⇒ (ii) ⇒ (iii) and (i) ⇒ (iv) ⇒ (iii) The converses in general are not true. In this paper, we consider the bifunctions f and g under the following conditions. Condition A

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(A1) f (x, ·) is convex, weakly lower semicontinuous and subdifferentiable on H for every fixed x ∈ H. (A2) f (·, y) is weakly upper semicontinuous on H for every fixed y ∈ H. (A3) f is β-strongly monotone on H × H. (A4) For each x, y ∈ H, there exists L > 0 such that kw − vk ≤ Lkx − yk,

∀w ∈ ∂f (x, ·)(x), v ∈ ∂f (y, ·)(y).

(A5) The function x 7→ ∂f (x, ·)(x) is bounded on the bounded subsets of H. Condition B (B1) g(x, ·) is convex, weakly lower semicontinuous and subdifferentiable on H for every fixed x ∈ H. (B2) g(·, y) is weakly upper semicontinuous on H for every fixed y ∈ H. (B3) g is pseudomonotone on C with respect to Ω, i.e., g(x, x∗ ) ≤ 0,

∀x ∈ C, x∗ ∈ Ω.

(B4) g is Lipschitz-type continuous, i.e., there exist two positive constants L1 , L2 such that g(x, y) + g(y, z) ≥ g(x, z) − L1 kx − yk2 − L2 ky − zk2 , ∀x, y, z ∈ H. (B5) g is jointly weakly continuous on H × H in the sense that, if x, y ∈ H and {xn }, {yn } ∈ H converge weakly to x and y, respectively, then g(xn , yn ) → g(x, y) as n → +∞. Example 2.3 ([40]). Let f, g : R × R → R be defined by f (x, y) = 5y 2 − 7x2 + 2xy and g(x, y) = 2y 2 − 7x2 + 5xy. It follows that f and g satisfy Condition A and Condition B, respectively. Lemma 2.4 ([3], Propositions 3.1, 3.2). If the bifunction g satisfies Assumptions (B1), (B2), and (B3), then the solution set Ω is closed and convex. Remark 2.5. Let the bifunction f satisfy Condition A and the bifunction g satisfy Condition B. If Ω 6= ∅, then the bilevel equilibrium problem (1.2) has a unique solution, see the details in [33]. Lemma 2.6 ([9]). Let φ : C → R be a convex, lower semicontinuous, and subdifferentiable function on C. Then x∗ is a solution to the convex optimization problem min{f (x) : x ∈ C} if and only if 0 ∈ ∂φ(x∗ ) + NC (x∗ ). The following lemmas will be used in the proof of the convergence result. Lemma 2.7 ([38]). Let {an } be a sequence of nonnegative real numbers, {αn } be a sequence in (0, 1), and {ξn } be a sequence in R satisfying the condition an+1 ≤ (1 − αn )an + αn ξn , where

P∞ n=0

∀n ≥ 0,

αn = ∞ and lim supn→∞ ξn ≤ 0. Then limn→∞ an = 0.

Lemma 2.8 ([25]). Let {an } be a sequence of real numbers that does not decrease at infinity, in the sense that there exists a subsequence {anj } of {an } such that anj < anj +1

998

for all

j ≥ 0.

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Also consider the sequence of integers {τ (n)}n≥n0 defined, for all n ≥ n0 , by τ (n) = max{k ≤ n | ak < ak+1 }. Then {τ (n)}n≥n0 is a nondecreasing sequence verifying lim τ (n) = ∞,

n→∞

and, for all n ≥ n0 , the following two estimates hold: aτ (n) ≤ aτ (n)+1

and

an ≤ aτ (n)+1 .

Lemma 2.9 ([40]). Suppose that f is β-strongly monotone on H and satisfies (A4). Let 0 < α < 1, 2β 0 ≤ η ≤ 1 − α, and 0 < µ < L 2 . For each x, y ∈ H, w ∈ ∂f (x, ·)(x), and v ∈ ∂f (y, ·)(y), we have k(1 − η)x − αµw − [(1 − η)y − αµv]k ≤ (1 − η − ασ)kx − yk, p where σ = 1 − 1 − µ(2β − µL2 ) ∈ (0, 1]. 3. Main Result In this section, we propose the algorithm for finding the solution of a bilevel equilibrium problem under the strong monotonicity of f and the pseudomonotonicity and Lipschitztype continuous conditions on g. Algorithm 3.1. Initialization: Choose x0 , x1 ∈ H, 0 < µ < {αn } ⊂ (0, 1), {²n } ⊂ [0, +∞) and {ηn } are such that  X∞  lim α = 0, αn = ∞,  n→∞ n  n=0  0 ≤ ηn ≤ 1 − αn ∀n ≥ 0, limn→∞ ηn = η < 1,  X∞    ²n < ∞.

2β L2 ,

θ ∈ [0, 1), the sequences

n=0

Select initial x0 , x1 ∈ C and set n ≥ 1. Step 1.: Given xn−1 and xn (n ≥ 1), choose θn such that 0 ≤ θn ≤ θ¯n , where  ¾ ½ ²n   min θ, if xn 6= xn−1 , kxn − xn−1 k θn =  θ if otherwise. Choose {λn } such that ¯ < min 0 < λ ≤ λn ≤ λ −

µ

1 + θn 1 + θn , 2L1 2L2

(3.1)

¶ .

Compute sn = xn + θn (xn − xn−1 ), ¾ ½ 1 2 yn = arg min λn g(xn , y) + ky − sn k , 2 y∈C ¾ ½ 1 2 zn = arg min λn g(yn , y) + ky − xn k . 2 y∈C Step 2. Compute wn ∈ ∂f (zn , ·)(zn ) and xn+1 = ηn xn + (1 − ηn )zn − αn µwn . Set n := n + 1 and return to Step 1.

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Remark 3.2. Some remarks on the algorithm are in order now. (1) Evidently, we have from (3.1) that X∞ θn kxn − xn−1 k < ∞,

(3.2)

n=0

due to θn kxn − xn−1 k ≤ θ¯n kxn − xn−1 k ≤ ²n . (2) When θn = 0, Algorithm 3.1 reduces to Algorithm 1 of [40]. Theorem 3.3. Let bifunctions f and g satisfy Condition A and Condition B, respectively. Assume that Ω 6= ∅. Then the sequence {xn } generated by Algorithm 3.1 converges strongly to the unique solution of the bilevel equilibrium problem (1.2). Proof. Under assumptions of two bifunctions f and g, we get the unique solution of the bilevel equilibrium problem (1.2), denoted by x∗ . Step 1: Show that kzn −x∗ k2 ≤ kxn −x∗ k2 −(1+θn −2λn L1 )kxn −yn k2 −(1+θn −2λn L2 )kyn −zn k2 −θn kxn −xn−1 k2 . (3.3) The definition of yn and Lemma 2.6 imply that ¾ ½ 1 2 0 ∈ ∂ λn g(xn , y) + ky − sn k (yn ) + NC (yn ). 2 ¯ ∈ NC (yn ) such that There are w ∈ ∂g(xn , ·)(yn ) and w λn w + yn − sn + w ¯ = 0.

(3.4)

Since w ¯ ∈ NC (yn ), we have hw, ¯ y − yn i ≤ 0 for all y ∈ C.

(3.5)

By using (3.4) and (3.5), we obtain λn hw, y − yn i ≥ hsn − yn , y − yn i for all y ∈ C. Since zn ∈ C, we have λn hw, zn − yn i ≥ hsn − yn , zn − yn i.

(3.6)

It follows from w ∈ ∂g(xn , ·)(yn ) that g(xn , y) − g(xn , yn ) ≥ hw, y − yn i for all y ∈ H.

(3.7)

By using (3.6) and (3.7), we get λn {g(xn , zn ) − g(xn , yn )} ≥ hsn − yn , zn − yn i.

(3.8)

Similarly, the definition of zn implies that ¾ ½ 1 0 ∈ ∂ λn g(yn , y) + ky − xn k2 (zn ) + NC (zn ). 2 There are u ∈ ∂g(yn , ·)(zn ) and u ¯ ∈ NC (x) such that λn u + zn − xn + u ¯ = 0.

(3.9)

Since u ¯ ∈ NC (zn ), we have h¯ u, y − zn i ≤ 0 for all y ∈ C.

(3.10)

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By using (3.9) and (3.10), we obtain λn hu, y − zn i ≥ hxn − zn , y − zn i for all y ∈ C. Since x∗ ∈ C, we have λn hu, x∗ − zn i ≥ hxn − zn , x∗ − zn i

(3.11)

It follows from u ∈ ∂g(yn , ·)(zn ) that g(yn , y) − g(yn , zn ) ≥ hu, y − zn i for all y ∈ H.

(3.12)

By using (3.11) and (3.12), we get λn {g(yn , x∗ ) − g(yn , zn )} ≥ hxn − zn , x∗ − zn i. Since x∗ ∈ Ω, we have g(x∗ , yn ) ≥ 0. If follows from the pseudomonotonicity of g on C with respect to Ω that g(yn , x∗ ) ≤ 0. This implies that hxn − zn , zn − x∗ i ≥ λn g(yn , zn ).

(3.13)

Since g is Lipschitz-type continuous, there exist two positive constants L1 , L2 such that g(yn , zn ) ≥ g(xn , zn ) − g(xn , yn ) − L1 kxn − yn k2 − L2 kyn − zn k2 .

(3.14)

By using (3.13) and (3.14), we get hxn − zn , zn − x∗ i ≥ λn {g(xn , zn ) − g(xn , yn )} − λn L1 kxn − yn k2 − λn L2 kyn − zn k2 . From (3.8) and the above inequality, we obtain 2hxn − zn , zn − x∗ i ≥ 2hsn − yn , zn − yn i − 2λn L1 kxn − yn k2 − 2λn L2 kyn − zn k2 .

(3.15)

By the definition of sn , we have that 2hsn − yn , zn − yn i = 2hxn + θn (xn − xn−1 ) − yn , zn − yn i = −2hxn − yn , yn − zn i + 2θn hxn − xn−1 , zn − yn i. We know that 2hxn − zn , zn − x∗ i = kxn − x∗ k2 − kzn − xn k2 − kzn − x∗ k2 −2hxn − yn , yn − zn i = −kxn − zn k2 + kxn − yn k2 + kyn − zn k2 2θn hxn − xn−1 , zn − yn i = θn (kxn − yn k2 − kxn − xn+1 k2 − kyn − zn k2 ). From (3.15), we can conclude that kzn −x∗ k2 ≤ kxn −x∗ k2 −(1+θn −2λn L1 )kxn −yn k2 −(1+θn −2λn L2 )kyn −zn k2 −θn kxn −xn−1 k2 . (3.16) Step 2: The sequences {xn }, {wn }, {zn } are bounded. n {yn } and o 1+θn 1+θn Since 0 < λn < a, where a = min 2L1 , 2L2 , we have (1 + θn − 2λn L1 ) > 0 and (1 + θn − 2λn L2 ) > 0. It follows from (3.3) and the above inequalities that kzn − x∗ k ≤ kxn − x∗ k

for all n ∈ N.

(3.17)

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By Lemma 2.9 and (3.17), we obtain kxn+1 − x∗ k = kηn xn + (1 − ηn )zn − αµwn − x∗ + ηn x∗ − ηn x∗ + αn µv − αn µvk = k(1 − ηn )zn − αn µwn − (1 − ηn )x∗ + αn µv + ηn (xn − x∗ ) − αn µvk ≤ k(1 − ηn )zn − αn µwn − [(1 − ηn )x∗ + αn µv]k + ηn kxn − x∗ k + αn µkvk ≤ (1 − ηn − αn σ)kzn − x∗ k + ηn kxn − x∗ k + αn µkvk ≤ (1 − ηn − αn σ)kxn − x∗ k + ηn kxn − x∗ k + αn µkvk ¶ µ µkvk ∗ , = (1 − αn σ)kxn − x k + αn τ σ

(3.18)

where wn ∈ ∂f (zn , ·)(zn ) and v ∈ ∂f (x∗ , ·)(x∗ ). This implies that ¾ ½ µkvk ∗ ∗ . kxn+1 − x k ≤ max kxn − x k, σ By induction, we obtain ¾ ½ µkvk . kxn − x∗ k ≤ max kx0 − x∗ k, σ Thus the sequence {xn } is bounded. By using (3.17), we have {zn }, and using Condition (A5), we can conclude that {wn } is also bounded. Step 3: Show that the sequence {xn } converges strongly to x∗ . Since x∗ ∈ Ω∗ , we have f (x∗ , y) ≥ 0 for all y ∈ Ω. Note that f (x∗ , x∗ ) = 0. Thus x∗ is a minimum of the convex function f (x∗ , ·) over Ω. By Lemma 2.6, we obtain 0 ∈ ∂f (x∗ , ·)(x∗ ) + NΩ (x∗ ). Then there exists v ∈ ∂f (x∗ , ·)(x∗ ) such that hv, z − x∗ i ≥ 0 for all z ∈ Ω.

(3.19)

Note that kx − yk2 ≤ kxk2 − 2hy, x − yi for all x, y ∈ H.

(3.20)

From Lemma 2.9 and (3.20), we obtain kxn+1 − x∗ k2 = kηn xn + (1 − ηn )zn − αµwn − x∗ k2 = k(1 − ηn )zn − αn µwn − [(1 − ηn )x∗ + αn µv] + ηn (xn − x∗ ) − αn µvk2 ≤ k(1 − ηn )zn − αn µwn − [(1 − ηn )x∗ + αn µv] + ηn (xn − x∗ )k2 − 2αn µhv, xn+1 − x∗ i © ª ≤ k(1 − ηn )zn − αn µwn − [(1 − ηn )x∗ + αn µv] + ηn (xn − x∗ )k2 − 2αn µhv, xn+1 − x∗ i ≤ [(1 − ηn − αn σ)kzn − x∗ k + ηn kxn − x∗ k]2 − 2αn µhv, xn+1 − x∗ i ≤ (1 − ηn − αn σ)kzn − x∗ k2 + ηn kxn − x∗ k2 − 2αn µhv, xn+1 − x∗ i ≤ (1 − ηn − αn σ)kxn − x∗ k2 + ηn kxn − x∗ k2 − 2αn µhv, xn+1 − x∗ i = (1 − αn σ)kxn − x∗ k2 − 2αn µhv, xn+1 − x∗ i

(3.21)

It follows that kxn+1 − x∗ k2 ≤ (1 − αn σ)kxn − x∗ k2 + 2αn µhv, x∗ − xn+1 i.

(3.22)

Let us consider two cases.

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Case 1: There exists n0 such that kxn − x∗ k is decreasing for n ≥ n0 . Therefore the limit of sequence kxn − x∗ k exists. By using (3.17) and (3.22), we obtain 0 ≤ kxn − x∗ k2 − kzn − x∗ k2 2αn µ αn σ kzn − x∗ k2 − hv, xn+1 − x∗ i ≤− 1 − ηn 1 − ηn 1 (kxn − x∗ k2 − kxn+1 − x∗ k2 ). + 1 − ηn Since limn→∞ = ηn < 1, limn→∞ αn = 0 and the limit of kxn − x∗ k exists, we have lim (kxn − x∗ k2 − kzn − x∗ k2 ) = 0.

(3.23)

n→∞

From 0 < λn < a and inequality (3.3), we get (1 + θn − a)kxn − yn k2 ≤ (1 + θn − 2λn L1 )kxn − yn k2 ≤ kxn − x∗ k2 − kzn − x∗ k2 . By using (3.23), we obtain limn→∞ kxn − yn k = 0. Next, we show that lim suphv, x∗ − xn+1 i ≤ 0.

(3.24)

n→∞

Take a subsequence {xnk } of {xn } such that lim suphv, x∗ − xn+1 i = lim suphv, x∗ − xnk i. n→∞

k→∞

Since {xnk } is bounded, we may assume that {xnk } converges weakly to some x ¯ ∈ H. Therefore lim suphv, x∗ − xn+1 i = lim suphv, x∗ − xnk i = hv, x∗ − x ¯i. n→∞

(3.25)

k→∞

Since limn→∞ kxn − yn k = 0 and xnk * x ¯, we have ynk * x ¯. Let us consider that lim ksn − yn k ≤ lim ksn − xn k + lim kxn − yn k.

n→∞

n→∞

n→∞

By the definition of sn , we have that lim ksn − xn k = lim kxn − θn (xn − xn−1 ) − xn k

n→∞

n→∞

= lim θn kxn − xn−1 k. n→∞

P∞

Using the assumption n=1 θn kxn − xn−1 k < ∞, it implies that limn→∞ θn kxn − xn−1 k = 0. Thus limn→∞ ksn − xn k = 0. Since limn→∞ ksn − xn k = 0 and xnk * x ¯, we have snk * x ¯. Since C is closed and convex, it is also weakly closed and thus x ¯ ∈ C. Next, we show that x ¯ ∈ Ω. From the definition of {yn } and Lemma 2.6, we obtain ¾ ½ 1 0 ∈ ∂ λn g(xn , y) + ksn − yk2 (yn ) + NC (yn ). 2 There exist w ¯ ∈ NC (yn ) and w ∈ ∂g(xn , ·)(yn ) such that λn w + yn − sn + w ¯ = 0.

(3.26)

Since w ¯ ∈ NC (yn ), we have hw, ¯ y − yn i ≤ 0 for all y ∈ C. From (3.26), we obtain λn hw, y − yn i ≥ hsn − yn , y − yn i for all y ∈ C.

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(3.27)

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Since w ∈ ∂g(xn , ·)(yn ), we have g(xn , y) − g(xn , yn ) ≥ hw, y − yn i for all y ∈ H.

(3.28)

Combining (3.27) and (3.28), we get λn {g(xn , y) − g(xn , yn )} ≥ hsn − yn , y − yn i for all y ∈ C.

(3.29)

Taking n = nk and k → ∞ in (3.29), the assumption of λn and (B5), we obtain g(¯ x, y) = 0 for all y ∈ C. This implies that x ¯ ∈ Ω. By inequality (3.19), we obtain hv, x ¯ − x∗ i ≥ 0. It follows from (3.25) that lim suphv, x∗ − xn+1 i ≤ 0.

(3.30)

n→∞

We can write inequality (3.22) in the following form: kxn+1 − x∗ k2 ≤ (1 − αn σ)kxn − x∗ k2 + αn σξn , ∗ where ξn = 2µ σ hv, x − xn+1 i. It follows from (3.30) that lim supn→∞ ξn ≤ 0. By Lemma 2.7, we can conclude that limn→∞ kxn − x∗ k2 = 0. Hence xn → x∗ as n → ∞. Case 2: There exists a subsequence {xnj } of {xn } such that kxnj − x∗ k ≤ kxnj+1 − x∗ k for all j ∈ N. By Lemma 2.8, there exists a nondecreasing sequence {τ (n)} of N such that limn→∞ τ (n) = ∞, and for each sufficiently large n ∈ N, we have

kxτ (n) − x∗ k ≤ kxτ (n)+1 − x∗ k

and

kxn − x∗ k ≤ kxτ (n)+1 − x∗ k.

(3.31)

Combining (3.18) and (3.31), we have kxτ (n) − x∗ k ≤ kxτ (n)+1 − x∗ k ≤ (1 − ητ (n) − ατ (n) σ)kzτ (n) − x∗ k + ητ (n) kxτ (n) − x∗ k + ατ (n) µkvk.

(3.32)

From (3.17) and (3.32), we get 0 ≤ kxτ (n) − x∗ k − kzτ (n) − x∗ k ≤ −

ατ (n) σ ατ (n) σ kzτ (n)−x∗ k + kvk. 1 − ητ (n) 1 − ητ (n)

(3.33)

Since limn→∞ αn = 0, limn→∞ ηn = η < 1, {zn } is bounded, and (3.33), we have limn→∞ (kxτ (n) − x∗ k − kzτ (n) − x∗ k) = 0. It follows from the boundedness of {xn } and {zn } that lim (kxτ (n) − x∗ k2 − kzτ (n) − x∗ k2 ) = 0.

n→∞

(3.34)

By using the assumption of {λn }, we get the following two inequalities: 1 + θn − 2λτ (n) L1 > 1 + θn − 2aL1 > 0 and 1 + θn − 2λτ (n) L2 > 1 + θn − 2aL2 > 0. From (3.3), we obtain kzτ (n) − x∗ k2 ≤ kxτ (n) − x∗ k2 − (1 + θn − 2λτ (n) L1 )kxτ (n) − yτ (n) k2 − (1 + θn − 2λτ (n) L2 )kyτ (n) − zτ (n) k2 ≤ kxτ (n) − x∗ k2 − (1 + θn − 2aL1 )kxτ (n) − yτ (n) k2 − (1 + θn − 2aL2 )kyτ (n) − zτ (n) k2 . This implies that 0 < (1 + θn − 2aL1 )kxτ (n) − yτ (n) k2 + (1 + θn − 2aL2 )kyτ (n) − zτ (n) k2 ≤ kxτ (n) − x∗ k2 − kzτ (n) − x∗ k2 .

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It follows from (3.34) and the above inequality that lim kxτ (n) − yτ (n) k = 0 and

n→∞

lim kyτ (n) − zτ (n) k = 0.

(3.35)

n→∞

Note that kxτ (n) − zτ (n) k ≤ kxτ (n) − yτ (n) k + kyτ (n) − zτ (n) k. From (3.35), we have lim kxτ (n) − zτ (n) k = 0.

(3.36)

n→∞

By using the definition of xn+1 and Lemma 2.9, we obtain kxτ (n)+1 − xτ (n) k2 = kητ (n) xτ (n) + (1 − ητ (n) )zτ (n) − ατ (n) µtτ (n) − xτ (n) k = k(1 − ητ (n) )zτ (n) − ατ (n) µtτ (n) − [(1 − ητ (n) )xτ (n) − ατ (n) wτ (n) ] − ατ (n) wτ (n) k ≤ k(1 − ητ (n) )zτ (n) − ατ (n) tτ (n) − [(1 − ητ (n) )xτ (n) − ατ (n) wτ (n) ]k + ατ (n) kwτ (n) k ≤ (1 − ητ (n) − ατ (n) σ)kzτ (n) − xτ (n) k + ατ (n) kwτ (n) k ≤ kzτ (n) − xτ (n) k + ατ (n) kwτ (n) k, where tτ (n) ∈ ∂f (zτ (n) , ·)(zτ (n) ) and wτ (n) ∈ ∂f (xτ (n) , ·)(xτ (n) ). Since limn→∞ αn = 0, the boundedness of {wτ (n) } and (3.36), we have limn→∞ kxτ (n)+1 − xτ (n) k = 0. As proved in the first case, we can conclude that lim suphv, x∗ − xτ (n)+1 i = lim suphv, x∗ − xτ (n) i ≤ 0. n→∞

(3.37)

n→∞

Combining (3.22) and (3.31), we obtain kxτ (n)+1 − x∗ k2 ≤ (1 − αn(τ ) σ)kxτ (n) − x∗ k2 + 2αn(τ ) µhv, x∗ − xτ (n)+1 i ≤ (1 − αn(τ ) σ)kxτ (n)+1 − x∗ k2 + 2αn(τ ) µhv, x∗ − xτ (n)+1 i. By using (3.31) again, we have kxn − x∗ k2 ≤ kxτ (n)+1 − x∗ k2 ≤

2µ hv, x∗ − xτ (n)+1 i. σ

From (3.37), we can conclude that lim supn→∞ kxn − x∗ k2 ≤ 0. Hence xn → x∗ as n → ∞. This completes the proof. ¥ 4. Numerical example In this section, we provide a numerical example to test our algorithm. All Matlab colds were performed on a computer with CPU Intel Core i7-7500U, up to 3.5GHz, 4GB of RAM under version MATLAB R2015b. In the following example, we use the standard Euclidean norm and inner product. Example 4.1. We compare our algorithm with Algorithm 1 proposed in Yuying et al. [40]. Let us consider a problem when H = Rn and C = {x ∈ Rn : −5 ≤ xi ≤ 5, ∀i ∈ {1, 2, ..., n}}. Let the bifunction g : Rn × Rn → R be defined by g(x, y) = hP x + Qy, y − xi

for all

x, y ∈ Rn ,

where P and Q are randomly symmetric positive semidefinite matrices such that P − Q is positive semidefinite. Then g is pseudomonotone on Rn . Next, we obtain that g is Lipschitz-type continuous with L1 = L2 = 21 kP − Qk. Furthermore, we define the bifunction f : Rn × Rn → R as f (x, y) = hAx + By, y − xi

1005

for all

x, y ∈ Rn ,

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Table 1: Comparison: proposed Algorithm 3.1 and Yuying et al. [40] with x0 = x1 ∈ {x ∈ Rn : xi = 1, ∀i = 1, 2, ..., n}.

n 5 10 50

Algorithm 3.1 θ = 0.6 θ = 0.9 No. of Iter. CPU (Time) No. of Iter. CPU (Time) 28 1.0178 29 1.0015 43 1.7310 38 1.3645 90 4.3222 88 4.6822

Yuying et al. Algorithm No. of Iter. 34 54 98

CPU (Time) 1.3618 1.9099 5.5028

Table 2: Comparison: proposed Algorithm 3.1 and Yuying et al. [40] with x0 = x1 ∈ {x ∈ Rn : xi = i, ∀i = 1, 2, ..., n}.

n 5 10 50

Algorithm 3.1 θ = 0.6 θ = 0.9 No. of Iter. CPU (Time) No. of Iter. CPU (Time) 30 1.0388 32 1.1074 50 1.8239 45 1.8472 108 6.6858 105 6.5254

Yuying et al. Algorithm No. of Iter. 37 61 116

CPU (Time) 1.3528 2.3260 6.7247

with A and B being positive definite matices defined by B = N T N + nIn

and

A = B + M T M + nIn ,

(4.1)

where M, N are randomly n × n matrices and In is the identity matrix. Moreover, ∂f (x, ·)(x) = {(A + B)x} and k(A + B)x − (A + B)yk ≤ kA + Bkkx − yk for all x, y ∈ Rn . Thus the mapping x → ∂f (x, ·)(x) is bounded and kA + Bk-Lipschitz continuous on every bounded subset of H. It is easy to see that all the conditions of Theorem 3.3 and of Theorem 3.1 in [40] are satisfied. New, we compare the performance of our algorithm and algorithm of Yuying et al. [40], we take 1 1 k+1 2 , the same starting point x0 = x1 ∈ {x ∈ λk = , αk = , ηk = ,µ= k+5 k+4 3(k + 4) kA + Bk2 Rn : xi = 1, ∀ i = 1, 2, ..., n} and x0 = x1 ∈ {x ∈ Rn : xi = i, ∀ i = 1, 2, ..., n} for all the algorithms. 1 For Algorithm 3.1, we choose ²k = 1.1 , θ ∈ [0, 1) and θk such that 0 ≤ θk ≤ θ¯k , where k  ¾ ½ 1   min θ, if xk 6= xk−1 , k 1.1 kxk − xk−1 k θk =  θ if otherwise. To terminate the algorithm, we used the stopping criteria kxk+1 − xk k < ε with ε = 10−6 is a tolerance. The results are reported in the Table 1 and Table 2, we can see that the number of iterations (No. of Iter.) by Algorithm 3.1 with different inertial parameters (θ = 0.6 and θ = 0.9) is less than that of Yuying et al. Algorithm [40], for two different starting points, we can see that in this example the starting points x0 = x1 ∈ {x ∈ Rn : xi = 1, ∀ i = 1, 2, ..., n} give better performance than x0 = x1 ∈ {x ∈ Rn : xi = i, ∀ i = 1, 2, ..., n}. Moreover, Figure 1 and Figure 2 illustrate the numerical behavior of both algorithms. In these figures, the value of errors kxk+1 −xk k is represented by the y-axis, number of iterations is represented by the x-axis.

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10 1 Yuying et al. Alg. Alg. 3.1, θ = 0.6 Alg. 3.1, θ = 0.9

10 0

Error

10 -1 10 -2 10 -3 10 -4 10 -5 10 -6

0

10

20

30

40

50

60

70

80

Number of Iterations Figure 1: Comparison of proposed Algorithm 3.1 and Yuying et al. [40] with x0 = (1, 1, ...., 1)T and n=50.

10 2 Yuying et al. Alg. Alg. 3.1, θ = 0.6 Alg. 3.1, θ = 0.9

Error

10 0

10 -2

10 -4

10 -6

0

10

20

30

40

50

60

70

80

90

100

Number of Iterations Figure 2: Comparison of proposed Algorithm 3.1 and Yuying et al. [40] with x0 = (1, 2, ...., 50)T and n=50.

5. Conclusions In this article, we introduced an iterative algorithm for finding the solution of a bilevel equilibrium problem in real Hilbert space. Under some suitable conditions imposed on parameters, we proved the strong convergence of the algorithm. We showed the efficiency of the proposed algorithm is verified by a numerical experiment and preliminary comparison. These numerical results have also confirmed that the algorithm with inertial effects seems to work better than without inertial effects.

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Conflict of Interests The authors declare that there is no conflict of interests regarding the publication of this paper. Acknowledgements The authors would like to thank Naresuan University and The Thailand Research Fund for financial support. Moreover, J. Munkong is also supported by Naresuan University and The Royal Golden Jubilee Program under Grant Ph.D/0219/2556, Thailand. References [1] P.N. Anh, T.T.H. Anh, N.D. Hien, Modified basic projection methods for a class of equilibrium problems, Numer. Algorithms 79(1), 139-152 (2018). [2] G.C. Bento, J.X. Cruz Neto, J.O. Lopes, P.A. Jr Soares, A. Soubeyran, Generalized proximal distances for bilevel equilibrium problems, SIAM J. Optim. 26, 810-830 (2016). [3] M. Bianchi, S. Schaible, Generalized monotone bifunctions and equilibrium problems, J. Optim. Theory Appl. 90, 31-43 (1996). [4] E. Blum, and W. Oettli, From optimization and variational inequality to equilibrium problems, The Mathematics Student, 63 (1994), pp. 127-149. [5] R.I. Bot, E.R. Csetnek, C. Hendrich, Inertial DouglasRachford splitting for monotone inclusion problems. Appl. Math. Comput. 256, 472-487 (2015). [6] J. Cea, Optimisation: theorie et algorithmes (Dunod, Paris, 1971); Polish translation: Optymalizacja: Teoria i algorytmy (PWN, Warszawa, 1976). [7] O. Chadli, Z. Chbani, H. Riahi, Equilibrium problems with generalized monotone bifunctions and applications to variational inequalities, J. Optim. Theory Appl. 105, 299-323 (2000). [8] Z. Chbani, H. Riahi, Weak and strong convergence of proximal penalization and proximal splitting algorithms for two-level hierarchical Ky Fan minimax inequalities, Optimization 64, 12851303 (2015). [9] P. Daniele, F. Giannessi, A. Maugeri, Equilibrium Problems and Variational Models, Kluwer Academic, Norwell (2003). [10] S. Dempe, Annotated bibliography on bilevel programming and mathematical programs with equilibrium constraints, Optimization 52, 333-359 (2003). [11] F. Deutsch, Best Approximation in Inner Product Spaces (Springer, New York, 2001). [12] B.V. Dinh, D.S. Kim, Extragradient algorithms for equilibrium problems and symmetric generalized hybrid mappings, Optim. Lett. 17(3), 537-553 (2017). [13] Q.L. Dong, H.B. Yuan, C.Y. Je, Th.M. Rassias, Modified inertial Mann algorithm and inertial CQ-algorithm for nonexpansive mappings. Optim. Lett. 12, 87-102 (2018). [14] A. Gal´antai, Projectors and Projection Methods (Kluwer Academic, Boston, 2004). [15] K. Goebel, S. Reich, Uniform convexity, hyperbolic geometry, and nonexpansive mappings. Marcel Dekker, New York (1984).

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[36] L.Q. Thuy, T.N. Hai, A projected subgradient algorithm for bilevel equilibrium problems and applications, J. Optim. Theory Appl., 175(2), 411-431 (2017). https://doi.org/10.1007/s10957-017-1176-2. [37] D.Q. Tran, L.D. Muu, V.H. Nguyen, Extragradient algorithms extended to equilibrium problems, Optimization 57, 749-776 (2008). [38] H.K. Xu, Iterative algorithms for nonlinear operators, J. Lond. Math. Soc. 66, 240-256 (2002). [39] Y. Yao, A. Petrusel, X. Qin, An improved algorithm based on Korpelevich’s method for variational inequalities in Banach spaces, J. Nonlinear Convex Anal. 19, 397-406 (2018). [40] T. Yuying, B. V. Dinh, D. S. Kim, S. Plubtieng, Extragradient subgradient methods for solving bilevel equilibrium problems, J. Inequal. Appl., 2018:327, 1 - 21 (2018).

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TABLE OF CONTENTS, JOURNAL OF COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 29, NO. 5, 2021

Differential Equations and Inclusions Involving Mixed Fractional Derivatives with Four-Point Nonlocal Fractional Boundary Conditions, Bashir Ahmad, Sotiris K. Ntouyas, and Ahmed Alsaedi,………………………………………………………………………………………817 𝐿2 -Primitive Process for Retarded Stochastic Neutral Functional Differential Equations in Hilbert Spaces, Yong Han Kang, Jin-Mun Jeong, and Seong Ho Cho,…………………………….838 Fuzzy Stability of Cubic Functional Equations with Extra Terms, Chang Il Kim,Giljun Han,862 Drygas Functional Equations with Extra Terms and Its Stability, Young Ju Jeon and Chang Il Kim,…………………………………………………………………………………………870 Theoretical and Numerical Discussion for the Mixed Integro-Differential Equations, M. E. Nasr and M. A. Abdel-Aty,……………………………………………………………………….880 Representation of the Matrix for Conversion between Triangular Bézier Patches and Rectangular Bézier Patches, P. Sabancigil, M. Kara, and N. I. Mahmudov,…………………………….893 Permeable Values with Applications in BE-Algebras, Jung Mi Ko, Young Hie Kim, and Sun Shin Ahn,……………………………………………………………………………………902 Rapid Gradient Penalty Schemes and Convergence for Solving Constrained Convex Optimization Problem in Hilbert Spaces, Natthaphon Artsawang,Kasamsuk Ungchittrakool,910 Approximation by Modified Lupaș Operators Based On (p,q)-Integers, Asif Khan, Zaheer Abbas, Mohd Qasim, and M. Mursaleen,……………………………………………………922 Additive-Quadratic Functional Inequalities in Fuzzy Normed Spaces and Stability, Young Ju Jeon and Chang Il Kim,…………………………………………………………………….934 Generalized Additive-Cubic Functional Equation and its Stability, Chang Il Kim,………..944 Toeplitz Duals of Fibonacci Sequence Spaces, Kuldip Raj, Suruchi Pandoh, and Kavita Saini,954 Completely Monotonic Functions Involving Bateman's G-function, Mansour Mahmoud, Ahmed Talat, Hesham Moustafa, and Ravi P. Agarwal,………………………………………………970 On Ramanujan's Asymptotic Formula for n!, Ahmed Hegazi, Mansour Mahmoud, and Hend Salah,…………………………………………………………………………………………987

TABLE OF CONTENTS, JOURNAL OF COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 29, NO. 5, 2021 (continues)

An Inertial Extragradient Subgradient Method for Solving Bilevel Equilibrium Problems, Jiraprapa Munkong, Kasamsuk Ungchittrakool, and Ali Farajzadeh,…………………………995

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J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 29, NO.6, 2021, COPYRIGHT 2021 EUDOXUS PRESS, LLC

Daubechies Wavelet Method for Second Kind Fredholm Integral Equations with Weakly Singular Kernel ∗ Xin Luo

†

Jin Huang‡

Abstract In this paper, the weakly singular Fredholm integral equations of the second kind are solved by the periodized Daubechies wavelets method. In order to obtain a good degree of accuracy of the numerical solutions, the Sidi-Israeli quadrature formulae are used to construct the approximation of the singular kernel functions. By applying the asymptotically compact theory, we prove the convergence of approximate solutions. In addition, the sidi transformation can be used to degrade the singularities when the kernel function is non-periodic. At last, numerical examples show the method is efficient and errors of the numerical solutions possess high accuracy order O (h3+α ), where h is the mesh size. Keyword : Daubechies wavelets; weakly singular kernel; Fredholm integral equation of the second; linear and nonlinear integral equations; convergence rate.

1

Introduction

Many problems in science and engineering such as Lapalace’s equation, problems in elasticity, conformal mapping, free surface flows and so on, result in Fredholm integral equations with singular or weakly singular (in general logarithmic) and periodic kernels [11]. Therefore, singular or weakly Singular Fredholm linear equations and its nonlinear counterparts are most frequently studied for decades. ∗

This work is supported by Project (NO. KYTZ201505) Supported by the Scientific Research Foundation of CUIT † College of Applied Mathematics, Chengdu University of Information Technology, Chengdu 610225, P.R. China, corresponding author: [email protected] ‡ School of Mathematical Sciences, University of Electronic Science and Technology of China, Chengdu 611731, P.R.China

1

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Generally, the weakly singular Fredholm integral equation of the second can be converted into the following form Z 1 u(x) − k(x, y)g(u(y))dy = f (x), x ∈ [0, 1], (1.1) 0

where k(x, y) = H1 (x, y)|x − y|α (ln |x − y|)β + H2 (x, y), α > −1, β ≥ 0,

(1.2)

u(x) is an unknown function and f ∈ L2 [0, 1], and Hj (x, y) (j = 1, 2) are continuous on [0, 1]. The integral equation (1.1) is linear when g(u(y)) = u(y), and when g(u(y)) 6= u(y) the equation is nonlinear. As is known, several different orthonormal basis functions, for example, Chebyshev polynomial [8], Fourier functions [2], and wavelets [3, 4, 5, 6, 7, 9, 10, 13, 14, 16, 17], can be used to approximate the solutions of integral equations. However, for large scale problems, the most attractive one among them may be the wavelet bases, in which the kernel can be transformed to a sparse matrix after discretization. This is mainly due to functions with fast oscillations, or even discontinuities, in localized regions may be approximated well by a linear combination of relatively few wavelets [3]. This paper is organized as follows: in Section 2, the periodized Daubechies wavelets is introduced for solving weakly singular Fredholm integral equations of the second in detail. In Section 3, the convergence and error analysis are investigated. In Section 4, numerical examples are provided to verify the theoretical results. Some useful conclusions are made in Section 5.

2 2.1

Periodized Daubechies wavelets method Multiresolution analysis and function expansions

Wavelets are attractive for the numerical solution of integrations, because their vanishing moments property leads to operator compression. Especially, Daubechies wavelets [12, 15] have many good properties and can deal with some types of kernels arising from boundary integral formulation of elliptic PDEs, and the coefficient are often numerically sparse. In fact, there are only O (n log n) significant elements. Supposed that ψ and φ be the the wavelet of genus N and Daubechies scaling function respectively. Thus their support are supp(φ) = supp(ψ) = [0, N −1]. For any j, k ∈ Z, we introduce the notations φj,k (x) = 2j/2 φ(2j x−k) and ψj,k (x) = 2j/2 ψ(2j x−k), then their periodic kin with period-1 can be described by X X φ˜j,k (x) = φj,k (x + n), ψ˜j,k (x) = ψj,k (x + n), x ∈ R, 0 ≤ k < 2j . (2.1) n∈Z

n∈Z

Here {φ˜j,k (x)}k∈Z and {ψ˜j,k (x)}k∈Z are orthogonal [17]. Defining the periodic spaces j −1 ˜ j = span{ψ˜j,k }2j −1 . A chain of spaces V˜0 ⊂ V˜1 · · · ⊂ V˜j = span{φ˜j,k }2k=0 and W k=0 2

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L2 [0, 1] can be constructed, which subject to the following conditions: (a) ∪j≥0 V˜j = ˜ j = V˜j+1 , W ˜ j ⊥ V˜j . L2 [0, 1], ∩j∈Z V˜j = {0}; (b) h(x) ∈ V˜j ⇔ h(2x) ∈ V˜j+1 ; (c) V˜j ⊕ W The Daubechies wavelets and scaling functions described above result in the wavelet theory (i.e., multiresolution analysis (MRA)) of L2 [0, 1]. Supposed that function p(x) ∈ L2 [0, 1] be approximated by scaling series at resolution J as J −1 2X p(x) = cJ,k φ˜J,k (x) = Φt (x)c, x ∈ [0, 1], (2.2) k=0

where

Φ(x) = [φ˜J,0 (x), φ˜J,1 (x), · · · , φ˜J,2J−1 (x)],

and

Z t

c = (cJ,0 , cJ,1 , · · · , cJ,2J−1 ) ,

cJ,k =

1

p(x)φ˜J,k (x)dx.

(2.3) (2.4)

0

First, we calculate the wavelet coefficient cJ,k for nonsingular function p(x) ∈ L [0, 1]. Let xi = i/2J , i = 0, 1, · · · , 2J−1 . Substituting x = xi into Eq. (2.2), we have J −1 J −1 2X 2X X J J/2 cJ,k φ˜J,k (i/2 ) = 2 cJ,k φj,k (2J n + i − k). (2.5) p(x) = 2

k=0

k=0

n∈Z

By using the relationship between supp(φ) and [0, 1], we know when J ≥ J0 only finite terms of the inner summation in (2.5) contribute the following result ½ 0 or 1, if 2J − N + 2 ≤ N − 1, n= 0, if 0 ≤ k ≤ 2J s − N + 1. Now we write (2.5) as the matrix form p = T c,

(2.6)

where p = [p(0), p(1/2J ), · · · , p((2J − 1)/2J )]t , T is the nonsingular matrix which entries are the function values of φ(x) at integers (i.e., φ(0),φ(1),· · · ,φ(N − 2)) appear in it, and hence it satisfies J

2 X i=1

J

Tij =

2 X

Tij = 2J/2 , i, j = 1, 2, · · · , 2J .

(2.7)

j=1

Consequently, the function k(x, y) ∈ L2 ([0, 1]×[0, 1]) in Eq.(1.1) can be approximated at resolution J as k(x, y) = Φt (x)QΦ(y), (2.8) where Q is the 2J × 2J coefficient matrix. Eq. (2.8) can be written as the following form Q = T −1 KT −t , (2.9) where K is the 2J × 2J kernel matrix with Ki,j = k(i/2J , j/2J ). 3

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Secondly, if the function p(x) is singular on [0, 1], some values of p(x) at the dyadic points xi = i/2J , i = 0, 1, · · · , 2J − 1 may be unbounded and then c can not be immediately solved from the Eq.(2.6). In order to avoid the Eq.(2.6) being invalid, we can use the method in the literature [17] to compute the values of p(x). Without loss of generality, we assumed that function p(x) ∈ L2 [0, 1] has only one singular point xi = i/2J , i ∈ {0, 1, · · · , 2J − 1}. Then the function value p(xi ) in Eq.(2.6) can be computed via on the following ( see [17]) Z p(i/2J ) = 2J

p(x)dx − 0

R1

where integration [11].

2.2

J −1 2X

1

0

p(j/2J ),

i ∈ {0, 1, · · · , 2J − 1},

(2.10)

j=0,j6=i

p(x)dx can be calculated by Sidi-Israeli quadrature formulae

Kernel function approximation and discretization of singular integral equation

Motivated by the Eq.(2.10) and by thinking k(x, y) as a one-dimensional function of variable x and y respectively, we also have Z J

J

1

k(x, y)dy −

k(x, m/2 ) = 2

0

J −1 2X

k(x, j/2J ),

m ∈ {0, 1, · · · , 2J − 1}. (2.11)

j=0,j6=m

The following Theorem 2.1 can be used to construct the kernel approximation of Eq.(1.1). Theorem 2.1 [11] Assume that the functions H1 (x, y) and H2 (x, y) are 2` times differentiable on [a, b]. Assume also that the functions k(x, y) are periodic with period e = (−∞, ∞)\{x + jT }∞ T = b − a, and that they are 2` times differentiable on R j=−∞ . α β If k(x, y) = H1 (x, y)|x − y| (ln |x − y|) + H2 (x, y), s > −1, β = 0, 1, then the quadrature rules of the following integral Z b k(x, y)dy, (2.12) I[k(x, y)] = a

are n X

In [k(x, y)] = h

0

k(x, yj ) + 2[βζ (−α) − ζ(−α)(ln h)β ]H1 (x, x)hα+1 + H2 (x, x)h,

j=1,yj 6=x

(2.13) and the quadrature errors are

En [k(x, y)] = 2

`−1 X

(2µ)

0

[βζ (−α − 2µ) − ζ(−α − 2µ)(ln h)β ]

µ=1

H1

(x, yj ) 2µ+α+1 h + o (h2` ), (2µ)! (2.14)

4

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where En [k(x, y] = I[k(x, y)] − In [k(x, y)], and the mesh size is h = (b − a)/n. Let n = 2J and by (2.13), we can get the Nystr¨om approximation for the kernel function k(x, y) ½ 0 2[βζ (−α) − ζ(−α)(ln h)β ]H1 (xi , xi )hα + H2 (xi , xi ), if i = j, kD (xi , yj ) = k(xi , yj ), if i 6= j. (2.15) Supposed that the kernel function k(x, y), u(x) and f (x) be approximated at resolution J as k(x, y) = Φt (x)QΦ(y), f (x) = Φt (x)b and u(x) = Φt (x)c,

(2.16)

where c = [c(0), c(1/2J ), · · · , c((2J − 1)/2J )]t is the expansion coefficient vector of u(x). By the orthonormality of periodized wavelets, the integration of the product of the same two scaling function vectors is achieved as Z 1 Φ(x)Φt (x)dx = I, (2.17) 0

where I is the 2J by 2J identity matrix. For the linear integral equation, we have Z 1 t Φ (x)c − Φt (x)QΦ(y)Φt (y)cdy = Φt (x)b. (2.18) 0

Substituting (2.15), (2.16) and (2.17) into (2.18), and by invoking (2.9), we get a linear system (I − KD (T t T )−1 )uD = f, (2.19) where f = [f (0), f (1/2J ), · · · , f ((2J − 1)/2J )]t and KD = T t QT . Similarly, the nonlinear case for Eq. (1.1) can be transformed into the following by the wavelet method uD − KD (T T t )−1 g(uD ) = f, (2.20) Eq. (2.20) is a system of nonlinear equations about u and can be computed by Newton iteration method.

3

Convergence and error analysis

In this section, we mainly study the convergence and error for the linear case of (1.1) by wavelet method. We write Eq. (1.1) as the operate form ˜ = f, (I − K)u where

Z ˜ (Ku)(x) =

(3.1)

1

k(x, y)u(y)dy,

(3.2)

0

5

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with the kernel k(x, y) = H1 (x, y)|x − y|α (ln |x − y|)β + H2 (x, y), α > −1, β ≥ 0,

(3.3)

˜ is defined by and the approximation of K ˜ n u)(x) = h (K

n X

k(x, yj )u(yj ) + ωn (x)u(x),

(3.4)

j=1,yj 6=x

where the weight function 0

ω(x) = 2[βζ (−α) − ζ(−α)(ln h)β ]H1 (x, x)hα+1 + H2 (x, x)h.

(3.5)

Supposed that the approximation of (3.1) is ˜ n )un (x) = g. (I − K

(3.6)

˜ n is defined by (3.4), then the operator Lemma 3.1 Supposed the the operator K ˜ ˜ i.e., sequence {Kn } is asymptotically compactly convergent to K, a.c ˜ ˜n → K K,

(3.7)

a.c

where → denotes the asymptotically compact convergence. ˜ be defined by Proof. Let the continuous kernel approximation of K ½ k(x, y), if |x − y| ≥ h, c kn (x, y) = H1 (x, x)hα (ln h)β + H2 (x, x), if |x − y| < h,

(3.8)

and the corresponding operator approximation be (Knc u)(x) = h

n X

knc (x, yj )u(yj ).

(3.9)

j=1

For any v ∈ C[0, 1], we have Z 1 c ˜ − Kn )vk = sup k(K |(k(x, y) − knc (x, y))v(y)|dy kvk∞ ≤1

Z

≤ Z0

1

0

|k(x, y) − knc (x, y)|dykvk∞

|H1 (x, y)|||x − y|α (ln |x − y|)β − hα (ln h)β |dykvk∞ |x−y|≤h Z ≤ max |H1 (x, y)| ||x − y|α (ln |x − y|)β − hα (ln h)β |dykvk∞ ≤

x,y∈C[0,1]

|x−y|≤h

β α

= O((lnh) h )kvk∞ , (3.10) 6

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hence, we can obtain ˜ − Knc k = O((lnh)β hα ) → 0, as h → 0. kK

(3.11)

On the other hand, we know ω(x) → 0 as h → 0 by (3.5), then ˜ n k → 0, as h → 0. kKnc − K

(3.12)

First, there exists a subsequence in {Knc yn } for any yn ⊂ C[0, 1] by (3.11),. Without loss of generality, assumed that Knc yn → z and by (3.12), then ˜ n yn − zk ≤ kK ˜ n yn − Knc yn k + kKnc yn − zk kK ˜ n − K c kkyn k + kK c yn − zk → 0, ≤ kK n

(3.13)

n

˜ n } is asymptotically compactly convergent. Secondly, that is to say, the sequence {K for any y ∈ C[0, 1], we have ˜ n y − Kyk ˜ ≤ kK ˜ n − K c kkyk + kK c y − Kyk ˜ → 0. kK n n

(3.14)

The proof of Lemma 3.1 is completed. ¤ ˜ n (I − o(h)E)} is asymptotically comCorollary 3.2 The operator sequence {K ˜ pactly convergent to K, i.e., a.c ˜ ˜ n (I − o(h)E) → K K.

(3.15)

where E is a matrix and every element in it is one. Proof. By Lemma 3.1, we know a.c ˜ ˜n → K, K

(3.16)

˜ n − Kk→0. ˜ kK

(3.17)

˜ n (I − o(h)E) − Kk ˜ ≤ kK ˜ n − Kk ˜ + kK ˜ n kko(h)Ek→0. kK

(3.18)

that is, Hence, we immediately have

The proof of Corollary 3.2 is completed. ¤ Let x = (i − 1)h, i = 1, 2, · · · , 2J , where h = 1/2J . Using the trapezoidal rule to approximate Eq.(2.17), then we have hT T t = I + o(h)E, where E is a matrix and every element in it is one. By (hT T t )−1 = I + o(h)E, (2.15) and (2.19), we get

which is equivalent to

˜ n (hT ht T )−1 )uD = f, (I − K

(3.19)

˜ n (I − o(h)E)uD = f. (I − K

(3.20)

Hence, by the Corollary 3.2 we get the following remark. Remark 1 According to the Corollary 3.2, the solutions uD of Eq.(3.20) by 7

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Daubechies wavelet method are convergent to the solutions un of Eq.(3.6) when h→0. Theorem 3.3 The solutions of Eq.(3.6) have asymptotic expansions hold at nodes un (x) = u(x) + σ1 (x)hα+3 + σ2 (x)hα+3 lnh + o(hα+5 lnh),

(3.21)

where σj (x) ∈ C[0, 1], j = 1, 2 are independent of h, and σ2 = 0 when β = 0 and α > −1, or β = 1 and α = 0. Proof. We construct the auxiliary equation ˜ = P (x), (I − K)σ

(3.22)

P (x) = [βζ (−α − 2) − ζ(−α − 2)(ln h)β ](H1 u)(2) h3+α .

(3.23)

where

0

By invoking Eq.(2.14), we have ˜ n − K)u(x) ˜ (K = −P (x) + o(h5+α ln h).

(3.24)

˜ n )(un − u − h3+α σ) = f − u + K ˜ n u + h3+α (I − K ˜ n )σ (I − K ˜ n − K)u ˜ + h3+α (I − K)σ ˜ + h3+α (K ˜ n − K)σ ˜ = (K

(3.25)

Using (3.22), we get

α+5

= o(h

lnh),

that is, un − u − h3+α σ = o(hα+5 lnh).

(3.26)

σ = −σ1 − σ2 (ln h)β ,

(3.27)

From (3.22), we obtain where 0 ˜ −1 (H1 u)(2) , and σ2 = ζ(−α−2)(I − K) ˜ −1 (H1 u)(2) . (3.28) σ1 = −βζ (−α−2)(I − K)

Substituting (3.28) into (3.26), and by ζ(−2) = 0 (see [1]), we know that (3.21) holds. The proof of Theorem 3.3 is completed. ¤ Remark 2 According to the Theorem 3.3 and Remark 1, the numerical solutions uD of Eq.(3.19) possess high accuracy order O (h3+α ) as h→0.

4

Numerical experiments

In this section, two numerical examples about the Fredholm equations ¯ ¯ are computed by Daubechies wavelet method. Let errnu (x) = ¯u(x) − un (x)¯ be the errors by Daubechies wavelet method using n (= 2J J = 3, · · · , 8) nodes, and let EOC = log(errn /err2n )/ log 2 be the estimated order of convergence. 8

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If the kernel function k(x, y) of Eq.(1.1) is not periodic, we can apply the Sidi transformation for Eq.(1.1) and make the kernel be periodic. The Sidi transformation is defined by (see [18]) Z t Z ¡ 1 ¢−1 γ ψγ (t) = (sinπτ ) dτ (sinπτ )γ dτ : [0, 1] → [0, 1], γ ≥ 1. 0

0

In the following three examples, the errors and error ratio of numerical solutions at the selected points x1 = 0, x2 = 0.25 and x3 = 0.5 by Daubechies wavelet method using transformation ψ6 (t) are listed in tables. Example 1. Consider the linear Fredholm equation of the first kind Z 1 u(x) + ln|x − y|u(y)dy = g(x) 0 2

2

where g(x) = x ln x/2 + (1 − x ) ln(1 − x)/2 + x/2 − 1/4 and the exact solution is u(x) = x. We use periodic Daubechies wavelet of genus D = 12 as basis functions to compute the errors for Example 1 using different resolutions. The plots of computed errors are shown in Figure 1 and the errors and error ratio of numerical solutions are listed in Table 1. From the results in Table 1, we can see EOC ≈ 3. Table 1: The Errors of u. J

3 2.058-02 EOC(x1 ) − errnu (x2 ) 2.328-02 EOC(x2 ) − u errn (x3 ) 2.278-02 EOC(x3 ) − errnu (x1 )

4 2.111-03 3.2857 2.485-03 3.228 9.764-05 7.866

5 3.214-04 2.715 3.607-04 2.784 1.752-05 2.478

6 3.935-05 3.030 4.401-05 3.035 2.222-06 2.979

7 4.896-06 3.007 5.481-06 3.006 2.778-07 3.000

8 6.116-07 3.001 6.847-07 3.001 3.474-08 3.000

−5

0.04

4 J=3 J=4 J=5

0.03

x 10

J=6 J=7 J=8

2

0.02

Errors

Errors

0 0.01

−2

0 −4 −0.01 −6

−0.02

−0.03

0

0.2

0.4

0.6

0.8

−8

1

0

0.2

0.4

0.6

0.8

1

Figure 1: The error distributions of Example 1 at different resolutions. 9

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Example 2. Solving the following non-periodic second kind Fredholm integral equation with algebraic singular kernel Z 1 u(x) + |x − y|−1/2 u(y)dy = g(x), 0

p

p where g(x) = x + 2(x (x) − x /3 + x (1 − x) + (1 − x)2/3 /3) and the exact solution is u(x) = x. We also use periodic Daubechies wavelet of genus D = 12 as basis functions to compute the errors using different resolutions. The plots of computed errors are shown in Figure 2 and the errors and error ratio of numerical solutions are listed in Table 2. From the results in Table 2, we can see EOC ≈ 2.5. 2/3

Table 2: The Errors of u. J 3 errnu (x1 ) 1.713-03 EOC(x1 ) − u errn (x2 ) 8.526-03 EOC(x2 ) − errnu (x3 ) 4.698-03 EOC(x3 ) −

4 3.964-05 5.433 3.899-04 4.451 2.975-04 3.981

5 1.638-05 1.275 3.847-05 3.341 6.878-05 2.113

6 2.627-06 2.640 4.843-06 2.990 1.216-05 2.499

7 4.505-07 2.544 8.529-07 2.505 2.150-06 2.500

8 7.898-08 2.512 1.508-07 2.500 3.800-07 2.500

−5

0.015

8 J=3 J=4 J=5

0.01

x 10

J=6 J=7 J=8

6

0.005

Errors

Errors

4 0

2

−0.005 0 −0.01 −2

−0.015

−0.02

0

0.2

0.4

0.6

0.8

−4

1

0

0.2

0.4

0.6

0.8

1

Figure 2: The error distributions of Example 2 at different resolutions. Example 3. Solving the following nonlinear second kind Fredholm integral equation with weakly singular kernel Z 1 ¯ ¯ ln¯x − y ¯g(u(y))dy = f (x), u(x) + 0

10

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where 1 1 x f (x) = (x − 0.5)2/3 + (x2 − x + 1/3) − x(x2 /3 − x/2 + 0.25) ln( )− ln(1 − x). 3 1−x 12 The exact solution is u(x) = (x − 0.5)2/3 . The periodic Daubechies wavelets of genus D = 12 as basis functions are used to compute the errors. The Newton iteration method is used for solve Example 3 and the initial vector of u0 is given by u0 = (1, 1, · · · , 1)t2J ×1 . After 4 iterations the errors are shown in Fig.3. The errors and error ratio of numerical solutions are listed in Table 3. From Table 3, we can see EOC ≈ 3. Table 3: The Errors of u. J

3 4.900-04 EOC(x1 ) − errnu (x2 ) 1.659-03 EOC(x2 ) − u errn (x3 ) 6.442-04 EOC(x3 ) − errnu (x1 )

4 4.755-05 3.365 1.565-04 3.407 2.102-04 1.616

5 4.371-06 3.443 1.714-05 3.190 6.570-05 1.678

6 5.481-07 2.996 2.141-06 3.001 8.145-06 3.012

−3

2

8 8.582-09 3.000 3.340-08 3.001 1.252-07 3.003

−6

x 10

10

x 10

J=3 J=4 J=5

1

J=6 J=7 J=8

5 Errors

0 Errors

7 6.865-08 2.997 2.673-07 3.002 1.004-06 3.019

−1

−2

0

−3

−4

0

0.2

0.4

0.6

0.8

−5

1

0

0.2

0.4

0.6

0.8

1

Figure 3: The error distributions of Example 3 at different resolutions.

5

Conclusions

In this paper, the Sidi-Israeli quadrature formula is used to construct the approximation of kernel functions and then the Daubechies wavelet method is used to solve Eq. (1.1). when the kernel functions are not periodic, we can apply the Sidi transformation for Eq.(1.1) and make the kernels be periodic. Because the wavelet integrations are completely avoided and the expansion coefficients obtained here are exact, which 11

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makes the wavelets method has a good degree of accuracy. In addition, the Daubechies wavelets method is used for linear Fredholm integration equation, the discrete matrix of the associated linear system can be transformed into a very sparse and symmetrical one. Accordingly, many preconditioners can be used to reduce the computational cost.

References [1] M. Abramowitz, I.A. Stegun, Handbook of Mathematical Functions, National Bureau of Standards, Applied Mathematics Series, No. 55, Government Printing Office, Washington, D.C., 1964. [2] B. Asady, M. Tavassoli Kajani, A. Hadi Vencheh, A. Heydari, Solving second kind integral equations with hybrid Fourier and blockC-pulse functions, Appl. Math. Comput. 160 (2005) 517–522. [3] O.M. Nielsen, Wavelets in Scientific Computing, Ph.D. Dissertation, Technical University of Denmark, Denmark, 1998. [4] W. Sweldens, R. Piessens, Quadrature formulae and asymptotic error expansions for wavelet approximations of smooth functions, SIAM J. Numer. Anal. 31 (1994) 1240–1264. [5] Pin-wen Zhang, Wavelet method for boundary integral equations, J. Comp. math. 18 (2000) 25–42 [6] K. Amaratunga, Wavelet-Galerkin solutions for one-dimensional partial differential equations, Int. J. Numer. Meth. Engng. 37 (1994) 2703–2716. [7] S. Amini, S. P. Nixon, Multiwavelet Galerkin boundary element solution of Laplace’s equation, Eng. Anal. Bound. Ele. 30 (2006) 116–123 [8] M. Tavassoli Kajani, A. Hadi Vencheh, Solving second kind integral equations with Hybrid Chebyshev and blockCpulse functions, Appl. Math. Comput. 163 (2005) 71–77. [9] Charles A. Micchelli, Yuesheng Xu and Yunhe Zhao, Wavelet Galerkin methods for second-kind integral equations, J. Comp. Appl. Math. 86 (1997) 251–270 [10] K. Maleknejad, F. Mirzaee, Using rationalized Haar wavelet for solving linear integral equations, Appl. Math. Comput. 160 (2005) 579–587. [11] A. Sidi and M. Israeli, Quadrature methods for periodic singular and weakly singular Fredholm integral equation, J. Sci. Comput. 3 (1988) 201–231. [12] I. Daubechies, Ten Lectures on Wavelets, SIAM, Philadelphia, PA, 1992. 12

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[13] Y. Mahmoudi, Wavelet Galerkin method for numerical solution of nonlinear integral equation, Appl. Math. Comput. 167 (2) (2005) 1119-C1129. [14] X.Z. Liang, M.C. Liu, X.J. Che, Solving second kind integral equations by Galerkin methods with continuous orthogonal wavelets, J. Comput. Appl. Math. 136 (2001) 149–161. [15] J.M. Restrepo, G.K. Leaf, Inner product computations using periodized Daubechies wavelets, Int. J. Numer. Meth. Engng. 40 (1997) 3557–3578. [16] J.G. Ren, Wavelet methods for boundary integral equations, Commun. Numer. Meth. Engng. 13 (1997) 373–385. [17] Jin-You Xiao, Li-Hua Wen, Duo Zhang, Solving second kind Fredholm integral equations by periodic wavelet Galerkin method, Appl. Math. Comput. 175 (2006) 508–518. [18] Sidi, A., A new variable transformation for numerical integration, Int Ser Numer Math. 112 (1993) 359–373.

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Some …xed point results in ordered complete dislocated quasi Gd metric space Abdullah Shoaib1 , Muhammad Arshad2 , Tahair Rasham3 Abstract: In this paper, we discuss the …xed points of mappings satisfying contractive type condition on a closed ball in an ordered complete dislocated quasi G metric space. The notion of dominated mappings is applied to approximate the unique solution of non linear functional equations. An example is given to show the validity of our work. Our results improve/generalize several well known recent and classical results. 2010 Mathematics Subjects Classi…cation: 46S70; 47H10; 54H25. Keywords and phrases: …xed point; contractive dominated mappings; closed ball; ordered complete dislocated quasi metric spaces.

1

Introduction and Preliminaries

Let T : X ! X be a mapping. A point x 2 X is called a …xed point of T if x = T x: Let x0 be an arbitrary chosen point in X. De…ne a sequence fxn g in X by a simple iterative method given by xn+1 = T xn ; where n 2 f0; 1; 2; 3; :::g: Such a sequence is called a picard iterative sequence and its convergence plays a very important role in proving existence of …xed point of a mapping T . A self mapping T on a metric space X is said to be a Banach contraction mapping if, d(T x; T y)

kd(x; y)

holds for all x; y 2 X where 0 k < 1: Recently, many results appeared in literature related to …xed point results in complete metric spaces endowed with a partial ordering . Ran and Reurings [17] proved an analogue of Banach’s …xed point theorem in metric space endowed with partial order and gave applications to matrix equations. Subsequently, Nieto et. al. [12] extended the results of [17] for non decreasing mappings and applied this results obtain a unique solution for a 1st order ordinary di¤erential equation with periodic boundary conditions. On the other hand in 2005, Mustafa and Sims in [14] introduce the notion of a generalized metric space as generalization the usual metric space. Mustafa and others studied …xed point theorems for mappings satisfying di¤erent contractive conditions for further useful results can be seen in [3, 8, 9, 10, 15, 16, 21]. Recently, Arshad et. al. [4] proved a result concerning the existence of …xed points of a mapping satisfying a contractive condition on closed ball in a complete dislocated metric space. For further results on closed ball we refer the reader to [5, 6, 7, 13, 20] and references their in. The dominated mapping [2] which satis…es the condition f x x occurs very naturally in several practical problems . For example x denotes the total quantity of food produced over a certain period of time and f (x) gives the quantity of food consumed over the same period in a certain town, then we must have f x x: In this paper we have obtained …xed point results for dominated self- mappings in an ordered complete dislocated quasi Gd metric space on a closed ball 1

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under contractive condition to generalize, extend and improve some classical …xed point results. We have used weaker contractive condition and weaker restrictions to obtain unique …xed point. Our results do not exists even yet in metric spaces. An example shows how this result can be used when the corresponding results cannot. De…nition 1 Let X be a nonempty set and let Gd : X function satisfying the

X

X ! R+ be a

following axioms: (i) If Gd (x; y; z) = Gd (y; z; x) = Gd (z; x; y) = 0,then x = y = z; (ii) Gd (x; y; z) Gd (x; a; a) + Gd (a; y; z) for all x; y; z; a 2 X (rectangle inequality). Then the pair (X; Gd ) is called the dislocated quasi Gd -metric space. It is clear that if Gd (x; y; z) = Gd (y; z; x) = Gd (z; x; y) = 0 then from (i) x = y = z: But if x = y = z then Gd (x; y; z) may not be 0: It is observed that if Gd (x; y; z) = Gd (y; z; x) = Gd (z; x; y) for all x; y; z 2 X; then (X; Gd ) becomes a dislocated Gd -metric space. Example 2 If X = R+ [ f0g then Gd (x; y; z) = x + maxfx; y; zg de…nes a dislocated quasi metric G on X. De…nition 3 Let (X; Gd ) be a Gd -metric space, and let fxn g be a sequence of points in X, a point x in X is said to be the limit of the sequence fxn g if limm;n!1 Gd (x; xn ; xm ) = 0; and one says that sequence fxn g is Gd -convergent to x:Thus, if xn ! x in a Gd -metric space (X; Gd ), then for any 2 > 0; there exists n; m 2 N such that Gd (x; xn ; xm ) < 2; for all n; m N: De…nition 4 Let (X; Gd ) be a Gd -metric space. A sequence fxn g is called Gd -Cauchy sequence if, for each 2 > 0 there exists a positive integer n? 2 N such that Gd (xn ; xm; xl ) < 2 for all n; l; m n? ; i.e. if Gd (xn ; xm ; xl ) ! 0 as n; m; l ! 1: De…nition 5 Gd -metric space (X; Gd ) is said to be Gd -complete if every Gd Cauchy sequence in (X; Gd ) is Gd -convergent in X: Proposition 6 Let (X; Gd ) be a Gd -metric space, then the following are equivalent: (1) (2) (3) (4)

fxn g is Gd convergent to x: Gd (xn ; xn ; x) ! 0 as n ! 1: Gd (xn ; x; x) ! 0 as n ! 1: Gd (xn ; xm ; x) ! 0 as m n ! 1:

De…nition 7 Let (X; Gd ) be a Gd -metric space then for x0 2 X, r > 0; the closed ball with centre x0 and radius r is, B(x0 ; r) = fy 2 X : Gd (x0 ; y; y)

rg:

2

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De…nition 8 [2] Let (X; ) be a partial ordered set. Then x; y 2 X are called comparable if x y or y x holds. De…nition 9 [2] Let (X; ) be a partially ordered set. A self mapping f on X is called dominated if f x x for each x in X: Example 10 [2] Let X = [0; 1] be endowed with usual ordering and f : X ! X be de…ned by f x = xn for some n 2 N. Since f x = xn x for all x 2 X, therefore f is a dominated map.

2

Fixed Points of Contractive Mapping

Theorem 11 Let (X; ; Gd ) be an ordered complete dislocated quasi Gd metric space, and T : X ! X be a dominated mapping. Suppose there exists a; b such that a + 3b < 1 and for all comparable elements x; y and z in B(x0 ; r); with x0 2 B(x0 ; r); r > 0;. Gd (T x; T y; T z)

a Gd (x; y; z) + b [Gd (x; T x; T x) +Gd (y; T y; T y) + Gd (z; T z; T z)] where

=

(2.1)

a+b 1 2b

and Gd (x0 ; T x0 ; T x0 )

(1

)r:

If for a nonincreasing sequence fxn g in B(x0 ; r), fxn g ! u implies that u and G(x0 ; T x0 ; T x0 ) + G(v; T v; T v) + G(v; T v; T v) G(x0 ; v; v) + G(T x0 ; T v; T v) + G(T x0 ; T v; T v)

(2.2) xn

(2.3)

then there exists a point x? in B(x0 ; r) such that Gd (x? ; x? ; x? ) = 0 and x? = T x? : Proof. Consider a picard sequence xn+1 = T xn with initial guess x0 : As xn+1 = T xn xn for all n 2 f0g [ N: By inequality (2:2), Gd (x0 ; x1 ; x1 ) r. It implies that x1 2 B(x0 ; r): Similarly x2 : : : xj 2 B(x0 ; r) for some j 2 N: Gd (xj ; xj+1 ; xj+1 )

(1

= Gd (T xj 1 ; T xj ; T xj ) a Gd (xj 1 ; xj ; xj ) +b[Gd (xj 1 ; T xj 1 ; T xj 1 ) + Gd (xj ; xj+1 ; xj+1 ) +Gd (xj ; xj+1 ; xj+1 )] 2b)Gd (xj ; xj+1 ; xj+1 ) (a + b)Gd (xj 1 ; xj ; xj ) (a + b) Gd (xj ; xj+1 ; xj+1 ) Gd (xj 1 ; xj ; xj ) (1 2b) .. . j Gd (xj ; xj+1 ; xj+1 ) Gd (x0 ; x1 ; x1 ): (2.4) 3

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Now by using the inequality (2:2) and (2:4) we have Gd (xj ; xj+1 ; xj+1 ) Gd (xj ; xj+1 ; xj+1 ) Gd (xj ; xj+1 ; xj+1 ) Gd (xj ; xj+1 ; xj+1 )

Gd (x0 ; x1 ; x1 ) + Gd (x1 ; x2 ; x2 ) + (1 )r + (1 )r + + j (1 r(1 )[1 + + 2 + + j] j+1 (1 ) (1 ) ) Gd (xj ; xj+1 ; xj+1 )

r(1

)

+ Gd (xj ; xj+1 ; xj+1 ) )r

r r:

Thus xj+1 2 B(x0 ; r). Hence xn 2 B(x0 ; r) for all n 2 N: Now inequality (2:4) can be written as in the form of n

Gd (xn ; xn+1 ; xn+1 )

Gd (x0 ; x1 ; x1 ) for all n 2 N:

(2.5)

By using inequality (2:5) we get Gd (xn ; xn+i ; xn+i )

Gd (xn ; xn+1 ; xn+1 ) + n

(1 (1

Gd (xn ; xn+i ; xn+i )

i

)

+ Gd (xn+i

1 ; xn+i ; xn+i )

Gd (x0 ; x1 ; x1 ) ! 0 as n ! 1

)

(2.6)

Notice that the sequence fxn g is Cauchy sequence in (B(x0 ; r) ; Gd ): Therefore there exist a point x? 2 B(x0 ; r): lim Gd (xn ; x? ; x? )

n!1

=

Gd (x? ; T x? ; T x? ) By assumption x?

xn

Gd (x? ; T x? ; T x? ) Gd (x? ; T x? ; T x? )

Gd (x? ; T x? ; T x? ) (1

2b)Gd (x? ; T x? ; T x? )

xn

lim Gd (x? ; x? ; xn ) = 0

n!1

Gd (x? ; xn ; xn ) + Gd (xn ; T x? ; T x? ) 1,

therefore,

Gd (x? ; xn ; xn ) + Gd (T xn 1 ; T x? ; T x? ) Gd (x? ; xn ; xn ) + a Gd (xn 1 ; x? ; x? ) +b[Gd (xn 1 ; T xn 1 ; T xn 1 ) + Gd (x? ; T x? ; T x? ) Gd (x? ; T x? ; T x? )] Gd (x? ; xn ; xn ) + a Gd (xn 1 ; x? ; x? ) +b[Gd (xn 1 ; T xn 1 ; T xn 1 ) + 2Gd (x? ; T x? ; T x? ) Gd (x? ; xn ; xn ) + a Gd (xn 1 ; x? ; x? ) +b Gd (xn 1 ; xn ; xn )

Taking limn!1 both sides and using (2:6) we have (1

2b)Gd (x? ; T x? ; T x? )

0 + a(0) + b(0) ) Gd (x? ; T x? ; T x? ) ) x? = T x? :

0 (2.7)

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Similarly Gd (T x? ; T x? ; x? ) = 0 and Gd (T x? ; x? ; T x? ) = 0 and hence x? = T x? :Now Gd (x? ; x? ; x? )

(1

a

Gd (T x? ; T x? ; T x? ) a Gd (x? ; x? ; x? ) +3bGd (x? ; T x? ; T x? ) 3b)Gd (x? ; x? ; x? ) 0 ) Gd (x? ; x? ; x? ) 0: =

This implies that Gd (x? ; x? ; x? ) = 0: Uniqueness: Let y ? be another point in B(x0 ; r) such that y? Gd (y ? ; y ? ; y ? )

(1

a

T y? : (2.8) ? ? ? ? ? ? Gd (T y ; T y ; T y ) a Gd (y ; y ; y ) +3b[Gd (y ? ; T y ? ; T y ? )] 3b)Gd (y ? ; y ? ; y ? ) 0 ) Gd (y ? ; y ? ; y ? ) 0: ) Gd (y ? ; y ? ; y ? ) = 0: = =

If x? and y ? are comparable then Gd (x? ; y ? ; y ? )

(1

Gd (T x? ; T y ? ; T y ? ) a Gd (x? ; y ? ; y ? ) +b[Gd (x? ; T x? ; T x? ) + 2Gd (y ? ; T y ? ; T y ? )] a)Gd (x? ; y ? ; y ? ) 0 ) Gd (x? ; y ? ; y ? ) = 0: =

Similarly, Gd (y ? ; y ? ; x? ) = 0: This shows that x? = y ? : If x? and y ? are not comparable then there exist a point v 2 B(x0 ; r) which is a lower bound of both x? and y ? : Now we will to prove that T n v 2 B(x0 ; r): Moreover by assumptions v x? xn x0 : Now by using (2:1), we have, Gd (T x0 ; T v; T v)

a Gd (x0 ; v; v) + b [Gd (x0 ; x1 ; x1 ) + 2Gd (v; T v; T v)]:

By using (2:3); we have (1

Gd (T x0 ; T v; T v) 2b)Gd (T x0 ; T v; T v) Gd (T x0 ; T v; T v) Gd (T x0 ; T v; T v)

a Gd (x0 ; v; v) + b [Gd (x0 ; v; v) + 2Gd (x1 ; T v; T v)] (a + b) Gd (x0 ; v; v) (a + b) Gd (x0 ; v; v) (1 2b) Gd (x0 ; v; v): (2.9)

Now, Gd (x0 ; T v; T v) Gd (x0 ; T v; T v) Gd (x0 ; T v; T v) Gd (x0 ; T v; T v)

Gd (x0 ; x1 ; x1 ) + Gd (x1 ; T v; T v) Gd (x0 ; x1 ; x1 ) + Gd (x0 ; v; v) by (2:9) (1 )r + r r: 5

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It follows that T v 2 B(x0 ; r): Now we will prove that T n v 2 B(x0 ; r): By using mathematical induction to apply inequality (2:1): Let T 2 v; T 3 v; T j v 2 B(x0 ; r) for some j 2 N: As T jv

Tj

1

v

v

x?

xn

x0 :

Then, Gd (T j v; T j+1 v; T j+1 v) Gd (T j v; T j+1 v; T j+1 v) (1

2b)Gd (T j v; T j+1 v; T j+1 v) Gd (T j v; T j+1 v; T j+1 v) Gd (T j v; T j+1 v; T j+1 v) Gd (T j v; T j+1 v; T j+1 v) Gd (T j v; T j+1 v; T j+1 v) Gd (T j v; T j+1 v; T j+1 v)

= Gd (T (T j 1 v); T (T j v); T (T j v)) a Gd (T j 1 v; T j v; T j v) + b [Gd (T j 1 v; T j v; T j v) +2Gd (T j v; T j+1 v; T j+1 v)] (a + b)Gd (T j 1 v; T j v; T j v) Gd (T j 1 v; T j v; T j v) 2 Gd (T j 2 v; T j 1 v; T j 1 v) 3 Gd (T j 3 v; T j 2 v; T j 2 v) .. . j Gd (T j j v; T j (j 1) v; T j (j 1) v) j Gd (v; T v; T v) (2.10)

Now, Gd (xj+1 ; T j+1 v; T j+1 v) Gd (xj+1 ; T j+1 v; T j+1 v)

Gd (T xj ; T (T j v); T (T j v)) a Gd (xj ; T j v; T j v) +b [Gd (xj ; T xj ; T xj ) + 2Gd (T j v; T j+1 v; T j+1 v)]:

By using (2:4) and (2:10) Gd (xj+1 ; T j+1 v; T j+1 v) Gd (xj+1 ; T j+1 v; T j+1 v)

a j Gd (x0 ; v; v) +b[ j Gd (x0 ; x1 ; x1 ) + 2 j Gd (v; T v; T v)] a j Gd (x0 ; v; v) +b j [Gd (x0 ; x1 ; x1 ) + 2Gd (v; T v; T v)]

By using the condition (2:3) Gd (xj+1 ; T j+1 v; T j+1 v) Gd (xj+1 ; T j+1 v; T j+1 v) Gd (xj+1 ; T j+1 v; T j+1 v)

a j Gd (x0 ; v; v) +b j [Gd (x0 ; v; v) + 2 Gd (x0 ; v; v)] j (a + b + 2b )Gd (x0 ; v; v) j+1 Gd (x0 ; v; v) (2.11)

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Now , Gd (x0 ; T j+1 v; T j+1 v) Gd (x0 ; T j+1 v; T j+1 v) Gd (x0 ; T j+1 v; T j+1 v) Gd (x0 ; T j+1 v; T j+1 v)

Gd (x0 ; xj+1; xj+1 ) + Gd (xj+1 ; T j+1 v; T j+1 v) Gd (x0 ; x1; x1 ) + + Gd (xj ; xj+1; xj+1 ) j+1 +Gd (xj+1 ; T v; T j+1 v) Gd (x0 ; x1; x1 ) + Gd (x0 ; x1; x1 ) + + j+1 Gd (x0 ; v; v) by (2:5) and (2:11) Gd (x0 ; x1; x1 )[1 + + 2 + + j ] + j+1 r as v 2 B(x0 ; r)

Gd (x0 ; T j+1 v; T j+1 v)

(1

Gd (x0 ; T j+1 v; T j+1 v)

r:

)r

(1 (1

j+1

)

)

+

j+1

r=r

It follows that T j+1 v 2 B(x0 ; r) and hence T j v 2 B(x0 ; r): Now the inequality (2:10) can be written as Gd (T n v; T n+1 v; T n+1 v)

n

Gd (v; T v; T v) ! 0 as n ! 1

(2.12)

Now, Gd (x? ; y ? ; y ? ) Gd (x? ; y ? ; y ? ) Gd (x? ; y ? ; y ? )

= Gd (T x? ; T y ? ; T y ? ) Gd (T x? ; T n+1 v; T n+1 v) + Gd (T n+1 v; T y ? ; T y ? ) a Gd (x? ; T n v; T n v) + b [Gd (x? ; T x? ; T x? ) +2Gd (T n v; T n+1 v; T n+1 v)] + a Gd (T n v; T y ? ; T y ? ) +b [Gd (T n v; T n+1 v; T n+1 v) + 2Gd (y ? ; T y ? ; T y ? )]

By using (2:7); (2:8) and (2:12) we have Gd (x? ; y ? ; y ? ) Gd (x? ; y ? ; y ? ) Gd (x? ; y ? ; y ? )

a Gd (x? ; T n v; T n v) + a Gd (T n v; y ? ; y ? ) a [Gd (T x? ; T n v; T n v) + Gd (T n v; T y ? ; T y ? )] a [a Gd (x? ; T n 1 v; T n 1 v) + b Gd (x? ; T x? ; T x? ) +2b Gd (T n 1 v; T n v; T n v) + a Gd (T n 1 v; y ? ; y ? ) +b Gd (T n 1 v; T n v; T n v) + 2b Gd (y ? ; T y ? ; T y ? )]:

By using (2:7); (2:8) and (2:12) we have Gd (x? ; y ? ; y ? ) Gd (x? ; y ? ; y ? )

a2 [Gd (x? ; T n 1 v; T n 1 v) + Gd (T n 1 v; y ? ; y ? )] a3 [Gd (x? ; T n 2 v; T n 2 v) + Gd (T n 2 v; y ? ; y ? )] .. . Gd (x? ; y ? ; y ? ) an [Gd (x? ; T v; T v) + Gd (T v; y ? ; y ? )] Gd (x? ; y ? ; y ? ) ! 0 as n ! 1 Gd (x? ; y ? ; y ? ) = 0 x? = y ? : 7

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This proves the uniqueness of the …xed point. Now we give an example of an ordered complete dislocated quasi Gd -metric space in which the contraction does not hold on the whole space rather it holds on a closed ball only. Example 12 Let X = R+ [ f0g be endowed with usual order and Gd : X X ! X be a complete dislocated quasi Gd metric space de…ned by, Gd (x; y; z) =

X

0 if x = y = z max f2x; y; zg otherwise:

Then (X; Gd ) is a Gd complete G dislocated quasi metric space. Let T : X ! X be de…ned by, Tx =

x 5

x

if x 2 [0; 23 ] if x 2 [ 32 ; 1)

1 3

:

Clearly, T is a dominated mappings. Take x0 = 13 , r = 32 , B(x0 ; r) = [0; 32 ] and 1 1 ; and b = 10 : = 14 ; a + 3b < 1; where a = 10 Gd (x0 ; T x0 ; T x0 ) 1 1 1 Gd ( ; T ; T ) 3 3 3 Since (1

)r

(1

)r 2 1 1 2 = maxf ; ; g = 3 15 15 3 1 3 9 = (1 ) = 4 2 8 2 9 ) 3 8 ) 16 27

Also if x; y and z 2 [ 32 ; 1): We assume that x > y; and y > z; then maxf2x

2 ;y 3

1 ;z 3

1 g 3

Gd (T x; T y; T z)

1 maxf2x; y; zg 10 1 1 1 [maxf2x; x ;x g 10 3 3 1 1 + maxf2y; y ;y g 2 2 1 1 + maxf2z; z ;z g] 2 2 a Gd (x; y; z) + b [Gd (x; T x; T x) +Gd (y; T y; T y) + Gd (z; T z; T z)]

So the contractive conditions does not holds in X: Now if x; y and z 2 B(x0 ; r)

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then, Gd (T x; T y; T z)

2x y z 1 ; ; g f2x; y; zg 5 5 5 10 1 x x y y + [maxf2x; ; g + maxf2y; ; g 10 5 5 5 5 z z + maxfz; ; g] 5 5 ) Gd (T x; T y; T z) a Gd (x; y; z) + b [Gd (x; T x; T x) +Gd (y; T y; T y) + Gd (z; T z; T z)]: =

maxf

Hence it satis…es all the requirements of Theorem11. If we take b = 0 in inequality (2:1) then we obtain the following corollary. Corollary 13 Let (X; ; G) be an ordered complete dislocated quasi G metric space, T : X ! X be a dominated mapping and x0 be any arbitrary point in X. Suppose there exists a 2 [0; 1) with, G(T x; T y; T z)

a G(x; y; z); for all x; y and z 2 Y = B(x0 ; r);

and G(x0; T x0; T x0 )

(1

a)r:

If for a nonincreasing sequence fxn g ! u implies that u xn . Then there exists a point x? in B(x0; r) such that x? = Sx? and G(x? ; x? ; x? ) = 0: Moreover if for any three points x; y and z in B(x0 ; r) such that there exists a point v 2 B(x0; r) such that v x; v y and v z; that is, every three of elements in B(x0 ; r) has a lower bound, then the point x? is unique. Similarly if we take a = 0 in inequality (2:1) then we obtain the following corollary. Corollary 14 Let (X; ; G) be an ordered complete dislocated quasi G-metric space T : X ! R be a mapping and x0 be an arbitrary point in X: Suppose there exists b 2 0; 31 with G(T x; T y; T z)

b (G(x; T x; T x) + G(y; T y; T y) + G(z; T z; T z))

for all comparable elements x; y; z 2 B(x0; r) and G(x0 ; T x0 ; T x0 )

(1

)r;

where = 1 b2b . If for non increasing sequence fxn g ! u implies that u xn : Then there exists a point x? in B(x0 ; r) such that x? = Sx? and G(x? ; x? ; x? ) = 0: Moreover, if for any three points x; y; z 2 B(xo ; r); there exists a point v in B(x0 ; r) such that v x and v y, v z:

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References [1] M. Abbas and B. E. Rhoades, Common …xed point results for noncommuting mappings without continuity in generalized metric spaces, Appl Maths and Computation, 215(2009), 262-269. [2] M. Abbas and S .Z. Nemeth, Finding solutions of implict complementarity problems by isotonicty of metric projection, Nonlinear Anal, 75(2012), 2349-2361. [3] R. Agarwal and E. Karapinar, Remarks on some coupled …xed point theorems in G-metric spaces, Fixed Point Theory and Appli, 2013(2013),15pages. [4] M. Arshad ,A. Shoaib, and I. Beg, Fixed Point of pair of contractive dominated mappings on a closed ball in an ordered complete dislocated metric space accepted in Fixed Point Theory and Appl, 2013(2013),15pages. [5] A. Azam, S. Hussain and M. Arshad, Common Fixed Points of Kannan Type Fuzzy Mappings on closed balls, Appl. Math. Inf. Sci. Lett. 1, 2 (2013), 7-10. [6] A. Azam, S. Hussain and M. Arshad, Common …xed points of Chatterjea type fuzzy mappings on closed balls, Neural Computing & Appl, 21(2012), S313–S317. [7] A. Azam, M. Waseem, M. Rashid, Fixed point theorems for fuzzy contractive mappings in quasi-pseudo-metric spaces, Fixed Point Theory Appl, 27(2013) 14pages. [8] Lj. Gaji ’c and M. Stojakovi ’c, On Ciri ’c generalization of mappings with a contractive iterate at a point in G-metric spaces, Appl Maths and computation, 219(2012), 435-441. [9] H. Hydi, W. Shatanawi, C. Vetro, On generalized weak G-contraction mappings in G-metric spaces, Compute. Math. Appl., 62 (2011), 4223-4229. [10] M. Jleli and B. Samet, Remarks on G-metric spaces and …xed point theorems Fixed Point Theory appl, 210 (2012). [11] H.K Nashine, Coupled common …xed point results in ordered G-metric spaces, J. Nonlinear Sc. Appl. 1(2012), 1-13. [12] J. J. Nieto and R. Rodrigguez-Lopez, Contractive mapping theorems in partially ordered sets and applications to ordinary di¤erential equations, Order, 22 (3) (2005), 223-239. [13] M. A. Kutbi, J. Ahmad, N. Hussain and M. Arshad, Common Fixed Point Results for Mappings with Rational Expressions, Abstr. Appl. Anal, 2013, Article ID 549518, 11 pages. 10

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[14] Z. Mustafa and B. Sims, A new approach to generalized metric spaces, Journal of Nonlinear and Convex Anal, 7(2006) 289-297. [15] Z. Mustafa, H. Obiedat, and F. Awawdeh, Some …xed point theorem for mappings on a complete G- metric space, Fixed point theory and appl, 2008(2008),12pages. [16] H. Obiedat and Z. Mustafa, Fixed point results on a non symmetric Gmetric spaces,Jordan Journal of Maths and Stats, 3(2010), 65-79. [17] A.C.M. Ran, M.C.B. Reurings, A …xed point theorem in partially ordered sets and some applications to matrix equations, Proc. Amer. Math. Soc., 132 (5) (2004), 1435-1443. [18] B. Samet, C. Vetro, F. Vetro, Remarks on G-metric spaces, Int. J. Anal., 2013(2013), 6pages. [19] W. Shatanawi, “Fixed point theory for contractive mappings satisfying maps in G-metric spaces, Fixed Point Theory and Appl, 2010(2010), 9pages. [20] A. Shoaib, M. Arshad and J. Ahmad, Fixed point results of locally cotractive mappings in ordered quasi-partial metric spaces, The Sci World Journal, 2013(2013), 14pages. [21] S. Zhou and F.Gu, Some new …xed points in G- metric spaces Journal of Hangzhou Normal University, 11(2010), 47-50.

1

Department of mathematics, Riphah International University, Islamabad44000, Pakistan. E-mail: [email protected]. 2;3 Department of mathematics, International Islamic University, H-10, Islamabad44000, Pakistan. E-mails: [email protected], [email protected].

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MIZOGUCHI- TAKAHASHI’S FIXED POINT THEOREM IN νGENERALIZED METRIC SPACES SALHA ALSHAIKEY *, SAUD M. ALSULAMI AND MONAIRAH ALANSARI

Abstract. Our main work is to prove Mizogchi- Takahashi theorem in νgeneralized metric space in the sense of Brancairi. In the same setting we prove two more theorems which are generalizations of the main one.

1. Introduction A metric is defined as a mapping d : X × X → [0, ∞), for any non-empty set X which satisfying the following axioms, for any x, y, z ∈ X (i) d(x, y) = 0 iff x = y (ii) d(x, y) = d(y, x) (iii) d(x, y) ≤ d(x, z) + d(z, y). We said that the pair (X, d) is a metric space. The theory of metric spaces form a basic environment for a lot of concepts in mathematics such as the fixed point theorems which have an important rules in various branches of mathematical analysis. One of the famous result of fixed point theorems is Banach Contraction Principle which state that, Theorem 1.1. [11](Banach Contraction Principle) Let (X, d) be a complete metric space. Let T : X → X be a self map on X such that d(T x, T y) ≤ rd(x, y), hold for any x, y ∈ X, where r ∈ [0, 1). Then T has a unique fixed point. Many authors explored the importance of this theorem and extended it in different directions. For examples, we refer the reader to the following papers [2, 9, 8, 6], and the references therein. In (1969) Nadler extended theorem 1.1 for multi-valued mapping. Recall that the set of all non- empty, closed and bounded subsets of X is denoted by CB(X) and let A, B be any sets in CB(X). A Hausdorff metric is defined as H(A, B) = max{sup d(a, B), sup d(b, A)} a∈A

b∈B

Theorem 1.2. [12](Nadler’s theorem) Let (X, d) be a complete metric space. Let T : X → CB(X) be a multi-valued map. Assume that H(T x, T y) ≤ rd(x, y), holds for each x, y ∈ X and r ∈ [0, 1). Then T has a fixed point. Key words and phrases. Mizoguchi-Takahashi’s space,Generalized of MT- theorem, Fixed point theory.

theorem,

ν-

generalized

metric

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Many attempts have been done to generalize Nadler’s theorem. One of these generalizations is Mizoguchi- Takahashi’s theorem which stats that: Theorem 1.3. [10] Let (X, d) be a complete metric space. Let T : X → CB(X) be a multi-valued mapping. Assume that H(T x, T y) ≤ β(d(x, y))d(x, y), hold for each x, y ∈ X, where β : [0, ∞) → [0, 1) is a function such that lim sups→t+ β(s) < 1. Then T has a fixed point. Remark 1.4. The function β in theorem 1.3, which satisfies lim sups→t+ β(s) < 1 is called Mizoguchi- Takahashi function (MT- function for short). Starting with Mizoguchi and Takahashi’s paper, many generalizations of their theorem have been established see [3, 13]. Recently, Eldred et al [4], claimed that Nadler’s and Mizoguchi- Takahashi’s theorems are equivalent. However, in [14], Suzuki proved that their claim is not true and he shown that Mizoguchi- Takahashi’s theorem (1.3) is a real extension of Nadler’s theorem. This is why we are interesting in such theorem. In another direction, in (2000) Branciari created a new concept of generalized metric spaces by modifying the triangle inequality to involve more points. Definition 1.5. [1] Let X be a non- empty set and d : X ×X → [0, ∞). For ν ∈ N, a pair (X, d) is called a ν- generalized metric space if the following hold: (M1) d(x, y) = 0 iff x = y (M2) d(x, y) = d(y, x) (M3) d(x, y) ≤ d(x, u1 ) + d(u1 , u2 ) + ... + d(uν , y), for any x, u1 , u2 , ...uν , y ∈ X, such that x, u1 , u2 , ...uν , y are all different. It is not difficult to show that the new space is not the same as the original one. Moreover, the new space is hard to deal with because it does not satisfy all topological properties that metric space has, see [15] for more details. Recently, in [16], Suzuki proved Nadler’s theorem in ν- generalized metric spaces. The main work of this paper is to prove Mizoguchi -Takahashi’s theorem in ν- generalized metric spaces. Firstly, we will list all the necessary definitions and some results that we will need. Then, we will be able to prove our main results. 2. Preliminary Definition 2.1. A point x ∈ X is said to be a fixed point of multi-valued map T if x ∈ T x. Definition 2.2. [1] Let (X, d) be a ν- generalized metric space. A sequence {xn }n∈N ∈ X is said to be Cauchy sequence if lim sup d(xn , xm ) = 0 n n>m

Definition 2.3. [16] A sequence {xn }n∈N is said to be ( all xn ’s are different and ∞ X d(xj , xj+1 ) < ∞

P , 6=)- Cauchy sequence if

j=1

Definition 2.4. [16] Let (X, d)P be a ν- generalized metric space. We said that, X P is a ( , 6=)- complete if every ( , 6=)- Cauchy sequence converges.

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Lemma 2.5. [16, 5] Let (X, d) be a ν- generalized metric space. P • Every converge ( , 6=)- Cauchy sequence is Cauchy. • Let {xn }n∈N be a Cauchy sequence converges to some y ∈ X and {yn } ∈ X be a sequence such that limn→∞ d(xn , yn ) = 0. Then, {yn } also converges to y. Lemma 2.6. [14] Let β : [0, ∞) → [0, 1) is a MT-function. Then, for all s ∈ [0, ∞), there exist rs ∈ [0, 1) and εs > 0 such that β(t) ≤ rs for all t ∈ [s, s + εs ) Lemma 2.7. [12] Let (X, d) be a metric space. For any A, B ∈ CB(X) and ε > 0, there exist a ∈ A and b ∈ B such that d(a, b) ≤ H(A, B) + ε 3. Main Result In this section we prove Mizoguchi -Takahashi’s theorem in ν -generalized metric spaces and some of its generalizations in the space. P Theorem 3.1. Let (X, d) be a ( , 6=) complete, ν- generalized metric space. and let T be a multi-valued map defined from X into CB(X) satisfies the following: (i) If {yn } ∈ T x and {yn } converges to y then y ∈ T x. (ii) For any x, y ∈ X, H(T x, T y) ≤ α(d(x, y))d(x, y), where α is MT-function. Then T has a fixed point. 1 + α(t) . It is not difficult 2 to show that α(t) < γ(t), for any t ∈ [0, ∞) and lims→t+ sup γ(s) < 1. Moreover, for each x, y ∈ X and v ∈ T x, there exist u ∈ T y such that Proof. Let define a function γ : [0, ∞) → [0, 1) as γ(t) =

d(v, u) ≤ γ(d(x, y))d(x, y). Putting v = y, we will get that d(y, u) ≤ γ(d(x, y))d(x, y) Define f (x) = inf{d(x, b) : b ∈ T x} and suppose that T does not have a fixed point ( i.e., for all x ∈ X, f (x) > 0). Let x1 ∈ X be arbitrary and choose x2 ∈ T x1 satisfying (1)

d(x1 , x2 )
0 such that γ(s) ≤ r for all s ∈ [δ, δ + ε). For any n ∈ N, we can choose µ ∈ N satisfying δ ≤ d(xn , xn+1 ) ≤ δ + ε with n ≥ µ. So, ∞ X

d(xn , xn+1 ) ≤

n=1

≤

µ X n=1 µ X

d(xn , xn+1 ) +

∞ X

d(xn , xn+1 )

n=µ+1

d(xn , xn+1 ) +

n=1

∞ X

rn d(xµ , xµ+1 )

n=1

< ∞. P P Thus {xn } is a ( , 6=)- Cauchy sequence in ( , 6=) complete ν- generalize metric space. Then, it is converge to some z ∈ X and by lemma (2.5), {xn } is a Cauchy sequence. From our assumption we choose {un } ∈ T z satisfy d(xn+1 , un ) ≤ H(T xn , T z) ≤ γ(d(xn , z))d(xn , z), for any n ∈ N. But {xn } converges to z, so d(xn+1 , un ) → 0 as n → ∞. Thus we have xn+1 → z and xn+1 → un . Therefore, by lemma(2.5) d(un , z) = 0 as n → ∞. So d(T z, z) = 0 implies f (z) = 0 which is a contradiction. Therefore, there exist z ∈ X such that f (z) = 0 and hence z ∈ T z is a fixed point.  Definition 3.2. [7] A multi- valued map T from X into CB(X) is called α- admissible if for any x ∈ X and y ∈ T x, α(x, y) ≥ 1 implies α(y, z) ≥ 1 for any z ∈ T y, where α : X × X → [0, ∞). The up coming lemma proved in [18], for single-valued map here, we prove it for multi- valued map.

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Lemma 3.3. Let (X, d) be a ν- generalized metric space. Let T be a multi-valued mapping from X into 2X and {xn }n∈N be a sequence in X defined by xn+1 ∈ T xn such that xn 6= xn+1 . Assume that d(xn , xn+1 ) ≤ δd(xn−1 , xn )

(6)

hold for any δ ∈ [0, 1). Then xn 6= xm ∀n 6= m ∈ N. Proof. We prove that xn+` 6= xn for all n ∈ N and ` ≥ 1. Suppose the contrary that is xn+` = xn for some n ∈ N and ` ≥ 1. By assumption, we have that xn+`+1 = xn+1 . Then from (6) we get (7) d(xn , xn+1 ) = d(xn+` , xn+`+1 ) ≤ δd(xn+`−1 , xn+` ) ≤ ... ≤ δ ` d(xn , xn+1 ) < d(xn , xn+1 ) which is contradiction. Thus, we get xm 6= xn for all m 6= n in N.



Let Φ be the family of all functions ϕ : [0, ∞) → [0, ∞) which satisfying the following conditions: (a) ϕ(s) = 0 iff s = 0. (b) ϕ is non-decreasing and lower semi-continuous s < ∞. (c) lims→0+ sup ϕ(s) P Theorem 3.4. Let (X, d) be a ( , 6=) complete ν- generalized metric space. Let T : X → CB(X) be an α- admissible multi-valued mapping satisfying: (i) There exist x0 ∈ X and x1 ∈ T x0 such that α(x0 , x1 ) ≥ 1 (ii) If (yn ) ∈ T x and (yn ) converge to y then y ∈ T x (iii) α(x, y)H(T x, T y) ≤ φ(d(x, y))d(x, y) for any x, y ∈ X, and φ is MT-function. Then T has a fixed point. 1 + φ(t) such that lims→t+ sup β(s) < 1. 2 Clearly φ(t) < β(t) for each t ∈ [0, ∞). Let x0 ∈ X and choose x1 ∈ T x0 such that 1 − φ(d(x0 , x1 )) α(x0 , x1 ) ≥ 1. Assume x0 6= x1 so, d(x0 , x1 ) > 0. Since T x1 6= ∅, 2 choose x2 ∈ T x1 such that Proof. Let β : [0, ∞) → [0, 1) as β(t) =

1 − φ(d(x0 , x1 )) d(x0 , x1 ) 2 1 − φ(d(x0 , x1 )) ≤ α(x0 , x1 )H(T x0 , T x1 ) + d(x0 , x1 ) 2 1 − φ(d(x0 , x1 )) ≤ φ(d(x0 , x1 ))d(x0 , x1 ) + d(x0 , x1 ) 2 ≤ β(d(x0 , x1 ))d(x0 , x1 ).

d(x1 , x2 ) ≤ H(T x0 , T x1 ) +

Since T is α- admissible, x1 ∈ T x0 and α(x0 , x1 ) ≥ 1 then, α(T x0 , T x1 ) ≥ 1 which 1 − φ(d(x1 , x2 )) implies α(x1 , x2 ) ≥ 1. Similarly assume x1 6= x2 we have d(x1 , x2 ) > 2

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SALHA ALSHAIKEY *, SAUD M. ALSULAMI AND MONAIRAH ALANSARI

0 and choose x3 ∈ T x2 such that 1 − φ(d(x1 , x2 )) d(x1 , x2 ) 2 1 − φ(d(x1 , x2 )) ≤ α(x1 , x2 )H(T x1 , T x2 ) + d(x1 , x2 ) 2 1 − φ(d(x1 , x2 )) d(x1 , x2 ) ≤ φ(d(x1 , x2 ))d(x1 , x2 ) + 2 ≤ β(d(x1 , x2 ))d(x1 , x2 ).

d(x2 , x3 ) ≤ H(T x1 , T x2 ) +

Similarly, using the same method of proving theorem (3.1), we have our result.  P Theorem 3.5. Let (X, d) be a ( , 6=) complete ν- generalized metric space. Let T : X → CB(X) be a multi-valued map satisfying: ϕ(H(T x, T y)) ≤ α(ϕ(d(x, y)))ϕ(d(x, y)), for each x, y ∈ X, where α is a MT- function and ϕ ∈ Φ. Then T has a fixed point. 1 + α(t) . Since ϕ is non- decreasing Proof. Let γ : [0, ∞) → [0, 1) defined by γ(t) = 2 function, then   max sup ϕ(d(v, T y)), sup ϕ(d(u, T x)) v∈T x

(8)

u∈T y

  = max ϕ( sup d(v, T y)), ϕ( sup d(u, T x)) v∈T x

u∈T y

= ϕ (H(T x, T y)) ≤ γ(ϕ(d(x, y)))ϕ(d(x, y)). There exist an element z ∈ T y such that ϕ(d(y, z)) ≤ γ(ϕ(d(x, y)))ϕ(d(x, y)), for each x ∈ X and y ∈ T x. Thus, in the same way a sequence {xn }n∈N ∈ X defined as xn+1 ∈ T xn is constructed such that (9)

ϕ(d(xn , xn+1 ) ≤ γ(ϕ(d(xn−1 , xn ))ϕ(d(xn−1 , xn ))

for all n ∈ N. Since γ(t) < 1 for any t ∈ [0, ∞), hence from (9) we get (10)

ϕ(d(xn , xn+1 ) < ϕ(d(xn−1 , xn )).

Clearly {ϕ(d(xn−1 , xn ))} is decreasing sequence of positive real numbers. Hence it is converge to some non- negative real number, say . By contradiction, it is easy to show that  = 0. Note that, ϕ is a non- decreasing function which implies to d(xn , xn+1 ) < d(xn−1 , xn ). Thus the sequence {d(xn , xn+1 )} is also decreasing. Hence P∞ by lemma (3.3), the terms of the sequence all are different. Now, show that n=0 d(xn , xn+1 ) < ∞. Note that the sequence {d(xn , xn+1 )} is decreasing and bounded. Thus, it is converges to a positive real number (say δ) which implies that ϕ(δ) ≤ ϕ(d(xn , xn+1 )). Thus, ϕ(δ) ≤ lim ϕ(d(xn , xn+1 )) =  = 0. n→∞

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Since ϕ(s) = 0 if and only if s = 0 then, δ = 0. By lemma (2.6), there exist r ∈ [0, 1) such that, ϕ(d(xn , xn+1 )) ≤ rϕ(d(xn−1 , xn )). Therefore, ∞ X

ϕ(d(xn , xn+1 )) ≤

n=1

≤

µ X n=1 µ X

ϕ(d(xn , xn+1 )) +

∞ X

ϕ(d(xn , xn+1 ))

n=µ+1

ϕ(d(xn , xn+1 )) +

n=1

∞ X

rn ϕ(d(xµ , xµ+1 ))

n=1

< ∞. By defintion of ϕ, we have d(xn , xn+1 ) s ≤ lim < ∞. ϕ(d(xn , xn+1 )) s→0+ ϕ(s) P P Thus, the sequence {xn } is a ( , 6=)- Cauchy sequence. Since X is a ( , 6=) complete ν- generalized metric space and by lemma (2.5), it is Cauchy and then it is converge to some z ∈ X. From the definition of ϕ and its increasing we conclude that, lim sup

n→∞

ϕd(z, T z)) ≤ lim inf ϕ(d(xn+1 , T z) ≤ lim inf ϕ(H(T xn , T z)) n→∞

n→∞

≤ lim inf γ(ϕ(d(xn , z)))ϕ(d(xn , z)) ≤ lim inf ϕ(d(xn , z)) n→∞

n→∞

= lim ϕ(s) = lim ϕ(d(xn , xn+1 )) = 0. s→0+

n→∞

Therefore, ϕ(d(z, T z)) = 0. Thus by the definition of ϕ and since T z closed we have z ∈ T z is a fixed point.  References [1] A. Branciari. A fixed point theorem of Banach- Caccioppoli type on a class of generalized metric spaces. Publ. Math. 2000, 57 (1). [2] A. Alotaibi; S. M. Alsulami. Coupled coincidence points for monotone operators in partially ordered metric spaces. Fixed point Theory and Applications. 2011, 2011:44. [3] A. Amini- Harandi; D. O’Regan. Fixed point theorems for set- valued contraction type maps in metric spaces. Fixed point theory and appl. 2010, 2010:390183. [4] A.Eldred; J. Anuradha; P. Veeramani. On equivalence of generalized multi- valued contractions and Nadler’s fixed point theorem. [5] B. Alamri; T. Suzuki; LA. Khan. Caristis fixed point theorem and Subrahmanyams fixed point theorem in ν- generalized metric. J. Funct. Spaces. 2015. [6] F. Shaddad; M. Noorani; S. M. Alsulami. Common fixed point results of Ciric- Suzukitype inequality for multivalued maps in compact metric spaces. Journal of Inequalities and Applications. 2014, 2014:7. [7] G. Minak; I. Altun. Some new generalizations of Mizoguchi-Takahashi type fixed point theorem. J. of Ineq. and appl. 2013, 2013:493. [8] L. Ciric ; S. M. Alsulami . Some fixed point results in ordered Gp- metric spaces. Fixed point Theory and Applications. 2013, (317): 2013:317. [9] N. Hussain; H. K. Nashine; Z. Kadelburg; S. M. Alsulami. Weakly isotone increasing mappings and endpoints in partially ordered metric spaces. Journal of Inequalities and Applications. 2012. [10] N. Mizugushi, Takahashi, W. fixed point theorem for multivalued Mapping On Complete Metric Spaces. Int. journal of Math Anal and appl. 1989. [11] S. Banach. Sur les oprations dans les ensembles abstraits et leur application aux quations intgrales. Fondamental Mathematica. 1922, 3,133-181. [12] S. Nadler. Multi-valued contraction mapping. Pacific Journal Of Mathematics. 1969, vol.30, No. 2.

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8

SALHA ALSHAIKEY *, SAUD M. ALSULAMI AND MONAIRAH ALANSARI

[13] T. Kamran. Mizoguchi- Takahashi’s type fixed point theorem. Com. and Math. with Appl. 2009, 507-511. [14] T. Suzuki. Mizoguchi- Takahshi’s fixed point theorem is a real generlization of Nadler’s. J. Math. Anal. Appl. 2008, 340,752-755. [15] T. Suzuki. Generalized metric spaces do not have the compatible topology. Abstr. Appl. Anal. 2014. [16] T. Suzuki. Nadler’s fixed point theorem in ν - generalized metric space. Fixed Point Theory Appl. 2017. [17] T. Suzuki; B. Alamri; LA. Khan. Some notes on fixed point theorems in ν- generalized metric spaces. Bull. KyushuInst. Technol. Pure Appl. Math. 2015, 615-23. [18] Z. Mitrovic; S. Radenovic. The Banach and Reich contractions in bν (s)- metric space. Mathematics Subject Classification. 2010. Department of Mathematics, King Abdulaziz University, P.O. Box: 138381, Jeddah 21323, Saudi Arabia E-mail address: [email protected].(*Corresponding author) E-mail address: [email protected] E-mail address: [email protected]

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On Multiresolution Analyses Of Multiplicity n Richard A. Zalik

∗

Abstract This paper studies multiresolution analyses in L2 (Rd ) that have more than one scaling function and are generated by an arbitrary dilation matrix. It provides a further analysis of a representation theorem obtained by the author for such MRA’s.

1

Introduction

The concept of multiresolution analysis of multiplicity n is due to Alpert [1, 2, 3] who introduced his now well known dyadic multiresolution analysis with an arbitrary number of filters in L2 (R). Alpert’s results motivated a number of papers, focused on the univariate case, such as Herv´e [12, 13], Donovan, Geronimo and Hardin [6, 7], Geronimo and Marcell´an [10], Goodman, Lee and Tang [9], Goodman and Lee [8], and Hardin, Kessler and Massopoust [11]. Multiresolution analyses of multiplicity 1 (i.e., with a single scaling function) with arbitrary expansive matrices in L2 (R) were studied by Lemari´e [15, 16] and Madych [17], among others, and we should also mention Wojtaszczyk’s excellent textbook [25]. Properties of low pass filters and scaling functions in this context were studied by San Antol´ın [19, 20, 21, 22] and Cifuentes, Kazarian and San Antol´ın [5]. Saliani [18] extended these results to multiresolution analyses of multiplicity n generated by an expansive matrix. These results were further extended by Soto–Bajo [24] to multiresolution analyses having an arbitrary (not necessarily finite) set of generator functions. In [4], Behera studied multiwavelet packets and frame packets of L2 (Rd ) associated with multiresolution analyses of multiplicity n generated by an expansive matrix. In [26] the author presented a representation theorem for such multiresolution analyses, in [27] he gave some simple examples for the case n = 1, and in [23] San Antol´ın and the author obtained representation theorems for vector valued wavelets. Some of the authors cited above have showed that by using more than one scaling function it is possible to ∗

Department of Mathematics and Statistics, Auburn University, AL 36849-5310, [email protected] 1055

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construct wavelets that have a set of properties not available for wavelets associated with a multiresolution analysis having a single scaling function; for instance in [6] they constructed wavelets associated with more than two scaling functions having compact support, arbitrary regularity, orthogonality, and symmetry. These results would indicate that the further study of multiresolution analyses of multiplicity n may lead to other interesting results. In what follows, Z will denote the set of integers, Z+ the set of strictly positive integers and R the set of real numbers; C will denote the set of complex numbers, and I will stand for the identity matrix. Boldface lowcase letters will denote elements of Rd ; x · y will stand for the standard dot product of the vectors x and y; the vector norm || · || is defined by ||x||2 := x · x. If A is a matrix ||A|| will denote the matrix norm induced by the vector norm || · ||. The inner product of two functions f, g ∈ L2 (Rd ) will be denoted by hf, gi, their bracket product by [f, g], and the norm of f by ||f ||; thus, Z hf, gi := f (t)g(t) dt, Rd

[f, g](t) :=

X

f (t + k)g(t + k),

k∈Zd

and ||f || :=

p

hf, f i.

The Fourier transform of a function f will be denoted by fb. If f ∈ L(Rd ), Z b f (x) := e−i2πx·t f (t) dt. Rd

Let A ∈ Cd×d and |a| := det(A). For every j ∈ Z and k ∈ Zd the dilation operator DjA and the translation operator Tk are defined on L2 (Rd ) by DjA f (t) := |a|j/2 f (Aj t) and Tk f (t) := f (t + k) respectively. Let T := [0, 1], and let Td denote the d–fold cartesian product of T. A function f will be called Zd –periodic if it is defined on Rd and Tk f = f for every k ∈ Zd . Let u = {u1 , · · · , um } ⊂ L2 (Rd ); then T (u) = T (u1 , · · · , um ) := {Tk u; u ∈ u, k ∈ Zd } and S(u) = S(u1 , · · · , um ) := span T (u), where the closure is in L2 (Rd ). S(u) is called a finitely generated shift–invariant space or FSI and the functions u` are called the generators of S(u). In this case we will also use the symbols T (u1 , · · · , un ) and S(u1 , · · · , un ) to denote S(u) and T (u) respectively. 1056

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We also define T (Aj ; u) = T (Aj ; u1 , · · · , um ) := {DjA Tk u` ; ` = 1, · · · m, k ∈ Zd }, and S(Aj ; u) = Sj (Aj ; u1 , · · · , um ) := span T (Aj ; u). Given a sequence of functions u := {u1 , · · · , um } in L2 (Rd ), by G[u1 , · · · , um ](x), Gu (x) or G(x) we will denote its Gramian matrix, viz.  m G(x) := [ub` , ubj ](x) . `,j=1

Let Λ ⊂ Z and u = {uk ; k ∈ Λ} ⊂ S ⊂ L2 (Rd ). If S is a shift–invariant space then u is called a basis generator of S, if for every f ∈ S there are Zd –periodic functions pk , uniquely determined by f (up to a set of measure 0), such that X fb = pk u ck . k∈Λ

In what follows we will assume that A is a fixed matrix preserving the lattice Zd , i.e. AZd ⊂ Zd . We will also assume that A is expansive, that is, there exist constants C > 0 and δ > 1 such that for every j ∈ Z+ and x ∈ Rd ||Aj x|| ≥ Cδ j ||x||. Lemma 1. A is expansive if and only if all its eigenvalues have modulus larger that 1. Proof. Suppose first that all the eigenvalues of A have modulus larger that 1. If A is a Jordan block, then A = λI+N, where N has 1’s on the superdiagonal and 0’s elsewhere, and from e.g. [14, Lemma 3.1.4] we deduce that d   X j k j−k A x= λ N x k j

k=0

whence the assertion readily follows, and therefore it also follows when A is in Jordan canonical form. In general, if Q is the Jordan form of A, then Q = B−1 AB and we have Cδ j ||y|| ≤ ||Qj y|| = ||B−1 Aj By|| ≤ ||B−1 || ||Aj By||. Setting x = By the assertion readily follows. Conversely, if A has an eigenvalue λ with modulus less or equal to 1 and v is an eigenvector for λ with ||v|| = 1, then Av = λI; hence ||Aj v|| = |λ|j ≤ |λ|, and ||Aj v|| remains bounded. So A is not expansive. The previous proof was suggested by Wayne Lawton. An elementary proof may be found in San Antol´ın’s thesis [19, Lema A.12].

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A multiresolution analysis (MRA) of multiplicity n in L2 (Rd ) (generated by A) is a sequence {Vj ; j ∈ Z} of closed linear subspaces of L2 (Rd ) such that: (i) Vj ⊂ Vj+1 for every j ∈ Z. (ii) For every j ∈ Z, f (t) ∈ Vj if and only if f (At) ∈ Vj+1 . (iii)

S

j∈Z Vj

is dense in L2 (Rd ).

(iv) There are functions u := {u1 , · · · , un } such that T (u) is an orthonormal basis of V0 . From [18, Lemma 17] we know that if {Vj ; j ∈ Z} is a multiresolution analysis, then \ Vj = {0}. (1) j∈Z

This generalizes a result of Cifuentes, Kazarian and San Antol´ın, which was established for multiresolution analyses of multiplicity 1 (cf. [5, Lemma 4]). It follows from the definition of multiresolution analysis that there are Zd –periodic functions p`,j ∈ L2 (Td ) such that the functions u` satisfy the scaling identity ∗

u b` (A x) =

n X

p`,j (x)b uj (x),

j, ` = 1, · · · , n

a.e.,

j=1

where A∗ is the transpose of A. The functions u` are called scaling functions for the multiresolution analysis, and the functions p`,j are called the low pass filters associated with u. A finite set of functions ψ = {ψ1 , · · · , ψm } ∈ L2 (Rd ) will be called an orthonormal wavelet system if the affine sequence {DjA Tk ψ` ; j ∈ Z, k ∈ Zd , ` = 1, · · · , m} is an orthonormal basis of L2 (Rd ). Let ψ := {ψ1 , · · · , ψm } be an orthogonal wavelet system in L2 (Rd ) generated by a matrix A, let M := {Vj ; j ∈ Z} be a multiresolution analysis and let Wj denote the orthogonal complement of Vj in Vj+1 . We say that ψ is associated with an MRA, if T (ψ) is an orthonormal basis of W0 .

2

Representation of orthonormal wavelets.

For k > 1 let diag {−eiω , 1, · · · , 1}k denote the k × k diagonal matrix with −eiω , 1, · · · , 1 as its diagonal entries. With the convention that Arg 0 = 0 we have Theorem 1. Let M := {Vj ; j ∈ Z} be a multiresolution analysis of multiplicity n with scaling functions u := {u1 , · · · , un }, generated by a matrix A that preserves the lattice 1058

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Zd . For 1 ≤ ` ≤ n, let {v`,1 , · · · , v`,|a| } be an orthonormal basis generator of S(A, u` ), let e := (1, 0, · · · , 0) ∈ Rk , and ub` (x) =

|a| X

b`,j (x)b v`,j (x),

(2)

j=1

b` (x) := (b`,1 (x), · · · , b`,|a| (x))T ,

δ` (x) := eiArg b`,1 (x) ,

q` (x) := b` (x) + δ` (x)e,

b (x) := (b v v1,1 (x), · · · , vb1,|a| (x), · · · , vbn,1 (x), · · · , vbn,|a| (x))T , and

h i Q` (x) := diag {−δ` (x), 1, · · · , 1}|a| I − 2q` (x)q` (x)∗ /q` (x)∗ q` (x) .

Let a := det A, m := |a|n, and let  m Q(x) = q`,k (x)

`,k=1

be the m × m block diagonal matrix 

Q1 (x)

0 ..

 Q1 (x) ⊕ Q2 (x) ⊕ · · · ⊕ Qm (x) = 

 .

.

0



Qn (x)

If (b y1 (x), · · · , ybm (x))T := Q(x)b v(x), then y(`−1)|a|+1 = u` ;

1 ≤ ` ≤ n,

(3)

and {y(`−1)|a|+k ; 1 ≤ ` ≤ n, 2 ≤ k ≤ |a|} is an orthonormal wavelet system associated with M . The preceding theorem was proved in [26, Theorem 9] but there was arguably a gap in the proof, which is bridged by the following Lemma 2. Let m = n(|a| − 1), let {Vj ; j ∈ Z} be a multiresolution analysis and assume that {u` ; ` = 1, · · · , n} is an orthonormal basis generator of V0 . Then V1 = S(A, u1 ) ⊕ S(A, u2 ) ⊕ · · · ⊕ S(A, un ). Proof. From [26, Theorem 3] we know that there exist functions v`,k , ` = 1, · · · n, such that {v`,1 , · · · v`,|a| } is an orthonormal basis generator of S(A, u` ). Therefore {v`,k ; 1 ≤ ` ≤ n, 1 ≤ k ≤ |a|} is an orthogonal basis generator of S(A, u1 ) ⊕ S(A, u2 ) ⊕ · · · ⊕ S(A, un ), which is a subspace of V1 . But [26, Theorem 3] also tells us that every Riesz generator of V1 has |a|n functions, and the assertion readily follows from [26, Theorem 1]. 1059

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We can actually say more: the following theorem elucidates the structure of a multiresolution analysis of multiplicity n: Theorem 2. Let {Vj ; j ∈ Z} be a multiresolution analysis and assume that {u` ; ` = 1, · · · , n} is an orthonormal basis generator of V0 . Then (a) If j > 0, Vj = S(Aj , u1 ) ⊕ S(Aj , u2 ) ⊕ · · · ⊕ S(Aj , un ). (b) L2 (Rd ) =

∞ [

S(Aj ; u1 )+

j=0

∞ [

S(Aj ; u2 )+ · · · · · · +

j=0

∞ [

S(Aj ; un )

j=0

Proof. From [26, Theorem 4] we know that every orthonormal basis generator of Vj , j > 0, must have n|a|j functions, and an argument similar to the one employed in the proof of Lemma 2 yields (a). To prove (b), let f ∈ L2 (Rd ) and let ε > 0 be given; then there is a j ∈ Z+ and a g ∈ Vj such that ||f − g|| < ε. Since a fortiori g belongs to the closed set ∞ [ j=0

+

S(Aj ; u1 )

∞ [

+······+

S(Aj ; u2 )

j=0

∞ [

S(Aj ; un )

j=0

and ε is arbitrary, the assertion follows. Theorem 3. Let m = n(|a| − 1), let the functions yk , k = 1 · · · |a|n be constructed as in Theorem 1, and let qj = yj+` for ` = 1, · · · n, j = `|a|, · · · , (` + 1)(|a| − 1). Then {w1 , · · · wr } is an orthonormal wavelet system if and only if r = m and there exists an orthogonal matrix Q(x) such that (w1 , · · · wm )T = Q(x)(q1 , · · · qm )T . Proof. From Theorem 1 we know that {q1 , · · · qm } is an orthonormal basis system or, equivalently, that it is an orthonormal basis generator of S(u). The assertion now readily follows from [26, Corollary 3 and Theorem 5].

References [1] B. Alpert, Sparse representation of smooth linear operators (1990). Ph. D. Thesis, Yale University. [2] B. Alpert, Wavelets and Other Bases for Fast Numerical Linear Algebra, in “Wavelets: A Tutorial in Theory and Applications” (C.K. Chui, Editor), Academic Press (1992), pp. 217–236. [3] B. Alpert, A class of bases in L2 for the sparse representation of integral operators, SIAM J. Math. Anal. 24 (1993), no. 1, 246–262, 1060

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[4] B. Behera, Multiwavelet packets and frame packets of L2 (Rd ). Proc. Indian Acad. Sci. Math. Sci. 111 (2001), no. 4, 439–463. [5] F. Cifuentes, K. S. Kazarian and A. San Antol´ın, Characterization of scaling functions in a multiresolution analysis. Proc. American Math. Society 133, No. 3 (2004) 1013–1023. [6] G. C. Donovan, J. S. Geronimo, and D. P. Hardin, Intertwining multiresolution analyses and the construction of piecewise-polynomial wavelets, SIAM J. Math. Anal. 27 (1996), no. 6, 1791–1815 [7] G. C. Donovan, J. S. Geronimo, and D. P. Hardin, Construction of orthogonal wavelets using fractal interpolation functions, SIAM J. Math. Anal. 27 (1996), no. 4, 1158–1192. [8] T. N. T. Goodman and S. L. Lee, Wavelets of multiplicitiy r. Trans. American Math. Soc. 342, No. 1 (1994), 307–324. [9] T. N. T. Goodman, S. L. Lee, and W. S. Tang, Wavelets in wandering subspaces, Trans. Amer. Math. Soc. 338, No. 2 (1993), 639–54. [10] J. S. Geronimo and F. Marcell´an, On Alpert Multiwavelets. Proc. American Math. Society 143, No. 6 (2015), 2479–2494. [11] D. P. Hardin, B. Kessler, and P. R. Massopust, Multiresolution analyses based on fractal functions. J. Approx. Theory 71, No. 1 (1992),104–120. [12] L. Herv´e, Th`ese, Laboratoire de Probabilit´es, Universit´e de Rennes–I, 1992. [13] L. Herv´e, Multi–Resolution Analysis of Multiplicity d: Applications to Dyadic Interpolation. Appl. Comput. Harm. Anal. 1, (1994) 299–315. [14] R. A. Horn and C. J. Johnson, Matrix Analysis, Cambridge University Press, Reprinted 1993. [15] P.G. Lemari´e–Rieusset, Fonctions d‘e´chelle pour les ondelettes de dimension n. C. R. Acad. Sci. Paris S´er. I Math. 316 (1993), no. 2, 145–148. [16] P.G. Lemari´e–Rieusset, Projecteurs invariants, matrices de dilatation, ondelettes de dimension n et analyses multi–r´esolutions. Revista Mat. Iberoamer. 10 (1994), 283–347. [17] W. Madych, Some Elementary Properties of Multiresolution Analyses of L2 (R). In “Wavelets: A Tutorial in Theory and Applications” (C. K. Chui, Ed.). Academic Press (1992), 259–294. [18] S. Saliani, On stable refinable function vectors with arbitrary support, J. Approximation Theory 154 (2008), 105–125. [19] A. San Antol´ın, Caracterizaci´on y propiedades de las funciones de escala y filtros de paso bajo de un an´alisis de multirresoluci´ on. Tesis, Universidad Aut´onoma de Madrid, 2007. [20] A. San Antol´ın, Characterization of low pass filters in a multiresolution analysis, Studia Math. 190 (2009), no. 2, 99–116. 1061

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[21] A. San Antol´ın, Lawton’s Condition on Regular Low Pass Filters, J. Concrete Applic. Mathematics 8, No. 3 (2010), 416–425. [22] A. San Antol´ın, On low pass filters in a frame multiresolution analysis. Tohoku Math. J. (2) 63 (2011), no. 3, 427–439. [23] A. San Antol´ın and R. A. Zalik, Matrix–valued wavelets and multiresolution analysis. J. Appl. Funct. Anal. 7 (2012), no. 1–2, 13–25. [24] M. Soto–Bajo, Closure of dilates of shift-invariant subspaces, Cent. Eur. J. Math. 11, No. 10 (2013), 1785-1799 . [25] P. Wojtaszczyk, A Mathematical Introduction to Wavelets. Cambridge University Press, Cambridge, 1997. [26] R. A. Zalik, Bases of translates and multiresolution analyses, Appl. Comput. Harm. Anal. 24 (2008), 41-57. Corrigendum. Appl. Comput. Harm. Anal. 29 (2010), 121. [27] R. A. Zalik, On Orthonormal Wavelet Bases, J. Computational Analysis and Applications 27 (2019), 790–797.

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Generalized g–Fractional vector Representation Formula and integral Inequalities for Banach space valued functions George A. Anastassiou Department of Mathematical Sciences University of Memphis Memphis, TN 38152, U.S.A. [email protected] Abstract Here we give a very general fractional Bochner integral representation formula for Banach space valued functions. We derive generalized left and righ fractional Opial type inequalities, fractional Ostrowski type inequalities and fractional Grüss type inequalities. All these inequalities are very general having in their background Bochner type integrals.

2010 AMS Mathematics Subject Classi…cation : 26A33, 26B40, 26D10, 26D15, 46B25, 46E40. Keywords and Phrases: Bamach space valued functions, vector generalized fractional derivative, Caputo fractional derivative, generalized vector fractional integral inequalities, Bochner integral, fractional vector representation formula.

1

Background

We need De…nition 1 ([2]) Let [a; b] R, (X; k k) a Banach space, g 2 C 1 ([a; b]) and increasing, f 2 C ([a; b] ; X), > 0. We de…ne the left Riemann-Liouville generalized fractional Bochner integral operator Z x 1 1 0 Ia+;g f (x) := (g (x) g (z)) g (z) f (z) dz; (1) ( ) a 8 x 2 [a; b], where is the gamma function. The last integral is of Bochner type. Since f 2 C ([a; b] ; X), then f 2 0 L1 ([a; b] ; X). By [2] we get that Ia+;g f 2 C ([a; b] ; X). Above we set Ia+;g f := f and see that Ia+;g f (a) = 0: 1

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When g is the identity function id, we get that Ia+;id = Ia+ , the ordinary left Riemann-Liouville fractional integral Z x 1 1 Ia+ f (x) = (x t) f (t) dt; (2) ( ) a 8 x 2 [a; b], Ia+ f (a) = 0: We need Theorem 2 ([2]) Let ;

> 0 and f 2 C ([a; b] ; X). Then

+ Ia+;g Ia+;g f = Ia+;g f = Ia+;g Ia+;g f:

(3)

We need De…nition 3 ([2]) Let [a; b] R, (X; k k) a Banach space, g 2 C 1 ([a; b]) and increasing, f 2 C ([a; b] ; X), > 0. We de…ne the right Riemann-Liouville generalized fractional Bochner integral operator Ib

;g f

(x) :=

1 ( )

Z

b

(g (z)

1

g (x))

g 0 (z) f (z) dz;

(4)

x

8 x 2 [a; b], where is the gamma function. The last integral is of Bochner type. Since f 2 C ([a; b] ; X), then f 2 L1 ([a; b] ; X). By [2] we get that Ib ;g f 2 C ([a; b] ; X). Above we set Ib0 ;g f := f and see that Ib ;g f (b) = 0: When g is the identity function id, we get that Ib right Riemann-Liouville fractional integral Ib f (x) =

1 ( )

Z

b

(t

x)

1

;id

= Ib , the ordinary

f (t) dt;

(5)

x

8 x 2 [a; b], with Ib f (b) = 0: We need Theorem 4 ([2]) Let ; Ib

> 0 and f 2 C ([a; b] ; X). Then ;g Ib ;g f

= Ib +;g f = Ib

;g Ib ;g f:

(6)

We will use De…nition 5 ([2]) Let > 0, d e = n, d e the ceiling of the number. Let f 2 C n ([a; b] ; X), where [a; b] R, and (X; k k) is a Banach space. Let g 2 C 1 ([a; b]) ; strictly increasing, such that g 1 2 C n ([g (a) ; g (b)]) : 2

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We de…ne the left generalized g-fractional derivative X-valued of f of order as follows: Z x 1 (n) n 1 0 (g (t)) dt; Da+;g f (x) := (g (x) g (t)) g (t) f g 1 (n ) a (7) 8 x 2 [a; b]. The last integral is of Bochner type. If 2 = N, by [2], we have that Da+;g f 2 C ([a; b] ; X). We see that n Ia+;g

f

1 (n)

g

g

(x) = Da+;g f (x) ; 8 x 2 [a; b] :

(8)

We set n Da+;g f (x) :=

f

g

1 (n)

g (x) 2 C ([a; b] ; X) , n 2 N,

(9)

0 Da+;g f (x) = f (x) ; 8 x 2 [a; b] :

When g = id, then Da+;g f = Da+;id f = D a f;

(10)

the usual left X-valued Caputo fractional derivative, see [3]. We will use De…nition 6 ([2]) Let > 0, d e = n, d e the ceiling of the number. Let f 2 C n ([a; b] ; X), where [a; b] R, and (X; k k) is a Banach space. Let g 2 C 1 ([a; b]) ; strictly increasing, such that g 1 2 C n ([g (a) ; g (b)]) : We de…ne the right generalized g-fractional derivative X-valued of f of order as follows: Z b n ( 1) (n) n 1 0 Db ;g f (x) := (g (t) g (x)) g (t) f g 1 (g (t)) dt; (n ) x (11) 8 x 2 [a; b]. The last integral is of Bochner type. If 2 = N, by [2], we have that Db ;g f 2 C ([a; b] ; X). We see that Ibn

n

;g

( 1)

f

g

1 (n)

g (x) = Db

;g f

(x) ; a

x

b:

(12)

We set Dbn

;g f

1 n

n

f

Db0

;g f

(x) := f (x) ; 8 x 2 [a; b] :

Db

;g f

(x) = Db

(x) := ( 1)

g

g (x) 2 C ([a; b] ; X) , n 2 N,

(13)

When g = id, then ;id f

(x) = Db f;

(14)

the usual right X-valued Caputo fractional derivative, see [3]. 3

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We make Remark 7 All as in De…nition 5. We have (by Theorem 2.5, p. 7, [5]) Z x 1 (n) n 1 0 (g (t)) dt (g (x) g (t)) g (t) f g 1 Da+;g f (x) (n ) a f

g

1 (n)

g

(n f

1;[a;b]

)

g(x)

n

(g (x)

g (t))

(g (x)

g (a))

1

dg (t) =

g(a)

1 (n)

g

Z

g

(n

1;[a;b]

+ 1)

n

:

(15)

That is f

1 (n)

g

Da+;g f (x)

(n

g

1;[a;b]

+ 1)

8 x 2 [a; b] : If 2 = N, then Da+;g f (a) = 0. Similarly, by De…nition 6 we derive Z b 1 n Db ;g f (x) (g (t) g (x)) (n ) x f

g

1 (n)

g

(n f

1;[a;b]

)

g (a))

g 0 (t)

f

g(b)

n

(g (t)

g (x))

(g (b)

g (x))

1 (n)

g

1

;

(16)

(g (t)) dt

dg (t) =

g(x)

1 (n)

g

Z

1

n

(g (x)

(n

g

1;[a;b]

+ 1)

n

:

(17)

That is f Db

;g f

g

(x)

8 x 2 [a; b] : If 2 = N, then Db

1 (n)

(n ;g f

g

1;[a;b]

+ 1)

(g (b)

n

g (x))

;

(18)

(b) = 0.

Notation 8 We denote by n Da+;g := Da+;g Da+;g :::Da+;g (n times), n 2 N; n Ia+;g := Ia+;g Ia+;g :::Ia+;g ;

Dbn ;g

:= Db

Ibn

:= Ib

(19) (20)

;g Db ;g :::Db ;g ,

(21)

;g Ib ;g :::Ib ;g ;

(22)

and ;g

(n times), n 2 N: 4

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We are motivated by the following generalized fractional Ostrowski type inequality: Theorem 9 ([2]) Let g 2 C 1 ([a; b]) and strictly increasing, such that g 1 2 C 1 ([g (a) ; g (b)]), and 0 < < 1, n 2 N, f 2 C 1 ([a; b] ; X), where (X; k k) is a Banach space. Let x0 2 [a; b] be …xed. Assume that Fkx0 := Dxk0 ;g f , for k = 1; :::; n; ful…ll Fkx0 2 C 1 ([a; x0 ] ; X) and Dxi 0 ;g f (x0 ) = 0, i = 2; :::; n: Similarly, we assume that Gxk0 := Dxk0 +;g f , for k = 1; :::; n; ful…ll Gxk0 2 1 C ([x0 ; b] ; X) and Dxi 0 +;g f (x0 ) = 0, i = 2; :::; n: Then Z b 1 1 f (x) dx f (x0 ) b a a (b a) ((n + 1) + 1) (n+1)

(g (b)

g (x0 ))

(g (x0 )

g (a))

(n+1)

(n+1)

(b

x0 ) Dx0 +;g f (n+1) ;g

(x0

a) Dx0

f

1;[x0 ;b]

+

1;[a;x0 ]

:

(23)

In this work we will present several generalized fractional Bochner integral inequalities. We mention the following g-left generalized X-valued Taylor’s formula: Theorem 10 ([2]) Let > 0, n = d e, and f 2 C n ([a; b] ; X), where [a; b] R and (X; k k) is a Banach space. Let g 2 C 1 ([a; b]), strictly increasing, such that g 1 2 C n ([g (a) ; g (b)]). Then f (x) = f (a) +

n X1

i

(g (x)

g (a))

i=1

1 ( )

Z

x

(g (x) n X1

g(x)

1 (i)

(g (a)) +

g 0 (t) Da+;g f (t) dt =

(g (x)

i

(g (x)

g (a)) i!

i=1

Z

1

g (t))

g

a

f (a) + 1 ( )

f

i!

z)

1

f

g

Da+;g f

1 (i)

1

g

g(a)

(g (a)) +

(24)

(z) dz; 8 x 2 [a; b] :

We mention the following g-right generalized X-valued Taylor’s formula: Theorem 11 ([2]) Let > 0, n = d e, and f 2 C n ([a; b] ; X), where [a; b] R and (X; k k) is a Banach space. Let g 2 C 1 ([a; b]), strictly increasing, such that g 1 2 C n ([g (a) ; g (b)]). Then f (x) = f (b) +

n X1 i=1

i

(g (x)

g (b)) i!

f

g

1 (i)

(g (b)) +

5

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Z

1 ( )

b

(g (t) n X1

g(b)

(z

i

(g (x)

g (b))

f

i!

i=1

Z

g 0 (t) Db

;g f

(t) dt =

x

f (b) + 1 ( )

1

g (x))

1

g (x))

Db

g

;g f

g

1 (i)

1

g(x)

(g (b)) +

(25)

(z) dz; 8 x 2 [a; b] :

For the Bochner integral excellent resources are [4], [6], [7] and [1], pp. 422428.

2

Main Results

We give the following representation formula: Theorem 12 All as in Theorem 10. Then Z b n X1 f g 1 (k) (g (y)) Z b 1 f (y) = f (x) dx (g (x) b a a k! (b a) a

k

g (y)) dx+R1 (y) ;

k=1

(26)

for any y 2 [a; b], where 1 ( ) (b

R1 (y) = "Z

b [a;y) (x)

y

x

a

Z

Z

b [y;b]

(x)

a

Z

y

jg (x)

x

jg (x)

1

g (t)j 1

g (t)j

a)

g 0 (t) Dy

;g f

(t) dt dx #

0

g (t) Dy+;g f (t) dt dx :

(27)

here A stands for the characteristic function set A, where A is an arbitrary set. One may write also that Z y Z y 1 1 0 R1 (y) = (g (t) g (x)) g (t) Dy ;g f (t) dt dx ( ) (b a) a x (28) # Z b Z x 1 0 + (g (x) g (t)) g (t) Dy+;g f (t) dt dx ; y

y

for any y 2 [a; b] : Putting things together, one has f (y) =

1 b

a

Z

a

b

f (x) dx

n X1 k=1

f

g 1 k! (b

(k)

(g (y)) a)

Z

b

(g (x)

k

g (y)) dx

a

6

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J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 29, NO.6, 2021, COPYRIGHT 2021 EUDOXUS PRESS, LLC

1 ( ) (b Z

"Z

a)

Z

b [a;y) (x)

a

jg (x)

x

Z

b

(x)

[y;b]

y

a

x

jg (x)

y

1

g (t)j 1

g (t)j

g 0 (t) Dy

;g f

(t) dt dx #

0

g (t) Dy+;g f (t) dt dx :

(29)

In particular, one has f (y)

Z

1 b

a

n X1

b

f (x) dx+

a

g 1 k! (b

f

k=1

Z

(k)

(g (y)) a)

b

k

(g (x)

g (y)) dx = R1 (y) ;

a

(30)

for any y 2 [a; b] : Proof. Here x; y 2 [a; b]. We keep y as …xed. By Theorem 10 we get: f (x) = f (y) + 1 ( )

n X1

f

1 (k)

g

k=1

Z

x

(g (x)

(g (y))

(g (x)

k! 1

g (t))

k

g (y)) +

g 0 (t) Dy+;g f (t) dt; for any x

(31) y:

y

By Theorem 11 we get: f (x) = f (y) +

n X1

f

1 (k)

g

Z

y

(g (t)

(g (x)

k!

k=1

1 ( )

(g (y))

1

g (x))

g 0 (t) Dy

;g f

k

g (y)) +

(t) dt; for any x

(32)

y:

x

By (31), (32) we notice that Z

b

f (x) dx =

a

Z

n X1

f (y) dx +

a

1 ( ) Z

y

a

f

Z

b

f (x) dx =

(33)

y

1 (k)

(g (y))

k!

(g (t)

Z

y

(g (x)

k

g (y)) dx+

a

1

g (x))

g 0 (t) Dy

;g f

(t) dt dx+

x

y

n X1

f

k=1

y

f (x) dx +

y

f (y) dx + 1 ( )

g

k=1

Z

b

Z

y

a

y

Z

Z

b

Z

g

1 (k)

(g (y))

k!

b

(g (x)

k

g (y)) dx+

y

x

(g (x)

Z

g (t))

1

g 0 (t) Dy+;g f (t) dt dx:

y

7

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Hence it holds Z b n X1 f 1 f (x) dx = f (y) + b a a

g 1 k! (b

k=1

1 ( ) (b a) Z b Z y

Z

Z

y

x

jg (x)

jg (x)

y

1 ( ) (b a) Z b Z y

Z

y

a

1

jg (x)

g (t)j

(g (x)

k

g (y)) dx+

a

(34)

g (t) Dy

(g (y)) a)

1

g (t)j 1

b

0

(k)

y

jg (x)

Z

;g f

(t) dt dx+ #

g 0 (t) Dy+;g f (t) dt dx :

g 1 k! (b

f

k=1

x

x

y

n X1

1

g (t)j

g (t)j

Therefore we obtain Z b 1 f (y) = f (x) dx b a a Z

(g (y)) a)

y

x

a

(k)

Z

b

(g (x)

k

g (y)) dx

a

g 0 (t) Dy

(35) ;g f

(t) dt dx+ #

g 0 (t) Dy+;g f (t) dt dx :

Hence the remainder Z y Z y 1 1 0 R1 (y) := g (t) Dy ;g f (t) dt dx jg (x) g (t)j ( ) (b a) a x # Z b Z x 1 0 + jg (x) g (t)j g (t) Dy+;g f (t) dt dx = y

1 ( ) (b

+

a)

Z

a

"Z

y

b

a

b [y;b]

[a;y) (x)

(x)

Z

y

x

Z

y

jg (x)

x

jg (x)

g (t)j

g (t)j

1

1

g 0 (t) Dy

;g f

(t) dt dx #

(36)

g 0 (t) Dy+;g f (t) dt dx :

The theorem is proved. Next we present a left fractional Opial type inequality: Theorem 13 All as in Theorem 10. Additionally assume that 1, g 2 1 1 (k) C ([a; b]), and f g (g (a)) = 0, for k = 0; 1; :::; n 1. Let p; q > 1 : 1 1 p + q = 1. Then Z x 1 (37) kf (w)k Da+;g f (w) g 0 (w) dw 1 a ( ) 2q 1 2 Z x Z w Z x p q q q p( 1) (g (w) g (t)) dt dw Da+;g f (w) dw (g 0 (w)) ; a

a

a

8 x 2 [a; b] :

8

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Proof. By Theorem 10, we have that Z x 1 1 0 f (x) = (g (x) g (t)) g (t) Da+;g f (t) dt; ( ) a

8 x 2 [a; b] :

(38)

Then, by Hölder’s inequality we obtain, Z

1 ( )

kf (x)k

x

p(

(g (x)

1)

g (t))

Z

1 p

dt

a

x

q

(g 0 (t))

Da+;g f (t)

q

1 q

dt

:

a

Call z (x) :=

Z

x

q

(g 0 (t))

(39) q

Da+;g f (t)

dt;

(40)

0;

(41)

0; 8 x 2 [a; b] :

(42)

a

z (a) = 0. Thus

q

z 0 (x) = (g 0 (x)) and

Da+;g f (x)

q

1

(z 0 (x)) q = g 0 (x)

Da+;g f (x)

Consequently, we get kf (w)k g 0 (w) Z

1 ( )

w

(g (w)

g (t))

1)

1 p

dt

(43) 1

(z (w) z 0 (w)) q ;

8 w 2 [a; b] :

a

Then

Z

x

kf (w)k

a

1 ( )

p(

Da+;g f (w)

Da+;g f (w) g 0 (w) dw

Z x Z w 1 p( 1) (g (w) g (t)) dt ( ) a a Z x Z w p( 1) (g (w) g (t)) dt dw a

1 ( )

1 p

1

(z (w) z 0 (w)) q dw Z

1 p

a

Z

Z

x

a

w

(g (w)

p(

g (t))

1)

Z

Z

x

a

Z

w

0

z (w) z (w) dw

=

a

(45) 1 p

dt dw p(

(g (w)

1 q

x

a

1 ( )

(44)

g (t))

1)

2

z (x) 2

1 q

=

1 p

dt dw

a

x 0

q

(g (t))

Da+;g f (t)

q

2 q

dt

2

1 q

:

(46)

a

The theorem is proved. We also give a right fractional Opial type inequality:

9

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Theorem 14 All as in Theorem 11. Additionally assume that 1, g 2 1 1 1 (k) C ([a; b]), and f g (g (b)) = 0, k = 0; 1; :::; n 1. Let p; q > 1 : p + 1q = 1. Then Z b 1 kf (w)k Db ;g f (w) g 0 (w) dw (47) 1 x 2q ( ) ! q2 ! ! p1 Z Z Z b

b

p(

(g (t)

b

1)

g (w))

w

x

all a

q

(g 0 (w))

dt dw

Db

;g f

(w)

q

dw

;

x

x

b:

Proof. By Theorem 11, we have that Z

1 ( )

f (x) =

b

(g (t)

1

g (x))

g 0 (t) Db

;g f

(t) dt; all a

x

b: (48)

x

Then, by Hölder’s inequality we obtain, Z

1 ( )

kf (x)k

b

p(

(g (t)

1)

g (x))

dt

x

Call z (x) :=

Z

b

q

(g 0 (t))

! p1

Db

Z

b

q

0

(g (t))

Db

;g f

(t)

x

;g f

q

dt

! q1

:

(49) q

(t)

dt;

(50)

0;

(51)

0;

(52)

0; 8 x 2 [a; b] :

(53)

x

z (b) = 0. Hence z 0 (x) =

q

Db

;g f

(x)

q

Db

;g f

(x)

(g 0 (x))

and

z 0 (x) = (g 0 (x)) and

q

q

1

( z 0 (x)) q = g 0 (x)

Db

;g f

(x)

Consequently, we get kf (w)k g 0 (w) 1 ( )

Z

b

p(

(g (t)

1)

g (w))

w

Then

Z

Z

b

x

Z

! p1

dt

;g f

(w) 1

(z (w) ( z 0 (w))) q ; 8 w 2 [a; b] :

(54)

b

x

1 ( )

Db

kf (w)k

b

(g (t)

Db

(w) g 0 (w) dw

;g f

p(

g (w))

1)

w

dt

! p1

(55) 1

( z (w) z 0 (w)) q dw

10

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Z

1 ( )

b

Z

b

p(

(g (t)

g (w))

!

1)

dt dw

w

x

Z

1 ( )

Z

b

1 2

p(

(g (t)

1)

g (w))

( )

Z

Z

b

dt dw

b

p(

(g (t)

b

q

0

(g (t))

;g f

0

z (w) z (w) dw

q

(t)

dt

x

! q1

= (56)

! p1

z 2 (x) 2 ! ! p1

1)

g (w))

Db

b

x 1 q

=

dt dw

w

x

Z

Z

!

w

x

1 q

b

! p1

! q2

:

(57)

The theorem is proved. Two extreme fractional Opial type inequalities follow (case p = 1, q = 1). Theorem 15 All as in Theorem 10. Assume that f k = 0; 1; :::; n 1. Then Z

x

Da+;g f

kf (w)k Da+;g f (w) dw

a

all a

x

Z

2 1

( + 1)

g

1 (k)

(g (a)) = 0,

x

(g (w)

g (a)) dw ; (58)

a

b:

Proof. For any w 2 [a; b], we have that Z w 1 f (x) = (g (w) g (t)) ( ) a and kf (x)k

1 ( ) =

Z

1

w

(g (w)

g (t))

g 0 (t) Da+;g f (t) dt;

1

g 0 (t) dt

Da+;g f

a

Da+;g f

1

( + 1)

(g (w)

g (a)) :

Da+;g f

2 1

(59)

1

(60)

Hence we obtain kf (w)k Da+;g f (w)

( + 1)

(g (w)

g (a)) :

(61)

Integrating (61) over [a; x] we derive (58). Theorem 16 All as in Theorem 11. Assume that f 0; 1; :::; n 1. Then Z

2 b

x

all a

kf (w)k Db x

;g f

(w) dw

Db

;g f

1

( + 1)

Z

g

b

x

(g (b)

1 (k)

(g (b)) = 0, k = !

g (w)) dw ; (62)

b: 11

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Proof. For any w 2 [a; b], we have Z

1 ( )

f (x) =

(g (t)

1

g (w))

g 0 (t) Db

;g f

(t) dt;

(63)

w

and

Z

1 ( )

kf (x)k

b

b

(g (t)

g (w))

1

(g (b)

g (w)) :

g 0 (t) dt

w

=

Db

;g f

1

( + 1)

!

Db

;g f 1

(64)

Hence we obtain 2

kf (w)k Db

;g f

Db

(w)

;g f

1

( + 1)

(g (b)

g (w)) :

(65)

Integrating (65) over [x; b] we derive (62). Next we present three fractional Ostrowski type inequalities: Theorem 17 All as in Theorem 10. Then f (y)

h

1 b

a

Z

b

f (x) dx +

a

n X1

g 1 k! (b

f

k=1

(k)

1 ( + 1) (b (g (y)

g (a)) (y

a) Dy

;g f 1

(g (y)) a)

Z

b

(g (x)

k

g (y)) dx

a

(66)

a)

+ (g (b)

g (y)) (b

8 y 2 [a; b] :

y) Dy+;g f

1

i

;

Proof. De…ne Dy+;g f (t) = 0, for t < y; and Dy ;g f (t) = 0, for t > y: Notice for 0
1 :

(

khk1

Dy

;g f 1

+ Dy+;g h

+ Dy+;g f !#

1

!

1

;

(84)

1, we get: k

(

"

(g (b) g (a)) 2 ( + 1)

g

1

;g h

g

!

1

g

Dy+;g h

(85)

1;[g(a);g(b)]

1 1;[g(a);g(b)]

+

!)

;

> 1q , we get:

g

g

1

g (a))

Dy+;g f

+

1;[g(a);g(b)]

= 1,

;g f

+

1;[g(a);g(b)]

(g (b)

n (f; h)k

Dy

Dy

1

g

(g (b)

a)

g (a))

1 q;[g(a);g(b)]

1) + 1) p

+

q;[g(a);g(b)]

+

(86)

1

2 ( ) (p ( 1

1 1+ p

Dy+;g f

Dy+;g h

g

g

1 q;[g(a);g(b)]

1 q;[g(a);g(b)]

!

!)

+

:

All right hand sides of (84)-(86) are …nite. Proof. By Theorem 10 we have h (y) f (y) = n X1 k=1

h (y) f

g k! (b

1 (k)

a)

(g (y))

Z

h (y) b a

b

(g (x)

Z

b

f (x) dx

a

k

g (y)) dx + h (y) R1 (f; y) ;

(87)

a

16

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and f (y) h (y) = n X1 k=1

1 (k)

f (y) h g k! (b

Z

(g (y))

a)

f (y) b a

Z

b

h (x) dx

a

b

k

(g (x)

g (y)) dx + f (y) R1 (h; y) ;

(88)

a

8 y 2 [a; b] : Then integrating (87) we …nd Z

Rb

b

a

h (y) f (y) dy =

a

n X1 k=1

1 k! (b a)

Z

b

a

Z

b

a

b

h (y) f

Z

h (y) dy

1 (k)

g

!

b

f (x) dx

a

(g (y)) (g (x)

k

g (y)) dxdy

a

+

Z

b

h (y) R1 (f; y) dy;

(89)

a

and integrating (88) we obtain Z

Rb

b

a

f (y) h (y) dy =

a

n X1 k=1

1 k! (b a)

Z

b

a

Z

f (y) dy

a

b

b

1 (k)

f (y) h g

Rb

h (x) dx

a

(g (y)) (g (x)

k

g (y)) dxdy

a

+

Z

b

f (y) R1 (h; y) dy:

(90)

a

Adding the last two equalities (89) and (90), we get: 2

Z

2

b

f (x) h (x) dx =

a

n X1 k=1

1 k! (b a) (g (x)

""Z

a

b

Z

b

h (y) f

g

Rb a

f (x) dx

1 (k)

b

Rb a

h (x) dx

a

(g (y)) + f (y) h g

1 (k)

(g (y))

a

i Z k g (y)) dxdy +

#

b

(h (y) R1 (f; y) + f (y) R1 (h; y)) dy:

(91)

a

Divide the last (91) by 2 (b a) to obtain (83). Then, we upper bound Kn (f; h) using Theorems 17, 18, 19, to obtain (84)(86), respectively. We use also that a norm is a continuous function. The theorem is proved. We make 17

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Remark 21 (in support of the proof of Theorem 20) Let > 0, We have Z x 1 n 1 0 Dy+;g f (x) = (g (x) g (t)) g (t) f g (n ) y 8 x 2 [y; b] ; and n

Dy

( 1) (n )

;g f (x) =

Z

y

n

(g (t)

1

g (x))

g 0 (t) f

2 = N, d e = n. 1 (n)

(g (t)) dt; (92)

1 (n)

g

(g (t)) dt;

x

(93) 8 x 2 [a; y] ; both are Bochner type integrals. By change of variables for Bochner integrals, see [6], Lemma B. 4.10 and [7], p. 158, we get: 1 (n

Dy+;g f (x) =

)

Z

g(x)

n

(g (x)

1

z)

f

1 (n)

g

(z) dz =

g(y)

Dg(y)+ f

g

1

(g (x)) ; 8 x 2 [y; b] ;

(94)

and n

Dy

( 1) (n )

;g f (x) =

Dg(y) Here Dg(y)+ ; Dg(y) tiation operators. Fix w : w x0

Z

f

g(y)

n

(z

f

1

g

g

)

) g

g Z

1

(95)

Dg(x0 )+ f

g(x0 )

n

(g (w)

g (x0 )

g (y0 ). Hence

Dx0 +;g f (w) =

(g (w))

1

z)

f

g

f

g

1

g

1 (n)

(g (w)) = (z) dz

(96)

g(y0 )

Z

g(x0 )

(g (w)

n

1

z)

1 (n)

(z) dz

g(y0 )

1 (n) 1;[g(a);g(b)]

)

1 (n)

(n

(z) dz =

(g (x)) ; 8 x 2 [a; y] :

y0 ; w; x0 ; y0 2 [a; b], then g (w)

(n f

1 (n)

g

are the left and right X-valued Caputo fractional di¤ eren-

Dg(y0 )+ f

1 (n

f

g(x)

Dy0 +;g f (w)

1 (n

1

g (x))

1;[g(a);g(b)]

+ 1)

Z

g(x0 )

(g (w)

z)

n

1

(g (x0 )

z)

dz =

g(y0 )

h

(g (y0 )

n

z)

n

i

! 0;

18

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as y0 ! x0 , then g (y0 ) ! g (x0 ), proving continuity of Dg(y)+ f

g

1

(g (x))

with respect to g (y), and of course continuity of Dy+;g f (x) in y 2 [a; b] : Similarly, it is proved that Dy ;g f (x) is continuous in y 2 [a; b], the proof is omitted. Remark 22 Some examples for g follow: g (x) = ex , x 2 [a; b] R; g (x) = sin x; g (x) = tan x; h i where x 2 + "; " , where " > 0 small. 2 2

Indeed, the above examples of g are strictly increasing and contimuous functions. One can apply all of our results here for the above speci…c choices of g. We choose to omit this job.

References [1] C.D. Aliprantis and K.C. Border, In…nite Dimensional Analysis, Springer, New York, 2006. [2] G.A. Anastassiou, Principles of General Fractional Analysis for Banach space valued functions, Bulletin of Allahabad Math. Soc., 32 (1) (2017), 71-145. [3] G. Anastassiou, Intelligent Computations: Abstract Fractional Calculus, Inequalities, Approximations, Springer, Heidelberg, New York, 2018. [4] Bochner integral, Encyclopedia of Mathematics, URL: http://www.encyclo pediaofmath.org/index.php?title=Bochner_integral&oldid=38659. [5] M. Kreuter, Sobolev Space of Vector-valued functions, Ulm Univ., Master Thesis in Math., Ulm, Germany, 2015. [6] K. Mikkola, Appendix B Integration and Di¤ erentiation in Banach Spaces, http://math.aalto.…/~kmikkola/research/thesis/contents/thesisb.pdf [7] J. Mikusinski, The Bochner integral, Academic Press, New York, 1978.

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On solutions of semilinear second-order impulsive functional differential equations Ah-ran Park1 and Jin-Mun Jeong2,∗ 1,2 Department

of Applied Mathematics, Pukyong National University Busan 48513, Republic of Korea

Abstract This paper deals with the regularity for solutions of second-order semilinear impulsive differential equations contained the nonlinear convolution with cosine families, and obtain a variation of constant formula for solutions of the given equations. Keywords:semilinear second-order equations, regularity for solutions, cosine family, sine family AMS Classification Primary 35F25; Secondary 35K55

1

Introduction

In this paper we are concerned with the regularity of the following second-order semilinear impulsive differential system  Rt 00  w (t) = Aw(t) + k(t − s)g(s, w(s))ds + f (t), 0 < t ≤ T,  0 0 w(0) = x0 , w (0) = y0 ,   ∆w(tk ) = Ik1 (w(tk )), ∆w0 (tk ) = Ik2 (w0 (t+ k = 1, 2, ..., m k )),

(1.1)

in a Banach space X. Here k belongs to L2 (0, T ) and g : [0, T ] × D(A) → X is a nonlinear mapping such that w 7→ g(t, w) satisfies Lipschitz continuous. In (1.1), Email: 1 [email protected], 2,∗ [email protected]( Corresponding author) This work was supported by a Research Grant of Pukyong National University(2021Year).

1

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2 the principal operator A is the infinitesimal generator of a strongly continuous cosine family C(t), t ∈ R. The impulsive condition ∆w(tk ) = Ik1 (w(tk )),

∆w0 (tk ) = Ik2 (w0 (t+ k )),

k = 1, 2, ..., m

is combination of traditional evolution systems whose duration is negligible in comparison with duration of the process, such as biology, medicine, bioengineering etc. In recent years the theory of impulsive differential systems has been emerging as an important area of investigation in applied sciences. The reason is that it is richer than the corresponding theory of classical differential equations and it is more adequate to represent some processes arising in various disciplines. The theory of impulsive systems provides a general framework for mathematical modeling of many real world phenomena(see [1, 2] and references therein). The theory of impulsive differential equations has seen considerable development. Impulsive differential systems have been studied in [3, 4, 5, 6], second-order impulsive integrodifferential systems in [7, 8], and Stochastic differential systems with impulsive conditions in [9, 10, 11]. In this paper, we allow implicit arguments about L2 -regularity results for semilinear hyperbolic equations with impulsive condition. These consequences are obtained by showing that results of the linear cases [12, 13] and semilinear case [14] on the L2 -regularity remain valid under the above formulation of (1.1). Earlier works prove existence of solution by using Azera Ascoli theorem. But we propose a different approach from that of earlier works to study mild, strong and classical solutions of Cauchy problems by using the properties of the linear equation in the hereditary part. This paper is organized as follows. In Section 2, we give some definition, notation and the regularity for the corresponding linear equations. In Section 3, by using properties of the strict solutions of linear equations in dealt in Section 2, we will obtain the L2 -regularity of solutions of (1.1), and a variation of constant formula of solutions of (1.1). Finally, we also give an example to illustrate the applications of the abstract results..

2

Preliminaries

In this section, we give some definitions, notations, hypotheses and Lemmas. Let X be a Banach space with norm denoted by || · ||. Definition 2.1. [15] A one parameter family C(t), t ∈ R, of bounded linear operators in X is called a strongly continuous cosine family if

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3 c(1) C(s + t) + C(s − t) = 2C(s)C(t),

for all s, t ∈ R,

c(2) C(0) = I, for each fixed x ∈ X.

c(3) C(t)x is continuous in t on R

If C(t), t ∈ R is a strongly continuous cosine family in X , then S(t), t ∈ R is the one parameter family of operators in X defined by Z t C(s)xds, x ∈ X, t ∈ R. (2.1) S(t)x = 0

The infinitesimal generator of a strongly continuous cosine family C(t), t ∈ R is the operator A : X → X defined by Ax =

d2 C(0)x. dt2

We endow with the domain D(A) = {x ∈ X : C(t)x is a twice continuously differentiable function of t} with norm ||x||D(A) = ||x|| + sup{||

d C(t)x|| : t ∈ R} + ||Ax||. dt

We shall also make use of the set E = {x ∈ X : C(t)x is a once continuously differentiable function of t} with norm

d C(t)x|| : t ∈ R}. dt It is not difficult to show that D(A) and E with given norms are Banach spaces. The following Lemma is from Proposition 2.1 and Proposition 2.2 of [1]. ||x||E = ||x|| + sup{||

Lemma 2.1. Let C(t)(t ∈ R) be a strongly continuous cosine family in X. The following are true : c(4) C(t) = C(−t) for all t ∈ R, c(5) C(s), S(s), C(t) and S(t) commute for all s, t ∈ R, c(6) S(t)x is continuous in t on R for each fixed x ∈ X,

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4 c(7) there exist constants K ≥ 1 and ω ≥ 0 such that ||C(t)|| ≤ Keω|t| for all t ∈ R, Z t1 eω|s| ds for all t1 , t2 ∈ R, ||S(t1 ) − S(t2 )|| ≤ K t2

c(8) if x ∈ E, then S(t)x ∈ D(A) and d d2 C(t)x = AS(t)x = S(t)Ax = 2 S(t)x, dt dt c(9) if x ∈ D(A), then C(t)x ∈ D(A) and d2 C(t)x = AC(t)x = C(t)Ax, dt2 c(10) if x ∈ X and r, s ∈ R, then Z s Z s S(τ )xdτ ∈ D(A) and A( S(τ )xdτ ) = C(s)x − C(r)x, r

r

c(11) C(s + t) + C(s − t) = 2C(s)C(t) for all s, t ∈ R, c(12) S(s + t) = S(s)C(t) + S(t)C(s) for all s, t ∈ R, c(13) C(s + t) = C(t)C(s) − S(t)S(s) for all s, t ∈ R, c(14) C(s + t) − C(t − s) = 2AS(t)S(s) for all s, t ∈ R. The following Lemma is from Proposition 2.4 of [15]. Lemma 2.2. Let C(t)(t ∈ R) be a strongly continuous cosine family in X with infinitesimal generator A. If f : R → X is continuously differentiable, x0 ∈ D(A), y0 ∈ E, and Z t S(t − s)f (s)ds, t ∈ R, w(t) = C(t)x0 + S(t)y0 + 0

then w(t) ∈ D(A) for t ∈ R, w is twice continuously differentiable, and w satisfies 00

0

w (t) = Aw(t) + f (t), t ∈ R, w(0) = x0 , w (0) = y0 .

(2.2)

Conversely, if f : R → X is continuous, w(t) : R → X is twice continuously differentiable, w(t) ∈ D(A) for t ∈ R, and w satisfies (2.2), then Z t w(t) = C(t)x0 + S(t)y0 + S(t − s)f (s)ds, t ∈ R. 0

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5 Proposition 2.1. Let f : R → X is continuously differentiable, x0 ∈ D(A), y0 ∈ E. Then Z t S(t − s)f (s)ds, t ∈ R

w(t) = C(t)x0 + S(t)y0 + 0

is a solution of (2.2) belonging to L2 (0, T ; D(A)) ∩W 1,2 (0, T ; E). Moreover, we have that there exists a positive constant C1 such that for any T > 0, ||w||L2 (0,T ;D(A)) ≤ C1 (1 + ||x0 ||D(A) + ||y0 ||E + ||f ||W 1,2 (0,T ;X) ).

3

(2.3)

Nonlinear equations

This section is to investigate the regularity of solutions of a second-order nonlinear impulsive differential system  Rt 00   w (t) = Aw(t) + 0 k(t − s)g(s, w(s))ds + f (t), 0 < t ≤ T, 0 (3.1) w(0) = x0 , w (0) = y0 ,   1 0 2 0 + ∆w(tk ) = Ik (w(tk )), ∆w (tk ) = Ik (w (tk )), k = 1, 2, ..., m in a Banach space X. Assumption (G) Let g : [0, T ] × D(A) → X be a nonlinear mapping such that t 7→ g(t, w) is measurable and (g1) ||g(t, w1 ) − g(t, w2 )||D(A) ≤ L||w1 − w2 ||, for a positive constant L. Assumption (I) Let Ik1 : D(A) → X, Ik2 : E → X be continuous and there exist positive constants L(Ik1 ), L(Ik2 ) such that (i1) ||Ik1 (w1 ) − Ik1 (w2 )|| ≤ L(Ik1 )||w1 − w2 ||D(A) , for each w1 , w2 ∈ D(A) ||Ik1 (w)|| ≤ L(Ik1 ), for w ∈ D(A) (i2) ||Ik2 (w10 ) − Ik2 (w20 )|| ≤ L(Ik2 )||w10 − w20 ||E , for each w10 , w20 ∈ E ||Ik2 (w0 )|| ≤ L(Ik2 )||, for w0 ∈ E. For w ∈ L2 (0, T : D(A)), we set Z t F (t, w) = k(t − s)g(s, w(s))ds 0

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6 where k belongs to L2 (0, T ). Then we will seek a mild solution of (3.1), that is, a solution of the integral equation Z

t

S(t − s){F (s, w) + f (s)}ds w(t) =C(t)x0 + S(t)y0 + 0 X X + C(t − tk )Ik1 (w(tk )) + S(t − tk )Ik2 (w0 (t+ k )), t ∈ R. 0 0 such that the functional differential equation (3.1) admits a unique solution w in L2 (0, T0 ; D(A)) ∩ W 1,2 (0, T0 ; E).

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7 Proof. Let us fix T0 > 0 so that 3/2

C2 ≡ω −1 KLT0 (eωT0 − 1)||k||L2 (0,T0 ) √ 3/2 + {ω −1 K(eωT0 − 1) + 1}T0 / 3L||KeωT0 + 1||||k||W 1,2 (0,T0 ) √ + {ω −1 K(eωT0 − 1) + 1}T0 / 2L||KeωT0 + 1||||k(0)|| X L(Ik1 )KewT0 + {w−1 K(ewT0 − 1) + 2}

(3.4)

0 0: Then

, n 2 N;

is a fuzzy sub-

i=1

1 k Q

(S)

[ai ; bi ] Dn+1 (f ) (n+1)!

i=1

k Q

f (t) d (t) [ai ;bi ]

i=1

i=1

k Q

Z

+1

[ai ; bi ]

(S)

Z

k Q

t [ai ;bi ]

i=1

0

B B B1 ^ @ a+b 2

f a+b 2 k Q [ai ; bi ] i=1

1 C C C A

n+1

d (t) :

(32)

l1

10

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Proof. By Theorem 15. We make Remark 18 By multinomial theorem we have that n+1 xkl1

kt X

=

i=1

n+1 r1 ; r2 ; :::; rk

r1 +r2 +:::+rk =n+1

k X

!n+1

jti

xi j r

r

x1 j 1 jt2

jt1

where n+1 r1 ; r2 ; :::; rk

=

=

rk

x2 j 2 ::: jtk

xk j

;

(n + 1)! : r1 !r2 !:::rk !

(33)

(34)

By (27), (28) we get (S)

Z

f (t) d (t)

(Q) ^ f (x)

Q

(S)

Z

Q

(S)

Z " Q

Dn+1 (f ) kt (n + 1)!

X

r1 +r2 +:::+rk =n+1

X

(S)

r1 +r2 +:::+rk =n+1

X

r1 +r2 +:::+rk =n+1

Z

Q

n+1

xkl1

d (t) k Y

Dn+1 (f ) r1 !r2 !:::rk !

i=1 k Y

Dn+1 (f ) r1 !r2 !:::rk !

Dn+1 (f ) + 1 (S) r1 !r2 !:::rk !

i=1

Z

Q

(by (33), (34))

=

jti

i=1

!#

ri

!

xi j

jti

k Y

ri

xi j

jti

ri

xi j

(2)

d (t) (5)

d (t)

!

d (t) :

(35)

We have proved the following multivariate Ostrowski-Sugeno general inequality: Theorem 19 Here all as in Theorem 15. Then Z f (x) 1 (S) f (t) d (t) 1^ (Q) (Q) Q X

r1 +r2 +:::+rk =n+1

We make

0 @

Dn+1 (f ) r1 !r2 !:::rk !

(Q)

+1

1

A (S)

Z

Q

k Y

i=1

jti

ri

xi j

!

d (t) :

(36)

11

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Remark 20 In case k = 2, n = 1, by (27), (28) we get Z (S) f (t) d (t) (Q) ^ f (x) Q

(S) (S)

Z

D2 (f ) kt 2

Q

Q

(S)

Z

Z

D2 (f ) h (t1 2

Q

2

x1 ) + 2 jt1

D2 (f ) (t1 2

(S)

Z

x1 ) d (t) + (S)

Q

Z

Q

Z

x1 j jt2

2

+ (S) D2 (f ) 1+ 2

2

xkl1 d (t) =

D2 (f ) (t2 2

x2 )

D2 (f ) jt1

x1 j jt2

x1 ) d (t)+(1 + D2 (f )) (S)

Z

Q

(S)

Z

d (t)

(37)

x2 j d (t)

2

Q

D2 (f ) + 1+ 2

i

x2 ) d (t)

2

(t1

2

x2 j + (t2

(t2

jt1

x1 j jt2

x2 j d (t)

2

x2 ) d (t) :

Q

We have proved Corollary 21 Let Q be a compact and convex subset of R2 . Let f 2 (C (Q; R+ ) @f @f \C 2 (Q)), x = (x1 ; x2 ) 2 Q be …xed: @t (x1 ; x2 ) = @t (x1 ; x2 ) = 0. Here is 1 2 a fuzzy subadditive measure with (Q) > 0. Then Z 1 f (x) (S) f (t) d (t) 1^ (Q) (Q) Q 1+

D2 (f ) 2

(Q)

(S)

Z

(t1

2

x1 ) d (t)+

Q

1+ +

D2 (f ) 2

(Q)

(1 + D2 (f )) (S) (Q)

(S)

Z

(t2

Z

Q

jt1

x1 j jt2

x2 j d (t) (38)

2

x2 ) d (t) :

Q

References [1] G.A. Anastassiou, Multivariate Ostrowski type inequalities, Acta Math. Hungar., 76 (4) (1997), 267-278. [2] G.A. Anastassiou, Quantitative Approximations, Chapman & Hall / CRC, Boca Raton, New York, 2001.

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[3] M. Boczek, M. Kaluszka, On the Minkowaki-Hölder type inequalities for generalized Sugeno integrals with an application, Kybernetica, 52(3) (2016), 329-347. [4] A. Ostrowski, Über die Absolutabweichung einer di¤ erentiebaren Funktion von ihrem Integralmittelwert, Comment. Math. Helv., 10 (1938), 226-227. [5] E. Pap, Null-Additive Set functions, Kluwer Academic, Dordrecht (1995). [6] D. Ralescu, G. Adams, The fuzzy integral, J. Math. Anal. Appl., 75 (1980), 562-570. [7] M. Sugeno, Theory of fuzzy integrals and its applications, PhD thesis, Tokyo Institute of Technology (1974). [8] Z. Wang, G.J. Klir, Fuzzy Measure Theory, Plenum, New York, 1992.

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SOME APPLICATIONS OF INTERVAL-VALUED SUBSETHOOD MEASURES WHICH ARE DEFINED BY INTERVAL-VALUED CHOQUET INTEGRALS IN TRADE EXPORTS BETWEEN KOREA AND ITS TRADING PARTNERS JACOB WOOD, LEE-CHAE JANG*, HYUN-MEE KIM

Department of Business James Cook University Singapore, 149 Sims Road, Singapore 387380 E-mail : [email protected] *Correspondence Author, Graduate School of Education, Konkuk University, Seoul 143-701, Republic of Korea E-mail : [email protected] Graduate School of Education, Kookmim University, Seoul 02707, Republic of Korea E-mail : [email protected]

Abstract. In this paper, we consider subsethood measures introduced by Fan et al. [3] and the interval-valued Choquet integral with respect to a fuzzy measure of interval-valued fuzzy sets. Based on such a focus, we define three types of interval-valued subsethood measures and provide four interval-valued fuzzy sets to animal product exports between Korea and four selected trading partners. In particular, we investigate a strong interval-valued subsethood measure defined by the interval-valued Choquet integral which represents the degree of trade surplus between Korea and three trading partners in terms of the model of trade transactions with the United States and Korea

1. Introduction

Zadeh[18] first developed fuzzy sets and Murofushi-Sugeno [11] have studied fuzzy measures and Choquet integrals. Subsequently, using set-valued analysis theory developed by Aumann[1], we studied interval-valued Choquet integrals and their related applications(see[5, 6, 7, 8, 9]). In particular, through the restudy of the interval-valued Choquet integral in 2004 by Zhang-Guo-Liu[21], this research has been developed in a much more systematical 1991 Mathematics Subject Classification. 28E10, 28E20, 03E72, 26E50 11B68. Key words and phrases. Choquet integral, fuzzy measure, subsethood measures, the degree of trade surplus in trade exports. 1

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manner. Xuechang [16], Zeng-Li[19] have examined fuzzy entropy, distance and similarity measures, the likes of which form three key concepts of fuzzy set theory. Ruan-Kerre [12] also introduced various fuzzy implication operators and the Choquet integral were suggested for the first time by Choquet [2]. Further studied by Murofushi-Sugeno [9], Jang-Kwon [10], and Jang [12] provide some interesting interpretations of fuzzy measures and the Choquet integral. Subjective probability and Choquet expected utility were studied as an application of Choquet integral and form another pivotal component of fuzzy sets and information theories(see[13, 14, 15, 20]). A subsethood measure refers to the degree to which a fuzzy set is a subset of another fuzzy set. Many researchers have contributed to the area of a fuzzy subsethood measure that is closely related to the various tools introduced above (see[6, 7, 8, 11]). Their efforts have considered axiomatizing the properties of a subsethoods measure. In this paper, we consider subsethood measures introduced by Fan et al. [3] and the interval-valued Choquet integral with respect to a fuzzy measure of interval-valued fuzzy sets. Based on such a focus, we define three types of interval-valued subsethood measures and provide four interval-valued fuzzy sets to animal product exports between Korea and four selected trading partners. In configuring the four interval-valued fuzzy sets, the original data(see[4]) used had to be slightly modified to produce Table A4. In order to calculate the interval-valued Choquet integral, the rules (46) and (48) were introduced. Furthermore, we also investigate a strong interval-valued subsethood measure defined by an interval-valued Choquet integral which represents the degree of trade surplus between Korea and 3 trading partners in terms of the model of trade transactions with the United States and Korea. The information for the above degree of surplus is of great significance in providing accurate comparative figures on the size of trade that exists between the four countries that trade with Korea.

2. Preliminaries and definitions

Throughout this paper, we write X to denote a set, F (X) = {A|A = {(x, mA (x))| x ∈ X}, mA : X −→ [0, 1] is a function}

(1)

stands for the set of fuzzy sets in X(see[18]). We note that mA expresses the membership of a fuzzy set A, Ac is the complement of A, that is, Ac = {(x, mAc (x))| mAc (x) = 1 − mA (x), x ∈ X}.

(2)

Recall that for A, B ∈ F (X), A ⊂ B if and only if mA (x) ≤ mB (x), for all x ∈ X, and for A ∈ F (X), [A] = {x ∈ X| mA (x) > 0}, n(A) is the P cardinal number of crisp set [A], and M (A) is the fuzzy cardinal of A, that is, M (A) = x∈X mA (x). Now, we introduce three types of subsethood measure in Fan et al. [3].

Definition 2.1. ([3]) Let c : F (X) × F (X) −→ [0, 1] be a function. (1) c is called a strong subsethood measure if c has the following properties; (S1 ) (S2 )

if A ⊂ B, then c(A, B) = 1; if A = 6 ∅ then c(A, B)

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(S3 )

if A ⊂ B ⊂ C, then c(C, A) ≤ c(B, A) and c(C, A) ≤ c(C, B).

3

(3)

(2) c is called a subsethood measure if c has the following properties; (C1 ) (C2 ) (C3 )

if A ⊂ B, then c(A, B) = 1; c(X, ∅) = 0 if A ⊂ B ⊂ C, then c(C, A) ≤ c(B, A) and c(C, A) ≤ c(C, B).

(4)

(3) c is called a weak subsethood measure if c has the following properties; (W1 ) (W2 ) (W3 )

c(∅, ∅) = 1, c(∅, ∅) = 1, c(A, B) = 1; and c(X, X) = 1 if A 6= ∅ rmand A ∩ B = ∅, then c(A, B) = 0; if A ⊂ B ⊂ C, then c(C, A) ≤ c(B, A) and c(C, A) ≤ c(C, B).

(5)

We also list the set-theoretical arithmetic operators for the set of subintervals of an unit interval [0, 1] in R. We denote I([0, 1]) = {a = [a− , a+ ]| a− , a+ ∈ [0, 1] and a− ≤ a+ }.

(6)

For any a ∈ [0, 1], we define a = [a, a].

Definition 2.2. ([5, 6, 7, 8, 9]) If a = [a− , a+ ], b = [b− , b+ ] ∈ I([0, 1]), and k ∈ [0, 1], then the addition, scalar multiplication, minimum, maximum, inequality, subset, multiplication, and division as follows; (1) a + b = [a− + b− , a+ + b+ ], (2) ka = [ka− , ka+ ], (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 6= b, (7) a ⊂ b if and only if b− ≤ a− and a+ ≤ b+ , (8) a ⊗ b = [a− b− , a+ b+ ], and (9) a  b = [a− /b− ∧ a+ /b+ , a− /b− ∨ a+ /b+ ].

From Definition 2.1 (9), the following theorem can be easily obtained. Theorem 2.1. (1) If a = [a− , a+ ] ∈ I([0, 1]), then a  a = 1. (2) If b = [b− , b+ ] ∈ I([0, 1]) and b− > 0, then 1  b = [1/b+ , 1/b− ].

Definition 2.3. ([5, 6, 8, 9, 21]) Let (X, Ω) be a measurable space. (1) A fuzzy measure on X is a real-valued function µ : Ω −→ [0, 1] satisfies (i) (ii)

µ(∅) = 0 µ(E1 ) ≤ µ(E2 ) whenever E1 , E2 ∈ Ω and E1 ⊂ E2 .

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(2) A fuzzy measure µ is said to be continuous from below if for any sequence {En } ⊂ Ω and E ∈ Ω, such that if En ↑ E, then

lim µ(En ) = µ(E).

n→∞

(8)

(3) A fuzzy measure µ is said to be continuous from above if for any sequence {En } ⊂ Ω and E ∈ Ω such that if

En ↓ E, then

lim µ(En ) = µ(E).

n→∞

(9)

(5) A fuzzy measure µ is said to be continuous if it is continuous from below and continuous from above.

Definition 2.4. ([5, 6, 8, 9, 21]) (1) Let A ∈ F (X). The Choquet integrals with respect to a fuzzy measure µ of a fuzzy set A on a set E ∈ Ω is defined by Z Z 1 µE,mA (r)dr, (10) Cµ,E (A) = (C) mA dµ = E

0

where µE,mA (r) = µ({x ∈ X| mA (x) > r} ∩ E) and the integral on the right-hand side is an ordinary one. (2) A measurable function is said to be integrable if Cµ (A) = Cµ,X (A) exists.

It is well known that if X is a finite set, that is, X = {x1 , x2 , · · · , xn }, and A ∈ F (X), then we have n X
Cµ (A) = mA (x(i) ) µ(E(i) ) − µ(E(i+1) ) , (11) i=1

where (·) indicate a permutation on {1, 2, · · · , n} such that mA (x(1) ) ≤ mA (x(2) ) ≤ · · · ≤ mA (x(n) ) and also E(i) = {(i), (i + 1), · · · , (n)} and E(n+1) = ∅.

Theorem 2.2. Let A, B ∈ F (X). (1) If A ≤ B, then Cµ (A) ≤ Cµ (B). (2) If we define (mA ∨ mB )(x) = mA (x) ∨ mB (x) for all x ∈ X, then Cµ (A ∨ B) ≥ Cµ (A) ∨ Cµ (B). (3) If we define (mA ∧ mB )(x) = mA (x) ∧ mB (x) for all x ∈ X, then Cµ (A ∧ B) ≥ Cµ (A) ∧ Cµ (B).

3. Three types of interval-valued subsethood measures defined by interval-valued Choquet integral

In this section, we consider the interval-valued Choquet integral and list some properties of them.

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5

Definition 3.1. ([5, 6, 8, 9, 21]) (1) The interval-valued Choquet integral of an interval-valued measurable function f = [f − , f + ] on E ∈ Ω is defined by Z  C µ,E (f ) = (C) f dµ = Cµ,E (f ) | f ∈ S(f ) , (12) E

where S(f ) is the family of measurable selection of f . (2) f is said to be integrable if C µ (f ) = C µ,X (f ) 6= ∅. (3) f is said to be Choquet integrably bounded if there is an integrable function g such that ||f (x)|| = supr∈f (x) |r| ≤ g(x), for all x ∈ X.

(13)

Theorem 3.1. ([5, 6, 21]) (1) If a closed set-valued measurable function f is integralble and if E1 ⊂ E2 and E1 , E2 ∈ Ω, then C µ,E1 (f ) ≤ C µ,E2 (f ). (2) If a fuzzy measure µ is continuous , and a closed set-valued measurable function f is Choquet integrably bounded, then C µ (f ) is a closed set. (3) If the fuzzy measure µ is continuous, and an interval-valued measurable function f = [f − , f + ] is Choquet integrably bounded, then we have C µ (f ) = [Cµ (f − ), Cµ (f + )].

(14)

Let IF (X) be the set of all interval-valued fuzzy sets which are defined by A = {(x, mA )|mA : X −→ I([0, 1])} .

(15)

By using Theorem 2.2 and Theorem 3.1(3), we easily obtain the following theorem.

Theorem 3.2. Let A, B ∈ IF (X). (1) If A ≤ B, then C µ (A) ≤ C µ (B). (2) If we define (mA ∨ mB )(x) = mA (x) ∨ mB (x) for all x ∈ X, then C µ (A ∨ B) ≥ C µ (A) ∨ C µ (B). (3) If we define (mA ∧ mB )(x) = mA (x) ∧ mB (x) for all x ∈ X, then C µ (A ∧ B) ≥ C µ (A) ∧ C µ (B).

We denote mA = [mA− , mA+ ] and define three types of interval-valued subsethood measures on IF (X) × IF (X) as follows:

Definition 3.2. Let c : IF (X) × IF (X) −→ I([0, 1]) be a function. (1) c is called a strong interval-valued subsethood measure if c has the following properties; (IS1 ) (IS2 ) (IS3 )

if A ⊂ B, then c(A, B) = 1; if A = 6 ∅ then c(A, B) if A ⊂ B ⊂ C, then c(C, A) ≤ c(B, A) and c(C, A) ≤ c(C, B).

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(2) c is called an interval-valued subsethood measure if c has the following properties; (IC1 ) (IC2 ) (IC3 )

if A ⊂ B, then c(A, B) = 1; c(X, ∅) = 0 if A ⊂ B ⊂ C, then c(C, A) ≤ c(B, A) and c(C, A) ≤ c(C, B).

(17)

(3) c is called a weak interval-valued subsethood measure if c has the following properties; (IW1 ) (IW2 ) (IW3 )

c(∅, ∅) = 1, c(∅, ∅) = 1, c(A, B) = 1; and c(X, X) = 1 if A 6= ∅ rmand A ∩ B = ∅, then c(A, B) = 0; if A ⊂ B ⊂ C, then c(C, A) ≤ c(B, A) and c(C, A) ≤ c(C, B).

(18)

Let IF ∗ (X) = {A ∈ IF (X)| A has the integrably bounded funstion mA }. Note that if X is a finite set, then IF (X) = IF ∗ (X). Finally, we give three types of interval-valued subsethood measures defined by the Choquet integral with respect to a fuzzy measure on IF ∗ (X). By Theorem 3.1 (3), we note that for A = [A− , A+ ], B = [B − , B + ] ∈ IF ∗ (X), C µ (A) = [Cµ (A− ), Cµ (A+ )], C µ (B) = [Cµ (B − ), Cµ (B + )], and C µ (A ∧ B) = [Cµ (A− ∧ B − ), Cµ (A+ ∧ B + )].

Theorem 3.3. Let X be a set. If we define an interval-valued function c1 : IF ∗ (X) × IF ∗ (X) −→ I([0, 1]), ( 1, if A = B = ∅, c1 (A, B) = (19) C µ (A∧B) , if not, C (A) µ

then c1 is a strong interval-valued subsethood measure on IF ∗ (X). Proof. (IS1 ) If A ≤ B and B = ∅, that is, A = B = ∅, then by the definition of c1 , we have c1 (AB) = 1. If A ≤ B and B = 6 ∅, then mA ≤ mB . Thus, we have mA− ≤ mB − and mA+ ≤ mB + . Hence, we get c1 (A, B)

= = =

(IS2 ) If A C µ (A∧B) = 0. C µ (A)

C µ (A ∧ B) C µ (A) [Cµ (A− ∧ B − ), Cµ (A+ ∧ B + )] [Cµ (A− ), Cµ (A+ )] [Cµ (A− ), Cµ (A+ )] = 1. [Cµ (A− ), Cµ (A+ )]

(20)

6= ∅ and A ∧ B = ∅, then we get 0 = m∅ = mA∧B and hence c1 (A, B) =

(IS3 ) If A ≤ B ≤ C, then we have mA ≤ mB ≤ mC

(21)

mC ∧ mA ≤ mB ∧ mA .

(22)

C µ (B) ≤ C µ (C), and C µ (C ∧ A)) ≤ C µ (B ∧ A).

(23)

and hence, by (21), we have Thus, by (21) and (22), we get

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Note that if C = ∅, then B = ∅. By using (23), we have ( 1, c1 (C, A) = C µ (C∧A) , ( C µ (C) 1, = C µ (B∧A) , C (B)

7

if C = ∅, if C 6= ∅ if B = ∅, if B 6= ∅

µ

= c1 (B, A).

(24)

From (21), we also get C µ (C ∧ A)) ≤ C µ (B ∧ B).

(25)

By using (25), we also have ( c1 (C, A)

= ( =

1,

if C = ∅,

C µ (C∧A) , C µ (C)

if C 6= ∅

1,

if B = ∅,

C µ (C∧B) , C µ (C)

if B 6= ∅

= c1 (C, B).

(26)

Therefore, c1 is a strong interval-valued subsethood measure.

Theorem 3.4. Let X be a set.If we define an interval-valued function c2 : IF ∗ (X) × IF ∗ (X) −→ I([0, 1]), ( 1, if A = B = ∅, c2 (A, B) = (27) C µ (B) , if not, C (A∨∧B) µ

then c2 is an interval-valued subsethood measure on IF ∗ (X). Proof. (IC1 ) If A = B = ∅, then c2 (A, B) = 1. Since A ≤ B, we have mA ≤ mB . Thus, we get c2 (A, B) =

C µ (B) = 1. C µ (A ∨ B)

(28)

(IC2 ) By the definition of c2 , we have c2 (X, ∅) =

C µ (∅) C µ (X ∨ ∅)

= 0.

(29)

(IC3 ) If A ≤ B ≤ C, then we have mA ≤ mB ≤ mC

(30)

C µ (C) ≥ C µ (B), C µ (C ∨ A) = C µ (C), and C µ (B ∨ A) = C µ (B).

(31)

and hence, by (30), we hsve

Therefore by using (31), we have c2 (C, A)

=

C µ (A) C µ (C ∨ A)

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= ≤ =

C µ (A) C µ (C) C µ (A) C µ (B) C µ (A) = c2 (B, A), C µ (B ∨ A)

(32)

and c2 (C, A)

= = ≤ =

C µ (A) C µ (C ∨ A) C µ (A) C µ (C) C µ (B) C µ (C) C µ (B) = c2 (C, B). C µ (C ∨ B)

(33)

Therefore, c2 is an interval-valued subsethood measure.

The following definition c3 has some problem because of the definition of a complement of interval-valued fuzzy set. So, we note that for interval-valued fuzzy sets A = [A− , A+ ], the mc modified complement A of A is defined by mAmc (x) = [mA+ (x), 1]. mc

Through this definition A

(34)

, we can take note of the followings: (i) mAmc = [mA+ , 1] (ii) mA∨Amc = [mA− , 1] mc mc (iii) if A ≤ B, then B ≤ A .

(35)

Theorem 3.5. Let X be a set. If we define an interval-valued function c3 : IF ∗ (X) × IF ∗ (X) −→ I([0, 1]), mc

c3 (A, B) =

C µ (A

) ∨ C µ (B)

mc

C µ (A ∨ A

∨B∨B

mc

(36)

)

then c3 is an interval-valued subsethood measure on IF ∗ (X). Proof. (IW1 ) By the definition of c3 , we get c3 (∅, ∅)

= = =

C µ (∅) ∨ C µ (∅) mc

C µ (∅ ∨ ∅ ∨ ∅ ∨ ∅ C µ (X) ∨ C µ (∅)

mc

C µ (∅ ∨ X ∨ ∅ ∨ X) C µ (X) = 1. C µ (X)

)

(37)

Similarly, we have c3 (∅, X) = 1 and c3 (X, X) = 1.

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9

(IW2 ) By the definition of c3 , we have c3 (X, ∅)

C µ (X

=

mc

) ∨ C µ (∅) mc

mc

C µ (X ∨ X ∨ ∅ ∨ ∅ C µ (∅) 0 = = 0. 1 C µ (X)

=

) (38)

(IW3 ) If A ≤ B ≤ C, then we have mA ≤ mB ≤ mC .

(39)

mC mc ≤ mB mc ≤ mAmc .

(40)

Thus, by (35)(i) and (39), we have

From (40), we get C µ (C ∨ C

mc

mc

∨A∨A

)

mc

= C µ (C ∨ A ) mc ≥ C µ (B ∨ A ) mc mc = C µ (B ∨ B ∨ A ∨ A )

(41)

Therefore by using (41), we have c3 (C, A)

= ≤

C µ (C

mc

) ∨ C µ (A)

mc

cm

C µ (C ∨ C ∨ A ∨ A ) mc C µ (B ) ∨ C µ (A) cm

C µ (B ∨ B = c3 (B, A).

cm

cm

∨A∨A

) (42)

Similarly, we have c3 (C, A) ≤ c3 (C, B).

(43)

Therefore, c3 is a weak interval-valued subsethood measure.

4. Applications

In this section, by using the of Harmonized system (HS) product code data for product categories (s1 , . . . , s5 ) between Korea and its trading partners (that is, Korea-United States, Korea-New Zealand, Korea-Turkey, and Korea-Indea) over the 2010-2013 period, we construct four interval-valued fuzzy sets related with four countries and calculate a strong interval-valued subsethood measure c1 . Note that the product code definitions have been provided by the UN Comtrade’s online data base(see[22]) and the relevant categories are defined as follows: s1 . Live animals: animal products. s2 . Meat and edible meat offal. s3 . Fish and crustacreans, mollusks and other aguatic invertebrates. s4 . Dairy produce: bird’s eggs; natural honey; edible products of animal origin, not elsewhere specified or included. s5 . Products of animal origin; not elsewhere specified or included.

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Firstly, we denote that s is year, a(s) is trade value, and u(a(s)) is the utility of a(s). By using the u(a(s)) for the trade values of animal product exports between Korea and selected trading partners for HS Product Codes i = 1, 2, 3, 4, 5 in Table A1 in [4], we can calculate the Choquet integral of an utility on the set of trade values (in USD) that represent Korea’s trading relationship with a particular country for years 2010, 2012, 2012, 2013. Let S = {s1 , s2 , s3 , s4 , s5 } and a(s) be the interval-valued trade value of s during four years and "r # r a− (s) a+ (s) u(a(s)) = (44) , 100141401 100141401 be an interval-valued utility of a(s). The following table A1 is used to create four intervalvalued fuzzy sets required to draw a strong subsethood measure c1 defined by the intervalvalued Choquet integral. Table A1: The u(a(s)) for the trade value of animal product exports between Korea and selected trading partners for HS Product Codes si for i = 1, 2, 3, 4, 5. TP

s s1 s USA 2 s3 s4 s5 s1 s2 NZ s3 s4 s5 s1 s2 TR s3 s4 s4 s1 s2 IND s3 s4 s5

a ¯(s)(USD) [144949, 364918] = a ¯(s(1) ) [144949, 997539] = a ¯(s(3) ) [74866073, 100141401] = a ¯(s(5) ) [3722326, 5016833] = a ¯(s(4) ) [1017895, 863858] = a ¯(s(2) ) [1589, 6650] = a ¯(s(2) ) (1) [0, 0] = a ¯(s ) [46632301, 91263506] = a ¯(s(5) ) [113751, 277350] = a ¯(s(3) ) [218022, 393025] = a ¯(s(4) ) [150, 6900] = a ¯(s(4) ) [0, 0] = a ¯(s(1) ) [199874, 2532837] = a ¯(s(5) ) (2) [0, 0] = a ¯(s ) [0, 0] = a ¯(s(3) ) [450, 1300] = a ¯(s(2) ) [12135, 50630] = a ¯(s(5) ) [1865, 8695] = a ¯(s(3) ) [12135, 30938] = a ¯(s(4) ) (2) [0, 0] = a ¯(s )

u ¯(¯ a(s)) [0.03542, 0.06037] [0.03542, 0.09981] [0.86464, 1.00000] [0.19280, 0.22382] [0.09288, 0.10082] [0.00398, 0.00815] [0.00000, 0.00000] [0.68240, 0.95464] [0.03370, 0.05263] [0.04666, 0.06265] [0.00122, 0.00830] [0.00000, 0.00000] [0.04468, 0.15904] [0.00000, 0.00000] [0.00000, 0.00000] [0.00212, 0.00360] [0.00992, 0.05551] [0.00432, 0.00932] [0.00992,0.02249] [0.00000, 0.00000]

We remark that in order to calculate the interval-valued Choquet integrals for four intervalvalued fuzzy sets, we modified four interval-valued trading values for the United States and the India (see Table 5 in [4]) as follows; [286892, 364918] = a(s1 ) and [30005, 997539] = a(s2 ) are changed by     286892 + 3005 286892 + 3005 , 364918 = a(s2 ) and , 997539 = a(s2 ), 2 2

(45)

(46)

and [2656, 50630] = a(s2 ) and [21614, 30938] = a(s4 )

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are changed by     2656 + 21614 2656 + 21614 , 50630 = a(s2 ) and , 30938 = a(s4 ), 2 2

11

(48)

From Table A1, we construct four interval-valued fuzzy sets from S to I([0, 1]), U = U (U SA), N = N (N Z), T = T (T R), and I = I(ID) as follows; U N T

= {(s1 , [0.03542, 0.06037], (s2 , [0.03542, 0.09981], (s3 , [0.86464, 1.00000]), (s4 , [0, 19280, 0.22382]), (s5 , [0.09288, 0.10082])},

(49)

= {(s1 , [0.00398, 0.00815], (s2 , [0.00000, 0.00000], (s3 , [0.68240, 0.95464]), (s4 , [0, 03370, 0.05263]), (s5 , [0.04666, 0.06265])},

(50)

= {(s1 , [0.00122, 0.00830], (s2 , [0.00000, 0.00000], (s3 , [0.04468, 0.15904]), (s4 , [0, 00000, 0.00000]), (s5 , [0.00000, 0.00000])},

(51)

= {(s1 , [0.00212, 0.00360], (s2 , [0.00992, 0.05551], (s3 , [0.00432, 0.00932]), (s4 , [0, 00992, 0.02249]), (s5 , [0.00000, 0.00000])},

(52)

and I

In order to calculate the interval-valued Choquet integral, the four interval-valued fuzzy sets((49), (50), (51), (52)) were made to be increasing interval-valued fuzzy sets as follows: U

= {(s(1) , [0.03542, 0.06037]), (s(2) , [0.03542, 0.09981]), (s(3) , [0.09288, 0.10082]), (s(4) , [0, 19280, 0.22382]), (s(5) , [0.86464, 1.00000])}, (53)

N

= {(s(1) , [0.00000, 0.00000]), (s(2) , [0.00398, 0.00815]), (s(3) , [0, 03370, 0.05263]), (s(4) , [0.04666, 0.06265]), (s(5) , [0.68240, 0.95464])}, (54)

T

= {(s(1) , [0.00000, 0.00000], (s(2) , [0, 00000, 0.00000], (s(3) , [0.00000, 0.00000]), (s(4) , [0.00122, 0.00830]), (s(5) , [0.04468, 0.15904])}, (55)

I

= {(s(1) , [0.00000, 0.00000], (s(2) , [0.00212, 0.00360], (s(3) , [0.00432, 0.00932]), (s(4) , [0, 00992, 0.02249]), (s(5) , [0.00992, 0.05551])}, (56)

and

Now, the more diversified export items, the higher fuzzy measure are defined as follows(see[4]): µ(E(6) ) = µ(∅) = 0, µ(E(5) ) = µ1 ({s(5) }) = 0.1, µ(E(4) ) = µ1 ({s(4) , s(5) }) = 0.2, µ(E(3) ) = µ1 ({s(3) , s(4) , s(5) }) = 0.4, µ(E(2) ) = µ1 ({s(2) , s(3) , s(4) , s(5) }) = 0.7, µ(E(1) ) = µ1 ({s(1) , s(2) , s(3) , s(4) , s(5) }) = 1. (57) By using two interval-valued fuzzy sets (53), (54) and the above fuzzy measure (57), we can calculate a strong interval-valued subsethood measure for c1 (U , N ) as follows: C µ (U ) = [Cµ (U − ), Cµ (U + )] C µ (U ∧ N ) = [Cµ (U − ∧ N − ), Cµ (U + ∧ N + )],

(58)

and c1 (U , N )

C µ (U ∧ N )  C µ (U−)  Cµ (U ∧ N − ) Cµ (U + ∧ N + ) Cµ (U − ∧ N − ) Cµ (U + ∧ N + ) ∧ , ∨ , (59) = Cµ (U − ) Cµ (U + ) Cµ (U − ) Cµ (U + ) =

where Cµ (U − )

= mU − (s(1) )(µ(E(1) ) − µ(E(2) ))

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JACOB WOOD, LEE-CHAE JANG*, HYUN-MEE KIM

Cµ (U + )

+mU − (s(2) )(µ(E(2) ) − µ(E(3) )) +mU − (s(3) )(µ(E(3) ) − µ(E(4) )) +mU − (s(4) )(µ(E(4) ) − µ(E(5) )) +mU − (s(5) )(µ(E(5) )) − µ(E(6) ),

(60)

= mU + (s(1) )(µ(E(1) ) − µ(E(2) )) +mU + (s(2) )(µ(E(2) ) − µ(E(3) )) +mU + (s(3) )(µ(E(3) ) − µ(E(4) )) +mU + (s(4) )(µ(E(4) ) − µ(E(5) )) +mU + (s(5) )(µ(E(5) ) − µ(E(6) )),

(61)

and Cµ (U − ∧ N − )

Cµ (U + ∧ N + )

= mU − ∧N − (s(1) )(µ(E(1) ) − µ(E(2) )) +mU − ∧N − (s(2) )(µ(E(2) ) − µ(E(3) )) +mU − ∧N − (s(3) )(µ(E(3) ) − µ(E(4) )) +mU − ∧N − (s(4) )(µ(E(4) ) − µ(E(5) )) +mU − ∧N − (s(5) )(µ(E(5) ) − µ(E(6) )) = (mU − (s(1) ) ∧ µN − (s(1) ))(µ(E(1) −)µ(E(2) )) +(mU − (s(2) ) ∧ mN − (s(2) ))(µ(E(2) ) − µ(E(3) )) +(mU − (s(3) ) ∧ mN − (s(3) ))(µ(E(3) ) − µ(E(4) )) +(mU − (s(4) ) ∧ mN − (s(4) ))(µ(E(4) ) − µ(E(5) )) +(mU − (s(5) ) ∧ mN − (s(5) ))(µ(E(5) ) − µ(E(6) )),

(62)

= mU + ∧N + (s(1) )(µ(E(1) ) − µ(E(2) )) +mU + ∧N + (s(2) )(µ(E(2) ) − µ(E(3) )) +mU + ∧N + (s(3) )(µ(E(3) ) − µ(E(4) )) +mU + ∧N + (s(4) )(µ(E(4) ) − µ(E(5) )) +mU + ∧N + (s(5) )(µ(E(5) ) − µ(E(6) )) = (mU + (s(1) ) ∧ mN + (s(1) ))(µ(E(1) ) − µ(E(2) )) +(mU + (s(2) ) ∧ mN + (s(2) ))(µ(E(2) ) − µ(E(3) )) +(mU + (s(3) ) ∧ mN + (s(3) ))(µ(E(3) ) − µ(E(4) )) +(mU + (s(4) ) ∧ mN + (s(4) ))(µ(E(4) ) − µ(E(5) )) +(mU + (s(5) ) ∧ mN + (s(5) ))(µ(E(5) ) − µ(E(6) )).

(63)

Thus, by using (58) and (59), we get the following table A2 for the strong interval-valued subsethood measure between United States and New Zealand.

Table A2: The c1 (U , N ) between United States and New Zealand. C µ (U ) C µ (U , N ) c1 (U , N ) [0.14557, 0.19060] [0.08084, 0.11470] [0.55534, 0.60178]

Given that c1 (U , V ) represents the degree of trade surplus for the trading relationship for Korea and USA, and Korea and New Zealand. Finally, we can calculate c1 (U , T ) and c1 (U , I) in Table A1.

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SOME APPLICATIONS OF INTERVAL-VALUED SUBSETHOOD MEASURES ...

13

Table A3: The c1 (U , T ) between United States and Turkey. C µ (U ) C µ (U , T ) c1 (U , T ) [0.14557, 0.19060] [0.00459, 0.01673] [0.03153, 0.08778]

Table A4: The c1 (U , I) between United States and India. C µ (U ) C µ (U , I) c1 (U , I) [0.14557, 0.19060] [0.00348, 0.01074] [0.02393, 0.05635]

Tables A2, A3, and A4, demonstrate the results c1 (U , N ), c1 (U , T ), c1 (U , I). They highlight the degree of trade surplus that exists with the three trading partners in terms of the model of trade transactions with United States and Korea.

5. Conclusions

Using the concept of intervals, we defined three types of interval-valued subsethood measures in Definitions 3.2, 3.3 and 3.4. From these definitions, we proposed three types of interval-valued subsethood measures defined by the interval-valued Choquet integrals with respect to a continuous fuzzy measure in Theorems 3.2, 3.3, and 3.4. The fuzzy measure µ in (57) means that if set E includes more categories between Korea and its trading partner, then µ(E) receives a higher score. Moreover, intervals are also a very useful tool to express the degree of trade surplus between Korea and its four trading partners analyzed over the 2010-2013 period. In order to illustrate some applications of a strong interval-valued subsethood measure, we provided the four interval-valued fuzzy sets which were aggregated in (49), (50), (51), and (52) to animal product exports between Korea and four selected trading partners from 2010 to 2013. By using these interval-valued fuzzy sets, we obtained the strong interval-valued subsethood measure c1 (U , N ), c1 (U , T ), c1 (U , I) which represent the degree of trade surplus between Korea and 3 trading partners in terms of the model of trade transactions with the United States and South Korea in Tables 2, 3, and 4. It was found that New Zealand was at least 0.55534 to 0.60178 times smaller than the United States, while Turkey and India were also smaller, with Turkey at least 0.03153 to 0.08778 times smaller and India being at least 0.2393 to 0.05635 times smaller than the United States between 2010 and 2013, in terms of the trade values of animal product exports that exists between Korea and selected trading partners. Data Availability: All the authors solemnly declare that there is no data used to support the fndings of this study.

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Competing interests: The authors declare that they have no competing interests. Funding: This research received no external funding.

References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22]

R. J. Aumann, Integrals of set-valued functions, J. Math. Analy. Appl., 12(1) (1965) 1-12. G. Choquet, Theory of capacities, Ann. Inst. Fourier, 5 (1953) 131-295. J. Fan, W. Xie, J. Pei, Subsethood measure: new definitiond, Fuzzy Sets Syst., 106 (1999) 201-209. L.C. Jang, J. Wood, The application of the Choquet integral expected utility in international trade, Advan. Stud. Contem. Math., 27(2) (2017) 159-173. L.C. Jang, B.M. Kil, Y.K. Kim, J.S. Kwon, Some properties of Choquet integrals of set-valued functions, Fuzzy sets Syst., 91 (1997) 95-98. L.C.Jang, J.S. Kwon, On the representation of Choquet integrals of set-valued functions and null sets, Fuzzy sets Syst., 112 (2000) 233-239. L.C. Jang, Subsethood measures defined by Choquet integrals, Int. J. Fuzzy Logic Intell. Syst., 8(2) (2008) 146-150. L.C. Jang, A note on convergence properties of interval-valued capacity functionals and Choquet integrals, Information Sciences, 183 (2012) 151-158. L.C. Jang, A note on the interval-valued generalized fuzzy integral by means of an interval-representable pseudo-multiplication and their convergence properties, Fuzzy Sets Syst., 222 (2013) 45-57. L. Mangelsdorff, M. Weber, Testing Choquet expected utility, J. Econ. Behavior and Organization, 25 (1994) 437-457. T. Murofushi, M. Sugeno, An interpretation of fuzzy measures and the Choquet integral as an integral with respect to a fuzzy measure, Fuzzy Sets Syst., 29 (1989) 201-227. D. Ruan, E. Kerre, Fuzzy implication operators and generalized fuzzy method of cases, Fuzzy Sets Syst. 54 (1993) 23-37. D. Schmeidler, Subjective probability and expected utility without additivity, Econometrica, 57(3) (1989) 571-587. J. Wood, L.C. Jang, A note on Choquet integrals and imprecise market premium functionals, Proc. Jangjeon Math.Soc., 18(4) (2015) 601-608. J. Wood, L.C. Jang, A study on the Choquet integral with respect to a capacity and its applications, Global J. Pure Applied Math., 12(2) (2016) 1593-1599. L. Xuechang, Entropy, distance maesure and similarity measure of fuzzy sets and their relations, Fuzzy Sets Syst., 52 (1992) 201-227. V.R. Young, Fuzzy subsethood , Fuzzy Sets Syst., 77 (1996) 371-384. L.A. Zadeh, Fuzzy sets , Informationm and Control, 8 (1965) 338-353. W. Zeng, H. Li, Relationship btween similarity maesure and entropy of interval-valued fuzzy sets , Fuzzy Sets Syst., 57 (2006) 1477-1484. D. Zhang, Subjective ambiguity, expected utility and Choquet expected utility, Econ. Theory, 20 (2002) 159-181. D. Zhang, C. Guo, D. Liu, Set-valued Choquet integrals revisited, Fuzzy Sets and Systems, 147(2004), 475-485. WTO (2016). WTO Regional Trade Database.

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M -fractional integral inequalities George A. Anastassiou Department of Mathematical Sciences University of Memphis Memphis, TN 38152, U.S.A. [email protected] Abstract Here we present M -fractional integral inequalities of Ostrowski and Polya types.

2010 AMS Mathematics Subject Classi…cation : 26A33, 26D10, 26D15. Keywords and phrases: M -fractional derivative, Ostrowski inequality, Polya inequality.

1

Introduction

We are inspired by the following results: Theorem 1 ([2], p. 498, [1], [5]) (Ostrowski inequality) Let f 2 C 1 ([a; b]), x 2 [a; b]. Then 1 b

a

Z

b

f (z) dz

f (x)

2

2

(x

a) + (b x) 2 (b a)

a

!

kf 0 k1 :

(1)

Inequality (1) is sharp. In particular the optimal function is f (z) := jz

xj (b

a) ,

> 1:

(2)

Theorem 2 ([6], [7, p. 62], [8], [9, p. 83]) (Polya integral inequality) Let f (x) be di¤ erentiable and not identically a constant on [a; b] with f (a) = f (b) = 0. Then there exists at least one point 2 [a; b] such that 0

jf ( )j >

4 (b

2

a)

Z

b

f (x) dx:

(3)

a

In this short work we present inequalities of types (1) and (3) involving the left and right fractional local general M -derivatives, see [3], [4].

1

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2

Background

We need De…nition 3 ([4]) Let f : [a; 1) ! R and t > a, a 2 R. For 0 < 1 we de…ne the left local general M -derivative of order of function f , denoted by ; DM;a f (t), by ; DM;a f (t) := lim

f tE

" (t

a)

1 P

k=0

tk ( k+1) ,

one parameter. ; If DM;a f (t) exists over (a; ),

;

"

"!0

8 t > a, where E (t) =

f (t) (4)

> 0, is the Mittag-Le- er function with

; 2 R and lim DM;a f (t) exists, then t!a+

; ; DM;a f (a) = lim DM;a f (t) :

(5)

t!a+

Theorem 4 ([4]) If a function f : [a; 1) ! R has the left local general M derivative of order 2 (0; 1], > 0, at t0 > a, then f is continuous at t0 : We need Theorem 5 ([4]) (Mean value theorem) Let f : [ ; ] ! R with [ ; ] ; such that (1) f is continuous on [ ; ] ; ; (2) there exists DM;a f on ( ; ) for some 2 (0; 1]: Then, there exists c 2 ( ; ) such that f( )

( + 1) (c c

; f ( ) = DM;a f (c)

a)

(

> a, 0 2 =

):

(6)

We need De…nition 6 ([3]) Let f : ( 1; b] ! R and t < b, b 2 R. For 0 < 1 we de…ne the right local general M -derivative of order of function f , denoted as ; M;b Df (t), by ; M;b Df

f tE (t) :=

" (b

lim

; M;b Df

f (t) ;

"

"!0

8 t < b. ; If M;b Df (t) exists over ( ; b),

t)

2 R and lim

;

t!b M;b

(b) = lim

;

t!b M;b

(7)

Df (t) exists, then

Df (t) :

(8)

2

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Theorem 7 ([3]) If a function f : ( 1; b] ! R has the right local general M -derivative of order 2 (0; 1], > 0, at t0 < b, then f is continuous at t0 : We also need Theorem 8 ([3]) (Mean value theorem) Let f : [ ; ] ! R with [ ; ] ; such that (1) f is continuous on [ ; ] ; ; (2) there exists M;b Df on ( ; ) for some 2 (0; 1]: Then, there exists c 2 ( ; ) such that f( )

; M;b Df

f( )=

( + 1) (b c

(c)

c)

(

< b, 0 2 =

):

(9)

; ; Fractional derivatives DM;a and M;b D possess all basic properties of the ordinary derivatives and beyond, see [3], [4].

3

Main Results

We present the following M -fractional Ostrowski type inequality: Theorem 9 Let a < < < b, 0 2 = [ ; ], f : [a; b] ! R, which is continuous ; ; over [ ; ]. We assume that DM;a , M;b D exist and are continuous over [ ; x0 ] and [x0 ; ], respectively, where x0 2 [ ; ], for some 2 (0; 1]: Then 1 2 4

; DM;a f (x)

x

Z

(x0

f (x) dx

f (x0 )

2

a) (x0

) +

( + 1) 2( )

; M;b Df

1;[ ;x0 ]

(x)

x

(b 1;[x0 ; ]

x0 ) (

3

2 x0 ) 5 :

(10)

Proof. Let x 2 [ ; x0 ], the by Theorem 5, there exists c1 2 (x; x0 ), such that ! ; DM;a f (c1 ) f (x0 ) f (x) = ( + 1) (c1 a) (x0 x) : (11) c1 Thus jf (x)

f (x0 )j =

; DM;a f (c1 )

c1

; DM;a f (x)

x

( + 1) (c1

( + 1) (x0 1;[ ;x0 ]

a) jx

a) jx

x0 j ;

x0 j (12)

3

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8 x 2 [ ; x0 ] : Let now x 2 [x0 ; ], then by Theorem 8, there exists c2 2 (x0 ; x), such that ! ; M;b Df (c2 ) f (x) f (x0 ) = ( + 1) (b c2 ) (x x0 ) : (13) c2 Thus jf (x)

; M;b Df

f (x0 )j = ; M;b Df

(x)

( + 1) (b

f (x) dx

Z

"Z 1

+

2 4

jf (x)

x0

jf (x)

f (x0 )j dx +

x

; M;b Df

2

jf (x)

( + 1) (b

x0 )

x

f (x0 )) dx

(15) #

(x0 1;[ ;x0 ]

(x)

(b 1;[x0 ; ]

x0 ) (

(by (12), (14))

f (x0 )j dx Z

a) Z

x0

(x0

x) dx 3

x0 ) dx5 =

(x

x0

1;[x0 ; ]

; M;b Df

(14)

f (x0 )j dx =

1;[ ;x0 ]

; ( + 1) 4 DM;a f (x) 2( ) x

x0 j

x0 j ;

(f (x)

( + 1) (x0

(x)

x

Z

x0

; DM;a f (x)

Z

1

f (x0 ) =

x0 ) jx

x0 ) jx

1;[x0 ; ]

1

1

( + 1) (b

c2

x 8 x 2 [x0 ; ] : We have that Z 1

(c2 )

2

a) (x0

) +

(16)

3

2 x0 ) 5 :

The theorem is proved. Next we give two M -fractional Polya type inequalities:

4

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Theorem 10 All as in Theorem 9 and f (x0 ) = 0. Then Z 2 4

; DM;a f (x)

x

f (x) dx

(x0

Z

( + 1) 2

jf (x)j dx 2

a) (x0

) +

; M;b Df

(x)

x

1;[ ;x0 ]

(b

3

2 x0 ) 5 :

x0 ) (

1;[x0 ; ]

(17)

Proof. Same as in the proof of Theorem 9, by setting f (x0 ) = 0: Corollary 11 (to Theorem 10, case of x0 = f

2 4

+ 2

+ 2

) All as in Theorem 9 and

= 0. Then Z

; f (x) DM;a

x

1;[ ;

+ 2

]

+ 2

Proof. Apply (17) for x0 =

a

+ 2

2

( + 1) ( 8

jf (x)j dx

+

)

; M;b Df

x

(x)

b 1;[

+ 2

;

]

+ 2 (18)

3

5:

:

References [1] G.A. Anastassiou, Ostrowski type inequalities, Proc. AMS 123, 3775-3781 (1995). [2] G.A. Anastassiou, Quantitative Approximations, Chapmann & Hall / CRC, Boca Raton, New York, 2001. [3] G. Anastassiou, About the right fractional local general M -derivative, Analele Univ. Oradea, Fasc. Mate., accepted for publication, 2019. [4] G. Anastassiou, On the left fractional local general M -derivative, submitted for publication, 2019. [5] A. Ostrowski, Über die Absolutabweichung einer di¤ ertentiebaren Funktion von ihrem Integralmittelwert, Comment. Math. Helv. 10 (1938), 226-227. [6] G. Polya, Ein mittelwertsatz für Funktionen mehrerer Veränderlichen, Tohoku Math. J., 19 (1921), 1-3. [7] G. Polya, G. Szegö, Aufgaben und Lehrsätze aus der Analysis, Volume I, Springer-Verlag, Berlin, 1925. (German) 5

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[8] G. Polya, G. Szegö, Problems and Theorems in Analysis, Volume I, Classics in Mathematics, Springer-Verlag, Berlin, 1972. [9] G. Polya, G. Szegö, Problems and Theorems in Analysis, Volume I, Chinese Edition, 1984.

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TABLE OF CONTENTS, JOURNAL OF COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 29, NO. 6, 2021

Daubechies Wavelet Method for Second Kind Fredholm Integral Equations with Weakly Singular Kernel, Xin Luo and Jin Huang,…………………………………………………1023 Some Fixed Point Results in Ordered Complete Dislocated Quasi 𝐺𝑑 Metric Space, Abdullah Shoaib, Muhammad Arshad, and Tahair Rasham,…………………………………………1036 Mizoguchi- Takahashi's Fixed Point Theorem in 𝜈-Generalized Metric Spaces, Salha Alshaikey, Saud M. Alsulami, and Monairah Alansari,……………………………………………….1047 On Multiresolution Analyses of Multiplicity n, Richard A. Zalik,…………………………1055 Generalized g-Fractional Vector Representation Formula and Integral Inequalities for Banach Space Valued Functions, George A. Anastassiou,…………………………………………1063 On Solutions of Semilinear Second-Order Impulsive Functional Differential Equations, Ah-ran Park and Jin-Mun Jeong,……………………………………………………………………1082 Caputo 𝜓-Fractional Ostrowski and Grüss Inequalities for Several Functions, George A. Anastassiou,…………………………………………………………………………………1097 Results on Sequential Conformable Fractional Derivatives with Applications, Mona Khandaqji and Aliaa Burqan,……………………………………………………………………………1115 Multivariate Ostrowski-Sugeno Fuzzy Inequalities, George A. Anastassiou,………………1126 Some Applications of Interval-Valued Subsethood Measures which Are Defined by IntervalValued Choquet Integrals in Trade Exports Between Korea and its Trading Partners, Jacob Wood, Lee-Chae Jang, and Hyun-Mee Kim,……………………………………………….1139 M-Fractional Integral Inequalities, George A. Anastassiou,……………………………….1153