JOURNAL OF COMPUTATIONAL ANALYSIS AND APPLICATIONS VOLUME 16, 2014 0651098283

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Volume 16, Number 1 ISSN:1521-1398 PRINT,1572-9206 ONLINE

January 2014

Journal of Computational Analysis and Applications EUDOXUS PRESS,LLC

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

28601, USA.

Journal of Computational Analysis and Applications(JoCAAA) is published by EUDOXUS PRESS,LLC,1424 Beaver Trail Drive,Cordova,TN38016,USA,[email protected] http//:www.eudoxuspress.com.Annual Subscription Prices:For USA and Canada,Institutional:Print $600,Electronic $350,Print and Electronic $750.Individual:Print $300,Electronic $100,Print &Electronic $350.For any other part of the world add $100 more(postages) to the above prices for Print.No credit card payments. Copyright©2014 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 1) 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. 2) 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

20)Margareta Heilmann Faculty of Mathematics and Natural Sciences University of Wuppertal Gaußstraße 20 D-42119 Wuppertal, Germany, [email protected] Approximation Theory (Positive Linear Operators) 21) Christian Houdre School of Mathematics Georgia Institute of Technology Atlanta,Georgia 30332 404-894-4398 e-mail: [email protected] Probability, Mathematical Statistics, Wavelets

3) Mark J.Balas Department Head and Professor Electrical and Computer Engineering Dept. College of Engineering University of Wyoming 1000 E. University Ave. Laramie, WY 82071 307-766-5599 e-mail: [email protected] Control Theory,Nonlinear Systems, Neural Networks,Ordinary and Partial Differential Equations, Functional Analysis and Operator Theory

22) Irena Lasiecka Department of Mathematical Sciences University of Memphis Memphis, TN 38152 P.D.E, Control Theory, Functional Analysis, [email protected]

4) Dumitru Baleanu Cankaya University, Faculty of Art and Sciences, Department of Mathematics and Computer Sciences, 06530 Balgat, Ankara, Turkey, [email protected] Fractional Differential Equations Nonlinear Analysis, Fractional Dynamics

5) Carlo Bardaro Dipartimento di Matematica e Informatica

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

25) 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

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.

26) 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

6) 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.

27)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 117943775 Phone: +1-631-632-1998, Email : [email protected]

7) 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

28) 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

8) 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 9) 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

29) Ervin Y.Rodin Department of Systems Science and Applied Mathematics Washington University,Campus Box 1040 One Brookings Dr.,St.Louis,MO 631304899 314-935-6007 e-mail: [email protected] Systems Theory, Semantic Control, Partial Differential Equations, Calculus of Variations, Optimization

10) Ding-Xuan Zhou Department Of Mathematics City University of Hong Kong

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and Artificial Intelligence, Operations Research, Math.Programming

83 Tat Chee Avenue Kowloon,Hong Kong 852-2788 9708,Fax:852-2788 8561 e-mail: [email protected] Approximation Theory, Spline functions,Wavelets

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

11) 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.

31) I. P. Stavroulakis Department of Mathematics University of Ioannina 451-10 Ioannina, Greece [email protected] Differential Equations Phone +3 0651098283

12) Oktay Duman TOBB University of Economics and Technology, Department of Mathematics, TR-06530, Ankara, Turkey, [email protected] Classical Approximation Theory, Summability Theory, Statistical Convergence and its Applications

32) 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

13) 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

33) Roberto Triggiani Department of Mathematical Sciences University of Memphis Memphis, TN 38152 P.D.E, Control Theory, Functional Analysis, [email protected]

14) Augustine O.Esogbue School of Industrial and Systems Engineering Georgia Institute of Technology Atlanta,GA 30332 404-894-2323 e-mail: [email protected] Control Theory,Fuzzy sets, Mathematical Programming, Dynamic Programming,Optimization

34) Gilbert G.Walter Department Of Mathematical Sciences University of Wisconsin-Milwaukee,Box 413, Milwaukee,WI 53201-0413 414-229-5077 e-mail: [email protected] Distribution Functions, Generalised Functions, Wavelets 35) Xin-long Zhou Fachbereich Mathematik, Fachgebiet Informatik Gerhard-Mercator-Universitat Duisburg

15) Christodoulos A.Floudas Department of Chemical Engineering Princeton University

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Princeton,NJ 08544-5263 609-258-4595(x4619 assistant) e-mail: [email protected] OptimizationTheory&Applications, Global Optimization

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

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

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

17) H.H.Gonska Department of Mathematics University of Duisburg Duisburg, D-47048 Germany 011-49-203-379-3542 e-mail:[email protected] Approximation Theory, Computer Aided Geometric Design 18) 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 19) 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

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37) Lotfi A. Zadeh Professor in the Graduate School and Director, Computer Initiative, Soft Computing (BISC) Computer Science Division University of California at Berkeley Berkeley, CA 94720 Office: 510-642-4959 Sec: 510-642-8271 Home: 510-526-2569 FAX: 510-642-1712 e-mail: [email protected] Fuzzyness, Artificial Intelligence, Natural language processing, Fuzzy logic 38) Ahmed I. Zayed Department Of Mathematical Sciences DePaul University 2320 N. Kenmore Ave. Chicago, IL 60614-3250 773-325-7808 e-mail: [email protected] Shannon sampling theory, Harmonic analysis and wavelets, Special functions and orthogonal polynomials, Integral transforms

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J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 16, NO.1, 2014, COPYRIGHT 2014 EUDOXUS PRESS, LLC

Switching design for the robust stability of nonlinear uncertain stochastic switched discrete-time systems with interval time-varying delay G. Rajchakit Major of Mathematics, Faculty of Science Maejo University, Chiangmai 50290, Thailand Corresponding author: [email protected] Abstract This paper is concerned with robust stability of nonlinear uncertain stochastic switched discrete time-delay systems with interval time-varying delay. The system to be considered is subject to interval time-varying delays, which allows the delay to be a fast time-varying function and the lower bound is not restricted to zero. Based on the discrete Lyapunov functional, a switching rule for the robust stability for the nonlinear uncertain stochastic switched discrete time-delay system with interval time-varying delay is designed via linear matrix inequalities.

Keywords. Switching design, nonlinear uncertain stochastic switched discrete system, time-varying delay, robust stability, Lyapunov function, linear matrix inequality.

1

Introduction

Time delay is often a source of instability and poor performance, and is encountered in various engineering systems, such as chemical processes and long transmission lines in pneumatic systems. Time-delay systems have received much attention in recent years, and various topics concerning time-delay systems have been addressed; see, e.g., [1-10], and the references cited therein. As an important class of hybrid systems, switched systems arise in many practical processes that cannot be described by exclusively continuous or exclusively discrete models, such as manufacturing, communication networks, automotive engineering control and chemical processes (see, e.g., [1–3] and the references therein). On the other hand, time-delay phenomena are very common in practical systems. A switched system with time-delay individual subsystems is called a switched time-delay system; in particular, when the subsystems are linear, it is then called a switched timedelay linear system. During the last decades, the stability analysis of switched linear continuous/discrete time-delay systems has attracted a lot of attention [4–8]. The main approach for stability analysis relies on the use of Lyapunov-Krasovskii functionals and linear matrix inequlity (LMI) approach for constructing a common Lyapunov function [9–11]. Although many important results have been obtained for switched linear continuous-time systems, there are few results concerning the stability of switched linear discrete systems with time-varying delays (see, e.g., [1–3] and the references therein). It was shown in [5, 7, 12] that when all subsystems are asymptotically stable, the switching system is asymptotically stable under an arbitrary switching rule. The asymptotic stability for switching linear discrete time-delay systems has been studied in [13], but the result was limited to constant delays. In [14], a class of switching signals has been identified for the considered switched discrete-time delay systems to be stable under the average dwell time scheme. To the best of the author’s knowledge, the stability for linear discrete-time systems with both time-varying delays and nonlinear uncertain stochastic discrete switch system has not been fully investigated (see, e.g., [14–20] and the references therein), which are important in both theories and applications. This motivates our research.

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RAJCHAKIT 10-19

J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 16, NO.1, 2014, COPYRIGHT 2014 EUDOXUS PRESS, LLC

This paper studies robust stability problem for nonlinear uncertain stochastic switched discrete-time delay systems with interval time-varying delays. Specifically, our goal is to develop a constructive way to design switching rule to robustly stable of the nonlinear uncertain stochastic switched discrete-time delay systems with interval time-varying delay. By using improved Lyapunov-Krasovskii functionals combined with LMIs technique, we propose new criteria for the robust stability of the nonlinear uncertain stochastic switched discrete-time delay system with interval time-varying delay. Compared to the existing results, our result has its own advantages. First, the time delay is assumed to be a time-varying function belonging to a given interval, which means that the lower and upper bounds for the time-varying delay are available, the delay function is bounded but not restricted to zero. Second, the approach allows us to design the switching rule for robust stability in terms of LMIs. The paper is organized as follows: Section 2 presents definitions and some well-known technical propositions needed for the proof of the main results. Switching rule for the robust stability is presented in Section 3.

2

Preliminaries

The following notations will be used throughout this paper. R+ denotes the set of all real non-negative numbers; Rn denotes the n-dimensional space with the scalar product of two vectors hx, yi or xT y; Rn×r denotes the space of all matrices of (n × r)− dimension. N + denotes the set of all non-negative integers; AT denotes the transpose of A; a matrix A is symmetric if A = AT . Matrix A is semi-positive definite (A ≥ 0) if hAx, xi ≥ 0, for all x ∈ Rn ; A is positive definite (A > 0) if hAx, xi > 0 for all x 6= 0; A ≥ B means A − B ≥ 0. λ(A) denotes the set of all eigenvalues of A; λmin (A) = min{Reλ : λ ∈ λ(A)}. Consider a nonlinear uncertain stochastic switched discrete-time delay systems with interval time-varying delay of the form x(k + 1) = (Aγ + ∆Aγ (k))x(k) + (Bγ + ∆Bγ (k))x(k − d(k)) + f (k, x(k − d(k))) + σ(x(k), x(k − d(k)), k)ω(k),

k ∈ N +,

x(k) = vk ,

k = −d2 , −d2 + 1, ..., 0,

(2.1)

where x(k) ∈ Rn is the state, γ(.) : Rn → N := {1, 2, . . . , N } is the switching rule, which is a function depending on the state at each time and will be designed. A switching function is a rule which determines a switching sequence for a given switching system. Moreover, γ(x(k)) = i implies that the system realization is chosen as the ith system, i = 1, 2, ..., N. It is seen that the system (2.1) can be viewed as an autonomous switched system in which the effective subsystem changes when the state x(k) hits predefined boundaries. Ai , Bi , i = 1, 2, ..., N are given constant matrices. The nonlinear perturbations f (k, x(k − d(k))) satisfies the following condition f T (k, x(k − d(k)))f (k, x(k − d(k))) ≤ β 2 xT (k − d(k))x(k − d(k)),

(2.2)

where β is positive constant. For simplicity, we denote f (k, x(k − d(k)) by f , respectively. The time-varying uncertain matrices ∆Ai (k) and ∆Bi (k) are defined by: ∆Ai (k) = Eia Fia (k)Hia ,

∆Bi (k) = Eib Fib (k)Hib ,

where Eia , Eib , Hia , Hib are known constant real matrices with appropriate dimensions. Fia (k), Fib (k) are unknown uncertain matrices satisfying T Fia (k)Fia (k) ≤ I,

FibT (k)Fib (k) ≤ I,

k = 0, 1, 2, ...,

(2.3)

where I is the identity matrix of appropriate dimention, ω(k) is a scalar Wiener process (Brownian Motion) on (Ω, F, P) with E[ω(k)] = 0, E[ω 2 (k)] = 1, E[ω(i)ω(j)] = 0(i 6= j), (2.4)

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J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 16, NO.1, 2014, COPYRIGHT 2014 EUDOXUS PRESS, LLC

and σ: Rn × Rn × R → Rn is the continuous function, and is assumed to satisfy that σ T (x(k), x(k − d(k)), k)σ(x(k), x(k − d(k)), k) ≤ ρ1 xT (k)x(k) + ρ2 xT (k − d(k))x(k − d(k), x(k), x(k − d(k) ∈ Rn ,

(2.5)

where ρ1 > 0 and ρ2 > 0 are known constant scalars. The time-varying function d(k) : N + → N + satisfies the following condition: 0 < d1 ≤ d(k) ≤ d2 , ∀k ∈ N + Remark 2.1. It is worth noting that the time delay is a time-varying function belonging to a given interval, in which the lower bound of delay is not restricted to zero. Definition 2.1. The nonlinear uncertain stochastic switched system with interval time-varying delay (2.1) is robustly stable in the mean square if there exists a positive definite scalar function V (k, x(k) : Rn × Rn → R and a switching rule γ(.) such that E[∆V (k, x(k))] = E[V (k + 1, x(k + 1)) − V (k, x(k))] < 0, along any trajectory of solution of the system (2.1) for all uncertainties which satisfy (2.3). Definition 2.2. The system of matrices {Ji }, i = 1, 2, . . . , N, is said to be strictly complete if for every x ∈ Rn \{0} there is i ∈ {1, 2, . . . , N } such that xT Ji x < 0. It is easy to see that the system {Ji } is strictly complete if and only if N [

αi = Rn \{0},

i=1

where αi = {x ∈ Rn :

xT Ji x < 0}, i = 1, 2, ..., N.

Proposition 2.1. [22] The system {Ji }, i = 1, 2, . . . , N, is strictly complete if there exist δi ≥ 0, i = PN 1, 2, . . . , N, i=1 δi > 0 such that N X δi Ji < 0. i=1

If N = 2 then the above condition is also necessary for the strict completeness. Proposition 2.2. (Cauchy inequality) For any symmetric positive definite marix N ∈ M n×n and a, b ∈ Rn we have +aT b ≤ aT N a + bT N −1 b. Proposition 2.3. [23] Let E, H and F be any constant matrices of appropriate dimensions and F T F ≤ I. For any ǫ > 0, we have EF H + H T F T E T ≤ ǫEE T + ǫ−1 H T H.

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J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 16, NO.1, 2014, COPYRIGHT 2014 EUDOXUS PRESS, LLC

3

Main results

Let us set  Wi11  ∗ Wi (S1 , S2 , P, Q) =   ∗ ∗

Wi12 Wi22 ∗ ∗

Wi13 Wi23 Wi33 ∗

 Wi14 Wi24  , Wi34  Wi44

Wi11 = Q − P, Wi12 = S1 − S1 Ai , Wi13 = −S1 Bi , Wi14 = −S1 − S2 Ai , T T T Wi22 = P + S1 + S1T + S1 Eib Eib S1 + Hia Hia , Wi23 = −S1 Bi ,

Wi24 = S2 − S1 ,

(3.1)

T T T T Wi33 = −Q + S2 Eib Eib S2 + Hia Hia + Hib Hib + ρ2 I,

Wi34 = −S2 Bi , T T Wi44 = −S2 − S2T + Hia Hia + Hib Hib , T T Ji (S1 , S2 , Q) = (d2 − d1 )Q − S1 Ai − ATi S1T + S1 Eia Eia S1 T T T T T + S1 Eib Eib S1 + S2 Eia Eia S2 + Hia Hia + ρ1 I,

αi = {x ∈ Rn : α ¯ 1 = α1 ,

xT Ji (S1 , S2 , Q)x < 0}, i = 1, 2, ..., N,

α ¯ i = αi \

i−1 [

α ¯j ,

i = 2, 3, . . . , N.

j=1

The main result of this paper is summarized in the following theorem. Theorem 3.1. The nonlinear uncertain stochastic switched system with interval time-varying delay (2.1) is robustly stable if there exist symmetric positive definite matrices P > 0, Q > 0 and matrices S1 , S2 satisfying the following conditions: PN PN (i) ∃δi ≥ 0, i = 1, 2, . . . , N, i=1 δi > 0 : i=1 δi Ji (S1 , S2 , Q) < 0. (ii) Wi (S1 , S2 , P, Q) < 0, i = 1, 2, ..., N. The switching rule is chosen as γ(x(k)) = i, whenever x(k) ∈ α ¯i. Proof. Consider the following Lyapunov-Krasovskii functional for any ith system (2.1) V (k) = V1 (k) + V2 (k) + V3 (k), where V1 (k) = xT (k)P x(k),

V2 (k) =

k−1 X

xT (i)Qx(i),

i=k−d(k)

V3 (k) =

−d 1 +1 X

k−1 X

xT (l)Qx(l),

j=−d2 +2 l=k+j+1

We can verify that λ1 kx(k)k2 ≤ V (k).

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(3.2)

RAJCHAKIT 10-19

J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 16, NO.1, 2014, COPYRIGHT 2014 EUDOXUS PRESS, LLC

Let us set ξ(k) = [x(k) x(k + 1) x(k − d(k)) fi (k, x(k − d(k))) ω(k)]T    0 0 0 0 P 0 0 P 0 0 I I   H= 0 0 0 0 , G =  0 0 0 0 0 0 0 0

and 0 0 I 0

 0 0 . 0 I

Then, the difference of V1 (k) along the solution of the system (2.1) and taking the mathematical expectation, we obtained E[∆V1 (k)] = E[xT (k + 1)P x(k + 1) − xT (k)P x(k)]   0.5x(k)  0   = E[ξ T (k)Hξ(k) − 2ξ T (k)GT   0 ]. 0

(3.3)

because of ξ T (k)Hξ(k) = x(k + 1)P x(k + 1),   0.5x(k)  0  T  2ξ T (k)GT   0  = x (k)P x(k). 0 Using the expression of system (2.1) 0 = −S1 x(k + 1) + S1 (Ai + Eia Fia (k)Hia )x(k) + S1 (Bi + Eib Fib (k)Hib )x(k − d(k)) + S1 f + S1 σω(k), 0 = −S2 x(k + 1) + S2 (Ai + Eia Fia (k)Hia )x(k) + S2 (Bi + Eib Fib (k)Hib )x(k − d(k)) + S2 f + S2 σω(k), 0 = −σ T x(k + 1) + σ T (Ai + Eia Fia (k)Hia )x(k) + σ T (Bi + Eib Fib (k)Hib )x(k − d(k)) + σ T f + σ T σω(k), we have E[−2ξ T (k)GT 



0.5x(k)

     [−S1 x(k + 1) + S1 (Ai + Eia Fia (k)Hia )x(k) + S1 (Bi + Eib Fib (k)Hib )x(k − d(k)) + S1 f      +S1 σω(k)]      ] ×  [−S x(k + 1) + S (A + E F (k)H )x(k) + S (B + E F (k)H )x(k − d(k)) + S f 2 2 i ia ia ia 2 i ib ib ib 2     +S σω(k)] 2       [−σ T x(k + 1) + σ T (Ai + Eia Fia (k)Hia )x(k) + σ T (Bi + Eib Fib (k)Hib )x(k − d(k)) + σ T f  +σ T σω(k)] 

0.5I  S A + S 1 i 1 Eia Fia (k)Hia = E[−ξ T (k)GT   S2 Ai + S2 Eia Fia (k)Hia σ T Ai + σ T Eia Fia (k)Hia

0 −S1 −S2 −σ T

14

0 S1 Bi + S1 Eib Fib (k)Hib S2 Bi + S2 Eib Fib (k)Hib σ T Bi + σ T Eib Fib (k)Hib

0 S1 S2 σT

 0 S1 σ   ξ(k) S2 σ  σT σ

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

0.5I  S1 Ai + S1 Eia Fia (k)Hia T  −ξ (k)  S2 Ai + S2 Eia Fia (k)Hia σ T Ai + σ T Eia Fia (k)Hia

0 −S1 −S2 −σ T

0 S1 Bi + S1 Eib Fib (k)Hib S2 Bi + S2 Eib Fib (k)Hib σ T Bi + σ T Eib Fib (k)Hib

0 S1 S2 σT

T 0 S1 σ   Gξ(k)]. S2 σ  σT σ

Therefore, from (3.3) it follows that T T E[∆V1 (k)] = E[xT (k)[−P − S1 Ai − S1 Eia Fia (k)Hia − ATi S1T − Hia Fia (k)Eia S1T ]x(k)

+ 2xT (k)[S1 − S1 Ai − S1 Eia Fia (k)Hia ]x(k + 1) + 2xT (k)[−S1 Bi − S1 Eib Fib (k)Hib ]x(k − d(k)) + 2xT (k)[−S1 − S2 Ai − S2 Eia Fia (k)Hia ]f (k, x(k − d(k))) + 2xT (k)[−S1 σ − σ T Ai − σ T Eia Fia (k)Hia ]ω(k) + x(k + 1)[P + S1 + S1T ]x(k + 1) + 2x(k + 1)[−S1 Bi − S1 Eib Fib (k)Hib ]x(k − d(k)) + 2x(k + 1)[S2 − S1 ]f (k, x(k − d(k))) + 2x(k + 1)[σ T − S1 σ]ω(k) + 2xT (k − d(k))[−S3 Bi − S2 Eib Fib (k)Hib ]f (k, x(k − d(k))) + 2xT (k − d(k))[−σ T Bi − σ T Eib Fib (k)Hib ]ω(k) + f (k, x(k − d(k)))T [−S2 − S2T ]f (k, x(k − d(k))) + 2f (k, x(k − d(k)))T (k)[−S2 σ − σ T ]ω(k) + ω T (k)[−σ T σ]ω(k)]. By asumption (2.4), we have T T E[∆V1 (k)] = E[xT (k)[−P − S1 Ai − S1 Eia Fia (k)Hia − ATi S1T − Hia Fia (k)Eia S1T ]x(k)

+ 2xT (k)[S1 − S1 Ai − S1 Eia Fia (k)Hia ]x(k + 1) + 2xT (k)[−S1 Bi − S1 Eib Fib (k)Hib ]x(k − d(k)) + 2xT (k)[−S1 − S2 Ai − S2 Eia Fia (k)Hia ]f (k, x(k − d(k))) + x(k + 1)[P + S1 + S1T ]x(k + 1) + 2x(k + 1)[−S1 Bi − S1 Eib Fib (k)Hib ]x(k − d(k)) + 2x(k + 1)[S2 − S1 ]f (k, x(k − d(k))) + 2xT (k − d(k))[−S2 Bi − S2 Eib Fib (k)Hib ]f (k, x(k − d(k))) + f (k, x(k − d(k)))T [−S2 − S2T ]f (k, x(k − d(k))) + ω T (k)[−σ T σ]ω(k)]. Applying Propositon 2.2, Propositon 2.3, condition (2.3) and asumption (2.5), the following estimations hold T T T T T T T −S1 Eia Fia (k)Hia − Hia Fia (k)Eia S1 ≤ S1 Eia Eia S1 + Hia Hia , T T T −2xT (k)S1 Eia Fia (k)Hia x(k + 1) ≤ xT (k)S1 Eia Eia S1 x(k) + x(k + 1)T Hia Hia x(k + 1), T T T −2xT (k)S1 Eib Fib (k)Hib x(k − d(k)) ≤ xT (k)S1 Eib Eib S1 x(k) + x(k − d(k))T Hib Hib x(k − d(k)), T T T −2xT (k)S2 Eia Fia (k)Hia f ≤ xT (k)S2 Eia Eia S2 x(k) + f T Hia Hia f,

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T T T S2 x(k − d(k)) + f T Hib Hib f, −2x(k − d(k))T (k)S2 Eib Fib (k)Hib f ≤ x(k − d(k))T (k)S2 Eib Eib T T T −2xT (k + 1)S1 Eib Fib (k)Hib x(k − d(k)) ≤ xT (k + 1)S1 Eib Eib S1 x(k + 1) + x(k − d(k))T Hib Hib x(k − d(k)),

−σ T (x(k), x(k − d(k)), k)σ(x(k), x(k − d(k)), k) ≤ ρ1 xT (k)x(k) + ρ2 xT (k − d(k))x(k − d(k). Therefore, we have T T T T E[∆V1 (k)] ≤ E[xT (k)[−P − S1 Ai − ATi S1T + S1 Eia Eia S1 + S1 Eib Eib S1 T T T + S2 Eia Eia S2 + Hia Hia + ρ1 I]x(k)

+ 2xT (k)[S1 − S1 Ai ]x(k + 1) + 2xT (k)[−S1 Bi ]x(k − d(k)) + 2xT (k)[−S1 − S2 Ai ]f (k, x(k − d(k))) (3.4)

T T T + x(k + 1)[P + S1 + S1T + S1 Eib Eib S1 + Hia Hia ]x(k + 1) + 2x(k + 1)[−S1 Bi ]x(k − d(k))

+ 2x(k + 1)[S2 − S1 ]f (k, x(k − d(k))) T T T T + xT (k − d(k))[S2 Eib Eib S3 + Hia Hia + Hib Hib + ρ2 I]x(k − d(k))

+ 2xT (k − d(k))[−S3 Bi ]f (k, x(k − d(k))) T T + f (k, x(k − d(k)))T [−S2 − S2T + Hia Hia + Hib Hib ]f (k, x(k − d(k)))].

The difference of V2 (k) is given by k X

E[∆V2 (k)] = E[

T

x (i)Qx(i) −

i=k+1−d(k+1) k−d X1

= E[

k−1 X

xT (i)Qx(i)]

i=k−d(k)

xT (i)Qx(i) + xT (k)Qx(k) − xT (k − d(k))Qx(k − d(k))

(3.5)

i=k+1−d(k+1) k−1 X

+

k−1 X

xT (i)Qx(i) −

i=k+1−d1

xT (i)Qx(i)].

i=k+1−d(k)

Since d(k) ≥ d1 we have k−1 X

xT (i)Qx(i) −

i=k+1−d1

k−1 X

xT (i)Qx(i) ≤ 0,

i=k+1−d(k)

and hence from (3.5) we have E[∆V2 (k)] ≤ E[

k−d X1

xT (i)Qx(i) + xT (k)Qx(k) − xT (k − d(k))Qx(k − d(k))].

(3.6)

i=k+1−d(k+1)

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The difference of V3 (k) is given by E[∆V3 (k)] = E[

−d 1 +1 X

k X

x (l)Qx(l) −

−d 1 +1 X

k−1 X

xT (l)Qx(l) + xT (k)Q(ξ)x(k)

T

j=−d2 +2 l=k+j

= E[

[

−d 1 +1 X

k−1 X

xT (l)Qx(l)]

j=−d2 +2 l=k+j+1

j=−d2 +2 l=k+j k−1 X

−

xT (l)Qx(l) − xT (k + j − 1)Qx(k + j − 1)]]

(3.7)

l=k+j

= E[

−d 1 +1 X

[xT (k)Qx(k) − xT (k + j − 1)Qx(k + j − 1)]]

j=−d2 +2

= E[(d2 − d1 )xT (k)Qx(k) −

k−d X1

xT (j)Qx(j)].

j=k+1−d2

Since d(k) ≤ d2 , and k−d X1

xT (i)Qx(i) −

k−d X1

xT (i)Qx(i) ≤ 0,

i=k+1−d2

i=k=1−d(k+1)

we obtain from (3.6) and (3.7) that E[∆V2 (k) + ∆V3 (k)] ≤ E[(d2 − d1 + 1)xT (k)Qx(k) − xT (k − d(k))Qx(k − d(k))].

(3.8)

Therefore, combining the inequalities (3.4), (3.8) gives E[∆V (k)] ≤ E[xT (k)Ji (S1 , S2 , Q)x(k) + ψ T (k)Wi (S1 , S2 , P, Q)ψ(k)],

(3.9)

where ψ(k) = [x(k) x(k + 1) x(k − d(k)) f (k, x(k − d(k)))]T ,   Wi11 Wi12 Wi13 Wi14  ∗ Wi22 Wi23 Wi24  , Wi (S1 , S2 , P, Q) =   ∗ ∗ Wi33 Wi34  ∗ ∗ ∗ Wi44 Wi11 = Q − P, Wi12 = S1 − S1 Ai , Wi13 = −S1 Bi , Wi14 = −S1 − S2 Ai , Wi14 = −S1 − S2 Ai , T T T Wi22 = P + S1 + S1T + S1 Eib Eib S1 + Hia Hia , Wi23 = −S1 Bi , Wi24 = S2 − S1 , T T T T S2 + Hia Hia + Hib Hib + ρ2 I, Wi33 = −Q + S2 Eib Eib

Wi34 = −S2 Bi , T T Wi44 = −S2 − S2T + Hia Hia + Hib Hib , T T T T S1 + S1 Eib Eib S1 Ji (S1 , S2 , Q) = (d2 − d1 )Q − S1 Ai − ATi S1T + 2S1 Eia Eia T T T + S2 Eia Eia S2 + Hia Hia + ρ1 I.

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Therefore, we finally obtain from (3.9) and the condition (ii) that E[∆V (k)] < E[xT (k)Ji (S1 , S2 , Q)x(k)],

∀i = 1, 2, ...., N, k = 0, 1, 2, ....

We now apply the condition (i) and Proposition 2.1., the system Ji (S1 , S2 , Q) is strictly complete, and the sets αi and α ¯ i by ( 3.1) are well defined such that N [

αi = Rn \{0},

i=1 N [

α ¯ i = Rn \{0},

α ¯i ∩ α ¯ j = ∅, i 6= j.

i=1

Therefore, for any x(k) ∈ Rn , k = 1, 2, ..., there exists i ∈ {1, 2, . . . , N } such that x(k) ∈ α ¯ i . By choosing switching rule as γ(x(k)) = i whenever x(k) ∈ α ¯ i , from the condition (3.9) we have E[∆V (k)] ≤ E[xT (k)Ji (S1 , S2 , Q)x(k)] < 0,

k = 1, 2, ...,

which, combining the condition (3.2), and Definition 2.1., concludes the proof of the theorem in the mean square. Remark 3.1. Note that the results proposed in [5, 7, 12] for switching systems to be asymptotically stable under an arbitrary switching rule. The asymptotic stability for switching linear discrete time-delay systems studied in [13] was limited to constant delays. In [21], a class of switching signals has been identified for the considered switched discrete-time delay systems to be stable under the averaged well time scheme.

4

Conclusion

This paper has proposed a switching design for the robust stability of nonlinear uncertain stochastic switched discrete time-delay systems with interval time-varying delays. Based on the discrete Lyapunov functional, a switching rule for the robust stability for the nonlinear uncertain stochastic switched discrete time-delay system with interval time-varying delay is designed via linear matrix inequalities. Acknowledgments. This work was supported by the Thai Research Fund Grant, the Higher Education Commission and Faculty of Science, Maejo University, Thailand.

References [1] D. Liberzon, A.S. Morse, Basic problems in stability and design of switched systems, IEEE Control Syst. Mag., 19(1999), 57-70. [2] A.V. Savkin and R.J. Evans, Hybrid Dynamical Systems: Controller and Sensor Switching Problems, Springer, New York, 2001. [3] Z. Sun and S.S. Ge, Switched Linear Systems: Control and Design, Springer, London, 2005. [4] F. Gao, S. Zhong and X. Gao, Delay-dependent stability of a type of linear switching systems with discrete and distributed time delays, Appl. Math. Computation, 196(2008), 24-39. [5] G Rajchakit, T Rojsiraphisal, M Rajchakit, Robust stability and stabilization of uncertain switched discrete-time systems, Advances in Difference Equations 2012, 2012:134. doi:10.1186/1687-1847-2012134

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[6] M Rajchakit, G Rajchakit, Mean square robust stability of stochastic switched discrete-time systems with convex polytopic uncertainties, Journal of Inequalities and Applications 2012, 2012:135. doi:10.1186/1029-242X-2012-135 [7] S. Boyd, L.E. Ghaoui, E. Feron and V. Balakrishnan, Linear Matrix Inequalities in System and Control Theory, SIAM, Philadelphia, 1994. [8] D.H. Ji, J.H. Park, W.J. Yoo and S.C. Won, Robust memory state feedback model predictive control for discrete-time uncertain state delayed systems, Appl. Math. Computation, 215(2009), 2035-2044. [9] K. Ratchagit and V.N. Phat, Stability criterion for discrete-time systems, J. Ineq. Appl., 2010(2010), 1-6. [10] G.S. Zhai, B. Hu, K. Yasuda, and A. Michel, Qualitative analysis of discrete- time switched systems. In: Proc. of the American Control Conference, 2002, 1880-1885. [11] Manlika Rajchakit, Piyapong Niamsup, Grienggrai Rajchakit, A switching rule for exponential stability of switched recurrent neural networks with interval time-varying delay, Advances in Difference Equations 2013, 2013:44. doi:10.1186/1687-1847-2013-44 [12] Dong, H, Wang, Z, Ho, DWC, Gao, H: , Robust H∞ filtering for Markovian jump systems with randomly occurring nonlinearities and sensor saturation: the finite-horizon case. IEEE Trans. Sig. Process. 59, 3048–3057 (2011) [13] K. Ratchagit, A switching rule for the asymptotic stability of discrete-time systems with convex polytopic uncertainties, Asian-European J. Math., 5(2012), 1250025 (12 pages). [14] Jun Li, Weigen Wu, Jimin Yuan, Qianrong Tan, and Xing Yin, Delay-Dependent Stability Criterion of Arbitrary Switched Linear Systems with Time-Varying Delay, Discrete Dynamics in Nature and Society, 2010(2010). [15] M. Rajchakit and G. Rajchakit, Mean Square Exponential Stability of Stochastic Switched System with Interval Time-Varying Delays, Abstract and Applied Analysis, vol. 2012, Article ID 623014, 12 pages, 2012. doi:10.1155/2012/623014 [16] Wang, Z, Wei, G, Feng, G: Reliable H∞ control for discrete-time piecewise linear systems with infinite distributed delays. Automatica 45, 2991–2994 (2009) [17] M. Rajchakit and G. Rajchakit, LMI approach to robust stability and stabilization of nonlinear uncertain discrete-time systems with convex polytopic uncertainties, Advances in Difference Equations, 2012(2012). [18] Wang, Y, Wang, Z, Liang, J: A delay fractioning approach to global synchronization of delayed complex networks with stochastic disturbances. Phys. Lett. A 372, 6066–6073 (2008) [19] Wang, Z, Wang, Y, Liu, Y: Global synchronization for discrete-time stochastic complex networks with randomly occurred nonlinearities and mixed time delays. IEEE Trans. Neural Netw. 21, 11–25 (2010) [20] V. N. Phat and K. Ratchagit, Stability and stabilization of switched linear discrete-time systems with interval time-varying delay, Nonlinear Analysis: Hybrid Systems, 5(2011), 605-612. [21] W.A. Zhang, Li Yu, Stability analysis for discrete-time switched time-delay systems, Automatica, 45(2009), 2265-2271. [22] F. Uhlig, A recurring theorem about pairs of quadratic forms and extensions, Linear Algebra Appl., 25(1979), 219-237. [23] R.P. Agarwal, Difference Equations and Inequalities, Second Edition, Marcel Dekker, New York, 2000.

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Stabilization of switched discrete-time systems with convex polytopic uncertainties G. Rajchakit Major of Mathematics, Faculty of Science Maejo University, Chiangmai 50290, Thailand Corresponding author: [email protected] Abstract This paper is concerned with robust stabilization of switched discrete time-delay systems with convex polytopic uncertainties. The system to be considered is subject to interval time-varying delays, which allows the delay to be a fast time-varying function and the lower bound is not restricted to zero. Based on the discrete Lyapunov functional, a switching rule for the robust stabilization for the system with convex polytopic uncertainties is designed via linear matrix inequalities.

Keywords. Switching design, convex polytopic uncertainties, discrete system, robust stabilization, Lyapunov function, linear matrix inequality. AMS (MOS) Subject Classification. 34D20, 93D20, 37C75.

1

Introduction

In many physical phenomena and practical applications, such as autonomous transmission systems, computer disc drivers, room temperature control, power electronics, chaos generators (see, e.g., [1–3] and the references therein), they are governed by more than one dynamical systems (differential or difference equations) governed by switching laws to determine which subsystem will be activated on a certain time interval. Such systems are called switched systems. On the other hand, time-delay phenomena are very common in practical systems. A switched system with time-delay individual subsystems is called a switched time-delay system; in particular, when the subsystems are linear, it is then called a switched time-delay linear system. During the last decades, the stability analysis of switched linear continuous/discrete time-delay systems has attracted a lot of attention [4–7]. The main approach for stability analysis relies on the use of LyapunovKrasovskii functionals and linear matrix inequlity (LMI) approach for constructing a common Lyapunov function [8–10]. Although many important results have been obtained for switched linear continuous-time systems, there are few results concerning the stability of switched linear discrete systems with time-varying delays. It was shown in [5, 7, 11] that when all subsystems are asymptotically stable, the switching system is asymptotically stable under an arbitrary switching rule. The asymptotic stability for switching linear discrete time-delay systems has been studied in [12], but the result was limited to constant delays. In [14], a class of switching signals has been identified for the considered switched discrete-time delay systems to be stable under the average dwell time scheme. To the best of our knowledge, the stabilization of discrete-time systems with both convex polytopic uncertainties and switch system, non-differentiable time-varying delays has not been fully studied yet (see, e.g., [1, 4–27] and the references therein), which are important in both theories and applications. This motivates our research. This paper studies robust stabilization problem for switched linear discrete systems with convex polytopic uncertainties with interval time-varying delays. Specifically, our goal is to develop a constructive way to design switching rule to robust stabilization the system. By using improved Lyapunov-Krasovskii functionals

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combined with LMIs technique, we propose new criteria for the robust stabilization of the system. Compared to the existing results, our result has its own advantages. First, the time delay is assumed to be a time-varying function belonging to a given interval, which means that the lower and upper bounds for the time-varying delay are available, the delay function is bounded but not restricted to zero. Second, the approach allows us to design the switching rule for robust stabilization in terms of of LMIs. The paper is organized as follows: Section 2 presents definitions and some well-known technical propositions needed for the proof of the main results. Switching rule for the robust stabilization is presented in Section 3.

2

Preliminaries

The following notations will be used throughout this paper. R+ denotes the set of all real non-negative numbers; Rn denotes the n-dimensional space with the scalar product of two vectors hx, yi or xT y; Rn×r denotes the space of all matrices of (n × r)− dimension. AT denotes the transpose of A; a matrix A is symmetric if A = AT . Matrix A is semi-positive definite (A ≥ 0) if hAx, xi ≥ 0, for all x ∈ Rn ; A is positive definite (A > 0) if hAx, xi > 0 for all x 6= 0; A ≥ B means A − B ≥ 0. λ(A) denotes the set of all eigenvalues of A; λmin (A) = min{Reλ : λ ∈ λ(A)}. Consider a linear switched control discrete-time systems with convex polytopic uncertainties with interval time-varying delay of the form x(k + 1) = Aγ(x(k)) (ζ)x(k) + Bγ(x(k)) (ζ)u(k), x(k) = vk , k = −d2 , −d2 + 1, ..., 0,

k = 0, 1, 2, ...

(2.1)

where x(k) ∈ Rn is the state, u(k) ∈ Rm , m ≤ n, is the control input, γ(.) : Rn → N := {1, 2, . . . , N } is the switching rule, which is a function depending on the state at each time and will be designed. A switching function is a rule which determines a switching sequence for a given switching system. We consider a delayed feedback control law u(k) = Cγ(x(k)) (ζ)x(k − d(k)), k = −h2 , ..., 0,

(2.2)

and Cγ(x(k)) (ζ) is the controller gain to be determined. Moreover, γ(x(k)) = i implies that the system realization is chosen as the ith system, i = 1, 2, ..., N. It is seen that the system (2.1) can be viewed as an autonomous switched system in which the effective subsystem changes when the state x(k) hits predefined boundaries. Ai (ζ), Bi (ζ), Ci (ζ), i = 1, 2, ..., N are given constant matrices. The system matrices are subjected to uncertainties and belong to the polytope Ω given by Ω = {[Ai , Bi , Ci ](ζ) :=

N X

ζj [Aij , Bij , Cij ],

j=1

N X

ζj = 1, ζj ≥ 0},

j=1

where Aij , Bij , Cij , i, j = 1, 2, ..., N, are given constant matrices with appropriate dimensions. The timevarying function d(k) satisfies the following condition: 0 < d1 ≤ d(k) ≤ d2 ,

∀k = 0, 1, 2, ....

Remark 2.1. It is worth noting that the time delay is a time-varying function belonging to a given interval, in which the lower bound of delay is not restricted to zero. Applying the feedback controller (2.2) to the system (2.1), the closed-loop discrete time-delay system is x(k + 1) = Ai (ζ)x(k) + Bi (ζ)Ci (ζ)x(k − d(k)),

21

k = 0, 1, 2, ...

(2.3)

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Definition 2.1. The system (2.1) is robustly stablilizable if there exist a switching function γ(.) and a delayed feedback control (2.2) such that the zero solution of the system (2.3) is asymptotically stable for all uncertainties in Ω. Definition 2.2. The system of matrices {Ji }, i = 1, 2, . . . , N, is said to be strictly complete if for every x ∈ Rn \{0} there is i ∈ {1, 2, . . . , N } such that xT Ji x < 0. It is easy to see that the system {Ji } is strictly complete if and only if N [

αi = Rn \{0},

i=1

where αi = {x ∈ Rn :

xT Ji x < 0}, i = 1, 2, ..., N.

Proposition 2.1. [28] The system {Ji }, i = 1, 2, . . . , N, is strictly complete if there exist δi ≥ 0, i = PN 1, 2, . . . , N, i=1 δi > 0 such that N X δi Ji < 0. i=1

If N = 2 then the above condition is also necessary for the strict completeness. Proposition 2.2. For real numbers ζj ≥ 0, j = 1, 2, ..., N , (N − 1)

N X

ζj2 − 2

N −1 X

PN

j=1 ζj

N X

= 1, the following inequality hold

ζj ζl ≥ 0.

j=1 l=j+1

j=1

Proof. The proof is followed from the completing the square: (N − 1)

N X

ζj2 − 2

N X

ζj ζl =

j=1 l=j+1

j=1

3

N −1 X

N −1 X

N X

(ζj − ζl )2 ≥ 0.

j=1 l=j+1

Main results

Let us set kxk k =

sup

kx(k + s)k,

s∈[−d2 ,0]

 Qj − Pj RjT − ATij Rj −RjT Bij Cij Wijj (P, Q, R) = Rj − RjT Aij Pj + Rj + RjT −RjT Bij Cij  , T T T T −Cij Bij Rj −Cij Bij Rj −Qj   Qj − Pj RjT − ATil Rj −RjT Bil Cil Wijl (P, Q, R) = Rj − RjT Ail Pj + Rj + RjT −RjT Bil Cil  , −Qj −CilT BilT Rj −CilT BilT Rj   T T T Ql − Pl Rl − Aij Rl −Rl Bij Cij Wilj (P, Q, R) = Rl − RlT Aij Pl + Rl + RlT −RlT Bij Cij  , T T T T −Cij Bij Rl −Cij Bij Rl −Ql  N N N 0 0 X X X 0 0 , P (ζ) = ζj Pj , Q(ζ) = ζj Qj , R(ζ) = ζj Rj , λ1 = λmin (P ), j=1 j=1 j=1 0 0 



R R = 0 0

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Jijj (R, Q) := (d2 − d1 )Qj − ATij Rj − RjT Aij , Jijl (R, Q) := (d2 − d1 )Qj − ATil Rj − RjT Ail , Jilj (R, Q) := (d2 − d1 )Ql − ATij Rl − RlT Aij , αijj ={x ∈ Rn : n

αijl ={x ∈ R : n

αijl ={x ∈ R : α ¯ 1jj =α1jj ,

xT Jijj (R, Q)x < 0, }, T

x Jijl (R, Q)x < 0, }, T

x Jilj (R, Q)x < 0, },

α ¯ ijj = αijj \

i−1 [

α ¯ ijj ,

i = 1, 2, ..., N, j = 1, 2, ..., N, i = 1, 2, ..., N, j = 1, 2, ..., N − 1; l = j + 1, ..., N, i = 1, 2, ..., N, j = 1, 2, ..., N − 1; l = j + 1, ..., N,

i = 2, 3, . . . , N, j = 1, 2, . . . , N,

(3.1)

i=1

α ¯ 1jl =α1jl ,

α ¯ ijl = αijl \

i−1 [

α ¯ ijl ,

i = 2, 3, . . . , N, j = 1, 2, ..., N − 1; l = j + 1, ..., N,

i−1 [

α ¯ ilj ,

i = 2, 3, . . . , N, j = 1, 2, ..., N − 1; l = j + 1, ..., N.

i=1

α ¯ 1lj =α1lj ,

α ¯ ilj = αilj \

i=1

The main result of this paper is summarized in the following theorem. Theorem 3.1. The switched control system with convex polytopic uncertainties (2.1) is stabilizable by the delayed feedback control (2.2) if there exist symmetric matrices Pi > 0, Qi > 0, R ≥ 0, i = 1, 2..., N and matrix Ri , i = 1, 2..., N satisfying the following conditions PN PN (i) ∃δi ≥ 0, i=1 δi > 0 : i=1 δi Jijj < 0, and Jijj + R < 0, j = 1, 2, . . . , N.

i = 1, 2, . . . , N,

PN PN (ii) ∃δi ≥ 0, i=1 δi > 0 : i=1 [δi Jijl + δi Jilj ] < 0, and Jijl + Jilj − i = 1, 2, . . . , N, j = 1, 2, . . . , N − 1, l = j + 1, . . . , N. (iii) Wijj + R < 0, (iv) Wijl + Wilj −

i = 1, 2, ..., N, 2 N −1 R

< 0,

2 N −1 R

< 0,

j = 1, 2, ..., N.

i = 1, 2, ..., N,

j = 1, 2, ..., N − 1;

l = j + 1, ..., N.

The switching rule is chosen as γ(x(k)) = i, whenever x(k) ∈ α ¯ ijl . Proof. Consider the following Lyapunov-Krasovskii functional for any ith system (2.1) V (k) = V1 (k) + V2 (k) + V3 (k), where V1 (k) = xT (k)P (ζ)x(k),

V2 (k) =

k−1 X

xT (i)Q(ζ)x(i),

i=k−d(k)

V3 (k) =

−d 1 +1 X

k−1 X

xT (l)Q(ζ)x(l),

j=−d2 +2 l=k+j+1

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We can verify that λ1 kx(k)k2 ≤ V (k).

(3.2)

T

Let us set ξ(k) = [x(k) x(k + 1) x(k − d(k))] , and   0 0 0 H = 0 P (ζ) 0 , 0 0 0

 P (ζ) G = R(ζ) 0

 0 0 R(ζ) 0 . 0 0

Then, the difference of V1 (k) along the solution of the system is given by ∆V1 (k) = xT (k + 1)P (ζ)x(k + 1) − xT (k)P (ζ)x(k)   0.5x(k) = ξ T (k)H(ζ)ξ(k) − 2ξ T (k)GT (ζ)  0  . 0

(3.3)

because of ξ T (k)H(ζ)ξ(k) = x(k + 1)P (ζ)x(k + 1). Using the expression of system (2.3) 0 = −x(k + 1) + Ai (ζ)x(k) + Bi (ζ)Ci (ζ)x(k − d(k)), we have  0.5x(k) −2ξ (k)G (ζ) −x(k + 1) + Ai (ζ)x(k) + Bi (ζ)Ci (ζ)x(k − d(k)) ξ(k) 0    0.5I 0 0 0.5I Ai (ζ)T T T T −I = −ξ (k)G (ζ) Ai (ζ) −I Bi (ζ)Ci (ζ) ξ(k) − ξ (k)  0 0 0 0 0 (Bi (ζ)Ci (ζ))T T



T

 0 0 G(ζ)ξ(k). 0

Therefore, from (3.3) it follows that ∆V1 (k) = ξ T (k)Wi (P (ζ), Q(ζ), R(ζ))ξ(k), where



  0 0 0 0.5I Wi (P (ζ), Q(ζ), R(ζ)) = 0 P (ζ) 0 − GT (ζ) Ai (ζ) 0 0 0 0   0.5I ATi (ζ) 0  0 −I 0 G(ζ). − 0 (Bi (ζ)Ci (ζ))T 0

0 −I 0

(3.4)  0 Bi (ζ)Ci (ζ) 0

The difference of V2 (k) is given by k X

∆V2 (k) =

xT (i)Q(ζ)x(i) −

i=k+1−d(k+1) k−d X1

=

k−1 X

xT (i)Q(ζ)x(i)

i=k−d(k)

xT (i)Q(ζ)x(i) + xT (k)Q(ζ)x(k) − xT (k − d(k))Q(ζ)x(k − d(k))

(3.5)

i=k+1−d(k+1)

+

k−1 X

i=k+1−d1

T

x (i)Q(ζ)x(i) −

k−1 X

xT (i)Q(ζ)x(i).

i=k+1−d(k)

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Since d(k) ≥ d1 we have k−1 X

k−1 X

xT (i)Q(ζ)x(i) −

i=k+1−d1

xT (i)Q(ζ)x(i) ≤ 0,

i=k+1−d(k)

and hence from (3.5) we have ∆V2 (k) ≤

k−d X1

xT (i)Q(ζ)x(i) + xT (k)Q(ζ)x(k) − xT (k − d(k))Q(ζ)x(k − d(k)).

(3.6)

i=k+1−d(k+1)

The difference of V3 (k) is given by ∆V3 (k) =

−d 1 +1 X

k X

x (l)Q(ζ)x(l) −

−d 1 +1 X

k−1 X

xT (l)Q(ζ)x(l) + xT (k)Q(ζ)(ξ)x(k)

T

j=−d2 +2 l=k+j

=

[

−d 1 +1 X

k−1 X

xT (l)Q(ζ)x(l)

j=−d2 +2 l=k+j+1

j=−d2 +2 l=k+j

−

k−1 X

xT (l)Q(ζ)x(l) − xT (k + j − 1)Q(ζ)x(k + j − 1)]

(3.7)

l=k+j

=

−d 1 +1 X

[xT (k)Q(ζ)x(k) − xT (k + j − 1)Q(ζ)x(k + j − 1)]

j=−d2 +2

= (d2 − d1 )xT (k)Q(ζ)x(k) −

k−d X1

xT (j)Q(ζ)x(j).

j=k+1−d2

Since d(k) ≤ d2 , and k−d X1

xT (i)Q(ζ)x(i) −

k−d X1

xT (i)Q(ζ)x(i) ≤ 0,

i=k+1−d2

i=k=1−d(k+1)

we obtain from (3.6) and (3.7) that ∆V2 (k) + ∆V3 (k) ≤ (d2 − d1 + 1)xT (k)Q(ζ)x(k) − xT (k − d(k))Q(ζ)x(k − d(k)).

(3.8)

Therefore, combining the inequalities (3.4), (3.8) gives ∆V (k) ≤ xT (k)Ji (R(ζ), Q(ζ))x(k) + ξ T (k)Wi (P (ζ), Q(ζ), R(ζ))ξ(k),

(3.9)

where 

Q(ζ) − P (ζ) Wi (P (ζ), Q(ζ), R(ζ)) = R(ζ) − RT (ζ)Ai (ζ) −CiT (ζ)BiT (ζ)R(ζ)

RT (ζ) − ATi (ζ)R(ζ) P (ζ) + R(ζ) + RT (ζ) −CiT (ζ)BiT (ζ)R(ζ)

 −RT (ζ)Bi (ζ)Ci (ζ) −RT (ζ)Bi (ζ)Ci (ζ) , −Q(ζ)

and Ji (R(ζ), Q(ζ)) = (d2 − d1 )Q(ζ) − ATi (ζ)R(ζ) − RT (ζ)Ai (ζ). Let us denote



Qj − Pj Wijj (P, Q, R) = Rj − RjT Aij T T −Cij Bij Rj

25

RjT − ATij Rj Pj + Rj + RjT T T −Cij Bij Rj

 −RjT Bij Cij −RjT Bij Cij  , −Qj

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RjT − ATil Rj Pj + Rj + RjT −CilT BilT Rj



Qj − Pj Wijl (P, Q, R) = Rj − RjT Ail −CilT BilT Rj  Ql − Pl Wilj (P, Q, R) = Rl − RlT Aij T T −Cij Bij Rl

 −RjT Bil Cil −RjT Bil Cil  , −Qj  −RlT Bij Cij −RlT Bij Cij  , −Ql

RlT − ATij Rl Pl + Rl + RlT T T −Cij Bij Rl

Jijj (R, Q) := (d2 − d1 )Qj − ATij Rj − RjT Aij , Jijl (R, Q) := (d2 − d1 )Qj − ATil Rj − RjT Ail , Jilj (R, Q) := (d2 − d1 )Ql − ATij Rl − RlT Aij , (ATi R)jl := ATil Rj + ATij Rl ,

(RT Ai )jl = RjT Ail + RlT Aij ,

(RT Bi Ci )jl = RjT Bil Cil + RlT Bij Cij , Pjl = Pj + Pl ,

T T (CiT BiT R)jl = CilT BilT Rj + Cij Bij Rl ,

Qjl = Qj + Ql ,

Rjl = Rj + Rl .

From the convex combination of the expression of P (ζ), Q(ζ), R(ζ), A(ζ), B(ζ), C(ζ), we have   Qj − Pj RjT − ATij Rj −RjT Bij Cij N X Wi (P (ζ), Q(ζ), R(ζ)) = ζj2 Rj − RjT Aij Pj + Rj + RjT −RjT Bij Cij  T T T T j=1 −Qj Bij Rj −Cij Bij Rj −Cij   T Qj − Pj + Ql − Pl Rjl − (ATi R)jl −(RT Bi Ci )jl N −1 X N X T Pjl + Rjl + Rjl −(RT Bi Ci )jl  + ζj ζl  Rjl − (RT Ai )jl T T T T j=1 l=j+1 −(Ci Bi R)jl −(Ci Bi R)jl −Qjl =

N X

ζj2 Wijj (P, Q, R) +

N −1 X

N X

ζj ζl [Wijl (P, Q, R) + Wilj (P, Q, R)].

j=1 l=j+1

j=1

Ji (R(ζ), Q(ζ)) =

N X

+

N −1 X

ζj2 (d2 − d1 )Qj − ATij Rj − RjT Aij

j=1

N X

ζj ζl (d2 − d1 )Qjl − (ATi R)jl − (RT Ai )jl

j=1 l=j+1

=

N X

N −1 X

ζj2 Jijj (Q, R) +

j=1

N X

ζj ζl [Jijl (Q, R) + Jilj (Q, R)].

j=1 l=j+1

Then the conditions (i)-(iv) give Wi (P (ζ), Q(ζ), R(ζ)) < −

N X

ζj2 R +

j=1

Ji (R(ζ), Q(ζ)) < −

N X

N −1 N 2 X X ζj ζl R ≤ 0, N − 1 j=1 l=j+1

ζj2 R +

j=1

N −1 N 2 X X ζj ζl R ≤ 0, N − 1 j=1 l=j+1

because of Proposition 2.2: (N − 1)

N X j=1

ζj2 − 2

N −1 X

N X

ζj ζl =

j=1 l=j+1

26

N −1 X

N X

(ζj − ζl )2 ≥ 0.

j=1 l=j+1

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Therefore, we finally obtain from (3.9) and the condition (iii), (iv) that ∆V (k) < xT (k)Ji (R(ζ), Q(ζ))x(k),

∀i = 1, 2, ...., N, k = 0, 1, 2, ....

We now apply the condition (i), (ii), and Proposition 2.1., the system Ji (R(ζ), Q(ζ)) is strictly complete, and the sets αijl and α ¯ ijl by ( 3.1) are well defined such that N [

αijl = Rn \{0},

i=1 N [

α ¯ ijl = Rn \{0},

α ¯ ijl ∩ α ¯ tjl = ∅, i 6= t.

i=1

Therefore, for any x(k) ∈ Rn , k = 0, 1, 2, ...., there exists i ∈ {1, 2, . . . , N } such that x(k) ∈ α ¯ ijl . By choosing switching rule as γ(x(k)) = i whenever x(k) ∈ α ¯ ijl , from the condition (3.9) we have ∆V (k) ≤ xT (k)Ji (R(ζ), Q(ζ))x(k) < 0,

k = 1, 2, ...,

which, combining the condition (3.2) and the Lyapunov stability theorem [29], concludes the proof of the theorem. Remark 3.1. Note that theresult sproposed in [4,5,6] for switching systems to be asymptotically stable under an arbitrary switching rule. The asymptotic stability for switching linear discrete time-delay systems studied in [9] was limited to constant delays. In [10], a class of switching signals has been identified for the considered switched discrete-time delay systems to be stable under the averaged well time scheme.

4

Conclusion

This paper has proposed a switching design for the robust stabilization of switched linear discrete-time systems with convex polytopic uncertainties with interval time-varying delays. Based on the discrete Lyapunov functional, a switching rule for the robust stabilization for the system with convex polytopic uncertainties is designed via linear matrix inequalities. Acknowledgments. This work was supported by the Thai Research Fund Grant, the Higher Education Commission and Faculty of Science, Maejo University, Thailand.

References [1] D. Liberzon, A.S. Morse, Basic problems in stability and design of switched systems, IEEE Control Syst. Mag., 19(1999), 57-70. [2] A.V. Savkin and R.J. Evans, Hybrid Dynamical Systems: Controller and Sensor Switching Problems, Springer, New York, 2001. [3] Z. Sun and S.S. Ge, Switched Linear Systems: Control and Design, Springer, London, 2005. [4] F. Gao, S. Zhong and X. Gao, Delay-dependent stability of a type of linear switching systems with discrete and distributed time delays, Appl. Math. Computation, 196(2008), 24-39. [5] C.H. Lien, K.W. Yu, Y.J. Chung, Y.F. Lin, L.Y. Chung and J.D. Chen, Exponential stability analysis for uncertain switched neutral systems with interval-time-varying state delay, Nonlinear Analysis: Hybrid systems, 3(2009),334–342.

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[6] V.N. Phat, T. Bormat and P. Niamsup, Switching design for exponential stability of a class of nonlinear hybrid time-delay systems, Nonlinear Analysis: Hybrid Systems, 3(2009), 1-10. [7] G Rajchakit, T Rojsiraphisal, M Rajchakit, Robust stability and stabilization of uncertain switched discrete-time systems, Advances in Difference Equations 2012, 2012:134. doi:10.1186/1687-1847-2012134 [8] S. Boyd, L.E. Ghaoui, E. Feron and V. Balakrishnan, Linear Matrix Inequalities in System and Control Theory, SIAM, Philadelphia, 1994. [9] D.H. Ji, J.H. Park, W.J. Yoo and S.C. Won, Robust memory state feedback model predictive control for discrete-time uncertain state delayed systems, Appl. Math. Computation, 215(2009), 2035-2044. [10] K. Ratchagit and V.N. Phat, Stability criterion for discrete-time systems, J. Ineq. Appl., 2010(2010), 1-6. [11] G.S. Zhai, B. Hu, K. Yasuda, and A. Michel, Qualitative analysis of discrete- time switched systems. In: Proc. of the American Control Conference, 2002, 1880-1885. [12] V.N. Phat, Robust stability and stabilizability of uncertain linear hybrid systems with state delays, IEEE Trans. CAS II, 52(2005), 94-98. [13] K. Ratchagit, A switching rule for the asymptotic stability of discrete-time systems with convex polytopic uncertainties, Asian-European J. Math., 5(2012), 1250025 (12 pages). [14] W.A. Zhang, Li Yu, Stability analysis for discrete-time switched time-delay systems, Automatica, 45(2009), 2265-2271. [15] V. N. Phat and K. Ratchagit, Stability and stabilization of switched linear discrete-time systems with interval time-varying delay, Nonlinear Analysis: Hybrid Systems, 5(2011), 605-612. [16] M Rajchakit, G Rajchakit, Mean square robust stability of stochastic switched discrete-time systems with convex polytopic uncertainties, Journal of Inequalities and Applications 2012, 2012:135. doi:10.1186/1029-242X-2012-135 [17] Huaiqin Wu, Ning Li, Kewang Wang, Guohua Xu, and Qiangqiang Guo, Global Robust Stability of Switched Interval Neural Networks with Discrete and Distributed Time-Varying Delays of Neural Type, Mathematical Problems in Engineering, 2012(2012). [18] Manlika Rajchakit, Piyapong Niamsup, Grienggrai Rajchakit, A switching rule for exponential stability of switched recurrent neural networks with interval time-varying delay, Advances in Difference Equations 2013, 2013:44. doi:10.1186/1687-1847-2013-44 [19] Binbin Du and Xiaojie Zhang, Delay-Dependent Stability Analysis and Synthesis for Uncertain Impulsive Switched System with Mixed Delays, Discrete Dynamics in Nature and Society, 2011(2011). [20] M. De la Sen and A. Ibeas, Stability Results of a Class of Hybrid Systems under Switched ContinuousTime and Discrete-Time Control, Discrete Dynamics in Nature and Society, 2009(2009). [21] M. Rajchakit and G. Rajchakit, Mean Square Exponential Stability of Stochastic Switched System with Interval Time-Varying Delays, Abstract and Applied Analysis, vol. 2012, Article ID 623014, 12 pages, 2012. doi:10.1155/2012/623014 [22] Jinxing Lin and Chunxia Fan, Exponential Admissibility and Dynamic Output Feedback Control of Switched Singular Systems with Interval Time-Varying Delay, Mathematical Problems in Engineering, 2010(2010).

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[23] M. Rajchakit and G. Rajchakit, LMI approach to robust stability and stabilization of nonlinear uncertain discrete-time systems with convex polytopic uncertainties, Advances in Difference Equations, 2012(2012). [24] Jun Li, Weigen Wu, Jimin Yuan, Qianrong Tan, and Xing Yin, Delay-Dependent Stability Criterion of Arbitrary Switched Linear Systems with Time-Varying Delay, Discrete Dynamics in Nature and Society, 2010(2010). [25] Liguo Zhang, Hongfeng Li, and Yangzhou Chen, Robust Stability Analysis and Synthesis for Switched Discrete-Time Systems with Time Delay, Discrete Dynamics in Nature and Society, 2010(2010). [26] M. De la Sen, On the Characterization of Hankel and Toeplitz Operators Describing Switched Linear Dynamic Systems with Point Delays, Abstract and Applied Analysis, 2009(2009). [27] K. Ratchagit and V. N. Phat, Robust Stability and Stabilization of Linear Polytopic Delay-Difference Equations with Interval Time-Varying Delays, Neural, Parallel, and Scientific Computations, 19(2011), 361-372. [28] F. Uhlig, A recurring theorem about pairs of quadratic forms and extensions, Linear Algebra Appl., 25(1979), 219-237. [29] R.P. Agarwal, Difference Equations and Inequalities, Second Edition, Marcel Dekker, New York, 2000.

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A preconditioner for block two-by-two symmetric indefinite matrices ∗ Chun Wen †, Ting-Zhu Huang‡, School of Mathematical Sciences, University of Electronic Science and Technology of China, Chengdu, Sichuan, 611731, P.R. China

Abstract A new preconditioner for the numerical solution of block two-by-two symmetric indefinite matrices is presented in this paper. The proposed preconditioner is constructed as the product of two fairly simple preconditioners: one is the famous block Jacobi preconditioner, and the other is the popular constraint preconditioner. Here, we call it the product preconditioner. Results concerning the eigenvalue distribution and form of the eigenvectors of the product preconditioned matrix are analyzed. Numerical experiments are used to illustrate the efficiency of the proposed product preconditioner. Key words: Product preconditioner; Symmetric indefinite matrices; Krylov subspace method AMSC(2010): 65F10; 65N22

1

Introduction

Recently, a large amount of work has been devoted to the problem of solving linear systems in saddle point form. Here, our concern is to construct a new preconditioner for the numerical solution of block two-by-two symmetric indefinite matrices whose (1,1) and (2,2) block are nonsingular. Often this kind of linear systems in saddle point form is likely to generate from a wide range of applications, such as the Helmholtz equation { ∆u + (2π)2 u = f (x, y), (x, y) ∈ Ω ∪ ℜ+ 2, (1) u = 0, (x, y) ∈ ∂(Ω ∪ ℜ+ 2 ), with radiation boundary condition lim r(

r→∞

∂u − ı2πu) = 0, ∂η

(2)

where Ω = [0, 1] × [−1, 0] is a unit square domain, ℜ+ 2 denotes the upper half-space and ı is the imaginary unit in (2), see [1, 2] for details. ∗ This research is supported by NSFC (61170311), Chinese Universities Specialized Research Fund for the Doctoral Program (20110185110020), Sichuan Province Sci. & Tech. Research Project (12ZC1802). † E-mail: [email protected] ‡ E-mail: [email protected]

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Chun Wen, Ting-Zhu Huang By using a finite difference discretization to the Helmholtz equation (1) on the uniform grid of Ω, we obtain the linear system in saddle point form ( )( ) ( ) A B x f Au = = = b, (3) B T −C y g where A ∈ Rn×n and C ∈ Rm×m are nonsingular, B ∈ Rn×m , u = [xT , y T ]T ∈ Rn+m and b = [f T , g T ]T ∈ Rn+m , with x, f ∈ Rn and y, g ∈ Rm , are the unknown and given right-hand side vectors, respectively. Then the coefficient matrix A ∈ R(n+m)×(n+m) is a nonsingular, symmetric and possibly indefinite matrix, and our main aim is to solve the linear system (3) of n + m linear equations with n + m unknowns. Iterative procedure is a convenient numerical solution method for computing the linear system (3). Often we have Uzawa’s algorithms [3, 4] and multigrid methods [5, 6]. In particular, Krylov subspace methods have become more and more popular for solving the linear system (3), such as the conjugate gradient (CG) and biconjugate gradient stabilized (Bi-CGSTAB) methods, minimal residual method (MINRES), generalized minimal residual (GMRES) and quasi-minimal residual (QMR) methods which have been considered in [7–14]. However, these iterative methods are all likely to suffer from slow convergence for some large linear systems which come from many practical applications like the computational fluid dynamics and structural mechanics. Thus it is necessary to use the idea of preconditioning such that the preconditioned matrix has a tightly clustered eigenvalues, see [1, 15–22] and the references therein. More precisely, we see that a kind of triangular preconditioner has been proposed by Elman and Silvester [14] and Elman [23] when the (2,2) block matrix C = 0. These triangular preconditioners were extended by Kay, Loghin and Wathen [24], Cao [25] and Simoncini [26] to the case where C is symmetric positive or negative semidefinite. In addition, Keller, Gould and Wathen [18] presented a constraint preconditioner for the case C = 0, in which they discussed the eigenvalue distribution and form of the eigenvectors of the constraint preconditioned matrix and its minimal polynomial. Thereafter, Dollar and Wathen [19] and Dollar [22] studied an approximation factorization constraint preconditioner by combining with the conjugate gradient method, and extended the idea of [18] by allowing the matrix C to be symmetric and positive semidefinite. Furthermore, we found block diagonal, triangular and constraint preconditioners had been discussed by Siefert and De Sturler [17], Murphy, Golub and Wathen [15], De Sturler and Liesen [16], and Cao [20, 21] for the numerical solution of nonsymmetric or generalized saddle point problems. More preconditioning techniques for solving the linear system in saddle point form can be found in an excellent survey written by Benzi, Golub and Liesen [1]. In this paper, we are concerned with investigating a new preconditioner for the symmetric indefinite linear system (3). The proposed preconditioner is constructed as the product of two fairly simple preconditioners: one is the famous block Jacobi preconditioner, and the other is the popular constraint preconditioner [22]. We call it the product preconditioner. The idea used to develop the product preconditioner can trace back to [27]. Benzi has used the idea in [27] to solve Markov chain problems, see [28, 29]. Results concerning the eigenvalue distribution and form of the eigenvectors of the product preconditioned matrix are given in this paper. Numerical experiments with preconditioned GMRES method [30] on certain problem serve to illustrate the efficiency and stability of the proposed product preconditioner. The remainder of this paper is organized as follows. In Section 2, we first briefly introduce the background material on stationary iterations and matrix splittings, and then construct the

31

CHUN WEN ET AL 30-41

J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 16, NO.1, 2014, COPYRIGHT 2014 EUDOXUS PRESS, LLC

A preconditioner for block two-by-two symmetric indefinite matrices product preconditioner. In Section 3, we analyze the eigensolution distribution of the product preconditioned matrix. Numerical experiments with various preconditioned GMRES methods are presented in Section 4. Finally, conclusions are made in Section 5.

2

Background and product preconditioner

In this section, we first briefly introduce the background material on stationary iterations and matrix splittings from [27, 28, 31], and then construct the product preconditioner.

2.1

Stationary iterations and matrix splittings

Consider the solution of a large sparse linear system of the form Au = b, where A is a square and nonsingular, symmetric indefinite matrix, and b is the given right-hand vector. Stationary iterative method is likely to be an attractive method by using a splitting of the coefficient matrix A, denoted as A = M − N, where M is a nonsingular matrix. Then the splitting gives rise to the stationary iterative method uk+1 = T uk + c,

k = 0, 1, · · · ,

(4)

where T = M −1 N is called the iterative matrix, c = M −1 b, and u0 is a given initial guess. It is well known that the iterative method (4) converges for any initial guess u0 if and only if its spectral radius ρ(T ) < 1 [31]. Recently, Benzi and Szyld have defined a related approach by the alternating iterations { uk+1/2 = M1−1 N1 uk + M1−1 b, k = 0, 1, · · · , (5) uk+1 = M2−1 N2 uk+1/2 + M2−1 b, in an excellent paper [27], where A = M1 − N1 = M2 − N2 are splittings of A, both M1 and M2 matrices are nonsingular, and u0 is defined as above. Not only the existence and uniqueness of splittings for stationary iterative methods with applications to alternating methods were proved, but also the convergence theory of some alternating iterations were analyzed in [27]. In addition, Benzi and Szyld have constructed a splitting A = M − N based on the nonsingular matrix M1 and M2 . The splitting is given by (see Eq. (10) in [27]) M −1 = M2−1 (M1 + M2 − A)M1−1 .

(6)

Evidently, the matrix M1 + M2 − A must be nonsingular for (6) to be well defined.

2.2

Product preconditioner

Now, we construct the product preconditioner as the multiplication of two fairly simple preconditioners from the derivation of the alternating iterations in [27]. The first preconditioner is the famous block Jacobi preconditioner ( ) A O Mbj = . (7) O −C

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CHUN WEN ET AL 30-41

J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 16, NO.1, 2014, COPYRIGHT 2014 EUDOXUS PRESS, LLC

Chun Wen, Ting-Zhu Huang

Note that Mbj is nonsingular since both A and C are invertible. The second preconditioner is the popular nonsingular constraint preconditioner ( ) G B Msc = B T −C

(8)

discussed in [22], where G ∈ Rn×n is an approximation of A, but is not equal to A. In practice, G is often taken to be the diagonal matrix formed with the diagonal entries of A, i.e., G = diag(diag(A)). Note that the Schur complement matrices −(C +B T A−1 B) and −(C +B T G−1 B) are nonsingular since matrix A in (3) and Msc in (8) are nonsingular (proof can be found in [20]). According to the alternating iterations (5) and equation (6), the product preconditioner Mps is given by −1 −1 −1 Mps = Msc (Mbj + Msc − A)Mbj , (9) (

where the matrix Mbj + Msc − A =

G O O −C

)

−1 is well defined. From equation (9), we have the product preconditioner is invertible. Hence, Mps

( Mps = Mbj (Mbj + Msc −

A)−1 Msc

=

A AG−1 B BT −C

) .

(10)

Also, we can rewrite ( ) ( )( ) A AG−1 B I O A AG−1 B Mps = = , BT −C B T A−1 I O −(C + B T G−1 B) then, we have ( −1 Mps

=

A−1 − G−1 B(C + B T G−1 B)−1 B T A−1 G−1 B(C + B T G−1 B)−1 (C + B T G−1 B)−1 B T A−1 −(C + B T G−1 B)−1

) .

−1 A can be expressed as Finally, the product preconditioned matrix Mps −1 Mps A

3

( =

I A−1 B − G−1 B(C + B T G−1 B)−1 (C + B T A−1 B) O (C + B T G−1 B)−1 (C + B T A−1 B)

) .

(11)

−1 Properties of the preconditioned matrix Mps A

In this section, we focus on analyzing the eigenvalue distribution and form of the eigenvectors −1 A. of the product preconditioned matrix Mps

3.1

Eigenvalue distribution

In this section, we consider the eigenvalue distribution of the product preconditioned matrix −1 A. It is well known that the convergence of an iterative method has close relation to the Mps distribution of the eigenvalues of the coefficient matrix for symmetric matrix systems. Hence, 33

CHUN WEN ET AL 30-41

J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 16, NO.1, 2014, COPYRIGHT 2014 EUDOXUS PRESS, LLC

A preconditioner for block two-by-two symmetric indefinite matrices

a desired eigenvalue distribution is wished to obtain by the applications of preconditioning techniques. We prove a result of this type as follows. Theorem 1. Let A ∈ R(n+m)×(n+m) defined in (3) be a nonsingular and symmetric indefinite matrix. Preconditioning A by the product preconditioner ( ) A AG−1 B Mps = , BT −C where G ∈ Rn×n is an approximation of A, G ̸= A, A ∈ Rn×n and C ∈ Rm×m are nonsingular, −1 A has B ∈ Rn×m . Then the product preconditioned matrix Mps • an eigenvalue at 1 with multiplicity n; • m eigenvalues which are defined by the generalized eigenvalue problem (C + B T A−1 B)y = λ(C + B T G−1 B)y. −1 A, and [xT , y T ]T ̸= 0 is the corresponding Proof. Suppose λ is the eigenvalue of Mps eigenvector. Besides, from (11), we have the preconditioned matrix ( ) I A−1 B − G−1 B(C + B T G−1 B)−1 (C + B T A−1 B) −1 Mps A = , O (C + B T G−1 B)−1 (C + B T A−1 B)

where A−1 B − G−1 B(C + B T G−1 B)−1 (C + B T A−1 B) is irrelevant to the results in Theorem 1. Hence, by making use of the related knowledge in linear algebra, we obtain the results in Theorem 1 immediately. 

3.2

Eigenvector distribution

To our knowledge, the termination of a Krylov subspace method is not only related to the distribution of eigenvalues of the preconditioned matrix, but also to the number of corresponding linearly independent eigenvectors. Hence, for completeness of this paper, we establish the rela−1 A and discuss tionship between eigenvalues and eigenvectors of the preconditioned matrix Mps its eigenvector distribution. The following analysis is similar to the discussions in [4, 18, 22]. We start this part from the generalized eigenvalue problem ( )( ) ( )( ) A B x A AG−1 B x = λ , (12) B T −C y BT −C y −1 A, and [xT , y T ]T ̸= 0 is the corresponding eigenvector. By where λ is the eigenvalue of Mps calculations, we obtain Ax + By = λAx + λAG−1 By (13)

and B T x − Cy = λ(B T x − Cy).

(14)

From (14), we obtain (1 − λ)(B T x − Cy) = 0. Hence, either λ = 1 or B T x − Cy = 0 holds true. In the former case, we have By = AG−1 By. (15) Evidently, equation (15) is satisfied by y = 0, and thus there are n linearly independent eigenvectors of the form (xT , 0T )T associated with the unit eigenvalue. On the other hand, there 34

CHUN WEN ET AL 30-41

J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 16, NO.1, 2014, COPYRIGHT 2014 EUDOXUS PRESS, LLC

Chun Wen, Ting-Zhu Huang may exist y ̸= 0 which satisfies (15). Then, without loss of generality, we suppose that there are i (0 ≤ i ≤ m) linearly independent eigenvectors of the form [xT , y T ]T , where the components y result from the eigenvalue problem By = AG−1 By. Now, suppose λ ̸= 1, then we have B T x − Cy = 0, which implies y = C −1 B T x since C is nonsingular. Substituting this into equation (13), we get the generalized eigenvalue problem (A + BC −1 B T )x = λ(A + AG−1 BC −1 B T )x,

(16)

where x is impossible to be equal to a zero vector. Since if x = 0, then we have y = 0, which is conflict with the known condition [xT , y T ]T ̸= 0. Therefore, we suppose there exist j (0 ≤ j ≤ n) linearly independent eigenvectors of the form [xT , y T ]T , where components x arise from the eigenvalue problem (16) with y = C −1 B T x. We conclude this subsection with the following theorem. Theorem 2. Let A ∈ R(n+m)×(n+m) defined in (3) be a nonsingular and symmetric indefinite matrix. Preconditioning A by the product preconditioner ( Mps =

A AG−1 B BT −C

) ,

where G ∈ Rn×n is an approximation of A, G ̸= A, A ∈ Rn×n and C ∈ Rm×m are nonsingular, −1 A has n + m eigenvalues as given in B ∈ Rn×m . Then the product preconditioned matrix Mps Theorem 1 and n + i + j linearly independent eigenvectors. There are • n eigenvectors of the form [xT , 0T ]T that correspond to case λ = 1; • ∃ i (0 ≤ i ≤ m) eigenvectors of the form [xT , y T ]T , where the components y construct a basis of the generalized eigenvalue problem By = AG−1 By and λ = 1; • ∃ j (0 ≤ j ≤ n) eigenvectors of the form [xT , y T ]T that correspond to case λ ̸= 1. Proof. According to the analysis above, we have obtained the specific form of the eigenvec−1 A. Now, our aim is to prove that the n+i+j eigenvectors tors of the preconditioned matrix Mps are linearly independent, that is, we need to show that (

(1)

x1 0

··· ···

(1)

xn 0



(1)

a1  ..  .

  +

(

(1)

an (

+

)

(3) x1 (3) y1

··· ···

(3) xj (3) yj

)



(2)

x1 ··· (2) y1 ···

(2)

)

xi (2) yi

  (3) a1  .    ..  =    (3) aj



(2)

a1  ..  .

  

(2)

ai

 0 ..  .  0

(17)

implies that the vectors a(k) (k = 1, 2, 3) are zero vectors. Multiplying (17) by the preconditioned −1 A, and recalling that the first matrix in (17) arises from the case λ = 1 (k = matrix Mps k 1, · · · , n), the second matrix from the case λk = 1 (k = 1, · · · , i), where the components y are 35

CHUN WEN ET AL 30-41

J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 16, NO.1, 2014, COPYRIGHT 2014 EUDOXUS PRESS, LLC

A preconditioner for block two-by-two symmetric indefinite matrices basis vectors of the generalized eigenvalue problem By = λAG−1 By, and the last matrix from the case λk ̸= 1 (k = 1, · · · , j). We have     ( ) a(1) ( ) a(2) 1 1 (2) (2) (1) (1) x1 · · · xi  ..   ..  x1 · · · xn  . +  .  (2) (2) 0 ··· 0 y1 · · · yi (2) (1) ai an     (3) (3) ( (3) ) λ a 0 (3)  1 . 1   .  x1 · · · xj   .. (18) + (3) (3)   =  ..  . y1 · · · yj (3) (3) 0 λj aj Subtracting (17) from (18), we obtain (

(3) x1 (3) y1

··· ···

(3) xj (3) yj

)



  (3) (3) (λ1 − 1)a1    ..  = .   (3) (3) (λj − 1)aj

 0 ..  . .  0

(3)

Since the components xk (k = 1, · · · , j) are linearly independent eigenvectors which arise (3) (3) from the generalized eigenvalue problem (16) and yk = C −1 B T xk (k = 1, · · · , j). Thus we have (3) (λk − 1)ak = 0, k = 1, · · · , j. (3)

As a result of the eigenvalues λk (k = 1, · · · , j) are nonunit. We obtain ak = 0 (k = 1, · · · , j). (2) In addition, we know the components yk (k = 1, · · · , i) are basis vectors of the equation (2) By = AG−1 By, which implies that yk (k = 1, · · · , i) are linearly independent. Thus we have (2) ak = 0 (k = 1, · · · , i). (2) (3) Therefore, substituting ak = 0 (k = 1, · · · , i) and ak = 0 (k = 1, · · · , j) into (17), then equation (17) simplifies to     ( ) a(1) 0 1 (1) (1)  ..   ..  x1 · · · xn  .  =  . . 0 ··· 0 (1) 0 an (1)

(1)

Clearly, ak = 0 (k = 1, · · · , n) follows from the linear independence of xk Summarizing the discussions above, we obtain a(k) = 0 (k = 1, 2, 3). 

4

(k = 1, · · · , n).

Numerical experiments

In this section, we report on numerical results obtained with a Matlab 7. 0.1 implementation on a Window-XP with 2.93GHz 64-bit processor and 2GB memory. The main goal is to test the product preconditioner (PS) defined in (10) and to compare it with the block diagonal preconditioner (BD) ( ) G O Mbd = , (19) O −(C + B T G−1 B) 36

CHUN WEN ET AL 30-41

J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 16, NO.1, 2014, COPYRIGHT 2014 EUDOXUS PRESS, LLC

Chun Wen, Ting-Zhu Huang

presented in [16, 17, 24, 25], the block triangular preconditioner (BT) ( Mbt =

G B O −(C + B T G−1 B)

) ,

(20)

considered in [11, 16, 23–26] and the constraint (SC) preconditioners given in (8) by the computing time (CPU), iteration step (IT) and relative residual error (RES). There are various strategies to choose G in PS, SC, BD and BT preconditioners. In our computations, we not only take G to be the diagonal matrix formed with the diagonal entries of A, i.e., G = diag(diag(A)), but also to be the tridiagonal matrix of the (1,1) block matrix of A, that is, G = tridiag(A). As a representative iterative solver we used GMRES [30] with the right preconditioning in our experiments. All iterations are started from the zero vector, and terminated when RES = ∥b − Au∥2 /∥b∥2 ≤ 10−9 . The test problem is the Helmholtz equation (1), together with radiation boundary condition (2), see [2, 9] for details. By using a finite difference discretization to equation (1) on the uniform grid of Ω, we obtain the nonsingular and symmetric indefinite linear system (3), where A ∈ Rn×n and C ∈ Rm×m are nonsingular, B ∈ Rn×m . To be more precise, we have matrix A = K ⊗ I + I ⊗ K + I ⊗ D,

B = −(I ⊗ en ),

C = I − hT,

with K = tridiag(−1, 2, −1) ∈ Rp×p , D = −4π 2 h2 I, I ∈ Rp×p an identity matrix, en = [0, 0, · · · , 0, 1]T ∈ Rp , h = 1/(p + 1), and T ∈ Rp×p a Toeplitz matrix which results from the generating function f (θ) = 2|θ|(θ2 − 1). Hence, we have n = p2 , m = p, and the order of the coefficient matrix A is n + m. Moreover, we choose the right-hand vector b = [f T , g T ]T ∈ Rn+m such that the exact solution of system (3) is [xT , y T ]T = [1, 1, · · · , 1]T , and GMRES(50) with at most 50 restarts is used in our experiments thus the number 2500 in Table 1 and Table 2 means that the corresponding preconditioned GMRES method does not converge in 2500 iterations. h n+m BD IT CPU RES BT IT CPU RES SC IT CPU RES PS IT CPU RES

1/32 992 123 0.4530 8.4763e-10 98 0.3440 6.9060e-10 98 0.3280 7.4747e-10 8 0.1250 9.2136e-10

1/48 2256 325 2.5470 9.7414e-10 165 1.3750 9.6006e-10 217 1.5930 9.4590e-10 9 0.4530 4.2030e-10

1/64 4032 794 10.7350 9.7214e-10 554 7.7040 9.6952e-10 341 4.4210 9.7868e-10 10 1.2650 9.2256e-11

1/80 6320 530 12.6410 9.9784e-10 785 18.0790 9.8453e-10 850 17.6250 9.9417e-10 10 2.7180 1.5617e-10

1/90 8010 1108 33.2660 9.9993e-10 731 22.9220 9.9223e-10 976 27.5470 9.9499e-10 10 4.0620 1.9037e-10

Table 1: IT, CPU and RES of the BD, BT, SC and PS preconditioned GMRES methods for this Helmholtz equation when G = tridiag(A).

37

CHUN WEN ET AL 30-41

J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 16, NO.1, 2014, COPYRIGHT 2014 EUDOXUS PRESS, LLC

A preconditioner for block two-by-two symmetric indefinite matrices

h n+m BD IT CPU RES BT IT CPU RES SC IT CPU RES PS IT CPU RES

1/32 992 261 0.4220 8.7031e-10 211 0.3280 9.5016e-10 200 0.3430 9.8438e-10 9 0.0780 6.2956e-11

1/48 2256 615 2.0320 9.9129e-10 603 2.0160 9.7251e-10 410 1.3900 9.6321e-10 9 0.1720 6.5957e-10

1/64 4032 1085 5.8590 9.9364e-10 969 5.2500 9.9961e-10 1115 6.3750 9.9669e-10 10 0.3430 1.2888e-10

1/80 6320 1316 11.6560 9.9653e-10 2003 18.4530 9.9769e-10 1025 9.5320 9.9912e-10 10 0.5320 2.1624e-10

1/90 8010 2500 28.7190 8.1554e-09 1403 16.2660 9.9885e-10 1470 18.0310 9.9463e-10 10 0.7190 2.6212e-10

Table 2: IT, CPU and RES of the BD, BT, SC and PS preconditioned GMRES methods for this Helmholtz equation when G = diag(diag(A)).

Figure 1: Comparisons of the eigenvalue distribution of the BD, BT, SC and PS preconditioned matrices for this Helmholtz equation when G = tridiag(A) and n + m = 992.

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CHUN WEN ET AL 30-41

J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 16, NO.1, 2014, COPYRIGHT 2014 EUDOXUS PRESS, LLC

Chun Wen, Ting-Zhu Huang

Figure 2: Comparisons of the eigenvalue distribution of the BD, BT, SC and PS preconditioned matrices for this Helmholtz equation when G = diag(diag(A)) and n + m = 992.

Table 1 supplies the IT, CPU and RES of the BD, BT, SC and PS preconditioned GMRES methods for this Helmholtz equation when G = tridiag(A). As we have seen from Table 1, the PS preconditioned GMRES method has given the best iteration counts. For the BD, BT and SC preconditioned GMRES methods, their iteration counts have been reduced by around 96%. In terms of the computing time, the PS preconditioned GMRES method costs much less than these of the BD, BT and SC preconditioned GMRES methods. In addition, the precision of the relative residual error for the PS preconditioned GMRES method is higher than these of the BD, BT and SC preconditioned GMRES methods, except for the case that n + m = 992. Table 2 provides the IT, CPU and RES of the BD, BT, SC and PS preconditioned GMRES methods for this Helmholtz equation when G = diag(diag(A)). From Table 2, it is not difficult to find that, for this approximate (1,1) block matrix G, all the iteration counts, computing time and the relative residual error of the PS preconditioned GMRES method are better than these of the BD, BT and SC preconditioned GMRES methods. Both the numerical results in Table 1 and Table 2 have shown that the PS preconditioned GMRES method is superior to the BD, BT and SC preconditioned GMRES methods in obtaining a considerable reduction of iteration counts. These results have confirmed our theoretical analysis in previous sections. That is, the convergence of a Krylov subspace method under 39

CHUN WEN ET AL 30-41

J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 16, NO.1, 2014, COPYRIGHT 2014 EUDOXUS PRESS, LLC

A preconditioner for block two-by-two symmetric indefinite matrices

preconditioning has relation to the spectral properties of the preconditioned matrix. For obtaining an intuitive comparison, Figure 1 and Figure 2 have plotted the eigenvalue distribution of the BD, BT, SC and PS preconditioned matrices for G = tridiag(A) and G = diag(diag(A)) with the chosen order of the nonsingular and symmetric indefinite linear system (3) is 992, respectively.

5

Conclusions

We have proposed and investigated a new preconditioner for the numerical solution of block twoby-two symmetric indefinite matrices whose (1,1) and (2,2) blocks are nonsingular. As we have seen in this paper, the proposed preconditioner is constructed as the product of two fairly simple preconditioners: one is the famous block Jacobi preconditioner, and the other is the popular constraint preconditioner. Here, we call it the product preconditioner, and denote it as PS preconditioner. Results concerning the eigenvalue distribution and form of the eigenvectors of −1 A are discussed in Section 3, respectively. Numerical experiments the preconditioned matrix Mps with preconditioned GMRES method on the problem (1) are used to illustrate the efficiency and stability of the proposed product preconditioner. Moreover, we have confirmed our theoretical analysis by comparing the IT, CPU and RES of the BD, BT, SC and PS preconditioned GMRES methods in Table 1 and Table 2.

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Chun Wen, Ting-Zhu Huang

[12] G.H. Golub, A.J. Wathen, An iteration for indefinite systems and its application to the Navier-Stokes equations, SIAM J. Sci. Comput., 19 (1998), pp. 530-539. [13] J. Peters, V. Reichelt, A. Reusken, Fast iterative solvers for discrete stokes equations, SIAM J. Sci. Comput., 27 (2005), pp. 646-666. [14] H.C. Elman, D.J. Silvester, Fast nonsymmetric iterations and preconditioning for Navier-Stokes equations, SIAM J. Sci. Comput., 17 (1996), pp. 33-46. [15] M.F. Murphy, G.H. Golub, A.J. Wathen, A note on preconditioning for indefinite linear systems, SIAM J. Sci. Comput., 21 (2000), pp. 1969-1972. [16] E. De Sturler, J. Liesen, Block-diagonal and constraint preconditioners for nonsysmmetric indefinite linear systems, Part I: theory, SIAM J. Sci. Comput., 26 (2005), pp. 1598-1619. [17] G. Siefert, E. De Sturler, Preconditioners for generalized saddle-point problems, SIAM J. Numer. Anal., 44 (2006), pp. 1275-1296. [18] C. Keller, N.I.M. Gould, A.J. Wathen, Constraint preconditioning for indefinite linear systems, SIAM J. Matrix Anal. Appl., 21 (2000), pp. 1300-1317. [19] H.S. Dollar, A.J. Wathen, Approximation factorization constraint preconditioners for saddle-point matrices, SIAM J. Sci. Comput., 27 (2006), pp. 1555-1572. [20] Z.-H. Cao, Constraint Schur complement preconditioners for nonsymmetric saddle point problems, Appl. Numer. Math., 59 (2009), pp. 151-169. [21] Z.-H. Cao, A note on constraint preconditioning for nonsymmetric indefinite matrices, SIAM J. Matrix Anal. Appl., 24 (2002), pp. 121-125. [22] H.S. Dollar, Constraint-style preconditioners for regularized saddle point problems, SIAM J. Matrix Anal. Appl., 29 (2007), pp. 672-684. [23] H.C. Elman, Preconditioning for the steady-state Navier-Stokes equations with low viscosity, SIAM J. Sci. Comput., 20 (1999), pp. 1299-1316. [24] D. Kay, D. Loghin, A. Wathen, A preconditioner for the steady-state Navier-Stokes equations, SIAM J. Sci. Comput., 24 (2002), pp. 237-256. [25] Z.-H. Cao, Positive stable block triangular preconditioners for symmetric saddle point problems, Appl. Numer. Math., 57 (2007), pp. 899-910. [26] V. Simoncini, Block triangular preconditioners for symmetric saddle-point problems, Appl. Numer. Math., 49 (2004), pp. 63-80. [27] M. Benzi, D.B. Szyld, Existence and uniqueness of splittings for stationary iterative methods with applications to alternating methods, Numer. Math., 76 (1997), pp. 309-321. [28] M. Benzi, B. U¸car, Product preconditioning for Markov chain problems, in Proceedings of the 2006 Markov Anniversary Meeting, A. N. Langville and W. J. Stewart, eds., Raleigh, NC, 2006, (Charleston, SC), Boson Books, pp. 239-256. [29] M. Benzi, B. U¸car, Block triangular preconditioners for M -matrices and Markov chains, ETNA, 26 (2007), pp. 209-227. [30] Y. Saad, M.H. Schultz, GMRES: A generalized minimal residual algorithm for solving nonsymmetric linear systems, SIAM J. Sci. Statist. Comput., 7 (1986), pp. 856-869. [31] Y. Saad, Iterative Methods for Sparse Linear Systems, 2nd ed., SIAM, Philadelphia, PA, 2003.

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J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 16, NO.1, 2014, COPYRIGHT 2014 EUDOXUS PRESS, LLC

HYERS-ULAM STABILITY OF A GENERAL DIAGONAL SYMMETRIC FUNCTIONAL EQUATION CHOONKIL PARK AND HAMID REZAEI∗

Abstract. Using the direct method and the fixed point method, we prove the Hyers-Ulam stability for the symmetric functional equation f (φ1 (x, y, z)) = φ2 (f (x), f (y), f (z)) in Banach spaces. As a consequence, we obtain some stability results in the sense of Hyers-UlamRassias.

1. Introduction The stability theory of functional equations originated from the well-known Ulam’s problem [15], concerning the stability of homomorphisms in metric groups. Hyers [7] gave a first affirmative partial answer to the question of Ulam for Banach spaces. Let X1 and X2 be Banach spaces. Assume that f : X1 → X2 satisfies ∥f (x + y) − f (x) − f (y)∥ ≤ ε for all x, y ∈ X1 and some ε > 0. Then there exists a unique additive mapping T : X1 → X2 such that ∥f (x) − T (x)∥ ≤ ε for all x ∈ X1 . Hyers’ Theorem was generalized by Aoki [1] for additive mappings and by Th.M. Rassias [14] for linear mappings, considering the Cauchy difference to be unbounded. Theorem 1.1. ([14]) Let X1 be a normed space and X2 a Banach space. Let f : X1 → X2 satisfy the inequality ∥f (x + y) − f (x) − f (y)∥ ≤ θ(∥x∥p + ∥y∥p )

(1.1)

for all x, y ∈ X1 , where θ > 0 and p ∈ [0, 1). Then there exists a unique additive mapping 2θ p A : X1 → X2 such that ∥f (x) − A(x)∥ ≤ 2−2 p ∥x∥ for all x ∈ X1 . A generalization of the Th.M. Rassias theorem was obtained by Gˇavruta [6] by replacing the unbounded Cauchy difference by a general control function in the spirit of the Th.M. Rassias’ approach. J.M. Rassias [13] followed the innovative approach of the Th.M. Rassias Theorem [14] in which he replaced the factor ∥x∥p + ∥y∥p by ∥x∥p ∥y∥q for p, q ∈ R with p + q = 1. The stability problems of several functional equations have been extensively investigated by a number of authors and there are many interesting results concerning this problem (see [3, 5, 8, 9]). In this paper, we introduce the following functional equation f (φ1 (x, y, z)) = φ2 (f (x), f (y), f (z)).

(1.2)

MSC(2010): Primary: 39B12; 39B52; Secondary: 47H10; 39B72. Keywords: functional equation, Hyers-Ulam-Rassias stability, fixed point. ∗ Corresponding author. 42

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Using the direct method and the fixed point method, we prove the Hyers-Ulam stability of the functional equation (1.2) in Banach spaces. 2. Hyers-Ulam stability of (1.2): direct method In this section, we prove the Hyers-Ulam stability of the functional equation (1.2), where φi : Xi × Xi × Xi → Xi , i = 1, 2, are mappings such that φi (φi (x, x, x), φi (y, y, y)) = φi (φi (x, y, z), φi (x, y, z), φi (x, y, z)).

(2.1)

Let us call such mappings diagonal symmetric on Xi . For example (1) Let X be a vector space, and φ : X × X × X → X be a function that φ(λx, λy, λz) = λφ(x, y, z)

(x, y, z ∈ X)

for every scalar λ and φ(x, x, x) = αx for some scalar α, then φ is diagonal symmetric on X. (2) Let X be a vector space, and φ : X ×X ×X → X defined by φ(x, y, z) = ax+by +cz +d, where a, b, c, d are scalars and x, y, z ∈ X. Then it is easy to check that φ is diagonal symmetric. Theorem 2.1. Assume that X1 is a normed space and X2 is a Banach space and that φ1 , φ2 are continuous diagonal symmetric mappings on X1 , X2 , respectively. Put Ti (x) := φi (x, x, x) for i = 1, 2 and suppose that T2 is an invertible bounded linear operator on X2 . Let β : X1 × X1 × X1 → [0, +∞) be a function with this property that there exists some 0 < λ < 1 such that ∥T2−1 ∥β(T1 x, T1 y, T1 z) ≤ λβ(x, y, z) for all x, y, z ∈ X1 . If f : X1 → X2 is a mapping satisfying ∥f (φ1 (x, y, z)) − φ2 (f (x), f (y), f (z))∥ < β(x, y, z)

(2.2)

for all x, y, z ∈ X1 , then there exists a unique mapping A : X1 → X2 such that ∥f (x) − A(x)∥ ≤

∥T2−1 ∥β(x, x, x) , 1−λ

A(φ1 (x, y, z)) = φ2 (A(x), A(y), A(z))

(2.3) (2.4)

for all x, y, z ∈ X1 . Proof. Letting z = y = x (2.2), we get ∥f T1 (x) − T2 f (x)∥ ≤ β(x, x) for all x ∈ X1 . It follows from (2.1) that φi (Ti x, Ti y, Ti z) = Ti (φi (x, y, z))

(2.5)

for all x, y, z ∈ Xi and i = 1, 2. Let qn (x) := T2−n f (T1n x) for all n ≥ 1 and all x ∈ X1 . Then ∥qn+1 (x) − qn (x)∥ = ∥T2−n−1 f (T1n+1 x) − T2−n f (T1n x)∥ ≤ ∥T2−n−1 ∥∥f T1 (T1n x) − T2 f (T1n x)∥ ≤ ∥T2−1 ∥n+1 β(T1n x, T1n x) ≤ ∥T2−1 ∥λn β(x, x, x). 43

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Here, in the last inequality, the contractive property of β is used. Hence ∥qn+1 (x) − qn (x)∥ ≤ ∥T2−1 ∥λn β(x, x, x), and so the sequence {qn (x)} is a Cauchy sequence for each x. Since X2 is complete, there exists a limit mapping A(x) := limn→∞ qn (x). Now by induction on n, we prove that ∥qn (x) − f (x)∥ ≤

n−1 ∑

∥T2−1 ∥λi β(x, x, x)

(2.6)

i=0

for all n ∈ N and all x ∈ X1 . Fix x ∈ X1 . Note that ∥q1 (x) − f (x)∥ = ∥T2−1 f (T1 (x)) − f (x)∥ ≤ ∥T2−1 ∥∥f (T1 (x)) − T2 (f (x))∥ ≤ ∥T2−1 ∥β(x, x, x). Now suppose (2.6) holds for a fixed n. Then ∥qn+1 (x) − f (x)∥ ≤ ∥qn+1 (x) − qn (x)∥ + ∥qn (x) − f (x)∥ ≤ ∥T2−1 ∥λn β(x, x) +

n−1 ∑

∥T2−1 ∥λi β(x, x, x)

i=0

=

n ∑

∥T2−1 ∥λi β(x, x, x).

i=0

Letting n → +∞ in (2.6), we get ∥A(x) − f (x)∥ ≤

∥T2−1 ∥β(x, x, x) 1−λ

for all x ∈ X1 . Now we prove that A satisfies (2.4). Replacing x, y, z in (2.2) with T1n x, T1n y, T1n z, respectively, we get ∥f (φ1 (T1n x, T1n y, T1n y)) − φ2 (f (T1n x), f (T1n y), f (T1n z))∥ ≤

(2.7)

β(T1n x, T1n y, T1n z).

It follows from (2.5) that φ1 (T1n x, T1n y, T1n z) = T1n (φ1 (x, y, z))

(2.8)

for all x, y, z ∈ X1 , and φ2 (T2n x, T2n y, T1n z) = T2n (φ2 (x, y, z)) for all x, y, z ∈ X2 . Replacing x, y, z by T2−n x, T2−n y, T1n z, respectively, in the last above relation, we get φ2 (x, y, z) = T2n (φ2 (T2−n x, T2−n y, T2−n z)) and then replacing x, y, z by f (T1n x), f (T1n y), f (T1n z), respectively, we get φ2 (f (T1n x), f (T1n y), f (T1n z)) = T2n (φ2 (T2−n (f (T1n x)), T2−n (f (T1n y)), T2−n (f (T1n z)))). By the definition of qn , we obtain φ2 (f (T1n x), f (T1n y), f (T1n z)) = T2n (φ2 (qn (x), qn (y), qn (z))). 44

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It follows from (2.7), (2.8) and (2.9) that ∥qn (φ1 (x, y, z)) − φ2 (qn (x), qn (y), qn (z))∥ = ∥T2−n f (T1n φ1 (x, y, z)) − φ2 (qn (x), qn (y), qn (z))∥ ≤ ∥T2−1 ∥n ∥f (T1n φ1 (x, y, z)) − T2n φ2 (qn (x), qn (y), qn (z))∥ = ∥T2−1 ∥n ∥f (φ1 (T1n x, T1n y, T1n z)) − φ2 (f (T1n x), f (T1n y), f (T1n z))∥ ≤ ∥T2−1 ∥n β(T1n x, T1n y, T1n z) ≤ λn β(x, x, x). Therefore, ∥qn (φ1 (x, y, z)) − φ2 (qn (x), qn (y), qn (z))∥ ≤ λn β(x, x, x) for all x, y ∈ X1 and all n ∈ N. Applying the continuity of φ, considering 0 < λ < 1, and letting n → +∞ in the last inequality, we obtain (2.4). Now we prove that A is a unique mapping satisfying (2.3) and (2.4). Assume that there exists another mapping A′ : X → X satisfying (2.3) and (2.4). Letting y = x in (2.4), we get AT1 (x) = T2 A(x) and A′ T1 (x) = T2 A′ (x) and more generally AT1n (x) = T2n A(x) and A′ T1n (x) = T2n A′ (x). Hence A(x) = T2−n A(T1n (x)) and A′ (x) = T2−n A′ (T1n (x)) for all x ∈ X and n ∈ N. By the triangle inequality, (2.3) and (2.10), we obtain ∥A(x) − A′ (x)∥ = ∥T2−n A(T1n x) − T2−n A′ (T1n x)∥ ≤ ∥T2−1 ∥n ∥A(T1n x) − A′ (T1n x)∥ ≤ ∥T2−1 ∥n (∥A(T1n x) − f (T1n x)∥ + ∥f (T1n x) − A′ (T1n x)∥) ( ∥T −1 ∥β(T1n x, T1n x) ) ≤ ∥T2−1 ∥n 2 2 1−λ −1 n ( ∥T2 ∥ β(T1n x, T1n x) ) ≤ 2∥T2−1 ∥ 1−λ n λ β(x, x) ≤ 2∥T2−1 ∥ 1−λ for all x ∈ X1 and all n ∈ N. Letting n → +∞, we get A(x) = A′ (x) for all x ∈ X1 .



The proof of the following theorem is similar and we omit it: Theorem 2.2. Assume that X1 is a normed space and X2 is a Banach space and that φ1 , φ2 are continuous diagonal symmetric mappings on X1 , X2 , respectively. Put Ti (x) := φi (x, x, x) for i = 1, 2 and suppose that T2 is a bounded linear operator on X2 and T1 is invertible on X1 . Let β : X1 × X1 × X1 → [0, +∞) be a function with this property that there exists some 0 < λ < 1 such that ∥T2 ∥β(T1−1 x, T1−1 y, T1−1 z) ≤ λβ(x, y, z) for all x, y, z ∈ X1 . If f : X1 → X2 is a mapping satisfying ∥f (φ1 (x, y, z)) − φ2 (f (x), f (y), f (z))∥ < β(x, y, z) 45

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for all x, y, z ∈ X1 , then there exists a unique mapping A : X1 → X2 such that ∥T1−1 ∥β(x, x, x) 1−λ A(φ1 (x, y, z)) = φ2 (A(x), A(y), A(z)) ∥f (x) − A(x)∥ ≤

for all x, y, z ∈ X1 . Corollary 2.3. Assume that X1 is a normed space and X2 is a Banach space and that φ1 , φ2 are continuous diagonal symmetric mappings on X1 , X2 , respectively. Put Ti (x) := φi (x, x, x) for i = 1, 2 and suppose that T2 is an invertible bounded linear operator on X2 . Let f : X1 → X2 be a mapping for which there exist some θ1 , θ2 > 0, and p ≥ 0 such that ∥f (φ1 (x, y, z)) − φ2 (f (x), f (y), f (z))∥ < θ1 (∥x∥p + ∥y∥p + ∥z∥p ) + θ2 (∥x∥p/3 ∥y∥p/3 ∥z∥p/3 ) for all x, y, z ∈ X1 . If ∥T2−1 ∥∥T1 ∥p < 1, then there exists a unique mapping A : X1 → X2 such that (2θ1 + θ2 )∥x∥p ∥f (x) − A(x)∥ ≤ θ∥T2−1 ∥ , 1 − ∥T2−1 ∥∥T1 ∥p A(φ1 (x, y, z)) = φ2 (A(x), A(y), A(z)) for all x, y, z ∈ X1 . Proof. Let β(x, y, z) := θ1 (∥x∥p + ∥y∥p + ∥z∥p ) + θ2 (∥x∥p/3 ∥y∥p/3 ∥z∥p/3 ) for x, y ∈ X1 , and λ := ∥T2−1 ∥∥T1 ∥p . Then ∥T2−1 ∥β(T1 x, T1 y) ≤ λβ(x, y, z) for all x, y, z ∈ X1 . This completes the proof.



Consider the following choices of φ1 , φ2 , T1 and T2 : (1) φ1 (x, y, z) = φ2 (x, y, z) = x+y+z and T1 (x) = T2 (x) = 2

3x 2 ;

(2) φ1 (x, y, z) = xyz, φ2 (x, y, z) = x + y + z,T1 (x) = x3 and T2 (x) = 3x; to deduce the following corollary: Corollary 2.4. The following functional equations has Hyers-Ulam stability in the sense of Theorem 2.1 and 2.2: (i) 2f ( x+y+z ) = f (x) + f (y) + f (z), f : X1 → X2 where X1 is a vector space and X2 is a 2 Banach space. (ii) f (xyz) = f (x) + f (y) + f (z), f : X1 → X2 where X1 is any abelian semigroup and X2 is a Banach space. Let A be a C ∗ -algebra and a ∈ A a self-adjoint element, i.e., a = a∗ . Then a is said to be positive if it is of the form a = bb∗ for some a ∈ A. The set of positive elements of A is denoted by A+ . Note that A+ is a closed convex cone (see [4]). It is well-known that for a positive element a and a positive integer n there exists a unique √ positive element x ∈ A+ such that a = xn . We denote x by n a. Then the functional 46

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equation (1.1) is Hyers-Ulam stable in the sense of Theorem 2.1 and 2.2 when the mapping φ : A+ × A+ × A+ → A+ is one of the following choices: √ (1) φ(x, y, z) = n ax2 + by 2 + cz 2 where a + b + c > 1, √ (2) φ(x, y, z) = n xn + y n + z n , 3. Hyers-Ulam stability of (1.2): fixed point method We now introduce one of the fundamental results of the fixed point theory. For a nonempty set X, we introduce the definition of the generalized metric on X. A function d : X × X → [0, ∞] is called a generalized metric on X if and only if d satisfies • d(x, y) = 0 if and only if x = y, • d(x, y) = d(y, x) for all x, y ∈ X, • d(x, z) ≤ d(x, y) + d(y, z), for all x, y, z ∈ X. Using the fixed point method, we prove the Hyers-Ulam stability of the functional equation (1.2) in Banach spaces. Theorem 3.1. ([10]) Let (X , d) be a generalized complete metric space. Assume that Λ : X → X is a strictly contractive operator with the Lipschitz constant L < 1, i.e., d(Λg, Λh) ≤ Ld(g, h) for all g, h ∈ X . If there exists a nonnegative integer n0 such that d(Λn0 +1 f, Λn0 f ) < +∞ for some f ∈ X , then the following statements are true: (1) The sequence {Λn f } converges to a fixed point A of Λ; (2) A is the unique fixed point of Λ in X ∗ = {g ∈ X : d(Λn0 f, g) < +∞}; (3) If g ∈ X ∗ , then d(g, A) ≤

1 d(Λg, g). 1−L

Radu [12] proved the Hyers-Ulam stability of the additive Cauchy equation (1.1) by using fixed point method (see [2]). In the following, Theorem 2.1 is proved by the fixed point method. Theorem 3.2. Let X1 , X2 , φ1 , φ2 , T1 , T2 , β be given as in Theorem 2.1. If f : X1 → X2 is a mapping satisfying (2.2), then there exists a unique mapping A : X1 → X2 satisfying (2.3) and AT1 (x) = T2 A(x) for all x ∈ X1 . Proof. Letting y = x in (2.2), we get ∥f T1 (x) − T2 f (x)∥ ≤ β(x, x, x) for all x ∈ X1 . Consider the set X := {f : f : X1 → X2 is a function } and define the generalized metric on X by { d(g, h) = inf µ ∈ (0, +∞) : ∥g(x) − h(x)∥ ≤ µβ(x, x) for all x ∈ X1 }. 47

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where, as usual, inf = +∞. It is easy to show that (X , d) is complete (see [11]). Now we consider the linear mapping Λ : X → X such that Λg(x) = T2−1 g(T1 x) for all x ∈ X1 . For given g, h ∈ X , ∥Λg(x) − Λf (x)∥ = ∥T2−1 g(T1 x) − T2−1 h(T1 x)∥ ≤ ∥T2−1 ∥β(T1 x, T1 x, T1 x) ≤ λβ(x, x, x) for all x ∈ X1 . By the definition of d, d(Λf, Λg) ≤ λd(f, g). Note that ∥f (x) − Λf (x)∥ = ∥f (x) − T2−1 f (T1 x)∥ ≤ ∥T2−1 ∥∥T1 f (x) − f (T1 x)∥ ≤ ∥T2−1 ∥β(x, x, x) for all x ∈ X1 , and so d(Λf, f ) ≤ ∥T2−1 ∥ < +∞. By the preceding theorem, there exists a mapping A : X1 → X2 satisfying the following conditions: (1) A is a fixed point of Λ, i.e., T2−1 AT1 = ΛA = A whence A(T1 (x)) = T2 (A(x)) for all x ∈ X1 . Moreover, A is a unique fixed point of Λ in the set X ∗ := {g ∈ X : d(f, g) < +∞} which implies that ∥f (x) − A(x)∥ ≤ µβ(x, x, x).

(2) d(Λn f, A) → 0 as n → +∞, i.e., A(x) = limn T2−n f (T1n x). (3) By (3) of the preceding theorem, we conclude that d(f, A) ≤

1 1 d(f, Λf ) < ∥T −1 ∥, 1−λ 1−λ 2

and so

∥T2−1 ∥β(x, x, x) , 1−λ as desired. In order to prove that A satisfies (2.4), we can proceed exactly as in the proof of Theorem 2.2 to show that A : X1 → X2 is indeed a mapping satisfying (2.4).  ∥f (x) − A(x)∥ ≤

References [1] T. Aoki, On the stability of the linear transformation in Banach spaces, J. Math. Soc. Japan 2 (1950), 64–66. [2] L. Cadariu and V. Radu, On the stability of the Cauchy functional equation: a fixed point approach, Grazer Math. Ber. 346 (2004), 43–52. [3] S. Czerwik, Functional Equations and Inequalities in Several Variables, World Scientific, Hackensacks, New Jersey, 2002. [4] J. Dixmier, C ∗ -Algebras, North-Holland Publ. Co., Amsterdam, New York and Oxford, 1977. [5] G.L. Forti, Elementary remarks on Ulam-Hyers stability of linear functional equations, J. Math. Anal. Appl. 328 (2007), 109–118. [6] P. Gˇ avruta, An answer to a question of Th.M. Rassias and J. Tabor on mixed stability of mappings, Bul. Stiint. Univ. Politeh. Timis. Ser. Mat. Fiz. 42 (1997), 1–6. 48

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[7] D.H. Hyers, On the stability of the linear functional equation, Proc. Natl. Acad. Sci. USA 27 (1941) 222–224. [8] D.H. Hyers, G. Isac and Th.M. Rassias, Stability of Functional Equations in Several Variables, Birkh¨ auser, Boston, Basel, Berlin, 1998. [9] S. Jung, Hyers-Ulam-Rassias Stability of Functional Equations in Nonlinear Analysis, Springer, New York, 2011. [10] B. Margolis and J.B. Diaz, A fixed point theorem of the alternative for contractions on a generalized complete metric space, Bull. Amer. Math. Soc. 74 (1968), 305–309. [11] 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. [12] V. Radu, The fixed point alternative and the stability of functional equations, Fixed Point Theory 4 (2003), 91–96. [13] J.M. Rassias, On approximation of approximately linear mappings by linear mappings, J. Funct. Anal. 46 (1982), 126–130. [14] Th.M. Rassias, On the stability of the linear mapping in Banach spaces, Proc. Amer. Math. Soc. 72 (1978), 297–300. [15] S.M. Ulam, A Collection of Mathematical Problems, Interscience Tracts in Pure and Applied Mathematics, No. 8, Interscience, New York, NY, USA, 1960. Choonkil Park Department of Mathematics, Research Institute for Natural Sciences, Hanyang University, Seoul 133-791, Korea E-mail address: [email protected] Hamid Rezaei Department of Mathematics, College of Sciences, Yasouj University, Yasouj-75914-74831, Iran E-mail address: [email protected]

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GENERAL DECAY OF SOLUTIONS FOR A SINGULAR NONLOCAL VISCOELASTIC PROBLEM WITH NONLINEAR DAMPING AND SOURCE YUN SUN, GANG LI, AND WENJUN LIU Abstract. This paper deals with a singular nonlocal viscoelastic problem with nonlinear damping and source terms. We establish a general decay rate result without imposing any restrictive growth assumption on the damping term.

1. Introduction In this paper, we investigate the following one-dimensional viscoelastic equation  ∫ t 1 1   u − g(t − s) (xux (x, s))x ds + h(ut ) = b|u|p−2 u, x ∈ (0, ℓ), t ∈ (0, ∞), (xu ) +  tt x x   x x 0   ∫ ℓ

t ∈ [0, ∞),

xu(x, t)dx = 0 u(ℓ, t) = 0,    0     u(x, 0) = u0 (x), ut (x, 0) = u1 (x),

(1.1)

x ∈ [0, ℓ],

where ℓ < ∞, b > 0, p > 2, g and h are specific functions which will be given later. This type of evolution problems are generally encountered when the data on the boundary can not be measured directly, but their average values are known. For the case of singular type, we can refer to [8, 9, 10, 11, 14] for the existence, uniqueness and blow-up results. Here, it is worth mentioning that many results concerning decay have been extensively studied for the case of classical conditions. Under the condition −ξ1 g(t) ≤ g ′ (t) ≤ −ξ2 g(t), the exponential or polynomial decay results were obtained in [3, 4, 5, 6]. Later, some authors relaxed these conditions by considering only g ′ (t) ≤ −ξg(t) or g ′ (t) ≤ −ξg r (t) , for all t ≥ 0 and some ξ > 0 (see [1, 2, 15]). In [12, 13], the condition has been replaced by g ′ (t) ≤ −ξ(t)g(t), where ξ(t) is a positive function. This allows the authors to obtain general rates of decay than just exponential or polynomial type. Motivated by [11, 13], we study problem (1.1) in this paper and intend to establish a general decay result under certain conditions, without imposing any restrictive growth assumption on the damping term. The paper is organized as follows. In Section 2 we present some assumptions and known results needed for our work. Section 3 is devoted to the proof of some lemmas and the decay result: Theorem 2.4 . 2. Preliminaries and main result In this section we first introduce some functional spaces and present some assumptions and known results which will be used throughout this paper, and then state our main result. (∫ )1 p ℓ p p p Let Lx = Lx (0, ℓ) be the weighted Banach space equipped with the norm ∥u∥p = 0 x|u| dx . In particular, when p = 2, we denote H = L2x (0, ℓ) to be the weighted Hilbert space of square 2010 Mathematics Subject Classification. 35B35; 35B40; 35L20. Key words and phrases. nonlocal viscoelastic problem; general decay; nonlinear damping. This paper was supported by the JSPS Innovation Program (CXLX12 0490).

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Y. SUN, G. LI, AND W. J. LIU

integrable functions having the finite norm ∥u∥H =

(∫

)1

ℓ 2 0 xu dx

2

. We take V = Vx1,1 (0, ℓ) ( )1 = ∥u∥2H + ∥ux ∥2H 2 , and

to be the weighted Hilbert space equipped with the norm ∥u∥V V0 = {v ∈ V such that v(ℓ) = 0}. For the functionals g and h we give the following assumptions as in [13]: (H1) g(t) : R+ → R+ is a C 1 function such that ∫ ∞ g(0) > 0, 1− g(s)ds = l > 0, 0

and there exists a nondecreasing differentiable function ξ(t) such that ∫ +∞ g ′ (t) ≤ −ξ(t)g(t), t ≥ 0 and ξ(t)dt = ∞ 0

(H2) h : R 7→ R is a nondecreasing C 0 function such that there exists a strictly increasing function h0 ∈ C 1 ([0, +∞)), with h0 (0) = 0, and positive constants c1 , c2 , and ϵ such that h0 (|s|) ≤ |h(s)| ≤ h−1 0 (s), c1 |s| ≤ |h(s)| ≤ c2 |s|,

∀ |s| ≤ ϵ,

(2.1)

∀ |s| ≥ ϵ.

(2.2)

Remark 1. Hypothesis (H2) implies that sh(s) > 0, for all s ̸= 0. Lemma 2.1. ([11]) For any v in V0 , we have ∫ ℓ ∫ ℓ 2 x(v(x)) dx ≤ C∗ x (vx (x))2 dx. 0

0

Lemma 2.2. ([11]) For any v in V0 , 2 < p < 4, we have ∫ ℓ x(v(x))p dx ≤ Cp ∥vx ∥pH , 0

where Cp is a constant depending on p only. Lemma 2.3. ([11, Theorem 2.3]) Suppose that 2 < p < 3 and (H1) and (H2) hold. Then for any u0 in V0 and u1 in H, problem (1.1) has a unique local solution u ∈ C(0, t∗ ; V0 ) ∩ C 1 (0, t∗ ; H) for t∗ > 0 small enough. Now we introduce the functionals for I(t) and E(t): ∫ ℓ ∫ t )∫ ℓ ( 2 x|u(t)|p dx, xux dx + (g ◦ ux )(t) − b I(t) := I(u(t)) = 1 − g(s) ds 0

0

∫ t )∫ ℓ 1( 1 E(t) := E(u(t)) = 1− g(s) ds xu2x dx + (g ◦ ux )(t) 2 2 0 0 ∫ ℓ ∫ ℓ b 1 − x|u(t)|p dx + xu2t dx, p 0 2 0 where (g ◦ ux )(t) =

∫ ℓ∫ 0

t

(2.3)

0

(2.4)

xg(t − s)|ux (x, t) − ux (x, s)|2 dsdx.

0

Remark 2. Multiplying Eq. (1.1) by xut and integrating over (0, ℓ), we can easily get ∫ ℓ ∫ ℓ 1 ′ 1 ′ 2 E (t) = (g ◦ ux )(t) − g(t) xux (x, t)dx − xut h(ut )dx ≤ 0, ∀ t ≥ 0. (2.5) 2 2 0 0 Our main result of this paper reads as follows. 51

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GENERAL DECAY OF SOLUTIONS FOR A SINGULAR NONLOCAL VISCOELASTIC PROBLEM

Theorem 2.4. Suppose that (H1) and (H2) hold, 2 < p < 3, if (u0 , u1 ) ∈ V0 × H such that ( ) p−2 2 bCp 2p β= E(u0 , u1 ) < 1, I(u0 ) > 0. (2.6) l (p − 2)l Then, there exists a constant C > 0 such that, for t large, the solution of (1.1) satisfies ( ( ))2 1 E(t) ≤ C H0−1 ∫ t where H0 (s) = sh0 (s). 0 ξ(s)ds

(2.7)

Moreover, if we define J(s) = h0s(s) , which is strictly increasing with J(0) = 0, then we can improve (2.7) to the following estimate: ( ( ))2 1 −1 E(t) ≤ C h0 . (2.8) ∫t 0 ξ(s)ds For the proof of the above theorem, we use the following lemma. Lemma 2.5. ([7]) Let E : R+ → R+ be a nonincreasing function and σ : R+ → R+ be a strictly increasing C 1 function, with σ(t) → +∞ as t → +∞. Assume that there exist p, q ≥ 0 and c > 0 such that ∫ ∞ cE(s) σ ′ (t)E(t)1+p dt ≤ cE(s)1+p + , 1 ≤ S < +∞. σq S Then there exist positive constants κ and ω such that E(t) ≤ κe−ωσ(t) κ E(t) ≤ 1+q σ(t) p

∀ t ≥ 1, if p = q = 0 ∀ t ≥ 1, if p > 0.

3. General decay of solutions In this section we prove our main result. For this purpose we establish several lemmas. Lemma 3.1. ([11, Lemma 4.1 and Lemma 4.2]) Under the assumptions of Theorem 2.4, we conclude that I(u(t)) > 0, ∀ t > 0 and the solution is global and bounded. Furthermore, the following inequality holds ( ) ∫ µ 2p 2 l xux dx ≤ E(u0 , u1 ), ∀t > 0. (3.1) p−2 0 Lemma 3.2. For all u ∈ V0 , there exists C∗ > 0 such that )2 ∫ ℓ (∫ t x g(t − s)(u(t) − u(s))ds dx ≤ (1 − l)C∗ (g ◦ ux )(t). 0

0

Proof. Using Cauchy-Schwarz’s inequality, (H1) and Lemma 2.1, we can easily obtain the result. We define the following functionals L(t) := N1 E(t) + N2 K(t) + χ(t), where

∫ K(t) := − ∫ χ(t) :=

∫

ℓ

xut 0

t

(3.2)

g(t − s)(u(t) − u(s))dsdx,

0

ℓ

xuut dx, 0

N1 and N2 are positive constants to be chosen later. 52

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Y. SUN, G. LI, AND W. J. LIU

Lemma 3.3. Suppose that (H1) holds and p > 2. Let u be the solution of problem (1.1). Then there exist positive constants α1 , α2 > 0 such that α1 E(t) ≤ L(t) ≤ α2 E(t).

(3.3)

Proof. Straightforward computations, Young’s inequality and Lemma 3.2 lead to [ ( ) ]∫ ℓ ( )∫ ℓ ∫ t N1 C∗ N1 N2 1 L(t) ≤ 1− g(s)ds + xu2x dx + + + xu2t dx 2 2 2 2 2 0 0 0 [ ] ∫ ℓ N1 N2 bN1 + + (1 − l)C∗ (g ◦ ux )(t) − x|u|p dx 2 2 p 0 (∫ ℓ ) ∫ ℓ ∫ b ℓ ≤α1 xu2x dx + xu2t dx + (g ◦ ux )(t) − x|u|p dx , p 0 0 0 for some α1 > 0. On the other hand, 1 L(t) ≥ (N1 l − C∗ ) 2

∫

ℓ

xu2x dx 0

1 + (N1 − N2 − 1) 2

bN1 1 + [N1 − N2 (1 − l)C∗ ] (g ◦ ux )(t) − 2 p

∫

(3.4)

ℓ

xu2t dx 0

∫

ℓ

x|u|p dx.

(3.5)

0

Choose N2 > 1 and then take N1 satisfying { } C∗ N1 > max , N2 + 1, N2 (1 − l)C∗ . l

(3.6)

Then we completes the proof. Lemma 3.4. Suppose that (H1) and (H2) hold and p > 2, let (u0 , u1 ) ∈ V0 × H be given. If u is the solution of (1.1), then we have ∫ ℓ ∫ ℓ ∫ ℓ ∫ l ℓ 2 2 ′ 2 x|u|p dx. (3.7) xh (ut )dx + b χ (t) ≤ − xut dx + C(g ◦ ux )(t) + C xux dx + 2 0 0 0 0 Proof. By exploiting problem (1.1) and integrating by parts, we get (∫ t )∫ ℓ ∫ ℓ ∫ ℓ ′ 2 2 χ (t) = xut dx − xux dx + g(s)ds xu2x dx 0

∫

0

∫

ℓ

t

xux

+

0

0

g(t − s) (ux (s) − ux (t)) dsdx −

∫

ℓ

ℓ

x|u|p dx.

xuh(ut )dx + b 0

0

0

∫

(3.8)

0

Using Young’s and Poincar´e’s inequalities and Lemma 3.2, we obtain ∫ ℓ ∫ t xux g(t − s) (ux (s) − ux (t)) dsdx 0

∫ ≤δ

0 ℓ

xu2x dx 0

∫

1 + 4δ

∫

(∫

ℓ

x 0

t

)2 g(t − s) (ux (s) − ux (t)) ds

dx

0

ℓ

C (g ◦ ux )(t), (3.9) δ 0 ∫ ℓ ∫ ℓ ∫ ℓ ∫ ℓ ∫ ℓ 1 1 2 2 2 − xuh(ut )dx ≤ δ xu dx + xh (ut )dx ≤ δC∗ xux dx + xh2 (ut )dx. (3.10) 4δ 4δ 0 0 0 0 0 ≤δ

xu2x dx +

Combining (3.8)-(3.10), and choosing δ small enough such that δ ≤ tained. 53

l 2(1+C∗ ) ,

then (3.7) is ob-

SUN ET AL 50-55

J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 16, NO.1, 2014, COPYRIGHT 2014 EUDOXUS PRESS, LLC

GENERAL DECAY OF SOLUTIONS FOR A SINGULAR NONLOCAL VISCOELASTIC PROBLEM

Lemma 3.5. Under the assumptions (H1) and (H2), suppose 2 < p < 3, then the functional K satisfies, along the solution, the estimate (∫ t )∫ ℓ ( )∫ ℓ ( ) δl2 C K ′ (t) ≤ − g(s)ds − δ xu2t dx + δ + xu2x dx + C + (g ◦ ux )(t) bCp δ 0 0 0 ∫ ℓ C − (g ′ ◦ ux )(t) + C xh2 (ut )dx, ∀ 0 < δ < 1. (3.11) δ 0 Proof. By direct computations and (1.1), we get ( )∫ ℓ ∫ t ∫ t ′ K (t) = 1 − g(s)ds xux g(t − s) (ux (t) − ux (s)) dsdx 0

∫ +

0

(∫

ℓ

t

x 0

)∫

t

−

xu2t dx

0 ∫ ℓ

0

∫ p−2

x|u|

∫

ℓ

g(s)ds

−b

)2

g(t − s) (ux (t) − ux (s)) ds

0

(∫

0

u

+

t

xut 0

t

xh(ut ) 0

∫

ℓ

dx − ∫

ℓ

∫

t

g ′ (t − s)(u(t) − u(s))dsdx

0

g(t − s)(u(t) − u(s))dsdx

0

g(t − s)(u(t) − u(s))dsdx.

(3.12)

0

0

By Young’s inequality and Lemma 3.2, we have ( )∫ ℓ ∫ t ∫ t ∫ ℓ C 1− g(s)ds xux g(t − s) (ux (t) − ux (s)) dsdx ≤ δ xu2x dx + (g ◦ ux )(t), (3.13) δ 0 0 0 0 )2 ∫ ℓ (∫ t x g(t − s) (ux (t) − ux (s)) ds dx ≤ C(g ◦ ux )(t), (3.14) ∫ −

0 ℓ

xut 0

∫

∫

ℓ

∫

xh(ut )

0 t

∫

′

g (t − s)(u(t) − u(s))dsdx ≤ δ

0 t

∫

g(t − s)(u(t) − u(s))dsdx ≤ C

xu2t dx −

0 ℓ

C ′ (g ◦ ux )(t), δ

xh2 (ut )dx + C(g ◦ ux )(t),

(3.15) (3.16)

0

0

0

ℓ

As for the sixth term, using Lemma 2.2, (2.6) and (3.1), we get ∫ ℓ ∫ t p−2 −b x|u| u g(t − s)(u(t) − u(s))dsdx 0 ℓ

0

(∫ ℓ )p−2 (∫ ℓ ) C C 2 2 x|u| dx + (g ◦ ux )(t) ≤ bδCp ≤bδ xux dx xux dx + (g ◦ ux )(t) 2δ 2δ 0 0 0 [ ]p−2 (∫ ℓ ) ∫ ℓ 2 C 2pE(u0 , u1 ) δl ≤bδCp xu2x dx ≤ xu2x dx + (g ◦ ux )(t). (3.17) (p − 2)l bCp 0 2δ 0 ∫

2p−2

Combining (3.12)-(3.17), the assertion of the lemma is established. l ( l 2 ) ≤ Now select N1 , N2 large so that (3.3) remains valid and l 2(1+C∗ ) . Set g0 = 4N2 1+ bC p ∫ t0 g(s)ds for some fixed t > 0. By combining (2.5), (3.2), (3.7) and (3.11), we take δ = 0 0 l ) and obtain, for all t ≥ t0 , ( l2 4N2 1+ bC

p

( )   ( )∫ ℓ 2 1 + l2 4CN 2 bCp l l L′ (t) ≤ − xu2x dx − N2 g0 − − 1 xu2t dx +  + C  (g ◦ ux )(t) 4 0 4 l 0 ( )  2 1 + l2 ∫ ℓ ∫ ℓ 4CN 2 bCp 1 ′ 2   + N1 − (g ◦ ux )(t) + (CN2 + C) xh (ut )dx + b x|u|p dx. 2 l 0 0 ∫

ℓ

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Y. SUN, G. LI, AND W. J. LIU

At this point, since N2 large enough, so we can have k := N2 g0 − 4CN22

(

l2 1+ bC p

)

l 4

− 1 > 0, then N1 large

enough so that (3.6) remains valid and 12 N1 − > 0. Thus, using (H1), it turns out l that ∫ ∫ ℓ ∫ ℓ ∫ ℓ l ℓ 2 ′ 2 2 L (t) ≤ − xux dx − k xut dx + C(g ◦ ux )(t) + C xh (ut )dx + b x|u|p dx, 4 0 0 0 0 which implies ∫ ℓ ′ E(t) ≤ −mL (t) + C(g ◦ ux )(t) + C xh2 (ut )dx, ∀ t ≥ t0 . (3.18) 0

∫t Proof of Theorem 2.4. (Sketch) Define ϕ(t) = 1 + 1 h 1 1 ds, ∀ t ≥ 1 and σ(t) = 0( s ) (∫ ) t −1 ϕ 0 ξ(s)ds , for ∀ t ≥ t1 ≥ t0 . Then continue as that of [13, Theorem 3.5] we can complete the proof. References [1] S. Berrimi and S. A. Messaoudi, Existence and decay of solutions of a viscoelastic equation with a nonlinear source, Nonlinear Anal. 64 (2006), no. 10, 2314–2331. [2] S. Berrimi and S. A. Messaoudi, Exponential decay of solutions to a viscoelastic equation with nonlinear localized damping, Electron. J. Differential Equations 2004 (2004), no. 88, 10 pp. [3] M. M. Cavalcanti, V. N. Domingos Cavalcanti and J. A. Soriano, Exponential decay for the solution of semilinear viscoelastic wave equations with localized damping, Electron. J. Differential Equations 2002 (2002), no. 44, 14 pp. [4] M. M. Cavalcanti et al., Existence and uniform decay rates for viscoelastic problems with nonlinear boundary damping, Differential Integral Equations 14 (2001), no. 1, 85–116. [5] M. M. Cavalcanti, V. N. Domingos Cavalcanti and J. Ferreira, Existence and uniform decay for a non-linear viscoelastic equation with strong damping, Math. Methods Appl. Sci. 24 (2001), no. 14, 1043–1053. [6] M. M. Cavalcanti and H. P. Oquendo, Frictional versus viscoelastic damping in a semilinear wave equation, SIAM J. Control Optim. 42 (2003), no. 4, 1310–1324. [7] P. Martinez, A new method to obtain decay rate estimates for dissipative systems with localized damping, Rev. Mat. Complut. 12 (1999), no. 1, 251–283. [8] S. Mesloub and A. Bouziani, On a class of singular hyperbolic equation with a weighted integral condition, Int. J. Math. Math. Sci. 22 (1999), no. 3, 511–519. [9] S. Mesloub and F. Mesloub, Solvability of a mixed nonlocal problem for a nonlinear singular viscoelastic equation, Acta Appl. Math. 110 (2010), no. 1, 109–129. [10] S. Mesloub and S. A. Messaoudi, A three point boundary value problem hyperbolic equation, Electron. J. Differ. Equ. 2002 (2002), no. 62, 13 pp. [11] S. Mesloub and S. A. Messaoudi, Global existence, decay, and blow up of solutions of a singular nonlocal viscoelastic problem, Acta Appl. Math. 110 (2010), no. 2, 705–724. [12] S. A. Messaoudi, General decay of the solution energy in a viscoelastic equation with a nonlinear source, Nonlinear Anal. 69 (2008), no. 8, 2589–2598. [13] S. A. Messaoudi and M. I. Mustafa, On the control of solutions of viscoelastic equations with boundary feedback, Nonlinear Anal. Real World Appl. 10 (2009), no. 5, 3132–3140. [14] S. Wu, Blow-up of solutions for a singular nonlocal viscoelastic equation, J. Partial Differ. Equ. 24 (2011), no. 2, 140–149. [15] Z. Zhang et al., A note on decay properties for the solutions of a class of partial differential equation with memory, J. Appl. Math. Comput. 37 (2011), no. 1-2, 85–102. (Y. Sun) College of Mathematics and Statistics, Nanjing University of Information Science and Technology, Nanjing 210044, China (G. Li) College of Mathematics and Statistics, Nanjing University of Information Science and Technology, Nanjing 210044, China (W. J. Liu) College of Mathematics and Statistics, Nanjing University of Information Science and Technology, Nanjing 210044, China E-mail address: [email protected]

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Measuring fuzziness of generalized fuzzy rough sets induced by pseudo-operations Zhan-hong Shi1,2,∗ , Zeng-tai Gong2,∗ 1. College of Science, Gansu Agricultural University, Lanzhou 730070, P.R. China 2. College of Mathematics and Statistics, Northwest Normal University, Lanzhou 730070, P.R. China

Abstract: Rough sets is a new mathematical tool to handle imprecision, vagueness and uncertainty in data analysis. But, in Pawlak’s rough set model, equivalence relation is a key and primitive notion and this equivalence relation seems to be a very stringent condition that limited the application domain of the rough sets. Various fuzzy generalizations of rough approximations have been made over the years. In this paper, we consider pseudo-operation of the following form: x ⊕ y = g −1 (g(x) + g(y)), where g is a positive strictly monotone generating function and g −1 is its pseudo-inverse. Using this type of pseudooperation, the pseudo-generalized fuzzy rough sets are presented and some properties of the pseudo fuzzy rough approximation operators are investigated. Moreover, we define a measure of fuzziness based on pseudo-generalized fuzzy rough sets with the new pseudo-lower and pseudo-upper approximations. Keywords: Fuzzy sets; Rough sets; Pseudo-operations; Approximation operators

1. Introduction The theory of rough set was originally proposed by Pawlak [1] as a mathematical approach to handle imprecision, vagueness and uncertainty in data analysis. By using the concepts of lower and upper approximations in rough set theory, knowledge hidden in information systems may be unraveled and expressed in the form of decision rules. However, in Pawlak’s rough set model, an equivalence relation is a key and primitive notion. This equivalence relation, however, seems to be a very stringent condition that may limit the application domain of the rough set model. Generalizations of rough set theory were considered by scholars in order to deal with complex practical problems [2-7]. There are at least two approaches for the development of definitions of lower and upper approximation operators, namely, the constructive and axiomatic approaches. In the constructive approach, some authors have extended equivalence relation to tolerance relations [8], similarity relations [9], ordinary binary relations [7,10], and others [11-13]. Meanwhile, some authors have relaxed the partition of universe to the covering and obtain the covering-based rough sets [4,1420]. In addition, generalizations of rough sets to the fuzzy environment have also been made [2,5,21-26]. By introducing the lower and upper approximations in fuzzy set theory, Dubois and ∗ Corresponding author. E-mail Address: [email protected] (Z.H. Shi); [email protected] (Z.T. Gong) 56

SHI ET AL 56-66

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Z.H. Shi, Z.T. Gong: Measuring fuzziness of generalized fuzzy rough sets induced by pseudo-operations

Prade [27] formulated rough fuzzy sets and fuzzy rough sets, they constructed a pair of lower and upper approximation operators for fuzzy sets with respect to fuzzy similarity relation by using the t-norm Min and its dual conorm Max. By using a residual implication (for short, R-implication) to define the lower approximation operator, Morsi and Yakout [28] generalized the fuzzy rough sets in the sense of Dubois and Prade. Later, Radzikowska and Kerre [29] proposed a more general approach to the fuzzification of a rough set. This approach is based on a border implication I (not necessarily a R-implication) and a triangular norm T . Recently, Mi et al. [30] presented the generalized fuzzy rough sets determined by a triangular norm, Ouyang et al. [31] discussed fuzzy rough sets based on tolerance relations. In the axiomatic approaches, a set of axioms is used to characterize the approximations. Lin and Liu [32] proposed six axioms on a pair of abstract operators on the power set of universe in the framework of topological spaces. Under these axioms, there exists an equivalence relation such that the lower and upper approximations are the same as the abstract operators. The most important axiomatic studies for crisp rough sets were made by Yao [7,10,33]. Recently, the research of the axiomatic approach has also been extended to approximation operators in the fuzzy environment [28,30,34-37]. In some problems with uncertainty in the theory of probabilistic metric spaces, fuzzy logics and fuzzy measures, the pseudo-operations such as pseudo-additions and pseudo-multiplications are used [38-40]. Pseudo-analysis [38-47] has been applied in different fields, e.g., measure theory, integration, convolution, Laplace transform, optimization, nonlinear differential and difference equations, economics, game theory, etc. Interestingly, by using the Aczel’s theorem [48], the pseudo-additions and pseudo-multiplications could be transferred into the corresponding results of reals such as the addition operator and multiplication operator. This can bring us the convenience of calculation. We note that there are some literatures about pseudo integrals [7,8,10,25,35], but little literatures about rough set model based on pseudo-operations. The main purpose of this paper is to present a general framework for the study of fuzzy rough approximation operators based on pseudo-operations. By using the pseudo-operations, the pseudo-lower and pseudo-upper approximation operators are defined. Meanwhile, some properties of the proposed pseudo fuzzy rough approximation operators are investigated. Connections between the new and the existing fuzzy rough approximation operators are also discussed. Compared with the previous rough set models based on triangular norms [28-30,37], the pseudo-generalized fuzzy rough set proposed in this paper has its advantage to calculate its lower and upper approximations conveniently. The remainder of this paper is organized as follows. In section 2, we recall some basic concepts of fuzzy sets, fuzzy relation, rough sets and pseudo-operations. In section 3, the pseudo-generalized fuzzy rough sets are presented. Some properties of the proposed pseudo fuzzy rough approximation operators are also investigated in this section. In section 4, the fuzziness of pseudo-generalized fuzzy rough sets is given. Section 5 presents conclusions. 2. Preliminaries 2.1 Fuzzy sets 57

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Z.H. Shi, Z.T. Gong: Measuring fuzziness of generalized fuzzy rough sets induced by pseudo-operations

Let U be a universe. Fuzzy set A is a mapping from U into the unit interval [0, 1]: A : U → [0, 1], where for each x ∈ U , we call A(x) the membership degree of x in A. If U = {x1 , x2 , · · · , xn }, n X then the fuzzy set A on U can be expressed by A(xi )/xi . Additionally, the fuzzy power set, i=1

i.e., the set of all fuzzy sets in the universe U is denoted by F(U ) [49]. For fuzzy sets A, B ∈ F(U ), A ⊆ B ⇔ A(x) ≤ B(x); (A ∩ B)(x) = A(x) ∧ B(x) = min{A(x), B(x)}; (A ∪ B)(x) = A(x) ∨ B(x) = max{A(x), B(x)}; (∼ A)(x) = 1 − A(x), where ∼ A is the complement of A. 2.2 Fuzzy relation Let U and W be two nonempty sets. The Cartesian product of U and W is denoted by U × W. A fuzzy relation R from U to W is a fuzzy subset of U × W , i.e., R ∈ F(U × W ), and R(x, y) is called the degree of relation between x and y. In particular, if U = W , we call R a fuzzy relation on U . Usually, a fuzzy relation can be expressed by a fuzzy matrix. 2.3 Rough sets In traditional Pawlak rough set theory, the pair (U, R) is called an approximation space (it is also called Pawlak approximation space), where U is a finite and non-empty set called the universe and R is an equivalence relation on U , i.e., R is reflexive, symmetrical and transitive. The relation R decomposes the set U into a disjoint class in such a way that two elements x and y are in the same class iff (x, y) ∈ R. Suppose R is an equivalence relation on U . With respect to R, we can define an equivalence class of an element x in U as follows: [x]R = {y|(x, y) ∈ R}. The quotient set of U by the relation R is denoted by U/R, and U/R = {X1 , X2 , · · · , Xm }. where Xi (i = 1, 2, · · · , m) is an equivalence class of R. Given an arbitrary set X ⊆ U , it may not be possible to describe X precisely in the approximation space (U, R). One may characterize X by a pair of lower and upper approximations defined as follows: RX = {x ∈ U |[x]R ⊆ X} = ∪{Y ∈ U/R|Y ⊆ X}; RX = {x ∈ U |[x]R ∩ X 6= φ} = ∪{Y ∈ U/R|Y ∩ X 6= φ}. The pair (RX, RX) is referred to as a rough set of X. 2.4 Pseudo-operations Throughout this paper, we only consider the case of pseudo-addition and present the fuzzy generalized rough sets using pseudo-addition. For the case of pseudo-multiplication, the discussion can be given similarly. 58

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Z.H. Shi, Z.T. Gong: Measuring fuzziness of generalized fuzzy rough sets induced by pseudo-operations

Definition 2.1 An operation ⊕ : [0, ∞]2 → [0, ∞] is called a pseudo-addition if it satisfies the following axioms: (1) Associativity: a ⊕ (b ⊕ c) = (a ⊕ b) ⊕ c for all a, b, c ∈ [0, ∞]. (2) Monotonicity: a ⊕ b ≤ c ⊕ d whenever 0 ≤ a ≤ c ≤ ∞, 0 ≤ b ≤ d ≤ ∞. (3) 0 is neutral element: a ⊕ 0 = 0 ⊗ a = a for all a ∈ [0, ∞]. (4) Continuity: for any sequences (an )n∈N , (bn )n∈N in [0, ∞]N such that lim an = a and n→∞ lim bn = b it holds lim an ⊕ bn = a ⊕ b. n→∞

n→∞

(5) Commutativity: a ⊕ b = b ⊕ a for all a, b ∈ [0, ∞]. Lemma 2.1 (Aczel’s theorem) Let g be a positive strictly monotone function defined on [a, b] ⊆ (−∞, +∞) such that 0 ∈ Ran(g). The generalized generated pseudo-addition ⊕ and the generalized generated pseudo-multiplication ¯ are given by x ⊕ y = g −1 (g(x) + g(y)), x ¯ y = g −1 (g(x)g(y)), where g −1 is pseudo-inverse function for function g: g −1 (y) = sup{x ∈ [a, b]|g(x) < y} if g is a non-decreasing function and g −1 (y) = sup{x ∈ [a, b]|g(x) > y} if g is a non-increasing function. Example 2.2 Suppose that g(x) = 1 − x (x ∈ [0, 1]), then its pseudo-inverse is ( 1 − x, x ∈ [0, 1], g −1 (x) = 0, x ∈ [1, +∞). And x ⊕ y = g −1 (g(x) + g(y)) = max{0, x + y − 1}, this is Lukasiewicz t-norm. 3. Construction of pseudo fuzzy rough approximation operators Definition 3.1 Let U and W be two nonempty sets, R a fuzzy relation from U to W , then (U, W, R) is called a fuzzy approximation space. g : [0, 1] → [0, +∞) is a strictly decreasing function such that g(1) = 0 and g(x) + g(y) ∈ Ran(g) ∪ [g(0+ ), +∞) for all (x, y) ∈ [0, 1]2 . Then for any A ∈ F(W ), the pseudo-lower approximation R⊕ (A) and the pseudo-upper approximation R⊕ (A) of A are defined as follows, respectively: ^ ^ {1 − R(x, y) ⊕ (1 − A(y))} = R⊕ (A)(x) = {1 − g −1 (g(R(x, y)) + g(1 − A(y)))}, x ∈ U ; y∈W

R⊕ (A)(x) =

_

y∈W

{R(x, y) ⊕ A(y)} =

_

y∈W −1

{g (g(R(x, y)) + g(A(y)))}, x ∈ U.

y∈W

The pair (R⊕ (A), R⊕ (A)) is called a pseudo-generalized fuzzy rough set. R⊕ and R⊕ are referred to as the pseudo-lower and pseudo-upper fuzzy rough approximation operators, respectively. Example 3.1 Suppose that (U, W, R) is a fuzzy approximation space, where U and W are two sets called object set and attribute set. Let U = {x1 , x2 , x3 }, W = {a1 , a2 , a3 , a4 }. R ∈ F(U × W ) is a fuzzy relation from U to W and R can be seen in Table 2: For a fuzzy attribute set A = 0.8/a1 + 0.3/a2 + 1/a3 + 0.9/a4 ∈ F(W ), 59

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Z.H. Shi, Z.T. Gong: Measuring fuzziness of generalized fuzzy rough sets induced by pseudo-operations

Table 1: A fuzzy approximation space a1 a2 a3 a4 x1 1 0.4 0 0.1 x2 0.3 0.9 0.7 0.6 x3 0.9 0.2 1 0 if we take a strictly decreasing function as g(x) = 1 − x (x ∈ [0, 1]), then the pseudo-lower approximation R⊕ (A) and the pseudo-upper approximation R⊕ (A) of A can be computed as follows: R⊕ (A)(x1 ) = min{1 − g −1 (0 + 0.8), 1 − g −1 (0.6 + 0.3), 1 − g −1 (1 + 1), 1 − g −1 (0.9 + 0.9)} = 0.8; R⊕ (A)(x2 ) = min{1−g −1 (0.7+0.8), 1−g −1 (0.1+0.3), 1−g −1 (0.3+1), 1−g −1 (0.4+0.9)} = 0.4; R⊕ (A)(x3 ) = min{1 − g −1 (0.1 + 0.8), 1 − g −1 (0.8 + 0.3), 1 − g −1 (0 + 1), 1 − g −1 (1 + 0.9)} = 0.9; R⊕ (A)(x1 ) = max{g −1 (0 + 0.2), g −1 (0.6 + 0.7), g −1 (1 + 0), g −1 (0.9 + 0.1)} = 0.8; R⊕ (A)(x2 ) = max{g −1 (0.7 + 0.2), g −1 (0.1 + 0.7), g −1 (0.3 + 0), g −1 (0.4 + 0.1)} = 0.7; R⊕ (A)(x3 ) = max{g −1 (0.1 + 0.2), g −1 (0.8 + 0.7), g −1 (0 + 0), g −1 (1 + 0.1)} = 1. That is, R⊕ (A) = 0.8/x1 + 0.4/x2 + 0.9/x3 , R⊕ (A) = 0.8/x1 + 0.7/x2 + 1/x3 . Remark 3.1 If R is a crisp binary relation from U to W , then the pseudo fuzzy rough approximation operators defined in Definition 3.1 are degenerated into the approximation operators defined in [36]. That is, for every A ∈ F(W ), x ∈ U, R⊕ (A)(x) = sup{A(y)|y ∈ Rs (x)}, R⊕ (A)(x) = inf{A(y)|y ∈ Rs (x)}, where Rs (x) = {y ∈ W |(x, y) ∈ R}. In fact, R_ ⊕ (A)(x) = {g −1 (g(R(x, y)) + g(A(y)))} y∈W

W = sup{g −1 (g(1) + g(A(y)))|y ∈ Rs (x)} sup{g −1 (g(0) + g(A(y)))|y ∈ / Rs (x)} −1 = sup{g (g(1) + g(A(y)))|y ∈ Rs (x)} = sup{g −1 (0 + g(A(y)))|y ∈ Rs (x)} = sup{A(y)|y ∈ Rs (x)}, R^ ⊕ (A)(x) = {1 − g −1 (g(R(x, y)) + g(1 − A(y)))} y∈W V = inf{1−g −1 (g(1)+g(1−A(y)))|y ∈ Rs (x)} inf{1−g −1 (g(0)+g(1−A(y)))|y ∈ / Rs (x)} −1 = inf{1 − g (g(1) + g(1 − A(y)))|y ∈ Rs (x)} = inf{1 − g −1 (0 + g(1 − A(y)))|y ∈ Rs (x)} = inf{A(y)|y ∈ Rs (x)}.

60

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Z.H. Shi, Z.T. Gong: Measuring fuzziness of generalized fuzzy rough sets induced by pseudo-operations

Remark 3.2 If R is a crisp binary relation on U and A is a crisp set on U , then the pseudo fuzzy rough approximation operators defined in Definition 3.1 are degenerated into the approximation operators defined in [7]. That is, for any A ∈ P (U ), x ∈ U, R⊕ (A) = {x ∈ U |Rs (x) ∩ A 6= φ}, R⊕ (A) = {x ∈ U |Rs (x) ⊆ A}. where Rs (x) = {y ∈ U |(x, y) ∈ R}. In fact, by Remark 3.2, we know that if A ∈ P (U ) then for any x ∈ U, x ∈ R⊕ (A) ⇔ R⊕ (A)(x) = 1 ⇔ ∃ y ∈ Rs (x) such that A(y) = 1, i.e., y ∈ A ⇔ Rs (x) ∩ A 6= φ, x ∈ R⊕ (A) ⇔ R⊕ (A)(x) = 1 ⇔ A(y) = 1 for every y ∈ Rs (x), i.e., y ∈ A ⇔ Rs (x) ⊆ A. Remark 3.3 If R is a crisp equivalence relation on U and A is a fuzzy set on U , then the pseudo fuzzy rough approximation operators defined in Definition 3.1 are degenerated into the approximation operators defined in [27]. That is, for every A ∈ F(U ), x ∈ U, R⊕ (A)(x) = sup{A(y)|y ∈ [x]R }, R⊕ (A)(x) = inf{A(y)|y ∈ [x]R }. In fact, if R is a crisp equivalence relation on U , then Rs (x) = [x]R . Remark 3.4 If R is a crisp equivalence relation on U and A is a crisp set on U , then the pseudo fuzzy rough approximation operators defined in Definition 3.1 are degenerated into the approximation operators defined in [1]. That is, for any A ∈ P (U ), x ∈ U, R⊕ (A) = {x ∈ U |[x]R ∩ A 6= φ}, R⊕ (A) = {x ∈ U |[x]R ⊆ A}.   0.8 0.9 0.6   Example 3.2 Let U = {x1 , x2 , x3 } be the universe of discourse, R =  0.7 0.9 0.1  be a 0.8 0.2 0.8 fuzzy relation on U . Suppose that A, B, C ∈ F(U ), and A = 0.4/x1 + 0.5/x2 + 0.8/x3 ; B = 0.6/x1 + 0.7/x2 + 0.2/x3 ; C = 0.6/x1 + 0.8/x2 + 0.9/x3 . Let g : [0, 1] → [0, +∞) given by g(x) = 1 − x be a generating function for pseudo-addition ⊕, then we can compute that R⊕ (A) = 0.6/x1 + 0.6/x2 + 0.6/x3 ; R⊕ (A) = 0.4/x1 + 0.4/x2 + 0.6/x3 ; R⊕ (B) = 0.6/x1 + 0.8/x2 + 0.4/x3 ; R⊕ (B) = 0.6/x1 + 0.6/x2 + 0.4/x3 ; R⊕ (C) = 0.9/x1 + 1/x2 + 0.9/x3 ; R⊕ (C) = 0.7/x1 + 0.7/x2 + 0.7/x3 . From computation above, we can find A ⊆ C implies that R⊕ (A) ⊆ R⊕ (C) and R⊕ (A) ⊆ R⊕ (C). Furthermore, A ∩ B = 0.4/x1 + 0.5/x2 + 0.2/x3 , 61

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Z.H. Shi, Z.T. Gong: Measuring fuzziness of generalized fuzzy rough sets induced by pseudo-operations

A ∪ B = 0.6/x1 + 0.7/x2 + 0.8/x3 . And R⊕ (A ∩ B) = 0.6/x1 + 0.6/x2 + 0.4/x3 ; R⊕ (A ∩ B) = 0.4/x1 + 0.4/x2 + 0.2/x3 ; R⊕ (A ∪ B) = 0.8/x1 + 0.8/x2 + 0.8/x3 ; R⊕ (A ∪ B) = 0.6/x1 + 0.6/x2 + 0.6/x3 . Thus, we notice that R⊕ (A ∩ B) = R⊕ (A) ∩ R⊕ (B), R⊕ (A ∪ B) = R⊕ (A) ∪ R⊕ (B); R⊕ (A ∪ B) ⊇ R⊕ (A) ∪ R⊕ (B), R⊕ (A ∩ B) ⊆ R⊕ (A) ∩ R⊕ (B). 4. Measuring fuzziness of pseudo-generalized fuzzy rough sets Let (U, W, R) be a fuzzy approximation space, where U and W are two nonempty sets, R is a fuzzy relation from U to W . For any A ∈ F(W ), the pseudo-generalized fuzzy rough set of A is (R⊕ (A), R⊕ (A)). Thus in the fuzzy approximation space (U, W, R), A is approximated by two fuzzy sets, one called the pseudo-lower approximation of A, and another called the pseudo-upper approximation of A. In this section, we suppose that U = W and give an approach to measuring the fuzziness of pseudo-generalized fuzzy rough sets. Definition 4.1 Let U be a universe of discourse, R be a fuzzy relation on U . For any x ∈ U and A ∈ F(U ), the degree of rough membership of x in A is defined by P y∈U [R(x, y) ⊕ A(y)] P . r(A)(x) = y∈U R(x, y) From Definition 4.1, we note that the fuzzy set A and fuzzy relation R on U can induce a new fuzzy set r(A) of U . Theorem 4.1 For any fuzzy sets A, B ∈ F(U ), (1) if A ⊆ B, then r(A) ⊆ r(B); (2) r(A ∩ B) ⊆ r(A) ∩ r(B), r(A ∪ B) ⊇ r(A) ∪ r(B). Proof (1) Since for any x ∈ U , A(x) ≤ B(x). By Definition 4.1, we have P P y∈U [R(x, y) ⊕ A(y)] y∈U [R(x, y) ⊕ B(y)] P P ≤ = r(B)(x). r(A)(x) = y∈U R(x, y) y∈U R(x, y) So r(A) ⊆ r(B). (2) For any A, B ∈ F(U ), we have A ∩ B ⊆ A and A ∩ B ⊆ B. It implies that r(A ∩ B) ⊆ r(A),

r(A ∩ B) ⊆ r(B).

Thus, r(A ∩ B) ⊆ r(A) ∩ r(B). r(A ∪ B) ⊇ r(A) ∪ r(B) can be proved in a similar way. 62

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Z.H. Shi, Z.T. Gong: Measuring fuzziness of generalized fuzzy rough sets induced by pseudo-operations

Definition 4.2 Let U be a universe of discourse, R be a fuzzy relation on U , A ∈ F(U ). The fuzziness of pseudo-generalized fuzzy rough set (R⊕ (A), R⊕ (A)) is defined by F R(A) = −

1 X r(A)(x) · log2 r(A)(x). |U | x∈U

Example 4.1 (Continue the Example 3.2) 

 0.8 0.9 0.6   In Example 3.2, fuzzy relation R =  0.7 0.9 0.1 , three fuzzy sets A, B, C are denoted 0.8 0.2 0.8 as follows, respectively: A = 0.4/x1 + 0.5/x2 + 0.8/x3 ; B = 0.6/x1 + 0.7/x2 + 0.2/x3 ; C = 0.6/x1 + 0.8/x2 + 0.9/x3 . Meanwhile, g(x) = 1 − x (x ∈ [0, 1]). Thus, we can compute that g −1 (0.2 + 0.6) + g −1 (0.1 + 0.5) + g −1 (0.4 + 0.2) 0.8 + 0.9 + 0.6 0.2 + 0.4 + 0.4 = 0.8 + 0.9 + 0.6 = 0.435.

r(A)(x1 ) =

In a similar way, we get

0.1 + 0.4 + 0 = 0.294, 0.7 + 0.9 + 0.1 0.2 + 0 + 0.6 = 0.444. r(A)(x3 ) = 0.8 + 0.2 + 0.8 r(A)(x2 ) =

That is, r(A) = 0.435/x1 + 0.294/x2 + 0.444/x3 . In addition, we can obtain that r(B) = 0.435/x1 + 0.529/x2 + 0.222/x3 , r(C) = 0.783/x1 + 0.588/x2 + 0.555/x3 , r(A ∩ B) = 0.261/x1 + 0.294/x2 + 0.111/x3 , r(A ∪ B) = 0.609/x1 + 0.529/x2 + 0.555/x3 . From computation above, we note that A ⊆ C ⇒ r(A) ⊆ r(C), r(A ∩ B) ⊆ r(A) ∩ r(B) and r(A ∪ B) ⊇ r(A) ∪ r(B) hold. Furthermore, we have 1 F R(A) = − (0.435 × log2 0.435 + 0.294 × log2 0.294 + 0.444 × log2 0.444) ≈ 0.521; 3 1 F R(B) = − (0.435 × log2 0.435 + 0.529 × log2 0.529 + 0.222 × log2 0.222) ≈ 0.497; 3 63

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1 F R(C) = − (0.783 × log2 0.783 + 0.588 × log2 0.588 + 0.555 × log2 0.555) ≈ 0.399; 3 1 F R(A ∩ B) = − (0.261 × log2 0.261 + 0.294 × log2 0.294 + 0.111 × log2 0.111) ≈ 0.459; 3 1 F R(A ∪ B) = − (0.609 × log2 0.609 + 0.529 × log2 0.529 + 0.555 × log2 0.555) ≈ 0.464. 3 From the results of Example 4.1, we note that F R(A) ≥ F R(C) whenever A ⊆ C, but for A ∩ B ⊆ A, F R(A ∩ B) ≤ F R(A). It can be shown that for any A, B ∈ F(U ), if A ⊆ B, F R(A) ≤ F R(B) or F R(A) ≥ F R(B) does not hold. 5. Conclusions At present, there are many researchers about pseudo-analysis. Pseudo-analysis has been applied in different fields. It is interesting to combine pseudo-operations and rough set in order to expand the application domain of pseudo-analysis and rough set. In this paper, we presented a generalized fuzzy rough set model based on pseudo-operation, constructed pseudo fuzzy rough approximation operations. Some properties of the proposed generalized fuzzy rough approximation operators also investigated. At the same time, the fuzziness of pseudo-generalized fuzzy rough sets is given. Acknowledgement This paper is supported by the National Natural Science Foundations of China (71061013, 61262022), the Natural Science Foundation of Gansu Province (1208RJZA251) and the Innovation Foundation of Gansu Agricultural University (GAU-CX1006). References [1] Z. Pawlak, Rough Sets, International Journal of Computer and Information Sciences 11 (5) (1982) 341-356. [2] Z.T. Gong, B.Z. Sun, D.G. Chen, Rough set theory for the interval-valued fuzzy information systems, Information Sciences 178 (2008) 1968-1985. [3] T.Y. Lin, Neighborhood systems and relational database, In Proceedings of 1988 ACM sixteenth annual computer science conference, February (1998) 23-25. [4] Z.H. Shi, Z.T. Gong, The further investigation of covering-based rough sets: uncertainty characterization, similarity measure and generalized models, Information Sciences 180 (2010) 3745-3763. [5] B.Z. Sun, Z.T. Gong, D.G. Chen, Fuzzy-rough set theory for the interval-valued fuzzy information systems, Information Sciences 178 (2008) 2794-2815. [6] W.Z. Wu, W.X. Zhang, Constructive and axiomatic approaches of fuzzy approximation operators, Information Sciences 159 (2004) 233-254. [7] Y.Y. Yao, Constructive and algebraic method of rough sets, Information Sciences 109 (1998) 21-47. [8] A. Skowron, J. Stepaniuk, Tolerance approximation spaces, Fundamenta Informaticae 27 (1996) 245-253. 64

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[32] T.Y. Lin, Q. Liu, Rough approximate operators: axiomatic rough set theory, in: W. Ziarko (Ed.), Rough Sets, Fuzzy Sets and Knowledge Discovery, Springer, Berlin, 1994, pp. 256-260. [33] Y.Y. Yao, A comparative study of fuzzy sets and rough sets, Information Sciences 109 (1998) 227242. [34] G.L. Liu, Axiomatic systems for rough sets and fuzzy rough sets, International Journal of Approximation Reasoning 48 (2008) 857-867. [35] Y.H. She, G.J. Wang, An axiomatic approach of fuzzy rough sets based on residuated lattices, Computers and Mathematics with Applications 58 (2009) 189-201. [36] W.Z. Wu, J.S. Mi, W.X. Zhang, Generalized fuzzy rough sets, Information Sciences 151 (2003) 263-282. [37] W.Z. Wu, Y. Leung, J.S. Mi, On characterizations of (I,T)-fuzzy rough approximation operators, Fuzzy Sets and Systems 154 (1) (2005) 76-102. [38] H. Ichihashi, M. Tanaka, K. Asai, Fuzzy integrals based on pseudo-additions and multiplications, Journal of Mathematical Analysis and Applications 130 (1988) 354-364. [39] J. Li, M. Radko, S. Peter, Pseudo-optimal measures, Information Sciences 180 (2010) 4015-4021. [40] E. Pap, Pseudo-additive measures and their applications, in: E. Pap (Ed.), Handbook of Measure Theory, Elsevier, North-Holland, Amsterdam, 2002, 1237-1260. [41] S.P. Ivana, G. Tatjana, D. Martina, Riemann-Stieltjes type integral based on generated pseudooperations, Novi Sad Journal of Mathematics 36 (2) (2006) 111-124. [42] K. Lendelova, On the pseudo-Lebesgue-Stieltjes integral, Novi Sad Journal of Mathematics 36 (2) (2006) 125-136. [43] E. Pap, N. Ralevic, Pseudo-Laplace transform, Nonlinear Analysis 33 (1998) 533-550. [44] E. Pap, I. Stajner, Generalized pseudo-convolution in the theory of probabilistic metric spaces, information, fuzzy numbers, optimization, system theory, Fuzzy Sets and Systems 102 (1999) 393415. [45] E. Pap, Generalization of the Jensen inequality for pseudo-integral, Information Sciences 180 (2010) 543-548. [46] E. Pap, Generalized real analysis and its applications, International Journal of Approximate Reasoning 47 (2008) 368-386. [47] M. Sugeno, T. Murofushi, Pseudo-additive measures and integrals, Journal of Mathematical Analysis and Applications 122 (1987) 197-222. [48] J. Aczel, Lectures on Functional Equations and their Applications, Academic Press, New York, 1966. [49] L.A. Zadeh, Fuzzy sets, Information and Control 8 (3) (1965) 338-353. [50] W.X. Zhang, Y. Liang, P. Xu, Uncertainty reasoning based on inclusion degree, Tsinghua University Press, Beijing, 2007.

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11

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J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 16, NO.1, 2014, COPYRIGHT 2014 EUDOXUS PRESS, LLC

Global analysis for delay virus infection model with multitarget cells a

A. M. Elaiwa;b and M. A. Alghamdia Department of Mathematics, Faculty of Science, King Abdulaziz University, P.O. Box 80203, Jeddah 21589, Saudi Arabia. b Department of Mathematics, Faculty of Science, Al-Azhar University (Assiut Branch), Assiut, Egypt. Email: [email protected] Abstract

This paper investigates the qualitative behavior of viral infection model with multitarget cells in vivo. The infection rate is given by Crowley-Martin functional response. By assuming that the virus attack n classes of uninfected target cells, we study a viral infection model of dimension 2n + 1 with distributed delay. To describe the latent period for the contacted target cells with viruses to begin producing viruses, two types of distributed delay are incorporated into the model. The basic reproduction number R0 of the model is de…ned which determines the dynamical behavior of the model. Utilizing Lyapunov functionals and LaSalle’s invariance principle, we have proven that if R0 1 then the uninfected steady state is globally asymptotically stable, and if R0 > 1 then the infected steady state is globally asymptotically stable. . Keywords: Viral infection; Global stability; Delay; Crowley-Martin functional response. AMS subject classi…cations. 92D25, 34D20, 34D23 :

1

Introduction

Mathematical models have proven their importance in understanding the dynamical behaviors of various viruses such as human immunode…ciency virus (HIV), hepatitis B virus (HBV), hepatitis C virus (HCV), etc. [1]. The interatcion of the virus and target cells has been formulated as ordinary di¤erential equations in several works (see e.g. [2], [3], [4], [12], [11], [5] and [6]). The basic mathematical model describing the dynamics of viral infection can be written in a general form as [6]: x_ = dx h(x; v); y_ = h(x; v) y; v_ = ky rv;

(1) (2) (3)

where x; y and v represent the populations of the uninfected target cells, infected cells and free virus particles, respectively. The uninfected cells are generated from sources within the body at rate . The parameter d is the death rate constant of the uninfected target cells. Eq. (2) describes the population dynamics of the infected cells and shows that they die with rate constant . The virus particles are produced by the infected cells with rate constant k, and are cleared from plasma with rate constant r. The function h(x; v) represents the incidence rate of infection and it has been considered in the viral infection models by di¤erent forms: Bilinear incidence rate [2], [3]: h(x; v) = xv: xv Saturated incidence rate [30]: h(x; v) = 1+bv : xv Holling type II functional response [34]: h(x; v) = 1+ax : xv Beddington-DeAngelis infection rate [28]: h(x; v) = 1+ax+bv : xv Crowley-Martin functional response [31], [32]: h(x; v) = (1+ax)(1+bv) ;where a; b 0 and is the rate constant characterizing infections of the cells. The Crowley-Martin type of functional response was …rst introduced by Crowley and Martin [33]. Model (1)-(3) is based on the assumption that, once the virus contacts a target cell, the cell begins producing new virus particles. More realistic models incorporate the delay between the time of viral entry into the target cell and the time the production of new virus particles, modeled with discrete time delay or distributed time delay using functional di¤erential equations. Many researchers have devoted their e¤ort in developing various mathematical models of viral infections with discrete or distributed delays and studying their qualitative behaviors (see e.g. [8], [10], [9], [27], [29], [24], [26], [22], [21], [34]). 1 67

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In the literature, most of the proposed mathematical models for viral infection assume that the virus has one class of target cells, (e.g. CD4+ T cells in case of HIV or hepatic cells in case of HCV and HBV) (see e.g. [2], [3] and the book Nowak and May [1]). In [7], [25], [13], [15], [18], [19], and [16], some HIV models with two classes of target cells, CD4+ T cells and macrophages have been proposed. The global stability of these models has been investigated in ([13], [15] and [16]). Because the interactions of some types of viruses in vivo is complex and is not known clearly, we would suppose that the virus may attack n classes of target cells where n 1 [14], [17]. In [17], models with discrete-time delays and saturated incidence rate have been studied. Elaiw [14] studied a class of virus infection models with multitarget cells without time delay. The purpose of this paper is to propose a viral infection model with multitarget cells and Crowley-Martin functional response and investigate its qualitative behavior. We incorporate distributed delay into the model which represents an intracellular latent period for the contacted uninfected target cells with virus to begin producing new virus particles. The global stability of this model is established using Lyapunov functionals and LaSalle’s invariance principle. We prove that the global dynamics of this model is determined by the basic reproduction number R0 . If R0 1, then the uninfected steady state is globally asymptotically stable (GAS) and if R0 > 1, then the infected steady state exists and is GAS.

2

Model with distributed time delays

In this section we propose a virus dynamics model with multitarget cells and multiple distributed intracellular delays. x_ i (t) = y_ i (t) =

di xi

i

i

Zi

fi ( )e

i xi (t)v(t)

(1 + ai xi (t))(1 + bi v(t)) mi

xi (t (1 + ai xi (t

;

)v(t ) ))(1 + bi v(t

))

d

i yi (t);

i = 1; :::; n

(4)

i = 1; :::; n

(5)

0

v(t) _ =

n X i=1

ki

Zi

gi ( )e

ni

yi (t

)d

rv(t);

(6)

0

where xi and yi represent the populations of the uninfected target cells and infected cells of class i, respectively, v is the population of the virus particles. To account for the time lag between viral contacting a target cell and the production of new virus particles, two distributed intracellular delays are introduced. It is assumed that the target cells of class i are contacted by the virus particles at time t become infected cells at time t, where is a random variable with a probability distribution fi ( ) over the interval [0; i ] and i is limit superior of this delay. The factor e mi accounts for the loss of target cells during delay period where mi is positive constant. On the other hand, it is assumed that, a cell infected at time t starts to yield new infectious virus at time t where is distributed according to a probability distribution gi ( ) over the interval [0; i ] and i is limit superior of this delay. The factor e ni account for the cells loss during this delay period where ni is positive constant. All the other parameters of the model have the same biological meaning as given in model (1)-(3). The probability distribution functions fi ( ) : [0; i ] ! R+ and gi ( ) : [0; i ] ! R+ are integral functions Ri Ri Ri Ri 0, with fi ( )d = gi ( )d = 1; i = 1; :::; n. De…ne Fi = fi ( )e mi d and Gi = gi ( )e ni d , mi 0

ni

0

0

0

0. It is clear that 0 < Fi 1 and 0 < Gi 1, i = 1; :::; n. The initial conditions for system (4)-(6) take the form

xj ( ) = 'j ( ); yj ( ) = 'j+n ( ); j = 1; :::; n; v( ) = '2n+1 ( ); 'j ( ) 0; 2 [ `; 0); 'j (0) > 0; j = 1; :::; 2n + 1;

(7)

where ` = maxf 1 ; :::; n ; 1 ; :::; n g, ('1 ( ); '2 ( ); :::; '2n+1 ( )) 2 C and C = C([ `; 0]; R2n+1 ) is the Banach + space of continuous functions mapping the interval [ `; 0] into R2n+1 . By the fundamental theory of functional + di¤erential equations [20], system (4)-(6) has a unique solution satisfying initial conditions (7).

2.1

Non-negativity and boundedness of solutions

In the following, we establish the non-negativity and boundedness of solutions of (4)-(6) with initial conditions (7). Let x = (x1 ; x2 ; :::; xn )T and y = (y1 ; y2 ; :::; yn )T . Proposition 2. Let (x(t); y(t); v(t)) be any solution of (4)-(6) satisfying the initial conditions (7), then x(t); y(t) and v(t) are all non-negative for t 0 and ultimately bounded. 2 68

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Proof. First, we prove that xi (t) > 0, i = 1; :::; n, for all t 0. Assume that xi (t) lose its non-negativity on some local existence interval [0; !] for some constant ! and let t1 2 [0; !] be such that xi (t1 ) = 0. From Eq. (4) we have x_ i (t1 ) = i > 0. Hence xi (t) < 0 for some t 2 (t1 "; t1 ), where " > 0 is su¢ ciently small. This leads to a contradiction and hence xi (t) > 0, for all t 0. Further, from Eqs. (5) and (6) we have yi (t) = yi (0)e

it

+

i

Zt

e

ki

Zt

i (t

)

v(t) = v(0)e

+

n X i=1

fi ( )e

mi

xi ( (1 + ai xi (

)v( ) ))(1 + bi v(

))

d d ;

0

0

rt

Zi

e

r(t

)

0

Zi

gi ( )e

ni

yi (

)d d ;

0

con…ming that yi (t) 0; i = 1; :::; n, and v(t) 0 for all t 2 [0; `]. By a recursive argument, we obtain yi (t) 0, i = 1; :::; n, and v(t) 0 for all t 0. Now we show the boundedness of the solutions of (4)-(6). Eqs. (4) imply that lim supt!1 xi (t) x0i , i R where x0i = i =di , and thus xi (t) is ultimately bounded. If follows that fi ( )e mi xi (t )d Fi x0i . Let Xi (t) =

Ri

0

fi ( )e

mi

xi (t

)d + yi (t), i = 1; :::; n, then

0

X_ i (t) =

Zi

fi ( )e

mi

+

Zi

fi ( )e

mi

di xi (t

i

i xi (t

)

(1 + ai xi (t

)v(t ) ))(1 + bi v(t

))

d

0

i xi (t

)v(t ) ))(1 + bi v(t

(1 + ai xi (t

))

d

i yi (t)

Fi

i Xi (t);

i

0

where

i

= minfdi ; i g. Hence lim supt!1 Xi (t)

we get lim supt!1 yi (t)

n X

ki Li

i=1

2.2

i Fi = i

. Since

Zi

gi ( )e

ni

d

rv =

n P

i=1

fi ( )e

mi

xi (t

)d > 0,

n X

ki Li Gi

rv;

i=1

0

L , where L =

Ri 0

Li . On the other hand, v(t) _

then lim supt!1 v(t)

Li , where Li =

ki Li Gi . r

Therefore, x(t); y(t) and v(t) are ultimately bounded.

Steady states

System (4)-(6) has an uninfected steady state E0 = (x0 ; y0 ; v 0 ), where x0i = dii , yi0 = 0; i = 1; :::; n and v 0 = 0. In addition to E0 , the system can has a positive infected steady state E1 (x ; y ; v ). The coordinates of the infected steady state, if they exist, satisfy the equalities: i xi v ; (1 + ai xi )(1 + bi v ) i xi v ; i yi = Fi (1 + ai xi )(1 + bi v ) n X rv = Gi ki yi : i

= di xi +

i = 1; :::; n;

(8)

i = 1; :::; n;

(9) (10)

i=1

The basic reproduction number of system (4)-(6) is given by R0 =

n X

Ri =

i=1

n X Fi Gi i ki x0i ; r(1 + ai x0i ) i=1 i

(11)

where Ri is the basic reproduction number for the dynamics of the interaction of the virus only with the target cells of class i. Lemma 1. If R0 > 1, then there exists a positive steady state E1 . 3 69

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J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 16, NO.1, 2014, COPYRIGHT 2014 EUDOXUS PRESS, LLC

Proof. To compute the steady states of model (4)-(6), we let the right-hand sides of Eqs. (4)-(6) equal zero, i

i xi v = 0; (1 + ai xi )(1 + bi v) Fi i xi v i yi = 0; (1 + ai xi )(1 + bi v) n X Gi ki yi rv = 0:

di xi

i = 1; :::; n;

(12)

i = 1; :::; n;

(13) (14)

i=1

Solving Eq. (12) with respect to xi , we get xi as a function of v as: q 2 0 [(1 + i v) ai x0i (1 + bi v)] + 4ai x0i (1 + bi v)2 a x (1 + b v) (1 + v) + i i i i + ; xi = 2ai (1 + bi v) q 2 ai x0i (1 + bi v) (1 + i v) [(1 + i v) ai x0i (1 + bi v)] + 4ai x0i (1 + bi v)2 xi = ; 2ai (1 + bi v)

(15)

(16)

where, i = bi + dii . + It is clear that if v > 0 then x+ i > 0 and xi < 0. Let us choose xi = xi . From Eqs. (12)-(14) we have n X ki Fi Gi

(

di xi )

i

rv = 0:

(17)

i

i=1

Since xi is a function of v, then we can de…ne a function S1 (v) as: S1 (v) =

n X ki Fi Gi i=1

(

di xi )

i

rv = 0:

i

It is clear that when v = 0, then xi = x0i and S1 (0) = 0 and when v = v = it in Eq. (15) we obtain xi > 0 and S1 (v) =

n X ki di Fi Gi

i=1

F i G i ki ir

i

> 0, then substituting

xi < 0:

i

i=1

Since S1 (v) is continuous for all v

n P

0; we have that S10 (0) =

n X ki i x0i Fi Gi (1 + ai x0i ) i=1 i

r = r(R0

1):

Therefore, if R0 > 1, then S10 (0) > 0. It follows that there exists v 2 (0; v) such that S1 (v ) = 0. From Eq. (15), we obtain xi > 0; i = 1; :::; n. Moreover, from Eq, (13) we get yi > 0; i = 1; :::; n.

2.3

Global stability

In this section, we prove the global stability of the uninfected and infected steady states of system (4)-(6) employing the method of Lyapunov functional which is used in [23] for SIR epidemic model with distributed delay. Next we shall use the following notation: z = z(t), for any z 2 fxi ; yi ; v; i = 1; :::; ng. We also de…ne a function H : (0; 1) ! [0; 1) as H(z) = z 1 ln z. It is clear that H(z) 0 for any z > 0 and H has the global minimum H(1) = 0. Theorem 1. (i) If R0 1, then E0 is GAS. (ii) If R0 > 1, then E1 is GAS. Proof. (i) De…ne a Lyapunov functional W1 as: 2 Zi Z n X ki Fi Gi 4 x0i xi 1 i ni W1 = H + yi + gi ( )e yi (t )d d 1 + ai x0i x0i Fi Fi G i i i=1 0 0 3 Zi Z x (t )v(t ) i i + fi ( )e mi d d 5 + v: Fi (1 + ai xi (t ))(1 + bi v(t )) 0

0

4 70

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The time derivative of W1 along the trajectories of (4)-(6) satis…es n

X ki Fi Gi dW1 1 = dt 1 + ai x0i i i=1 +

i

Fi

Zi

fi ( )e

mi

Zi

fi ( )e

mi

x0i xi

1

xi (t (1 + ai xi (t

i xi v (1 + ai xi )(1 + bi v)

di xi

i

)v(t ) ))(1 + bi v(t

))

d

i

yi

Fi

0

+

i

Fi

xi v (1 + ai xi )(1 + bi v) 3

0

+

i

Fi G i

Zi

gi ( )e

ni

(yi

))d 5 +

yi (t

0

=

n X ki Fi Gi i

i=1

+ =

i

0 i xi v

ai x0i ) (1

+ ai xi )(1 + bi v) (1 + n X ki Fi Gi i xi x0i (1 + ai x0i )

i

i=1

n X ki Fi Gi di xi x0i 0 i xi (1 + ai xi ) i=1

=

n X

=

2

ki Fi Gi di xi x0i 0 i xi (1 + ai xi )

i=1

n X

ki

i=1

x0i

xi x0i

2

1 + ai x0i

xi (t (1 + ai xi (t

xi

Zi

gi ( )e

x0i

xi

2

+

2

yi (t

)d

rv

i xi v (1 + ai x0i ) (1 + ai xi )(1 + bi v)

(1 +

0 i xi v ai x0i ) (1

n X

rv

+ bi v)

Fi Gi ki i x0i v r(1 + ai x0i )(1 + bi v) i=1 i ! rbi Ri v 2 + + (R0 1) rv: 1 + bi v

+r

ni

d

))

0

i xi v (1 + ai xi )(1 + bi v)

+

)v(t ) ))(1 + bi v(t

rv

rv

(18)

1 If R0 1, then dW 0 for all xi ; v > 0. By Theorem 5.3.1 in [20], the solutions of system (4)-(6) limit to M , dt dW1 0 1 the largest invariant subset of dW dt = 0 . Clearly, it follows from (18) that dt = 0 if and only if xi = xi and v = 0. Noting that M is invariant, for each element of M we have v = 0, then v_ = 0. From Eq. (6) we drive n Ri P 0 1 that 0 = v_ = gi ( )e ni ki yi (t )d . This yields yi = 0 and hence dW dt = 0 if and only if xi = xi , yi = 0

i=1 0

and v = 0. From LaSalle’s invariance principle, E0 is GAS. (ii) We construct the following Lyapunov functional 2 Zxi n X ki Fi Gi 6 xi (1 + ai ) 1 W2 = d + yi H 4xi xi (1 + ai xi ) Fi i i=1 xi

+

+

1 i xi v Fi (1 + ai xi )(1 + bi v ) i yi

Fi Gi

Zi

gi ( )e

ni

0

Z

Zi

fi ( )e

mi

0

0

yi (t yi

H

0

)

Di¤erentiating with respect to time yields n

X ki Fi Gi dW2 = dt i i=1 1 + Fi +

i

Fi

1 Zi

yi yi

fi ( )e

@ mi

H

i

Zi

fi ( )e

xi (t )v(t )(1 + ai xi )(1 + bi v ) xi v (1 + ai xi (t ))(1 + bi v(t ))

3

d d 5+v H

xi (1 + ai xi ) xi (1 + ai xi )

1 0

Z

mi

yi yi

i

di xi

xi (t (1 + ai xi (t

v v

:

i xi v (1 + ai xi )(1 + bi v)

)v(t ) ))(1 + bi v(t

))

d

i yi

0

xi v (1 + ai xi )(1 + bi v)

d d

xi (t (1 + ai xi (t

)v(t ) ))(1 + bi v(t

1 A

))

0

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xi (t )v(t )(1 + ai xi )(1 + bi v) xi v ln (1 + ai xi )(1 + bi v ) xi v(1 + ai xi (t ))(1 + bi v(t )) 3 Zi y (t ) i i gi ( )e ni yi yi (t ) + yi ln + d 5 Fi G i yi 0 0 1 Zi n X v @ + 1 ki gi ( )e ni yi (t )d rv A : v i=1 +

d

(19)

0

Collecting terms of (19) we obtain n

X ki Fi Gi dW2 = dt i i=1 +

+

xi (1 + ai xi ) xi (1 + ai xi )

1

i vxi

i

(1 + ai xi )(1 + bi v)

Fi

i yi + Fi Gi

ni

gi ( )e

di xi )

i

mi

fi ( )e

yi xi (t yi (1 + ai xi (t

)v(t ) ))(1 + bi v(t

))

d +

i

Fi

yi

0

Zi

1 i xi v Fi (1 + ai xi )(1 + bi v ) Zi

Zi

(

fi ( )e

mi

0

yi (t yi

ln

)

0

ln 3

xi (t )v(t )(1 + ai xi )(1 + bi v) xi v(1 + ai xi (t ))(1 + bi v(t )) Z n v X ki gi ( )e v i=1

d

i

d 5

rv

ni

yi (t

)d + rv :

0

Using the infected steady state conditions (8)-(10), we obtain n

X ki Fi Gi dW2 = dt i i=1 Z

i y Fi2 i

+3

i

Fi

i

mi

fi ( )e

0

yi +

i yi + Fi Gi

xi (1 + ai xi ) xi (1 + ai xi )

1

Zi

i y Fi2 i

gi ( )e

Zi

(di xi

i

di xi )

Fi

yi

xi (1 + ai xi ) v(1 + bi v ) i + yi xi (1 + ai xi ) Fi v (1 + bi v)

xi (t )v(t )yi (1 + ai xi )(1 + bi v ) d xi v yi (1 + ai xi (t ))(1 + bi v(t ))

fi ( )e

mi

ln

xi (t )v(t )(1 + ai xi )(1 + bi v) xi v(1 + ai xi (t ))(1 + bi v(t ))

0

ni

ln

yi (t yi

)

v i yi Fi v

d

i

Fi Gi

yi

0

=

n X ki Fi Gi i=1 i

Fi i

Fi2

yi

i

Fi Gi

xi (1 + ai xi ) xi (1 + ai xi )

1

xi (1 + ai xi ) xi (1 + ai xi )

yi H Zi

fi ( )e

mi

0

yi

Zi 0

Zi

gi ( )e

0

i

gi ( )e

ni

d

i

Fi

(di xi

yi H

di xi ) +

i

Fi

yi

1

ni

v yi (t vyi

)

3

d 5

v v(1 + bi v ) 1 + bi v + + v v (1 + bi v) 1 + bi v

1 + bi v 1 + bi v

xi (t )v(t )yi (1 + ai xi )(1 + bi v ) xi v yi (1 + ai xi (t ))(1 + bi v(t )) 3 v yi (t ) H d 5 vyi

H

d

6 72

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J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 16, NO.1, 2014, COPYRIGHT 2014 EUDOXUS PRESS, LLC n X ki Fi Gi

=

+

+

i

Fi

"

i=1

i

yi H

1 + bi v 1 + bi v

i

Fi G i

yi

Zi

2

2

v ) di (xi xi ) i yi bi (v i + + yi H xi (1 + ai xi ) Fi v (1 + bi v) (1 + bi v ) Fi

gi ( )e

+

ni

i y Fi2 i

H

0

Zi

fi ( )e

0

v yi (t vyi

)

mi

H

xi (1 + ai xi ) xi (1 + ai xi )

xi (t )v(t )yi (1 + ai xi )(1 + bi v ) ))(1 + bi v(t )) xi v yi (1 + ai xi (t

d

3

d 5:

2 It is easy to see that if xi ; yi ; v > 0; i = 1; :::; n, then dW 0. By Theorem 5.3.1 in [20], the solutions of dt dW2 2 system (4)-(6) limit to M , the largest invariant subset of dt = 0 . It can be seen that dW dt = 0 if and only if xi = xi ; v = v , and H = 0 i.e.

v yi (t xi (t )v(t )yi (1 + ai xi )(1 + bi v ) = ))(1 + bi v(t )) xi v yi (1 + ai xi (t vyi If v = v , then from Eq. (20) we have yi = yi ; and hence implies global stability of E1 .

3

dW2 dt

)

= 1 for all

2 [0; `]:

(20)

equal to zero at E1 . LaSalle’s invariance principle

Conclusion

In this paper, we have investigated mathematical model of virus dynamics with distributed delay. We have assumed that the virus attack n classes of target cells. The infection rate is given by Crowley-Martin functional response. By de…ning the delay-dependent basic reproduction number R0 , we have discussed the existence of the steady states. The global stability of the uninfected and infected steady states of the model has been established using suitable Lyapunov functionals and LaSalle’s invariant principle. We have proven that, if R0 < 1, then the uninfected steady state is GAS and if R0 > 1, then infected steady state is GAS.

4

Acknowledgements

This article was funded by the Deanship of Scienti…c Research (DSR), King Abdulaziz University, Jeddah. The authors, therefore, acknowledge with thanks DSR technical and …nancial support.

References [1] M.A. Nowak, and R.M. May, “Virus dynamics: Mathematical Principles of Immunology and Virology,” Oxford Uni., Oxford, 2000. [2] M.A. Nowak, C.R.M. Bangham, Population dynamics of immune responses to persistent viruses, Science, 272 (1996), 74-79. [3] M. A. Nowak, S. Bonhoe¤er, A. M. Hill, Viral Dynamics in Hepatitis B Virus Infection, Proc. Natl. Acad. Sci. USA, 93 (1993), 4398-4402. [4] L. Wang, M. Y. Li, and D. Kirschner, Mathematical analysis of the global dynamics of a model for HTLV-I infection and ATL progression, Math. Biosc., 179 (2002), 207-217. [5] A. Korobeinikov, Global properties of basic virus dynamics models, Bull. Math. Biol., 66 (2004), 879-883. [6] A. Korobeinikov, Global asymptotic properties of virus dynamics models with dose-dependent parasite reproduction and virulence, and non-linear incidence rate, Math. Med. Biol., 26 (2009), 225-239. [7] D. S. Callaway, and A. S. Perelson, HIV-1 infection and low steady state viral loads, Bull. Math. Biol., 64 (2002), 29-64. [8] R.V. Culshaw, and S. Ruan, A delay-di¤erential equation model of HIV infection of CD4+ T-cells, Math. Biosci., 165 (2000), 27-39. [9] P. W. Nelson, J. Murray, and A. Perelson, A model of HIV-1 pathogenesis that includes an intracellular delay, Math. Biosci., 163 (2000), 201-215. 7 73

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[10] N. M. Dixit, and A. S. Perelson, Complex patterns of viral load decay under antiretroviral therapy: In‡uence of pharmacokinetics and intracellular delay, J. Theoret. Biol., 226 (2004), 95-109. [11] E. Gyurkovics, and A. M. Elaiw, A Stabilizing sampled-data ` step receding horizon control with application to a HIV/AIDS model, Di¤er. Eqs. Dynamic. Systems 14 (3-4) (2006), 323-352. [12] A. M. Elaiw, K. Kiss, M.A.L. Caetano, Stabilization of HIV/AIDS model by receding horizon control, J. of Appl. Math. Computing 18 (1-2) (2005), 95-112. [13] A. M. Elaiw, Global properties of a class of HIV models, Nonlinear Anal. Real World Appl., 11 (2010), 2191-3286. [14] A. M. Elaiw, Global properties of a class of virus infection models with multitarget cells, Nonlinear Dynam., 69 (2012) 423-435. [15] A. M. Elaiw, and X. Xia, HIV dynamics: Analysis and robust multirate MPC-based treatment schedules, J. Math. Anal. Appl., 356 (2009), 285-301. [16] A.M. Elaiw, I. A. Hassanien, and S. A. Azoz, Global stability of HIV infection models with intracellular delays, J. Korean Math. Soc., 49(4) (2012), 779-794. [17] A. M. Elaiw and M. A. Alghamdi, Global properties of virus dynamics models with multitarget cells and discrete-time delays, Discrete Dynamics in Nature and Society, 2011, Article ID 201274. [18] A.M. Elaiw, Global dynamics of an HIV infection model with two classes of target cells and distributed delays, Discrete Dynamics in Nature and Society, vol. 2012, Article ID 253703, 13 pages, 2012. [19] A.M. Elaiw and S.A. Azoz, Global properties of a class of HIV infection models with Beddington-DeAngelis functional response, Mathematical Methods in the Applied Sciences, 36 (2013), 383-394. [20] J.K. Hale, and S. Verduyn Lunel, “Introduction to functional di¤erential equations,”Springer-Verlag, New York, 1993. [21] G. Huang, Y. Takeuchi, and W. Ma, Lyapunov functionals for delay di¤erential equations model of viral infection, SIAM J. Appl. Math., 70 (2010), 2693-2708. [22] M. Y. Li, and H. Shu, Impact of intracellular delays and target-cell dynamics on in vivo infection, SIAM J. Appl. Math., 70 (2010), 2434-2448. [23] C. C. McCluskey, Complete global stability for an SIR epidemic model with delay-distributed or discrete, Nonlinear Anal. Real World Appl., 11 (2010), 55-59. [24] P. W. Nelson, and A. S. Perelson, Mathematical analysis of delay di¤erential equation models of HIV-1 infection, Math. Biosci., 179 (2002), 73-94. [25] A. S. Perelson, and P. W. Nelson, Mathematical analysis of HIV-1 dynamics in vivo, SIAM Rev., 41 (1999), 3-44. [26] R. Xu, Global dynamics of an HIV-1 infection model with distributed intracellular delay, Comput. Math. Appl., 61 (2011), 2799-2805. [27] S. Liu and L. Wang, Global stability of an HIV-1 model with distributed intracellular delays and a compination therapy, Math. Biosc. Eng., 7 (2010) 675-685. [28] G. Huang, W. Ma, Y. Takeuchi, Global properties for virus dynamics model with Beddington-DeAngelis functional response, Appl. Math. Lett., 22 (2009), 1690-1694. [29] G. Huang, W. Ma, Y. Takeuchi, Global analysis for delay virus dynamics model with Beddington-DeAngelis functional response, Appl. Math. Lett., 24, no. 7 (2011), 1199–1203. [30] X. Song, A.U. Neumann, Global stability and periodic solution of the viral dynamics, J. Math. Anal. Appl., 329 (2007), 281-297. [31] S. Xu, Global stability of the virus dynamics model with Crowley-Martin functional response, Electronic Journal of Qualitative Theory of Di¤erential Equations, 2012 no. 9, (2012), 1-9.

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[32] X. Y. Zhou, J. A. Cui, Global stability of the viral dynamics with Crowley-Martin functional response. Bull. Korean. Math. Soc., 48 (2011), 555-574. [33] P. H. Crowley, E. K. Martin, Functional responses and interference within and between year classes of a dragon‡y population. J. North. Am. Benth. Soc., 8 (1989), 211-221. [34] G. Huang, H. Yokoi, Y. Takeuchi, T. Kajiwara, T. Sasaki, Impact of intracellular delay, immune activation delay and nonlinear incidence on viral dynamics, Japan J. Indust. Appl. Math., 28 (2011), 383-411.

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The parameter reduction of soft sets and its algorithm ∗ Zhaowen Li†

Ninghua Gao‡

February 21, 2013 Abstract: Soft set theory is a new mathematical tool to deal with uncertain problems. In this paper, we prove the fact that there exists a one-to-one correspondence between “the set of all soft sets” and “the set of all 2-value information systems”. Base on this fact, we investigate the parameter reduction of soft sets by means of the knowledge reduction in rough set theory and give an algorithm. Parameters of soft sets are classified and the core of soft sets are obtained. Keywords: Soft sets; Rough sets; Information systems; One-to-one correspondences; Parameter reductions; Cores.

1

Introduction

In 1999, Molodtsov [6] proposed soft set theory as a new mathematical tool for dealing with uncertainties which is free from the difficulties affecting existing method. As reported in [6, 7], a wide range of applications of soft sets have been developed in many different fields, including the smoothness of functions, game theory, operations research, Riemann integration, Perron integration, probability theory and measurement theory. Presently, works on theory of soft sets are progressing rapidly. Maji et al. [8, 9] further studied the theory of soft sets, used this theory to solve some decision making problems. Jiang et al. [4] extended soft sets with description logics. Ge et al. [3] discussed relationships between soft sets and topological spaces. Rough set theory was initiated by [10] for dealing with vagueness and granularity in information systems. This theory handles the approximation of an arbitrary subset of a universe by two definable or observable subsets called lower and upper approximations. It has been successfully applied to machine learning, ∗ This work is supported by the National Natural Science Foundation of China (No.11061004). † Corresponding Author, School of Science, Guangxi University for Nationalities, Nanning, Guangxi 530006, P.R.China. [email protected] ‡ School of Science, Guangxi University for Nationalities, Nanning, Guangxi 530006, P.R.China. [email protected]

1

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intelligent information systems, inductive reasoning, pattern recognition, mereology, image processing, signal analysis, knowledge discovery, decision analysis, expert information systems and many other fields (see [11]). Soft set itself has classification ability. The parameter reduction of soft sets means reducing the number of parameters for a soft set to the minimum without distorting its original classification ability. Thus, the parameter reduction of soft sets is a very important problem in soft set theory. Maji et al. [9] introduce parameter reduction of soft sets. Unfortunately some errors in [9] were pointed out by Chen et al. [2]. They present a new definition of parameterization reduction in soft sets. In [5], Kong et al. pointed out some odd situations which may occur when method of reduction of parameters in case of soft sets given in [2] is applied. So they introduced the concept of reduction of normal parameters. In [1], it has been seen that there is a very close relationship between soft sets and rough sets. The purpose of this paper is to investigate further the parameter reduction of soft sets with the help of rough set theory. We prove the fact that there exists a one-to-one correspondence between “the set of all soft sets” and “the set of all 2-value information systems”. Base on this fact, we can do consider the parameter reduction of soft sets by means of the knowledge reduction in rough set theory.

2

Preliminaries

2.1

Soft sets

Definition 2.1 ([6]). Let U be an initial universe and let A be a set of parameters. A pair (f, A) is called a soft set over U , if f is a mapping given by f : A → 2U where 2U is the power set of U . We denote (f, A) by fA . In other words, a soft set over U is a parameterized family of subsets of the universe U . For e ∈ A, f (e) may be considered as the set of e-approximate elements of the soft set fA . Example 2.2. Let U = {h1 , h2 , h3 , h4 , h5 } be a universe consisting of five stores. Let A = {a1 , a2 , a3 , a4 , a5 , a6 , a7 } be is a set of status of stores where a1 , a2 , a3 , a4 , a5 , a6 and a7 represent respectively the parameters “high empowerment of sales personnel”, “medium empowerment of sales personnel”, “low empowerment of sales personnel”, “good perceived quality of merchandise”, “average perceived quality of merchandise”, “high traffic location” and “low traffic location”, respectively. We define fA as follows f (a1 ) = {h1 }, f (a2 ) = {h2 , h3 , h5 }, f (a3 ) = {h4 }, f (a4 ) = {h1 , h2 , h3 }, f (a5 ) = {h4 , h5 }, f (a6 ) = {h1 , h2 , h3 }, f (a7 ) = {h4 , h5 }. Soft sets fA can be described as the following Table 1. If hi ∈ f (aj ), then hij = 1; otherwise hij = 0, where hij are the entries in Table 1. 2

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Table 1: Tabular representation of the soft set fA a1 1 0 0 0 0

h1 h2 h3 h4 h5

a2 0 1 1 0 1

a3 0 0 0 1 0

a4 1 1 1 0 0

a5 0 0 0 1 1

a6 1 1 1 0 0

a7 0 0 0 1 1

Definition 2.3. Let fA be a soft set over U . fA is called non-trivial, if for any a ∈ A, f (a) 6= ∅ and f (a) 6= U. In this paper, we only consider non-trivial soft sets.

2.2

Information systems

Definition 2.4 ([11, 12]). Let U be a finite set of objects and let A be a finite set of attributes. The pair (U, A, V, g) is called an information system ( a knowledge representation system ), if g is an information function from U × A to V = S Va where every Va = {g(x, a) : a ∈ A and x ∈ U } is the values of the a∈A

attribute a. Definition 2.5. An information system (U, A, V, g) is called 2-value, if V = {0, 1}. Example 2.6. Let U = {h1 , h2 , h3 , h4 } be a universe consisting of four patients, and let A = {a1 , a2 , a3 } be a set of attributes where a1 , a2 and a3 represent respectively the attributes “ headache”, “ muscle pain” and “ fever”. Now, we consider an information system (U, A, V, g), which describes the “ symptoms of patients”. For instance, “g(h1 , a1 ) = yes” means “h1 suffers from headache” and its functional value is yes; “g(h3 , a2 ) = no ”means “h3 has no muscle pain” and its functional value is no; “g(h3 , a3 ) = no” means “h3 doesn’t have a fever” and its functional value is no. We define g(h1 , a1 ) =yes, g(h1 , a2 ) =yes, g(h1 , a3 ) =no; g(h2 , a1 ) =yes, g(h2 , a2 ) =yes, g(h2 , a3 ) =yes; g(h3 , a1 ) =yes, g(h3 , a2 ) =yes, g(h3 , a3 ) =no; g(h4 , a1 ) =no, g(h4 , a2 ) =yes, g(h4 , a3 ) =no. Let hij be the entries. If g(hi , aj ) =yes, then hij = 1; if g(hi , aj ) =no, then hij = 0. A 2-value information system (U, A, V, g) can be described as the following Table 2. S In Table 2, Va1 = {0, 1}, Va2 = {0, 1}, Va3 = {0, 1}, V = a∈A Va = {0, 1}. Let (U, A, V, g) be an information system and let P ⊆ A. We denote ind(P ) = {(x, y) ∈ U × U : g(x, a) = g(y, a) f or any a ∈ P }. 3

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Table 2: The 2-value information system (U, A, V, g) a1 1 1 1 0

h1 h2 h3 h4

a2 1 1 1 1

a3 0 1 0 0

Obviously, ind(P ) is an equivalence relation on U , which is called the equivalence relation induced by P . Sometimes, we replace respectively ind(P ) and U/ind(P ) by P and U/P where U/ind(P ) = {[x]ind(P ) : x ∈ U }. Specially, we replace ind({a}) by a for a ∈ A. Theorem 2.7. Let S = fA be a soft set over U and let IS = (U, A, V, gs ) be a 2-value information system induced by S. Then for any a ∈ A, U/a = {f (a), U − f (a)}. Proof. Since a = {(x, y) ∈ U × U : gs (x, a) = gs (y, a)}, gs (x, a) = gs (y, a) = 1 or gs (x, a) = gs (y, a) = 0. This implies that {x, y} ⊂ f (a) or {x, y} ⊂ U − f (a). Thus U/a = {f (a), U − f (a)}.

2.3

The relationship between soft sets and information systems

Definition 2.8. Let S = fA be a soft set over U . Then IS = (U, A, V, gs ) is called a 2-value information system induced by S where gs : U × A → V . For any x ∈ U and a ∈ A, ½ 1, x ∈ f (a), gs (x, a) = 0, x 6∈ f (a). Definition 2.9. Let I = (U, A, V, g) be a 2-value information system. Then SI = (fI , A) is called a soft set over U induced by I where fI : A → 2U and for any x ∈ U and a ∈ A, fI (a) = {x ∈ U : g(x, a) = 1}. Lemma 2.10. Let S = fA be a soft set over U , let IS = (U, A, V, gs ) be a 2-value information system induced by S over U and let SIS = (fIS , A) be a soft set over U induced by IS . Then S = SIS .

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Proof. By Definition 2.9, for any a ∈ A, fIS (a) = {x ∈ U : gs (x, a) = 1}. By Definition 2.8, for any x ∈ U and a ∈ A, ½ 1, x ∈ f (a), gs (x, a) = 0, x 6∈ f (a). This implies that gs (x, a) = 1 ⇔ x ∈ f (a). So, for ∀x ∈ U, a ∈ A, f (a) = fIS (a). Hence fA = (fIS , A). This implies that S = SIS . Lemma 2.11. Let I = (U, A, V, g) be a 2-value information system, Let SI = (fI ,A ) be a soft set over U induced by I and let ISI = (U, A, V, gsI ) be a 2-value information system induced by SI . Then I = ISI . Proof. By Definition 2.8, for any x ∈ U and a ∈ A, ½ 1, x ∈ fI (a), gsI (x, a) = 0, x 6∈ fI (a). For any x ∈ U and a ∈ A, by Definition 2.9, fI (a) = {x ∈ U : g(x, a) = 1}. Since I = (U, A, V, g) is a 2-value information system, g(x, a) = 0 for x 6∈ fI (a), This implies that ½ 1, x ∈ fI (a), g(x, a) = 0, x 6∈ fI (a). So for any x ∈ U and a ∈ A, gsI (x, a) = g(x, a). Hence gsI = g. This implies that that I = ISI . Theorem 2.12. Let Σ = {S : S = fA is a sof t set over U } and Γ = {I : I = (U, A, V, g)isa2 − valueinf ormationsystem}. Then there exists a one-to-one correspondence between Σ and Γ. Proof. Two mappings F : Σ → Γ and G : Γ → Σ are defined as follows: F (S) = IS f or ∀S ∈ Σ; G(I) = SI f or ∀I ∈ Γ. By Lemma 2.9, G ◦ F = iΣ where G ◦ F is the composition of F and G, and iΣ is the identity mapping on Σ. By Lemma 2.10, F ◦ G = iΓ where G ◦ F is the composition of G and F , and iΓ is the identity mapping on Γ. Hence F and G are both a one-to-one correspondence. This prove that there exists a one-to-one correspondence between Σ and Γ.

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3

The parameter reduction of soft sets

Soft sets and rough sets are two different concepts to deal with uncertainty. Both of these concepts help in decision-making problems. Soft set itself has classification ability. The parameter reduction of soft sets means reducing the number of parameters for a soft set to the minimum without distorting its original classification ability. Specific approach is first classifying the parameter according to the importance of parameters and then finding the minimum set of parameters (ie., the core for a soft set) without distorting the original classification ability of soft sets. Reduction of parameters of soft sets plays a vital role in decision-making problems. Reduction of parameters can save expensive tests and time. Since there exists a one-to-one correspondence between “the set of all soft sets” and “the set of all 2-value information systems” ( see Theorem 2.12 ), we can do the parameter reduction of soft sets with the help of the knowledge reduction in rough set theory. Definition 3.1. Let fA be a soft set over U . (1) A∗ ⊆ A is called a parameter reduction of fA (brief. a fA -parameter reduction), if ind(A) = ind(A∗ ) and ind(A) 6= ind(B) for any B ( A∗ . (2) The intersection set of all fA -parameter reductions is called the core of fA . We denote it by core(fA ). In this paper, we denote the set of all fA -parameter reductions by pr(fA ). Proposition 3.2. Let fA be a soft set over U . Then pr(fA ) 6= ∅. Proof. (1) If ind(A) 6= ind(A − {a}) for any a ∈ A, then A itself is a fA parameter reduction. (2) If ind(A) = ind(A−{a}) for some a ∈ A, then we consider B1 = A−{a}. If ind(A) 6= ind(B1 − {b1 }) for any b1 ∈ B1 , B1 is a fA -parameter reduction. Otherwise, we consider B1 −{b1 } again and repeat the above mentioned process. Since A is a finite set, we can find a fA -parameter reduction. Thus, pr(fA ) 6= ∅. Definition 3.3. Let fA be a soft set over U and let pr(fA ) = {Ci : 1 ≤ i ≤ n}. Then n T (1) a ∈ A is called core, if a ∈ Ci = core(fA ). i=1

(2) a ∈ A is called relative indispensable, if a ∈

n S i=1

Ci − core(fA ).

(3) a ∈ A is called absolutely dispensable, if a ∈ A −

n S i=1

Ci .

(4) a ∈ A is called dispensable, if a ∈ A − core(fA ). Obviously, a ∈ A is dispensable if and only if a is relative indispensable or absolutely dispensable.

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Definition 3.4. Let A, B ⊂ 2U . A is called a refinement of B, if for any A ∈ A, there exists B ∈ B such that A ⊆ B. We denote it by A ≤ B. Lemma 3.5. Let R and ρ be two equivalence relations on U . If R ⊆ ρ, then U/R ≤ U/ρ. Proof. Suppose that A ∈ X/R. Since R is an equivalence relation on X, there exists x ∈ X, such that A = [x]R . Suppose that y ∈ [x]R . Then xRy. This implies that (x, y) ∈ R. Since R ⊆ ρ, (x, y) ∈ ρ. This implies that y ∈ [x]ρ . Then [x]R ⊆ [x]ρ . Pick B = [x]ρ . Then A ⊆ B and so X/R ≤ X/ρ. The following Theorem 3.6 and Corollary 3.7 give the parameter reduction of soft sets. Theorem 3.6. Let fA be a soft set over U . Then (1) |pr(fA )| = 1 if and only if core(fA ) ∈ pr(fA ). (2) a ∈ core(fA ) if and only if U/ind(A) 6= U/ind(A − {a}). (3) a ∈ A is dispensable if and only if U/ind(A) = U/ind(A − {a}). Proof. (1) Sufficiency. Let core(fA ) ∈ pr(fA ). Note that pr(fA ) = {Ci : 1 ≤ i ≤ n}. We only need to prove n = 1. 1) Suppose n = 2. Then there are only two different fA -parameter reductions C1 and C2 . a) If C1 ( C2 . Since C2 ∈ pr(fA ), ind(A) 6= ind(C1 ). Then C1 6∈ pr(fA ). This is a contradiction. b) If C2 ( C1 . We can similarly prove that this implies a contradiction. c) If C1 * C2 and C2 * C1 . Obviously, core(fA ) = C1 ∩ C2 and core(fA ) ( C1 . Since C1 ∈ pr(fA ), ind(A) 6= ind(core(fA )). Then core(fA ) 6∈ pr(fA ). This is also a contradiction. 2) Suppose n ≥ 3. This is similar to the proof of 1). Thus |pr(fA )| = 1. Necessity. This is obvious. (2) Sufficiency. Suppose that U/ind(A) 6= U/ind(A − {a}). We claim that a ∈ Ci for any 1 ≤ i ≤ n. Otherwise. a 6∈ Ci0 for some Ci0 . This implies that U/ind(A) = U/ind(Ci0 ). Since ind(Ci0 ) ⊇ ind(A − {a}) ⊇ ind(A), by Lemma 3.5, U/ind(Ci0 ) ≥ U/ind(A−{a}) ≥ U/ind(A). So U/ind(A) = U/ind(A−{a}), a contradiction. This implies that a ∈ core(fA ). Necessity. Suppose that U/ind(A) = U/ind(A − {a}). Since pr(fA ) 6= ∅, there exists B10 ⊆ A − {a} such that B10 ∈ pr(fA ). So a 6∈ core(fA ). This is a contradiction. Thus U/ind(A) 6= U/ind(A − {a}). (3) Sufficiency. Suppose that U/ind(A) = U/ind(A − {a}). Since A − {a} is a finite set, there exists B2 ⊆ A − {a} such that B2 ∈ pr(fA ). So a 6∈ core(fA ). This implies that a ∈ A − core(fA ). 7

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Thus a is a dispensable parameter. Necessity. Suppose that U/ind(A) 6= U/ind(A−{a}). Similar to the proof of (2), we have a ∈ core(fA ). Then a 6∈ A − core(fA ). Note that a is a dispensable parameter. Then a ∈ A − core(fA ). This implies a contradiction. Thus U/ind(A) = U/ind(A − {a}). Corollary 3.7. core(fA ) = {a ∈ A : U/ind(A) 6= U/ind(A − {a})}.

4

Algorithms

Algorithms 4.1. Let fA be a soft set over U . The algorithm of parameter reduction is shown as follows: Input: A soft set fA . Output: pr(fA ) and core(fA ). Step 1. Calculate U/ind(A) and U/ind(A − {a}) for any a ∈ A; Step 2. If U/ind(A) 6= U/ind(A − {a}) for any a ∈ A, then pr(fA ) = {A} and core(fA ) = A; Step 3. If U/ind(A) = U/ind(A − {a}) for some a ∈ A, then we consider B1 = A − {a}. If U/ind(A) 6= U/ind(B1 − {b1 }) for any b1 ∈ B1 , then B1 ∈ pr(fA ); Otherwise, we consider B1 − {b1 } again; Step 4. Output pr(fA ) and core(fA ). Example 4.2. Let U = {h1 , h2 , h3 , h4 , h5 }, A = {a1 , a2 , a3 , a4 } and let fA be a soft set over U , defined as follows f (a1 ) = {h1 , h2 , h5 }, f (a2 ) = ∅, f (a3 ) = {h3 }, f (a4 ) = {h3 , h4 }. By Theorem 2.7, we have U/a1 = {f (a1 ), U −f (a1 )} = {{h1 , h2 , h5 }, {h3 , h4 }}, U/a2 = {f (a2 ), U − f (a2 )} = {{h1 , h2 , h3 , h4 , h5 }}, U/a3 = {f (a3 ), U − f (a3 )} = {{h3 }, {h1 , h2 , h4 , h5 }}, U/a4 = {f (a4 ), U − f (a4 )} = {{h3 , h4 }, {h1 , h2 , h5 }}. And U/A = {{h1 , h2 , h5 }, {h3 }, {h4 }}. U/ind(A−{a1 }) = {{h1 , h2 , h5 }, {h3 }, {h4 }} = U/ind(A). U/ind(A − {a2 }) = {{h1 , h2 , h5 }, {h3 }, {h4 }} = U/ind(A). U/ind(A − {a3 }) = {{h1 , h2 , h5 }, {h3 , h4 }} 6= U/ind(A). U/ind(A − {a4 }) = {{h1 , h2 , h5 }, {h3 }, {h4 }} = U/ind(A). This implies that U/ind({a2 , a3 , a4 }) = U/ind({a1 , a3 , a4 }) = U/ind({a1 , a2 , a3 }) = U/ind(A). Since U/ind({a2 , a3 , a4 }) = U/ind({a3 , a4 }), U/ind({a3 , a4 }) 6= U/ind({a3 }) and U/ind({a3 , a4 }) 6= U/ind({a4 }), {a3 , a4 } is a fA -parameter reduction. Since U/ind({a1 , a3 , a4 }) = U/ind({a1 , a3 }), U/ind({a1 , a3 }) 6= U/ind({a1 }) and U/ind({a1 , a3 }) 6= U/ind({a4 }), {a1 , a3 } also is a fA -parameter reduction. Obviously, pr(fA ) = {{a3 , a4 }, {a1 , a3 }}, core(fA ) = {a3 , a4 } ∩ {a1 , a3 } = {a3 }. 8

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Example 4.3. In Example 4.2, we have (1) a3 is core. (2) a1 and a4 are relative indispensable. (3) a2 is absolutely dispensable. (4) a1 , a2 and a4 are dispensable.

References [1] M.Irfan Ali, A note on soft sets, rough soft sets and fuzzy soft sets, Appl. Soft Comput., 13(2011), 3329-3332. [2] D.Chen, E.C.C.Tsang, D.S.Yeung, X.Wang, The parameterization reduction of soft sets and its applications, Comput. Math. Appl., 49(2005), 757763. [3] X.Ge, Z.Li, Y.Ge, Topological spaces and soft sets, J. Comput. Anal. Appl., 13(2011), 881-885. [4] Y.Jiang, Y.Tang, Q.Chen, J.Wang, S.Tang, Extending soft sets with description logics, Comput. Math. Appl., 59(2010), 2087-2096. [5] Z.Kong, L.Gao, L.Wang, S.Li, The normal parameter reduction of soft sets and its algorithm, Comput. Math. Appl., 56(2008), 3029-3037. [6] D.Molodtsov, Soft set theory-first result, Comput. Math. Appl., 37(1999), 19-31. [7] D. Molodtsov, The theory of soft sets, URSS Publishers, Moscow, 2004. [8] P.K.Maji, R.Biswas, A.R.Roy, Soft set theory, Comput. Math. Appl., 45(2003), 555-562. [9] P.K.Maji, A.R.Roy, An application of soft sets in a decision making problem, Comput. Math. Appl., 44(2002), 1077-1083. [10] Z. Pawlak, Rough sets, Int. J. Comput. Inform. Sci., 11(1982), 341-356. [11] Z. Pawlak, Rough Sets: Theoretical aspects of reasoning about data, Kluwer Academic Publishers., Dordrecht, 1991. [12] W.Zhang, W.Wu, J.Liang, D.Li, Rough sets theorey and methods, Chinese Scientific Publishers, Beijing, 2001.

9

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FUNCTIONAL INEQUALITIES ASSOCIATED WITH BI-CAUCHY ADDITIVE FUNCTIONAL EQUATIONS GANG LU, CHOONKIL PARK, AND DONG YUN SHIN∗ Abstract. In this paper, we prove the Hyers-Ulam stability for the following functional inequalities: ∥f (x1 , y1 ) + f (x2 , y2 ) + f (x3 , y3 )∥ ≤ ∥f (x1 + x2 + x3 , y1 + y2 + y3 )∥,

( x1 + x2 + x3 y1 + y2 + y3 )

∥f (x1 , y1 ) + f (x2 , y2 ) + f (x3 , y3 )∥ ≤ 2f ,

, 2 2

( ) x + x y + y

1 2 1 2 ∥f (x1 , y1 ) + f (x2 , y2 ) + 2f (x3 , y3 )∥ ≤ 2f + x3 , + y3 2 2 in Banach spaces.

(1) (2) (3)

1. Introduction and preliminaries The stability problem of functional equations originated from a question of Ulam [1] concerning the stability of group homomorphisms: Let (G1 , ∗) be a group and let (G2 , ⋄, d) be a metric group with the metric d(·, ·). Given ϵ > 0, does there exist a δ(ϵ) > 0 such that if a mapping h : G1 → G2 satisfies the inequality d(h(x ∗ y), h(x) ⋄ h(y)) < δ for all x, y ∈ G1 , then there is a homomorphism H : G1 → G2 with d(h(x), H(x)) < ϵ for all x ∈ G1 ? If the answer is affirmative, we would say that the question of homomorphism H(x ∗ y) = H(x) ⋄ H(y) is stable. The concept of stability for a functional equation arises when we replace the functional equation by an inequality which acts as a perturbation of the equation. Thus the stability question of functional equation is that how do the solutions of the inequality differ from those of the given functional equation? Hyers [2] gave a first affirmative answer to the question of Ulam for Banach spaces. Let X and Y be Banach spaces. Assume that f : X → Y satisfies ∥f (x + y) − f (x) − f (y)∥ ≤ ϵ for all x, y ∈ X and some ϵ ≥ 0. Then there exists a unique additive mapping T : X → Y such that ∥f (x) − T (x)∥ ≤ ϵ for all x ∈ X. Let X and Y be Banach spaces with norms ∥ · ∥ and ∥ · ∥, respectively. Consider f : X → Y to be a mapping such that f (tx) is continuous in t ∈ R for each fixed x ∈ X. Th.M. Rassias 2010 Mathematics Subject Classification. Primary 39B62, 39B52, 46B25. Key words and phrases. Bi-Cauchy additive mapping; Hyers-Ulam stability; Banach space. ∗ The corresponding author.

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G. LU, C. PARK, AND D.Y. SHIN

[3] introduced the following inequality, that we call Cauchy-Rassias inequality : Assume that there exist constants λ ≥ 0 and p ∈ [0, 1) such that ∥f (x + y) − f (x) − f (y)∥ ≤ λ(∥x∥p + ∥y∥p ) for all x, y ∈ X. Th.M. Rassias [3] showed that there exists a unique R-linear mapping T : X → Y such that 2λ ∥f (x) − T (x)∥ ≤ ∥x∥p 2 − 2p for all x ∈ X. Beginning around the year 1980 the topic of approximate homomorphisms, or the stability of the equation of homomorphism, was studied by a number of mathematicians. Gˇ avruta [4] generalized the Rassias’ result. A square norm on an inner product space satisfies the important parallelogram equality ∥x + y∥2 + ∥x − y∥2 = 2∥x∥2 + 2∥y∥2 The functional equation f (x + y) + f (x − y) = 2f (x) + 2f (y) is called a quadratic functional equation. In particular, every solution of the quadratic functional equation is said to be a quadratic mapping. A Hyers-Ulam stability problem for the quadratic functional equation was proved by Skof [5] for mappings f : X → Y , where X is a normed space and Y is a Banach space. Cholewa [6] noticed that the theorem of Skof is still true if the relevant domain X is replace by an Abelian group. In [7], Czerwik proved the Hyers-Ulam stability of the quadratic functional equation. Borelli and Forti [8] generalized the stability result. The stability problems of several functional equations have been extensively investigated by a number of authors and there are many interesting results concerning this problem. A large list of references can be found in [9]–[28]. In this paper, let X be a vector space and Y a Banach space. A mapping f : X → Y is called a Cauchy additive mapping if f satisfies the functional equation f (x + y) = f (x) + f (y). For a given mapping f : X × X → Y , we define f (x1 + x2 , y1 + y2 ) = f (x1 , y1 ) + f (x2 , y2 )

(1.1)

for all (x1 , y1 ), (x2 , y2 ) ∈ X × X. A mapping f : X × X → Y is called a bi-Cauchy mapping if f satisfies the functional equation (1.1). We investigate the functional inequalities (1), (2) and (3) and prove the Hyers-Ulam stability of the functional inequalities (1), (2) and (3). 2. Hyers-Ulam stability of the functional inequality (1) Proposition 2.1. Let f : X × X → Y be a mapping such that ∥f (x1 , y1 ) + f (x2 , y2 ) + f (x3 , y3 )∥ ≤ ∥f (x1 + x2 + x3 , y1 + y2 + y3 )∥

(2.1)

for all (x1 , y1 ), (x2 , y2 ), (x3 , y3 ) ∈ X × X. Then the mapping f : X → Y is bi-Cauchy additive. Proof. Letting (x1 , y1 ) = (x2 , y2 ) = (x3 , y3 ) = (0, 0) in (2.1), we have ∥3f (0, 0)∥ ≤ ∥f (0, 0)∥ and so f (0, 0) = 0. Letting x1 = x, x2 = −x, x3 = 0, y1 = y, y2 = −y, y3 = 0 in (2.1), we get ∥f (x, y) + f (−x, −y)∥ ≤ 0 and so f (x, y) = −f (−x, −y) for all (x, y) ∈ X × X.

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BI-CAUCHY ADDITIVE FUNCTIONAL EQUATIONS

Next, we show that f is a bi-Cauchy additive mapping. ∥f (x1 , y1 ) + f (x2 , y2 ) − f (x1 + x2 , y1 + y2 )∥ = ∥f (x1 , y2 ) + f (x2 , y2 ) + f (−x1 − x2 , −y1 − y2 )∥ ≤ ∥f (0, 0)∥ = 0 and so f (x1 +x2 , y1 +y2 ) = f (x1 , y1 )+f (x2 , y2 ) for all (x1 , y1 ), (x2 , y2 ) ∈ X ×X, as desired.



Theorem 2.2. Assume that a mapping f : X × X → Y satisfies the inequality ∥f (x1 , y1 ) + f (x2 , y2 ) + f (x3 , y3 )∥ ≤ ∥f (x1 + x2 + x3 , y1 + y2 + y3 )∥ + ϕ((x1 , y1 ), (x2 , y2 ), (x3 , y3 )),

(2.2)

where ϕ : (X × X)3 → [0, ∞) satisfies e 1 , y1 ), (x2 , y2 ), (x3 , y3 )) := ϕ((x

∞ ∑ j=1

2j ϕ

(( x

1 , j 2

y1 ) ( x2 y2 ) ( x3 y3 )) , j, j , j, j l and all (x, y) ∈ X ×X. It follows from (2.3) and (2.5) that the sequence {2k f ( 2xk , 2yk )} is a Cauchy sequence for all (x, y) ∈ X × X. Since Y is

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G. LU, C. PARK, AND D.Y. SHIN

complete, the sequence {2k f ( 2xk , 2yk )} converges. So we can define the mapping A : X × X → Y by (x y) A(x, y) := lim 2k f k , k k→∞ 2 2 for all (x, y) ∈ X × X. Moreover, letting l = 0 and passing the limit m → ∞, we get (2.4). Now, we show that A(x, y) is a bi-Cauchy additive mapping. It follows from (2.2) and (2.3) that ∥A(x, y) + A(−x, −y)∥

( ( )

x y) −x −y k = lim 2 f k , k + f , k + f (0, 0)

k k→∞ 2 2 2 2 [ ( (( ) ) )] (

x x y ) −x −y −x y −y k

≤ lim 2 f + k + 0, k + k + 0 + ϕ , , , (0, 0) , k→∞ 2k 2 2 2 2k 2k 2k 2k =0 and so A(x, y) = −A(−x, −y) for any (x, y) ∈ X × X. ∥A(x1 , y1 ) + A(x2 , y2 ) − A(x1 + x2 , y1 + y2 )∥ = ∥A(x1 , y1 ) + A(x2 , y2 ) + A(−x1 − x2 , −y1 − y2 )∥

( ) ( (x y )

x1 y 1 ) −x1 − x2 −y1 − y2 2 2 k

= lim 2 f k , k + f k , k + f ,

k→∞ 2 2 2 2 2k 2k ) [ (

y2 −y1 − y2 x1 x2 −x1 − x2 y1

+ k + , k+ k+ ≤ lim 2k f

k k k→∞ 2 2 2 2 2 2k (( ( ))] x1 y1 ) ( x2 y2 ) −x1 − x2 −y1 − y2 +ϕ , , k, k , , 2k 2k 2 2 2k 2k =0 for all (x1 , y1 ), (x2 , y2 ) ∈ X × X. Thus the mapping A : X × X → is bi-Cauchy additive. Next, we prove the uniqueness of A. Suppose that T : X × X → Y is another additive mapping satisfying (2.4). We may obtain

(x y) ( x y )

∥A(x, y) − T (x, y)∥ = lim 2k A k , k − T , k k k→∞ 2 2

( 2x 2y ) ( x y )

k ≤ lim 2 A k , k − f k , k k→∞ 2 2 2 2

(x y) ( x y )

k + lim 2 T , − f ,

k→∞ 2k 2k 2k 2k [ (( x ) ( y x y ) ( x y )) , , , , − , − ≤ lim 2 ϕe k→∞ 2k+1 2k+1 2k+1 2k+1 2k 2k (( x y ) ( x ) )] y + ϕe , , − k , − k , (0, 0) 2k 2k 2 2 = 0 for all (x, y) ∈ X × X. Thus we can conclude that A(x, y) = T (x, y) for all (x, y) ∈ X × X. This complete the proof. 

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BI-CAUCHY ADDITIVE FUNCTIONAL EQUATIONS

3. Hyers-Ulam stability of the functional inequality (2) Proposition 3.1. Let f : X × X → Y be a mapping such that

( )

x1 + x2 + x3 y1 + y2 + y3

∥f (x1 , y1 ) + f (x2 , y2 ) + f (x3 , y3 )∥ ≤ 2f ,

2 2

(3.1)

for all (x1 , y1 ), (x2 , y2 ), (x3 , y3 ) ∈ X × X. Then the mapping f : X × X → Y is bi-Cauchy additive. Proof. Letting x1 = x2 = x3 = 0, y1 = y2 = y3 = 0 in (3.1), we get ∥3f (0, 0)∥ ≤ ∥2f (0, 0)∥. So f (0, 0) = 0. Letting x1 = x, y1 = y, x2 = −x, y2 = −y and x3 = y3 = 0 in (3.1), we get ∥f (x, y) + f (−x, −y) + f (0, 0)∥ ≤ ∥2f (0, 0)∥ = 0 for all (x, y) ∈ X × X. So f (−x, −y) = −f (x, y) for all (x, y) ∈ X × X. Letting x3 = −x1 − x2 , y3 = −y1 − y2 in (3.1), we obtain ∥f (x1 , y1 ) + f (x2 , y2 ) − f (x1 + x2 , y1 + y2 )∥ = ∥f (x1 , y1 ) + f (x2 , y2 ) + f (−x1 − x2 , −y1 − y2 )∥ ≤ ∥2f (0, 0)∥ = 0 for all (x1 , y1 ), (x2 , y2 ) ∈ X × X. Thus f (x1 , y1 ) + f (x2 , y2 ) = f (x1 + x2 , y1 + y2 ) for all (x1 , y1 ), (x2 , y2 ) ∈ X × X, as desired.



Theorem 3.2. Assume that a mapping f : X × X → Y satisfies the inequality

( )

x1 + x2 + x3 y1 + y2 + y3

, ∥f (x1 , y1 ) + f (x2 , y2 ) + f (x3 , y3 ) ≤ 2f

2 2

(3.2)

+ ϕ((x1 , y1 ), (x2 , y2 ), (x3 , y3 )) where ϕ : (X × X)3 → [0, ∞) satisfies e 1 , y1 ), (x2 , y2 ), (x3 , y3 )) := ϕ((x

∞ ∑ j=1

2j ϕ

(( x

1 , j 2

y1 ) ( x2 y2 ) ( x3 y3 )) , j, j , j, j 0 are the ”forward”and the ”backward” deviations of random variable zj , j = 1, . . . , N , respectively. For the stochastic linear constraint (2.15), the worst-case convex support of the uncertain parameter can be specified as follows, N o n X W = r : ∃z ∈ RN , r = r 0 + ∆r j ˜zj , −z ≤ z ≤ z

(3.3)

j=1

Therefore, under affine data perturbation, the worse-case uncertainty set is a parallelotope in which the feasible solution is characterized by Soyster [18], which, of course, is a very conservative. To derive a less conservative approximation, we need to choose the budget of uncertainty, Ω, appropriately. The natural uncertainty set to consider is the intersection of a norm uncertainty set, VΩ and the worst-case support set, W as follows. N n o X SΩ = r : ∃z ∈ RN , r = r 0 + ∆r j ˜zj , z ∈ AΩ (z ), −z ≤ z ≤ z

(3.4)

j=1

As the budget of uncertainty Ω increases, the norm uncertainty set, VΩ expands radially from the point r 0 until it engulfs the set W. In this case, the uncertainty set SΩ = W. Hence, for any choice of Ω, the uncertainty set SΩ is always less conservative than the worst-case uncertainty set W. We call the uncertainty SΩ as genal affine data perturbation uncertainty. We will show an equivalent formulation of the corresponding robust counterpart of (2.15) under the generalized uncertainty set, SΩ . The dual norm kuk∗ is defined as kuk∗ = max u 0 x (3.5) {kx k≤1}

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Theorem 3.1 The robust counterpart of (2.15) in which UΩ = SΩ is equivalent to  ∃u, λ, s ∈ RN , h ∈ R     −r00 x + Ωh + λ0 z + s0 z ≤ −µ    kuk∗ ≤ h, (3.6) uj ≥ −pj (∆r0j x + λj − sj ), ∀j = {1, . . . , N }     u ≥ qj (∆r0j x + λj − sj ), ∀j = {1, . . . , N }    j u, λ, s ≥ 0. Proof: From (2.15), we have max

− xT r ≤ −µ

(3.7)

Under the condition UΩ = SΩ , the robust counterpart of (3.7) is as follows, −r T0 x + max z0 y ≤ −µ {z∈C}

(3.8)

where n o C = (v , w ) : kP−1 v + Q−1 w k ≤ Ω, −z ≤ z ≤ z , v , w ≥ 0 0

and yj = −∆rj x . Since C is a compact convex set with nonempty interior, we can use strong duality to obtain the equivalent representation. Observe that max

{(v ,w ):kP−1 v +Q−1 w k≤Ω,−z ≤z ≤z ,v ,w ≥0}

(v − w )0 y

( =

) 0

0

(v − w ) y + r (z − v + w ) + s (z + v − w )

max

min r, s≥0

0

{(v ,w ):kP−1 v +Q−1 w k≤Ω,v ,w ≥0}

( =

) 0

0

0

(y − r + s ) v − (y − r + s ) w + r z + s z

max

min r, s≥0

0

{(v ,w ):kP−1 v +Q−1 w k≤Ω,v ,w ≥0}

( = =

min r, s≥0

) max

0

0

0

( − r + s ) v − Q (y − r + s ) w + r z + s z

P y

{(v ,w ):kv +w k≤Ω,v ,w ≥0}

0

min Ωkuk∗ + r0 z + s0 z r, s≥0

where uj

= max{pj (yj − rj + sj ), −qj (yj − rj + sj )} 0

0

= max{−pj (∆rj x + rj − sj ), qj (∆rj x + rj − sj )}. Hence the robust counterpart is the same as −r T0 x + Ωkuk∗ + r0 z + s0 z ≤ −µ

(3.9)

Adding an auxiliary variable h ∈ R, we can easily obtain the equivalent formulation of (3.9), that is (3.6).

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The complete formulation and complexity class of the robust counterpart depends on the representation of the dual norm constraint, kuk∗ ≤ h. In this paper, we select the l1 norm. So the kuk∗ ≤ h is equivalent to uj ≤ h, ∀j ∈ N.

(3.10)

By (2.10)-(2.14) and (3.10), the robust portfolio selection problem can be written as the following linear programming problem with variables (x, z, u, λ, s, v, θ, α, h) ∈ Rn × RT × RN × RN × RN × RN × R × R × R : min

θ

s.t.

α+

J

X 1 zj ≤ θ T (1 − β) i=1

zj ≥ −xT y[j] − α,

j = 1, . . . , T

T

x 1 = 1, x≥0 0 −r 0 x + Ωh + λ0 z + s0 z ≤ −µ uj ≤ h, ∀j ∈ N uj ≥ −pj (∆r0j x + λj − sj ), ∀j = {1, . . . , N } uj ≥ qj (∆r0j x + λj − sj ), ∀j = {1, . . . , N }

(3.11)

N z, u, λ, s ≥ 0, v ∈ R+ , p ∈ R+

4.

Empirical Results

In this section, we apply the robust portfolio optimization methods discussed in the previous sections to real market data and compare the behavior of the solutions obtained by the robust optimization technique. In all tables and figure, the methods have the following meanings: • ”CVaR” stands for the initial CVaR method in [7]. • ”BCVAR” stands for the robust mean-CVaR Portfolio optimization under box uncertainty set in [10]. • ”ECVaR” stands for the robust mean-CVaR Portfolio optimization under ellipsoidal uncertainty set in [11]. • ”ACVaR” stands for the robust mean-CVaR Portfolio optimization (3.11) under a genal affine data perturbation uncertainty set. We utilize MatLab7.0 for solving models CVaR, BCVAR, and ACVaR which are linear programming problems. The model ECVaR is an SOCP and solved by SeDuMi1.02 [21]. We consider a portfolio of 10 small cap stocks from 5 different industry categories of the S&P 600 index(Table 2), and use historical returns from May,

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1998 to June, 2006. There are a total of 2,000 observations for each stock. Table 2: List of Stocks and Corresponding Industries Industry discretionary Consumer discretionary Financials Industrials Information technology Healthcare

Company name (ticker) Aztar Corp. (AZR), Hancock Fabrics Inc. (HKF) Downey S & L Assn. (DSL), HARB AAR Corp. (AIR), CDI Corp. (CDI) FEI Company (FEIC), Exar Corp. (EXAR) BioLase Technology (BLTI), BDR

In our first experiment, using the data presented above, we generated the classical and robust efficient frontiers. The parameters for all optimization models are set as follows: • For the CVaR formulation, mean return r is given by the sample mean. • For the BCVaR formulation, we assume that mean return r0 is given by the sample mean, and that ri is determined by the standard deviation of the stock i’ sample return. • According to the ECVaR formulation,we assume that mean return r0 is given by the sample mean. For simplicity, the scaling matrix of the ellipsoid P is assumed to be a diagonal matrix ρI, where ρ is a nonnegative parameter. • For the ACVaR formulation, we assume that mean return r0 is given by the sample mean, and that ∆ri is determined by the standard deviation of the stock i’ sample return, and assume they are also stochastically independent (N=10). We set Ω = 0.8 , z = z = 1 and pj = 1.5, qj = 2 in our Numerical experiments. 3

2.5

CVaR

Portflio return %

ECVaR

ACVaR

2

1.5 BCVaR 1

0.5

0

0

5

10

15

20 25 Portflio CVaR

30

35

40

45

Figure 1-portfolio efficient frontiers for the different optimization formulations with β = 1%.

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As shown in Figure 1, it is apparent that ACVaR outperforms both ECVaR and BCVaR in terms of realized CVaR. As expected, CVaR is dominated by ACVaR. We also can see that the robust optimal portfolios are somewhat conservative in comparison to that of the CVaR model. But in the next experiment, we will see the robust optimal portfolios can result in more stable portfolio returns. In our second experiment, we study the cumulative portfolio wealth if a portfolio manager employs a simple buy-and-hold strategy. The entire data sequence is divided into investment periods of length T = 200 days. In all there are p = 10 time periods. For each period p, first, we consider moving windows of n = 10 days and compute the parameters for all optimization models as experiment 1. Once all the parameters are set, the portfolio xpCV aR , xpBCV aR , xpECV aR , p xACV aR for period p can be computed by solving the portfolio selection model CVaR, BCVaR, ECVaR, and ACVaR respectively. The portfolio xpCV aR , xpBCV aR , xpECV aR , xpACV aR are held constant for the period p and then rebalanced to the p+1 p+1 p+1 portfolio xp+1 CV aR , xBCV aR , xECV aR , xACV aR for period p + 1. p p p p Let WCV aR , WBCV aR , WECV aR , WACV aR denote the wealth at the end of period p of an investor with initial wealth w0 = 1 . Because these strategies require a block of data of length T = 200 to estimate all of parameters, the first investment period p = 1 starts from the time instant T + 1. Therefore, 10 time periods of length ”T = 200” only have 9 investment periods. 4.5

4

3.5

Wealth

ACVaR 3

2.5

CVaR ECVaR

2 BCVaR

1.5

1

0

1

2

3

4 5 Investment period

6

7

8

9

Figure 2-The wealth resulting from the four strategies with window n = 10 at each investment period. It is clear that the wealth generated by the ACVaR model is much better than other models at the end of investment period. But in Figure 2 the wealth generated by the ACVaR model is a litter lower than other models at early investment periods. Therefore, it is not guaranteed that the ACVaR model always has an advantage of other three models. On the other hand, the optimal

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portfolio allocation based on the ACVaR approach tends to result in stable returns, whereas, for example, the behavior of the optimal portfolio obtained with the CVaR approach is erratic.

5.

Conclusion

under a genal affine data perturbation uncertainty set, we propose a computationally tractable robust optimization method for minimizing the CVaR of a portfolio. The remarkable characteristic of the new method is that, using L1 norm, the robust optimization model retains the complexity of original portfolio optimization problem, i.e., the robust counterpart problem is still a linear programming problem. This fact has important theoretical and practical implications. Since the computational complexity of an LP is simplest in all of program problems, it follows that robust portfolio optimization is able to provide protection against parameter fluctuations at light computational cost. Moreover, the LP problem is maybe the best known and the most frequently solved optimization problem in the real world. The numerical experiments presented in this paper suggest that the behavior of portfolios can be improved by using the robust CVaR model under a genal affine data perturbation uncertainty set. And the robustness is achieved at relatively high performance and low cost.

References [1] H.M. Markowitz, Portfolio selection. J. Finance. 1952, 7: 77-91. [2] T.J. Linsmeier, N. D. Pearson, Risk Measurement: An introduction to value-at-risk. Technical report 96-04, OFOR, University of Illinois, Urbana-Champaign, IL, 1996. [3] P. Artzner, F. Delbaen, J. M. Eber, D. Heath, Coherence measures of risk. Math. Finance. 1999, 9: 203-228. [4] Dowd, K.: Measuring Market Risk. JohnWiley and Sons, NewYork 2002. [5] Ogryczak W, Ruszcz´ ynski A. Dual stochastic dominance and related meanCrisk models. SIAM Journal on Optimization. 2002, 13: 60-78. [6] R.T. Rockafellar, S. Uryasev, Optimization of conditional Valueat-Risk. J. Risk. 2000, 2: 21-41. [7] R.T. Rockafellar, S. Uryasev, Conditional Value-at-Risk for general loss distributions. J. Banking and Finance. 2002, 26: 1443-1471. [8] F. Black, R. Litterman, Global portfolio optimization. Financial Analysts J. 1992, 48: 28-43.

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[9] V.K. Chopra, W.T. Ziemba, The effect of errors in means, variance and covariances on optimal portfolio choice. J. Portfolio Manag. winter, 1993, 6-11. [10] A.G. Quaranta, A. Zaffaroni: Robust optimization of conditional value at risk and portfolio selection. J. Banking and Finance. 2008, 32: 2046-2056. [11] X.M. An and G.M. Luo, Robust Portfolio Selection under Ellipsoidal Uncertainty, Journal of Hunan University(Natural Sciences), 2010, 37: 89-92. [12] F.H. Wen, X.G. Yang, Skewness of Return Distribution and Coefficient of Risk Premium, Journal of Systems Science and Complexity. 2009, 22: 360-371. [13] F.H. Wen, Z.F. Liu, A Copula-based Correlation Measure and Its Application in Chinese Stock Market, International Journal of Information Technology & Decision Making. 2009, 8: 1-15. [14] M.S. Lobo and S. Boyd. The worst-case risk of a portfolio, Technical Report, http://faculty.fuqua.duke.edu/∼ mlobo/bio/researchfiles/rskbnd.pdf, 2000. [15] D. Goldfarb, G. Iyengar, Robust portfolio selection problems. Mathematics of Operations Research. 2003, 28: 1-38. [16] El Ghaoui, L., Oks, M., Oustry, F.: Worst-Case Value-at-Risk and Robust Portfolio Optimization: A Conic Programming Approach. Operations Research, 2003, 51: 543-556 [17] S.S. Zhu, M. Fukushima, Worst-Case Conditional Value-at-Risk with application to robust portfolio management. Operations Research. 2009, 57: 1155-1168. [18] A.L. Soyster, Convex programming with set-inclusive constraints and applications to inexact linear programming. Operations Research. 1973, 21: 1154-1157. [19] Z.F. Dai, D.H. Li, F.H. Wen. Robust Conditional value-at-risk optimization for Asymmetrically Distributed Asset Returns. Pacific Journal of Optimization, 2012, 8(3): 429-445. [20] X. Chen, M. Sim and P. Sun, A robust optimization perspective of stochastic programming, Operations Research, 2007, 55: 1058-1077. [21] J. Sturm, Using SeDuMi 1.02, a MATLAB toolbox for optimization over symmetric cones. Optim. Methods and Software. 1999, 11: 625-653.

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RANDOM DERIVATIONS ON RANDOM NORMED ALGEBRAS JUNG RYE LEE, CHOONKIL PARK, AND DONG YUN SHIN∗ Abstract. Using the fixed point method, we prove the Hyers-Ulam stability of random derivations in random normed algebras associated with the Cauchy functional equation.

1. Introduction Fuzzy set theory is a powerful tool set for modeling uncertainty and vagueness in various problems arising in the field of science and engineering. It has also very useful applications in various fields, e.g., population dynamics [4], chaos control [13], computer programming [15], etc. Recently, the fuzzy topology has proved to be a very useful tool to deal with such situations where the use of classical theories breaks down. In the sequel, we adopt the usual terminology, notations and conventions of the theory of random normed spaces, as in [7, 25, 26, 31, 32]. Throughout this paper, ∆+ is the space of distribution functions, that is, the space of all mappings F : R ∪ {−∞, ∞} → [0, 1] such that F is left-continuous and non-decreasing on R, F (0) = 0 and F (+∞) = 1. D+ is a subset of ∆+ consisting of all functions F ∈ ∆+ for which l− F (+∞) = 1, where l− f (x) denotes the left limit of the function f at the point x, that is, l− f (x) = limt→x− f (t). The space ∆+ is partially ordered by the usual point-wise ordering of functions, i.e., F ≤ G if and only if F (t) ≤ G(t) for all t in R. The maximal element for ∆+ in this order is the distribution function ε0 given by   0, if t ≤ 0, ε0 (t) =  1, if t > 0.

Definition 1.1. ([31]) A mapping T : [0, 1] × [0, 1] → [0, 1] is a continuous triangular norm (briefly, a continuous t-norm) if T satisfies the following conditions: (a) T is commutative and associative; (b) T is continuous; (c) T (a, 1) = a for all a ∈ [0, 1]; (d) T (a, b) ≤ T (c, d) whenever a ≤ c and b ≤ d for all a, b, c, d ∈ [0, 1]. Typical examples of continuous t-norms are TP (a, b) = ab, TM (a, b) = min(a, b) and TL (a, b) = max(a + b − 1, 0) (the Lukasiewicz t-norm). Recall (see [16, 17]) that if T n xi is defined is a t-norm and {xn } is a given sequence of numbers in [0, 1], then Ti=1 n−1 ∞ n 1 recurrently by Ti=1 xi = x1 and Ti=1 xi = T (Ti=1 xi , xn ) for n ≥ 2. Ti=n xi is defined as 2010 Mathematics Subject Classification. Primary 47H10, 39B52, 37H10, 60H25, 39B72, 47B47, 17B40, 54E70. Key words and phrases. Random normed algebra; Fixed point; Hyers-Ulam stability; Cauchy functional equation; Random derivation. ∗ Corresponding author.

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J. LEE, C. PARK, AND D. SHIN ∞ Ti=1 xn+i−1 . It is known ([17]) that for the Lukasiewicz t-norm the following implication holds: ∞

lim (TL )∞ i=1 xn+i−1 = 1 ⇐⇒ n→∞

∑

(1 − xn ) < ∞.

n=1

Definition 1.2. ([32]) A random normed space (briefly, RN-space) is a triple (X, µ, TM ), where X is a vector space and µ is a mapping from X into D+ such that the following conditions hold: (RN1 ) µx (t) = ε0 (t) for all t > 0 if and only if x = 0; t (RN2 ) µαx (t) = µx ( |α| ) for all x ∈ X, α ̸= 0; (RN3 ) µx+y (t + s) ≥ TM (µx (t), µy (s)) for all x, y ∈ X and all t, s > 0. Every normed space (X, ∥ · ∥) defines a random normed space (X, µ, TM ), where t µx (t) = t + ∥x∥ for all t > 0. This space is called the induced random normed space. Definition 1.3. Let (X, µ, T ) be an RN-space. (1) A sequence {xn } in X is said to be convergent to x in X if, for every ϵ > 0 and λ > 0, there exists a positive integer N such that µxn −x (ϵ) > 1 − λ whenever n ≥ N . (2) A sequence {xn } in X is called a Cauchy sequence if, for every ϵ > 0 and λ > 0, there exists a positive integer N such that µxn −xm (ϵ) > 1 − λ whenever n ≥ m ≥ N . (3) An RN-space (X, µ, T ) is said to be complete if and only if every Cauchy sequence in X is convergent to a point in X. Theorem 1.4. ([31]) If (X, µ, T ) is an RN-space and {xn } is a sequence such that xn → x, then limn→∞ µxn (t) = µx (t) almost everywhere. Definition 1.5. A random normed algebra is a random normed space with algebraic structure such that (RN4 ) µxy (ts) ≥ µx (t)µy (s) for all x, y ∈ X and all t, s > 0. Example 1.6. Every normed algebra (X, ∥.∥) defines a random normed algebra (X, µ, TM ), where t µx (t) = t + ∥x∥ for all t > 0. This space is called the induced random normed algebra. Definition 1.7. Let (X, µ, TM ) be a random normed algebra. An R-linear mapping f : X → X is called a random derivation if f (xy) = f (x)y + xf (y) for all x, y ∈ X. 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. We recall a fundamental result in fixed point theory. Theorem 1.8. ([6, 9]) Let (X, d) be a complete generalized metric space and let J : X → X be a strictly contractive mapping with Lipschitz constant L < 1. Then for each given element x ∈ X, either d(J n x, J n+1 x) = ∞ for all nonnegative integers n or there exists a positive integer n0 such that

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(1) (2) (3) (4)

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

The stability problem of functional equations originated from a question of Ulam [33] concerning the stability of group homomorphisms. Hyers [18] gave a first affirmative partial answer to the question of Ulam for Banach spaces. Hyers’ Theorem was generalized by Aoki [1] for additive mappings and by Th.M. Rassias [30] for linear mappings by considering an unbounded Cauchy difference. The paper of Th.M. Rassias [30] has provided a lot of influence in the development of what we call Hyers-Ulam stability of functional equations. A generalization of the Th.M. Rassias theorem was obtained by G˘avruta [14] by replacing the unbounded Cauchy difference by a general control function in the spirit of Th.M. Rassias’ approach. The stability problems of several functional equations have been extensively investigated by a number of authors and there are many interesting results concerning this problem (see [2, 3, 5, 8, 10, 11, 19, 21, 22, 23, 29]). In 1996, G. Isac and Th.M. Rassias [20] were the first to provide applications of stability theory of functional equations for the proof of new fixed point theorems with applications. By using fixed point methods, the stability problems of several functional equations have been extensively investigated by a number of authors (see [25, 27, 28]). The Hyers-Ulam stability of different functional equations in random normed and fuzzy normed spaces has been recently studied in [24, 26]. Using the fixed point method, we prove the Hyers-Ulam stability of random derivations in random normed algebras, associated with the Cauchy functional equation f (x + y) = f (x) + f (y). Throughout this paper, assume that (X, µ, TM ) is a complete random normed algebra. 2. Hyers-Ulam stability of random derivations in random normed algebras Using the fixed point method, we prove the Hyers-Ulam stability of random derivations associated with the Cauchy functional equation. Theorem 2.1. Let φ : X 2 → [0, ∞) be a function such that there exists a constant 0 < L < 12 with L φ (2x, 2y) 2 for all x, y ∈ X. Let f : X → X be a mapping satisfying φ(x, y) ≤

t , t + φ(x, y) t µf (xy)−f (x)y−xf (y) (t) ≥ t + φ(x, y)

µf (rx+ry)−rf (x)−rf (y) (t) ≥

for all r ∈ R, all x, y ∈ X and all t > 0. Then

(

x D(x) := lim 2 f n n→∞ 2

(2.1) (2.2)

)

n

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exists for each x ∈ X and defines a random derivation D : X → X such that µf (x)−D(x) (t) ≥

(2 − 2L)t (2 − 2L)t + Lφ(x, x)

(2.3)

for all x ∈ X and all t > 0. Proof. Letting y = x and r = 1 in (2.1), we get µf (2x)−2f (x) (t) ≥

t t + φ(x, x)

(2.4)

for all x ∈ X and all t > 0. So µf (x)−2f ( x ) (t) ≥ 2

t t+φ

(

x x , 2 2

) ≥

2t 2t + Lφ (x, x)

(2.5)

for all x ∈ X and all t > 0. Consider the set

S := {g : X → X} and introduce the generalized metric on S: t d(g, h) = inf{ν ∈ R+ : µg(x)−h(x) (νt) ≥ , ∀x ∈ X, ∀t > 0}, t + φ(x, x) where, as usual, inf ϕ = +∞. It is easy to show that (S, d) is complete (see the proof of [26, Lemma 2.1]). Now we consider the linear mapping J : S → S such that ( ) x Jg(x) := 2g 2 for all x ∈ X. Let g, h ∈ S be given such that d(g, h) = ε. Then t µg(x)−h(x) (εt) ≥ t + φ(x, x) for all x ∈ X and all t > 0. Hence

(

L µJg(x)−Jh(x) (Lεt) = µ2g( x )−2h( x ) (Lεt) = µg( x )−h( x ) εt 2 2 2 2 2 ≥

Lt 2

Lt 2(

+φ

x x , 2 2

) ≥

Lt 2

+

Lt 2 L φ(x, x) 2

=

)

t t + φ(x, x)

for all x ∈ X and all t > 0. So d(g, h) = ε implies that d(Jg, Jh) ≤ Lε. This means that d(Jg, Jh) ≤ Ld(g, h) for all g, h ∈ S. It follows from (2.5) that

(

µf (x)−2f ( x ) 2

)

L t t ≥ 2 t + φ(x, x)

for all x ∈ X and all t > 0. So d(f, Jf ) ≤ L2 . By Theorem 1.8, there exists a mapping D : X → X satisfying the following:

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(1) D is a fixed point of J, i.e., ( )

x 1 = D(x) 2 2 for all x ∈ X. The mapping D is a unique fixed point of J in the set D

(2.6)

M = {g ∈ S : d(f, g) < ∞}. This implies that D is a unique mapping satisfying (2.6) such that there exists a ν ∈ (0, ∞) satisfying t µf (x)−D(x) (νt) ≥ t + φ(x, x) for all x ∈ X and all t > 0; (2) d(J n f, D) → 0 as n → ∞. This implies the equality (

x lim 2 f n n→∞ 2

)

n

for all x ∈ X; (3) d(f, D) ≤

1 d(f, Jf ), 1−L

= D(x)

which implies the inequality d(f, D) ≤

L . 2 − 2L

This implies that the inequality (2.3) holds. By (2.1), µ2n f ( xn + yn )−2n f ( xn )−2n f ( yn ) (2n t) ≥ 2

2

2

2

t

t+φ

(

x , y 2n 2n

)

for all x, y ∈ X, all t > 0 and all n ∈ N. So µ2n f ( xn + yn )−2n f ( xn )−2n f ( yn ) (t) ≥ 2

2

2

2

t 2n t 2n

+

for all x, y ∈ X, all t > 0 and all n ∈ N. Since limn→∞ and all t > 0,

Ln φ (x, y) 2n t 2n Ln t + 2n φ(x,y) 2n

= 1 for all x, y ∈ X

µD(x+y)−D(x)−D(y) (t) = 1 for all x, y ∈ X and all t > 0. Thus the mapping D : X → X is Cauchy additive. Let y = 0 in (2.1). By (2.1), µ2n f ( rxn )−2n rf ( xn ) (2n t) ≥ 2

2

t

t+φ

(

x ,0 2n

)

for all r ∈ R, all x ∈ X, all t > 0 and all n ∈ N. So µ2n f ( rxn )−2n rf ( xn ) (t) ≥ 2

2

108

t 2n t 2n

+

Ln φ (x, 0) 2n

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J. LEE, C. PARK, AND D. SHIN

for all r ∈ R, all x ∈ X, all t > 0 and all n ∈ N. Since limn→∞ x ∈ X and all t > 0,

t 2n t Ln + 2n φ(x,0) 2n

= 1 for all

µH(rx)−rH(x) (t) = 1 for all r ∈ R, all x ∈ X and all t > 0. Thus the additive mapping D : X → X is R-linear. By (2.2), t ( ) µ4n f ( xn · yn )−2n f ( xn )·y−x·2n f ( yn ) (4n t) ≥ 2 2 2 2 t+φ x , y 2n

2n

for all x, y ∈ X, all t > 0 and all n ∈ N. So µ4n f ( xn · yn )−2n f ( xn )·y−x·2n f ( yn ) (t) ≥ 2

2

2

2

t 4n t 4n

+

for all x, y ∈ X, all t > 0 and all n ∈ N. Since limn→∞ and all t > 0,

Ln φ (x, y) 2n

t 4n Ln t + 2n φ(x,y) 4n

= 1 for all x, y ∈ X

µD(xy)−D(x)y−xD(y) (t) = 1 for all x, y ∈ X and all t > 0. Thus the mapping D : X → X satisfies D(xy) = D(x)y + xD(y) for all x, y ∈ X. Therefore, there exists a unique random derivation D : X → X satisfying (2.3).  Theorem 2.2. Let φ : X 2 → [0, ∞) be a function such that there exists a constant 0 < L < 1 with ( ) x y φ(x, y) ≤ 2Lφ , 2 2 for all x, y ∈ X. Let f : X → X be a mapping satisfying (2.1) and (2.2). Then 1 D(x) := lim n f (2n x) n→∞ 2 exists for each x ∈ X and defines a random derivation D : X → X such that µf (x)−D(x) (t) ≥

(2 − 2L)t (2 − 2L)t + φ(x, x)

for all x ∈ X and all t > 0. Proof. Let (S, d) be the generalized metric space defined in the proof of Theorem 2.1. Consider the linear mapping J : S → S such that 1 Jg(x) := g (2x) 2 for all x ∈ X. It follows from (2.4) that ( ) 1 t µf (x)− 1 f (2x) t ≥ 2 2 t + φ(x, x) for all x ∈ X and all t > 0. So d(g, Jg) ≤ 12 . The rest of the proof is similar to the proof of Theorem 2.1.

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RANDOM DERIVATIONS ON RN-ALGEBRAS

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

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[21] S. Jung, Hyers-Ulam stability of linear differential equations of first order, Appl. Math. Lett. 17 (2004), 1135–1140. [22] S. Jung, Hyers-Ulam stability of linear differential equations of first order II, Appl. Math. Lett. 19 (2006), 854–858. [23] S. Jung, Hyers-Ulam stability of linear partial differential equations of first order, Appl. Math. Lett. 22 (2009), 70–74. [24] D. Mihet¸, The probabilistic stability for a functional equation in a single variable, Acta Math. Hungar. 123 (2009), 249–256. [25] D. Mihet¸, The fixed point method for fuzzy stability of the Jensen functional equation, Fuzzy Sets and Systems 160 (2009), 1663–1667. [26] D. Mihet¸ and V. Radu, On the stability of the additive Cauchy functional equation in random normed spaces, J. Math. Anal. Appl. 343 (2008), 567–572. [27] M. Mirzavaziri and M.S. Moslehian, A fixed point approach to stability of a quadratic equation, Bull. Braz. Math. Soc. 37 (2006), 361–376. [28] V. Radu, The fixed point alternative and the stability of functional equations, Fixed Point Theory 4 (2003), 91–96. [29] J.M. Rassias and H. Kim, Approximate homomorphisms and derivations between C ∗ -ternary algebras, J. Math. Phys. 49 063507 (2008). [30] Th.M. Rassias, On the stability of the linear mapping in Banach spaces, Proc. Amer. Math. Soc. 72 (1978), 297–300. [31] B. Schweizer and A. Sklar, Probabilistic Metric Spaces, North-Holland, New York, 1983. [32] A.N. Sherstnev, On the notion of a random normed space, Dokl. Akad. Nauk SSSR 149 (1963), 280–283 (in Russian). [33] S.M. Ulam, A Collection of the Mathematical Problems, Interscience Publ. New York, 1960. Jung Rye Lee Department of Mathematics, Daejin University, Kyeonggi 487-711, South Korea E-mail address: [email protected] Choonkil Park Department of Mathematics, Research Institute for Natural Sciences, Hanyang University, Seoul 133-791, South Korea E-mail address: [email protected] Dong Yun Shin Department of Mathematics, University of Seoul, Seoul 130-743, South Korea E-mail address: [email protected]

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A note on the q-extension of second kind Euler numbers and polynomials C. S. Ryoo Department of Mathematics, Hannam University, Daejeon 306-791, Korea

Abstract : In this paper, by using the p-adic integral on Zp , we construct a new type of the q-extension of the second kind Euler numbers En,q and polynomials En,q (x). From these numbers and polynomials, we establish some interesting identities and relations. By using the q-extension of the second kind Euler numbers En,q and polynomials En,q (x), the q-Euler zeta function and Hurwitz-type q-Euler zeta functions are defined. Key words : the second kind Euler numbers and polynomials, the q-extension of the second kind Euler numbers and polynomials 2000 Mathematics Subject Classification : 11B68, 11S40, 11S80 1. Introduction Throughout this paper, we always make use of the following notations: N = {1, 2, 3, · · · } denotes the set of natural numbers, 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 1 assume that |q − 1|p < p− p−1 so that q x = exp(x log q) for |x|p ≤ 1.Throughout this paper we use the notation: [x]q =

1 − qx , cf. [1, 2, 3, 4, 5, 6] . 1−q

For g ∈ U D(Zp ) = {g|g : Zp → Cp is uniformly differentiable function}, Kim[1, 2] defined the p-adic integral on Zp as follows:  I1 (g) = g(x)dμ−1 (x) = lim

N →∞

Zp



g(x)(−1)x .

(1.1)

0≤x 1. ζq (s) = 2

∞  (−1)n q n . [2n + 1]sq n=1

(3.1)

Note that ζq (s) is a meromorphic function on C. Note that, if q → 1, then ζq (s) = ζ(s) which is the Euler zeta functions(see [6]). Relation between ζq (s) and Ek,q is given by the following theorem. Theorem 8. For k ∈ N, we have ζq (−k) = Ek,q . Observe that ζq (s) function interpolates Ek,q numbers at non-negative integers. By using (2.9), we note that  ∞   dk  Fq (t, x) =2 (−1)m q m [2x + 1 + m]kq (3.2) dtk t=0 m=0

and



d dt

k   ∞

tn En,q (x) n! n=0

    

= Ek,q (x), for k ∈ N.

(3.3)

t=0

By (3.2) and (3.3), we are now ready to define the Hurwitz q-Euler zeta functions. Definition 9. Let s ∈ C with Res > 1. ζq (s, x) = 2

∞ 

(−1)n q n . [n + 2x + 1]sq n=0

(3.4)

Note that ζq (s, x) is a meromorphic function on C. Obverse that, if q → 1, then ζq (s, x) = ζ(s, x) which is the Hurwitz Euler zeta functions(see [6]). Relation between ζq (s, x) and Ek,q (x) is given by the following theorem. Theorem 10. For k ∈ N, we have ζq (−k, x) = Ek,q (x). Observe that ζq (−k, x) function interpolates Ek,q (x) numbers at non-negative integers. REFERENCES 1. Kim, T.(2007). q-Euler numbers and polynomials associated with p-adic q-integrals, J. Nonlinear Math. Phys., v. 14, pp. 15-27. 2. Kim, T.(2002). q-Volkenborn integration, Russ. J. Math. Phys., v. 9, pp. 288-299. 3. Kim, T., Jang, L. C., Pak, H. K.(2001).

A note on q-Euler and Genocchi numbers , Proc.

Japan Acad. , v.77 A, pp. 139-141. 4. Ryoo, C.S. (2010).

Calculating zeros of the second kind Euler polynomials, Journal of

Computational Analysis and Applications, v.12, pp. 828-833. 5. Ryoo, C.S.(2009). Calculating zeros of the q-Euler polynomials, Proceeding of the Jangjeon Mathematical Society, v.12, pp. 253-259. 6. Ryoo, C.S. (2011). A note on the q-Hurwitz Euler zeta functions, Journal of Computational Analysis and Applications, v.13, pp. 1012-1018.

116

RYOO 112-116

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SOME PROPERTIES OF BAZILEVIC FUNCTIONS RELATED WITH CONIC DOMAINS KHALIDA INAYAT NOOR, MOHSAN RAZA∗ AND KAMRAN YOUSAF Abstract. The aim of this paper is to study the Bazilevic functions associated with conic domains. Some properties of analytic functions related with Bazilevic functions by using the concept of convolution are examined. We investigate some results concerned with integral preserving property and radius problems which generalize the already proved results.

1. Introduction Let A be the class of analytic functions ∞ X F (z) = z + an z n ,

(1.1)

n=2

defined in the open unit disc E = {z : |z| < 1}. For any two analytic functions f and g with ∞ ∞ X X n bn z and g (z) = cn z n , z ∈ E, f (z) = n=0

n=0

the convolution (Hadamard product) is given by ∞ X (f ∗ g) (z) = bn cn z n , z ∈ E. n=0

A function f ∈ A is starlike univalent function of order ρ, if and only if zf 0 (z) > ρ, 0 ≤ ρ < 1, z ∈ E. Re f (z) This class of functions is denoted by S ∗ (ρ) . Kanas and Wisnowska [7] studied k − U CV, the class of k-uniformly convex and k − ST , the corresponding class of k-starlike functions. A function f ∈ A is said to be in the class k − U CV of k-uniformly convex function, if   zf 00 (z) zf 00 (z) (1.2) ≥k , k ≥ 0, z ∈ E. Re 1 + 0 f (z) f 0 (z) Similarly a function f ∈ A is said to be in the class denoted by k − ST , if and only if   0 zf 0 (z) zf (z) ≥k − 1 , k ≥ 0, z ∈ E. Re (1.3) f (z) f (z) ∗

Corresponding author 2010 Mathematics Subject Classification. 30C45, 30C50. Key words and phrases. Bazilevic functions, Convolution, Conic Domains.

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Geometric interpretation. The function f ∈ k − U CV and f ∈ k − ST, if and 00 (z) 0 (z) only if zff 0 (z) + 1 and zff (z) , respectively, take all values in the conic domain Ωk , which is included in the right half plane such that p Ωk = {u + iv : u > k (u − 1)2 + v 2 }, 00

0

(z) (z) with p(z) = zff 0 (z) + 1 or p(z) = zff (z) and considering the functions which map E onto the conic domain Ωk such that 1 ∈ Ωk , we may rewrite the conditions (1.2) or (1.3) in the form p(z) ≺ qk (z). The domain Ωk,ρ is such that

Ωk,ρ = (1 − ρ) Ωk + ρ,

0 ≤ ρ < 1.

The function qk,ρ plays the role of extremal for these classes and is given by  1+(1−ρ)z  , k = 0,  1−z       √ 2   2γ(1−ρ) 1+ z   log 1−√z , k = 1, 1 + π2      2
√  2(1−ρ) (1.4) qk,ρ (z) = 2 arccos k arctan h z , 0 < k < 1, 1 + 2 sinh  1−k π         u(z)  √  t  R  π  1 + (1−ρ) √1 sin  2R(t) dx + (1−ρ) , k > 1, √  2 k2 −1 k2 −1  2 1−x 1−(tx)  0  0  √ πR (t) z−√ t where u(z) = 1− , t ∈ (0, 1), z ∈ E and t is chosen such that k = cosh , 4R(t) tz 0 with R(t) is Legendre’s complete elliptic integral of the first kind and R (t) is complementary integral of R(t). By virtue of (1.4) and the properties of the domains Ωk,ρ , we have p ≺ qk,ρ implies k+ρ Re p(z) > Re qk,ρ (z) > . k+1 A function p, analytic in E with p(0) = 1, is said to be in the class k − P (ρ) ⊂ P, if it is subordinate to qk,ρ in E. That is p ∈ k − P (ρ), if and only if p ≺ qk,ρ , where qk,ρ is given by (1.4) and p(E) ⊂ qk,ρ (E). It is noted that 0 − P (0) = P, the class of analytic functions with positive real part and p ∈ 0 − P (ρ) = P (ρ) implies that Rep(z) > ρ, z ∈ E. Recently Noor [11] has extended the class k − P (ρ) and defined the following subclass of caratheodory class P. Definition 1.1. Let p be analytic in E with p(0) = 1. Then p ∈ k − Pm (ρ), if and only if     m 1 m 1 p(z) = + p1 (z) − − p2 (z) , p1 (z), p2 (z) ∈ k − P (ρ), 4 2 4 2 for m ≥ 2, 0 ≤ ρ < 1, k ∈ [0, ∞), z ∈ E. We note that k − P2 (ρ) = k − P (ρ) and 0 − Pm (0) = Pm , the well-known class defined in [12]. 118

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Definition 1.2. [11] Let f ∈ A. Then f ∈ k − U Rm (ρ), 0 ≤ ρ < 1, k ∈ [0, ∞) and m ≥ 2, if and only if zf 0 (z) ∈ k − Pm (ρ), z ∈ E. f (z) Noor called k − U Rm (ρ), the class of functions of k-uniform bounded boundary rotation m with order ρ. It can easily be seen that 0 − U Rm (0) = Rm , the class of functions of bounded boundary rotation. It is also noted that 1−U R2 (0) = U ST, the class of unifromly starlike functions. Now using the concepts of class k − Pm and the class of uniformly starlike functions, we define the following: Definition 1.3. Let F ∈ A, α, β ∈ R, α > 0, f ∈ U ST. Then F ∈ k − U Bm (α, β) , if and only if  0  zF (z) F α+iβ−1 (z) ∈ k − Pm , z ∈ E. (1.5) z iβ f α (z) Remark 1.4. From (1.5) it can easily be seen that F ∈ k − U Bm (α, β) can be represented by the following integral representation 1   α+iβ Zz F (z) = (α + iβ) h(t)f α (t)tiβ−1 dt h ∈ k − Pm , f ∈ U ST, z ∈ E. (1.6) 0

We note that, with m = 2, k = 0, the class k − U Bm (α, β) reduces to the class of Bazilevic functions introduced in [3], where he showed that a Bazilevic function is univalent in E and has the integral representation given by (1.6) . For recent work of the above mentioned classes, we refer [1,2,6,9,13]. We need the following lemmas which will be used in our main results. 2. Preliminary Results  α , where α > 0 also belongs to U ST Lemma 2.1. Let g ∈ U ST. Then z g(z) z in E. Proof. Let

α g (z) G1 (z) = z . z Taking logarithmic differentiation of both sides we have zG01 (z) zg 0 (z) + (1 − α) = α G1 (z) g (z) = αh0 (z) + (1 − α) . 

Since h0 ∈ 1 − P, p0 (z) = 1 ∈ 1 − P and 1 − P is convex set, see [11], therefore G1 (z) ∈ 1 − P. Remark 2.2. This result can easily be extended to the class k − U Rm using the fact that k − Pm is convex set, see [11]. 119

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Lemma 2.3. [15] Let f ∈ C and g ∈ S ∗ . Then for any analytic function F with F (0) = 1 in E f ∗ Fg (E) ⊂ coF (E) , f ∗g where coF (E) denotes the convex hull of F (E) (the smallest convex set which contains F (E)). Lemma 2.4. [10] Let u = u1 + iu2 , v = v1 + iv2 and ψ (u, v) be a complex valued function satisfying the conditions: (i) ψ (u, v) is continuous in a domain D ⊂ C2 , (ii) (1, 0) ∈ D and Reψ (1, 0) > 0, (iii) Reψ (iu2 , v1 ) ≤ 0, whenever (iu2 , v1 ) ∈ D and v1 ≤ − 12 (1 + u22 ) . If h (z) = 1 + c1 z + · · · is a function analytic in E such that (h(z), zh0 (z)) ∈ D and Reψ (h(z), zh0 (z)) > 0 for z ∈ E, then Reh(z) > 0 in E. 3. Main Results Theorem 3.1. Let α ∈ R, α > 0 and c ∈ C, Re c ≥ 0 and let f ∈ U ST. Then   α1 Zz g(z) = (c + 1) z −c tc−1 f α (t)dt ∈ U ST, z ∈ E. (3.1) 0

Proof. We can write (3.1) by using convolution as  α 1 f (z) φα+c (z) α g (z) = z ∗ , (3.2) z z P α+c n where φα+c (z) = ∞ n=1 α+c+n−1 z is convex in E, see [14]. Now from (3.2) , we get   0

φα+c (z) ∗ z

f (z) z

α

zf 0 (z)

f (z) zg (z)  α . = g (z) φα+c (z) ∗ z f (z) z α 
∗ 1 Since, by Lemma 2.1, z f (z) ∈ U ST ⊂ S ⊂ S ∗ , φα+c (z) is convex, it z 2 follows from Lemma 2.3 that  α φα+c (z) ∗ z f (z) H (z) z zf 0 (z)  α . (E) ⊂ coH (E) , H (z) = f (z) φα+c (z) ∗ z f (z) z

This proves that

zg 0 (z) g(z)

∈ 1 − P and thus g ∈ U ST.

Theorem 3.2. Let F ∈ k − U Bm (α, β) , φ ∈ C. Then 1 " # α+iβ α+iβ F (z) φ (z) G (z) = z ∗ ∈ k − U Bm (α, β) z z

(3.3)

in E. 120

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Proof. Since F ∈ k − U Bm (α, β) , there exists f ∈ U ST such that  0  zF (z) F α+iβ−1 (z) ∈ k − Pm , z ∈ E. (3.4) z iβ f α (z) Define  α 1 f (z) φ (z) α g (z) = z ∗ . (3.5) z z Then g ∈ U ST by Theorem 3.1. Therefore from (3.3) , (3.4) and (3.5) , it follows that  α 0 α+iβ−1 f (z) zF (z)F (z) φ(z) ∗ z z z iβ f α (z) zG0 (z)Gα+iβ−1 (z)  α = f (z) z iβ g α (z) φ(z) ∗ z z

φ (z) ∗ f1 (z)H0 (z) , φ(z) ∗ f1 (z)  α 0 α+iβ−1 (z) f (z) where H0 = zF (z)F ∈ k − P , f (z) = z ∈ S ∗ . Now using Lemma m 1 iβ α z z f (z) 2.3, we obtain the desired result. =

Applications of Theorem 2.3 The class k −U Bm (α, β) is invariant under the following integral representation 1   α+iβ Zz F1 (z) = (α + iβ + c)z −c tc−1 F α+iβ (t)dt , 0

where Re c ≥ 0 and F (z) ∈ k − U Bm (α, β). In fact we can write 1 " # α+iβ α+iβ F (z) φα+iβ+c (z) F1 (z) = z ∗ , z z and since φα+iβ+c is convex in E, the result is immediate from Theorem 3.2. We note the following special cases. (i) For β = 0, α = 1, we have Zz F1 (z) = (1 + c)z −c tc−1 F (t)dt, Re c ≥ 0, (3.6) 0

and this is the generalized Bernadi integral operator [4]. When k = 0, m = 2, we have the result for the class K of close-to-convex functions [4]. (ii) In (3.6), by taking c = 1, we obtain Libera operator [8] and c = 0 leads us to the well-known Alexander operator. (iii) When β = c = 0 implies that α Zz  F (t) α [F1 (z)] = α dt, α > 0, (3.7) t 0

a generalized form of Alexander operator. 121

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With m = 2, k = 0, we note that the class of Bazilevic functions of type α (see, [16]) is invariant under the integral operator defined in (3.6). Class Bm (α, β, γ) We now deal the case k = 0, when F ∈ A, α, β ∈ R, α > 0, f ∈ S ∗ . Then F ∈ Bm (α, β, γ), if and only if zF 0 (z)F α+iβ−1 (z) ∈ Pm (γ), 0 ≤ γ < 1. z iβ f α (z) Theorem 3.3. Let F ∈ Bm (α, β, γ) and set 1   α+iβ Zz G(z) = (α + iβ + c)z −c tc−1 F α+iβ (t)dt , c ≥ 0.

(3.8)

0

Then G ∈ Bm (α, β, γ1 ), where 2

zg 0 (z) 2γC1 + c + αh1 zg 0 (z) , C1 = α + c + iβ , γ1 = , h1 = Re 2C1 + c + αh1 g (z) g (z) and g is integral representation of f ∈ S ∗ and is starlike. Proof. Since F ∈ Bm (α, β, γ) so there exists f ∈ S ∗ such that zF 0 (z)F α+iβ−1 (z) ∈ Pm (γ), z ∈ E. z iβ f α (z)

(3.9)

zG0 (z)Gα+iβ−1 (z) = h(z), z iβ g α (z)

(3.10)

Set

where 

α + iβ + c g(z) = z c+iβ

Z

z c+iβ−1 α

t

f (t)dt

 α1

∈ S∗

(3.11)

0 zg 0 (z) g(z)

by Theorem 3.1. Since g ∈ S ∗ we set = h0 (z) = h1 + ih2 , h0 ∈ P in E. Now from (3.8) – (3.11) , we obtain after some computations   zh0 (z) h(z) + ∈ Pm (γ). (3.12) αh0 (z) + c + iβ Writing h(z) = (1 − γ1 )p(z) + γ1 . It follows from (3.12) that for i = 1, 2,   (1 − γ1 ) zp0i (z) + γ1 − γ ∈ P, z ∈ E. (1 − γ1 ) pi (z) + αh0 (z) + c + iβ We construct the functional ψ(u, v) by taking u = pi (z), v = zp0i (z) as follows: (u, v) = (1 − γ1 )u +

(1 − γ1 )v + (γ1 − γ). αh0 (z) + c + iβ 122

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The first two conditions of Lemma 2.4 can easily be verified. We check condition (iii) as below: (1 − γ1 )(c + αh1 )v |αh0 (z) + c + iβ|2 (1 − γ1 )(c + αh1 )(1 + u22 ) (γ1 − γ) − 2 |αh0 (z) + c + iβ|2 2(γ1 − γ)C1 − (1 − γ1 )(c + αh1 )(1 + u22 ) 2C1 2(γ1 − γ)C1 − (1 − γ1 )(c + αh1 ) + (γ1 − 1)(c + αh1 )u22 2C1 2 A + Bu2 , (3.13) 2C1

Re ψ(iu2 , v1 ) = (γ1 − γ) + ≤ = = =

where A = 2(γ1 − γ)C1 − (1 − γ1 )(c + αh1 ) and B = (γ1 − 1)(c + αh1 ). Since C1 = |αh0 (z) + c + iβ|2 > 0 and the right hand side of (3.13) is less than or equal to zero, if A ≤ 0 and B ≤ 0. Now from B ≤ 0, we have γ1 < 1 and from A ≤ 0, we obtain the value of γ1 given by 2γC1 + c + αh1 . 2C1 + c + αh1 Thus all the conditions of Lemma 2.4 are satisfied and pi ∈ P which mean hi ∈ P (γ1 ) and hence h ∈ Pm (γ1 ) . This proves our result. γ1 =

Theorem 3.4. Let G ∈ B2 (α, β, 0), where 1   α+iβ Zz G(z) = (α + iβ + c)z −c tc−1 F α+iβ (t)dt , c ≥ 0. 0

Then F ∈ B2 (α, β, 0), for |z| < r0 and ( √ r0 =

1 , 2

−(α+1)+ c2 +2α+1 , c−α

c > α, c = α = 1.

(3.14)

Proof. Since G ∈ B2 (α, β, 0), so there exists g ∈ S ∗ such that zG0 (z)Gα+iβ−1 (z) = h(z), z iβ g α (z) where 

α + iβ + c g(z) = z c+iβ ∗

Z

z c+iβ−1 α

t

f (t)dt

 α1

∈ S∗

0 zg 0 (z) g(z)

by Theorem 3.1. Since g ∈ S we set = h0 (z) ∈ P in E. Now from (3.8), (3.10) and (3.11) , we obtain after some computation zF 0 (z)F α+iβ−1 (z) zh0 (z) . = h(z) + z iβ f α (z) αh0 (z) + c + iβ 123

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This implies that   zF 0 (z)F α+iβ−1 (z) zh0 (z) Re = Re h(z) + z iβ f α (z) αh0 (z) + c + iβ zh0 (z) . ≥ Re h(z) − αh0 (z) + c + iβ Using the well-known distortion results for class P, we have   zF 0 (z)F α+iβ−1 (z) 2r 1 Re ≥ Re h(z) 1 − . z iβ f α (z) 1 − r2 αh0 (z) + c + iβ Since h0 ∈ P, we have |αh0 (z) + c + iβ| ≥ Re {αh0 (z) + c + iβ}   1−r ≥ α +c 1+r α (1 − r) + c (1 + r) = . 1+r It follows easily that   zF 0 (z)F α+iβ−1 (z) 2r Re ≥ Re h(z) 1 − z iβ f α (z) (1 − r) {α (1 − r) + c (1 + r)}   (α + c) − 2 (α + 1) r + (α − c) r2 = Re h(z) . (1 − r) {α (1 − r) + c (1 + r)} Hence F ∈ B2 (α, β, 0), for |z| < r0 , where r0 is given in (3.14) . This completes the proof. For α = 1 and β = 0, we have the result proved by Bernardi [5] for Bernardi operator. Corollary 3.5. Let G ∈ K, where z

G(z) = (1 + c) z

−c

Z

tc−1 F (t)dt.

0

Then F ∈ K, for |z| < r0 and  r0 =

√ −2+ c2 +3 , c−1 1 , 2

c > 1, c = 1.

References [1] Arif, M., Noor, K. I., Raza, M. On a class of analytic functions related with generalized Bazilevic type functions. Comput. Math. Appl., 61, 2456–2462 (2011). [2] Arif, M., Raza, M., Noor, K. I., Malik, S. N. On strongly Bazilevic functions associated with generalized Robertson functions. Math. Comput. Mod., 54, 1608-1612 (2011). [3] Bazilevic, I. E. On a class of integrability in quadratures of the Loewner-Kufarev equation. Mat. Sb., 37, 471-476 (1955). [4] Bernardi, S. D. Convex and Starlike Univalent Functions. Trans. Amer. Math. Soc., 135, 429–446 (1969).

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BAZILEVIC FUNCTIONS RELATED WITH CONIC DOMAINS

[5] Bernardi, S. D. The radius of univalence of certain analytic functions. Proc. Amer. Math. Soc., 24, 312–318 (1970) . [6] Irmak, H., Bulboac˘ a, T., Tuneski, N. Some relations between certain classes consisting of α-convex type and Bazilevi´c type functions. Appl. Math. Lett., 24, 2010–2014 (2011). [7] Kanas, S., Wisniowska, A. Conic regions and k-uniform convexity. J. Comput. Appl. Math., 105, 327–336 (1999). [8] Libra, R. J. Some Classes of Regular Univalent Functions. Proc. Amer. Math. Soc., 16, 755–758 (1965). [9] Malik, S. N., Raza, M., Arif, M., Hussain, S. Coeffecient estimates for some sub classes of analytic functions related with conic domains. An. St. Univ. Ovidius Constanta, (2012) (in press). [10] Miller, S. S. Differential inequalities and Caratheodory functions. Bull. Amer. Math. Soc., 81, 79–81 (1975). [11] Noor, K. I. On a generalization of uniformly convex and related functions. Comp. Math. Appl., 61(1), 117-125 (2011). [12] Pinchuk, B. Functions with bounded boundary rotation. Israel J. Math., 10, 7–16 (1971). [13] Raza, M., Noor, K. I. A class of Bazilevic type functions defined by convolution operator, J. Math. Inequal., 5(2), 253-261 (2011). [14] Ruscheweyh, S. Eine Invarianzeigenschaft der Basileviˇc-funktionen. Math. Z., 134, 215-219 (1973). [15] Ruscheweyh, S., Sheil-Small, T. Hadmard products of schlicht functions and the polyaShoenberg conjecture. Comm. Math. Helv., 48, 119-135 (1973). [16] Singh, R. On Bazileviˇc functions, Proc. Amer. Math. Soc., 38, 261-271 (1973). KHALIDA INAYAT NOOR Department of Mathematics, COMSATS Institute of Information Technology, Islamabad Pakistan. E-mail address: [email protected] MOHSAN RAZA Department of Mathematics, GC University Faisalabad, Pakistan. E-mail address: [email protected] KAMRAN YOUSAF Department of Mathematics, COMSATS Institute of Information Technology, Islamabad Pakistan. E-mail address: [email protected].

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obius invariant Some characterizations in some M¨ spaces A. El-Sayed Ahmed Sohag University, Faculty of Science, Department of Mathematics, Sohag 82524, Egypt and Taif University, Faculty of Science, Mathematics Department Box 888 El-Hawiyah, El-Taif 5700, Saudi Arabia and e-mail: [email protected] A. Kamal Majmaah University, Faculty of Science and Humanities, Mathematics Department Box 66 Ghat 11914, Saudi Arabia e-mail: alaa [email protected] Aydah Ahmadi King Khalid University, Faculty for girls, Math.Dept, KSA

Abstract. We give two characterizations of the M¨ obius invariant QK (p, q) spaces, one in terms of a double integral and the other in terms of the mean oscillation in the Bergman metric. Both characterizations avoid the use of derivatives.

1

Introduction

Let ∆ = {z ∈ C : |z| < 1} be the unit disk of the complex plane C. The Green’s function in the unit disk ∆ with a−z singularity at a ∈ ∆ is given by g(z, a) = log |ϕa1(z)| , where ϕa (z) = 1−¯ . For 0 < r < 1, let ∆(a, r) = {z ∈ ∆ : az |ϕa (z)| < r} be the pseudo-hyperbolic disk with the center a ∈ ∆ and radius r. Through this paper, we assume that K : [0, ∞) → [0, ∞) is a right continuous and nondecreasing function. For 0 < p < ∞ and −2 < q < ∞, we say that a function f analytic in ∆ belongs to the space QK (p, q) if

Z kf kpK,p,q

=

¯ 0 ¯p ¡ ¢ ¯f (z)¯ 1 − |z|2 q K(1 − |ϕa (z)|2 )dA(z) < ∞,

sup a∈∆

∆

where dA(z) is the Euclidean area element on ∆. It is clear that QK (p, q) is a Banach space with the norm kf k = |f (0)| + kf kK,p,q where p ≥ 1. If q + 2 = p, QK (p, q) is M¨ obius invariant, i.e., kf ◦ ϕa k = kf kK,p,q for all a ∈ ∆. Now we consider some special cases. If p = 2, and q = 0, we obtain that QK (p, q) = QK (cf. [4, 9]). If K(t) = ts , then QK (p, q) = F (p, q, s) (cf. [11]) that F (p, q, s) is contained in q+2 − Bloch space. p The space QK,0 (p, q) consists of analytic function f in ∆ with the property that

Z |f 0 (z)|p (1 − |z|2 )q K(1 − |ϕa (z)|2 )dA(z) = 0.

lim

|a|→1−

∆

AMS: 47B38 46E15. Key words and phrases : Berezin transform,QK (p, q) spaces, M¨ obius invariant spaces.

1

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It can be checked that QK,0 (p, q) is a closed subspace in QK (p, q). The following identity is easily verified: (1 − |a|2 )(1 − |z|2 ) = (1 − |z|2 )|ϕ0a (z)|. |1 − a ¯z|2

1 − |ϕa (z)|2 =

For a ∈ ∆, the substitution z = ϕa (w) results in the Jacobian change in measure given by dA(w) = |ϕa (z)|2 dA(z). For a Lebesgue integrable or a non-negative Lebesgue measurable function h on ∆ we thus have the following change-of-variable formula:

Z

µ

Z h(ϕa (w))dA(w) = ∆(0,r)

h(z) ∆(a,r)

1 − |ϕa (z)|2 1 − |z|2

¶2 dA(z) .

Note that ϕa (ϕa (z)) = z and thus ϕ−1 a (z) = ϕa (z). For a, z ∈ ∆ and 0 < r < 1, the pseudo-hyperbolic disk ∆(a, r) is defined by ∆(a, r) = {z ∈ ∆ : |ϕa (z)| < r}. We will also need to use the so-called Berezin transform. More specifically, for any function f ∈ L1 (∆, dA), we define a function Bf by

Z

(1 − |z|2 )2 f (w)dA(w), |1 − zw|4

Bf (z) = ∆

z∈∆

we call Bf the Berezin transform of f. By a change of variables, we can also write

Z

Bf (z) =

f ◦ ϕz (w)dA(w),

z∈∆

∆

see [1, 2, 3, 6] and [12] for basic properties of the Berezin transform. If the function K is only defined on (0, 1], then we extend it to (0, ∞) by setting K(t) = K(1) for t > 1. We can then define on auxiliary function as follows: ϕK (s) = sup 0 0 (independent of K) such that

Z

¯ 0 ¯p ¡ ¢ ¯f (z)¯ 1 − |z|2 p−2 K(1 − |z|2 )dA(z) ≤ CI(f )

∆

for all analytic functions f in ∆, where

¯p Z Z ¯¯ ¢p−2 f (z) − f (w)¯ ¡ ¯ 1 − |z|2 K(1 − |z|2 )d(z)dA(w). I(f ) = 4 ¯ 1 − z w| ¯ ∆ ∆

2

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Proof. We write the double integral I(f ) as an iterated integral

Z

¢2 Z ¡ ¯ 1 − |z|2 ¯ ¯f (z) − f (w)¯p dA(w) ¯ 4 ¯ 1 − z w| ¯ ∆

K(1 − |z|2 ) dA(z) (1 − |z|2 )4−p

I(f ) = ∆

Making a change of variables in the inner integral, we obtain

Z

∆

It is well known that

Z

K(1 − |z|2 ) dA(z) (1 − |z|2 )4−p

I(f ) =

Z

¯ ¯ ¯f (ϕz (w)) − f (z)¯p dA(w).

(2)

∆

Z |g(w) − g(0)|p dA(w) ∼

∆

|g 0 (w)|p (1 − |w|2 )p dA(w),

(3)

∆

for analytic functions g in ∆. Applying (3) to the inner integral in (2) with the function g(w) = f (ϕz (w)), we deduce that

Z

I(f ) ∼ ∆

K(1 − |z|2 ) dA(z) (1 − |z|2 )4−p

Z

¯ ¯ ¡ ¢ ¯(f ◦ ϕz )0 (w)¯p 1 − |w|2 p dA(w). ∆

Therefore, by the chain rule and a change of variables, we get

Z

I(f )

¡

∼

1 − |z|

¢

2 p−2

Z

¯ 0 ¯p (1 − |w|2 )p ¯f (w)¯ dA(w).

2

K(1 − |z| )dA(z)

∆

(4)

|1 − z w| ¯4

∆

Fix any positive radius R. Then there exists a constant C > 0 such that

Z

¡

I(f ) ≥ C

1 − |z|

Z

¢

2 p−2

¯ 0 ¯p (1 − |w|2 )p ¯f (w)¯ dA(w).

2

K(1 − |z| )dA(z)

|1 − z w| ¯4

∆(z,R)

∆

It is well known that (see e.g [8]) (1 − |w|2 ) 1 1 . ∼ ∼ p |1 − z w| ¯2 (1 − |z|2 ) |∆(z, R)| for w ∈ ∆(z, R). It is follows that there exists a positive constant C such that

Z

2 p−2

I(f ) ≥ C

(1 − |z| )

K(1 − |z| )dA(z)

∆

Then,

Z

Z

1

2

|∆(z, R)|

p 2

¯ 0 ¯p ¯f (w)¯ dA(w).

∆(z,R)

¯ 0 ¯p ¯f (z)¯ (1 − |z|2 )p−2 K(1 − |z|2 )dA(z).

I(f ) ≥ C ∆

This complete the proof of the theorem. Theorem 2.2 Let p > 2. If the function K satisfies condition (1) and suppose that K(t) ≈ tn K(t); 0 < t < 1, n ≥ 0. Then there exists a constant C > 0 such that

Z

¯ 0 ¯p ¡ ¢ ¯f (z)¯ 1 − |z|2 p−2 K(1 − |z|2 )dA(z) ≥ CI(f )

∆

for all analytic functions f in ∆, where I(f ) is as given in Lemma 2.1. Proof. By Fubini’s theorem, we can rewrite (4) as

Z I(f ) ∼

Z ∼ ∆

∆

¯ 0 ¯p ¯f (w)¯ (1 − |w|2 )p−2 dA(w)

2 p−2

(1 − |z| )

Z ¡ ∆

¡

Z

¯ 0 ¯p ¯f (w)¯ (1 − |w|2 )p−2 dA(w)

∆

1 − |w|2

¢2

|1 − z w| ¯4

1 − |w|2

¢2

|1 − z w| ¯4

K(1 − |z|2 )dA(z).

K(1 − |z|2 )dA(z).

(5)

The inner integral above is nothing but the Berezin transform of the function K(1 − |z|2 ) at the point w. The desired estimate now follows from Lemma 2.1 We can now prove the main result of this section

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Theorem 2.3 Suppose K satisfies condition (1) and satisfies all conditions of Theorems 2.1 and 2.2, then an analytic function f in ∆ belongs to QK (p, p − 2) if and only if

¯p Z Z ¯¯ f (z) − f (w)¯ ¡ ∆

|1 −

∆

z w| ¯4

1 − |z|2

¢p−2

K(1 − |z|2 ))dA(z)dA(w) < ∞.

(6)

Proof. f ∈ QK (p, p − 2) if and only if

Z

¯p ¡

|f 0 (z)¯ 1 − |z|2

sup a∈∆

¢p−2

K(1 − |ϕa (z)|)dA(z) < ∞.

∆

By a change of variables, we have f ∈ QK (p, p − 2) if and only if

Z

¯ ¯ ¡ ¢ ¡ ¢ ¯(f ◦ ϕa )0 (z)¯p 1 − |ϕa (z)|2 p−2 K 1 − |z|2 dA(z)

sup a∈∆

∆

Replacing f by f ◦ ϕa in Theorems 2.1 and 2.2, we conclude that f ∈ QK (p, p − 2) iff sup a∈∆

¯p Z Z ¯¯ f ◦ ϕa (z) − f ◦ ϕa (w)¯ ¡ ∆

|1 − z w| ¯4

∆

1 − |z|2

¢p−2 ¡

¢

K 1 − |z|2 dA(z)dA(w) < ∞.

Changing variables and simplifying the result, we find that the double integral above is the same as sup a∈∆

¯p Z Z ¯¯ f (z) − f (w)¯ ¡ ∆

|1 −

∆

1 − |z|2

z w| ¯4

¢p−2

K(1 − |ϕa (z)|2 )dA(z)dA(w) < ∞.

Therefore, f ∈ QK (p, p − 2) iff the condition (6) holds.

3

Bergman metric and QK (p, q) spaces

In this section we give two closely related characterizations of QK (p, q) spaces, one in terms of the Berezin transform and the other in terms of certain class of analytic functions in Bergman metric. Given a function f ∈ Lp (∆, dA) it is customary to write 1

S(f )(z) = (B(|f |p ) − |Bf (z)|p ) p . It easy to check that

Z (S(f )(z))p

|f ◦ ϕz (w) − Bf (z)|p dA(w)

=

Z

∆

|f (w) − Bf (z)|p

= ∆

(1 − |z|2 )2 dA(w). |1 − z w| ¯4

If the function f is analytic, then it is easy to see that Bf = f, so that

Z

p

(S(f )(z))

|f ◦ ϕz (w) − f (z)|p dA(w)

=

Z

∆

|f (w) − f (z)|p

= ∆

(1 − |z|2 )2 dA(w). |1 − z w| ¯4

We can now reformulate Theorem 3.1 as follows Theorem 3.1 If K satisfies condition (1), then an analytic function f in ∆ belongs to QK (p, p − 2) iff

Z

¡

(S(f )(z))p 1 − |z|2

sup a∈∆

¢p−2

K(1 − |ϕa (z)|2 )dτ (z) < ∞,

(7)

∆

where dτ (z) =

dA(z) (1 − |z|2 )2

is the M¨ obius invariant measure on the unit disk.

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Proof. From Theorem 3.1 Ia (f ) =

¯p Z Z ¯¯ f (z) − f (w)¯ ¡ ∆

|1 − z w| ¯4

∆

1 − |z|2

¢q

K(1 − |ϕa (z)|2 )dA(z)dA(w)

we rewrite it as an iterated integral

Z

¡

Ia (f ) =

¢

2 p

1 − |z|

2

K(1 − |ϕa (z)| ) dτ (z)

¯p Z ¯¯ f (z) − f (w)¯

∆

∆

|1 − z w| ¯4

dA(w)

or

Z

¡

1 − |z|2

Ia (f ) =

¢p−2

Z K(1 − |ϕa (z)|2 ) dτ (z) ∆

∆

¡ ¢2 ¯ ¯ 1 − |z|2 ¯f (z) − f (w)¯p dA(w) 4 |1 − z w| ¯

According to the calculations preceding this theorem, we have

Z

¡

(S(f )(z))p 1 − |z|2

Ia (f ) =

¢p−2

K(1 − |ϕa (z)|2 )dτz

∆

This proves the desired result. Now, fix a positive radius R and denote by 1 AR (f )(z) = |D(z, R)|

Z f (w)dA(w) ∆(z,R)

the a verge of f over the Bergman metric ball D(z, R). For p ≥ 1, we define

· SR (f )(z) = It is easy to verify that

1 |D(z, R)|p

Z p

¸ p1

|f (w) − AR (f )(z)| dA(w)

.

∆

(SR (f )(z))p = AR (|f |p )(z) − |AR (f )(z)|p .

Now, we prove the following theorem: Theorem 3.2 If K satisfies condition (1), then an analytic function f in ∆ belongs to QK (p, p − 2) if and only if

Z

¡

(SR (f )(z))p 1 − |z|2

sup a∈∆

¢p−2

K(1 − |ϕa (z)|2 )dτ (z) < ∞,

(8)

∆

where R is any fixed positive radius. Proof. There exists a positive constant C which is depending on R only such that SR (f )(z) ≤ C S(f )(z),

z ∈ ∆,

p

where f is any function in L (∆, dA). Therefore, condition (6) implies condition (7). On the other hand, since D(0, R) is an Euclidean disk centered at the origin, we can find a positive constant C which is depending on R only such that

Z 0

p

|f (w) − C|p dA(w)

|f (0)| ≤ C D(0,R)

for all analytic f in ∆ and all complex constants C. Replace f by f ◦ ϕz and replace C by AR (f )(z) then

Z (1 − |z|2 )p |f 0 (z)|p ≤ C

|f ◦ ϕz (w) − AR (f )(z)|p dA(w) D(0,R)

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Make an obvious change of variables on the right hand side, we obtain

¡

Z 2 p

0

p

(1 − |z| ) |f (z)| ≤ C

|f (w) − AR (f )(z)| D(z,R)

Since

¡

1 − |z|2

¢2

|1 − z w| ¯4

p

1 − |z|2

¢2

|1 − z w| ¯4

dA(w).

1 1 ∼ ¡ ¢ ∼ |D(z, R)| 2 2 1 − |z|

for w ∈ ∆(z, R), we can find another positive constant C such that (1 − |z|2 )p |f 0 (z)|p ≤ C (SR (f )(z))p , It follows that for each a ∈ ∆ that

Z

0

sup a∈∆

p

¡

¢

2 q

|f (z)| 1 − |z|

Z

¡

2

(SR (f )(z))p 1 − |z|2

K(1 − |ϕa (z)| )dA(z) ≤ C sup a∈∆

∆

z∈∆

¢p−2

K(1 − |ϕa (z)|2 )dτ (z).

∆

This shows that the condition (7) implies f ∈ QK (p, p − 2). Recall from [5] that a positive Borel measure µ on ∆ is called a K-Carleson measure if

µ

Z

K

sup I

S(I)

¶

1 − |z| dµ(z) < ∞, |I|

where the supremum is taken over all sub-arcs I ⊂ ∂∆. Here, for a sub-arcs I of ∂∆, |I| is the length of I and S(I) = {rξ : ξ ∈ I, 1 − |I| < r < 1} is the corresponding Carlesson square. Also, A positive Borel measure µ on ∆ is called a vanishing K− Carleson measure if

µ

Z

lim

|z|→1−

K ∆

¶

1 − |z| dµ(z) = 0. |I|

Theorem 3.3 Suppose K satisfies the following two conditions: (a) There exists a constant C > 0 such that K(2t) ≤ CK(t) for all t > 0. (b) The auxiliary function ϕk has the property that

Z

1

ϕk (s) 0

ds < ∞. s

Let µ be a positive Borel measure on ∆. Then µ is a K-Carleson measure if and only if

Z

K(1 − |ϕa (z)|2 )dµ(z) < ∞.

sup a∈∆

∆

Proof. Since QK (p, q) is defined by the condition

Z

a∈∆

¡

|f 0 (z)|p 1 − |z|2

sup ∆

¢q

K(1 − |ϕa (z)|2 )dA(z) < ∞,

¡

¢q

we see that f ∈ QK (p, q) if and only if the measure 1 − |z|2 |f 0 (z)|p dA(z) is a K-Carleson measure. The following Corollary gives two analogous characterizations. Corollary 3.1 Suppose K satisfies condition (1) and conditions (a) and (b) in Theorem 3.3. Let R > 0 be a fixed radius. Then the following conditions are equivalent for an analytic function f in ∆. (a) The function f belong to QK (p, p − 2). (b) The measure dµ(z) = (S(f )(z))p (1 − |z|2 )p−2 dτ (z) is a K-Carleson measure. (c) The measure dν(z) = (SR (f )(z))p (1 − |z|2 )p−2 dτ (z) is a K-Carleson measure.

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Proof. This is a direct consequence of Theorems 3.1,3.2, and 3.3. The little-oh version of the above result can be stated as follows: Theorem 3.4 Suppose K satisfies condition (1) and R > 0 is a fixed, then the following conditions are equivalent for all analytic functions f in ∆. (1) f ∈ QK,0 (p, p − 2) (2)

lim

¯p Z Z ¯¯ f (z) − f (w)¯ ¡

|z|→1−

∆

∆

Z Z (3)

∆

Z Z (4)

¡

(S(f )(z))p 1 − |z|2

lim

|z|→1−

¢p−2

¡

(SR (f )(z))p 1 − |z|2 ∆

¢p−2

K(1 − |ϕa (z)|2 )dA(z)dA(w) = 0

K(1 − |ϕa (z)|2 )dτ (z) = 0

∆

lim

|z|→1−

1 − |z|2

|1 − z w| ¯4

¢p−2

K(1 − |ϕa (z)|2 )dτ (z) = 0.

∆

References [1] Y. Ameur, N. Makarov and H. Hedenmalm, Berezin transform in polynomial Bergman spaces, Commun. Pure Appl. Math. 63(12)(2010), 1533 - 1584. [2] O. Blasco and S. P´erez-Esteva, Schatten-Herz operators, Berezin transform and mixed norm spaces, Integral Equations Oper. Theory 71(1)(2011), 65 - 90. [3] M. Engliˇs, and R. Ot´ ahalov´ a, Covariant derivatives of the Berezin transform, Trans. Am. Math. Soc. 363(10)(2011), 5111 - 5129. [4] M. Ess´en and H. Wulan, On analytic and meromorphic functions and spaces of QK type, Illinois J. Math. 46(2002), 1233 - 1258. [5] M. Ess´en, H. Wulan, and J. Xiao, Function-theoretic aspects of mobius invariant QK spaces, J. Funct. Anal. 230 (2006), 78 115. [6] H. Hedenmalm, B. Korenblum, and K. Zhu, Theory of Bergman Spaces, Springer, New York, 2000. [7] B. Korenblum and K. Zhu, Complemented invariant subspaces in Bergman spaces, J. London Math. Soc. 71(2005), 467 - 480. [8] K. Stroethoff, Besov-type characterisations for the Bloch space, Bull. Austral. Math. Soc. 39 (1989), 405 420. [9] H. Wulan and K. Zhu, QK spaces via higher derivative, The Rocky Mountain Journal of Mathematics, (332)(1)(2008), 329 - 350. [10] H. Wulan and K. Zhu, Derivative - free characterizations of QK spaces, J. Aust. Math. Soc. 82(2007), 283 295. [11] R. Zhao, On a general family of function spaces, Ann. Acad. Sci. Fenn. Math. Diss., 105(1996). [12] K. Zhu, Operator theory in function spaces, Marcel Dekker, New York, 1990.

7

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Coefficient bounds for certain subclasses of close-to-convex functions of Janowski type Wasim Ul-Haq1,a , Attiya Nazneena , Muhammad Arifa , Nasir Rehmanb February 26, 2013 In this work, we aim to determine the coefficient estimates for functions in certain subclasses of close-to-convex functions of Janowski type and related functions of complex order, which are here defined by means of Cauchy-Euler type non-homogeneous differential equation. Several interesting consequences of our results are also observed.

Key words: Analytic functions; Close-to-convex; Coefficient Estimates. Subject classification: 30C45, 30C50. † 1

E-mail addresses: [email protected] (W. Ul-Haq), [email protected] (A. Nazneen), [email protected] [email protected] (N. Rehman)

1

Introduction

We denote by A the class of functions f (z) which are analytic in the open unit disc E = {z : |z| < 1} and of the form f (z) = z +

∞ X

an z n .

(1)

n=2 1 Corresponding

author.

a. Department of Mathematics, Abdul Wali Khan University Mardan, Khyber Pakhtunkhwa, Pakistan b. Department of Mathematics and Statistics, Allama Open Iqbal University, Islamabad,Pakistan

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Let S denote the class of all functions in A which are univalent. Also let Sγ∗ , Cγ , Kγ and Qγ be the subclasses of A consisting of all functions which are starlike, convex, close-to-convex and quasi convex of complex order γ (γ 6= 0) respectively, for details see [3, 5, 7]. We note that for 0 < γ ≤ 1, these classes coincide with the well known classes of starlike, convex and close-to-convex of order 1−γ. Recently Altinta¸s et al.[1] considered the following class of functions denoted by SC(γ, λ, A, B) and defined as:   o n 0 0 SC(γ, λ, A, B) =

f (z) ∈ A : 1 +

1 γ

z[(1 − λ)f (z) + λzf (z)] − 1, (1 − λ)f (z) + λzf 0 (z)

≺

1 + Az ,z ∈ E 1 + Bz

,

(2)

where −1 ≤ B < A ≤ 1, 0 ≤ λ ≤ 1, γ ∈ C − {0}. Note that the classes SC(1, 0, A, B) = S ∗ [A, B] and SC(1, 1, A, B) = C[A, B] were introduced by Janowski [4] and are called classes of Janowski starlike and Janowski convex functions respectively. Also SC(γ, 0, 1, −1) = Sγ∗ , SC(γ, 1, 1, −1) = Cγ . Throughout the entire paper onward we assume A ≤ 1, 0 ≤ λ ≤ 1, γ ∈ C − {0} unless otherwise KQ(γ, λ, A, B) be the class of functions f (z) ∈ g(z) ∈ SC(1, λ, A, B) such that   1 z[(1 − λ)f (z) + λzf 0 (z)]0 − 1 ≺ 1+ γ (1 − λ)g(z) + λzg 0 (z)

the restrictions −1 ≤ B < mentioned. Now we denote A if there exist a function 1 + Az , z ∈ E. 1 + Bz

As special choices we have the following relationships KQ(1, 0, A, B) = K[A, B], KQ(1, 1, A, B) = Q[A, B], see [Noor, [6]] KQ(γ, 0, 1, −1) = Kγ , KQ(γ, 1, 1, −1) = Qγ . Motivated from the recent work of Srivastava et al. [9] and Altinta¸s et al. [2] the main purpose of our investigation is to derive coefficient estimates of a subfamily DK(γ, λ, A, B, m; µ) of A, which consists of functions f (z) in A satisfying the following Cauchy Euler type non homogenous differential equation m

m−1 w +···m Cm w dz m−1

w m z m ddzm + C1 (µ+m−1)z m−1 d

Qm−1 j=0

(µ+j)=h(z)

Qm−1 j=0

(µ+j+1),

(3)

where w = f (z), h(z) ∈ KQ(γ, λ, A, B), µ ∈ R − (−∞, −1], m ∈ N ∗ = {2, 3, · · ·} for details we refer to [2, 8, 9, 10, 11]. The following result which is due to Altinta¸s et al. [2] is essential in deriving our main results. Lemma 1. [2]. Let f (z) ∈ SC(γ, λ, A, B) and be of the form (1). Then i Qn−2 h 2|γ|(A−B) j=0 j + 1−B , n ∈ N ∗. |an | ≤ (n − 1)![1 + λ(n − 1)]

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2

Coefficient Estimates for functions in the class KQ(γ, λ, A, B)

We first establish the below result for the functions in the class KQ(γ, λ, A, B). Theorem 1. Let f (z) ∈ KQ(γ, λ, A, B) and be defined by (1). Then  2(A−B)  Q Qn−2  2(A−B)  j+ j+ Pn−1 n−k−2 1−B 1−B 2|γ| j=0 j=0 A−B + n[1+λ(n−1)] 1−B , n∈N ∗ . (4) |an |≤ n![1+λ(n−1)] (n−k−1)! k=1 ∞ P

Proof. Since f (z) ∈ KQ(γ, λ, A, B), then there exists g(z) = z +

bn z n

n=2

belonging to the class SC(1, λ, A, B) such that   1 zF 0 (z) 1 + Az 1+ −1 ≺ , f orz ∈ E, γ G(z) 1 + Bz P∞ P∞ where F (z) = z + n=2 An z n and G(z) = z + n=2 Bn z n , with An = [1 + λ(n − 1)]an , Bn = [1 + λ(n − 1)]bn . Let 1 1+ γ Since q(z) ≺



1+Az 1+Bz ,

(5)

 ∞ X zF 0 (z) cn z n , f orz ∈ E, − 1 = q(z) = 1 + G(z) n=1

(6)

z ∈ E, we find that by definition of subordination

q(z) =

1 + Aw(z) , w(0) = 0; |w(z)| < 1. 1 + Bw(z)

Therefore, we have q(z) − 1 < 1, q(z) = u + iv, |w(z)| = A − Bq(z) which further implies that 2u(1 − AB) > 1 − A2 + (1 − B 2 )(u2 + v 2 ). 2

Also, since |q(z)| ≥ (Req(z))2 , we have (1 − B 2 )u2 − 2u(1 − AB) + 1 − A2 < 0 =⇒ Req(z) >

1−A . 1−B

(7)

From (6) and (7), we find that  |cn | ≤ 2

A−B 1−B

 , n ∈ N.

(8)

Then from (6), we obtain " nAn = Bn + γ cn−1 +

n−1 X

# ck Bn−k , n ≥ 2

k=1

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Now using Lemma 1 together with (5) and (8), we have i i Qn−2 h Qn−k−2 h 2(A−B) n−1 j + 2(A−B) j=0 j + 1−B 1−B 2 |γ| A − B X j=0 |An | ≤ + , n! n 1−B (n − k − 1)! k=1

and hence from the relation between F (z) and f (z) as in (5), we obtain the desired result. By assigning different specific values to the involved parameters A, B, γ, λ in Theorem 1, we deduce the following interesting results. Corollary 1. Let f (z) ∈ KQ(1, 0, A, B) = K[A, B] and be defined by (1). Then i i Qn−k−2 h Qn−2 h 2(A−B) 2(A−B) n−1 j + j + X j=0 j=0 1−B 1−B 2 A−B + n ∈ N ∗. |an | ≤ n! n 1−B (n − k − 1)! k=1

Corollary 2. Let f (z) ∈ KQ(γ, λ, 1, −1) and be defined by (1). Then |an | ≤

1 [1 + (n − 1) |γ|] , n ∈ N ∗ . [1 + λ(n − 1)]

Corollary 3 [3]. Let f (z) ∈ KQ(γ, 0, 1, −1) = K(γ) and be defined by (1). Then |an | ≤ 1 + (n − 1) |γ| , n ∈ N ∗ . Corollary 4 [5]. Let f (z) ∈ KQ(γ, 1, 1, −1) = Q(γ) and be defined by (1). Then for n ∈ N ∗ = {2, 3, 4, . . .}. |an | ≤

1 + (n − 1) |γ| , n ∈ N ∗. n

For γ = 1 in Corollary 2 and Corollary 3, we obtain the well-known coefficient estimates for close-to-convex and quasi convex functions. Corollary 5. Let f (z) ∈ KQ(1 − α, λ, 1 − 2β, −1) and be defined by (1). Then for n ∈ N ∗ Qn−2 |an | ≤

j=0

[j + 2(1 − β)]

n![1 + λ(n − 1)]

n−1 2(1 − α)(1 − β) X + n[1 + λ(n − 1)] k=1

Qn−k−2 j=0

[j + 2(1 − β)]

(n − k − 1)!

.

Corollary 6. Let f (z) ∈ KQ(1 − α, 0, 1, −1) = K(1 − α) and be defined by (1). Then |an | ≤ n(1 − α) + α, n ∈ N ∗ = {2, 3, 4, . . .}. Corollary 7. Let f (z) ∈ KQ(1 − α, 1, 1, −1) = Q(1 − α) and be defined by (1). Then |an | ≤ 1 − α +

α , n ∈ N ∗ = {2, 3, 4, . . .}. n

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3

Coefficient Estimates of the class BK(γ, λ, A, B; µ)

The theorem below is our main coefficient estimates for functions in the class DK(γ, λ, A, B, m; µ). Theorem 2. Let f (z) ∈ DK(γ, λ, A, B, m; µ) and be defined by (1). Then for n ∈ N ∗ = {2, 3, 4, . . .}  2(A−B)   Q Qm−1  Qn−2  2(A−B)  j+ j+ (µ+j+1) Pn−1 n−k−2 1−B 1−B 2|γ| j=0 j=0 j=0 A−B + . |an |≤ Qm−1 n![1+λ(n−1)] n[1+λ(n−1)] 1−B (n−k−1)! k=1 j=0

(µ+j+n)

(9)

P∞ Proof. Let h(z) = z + n=2 bn z n ∈ KQ(γ, λ, β),so that Qm−1 j=0 (µ + j + 1) bn , n ∈ N ∗ , µ ∈ R − (−∞, −1]. an = Qm−1 (µ + j + n) j=0

Hence, by using Theorem 1, we immediately obtain the required inequality (9). Corollary 8. Let f (z) ∈ DK(γ, λ, 1 − 2β, −1, 2; µ) and be defined by (1). Then for n ∈ N ∗ = {2, 3, 4, . . .} Q  Qn−2  [j+2(1−β)] [j+2(1−β)] Pn−1 n−k−2 (1+µ)(2+µ) 2|γ|(1−β) j=0 j=0 |an |≤ (n+1+µ)(n+µ)

n![1+λ(n−1)]

+ n[1+λ(n−1)]

k=1

(n−k−1)!

.

References ¨ Ozkan, ¨ [1] O. Altınta¸s, O. H.M. Srivastava, Neighborhoods of a class of analytic functions with negative coefficients, Appl. Math. Lett. 13 (1995) 63–67. [2] O. Altınta¸s, H. Irmak, S. Owa, H.M. Srivastava, Coefficient bounds for some families of starlike and convex functions of complex order, Appl. Math. Lett. 20 (2007) 1218-1222. [3] H. S. Al-Amiri, T. S Fernando, On close-to-convex functions of complex order, Internat. J. Math. and Math. Sci. 13(1990)321-330. [4] W. Janowski, Some extremal problems for certain families of analytic functions, Ann. Polon. Math. 28 (1973), 297–326. [5] O. S. Kown, S. Owa, .On quasi convex functions of complex order, Soochow J. Math. 20 (1994)241-250. [6] K. I. Noor, Radius problems for a subclass of close-to-convex univalent functions, Internat. J. Math. Math. Sci., 15(4)(1992)719-726. [7] K. I. Noor; Quasi-convex functions of complex order, PanAmer. Math. J. 3(2)(1993)81-90. [8] H. M. Srivastava , S. Owa, Certain classes of analytic functions with varying arguments, J. Math. Appl. 136(1988)217-228.

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[9] H.M. Srivastava, O. Altınta¸s, S. K. Serenbay, Coefficient bounds for certain subclasses of starlike functions of complex order, Appl. Math. Lett. 24(2011)1359-1363. [10] H.M. Srivastava, Q.-H. Xu, G.-P. Wu, Coefficient estimates for certain subclasses of spiral-like functions of complex order, Appl. Math. Lett. 23 (2010)763–768.Tokyo, 1983. [11] H.M. Srivastava, S. Owa (Eds.), Current Topics in Analytic Function Theory, World Scientific Publishing Company, Singapore, New Jersey, London and Hong Kong, 1992. [12] P. Wiatrowski, On the coefficient of some family of holomorphic functions, Zeszyty Nauk. Uniw. Lodz Nauk. Mat.-Przyrod Ser. 39(2)(1970)75-85.

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Five-order Extrapolation Algorithms for Laplace Equation with Linear Boundary Condition∗ Pan Cheng†, Zhi lin, Peng Xie School of Science, Chongqing Jiaotong University, Chongqing ,400074, PR.China

Abstract Laplace equation with linear boundary condition will be converted into a boundary integral equation(BIE) with logarithmic singularity following potential theory. In this paper, a Sidi quadrature formula is introduced to approximate the logarithmic singularity integral operator with O(h3 ) approximate accuracy order. A similar approximate equation is also constructed for the logarithmic singular operator, which is based on coarse grid with mesh width 2h. So an extrapolation algorithm is applied to approximate the logarithmic operator and the accuracy order is improved to O(h5 ). Moreover, the accuracy order is based on fine grid h. The convergence and stability are proved based on Anselone’s collective compact and asymptotic compact theory. Furthermore, an asymptotic expansion with odd powers of the errors is presented with convergence rate O(h5 ). Using h5 −Richardson extrapolation algorithms(EAs), not only the approximation accuracy order can be improved to O(h7 ), but also an a posteriori error estimate can be obtained for constructing a self-adaptive algorithm. numerical examples are shown to verify its efficiency. Keywords: boundary integral equation, Richardson extrapolation algorithm, Laplace equation, a posteriori error estimate 2000 MSC: 65N25, 65N38

1

Introduction

Laplace equation with linear boundary condition is defined as follows: to find non-zero deformation u ˜ in the domain Ω and on the boundary Γ satisfying  u = 0, in Ω,  △˜ (1) ˜  ∂u = −c˜ u(x) + f˜(x), on Γ, ∂n where Ω ⊂ R2 is a bounded, simply connected domain with a smooth boundary Γ, ∂/(∂n) is an normal outward derivative on Γ, c is a positive constant, and f˜(x) is a given function. By means of potential theory, Eq.(1) will be transformed into a boundary integral equation(BIE) as follows[1,2,3] : ∫ ∫ ∂u ˜(x) ∗ (2) dsx , y ∈ Γ, α(y)˜ u(y) − k (y, x)˜ u(x)dsx = h∗ (y, x) ∂n x Γ Γ ∗ Project is supported by the national natural science foundation of China (11271389), and is supported by natural science foundation project of CQ (CSTC2013JCYJA00017, CSTC2011AC6104) † Email-address: cheng [email protected](Pan Cheng) −

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where α(y) = θ(y)/(2π) is related to the interior angle θ(y) of Ω at point y ∈ Γ, in particular, when y is on a smooth part of the boundary Γ, α(y) = 1/2, and h∗ (y, x) is the fundamental solution:  1  ln |x − y|,  h∗ (y, x) = − 2π (3) ∗   k ∗ (y, x) = − ∂h (y, x) , ∂n where |x − y| is the distance between points x and y. The left terms in Eq.(2) are smooth integrals and the right hand side term is characterized as a logarithmic singularity. Various numerical methods have been proposed for dealing with the singularity, such as Galerkin methods in Stephan and Wendland[4] , Chandler[5] , Sloan and Spence[6] , and Amini and Nixon[7] , collocation methods in Elschner and Graham[8] and Yan[9] , quadrature methods in Sidi and Israeli[10] , Saranen[11] , Huang and L¨ u[12,13] and [14] combined Trefftz methods in Li . Extrapolation algorithms (EAs) based on asymptotic expansion about errors are effective parallel algorithms, which possesses high accuracy degree, good stability and almost optimal computational complexity. Cheng et al.[15,16] harnessed extrapolation algorithms to obtain high accuracy order for Steklov eigenvalue in Laplace equations with smoothed and polygonal boundary condition. Huang and L¨ u established extrapolation algorithms for solving the Steklov eigenvalue problems[3] ,the Helmholtz equations[17] and the Laplace equations[18] with accuracy order O(h3 ). After the Extrapolation algorithms, the accuracy order of the approximate solution will be improved to O(h5 ). A quadrature method[19,20] is presented for solving the boundary integral equation, in which the generation of the discrete matrixes does not require any calculations of singular integrals. The logarithmic integral kernel is approximated by extrapolation algorithms derived from Sidi’s quadrature rule. An asymptotic expansion about the error is obtained with convergence rate O(h5 ). Note that the five order approximate solution is obtained directly and is based on fine grid h. Although there are some papers[17−20] also obtain the same accuracy order, there are three main priority for our paper: firstly, those accuracy orders are based on fine grid; secondly, because the accuracy order is not derived from the extrapolation algorithms but from the directly calculation, so there are not any errors generated from the extrapolation algorithms; finally, when an linear equation with n order is solved, there are n approximate solutions uh can be obtained on boundary Γ with accuracy order O(h5 ), while not n/2 values from extrapolation method. This paper is organized as follows: In Section 2 a Sidi’s quadrature method is recombined to approximate integral equations for solving the approximate solution; In Section 3 an asymptotically compact theory is provided for stability and convergence, and an asymptotic expansion for approximate solution is shown with convergence rate O(h5 ); In Section 4 the Richardson extrapolation algorithms are applied to improve the accuracy order to O(h7 ); In Section 5 numerical examples illustrate the calculate progress.

2

Five order approximate methods

Assume that Γ is a smooth closed curve described by a regular parameter mapping x(s) = (x1 (s), x2 (s)) : [0, 2π] → Γ, satisfying |x′ (s)|2 = |x′1 (s)|2 + |x′2 (s)|2 > 0. Let C 2m [0, 2π] denote the set of 2m times differentiable periodic functions with the periodic 2π and xi (s) ∈ C 2m [0, 2π], i = 1, 2. Define the following integral operators on C 2m [0, 2π]:  ∫ 2π  ∂u(t)   (Ku)(s) = 2 dt k(t, s)  ∂n 0 ∫ 2π    h(t, s)u(t)dt.  (Hu)(s) = 2 0

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CHENG ET AL. FIVE ORDER EXTRAPOLATION ALGORITHMS

where u(t) = u ˜(x1 (t), x2 (t)), k(t, s) = k ∗ (x(t), x(s)) |x′ (t)| and h(t, s) = h∗ (x(t), x(s))|x′ (t)|. Because 1 ln |x(t) − x(s)||x′ (t)|, h(t, s) = − 2π so h(t, s) is a logarithmic weak singular kernel and k(t, s) is a smooth kernel. Then Eq.(2) is equivalent to (I − K)u − cHu = Hf (4) where I is an identity operator, and f = f˜(x(t)). Let h = 2π/n (n ∈ N is supposed to be an even number and so n/2 ∈ N ) be the mesh width and tj = sj = jh, (j = 0, 1, . . . , n − 1) be the nodes. In order to approximate the integral operators K and H, a Lemma is obtained: ∫ 2π Lemma 1:[19] Consider the integral 0 G(x)dx with integral kernel G(x). Assume that the functions g(x), g˜(x) are 2m times differentiable on [0, 2π]. Also assume that the integral kernel G(x) are periodic function with period 2π. Then the following conclusion can be drawn: ∑n (a). If G(x) = g(x)/(x − t) + g˜(x), and Qn [G] = h j=1,xj ̸=t G(xj ), then En [G] = h[˜ g (t) + g ′ (t)] + O(h2m ) as h −→ 0, ∫ 2π where En [G] = 0 G(x)dx − Qn [G] in all cases; ∑n g (t) − (b). If G(x) = g(x)(x − t)s + g˜(x), s > −1, and Qn [G] = h j=1,xj ̸=t G(xj ) + h˜ s+1 2ζ(−s)g(t)h , then En [G] = −2

m−1 ∑ µ=1

ζ(−s − 2µ) (2µ) g (t)h2µ+s+1 + O(h2m ), as h → 0; (2µ)!

where ς(t) is the Riemann zeta function. ∑n (c). If G(x) = g(x)(x − t)s log |x − t| + g˜(x), s > −1, and Qn [G] = h j=1,xj ̸=t G(xj ) + h˜ g (t) + 2[ζ ′ (−s) − ζ(−s) log h]g(t)hs+1 , then En [G] = −2

m−1 ∑

[ζ ′ (−s − 2µ) − ζ(−s − 2µ) log h]

µ=1

g (2µ) (t) 2µ+s+1 h + O(h2m ), as h → 0; (2µ)!

Especially, when s = 0, then ζ ′ (0) = −(1/2) log(2π), and we have Qn [G] = h

n ∑

G(xj ) + h˜ g (t) + log

j=1,xj =t

then En [G] = 2

m−1 ∑

ζ ′ (−2µ)

µ=1

(h) g(t)h, 2π

g (2µ) (t) 2µ+1 h + O(h2m ), as h → 0. (2µ)!

Since K is a smooth integral operator with period 2π, we obtain a high accuracy approximation when set g(x) ≡ 0 in case (a) of Lemma 1: (Kh u)(s) = h

n−1 ∑

k(tj , s)u(tj ),

(5)

(Ku)(s) − (Kh u)(s) = O(h2m ).

(6)

j=0

with the error estimate

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For the logarithmic weak singular operator H, the continuous approximation of its kernel hn (t, τ ) is defined as:  |t − s| ≥ h,  h(t, s), (h ) hn (t, s) = (7)  ln |x′ (s)| , |t − s| < h, 2π so its approximation operator can be obtained when set g˜(x) ≡ 0 and s = 0 in case (c) of Lemma 1: n−1 ∑ (Hh u)(s) = h hn (tj , s)u(tj ), (8) j=0

which has the following error estimate: (Hu)(s) − (Hh u)(s) = 2h3

m−1 ∑ ς ′ (−2µ) ς ′ (−2) (2) u +2 u(2µ) (s)h2µ+1 + O(h2m ). 2! (2µ)! µ=2

(9)

We can find that there is an asymptotic expansion with accuracy order O(h3 ) for the logarithmic singular operator. In order to improve the accuracy order from O(h3 ) to O(h5 ), a coarse grid 2h = 2π/(n/2) = 4π/n is obtained. The approximate operator based on coarse grid 2h is shown as: n−1 ∑ (H2h u)(s) = 2h hn (tj , s)u(tj )ϑj , j=0

{

where ϑj =

0, 1,

j is an odd number, j is an even number.

The error estimate is: (Hu)(s) − (H2h u)(s) = 2(2h)3 +2

m−1 ∑ µ=2

ς ′ (−2) (2) u 2!

ς ′ (−2µ) (2µ) u (s)(2h)2µ+1 + O((2h)2m ). (2µ)!

(10)

An extrapolation algorithm is used to counteract the item O(h3 ) in Eqs (9) and (10): (Jh u)(s) =

8 1 (Hh u)(s) − (H2h u)(s). 7 7

The error for the approximate operator will be improved from O(h3 ) to O(h5 ): (Hu)(s) − (Jh u)(s) =

m−1 ∑

ηµ h2µ+1 + O(h2m ),

(11)

µ=2

where ηµ is some coefficients combination of the item h2µ+1 . So the accuracy order is not only improved to O(h5 ), but also built on the fine grid h. Thus we obtain the numerical approximate equations of Eq.(4): (I − Kh )uh − cJh uh = Jh fh ,

(12)

where Kh and Jh are discrete matrices of order n corresponding to the operators K and H, respectively.

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3

Asymptotical compact convergence

According to the logarithmic capacity theory[3] , the eigenvalues of K and Kh do not include 1. Then the Eqs. (4) and (12) can be rewritten as follows: find u ∈ C[0, 2π] satisfying (I − L)u = φ, and find uh satisfying −1

where L = c(I −K)

(13)

(I − Lh )uh = φh , −1

H, Lh = c(I −Kh )

(14) −1

Jh , φ = (I −K)

−1

Hf and φh = (I −Kh )

Jh fh .

Theorem 1. The approximate operator sequence {Lh } is an asymptotical compact[21,22] sequence and convergent to L in C[0, 2π], i,e. a.c

Lh → L,

(15)

a.c

where → means the asymptotically compact convergence. This proof can be obtained similarly as the proofs in the papers[15,16]. Corollary[13,15] 1. Under the assumption of Theorem 1, we obtain { ∥(Lh − L)L∥ → 0 ∥(Lh − L)Lh ∥ → 0, as h → 0.

4

Asymptotic expansions of the approximate solutions

Theorem 2. Suppose u(s) ∈ C (2m) [0, 2π], then we have the following asymptotic expansion (Lh − L)u(s) =

m−1 ∑

ψj (s)h2j+1 + O(h2m ),

(16)

j=2

where ψj (s) ∈ C (2m−2j) , j = 2, . . . , m − 1, are functions independent of h. Proof. According to properties of the approximate operators, there is (Ku)(s) − (Kh u)(s) = O(h2m ). and (Hu)(s) − (Jh u)(s) =

m−1 ∑

(17)

ηj (s)h2j+1 + O(h2m ).

(18)

j=2

We consider the relationship between Lh and L: (Lh − L)u = c(I − Kh )−1 Jh u − c(I − K)−1 Ju = c(I − Kh )−1 Jh u − (I − K)−1 Jh u + c(I − K)−1 Jh u − (I − K)−1 Ju = c[(I − Kh )−1 − (I − K)−1 ]Jh u + c(I − K)−1 (Jh − H)u = c(I − K)−1 (Kh − K)(I − Kh )−1 Jh u + c(I − K)−1 (Jh − H)u Substituting the errors of Eqs. (19) and (20) into the above equation, and setting ψj (s) = c(I − K)−1 ηj (s), we complete the proof of Theorem 2. 

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Theorem 3. Suppose x(t), g(t) ∈ C 2m [0, 2π], Then there exists functions ω ¯ l ∈ C 2m−2l [0, 2π], l = 1, . . . , m independent of h, such that m−1 ∑

(u − uh )|t=tj =

h2l+1 ω ¯ l |t=tj + O(h2l )

(19)

l=2 a.c

Proof. Because (I − K)−1 is exist, and Jh → H, so there is an asymptotic expansion for function φ: (φ − φh )|t=tj = h5 ω2 |t=tj + h7 ω3 |t=tj + . . . + O(h2m ), (20) where ωl ∈ C 2m−2l [0, 2π], l = 2, . . . , m − 1. Because u and uh satisfy Eqs. (13) and (14) respectively, we obtain (I − Lh )(uh − u)|t=tj [ ] = (I − Lh )uh − (I − L)u + (I − L)u − (I − Lh )u

t=tj

(21)

= (φh − φ)|t=tj + (L − Lh )u|t=tj = h5 ϕ2 |t=tj + . . . + O(h2m ), where ϕl = ωl + ψl , l = 2, . . . , m − 1. Define an auxiliary equation (I − L)¯ ω l = ϕl ,

l = 2, . . . , m − 1,

(22)

and its approximate equation (I − Lh )¯ ωlh = ϕlh ,

l = 2, . . . , m − 1.

(23)

Substituting Eq. (25) into Eq. (23), we obtain (I − Lh )(uh − u −

m−1 ∑

h2l+1 ω ¯ lh )|t=tj = O(h2m ).

(24)

l=2

Noticing ω ¯ lh ∈ C 2m−2l [0, 2π], we have (¯ ωl − ω ¯ lh )(ti ) = O(h2m−2l ).

(25)

When substitute ω ¯ lh by ω ¯ l and consider the asymptotic compact properties[21] , we obtain (

uh − u −

m−1 ∑

) h2l+1 ω ¯ l |t=tj = O(h2m ),

(26)

l=2

so the proof is completed.  The asymptotic expansion in Eq. (21) implies that the Richardson extrapolation[23] can be applied to improve the accuracy order. A higher accuracy order O(h7 ) can be obtained by computing some approximation on Γ in parallel. It can be described as follows: Taking h and h/2 to solve Eq. (12) in parallel, we obtain that uh (ti ), uh/2 (ti ) are the solutions on Γ. According to the asymptotic expansion, we obtain u∗h (ti ) =

1 (32uh/2 (ti ) − uh (ti )), 31

(27)

and the error is |u∗h (ti ) − u(ti )| = O(h7 ).

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Moreover, using |u∗h (ti ) − u(ti )| = O(h7 ), we obtain a posteriori error estimate |u(ti ) − uh/2 (ti )| ≤ |u(ti ) − + ≤ Note that the upper limitation algorithms.

5

1 (32uh/2 (ti ) − uh (ti ))| 32

1 |uh/2 (ti ) − uh (ti )| 31

1 |uh/2 (ti ) − uh (ti )| + O(h7 ). 31

1 31 |uh/2 (ti )

− uh (ti )| can be used to construct self-adaptive

Numerical examples

In this section, we consider some computational aspects of the approximate equation and present two examples to illustrate the accelerated convergence of the extrapolation algorithms. Example 1[24] : Consider the boundary value problem satisfying  u = 0, in Ω,  △˜ ˜  ∂u = −˜ u(x) + f˜(x), on Γ, ∂n

(28)

where f˜(x) = 1 and Ω is the region ( x )2 1

a

+

( x )2 2

b

< 1,

(29)

with (a, b) = (1, 2). The boundary Γ can be described as: x1 = cos t, x2 = 2 sin t, 0 ≤ t ≤ 2π. So the analyzed solution will be obtained as u(x) ≡ 1. This problem is calculated in paper [24] by Nystr¨om method. The results is listed in Table 1 and it shows that the convergent rate is three order. The denotes in Table 1 represent the following means: ti = 2iπ/10, with i = 1, . . . , 10; ei is the errors at ti ; and i (h) rate = log2 eie(h/2) . Table 1: Errors of the Nystr¨om solutions in paper [24]. ti 0.628319 1.256637 1.884956 2.513274 3.141593 3.769911 4.398230 5.026548 5.654867 6.283185

ei with h = 2π 10 0.121881E-02 -0.241312E-02 -0.241325E-02 0.121870E-02 0.163276E-02 0.121862E-02 -0.241311E-02 -0.241343E-02 0.121874E-02 0.163256E-02

ei with h = 2π 20 0.131313E-03 -0.350908E-03 -0.350658E-03 0.131397E-03 0.189617E-03 0.132229E-03 -0.351662E-03 -0.351146E-03 0.131003E-03 0.189412E-03

rate 3.21 2.78 2.78 3.21 3.11 3.20 2.78 2.78 3.22 3.11

ei with h = 2π 40 0.162984E-04 -0.439971E-04 -0.442431E-04 0.162478E-04 0.236295E-04 0.171674E-04 -0.435820E-04 -0.437735E-04 0.163326E-04 0.236443E-04

rate 3.01 3.00 2.99 3.02 3.00 2.95 3.01 3.00 3.00 3.00

We calculate the boundary numerical solutions uh on Γ following Eq. (12). The boundary is divided into 5 ∗ 2n with n = 0, 1, 2, . . . pieces For convenience, we introduce some denotes:

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Figure 1: Boomerang-shaped domain for numerical example 2. eh (P ) = |uh (P ) − u(P )| is the error of the displacement; rh (P ) = log2 eh (P )/eh/2 (P ) is the error ratio; e¯h (P ) = |u∗h (P ) − u(P )| is the error after Richardson extrapolation, and 1 ph (P ) = 31 |uh/2 (P ) − uh (P )| is a posteriori error estimate. Table 2 lists the approximate values of uh (P ) at points P1 = (a cos 0, b sin 0), P2 = (a cos(π/5), b sin(π/5)) and P3 = (a cos(2π/5), b sin(2π/5)). Table 2: The errors, errors ratio of eh , rh and a posteriori error estimate ph , at points P = P1 , P2 , P3 . n eh (P1 ) rh (P1 ) ph (P1 ) eh (P2 ) rh (P2 ) ph (P2 ) eh (P3 ) rh (P3 ) ph (P3 )

5 2.043E-04

7.203E-04

4.726E-04

10 6.117E-06 5.062 6.133E-06 2.126E-05 5.102 2.226E-05 1.378E-05 5.100 1.389E-05

20 1.848E-07 5.049 1.852E-07 6.419E-07 5.050 6.428E-07 4.096E-07 5.073 4.106E-07

40 5.634E-09 5.036 5.634E-09 1.959E-08 5.034 1.966E-08 1.256E-08 5.028 1.257E-08

80 1.728E-10 5.027 1.727E-10 6.061Ee-10 5.014 6.061Ee-10 3.886E-10 5.014 3.886E-10

From Table 2, we can numerically see rh ≈ 5, that means the convergent rate is almost five order, which agrees with Theorem 3 very well. Table 3. the errors eh (θ), e˜h (θ) and errors ratio rh (θ) when θ1 = 0, θ2 = π/5 on Γ. n eh (θ1 ) rh (θ1 ) ph (θ1 ) eh (θ2 ) rh (θ2 ) ph (θ2 )

5 6.138E-4

5.413E-4

10 1.779E-5 5.109 1.791E-5 1.574E-5 5.104 1.632E-5

20 5.264E-7 5.079 5.288E-7 4.690E-7 5.068 4.728E-7

40 1.585E-8 5.053 1.585E-8 1.413E-8 5.052 1.403E-8

80 4.863E-10 5.027 4.854E-10 4.351E-10 5.022 4.350E-10

Example 2[15] : Consider another boundary value problem with a non-convex boomerangshaped cross section boundary. Similar problem is discussed for Helohmotz equation with nonlinear boundary condition in the same domain in paper [15]. The boundary Γ is illustrated in Fig.1 and described by the parametric representation: x(t) = (x1 (t), x2 (t)) = (cos t + 0.65 cos 2t + 0.65, 1.5 sin t),

146

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CHENG ET AL. FIVE ORDER EXTRAPOLATION ALGORITHMS √ We set c = 2 and f = (1.5 cos t+sin t+1.3 sin 2t)/ w +2(cos(t)+0.65 cos(2t)+1.5 sin(t)) with w = (1.5 cos t)2 +(sin t+1.3 sin 2t)2 . Then the analytic solution is u(t) = x1 (t)+x2 (t) = cos t + 0.65 cos 2t + 0.65 + 1.5 sin t, t ∈ [0, 2π]. In Table 3 we list some errors of the uh (y) on Γ computed by formulae (14) and then the uh at arbitrary point in Ω can be obtained following Eq.(15). We also use the denotes as used in Table 1. Evidently, from Table 3, a similar conclusion can be obtained as example 1 done.

Conclusion Generally, there are three main advantages for the Sidi’s quadrature method: (1) Evaluating each element of discretization matrices is very simple and straightforward, which does not require any singular integrals; (2)We can obtain a high accuracy order O(h5 ) and an asymptotic expansion of the errors with odd powers, which are based on fine grid h. Harnessing the Richardson extrapolation algorithms, a higher accuracy order O(h7 ) can be obtained. (3)The accuracy order O(h5 ) of the approximate solution is obtained directly, which avoid the errors derived from the extrapolation algorithms as some articles have done.

References [1] C.A. Brebbia, J. Telles, L. Wrobel, Boundary element techniques : theory and applications in engineering. Springer-Verlag. New York, 1984. [2] Pan Cheng, Jin Huang and Guang Zeng, High accuracy eigensolution and its extrapolation for potential equations Appl. Math. Mech., 31(2010), 1527-1536. [3] P.Cheng, J.Huang, G.Zeng, Splitting extrapolation algorithms for solving the boundary integral equations of Steklov problems on polygons by mechanical quadrature methods, Eng. Anal. Bound. Elem., 35(2011), 1136-1141. [4] E.P. Stephan, W.L. Wendland, An augment Galerkin procedure for the boundary integral method applied to two-dimensional screen and crack problems. Appl. Anal., 18(1984),183-219. [5] G. Chandler, Galerkin’s method for boundary integral equations on polygonal domains. J. Aust. Math. Soc. Sor. B.V, 26(1984),1-13. [6] I.H. Sloan, A. Spence, The Galerkin method for integral equations of first-kind with logarithmic kernel: theory. IMA J. Numer. Anal., 8(1988), 105-122. [7] S. Amini, S.P. Nixon, Preconditioned multiwavelet Galerkin boundary element solution of Laplace’s equation, Eng. Anal. Bound. Elem., 7(2006), 523-530. [8] J. Elschner, I. Graham, An optimal order collocation method for first kind boundary integral equations on polygons. Numer. Math., 70(1995), 1-31. [9] Y. Yan, The collocation method for first-kind boundary integral equations on polygonal regions. Math. Comput., 54(1990), 139-154. [10] A. Sidi, M. Israeli, Quadrature methods for periodic singular Fredholm integral equations, J. Sci. Comput., 3(1988), 201-231. [11] J. Saranen, The modified quadrature method for logarithmic kernel integral equations on closed curves. J. Int. Eqns. Appl., 3(1991), 575-600.

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[12] P.Cheng, J. Huang, Extrapolation algorithms for solving n onlinear boundary integral equations by mechanical quadrature methods, Numerical Algorithm 58(2011), 545-554. [13] P.Cheng, X. Luo, Z. Wang, J. Huang, Mechanical quadrature methods and extrapolation algorithms for boundary integral equations with linear boundary conditions in elasticity. J. of elasticity, 108(2012), 193-207. [14] J. Huang, Z.C. Li , I.L. Chen, A.H.D. Cheng, Advanced quadrature methods and splitting extrapolation algorithms for first kind boundary integral equations of Laplace’s equation with discontinuity solutions. Eng. Anal. Bound. Elem., 34(2010), 1003-1008. [15] T. L¨ u, J. Huang, High accuracy Nystr¨om approximations and their extrapolation for solving weakly singular integral equations of the second kind, J. Chinese Comp. Phy., 3(1997), 349-355. [16] Z.C. Li, Combinations of method of fundamental solutions for Laplace’s equation with singularities. Eng. Anal. Bound. Elem., 10(2008), 856-869. [17] P.Cheng, J.Huang, Z.Wang, Nystrom methods and extrapolation for solving Steklov eigensolutions and its application in elasticity Numer. Meth. Part. Diff. Equ., 28(2012), 2021-2040. [18] P.Cheng, J.Huang, G.Zeng, High Accuracy Eigensolutions in Elasticity for Boundary Integral Equations by Nystrom Method, Inte. J. Math. Comp. Sci., 7(2011)166-170. [19] J.Huang, Z.Wang, Extrapolation algorithm for solving mixed boundary integral equations of the Helmholtz equation by mechanical quadrature methods. SIAM J. Sci. Comp., Vol. 31(6)(2009), 4115-4129. [20] J. Huang, T. L¨ u, Z.C. Li, The mechanical quadrature methods and their splitting extrapolations for boundary integral equations of first kind on open arcs. Appl. Numer. Math., 59(2009),2908-2922. [21] A.Sidi, M.Israeli, Quadrature methods for periodic singular and weakly singular Fredholm integral equations, J. Sci. Comp., 3(1988), 201-231. [22] A.Sidi. Practical Extrapolation Methods: Theory and Applications, Cambridge University Press, 2003. [23] P.M.Anselone, Collectively compact operator approximation theory, Prentice-Hall, Englewood Cliffs, New Jersey, 1971. [24] P.M.Anselone, Singularity subtraction in the numerical solution of integral equations, J. Austral. Math. Soc. (Series B), 22(1981), 408-418. [25] C.B.Lin, T.L¨ u, T.M.Shih, The splitting extrapolation method, World Scientific, Singapore, 1995. [26] Y.S.Xu, Y.H.Zhao, An extrapolation method for a class of boundary integral equations, SIAM J. Math. Comp., 65(1996), 587-610.

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Sufficient conditions for univalence obtained by using first order nonlinear strong differential subordinations Georgia Irina Oros Department of Mathematics, University of Oradea Str. Universita˘¸tii, No.1, 410087 Oradea, Romania E-mail: [email protected] Abstract The concept of differential subordination was introduced in [3] by S.S. Miller and P.T. Mocanu and the concept of strong differential subordination was introduced in [1] by J.A. Antonino and S. Romaguera. This last concept was applied in the special case of Briot-Bouquet strong differential subordination. In [5] the authors have developed the general theory of strong differential subordinations following the general theory introduced in [3]. In [6], the special case of first order linear strong differential subordinations was studied. Now, we study another special case, the first order nonlinear strong differential subordinations.

Keywords: analytic function, differential subordination, strong subordination, first order linear, first order nonlinear. 2000 Mathematical Subject Classification: 30C45, 34A30.

1

Introduction

Let H = H(U ) denote the class of functions analytic in U . For n a positive integer and a ∈ C, let H[a, n] = {f ∈ H : f (z) = a + an z n + an+1 z n+1 + . . . , z ∈ U }. Let A be the class of functions f of the form f (z) = z + a2 z 2 + a3 z 3 + . . . , z ∈ U. In adition, we need the classes of convex, alpha-convex, close-to-convex and starlike (univalent) functions 0 0 (z) + given respectively by K = {f ∈ A : Re zf 00 (z)/f 0 (z) + 1 > 0}, Mα = {f ∈ A : f (z)fz (z) 6= 0, Re (1 − α) zff (z) ´ ³ zf 00 (z) α 1 + f 0 (z) > 0, z ∈ U }, C = {f ∈ A : Re f 0 (z) > 0, z ∈ U }, and S ∗ = {f ∈ A : Re zf 0 (z)/f (z) > 0}. Definition 1.1 [1], [2], [3] Let H(z, ξ) be analytic in U ×U and f (z) analytic and univalent in U . The function H(z, ξ) is strongly subordinate to f (z), written H(z, ξ) ≺≺ f (z) if for each ξ ∈ U , H(z, ξ) is subordinate to f (z). Remark 1.1 (i) Since f (z) is analytic and univalent Definition 1.1 is equivalent to H(0, ξ) = f (0) and H(U × U ) ⊂ f (U ). (ii) If H(z, ξ) ≡ H(z) then the strong subordination becomes the usual notion of subordination. Definition 1.2 [4], [5, Definition 2.2.b, p. 21] We denote ¾by Q the set of functions q that are analytic and ½ injective in U \ E(q), where E(q) = ζ ∈ ∂U ; lim q(z) = ∞ and are such that q 0 (ζ) 6= 0 for ζ ∈ ∂U \ E(q). z→ζ

The subclass of Q for which f (0) = a is denoted by Q(a).

Definition 1.3 [5, Definition 4] Let Ω be a set in C, q ∈ Q and n a positive integer. The class of admissible functions ψ n [Ω, q] consists of those functions ψ : C2 × U × U → C that satisfy the admissibility condition: ψ(r, s; z, ξ) 6∈ Ω

(A)

whenever r = q(ζ), s = mζq 0 (ζ), z ∈ U , ξ ∈ U , ζ ∈ ∂U \ E(f ) and m ≥ n ≥ 1.

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Remark 1.2 The function q(z) = M Mz+a M+az , with M > 0 and |a| < M , satisfies ∆ = q(U ) = UM = U (0, M ), q(0) = a, E(q) = ∅ and q ∈ Q. If a = 0, then (A) simplifies to ψ(M eiθ , Keiθ ; z, ξ) 6∈ Ω

(A’)

whenever K ≥ nM , z ∈ U , ξ ∈ U and θ ∈ R. Remark 1.3 The function q(z) = If a = 1, then (A) simplifies to when ρ, σ ∈ R, σ ≤

− n2 (1

a+az 1−z

with Re a > 0, satisfies q(U ) = ∆, q(0) = a, E(q) = {1} and q ∈ Q. ψ(ρi, σ, z, ξ) 6∈ Ω,

2

(A”)

+ ρ ), z ∈ U , ξ ∈ U and n ≥ 1.

Lemma 1.1 [3], [4, Lemma 2.2.d, p. 24] Let q ∈ Q(a), with q(0) = a and p(z) = a + an z n + . . . analytic in U , with p(z) 6≡ a, n ≥ 1. If p is not subordinate to q, then there exist points z0 = r0 eiθ0 ∈ U and ζ 0 ∈ ∂U \ E(q), and an m ≥ n ≥ 1 for which p(Ur0 ) ⊂ q(U ) (i) p(z0 ) = q(ζ 0 ) (ii) z0 p0 (z0 ) = mζ 0 q 0 (ζ 0 ). Definition 1.4 [6] A strong differential subordination of the form A(z, ξ)zp0 (z) + B(z, ξ)p(z) ≺≺ h(z), z ∈ U, ξ ∈ U , where A(z, ξ)zp0 (z) + B(z, ξ)p(z) is analytic in U for all ξ ∈ U and h is an analytic and univalent function in U, is called first order linear strong differential subordination.

2

Main results

Definition 2.1 A strong differential subordination of the form A(z, ξ)zp0 (z) + B(z, ξ)p(z) + C(z, ξ)p2 (z) + D(z, ξ) ≺≺ h(z),

(1)

where A(z, ξ)zp0 (z) + B(z, ξ)p(z) + C(z, ξ)p2 (z) + D(z, ξ) is analytic in U for all ξ ∈ U and h is an analytic and univalent function in U, is called first order nonlinear strong differential subordination. Remark 2.1 If C(z, ξ) = D(z, ξ) = 0 then (1) becomes a linear strong differential subordination studied in [6]. Remark 2.2 If A(z, ξ) = A(z), B(z, ξ) = B(z), C(z, ξ) = C(z), D(z, ξ) = D(z) then (1) becomes a nonlinear differential subordination studied in [7]. Next, we find conditions for the functions p, A, B, C, D and h such that (1) holds. Theorem 2.1 Let p ∈ H[0, n], A, B, C : U × U → C with Re A(z, ξ) ≥ 0,

Re [A(z, ξ) + B(z, ξ)] ≥ 1 + M |C(z, ξ)|

(2)

and A(z, ξ)zp0 (z) + B(z, ξ)p(z) + C(z, ξ)p2 (z) an analytic function in U for all ξ ∈ U . Then A(z, ξ)zp0 (z) + B(z, ξ)p(z) + C(z, ξ)p2 (z) ≺≺ M z

(3)

implies p(z) ≺ M z, M > 0, z ∈ U. Proof. Let ψ(r, s; z, ξ) : C2 × U × U → C given by Definition 1.3. For r = p(z), s = zp0 (z), z ∈ U we have ψ(r, s; z, ξ) = A(z, ξ)s + B(z, ξ)s + C(z, ξ)r 2 .

(4)

Then (3) becomes ψ(r, s; z, ξ) ≺≺ M z,

z ∈ U, ξ ∈ U .

(5)

If we consider h(z) = M z, M > 0 then h(U ) = U (0, M ) and (5) is equivalent to ψ(r, s; z, ξ) ∈ U (0, M ),

z ∈ U, ξ ∈ U .

(6)

Suppose that p is not subordinated to h(z) = M z. Then, from Lemma 1.1, we have that there exist z0 ∈ U , z0 = r0 eiθ0 , θ0 ∈ R and ζ 0 ∈ ∂U with |ζ 0 | = 1, such that p(z0 ) = h(ζ 0 ) = M eiθ0 , z0 p0 (z0 ) = mζ 0 h0 (ζ 0 ) = Keiθ0 , K ≥ nM . 150

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By replacing r with p(z0 ) = h(ζ 0 ) = M eiθ0 and s with z0 p0 (z0 ) = mζ 0 h0 (ζ 0 ) = Keiθ0 in (4) and taking into account the conditions from (2), we have |ψ(p(z0 ), z0 p0 (z0 ); z0 , ξ)| = |ψ(M eiθ0 , Keiθ0 ; z0 , ξ)| = |A(z0 , ξ)Keiθ0 + B(z0 , ξ)M eiθ0 + C(z0 , ξ)M 2 e2iθ0 | = |A(z0 , ξ)K+B(z0 , ξ)M +C(z0 , ξ)M 2 e2iθ0 | ≥ |A(z0 , ξ)K+B(z0 , ξ)M |−M 2 |C(z0 , ξ)| ≥ Re [A(z0 , ξ)K+B(z0 , ξ)M ]− M 2 |C(z0 , ξ)| ≥ KRe A(z0 , ξ) + M Re B(z0 , ξ) − M 2 |C(z0 , ξ)| ≥ nM Re A(z0 , ξ) + M Re B(z0 , ξ) − M 2 |C(z0 , ξ)| ≥ M Re [A(z0 , ξ) + B(z0 , ξ)] − M 2 |C(z0 , ξ)| ≥ M, which contradicts (6). This means the assumption made is false, hence p(z) ≺ M z, M > 0, z ∈ U . √ Example 2.1 Let A(z, ξ) = z + ξ + 4, B(z, ξ) = 3z − 2ξ + 12 − 8i, C(z, ξ) = 2z − 3ξ + 1 − 3i, M = 12 . Since z ∈ U , ξ ∈ U , we have Re A(z, ξ) ≥ 2, Re B(z, ξ)| ≥ 7, |C(z, ξ)| ≤ 16, Re [A(z, ξ) + B(z, ξ)] ≥ 9. From√Theorem 2.1, we obtain: If p ∈ [0, n], n ∈ N, and (z + ξ + 4)zp0 (z) + (3z − 2ξ + 12 − 8i)p(z) + (2z − (z + ξ + 4)zp0 (z) + (3z − 2ξ + 12 − 8i)p(z) 2ξ + 1 − 3i)p2 (z) √ is a2functionzof z, analytic in U for all ξ ∈ U , then z +(2z − 3ξ + i − 3i)p (z) ≺≺ 2 , z ∈ U, ξ ∈ U , implies p(z) ≺ 2 , z ∈ U. Theorem 2.2 Let p ∈ [0, n], A, B, C, D : U × U → C with Re A(z, ξ) ≥ 0,

Re C(z, ξ) ≥ 0,

n Re A(z, ξ) ≥ Re D(z, ξ) 2

(7)

q£ ¤ £n ¤ n and Im B(z, ξ) ≤ 2 2 Re A(z, ξ) + Re C(z, ξ) 2 Re A(z, ξ) − Re D(z, ξ) .

If A(z, ξ)zp0 (z)+B(z, ξ)p(z)+C(z, ξ)p2 (z)+D(z, ξ) is analytic in U for all ξ ∈ U and satisfies the inequality Re [A(z, ξ)zp0 (z) + B(z, ξ)p(z) + C(z, ξ)p2 (z) + D(z, ξ)] > 0

(8)

then Re p(z) > 0, z ∈ U. Proof. Let ψ(r, s; z, ξ) : C2 × U × U → C given by Definition 1.3. For r = p(z), s = zp0 (z), z ∈ U we have ψ(r, s; z, ξ) = A(z, ξ)s + B(z, ξ)r + C(z, ξ)r2 + D(z, ξ),

z ∈ U, ξ ∈ U .

(9)

Then (8) becomes Re ψ(r, s; z, ξ) > 0, If we consider h(z) =

1+z 1−z

z ∈ U, ξ ∈ U .

(10)

then h(U ) = {w ∈ C; Re w > 0} and (10) is equivalent to ψ(r, s; z, ξ) ≺≺

1+z , 1−z

z ∈ U, ξ ∈ U .

(11)

1+z . Then, from Lemma 1.1, we have that there exist z0 = r0 eiθ0 , Suppose that p is not subordinated to h(z) = 1−z θ0 ∈ R and ζ 0 ∈ ∂U such that p(z0 ) = h(ζ 0 ) = ρi, ρ ∈ R, z0 p0 (z0 ) = mζ 0 h0 (ζ 0 ) = σ, σ ∈ R, σ ≤ − n2 (1 + ρ2 ). By replacing r with ρi and s with σ in (9) and using the conditions given by (7) we obtain Re ψ(p(z0 ), z0 p0 (z0 ); z0 , ξ) = Re ψ(ρi, σ; z0 , ξ) = Re [A(z0 , ξ)σ+B(z0 , ξ)ρi−ρ2 C(z0 , ξ)+D(z0 , ξ)] = σRe A(z0 , ξ)− n 2 2 2 ρIm B(z £ 0 , ξ)−ρ Re C(z0 , ξ)+Re D(z ¤ 0 , ξ) ≥ − 2 (1+ρ n)Re A(z0 , ξ)−ρIm B(z0 , ξ)−ρ Re C(z0 , ξ)+Re D(z0 , ξ) 2 n ≥ −ρ 2 Re A(z0 , ξ) + Re C(z0 , ξ) − ρIm B(z0 , ξ) − 2 Re A(z0 , ξ) + Re D(z0 , ξ) ≤ 0, 1+z , z ∈ U , which is equivalent which contradicts (10). This means the assumption made is false, hence p(z) ≺ 1−z to Re p(z) > 0, z ∈ U .

Theorem 2.3 Let p ∈ H[1, n], A, B, C, D : U × U → C with Re A(z, ξ) ≥ 0,

Re C(z, ξ) ≥ 0,

n Re A(z, ξ) ≥ Re D(z, ξ) + 1 2

(12)

q£ ¤£ n ¤ n and Im B(z, ξ) ≤ 2 2 Re A(z, ξ)+Re C(z, ξ) 2 Re A(z, ξ)−Re D(z, ξ)−1 .

If A(z, ξ)zp0 (z)+B(z, ξ)p(z)+C(z, ξ)p2 (z)+D(z, ξ) is analytic in U for all ξ ∈ U and satisfies the nonlinear strong differential subordination A(z, ξ)zp0 (z) + B(z, ξ)p(z) + C(z, ξ)p2 (z) + D(z, ξ) ≺≺ z then p(z) ≺

1+z 1−z ,

(13)

z ∈ U. 151

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Proof. Let ψ given by (9). For r = p(z), s = zp0 (z), (13) becomes ψ(r, s; z, ξ) ≺≺ z,

z ∈ U, ξ ∈ U .

(14)

If we consider h(z) = z, z ∈ U , then q(U ) = U and from (14) we have ψ(r, s; z, ξ) ∈ U,

z ∈ U, ξ ∈ U ,

(15)

which is equivalent to −1 < Re ψ(r, s; z, ξ) < 1,

z ∈ U, ξ ∈ U .

(16)

1+z Suppose that p is not subordinated to q(z) = 1−z . Then, from Lemma 1.1 we have that there exist iθ0 z0 = r0 e , θ0 ∈ R and ζ 0 ∈ ∂U , such that p(z0 ) = q(ζ 0 ) = ρi, z0 p0 (z0 ) = mζ 0 q 0 (ζ 0 ) = σ, σ ∈ R, σ ≤ − n2 (1+ρ2 ). By replacing r with ρi and s with σ in (9) and using the conditions given by (12), we have: Re ψ(r, s; z0 , ξ) = Re ψ(ρi, σ; z0 , ξ) = Re [A(z0 , ξ)σ + B(z0 , ξ)ρi − C(z0 , ξ)ρ2 + D(z0 , ξ)] = σRe A(z0 , ξ) − n 2 2 2 ρIm B(z £ 0 , ξ)−ρ Re C(z0 , ξ)+Re D(z ¤ 0 , ξ) ≤ − 2 (1+ρ n)Re A(z0 , ξ)−ρIm B(z0 , ξ)−ρ Re C(z0 , ξ)+Re D(z0 , ξ) 2 n ≤ −ρ 2 Re A(z0 , ξ) + Re C(z0 , ξ) − ρIm B(z0 , ξ) − 2 Re A(z0 , ξ) + Re D(z0 , ξ) ≤ −1, which contradicts (15). That means the assumption made was false, hence p(z) ≺ 1+z 1−z , z ∈ U .

References [1] José A. Antonino and Salvador Romaguera, Strong differential subordination to Briot-Bouquet differential equations, Journal of Differential Equations 114(1994), 101-105. [2] José A. Antonino, Strong differential subordination and applications to univalency conditions, J. Korean Math. Soc., 43(2006), no.2, 311-322. [3] S.S. Miller and P.T. Mocanu, Differential subordinations and univalent functions, Michigan Math. J., 28(1981), no.2, 157-172. [4] S.S. Miller and P.T. Mocanu, Differential subordinations. Theory and applications, Pure and Applied Mathematics, Marcel Dekker, Inc., New York, 2000. [5] Georgia Irina Oros and Gheorghe Oros, Strong differential subordination, Turkish Journal of Mathematics, 33 (2009), 249-257. [6] Georgia Irina Oros, First order strong differential superordination, General Mathematics, vol. 15, No. 2-3 (2007), 77-87. [7] Georgia Irina Oros and Gheorghe Oros, On a first order nonlinear differential subordinations I, Analele Universita˘¸tii Oradea, Fasc. Matematica˘, Tom. IX, 2003, 65-70.

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A note on the symmetric properties for the second kind twisted q-Euler polynomials C. S. Ryoo Department of Mathematics, Hannam University, Daejeon 306-791, Korea

Abstract : In this paper, we introduce the second kind twisted q-Euler numbers and polynomials. By using these numbers and polynomials, we give some interesting relations between the power sums and the the second kind twisted Euler polynomials. Key words : the second kind Euler numbers and polynomials, the second kind twisted Euler numbers and polynomials, the second kind twisted q-Euler numbers and polynomials, alternating sums 1. Introduction Euler numbers, Euler polynomials, q-Euler numbers, q-Euler polynomials, the second kind Euler number and the second kind Euler polynomials were studied by many authors. Euler numbers and polynomials posses many interesting properties and arising in many areas of mathematics and physics(see for details [1-9]). In this paper, we introduce the second kind twisted q-Euler numbers and polynomials. In this paper, by using the symmetry of p-adic q-integral on Zp , we give recurrence identities the second twisted q-Euler polynomials and the power sums. Throughout this paper, we always make use of the following notations: N = {1, 2, 3, · · · } denotes the set of natural 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 . 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 g(x) − g(y) Fg (x, y) = x−y have a limit l = g  (a) as (x, y) → (a, a). For g ∈ U D(Zp ), Kim defined the fermionic p-adic integral on Zp (see [1])   g(x)dμ−1 (x) = lim g(x)(−1)x . (1.1) I−1 (g) = lim Iq (g) = q→−1

N →∞

Zp

0≤x 0. Therefore, we get

z (f ? g)0 (z) 1 < eiλ (p − α) cos λ (f ? g) (z)

 

or



< eiλ



z (f ? g)0 (z) (f ? g) (z)

 − α cos λ =
0 p(z)

(15)

 > α cos λ.

(16)

and this implies that f (z) ∈ Qλ (p, n, α; g (z)) . By taking g (z) is an identity function and Koebe p-valent functions with λ = 0 in Theorem 2.1, we obtain Corollary 2.2 and Corollary 2.3 respectively proved by Goyal et.al [1]. Corollary 2.2. If f (z) ∈ A(p, n) satisfies

   1  1−p 1−α f 0 (z) n+1 f (z) p−α p−α p−α − p + α < p z − αz (p − α) (z ∈ U), z f (z) (n + 1)2 + 1 for 0 ≤ α < p, Corollary 2.3. If f (z) ∈ A(p, n) satisfies

then

f (z)

∈

(17) Sp∗ (n, α) .

 1  p−α α+1−p  00 n+1 (f 0 (z)) 0 zf (z) + (1 − α) f (z) − p + α < p (p−α) (z ∈ U), p−1 pz (n + 1)2 + 1 (18) for 0 ≤ α < p, then f (z) ∈ Cp (n, α) . Further If we take n = 1 and p = 1 in Corollary 2.2 and Corollary 2.3, we get the following result proved by Uyanik et al [5]. Corollary 2.4. If f (z) ∈ A satisfies

  1   2 f (z) 1−α zf 0 (z) − α − 1 + α < √ (1 − α) (z ∈ U), z f (z) 5

(19)

for 0 ≤ α < 1, then f (z) ∈ S ∗ (α) .

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Corollary 2.5. If f (z) ∈ A satisfies

n α 0
1−α f 0 (z) + f (z)

1 2 zf 00 (z) − 1 < √ (z ∈ U), 1−α 5



o

(20)

for 0 ≤ α < 1, then f (z) ∈ C (α) . Remark 2.1. If we put α = 0 and p = 1 in Corollary 2.4 and Corollary 2.5, we get the result proved by Mocanu [6] and Nunokawa et al [7] respectively. Theorem 2.6. If p(z), given by (9), satisfies

00 p (z) < p n + 1

(n + 1)2 + 1

(z ∈ E) ,

(21)

then f (z) ∈ Qλ (p, n, α; g (z)). Proof. From (9) , we have p(z) ∈ A(n). Also z

Z =

0 p (z) − 1

Z p (t)dt ≤

00 iθ h (ρe ) dρ

(22)

0

0

n+1

≤

|z|

00

p

(n + 1)2 + 1

|z| ≤ p

n+1 (n + 1)2 + 1

,

(23)

where we have used (21). This proves that p(z) satisfies the condition of Lemma 2.1 and therefore p(z) ∈ S ∗ (n, 0), which leads f (z) ∈ Qλ (p, n, α; g (z)) . Theorem 2.7. If f (z) ∈ A(p, n) satisfies

   eiλ  eiλ  (p−α)  0 cos λ (f ? g) (z) (p−α) cos λ p   (f ? g) (z) − zp (f ? g) (z) z ≤

(n + 1) (p − α) cos λ 2

p

(n + 1)2 + 1

,

(24)

then f (z) ∈ Qλ (p, n, α; g (z)) . Proof. Let us define a function p(z) by z

Z



p(z) = 0

(f ? g) (t) tp

eiλ  (p−α) cos λ

Then 0



zp (z) = z

(f ? g) (z) zp

dt.

(25)

eiλ  (p−α) cos λ

.

(26)

Let g(z) = zp0 (z). Then g(z) ∈ A (n). Consider

Z 0 g (z) − 1 = p0 (z) + zp00 (z) − 1 ≤ p0 (z) − 1 + zp00 (z) =

0

|z|

Z ≤ 0

Z ≤ 0

z

p (t)dt + zp00 (z) 00

eiλ eiλ (p − α) cos λ H (z) dt + (p − α) cos λ H (z) |z| |z|

(n + 1)

p

2

(n + 1)2 + 1

(n + 1) (n + 1) dt + p |z| < p . 2 (n + 1)2 + 1 (n + 1)2 + 1

162

(27)

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with

 H (z) =

(f ? g) (z) zp

  eiλ eiλ  (p−α)  (p−α)  0 cos λ cos λ (f ? g) (z) p  − . (f ? g) (z)

z

(28)

Therefore, by using Lemma 2.1, we have g(z) = zp0 (z) ∈ S ∗ (n, 0).

(29)

This means that p (z) ∈ C (n, 0), which implies that f (z) ∈ Qλ (p, n, α; g (z)) . 3. Generalized Alexander Integral Operator For f (z), g (z) ∈ A(p, n), we consider

Z

z

G(z) = 0



(f ∗ g) (t) tp

γ dt = z +

γap+n bp+n n+1 z + ... n+1

(30)

z Clearly G(z) ∈ A(n) and when p = 1, γ = 1, g (z) = 1−z , then (30) reduces to the well-known Alexander integral operator [8]. Theorem 3.1. If γ ≥ p1 and f (z), g (z) ∈ A(p, n) satisfies

γeiλ γ ((f ∗ g) (z)) cos λ pγeiλ z cos λ +1

z (f ∗ g)0 (z)




(f ∗ g) (z)

! (n + 1) cos λ −p ≤ p , 2 (n + 1)2 + 1

(31)

then f (z) ∈ Qλ (p, n, 0; g (z)) . Proof. From (30), we get G0 (z) =



(f ∗ g) (z) zp

γ .

(32)

Differentiating (32), logarithmically, we get G00 (z) =γ G0 (z)

(f ∗ g)0 (z) p − (f ∗ g) (z) z



 .

(33)

Then by simple computation, we have,

iλ     γe 0 cos λ 00  0  eiλ −1 (f ∗ g) (z) (f ∗ g) (z) p G (z) G (z) cos λ = γ − p z (f ∗ g) (z) z (n + 1) cos λ

≤ 2

p

(n + 1)2 + 1

,

where we have used (31). Therefore

00  0  eiλ −1 (n + 1) cos λ G (z) G (z) cos λ ≤ p 2 (n + 1)2 + 1 By using Theorem 2.7 with p = 1, α = 0 and g (z) = Cλ (1, n, 0) . From (33), we can write

 < e

iλ



zG00 (z) 1+ G0 (z)

 = γ

1 p− γ

 cos λ (sinceG (z) ∈ Cλ (1, n, 0))

(36)

which shows that f (z) ∈ Qλ (p, n, 0; g (z)) , where γ ≥ p1 .

References [1] S. P. Goyal, S. K. Bansal, P. Goswami, Extension of sufficient conditions for starlikeness and convexity of order α for multivalent function, Appl. Math. Lett., 25(11)(2012), 1993-1998. ˇ [2] L. Spacek, Prispˇevek k teorii funkei prostych, Capopis Pest. Mat. Fys., 62(1933), 12-19. [3] M. S. Robertson, Univalent functions f (z) for wich zf 0 (z) is spiral-like, Mich. Math. J., 16(1969), 97-101. [4] P. T. Mocanu, Some simple criteria for starlikeness and convexity, Libertas Math., 13(1993), 27-40. [5] N. Uyanik, M. Aydogan, S. Owa, Extension of sufficient conditions for starlikeness and convexity of order α, Appl. Math. Lett., 24(9)(2011), 1393-1399. [6] P. T. Mocanu, Some starlikeness conditions for analytic functions, Rev. Roumaine Math., Pures Appl., 33(1988), 117-124. [7] M. Nunokawa, S. Owa, Y. Polattoglu, M. Caglar, E. Y. Duman, Some sufficient conditions for starlikeness and convexity, Turk. J. Math., 34(2010), 333-337. [8] J. W. Alexander, Function which map the interior of the unit circle upon simple regions, Ann. Math., 17(1915), 12-22.

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ADDITIVE FUNCTIONAL INEQUALITIES IN PARANORMED SPACES SEO-YUN YANG AND CHOONKIL PARK∗ Abstract. In this paper, we investigate the following additive functional inequalities

(

)

1

f (x) + 1 f (y) + f (z) + f (w) ≤ f x + y + z + w ,

s

s s

(

)



1

f (x) + 1 f (y) + 1 f (z) + f (w) ≤ f x + y + z + w



s s s s in paranormed spaces for a fixed integer s greater than 1. Furthermore, we prove the Hyers-Ulam stability of the above additive functional inequalities in paranormed spaces.

1. Introduction and preliminaries The concept of statistical convergence for sequences of real numbers was introduced by Fast [3] and Steinhaus [26] independently and since then several generalizations and applications of this notion have been investigated by various authors (see [5, 14, 16, 17, 25]). This notion was defined in normed spaces by Kolk [15]. We recall some basic facts concerning Fr´ echet spaces. Definition 1.1. [28] Let X be a vector space. A paranorm P : X → [0, ∞) is a function on X such that (1) P (0) = 0; (2) P (−x) = P (x) ; (3) P (x + y) ≤ P (x) + P (y) (triangle inequality) (4) If {tn } is a sequence of scalars with tn → t and {xn } ⊂ X with P (xn − x) → 0, then P (tn xn − tx) → 0 (continuity of multiplication). The pair (X, P ) is called a paranormed space if P is a paranorm on X. The paranorm is called total if, in addition, we have (5) P (x) = 0 implies x = 0. A Fr´ echet space is a total and complete paranormed space. The stability problem of functional equations originated from a question of Ulam [27] concerning the stability of group homomorphisms. Hyers [10] gave a first affirmative partial answer to the question of Ulam for Banach spaces. Hyers’ Theorem was generalized by Aoki [1] for additive mappings and by Th.M. Rassias [21] for linear mappings by considering an unbounded Cauchy difference. A generalization of the Th.M. Rassias theorem was obtained by G˘avruta [7] by replacing the unbounded Cauchy difference by a general control function in the spirit of Th.M. Rassias’ approach. 2010 Mathematics Subject Classification. Primary 35A17; 39B52; 39B72. Key words and phrases. Jordan-von Neumann functional equation, Hyers-Ulam stability, paranormed space; functional inequality. ∗ Corresponding author.

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In 1990, Th.M. Rassias [22] during the 27th International Symposium on Functional Equations asked the question whether such a theorem can also be proved for p ≥ 1. In 1991, Gajda [6] following the same approach as in Th.M. Rassias [21], gave an affirmative solution to this question for p > 1. It was shown by Gajda [6], as well as by Th.M. Rassias ˇ and Semrl [23] that one cannot prove a Th.M. Rassias’ type theorem when p = 1 (cf. the books of P. Czerwik [2], D.H. Hyers, G. Isac and Th.M. Rassias [11]). In 1982, J.M. Rassias [20] followed the innovative approach of the Th.M. Rassias’ theorem [21] in which he replaced the factor ∥x∥p + ∥y∥p by ∥x∥p · ∥y∥q for p, q ∈ R with p + q ̸= 1. G˘avruta [7] provided a further generalization of Th.M. Rassias’ Theorem. During the last two decades a number of papers and research monographs have been published on various generalizations and applications of the Hyers-Ulam stability to a number of functional equations and mappings (see [12, 13, 18]). In [8], Gil´anyi showed that if f satisfies the functional inequality ∥2f (x) + 2f (y) − f (xy −1 )∥ ≤ ∥f (xy)∥ then f satisfies the Jordan-von Neumann functional equation

(1.1)

2f (x) + 2f (y) = f (xy) + f (xy −1 ). See also [24]. Fechner [4] and Gil´anyi [9] proved the Hyers-Ulam stability of the functional inequality (1.1). Park, Cho and Han [19] proved the Hyers-Ulam stability of the following functional inequalities

( )

x + y + z

∥f (x) + f (y) + f (z)∥ ≤ 2f

, 2 ∥f (x) + f (y) + f (z)∥ ≤ ∥f (x + y + z)∥,

( )

x+y

∥f (x) + f (y) + 2f (z)∥ ≤ 2f + z

. 2 We proved the Hyers-Ulam stability of the following functional inequalities

( )

1

1 x+y

f (x) + f (y) + f (z) + f (w) ≤ f + z + w

, (1.2)

s s s

(

)



1 1 1 x+y+z

f (x) + f (y) + f (z) + f (w) ≤ f + w

(1.3)

s s s s for a fixed integer s greater than 1. In Section 2, we prove the Hyers-Ulam stability of the functional inequality (1.2) in paranormed spaces. In Section 3, we prove the Hyers-Ulam stability of the functional inequality (1.3) in paranormed spaces. Throughout this paper, assume that (X, P (·)) is a total paranormed space and that (Y, ∥ · ∥) is a Banach space. 2. Hyers-Ulam stability of the functional inequality (1.2) In this section, we prove the Hyers-Ulam stability of the functional inequality (1.2) in paranormed spaces. Proposition 2.1. Let f : X → Y be a mapping such that

( )

1

1 x+y

f (x) + f (y) + f (z) + f (w) ≤ f + z + w



s s s

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(2.1)

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FUNCTIONAL INEQUALITIES IN PARANORMED SPACES

for all x, y, z, w ∈ X. Then f is additive. Proof. Letting x = y = z = w = 0 in (2.1), we get (

)



2

2 + 2 ∥f (0)∥ =

f (0) + 2f (0)

≤ ∥f (0)∥ s s

and so f (0) = 0. Letting x = y = 0 and w = −z in (2.1), we get f (−z) = −f (z) for all z ∈ X. Letting x = −sz and y = w = 0 in (2.1), we get ( )

f (sz) = f (sz)

&

z s

f

1 = f (z) s

for all z ∈ X. Letting z = − x+y and w = 0 in (2.1), we get s f (x + y) = f (x) + f (y) for all x, y ∈ X. Thus f is additive.



Note that P (sx) ≤ sP (x) for all x ∈ X. Theorem 2.2. Let r be a positive real number with r < 1, and f : X → Y be an odd mapping such that

( )

1

1 x+y

f (x) + f (y) + f (z) + f (w) ≤ f + z + w

(2.2)

s s s + P (x)r + P (y)r + P (z)r + P (w)r for all x, y, z, w ∈ X. Then there exists a unique additive mapping h : X → Y such that ( r ) s +1 ∥f (x) − h(x)∥ ≤ s P (x)r (2.3) s − sr for all x ∈ X. Proof. Letting y = w = 0 and z = − xs in (2.2), we get

( ) ( ) ( )

1 x



1 x

x r r

f (x) − f = f (x) + f − ≤ P (x) + P −



s

and so

s

s

s

s



1

f (sx) − f (x) ≤ P (sx)r + P (−x)r ≤ (sr + 1)P (x)r

s for all x ∈ X. Hence



1

1 m

f (sl x) − f (s x)

l

≤ m

s

s

m−1 ∑

1

f (sj x) −

j

j=l

s

≤ (sr + 1)

m−1 ∑ j=l

167

1 sj+1

srj P (x)r sj

j+1

f (s

x)

(2.4)

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for all nonnegative integers m and l with m > l and all x ∈ X. It follows from (2.4) that the sequence { s1n f (sn x)} is a Cauchy sequence for all x ∈ X. Since Y is complete, the sequence { s1n f (sn x)} converges. So one can define the mapping h : X → Y by 1 f (sn x) n s for all x ∈ X. Moreover, letting l = 0 and passing the limit m → ∞ in (2.4), we get (2.3). It follows from (2.2) that

1

1

h(x) + h(y) + h(z) + h(w)

s s

1

1 1 n n n n = lim n f (s x) + f (s y) + f (s z) + f (s y) n→∞ s s s

( ( ))

x + y 1

snr n + z + w

+ lim n (P (x)r + P (y)r + P (z)r + P (w)r ) ≤ lim n f s n→∞ s n→∞ s s)

(

x+y =

h + z + w

s for all x, y, z, w ∈ X. So

( )

1

1 x+y

h(x) + h(y) + h(z) + h(w) = h + z + w



s s s for all x, y, z, w ∈ X. By Proposition 2.1, the mapping h : X → Y is additive. Now, let T : X → Y be another additive mapping satisfying (2.3). Then we have 1 ∥h(x) − T (x)∥ = n ∥h (sn x) − T (sn x)∥ s 1 ≤ n (∥h (sn x) − f (sn x)∥ + ∥T (sn x) − f (sn x)∥) s 2s(sr + 1)snr ≤ P (x)r , (s − sr )sn h(x) := n→∞ lim

which tends to zero as n → ∞ for all x ∈ X. So we can conclude that h(x) = T (x) for all x ∈ X. This proves the uniqueness of h. Thus the mapping h : X → Y is a unique additive mapping satisfying (2.3).  3. Hyers-Ulam stability of the functional inequality (1.3) In this section, we prove the Hyers-Ulam stability of the functional inequality (1.3) in paranormed spaces. Proposition 3.1. Let f : X → Y be a mapping such that

(

)



1 1 1 x+y+z

f (x) + f (y) + f (z) + f (w) ≤ f + w



s s s s for all x, y, z, w ∈ X. Then f is additive.

(3.1)

Proof. Letting x = y = z = w = 0 in (3.1), we get

) (

3

3

+ 1 ∥f (0)∥ = f (0) + f (0)

≤ ∥f (0)∥ s s

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FUNCTIONAL INEQUALITIES IN PARANORMED SPACES

and so f (0) = 0. Letting y = z = x and w = −x in (3.1), we get f (−x) = −f (x) for all x ∈ X. Letting w = − xs and y = z = 0 in (3.1), we get (

1 f (x) = f s for all x ∈ X. Letting z = −x − y and w = 0

)

1 x s in (3.1), we get

f (x + y) = f (x) + f (y) for all x, y ∈ X. Thus f is additive.



Note that P (sx) ≤ sP (x) for all x ∈ X. Theorem 3.2. Let r be a positive real number mapping such that

1

1 1

f (x) + f (y) + f (z) + f (w) ≤

s s s +

with r < 1, and let f : X → Y be an odd

( )

x+y+z

f

+ w

s r P (x) + P (y)r + P (z)r + P (w)r

(3.2)

for all x, y, z, w ∈ X. Then there exists a unique additive mapping h : X → Y such that ) ( r s +1 P (x)r (3.3) ∥f (x) − h(x)∥ ≤ s s − sr for all x ∈ X. Proof. Letting y = x = 0 and z = − xs in (3.2), we get

( )

1 x

f (x) − f

=

s

and so

s

( ) ) (

1 x

x r r

f (x) + f −

≤ P (x) + P −

s

s

s



1

f (sx) − f (x) ≤ P (sx)r + P (−x)r ≤ (sr + 1)P (x)r

s for all x ∈ X. Hence



1

1 m

f (sl x) − f (s x)

l

≤ m

s

s

m−1 ∑

j=l



1 1 j j+1 f (s x) − f (s x)

sj sj+1

≤ (sr + 1)

m−1 ∑ j=l

srj P (x)r sj

(3.4)

for all nonnegative integers m and l with m > l and all x ∈ X. It follows from (3.4) that the sequence { s1n f (sn x)} is a Cauchy sequence for all x ∈ X. Since Y is complete, the sequence { s1n f (sn x)} converges. So one can define the mapping h : X → Y by h(x) := n→∞ lim

1 f (sn x) sn

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for all x ∈ X. Moreover, letting l = 0 and passing the limit m → ∞ in (3.4), we get (3.3). It follows from (3.2) that

1

1

h(x) + h(y) + h(z) + h(w)

s s

1

1 1 n n n n = n→∞ lim n f (s x) + f (s y) + f (s z) + f (s y) s s s

( ( ))

1

x + y snr ≤ lim n f sn + z + w

+ lim n (P (x)r + P (y)r + P (z)r + P (w)r ) n→∞ s n→∞ s s)

(

x + y =

h + z + w

s for all x, y, z, w ∈ X. So

( )

1

1 x+y

h(x) + h(y) + h(z) + h(w) = h

+ z + w



s s s for all x, y, z, w ∈ X. By Proposition 3.1, the mapping h : X → Y is additive. Now, let T : X → Y be another additive mapping satisfying (3.3). Then we have 1 ∥h(x) − T (x)∥ = n ∥h (sn x) − T (sn x)∥ s 1 ≤ n (∥h (sn x) − f (sn x)∥ + ∥T (sn x) − f (sn x)∥) s 2s(sr + 1)snr ≤ P (x)r , (s − sr )sn which tends to zero as n → ∞ for all x ∈ X. So we can conclude that h(x) = T (x) for all x ∈ X. This proves the uniqueness of h. Thus the mapping h : X → Y is a unique additive mapping satisfying (3.3).  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-2012R1A1A2004299). References [1] T. Aoki, On the stability of the linear transformation in Banach spaces, J. Math. Soc. Japan 2 (1950), 64–66. [2] P. Czerwik, Functional Equations and Inequalities in Several Variables, World Scientific Publishing Company, New Jersey, Hong Kong, Singapore and London, 2002. [3] H. Fast, Sur la convergence statistique, Colloq. Math. 2 (1951), 241–244. [4] W. Fechner, Stability of a functional inequalities associated with the Jordan-von Neumann functional equation, Aequationes Math. 71 (2006), 149–161. [5] J.A. Fridy, On statistical convergence, Analysis 5 (1985), 301–313. [6] Z. Gajda, On stability of additive mappings, Internat. J. Math. Math. Sci. 14 (1991), 431–434. [7] P. Gˇavruta, A generalization of the Hyers-Ulam-Rassias stability of approximately additive mappings, J. Math. Anal. Appl. 184 (1994), 431–436. [8] A. Gil´anyi, Eine zur Parallelogrammgleichung ¨ aquivalente Ungleichung, Aequationes Math. 62 (2001), 303–309.

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FUNCTIONAL INEQUALITIES IN PARANORMED SPACES

[9] A. Gil´anyi, On a problem by K. Nikodem, Math. Inequal. Appl. 5 (2002), 707–710. [10] D.H. Hyers, On the stability of the linear functional equation, Proc. Natl. Acad. Sci. U.S.A. 27 (1941), 222–224. [11] D.H. Hyers, G. Isac and Th.M. Rassias, Stability of Functional Equations in Several Variables, Birkh¨ auser, Basel, 1998. [12] K. Jun and Y. Lee, A generalization of the Hyers-Ulam-Rassias stability of the Pexiderized quadratic equations, J. Math. Anal. Appl. 297 (2004), 70–86. [13] S. Jung, Hyers-Ulam-Rassias Stability of Functional Equations in Mathematical Analysis, Hadronic Press lnc., Palm Harbor, Florida, 2001. [14] S. Karakus, Statistical convergence on probabilistic normed spaces, Math. Commun. 12 (2007), 11–23. [15] E. Kolk, The statistical convergence in Banach spaces, Tartu Ul. Toime. 928 (1991), 41–52. [16] M. Mursaleen, λ-statistical convergence, Math. Slovaca 50 (2000), 111–115. [17] M. Mursaleen and S.A. Mohiuddine, On lacunary statistical convergence with respect to the intuitionistic fuzzy normed space, J. Computat. Anal. Math. 233 (2009), 142–149. [18] C. Park, Homomorphisms between Poisson JC ∗ -algebras, Bull. Braz. Math. Soc. 36 (2005), 79–97. [19] C. Park, Y. Cho and M. Han, Functional inequalities associated with Jordan-von Neumann-type additive functional equations, J. Inequal. Appl. 2007 (2007), Article ID 41820, 13 pages. [20] J.M. Rassias, On approximation of approximately linear mappings by linear mappings, J. Funct. Anal. 46 (1982) 126–130. [21] Th.M. Rassias, On the stability of the linear mapping in Banach spaces, Proc. Amer. Math. Soc. 72 (1978), 297–300. [22] Th.M. Rassias, Problem 16; 2, Report of the 27th International Symp. on Functional Equations, Aequationes Math. 39 (1990), 292–293; 309. ˇ [23] Th.M. Rassias and P. Semrl, On the behaviour of mappings which do not satisfy Hyers-Ulam stability, Proc. Amer. Math. Soc. 114 (1992), 989–993. [24] J. R¨atz, On inequalities associated with the Jordan-von Neumann functional equation, Aequationes Math. 66 (2003), 191–200. ˇ at, On the statistically convergent sequences of real numbers, Math. Slovaca 30 (1980), 139– [25] T. Sal´ 150. [26] H. Steinhaus, Sur la convergence ordinaire et la convergence asymptotique, Colloq. Math. 2 (1951), 73–34. [27] S.M. Ulam, A Collection of the Mathematical Problems, Interscience Publ. New York, 1960. [28] A. Wilansky, Modern Methods in Topological Vector Space, McGraw-Hill International Book Co., New York, 1978. Seo-Yun Yang Department of Mathematics, Hanyang University, Seoul 133-791, Korea E-mail address: [email protected] Choonkil Park Department of Mathematics, Research Institute for Natural Sciences, Hanyang University, Seoul 133-791, Korea E-mail address: [email protected]

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SOME IDENTITIES FOR BERNOULLI POLYNOMIALS INVOLVING CHEBYSHEV POLYNOMIALS DAE SAN KIM, TAEKYUN KIM AND SANG-HUN LEE

Abstract. In this paper we derive some new and interesting identities for Bernoulli, Euler and Hermite polynomials associated with Chebyshev polynomials.

1. Introduction The Bernoulli number are defined by the generating function to be (1)

∞ X t Bn n Bt = e = t , et − 1 n! n=0

(see [3,13,14]),

with the usual convention about replacing B n by Bn . As is well known, the Bernoulli polynomials are given by n   X n n B (x) = (B + x) = Bn−l xl , (see [1-8]). (2) n l l=0

From (1), we note that the recurrence relation for the Bernoulli numbers is given by B0 = 1,

(B + 1)n − Bn = δ1,n ,

(see [6-8]),

where δm,n is the Kronecker symbol. By (2), we get n−1 X n − 1 dBn (x) =n Bn−1−l xl = nBn−1 (x). (3) dx l l=0

Thus, by (3), we see that Z Bn+1 (x) (4) + C, Bn (x)dx = n+1

(see [3]),

where C is a some constant. The Euler polynomials are defined by the generating function to be (5)

∞ X 2 tn xt E(x)t e = e = En (x) , t e +1 n! n=0

with the usual convention about replacing E n (x) by En (x), (see [1,2,4,10,11]). In the special case, x = 0, En (0) = En are called the n-th Euler numbers. 1

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DAE SAN KIM, TAEKYUN KIM AND SANG-HUN LEE

It is well known [6, 15] that Hermite polynomials are given by the generating function to be ∞ X tn 2xt−t2 H(x)t e = e = Hn (x) , (6) n! n=0 with the usual convention about replacing H n (x) by Hn (x). From (6),we have dHn (x) = 2nHn−1 (x), Hn (x) = (−1)n Hn (−x). dx By (1) and (2), we easily get n   X n B (x) = En−k (x), (see [1-15]), n (8) k (7)

k=0 k6=1

En (x) = −2

(9)

n   X n El+1 l=0

l

l+1

En−l (x),

and (10)

xn =

 n 
1 X n+1 1 Bl (x). Bn+1 (x + 1) − Bn+1 (x) = l n+1 n+1 l=0

The Chebyshev polynomial Tn (x) of the first kind is a polynomial in x of degree n, defined by the relation Tn (x) = cos nθ,

(11)

when x = cos θ,

(see [9]).

If the range of the variable x is the interval [−1, 1], then the range of the corresponding variable θ can be taken as [0, π]. It is known that cos nθ is a polynomial of degree n in cos θ, and indeed we are familiar with elementary formulas cos 3θ = 4 cos3 θ − 3 cos θ, cos 4θ = 8 cos4 θ − 8 cos2 θ + 1, · · · . Thus, by (11), we get T0 (x) = 1,

T1 (x) = x,

T2 (x) = 2x2 − 1,

T3 (x) = 4x3 − 3x,

T4 (x) = 8x4 − 8x2 + 1, · · · . The Chebyshev polynomial Un (x) of the second kind is a polynomial of degree n in x defined by (12)

Un (x) = sin (n + 1)θ/ sin θ,

when x = cos θ,

(see [9]).

Thus, from (12), we have U0 (x) = 1,

U1 (x) = 2x,

U2 (x) = 4x2 − 1,

U3 (x) = 8x3 − 4x, · · · .

By (11), we see that Tn (x) is a polynomial of degree n with integral coefficients and the leading coefficient 2n−1 (n ≥ 1) and 1 (n = 0). It is not difficult to show that Un (x) is a polynomial of degree n with integral coefficients and the leading coefficient 2n (n ≥ 0). Tn (x) is a solution of (1 − x2 )y 00 − xy 0 + n2 y = 0 and Un (x) is a solution of (1 − x2 )y 00 − 3xy 0 + n(n + 2)y = 0. It is well known [9] that the generating functions of Tn (x) and Un (x) are given by ∞ X 1 − xt = Tn (x)tn , (13) 1 − 2xt + t2 n=0

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IDENTITIES FOR BERNOULLI POLYNOMIALS INVOLVING CHEBYSHEV POLYNOMIALS 3

and (14)

∞ X 1 = Un (x)tn , 1 − 2xt + t2 n=0

for |x| ≤ 1, |t| < 1.

From (11) and (12), we have Z

1

(15) −1

  0, Tn (x)Tm (x) π √ , dx =  2 1 − x2 π,

if n 6= m if n = m > 0 , if n = m = 0

and Z

1

(16)

(1 − x2 )1/2 Un (x)Um (x)dx =

−1

π δn,m , 2

(see [9]).

The equations (15) and (16) are used to derive our main result in this paper. The Rodrigues’ formulae for Tn (x) and Un (x) are known as follows:  n  d (−1)n 2n n! 2 1/2 2 n−1/2 (1 − x ) (1 − x ) (17) Tn (x) = , (2n)! dxn and (18)

 n  d (−1)n 2n (n + 1)! 2 −1/2 2 n+1/2 (1 − x ) (1 − x ) . Un (x) = (2n + 1)! dxn

The equations (17) and (18) are also used to derive our result related to orthogonality of Chebyshev polynomials. From (11) and (12), we can easily derive the following equations (19) and (20): √ √ (x + x2 − 1)n + (x − x2 − 1)n (19) Tn (x) = , 2 and (20)

Un (x) =

(x +

√ √ x2 − 1)n+1 − (x − x2 − 1)n+1 √ . 2 x2 − 1

By the definitions of Tn (x) and Un (x), we easily get (21)

dTn (x) = nUn−1 (x), dx

From (21), we have Z Tn+1 (x) (22) Un (x)dx = , n+1

dUn (x) (n + 1)Tn+1 (x) − xUn (x) = . dx x2 − 1

Z Tn (x)dx =

nTn+1 (x) xTn (x) − . n2 − 1 n−1

In this paper we derive some new and interesting identities for Bernoulli, Euler and Hermite polynomials arising from the orthogonality of the Chebyshev polynomials for the inner product space with weighted inner product.

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DAE SAN KIM, TAEKYUN KIM AND SANG-HUN LEE

2. Some identities for Bernoulli, Euler and Hermite polynomials involving Chebyshev polynomials Let Pn = {p(x) ∈ Q[x] | deg p(x) ≤ n}. Then Pn is an inner product space with the weighted inner product Z 1 p(x)q(x) √ hp(x), q(x)i = dx, where p(x), q(x) ∈ Pn . 1 − x2 −1 From (15), we note that {T0 (x), T1 (x), · · · , Tn (x)} is an orthogonal basis for Pn . Let us assume p(x) ∈ Pn . Then p(x) is generated by {T0 (x), T1 (x), · · · , Tn (x)} to be n X p(x) = Ck Tk (x). (23) k=0

By (15) and (23), we get  Z Z  δk (−1)k 2k k! 1 dk δk 1 Tk (x)p(x) 2 k−1/2 √ dx = (1 − x ) p(x)dx, Ck = k π −1 π (2k)! 1 − x2 −1 dx (24)  1, if k = 0 where δk = 2, if k > 0. Let us take p(x) = xn ∈ Pn . From (24), we have  Z  (−1)k 2k k!δk 1 dk 2 k−1/2 (1 − x ) xn dx Ck = k π(2k)! −1 dx (25) Z 1 (−1)k 2k k! n! k = δk (−1) (1 − x2 )k−1/2 xn−k dx. π(2k)! (n − k)! −1 It is easy to show that (26) Z

1

(1 − x2 )k−1/2 xn−k dx =

−1

=

Z

1

(1 − y)k−1/2 y

n−k+1 −1 2

dy

0

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

By (25) and (26), we get ( (27)

(1 + (−1)n−k ) 2

Ck =

0, n!δk , n−k 2n ( n+k 2 )!( 2 )!

if n − k ≡ 1 (mod 2) if n − k ≡ 0 (mod 2).

From (27), we note that (28)

xn =

n X

Ck Tk (x) =

k=0

n! 2n−1

where n ≡ 1 (mod 2). For n ≡ 0 (mod 2), we have ( n! T0 (x) n x = n
2 + 2 (29) 2 ( n )! 2

X 1≤k≤n k≡1 (mod 2)

X 2≤k≤n k≡0 (mod 2)

175

Tk (x) , n+k ( 2 )!( n−k 2 )!

) Tk (x) . n−k ( n+k 2 )!( 2 )!

KIM ET AL 172-180

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IDENTITIES FOR BERNOULLI POLYNOMIALS INVOLVING CHEBYSHEV POLYNOMIALS 5

Let us take p(x) = Bn (x) ∈ Pn . Then  Z  (−1)k 2k k!δk 1 d
k Ck = (1 − x2 )k−1/2 Bn (x)dx π(2k)! dx −1 Z 1 k k (−1) 2 k!δk n! k (1 − x2 )k−1/2 Bn−k (x)dx = (−1) (30) π(2k)! (n − k)! −1 Z 1 n−k X n − k  2k k!δk n! = Bn−k−l (1 − x2 )k−1/2 xl dx. π(2k)! (n − k)! l −1 l=0 R1 2 k−1/2 l Now, we compute −1 (1 − x ) x dx. Z 1 Z 1 2 k−1/2 l l (1 − x ) x dx = (1 + (−1) ) (1 − x2 )k−1/2 xl dx −1 0 ( (31) 0, if l ≡ 1 (mod 2) = l!(2k)!π , if l ≡ 0 (mod 2). 22k+l ( 2k+l )!( l )!k! 2

2

By (30) and (31), we get Ck = (32)

n! (2k)!π 2k k!δk × × 2k π(2k)! (n − k)! 2 k!

n!δk = k 2 (n − k)!

X



0≤l≤n−k l≡0 (mod 2)

 n−k l! Bn−k−l l 2k+l l l 2 ( 2 )!( 2 )!

n−k Bn−k−l l! l . 2k+l l 2 ( 2 )!( 2l )!




X 0≤l≤n−k l≡0 (mod 2)

Therefore, by (32), we obtain the following theorem. Theorem 2.1. For n ∈ Z+ , we have Bn (x) = n!

X 0≤k≤n

δk 2k (n − k)!

X 0≤l≤n−k l≡0 (mod 2)

n−k Bn−k−l l! l l 2l ( 2k+l 2 )!( 2 )!




! Tk (x).

By the same method, we can derive the following identity: !
n−k X X En−k−l l! δk l En (x) = n! Tk (x). l 2k (n − k)! 2l ( 2k+l 2 )!( 2 )! 0≤k≤n

0≤l≤n−k l≡0 (mod 2)

Let us take p(x) = Hn (x) ∈ Pn . From (24), we have  Z  (−1)k 2k k!δk 1 dk 2 k−1/2 Ck = (1 − x ) Hn (x)dx k π(2k)! −1 dx Z 1 (−1)k 2k k!δk n! k k = × (−1) 2 (1 − x2 )k−1/2 Hn−k (x)dx (33) (2k)!π (n − k)! −1  Z 1 n−k  22k k!δk n! X n − k = Hn−k−l 2l (1 − x2 )k−1/2 xl dx, (2k)!(n − k)!π l −1 l=0

where Hn−k−l is the (n − k − l)th Hermite number.

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DAE SAN KIM, TAEKYUN KIM AND SANG-HUN LEE

By (31) and (33), we get (34)

X

Ck = n!δk

0≤l≤n−k l≡0 (mod 2)

Hn−k−l . l (n − k − l)!( 2k+l 2 )!( 2 )!

Therefore, by (34), we obtain the following theorem. Theorem 2.2. For n ∈ Z+ , we have X

Hn (x) = n!

0≤k≤n

δk

X 0≤l≤n−k l≡0 (mod 2)

! Hn−k−l Tk (x). l (n − k − l)!( 2k+l 2 )!( 2 )!

Let P∗n = {p(x) ∈ Q[x] | deg p(x) ≤ n}. Then P∗n is an inner product space with R1 √ the weighted inner product hp(x), q(x)i = −1 1 − x2 p(x)q(x)dx, where p(x), q(x) ∈ Pn . Then {U0 (x), U1 (x), · · · , Un (x)} is an orthogonal basis for the inner product space P∗n . For p(x) ∈ P∗n , let p(x) =

(35)

n X

Ck Uk (x),

k=0

where

(36)

Z 2 1 2 Ck = hp(x), Uk (x)i = (1 − x2 )1/2 Uk (x)p(x)dx π π −1  Z  (−1)k 2k+1 (k + 1)! 1 dk 2 k+1/2 (1 − x ) p(x)dx. = k (2k + 1)!π −1 dx

Let us assume that p(x) = xn ∈ P∗n . Then, by (36), we get  Z  (−1)k 2k+1 (k + 1)! 1 dk 2 k+1/2 Ck = (1 − x ) xn dx k (2k + 1)!π −1 dx (37) Z (−1)k 22k+1 (k + 1)! (−1)k n! 1 = × (1 − x2 )k+1/2 xn−k dx. (2k + 1)!π (n − k)! −1 It is easy to show that (38) Z 1 Z 1 (1 − x2 )k+1/2 xn−k dx = (1 + (−1)n−k ) (1 − x2 )k+1/2 xn−k dx −1 0 ( 0, if n − k ≡ 1 (mod 2) = (n−k)!(2k+2)!π , if n − k ≡ 0 (mod 2). 2n+k+2 ( n+k+2 )!( n−k )!(k+1)! 2

2

Therefore, by (37) and (38), we obtain the following proposition. Proposition 2.3. For n ∈ Z+ , we have X n! k+1 xn = n Uk (x). 2 ( n+k+2 )!( n−k 2 2 )! 0≤k≤n k≡n (mod 2)

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IDENTITIES FOR BERNOULLI POLYNOMIALS INVOLVING CHEBYSHEV POLYNOMIALS 7

Let us consider p(x) = Bn (x) ∈ P∗n . From (36), we have  Z  (−1)k 2k+1 (k + 1)! 1 dk 2 k+1/2 Ck = (1 − x ) Bn (x)dx k (2k + 1)!π −1 dx Z (−1)k 2k+1 (k + 1)! (−1)k n! 1 (1 − x2 )k+1/2 Bn−k (x)dx = × (39) (2k + 1)!π (n − k)! −1 Z 1 n−k X n − k  2k+1 (k + 1)! n! = × Bn−k−l (1 − x2 )k+1/2 xl dx. (2k + 1)!π (n − k)! l −1 l=0

It is not difficult to show that Z 1 Z 1 2 k+1/2 l l (1 − x2 )k+1/2 xl dx (1 − x ) x dx = (1 + (−1) ) 0 −1 ( (40) 0, if l ≡ 1 (mod 2) = (2k+2)!l!π , if l ≡ 0 (mod 2). 22k+2+l ( 2k+2+l )!(k+1)!( l )! 2

2

By (39) and (40), we get (41)

Ck =

(k + 1)n! 2k

X 0≤l≤n−k l≡0 (mod 2)

Bn−k−l . (n − k − l)!2l ( 2k+l+2 )!( 2l )! 2

Therefore, by (41), we obtain the following theorem. Theorem 2.4. For n ∈ Z+ , we have X

Bn (x) = n!

0≤k≤n

k+1 2k

X 0≤l≤n−k l≡0 (mod 2)

! Bn−k−l Uk (x). 2l (n − k − l)!( 2k+l+2 )!( 2l )! 2

By the same method, we can derive the following identity: X

En (x) = n!

0≤k≤n

k+1 2k

X 0≤l≤n−k l≡0 (mod 2)

! En−k−l Uk (x). )!( 2l )! 2l (n − k − l)!( 2k+l+2 2

Pn Let us take p(x) = Hn (x) ∈ P∗n . Then Hn (x) = k=0 Ck Uk (x), with  Z  (−1)k 2k+1 (k + 1)! 1 dk 2 k+1/2 (1 − x ) Hn (x)dx Ck = k (2k + 1)!π −1 dx  Z 1 n−k  22k+1 (k + 1)!n! X n − k l = 2 Hn−k−l (1 − x2 )k+1/2 xl dx (42) (2k + 1)!π(n − k)! l −1 l=0 X Hn−k−l 1 = n!(k + 1) × 2k+l+2 l . (n − k − l)! ( 2 )!( 2 )! 0≤l≤n−k l≡0 (mod 2)

Thus, by (42) and (43), we get Hn (x) = n!

X 0≤k≤n

(k + 1)

X 0≤l≤n−k l≡0 (mod 2)

178

! Hn−k−l Uk (x). (n − k − l)!( 2k+l+2 )!( 2l )! 2

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8

DAE SAN KIM, TAEKYUN KIM AND SANG-HUN LEE

ACKNOWLEDGEMENTS. 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 2012R1A1A2003786.

References [1] S. Araci, M. Acikgoz, A note on the Frobenius-Euler numbers and polynomials associated with Bernstein polynomials, Adv. Stud. Contemp. Math. 22 (2012), no. 3, 399-406. [2] A. Bayad, T. Kim, Identities involving values of Bernstein, q-Bernoulli, and q-Euler polynomials, Russ. J. Math. Phys. 18 (2011), no. 2, 133-143. [3] L. Carlitz, Note oin the integral of the product of several Bernoulli polynomials, J. London Math. Soc. 34(1959), 362-363. [4] M. Can, M. Cenkci, V. Kurt, Y. Simsek, Twisted Dedekind type sums associated with Barnes’ type multiple Frobenius-Euler l-functions, Adv. Stud. Contemp. Math. 18 (2009), no. 2, 135-160. [5] L. Fox, I. B. Parker, Chebyshev polynomials in numerical analysis, Oxford University Press, London-New York-Toronto, Ont. 1968. [6] D. S. Kim, T. Kim, S.-H. Rim, S, H. Lee, Hermite polynomials and their applications associated with Bernoulli and Euler numbers, Discrete Dynamics in Nature and Society 2012(2012), Article ID 974632, 13 pp. [7] D. S. Kim, T. Kim, S.-H. Lee, Y.-H. Kim, Some identities for the product of two Bernoulli and Euler polynomials,Advances in Difference Equations 2012(2012) Article ID 2012:95, 14 pp. [8] T. Kim, Identities involving Frobenius-Euler polynomials arising from nonlinear differential equations, J. Number Theory ( Article in Press), 2012. [9] D. S. Kim, T. Kim, Extended Laguerre polynomials associated with Hermite, Bernoulli, and Euler numbers and polynomials, Abstr. Appl. Anal. 2012(2012), Art. ID 957350, 15 pp. [10] D. S. Kim, T. Kim, Bernoulli basis and the product of several Bernoulli polynomials, Int. J. Math. Math. Sci. 2012(2012), Art. ID 463659, 12 pp. [11] T. Kim, C. S. Ryoo, H. Yi A note on q-Bernoulli numbers and q-Bernstein polynomials, Ars Combin. 104 (2012), 437-447. [12] J. C. Mason, D.C. Handscomb, Chebyshev Polynomials, Chapman Hall CRC , A CRC Press Company Boca Raton London New York Washington, D.C., 2003. [13] H. Ozden, I. N. Cangul, Y. Simsek, Remarks on q-Bernoulli numbers associated with Daehee numbers, Adv. Stud. Contemp. Math. 18 (2009), no. 1, 41-48. [14] S.-H. Rim, J. Jeong, On the modified q-Euler numbers of higher order with weight, Adv. Stud. Contemp. Math. 22 (2012), no. 1, 93-98. [15] C. S. Ryoo, Some identities of the twisted q-Euler numbers and polynomials associated with q-Bernstein polynomials, Proc. Jangjeon Math. Soc. 14 (2011), no. 2, 239-248.

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IDENTITIES FOR BERNOULLI POLYNOMIALS INVOLVING CHEBYSHEV POLYNOMIALS 9

Dae San Kim Department of Mathematics, Sogang University, Seoul 121-742, Republic of Korea TaeKyun Kim Department of Mathematics, Kwangwoon University, Seoul 139-701, Republic of Korea E-mail: [email protected], [email protected] Sang-Hun Lee Division of General Education, Kwangwoon University, Seoul 139-701, Republic of Korea

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An economical aggregation algorithm for algebraic multigrid (AMG)∗ Liang-Jian Deng†, Ting-Zhu Huang‡, Xi-Le Zhao§ Liang Zhao¶, Si Wang∥ School of Mathematical Sciences, University of Electronic Science and Technology of China, Chengdu, Sichuan, 611731, P.R. China

Abstract Aggregation-based AMG method is a widely studied technique of robustness for large-scale linear systems. Some previous aggregation algorithms, belonging to a part of aggregation-based AMG method, exhibit certain excellent properties. These aggregation methods, however, have to aggregate every grid points so that these methods lead expensive computation with grid points increasing. In the paper, a property that the aggregations hold particular structure associated with certain condition is discovered to damp the computation of aggregation algorithms. Meanwhile, this property is under the condition of the system matrix derived from the 9-point Finite Difference Method (FDM) and the particular setting of grid number. Furthermore, the conclusions about multilevel, such as the setting rule of grid number and corresponding theoretical analysis, are obtained from the extension of two level issues. Computational experiments demonstrate that the CPU time of new aggregation algorithm which generates the same aggregations with previous aggregation algorithms, keeps on a low level evidently, even for the linear systems of millions grade. Key words: Aggregation-based AMG; Aggregation algorithms; Economical computation; Poisson-like equations; Helmholtz-like equations; Millions grade problems

1

Introduction

Several methods can be utilized to solve the large-scale sparse linear systems Ax = b,

(1)

∗ This research is supported by NSFC (61170311), Chinese Universities Specialized Research Fund for the Doctoral Program (20110185110020), Sichuan Province Sci. & Tech. Research Project (12ZC1802), the Fundamental Research Funds for the Central Universities. † E-mail: [email protected] ‡ E-mail: [email protected] § E-mail: [email protected] ¶ E-mail: [email protected] ∥ E-mail: [email protected]

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where A ∈ RN ×N arises from the discretization of a scalar second-order elliptic partial differential equation (PDE). AMG method for large-scale linear systems is among the most efficient and convenient iterative methods [1, 2, 3, 4, 5, 6, 7, 8]. AMG method is composed with two parts: one is the setup phase and the other is the solve phase. Setup phase is associated with three parts on each level, i.e., defining coarse grids (aggregations), constructing transfer operators (i.e., prolongation and restriction operators), computing the linear systems on the coarse level, respectively. Solve phase involves a recursive process with solving the linear systems level by level and contains three parts mainly, i.e., the smoothing steps, the transferring of linear systems among levels and solving linear systems on the coarsest level, respectively. AMG method is a recursive method of efficiency for large-scale linear systems with mainly recursive forms: V-cycle and W-cycle, for instance, in [9, 10, 11, 12]. It projects the large-scale problems, level by level, to the small-scale problems until the problems can be solved as accurate as possible. The most important issue is making the computed solutions approximate to the true solutions. We transfer the linear systems of different levels through the restriction and prolongation operators R and P . AMG method has the following relationship among levels Ai+1 = Ri Ai Pi ,

i = 1, 2, · · · , lmax − 1,

(2)

where lmax labels the number of levels. Ri is the restriction operator from the i -th level to the (i + 1)-th level while Pi , Pi = RiT , is prolongation operator from the (i + 1)-th level to the i -th level. The subscripts of Ai denote corresponding belonging levels, and the levels range from fine to coarse with i increasing. AMG method mainly refers to classical AMG method, aggregation-based AMG method, adaptive AMG method and AMGe method [6, 7, 13, 14]. Classical AMG method is introduced by Brandt, McCormick, Ruge and St¨ uben [15, 16], it has been employed to solve linear systems whose coefficient matrices are M -matrices. For aggregationbased AMG method, the most critical step is the construction of the prolongation operators by the aggregation algorithms [17, 18, 19, 20, 21, 22] based on different definitions of strength of Connection. Adaptive AMG method utilizes a multigrid algorithm to enhance the efficiency of the prolongation, aiming to earn a more efficient AMG algorithm [23, 24]. The AMGe method, located in [25, 26], was first introduced as a measure to improve the robustness of AMG for the finite element problems. It is different from standard AMG method for requiring access to local element stiffness matrices (in addition to the assembled global stiffness matrices). The main differences among these AMG methods, e.g., classical AMG method, aggregation-based AMG method, adaptive AMG method and AMGe method, can be discriminated by the constructions of transfer operators and coarse grids, respectively. Particularly, transfer operators of aggregation-based AMG method can be generated by a classical aggregation algorithm, corresponding details in [17]. The motivation of our work is to acquire the aggregations, same as the aggregations of the algorithm in [17], with cheaper computation by utilizing the property discovered in this paper. In this paper, we mainly focus on the setup phase and establish a novel construction, aiming to reduce the computation of constructing aggregations. During the process of generating aggregations, an excellent discovery is found that the aggregations , obtained 2 182

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under the condition of the square grid number satisfying (3i + 4) × (3i + 4), i ∈ N , are symmetric. Besides, the system matrix on the finest level should be derived from the discretization of a scalar second-order PDE with 9-point FDM. Then we make use of these properties, the symmetry of aggregations and the relationship among subscripts of the grid points, to construct a new aggregation algorithm to decrease the computation. Computational experiments present that the new aggregation algorithm gains a lower consuming-time, besides, the same aggregations compared with previous aggregation algorithm in [17]. Particularly, we have to emphasize that this paper is mainly to improve the aggregation algorithm in the setup phase, aiming to gain more economical computation. The solve phase, meanwhile, keeps unchanged while our proposed method is applied in the setup phase. The paper is organized as follows. In section 2, we introduce the basic scheme of AMG method and the classical aggregation algorithm. Section 3 is about the new aggregation algorithm based on 9-point FDM. Meanwhile, some theoretical analysis and conclusions on the parameter and grid number on finest level, respectively, are presented in this section. Section 4 shows the capability of our aggregation algorithm on some numerical experiments about 2D Poisson-like equation and 2D Helmholtz-like equation. A compact conclusion will be presented in section 5.

2

Aggregation-based AMG methods

Aggregation-based AMG method clusters the fine grid of unknowns to aggregations representing the unknowns on the coarse level. Different with other methods, aggregationbased AMG method constructs the transfer operators mentioned in section 1 by aggregating the unknowns on each level. The coarsening part in classic AMG method is realized mainly by the aggregation algorithm (i.e., the setup phase mentioned in section 1) generating prearranged conditions for solve phase, e.g., aggregations, transfer operators and linear systems on coarse level. Meanwhile, it is necessary to introduce the basic AMG scheme ( See the following forma). Where A1 = A ∈ RN ×N , b1 = b ∈ RN , yi = MGM(x0 , bi , i) If (i = m) Then ym = Solve(Am , bm , em ) Else xi = Smooth(Ai , bi , x0 ) ri+1 = Ri (bi − Ai xi ) Ai+1 = Ri Ai Pi di+1 = MGM(0, ri+1 , i + 1) x ˆi = xi + Pi di+1 xi = Smooth(Ai , bi , x ˆi ) x0 = 0 ∈ RN and transfer operators Ri = PiT . The above recursive process is called V-cycle while another recursive type of AMG is called W-cycle doing twice on the fifth row. Aggregation-based AMG is divide into two parts: one is setup phase and the other is solve phase mentioned in above section. The setup phase may be considered as the prearranged section of the solve phase for solving the linear system (1), i.e., aggregations, transfer operators (i.e., Ri and Pi ) and coarse linear systems Ai on each 3 183

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level, respectively, so the setup phase part acts actually an important role in the whole process of AMG.

2.1

The classical aggregation algorithm

Before giving the new algorithm, it is necessary to introduce the classical aggregation algorithm coming from [17, 27]. The following content is about the graph GAl (Vl , El ) of the system matrices Al on the l-th level. We have to emphasize that the goal of illustrating this classical algorithm is to present that our new algorithm generates the same aggregations with the classical algorithm. The system matrix A, generating the graph GA (V, E), is generally gained by handling the PDE with different methods of discretization, e.g., 5-point FDM and 9-point FDM, etal. In this section, some definitions about graph theory are summarized again. Definition 1 ([27]). Corresponding to a sparse matrix A with symmetric sparsity pattern (i.e., ai,j ̸= 0 ⇔ aj,i ̸= 0), let GA (V, E) be the graph that consists of a set V = {v1 , v2 , v3 , · · · , vn } of n ordered vertices (nodes, unknowns), and a set of edges E such that the edge ei,j ∈ E exists (connecting vi and vj ) if and only if ai,j ̸= 0, i ̸= j. For a vertex vi , the set of neighbor vertices Ni is defined in the following form, Ni = {vj ∈ V |ei,j ∈ E} ,

(3)

|Ni | denotes the number of the elements in the set Ni . The degree of a vertex vi is deg(vi ) = |Ni |. For example, if the matrix is   4 −1 −1 −1  −1 4 0 −1  , A=  −1 0 4 −1  −1 −1 −1 4 and the GA (V, E) of the matrix A is shown in Figure 1.

Figure 1: The matrix graph The following contents introduce the classical aggregation algorithm and the con{ }Nl+1 (i.e., the i-th aggregation on the l-th level), only struction of aggregations Ali i=1 depending on the l-th level system matrix Al . For a given parameter θ ∈ (0, 1], the strongly coupled neighborhood of the node vi on the l-th level is defined as { } √ Nil (θ) = vj ∈ Vl ||ai,j | ≥ θ ai,i aj,j . (4) 4 184

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The classical aggregation algorithm, proposed by P. Vanˇ ek, J. Mandel and M. Brezina [17] and utilized by Wagner [27], is presented in the following part. Algorithm 1 Let a Nl × Nl matrix Al with the corresponding graph GAl (Vl , El ) and θ ∈ (0, 1] be given. The following Aggregation (GAl (Vl , El )) generates a disjoint { }Nl+1 covering Ali i=1 of the set V = {v1 , v2 , v3 , · · · , vNl }. Aggregation(GAl (Vl , El )) { initialization: U = {vi ∈ Vl |Nil (0) ̸= {vi }}; j=0; step 1: f or(vi ∈ U ) { if (N{il (θ) ⊂ U ) } j + +; Alj = Nil (θ); U = U \Alj ; } end step 2: elz = Alz ; end f or(z ≤ j) A f or(vi ∈ U ) { f or(z ≤ j) { elz ̸= {}) if (Nil (θ) ∩ A } end } end

{ l } Az = Alz ∪ {vi }; U = U \{vi }; break;

step 3: f or(vi ∈ U ) { j + +; Alj = Nil (θ) ∩ U ; U = U \Alj ; } end }

In the part of initialization, the set U does not contain all nodes, meanwhile, isolated nodes are not aggregated. In step 1, disjoint strongly coupled neighborhoods are selected as the initial approximation of the covering. Step 2 adds remaining nodes vi ∈ U to one of the sets Alz to which the node vi is strongly connected if any such set exists. Finally, in step 3, the still remaining nodes vi ∈ U are clustered into aggregations that consist of subsets of strongly coupled neighborhoods. The above algorithm acts crucial role for AMG method due to generating the prearranged information that mentioned at the beginning of this section. The above algorithm, however, runs slowly because it has to aggregate every point in the domain 5 185

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and judge whether the points belong to certain aggregation. Can we accelerate the process of above algorithm by some particular constructions? Fortunately, next section will introduce the new discovery about the 9-point FDM based aggregation algorithm. We draw this inspiration of the discovery to develop a completely different algorithm with Algorithm 1.

3

The new aggregation algorithm

In this section, the discovery about the aggregations is illustrated clearly, meanwhile, the aggregation algorithm, according to the discovery, obtains the aggregations without through Algorithm 1 entirely but a new way of more economical computation. In classical algorithm (i.e., Algorithm 1), the final aggregations have to be gained by aggregating every point while the new way only needs to satisfy the particular condition about number of grids. The new aggregation algorithm is based on the following definition of strongly coupled neighborhood, i.e., the eq. (4). If the problems are from the discretization of 9-point FDM, the Ni (θ), strongly coupled neighborhood of the node vi , contains eight nodes around the vi , e.g., N10 (θ) = {v2 , v3 , v4 , v9 , v11 , v16 , v17 , v18 } (Figure 2). Besides, the parameter θ must ensure existent according to the results in subsection 3.2.

Figure 2: The instruction figure

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Figure 3: The process of constructing aggregations according to Algorithm 1 with 9-point FDM

3.1

The discovery for constructing aggregations based on 9-point FDM

To demonstrate our new aggregation algorithm clearly, we give the Figure 3 with small grid number n. From Figure 3, we learn that 1. When n = 2 + 3x, (x = 0, 1, 2, · · · ), the aggregations are symmetrical about back diagonal direction, and the number of aggregations is (1 + x)2 . 2. When n = 4 + 3x, (x = 0, 1, 2, · · · ), the aggregations are symmetrical about horizontal direction and vertical direction (See Figure 2), and the number of final aggregations is (2 + x)2 . Particularly, when n = 4 + 3x, (x = 0, 1, 2, · · · ) (See • of Figure 3), there is the property of symmetry so that the aggregations can be gained by fixed scheme easily when the grid number is set as n = 4 + 3x, (x = 0, 1, 2, · · · ) (See details in the following algorithm). We note the subscripts from left to right and then from down to up (See Figure 2) to illustrate our algorithm clearly. Based on the discovery, the finally aggregation algorithm is given as follows. Algorithm 2. Consider matrix A ∈ RNl ×Nl (Nl = n2l , nl = 4 + 3x, x = 0, 1, 2, · · · ) and corresponding graph GAl (Vl , El ) and θ ∈ (0, 1] being given. Then Aggregation { }Nl+1 (GAl (Vl , El )) generates a disjoint covering Ali i=1 of the set V = {v1 , v2 , v3 , · · · , vNl }. Aggregation(GAl (Vl , El )) { /* firstly, we have the relation: nl = 4 + 3x, (x = 0, 1, 2, · · · ),Alk,j = Al(k−1)(x+2)+j (See the following paragraph)*/ /*step 1: get four angle’s aggregations (See Figure 4 (a)) */ Get Al1,1

Al1,(x+2)

Al(x+2),1

Al(x+2),(x+2) 8 188

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/* step 2: get the aggregations of upper boundary and lower boundary (See Figure 4 (b) (c)) */ for j=2:(x+1) /* get the aggregations of lower boundary */ Al1,j = {V3(j−1) , V3(j−1)+1 , V3(j−1)+2 , V3(j−1)+n , V3(j−1)+n+1 , V3(j−1)+n+2 }; /* get the aggregations of upper boundary */ Al(x+2),j = {V3(j−1)+(3x+2)·n , V3(j−1)+(3x+2)·n+1 , V3(j−1)+(3x+2)·n+2 , V3(j−1)+(3x+2)·n+n , V3(j−1)+(3x+2)·n+n+1 , V3(j−1)+(3x+2)·n+n+2 }; end /* step 3: get the aggregations of left boundary and right boundary and central zone (See Figure 4 (d) (e) (f)) */ for k=2:(x+1) /* get the aggregations of left boundary (See Figure 4 (d))*/ Alk,1 = {V(3k−4)·n+1 , V(3k−4)·n+2 , V(3k−4)·n+n+1 , V(3k−4)·n+n+2 , V(3k−4)·n+2n+1 , V(3k−4)·n+2n+2 }; /* get the aggregations of right boundary (See Figure 4 (e))*/ Alk,(x+2) = {V(3k−4)·n+3x+3 , V(3k−4)·n+3x+4 , V(3k−4)·n+3x+3+n , V(3k−4)·n+3x+4+n , V(3k−4)·n+3x+3+2n , V(3k−4)·n+3x+4+2n }; /* get the aggregations of central zone (See Figure 4 (f))*/ for

j=2:(x+1) Alk,j = {V(3k−4)·n+3(j−1) , V(3k−4)·n+3(j−1)+1 , V(3k−4)·n+3(j−1)+2, V(3k−4)·n+3(j−1)+n , V(3k−4)·n+3(j−1)+n+1 , V(3k−4)·n+3(j−1)+n+2, V(3k−4)·n+3(j−1)+2n , V(3k−4)·n+3(j−1)+2n+1 , V(3k−4)·n+3(j−1)+2n+2 }; end end } where the aggregation algorithm is under the condition that PDEs are discretilized by 9-point FDM when n = 4 + 3x, (x = 0, 1, 2, · · · ). We utilize a useful formula Ak,j = A(k−1)(x+2)+j , matching the Algorithm 2 for two-dimensional Ak,j . The formula is easy to be proved. Seeing Figure 5, we learn that A1 = A1,1 , A2 = A1,2 , A3 = A1,3 , etc. By n = 4 + 3x, (x = 0, 1, 2, · · · ), the aggregations’ number of every row is x + 2 and 9 189

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the total number of k −1 rows is (k −1)(x+2). So Ak,j = A(k−1)(x+2)+j is proved easily. We can obtain the aggregations of boundary and central zone easily through step 2 and step 3 of the above Algorithm 2, respectively. Finally, this algorithm generates the same aggregations with classical algorithm (i.e., Algorithm 1).

Figure 4: The instruction figure

Figure 5: The instruction figure of formula Ak,j = A(k−1)(x+2)+j

3.2

About the parameter θ

The parameter θ ∈ (0, 1] in equation (4) plays a significant role, because θ can decide the node vj whether belongs to certain strongly coupled neighborhood Ni (θ) of node vi . For example, if the parameter θ is smaller enough, then the corresponding strongly coupled neighborhood of node vi will contain more nodes. Moreover, maybe the finally aggregations by the Algorithm 2 are changed obviously with slightly change of θ ∈ (0, 1], so it is necessary to discuss the parameter in equation (4). Due to the discretization method (i.e., 9-point FDM) of corresponding problems, we hope the Ni (θ), the strongly coupled neighborhood of node vi , contains the corresponding eight nodes of around vi . We will demonstrate the existence of parameter θ firstly for this condition. Theorem 1. Let the strongly coupled neighborhood of node vi be defined as equation (4). Consider the coefficient matrix A ∈ RN ×N arising from 9-point FDM, if aij ̸= 10 190

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0, (i ̸= j), then, there must exist θ ∈ (0, 1], such that Ni (θ) contains the corresponding eight nodes of around vi , i = 1, 2, 3, · · · , N . Proof. We learn that the coupled neighborhood of vi is defined as (4) { } √ Nil (θ) = vj ∈ Vl ||ai,j | ≥ θ ai,i aj,j . According to known conditions by 9-point FDM, all diagonal elements of matrix A are the nonzero (i.e, aii ̸= 0), so above definition can be written as follows |ai,j | 0 0} ,

(5)

such that Ni (θ) contains the corresponding eight nodes of around vi , i = 1, 2, 3, · · · , N . Proof. Due to the 9-point FDM, the coefficient matrix A ∈ RN ×N is a nine diagonal matrix that every row of A has only nine nonzero elements including ai,i ̸= 0. According to the new definition (5) and 9-point FDM, it is easy to know that Ni (θ) contains the corresponding eight nodes of around vi , i = 1, 2, 3, · · · , N . 2

3.3

Extending to multilevel

This section mainly makes a discussion about extending the proposed Algorithm 2 to multilevel. According to section 3.1, if the grid number on the fine level is (3x + 4) × (3x+4) (x = 0, 1, 2, · · · ), then the grid number on the next coarse level is (x+2)×(x+2) under the condition that one aggregation on the fine level generates only one grid node on the coarse level. In order to extend the relationship from two level to multilevel, two significant conclusions are given in the following analysis. Theorem 2. Let the number of levels of multigrid be L and assuming the gird number is nL = 3i + 4 (i.e., the square grid is (3i + 4) × (3i+4)) on the coarsest level. If the girds number n on the finest level satisfies the following equation n=(

L−1 ∑

3j · 2) + 3L · i + 4,

(6)

j=1

then the Algorithm 2 can be extended to multilevel. 11 191

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Proof. Since the number of levels is L and the gird number on the coarsest level L is nL = 3i + 4. According to the conclusions of section 3.1, the grid number on the (L − 1)-th level should be nL−1 = 3(nL − 2) + 4 = 3(3 · i + 4 − 2) + 4 = 32 · i + 3 · 2 + 4, and the grid number on the (L − 2)-th level should be nL−2 = 3(nL−1 − 2) + 4 = 3(32 · i + 3 · 2 + 4 − 2) + 4 =3 ·i+3 ·2+3·2+4=( 3

2

2 ∑

3j · 2) + 33 · i + 4,

j=1

and similar to above, the grids number on the (L − 3)-th level should be nL−3 = 3(nL−2 − 2) + 4 = 3(33 · i + 32 · 2 + 3 · 2 + 4 − 2) + 4 =3 ·i+3 ·2+3·2+4=( 3

2

3 ∑

3j · 2) + 34 · i + 4,

j=1

it is easy to extend to the finest level by mathematical induction, the grids number on the finest level should satisfy L−2 ∑

n1 = 3(n2 − 2) + 4 = 3((

3 · 2) + 3 j

L−1

· i + 4 − 2) + 4 = (

j=1

L−1 ∑

3j · 2) + 3L · i + 4,

j=1

and n = n1 is the grids number satisfying Algorithm 2 on the finest level and it also extends the Algorithm 2 to multilevel. 2 Theorem 2 requires the grid number on coarsest level being nL = 3i + 4, i = 0, 1, 2, · · · , moreover, we can also extend to the arbitrary grids number nL = i, i = 0, 1, 2, · · · , on coarsest level. Corollary 2. For arbitrary grid number nL = i, i = 0, 1, 2, · · · , on the coarsest level, if the number of levels is L, then the gird number n on the finest level satisfy n=(

L−2 ∑

3j · 2) + 3L−1 · (i − 2) + 4,

(7)

j=1

then the Algorithm 2 can be extended to multilevel. Proof. It is easy to gain the Corollary 2 by replacing nL = 3i + 4 in Theorem 2 with nL = i. 2

4

Computational experiments

All experimental problems are discretized by 9-point FDM. Before our experiments, we will introduce the 9-point FDM briefly.

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Figure 6: The instruction figure We can get the 9-point FDM from the 5-point FDM in which one point (i, j) is only relevant to its adjacent four points (See Figure 6), i.e., (i − 1, j), (i + 1, j), (i, j − 1), (i, j + 1). For example, if the elliptic PDE in a square domain is Poisson equation −∆u = −(

∂2u ∂2u + 2 ) = f (x, y), (x, y) ∈ R[a,b]×[a,b] , ∂x2 ∂y

(8)

if h1 = h2 = (b − a)/(n + 1), then the 5-point FDM scheme can be obtained as follows −∆h ui,j =

1 (−ui,j+1 − ui,j−1 − ui+1,j − ui−1,j + 4ui,j ) = fi,j , h2

(9)

We define the vector uh = [u11 , u21 , · · · , un,1 ; · · · ; u1,n , u2,n , · · · , un,n ]T and assume zero boundary, then the finally linear system is obtained by (9) 1 Huh = g, h2 where



B −I  −I B −I   .. .. H= . .   −I





(10)

4 −1   −1 4 −1     .. .. .. .. , B =   . . . .    B −I  −1 4 −1 −I B −1 4

    ,  

where the right hand vector is known beforehand and I is the identity matrix. Then we rotate the coordinate system with π/4 so that the point (i, j) is relevant to its adjacent four points (See Figure 6), i.e., (i − 1, j + 1), (i − 1, j − 1), (i + 1, j − 1), (i + 1, j + 1). By this rotation, another 5-point FDM scheme is as follows ¯ h ui,j = −∆

1 (−ui+1,j+1 − ui+1,j−1 − ui−1,j+1 − ui−1,j−1 + 4ui,j ) = fi,j , 2h2

(11)

and it also gains the similar linear system with (10) but the wider bandwidth of coefficient matrix. 13 193

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Combining with the above two 5-point FDM scheme (9) and (11), the finally 9-point FDM scheme of Poisson equation [10] is determined, 2 1¯ h2 −( ∆h + ∆ ∆h fij , h )uij = fij + 3 3 12

(12)

where this scheme has smaller truncation error of O(h4 ) than 5-point FDM scheme. Besides, some notations are necessary to be introduced. The ti where i = 1, 2, · · · , L− 1, is just the CPU time of constructing aggregations by Algorithm 2 on the i-th level. Ti , i = 1, 2, · · · , L − 1, represents the total CPU time of generating prolongation operators by the following equation (13) and the coefficient matrices by equation (2) on the i-th level, respectively. { 1, i ∈ Alj , l Pij = (13) 0, otherwise. Moreover, the dimension N of coefficient matrix on the finest level, i.e., n2 , is computed by equation (6). Next, we will present two examples to demonstrate the efficiency of our algorithm.

4.1

Example 1: Poisson-like equation

First example is a 2D Poisson-like equation containing two scalars α, β ∈ R, it can be written in the form of −(α

∂2u ∂2u + β ) = f (x, y), (x, y) ∈ Ω = [a, b] × [a, b], ∂x2 ∂y 2

(14)

where the f (x, y) ∈ R is an arbitrary function and the boundary condition is u(a, y) = u(b, y) = u(x, a) = u(x, b) = C ∈ R.

(15)

It is easy to obtain the finally coefficient matrix A ∈ RN ×N arising from the 9-point FDM of equation (14). A is nine diagonal matrix 

B  R   A=  

R B .. .



R .. . R

..

. B R



e   k     ,B =     R  B

k e .. .



k .. . k

.. e k

.



p   q     ,R =     k  e

q p .. .

 q .. q

.

..

. p q

   ,  q  p

where e = 12(α + β) − 4αβ, k = 2αβ − 6α, p = 2αβ − 6β, q = −αβ. In Table 1, we choose the L being 3 levels and set the grid number nL on the coarsest level being 64, 94 and 124, respectively. According to Theorem 2, the dimensions of linear systems on the finest level are 322624, 702244, 1227664, respectively. The CPU time t, constructing aggregation on each level, is very short and not exceeding 0.4s for the large-scale matrix with dimension 1227664 while the classical algorithm exceeds 1000s. The total time for the dimension with 322624, 702244, 1227664 is about 10.824s, 49.501s, 148.483s, respectively. 14 194

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Table 1: CPU time for Poisson-like equation by our method with 3 levels t N 322624 702244 1227664

t1 0.102 0.192 0.320

t2 0.009 0.018 0.033

T T1 T2 10.525 0.188 48.601 0.690 146.160 1.970

Total t+T 10.824 49.501 148.483

Table 2: CPU time for Poisson-like equation by our method with 4 levels t N 237169 795664 1682209

t1 0.081 0.215 0.436

t2 0.006 0.021 0.045

t3 0.0008 0.003 0.005

T1 5.65 62.150 274.050

T T2 0.108 0.849 3.644

T3 0.007 0.027 0.077

Total t+T 5.853 63.265 278.257

In Table 2, L is 4 and the grid number nL is 19, 34, 49, respectively. The dimensions of linear systems on the finest level are 237169, 795664, 1682209, respectively, according to Theorem 2. The consuming time t for constructing aggregations on each level is not exceeding 0.5s for the large-scale matrix with dimension 1682209 while the classical algorithm can not compute the consuming time. Total time for the dimension with 237169, 795664, 1682209 is about 5.853s, 63.265s, 278.257s, respectively. The two tables with different maximal levels illustrate that the Algorithm 2 is indeed a novel and fast method for the setup phase of aggregation-base AMG method. From Table 1 and Table 2, it is easy to learn that the total time does not only contain the time t, constructing aggregations by Algorithm 2, but also the time T , constructing prolongation operators and generating the system matrices on each level. Furthermore, the mainly cost of total time is clearly T1 , because the matrices, keeping largest dimension on the finest level, are referred to vast matrix-matrix multiplication according to equation (2). The time on other levels are shorter seriously than the finest level and decreased evidently.

4.2

Example 2: Helmholtz equation

The second example containing a scalars ω ∈ R is a 2D Helmholtz equation, the form of this equation is as follows −(

∂2u ∂2u + 2 ) − ω 2 u = f (x, y), (x, y) ∈ Ω = [a, b] × [a, b], ∂x2 ∂y

(16)

where the f (x, y) ∈ R can be also an arbitrary function and the boundary condition is u(a, y) = u(b, y) = u(x, a) = u(x, b) = C ∈ R,

(17)

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Table 3: Time consuming for Helmholtz equation by our method with 4 levels

N 237169 795664 1682209

t1 0.085 0.275 0.442

t t2 0.008 0.026 0.055

t3 0.001 0.004 0.006

T1 5.84 62.450 280.150

T T2 0.119 0.836 3.744

T3 0.009 0.030 0.087

Total t+T 6.062 63.621 284.484

where ω ∈ R is a determined scalar, h = (b − a)/(n + 1). Similar to section 4.1, A is also a nine diagonal matrix   B R  R B R      .. .. .. A= , . . .    R B R  R B and



20 − 2h2 ω 2  −4   B=  

−4 20 − 2h2 ω 2 .. .

 −4 .. . −4

..

. 20 − 2h2 ω 2 −4

−4 20 − 2h2 ω 2



−4   −1     ,R =     

−1 −4 .. .

 −1 .. . −1

..

. −4 −1

   ,  −1  −4

In this example, ω, set to be 0.2, is utilized for all experiments. Actually the linear system generated by 9-point FDM has the same form with example 1, so the CPU time for the same dimension problem is almost not different. In Table 3, similar to Table 2, L is set to be 4 and nL is also 19, 34, 49, respectively. The dimensions of linear systems on the finest level are 237169, 795664, 1682209, respectively. From Table 3, it is easy to learn that our algorithm speeds much less time while the classical one can not finish the setup phase within 1000s. Besides, the number of nonzero elements (NNZ) of system matrices on each level is presented in Table 4 where the NNZ1, NNZ2, NNZ3 and NNZ4 represent the number of nonzero elements on level 1, 2, 3 and 4, respectively. Furthermore, we will try some larger scale system matrices to illustrate our Algorithm 2 all alone, i.e., the prolongation operators and system matrices on each level is out of range in the following test. L and is chosen to be 3 and nL is set to be 229, 379, 604, 754, 904, respectively, i.e., the dimensions of the system matrices on the finest level are 4214809, 11580409, 29463184, 45941284 and 66064384, respectively. The finally shown results are in Table 5 clearly and indeed quite attractive.

5

Conclusion

This paper describes a new algorithm for constructing the aggregations in the setup phase of aggregation-based AMG method. The new algorithm, utilizing some particular 16 196

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Table 4: NNZ on each level for Helmholtz equation by our method with 4 levels N 237169 795664 1682209

NNZ1 2128681 7150276 15124321

NNZ2 237169 795664 1682209

NNZ3 26569 88809 187489

NNZ4 3025 10000 21025

Table 5: Time consuming of constructing aggregations by Algorithm 2 with 3 levels N t1 t2 Total

4214809 1.028 0.091 1.119

11580409 2.815 0.260 3.075

29463184 7.002 0.713 7.715

45941284 10.844 0.112 10.956

66064384 15.553 1.620 17.173

settings, e.g., the particular grid number on the finest level according to Theorem 2 and the discretization with 9-point FDM, is different with the any previous aggregation algorithms. During the process of constructing aggregations, the symmetry of the aggregations was discovered if the number of square grid satisfies the conditions of equation (5), (6) and (7). Moreover, some theoretical and practical conclusions such as Theorem 1, etal., were also illustrated in this paper. Computational experiments for Poisson-like equation and Helmholtz-like equation presented that the new aggregation algorithm captured the perfect results in the CPU time even for millions grade problems.

References [1] A. Brandt, Algebraic multigrid theroy: the symmetric case, Appl. Math. Comput., 1986, pp: 23-56 [2] K. St¨ uben, A review of algebraic multigrid, J. Comput. Appl. Math., 1999, pp: 281-309 [3] P. Vanˇ ek, M. Brezina and R. Tezaur, Two-grid method for linear elasticity on unstructured meshes, SIAM J. Sci. Comput., 1999, pp: 900-923. [4] U. Trottenberg, C. W. Oosterlee and A. Sch¨ uller, Multigrid, Academic Press, London, 2001 [5] R. D. Falgout and P. S. Vassilevski, On generalizing the AMG framwork, SIAM J. Numerical Anal., 2004, pp: 1669-1693 [6] R. D. Falgout, An introduction to algebraic multigrid, Computing in Science and Engineer, 2006, pp: 24-33 [7] K. St¨ uben, Multigrid, chapter An introduction to algebraic multigrid, Academic Press, 2001, pp: 413-532 [8] J. K. Kraus, Algebraic multigrid based on computational molecules, 2: linear elasticity problems, SIAM J. Sci. Comput., 2008, pp: 505-524

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[9] W. L. Briggs, V. E. Henson and S. F. McCormick, A Multigrid Tutorial (Second Edition), SIAM, Philadelphia, 2000 [10] Y. Saad, Iterative Methods for Sparse Linear Systems (Second Edition), University of Minnesota, Minneapolis, Minnesota, 2003 [11] A. Thekale, T. Gradl, K. Klamroth and U. R¨ ude, Optimizing the number of multigrid cycles in the full multigrid algorithm, Numerical Lin. Alg. Appl., 2010, pp: 199-210 [12] A. Napov and Y. Notay, When does two-grid optimality carry over to the V-cycle?, Numerical Lin. Alg. Appl., 2010, pp: 273-290 [13] L. N. Olson, J. Schroder and R. S. Tuminaro, A new perspective on strength measures in algebraic multigrid, Numerical Lin. Alg. Appl., 2010, pp: 713-733 [14] A. H. Baker, Tz. V. Kolev and U. M. Yang, Improving algebraic multigrid interpolation operators for linear elasticity problems, Numerical Lin. Alg. Appl., 2010, pp: 495-517 [15] A. Brandt, S. F. Mccormick and J. W. Ruge, Algebraic Multigrid (AMG) for Sparse Matrix Equations in Sparsity and Its Applications, Cambridge University Press, Cambridge, 1984 [16] A. Brandt, S. F. Mccormick and J. W. Ruge, Multigrid Methods and Applications, vol.4 of Computation Mathematics, Springer Verlag, Berlin, 1985 [17] P. Vanˇ ek, J. Mandel and M. Brezina, Algebraic multigrid based on smoothed aggregation for second and fourth order problems, Computing, 1996, pp: 179-196. [18] M. Brezina, R. Falgout, S. MacLachlan, T. Manteuffel, S. McCormick and J. Ruge, Adaptive smoothed aggregation(αSA) multigrid, SIAM Review, 2005, pp: 317-346 [19] L. N. Olson and J. B. Schroder, Smoothed aggregation for Helmholtz problems, Numerical Lin. Alg. Appl., 2010, pp: 361-386 [20] Y. Notay, Aggregation-based algebraic multilevel preconditioning, SIAM J. Matrix Anal. Appl., 2006, pp: 998-1018 [21] A. Napov, Y. Notay, Algebraic analysis of aggregation-based multigrid, Numerical Lin. Alg. Appl., 2010, pp: 539-564 [22] A. C. Muresan and Y. Notay, Analysis of aggregation-based multigrid, SIAM J. Sci. Comput., 2008, pp: 1082-1103 [23] M. Brezina, R. Falgout, S. MacLachlan, T. Manteuffel and S. McCormick and J.Ruge, Adaptive algebraic multigrid, SIAM J. Sci. Comput., 2006, pp: 1261-1286 [24] M. Brezina, R. Falgout, S. Maclachlan, T. Manteuffel, S. Mccormick and J. RUGE, Adaptive smoothed aggregation (αSA), SIAM J. Sci. Comput., 2004, pp: 1896-1920 [25] M. Brezina, A. J. Cleary, R. D. Falgout, V. E. Henson, J. E. Jones, T. A. Manteuffel, S. F. McCormick and J. W. Ruge, Algebraic multigrid based on element interpolation (AMGe), SIAM J. Sci. Comput., 2000, pp: 1570-1592 [26] V. E. Henson and P. S. Vassilevski, Element-free AMGe:general algorithms for computing interpolation weights in AMG, SIAM J. Sci. Comput., 2001, pp: 629-650 [27] C. Wagner, Introduction to algebraic multigrid, Technical report, University of Heidelberg, Germany, 1999

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TABLE OF CONTENTS, JOURNAL OF COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 16, NO. 1, 2014 Switching Design for the Robust Stability of Nonlinear Uncertain Stochastic Switched DiscreteTime Systems with Interval Time-Varying Delay, G. Rajchakit,……………………………10 Stabilization of Switched Discrete-Time Systems with Convex Polytopic Uncertainties, G. Rajchakit,…………………………………………………………………………………….20 A Preconditioner for Block Two-By-Two Symmetric Indefinite Matrices, Chun Wen, and Ting-Zhu Huang,…………………………………………………………………………….30 Hyers-Ulam Stability of a General Diagonal Symmetric Functional Equation, Choonkil Park, and Hamid Rezaei,……………………………………………………………………………….42 General Decay of Solutions for a Singular Nonlocal Viscoelastic Problem with Nonlinear Damping and Source, Yun Sun, Gang Li, and Wenjun Liu,……………………………….50 Measuring Fuzziness of Generalized Fuzzy Rough Sets Induced by Pseudo-Operations, Zhan-hong Shi, and Zeng-tai Gong,…………………………………………………………56 Global Analysis for Delay Virus Infection Model with Multitarget Cells, A.M. Elaiw, and M.A. Alghamdi,………………………………………………………………………………67 The Parameter Reduction of Soft Sets and Its Algorithm, Zhaowen Li, and Ninghua Gao,.76 Functional Inequalities Associated With Bi-Cauchy Additive Functional Equations, Gang Lu, Choonkil Park, and Dong Yun Shin,……………………………………………………………85 Robust CVaR-based Portfolio Optimization Under a Genal Affine Data Perturbation Uncertainty Set, Zhifeng Dai, and Fenghua Wen,……………………………………………………………93 Random Derivations on Random Normed Algebras, Jung Rye Lee, Choonkil Park, and Dong Yun Shin,………………………………………………………………………………………104 A Note on the q-Extension of Second Kind Euler Numbers and Polynomials, C. S. Ryoo,…112 Some Properties of Bazilevic Functions Related With Conic Domains, Khalida Inayat Noor, Mohsan Raza, and Kamran Yousaf,………………………………………………………….117 Some Characterizations in Some Möbius Invariant Spaces, A. El-Sayed Ahmed, A. Kamal, and Aydah Ahmadi,……………………………………………………………………………….126

TABLE OF CONTENTS, JOURNAL OF COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 16, NO. 1, 2014 (continued)

Coefficient Bounds for Certain Subclasses of Close-to-Convex Functions of Janowski Type, Wasim Ul-Haq, Attiya Nazneen, Muhammad Arif, and Nasir Rehman,………………………133 Five-order Extrapolation Algorithms for Laplace Equation with Linear Boundary Condition, Pan Cheng, Zhi lin, and Peng Xie,……………………………………………………………139 Sufficient Conditions for Univalence Obtained by Using First Order Nonlinear Strong Differential Subordinations, Georgia Irina Oros,…………………………………………… 149 A Note on the Symmetric Properties for the Second Kind Twisted q-Euler Polynomials, C.S. Ryoo,……………………………………………………………………………………153 Sufficient Conditions for Functions to be in a Class of p-Valent Analytic Functions, M. Arif, M. Ayaz, W. Haq, and J. Iqbal,………………………………………………………………159 Additive Functional Inequalities in Paranormed Spaces, Seo-Yun Yang, and Choonkil Park,.165 Some Identities for Bernoulli Polynomials Involving Chebyshev Polynomials, Dae San Kim, Taekyun Kim, and Sang-Hun Lee,……………………………………………………………172 An Economical Aggregation Algorithm for Algebraic Multigrid (AMG), Liang-Jian Deng, Ting-Zhu Huang, Xi-Le Zhao, Liang Zhao, and Si Wang,……………………………………181

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J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 16, NO.2, 2014, COPYRIGHT 2014 EUDOXUS PRESS, LLC

Relationship between lower and higher order anti-periodic boundary value problems and existence results Bashir Ahmad, Ahmed Alsaedi, Afrah Assolami

1

Department of Mathematics, Faculty of Science, King Abdulaziz University, P.O. Box 80203, Jeddah 21589, Saudi Arabia e-mail: bashirahmad− [email protected] (B. Ahmad), [email protected] (A. Alsaedi), [email protected] (A. Assolami)

Abstract In this paper, we develop a relationship between lower and higher order classical anti-periodic boundary value problems. Some existence results for a 5th−order anti-periodic boundary value problem of nonlinear ordinary differential equations are also presented. Our results are based on some standard tools of fixed point theory. The paper concludes with illustrative examples.

Key words and phrases: Ordinary differential equations; anti-periodic boundary conditions; existence; fixed point theorems AMS (MOS) Subject Classifications: 334A34, 34B15, 34B27

1

Introduction

Anti-periodic boundary value problems occur in the mathematical modelling of a variety of physical processes and have recently received considerable attention. Examples include anti-periodic trigonometric polynomials in the study of interpolation problems [10], anti-periodic wavelets [8], difference equations [7, 20], ordinary, partial and abstract differential equations [1, 2, 12, 13, 14, 16, 19, 21, 22], fractional differential equations [5, 6] and impulsive differential equations [3, 4, 11, 15], etc. For some more application of anti-periodic boundary conditions in physics, see [9, 17] and the references therein. The objective of this paper is to study a relationship between solutions of lower and higher order classical anti-periodic boundary value problems. For this purpose, we consider a 5th−order anti-periodic boundary value problem and show that the solution for a 4th−order anti-periodic problem follows from that of 5th−order problem, the solution for 3rd−order anti-periodic problem can be deduced from that of 4th−order problem, and so on by applying a typical strategy. Expressions for Green’s functions of anti-periodic problems are also presented. Besides this, we develop the existence theory for 5th−order anti-periodic boundary value problems and illustrate it with some examples. Precisely, we consider the following problem:   D(5) x(t) = f (t, x(t)), t ∈ [0, T ], T > 0,  x(0) = −x(T ), x0 (0) = −x0 (T ), x00 (0) = −x00 (T ), x000 (0) = −x000 (T ), x(iv) (0) = −x(iv) (T )

(1)

where f is a given continuous function. 210

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2

Linear Problem

Lemma 2.1 For any y ∈ C[0, T ], the unique solution for a linear 5th−order antiperiodic boundary value problem   D(5) x(t) = y(t), t ∈ [0, T ], (2)  x(0) = −x(T ), x0 (0) = −x0 (T ), x00 (0) = −x00 (T ), x000 (0) = −x000 (T ), x(iv) (0) = −x(iv) (T ) is

T

Z x(t) =

G5 (t, s)y(s)ds, 0

Where G5 (t, s) is the Green’s function given by  −s)3 −s)2 2(t−s)4 −(T −s)4  + (T −2t)(T + t(T −t)(T  2(4!) 4(3!) 4(2!)    3 4  −tT 3 ) (6t2 T −4t3 −T 3 )(T −s)   + (2T t −t , 0 < s < t < T,  + 48 48 G5 (t, s) =   −s)4 −s)3 −s)2   − (T2(4!) + (T −2t)(T + t(T −t)(T  4(3!) 4(2!)     (6t2 T −4t3 −T 3 )(T −s) (2T t3 −t4 −tT 3 ) + + , 0 < t < s < T. 48 48

(3)

Proof. It is well known that the integral representation for the solution of equation D(5) x(t) = y(t) can be written as Z t (t − s)4 (4) x(t) = y(s)ds − bo − b1 t − b2 t2 − b3 t3 − b4 t4 . 4! 0 where bo , b1 , b2 , b3 , b4 ∈ R are arbitrary constants. Using the boundary conditions of the problem (2) in (4), we find that T

1 bo = 2

Z

1 2

Z

1 4

Z

b1 = b2 = b4 =

0 T

0

1 48

T

0

T

(T − s)4 T y(s)ds − 4! 4

Z

(T − s)3 T y(s)ds − 3! 4

Z

(T − s)2 T y(s)ds − 2! 8

Z

0 T

0

T

(T − s)3 T3 y(s)ds + 3! 48

Z

(T − s)2 T3 y(s)ds + 2! 48

Z

T

(T − s)y(s)ds, b3 = 0

(T − s)y(s)ds, 0

1 12

T

y(s)ds, 0

Z

T

(T − s)y(s)ds − 0

T 24

Z

T

y(s)ds, 0

T

Z

y(s)ds. 0

Substituting the values of bo , b1 , b2 , b3 and b4 in (4), we obtain Z Z (t − s)4 1 T (T − s)4 (T − 2t) T (T − s)3 y(s)ds − y(s)ds + y(s)ds 4! 2 0 4! 4 3! 0 0 Z Z t(T − t) T (T − s)2 (6t2 T − 4t3 − T 3 ) T + y(s)ds + (T − s)y(s)ds 4 2! 48 0 0 Z (2T t3 − t4 − tT 3 ) T + y(s)ds. 48 0 Z

x(t)

t

=

(5)

Alternatively, (5) can be written in term of Green’s function as Z x(t) =

T

G5 (t, s)y(s)ds, 0

where G5 (t, s) is given by (3). This completes the proof. 211

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2.1

Relationship between higher-order and lower-order anti-periodic problems

It is found that there exists a relationship between higher-order and lower-order anti-periodic boundary p −s)p−1 value problems. For instance, by dropping the last term of (5) and replacing (T −s) by (T(p−1)! in the p! resulting expression of (5), we obtain Z Z 1 T (T − s)3 (T − 2t) T (T − s)2 (t − s)3 y(s)ds − y(s)ds + y(s)ds 3! 2 0 3! 4 2! 0 0 Z Z t(T − t) T (T − s) (6t2 T − 4t3 − T 3 ) T + y(s)ds, y(s)ds + 4 1! 48 0 0 Z

x(t)

t

=

(6)

which is the solution of 4th−order anti-periodic boundary value problem:  (4) D x(t) = y(t), t ∈ [0, T ], x(0) = −x(T ), x0 (0) = −x0 (T ), x00 (0) = −x00 (T ), x000 (0) = −x000 (T ),

(7)

In this case, Green’s function G4 (t, s) is  3 3 2 2 3 3  2(t−s) −(T −s) + (T −2t)(T −s) + t(T −t)(T −s) + (6t T −4t −T ) , 0 < s < t < T, 2(3!) 4(2!) 4 48 G4 (t, s) = 2 3 −s)2 −s) −T 3 ) −s)3  + (T −2t)(T + t(T −t)(T + (6t T −4t , 0 < t < s < T. − (T2(3!) 4(2!) 4 48 p−1

p

−s) by (T(p−1)! in the remaining terms Similarly, by dropping the last term of (6) and replacing (T −s) p! of (6), we obtain the solution of a third-order anti-periodic boundary value problem given by

Z x(t)

t

= 0

Z =

(t − s)2 1 y(s)ds − 2! 2

Z 0

T

(T − s)2 (T − 2t) y(s)ds + 2! 4

Z 0

T

(T − s) t(T − t) y(s)ds + 1! 4

Z

T

y(s)ds 0

T

G3 (t, s)y(s)ds, 0

(8) where G3 (t, s) =

 

2(t−s)2 −(T −s)2 2(2!) 2



−s) − (T2(2!) +

+

(T −2t)(T −s) 4

(T −2t)(T −s) 4

+

+

t(T −t) , 4

t(T −t) , 4

p

0 < s < t < T,

0 < t < s < T. p−1

−s) If we discard the last term of (8) and replacing (T −s) by (T(p−1)! in the remaining terms of (8), p! then we get the solution of a second-order anti-periodic boundary value problem, which is given by t

Z x(t) = 0

1 (t − s) y(s)ds − 1! 2

Z

T

0

(T − s) (T − 2t) y(s)ds + 1! 4

and the associated Green’s function G2 (t, s) is ( 2(t−s)−(T −s) G2 (t, s) =

2 (T −s) − 2

+

(T −2t) , 4 (T −2t) , 0 4

+

Z

T

y(s)ds,

(9)

0

0 < s < t < T, < t < s < T.

Thus, the above strategy is quite useful to write down the solutions for lower-order anti-periodic problems once the solution of a higher-order anti-periodic problem is available.

3

Some existence results

Let C = C([0, T ], R) denotes the Banach space of all continuous functions from [0, T ] → R endowed with the norm defined by kxk = sup{|x(t)|, t ∈ [0, T ]}.

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Define an operator U : C → C as Z (t − s)4 1 T (T − s)4 (Ux)(t) = f (s, x(s))ds − f (s, x(s))ds 4! 2 0 4! 0 Z T Z (T − 2t) (T − s)3 t(T − t) T (T − s)2 + f (s, x(s))ds + f (s, x(s))ds 4 3! 4 2! 0 0 Z Z (2T t3 − t4 − tT 3 ) T (6t2 T − 4t3 − T 3 ) T (T − s)f (s, x(s))ds + f (s, x(s))ds, t ∈ [0, T ]. + 48 48 0 0 (10) Observe that the problem (1) has solutions if and only if the operator U has fixed points. Z

t

To prove the existence of solutions for (1), we recall some known results.

Theorem 3.1 ([18]) let X be a Banach space. Assume that T : X → X is completely continuous operator and the set V = {u ∈ X|u = µT u, 0 < µ < 1} is bounded. Then T has a fixed point in X Theorem 3.2 ([18]) Let X be a Banach space. Assume that Ω is an open bounded subset of X with θ ∈ Ω and let T : Ω → X be a completely continuous operator such that kT uk ≤ kuk,

∀u ∈ ∂Ω.

Then T has a fixed point in Ω. Theorem 3.3 ([18]) Let M be a closed convex and nonempty subset of a Banach space X. Let A, B be the operators such that (i) Ax + By ∈ M whenever x, y ∈ M ; (ii) A is compact and continuous; (iii) B is a contraction mapping. Then there exists z ∈ M such that z = Az + Bz. Now we are in a position to present some existence results for problem (1). Theorem 3.4 Assume that there exists a positive constant L1 such that |f (t, x)| ≤ L1 for t ∈ [0, T ], x ∈ C. Then the problem (1) has at least one solution. Proof. First of all, we show that the operator U defined by 10 is completely continuous. Observe that continuity of the operator U follows from the continuity of f. Let Ω ⊂ C be bounded. Then, ∀x ∈ Ω, it follows by the assumption |f (t, x)| ≤ L1 that Z (t − s)4 1 T (T − s)4 |f (s, x(s))|ds + |f (s, x(s))|ds 4! 2 0 4! t∈[0,T ] 0 Z T Z T 1 (T − s)3 1 (T − s)2 + |T − 2t| |f (s, x(s))|ds + |t(T − t)| |f (s, x(s))|ds 4 3! 4 2! 0 0 Z Z o |6t2 T − 4t3 − T 3 | T |2T t3 − t4 − tT 3 | T + (T − s)|f (s, x(s))|ds + |f (s, x(s))|ds 48 48 0 0 Z T Z T n1 Z t 1 |T − 2t| ≤ L1 supt∈[0,T ] (t − s)4 ds + (T − s)4 ds + (T − s)3 ds 4! 0 2(4!) 0 4(3!) 0 Z Z Z |t(T − t)| T |6t2 T − 4t3 − T 3 | T |2T t3 − t4 − tT 3 | T o + (T − s)2 ds + (T − s)ds + ds 4(2!) 48 48 0 0 0

k(Ux)k ≤

≤

sup

nZ

193T 5 L1 3840

t

= L2 , (11) 213

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Anti-periodic boundary value problems

which implies that k(Ux)k ≤ L2 . Furthermore, Z n Z t (t − s)3 1 T (T − s)3 0 k(Ux) k = sup |f (s, x(s))|ds + |f (s, x(s))|ds 3! 2 0 3! t∈[0,T ] 0 Z T Z |T − 2t| (T − s)2 |t(T − t)| T + (T − s)|f (s, x(s))|ds |f (s, x(s))|ds + 4 2! 4 0 0 Z o |6T t2 − 4t3 − T 3 | T |f (s, x(s))|ds + 48 0 Z Z n Z t (t − s)3 1 T (T − s)3 |T − 2t| T (T − s)2 ≤ L1 sup ds + ds + ds 3! 2 0 3! 4 2! t∈[0,T ] 0 0 Z T Z |6T t2 − 4t3 − T 3 | T o |t(T − t)| (T − s)ds + ds + 4 48 0 0 ≤

(12)

4

L1 15T 96 = L3 .

Hence, for t1 , t2 ∈ [0, T ], we have Z

t2

|(Ux)(t2 ) − (Ux)(t1 )| ≤

|(Ux)0 (s)|ds ≤ L3 (t2 − t1 ).

t1

Thus, by the foregoing arguments, one can infer that the operator U is equicontinuous on [0, T ]. Hence, by the Arzela-Ascoli theorem, the operator U : C → C is completely continuous. Next, we consider the set V = {x ∈ C | x = µUx, 0 < µ < 1}, and show that it is bounded. Let x ∈ V, then x = µUx, 0 < µ < 1. For any t ∈ [0, T ], we have Z t Z Z (t − s)4 1 T (T − s)4 (T − 2t) T (T − s)3 x(t) = y(s)ds − y(s)ds + y(s)ds 4! 2 0 4! 4 3! 0 0 Z Z Z (6t2 T − 4t3 − T 3 ) T t(T − t) T (T − s)2 (2T t3 − t4 − tT 3 ) T y(s)ds + + (T − s)y(s)ds + y(s)ds 4 2! 48 48 0 0 0 Z T = G5 (t, s)y(s)ds 0

(13) Z 1 T (T − s)4 (t − s)4 |f (s, x(s))|ds + |f (s, x(s))|ds |x(t)| = µ|(Ux)(t)| ≤ 4! 2 0 4! 0 Z T Z |T − 2t| (T − s)3 |t(T − t)| T (T − s)2 + |f (s, x(s))|ds + |f (s, x(s))|ds 4 3! 4 2! 0 0 Z Z |6t2 T − 4t3 − T 3 | T |2T t3 − t4 − tT 3 | T (T − s)|f (s, x(s))|ds + |f (s, x(s))|ds + 48 48 0 0 h R RT t 1 1 ≤ L1 4! (t − s)4 ds + 2(4!) (T − s)4 ds 0 0 Z Z |T − 2t| T |t(T − t)| T 3 + (T − s) ds + (T − s)2 ds 4(3!) 0 4(2!) 0 Z Z |6t2 T − 4t3 − T 3 | T |2T t3 − t4 − tT 3 | T i + (T − s)ds + ds 48 48 0 0 n 2|t5 | + T 5 |T − 2t|T 4 |t(T − t)|T 3 |6t2 T − 4t3 − T 3 |T 2 + + + ≤ max 2(5!) 4(4!) 4(3!) 48(2!) t∈[0,T ] |2T t3 − t4 − tT 3 |T o + L1 = M1 . 48 Thus, kxk ≤ M1 for any t ∈ [0, T ]. So, the set V is bounded. Thus, by the conclusion of Theorem 3.1, the operator U has at least one fixed point, which means that (1) has at least one solution. Z

t

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B. Ahmad, A. Alsaedi and A. Assolami

Theorem 3.5 Let f : [0, T ] × R → R, and lim

x→0

f (t,x) x

= 0. Then the problem (1) has at least one

solution. Proof. By the assumption lim

x→0

f (t,x) x

= 0, there exists a constant r > 0 such that |f (t, x)| ≤ δ|x|

for 0 < |x| < r, where δ > 0 satisfies the condition

maxt∈[0,T ]

n

2|t5 |+T 5 2(5!)

+

|T −2t|T 4 4(4!)

+

|t(T −t)|T 3 4(3!)

|6t2 T −4t3 −T 3 |T 2 48(2!)

+

+

|2T t3 −t4 −tT 3 |T 48

o δ ≤ 1.

(14)

Define Ω1 = {x ∈ C | kxk < r} and take x ∈ C such that kxk = r, that is, x ∈ ∂Ω. As before, it can be shown that U is completely continuous and n 5 5 o 4 2 3 3 4 3 |+T −t)|T 3 −T 3 |T 2 |Ux(t)| ≤ maxt∈[0,T ] 2|t2(5!) + |T −2t|T + |t(T4(3!)) + |6t T −4t + |2T t −t48−tT |T δkxk, 4(4!) 48(2!) (15) which, in view of (14), yields kUxk ≤ kxk, x ∈ ∂Ω. Therefore, by Theorem 3.2, the operator U has at least one fixed point which corresponds to at least one solution of problem (1). Our next existence result is based on Krasnoselskii’s fixed point theorem [18]. Theorem 3.6 Let f : [0, 1] × R → R be a continuous function satisfying the assumptions (A1 ) |f (t, x) − f (t, y)| ≤ L|x − y|, ∀t ∈ [0, T ], x, y ∈ R; (A2 ) |f (t, x)| ≤ µ(t), ∀(t, x) ∈ [0, 1] × R, and µ ∈ C([0, T ], R+ ). Then the problem (1) has at least one solution on [0, T ] if L
ρM/(1 − ρL) with ρ = 193T 5 /3840, we show that UBr ⊂ Br , where Br = {x ∈ C : kxk ≤ r}. For x ∈ Br , we have Z t (t − s)4 |(Ux)(t)| ≤ |f (s, x(s)) − f (s, 0) + f (s, 0)|ds 4! 0 Z 1 T (T − s)4 + |f (s, x(s)) − f (s, 0) + f (s, 0)|ds 2 0 4! Z T 1 (T − s)3 + |T − 2t| |f (s, x(s)) − f (s, 0) + f (s, 0)|ds 4 3! 0 Z T 1 (T − s)2 + |t(T − t)| |f (s, x(s)) − f (s, 0) + f (s, 0)|ds 4 2! 0 Z T 1 + |6t2 T − 4t3 − T 3 | (T − s)|f (s, x(s)) − f (s, 0) + f (s, 0)|ds (17) 48 0 3 4 3 Z T |2T t − t − tT | + |f (s, x(s)) − f (s, 0) + f (s, 0)|ds 48 0 h R RT t 1 1 ≤ (Lr + M ) 4! (t − s)4 ds + 2(4!) (T − s)4 ds 0 0 Z Z |t(T − t)| T |T − 2t| T 3 (T − s) ds + (T − s)2 ds + 4(3!) 0 4(2!) 0 Z T Z i 1 |2T t3 − t4 − tT 3 | T 2 3 3 + |6t T − 4t − T | (T − s)ds + (T − s)ds 48 48 0 0 ≤

5

(Lr + M ) 193T 3840 ≤ r,

which implies that k(Uxk ≤ r. Thus, Ux ∈ Br ∀x ∈ Br . Hence UBr ⊂ Br . Now, for x, y ∈ C and for each t ∈ [0, T ], we obtain k(Ux) − (Uy)k Z n Z t (t − s)4 1 T (T − s)4 |f (s, x(s)) − f (s, y(s))|ds + |f (s, x(s)) − f (s, y(s))|ds ≤ max 4! 2 0 4! t∈[0,T ] 0 Z T 1 (T − s)3 + |T − 2t| |f (s, x(s)) − f (s, y(s))|ds 4 3! 0 216

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B. Ahmad, A. Alsaedi and A. Assolami T

(T − s)2 |f (s, x(s)) − f (s, y(s))|ds 2! 0 Z T 1 + |6t2 T − 4t3 − T 3 | (T − s)|f (s, x(s)) − f (s, y(s))|ds 48 0 Z T o 1 + |2T t3 − t4 − tT 3 | |f (s, x(s)) − f (s, y(s))|ds 48 0 Z T Z n1 Z t 1 |T − 2t| T (t − s)4 ds + (T − s)4 ds + (T − s)3 ds ≤ Lkx − yk max 2(4!) 0 4(3!) 0 t∈[0,T ] 4! 0 Z Z T Z T o |t(T − t)| T 1 1 2 2 3 3 3 4 3 + (T − s) ds + |6t T − 4t − T | (T − s)ds + |2T t − t − tT | ds 4(2!) 48 48 0 0 0 193T 5 L kx − yk, ≤ 3840 1 + |t(T − t)| 4

Z

5

L which depends only on the parameters T, L involved in the problem. As 193T 3840 < 1, therefore U is a contraction. Thus, the conclusion of the theorem follows by the contraction mapping principle (Banach fixed point theorem). This completes the proof.

Example 3.8 Consider the following antiperiodic boundary value problem h i  2 − cos2 x(t)  e 1 + 3 cos 2t + 2 ln(2 + 3 sin x(t))   (5)   ,  D x(t) = 3 + sin x(t) x(0) = −x(1), x0 (0) = −x0 (1), x00 (0) = −x00 (1),       000 x (0) = −x000 (1), x(iv) (0) = −x(iv) (1),

0 < t < 1, (18)

Clearly |f (t, x)| ≤ 2 + ln 5 = L1 . So the hypothesis of Theorem 3.4 holds. Hence, by the conclusion of Theorem 3.4, there exists at least one solution for problem (18). Example 3.9 Consider the problem  √ 1 D(5) x(t) = (5 + x3 (t)) 2 + 2(t + 1)(x − sin x(t)) − 5, 0 < t < 1,     x(0) = −x(1), x0 (0) = −x0 (1), x00 (0) = −x00 (1),    000  x (0) = −x000 (1), , x(iv) (0) = −x(iv) (1). Since

1

(5 + x3 ) 2 + 2(t + 1)(x − sinx) − lim x→0 x therefore, the conclusion of Theorem 3.5 applies to problem (19).

√ 5

(19)

= 0,

Example 3.10 Consider the problem  p |x|  (5)  + sin(t 1 + t2 ), 0 < t < 1, D x(t) =    1 + |x| x(0) = −x(1), x0 (0) = −x0 (1), x00 (0) = −x00 (1),      000 x (0) = −x000 (1), , x(iv) (0) = −x(iv) (1).

(20)

Obviously L = 1 as |x| |y| |f (t, x) − f (t, y)| = − ≤ |x − y|. 1 + |x| 1 + |y| Since T = 1 in this case, therefore, L < 3840/193. Hence, by Theorem 3.7, problem (20) has a unique solution on [0, 1]. 217

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Anti-periodic boundary value problems

Example 3.11 Consider the problem p   D(5) x(t) = L( 1 + t2 cos t + tan−1 x + (t + 1)2 ), 0 < t < 1    x(0) = −x(1), x0 (0) = −x0 (1), x00 (0) = −x00 (1),

   

(21)

x000 (0) = −x000 (1) , x(iv) (0) = −x(iv) (1)

It can easily be found that |f (t, x) − f (t, y)| ≤ L|x − y|, where L < 3840/161. Thus, the conclusion of Theorem 3.6 applies to the problem (21). Acknowledgements. This paper was funded by King Abdulaziz University under grant No. (9/34/Gr). The authors, therefore, acknowledge technical and financial support of KAU.

References [1] A.R. Aftabizadeh, S. Aizicovici, N.H. Pavel, On a class of second-order anti-periodic boundary value problems. J. Math. Anal. Appl. 171 (1992), 301-320. [2] A.R. Aftabizadeh, Y.K. Huang, N.H. Pavel, Nonlinear third-order differential equations with antiperiodic boundary conditions and some optimal control problems. J. Math. Anal. Appl. 192 (1995), 266-293. [3] B. Ahmad, J. J. Nieto, Existence and approximation of solutions for a class of nonlinear impulsive functional differential equations with anti-periodic boundary conditions. Nonlinear Anal. 69 (2008), 3291-3298. [4] B. Ahmad, Existence of solutions for second order nonlinear impulsive boundary value problems. Elect. J. Differential Equations, 2009(2009), No. 68, 1-7. [5] B. Ahmad, J.J. Nieto, Existence of solutions for anti-periodic boundary value problems involving fractional differential equations via Leray-Schauder degree theory. Topol. Methods Nonlinear Anal. 35 (2010), 295-304. [6] R.P. Agarwal, B. Ahmad, Existence theory for anti-periodic boundary value problems of fractional differential equations and inclusions. Comput. Math. Appl. 62 (2011), 1200-1214. [7] A. Cabada, D.R. Vivero, Existence and uniqueness of solutions of higher-order antiperiodic dynamic equations. Adv. Difference Equ. 4 (2004), 291-310. [8] H.L. Chen, Antiperiodic wavelets. J. Comput. Math. 14 (1996), 32-39. [9] Y. Chen, J.J. Nieto, D. O’Regan, Antiperiodic solutions for fully nonlinear first-order differential equations. Math. Comput. Modelling 46 (2007), no. 9-10, 1183-1190. [10] F.J. Delvos, L. Knoche, Lacunary interpolation by antiperiodic trigonometric polynomials. BIT 39 (1999), 439-450. [11] W. Ding, Y. Xing, M. Han, Antiperiodic boundary value problems for first order impulsive functional differential equations. Appl. Math. Comput. 186 (2007), 45-53. [12] D. Franco, J.J. Nieto, D. O’Regan, Anti-periodic boundary value problem for nonlinear first order ordinary differential equations. Math. Inequal. Appl. 6 (2003), 477-485. [13] D. Franco, J.J. Nieto, Existence of solutions for first order ordinary differential equations with nonlinear boundary conditions. Appl. Math. Comput. 153 (2004), 793-802. [14] B. Liu, An anti-periodic LaSalle oscillation theorem for a class of functional differential equations. J. Comput. Appl. Math. 223 (2009), 1081-1086. 218

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[15] Z. Luo, J. Shen, J.J. Nieto, antiperiodic boundary value problem for first-order impulsive ordinary differential equations. Comput. Math. Appl. 49 (2005), 253-261. [16] M. Nakao, Existence of an anti-periodic solution for the quasilinear wave equation with viscosity. J. Math. Anal. Appl. 204 (1996), 754764. [17] J. Shao, Anti-periodic solutions for shunting inhibitory cellular neural networks with time-varying delays. Phys. Lett. A 372 (2008), 5011-5016. [18] D.R. Smart, Fixed Point Theorems. Cambridge University Press, 1980. [19] P. Souplet, Optimal uniqueness condition for the antiperiodic solutions of some nonlinear parabolic equations. Nonlinear Anal. 32 1998), 279286. [20] Y. Wang, Y.M. Shi, Eigenvalues of second-order difference equations with periodic and antiperiodic boundary conditions. J. Math. Anal. Appl. 309 (2005), 56-69. [21] K. Wang, Y. Li, A note on existence of (anti-)periodic and heteroclinic solutions for a class of second-order odes. Nonlinear Anal. 70 (2009), 1711-1724. [22] Y. Yin, Anti-periodic solutions of some semilinear parabolic boundary value problems. Dynam. Contin. Discrete Impuls. Systems 1 (1995), 283-297.

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The improved  G '/ G  -expansion method to the (2+1)-dimensional breaking soliton equation Hasibun Naher a ,b 1, and Farah Aini Abdullah a a b

School of Mathematical Sciences, Universiti Sains Malaysia, 11800 Penang, Malaysia

Department of Mathematics and Natural Sciences, BRAC University, 66 Mohakhali, Dhaka 1212, Bangladesh Email: [email protected], [email protected] Abstract

In this article, we generate abundant traveling wave solutions of partial differential equation, namely, the (2+1)dimensional breaking soliton equation involving parameter by applying the improved method. In this method, G  G  G  0 together with

F   

U



f  U

where

 G '/ G  -expansion

s f  G '/ G 

f

is implemented,

s f  f  0, 1,  2,...,  U  ,  and  are constants. In addition, the obtained analytical solutions are

illustrated in three different families including solitons and periodic solurions. Further, it is vital mentioning that, for a special case, some of our solutions are in good contract with those gained by other authors. Keywords: The improved

 G '/ G  -expansion method, the breaking soliton equation, solitary solutions, periodic

solutions, nonlinear evolution equations. AMS Subject Classification: 35Q51, 35Q53, 37K10 1. Introduction Nonlinear partial differential equations (PDEs) have become a useful tool for describing complex physical phenomena of mathematical physics, engineering sciences and other scientific real time application fields. Consequently, the study of analytical solutions of PDEs has now become an imperative area to researchers. In the recent past, new exact solutions may help to reveal new phenomena. A wide range of powerful methods are being introduced to obtain analytical solutions, such as, the Hirota’s bilinear transformation method [1], the inverse scattering method [2], the Backlund transformation method [3], the Jacobi elliptic function expansion method [4], the tanh-coth method [5,6], the direct algebraic method [7], the F-expansion method [8,9], the Cole-Hopf transformation method [10], the Exp-function method [11-15] and others [16-20]. Recently, Wang et al. [21] introduced a method, called the basic

 G / G  -expansion

method to construct

traveling wave solutions of some nonlinear evolution equations. In this method, they employed

1

Corresponding author. Tel.: +60103934805 Fax: +60 4 6570910 Email : [email protected]

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i

m

u     ai  G '/ G  as traveling wave solutions, where am  0. Subsequently, many researchers studied i 0

different nonlinear partial differential equations by using this method, such as, [22-27]. In recent times, this basic

 G / G  -expansion

method has been extended by Zhang [28], which is called the improved

expansion method. In this improved method,

F   

U



f  U

solutions, where either

s f  G '/ G 

 G / G  -

f

is applied, as traveling wave

sU or sU may be zero, but both sU and sU cannot be zero at a time. Consequently,

many researchers implemented this powerful method for solving various differential equations to obtain abundant and more general exact traveling wave solutions, for example [29-35]. Many researchers used different methods to investigate the (2+1)-dimensional breaking soliton equation. For instance, Ping [36] constructed exact solutions of this equation by using improved Riccati equation method. In Ref [37], Wazwaz implemented modified Hirota bilinear method to obtain analytical solutions of the same equation whilst Peng [38] executed modified mapping method of the same equation for establishing exact solutions. Bekir and Uygun [39] studied this equation for obtaining traveling wave solutions via the basic

 G '/ G  -expansion

method. In this basic

 G '/ G  -expansion

i

m

method,

u     ai  G '/ G 

where

i 0

am  0, is considered as traveling wave solutions instead of F   

U



f  U

s f  G '/ G 

f

where either

sU or

sU may be zero, but both sU and sU cannot be zero at a time. The importance of this present work is, the (2+1)-dimensional breaking soliton equation is investigated to construct abundant traveling wave solutions including solitons, periodic and rational solutions by applying the improved

 G '/ G  -expansion method.  G '/ G  -expansion method

2. The improved

Consider the general nonlinear partial differential equation:

H  u, ut , ux , u y , uxt , u yt , uxy , ut t , uxx , u yy ,...  0, where

(1)

u  u  x, y, t  is an unknown function, H is a polynomial in u  x, y, t  and the subscripts stand for the

partial derivatives. The main steps of the method [28] are: Step 1. We suppose that combining the real variables x, y and

u  x, y, t   F   , where

  x  y W t,

t by a complex variable  : (2)

W is the speed of the traveling wave. Now using transformation Eq. (2), Eq. (1) is transformed into an

ordinary differential equation (ODE) for

u  F   :

A  F , F ', F , F ,...  0,

(3)

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where A is a function of

F   and the superscripts indicate the ordinary derivatives with respect to  .

Step 2. According to possibility, Eq. (3) can be integrated term by term one or more times, yields constant(s) of integration. The integral constant may be zero, for simplicity. Step 3. Suppose that the traveling wave solution of Eq. (3) can be expressed in the form [28]: U

F   



f  U

with

s f  G '/ G 

f

(4)

G  G   satisfies the second order linear ODE: G  G  G  0,

where

(5)

s f  f  0, 1,  2,...,  U  ,  and  are constants.

Step 4. To determine the positive integer U , taking the homogeneous balance between the highest order nonlinear terms and the highest order derivatives appearing in Eq. (3). Step 5. Substituting Eqs. (4) and (5) into Eq. (3) with the value of U obtained in Step 4. Equating the coefficients of for

 G '/ G  ,  m  0, 1,  2,... , m

then setting each coefficient to zero, we obtain a set of algebraic equations

s f  f  0, 1,  2,...,  U  ,W ,  and .

Step 6. Solve the system of algebraic equations which are obtained in step 5 with the aid of algebraic software Maple and we obtain values for

s f  f  0, 1,  2,...,  U  , W ,  and .

Then, substitute obtained values in Eq. (4) along with Eq. (5) with the value of U , we can obtain the traveling wave solutions of Eq. (1). 3. Application of the method In this section, the (2+1)-dimensional breaking soliton equation has been investigated by applying the improved

 G '/ G  -expansion method for finding abundant new traveling wave solutions. 3.1 The (2+1)-dimensional breaking soliton equation Let us consider the (2+1)-dimensional breaking soliton equation followed by Bekir and Uygun [39]:

ut   uxxy  4 uvx  4 ux v  0,

(6)

u y  vx Making use the traveling wave transformation Eq. (2) into the Eq. (6), which yields:

WF    F   4 Fv  4 F v  0, F   v.

(7)

Integrating the second equation in the system and neglecting constants of integration, we find

F  v.

(8)

Substituting Eq. (8) into the first equation of the system, therefore, integrating with respect

WF  4 F 2   F   0.



once yields: (9)

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Taking the homogeneous balance between the nonlinear term

F 2 and the highest order derivative F  in Eq.

(9), we obtain U  2. Therefore, the solution of Eq. (9) is of the form:

F    s2  G '/ G   s1  G '/ G   s0  s1  G '/ G   s2 G '/ G  , 2

where

1

2

(10)

s2 , s1 , s0 , s1 and s2 are constants to be determined.

Substituting Eq. (10) together with Eq. (5) into the Eq. (9), the left-hand side of Eq. (9) is converted into a polynomial of

 G '/ G  ,  m  0, 1,  2,... . According to Step 5, collecting all terms with the same power of m

 G '/ G  . Then, setting each coefficient of the resulted polynomial to zero, algebraic equations (for simplicity, which are not displayed) for

we obtain a set of simultaneous

s2 , s1 , s0 , s1 , s2 ,W ,  and . Solving the

system of obtained algebraic equations with the help of algebraic software Maple, we obtain four different values. Case 1:

s2  0, s1  0, s0  where

3 3 3 , s1  , s2  , W     2  4  , 2 2 2

(11)

 and  are free parameters.

Case 2:

s2  0, s1  0, s0  where

1 2 3 3   2  , s1  , s2  , W     2  4  ,  4 2 2

(12)

 and  are free parameters.

Case 3:

s2 

3 2 3 3 , s1  , s0  , s1  0, s2  0, W     2  4  , 2 2 2

where

 and  are free parameters.

(13)

Case 4:

3 2 3 1 s2  , s1  , s0    2  2  , s1  0, s2  0, W     2  4  , 2 2 4 where

(14)

 and  are free parameters.

Substituting the general solution Eq. (5) into Eq. (10), we obtain three different families of traveling wave solutions of Eq. (9): Family 1: Hyperbolic function solutions: When

 2  4  0, we obtain

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1 1   H sinh  2  4   L cosh  2  4     1 2 2 2 F    s2     4  1 1 2  2 H cosh  2  4   L sinh  2  4     2 2 1  H sinh   1 2 2  s1     4 1 2 2  H cosh  2 1  H sinh   1 2  s1    2  4 1 2  2 H cosh  2

1 2 1 2   4   L sinh 2 1  2  4   L cosh 2 1  2  4   L sinh 2  2  4   L cosh

2

1

  2  4     s0 2   4      2  4     2  4   

(15.1)

2

1 1   H sinh  2  4   L cosh  2  4     1 2 2 2  s2     4  , 1 1 2  2 H cosh  2  4   L sinh  2  4    2 2 

1 1   H sinh  2  4   L cosh  2  4     1 2 2 v    s2    2  4  1 1 2 2 2  H cosh   4   L sinh  2  4    2 2  1  H sinh   1 2  s1    2  4 1 2  2 H cosh  2 1  H sinh   1 2  s1    2  4 1 2 2  H cosh  2

1   4   L cosh 2 1  2  4   L sinh 2 1  2  4   L cosh 2 1 2   4   L sinh 2 2

2

1

   4     s0 2   4    2

   4    2   4   

(15.2)

2

2

1 1   H sinh  2  4   L cosh  2  4     1 2 2  s2    2  4  , 1 1 2 2 2 2  H cosh   4   L sinh   4    2 2  where

H and L are arbitrary constants. If H , L,  and  take particular values, various known results in the

literature can be rediscovered. Family 2: Trigonometric function solutions: When

 2  4  0, we obtain

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1 1   4   2   L cos 4   2    H sin   1 2 2 2 F    s2   4    1 1 2  2 H cos 4   2   L sin 4   2     2 2

2

1

1 1    H sin 4   2   L cos 4   2     1 2 2 2  s1   4     s0 1 1 2 2 2 2  H cos 4     L sin 4      2 2  (16.1) 1 1    H sin 4   2   L cos 4   2     1 2 2 2 s1   4    1 1 2 2  H cos 4   2   L sin 4   2    2 2  1 1    H sin 4   2   L cos 4   2     1 2 2 2  s2   4    1 1 2  2 H cos 4   2   L sin 4   2    2 2 

1 1   4   2   L cos 4   2    H sin   1 2 2 v    s2   4   2  1 1 2 2 2  H cos 4     L sin 4   2    2 2 

2

2

1

1 1    H sin 4   2   L cos 4   2     1 2 2  s1   4   2   s0 1 1 2 2 2 2  H cos 4     L sin 4      2 2  1 1    H sin 4   2   L cos 4   2     1 2 2 s1   4   2  1 1 2 2 2 2  H cos 4     L sin 4      2 2  1 1    H sin 4   2   L cos 4   2     1 2 2  s2   4   2  1 1 2 2 2  H cos 4     L sin 4   2    2 2  where

(16.2)

2

H and L are arbitrary constants. If H , L,  and  take particular values, various known results in the

literature can be rediscovered. Family 3: Rational function solution: When

 2  4  0, we obtain 2

1

      L  L  L  F    s2       s1    s0  s1   H  L  H  L  H  L   2  2  2 2

  L   s2    , H  L   2

(17.1)

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2

1

      L  L  L  v    s2       s1    s0  s1   H  L  H  L  H  L   2  2  2 2

  L   s2    , H  L   2

(17.2)

Substituting Eqs. (11), (12), (13) and (14) together with the general solution Eq. (5) into the Eq. (10), yields the hyperbolic function solution Eqs. (15.1) and (15.2), our traveling wave solutions become respectively (if H  0 but L  0 ): 2 3   2  4   1   2 F1    1  coth   4       , 8 2     2 3   2  4   1   2 v1      4    , 1  coth  8 2    

where

  x  y     2  4  t.

F2  

 

v2  

 

where

2

2

2  4   1   2   4    , 1  3coth  8 2    2  4   1   2 1  3coth   4       , 8 2    

  x  y     2  4  t.

 3    F3      2   2   3    v3      2   2  where

2

    2  4 1 coth  2  4         2 2 2  

1    2  4 1 2 coth   4    1 ,   2 2   1    2  4 1 coth  2  4    1 ,   2 2  

  x  y     2  4  t.

 3    F4      2   2   3    v4      2   2  where

2

    2  4 1 coth  2  4         2 2 2  

2

    2  4 1 coth  2  4         2 2 2   2

    2  4 1 coth  2  4         2 2 2  

  x  y     2  4  t.

1   1 2  2  4 1 2 coth   4        2  ,   4 2 2   1   1 2  2  4 1 2 coth   4        2  ,   4 2 2  

Again, substituting Eqs. (11), (12), (13) and (14) together with the general solution Eq. (5) into the Eq. (10), we obtain the hyperbolic function solution Eq. (15.1) and (15.2), exact solutions become respectively (if L  0 but

H  0 ):

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F5   

2 3   2  4   1   2 1  tanh   4       , 8 2    

2 3   2  4   1   2 v5      4    . 1  tanh  8 2    

F6  

 

v6  

 

2

2

2  4   1   2   4    , 1  3tanh  8 2    2  4   1   2   4    . 1  3tanh  8 2   

2 1       3     2  4 1  2  4 1 2 2 F7      tanh   4       tanh   4    1 ,   2   2   2 2 2 2 2      2 1       3     2  4 1  2  4 1 v7      tanh  2  4       tanh  2  4    1 .   2   2   2 2 2 2 2      2 1        3    2  4  2  4 1 1 1 F8      tanh  2  4       tanh  2  4       2  2  ,   2   4 2   2 2 2 2 2      2 1       1 2 3     2  4 1  2  4 1 2 2 v8      tanh   4       tanh   4        2 .   2   4 2   2 2 2 2 2      Substituting Eqs. (11), (12), (13) and (14) together with the general solution Eq. (5) into the Eq. (10), yields the

trigonometric function solution Eq. (16.1) and (16.2), we obtain following solutions respectively (if H  0 but

L  0 ): 2 3  4   2   1   2 F9    4      , 1  cot  8 2   

v9    where

2 3  4   2   1   2 1  cot 4         , 8 2    

  x  y     2  4  t.

2   4   2   1   2 F10    1  3cot  4      ,  8 2     2   4   2   1   2 v10    4      , 1  3cot  8 2   

where

  x  y     2  4  t.

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 3    F11      2   2   3      v11    2   2  where

2

   4   2 1  cot 4   2        2 2 2  

  x  y     2  4  t.

 3    F12      2   2   3    v12      2   2  where

2

   4   2 1 cot 4   2         2 2 2  

2

   4   2 1 cot 4   2         2 2 2   2

   4   2 1 cot 4   2         2 2 2  

  x  y     2  4  t.

1   4   2 1 2 cot 4      1 ,   2 2   1   4   2 1 cot 4   2    1 ,   2 2  

   1  2   2  ,  4  1   1 4   2 1 cot 4   2      2   2  ,   4 2 2    4   2 1 cot 4   2    2 2 

1

Also, substituting Eqs. (11), (12), (13) and (14) together with the general solution Eq. (5) into the Eq. (10), yields the trigonometric function solution Eq. (16.1) and (16.2), our solutions become respectively (if L  0 but

H  0 ): 2 3  4   2   1   2 F13    4      , 1  tan  8 2    2 3  4   2   1   2 v13    4      . 1  tan  8 2    2   4   2   1   2 F14    4      , 1  3tan  8 2     2   4   2   1   2 v14    4      . 1  3tan  8 2   

 3    F15      2   2   3    v15      2   2 

2

   4   2 1 tan 4   2         2 2   2 2

   4   2 1 tan 4   2         2 2   2

1   4   2 1 2 tan 4      1 ,   2 2   1   4   2 1 2 tan 4      1 .   2 2  

2 1       1 3    4   2 1 4   2 1 2 2 F16      tan 4         tan 4        2   2  ,   2   4 2   2 2 2 2 2      2 1       1 3    4   2 1 4   2 1 2 2 v16      tan 4         tan 4        2   2 .   2   4 2   2 2 2 2 2      Substituting Eqs. (11), (12), (13) and (14) together with the general solution Eq. (5) into the Eq. (10), we obtain

the rational function solution Eqs. (17.1) and (17.2), our wave solutions become respectively (if

 2  4  0 ):

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 2L  3 F17       2  4     8   H  L   2L  3 v17       2  4     8   H  L 

2

 ,  

2

 .  

 2L  1 F18       2  4   3   8   H  L   2L  1 v18       2  4   3   8   H  L  F19   

2

 ,  

2

 .  

2 1    3    L  L        1 ,     2   2 H  L  H  L   2 

2 1    3    L  L  v19          1  .     2   2 H  L  H  L   2 

  3    L  L  F20           2   2 H  L  H  L   2 2

  3    L  L  v20           2   2 H  L  H  L   2 2

1

 1     2  2  ,   4

1

 1     2  2  .  4 

4. Results and discussion It is important to point out that some of our traveling wave solutions are in good contract with existing results which are depicted in the table. Moreover, some of obtained solutions are shown in figure 1 to figure 8. 4.1 Table. Comparison between Bekir and Uygun [39] solutions and Newly obtained solutions Bekir and Uygun [39] solutions i. If

New solutions

C1  0, C2  0,   2 and   3 solution Eq.

i.If

v1    v1   , solutions F1   and v1  

(4.13) (from section 4) becomes:

3 1  u1    1  coth 2   and 8 2 

become:

3 1  v1    1  coth 2   . 8 2  ii. If

  2,   3, F1    u1   and

3 1  u1    1  coth 2   and 8 2 

3 1  v1    1  coth 2   . 8 2 

C1  0, C2  0,   2 and   3 solution Eq.

ii. If

  2,   3, F2    u2   and

v2    v2   , solutions F2   and v2  

(4.14) (from section 4) becomes:

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1 1  u2    1  3coth 2   and 8 2 

become:

1 1  v2    1  3coth 2   . 8 2 

1 1  v2    1  3coth 2   . 8 2 

iii. If

C1  0, C2  0,   3 and   4 solution Eq.

iii. If

1 1  u2    1  3coth 2   and 8 2 

  3,   4, F5    u1   and

v5    v1   , solutions F5   and v5  

(4.13) (from section 4) becomes:

u1   

3 1  tanh 2   and  2

become:

u1   

v1   

3 1  tanh 2   .  2

v1   

3 1  tanh 2   .  2

iv. If

C1  0, C2  0,   3 and   4 solution Eq.

iv. If

1 u2    1  3tanh 2   and 2

v. If

C1  0, C2  0,   2 and   3 solution Eq.

u2   

v2   

1 1  3tanh 2   .  2

v. If

  v    3 1  cot  2

3

vi. If

 and 2   . 2

become:





2 .



2



and



C1  0, C2  0,   2 and   3 solution Eq. vi. If   3,   2, F10    u4   and

  v     1  3cot 

u4     1  3cot

2

2

vii. If



u3    3 1  cot 2

v3    3 1  cot 2

v10    v4   , solutions F10   and v10  

(4.16) (from section 4) becomes:

4

  3,   2, F9    u3   and

v9    v3   , solutions F9   and v9  

(4.15) (from section 4) becomes:

u3    3 1  cot

1 1  3tanh 2   and  2

become:

1 1  3tanh 2   .  2

2

  3,   4, F6    u2   and

v6    v2   , solutions F6   and v6  

(4.14) (from section 4) becomes:

v2   

3 1  tanh 2   and  2

 and 2  . 2

become:



u4     1  3cot 2



v4     1  3cot 2





2



and



2 .

C1  0, C2  0,   1 and   1 solution Eq. vii. If   1,   1, F13    u3   and

v13    v3   , solutions F13   and v13  

(4.15) (from section 4) becomes:

 3  9 u3     1  tan 2     and  8  2  

become:

 3  9 u3     1  tan 2     and  8   2 

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 3  9 v3     1  tan 2     .  8   2  viii. If

 3  9 v3     1  tan 2     .  8   2 

C1  0, C2  0,   1 and   1 solution

viii. If

  1,   1, F14    u4   and

v14    v4   , solutions F14   and v14  

Eq. (4.16) (from section 4) becomes:

 3  3 u4     1  3tan 2     and  8   2 

become:

 3  3 v4     1  3tan 2     .  8   2 

 3  3 v4     1  3tan 2     .  8  2  

Beyond the table, we obtain new exact traveling wave solutions and

 3  3 u4     1  3tan 2     and  8   2 

F3 , F4 , F7 , F8 , F11 , F12 , F15 , F16 , F17 , F18 , F19

F20 which are not being established in the previous literature.

4.2 Graphical descriptions of the solutions The graphical illustrations of some solutions are described in the following figures with the aid of commercial software Maple:

Fig. 1: Periodic solution for   6,   10,   1.1014

Fig. 2: Solitons solution for   5,   6,   1

Fig. 3: Solitons solution for   3,   2,  5

Fig. 4: Solitons solution for   7,   12,  0.25

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Fig. 5: Periodic solution for   1,   2,  1.109

Fig. 6: Solitons solution for   2,   0.75,   0.5

Fig. 7: Periodic solution for   6,   10,  0.001

Fig. 8: Solitons solution for   6,   10,  0.75

5. Conclusions In this article, the improved

 G '/ G  -expansion method has been successfully applied for constructing abundant

traveling wave solutions including solitons, periodic and rational solutions of the nonlinear evolution equation, namely, the (2+1)-dimensional breaking soliton equation. Furthermore, it is more imperative declaring that some of our solutions are being coincided with existing results, if parameter taken particular values. Moreover, the obtained solutions show that the performance of this method is reliable, effective and more general than the basic

 G '/ G  -expansion method because it can establish many new solutions at a time. Therefore, this straightforward and powerful method can be more successfully implemented to investigate a different class of nonlinear partial differential equations which frequently arise in engineering sciences, mathematical physics and other scientific real time application fields. Acknowledgement: This article is supported by the USM short term grant (Ref. No. 304/PMATHS/6310072) and authors would like to express their thanks to the School of Mathematical Sciences, USM for providing related research facilities.

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A Multiple Attribute Group Decision Making Method based on Generalized Interval-valued Trapezoidal Fuzzy Numbers Chao Liu,

Peide Liu?

School of Management Science and Engineering, Shandong University of Finance and Economics, Jinan Shandong 250014, China [email protected]

Abstract. A ranking approach based on grey correlative coefficient is presented to solve the multiple attribute decision making problems in which the attribute values and the weights take the form of generalized interval-valued trapezoidal fuzzy numbers (GIVTFN). Firstly, the concept, the operational rules and the distance of GIVTFN are given, and the method of linguistic variables converted into GIVTFN is introduced. Secondly, the normalization method of the decision Matrix based on the GIVTFN is proposed, and a grey relational decision making method based on the GIVTFN is presented and decision making steps are illustrated in detail, and the alternatives is ranked based on the grey correlative coefficient. Finally, an illustrate example is given to show the effectiveness of the proposed method. Keywords: interval-valued fuzzy number; grey correlative coefficient; multiple attribute group decision making

1

Introduction

Since the object things are fuzzy, uncertainty and Human thinking is ambiguous, the majority of multi-attribute decision-making is uncertain, which is called fuzzy multiple attribute decision-making (FMADM). Since Bellmanhe and Zadeh [2] firstly proposed the fuzzy decision making model based on the theory of fuzzy mathematics, the research on FMADM has been receiving more and more attentions, and many achievements have been made based on the various fuzzy attribute values, such as interval numbers, triangular fuzzy numbers, and trapezoidal fuzzy numbers etc. Wei and Wei [14], Men and Ji [9] proposed the grey relational analysis method with various attribute values respectively, such as interval numbers and triangular fuzzy numbers. The concept of interval-valued fuzzy set is firstly proposed by Gorzlczany [4] and Turksen [10], and then Wang and Li [11, 12] gave the extended operations of interval-valued fuzzy numbers, and proposed the concept and properties ?

The corresponding author

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2

Chao Liu,

Peide Liu

of similarity coefficient of the interval-valued fuzzy numbers. Hong and Lee [5] proposed the distance of interval-valued fuzzy numbers. Ashtiani, et al [1] proposed the extended TOPSIS method based on the interval-valued triangular fuzzy numbers. Liu [6] proposed an extended TOPSIS method for multiple attribute group decision making based on generalized interval-valued trapezoidal fuzzy numbers (GIVTFN). Wei and Chen [13] proposed similarity measures between GIVTFNs for risk analysis. Liu [7] proposed some aggregation operators, such as the generalized interval-valued trapezoidal fuzzy number weighted aggregation operator (ITWA), the generalized interval-valued trapezoidal fuzzy number ordered weighted aggregation operator (ITOWA), and the generalized interval-valued trapezoidal fuzzy numbers hybrid aggregation operator (ITHA), to solve the FMAGM This paper proposed a decision making method based on the grey correlative coefficient for solving the MADM problems which the attribute weights and values are given with the form of GIVTFN.

2 2.1

The basic concept of the GIVTFN The GIVTFN

(1) The definition of the GIVTFN [13] L L L Wang and Li [12] proposed the GIVTFN A˜˜ = [A˜˜L , A˜˜U ] = [(aL ˜ 1 , a2 , a3 , a4 ; wA ˜L ), U U L L L L U U (a1 , a2 , a3 , a4 ; wA˜˜U )] shown in Fig. 1. Where, 0 ≤ a1 ≤ a2 ≤ a3 ≤ a4 ≤ 1, ˜˜L ˜˜U U U U 0 ≤ aU ˜ ˜ 1 ≤ a2 ≤ a3 ≤ a4 ≤ 1, 0 ≤ wA ˜U ≤ 1 and A ⊂ A . As shown in ˜L ≤ wA Fig. 1, we can conclude that the GIVTFN A˜˜ consists of the lower value A˜˜L and ˜U . the upper value A˜

Fig. 1. generalized interval-valued trapezoidal fuzzy numbers

(2) The operational rules of the GIVTFNs [13] ˜ = [A˜ ˜L , A˜˜U ] = [(aL , aL , aL , aL ; w ˜ ), (aU , aU , aU , aU ; w ˜ )], Suppose that A˜ 4 3 4 1 2 3 1 2 ˜L ˜U A A ˜ ˜ U U U U L ˜ U L L L L ˜ ˜ ˜ B = [B , B ] = [(b1 , b2 , b3 , b4 ; wB˜˜ L ), (b1 , b2 , b3 , b4 ; wB˜˜ U )] are the two GIVTFNs, Then the operational rules are defined shown as follows: ˜ ˜⊕B ˜ = [(aL + bL , aL + bL , aL + bL , aL + bL ; min(w ˜ , w ˜ )), A˜ 1 1 2 2 3 3 4 4 ˜L ˜L A B U U U U U U U (aU ˜ ˜ 1 + b1 , a2 + b2 , a3 + b3 , a4 + b4 ; min(wA ˜U , wB ˜ U ))]

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˜ L L L L L L L ˜⊗B ˜ = [(aL A˜ ˜ ˜ 1 × b1 , a2 × b2 , a3 × b3 , a4 × b4 ; min(wA ˜L , wB ˜ L )),

(2)

U U U U U U U (aU ˜ ˜ 1 × b1 , a2 × b2 , a3 × b3 , a4 × b4 ; min(wA ˜U , wB ˜ U ))]

L L L U U U U ˜ = [(λaL λA˜ ˜ ˜ 1 , λa2 , λa3 , λa4 ; wA ˜L ), (λa1 , λa2 , λa3 , λa4 ; wA ˜U )], λ > 0

(3)

2.2

The distance between two GIVTFNs ˜ = [A˜ ˜L , A˜ ˜˜ = ˜U ] = [(aL , aL , aL , aL ; w ˜ ), (aU , aU , aU , aU ; w ˜ )], B Suppose that A˜ 1 2 3 4 1 2 3 4 ˜L ˜U A A ˜ ˜ ˜ L, B ˜ U ] = [(bL , bL , bL , bL ; w ), (bU , bU , bU , bU ; w )] are the two GIVTFNs, [B 1

2

3

4

˜ ˜L B

1

2

3

4

˜ ˜U B

˜˜ ˜˜ is calculated as follows: then the distance of two GIVTFNs (Aand B) q ˜ ˜ ˜ B) ˜ = (y ˜ − y ˜ )2 + (x ˜ − x ˜ )2 + (y ˜ − y ˜ )2 + (x ˜ − x ˜ )2 /4 d(A, ˜L ˜L ˜L ˜L ˜U ˜U ˜U ˜U A B A B A B A B (4) where (xA˜˜L , yA˜˜L ), (xA˜˜U , yA˜˜U ), (xB˜˜ L , yB˜˜ L ), (xB˜˜ U , yB˜˜ U ) are the coordinate of COG points defined by Chen and Chen [3] for generalized trapezoidal fuzzy numbers ˜ ˜ ˜ U respectively. ˜ L, B ˜U , B ˜L , A˜ A˜ ˜ ˜ ˜ B) ˜ satisfies the following properties: d(A, ˜˜ ≤ 1. ˜˜ B) ˜ ˜ are the normalized GIVTFNs, then 0 ≤ d(A, ˜ and B (i) if A˜ ˜ ˜ ˜ ˜ = 0. ˜ B) ˜ ⇔ d(A, ˜=B (ii) A˜ ˜ ˜ ˜ ˜ ˜ ˜ A). ˜ = d(B, ˜ B) (iii) d(A, ˜˜ ˜˜ B). ˜ ˜ ˜ ˜ ˜ ≥ d(A, ˜ B) ˜ + d(C, ˜ C) (iv) d(A, 2.3

The method converted linguistic terms into the GIVTFNs

In the real decision making, it is difficult to adopt the form of GIVTFN to give the attribute values and weights directly by the decision makers. However, we can adopt the form of linguistic terms easily. Wei and Chen [13] proposed a method from 9-member linguistic terms to the GIVTFNs (see Table 1) Table 1. 9-member linguistic term sets to GIVTFNs linguistic terms linguistic terms GIVTFNs (the attribute values) (weights) Absolutely-poor(AP) Absolutely-low(AL) [(0.00,0.00,0.00,0.00;0.8),(0.00,0.00,0.00,0.00;1.0)] Very-poor(VP) Very-low (VL) [(0.00,0.00,0.02,0.07;0.8),(0.00,0.00,0.02,0.07;1.0)] poor (P) low (L) [(0.04,0.10,0.18,0.23;0.8),(0.04,0.10,0.18,0.23;1.0)] Medium-poor(MP) Medium-low(ML) [(0.17,0.22,0.36,0.42;0.8),(0.17,0.22,0.36,0.42;1.0)] Medium (M) Medium (M) [(0.32,0.41,0.58,0.65;0.8),(0.32,0.41,0.58,0.65;1.0)] Medium-good(MG) Medium-high(MH) [(0.58,0.63,0.80,0.86;0.8),(0.58,0.63,0.80,0.86;1.0)] good(G) high(H) [(0.72,0.78,0.92,0.97;0.8),(0.72,0.78,0.92,0.97;1.0)] Very-good(VG) very-high (VH) [(0.93,0.98,1.00,1.00;0.8),(0.93,0.98,1.00,1.00;1.0)] Absolutely-good(AG) Absolutely-high(AH) [(1.00,1.00,1.00,1.00;0.8),(1.00,1.00,1.00,1.00;1.0)]

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Group decision making method

3.1

Description the decision making problems

Let A = {A1 , A2 , . . . , Am } be the set of alternatives, C = {C1 , C2 , . . . , Cn } be the set of attributes, and E = {e1 , e2 , . . . , eq } be the set of decision makers. Suppose L L L L U U U U U ˜ that a ˜ijk = [(aL ijk1 , aijk2 , aijk3 , aijk4 ; wijk ), (aijk1 , aijk2 , aijk3 , aijk4 ; wijk )] is the attribute value for the alternative Ai with respect to the attribute Cj given by ˜ ˜˜ kj = [(ω L , ω L , ω L , ω L ; η L ), the decision maker ek , and a ˜ijk is a GIVTFN, ω kj1 kj2 kj3 kj4 kj U U U U U (ωkj1 , ωkj2 , ωkj3 , ωkj4 ; ηkj )] is the weight of attribute Cj given by the decision ˜ maker ek , and ω ˜ kj is also a GIVTFN. Let λ = (λ1 , λ2 , . . . , λq ) be the vector of q decision makers, where λk is a real number, and Σk=1 λk = 1. Then we use the attribute weights, the decision maker weights, and the attribute values to rank the alternatives. 3.2

Normalize the decision-making information

In order to eliminate the impact of different physical dimension to the decisionmaking result, we need normalize the decision-making information. Consider that there are generally benefit attributes (I1 ) and cost attributes (I2 ). The normalizing method is shown as follows: L L L L U U U U U ˜ x ˜ijk = [(xL ijk1 , xijk2 , xijk3 , xijk4 ; wijk ), (xijk1 , xijk2 , xijk3 , xijk4 ; wijk )]

= [((

L L L U U U aU aL ijk1 aijk2 aijk3 aijk4 ijk1 aijk2 aijk3 aijk4 L U , , , ; wijk ), ( , , , ; wijk ))] mjk mjk mjk mjk mjk mjk mjk mjk

(5)

for benefit attributes, where mjk = maxi (aU ijk4 ). L L L L U U U U U ˜ x ˜ijk = [(xL ijk1 , xijk2 , xijk3 , xijk4 ; wijk ), (xijk1 , xijk2 , xijk3 , xijk4 ; wijk )] njk njk njk njk njk njk njk njk L U ), ( U , U , U , U ; wijk ))] = [(( L , L , L , L ; wijk aijk1 aijk2 aijk3 aijk4 aijk1 aijk2 aijk3 aijk4

(6)

for cost attributes, where njk = mini (aL ijk1 ). 3.3

Aggregate the evaluation information of each decision maker into the collective information

According to the different attribute values and weights given by different experts, we can get the collective attribute values and weights. The steps are shown as shown as follows: q L L L L U U U U U ˜ij = [(xL x ˜ (λk x ˜˜ijk ) ij1 , xij2 , xij3 , xij4 ; wij ), (xijk1 , xijk2 , xijk3 , xijk4 ; wij )] = Σ k=1

q L L L L U U U U U = Σk=1 {λk × [(xL ijk1 , xijk2 , xijk3 , xijk4 ; wijk ), (xijk1 , xijk2 , xijk3 , xijk4 ; wijk )]} q q q q L L L L = [(Σk=1 (λk xL ijk1 ), Σk=1 (λk xijk2 ), Σk=1 (λk xijk3 ), Σk=1 (λk xijk4 ); min(wijk )),

k q q q q U U U U U (Σk=1 (λk xijk1 ), Σk=1 (λk xijk2 ), Σk=1 (λk xijk3 ), Σk=1 (λk xijk4 ); min(wijk ))] k

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q L L L L U U U U ˜ ˜˜ kj ) ω ˜ j = [(ωj1 , ωj2 , ωj3 , ωj4 ; ηjL ), (ωj1 , ωj2 , ωj3 , ωj4 ; ηjU )] = Σk=1 (λk × ω q L L L L L U U U U U = Σk=1 (λk × [(ωkj1 , ωkj2 , ωkj3 , ωkj4 ; ηkj ), (ωkj1 , ωkj2 , ωkj3 , ωkj4 ; ηkj )]) q q q q L L L L L = [(Σk=1 (λk ωkj1 ), Σk=1 (λk ωkj2 ), Σk=1 (λk ωkj3 ), Σk=1 (λk ωkj4 ); min(ηkj )) (8) k q q q q U U U U U , (Σk=1 (λk ωkj1 ), Σk=1 (λk ωkj2 ), Σk=1 (λk ωkj3 ), Σk=1 (λk ωkj4 ); min(ηkj ))] k

3.4

Construct the weighted matrix

˜ = [v˜ Let V˜ ˜ij ]m×n be the weighted matrix, then L L L L L U U U U U ˜˜ij ⊗ ω ˜˜ j v˜ ˜ij = [(vij1 , vij2 , vij3 , vij4 ; $ij ), (vij1 , vij2 , vij3 , vij4 ; $ij )] = x L L L L L L L L = [(xL ij1 ωj1 , xij2 ωj2 , xij3 ωj3 , xij4 ωj4 ; min(wij , ηj )),

(9)

U U U U U U U U (xU ij1 ωj1 , xij2 ωj2 , xij3 ωj3 , xij4 ωj4 ; min(wij , ηj ))]

3.5

The decision making method based on grey relational theory

(1) Determine the positive ideal solution and the negative ideal solution of the evaluation objects. Suppose that the positive ideal solution and the negative ˜ + = [v˜ ˜j+ ]1×n , V˜˜ − = [v˜˜j− ]1×n , then ideal solution are V˜ L+ L+ L+ L+ U+ U+ U+ U+ v˜ ˜j+ = [(vj1 , vj2 , vj3 , vj4 ; $jL+ ), (vj1 , vj2 , vj3 , vj4 ; $jU + )] L L L L L = [(max(vij1 ), max(vij2 ), max(vij3 ), max(vij4 ); max($ij )), i

i i i i U U U U U (max(vij1 ), max(vij2 ), max(vij3 ), max(vij4 ); max($ij ))] i i i i i

(10)

L− L− L− L− U− U− U− U− v˜ ˜j− = [(vj1 , vj2 , vj3 , vj4 ; $jL− ), (vj1 , vj2 , vj3 , vj4 ; $jU − )] L L L L L = [(min(vij1 ), min(vij2 ), min(vij3 ), min(vij4 ); min($ij )), i

i i i i U U U U U (min(vij1 ), min(vij2 ), min(vij3 ), min(vij4 ); min($ij ))] i i i i i

(11)

(2) Calculate the grey correlative degree of ith alternative and the positive ideal solution with respect to the jth attribute [8]. The grey correlative coefficient of ith alternative and the positive ideal solution with respect to the jth attribute is + rij =

m + ξM , ξ ∈ (0, 1) ∆+ ij + ξM

(12)

+ ˜ where ∆+ ˜j+ , v˜ ˜ij ), m = |{z} min |{z} min ∆+ max max ij = d(v ij , M = |{z} |{z} ∆ij . ξ is a resolution i

j

i

j

coefficient, generally, ξ = 0.5. The grey correlative degree of the ith alternative and the positive ideal solutions is 1 n + Ri+ = Σj=1 rij , (i = 1, 2, . . . , m) (13) n

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(3) Calculate the grey correlative degree of ith alternative and the negative ideal solution with respect to the jth attribute [8]. The grey correlative coefficient of ith alternative and the negative ideal solution with respect to the jth attribute is m + ξM , ξ ∈ (0, 1) ∆− ij + ξM

− rij =

(14)

− ˜ where ∆− ˜j− , v˜ ˜ij ), m = |{z} min |{z} min ∆− max max ij = d(v ij , M = |{z} |{z} ∆ij , ξ is a resolui

j

i

j

tion coefficient, generally, ξ = 0.5. The grey correlative degree of the ith alternative and the negative ideal solutions is 1 n − rij , (i = 1, 2, . . . , m) (15) Ri− = Σj=1 n (4) Calculate the grey correlative similarity coefficient of each alternative. Ci =

Ri+ , (i = 1, 2, . . . , m) + Ri−

(16)

Ri+

The grey correlative similarity coefficient Ci satisfies the property: 0 < Ci < 1. (5) Rank the alternatives Based on the grey correlative similarity coefficient, we can rank all alternatives. The bigger the grey correlative similarity coefficient is, the better prior the alternative is, or vice versa.

4

An Illustrative Example

Suppose that a Telecommunication Company intends to choose a manager for R&D department from four volunteers named A1, A2, A3 and A4 (The data came from [7]) . The decision making committee assesses the four concerned volunteers based on five attributes, including: (1) proficiency in identifying research areas (C1), (2) proficiency in administration (C2), (3) personality (C3), (4) past experience (C4) and (5) self-confidence (C5). The number of the committee members is three, labeled asDM1, DM2, DM3 respectively. Each decision maker has presented his assessment based on linguistic terms for the importance of each attribute and the evaluation information of four volunteers shown in Tables2, 3, 4 and 5 respectively. Decision steps are shown as follows: Table 2. the attribute weights given by three DMs DM1 DM2 DM3

c1 VH VH VH

c2 H H MH

c3 H MH MH

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c4 VH H VH

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Table 3. the evaluation information of four volunteers given by DM1 c1 VG G VG G

a1 a2 a3 a4

c2 VG VG MG F

c3 VG VG G F

c4 VG VG G G

c5 VG MG G MG

Table 4. the evaluation information of four volunteers given by DM2 c1 G G G VG

a1 a2 a3 a4

c2 MG VG G F

c3 G VG MG MG

c4 G VG VG F

c5 VG MG G G

(1) Convert the linguistic terms into the GIVTFNs, we can get the decision data expressed by interval-valued trapezoidal fuzzy numbers. (See [7]). (2) Aggregate the individual preferences in order to obtain a collective preference value for each alternative: 

[(0.743, 0.797, 0.907, 0.943; 0.800), (0.743, 0.797, 0.907, 0.943; 1.000)],  [(0.673, 0.730, 0.880, 0.933; 0.800), (0.673, 0.730, 0.880, 0.933; 1.000)], ˜ [x ˜ij ]4×5 =  [(0.860, 0.913, 0.973, 0.990; 0.800), (0.860, 0.913, 0.973, 0.990; 1.000)], [(0.743, 0.797, 0.907, 0.990; 0.800), (0.743, 0.797, 0.907, 0.943; 1.000)], [(0.610, 0.673, 0.793, 0.837; 0.800), (0.610, 0.673, 0.793, 0.837; 1.000)], [(0.813, 0.863, 0.933, 0.953; 0.800), (0.813, 0.863, 0.933, 0.953; 1.000)], [(0.743, 0.797, 0.907, 0.943; 0.800), (0.743, 0.797, 0.907, 0.943; 1.000)], [(0.523, 0.600, 0.720, 0.767; 0.800), (0.523, 0.600, 0.720, 0.767; 1.000)], [(0.790, 0.847, 0.947, 0.980; 0.800), (0.790, 0.847, 0.947, 0.980; 1.000)], [(0.860, 0.913, 0.973, 0.990; 0.800), (0.860, 0.913, 0.973, 0.990; 1.000)], [(0.743, 0.797, 0.907, 0.943; 0.800), (0.743, 0.797, 0.907, 0.943; 1.000)], [(0.493, 0.557, 0.727, 0.790; 0.800), (0.493, 0.557, 0.727, 0.790; 1.000)], [(0.860, 0.913, 0.973, 0.990; 0.800), (0.860, 0.913, 0.973, 0.990; 1.000)], [(0.813, 0.863, 0.933, 0.953; 0.800), (0.813, 0.863, 0.933, 0.953; 1.000)], [(0.860, 0.913, 0.973, 0.990; 0.800), (0.860, 0.913, 0.973, 0.990; 1.000)], [(0.657, 0.723, 0.833, 0.873; 0.800), (0.657, 0.723, 0.833, 0.873; 1.000)],  [(0.930, 0.980, 1.000, 1.000; 0.800), (0.930, 0.980, 1.000, 1.000; 1.000)] [(0.627, 0.680, 0.840, 0.897; 0.800), (0.627, 0.680, 0.840, 0.897; 1.000)]  [(0.673, 0.730, 0.880, 0.933; 0.800), (0.673, 0.730, 0.880, 0.933; 1.000)]  [(0.540, 0.607, 0.767, 0.827; 0.800), (0.540, 0.607, 0.767, 0.827; 1.000)]

Table 5. the evaluation information of four volunteers given by DM3 a1 a2 a3 a4

c1 MG MG VG MG

c2 F MG VG VG

c3 G G VG MG

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c4 VG MG VG VG

c5 VG G MG F

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[[(0.930, 0.980, 1.000, 1.000; 0.800), (0.930, 0.980, 1.000, 1.000; 1.000)], [(0.627, 0.680, 0.840, 0.897; 0.800), (0.627, 0.680, 0.840, 0.897; 1.000)], ˜ [ω ˜ j ]5 = [(0.627, 0.680, 0.840, 0.897; 0.800), (0.627, 0.680, 0.840, 0.897; 1.000)], [(0.930, 0.980, 1.000, 1.000; 0.800), (0.930, 0.980, 1.000, 1.000; 1.000)], [(0.320, 0.410, 0.580, 0.650; 0.800), (0.320, 0.410, 0.580, 0.650; 1.000)]] (3) Calculate the weighted decision making matrix: 

[(0.691, 0.781, 0.907, 0.943; 0.800), (0.691, 0.781, 0.907, 0.943; 1.000)],  [(0.626, 0.715, 0.880, 0.933; 0.800), (0.626, 0.715, 0.880, 0.933; 1.000)], ˜ [v˜ij ]4×5 =  [(0.800, 0.895, 0.973, 0.990; 0.800), (0.800, 0.895, 0.973, 0.990; 1.000)], [(0.691, 0.781, 0.907, 0.943; 0.800), (0.691, 0.781, 0.907, 0.943; 1.000)], [(0.382, 0.458, 0.666, 0.750; 0.800), (0.382, 0.458, 0.666, 0.750; 1.000)], [(0.510, 0.587, 0.784, 0.855; 0.800), (0.510, 0.587, 0.784, 0.855; 1.000)], [(0.466, 0.542, 0.762, 0.846; 0.800), (0.466, 0.542, 0.762, 0.846; 1.000)], [(0.328, 0.408, 0.605, 0.687; 0.800), (0.328, 0.408, 0.605, 0.687; 1.000)], [(0.495, 0.576, 0.795, 0.879; 0.800), (0.495, 0.576, 0.795, 0.879; 1.000)], [(0.539, 0.621, 0.818, 0.888; 0.800), (0.539, 0.621, 0.818, 0.888; 1.000)], [(0.466, 0.542, 0.762, 0.846; 0.800), (0.466, 0.542, 0.762, 0.846; 1.000)], [(0.309, 0.379, 0.610, 0.708; 0.800), (0.309, 0.379, 0.610, 0.708; 1.000)], [(0.800, 0.895, 0.973, 0.990; 0.800), (0.800, 0.895, 0.973, 0.990; 1.000)], [(0.756, 0.846, 0.933, 0.953; 0.800), (0.756, 0.846, 0.933, 0.953; 1.000)], [(0.800, 0.895, 0.973, 0.990; 0.800), (0.800, 0.895, 0.973, 0.990; 1.000)], [(0.611, 0.709, 0.833, 0.873; 0.800), (0.611, 0.709, 0.833, 0.873; 1.000)],  [(0.298, 0.402, 0.580, 0.650; 0.800), (0.298, 0.402, 0.580, 0.650; 1.000)] [(0.201, 0.279, 0.487, 0.583; 0.800), (0.201, 0.279, 0.487, 0.583; 1.000)]  [(0.215, 0.299, 0.510, 0.607; 0.800), (0.215, 0.299, 0.510, 0.607; 1.000)]  [(0.173, 0.249, 0.445, 0.537; 0.800), (0.173, 0.249, 0.445, 0.537; 1.000)]

(4) Determine the positive ideal solution and the negative ideal solution: [[(0.800, 0.895, 0.973, 0.990; 0.800), (0.800, 0.895, 0.973, 0.990; 1.000)], [(0.510, 0.587, 0.784, 0.855; 0.800), (0.510, 0.587, 0.784, 0.855; 1.000)], ˜ V˜ + = [(0.539, 0.621, 0.818, 0.888; 0.800), (0.539, 0.621, 0.818, 0.888; 1.000)], [(0.800, 0.895, 0.973, 0.990; 0.800), (0.800, 0.895, 0.973, 0.990; 1.000)], [(0.298, 0.402, 0.580, 0.650; 0.800), (0.298, 0.402, 0.580, 0.650; 1.000)]] [[(0.626, 0.715, 0.880, 0.933; 0.800), (0.626, 0.715, 0.880, 0.933; 1.000)], [(0.328, 0.408, 0.605, 0.687; 0.800), (0.328, 0.408, 0.605, 0.687; 1.000)], ˜ − = [(0.309, 0.379, 0.601, 0.708; 0.800), (0.309, 0.379, 0.610, 0.708; 1.000)], V˜ [(0.611, 0.709, 0.833, 0.873; 0.800), (0.611, 0.709, 0.833, 0.873; 1.000)], [(0.173, 0.249, 0.445, 0.537; 0.800), (0.173, 0.249, 0.445, 0.537; 1.000)]] (5) Calculate the grey correlative coefficient matrix:  0.5610 0.4722 0.7826 1.0000 1.0000 0.4608 1.0000 1.0000 0.7178 0.5338 R+ =  1.0000 0.7825 0.6328 1.0000 0.5947   0.5610 0.3766 0.3333 0.4045 0.4516  0.7207 0.6501 0.3673 0.4045 0.4516 1.0000 0.3766 0.3333 0.4809 0.7451 R− =  0.4608 0.4205 0.4132 0.4045 0.6521  0.7207 1.0000 1.0000 1.0000 1.0000 (6) Calculate the grey correlative similarity coefficient of each alternative: C = (0.5953, 0.5584, 0.6304, 0.3106)

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A Multiple Attribute Group Decision

9

(7) Rank the alternatives: Based on the grey correlative similarity coefficient, we can rank the alternatives: a3  a1  a2  a4 . (8) Analysis: In this example, the proposed method produces the same ranking as [1] and [7], which proves the method in this paper is effective. Comparing with [1] and [7], the advantages proposed in this paper are more general and simpler in dealing with more complex problems of fuzzy multiple attribute decision making.

5

Conclusion

Fuzzy multiple attribute decision making (FMADM) problems widely exist in the real decision-making, and the GIVTFN can be precisely express the attribute values and weights of the FMADM problems. In this paper, we proposed a decision making method based on the grey correlative coefficient to solve the MADM problems in which the attribute weights and values are given by the GIVTFN, and decision making steps were given in detail. Comparing with [1] and [7], the advantages proposed in this paper are more general and simpler in dealing with the fuzzy multiple attribute decision making. This method enriches and develops the theory and method of FMADM, and gives a new idea for solving the FMADM problems. In the future, we will research the applications of the propose method. Acknowledgement This paper is supported by the National Natural Science Foundation of China (No. 71271124, No.61273230), the National Social Science Foundation of China (No. 11BJY147), the New Century Excellent Talents Support Program of Ministry of Education of China (No. NCET-12-1027), and the Natural Science Foundation of Shandong Province (No.ZR2011FM036).

References 1. B. Ashtiani, F. Haghighirad, A. Makui, G. Montazer, Extension of fuzzy TOPSIS method based on interval-valued fuzzy sets, Applied Soft Computing, 9, 457461(2009). 2. R.E. Bellman, L.A. Zadeh, Decision-making in a fuzzy environment, Management Science, 17, 141-164 (1970). 3. S.J. Chen, S.M. Chen, A new method for handing multicriteria fuzzy decisionmaking problems using FN-IOWA operators, Cybernetics and Systems, 34,109137(2003). 4. M.B. Gorzalczany, A method of inference in approximate reasoning based on interval-valued fuzzy sets, Fuzzy Sets and Systems, 21, 1-17 (1987). 5. D.H. Hong, S. Lee, Some algebraic properties and a distance measure for intervalvalued fuzzy numbers, Information Sciences, 148, 1-10 (2002).

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Chao Liu,

Peide Liu

6. P.D. Liu, An Extended TOPSIS Method for Multiple Attribute Group Decision Making based on Generalized Interval-valued Trapezoidal Fuzzy Numbers, Informatica, 35, 185-196 (2011). 7. P.D. Liu, A weighted aggregation operators multi-attribute group decision-making method based on interval-valued trapezoidal fuzzy numbers, Expert Systems with Applications, 38, 1053-1060 (2011). 8. S.F. Liu, T.B. Guo, Y.G. Dang, Grey system theory and application, Beijing: Science Press, 1999. 9. F. Men, S.Q. Ji, An improved FMEA based on Fuzzy Set theory and Grey Relational Theory, Industrial engineering and management, 2, 55-59 (2008). 10. I.B. Turksen, Interval-valued strict preference with Zadeh triples, Fuzzy Sets and Systems, 78, 183-195 (1996). 11. G. Wang, X. Li, The applications of interval-valued fuzzy numbers and intervaldistribution numbers, Fuzzy Sets and Systems, 98, 331-335 (1998). 12. G. Wang, X. Li, Correlation and information energy of interval-valued fuzzy numbers, Fuzzy Sets and Systems, 103, 69-175 (2001). 13. S.H. Wei, S.M. Chen, Fuzzy risk analysis based on interval-valued fuzzy numbers, Expert Systems with Applications, 36, 2285-2299 (2009). 14. G.W. Wei, Y. Wei, Model of Grey Relational Analysis for Interval Multiple Attribute Decision Making with Preference Information on Alternatives, Chinese Journal of Management Science, 16, 158-162 (2008).

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Note on the second kind Barnes’ type multiple q-Euler polynomials C. S. Ryoo Department of Mathematics, Hannam University, Daejeon 306-791, Korea Abstract : In this paper we introduce the second kind Barnes-type multiple q-Euler numbers and polynomials, by using fermionic p-adic invariant integral on Zp . We give some interesting properties. Key words : The second kind q-Euler numbers and polynomials, the second kind Barnes-type multiple q-Euler numbers and polynomials 2000 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, 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) =

pN −1 Zp



g(x)dμ−1 (x) = lim

N →∞

From (1.1), we note that   g(x + 1)dμ−1 (x) + Zp

Zp

g(x)(−1)x , see [1, 2, 3].

(1.1)

x=0

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(see [4]):  tn 2et = En . e2t + 1 n=0 n! ∞

(1.3)

We introduce the second kind Euler polynomials En (x) as follows: tn 2et xt  e = En (x) . e2t + 1 n! n=0 ∞

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(1.4)

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In [4], we studied the second kind Euler numbers En and polynomials En (x) and investigate their properties. The main aim of this paper is to study the second kind Barnes-type multiple q-Euler polynomials, by using fermionic p-adic invariant integral on Zp . 2. The second kind Barnes-type multiple q-Euler polynomials In this section, we assume that w1 , . . . , wk ∈ Zp and a1 , . . . , ak ∈ Z. We introduce the second kind Barnes-type multiple q-Euler polynomials, En,q (w1 , . . . , wk ; a1 , . . . , ak | x). For k ∈ N, we define the second kind Barnes-type multiple q-Euler polynomials as follows: 



q a1 x1 +···+ak xk e(x+2w1 x1 +···+2wk xk +k)t dμ−1 (x1 ) · · · dμ−1 (xk ) ··· Z Z  p  p k−times 2k ekt ext = a1 2w1 t (q e + 1)(q a2 e2w2 t + 1) · · · (q ak e2wk t + 1) ∞  tn = En,q (w1 , . . . , wk ; a1 , . . . , ak | x) . n! n=0

(2.1)

In the special case, x = 0, En,q (w1 , . . . , wk ; a1 , . . . , ak | 0) = En,q (w1 , . . . , wk ; a1 , . . . , ak ) are called the second kind n-th Barnes-type multiple q-Euler numbers. Theorem 1. For positive integers n and k, we have En,q (w1 , . . . , wk ; a1 , . . . , ak | x)   q a1 x1 +···+ak xk (x + 2w1 x1 + · · · + 2wk xk + k)n dμ−1 (x1 ) · · · dμ−1 (xk ). ··· = Z Z  p  p k−times By using the above Theorem 1, we have the following corollary. Corollary 2. For positive integers n, we have En,q (w1 , . . . , wk ; a1 , . . . , ak )    k q i=1 ai xi (2w1 x1 + · · · + 2wk xk + k)n dμ−1 (x1 ) · · · dμ−1 (xk ). ··· = Z Z  p  p k−times

(2.2)

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

n   n l=0

247

l

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

(2.3)

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where

n k

is a binomial coefficient.

In the special case, (w1 , . . . , wk ; a1 , . . . , ak ) = (1, . . . , 1; 1, . . . , 1), we have       k−times k−times (k) (x), En,q (w1 , . . . , wk ; a1 , . . . , ak | x) = En,q (k)

where En,q (x) denotes the second kind q-Euler polynomials of higher order(see [5]). We define distribution relation of the second kind Barnes-type multiple q-Euler polynomials as follows: For m ∈ N with m ≡ 1( mod 2), we obtain ∞ 

tn En,q (w1 , . . . , wk ; a1 , . . . , ak | x) n! n=0 =

2k ekmt (q a1 m e2w1 mt + 1)(q a2 m e2w2 mt + 1) · · · (q ak m e2wk mt + 1) ⎛ ⎞ x + 2w1 l1 + · · · + 2wk lk + k − mk ⎝ ⎠(mt) m−1  k m l1 +···+lk a l i i (−1) q i=1 e . × l1 ,...,lk =0

From the above, we obtain ∞ 

k tn  n  En,q (w1 , . . . , wk ; a1 , . . . , ak | x) = m (−1)l1 +···+lk q i=1 ai li n! n=0 n=0 l1 ,...,lk =0  x + 2w1 l1 + · · · + 2wk lk + k − mk tn . × En,qm w1 , . . . , wk ; a1 , . . . , ak | m n! m−1

∞

By comparing coefficients of theorem.

tn in the above equation, we arrive at the following n!

Theorem 3 (Distribution relation). For m ∈ N with m ≡ 1( mod 2), we have m−1 

n

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

k

i=1

a i li

l1 ,...,lk =0

 × En,qm

(−1)l1 +···+lk q

x + 2w1 l1 + · · · + 2wk lk + k − mk w 1 , . . . , w k ; a1 , . . . , a k | m

From (2.1), we derive   q a1 x1 +···+ak xk e(x+2w1 x1 +···+2wk xk +k)t dμ−1 (x1 ) · · · dμ−1 (xk ) ··· Z Z  p  p k−times ∞  k k =2 (−1)m1 +···+mk q i=1 ai mi e(x+2w1 m1 +···+2wk mk +k)t .

.

(2.4)

m1 ,...mk =0

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From (2.2) and (2.4), we note that En,q (w1 , . . . , wk ; a1 , . . . , ak | x) ∞  k k =2 (−1)m1 +···+mk q i=1 ai mi (x + 2w1 m1 + · · · + 2wk mk + k)n .

(2.5)

m1 ,...mk =0

By using binomial expansion and (2.1), we have the following addition theorem. Theorem 4(Addition theorem). The second kind Barnes-type multiple q-Euler polynomials En,q (w1 , . . . , wk ; a1 , . . . , ak | x) satisfies the following relation: En,q (w1 , . . . , wk ; a1 , . . . , ak | x + y) n   n = El,q (w1 , . . . , wk ; a1 , . . . , ak | x)y n−l . l l=0

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

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

(3.1)

Hence, by (3.1), we obtain ∞ 

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

n=0

=2

∞ 

k

(−1)m1 +···+mk q

k

i=1

tn n!

ai mi (x+2w1 m1 +···+2wk mk +k)t

e

.

m1 ,...mk =0

By applying derivative operator,

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

En,q (w1 , . . . , wk ; a1 , . . . , ak | x) ∞  k k =2 (−1)m1 +···+mk q i=1 ai mi (x + 2w1 m1 + · · · + 2wk mk + k)n .

(3.2)

m1 ,...mk =0

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By (3.2), we define the second kind Barnes-type multiple q-Euler zeta function ζq (w1 , . . . , wk ; a1 , . . . , ak | s, x) as follows: Definition 1. For s, x ∈ C with Re(x) > 0, a1 , . . . , ak ∈ C, we define ζq (w1 , . . . , wk ; a1 , . . . , ak | s, x) ∞ 

= 2k

m1 ,...,mk

k

(−1)m1 +···+mk 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 5. For positive integer l, we have ζq (w1 , . . . , wk ; a1 , . . . , ak | −l, x) = El,q (w1 , . . . , wk ; a1 , . . . , ak | x). By (2.6), we define the second kind multiple q-Euler zeta function ζq (w1 , . . . , wk ; a1 , . . . , ak | s) as follows: Definition 2. For s ∈ C, we define ∞ 

k

ζq (w1 , . . . , wk ; a1 , . . . , ak | s) = 2

m1 ,...,mk

k

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

(3.4)

For s = −l in (3.4) and using (2.6), we arrive at the following theorem. Theorem 6. For positive integer l, we have ζq (w1 , . . . , wk ; a1 , . . . , ak | −l) = El,q (w1 , . . . , wk ; a1 , . . . , ak ). REFERENCES 1. T. Kim, On Euler-Barnes multiple zeta function, Russ. J. Math. Phys., 10(2003), 261-267. 2. T. Kim, Barnes-type multiple q-zeta functions and q-Euler polynomials, J. Phys. A : Math. Theor., 43(2010) 255201(11pp). 3. T. Kim, q-Volkenborn integration, Russ. J. Math. Phys., 9(2002), 288-299. 4. C. S. Ryoo, Calculating zeros of the second kind Euler polynomials, J. Comput. Anal. Appl. 12 (2010), 828-833. 5. C. S. Ryoo, A Note on the second kind q-Euler polynomials of higher order, Applied Mathematics Sciences, 5 (2011), 3421-3427. 250

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The Approximation and Growth Problem of Dirichlet Series of Infinite Order ∗ Hua Wanga and Hong-Yan Xub† a

Department of Informatics and Engineering, Jingdezhen Ceramic Institute, Jingdezhen, Jiangxi, 333403, China e-mail: [email protected]

b

Department of Informatics and Engineering, Jingdezhen Ceramic Institute, Jingdezhen, Jiangxi, 333403, China e-mail: [email protected]

Abstract By introducing the concept of βU -order, we first investigate the growth of Dirichlet series of infinite order which convergence in the half plane, and a necessary and sufficient conditions on the growth of Dirichlet series with finite βU -order has been obtained. We also investigate the error in approximating Dirichlet series of finite order βU -order in the half plane by Dirichlet polynomials. Some relations between the error and growth of Dirichlet series of finite βU -order have been obtained. Key words: growth, βU -order, approximation, Dirichlet series. 2010 Mathematics Subject Classification: 30B50, 30D15.

1

Introduction and Basic Notes Consider Dirichlet series f (s) =

∞ X

an eλn s ,

s = σ + it,

(1)

n=1

where 0 ≤ λ1 < λ2 < · · · < λn < · · · , λn → ∞ as n → ∞;

(2)

s = σ + it (σ, t are real variables); an are nonzero complex numbers and lim sup(λn+1 − λn ) = h < +∞,

(3)

n→+∞

lim sup n→+∞

log+ |an | = 0, λn

(4)

then from (2), by using the similar method in [20] or [16, 17], we can get lim sup n→∞

n = E < +∞, λn

lim sup n→∞

log n = 0. λn

(5)

∗ This work was supported by the NNSF of China(11301233, 61202313), the Natural Science Foundation of Jiang-Xi Province in China(Grant No. 2010GQS0119, No.20132BAB211001). † Corresponding author

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Then the abscissas of convergence and absolute convergence is 0, that is, f (s) is an analytic function in the left half plane H = {s = σ + it : σ < 0, t ∈ R}. We denote D to be the class of all functions f (s) satisfying (2)-(4) and analytic in Res < 0, denote Dα to be the class of all functions f (s) satisfying (2)-(3) and analytic in Re ≤ α where −∞ < α < +∞. Thus, if −∞ < α < 0 and f (s) ∈ D, then f (s) ∈ Dα ; if 0 < α < +∞ and f (s) ∈ Dα , then f (s) ∈ D. We denote Πk to be the class of all exponential polynomial of degree almost k, that is,   k X  Πk = bj eλj s : (b1 , b2 , . . . , bk ) ∈ Ck .   j=1

For f (s) ∈ D, M (σ, f ) =

max

−∞ 0). Then and if the real function M (x) satisfies lim supx→∞ β(log log x we have β(ϕ(x) log M (x)) lim sup = ν. log x x→∞

Proof: The proof of this lemma can be found in [19], for the convenience of the reader, we give the process of proof of this lemma as follows. Two following cases will be considered. Case 1. If ϕ(x) is not a constant. From the assumptions of Lemma 2.1, we can get that ϕ(x) → ∞ as x → ∞. Then, for sufficiently large x, we have ϕ(x) > 1. From β(x) ∈ F , we have limx→∞ log M (x) = ∞. Then from the Cauchy mean value theorem, there exists ξ(log M (x) < ξ < β(x) log M (x)) satisfying β(ϕ(x) log M (x)) − β(log M (x)) β 0 (ξ) = = ξβ 0 (ξ), log(ϕ(x) log M (x)) − log log M (x) (log ξ)0 that is, β(ϕ(x) log M (x)) = β(log M (x)) + log ϕ(x)ξβ 0 (ξ).

(8)

lim supx→∞ loglogϕ(x) x

Since xβ 0 (x) = o(1) as x → +∞ and = %, (0 ≤ % < ∞), by (8), we can get the conclusion of Lemma 2.1. Case 2. If ϕ(x) is a constant. By using the same argument as in Case 1, we can prove that the conclusion of Lemma 2.1 is true. Thus, this completes the proof of Lemma 2.1. The following lemma is very important in study of the growth of analytic functions represented by Dirichlet series convergent in the right half plane, which show the relation between M (σ, f ) and m(σ, f ) of such functions. Lemma 2.2 ([20]) If Dirichlet series (1) satisfies (2) (3), then for any given ε ∈ (0, 1) and for σ(< 0) sufficiently reaching 0, we have 1 m(σ, f ) ≤ M (σ, f ) ≤ K(ε)(− )m((1 − ε)σ, f ), σ where K(ε) is a positive constant depending on ε and f (s).

3

The proof of Theorem 1.3

We first prove ” ⇐= ” of Theorem 1.3. Suppose that lim sup n→+∞

β(log+ |an |)   = T, log U log+λn|an |

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(9)

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then for any ε(> 0) and sufficiently large n, we have    λn log+ |an | < γ (T + ε) log U , log+ |an | where γ(x) is the inverse function of β(x). Let V (x) is the inverse function of U (x), then    λn 1 + > V exp β(log |an |) , T +ε log+ |an | and

λn

log+ |an |
I, then from (10) and (12), we can get +

λn σ

log |an |e

! −1 1 + ≤ λn β(log |an |) +σ T +ε !    −1 1 β(I) +σ ≤ λn V exp T +ε σ = λn < 0. 1 + log2 U (− σ1 )    V exp

(14)

From (13) and (14), we have   1 log m(σ, f ) ≤ γ (T + 2ε) log U (− ) . σ

256

(15)

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From (15) and Lemma 2.2, and since ε is arbitrary, it follows lim sup σ→0−

β(log M (σ, f )) β(log+ |an |) = T. ≤ lim sup 1 λn log U (− σ ) ) n→+∞ log U ( log+ |a | n

Suppose that lim sup σ→0−

Take δ = have

β(log M (σ, f )) β(log+ |an |) = J < lim sup = T. λn log U (− σ1 ) ) n→+∞ log U ( log+ |a |

(16)

n

T −J 5 ,

then for any positive integer n and sufficiently small σ(< 0), we   1 (17) log+ |an |eλn σ ≤ log M (σ, f ) < γ (J + δ) log U (− ) , σ

and from (16), there exists a subsequence {n(ν)} satisfying +

β(log |an(ν) |) >

(T − δ) log U (

!

λn(ν) log+ |an(ν) |

) .

(18)

Choose the sequence {σν } satisfying   log+ |an(ν) | 1 γ (J + δ) log U (− ) = λ λ σν 1 + log U ( log+n(ν) ) log2 log U ( log+n(ν) |a | |a

. (19)

n(ν) |

n(ν)

)

From (17), if follows log+ |an(ν) |eλn(ν) σν
0, for sufficiently large n, we have    λn + , log An < γ (T + τ ) log U log+ An

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where γ(x) is the inverse functions of β(x). Let V (x) and U (x) be two reciprocally inverse functions, then we have    1 λn + V exp β(log An ) < , T +τ log+ An    −1 1 log+ An ≤ λn V exp β(log+ An ) . T +τ Thus, we have +

λn σ

log (An e

) ≤ λn

   V exp

! −1 1 + β(log An ) +σ . T +τ

(23)

For any fixed and sufficiently small σ < 0, set G=γ

(T + τ ) log U

1 1
− − σ σ log2 U − σ1

!! ,

that is,  1 exp β(G) . (24) T +τ  n o 1 If log+ An ≤ G, for sufficiently large n, let V exp T +τ β(log+ An ) ≥ 1, from σ < 0,(23),(24) and the definition of U (x), we have !    −1 1 + + λn σ log An e ≤ λn V exp β(log An ) +σ T +τ !! 1 1
≤ G = γ (T + τ ) log U − − σ σ log2 U − σ1     1 ≤ γ (T + τ ) log (1 + o(1))U − . (25) σ 1 1
=V + −σ −σ log2 U − σ1





If log+ An > G, from (23) and (24), we have +

λn σ

log An e

! −1 1 ≤ λn V2 exp β(G) +σ T +τ   !−1 1 1
≤ λn  + + σ  < 0. −σ −σ log2 U − σ1 





(26)

For sufficiently large n, from (25) and (26), we have     1 + λn σ log An e ≤ γ (T + τ ) log (1 + o(1))U − . σ From the definition of En (f, α), there exists p(s) ∈ Πn−1 and a real constant K2 (> 1) such that ||f − p||α ≤ K2 En−1 (f, α). (27) Since f (s) ∈ Dα and from [20, P.16], for any real numbers t0 , ϑ(6= 0), we have Z Z 1 R ϑit 1 R lim e dt = 0, an eαλn = lim f (α + it)e−λn it dt. (28) R→+∞ R t R→∞ R t 0 0

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From (28), for any real number x 6= 0, we have 1 lim R→∞ R

Z

R

ex(α+it) dt = 0.

(29)

t0

Thus, from (28) and (29), for any p1 (s) ∈ Πn−1 , we have 1 R→∞ R

an eαλn = lim

Z

R

[f (α + it) − p1 (α + it)]e−λn it dt,

t0

that is, |an |eαλn ≤ ||f − p1 ||α .

(30)

From (27) and (30), we can get |an |eαλn ≤ K2 En−1 (f, α),

(31)

where K2 > 1 is a real constant. Since An = En−1 (f, α)e−αλn and τ is arbitrary, from (31), by Lemma 2.1 and Theorem 1.2, we can get lim sup σ→0−

β(log+ M (σ, f )) ≤ T. log U (− σ1 )

Suppose that lim sup σ→0+

β(log+ M (σ, f )) = η < T. log U (− σ1 )

Thus, there exists any real number ε(0 < ε < T −η 4 ), for any sufficient small σ < 0, from Lemma 2.2, we have   1 log M (σ, f ) ≤ γ (η + ε) log U (− ) . (32) σ For any sufficiently small σ < 0 and −∞ < α < σ < 0, we have En−1 (f, α) ≤ ||f − pn−1 ||α ≤

∞ X

|ak |eλk α ≤ M (σ, f )

∞ X

eλn (α−σ) ,

(33)

k=n

k=n

Pn−1 where pn−1 (s) = k=1 ak eλk s . From (3), we take 0 < h0 < h satisfying λn+1 − λn ≥ h0 for a sub-sequence of {n}. Thus, for sufficiently small σ < 0 such that σ ≥ α2 , from (33) we have En−1 (f, α) ≤ M (σ, f )eλn (α−σ)

∞ X

e(λk −λn )(α−σ)

k=n α

0

≤ M (σ, f )eλn (α−σ) e− 2 h n

∞ X

0

α

e2h k

k=n



= M (σ, f )eλn (α−σ) 1 − e

α 0 2h

−1

.

Then for sufficiently small σ < 0 and −∞ < α < σ < 0, we have M (σ, f ) ≥ K3 En−1 (f, α)e−λn (α−σ) = K3 An eλn σ ,

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(34)

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α

0

where K3 = 1 − e 2 h . From (32) and (34), noting the properties of the function γ(x), we have   1 log+ An eλn σ ≤ log M (σ, f ) ≤ γ (η + 2ε) log U (− ) . (35) σ From (22), there exists a subsequence {λn(p) }, for sufficiently large p, we have ! λn(p) + . (36) β(log An(p) ) > (T − ε) log U log+ An(p) Take a sequence {σp } satisfying   log+ An(p) 1 γ (η + 2ε) log U (− ) = λ λ σp 1 + log U ( log+n(p) ) log2 log U ( log+n(p) A A n(p)

.

(37)

)

n(p)

From (35) and (37), we get  1 log An(p) + λn(p) σp ≤ γ (η + 2ε) log U (− ) σp 

+

=

log+ An(p) λ

,

λ

1 + log U ( log+n(p) A

n(p)

) log2 log U ( log+n(p) A

n(p)

)

that is, λn(p) 1 − ≤ σp log+ An(p)

1 + log U (

λn(p) log+ An(p)

2

) log log U (

!

λn(p) log+ An(p)

) .

Thus, we have U (− ≤U

1 ) σp λn(p)

log+ An(p)

≤U 1+o(1)

1 + log U (

λn(p)

λn(p) log+ An(p)

2

) log log U (

λn(p) log+ An(p)

!! )

(38)

!

log+ An(p)

.

From (37) and (38), we have log+ An(p) !   λn(p) λn(p) 1 2 =γ (η + 2ε) log U ( ) 1 + log U ( + ) log log U ( + ) . σp log An(p) log An(p) Thus, from the and (38), thereexists a real number  Cauchy mean value  theorem  λ 1 ξ between γ (η + 2ε) log U ( σp ) and γ (η + 2ε) log U ( σ1p ) (1+log U ( log+n(p) ) A n(p)

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λ

log2 log U ( log+n(p) A

n(p)

)) such that


β log+ An(p) !   λn(p) λn(p) 1 2 =β γ (η + 2ε) log U ( ) (1 + log U ( + ) log log U ( + )) σp log An(p) log An(p) !! λn(p) =β γ (η + 2ε)(1 + o(1)) log U ( + ) log An(p) ! λn(p) λn(p) 2 ) log log U ( + ) ξβ 0 (ξ), + log 1 + log U ( + log An(p) log An(p) Since lim

 λ log 1 + log U ( log+n(p) A

λ

n(p)

) log2 log U ( log+n(p) A

n(p)

λ

p→∞

log U ( log+n(p) A

n(p)

 ) = 0,

)

then for sufficiently large p, we have
β log+ An(p) =(η + 2ε)(1 + o(1)) log U ( + K3 ξβ 0 (ξ) log U ( where K3 is a constant. From (36),(39) and 0 < ε < get lim sup σ→0−

T −η 4 ,

λn(p) log+ An(p)

λn(p) log+ An(p)

)

(39)

),

we can get a contradiction. Thus, we can

β(log+ M (σ, f )) = T. log U (− σ1 )

Hence, the sufficiency of Theorem 1.6 is completed. We can prove the necessity of Theorem 1.6 by using the similar argument as in the proof of the sufficiency of Theorem 1.6. Thus, the proof of Theorem 1.6 is completed.

References [1] P.V. Filevich, M.N. Sheremeta, Regularly increasing entire Dirichlet series, Mathematical Notes 74 (2003), 110-122; Translated from Matematicheskie Zametki 74 (2003), 118-131. [2] Z. S. Gao , The growth of entire functions represented by Dirichlet series, Acta Mathematica Sinica 42 (1999), 741-748(in Chinese). [3] Z. D. Gu, D. C. Sun, The reuglar growth of Dirichlet series on the whole plane, Acta Mathematica Scientia 31 (2011), 991-997(in Chinese). [4] L. Kinnunen, Linear differential equations with solutions of finite iterated order, Southeast Asian Bull. Math. 22 (4) (1998), 385-405. [5] Y.Y. Kong, On some q-order and q-types of Dirichlet-Hadamard function, Acta Mathematica Sinica, Chinese Series. 52(6) (2009), 1165-1172. [6] M.S. Liu, The regular growth of Dirichlet series of finite order in the half plane, J. Sys. Sci. & Math. Scis. 22(2) (2002), 229-238. [7] A. Mishkelyavichyus, A Tauberian theorem for the Laplace-Stieltjes integral and the Dirichlet series (Russian) Litovsk. Mast. Sb. 29(4) (1989), 745-753.

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[8] A. Nautiyal, On the coefficients of analytic Dirichlet series of fast growth, Indian J. Pure Appl. Math. 15(10),1984, 1102-1114. [9] J.H. Ning, C.F. Yi, W.P. Huang, Regular growth of the generalized Dirichlet series, Acta Mathematica Scientia 31 (2012), 379-386(in Chinese). [10] L.N. Shang, Z.S. Gao, Entire functions defined by Dirichlet series, J. Math. Anal. Appl. 339, (2008), 853-862. [11] D. C. Sun, The existence theorem of Nevanlinna direction, Chin. Ann. of Math. 7A (1986), 212-221 (in Chinese). [12] D.C. Sun, On the distribution of values of random Dirichlet series II, Chinese Ann. Math. Ser. B 11(1) (1990), 33-44. [13] D.C. Sun, The growth of Dirichlet series, J.Analysis, 3, 1995, 73-86. [14] D.C. Sun, T.W. Chen, Random Dirichlet series of infinite order, Acta Mathematica Sinica 44 (2001), 259-268(in Chinese). [15] D.C. Sun, Z.S. Gao, The growth of Dirichlet series in the half plane, Acta Mathematica Scientia. 22A(4) (2002), 557-563. [16] G. Valiron, Entire function and Borel’s directions, Proc. Nat. Acad. Sci. USA. 20 (1934), 211-215. [17] G. Valiron, Fonction entireres d’order fini et fonction m´eromorphes, Genere, L’Enseigenment Mathematique 1960. [18] H.Y. Xu, C.F. Yi, The approximation problem of Dirichlet series of finite order in the half plane, Acta Mathematica Sinica 53 (3) (2010), 617-624(in Chinese). [19] H.Y. Xu, C.F. Yi, The growth and approoximation of Dirichlet series of infinite order, Advances in Mathematics, in press(in Chinese). [20] J.R. Yu, X.Q. Ding, F.J. Tian, On The Distribution of Values of Dirichlet Series And Random Dirichlet Series, Wuhan: Press in Wuhan University, 2004(in Chinese). [21] J.R. Yu, Dirichlet series and the random Dirichlet series, Beijing, Science Press, 1997. [22] J.R. Yu, Some properties of random Dirichlet series (in Chinese), Acta Math. Sinica 21 (1978), 97-118. ´ [23] J.R. Yu, Sur les droites de Borel de certaines fonction enti`eres, Annales Ecole Norm. Sup. 68(3) (1951), 65-104. [24] J.R. Yu, D.C. Sun, On the distribution of values of random Dirichlet series(I), Lectures on Comp. Anal. Singapore, World Scientific, 1988.

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WEAK AND STRONG CONVERGENCE THEOREMS OF PROXIMAL POINT ALGORITHM FOR SOLVING GENERALIZED MIXED EQUILIBRIUM PROBLEMS AND FINDING ZEROES OF MAXIMAL MONOTONE OPERATORS IN BANACH SPACES WITHUN PHUENGRATTANA1 , SUTHEP SUANTAI1 , KRIENGSAK WATTANAWITOON2 , UAMPORN WITTHAYARAT3 AND POOM KUMAM3,∗

Abstract. Based on the results proposed by Li and Song [Modified proximal-point algorithm for maximal monotone operators in Banach spaces], J. Optim. Theory appl. 138 (2008) 45-64.], we modify and generate our new iterative scheme for solving generalized mixed equilibrium problems and finding zeroes of maximal monotone operators in a Banach space under the appropriate conditions. We also prove strong and weak convergence theorems of this proximal point algorithm and give an example with numerical test which corresponding to our main results. Furthermore, we also consider the convex minimization problem and the problem of finding a zero point of an α-inverse strongly monotone operator as its appplications.

1. Introduction Let E be a Banach space, C be a closed convex subset of E, and let S be a mapping from C into itself. A point p in C is said to be an asymptotic fixed point of S (see [22]) if C contains a sequence {xn } which converges weakly to p such that limn→∞ ∥xn − Sxn ∥ = 0. The set of asymptotic fixed points of S will be denoted by F[ (S). A mapping S from C into itself is said to be relatively nonexpansive [ [21, 26, 37] if F (S) = F (S) and ϕ(p, Sx) ≤ ϕ(p, x) for all x ∈ C and p ∈ F (S). The asymptotic behavior of a relatively nonexpansive mapping was studied in [6, 7]. A mapping S is said to be ϕ-nonexpansive, if ϕ(Sx, Sy) ≤ ϕ(x, y) for x, y ∈ C. A mapping S is said to be quasi ϕ-nonexpansive if F (S) ̸= ∅ and ϕ(p, Sx) ≤ ϕ(p, x) for x ∈ C and p ∈ F (S). Let E be a Banach space with norm ∥ · ∥, C be a nonempty closed convex subset of E and let E ∗ be the dual of E. Let Θ : C × C −→ R be a bifunction, φ : C −→ R be a real-valued function, and B : C −→ E ∗ be a nonlinear mapping. The generalized mixed equilibrium problem, which is to find x ∈ C such that Θ(x, y) + ⟨Bx, y − x⟩ + φ(y) − φ(x) ≥ 0, ∀y ∈ C. (1.1) The solutions set to (1.1) is denoted by Ω, i.e., Ω = {x ∈ C : Θ(x, y) + ⟨Bx, y − x⟩ + φ(y) − φ(x) ≥ 0,

∀y ∈ C}.

(1.2)

If B = 0, the problem (1.1) reduce into the mixed equilibrium problem for Θ, denoted by M EP (Θ, φ), which is to find x ∈ C such that Θ(x, y) + φ(y) − φ(x) ≥ 0,

∀y ∈ C.

(1.3)

If Θ ≡ 0, the problem (1.1) reduce into the mixed variational inequality of Browder type, denoted by V I(C, B, φ), is to find x ∈ C such that ⟨Bx, y − x⟩ + φ(y) − φ(x) ≥ 0,

∀y ∈ C.

(1.4)

If B = 0 and φ = 0 the problem (1.1) reduce into the equilibrium problem for Θ, denoted by EP (Θ), is to find x ∈ C such that Θ(x, y) ≥ 0, ∀y ∈ C. (1.5) The above formulation (1.5) was shown in [5] to cover monotone inclusion problems, saddle point problems, variational inequality problems, minimization problems, optimization problems, variational 2000 Mathematics Subject Classification. :47H09, 47H10. Key words and phrases. Strong Convergence, Weak Convergence, Proximal Point Algorithm, Generalized Mixed Equilibrium Problems, Maximal Monotone Operators. ∗ Corresponding author: [email protected]. 264 KUMAM ET AL 264-281 1

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inequality problems, vector equilibrium problems, Nash equilibria in noncooperative games. In addition, there are several other problems, for example, the complementarity problem, fixed point problem and optimization problem, which can also be written in the form of an EP (Θ). In other words, the EP (Θ) is an unifying model for several problems arising in physics, engineering, science, optimization, economics, etc. In the last two decades, many papers have appeared in the literature on the existence of solutions of EP (Θ); see, for example [5, 11, 19, 29] and references therein. Some solution methods have been proposed to solve the EP (Θ); see, for example, [8, 11, 24, 25, 27, 28, 29] and references therein. In 2005, Combettes and Hirstoaga [8] introduced an iterative scheme of finding the best approximation to the initial data when EP (Θ) is nonempty and they also proved a strong convergence theorem. Let E be a Banach space with norm ∥ · ∥, C be a nonempty closed convex subset of E and let E ∗ denote the dual of E. Let A be a monotone operator of C into E ∗ . The variational inequality problem is to find a point x ∈ C such that ⟨Ax, y − x⟩ ≥ 0 for all y ∈ C.

(1.6)

The set of solutions of the variational inequality problem is denoted by V I(C, A). Such a problem is connected with the convex minimization problem, the complementarity problem, the problem of finding a point u ∈ E satisfying 0 = Au and so on. An operator A of C into E ∗ is said to be inverse-strongly monotone, if there exists a positive real number α such that ⟨x − y, Ax − Ay⟩ ≥ α∥Ax − Ay∥2

(1.7)

for all x, y ∈ C. In such a case, A is said to be α-inverse-strongly monotone. If an operator A of C into E ∗ is α-inverse-strongly monotone, then A is Lipschitz continuous, that is ∥Ax − Ay∥ ≤ α1 ∥x − y∥ for all x, y ∈ C. In Hilbert space H, Iiduka et al. [13] proved that the sequence {xn } defined by: x1 = x ∈ C and xn+1 = PC (xn − λn Axn ),

(1.8)

where PC is the metric projection of H onto C and {λn } is a sequence of positive real numbers, converges weakly to some element of V I(C, A). In 2008, Iiduka and Takahashi [12] introduced the following iterative scheme for finding a solution of the variational inequality problem for an inverse-strongly monotone operator A that satisfies the following conditions in a 2-uniformly convex and uniformly smooth Banach space E: (C1) A is inverse-strongly monotone, (C2) V I(C, A) ̸= ∅, (C3) ∥Ay∥ ≤ ∥Ay − Au∥ for all y ∈ C and u ∈ V I(C, A). Let x1 = x ∈ C and xn+1 = ΠC J −1 (Jxn − λn Axn ) (1.9) for every n = 1, 2, 3, . . ., where ΠC is the generalized metric projection from E onto C, J is the duality mapping from E into E ∗ and {λn } is a sequence of positive real numbers. They proved that the sequence {xn } generated by (1.9) converges weakly to some element of V I(C, A). Consider the problem of finding: v ∈ E such that 0 ∈ T (v),

(1.10)

∗

where T is an operator from E into E . Such v ∈ E is called a zero point of T . When T is a maximal monotone operator, a well-know methods for solving (1.10) in a Hilbert space H is the proximal point algorithm: x1 = x ∈ H and, xn+1 = Jrn xn , n = 1, 2, 3, . . . , (1.11) where {rn } ⊂ (0, ∞) and Jrn = (I + rn T )−1 J, then Rockafellar [23] proved that the sequence {xn } converges weakly to an element of T −1 (0). In 2000, Kamimura and Takahashi [16] proved the following strong convergence theorem in Hilbert spaces, by the following algorithm xn+1 = αn x + (1 − αn )Jrn xn , n = 1, 2, 3, . . . ,

(1.12)

−1

where Jr = (I + rT ) J, then the sequence {xn } converges strongly to PT −1 (0) (x), where PT −1 (0) is the projection from H onto T −1 (0). These results were extended to more general Banach spaces see 265 [15, 17].

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WEAK AND STRONG CONVERGENCE THEOREMS OF PROXIMAL POINT ALGORITHM

3

In 2003, Kohsaka and Takahashi [17] introduced the following iterative sequence for a maximal monotone operator T in a smooth and uniformly convex Banach space: x1 = x ∈ E and xn+1 = J −1 (αn Jx + (1 − αn )J(Jrn xn )), n = 1, 2, 3, . . . , ∗

(1.13)

−1

where J is the duality mapping from E into E and Jr = (I + rT ) J. In 2004, Kamimura et al. [14] considered the algorithm (1.14) in a uniformly smooth and uniformly convex Banach space E, namely xn+1 = J −1 (αn Jxn + (1 − αn )J(Jrn xn )), n = 1, 2, 3, . . . .

(1.14)

−1

They proved that the algorithm (1.14) converges weakly to some element of T 0. In 2008, Li and Song [18] proved a strong convergence theorem in a Banach space, by the following algorithm: x1 = x ∈ E and yn = J −1 (βn J(xn ) + (1 − βn )J(Jrn xn )), xn+1 = J −1 (αn Jx + (1 − αn )J(yn )),

(1.15)

with ∑∞ the coefficient sequences {αn }, {βn } ⊂ [0, 1] and {rn } ⊂ (0, ∞) satisfying limn−→∞ αn = 0, n=1 αn = ∞, lim n−→∞ βn = 0, and lim n−→∞ rn = ∞, where J is the duality mapping from E into E ∗ and Jr = (I + rT )−1 J. Then they proved that the sequence {xn } converges strongly to ΠC x, where ΠC is the generalized projection from E onto C. In this paper, motivated and inspired by Kamimura et al. [14], Li and Song [18], Iiduka and Takahashi [12] and Zhang [38], we introduce the new hybrid algorithm defined by: x1 = x ∈ C  un ∈ C such thatΘ(un , y) + ⟨Bun , y − un ⟩ + φ(y) − φ(un )     + r1n ⟨y − un , Jun − Jxn ⟩ ≥ 0, ∀y ∈ C,  (1.16) zn = ΠC J −1 (Jun − λn Aun ),  −1  y = J (β J(x ) + (1 − β )J(J z )),  n n n rn n   n xn+1 = ΠC J −1 (αn J(x) + (1 − αn )J(yn )). Under appropriate difference conditions, we will prove that the sequence {xn } generated by algorithms (1.16) converges strongly to the point ΠΩ∩V I(C,A)∩T −1 (0) x and converges weakly to the point limn−→∞ ΠΩ∩V I(C,A)∩T −1 (0) xn . The results presented in this paper extend and improve the corresponding ones announced by Kamimura et al. [14], Li and Song [18] and some authors in the literature. Furthermore, in the last section, we will state example for that satisfies the condition (C1)-(C3) and also we consider the minimization problem and the complementarity problem. 2. Preliminaries A Banach space E is said to be strictly convex if ∥ x+y 2 ∥ < 1 for all x, y ∈ E with ∥x∥ = ∥y∥ = 1 and x ̸= y. Let U = {x ∈ E : ∥x∥ = 1} be the unit sphere of E. Then the Banach space E is said to be smooth provided ∥x + ty∥ − ∥x∥ lim t→0 t exists for each x, y ∈ U. It is also said to be uniformly smooth if the limit is attained uniformly for x, y ∈ E. The modulus of convexity of E is the function δ : [0, 2] → [0, 1] defined by x+y δ(ε) = inf{1 − ∥ ∥ : x, y ∈ E, ∥x∥ = ∥y∥ = 1, ∥x − y∥ ≥ ε}. (2.1) 2 A Banach space E is uniformly convex if and only if δ(ε) > 0 for all ε ∈ (0, 2]. Let p be a fixed real number with p ≥ 2. A Banach space E is said to be p-uniformly convex if there exists a constant c > 0 such that δ(ε) ≥ cεp for all ε ∈ [0, 2]; see [3, 31] for more details. Observe that every p-uniform convex is uniformly convex. One should note that no a Banach space is p-uniform convex for 1 < p < 2. It is well known that a Hilbert space is 2 -uniformly convex and uniformly smooth. For each p > 1, the ∗ generalized duality mapping Jp : E → 2E is defined by Jp (x) = {x∗ ∈ E ∗ : ⟨x, x∗ ⟩ = ∥x∥p , ∥x∗ ∥ = ∥x∥p−1 }

(2.2)

for all x ∈ E. In particular, J = J2 is called the normalized duality mapping. If E is a Hilbert space, then J = I, where I is the identity mapping. It is also known that if E is uniformly smooth, then J is uniformly norm-to-norm continuous on each bounded subset of E. 266 We know the following (see [30]):

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(1) if E is smooth, then J is single-valued; (2) if E is strictly convex, then J is one-to-one and ⟨x − y, x∗ − y ∗ ⟩ > 0 holds for all (x, x∗ ), (y, y ∗ ) ∈ J with x ̸= y; (3) if E is reflexive, then J is surjective; (4) if E is uniformly convex, then it is reflexive; (5) if E ∗ is uniformly convex, then J is uniformly norm-to-norm continuous on each bounded subset of E. The duality J from a smooth Banach space E into E ∗ is said to be weakly sequentially continuous [10] if xn ⇀ x implies Jxn ⇀∗ Jx, where ⇀∗ implies the weak∗ convergence. Lemma 2.1. ([4, 35]). If E be a 2-uniformly convex Banach space. Then, for all x, y ∈ E we have 2 ∥Jx − Jy∥, c2 where J is the normalized duality mapping of E and 0 < c ≤ 1. ∥x − y∥ ≤

The best constant

1 in Lemma is called the 2 -uniformly convex constant of E; see [3]. c

Lemma 2.2. ([4, 36]). If E be a p-uniformly convex Banach space and let p be a given real number with p ≥ 2. Then for all x, y ∈ E, Jx ∈ Jp (x) and Jy ∈ Jp (y) ⟨x − y, Jx − Jy⟩ ≥

cp 2p−2 p

where Jp is the generalized duality mapping of E and

∥x − y∥p ,

1 is the p-uniformly convexity constant of E. c

Lemma 2.3. (Xu [35]). Let E be a uniformly convex Banach space. Then for each r > 0, there exists a strictly increasing, continuous and convex function g : [0, ∞) −→ [0, ∞) such that g(0) = 0 and ∥λx + (1 − λy)∥2 ≤ λ∥x∥2 + (1 − λ)∥y∥2 − λ(1 − λ)g(∥x − y∥)

(2.3)

for all x, y ∈ {z ∈ E : ∥z∥ ≤ r} and λ ∈ [0, 1]. Let E be a smooth, strictly convex and reflexive Banach space and let C be a nonempty closed convex subset of E. Throughout this paper, we denote by ϕ the function defined by ϕ(x, y) = ∥x∥2 − 2⟨x, Jy⟩ + ∥y∥2 ,

for x, y ∈ E.

(2.4)

Following Alber [1], the generalized projection ΠC : E → C is a map that assigns to an arbitrary point x ∈ E the minimum point of the functional ϕ(x, y), that is, ΠC x = x ¯, where x ¯ is the solution to the minimization problem ϕ(¯ x, x) = inf ϕ(y, x) (2.5) y∈C

existence and uniqueness of the operator ΠC follows from the properties of the functional ϕ(x, y) and strict monotonicity of the mapping J. It is obvious from the definition of function ϕ that (see [1]) (∥y∥ − ∥x∥)2 ≤ ϕ(y, x) ≤ (∥y∥ + ∥x∥)2 ,

∀x, y ∈ E.

(2.6)

If E is a Hilbert space, then ϕ(x, y) = ∥x − y∥2 . If E is a reflexive, strictly convex and smooth Banach space, then for x, y ∈ E, ϕ(x, y) = 0 if and only if x = y. It is sufficient to show that if ϕ(x, y) = 0 then x = y. From (2.6), we have ∥x∥ = ∥y∥. This implies that ⟨x, Jy⟩ = ∥x∥2 = ∥Jy∥2 . From the definition of J, one has Jx = Jy. Therefore, we have x = y; see [9, 30] for more details. Lemma 2.4. (Kamimura and Takahashi [15]). Let E be a uniformly convex and smooth real Banach space and let {xn }, {yn } be two sequences of E. If ϕ(xn , yn ) → 0 and either {xn } or {yn } is bounded, then ∥xn − yn ∥ → 0. Lemma 2.5. (Alber [1]). Let C be a nonempty closed convex subset of a smooth Banach space E and x ∈ E. Then, x0 = ΠC x if and only if ⟨x0 − y, Jx − Jx 267 0 ⟩ ≥ 0,

∀y ∈ C.

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Lemma 2.6. (Alber [1]). Let E be a reflexive, strictly convex and smooth Banach space, let C be a nonempty closed convex subset of E and let x ∈ E. Then ϕ(y, ΠC x) + ϕ(ΠC x, x) ≤ ϕ(y, x),

∀y ∈ C.

Let E be a strictly convex, smooth and reflexive Banach space, let J be the duality mapping from E into E ∗ . Then J −1 is also single-valued, one-to-one, and surjective, and it is the duality mapping from E ∗ into E. Define a function V : E × E ∗ −→ R as follows (see [17]): V (x, x∗ ) = ∥x∥2 − 2⟨x, x∗ ⟩ + ∥x∗ ∥2

(2.7)

for all x ∈ Ex ∈ E and x∗ ∈ E ∗ . Then, it is obvious that V (x, x∗ ) = ϕ(x, J −1 (x∗ )) and V (x, J(y)) = ϕ(x, y). Lemma 2.7. (Kohsaka and Takahashi [17, Lemma 3.2]). Let E be a strictly convex, smooth and reflexive Banach space, and let V be as in (2.7). Then V (x, x∗ ) + 2⟨J −1 (x∗ ) − x, y ∗ ⟩ ≤ V (x, x∗ + y ∗ )

(2.8)

for all x ∈ E and x∗ , y ∗ ∈ E ∗ . Let E be a reflexive, strictly convex and smooth Banach space. Let C be a closed convex subset of E. Because ϕ(x, y) is strictly convex and coercive in the first variable, we know that the minimization problem inf y∈C ϕ(x, y) has a unique solution. The operator ΠC x := arg miny∈C ϕ(x, y) is said to be the generalized projection of x on C. A set-valued mapping T : E −→ E ∗ with domain D(T ) = {x ∈ E : T (x) ̸= ∅} and range R(T ) = ∗ {x ∈ E ∗ : x∗ ∈ T (x), x ∈ D(T )} is said to be monotone if ⟨x − y, x∗ − y ∗ ⟩ ≥ 0 for all x∗ ∈ T (x), y ∗ ∈ T (y). We denote the set {s ∈ E : 0 ∈ T x} by T −1 (0). T is maximal monotone if its graph G(T ) is not properly contained in the graph of any other monotone operator. If T is maximal monotone, then the solution set T −1 (0) is closed and convex. Let E be a reflexive, strictly convex and smooth Banach space, it is knows that T is a maximal monotone if and only if R(J + rT ) = E ∗ for all r > 0. Define the resolvent of T by Jr x = xr . In other words, Jr = (J + rT )−1 J for all r > 0. Jr is a single-valued mapping from E to D(T ). Also, T −1 (0) = F (Jr ) for all r > 0, where F (Jr ) is the set of all fixed points of Jr . Define, for r > 0, the Yosida approximation of T by Tr = (J − JJr )/r. We know that Tr x ∈ T (Jr x) for all r > 0 and x ∈ E. Lemma 2.8. (Kohsaka and Takahashi [17, Lemma 3.1]). Let E be a smooth, strictly convex and reflexive Banach space, let A ⊂ E × E ∗ be a maximal monotone operator with T −1 (0) ̸= ∅, let r > 0 and let Jr = (J + rT )−1 J. Then ϕ(x, Jr y) + ϕ(Jr y, y) ≤ ϕ(x, y) for all x ∈ T

−1

0 and y ∈ E.

Let A be an inverse-strongly monotone mapping of C into E ∗ which is said to be hemicontinuous if for all x, y ∈ C, the mapping F of [0, 1] into E ∗ , defined by F (t) = A(tx + (1 − t)y), is continuous with respect to the weak∗ topology of E ∗ . We define by NC (v) the normal cone for C at a point v ∈ C, that is, NC (v) = {x∗ ∈ E ∗ : ⟨v − y, x∗ ⟩ ≥ 0, ∀y ∈ C}. (2.9) Theorem 2.9. (Rockafellar [23]). Let C be a nonempty, closed convex subset of a Banach space E and A a monotone, hemicontinuous operator of C into E ∗ . Let T ⊂ E × E ∗ be an operator defined as follows: { Av + NC (v), v ∈ C; Tv = (2.10) ∅, otherwise. Then T is maximal monotone and T −1 (0) = V I(C, A). Lemma 2.10. (Tan and Xu [33]). Let {an } and {bn } be two sequence of nonnegative real numbers satisfying the inequality an+1 = an + bn for all n ≥ 0. ∑∞ If n=1 bn < ∞, then limn−→∞ an exists. 268

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W. PHUENGRATTANA, S. SUANTAI, K. WATTANAWITOON, U. WITTHAYARAT AND P. KUMAM

Lemma 2.11. (Xu [34]). Let {sn } be a sequence of nonnegative real numbers satisfying sn+1 = (1 − αn )sn + αn tn + rn n ≥ 1, ∑∞ where {αn }, {tn }, and {rn } satisfy {αn } ⊂ [0, 1], n=1 αn = ∞, lim supn−→∞ tn ≤ 0 and rn ≥ 0, ∑ ∞ n=1 rn < ∞. Then limn−→∞ sn = 0. For solving the mixed equilibrium problem, let us assume that the bifunction Θ : C × C → R and φ : C → R is convex and lower semi-continuous satisfies the following conditions: (A1) Θ(x, x) = 0 for all x ∈ C; (A2) Θ is monotone, i.e., Θ(x, y) + Θ(y, x) ≤ 0 for all x, y ∈ C; (A3) for each x, y, z ∈ C, lim sup Θ(tz + (1 − t)x, y) ≤ Θ(x, y); t↓0

(A4) for each x ∈ C, y 7→ Θ(x, y) is convex and lower semi-continuous. Lemma 2.12. (Blum and Oettli [5]). Let C be a closed convex subset of a uniformly smooth, strictly convex and reflexive Banach space E and let Θ be a bifunction of C × C into R satisfying (A1)-(A4). Let r > 0 and x ∈ E. Then, there exists z ∈ C such that 1 Θ(z, y) + ⟨y − z, z − x⟩ ≥ 0 for all y ∈ C. r Lemma 2.13. (Takahashi and Zembayashi [32]). Let C be a closed convex subset of a uniformly smooth, strictly convex and reflexive Banach space E and let Θ be a bifunction from C × C to R satisfying (A1)-(A4). For all r > 0 and x ∈ E, define a mapping Tr : E −→ C as follows: 1 Tr x = {z ∈ C : Θ(z, y) + ⟨y − z, Jz − Jx⟩ ≥ 0, ∀y ∈ C} (2.11) r for all x ∈ E. Then, the followings hold: (1) Tr is single-valued; (2) Tr is a firmly nonexpansive-type mapping, i.e., for all x, y ∈ E, ⟨Tr x − Tr y, JTr x − JTr y⟩ ≤ ⟨Tr x − Tr y, Jx − Jy⟩; (3) F (Tr ) = EP (Θ); (4) EP (Θ) is closed and convex. Lemma 2.14. (Takahashi and Zembayashi [32]). Let C be a closed convex subset of a smooth, strictly convex, and reflexive Banach space E, let Θ be a bifunction from C × C to R satisfying (A1)-(A4) and let r > 0. Then, for x ∈ E and q ∈ F (Tr ), ϕ(q, Tr x) + ϕ(Tr x, x) ≤ ϕ(q, x). Lemma 2.15. (Zhang [38]). Let C be a closed convex subset of a smooth, strictly convex and reflexive Banach space E. Let B : C −→ E ∗ be a continuous and monotone mapping, φ : C → R is convex and lower semi-continuous and Θ be a bifunction from C × C to R satisfying (A1)-(A4). For r > 0 and x ∈ E, then there exists u ∈ C such that 1 Θ(u, y) + ⟨Bu, y − u⟩ + φ(y) − φ(u) + ⟨y − u, Ju − Jx⟩ ≥ 0, ∀y ∈ C. r Define a mapping Kr : C −→ C as follows: 1 Kr (x) = {u ∈ C : Θ(u, y) + ⟨Bu, y − u⟩ + φ(y) − φ(u) + ⟨y − u, Ju − Jx⟩ ≥ 0, ∀y ∈ C} (2.12) r for all x ∈ E. Then, the followings hold: (i) Kr is single-valued; (ii) Kr is firmly nonexpansive, i.e., for all x, y ∈ E, ⟨Kr x − Kr y, JKr x − JKr y⟩ ≤ ⟨Kr x − Kr y, Jx − Jy⟩; (iii) F (Kr ) = Ω; (iv) Ω is closed and convex; (v) ϕ(p, Kr z) + ϕ(Kr z, z) ≤ ϕ(p, z) ∀p ∈ F (Kr ), z ∈ E. Remark 2.16. (Zhang [38]). It follows from Lemma 2.13 that the mapping Kr : C −→ C defined by (2.12) is a relatively nonexpansive mapping. Thus, 269it is quasi-ϕ-nonexpansive.

KUMAM ET AL 264-281

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3. Strong Convergence Theorem In this section, we prove a strong convergence theorem for finding a common element of the set of solutions of mixed equilibrium problems, the set of solution of the variational inequality problem and the zero point of a maximal monotone operators in a Banach space by using the shrinking hybrid projection method. Theorem 3.1. Let E be a 2-uniformly convex and uniformly smooth Banach space, let C be a nonempty closed convex subset of E. Let Θ be a bifunction from C × C to R satisfying (A1)-(A4) let φ : C −→ R be a proper lower semicontinuous and convex function, let T : E −→ E ∗ be a maximal monotone operator satisfying D(T ) ⊂ C. Let Jr = (J + rT )−1 J for r > 0 and let B : C −→ E ∗ be a continuous and monotone mappings, with F := Ω ∩ V I(C, A) ∩ T −1 (0) ̸= ∅. Assume that A an operator of C into E ∗ that satisfies the conditions (C1)-(C3). Let {xn } be a sequence generated by x1 = x ∈ C and,  un ∈ C such that Θ(un , y) + ⟨Bun , y − un ⟩ + φ(y) − φ(un )     + r1n ⟨y − un , Jun − Jxn ⟩ ≥ 0, ∀y ∈ C,  (3.1) zn = ΠC J −1 (Jun − λn Aun ),  −1   yn = J (βn J(xn ) + (1 − βn )J(Jrn zn )),   xn+1 = ΠC J −1 (αn J(x) + (1 − αn )J(yn )), for all n ∈ N, where ΠC is the generalized projection from E onto C, J is the duality mapping on E. ∑∞ The coefficient sequence {αn }, {βn } ⊂ (0, 1], {rn } ⊂ (0, ∞) satisfying limn−→∞ αn = 0, n=1 αn = ∞, c2 α 1 lim supn−→∞ βn < 1, lim inf n−→∞ rn > 0 and {λn } ⊂ [a, b] for some a, b with 0 < a < b < , is 2 c the 2-uniformly convexity constant of E. Then the sequence {xn } converges strongly to ΠF x. Proof . Let H(un , y) = Θ(un , y) + ⟨Bun , y − un ⟩ + φ(y) − φ(un ), y ∈ C and Krn = {u ∈ C : H(un , y) + 1 −1 (Jun −λn Aun ), rn ⟨y−un , Jun −Jxn ⟩ ≥ 0, ∀y ∈ C}. We first show that {xn } is bounded. Put vn = J −1 let p ∈ F := Ω ∩ V I(C, A) ∩ T (0) and un = Krn xn . Since Jrn are relatively nonexpansive mappings. By (3.1) ϕ(p, un ) = ϕ(p, Krn xn ) ≤ ϕ(p, xn )

(3.2)

and Lemma 2.7, the convexity of the function V in the second variable, we have ϕ(p, zn ) =

ϕ(p, ΠC vn )

≤

ϕ(p, vn ) = ϕ(p, J −1 (Jun − λn Aun ))

≤ =

V (p, Jun − λn Aun + λn Aun ) − 2⟨J −1 (Jun − λn Aun ) − p, λn Aun ⟩ V (p, Jun ) − 2λn ⟨vn − p, Aun ⟩

=

ϕ(p, un ) − 2λn ⟨un − p, Aun ⟩ + 2⟨vn − un , −λn Aun ⟩.

(3.3)

Since p ∈ V I(C, A) and A is α-inverse-strongly monotone, we have − 2λn ⟨un − p, Aun ⟩ = ≤

−2λn ⟨un − p, Aun − Ap⟩ − 2λn ⟨un − p, Ap⟩ −2αλn ∥Aun − Ap∥2 ,

(3.4)

and by Lemma 2.1, we obtain 2⟨vn − un , −λn Aun ⟩

= 2⟨J −1 (Jun − λn Aun ) − un , −λn Aun ⟩

≤ 2∥J −1 (Jun − λn Aun ) − un ∥∥λn Aun ∥ 4 ≤ ∥Jun − λn Aun − Jun ∥∥λn Aun ∥ c2 4 2 = λ ∥Aun ∥2 c2 n 4 2 ≤ λ ∥Aun − Ap∥2 . c2 n Substituting (3.4) and (3.5) into (3.3), we get ϕ(p, zn ) ≤

2 ϕ(p, un ) − 2αλn ∥Au 270 n − Ap∥ +

(3.5)

4 2 λ ∥Aun − Ap∥2 c2 n KUMAM ET AL 264-281

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W. PHUENGRATTANA, S. SUANTAI, K. WATTANAWITOON, U. WITTHAYARAT AND P. KUMAM

≤

ϕ(p, un ) + 2λn (

≤

ϕ(p, un )

≤

ϕ(p, xn ).

2 λn − α)∥Aun − Ap∥2 c2 (3.6)

By Lemma 2.7, Lemma 2.8 and (3.6), we have ϕ(p, yn )

=

ϕ(p, J −1 (βn J(xn ) + (1 − βn )J(Jrn zn )))

= V (p, βn J(xn ) + (1 − βn )J(Jrn zn )) ≤ βn V (p, J(xn )) + (1 − βn )V (p, J(Jrn zn )) = βn ϕ(p, xn ) + (1 − βn )ϕ(p, Jrn zn ) ≤ βn ϕ(p, xn ) + (1 − βn )(ϕ(p, zn ) − ϕ(Jrn zn , zn )) ≤ βn ϕ(p, xn ) + (1 − βn )ϕ(p, zn ) ≤ βn ϕ(p, xn ) + (1 − βn )ϕ(p, xn ) = ϕ(p, xn ),

(3.7)

it follows that ϕ(p, ΠC J −1 (αn J(x1 ) + (1 − αn )J(yn )))

ϕ(p, xn+1 ) =

≤ ϕ(p, J −1 (αn J(x1 ) + (1 − αn )J(yn ))) = V (p, αn J(x1 ) + (1 − αn )J(yn )) ≤ αn V (p, J(x1 )) + (1 − αn )V (p, J(yn )) = αn ϕ(p, x1 ) + (1 − αn )ϕ(p, yn ) ≤ αn ϕ(p, x1 ) + (1 − αn )ϕ(p, xn )

(3.8)

for all n ∈ N. Hence, by induction, we have that ϕ(p, xn ) ≤ ϕ(p, x1 ) for all n ∈ N. Since (∥xn ∥−∥p∥)2 ≤ ϕ(p, xn ). It implies that {xn } is bounded and {yn }, {zn }, {Jrn zn } are also bounded. From (3.6), (3.7) and (3.8), we have ϕ(p, xn+1 ) ≤ ≤

αn ϕ(p, x1 ) + (1 − αn )(βn ϕ(p, xn ) + (1 − βn )(ϕ(p, xn ) − ϕ(Jrn zn , zn ))) αn ϕ(p, x1 ) + (1 − αn )ϕ(p, xn ) − (1 − αn )(1 − βn )ϕ(Jrn zn , zn )

and then (1 − αn )(1 − βn )ϕ(Jrn zn , zn ) ≤ αn ϕ(p, x1 ) + (1 − αn )ϕ(p, xn ) − ϕ(p, xn+1 ) for all n ∈ N. Since limn−→∞ αn = 0, lim supn−→∞ βn < 1, it follows that limn−→∞ ϕ(Jrn zn , zn ) = 0. Applying Lemma 2.4, we have lim ∥Jrn zn − zn ∥ = 0. (3.9) n−→∞

Since J is uniformly norm-to-norm continuous on bounded sets, we obtain lim ∥JJrn zn − Jzn ∥ = 0.

(3.10)

n−→∞

By (3.2), (3.6), (3.7) and (3.8) again, we note that ϕ(p, xn+1 )

≤ αn ϕ(p, x1 ) + (1 − αn )[βn ϕ(p, xn ) + (1 − βn )[(ϕ(p, xn ) − 2λn (α − ≤ αn ϕ(p, x1 ) + (1 − αn )ϕ(p, xn ) − (1 − αn )(1 − βn )2λn (α −

2 λn )∥Aun − Ap∥2 ]] c2

2 λn )∥Aun − Ap∥2 c2

and hence 2λn (α −

2 1 λn )∥Aun − Ap∥2 ≤ (αn ϕ(p, x1 ) + (1 − αn )ϕ(p, xn ) − ϕ(p, xn+1 )) 2 c (1 − αn )(1 − βn )

for all n ∈ N. Since 0 < a ≤ λn ≤ b
0, it follows from (3.10) that 1 ∥Jzn − J(Jrn zn )∥ = 0. n−→∞ n−→∞ rn If (z, z ∗ ) ∈ T , then it holds from the monotonicity of A that lim ∥Arn zn ∥ = lim

(3.17)

⟨z − zni , z ∗ − Arni zni ⟩ ≥ 0 for all i ∈ N. Letting i −→ ∞, we get ⟨z − u, z ∗ ⟩ ≥ 0. Then, the maximality of T implies u ∈ T −1 (0). Next, we show that u ∈ V I(C, A). Let L ⊂ E × E ∗ be an operator as follows: { Av + NC (v), v ∈ C; Lv = ∅, otherwise. By Theorem 2.9, L is maximal monotone and L−1 (0) = V I(C, A). Let (v, w) ∈ G(L). Since w ∈ Lv = Av + NC (v), we get w − Av ∈ NC (v). From zn ∈ C, we have ⟨v − zn , w − Av⟩ ≥ 0. On the other hand, since zn = ΠC J

−1

(3.18)

(Jun − λn Aun ). Then by Lemma 2.5, we have

272 ⟨v − zn , Jzn − (Ju n − λn Aun )⟩ ≥ 0,

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W. PHUENGRATTANA, S. SUANTAI, K. WATTANAWITOON, U. WITTHAYARAT AND P. KUMAM

thus ⟨v − zn ,

Jun − Jzn − Auxn ⟩ ≤ 0. λn

(3.19)

It follows from (3.18) and (3.19) that ⟨v − zn , w⟩

≥

⟨v − zn , Av⟩

≥

⟨v − zn , Av⟩ + ⟨v − zn ,

Jun − Jzn − Aun ⟩ λn Jun − Jzn = ⟨v − zn , Av − Aun ⟩ + ⟨v − zn , ⟩ λn = ⟨v − zn , Av − Azn ⟩ + ⟨v − zn , Azn − Aun ⟩ + ⟨v − zn ,

≥ ≥

∥zn − un ∥ ∥Jun − Jzn ∥ − ∥v − zn ∥ α a ∥zn − un ∥ ∥Jun − Jzn ∥ −M ( + ), α a

Jun − Jzn ⟩ λn

−∥v − zn ∥

where M = supn≥1 {∥v − zn ∥}. From (3.12) and (3.13), we obtain ⟨v − u, w⟩ ≥ 0. By the maximality of L, we have u ∈ L−1 (0) and hence u ∈ V I(C, A). Next, we show that u ∈ Ω. From (3.16) and J is uniformly norm-to-norm continuous on bounded set, we obtain lim ∥Jun − Jxn ∥ = 0. (3.20) n−→∞

From the assumption lim inf n−→∞ rn > a, we get lim

n→∞

∥Jun − Jxn ∥ = 0. rn

Noticing that un = Krn xn , we have H(un , y) +

1 ⟨y − un , Jun − Jxn ⟩ ≥ 0, rn

∀y ∈ C.

Hence, H(uni , y) +

1 ⟨y − uni , Juni − Jxni ⟩ ≥ 0, rni

∀y ∈ C.

From the (A2), we note that ∥y − uni ∥

∥Juni − Jxni ∥ 1 ≥ ⟨y − uni , Juni − Jxni ⟩ ≥ −H(uni , y) ≥ H(y, uni ), rni rni

∀y ∈ C.

Taking the limit as n → ∞ in above inequality and from (A4) and uni ⇀ u, we have H(y, u) ≤ 0, ∀y ∈ C. For 0 < t < 1 and y ∈ C, define yt = ty + (1 − t)u. Noticing that y, u ∈ C, we obtains yt ∈ C, which yields that H(yt , u) ≤ 0. It follows from (A1) that 0 = H(yt , yt ) ≤ tH(yt , y) + (1 − t)H(yt , x ˆ) ≤ tH(yt , y). That is, H(yt , y) ≥ 0. Let t ↓ 0, from (A3), we obtain H(u, y) ≥ 0, ∀y ∈ C. This implies that u ∈ Ω. Hence u ∈ F := Ω ∩ V I(C, A) ∩ T −1 (0). Finally, we show that u = ΠF x. Indeed from xn = ΠCn x and Lemma 2.5, we have ⟨Jx − Jxn , xn − z⟩ ≥ 0, ∀z ∈ Cn . Since F ⊂ Cn , we also have ⟨Jx − Jxn , xn − p⟩ ≥ 0, ∀p ∈ F.

(3.21)

Taking limit n −→ ∞, we obtain ⟨Jx − Ju, u − p⟩ ≥ 0, ∀p ∈ F. By again Lemma 2.5, we can conclude that u = Π273 F x. This completes the proof.



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Corollary 3.2. Let E be a 2-uniformly convex and uniformly smooth Banach space, let C be a nonempty closed convex subset of E. Let T : E −→ E ∗ be a maximal monotone operator satisfying D(T ) ⊂ C. Let Jr = (J + rT )−1 J for r > 0 and let A be an α-inverse-strongly monotone operator of C into E ∗ , with F := V I(C, A) ∩ T −1 (0) ̸= ∅. Assume that A is an operator of C into E ∗ which satisfies the condition (C1) − (C3). Let {xn } be a sequence generated by x1 = x ∈ C and,   zn = ΠC J −1 (Jxn − λn Axn ), yn = J −1 (βn J(xn ) + (1 − βn )J(Jrn zn )), (3.22)  xn+1 = ΠC J −1 (αn J(x1 ) + (1 − αn )J(yn )), for all n ∈ N, where ΠC is the generalized projection from E onto C, J is the duality mapping on E. ∑∞ The coefficient sequence {αn }, {βn } ⊂ (0, 1], {rn } ⊂ (0, ∞) satisfying limn−→∞ αn = 0, n=1 αn = ∞, c2 α 1 lim supn−→∞ βn < 1, lim inf n−→∞ rn > 0 and {λn } ⊂ [a, b] for some a, b with 0 < a < b < , is 2 c the 2-uniformly convexity constant of E. Then the sequence {xn } converges strongly to ΠF x. 4. Weak Convergence Theorem We next prove a weak convergence theorem under difference condition on data. First we prove the generalized projection sequence {ΠF xn } of {xn } is strongly convergent. Theorem 4.1. Let E be a 2-uniformly convex and uniformly smooth Banach space, let C be a nonempty closed convex subset of E. Let Θ be a bifunction from C × C to R satisfying (A1)-(A4) let φ : C −→ R be a proper lower semicontinuous and convex function, let T : E −→ E ∗ be a maximal monotone operator satisfying D(T ) ⊂ C. Let Jr = (J + rT )−1 J for r > 0 and let A be an α-inverse-strongly monotone operator of C into E ∗ and let B : C −→ E ∗ be a continuous and monotone mappings, with F := Ω ∩ V I(C, A) ∩ T −1 (0) ̸= ∅. Assume that A is an operator of C into E ∗ which satisfies the condition (C1) − (C3). Let {xn } be a sequence generated by x1 = x ∈ C and,  un ∈ C such that Θ(un , y) + ⟨Bun , y − un ⟩ + φ(y) − φ(un )     + r1n ⟨y − un , Jun − Jxn ⟩ ≥ 0, ∀y ∈ C,  (4.1) zn = ΠC J −1 (Jun − λn Aun ),  −1  y = J (β J(x ) + (1 − β )J(J z )),  n n n n r n n   xn+1 = ΠC J −1 (αn J(x1 ) + (1 − αn )J(yn )), for all n ∈ N, where ΠC is the generalized projection from E onto C, ∑J∞is the duality mapping on E. The coefficient sequence {αn }, {βn } ⊂ (0, 1], {rn } ⊂ (0, ∞) satisfying n=1 αn < ∞, lim supn−→∞ βn < 1, c2 α 1 lim inf n−→∞ rn > 0 and {λn } ⊂ [a, b] for some a, b with 0 < a < b < , is the 2-uniformly convexity 2 c constant of E. Then the sequence {ΠF xn } converges strongly to an element v of F , which is a unique element of F satisfying lim ϕ(v, xn ) = min lim ϕ(y, xn ). n−→∞

y∈F n−→∞

Proof . Let H(un , y) = Θ(un , y) + ⟨Bun , y − un ⟩ + φ(y) − φ(un ), y ∈ C and Krn = {u ∈ C : H(un , y) + 1 −1 (Jun −λn Aun ), rn ⟨y −un , Jun −Jxn ⟩ ≥ 0. ∀y ∈ C}. We first show that {xn } is bounded. Put vn = J −1 let p ∈ F := Ω ∩ V I(C, A) ∩ T (0) and un = Krn xn . Since Jrn are relatively nonexpansive mappings. By (3.8), we have that, for all n ∈ N ∑∞

ϕ(p, xn+1 ) ≤ αn ϕ(p, x1 ) + (1 − αn )ϕ(p, xn ).

(4.2)

From n=1 αn < ∞ and Lemma 2.10, we deduce that limn−→∞ ϕ(p, xn ) exists. This implies that {ϕ(p, xn )} is bounded. So {xn } is bounded. Define a function g : F −→ [0, ∞) as follows: g(p) = lim ϕ(p, xn ), ∀p ∈ F. n−→∞

Then, by the same argument as in proof of [14, Theorem 3.1], we obtain g is a continuous convex function and if ∥zn ∥ −→ ∞ then g(zn ) −→ ∞. Hence, by [30, Theorem 1.3.11], there exists a point v ∈ F such that g(v) = min g(y)(:= l). 274 y∈F

(4.3) KUMAM ET AL 264-281

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Put wn = ΠF xn for all n ≥ 0. We next prove that wn −→ v as n −→ ∞. Suppose on the contrary that there exists ϵ0 > 0 such that, for each n ∈ N, there is n′ ≥ n satisfying ∥wn′ − v∥ ≥ ϵ0 . Since v ∈ F , we have ϕ(wn , xn ) = ϕ(ΠF xn , xn ) ≤ ϕ(v, ΠF xn ) + ϕ(ΠF xn , xn ) ≤ ϕ(v, xn )

(4.4)

for all n ≥ 0. This implies that lim sup ϕ(wn , xn ) ≤ lim ϕ(v, xn ) = l. n−→∞

n−→∞

(4.5)

Since (∥v∥ − ∥ΠF xn ∥)2 ≤ ϕ(v, wn ) ≤ ϕ(v, xn ) for all n ≥ 0 and {xn } is bounded, we get {wn } is also bounded. By Lemma 2.3, there exists a stricly increasing, continuous and convex function K : [0, ∞) −→ [0, ∞) such that K(0) = 0 and ∥

1 1 1 wn + v 2 ∥ ≤ ∥wn ∥2 + ∥v∥2 − K(∥wn − v∥), 2 2 2 4

1 for all n ≥ 0. Now, choose σ satisfying 0 < σ < K(ϵ0 ). Hence, there exists n0 ∈ N such that 4 ϕ(wn , xn ) ≤ l + σ, ϕ(v, xn ) ≤ l + σ,

(4.6)

(4.7)

for all n ≥ 0. Thus there exists k ≥ n0 satisfying the following: ϕ(wk , xk ) ≤ l + σ, ϕ(v, xk ) ≤ l + σ, ∥wk − v∥ ≥ ϵ0 .

(4.8)

From (4.2), (4.6) and (4.8), we obtain ϕ(

wk + v , xn+k ) ≤ 2 = ≤ = ≤

wk + v , xk ) 2 wk + v wk + v 2 ∥ − 2⟨ , Jxk ⟩ + ∥xk ∥2 ∥ 2 2 1 1 1 ∥wk ∥2 + ∥v∥2 − K(∥wk − v∥) − ⟨wk + v, Jxk ⟩ + ∥xk ∥2 2 2 4 1 1 1 ϕ(wk , xk ) + ϕ(v, xk ) − K(∥wk − v∥) 2 2 4 1 l + σ − K(ϵ0 ), 4

ϕ(

(4.9)

for all n ≥ 0. Hence l ≤ lim ϕ( n−→∞

wk + v wk + v 1 , xn ) = lim ϕ( , xn+k ) ≤ l + σ − K(ϵ0 ) < l + σ − σ = l. n−→∞ 2 2 4

(4.10)

This is a contradiction. So, {wn } converges strongly to v ∈ F := Ω ∩ V I(C, A) ∩ T −1 (0). Consequently, v ∈ F is the unique element of F such that lim ϕ(v, xn ) = min lim ϕ(y, xn ).

n−→∞

This completes the proof.

y∈F n−→∞

(4.11) 

Theorem 4.2. Let E be a 2-uniformly convex and uniformly smooth Banach space, let C be a nonempty closed convex subset of E. Let T : E −→ E ∗ be a maximal monotone operator satisfying D(T ) ⊂ C. Let Jr = (J + rT )−1 J for r > 0 and let A be an α-inverse-strongly monotone operator of C into E ∗ , with F := V I(C, A) ∩ T −1 (0) ̸= ∅. Assume that A is an operator of C into E ∗ which satisfies the condition (C1) − (C3). Let {xn } be a sequence generated by x1 = x ∈ C and,   zn = ΠC J −1 (Jxn − λn Axn ), yn = J −1 (βn J(xn ) + (1 − βn )J(Jrn zn )), (4.12)  xn+1 = ΠC J −1 (αn J(x1 ) + (1 − αn )J(yn )), for all n ∈ N, where ΠC is the generalized projection from E onto C, ∑J∞is the duality mapping on E. The coefficient sequence {αn }, {βn } ⊂ (0, 1], {rn } ⊂ (0, ∞) satisfying n=1 αn < ∞, lim supn−→∞ βn < 1, c2 α 1 , is the 2-uniformly convexity lim inf n−→∞ rn > 0 and {λn } ⊂ [a, b] for some a, b275 with 0 < a < b < 2 c KUMAM ET AL 264-281

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13

constant of E. Then the sequence {ΠF xn } converges strongly to an element v of F , which is a unique element of F satisfying lim ϕ(v, xn ) = min lim ϕ(y, xn ). n−→∞

y∈F n−→∞

Now, we prove a weak convergence theorem for the algorithm (4.13) below under different condition on data. Theorem 4.3. Let E be a 2-uniformly convex and uniformly smooth Banach space, let C be a nonempty closed convex subset of E. Let Θ be a bifunction from C × C to R satisfying (A1)-(A4) let φ : C −→ R be a proper lower semicontinuous and convex function, let T : E −→ E ∗ be a maximal monotone operator satisfying D(T ) ⊂ C. Let Jr = (J + rT )−1 J for r > 0 and let A be an α-inverse-strongly monotone operator of C into E ∗ and let B : C −→ E ∗ be a continuous and monotone mappings, with F := Ω ∩ V I(C, A) ∩ T −1 (0) ̸= ∅. Assume that A is an operator of C into E ∗ which satisfies the condition (C1) − (C3). Let {xn } be a sequence generated by x1 = x ∈ C and,    un ∈ C such1 that Θ(un , y) + ⟨Bun , y − un ⟩ + φ(y) − φ(un )   + rn ⟨y − un , Jun − Jxn ⟩ ≥ 0, ∀y ∈ C,  (4.13) zn = ΠC J −1 (Jun − λn Aun ),  −1  y = J (β J(x ) + (1 − β )J(J z )),  n n n n r n n   xn+1 = ΠC J −1 (αn J(x1 ) + (1 − αn )J(yn )), for all n ∈ N, where ΠC is the generalized projection from E onto C, ∑J∞is the duality mapping on E. The coefficient sequence {αn }, {βn } ⊂ (0, 1], {rn } ⊂ (0, ∞) satisfying n=1 αn < ∞, lim supn−→∞ βn < 1, c2 α 1 lim inf n−→∞ rn > 0 and {λn } ⊂ [a, b] for some a, b with 0 < a < b < , is the 2-uniformly 2 c convexity constant of E. Then the sequence {xn } converges weakly to an element v of F , where v = limn−→∞ ΠF xn . Proof . As in Proof of Theorem 3.1, we have {xn } is bounded, there exists a subsequence {xni } of {xn } such that xni ⇀ u ∈ C and hence u ∈ F := Ω ∩ V I(C, A) ∩ T −1 (0). By Theorem 4.1, the {ΠF xn } converges strongly to a point v ∈ F which is a unique element of F such that lim ϕ(v, xn ) = min lim ϕ(y, xn ).

n−→∞

y∈F n−→∞

(4.14)

By the uniform smoothness of E, we also have limn−→∞ ∥JΠF xni − Jv∥ = 0. Finally, we prove u = v. From Lemma 2.5 and u ∈ F , we have ⟨ΠF xni − u, Jxni − JΠF xni ⟩ ≥ 0. Since J is weakly sequentially continuous, uni ⇀ u and un − xn −→ 0, then ⟨v − u, Ju − Jv⟩ ≥ 0. On the other hand, since J is monotone, we have ⟨v − u, Ju − Jv⟩ ≤ 0. Hence, ⟨v − u, Ju − Jv⟩ = 0. Since E is strict convexity, it follows that u = v. Therefore the sequence {xn } converges weakly to v = limn−→∞ ΠF xn . This completes the proof.  Theorem 4.4. Let E be a 2-uniformly convex and uniformly smooth Banach space, let C be a nonempty closed convex subset of E. Let T : E −→ E ∗ be a maximal monotone operator satisfying D(T ) ⊂ C. Let Jr = (J + rT )−1 J for r > 0 and let A be an α-inverse-strongly monotone operator of C into E ∗ , with F := V I(C, A) ∩ T −1 (0) ̸= ∅. Assume that A is an operator of C into E ∗ which satisfies the condition (C1) − (C3). Let {xn } be a sequence generated by x1 = x ∈ C and,   zn = ΠC J −1 (Jxn − λn Axn ), yn = J −1 (βn J(xn ) + (1 − βn )J(Jrn zn )), (4.15)  xn+1 = ΠC J −1 (αn J(x1 ) + (1 − αn )J(yn )), for all n ∈ N, where ΠC is the generalized projection from E onto C, ∑J∞is the duality mapping on E. The 276 coefficient sequence {αn }, {βn } ⊂ (0, 1], {rn } ⊂ (0, ∞) satisfying n=1 αn < ∞, lim supn−→∞ βn < 1,

KUMAM ET AL 264-281

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W. PHUENGRATTANA, S. SUANTAI, K. WATTANAWITOON, U. WITTHAYARAT AND P. KUMAM

c2 α 1 , is the 2-uniformly 2 c convexity constant of E. Then the sequence {xn } converges weakly to an element v of F , where v = limn−→∞ ΠF xn . lim inf n−→∞ rn > 0 and {λn } ⊂ [a, b] for some a, b with 0 < a < b
0 and let B : C −→ E ∗ be a continuous and monotone mappings, with F := Ω ∩ V I(C, A) ∩ T −1 (0) ̸= ∅.Assume that f is a functional on E that satisfies the following conditions: (1) f is continuously Fr´ echet differentiable, convex functional on E and ∇f is α-inverse strongly monotone, (2) ∥∇f |C (y)∥ ≤ ∥∇f |C (y) − ∇f |C (u)∥ for all y ∈ C and u ∈ S. Suppose that x1 = x ∈ C and {xn } be a sequence generated by  un ∈ C such that Θ(un , y) + ⟨Bun , y − un ⟩ + φ(y) − φ(un )     + r1n ⟨y − un , Jun − Jxn ⟩ ≥ 0, ∀y ∈ C,  (5.1) zn = ΠC J −1 (Jun − λn ∇f |C un ),  −1  y = J (β J(x ) + (1 − β )J(J z )),  n n n n r n n   xn+1 = ΠC J −1 (αn J(x) + (1 − αn )J(yn )), for all n ∈ N, where ΠC is the generalized projection from E onto C, J is the duality mapping on E. ∑∞ The coefficient sequence {αn }, {βn } ⊂ (0, 1], {rn } ⊂ (0, ∞) satisfying limn−→∞ αn = 0, n=1 αn = ∞, c2 α 1 lim supn−→∞ βn < 1, lim inf n−→∞ rn > 0 and {λn } ⊂ [a, b] for some a, b with 0 < a < b < , is 2 c the 2-uniformly convexity constant of E. Then the sequence {xn } converges strongly to ΠF x. Proof. By Theorem 3.1, we put A = ∇f |C and from Lemma 5.1 and the condition (1) of Theorem 5.2 that ∇f |C is an α-inverse strongly monotone operator of C into E ∗ . So, we obtain that {xn } converges strongly to ΠF x0 .  Next, we will consider the zero point of an α-inverse strongly monotone operator of E into E ∗ . Without loss of generality, we let C = E then the condition (C3) of the operator A in Theorem 3.1 holds. Theorem 5.3. Let E be a 2-uniformly convex and uniformly smooth Banach space, let Θ be a bifunction from E × E to R satisfying (A1)-(A4) let φ : E −→ R be a proper lower semicontinuous and convex function, let T : E −→ E ∗ be a maximal monotone operator satisfying D(T ) ⊂ C. Let Jr = (J +rT )−1 J for r > 0 and let B : E −→ E ∗ be a continuous and monotone mappings, with F := Ω ∩ V I(E, A) ∩ T −1 (0) ̸= ∅.Assume that A is an operator of E into E ∗ that satisfies the following conditions: (1) A is α-inverse strongly monotone, 277 (2) A−1 (0) = {u ∈ E : Au = 0} ̸= ∅.

KUMAM ET AL 264-281

J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 16, NO.2, 2014, COPYRIGHT 2014 EUDOXUS PRESS, LLC

WEAK AND STRONG CONVERGENCE THEOREMS OF PROXIMAL POINT ALGORITHM

Suppose that x1 = x ∈ C and {xn } be a sequence generated by  un ∈ C such that Θ(un , y) + ⟨Bun , y − un ⟩ + φ(y) − φ(un )     + r1n ⟨y − un , Jun − Jxn ⟩ ≥ 0, ∀y ∈ C,  −1 zn = J (Jun − λn Aun ),   y = J −1 (βn J(xn ) + (1 − βn )J(Jrn zn )),    n xn+1 = J −1 (αn J(x) + (1 − αn )J(yn )),

15

(5.2)

for all n ∈ N, where ΠC is the generalized projection from E onto C, J is the duality mapping on E. ∑∞ The coefficient sequence {αn }, {βn } ⊂ (0, 1], {rn } ⊂ (0, ∞) satisfying limn−→∞ αn = 0, n=1 αn = ∞, c2 α 1 lim supn−→∞ βn < 1, lim inf n−→∞ rn > 0 and {λn } ⊂ [a, b] for some a, b with 0 < a < b < , is 2 c the 2-uniformly convexity constant of E. Then the sequence {xn } converges strongly to z = ΠF x. Proof. In Theorem (3.1), we put C = E. Therefore ΠE = I, then we have J −1 (Jun − λn Aun ) = ΠE J −1 (Jun − λn Aun ) for every n = 1, 2, ... and we also have V I(E, A) = A−1 0 and ∥Ay∥ = ∥Ay − 0∥ = ∥Ay − Au∥, ∀y ∈ E and u ∈ A−1 (0). Hence, by Theorem 3.1, {xn } converges strongly to some element z in Ω ∩ A−1 (0) ∩ T −1 (0)



6. Examples and Numerical results In this section, we give examples and numerical results which support our strong convergence theorem. Example 6.1. Let E = R and C = [−1, 1]. Let Θ(x, y) = −5x2 + xy + 4y 2 , Bx = 4x and φx = x2 . Find x ∈ C such that Θ(x, y) + ⟨Bx, y − x⟩ + φ(y) ≥ φ(x), ∀y ∈ [−1, 1]. Solution. We can see that Θ, B and φ are satisfied all conditions in Theorem 3.1. For each r > 0 and x ∈ [−1, 1], by Lemma(2.12), we can guarantee that there exists z ∈ [−1, 1] such that for each y ∈ [−1, 1], 1 Θ(z, y) + ⟨Bz, y − z⟩ + φ(y) + ⟨y − z, z − x⟩ ≥ φ(z) r 1 ⇔ −5z 2 + zy + 4y 2 + ⟨4z, y − z⟩ + y 2 + ⟨y − z, z − x⟩ ≥ z 2 r 2 2 2 ⇔ 5ry + (5rz + z − x)y + (−10z r − z + zx) ≥ 0 Put H(y) = 5ry 2 + (5rz + z − x)y + (−10z 2 r − z 2 + zx). We can see that H is a quadratic function of y with the coefficient a = 5r, b = (5rz + z − x) and c = (−10z 2 r − z 2 + zx). Next, we will compute the discriminant △ of H as shown in the following: △

= b2 − 4ac = (5rz + z − x)2 − 4(5r)(−10z 2 r − z 2 + zx) = x2 − 2(15rz + z)x + (15rz + z)2 = (x − (15rz + z))2 .

We know that H(y) ≥ 0 for all y ∈ [−1, 1] if it has at most one solution in [−1, 1]. So △ ≤ 0 and hence x . x = 15rz + z. Now we have z = Kr x = 15r + 1 Next, we consider our main algorithm in the strong convergence theorem. Let {xn }∞ n=1 be the sequence generated by x1 = x ∈ [−1, 1] and  un ∈ C such that Θ(un , y) + ⟨Bun , y − un ⟩ + φ(y) − φ(un )     + r1n ⟨y − un , Jun − Jxn ⟩ ≥ 0, ∀y ∈ C,  (6.1) zn = ΠC J −1 (Jun − λn Aun ),  −1  y = J (β J(x ) + (1 − β )J(J z )),  n n n rn n   n 278 xn+1 = ΠC J −1 (αn J(x) + (1 − αn )J(yn )),

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Algorithm 1. Let Ax = x, J = I and Jrn = I then the above algorithm is equivalent to the following: xn+1 = αn x + (1 − αn )[βn xn + (1 − βn )(1 − λn )Krn xn ]. Algorithm 2. Let Ax =

x 1+x , J

(6.2)

= I and Jrn = I, then we have

xn+1 = αn x + (1 − αn )[βn xn + (1 − βn )(

xn (16n + 1) xn (n + 1) − λn )]. 16n + 1 xn (n + 1) + 16n + 1

(6.3)

Next, we give the numerical test for both algorithms (6.2,6.3). 1 n Let αn = 100n , βn = λn = 12 and rn = n+1 . Choose x1 = x = 1 then the algorithm (6.2) becomes xn+1 =

1 33n + 3 1 + (1 − )[ ]xn , ∀n ≥ 1 100n 100n 64n + 4

and the algorithm (6.3) becomes xn+1 =

xn (n + 1) 1 1 1 xn (n + 1) 1 + (1 − )[ xn + ( − )], ∀n ≥ 1. 100n 100n 2 2 16n + 1 2xn (n + 1) + 32n + 2

Numerical Result n 1 2 3 4 5 6 7 8 9 .. .

xn by Algorithm 1. 1.0000 0.5341 0.2828 0.1500 0.0802 0.0435 0.0242 0.0139 0.0084 .. .

n 1 2 3 4 5 6 7 8 9 .. .

xn by Algorithm 2. 1.0000 0.5770 0.3245 0.1815 0.1017 0.0575 0.0329 0.0193 0.0117 .. .

413 414 415

0.0001 0.0001 0.0000

435 436 437

0.0001 0.0001 0.0000

Figure 1. This table shows the value of sequence {xn } on each iteration steps. 279

KUMAM ET AL 264-281

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WEAK AND STRONG CONVERGENCE THEOREMS OF PROXIMAL POINT ALGORITHM

17

Figure 2. This figure shows the graph of the above table, we can see that xn converges to zero. 7. Acknowledgements The third author would like to give a special thank to the Hands-on Research and Development Project, Rajamangala University of Technology Lanna (UR1-003) for the financial support during the preparation of this manuscript. Furthermore, the last author was supported by the Higher Education Commission and the Thailand Research Fund. References 1. Ya. I. Alber, Metric and generalized projection operators in Banach spaces: properties and applications, Theory and Applications of Nonlinear Operators of Accretive and Monotone Type, A.G. Kartsatos (Ed), Marcel Dekker, New York., 178 (1996) 15–50. 2. J.B. Baillon, G. Haddad, Quelques propri´ et´ es des op´ erateurs angle-born´ es et η-cycliquement monotones, Israel J. Math. 26(1977) 137–150. 3. K. Ball, E. A. Carlen and E. H. Lieb, Sharp uniform convexity and smoothness inequalities for trace norm, Invent. Math. 115 (1994) 463–482. 4. B. Beauzamy, Introduction to Banach spaces, and their Geometry, 2nd ed., Noth Holland, 1995. 5. E. Blum and W. Oettli, From optimization and variational inequalities to equilibrium problems, Math. Student., 63 (1994) 123–145. 6. D. Butnariu, S. Reich and A.J. Zaslavski, Asymptotic behavior of relatively nonexpansive operators in Banach spaces, J. Appl. Anal. 7 (2001) 151–174. 7. Y. Cens or, S. Reich, Iterations of paracontractions and firmly nonexpansive operators with applications to feasibility and optimization, Optimization 37 (1996) 323–339. 8. P. L. Combettes and S.A. Hirstoaga, Equilibrium programming in Hilbert spaces, J. Nonlinear Convex Anal., 6 (2005) 117–136. 9. I. Cioranescu, Geometry of Banach spaces, Duality Mappings and Nonlinear Problems, Kluwer, Dordrecht, 1990. 10. J. Diestel, Geometry of Banach spaces-selected topics, vol. 485 of Lecture Notes in Mathematics, Springer, Berlin, Germany, 1975. 11. S. D. Flam and A. S. Antipin, Equilibrium progamming using proximal-link algolithms, Math. Program., 78 (1997) 29–41. 12. H. Iiduka and W. Takahashi, Weak convergence of a projection algorithm for variational inequalities in a Banach space, J. Math. Anal. Appl. 339 (2008) 668–679. 13. H. Iiduka, W. Takahashi and M. Toyoda, Approximation of solutions of variational inequalities for monotone mappings, Panamer. Math. J. 14 (2004) 49–61. 14. S. Kamimura, F. Kohsaka and W. Takahashi, Weak and strong convergence of a maximal monotone operators in a Banach space, Set-Valued Analysis, 12 no.4 (2004) 417–429. 15. S. Kamimura and W. Takahashi, Strong convergence of a proximal-type algorithm in a Banach space, SIAM J. Optim. 13 no.3 (2002) 938–945. 16. S. Kamimura and W. Takahashi, Approximating solutions of maximal monotone operators in Hilbert spaces, J. Approx. Theory. 160 (2000) 226–240. 17. F. Kohsaka and W. Takahashi, Strong convergence of an iterative sequence for maximal monotone operators in a 280 Banach space, Abstr. Appl.Anal. 3 (2004) 239–249. KUMAM ET AL 264-281

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W. PHUENGRATTANA, S. SUANTAI, K. WATTANAWITOON, U. WITTHAYARAT AND P. KUMAM

18. L. Li and W. Song, Modified Proximal-Point Algorithm for maximal monotone operators in Banach spaces, J. Optim. Theory Appl. 138 (2008) 45–64. 19. A. Moudafi and M. Thera, Proximal and dynamical approaches to equilibrium problems. in: Lecture note in Economics and Mathematical Systems, Springer-Verlag, New York, 477 (1999), 187–201. 20. S. Mutsushita and W. Takahashi, A strong convergence theorem for relatively nonexpansive mappings in a Banach space, J. Approx. Theory, 134 (2005), 257–266. 21. W. Nilsrakoo and S. Saejung, Strong convergence to common fixed points of countable relatively quasi-nonexpansive mappings, Fixed Point Theory and Appl. (2008), Article ID 312454, 19 pages. 22. S. Reich, A weak convergence theorem for the alternating method with Bregman distance, in: A.G. Kartsatos (Ed.), Theory and Applicationsof Nonlinear Operatorsof Accretive and Monotone Type, Marcel Dekker, New York, 1996, pp. 313–318. 23. R.T. Rockafellar, Monotone operators and the proximal point algorithm, SIAM J. Control Optim. 14 (1976) 877–898. 24. S. Saewan and P. Kumam, A hybrid iterative scheme for a maximal monotone operator and two countable families of relatively quasi-nonexpansive mappings for generalized mixed equilibrium and variational inequality problems, Abstract and Applied Analysis, Volume 2010, Article ID 123027, 31 pages 25. S. Saewan, P. Kumam and K. Wattanawitoon, Convergence theorem based on a new hybrid projection method for finding a common solution of generalized equilibrium and variational inequality problems in Banach spaces, Abstract and Applied Analysis, Volume 2010, Article ID 734126, 26 pages. 26. Y. Su, D. Wang and M. Shang, Strong convergence of monotone hybrid algorithm for hemi-relatively nonexpansive mappings, Fixed Point Theory and Appl., (2008) Article ID 284613, 8 pages. 27. A. Tada and W. Takahashi, Strong convergence theorem for an equilibrium problem and a nonexpansive mapping, in: W. Takahashi, T. Tanaka (Eds.), Nonlinear Analysis and Convex Analysis, Yokohama Publishers, Yokohama, 2006, pp. 609-617. 28. A. Tada and W. Takahashi, Weak and strong convergence theorems for a nonexpansive mappings and an equilibrium problem, J. Optim. Theory Appl., 133 (2007),359–370. 29. S. Takahashi and W. Takahashi, Viscosity approximation methods for equilibrium problems and fixed point problems in Hilbert spaces, J. Math. Anal. Appl., 331 (2007), 506–515. 30. W. Takahashi, Nonlinear Functional Analysis, Fixed Point Theory and Its Application, Yokohama-Publishers, Yokohama, Japan 2000. 31. Y. Takahashi, K. Hashimoto and M. Kato, On shap uniform convexity, smoothness, and strong type, cotype inequalities, J. Nonlinear Convex Anal. 3 (2002) 267–281. 32. W. Takahashi and K. Zembayashi, Strong and weak convergence theorems for equilibrium problems and relatively nonexpansive mappings in Banach spaces, Nonlinear Anal. 70 (2009), 45–57. 33. K. K. Tan and H.K. Xu, Approximating fixed points of nonexpansive mappings by the Ishikawa iteration process, J.Math.Anal.Appl. 178 (1993) 301–308. 34. H. K. Xu, Iterative algorithms for nonlinear operators, J. Lond. Math. Soc., 66 (2002), 240–256. 35. H.K. Xu, Inequalities in Banach spaces with applications, Nonlinear Anal., 16 (2009) 1127–1138. 36. C. Zalinescu, On uniformly convex functions, J.Math.Anal.Appl. 95 (1983) 344–374. 37. H. Zegeye and N. Shahzad, Strong convergence for monotone mappings and relatively weak nonexpansive mappings, Nonlinear Anal., 70 (2009) 2707–2716. 38. S. Zhang, Generalized mixed equlibrium problem in Banach spaces, Appl. Math. Mech. -Engl. Ed., 30 (2009) 1105– 1112. (Withun Phuengrattana) Department of Mathematics, Faculty of Science, Chiang Mai University, Chiang Mai 50200, Thailand. E-mail address: withun [email protected] (Suthep Suantai) Department of Mathematics, Faculty of Science, Chiang Mai University, Chiang Mai 50200, Thailand. E-mail address: [email protected] (Kriengsak Wattanawitoon) Department of Mathematics and Statistics, Faculty of Science and Agricultural Technology, Rajamangala University of Technology Lanna Tak, Tak 63000, Thailand. E-mail address: [email protected] (Uamporn Witthayarat) Department of Mathematics, Faculty of Science, King Mongkut’s University of Technology Thonburi, Bangkok 10140, Thailand. E-mail address: n [email protected] (Poom Kumam) Department of Mathematics, Faculty of Science, King Mongkut’s University of Technology Thonburi, Bangkok 10140, Thailand. E-mail address: [email protected]

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On a class of two dimensional (w, q)-Bernoulli and (w, q)-Euler polynomials: Properties and location of zeros ¨ N.I. Mahmudov, A. Akkele¸s and A. Oneren Eastern Mediterranean University Gazimagusa, TRNC, Mersiin 10, Turkey Email: [email protected] Abstract The main purpose of this paper is to introduce and investigate two dimensional (w, q)-Bernoulli and (w, q)-Euler polynomials. The q-analogues of well-known formulas are derived. The q-analogue of the Srivastava–Pint´er addition theorem is obtained. Furthermore we explore the shapes of the q-Bernoulli numbers and the q-Bernoulli polynomials. We describe the structure of the roots of the q-Bernoulli polynomials for values of the index n using a computer.

1

Introduction

Throughout this paper, we always make use of the following notation: N denotes the set of natural numbers, N0 denotes the set of nonnegative integers, R denotes the set of real numbers, C denotes the set of complex numbers. The q-shifted factorial is defined by (a; q)0 = 1,

(a; q)n =

n−1 Y

¡ ¢ 1 − qj a ,

n ∈ N,

(a; q)∞ =

j=0

∞ Y ¡

¢ 1 − qj a ,

|q| < 1, a ∈ C.

j=0

The q-number and q-factorial is defined by [a]q =

1 − qa 1−q

(q 6= 1) ;

[0]q ! = 1;

[n]q ! = [1]q [2]q ... [n]q

n ∈ N, a ∈ C

respectively. The q-polynomial coefficient is defined by · ¸ (q; q)n n . = k q (q; q)n−k (q; q)k n

The q-analogue of the function (x + y) is defined by n

(x + y)q :=

¸ n · X 1 n q 2 k(k−1) xn−k y k , k q

n ∈ N0 .

k=0

The q-binomial formula is known as n

(1 − a)q = (a; q)n =

n−1 Y

n ¡ ¢ X 1 − qj a =

j=0

k=0

·

n k

282

¸ 1

k

q 2 k(k−1) (−1) ak . q

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In the standard approach to the q-calculus two exponential functions are used: eq (z) =

∞ ∞ X Y zn 1 = , [n]q ! (1 − (1 − q) q k z) n=0

0 < |q| < 1, |z|
0; is said to be m convex, where m 2 [0; 1] ; if we have f (tx + m (1

t) y)

tf (x) + m (1

t) f (y)

1

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for all x; y 2 [0; b] and t 2 [0; 1] : We say that f is m concave if m convex.

f is

In [4], Mihe¸san gave de…nition of ( ; m) convexity as following; De…nition 2 The function f : [0; b] ! R; b > 0 is said to be ( ; m) convex, where ( ; m) 2 [0; 1]2 ; if we have f (tx + m(1

t)y)

t f (x) + m(1

t )f (y)

for all x; y 2 [0; b] and t 2 [0; 1]: Denote by Km (b) the class of all ( ; m) convex functions on [0; b] for which f (0) 0: If we choose ( ; m) = (1; m), it can be easily seen that ( ; m) convexity reduces to m convexity and for ( ; m) = (1; 1); we have ordinary convex functions on [0; b]: For the recent results based on the above de…nitions see the papers [2]-[9]. De…nition 3 ([1]) A function f : [0; b] ! (0; 1) is said to be m logarithmically convex if the inequality f (tx + m (1

t

t) y)

m(1 t)

[f (x)] [f (y)]

(2)

holds for all x; y 2 [0; b], m 2 (0; 1], and t 2 [0; 1]. Obviously, if putting m = 1 in De…nition 3, then f is just the ordinary logarithmically convex function on [0; b]. De…nition 4 ([1]) A function f : [0; b] ! (0; 1) is said to be ( ; m) logarithmically convex if t m(1 t ) f (tx + m (1 t) y) [f (x)] [f (y)] (3) holds for all x; y 2 [0; b], ( ; m) 2 (0; 1]

(0; 1] ; and t 2 [0; 1].

Clearly, when taking = 1 in De…nition 4, then f becomes the standard m-logarithmically convex function on [0; b]. We give some necessary de…nitions and mathematical preliminaries of fractional calculus theory which are used throughout this paper. De…nition 5 Let f 2 L1 [a; b]: The Riemann-Liouville integrals Ja+ f and Jb f of order > 0 with a 0 are de…ned by Ja+ f (x) =

1 ( )

Zx

(x

t)

1

f (t)dt; x > a

Zb

(t

x)

1

f (t)dt; x < b

a

and Jb f (x) =

1 ( )

x

respectively where ( ) =

1 R

e tu

0

1

du: Here is Ja0+ f (x) = Jb0 f (x) = f (x): 2

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In the case of = 1, the fractional integral reduces to the classical integral. For some recent results connected with fractional integral inequalities see [11][18]. The aim of this study is to establish some Ostrowski type inequalities for the class of functions whose derivatives in absolute value are m and ( ; m) logarithmically convex functions via Riemann-Liouville fractional integral.

2

THE NEW RESULTS

In order to prove our results, we need the following lemma that has been obtained in [11]: Lemma 1 ([11]) Let f : [a; b] ! R be a di¤ erentiable mapping on (a; b) with a < b: If f 0 2 L [a; b] ; then for all x 2 [a; b] and > 0 we have: ( + 1) a) + (b x) f (x) Jx f (a) + Jx+ f (b) b a b a +1 Z 1 +1 Z 1 (x a) (b x) t f 0 (tx + (1 t) a) dt + t f 0 (tx + (1 b a b a 0 0

(x

=

where ( ) =

1 R

e tu

1

t) b) dt

du:

0

Theorem 2 Let f : [0; 1) ! (0; 1) be di¤ erentiable mapping with a; b 2 [0; 1) such that a < b: If jf 0 (x)j is ( ; m) logarithmically convex function with jf 0 (x)j M; f 0 2 L [a; b] ; ( ; m) 2 (0; 1] (0; 1] and > 0; then the following inequality for fractional integrals holds: (x

a) + (b b a

1 2 +1

( + 1) Jx f (a) + Jx+ f (b) b a # +1 +1 a) + (b x) 2 (b a)

x)

f (x) " (x + K1 ( ; m; t)

where K1 ( ; m; t) =

8 >
:

(4)

:

3

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Proof. write

By Lemma 1 and since jf 0 j is ( ; m) logarithmically convex; we can (x

a) + (b b a +1 Z 1 (x a) t b a 0 +1 Z 1 (x a) t b a 0 +1 Z 1 (x a) t b a 0

x)

( + 1) Jx f (a) + Jx+ f (b) b a +1 Z 1 (b x) jf 0 (tx + (1 t) a)j dt + t jf 0 (tx + (1 t) b)j dt b a 0 +1 Z 1 a m(1 t ) (b x) b t t 0 0 dt + jf (x)j f t jf 0 (x)j f 0 m b a m 0 +1 Z 1 (b x) M m+t (1 m) dt + t M m+t (1 m) dt: b a 0 f (x)

c2 +d2 2 ;

By using the elemantery inequality cd

m(1 t )

we have (5)

( + 1) a) + (b x) f (x) Jx f (a) + Jx+ f (b) b a b a +1 Z 1 2 +1 Z 1 2 (b x) (x a) t + M 2(m+t (1 m)) t + M 2(m+t dt + b a 2 b a 2 0 0 # " Z 1 +1 +1 1 (x a) + (b x) + : M 2(m+t (1 m)) dt 2 +1 2 (b a) 0 (x

=

(1 m))

dt

If we choose M = 1; then Z

1

M 2(m+t

(1 m))

dt = 1:

0

If M < 1; then M 2(m+t Z

(1 m))

M 2(m+

1

M 2(m+

t(1 m))

dt =

0

t(1 m))

, thus

M 2m M 2 2 m 1 (2 2 m) ln M

which completes the proof. Corollary 3 Let f : [0; 1) ! (0; 1) be di¤ erentiable mapping with a; b 2 [0; 1) such that a < b: If jf 0 (x)j is m logarithmically convex function with jf 0 (x)j M; f 0 2 L [a; b] ; m 2 (0; 1] and > 0; then the following inequality for fractional integrals holds: (x

a) + (b b a

x)

1 M2 + 2 + 1 2 ln M

( + 1) Jx f (a) + Jx+ f (b) b a " # +1 +1 M 2m (x a) + (b x) : 2m ln M 2 (b a) f (x)

(6)

4

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dt

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Proof. If we take

= 1 in (4), we get the required result.

Corollary 4 Let f : [0; 1) ! (0; 1) be di¤ erentiable mapping with a; b 2 [0; 1) such that a < b: If jf 0 (x)j is logarithmically convex function with jf 0 (x)j M; f 0 2 L [a; b] and > 0; then the following inequality for fractional integrals holds: (x

a) + (b b a

x) "

1 + M2 2 +1 Proof. If we take

( + 1) Jx f (a) + Jx+ f (b) b a # +1 +1 a) + (b x) : 2 (b a)

f (x)

(x

(7)

= m = 1 in (5), we get the required result.

Corollary 5 Let f : [0; 1) ! (0; 1) be di¤ erentiable mapping with a; b 2 [0; 1) such that a < b: If jf 0 (x)j is logarithmically convex function with jf 0 (x)j M and f 0 2 L [a; b] ; then the following inequality holds: " # Z b 2 2 1 1 (x a) + (b x) 2 f (x) +M : f (u)du b a a 3 2 (b a) Proof. If we choose

= 1 in (7), we get the required result.

Theorem 6 Let f : [0; 1) ! (0; 1) be di¤ erentiable mapping with a; b 2 q [0; 1) such that a < b: If jf 0 (x)j is ( ; m) logarithmically convex function q 0 0 with jf (x)j M; f 2 L [a; b] ; ( ; m) 2 (0; 1] (0; 1] and > 0; then the following inequality for fractional integrals holds: (8) (x

a) + (b b a

x)

q

q 1 (q p) + q where q > 1; 0

p

K2 ( ; m; t) =

( + 1) Jx f (a) + Jx+ f (b) b a " +1 1 (x a) + (b x) (K2 ( ; m; t)) q (b a)

f (x) 1 q

1

q and 8 qm > > M (
> :

p+1;ln M q

(m

1)

1 p+1

)

p+1

(m

1)

))

+1

#

;M < 1 : ;M = 1

Proof. From Lemma 1 and by using the properties of modulus, we have (x

a) + (b x) ( + 1) f (x) Jx f (a) + Jx+ f (b) b a b a +1 Z 1 +1 Z 1 (x a) (b x) 0 t jf (tx + (1 t) a)j dt + t jf 0 (tx + (1 b a b a 0 0

t) b)j dt:

5

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By applying the Hölder inequality for q > 1; 0

p

q; we get

(x

a) + (b x) ( + 1) f (x) Jx f (a) + Jx+ f (b) b a b a 20 1qq1 0 1 1 q1 Z1 Z +1 q p (x a) q 6@ @ t p jf 0 (tx + (1 t) a)j dtA t ( q 1 ) dtA 4 b a 0

+

(b b

x) a

+1

0

0 1 Z q @ t (q

p 1

0

It is easy to see that

Z1

q t (q

p 1

1qq1 0

) dtA

Z1

@ t

p

1 q1 3 q 7 t) b)j dtA 5 :

jf 0 (tx + (1

0

) dt =

q 1 p) + q

(q

1

:

0

q

Hence, by ( ; m) logarithmically convexity of jf 0 j ; we have (9) (x

(x b

a) + (b b a a) a

x)

+1

( + 1) Jx f (a) + Jx+ f (b) b a 0 1 q 1 Z q @ t p jf 0 (x)jqt f 0 a 1 m

f (x) q 1 p) + q

(q

qm(1 t )

0

+

(b b

x) a

q

+1

(q

q 1 p) + q

1

q

0

Z1

@ t p t jf 0 (x)jqt

1

b m

f0

0

(x

=

b

a) a

q

+1

q 1 p) + q

(q

0

1 q

Z1

@ t p M qm+qt

1

(1 m)

0

+

(b b

x) a

q

+1

(q

q 1 p) + q

1

q

1

0

Z1

@ t p M qm+qt

Z1

t p dt =

dtA

qm(1 t )

1 q1

dtA

dtA

1 q1

(1 m)

dtA :

p + 1; ln M q

(m 1)

0

If we choose M = 1; then

1 q1

1 q1

1 : p+1

0

If M < 1; then M qm+qt Z1

t p M qm+q

t(1 m)

(1 m)

dt =

M qm+q

M qm

t(1 m)

, thus

( p + 1) ln M q

0

(m 1)

p+1

6

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which completes the proof. Corollary 7 Let f : [0; 1) ! (0; 1) be di¤ erentiable mapping with a; b 2 q [0; 1) such that a < b: If jf 0 (x)j is m logarithmically convex function with q jf 0 (x)j M; f 0 2 L [a; b] ; m 2 (0; 1] and > 0; then the following inequality for fractional integrals holds: (x

a) + (b b a

x)

q

q 1 (q p) + q where q > 1; 0

( + 1) Jx f (a) + Jx+ f (b) b a " +1 1 (x a) + (b x) (K2 (1; m; t)) q (b a)

f (x) 1 q

1

q and 8 qm M ( > > < K2 (1; m; t) = > > :

+1

#

p

Proof. If we set

(

( p+1)

(ln M q(m

p+1;ln M q(m 1)

)

1)

))

p+1

;M < 1 :

1 p+1

;M = 1

= 1 in 8, the proof is completed.

Corollary 8 Let f : [0; 1) ! (0; 1) be di¤ erentiable mapping with a; b 2 q q [0; 1) such that a < b: If jf 0 (x)j is logarithmically convex function with jf 0 (x)j 0 M; f 2 L [a; b] and > 0; then the following inequality for fractional integrals holds: (x

a) + (b b a

x)

f (x) q

Mm where q > 1; 0

q 1 (q p) + q p

Proof. If we set

1 q

1

( + 1) Jx f (a) + Jx+ f (b) b a 1 " +1 q 1 (x a) + (b x) p+1 (b a)

+1

#

q. = m = 1 in 9, the proof is completed.

Corollary 9 Let f : [0; 1) ! (0; 1) be di¤ erentiable mapping with a; b 2 q [0; 1) such that a < b: If jf 0 (x)j is ( ; m) logarithmically convex function q 0 0 with jf (x)j M; f 2 L [a; b] and ( ; m) 2 (0; 1] (0; 1] ; then the following inequality holds: 1

f (x)

b

a

Z

2q

1 p

f (u)du 1

q

1

(10)

a

q

q

b

(K1 ( ; m; t))

1 q

"

(x

2

a) + (b (b a)

2

x)

#

7

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where q > 1; 0

p

q and 8 qm M ( > > < K3 ( ; m; t) = > > :

Proof. If we set

(p+1;ln M q 1) p+1 )

(p+1)

(

(m

1)

))

;M < 1

ln M q (m

:

1 p+1

;M = 1

= 1 in 8, the proof is completed.

Corollary 10 Let f : [0; 1) ! (0; 1) be di¤ erentiable mapping with a; b 2 q [0; 1) such that a < b: If jf 0 (x)j is ( ; m) logarithmically convex function q with jf 0 (x)j M; f 0 2 L [a; b] and ( ; m) 2 (0; 1] (0; 1] ; then the following inequality holds: 1

f (x)

b

q

1 2 where q > 1; 0

a

Z

b

f (u)du

a

1 q

q and 8 > < K4 ( ; m; t) = > :

(K1 ( ; m; t))

1 q

"

(x

2

a) + (b (b a)

2

x)

#

p

M qm ( (2)

(2;ln M q (m 1) 2 )

(

1)

))

ln M q (m 1 2

;M < 1 : ;M = 1

Proof. If we set p = 1 in 10, the proof is completed.

References [1] R.-F. Bai, F. Qi and B.-Y. Xi, Hermite-Hadamard type inequalities for the m and ( ; m) logarithmically convex functions, Filomat 27 (2013), 1-7. [2] M.K. Bakula, J. Peµcari´c and M. Ribibi´c, Companion inequalities to Jensen’s inequality for m convex and ( ; m) convex functions, J. Inequal. Pure and Appl. Math., 7 (5) (2006), Article 194. [3] S.S. Dragomir and G. Toader, Some inequalities for m convex functions, Studia University Babes Bolyai, Mathematica, 38 (1) (1993), 21-28. [4] V.G. Mihe¸san, A generalization of the convexity, Seminar of Functional Equations, Approx. and Convex, Cluj-Napoca (Romania) (1993). [5] G. Toader, Some generalization of the convexity, Proc. Colloq. Approx. Opt., Cluj-Napoca, (1984), 329-338. [6] G. Toader, On a generalization of the convexity, Mathematica, 30 (53) (1988), 83-87. 8

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[7] A. Ostrowski, Über die Absolutabweichung einer di¤ erentierbaren Funktion von ihren Integralmittelwert, Comment. Math. Helv., 10, 226-227, (1938). [8] M.E. Özdemir, H. Kavurmaci, E. Set, Ostrowski’s type inequalities for ( ; m) convex functions, KYUNGPOOK Math. J. 50 (2010) 371–378. [9] H. Kavurmaci, M.E. Özdemir and M. Avci, New Ostrowski type inequalities for m convex functions and applications, Hacettepe Journal of Mathematics and Statistics, Volume 40 (2) (2011), 135 –145. [10] M. Alomari and M. Darus, Some Ostrowski type inequalities for convex functions with applications, RGMIA Res. Rep. Coll., (2010) 13, 2, Article 3. [ONLINE: http://ajmaa.org/RGMIA/v13n2.php]. [11] E. Set, New inequalities of Ostrowski type for mappings whose derivatives are s-convex in the second sense via fractional integrals, Comput. Math. Appl., 63 (2012) 1147-1154. [12] S. Belarbi and Z. Dahmani, On some new fractional integral inequalities, J. Ineq. Pure and Appl. Math., 10(3), Art. 86 (2009). [13] Z. Dahmani, New inequalities in fractional integrals, International Journal of Nonlinear Science, 9(4), 493-497 (2010). [14] Z. Dahmani, On Minkowski and Hermite-Hadamard integral inequalities via fractional integration, Ann. Funct. Anal. 1(1), 51-58 (2010). [15] Z. Dahmani, L. Tabharit and S. Taf, Some fractional integral inequalities, Nonl. Sci. Lett. A., 1(2), 155-160 (2010). [16] M.Z. Sar¬kaya, E. Set, H. Yaldiz and N. Ba¸sak, Hermite-Hadamard’s inequalities for fractional integrals and related fractional inequalities, Mathematical and Computer Modelling, In Press. [17] Z. Dahmani, L. Tabharit and S. Taf, New generalizations of Grüss inequality using Riemann-Liouville fractional integrals, Bull. Math. Anal. Appl., 2(3), 93-99 (2010). [18] M.E. Özdemir, H. Kavurmac¬and M. Avc¬, New inequalities of Ostrowski type for mappings whose derivatives are ( ; m)-convex via fractional integrals, RGMIA Research Report Collection, 15, Article 10, 8 pp (2012).

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ON A CLASS OF OPERATORS FROM BERGMAN-TYPE SPACES TO WEIGHTED-TYPE SPACES LI KE AND JIANG ZHI-JIE

Abstract. Let D = {z ∈ C : |z| < 1} be the open unit disk in the complex plane C and H(D) be the space of all analytic functions on D. For ϕ an analytic self-map of D and g ∈ H(D), we define two operators on H(D) by DWϕ,g f = (g · f ◦ ϕ)0 and Wϕ,g Df = (g · f 0 ◦ ϕ). By using some growth properties of the inducing maps ϕ and g to the boundary of D, we obtain an asymptotical expression of the essential norms for operators DWϕ,g and Wϕ,g D from Bergman-type to weighted-type spaces in this paper.

1. Introduction Let D be the open unit disk in the complex plane C and H(D) be the set of all analytic functions on D. If u is a positive continuous function on [0, 1) and there exist positive numbers δ ∈ [0, 1), s and t, 0 < s < t, such that u(r)/(1 − r)s is decreasing on [δ, 1) and limr→1− u(r)/(1 − r)s = 0; u(r)/(1 − r)t is increasing on [δ, 1) and limr→1− u(r)/(1 − r)t = ∞, then u is called a normal weight function (see [1]). For such normal weights, one can consider the following examples: u(r) = (1 − r2 )α , α ∈ (0, ∞), u(r) = (1 − r2 )α {log 2(1 − r2 )−1 }β , α ∈ (0, 1), β ∈

α − 1  log 2, 0 , 2

and  α − 1 log 2, 0 . 2 For 0 < p < ∞ and the normal weight function u, the Bergman-type space Apu on D is defined by Z n o u(|z|)p Apu = f ∈ H(D) : kf kp = |f (z)|p dA(z) < ∞ . 1 − |z| D u(r) = (1 − r2 )α {log log e2 (1 − r2 )−1 }γ , α ∈ (0, 1), γ ∈

When 1 ≤ p < ∞, Apu is a Banach space with the norm k · k. When 0 < p < 1, it is a Fr´echet space with the translation invariant metric d(f, g) = kf − gkp . For this space and some operators, see, e.g., [1] and [2]. Let 0 < α < ∞, and let Hα∞ be the weighted Banach space of analytical functions on D satisfying kf kHα∞ = sup(1 − |z|2 )α |f (z)| < ∞. z∈D

2000 Mathematics Subject Classification. Primary 47B38; Secondary 47B33, 47B37. Key words and phrases. Bergman-type space, weighted-type space, essential norm. 1

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Let ν be a radial bounded continuous positive function D. As the generalization of the weighted Banach space, the weighted-type space Hν∞ consists of all f ∈ H(D) such that kf kHν∞ = sup ν(z)|f (z)| < ∞. z∈D

Hν∞

It is known that is a Banach space under the norm k · kHν∞ . Let ϕ be an analytic self-map of D and g ∈ H(D). For f ∈ H(D), the weighted composition operator Wϕ,g is defined by Wϕ,g f (z) = g(z) · f (ϕ(z)). When g ≡ 1 on D, Wϕ,1 := Cϕ is called the composition operator. When ϕ(z) = z, Wz,g := Mg is the multiplication operator. During the past few decades, weighted composition operators have been studied extensively on spaces of analytic functions on D. For some recent results, see, e.g., [3]-[10]. Using the weighted composition operator Wϕ,g , we define the following two operators: DWϕ,g f (z) = (g · f ◦ ϕ)0 (z) and Wϕ,g Df (z) = (g · f 0 ◦ ϕ)(z). The present author introduced the definitions of DWϕ,g and Wϕ,g D and studied them in [11]. If g ≡ 1 on D, then DWϕ,1 = DCϕ and Wϕ,1 D = Cϕ D are called the products of differentiation and composition. Hibschweiler and Portnoy [12] considered the bounded and compact DCϕ and Cϕ D on Hardy and weighted Bergman spaces. Ohno [13] also studied these problems for them on Bloch and little Bloch spaces. It is well-known that by calculating the essential norm one can also give a characterization of compactness of the linear operator. Recently, we have obtained some asymptotical expression of the essential norms of the weighted composition operators from weighted Bergman spaces to weighted-type spaces on the unit disk and unit ball in [14] and [15]. It is a natural problem how to express the essential norms of DWϕ,g and Wϕ,g D from the Bergman-type spaces to the weighted-type spaces. In this paper, we shall consider this problem. At the last of this section, we recall the definition of the essential norm of the bounded linear operators. Let X and Y be Banach spaces and T : X → Y be a bounded linear operator. The essential norm of the operator T : X → Y is defined by kT ke,X→Y = inf{kT − Kk : K ∈ K}, where K denotes the set of all compact linear operators from X to Y . By this definition, we have that the bounded linear operator T : X → Y is compact if and only if kT ke,X→Y = 0. Throughout this paper, constants are denoted by C, they are positive and may differ from one occurrence to the other. The notation a  b means that there is a positive constant C such that a/C ≤ b ≤ Ca.

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2. The main results The following lemma is also right for the bounded operator Wϕ,g D : Apu → Hν∞ . Since the proof is standard, it is omitted (see, e.g., Proposition 3.11 in [16]). Lemma 2.1. Suppose that ϕ is an analytic self-map of D, g ∈ H(D) and the operator DWϕ,g : Apu → Hν∞ is bounded, then the operator DWϕ,g : Apu → Hν∞ is compact if and only if for bounded sequence (fn )n∈N in Apu such that fn → 0 uniformly on every compact subset of D as n → ∞, it follows that lim kDWϕ,g fn kHν∞ = 0.

n→∞

For the cases of n = 0 and n = 1, the following lemma had been obtained in [2]. However, here we shall give the proof for n ∈ N+ . Lemma 2.2. For n ∈ N+ , there is a positive constant C independent of f ∈ Apu such that for every z ∈ D the following inequality holds kf k |f (n) (z)| ≤ C 1 . u(|z|)(1 − |z|2 )n+ p Proof. For f ∈ Apu , by Lemma 2.1 in [2] we have that |f (z)| ≤ C

kf k 1

u(|z|)(1 − |z|2 ) p

.

(1)

Since u is normal, it is not difficult to see that for each w ∈ {z + 1−|z| 2 ξ : ξ ∈ ∂D} = ), the following relationship holds D(z, 1−|z| 2 u(|z|)  u(w),

(2)

where D(z, 1−|z| 2 ) denotes the disk with center z and radius Since 1 − |z + we get that

1−|z| 2 2 ξ|

2 n

(1 − |z| ) |f

2

≥

1−|z| 4

(n)

n!22n+1 (z)| ≤ π

1−|z| 2 .

for all ξ ∈ ∂D, by the Cauchy integral formula
f z + 1 − |z| ξ |dξ|. 2 ∂D

Z

(3)

By (1), (2) and (3) we have that n!22n+1 (z)| ≤ π


f z + 1 − |z| ξ |dξ| 2 ∂D kf k ≤C 1 , u(|z|)(1 − |z|2 ) p from which the desired result follows. 2 n

(1 − |z| ) |f

(n)

Z

In fact, the next result was obtained in [2]. Lemma 2.3. Suppose that ϕ is an analytic self-map of D and g ∈ H(D), then the operator DWϕ,g : Apu → Hν∞ is bounded if and only if the following conditions are satisfied: (i) M0 := sup z∈D

ν(z)|g 0 (z)| 1

u(|ϕ(z)|)(1 − |ϕ(z)|2 ) p

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(ii) M1 := sup z∈D

ν(z)|g(z)||ϕ0 (z)|

< ∞.

1

u(|ϕ(z)|)(1 − |ϕ(z)|2 )1+ p

Now we begin to formulate and proof the main result of this paper. Theorem 2.4. Suppose that ϕ is an analytic self-map of D, g ∈ H(D) and DWϕ,g : Apu → Hν∞ is bounded, then kDWϕ,g ke,Apu →Hν∞  lim sup j→∞

ν(zj )|g 0 (zj )| u(|ϕ(zj )|)(1 − |ϕ(zj )|2 )

1 p

+ lim sup j→∞

ν(zj )|g(zj )||ϕ0 (zj )| 1

u(|ϕ(zj )|)(1 − |ϕ(zj )|2 )1+ p

Proof. Suppose that {ϕ(zj )}j∈N is a sequence in D such that |ϕ(zj )| → 1− as j → ∞. For this sequence {ϕ(zj )}j∈N , by taking {fϕ(zj ) } the functions in the proof of Lemma 2.3, we have seen that maxj∈N kfϕ(zj ) k ≤ C and fϕ(zj ) → 0 uniformly on compacta of D as j → ∞. Hence for every compact operator K : Apu → Hν∞ , we have kKfϕ(zj ) kHν∞ → 0 as j → ∞. Thus it follows that kDWϕ,g − Kk = sup k(DWϕ,g − K)f kHν∞ kf k=1

k(DWϕ,g − K)fϕ(zj ) kHν∞ kfϕ(zj ) k

≥ lim sup j→∞

kDWϕ,g fϕ(zj ) kHν∞ − kKfϕ(zj ) kHν∞ kfϕ(zj ) k

≥ lim sup j→∞

≥ C −1 lim sup j→∞

ν(zj )|g(zj )||ϕ0 (zj )| 1

u(|ϕ(zj )|)(1 − |ϕ(zj )|2 )1+ p

.

(4)

By taking the infimum in (4) over the set of all compact operators K : Apu → Hν∞ , we obtain that ν(zj )|g(zj )||ϕ0 (zj )| kDWϕ,g ke,Apu →Hν∞ ≥ C −1 lim sup (5) 1 . j→∞ u(|ϕ(zj )|)(1 − |ϕ(zj )|2 )1+ p By using the similar method, we also can prove that kDWϕ,g ke,Apu →Hν∞ ≥ C −1 lim sup j→∞

Consequently, we have obtained that  kDWϕ,g ke,Apu →Hν∞ ≥C lim sup j→∞

+ lim sup j→∞

ν(zj )|g 0 (zj )| 1

u(|ϕ(zj )|)(1 − |ϕ(zj )|2 ) p

.

ν(zj )|g 0 (zj )| 1

u(|ϕ(zj )|)(1 − |ϕ(zj )|2 ) p  ν(zj )|g(zj )||ϕ0 (zj )| 1

u(|ϕ(zj )|)(1 − |ϕ(zj )|2 )1+ p

.

(6)

Now suppose that {rj }j∈N is a positive sequence which increasingly converges to 1. For each rj , consider the operator DWrj ϕ,g . By Lemma 2.3, the boundedness of DWϕ,g : Apu → Hν∞ implies that the operator DWrj ϕ,g : Apu → Hν∞ is bounded. Since |rj ϕ(z)| ≤ rj < 1, by Lemma 2.1 we have that the operator DWrj ϕ,g : Apu → Hν∞ is also compact. Hence we have that kDWϕ,g − DWrj ϕ,g k = sup k(DWϕ,g − DWrj ϕ,g )f kHν∞ kf k=1

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0
0 = sup sup ν(z) g(z)(f ◦ ϕ(z)) − g(z)(f ◦ rj ϕ(z)) kf k=1 z∈D

= sup sup ν(z) g 0 (z)f (ϕ(z)) + g(z)f 0 (ϕ(z))ϕ0 (z) kf k=1 z∈D

− g 0 (z)f (rj ϕ(z)) − rj g(z)f 0 (rj ϕ(z))ϕ0 (z) ≤ sup sup ν(z)|g 0 (z)| f (ϕ(z)) − f (rj ϕ(z)) kf k=1 z∈D

+ sup sup ν(z)|g(z)||ϕ0 (z)| f 0 (ϕ(z)) − rj f 0 (rj ϕ(z)) kf k=1 z∈D

≤ sup sup ν(z)|g 0 (z)| f (ϕ(z)) − f (rj ϕ(z)) kf k=1 z∈D

+ (1 − rj ) sup sup ν(z)|g(z)||ϕ0 (z)| f 0 (rj ϕ(z)) kf k=1 z∈D

+ sup sup ν(z)|g(z)||ϕ0 (z)| f 0 (ϕ(z)) − f 0 (rj ϕ(z)) kf k=1 z∈D

sup ν(z)|g 0 (z)| f (ϕ(z)) − f (rj ϕ(z))

≤ sup

kf k=1 |ϕ(z)|≤δ

sup ν(z)|g 0 (z)| f (ϕ(z)) − f (rj ϕ(z))

+ sup

kf k=1 |ϕ(z)|>δ

ν(z)|g(z)||ϕ0 (z)|

+ C(1 − rj ) sup

1

u(rj |ϕ(z)|)(1 − rj2 |ϕ(z)|2 )1+ p sup ν(z)|g(z)||ϕ0 (z)| f 0 (ϕ(z)) − f 0 (rj ϕ(z)) z∈D

+ sup

kf k=1 |ϕ(z)|≤δ

sup ν(z)|g(z)||ϕ0 (z)| f 0 (ϕ(z)) − f 0 (rj ϕ(z))

+ sup

kf k=1 |ϕ(z)|>δ

sup f (ϕ(z)) − f (rj ϕ(z))

≤ kDWϕ,g 1kHν∞ sup

kf k=1 |ϕ(z)|≤δ

+C

sup |ϕ(z)|>δ

+C

sup |ϕ(z)|>δ

ν(z)|g 0 (z)| 1

u(|ϕ(z)|)(1 − |ϕ(z)|2 ) p ν(z)|g 0 (z)| 1

u(rj |ϕ(z)|)(1 − rj2 |ϕ(z)|2 ) p

+ C(1 − rj ) sup

ν(z)|g(z)||ϕ0 (z)| 1

u(|ϕ(z)|)(1 − |ϕ(z)|2 )1+ p + sup ν(z)|g(z)||ϕ0 (z)| sup sup |f 0 (ϕ(z)) − f 0 (rj ϕ(z)) z∈D

z∈D

+C

kf k=1 |ϕ(z)|≤δ

sup |ϕ(z)|>δ

+C

sup |ϕ(z)|>δ

ν(z)|g(z)||ϕ0 (z)| 1

u(|ϕ(z)|)(1 − |ϕ(z)|2 )1+ p ν(z)|g(z)||ϕ0 (z)| 1

u(rj |ϕ(z)|)(1 − rj2 |ϕ(z)|2 )1+ p

.

(7)

We consider Ij0 := sup

sup |f (ϕ(z)) − f (rj ϕ(z))|.

kf k=1 |ϕ(z)|≤δ

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By using the mean value theorem and the subharmonicity of f and Lemma 2.2 we have sup (1 − rj )|ϕ(z)| sup |f 0 (z)|

Ij0 ≤ sup

kf k=1 |ϕ(z)|≤δ

≤

|z|≤δ

C(1 − rj ) 1

max u(r)(1 − δ 2 )1+ p

.

(8)

0≤r≤δ

By (8), we obtain that Ij0 → 0 as j → ∞. Using the same method, we also have that Ij1 = sup sup |f 0 (ϕ(z)) − f 0 (rj ϕ(z))| → 0 as j → ∞. Letting j → ∞ in kf k=1 |ϕ(z)|≤δ

(7), from the above discussions and the boundedness of DWϕ,g : Apu → Hν∞ we obtain that ν(z)|g 0 (z)| kDWϕ,g − DWrj ϕ,g k ≤ 2C sup 1 |ϕ(z)|>δ u(|ϕ(z)|)(1 − |ϕ(z)|2 ) p ν(z)|g(z)||ϕ0 (z)| + 2C sup 1 1+ p |ϕ(z)|>δ u(|ϕ(z)|)(1 − |ϕ(z)|2 ) as j → ∞. Since kDWϕ,g ke,Apu →Hν∞ ≤ kDWϕ,g − DWrj ϕ,g k, we end the proof. Corollary 2.5. Suppose that ϕ is an analytic self-map of D, g ∈ H(D) and DWϕ,g : Apu → Hν∞ is bounded, then DWϕ,g : Apu → Hν∞ is compact if and only if the following conditions are satisfied: (i) lim

ν(z)|g 0 (z)|

|ϕ(z)|→1−

= 0,

1

u(|ϕ(z)|)(1 − |ϕ(z)|2 ) p

(ii) lim

|ϕ(z)|→1−

ν(z)|g(z)||ϕ0 (z)| 1

u(|ϕ(z)|)(1 − |ϕ(z)|2 )1+ p

= 0.

Similar to Theorem 2.4, we can prove the following result. Theorem 2.6. Suppose that ϕ is an analytic self-map of D, g ∈ H(D) and Wϕ,g D : Apu → Hν∞ is bounded, then kWϕ,g Dke,Apu →Hν∞  lim sup j→∞

ν(zj )|g(zj )| 1

u(|ϕ(zj )|)(1 − |ϕ(zj )|2 )1+ p

.

By Theorem 2.6, we have Corollary 2.7. Suppose that ϕ is an analytic self-map of D, g ∈ H(D) and Wϕ,g D : Apu → Hν∞ is bounded, then Wϕ,g D : Apu → Hν∞ is compact if and only if lim

|ϕ(z)|→1−

ν(z)|g(z)| 1

u(|ϕ(z)|)(1 − |ϕ(z)|2 )1+ p

= 0.

Acknowledgements.The first author is supported by the Natural Science Foundation of SUSE (Grant No.2012KY06).The second author is supported by the Introduction of Talent Project of SUSE(Grant No.2011RC13)and the Science Foundation of Sichuan Province(Grant No.11ZA120).

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References [1] Jiang, Z. J., On Volterra compositon operators from Bergman-type space to Bloch-type space, Czechoslovak Mathematicial Journal, 2011, 136:993-1005. [2] Li, S. and Stevi´ c, S., Weighted composition operators from Bergman-type spaces into Bloch spaces, Proc. Indian Acad. Sci., 2007, 117:371-385. [3] Jiang, Z. J. and Bai, H. B., Weighted composition operators on Hardy space H p (BN )(In Chinese), Advances in Mathematics, 2008, 37:749-754. [4] Jiang, Z. J., Weighted composition operators from Bergman-type spaces to Bers-type spaces, Acta Mathematics Sinica, (Chinese Series), 2010, 53:67-74. [5] Stevi´ c, S. and Jiang, Z. J., Differences of weighted composition operators on the unit polydisk, Siberian Mathematics Journal, 2011, 52:358-371. [6] Stevi´ c, S., Norm of weighted composition operators from Bloch space to Hµ∞ on the unit ball, Ars. Combin., 2008, 88:125-127. [7] Li, S. and Stevi´ c, S., Weighted composition operators from Zygmund spaces into Bloch spaces, Appl. Math. Comput., 2008, 206:825-831. [8] Ohno, S., Weighted composition operators between H ∞ and the Bloch space, Taiwan. J. Math. Soc., 2001, 5:555-563. [9] Sharma, A. K. and Abbas, Z., Weighted composition operators between weighted BergmanNevanlinna and Bloch-type spaces, Applied Mathematical Sciences., 2010, 41:2039-2048. [10] Zhou, Z. H. and Chen, R. Y., Weighted composition operators from F (p, q, s) to Bloch space on the unit ball, Internat. J. Math., 2008, 19:899-926. [11] Jiang, Z. J., On a class of opertors from weighted Bergman spaces to some spaces of analytic functions, Taiwanese Journal of Mathematics, 2011, 15:2095-2121. [12] Hibschweiler, R. A. and Portnoy, N., Composition followed by differentiation between Bergman and Hardy spaces, Rocky Mountain J. Math., 2005, 35:843-855. [13] Ohno, S., Products of composition and differentiation on Bloch spaces, Bull. Korean Math. Soc., 2009, 46:1135-1140. [14] Jiang, Z. J. and Stevi´ c, S., Compact differences of weighted composition operators from weighted Bergman spaces to weighted-type spaces, Appl. Math. Comput., 2010, 217:35223530. [15] Stevi´ c, S. and Jiang, Z. J., Compactness of differences of weighted composition operators from weighted Bergman spaces to weighted-type spaces on the unit ball, Taiwanese Journal of Mathematics, 2011, 15:2647-2665. [16] Cowen, C. C. and MacCluer, B. D., Composition operators on spaces of analytic functions, CRC Press, 1995. Li Ke, School of Science, Sichuan University of Science and Engineering, Zigong, Sichuan, 643000, P. R. China E-mail address: [email protected] Jiang Zhi-Jie, Institute of Nonlinear Science and Engineering Computing, Sichuan University of Science and Engineering, Zigong, Sichuan, 643000, P. R. China E-mail address: [email protected]

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´ GENERALIZED JORDAN HOMOMORPHISMS IN FRECHET ALGEBRAS MADJID ESHAGHI GORDJI1 , M. BAVAND SAVADKOUHI2 , CHOONKIL PARK∗3 AND JUNG RYE LEE4 Abstract. A linear mapping h : A → B is called a generalized Jordan homomorphism if there exists a homomorphism h′ : A → B such that h(a2 ) = h(a)h′ (a) for all a ∈ A. In this paper, we investigate generalized Jordan homomorphisms in Fr´echet algebras, associated with the following functional equation ( ) ( ) a+b a−b f +f = f (a). 2 2 Moreover, we prove the Hyers-Ulam stability of generalized Jordan homomorphisms in Fr´echet algebras.

1. Introduction The stability problem of functional equations originated from a question of Ulam [30] in 1940, concerning the stability of group homomorphisms. In 1941, Hyers [20] gave a first affirmative answer to the question of Ulam for Banach spaces. In 1978, Th. M. Rassias [26] provided a generalization of Hyers’ Theorem which allows the Cauchy difference to be unbounded. The stability problems of several functional equations have been extensively investigated by a number of authors and there are many interesting results concerning this problem (see [3, 4, 6, 7, 10, 11, 12, 15, 21, 24, 25, 28]). Definition 1.1. A topological vector space X is a Fr´echet space if it satisfies the following three properties: (1) it is complete as a uniform space, (2) it is locally convex, (3) its topology can be induced by a translation invariant metric, i.e., a metric d : X × X → R such that d(x, y) = d(x + a, y + a) for all a, x, y ∈ X. 2010 Mathematics Subject Classification. 39B52; 46Hxx; 46K05; 39B82; 47B47. Key words and phrases. Hyers-Ulam stability; generalized homomorphism; generalized Jordan homomorphism; Fr´echet algebra. ∗ Corresponding author. 391

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M. ESHAGHI GORDJI, M.B. SAVADKOUHI, C. PARK, J. LEE

For more detailed definitions of such terminologies, we can refer to [8]. Note that a ternary algebra is called ternary Fr´echet algebra if it is a Fr´echet space with a metric d. Fr´echet algebras, named after Maurice Fr´echet, are special topological algebras as follows. Note that the topology on A can be induced by a translation invariant metric, i.e. a metric d : X × X → R such that d(x, y) = d(x + a, y + a) for all a, x, y ∈ X. Trivially, every Banach algebra is a Fr´echet algebra as the norm induces a translation invariant metric and the space is complete with respect to this metric. Definition 1.2. Let A and B be two algebras. A linear mapping h : A → B is called a generalized Jordan homomorphism if there exists a homomorphism h′ : A → B such that h(a2 ) = h(a)h′ (a) for all a, b ∈ A. For example, every Jordan homomorphism (resp., homomorphism) is a generalized Jordan homomorphism (resp., generalized homomorphism), but the converse is false, in general. For instance, let A be an algebra over C and let h : A → A be a non-zero Jordan homomorphism (resp., homomorphism) on A. Then we have ih(a2 ) = ih(a)2 = ih(a)h(a), (resp., ih(ab) = ih(a)h(b) = ih(a)h(b)). This means that ih is a generalized Jordan homomorphism (resp., generalized homomorphism). It is easy to see that ih is not a Jordan homomorphism (resp., homomorphism). Th.M. Rassias [27], Gajda [19] and Bourgin [5] proved the stability problem of ring homomorphisms between unital Banach algebras. Badora [1] proved the Hyers-Ulam stability of ring homomorphisms, which generalizes the result of Bourgin. Miura al et. [22] proved the Hyers-Ulam stability of Jordan homomorphisms. For more details about the results concerning stability of functional equations on Banach algebras, the reader refer to [2, 13, 14, 16, 17, 18]. Recently, Eshaghi Gordji and Bavand Savadkouhi [9] proved the Hyers-Ulam stability of generalized homomorphisms in quasi-Banach algebras. In this paper, we prove the Hyers-Ulam stability of generalized Jordan homomorphisms in Fr´echet algebras.

2. Hyers-Ulam stability of generalized Jordan homomorphisms in ´chet algebras Fre In this section, we prove the Hyers-Ulam stability of generalized Jordan homomorphisms in Fr´echet algebras. 392

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´ GENERALIZED JORDAN HOMOMORPHISMS IN FRECHET ALGEBRAS

Lemma 2.1. ([23]) Let X and Y be linear spaces and let f : X → Y be an additive mapping such that f (µx) = µf (x) for all x ∈ X and all µ ∈ T1 := {µ ∈ C : |µ| = 1}. Then the mapping f is C-linear. Theorem 2.2. Let A be a Fr´echet algebra with metric d and unit e, and B a Banach algebra with norm ∥ · ∥ and unit I. Let f, g : A → B be mappings with f (0) = 0, g(0) = 0 and g(e) = I for which there exists a function ϕ : A3 → [0, ∞) such that ˜ y, z) := ϕ(x,

∞ ∑ 1 ϕ(2j x, 2j y, 2j z) < ∞, j 2 j=1

(

) ( )

µx + µy µx − µy 2

f

≤ ϕ(x, y, z), + z + f − µf (x) + f (z)g(z)

2 2

(

)

µxy + µz

2g

≤ ϕ(x, y, z) − µg(x)g(y) − µg(z)

2

(2.1)

(2.2)

(2.3)

for all x, y, z ∈ A and all µ ∈ T1 . Then there exist a unique generalized Jordan homomorphism H : A → B and a unique homomorphism G : A → B such that H(x2 ) = H(x)G(x), ˜ 0, 0), ∥f (x) − H(x)∥ ≤ ϕ(x,

(2.4)

˜ e, 0) ∥g(x) − G(x)∥ ≤ ϕ(x,

(2.5)

for all x ∈ A. Proof. Putting y = z = 0 and µ = 1 in (2.2), we get

(x)

− f (x) ≤ ϕ(x, 0, 0)

2f 2 and so



f (2x)

ϕ(2x, 0, 0)

≤ − f (x)

2

2

(2.6)

for all x ∈ A. Using the Rassias’ method on (2.6) ([19]), one can use induction on n to show that

n

f (2n x)

∑ ϕ(2j x, 0, 0)

≤ − f (x) (2.7)

2n

j 2 j=1 393

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M. ESHAGHI GORDJI, M.B. SAVADKOUHI, C. PARK, J. LEE

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

n+m ∑ ϕ(2j x, 0, 0)

f (2n+m x) f (2m x)

≤ −

2n+m 2m j=m+1 2j for all nonnegative integers n and m with n ≥ m and all x ∈ A. It follows from (2.1) n that the sequence { f (22n x) } is Cauchy. Due to the completeness of B, this sequence is convergent. So one can define the mapping H : A → B by f (2n x) (2.8) n→∞ 2n for all x ∈ A. Letting z = 0 and replacing x, y by 2n x, 2n y, respectively, in (2.2) and multiplying both sides by 21n , we get

(

) ( )

H µx + µy + H µx − µy − µH(x)

2 2

( n

) ( n )

2 µ(x − y) 2 µ(x + y) 1 n

+ f − µf (2 x) = lim n f

n→∞ 2 2 2 ϕ(2n x, 2n y, 0) ≤ lim n→∞ 2n for all x, y ∈ A, µ ∈ T1 and all nonnegative integers n. Taking the limit as n → ∞, we obtain ( ) ( ) µx + µy µx − µy H +H = µH(x) (2.9) 2 2 H(x) := lim

for all x, y ∈ A and all µ ∈ T1 . Letting µ = 1 in (2.9), we can easily show that H is additive. Letting y = 0 in (2.9), we get ( µx ) H(µx) = 2H = µH(x) 2 for all x ∈ A and all µ ∈ T1 . By Lemma 2.1, the mapping H : A → B is C-linear. Moreover, it follows from (2.7) and (2.8) that ˜ 0, 0) ∥f (x) − H(x)∥ ≤ ϕ(x, for all x ∈ A. Putting y = e, z = 0 and µ = 1 in (2.3), we get

(x)

− g(x) ≤ ϕ(x, e, 0)

2g 2 and so

g(2x)

ϕ(2x, e, 0)

≤ − g(x)

2

2 394

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for all x ∈ A. By the same method as above, one can show that there is a C-linear mapping G : A → B satisfying (2.5). The mapping G : A → B is given by g(2n x) n→∞ 2n

G(x) = lim

for all x ∈ A. Putting z = 0 and µ = 1 in (2.3), we get

( xy )

− g(x)g(y) ≤ ϕ(x, y, 0)

2g 2 and so

( n )

1 4 xy n n

2g

≤ 1 ϕ(2n x, 2n y, 0) ≤ 1 ϕ(2n x, 2n y, 0), − g(2 x)g(2 y)

4n 4n 2 2n ( ) = G(x)G(y) for all x, y ∈ A. So which tends to zero as n → ∞. Thus G(xy) = 2G xy 2 G : A → B is a homomorphism. Putting x = y = 0 and µ = 1 in (2.2), we get

( 2)

f z − f (z)g(z) ≤ ϕ(0, 0, z) and so

( ) 1

f 4n z 2 − f (2n z)g(2n z) ≤ 1 ϕ(0, 0, 2n z) ≤ 1 ϕ(0, 0, 2n z), 4n 4n 2n which tends to zero as n → ∞. Thus H(z 2 ) = H(z)G(z) for all z ∈ A. So H : A → B is a generalized Jordan homomorphism. Now, let H ′ : A → B be another generalized Jordan homomorphism satisfying (2.4). Then we have 1 ∥H(2n x) − H ′ (2n x)∥ 2n 1 ≤ n (∥H(2n x) − f (2n x)∥ + ∥f (2n x) − H ′ (2n x)∥) 2 2 ˜ n ≤ n ϕ(2 x, 0, 0) 2

∥H(x) − H ′ (x)∥ =

which tends to zero as n → ∞ for all x ∈ A. So we can conclude that H(x) = H ′ (x) for all x ∈ A. This proves the uniqueness of H. Thus the mapping H : A → B is a unique generalized Jordan homomorphism satisfying (2.4). Similarly, one can prove the uniqueness of G.  395

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M. ESHAGHI GORDJI, M.B. SAVADKOUHI, C. PARK, J. LEE

Corollary 2.3. Let A be a Fr´echet algebra with metric d and unit e, and B a Banach algebra with norm ∥ · ∥ and unit I. Let 0 < p < 1. Let f, g : A → B be mappings with f (0) = 0, g(0) = 0 and g(e) = I such that

(

) ( )

µx + µy µx − µy 2

f

≤ d(x, 0)p + d(y, 0)p + d(z, 0)p , + z + f − µf (x) + f (z)g(z)

2 2

(

)

µxy + µz p p p

2g − µg(x)g(y) − µg(z)

≤ d(x, 0) + d(y, 0) + d(z, 0)

2 for all x, y, z ∈ A and all µ ∈ T1 . Then there exist a unique generalized Jordan homomorphism H : A → B and a unique homomorphism G : A → B such that H(x2 ) = H(x)G(x), ∥f (x) − H(x)∥ ≤

2p d(x, 0)p , 2 − 2p

2p (d(x, 0)p + d(e, 0)p ) ∥g(x) − G(x)∥ ≤ 2 − 2p for all x ∈ A. Proof. Note that d(2x, 0) ≤ 2d(x, 0). It follows from Theorem 2.2 by putting ϕ(x, y, z) = d(x, 0)p + d(y, 0)p + d(z, 0)p for all x, y, z ∈ A.  Theorem 2.4. Let A be a Banach algebra with norm ∥ · ∥ and unit I, and B a Fr´echet algebra with metric d and unit e. Let f, g : A → B be mappings with f (0) = 0, g(0) = 0 and g(I) = e for which there exists a function ϕ : A3 → [0, ∞) such that ∞ (x y z) ∑ j 4 ϕ j , j , j < ∞, (2.10) 2 2 2 j=0 ( ( ) ( ) ) µx − µy µx + µy 2 d f +z +f , µf (x) − f (z)g(z) ≤ ϕ(x, y, z), 2 2 ) ( ( ) µxy + µz , µg(x)g(y) + µg(z) ≤ ϕ(x, y, z) d 2g 2

(2.11)

(2.12)

for all x, y, z ∈ A and all µ ∈ T1 . Then there exist a unique generalized Jordan homomorphism H : A → B and a unique homomorphism G : A → B such that H(x2 ) = H(x)G(x), ˜ 0, 0), d(f (x), H(x)) ≤ ϕ(x, 396

(2.13) PARK ET AL 391-401

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´ GENERALIZED JORDAN HOMOMORPHISMS IN FRECHET ALGEBRAS

˜ I, 0) d(g(x), G(x)) ≤ ϕ(x,

(2.14)

for all x ∈ A. Here ˜ y, z) := ϕ(x,

∞ ∑ j=0

2j ϕ

(x y z) , , 2. Let f, g : A → B be mappings with f (0) = 0, g(0) = 0 and g(I) = e such that ( ( ) ( ) ) µx + µy µx − µy 2 d f +z +f , µf (x) − f (z)g(z) ≤ θ(∥x∥p + ∥y∥p + ∥z∥p ), 2 2 ( d 2g

(

µxy + µz 2

)

) − µg(x)g(y), µg(z) ≤ θ(∥x∥p + ∥y∥p + ∥z∥p )

for all x, y, z ∈ A and all µ ∈ T1 . Then there exist a unique generalized Jordan homomorphism H : A → B and a unique homomorphism G : A → B such that H(x2 ) = H(x)G(x), d(f (x), H(x)) ≤

d(g(x), G(x)) ≤

2p θ ∥x∥p , 2p − 2

2p θ(∥x∥p + 1) 2p − 2

for all x ∈ A. Proof. Note that d(2x, 0) ≤ 2d(x, 0). It follows from Theorem 2.4 by putting ϕ(x, y, z) = θ(∥x∥p + ∥y∥p + ∥z∥p ) for all x, y, z ∈ A.  399

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M. ESHAGHI GORDJI, M.B. SAVADKOUHI, C. PARK, J. LEE

Acknowledgement C. Park was supported by Basic Science Research Program through the National Research Foundation of Korea funded by the Ministry of Education, Science and Technology (NRF-2012R1A1A2004299). References [1] R. Badora, On approximate derivations, Math. Inequal. Appl. 9 (2006), 167–173. [2] M. Bavand Savadkouhi, M. Eshaghi Gordji, J.M. Rassias and N. Ghobadipour, Approximate ternary Jordan derivations on Banach ternary algebras, J. Math. Phys. 50 (2009), Article ID 042303, 9 pages. [3] M. Bessenyei, G. Horv´ath and C.G. K´ezi, Functional equations on finite groups of substitutions, Expo. Math. (to appear). [4] M. Bessenyei and C.G. K´ezi, Functional equations and groups substitutions, Linear Algebra and its Applications 434 (2011), 1525–1531. [5] D.G. Bourgin, Approximately isometric and multiplicative transformations on continuse function rings, Duke Math, J. 16 (1949), 358–397. [6] I. Chang, Stability of higher ring derivations in fuzzy Banach algebras, J. Computat. Anal. Appl. 14 (2012), 1059–1066. [7] I. Cho, D. Kang and H. Koh, Stability problems of cubic mappings with the fixed point alternative, J. Computat. Anal. Appl. 14 (2012), 132–142. [8] M. Eshaghi Gordji, Stability of an additive-quadratic functional equation of two variables in F-spaces, J. Nonlinear Sci. Appl. 2 (2009), 251–259. [9] M. Eshaghi Gordji and M. Bavand Savadkouhi, Approximation of generalized homomorphisms in quasi-Banach algebras, Analele Univ. Ovidius Constata, Math. Series 17 (2009), 203–214. [10] M. Eshaghi Gordji, M. Bavand Savadkouhi and M. Bidkham, Stability of a mixed type additive and quadratic functional equation in non-Archimedean spaces, J. Computat. Anal. Appl. 12 (2010), 454–462. [11] M. Eshaghi Gordji and A. Bodaghi, On the stability of quadratic double centralizers on Banach algebras, J. Computat. Anal. Appl. 13 (2011), 724–729. [12] M. Eshaghi Gordji, R. Farokhzad Rostami and S.A.R. Hosseinioun, Nearly higher derivations in unital C ∗ -algebras, J. Computat. Anal. Appl. 13 (2011), 734–742. [13] M. Eshaghi Gordji, M.B. Ghaemi, S. Kaboli Gharetapeh, S. Shams and A. Ebadian, On the stability of J ∗ -derivations, J. Geom. Phys. 60 (2010), 454–459. [14] M. Eshaghi Gordji and N. Ghobadipour, Stability of (α, β, γ)-derivations on Lie C ∗ -algebras, Internat. J. Geom. Meth. Mod. Phys. 7 (2010), No. 7, 1–10. [15] M. Eshaghi Gordji, S. Kaboli Gharetapeh, T. Karimi, E. Rashidi and M. Aghaei, Ternary Jordan derivations on C ∗ -ternary algebras, J. Computat. Anal. Appl. 12 (2010), 463–470. [16] M. Eshaghi Gordji and A. Najati, Approximately J ∗ -homomorphisms: A fixed point approach, J. Geom. Phys. 60 (2010), 809–814. [17] M. Eshaghi Gordji, J.M. Rassias and N. Ghobadipour, Generalized Hyers-Ulam stability of the generalized (n, k)-derivations, Abs. Appl. Anal. 2009 (2009), Article ID 437931, 8 pages. 400

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[18] R. Farokhzad and S. A. R. Hosseinioun, Perturbations of Jordan higher derivations in Banach ternary algebras: An alternative fixed point approach, Int. J. Nonlinear Anal. Appl. 1 (2010), 42–53. [19] Z. Gajda, On stability of additive mappings, Internat. J. Math. Math. Sci. 14 (1991), 431–434. [20] D.H. Hyers, On the stability of the linear functional equation, Proc. Natl. Acad. Sci. USA 27 (1941), 222–224. [21] H.A. Kenary, J. Lee and C. Park, Non-Archimedean stability of an AQQ-functional equation, J. Computat. Anal. Appl. 14 (2012), 211–227. [22] T. Miura, S.E. Takahashi and G. Hirasawa, Hyers-Ulam-Rassias stability of Jordan homomorphisms on Banach algebras, J. Inequal. Appl 4 (2004), 435–441. [23] C. Park, Homomorphisms between Poisson JC ∗ -algebras, Bull. Braz. Math. Soc. 36 (2005), 79–97. [24] C. Park, Y. Cho and H.A. Kenary, Orthogonal stability of a generalized quadratic functional equation in non-Archimedean spaces, J. Computat. Anal. Appl. 14 (2012), 526–535. [25] C. Park, S. Jang and R. Saadati, Fuzzy approximate of homomorphisms, J. Computat. Anal. Appl. 14 (2012), 833–841. [26] Th.M. Rassias, On the stability of the linear mapping in Banach spaces, Proc. Amer. Math. Soc. 72 (1978), 297–300. [27] Th.M. Rassias, Problem 16; 2, Report of the 27th International Symp. on Functional Equations, Aequationes Math. 39 (1990), 292–293; 309. [28] S. Shagholi, M. Eshaghi Gordji and M. Bavand Savadkouhi, Stability of ternary quadratic derivations on ternary Banach algebras, J. Computat. Anal. Appl. 13 (2011), 1097–1105. [29] S. Shagholi, M. Eshaghi Gordji and M. Bavand Savadkouhi, Nearly ternary cubic homomorphism in ternary Fr´echet algebras, J. Computat. Anal. Appl. 13 (2011), 1097–1105. [30] S.M. Ulam, Problems in Modern Mathematics, Chapter VI, Science ed., Wiley, New York, 1940. 1,2

Department of Mathematics, Semnan University, P.O. Box 35195-363, Semnan, Iran Research Group of Nonlinear Analysis and Applications (RGNAA), Semnan , Iran; Centre of Excellence in Nonlinear Analysis and Applications (CENAA), Semnan University, Iran, 3 4

4

Research Institute for Natural Sciences, Hanyang University, Seoul 133-791, Korea, Department of Mathematics, Daejin University, Kyeonggi 487-711, Korea E-mail address: 1 [email protected]; 2 [email protected]; 3 [email protected]; [email protected]

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402

TABLE OF CONTENTS, JOURNAL OF COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 16, NO. 2, 2014 Relationship Between Lower and Higher Order Anti-Periodic Boundary Value Problems and Existence Results, Bashir Ahmad, Ahmed Alsaedi, and Afrah Assolami,……………………210 The Improved (G'/G)-Expansion Method to the (2+1)-Dimensional Breaking Soliton Equation, Hasibun Naher, and Farah Aini Abdullah,…………………………………………………….220 A Multiple Attribute Group Decision Making Method based on Generalized Interval-valued Trapezoidal Fuzzy Numbers, Chao Liu, and Peide Liu,……………………………………..236 Note on the Second Kind Barnes’ Type Multiple q-Euler Polynomials, C. S. Ryoo,……….246 The Approximation and Growth Problem of Dirichlet Series of Infinite Order, Hua Wang, and Hong-Yan Xu,……………………………………………………………………………….251 Weak and Strong Convergence Theorems of Proximal Point Algorithm for Solving Generalized Mixed Equilibrium Problems and Finding Zeroes of Maximal Monotone Operators in Banach Spaces, Withun Phuengrattana, Suthep Suantai, Kriengsak Wattanawitoon, Uamporn Witthayarat, and Poom Kumam,…………………………………………………………….264 On a class of Two Dimensional (w,q)-Bernoulli and (w,q)-Euler Polynomials: Properties and Location of Zeros, N.I. Mahmudov, A. Akkeleș, and A. Öneren,………………………….282 Some Identities of Polynomials Arising From Umbral Calculus, Dae San Kim, Taekyun Kim, and Seog-Hoon Rim,…………………………………………………………………………293 An Incomplete Discontinuous Galerkin Finite Element Method for Second Order Elliptic Problem, Fuzheng Gao, and Feng Qiao,…………………………………………………….307 Leader-Following Consensus of Multi-Agent Systems with Memory, Xin-Lei Feng, and Ting-Zhu Huang,…………………………………………………………………………….315 Tripled Coincidence Points for Mixed Comparable Mappings in Partially Ordered Metric Spaces, Jiandong Yin,…………………………………………………………………………………329 Fixed Point Theorems for a Banach Type Contraction on Tvs-cone Metric Spaces Endowed with a Graph, Prasit Cholamjiak,……………………………………………………………………338 Approximate (m,n)-Cauchy-Jensen Mappings in Quasi-β-Normed Spaces, John Michael Rassias, and Hark-Mahn Kim,…………………………………………………………………………346

TABLE OF CONTENTS, JOURNAL OF COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 16, NO. 2, 2014 (continued)

Fixed Point Method for Intuitionistic Fuzzy Stability of Mixed Type Cubic-Quartic Functional Equation, Saud M. Alsulami,…………………………………………………………………359 Isometries of P-Nuclear Operators, Abdelrahman Yousef, and Roshdi Khalil,………………368 Inequalities of Ostrowski's type for m- and (α,m)- Logarithmically Convex Functions via Riemann-Liouville Fractional Integrals, Ahmet Ocak Akdemir,………………………………375 On a Class of Operators from Bergman-Type Spaces to Weighted-Type Spaces, Li Ke, and Jiang Zhi-Jie,……………………………………………………………………………………384 Generalized Jordan Homomorphisms in Fréchet Algebras, Madjid Eshaghi Gordji, M. Bavand Savadkouhi, Choonkil Park, and Jung Rye Lee,………………………………………………391

Volume 16, Number 3 ISSN:1521-1398 PRINT,1572-9206 ONLINE

April 2014

Journal of Computational Analysis and Applications EUDOXUS PRESS,LLC

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

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Editorial Board Associate Editors of Journal of Computational Analysis and Applications 1) 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. 2) 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

20)Margareta Heilmann Faculty of Mathematics and Natural Sciences University of Wuppertal Gaußstraße 20 D-42119 Wuppertal, Germany, [email protected] Approximation Theory (Positive Linear Operators) 21) Christian Houdre School of Mathematics Georgia Institute of Technology Atlanta,Georgia 30332 404-894-4398 e-mail: [email protected] Probability, Mathematical Statistics, Wavelets

3) Mark J.Balas Department Head and Professor Electrical and Computer Engineering Dept. College of Engineering University of Wyoming 1000 E. University Ave. Laramie, WY 82071 307-766-5599 e-mail: [email protected] Control Theory,Nonlinear Systems, Neural Networks,Ordinary and Partial Differential Equations, Functional Analysis and Operator Theory

22) Irena Lasiecka Department of Mathematical Sciences University of Memphis Memphis, TN 38152 P.D.E, Control Theory, Functional Analysis, [email protected]

4) Dumitru Baleanu Cankaya University, Faculty of Art and Sciences, Department of Mathematics and Computer Sciences, 06530 Balgat, Ankara, Turkey, [email protected] Fractional Differential Equations Nonlinear Analysis, Fractional Dynamics

5) Carlo Bardaro Dipartimento di Matematica e Informatica

407

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An note on Sr -covering approximation spaces Bin Qin†

∗

Xun Ge‡

February 21, 2013 Abstract: In this paper, we prove that a covering approximation space (U, C) is an Sr -covering approximation space if and only if {N (x) : x ∈ U } forms a partition of the universe of discourse U . Furthermore, we give some simple characterizations for Sr -space (U, C) by using only a single covering approximation operator and by using only elements of covering C. Results of this paper answer affirmatively an open problem posed by Z.Yun et al. in [16]. Keywords: Universe of discourse; Covering approximation space; Sr -covering approximation space; Covering lower (upper) approximation operation; Neighborhood; Partition.

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Introduction

Rough set theory, which was first proposed by Z.Pawlak in [4], is a useful tool in researches and applications of process control, economics, medical diagnosis, biochemistry, environmental science, biology, chemistry, psychology, conflict analysis and other fields [2, 3, 5, 6, 10, 14, 15, 18, 19]. In the classical rough set theory, Pawlak approximation spaces are based on partitions of the universe of discourse U , but this requirement is not satisfied in some situations [20]. In the past years, Pawlak approximation spaces have been extended to covering approximation spaces [1, 8, 12, 13, 16, 21]. Definition 1.1 ([21]). Let U , the universe of discourse, be a finite set and C be a family of nonempty subsets of U . S (1) C is called a covering of T U if {K : K ∈ C} = U . Furthermore, C is called a partition of U if also K K 0 = ∅ for all K, K 0 ∈ C, where K 6= K 0 . (2) The pair (U, C) is called a covering approximation space (resp. a Pawlak approximation space) if C is a covering (resp. a partition) of U . ∗ This work is supported by the National Natural Science Foundation of China (No.11226085), the Natural Science Foundation of Guangxi Province in China (2012GXNSFDA276040) and the Science Foundation of Guangxi College of Finance and Economics(No.2012ZD001). † School of Information and Statistics, Guangxi College of Finance and Economics, Nanning, Guangxi 530003, China, [email protected] ‡ Corresponding Author, School of Mathematical Sciences, Soochow University, Suzhou 215006, China, [email protected]

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T (3) {K : x ∈ K ∈ C} is called the neighborhood of x and denoted as N eighborC (x). When there is no confusion, we omit C at the lowercase and abbreviate N eighborC (x) to N (x). For a covering approximation spaces (U, C), it is interesting to study the condition for {N (x) : x ∈ U } to form a partition of universe U . In particular, it is an important issue in covering approximation spaces theory to characterize this condition by covering lower (upper) approximation operations [8, 16]. In order to give a more detailed description for this issue, we present some covering lower (upper) approximation operations as follows. Definition 1.2 ([16]). Let (U, C) be a covering approximation space and X ⊆ U . Put S (1) C2 (X) = {K : K ∈ C ∧ K ⊆ X}, C2 (X) = U − C2 (U − X); T (2) C3 (X) = {x ∈ U : N (x) ⊆ X}, C3 (X) = {x ∈ U : N (x) X 6= ∅}; (3) C4 (X) = {x ∈ U : ∃u(u ∈ N (x) ∧ N (u) ⊆ X)}, C4 (X) = {x ∈ U : ∀u(u ∈ N (x) → N (u) ∩ X 6= ∅)}; S (4) C5 (X) T = {x ∈ U : ∀u(x ∈ N (u) → N (u) ⊆ X)}, C5 (X) = {N (x) : x ∈ U ∧ N (x) X 6= ∅}; S (5) C6 (X) = {x ∈ U : ∀u(x ∈ N (u) → u ∈ X)}, C6 (X) = {N (x) : x ∈ X}. Then Ci (resp. Ci ) is called a covering lower (resp. upper) approximation operation and Ci (X) (resp. Ci (X)) is called covering lower (resp. upper) approximation of X. Here, i = 2, 3, 4, 5, 6. Remark 1.3. In [8], Ci and Ci are denoted by Ci−1 and Ci−1 respectively. Here, i = 2, 3, 4, 5, 6. K.Qin et al. gave the following theorem. Theorem 1.4 ([8]). Let (U, C) be a covering approximation space. Then the following are equivalent. (1) (2) (3) (4) (5) (6) (7)

{N (x) : x ∈ U } forms a partition of U . C5 (X) = C6 (X) for each X ⊆ U . C5 (X) = C4 (X) for each X ⊆ U . C3 (X) = C4 (X) for each X ⊆ U . C6 (X) = C4 (X) for each X ⊆ U . C3 (X) = C6 (X) for each X ⊆ U . C5 (X) = C3 (X) for each X ⊆ U .

Recently, taking Theorem 1.4 into account, Z. Yun et al. [16] investigated the following question. Question 1.5 ([16]). Can we characterize the conditions under which {N (x) : x ∈ U } forms a partition of U by using only a single covering approximation operator among C2 -C6 ? 2

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The following results were obtained. Theorem 1.6 ([16]). Let (U, C) be a covering approximation space. Then the following are equivalent. (1) (2) (3) (4)

{N (x) : x ∈ U } forms a partition of U . C3 (C3 (X)) = C3 (X) for each X ⊆ U . C6 (C6 (X)) = C6 (X) for each X ⊆ U . C4 (X) ⊆ X for each X ⊆ U .

Theorem 1.7 ([16]). Let (U, C) be a covering approximation space. (1) If C2 (C2 (X)) = C2 (X) for each X ⊆ U , then {N (x) : x ∈ U } forms a partition of U , not vice versa. (2) If {N (x) : x ∈ U } forms a partition of U , then C5 (C5 (X)) = C5 (X) for each X ⊆ U , not vice versa. As an open problem, the following question is raised in the end of [16]. Question 1.8 ([16]). How to give sufficient and necessary conditions for {N (x) : x ∈ U } to form a partition of U by using only a single covering approximation operator Ci (i = 2, 5)? In this paper, we investigate Question 1.5 and Question 1.8 by Sr -covering approximation spaces. Here, Sr -covering approximation spaces was introduced by X.Ge in [1]. Definition 1.9 ([1]). A covering approximation space (U, C) is called an Sr space (Sr -space is the abbreviation of Sr -covering S approximation space) if x ∈ K ∈ C implies D(x) ⊂ K, where D(x) = U − (C − Cx ). In this paper, we gives a ”nice” characterization for Sr -space. By this result, we translate the condition for {N (x) : x ∈ U } to form a partition of the universe of discourse U into Sr -space (U, C) in Question 1.5 and Question 1.8. Furthermore, we obtain some simple characterizations for Sr -space (U, C) by using only a single covering approximation operator and by using only elements of covering C, which answer Question 1.5 and Question 1.8 and improve some results obtained in [16].

2

Preliminaries

For a covering approximation space (U, C), we say that {N (x) : xT∈ U } forms a partition of U if for every pair x, y ∈ U , N (x) = N (y) or N (x) N (y) = ∅. Before our discussion, we give some notations. Note 2.1. Let (U, C) be a covering approximation space. Throughout this paper, we use the following notations, where x ∈ U , X ⊆ U and F ⊆ 2U . T T (1) S F = S{F : F ∈ F}. (2) F = {F : F ∈ F}. 3

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(3) Cx = {KT: x ∈ K ∈ C}. (4) N (x) = CxS . (5) D(x) = U − (C − Cx ). Remark 2.2. It is clear that x ∈ N (x) and x ∈ D(x). Note that x ∈ K ∈ C if and only if K ∈ Cx , we also replace x ∈ K ∈ C by K ∈ Cx in this paper. The following three lemmas are known. Lemma 2.3 ([8, 9]). Let (U, C) be a covering approximation space and X, Y ⊆ U . Then the following hold. (1) (2) (3) (4) (5)

Ci (U ) = U = Ci (U ), Ci (∅) = ∅ = Ci (∅) for i = 2, 3, 4, 5, 6. Ci (X) ⊆ X ⊆ Ci (X) for i = 2, 3, 5, 6. X ⊆ Y ⊆ U =⇒ Ci (X) ⊆ Ci (Y ), Ci (X) ⊆ Ci (Y ) for i = 2, 3, 4, 5, 6. T T S S Ci (X Y ) = Ci (X) Ci (Y ), Ci (X Y ) = Ci (X) Ci (Y ) for i = 3, 5, 6. Ci (X) = U − Ci (U − X), Ci (X) = U − Ci (U − X) for i = 2, 3, 4, 5, 6.

Lemma 2.4 ([8]). Let (U, C) be a covering approximation space. Then the following are equivalent. (1) {N (x) : x ∈ U } forms a partition of U . (2) For every pair x, y ∈ U , x ∈ N (y) =⇒ y ∈ N (x). Lemma 2.5 ([1]). Let (U, C) be a covering approximation space and x, y ∈ U . Then the following are equivalent. (1) (2) (3) (4) (5)

x ∈ N (y). Cy ⊆ Cx . N (x) ⊆ N (y). D(y) ⊆ D(x). y ∈ D(x).

Proposition 2.6. Let (U, C) be a covering approximation space. Then the following are equivalent. (1) (U, C) is an Sr -spaces. (2) {N (x) : x ∈ U } forms a partition of U . Proof. (1) =⇒ (2): Suppose that (U, C) is an Sr -spaces. Let x, y ∈ U and x ∈ N (y). Then y ∈ D(x) by Lemma 2.5. For each K ∈ Cx , D(x) ⊆ K, we have y ∈ K. This proves that y ∈ N (x). By Lemma 2.4, {N (x) : x ∈ U } forms a partition of U . (2) =⇒ (1): Suppose that {N (x) : x ∈ U } forms a partition of U . Let K ∈ C and x ∈ K. Then N (x) ⊆ K. If y ∈ D(x), then x ∈ N (y) by Lemma 2.5. By Lemma 2.4, y ∈ N (x) ⊆ K. This proves that D(x) ⊆ K. So (U, C) is an Sr -space. Proposition 2.6 gives a ”nice” characterization for Sr -space, which is help for us to further comprehend [1, Remark 1.2]). 4

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The main results

Theorem 3.1. Let (U, C) be a covering approximation space. Then the following are equivalent. (1) (U, C) is an Sr -space. (2) C2 ({x}) ⊆ K for each x ∈ U and each K ∈ Cx . Proof. (1) =⇒ (2): Suppose that (U, C) is an Sr -space. Let x ∈ U and K ∈ Cx . Then D(x) ⊆ K. If y ∈ C2 ({x}) = U − C2 (U − {x}), then y 6∈ C2 (U − {x}). So, 0 0 0 for each K 0 ∈ C, if K 0 ⊆ U −{x} S then y 6∈ K . That is, for each K ∈SC, if x 6∈ K 0 then y 6∈ K , and hence y 6∈ (C − Cx ). It follows that y ∈ U − (C − Cx ) = D(x) ⊆ K. This proves that C2 ({x}) ⊆ K. U and K ∈ Cx . Then C2 ({x}) ⊆ (2) =⇒ (1): Suppose that S (2) holds. Let x ∈ S K. If y ∈ D(x) = U − (C − Cx ), then y 6∈ (C − Cx ). So y 6∈ K for each K ∈ C − Cx . That is, for each K ∈ C, if x 6∈ K then y 6∈ K. Note that x 6∈ K if and only if K ⊆ U − {x}. Thus, y 6∈ C2 (U − {x}). It follows that y ∈ U − C2 (U − {x}) = C2 ({x}) ⊆ K. This proves that D(x) ⊆ K. So (U, C) is an Sr -space. Let (U, C) be a covering approximation space. It is clear that if C2 (C2 (X)) = C2 (X) for each X ⊆ U . Then C2 (K) = K for each K ∈ C. So the following corollary improves Theorem 1.7(1), and the proof is quite simple. Corollary 3.2. Let (U, C) be a covering approximation space. If C2 (K) = K for each K ∈ C, then (U, C) is an Sr -space. Proof. Let C2 (K) = K for each K ∈ C. If x ∈ U and K ∈ Cx , then C2 ({x}) ⊆ C2 (K) = K from Lemma 2.3(3). By Theorem 3.1, (U, C) is an Sr -space. Remark 3.3. [16, Example 3.9] and Proposition 2.6 show that Corollary 3.2 can not be reversed. What are sufficient and necessary conditions such that C2 (K) = K for each K ∈ C? The following proposition gives an answer. Proposition 3.4. Let (U, C) be a covering approximation space. Then C2 (K) = K for each K ∈ C if and only if the following hold. (1) (U, C) is an S Sr -space. (2) C2 (K) = {C2 ({x}) : x ∈ K} for each K ∈ C. Proof. Necessity: Let C2 (K) = K for each K ∈ C. By Corollary 3.2, (U, C) is an Sr -space.S Let K ∈ C. By Lemma 2.3(3), C2 ({x}) ⊆ C2 (K) for each x ∈ K. Thus {C2 ({x}) : x ∈ K} ⊆ C2 (K). On the other S hand, by Lemma each x ∈ K, so C2 (K) = K ⊆ {C2 ({x}) : x ∈ K}. 2.3(2), x ∈ C2 ({x}) for S Consequently, C2 (K) = {C2 ({x}) : x ∈ K}. Sufficiency: Suppose that (1) and (2) hold. SLet K ∈ C. By Theorem 3.1, C2 ({x}) ⊆ K for each x ∈ K. Thus, C2 (K) = {C2 ({x}) : x ∈ K} ⊆ K. On the other hand, K ⊆ C2 (K) by Lemma 2.3(2). So C2 (K) = K. 5

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Similarly, the following proposition is obtained, which gives sufficient and necessary conditions such that C2 (C2 (X)) = C2 (X) for each X ⊆ U . We omit its proof. Proposition 3.5. Let (U, C) be a covering approximation space. Then C2 (C2 (X)) = C2 (X) for each X ⊆ U if and only if the following hold. (1) (U, C) is an S Sr -space. (2) C2 (X) = {C2 ({x}) : x ∈ X} for each union X of elements of C. Lemma 3.6. Let (U, C) be a covering approximation space. Then the following are equivalent. (1) (U, C) is an Sr -space. (2) C3 ({x}) ⊆ K for each x ∈ U and each K ∈ Cx . Proof. (1) =⇒ (2): Suppose that (U, C) is an Sr -space. LetTx ∈ U and K ∈ Cx . Then D(x) ⊆ K. If y ∈ C3 ({x}) = {z ∈ U : N (z) {x} 6= ∅}, then T N (y) {x} 6= ∅, so x ∈ N (y). By Lemma 2.5, y ∈ D(x) ⊆ K. This proves that C3 ({x}) ⊆ K. (2) =⇒ (1): Suppose that (2) holds. Let x ∈ U and K ∈ T Cx . Then C3 ({x}) ⊆ K. If y ∈ D(x), then x T ∈ N (y) from Lemma 2.5, i.e., N (y) {x} 6= ∅. It follows that y ∈ {z ∈ U : N (z) {x} 6= ∅} = C3 ({x}) ⊆ K. This proves that D(x) ⊆ K. So (U, C) is an Sr -space. Theorem 3.7. Let (U, C) be a covering approximation space. Then the following are equivalent. (1) (U, C) is an Sr -space. (2) C3 (K) = K for each K ∈ C. Proof. (1) =⇒ (2): Suppose that (U, C) is an Sr -space. Let K S ∈ C. By Lemma 3.6, C3 ({x}) ⊆ K for each x ∈ K. By Lemma 2.3(4), C3 (K) = {C3 ({x}) : x ∈ K} ⊆ K. On the other hand, by Lemma 2.3(3), K ⊆ C3 (K). Consequently, C3 (K) = K. (2) =⇒ (1): Suppose that (2) holds. Let x ∈ U and K ∈ Cx . Then C3 (K) = K. By Lemma 2.3(3), C3 ({x}) ⊆ C3 (K) = K. By Lemma 3.6, (U, C) is an Sr -space. The following shows that “C4 (X) ⊆ X” in Theorem 1.6(4) can not be replaced by “C4 (X) = X” Example 3.8. There exists a covering approximation space (U, C) such that (U, C) is an Sr -space and C4 (X) 6= X for some X ⊆ U . Proof. Let U = {a, b.c} and C = {{a, b}, {c}}. Then (U, C) is a Pawlak approximation space. It is known that each Pawlak approximation space is an Sr -space (see [1, Remark 3.4]). Put X = {a, c}. It is not difficult to check that C4 (X) = {c}. So C4 (X) 6= X. However, we have the following. 6

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Theorem 3.9. Let (U, C) be a covering approximation space. Then the following are equivalent. (1) (U, C) is an Sr -space. (2) C4 (K) = K for each K ∈ C. Proof. (1) =⇒ (2): Suppose that (U, C) is an Sr -space. Let K ∈ C. By Theorem 1.6 and Proposition 2.6, C4 (K) ⊆ K. On the other hand, Let x ∈ K. Then x ∈ N (x) and N (x) ⊆ K. By the definition of C4 (K), x ∈ C4 (K). This proves that K ⊆ C4 (K). Consequently, C4 (K) = K. (2) =⇒ (1): Suppose that (2) holds. Let x ∈ U and K ∈ Cx , then C4 (K) ⊆ K. If y ∈ D(x), then x ∈ N (y) from Lemma 2.5. Note that N (x) ⊆ K. So y ∈ {z ∈ U : ∃u(u ∈ N (z) ∧ N (u) ⊆ K)} = C4 (K) ⊆ K. This proves that D(x) ⊆ K. So (U, C) is an Sr -space. Lemma 3.10. Let (U, C) be a covering approximation space and X ⊂ U . Then C5 (X) = X if and only if C5 (X) = X. S (x) : x ∈ Proof. Necessity: Suppose that C5 (X) = X. Let y ∈ C5 (X) = {N T T U ∧N (x) X 6= ∅}. Then there is z ∈ U such that y ∈ N (z) and N (z) X 6= ∅. T Pick v ∈ N (z) X, then v ∈ X = C5 (X) = {x ∈ U : ∀u(x ∈ N (u) =⇒ N (u) ⊆ X)}. It follows that N (z) ⊆ X since v ∈ N (z). So y ∈ N (z) ⊆ X. This proves that C5 (X) ⊆ X. By Lemma 2.3(2), X ⊆ C5 (X). Consequently, C5 (X) = X. Sufficiency: Suppose that C5 (X) = X. By Lemma 2.3(2), C5 (X) ⊆ X. It suffices to prove that X ⊆ C5 (X). If X 6⊆ C5 (X), then there is y ∈ X such that y 6∈ C5 (X) = {x ∈ U : ∀u(x ∈ N (u) =⇒ N (u) ⊆ X)}. So there is v ∈ U T such that y ∈ T N (v) 6⊆ X. Pick z ∈SN (v) such that z 6∈ X. T Note that y ∈ N (v) X. So N (v) X 6= ∅. Thus z ∈ {N (x) : x ∈ U ∧ N (x) X 6= ∅} = C5 (X) = X. This contradicts that z 6∈ X. Lemma 3.11. Let (U, C) be a covering approximation space. Then the following are equivalent. (1) (U, C) is an Sr -space. (2) C5 ({x}) ⊆ K for each x ∈ U and each K ∈ Cx . Proof. (1) =⇒ (2). Suppose that (U,S C) is an Sr -space. Let x ∈ U and K ∈ Cx , then D(x) ⊆ K. If y ∈ C5 ({x}) = {N (z) : z ∈ U ∧ x ∈ N (z)}, then there exists z ∈ U such that x ∈ N (z) and y ∈ N (z). By Lemma 2.5, z ∈ D(x) ⊆ K, hence N (z) ⊆ K. It follows that y ∈ N (z) ⊆ K. This proves that C5 ({x}) ⊆ K. (2) =⇒ (1). Suppose that (2) holds. Let x ∈ U and K ∈ Cx , then S C5 ({x}) ⊆ K. If y ∈ D(x), then x ∈ N (y) from Lemma 2.5. So N (y) ⊆ {N (z) : z ∈ U ∧ x ∈ N (z)} = C5 ({x}) ⊆ K. It follows that y ∈ N (y) ⊆ K. This proves that D(x) ⊆ K. So (U, C) is an Sr -space. Theorem 3.12. Let (U, C) be a covering approximation space. Then the following are equivalent. (1) (U, C) is an Sr -space. 7

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(2) C5 (K) = K for each K ∈ C. (3) C5 (K) = K for each K ∈ C. Proof. (1) =⇒ (2): Suppose that (U, C) is an Sr -space. Let K ∈ C.SBy Lemma 3.11, C5 ({x}) ⊆ K for each x ∈ K. By Lemma 2.3(4), C5 (K) = {C5 ({x}) : x ∈ K} ⊆ K. On the other hand, by Lemma 2.3(2), K ⊆ C5 (K). Consequently, C5 (K) = K. (2) =⇒ (1): Suppose that (2) holds. Let x ∈ U and K ∈ Cx , then C5 (K) = K. By Lemma 2.3(3), C3 ({x}) ⊆ C5 (K) = K. By Lemma 3.11, (U, C) is an Sr -space. (2) ⇐⇒ (3): It holds by Lemma 3.10. Theorem 3.13. Let (U, C) be a covering approximation space. Then the following are equivalent. (1) (U, C) is an Sr -space. (2) C6 (K) = K for each K ∈ C. Proof. (1) =⇒ (2): Suppose that (U, C) is an Sr -space. Let K ∈ C. Then C6 (K) ⊆ K by Lemma 2.3(2). It suffices to prove that K ⊆ C6 (K). Let x ∈ K, then D(x) ⊆ K since (U, C) is an Sr -space. For each u ∈ U , if x ∈ N (u), then u ∈ D(x) by Lemma 2.5. It follows that u ∈ K. So x ∈ {z ∈ U : ∀u(z ∈ N (u) → u ∈ K)} = C6 (K). This proves that K ⊆ C6 (K). (2) =⇒ (1): Suppose that (2) holds. Let x ∈ U and K ∈ Cx , then C6 (K) = K. If y ∈ D(x), then x ∈ N (y) from Lemma 2.5. x ∈ K = C6 (K) = {x ∈ U : ∀u(x ∈ N (u) → u ∈ K)}, so x ∈ N (u) implies u ∈ K for each u ∈ U . It follows that y ∈ K since x ∈ N (y). This proves that D(x) ⊆ K. So (U, C) is an Sr -space.

4

Conclusions

This paper answers an open problem posed by Z.Yun et al. in [16]. We give some simple characterizations for Sr -space (U, C) by using only a single covering approximation operator and by using only elements of covering. The main results are summarized as follows. Theorem 4.1. Let (U, C) be a covering approximation space. Then the following are equivalent. (1) (2) (3) (4) (5) (6) (7) (8)

(U, C) ia an Sr -space. {N (x) : x ∈ U } forms a partition of U . C2 ({x}) ⊆ K for each x ∈ U and each K ∈ Cx . C3 (K) = K for each K ∈ C. C4 (K) = K for each K ∈ C. C5 (K) = K for each K ∈ C. C5 (K) = K for each K ∈ C. C6 (K) = K for each K ∈ C.

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In the previous sections, covering approximation operators C2 -C6 are used for our discussion. However, there are also other useful covering approximation operators, which play an important role in research of covering approximation spaces [7, 11, 17, 20, 21]. Definition 4.2 ([20]). Let (U, C) be a covering approximation space and x ∈ U . M d(x) = {K : K ∈ Cx ∧ (∀S ∈ Cx ∧ S ⊆ K → K = S)} is called the minimal description of x. Definition 4.3 ([20]). Let (U, C) be a covering approximation space and X ⊆ U . Put S (1) CL(X) = {K : KS∈ C ∧ K ⊆ X}; (2) F H(X) = S CL(X) {M d(x) :Tx ∈ X − CL(X)}; (3) SH(X) = S{K : K ∈ C ∧ K X 6= ∅}; (4) T H(X) = {M d(x) S : x ∈ X}; T (5) RH(X) = CL(X)S {K : K ∈ C ∧ K (X − CL(X)) 6= ∅}; (6) IH(X) = CL(X) {N (x) : x ∈ X − CL(X)}. CL is called covering lower approximation operation. F H, SH, T H, RH and IH are called the first, the second, the third, the fourth, and the fifth covering upper approximation operations, respectively. Can we characterize the conditions under which (U, C) is an Sr -space by using only a single covering approximation operator in Definition 4.3? It is an interesting question and is still worthy to be considered in research of covering approximation spaces.

References [1] X.Ge, An application of covering approximation spaces on network security, Computers and Mathematics with Applications, 60(2010), 1191-1199. [2] W.Li, N.Zhong, Y.Y.Yao, J.Liu, C.Liu, Developing intelligent applications in social e-mail networks, Lecture Notes in Artificial Intelligence, 4259(2006), 776-785. [3] S.K.Pal, B.U.Shankar, P.Mitra, Granular computing, rough entropy and object extraction, Pattern Recognition Letters, 26(2005), 2509-2517. [4] Z.Pawlak, Rough sets, International Journal of Computer and Information Science, 11(1982), 341-356. [5] Z.Pawlak, Rough Sets: Theoretical Aspects of Reasoning About Data, Kluwer Academic Publishers, Boston, 1991. [6] L.Polkowski, A.Skowron (Eds.), Rough sets and current trends in computing, Vol. 1424, Springer, Berlin, 1998.

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[7] J.A.Pomykala, Approximation operations in approximation space, Bulletin of the Polish Academy of Sciences, 35(9-10)(1987), 653-662. [8] K.Qin, Y.Gao, Z.Pei, On Covering rough sets, Lecture Notes in Artificial Intelligence, 4481(2007), 34-41. [9] P.Samanta, M.Chakraborty, Covering based approaches to rough sets and implication lattices, Lecture Notes in Artificial Intelligence, 5908(2009), 127-134. [10] A.Skowron, J.F.Peters, Rough-Granular Computing, in: Handbook of Granular Computing, Edited by W. Pedrycz, A. Skowron, V. Kreinovich, John Wiley & Sons, Ltd., 2008, 285-328. [11] E.Tsang, D.Cheng, J.Lee, D.Yeung, On the upper approximations of covering generalized rough sets, in: Proceedings of the 3rd International Conference Machine Learning and Cybernetics, 2004, pp. 4200-4203. [12] E.Tsang, D.Chen, D.Yeung, Approximations and reducts with covering generalized rough sets, Computers and Mathematics with Applications, 56(2008), 279-289. [13] Y.Y.Yao, On generalizing rough set theory, Lecture Notes in Artificial Intelligence, 2639(2003), 44-51. [14] Y.Y.Yao, N.Zhong, Potential applications of granular computing in knowledge discovery and data mining, in: Proceedings of World Multiconference on Systemics, Cybernetics and Informatics, 1999, pp. 573-580. [15] D.Yeung, D.Chen, E.Tsang, J.Lee, W.Xizhao, On the generalization of fuzzy rough sets, IEEE Transactions on Fuzzy Systems, 13(3)(2005), 343361. [16] Z.Yun, X.Ge, X.Bai, Axiomatization and conditions for neighborhoods in a covering to form a partition, Information Sciences, 181(2011), 1735-1740. [17] W.Zakowski, Approximations in the space (u, Π), Demonstratio Mathematica, 16(1983), 761-769. [18] N.Zhong, J.Dong, S.Ohsuga, Using rough sets with heuristics to feature selection, J. Intell. Inform. Syst., 16(3)(2001), 199-214. [19] N.Zhong, Y.Y.Yao, M.Ohshima, Peculiarity oriented multidatabase mining, IEEE Transactions on Knowledge and Data Engineering, 15(4)(2003), 952-960. [20] W.Zhu, Relationship between generalized rough sets based on binary relation and covering, Information Sciences, 179(2009), 210-225. [21] W.Zhu, F.Wang, On three types of covering rough sets, IEEE Transactions on Knowledge and Data Engineering, 19(2007), 1131-1144.

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Difference of Generalized Composition Operators from H ∞ to the Bloch Space Geng-Lei Li Department of Mathematics, Tianjin Polytechnic University,Tianjin 300387,P.R. China, [email protected] 1

Abstract We characterized the difference of generalized composition operator on the bounded analytic function space to the Bloch space in the disk. The boundedness and compactness of it were investigated.

1

Introduction

Let D be the unit disk of the complex plane, and S(D) be the set of analytic self-maps of D. The algebra of all holomorphic functions with domain D will be denoted by H(D). The Bloch space B consists of all f ∈ H (D) such that 2

kf kB = sup(1 − |z| ) |f 0 (z)| < ∞, z∈D

then k·kB is a complete semi-norm on B, which is M¨obius invariant. The space B becomes a Banach space with the norm kf k = |f (0)| + kf kB . Denote H ∞ (D) by H ∞ ,the space of all bounded analytic functions in the unit disk with the norm kf k∞ = sup |f (z)|. z∈D

Let ϕ be an analytic self-map of D, and g ∈ H(D) , the generalized composition operator Cϕg induced by ϕ and g is defined by (Cϕg f )(z)

Z =

z

f 0 (ϕ(ξ))g(ξ)dξ,

0

for z ∈ D and f ∈ H(D). The definition of the generalized composition was first introduced by S. Li, S. Stevi´c in [9], and in the paper, the boundedness and compactness of the generalized composition operator on Zygmund spaces and Bloch type spaces were investigated by them. In the past few decades, boundedness, compactness, isometries and essential norms of composition and closely related operators between various spaces of holomorphic functions have been studied by many authors, see, e.g., [1, 2, 6, 14, 18, 19, 21, 22]. Recently, many papers focused on studying the mapping properties of the difference of two composition operators, i.e., (Cϕ − Cψ )(f ) = f ◦ ϕ − f ◦ ψ. 1 The authors were supported in part by the National Natural Science Foundation of China (Grant Nos. 11126165) 2010 Mathematics Subject Classification. Primary: 47B38; Secondary: 30H30, 30H05, 47B33, 47G10. Key words and phrases.generalized composition operator, Bloch space, compactness, difference..

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Differences of composition operators were studies first on hardy space H 2 (D) (see,e.g[3]). In [13], MacCluer, Ohno and Zhao, characterized the compactness of the difference of two composition operators on H ∞ (D) in terms of the Poincar´e distance. A fewer years later, these results were extended to the setting of H ∞ (Bn ) by Toews [20] . In [23], Z. H. Zhou and L. Zhang discussed the differences of the products of integral type and composition operators from H ∞ to the Bloch space, more results ,for example, can be seen in [4, 5, 8, 15, 16, 17]. Building on those foundation, this paper continues the research of this part, and discusses the difference of two generalized composition operators from the bounded analytic function space to the Bloch space in the disk.

2

Notation and Lemmas

First, we will introduce some notation and state a couple of lemmas. For a ∈ D, the involution ϕa which interchanges the origin and point a, is defined by a−z . 1 − az

ϕa (z) =

For z, w in D, the pseudo-hyperbolic distance between z and w is given by z−w , ρ(z, w) = |ϕz (w)| = 1 − zw and the hyperbolic metric is given by Z β (z, w) = inf γ

γ

|dξ| 2

1 − |ξ|

=

1 1 + ρ(z, w) log , 2 1 − ρ(z, w)

where γ is any piecewise smooth curve in D from z to w. The following lemma is well known in [24]. Lemma 1. For all z, w ∈ D, we have 1 − ρ2 (z, w) =

(1 − |z|2 )(1 − |w|2 ) 2

|1 − zw|

.

A little modification of Lemma 1 in [7] shows the following lemma. Lemma 2. There exists a constant C > 0 such that     2 2 1 − |z| f 0 (z) − 1 − |w| f 0 (w) ≤ C kf kB · ρ(z, w) for all z, w ∈ D and f ∈ B. Lemma 3. Assume that f ∈ H ∞ (D), then for each n ∈ N , there is a positive constant C independent of f such that sup (|1 − |z|)n f (n) (z) < C||f ||∞ . z∈D

Remark The Lemma 3 can be concluded from [11]. 1−|z|2 ∗ Throughout the remainder of this paper, we will denote 1−|ϕ(z)| 2 by the ϕ and constants are denoted by C , they are positive and not necessarily the same in each appearance.

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3

Main theorems

Theorem 1. Let ϕ1 , ϕ2 be analytic self-maps of the unit disk and g1 , g2 ∈ H(D). Then the following statements are equivalent. (i) Cϕg11 − Cϕg22 : H ∞ → B is bounded; (ii) sup |ϕ∗1 (z)| |g1 (z)| ρ(ϕ1 (z), ϕ2 (z)) < ∞ (1) z∈D

sup |ϕ∗2 (z)| |g2 (z)| ρ(ϕ1 (z), ϕ2 (z)) < ∞

(2)

sup |ϕ∗1 (z)g1 (z) − ϕ∗2 (z)g2 (z)| < ∞.

(3)

z∈D

and z∈D

Proof. We first prove (ii) ⇒ (i). Assume that (1), (2), (3) hold. As the definition of Cϕg , obviously, (Cϕg11 − Cϕg22 )f (0) = 0 By Lemma 2 and Lemma3, for every f ∈ H ∞ , we have ||Cϕg11 − Cϕg22 ||B 2

sup(1 − |z| ) |f 0 (ϕ1 (z))g1 (z) − f 0 (ϕ2 (z))g2 (z)| z∈D 2 2 = sup (1 − |ϕ1 (z)| )ϕ∗1 (z)f 0 (ϕ1 (z))g1 (z) − (1 − |ϕ2 (z)| )ϕ∗2 (z)f 0 (ϕ2 (z))g2 (z) z∈D 2 2 ≤ sup |ϕ∗1 (z)g1 (z)| (1 − |ϕ1 (z)| )f 0 (ϕ1 (z)) − (1 − |ϕ2 (z)| )f 0 (ϕ2 (z))

=

z∈D

+

2

sup(1 − |ϕ2 (z)| ) |f 0 (ϕ2 (z))| |g1 (z)ϕ∗1 (z)) − g2 (z)ϕ∗2 (z))| z∈D

≤ C sup |ϕ∗1 (z)g1 (z)| ρ(ϕ1 (z), ϕ2 (z)) kf kB z∈D

+

sup |g1 (z)ϕ∗1 (z)) − g2 (z)ϕ∗2 (z))| kf kB z∈D

≤ C kf k∞ . That is Cϕg11 − Cϕg22 is bounded. Next we show that (i) implies (ii). We assume Cϕg11 − Cϕg22 : H ∞ → B is bounded. For every ω ∈ D, we take the test function fϕ1 ,ω (z) =

ϕ1 (ω) − z 1 − ϕ1 (ω)z

.

We can obtain easily that fϕ1 ,ω ∈ H ∞ and ||fϕ1 ,ω ||∞ ≤ 1. Therefore, we have C ≥ ||(Cϕg11 − Cϕg22 )fϕ1 ,ω ||B 2 = sup(1 − |z| ) fϕ0 1 ,ω (ϕ1 (z))g1 (z) − fϕ0 1 ,ω (ϕ2 (z))g2 (z) z∈D

2 ≥ (1 − |ω| ) fϕ0 1 ,ω (ϕ1 (ω))g1 (ω) − fϕ0 1 ,ω (ϕ2 (ω))g2 (ω) (1 − |ϕ1 (ω)|2 )(1 − |ϕ2 (ω)|2 ) ∗ ∗ ϕ2 (ω)g2 (ω) = ϕ1 (ω)g1 (ω) − (1 − ϕ1 (ω)ϕ2 (ω))2 ∗ 2 ∗ ≥ |ϕ1 (ω)g1 (ω)| − (1 − ρ(ϕ1 (ω), ϕ2 (ω)) |ϕ2 (ω)g2 (ω)| .

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This is ∗ |ϕ1 (ω)g1 (ω)| − (1 − ρ(ϕ1 (ω), ϕ2 (ω))2 |ϕ∗2 (ω)g2 (ω)| ≤ C. Similarly,letting the test function fϕ2 ,ω (z) =

ϕ2 (ω)−z , 1−ϕ2 (ω)z

(4)

we can obtain

∗ |ϕ2 (ω)g2 (ω)| − (1 − ρ(ϕ1 (ω), ϕ2 (ω))2 |ϕ∗1 (ω)g1 (ω)| ≤ C.

(5)

We take the test functions as follow: f (z) = fϕ21 ,ω (z) = (

ϕ1 (ω) − z 1 − ϕ1 (ω)z

)2 , g(z) = fϕ22 ,ω (z) = (

ϕ2 (ω) − z 1 − ϕ2 (ω)z

)2 .

(6)

The following conclusions can be easily concluded 2(1 − ρ(ϕ1 (ω), ϕ2 (ω))2 )ρ(ϕ1 (ω), ϕ2 (ω)) |ϕ∗2 (ω)g2 (ω)| ≤ C,

(7)

2(1 − ρ(ϕ1 (ω), ϕ2 (ω))2 )ρ(ϕ1 (ω), ϕ2 (ω)) |ϕ∗1 (ω)g1 (ω)| ≤ C.

(8)

If ρ(ϕ1 (ω), ϕ2 (ω)) ≤

1 2

,then by (8), we have |ϕ∗1 (z)g1 (z)| ρ(ϕ1 (z), ϕ2 (z)) < C.

If ρ(ϕ1 (ω), ϕ2 (ω)) >

1 2

,then by (7), we have (1 − ρ(ϕ1 (ω), ϕ2 (ω))2 ) |ϕ∗2 (ω)g2 (ω)| ≤ C,

then, |ϕ∗1 (ω))g1 (ω)| ≤ C is followed by (4), so |ϕ∗1 (ω)| |g1 (ω)| ρ(ϕ1 (ω), ϕ2 (ω)) < C. We can get (1) by use of the arbitrary of ω. Analogously, (2) was also can be obtained. Finally, in order to prove the condition (3), using Lemma 2 and Lemma 3, we have C ≥ ||(Cϕg11 − Cϕg22 )fϕ1 ,ω ||B ≥ |g1 (ω)ϕ∗1 (ω) − g2 (ω)ϕ∗2 (ω)| 2 2 (1 − |ϕ (ω)| )(1 − |ϕ (ω)| ) 1 2 − |g2 (ω)ϕ∗2 (ω)| 1 − 2 (1 − ϕ1 (ω)ϕ2 (ω)) ≥ |g1 (ω)ϕ∗1 (ω) − g2 (ω)ϕ∗2 (ω)| − |g2 (ω)ϕ∗2 (ω)| (1 − |ϕ1 (ω)|2 )fϕ0 1 ,ω (ϕ1 (ω)) − (1 − |ϕ2 (ω)|2 )fϕ0 1 ,ω (ϕ2 (ω)) ≥ |g1 (ω)ϕ∗1 (ω) − g2 (ω)ϕ∗2 (ω)| − C |g2 (ω)ϕ∗2 (ω)| ρ(ϕ1 (ω), ϕ2 (ω)).

Then, sup |ϕ∗1 (z)g1 (z) − ϕ∗2 (z)g2 (z)| < ∞.

z∈D

This is completes the proof of this theorem. By the studying similarly to the proof of Theorem 3.2 in the paper [7], the following theorem can be obtained.

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Theorem 2. Let ϕ1 , ϕ2 be analytic self-maps of the unit disk and g1 , g2 ∈ H(D), Cϕg11 , Cϕg22 : H ∞ → B are bounded but not compact, Then the following statements are equivalent. (i) Cϕg11 − Cϕg22 : H ∞ → B is compact; (ii) Both (a) and (b) hold: (a)Γ∗ (ϕ1 ) = Γ∗ (ϕ2 ) 6= ∅, thenΓ∗ (ϕ1 ) ⊂ Γ(ϕ1 ) ∩ Γ(ϕ2 ) (b)F or{zn } ∈ Γ(ϕ1 ) ∩ Γ(ϕ2 ), lim |ϕ∗1 (zn )| |g1 (zn )| ρ(ϕ1 (zn ), ϕ2 (zn )) = 0

n→∞

lim |ϕ∗2 (zn )| |g2 (zn )| ρ(ϕ1 (zn ), ϕ2 (zn )) = 0

n→∞

and lim |ϕ∗1 (zn )g1 (zn ) − ϕ∗2 (zn )g2 (zn )| = 0

n→∞

(iii) lim ||(Cϕg11 − Cϕg22 )ϕλ ||B = 0

|λ|→1

and lim ||(Cϕg11 − Cϕg22 )(ϕλ )2 ||B = 0.

|λ|→1

Here, Γ(ϕ1 ) is the set of sequence {zn } in D such that |ϕ1 (zn )| → 1. Γ∗ (ϕ1 ) is the set of sequence {zn } in D such that |ϕ1 (zn )| → 1 and ϕ∗1 (zn )g1 (zn ) does not approach the 0. Next, the other major theorem will be given Theorem 3. Let ϕ1 , ϕ2 be analytic self-maps of the unit disk and g1 , g2 ∈ H(D), Cϕg11 , Cϕg22 : H ∞ → B are bounded, Then the following statements are equivalent. (i) Cϕg11 − Cϕg22 : H ∞ → B is compact; (ii) lim |ϕ∗1 (z)| |g1 (z)| ρ(ϕ1 (z), ϕ2 (z)) = 0 |ϕ1 (z)|→1

lim

|ϕ2 (z)|→1

|ϕ∗2 (z)| |g2 (z)| ρ(ϕ1 (z), ϕ2 (z)) = 0

and lim

|ϕ1 (z)|,|ϕ2 (z)|→1

|ϕ∗1 (z)g1 (z) − ϕ∗2 (z)g2 (z)| = 0.

Proof. We first prove (i) ⇒ (ii).We assume that Cϕg11 − Cϕg22 : H ∞ → B is compact, then, Cϕg11 , Cϕg22 are compact or noncompact. If they are compact, the following conclusions are obtained obviously by the Theorem 2 in [12], lim |ϕ∗1 (z)| |g1 (z)| = 0, lim |ϕ∗2 (z)| |g2 (z)| = 0, |ϕ1 (z)|→1

|ϕ2 (z)|→1

then, the (ii) holds by them. If they are all noncompact, for a sequence {zn }, such that |ϕ1 (zn )| → 1, if |ϕ∗1 (zn )| |g1 (zn )| → 0, then, lim |ϕ∗1 (zn )| |g1 (zn )| ρ(ϕ1 (zn ), ϕ2 (zn )) = 0;

n→∞

if lim |ϕ∗1 (zn )| |g1 (zn )| 6= 0,

n→∞

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then {zn } ∈ Γ∗ (ϕ1 ). By Theorem 2, {zn } ∈ Γ∗ (ϕ1 ) ⊂ Γ(ϕ1 ) ∩ Γ(ϕ2 ), and lim |ϕ∗1 (zn )| |g1 (zn )| ρ(ϕ1 (zn ), ϕ2 (zn )) = 0.

n→∞

Hence, lim

|ϕ1 (z)|→1

|ϕ∗1 (z)| |g1 (z)| ρ(ϕ1 (z), ϕ2 (z)) = 0.

According to similarly proof, we can get lim

|ϕ2 (z)|→1

|ϕ∗2 (z)| |g2 (z)| ρ(ϕ1 (z), ϕ2 (z)) = 0.

For {zn } such that |ϕ1 (zn )| , |ϕ2 (zn )| → 1, using Theorem 2, we have lim |ϕ∗1 (zn )g1 (zn ) − ϕ∗2 (zn )g2 (zn )| = 0.

n→∞

Due to the arbitrary of {zn }, we have lim

|ϕ1 (z)|,|ϕ2 (z)|→1

|ϕ∗1 (z)g1 (z) − ϕ∗2 (z)g2 (z)| = 0.

This is completes the proof of (i) ⇒ (ii). (ii) ⇒ (i) If the operators Cϕg11 , Cϕg22 are all noncompact, (i) holds obviously by Theorem 2. If one of the operators Cϕg11 , Cϕg22 is compact, we may also assume that Cϕg11 is compact, then by the Theorem 2 in [10], we have lim

|ϕ1 (z)|→1

|ϕ∗1 (z)| |g1 (z)| = 0.

Let {zn } be an arbitrary sequence in D, such that |ϕ2 (zn )| → 1 as n → ∞. If |ϕ1 (zn )| approach 1, since lim

|ϕ1 (z)|,|ϕ2 (z)|→1

|ϕ∗1 (z)g1 (z) − ϕ∗2 (z)g2 (z)| = 0,

We obtain lim |ϕ∗2 (zn )| |g2 (zn )| = 0.

n→∞

If |ϕ1 (zn )| does not approach 1, then ρ(ϕ1 (z), ϕ2 (z)) does not approach 0, since, lim

|ϕ2 (z)|→1

|ϕ∗2 (z)| |g2 (z)| ρ(ϕ1 (z), ϕ2 (z)) = 0.

We also obtain lim |ϕ∗2 (zn )| |g2 (zn )| = 0.

n→∞

Due to the arbitrary of {zn }, we have lim

|ϕ2 (z)|→1

|ϕ∗2 (z)| |g2 (z)| = 0.

Therefore, Cϕg22 is a compact operator, therefore, Cϕg11 − Cϕg22 is compact. This is completes the proof of this theorem.

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References [1] E. Berkson, Composition operators isolated in the uniform operator topology, Proc. Amer. Math Soc., 81, 230-232 (1981). [2] J. Bonet, M.Lindstr¨ om and E. Wolf, Isometric weighted composition operators on weighted Banach spaces of type H ∞ , Proc. Amer. Math. Soc., 136(12), 4267-4273 (2008). [3] C.C.Cowen, B.D. Macclur, Composition Operators on Spces of Analytic Functions, CRC PRESS, 1995. [4] Z.S. Fang and Z.H. Zhou,Differences of composition operators on the space of bounded analytic functions in the polydisc, Abstract and Applied Analysis, 2008, Article ID 983132, 10 pages, (2008). [5] Z.S. Fang and Z.H. Zhou,Differences of composition operators on the space of bounded analytic functions in the polydisc, Bull. Aust. Math. Soc., 79, 465-471 (2009). [6] T. Hosokawa, K. Izuchi and D. Zheng, Isolated points and essential components of composition operators on H ∞ , Proc. Amer. Math. Soc., 130, 1765-1773 (2002). [7] T. Hosokawa and S. Ohno, Differences of composition operators on the Bloch space, J. Operator Theory, 57,229-242 (2007). [8] M. Lindstr¨ om and E. Wolf. Essential norm of the difference of weighted composition operators , Monatsh. Math., 153, 133-143 (2008). [9] S. Li and S. Stevi´c, Generalized composition operators on Zygmund spaces and Bloch type spaces, J. Math. Anal. Appl., 338,1282-1295 (2008). [10] S. Li and S. Stevi´c, Product of integral-type operators and composition operators between Bloch type spaces, J. Math. Anal. Appl., 349, 596-610 (2009). [11] S. Li and S. Stevi´c, Products of Volterra type operators from H ∞ and the Bloch type spaces to Zygmund spaces, J. Math. Anal. Appl., 345, 40-52 (2008). [12] S. Li and S. Stevi´c, Product of composition and integral type operators from H ∞ to the Bloch spaces, Complex variables and Elliptic Equations, 53, 463-474 (2008). [13] B. MacCluer, S. Ohno and R. Zhao, Topological structure of the space of composition operators on H ∞ , Integr. Equ. Oper. Theory, 40(4), 481-494 (2001). [14] A. Montes-Rodr´ıguez, Weighted composition operators on weighted Banach spaces of analytic functions, J. London Math. Soc., 61(3), 872-884 (2000). [15] J. Moorhouse, Compact difference of composition operators, J. Funct. Anal., 219, 70-92 (2005). [16] Pekka J. Nieminen. compact differences of composition operators on Bloch and Lipschitz spaces, CMFT, 7 (2), 325-344 (2007). [17] E. Wolf. Differences of composition operators between weighted Banach spaces of holomorphic functions on the unit polydisk, Result. Math., 51, 361-372 (2008). [18] J. H. Shapiro, Composition operators and classical function theory, Spriger-Verlag, 1993.

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[19] J. H. Shi and L. Luo, Composition operators on the Bloch space, Acta Math. Sinica, 16, 85-98 (2000). [20] C. Toews, Topological components of the set of composition operators on H ∞ (BN ), Integr. Equ. Oper. Theory, 48, 265-280 (2004). [21] Z. H. Zhou and R. Y. Chen, Weighted composition operators fom F (p, q, s) to Bloch type spaces, International Jounal of Mathematics, 19(8), 899-926 (2008). [22] Z. H. Zhou and J. H. Shi, Compactness of composition operators on the Bloch space in classical bounded symmetric domains, Michigan Math. J., 50, 381-405 (2002). [23] Z. H. Zhou and L. Zhang, Differences of the products of integral type and composition operators from H ∞ to the Bloch space, Complex Variables and Elliptic Equations, DOI: 10.1080/17476933.2011.636150 [24] K. H. Zhu, Spaces of holomorphic functions in the unit ball, Springer-Verlag (GTM 226), 2004.

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Isometries among the generalized composition operators on Bloch type spaces Geng-Lei Li Department of Mathematics, Tianjin Polytechnic University,Tianjin 300387,P.R. China, [email protected] 1

Abstract In this paper, we characterize the isometries among the generalized composition operators on Bloch type spaces in the disk.

1

Introduction

Let D be the unit disk of the complex plane, and S(D) be the set of analytic self-maps of D. The algebra of all holomorphic functions with domain D will be denoted by H(D). We recall that the Bloch type space B α (α > 0) consists of all f ∈ H (D) such that 2

kf kBα = sup(1 − |z| )α |f 0 (z)| < ∞, z∈D

then k·kBα is a complete semi-norm on B α , which is M¨obius invariant. It is well known that B α is a Banach space under the norm kf k = |f (0)| + kf kBα . Let ϕ be an analytic self-map of D, and g ∈ H(D) , the generalized composition operator Cϕg induced by ϕ and g is defined by (Cϕg f )(z) =

z

Z

f 0 (ϕ(ξ))g(ξ)dξ,

0

for z ∈ D and f ∈ H(D). The definition of generalized composition operator was first introduced by S. Li, S. Stevi´c in [20], and in the paper, the boundedness and compactness of the generalized composition operator on Zygmund spaces and Bloch type spaces were investigated by them. If we use the derivative of some function g to instead of g in operator Cϕg , we can get a new integral operator Lϕ g , which is also called generalized composition operator. Let ϕ ∈ S(D) and g ∈ H(D), the operator Lϕ g induced by ϕ and g is defined by (Lϕ g f )(z) =

Z

z

f 0 (ϕ(ξ))g 0 (ξ)dξ,

0

for z ∈ D and f ∈ H(D). More results about boundedness, compactness, differences and essential norms of composition and closely related operators between various spaces of holomorphic functions have 1 The authors were supported in part by the National Natural Science Foundation of China (Grant Nos. 10971153, 10671141) 2010 Mathematics Subject Classification. Primary: 47B38; Secondary: 30H30, 30H05, 47B33, 47G10. Key words and phrases.generalized composition operator, Bloch type space, isometry.

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G.L.Li:Isometries among the generalized composition operators

been studied by many authors, see, e.g., [12, 18, 19, 21, 25, 27]. Recently, many papers focused on studying isometries of the composition operators on various spaces of holomorphic functions. Let X and Y be two Banach spaces, and recall that a linear isometry is a linear operator T from X to Y such that kT f kY = kf kX for all f ∈ X. In [3], Banach showed great interet in the form of an isometry on a specific Banach space. In most cases the isometries of a space of analytic functions on the disk or the ball have the canonical form of weighted composition operators, which is also true for most symmetric function spaces. For example, the surjective isometries of Hardy and Bergman spaces are certain weighted composition operators. See [13, 14, 15]. The description of all isometric composition operators is known for the Hardy space H 2 (see [8]). An analogous statement for the Bergman space A2α with standard radial weights has recently been obtained in [7], and there is a unified proof for all Hardy spaces and also for arbitrary Bergman spaces with reasonable radial weights [24]. In [9], Colonna gave a characterization of the isometric composition operators on the Bloch space in terms of the factorization of the symbol in H ∞ , which shows that there is a very large class of isometries besides the rotations. By contrast, in [26], Zorboska showed that in the case α 6= 1, the isometries of the composition operators on B α are the operators whose symbol is a rotation. Continued the work of isometry, in 2008, Bonet, Lindstr¨om and Wolf [4] studied isometric weighted composition operators on weighted Banach spaces of type H ∞ . Cohen and Colonna [6] discussed the spectrum of an isometric composition operators on the Bloch space of the polydisk. In 2009, Allen and Colonna [1] investigated the isometric composition operators on the Bloch space in C n . They [2] also discussed the isometries and spectra of multiplication operators on the Bloch space in the disk. Isometries of weighted spaces of holomorphic functions on unbounded domains were discussed by Boyd and Rueda in [5]. In 2010, Li and Zhou discussed the isometries on products of composition and integral operators on Bloch type space in [10].more results ,for example, can be seen in [11, 16, 17, 22, 23]. The paper continues the research of it, and discusses the isometries among the generalized composition operators on Bloch type space in the disk.

2

Notation and Lemmas

First, we will introduce some notations and state a couple of lemmas. For a ∈ D, the involution ϕa which interchanges the origin and point a, is defined by ϕa (z) =

a−z . 1 − az

For z, w in D, the pseudohyperbolic distance between z and w is given by z−w . ρ(z, w) = |ϕz (w)| = 1 − zw The following lemma is well known [25]. Lemma 1. For all z, w ∈ D, we have 1 − ρ2 (z, w) =

(1 − |z|2 )(1 − |w|2 ) 2

|1 − zw|

.

For ϕ ∈ S(D), the Schwarz-Pick lemma shows that ρ (ϕ(z), ϕ(w)) ≤ ρ(z, w), and if equality holds for some z 6= w, then ϕ is an automorphism of the disk. It is also well known that for ϕ ∈ S(D), Cϕ is always bounded on B. A little modification of Lemma 1 in [4] shows the following lemma.

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G.L.Li:Isometries among the generalized composition operators

Lemma 2. There exists a constant C > 0 such that  α  α 2 2 f 0 (z) − 1 − |w| f 0 (w) ≤ C kf kBα · ρ(z, w) 1 − |z| for all z, w ∈ D and f ∈ B α . Throughout the rest of this paper, C will denote a positive constant, the exact value of which will vary from one appearance to the next.

3

Main theorems

Theorem 1. Let ϕ be analytic self maps of the unit disk and g ∈ H(D) . Then the operator Cϕg : B α → B β is an isometry in the semi-norm if and only if the following conditions hold: (A)

sup z∈D

(B) 0 and

(1−|z|2 )β (1−|ϕ(z)|2 )α

|g(z)| ≤ 1;

For every a ∈ D, there at least exists a sequence {zn } in D, such that lim ρ(ϕ(zn ), a) = n→∞

2 β lim (1−|zn | )2 α n→∞ (1−|ϕ(zn )| )

|g(zn )| = 1.

Proof. We prove the sufficiency first. By condition (A), for every f ∈ B α , we have ||Cϕg f ||Bβ

=

2

sup (1 − |z| )β |f 0 (ϕ(z))| |g(z)| z∈D

=

sup

(1 − |z|2 )β

≤

2

2

(1 − |ϕ(z)| )α kf kBα . z∈D

|g(z)| (1 − |ϕ(z)| )α |f 0 (ϕ(z))|

Next we show that the property (B) implies ||Cϕg f ||Bβ ≥ ||f ||Bα Given any f ∈ B α , then ||f ||Bα = lim (1−|am |2 )α |f 0 (am )| for some sequence {am } ⊂ D. m→∞

For any fixed m, by property (B), there is a sequence {zkm } ⊂ D such that 2

β (1 − |z m k | )

ρ(ϕ(zkm ), am ) → 0 and

2

(1 − |ϕ(zkm )| )α

|g(zkm )| → 1

as k → ∞. By Lemma 2, for all m and k, (1 − |ϕ(zkm )|2 )α f 0 (ϕ(zkm )) − (1 − |am |2 )α f 0 (am ) ≤ C||f ||Bα · ρ(ϕ(zkm ), am ). Hence (1 − |ϕ(zkm )|2 )α |f 0 (ϕ(zkm ))| ≥ (1 − |am |2 )α |f 0 (am )| − C||f ||Bα · ρ(ϕ(zkm ), am ) Therefore, ||Cϕg f ||Bβ

=

sup z∈D

(1 − |z|2 )β 2

(1 − |ϕ(z)| )α

2

|g(z)| (1 − |ϕ(z)| )α |f 0 (ϕ(z))| 2

≥ lim sup k→∞

=

β (1 − |z m k | ) 2

(1 − |ϕ(zkm )| )α

|g(zkm )| (1 − |ϕ(zkm )|2 )α |f 0 (ϕ(zkm ))|

(1 − |am |2 )α |f 0 (am )|.

The inequality ||Cϕg f ||Bβ ≥ ||f ||Bα follows by letting m → ∞.

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G.L.Li:Isometries among the generalized composition operators

From the above discussions, we have ||Cϕg f ||Bβ = ||f ||Bα , which means that Cϕg is an isometry operator in the semi-norm from B α to B β . Now we turn to the necessity. For any a ∈ D, we begin by taking test function Z z (1 − |a|2 )α dt. (1) fa (z) = ¯t)2α 0 (1 − a It is clear that fa0 (z) =

(1−|a|2 )α (1−¯ az)2α .

Using Lemma 1, we have (1 − |z|2 )α (1 − |a|2 )α = (1 − ρ2 (a, z))α . |1 − a ¯z|2α

(1 − |z|2 )α |fa0 (z)| =

(2)

So kfa kBα = sup(1 − |z|2 )α |fa0 (z)| ≤ 1.

(3)

z∈D

(1−|a|2 )2α (1−|a|2 )2α

On the other hand, since (1 − |a|2 )α |fa0 (a)| = isometry assumption, for any a ∈ D, we have 1

= 1, we have kfa kBα = 1. By

= ||fϕ(a) ||Bα = ||Cϕg fϕ(a) ||Bβ =

sup z∈D

(1 − |z|2 )β 2

(1 − |ϕ(z)| )α

0 2 |g(z)| (1 − |ϕ(z)| )α fϕ(a) (ϕ(z))

2

≥

(1 − |a| )β 2

(1 − |ϕ(a)| )α

|g(a)| .

So (A) follows by the arbitrariness of a. Since ||fa ||Bα = ||Cϕg fa ||Bβ = 1, there exists a sequence {zm } ⊂ D such that d(Cϕg fa ) 2 β ((1 − |zm | ) (zm ) = (1 − |zm |2 )β |fa0 (ϕ(zm ))||g(zm )| → 1 dz

(4)

as m → ∞. It follows from (A) that (1 − |zm |2 )β |fa0 (ϕ(zm ))||g(zm )| (1 − |zm |2 )β 2 α 0 = 2 α |g(zm )| (1 − |ϕ(zm )| ) |fa (ϕ(zm ))| (1 − |ϕ(zm )| ) 2

≤ (1 − |ϕ(zm )| )α |fa0 (ϕ(zm ))| .

(5) (6)

Combining (4) and (6), it follows that 2

1 ≤ lim inf (1 − |ϕ(zm )| )α |fa0 (ϕ(zm ))| m→∞

2

≤ lim sup(1 − |ϕ(zm )| )α |fa0 (ϕ(zm ))| ≤ 1. m→∞

The last inequality follows by (2) since ϕ(zm ) ∈ D. Consequently, 2

lim (1 − |ϕ(zm )| )α |fa0 (ϕ(zm ))| = lim (1 − ρ2 (ϕ(zm ), a))α = 1.

m→∞

m→∞

(7)

That is, lim ρ(ϕ(zm ), a) = 0. m→∞

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Combining (4), (5) and (7), we know lim

m→∞

(1 − |zm |2 )β 2

(1 − |ϕ(zm )| )α

|g(zm )| = 1.

This completes the proof of this theorem. Corollary 1. Let U denote unitary transformation in the unit disk, then CU1 : B α → B β is an isometry in the semi-norm.  If we use the derivative of some function g to instead of g in operator Cϕg , by the above theorem. we can easily get the following result about the operator Lϕ g. Theorem 2. Let ϕ be analytic self maps of the unit disk and g ∈ H(D) . Then the operator Cgϕ : B α → B β is an isometry in the semi-norm if and only if the following conditions hold: (C)

sup z∈D

(D) 0 and

(1−|z|2 )β (1−|ϕ(z)|2 )α

|g 0 (z)| ≤ 1;

For every a ∈ D, there at least exists a sequence {zn } in D, such that lim ρ(ϕ(zn ), a) =

2 β lim (1−|zn | )2 α n→∞ (1−|ϕ(zn )| )

n→∞

0

|g (zn )| = 1.

Remark If α = 1, β = 1, then B α and B β will be Bloch space B. There are similar results on the Bloch space B corresponding to Theorems 1 and 2.

References [1] R. F. Allen and F. Colonna, Isometric composition operators on the Bloch space in C n , J. Math. Anal. Appl., 355 (2009), 675-688. [2] R. F. Allen and F. Colonna, Isometries and spectra of multiplication operators on the Bloch space, Bull. Aust. Math. Soc., 79 (2009),147-160. [3] S. Banach, Theorie des Operations Lineares, Chelsea, Warzaw, 1932. [4] J.Bonet, M. Lindstr¨ om and E. Wolf, Isometric weighted composition operators on weighted Banach spaces of type H ∞ , Proc. Amer. Math. Soc., 136(12)(2008), 42674273. [5] C. Boyd and P. Rueda, Isometries of weighted spaces of holomorphic functions on unbounded domains, Proceedings of the Royal Society of Edinburgh Section Amathematics, 139A (2009), 253-271. [6] J.M.Cohen and F. Colonna, Isometric Composition Operators on the Bloch Space in the Polydisk, Contemporary Mathematics, Banach Spaces of Analytic Functions, 454 (2008), 9-21. [7] B. J. Carswell and C. Hammond, Composition operators with maximal norm on weighted Bergman spaces, Proc. Amer. Math. Soc., 134 (2006), 2599-2605. [8] B. A. Cload, Composition operators: hyperinvariant subspaces, quasi-normals and isometries, Proc. Amer. Math. Soc., 127 (1999),1697-1703. [9] F. Colonna, Characterisation of the isometric composition operators on the Bloch space, Bull. Austral. Math. Soc. 72 (2005), 283-290.

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[10] Geng-Lei Li and Ze-Hua Zhou,Isometries on Products of composition and integral operators on Bloch type space, Journal of Inequalities and Applications, 2010 (2010), Article ID 184957: 9 pages. (SCI No. 664HU) [11] W. Hornor and J. E. Jamison, Isometries of some Banach spaces of analytic functions, Integral Equations Operator Theory ,41(2001), 410-425. [12] T. Hosokawa, S. Ohno, Differences of composition operators on the Bloch space, J. Operator Theory, 57(2007), 229-242. [13] C.J. Kolaski, Isometries of weighted Bergman spaces, Can.J. Math., 34 (1982), 910-915. [14] C. J. Kolaski, Isometries of some smooth spaces of analytic functions, Complex Variables, 10 (1988), 115-122. [15] A. Koranyi and S. Vagi, On isometries of H p of bounded symmetruc domains, Cann. J. Math., 28(1976), 334-340. [16] S. Y. Li, Composition operators and isometries on holomorphic function spaces over domains in C n . AMS/IP Stud Adv Math. 39 (2007), 161-174. [17] S. Y. Li and Y. B. Ruan, On characterizations of isometries on function spaces, Science in China Series A: Mathematics, 51(4)(2008),620-631. [18] S. Li, S. Stevi´c, Products of composition and integral type operators from H ∞ to the Bloch space, Complex variables and EllipticEquations, 53 (2008), 463-474. [19] S. Li, S. Stevi´c, Products of integral-type operators and composition operators between Bloch-type spaces, J. Math. Anal. Appl., 349(2009),596-610. [20] S. Li, S. Stevi´c, Generalized composition operators on Zygmund spaces and Bloch type spaces, J. Math. Anal. Appl., 338(2008),1282-1295. [21] M. Lindstr¨ om and E. Wolf. Essential norm of the difference of weighted composition operators, Monatsh .Math., 153(2008), 133-143. [22] A. Matheson, Isometries into function algebras, Houston J. Math., 30 (2004), 219-230 . [23] M. J. Mart´tn and D. Vukoti´c, Isometries of the Bloch space among the composition operators, Bull. London Math. Soc., 39 (2007), 151-155. [24] M. J. Mart´tn and D. Vukoti´c, Isometries of some classical function spaces among the composition operators, in Recent advances in Operator-related function theory, A. L. Matheson, M. I. Stessin, and R. M. Timoney,Eds., vol. 393 of Contemporary Mathematics,133-138, American Mathematical Society, Providence, RI,USA, 2006. [25] K. H. Zhu, Spaces of holomorphic functions in the unit ball, Springer-Verlag (GTM 226), 2004. [26] N. Zorboska, Isometric composition operators on the Bloch-type spaces, C. R. Math. Acad. Sci. Soc. 29(3)(2007), 91-96. [27] Z. H. Zhou and J. H. Shi, Compactness of composition operators on the Bloch space in classical bounded symmetric domains, Michigan Math. J., 50(2002),381-405.

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Coupled Fixed Point Theorems for Generalized Symmetric Contractions in Partially Ordered Metric Spaces and applications M. Jain1 , K. Tas

∗2

, B.E. Rhoades 3 , and N. Gupta

4

1 Department

of Mathematics, Ahir College, Rewari 123401, India , E-mail: manish [email protected] of Mathematics and Computer Science, Cankaya University, 06530 Ankara, Turkey, E-mail: [email protected] Department of Mathematics,Indiana University, Bloomington, IN 47405-7106, U.S.A. , E-mail: [email protected] 4 HAS Department, YMCAUST, Faridbad, India, E-mail: [email protected]

2 Department 3

February 20, 2013

Abstract In the setting of partially ordered metric spaces, we introduce the notion of generalized symmetric g-Meir-Keeler type contractions and use the notion to establish the existence and uniqueness of coupled common fixed points. Our notion extends the notion of generalized symmetric Meir-Keeler contractions given by Berinde et. al. [V. Berinde, and M. Pacurar, Coupled fixed point theorems for generalized symmetric Meir-Keeler contractions in ordered metric spaces, Fixed Point Theory and Appl., 2012, 2012:115, doi:10.1186/16871812-2012-115] to a pair of mappings. We also give some applications of our main results. AMS Subject Classification: 47H10, 46T99, 54H25 . Key Words : partially ordered metric space, fixed point, generalized symmetric contractions, coupled fixed point.

1

Introduction

Banach [1] in his classical work gave the following contractive theorem: Theorem 1.1. Let (X, d) be a metric space and T : X → X be a self mapping. If (X, d) is complete and T is a contraction, that is, there exists a constant k ∈ [0, 1) such that d(T x, T y) ≤ kd(x, y), ∀x, y ∈ X

(1.1)

then, T has a unique fixed point u ∈ X and for any x0 ∈ X, the Picard iteration {T n (x0 )} converges to u. This contraction principle proved to be a very powerful tool in nonlinear analysis, and different authors have generalized it in many ways. One can refer to the works noted in references [2]- [17]. Meir and Keeler [9] generalized the contraction principle due to Banach by considering a more general contractive condition in their work as follows: Theorem 1.2. [9] Let (X, d) be a complete metric space and T : X → X be a given mapping. Suppose that, for any  > 0, there exists δ() > 0 such that  ≤ d(x, y) <  + δ() ⇒ d(T (x), T (y)) <  ∗ Corresponding

(1.2)

author

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for all x, y ∈ X. Then T admits a unique fixed point x0 ∈ X and for all x ∈ X, the sequence {T n (x)} converges to x0 . By extending the Banach contraction principle to partially ordered sets, Turinici [16] laid the foundation for a new trend in fixed point theory. Ran and Reurings [17] developed some applications of Turinici’s theorem to matrix equations. The work of Bhaskar and Lakshmikantham [18] is worth mentioning, as they introduced the new notion of fixed points for mappings having domain the product space X × X, which they called coupled fixed points, and thereby proved some coupled fixed point theorems for mappings satisfying the mixed monotone property in partially ordered metric spaces. As an application, they discussed the existence and uniqueness of a solution for a periodic boundary value problem. Lakshmikantham and Ciric [19] extended the notion of the mixed monotone property to the mixed g-monotone property and generalized the results of Bhaskar and Lakshmikantham [18] by establishing the existence of coupled coincidence points, using a pair of commutative maps.This proved to be a milestone in the development of fixed point theory with applications to partially ordered sets. Since then much work has been done in this direction by different authors. For more details the reader may consult [20]-[31]. Gordji et. al. [32], extended the results of Bhaskar and Lakshmikantham [18], and Samet [33] by introducing the concept of generalized g-Meir-Keeler type contractions. Abdeljawad et. al. [34] and Jain et. al. [36] proved some interesting results in partially ordered partial metric spaces and remarked that the metric space case of their results, proved recently in Gordji et. al. [32] has gaps. They claimed that some of the results proved by Gordji et. al.[32] cannot be true if obtained via nonstrongly minihedral cones. On the other hand, Berinde et. al. [35] with their outstanding new approach introduced the notion of generalized symmetric Meir-Keeler contractions and complemented the results due to Samet [33]. In this paper, we introduce the notion of generalized symmetric g-Meir-Keeler type contractions that extends the concept of generalized symmetric Meir-Keeler contractions given by Berinde et. al. [35] to a pair of mappings. Following Abdeljawad et. al. [34], we establish the existence and uniqueness of coupled common fixed points for mixed g-monotone mappings satisfying generalized symmetric conditions in partially ordered metric spaces. To validate our results we also give some applications. Before we proceed, we first summarize some basic results and definitions useful in our study. Definition 1.3. [18] Let (X, ≤) be a partially ordered set and F : X × X → X. The mapping F is said to have the mixed monotone property if F (x, y) is monotone non-decreasing in x and monotone non-increasing in y; that is, for any x, y ∈ X, x1 , x2 ∈ X, x1 ≤ x2

implies F (x1 , y) ≤ F (x2 , y)

y1 , y2 ∈ X, y1 ≤ y2

implies F (x, y1 ) ≥ F (x, y2 )

and Definition 1.4. [18] An element (x, y) ∈ X × X, is called a coupled fixed point of the mapping F : X × X → X if F (x, y) = x and F (y, x) = y. Definition 1.5. [19] Let (X, ≤) be a partially ordered set and F : X × X → X and g : X → X. We say F has the mixed g-monotone property if F (x, y) is monotone g-nondecreasing in its first argument and is monotone g-nonincreasing in its second argument; that is, for any x, y ∈ X, x1 , x2 ∈ X, g(x1 ) ≤ g(x2 ) implies F (x1 , y) ≤ F (x2 , y) and y1 , y2 ∈ X, g(y1 ) ≤ g(y2 ) implies F (x, y1 ) ≥ F (x, y2 ) Definition 1.6. [19] An element (x, y) ∈ X ∈ X, is called a coupled coincidence point of the mappings F : X × X → X and g : X → X if F (x, y) = gx and F (y, x) = gy. Definition 1.7. [19] An element (x, y) ∈ X ∈ X, is called a coupled common fixed point of the mappings F : X × X → X and g : X → X if x = gx = F (x, y) and y = gy = F (y, x). 439

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Definition 1.8. [19] Let X be a non-empty set and F : X × X → X and g : X → X. We say that F and g are commutative if gF (x, y) = F (gx, gy) for all x, y ∈ X. Later, Choudhury and Kundu[20] introduced the notion of compatibility in the context of coupled coincidence point problems and used this notion to improve the results noted in [19]. Definition 1.9. [20] The mappings F : X × X → X and g : X → X are said to be compatible if limn→∞ d(g(F (xn , yn ), F (gxn , gyn )) = 0 and limn→∞ d(g(F (yn , xn ), F (gyn , gxn )) = 0 whenever {xn } and {yn } are sequences in X such that limn→∞ F (xn , yn ) = limn→∞ gxn = x and limn→∞ F (yn , xn ) = limn→∞ gyn = y for some x, y ∈ X. Recently, Gordji et. al. [32] replaced the mixed g-monotone property with the mixed strict g-monotone property and extended the results of Bhaskar and Lakshmikantham [18]. Definition 1.10. [32] Let (X, ≤) be a partially ordered set and F : X ×X → X and g : X → X. We say F has the mixed strict g-monotone property if for any x, y ∈ X, x1 , x2 ∈ X, g(x1 ) < g(x2 ) implies F (x1 , y) < F (x2 , y) and y1 , y2 ∈ X, g(y1 ) < g(y2 ) implies F (x, y1 ) > F (x, y2 ) Here if we replace g with identity mapping in Definition 1.10, we get the definition of mixed strict monotone property of F. Theorem 1.11. [36] Let (X, ≤) be a partially ordered set and suppose there exists a metric d on X such that (X, d) is a complete metric space. Let F : X × X → X be a mapping having the mixed monotone property on X such that there exist two elements x0 , y0 ∈ X with x0 ≤ F (x0 , y0 ) and y0 ≥ F (y0 , x0 ). Suppose that there exists a real number k ∈ [0, 1) such that d(F (x, y), F (u, v)) + d(F (y, x), F (v, u)) ≤ k[d(x, u) + d(y, v)]

(1.3)

for all x, y, u, v ∈ X with x ≥ u, y ≤ v. Suppose that either (a) F is continuous or (b) X has the following property: (i) if a non-decreasing sequence {xn } → x, then xn ≤ x for all n > 0; (ii) if a non-decreasing sequence {yn } → y, then y ≤ yn for all n > 0; Then F has a coupled fixed point in X. We now introduce our notion. Definition 1.12. Let (X, ≤) be a partially ordered set and d be a metric on X. Let F : X × X → X and g : X → X be two mappings. We say that F is a generalized symmetric g-Meir-Keeler type contraction if, for any  > 0, there exists a δ() > 0 such that , for all x, y, u, v ∈ X with g(x) ≤ g(u) and g(y) ≥ g(v) ( or g(x) ≥ g(u) and g(y) ≤ g(v)), ≤

1 [d(g(x), g(u)) + d(g(y), g(v))] <  + δ() 2

implies 1 [d(F (x, y), F (u, v)) + d(F (y, x), F (v, u))] <  2

(1.4)

If, in Definition 1.12, we replace g by the identity mapping, we obtain the definition of a generalized symmetric Meir-Keeler type contraction due to Berinde et. al. [35]. Definition 1.13. [35] Let (X, ≤) be a partially ordered set and d be a metric on X. Let F : X × X → X be the given mapping. We say that F is a generalized symmetric Meir-Keeler type contraction if for any  > 0, there exists a δ() > 0 such that , for all x, y, u, v ∈ X with x ≤ u and y ≥ v ( or x ≥ u and y ≤ v), 440

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≤

1 [d(x, u) + d(y, v)] <  + δ() 2

implies 1 [d(F (x, y), F (u, v)) + d(F (y, x), F (v, u))] <  2

(1.5)

Proposition 1.14. Let (X, d, ≤) be a partially ordered metric space and F : X × X → X be a given mapping. If contractive condition (1.3) is satisfied for 0 < k < 1 , then F is a generalized symmetric Meir-Keeler type contraction. Proof. Assume that (1.3) is satisfied for 0 < k < 1. For all  > 0 , it is easy to check that (1.5)
is satisfied with δ() = k1 − 1 . Lemma 1.15. Let (X, ≤) be a partially ordered set and d be a metric on X. Let F : X × X → X and g : X → X be two mappings. If F is a generalized symmetric g-Meir-Keeler type contraction, then we have d(F (x, y), F (u, v)) + d(F (y, x), F (v, u)) < d(g(x), g(u)) + d(g(y), g(v))

(1.6)

for all x, y, u, v ∈ X with g(x) < g(u), g(y) ≥ g(v) (or g(x) ≤ g(u), g(y) > g(v)). Proof. Without loss of generality, we may assume that g(x) < g(u), g(y) ≥ g(v) where x, y, u, v ∈ X. Then d(g(x), g(u)) + d(g(y), g(v)) > 0. Since F is a generalized symmetric g-Meir- Keeler type contraction, for  = ( 12 )[d(g(x), g(u)) + d(g(y), g(v))], there exists a δ() > 0 such that , for all x0 , y0 , u0 , v0 ∈ X with g(x0 ) < g(u0 ) and g(y0 ) ≥ g(v0 ), ≤

1 [d(g(x0 ), g(u0 )) + d(g(y0 ), g(v0 ))] <  + δ() 2

implies 1 [d(F (x0 , y0 ), F (u0 , v0 )) + d(F (y0 , x0 ), F (v0 , u0 ))] <  2 Then the result follows by choosing x = x0 , y = y0 , u = u0 , v = v0 ; that is, d(F (x, y), F (u, v)) + d(F (y, x), F (v, u)) < d(g(x), g(u)) + d(g(y), g(v))

2

Existence of Coupled Coincidence Points

We now establish our first main result. Theorem 2.1. Let (X, ≤, d) be a partially ordered metric space. Suppose that X has the following properties: (i) if {xn } is a sequence such that xn+1 > xn for each n = 1, 2, . . . and xn → x, then xn < x for each n = 1, 2, . . .. (ii) if {yn } is a sequence such that yn+1 < yn for each n = 1, 2, . . . and yn → y, then yn > y for each n = 1, 2, . . .. Let F : X × X → X and g : X → X be mappings such that F (X × X) ⊆ g(X) and g(X) is a complete subspace of (X, d). Also, suppose that (a) F has the mixed strict g-monotone property; (b) F is a generalized symmetric g-Meir-Keeler type contraction; (c) there exists x0 , y0 ∈ X such that g(x0 ) < F (x0 , y0 ) and g(y0 ) ≥ F (y0 , x0 )(or g(x0 ) ≤ F (x0 , y0 ) and g(y0 ) > F (y0 , x0 )). Then, there exist x, y ∈ X such that g(x) = F (x, y) and g(y) = F (y, x). 441

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Proof. Without loss of generality, we may assume that there exist x0 , y0 ∈ X such that g(x0 ) < F (x0 , y0 ) and g(y0 ) ≥ F (y0 , x0 ). Since F (X × X) ⊆ g(X), we can choose x1 , y1 ∈ X such that g(x1 ) = F (x0 , y0 ), g(y1 ) = F (y0 , x0 ). Again we can choose x2 , y2 ∈ X such that g(x2 ) = F (x1 , y1 ), g(y2 ) = F (y1 , x1 ). Continuing this process, we construct sequences {gxn } and {gyn } such that g(xn+1 ) = F (xn , yn ), g(yn+1 ) = F (yn , xn ), ∀n ≥ 0 (2.1) Using conditions (a), (c) and mathematical induction, it is easy to see that g(x0 ) < g(x1 ) < g(x2 ) < . . . < g(xn ) < g(xn+1 ) < . . .

(2.2)

g(yn+1 ) < g(yn ) < . . . < g(y2 ) < g(y1 ) < g(y0 ).

(2.3)

δn := d(g(xn ), g(xn+1 )) + d(g(yn ), g(yn+1 ))

(2.4)

and Denote by Using (2.1) of Lemma 1.15, and condition (b), we have δn := d(g(xn ), g(xn+1 )) + d(g(yn ), g(yn+1 )) = d(F (xn−1 , yn−1 ), F (xn , yn )) + d(F (yn−1 , xn−1 ), F (yn , xn )) < d(g(xn−1 ), g(xn )) + d(g(yn−1 ), g(yn )) = δn−1

(2.5)

Thus, the sequence {δn } is a decreasing sequence. Therefore there exists some δ ∗ ≥ 0 such that limn→∞ δn = δ ∗ . We claim that δ ∗ = 0. Suppose, to the contrary, that δ ∗ 6= 0. Then there exists a positive integer m such that, for any n ≥ m, we have ≤

1 δn = [d(g(xn ), g(xn+1 )) + d(g(yn ), g(yn+1 ))] <  + δ() 2 2

(2.6)

where  = δ ∗ /2 and δ() is chosen by condition (b). In particular, for n = m, we have δm 1 = [d(g(xm ), g(xm+1 )) + d(g(ym ), g(ym+1 ))] <  + δ() 2 2 Then, by condition (b), it follows that ≤

1 [d(F (xm , ym ), F (xm+1 , ym+1 )) + d(F (ym , xm ), F (ym+1 , xm+1 ))] <  2

(2.7)

(2.8)

and hence, from (2.1), we have 1 [d(g(xm+1 ), g(xm+2 )) + d(g(ym+1 ), g(ym+2 ))] <  2

(2.9)

a contradiction to (2.6) for n = m + 1. Thus we must have δ ∗ = 0 and hence lim δn = lim [d(g(xn ), g(xn+1 )) + d(g(yn ), g(yn+1 ))] = 0

n→∞

n→∞

(2.10)

We now prove that {g(xn )} and {g(yn )} are Cauchy sequences. Take an arbitrary  > 0. Then, by (2.10), it follows that there exists some k ∈ N such that 1 [d(g(xk ), g(xk+1 )) + d(g(yk ), g(yk+1 ))] < δ() 2

(2.11)

Without loss of generality, assume that k has been chosen so large that δ() ≤  and define the set ∧ := {(g(x), g(y)) : (x, y) ∈ X 2 , d(g(x), g(xk )) + d(g(y), g(yk )) < 2( + δ()), and g(x) > g(xk ), g(y) ≤ g(yk )} 442

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We claim that (g(x), g(y)) ∈ ∧ implies that (F (x, y), F (y, x)) ∈ ∧

(2.13)

where x, y ∈ X. Take (g(x), g(y)) ∈ ∧. Then, using the triangle inequality and (2.11), we have 1 [d(g(xk ), F (x, y)) 2

+ d(g(yk ), F (y, x))] ≤ + = +

1 [d(g(xk ), g(xk+1 )) + d(g(xk+1 ), F (x, y))] 2

1 [d(g(yk ), g(yk+1 )) + d(g(yk+1 ), F (y, x))] 2 1 [d(g(xk ), g(xk+1 )) + d(g(yk ), g(yk+1 ))] 2 1 [d(g(xk+1 ), F (x, y)) + d(g(yk+1 ), F (y, x))] 2

1 < δ() + [d(F (x, y), F (xk , yk )) + d(F (y, x), F (yk , xk ))] (2.14) 2 We distinguish two cases. First Case: 12 [d(g(xk ), F (x, y)) + d(g(yk ), F (y, x))] ≤ . By Lemma 1.15 and Definition of ∧ , the inequality (2.14) becomes 1 [d(g(xk ), F (x, y)) 2

+ d(g(yk ), F (y, x))] ≤ δ() +
g(xk ), F (y, x) > g(yk )

(2.19)

Also, F (X × X) ⊆ g(X). Consequently, we have (F (x, y), F (y, x)) ∈ ∧ ; that is (2.13) holds. By (2.11), we have (g(xk+1 ), g(yk+1 )) ∈ ∧.Then, using (2.13), we have (g(xk+1 ), g(yk+1 )) ∈ ∧

⇒ d(F (xk+1 , yk+1 ), F (yk+1 , xk+1 )) = (g(xk+2 ), g(yk+2 ) ∈ ∧ ⇒ d(F (xk+2 , yk+2 ), F (yk+2 , xk+2 )) = (g(xk+3 ), g(yk+3 ) ∈ ∧ ⇒ . . . ⇒ (g(xn ), g(yn )) ∈ ∧ ⇒ . . . 443

(2.20)

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Then, for all n > k, we have (g(xn ), g(yn )) ∈ ∧ . This implies that, for all n, m > k, we have d(g(xn ), g(xm ))

+ + = ≤

d(g(yn ), g(ym )) ≤ d(g(xn ), g(xk )) + d(g(xk ), g(xm )) d(g(yn ), g(yk )) + d(g(yk ), g(ym )) [d(g(xn ), g(xk )) + d(g(yn ), g(yk ))] + [d(g(xk ), g(xm )) + d(g(yk ), g(ym ))] 4( + δ()) ≤ 8

Therefore, the sequences {g(xn )} and {g(yn )} are Cauchy. Since (g(X), d) is complete, there exist x, y ∈ X such that lim d(g(xn ), g(x)) = 0, lim d(g(yn ), g(y)) = 0

n→∞

n→∞

(2.21)

Since the sequences {g(xn )} and {g(yn )} are monotone increasing and monotone decreasing, respectively, by conditions (i) and (ii), we have g(xn ) < g(x),

g(yn ) > g(y)

(2.22)

for each n ≥ 0. Therefore, by (2.22) and Lemma 1.15, along with condition (b), we obtain d(g(xn+1 ), F (x, y))

+ d(g(yn+1 ), F (y, x)) = d(F (xn , yn ), F (x, y)) + d(F (yn , xn ), F (y, x))

< d(g(xn ), g(x)) + d(g(yn ), g(y))

(2.23)

Letting n → ∞ in (2.23) and using (2.21), we get d(g(x), F (x, y)) + d(g(y), F (y, x)) ≤ lim [d(g(xn ), g(x)) + d(g(yn ), g(y))] n→∞

(2.24)

which yields F (x, y) = g(x), F (y, x) = g(y). This completes the proof. Corollary 2.2. Let (X, ≤, d) be a partially ordered metric space. Suppose that (X, d) is complete and has the following properties: (i) if {xn } is a sequence such that xn+1 > xn for each n = 1, 2, . . . and xn → x, then xn < x for each n = 1, 2, . . .. (ii) if {yn } is a sequence such that yn+1 < yn for each n = 1, 2, . . . and yn → y, then yn > y for each n = 1, 2, . . .. Let F : X × X → X be a mapping. Also, suppose that (d) F has the mixed strict monotone property; (e) F is a generalized symmetric Meir-Keeler type contraction; (f ) there exists x0 , y0 ∈ X such that x0 < F (x0 , y0 ) and y0 ≥ F (y0 , x0 )(or x0 ≤ F (x0 , y0 ) and y0 > F (y0 , x0 )). Then, there exist x, y ∈ X such that x = F (x, y) and y = F (y, x). Remark 2.3. If, in Theorem 2.1 condition (c) is replaced by the following condition: (g) there exist x0 , y0 ∈ X such that g(x0 ) > F (x0 , y0 ) and g(y0 ) ≤ F (y0 , x0 ) (or g(x0 ) ≥ F (x0 , y0 ) and g(y0 ) < F (y0 , x0 ) ), then we also get the existence of some x, y ∈ X such that g(x) = F (x, y) and g(y) = F (y, x). And, if in Corollary 2.2, condition (f) is replaced by the following condition: (h) there exist x0 , y0 ∈ X such that x0 > F (x0 , y0 ) and y0 ≤ F (y0 , x0 ) (or x0 ≥ F (x0 , y0 ) and y0 < F (y0 , x0 ) ), then we also get the existence of some x, y ∈ X such that x = F (x, y) and y = F (y, x). Remark 2.4. Corollary 2.2, along with Remark 2.3, improves on the result of Berinde et. al. ([35], Theorem 2) by removing the continuity assumption on the mixed monotone operator F .

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3

Existence and Uniqueness of Coupled Fixed Points

In this section we prove the existence and uniqueness of coupled fixed points. Before we proceed, we need to consider the following. For a partially ordered set (X, ≤), we endow X × X with the following order ≤g (u, v) ≤g (x, y) ⇒ g(u) < g(x), g(y) ≤ g(v), ∀(x, y), (u, v) ∈ X × X

(3.1)

In this case, we say that (u, v) and (x, y) are g -comparable if either (u, v) ≤g (x, y) or (x, y) ≤g (u, v) . If g = IX , then we simply say that (u, v) and (x, y) are comparable and denote this fact by (u, v) ≤ (x, y). Lemma 3.1. Let F : X × X → X and g : X → X be compatible maps and suppose there exists an element (x, y) ∈ X × X such that g(x) = F (x, y) and g(y) = F (y, x). Then gF (x, y) = F (g(x), g(y)) and gF (y, x) = F (g(y), g(x)). Proof. Since the pair (F, g) is compatible, it follows that lim d(gF (xn , yn ), F (g(xn ), g(yn ))) = 0

n→∞

and lim d(gF (yn , xn ), F (g(yn ), g(xn ))) = 0

n→∞

whenever {xn } and {yn } are sequences in X such that limn→∞ F (xn , yn ) = limn→∞ g(xn ) = a, limn→∞ F (yn , xn ) = limn→∞ g(yn ) = b for some a, b ∈ X. Taking xn = x, yn = y and using the fact that g(x) = F (x, y), g(y) = F (y, x) , it follows immediately that d(gF (x, y), F (g(x), g(y))) = 0 and d(gF (y, x), F (g(y), g(x))) = 0. Hence, gF (x, y) = F (g(x), g(y)) and gF (y, x) = F (g(y), g(x)). Theorem 3.2. In Theorem 2.1, assume, in addition, that, for all non g-comparable points (x, y), (x∗ , y ∗ ) ∈ X × X, there exists a point (a, b) ∈ X × X such that (F (a, b), F (b, a)) is comparable to both (g(x), g(y)) and (g(x∗ ), (y ∗ )) . Also assume that F and g are compatible. Then,F and g have a unique coupled common fixed point; that is, there exists a point (u, v) ∈ X × X such that u = g(u) = F (u, v), v = g(v) = F (v, u) (3.2) Proof. From Theorem 2.1 it follows that the set of coupled coincidence points of F and g is non-empty. We shall first show that, if (x, y) and (x∗ , y ∗ ) are coupled coincidence points, that is, if g(x) = F (x, y) , g(y) = F (y, x) and g(x∗ ) = F (x∗ , y ∗ ) , g(y ∗ ) = F (y ∗ , x∗ ) , then g(x) = g(x∗ )andg(y) = g(y∗)

(3.3)

For this, we distinguish the following two cases. First Case. (x, y) is g-comparable to (x∗ , y ∗ ) with respect to the ordering in X × X, where F (x, y) = g(x), F (y, x) = g(y), F (x∗ , y ∗ ) = g(x∗ ), F (y ∗ , x∗ ) = g(y ∗ )

(3.4)

Without loss of generality, we may assume that g(x) = F (x, y) < F (x∗ , y ∗ ) = g(x∗ ), g(y) = F (y, x) ≥ F (y ∗ , x∗ ) = g(y ∗ )

(3.5)

Using Lemma 1.15 we have 0 < d(g(x), g(x∗ )) + d(g(y ∗ ), g(y)) = d(F (x, y), F (x∗ , y ∗ )) + d(F (y ∗ , x∗ ), F (y, x)) < d(g(x), g(x∗ )) + d(g(y ∗ ), g(y)) a contradiction. Therefore, we have (g(x), g(y)) = (g(x∗ ), g(y ∗ )) . Hence (3.3) holds. Second Case. (x, y) is not g-comparable to (x∗ , y ∗ ). 445

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By assumption, there exists a point (a, b) ∈ X × X such that (F (a, b), F (b, a)) is comparable to both (g(x), g(y)) and (g(x∗ ), g(y ∗ )). Then we have g(x) = F (x, y) < F (a, b),

F (x∗ , y ∗ ) = g(x∗ ) < F (a, b),

(3.6)

g(y) = F (y, x) ≥ F (b, a),

F (y ∗ , x∗ ) = g(y ∗ ) ≥ F (b, a),

(3.7)

and ∗

Further, setting x = x0 , y = y0 , a = a0 , b = b0 and x = 2.1, we obtain

x∗0 , y ∗

=

y0∗

as in the proof of Theorem

g(xn+1 ) = F (xn , yn ),

g(yn+1 ) = F (yn , xn ), ∀n = 0, 1, 2, . . .

g(an+1 ) = F (an , bn ),

g(bn+1 ) = F (bn , an ), ∀n = 0, 1, 2, . . .

g(x∗n+1 )

∗ g(yn+1 )

=

F (x∗n , yn∗ ),

=

F (yn∗ , x∗n ), ∀n

(3.8)

= 0, 1, 2, . . .

Since (F (x, y), F (y, x)) = (g(x), g(y)) = (g(x1 ), g(y1 )) is comparable with (F (a, b), F (b, a)) = (g(a1 ), g(b1 )), we have g(x) < g(a1 ) and g(y) ≥ g(b1 ).Using the fact that F has the mixed strict g -monotone property, g(x) < g(an ) and g(bn ) < g(y) for all n ≥ 2. Thus, by Lemma 1.15, we have 0 < d(g(x), g(an+1 )) + d(g(y), g(bn+1 )) = d(F (x, y), F (an , bn )) + d(F (y, x), F (bn , an )) < d(g(x), g(an )) + d(g(y), g(bn ))

(3.9)

Let αn = d(g(x), g(an )) + d(g(y), g(bn )). Then, by (3.9), it follows that {αn } is a decreasing sequence, and hence converges to some α ≥ 0. We claim that α = 0. Suppose, to the contrary, that α > 0. Then there exists a positive integer p such that, for n ≥ p, we have ≤ where  =

α 2

1 an = [d(g(x), g(an )) + d(g(y), g(bn ))] <  + δ(), 2 2

(3.10)

and δ() is chosen by condition (b) of Theorem 2.1. In particular, for n = p, ≤

ap 1 = [d(g(x), g(ap )) + d(g(y), g(bp ))] <  + δ(), 2 2

(3.11)

Then, by condition (b) of Theorem 2.1, we have 1 [d(F (x, y), F (ap , bp )) + d(F (y, x), F (bp , ap ))] < , 2

(3.12)

and hence

1 [d(g(x), g(ap+1 )) + d(g(y), g(bp+1 ))] < , 2 a contradiction to (3.10) for n = p + 1. Thus α = 0, and hence lim αn = lim [d(g(x), g(an )) + d(g(y), g(bn ))] = 0

n→∞

n→∞

(3.13)

(3.14)

Similarly, it follows that lim [d(g(x∗ ), g(an )) + d(g(y ∗ ), g(bn ))] = 0

n→∞

(3.15)

Using the triangle inequality, we get d(g(x), g(x∗ )) + d(g(y), g(y ∗ )) ≤ d(g(x), g(an )) + d(g(an ), g(x∗ )) + d(g(y), g(bn )) + d(g(bn ), g(y ∗ )) = [d(g(x), g(an )) + d(g(y), g(bn ))] + [d(g(x∗ ), g(an )) + d(g(y ∗ ), g(bn ))] → 0 446

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as n → ∞. Hence, it follows that d(g(x), g(x∗ )) = 0 and d(g(y), g(y ∗ )) = 0. Therefore, (3.3) holds immediately. Thus, in both the cases, we have proved that (3.3) holds. Now, since g(x) = F (x, y), g(y) = F (y, x) and the pair (F, g) is compatible, by Lemma 3.1, it follows that g(g(x)) = gF (x, y) = F (gx, gy)

and

g(g(y)) = gF (y, x) = F (gy, gx).

(3.17)

Denote g(x) = z, g(y) = w. Then by (3.17), g(z) = F (z, w)

and

g(w) = F (w, z).

(3.18)

Thus (z, w) is a coupled coincidence point. Then by (3.3) with x∗ = z and y∗ = w , it follows that g(z) = g(x) and g(w) = g(y), that is, g(z) = z

and

g(w) = w.

(3.19)

By (3.18) and (3.19), z = g(z) = F (z, w) and w = g(w) = F (w, z). Therefore,(z, w) is a coupled common fixed point of F and g. To prove uniqueness, assume that (p, q) is another coupled common fixed point of F and g. Then by (3.3) we have p = g(p) = g(z) = z and q = g(q) = g(w) = w. This completes the proof. Corollary 3.3. Suppose that all the hypotheses of Corollary 2.2 hold, and further, for all (x, y), (x∗ , y ∗ ) ∈ X × X, there exists a point (a, b) ∈ X × X that is comparable to (x, y) and (x∗ , y ∗ ). Then F has a unique coupled fixed point.

4

Results of Integral Type

Inspired by the work of Suzuki [37], we prove the following result, which will be useful in developing some applications of the main results proved in Section 2. Theorem 4.1. Let (X, d, ≤) be a partially ordered metric space. Let F : X × X → X and g : X → X be two given mappings. Assume that there exists a function θ : [0, +∞) → [0, +∞) satisfying the following conditions: (I) θ(0) = 0 and θ(t) > 0 for any t > 0; (II) θ is increasing and right continuous; (III) for any  > 0, there exists δ() > 0 such that, for all x, y, u, v ∈ X with g(x) ≤ g(u) and g(y) ≥ g(v), 1  ≤ θ( [d(g(x), g(u)) + d(g(y), g(v))]) <  + δ() 2 1 implies θ( [d(F (x, y), F (u, v)) + d(F (y, x), F (v, u))]) <  2

(4.1)

Then F is a generalized symmetric g-Meir-Keeler type contraction. Proof. For any  > 0 it follows from (I) that θ() > 0, and so there exists an α > 0 such that, for all u, v, u∗ , v ∗ ∈ X with g(u) ≤ g(u∗ ) and g(v) ≥ g(v ∗ ), 1 θ() ≤ θ( [d(g(u), g(u∗ )) + d(g(v), g(∗ v))]) < θ() + α 2 1 implies θ( [d(F (u, v), F (u∗ , v ∗ )) + d(F (v, u), F (v ∗ , u∗ ))]) < θ() 2 447

(4.2)

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By the right continuity of θ , there exists δ > 0 such that θ( + δ) < θ() + α. For any x, y, u, v ∈ X such that g(x) ≤ g(u), g(y) ≥ g(v) and ≤

1 [d(g(x), g(u)) + d(g(y), g(v))] <  + δ. 2

(4.3)

Then, since θ is an increasing function, we get the following: 1 θ() ≤ θ( [d(g(x), g(u)) + d(g(y), g(v))]) < θ( + α) < θ() + α. 2

(4.4)

By (4.2), we have 1 θ( [d(F (x, y), F (u, v)) + d(F (y, x), F (v, u))]) < θ() 2 and hence, 1 [d(F (x, y), F (u, v)) + d(F (y, x), F (v, u))]) < . 2 Therefore, it follows that F is a generalized symmetric g-Meir-Keeler type contraction. This completes the proof. The following result is an immediate consequence of Theorems 2.1 and 4.1. Corollary 4.2. Let (X, d, ≤) be a partially ordered metric space. Given F : X × X → X and g : X → X such that F (X × X) ⊂ g(X), g(X) is a complete subspace and the following hypotheses hold: (IV) F has the mixed strict g-monotone property; (V) for every  > 0 , there exists δ() > 0 such that (1/2)[d(g(x),g(u))+d(g(y),g(v))]

Z ≤

φ(t)dt <  + δ() 0

Z

(1/2)[d(F (x,y),F (u,v))+d(F (y,x),F (v,u))]

implies

φ(t)dt < 

(4.5)

0

for all gx R≤ gu and gy ≥ gv , where φ : [0, +∞) → [0, +∞) is a locally integrable function s satisfying 0 φ(t)dt > 0 for all s > 0; (VI) there exist x0 , y0 ∈ X such that g(x0 ) < F (x0 , y0 ) and g(y0 ) ≥ F (y0 , x0 ). Assume that the hypotheses (i) and (ii) given in Theorem 2.1 hold. Then,F and g have a coupled coincidence point. Corollary 4.3. Let (X, d, ≤) be a partially ordered metric space. Given F : X × X → X and g : X → X such that F (X × X) ⊂ g(X), g(X) is a complete subspace and the following hypotheses hold: (VII) F has the mixed g-monotone property; (VIII) for all gx ≤ gu and gy ≤ gv Z

(1/2)[d(g(x),g(u))+d(g(y),g(v))]

Z

(1/2)[d(F (x,y),F (u,v))+d(F (y,x),F (v,u))]

φ(t)dt ≤ k 0

φ(t)dt

(4.6)

0

k ∈ (0, 1) and φ is a locally integrable function from [0, +∞) into itself satisfying Rwhere s φ(t)dt > 0 for all s > 0; 0 (IX) there exist x0 , y0 ∈ X such that g(x0 ) < F (x0 , y0 ) and g(y0 ) ≥ F (y0 , x0 ). Assume that the hypotheses (i) and (ii) given in Theorem 2.1 hold. Then,F and g have a coupled coincidence point. Proof. For each  > 0 , take δ() = ( k1 − 1) and apply Corollary 4.2. 448

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5

Applications to Integral Equations

As an application of the results proved in Sections 2 and 3, we study the existence of solutions for the following system of integral equations: Z b x(t) = (K1 (t, s) + K2 (t, s))(f (s, x(s)) + g(s, y(s)))ds + h(t) a b

Z y(t)

=

(K1 (t, s) + K2 (t, s))(f (s, y(s)) + g(s, x(s)))ds + h(t)

(5.1)

a

where t ∈ I = [a, b]. Let Φ : [0, +∞) → [0, +∞) denote the class of functions φ : [0, ) → [0, ) which satisfies the following conditions: (i) φ is increasing; (ii) for each x ≥ 0, there exists a k ∈ (0, 1) such that φ(x) ≤ ( k2 )x We assume that K1 , K2 , f, g satisfy the following conditions. Assumption 5.1. (i) K1 (t, s) ≥ 0 and K2 (t, s) ≤ 0 for all t, s ∈ [a, b]; (ii) There exist λ, µ > 0 and φ ∈ Φ such that for all x, y ∈ R, x > y, 0 < f (t, x) − f (t, y) ≤ λφ(x − y)

(5.2)

− µφ(x − y) ≤ g(t, x) − g(t, y) < 0;

(5.3)

and (iii) Z t∈I

b

(K1 (t, s) − K2 (t, s))ds ≤ 1;

(λ + µ) sup

(5.4)

a

Definition 5.2. An element (α, β) ∈ X × X with X = C(I, R) is called a coupled lower and upper solution of the integral equation (5.1) if for all t ∈ I , Z b Z b α(t) < (K1 (t, s)(f (s, α(s)) + g(s, β(s)))ds + K2 (t, s))(f (s, β(s)) + g(s, α(s)))ds + h(t) a

a

and Z β(t) ≥

b

Z (K1 (t, s)(f (s, β(s)) + g(s, α(s)))ds +

a

b

K2 (t, s))(f (s, α(s)) + g(s, β(s)))ds + h(t) a

Theorem 5.3. Consider the integral equation (5.1) with K1 , K2 ∈ C(I, R), f, g ∈ C(I × R, R) and h ∈ C(I, R). Suppose that there exists a coupled lower and upper solution (α, β) of (5.1) with α ≤ β and that Assumption 5.1 is satisfied. Then the integral equation (5.1) has a solution. Proof. Consider the natural order relation on X = C(I, R) ; that is, for x, y ∈ C(I, R) x ≤ y ⇒ x(t) ≤ y(t), ∀t ∈ I It is well known that X is a complete metric space with respect to the sup metric d(x, y) = sup |x(t) − y(t)|, x, y ∈ C(I, R). t∈I

Suppose that {un } is a strictly increasing sequence in X that converges to a point u ∈ X. Then for every t ∈ I, the sequence of real numbers u1 (t) < u2 (t) < . . . < un (t) < . . . converges to u(t). Therefore, for all t ∈ I, n ∈ N, un (t) < u(t). Hence, un < u for all n. Similarly, it can be verified that, if for all t ∈ I, v(t) is a limit of a strictly decreasing sequence 449

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vn (t) in X, then v(t) < vn (t) for all n and hence v < vn . Therefore conditions (i) and (ii) of Corollary 2.1 hold. Also, X × X = C(I, R) × C(I, R) is a partially ordered set under the following order relation in X × X (x, y), (u, v) ∈ X × X, (x, y) ≤ (u, v) ⇒ x(t) ≤ u(t)

and y(t) ≥ v(t), ∀t ∈ I.

For any x, y ∈ X , max{x(t), y(t)} and min{x(t), y(t)}, for each t ∈ I, are in X and are the upper and lower bounds of x, y, respectively. Therefore, for every (x, y), (u, v) ∈ X × X, there exists a (max{x, u}, min{y, v)}) ∈ X × X that is comparable to (x, y) and (u, v). Define F : X × X → X by Rb Rb F (x, y)(t) = a K1 (t, s)(f (s, x(s)) + g(s, y(s)))ds + a K2 (t, s)(f (s, y(s)) + g(s, x(s)))ds + h(t) for all t ∈ [a, b]. We now show that F has the mixed strict monotone property. For x1 (t) < x2 (t) for all t ∈ [a, b] we have b

Z F (x1 , y)(t) − F (x2 , y)(t) =

K1 (t, s)(f (s, x1 (s)) + g(s, y(s)))ds a

b

Z +

K2 (t, s)(f (s, y(s)) + g(s, x1 (s)))ds + h(t) a b

Z −

K1 (t, s)(f (s, x2 (s)) + g(s, y(s)))ds a b

Z −

K2 (t, s)(f (s, y(s)) + g(s, x2 (s)))ds − h(t) a b

Z

K1 (t, s)(f (s, x1 (s)) − f (s, x2 (s)))ds

= a b

Z

K2 (t, s)(g(s, x1 (s)) − g(s, x2 (s)))ds < 0

+ a

by Assumption 5.1. Hence F (x1 , y)(t) < F (x2 , y)(t), ∀t ∈ I ; that is, F (x1 , y) < F (x2 , y). Similarly, if y1 > y2 , that is, y1 (t) > y2 (t), for all t ∈ [a, b], we have Z F (x, y1 )(t) − F (x, y2 )(t) =

b

K1 (t, s)(f (s, x(s)) + g(s, y1 (s)))ds a

Z

b

+

K2 (t, s)(f (s, y1 (s)) + g(s, x(s)))ds + h(t) a

Z

b

−

K1 (t, s)(f (s, x(s)) + g(s, y2 (s)))ds a

Z

b

−

K2 (t, s)(f (s, y2 (s)) + g(s, x(s)))ds − h(t) a

Z

b

K1 (t, s)(g(s, y1 (s)) − g(s, y2 (s)))ds

= a

Z

b

K2 (t, s)(f (s, y1 (s)) − f (s, y2 (s)))ds < 0

+ a

by Assumption 5.1. Hence F (x, y1 )(t) < F (x, y2 )(t), ∀t ∈ I ; that is, F (x, y1 ) < F (x, y2 ). Therefore F satisfies mixed strict monotone property. Next, we verify that F satisfies (1.3). For x ≥ u, y ≤ v, that is, x(t) ≥ u(t), y(t) ≤ v(t) for all

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t ∈ I, we have Z F (x, y)(t) − F (u, v)(t)

b

K1 (t, s)(f (s, x(s)) + g(s, y1 (s)))ds

= a

Z

b

+

K2 (t, s)(f (s, y(s)) + g(s, x(s)))ds a

Z

b

−

K1 (t, s)(f (s, u(s)) + g(s, v(s)))ds a

Z

b

−

K2 (t, s)(f (s, v(s)) + g(s, u(s)))ds a

Z

b

K1 (t, s)(f (s, x(s)) − f (s, u(s)) − g(s, y(s)) − g(s, v(s)))]ds

= a

Z

b

K2 (t, s)[(f (s, y(s)) − f (s, v(s))) − g(s, x(s)) − g(s, u(s)))ds

+ a

Z

b

K1 (t, s)[(f (s, x(s)) − f (s, u(s)) − (g(s, v(s)) − g(s, y(s)))]ds

= a

Z

b

−

K2 (t, s)[f (s, v(s)) − f (s, y(s)) − (g(s, x(s)) − g(s, u(s))]ds a

Z

b

≤

K1 (t, s)[λφ(x(s) − u(s)) + µφ(v(s) − y(s))]ds a

Z −

b

K2 (t, s)[λφ(v(s) − y(s)) + µφ(x(s) − u(s))]ds

(5.5)

a

Since the function φ is increasing and x ≥ u and y ≤ v, we have φ(x(s) − u(s)) ≤ φ(sup |x(t) − u(t)|) = φ(d(x, u)) t∈I

and φ(v(s) − y(s)) ≤ φ(sup |v(t) − y(t)|) = φ(d(v, y)) t∈I

Hence, using (5.5) and the fact that K2 (t, s) ≤ 0 , we obtain Z |F (x, y)(t) − F (u, v)(t)|

b

≤

K1 (t, s)[λφ(d(x, u)) + µφ(d(v, y))]ds a

Z

b

−

K2 (t, s)[λφ(d(v, y)) + µφ(d(x, u))]ds

(5.6)

a

Since all of the quantities on the right hand side of (5.5) are non-negative, inequality (5.6) is satisfied. Similarly, we can show that Z |F (y, x)(t) − F (v, u)(t)|

b

≤

K1 (t, s)[λφ(d(v, y)) + µφ(d(x, u))]ds a

Z −

b

K2 (t, s)[λφ(d(x, u)) + µφ(d(v, y))]ds

(5.7)

a

Summing (5.6) and (5.7), dividing by 2 , and then taking the supremum with respect to t we

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get, by using (5.4) that d(F (x, y) + F (u, v)) + d(F (y, x) + F (v, u)) 2 Z b φ(d(v, y)) + φ(d(x, u)) ≤ (λ + φ) sup (K1 (t, s) − K2 (t, s))ds. 2 t∈I a φ(d(v, y)) + φ(d(x, u)) ≤ 2 Since φ is increasing, φ(d(x, u)) ≤ φ(d(x, u) + d(v, y)),

φ(d(v, y)) ≤ φ(d(x, u) + d(v, y))

and hence k φ(d(v, y)) + φ(d(x, u)) ≤ φ(d(x, u) + d(v, y)) ≤ ( )[d(x, u) + d(v, y)] 2 2 by the definition of φ. Thus d(F (x, y) + F (u, v)) + d(F (y, x) + F (v, u)) k ≤ ( )[d(x, u) + d(v, y)] 2 2 which reduces to the symmetric contractive condition (1.3). Then, by Proposition 1.14, F is a generalized symmetric Meir-Keeler type contraction. Finally, let (α, β) be a coupled lower and upper solution of the integral equation (5.1), then we have α(t) < F (α, β)(t) and β(t) ≥ F (β, α)(t) for all t ∈ [a, b] , that is, α < F (α, β) and β ≥ F (β, α). Therefore, Corollaries 2.2 and 3.2 yield that F has a unique coupled fixed point (x, y) and hence the system (5.1) has a unique solution.

References [1] S. Banach, Surles operations dans les ensembles et leur application aux equation sitegrales. Fund. Math. 3, 133-181 (1922). [2] R. P. Agarwal, M. Meehan, and D. ORegan, Fixed Point Th. Appl., Camb. Univ. Press, 2001. [3] R. P. Agarwal, M. A. El-Gebeily, and D. ORegan, Generalized contractions in partially ordered metric spaces, Applicable Anal., vol. 87, no. 1, pp. 109-116, 2008. [4] D. W. Boyd and J. S. W. Wong, On nonlinear contractions, Proc. Amer. Math. Soc., Vol. 20, pp. 458-464, 1969. [5] A. Branciari, A xed point theorem for mappings satisfying a general contractive condition of integral type, Internat. Jour. Math. Math. Sc., Vol. 29, no. 9, pp. 531-536, 2002. [6] L. B. Ciri’c , A generalization of Banachs contraction principle, Proc. Amer. Math. Soc., Vol. 45, pp. 267-273, 1974. [7] J. Dugundji and A. Granas, Fixed Point Th., Springer, New York, NY, USA, 2003. [8] D. J. Guo and V. Lakshmikantham, Nonlinear Problems in Abstract Cones, Academic Press, Boston, Mass, USA, 1988. [9] A. Meir and E. Keeler, A theorem on contraction mappings, Jour. Math. Anal. Appl., Vol. 28, pp. 326-329, 1969. 452

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[10] J. J. Nieto and R. Rodriguez-Lopez, Contractive mapping theorems in partially ordered sets and applications to ordinary dierential equations, Order, Vol. 22, no. 3, pp. 223-239, 2005. [11] B. E. Rhoades, A comparison of various denitions of contractive mappings, Trans. Amer. Math. Soc., vol. 226, pp. 257-290, 1977. [12] D. R. Smart, Fixed Point Theorems, Camb. Univ. Press, London, UK, 1974. [13] T. Suzuki, Meir-Keeler contractions of integral type are still Meir-Keeler contractions, Internat. Jour. Math. Math. Sc., Article ID 39281, 6 pages, 2007. [14] T. Suzuki, A generalized Banach contraction principle that characterizes metric completeness, Proc. Amer. Math. Soc., Vol. 136, no. 5, pp. 1861-1869, 2008. [15] E. Zeidler, Nonlinear Functional Analysis and Its Applications I: Fixed-Point Theorems, Springer, Berlin, Germany, 1986. [16] M. Turinici, Abstract comparison principles and multivariable Gronwall-Bellman inequalities, Jour. Math. Anal. Appl., Vol. 117, no. 1, pp. 100-127, 1986. [17] A. C. M. Ran and M. C. B. Reurings, A xed point theorem in partially ordered sets and some applications to matrix equations, Proc. Amer. Math. Soc., Vol. 132, no. 5, pp. 1435-1443, 2004. [18] T. G. Bhaskar and V. Lakshmikantham, Fixed point theorems in partially ordered metric spaces and applications, Nonlinear Anal.: TMA, Vol. 65, no. 7, pp. 1379-1393, 2006. [19] V. Lakshmikantham and Lj. B. Ciri’c, Coupled xed point theorems for nonlinear contractions in partially ordered metric spaces, Nonlinear Anal., Vol. 70, 4341-4349, 2009. [20] Binayak S. Choudhury and A. Kundu, A coupled coincidence point result in partially ordered metric spaces for compatible mappings. Nonlinear Anal., Vol. 73, 2524-2531 (2010). [21] A. Alotaibi, S. M. Alsulami, Coupled coincidence points for monotone operators in partially ordered metric spaces, Fixed Point Theory and Appl., 2011, 44 (2011). [22] N. V. Luong, N. X. Thuan, Coupled xed point in partially ordered metric spaces and applications. Nonlinear Anal., Vol. 74, 983992 (2011). [23] N. Hussain, A. Latif, M. H. Shah, Coupled and tripled coincidence point results without compatibility, Fixed Point Theory and Appl., 2012, 77(2012), doi: 10.1186/1687-18122012-77. [24] R. Saadati, S.M. Vaezpour, P. Vetro, B.E. Rhoades, Fixed point theorems in generalized partially ordered G-metric spaces, Math. Comput. Modelling 52 (2010) 797-801. [25] B. Samet, Calogero Vetro, Coupled fixed point theorems for multi-valued nonlinear contraction mappings in partially ordered metric spaces, Nonlinear Anal.: TMA, Vol. 74 (12) (2011) 4260-4268. [26] W. Shatanawi, Partially ordered cone metric spaces and coupled fixed point results Comput. Math. Appl., Vol. 60 (8) (2010) 2508-2515. [27] W. Shatanawi, B. Samet, M. Abbas, Coupled fixed point theorems for mixed monotone mappings in ordered partial metric spaces, Math. Comput. Modelling, doi:10.1016/j.mcm.2011.08.042.

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[28] W. Sintunavarat, Y. J. Cho, P. Kumam, Coupled coincidence point theorems for contractions without commutative condition in intuitionistic fuzzy normed spaces, Fixed Point Theory and Appl., 2011, 2011:81. [29] W. Sintunavarat, Y. J. Cho, P. Kumam, Coupled fixed point theorems for weak contraction mapping under F-invariant set, Abstract and Appl. Anal., Vol. 2012, Article ID 324874, 15 pages, 2012. [30] W. Sintunavarat, P. Kumam, Coupled best proximity point theorem in metric spaces, Fixed Point Theory and Appl., 2012, 2012:93. [31] H. Aydi, E. Karapnar, and W. Shatanawi, Coupled xed point results for (ψ, φ) -weakly contractive condition in ordered partial metric spaces, Comput. Math. Appl., Vol. 62, no. 12, pp. 44494460, 2011 [32] M. E. Gordji, Y. J. Cho, S. Ghods, M. Ghods, and M. H. Dehkordi, Coupled xed point theorems for contractions in partially ordered metric spaces and applications, Mathematical Problems in Engineering, Vol. 2012, Article ID 150363, 20 pages, 2012. [33] B. Samet, Coupled fixed point theorems for a generalized Meir-Keeler contraction in partially ordered metric spaces, Nonlinear Anal., Vol. 72, no. 12, pp. 45084517, 2010. [34] Thabet Abdeljawad, Hassen Aydi, and Erdal Karapnar, Coupled Fixed Points for MeirKeeler Contractions in Ordered Partial Metric Spaces, Mathematical Problems in Engineering, Vol. 2012, Article ID 327273, 20 pages, doi:10.1155/2012/327273. [35] V. Berinde, and M. Pacurar, Coupled fixed point theorems for generalized symmetric Meir-Keeler contractions in ordered metric spaces, Fixed Point Theory and Appl., 2012, 2012:115, doi:10.1186/1687-1812-2012-115. [36] M. Jain, K. Tas, S. Kumar, N. Gupta, Coupled common fixed points involving a (φ, ψ) contractive condition for mixed g-monotone operators in partially ordered metric spaces, Journal of Inequalities and Applications 2012, 2012:285 doi:10.1186/1029-242X-2012-285. [37] T. Suzuki, Meir-Keeler contractions of integral type are still Meir-Keeler contractions, Internat. Jour. Math. Math. Sc., Vol. 2007, Article ID 39281, 6 pages, 2007.

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Pointwise Superconvergence Patch Recovery for the Gradient of the Linear Tetrahedral Element Jinghong Liu∗and Yinsuo Jia† We consider the finite element approximation to the solution of a self-adjoint, second-order elliptic boundary value problem in three dimensions over a fully uniform mesh of piecewise linear tetrahedral elements. First, the supercloseness of the gradients between the piecewise linear finite element solution uh and the linear interpolation uI is derived by using a weak estimate and an estimate of the discrete derivative Green’s function. We then analyze a superconvergence patch recovery scheme for the gradient of the finite element solution, showing that the recovered gradient of uh is superconvergent to the gradient of the true solution u.

1

Introduction

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

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recovered gradient with O(h2−ε ) order accuracy. Unlike the results in [11–13], this article will show a pointwise superconver4 gence estimate with O(h2 | ln h| 3 ) order accuracy for the recovered gradient by using SPR. In this article, we shall use the letter C to denote a generic constant which may not be the same in each occurrence and also use the standard notations for the Sobolev spaces and their norms.

2

Model Problem and Finite Element Space

Suppose Ω ⊂ R3 is a rectangular block with boundary, ∂Ω, consisting of faces parallel to the x-, y-, and z-axes. We consider the self-adjoint, variable coefficients second-order elliptic problem Lu ≡ −

3 ∑

∂j (aij ∂i u) = f in Ω,

u = 0 on ∂Ω.

(2.1)

i,j=1

Here we assume f is smooth enough, and A = (aij ) is a 3 × 3 symmetric matrix function in (L∞ (Ω))3×3 and uniformly positive definite. Set ∂1 u = ∂u ∂x , ∂u ∂2 u = ∂u , and ∂ u = . Thus, the variational formulation of (2.1) is 3 ∂y ∂z a(u , v) = (f , v) ∀ v ∈ H01 (Ω), where

∫ a(u , v) ≡

3 ∑

(2.2)

aij ∂i u∂j v dxdydz

Ω i,j=1

∫

and (f , v) =

f v dxdydz. Ω

To discretize the problem (2.2), one proceeds as follows. The domain Ω is firstly partitioned into cubes of side h, and each of these is then subdivided into six tetrahedra (see Fig. 1). We denote by T h this partition.

Figure 1: A tetrahedral partition

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For this fully uniform mesh of tetrahedral elements, let S0h (Ω) ⊂ H01 (Ω) be the piecewise linear tetrahedral finite element space, and uI ∈ S0h (Ω) the Lagrange interpolant to the solution u of (2.2). Discretizing (2.2) using S0h as approximating space means finding uh ∈ S0h such that a(uh , v) = (f , v) for all v ∈ S0h . Here uh is a finite element approximation to u. Thus we have the Galerkin orthogonality relation a(u − uh , v) = 0 ∀ v ∈ S0h (Ω).

(2.3)

To derive the main result of this article, for every Z ∈ Ω, we need to introduce the discrete derivative Green’s function ∂Z,ℓ GhZ ∈ S0h (Ω) defined by a(v, ∂Z,ℓ GhZ ) = ∂ℓ v(Z) ∀ v ∈ S0h (Ω).

(2.4)

Here, for any direction ℓ ∈ R3 , |ℓ| = 1, ∂Z,ℓ GhZ and ∂ℓ v(Z) stand for the following onesided directional derivatives, respectively. ∂Z,ℓ GhZ =

GhZ+∆Z − GhZ v(Z + ∆Z) − v(Z) , ∂ℓ v(Z) = lim , ∆Z = |∆Z|ℓ. |∆Z| |∆Z| |∆Z|→0 |∆Z|→0 lim

Remark 1. Since ∆Z = |∆Z|ℓ, that is, ∆Z is of the same direction as ℓ. Thus, provided that the direction ℓ is given, the above limits exist. Hence, no matter what direction is given, the above definition has good meaning.

3

Gradients Recovered by SPR and Superconvergence

SPR is a gradient recovery method introduced by Zienkiewicz and Zhu. This method is now widely used in engineering practices for its robustness in a posterior error estimation and its efficiency in computer implementation. For v ∈ S0h (Ω), we denote by Rh the SPR-recovery operator and begin by defining the point values of Rh v at the element nodes. After the recovered gradient values of all nodes are obtained, we give a linear interpolation by using these values, namely SPR-recovery gradient Rh v. Obviously Rh v ∈ S0h (Ω). Let us firstly assume N is a interior node of the partition T h , and denote by ω the element patch around N containing 24 tetrahedra. Under the local coordinate system centered N , we let Qi be the barycenter of a tetrahedron ei ⊂ ω, i = 1, 2, · · · , 24. SPR uses the discrete least-squares fitting to seek linear vector function p ∈ (P1 (ω))3 , such that 24 ∑ [p(Qi ) − ∇v(Qi )]q(Qi ) = 0 ∀q ∈ P1 (ω),

(3.1)

i=1

where v ∈ S0h (Ω). The existence and uniqueness of the minimizer in (3.1) can be found in [14].

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We define Rh v(N ) = p(0). Then the following Lemma 3.1 and Lemma 3.2 hold. Lemma 3.1. Let ω be the element patch around an interior node N , and u ∈ W 3,∞ (ω). For uI ∈ S0h (Ω) the interpolant to u, we have |∇u(N ) − Rh uI (N )| ≤ Ch2 ∥u∥3,∞,ω . Lemma 3.2. The recovery operator Rh satisfies 1 ∑ ∇v(Qi ). 24 i=1 24

Rh v(N ) =

Lemma 3.3. For v ∈ S0h (Ω), we have the weak estimate |a(u − uI , v)| ≤ Ch2 ∥u∥3,∞,Ω |v|1,1,Ω . Lemma 3.4. For ∂Z,ℓ GhZ the discrete derivative Green’s function defined in (2.4), we have the following estimate ∂Z,ℓ GhZ ≤ C| ln h| 43 . 1,1 Remark 2. The proofs of Lemma 3.1 and Lemma 3.2 can be seen in [11], Lemma 3.3 in [13], and Lemma 3.4 in [15]. Theorem 3.1. For uI and uh , the linear interpolant and the linear tetrahedral finite element approximation to u, respectively. Thus we have the following supercloseness estimate 4

|uh − uI |1,∞,Ω ≤ Ch2 | ln h| 3 ∥u∥3,∞,Ω . Proof. For every Z ∈ Ω and any direction ℓ, from (2.3) and (2.4), ∂ℓ (uh − uI ) (Z) = a(uh − uI , ∂Z,ℓ GhZ ) = a(u − uI , ∂Z,ℓ GhZ ). Hence, using Lemma 3.3,

|∂ℓ (uh − uI ) (Z)| ≤ Ch2 ∥u∥3,∞,Ω ∂Z,ℓ GhZ 1,1,Ω ,

which combined with Lemma 3.4 completes the proof of Theorem 3.1. Theorem 3.2. For uI ∈ S0h (Ω) the linear interpolant to u, the solution of (2.2), and Rh the gradient recovered operator by SPR, we have the superconvergent estimate |∇u − Rh uI |0,∞,Ω ≤ Ch2 ∥u∥3,∞,Ω . Proof. Denote by F : eˆ → e an affine transformation. Obviously, there exists an element e ∈ T h , using the triangle inequality and the Sobolev Embedding Theorem [16], and considering Lemma 3.2, such that |∇u − Rh uI |0, ∞, Ω

= ≤ ≤ ≤ ≤

|∇u − Rh uI |0, ∞, e Ch−1 [|∇ˆ u − Rh uˆI |0, ∞, eˆ

] u|0, ∞, eˆ + |Rh uˆI |0, ∞, eˆ Ch−1 |∇ˆ [ ] u|0, ∞, χˆ + |uˆI |1, ∞, χˆ Ch−1 |∇ˆ Ch−1 ∥ˆ u∥3, ∞, χˆ ,

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where χ ˆ is a small patch of elements surrounding the tetrahedron, eˆ. Due to the fact that, for u ˆ quadratic over χ, ˆ ∇ˆ u − Rh uˆI = 0 in eˆ, so, from the Bramble-Hilbert Lemma [17], |∇u − Rh uI |0, ∞, Ω ≤ Ch−1 |ˆ u|3, ∞, χˆ ≤ Ch2 |u|3, ∞, Ω , which completes the proof of Theorem 3.2. Theorem 3.3. For uh ∈ S0h (Ω) the linear finite element approximation to u, the solution of (2.2), and Rh the gradient recovered operator by SPR, we have the superconvergent estimate 4

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

≤ |Rh (uh − uI )|0, ∞, Ω + |∇u − Rh uI |0, ∞, Ω ≤ |uh − uI |1, ∞, Ω + |∇u − Rh uI |0, ∞, Ω ,

which combined with Theorems 3.1 and 3.2 completes the proof of Theorem 3.3. Acknowledgments This work is supported by National Natural Science Foundation of China under Grant 11161039. References [1] I. Babu˘ska and T. Strouboulis,The finite element method and its reliability, Numerical Mathematics and Scientific Computation, Oxford Science Publications, 2001. [2] C. M. Chen, Construction theory of superconvergence of finite elements, Hunan Science and Technology Press, Changsha, China, 2001 (in Chinese). [3] C. M. Chen and Y. Q. Huang, High accuracy theory of finite element methods, Hunan Science and Technology Press, Changsha, China, 1995 (in Chinese). [4] Q. Lin and N. N. Yan,Construction and analysis of high efficient finite elements, Hebei University Press, Baoding, China, 1996 (in Chinese). [5] L. B. Wahlbin,Superconvergence in Galerkin finite element methods, Springer Verlag, Berlin, 1995. [6] Q. D. Zhu and Q. Lin,Superconvergence theory of the finite element methods, Hunan Science and Technology Press, Changsha, China, 1989 (in Chinese).

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[7] O. C. Zienkiewicz and J. Z. Zhu, A simple estimator and adaptive procedure for practical engineering analysis, International Journal for Numerical Methods in Engineering, vol. 24, pp. 337–357, 1987. [8] O. C. Zienkiewicz and J. Z. Zhu, The superconvergence patch recovery and a posteriori error estimates. Part 1: The recovery techniques, International Journal for Numerical Methods in Engineering, vol. 33, pp. 1331–1364, 1992. [9] O. C. Zienkiewicz and J. Z. Zhu, The superconvergence patch recovery and a posteriori error estimates. Part 2: Error estimates and adaptivity, International Journal for Numerical Methods in Engineering, vol. 33, pp. 1365–1382, 1992. [10] Z. M. Zhang and A. Naga, A new finite element gradient recovery method: Superconvergence property, SIAM Journal on Scientic Computing, vol. 26, pp. 1192–1213, 2005. [11] J. Chen and D. S. Wang, Three-dimensional finite element superconvergent gradient recovery on Par6 patterns, Numerical Mathematics: Theory, Methods and Applications, vol. 3, no. 2, pp. 178–194, 2010. [12] L. Chen, Superconvergence of tetrahedral linear finite elements, International Journal of Numerical Analysis and Modeling, vol. 3, no. 3, pp. 273–282, 2006. [13] G. Goodsell, Pointwise superconvergence of the gradient for the linear tetrahedral element, Numerical Methods for Partial Differential Equations, vol. 10, pp. 651–666, 1994. [14] B. Li and Z. M. Zhang, Analysis of a class of superconvergence patch recovery techniques for linear and bilinear Finite elements,Numerical Methods for Partial Differential Equations, vol. 15, pp. 151–167, 1999. [15] J. H. Liu and Q. D. Zhu, The estimate for the W 1,1 -seminorm of discrete derivative Green’s function in three dimensions, Journal of Hunan University of Arts and Sciences, vol. 16, pp. 1–3, 2004 (in Chinese). [16] R. A. Adams, Sobolev Spaces, Academic Press, New York, 1975. [17] J. H. Bramble and S. R. Hilbert, Estimation of Linear Functionals on Sobolev Spaces with Applications to Fourier Transforms and Spline Interpolation, SIAM Journal on Numerical Analysis, vol. 7, pp. 112, 1970.

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Hyers-Ulam stability of quadratic functional equations in paranormed spaces

Jinwoo Choi Mathematics Branch, Seoul Science High School, Seoul 110-530, Korea e-mail: [email protected]

Joon Hyuk Yang∗ Mathematics Branch, Seoul Science High School, Seoul 110-530, Korea e-mail: [email protected]

Choonkil Park Department of Mathematics, Research Institute for Natural Sciences, Hanyang University, Seoul 133-791, Korea e-mail: baak@@hanyang.ac.kr Abstract. Lin [18, 19] introduced and investigated the following quadratic functional equations ( n ) ( n ) n ∑ ∑ ∑ xi + cf f xi − (n + c − 1)xj (0.1) i=1

j=2

i=1



= (n + c − 1) f (x1 ) + c

Q

( n ∑ i=1

f (xi ) +

i=2

) di xi

n ∑

+

∑

di dj Q(xi − xj ) =

(n−1 ∑

n ∑ i l and all x ∈ Y . It follows from (2.3) that the sequence {v 2k f ( vxk )} is a Cauchy sequence for all x ∈ Y . Since X is complete, the sequence {v 2k f ( vxk )} converges. So the mapping R : Y → X can be defined as (x) R(x) := lim v 2k f k→∞ vk for all x ∈ Y . By (2.1), ( (x ( (x xn )) xn )) 1 1 P (DR(x1 , · · · , xn )) = lim P v 2k DR k , · · · , k ≤ lim v 2k P DR k , · · · , k k→∞ k→∞ v v v v ( n ) ∑n r ∑ xi θ ( i=1 ||xi ∥| ) ≤ lim v 2k θ || k ||r = lim =0 k→∞ k→∞ v v (r−2)k i=1 for all x1 , · · · , xn ∈ Y . So DR(x1 , · · · , xn ) = 0. So the mapping R : Y → X is quadratic. Moreover, letting l = 0 and passing the limit m → ∞ in (2.3), we get (2.2). So there exists a quadratic mapping R : Y → X satisfying (2.2). Now, let T : Y → X be another quadratic mapping satisfying (2.2). Then we have ( (x) ( x )) P (R(x) − T (x)) = P v 2q R q − v 2q T q ( ( v( x ) ( xv))) ( ( (x) ( x ))) 2q ≤ P v R q −f + P v 2q T −f q q v v v vq 2q v 2θ ∥x∥r · rq , ≤ r 2 v −v v which tends to zero as q → ∞ for all x ∈ Y . So we can conclude that R(x) = T (x) for all x ∈ Y . This proves the uniqueness of R. Thus the mapping R : Y → X is the unique quadratic mapping satisfying (2.2).  Theorem 2.2. Let r, θ be positive real numbers with r < 2. Let f : X → Y be a mapping such that ||Df (x1 , · · · , xn )|| ≤ θ 463

n ∑

P (xi )r

(2.4)

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J. Choi, J. H. Yang, C. Park for all x1 , · · · , xn ∈ X. Then there exists a unique quadratic mapping R : X → Y such that ||f (x) − R(x)|| ≤

θ P (x)r v2 − vr

(2.5)

for all x ∈ X. Proof. Putting x2 = x and x1 = x3 = x4 = · · · = xn = 0 in (2.4), we get ||f (vx) − v 2 f (x)|| ≤ θP (x)r and so



f (x) − 1 f (vx) ≤ θ 1 P (x)r

2 v v2

for all x ∈ X Hence

m−1

1 ( l )

∑ 1 v rj P (x)r

f v x − 1 f (v m x) ≤ θ 2

v 2l

2m v v v 2j

(2.6)

j=l

holds for all non-negative integers l and m with m > l and all x ∈ X. It follows from (2.6) that the sequence { v12k f (v k x)} is a Cauchy sequence for all x ∈ X. Since Y is complete, the sequence { v12k f (v k x)} converges. So the mapping R : X → Y can be defined as R(x) := lim

k→∞

1 ( k ) f v x v 2k

for all x ∈ X. By (2.4),

1

1 k k

||DR(x1 , · · · , xn )|| = lim 2k DR(v x1 , · · · , v xn ) lim 2k ∥DR(v k x1 , · · · , v k xn )∥

≤ k→∞ k→∞ v v n v rk ∑ θ P (xi )r = 0 k→∞ v 2k i=1

≤ lim

for all x1 , · · · , xn ∈ X. So DR(x1 , · · · , xn ) = 0. So the mapping R : X → Y is quadratic. Moreover, letting l = 0 and passing the limit m → ∞ in (2.6), we get (2.5). So there exists a quadratic mapping R : X → Y satisfying (2.5). Now, let T : X → Y be another quadratic mapping satisfying (2.5). Then we have

1

1 q q

||R(x) − T (x)|| = 2q R (v x) − 2q T (v x) v v



1

1

q q q q

≤ 2q (R (v x) − f (v x)) + 2q (T (v x) − f (v x))

v v rq 2θ v ≤ P (x)r · 2q , v2 − vr v which tends to zero as q → ∞ for all x ∈ X. So we can conclude that R(x) = T (x) for all x ∈ X. This proves the uniqueness of R. Thus the mapping R : X → Y is the unique quadratic mapping satisfying (2.5).  464

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Stability of quadratic functional equations 3. Hyers-Ulam stability of the functional equation (0.2) in paranormed spaces For a given mapping f , we define ( n ) ∑ DQ(x1 , · · · , xn ) := Q d i xi + i=1

∑

di dj Q (xi − xj ) −

n ∑ i=1

1≤i 2. Let Q : Y → X be a mapping such that n ∑ P (DQ(x1 , · · · , xn )) ≤ θ ∥xi ∥r (3.1) i=1

for all x1 , · · · , xn ∈ Y . Then there exists a unique quadratic mapping R : Y → X such that nθ P (Q(x) − R(x)) ≤ r ∥x∥r d − d2 for all x ∈ Y . Proof. Putting x1 = · · · = xn =

x d

(3.2)

in (3.1), we get ( ( x )) θ(n∥x∥r ) P Q(x) − d2 Q ≤ d dr

for all x ∈ X Hence ( (x) ( x )) m−1 ∑ θ(n∥x∥r ) P d2l Q l − d2m Q m ≤ d d d(r−2)j+r

(3.3)

j=l

holds for all non-negative integers l and m with m > l and all x ∈ Y . It follows from (3.3) that the sequence {d2k Q( dxk )} is a Cauchy sequence for all x ∈ Y . Since X is complete, the sequence {d2k Q( dxk )} converges. So the mapping R : Y → X can be defined as (x) R(x) := lim d2k Q k k→∞ d for all x ∈ Y . By (3.1), ( (x ( (x xn )) xn )) 1 1 P (DR(x1 , · · · , xn )) = lim P d2k DR k , · · · , k ≤ lim d2k P DR k , · · · , k k→∞ k→∞ d d d d ) ( n ∑ n

r ∑ xi r θ ( ||x ∥| ) i i=1 ≤ lim d2k θ = lim =0

k (r−2)k k→∞ k→∞ d d i=1 for all x1 , · · · , xn ∈ Y . So DR(x1 , · · · , xn ) = 0. So the mapping R : Y → X is quadratic. Moreover, letting l = 0 and passing the limit m → ∞ in (3.3), we get (3.2). So there exists a quadratic mapping R : Y → X satisfying (3.2). Now, let T : Y → X be another quadratic mapping satisfying (3.2). Then we have ( x )) ( (x) P (R(x) − T (x)) = P d2q R q − d2q T q ( ( d( x ) ( dx ))) ( ( (x) ( x ))) 2q ≤ P d2q R q − Qf + P d T − Q d dq dq dq 2q 2nθ d ≤ ∥x∥r · rq , dr − d2 d 465

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J. Choi, J. H. Yang, C. Park which tends to zero as q → ∞ for all x ∈ Y . So we can conclude that R(x) = T (x) for all x ∈ Y . This proves the uniqueness of R. Thus the mapping R : Y → X is the unique quadratic mapping satisfying (3.2).  Theorem 3.2. Let r, θ be positive real numbers with r < 2. Let Q : X → Y be a mapping such that n ∑ ||DQ(x1 , · · · , xn )|| ≤ θ P (xi )r (3.4) i=1

for all x1 , · · · , xn ∈ X. Then there exists a unique quadratic mapping R : X → Y such that nθ ||Q(x) − R(x)|| ≤ 2 P (x)r d − dr for all x ∈ X. Proof. Putting x1 = · · · = xn = x in (3.4), we get ||Q(dx) − d2 Q(x)|| ≤ nθP (x)r and so for all x ∈ X Hence



Q(x) − 1 Q(dx) ≤ nθ P (x)r

d2 d2

m−1 ∑ drj

1 ( l )

Q d x − 1 Q (dm x) ≤ nθ P (x)r

d2l

2m 2 d d d2j

(3.5)

j=l

holds for all non-negative integers l and m with m > l and all x ∈ X. It follows from (3.5) that the sequence { d12k Q(dk x)} is a Cauchy sequence for all x ∈ X. Since Y is complete, the sequence { d12k Q(dk x)} converges. So the mapping R : X → Y can be defined as ( ) 1 R(x) := lim 2k Q dk x k→∞ d for all x ∈ X.  The rest of the proof is similar to the proofs of Theorems 2.2 and 3.1. Acknowledgments J. Choi and J. H. Yang were supported by R & E Program in 2012, and C. Park was supported by Basic Science Research Program through the National Research Foundation of Korea funded by the Ministry of Education, Science and Technology (NRF-2012R1A1A2004299). References [1] J. Aczel and J. Dhombres, Functional Equations in Several Variables, Cambridge Univ. Press, Cambridge, 1989. [2] T. Aoki, On the stability of the linear transformation in Banach spaces, J. Math. Soc. Japan 2 (1950), 64–66. [3] P.W. Cholewa, Remarks on the stability of functional equations, Aequationes Math. 27 (1984), 76–86. [4] S. Czerwik, On the stability of the quadratic mapping in normed spaces, Abh. Math. Sem. Univ. Hamburg 62 (1992), 59–64. [5] P. Czerwik, Functional Equations and Inequalities in Several Variables, World Scientific Publishing Company, New Jersey, Hong Kong, Singapore and London, 2002. 466

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Stability of quadratic functional equations [6] M. Eshaghi Gordji and M.B. Savadkouhi, Stability of a mixed type cubic-quartic functional equation in non-Archimedean spaces, Appl. Math. Letters 23 (2010), 1198–1202. [7] H. Fast, Sur la convergence statistique, Colloq. Math. 2 (1951), 241–244. [8] J.A. Fridy, On statistical convergence, Analysis 5 (1985), 301–313. [9] Z. Gajda, On stability of additive mappings, Internat. J. Math. Math. Sci. 14 (1991), 431–434. [10] P. Gˇavruta, A generalization of the Hyers-Ulam-Rassias stability of approximately additive mappings, J. Math. Anal. Appl. 184 (1994), 431–436. [11] D.H. Hyers, On the stability of the linear functional equation, Proc. Natl. Acad. Sci. U.S.A. 27 (1941), 222–224. [12] D.H. Hyers, G. Isac and Th.M. Rassias, Stability of Functional Equations in Several Variables, Birkh¨ auser, Basel, 1998. [13] G. Isac and Th.M. Rassias, On the Hyers-Ulam stability of ψ-additive mappings, J. Approx. Theory 72 (1993), 131–137. [14] K. Jun and Y. Lee, A generalization of the Hyers-Ulam-Rassias stability of the Pexiderized quadratic equations, J. Math. Anal. Appl. 297 (2004), 70–86. [15] S. Jung, Hyers-Ulam-Rassias Stability of Functional Equations in Mathematical Analysis, Hadronic Press lnc., Palm Harbor, Florida, 2001. [16] S. Karakus, Statistical convergence on probabilistic normed spaces, Math. Commun. 12 (2007), 11–23. [17] E. Kolk, The statistical convergence in Banach spaces, Tartu Ul. Toime. 928 (1991), 41–52. [18] C.S. Lin, A-bilinear forms and generalized A-quadratic forms on unitary left A-modules, Bull. Austral. Math. Soc. 39 (1989), 49–53. [19] C.S. Lin, Sesquilinear and quadratic forms on modules over ∗-algebra, Publ. de L’Inst. Math. 51 (1992), 81–86. [20] M. Mursaleen, λ-statistical convergence, Math. Slovaca 50 (2000), 111–115. [21] M. Mursaleen and S.A. Mohiuddine, On lacunary statistical convergence with respect to the intuitionistic fuzzy normed space, J. Computat. Anal. Math. 233 (2009), 142–149. [22] C. Park, Homomorphisms between Poisson JC ∗ -algebras, Bull. Braz. Math. Soc. 36 (2005), 79–97. [23] Th.M. Rassias, On the stability of the linear mapping in Banach spaces, Proc. Amer. Math. Soc. 72 (1978), 297–300. [24] Th.M. Rassias, Problem 16; 2, Report of the 27th International Symp. on Functional Equations, Aequationes Math. 39 (1990), 292–293; 309. [25] Th.M. Rassias (ed.), Functional Equations and Inequalities, Kluwer Academic, Dordrecht, 2000. [26] Th.M. Rassias, On the stability of functional equations in Banach spaces, J. Math. Anal. Appl. 251 (2000), 264–284. [27] Th.M. Rassias, On the stability of functional equations and a problem of Ulam, Acta Math. Appl. 62 (2000), 23–130. ˇ [28] Th.M. Rassias and P. Semrl, On the behaviour of mappings which do not satisfy Hyers-Ulam stability, Proc. Amer. Math. Soc. 114 (1992), 989–993. ˇ at, On the statistically convergent sequences of real numbers, Math. Slovaca 30 (1980), 139– [29] T. Sal´ 150. [30] F. Skof, Propriet` a locali e approssimazione di operatori, Rend. Sem. Mat. Fis. Milano 53 (1983), 113–129. [31] H. Steinhaus, Sur la convergence ordinaire et la convergence asymptotique, Colloq. Math. 2 (1951), 73–34. [32] S.M. Ulam, A Collection of the Mathematical Problems, Interscience Publ. New York, 1960. [33] A. Wilansky, Modern Methods in Topological Vector Space, McGraw-Hill International Book Co., New York, 1978. 467

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Union soft sets applied to commutative BCI-ideals Young Bae Jun1 , Seok Zun Song2 and Sun Shin Ahn3,∗ 1

Department of Mathematics Education (and RINS), Gyeongsang National University, Chinju, 660-701, Korea 2 Department of Mathematics, Jeju National University, Jeju 690-756, Korea 3

Department of Mathematics Education, Dongguk University, Seoul 100-715, Korea

Abstract. The aim of this article is to lay a foundation for providing a soft algebraic tool in considering many problems that contain uncertainties. In order to provide these soft algebraic structures, the notion of union soft commutative BCI-ideals is introduced, and related properties are investigated. Relations between a union soft commutative BCI-ideal and a (closed) union soft BCI-ideal are displayed. Conditions for a union soft BCIideal to be a union soft commutative BCI-ideal are established. Characterizations of a union soft commutative BCI-ideal are considered, and a new union soft commutative BCI-ideal from an old one is constructed.

1. Introduction The real world is inherently uncertain, imprecise and vague. Various problems in system identification involve characteristics which are essentially non-probabilistic in nature [26]. In response to this situation Zadeh [27] introduced fuzzy set theory as an alternative to probability theory. Uncertainty is an attribute of information. In order to suggest a more general framework, the approach to uncertainty is outlined by Zadeh [28]. To solve complicated problem in economics, engineering, and environment, we can’t successfully use classical methods because of various uncertainties typical for those problems. There are three theories: theory of probability, theory of fuzzy sets, and the interval mathematics which we can consider as mathematical tools for dealing with uncertainties. But all these theories have their own difficulties. Uncertainties can’t be handled using traditional mathematical tools but may be dealt with using a wide range of existing theories such as probability theory, theory of (intuitionistic) fuzzy sets, theory of vague sets, theory of interval mathematics, and theory of rough sets. However, all of these theories have their own difficulties which are pointed out in [23]. Maji et al. [19] and Molodtsov [23] suggested that one reason for these difficulties may be due to the inadequacy of the parametrization tool of the theory. To overcome these difficulties, Molodtsov [23] introduced the concept of soft set as a new mathematical tool for dealing with uncertainties that is free from the difficulties that have troubled the usual theoretical approaches. Molodtsov pointed out several directions for the applications of soft sets. Worldwide, there has been a rapid growth in interest in soft set theory 0

2010 Mathematics Subject Classification: 06F35; 03G25; 06D72. Keywords: Exclusive set, Union soft algebra, (Closed) union soft BCI-ideal, Union soft c-BCI-ideal. ∗ The corresponding author. Tel: +82 2 2260 3410, Fax: +82 2 2266 3409 0 E-mail: [email protected] (Y. B. Jun); [email protected] (S. Z. Song); [email protected] (S. S. Ahn) 0

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and its applications in recent years. Evidence of this can be found in the increasing number of high-quality articles on soft sets and related topics that have been published in a variety of international journals, symposia, workshops, and international conferences in recent years. Maji et al. [19] described the application of soft set theory to a decision making problem. Maji et al. [18] also studied several operations on the theory of soft sets. Akta¸s and C ¸ a˘gman [2] studied the basic concepts of soft set theory, and compared soft sets to fuzzy and rough sets, providing examples to clarify their differences. They also discussed the notion of soft groups. Jun and Park [17] studied applications of soft sets in ideal theory of BCK/BCI-algebras. Jun et al. [14, 15] introduced the notion of intersectional soft sets, and considered its applications to BCK/BCIalgebras. Also, Jun [10] discussed the union soft sets with applications in BCK/BCI-algebras. We refer the reader to the papers [1, 3, 5, 6, 7, 9, 11, 12, 13, 16, 24, 25, 29] for further information regarding algebraic structures/properties of soft set theory. In this paper, we discuss applications of the union soft sets in a commutative BCI-ideals of BCI-algebras. We introduce the notion of union soft commutative BCI-ideals, and investigated related properties. We consider relations between a union soft commutative BCI-ideal and a (closed) union soft BCI-ideal. We provide conditions for a union soft BCI-ideal to be a union soft commutative BCI-ideal, and establish characterizations of a union soft commutative BCIideal. We construct a new union soft commutative BCI-ideal from an old one. 2. Preliminaries A BCK/BCI-algebra is an important class of logical algebras introduced by K. Is´eki and was extensively investigated by several researchers. An algebra (X; ∗, 0) of type (2, 0) is called a BCI-algebra if it satisfies the following conditions: (I) (II) (III) (IV)

(∀x, y, z ∈ X) (((x ∗ y) ∗ (x ∗ z)) ∗ (z ∗ y) = 0), (∀x, y ∈ X) ((x ∗ (x ∗ y)) ∗ y = 0), (∀x ∈ X) (x ∗ x = 0), (∀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 axioms: (a1) (a2) (a3) (a4)

(∀x ∈ X) (x ∗ 0 = x), (∀x, y, z ∈ X) (x ≤ y ⇒ x ∗ z ≤ y ∗ z, z ∗ y ≤ z ∗ x), (∀x, y, z ∈ X) ((x ∗ y) ∗ z = (x ∗ z) ∗ y), (∀x, y, z ∈ X) ((x ∗ z) ∗ (y ∗ z) ≤ x ∗ y)

where x ≤ y if and only if x ∗ y = 0. In a BCI-algebra X, the following hold: (b1) (∀x, y ∈ X) (x ∗ (x ∗ (x ∗ y)) = x ∗ y) , (b2) (∀x, y ∈ X) (0 ∗ (x ∗ y) = (0 ∗ x) ∗ (0 ∗ y)) . 469

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A BCI-algebra X is said to be commutative (see [22]) if (∀x, y ∈ X) (x ≤ y ⇒ x = y ∗ (y ∗ x)) .

(2.1)

Proposition 2.1. A BCI-algebra X is commutative if and only if it satisfies: (∀x, y ∈ X) (x ∗ (x ∗ y) = y ∗ (y ∗ (x ∗ (x ∗ y)))) .

(2.2)

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 I of a BCI-algebra X is called a BCI-ideal of X if it satisfies: 0 ∈ I,

(2.3)

(∀x ∈ X) (∀y ∈ I) (x ∗ y ∈ I ⇒ x ∈ I) .

(2.4)

A BCI-ideal I of a BCI-algebra X satisfies: (∀x ∈ X)(∀y ∈ I) (x ≤ y ⇒ x ∈ I)

(2.5)

A BCI-ideal I of a BCI-algebra X is said to be closed if it satisfies: (∀x ∈ X) (x ∈ I ⇒ 0 ∗ x ∈ I) A subset I of a BCI-algebra X is called a commutative BCI-ideal (briefly, c-BCI-ideal) of X (see [20]) if it satisfies (2.3) and (x ∗ y) ∗ z ∈ I, z ∈ I ⇒ x ∗ ((y ∗ (y ∗ x)) ∗ (0 ∗ (0 ∗ (x ∗ y)))) ∈ I

(2.6)

for all x, y, z ∈ X. Proposition 2.2 ([20]). A BCI-ideal I of a BCI-algebra X is commutative if and only if x∗y ∈ I implies x ∗ ((y ∗ (y ∗ x)) ∗ (0 ∗ (0 ∗ (x ∗ y)))) ∈ I. Proposition 2.3 ([20]). Let I be a closed BCI-ideal of a BCI-algebra X. Then I is commutative if and only if it satisfies: (∀x, y ∈ X) (x ∗ y ∈ I ⇒ x ∗ (y ∗ (y ∗ x)) ∈ I) Observe that every c-BCI-ideal is a BCI-ideal, but the converse is not true (see [20]). We refer the reader to the books [8, 21] for further information regarding BCK/BCI-algebras. A soft set theory is introduced by Molodtsov [23], and C ¸ aˇgman et al. [4] provided new definitions and various results on soft set theory. In what follows, let U be an initial universe set and E be a set of parameters. We say that the pair (U, E) is a soft universe. Let P(U ) denotes the power set of U and A, B, C, · · · ⊆ E. Definition 2.4 ([4, 23]). A soft set FA over U is defined to be the set of ordered pairs FA := {(x, fA (x)) : x ∈ E, fA (x) ∈ P(U )} , where fA : E → P(U ) such that fA (x) = ∅ if x ∈ / A. 470

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The function fA is called the approximate function of the soft set FA . The subscript A in the notation fA indicates that fA is the approximate function of FA . In what follows, denote by S(U ) the set of all soft sets over U. Let FA ∈ S(U ) and let τ ⊆ U. Then the τ -exclusive set of FA is defined to be the set e(FA ; τ ) := {x ∈ A | fA (x) ⊆ τ } . Obviously, we have the following properties: (1) e(FA ; U ) = A. (2) fA (x) = ∩ {τ ⊆ U | x ∈ e(FA ; τ )} . (3) (∀τ1 , τ2 ⊆ U ) (τ1 ⊆ τ2 ⇒ e(FA ; τ1 ) ⊆ e(FA ; τ2 )) . 3. Union soft c-BCI-ideals Definition 3.1 ([10]). Let (U, E) = (U, X) where X is a BCI-algebra. Given a subalgebra A of E, let FA ∈ S(U ). Then FA is called a union soft BCI-ideal (briefly, U-soft BCI-ideal) over U if the approximate function fA of FA satisfies: (∀x ∈ A) (fA (0) ⊆ fA (x)) ,

(3.1)

(∀x, y ∈ A) (fA (x) ⊆ fA (x ∗ y) ∪ fA (y)) .

(3.2)

Definition 3.2. Let (U, E) = (U, X) where X is a BCI-algebra. Given a subalgebra A of E, let FA ∈ S(U ). Then FA is called a union soft commutative BCI-ideal (briefly, U-soft c-BCI-ideal) over U if the approximate function fA of FA satisfies (3.1) and fA (x ∗ ((y ∗ (y ∗ x)) ∗ (0 ∗ (0 ∗ (x ∗ y))))) ⊆ fA ((x ∗ y) ∗ z) ∪ fA (z)

(3.3)

for all x, y, z ∈ A. Example 3.3. Let (U, E) = (U, X) where X = {0, a, 1, 2, 3} is a BCI-algebra with the following Cayley table: ∗ 0 a 1 2 3 0 0 0 3 2 1 a a 0 3 2 1 1 1 1 0 3 2 2 2 2 1 0 3 3 3 3 2 1 0 Let τ1 , τ2 and τ3 be subsets of U such that τ1 ( τ2 ( τ3 . Define a soft set FE over U as follows: FE = {(0, τ1 ), (a, τ2 ), (1, τ3 ), (2, τ3 ), (3, τ3 )} . Routine calculations show that FE is a U-soft c-BCI-ideal over U. Theorem 3.4. Let (U, E) = (U, X) where X is a BCI-algebra. Then every U-soft c-BCI-ideal is a U-soft BCI-ideal. 471

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Proof. Let FA be a U-soft c-BCI-ideal over U where A is a subalgebra of E. Taking y = 0 in (3.3) and using (a1) and (III) imply that fA (x) = fA (x ∗ 0) = fA (x ∗ ((0 ∗ (0 ∗ x)) ∗ (0 ∗ (0 ∗ (x ∗ 0))))) ⊆ fA ((x ∗ 0) ∗ z) ∪ fA (z) = fA (x ∗ z) ∪ fA (z) for all x, z ∈ A. Therefore FA is a U-soft BCI-ideal over U.



The following example shows that the converse of Theorem 3.4 is not true. Example 3.5. Let (U, E) = (U, X) where X Cayley table: ∗ 0 0 0 1 1 2 2 3 3 4 4

= {0, 1, 2, 3, 4} is a BCI-algebra with the following 1 0 0 2 3 4

2 0 1 0 3 4

3 0 0 0 0 3

4 0 0 0 0 0

Let τ1 , τ2 and τ3 be subsets of U such that τ1 ( τ2 ( τ3 . Define a soft set FE over U as follows: FE = {(0, τ1 ), (1, τ2 ), (2, τ3 ), (3, τ3 ), (4, τ3 )} . Routine calculations show that FE is a U-soft BCI-ideal over U. But it is not a U-soft c-BCIideal over U since fE (2 ∗ ((3 ∗ (3 ∗ 2)) ∗ (0 ∗ (0 ∗ (2 ∗ 3))))) = τ3 * τ1 = fE ((2 ∗ 3) ∗ 0) ∪ fE (0). We provide conditions for a U-soft BCI-ideal to be a U-soft c-BCI-ideal. Theorem 3.6. Let (U, E) = (U, X) where X is a BCI-algebra. For a subalgebra A of E, let FA ∈ S(U ). Then the following are equivalent: (1) FA is a U-soft c-BCI-ideal over U. (2) FA is a U-soft BCI-ideal over U and its approximate function fA satisfies: (∀x, y ∈ A) (fA (x ∗ ((y ∗ (y ∗ x)) ∗ (0 ∗ (0 ∗ (x ∗ y))))) ⊆ fA (x ∗ y)) .

(3.4)

Proof. Assume that FA is a U-soft c-BCI-ideal over U. Then FA is a U-soft BCI-ideal over U (see Theorem 3.4). If we take z = 0 in (3.3) and use (a1) and (3.1), then we have (3.4). Conversely, let FA be a U-soft BCI-ideal over U such that its approximate function fA satisfies (3.4). Then fA (x ∗ y) ⊆ fA ((x ∗ y) ∗ z) ∪ fA (z) for all x, y, z ∈ A by (3.2), which implies from (3.4) that fA (x ∗ ((y ∗ (y ∗ x)) ∗ (0 ∗ (0 ∗ (x ∗ y))))) ⊆ fA ((x ∗ y) ∗ z) ∪ fA (z) for all x, y, z ∈ A. Therefore FA is a U-soft c-BCI-ideal over U. 472

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Definition 3.7 ([10]). Let (U, E) = (U, X) where X is a BCI-algebra. Given a subalgebra A of E, let FA ∈ S(U ). A U-soft BCI-ideal FA over U is said to be closed if the approximate function fA of FA satisfies: (∀x ∈ A) (fA (0 ∗ x) ⊆ fA (x)) .

(3.5)

Lemma 3.8 ([10]). Let (U, E) = (U, X) where X is a BCI-algebra. Given a subalgebra A of E, let FA ∈ S(U ). (1) If FA is a U-soft BCI-ideal over U, then the approximate function fA satisfies the following condition: (∀x, y, z ∈ A) (x ∗ y ≤ z ⇒ fA (x) ⊆ fA (y) ∪ fA (z)) .

(3.6)

(2) If the approximate function fA of FA satisfies (3.1) and (3.6), then FA is a U-soft BCIideal over U. Theorem 3.9. Let (U, E) = (U, X) where X is a BCI-algebra. For a subalgebra A of E, let FA be a closed U-soft BCI-ideal over U. Then the following are equivalent: (1) FA is a U-soft c-BCI-ideal over U. (2) The approximate function fA of FA satisfies: (∀x, y ∈ A) (fA (x ∗ (y ∗ (y ∗ x))) ⊆ fA (x ∗ y)) .

(3.7)

Proof. Assume that FA is a U-soft c-BCI-ideal over U. Note that (x ∗ (y ∗ (y ∗ x))) ∗ (x ∗ ((y ∗ (y ∗ x)) ∗ (0 ∗ (0 ∗ (x ∗ y))))) ≤ ((y ∗ (y ∗ x)) ∗ (0 ∗ (0 ∗ (x ∗ y)))) ∗ (y ∗ (y ∗ x)) = ((y ∗ (y ∗ x)) ∗ (y ∗ (y ∗ x))) ∗ (0 ∗ (0 ∗ (x ∗ y))) = 0 ∗ (0 ∗ (0 ∗ (x ∗ y))) = 0 ∗ (x ∗ y) for all x, y ∈ A. Using Lemma 3.8(1), (3.4) and (3.5), we have fA (x ∗ (y ∗ (y ∗ x))) ⊆ fA (x ∗ ((y ∗ (y ∗ x)) ∗ (0 ∗ (0 ∗ (x ∗ y))))) ∪ fA (0 ∗ (x ∗ y)) ⊆ fA (x ∗ y) ∪ fA (0 ∗ (x ∗ y)) = fA (x ∗ y) for all x, y ∈ A. Now suppose that the approximate function fA of FA satisfies (3.7). Since (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)),

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it follows from Lemma 3.8(1), (3.5) and (3.7) that fA (x ∗ ((y ∗ (y ∗ x)) ∗ (0 ∗ (0 ∗ (x ∗ y))))) ⊆ fA (x ∗ (y ∗ (y ∗ x))) ∪ fA (0 ∗ (0 ∗ (x ∗ y))) ⊆ fA (x ∗ y) ∪ fA (0 ∗ (0 ∗ (x ∗ y))) = fA (x ∗ y) for all x, y ∈ A. By Theorem 3.6, FA is a U-soft c-BCI-ideal over U.



Theorem 3.10. Let (U, E) = (U, X) where X is a commutative BCI-algebra. Then every closed U-soft BCI-ideal is a U-soft c-BCI-ideal. Proof. Let FA be a closed U-soft BCI-ideal over U where A is a subalgebra of E. Using (a3), (b1), (I), (III) and Proposition 2.1, we have (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))) = (y ∗ x) ∗ (y ∗ (x ∗ (x ∗ y))) ≤ (x ∗ (x ∗ y)) ∗ x = 0 ∗ (x ∗ y) It follows from Lemma 3.8(1) and (3.5) that fA (x ∗ (y ∗ (y ∗ x))) ⊆ fA (x ∗ y) ∪ fA (0 ∗ (x ∗ y)) = fA (x ∗ y), for all x, y ∈ A. Therefore, by Theorem 3.9, FA is a U-soft c-BCI-ideal over U.



Using the notion of τ -exclusive sets, we consider a characterization of a U-soft c-BCI-ideal. Lemma 3.11 ([10]). Let (U, E) = (U, X) where X is a BCI-algebra, Given a subalgebra A of E, let FA ∈ S(U ). Then the following are equivalent. (1) FA is a U-soft BCI-ideal over U. (2) The nonempty τ -exclusive set of FA is a BCI-ideal of A for any τ ⊆ U. Theorem 3.12. Let (U, E) = (U, X) where X is a BCI-algebra, Given a subalgebra A of E, let FA ∈ S(U ). Then the following are equivalent. (1) FA is a U-soft c-BCI-ideal over U. (2) The nonempty τ -exclusive set of FA is a c-BCI-ideal of A for any τ ⊆ U. Proof. Assume that FA is a U-soft c-BCI-ideal over U. Then FA is a U-soft BCI-ideal over U by Theorem 3.4. Hence e(FA ; τ ) is a BCI-ideal of A for all τ ⊆ U by Lemma 3.11. Let τ ⊆ U and x, y ∈ A be such that x ∗ y ∈ e(FA ; τ ). Then fA (x ∗ y) ⊆ τ, and so fA (x ∗ ((y ∗ (y ∗ x)) ∗ (0 ∗ (0 ∗ (x ∗ y))))) ⊆ fA (x ∗ y) ⊆ τ by Theorem 3.6. Thus x ∗ ((y ∗ (y ∗ x)) ∗ (0 ∗ (0 ∗ (x ∗ y)))) ∈ e(FA ; τ ). 474

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Young Bae Jun, Seok Zun Song and Sun Shin Ahn

It follows from Proposition 2.2 that e(FA ; τ ) is a c-BCI-ideal of A. Conversely, suppose that the nonempty τ -exclusive set of FA is a c-BCI-ideal of A for any τ ⊆ U. Then e(FA ; τ ) is a BCI-ideal of A for all τ ⊆ U. Hence FA is a U-soft BCI-ideal over U by Lemma 3.11. Let x, y ∈ A be such that fA (x ∗ y) = τ. Then x ∗ y ∈ e(FA ; τ ), and so x ∗ ((y ∗ (y ∗ x)) ∗ (0 ∗ (0 ∗ (x ∗ y)))) ∈ e(FA ; τ ) by Proposition 2.2. Hence fA (x ∗ ((y ∗ (y ∗ x)) ∗ (0 ∗ (0 ∗ (x ∗ y))))) ⊆ τ = fA (x ∗ y). It follows from Theorem 3.6 that FA is a U-soft c-BCI-ideal over U.



The c-BCI-ideals e(FA ; τ ) in Theorem 3.12 are called the exclusive c-BCI-ideals of FA . Theorem 3.13. Let (U, E) = (U, X) where X is a BCI-algebra. Let FE , GE ∈ S(U ) such that (i) (∀x ∈ E) (fE (x) ⊆ gE (x)) , (ii) FE and GE are U-soft BCI-ideals over U. If FE is closed and GE is a U-soft c-BCI-ideal over U, then FE is also a U-soft c-BCI-ideal over U. Proof. Assume that FE is closed and GE is a U-soft c-BCI-ideal over U. Let τ be a subset of U such that e(FE ; τ ) ̸= ∅ ̸= e(GE ; τ ). Then e(FE ; τ ) and e(GE ; τ ) are BCI-ideals of E and obviously e(FE ; τ ) ⊇ e(GE ; τ ). Let x ∈ e(FE ; τ ). Then fE (x) ⊆ τ, and so fE (0 ∗ x) ⊆ fE (x) ⊆ τ since FE is closed. Thus 0 ∗ x ∈ e(FE ; τ ), and thus e(FE ; τ ) is a closed BCI-ideal of E. Since GE is a U-soft c-BCI-ideal over U, it follows from Theorem 3.12 that e(GE ; τ ) is a c-BCI-ideal of E. Let x, y ∈ E be such that x ∗ y ∈ e(FE ; τ ). Then 0 ∗ (x ∗ y) ∈ e(FE ; τ ). Since (x ∗ (x ∗ y)) ∗ y = 0 ∈ e(GE ; τ ), it follows from Proposition 2.2 that (x ∗ (x ∗ y)) ∗ (y ∗ (y ∗ (x ∗ (x ∗ y)))) = (x ∗ (x ∗ y)) ∗ ((y ∗ (y ∗ (x ∗ (x ∗ y)))) ∗ (0 ∗ (0 ∗ ((x ∗ (x ∗ y)) ∗ y)))) ∈ e(GE ; τ ) ⊆ e(FE ; τ ) so from (a3) that (x ∗ (y ∗ (y ∗ (x ∗ (x ∗ y))))) ∗ (x ∗ y) ∈ e(FE ; τ ). Hence x ∗ (y ∗ (y ∗ (x ∗ (x ∗ y)))) ∈ e(FE ; τ ) by (2.4). Note that (x ∗ (y ∗ (y ∗ x))) ∗ (x ∗ (y ∗ (y ∗ (x ∗ (x ∗ y))))) ≤ (y ∗ (y ∗ (x ∗ (x ∗ y)))) ∗ (y ∗ (y ∗ x)) ≤ (y ∗ x) ∗ (y ∗ (x ∗ (x ∗ y))) ≤ (x ∗ (x ∗ y)) ∗ x = 0 ∗ (x ∗ y) ∈ e(FE ; τ ). Using (2.5) and (2.4), we have x ∗ (y ∗ (y ∗ x)) ∈ e(FE ; τ ). Hence e(FE ; τ ) is a c-BCI-ideal of E by Proposition 2.3. Therefore FE is a U-soft c-BCI-ideal over U by Theorem 3.12.  475

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Union soft sets applied to commutative BCI-ideals

Theorem 3.14. Let (U, E) = (U, X) and FA ∈ S(U ) where X is a BCI-algebra and A is a subalgebra of E. For a subset τ of U, define a soft set FA∗ over U by { fA (x) if x ∈ e(FA ; τ ), ∗ fA : E → P(U ), x 7→ U otherwise. If FA is a U-soft c-BCI-ideal over U, then so is FA∗ . Proof. If FA is a U-soft c-BCI-ideal over U, then e(FA ; τ ) is a c-BCI-ideal of A for any τ ⊆ U. Hence 0 ∈ e(FA ; τ ), and so fA∗ (0) = fA (0) ⊆ fA (x) ⊆ fA∗ (x) for all x ∈ A. Let x, y, z ∈ A. If (x ∗ y) ∗ z ∈ e(FA ; τ ) and z ∈ e(FA ; τ ), then x ∗ ((y ∗ (y ∗ x)) ∗ (0 ∗ (0 ∗ (x ∗ y)))) ∈ e(FA ; τ ) and so fA∗ (x ∗ ((y ∗ (y ∗ x)) ∗ (0 ∗ (0 ∗ (x ∗ y))))) = fA (x ∗ ((y ∗ (y ∗ x)) ∗ (0 ∗ (0 ∗ (x ∗ y))))) ⊆ fA ((x ∗ y) ∗ z) ∪ fA (z) = fA∗ ((x ∗ y) ∗ z) ∪ fA∗ (z). If (x ∗ y) ∗ z ∈ / e(FA ; τ ) or z ∈ / e(FA ; τ ), then fA∗ ((x ∗ y) ∗ z) = U or fA∗ (z) = U. Hence fA∗ (x ∗ ((y ∗ (y ∗ x)) ∗ (0 ∗ (0 ∗ (x ∗ y))))) ⊆ U = fA∗ ((x ∗ y) ∗ z) ∪ fA∗ (z). This shows that FA∗ is a U-soft c-BCI-ideal over U.



Theorem 3.15. Let (U, E) = (U, X) where X is a BCI-algebra. Then any c-BCI-ideal of E can be realized as an exclusive c-BCI-ideal of some U-soft c-BCI-ideal over U. Proof. Let A be a c-BCI-ideal of E. For any subset τ ( U, let FA be a soft set over U defined by { τ if x ∈ A, fA : E → P(U ), x 7→ U if x ∈ / A. Obviously, fA (0) ⊆ fA (x) for all x ∈ E. For any x, y, z ∈ E, if (x ∗ y) ∗ z ∈ A and z ∈ A then x ∗ ((y ∗ (y ∗ x)) ∗ (0 ∗ (0 ∗ (x ∗ y)))) ∈ A. Hence fA ((x ∗ y) ∗ z) ∪ fA (z) = τ = fA (x ∗ ((y ∗ (y ∗ x)) ∗ (0 ∗ (0 ∗ (x ∗ y))))). If (x ∗ y) ∗ z ∈ / A or z ∈ / A then fA ((x ∗ y) ∗ z) = U or fA (z) = U. It follows that fA (x ∗ ((y ∗ (y ∗ x)) ∗ (0 ∗ (0 ∗ (x ∗ y))))) ⊆ U = fA ((x ∗ y) ∗ z) ∪ fA (z). Therefore FA is a U-soft c-BCI-ideal over U, and clearly e(FA ; τ ) = A. This completes the proof.  Acknowledgments The Second Author was supported by Basic Science Research Program through the National Research Foundation of Korea(NRF) funded by the Ministry of Education, Science and Technology(No. 2012R1A1A2042193). 476

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Young Bae Jun, Seok Zun Song and Sun Shin Ahn

References [1] U. Acar, F. Koyuncu and B. Tanay, Soft sets and soft rings, Comput. Math. Appl. 59(2010) 3458-3463. [2] H. Akta¸s and N. C ¸ a˘gman, Soft sets and soft groups, Inform. Sci. 177(2007) 2726-2735. [3] A. O. Atag¨ un and A. Sezgin, Soft substructures of rings, fields and modules, Comput. Math. Appl. 61 (2011) 592-601. [4] N. C ¸ aˇgman, F. C ¸ itak and S. Engino˘glu, Soft set theory and uni-int decision making, Eur. J. Oper. Res. 207 (2010) 848-855. [5] N. C ¸ aˇgman and S. Engino˘glu, FP-soft set theory and its applications, Ann. Fuzzy Math. Inform. 2 (2011) 219-226. [6] F. Feng, Soft rough sets applied to multicriteria group decision making, Ann. Fuzzy Math. Inform. 2 (2011) 69-80. [7] F. Feng, Y. B. Jun and X. Zhao, Soft semirings, Comput. Math. Appl. 56 (2008) 2621-2628. [8] Y. Huang, BCI-algebra, Science Press, Beijing 2006. [9] Y. B. Jun, Soft BCK/BCI-algebras, Comput. Math. Appl. 56 (2008) 1408-1413. [10] Y. B. Jun, Union soft sets with applications in BCK/BCI-algebras, Inform. Sci. (submitted). [11] Y. B. Jun, H. S. Kim and J. Neggers, Pseudo d-algebras, Inform. Sci. 179 (2009) 1751-1759. [12] Y. B. Jun, K. J. Lee and A. Khan, Soft ordered semigroups, Math. Logic Q. 56 (2010) 42-50. [13] Y. B. Jun, K. J. Lee and C. H. Park, Soft set theory applied to ideals in d-algebras, Comput. Math. Appl. 57 (2009) 367-378. [14] Y. B. Jun, M¿ S. Kang and K. J. Lee, Intersectional soft sets and applications to BCK/BCI-algebras, Commun. Koean Math. Soc. 28 (2013) 11-24. [15] Y. B. Jun, K. J. Lee and E. H. Roh, Intersectional soft BCK/BCI-ideals, Ann. Fuzzy Math. Inform. 4 (2012) 1-7. [16] Y. B. Jun, K. J. Lee and J. Zhan, Soft p-ideals of soft BCI-algebras, Comput. Math. Appl. 58 (2009) 2060-2068. [17] Y. B. Jun and C. H. Park, Applications of soft sets in ideal theory of BCK/BCI-algebras, Inform. Sci. 178 (2008) 2466-2475. [18] P. K. Maji, R. Biswas and A. R. Roy, Soft set theory, Comput. Math. Appl. 45 (2003) 555-562. [19] P. K. Maji, A. R. Roy and R. Biswas, An application of soft sets in a decision making problem, Comput. Math. Appl. 44 (2002) 1077-1083. [20] J. Meng, An ideal characterization of commutative BCI-algebras, Pusan Kyongnam Math. J. 9 (1993) 1-6. [21] J. Meng and Y. B. Jun, BCK-algebras, Kyungmoon Sa Co. Seoul 1994. [22] J. Meng and X. L. Xin, Commutative BCI-algebras, Math. Japon. 37 (1992) 569-572. [23] D. Molodtsov, Soft set theory - First results, Comput. Math. Appl. 37 (1999) 19-31. ¨ urk, Soft WS-algebras, Commun. Korean Math. Soc. 23 (2008) 313-324. [24] C. H. Park, Y. B. Jun and M. A. Ozt¨ [25] S. Z. Song, K. J. Lee and Y. B. Jun, Intersectional soft sets applied to subalgebras/ideals in BCK/BCIalgebras, Knowledge-Based Systems (submitted). [26] L. A. Zadeh, From circuit theory to system theory, Proc. Inst. Radio Eng. 50 (1962) 856-865. [27] L. A. Zadeh, Fuzzy sets, Inform. Control 8 (1965) 338-353. [28] L. A. Zadeh, Toward a generalized theory of uncertainty (GTU) - an outline, Inform. Sci. 172 (2005) 1-40. [29] J. Zhan and Y. B. Jun, Soft BL-algebras based on fuzzy sets, Comput. Math. Appl. 59 (2010) 2037-2046.

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SOME INEQUALITIES WHICH HOLD FOR STARLIKE LOG-HARMONIC MAPPINGS OF ORDER α 1 ¨ H. Esra Ozkan , Melike Aydo˜ gan2

Department of Mathematics and Computer Sciences, Istanbul K¨ ult¨ ur University, 34156, Bakirkoy, Turkey Department of Mathematics, Isik University, 34980, Sile, Istanbul, Turkey 1

2

[email protected]

[email protected] Abstract

Let H(D) be the linear space of all analytic functions defined on the open disc D = { z| |z| < 1}. A log-harmonic mappings is a solution of the nonlinear elliptic partial differential equation f fz = w fz f where w(z) ∈ H(D) is second dilatation such that |w(z)| < 1 for all z ∈ D. It has been shown that if f is a non-vanishing log-harmonic mapping, then f can be expressed as f (z) = h(z)g(z) where h(z) and g(z) are analytic function in D. On the other hand, if f vanishes at z = 0 but it is not identically zero then f admits following representation f (z) = z |z|2β h(z)g(z) where Reβ > − 12 , h and g are analytic in D, g(0) = 1, h(0) 6= 0. Let f = z |z|2β hg be a univalent log-harmonic mapping. ———————————————————————————————2000 AMS Mathematics Subject Classification 30C45. Keywords and phrases: Starlike log-harmonic functions, univalent functions, distortion theorem, Marx-Strohhacker inequality 1

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We say that f is a starlike log-harmonic mapping of order α if ∂(arg f (reiθ )) zfz − zfz = Re > α , 0 ≤ α < 1. (∀z ∈ U ) ∂θ f ∗ and denote by Slh (α) the set of all starlike log-harmonic mappings of order α. The aim of this paper is to define some inequalities of starlike log-harmonic functions of order α (0 ≤ α ≤ 1).

I. Introduction Let Ω be the family of functions φ(z) regular in the unit disc D and satisfying the conditions φ(0) = 0, |φ(z)| < 1 for all z ∈ D. Next, denote by P (α)(0 ≤ α < 1) the family of functions p(z) = 1 + p1 z + ... regular in D and such that p(z) in P (α) if and only if p(z) =

1 + (1 − 2α) φ(z) 1 − φ(z)

for some functions z ∈ Ω and every z ∈ D. Let S1 (z) and S2 (z) be analytic functions in the open unit disc, with S1 (0) = S2 (0), if S1 (z) = S2 (φ(z)) then we say that S1 (z) is subordinate to S2 (z), where φ(z) ∈ Ω([4]), and we write S1 (z) ≺ S2 (z). Let H(D) be the linear space of all analytic functions defined on the open disc D = {z| |z| < 1}. A log-harmonic mappings is a solution of the nonlinear elliptic partial differential equation f fz = w fz f where w(z) ∈ H(D) is second dilatation such that |w(z)| < 1 for all z ∈ D. It has been shown that if f is a non-vanishing log-harmonic mapping, then f can be expressed as f (z) = h(z)g(z) where h(z) and g(z) are analytic function in D. 2

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On the other hand, if f vanishes at z = 0 but it is not identically zero then f admits following representation f (z) = z |z|2β h(z)g(z) where Reβ > − 21 , h and g are analytic in D, g(0) = 1, h(0) 6= 0. Let f = z |z|2β hg be a univalent log-harmonic mapping. We say that f is a starlike logharmonic mapping of order α if ∂(arg f (reiθ )) zfz − zfz = Re > α , 0 ≤ α < 1. (∀z ∈ U ) ∂θ f ∗ and denote by Slh (α) the set of all starlike log-harmonic mappings of order α([3]). If α = 0, we get the class of starlike log-harmonic mappings. Also, let ∗ ST (α) = {f ∈ Slh (α) and f ∈ H(U )} . ∗ If f ∈ Slh (0) then F (ς) = log(f (eς )) is univalent and harmonic on the half plane { ς | Re {ς} < 0}. It is known that F is closely related with the theory of nonparametric minimal surfaces over domains of the form −∞ < u < u0 (v) , u0 (v + 2π) = u0 (v), see ([1],[2]).

In this paper, we obtain Marx-Strohhacker Inequality and new distortion theorems using the subordination prinsiple for the starlike log-harmonic mappings of order α, previously studied by Z. Abdulhadi and Y. Abu Muhanna [3] who obtained the representation theorem and a different distortion theorem for the same class.

3

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II. Main Results Theorem 2.1.Let f (z) = zh(z)g(z) be an analytic logaritmic harmonic function in the open unit disc U . If f (z) satisfies the condition h0 (z) g 0 (z) 2(1 − α)z z −z ≺ = F (z) h(z) g(z) 1−z

(1)

∗ (α). then f ∈ Slh Proof. We define the function by

h = (1 − φ(z))−2(1−α) g

(2)

where (1 − φ(z))−2(1−α) has the value 1 at z = 0. Then w(z) is analytic and φ(0) = 0. If we take the logarithmic derivative of (2) and the after brief calculations, we get z

h0 (z) g 0 (z) 2(1 − α)zφ0 (z) −z ≺ h(z) g(z) 1 − φ(z)

Now it is easy to realize that the subordination (1) is equivalent to |φ(z)| < 1 for all z ∈ U . Indeed assume the contrary: then there is a z1 ∈ U such that |φ(z1 )| = 1, so by I.S. Jack Lemma z1 φ0 (z1 ) = kφ(z1 ) for some k ≥ 1 and for such z1 ∈ U , we have z1

2(1 − α)kφ(z1 ) h0 (z1 ) g 0 (z1 ) − z1 ≺ = F (φ(z1 )) ∈ / F (U ) h(z1 ) g(z1 ) 1 − φ(z1 )

but this contradicts (1); so our assumption is wrong, i.e, |φ(z)| < 1 for all z ∈ u. By using condition (1) we get 1+z

h0 (z) g 0 (z) 1 + (1 − 2α)φ(z) −z = h(z) g(z) 1 − φ(z)

.

(3)

∗ The equality (3) shows that f (z) ∈ Slh (α).

4

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Corollary 2.2.For the starlike logharmonic functions of order α, we have Marx-Strohhacker Inequality is   1 g 2(1−α) 1 − 0 and for every z ∈ D we can find fz ∈ Hv∞ with kfz kv ≤ 1 such that |fz (z)| = ve(z) . We say that a weight v is radial if v(z) = v(|z|) for every z ∈ D. A positive continuous function v is called normal if there exist δ ∈ [0, 1) and s, t(0 < s < t) such that for every z ∈ D with |z| ∈ [δ, 1), v(|z|) is decreasing on [δ, 1) and (1 − |z|)s

v(|z|) = 0; |z|→1 (1 − |z|)s

v(|z|) is increasing on [δ, 1) and (1 − |z|)t

v(|z|) = ∞. |z|→1 (1 − |z|)t

lim

lim

A radial, non-increasing weight is called typical if lim v(z) = 0. When studying the struc|z|→1

ture and isomorphism classes of the space Hv∞ (see [6, 7]), Lusky introduced the following condition (L1) (renamed after the author) for radial weights: v(1 − 2−n−1 ) > 0, n∈N 1 − 2−n

(L1) inf

which will play a great role in this article. Moreover, radial weights with (L1) (for example, see [2]) are essential, that is, we can find a constant k > 0 such that v(z) ≤ ve(z) ≤ kv(z) for every z ∈ D. a−z Now, let ϕa (z) = 1−¯ obius transformation that interchanges a and 0. az , z ∈ D, be the M¨ We will use the fact that derivative of ϕa is given by

ϕ0a (z) = −

1 − |a|2 for every z ∈ D. (1 − a ¯z)2

Our aim in this note is to characterize boundedness and compactness of operator Dnφ from weighted Banach spaces of holomorphic functions to weighted Bloch spaces in terms of the involved weights as well as the inducing map. For n = 0 and n = 1, as corollaries we get a characterization of boundedness and compactness of Cφ and Cφ D that act from weighted Banach spaces of holomorphic functions to weighted Bloch spaces. Throughout this paper, we will use the symbol C to denote a finite positive number, and it may differ from one occurrence to the other.

2

Background and Some Lemmas

Now let us state a couple of lemmas, which are used in the proof of the main results in the next sections. The first lemma is taken from [9].

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Chen, Zeng and Zhou: N-differentiation composition operators

Lemma 1. Let v be a radial weight satisfying condition (L1). There is a constant C > 0 (depending only on the weight v) such that for all f ∈ Hv∞ , |f (n) (z)| ≤ C

kf kv , v(z)(1 − |z|2 )n

(1)

for every z ∈ D and n ∈ N. Proof. We will prove the theorem by mathematical induction. For n = 1, see Lemma 2 in [9]. If (1) is true for n − 1. Then for n, let u(z) = v(z)(1 − |z|2 )n−1 , since |f (n−1) (z)| ≤ C

kf kv , v(z)(1 − |z|2 )n−1

then f (n−1) ∈ Hu∞ . For f (n−1) using the result of n = 1 the lemma is proved. The following result is well-known (see, e.g. [3, 8]) Lemma 2. Suppose that w is a normal weight and v is a radial weight satisfying (L1). Then the operator Dφn : Hv∞ → Bw (or Bw,0 ) is compact if and only if whenever {fm } is a bounded sequence in Hv∞ with fm → 0 uniformly on compact subsets of D, and then kDφn fm kBw → 0. The following lemma can be proved similarly to Lemma 1 in [4] (see, also [5]). It will be useful to give a criterion for compactness in Bw,0 . Lemma 3. Assume w is normal. A closed set K in Bw,0 is compact if and only if it is bounded and satisfies lim sup w(z)|f 0 (z)| = 0.

(2)

|z|→1 f ∈K

3

The Boundedness of Dφn : Hv∞ → Bw (or Bw,0 )

In this section we formulate and prove results regarding the boundedness of the operator Dφn : Hv∞ → Bw (or Bw,0 ). Theorem 1. Suppose that w be arbitrary weight, v be a radial weight satisfying condition (L1), then Dφn : Hv∞ → Bw is bounded if and only if sup z∈D

w(z)|φ0 (z)| < ∞, v(φ(z))(1 − |φ(z)|2 )n+1

(3)

Proof. First, we assume that the operator Dφn : Hv∞ → Bw is bounded. Fix a point a ∈ D, and consider the function fa (z) = ϕn+1 (z)ga (z) for every z ∈ D, a where ga is a function in the unit ball of Hv∞ such that ga (a) =

1 v e(a) .

Then

kfa kv = sup v(z)|fa (z)| ≤ sup v(z)|ga (z)| ≤ 1. z∈D

z∈D

It is easy to check that (ϕn+1 )(k) (a) = 0, a

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Chen, Zeng and Zhou: N-differentiation composition operators

(ϕn+1 )(n+1) (a) = a So fa(n+1) (a) =

n+1 X

(−1)n+1 (n + 1)! . (1 − |a|2 )n+1

k Cn+1 (ϕn+1 )(k) (a)ga(n+1−k) (a) = a

k=0

(−1)n+1 (n + 1)! . (1 − |a|2 )n+1 ve(a)

Then by the boundedness of Dφn : Hv∞ → Bw , we have (n+1)

∞ > kDφn fφ(a) kBw ≥ sup w(z)|fφ(a) (φ(z))φ0 (z)| z∈D

(n+1)

≥ w(a)|fφ(a) (φ(a))φ0 (a)| =

(n + 1)!w(a)|φ0 (a)| . (1 − |φ(a)|2 )n+1 ve(φ(a))

Since v has (L1), the weights v and ve are equivalent then ve can be replaced by v, and combine with the arbitrariness of a ∈ D, we obtain (3). Conversely, an application of Lemma 1 yields w(z)|f (n+1) (φ(z))φ0 (z)| ≤ C

w(z)|φ0 (z)| kf kv , v(φ(z))(1 − |φ(z)|2 )n+1

(4)

and |f (n) (φ(0))| ≤ C

kf kv . v(φ(0))(1 − |φ(0)|2 )n

Combine with this and taking the supremum in (4) over D, then employing condition (3), we see that Dφn : Hv∞ → Bw must be bounded. By the similar proof of Theorem 1 we see that the following result is true. Theorem 2. Suppose that w be arbitrary weight, v be a radial weight satisfying condition (L1), then Dφn : Hv∞ → Bw,0 is bounded if and only if w(z)|φ0 (z)| = 0. |z|→1 v(φ(z))(1 − |φ(z)|2 )n+1 lim

(5)

Especially, for n = 0, we obtain necessary and sufficient conditions for the boundedness of the operators Cφ : Hv∞ → Bw (or Bw,0 ). Corollary 1. Suppose that w be arbitrary weight, v be a radial weight satisfying condition (L1), then the following statements hold: (i) Cφ : Hv∞ → Bw is bounded if and only if sup z∈D

w(z)|φ0 (z)| < ∞. v(φ(z))(1 − |φ(z)|2 )

(ii) Cφ : Hv∞ → Bw,0 is bounded if and only if w(z)|φ0 (z)| = 0. |z|→1 v(φ(z))(1 − |φ(z)|2 ) lim

For n = 1, Dφn is the operator Cφ D, then we have the following corollary .

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Chen, Zeng and Zhou: N-differentiation composition operators

Corollary 2. Suppose that w be arbitrary weight, v be a radial weight satisfying condition (L1), then the following statements hold: (i) Cφ D : Hv∞ → Bw is bounded if and only if sup z∈D

(ii) Cφ D :

Hv∞

w(z)|φ0 (z)| < ∞. v(φ(z))(1 − |φ(z)|2 )2

→ Bw,0 is bounded if and only if w(z)|φ0 (z)| = 0. |z|→1 v(φ(z))(1 − |φ(z)|2 )2 lim

4

The Compactness of Dφn : Hv∞ → Bw (or Bw,0 )

In this section, we turn our attention to the question of compactness. Theorem 3. Suppose that w be arbitrary weight, v be a radial weight satisfying condition (L1). Then Dφn : Hv∞ → Bw is compact if and only if lim

sup

r→1 |φ(z)|>r

w(z)|φ0 (z)| = 0. v(φ(z))(1 − |φ(z)|2 )n+1

(6)

Proof. First, we assume that the operator Dφn : Hv∞ → Bw is compact. Let {zm }m ⊂ D be a sequence with |φ(zm )| → 1 such that lim

sup

r→1 |φ(z)|>r

w(zm )|φ0 (zm )| w(z)|φ0 (z)| = lim . m→∞ v(φ(zm ))(1 − |φ(zm )|2 )n+1 v(φ(z))(1 − |φ(z)|2 )n+1

By passing to a subsequence and still denoted by {zm }m , we assume that there is N ∈ N, such that |φ(zm )|m ≥ 12 for every m ≥ N . For every m ∈ N, we consider functions fm (z) = z m ϕn+1 φ(zm ) (z)gφ(zm ) (z) for every z ∈ D, 1 where gφ(zm ) is a function in the unit ball of Hv∞ such that |gφ(zm ) (φ(zm ))| = ve(φ(z . m )) ∞ Again since v has (L1), ve may be replaced by v. Obviously, {fm }m ⊂ Hv is a bounded sequence that tends to zero uniformly on the compact subsets of D. Hence by Lemma 2, we have that kDφn fm kBw → 0. Moreover, (k) (z m ϕn+1 (φ(zm )) = 0, φ(zm ) ) (n+1) (z m ϕn+1 (φ(zm )) = φ(zm ) )

Since (n+1) fm (φ(zm ))

=

n+1 X

k = 0, 1, ..., n;

(−1)n+1 (n + 1)!φm (zm ) . (1 − |φ(zm )|2 )n+1 (n+1−k)

k (k) Cn+1 (z m ϕn+1 gφz φ(zm ) )

m

(φ(zm )).

k=0 (n+1)

Therefore |fm

(φ(zm ))| = 0 ← = ≥

(n+1)!|φ(zm )|m v e(φ(zm ))(1−|φ(zm )|2 )n+1 ,

and for m ≥ N

(n+1) kDφn fm kBw ≥ w(zm )|fm (φ(zm ))φ0 (zm )|

(n + 1)!w(zm )|φ0 (zm )||φ(zm )|m ve(φ(zm ))(1 − |φ(zm )|2 )n+1 1 w(zm )|φ0 (zm )| , 2 v(φ(zm ))(1 − |φ(zm )|2 )n+1

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Chen, Zeng and Zhou: N-differentiation composition operators

and the claim follows. Conversely, suppose that (6) holds. Let {fm }m ⊂ Hv∞ be a bounded sequence which converges to zero uniformly on the compact subsets of D, we may assume that kfm kv ≤ 1 for every m ∈ N. By Lemma 2 we have to show that kDφn fm kBw → 0 if m → ∞. Let us fix ε > 0. By hypothesis there is 0 < r < 1 such that ε w(z)|φ0 (z)| < if |φ(z)| > r, v(φ(z))(1 − |φ(z)|2 )n+1 2C where C is the constant given in Lemma 1. Thus, if |φ(z)| > r, by Lemma 1, (n+1) w(z)|φ0 (z)||fm (φ(z))| ≤ C

w(z)|φ0 (z)| ε kfm kv < . v(φ(z))(1 − |φ(z)|2 )n+1 4

(7)

Now, we can find M > 0 such that sup w(z)|φ0 (z)| ≤ M .

(8)

|φ(z)|≤r

Moreover, since {fm }m converges to 0 uniformly on compact subsets of D as m → ∞. (n+1) }m also converges to 0 uniformly on compact Cauchy’s integral formula gives that {fm subsets of D as m → ∞. So there is N1 ∈ N such that ε for every m ≥ N1 . 4M

(n+1) sup |fm (φ(z))| ≤

|φ(z)|≤r

(9)

(n)

Also, {fm (φ(0))}m converges to 0 as m → ∞, then there exists N2 > 0 such that (n) |fm (φ(0))| < 2ε for every m > N2 . Finally, together with (7) (8) and (9) we can conclude that kDφn fm kBw

(n) (n+1) = |fm (φ(0))| + sup w(z)|φ0 (z)||fm (φ(z))| z∈D

≤

(n) |fm (φ(0))|

(n+1) + sup w(z)|φ0 (z)| sup |fm (φ(z))| |φ(z)|≤r 0

+ sup w(z)|φ |φ(z)|>r


0 there exists a r ∈ (0, 1) such that w(z)|φ0 (z)| < ε if r < |φ(z)| < 1. v(φ(z))(1 − |φ(z)|2 )n+1

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Chen, Zeng and Zhou: N-differentiation composition operators

n+1

z On the other hand, since h(z) = (n+1)! ∈ Hv∞ , from the compactness of Dφn : Hv∞ → Bw,0 , it follows that φ ∈ Bw,0 . Then there exists a ρ ∈ (r, 1) such that

w(z)|φ0 (z)| < ε inf v(t)(1 − |t|2 )n+1 if ρ < |z| < 1,

(11)

t∈[0,r]

Therefore, when ρ < |z| < 1 and r < |φ(z)| < 1, we have that w(z)|φ0 (z)| < ε. v(φ(z))(1 − |φ(z)|2 )n+1

(12)

If ρ < |z| < 1 and |φ(z)| ≤ r, combine with (11), we have that w(z)|φ0 (z)| w(z)|φ0 (z)| ≤ < ε. 2 n+1 v(φ(z))(1 − |φ(z)| ) inf v(t)(1 − |t|2 )n+1

(13)

t∈[0,r]

Inequalities (12) and (13) imply (10) holds. Conversely, assume that (10) holds. Then (3) holds, which along with (4) implies that the set Dφn ({f ∈ Hv∞ : kf kv ≤ 1}) is bounded in Bw,0 . By Lemma 3 we see that Dφn : Hv∞ → Bw,0 is compact if and only if sup w(z)|f (n+1) (φ(z))φ0 (z)| = 0..

lim

|z|→1 kf kv ≤1

(14)

Taking the supremum in (4) over the unit ball of Hv∞ , then letting |z| → 1, we obtain (14), from which the compactness of Dφn : Hv∞ → Bw,0 follows. Noticing the results of Theorem 2 and Theorem 4, we conclude that the boundedness and compactness of the operator Dφn : Hv∞ → Bw,0 is equivalent. Similarly, for n = 0, we obtain necessary and sufficient conditions for the compactness of the operators Cφ : Hv∞ → Bw (or Bw,0 ). Corollary 3. Suppose that w be a normal weights, v be a radial weight satisfying condition (L1). Then the following statements hold: (i) Cφ : Hv∞ → Bw is compact if and only if lim

sup

r→1 |φ(z)|>r

w(z)|φ0 (z)| = 0. v(φ(z))(1 − |φ(z)|2 )

(ii) Cφ : Hv∞ → Bw,0 is compact if and only if w(z)|φ0 (z)| = 0. |z|→1 v(φ(z))(1 − |φ(z)|2 ) lim

And for n = 1, Dφn is the operator Cφ D. Corollary 4. Suppose that w be a normal weights, v be a radial weight satisfying condition (L1). Then the following statements hold: (i) Cφ D : Hv∞ → Bw is compact if and only if lim

sup

r→1 |φ(z)|>r

w(z)|φ0 (z)| = 0. v(φ(z))(1 − |φ(z)|2 )2

(ii) Cφ D : Hv∞ → Bw,0 is compact if and only if w(z)|φ0 (z)| = 0. |z|→1 v(φ(z))(1 − |φ(z)|2 )2 lim

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Chen, Zeng and Zhou: N-differentiation composition operators

References [1] K.D. Bierstedt, J. Bonet, J. Taskinen, Associated weights and spaces of holomorphic functions, Studia Math., 127, 137-168 (1988). [2] J. Bonet, P. Doma´ nski and M. Lindstr¨om, Essential norm and weak compactness of composition opterators on weighted Banach spaces of analytic functions, Canad. Math. Bull., 42(2),139-148 (1999). [3] K. Avetisyan, S. Stevi´c, Extended Ces`aro opterators between different Hardy spaces, Appl. Math. Comput., 207, 346-350 (2009). [4] K. Madigan, A. Matheson, Compact compositon operators on the Bloch space, Trans. Amer. Math. Soc., 347(7), 2679-2687 (1995). [5] A. Montes-Rodriguez, Weighted compositon operators on weighted Banach spaces of analytic functions, J. London Math. Soc., 61(3), 872-884 (2000). [6] W. Lusky, On weighted spaces of harmonic and holomorphic functions, J. London. Math. Soc., 51(2), 309-320 (1995). [7] W. Lusky, On the isomorphism classes of weighted spaces of harmonic and holomorphic functions, Studia. Math., 175(1), 19-45 (2006). [8] Z. Hu, Extended Ces` aro opterators on mixed-norm spaces, Proc. Amer. Math. Soc., 131(7), 2171-2179 (2003). [9] E. Wolf, Composition followed by differentiation between weighted Banach spaces of holomorphic functions, Revista Da La Real Academia De Ciencias Exactas, Fisicas Y Naturales. Serie A. Matematics, 105(2), 315-322 (2011). [10] Y. Wu, H. Wulan, Products of differentiation and composition operators on the Bloch space, Collect. Math., 63, 93-107 (2012). [11] C.C. Cowen and B.D. MacCluer, Composition Operators on Spaces of Analytic Functions, CRC Press, Boca Raton, FL, 1995. [12] J.N. Dai and C.H. Ouyang, Difference of weighted composition optertors on Hα∞ (BN ), J. Inequal. Appl., 2009, 19p., Article ID 127431,(2009). [13] S. Stevi´c, Composition opterators between H ∞ and the α-Bloch spaces on the polydisc, Z. Anal. Anwendungen, 25(4), 457-466 (2006). [14] D. Garc´ıa, M. Maestre and P.S. Peris, Composition opterators between weighted spaces of holomorphic functions on Banach spaces, Ann. Aca. Sci. Fen. Math., 29(7) , 81-98 (2004). [15] Z.H. Zhou, Y.X. Liang and X.T Dong, Differences of weighted composition operators from Hardy space to weighted-type spaces on the unit ball, Ann. Polon. Math., 104(3) , 309-319 (2012). [16] K.H. Zhu, Spaces of Holomorphic Functions in the Unit Ball, Graduate Texts in the Mathematics 226, Springer, New York, 2004.

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FUZZY n-JORDAN ∗-DERIVATIONS ON INDUCED FUZZY C ∗ -ALGEBRAS CHOONKIL PARK1 , KHATEREH GHASEMI∗2 , SHAHRAM GHAFFARY GHALEH3 Abstract. Using the fixed point method, we prove the fuzzy version of the Hyers-Ulam stability of n-Jordan ∗-derivations on induced fuzzy C ∗ -algebras associated with the following functional equation ) ( ) ( ) ( b−a a − 3c 3a − b + 3c f +f +f = f (a). 3 3 3

1. Introduction and Preliminaries The stability of functional equations originated from a question of Ulam [36] concerning the stability of group homomorphisms in 1940. More precisely, he proposed the following problem: Given a group G, a metric group (G ′ , d) and ϵ > 0, does there exist a δ > 0 such that if a function f : G → G ′ satisfies the inequality d(f (xy), f (x)f (y)) < δ for all x, y ∈ G, then there exists a homomorphism T : G → G ′ such that d(f (x), T (x)) < ϵ for all x ∈ G? In 1941, Hyers [16] gave a partial solution of the Ulam’s problem for the case of approximate additive mappings under the assumption that G and G ′ are Banach spaces. In 1950, Aoki [1] generalized the Hyers’ theorem for approximately additive mappings. In 1978, Th. M. Rassias [33] generalized the theorem of Hyers by considering the stability problem with unbounded Cauchy differences. The stability problems of several functional equations have been extensively investigated by a number of authors and there are many interesting results concerning this problem (see [7, 9, 11, 12, 13, 14, 19, 30, 31, 34, 35]). 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. We recall a fundamental result in fixed point theory. Theorem 1.1. ([4, 10]) Let (X, d) be a complete generalized metric space and let J : X → X be a strictly contractive mapping with Lipschitz constant L < 1. Then for each given element x ∈ X, either d(J n x, J n+1 x) = ∞ for all nonnegative integers n or there exists a positive integer n0 such that (1) d(J n x, J n+1 x) < ∞, ∀n ≥ n0 ; n (2) the sequence {J x} converges to a fixed point y ∗ of J; (3) y ∗ is the unique fixed point of J in the set Y = {y ∈ X | d(J n0 x, y) < ∞}; 2010 Mathematics Subject Classification. Primary 46S40; 47S40; 39B52; 47H10; 46L05. Key words and phrases. Fuzzy n-Jordan ∗-derivation; induced fuzzy C ∗ -algebra; Hyers-Ulam stability. ∗ Corresponding author.

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C. PARK, K. GHASEMI, S. GHAFFARY GHALEH

(4) d(y, y ∗ ) ≤

1 1−L d(y, Jy)

for all y ∈ Y .

In 1996, G. Isac and Th.M. Rassias [17] were the first to provide applications of stability theory of functional equations for the proof of new fixed point theorems with applications. By using fixed point methods, the stability problems of several functional equations have been extensively investigated by a number of authors (see [5, 6, 27, 28, 32]). Katsaras [18] defined a fuzzy norm on a vector space to construct a fuzzy vector topological structure on the space. Some mathematics have defined fuzzy normed on a vector space from various points of view [15, 21, 23, 24, 25, 29, 37]. In particular, Bag and Samanta [3] following Cheng and Mordeson [8], gave an idea of fuzzy norm in such a manner that the corresponding fuzzy metric is of Kramosil and Michalek type [20]. They established a decomposition theorem of a fuzzy norm into a family of crisp norms and investigated some properties of fuzzy normed spaces [2]. We use the definition of fuzzy normed spaces given in [3, 23, 24] to investigate a fuzzy version of the Hyers-Ulam stability of n-Jordan ∗-derivations in induced fuzzy C ∗ -algebras associated with the following functional equation ( ) ( ) ( ) b−a a − 3c 3a − b + 3c f +f +f = f (a). 3 3 3 Definition 1.2. ([3, 23, 24, 25]) Let X be a complex vector space. A function N : X ×R → [0, 1] is called a fuzzy norm on X if for all x, y ∈ X and all s, t ∈ R, N1 : N (x, t) = 0 for t ≤ 0 N2 : x = 0 if and only if N (x, t) = 1 for all t > 0 t N3 : N (cx, t) = N (x, |c| ) if c ∈ C − {0} N4 : N (x + y, s + t) ≥ min{N (x, s), N (y, t)} N5 : N (x, ·) is a non-decreasing function of R and limt→∞ N (x, t) = 1 N6 : for x ̸= 0, N (x, .) is continuous on R. The pair (X , N ) is called a fuzzy normed vector space. Definition 1.3. ([3, 23, 24, 25]) Let (X , N ) be a fuzzy normed vector space. (1) 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 } and we denote it by N − limn→∞ xn = x. (2) A sequence {xn } in X is called Cauchy if for each ϵ > 0 and each t > 0 there exists an n0 ∈ N such that for all n ≥ n0 and all p > 0, we have N (xn+p − xn , t) > 1 − ϵ. It is well-known that every convergent sequence in a fuzzy normed vector space is Cauchy. If each Cauchy sequence is convergent, then the fuzzy norm is said to be complete and the fuzzy normed vector space is called a fuzzy Banach space. We say that a mapping f : X → Y between fuzzy normed vector space X , Y is continuous at point x0 ∈ X if for each sequence {xn } converging to x0 in X , then the sequence {f (xn )} converges to f (x0 ). If f : X → Y is continuous at each x ∈ X , then f : X → Y is said to be continuous on X (see [2]). Definition 1.4. Let X be a ∗-algebra and (X , N ) a fuzzy normed space. (1) The fuzzy normed space (X , N ) is called a fuzzy normed ∗-algebra if N (xy, st) ≥ N (x, s) · N (y, t)

495

&

N (x∗ , t) = N (x, t)

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FUZZY n-JORDAN ∗-DERIVATIONS

(2) A complete fuzzy normed ∗-algebra is called a fuzzy Banach ∗-algebra. Example 1.5. Let (X , ∥ · ∥) be a normed ∗-algebra. let { t t+∥x∥ , t > 0, x ∈ X N (x, t) = 0, t ≤ 0, x ∈ X . Then N (x, t) is a fuzzy norm on X and (X , N (x, t)) is a fuzzy normed ∗-algebra. Definition 1.6. Let (X , ∥ · ∥) be a C ∗ -algebra and N a fuzzy norm on X . (1) The fuzzy normed ∗-algebra (X , N ) is called an induced fuzzy normed ∗-algebra (2) The fuzzy Banach ∗-algebra (X , N ) is called an induced fuzzy C ∗ -algebra. Definition 1.7. Let (X , N ) be an induced fuzzy normed ∗-algebra. Then a C-linear mapping D : (X , N ) → (X , N ) is called a fuzzy n-Jordan ∗-derivation if D(an ) = D(a)an−1 + aD(a)an−2 + . . . + an−2 D(a)a + an−1 D(a)

&

D(a∗ ) = D(a)∗

for all a ∈ X . Throughout this paper, assume that (X , N ) is an induced fuzzy C ∗ -algebra. 2. Main results Lemma 2.1. Let (Z, N ) be a fuzzy normed vector space and let f : X → Z be a mapping such that ) ( ) ( ) ) ( ) ( ( x − 3z 3x − y + 3z t y−x +f +f , t ≥ N f (x), (2.1) N f 3 3 3 2 for all x, y, z ∈ X and all t > 0. Then f is additive, i.e., f (x+y) = f (x)+f (y) for all x, y ∈ X . Proof. Letting x = y = z = 0 in (2.1), we get ) ( ) ( t t ≥ N f (0), N (3f (0), t) = N f (0), 3 2 for all t > 0. By N5 and N6 , N (f (0), t) = 1 for all t > 0. It follows from N2 that f (0) = 0. Letting y = x = 0 in (2.1), we get ( ) t N (f (0) + f (−z) + f (z), t) ≥ N f (0), =1 2 for all t > 0. It follows from N2 that f (−z) + f (z) = 0 for all z ∈ X . So f (−z) = −f (z) for all z ∈ X . Letting x = 0 and replacing y, z by 3y, −z, respectively, in (2.1), we get ) ( t N (f (y) + f (z) + f (−y − z), t) ≥ N f (0), =1 2 for all t > 0. It follows from N2 that f (y) + f (z) + f (−y − z) = 0

(2.2)

for all y, z ∈ X . Thus f (y + z) = f (y) + f (z) for all y, z ∈ X , as desired.



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C. PARK, K. GHASEMI, S. GHAFFARY GHALEH

Using the fixed point method, we prove the Hyers-Ulam stability of fuzzy n-Jordan ∗derivations on induced fuzzy C ∗ -algebras. Theorem 2.2. Let φ : X 3 → [0, ∞) be a function such that there exists an L < (x y z ) L φ , , ≤ φ(x, y, z) 3 3 3 3

3 3n

for all x, y, z ∈ X . Let f : X → X be a mapping such that ( ( ) ( ) ( ) ) y−x x − 3z 3x − y + 3z N µf + µf + µf − f (µx), t 3 3 3 t ≥ , t + φ(x, y, z) N (f (wn ) − f (w)wn−1 − wf (w)wn−2 − · · · − wn−2 f (w)w − wn−1 f (w) t +f (v ∗ ) − f (v)∗ , t) ≥ t + φ(w, v, 0)

with (2.3)

(2.4)

(2.5)

for all x, y, z, w, v ∈ X , all t > 0 and all µ ∈ T1 := {λ ∈ C : |λ| = 1}. Then D(x) = N − limn→∞ 3n f ( 3xn ) exists for each x ∈ X and defines a fuzzy n-Jordan ∗-derivation D : X → X such that N (f (x) − D(x), t) ≥

(1 − L)t (1 − L)t + φ(x, 2x, 0)

(2.6)

for all x ∈ X and all t > 0. Proof. Letting µ = 1, y = 2x and z = 0 in (2.4), we get ) ( (x) t − f (x), t ≥ N 3f 3 t + φ (x, 2x, 0)

(2.7)

for all x ∈ X . Consider the set S := {g : X → X } and introduce the generalized metric on S: d(g, h) = inf{α ∈ R+ : N (g(x) − h(x), αt) ≥

t , ∀x ∈ X , ∀t > 0}, t + φ (x, 2x, 0)

where, as usual, inf ϕ = +∞. It is easy to show that (S, d) is complete (see the proof of [22, Lemma 2.1]). Now we consider the linear mapping J : S → S such that (x) Jg(x) := 3g 3 for all x ∈ X . Let g, h ∈ S be given such that d(g, h) = ε. Then N (g(x) − h(x), εt) ≥

497

t t + φ (x, 2x, 0)

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FUZZY n-JORDAN ∗-DERIVATIONS

for all x ∈ X and all t > 0. Hence

( ( ) (x) ) (x) L ) ( (x) x − 3h , Lεt = N g −h , εt N (Jg(x) − Jh(x), Lεt) = N 3g 3 3 3 3 3 Lt

≥ =

Lt 3

+φ

(3x

2x 3, 3 ,0

)≥

Lt 3 Lt 3

+ L3 φ (x, 2x, 0)

t t + φ (x, 2x, 0)

for all x ∈ X and all t > 0. So d(g, h) = ε implies that d(Jg, Jh) ≤ Lε. This means that d(Jg, Jh) ≤ Ld(g, h) for all g, h ∈ S. It follows from (2.7) that d(f, Jf ) ≤ 1. By Theorem 1.1, there exists a mapping D : X → X satisfying the following: (1) D is a fixed point of J, i.e., (x) 1 D = D(x) 3 3 for all x ∈ X . The mapping D is a unique fixed point of J in the set

(2.8)

M = {g ∈ S : d(f, g) < ∞}. This implies that D is a unique mapping satisfying (2.8) such that there exists a α ∈ (0, ∞) satisfying t N (f (x) − D(x), αt) ≥ t + φ (x, 2x, 0) for all x ∈ X ; (2) d(J k f, D) → 0 as k → ∞. This implies the equality N - lim 3k f k→∞

for all x ∈ X ; (3) d(f, D) ≤

1 1−L d(f, Jf ),

(x) = D(x) 3k

which implies the inequality d(f, D) ≤

1 . 1−L

This implies that the inequality (2.7) holds. It follows from (2.3) that ∞ ∑

3k φ(

k=0

for all x, y, z ∈ X . By (2.4),

( N

( k

3 µf

y−x 3k+1

x y z , , ) 0 and all µ ∈ T1 . So ( ( ) ( ) ( ) y−x x − 3z 3x − y + 3z k k N 3k µf + 3 µf + 3 µf 3k+1 3k+1 3k+1 t ( µx ) ) t 3k −3k f = ,t ≥ t x y x y z k 3k t + 3 φ( , , z) + φ( 3k , 3k , 3k ) 3k 3k 3k 3k for all x, y, z ∈ X , all t > 0 and all µ ∈ T1 . Since limk→∞

t

t+3k φ(

x y , , z 3k 3k 3k

)

= 1 for all x, y, z ∈ X

and all t > 0, ) ( ) ( ) ) ( ( x − 3z 3x − y + 3z y−x N µD + µD + µD − D(µx), t = 1 3 3 3 for all x, y, z ∈ X , all t > 0 and all µ ∈ T1 . Thus ( ) ( ) ( ) y−x x − 3z 3x − y + 3z µD + µD + µD = D(µx) 3 3 3

(2.9)

for all x, y, z ∈ X , all t > 0 and all µ ∈ T1 . Letting x = y = z = 0 in (2.9), we get D(0) = 0. Let µ = 1 and x = 0 in (2.9). By the same reasoning as in the proof of Lemma 2.1, one can easily show that D is additive. Letting y = 2x and z = 0 in (2.9), we get (x) µD(x) = 3µD = D(µx) 3 for all x ∈ X and all µ ∈ T1 . By [26, Theorem 2.1], the mapping D : X → X is C-linear. By (2.5) and letting v = 0 in (2.5), we have ( ( n) (w) (w) (w) w nk n−2 nk n−1 nk n−2 N 3nk f · − 3 w f − 3 f w − 3 wf w − · · w 3nk 3k 3k 3k ) (w) t −3nk wn−1 f k , 3nk t ≥ 3 t + φ( 3wk , 0, 0) for all w ∈ X and all t > 0. So ( ( n) (w) (w) (w) w nk nk n−2 nk n−1 nk n−2 N 3 f · − 3 w f − 3 f w − 3 wf w − · · w 3nk 3k 3k 3k t (w) ) t 3nk = −3nk wn−1 f k , t ≥ t w n−1 3 t + (3 L)k φ(w, 0, 0) + φ( 3k , 0, 0) 3nk for all w ∈ X and all t > 0. Since limk→∞

t t+(3n−1 L)k φ(w,0,0)

= 1 for all w ∈ X and all t > 0,

N (D(wn ) − D(w)wn−1 − wD(w)wn−2 · · · wn−2 D(w)w − wn−1 D(w), t) = 1 for all w ∈ X and all t > 0. Thus D(wn ) − D(w)wn−1 − wD(w)wn−2 · · · wn−2 D(w)w − wn−1 D(w) = 0 for all w ∈ X . By (2.5) and letting w = 0 in (2.5), we have ( ( ∗) ) ( v )∗ v t k k N 3k f − 3 f , 3 t ≥ t + φ(0, 3vk , 0) 3k 3k

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for all v ∈ X and all t > 0. So ( ( ∗) ( v )∗ ) v k k N 3 f − 3 f k ,t ≥ 3k 3 for all v ∈ X and all t > 0. Since limk→∞

t 3k t 3k

+ φ(0,

t t+3k φ(0,

v 3k

,0)

v , 0) 3k

=

t t+

3k φ(0, 3vk , 0)

= 1 for all v ∈ X and all t > 0,

N (D(v ∗ ) − D(v)∗ , t) = 1 for all x ∈ X and all t > 0. Thus D(v ∗ ) − D(v)∗ = 0 for all v ∈ X . Therefore, the mapping D : X → X is a fuzzy n-Jordan ∗-derivation.



Corollary 2.3. Let θ ≥ 0 and let p be a real number with p > n. Let X be a normed vector space with norm ∥ · ∥. Let f : X → X be a mapping satisfying ( ( ) ( ) ( ) ) y−x x − 3z 3x − y + 3z N µf + µf + µf − f (µx), t 3 3 3 t , (2.10) ≥ p t + θ(∥x∥ + ∥y∥p + ∥z∥p ) N (f (wn ) − f (w)wn−1 − wf (w)wn−2 − · · · − wn−2 f (w)w − wn−1 f (w) t +f (v ∗ ) − f (v)∗ , t) ≥ t + θ(∥w∥p + ∥v∥p )

(2.11)

for all x, y, z, w, v ∈ X , all t > 0 and all µ ∈ T1 . Then D(x) = N − limn→∞ 3n f ( 3xn ) exists for each x ∈ X and defines a fuzzy n-Jordan ∗-derivation D : X → X such that N (f (x) − D(x), t) ≥

(3p − 3)t (3p − 3)t + 3p θ∥x∥p

for all x ∈ X and all t > 0. Proof. The proof follows from Theorem 2.2 by taking φ(x, y, z) = θ(∥x∥p + ∥y∥p + ∥z∥p ) 

and L = 31−p .

Theorem 2.4. Let φ : X 3 → [0, ∞) be a function such that there exists an L < 1 with (x y z ) φ(x, y, z) ≤ 3Lφ , , 3 3 3 for all x, y, z ∈ X . Let f : X → X be a mapping satisfying (2.4) and (2.5). Then D(x) = N − limn→∞ 31n f (3n x) exists for each x ∈ X and defines a fuzzy n-Jordan ∗-derivation D : X → X such that (1 − L)t N (f (x) − D(x), t) ≥ (2.12) (1 − L)t + Lφ(x, 0, 0) for all x ∈ X and all t > 0. Proof. Let (S, d) be the generalized metric space defined in the proof of Theorem 2.2. Consider the linear mapping J : S → S such that 1 Jg(x) := g (3x) 3 for all x ∈ X .

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It follows from (2.7) that ( ) 1 1 t t N f (x) − f (3x), t ≥ ≥ 3 3 t + φ(3x, 0, 0) t + 3Lφ(x, 0, 0) for all x ∈ X and all t > 0. So d(f, Jf ) ≤ L. Hence d(f, D) ≤

L , 1−L

which implies that the inequality (2.12) holds. The rest of the proof is similar to the proof of Theorem 2.2.



Corollary 2.5. Let θ ≥ 0 and let p be a positive real number with p < 1. Let X be a normed vector space with norm ∥ · ∥. Let f : X → X be a mapping satisfying (2.10) and (2.11). Then D(x) = N − limn→∞ 31n f (3n x) exists for each x ∈ X and defines a fuzzy n-Jordan ∗-derivation D : X → X such that (3 − 3p )t N (f (x) − D(x), t) ≥ (3 − 3p )t + 3p θ∥x∥p for all x ∈ X and all t > 0. Proof. The proof follows from Theorem 2.4 by taking φ(x, y, z) = θ(∥x∥p + ∥y∥p + ∥z∥p ) 

and L = 3p−1 . Acknowledgement

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 (NRF2012R1A1A2004299). References 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12.

Aoki, T., On the stability of the linear transformation in Banach spaces, J. Math. Soc. Japan 2 (1950) 64–66. Bag, T., Samanta, S.K., Fuzzy bounded linear operators, Fuzzy Set and Systems 151 (2005) 513–547. Bag, T., Samanta, S.K., Finite dimensional fuzzy normed linear space, J. Fuzzy Math. 11 (2003) 687–705. C˘ adariu, L., V. Radu, V., Fixed points and the stability of Jensen’s functional equation, J. Inequal. Pure Appl. Math. 4, no. 1, Art. ID 4 (2003). C˘ adariu, L., Radu, V., On the stability of the Cauchy functional equation: a fixed point approach, Grazer Math. Ber. 346 (2004) 43–52. C˘ adariu, L., Radu, V., Fixed point methods for the generalized stability of functional equations in a single variable, Fixed Point Theory and Applications 2008, Art. ID 749392 (2008). I. Chang, Stability of higher ring derivations in fuzzy Banach algebras, J. Computat. Anal. Appl. 14 (2012), 1059–1066. Cheng, S.C., Mordeson J.M., Fuzzy linear operators and fuuzy normed linear spaces, Cacutta Math. Soc. 86 (1994) 429–436. I. Cho, D. Kang, H. Koh, Stability problems of cubic mappings with the fixed point alternative, J. Computat. Anal. Appl. 14 (2012), 132–142. Diaz, J., Margolis, B., A fixed point theorem of the alternative for contractions on a generalized complete metric space, Bull. Amer. Math. Soc. 74 (1968) 305–309. M. Eshaghi Gordji, M. Bavand Savadkouhi, M. Bidkham, Stability of a mixed type additive and quadratic functional equation in non-Archimedean spaces, J. Computat. Anal. Appl. 12 (2010), 454–462. M. Eshaghi Gordji, A. Bodaghi, On the stability of quadratic double centralizers on Banach algebras, J. Computat. Anal. Appl. 13 (2011), 724–729.

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13. M. Eshaghi Gordji, R. Farokhzad Rostami, S.A.R. Hosseinioun, Nearly higher derivations in unital C ∗ algebras, J. Computat. Anal. Appl. 13 (2011), 734–742. 14. M. Eshaghi Gordji, S. Kaboli Gharetapeh, T. Karimi, E. Rashidi, M. Aghaei, Ternary Jordan derivations on C ∗ -ternary algebras, J. Computat. Anal. Appl. 12 (2010), 463–470. 15. Felbin, C., Finite dimensional fuzzy normed linear spaces, Fuzzy Sets and Systems 48 (1992) 239–248. 16. Hyers, D.H., On the stability of the linear functional equation, Proc. Natl. Acad. Sci. USA 27 (1941) 222–224. 17. Isac, G., Rassias, Th.M., Stability of ψ-additive mappings: Appications to nonlinear analysis, Internat. J. Math. Math. Sci. 19 (1996) 219–228. 18. Katsaras, A.K., Fuzzy topological vector spaces II, Fuzzy Sets and Systems 12 (1984) 143–154. 19. H.A. Kenary, J. Lee, C. Park, Non-Archimedean stability of an AQQ-functional equation, J. Computat. Anal. Appl. 14 (2012), 211–227. 20. Kramosil, I., Michalek, J., Fuzzy metric and statistical metric spaces, Kybernetica 11 (1975) 326–334. 21. Krishna, S.V., Sarma, K.K.M., Separation of fuzzy normed linear spaces Fuzzy Sets and Systems 63 (1994) 207–217. 22. Mihet¸, D., Radu, V., On the stability of the additive Cauchy functional equation in random normed spaces, J. Math. Anal. Appl. 343 (2008) 567–572. 23. Mirmostafaee, A.K., Mirzavaziri, M., Moslehian M.S., Fuzzy stability of the Jensen functional equation Fuzzy Sets and Systems 159 (2008) 730–738. 24. Mirmostafaee, A.K., Moslehian, M.S., Fuzzy versions of Hyers-Ulam-Rassias theorem, Fuzzy Sets and Systems 159 (2008) 720–729. 25. Mirmostafaee, A.K., Moslehian, M.S., Fuzzy approximately cubic mappings, Inform. Sca. 178 (2008) 3791– 3798. 26. Park, C., Homomorphisms between Poisson JC ∗ -algebras, Bull. Braz. Math. Soc. 36 (2005) 79–97. 27. Park, C., Fixed points and Hyers-Ulam-Rassias stability of Cauchy-Jensen functional equations in Banach algebras, Fixed Point Theory and Applications 2007, Art. ID 50175 (2007). 28. Park, C., Generalized Hyers-Ulam-Rassias stability of quadratic functional equations: a fixed point approach, Fixed Point Theory and Applications 2008, Art. ID 493751 (2008). 29. Park, C., Fuzzy stability of additive functional inequalities with the fixed point alternative, J. Inequal. Appl. 2009, Art. ID 410576 (2009). 30. C. Park, Y. Cho, H.A. Kenary, Orthogonal stability of a generalized quadratic functional equation in nonArchimedean spaces, J. Computat. Anal. Appl. 14 (2012), 526–535. 31. C. Park, S. Jang, R. Saadati, Fuzzy approximate of homomorphisms, J. Computat. Anal. Appl. 14 (2012), 833–841. 32. Radu, V., The fixed point alternative and the stability of functional equations, Fixed Point Theory 4 (2003) 91–96. 33. Rassias, Th.M., On the stability of the linear mapping in Banach spaces, Proc. Amer. Math. Soc. 72 (1978) 297–300. 34. S. Shagholi, M. Eshaghi Gordji, M. Bavand Savadkouhi, Stability of ternary quadratic derivations on ternary Banach algebras, J. Computat. Anal. Appl. 13 (2011), 1097–1105. 35. S. Shagholi, M. Eshaghi Gordji, M. Bavand Savadkouhi, Nearly ternary cubic homomorphism in ternary Fr´echet algebras, J. Computat. Anal. Appl. 13 (2011), 1106–1114. 36. Ulam, S.M., Problems in Modern Mathematics, Chapter VI. Science ed. Wily, New York, 1940. 37. Xiao, J.Z., Zhu, X.H., Fuzzy normed spaces of opeators and its completeness, Fuzzy Sets and Systems 133 (2003) 389–399. 1

Department of Mathematics, Research Institute for Natural Sciences, Hanyang University, Seoul 133-791, Korea E-mail address: [email protected] 2

Department of Mathematics, Payame Noor University of Khash, Khash, Iran E-mail address: [email protected] 3

Department of Mathematics, Payame Noor University of Zahedan, Zahedan, Iran E-mail address: [email protected]

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Hyers-Ulam stability of a Tribonacci functional equation in 2-normed spaces Majid Eshaghi Gordji1 , Ali Divandari2 , Mohsen Rostamian3 , Choonkil Park4 and Dong Yun Shin∗5 1

Department of Mathematics, Semnan University, P.O. Box 35195-363, Semnan, Iran; Center of Excellence in Nonlinear Analysis and Applications (CENAA), Semnan University, Iran 2,3 Department of Mathematics, Semnan University, Iran 4 Department of Mathematics, Research Institute for Natural Sciences, Hanyang University, Seoul 133-791, Korea 5 Department of Mathematics, University of Seoul, Seoul 130-743, Korea Abstract. In this paper, we investigate the Hyers-Ulam stability of the Tribonacci functional equation f (x) = f (x − 1) + f (x − 2) + f (x − 3) in 2-Banach spaces. Keywords: Hyers-Ulam stability, 2-Banach space, Fibonacci functional equation, Tribonacci functional equation.

1. Introduction and preliminaries The concept of 2-normed spaces was first introduced by S. G¨ahler [8]. Let X be a complex vector space of a dimension greater than one. Suppose that ∥·, ·∥ is a real valued mapping on X × X satisfying the following conditions N1: ∥b, a∥ = ∥a, b∥ N2: ∥a, b∥ = 0 ⇔ a and b are linearly dependent N3: ∥αa, b∥ = |α| ∥a, b∥ N4: ∥a + a ˜, b∥ ≤ ∥a, b∥ + ∥˜ a, b∥ for all a, b ∈ X and α ∈ C. Then ∥·, ·∥ is called a 2-norm on X and the pair (X, ∥·, ·∥) is called a 2-normed space. Some of the basic properties of 2-norms are that they are non-negative and ∥a, b + αa∥ = ∥a, b∥ for all a, b ∈ X and α ∈ C. As an example of a 2-normed space, we may take an inner product space (X, < ·, · >), and define the standard 2-norm on X by < a, a > < a, b > . ∥a, b∥ = < b, a > < b, b > A sequence {xn } in a 2-normed space (X; ∥·, ·∥) is said to converge to some x ∈ X in the 2-norm if ∥x − xn , u∥ → 0 as n → ∞ for all u ∈ X. A sequence {xn } in a 2-normed 0

E-mails:1 [email protected]; 2 [email protected]; [email protected]; [email protected]; 5 [email protected]. ∗ Corresponding author.

3

4

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space (X, ∥·, ·∥) is said to be Cauchy with respect to the 2-norm if lim ∥xn − xm , u∥ = 0

n,m→∞

for all u ∈ X. If every Cauchy sequence in X converges to some x ∈ X, then X is said to be complete with respect to the 2-norm. Any complete 2-normed space is said to be 2-Banach space. Throughout this paper, we denote by Tn the nth Tribonacci number for n ∈ N. In particular, we define T0 = 0, T1 = T2 = 1 and Tn = Tn−1 +Tn−2 +Tn−3 for n ≥ 3. Similar application of Pascal’s triangle in the Fibonacci numbers can be applied to calculate the Tribonacci numbers. 1 1 1 1 1 2 1 3 1 4 1 1 5 5 7 1 7 13 † 7 1 13 . 25 9 1 .. 1 9 25 (a) Numbers in the nth row are the sum of three neighbours: 25 = 13 + 5 + 7. (b) Sums of shallow diagonals giving Tribonacci numbers: 4= 1+3.

Let X be 2-Banach space. A function f : R → X is called a Tribonacci function if it satisfies f (x) = f (x − 1) + f (x − 2) + f (x − 3). (1.1) The stability of functional equations originated from a question of Ulam [15] in 1940. In the next year, Hyers [9] proved the problem for the Cauchy functional equation. The stability problems of several functional equations have been extensively investigated by a number of authors and there are many interesting results concerning this problem (see [1, 2, 3, 4, 5, 6, 10, 11, 12, 13, 14]). Recently, Bidkham and et al. [7] investigated the solution and the Hyers-Ulam stability of (1.1) in normed spaces. In this paper, we establish the Hyers-Ulam stability of (1.1) in 2-normed spaces. We denote the roots of the equation x3 − x2 − x − 1 = 0 By α, β and γ. β and γ are complex, |β| = |γ| and α is greater than one. We have α + β + γ = 1 , αβ + αγ + βγ = −1, αβγ = 1.

(1.2)

2. Main result As we shall see in the following theorem, the general solution of the Tribonacci functional equation is strongly related to the Tribonacci numbers Tn . Theorem 2.1. ([7]) Let X be a real vector space. A function f : R → X is a Tribonacci function if and only if there exists a function g : [−2, 2] → X such that ′ g(x − [x] − 1) + T[x]+1 g(x − [x] − 2), f (x) = T[x]+2 g(x − [x]) + T[x] ′ where T[x] = T[x]+3 − T[x]+2 for all x ∈ R.

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In the following theorem, we prove the Hyers-Ulam stability of the Tribonacci functional equation (1.1) in 2-Banach spaces. We try to prove this theorem under condition ∥f (x) − [f (x − 1) + f (x − 2) + f (x − 3)], z∥ ≤ ϵ for all x ∈ R and z ∈ X, but this condition is very heavy and often inaccessible. In the following we offer a condition to obtain a best result. Theorem 2.2. Let (X, ∥·, ·∥) be a real 2-Banach space. If a function f : R → X satisfies the inequality ∥f (x), f (x − 1) + f (x − 2) + f (x − 3)∥ ≤ ϵ for all x ∈ R and some ϵ > 0, then there exists a Tribonacci function G : R → X such that 1 2(1 + |β|) + |β|2 ∥f (x), G(x)∥ ≤ 2 [ ]ϵ |α (β − γ) + β 2 (γ − α) + γ 2 (α − β)| 1 − |β|2 for all x ∈ R. Proof. By (1.2), it follows from (1.1) that ∥f (x), (α + β + γ)f (x − 1) − (αβ + αγ + βγ)f (x − 2) + αβγ f (x − 3)∥ ≤ ϵ for all x ∈ R. If we replace x by x − r and x + r in the last inequality, then we have ∥f (x − r), α[f (x − r − 1) − γf (x − r − 2)] +β [f (x − r − 1) − (α + γ)f (x − r − 2) + αγf (x − r − 3)] + γf (x − r − 1)∥ ≤ ϵ, ∥f (x − r), α[f (x − r − 1) − βf (x − r − 2)] +γ [f (x − r − 1) − (α + β)f (x − r − 2) + αβf (x − r − 3)] + βf (x − r − 1)∥ ≤ ϵ, ∥f (x + r), α[f (x + r − 1) − γf (x + r − 2)] +β [f (x + r − 1) − (α + γ)f (x + r − 2) + αγf (x + r − 3)] + γf (x + r − 1)∥ ≤ ϵ for all x ∈ R and all r ∈ Z. Hence we have ∥f (x − r), β r α[f (x − r − 1) − γf (x − r − 2)] + β r+1 [f (x − r − 1) − (α + γ)f (x − r − 2) + αγf (x − r − 3)] + β r γf (x − r − 1)∥ ≤ |β r |ϵ, ∥f (x − r), γ β[f (x − r − 1) − αf (x − r − 2)] + γ r

r+1

(2.1)

[f (x − r − 1)

− (α + β)f (x − r − 2) + αβf (x − r − 3)] + γ r αf (x − r − 1)∥ ≤ |γ r |ϵ,

(2.2)

∥f (x + r), α−r β[f (x + r − 1) − γf (x − r − 2)] + α−r+1 [f (x + r − 1) − (β + γ)f (x + r − 2) + βγf (x + r − 3)]α−r βf (x − r − 1)∥ ≤ |α−r |ϵ

(2.3)

for all x ∈ R and all r ∈ Z. Then we have ∥f (x), α[f (x−1)−γf (x−2)]+γf (x−1)+β n [f (x−n)−(α+γ)f (x−n−1)+αγf (x−n−2)]∥ ≤

n=1 ∑

∥f (x−r), β r α[f (x−r −1)−γf (x−r −2)]+β r+1 [f (x−r −1)−(α +γ)f (x−r −2)

r=0

+ αγf (x − r − 3)] + β r γf (x − r − 1)∥ ≤ 505

n−1 ∑

|β r |ϵ,

(2.4)

r=0

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∥f (x), β[f (x−1)−αf (x−2)]+αf (x−1)+γ n [f (x−n)−(α+β)f (x−n−1)+αβf (x−n−2)]∥ ≤

n=1 ∑

∥f (x−r), γ r β[f (x−r −1)−αf (x−r −2)]+γ r+1 [f (x−r −1)−(α +β)f (x−r −2)

r=0

+ αβf (x − r − 3)] + γ αf (x − r − 1)∥ ≤ r

n−1 ∑

|γ r |ϵ,

(2.5)

r=0

∥f (x), γ[f (x−1)−βf (x−2)]+γf (x−1)+α−n [f (x−n)−(β+γ)f (x−n−1)+βγf (x−n−2)]∥ ≤

n=1 ∑

∥f (x+r), α−r β[f (x+r−1)−γf (x−r−2)]+α−r+1 [f (x+r−1)−(β +γ)f (x+r−2)

r=0

+ βγf (x + r − 3)] + α−r βf (x − r − 1)∥ ≤

n−1 ∑

|α−r |ϵ

(2.6)

r=0

for all x ∈ R and all r ∈ Z. By (2.1), (2.2) and (2.3), we obtain that {β n [f (x − r − 1) − (α + γ)f (x − r − 2) + αγf (x − r − 3)]}, {γ n [f (x − r − 1) − (α + β)f (x − r − 2) + αβf (x − r − 3)]}, {α−n [f (x + r − 1) − (β + γ)f (x + r − 2) + βγf (x + r − 3)]} are Cauchy sequences for any fixed x ∈ R. Hence we can define the functions G1 : R → X, G2 : R → X and G3 : R → X by G1 = lim β n [f (x − r − 1) − (α + γ)f (x − r − 2) + αγf (x − r − 3)], n→∞

G2 = lim γ n [f (x − r − 1) − (α + β)f (x − r − 2) + αβf (x − r − 3)], n→∞

G3 = lim α−n [f (x + r − 1) − (β + γ)f (x + r − 2) + βγf (x + r − 3)] n→∞

for all x ∈ R and all r ∈ Z. Using the above definition of G1 , G2 and G3 , we show that there are Tribonacci functions G1 (x − 1) + G1 (x − 2) + G1 (x − 3) = β −1 limn→∞ β n+1 [f (x − (n + 1)) − (α + γ)f (x − (n + 1) − 1) + αγf (x − (n + 1) − 2)] +β −2 limn→∞ β n+2 [f (x − (n + 2)) − (α + γ)f (x − (n + 2) − 1) + αγf (x − (n + 2) − 2)] +β −3 limn→∞ β n+3 [f (x − (n + 3)) − (α + γ)f (x − (n + 3) − 1) + αγf (x − (n + 3) − 2)] = β −1 G1 (x) + β −2 G1 (x) + β −3 G1 (x) = G1 (x), G2 (x − 1) + G2 (x − 2) + G2 (x − 3) = γ −1 limn→∞ γ n+1 [f (x − (n + 1)) − (α + β)f (x − (n + 1) − 1) + αβf (x − (n + 1) − 2)] 506

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J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 16, NO.3, 2014, COPYRIGHT 2014 EUDOXUS PRESS, LLC

Tribonacci functional equation in 2-normed spaces

+γ −2 limn→∞ γ n+2 [f (x − (n + 2)) − (α + β)f (x − (n + 2) − 1) + αβf (x − (n + 2) − 2)] +γ −3 limn→∞ γ n+3 [f (x − (n + 3)) − (α + β)f (x − (n + 3) − 1) + αβf (x − (n + 3) − 2)] = γ −1 G2 (x) + γ −2 G2 (x) + γ −3 G2 (x) = G2 (x), G3 (x − 1) + G3 (x − 2) + G3 (x − 3) = α−1 limn→∞ α−n+1 [f (x − (n + 1)) − (β + γ)f (x − (n + 1) − 1) + βγf (x − (n + 1) − 2)] +α−2 limn→∞ α−n+2 [f (x − (n + 2)) − (β + γ)f (x − (n + 2) − 1) + βγf (x − (n + 2) − 2)] +α−3 limn→∞ α−n+3 [f (x − (n + 3)) − (β + γ)f (x − (n + 3) − 1) + βγf (x − (n + 3) − 2)] = α−1 G3 (x) + α−2 G3 (x) + α−3 G3 (x) = G3 (x) for all x ∈ R. It follows from (2.4), (2.5) and (2.6) that ∥f (x), (α + γ)f (x − 1) − αγf (x − 2) + G2 (x)∥ ≤ ∥f (x), (α + β)f (x − 1) − αβf (x − 2) + G2 (x)∥ ≤ ∥f (x), (β + γ)f (x − 1) − βγf (x − 2) + G3 (x)∥ ≤

1 ϵ, 1 − |β|

(2.7)

1 1 ϵ= ϵ, 1 − |γ| 1 − |β|

(2.8)

|α−1 | |β 2 | ϵ = ϵ 1 − |α−1 | 1 − |β 2 |

(2.9)

for all x ∈ R. Now, put ∆ = α2 (β − γ) + β 2 (γ − α) + γ 2 (α − β), and define γ 2 (α − β) α2 (β − γ) β 2 (γ − α) G1 (x) + G2 (x) + G3 (x) ∆ ∆ ∆ for all x ∈ R. By (2.7), (2.8) and (2.9), we have G(x) :=

∥f (x), G(x)∥ β 2 (γ − α) γ 2 (α − β) α2 (β − γ) = ∥f (x), G1 (x) + G2 (x) + G3 (x)∥ ∆ ∆ ∆ 1 ≤ [∥f (x), β 2 (γ 2 − α2 )f (x − 1) − β 2 (γ − α)αγf (x − 2) + β 2 (γ − α)G1 ∥ |∆| +∥f (x), γ 2 (α2 − β 2 )f (x − 1) − γ 2 (α − β)αβf (x − 2) + γ 2 (α − β)G2 ∥ +∥f (x), α2 (β 2 − γ 2 )f (x − 1) − α2 (β − γ)βγf (x − 2) + α2 (β − γ)G3 ∥] 1 2 |β|2 ≤ [ + ]ϵ |∆| 1 − |β| 1 − |β|2 1 2(1 + |β|) + |β|2 ≤ [ ]ϵ |∆| 1 − |β|2 for all x ∈ R.

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J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 16, NO.3, 2014, COPYRIGHT 2014 EUDOXUS PRESS, LLC

M. Eshaghi Gordji, A. Divandari, M. Rostamian, C. Park, D. Shin

On the other hand, it is easy to show that G is a Tribonacci function and this completes the proof.  Acknowledgments C. Park was supported by Basic Science Research Program through the National Research Foundation of Korea funded by the Ministry of Education, Science and Technology (NRF-2012R1A1A2004299). D. Y. Shin was supported by Basic Science Research Program through the National Research Foundation of Korea funded by the Ministry of Education, Science and Technology (NRF-2010-0021792). References [1] I. Chang, Stability of higher ring derivations in fuzzy Banach algebras, J. Computat. Anal. Appl. 14 (2012), 1059–1066. [2] I. Cho, D. Kang, H. Koh, Stability problems of cubic mappings with the fixed point alternative, J. Computat. Anal. Appl. 14 (2012), 132–142. [3] M. Eshaghi Gordji, M. Bavand Savadkouhi, M. Bidkham, Stability of a mixed type additive and quadratic functional equation in non-Archimedean spaces, J. Computat. Anal. Appl. 12 (2010), 454–462. [4] M. Eshaghi Gordji, A. Bodaghi, On the stability of quadratic double centralizers on Banach algebras, J. Computat. Anal. Appl. 13 (2011), 724–729. [5] M. Eshaghi Gordji, R. Farokhzad Rostami, S.A.R. Hosseinioun, Nearly higher derivations in unital C ∗ -algebras, J. Computat. Anal. Appl. 13 (2011), 734–742. [6] M. Eshaghi Gordji, S. Kaboli Gharetapeh, T. Karimi, E. Rashidi, M. Aghaei, Ternary Jordan derivations on C ∗ -ternary algebras, J. Computat. Anal. Appl. 12 (2010), 463–470. [7] M. Eshaghi Gordji, M. Naderi and W. Y. Lee, Solution and stability of Tribonacci functional equation, Discrete Dynamics in Nature and Society 2012, Article ID 207356, 11 pages (2012). [8] S. G¨ahler, 2-metriche Raume und ihre topologische struktur, Math. Nachr. 26 (1963), 115–148. [9] D.H. Hyers, On the stability of the linear functional equation, Proc. Natl. Acad. Sci. USA 27 (1941), 221–224. [10] H.A. Kenary, J. Lee, C. Park, Non-Archimedean stability of an AQQ-functional equation, J. Computat. Anal. Appl. 14 (2012), 211–227. [11] C. Park, Y. Cho, H.A. Kenary, Orthogonal stability of a generalized quadratic functional equation in non-Archimedean spaces, J. Computat. Anal. Appl. 14 (2012), 526–535. [12] C. Park, S. Jang, R. Saadati, Fuzzy approximate of homomorphisms, J. Computat. Anal. Appl. 14 (2012), 833–841. [13] S. Shagholi, M. Eshaghi Gordji, M. Bavand Savadkouhi, Stability of ternary quadratic derivations on ternary Banach algebras, J. Computat. Anal. Appl. 13 (2011), 1097–1105. [14] S. Shagholi, M. Eshaghi Gordji, M. Bavand Savadkouhi, Nearly ternary cubic homomorphism in ternary Fr´echet algebras, J. Computat. Anal. Appl. 13 (2011), 1106–1114. [15] S.M. Ulam, Problems in Modern Mathematics, Chapter VI, Science ed., Wiley, New York, 1940.

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J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 16, NO.3, 2014, COPYRIGHT 2014 EUDOXUS PRESS, LLC

An identity of the twisted q-Euler polynomials with weak weight α associated with the p-adic q-integrals on Zp C. S. Ryoo Department of Mathematics, Hannam University, Daejeon 306-791, Korea Abstract : In [7], we studied the twisted q-Euler numbers and polynomials with weak weight α. By using these numbers and polynomials, we investigate the alternating sums of powers of consecutive integers. By applying the symmetry of the fermionic p-adic q-integral on Zp , we give recurrence identities the twisted q-Euler polynomials with weak weight α. 2000 Mathematics Subject Classification - 11B68, 11S40, 11S80. Key words : Euler numbers and polynomials, q-Euler numbers and polynomials, q-Euler numbers and polynomials, alternating sums, the twisted q-Euler polynomials with weak weight α. 1. Introduction The Euler numbers and polynomials possess many interesting properties and arising in many areas of mathematics and physics(see [1-12]). Throughout this paper, we always make use of the following notations: 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 1 assume that |q − 1|p < p− p−1 so that q x = exp(x log q) for |x|p ≤ 1. Throughout this paper we use the notation: 1 − qx 1 − (−q)x , [x]−q = (cf. [1-6]) . [x]q = 1−q 1+q Hence, limq→1 [x] = x for any x with |x|p ≤ 1 in the present p-adic case. For g ∈ U D(Zp ) = {g|g : Zp → Cp is uniformly differentiable function}, the p-adic q-integral was defined by Kim as follows: 

N

p −1 [2]q I−q (g) = g(x)dμ−q (x) = lim g(x)(−q)x , see [1-5] . pN N →∞ 1 + q Zp x=0

(1.1)

If we take g1 (x) = g(x + 1) in (1.1), then we easily see that qI−q (g1 ) + I−q (g) = [2]q g(0).

(1.2)

N

Let Tp = ∪N ≥1 CpN = limN →∞ CpN , where CpN = {ζ|ζ p = 1} is the cyclic group of order pN . For ζ ∈ Tp , we denote by φζ : Zp → Cp the locally constant function x −→ ζ x . In [7], we defined the twisted q-Euler numbers and polynomials with weak weight α and investigate their properties. For α ∈ Z, q ∈ Cp with |1 − q|p ≤ 1, and ζ ∈ Tp , the twisted q-Euler  (α) (x) with weak weight α are defined by polynomials E n,q,ζ

(α) Fq,ζ (x, t) =

∞ 

n

 (α) (x) t = [2]qα ext . E n,q,ζ n! ζq α et + 1 n=0 509

(1.3)

RYOO 509-514

J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 16, NO.3, 2014, COPYRIGHT 2014 EUDOXUS PRESS, LLC

 (α) with weak weight α are defined by the generating function: The twisted q-Euler numbers E n,q,ζ (α) Fq,ζ (t) =

∞ 

n

 (α) t = [2]qα . E n,q,ζ n! ζq α et + 1 n=0

(1.4)

 (α) and polynomials E  (α) (x) with The following elementary properties of the q-Euler numbers E n,q,ζ n,q,ζ weak weight α are readily derived form (1.1), (1.2), (1.3) and (1.4) (see, for details, [7]). We, therefore, choose to omit details involved. Theorem 1(Witt formula). For α ∈ Z, q ∈ Cp with |1 − q|p < 1, and ζ ∈ Tp , we have   (α) n x n  (α) =  α E ζ x dμ (x), E (x) = ζ y (x + y) dμ−qα (y). −q n,q,ζ n,q,ζ Zp

Zp

Theorem 2. For any positive integer n, we have n    n  (α) n−k  (α) (x) = Ek,q,ζ x . E n,q,ζ k k=0

In this paper, by using the symmetry of p-adic q-integral on Zp , we obtain the recurrence identities the twisted q-Euler polynomials with weak weight α.

2. The alternating sums of powers of consecutive q-integers Let q be a complex number with |q| < 1 and ζ be the pN -th root of unity. By using (1.3), we give the alternating sums of powers of consecutive q-integers as follows: ∞ 

∞ n   (α) t = [2]qα α = [2] (−1)n ζ n q αn ent . E q n,q,ζ α et + 1 n! ζq n=0 n=0

From the above, we obtain −

∞ 

(−1)n ζ n q αn e(n+k)t +

n=0

∞ 

(−1)n−k ζ n−k q α(n−k) ent =

n=0

k−1 

(−1)n−k ζ n−k q α(n−k) ent .

n=0

Thus, we have − [2]qα

∞ 

(−1)n ζ n q αn e(n+k)t + [2]qα (−1)−k ζ −k q −αk

n=0

∞ 

(−1)n ζ n q αn ent

n=0 −k −k −αk

= [2]qα (−1)

ζ

q

k−1 

(2.1)

n n αn nt

(−1) ζ q

e .

n=0

By using (1.3)and (1.4), and (2.1), we obtain   ∞ ∞ k−1 j    tj tj (α) (α) t −k −k −αk −k −k −αk n n αn j   − . (−1) ζ q Ej,q,ζ (k) +(−1) ζ q Ej,q,ζ = [2]qα (−1) ζ q n j! j! j! n=0 j=0 j=0 j=0 ∞ 

By comparing coefficients of k−1 

tj in the above equation, we obtain j!

(−1)n ζ n q αn nj =

 (α)  (α) (k) + E (−1)k+1 ζ k q αk E j,q,ζ j,q,ζ [2]qα

n=0

.

By using the above equation we arrive at the following theorem: 510

RYOO 509-514

J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 16, NO.3, 2014, COPYRIGHT 2014 EUDOXUS PRESS, LLC

Theorem 3. Let k be a positive integer and q ∈ C with |q| < 1. Then we obtain k−1 

(α) Tj,q,ζ (k − 1) =

(−1)n ζ n q αn nj =

 (α)  (α) (k) + E (−1)k+1 ζ k q αk E j,q,ζ j,q,ζ [2]qα

n=0

.

Remark 4. For ζ = 1, we have lim Tj,q,ζ (k − 1) = (α)

q→1

k−1 

(−1)n nj =

n=0

(−1)k+1 Ej (k) + Ej , 2

where Ej (x) and Ej denote the Euler polynomials and Euler numbers, respectively. Next, we assume that q ∈ Cp and ζ ∈ Tp . We obtain recurrence identities the q-Euler polynomials and the q-analogue of alternating sums of powers of consecutive integers. By using (1.1), we have n−1  (−1)n−1−l q l g(l), q n I−q (gn ) + (−1)n−1 I−q (g) = [2]q l=0

where gn (x) = g(x + n). If n is odd from the above, we obtain q n I−q (gn ) + I−q (g) = [2]q

n−1 

(−1)n−1−l q l g(l) (cf. [1-5]).

(2.2)

l=0

It will be more convenient to write (2.2) as the equivalent integral form q αn



 g(x + n)dμ−qα (x) +

Zp

g(x)dμ−qα (x) = [2]qα

Zp

n−1 

(−1)k q αk g(k).

(2.3)

k=0

Substituting g(x) = ζ x ext into the above, we obtain ζ n q αn

 Zp

ζ x e(x+n)t dμ−qα (x) +

 Zp

ζ x ext dμ−qα (x) = [2]qα

Zp

x (x+n)t

ζ e Zp

(2.4)

[2]qα , ζq α et + 1

(2.5)

[2]qα . dμ−qα (x) = e α ζq et + 1 nt

By using (2.4) and (2.5), we have   n αn x (x+n)t α ζ q ζ e dμ−q (x) + Zp

(−1)j ζ j q αj ejt .

j=0

After some elementary calculations, we have  ζ x ext dμ−qα (x) = 

n−1 

Zp

ζ x ext dμ−qα (x) =

[2]qα (1 + ζ n q αn ent ) . ζq α et + 1

From the above, we get [2]qα Zp ζ x ext dμ−qα (x) [2]qα (1 + ζ n q αn ent ) = . ζq α et + 1 ζ nx q α(n−1)x entx dμ−qα (x) Zp By substituting Taylor series of ext into (2.4), we obtain    ∞  n αn x m ζ q ζ (x + n) dμ−qα (x) + m=0

=

∞  m=0

⎛ ⎝[2]qα

Zp

n−1 

⎞ (−1)j ζ j q αj j m ⎠

j=0

511

Zp

(2.6)

 ζ x xm dμ−qα (x)

tm m!

tm . m!

RYOO 509-514

J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 16, NO.3, 2014, COPYRIGHT 2014 EUDOXUS PRESS, LLC

By comparing coefficients m    m

n αn

ζ q

k

k=0

n

tm in the above equation, we obtain m!

m−k





x k

Zp

ζ x dμ−qα (x) +

Zp

ζ x xm dμ−qα (x) = [2]qα

n−1 

(−1)j ζ j q αj j m .

j=0

By using Theorem 3, we have   m    m m−k (α) n αn x k n ζ q ζ x dμ−qα (x) + ζ x xm dμ−qα (x) = [2]qα Tm,q,ζ (n − 1). k Zp Zp

(2.7)

k=0

By using (2.6) and (2.7), we arrive at the following theorem: Theorem 5. Let n be odd positive integer. Then we have ∞   tm  ζ x ext dμ−qα (x) Zp (α) . = Tm,q,ζ (n − 1) nx α(n−1)x ntx m! ζ q e dμ−qα (x) m=0 Zp Let w1 and w2 be odd positive integers. By (2.5), Theorem 5, and after some elementary calculations, we obtain the following theorem. Theorem 6. Let w1 and w2 be odd positive integers. Then we have ∞  m ζ w2 x ew2 xt dμ−qw2 α (x) [2]qw2 α   (α) Zp m t . = T (w − 1)w w w 2 [2]qα m=0 m,q 2 ,ζ 2 m! ζ w1 w2 x q α(w1 w2 −1)x ew1 w2 tx dμ−qα (x) Zp

(2.8)

By (1.1), we obtain ζ w1 x1 +w2 x2 e(w1 x1 +w2 x2 +w1 w2 x)t dμ−qw1 α (x1 )dμ−qw2 α (x2 ) Zp Zp ζ w2 x2 q α(w1 w2 −1)x ew1 w2 xt dμ−qα (x) Zp ew1 w2 xt Zp ζ w1 x1 ew1 x1 t dμ−qw1 α (x1 ) Zp ζ w2 x2 ew2 x2 t dμ−qw2 α (x2 ) . = ζ w1 w2 x q α(w1 w2 −1)x ew1 w2 xt dμ−qα (x) Zp

(2.9)

By using (2.8) and (2.9), after elementary calculations, we obtain    ζ w2 x2 ex2 w2 t dμ−qw2 α (x2 ) Zp w1 x1 (w1 x1 +w1 w2 x)t ζ e dμ−qw1 α (x1 ) a= ζ w1 w2 x q α(w1 w2 −1)x ew1 w2 xt dμ−qα (x) Zp Zp    ∞ ∞ m m  (α) w2 α  [2] t t q (α) m m  w1 w1 (w2 x)w . E T w w (w1 − 1)w2 = 1 m,q ,ζ m! [2]qα m=0 m,q 2 ,ζ 2 m! m=0

(2.10)

By using Cauchy product in the above, we have ⎛ ⎞ ∞ m   m   w α m [2]  (α)w1 w1 (w2 x)wj T(α) w2 w2 (w1 − 1)wm−j ⎠ t . ⎝ q 2 E a= 1 m−j,q ,ζ 2 j,q ,ζ [2]qα j=0 j m! m=0 By using the symmetry in (2.10), we obtain  ζ

a=  =

Zp ∞ 

m=0

w2 x2 (w2 x2 +w1 w2 x)t

e



dμ−qw2 α (x2 )

 (α) w2 w2 (w1 x)wm t E 2 m,q ,ζ

m

m!





ζ w1 x1 ex1 w1 t dμ−qw1 α (x1 ) Zp ζ w1 w2 x q α(w1 w2 −1)x ew1 w2 xt dμ−qα (x) Zp

∞ [2]qw1 α  (α) tm Tm,qw1 ,ζ w1 (w2 − 1)w1m [2]qα m=0 m!

512

(2.11)





.

RYOO 509-514

J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 16, NO.3, 2014, COPYRIGHT 2014 EUDOXUS PRESS, LLC

Thus we obtain

⎞ m   m  w α m [2]  (α)w2 w2 (w1 x)wj T(α) w1 w1 (w2 − 1)wm−j ⎠ t . ⎝ q 1 E a= 2 m−j,q ,ζ 1 j,q ,ζ [2]qα j=0 j m! m=0 ∞ 

⎛

By comparing coefficients theorem.

(2.12)

tm in the both sides of (2.11) and (2.12), we arrive at the following m!

Theorem 7. Let w1 and w2 be odd positive integers. Then we obtain m    m

 (α)w1 w1 (w2 x)wj T(α) w2 w2 (w1 − 1)wm−j E 1 m−j,q ,ζ 2 j,q ,ζ j j=0 m    m  (α) (α) Ej,qw2 ,ζ w2 (w1 x)w2j Tm−j,qw1 ,ζ w1 (w2 − 1)w1m−j , = [2]qw1 α j j=0

[2]qw2 α

 (α) (x) and T(α) (k) denote the twisted q-Euler polynomials with weak weight α and the where E k,q,ζ m,q,ζ q-analogue of alternating sums of powers of consecutive integers, respectively. By using Theorem 2, we have the following corollary: Corollary 8. Let w1 and w2 be odd positive integers. Then we obtain [2]qw1 α

j    m   j m j=0 k=0

= [2]qw2 α

k

j

 (α) w2 T(α) w1 w1 (w2 − 1) w1m−k w2j xj−k E k,q,ζ m−j,q ,ζ

j    m   j m j=0 k=0

j

k

 (α) w1 T(α) w2 w2 (w1 − 1). w1j w2m−k xj−k E k,q,ζ m−j,q ,ζ

By using (2.9), we have    w1 w2 xt w1 x1 x1 w1 t a= e ζ e dμ−qw1 α (x1 ) Zp



ζ w2 x2 ex2 w2 t dμ−qw2 α (x2 ) Zp ζ w1 w2 x q α(w1 w2 −1)x ew1 w2 xt dμ−qα (x) Zp   w2 (w1 t) x1 +w2 x+j w1 ζ w1 x1 e dμ−qw1 α (x1 )



 w1 −1 [2]qw2 α  j w2 j w2 αj = (−1) ζ q [2]qα j=0 Zp ⎛ ⎞   w ∞ 1 −1 n  w α w [2]  (α)w1 w1 w2 x + j 2 w1n ⎠ t . ⎝ q 2 = (−1)j ζ w2 j q w2 αj E ,ζ n,q [2]qα j=0 w1 n! n=0

(2.13)

By using the symmetry property in (2.13), we also have     ζ w1 x1 ex1 w1 t dμ−qw1 α (x1 ) Zp w1 w2 xt w2 x2 x2 w2 t a= e ζ e dμ−qw2 α (x2 ) ζ w1 w2 x q α(w1 w2 −1)x ew1 w2 xt dμ−qα (x) Zp Zp 



w1  w2 −1 (w2 t) x2 +w1 x+j [2]qw1 α  j w1 j w1 αj w2 x2 w2 = (−1) ζ q ζ e dμ−qw2 α (x2 ) [2]qα j=0 Zp ⎛ ⎞   w2 −1 ∞  w1 α  [2] tn w q 1 (α)  w2 w2 w1 x + j ⎝ = w2n ⎠ . (−1)j ζ w1 j q w1 αj E n,q ,ζ [2]qα j=0 w2 n! n=0 By comparing coefficients

(2.14)

tn in the both sides of (2.13) and (2.14), we have the following theorem. n!

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J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 16, NO.3, 2014, COPYRIGHT 2014 EUDOXUS PRESS, LLC

Theorem 9. Let w1 and w2 be odd positive integers. Then we have   w 1 −1 w2 (α) j w j w αj 2 2  w1n (−1) ζ q En,qw1 ,ζ w1 w2 x + j [2]qw2 α w 1 j=0 =[2]qw1 α

w 2 −1

j w1 j w1 αj

(−1) ζ

q

j=0

 (α)w2 w2 E n,q ,ζ



w1 w1 x + j w2



(2.15)

w2n .

Remark 10. Let w1 and w2 be odd positive integers. If q → 1 and ζ = 1, we have     w w 1 −1 2 −1 w2 w1 j n j w1 = w2n . (−1) En w2 x + j (−1) En w1 x + j w w 1 2 j=0 j=0 Substituting w1 = 1 into (2.15), we arrive at the following corollary. Corollary 11. Let w2 be odd positive integer. Then we obtain   w 2 −1 x+j j j αj  (α)  (α) (x) = [2]qα w2n . (−1) ζ q E E n,q,ζ n,q w2 ,ζ w2 [2]qw2 α j=0 w2 REFERENCES 1. T. Kim, q-Volkenborn integration, Russ. J. Math. Phys., 9(2002), 288-299. 2. T. Kim, q-Euler numbers and polynomials associated with p-adic q-integrals, J. Nonlinear Math. Phys., 14(2007), 15-27. 3. T. Kim, An identity of the symmetry for the Frobenius-Euler polynomials associated with the Fermionic p-adic invariant q-integral on Zp , Rocky Mountain Journal of Mathematics, 41, (2011), 239-247. 4. S-H. Rim, T. Kim and C.S. Ryoo, On the alternating sums of powers of consecutive q-integers, Bull. Korean Math. Soc., 43(2006), 611-617. 5. T. Kim, J. Choi, Y-.H. Kim, C. S. Ryoo, A Note on the weighted p-adic q-Euler measure on Zp , Advan. Stud. Contemp. Math., 21(2011), 35-40. 6. C. S. Ryoo, T. Kim, L.-C. Jang, Some relationships between the analogs of Euler numbers and polynomials, J. Inequal.Appl., 2007(2007), Art ID 86052, 22pp. 7. C. S. Ryoo, A note on the twisted q-Euler numbers and polynomials with weak weight α, Adv. Studied Theor. Phys., 6(2012), 1109-1116. 8. C. S. Ryoo, A note on the weighted q-Euler numbers and polynomials, Advan. Stud. Contemp. Math., 21(2011), 47-54. 9. C.S. Ryoo, A numerical computation of the structure of the roots of the second kind q-Euler polynomials, Journal of Computational Analysis and Applications, 14(2012), pp. 321-327. 10. C.S. Ryoo, Calculating zeros of the second kind Euler polynomials, Journal of Computational Analysis and Applications, 12(2010), 828-833. 11. Y. Simsek, Theorem on twisted L-function and twisted Bernoulli numbers, Advan. Stud. Contemp. Math., 12(2006), 237-246. 12. Y. Simsek, O. Yurekli, V. Kurt, On interpolation functions of the twisted generalized FrobeniusEuler numbers, Adv. Stud. Contemp. Math., 14(2007), 49-68.

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Two-Level Hierarchical Basis Preconditioner for Elliptic Equations with Jump Coefficients∗ Zhiyong Liu†, Yinnian He‡ November 1, 2013 Abstract. This paper provides a proof of robustness of the two-level hierarchical basis preconditioner for the linear finite element approximation of second order elliptic problems with strongly discontinuous coefficients. As a result, we prove that the convergence rate of the conjugate gradient method with two-level preconditioner is uniform with respect to large jumps and mesh sizes. Key words. Jump Coefficients, Conjugate Gradient, Effective Condition Number, Two-Level Hierarchical Basis AMS subject classifications. 65N30, 65N55, 65F10

1

Introduction

In this paper, we will discuss the two-level hierarchical basis preconditioned conjugate gradient methods for the linear finite element approximation of the second order elliptic boundary value problem  −∇ · (ω∇u) = f    u = gD    −ω ∂u = g N ∂n

in Ω on ΓD

( 1.1)

on ΓN

where Ω ∈ Rd (d = 1, 2 or 3) is a polygonal or polyhedral domain with ∗

This work was supported by the Natural Science Foundations of China (No.11271298 and No.11362021). † Center for Computational Geosciences, School of Mathematics and Statistics, Xi’an Jiaotong University, Xi’an 710049, P.R. China.(Corresponding author: [email protected]) ‡ Center for Computational Geosciences, School of Mathematics and Statistics, Xi’an Jiaotong University, Xi’an 710049, P.R. China.([email protected])

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Dirichlet boundary ΓD and Neumann boundary ΓN . The coefficient ω = ω(x) is a positive and piecewise constant function. More precisely, we assume that there are M open disjointed polygonal or polyhedral regions Ωm (m = 1, · · ·, M ) satisfying ∪M m=1 Ωm = Ω with ωm = ω|Ωm , m = 1, · · ·, M where each ωm > 0 is a constant. The analysis can be carried through to a more general case when ω(x) varies moderately in each subdomain. We assume that the subdomain Ωm : m = 1, · · ·, M are given and fixed but may possibly have complicated geometry. We are concerned with the robustness of the preconditioned conjugate gradient method in regard to both the meshsize and jump coefficients. This model problem is relevant to many applications, such as groundwater flow [1, 15], fluid pressure prediction [19], electromagnetics [13], semiconductor modeling [9], electrical power network modeling [14] and fuel cell modeling [20, 21], where the coefficients have large discontinuities across interfaces between subdomains with different material properties. The goal of the current paper is to provide proof of the robustness of the two-level hierarchical basis preconditioner (Two-Level-PCG). The rest of the paper is organized as follows. To the paper is comprehensive and self-contained, we refer directly to parts of contents in [23] and [25].(Section 2 in the paper). In Section 2, we introduce some basic notation, the PCG algorithm and some theoretical foundations. In Section 3, we introduce the two-level hierarchical basis method and preconditioner. In Section 4, we analyze the eigenvalue distribution of the two-level preconditioned system and prove the convergence rate of the PCG algorithm. Section 5 is the conclusions. Following [22], short notation x . y means x ≤ Cy; and x∼y means cx ≤ y ≤ Cx.

2 2.1

Preliminaries Notation

We introduce the bilinear form a(u, v) =

M X

1 ωm (∇u, ∇v)L2 (Ωm ) , ∀u, v ∈ HD (Ω),

m=1

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1 (Ω) = {v ∈ H 1 (Ω) : v| 1 where HD ΓD = 0}, and introduce the H -norm and seminorm with respect to any subregion Ωm by 1

kuk1,Ωm = (kuk20,Ωm + |u|21,Ωm ) 2 .

|u|1,Ωm = k∇uk0,Ωm , Thus, a(u, u) =

M X

ωm |u|21,Ωm := |u|21,ω .

m=1

We also need the weighted

L2 -inner

(u, v)0,ω =

product

M X

ωm (u, v)L2 (Ωm )

m=1

and the weighted L2 - and H 1 -norms 1

2 , kuk0,ω = (u, u)0,ω

1

kuk1,ω = (kuk20,ω + |u|21,ω ) 2 .

For any subset O ⊂ Ω, let |u|1,ω,O and kuk0,ω,O be the restrictions of |u|1,ω and kuk0,ω on the subset O, respectively. For the distribution of the coefficients, we introduce the index set I = {m : meas(∂Ωm ∩ ΓD ) = 0} where meas(·) is the d − 1 measure. In other words, I is the index set of all subregions which do not touch the Dirichlet boundary. We assume that the cardinality of I is m0 . We shall emphasize that m0 is a constant which depends only on the distribution of the coefficients.

2.2

The Discrete Systems

Given a quasi-uniform triangulation Th with the meshsize h, let 1 Vh = {v ∈ HD (Ω) : v|τ ∈ P1 (τ ), ∀τ ∈ Th }

be the piecewise linear finite element space, where P1 denotes the set of linear polynomials. The finite element approximation of (1.1) is the function u ∈ Vh , such that Z a(u, v) = (f, v) + gN v, ∀v ∈ Vh . ΓN

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We define a linear symmetric positive definite operator A : Vh → Vh by (Au, v)0,ω = a(u, v). The related inner product and the induced energy norm are denoted by p (·, ·)A := a(·, ·), k · kA := a(·, ·). Then we have the following operator equation, Au = F.

2.3

( 2.1)

Preconditioned Conjugate Gradient (PCG) Methods

The well known conjugate gradient method is the basis of all the preconditioning techniques to be studied in this paper. The PCG methods can be viewed as a conjugate gradient method applied to the preconditioned system BAu = BF. Here, B is an SPD operator, known as a preconditioner of A. Note that BA is symmetric with respect to the inner product (·, ·)B −1 (or (·, ·)A ). For the implementation of the PCG algorithm, we refer to the monographs [2, 17, 18]. Let uk , k = 0, 1, · · ·, be the solution sequence of the PCG algorithm. It is well known that Ãp !k k(BA) − 1 ku − uk kA ≤ 2 p ku − u0 kA , ( 2.2) k(BA) + 1 which implies that the PCG method generally converges faster with a smaller condition number k(BA). Even though the estimate given in (2.2) is sufficient for many applications, in general it is not sharp. One way to improve the estimate is to look at the eigenvalue distribution of BA(see [2, 12] for more details). More specifically, suppose that we can divide σ(BA), the spectrum of BA, into two sets, σ0 (BA) and σ1 (BA), where σ0 consists of all ”bad” eigenvalues and the remaining eigenvalues in σ1 are bounded above and below, then we have the following theorem. Theorem 2.1 Suppose that σ(BA) = σ0 (BA) ∪ σ1 (BA) such that there are m elements in σ0 (BA) and λ ∈ [a, b] for each λ ∈ σ1 (BA). Then Ãp !k−m b/a − 1 ku − uk kA ≤ 2K p ku − u0 kA , ( 2.3) b/a + 1

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Y

max

λ∈σ1 (BA)

µ∈σ0 (BA)

5

¯ ¯ ¯ ¯ ¯1 − λ ¯ . ¯ µ¯

If there are only m small eigenvalues in σ0 , say 0 < λ1 ≤ λ2 · ·· ≤ λm ¿ λm+1 ≤ · · · ≤ λn , then

¯ µ ¶m m ¯ Y ¯ ¯ λ n ¯1 − ¯ ≤ λn − 1 = (k(BA) − 1)m . K= ¯ λi ¯ λ1 i=1

In this case, the convergence rate estimate (2.3) becomes ku − uk kA ≤ 2(k(BA) − 1)m ku − u0 kA

Ãp

b/a − 1

!k−m

p

b/a + 1

.

( 2.4)

Based on (2.4), given a tolerance 0 < ² < 1, the number of iterations of the k kA PCG algorithm needed for ku−u ku−u0 kA < ² is given by ! Ãp µ ¶ ¶ b/a + 1 2 . k ≥ m + log + mlog(k(BA) − 1) /log p ² b/a − 1 µ

( 2.5)

Observing the convergent estimate (2.4), if there are only a few small eigenvalues of BA in σ0 (BA), then √ the convergent rate of the PCG methods b/a+1 will be dominated by the factor √ , i.e., by b/a where b = λn (BA) and b/a−1

a = λm+1 (BA). We define this quantity as the ”effective condition number”. Definition. ([23]) Let V be a Hilbert space. The m-th effective condition number of an operator A : V → V is defined by km+1 (A) =

λmax (A) λm+1 (A)

where λm+1 (A) is the (m+1)-th minimal eigenvalue of A. To estimate the effective condition number, we need to estimate λm+1 (A). A fundamental tool is the following Courant-Fisher min-max theorem. Theorem 2.2 The eigenvalues of a SPD operator A : V → V are characterized by the relation λm+1 (A) =

min

max

S,dim(S)=n−m. x∈S,x6=0

519

(Ax, x) . (x, x)

( 2.6)

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Especially, for any subspace V0 ⊂ V with dim(V0 ) = n − m, the following estimation of λm+1 (A) holds: λm+1 (A) ≥ min

06=x∈V0

3

(Ax, x) . (x, x)

( 2.7)

Two-Level Hierarchical Basis Preconditioner

The classical two-level hierarchical basis method was proposed and developed by Axelsson, Bank, Dupont, and Yserentant [3, 4, 5, 6, 24]. As usual, we assume that V is decomposed as a direct sum V = SVs ⊕ P Vc .

( 3.1)

for some components Vs and Vc isomorphic to Rns and Rnc respectively, ¶ µ with I n = ns + nc . A typical and simple example to keep in mind is S = 0 ¶ µ W for some W such that the square matrix (S, P ) is unit and P = I upper triangular, and hence invertible.

3.1

Some notation

Two ingredients (the space decomposition (3.1) and the smoother M ) are important in the two-level hierarchical basis method. Various restrictions of M and A to the subspaces mentioned before will be needed. We first define the exact coarse grid matrix Ac and its hierarchical complement As as follows Ac = P T AP, As = S T AS. Later we will see, in the case of a two level hierarchical basis preconditioner, one needs M to be well-defined only on the first component SVs . In that case, we refer to M as Ms . Then Ms = S T M S. In order to define the hierarchical basis preconditioner, we also need two symmetrized version of the smoother M : f = M T (M T + M − A)−1 M, M

( 3.2)

M = M (M T + M − A)−1 M T .

( 3.3)

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v3

v3

θ3

θ3

v1

θ1

7

θ2

v2

v1

θ1

θ2

v2

Figure 1: Quadratic element(left) and piecewise linear element(right). If we assume that A = D − L − LT where D, L, LT are the diagonal, lower triangle, and upper triangle part of A, and let M = D − LT , then −1 fs−1 = (D − LT )−1 M s Ds (D − L)s ,

( 3.4)

where Ds = S T DS, (D−LT )s = S T (D−LT )S, and (D−L)s = S T (D−L)S.

3.2

The Element Stiffness Matrix for The Hierarchical Basis

In this subsection, we consider the stiffness matrix for the hierarchical basis in each element. Following Braess [7], and Bank [6], simply we let ω = 1 in (1.1) and let t be a triangle with vertices vi , edges ei , and angles θi , 1 ≤ i ≤ 3. Here, we consider two kinds of different hierarchical basis: the quadratic element and piecewise linear element. For the space of continuous quadratic finite elements (illustrated on the left in Figure 1), we let φi , 1 ≤ i ≤ 3 denote the linear basis functions for element t. Then on element t, the subspace P Vc will be the span of hφi i3i=1 .

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And the subspace SVs is composed of the quadratic bump functions hψi i3i=1 , where ψi = 4φj φk , and (i, j, k) is a cyclic permutation of (1, 2, 3). For the space of continuous piecewise linear polynomials on a refined mesh (illustrated on the right in Figure 1), let P Vc be defined as the quadratic finite elements above. But the subspace SVs contains the continuous piecewise polynomials on the fine grid that are zero at the vertices of t. Then on element t, the subspace SVs = hφbi i3i=1 , where φbi is the standard nodal piecewise linear basis functions associated with the midpoint of edge ei of t. Following [6] and [7], we can establish the relation Li = cotθi = −2|t|∇φj · ∇φk ,

( 3.5)

where |t| is measure of element t, it is about hd , d = 1, 2, 3. Then the element stiffness matrix for the quadratic hierarchical basis can be shown to be ¶ µ ∗ ∗ t , ( 3.6) AQ = ∗ Ats where Ats is the restriction of As on the element t, and   L1 + L2 + L3 −L3 −L2 4 . −L3 L1 + L2 + L3 −L1 Ats =  3 −L2 −L1 L1 + L2 + L3 The diagonal of Ats is   L1 + L2 + L3 0 0 4 . 0 L1 + L2 + L3 0 Dst =  3 0 0 L1 + L2 + L3

( 3.7)

( 3.8)

The element stiffness matrix for the piecewise linear hierarchical basis is given by µ ¶ ∗ ∗ t . ( 3.9) AL = ∗ Ats In this case, 

and

 L1 + L2 + L3 −L3 −L2 , −L3 L1 + L2 + L3 −L1 Ats =  −L2 −L1 L1 + L2 + L3

( 3.10)

 L1 + L2 + L3 0 0 . 0 L1 + L2 + L3 0 Dst =  0 0 L1 + L2 + L3

( 3.11)



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3.3

9

The Two-Level Hierarchical Basis Preconditioner

In this subsection, we can define the two-level hierarchical basis preconditioner using some notations in above subsections. Let ¶µ µ ¶µ ¶ I Ms−T S T AP I 0 Ms 0 b . ( 3.12) BT L = P T ASMs−1 I 0 I 0 Ac Then, the two-level hierarchical basis preconditioner is defined by BT−1L =

4

¡

S, P

¢

b −1 B TL

¡

S, P

¢T

.

( 3.13)

The Condition Number Analysis of BT−1L A

Following [11], for the two-level hierarchical basis preconditioner BT−1L , we have following estimate. Lemma 4.1 Assume that (Ms + MsT − As ) is S.P.D, for any v ∈ V , the following bounds hold: 1 T v BT L v ≤ v T Av ≤ v T BT L v, K

K . sup w

1 fs−1 As ) λ(M

.

( 4.1)

f−1 is given by (3.4), then we have the following relationship between If M s the symmetric Gauss-Seidel preconditioner and the Jacobi preconditioner. Lemma 4.2 For any v ∈ V , we have 1 T fs v ≤ v T Ds v. v Ds v ≤ v T M 4 Proof:

( 4.2)

fs = (D − L)s D−1 (D − LT )s . M s

Following the Schwarz inequality we can prove the second inequality. Then, we prove the first inequality. Using the fact that Ds and As are S.P.D, then for any v ∈ V we have 1 1 ((D − L)s v, v)A = ((As + Ds )v, v)A ≥ (Ds v, v)A . 2 2 Taking v = (D − LT )−1 s w, we have for all w ∈ V : 1 −T −1 −T −1 (Ds (D−L)−T s w, (D−L)s w)A ≤ ((D−L)s w, w)A = (Ds (D−L)s w, Ds w)A . 2

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On the other hand, 1 f−1 f−1 w, w)1/2 (D−1 w, w)1/2 . (Ms w, w)A ≤ (M s s A A 2 Consequently,

f−1 w, w)A ≤ 4(D−1 w, w)A , (M s s

and

fs v, v)A ≥ 1 (Ds v, v)A . (M 4 The proof of Lemma 4.2 can be found in [26]. Following lemma provides the eigenvalue estimate of Jacobi preconditioner. Lemma 4.3 For any v ∈ V , we have v T Ds v∼h−2 kvk20,ω . Proof: Note that on each element, we have 3 X

Li = −2hd (∇φ1 ∇φ2 + ∇φ1 ∇φ3 + ∇φ2 ∇φ3 )∼hd−2 .

i=1

Consequently, following [23] we have v T Ds v∼hd−2 (v, v)l2 ,ω ∼h−2 kvk20,ω . This completes the proof. In order to research the effective condition number for the Jacobi preconditioner, we need to define the space ½ ¾ Z eh = v ∈ Vh : V v = 0, m ∈ I . Ωm

On this space, the following Poincare-Friedrichs inequality holds: kvk0,ω . k∇vk0,ω ,

eh . ∀v ∈ V

( 4.3)

Then, we have following important lemma. Lemma 4.4 Assume that the triangulation Th is quasi-uniform, then we have h2 J(ω)−1 v T Ds v . v T As v, ∀v ∈ Rn , ( 4.4) and h2 v T Ds v . v T As v, where J(ω) =

eh . ∀v ∈ V

( 4.5)

maxm ωm minm ωm .

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Proof: In fact, we have a(v, v) ≥ min{ωm }|v|21,Ω & min{ωm }kvk20,Ω ≥ m

m

minm {ωm } −2 2 h (h )kvk20,Ω . maxm {ωm }

Applying Lemma 4.3 and inequality (4.3), we also have v T Ds v .

X

h−2 kvk20,ω,τ = h−2

M X

ωm kvk20,Ωm . h−2 |v|21,ω = h−2 v T As v.

m=1

τ ∈Th

This completes the proof. Followed by Lemmas 4.1-4.4, we have the following results regarding the condition number of BT−1L A. Theorem 4.1 For the hierarchical basis preconditioner BT−1L defined by (3.13), the condition number and m0 -th effective condition number satisfies: k(BT−1L A) ≤ J(ω)h−2 ,

km0 +1 (BT−1L A) ≤ h−2 .

Theorem 4.2 For the hierarchical basis preconditioned conjugate gradient methods, we have the following convergence rate µ ¶k−m0 ku − uk kA 2 −2 m0 ≤ 2(C1 J(ω)h − 1) 1− , k ≥ m0 . ( 4.6) ku − u0 kA 1 + C2 h−1

5

Conclusions

In this paper, we provided a proof of robustness of the two-level hierarchical basis preconditioner for the linear finite element approximation of second order elliptic problems with strongly discontinuous coefficients. We discussed the eigenvalue distribution of the Two-Level-preconditioner and found that only a few small eigenvalues infected by the large jump.

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J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 16, NO.3, 2014, COPYRIGHT 2014 EUDOXUS PRESS, LLC

A New Fourth-Order Explicit Finite Difference Method for the Solution of Parabolic Partial Differential Equation with Nonlocal Boundary Conditions M. Ghoreishi∗, A.I.B.Md.Ismail∗ and A. Rashid†

Abstract In this paper, a new fourth-order explicit finite difference method is proposed for solving linear and nonhomogeneous parabolic partial differential equation with nonlocal boundary conditions. The advantage of the explicit finite difference methods is easier to implement than the implicit methods. Moreover, the explicit method need lesser CPU time than the implicit schemes. Numerical results show that the proposed method is very accurate and effective. Key words: Finite difference method, Fourth-order explicit method, Nonlocal boundary conditions, nonhomogeneous parabolic partial differential equation.

1

Introduction

Many physical phenomena can be modelled by parabolic partial differential equations which involve integral terms in the boundary conditions. These boundary conditions are called nonlocal boundary conditions. One-dimensional parabolic equation with nonlocal boundary conditions have important applications in electro-chemistry, porous media flow, thermo-elasticity, heat conduction and several others. The existence, uniqueness and theoretical aspects of these equations have been studied by [17, 20, 35]. Generally, it is difficult to find the analytical solution of parabolic partial differential equations with nonlocal boundary conditions. Approximate and numerical techniques for obtaining approximate solution of these equations have been developed by many researchers [5, 6, 7, 8, 9, 11, 19, 23, 27, 29, 28]. Some standard numerical methods have been used for the solution of one dimensional diffusion equation with nonlocal boundary conditions such as finite difference method, finite element method, adomian decomposition method (ADM), Chebyshev spectral collocation method, reducing kernel space method and method of lines [1, 13, 21, 24, 25, 26, 30]. In this paper a method based on explicit finite difference method is introduced and applied to obtain the numerical solution of the following parabolic equation: ∂u ∂2u = + q(x, t), ∂t ∂x2

0 ≤ x ≤ 1 , 0 ≤ t ≤ T,

(1.1)

with initial condition u(x, 0) = f (x),

0 ≤ x ≤ 1,

and subject to the boundary conditions Z 1 u(0, t) = φ(x, t)u(x, t)dx + g1 (t),

(1.2)

0 < t ≤ T,

(1.3)

0 < t ≤ T,

(1.4)

0

Z u(1, t) =

1

ψ(x, t)u(x, t)dx + g2 (t), 0

∗ School

of Mathematical Science, Universiti Sains Malaysia, 11800, Penang, Malaysia. of Mathematics, Gomal University, Dera Ismail Khan, Pakistan, E-Mail:

† Department

[email protected]

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J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 16, NO.3, 2014, COPYRIGHT 2014 EUDOXUS PRESS, LLC

where q(x, t), f (x), g1 (t), g2 (t), φ(x, t) and ψ(x, t) are known functions. Many authors applied various type of finite difference methods to obtain the numerical solution of equations (1.1)–(1.4). Dehghan [15, 16] applied the forward Euler, backward Euler, BTCS (backward in time and centered in space) schemes, Crandall’s implicit formula, FTCS (forward in time and centered in space) method for the heat equation. The nonlocal boundary conditions have been approximated by Trapezoidal rule and fourth-order Simpson composite formula. Zhou et al. [36], Mu and Du [25] introduced an efficient technique based on reproducing kernel space to solve the partial differential equations with nonlocal boundary conditions. The BTCS and explicit Crandall’s formula have been developed by Martin-Vaquero and Vigo-Aguiar [32, 31] to solve the above mentioned equations . The aim of this paper is to describe an efficient technique based on explicit finite difference method to find out the numerical solution of parabolic equation with nonlocal boundary conditions. The new method is of fourth order and it is compared with BTCS, Crank-Nicolson and Crandall’s formula. The n+1 n+1 n basic idea of this approach is to write q(x, t) as a linear combination of qi−1 , qin+1 , qi+1 , qi−1 , qin and n qi+1 . The objective of this technique is to improve the results obtained by many researchers in some n+1 n+1 n n papers [3, 10, 16, 26, 31, 32, 36]. We considered that coefficient qi+1 , qi−1 and qi−1 , qi+1 are not equal. The nonlocal boundary conditions are solved by higher order Integration rules. This paper is organized as follows: In section 2, the new fourth-order explicit technique is presented, the composite Simpson rule and sixth-order formula for the nonlocal boundary conditions are also introduced. Numerical results are presented in section 3. Finally conclusion is given in section 4.

2

Explicit Finite Difference Method

The domain [0, 1] × [0, T ] is divided into an M × N mesh with a spatial size of h = 1/M and temporal size k = T /N . The grid points (xi , tn ) are defined by xi = ih, tn = nk,

i = 0, 1, ..., M, n = 0, 1, ..., N,

where M and N are integers. The notation uni , qin , φni , ψin , g1n and g2n represents, respectively, the finite difference approximations of u(xi , tn ), q(xi , tn ), φ(xi , tn ), ψ(xi , tn ), g1 (tn ) and g2 (tn ). The FTCS (forward in time and centrad in space) finite difference scheme for the heat equation (1.1) can be written as un+1 = runi−1 + (1 − 2r)uni + runi+1 + kqin , (2.1) i for i = 1, ..., M − 1, n = 0, 1, ..., N − 1 and r = [16]:

k h2 .

The stability condition for this method is proved in

1 . 2 The local truncation error of this method can be written as [31, 33]: r≤

τ = (ut − uxx − q) +

6rutt − uxxxx 2 60r2 uttt − uxxxxxx 4 h + h + O(h6 ). 12 360

(2.2)

Now it can be verified that utt = uxxxx + qxx + qt .

(2.3)

By substituting (2.3) into (2.2) the truncation error can be obtained as τ=

6rqt + 6rqxx + (6r − 1)uxxxx 2 60r2 uttt − uxxxxxx 4 h + h + O(h6 ). 12 360

(2.4)

n+1 n+1 It is clear that (2.4) is second order. Now, we write qin using a linear combination of qi−1 , qin+1 , qi+1 , n n qi−1 , qin and qi+1 , then we have

un − 2uni + uni−1 un+1 − uni n+1 n+1 n n i = i+1 + a1 qi−1 + a2 qin+1 + a3 qi+1 + a4 qi−1 + a5 qin + a6 qi+1 . k h2

(2.5)

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J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 16, NO.3, 2014, COPYRIGHT 2014 EUDOXUS PRESS, LLC

By using the Taylor’s expansion in (2.5), we can obtain à 6 ! X h2 ai − 1 + h(a3 − a1 − a4 + a6 )qx + rh2 (a1 + a2 + a3 )qt + (a1 + a3 + 2 i=1

(2.6)

h3 a4 + a6 )qxx − (a6 − a4 )qxxx + rh3 (a3 − a1 )qxt + A(u(x, t), q)h4 + O(h6 ) = 0. 6 where

−15qtt + 10utt − 15qxxxx − 6uxxxxxx + (60 − 180a5 )qxxt . 2160 By substituting (2.4), (2.6) and (2.7) into (2.5), we get the following system of linear equations A(u(x, t), q) =

a1 + a2 + a3 + a4 + a5 + a6 = 1, a3 − a1 − a4 + a6 = 0, 1 a1 + a2 + a3 = , 2 a6 − a4 = 0, a3 − a1 = 0, a1 + a3 + a4 + a6 = r,

(2.7)

(2.8)

and r = 1/6. By selecting a5 = m thus equation (2.5) can be written as k n+1 n+1 [(6m − 2)(qi−1 + qi+1 ) 12 n n + (10 − 12m)qin+1 − (6m − 3)(qi−1 + qi+1 ) + 12mqin ].

un+1 = runi−1 + (1 − 2r)uni + runi+1 + i

(2.9)

It should be noted that this technique is fourth order accurate when r = 1/6. By considering (2.9), we can consider many values for m such that the solution of these equation become converges to the exact solution. Our goal in this paper is to improve the results obtained in the literature. To find the optimal value of m, we can apply the following algorithm. 1. Step 1: We consider,

m1 =

a1 , b1

m2 =

a2 b2

2. Step 2: We calculate, Em1 and Em2 3. Step 3: If Em1 < Em2 then, m2 =

a1 + a2 , b 1 + b2

else m1 =

a1 + a2 b 1 + b2

4. Step 4: If |Emi | < l then m = mi is optimal i = 1 or 2, else we repeat Step 1. Equation (2.9) has M −1 linear equations and M +1 unknowns. Thus two more equations are needed. The integral in the boundary conditions can be approximated by composite Simpson rule and sixth order formula. 2.1 Composite Simpson formula The Simpson composite formula for solving the nonlocal boundary conditions (1.3) and (1.4) can be written as [16, 31]: Z un+1 0

=

1

φ(x, tn+1 )u(x, tn+1 )dx + g1n+1 =

0 M/2

+4

X i=1

h ¡ n+1 n+1 φ u0 3 0 (2.10)

M/2−1 n+1 φn+1 2i−1 u2i−1

+2

X

n+1 φn+1 2i u2i

+

n+1 φn+1 M uM )

+

g1n+1

4

+ O(h ),

i=1

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J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 16, NO.3, 2014, COPYRIGHT 2014 EUDOXUS PRESS, LLC

and Z un+1 = M

1

ψ(x, tn+1 )u(x, tn+1 )dx + g2n+1 =

0 M/2

X

+4

h n+1 n+1 (ψ u0 3 0 (2.11)

M/2−1 n+1 n+1 ψ2i−1 u2i−1

X

+2

i=1

n+1 n+1 ψ2i u2i

+

n+1 n+1 ψM uM )

+

g2n+1

4

+ O(h ).

i=1

Thus M/2

(hφn+1 0

−

3)un+1 0

+ 4h

X

M/2−1 n+1 φn+1 2i−1 u2i−1

X

+ 2h

i=1

X

(2.12)

n+1 n+1 n+1 ψ2i u2i + (hψM − 3)un+1 = −3g1n+1 , M

(2.13)

i=1

M/2

hψ0n+1 un+1 + 4h 0

n+1 n+1 φn+1 + hφn+1 = −3g1n+1 , 2i u2i M uM

M/2−1 n+1 n+1 ψ2i−1 u2i−1 + 2h

i=1

X i=1

Combining (2.12) and (2.13) with (2.9) gives (M + 1) × (M + 1) linear system of equations. We can obtain n+1 F1 (Φ, U )(hψM − 3) − hF2 (Ψ, U )φn+1 M , un+1 = 0 J(Φ, Ψ, U ) F2 (Ψ, U )(hφn+1 − 3) − hF1 (Φ, U )ψ0n+1 0 , J(Φ, Ψ, U )

un+1 = M where





M/2

F1 (Φ, U ) = −4h   F2 (Ψ, U ) = −4h 

X



M/2

X



n+1  − 3g1n+1 , φn+1 2i u2i

i=1



M/2−1

n+1 n+1  ψ2i−1 u2i−1 − 2h 

X

n+1 n+1  − 3g2n+1 , ψ2i u2i

i=1

i=1

and



M/2−1

n+1  − 2h  φn+1 2i−1 u2i−1

i=1

X



n+1 n+1 J(Φ, Ψ, U ) = (hφn+1 − 3)(hψM − 3) − h2 φn+1 6= 0. 0 M ψ0

2) Sixth-order formula The sixth-order integration formula can be used to approximate numerically the integral present in the boundary conditions (1.3) and (1.4). We can write the sixth-order formula as [31]: Z 1 n+1 u0 = u(0, tn+1 ) = φ(x, tn+1 )u(x, tn+1 )dx + g1 (tn+1 ) 0

=

M/2 M/4−1 X X 2h n+1 n+1 n+1 n+1 [7φ0 u0 + 32 φn+1 u + 12 φn+1 2i−1 2i−1 4i+2 u4i+2 45 i=1 i=0

(2.14)

M/4−2

X

+ 14

n+1 n+1 n+1 n+1 φn+1 + O(h6 ), 4i+4 u4i+4 + 7φM uM ] + g1

i=0

and Z un+1 = u(1, tn+1 ) = M =

1

ψ(x, tn+1 )u(x, tn+1 )dx + g1 (tn+1 ) 0 M/2 M/4−1 X X 2h n+1 n+1 n+1 n+1 [7ψ0n+1 un+1 + 32 φ u + 12 ψ4i+2 u4i+2 0 2i−1 2i−1 45 i=1 i=0

(2.15)

M/4−2

+ 14

X

n+1 n+1 n+1 n+1 ψ4i+4 u4i+4 + 7ψM uM ] + g1n+1 + O(h6 ),

i=0

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RASHID ET AL 528-539

J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 16, NO.3, 2014, COPYRIGHT 2014 EUDOXUS PRESS, LLC

where M should be a multiple of 4. So n+1 (14hφn+1 − 45)un+1 + 14hφn+1 = F1 (Φ, U ), 0 0 M uM

(2.16)

n+1 14hψ0n+1 un+1 + (14hψM − 45)un+1 = F2 (Ψ, U ), 0 M

(2.17)

where M/2

F1 (Φ, U ) = −64h

X

M/4−1

X

n+1 φn+1 2i−1 u2i−1 − 24h

i=1

i=0

M/2

F2 (Ψ, U ) = −64h

X

M/4−2 n+1 φn+1 4i+2 u4i+2 − 28h

i=1

X

n+1 n+1 φn+1 , 4i+4 u4i+4 − 45g1

i=0

M/4−1 n+1 n+1 ψ2i−1 u2i−1 − 24h

X

M/4−2 n+1 n+1 ψ4i+2 u4i+2 − 28h

i=0

X

n+1 n+1 ψ4i+4 u4i+4 − 45g1n+1 ,

i=0

Combining (2.16), (2.17) with (2.9) gives (M + 1) × (M + 1) linear system of equations. We can obtain

where

un+1 = 0

n+1 F1 (Φ, U )(14hψM − 45) − 14hF2 (Ψ, U )φn+1 M , J(Φ, Ψ, U )

un+1 = M

F2 (Ψ, U )(14hφn+1 − 45) − 14hF1 (Φ, U )ψ0n+1 0 , J(Φ, Ψ, U )

n+1 n+1 n+1 J(Φ, Ψ, U ) = 196h2 (φn+1 ψM − φn+1 ) − 630h(φn+1 + ψM ) + 2025. 0 0 M ψ0

It should be noted that the system (2.8) does not have unique solution. Thus it can help us to obtain the optimal value of m whilst it was not considered in [31]. To check the accuracy of present method, we compared our results with the results obtained in [31]. It should also be noted that the explicit finite difference methods are easier to implement than the implicit schemes or Crank-Nicolson method, because in explicit schemes there is only one unknown is involved in the finite difference formula. Moreover, implicit finite difference schemes require the solution of a large number of simultaneous linear algebraic equations at each steps resulting in an extensive amount of CPU time utilized compared to explicit finite difference methods for the same values of s and h.

3

Illustrative Examples

In this section, the new explicit finite difference method (NFTCS) applied to linear and nonhomogeneous parabolic partial differential equation (1.1) with nonlocal boundary conditions (1.3)–(1.4). The results show that the described method is very accurate, capable and powerful. The numerical results indicate that the approximate solution convergence to the exact solution as h tends to zero. The Simpson formula and sixth-order formula are used to approximate the integral in the examples. The MATHEMATICA software is used to find the approximate solution and CPU time. For describing the error, we define relative error ER and the absolute error EA as follows: ER (u(x, t)) =

|u(ih, jk)approx − u(ih, jk)exact | |u(ih, jk)exact |

and EA (u(x, t)) = |u(ih, jk)approx − u(ih, jk)exact | where u(ih, jk)approx is the approximate solution and u(ih, jk)exact is the exact solution. Example 1: We consider the nonhomogeneous parabolic partial differential equation [16, 31, 32, 33, 36] ∂2u ∂u = + q(x, t), ∂t ∂x2

0 ≤ x ≤ 1 , 0 ≤ t ≤ T,

(3.1)

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with the following initial and boundary conditions f (x) = x2 , φ(x, t) = x, ψ(x, t) = x, −1 g1 (t) = , 4(t + 1)2 3 , g2 (t) = 4(t + 1)2 −2(x2 + t + 1) , q(x, t) = (t + 1)3

0 < x < 1, 0 < t < 1, 0 < t < 1,

0 < x < 1, 0 < x < 1,

0 < t < 1, 0 < t < 1, 0 ≤ t ≤ 1,

It can be verified that the exact solution is

µ

u(x, t) =

x t+1

0 < x < 1.

¶2 .

By applying the algorithm which is introduced in section 2, we take m = 0.427487 in (2.9) and obtained results are shown in tables 1 and 2 for various values of x and h. The Simpson composite rule and sixth-order formula are used to approximate the nonlocal boundary conditions (1.3) and (1.4). It can be seen that the errors are very small. In the last row in each table, we have obtained the CPU time consumed in the implementation of NFTCS for various step size h and x at t = 1. As expected, the CPU time increases as the step size h decrease. Table 1: Absolute error NFTCS at t = 1 by using the Simpson formula. x/h

0.25

0.125

0.0625

0.03125

0 0.25

1.97727×10−9 8.07057×10−8

2.73501×10−13 4.87357×10−9

4.75005×10−14 3.03945×10−10

1.46390×10−14 1.89884×10−11

0.5 0.75 1

3.92653×10−8 3.27403×10−8 1.97727×10−9

2.26709×10−9 2.21648×10−9 2.73392×10−13

1.40977×10−10 1.39182×10−10 4.75009×10−14

8.80097×10−12 8.70845×10−12 1.43112×10−14

CPU

0.078

2.248

76

1006.45

Table 2: Absolute error NFTCS at t = 1 by using the sixth-order formula. x/h

0.25

0.125

0.0625

0.03125

0

2.32131×10−10

7.89537×10−13

5.05059×10−14

3.47797×10−14

0.25 0.5 0.75 1

7.77293×10−8 3.60167×10−8 3.57167×10−8 2.32131×10−10

4.86301×10−9 2.25557×10−9 2.22705×10−9 7.89538×10−13

3.03905×10−10 1.40933×10−10 1.39223×10−10 5.05068×10−14

1.89883×10−11 8.80078×10−11 8.70862×10−12 3.47777×10−14

CPU

0.078

1.391

53.687

807.840

The relative error ER for u(0.5, 1) is obtained for different step size h and compared the results obtain by [31] (scheme FTCS4). We used the algorithm introduced in section 2 with optimal value of m. The Simpson formula (NFTCS4) and sixth-order formula (NFTCS6) have been used for approximating the integrals in the nonlocal boundary conditions. Table 3: Relative error ER for u(0.5, 1) at various spatial length. h 0.25 0.125 0.0625 0.03125

FTCS4 [31]

NFTCS4

NFTCS6

0.000127462 7.96658×10−6 4.97912×10−7 3.11195×10−8

6.28724×10−7

5.76746×10−7 3.61190×10−8 2.25683×10−9 1.40812×10−10

3.63034×10−8 2.25753×10−9 1.40816×10−10

In table 4, we compared the relative error ER at x = 0.5 and t = 1 by using NFTCS described in this paper and FTCS, explicit Crandall’s formula (ECF) and implicit Crandall’s formula (ICF) obtained in [31]. It is observed that the our results are better than the obtained results in [31]. 6 533

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Table 4: Comparison between relative error ER in [31] and our solution for u(0.5, 1). FTCS [31]

ECF [31]

ICF [31]

Our method

1/8

h

7.96658×10−6

1/16

4.97912×10−7

6.19972×10−6 3.87463×10−7

1.32985×10−5 8.31040×10−7

3.63034×10−8 2.25753×10−9

1/32

3.11195×10−8

2.42163×10−8

5.19395×10−8

4.37617×10−11

In Figure 1, we show the absolute error EA using the NFTCS at x = 0.125 and x = 1 with h = 1/8, while the Simpson formula is used for approximating the integrals in the boundary conditions. Similarly, the graph of absolute error NFTCS at x = 0.0625 and x = 1 with h = 1/16 is shown in Figure 2. Absolute error for x=0.125

Absolute error for x=1

5. ´ 10-8 5. ´ 10-8 4. ´ 10-8 4. ´ 10-8 3. ´ 10

Error

Error

-8

3. ´ 10-8

2. ´ 10-8

2. ´ 10-8

1. ´ 10

-8

1. ´ 10-8

0

0 0

100

200

300

0

100

200

k

300

k

Figure 1: The absolute error between solution obtained by using NFTCS and the exact solution for x = 0.125 and x = 1 with h = 1/8, while the Simpson formula is used for approximating the integrals in boundary conditions.

Absolute error for x=0.0625

Absolute error for x=1 3.5 ´ 10-9

2.5 ´ 10

-9

3. ´ 10-9 2. ´ 10-9

2.5 ´ 10-9 Error

Error

1.5 ´ 10-9

2. ´ 10-9

1.5 ´ 10-9

1. ´ 10-9

1. ´ 10-9 5. ´ 10-10

5. ´ 10-10

0

0 0

500

1000

1500

0

k

500

1000

1500

k

Figure 2: The absolute error between solution obtained by using NFTCS and the exact solution for x = 0.0625 and x = 1 with h = 1/16, while the Simpson formula is used for approximating the integrals in boundary conditions. The consumed CPU time of three numerical schemes FTCS [31], FTCS4, and FTCS6 is obtained and the graph of CPU time is shown in Figure 3. It is clear from the Figure 3 that with the same step size h, our method consumed less CPU time than other two numerical schemes. Example 2: To check the performance of the explicit finite difference scheme described in section 2, we take the

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CPU time of relation error for uH0.5,1L

2500 : FTCS @31D 2000 CPU Time

: NFTCS4 1500

: NFTCS6 1000

500

0 0

5. ´ 10-7

1. ´ 10-6

1.5 ´ 10-6

Relative error

Figure 3: The CPU time spent to find the relative error ER for u(0.5, 1).

heat equation (1.1) with the initial condition and the boundary conditions as follows [12, 24, 18]: f (x) = x2 − x +

δ , 6(1 + δ)

φ(x, t) = −δ, ψ(x, t) = −δ, g1 (t) = 0, g2 (t) = 0, µ q(x, t) = − x2 − x +

0 < x < 1,

¶ δ + 2 e−t , 6(1 + δ)

0 < x < 1, 0 < x < 1,

0 ≤ t ≤ 1, 0 ≤ t ≤ 1, 0 < x < 1, 0 < x < 1,

0 < x < 1,

0 ≤ t ≤ 1,

with δ = 0.0144. It can be verified that the exact solution is µ ¶ δ u(x, t) = x2 − x + e−t . 6(1 + δ) We take m = 0.5166667, in equation (2.9) and the obtained results are shown in table 5 and table 6. The Simpson composite rule and sixth-order formula are used to approximate the nonlocal boundary conditions (1.3) and (1.4). It can be seen that the errors are very small with different value of x and t. In the last row in each table, we observed that the consumed CPU time increase with the decrease of step size h . Table 5: Absolute error NFTCS at t = 1 by using the Simpson formula. x/h

0.25

0.125

0.0625

0.03125

0 0.25 0.5 0.75 1

7.11516×10−12 1.37993×10−9 2.54796×10−9 1.37993×10−9 7.11516×10−12

5.71649×10−13 8.67588×10−11 1.58515×10−10 8.67587×10−11 5.71649×10−13

3.62120×10−14 5.42373×10−12 9.90530×10−12 5.42373×10−12 3.62121×10−14

2.23064×10−15 3.36300×10−13 6.22738×10−13 3.36300×10−13 2.23064×10−15

CPU

0.093

1.407

53.578

1670.780

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Table 6: Absolute error NFTCS at t = 1 by using the Sixth-order formula. x/h

0.25

0.125

0.0625

0.03125

0

9.18984×10−12

0.25

1.37764×10−9

5.79563×10−13 8.67501×10−11

3.62435×10−14 5.42373×10−12

2.23118×10−15 3.36300×10−13

0.5

2.55032×10−9

1.58524×10−10

9.90531×10−12

6.22738×10−13

0.75 1

1.37764×10−9

8.67501×10−11

5.42373×10−12

9.18984×10−12

5.79563×10−13

3.62435×10−14

3.36300×10−13 2.23117×10−15

CPU

0.079

1.406

52.985

1706.580

In table 7, we present a comparison between the numerical solution of this problem by using new explicit finite difference method and those obtained by the method described in [31]. we observed that the relative error ER of present method is better than the method described in [31]. Further more the Simpson formula (NFTCS4) and sixth-order formula (NFTCS6) have been used for approximating the integrals in the nonlocal boundary conditions. Table 7: Relative error ER for u(0.5, 1) at various spatial length. h

Method in [31]

NFTCS4

NFTCS6

0.25

3.12655×10−5

0.125 0.0625

1.95407×10−6 1.22130×10−7

2.79690×10−8 1.74002×10−9 1.08731×10−10

2.79949×10−8 1.74012×10−9 1.08731×10−10

0.03125

7.63314×10−9

6.83580×10−12

6.83580×10−12

In Figure 4, we show the absolute error EA using the NFTCS at x = 0.125 and x = 1 with step size h = 1/8 when the Simpson formula is used for approximating the integrals in the boundary conditions. Similarly, we show the absolute error NFTCS at x = 0.0625 and x = 1 with step size h = 1/16 in Figure 5. Absolute error for x=0.125

Absolute error for x=1 1.5 ´ 10

-12

2. ´ 10-10

1. ´ 10-12 Error

Error

1.5 ´ 10-10 1. ´ 10-10

5. ´ 10-13 5. ´ 10-11 0

0 0

100

200

300

0

k

100

200

300

k

Figure 4: The absolute error between solution obtained by using NFTCS and the exact solution for x = 0.125 and x = 1 with h = 1/8, while the Simpson formula is used for approximating the integrals in boundary conditions. The consumed CPU time to obtain the numerical solution of the present method with different algorithm to to solve the non local boundary conditions are shown Figure 6. It is clear that for the all small value of step size h, the consumed CPU time in applying the our methods and consumed CPU time by the algorithm proposed in [31] are almost same.

4

Conclusion

Several approaches have been developed for obtaining the numerical solution of heat equation with non-local boundary conditions. A new fourth-order explicit finite difference method has been applied to 9 536

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Absolute error for x=0.0625

Absolute error for x=1 1. ´ 10-13

3. ´ 10

-11

2. ´ 10-11

6. ´ 10-14

Error

8. ´ 10-14

Error

2.5 ´ 10-11

1.5 ´ 10

-11

4. ´ 10-14 1. ´ 10-11 2. ´ 10-14

5. ´ 10-12 0

0 0

500

1000

1500

0

500

1000

k

1500

k

Figure 5: The absolute error between solution obtained by using NFTCS and the exact solution for x = 0.0625 and x = 1 with h = 1/16, while the Simpson formula is used for approximating the integrals in boundary conditions. CPU time of relative error for uH0.5,1L

1500

CPU Time

: FTCS @31D

: NFTCS4

1000

: NFTCS6 500

0 0

5. ´ 10-8

1. ´ 10-7

1.5 ´ 10-7

2. ´ 10-7

2.5 ´ 10-7

3. ´ 10-7

Relative error

Figure 6: The CPU time spent to find the relation error ER for u(0.5, 1) in table 7.

obtain numerical solution of one dimensional linear and non-homogeneous parabolic partial differential equation with nonlocal boundary conditions in this paper. The present method is also capable for solving parabolic type partial differential equations with non-local boundary conditions.

References [1] A. Whye-Teong, Numerical solution of a non-classical parabolic problem: An integro-differential approach, Appl. Math. Comput., 175 (2006), 969–979. [2] N. Borovykh, Stability in the numerical solution of the heat equation with nonlocal boundary conditions, Appl. Numer. Math., 42 (2002), 17–27. [3] A. Boutayeb and A. Chetouani, A numerical comparison of different methods applied to the solution of problems with non-local boundary conditions, Appl. Math. Sci., 1(44) (2007), 2173–2185. [4] A. Boutayeb and A. Chetouani, Global extrapolation of numerical methods for solving a parabolic problem with nonlocal boundary conditions, Int. J. Comput. Math., 80 (2003), 789–797.

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[5] B. Cahlon, D.M. Kulkarni and P. Shi, Stepwise stability for the heat equation with a nonlocal constraint, SIAM J. Numer. Anal. 32 (1995), 571–593. [6] J. R. Cannon, The solution of the heat equation subject to the specification of energy, Quart. Appl. Math., 21 (1963) 155–160. [7] J. R. Cannon and H. M. Yin, On a class of non-classic parabolic problems, J. Differen. Equat., 79 (1989), 266–288. [8] V. Capasso, K. Kunisch, A reaction-diffusion system arising in modelling man-environment diseases, Quart. Appl. Math., 46 (1988), 431–449. [9] Y. Choi and K. Chan, A parabolic equation with nonlocal boundary conditions arising from electrochemistry. Non. Anal. Theor. Math. Appl., 18 (1992), 317–327. [10] D.S. Daoud, Determination of the source parameter in a heat equation with a non-local boundary conditions, J. Comput. Appl. Math., 221 (2008), 261–272. [11] W.A. Day, Parabolic equations and thermodynamics, Quart. Appl. Math., 50 (1992), 523–533. [12] W.A. Day, Existence of a property of solutions of the heat equation subject to linear thermoelasticity and other theories, Quart. Appl. Math., 40 (1982), 319–330. [13] M. Dehghan, The use of Adomian decomposition method for solving the one dimensional parabolic equation with non-local boundary specification, Int. J. Comput. Math., 81 (2004) 25–34. [14] M. Dehghan, Numerical solution of a parabolic equation with non-local boundary specifications, Appl. Math. Comput., 145 (2003), 185–194. [15] M. Dehghan, The one-dimensional heat equation subject to a boundary integral specification, Chaos Solit. Fract., 32 (2007), 661–675. [16] M. Dehghan, Efficient techniques for the second-order parabolic equation subject to nonlocal specifications, Appl. Numer. Math., 52 (2005), 39–62. [17] K. Deng, Comparison Principle for some nonlocal problems, Quart. Appl. Math., 50 (1992), 517–522. [18] G. Ekolin, Finite difference methods for a non-local boundary value problem for the heat equation. BIT, 31 (1991), 245–55. [19] R. E. Ewing and T. Lin, A class of parameter estimation techniques of fluid flow in porous media, Adv. Water Resour., 14 (1993), 89–97. [20] A. Friedman, Monotonic decay of solutions of parabolic equations with nonlocal boundary conditions, Quart. Appl. Math., 44 (1986), 401–407. [21] A. Golbabai and M. Javidi, A numerical solution for non-classical parabolic problem based on Chebyshev spectral collocation method, Appl. Math. Comput., 190 (2007), 179–185. [22] F. Ivanauskas, T. Mekauskas and M. Sapagovas, Stability of difference schemes for two-dimensional parabolic equations with non-local boundary conditions, Appl. Math. Comput., 215 (2009), 2716– 2732. [23] N.I. Kamynin, A boundary value problem in the theory of the heat conduction with non-classical boundary condition, USSR Comput. Math. Math. Phys., 4 (1964), 33–59. [24] L. Yunkang, Numerical solution of the heat equation with nonlocal boundary conditions, J. Comput. Appl. Math., 110 (1999), 115–127. [25] L. Mu and H. Du, The solution of a parabolic differential equation with non-local boundary conditions in the reproducing kernel space, Appl. Math. Comput., 202 (2008), 708–714.

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[26] M. A. Rehman and M. S. A. Taj, Fourth-Order method for non-homogeneous heat equation with nonlocal boundary conditions, Appl. Math. Sci., 3(37) 2009, 1811–1821. [27] A. Samarskii, Some problems in differential equations theory. J. Differ. Equat., 16 (1980), 1221–1228. [28] P. Shi, Weak solution to evolution problem with a nonlocal constraint, SIAM J. Numer. Anal., 24 (1993), 46–58. [29] P. Shi and M. Shillor, Design of contact patterns in one-dimensional thermoelasticity, Theoretical aspects of industrial design. Philadelphia, PA: Society for Industrial and Applied Mathematics, 1992. [30] Z.Z. Sun, A second-order accurate finite difference scheme for a class of nonlocal parabolic equations with natural boundary conditions, J. Comput. Appl. Math., 76 (1996) 137–146. [31] J. Martn-Vaquero, Two-level fourth-order explicit schemes for diffusion equations subject to boundary integral specifications, Chao. Solit. Fract. 42 (2009), 2364–2372. [32] J. Martin-Vaquero and J. Vigo-Aguiar, A note on efficient techniques for the second-order parabolic equation subject to non-local conditions, Appl. Numer. Math., 59 (2009), 1258–1264. [33] J. Martin-Vaquero, J. Vigo-Aguiar, On the numerical solution of the heat conduction equations subject to nonlocal conditions, Appl. Numer. Math., 59 (2009), 2507–2514 [34] S. Wang and Y. Lin, A numerical method for the diffusion equation with nonlocal boundary specifications. Int. J. Eng. Sci., 28 (1990), 543–600. [35] Y.F. Yin, On nonlinear parabolic equations with nonlocal boundary condition, J. Math. Anal. Appl., 185 (1994), 54–60. [36] Y. Zhoua, M. Cui and Y. Lin, Numerical algorithm for parabolic problems with non-classical conditions, J. Comput. Appl. Math., 230 (2009), 770–780.

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Numerical Solutions of Fourth Order Lidstone Boundary Value Problems Using Discrete Quintic Splines Fengmin Chen and Patricia J. Y. Wong School of Electrical and Electronic Engineering Nanyang Technological University 50 Nanyang Avenue, Singapore 639798, Singapore E-mail: [email protected], [email protected] Abstract. In this paper, the numerical treatment of a fourth order Lidstone boundary value problem is proposed with the use of a discrete quintic spline based on central differences. It is shown that the method is of order 4 if a parameter takes a specific value, else it is of order 2. A well known numerical example is presented to illustrate our method as well as to compare the performance with other numerical methods proposed in the literature. Keywords : Discrete quintic spline, central difference, Lidstone boundary value problem, numerical solution, fourth order.

1

Introduction

We consider the fourth order Lidstone boundary value problem y (4) (x) = f (x)y(x) + g(x), a ≤ x ≤ b y(a) = A1 ,

y(b) = B1 ,

y 00 (a) = A2 ,

y 00 (b) = B2

(1.1)

where f (x) and g(x) are continuous on [a, b] and Ai , Bi , i = 1, 2 are arbitrary real finite constants. Lidstone boundary value problems have received a lot of attention in the literature, notably on the existence of positive solutions, see for example [1, 7, 22] and the references cited therein. The fourth order Lidstone boundary value problem (1.1) considered arises from the physical problem of bending a rectangular simply supported beam resting on an elastic foundation [14, 17], here y is the vertical deflection of the plate. The use of polynomial splines in the numerical treatment of (1.1) has gathered substantial interests over the years. Usmani and Warsi [20] have used quintic and sextic splines respectively to develop second and fourth order convergent methods for (1.1). Thereafter, quartic splines

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Numerical Solutions of 4th Order Lidstone BVP

are employed by Usmani [19] to formulate second order convergent method. Also, during their investigation on fourth order obstacle boundary value problems, Al-Said and Noor [2] and Al-Said et al [3] have respectively used cubic and quartic splines to obtain second order convergent methods for (1.1). Recently, nonpolynomial spline functions have been proposed by Ramadan et al [12] to obtain second and fourth order convergent methods for (1.1), these methods reduced to those of [2, 19, 20] when certain parameters take certain values. A related problem to (1.1) arises from the bending of a long uniformly loaded rectangular plate supported over the entire surface by an elastic foundation and rigidly supported along the edges [14, 17], here the boundary conditions are the conjugate type y(a) − A1 = y(b) − B1 = y 0 (a) − A2 = y 0 − B2 = 0. For this problem, second order convergent methods based on quintic splines have been established in [13, 16, 18], while fourth order convergent method based on sextic splines has been discussed in [18]. The general observation from all these research is that spline methods usually give better (or comparable) approximation than finite difference methods and shooting type methods. Motivated by all the above research especially the use of splines in solving (1.1), in this paper we shall employ a discrete quintic spline to get a numerical solution of (1.1). Our proposed method is fourth-order convergent when a parameter takes certain value, else it is second-order convergent. Through a well know numerical example, we illustrate that our method outperforms other spline methods for solving (1.1) in the literature [2, 3, 12, 19, 20]. Discrete splines were first introduced by Mangasarian and Schumaker [11] in 1971 as solutions to constrained minimization problems in real Euclidean space, which are discrete analogs of minimization problems in Banach space whose solutions are generalized splines. Subsequent investigations on discrete splines can be found in the work of Schumaker [15], Astor and Duris [4], Lyche [9, 10] and Wong et al [5, 6, 21]. Following [9, 10], the discrete spline we use will involve central differences. The plan of the paper is as follows. In section 2, we shall derive our method. The matrix form of the method is presented in section 3 and its convergence analysis is performed. In section 4, we present a well known example and compare the performance of our method with other methods in the literature.

2

Numerical Method for (1.1)

Suppose P : a = x0 < x1 < · · · < xn = b is a uniform mesh of [a, b] with xi − xi−1 = p, 1 ≤ i ≤ n, i.e., the step size p = b−a n . Let h ∈ (0, p] be a given constant. We recall the central difference operator

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Dh applying to a function F (x) gives F (x + h) − F (x − h) ; 2h F (x + h) − 2F (x) + F (x − h) {2} ; Dh F (x) = h2 F (x + 2h) − 2F (x + h) + 2F (x − h) − F (x − 2h) {3} Dh F (x) = ; 2h3 F (x + 2h) − 4F (x + h) + 6F (x) − 4F (x − h) + F (x − 2h) {4} . Dh F (x) = h4 {0}

{1}

Dh F (x) = F (x);

Dh F (x) =

We also use the basic polynomials x{j} introduced by [10] x{j} = xj , j = 0, 1, 2; x{4} = x2 (x2 − h2 ),

x{3} = x(x2 − h2 ),

x{5} = x(x2 − h2 )(x2 − 4h2 ).

{1}

{1}

It is noted that Dh x{j} = jx{j−1} , j = 0, 1, 2, 3, 5 and Dh x{4} = 2x(2x2 + h2 ). Definition 1. Let S(x; h) be a piecewise continuous function defined over [a, b] (with mesh P ) and Si (x) be its restriction in [xi−1 , xi ], 1 ≤ i ≤ n passing through the points (xi−1 , si−1 ) and (xi , si ). We say S(x; h) is a discrete quintic spline if Si (x), 1 ≤ i ≤ n is a polynomial of degree 5 or less and (Si+1 − Si )(xi + jh) = 0,

j = −2, −1, 0, 1, 2,

1 ≤ i ≤ n − 1.

(2.1)

The above definition is in the spirit of discrete cubic spline studied in [10]. In fact, in terms of central differences, the condition (2.1) has the following equivalent form {j}

{j}

Dh Si (xi ) = Dh Si+1 (xi ),

j = 0, 1, 2, 3, 4,

1 ≤ i ≤ n − 1.

(2.2)

Throughout, we shall use the notations (k)

yi

= y (k) (xi ), {2}

Mi = Dh Si (xi ),

fi = f (xi ),

gi = g(xi ),

si = Si (xi ),

{4}

0 ≤ i ≤ n.

Fi = Dh Si (xi ),

We propose si ’s to be the numerical solution of (1.1) at the mesh points, i.e., yi ∼ = si ,

0 ≤ i ≤ n.

(2.3)

Discretizing (1.1) and noting the Lidstone boundary conditions, we set s0 = y0 = A1 ,

sn = y n = B 1 ,

Mn = yn00 = B2 ,

M0 = y000 = A2 ,

Fi = fi si + gi ,

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0 ≤ i ≤ n.

(2.4)

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Numerical Solutions of 4th Order Lidstone BVP

We shall now obtain an explicit expression of Si (x) in terms of its central dif¯ (x) satisfy the following ¯ i (x) and h ferences. To begin, let the functions hi (x), h i for 0 ≤ i, j ≤ n : hi (xj ) = δij ,

{2}

{4}

Dh hi (xj ) = Dh hi (xj ) = 0,

{2} ¯ Dh h i (xj ) = δij ,

¯ i (xj ) = D{4} h ¯ i (xj ) = 0, h h

{4} ¯ Dh h i (xj ) = δij ,

¯ (x ) = D{2} h ¯ (x ) = 0. h i j i j h

By direct computation, we obtain the explicit expressions: x − xi−1 , x ∈ [xi−1 , xi ], 1 ≤ i ≤ n hi (x) = p xi+1 − x , x ∈ [xi , xi+1 ], 0 ≤ i ≤ n − 1 = p

¯ i (x) h

=

0,

otherwise;

=

(p2 − h2 )(x − xi−1 ) (x − xi−1 ){3} − , 6p 6p x ∈ [xi−1 , xi ], 1 ≤ i ≤ n

=

(xi+1 − x){3} (p2 − h2 )(xi+1 − x) − , 6p 6p x ∈ [xi , xi+1 ], 0 ≤ i ≤ n − 1

¯ (x) h i

=

0,

=

(x − xi−1 ){5} (p2 − h2 )(x − xi−1 ){3} − 120p 36p +

otherwise;

(x − xi−1 )(p2 − h2 )(7p2 + 2h2 ) , 360p x ∈ [xi−1 , xi ], 1 ≤ i ≤ n {5}

=

(xi+1 − x) 120p +

−

(p2 − h2 )(xi+1 − x){3} 36p

(xi+1 − x)(p2 − h2 )(7p2 + 2h2 ) , 360p x ∈ [xi , xi+1 ], 0 ≤ i ≤ n − 1

=

0,

otherwise.

Clearly, Si (x), the restriction of S(x; h) in [xi−1 , xi ], can be expressed as ¯ (x) ¯ i−1 (x) + Mi h ¯ i (x) + Fi−1 h Si (x) = si−1 hi−1 (x) + si hi (x) + Mi−1 h i−1 ¯ (x), +Fi h i

x ∈ [xi−1 , xi ], 1 ≤ i ≤ n. (2.5)

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{1}

{1}

Using (2.5), the ‘continuity’ requirement Dh Si (xi ) = Dh Si+1 (xi ), 1 ≤ i ≤ n − 1 leads to the equation (p2 − h2 )Mi−1 + 2(h2 + 2p2 )Mi + (p2 − h2 )Mi+1 = 6(si−1 − 2si + si+1 ) +  +(2h2 + 7p2 )Fi+1 .

(p2 − h2 )  2 (2h + 7p2 )Fi−1 + 4(4p2 − h2 )Fi 60

{3} Dh Si (xi )

{3} Dh Si+1 (xi ),

(2.6) 1 ≤ i ≤ n−1

Further, the ‘continuity’ requirement = yields  1 2 (p − h2 )Fi−1 + 2(h2 + 2p2 )Fi + (p2 − h2 )Fi+1 . Mi−1 − 2Mi + Mi+1 = 6 (2.7) Using (2.6) and (2.7) in a lengthy algebraic procedure, we are able to eliminate M ’s and get the ‘F -equation’ as a1 Fi−2 + a2 Fi−1 + a3 Fi + a2 Fi+1 + a1 Fi+2 = si−2 − 4si−1 + 6si − 4si+1 + si+2 ,

2≤i≤n−2

(2.8)

where 2(p2 − h2 )(8h2 + 13p2 ) (p2 − h2 )(p2 − 4h2 ) , a2 = , 120 120 (2.9) 6(4h4 + 5h2 p2 + 11p4 ) a3 = . 120 Upon substituting Fj = fj sj + gj into (2.8), we see that (2.8) gives (n − 3) equations with (n − 1) unknowns si , 1 ≤ i ≤ n − 1. a1 =

In order to solve for the unknown si ’s, we need two more equations which we write as b1 F0 + b2 F1 + b3 F2 + b4 F3 = p2 M0 + b5 s0 + b6 s1 + b7 s2 + b8 s3

(2.10)

and c1 Fn−3 + c2 Fn−2 + c3 Fn−1 + c4 Fn = p2 Mn + c5 sn−3 + c6 sn−2 + c7 sn−1 + c8 sn (2.11) where bi and ci , 1 ≤ i ≤ 8 are real numbers. We require the local truncation errors in both (2.10) and (2.11) to be O(p8 ) (the reason will be clear when we perform the convergence analysis in section 3). To fulfill this, we carry out Taylor (k) series expansion in (2.10) about x0 and set the coefficients of s0 , 0 ≤ k ≤ 7 to zeros. This yields 8 equations which we can solve to get bi , 1 ≤ i ≤ 8. Similarly, (k) in (2.11) we expand about xn and set the coefficients of sn , 0 ≤ k ≤ 7 to zeros, then we solve 8 equations to get ci , 1 ≤ i ≤ 8. The resulting (2.10) and (2.11) are given as follows p4 (28F0 + 245F1 + 56F2 + F3 ) − p2 M0 = −2s0 + 5s1 − 4s2 + s3 , 360

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(2.12)

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Numerical Solutions of 4th Order Lidstone BVP

p4 (Fn−3 + 56Fn−2 + 245Fn−1 + 28Fn ) − p2 Mn = sn−3 − 4sn−2 + 5sn−1 − 2sn . 360 (2.13) Once again, we substitute Fj = fj sj + gj into (2.12) and (2.13) to give two equations in si , i = 1, 2, 3, n − 3, n − 2, n − 1. Noting (2.4) the values of s0 , sn , M0 and Mn are already known, hence we can now solve (2.8), (2.12), (2.13) to obtain the values of si , 1 ≤ i ≤ n − 1. The solvability of the linear system will be discussed in section 3.

3

Convergence Analysis

In this section, we shall establish the existence of a unique solution for (2.8), (2.12), (2.13) and also conduct a convergence analysis for the method presented in section 2. To begin, we define the norms of a column vector T = [ti ] and a matrix A = [aij ] as follows: kT k = max |ti | i

and

kAk = max i

X

|aij |.

j

Let ei = yi − si , 1 ≤ i ≤ n − 1 be the errors. Let Y = [yi ], S = [si ], W = [wi ], T = [ti ] and E = [ei ] be (n − 1)-dimensional column vectors. The system (2.8), (2.12), (2.13) can be written as AS = W

(3.1)

where A = A0 + Q,

Q = BF,

F = diag(fi ), i = 1, 2, . . . , n − 1,

(3.2)

A0 and B are (n − 1) × (n − 1) five-band symmetric matrices given by 

5

−4

1



 −4 6 −4 1    1 −4 6 −4 1   .. A0 =  .    1 −4 6 −4 1    1 −4 6 −4 1

545

−4

      ,      

(3.3)

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

− 245p 360

       B=      

4

4

4



− 56p 360

p − 360

−a2

−a3

−a2

−a1

−a1

−a2

−a3

−a2 ..

−a1

−a1

.

−a2

−a3

−a2

−a1

−a1

−a2

−a3

−a2

4

p − 360

4

− 56p 360

− 245p 360

4

             

(3.4)

and for the vector W = [wi ], we have  p4   2s0 − p2 M0 + (28f0 s0 + 28g0 + 245g1 + 56g2 + g3 ), i=1   360     −s0 + a1 f0 s0 + a1 g0 + a2 g1 + a3 g2 + a2 g3 + a1 g4 , i=2      a g + a g + a g + a g + a g , 3 ≤ i ≤ n −3  1 i−2 2 i−1 3 i 2 i+1 1 i+2  −sn + a1 gn−4 + a2 gn−3 + a3 gn−2 + a2 gn−1 + a1 gn + a1 fn sn , wi =    i=n−2     4   p   (gn−3 + 56gn−2 + 245gn−1 + 28gn + 28fn sn ), 2sn − p2 Mn +   360   i = n − 1. (3.5) From (3.1) we have A(Y − E) = W or AY = W + T

(3.6)

T = AE.

(3.7)

where For 2 ≤ i ≤ n − 2, the i-th equation of the linear system (3.7) is (4)

(4)

(4)

(4)

(4)

yi−2 − 4yi−1 + 6yi − 4yi+1 + yi+2 = a1 yi−2 + a2 yi−1 + a3 yi + a2 yi+1 + a1 yi+2 + ti where ti ’s are the local truncation errors given by ti =

p4 (p2 − 3h2 ) (6) p4 (4p4 − 15p2 h2 + 8h4 ) (8) yi + yi + O(p9 ). 12 240

(3.8)

For i = 1, n − 1, the i-th equations of the linear system (3.7) are respectively −2y0 + 5y1 − 4y2 + y3 =

 p4  (4) (4) (4) (4) − p2 y000 + t1 28y0 + 245y1 + 56y2 + y3 360

and yn−3 − 4yn−2 + 5yn−1 − 2yn  p4  (4) (4) (4) = yn−3 + 56yn−2 + 245yn−1 + 28yn(4) − p2 yn00 + tn−1 360

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Numerical Solutions of 4th Order Lidstone BVP

where t1 and tn−1 are the local truncation errors given by t1 = tn−1 = −

241 8 (8) p yi + O(p9 ). 60480

Remark 1. For the special case h = ti = −

√p , 3

(3.9)

it is clear from (3.8) that

p8 (8) y + O(p9 ), 2160 i

2 ≤ i ≤ n − 2.

Thus, taking (3.9) into consideration, we have kT k =

241 8 p L 60480

(3.10)

where L = maxx |y (8) (x)|. Lemma 1. [2] The matrix A0 is invertible and kA−1 0 k≤

5(b − a)4 + 4(b − a)2 p2 5n4 + 4n2 = . 384 384p4

(3.11)

Lemma 2. [8] Let D be a square matrix such that kDk < 1. Then, (I + D) is nonsingular and 1 k(I + D)−1 k ≤ . 1 − kDk Our first result guarantees the existence of a unique solution for (2.8), (2.12), (2.13). Theorem 1. The system (3.1) has a unique solution if 489 ˆ Kf < 1 480 where K =

5(b−a)4 +4(b−a)2 p2 384

(3.12)

and fˆ = max1≤i≤n−1 |fi |.

Proof. If (3.1) has a unique solution, then it can be written as −1 −1 −1 S = A−1 W = (A0 + Q)−1 W = [A0 (I + A−1 W = (I + A−1 A0 W. 0 Q)] 0 Q) (3.13) −1 From Lemma 1 the inverse A−1 exists, hence it remains to show that (I +A 0 0 Q) is nonsingular. 489 4 p . Since Q = BF, we find From (3.4), a direct computation gives kBk ≤ 480

kQk ≤ kBk kF k ≤

547

489 4 ˆ p f. 480

(3.14)

WONG ET AL 540-551

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F. Chen and P. J. Y. Wong

It follows from (3.11) and (3.14) that 5(b − a)4 + 4(b − a)2 p2 384p4



 489 4 ˆ 489 ˆ p f = Kf < 1 480 480 (3.15) where we have used (3.12) in the last inequality. Since kA−1 0 Qk < 1, we conclude from Lemma 2 that (I+A−1 0 Q) is nonsingular. Hence, (3.1) has a unique solution given by (3.13).  −1 kA−1 0 Qk ≤ kA0 k kQk ≤

The next result gives the order of convergence of our method. Theorem 2. Suppose

489 ˆ 480 K f
max |2α(c + d) − 2b(β + γ)|, |2γ(b + c) − 2d(α + β)| , |2β(b + d) − 2c(α + γ)|}

(7)

Then the positive equilibrium point of Equation (1) is locally asymptotically stable. Proof. It is follows by theorem (1.1) that equation (5) is asymptotically stable if all roots of equation (6) lie in the open disc, |λ| < 1 that is if |a3 | + |a2 | + |a1 | + |a0 | < 1 [(bβ − cα) + (bγ − dα)](1 − a) a + (α + β + γ)(b + c + d) [−(bβ − cα) + (cγ − dβ)](1 − a) [−(bγ − dα) − (cγ − dβ)](1 − a) + 0, B2 > 0 , and B3 > 0 . In this case we see from equation (8) that (bβ − cα) + (bγ − dα) − (bβ − cα) + (cγ − dβ) − (bγ − dα) − (cγ − dβ) < (α + β + γ)(b + c + d) if and only if (α + β + γ)(b + c + d) > 0 which is always true. 2. B1 > 0, B2 > 0 , and B3 < 0 . It follows from equation (8) that (bβ − cα) + (bγ − dα) − (bβ − cα) + (cγ − dβ) + (bγ − dα) + (cγ − dβ) < (α + β + γ)(b + c + d) if and only if (α + β + γ)(b + c + d) > 2γ(b + c) − 2d(α + β) which is satisfied by Condition (7). Also, we can prove the other cases. The proof is complete.

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T. F. Ibrahim

3

Boundedness of Solutions of Equation (1)

Here we study the boundedness nature and persistence of solutions of Equation (1). Theorem 3.1. Every solution of Equation (1) is bounded and persists if a < 1. Proof. Let {xn }∞ n=−t be a solution of Equation (1). It follows from Equation (1) that bxn−l + cxn−s + dxn−r xn+1 = axn−k + αxn−l + βxn−s + γxn−r = axn−k +

bxn−l cxn−s + αxn−l + βxn−s + γxn−r αxn−l + βxn−s + γxn−r +

dxn−r αxn−l + βxn−s + γxn−r

Then xn+1 ≤ axn−k +

bxn−l c d cxn−s dxn−r b + + = xn−k + + + αxn−l βxn−s γxn−r α β γ

for all

n≥1

By using a comparison, we see that lim sup xn ≤

n→∞

bβγ + cαγ + dαβ =M αβγ(1 − a)

(9)

Thus the solution is bounded. Now we wish to show that there exists m > 0 such that xn ≥ m for all n ≥ 1. The transformation xn = y1n will reduce Equation (1) to the equivalent form yn+1 = yn−k (αyn−s yn−r + βyn−l yn−r + γyn−l yn−s ) a(αyn−s yn−r + βyn−l yn−r + γyn−l yn−s ) + yn−k (byn−s yn−r + cyn−l yn−r + dyn−l yn−s ) It follows that yn+1 ≤

yn−k (αyn−s yn−r + βyn−l yn−r + γyn−l yn−s ) yn−k (byn−s yn−r + cyn−l yn−r + dyn−l yn−s )

αyn−s yn−r βyn−l yn−r γyn−l yn−s α β γ + + = + + byn−s yn−r cyn−l yn−r dyn−l yn−s b c d αcd + βbd + γbc = = H for all n ≥ 1 bcd Thus we obtain ≤

xn =

1 1 bcd ≥ = = m for all yn H αcd + βbd + γbc

n≥1

From Equations (9) and (10) we see that m ≤ xn ≤ M for all Therefore every solution of Equation (1) is bounded and persists.

556

(10) n ≥ 1 .

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Periodicity and Global Attractivity of Difference Equation

Theorem 3.2. Every solution of Equation (1) is unbounded if a > 1. Proof. Let {xn }∞ n=−t be a solution of Equation (1). Then from Equation (1) we see that

xn+1 = axn−k +

bxn−l + cxn−s + dxn−r > axn−k αxn−l + βxn−s + γxn−r

for all

n≥1

We see that the right hand side can write as follows yn+1 = ayn−k ⇒ ykn+i = an yk+i ,

i = 0, 1, ..., k,

and this equation is unstable because a > 1, and lim yn = ∞ Then by using ratio test {xn }∞ n=−t is unbounded from above.

4

n→∞

Existence of Periodic Solutions

In this section we study the existence of periodic solutions of equation (1). The following theorem states the necessary and sufficient conditions that this equation has periodic solutions of prime period two and there is clear that there exists a sixteen cases of the indexes s, l, k, r as we see in the following theorem and lemmas. Theorem 4.1. Equation (1) has positive prime period two solutions if and only if one of the following statements holds 1. (b+d−c)(α+γ −β)(1+a)+4(aβ(b+d)+c(α+γ)) > 0,α+γ > β,b+d > c and l, r-odd, k, s-even. 2. (c + d − b)(β + γ − α)(1 + a) + 4(aα(c + d) + b(β + γ)) > 0, β + γ > α, c + d > b and k, r-odd, l, s-even. 3. (b + c − d)(α + β − γ)(1 + a) + 4(aγ(b + c) + d(α + β)) > 0, α + β > γ, b + c > d and k, l-odd, r, s-even. 4. (b − c − d)(α − β − γ)(1 + a) + 4(ab(β + γ) + α(c + d)) > 0, α > β + γ, b > c + d and l-odd, k, s, r-even. 5. (c − b − d)(β − α − γ)(1 + a) + 4(ac(α + γ) + β(b + d)) > 0, β > α + γ, c > b + d and k-odd, l, s, r-even. 6. (d − b − c)(γ − α − β)(1 + a) + 4(ad(β + α) + γ(b + c)) > 0 , γ > α + β, d > b + c and r-odd, l, k, s-even. 7. (c + d − b)(α − β − γ) − 4b(β + γ) > 0, a < 1, α > β + γ, c + d > b and k, s, r-odd, l-even 8. (b + d − c)(β − α − γ) − 4c(α + γ) > 0, a < 1, β > α + γ, b + d > c and l, s, r-odd, k-even.

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T. F. Ibrahim

9. (b + c − d)(γ − α − β) − 4d(α + β) > 0, a < 1, γ > α + β, b + c > d and k,s,l-odd, r-even. 10. (d − b − c)(α + β − γ) − 4γ(b + c) > 0, a < 1, α + β > γ, d > b + c and s, r-odd, l, k-even 11. (c − b − d)(α + γ − β) − 4β(b + d) > 0, a < 1, α + γ > β, c > b + d and s, k-odd, l, r-even. 12. (b − c − d)(β + γ − α) − 4α(c + d) > 0, a < 1, β + γ > α, b > c + d and s, l-odd, r, k-even. Proof. We will prove the theorem when Condition (1) is true and the proof of the other cases are similar and so we will be omitted. First suppose that there exists a prime period two solution ..., p, q, p, q, ..., of equation (1). We will prove that Condition (1) holds.We see from equation (1) that p = aq +

ep + cq bp + cq + dp = aq + αp + βq + γp f p + βq

where e = b + d, f = α + γ, and q = ap +

bq + cp + dq eq + cp = ap + αq + βp + γq f q + βp

Then f p2 + βpq = af pq + aβq 2 + ep + cq,

(11)

f q 2 + βpq = af pq + aβp2 + eq + cp,

(12)

Subtracting (11) from (12) gives f (p2 − q 2 ) = −aβ(p2 − q 2 ) + (e − c)(p − q) . Since p 6= q, it follows that e−c (13) p+q = f + aβ Again, adding (11) and (12) yields (f − aβ)(p2 + q 2 ) + 2(β − af )pq = (e + c)(p + q)

(14)

It follows by (13),and (14) that pq =

(eaβ + cf )(e − c) (f + aβ)2 (β − f )(1 + a)

(15)

Now it is clear from equation (13) and equation (15) that p and q are the two distinct roots of the quadratic equation (f + aβ)t2 − (e − c)t +

(eaβ + cf )(e − c) = 0, (f + aβ)(β − f )(1 + a)

558

(16)

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Periodicity and Global Attractivity of Difference Equation

)(e−c) > 0 . Therefore Inequality (1) holds. and so (e − c)2 − 4(eaβ+cf (β−f )(1+a) Conversely , suppose that Inequality (1) is true. We will show that equation (1) has a prime period two solution. Assume that

p= where ζ =

q (e − c)2 −

e−c+ζ , 2(f + aβ)

4(eaβ+cf )(e−c) (β−f )(1+a)

and

q=

e−c−ζ 2(f + aβ)

. We see from Inequality (1) that

(e − c)(f − β)(1 + a) + 4(eaβ + cf ) > 0, e > c, f > β, )(e−c) . Therefore p and q are distinct which equivalents to (e − c)2 > 4(eaβ+cf (β−f )(1+a) real numbers. Set x−3 = p, x−2 = q, x−1 = p and x0 = q. We wish to show that x1 = x−1 = p and x2 = x0 = q .It follows from equation (1) that

e ep + cq e−c−ζ
x1 = aq + =a + f p + βq 2(f + aβ) f

e−c−ζ 2(f +aβ)
e−c−ζ 2(f +aβ)




+c +β

e−c−ζ 2(f +aβ)
e−c−ζ 2(f +aβ)




Dividing the denominator and numerator by 2(f + aβ) gives x1 = a

e−c−ζ
(e − c)[(e + c) + ζ] + 2(f + aβ) (f + β)(e − c) + (f − β)ζ

Multiplying the denominator and numerator of the right side by (f + β)(e − c) − (f − β)ζ gives x1 = a

e−c−ζ
2(f + aβ)

 ) (e − c) 2(e − c)[f c + βe − 2(eaβ+cf ] + 2ζ(βe − cf ) 1+a +  )(eaβ+cf )  4(e − c) βf (e − c) + (β−f(1+a) Multiplying the denominator and numerator of the right side by (1 + a) we obtain x1 =

e−c+ζ ae − ac − aζ + (e − c)(1 − a) + ζ(1 + a) = = p. 2(f + aβ) 2(f + aβ)

Similarly as before one can easily show that x2 = q. Then it follows by induction that x2n = q and x2n+1 = p for all n ≥ −1.Thus equation (1) has the prime period two solution ..., p, q, p, q, ..., where p and q are the distinct roots of the quadratic equation (16) and the proof is complete. Lemma 4.2. If l, k, s, r-even. Then there exists a prime period two solutions if and only if a = −1.

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T. F. Ibrahim

Proof. First suppose that there exists a prime period two solution ..., p, q, p, q, ..., then we see from equation (1) that when l, k, s, r-even p = aq +

b+c+d , α+β+γ

(17)

q = ap +

b+c+d α+β+γ

(18)

Subtracting equation (17) from equation (18) gives p − q = −a(p − q). Since p 6= q, it follows that a = −1. Again, adding equation (17) and equation (18) b+c+d . If we take yields p + q = α+β+γ p=

b+c d ,q = , if α+β+γ α+β+γ

b + c 6= d.

Set x−s = q, x−l = p , x−k = q,...,x−2 = q, x−1 = p and x0 = q.We wish to show that x1 = x−1 = p and x2 = x0 = q. It follows from equation (1) that bq+cq+dq = p . Similarly as before one can easily show that x2 = q. x1 = aq + αq+βq+γq Then it follows by induction that x2n = q and x2n+1 = p for all n ≥ −1. Thus equation (1) has the prime period two solution and the proof is complete. Lemma 4.3. If l, k, r-odd, s-even. Then there exists a positive prime period two solutions if and only if a = −1. Lemma 4.4. If l, k, s, r-odd (or l, k, r-even, s-odd). Then there no prime period two solution.

5

Global Attractor of the Equilibrium Point of Equation (1)

In this section we investigate the global asymptotic stability of equation (1). Lemma 5.1. For any values of the quotient αb , βc and γd the function f (u, v, w, t) defined by equation (4) has the monotonicity behavior in its three arguments. Proof. The proof follows by some computations and it will be omitted. Remark 5.2. It follows from equation (1), when αb = βc = γd , that xn+1 = α axn−k + λ for all n ≥ −t and for some constant λ.Whenever the quotients A , β γ and are not equal, we get the following result. B C Theorem 5.3. The equilibrium point x is a global attractor of equation (1) if one of the following statements holds (1)

b c d ≥ ≥ α β γ

and

d≥b+c

(19)

(2)

b d c ≥ ≥ α γ β

and

c≥b+d

(20)

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Periodicity and Global Attractivity of Difference Equation

c β c (4) β d (5) γ d (6) γ (3)

b α d ≥ γ c ≥ β b ≥ α ≥

d γ b ≥ α b ≥ α c ≥ β ≥

and

d≥b+c

(21)

and

b≥c+d

(22)

and

b≥c+d

(23)

and

c≥b+d

(24)

Proof. Let {xn }∞ n=−t be a solution of equation (1) and again let f be a function defined by equation (4). We will prove the theorem when case (1) is true and the proof of the other cases are similar and so we will be omitted. Assume that (19) is true, then it is easy from the equations after equation (4) to see that the function f (u, v, w, t) is non-decreasing in u, v and non-increasing in t and it is not clear what is going on with w. So we consider the following two cases:Case(1) Assume that the function f (u, v, w, t)is non-decreasing in w. Suppose that (m, M ) is a solution of the system M = f (M, M, M, m) and m = g(m, m, m, M ). Then from equation (1), we see that (α + β)(1 − a)M 2 + γ(1 − a)M m = (b + c)M + dm, (α + β)(1 − a)m2 + γ(1 − a)M m = (b + c)m + dM Subtracting this two equations we obtain  (M − m) (α + β)(1 − a)(M + m) + (d − b − c) = 0, under the conditions d ≥ b + c, a < 1, we see that M = m. It follows by theorem (1.2) that x is a global attractor of equation (1) and then the proof is complete. Case(2) Assume that the function f (u, v, w, t)is non-increasing in w. Suppose that (m, M ) is a solution of the system M = f (M, M, m, m) and m = g(m, m, M, M ).Then from equation (1), we see that M (1 − a) =

bM + cm + dm , αM + βm + γm

m(1 − a) =

bm + cM + dM αm + βM + γM

then under the conditions d ≥ b + c, a < 1, we see that M = m. It follows by theorem (1.2) that x is a global attractor of equation (1) and then the proof is complete.

6

Numerical examples

For confirming the results of this paper, we consider numerical examples which represent different types of solutions to equation (1). Example 6.1. See Fig.1, since l = 0, k = 1, s = 2, r = 1, x−2 = 3, x−1 = 0.4, x0 = 0.2, a = 0.8, b = 0.4, c = 0.8, d = 2, α = 0.1, β = 1, γ = 0.8.

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T. F. Ibrahim

plot of x(n+1)= ax(n−k)+(bx(n−l)+cx(n−s)+dx(n−r))/(alpha x(n−l)+beta x(n−s)+ gama x(n−r)) 9 8 7

x(n)

6 5 4 3 2 1 0 0

10

20

30

40 n

50

60

70

80

Example 6.2. See Fig.2, since l = 3, k = 0, s = 1, r = 2, x−3 = 3, x−2 = 7, x−1 = 2, x0 = 1.5, a = 1.2, b = 3, c = 5, d = 2, α = 1, β = 2.1, γ = 1.1. plot of x(n+1)= ax(n−k)+(bx(n−l)+cx(n−s)+dx(n−r))/(alpha x(n−l)+beta x(n−s)+ gama x(n−r)) 1600 1400 1200

x(n)

1000 800 600 400 200 0 0

5

10

15 n

20

25

30

Example 6.3. See Fig.3, since l = 1, k = 0, s = 2, r = 3, x−3 = x−1 = p, x−2 = x0 = q, a = 0.6, b = 7, c = 3, d = 9, α = 3.8, β = 0.2, γ = 1.2.

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Periodicity and Global Attractivity of Difference Equation

plot of x(n+1)= ax(n−k)+(bx(n−l)+cx(n−s)+dx(n−r))/(alpha x(n−l)+beta x(n−s)+ gama x(n−r)) 3 2.5 2

x(n)

1.5 1 0.5 0 −0.5 −1 0

7

2

4

6

8

10 n

12

14

16

18

20

ACKNOWLEDGEMENT

This article was funded by the Deanship of Scientific Research (DSR), the King khalid University, Abha under Project No. KKU.S.180.33 . The author, therefore, acknowledge with thanks DSR technical and financial support.

References [1] C. Cinar, On the positive solutions of the difference equation xn+1 = (axn−1 )/(1 + bxn xn−1 ), Appl. Math. Comp., 156 (2004), 587-590. [2] E. M. Elabbasy, H. El-Metwally and E. M. Elsayed, On the difference equation xn+1 = axn − (bxn )/(cxn − dxn−1 ), Adv. Differ. Equ., Volume 2006, Article ID 82579, 1–10. [3] 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. [4] E. M. Elabbasy, H. El-Metwally and E. M. Elsayed, On the difference αxn−k equations xn+1 = , J. Conc. Appl. Math., 5 (2) (2007), 101-113. k Q β+γ

xn−i

i=0

[5] E. M. Elsayed, Solution and attractivity for a rational recursive sequence, Discrete Dynamics in Nature and Society, Volume 2011, Article ID 982309, 17 pages.

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T. F. Ibrahim

[6] E. M. Elsayed, On the solution of some difference equations, European Journal of Pure and Applied Mathematics, 4 (3) (2011), 287-303. [7] E. M. Elsayed, Solutions of rational difference system of order two, Math. Comp. Mod., 55 (2012), 378–384. [8] E. M. Elsayed, Behavior and expression of the solutions of some rational difference equations, Journal of Computational Analysis and Applications, To appear in vol 15 (2013). [9] V. L. Kocic and G. Ladas, Global Behavior of Nonlinear Difference Equations of Higher Order with Applications, Kluwer Academic Publishers, Dordrecht, 1993. [10] M. R. S. Kulenovic and G. Ladas, Dynamics of Second Order Rational Difference Equations with Open Problems and Conjectures, Chapman and Hall / CRC Press, 2001. [11] M. Saleh and M. Aloqeili, On the difference equation yn+1 = A + ((yn )/(yn−k )) with A < 0, Appl. Math. Comp., 176 (1) (2006), 359–363. [12] D. Simsek, C. Cinar and I. Yalcinkaya, On the recursive sequence xn+1 = ((xn−3 )/(1 + xn−1 )), Int. J. Contemp. Math. Sci., 1 (10) (2006), 475-480. [13] C. Wang, S. Wang, Z. Wang, H. Gong, R. Wang, Asymptotic stability for a class of nonlinear difference equation, Dis. Dyn. Nat. Soc., Volume 2010, Article ID 791610, 10pages. [14] C. Wang, Q. Shi, S. Wang. Asymptotic behavior of equilibrium point for a family of rational difference equation. Advances in Difference Equations, Volume 2010, Article ID 505906. 10 pages. [15] I. Yalnkaya, and C. Cinar, On the dynamics of the difference equation xn+1 = ((axn−k )/(b + cxpn )), Fasciculi Mathematici, 42 (2009), 133-139. [16] I. Yalnkaya, On the difference equation xn+1 = α+((xn−m )/(xkn )), Discrete Dynamics in Nature and Society, Vol. 2008, Article ID 805460, 8 pages, doi: 10.1155/2008/ 805460. [17] I. Yalnkaya, C. Cinar and M. Atalay, On the solutions of systems of difference equations, Advances in Difference Equations, Vol. 2008, Article ID 143943, 9 pages, doi: 10.1155/2008/143943. [18] I. Yalnkaya, On the global asymptotic behavior of a system of two nonlinear difference equations, ARS Combinatoria, 95 (2010), 151-159. [19] E. M. E. Zayed and M. A. El-Moneam, On the rational recursive sequence xn+1 = axn −((bxn )/(cxn −dxn−k )), Comm. Appl. Nonli. Anal., 15 (2008), 47-57.

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Quadratic derivations on non-Archimedean Banach algebras 1

Choonkil Park, 2 S. Shagholi, 3 A. Javadian, 4 M.B. Savadkouhi and 5 Madjid Eshaghi Gordji 1

Research Institute for Natural Sciences, Hanyang University, Seoul 133-791, Korea Department of Mathematics, Semnan University, P. O. Box 35195-363, Semnan, Iran Center of Excellence in Nonlinear Analysis and Applications (CENAA), Semnan University, Iran 3 Department of Physics, Semnan University,P. O. Box 35195-363, Semnan, Iran 4 Department of Mathematics, Islamic Azad University, Someh Sara Branch, Someh Sara, Iran 2,5

Abstract. Let A be an algebra and X be an A-module. A quadratic mapping D : A → X is called a quadratic derivation if D(ab) = D(a)b2 + a2 D(b) for all a1 , a2 ∈ A. We investigate the Hyers-Ulam stability of quadratic derivations from a non-Archimedean Banach algebra A into a non-Archimedean Banach A-module.

1. Introduction A definition of stability in the case of homomorphisms between metric groups was proposed by a problem by Ulam [32] in 1940. In 1941, Hyers [17] gave a first affirmative answer to the question of Ulam for Banach spaces. Hyers’ Theorem was generalized by Rassias [27] for linear mappings by considering an unbounded Cauchy difference (see [3, 4, 8, 10, 18, 19, 22, 25, 29]). The functional equation f (x + y) + f (x − y) = 2f (x) + 2f (y). (1.1) is related to symmetric bi-additive function. It is natural that this equation is called a quadratic functional equation. In particular, every solution of the quadratic equation (1.1) is said to be a quadratic mapping. It is well known that a mapping f between real vector spaces is quadratic if and only if there exits a unique symmetric bi-additive mapping B such that f (x) = B(x, x) for all x (see [1, 20]). The bi-additive mapping B is given by 1 (f (x + y) − f (x − y)). 4 The Hyers-Ulam stability problem for the quadratic functional equation (1.1) was proved by Skof for mappings f : A → B, where A is a normed space and B is a Banach space (see [31]). Cholewa [6], Czerwik [7] and Grabiec [16] have generalized the results of stability of quadratic mappings. Borelli and Forti [5] generalized the stability result as follows (cf. [23, 24]): Let G be an Abelian group, and X a Banach space. Assume that a mapping f : G → X satisfies the functional inequality B(x, y) =

∥f (x + y) + f (x − y) − 2f (x) − 2f (y)∥ ≤ φ(x, y) for all x, y ∈ G, where φ : G × G → [0, ∞) is a function such that Φ(x, y) :=

∞ ∑ 1 φ(2i x, 2i y) < ∞ i+1 4 i=0

for all x, y ∈ G. Then there exists a unique quadratic mapping Q : G → X with the property ∥f (x) − Q(x)∥ ≤ Φ(x, x) 0

2010 Mathematics Subject Classification: 39B82, 39B52, 46H25. Keywords: non-Archimedean Banach algebra; non-Archimedean Banach module; quadratic functional equation; Hyers-Ulam stability. 0 E-mail: [email protected], s [email protected], [email protected], [email protected], [email protected] Corresponding author: M. Eshaghi Gordji; [email protected] 0

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C. Park, S. Shagholi, A. Javadian, M.B. Savadkouhi, M. Eshaghi Gordji for all x ∈ G. Let K be a field. A non-Archimedean absolute value on K is a function | · | : K → R such that for any a, b ∈ K we have (i) |a| ≥ 0 and equality holds if and only if a = 0, (ii) |ab| = |a||b|, (iii) |a + b| ≤ max{|a|, |b|}. The condition (iii) is called the strict triangle inequality. By (ii), we have |1| = | − 1| = 1. Thus, by induction, it follows from (iii) that |n| ≤ 1 for each integer n. We always assume, in addition, that | · | is nontrivial, i.e., that there is an a0 ∈ K such that |a0 | ̸∈ {0, 1}. Let X be a linear space over a scalar field K with a non-Archimedean nontrivial valuation | · |. A function ∥ · ∥ : X → R is a non-Archimedean norm (valuation) if it satisfies the following conditions: (N A1) ∥x∥ = 0 if and only if x = 0; (N A2) ∥rx∥ = |r|∥x∥ for all r ∈ K and x ∈ X; (N A3) the strong triangle inequality (ultrametric); namely, ∥x + y∥ ≤ max{∥x∥, ∥y∥}

(x, y ∈ X).

Then (X, ∥ · ∥) is called a non-Archimedean space. It follows from (N A3) that ∥xm − xℓ ∥ ≤ max{∥xȷ+1 − xȷ ∥ : ℓ ≤ ȷ ≤ m − 1}

(m > ℓ).

Therefore, a sequence {xm } is Cauchy in X if and only if {xm+1 − xm } converges to zero in a non-Archimedean space. By a complete non-Archimedean space we mean one in which every Cauchy sequence is convergent. A non-Archimedean Banach algebra is a complete non-Archimedean algebra A which satisfies ∥ab∥ ≤ ∥a∥∥b∥ for all a, b ∈ A. A non-Archimedean Banach space X is a non-Archimedean Banach A-bimodule if X is an A-bimodule which satisfies max{∥xa∥, ∥ax∥} ≤ ∥a∥∥x∥ for all a ∈ A, x ∈ X. For more detailed definitions of non-Archimedean Banach algebras, we can refer to [30]. Let A be a normed algebra and let X be a Banach A-module. We say that a mapping D : A → X is a quadratic derivation if D is a quadratic mapping satisfying D(x1 x2 ) = D(x1 )x22 + x21 D(x2 )

(1.2)

for all x1 , x2 ∈ A. Recently, the stability of derivations has been investigated by a number of mathematicians including [2, 11, 12, 13, 14, 15, 21, 26, 28] and references therein. More recently, Eshaghi Gordji [9] established the stability of ring derivations on non-Archimedean Banach algebras. In this paper, we investigate the approximately quadratic derivations on non-Archimedean Banach algebras. 2. Main results In the following we suppose that A is a non-Archimedean Banach algebra and X is a non-Archimedean Banach A-bimodule. Assume that |2| ̸= 1. Theorem 2.1. Let f : A → X be a given mapping with f (0) = 0 and let φ1 : A × A → R+ and φ2 : A × A → R+ be functions such that ∥f (x1 x2 ) − f (x1 )x22 − x21 f (x2 )∥ ≤ φ1 (x1 , x2 ), (2.1) ||f (x + y) + f (x − y) − 2f (x) − 2f (y)|| ≤ φ2 (x, y)

(2.2)

for all x1 , x2 , x, y ∈ A. Assume that for each x ∈ A { } 1 φ2 (2k x, 2k x) lim max : 0 ≤ k ≤ n − 1 n→∞ |2|2k |2|2 denoted by Ψ(x, x), exists. Suppose lim

n→∞

φ1 (2n x1 , 2n x2 ) φ2 (2n x, 2n y) = lim =0 n→∞ |2|4n |2|2n

for all x1 , x2 , x, y ∈ A. Then there exists a unique quadratic derivation D : A → X such that ∥D(x) − f (x)∥ ≤ Ψ(x, x) 566

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Quadratic derivations on non-Archimedean Banach algebras for all x ∈ A. Proof. Setting y = x in (2.2), we get ∥f (2x) − 4f (x)∥ ≤ φ2 (x, x) for all x ∈ A, and then dividing by |2| in (2.4), we obtain

f (2x)

≤ φ2 (x, x) − f (x)

22

|2|2

(2.4)

2

for all x ∈ A. Replacing x by 2x and then dividing by |2|2 in (2.5), we obtain

f (22 x) f (2x)

≤ φ2 (2x, 2x) .

−

24 22 |2|4 Combining (2.5), (2.6) and the strong triangle inequality (NA3) yields

{ }

f (22 x)

≤ max φ2 (2x, 2x) , φ2 (x, x) . − f (x)

24

|2|4 |2|2 Following the same argument, one can prove by induction that

{ }

f (2n x)

1 φ2 (2k x, 2k x)

≤ max − f (x) : 0 ≤ k ≤ n − 1 .

22n

|2|2 |2|2k

(2.5)

(2.6)

(2.7)

(2.8)

Replacing x by 2n−1 x and dividing by |2|2(n−1) in (2.5), we find that

n−1

f (2n x) f (2n−1 x) x, 2n−1 x)

≤ φ2 (2 −

22n

2(n−1) |2|2n 2 n

x) } is a Cauchy sequence. Since X is complete, it follows for all positive integers n and all x ∈ A. Hence { f (2 22n n f (2 x) f (2n x) that { 22n } is convergent. Set D(x) = limn→∞ 22n . By taking the limit as n → ∞ in (2.8), we see that ∥D(x) − f (x)∥ ≤ Ψ(x, x) and (2.3) holds for all x ∈ A. In order to show that D satisfies (1.2), replacing x1 , x2 by 2n x1 , 2n x2 in (2.1), and dividing both sides of (2.1) by |2|4n , we get

n

f (2n x1 · 2n x2 )

f (2n x1 ) φ1 (2n x1 , 2n xn ) n 2 n 2 f (2 x2 )

− · (2 x ) − (2 x ) . ≤ . 2 1

4n 4n 4n 2 2 2 |2|4n

Taking the limit as n → ∞, we find that D satisfies (1.2). Replacing x by 2n x and y by 2n y in (2.2) and dividing by |2|2n , we get

n n

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

≤ φ2 (2 x, 2 y) . + − 2 − 2

2n 2n 2n 2n 2n 2 2 2 2 |2| Taking the limit as n → ∞, we find that D satisfies (1.1). Now, suppose that there is another such mapping D′ : A → X satisfying D′ (x+y)+D′ (x−y) = 2D′ (x)+2D′ (y) and ∥D′ (x) − f (x)∥ ≤ Ψ(x, x). Then for all x ∈ A, we have 1 ∥D(2n x) − D′ (2n x)∥ |2|2n 1 ≤ lim max{∥D(2n x) − f (2n x)∥, ∥D′ (2n x) − f (2n x)∥} n→∞ |2|2n

∥D(x) − D′ (x)∥ = lim

n→∞

≤ lim lim

n→∞ k→∞

φ2 (2j x, 2j x) 1 max{ : n ≤ j ≤ k + n − 1} = 0. 2 |2| |2|2j

′



It follows that D(x) = D (x).

Corollary 2.2. Let θ1 and θ2 be nonnegative real numbers, and let p be a real number such that p > 4. Suppose that a mapping f : A → X satisfies ∥f (x1 x2 ) − f (x1 )x22 − x21 f (x2 )∥ ≤ θ1 (∥x1 ∥p + ∥x2 ∥p ), ∥f (x + y) + f (x − y) − 2f (x) − 2f (y)∥ ≤ θ2 (∥x∥p + ∥y∥p ) 567

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C. Park, S. Shagholi, A. Javadian, M.B. Savadkouhi, M. Eshaghi Gordji for all x1 , x2 , x, y ∈ A. Then there exists a unique quadratic derivation D : A → X such that } { θ2 ∥x∥p 0 ≤ k ≤ n − 1 ∥D(x) − f (x)∥ ≤ lim max n→∞ |2|.|2|k(2−p) for all x ∈ A. Proof. Let φ1 : A × A → R+ and φ2 : A × A → R+ be functions such that φ1 (x1 , , x2 ) = θ1 (∥x1 ∥p + ∥x2 ∥p ) and φ2 (x, y) = θ2 (∥x∥p + ∥y∥p ) for all x1 , x2 , x, y ∈ A. Then we have lim

n→∞

φ2 (2n x, 2n y) = lim θ2 · |2|n(p−2) · (∥x∥p + ∥y∥p ) = 0 n→∞ |2|2n

θ1 |2|pn φ1 (2n x1 , 2n x2 ) = lim (∥x1 ∥p + ∥x2 ∥p ) = 0 n→∞ |2|4n n→∞ |2|4n Applying Theorem 2.1, we conclude the required result. lim

(x, y ∈ A), (x1 , x2 ∈ A). 

Theorem 2.3. Let f : A → X be a mapping and let φ1 : A × A → R+ , φ2 : A × A → R+ be functions such that ∥f (x1 x2 ) − f (x1 )x22 − x21 f (x2 )∥ ≤ φ1 (x1 , x2 ),

(2.9)

∥f (x + y) + f (x − y) − 2f (x) − 2f (y)∥ ≤ φ2 (x, y)

(2.10)

for all x1 , x2 , x, y ∈ A. Assume that for each x ∈ A { ( x } x ) lim max |2|2k φ2 k+1 , k+1 : 0 ≤ k ≤ n − 1 n→∞ 2 2 denoted by Ψ(x, x), exists. Suppose

(x y ) x2 ) = lim |2|2n φ2 n , n = 0 n n→∞ n→∞ 2 2 2 for all x1 , x2 , x, y ∈ A. Then there exists a unique quadratic derivation D : A → X such that lim |2|4n φ1

(x

1 , 2n

∥D(x) − f (x)∥ ≤ Ψ(x, x)

(2.11)

∥f (2x) − 4f (x)∥ ≤ φ2 (x, x).

(2.12)

( x ) (x x)

, .

f (x) − 4f

≤ φ2 2 2 2

(2.13)

for all x ∈ A. Proof. Setting y = x in (2.10), we obtain

Replacing x by

x 2

in (2.12), one obtains

in (2.13) and multiplying by |2|2 , we obtain that

(x) ( x ) (x x)

2

2 − 24 f ≤ |2| φ , .

2 f

2 2 22 22 22 By using (2.13), (2.14) and strong triangle inequality (NA3), we get

( x ) { (x x) ( x x )}

4 , , |2|2 φ2 2 , 2

f (x) − 2 f

≤ max φ2 2 2 2 2 2 2 for x ∈ A. Next we prove by induction that

( x ) { ( x } x )

2n 2k ≤ max |2| φ , : 0 ≤ k ≤ n − 1 .

f (x) − 2 f

2 2n 2k+1 2k+1 Again replacing x by

Replacing x by

x 2n−1

x 2

and multiplying by |2|2(n−1) in (2.13), we obtain

( x ) ( x ) (x x)

2(n−1)

2(n−1) f − 22n f φ2 n , n

2

≤ |2| n−1 n 2 2 2 2

(2.14)

(2.15)

(2.16)

(2.17)

for all x ∈ A. Hence {22n f ( 2xn )} is a Cauchy sequence. Since X is complete, it follows that {22n f ( 2xn )} is convergent. Set D(x) = limn−→∞ {22n f ( 2xn )}. By taking the limit as n → ∞ in (2.16), we see that ∥f (x) − D(x)∥ ≤ Ψ(x, x) and (2.11) holds for all x ∈ A. 568

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Quadratic derivations on non-Archimedean Banach algebras Replacing x1 , x2 by 2xn1 , 2xn2 in (2.9) and multiplying by |2|4n , we get

( x ) ( x )2 ( ) ( )2 ( x )

4n ( x1 x2 ) 1 2 2 4n 4n x1

≤ 24n φ1 x1 , x2 .

2 f − 2 f · − 2 f

n n n n n n n n 2 2 2 2 2 2 2 2 Taking the limit as n → ∞, we find that D satisfies (1.2). Replacing x by 2xn and y by 2yn in (2.10) and multiplying by |2|2n , we have

(x (x) ( y ) (x (x y ) y ) y )

2n 2n − 22n · 2f + n + 22n f − n − 22n · 2f

2 f

≤ |2| φ2 n , n . n n n n 2 2 2 2 2 2 2 2 Taking the limit as n → ∞, we find that D satisfies (1.1). Now, suppose that there is another such mapping D′ : A → X satisfying D′ (x + y) + D′ (x − y) = 2D′ (x) + 2D′ (y) and ∥D′ (x) − f (x)∥ ≤ Ψ(x, x). Then for all x ∈ A, we have

( ) ( y ) x

∥D(x) − D′ (x)∥ = lim |2|2n D n − D′ n n→∞ 2 2 { ( x ) ( x ) ( x ) ( x ) }

′

≤ lim |2|2n max D n − f −f

, D

n n n→∞ 2 2 2 2n { ( x } x ) ≤ lim lim max φ2 j+1 , j+1 : n ≤ j ≤ k + n − 1 = 0 n→∞ k→∞ 2 2 ′  and so D(x) = D (x) for all x ∈ A. Corollary 2.4. Let θ1 and θ2 be nonnegative real numbers, and let p be a positive real number such that p < 2. Suppose that a mapping f : A → X satisfies ∥f (x1 x2 ) − f (x1 )x22 − x21 f (x2 )∥ ≤ θ1 (∥x1 ∥p + ∥x2 ∥p ), ∥f (x + y) + f (x − y) − 2f (x) − 2f (y)∥ ≤ θ2 (∥x∥p + ∥y∥p ) for all x1 , x2 , x, y ∈ A. Then there exists a unique quadratic derivation D : A → X such that ∥D(x) − f (x)∥ ≤ lim max{θ2 ∥x∥p .|2|(k+1)(1−p) 0 ≤ k ≤ n − 1} n→∞

for all x ∈ A. Proof. Let φ1 : A × A → R+ and φ2 : A × A → R+ be functions such that φ1 (x1 , , x2 ) = θ1 (∥x1 ∥p + ∥x2 ∥p ) and φ2 (x, y) = θ2 (∥x∥p + ∥y∥p ) for all x1 , x2 , x, y ∈ A. We have (x y ) lim |2|2n φ2 n , n = lim (|2|n(2−p) )θ2 (∥x∥p + ∥y∥p ) = 0 (x, y ∈ A), n→∞ n→∞ 2 2 (x x ) 1 2 4n lim |2| φ1 n , n = lim |2|n(4−p) θ1 (∥x1 ∥p + ∥x2 ∥p ) = 0 (x1 , x2 ∈ A). n→∞ n→∞ 2 2 Applying Theorem 2.4, we conclude the required result.  Acknowledgments C. Park was supported by Basic Science Research Program through the National Research Foundation of Korea funded by the Ministry of Education, Science and Technology (NRF-2012R1A1A2004299).

References [1] J. Aczel and J. Dhombres, Functional equations in several variables, Cambridge Univ. Press, 1989. [2] M. Bavand Savadkouhi, M. Eshaghi Gordji, J. M. Rassias and N. Ghobadipour, Approximate ternary Jordan derivations on Banach ternary algebras, J. Math. Phys. 50 (2009), Article ID 042303. [3] C. Borelli, On Hyers-Ulam stability for a class of functional equations, Aequationes Math. 54 (1997), 74–86. [4] D.G. Bourgin, Class of transformations and bordering transformations, Bull. Amer. Math. Soc. 27 (1951), 223–237. [5] C. Borelli and G.L. Forti, On a general Hyers-Ulam stability result, Internat. J. Math. Math. Sci. 18 (1995), 229–236. [6] P.W. Cholewa, Remarks on the stability of functional equations, Aequationes Math. 27 (1984), 76–86. [7] S. Czerwik, On the stability of the quadratic mapping in normed spaces, Abh. Math. Sem. Univ. Hamburg 62 (1992), 59–64. 569

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C. Park, S. Shagholi, A. Javadian, M.B. Savadkouhi, M. Eshaghi Gordji [8] A. Ebadian, N. Ghobadipour and M. Eshaghi Gordji, A fixed point method for perturbation of bimultipliers and Jordan bimultipliers in C ∗ -ternary algebras, J. Math. Phys. 51 (2010), Article ID 103508. [9] M. Eshaghi Gordji, Nearly ring homomorphisms and nearly ring derivations on non-Archimedean Banach algebras, Abs. Appl. Anal. 2010, Article ID 393247 (2010). [10] M. Eshaghi Gordji and A. Bodaghi, On the stability of quadratic double centralizers on Banach algebras, J. Comput. Anal. Appl. 13 (2011), 724–729. [11] M. Eshaghi Gordji, R. Farokhzad Rostami and S.A.R. Hosseinioun, Nearly higher derivations in unital C ∗ algebras, J. Comput. Anal. Appl. 13 (2011), 734–742. [12] M. Eshaghi Gordji, S. Kaboli Gharetapeh, T. Karimi, E. Rashidi and M. Aghaei, Ternary Jordan derivations on C ∗ -ternary algebras, J. Comput. Anal. Appl. 12 (2010), 463–470. [13] M. Eshaghi Gordji, J.M. Rassias and N. Ghobadipour, Generalized Hyers-Ulam stability of the generalized (n, k)–derivations, Abs. Appl. Anal. 2009, Article ID 437931 (2009). [14] R. Farokhzad and S.A.R. Hosseinioun, Perturbations of Jordan higher derivations in Banach ternary algebras: An alternative fixed point approach, Int. J. Nonlinear Anal. Appl. 1 (2010), No. 1, 42–53. [15] N. Ghobadipour, A. Ebadian, Th.M. Rassias and M. Eshaghi Gordji, A perturbation of double derivations on Banach algebras, Commun. Math. Anal. 11 (2011), 51–60. [16] A. Grabiec, The generalized Hyers-Ulam stability of a class of functional equations, Publ. Math. Debrecen 48 (1996) 217–235. [17] D.H. Hyers, On the stability of the linear functional equation, Proc. Natl. Acad. Sci. USA 27 (1941), 222–224. [18] D.H. Hyers, G. Isac and Th.M. Rassias, Stability of Functional Equations in Several Variables, Birkh¨ auser, Boston, Basel, Berlin, 1998. [19] G. Isac and Th.M. Rassias, On the Hyers-Ulam stability of ψ-additive mappings, J. Approx. Theory. 72 (1993), 131–137. [20] Pl. Kannappan, Quadratic functional equation and inner product spaces, Results Math. 27 (1995), 368–372. [21] H. Kim and I. Chang, stability of the functional equations related to a multiplicative derivation, J. Appl. Computing. 11 (2003), 413–421. [22] A. Najati and Th.M. Rassias, Stability of a mixed functional equation in several variables on Banach modules, Nonlinear Anal.–TMA 72 (2010), 1755–1767. [23] C. Park, Generalized quadratic mappings in several variables, Nonlinear Anal.–TMA 57 (2004), 713–722. [24] C. Park, On the stability of the quadratic mapping in Banach modules, J. Math. Anal. Appl. 276 (2002), 135–144. [25] C. Park and Th.M. Rassias, Hyers-Ulam stability of a generalized Apollonius type quadratic mapping, J. Math. Anal. Appl. 322 (2006), 371–381. [26] C. Park and Th.M. Rassias, Homomorphisms and derivations in proper JCQ∗-triples, J. Math. Anal. Appl. 337 (2008), 1404–1414. [27] Th.M. Rassias, On the stability of the linear mapping in Banach spaces, Proc. Amer. Math. Soc. 72 (1978), 297–300. [28] S. Shagholi, M. Eshaghi Gordji and M. Bavand Savadkouhi, Stability of ternary quadratic derivations on ternary Banach algebras, J. Comput. Anal. Appl. 13 (2011), 1097–1105. [29] S. Shagholi, M. Eshaghi Gordji and M. Bavand Savadkouhi, Nearly ternaty cubic homomorphisms in ternary Fr´echet algebras, J. Comput. Anal. Appl. 13 (2011), 1106–1114. [30] N. Shilkret, Non-Archimedian Banach algebras, Thesis (Ph.D.)-Polytechnic Univ., ProQuest LLC, 1968. [31] F. Skof, Propriet locali e approssimazione di operatori, Rend. Sem. Mat. Fis. Milano. 53 (1983), 113–129. [32] S.M. Ulam, A collection of Mathematical problems, Interscience Publ, New York., 1960.

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Soft q-ideals of soft BCI-algebras Yun Sun Hwang1 and Sun Shin Ahn2∗ 1

Department of Mathematics Education, Dongguk University, Seoul 100-715, Korea

2

Department of Mathematics Education, Dongguk University, Seoul 100-715, Korea

Abstract. The notion of soft q-ideals and q-idealistic soft BCI-algebras are introduced, and several properties of them are investigated. Characterizations of a (fuzzy) q-ideals in BCI-algebras are considered. Relations between fuzzy q-ideals and p-idealistic soft BCI-algebras are discussed.

1. Introduction D. Molodtsov ([2]) introduced introduced the concept of soft set as a new mathematical tool for dealing with uncertainties that is free from the difficulties that have troubled the usual theoretical applications. Y. B. Jun ([6]) applied first the notion of soft sets by Moldtsov to the theory of BCK-algebras. Y. B. Jun and C. H. Park ([8]) dealt with the algebraic structure of BCK/BCIalgebras by applying soft set theory. They discussed the algebraic properties of soft sets in BCK/BCI-algebras. In [7], Y. B. Jun, K. J. Lee and J. Zhan introduced the notion of soft p-ideals and p-idealistic soft BCI-algebras, and investigated their properties. Y. S. Hwang and S. S. Ahn ([5]) defined the notion of vague q-ideal of a BCI-algebra and studied several properties of them. In this paper, we introduced the notion of soft q-ideals and q-idealistic soft BCI-algebras, and investigate several properties of them. We also consider characterizations of a (fuzzy) q-ideals in BCI-algebras and study relations between fuzzy q-ideals and p-idealistic soft BCI-algebras. 2. Preliminaries

We review some definitions and properties that will be useful in our results. By a BCI-algebra we mean an algebra (X, ∗, 0) of type (2,0) satisfying the following conditions: (a1) (a2) (a3) (a4)

(∀x, y, z ∈ X) (((x ∗ y) ∗ (x ∗ z)) ∗ (z ∗ y) = 0), (∀x, y ∈ X) ((x ∗ (x ∗ y)) ∗ y = 0), (∀x ∈ X) (x ∗ x = 0), (∀x, y ∈ X) (x ∗ y = 0, y ∗ x = 0 ⇒ x = y).

0

2010 Mathematics Subject Classification: 06F35; 03G25. Keywords: Soft set; (q-idealistic) soft BCI-algebra; Soft ideal; Soft q-ideal. The corresponding author. 0 E-mail: [email protected] (Y. S. Hwang); [email protected] (S. S. Ahn) 0

∗

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Yun Sun Hwang and Sun Shin Ahn

In any BCI-algebra X one can define a partial order “≤” by putting x ≤ y if and only if x ∗ y = 0. A BCI-algebra X has the following properties: (b1) (b2) (b3) (b4) (b5) (b6)

(∀x ∈ X) (x ∗ 0 = x). (∀x, y, z ∈ X) ((x ∗ y) ∗ z = (x ∗ z) ∗ y). (∀x, y ∈ X) (0 ∗ (x ∗ y) = (0 ∗ x) ∗ (0 ∗ y)). (∀x, y ∈ X) (x ∗ (x ∗ (x ∗ y)) = x ∗ y). (∀x, y, z ∈ X) (x ≤ y ⇒ x ∗ z ≤ y ∗ z, z ∗ y ≤ z ∗ x). (∀x, y, z ∈ X) ((x ∗ z) ∗ (y ∗ z) ≤ x ∗ y).

A non-empty subset S of a BCI-algebra X is called a subalgebra of X if x ∗ y ∈ S whenever x, y ∈ S. A non-empty subset A of a BCI-algebra X is called an ideal of X if it satisfies: (c1) 0 ∈ A, (c2) (∀x ∈ X)(∀y ∈ A)(x ∗ y ∈ A ⇒ x ∈ A). Note that every ideal A of a BCI-algebra X satisfies: (∀x ∈ X) (∀y ∈ A) (x ≤ y ⇒ x ∈ A). A non-empty subset A of a BCI-algebra X is called a q-ideal of X if it satisfies (c1) and (c3) (∀x, y, z ∈ X)(x ∗ (y ∗ z) ∈ A, y ∈ A ⇒ x ∗ z ∈ A). Note that any q-ideal is an ideal, but the converse is not true in general. We refer the reader to the book [4] for further information regarding BCI-algebras. Molodtsov ([2]) defined the soft set in the following way: Let U be an initial set and E be a set of parameters. Let P(U ) denote the power set of U and A ⊂ E. Definition 2.1.([2]) A pair (F , A) is called a soft set over U , where F is a mapping given by F : A → P(U ). In other words, a soft set over U is a parameterized family of subsets of the universe U . For  ∈ A, F () may be considered as the set of -approximate elements of the soft set (F , A). Clearly, a soft set is not a set. For illustration, Molodtsov considered several examples in [2]. Definition 2.2.([3]) Let (F , A) and (G , B) be two soft sets over a common universe U . The intersection of (F , A) and (G , B) is defined to be the soft set (H , C) satisfying the following conditions: (i) C = A ∩ B, (ii) (∀e ∈ C)(H (e) = F (e) or G (e), (as both are same sets)). ˜ (G , B) = (H , C). In this case, we write (F , A)∩ Definition 2.3.([3]) Let (F , A) and (G , B) be two soft sets over a common universe U . The union of (F , A) and (G , B) is defined to be the soft set (H , C) satisfying the following conditions: 572

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Soft q-ideals of soft BCI-algebras

(i) C = A ∪ B, (ii) ∀e ∈ C H (e) =

   F (e)

if e ∈ A\B

G (e) if e ∈ B\A   F (e) ∪ G (e) if e ∈ A ∩ B.

˜ (G , B) = (H , C). In this case, we write (F , A)∪ Definition 2.4.([3]) If (F , A) and (G , B) are two soft sets over a common universe U , then ˜ (G , B) is defined by (F , A)∧ ˜ (G , B) = (H , A × B), “(F , A)AN D(G , B)” denoted by (F , A)∧ where H (α, β) = F (α) ∩ G (β) for all (α, β) ∈ A × B. Definition 2.5.([3]) If (F , A) and (G , B) are two soft sets over a common universe U , then ˜ (G , B) is defined by (F , A)∨ ˜ (G , B) = (H , A × B), “(F , A)OR(G , B)” denoted by (F , A)∨ where H (α, β) = F (α) ∪ G (β) for all (α, β) ∈ A × B. Definition 2.6.([3]) For two soft sets (F , A) and (G , B) over a common universe U , we say that ˜ , B), if it satisfies: (F , A) is a soft subset of (G , B), denoted by (F , A)⊂(G (i) A ⊂ B, (ii) For every  ∈ A, F () and G () are identical approximations. The most appropriate theory for dealing with uncertainties is the theory of fuzzy sets developed by Zadeh ([11]). 3. Soft q-ideals In what follows let X and A be a BCI-algebra and a nonempty set, respectively, and R will refer to an arbitrary binary relation between an element of A and an element of X, that is, R is a subset of A × X without otherwise specified. A set-valued function F : A → P(X) can be defined as F (x) = {y ∈ X|(x, y) ∈ R} for all x ∈ A. The pair (F , A) is then a soft set over X. Definition 3.1.([8]) Let S be a subalgebra of X. A subset I of X is called an ideal of X related to S (briefly, S-ideal of X), denoted by I / S, if it satisfies: (i) 0 ∈ I, (ii) (∀x ∈ S)(∀y ∈ I)(x ∗ y ∈ I ⇒ x ∈ I). Definition 3.2. Let S be a subalgebra of X. A subset I of X is called a q-ideal of X related to S (briefly, S-q-ideal of X), denoted by I /q S, if it satisfies: (i) 0 ∈ I, (ii) (∀x, z ∈ S)(∀y ∈ I)(x ∗ (y ∗ z) ∈ I ⇒ x ∗ z ∈ I). 573

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Yun Sun Hwang and Sun Shin Ahn

Example 3.3. Let X := {0, 1, a, b} be a BCI-algebra ([4]) in which the ∗-operation is given by the following table: ∗ 0 1 a b 0 0 0 a a 1 1 0 a a a a a 0 0 b b a 1 0 Then S := {0, 1, a} is a subalgebra of X and I := {0, 1} is an S-q-ideal of X. Note that every S-q-ideal of X is an S-ideal of X (∵ Put z := 0 in Definition 3.2(ii)). But the converse is not true in general as seen in the following example. Example 3.4. Let X := {0, 1, 2, a, b} be a BCI-algebra ([4]) in which the ∗-operation is given by the following table: ∗ 0 1 2 a b 0 0 0 0 b a 1 1 0 0 b a 2 2 1 0 b a a a a a 0 b b b b b a 0 Then S := {0, a, b} is a subalgebra of X and {0} is an S-ideal of X, but not an S-q-ideal of X, since a ∗ (0 ∗ b) = a ∗ a = 0 ∈ {0}, 0 ∈ {0}, and a ∗ b = b ∈ / {0}. Definition 3.5.([6]) Let (F , A) be a soft set over X. Then (F , A) is called a soft BCI-algebra over X if F (x) is a subalgebra of X for all x ∈ A. Definition 3.6.([8]) Let (F , A) be a soft set over X. A soft set (G , I) over X is called a soft ideal of (F , A), denoted by (G , I)˜/(F , A) if it satisfies: (i) I ⊂ A, (ii) (∀x ∈ I)(G (x) / F (x)). Definition 3.7. Let (F , A) be a soft set over X. A soft set (G , I) over X is called a soft q-ideal of (F , A), denoted by (G , I)˜/q (F , A) if it satisfies: (i) I ⊂ A, (ii) (∀x ∈ I)(G (x) /q F (x)). Example 3.8. Consider a BCI-algebra X = {0, 1, a, b} which is given in Example 3.3. Let (F , A) be a soft set over X, where A := {0, 1, a} ⊂ X and F : A → P(X) is a set-valued function defined by F (x) = {0} ∪ {y ∈ X|y ∗ (y ∗ x) ∈ {0, 1, a}} for all x ∈ A. Then F (0) = F (1) = F (a) = X, which are subalgebras of X. Hence (F , A) is a soft BCI-algebra over X. Let I := {0, 1} and G : I → (X) be a set-valued function defined by G (x) = {0} ∪ {y ∈ X|x ≤ y} 574

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Soft q-ideals of soft BCI-algebras

for all x ∈ I. Then G (0) = {0, 1} /q F (0) and G (1) = {0, 1} /q F (1). Hence (G , I) is a soft q-ideal of (F , A). Note that every soft q-ideal is a soft ideal. But the converse is not true in general as seen in the following example. Example 3.9. Let X := {0, 1, a, b} be a BCI-algebra ([4]) in which the ∗-operation is given by the following table: ∗ 0 1 a b 0 0 0 b a 1 1 0 b a a a a 0 b b b b a 0 For A = {0, 1}, define a set-valued function F (x) : A → P(X) by F (x) = {0} ∪ {y ∈ X|y ∗ (y ∗ x) = 0} for all x ∈ A. Then F (0) = X and F (1) = {0, a, b} are subalgebras of X. Hence (F , A) is a soft BCI-algebra over X. For I := {0}, let G : I → P(X) be a set-valued function defined by G (x) = {0} ∪ {y ∈ X|x ≤ y} for all x ∈ I. Then G (0) = {0, 1}. Hence G (0) / F (0), but G (0) 6q F (0) since a ∗ (0 ∗ b) = 0, 0 ∈ {0, 1} and a ∗ b = b ∈ / {0, 1}. Theorem 3.10. Let (F , A) be a soft BCI-algebra over X. For any soft sets, (G1 , I1 ) and (G, I2 ) over X where I1 ∩ I2 6= ∅, we have ˜ (G2 , I2 )˜/q (F , A). (G1 , I1 )/˜q (F , A), (G2 , I2 )/˜q (F , A) ⇒ (G1 , I1 )∩

Proof. By Definition 2.2, we can write ˜ (G2 , I2 ) = (G , I), (G1 , I1 )∩ where I = I1 ∩ I2 and G (x) = G1 (x) or G2 (x) for all x ∈ I. Obviously, I ⊂ A and G : I → P(X) is a mapping. Hence (G , I) is a soft set over X. Since (G1 , I1 )/˜q (F , A) and (G2 , I2 )/˜q (F , A), we know that G (x) = G1 (x)˜/q F (x) or G (x) = G2 (x)˜/q F (x) for all x ∈ I. Hence ˜ (G2 , I2 ) = (G , I)˜/q (F , A). (G1 , I1 )∩ This completes the proof.



Corollary 3.11. Let (F , A) be a soft BCI-algebra over X. For any soft sets, (G1 , I) and (G2 , I) over X, we have ˜ (G2 , I)˜/q (F , A). (G1 , I)/˜q (F , A), (G2 , I)/˜q (F , A) ⇒ (G1 , I)∩ 575

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Yun Sun Hwang and Sun Shin Ahn

Proof. Straightforward.



Theorem 3.12. Let (F , A) be a soft BCI-algebra over X. For any soft sets, (G , I) and (H , J) over X in where I ∩ J = ∅, we have ˜ (H , J)˜/q (F , A). (G , I)/˜q (F , A), (H , J)/˜q (F , A) ⇒ (G , I)∪

Proof. Assume that (G , I)˜/q (F , A) and (H , J)˜/q (F , A). By Definition 2.3, we can write ˜ (H , J) = (K , U ) where U = I ∪ J and for every x ∈ U , (G , I)∪   if x ∈ I\J  G (x) K (x) = H (x) if x ∈ J\I   G (x) ∪ H (x) if x ∈ I ∩ J. Since I ∩ J = ∅, either x ∈ I\J or x ∈ J\I for all x ∈ U . If x ∈ I\J, then K (x) = G (x) /q F (x) since (G , I)/˜q (F , A). If x ∈ J\I, then K (x) = H (x) /q F (x) since (H , J)/˜q (F , A). Thus K (x) /q F (x) for all x ∈ U , and so (G , I)(H , J) = (K , /˜q (F , A).  If I and J are not disjoint in Theorem 3.12, then Theorem 3.12 is not true in general as seen in the following example. Example 3.13. Consider a BCI-algebra X = {0, 1, a, b} which is given in Example 3.3. Let (F , A) be a soft set over X, where A := {0, 1, a} ⊂ X and F : A → P(X) is a set-valued function defined by F (x) = {0} ∪ {y ∈ X|y ∗ (y ∗ x) ∈ {0, 1, a}} for all x ∈ A. Then (F , A) is a soft BCI-algebra over X (see Example 3.8). Let I := {0, 1} and G : I → (X) be a set-valued function defined by G (x) = {0} ∪ {y ∈ X|x ≤ y} for all x ∈ I. Then (G , I) is a soft q-ideal of (F , A) (see Example 3.8). Let J := {0} and H : J → P(X) be defined by H (x) = {x, a}. Then H (0) = {0, a} /q F (0). But G (0) ∪ H (0) = {0, 1, a} 6q F (0) since b ∗ (1 ∗ 0) = a, 1 ∈ {0, 1, a} and b ∗ 0 = b ∈ / {0, 1, a}. 4. q-idealistic soft BCI-algebras Definition 4.1.([8]) Let (F , A) be a soft set over X. Then (F , A) is called an idealistic soft BCI-algebra over X if F (x) is an ideal of X for all x ∈ A. Definition 4.2. Let (F , A) be a soft set over X. Then (F , A) is called a q-idealistic soft BCI-algebra over X if F (x) is a q-ideal of X for all x ∈ A. 576

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Soft q-ideals of soft BCI-algebras

Example 4.3. Let X := {0, a, b, c} be a BCI-algebra ([9]) in which the ∗-operation is given by the following table: ∗ 0 a b c 0 0 a b c a a 0 c b b b c 0 a c c b a 0 Let A = X and G : A → P(X) be a set-valued function defined by G (x) = {0, x} for all x ∈ A. Then G (0) = {0}, G (a) = {0, a} and G (c) = {0, c}, which are ideals of X. Hence (G , A) is an idealistic soft BCI-algebra over X ([8]). Note that G (x) is a q-ideal of X for all x ∈ A. Hence (G , A) is a q-idealistic soft BCI-algebra over X. For any element x of a BCI-algebra X, we define the order of X, denoted by o(x), as o(x) = min{n ∈ N|0 ∗ xn = 0} where 0 ∗ xn = (· · · ((0 ∗ x) ∗ x) ∗ · · · ) ∗ x in which x appears n-times. Example 4.4. Let X := {0, a, b, c, de, f, g} be a BCI-algebra ([1]) in which the ∗-operation is given by the following table: ∗ 0 a b c d e f g 0 0 0 0 0 d d d d a a 0 0 0 e d d d b b b 0 0 f f d d c c b a 0 g f e d d d d d d 0 0 0 0 e e d d d a 0 0 0 f f f d d b b 0 0 g g f e d c b a 0 Let (F , A) be a soft set over X, where A = {a, b, c} ⊂ X and F : A → P(X) is a set-valued function defined as follows: F (x) = {y ∈ X|o(x) = o(y)} for all x ∈ A. Then F (a) = F (b) = F (c) = {0, a, b, c} is an ideal of X. Hence (F , A) is an idealistic soft BCI-algebra over X ([6]). If we take B := {a, b, d, f } ⊂ X and define a set-valued function G : B → P(X) by G (x) = {0} ∪ {y ∈ X|o(x) = o(y)} for all x ∈ B, then (G , B) is not a q-idealistic soft BCI-algebra over X. In fact, since f ∗ (g ∗ e) = d, g ∈ {0, d, e, f, g} and f ∗ e = b ∈ / {0, d, e, f, g}, G (d) = {0, d, e, f, g} is not a q-ideal of X. Obviously, every q-idealistic soft BCI-algebra over X is an idealistic soft BCI-algebra over X, but the converse is not true in general as seen in the following example. 577

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J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 16, NO.3, 2014, COPYRIGHT 2014 EUDOXUS PRESS, LLC

Yun Sun Hwang and Sun Shin Ahn

Example 4.5. Consider a BCI-algebra X := Y ×Z, where (Y, ∗, 0) is a BCI-algebra over X and (Z, −, 0) is the adjoint BCI-algebra of the additive group (Z, +, 0) integers. Let F : X → P(X) be a set-valued function defined as follows: ( Y × N0 if x ∈ N0 F (y, n) = {(0, 0)} otherwise for all (y, n) ∈ X, where N0 is the set of all non-negative integers. Then (F , X) is an idealistic soft BCI-algebra over X([8]). But it is not a q-idealistic soft BCI-algebra over X since {(0, 0)} is not a q-ideal of X. In fact, (0, 3) ∗ ((0, 0) ∗ (0, −3)) = (0, 0) ∈ {(0, 0)} and (0, 3) ∗ (0, −3) = (0, 6) ∈ / {(0, 0)}. Proposition 4.6. Let (F , A) and (G , B) be soft sets over X where B ⊆ A ⊆ X. If (F , A) is a q-idealistic soft BCI-algebra over X, then so is (G , B). Proof. Straightforward.



The converse of Proposition 4.6 is not true in general as seen in the following example. Example 4.7. Consider a q-idealistic soft BCI-algebra (F , A) over X which is described in Example 4.4. If we take B := {a, b, c, d} ⊇ A = {a, b, c}, then (F , B) is not a q-idealistic soft BCI-algebra over X since F (d) = {d, e, f, g} is not a q-ideal of X. Theorem 4.8. Let (F , A) and (G , B) be two q-idealistic soft BCI-algebra over X. If A∩B 6= ∅, ˜ (G , B) is a q-idealistic soft BCI-algebra over X. then the intersection (F , A)∩ ˜ (G , B) = (H , C), where C = A ∩ B and Proof. Using Definition 2.2, we can write (F , A)∩ H (x) = F (x) or G (x) for all x ∈ C. Note that H : C → P(X) is a mapping, and therefore (H , C) is a soft set over X. Since (F , A) and (G , B) are q-idealistic soft BCI-algebras over X, it follows that H (x) = F (x) is a q-ideal of X, or H (x) = G (x) is a q-ideal of X for all x ∈ C. ˜ (G , B) is a q-idealistic soft BCI-algebra over X. Hence (H , C) = (F , A)∩  Corollary 4.9. Let (F , A) and (G , A) be two q-idealistic soft BCI-algebra over X. Then the ˜ (G , A) is a q-idealistic soft BCI-algebra over X. intersection (F , A)∩ Proof. Straightforward.



Theorem 4.10. Let (F , A) and (G , B) be two q-idealistic soft BCI-algebra over X. If A∩B = ∅, ˜ (G , B) is a q-idealistic soft BCI-algebra over X. then the union (F , A)∪ ˜ (G , B) = (H , C), where C = A ∪ B and for every Proof. Using Definition 2.3, we write (F , A)∪ x ∈ C,   if x ∈ A\B  F (x) H (x) = G (c) if x ∈ B\A   F (x) ∪ G (x) if x ∈ A ∩ B. Since A ∩ B = ∅, either x ∈ A\B or x ∈ B\A for all x ∈ C. If x ∈ A\B, then H (x) = F (x) is a q-ideal of X since (F , A) is a q-idealistic soft BCI-algebra over X. If x ∈ B\A, then 578

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Soft q-ideals of soft BCI-algebras

H (x) = G (x) is a q-ideal of X since (G , B) is a q-idealistic soft BCI-algebra over X. Hence ˜ (G , A) is a q-idealistic soft BCI-algebra over X. (F , A)∪  ˜ (G , B) Theorem 4.11. If (F , A) and (G , B) are q-idealistic soft BCI-algebra over X, then (F , A)∧ is a q-idealistic soft BCI-algebra over X. Proof. By Definition 2.4, ˜ (G , B) = (H , A × B), (F , A)∧ where H (x, y) = F (x) ∩ G (y) for all (x, y) ∈ A × B. Since F (x) and G (y) are q-ideals of X, the intersection F (x) ∩ G (y) is also a q-ideal of X. Hence H (x, y) is a q-ideal of X for all ˜ (G , B) is a q-idealistic soft BCI-algebra over X. (x, y) ∈ A × B, and therefore (F , A)∧  Definition 4.12. A q-idealistic BCI-algebra (F , A) over X is said to be trivial (resp., whole) if F (x) = {0} (resp., F (x) = X) for all x ∈ A. Example 4.13. Let X = {0, a, b, c} be a BCI-algebra which is given Example 4.3. Let (F , A) be a soft set over X, where A := {a, b, c} ⊂ X, and let F : A → P(X) be a set-valued function defined by F (x) = {y ∈ X|o(x) = o(y)} for all x ∈ X. Then F (a) = F (b) = F (c) = X. It is check that X /q X. Hence (F , X \ {0}) is a whole q-idealistic soft BCI-algebra over X. Let G : {0} → P(X) be a set-valued function defined by G (x) = x for all x ∈ {0}. Then G (0) = {0}. It is check that {0} /q X. Hence (G , {0}) is a trivial q-idealistic soft BCI-algebra over X. Definition 4.14.([10]) A fuzzy set µ in X is a fuzzy q-ideal of X if it satisfies the following assertions: (i) (∀x ∈ X)(µ(0) ≥ µ(x)), (ii) (∀x, y, z ∈ X)(µ(x ∗ z) ≥ min{µ(x ∗ (y ∗ z)), µ(y)}). Lemma 4.15. A fuzzy set µ in X is a fuzzy q-ideal of X if and only if it satisfies: (∀t ∈ [0, 1])(U (µ; t) 6= ∅ ⇒ U (µ; t) is a q-ideal of X).

Proof. Straightforward.



Theorem 4.16. For every fuzzy q-ideal of X, there exists a q-idealistic soft BCI-algebra (F , A) over X. Proof. Let µ be a fuzzy q-ideal of X. Then U (µ; t) := {x ∈ X|µ(x) ≥ t} is a q-ideal of X for all t ∈ Im(µ). If we take A = Im(µ) and consider a set-valued function F : A → P(X) given by F (t) = U (µ; t) for all t ∈ A, then (F , A) is a q-idealistic soft BCI-algebra over X.  Conversely, the following theorem is straightforward. Theorem 4.17. For any fuzzy set µ in X, if a q-idealistic soft BCI-algebra (F , A) over X is given by A = Im(µ) and F (t) = U (µ; t) for all t ∈ A, then µ is a fuzzy q-ideal of X. 579

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Yun Sun Hwang and Sun Shin Ahn

Let µ be a fuzzy set in X and let (F , A) be a soft set over X in which A = Im(µ) and F : A → P(X) is a set-valued function defined by (4.1) (t ∈ A)(F (t) = {x ∈ X|µ(x) + t > 1}). Then there exists t ∈ A such that F (t) is not a q-ideal of X as seen in the following example. Example 4.18. For any BCI-algebra X, define a fuzzy set µ in X by µ(0) = t0 < 0.5 and µ(x) = 1 − t0 for all x 6= 0. Let A = Im(µ) and F : A → P(X) be a set-valued function given by (4.1). Then F (1 − t0 ) = X\{0}, which is not a q-ideal of X. Theorem 4.19. Let µ be a fuzzy set in X and let (F , A) be a soft over X in which A = [0, 1] and F : A → P(X) is given by (4.1). Then the following assertions are equivalent: (1) µ is a fuzzy q-ideal of X, (2) for every t ∈ A with F (t) 6= ∅, F (t) is a q-ideal of X. Proof. Assume that µ is a fuzzy q-ideal of X. Let t ∈ A be such that F (t) = 6 ∅. If we select x ∈ F (t), then µ(0) + t ≥ µ(x) + t > 1, and so 0 ∈ F (t). Let t ∈ A and x, y, z ∈ X be such that y ∈ F (t) and x ∗ (y ∗ z) ∈ F (t). Then µ(y) + t > 1 and µ(x ∗ (y ∗ z)) + t > 1. Since µ is a fuzzy q-ideal of X, it follows that µ(x ∗ z) + t ≥ min{µ(x ∗ (y ∗ z)), µ(y)} + t = min{µ(x ∗ (y ∗ z)) + t, µ(y) + t} > 1, so that x ∗ z ∈ F . Hence F (t) is a q-ideal of X with F (t) 6= ∅. Conversely, suppose that (2) is valid. If there exists a ∈ X such that µ(0) < µ(a), then we can select ta ∈ A such that µ(0) + ta ≤ 1 < µ(a) + ta . It follows that a ∈ F (ta ) and 0 ∈ / F (ta ), which is a contradiction. Hence µ(0) ≥ µ(x) for all x ∈ X. Now, assume that µ(a ∗ c) < min{µ(a ∗ (b ∗ c)), µ(b)} for some a, b, c ∈ X. Then µ(a ∗ c) + s0 ≤ 1 < min{µ(a ∗ (b ∗ c)), µ(b)} + s0 , for some s0 , which implies a ∗ (b ∗ c) ∈ F (s0 ) and b ∈ F (s0 ), but a ∗ c ∈ F (s0 ). This is a contradiction. Therefore µ(x ∗ z) ≥ min{µ(x ∗ (y ∗ z)), µ(y)}, for all x, y, z ∈ X, and thus µ is a fuzzy q-ideal of X.



Corollary 4.20. Let µ be a fuzzy set in X such that µ(x) > 0.5 for some x ∈ X, and let (F , A) be a soft set over X in which A := {t ∈ Im(µ)|t > 0.5} and F : A → P(X) is given by (4.1). If µ is a fuzzy q-ideal of X, then (F , A) is a q-idealistic soft BCI-algebra over X. Proof. Straightforward.

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Soft q-ideals of soft BCI-algebras

Theorem 4.21. Let µ be a fuzzy set in X and let (F , A) be a soft set over X in which A = (0.5, 1] and F : A → P(X) is defined by (∀t ∈ A)(F (t) = U (µ; t)). Then F (t) is a q-ideal of X for all t ∈ A with F (t) 6= ∅ if and only if the following assertions are valid: (1) (∀x ∈ X)(max{µ(0), 0.5} ≥ µ(x)), (2) (∀x, y, z ∈ X)(max{µ(x ∗ z), 0.5} ≥ min{µ((x ∗ (y ∗ z)), µ(y)}). Proof. Assume that F (t) is a q-ideal of X for all t ∈ A with F (t) 6= ∅. If there exists x0 ∈ X such that max{µ(0), 0.5} < µ(x0 ), then we can select t0 ∈ A such that max{µ(0), 0.5} < t0 ≤ µ(x0 ). It follows that µ(0) < t0 , so that x0 ∈ F (t0 ) and 0 ∈ / F (t0 ). This is a contradiction, and so (1) is valid. Suppose that there exist a, b, c ∈ X such that max{µ(a ∗ c), 0.5} < min{µ(a ∗ (b ∗ c)), µ(b)}. Then max{µ(a ∗ c), 0.5} < u0 ≤ min{µ(a ∗ (b ∗ c)), µ(b)}. for some u0 ∈ A. Thus a ∗ (b ∗ c) ∈ F (u0 ) and b ∈ F (u0 ), but a ∗ c ∈ / F (u0 ). This is a contradiction, and so (2) is valid. Conversely, suppose that (1) and (2) are valid. Let t ∈ A with F (t) 6= ∅. For any x ∈ F (t), we have max{µ(0), 0.5} ≥ µ(x) ≥ t > 0.5 and so µ(0) ≥ t, i.e., 0 ∈ F (t). Let x, y, z ∈ X be such that y ∈ F (t) and x ∗ (y ∗ z) ∈ F (t). Then µ(y) ≥ t and µ(x ∗ (y ∗ z)) ≥ t. It follows from the second condition that max{µ(x ∗ z), 0.5} ≥ min{µ(x ∗ (y ∗ z)), µ(y)} ≥ t > 0.5, so that µ(x ∗ z) ≥ t, i.e., x ∗ z ∈ F (t). Therefore F (t) is a q-ideal of X for all t ∈ A with F (t) 6= ∅. 

References [1] M. A. Chaudhry, Weakly positive implicative and weakly implicative BCI-algebras, Math. Jpn. 35(1990), 141-151. [2] D. Molodtsov, Soft set theory-First results, Comput. Math. Appl. 37(1999), 19-31. [3] P. K. Maji, R. Biswas and A. R. Roy, Soft set theory, Comput. Math. Appl. 45(2003), 555-562. [4] Y. Huang, BCI-algebras, Sciences Press, Bejing, 2006. [5] Y. S. Hwang and S. S. Ahn, Vague q-ideals in BCI-algebras, Honam Math. J. 34(2012), 549-557. [6] Y. B. Jun, Soft BCK/BCI-algebras, Comput. Math. Appl. 56(2008), 1408-1413. [7] Y. B. Jun , K. J. Lee and J. Zhan, Soft p-ideals of soft BCI-algebras, Comput. Math. Appl. 58(2009), 2060-2068. [8] Y. B. Jun and C. H. Park, Applications of soft sets in ideal theory of BCK/BCI-algebras, Inform. Sci. 178(2008), 2466-2475. 581

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Yun Sun Hwang and Sun Shin Ahn [9] Y. L. Liu, J. Meng, X. H. Zhang and Z. C. Yue, q-ideals and a-ideals in BCI-algebras, Southeast Asian Bull. Math. 24(2000), 243-253. [10] S. K. Sung and S. S. Ahn, Fuzzy q-ideals of BCI-algebras with degrees in the interval (0, 1], J. Chungcheong Math. Soc. 25(2012), 241-251. [11] L. A. Zadeh, Fuzzy sets, Inform. Control 8(1965), 338-353.

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Convergence of parallel multisplitting USAOR methods for block H −matrices linear systems Xue-Zhong Wang School of Mathematics and Statistics, Hexi University, Zhangye, Gansu, 734000 P.R. China

Abstract: In this paper, We present parallel multisplitting blockwise relaxation methods for solving the large sparse blocked linear systems, which come from the discretizations of many discrential equations, and study the convergence of our methods associated with USAOR multisplitting when the coefficient matrices of the blocked linear systems are block H -matrices. A lot of numerical experiments show that our methods are applicable and efficient. Key words: Block matrix multisplitting; Blockwise relaxation parallel multisplitting method; Convergence; Block H -matrix., 2000 MR Subject Classification: 65F10, 65F50

1. Introduction For the linear system Ax = b,

(1.1)

where A is an n × n square matrix, and x and b are n-dimensional vectors. O’Leary and White [6]invented the matrix multisplitting method in 1985 for solving parallely the large sparse linear systems on the multiprocessor systems and was further studied by many authors. For example, Neumann and Plemmons [5] developed some more refined convergence results for one of the cases considered in [6], Elsner [7] established the comparison theorems about the asymptotic convergence rate of this case, Frommer and Mayer [8] discussed the successive overrelaxation (SOR) method in the sense of multisplitting, White [9,10] studied the convergence properties of the above matrix multisplitting methods for the symmetric positive definite matrix class, as well as matrix multisplitting methods as preconditioners, respectively, Bai [4] established the convergence theory of a class of asynchronous multisplitting blockwise relaxation methods, Zhang, Huang, et, al. [3] present local relaxed parallel multisplitting method and global relaxed parallel multisplitting method for H -matrices and so on. On the other hand, Since the finite element or the finite difference discretizations of many partial differential equations usually result in the large sparse systems of linear equations of regularly blocked structures, recently, [1,4] further generalized the matrix multisplitting concept of O’Leary and White [6] to a blocked form and proposed a class of parallel matrix multisplitting blockwise relaxation methods. This class of methods, besides enjoying all the advantages of the existing pointwise parallel matrix multisplitting methods discussed in [6,12], possesses better convergence properties and robuster numerical behaviours. Therefore, the parallel matrix multisplitting blockwise relaxation methods for the solution of large and sparse nonsingular blocked linear system have become more and more obvious. In the following, we recall the mathematical descriptions of the blocked linear system and the BMM introduced in [1,4]. PN Let N (≤ n) and n i (≤ n)(i = 1, 2, . . . , N ) be given positive integers satisfying i =1 n i = n, and denote Vn (n 1 , . . . , n N ) = {x ∈ Rn |x = (x 1T , . . . , x NT )T , x i ∈ Rn i }, Ln (n 1 , . . . , n N ) = {A ∈ Rn ×n |A = (A i j )N ×N , A i j ∈ Rn i ×n j }, When the context is clear we will simply use Ln for Ln (n 1 , . . . , n N ) and Vn for Vn (n 1 , . . . , n N ). Then, the blocked linear system to be solved can be expressed as the form Ax = b,

A ∈ Ln ,

x ,b ∈ Vn

(1.2)

Email address: [email protected] (Xue-Zhong Wang)

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Convergence of parallel multisplitting USAOR methods for block H −matrices linear systems where A ∈ Ln is nonsingular and b ∈ Vn are general known coefficient matrix and right-hand vector, respectively, and x ∈ Vn is the unknown vector. If blocked matrices M k , N k , E k ∈ Ln (k = 1, 2, . . . , α) satisfy 1. A = M k − N k , M k nonsingular, k = 1, 2, . . . , α, (k ) (k ) 2. E k = d i a g (E 11 , . . . , E N N ), k = 1, 2, . . . , α, Pα (k ) 3. k =1 ||E i i || = 1, i = 1, 2, . . . , N , then we call the collection of triples (M k , N k , E k )(k = 1, 2, . . . , α) is a BMM of the blocked matrix A ∈ Ln , where || · || denotes the consistent matrix norm. Suppose that we have a multiprocessor with α processors connected to a host processor, that is, the same number of processors as splittings, and that all processors have the last update vector x k , then the k th processor only computes those entries of the vector M k−1 N k x k + M k−1b, (k )

which correspond to the block diagonal entries E i i of the blocked matrix E k . The processor then scales these entries so as to be able to deliver the vector E K (M k−1 N k x k + M k−1b ) to the host processor, performing the parallel multisplitting scheme x m +1 =

α X

E K M k−1 N k x m +

k =1

α X

E K M k−1b = Hx m + G b, m = 0, 1, 2, . . .

k =1

Under reasonable restrictions on the relaxation parameters and the multiple splittings, we establish local parallel multisplitting blockwise relaxation method, global parallel multisplitting blockwise relaxation method and global nonstationary parallel multisplitting blockwise relaxation method for solving the large sparse blocked linear systems and study the convergence of our methods associated with USAOR multisplitting when the coefficient matrices of the blocked linear systems are block H -matrices. 2. Establishments of the methods Given a positive integer α(α ≤ N ), we Sαseparate the number set {1, 2, . . . N } into a nonempty subsets J k (k = 1, 2, . . . , α) such that J k ⊆ {1, 2, . . . , N } and k =1 J k = {1, 2, . . . , N }. Note that there may be overlappings among the subsets J 1 , J 2 , . . . , J α . Corresponding to this separation, we introduce matrices D =diag(A 11 , . . . , A N N ) ∈ Ln¨ , (k ) L i j for i , j ∈ J k and i > j , (k ) (k ) L k = (Li j ) ∈ Ln , Li j = 0 otherwise, ¨ (k ) for i , j , Ui j (k ) (k ) Uk = (Ui j ) ∈ Ln , Ui j = 0 otherwise, ¨ (k ) Eii for i ∈ J k , (k ) (k ) (k ) E k =diag(E 11 , . . . , E N N ) ∈ Ln , E i i = i , j = 1, 2, . . . , N ; k = 1, 2, . . . , α. 0 otherwise, Obviously, D is a blocked diagonal matrix, L k (k = 1, 2, . . . , α) are blocked strictly lower triangular matrices, Uk (k = 1, 2, . . . , α) are general blocked matrices, and E k (k = 1, 2, . . . , α) are blocked diagonal matrices. If they satisfy 1. D is nonsingular; 2. A D − L k − Uk , k = 1, 2, . . . , α; P= α 3. k =1 E k = I , then the collection of triples (D − L k ,Uk , E k ) and (D − Uk , L k , E k ) (k = 1, 2, . . . , α) are BMM of the blocked matrix A ∈ Ln . Here, I denotes the identity matrix of order n × n. We will present local parallel multisplitting blockwise relaxation USAOR method (LBUSAOR) and global parallel multisplitting blockwise relaxation USAOR method (GBUSAOR).

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Xue-Zhong Wang Algorithm 2.1. (local parallel multisplitting blockwise relaxation method) Given the initial vector For m = 0, 1, 2, . . . repeat (I) and (II), until convergence. (I) For k = 1, 2, . . . , α, (parallel) solving y k : M k y k = N k x m + b. (II) Computing x m +1 =

α X

E k yk . k =1

Algorithm 2.1 associated with LBUSAOR method can be written as x m +1 = H L BUSAOR x m + G L BUSAOR b, m = 0, 1, · · · , where H L BUSAOR Uω2 r2 (k ) L ω1 r1 (k ) G L BUSAOR

= = = =

(2.1)

Pα

k =1 E k Uω2 r2 (k )L ω1 r1 (k ), (D − r2Uk )−1 {(1 − ω2 )D + (ω2 − r2 )Uk + ω2 L k }, −1 (D Pα− r1 L k ) {(1 − ω1 )D + (ω1 − r1 )L k + ω1Uk }, −1 k =1 E k (D − r 2Uk ) {(ω1 + ω2 − ω1 ω2 )D + ω2 (ω1 − r 1 )L k +ω1 (ω2 − r2 )Uk }(D − r1 L k )−1

(2.2)

By using a suitable positive relaxation parameter β , we will establish global parallel multisplitting blockwise relaxation USAOR method which is based on Algorithm 2.1. Algorithm 2.2. (global parallel multisplitting blockwise relaxation method) Given the initial vector For m = 0, 1, 2, . . . repeat (I) and (II), until convergence. (I) For k = 1, 2, . . . , α, (parallel) solving y k : M k y k = N k x m + b. (II) Computing x m +1 = β

α X

E k y k + (1 − β )x m .

k =1

Algorithm 2.2 associated with GBUSAOR method can be written as x m +1 = HG BUSAOR x m + βG L BUSAOR b, m = 0, 1, · · · ,

(2.3)

where HG BUSAOR = β H L BUSAOR + (1 − β )I . In the standard multisplitting method each local approximation is updated exactly once using the same previous iterate x m . On the other hand, it is possible to update the local approximations more than once, using different iterates computed earlier. In this case, we call this method a nonstationary multisplitting method [15,16,17]. The main idea of the nonstationary method is that at the mth iteration each processor k solves the system q (m , k ) times, using in each time the new calculated vector to update the right-hand side; i.e., we have the following algorithm: Algorithm 2.3. (global nonstationary parallel multisplitting blockwise relaxation method) Given the initial vector For m = 0, 1, 2, . . . repeat (I) and (II), until convergence. (i ) (I) For i = 1, 2, . . . ,q (m , k ), (parallel) solving y k : (i )

(i −1)

M k yk = N k yk

585

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Convergence of parallel multisplitting USAOR methods for block H −matrices linear systems (II) Computing x m +1 = β

α X

q (m ,k )

E k yk

+ (1 − β )x m .

k =1

Algorithm 2.3 associated with GNBUSAOR method can be written as x m +1 = HG N BUSAOR x m + βG G N BUSAOR b, m = 0, 1, · · · , where HG N BUSAOR Pωr Q ξη

= = =

G G N BUSAOR

=

Pα β k =1 E k (Pωr Q ξη )q (m ,k ) + (1 − β )I (D − rk Uk )−1 {(1 − ωk )D + (ωk − rk )Uk + ωk L k } = M r−1 N ωr , (D − ηk L k )−1 {(1 − ξk )D + (ξk − ηk )L k + ξk Uk } = M η−1 N ξη , Pq (m ,k )−1 Pα β k =1 E k i =1 (M η M r )−1 (N ωr N ξη )i (M η M r )−1 ωk ξk .

(2.4)

(2.5)

It follows that when q (m , k ) = 1, ωk = ω2 , rk = r2 , ξk = ω1 and ηk = r1 for 1 < k < α, m = 0, 1, 2...., Algorithm 2.3 reduces to Algorithm 2.2. 3. Preliminaries We shall use the following notations and lemmas. A matrix A = (a i j ) is called a Z -matrix if for any i , j , a i j ≤ 0. A Z -matrix is a nonsingular M -matrix if A is nonsingular and if A −1 ≥ 0. Additionally, we denote the spectral radius of A by ρ(A). It is well-known that if A ≥ 0 and there exists a vector x > 0 such that Ax < αx , then ρ(A) < α. Let Ln,I (n 1 , . . . , n N ) = {M = (M i j ) ∈ Ln |M i i ∈ Rn i ×n i nonsingular, i = 1, . . . , N }. We will review the concepts of strictly block diagonally dominant matrix and the block H-matrix. Let A ∈ Ln ,I . Then its block comparison matrix 〈A〉 is defined by ¨ −1 kA −1 i = j, ij k , i , j = 1, . . . , N 〈A〉i j = −kA i j k, i , j, where k · k is a consistent matrix norm. If −1 kA −1 > ii k

X kA i j k, j = 1, 2, . . . , N . i ,j

Then A is said to be a strictly block diagonally dominant matrix [13], if there exists a positive diagonally matrix X such that AX is a strictly block diagonally dominant matrix, then A is said to be a block H -matrix [14]. Clearly, a strictly block diagonally dominant matrix is certainly a block H -matrix. Definition 3.1 [1]. Let M ∈ Ln . We call [M ] = (kM i j k) ∈ RN ×N the block absolute value of the blocked matrix M . The block absolute value [x ] ∈ RN of a blocked vector x ∈ Vn is defined in an analogous way. These kinds of block absolute values have the following important properties. Lemma 3.1 [1]. Let L, M ∈ Ln , x , y ∈ Vn and r ∈ R1 . Then 1. |[L] − [M ]| ≤ [L + M ] ≤ [L] + [M ] (|[x ] − [y ]| ≤ [x + y ] ≤ [x ] + [y ]); 2. [LM ] ≤ [L][M ] ([x y ] ≤ [x ][y ]); 3. [r M ] ≤ |r |[M ] ([r x ] ≤ |r |[x ]); 4. ρ(M ) ≤ ρ(|M |) ≤ ρ([M ]) (here, k · k is either k · k∞ or k · k1 ). Lemma 3.2 [1]. Let M ∈ Ln,I be a strictly block diagonally dominant matrix, then 1. M is nonsingular; 2. [(M )−1 ] ≤ 〈M 〉−1 ; 3. ρ(J (〈M 〉)) < 1.

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Xue-Zhong Wang 4. Convergence For Algorithms 1, 2 and 3, we give convergence theorems for block diagonally dominant matrices and block H −matrices. Theorem 4.1. Let A be a strictly block diagonally dominant matrix, and the collection of triples (D − L k ,Uk , E k ) and (D − Uk , L k , E k ) (k = 1, 2, . . . , α) are BMM of the blocked matrix A ∈ Ln . Assume that 〈A〉 = 〈D〉 − [L k ] − [Uk ] = 〈D〉 − [B ],

k = 1, 2, . . . , α,

0 < ω1 , ω2
0, and 2 〈D〉 − r [B ], (k = 1, 2, . . . , α) are strictly block diagonally dominant matrix for 0 < r < 1+ρ which follow from A is 2 a strictly block diagonally dominant matrix. Since 〈D〉 − r [B ] ≤ 〈D〉 − r [Uk ] ≤ 〈D〉 for 0 < r < 1+ρ , k = 1, 2, . . . , α, and 〈A〉 is a strictly diagonally dominant matrix, we have 〈D〉 − r [B ] and 〈D〉 are strictly diagonally dominant M 2 matrices, for 0 < r < 1+ρ , k = 1, 2, . . . , α. Therefore, 〈D〉 − r [Uk ] are strictly diagonally dominant M -matrices, and 2 then D − rUk are strictly block diagonally dominant matrices, for 0 < r < 1+ρ , k = 1, 2, . . . , α. Let L¯k = D −1 L k and U¯k = D −1Uk , then I −r L¯k and I −r U¯k are also strictly block diagonally dominant matrices,

for 0 < r
0. By the Perron-Frobenius theorem for any ε > 0, there is a vector x ε > 0 such that J ε (〈A〉)x ε = ρ( J ε (〈A〉))x ε = ρε x ε ,

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Convergence of parallel multisplitting USAOR methods for block H −matrices linear systems where ρε = ρ( J ε (〈A〉)). Moreover, if ε > 0 is small enough, we have ρε < 1 by continuity of the spectral radius. Thus, we can get |1 − ωi | + ωi ρε < 1 for 0 < ωi < and then [H L BUSAOR ]x ε

2 , 1+ρ

i = 1, 2,

Pα

Pα ≤ k =1 [E k ]T (ω2 )T (ω1 )x ε (|1 − ω2 |I + ω2 J ε (〈A〉))(|1 − ω1 |I + ω1 J ε (〈A〉))x ε (|1 − ω1 | + ω1 ρε )(|1 − ω2 | + ω2 ρε )x ε x ε,

≤ ≤ =
(2/|β|)(|aT |Jm y)Jm |a|. Select a positive scalar α such that (hBi − Jm |C|)y > αJm |a| > (2/β)(|aT |Jm y)Jm |a|, we obtain that 

hBi

   −|aT |J m   −|C|

−Jm |a|

−Jm |C|Jm T

|β|

−|a |

−|a|

Jm hBiJm



y

    

α

    > 0,  

Jm y

which mean that the matrix hAi is an M -matrix. Thus we yield the desired results.

4. The iterative methods for circulant matrices 4.1 Construction of an arithmetic mean splitting We will give a new splitting scheme of a circulant matrix A, which is called the arithmetic mean splitting. Let the circulant matrix A = M − N be a random convergent splitting of the matrix A. An iterative sequence derived from the splitting is defined by xk+1 = M −1 N xk + M −1 b.

(4.1.1)

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Since A = GT AG, we can get another convergent iterative sequence: xk+1 = GT M −1 N Gxk + GT M −1 Gb.

(4.1.2)

In terms of (4.1.1) and (4.1.2), we can obtain a new iterative sequence:

Denote F = written as

xk+1 =

1 1 (M −1 N + GT M −1 N G)xk + (M −1 + GT M −1 G)b. 2 2

1 −1 2 (M

+ GT M −1 G) and H =

1 −1 N 2 (M

(4.1.3)

+ GT M −1 N G), then (4.1.3) can be

xk+1 = Hxk + F b.

(4.1.4)

If det(F)6= 0, a new splitting of A can be expressed as A = F −1 − F −1 H.

(4.1.5)

(4.1.5) is called the arithmetic mean splitting of the matrix A. By (4.1.3) we can derive a splitting from a random convergent splitting of the circulant matrix A. Theorem 4.1.1 Let A be a circulant matrix and A=M −N be a weak regular splitting, then A = GT AG = GT M G − GT N G is also a weak regular splitting. Proof Since A = M − N is a weak regular splitting, there holds M −1 ≥ 0 and M −1 N ≥ 0. Consider that G is a permutation matrix and use the fact G−1 = GT , then (GT M G)−1 = GT M −1 G ≥ 0, and (GT M G)−1 (GT N G) = GT M −1 N G ≥ 0, Therefore A = GT M G − GT N G is a weak regular splitting. According to Lemma 2.8 and Theorem 4.1.1, we can get the following iterative convergent theorem. Theorem 4.1.2 Let A be a circulant M-matrix, and A = M − N be a weak regular of A, then the iterative sequence xk+1 =

1 1 (M −1 N + GT M −1 N G)xk + (M −1 + GT M −1 G)b 2 2

is convergent. Using Lemma 2.9 we can get the following result. Theorem 4.1.3 Let A be a circulant H-matrix, A = F − Q be a splitting of the matrix A and hAi = hF i − |Q|, then the iterative sequence xk+1 =

1 −1 1 (F J + GT F −1 JG)xk + (F −1 + GT F −1 G)b 2 2

is convergent.

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4.2 Two new splittings of circulant M -matrices Now we will present two new splittings of the circulant matrix A and investigate their convergence. (1) Opposite triangular splitting I: (i) For n = 2m, A = F1 − Q1 , where     ∗ ∗ ˆ1 Cˆ1 B B C 1    1  F1 =  ,  , Q1 =  ˆ1 C1∗ B1∗ Cˆ1 B ˆ1 and Cˆ1 is the left lower triangular matrix of B and C respectively, and here B strictly right upper triangular matrix of −B and −C respectively. (ii) For n = 2m + 1, A = F2 − Q2 , where    ˆ2 Jm a Jm Cˆ2 Jm B2∗ 0 Jm C2∗ Jm B        , Q2 =  −aT J F2 =  −aT β 0 m 0   0     ˆ 2 Jm C2∗ 0 Jm B2∗ Jm Cˆ2 a Jm B

B1∗ and C1∗ is the

   ,  

ˆ2 is the left lower triangular matrix of B, Cˆ2 is the left upper triangular matrix of C, B ∗ here B 2 is the strictly right upper triangular matrix of −B, and C2∗ is the strictly right lower triangular matrix of −C. (2) Opposite triangular splitting II: (i) For n = 2m, A = R1 − V1 , where 



  E1 R1 =  H1

E1∗

H1    , V1 =  H1∗ E1

H1∗ E1∗

  ,

here E1 and H1 is the right upper triangular matrix of B and C respectively, and E1∗ and H1∗ is the strictly left lower triangular matrix of −B and −C respectively. (ii) For n = 2m + 1,A = R2 − V2 , where 

E2

  T R2 =   a Jm  H2

0

Jm H2 Jm T

β

a

0

Jm E2 Jm





E2∗

     , V2 =  0     H2∗

−Jm a

Jm H2∗ Jm

0

0

−a

Jm E2∗ Jm

   ,  

here E2 is the right upper triangular matrix of B, H2 is the right lower triangular matrix of C, E2∗ is the strictly left lower triangular matrix of −B, and H2∗ is the strictly left upper triangular matrix of −C. In terms of the opposite triangular splitting I, we can get the following two SOR iterative sequences [5], at the same time, we can also get the similar conclusion by means of the opposite triangular splitting II. (i) For n = 2m, (a) Global SOR sequence F1 xk+1 = ((1 − ω)F1 + ωQ1 )xk + ωb.

(4.2.1)

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(b) Part SOR sequence F1 xk+1 = Q1 xk + b.

(4.2.2)

P T F1 P P T xk+1 = P T Q1 P P T xk + P T b,

(4.2.3)

Thus we have



ˆ ˆ  B1 − C1 Fˆ1 = P T F1 P =  0



ˆ 0   T1 = ˆ1 + Cˆ1 B 0





B1∗ − C1∗ ˆ 1 = P T Q1 P =  G  0



0 B1∗ + C1∗



ˆ   H1 = 0

 0  , Tˆ2

 0  , ˆ H2

ˆ1 − Cˆ1 and Tˆ2 = B ˆ1 + Cˆ1 are m × m left lower triangular matrices, H ˆ 1 = B∗ − C ∗ where Tˆ1 = B 1 1 ˆ 2 = B ∗ + C ∗ are m × m strictly right upper triangular matrices. and H 1 1 Let P T xk+1 = yk+1 , P T xk = yk , P T b = ˆb, then (4.2.3) becomes         (1) (1) ˆb(1) ˆ1 0 ˆ1 0 y H y T   k     k+1    . +  =   (2) (2) ˆb(2) ˆ2 yk 0 H yk+1 0 Tˆ2 Thus

  ˆ 1 y (1) + ˆb1 ,  Tˆ1 y (1) = H k+1 k

(4.2.4)

  Tˆ2 y (2) = H ˆ 2 y (2) + ˆb2 . k+1 k From (4.2.4), we can get Part SOR sequence:   ˆ 1 )y (1) + ω1ˆb1 ,  Tˆ1 y (1) = ((1 − ω1 )Tˆ1 + ω1 H k+1 k

(4.2.5)

  Tˆ2 y (2) = ((1 − ω2 )Tˆ2 + ω2 H ˆ 2 )y (2) + ω1ˆb2 . k+1 k (ii) For n = 2m + 1 (aa) Global SOR sequence F2 xk+1 = ((1 − ω)F2 + ωQ2 )xk + ωb.

(4.2.6)

(bb) Part SOR sequence F2 xk+1 = Q2 xk + b. We can get the similar Part SOR iterative sequence:    T ∗ y (1) = ((1 − ω1 )T ∗ + ω1 H ∗ )y (1) + ω1 b∗ , 1 1 k 1 1 k+1

(4.2.7)

(4.28)

  T ∗ y (2) = ((1 − ω2 )T ∗ + ω2 H ∗ )y (2) + ω1 b∗ , 2 k+1 2 2 k 2

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



ˆ2 − Jm Cˆ2 and T ∗ =  where T1∗ = B  √ 2

matrices, H1∗ = B2∗ − Jm C2∗ and H2∗

β

0

  are m × m left lower triangular

ˆ2 + Jm Cˆ2 2Jm a B   √ T 0 − 2a J m   =   are m × m strictly right upper 0 B2∗ + Jm C2∗

triangular matrices. Now we will discuss the convergence of the two splittings of circulant matrices and the SOR iterative sequence above. Theorem 4.2.1 Let A be a circulant M -matrix, and A = F − Q be opposite triangular splitting I or II of the matrix A, then ρ(F −1 Q) < 1. Proof It can easily get that A ≤ F ≤ |D|, where D is the diagonal part of the matrix A. By Lemma 2.6, F is also an M -matrix, then F −1 ≥ 0. On the other hand, it is evident that Q ≥ 0. By the definition of the regular splitting, A = F − Q is a regular splitting of the matrix A. Using Lemma 2.2 we have ρ(F −1 Q) < 1. Theorem 4.2.2 Let A be a circulant M -matrix, and A = F1 − Q1 be opposite triangular splitting I of A,  then  (1) if ω ∈ 0, 1+ρ(F2−1 Q ) ; Global SOR sequence is convergent, 1 1     2 (2) if ω1 ∈ 0, 1+ρ(Tˆ−1 Hˆ ) , ω2 ∈ 0, 1+ρ(Tˆ2−1 Hˆ ) , Part SOR sequence is convergent. 1 2 1 2   −1 Proof (1) By Theorem 4.2.1, ρ(F1 Q1 ) < 1. Using Lemma 6 in [6], when ω ∈ 0, 1+ρ(F2−1 Q ) , 1

1

ρ(H(ω)) < 1, where H(ω) = (1 − ω)I + ωF1−1 Q1 is the iterative matrix of global SOR sequence. ˆ 2) < ˆ 1 ) < 1, and ρ(Tˆ−1 H ˆ 1 ) < 1. From (4.2.3), ρ(Tˆ−1 H ˆ −1 Q (2) Since ρ(F1−1 Q1 ) < 1, then ρ(F 2 1   1  1. By Lemma 6 of [6], when ω1 ∈

0, 1+ρ(Tˆ2−1 Hˆ 1

1)

and ω2 ∈

0, 1+ρ(Tˆ2−1 Hˆ 2

2)

, the Part SOR

sequence is convergent. Similarly, (4.2.7) and (4.2.8) are convergent. It is easy to find that the proof of the case of n = 2m + 1 is similar to above. Using the same methods, we can obtain the related results of the splitting II. We will make a comparison of convergence rate of the iterative sequences. From Lemma 2.10, we can get the following two theorems. 0 Theorem 4.2.3 Let A be a circulant M -matrix, A = D − (L + U ) be Jacobi s splitting of A and A = F − Q be opposite triangular splitting I of A, then ρ(F −1 Q) ≤ ρ(D−1 (L + U )). Proof It can easily get that A−1 ≥ 0, A = F − Q and A = D − (L + U ) are the regular splittings of the matrix A and 0 ≤ Q ≤ L + U , then by Lemma 2.7 there holds ρ(F −1 Q) ≤ ρ(D−1 (L + U )). Example 4.2.4 Consider the circulant M -matrix 

 2

−1

   2  0 A=   −0.5 0   −1 −0.5

−0.5 −1 2 0

0    −0.5  .  −1    2

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0

The iterative matrices of Jacobi s splitting and the opposite triangular splitting I of the matrix A can be expressed by GJ and GI , respectively. We have ρ(GJ ) = 0.7500 and ρ(GI ) = 0.4444. Thus ρ(GI ) < ρ(GJ ). 0 Theorem 4.2.5 Let A be a circulant M -matrix, A = D −(L+U ) and A = R−V be Jacobi s splitting and opposite triangular splitting II of the matrix A respectively, then ρ(R−1 V ) ≤ ρ(D−1 (L + U )). Proof The proof is similar to that of Theorem 4.2.3. Example 4.2.6 Consider the circulant M -matrix  8 −1 −2 −4        −4 8 −1 −2  .  A=   −2 −4 8 −1      −1 −2 −4 8 

Let GII be the iterative matrix of opposite triangular splitting II. We get ρ(GJ ) = 0.8750, and ρ(GII ) = 0.6944. Thus ρ(R−1 V ) ≤ ρ(D−1 (L + U )). Example 4.2.7 Consider the circulant M-matrix  8 −1 −2 −3        −3 8 −1 −2  .  A=   −2 −3 8 −1      −1 −2 −3 8 

Let the iterative matrix of Gauss-Seidel splitting be GG . We get ρ(GG ) = 0.5111, and ρ(GI ) = ρ(GII ) = 0.4444. Then ρ(F1−1 J1 ) = ρ(F2−1 J2 ) ≤ ρ(D − L)−1 U ), which mean that in this example, the opposite triangular splitting I and II have a better convergence rate than that of Gauss-Seidel splitting. In fact, we can get the similar conclusion for the case of n = 2m + 1. 4.3 Several splittings of circulant H-matrices In this subsection, we also give two new splittings which are similar to those in Subsection 4.2. Now we only discuss their convergence, their costs of computation and store are analogous with those of Subsection 4.2. Theorem 4.3.1 Let A be a circulant H-matrix, A = F − Q be opposite triangular splitting I(II) of the matrix A, then ρ(F −1 Q) < 1. Proof There holds hAi = hF i − |Q|, by Lemma 2.9, thus we get ρ(F −1 Q) < 1. Theorem 4.3.2 Let A be a circulant H-matrix, A = F − Q be opposite triangular splitting I(II) of A, then   (1) if ω ∈ 0, 1+ρ(F2−1 Q ) , then Global SOR sequence is convergent; 1 1     2 (2) if ω1 ∈ 0, 1+ρ(Tˆ−1 Hˆ ) , ω2 ∈ 0, 1+ρ(Tˆ2−1 Hˆ ) , then Part SOR sequence is convergent. 1

1

2

2

Proof The proof is similar to Theorem 4.2.2.

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4.4 Three algorithms for the solution of Ax = b Finally, we will construct three algorithms for the linear system Ax = b. The following algorithms 1 and 2 are based on the opposite triangular splittings in Subsections 4.2 and 4.3, and GMRES(m) algorithm is applied when the matrix A is very large and sparse. Algorithm 1 (opposite triangular splitting I) Step 1: Select an arbitrary starting point x0 and a stopping criteria ε. Step 2: Let A = F − Q be the opposite triangular splitting I. Its iterative sequence is F xk = Qxk−1 + b, where A is a circulant M -matrix or a circulant H-matrix. By Lemma 2.5, there exists an orthogonal matrix P such that P T F P P T xk+1 = P T QP P T xk + P T b. ˆ = P T QP are a left lower triangular matrix and a It is easy to know that Fˆ = P T F P and Q right strictly upper triangular matrix, respectively. Let x ˆk = P T xk = (ˆ xk,1 x ˆk,2 , · · · , x ˆk,n )T , ˆb = P T b = (ˆb1ˆb2 , · · · , ˆbn )T . Step 3: For k = 1, 2, · · · , and for j = 1 to n, construct x ˆk,i

1 = ˆ Fi,i

ˆbi −

i−1 X

Fˆi,s x ˆk,s +

s=1

n X

! ˆ i,s x Q ˆk−1,s

.

s=i+1

Step 4: If kˆ xk − x ˆk−1 k < ε, then stop, let x = P x ˆk , which is an approximate solution to the linear system Ax = b; Otherwise set k = k + 1 and return to step 3. Algorithm 2 (opposite triangular splitting II) Step 1: Select an arbitrary starting point x0 and a stopping criteria ε. Step 2: Let A = R − V be the opposite triangular splitting II and its iterative sequence be Rxk = V xk−1 + b, where A is a circulant M -matrix or a circulant H-matrix. By Lemma 2.5, there exists an orthogonal matrix P : P T RP P T xk+1 = P T V P P T xk + P T b. ˆ = P T RP and Vˆ = P T V P are an right upper triangular matrix and It is easy to know that R a strictly left lower triangular matrix respectively. Let x ˆk = P T xk = (ˆ xk,1 x ˆk,2 , · · · , x ˆk,n )T , ˆb = P T b = (ˆb1ˆb2 , · · · , ˆbn )T . Step 3: For k = 1, 2, · · · , and for j = 1 to n, construct x ˆk,i

1 = ˆ Ri,i

ˆbi +

i−1 X

Vˆi,s x ˆk−1,s −

s=1

n X

! ˆ i,s x R ˆk,s

.

s=i+1

Step 4: If kˆ xk − x ˆk−1 k < ε, then stop, let x = P x ˆk , which is an approximate solution to the linear system Ax = b; Otherwise set k = k + 1 and return to step 3.

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When the circulant matrix A is very large and sparse, the GMRES(m) algorithm is very useful to solve the linear system Ax = b. Using the circulant property of the matrix A, we can reduce a large of the cost of computation and store by means of GMRES(m) algorithm. Algorithm 3 (GMRES(m) algorithm) Step 1: Reduce the linear system Ax = b to P T AP P T x = P T b. Let A˜ = P T AP, x ˜ = P T x, and ˜b = P T b. ˜x0 and v1 = r0 /kr0 k2 . Step 2: Choose x ˜0 ∈ Rn , calculate r0 = ˜b − A˜ ˜ Step 3: Choose an appropriate m, obtain {vi }m i=1 and Hm by the Arnoldi process. ˜ Step 4: Calculate ym = miny∈Rk kβe1 − Hm yk2 . Step 5: Obtain x ˜m = x ˜ 0 + V m ym . ˜xm k. For a given ε > 0, if krm k < ε, then stop, and we can Step 6: Calculate krm k = k˜b − A˜ obtain the approximate solution: x = P x ˜. Step 7: Otherwise let x ˜0 = x ˜m , and v1 = rm /krm k2 , return to step 3.

References [1] P. J. Davis, Circulant Matrices, second ed., Chelsea Publishing, New York, 1994. [2] R. M. Gray, Toeplitz and circulant matrices: a review. Available from: . [3] A. Berman and R. Plemmons, Nonnegative Matrices in the Mathematical Sciences, Academic Press, New York, 1979. [4] J. L. Chen and X. H. Chen, Special matrices, Tsinghua University Press, Beijing, 2001. [5] O. Rojo, H. Rojo,Some results on symmetric circulant matrices and on symmetric centrosymmetric matrices, Linear Algebra and its Applications, 392(2004):211-233. [6] C. B. Lu and C. Q. Gu, The computation of the square roots of circulant matrices, Appl. Math. Comput. 217 (2011), 6819-6829. [7] J. Janas and M. Moszy´ nski, Spectral properties of Jacobi matrices by asymptotic analysis, Journal of Approximation Theory 120(2003):309-336. [8] Y. X. Peng, X. Y. Hu and L. Zhang, An iteration method for the symmetric solutions and the optimal approximation solution of the matrix equation AXB = C, Applied Mathematics and Computation, 160 (2005): 763-777. [9] L. Elsner, Comparisons of weak regular splitting and multisplitting methods, Numer. Math., 56 (1989):283-289. [10] Y. Mei, Computing the square roots of a class of circulant matrices, Journal of Applied Mathematics, (2012), 1-15. [11] J. W. Demmel, Applied Numerical Linear Algebra, Society for Industrial and Applied Mathematics, Philadelphia, 1997.

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TABLE OF CONTENTS, JOURNAL OF COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 16, NO. 3, 2014 An Note on Sr-covering Approximation Spaces, Bin Qin, and Xun Ge,…………………….414 Difference of Generalized Composition Operators from H∞ to the Bloch Space, Geng-Lei Li,424 Isometries Among the Generalized Composition Operators on Bloch type Spaces, Geng-Lei Li,….432 Coupled Fixed Point Theorems for Generalized Symmetric Contractions in Partially Ordered Metric Spaces and Applications, M. Jain, K. Tas, B.E. Rhoades, and N. Gupta,……………438 Pointwise Superconvergence Patch Recovery for the Gradient of the Linear Tetrahedral Element, Jinghong Liu, and Yinsuo Jia,……………………………………………………………….455 Hyers-Ulam Stability of Quadratic Functional Equations in Paranormed Spaces, Jinwoo Choi, Joon Hyuk Yang, and Choonkil Park,………………………………………………………461 Union Soft Sets Applied to Commutative BCI-Ideals, Young Bae Jun, Seok Zun Song, and Sun Shin Ahn,………………………………………………………………………………468 Some Inequalities Which Hold for Starlike Log-Harmonic Mappings of Order α, H. Esra Özkan, and Melike Aydoğan,………………………………………………………………………478 N-differentiation Composition Operators from Weighted Banach Spaces of Holomorphic Function to Weighted Bloch Spaces, Cui Chen, Hong-Gang Zeng, and Ze-Hua Zhou,……486 Fuzzy n-Jordan *-Derivations on Induced Fuzzy C*-Algebras, Choonkil Park, Khatereh Ghasemi, and Shahram Ghaffary Ghaleh,……………………………………………………494 Hyers-Ulam Stability of a Tribonacci Functional Equation in 2-Normed Spaces, Majid Eshaghi Gordji, Ali Divandari, Mohsen Rostamian, Choonkil Park, and Dong Yun Shin,………….503 An Identity of the Twisted q-Euler Polynomials with Weak Weight α Associated with the p-Adic q-Integrals on Zp, C. S. Ryoo,………………………………………………………………509 Two-Level Hierarchical Basis Preconditioner for Elliptic Equations with Jump Coefficients, Zhiyong Liu, and Yinnian He,………………………………………………………………515 A New Fourth-Order Explicit Finite Difference Method for the Solution of Parabolic Partial Differential Equation with Nonlocal Boundary Conditions, M. Ghoreishi, A.I.B.Md.Ismail, and A. Rashid,……………………………………………………………………………………528

TABLE OF CONTENTS, JOURNAL OF COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 16, NO. 3, 2014 (continued)

Numerical Solutions of Fourth Order Lidstone Boundary Value Problems Using Discrete Quintic Splines, Fengmin Chen, and Patricia J. Y. Wong,……………………………………………540 Periodicity and Global Attractivity of Difference Equation of Higher Order, T. F. Ibrahim,..552 Quadratic Derivations on non-Archimedean Banach Algebras, Choonkil Park, S. Shagholi, A. Javadian, M.B. Savadkouhi, and Madjid Eshaghi Gordji,……………………………………565 Soft q-Ideals of Soft BCI-Algebras, Yun Sun Hwang, and Sun Shin Ahn,…………………571 Convergence of Parallel Multisplitting USAOR Methods for Block H-Matrices Linear Systems, Xue-Zhong Wang,……………………………………………………………………………583 The Properties and Iterative Algorithms of Circulant Matrices, Chengbo Lu,………………592

Volume 16, Number 4 ISSN:1521-1398 PRINT,1572-9206 ONLINE

May 2014

Journal of Computational Analysis and Applications EUDOXUS PRESS,LLC

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

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Editorial Board Associate Editors of Journal of Computational Analysis and Applications 1) 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. 2) 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

20)Margareta Heilmann Faculty of Mathematics and Natural Sciences University of Wuppertal Gaußstraße 20 D-42119 Wuppertal, Germany, [email protected] Approximation Theory (Positive Linear Operators) 21) Christian Houdre School of Mathematics Georgia Institute of Technology Atlanta,Georgia 30332 404-894-4398 e-mail: [email protected] Probability, Mathematical Statistics, Wavelets

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Disjoint mixing weighted backward shifts on the space of all complex valued square summable sequences Liang Zhang Department of Mathematics, Tianjin University,Tianjin 300072,P.R. China [email protected] Ze-Hua Zhou∗ Department of Mathematics, Tianjin University,Tianjin 300072,P.R. China [email protected]; [email protected] 1

Abstract We characterize when N -tuples Bar11 , · · · , BarN N of powers of unilateral weighted backward shift operators defined the space of all complex valued square summable sequences exhibit the d-mixing properties, and give several equivalent conditions of d-mixing properties. At the same time, d-mixing powers of bilateral weighted backward shifts on Banach sequence spaces are also discussed in this paper.

1

Introduction

Let N be the set of all positive integral numbers, K a real or complex scalar field, and the space of all sequences KN = {(xn )n ; xn ∈ K, n ∈ N}. Let 1 ≤ p < ∞. Then the space ( ) ∞ X lp := x = (xn )n ∈ KN ; |xn |p < ∞ n=1 ∞ P

 of p-summable sequences, endowed with the norm kxk :=

p

|xn |

1/p , is a Banach

n=1

space. In particular, l2 is a Hilbert space with inner product defined by hx, yi :=

∞ P

xn yn .

n=1

Occasionally we let the index start with 0. The finite sequences, that is, sequences of the form (x1 , · · · , xn , 0, · · · ), n ≥ 1, constitute a dense subset. Considering only the finite sequences with entries from Q or Q + iQ we see that any lp , 1 ≤ p < ∞, is separable. The space lp (Z) of p-summable sequences, indexed over Z, is defined analogously. The space   l∞ :=

x = (xn )n ∈ KN ; sup |xn | < ∞ n∈N

of bounded sequences, endowed with the sup-norm kxk := sup |xn |, is a Banach space. Since n∈N

it is not separable it will be of less interest to us. Instead, its closed subspace n o c0 := x = (xn )n ∈ KN ; lim xn = 0 n→∞

of null sequences is a separable Banach space under the induced norm. 1 The authors were supported in part by the National Natural Science Foundation of China (Grant Nos. 11371276; 11301373; 11201331; 10971153). ∗ Corresponding author. 2010 Mathematics Subject Classification. Primary: 47A16; Secondary: 47B37. Key words and phrases. Hypercyclic operator; disjoint mixing operator; weighted backward shifts; sequence spaces.

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Definition 1.1. A Fr´echet space is a vector space X, endowed with a separating increasing sequence (pn )n (by considering maxkn pk , if necessary) of seminorms, which is complete in the metric given by d(x, y) =

∞ X 1 min{1, pn (x − y)}, x, y ∈ X. n 2 n=1

Let X denote separable infinite dimensional Fr´echet space, and by L(X) denote the space of linear and continuous operators on a separable, infinite dimensional Fr´echet space X. Definition 1.2. An operator T : X → X is called hypercyclic if there is some x ∈ X whose orbit under T is dense in X. In such a case, x is called a hypercyclic vector for T . The set of hypercyclic vectors for T is denoted by HC(T ). The first examples of hypercyclic operators were on the space X = H(C) of entire functions on the complex plane C, endowed with the topology of uniform convergence on compact subsets of C. In 1929, Birkhoff [1] showed that the operators of translation on this space are hypercyclic. In 1952, MacLane [15] showed that the operators of differentiation are also hypercyclic. In 1991, Godefroy and Shapiro [13] provided a comprehensive extension of these two results to all convolution operators (but scalar multiples of the identity). Some other related classical results have been characterized, such as in [2, 6, 7, 12, 13, 9, 10, 20] and the related references therein. The first example of a hypercyclic operator acting on a Banach space was given by Rolewicz [16]. The example is the backward shift operator on lp , scaled by a constant greater than 1. The backward shift operator B on the space l2 of all complex valued square summable sequences is defined by B(a0 , a1 , a2 , · · · ) = (a1 , a2 , a3 , · · · ). Since B is a contraction, B itself cannot be hypercyclic. It was shown in 1969 by Rolewicz [16] that if B is the backward shift, then λB is hypercyclic if and only if |λ| > 1. It then follows easily that B itself is supercyclic. It was shown later in 1974 by Hilden and Wallen [14] that any (unilateral) backward weighted shift is supercyclic. In [18] and [19], Salas characterized the bilateral weighted shifts that are hypercyclic and those that are supercyclic in terms of their weight sequence. And, in [8], bilateral weighted backward shifts on l2 spaces are also discussed and hypercyclic and supercyclic properties are characterized, respectively. Recently, there have been an increasing interest in studying the disjoint hypercyclicity acting on different spaces of holomorphic functions. In [3] and [4], disjoint hypercyclic unilateral weighted backward shifts on c0 (N) or lp (N) are characterized and disjoint hypercyclic properties of unilateral weighted backward shifts are also discussed. In 2012, in [5], the authors have showed that every separable infinite-dimensional Fr´echet space supports an arbitrarily large finite and commuting disjoint mixing collection of operators. When this space is a Banach space, it supports an arbitrarily large finite disjoint mixing collection of C0 -semigroups. In this paper, we will discuss disjoint mixing powers of unilateral weighted backward shifts on l2 (N) and characterize the equivalent conditions of disjoint mixing powers of bilateral weighted backward shifts on l2 (Z).

2

Preliminary definitions

∞ Definition 2.1. We say that N ≥ 2 sequences of operators (T1,j )∞ j=1 , · · · , (TN,j )j=1 on separable infinite dimensional Fr´echet space X, are d-topologically transitive (respectively, dmixing), provided for every non-empty open subsets V0 , V1 , · · · , VN of X, there exists m ∈ N,

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Zhang and Zhou: Disjoint mixing weighted backward shifts

so that −1 −1 ∅ 6= V0 ∩ T1,m (V1 ) ∩ · · · ∩ TN,m (VN ) −1 −1 ( respectively, so that ∅ 6= V0 ∩ T1,j (V1 ) ∩ · · · ∩ TN,j (VN ), for ∀j ≥ m); Also, we say that T1 , · · · , TN are d-topologically transitive ( respectively d-mixing), provided (T1j )∞ j=1 , · · · , j ∞ (TN )j=1 are d-topologically transitive sequences (respectively, d-mixing sequences).

Definition 2.2. We say that N ≥ 2 hypercyclic operators T1 , T2 , · · · , TN acting on separable infinite dimensional Fr´echet space X are disjoint , or diagonally hypercyclic (in short, dhypercyclic), provided there is some vector (z, z, · · · , z) in the diagonal of X N = X × X × · · · × X such that {(z, z, · · · , z), (T1 z, T2 z, · · · , TN z), (T12 z, T22 z, · · · , TN2 z), · · · } (z ∈ X) is dense in X N . We call the vector z ∈ X a d-hypercyclic vector associated to the operators T1 , T2 , · · · , TN . ∞ Definition 2.3. We say that N ≥ 2 sequences of operators (T1,j )∞ j=1 , · · · , (TN,j )j=1 on separable infinite dimensional Fr´echet space X are d-universal (respectively densely d-universal) if {(T1,j z, · · · , TN,j z) : j ∈ N}

is dense in X N for some vector z ∈ X (respectively for each vector z in a given dense subset ∞ of X ). We call such vector z a d-universal vector for (T1,j )∞ j=1 , · · · , (TN,j )j=1 . Also, we ∞ ∞ say that (T1,j )j=1 , · · · , (TN,j )j=1 are hereditarily universal (respectively hereditarily densely d-universal) provided for each increasing sequence of positive integers (nk ) the sequences ∞ (T1,nk )∞ k=1 , · · · , (TN,nk )k=1 are d-universal (respectively densely d-universal).

3

Disjoint mixing unilateral weighted backward shifts on l2 (N)

In this section, X = l2 (N) over the real or complex scalar field K and, given a bounded sequence a = (ak )k of non-zero weights, let Ba : X → X be the unilateral weighted shift: Ba

x = (x0 , x1 , · · ·) → (a1 x1 , a2 x2 , · · ·) Theorem 3.1. Let X = l2 (N), and let integers 1 ≤ r1 < r2 < · · · < rN be given. For each ∞ 1 ≤ l ≤ N, let al = (al.k )k=1 be a weight sequence and Bal : X → X be the corresponding unilateral backward shift Ba

l x = (x0 , x1 , · · ·) → (al,1 x1 , al,2 x2 , · · ·)

And, let Bar11 , · · · , BarN be mixing on X = l2 (N). Then the following are equivalent: N r1 rN (a) Ba1 , · · · , BaN are d-mixing. (b) The following conditions hold: lim |al,1 · · · al,rl n | = +∞, (1 ≤ l ≤ N )

n→+∞

|al,1 · · · al,rl n | = +∞, (1 ≤ s < l ≤ N ) . lim n→+∞ as,1+(r −r )n · · · as,r n l

l

s

(c) Bar11 , · · · , BarN satisfy the d-Hypercyclicity Criterion with respect to some syndetic N sequence. (d) Bar11 , · · · , BarN are hereditarily densely d-hypercyclic with respect to the sequence (n). N

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To prove the theorem, let us state d-Hypercyclicity Criterion and a couple of results, which are used in the proofs of the main theorems. Definition 3.2. Let (nk ) be a strictly increasing sequences of positive integers. We say that T1 , T2 , · · · ., TN ∈ L (X) satisfy the d-Hypercyclicity Criterion with respect to (nk ) provided there exist dense subsets X0 , X1 , · · · , XN of X and mappings Sl,k : Xl → X(1 ≤ l ≤ N, k ∈ N) satisfying (i) Tlnk → 0 pointwise on X0 ,

(3.1)

(ii) Sl,k → 0 pointwise on Xl , and

(3.2)

k→∞

(iii)

k→∞ (Tlnk Si,k

− δi,l IdXl ) → 0 pointwise on Xl (1 ≤ i ≤ N ). k→∞

(3.3)

In general, we say that T1 , T2 , · · · ., TN ∈ L (X) satisfy the d-Hypercyclicity Criterion if there exists some sequence (nk ) for which the above is satisfied. Lemma 3.3. [4, Proposition 7] Let T1 , T2 , · · · ., TN satisfy the d-Hypercyclicity Criterion ∞ ∞ with respect to a sequence {nk }. Then the sequences {T1nk }k=1 , · · · , {TNnk }k=1 are d-mixing. In particular, T1 , T2 , · · · ., TN are d-hypercyclic. An increasing sequence of positive integers {nk } is syndetic if sup{nk+1 − nk } < ∞. k

We say that T satisfies the Hypercyclicity Criterion for a syndetic sequence if the sequence {nk } is syndetic in the above criterion . Notice that a large class of hypercyclic operators satisfies the Hypercyclicity Criterion for a syndetic sequence, for instance: λB where |λ| > 1 and B is the backward shift on `2 = `2 (N) (the Hilbert space of square summable sequences). Theorem 3.4. Let T1 , T2 , · · · , TN be operators acting on separable infinite dimensional Fr´echet space X. Assume that T1 , T2 , · · · , TN satisfy the d-Hypercyclicity Criterion with respect to some syndetic sequence {nk }. Then T1 , T2 , · · · , TN are d-mixing. Proof. Let V0 , V1 , · · · , VN be open and non-empty subsets of X. Since the sequence {nk } in the d-Hypercyclicity Criterion is syndetic, there is some positive integer m such that nk+1 − nk ≤ m, ∀k ≥ 0. For i = 0, 1, · · · , m and l = 1, · · · , N , consider open sets Vl,i such that T i (Vl,i ) = Vl . Pick y0 ∈ V0 ∩ X0 and take ε > 0 such that the ball B (y0 , (N + 1) ε) ⊂ V0 . Also, for each i = 0, 1, · · · , m and l = 1, · · · , N , take yl,i ∈ Vl,i ∩ Xl and we may assume that ε is small enough such that the ball B (yl,i , (N + 1) ε) ⊂ Vl,i . In what follows we write where kxk = d (x, 0) is the complete invariant metric of the Fr´echet space. By (3.1), (3.2) and (3.3), there exists k0 ∈ N so that Tlnk y0 , Sl,k yl,i and (Tlnk Sj,k yj,i − δj,l yj,i ) belong to B (0, ε) for ∀k ≥ k0 , 1 ≤ j ≤ N, and 0 ≤ i ≤ m. Set N1 = nk0 , and let n ≥ N1 . It is followed that there is some nk with k ≥ k0 and 0 ≤ r ≤ m such that n = nk + r. PN Then yn := y0 + j=1 Sj,nk yj,r ∈ V0 and Tln yn = Tln y0 +

XN j=1

Tln Sj,nk yj,r ⊂ Tlr (B (yl,r , (N + 1) ε)) ⊂ Vl (1 ≤ l ≤ N ) .

That is, V0 ∩ T1−n (V1 ) ∩ ... ∩ TN−n (VN ) 6= ∅ for each n ≥ N1 . So T1 , T2 , · · · , TN are d-mixing.

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Now, we can prove Theorem 3.1. Proof. of Theorem 3.1. δ are d-mixing, (a) ⇒ (b). Let ε > 0 and pick 0 < δ < 1 with 1−δ < ε. Since Bar11 , · · · , BarN N there exists m ∈ N, for all j ≥ m, all 1 ≤ l ≤ N so that N

l V0 ∩ ∩ Ba−jr (Vl ) 6= φ. l

l=1

In particular, let V0 = V1 = · · · = VN = {x ∈ X; kx − e0 k < δ}. Then there exists a vector x = (x0 , x1 , · · ·) and m ∈ N (m > 1), for all n ≥ m > 1, for k ≥ r1 n, we have |xk | < δ

rn

Bal x − e0 < δ (1 ≤ l ≤ N ) l

(3.4)

Then for l = 1, · · · , N , we get 1 − δ < |al,1 · · · al,rl n xrl n | < 1 + δ, |al,i+1 · · · al,i+rl n xi+rl n | < δ,

i > 0.

(3.5)

Now, let 1 ≤ l ≤ N be fixed. Combining (3.4) with (3.5), we get |al,1 , · · · al,rl n | >

1−δ 1 > . δ ε

And, for 1 ≤ s < l ≤ N , it follows from (3.4) and (3.5) that |al,1 · · · al,rl n | as,(r −r )n+1 · · · as,r n l

s

=

l

>

|al,1 · · · al,rl n xrl n | · · · as,rl n x(rl −rs )n+rs n s )n+1 l 1−δ 1 > . δ ε

as,(r −r

(b) ⇒ (c). Now, let X0 = span {e0 , e1 · · ·} . Notice that X0 is dense in X, and that Barll n → 0 pointwise on X0 (1 ≤ l ≤ N ), consider the mappings Sl,n : X0 → X given by n→∞

 rn l z }| {  Sl,n (x0 , x1 · · ·) = 0, · · · , 0,

 x0 xj ,··· , , · · · al,1 al,2 · · · al,rl n al,1+j al,2+j · · · al,rl n+j

Therefore, Barll n Sl,n = IdX0 and since (b) holds, Sl,n → 0 pointwise on X0 (1 ≤ l ≤ N ). n→∞ Now, for 1 ≤ s < l ≤ N and since rs < rl , we get Barll n Ss,n → 0. And Barsl n Sl,n → 0 on n→∞ n→∞ X0 . Then, Bar11 , · · · , BarN satisfy the d-Hypercyclicity Criterion with respect to the sequence N (n). (c) ⇒ (a). By Theorem 3.4, it is obvious. (d) ⇔ (a). An application of the Baire theorem shows that if X is Baire and second countable, then a sequence {(T1,n , T2,n , · · · , Tk,n )}n∈Z+ is d-transitive if and only it is densely d-universal. Clearly, a sequence {(T1,n , T2,n , · · · , Tk,n )}n∈Z+ is d-mixing if and only if its every subsequence is d-transitive. Again, we know that Bar11 , · · · , BarN are d-mixing if and N only if Bar11 , · · · , BarN are hereditarily densely d-hypercyclic with respect to the full sequence N (n).

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4

Disjoint mixing bilateral weighted backward shifts on l2 (Z)

In this section, we characterize the equivalent conditions of disjoint mixing bilateral weighted backward shifts on l2 (Z). Theorem 4.1. Let X = l2 (Z). For each 1 ≤ l ≤ N, let al = (al,j )j∈Z be a bounded bilateral sequence of non-zero scalars, and Bal : X → X be the corresponding backward shift given by Bal ek = al,k ek−1 (k ∈ Z) be mixing on X = l2 (Z). Then, For any integers 1 ≤ r1 < r2 < · · · < rN , let Bar11 , · · · , BarN N the following are equivalent: are d-mixing. (a) Bar11 , · · · , BarN N (b) The following conditions hold:  Qrl p al,i | = +∞  lim | i=1 p→+∞ Q ,1≤l≤N (4.1)  lim 0i=1−r p al,i = 0 l p→+∞

      

Q lp al,i | | ri=1 Q =∞ r p l p→+∞ i=(r as,i l −rs )p+1 Q rs p i=1−(r −rs )p al,i l Q =0 lim rs p | i=1 as,i | p→+∞

lim

,1≤s 0 and pick 0 < δ < 1 with d-mixing, there exists a vector x = (x0 , x1 · · ·) so that

δ 1−δ

< ε. Since Bar11 , · · · , BarN are N

|x − e0 | < δ.

(4.3)

and m ∈ N (m > 0), for all p ≥ m > 0, for k ≥ r1 p, we have |xk | < δ,

rp

Bal x − e0 < δ (1 ≤ l ≤ N ) . l

(4.4)

It follows from (4.3) that 

|x0 − 1| < δ, , |xi | < δ, if |i| > 0.

(4.5)

Now, by the definition of bilateral shifts, (4.4) and (4.5), we get r p l Y 1−δ 1 > al,i > δ ε i=1

Similarly, since |prl | > 0, we have Y 0 δ a

Q > > rl p δ δ ε i=(rl −rs )p+1 as,i Similarly, if 1 ≤ s < l ≤ N , we get r p rs p s Y Y < ε a a s,i l,i i=1−(rl −rs )p i=1 (b) ⇒ (c). By assumption, (4.1) and (4.2) hold. Now, let
n X0 := span{ek : k ∈ Z}. By the → 0 pointwise on X0 . And for definition of bilateral shifts, (4.1) and (4.2), we get Barll n→∞ each 1 ≤ l ≤ N , let Sl,n : X0 → X be the linear map given by 1

ek+rl n (k ∈ Z) . al,k+1 al,k+2 · · · al,k+rl n
n By (4.1), Sl,n → 0 pointwise on X0 . And Barll Sl,n = IdX0 . At the same time, for n→∞ 1 ≤ s < l ≤ N , we know Sl,n ek =


n Barll

Qk+rs n Ss,n ek =

Therefore, By (4.2), Barll
n Barss By (4.2), Barss


n


n

i=k−(rl −rs )n+1 Qk+rs n i=k+1 as,i

al,i

ek+rs n−rl n , (k ∈ Z) .

Ss,n → 0. Similarly, for 1 ≤ l < s ≤ N , we have n→∞

Qk+rl n Sl,n ek =

i=k+(rl −rs )n+1 Qk+rl n i=k+1 al,i

as,i

ek+rl n−rs n , (k ∈ Z) .

Sl,n → 0 pointwise on X0 . So Bar11 , · · · , BarN satisfy the d-Hypercyclicity N n→∞

Criterion with respect to the sequence (n). (c) ⇒ (a). By Theorem 3.4, Bar11 , · · · , BarN are d-mixing. N (d) ⇔ (a). It is obvious.

References [1] G.D. Birkhoff, D´emonstration dun th´eor`eme ´el´ementaire sur les fonctions enti`eres, C.R. Acad. Sci. Paris 189(1929) 473-475. [2] P. S. Bourdon, J. H. Shapiro, Cyclic phenomena for composition operators, Mem. Amer. Math. Soc. 596 (1997). [3] J. B`es, A. Peris, Disjointness in hypercyclicity, J. Math. Anal. Appl. 336 (2007), 297315. ¨ Maritin, A. Peris, Disjoint hypercyclic linear fractional composition opera[4] J. B`es, O. tors, J. Math. Anal. Appl. 381 (2011), 843-856. ¨ Maritin, A. Peris c, S. Shkarind, Disjoint mixing operators, J. Funct. Anal. [5] J.B`es, O. 263 (2012), 1283-1322.

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Zhang and Zhou: Disjoint mixing weighted backward shifts

[6] K. C. Chan, J. H. Shapiro, The cyclic behavior of translation operators on Hilbert spaces of entire functions, Indiana Univ. Math. J. 40 (1991), 1421-1449. [7] R. Y. Chen and Z.H.Zhou, Hypercyclicity of weighted composition operators on the unit ball of CN , J. Korean Math. Soc. 48(5) (2011), 969-984. [8] N. Feldman, Hypercyclicity and Supercyclicity for invertible bilateral weighted shifts, Proc. Amer. Math. Soc. 131(2)(2003), 479-485. [9] K. G. Grosse-Erdmann, Universal families and hypercyclic operators, Bull. Amer. Math. Soc. 36(3) (1999), 345-381. [10] K. G. Grosse-Erdmann, Recent developments in hypercyclicity, Rev. R. Acad. Cien. Serie A. Mat. 97 (2) (2003), 273-286. [11] K. G. Grosse-Erdmann, A. P. Manguillot, Linear Chaos, Springer, New York, 2011. [12] R. M. Gethner, J. H. Shapiro, Universal vectors for operators on spaces of holomorphic functions, Proc. Amer. Math. Soc. 100 (1987), 281-288. [13] G. Godefroy, J. H. Shapiro, Operators with dense invariant cyclic vector manifolds, J. Funt. Anal. 98 (1991), 229-269. [14] H. M. Hilden, L. J. Wallen, Some cyclic and non-cyclic vectors of certain operators, Ind. Univ. Math. J. 23 (1974), 557-565. [15] G. R. MacLane, Sequences of derivatives and normal families, J. Anal. Math. 2 (1952), 72-87. [16] S. Rolewicz, On orbits of elements, Studia Math. 32 (1969), 17-22. [17] H. Rezaei, Notes on dynamics of the adjoint of a weighted composition operator, Taiwanese J. Math. 14(4) (2010), 1377–1384. [18] H. N. Salas, Hypercyclic weighted shifts, Trans Amer. Math. Soc. 347 (1995), 993-1004. [19] H. N. Salas, Supercyclicity and weighted shifts, Studia Math. 135(1) (1999), 55-74. [20] B. Yousefi, H. Rezaei, Some necessary and sufficient conditions for Hypercyclicity Criterion, Proc. India Acad. Sci. (Math. Sci.) 115(2) (2005), 209-216.

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SOME IDENTITIES INVOLVING ASSOCIATED SEQUENCES OF SPECIAL POLYNOMIALS TAEKYUN KIM1 AND DAE SAN KIM2

Abstract. In this paper, we study some properties of associated sequences of special polynomials. From the properties of associated sequences of polynomials, we derive some interesting identities of special polynomials.

1. Introduction For r ∈ R, the Bernoulli polynomials of order r are defined by the generating function to be )r ( ∞ ∑ tn t xt e = (1.1) Bn(r) (x) , (r ∈ R), (see [12,13,14,18,21]). t e −1 n! n=0 (r)

(r)

In the special case, x = 0, Bn (0) = Bn are called the n-th Bernoulli numbers of order r. It is also well known that the Euler polynomials of order r are defined by the generating function to be ( )r ∞ ∑ 2 tn xt e = (1.2) En(r) (x) , (r ∈ R), (see [9,10,11,19,20]). t e +1 n! n=0 (r)

(r)

Let x = 0. Then En (0) = En are called the n-th Euler numbers of order r. Let F be the set of all formal power series in the variable t over C with { } ∞ ∑ ak k F = f (t) = t ak ∈ C . (1.3) k! k=0

Let P be the algebra of polynomials in the variable x over C and P∗ be the vector space of all linear functionals on P. The action of the linear functional L on a polynomial p(x) is defined by ⟨L | p(x)⟩ and the vector space structure on P∗ is derived by ⟨L + M |p(x)⟩ = ⟨L|p(x)⟩ + ⟨M |p(x)⟩, ⟨cL|p(x)⟩ = c ⟨L|p(x)⟩, where c is a complex constant. ∑∞ For f (t) = k=0 ak!k tk ∈ F, we define a linear functional on P by setting ⟨f (t)|xn ⟩ = an , (n ≥ 0), (see [3,8,17]). By (1.3) and (1.4), we get ⟨ k n⟩ t |x = n!δn,k , (n, k ≥ 0), (see [4,5,7,10,17,18]),

(1.4) (1.5)

where δn,k is the Kronecker symbol. ∑∞ ⟨L|xk ⟩ Let fL (t) = k=0 k! tk . Then, by (1.5), we get ⟨fL (t)|xn ⟩ = ⟨L|xn ⟩. So, we see that fL (xt) = L. The map L 7→ fL (t) is a vector space isomorphism from P∗ 1991 Mathematics Subject Classification. 05A10, 05A19. Key words and phrases. Bernoulli polynomial, Euler polynomial, Abel polynomial. 1

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onto F. Henceforth, F is thought of as both a formal power series and a linear functional (see [4, 6, 8, 17, 18]). We call F the umbral algebra. The umbral calculus is the study of umbral algebra (see [4, 10, 17]). The order o(f (t)) of the non-zero power series f (t) is the smallest integer k for which the coefficient of tk does not vanish. If o(f (t)) = 1, then f (t) is called a delta series and if o(f (t)) = 0, then f (t) is called an invertible series. Let o(f (t)) = 1 and o(g(t)) = 0. Then ⟨ ⟩ there exists a unique sequence Sn (x) of polynomials such that g(t)f (t)k |Sn (x) = n!δn,k where n, k ≥ 0. The sequence Sn (x) is called Sheffer sequence for (g(t), f (t)), which is denoted by Sn (x) ∼ (g(t), f (t)). If Sn (x) ∼ (1, f (t)), then Sn (x) is called the associated sequence for f (t). By (1.5), we see that ⟨eyt | p(x)⟩ = p(y). Let f (t) ∈ F and p(x) ∈ P. Then we have ⟩ ⟩ ∞ ⟨ ∞ ⟨ k ∑ ∑ f (t)|xk k t |p(x) k f (t) = t , p(x) = x , (see [10,17]), (1.6) k! k! k=0

k=0

and ⟨f1 (t)f2 (t) · · · fm (t)| x ⟩ = n

(

∑ i1 +···+im =n

n i1 , . . . , im

)

⟩ ⟩ ⟨f1 (t)| xi1 · · · ⟨fm (t)| xi1 , (1.7)

where f1 (t), f2 (t), · · · , fm (t) ∈ F, (see [4, 10, 17]). From (1.6), we have ⟩ ⟨ ⟩ ⟨ p(k) (0) = tk |p(x) , 1 p(k) (x) = p(k) (0).

(1.8)

Thus, by (1.8), we get dk p(x) , (k ≥ 0), (see [17]). dxk For Sn (x) ∼ (g(t), f (t)), we have the following equations: n ( ) ∑ n Sn (x + y) = pk (y)Sn−k (x), where pk (y) = g(t)Sk (y), k tk p(x) = p(k) (x) =

(1.9)

(1.10)

k=0

and ∞

∑ Sk (y) 1 y f¯(t) e = tk , for all y ∈ C (see [10,15,16,17,18]), k! g(f¯(t)) k=0

(1.11)

where f¯(t) is the compositional inverse of f (t). Let pn (x) ∼ (1, f (t)) and qn (x) ∼ (1, g(t)). Then the transfer formula for associated sequence implies that, for n ∈ N, )n ( f (t) x−1 pn (x), (see [11,17,22]). (1.12) qn (x) = x g(t) Now we introduce several important sequences which are used to derive our results in this paper (see [10, 11, 17]): (The Poisson-Charlier sequences) n ( ) ( ) ∑ t n Cn (x; a) = (−1)n−k a−k (x)k ∼ ea(e −1) , a(et − 1) , k k=0

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where a ̸= 0, (x)n = x(x − 1) · · · (x − n + 1), n ∑ k=0

tk Cn (k; a) e−t = k!

(

t−a a

)n , (a ̸= 0), n ∈ N ∪ {0},

(1.13)

(The Abel sequences) ( ) An (x; b) = x(x − bn)n−1 ∼ 1, tebt , (b ̸= 0),

(1.14)

(The Mittag-Leffler sequences) ( ) n ( ) ∑ n et − 1 k , Mn (x) = (n − 1)n−k 2 (x)k ∼ 1, t k e +1

(1.15)

k=0

(The exponential sequences) ϕn (x) =

n ∑

S2 (n, k)xk ∼ (1, log(1 + t)) ,

(1.16)

k=0

and (The Laguerre sequences)

Ln (x) =

) n ( ∑ n − 1 n! k=1

k − 1 k!

( (−x) ∼ k

t 1, t−1

) .

(1.17)

In this paper, we study some properties of associated sequences of special polynomials. From the properties of associated sequences of specials polynomials, we derive some interesting identities involving associated sequences of special polynomials.

2. Associated sequences of special polynomials. As is well known, the Bessel differential equation is given by ′′

′

x2 y + 2(x + 1)y + n(n + 1)y = 0, (see [1,2]).

(2.1)

From (2.1), we have the solution of (2.1) as follows: yn (x) =

n ∑ (n + k)! ( x )k , (see [1,2]). (n − k)!k! 2

(2.2)

k=0

Let us consider the following associated sequences: ( ) t2 pn (x) ∼ 1, t − , xn ∼ (1, t), (see [1,2,10,17]). 2

628

(2.3)

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From (1.12) and (2.3), for n ∈ N, we have ( )n ( )−n t t −1 n pn (x) = x x x = x 1 − xn−1 2 2 t − t2 ( )k ) ∞ ( ∑ t −n k =x (−1) xn−1 k 2 k=0 n−1 ∑ (n + k − 1) ( 1 )k =x (n − 1)k xn−1−k 2 k k=0 n−1 ∑ (n + k − 1)! ( 1 )k xn−k = k!(n − 1 − k)! 2 k=0 ( )n−k n ∑ (2n − k − 1)! 1 = xk . (n − k)!(k − 1)! 2

(2.4)

k=1

By (2.2) and (2.4), we get

) ( ) ( t2 1 ∼ 1, t − . pn (x) = x yn−1 x 2 n

(2.5)

From (1.11) and (2.5), we can derive the following generating function of pn (x): ( ∞ 1) ∑ tk x 1−(1−2t) 2 , (2.6) pk (x) = e k! k=0

and, by (1.10), we get ( ) ∑ ( ) ( ) n ( ) 1 1 n k n−k 1 (x + y)n yn−1 = yn−k−1 . x y yk−1 x+y x y k

(2.7)

k=0

By (1.12) and (2.3), we get ( )n 2 ( )n t − t2 t−2 n x =x x−1 pn (x) = x x−1 pn (x). t −2

(2.8)

Thus, by (1.13), (2.4) and (2.8), we get ( )2 ∞ ∑ ( ) t−2 tk (−1)n xn−1 = x−1 pn (x) = Cn (k; 2) e−t x−1 pn (x) 2 k! k=0

=

n−1 ∑ k=0

Cn (k; 2)

tk (x − 1)−1 pn (x − 1) k!

( )n−l n 1 tk ∑ (2n − l − 1)! (x − 1)l−1 = Cn (k; 2) k! (l − 1)!(n − l)! 2 l=1 k=0 ( ) ( )n−l n−1 n ∑ ∑ (2n − l − 1)! l − 1 1 = Cn (k; 2) (x − 1)l−1−k (l − 1)!(n − l)! 2 k k=0 l=k+1 ( ) ( )n−m−k−1 n−1 ∑ n−m−1 ∑ m+k (2n − m − k − 2)! 1 = Cn (k; 2) (x − 1)m . k (m + k)!(n − m − k − 1)! 2 m=0 k=0 (2.9) n−1 ∑

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From (2.8), we have ( )n )k n ( )( ∑ t n t n−1 −1 x = 1− − x pn (x) = x−1 pn (x) 2 k 2 k=0 ( )n+k−l ( ) n−1 n ∑ ∑ 1 n (2n − l − 1)! (−1)k = (l − 1)k xl−1−k (l − 1)!(n − l)! 2 k k=0 l=k+1 (2.10) ( )n−m−1 n−1 ∑ n−m−1 ∑ ( n) 1 (2n − m − k − 2)!(−1)k m = (m + k)k x (m + k)!(n − m − 1 − k)! 2 k m=0 k=0 } { ( )n−m−1 ( ) n−1 ∑ n−m−1 ∑ n (2n − m − k − 2)! 1 k xm . = (−1) 2 k m!(n − m − k − 1)! m=0 k=0

Therefore, by (2.9) and (2.10), we obtain the following theorem. Theorem 2.1. For n ∈ N, we have ( ) ( )n−m−k−1 n−1 ∑ n−m−1 ∑ m+k (2n − m − k − 2)! 1 n n−1 (x−1)m . (−1) x = Cn (k; 2) k (m + k)!(n − m − k − 1)! 2 m=0 k=0

Moreover, n−m−1 ∑

(−1)k

k=0

( )n−m−1 ( ) 1 n (2n − m − k − 2)! = 0, 2 k m!(n − m − k − 1)!

where 0 ≤ m ≤ n − 2. Let us consider the following associated sequences: ( ) ( ) t pn (x) ∼ 1, tec(e −1) , c ̸= 0, An (x; b) = x(x − bn)n−1 ∼ 1, tebt , b ̸= 0. (2.11) By (1.12) and (2.11), we get ( )n ∞ ∑ t (−nc)k t −1 n pn (x) = x x x = x (e − 1)k xn−1 . t −1) c(e k! te k=0

(2.12)

We recall that Newton’s difference operator ∆ is defined by ∆f (x) = f (x+1)−f (x). For n ∈ N, we easily see that n ( ) ∑ n n ∆ p(x) = (−1)n−k p(x + k). (2.13) k k=0

By (2.13), we get k ( ) k ( ) ∑ ∑ k k k−l lt (e − 1) p(x) = (−1) e p(x) = (−1)l−k p(x + l) = ∆k p(x). l l l=0 l=0 (2.14) In particular, if we take p(x) = xn−1 , then we have k ( ) ∑ k (et − 1)k xn−1 = (−1)k−j (x + j)n−1 . (2.15) j j=0 t

k

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From (2.12) and (2.15), we have pn (x) = x

n−1 k ∑∑ k=0 j=0

( ) ) ( t (−1)j (nc)k k (x + j)n−1 ∼ 1, tec(e −1) , c ̸= 0. k! j

(2.16)

Therefore, by (2.16), we obtain the following lemma. ( ) t Lemma 2.2. For c ̸= 0 and n ∈ N, let pn (x) ∼ 1, tec(e −1) . Then we have pn (x) = x

n−1 k ∑∑ k=0 j=0

( ) (−1)j (nc)k k (x + j)n−1 . k! j

From the definition of Abel sequences and (2.11), we note that ( )n c(et −1) te x−1 pn (x) An (x; b) = x(x − bn)n−1 = x tebt (2.17) ( ∞ (−b) )n )n ( ∑a (−c) l −1 −bt−c(1−et ) −1 l x pn (x) = x =x e t x pn (x), l! l=0 ( ) (β) where an (x) ∼ (1 − t)−β , log(1 − t) is the actuarial polynomial with the generating function given by ∞ (β) ∑ a (x) l

l=0

l!

t

tl = eβt+x(1−e ) .

By Lemma 2.2 and (2.17), we get An (x; b) { ∞ ∑ =x

} ) m tm (−b) (−b) a (−c) · · · aln (−c) l1 , . . . , ln l1 m! m=0 l1 +···+ln =m   ( ) k n−1  ∑∑ (−1)j (nc)k k × (x + j)n−1   k! j k=0 j=0   )( ) ∏ ( ) )( n−1 n−1 k ( n ∑ ∑ ∑∑ n−1 m k  (−1)j (nc)k (−b)  alj (−c) =x (x + j)n−1−m . m l , . . . , l j k! 1 n m=0 j=1 j=0 ∑

(

l1 +···+ln =m k=0

(2.18) Therefore, by (2.18), we obtain the following theorem. Theorem 2.3. For n ≥ 1, b ̸= 0, c ̸= 0, we have An (x; b)   ) ( )( )( ) ∏ n−1 n−1 k ( n ∑ ∑ ∑∑ (−1)j (nc)k n−1 m k  (−b)  =x (x + j)n−1−m . alj (−c) k! m l , . . . , l j 1 n m=0 j=0 j=1 l1 +···+ln =m k=0

For (2.17), we note that

)n ( t An (x; b) = x ec(e −1) e−nbt x−1 pn (x) )n ( t = x ec(e −1) (x − nb)−1 pn (x − nb).

631

(2.19)

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By (1.16) and Lemma 2.2, we easily see that the generating function of exponential sequences is given by ∞ ∑ t tk (2.20) ϕk (x) = ex(e −1) . k! k=0

From (2.19) and (2.20), we have An (x; b)  ∞ ∑ =x 

   ) ∏ n m m t  ϕlj (c) (x − nb)−1 pn (x − nb)  l , . . . , l m! 1 n m=0 l1 +···+ln =m j=1   ( ) ( ) ∏ ∞ n n−1 k ∑ ∑ m tm ∑ ∑ (−1)j (nc)k k  (x − nb + j)n−1 =x ϕlj (c) . , l m! k! j l , . . n 1 m=0 l1 +···+ln =m j=1 k=0 j=0   )( ) ∏ )( n k ( n−1 n−1 ∑ ∑ ∑∑ k  (−1)j (nc)k n−1 m =x ϕlj (c) (x − nb + j)n−m−1 . . , l j k! m l , . . n 1 m=0 j=1 j=0 ∑

(

l1 +···+ln =m k=0

(2.21) Therefore, by (2.21), we obtain the following corollary. Corollary 2.4. For n ≥ 1, b ̸= 0, c ̸= 0, we have An (x; b) =x

n−1 ∑

∑

  )( )( ) ∏ n n−1 m k  (−1)j (nc)k ϕlj (c) (x − nb + j)n−m−1 . m l1 , . . . , l n j k! j=1

n−1 k ( ∑∑

m=0 l1 +···+ln =m k=0 j=0

Note that xn ∼ (1, t). By (1.12), (1.13) and (1.17), we get ( )n t Ln (x) = x x−1 xn = x(t − 1)n xn−1 t t−1

(n−1 ∑

) n−1 ∑ tk −t tk Cn (k; 1) e =x xn−1 = x Cn (k; 1) (x − 1)n−1 k! k! k=0 k=0 ( ) ( n−1 n−1 ∑ ∑ n − 1) n−1 n−1−k =x Cn (k; 1) (x − 1) =x Cn (n − 1 − k; 1)(x − 1)k . k k k=0 k=0 (2.22) Therefore, by (2.22), we obtain the following theorem. Theorem 2.5. For n ≥ 1, we have n−1 ∑ (n − 1) Ln (x) = x Cn (n − 1 − k; 1)(x − 1)k . k k=0

Mott considered the associated sequences for f (t) = sequence is given by ( ) −2t Sn (x) ∼ 1, . 1 − t2

632

−2t 1−t2 .

That is, the Mott (2.23)

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8

From (2.23), we note that the generating function of Mott sequences is given by ( ( )) √ ∞ ∑ tk 1 − 1 + t2 Sk (x) = exp x . k! t k=0

By (1.12), (1.17) and (2.23), we get ( t )n

)n t−1 x−1 Ln (−x) 2 (n−1 ) ∑ tk −t n −1 −n −n x−1 Ln (−x) = 2 x(t − 1) x Ln (−x) = 2 x Cn (k; 1) e k! x−1 Ln (−x) = x

t+1 −2t 1−t2

Sn (x) = x

(

k=0

n−1 ∑

k

t (x − 1)−1 Ln (1 − x) k! k=0 ) n−1 n ( ∑ 1 ∑ n − 1 n! k = 2−n x t (x − 1)l−1 Cn (k; 1) k! l − 1 l! l=1 k=0 ( )( ) n−1 n n! ∑ ∑ n − 1 l − 1 Cn (k; 1) = n x(x − 1)l−1−k . 2 l−1 k l!

= 2−n x

Cn (k; 1)

k=0 l=1

(2.24) Thus, by (2.24), we obtain the following lemma. ( ) −2t Lemma 2.6. For n ∈ N, let Sn (x) ∼ 1, 1−t . Then we have 2 )( ) n−1 n ( n! ∑ ∑ n − 1 l − 1 Cn (k; 1) Sn (x) = n x(x − 1)l−1−k . l−1 k 2 l! k=0 l=1

As is known, we have ( (an) xBn−1 (x)

∼

( 1, t

et − 1 t

(

)a ) ,

(bn) xEn−1 (x)

∼

( 1, t

et + 1 2

)b ) ,

(2.25)

where a, b are positive integers (see [10, 11, 17]). For n ≥ 1, by (1.12) and (2.25), we get  ( t )a  n t e −1 t (bn) (an) xEn−1 (x) = x  ( t )b  x−1 Bn−1 (x) e +1 t 2 (2.26) )−bn ( t )an ( t e +1 e −1 (an) =x Bn−1 (x). 2 t Thus, by (2.26), we get (

et + 1 2

)bn

( (bn) En−1 (x)

=

633

et − 1 t

)an (an)

Bn−1 (x).

(2.27)

KIM ET AL 626-642

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ASSOCIATED SEQUENCES OF SPECIAL POLYNOMIALS bn ( ) ∑ ( )bn (bn) bn kt (bn) LHS of (2.27) = 2−bn et + 1 En−1 (x) = 2−bn e En−1 (x) k k=0 bn ( ) ∑ bn (bn) = 2−bn En−1 (x + k). k

9

(2.28)

k=0

( )an ∞ ∑ 1 tl (an) RHS of (2.27) = (an)! S2 (l, an) Bn−1 (x) t l! l=an

=

n−1 ∑ l=0

(an)! (an) S2 (l + an, an)(n − 1)l Bn−1−l (x) (l + an)!

= (n − 1)!

n−1 ∑ l=0

(an)! (an) S2 (l + an, an)Bn−1−l (x), (l + an)!(n − 1 − l)! (2.29)

where S2 (n, k) is the Stirling number of the second kind. Therefore, by (2.28) and (2.29), we obtain the following theorem. Theorem 2.7. For n ≥ 1, a, b ∈ N ∪ {0}, we have bn ( ) ∑ bn k=0

k

(bn)

En−1 (x+k) = 2bn (n−1)!

n−1 ∑

(an)! (an) S2 (l+an, an)Bn−1−l (x). (l + an)!(n − 1 − l)!

l=0

The Pidduck sequences is given by ( Pn (x) ∼

2 et − 1 , et + 1 et + 1

) .

(2.30)

From (2.30), we can derive the generating function of the Pidduck sequences as follows: ( )x ∞ ∑ tk 1+t Pk (x) = (1 − t)−1 . (2.31) k! 1−t k=0 ( ) Let Sn (x) ∼ 1, et2t +1 . Then, from (1.12), (1.15) and (2.30), we have ( 2t )n 2 t +1 Mn (x) = t Pn (x) = x eet −1 x−1 Sn (x) e +1 t e +1 ( )n t = 2n x t x−1 Sn (x). e −1 By (1.12), we easily get ( Sn (x) = x −n

=2

2t et +1

)−n x−1 xn = 2−n x(et + 1)n xn−1

t x

(2.32)

n ( ) ∑ n j=0

j

jt n−1

e x

634

−n

=2

n ( ) ∑ n x (x + j)n−1 . j j=0

(2.33)

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From (2.32) and (2.33), we have ( Mn (x) = 2n x

t et − 1

)n

 2−n

n ( ) ∑ n

j

j=0

 (x + j)n−1 

)n n ( ) n ( ) ( ∑ ∑ n n t (n) n−1 (x + j) = xBn−1 (x + j). = x t e − 1 j j j=0 j=0

(2.34)

By (2.32) and (2.34), we get n ( ) ∑ 1 t 1 n (n) (e + 1)Mn (x) = (et + 1) xBn−1 (x + j) 2 2 j j=0 ( ) n } { 1∑ n (n) (n) = (x + 1)Bn−1 (x + 1 + j) + xBn−1 (x + j) 2 j=0 j   ) ( ) ) n ((   ∑ 1 n n (n) (n) (n) = (x + 1) + x Bn−1 (x + j) + (x + 1)Bn−1 (x + n + 1) + xBn−1 (x)  2  j=1 j−1 j

Pn (x) =

) ( )} n+1 {( 1∑ n+1 n (n) = x+ Bn−1 (x + j). 2 j=0 j j−1 (2.35) Therefore, by (2.35), we obtain the following theorem. Theorem 2.8. For n ≥ 1, we have ) ( )} n+1 {( 1∑ n+1 n (n) Pn (x) = x+ Bn−1 (x + j). 2 j=0 j j−1 Let us consider the following two associated sequences: ( Sn (x) ∼

2t 1, t e +1

)

( ) et − 1 , Mn (x) ∼ 1, t . e +1

(2.36)

For n ≥ 1, by (1.12), we get ( et −1 )n Sn (x) = x

et +1 2t et +1

= 2−n x

(

x−1 Mn (x)

et − 1 t

635

)n

(2.37) x−1 Mn (x).

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By (2.34) and (2.37), we get ∞ 1 ∑ tl Sn (x) = 2−n x n n! S2 (l, n) x−1 Mn (x) t l! l=n

n−1 ∑

n!S2 (l + n, n) l −1 t (x Mn (x)) (l + n)! l=0 n−1 n ( ) ∑ ∑ n! n (n) = 2−n x S2 (l + n, n) (n − 1)l Bn−1−l (x + j) (l + n)! j j=0 l=0 (n) n−1 n ∑∑ j S2 (l + n, n) (n) Bn−1−l (x + j). = 2−n xn!(n − 1)! (l + n)!(n − l − 1)! j=0 = 2−n x

(2.38)

l=0

Therefore, by (2.33) and (2.36), we obtain the following theorem. Theorem 2.9. For n ≥ 1, we have (n) n ( ) n−1 n ∑ ∑∑ n j S2 (l + n, n) (n) n−1 (x + j) = n!(n − 1)! Bn−1−l (x + j). j (l + n)!(n − l − 1)! j=0 j=0 l=0

Moreover,

(n) n ( ) n−1 n ∑ ∑∑ n n−1 j S2 (l + n, n) (n) j = n!(n − 1)! Bn−1−l (j). j (l + n)!(n − l − 1)! j=0 j=0 l=0

By (1.15), we get x−1 Mn (x) =

n ( ) k−1 ∑ ∑ n (n − 1)n−k 2k S1 (k − 1, j)(x − 1)j , k j=0

(2.39)

k=1

where S1 (k, j) is the Stirling number of the first kind. From (2.38) and (2.39), we can derive n−1 ∑ n! Sn (x) = 2−n x S2 (l + n, n)tl (x−1 Mn (x)) (l + n)! l=0 n−1 n ( ) k−1 ∑ n! ∑ ∑ n = 2−n x S2 (l + n, n)tl (n − 1)n−k 2k S1 (k − 1, j)(x − 1)j (l + n)! k j=0 l=0 k=1 (n) k n−1 n−1 n ∑∑ ∑ k 2 S2 (l + n, n)S1 (k − 1, j) l = 2−n xn!(n − 1)! t (x − 1)j (l + n)!(k − 1)! j=0 l=0 k=j+1 (n) k j n−1 n ∑∑ ∑ −n k 2 S2 (l + n, n)S1 (k − 1, j) = 2 xn!(n − 1)! (j)l (x − 1)j−l . (l + n)!(k − 1)! j=0 l=0 k=j+1

(2.40) Therefore, by (2.33) and (2.40), we obtain the following theorem. Theorem 2.10. For n ≥ 1, we have (n) k j n ( ) n−1 n ∑ ∑∑ ∑ n k 2 S2 (l + n, n)S1 (k − 1, j)j! (x−1)j−l . (x+j)n−1 = n!(n−1)! (l + n)!(k − 1)!(j − l)! j j=0 j=0 l=0 k=j+1

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Remark. From (2.34), we note that x−1 Mn (x) =

n ( ) ∑ n

k

k=0

(x + k − 1)n−1 .

(2.41)

By (2.38) and (2.41), we get n−1 ∑

n! S2 (l + n, n)tl (x−1 Mn (x)) (l + n)! l=0 ( n ( ) ) n−1 ∑ n! ∑ n −n l S2 (l + n, n)t =2 x (x + k − 1)n−1 (l + n)! k l=0 k=0 j ∑ n−1 n (n) ∑∑ −n l S2 (l + n, n)S1 (n − 1, j) (j)l (x + k − 1)j−l = 2 xn! (l + n)! j=0 l=0 k=0 j ∑ n−1 n (n) ∑∑ −n k S2 (l + n, n)S1 (n − 1, j)j! = 2 xn! (x + k − 1)j−l . (l + n)!(j − l)! j=0

Sn (x) = 2−n x

(2.42)

l=0 k=0

So, by (2.33) and (2.42), we get j ∑ n (n) n ( ) n−1 ∑ ∑∑ n n−1 k S2 (l + n, n)S1 (n − 1, j)j! (x + j) = n! (x + k − 1)j−l . j (l + n)!(j − l)! j=0 j=0 l=0 k=0

(a) Nn (x)

(2.43) of order a is defined by the generating function

The Narumi polynomials to be ( )a ∞ (a) ∑ Nk (x) k log(1 + t) t = (1 + t)x . k! t

(2.44)

k=0

Thus, from (2.44), we see that (( t )a ) e −1 Nn(a) (x) ∼ , et − 1 , (see [17,18]). t (a)

(2.45)

(a)

In the special case, x = 0, Nk (0) = Nk are called the k-th Narumi numbers of (1) (1) order a. If a = 1 in (2.45), then we will write Nn (x) and Nn for Nn (x) and Nn . By (1.12) and (1.16), we get ( )n ( )n t t ϕn (x) = x x−1 xn = x xn−1 log(1 + t) log(1 + t) (∞ ) n−1 ∑ N (−n) (0) ∑ N (−n) k k k =x t xn−1 = x (n − 1)k xn−1−k (2.46) k! k! k=0 k=0 ) n−1 n ( ∑ (n − 1) (−n) ∑ n−1 (−n) n−k = Nk x = Nn−k xk . k k−1 k=0

k=1

Therefore, by (1.16) and (2.46), we obtain the following lemma. Lemma 2.11. For n, k ∈ N with k ≤ n, we have ( ) n−1 (−n) S2 (n, k) = Nn−k . k−1

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13

By (1.12), (1.16) and (1.17), we get ( )n t ϕn (x) = x x−1 Ln (−x) (1 + t) log(1 + t) ( )n ) n ( ∑ t n − 1 n! l−1 x =x (1 + t)−n log(1 + t) l − 1 l! l=1 (∞ ) n ( ∑ N (−n) (−n) ∑ n − 1) n! k k =x t xl−1 k! l − 1 l! k=0 l=1 ( ) n−1 n ∑ N (−n) (−n) ∑ n − 1 n! k =x (l − 1)k xl−1−k k! l − 1 l! k=0 l=k+1 (n−1)(l−1) n−1 n ∑ ∑ l−1 k (−n) = n! Nk (−n)xl−k l! k=0 l=k+1 ( n−1 )(k+m−1) n−1 ∑ n−k ∑ k+m−1 (−n) k = n! Nk (−n)xm (k + m)! k=0 m=1 } {n−m ( n−1 )(k+m−1) n ∑ ∑ k+m−1 (−n) k = n! Nk (−n) xm . (k + m)! m=1

(2.47)

k=0

From (1.16) and (2.47), we have S2 (n, m) = n!

n−m ∑

(

n−1 k+m−1

)(k+m−1) k

(k + m)!

k=0

(−n)

Nk

(−n),

(2.48)

where 1 ≤ m ≤ n. Therefore, by Lemma 2.11 and (2.48), we obtain the following theorem. Theorem 2.12. For m, n ∈ N with m ≤ n, we have ( n−1 )(k+m−1) ( ) n−m ∑ k+m−1 n−1 (−n) (−n) k Nn−m = n! Nk (−n). m−1 (k + m)! k=0

It is well known that ( )n ∞ ∑ t tk (k−n+1) (1 + t)x−1 = Bk (x) , (see [17]). log(1 + t) k!

(2.49)

k=0

Thus, by (2.44) and (2.49), we get ( )n ∞ ∞ ∑ ∑ tk t tk (k−n+1) (−n) Bk (x) = (1 + t)x−1 = Nk (x − 1) . k! log(1 + t) k! k=0

(2.50)

k=0

By comparing the coefficients on the both sides of (2.50), we get (k−n+1)

Bk

(−n)

(x) = Nk

(x − 1).

(2.51)

Therefore, by (2.51), we obtain the following corollary. Corollary 2.13. For m, n ∈ N with m ≤ n, we have ( n−1 )(k+m−1) ( ) n−m ∑ k+m−1 n−1 (−m+1) (k−n+1) k B (1) = n! ( Bk (−n + 1). m − 1 n−m (k + m)! k=0

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Let us consider the following associated sequence: Sn (x) ∼ (1, t(1 + t)a ) , a ̸= 0. Then, by (1.12) and (2.52), we get ( )n t x−1 xn = x(1 + t)−an xn−1 Sn (x) = x t(1 + t)a n−1 n−1 ∑ (−an) ∑ (−an) =x tl xn−1 = (n − 1)l xn−l l l l=0 l=0 ) n ( ∑ −an = (n − 1)n−l xl . n−l

(2.52)

(2.53)

l=1

For n ≥ 1, from (1.16) and (2.52), we have ( )n t(1 + t)a x−1 Sn (x) ϕn (x) = x log(1 + t) ( )n t =x (1 + t)an x−1 Sn (x) log(1 + t) (n−1 (−n) ) ∑N (an) k =x tk x−1 Sn (x). k!

(2.54)

k=0

By (2.53) and (2.54), we get ) n ( (an) ∑ −an (n − 1)n−l tk xl−1 k! n−l k=0 l=1 ( ) n n−1 (−n) ∑N (an) ∑ −an k =x (n − 1)n−l (l − 1)k xl−1−k k! n−l l=k+1 k=0 (−an)(l−1) n−1 n ∑ ∑ (−n) n−l k = (n − 1)! Nk (an)xl−k (l − 1)! k=0 l=k+1 ( −an )(k+m−1) n−1 ∑ n−k ∑ n−k−m (−n) k = (n − 1)! Nk (an)xm (k + m − 1)! k=0 m=1 {n−m ( −an )(k+m−1) } n ∑ ∑ n−k−m (−n) k = (n − 1)! Nk (an) xm (k + m − 1)! m=1 k=0 {n−m ( −an )(k+m−1) } n ∑ ∑ n−k−m (k−n+1) k = (n − 1)! Bk (an + 1) xm . (k + m − 1)! m=1

ϕn (x) = x

n−1 ∑

(−n)

Nk

(2.55)

k=0

Therefore, by (1.16) and (2.55), we obtain the following theorem. Theorem 2.14. For m, n ∈ N with m ≤ n, we have ( −an )(k+m−1) ( ) n−m ∑ n−k−m n−1 (−m+1) (k−n+1) k B (1) = (n − 1)! Bk (an + 1). m − 1 n−m (k + m − 1)! k=0

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Remarks (I). For n ≥ 1, we have ( )n log(1 + t) xn = x x−1 ϕn (x) t } {n−m ( n ∑ ∑ k + m − 1) (n) = S2 (n, k + m)Nk xm . k m=1

15

(2.56)

k=0

By comparing the coefficients on the both sides of (2.56), we get n−m ∑( k=0

) k+m−1 (n) S2 (n, k + m)Nk = δm,n , k

(2.57)

where 1 ≤ m ≤ n. (II). For n ≥ 1, we have ( )n log(1 + t) Ln (−x) = x x−1 ϕn (x) t 1+t

} {n−m ( ∑ k + m − 1) (n) = S2 (n, k + m)Nk (n) xm . k m=1 n ∑

(2.58)

k=0

By (1.17) and (2.58), we get (

) n−m ∑ (k + m − 1) n − 1 n! (n) = S2 (n, k + m)Nk (n), m − 1 m! k

(2.59)

k=0

where 1 ≤ m ≤ n. As is known, the Laguerre polynomials of order α are given by Sheffer sequences to be ( ) t (α) −α−1 Ln (x) ∼ (1 − t) , . (2.60) t−1 Thus, by the definition of Sheffer sequence, we get ⟩ ⟨ ( )k t (1 + t)−α−1 L(α) (−x) = n!δn,k (n, k ≥ 0). t+1 n

(2.61)

By (2.61), we easily see that Ln (−x) = (1 + t)−α−1 L(α) n (−x) ∼

( 1,

t 1+t

) .

(2.62)

From (2.58) and (2.62), we have (1 + t)−α−1 L(α) n (−x) = Ln (−x) { } ( n n−m ∑ ∑ k + m − 1) (n) = S2 (n, k + m)Nk (n) xm . k m=1

(2.63)

k=0

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Thus, by (2.63), we get L(α) n (−x)

} {n−m ( ∑ k + m − 1) (n) = (1 + t) S2 (n, k + m)Nk (n) xm k m=1 k=0 { n n−m ( } n ∑ ∑ ∑ k + m − 1)(α + 1) (n) = (m)m−l S2 (n, k + m)Nk (n) xl . k m−l l=0 m=l k=0 (2.64) α+1

n ∑

It is known that L(α) n (−x) =

) n ( ∑ n + α n! l=0

n−l

l!

xl .

(2.65)

By (2.64) and (2.65), we get ( ) n n−m ∑ ∑ (k + m − 1)(α + 1) n + α n! (n) = (m)m−l S2 (n, k + m)Nk (n), k m−l n − l l! m=l k=0

where 0 ≤ l ≤ n. Finally, we consider the following associated sequences: ) n ( ∑ −an Sn (x) = (n − 1)n−k xk ∼ (1, t(1 + t)a ) , a ̸= 0. n−k

(2.66)

k=1

Thus, by (1.12) and (2.66), we get ( )n log(1 + t) Sn (x) = x x−1 ϕn (x) t(1 + t)a ( )n log(1 + t) =x (1 + t)−an x−1 ϕn (x) t Nk (−an) k ∑ t S2 (n, l)xl−1 k! k=0 l=0 {n−m ( } n ∑ ∑ k + m − 1) (n) = S2 (n, k + m)Nk (−an) xm . k m=1 =x

n−1 ∑

n

(n)

(2.67)

k=0

From (2.66) and (2.67), we have ( ) n−m ∑ (k + m − 1) −an (n) (n − 1)n−m = S2 (n, k + m)Nk (−an), n−m k k=0

where m, n ∈ N with m ≤ n and a ̸= 0. ACKNOWLEDGEMENTS. This paper is supported in part by the Research Grant of Kwangwoon University in 2013. References [1] S. Araci, M. Acikgoz, A note on the Frobenius-Euler numbers and polynomials associated with Bernstein polynomials, Adv. Stud. Contemp. Math. 22 (2012), no. 3, 399-406. [2] S. Araci, M. Acikgoz, Extended q-Dedekind-type Daehee-Changhee sums associated with Extended q-Euler polynomials, arXiv:1211.1233 [3] L. Carlitz, Some generating functions for Laguerre polynomials, Duke Math. J., 35 (1968) 825-827. [4] L. Carlitz, A note on the Bessel polynomials, Duke Math. J., 24 (1957) 151-162.

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[5] B. Diarra, Ultrametric umbral calculus in characteristic p, Bull. Belg. Math. Soc. Simon Stevin, 14 (2007) 845-869. [6] T. Ernst, Examples of a q-umbral calculus, Adv. Stud. Contemp. Math. 16 no. 1, (2008) 1-22. [7] Q. Fang and T. Wang, Umbral calculus and invariant sequences, Ars Combin., 101 (2011), 257-264. [8] D. S. Kim, T.Kim, S.-H. Lee and S.-H. Rim, Frobenius-Euler polynomials and umbral calculus in the p-adic case, Adv. Difference Equ. 2012, 2012:222. [9] D. S. Kim and T. Kim, Some new identities of Frobenius-Euler numbers and polynomials, J. of Ineq. and Appl., 2012, 2012:307. [10] D. S. Kim and T. Kim, Applications of Umbral Calculus Associated with p-Adic Invariant Integrals on Zp , Abstract and Applied Analysis 2012 (2012), Article ID 865721, 12 pages. [11] D. S. Kim, T.Kim, S.-H. Lee and S.-H. Rim, Some identities of Frobenius-Euler polynomials arising from umbral calculus, Adv. Difference Equ. 2012, 2012:196. [12] D. S. Kim, T. Kim, S.-H. Lee and Y.-H. Kim, Some identities for the product of two Bernoulli and Euler polynomials, Adv. Difference Equ. 2012, 2012:95. [13] T. Kim, S.-H. Rim, D. V. Dolgy and S.-H. Lee, Some identities on Bernoulli and Euler polynomials arising from the orthogonality of Laguerre polynomials, Adv. Difference Equ. 2012, 2012:201. [14] T. Kim, Some identities on the q-Euler polynomials of higher order and q-Stirling numbers by the fermionic p-adic integral on Zp , Russ. J. Math. Phys. 16 (2009) no. 4, 484-491. [15] T. Kim, On the weighted q-Bernoulli numbers and polynomials, Adv. Stud. Contemp. Math. 21 (2011), no. 2, 207-215. [16] A. K. Kwa´ sniewski, More on the Bernoulli-Taylor formula for extended umbral calculus, Adv. Appl. Clifford Algebr., 16 (2006), no. 1, 29-39. [17] T. J. Robinson, Formal calculus and umbral calculus, Electron. J. Combin., 17 (2010), no. 1, Research Paper 95, 31 pp. [18] S. Roman, More on the umbral calculus, with emphasis on the q-umbral calculus, J. Math. Anal. Appl., 107 (1985), 222-254. [19] S. Roman, The umbral calculus, Dover Publ. Inc. New York, 2005. [20] C. Ryoo, Some relations between twisted q-Euler numbers and Bernstein polynomials, Adv. Stud. Contemp. Math., 21 (2011), no. 2, 271-223. [21] Y. Simsek, Generating functions of the twisted Bernoulli numbers and polynomials associated with their interpolation functions, Adv. Stud. Contemp. Math., 16 (2008), no. 2, 251-278 . [22] X.-H. Sun, On umbral calculus, I, J. Math. Anal. Appl., 224 (2000), no. 2, 279-290. 1 Department of Mathematics, Kwangwoon University, Seoul 139-701, Republic of Korea. E-mail address: [email protected], [email protected] 2

Department of Mathematics, Sogang University, Seoul 121-742, Republic of Korea.

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KIM ET AL 626-642

J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 16, NO.4, 2014, COPYRIGHT 2014 EUDOXUS PRESS, LLC

SOME NEW INTEGRAL INEQUALITIES OF THE TYPE OF HERMITE-HADAMARD’S FOR THE MAPPINGS WHOSE ABSOLUTE VALUES OF THEIR DERIVATIVE ARE CONVEX MUHAMMAD IQBAL, MUHAMMAD IQBAL BHATTI, AND MUHAMMAD MUDDASSAR

Abstract. In this paper, We establish various inequalities for some differentiable mappings that are linked with the illustrious Hermite-Hadamard integral inequality for mappings whose absolute values of derivatives are convex. The new integral inequalities are then applied to some special means and numerical integration to get some better estimate than some already presented.

1. Introduction The role of mathematical inequalities within the mathematical branches as well as in its enormous applications should not be underestimated. The appearance of the new mathematical inequality often puts on firm foundation for the heuristic algorithms and procedures used in applied sciences. Among others one of the main inequality, which gives us an explicit error bounds in the trapezoidal and midpoint rules of a smooth function, called Hermit-Hadamard’s inequality defined as [9, p. 53]:   Z b 1 f (a) + f (b) a+b f ≤ f (x) dx ≤ , (1) 2 b−a a 2 where f : [a, b] → R is a convex function. Both inequalities hold in the reversed direction for f to be concave. We note that Hermit-Hadamard’s inequality (1) may be regarded as a refinement of the concept of convexity and it follows easily from Jensens inequality. Inequality (1) has received renewed attention in recent years and a remarkable variety of refinements and generalizations have been found [1, 2, 3, 4, 8] and the references cited therein. One of the refinements of the celebrated Hermit-Hadamard’s inequality (1) is given in [9, Page 55] as follows:

  Z b f 3a+b + f a+3b a+b 1 4 4 f ≤ ≤ f (u)du 2 2 b−a a     1 a+b f (a) + f (b) f (a) + f (b) ≤ f + ≤ , (2) 2 2 2 2 where f : [a, b] → R is convex function. M. A. Latif et al. [7] discussed some new estimations regarding 2nd term and rd 3 term in (2) for differentiable functions whose absolute values are convex. Date: December 3, 2012. 2000 Mathematics Subject Classification. 26A51, 26D15, 26D10. Key words and phrases. Convex function, Hermite-Hadamard inequality, H¨ older inequality, Power-mean inequality, Special means, Quadrature formula. 1

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J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 16, NO.4, 2014, COPYRIGHT 2014 EUDOXUS PRESS, LLC

2

M. IQBAL, M. I. BHATTI, AND M. MUDDASSAR

In this paper, some new Hermite-Hadamard type inequalities involving differentiable functions whose absolute values are convex and concave. Our established results provides estimates of left Hermite-Hadamard inequality (1) as obtained in [3, 5, 6] and regarding 2nd and 3rd term of inequality (2) as obtained in [7] at end-points and mid-point of interval [a, b], respectively. This work is organized in the following way. After this Introduction, in Section 2 main results are given. In Section 3 some applications for some special means are provided. In the last Section 4, error is estimated for the generalized quadrature formula. 2. Main Results In order to prove our main theorems, we first prove the following lemma: Lemma 1. Let f : I ⊆ R → R be a differentiable function on I ◦ ,the interior of I where a, b ∈ I with a < b. If f 0 ∈ L[a, b], then     Z b x−a a+x b−x b+x 1 f (u)du = f + f − b−a 2 b−a 2 b−a a Z 1      Z 1 (x − a)2 a+x a+x 0 0 dt+ (t − 1)f tx + (1 − t) dt + (1 − t)f ta + (1 − t) 4(b − a) 0 2 2 0 Z      Z 1 1 (b − x)2 b+x b+x 0 0 dt + dt , (1 − t)f tx + (1 − t) (t − 1)f tb + (1 − t) 4(b − a) 2 2 0 0 for all x ∈ [a, b]. Proof. Integrating by parts and making use of the substitution u = ta+(1−t) a+x 2 , we have   Z (x − a)2 1 a+x (1 − t)f 0 ta + (1 − t) dt 4(b − a) 0 2 (
1  )  Z 1 2 (x − a)2 2(1 − t)f ta + (1 − t) a+x a+x 2 dt = (−1)f ta + (1 − t) − a−x 4(b − a) a−x 2 0 0   Z a+x 2 x−a a+x 1 = f − f (u)du. 2(b − a) 2 b−a a Analogously:     Z Z x (x − a)2 1 a+x x−a a+x 1 0 (t − 1)f tx + (1 − t) dt = f − f (u)du, 4(b − a) 0 2 2(b − a) 2 b − a a+x 2     Z Z b+x 2 (b − x)2 1 b+x b−x b+x 1 (1 − t)f 0 tx + (1 − t) dt = f − f (u)du, 4(b − a) 0 2 2(b − a) 2 b−a x and (b − x)2 4(b − a)

Z 0

1

    Z b b+x b−x b+x 1 f (u)du. (t − 1)f 0 tb + (1 − t) dt = f − 2 2(b − a) 2 b − a b+x 2

Adding above equalities, we get the desired equality. This completes the proof of the lemma.

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IQBAL ET AL 643-653

J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 16, NO.4, 2014, COPYRIGHT 2014 EUDOXUS PRESS, LLC

NEW HERMITE-HADAMARD TYPE INEQUALITIES

3

Theorem 1. Let f : I ⊆ R → R be a differentiable function on I ◦ such that f 0 ∈ L[a, b], where a, b ∈ I with a < b. If |f 0 | is convex on [a, b], then Z b x − a a + x b − x b + x 1 f (u)du ≤ f + f − b−a 2 b−a 2 b−a a " # " #

0 2 |f 0 (x)|+4|f 0 b+x |+|f 0 (b)| |+|f (x)| (x − a)2 |f 0 (a)|+4|f 0 a+x (b − x) 2 2 + , b−a 24 b−a 24 for each x ∈ [a, b]. Proof. By using the convexity of |f 0 |, the properties of modulus on lemma 1, we have Z b x − a a + x b − x b + x 1 f (u)du f + f − b−a 2 b−a 2 b−a a       Z 0 a + x 0 a + x (x − a)2 1 ≤ (1 − t) f ta + (1 − t) + f tx + (1 − t) 2 dt 4(b − a) 0 2 (b − x)2 + 4(b − a)

1

      0 b + x b + x 0 f tb + (1 − t) + (1 − t) f tx + (1 − t) (3) dt 2 2 0     Z a + x (x − a)2 1 0 + t|f (x)| dt ≤ (1 − t) t|f 0 (a)| + 2(1 − t) f 0 4(b − a) 0 2     Z 0 b+x (b − x)2 1 0 0 + (1 − t) t|f (x)| + 2(1 − t) f + t|f (b)| dt 4(b − a) 0 2 " # " #

+|f 0 (x)| |+|f 0 (b)| (b − x)2 |f 0 (x)|+4|f 0 b+x (x − a)2 |f 0 (a)|+4 f 0 a+x 2 2 = + , b−a 24 b−a 24 Z

which completes the proof. Corollary 1. Under the conditions of theorem 1, the followings hold: Z b x − a a + x b − x b + x 1 f + f − f (u)du b−a 2 b−a 2 b−a a ≤

(x − a)2 [|f 0 (a)| + |f 0 (x)|] + (b − x)2 [|f 0 (x)| + |f 0 (b)|] , 8(b − a)

(4)

for each x ∈ [a, b]. Remark 1. By setting x = a (or x = b), inequality (4) reduces to [5, Theorem 2.2]. Remark 2. By setting x = (a+b)/2, theorem 1 reduces to [7, Theorem 1]. Moreover inequality (4) reduces to:    Z b b − a f 3a+b
+f a+3b
1 a + b 0 0 4 4 − f (u)du ≤ |f (a)|+2 f 0 +|f (b)| , (5) 2 b−a a 32 2 which gives sharp bound than as in [7, Corollary 1].

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J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 16, NO.4, 2014, COPYRIGHT 2014 EUDOXUS PRESS, LLC

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M. IQBAL, M. I. BHATTI, AND M. MUDDASSAR

Theorem 2. Let f : I ⊆ R → R be a differentiable function on I ◦ such that p f 0 ∈ L[a, b], where a, b ∈ I with a < b . If |f 0 | p−1 is convex on [a, b] for some fixed p p > 1 with q = p−1 , then  1/p Z b x − a a + x b − x b + x 2 1 ≤ f (u)du f + f − × b−a 2 b−a 2 b−a a p+1 ( "        ) 0 a + x q 1/q 0 a + x q 1/q (x − a)2 0 q 0 q + + |f (x)| + f |f (a)| + f 8(b − a) 2 2 (        )# 0 b + x q 1/q 0 b + x q 1/q (b − x)2 0 q 0 q + |f (b)| + f , |f (x)| + f 8(b − a) 2 2 for each x ∈ [a, b]. Proof. Using the well-known H¨older integral inequality in (3), we have     Z b x − a a + x b − x b + x 1 f (u)du + f − b−a f 2 b−a 2 b−a a Z 1 1/p Z 1   q 1/q 0 (x − a)2 f ta + (1 − t) a + x dt ≤ (1 − t)p dt 4(b − a) 2 0 0 Z 1  q 1/q 1/p Z 1  0 (x − a)2 f tx + (1 − t) a + x dt + (1 − t)p dt 4(b − a) 2 0 0 Z 1  q 1/q 1/p Z 1  0 (b − x)2 f tx + (1 − t) b + x dt + (1 − t)p dt 4(b − a) 2 0 0 Z 1  q 1/q 1/p Z 1  0 (b − x)2 f tb + (1 − t) b + x dt . + (1 − t)p dt 4(b − a) 2 0 0 p

By convexity of |f 0 | p−1 and the Hermite-Hadamard’s inequality, we have  1/p Z b x − a a + x b − x b + x 2 1 f + f − × f (u)du ≤ b−a 2 b−a 2 b−a a p+1 " (    q 1/q   q 1/q ) a + x (x − a)2 a + x |f 0 (a)|q + f 0 + |f 0 (x)|q + f 0 + 8(b − a) 2 2 (        )# 0 b + x q 1/q 0 b + x q 1/q (b − x)2 0 q 0 q + |f (b)| + f , |f (x)| + f 8(b − a) 2 2 which completes the proof. Corollary 2. Under the conditions of theorem 2, the followings hold: Z b x − a a + x b − x b + x 1 f + f − f (u)du b−a 2 b−a 2 b−a a   2 0 0 2 0 0 (x − a) (|f (a)|+|f (x)|)+(b − x) (|f (x)|+|f (b)|) ≤ , 22(p−1)/p (b − a)(p + 1)1/p

(6)

for each x ∈ [a, b].

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IQBAL ET AL 643-653

J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 16, NO.4, 2014, COPYRIGHT 2014 EUDOXUS PRESS, LLC

NEW HERMITE-HADAMARD TYPE INEQUALITIES

5

Proof. proof 0 a+x
The follows
from theorem 2, simply applying convexity on factors f q and f 0 b+x q and the fact 2 2 n X

(uk + vk )s ≤

k=1

n X

(uk )s +

k=1

n X

(vk )s , uk , vk ≥ 0; 1 ≤ k ≤ n; 0 ≤ s < 1.

k=1

Remark 3. By setting x = a (or x = b), inequality (6) reduces to [5, Theorem 2.4]. Remark 4. By setting x = (a + b)/2, theorem 2 reduces to [7, Theorem 2]. Theorem 3. Let f : I ⊆ R → R be a differentiable function on I ◦ such that f 0 ∈ L[a, b], where a, b ∈ I with a < b . If |f 0 |q is convex on [a, b] for some fixed q ≥ 1, then   Z b 1/q x − a a + x b − x b + x 1 1 f + f − f (u)du ≤ × b−a 2 b−a 2 b−a a 3 " (        ) 0 a + x q 1/q 0 a + x q 1/q (x − a)2 0 q 0 q + |f (x)| + 2 f + |f (a)| + 2 f 8(b − a) 2 2 (    q 1/q   q 1/q )# (b − x)2 b + x b + x |f 0 (x)|q + 2 f 0 + |f 0 (b)|q + 2 f 0 , 8(b − a) 2 2

for each x ∈ [a, b]. Proof. have

Using the well-known Power-mean integral inequality for q ≥ 1 in (3), we

    Z b x − a a+x b−x b+x 1 f + f − f (u)du b−a 2 b−a 2 b−a a Z  Z  q 1/q  1−1/q 1 1 0 (x − a)2 a + x (1 − t)dt ≤ (1 − t) f ta + (1 − t) dt 4(b − a) 2 0 0  Z 1 1−1/q Z 1  q 1/q 0 (x − a)2 a + x + (1 − t)dt (1 − t) f tx + (1 − t) dt 4(b − a) 2 0 0  Z 1 1−1/q Z 1  q 1/q 0 (b − x)2 b + x (1 − t)dt (1 − t) f tx + (1 − t) + dt 4(b − a) 2 0 0  Z 1 1−1/q Z 1  q 1/q 0 (b − x)2 b + x + (1 − t)dt (1 − t) f tb + (1 − t) . dt 4(b − a) 2 0 0

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By convexity of |f 0 |q   q 0 |f 0 (a)|q + 2 f 0 a + x (1 − t) f ta + (1 − t) dt ≤ 2 6 0   q Z 1 0 q 0 |f (x)| + 2 f 0 a + x (1 − t) f tx + (1 − t) dt ≤ 2 6 0   q Z 1 0 q 0 |f (x)| + 2 f 0 b+x (1 − t) f tx + (1 − t) dt ≤ 2 6 0 0   q Z 1 0 q f 0 |f (b)| + 2 b + x dt ≤ (1 − t) f tb + (1 − t) 2 6 0

Z

1


q

a+x 2

,


a+x q 2

b+x 2


q


b+x q 2

, ,

.

Combining all the obtained inequalities, we get   Z b 1/q x − a a + x b − x b + x 1 1 f (u)du ≤ × f + f − b−a 2 b−a 2 b−a a 3 " (        ) 0 a + x q 1/q 0 a + x q 1/q (x − a)2 0 q 0 q + |f (a)| + 2 f + |f (x)| + 2 f 8(b − a) 2 2 (        )# 0 b + x q 1/q 0 b + x q 1/q (b − x)2 0 q 0 q , + |f (b)| + 2 f |f (x)| + 2 f 8(b − a) 2 2 which completes the proof. Corollary 3. Under the conditions of theorem 3, the followings hold: Z b x − a a + x b − x b + x 1 f + f − f (u)du b−a 2 b−a 2 b−a a ≤

3(q−1)/q [(x − a)2 (|f 0 (a)| + |f 0 (x)|) + (b − x)2 (|f 0 (b)| + |f 0 (x)|)] , 8(b − a)

(7)

for each x ∈ [a, b]. Remark 5. By setting x = a (or x = b), inequality (7) reduces to [6, Theorem 2.1]. Remark 6. By setting x =

a+b 2 ,

theorem 3 reduces to [7, Theorem 3].

Theorem 4. Let f : I ⊆ R → R be a differentiable function on I ◦ such that f 0 ∈ L[a, b], where a, b ∈ I with a < b . If |f 0 |q is concave on [a, b] for some fixed q > 1, then  1/p  Z b x − a a + x b − x b + x 1 1 (x − a)2 f + f − f (u)du ≤ b−a 2 b−a 2 b−a a p+1 4(b − a)             2 0 3a + x 0 a + 3x f + f + (b − x) f 0 b + 3x + f 0 3b + x , 4 4 4(b − a) 4 4 for each x ∈ [a, b].

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IQBAL ET AL 643-653

J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 16, NO.4, 2014, COPYRIGHT 2014 EUDOXUS PRESS, LLC

NEW HERMITE-HADAMARD TYPE INEQUALITIES

7

Proof. Using the well-known H¨older integral inequality for q > 1 in inequality (3), we have Z b x − a a + x b − x b + x 1 f (u)du f + f − b−a 2 b−a 2 b−a a 1/p Z 1  Z 1  q 1/q 0 (x − a)2 f ta + (1 − t) a + x dt ≤ (1 − t)p dt 4(b − a) 2 0 0 1/p Z 1  Z 1  q 1/q 0 (x − a)2 f tx + (1 − t) a + x dt + (1 − t)p dt 4(b − a) 2 0 0 1/p Z 1  Z 1  q 1/q 0 (b − x)2 f tx + (1 − t) b + x dt + (1 − t)p dt 4(b − a) 2 0 0 1/p Z 1  Z 1  q 1/q 0 (b − x)2 f tb + (1 − t) b + x dt + . (1 − t)p dt 4(b − a) 2 0 0 By concavity of |f 0 |q and using the Hermite-Hadamard inequality (1), we have   q  q Z 1  0 f ta + (1 − t) a + x dt ≤ f 0 3a + x , 2 4 0   q  q Z 1  0 f tx + (1 − t) a + x dt ≤ f 0 a + 3x , 2 4 0   q  q Z 1  0 f tx + (1 − t) b + x dt ≤ f 0 b + 3x , 2 4 0   q  q Z 1  0 f tb + (1 − t) b + x dt ≤ f 0 3b + x . 2 4 0 Combining all the above inequalities gives the desired result. Remark 7. By setting x = a (or x = b), theorem 4 reduces to [3, Theorem 5]. Remark 8. By setting x = (a + b)/2 theorem 4 reduces to [7, Theorem 4]. Theorem 5. Let f : I ⊆ R → R be a differentiable function on I ◦ such that f 0 ∈ L[a, b], where a, b ∈ I with a < b . If |f 0 |q is concave on [a, b] for some fixed q ≥ 1 and |f 0 | is linear map, then     Z b x − a a + x b − x b + x 1 f + f − f (u)du b−a 2 b−a 2 b−a a ≤

(x − a)2 |f 0 (a + x)| + (b − x)2 |f 0 (b + x)| , 8(b − a)

for each x ∈ [a, b]. Proof. First, we note that by the concavity of |f 0 |q on [a, b] and the power-mean inequality, we note that |f 0 (tx + (1 − t)y)|q ≥ t|f 0 (x)|q + (1 − t)|f 0 (y)|q ≥ (t|f 0 (x)| + (1 − t)|f 0 (y)|)q and hence |f 0 (tx + (1 − t)y)| ≥ t|f 0 (x)| + (1 − t)|f 0 (y)|

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for all t ∈ [0, 1] and x, y ∈ [a, b]. This shows that |f 0 | is also concave on [a, b]. Now, using the Jensen’s integral inequality in inequality (3), we have Z b x − a a + x b − x b + x 1 f (u)du f + f − b−a 2 b−a 2 b−a a " (
!  Z 1 (x − a)2 0 (1 − t) ta + (1 − t) a+x 2 (1 − t)dt ≤ f + R1 4(b − a) (1 − t)dt 0 0 ! ) (

! a+x (b − x)2 0 (1 − t) tx + (1 − t) b+x 0 (1 − t) tx + (1 − t) 2 2 f + f R1 R1 4(b − a) (1 − t)dt (1 − t)dt 0 0 ! )#
(1 − t) tb + (1 − t) b+x 2 + f 0 R1 (1 − t)dt 0       2 (x − a) 0 2a + x 0 a + 2x ≤ f + f + 8(b − a) 3 3       (b − x)2 0 2x + b 0 x + 2b f + f , 8(b − a) 3 3 which completes the proof. Remark 9. By setting x = a (or x = b), theorem 5 reduces to [6, Theorem 2.2]. Remark 10. By setting x = (a + b)/2 in theorem 5, the followings hold Z b b−a f 3a+b
+ f a+3b
1 4 4 − f (u)du ≤ |f 0 (a + b)|, 2 b−a a 16

(8)

which gives sharp bound than as was obtained in [7, Corollary 5]. 3. Applications to Some Special Means We now consider the applications to the following special means. The arithmetic mean a+b A(a, b) = , a, b ∈ R 2 The harmonic mean 2ab , a, b ∈ R\{0} H(a, b) = a+b The logarithmic mean  a if a = b L(a, b) = b−a if a 6= b , a, b > 0 lnb−lna Generalized logarithmic mean  a    n1 Ln (a, b) = bn+1 −an+1  (n+1)(b−a)

if a = b if a 6= b

, n ∈ Z\{−1, 0}; a, b > 0

Now, using the results of Section 2, some new inequalities are derived for the above means.

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J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 16, NO.4, 2014, COPYRIGHT 2014 EUDOXUS PRESS, LLC

NEW HERMITE-HADAMARD TYPE INEQUALITIES

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Proposition 1. Let a, b ∈ R, a < b, 0 ∈ / [a, b] and n ∈ Z, |n| ≥ 2, then  n  n   
3a + b a + 3b b−a n A , − Ln (a, b) ≤ |n| [ A |a|n−1 , |b|n−1 4 4 16  +An−1 (a, b) Proof. Follows by inequality (5), setting f (x) = xn , x ∈ R, n ∈ Z. Proposition 2. Let a, b ∈ R, a < b, 0 ∈ / [a, b]. Then for all q ≥ 1  1−(1/q)    −1 3a + b a + 3b
H ≤ 3 , − L(a, b) (b − a)A |a|−2 , |b|−2 . 4 4 8 Proof. Follows by corollary 3 with x = (a + b)/2, setting f (x) =

1 x

.

4. The quadrature formula Let d : a = x0 < x1 < x2 < ... < xn = b be a division of the interval [a, b] and consider the quadrature formula Z b f (x)dx = Q(f, d) + E(f, d) (9) a

where    n−1   1X 3xi + xi+1 3xi + xi+1 Q(f, d) = f +f (xi+1 − xi ) 2 i=0 4 4 and E(f, d) denotes the approximation error. Here, we derive some error estimates for quadrature formula (9). Proposition 3. Let f : I ⊆ R → R be differentiable function on I ◦ such that f 0 ∈ L[a, b], where a, b ∈ I with a < b and |f 0 | is convex on [a, b], then in (9), for every division d of [a, b], we have |E(f, d)| ≤

    n−1 xi + xi+1 1 X (xi+1 − xi )2 |f 0 (xi )| + 2 f 0 + |f 0 (xi+1 )| 32 i=0 2

Proof. On applying inequality (5) on the subinterval [xi , xi+1 ](i = 0, 1, 2, ..., n − 1) of the division d, we have     f 0 3xi +xi+1 + f 0 xi +3xi+1 Z xi+1 4 4 1 − f (x)dx 2 xi+1 − xi xi     (xi+1 − xi ) xi + xi+1 0 ≤ |f 0 (xi )| + 2 f 0 + |f (x | . i+1 32 2 Now       n−1 i+1 i+1 Z  + f 0 xi +3x f 0 3xi +x X  xi+1 4 4 (xi+1 − xi ) |E(f, d)| = f (x)dx −  xi  2 i=0

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M. IQBAL, M. I. BHATTI, AND M. MUDDASSAR

    i+1 i+1 Z xi+1 f 0 3xi +x + f 0 xi +3x 4 4 1 ≤ f (x)dx − (xi+1 − xi ) x − x 2 i+1 i x i i=0     n−1 1 X xi + xi+1 0 (xi+1 − xi )2 |f 0 (xi )| + 2 f 0 ≤ + |f (x )| i+1 32 i=0 2 n−1 X

which completes the proof of the proposition. Proposition 4. Let f : I ⊆ R → R be differentiable function on I o such that f 0 ∈ L[a, b], where a, b ∈ I with a < b and |f 0 |p/p−1 is convex on [a, b], where p > 1. Then in (9), for every division d of [a, b], we have  |E(f, d)| ≤

4 p+1

1/p n−1 X i=0

(xi+1 − xi )2 [ |f 0 (xi )|+ 16    xi + xi+1 0 + |f (x )| 2 f 0 i+1 2

Proof. The proof is similar to that of proposition 3, applying the subinterval [xi , xi+1 ](i = 0, 1, 2, ..., n − 1) of the division d, with x = (xi + xi+1 )/2 on corollary 2. Proposition 5. Let f : I ⊆ R → R be a differentiable function on I ◦ such that f 0 ∈ L[a, b], where a, b ∈ I with a < b . If |f 0 |q is concave on [a, b] for some fixed q ≥ 1 and |f 0 | is linear map, then for every division d of [a, b], then the following inequality holds: |E(f, d)| ≤

n−1 X i=0

(xi+1 − xi )2 0 |f (xi + xi+1 )| 16

Proof. The proof is similar to that of proposition 3 and using inequality (8). References ¨ [1] M. Avci, H. Kavurmaci, and M. E. Ozdemir, New inequalities Of Hermite-Hadamard type via s−convex functions in the second sense, Appl. Math. Comput., 217(2011) 5171−5176. [2] S.S. Dragomir and C.E.M. Pearce, Selected topics on Hermite−Hadamard inequalities and applications, RGMIA Monographs, Victoria University, 2000. Online: http://www.sta.vu.edu.au/RGMIA/monographs/hermite− hadamard.html. [3] S. Hussain, M.I. Bhatti and M. Iqbal, Hadamard-type inequalities for s-convex functions I, Punjab Univ. Jour. Math., 41 (2009) 51-60. ¨ [4] H. Kavurmaci, M. Avci, and M. E. Ozdemir, New inequalities Of Hermite-Hadamard type for convex functions with applications, Jour. Ineq. Appl.,(2011). [5] U. S. Kirmaci, Inequalities for differentiable mappings and applications to special means of real numbers and to midpoint formula, Applied Mathematics and Computation 147 (2004) 137 − 146. ¨ [6] U. S. Kirmaci and M.E. Ozdemir, IOn some inequalities for differentiable mappings and applications to special means of real numbers and to midpoint formula, Applied Mathematics and Computation 153 (2004) 361 − 368. [7] M. A. Latif and S. S. Dragomir, New inequalities of Hermite-Hadamard type for functions whose derivatives in absolute value are convex with applications to special means and to general quadrature formula, RGMIA Research Report Collection, 14(2011) Preprint. Available Online: http://ajmaa.org/RGMIA/papers/v14/v14a103.pdf. [8] C.E.M. Pearce and J.E. Peˇ cari´ c, Inequalities for differentiable mappings with application to special means and quadrature formula, Appl. Math. Lett., 13 (2000) 51 − 55.

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11

[9] C.Niculescu and L. E. Persson, Convex functions and their applications, Springer, Berlin Heidelberg NewYork, (2004). On-line: http://web.cs.dal.ca/ jborwein/Preprints/Books/CUP/CUPold/np-convex.pdf. University of Engineering and Technology, Lahore, Pakistan E-mail address: iqbal− [email protected] E-mail address: [email protected] E-mail address: [email protected]

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A GENERALIZED ADDITIVE FUNCTIONAL INEQUALITY IN BANACH SPACES CHOONKIL PARK, GANG LU, AND DONG YUN SHIN∗ Abstract. In this paper, we investigate the Hyers-Ulam stability of the following function inequality ∥af (x) + bf (y) + cf (z)∥ ≤ ∥f (αx + βy + γz)∥

(1 < |a + b + c|)

in Banach spaces.

1. Introduction and preliminaries The stability problem of functional equations originated from a question of Ulam [1] in 1940, concerning the stability of group homomorphisms. Let (G1 , .) be a group and let (G2 , ∗) be a metric group with the metric d(., .). Given ϵ > 0, does there exist a δ0, such that if a mapping h : G1 → G2 satisfies the inequality d(h(x.y), h(x) ∗ h(y)) < δ for all x, y ∈ G1 , then there exists a homomorphism H : G1 → G2 with d(h(x), H(x)) < ϵ for all x ∈ G1 ? In the other words, Under what condition does there exists a homomorphism near an approximate homomorphism? The concept of stability for functional equation arises when we replace the functional equation by an inequality which acts as a perturbation of the equation. In 1941, Hyers [2] gave the first affirmative answer to the question of Ulam for Banach spaces. Let f : E → E ′ be a mapping between Banach spaces such that ∥f (x + y) − f (x) − f (y)∥ ≤ δ for all x, y ∈ E, and for some δ > 0. Then there exists a unique additive mapping T : E → E ′ such that ∥f (x) − T (x)∥ ≤ δ for all x ∈ E. Moreover, if f (tx) is continuous in t ∈ R for each fixed x ∈ E, then T is R-linear. In 1978, Th.M. Rassias [3] proved the following theorem. Theorem 1.1. Let f : E → E ′ be a mapping from a normed vector space E into a Banach space E ′ subject to the inequality ∥f (x + y) − f (x) − f (y)∥ ≤ ϵ(∥x∥p + ∥y∥p )

(1.1)

for all x, y ∈ E, where ϵ and p are constants with ϵ > 0 and p < 1. Then there exists a unique additive mapping T : E → E ′ such that 2ϵ ∥f (x) − T (x)∥ ≤ ∥x∥p (1.2) 2 − 2p 2010 Mathematics Subject Classification. Primary 39B62, 39B52, 46B25. Key words and phrases. Hyers-Ulam stability; additive functional inequality; Banach space; additive mapping. ∗ Corresponding author.

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for all x ∈ E. If p < 0 then inequality (1.1) holds for all x, y ̸= 0, and (1.2) for x ̸= 0. Also, if the function t 7→ f (tx) from R into E ′ is continuous in t ∈ R for each fixed x ∈ E, then T is R-linear. In 1991, Gajda [4] answered the question for the case p > 1, which was raised by Th.M. Rassias. On the other hand, J.M. Rassias [5] generalized the Hyers-Ulam stability result by presenting a weaker condition controlled by a product of different powers of norms. Theorem 1.2. ([6, 7]) If it is assumed that there exist constants Θ ≥ 0 and p1 , p2 ∈ R such that p = p1 + p2 ̸= 1, and f : E → E ′ is a mapping from a norm space E into a Banach space E ′ such that the inequality ∥f (x + y) − f (x) − f (y)∥ ≤ Θ∥x∥p1 ∥y∥p2 for all x, y ∈ E, then there exists a unique additive mapping T : E → E ′ such that Θ ∥f (x) − T (x)∥ ≤ ∥x∥p , 2 − 2p for all x ∈ E. If, in addition, f (tx) is continuous in t ∈ R for each fixed x ∈ E, then T is R-linear More generalizations and applications of the Hyers-Ulam stability to a number of functional equations and mappings can be found in [8]–[22]. In [23], Park et al. investigated the following inequalities

( )

x+y+z

, ∥f (x) + f (y) + f (z)∥ ≤ 2f

2 ∥f (x) + f (y) + f (z)∥ ≤ ∥f (x + y + z)∥,

( )

x+y

∥f (x) + f (y) + 2f (z)∥ ≤ 2f +z

2 in Banach spaces. Recently, Cho et al. [24] investigated the following functional inequality

) (

x+y+z

∥f (x) + f (y) + f (z) ≤ Kf (0 < |K| < |3|)

K in non-Archimedean Banach spaces. Lu and Park [25] investigated the following functional inequality

) (∑ N

∑



N (x )



i i=1 f (xi ) ≤ Kf (0 < |K| ≤ N )



K i=1

in Fr´ echet spaces. In [26], Lu and Park investigated the following functional inequalities

( )

x+y+z

(0 < |K| < 3), ∥f (x) + f (y) + f (z)∥ ≤ Kf

K

( )

x + y ∥f (x) + f (y) + Kf (z)∥ ≤ +z

Kf

K

(0 < K ̸= 2)

(1.3) (1.4)

and proved the Hyers-Ulam stability of the functional inequalities (1.3) and (1.4) in Banach spaces.

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GENERALIZED FUNCTIONAL INEQUALITY IN BANACH SPACES

Li et al. [27] considered the following functional inequalities

( )

ax + by + cz

(0 < |K| < |a + b + c|),

∥af (x) + bf (y) + cf (z)∥ ≤ Kf

K

) (

ax + by

∥af (x) + bf (y) + Kf (z)∥ ≤ Kf +z

K

(0 < K < |a + b + K|),

(1.5)

(1.6)

where a, b, c are nonzero real numbers, in quasi-Banach spaces. In this paper, we consider the following functional inequality ∥af (x) + bf (y) + cf (z)∥ ≤ ∥f (αx + βy + γz)∥ ,

(1 < |a + b + c|),

(1.7)

where a, b, c and α, β, γ are nonzero real number, and prove the Hyers-Ulam stability of the functional inequality (1.7) in Banach spaces. 2. Hyers-Ulam stability of the functional inequality (1.7) Throughout this section, assume that X is a normed space and that Y is a Banach space. Proposition 2.1. Let f : X → Y be a mapping such that ∥af (x) + bf (y) + cf (z)∥ ≤ ∥f (αx + βy + γz)∥

(2.1)

for all x, y, z ∈ X. Then the mapping f : X → Y is additive. Proof. Letting x = y = z = 0 in (2.1), we get ∥(a + b + c)f (0)∥ ≤ ∥f (0)∥. So f (0) = 0. Letting z = 0 and y = − αβ x in (2.1), we get

) (

af (x) + bf − α x ≤ ∥f (0)∥ = 0

β for all x ∈ X. So f (x) = − ab f (− αβ x) for all x ∈ X. Replacing x by −x and letting y = 0 and z = αγ x in (2.1), we get

( )

af (−x) + cf α x ≤ ∥f (0)∥ = 0

γ for all x ∈ X. So f (−x) = − ac f ( αγ x) for all x ∈ X. Then we get

( ) ( )

b α c α

∥f (x) + f (−x)∥ = − f − x − f x

a β a γ

( ) ( ) 1 α α

= af (0) + bf − x + cf x

|a| β γ

( )

1

f α · 0 − β α x + γ α x = 0 ≤

|a| β γ

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and so f (−x) = −f (x) for all x ∈ X. ∥f (x) + f (y) − f (x + y)∥ = ∥f (x) + f (y) + f (−x − y)∥



a α b α c αx + αy = )

− a f (− α x) − a f (− β y) − a f ( γ

1

af (− α x) + bf (− α y) + cf ( αx + αy ) =

|a| α β γ

( )

1

f α · (− α x) + β · (− α x) + γ · α(x + y) = 0 =

|a| α β γ for all x, y ∈ X. Thus f (x + y) = f (x) + f (y) for all x, y ∈ X, as desired.



Theorem 2.2. Assume that a mapping f : X → Y satisfies the inequality ∥af (x) + bf (y) + cf (z)∥ ≤ ∥f (αx + βy + γz)∥ + ϕ(x, y, z),

(2.2)

where ϕ : X 3 → [0, ∞) satisfies ϕ(0, 0, 0) = 0 and (( ) ( )j ( )j ) ∞ ( ) j ∑ j α α α c e y, z) := ϕ(x, ϕ x, y, z l and all x ∈ X. It means that the sequence {( ac )n f (( αγ )n x)} is a Cauchy sequence for all x ∈ X. Since Y is complete, the sequence {( ac )n f (( αγ )n x)} converges. We define the mapping A : X → Y by A(x) = limn→∞ {( ac )n f (( αγ )n x)} for all x ∈ X. Moreover, letting l = 0 and passing the limit m → ∞, we get (2.3). Next, we show that A : X → Y is an additive mapping.

( n ) ( ) c α x −αn x

∥A(x) + A(−x)∥ = lim ( )n f + f

n→∞ a γn γn [ ( n ) ( ) b α αn x α x c n

+ f − · n ≤ lim ( ) f

n→∞ a γn a β γ

( ( ) )

α αn x c αn x

f · + + f −

γn a γ γn

( ) ( ) ] nx nx

b α α c α α

+

a f − β · γn + a f γ · γn [ ( n ) ( ) c n α x α αn x 1 αn x αn+1 x lim ( ) ϕ ≤ ,− , 0 + ϕ − n , 0, n+1 |a| n→∞ a γn β γn γ γ )] ( n n+1 x αα x α , n+1 =0 + ϕ 0, − n β γ γ and so A(−x) = −A(x) for all x ∈ X.

( n ) ( n ) ( n ) α y α (x + y) c n α x

∥A(x) + A(y) − A(x + y)∥| = lim ( ) f +f −f

n→∞ a γn γn γn [ ( n ) ( ) n

c α x b αα x

= lim ( )n f + f −

n n→∞ a γ a β γn

( n ) ( )

c αn+1 y α y

+ f − n+1 + f

γn a γ

( n ) ( ) ( ) ]

α (x + y) b α αn x c αn+1 y

+ f + f − + f − n+1

γn a β γn a γ [ ( n ) ( ) 1 c α x α αn x αn y α αn x , − ), 0 + ϕ , 0, − ) ≤ lim ( )n ϕ ( ( |a| n→∞ a γn β γn γn γ γn ( n )] n n α (x + y) α α x α α x +ϕ , − ( n ), − ( n ) =0 n γ β γ γ γ

for all x, y ∈ X. Thus the mapping A : X → Y is additive.

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Now, we prove the uniqueness of A. Assume that T : X → Y is another additive mapping satisfying (2.3). Then we obtain

( ) ( ) α n c n α n

A ( ∥A(x) − T (x)∥ = ( ) ) x − T ( ) x

a γ γ [ ( ) ( )

c α α ≤ ( )n A ( )n x − f ( )n x

a γ γ

( ) ( ) ]

α n α n T ( + ) x − f ( ) x

γ γ [ ( ) ] 2 e α α α e ≤ ϕ x, − x, 0 + ϕ(0, − x, x) |a| β β γ which tends to zero as n → ∞ for all x ∈ X. Then we can conclude that A(x) = T (x) for all x ∈ X. This complete the proof.  Corollary 2.3. Assume that 1 ≤ ac < αγ or −1 ≥ ac > with p > 1. Let f : X → Y be a mapping satisfying

α γ.

Let p and θ be positive real numbers

∥af (x) + bf (y) + cf (z)∥ ≤ ∥f (αx + βy + γz)∥ + θ(∥x∥p + ∥y∥p + ∥z∥p ) for all x, y, z ∈ X. Then there exists a unique additive mapping A : X → Y such that p p ) α p ( α α γ 1 1 + + 2 p ∥f (x) − A(x)∥ ≤ θ∥x∥p α c |a| γ β γ − a for all x ∈ X. Proof. Defining ϕ(x, y, z) := θ(∥x∥p + ∥y∥p + ∥z∥p ) for all x, y, z ∈ X in Theorem 2.2, we get the desired result.  Acknowledgments 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 (NRF2012R1A1A2004299), G. Lu was supported by supported by Doctoral Science Foundation of Liaoning Province, China, by Hall of Liaoning Province Science and Technology (No. 20121055), and D. Y. Shin was supported by Basic Science Research Program through the National Research Foundation of Korea funded by the Ministry of Education, Science and Technology (NRF-2010-0021792). References [1] S.M. Ulam, A Collection of the Mathematical Problems, Interscience Publ., New York, 1960. [2] D.H. Hyers, On the stability of the linear functional equation, Proc. Natl. Acad. Sci. USA 27 (1941), 222–224. [3] Th.M. Rassias, On the stability of the linear mapping in Banach spaces, Proc. Amer. Math. Soc. 72 (1978), 297–300. [4] Z. Gajda, On stability of additive mappings, Internat. J. Math. Math. Sci. 14 (1991), 431–434. [5] J.M. Rassias, On approximation of approximately linear mappings by linear mappings, Bull. Sci. Math. 108 (1984), 445–446. [6] J.M. Rassias, On approximation of approximately linear mappings by linear mappings, J. Funct. Anal. 46 (1982), 126–130.

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[7] J.M. Rassias, On a new approximation of approximately linear mappings by linear mappings, Discuss. Math. 7 (1985), 193–196. [8] S. Jung, Hyers-Ulam-Rassias Stability of Functional Equations in Mathematical Analysis, Hadronic Press Inc., Palm Harbor, Florida, 2001. [9] G. Lu, C. Park, Hyers-Ulam stability of additive set-valued functional equations, Appl. Math. Lett. 24 (2011), 1312–1316. [10] I. Chang, Stability of higher ring derivations in fuzzy Banach algebras, J. Computat. Anal. Appl. 14 (2012), 1059–1066. [11] I. Cho, D. Kang, H. Koh, Stability problems of cubic mappings with the fixed point alternative, J. Computat. Anal. Appl. 14 (2012), 132–142. [12] M. Eshaghi Gordji, M. Bavand Savadkouhi, M. Bidkham, Stability of a mixed type additive and quadratic functional equation in non-Archimedean spaces, J. Computat. Anal. Appl. 12 (2010), 454–462. [13] M. Eshaghi Gordji, A. Bodaghi, On the stability of quadratic double centralizers on Banach algebras, J. Computat. Anal. Appl. 13 (2011), 724–729. [14] M. Eshaghi Gordji, R. Farokhzad Rostami, S.A.R. Hosseinioun, Nearly higher derivations in unital C ∗ algebras, J. Computat. Anal. Appl. 13 (2011), 734–742. [15] M. Eshaghi Gordji, S. Kaboli Gharetapeh, T. Karimi, E. Rashidi, M. Aghaei, Ternary Jordan derivations on C ∗ -ternary algebras, J. Computat. Anal. Appl. 12 (2010), 463–470. [16] H.A. Kenary, J. Lee, C. Park, Non-Archimedean stability of an AQQ-functional equation, J. Computat. Anal. Appl. 14 (2012), 211–227. [17] C. Park, Y. Cho, H.A. Kenary, Orthogonal stability of a generalized quadratic functional equation in nonArchimedean spaces, J. Computat. Anal. Appl. 14 (2012), 526–535. [18] C. Park, S. Jang, R. Saadati, Fuzzy approximate of homomorphisms, J. Computat. Anal. Appl. 14 (2012), 833–841. [19] S. Shagholi, M. Eshaghi Gordji, M. Bavand Savadkouhi, Stability of ternary quadratic derivations on ternary Banach algebras, J. Computat. Anal. Appl. 13 (2011), 1097–1105. [20] S. Shagholi, M. Eshaghi Gordji, M. Bavand Savadkouhi, Nearly ternary cubic homomorphism in ternary Fr´echet algebras, J. Computat. Anal. Appl. 13 (2011), 1106–1114. [21] C. Park, Homomorphisms between Poisson JC ∗ -algebra, Bull. Braz. Math. Soc. 36 (2005), 79–97. [22] C. Park, Hyers-Ulam-Rassias stability of homomorphisms in quasi-Banach algebras, Bull. Sci. Math. 132 (2008), 87–96. [23] C. Park, Y. Cho, M. Han, Functional inequalities associated with Jordan-von Neumann type additive functional equations, J. Inequal. Appl. 2007, Art. ID 41820 (2007). [24] Y. Cho, C. Park, R. Saadati, Functional inequalities in non-Archimedean Banach spaces, Appl. Math. Lett. 23 (2010), 1238–1242. [25] G. Lu, Y. Jiang, C. Park, Functional inequality in Fr´ echet spaces, J. Computat. Anal. Appl. 15 (2013), 369–373. [26] G. Lu, C. Park, Additive functional inequalities in Banach spaces, J. Inequal. Appl. 2012, Art. No. 294 (2012). [27] L. Li, G. Lu, C. Park, D. Shin, Additive functional inequalities in generalized quasi-Banach spaces, J. Computat. Anal. Appl. 15 (2013), 1165–1175. Choonkil Park Department of Mathematics, Research Institute for Natural Sciences, Hanyang University, Seoul 133-791, Korea E-mail address: [email protected] Gang Lu Department of Mathematics, School of Science, ShenYang University of Technology, Shenyang 110178, P.R. China E-mail address: [email protected] Dong Yun Shin Department of Mathematics, University of Seoul, Seoul 130-743, Korea E-mail address: dyshin@@uos.ac.kr

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An Efficient Spectral Collocation Algorithm for Solving Neutral Functional-Differential Equations L.M. Assas1 , A.H. Bhrawy1,2,∗ , M.A. Alghamdi1 1 2

Department of Mathematics, Faculty of Science, King Abdulaziz University, Jeddah, Saudi Arabia Department of Mathematics, Faculty of Science, Beni-Suef University, Beni-Suef, Egypt ∗ E-mail: [email protected]

Abstract. In this article, a spectral collocation method based on the Chebyshev polynomials is investigated for the approximate solution of a class of neutral functional-differential equations with variable coefficients, which have many applications in mathematical physics. A Chebyshev collocation method based on Chebyshev Gauss-Lobatto quadrature points is utilized to reduce the solution of such problem to a system of algebraic equations. In addition, accurate approximation is obtained by selecting few Chebyshev Gauss-Lobatto collocation points. Comparing the numerical results with those of known techniques shows that the present method is better in terms of accuracy over the other methods mentioned in this paper. keyword: Neutral functional-differential equations; Proportional delays; Collocation method; Shifted Chebyshev-Gauss-Lobatto quadrature.

1

Introduction

In this paper, we discuss the numerical solution of the neutral functional-differential equations (NFDEs) with proportional delays (u(x) + a(x)u(γm x))(m) = βu(x) +

m−1 ∑

bn (x)u(n) (γn x) + f (x), x ≥ 0,

(1.1)

n=0

with the initial conditions m−1 ∑

i = 0, 1, · · · , m − 1,

ηin u(n) (0) = λi ,

(1.2)

n=0

where a and bn (n = 0, 1, · · · , m − 1) are analytical functions, and β, pn , ηin , λi are constants with 0 < pn < 1(n = 0, 1, · · · , m). In fact, NFDEs play an important role in the mathematical modeling of real-world phenomena (see, [12, 19]). Over the years, it was found that most of delay differential equations cannot be solved exactly. Therefore, within the past few years, several fast and accurate numerical methods have been proposed for implementing approximations of such equations (see, for instance, [7, 15, 22]). In [13] and [14], the authors proposed the rational approximation and the spectral collocation approach to obtain numerical solutions of delay differential equations, respectively. In [26], Yalcinbas et al. developed the Hermite collocation approximation for tackling a class of delay differential equation with variable coefficients. Two efficient algorithms for solving pantograph equations are given in [27, 29], meanwhile, Yuzbasi et al. developed the Bessel collocation method for solving such equations in [28]. Recently, Chen and Wang [6] investigated the variational iteration method for the solution of NFDEs with proportional delays. Hu et al. [20] introduced linear multi-step scheme to present numerical solutions for NFDEs. The reproducing kernel Hilbert space method has been applied in [17] for introducing a numerical solution of NFDEs (1.1)-(1.2). Wang and his collaborators obtained numerical algorithms for NFDEs by using continuous Runge-Kutta methods [23], and one-leg θ-method [24, 25]. Our main aim of this paper is to propose an orthogonal collocation approach for the numerical solution of the neutral functional-differential equations with variable coefficients on the interval [0, L]. This approach is based on expanding the approximate solution as the members of a complete set of Chebyshev polynomials, and then the (N − m + 1) nodes of the shifted Chebyshev-Gauss-Lobatto quadrature are satisfied Eq. (1.1) to produce (N − m + 1) algebraic equations. These equations together with m additional algebraic equations from Eq. (1.1), constitute (N + 1) linear algebraic system of equations. The structure of the resulted matrix system is discussed. The main attractive property of the applying the Chebyshev collocation method is that the Gauss type quadrature nodes and weights of Chebyshev polynomials are explicitly and exactly known. This supplies a very compelling motivation for the use of Chebyshev polynomials. Finally, we implemented three numerical examples to demonstrate that the Chebyshev-Gauss-Lobatto collocation method is better in terms of accuracy over the other methods mentioned in this paper [1, 6, 24, 25, 17, 21]. 661

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1

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The paper is organized as follows. Section 2 is for preliminary needed hereafter, In Section 3, we design the shifted Chebyshev Gauss-Lobatto collocation technique for NFDEs with proportional delays. In Section 4, we present some numerical results demonstrating the efficiency of suggested numerical algorithm. Concluding remarks are given in Section 5.

2

Preliminaries

This section is devoted to the study of the properties of Chebyshev orthogonal polynomials [16, 3, 9, 10]. −1 The well-known Chebyshev polynomials are orthogonal with respect to the weight function ω(t) = (1 − t2 ) 2 in the interval (−1, 1) and can be determined with the aid of the three-term recurrence relation reads: Ti+1 (t) = 2tTi (t) − Ti−1 (t),

i = 1, 2, · · · ,

where T0 (t) = 1 and T1 = t. The Chebyshev polynomials are eigenfunctions of the Sturm-Liouville problem: ( )′ 1 1 (1 − t2 ) 2 (1 − t2 ) 2 Ti (t) + i2 Ti (t) = 0,

t ∈ [−1, 1].

Now, we present the so-called shifted Chebyshev polynomials defined on the interval (0, L), by introducing the change of variable t = 2x L − 1, denoting by TL,i (x) the shifted Chebyshev polynomials which can be evaluated from the recurrence formula: TL,i+1 (x) = 2(

2x − 1)TL,i (x) − TL,i−1 (x), L

i = 1, 2, · · · ,

According to the properties of the standard Chebyshev polynomials, we deduce that TL,i (0) = (−1)i ,

TL,i (L) = 1,

(−1) i(i + q − 1)! √ π, q ≤ i. Γ(q + 12 )(i − q)!Lq i−q

Dq TL,i (0) =

(2.1)

1 , then we define the weighted space L2ωL [0, L]. The shifted Chebyshev Lx − x2 polynomials form a complete L2ωL [0, L]-orthogonal system, i.e., Next, let ωL (x) = √

∫

L

TL,k (x)TL,j (x)ωL (x)dx = hk δk,j , 0

where

{ hk =

Ck 2 π,

0,

k = j, k= ̸ j,

C0 = 2,

Ck = 1,

k ≥ 1.

(2.2)

Lemma 2.1. The high-order derivatives of shifted Chebyshev polynomial can be expressed in terms of the shifted Chebyshev polynomials themselves as q

D TL,k (x) =

k−q ∑

Cq (k, i)TL,i (x),

k ≥ q,

(2.3)

i=0 (k+i−q) even

where Cq (k, i) =

22q k(p − i + q − 1)! (p + q − 1)! , Lq ci (q − 1)! (p − i)! p!

(2.4)

and 2p = k + i − q, c0 = 2, ci = 1; i ≥ 1. For the proof of the previous relation see, [8].2

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3

Shifted Chebyshev-Gauss collocation method

In the collocation methods [2, 5, 11, 18], one needs to exactly satisfy the differential equation at specified collocation points in the domain of solution. Generally, the distribution of the collocation nodes can be freely chosen, but an accurate approximations are obtained by selecting the collocation nodes as the zeros of the orthogonal polynomials. For shifted Chebyshev polynomials, two commonly used quadrature and collocation nodes, namely: (i) shifted Chebyshev-Gauss nodes (in the interior of the domain), and (ii) shifted Chebyshev-Gauss-Lobatto nodes (in the interior and at the two endpoints of the domain). Now, we will present the shifted Chebyshev-Gauss-Lobatto type quadratures. Let xN,j , 0 6 j 6 N , be the nodes of the standard Chebyshev-Gauss-Lobatto interpolation on (−1, 1) and ϖN,j , 0 6 j 6 N, be the corresponding weights. Throughout this paper, we assume that xL,N,j , 0 6 j 6 N stands for the nodes of the shifted Chebyshev-Gauss-Lobatto interpolation on the interval (0, L). Thus xL,N,j = L2 (xN,j + 1), and their corresponding wights are ϖL,N,j = ϖN,j , 0 6 j 6 N . Let SN (0, L) be the set of all polynomials of degree less than or equal to N . In virtue of the property of the standard Chebyshev-Gauss-Lobatto quadrature, one gets for any ϕ ∈ S2N −1 (0, L), ∫L

∫1 √

ωL (x)ϕ(x)dx = −1

0

=

N ∑

1 ϕ 1 − x2 (

ϖN,j ϕ

j=0

(

) L (x + 1) dx 2

L (xN,j + 1) 2

) =

N ∑

(3.1) ϖL,N,j ϕ(xL,N,j ).

j=0

TL Associating with this quadrature rule, we denote by IN the shifted Chebyshev-Gauss-Lobatto interpolation, TL IN u(xL,N,j ) = u(xL,N,j ),

0 ≤ k ≤ N.

In this section, we use the shifted Chebyshev-Gauss collocation method to solve numerically the model problem of (1.1), (1.2). We set SN (0, L) = span{TL,0 (x), TL,1 (x), . . . , TL,N (x)}.

(3.2)

The shifted Chebyshev-Gauss collocation method for solving (1.1) and (1.2) is to seek uN (x) ∈ SN (0, L), such that (u(xL,N −m,k ) + a(xL,N −m,k )u(γm xL,N −m,k ))(m) = βu(xL,N −m,k ) +

m−1 ∑

bn (xL,N −m,k )u(n) (γn xL,N −m,k )

n=0

+ f (xL,N −m,k ), m−1 ∑

(3.3)

k = 0, 1, · · · , N − m,

ηin u(n) (0) = λi ,

i = 0, 1, · · · , m − 1,

n=0

where the xL,N −m,k ; k = 1, 2, . . . , N − m − 1 are distinct and lie between 0 and L, xL,N −m,0 = 0 and xL,N −m,N −m = L. For simplicity in presentation and without loss of generality, assume a(x) ≡ 1. We now derive the collocation algorithm for solving (1.1) and (1.2). To do this, consider the solution is approximated by a truncated Chebyshev expansion uN (x) =

N ∑

a = (a0 , a1 , . . . , aN )T .

aj TL,j (x),

(3.4)

j=0

Let us firstly introduce the following corollary which will be of fundamental importance in what follows Corollary 3.1. The q-th order derivative of shifted Chebyshev polynomials with proportional delay can be written as k−q ∑ Dq TL,k (γr x) = γrq Cq (k, i)TL,i (γr x), k ≥ q, (3.5) i=0 (k+i−q) even

where Cq (k, i) is defined in (2.4). 2 663

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Now, we approximate u(x) and uq (x), q = 1, 2, . . . , m, as (3.4) and (2.3), and in virtue of Corollary 3.1 for obtaining uq (γr x), q = 1, 2, . . . , m, then Eq. (1.1) can be written as N ∑

aj D(m) TL,j (x) +

j=0

N ∑

aj D(m) TL,j (γm x) = β

j=0

N ∑

aj TL,j (x)

j=0

+

m−1 N ∑∑

(3.6) bn (x)aj D(n) TL,j (γn x) + f (x).

n=0 j=0

According to (2.3), we deduce that   j−m N ∑ ∑  (Cm (j, ρ)TL,ρ (x) + (γm )m Cm (j, ρ)TL,ρ (γm x)) aj   ρ=0 (j+ρ−m) even

j=0

=β

N ∑

aj TL,j (x) +

j=0

(3.7)

m−1 N ∑∑

j−n ∑

n=0 j=0

ρ=0 (j+ρ−n) even

aj bn (x)(γn )n Cn (j, ρ)TL,ρ (γn x) + f (x).

Also, by substituting Eq. (3.4) in Eq. (1.2) we obtain m−1 N ∑∑

ηin aj D(n) TL,j (0) = λi .

(3.8)

n=0 j=0

To find the solution uN (x), we first collocate Eq. (3.7) at the (N − m + 1) shifted Chebyshev roots, yields   j−m N ∑ ∑  (Cm (j, ρ)TL,ρ (xL,N −m,k ) + (γm )m Cm (j, ρ)TL,ρ (γm xL,N −m,k )) aj  j=0

ρ=0 (j+ρ−m) even

=β

N ∑

aj TL,j (xL,N −m,k )

(3.9)

j=0

+

m−1 N ∑∑

j−n ∑

n=0 j=0

ρ=0 (j+ρ−n) even

aj bn (xL,N −m,k )(γn )n Cn (j, ρ)TL,ρ (γn xL,N −m,k ) + f (xL,N −m,k ), k = 0, 1, . . . , N − m.

Next, Eq. (3.8), after using (2.1), can be written as m−1 N ∑∑

(−1)j−n

n=0 j=0

√ j(j + n − 1)! π ηin aj = λi , Γ(n + 12 )(j − n)!Ln

i = 0, 1, . . . , m − 1.

(3.10)

Let us denote a = (a0 , a1 , · · · , aN )T , fk = f (xL,N −m,k ),

k = 0, 1, · · · , N − m,

f = (f0 , f1 , · · · , fN −m , λ0 , · · · , λm−1 )T . The matrix system associated with (3.9) and (3.10) becomes m (A + γm B + βC +

m−1 ∑

γnn Dn + E) a = f,

(3.11)

n=0

where the matrices A, B, C, Di , i = 1, 2, . . . , m − 1 and E are given explicitly in the following theorem. Theorem 3.2. If we denote A = (akj )0 0. Choose an i ∈ N such that 1i < r and 1i < t. Then we can find an n0 ∈ N such that (xn , xm ) ∈ Ui for all n, m ≥ n0 . Therefore, M (xn , xm , t) ≥ M (xn , xm , 1i ) > 1 − 1i > 1 − r and N (xn , xm , t) ≤ N (xn , xm , 1i ) < 1 − 1i < 1 − r whenever n, m ≥ n0 . This show that {xn } is a Cauchy sequence in the complete intuitionistic fuzzy metric space (X, M, N, ∗, ♢). We finish the proof. Theorem 3.4 Let (X, τ ) be a topological space. Then (X, τ ) is completely metrizable if and only if it admits a compatible complete intuitionistic fuzzy metric. Proof Assume that (X, τ ) is completely metrizable. Let d be a complete metric on X compatible with τ . From Remark 2.8 and Remark 2.10, it follows that the intuitionistic fuzzy metric (Md , Nd ) induced by d is complete and it is compatible with τ . The converse follows immediately from Lemma 3.3. We are done. Lemma 3.5 [12] An intuitionistic fuzzy metric space (X, M, N, ∗, ♢) is compact if and only if (X, τ(M,N ) ) is compact.

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Lemma 3.6 [12] A subset of an intuitionistic fuzzy metric space (X, M, N, ∗, ♢) is compact if and only if it is precompact and complete. Definition 3.7 [12] A topological space is called a topologically complete intuitionistic fuzzy metrizable space if there exists a complete intuitionistic fuzzy metric inducing the given topology on it. Lemma 3.8 [12] An open subspace of a complete intuitionistic fuzzy metrizable space is a topologically complete intuitionistic fuzzy metrizable space. Theorem 3.9 Let (X, τ ) be a metrizable topological space. Then (X, τ ) is compact if and only if every compatible intuitionistic fuzzy metric is complete. Proof Suppose that (X, τ ) is compact. Then, by Theorem 3.2, Lemma 3.5 and Lemma 3.6, we deduce that every compatible intuitionistic fuzzy metric is complete. Conversely, suppose that each intuitionistic fuzzy metric on X compatible with τ is complete. Let d be a metric on X compatible with τ . The standard intuitionistic fuzzy metric (Md , Nd ) is complete. It follows from Remark 2.10 that d is complete. By Niemytzki-Tychonoff theorem (see [4]), we immediately conclude that (X, τ ) is compact. The proof is finished. Theorem 3.10 Let (X, τ ) be a second countable topological space. If {Kn } is ∞ ∪ an increasing compact subset sequence in X, X = Kn and Kn ⊂ intKn+1 (n ∈ n=1

N), then X is a topologically complete intuitionistic fuzzy metrizable space. Proof Let {Un |n ∈ N} be a countable base on X. Since X is locally compact, we can observe that X has one-point compactification X ∪ {∞}. It is trivial to verify that {Un |n ∈ N} ∪ {{∞} ∪ (X − Kn ) : n ∈ N} is a countable base on the compact space X ∪ {∞}. Note that X ∪ {∞} is Hausdorff, so it is regular. It follows from Urysohn metrization theorem that X ∪{∞} is a compact metrizable space. Take a metric d on X ∪ {∞}. By Remark 2.8 and Theorem 3.9, we deduce that (X ∪ {∞}, Md , Nd , ∗, ♢) is a complete intuitionistic fuzzy metric space. Observe that X is an open subspace in X ∪ {∞}. According to Lemma 3.8, we immediately conclude that X is a topologically complete intuitionistic fuzzy metrizable space. Let (Xi , Mi , Ni , ∗, ♢)(i = 1, 2) be two intuitionistic fuzzy metric spaces. We define two maps M, N : (X1 × X2 ) × (X1 × X2 ) × (0, ∞) → [0, 1] by M ((x1 , x2 ), (y1 , y2 ), t) = min{M1 (x1 , y1 , t), M2 (x2 , y2 , t)} and N ((x1 , x2 ), (y1 , y2 ), t) = max{N1 (x1 , y1 , t), N2 (x2 , y2 , t)} for all (x1 , x2 ), (y1 , y2 ) ∈ X1 × X2 and t > 0. Theorem 3.11 (X1 × X2 , M, N, ∗, ♢) is an intuitionistic fuzzy metric space. Proof It is straightforward to verify that conditions (b)-(d), (f),(g)-(i) and (k) in Definition 2.3 are satisfied. 5

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Let us verify condition (a) in Definition 2.3. Let (x1 , x2 ), (y1 , y2 ) ∈ X1 × X2 and t > 0. Suppose, without loss of generality, that N ((x1 , x2 ), (y1 , y2 ), t) = max{N1 (x1 , y1 , t), N2 (x2 , y2 , t)} = N1 (x1 , y1 , t). If M1 (x1 , y1 , t) ≤ M2 (x2 , y2 , t), then M ((x1 , x2 ), (y1 , y2 ), t)+N ((x1 , x2 ), (y1 , y2 ), t) = M1 (x1 , y1 , t)+N1 (x1 , y1 , t) ≤ 1. If M2 (x2 , y2 , t) ≤ M1 (x1 , y1 , t), then M ((x1 , x2 ), (y1 , y2 ), t) + N ((x1 , x2 ), (y1 , y2 ), t) = M2 (x2 , y2 , t) + N1 (x1 , y1 , t) ≤ M1 (x1 , y1 , t) + N1 (x1 , y1 , t) ≤ 1. Thus, in any case, we obtain M ((x1 , x2 ), (y1 , y2 ), t) + N ((x1 , x2 ), (y1 , y2 ), t) ≤ 1. We are now going to verify condition (e) in Definition 2.3. Let (x1 , x2 ), (y1 , y2 ), (z1 , z2 ) ∈ X1 × X2 and t, s > 0. Without loss of generality, one may assume that M ((x1 , x2 ), (y1 , y2 ), t) = min{M1 (x1 , y1 , t), M2 (x2 , y2 , t)} = M1 (x1 , y1 , t). If M1 (y1 , z1 , s) ≤ M2 (y2 , z2 , s), then M ((x1 , x2 ), (y1 , y2 ), t)∗M ((y1 , y2 ), (z1 , z2 ), s) = M1 (x1 , y1 , t)∗M1 (y1 , z1 , s) ≤ M1 (x1 , z1 , t+s). If M2 (y2 , z2 , s) ≤ M1 (y1 , z1 , s), then M ((x1 , x2 ), (y1 , y2 ), t) ∗ M ((y1 , y2 ), (z1 , z2 ), s) = M1 (x1 , y1 , t) ∗ M2 (y2 , z2 , s) ≤ M2 (x2 , y2 , t) ∗ M2 (y2 , z2 , s) ≤ M2 (x2 , z2 , t + s). So, we have M ((x1 , x2 ), (y1 , y2 ), t)∗M ((y1 , y2 ), (z1 , z2 ), s) ≤ min{M1 (x1 , z1 , t+s), M2 (x2 , z2 , t+s)} = M ((x1 , x2 ), (z1 , z2 ), t + s). Using the same method as above, we may verify condition (j) in Definition 2.3. Theorem 3.12 Let (Xi , Mi , Ni , ∗, ♢)(i = 1, 2) be two precompact intuitionistic fuzzy metric spaces. Then (X1 × X2 , M, N, ∗, ♢) is a precompact intuitionistic fuzzy metric space. Proof Let 0 < r < 1 and t > 0. Then, by hypothesis, there ∪ exist a finite subset A of X1 and a finite subset B of X2 such that X1 = B(M1 ,N1 ) (a, r, t) and a∈A ∪ X2 = B(M2 ,N2 ) (b, r, t), respectively. So b∈B

X1 × X2 =

∪

B(M,N ) ((a, b), r, t).

(a,b)∈A×B

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In fact, let (x, y) ∈ X1 × X2 and (a, b) ∈ A × B. Then M1 (x, a, t) > 1 − r and M2 (y, b, t) > 1 − r and N1 (x, a, t) < r and N2 (y, b, t) < r. Hence M ((x, y), (a, b), t) > 1 − r We are done.

and

N ((x, y), (a, b), t) < r.

According to Definition 2.9, we get Theorem 3.13 Let (Xi , Mi , Ni , ∗, ♢)(i = 1, 2) be two complete intuitionistic fuzzy metric spaces. Then (X1 × X2 , M, N, ∗, ♢) is a complete intuitionistic fuzzy metric space. The proof of Theorem 3.13 is straightforward and so it is omitted. From Lemma 3.6 and Theorem 3.12,3.13, we obtain immediately the following corollary. Corollary 3.14 Let (Xi , Mi , Ni , ∗, ♢)(i = 1, 2) be two compact intuitionistic fuzzy metric spaces. Then (X1 ×X2 , M, N, ∗, ♢) is a compact intuitionistic fuzzy metric space.

4

Conclusion

We have shown that a topological space is (completely) metrizable if and only if it admits a compatible (complete) intuitionistic fuzzy metric. A topologically complete intuitionistic fuzzy metrizable space was explored. At last, we have established the intuitionistic fuzzy product metric space of two intuitionistic fuzzy metric spaces and studied several properties of the intuitionistic fuzzy product metric space, as precompactness ,completeness and compactness.

References [1] C. Alaca, H. Efe and C. Yildiz, On the completion of intuitionistic fuzzy metric spaces, Chaos,Solitons & Fractals, To Appear. [2] K. Atanassov, In intuitionstic fuzzy sets, In: V. Sgurev, editor, VII ITKR’s Session, Sofia June, 1983 (Central Sci. and Techn. Library, Bulg. Academy of Sciences, 1984). [3] H. Efe and C. Yildiz, On the Hausdorff intuitionistic fuzzy metric on compact sets, International Journal of Pure and Applied Mathematics 31 (2) (2006) 143-155. [4] R. Engelking, General Topology, PWN-Polish Science Publishers,warsaw, 1977. [5] M. A. Erceg, Metric spaces in fuzzy set theory, J. Math. Anal. Appl. 69 (1979) 205-230. 7

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[6] V. Gregori, S. Romaguera, and P. Veeramani, A note on intuitionistic fuzzy metric spaces, Chaos,Solitons & Fractals 28 (2006) 902-905. [7] A. George and P. Veeramani, On some resules in fuzzy metric spaces, Fuzzy Sets and Systems 64 (1994) 395-399. [8] O. Kaleva and S. Seikkala, on fuzzy metric spaces, Fuzzy Sets and Systems 12 (1984) 215-229. [9] I.Kramosil and J.Michalek, Fuzzy metric and statistical metric spaces, Kybernetika 11 (1975) 326-334. [10] J. H. Park, Intuitionistic fuzzy metric spaces, Chaos,Solitons & Fractals 22 (2004) 1039-1046. [11] J. H. Park ,Y. B. Park and R. Saadati, Some results in intuitionistic fuzzy metric spaces, Journal of Computational Analysis and Applications 10 (4) (2008) 441-451. [12] R. Saadati and J. H. Park, On the intuitionistic fuzzy topological spaces, Chaos,Solitons & Fractals 27 (2006) 331-344. [13] B. Schweizer and A. Sklar, Statistical metric spaces, Pacific J Math 10 (1960) 314-334.

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A Certain Class of Harmonic Mappings Related to Functions of Bounded Boundary Rotation Ya¸sar Polato˜glu1, Emel Yavuz Duman2 , Melike Aydo˜gan3 Abstract Let V (k) be the class of functions with bounded boundary rotation and let SH be the class of sense-preserving harmonic mappings. In the present paper we investigate a certain class of harmonic mappings related to the function of bounded boundary rotation.

1

Introduction

Let Ω be the family of functions φ(z) regular in the open unit disc D = {z ∈ C||z| < 1} and satisfying the conditions φ(0) = 0, |φ(z)| < 1 for all z ∈ D. Next, denote by P the family of functions p(z) = 1 + p1 z + p2 z 2 + · · · regular in D and such that p(z) is in P if and only if p(z) =

1 + φ(z) 1 − φ(z)

(1.1)

for some function φ(z) ∈ Ω and every z ∈ D. Moreover, let A be the class of functions in the open unit disc D that are normalized with h(0) = h (0) − 1 = 0, then a function h(z) ∈ A is called convex on starlike if it maps D into a convex or starlike region, respectively. Corresponding classes are denoted by C and S ∗ . It is well known that C ⊂ S ∗ , 2010 Mathematics Subject Classification: 30C45, 30C55 Key words and phrases: Bounded boundary rotation, harmonic mappings, complex dilatation, growth and distortion theorems

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that both are subclasses of the univalent functions and have the following analytical representations   h (z) h(z) ∈ C if and only if Re 1 + z  > 0, z ∈ D, (1.2) h (z) and



h (z) h(z) ∈ S if and only if Re z h(z) ∗

 > 0, z ∈ D.

(1.3)

More on these classes can be found in [1]. Let h(z) be an element of A. If there is a function s(z) in C and a real β such that    h (z) Re iβ  > 0, z ∈ D (1.4) e s (z) then h(z) is called a close-to-convex function in D, and the class of such functions is denoted by CC. Further, let h(z), g(z) ∈ A. Then we say that h(z) is subordinate to g(z) and we write h(z) ≺ g(z). If there exists a function φ(z) ∈ Ω such that h(z) = g(φ(z)) for all z ∈ D. Specially if g(z) is univalent in D, then h(z) ≺ g(z) if and only if h(0) = g(0), h(D) ⊂ g(D), implies h(Dr ) ⊂ g(Dr ), where Dr = {z||z| < r, 0 < r < 1} (Subordination and Lindelof Principle [1]). In the terms of subordination we have ∞   1 + z P = p(z) = 1 + , pn z n |p(z) regular in D, p(z) ≺ 1 − z n=1

 1 + z h (z) ≺ , S ∗ = h(z) ∈ A|z h(z) 1−z    h (z) 1 + z C = h(z) ∈ A| 1 + z  ≺ , h (z) 1−z and

(1.5)

(1.6) (1.7)



 1+z h (z) CC = h(z), s(z) ∈ A| iβ  ≺ , s(z) ∈ C . e s (z) 1−z

(1.8)

Finally, a function analytic and locally univalent in a given simply connected domain is said to be of bounded boundary rotation if its range has bounded boundary rotation which is defined as the total variation of the direction angle of the tangent to the boundary curve under a complete circuit. 2

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Let V (k) denote the class of functions h(z) ∈ A which maps D conformally onto an image domain of boundary rotation at most kπ. The class of functions of bounded boundary rotation was introduced by Loewner [4] in 1917, and was developed by Paatero [6], [7] who systematically developed their properties and made an exhaustive study of the class V (k). Paatero has shown that h(z) ∈ V (k) if and only if   2π   h (z) = exp − log(1 − e−it z)dμ(t) (1.9) 0

where μ(t) is real valued function of bounded variation for which 



2π

2π

dμ(t) = 2 and 0

0

|dμ(t)| ≤ k.

For fixed k ≥ 2, it can also be expressed as   2π   (zh (z))   dθ ≤ kπ, z = reiθ . Re   (z)  h 0

(1.10)

(1.11)

Clearly if k1 ≤ k2 , then V (k1 ) ⊂ V (k2 ), that is the class V (k) obviously expands on k increases. V (2) is the class of C of convex univalent functions. Paatero showed that V (4) ⊂ S, where S is the class of normalized univalent functions [1]. Later Pinchuk [8] proved that functions in V (k) are close-toconvex in D if 2 ≤ k ≤ 4. More details on the functions with bounded boundary rotation can be found in [5]. A planar harmonic mapping in the open unit disc D is a complex-valued harmonic function f which maps D onto the some planar domain f (D). Since D is a simply connected domain, the mapping f has a canonical decomposition f = h(z) + g(z), where h(z) and g(z) are analytic in D and have the following power series expansions h(z) =

∞ 

an z n , g(z) =

n=0

∞ 

bn z n , z ∈ D.

n=0

where an , bn ∈ C, n = 0, 1, 2, 3, · · · . As in usual we call h(z) is analytic part of f and g(z) is co-analytic part of f . An elegant and complete account of the theory of the theory of harmonic mappings is given in Duren’s monograph [2]. 3

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Lewy [2] proved in 1936 that the harmonic mapping f is locally univalent in D if and only if its jacobien Jf = |h (z)|2 − |g  (z)|2 is different from zero in D. In view of this result, locally univalent harmonic mappings in the open unit disc are either sense-reversing if |g  (z)| > |h (z)| in D or sensepreserving if |g  (z)| < |h (z)| in D. Throughout this paper we will restrict ourselves to the study of sense-preserving harmonic mappings. We also note that f = h(z) + g(z) is sense-preserving in D if and only if h (z) does  not

g (z) vanish in the unit disc D, and the second complex dilatation w(z) = h (z) has the property |w(z)| < 1 in D. The class of all sense-preserving harmonic mappings of the open unit disc D with a0 = b0 = 0 and a1 = 1 and will be denoted by SH . Thus SH contains the standard class S of analytic univalent functions. The family of all mappings f ∈ SH with the additional property that g  (0) = 0, i.e., b1 = 0 0 0 is denoted by SH . Thus it is clear that S ⊂ SH ⊂ SH [2]. Now, we consider the following class of harmonic mappings α    g  (z) k 1+z iβ ≺ e b1 , α = −1, β ∈ R, h(z) ∈ C SHV (k) = f = h(z)+g(z) |  h (z) 1−z 2 (1.12) the aim of this paper is to investigate the class SHV (k) . For this aim we need the following lemma and theorems. Lemma 1.1. [3] Let φ(z) be regular in the open unit disc D. Then if |φ(z)| attains its maximum value on the circle |z| = r at the point z1 , one has z1 .φ (z) = kφ(z1 ) for some k ≥ 1. Theorem 1.2. [1] If h(z) is in V (k), then there is a p(z) such that h (z) = eiβ (p(z))α .g  (z) (1.13) ∞ where β is real, g(z) is in C and p(z) = n=0 pn z n has positive real part in D. Here eiβ pα0 = 1, α = k2 − 1. Theorem 1.3. [1] Let h(z) be an element of C, then r r ≤ |h(z)| ≤ 1+r 1−r and r r ≤ |h (z)| ≤ 2 (1 + r) (1 − r)2 for all |z| = r < 1. 4

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2

Main Results

Theorem 2.1. Let f = h(z) + g(z) be an elemet of SHV (k) , then α  g(z) 1+z iβ ≺ e b1 , z∈D h(z) 1−z where β ∈ R and α =

k 2

− 1.

Proof. Since f = h(z) + g(z) ∈ SHV (k) , then we have α  1+z g (z) iβ ≺ e b1 , h (z) 1−z and



g  (z) Re z  h (z)

 >

(2.1)

1 h (z) 1 h(z) ⇒z = ⇒  = (1 − φ(z)). 2 h(z) 1 − φ(z) zh (z)

(2.2)

On the other hand, if we investigate the properties of the linear transα formation w(z) = 1+z , α = k2 − 1, k ≥ 2 and using the subordina1−z > 1, 0 < 1−r < 1, tion and Lindelof Principle with 0 < r < 1, 1+r 1−r 1+r

1+z α α p(z) ∈ P ⇒ (p(z)) ≺ 1−z , we get  α     α  1−r 1+r 1−r 1+r α ≤ |p(z)| ≤ ≤ ≤ . (2.3) 1+r 1+r 1−r 1−r See Figure 1. Now, we define the function φ(z) by  k −1  1 + φ(z) 2 g(z) iβ = e b1 p0 h(z) 1 − φ(z) k

−1

where f = h(z) + g(z) ∈ SHV (k) , β ∈ R, eiβ p02 = 1. Therefore we have g  (z) | = b1 , 1 = 1+φ(0) ⇒ φ(0) = 0, φ(z) analytic, and h (z) z=0 1−φ(0)   g(z) 2zφ (z) g (z) = 1+ w(z) =  h (z) h(z) 1 + φ(z)  k2 −1    2zφ (z) 1 + φ(z) iβ . 1+ = e b0 1 − φ(z) 1 + φ(z)

(2.4)

5

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v

(P (Dr ))α P (Dr )

α α

1−r 1+r

1+r 1−r

1−r α

1+r α u 1−r

1+r

Figure 1: Now, it is easy to realize that the subordination 1 + z k −1 g (z) ≺ eiβ b1 ( )2  h (z) 1−z (from the definition of SHV (k) ) is equivalent to |φ(z)| < 1 for all z ∈ D. Indeed assume the contrary that there exists a z1 ∈ D such that |φ(z1 )| = 1. Then by I. S. Jack’s lemma (Lemma 1.1) z1 φ (z1 ) = kφ(z1 ), k ≥ 1 such z1 we have   g(z1 ) 2kφ(z1 ) g  (z1 ) = 1+ w(z1 ) =  h (z1 ) h(z1 ) 1 + φ(z1 )  k2 −1    2kφ(z1 ) 1 + φ(z1 ) iβ 1+ = e b1 p0 1 − φ(z1 ) 1 + φ(z1 ) = eiβ b1 w(φ(z1 )) ∈ / w(D) because |φ(z1 )| = 1, k ≥ 1 and the relations (2.3). But this is a contradiction to the condition of the definition of SHV (k) and so assumption is wrong, i.e., |φ(z)| < 1 for all z ∈ D. Corollary 2.2. Let f = h(z) + g(z) be an element of SHV (k) , then k

(1 − r) 2 −1 k

(1 + r) 2 +1

k



≤ |g (z)| ≤

(1 + r) 2 −1 k

(1 − r) 2 +1

,

(2.5)

6

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and

k

r(1 − r) 2 −1 k

(1 + r) 2

k

≤ |g(z)| ≤

r(1 + r) 2 −1

(2.6)

k

(1 − r) 2

for all |z| = r < 1. Proof. Since f = h(z) + g(z)is ∈ SHV (k) , thus using Theorem 2.1 then we can write   1 + r k −1 1 − r k −1  g (z)  )2 ≤    ≤ ( )2 , ( 1+r h (z) 1−r and   1 + r k −1 1 − r k −1  g(z)  )2 ≤  ≤ ( )2 ( 1+r h(z)  1−r for all |z| = r < 1. In this step, if we use Theorem 1.3 we get (2.5) and (2.6). We also note that the inequality (2.5) well known which was proved by Paatero [6]. Corollary 2.3. Let f = h(z) + g(z) be an element of SHV (k) , then F (−r) ≤ Jf ≤ F (r),

(2.7)

and

     1 (k−4)   1 (k−4) 1 1 1 1 1+r 2 1+r 2 1− ≤ |f | ≤ − −1 + 4−k 1−r 1+r k−4 1−r 1−r (2.8) where  k−2   1 1−r F (r) = 1− (1 − r)4 1+r

for all |z| = r < 1. Proof. Since 

then |h (z)|

 2

1−



1−r 1+r

1+r 1−r

 k2 −1

k−2 

     k2 −1  g (z)  1 + r ≤ |w(z)| =    ≤ , h (z) 1−r  2

2

≤ |h (z)| (1−|w(z)| ) ≤ |h (z)|

2



1−

k−2  1−r . 1+r (2.9)

7

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Using Theorem 1.3 in the equality (2.9) we get (2.7). Similarly (|h (z)| − |g (z)|) |dz| ≤ |df | ≤ (|h (z)| + |g  (z)|) |dz| ⇒ |h (z)| (1 − |w(z)|) |dz| ≤ |df | ≤ |h (z)| (1 + |w(z)|) |dz| ⇒  k2 −1  k2 −1     1 1 1+r 1+r 1− 1+ dr ≤ |df | dr (1 + r)2 1−r (1 − r)2 1−r which gives (2.8).

References [1] A. W. Goodman, Univalent Functions, Vol. I, II, Polygonal publishing Hours, Washington, New-Jersey, (1983). [2] P. Duren, Harmonic Mappings in the plane, Vol. 156, of Cambridge Tract in Mathematics, Cambridge University press, Cambridge, UK, 2004. [3] I. S. Jack, Functions starlike and convex of order α, J. London Math. Soc. 34(1959),215-216. [4] C. Loewner, Untersuchungen Uberdie Verzerrung bie Konformen Abbildungen des Einheitskeisez, |z| < 1, die durch Funktionen mit nicht verschwinden der Ableitung geliefert Werden,Ber. Verh. Sachs. Ges. Wiss. Leipzig, 69 (1917), 89-106. [5] K. I. Noor, B. Malik and S. Mustafa, A survey on functions of Bounded Boundary and Bounded Radius Rotation, Applied Mathematics, E-Notes 12, (2012), 136-152. [6] V. Paatero, Uber die Konforme Abbildungen Von Gebietten, deren Rander von Beschrankter Drehung , Sind. Ann. Acad. Sci. Feen. ser. A ,33 (1931). [7] V. Paatero, Uber Gebiete von beschrankter randdrehung, Ann. Acad. Sci. Fenn. Ser A. , 37 (1933). [8] B. Pinchuk, Functions with bounded boundary rotation , Isr. J. Math 10, (1971), 7-16. 8

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˜ lu Yas¸ar Polatog Department of Mathematics and Computer Sciences, ˙ ˙ Istanbul K¨ ult¨ ur University, Istanbul, Turkey e-mail: [email protected] Emel Yavuz Duman Department of Mathematics and Computer Science, ˙ Istanbul K¨ ult¨ ur University ˙ Istanbul, Turkey e-mail: [email protected] ˜ an Melike Aydog Department of Mathematics, ˙ I¸sık University, Me¸srutiyet Koyu, S¸ile Istanbul, Turkey e-mail: [email protected]

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Fuzzy norms on BCK-algebras and non-negativity of norms in algebras† Jeong Soon Han1 and Keum Sook So2,∗ 1

Department of Applied Mathematics, Hanyang University, Ahnsan, 426-791, Korea 2

Department of Mathematics, Hallym University, Chuncheon, 200-702, Korea

Abstract. In this paper, we discuss some fuzzy norms on BCK-algebras, and we find several conditions for norms to be non-negative in algebras. Finally we discuss fuzzy stable norms on several algebras.

1. Introduction Y. Imai and K. Is´eki introduced two classes of abstract algebras: BCK-algebras and BCI-algebras ([7, 8]). It is known that the class of BCK-algebras is a proper subclass of the class of BCI-algebras. BCK-algebras have some connections with other areas: A. Dvureˇcenskij and M. G. Graziano ([2]), C. S. Hoo ([6]), J. M. Font, A. J. Rodr´ıgez and A. Torrens ([3]) discussed BCK-algebras in connection with the areas of lattice ordered groups, M V -algebras and Wajsberg algebras. D. Mundici ([12]) proved that M V -algebras are categorically equivalent to bounded commutative BCK-algebras, and J. Meng ([11]) proved that implicative commutative semigroups are equivalent to a class of BCK-algebras. J. Neggers and H. S. Kim ([15]) introduced a new notion which appears to be of some interest, i.e., that of a B-algebra, and studied some of its properties. J. Neggers and H. S. Kim ([14]) introduced the notion of d-algebras which is another useful generalization of BCK-algebras, and then investigated several relations between d-algebras and BCK-algebras as well as several other relations between d-algebras and oriented digraphs. After that some further aspects were studied (see [9, 10, 13]). P. J. Allen et al. ([1]) developed a theory of companion d-algebras in sufficient detail to demonstrate considerable parallelism with the theory of BCK-algebras as well as obtaining a collection of results of a novel type. J. S. Han et al. ([4]) introduced several triangular norms in an arbitrary algebra, and investigated some conditions for the kernel Ker5 of a triangular norm 5 to be a d∗ -ideal, and as an application they constructed a quotient d-algebra. J. S. Han et al. ([5]) introduced the notion of an action YX as a generalization of the notion of a module, and obtained that the set of all fuzzy norms on YX forms a commutative monoid. In this paper, we discuss some fuzzy norms on BCK-algebras, and we find several conditions for norms to be non-negative in algebras. Finally we discuss fuzzy stable norms on several algebras.

0

2010 Mathematics Subject Classification: 08A72; 06F35. Keywords: norm, fuzzy norms, BCK-algebra, non-negative, B-algebra, stable. Correspondence: Tel: +82 33 248 2011, Fax: +82 33 256 2011 (K. S. So). 0 E-mail: [email protected]; [email protected] 0

∗

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Jeong Soon Han and Keum Sook So 2. Preliminaries An algebra (or a groupoid) (X, ∗) is a non-empty set X equipped with a binary operation ∗ on X. An algebra (X, ∗) is called a d-algebra ([14]) if there is a constant 0 satisfying the following axioms: (I) x ∗ x = 0, (II) 0 ∗ x = 0, (III) x ∗ y = 0 and y ∗ x = 0 imply x = y for all x, y ∈ X. We denote it by (X, ∗, 0) or X for brevity. In X we can define a binary relation “ ≤ ” by x ≤ y if and only if x ∗ y = 0. A BCK-algebra is a d-algebra X satisfying the following additional axioms: (IV) ((x ∗ y) ∗ (x ∗ z)) ∗ (z ∗ y) = 0, (V) ((x ∗ (x ∗ y)) ∗ y = 0 for all x, y, z ∈ X. J. S. Han et al. ([4]) introduced several (triangular) norms, and obtained some properties. Given an algebra (X, ∗), we consider mappings 5 : X → R such that one of the following identities holds: (T 1) 5(x) ≤ 5(y) + 5(x ∗ y), (T 2) 5(x ∗ y) ≤ 5(x) + 5(y), (T 3) 5(x ∗ z) ≤ 5(x ∗ y) + 5(y ∗ z), (T 4) 5((x ∗ z) ∗ (y ∗ z)) + 5((z ∗ x) ∗ (z ∗ y)) ≤ 5(x ∗ y) + 5(y ∗ x), for any x, y, z ∈ X. All of (T 1), (T 2), (T 3), (T 4) are versions of the “usual” triangle inequality, and thus we shall refer to a mapping 5 : X → R which satisfies an inequality (T i) above as a triangular norm of type (T i) when Ker5 := {x ∈ X | 5 (x) = 0} 6= ∅ (i = 1, 2, 3, 4). If we don’t require that Ker5 6= ∅, then we shall refer to it as a norm of type (T i) (or a (T i)-norm). A mapping 5 : X → R is said to be stable if 5(x ∗ y) ≤ 5(x) for any x, y ∈ X. In particular, if we replace R by [0, 1], we may consider 5 to be a fuzzy (triangular) norm of type (T i) (i = 1, 2, 3, 4). Example 2.1. ([4]) Let X be the power set of a finite set F , i.e., x ∈ X means x ⊆ F . If 5(x) = | x |, the cardinality of x, and x ∗ y := x − y, the collection of all elements of x not in y, then 5(∅) = 0, i.e., Ker5 6= ∅. Also, 5(x ∗ y) ≤ 5(x), i.e., 5 is stable, whence certainly 5(x ∗ y) ≤ 5(x) + 5(y), i.e., it is a triangular norm of type (T 2). Note that x = (x ∗ y) ∪ (x ∩ y), 5(x) = 5(x ∗ y) + 5(x ∩ y) and 5(x ∩ y) = 5(y ∗ xC ) ≤ 5(y), so that 5(x) ≤ 5(y)+5(x∗y) as well, and 5 is a triangular norm of type (T 1). Finally, x∗z ⊆ (x∗y)∪(y ∗z) means that 5(x ∗ z) ≤ 5(x ∗ y) + 5(y ∗ z), i.e., 5 is a triangular norm of type (T 3). Since [(x ∗ z) ∗ (y ∗ z)] ∪ [(z ∗ x) ∗ (z ∗ y)] ⊆ (x ∗ y) ∪ (y ∗ x), we have 5((x ∗ z) ∗ (y ∗ z)) + 5((z ∗ x) ∗ (z ∗ y)) ≤ 5(x ∗ y) + 5(y ∗ x), i.e., 5 : X → R is a stable and triangular norm of types (T 1) ∼ (T 4). Example 2.2. ([4]) If X = R and x ∗ y := max{0, x − y}, ∀x, y ∈ X, then 5(x) = | x | satisfies (T 2). Indeed, if x ≤ y, then x ∗ y = 0 and 5(x ∗ y) = 0 ≤ 5(x) + 5(y). If x > y, then 5(x ∗ y) = x − y ≤ | x − y | ≤ |x| + |y| = 5(x) + 5(y). On the other hand, if x = −5, y = −1, x − y = −4 and x ∗ y = 0, 5(x) = 5 6≤ 5(y) + 5(x ∗ y), so that (T 1) fails to hold. Also, if z = 0 and if (T 3) holds, then 5(x) ≤ 5(x ∗ y) + 5(y), which means (T 1) holds. Thus (T 3) fails, since (T 1) fails. Finally, if x = 5, y = −1, x ∗ y = 6 and 5(x ∗ y) > 5(x), i.e., the mapping 5 is not stable.

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Fuzzy norms on BCK-algebras and non-negativity of norms in algebras Example 2.3. ([4]) Let X := {0, 1, 2, 3} be an algebra with the following Cayley table: ∗ 0 1 2 3

0 0 1 2 3

1 0 0 2 3

2 0 0 0 3

3 0 1 0 0

Then it is a d-algebra. If we define a map 5 : X → [0, 1] by 0 = 5(0) < 5(1) < 5(2) < 5(3) ≤ 1, then it is easy to show that 5 is a fuzzy triangular norm of types (T 1) ∼ (T 3), but not of type (T 4), since 5((3 ∗ 1) ∗ (2 ∗ 1)) + 5((1 ∗ 3) ∗ (1 ∗ 2)) > 5(3 ∗ 2) + 5(2 ∗ 3).

3. Fuzzy norms on BCK-algebras Proposition 3.1. Let (X, ∗) be an algebra and let z0 ∈ X such that x ∗ z0 = x for all x ∈ X. Then every fuzzy (T 3)-norm on (X, ∗) is a fuzzy (T 1)-norm. Proof. If 5 is a fuzzy (T 3)-norm on X, then for any x, y ∈ X, we have 5(x ∗ z0 ) ≤ 5(x ∗ y) + 5(y ∗ z0 ) It follows that 5(x) ≤ 5(x ∗ y) + 5(y), i.e., 5 is a fuzzy (T 1)-norm on (X, ∗).



Corollary 3.2. If (X, ∗, 0) is a BCK-algebra, then every fuzzy (T 3)-norm on X is a fuzzy (T 1)-norm on X. Proof. In a BCK/BCI/BF -algebra (X, ∗, 0), x ∗ 0 = x holds for any x ∈ X.



Proposition 3.3. Let (X, ∗, 0) be a BCK-algebra and let 5 be a fuzzy (T 1)-norm with 5(0) = 0. If x ≤ y in X, then 5(x) ≤ 5(y). Proof. Let 5 be a fuzzy (T 1)-norm with 5(0) = 0. If x ≤ y in X, then 5(x) ≤ 5(y)+5(x∗y) = 5(y)+5(0) = 5(y).



Proposition 3.4. Let (X, ∗, 0) be a BCK-algebra and let 5 be a fuzzy (T 3)-norm on X with 5(0) = 0. If x ≤ y in X, then 5(x ∗ z) ≤ 5(y ∗ z) for any z ∈ X. Proof. Given z ∈ X, we have 5(x ∗ z) ≤ 5(x ∗ y) + 5(y ∗ z) = 5(0) + 5(y ∗ z) = 5(y ∗ z), proving the proposition.

 4. Non-negative norms

A map 5 : X → R is said to be non-negative if 5(x) ≥ 0 for all x ∈ X. Fortunately as we shall see, it is easy for norms to be non-negative. Proposition 4.1. Let (X, ∗) be an algebra. If 5 is a norm of types (T 1) and (T 2), then 5 is non-negative. Proof. Assume 5 is a norm of types (T 1) and (T 2). If we let y := x in (T 1), then 5(x) ≤ 5(x) + 5(x ∗ x) for any x ∈ X. It follows that 0 ≤ 5(x∗x) for any x ∈ X. Since 5 is a (T 2)-norm, we obtain 5(x∗x) ≤ 5(x)+5(x). Hence 0 ≤ 5(x ∗ x) ≤ 2 5 (x), proving that 5(x) ≥ 0.

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Jeong Soon Han and Keum Sook So Note that the converse of Proposition 4.1 need not be true in general. In Example 3.2, the mapping 5(x) = |x| is non-negative, but it is not a (T 1)-norm on X. Proposition 4.2. Let (X, ∗) be an algebra. If 5 is a norm of types (T 2) and (T 3), then 5 is non-negative. Proof. Let 5 be a norm of types (T 2) and (T 3). If we let z := x, y := x in (T 3), then 5(x∗x) ≤ 5(x∗x)+5(x∗x) for any x ∈ X, which means that 0 ≤ 5(x ∗ x) for any x ∈ X. If we let y := x in (T 2), then 5(x ∗ x) ≤ 5(x) + 5(x) = 2 5 (x). Hence 0 ≤

1 2

5 (x ∗ x) ≤ 5(x), proving that 5 is non-negative.



In Proposition 4.1, it was necessary to introduce two types of norms in a groupoid so that the norm is nonnegative. If we add some condition(s) in a groupoid, then we can find each types of norms can be non-negative.

Proposition 4.3. Let (X, ∗, 0) be an algebra with 0 ∗ x = 0 for all x ∈ X. (i). If 5 is either a (T 1)-norm or a (T 2)-norm on X, then 5 is non-negative, (ii). If 5 is a (T 3)-norm on X, then 5(y ∗ z) ≥ 0 for all y, z ∈ X. Proof. (i). If we let x := 0 in (T 1), then 5(0) ≤ 5(y) + 5(0 ∗ y) = 5(y) + 5(0) and hence 0 ≤ 5(y) for all y ∈ X. If we let x := 0 in (T 2), then 5(0) = 5(0 ∗ y) ≤ 5(0) + 5(y) and hence 0 ≤ 5(y) for all y ∈ X. (ii). If we let x := 0 in (T 3), then 5(0 ∗ z) ≤ 5(0 ∗ y) + 5(y ∗ z) for any y, z ∈ X. Since 0 ∗ x = 0 for all x ∈ X, we obtain 5(0) ≤ 5(0) + 5(y ∗ z), proving that 0 ≤ 5(y ∗ z) for all y, z ∈ X.



Example 4.4. Every (T 1)-norm or (T 2)-norm on a BCK-algebra is non-negative. Moreover, every (T 3)-norm on a BCK-algebra is also non-negative, since x ∗ 0 = x for all x ∈ X. Corollary 4.5. Let (X, ∗, 0) be an algebra with 0 ∗ x = 0 for all x ∈ X. Assume that there exist y, z ∈ X such that x = y ∗ z for any x ∈ X. Then every (T 3)-norm is non-negative. Proof. It follows immediately from Proposition 4.3-(ii).



Proposition 4.6. Let (X, ∗, 0) be an algebra with x ∗ x = x for all x ∈ X. If 5 is a (T 2)-norm on X, then it is non-negative. Proof. If 5 is a (T 3)-norm on X, then 5(x) = 5(x ∗ x) ≤ 5(x ∗ x) + 5(x ∗ x) = 2 5 (x) and hence 0 ≤ 5(x) for all x ∈ X.



Theorem 4.7. Let (X, ·, e) be a group all of whose elements are of finite order. Define a binary operation “∗” on X by x ∗ y := x · y −1 for all x, y ∈ X. Then every (T 1)-norm on (X, ∗) is non-negative. Proof. If 5 is a (T 1)-norm on (X, ∗), then 5(x) ≤ 5(y) + 5(x ∗ y) for all x, y ∈ X. Since x ∗ y = x · y −1 , we obtain (1)

5(x) ≤ 5(y) + 5(x · y −1 )

If we let y := e, the identity of X in (1), then we have 5(x) ≤ 5(e) + 5(x · e−1 ) = 5(e) + 5(x), which proves that 0 ≤ 5(e).

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Fuzzy norms on BCK-algebras and non-negativity of norms in algebras Given x ∈ X, we let the order of x be finite, i.e., o(x) := n < ∞. Then we have 5(xn ) ≤ 5(x) + 5(xn ∗ x) = 5(x) + 5(xn · x−1 ) = 5(x) + 5(xn−1 ) By induction, we obtain 0 ≤ 5(e) = 5(xn ) ≤ n 5 (x) for any x ∈ X. This proves the theorem.



The non-negativity of a (T 2)-norm on an arbitrary group holds in general as follows: Theorem 4.8. Let (X, ·, e) be a group. Define a binary operation “∗” on X by x ∗ y := x · y −1 for all x, y ∈ X. Then every (T 2)-norm on (X, ∗) is non-negative. Proof. Since 5 is a (T 2)-norm on (X, ∗), we have 5(x ∗ y) ≤ 5(x) + 5(y) for any x, y ∈ X. It follows that (2)

5(x · y −1 ) ≤ 5(x) + 5(y)

If we let Y := x in (2), then 5(x · x−1 ) ≤ 5(x) + 5(x), i.e., 5(e) ≤ 2 5 (x) for all x ∈ X. If we let y := e in (2), then 5(x) = 5(x · e−1 ) ≤ 5(x) + 5(e) and hence 0 ≤ 5(e). Hence 0 ≤ 5(x) for all x ∈ X, proving the theorem.



J. Neggers and H. S. Kim introduced the notion of B-algebras. A B-algebra ([15]) is a non-empty set X with a constant 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 all x, y, z in X. Proposition 4.9. Let (X, ∗, 0) be a B-algebra. If 5 is a (T 1)-norm on X with 5(x) = 5(0 ∗ x) for any x ∈ X, then it is non-negative. Proof. Let 5 be a (T 1)-norm on X with 5(x) = 5(0 ∗ x) for any x ∈ X. Then (3)

5(x) ≤ 5(y) + 5(x ∗ y)

for any x, y ∈ X. If we let x := 0, then 5(0) ≤ 5(y) + 5(0 ∗ y) = 2 5 (y) for any y ∈ X. If we let x := y in (3), then 5(y) ≤ 5(y) + 5(y ∗ y) = 5(y) + 5(0), i.e., 0 ≤ 5(0). This means that 0 ≤≤ 5(0) ≤ 2 5 (y), proving the proposition.



Proposition 4.10. If (X, ∗, 0) is a B-algebra, then every (T 2)-norm of X is non-negative. Proof. If 5 is a (T 2)-norm on X, then for any x, y ∈ X, we have (4)

5(x ∗ y) ≤ 5(x) + 5(y)

If we let x := y in (4), then 5(0) = 5(y ∗ y) ≤ 5(y) + 5(y) = 2 5 (y), i.e., 5(0) ≤ 2 5 (y) for all y ∈ X. If we let y := 0 in (4), then 5(x) = 5(x ∗ 0) ≤ 5(x) + 5(0), which shows that 0 ≤ 5(0). Hence 5 is non-negative. 691

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Jeong Soon Han and Keum Sook So 5. Fuzzy stable norms In this section we discuss the notion of fuzzy stable norms in groupoids. Proposition 5.1. Let (X, ∗) be a groupoid. If 5 is a fuzzy stable (T 1)-norm on (X, ∗), then it is a fuzzy (T 2)-norm on (X, ∗). Proof. Since 5 is stable, we have 5(x ∗ y) ≤ 5(x) ≤ 5(x) + 5(y) for all x, y ∈ X, i.e., 5 is a fuzzy (T 2)-norm on (X, ∗).



Proposition 5.2. Let (X, ·, e) be a group. If 5 is a fuzzy stable norm on (X, ·), then 5 is a constant mapping on X. Proof. Since 5 is a fuzzy stable norm on (X, ·), we have 5(x · y) ≤ 5(x)

(5)

for all x, y ∈ X. If we let x := e in (5), then 5(e · y) ≤ 5(e) for all y ∈ X, i.e., 5(y) ≤ 5(e). If we let y := x−1 in (5), then 5(e) = 5(x · x−1 ) ≤ 5(x) for all x ∈ X. Hence 5(x) = 5(e), i.e., 5 is a constant mapping on (X, ·).



Theorem 5.3. Let (X, ∗, 0) be a BCK-algebra. If 5 is a fuzzy (T 1)-norm on (X, ∗, 0) with 5(0) = 0, then it is a fuzzy stable (T 3)-norm on (X, ∗, 0). Proof. If (X, ∗, 0) is a BCK-algebra, then (x ∗ y) ∗ x = 0 for all x, y ∈ X. Since 5 is a fuzzy (T 1)-norm on (X, ∗, 0), we have 5(x ∗ y) ≤ 5(x) + 5((x ∗ y) ∗ x) = 5(x) + 5(0) = 5(x) for all x, y ∈ X, i.e., 5 is stable. Since ((x ∗ y) ∗ (x ∗ z)) ∗ (z ∗ y) = 0 holds always in any BCK-algebra and 5 is a fuzzy (T 1)-norm on X, we obtain 5((x ∗ y) ∗ (x ∗ z)) ≤ 5(z ∗ y) + 5(((x ∗ y) ∗ (x ∗ z)) ∗ (z ∗ y)) = 5(z ∗ y) + 5(0) = 5(z ∗ y) for any x, y, z ∈ X. It follows from 5 is a fuzzy (T 1)-norm on X that 5(x ∗ y) ≤ 5(x ∗ z) + 5((x ∗ y) ∗ (x ∗ z)) ≤ 5(x ∗ z) + 5(z ∗ y) = 5(z ∗ y) for all x, y, z ∈ X, proving that 5 is a fuzzy (T 3)-norm on (X, ∗, 0).



Acknowledgments This research was supported by Hallym University Research Fund 2012 (HRF-201212-009). 692

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Fuzzy norms on BCK-algebras and non-negativity of norms in algebras

References [1] P. J. Allen, H. S. Kim and J. Neggers, Companion d-algebras, Math. Slovaca 57 (2007), 93–106. [2] A. Dvureˇcenskij and M. G. Graziano, Commutative BCK-algebras and lattice ordered groups, Math. Japonica 32 (1999), 227–246. [3] J. M. Font, A. J. Rodr´ıgez and A. Torrens, Wajsberg algebras, Stochastica 8 (1984), 5–31. [4] J. S. Han, H. S. Kim and J. Neggers, Constructions of quotient algebras via triangular norms, Math. Slovaca 60 (2010), 1-10. [5] J. S. Han, H. S. Kim and J. Neggers, Actions, norms, subactions and kernels of (fuzzy) norms, Iranian J. Fuzzy Systems 7 (2010), 141-147. [6] C. S. Hoo, Unitary extensions of M V and BCK-algebras, Math. Japonica 37 (19xx), 585–590. [7] K. Is´eki, On BCI-algebras, Math. Seminar Notes 8 (1980), 125–130. [8] K. Is´eki and S. Tanaka, An introduction to theory of BCK-algebras, Math. Japonica 23 (1978), 1–26. [9] Y. B. Jun, J. Neggers and H. S. Kim, Fuzzy d-ideals of d-algebras, J. Fuzzy Math. 8 (2000), 123-130. [10] Y. C. Lee and H. S. Kim, On d-subalgebras of d-transitive d∗ -algebras, Math. Slovaca 49 (1999), 27–33. [11] J. Meng, Implicative commutative semigroups are equivalent to a class of BCK-algebras, Semigroup Forum 50 (1995), 89–96. [12] D. Mundici, M V -algebras are categorically equivalent to bounded commutative BCK-algebras, Math. Japonica 31 (1986), 889–894. [13] J. Neggers, Y. B. Jun and H. S. Kim, On d-ideals in d-algebras, Math. Slovaca 49 (1999), 243–251. [14] J. Neggers and H. S. Kim, On d-algebras, Math. Slovaca 49 (1999), 19–26. [15] J. Neggers and H. S. Kim, On B-algebras, Mate. Vesnik 54 (2002), 21–29.

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´ EXTENDED CESARO OPERATOR FROM HARDY SPACE TO ZYGMUND-TYPE SPACE ON THE UNIT BALL YU-XIA LIANG AND HONG-GANG ZENG∗

Abstract. In this paper, we characterize the boundedness and compactness of extended Ces´ aro operator from Hardy space to Zygmund-type space on the unit ball of Cn .

1. Introduction Let H(Bn ) be the class of all holomorphic functions on Bn , where Bn is the unit ball in the n-dimensional complex space Cn . Let dv denote the Lebesegue measure on Bn normalized so that v(Bn ) = 1, and dσ the normalized rotation invariant measure on the boundary Sn (= ∂Bn ) of Bn . For f ∈ H(Bn ), let n ∑ ∂f ℜf (z) = zj (z) ∂z j j=1 be the radial derivative of f . The Bloch space B(= B(Bn )) is defined as the space of holomorphic functions such that ∥f ∥B = |f (0)| + sup{(1 − |z|2 )|ℜf (z)| : z ∈ Bn } < ∞. Let B0 (= B0 (Bn )) denote the subspace of B consisting of those f ∈ B for which 2

(1 − |z| ) |ℜf (z)| → 0, |z| → 1. This space is called the little Bloch space. Moreover, for α > 0, we say f ∈ Bα if f ∈ H(Bn ) and ∥f ∥Bα = |f (0)| + sup{(1 − |z|2 )α |ℜf (z)| : z ∈ Bn } < ∞. For 0 < p < ∞, Hardy space H p (Bn ) consists of all holomorphic functions f ∈ H(Bn ) such that ∫ p p |f (rζ)|p dσ(ζ) < ∞. (1) ∥f ∥H p = sup Mp (f, r) = sup 0 0. xn d[g(x)]n Proof

For n = 0, the theorem is obviously true.

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Zeng-tai Gong and Ting Xie: Pseudo-differentiability, Pseudo-integrability and Nonlinear Differential Equations

Suppose that the theorem is true for n − 1 , i.e., d(n−1)⊕ f (x) = g −1 xn−1



dn−1 g(f (x)) d[g(x)]n−1

 .

Then

! d(n−1)⊕ f dxn−1  n−1   g(f (x)) d⊕ −1 d = g dx d[g(x)]n−1   n−1  d d g(f (x)) = g −1 dg(x) d[g(x)]n−1   n −1 d g(f (x)) . =g d[g(x)]n By mathematical induction, the theorem is proved. Definition 4.3 Let ⊕ be continuous and strictly increasing. Let f (x) be a bounded function defined on [c, d]. If for any partition of [c, d] d(n)⊕ f (x) d⊕ = xn dx

P : c = x0 < x1 < x2 < · · · < xn = d, denote λ = max (xi xi−1 ), if for any ξi ∈ [xi−1 , xi ], the limit 16i6n

lim

λ→0

n M

f (ξi ) (xi xi−1 )

i=1

exists, then f (x) is said to be pseudo-integrable on [c, d], and its pseudo-integral value equals to the limit R (⊕, ) value, denoted by [c,d] f (x)dx. Theorem 4.4 Let f1 and f2 be two pseudo-integrable functions on [c, d] and with the values in [a, b]. Then for λ1 , λ2 ∈ [a, b], λ1 f1 ⊕ λ2 f2 is also generalized integrable on [c, d] and Z (⊕, ) Z (⊕, ) Z (⊕, ) (λ1 f1 ⊕ λ2 f2 )dx = λ1 f1 dx ⊕ λ2 f2 dx. [c,d]

Proof

[c,d]

[c,d]

For any partition of [c, d] P : c = x0 < x1 < x2 < · · · < xn = d

and for any ξi ∈ [xi−1 , xi ], we have n M


λ1 f1 (ξi ) ⊕ λ2 f2 (ξi ) (xi xi−1 )

i=1

=

n M


λ1 f1 (ξi ) (xi xi−1 ) ⊕ λ2 f2 (ξi ) (xi xi−1 )

i=1 n M

=λ1

! f1 (ξi ) (xi xi−1 )

⊕ λ2

i=1

n M

! f2 (ξi ) (xi xi−1 ) .

i=1

Let λ = max (xi xi−1 ) → 0, since f1 and f2 are pseudo-integrable on [c, d], we have 16i6n

lim

n M

λ→0


λ1 f1 (ξi ) ⊕ λ2 f2 (ξi ) (xi xi−1 )

i=1

=λ1

lim

λ→0

Z

n M

! f1 (ξi ) (xi xi−1 )

i=1 (⊕, )

=λ1

Z

lim

λ→0

! f2 (ξi ) (xi xi−1 )

i=1

(⊕, )

f1 dx ⊕ λ2 [c,d]

⊕ λ2

n M

f2 dx. [c,d]

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Zeng-tai Gong and Ting Xie: Pseudo-differentiability, Pseudo-integrability and Nonlinear Differential Equations

According to Definition 4.3, λ1 f1 ⊕ λ2 f2 is pseudo-integrable on [c, d] and Z

(⊕, )

(⊕, )

Z

Z

(⊕, )

f1 dx ⊕ λ2

(λ1 f1 ⊕ λ2 f2 )dx = λ1

f2 dx. [c,d]

[c,d]

[c,d]

Theorem 4.5 Let f (x) be pseudo-integrable on [c, d]. Then Z

(⊕, )

f dx = g

−1

d

Z

 g(f (x))dg(x) ,

c

[c,d]

when the right part is meaningful. Proof Z (⊕, ) n M f dx = lim f (ξi ) (xi xi−1 ) λ→0

[c,d]

i=1

= lim [f (ξ1 ) (x1 x0 ) ⊕ f (ξ2 ) (x2 x1 ) ⊕ · · · ⊕ f (ξn ) (xn xn−1 )] λ→0

= lim {g −1 [g(f (ξ1 )) · g(x1 x0 )] ⊕ g −1 [g(f (ξ2 )) · g(x2 x1 )] ⊕ · · · λ→0 −1

⊕g

= lim g

[g(f (ξn )) · g(xn xn−1 )]}

−1

λ→0 −1

[g(f (ξ1 )) · g(x1 x0 ) + g(f (ξ2 )) · g(x2 x1 ) + · · · + g(f (ξn )) · g(xn xn−1 )]

[ lim g(f (ξ1 )) · (g(x1 ) − g(x0 )) + g(f (ξ2 )) · (g(x2 ) − g(x1 )) + · · · λ0 →0
+ g(f (ξn )) · (g(xn ) − g(xn−1 )) ] Z d  −1 =g g(f (x))dg(x) ,

=g

c 0

where λ = max |g(xi ) − g(xi−1 )|. 16i6n

Remark 4.5

For 1 6 i 6 n, we have xi xi−1 → 0 ⇐⇒ d(xi , xi−1 ) → 0 ⇐⇒ |g(xi ) − g(xi−1 )| → 0,

therefore max (xi xi−1 ) → 0 ⇐⇒ max |g(xi ) − g(xi−1 )| → 0,

16i6n

16i6n

namely,

0

λ → 0 ⇐⇒ λ → 0. Remark 4.6

In [8, 9, 10, 19], the authors directly defined the g-integral as follows. Z

(⊕, )

f dx = g −1

[c,d]

Z

d

 g(f )dx .

c

However, it may be more natural to define as the way proposed in this paper by Definition 4.3 and obtain Theorem 4.5. Additionally, the definition proposed in this paper coincides with the definition of integral R  R (⊕, ) d −1 with respect to a decomposable measure m proposed in [11-18], i.e., [c,d] f dm = g g(f )dg ◦ m . c Theorem 4.6 Suppose that f is continuous on [c, d]. Then we have ! Z (⊕, ) d⊕ f (t)dt = f (x) dt [c,x] for any x ∈ [c, d]. 718

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Proof

From Theorem 4.2 and 4.5, we have for any x ∈ [c, d]   R (⊕, ) ! Z (⊕, ) dg f (t)dt ⊕ d [c,x]  f (t)dt = g −1  dt dg(t) [c,x]   Rx −1 d( c g(f (t))dg(t)) =g dg(t) = g −1 (g(f (x))) = f (x),

where we have used the fundamental theorem of the usual calculus. Theorem 4.7 (Newton-Leibniz formula) Suppose that f has continuous pseudo-derivative on [c, d]. Then we have for any x ∈ [c, d] Z (⊕, ) ⊕ d f dt = f (x) f (c) dt [c,x] for any x ∈ [c, d]. Proof From Theorem 4.2 and 4.5, we have for any x ∈ [c, d] Z

(⊕, )

[c,x]

d⊕ f (t) dt = g −1 dt

Z

x



d⊕ f (t) dt



 g dg(t) c  Z x dg(f (t)) −1 dg(t) =g dg(t) c Z x  = g −1 dg(f (t)) c
= g −1 g(f (x)) − g(f (c)) = f (x) f (c).

5 Applications Example 5.1

Consider the following ordinary differential equation of the first order for s ∈ [0, +∞) 0

ln y − y + xs + x = 0.

(1)

Let g(x) = e−x , then Equation (1) can be represented as the following pseudo-differential equation d⊕ y = xs . dx Pseudo-integrating the preceding equation correspondingly, we have Z

(⊕, )

xs dx ⊕ C1

y=

 =g g(x )dg(x) + g(C1 )  −x−xs  e = −ln +C , 1 + sxs−1 −1

Z

s

where C = C2 + e−C1 . Example 5.2 Consider the following equation for p > 0 and s ∈ [0, +∞) 0

(y )1/p y 1−1/p − xs+(1−1/p) = 0. 719

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Zeng-tai Gong and Ting Xie: Pseudo-differentiability, Pseudo-integrability and Nonlinear Differential Equations

Let g(x) = xp , then Equation (2) can be represented as the following pseudo-differential equation d⊕ y = xs . dx Pseudo-integrating the preceding equation correspondingly, we have Z (⊕, ) xs dx ⊕ C1 y= Z  g(xs )dg(x) + g(C1 ) = g −1 =

xp(s+1) +C s+1

!1/p ,

where C = C2 + C1p . Example 5.3 Consider the following equation
0 (x − 1)3 (1 − y)y + xy 2 + (x − 1)3 − 2x y + x = 0. Let g(x) =

x 1−x ,

(3)

then Equation (3) can be represented as the following pseudo-differential equation d⊕ y = x. dx

Pseudo-integrating the preceding equation correspondingly, we have Z (⊕, ) y= xdx ⊕ C1 Z  −1 =g g(x)dg(x) + g(C1 ) =

2C(1 − x)2 + x2 , 2(1 − x)2 + 2C(1 − x)2 + x2

C1 where C = C2 + 1−C . 1 Example 5.4 Solve the following equation for λ > 0 0

λ(1 + λ)y y − (1 + λ)2x + (1 + λ)x = 0. Let g(x) = equation

(1+λ)x −1 , λ

(4)

λ > 0, then Equation (4) can be represented as the following pseudo-differential

d⊕ y = x. dx Pseudo-integrating the preceding equation correspondingly, we have Z (⊕, ) y= xdx ⊕ C1  Z −1 =g g(x)dg(x) + g(C1 )
ln ((1 + λ)x − 1)2 /2λ + C = , ln(1 + λ)

where C = C2 λ + (1 + λ)C1 . Example 5.5 Let Ω = [−L/2, L/2], L > 0 given. We consider as an important nonlinear partial differential equation the Kuramoto-Sivashinsky equation (see [5, 6, 20]) 1 ut + λuxxxx + uxx + u2x = 0 2 720

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Zeng-tai Gong and Ting Xie: Pseudo-differentiability, Pseudo-integrability and Nonlinear Differential Equations

for u = u(x, t) and x ∈ Ω, where λ is the given positive constant. Let g(x) = e−x/c , then k u = k + u, u ⊕ u = u − c ln 2. If u is a solution of the KuramotoSivashinsky equation, then put k u and u ⊕ u in the Kuramoto-Sivashinsky equation, respectively, we obtain an identity, thus k u and u ⊕ u are also solutions of the Kuramoto-Sivashinsky equation. Therefore (k u) ⊕ (k u) is a solution of the Kuramoto-Sivashinsky equation. We find new solutions of the Kuramoto-Sivashinsky equation.

References [1] J. Aczel, Lectures on Functional Equations and their Applications, Academic Press, New York, 1966. [2] D. Dubois, M. Prade, Fuzzy Sets and Systems: Theory and Applications, Academic Press, New York, 1980. [3] D. Dubois, M. Prade, A class of fuzzy measures based on triangular norms, Int.J. Gen. System 8 (1982) 43-61. [4] H. Ichihashi, H. Tanaka, K. Asai, Fuzzy integrals based on pseudo-additions and multiplications, Journal of Mathematical Analysis and Applications 130 (1988) 354-364. [5] Y. Kuramoto, Diffusion induced chaos in reactions systems, Progr. Theoret. Phys. Suppl. 64 (1978) 346-367. [6] Y. Kuramoto, T. Tsuzuki, Persistent propagation of concentration waves in dissipative media far from thermal equilibrium, Prog. Theor. Phys. 55 (1976) 356-369. [7] C. H. Ling, Representation of associative functions, Publ. Math. Debrecen 12 (1965) 189-212. [8] A. Markov´ a, A note on g-integral, Tatra Mountains Math. Publ. 8 (1996) 71-76. [9] A. Markov´ a, B. Rieˇ can, On the double g-integral, Novi Sad J. Math. (1996) 67-70. [10] R. Mesiar, Pseudo-linear integrals and derivatives based on a generator g, Tatra Mountains Math. Publ. 8 (1996) 67-70. [11] E. Pap, Integral generated by decomposable measure, Univ. u Novom Sadu Zb. Rad. Prirod.-Mat. Fak. Ser. Mat. 20 (1990) 135-144. [12] E. Pap, g-calculus, Univ. u Novom Sadu Zb. Rad. Prirod.-Mat. Fak. Ser. Mat. 23 (1993) 145-150. [13] E. Pap, Nonlinear difference equations and neural nets, BUSEFAL 60 (1994) 7-14. [14] E. Pap, Null-Additive Set Functions, Kluwer Academic Publishers, Dordrecht, Boston, London, 1995. [15] E. Pap, Decomposable measures and nonlinear equations, Fuzzy Sets Syst. 92 (1997) 205-221. [16] E. Pap, N. Ralevi´ c, Pseudo-Laplace Transform, Nonlinear Anal. 33 (1998) 533-550. ˇ [17] E. Pap, I. Stajner, Generalized pseudo-convolution in the theory of probabilistic metric spaces, information, fuzzy numbers, optimization, system theory, Fuzzy Sets Syst. 102 (1999) 393-415. [18] E. Pap, Pseudo-additive measures and their applications, in: E. Pap (Ed.), Handbook of Measure Theory, Elsevier, North-Holland, Amsterdam, 2002, pp. 1237-1260. [19] N. Ralevi´ c, Some new properties of g-calculus, Univ. u Novom Sadu Zb. Rad. Prirod.-Mat. Fak. Ser. Mat. 24 (1994) 139-157.

.

[20] G. Sivashinsky, Nonlinear analysis of hydrodynamic instability in laminar flame. . Derivation of basic equations, Acta Astron. 4 (1977) 1117-1206. [21] M. Sugeno, T. Murofushi, Pseudo-additive measures and intgerals, J. Math. Anal. Appl. 122 (1987) 197-222.

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A GENERAL THEOREM ASSOCIATED WITH THE BRIOT–BOUQUET DIFFERENTIAL SUBORDINATION ¼ ADEL A. ATTIYA AND TEODOR BULBOACA Abstract. Recently, many authors used the techniques of di¤erential subordination to …nd the best dominant of di¤erent subordination relation for certain subclasses of analytic functions, associated with the Briot–Bouquet di¤erential subordination. In this paper we …nd some general results, which determine the best dominant of a general form of di¤erential subordination relation for the above mentioned Briot–Bouquet di¤erential subordination.

1. Basic definitions and preliminaries We denote by H(U) the class of all analytic functions in the open unit disc U = fz 2 C : jzj < 1g, and let A be the class of functions ' 2 H(U) which satisfy '(0) = 1. To prove our main results, we need the following de…nitions and the lemmas. De…nition 1.1. [15, p. 36] Let f and F be two analytic functions. The function f is said to be subordinate to F , written as f (z) F (z), if there exists a function w analytic in U, with w(0) = 0 and jw(z)j < 1, and such that f (z) = F (w(z)). It is well-known that if the function F is univalent, then f (z) F (z) if and only if f (0) = F (0) and f (U) F (U). De…nition 1.2. [7, p. 16] Let : C2 U ! C, and let h be univalent in U. If p is an analytic function in U, and (p(z); z p0 (z); z) is also analytic in U, then we say that p satis…es a …rst order di¤erential subordination if (1.1)

(p(z); z p0 (z); z)

h(z):

The univalent function q is called to be a dominant of the di¤erential subordination (1.1), if p(z) q(z) for all p satisfying (1.1). If q~(z) q(z) for all the dominants q of (1.1), then we say that q~ is the best dominant of (1.1). 2010 Mathematics Subject Classi…cation. 30C45, 30C10. Key words and phrases. Analytic functions, di¤erential subordination, Briot– Bouquet di¤erential equation, Briot–Bouquet di¤erential subordination, operators in geometric function theory. 1

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¼ ADEL A. ATTIYA AND TEODOR BULBOACA

2

Lemma 1.1 (Wilken and Feng [17]; see also [7]). Let be a positive measure on the unit interval [0; 1]. Let g(z; t) be a complex valued function de…ned on U [0; 1] such that g( ; t) is analytic in U for each t 2 [0; 1], and such that g(z; ) is -integrable on [0; 1], for all z 2 U. In addition, suppose that Re g(z; t) > 0, g( r; t) is real, and Re

1 g(z; t)

1 ; for all jzj g( r; t)

r < 1; t 2 [0; 1]:

If the function G is de…ned by Z 1 g(z; t) d (t); G(z) = 0

then Re

1 G(z)

1 ; for all jzj G( r)

r < 1:

Each of the following identities asserted by Lemma 1.2 is well known: Lemma 1.2. [5, Chapter 9] For all real or complex numbers a, b and c, with c 2 = Z0 , the following equalities hold: Z 1 (b) (c b) tb 1 (1 t)c b 1 (1 tz) a d t = (1.2) 2 F1 (a; b; c; z); (c) 0 where Re c > Re b > 0;

(1.3)

2 F1 (a; b; c; z)

= (1

z)

a

2 F1

a; c

b; c;

z z

;

1

and (1.4)

2 F1 (a; b; c; z)

= 2 F1 (b; a; c; z):

The next lemma is a special case of a more general result [6, Corollary3.2.]: Lemma 1.3. [6, Example 1.] If complex number satis…es

1

A 0, and the

(1 A) ; 1 B

Re

then the following di¤erential equation 0

q(z) +

zq (z) q(z) +

=

1 + Az ; 1 + Bz

723

with q(0) = 1;

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A GENERAL THEOREM ASSOCIATED WITH . . .

has a univalent solution in U given by 8 z + (1 + Bz) (A B)=B > > > > Zz > > > > t + 1 (1 + Bt) (A B)=B d t > > < 0 q(z) = z + exp( Az) > > > ; > Zz > > > > > t + 1 exp( At) d t > :

3

; if B 6= 0

if B = 0:

0

Moreover, if the function ' is analytic in U and satis…es the following subordination 0

z' (z) '(z) + '(z) +

(1.5) then

'(z)

q(z)

1 + Az ; 1 + Bz 1 + Az ; 1 + Bz

and q is the best dominant of (1.5). 2. Main results Theorem 2.1. Let A, B and ditions (2.1) and let

1

be real numbers which satisfy the con-

B 0;

be a complex number with

(1 A) : 1 B 1: If ' 2 A and satis…es (1.5), then

(2.2)

(2.3)

Re

'(z)

where

(2.4)

M (z) =

q(z) =

8 1 Z > > > > t > >
> > > t > > :

exp ( (t

1)Az) d t;

if B = 0;

0

and q is the best dominant of (2.3). 2: Suppose, in addition, that is a real number, with (2.5)

>

1

A ; B

and

724

1

B < 0:

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¼ ADEL A. ATTIYA AND TEODOR BULBOACA

4

Then (2.6) +

Re '(z) >

(B A) 1; ; + B

2 F1

+ 1;

1

B B

; z 2 U;

1

and the right hand side of (2.6) cannot be replaced by a larger one. Proof. Since the function ' satis…es (1.5), from Lemma 1.3 we deduce that 1 + Az '(z) q(z) ; 1 + Bz where q is given in (2.3), and it is the best dominant of the given subordination. This proves the assertion (2.3) of the theorem. In order to prove (2.6) it is su¢ cient to show that (2.7)

inf fRe q(z) : z 2 Ug = q( 1):

(B A) , since B 1, then from (2.4) by using (1.2), B (1.3), and (1.4) for B 6= 0, we obtain that (A B) Z1 B 1 + Btz + 1 M (z) = t dt 1 + Bz

Putting a =

0

a

= (1 + Bz)

Z1

t

+

1

(1 + Btz)

a

dt

0

(2.8)

( + ) = 2 F1 1; a; + ( + + 1)

+ 1;

Bz Bz + 1

:

In view of the formula (1.2), this last relation holds whenever Re( + + 1) > Re a > 0, and under our assumption these inequalities are equivalent to (2.5). We remark that the condition (2.2) implies + > 0, and also the convergence of the integrals from the formula (2.8). Now, to prove (2.7) we need to show that 1 1 Re ; z 2 U: M (z) M ( 1) By using the relations (1.2) and (2.8) we have Z1 M (z) = h(z; t) d (t); 0

where h(z; t) = and d (t) =

1 + Bz 1 + (1 t)Bz ( + ) (a) ( + + 1

725

(z 2 U; 0 a)

ta 1 (1

t)

t

1);

+

a

d t;

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A GENERAL THEOREM ASSOCIATED WITH . . .

5

is a positive measure on [0; 1]. We note that Re h(z; t) > 0, h( r; t) is real whenever r 2 [0; 1), and for 1 B < 0 we have that 1 1 + (1 t)Bz 1 + (1 t)Br 1 Re = Re = : h(z; t) 1 + Bz 1 + Br h( r; t) Therefore, by using Lemma 1.1 we have 1 1 ; jzj r < 1; Re M (z) M ( r) which, upon letting r ! 1 , yields 1 1 Re > ; z 2 U: M (z) M ( 1) Since q is the best dominant of (2.3), it follows that the constant from the right-hand side of (2.6) cannot be replaced by a larger one. Corollary 2.1. Let A, B, and be constrained by the conditions (2.1), (2.2), and (2.5). Let Hs : H(U) ! H(U) be an operator such 0 zHs+1 (f )(z) that is analytic in U, with Hs+1 (f )(z) 0

zHs+1 (f )(z) Hs+1 (f )(z)

=

+k+ ;

z=0

and satis…es (2.9)

0

zHs+1 (f )(z) = kHs+1 (f )(z) + mHs (f )(z);

for some k; m 2 C and for all f 2 H(U). Also, let de…ne the class Rs;k; (A; B) by Rs;k; (A; B) =

f 2 H(U) :

1

1: If f 2 Rs;k; (A; B), then (2.10) 1 zH0s+1 (f )(z) (k + ) Hs+1 (f )(z)

zH0s (f )(z) Hs (f )(z)

q(z) =

1

1 + Az 1 + Bz

(k + )

1 M (z)

:

1 + Az ; 1 + Bz

where M is given by (2.4), and q is the best dominant of (2.10), hence Rs;k; (A; B) 2: Moreover, if then

Rs+1;k; (A; B):

is a real number,

1

B < 0, and

>

1

A , B

Hs+1 (Rs;k; (A; B)) S ( ); (where S ( ) represents the class of starlike functions of order ) and Rs;k; (A; B)

Rs+1;k;

1

726

2(

Re k)

; 1 ;

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¼ ADEL A. ATTIYA AND TEODOR BULBOACA

6

where =( + ) The constant

(B A) 1; ; + B

2 F1

+ 1;

1

B B

+ Re k:

1

cannot be replaced by a larger one.

Proof. Putting '(z) =

1

zH0s+1 (f )(z) Hs+1 (f )(z)

(k + ) ;

then ' 2 A, and using the identity (2.9), we have mHs (f )(z) = '(z) + : Hs+1 (f )(z)

Carrying out logarithmic di¤erentiation in the above relation and using Theorem 2.1, we deduce the result of the corollary. Corollary 2.2. Let A, B, and be constrained by the conditions (2.1), (2.2), and (2.5). Let Hs : H(U) ! H(U) be an operator such 0 zHs (f )(z) that is analytic in U, with Hs (f )(z) 0

zHs (f )(z) Hs (f )(z)

=

+k+ ;

z=0

and satis…es 0

(2.11)

zHs (f )(z) = kHs+1 (f )(z) + mHs (f )(z);

for some k; m 2 C and for all f 2 H(U). Also, let de…ne the class Ts;m; (A; B) by Ts;m; (A; B) =

f 2 H(U) :

1

zH0s (f )(z) Hs (f )(z)

1 + Az 1 + Bz

(m + )

1: If f 2 Ts+1;m; (A; B), then (2.12) 1 zH0s (f )(z) 1 1 (m + ) q(z) = Hs (f )(z) M (z)

:

1 + Az ; 1 + Bz

where M is given by (2.4), and q is the best dominant of (2.12), hence Ts+1;m; (A; B) 2: Moreover, if then

is a real number,

Ts;m; (A; B): 1

B < 0, and

>

1

A , B

Hs (Ts+1;m; (A; B)) S ( ); (where S ( ) represents the class of starlike functions of order ) and Ts+1;m; (A; B)

Ts;m;

1

727

2(

Re m)

; 1 ;

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A GENERAL THEOREM ASSOCIATED WITH . . .

7

where =( + ) The constant

2 F1

(B A) 1; ; + B

+ 1;

1

B B

1

+ Re m:

cannot be replaced by a larger one.

Proof. If we let '(z) =

zH0s (f )(z) Hs (f )(z)

1

(m + ) ;

then ' 2 A, and by using the identity (2.11) we have kHs+1 (f )(z) = '(z) + : Hs (f )(z)

Calculating the logarithmic derivative of the above relation and using Theorem 2.1, we deduce immediately the above result. Remarks 2.1. i) Putting f (z) = z p +

1 P

t=p+1

at z t (p 2 N), k =

n,

m = n + p, = p , and = + n in Corollary 2.1, we get the result due to Patel and Cho [10]. 1 P zf 0 (z) ii) Putting f (z) = z p + at z t (p 2 N), '(z) = pf (z)1 g(z) t=p+1 p g 2 Sp , = , and = 0 in Theorem 2.1, we get the result due to

Patel [9].

iii) Putting f (z) = z +

1 P

at z t , Hs = Js;b (Srivastava-Attiya operator

t=2

see([16], [4])), k = b, m = 1 + b, = 1 , and = + b in Corollary 2.1, we get the result due to Kutbi and Attiya [3]. 1 P f 0 (z) iv) Putting f (z) = z p + at z t (p 2 N), '(z) = + (1 ) p 1, pz t=p+1 p = , and = p in Theorem 2.1, we get the result due to Patel and Cho [11]. v) Putting f (z) = z p +

1 P

t=p+1

at z t (p 2 N) and Hs = Tp (a; c) (see

[12]), k = + p, m = , =p , and = + in Corollary 2.2, we get the result due to Patel et al. [12]. 1 P vi) Putting f (z) = z p + at z t (p 2 N), Hs = Hl;m p (a1 ) (Dziokt=p+1

Srivastava operator), k = a1 , m = p a1 , = p , and = a1 + p in Corollary 2.2, we get the result due to Patel et al. [14]. 1 P vii) Putting f (z) = z p + at z t (p 2 N) and Hs = z ;p (see [13]), t=p+1

k= =p , and m = Patel and Mishra [13].

in Corollary 2.2, we get the result due to

728

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8

¼ ADEL A. ATTIYA AND TEODOR BULBOACA

Example 2.1. i) Cho et al. [2] (see also [8]) introduced the operator Ip (a; c) de…ned by Ip (a; c)(f )(z) = z p 2 F1 (c; + p; a; z) f (z) (a 2 R, 1 P c 2 R n Z0 , p; n 2 N, and f (z) = z p + at z t ). In [2] the authors t=p+n

showed that the operator Ip (a; c)(f ) satis…es the identity (2.9) with k = p a and m = a. Then we may use the Corollary 2.1 to generalize their result, by replacing the operator Hs with Ip (a; c). ii) Cho et al. [1] introduced the operator Ik;p (see[1]), where they proved that the operator Ik;p satis…es the identity (2.11) with k = + p and m = . In order to generalize their result, we can apply Corollary 2.2 by replacing the operator Hs with Ik;p .

References [1] N. E. Cho, I. H. Kim and H. M. Srivastava, Sandwich-type theorems for multivalent functions associated with the Srivastava–Attiya operator, Appl. Math. Comput., 217 (2010), no. 2, 918— 928. [2] N. E. Cho, O. S. Kwon and H. M. Srivastava, Inclusion relationships and argument properties for certain subclasses of multivalent functions associated with a family of linear operators, J. Math. Anal. Appl., 292 (2004), no. 2, 470–483. [3] M. A. Kutbi and A. A. Attiya, Di¤erential subordination result with the Srivastava–Attiya integral operator, J. Ineq. Appl., 2010 (2010), 1–10. [4] M.A. Kutbi and A.A. Attiya, Di¤erential subordination results for certain integrodi¤erential operator and it’s applications, Abs. Appl. Anal., 2012(2012), 13 pp. [5] N. N. Lebedev, Special Functions and Their Applications. Revised edition, translated from the Russian and edited by Richard A. Silverman. Unabridged and corrected republication, Dover Publications Inc., New York, 1972. [6] S. S. Miller and P. T. Mocanu, Univalent solutions of Briot–Bouquet di¤erential subordinations, J. Di¤er. Equations, 56 (1985), 297–309. [7] S. S. Miller and P. T. Mocanu, Di¤erential Subordinations. Theory and Applications, Series in Pure and Applied Mathematics, No. 225., Marcel Dekker, Inc., New York, 2000. [8] A. O. Mostafa and M. K. Aouf, Some applications of di¤erential subordination of p–valent functions associated with Cho–Kwon–Srivastava operator, Acta Math. Sinica, 25 (2009), no. 9, 1483–1496. [9] J. Patel, On certain subclass of p–valently Bazilevi´c functions., J. Inequal. Pure Appl. Math., 6 (2005), no. 1, Article 16, 1–13. [10] J. Patel and N. E. Cho, Some classes of analytic functions involving Noor integral operator. J. Math. Anal. Appl., 312 (2005), no. 2, 564–575. [11] J. Patel and N. E. Cho, On certain su¢ cient conditions for close-to-convexity, Appl. Math. Comput., 187 (2007), no. 1, 369–378. [12] J. Patel, N. E. Cho and H. M. Srivastava, Certain subclasses of multivalent functions associated with a family of linear operators, Math. Comput. Modelling, 43 (2006), 320–338. [13] J. Patel and K. Mishra, On certain subclasses of multivalent functions associated with the extended fractional di¤erintegral operator, J. Math. Anal. Appl., 332 (2007),109–122.

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A GENERAL THEOREM ASSOCIATED WITH . . .

9

[14] J. Patel, A. K. Mishra and H. M. Srivastava, Classes of multivalent analytic functions involving the Dziok–Srivastava operator, Comput. Math. Appl., 54 (2007), 599–616. [15] Ch. Pommerenke, Univalent Functions, Vanderhoeck and Ruprecht, Göttingen, 1975. [16] H. M. Srivastava and A. A. Attiya, An integral operator associated with the Hurwitz–Lerch Zeta function and di¤erential subordination, Integr. Transf. Spec. Funct., 18(2007), no. 3, 207–216. [17] D. R. Wilken and J. Feng, A remark on convex and starlike functions, J. London Math. Soc., (2)21 (1980), 287–290. Department of Mathematics, Faculty of Science, University of Mansoura, Mansoura 35516, Egypt Current address: Department of Mathematics, College of Science, University of Hail, Hail, Saudi Arabia E-mail address: [email protected] Faculty of Mathematics and Computer Science, BabeS¸-Bolyai University, 400084 Cluj-Napoca, Romania E-mail address: [email protected]

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1

An SVD Free Construction of an Indicator Function as an Imaging Algorithm K. Kim∗ , K. H. Leem† , and G. Pelekanos† , Abstract — A novel SVD free method is presented to reconstruct the shape of a sound-soft obstacle from the knowledge of the time harmonic incident electromagnetic wave and the far field pattern of the scattered wave. The approach presented is based on Kirsch’s factorization method and constructs a simple indicator function which is used to visualize the scattering profile. The method is baptized as Singular Value Decomposition Free Indicator (SVDFI) and its performance is evaluated by comparing our reconstructions with those obtained via Morozov’s discrepancy principle (MDP). Numerical results that illustrate the effectiveness of SVDFI on reconstruction problems involving both simulated and real data are reported and analyzed. Keywords: factorization method, Helmholtz equation, inverse problems, singular value decomposition.

1

Introduction

The linear sampling method introduced by Colton and Kirsch [4] is one of the major visualization algorithms for solving inverse obstacle scattering problems in the resonance region. The method creates a binary criterion for points from the grid to be inside the scatterer and is very fast since it reconstructs the obstacle directly from the given data without requiring the solution of a forward problem. One of its disadvantages is that it requires lots of data available since one theoretically needs to know the far field for all incident and observation directions on the unit sphere Ω. The linear sampling method states that the norm of the solution g of the far field equation F g = e−ikx···z tends to infinity as the sampling point z moves away from the obstacle. However, since the far field operator F is compact, the right-hand side is almost never in the range of F and the far field equation is not solvable in general. Therefore one has to resort in finding regularized solutions ∗

Department of Mathematics, Yeungnam University, 719-749, Gyeongsangbuk-do, South Korea, E-mail:[email protected] † Department of Mathematics and Statistics, Southern Illinois University, Edwardsville, IL 62026, USA. E-mails: [email protected], [email protected] ‡ This research was supported by the Yeungnam University research grants in 2009

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2 g. Kirsch in [5, 6], improved the method through a certain factorization of the far field operator F of the form (F ⋆ F )1/4 which replaced the far field equation with (F ⋆ F )1/4 g = e−ikx···z . Then he was able to prove that a point z is located in the interior of the object if and only if the right-hand side belongs to the range of the operator (F ⋆ F )1/4 . In this work we propose a new visualization technique, inspired in part from the Multiple Signal Classification (MUSIC) algorithm [2]. Our method constructs an indicator function I(z) without using singular value decomposition. The profile of the object is then reconstructed by noting that the indicator function gets large values for z inside the object and it gets smaller for z outside. Our method is fast and simple and does not really solve the far field equation with respect to g for several right hand sides. In addition, no choice of cut-off for the indicator function is needed and a priori knowledge of the noise level in the data is not required. We organize our paper as follows. Section 2 will be devoted to the formulation of the problem and a brief description of the factorization method. Subsequently, Section 3 will deal with the formulation of the SVDFI within the framework of the factorization method. In order to show the effectiveness of our method, in Section 4, we will present numerical examples for the case of impenetrable and penetrable scatterers and we will compare the reconstructions obtained via SVDFI with the ones obtained by means of the Morozov’s discrepancy principle (MDP). In our experiments we will use simulated data obtained by means of the Nystr¨ om method [3] as well as real data. The real data are made available by the Electromagnetics Technology Division, Sensors Directorate, Air Force Research Laboratory, Hanscom Air Force Base, Massachusetts and are known by the name of The Ipswich Data [7].

2

Formulation of the problem and the linear sampling method

It is well known that the propagation of time-harmonic acoustic fields in a homogeneous medium, in the presence of a sound soft obstacle D, is modeled by the exterior boundary value problem (direct obstacle scattering problem) ¯ △2 u(x) + k 2 u(x) = 0, x ∈ R2 \ D

(2.1)

u(x) + u (x) = 0, x ∈ ∂D

(2.2)

i

where k is a real positive wave number and ui is a given incident field, that in the presence of D will generate the scattered field u. In addition, the scattered field u

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3 will satisfy the Sommerfeld radiation condition √ ∂u r( − iku) = 0 r→∞ ∂r lim

(2.3)

¯ and the limit is taken uniformly for all directions x r = |x|, x ∈ R2 \ D, ˆ = x/|x|. The Green’s formula implies that the solution u of the direct obstacle scattering problem above has the asymptotic behavior [3] eikr u(x) = u∞ (ˆ x) √ + O(r−3/2 ) r for some analytic function u∞ , called the far- field pattern of u, given by ∫ −eiπ/4 ∂u √ u∞ (ˆ x) = (y)e−ikˆx·y ds(y) ∂n 8πk ∂D

(2.4)

(2.5)

for x ˆ = x/|x| on the unit sphere Ω. In the case of the inverse problem, it represents the measured data. In particular, the inverse problem that will be considered here, is the problem of finding the shape of D from a complete knowledge of the far-field pattern. We now define the far-field equation eiπ/4 −ikˆx·z e (F gz )(ˆ x) = Φ∞ (ˆ x, z) = √ , z ∈ R2 8πk

(2.6)

where Φ∞ (ˆ x, z) is the far-field pattern of the fundamental solution of the Helmholtz equation given by i (1) (2.7) Φ(x, z) = H0 (k|x − z|), x ̸= z 4 (1)

in which H0 is the Hankel function of order zero and of the first kind. Moreover F : L2 (Ω) → L2 (Ω) is given by ∫ ˆ g(d) ˆ ds(d), ˆ (F g)(ˆ x) = u∞ (ˆ x; d) dˆ ∈ Ω

(2.8)

Ω

It is well known that the first version of the linear sampling method [4] solves the linear operator equation (2.6) based on the numerical observation that its solution will have a large L2 − norm outside and close to ∂D. Hence, reconstructions are obtained by plotting the norm of the solution. However, the problem is that the right-hand side does not in general belong to the range of the operator F . Kirsch [5] was able to overcome this difficulty with the introduction of a new version of the linear sampling method based on appropriate factorization of the

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4

10 0.7

10 8

0.6

10

6 4

0.5

10

I(z)

2 0 −2

0.4

10

0.3

10

−4 0.2

10

−6 −8 −10 −10

0.1

10 −5

0

5

0

10

2000

4000

6000

8000

10000

z

(a)

(b)

Figure 1: (a) Original profile of a kite, (b) Indicator function I(z). far-field operator F . In this method, Kirsch is elegantly using the spectral properties of the operator F to characterize the obstacle. In particular, the following linear operator equation is now used in place of equation (2.6) eiπ/4 −ikˆx·z e (F ⋆ F )1/4 gz = √ 8πk

(2.9)

and the spectral properties of F are used for the reconstructions. To be more specific, since F is normal and compact, which guarantees the existence of a singular system {σjc , ucj , vjc }, j ∈ N, of F with vjc = sj ucj and sj ∈ C with |sj | = 1, then the characterization of the object depends on a range test as described in the following theorem due to Kirsch [5]. Theorem 2.1 For any z ∈ R2 assume that k 2 is not a Dirichlet eigenvalue of −∆2 in D i.e. the corresponding homogeneous problem has only the trivial solution. Then a point z ∈ R2 belongs to D if and only if the series ∞ ∑ |(Φ∞ , vjc )|2 j=1

σjc

(2.10)

x, z) belongs to the range of the opconverges, or equivalently, if and only if Φ∞ (ˆ erator (F ∗ F )1/4 .

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5

10 0.8

10 8 6

0.7

10

4 0.6

2 I(z)

10

0

0.5

10

−2 −4

0.4

10 −6 −8

0.3

10 −10 −10

−5

0

5

10

0

2000

4000

6000

8000

10000

z

(a)

(b)

Figure 2: (a) Original profile of two circles, (b) Indicator function I(z)

3

The new imaging algorithm

In this section we propose a new visualization algorithm that does not use the of gϵ (., z) as an indicator of the domain D. Our approach makes use of the finite dimensionality present in any numerical computation and avoids the necessity of using the SVD of the discretized far field operator. Here, the direct scattering problem is solved for N directions of incidence distributed uniformly on the unit circle, x ˆj = (cos(2πj/N ), sin(2πj/N )T , j = 0, · · · , N − 1, and the far field pattern of the scattered field is determined for those same N directions. We then use these data and a composite trapezoidal rule to approximate the far field operator F given by ( 2.8) by its discrete analogue FD : CN → CN for u ∈ CN , L2 -norm

N −1 2π ∑ (FD u)j = u∞ (xˆj ; xˆk ) uk j = 0 · · · , N − 1 N

(3.11)

k=0

Arens et al proved in [1], through the use of a semi-discrete operator and some arguments from perturbation theory, that the first few eigenvalues and eigenspaces of FD are good approximations to those of F . Using the discrete operator above for each z we form the following N ×N linear system of the form (FD⋆ FD )1/4 g(z) = b∞ (z) = (Φ∞ (ˆ x0 , z), · · · , Φ∞ (ˆ xN −1 , z))T

(3.12)

We need to emphasize here that we are not directly solving the system above for g, but our imaging algorithm is rather based on the following simple observation.

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6 If z ∈ D, then b∞ (z) ∈ R((FD⋆ FD )1/4 ) and hence b∞ (z) ∈ / N (((FD⋆ FD )1/4 )⋆ ), therefore the indicator function I(z) =

∥((FD⋆ FD )1/4 )⋆ b∞ (z)∥L2 ∥b∞ (z)∥L2

(3.13)

is larger for points inside the object and smaller for points outside. Figure 1 below

(a)

(b)

Figure 3: Reconstructions of a kite with 9% error, (a) via SVDFI, (b) via MDP. shows an example of a kite excited by 60 incident waves, located in a grid of 100 × 100 points. The measurement error is 9%. Therefore the far-field matrix of equation (3.12) is 60 × 60 and the right hand side is 60 × 10, 000. We observe that the indicator function I(z) is large when z is located inside the object and rather small for z outside. Similarly 2 shows two circles excited by 100 incident waves, located in a grid of 100 × 100 points. The measurement error is 13%. Notice that the indicator function now exhibits two picks due to the presence of two objects.

4

Numerical Results

In order to simulate perturbed data, we generate Gaussian random matrices E1 , E2 and use a far-field matrix defined by FeD = FD + δ(E1 + i E2 )∥Fq ∥2 , δ > 0, where δ is given and where FD is constructed by using the Nystr¨om method [3]. As indicated in [3], the Nystr¨om method not only requires less computational effort

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7 compared to collocation and Galerkin methods but it is also generically stable in the sense that it preserves the condition of the integral equation. We first report an image reconstruction experiment of a sound-soft obstacle. In this case the object

(a)

(b)

(c)

Figure 4: Reconstructions of two circles with 13% error, (a) via SVDFI, (b) via MDP with correct error level, (c) via MDP with underestimated error level. SVDFI

SVDFI

10

10

8

8

6

6

4

4

2

2

0

0

−2

−2

−4

−4

−6

−6

−8 −10 −10

−8 −5

0

5

10

−10 −10

(a)

−5

0

5

10

(b)

Figure 5: SVDFI reconstructions of, (a) the aluminum triangle and (b) the ”mystery” object. to be reconstructed is a kite located in a grid of 100 × 100 points, the far-field matrix FeD is 60 × 60 (i.e., we use 60 incident and observed directions), and the relative noise level in FeD is 1% (which implied a relative noise level in (FeD⋆ FeD )1/4 of approximately 9%). Reconstruction of the kite via our method is shown in Figures 3-(a). For comparison purposes, in 3-(b) we have included a reconstruction obtained through

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8 the factorization method with Morozov discrepancy principle (MDP). Even though the reconstructions are comparable, notice that the Morozov discrepancy principle requires a-priori knowledge of the noise level in the data whether our method does not. We now continue our numerical experiments by attempting to reconstruct two sound-soft obstacles. In this case the objects to be reconstructed are two circles located in a grid of 100 × 100, the far-field matrix FeD is 100 × 100, and the relative noise level in FeD is 2% (which implied a relative noise level in (FeD⋆ FeD )1/4 of 13%.). In this experiment we consider the MDP in two distinct circumstances: (i) when the exact error norm ∥(FeD⋆ FeD )1/4 − (FD⋆ FD )1/4 ∥2 = ϵ is used as input data and (ii), when the error norm is underestimated and set to 0.03ϵ. The results are depicted in Figure 4. We note that while both our method and MDP (case (i)) produce reasonable reconstructions, MDP (case (ii)) yields reconstruction of poor quality due to the lack of a prior information about the noise level, and we see that our method may outperform MDP if the noise level is not correctly estimated. We will now consider real data sets (The Ipswich Data) produced by using an echo-free chamber, a fixed transmitter and a receiver rotating around the scatterer. The incident and observation angles are 36 for both experiments. Initially we will attempt to reconstruct an aluminum triangle (IPS009) whose outer circle has radius equal to 6 cm and using SVDFI. Figure 5-(a) shows that SVDFI yields a good reconstruction. Consequently we attempt the reconstruction of the ”mystery” object given by the data set (IPS007), with a priori information that the object is penetrable and lies inside a circle of radius 7.5 cm. The reconstruction results appear in figure 5-(b) and as clearly indicated by the SVDFI the object was a circular tube, with a smaller one in its interior. It is worthwhile to mention here that reconstructions via the Morozov’s discrepancy principle are difficult to obtain for real data cases due to the fact that the level of error in the experimental far-field matrix is not available or difficult to estimate.

References [1] T. Arens, A. Lechleiter and D. R. Luke, MUSIC for Extended Scatterers as an Instance of the Factorization Method, SIAM J. Appl. Math., 70 (2009) 1283-1304. [2] M. Cheney, The linear sampling method and the MUSIC algorithm, Inverse Problems 17 (2001), 591-596. [3] D. Colton, R. Kress, Inverse Acoustic and Electromagnetic Scattering Theory, Springer-Verlag, New York, 1992.

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9 [4] D. Colton, A. Kirsch, A simple method for solving the inverse scattering problems in the resonance region, Inverse Problems 12 (1996) 383-393. [5] A. Kirsch, Characterization of the shape of a scattering obstacle using the spectral data of the far-field operator, Inverse Problems, 14 (1998) 1489-1512. [6] A. Kirsch, N. Grinberg, The Factorization Method for Inverse Problems, Oxford University Press, Oxford, 2008. [7] R. McGahan, R. Kleinman, Third annual special session on image reconstruction using real data, part 1. IEEE Antennas Propag. Mag. 41(1999) 34-36.

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Applications of coupled N -structures in BCC-algebras Young Bae Jun1 and Sun Shin Ahn2∗ 1

Department of Mathematics Education (and RINS), Gyeongsang National University, Chinju, 660-701, Korea 2

Department of Mathematics Education, Dongguk University, Seoul 100-715, Korea

Abstract. The notions of a N -subalgebra, a BCK-N -ideal, a (strong) BCC-N -ideal of BCC-algebras are introduced, and related properties are investigated. Characterizations of a coupled N -subalgebra, a coupled BCK-N -ideals, and a coupled (strong) BCC-N -ideals of BCC-algebras are given. Relations among a coupled N -subalgebra, a coupled BCK-N -ideal and a coupled (strong) BCC-N of BCC-algebras are discussed.

1. Introduction Imai and Is´eki introduced two classes of abstract algebras: BCK-algebras and BCI-algebras ([7], [8]). It is known that the class of BCK-algebras is a proper subclass of the class of BCIalgebras. The class of all BCK-algebras is a qusivariety. K. Is´eki posed an interesting problem (solved by A. Wro´ nski [12]) whether the class of BCK-algebras is a variety. In connection with this problem, Y. Komori [11] introduced a notion of BCC-algebra, and W. A. Dudek redefined the notion of BCC-algebra by using a dual form of the ordinary definition in the sense of Y. Kormori. In [6], W. A. Dudek and X. H. Zhang introduced a new notion of ideals in BCCalgebras and described connections between such ideals and congruences. Dudek et al. ([3]) considered the fuzzification of ideals in BCC-algebras. Jun et.al ([9]), introduced the notion of coupled N -structures and its applications in BCK/BCI-algebras were discussed. In this paper, we introduce the notions of a coupled N -subalgebra, a coupled BCK-N -ideal, and a coupled (strong) BCC-N -ideals of BCC-algebras are introduced, and related properties are investigated. Characterizations of a coupled N -subalgebras, a coupled BCK-N -ideals and a coupled (strong) BCC-N -ideals of BCC-algebras are given. Relations among a coupled N subalgebra, a coupled BCK-N -ideal and a coupled (strong) BCC-N -ideal of BCC-algebras are discussed. 2. Preliminaries

For an abstract algebra (X, ∗, 0) of type (2, 0), consider the following axioms: 0

2010 Mathematics Subject Classification: 06D72; 06F35; 03G25. Keywords: Coupled N -structure; Coupled N -subalgebra; Coupled BCK-N -ideal; Coupled (strong) BCCN -ideal. ∗ The corresponding author. 0 E-mail: [email protected] (Y. B. Jun); [email protected] (S. S. Ahn) 0

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(a1) (a2) (a3) (a4) (a5) (a6) (a7) (a8)

(∀x, y, z ∈ X) (((x ∗ y) ∗ (z ∗ y)) ∗ (x ∗ z) = 0) , (∀x ∈ X) (x ∗ x = 0) , (∀x ∈ X) (0 ∗ x = 0) , (∀x ∈ X) (x ∗ 0 = x) , (∀x, y ∈ X) (x ∗ y = 0, y ∗ x = 0 ⇒ x = y) , (∀x, y ∈ X) ((x ∗ (x ∗ y)) ∗ y = 0) , (∀x, y, z ∈ X) ((x ∗ y) ∗ z = (x ∗ z) ∗ y) , (∀x, y, z ∈ X) (((x ∗ y) ∗ (x ∗ z)) ∗ (z ∗ y) = 0) ,

If (X, ∗, 0) satisfies axioms (a1), (a2), (a4) and (a5), then we say (X, ∗, 0) is a weak BCCalgebra. By a BCC-algebra we mean a weak BCC-algebra X satisfying the axiom (a3). If (X, ∗, 0) satisfies axioms (a8), (a6), (a2) and (a5), then we say (X, ∗, 0) is a BCI-algebra. By a BCK-algebra we mean a BCI-algebra X satisfying the axiom (a3). On any BCC-algebra (similarly as in the case of BCK-algebras) one can define the natural order ≤ by putting (∀x, y ∈ X) (x ≤ y ⇔ x ∗ y = 0) .

(2.1)

It is not difficult to verify that this order is partial and 0 is its smallest element. Any BCK-algebra is a BCC-algebra, but there are BCC-algebras which are not BCK-algebras (see [1]). Note that a BCC-algebra X is a BCK-algebra if and only if it satisfies the axiom (a7). Any BCC-algebra (X, ∗, 0) satisfies the following conditions: (b1) (∀x, y ∈ X) (x ∗ y ≤ x) , (b2) (∀x, y, z ∈ X) ((x ∗ y) ∗ (z ∗ y) ≤ x ∗ z) , (b3) (∀x, y, z ∈ X) (x ≤ y ⇒ x ∗ z ≤ y ∗ z, z ∗ y ≤ z ∗ x) . A non-empty subset S of a BCC-algebra X is called a BCC-subalgebra(briefly, subalgebra) of X if x ∗ y ∈ S whenever x ∈ S and y ∈ S. Let I be a subset of X with 0 ∈ I. We say that I is a (c1) BCK-ideal of X (see [5]) if it satisfies: (∀x, y ∈ X) (y ∈ I, x ∗ y ∈ I ⇒ x ∈ I) .

(2.2)

(c2) BCC-ideal of X (see [5]) if it satisfies: (∀x, y, z ∈ X) (y ∈ I, (x ∗ y) ∗ z ∈ I ⇒ x ∗ z ∈ I) .

(2.3)

(c3) strong BCC-ideal of X (see [10]) if it satisfies: (∀x, y, z ∈ X) (y ∈ I, (x ∗ y) ∗ z ∈ I ⇒ x ∈ I) .

(2.4)

3. Coupled N -structures applied to subalgebras and ideals in BCC-algebras Definition 3.1.([9]) A coupled N -structure C in a nonempty set X is an object of the form C = {hx; fC , gC i : x ∈ X} where fC and gC are N -functions on X such that −1 ≤ fC (x) + gC (x) ≤ 0 for all x ∈ X. 741

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A coupled N -structure C = {hx; fC , gC i : x ∈ X} in X can be identified to an ordered pair (fC , gC ) in F(X, [−1, 0]) × F(X, [−1, 0]). For the sake of simplicity, we shall use the notation C = (fC , gC ) instead of C = {hx; fC , gC i : x ∈ X} . For a coupled N -structure C = (fC , gC ) in X and t, s ∈ [−1, 0] with t + s ≥ −1, the set N {(fC , gC ); (t, s)} = {x ∈ X | fC (x) ≤ t, gC (x) ≥ s} is called an N (t, s)-level set of C = (fC , gC ). An N (t, t)-level set of C = (fC , gC ) is called an N -level set of C = (fC , gC ). Definition 3.2.([9]) A coupled N -structure C = (fC , gC ) in a BCC-algebra X is called a coupled N -subalgebra of X if it satisfies: _ ^ fC (x ∗ y) ≤ {fC (x), fC (y)} and gC (x ∗ y) ≥ {gC (x), gC (y)} (3.1) for all x, y ∈ X. Lemma 3.3.([9]) Every coupled N -subalgebra C = (fC , gC ) of a BCC-algebra X satisfies fC (0) ≤ fC (x) and gC (0) ≥ gC (x) for all x ∈ X. Proposition 3.4. If every N -subalgebra C = (fC , gC ) of a BCC-algebra X satisfies the inequalities fC (x ∗ y) ≤ fC (y) and gC (x ∗ y) ≥ gC (y) for any x, y ∈ X, then fC and gC are constant functions. Proof. Let x ∈ X. Using (a4) and assumption, we have fC (x) = fC (x ∗ 0) ≤ fC (0) and gC (x) = gC (x ∗ 0) ≥ gC (0). It follows from Lemma 3.3 that fC (x) = fC (0) and gC (x) = gC (0). Hence fC and gC are constant functions.  Definition 3.5.([9]) A coupled N -structure C = (fC , gC ) in a BCC-algebra X is called a coupled BCK-N -ideal of X if it satisfies: (c81) fC (0) ≤ fC (x) and gC (0) ≥ gC (x), W V (c82) fC (x) ≤ {fC (x ∗ y), fC (y)} and gC (x) ≥ {gC (x ∗ y), gC (y)} , for all x, y ∈ X. Example 3.6. Let X = {0, a, b, c, d} be a BCC-algebra([2]), which is not a BCK/BCI-algebra, with the following Cayley table: ∗

0

a

b

c

d

0 a b c d

0 a b c d

0 0 b c c

0 a 0 a d

0 0 0 0 c

0 0 0 0 0

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(1) Let C = (fC , gC ) be a coupled N -structure in X defined by  C = h0; −0.8, −0.2i, ha; −0.6, −0.3i, hb; −0.6, −0.3i, hc; −0.1, −0.5i, hd; −0.1, −0.5i .

Then C = (fC , gC ) is a coupled N -subalgebra, but not a coupled BCK-N -ideal of X since _ fC (c) = −0.1  −0.6 = {fC (c ∗ b) = fC (a), fC (b)} and/or gC (c) = −0.5  −0.3 =

^

{gC (c ∗ b) = gC (a), gC (b)} .

(2) Let D = (fD , gD ) be a coupled N -structure in X defined by  D = h0; −0.7, −0.1i, ha; −0.7, −0.1i, hb; −0.5, −0.4i, hc; −0.5, −0.4i, hd; −0.5, −0.4i . It is easy to show that D = (fD , gD ) is both a coupled N -subalgebra and a coupled BCK-N -ideal of X. Proposition 3.7.([9]) Every coupled BCK-N -ideal of a BCC-algebra X satisfies the following assertions: W V (i) (∀x, y, z ∈ X)(x ∗ y ≤ z ⇒ fC (x) ≤ {fC (y), fC (z)} , gC (x) ≥ {gC (y), gC (z)}). (ii) (∀x, y ∈ X)(x ≤ y ⇒ fC (x) ≤ fC (y), gC (x) ≥ gC (y)). Definition 3.8. A coupled N -structure C = (fC , gC ) in a BCC-algebra X is called a coupled BCC-N -ideal of X if it satisfies (c81) and W V (c83) fC (x ∗ z) ≤ {fC ((x ∗ y) ∗ z), fC (y)} and gC (x ∗ z) ≥ {gC ((x ∗ y) ∗ z), gC (y)} , for all x, y ∈ X. Example 3.9. (1) Consider a BCC-algebra X = {0, a, , b, c, d} and a coupled N -structure D = (fD , gD ) as in Example 3.6(2). Then D = (fD , gD ) is both a coupled N -subalgebra of X and a coupled BCK-N -ideal of X, but not a coupled BCC-N -ideal of X since _ fD (d ∗ c) = fD (c) = −0.5  −0.7 = {fD ((d ∗ a) ∗ c) = fD (0), fD (a)} and/or gD (d ∗ c) = gD (c) = −0.4  −0.1 =

^

{gD ((d ∗ a) ∗ c) = gD (0), gD (a)} .

(2) Let X := {0, 1, 2, 3, 4, 5} be a BCC-algebra ([2]), which is not a BCK/BCI-algebra, with the following Cayley table: 743

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∗

0

1

2

3

4

5

0 1 2 3 4 5

0 1 2 3 4 5

0 0 2 2 4 5

0 0 0 1 4 5

0 0 0 0 4 5

0 0 0 1 0 5

5 1 1 1 1 0

Let C = (fC , gC ) be a coupled N -structure in X defined by  C= h0; −0.8, −0.2i, h1; −0.6, −0.3i, h2; −0.6, −0.3i, h3; −0.6, −0.3i, h4; −0.6, −0.3i, h5; −0.2, −0.5i . It is easy to check that C = (fC , gC ) is both a coupled N -subalgebra of X and a coupled BCCN -ideal of X. Theorem 3.10.([9]) For a coupled N -structure C = (fC , gC ) in a BCC-algebra X, the following are equivalent: (1) C = (fC , gC ) is a coupled N -ideal of X. (2) The nonempty N (t, s)-level set N {(fC , gC ); (t, s)} is a BCK-ideal of X for all t, s ∈ [−1, 0] with t + s ≥ −1. Theorem 3.11. For a coupled N -structure C = (fC , gC ) in a BCC-algebra X, the following are equivalent: (1) C = (fC , gC ) is a coupled BCC-N -ideal of X. (2) The nonempty N (t, s)-level set N {(fC , gC ); (t, s)} is a BCC-ideal of X for all t, s ∈ [−1, 0] with t + s ≥ −1. Proof. Assume that C = (fC , gC ) is a coupled BCC- N -ideal of X. Let t, s ∈ [−1, 0] be such that t + s ≥ −1. Obviously, 0 ∈ N {(fC , gC ); (t, s)}. Let x, y, z ∈ X be such that (x ∗ y) ∗ z, y ∈ N {(fC , gC ); (t, s))}. Then fC ((x ∗ y) ∗ z) ≤ t, fC (y) ≤ t and gC ((x ∗ y) ∗ z) ≥ s, gC (y) ≥ s. It follows W V from (c83) that fC (x∗z) ≤ {fC ((x∗y)∗z), fC (y)} ≤ t and gC (x∗z) ≥ {gC ((x∗y)∗z), gC (y)} ≥ s, which imply that x ∗ z ∈ N {(fC , gC ); (t, s)}. Hence the nonempty N (t, s)-level set of C = (fC , gC ) is a BCC-ideal of X for all t, s ∈ [−1, 0] with t + s ≥ −1. Conversely, suppose that the nonempty N (t, s)-level set of C = (fC , gC ) is a BCC-ideal of X for all t, s ∈ [−1, 0] with t + s ≥ −1. Since 0 ∈ N {(fC , gC ); (t, s)}, the condition (c81) is W valid. Assume that there exist a, b, c ∈ X such that fC (a ∗ c) > {fC ((a ∗ b) ∗ c), fC (b)} or V W gC (a ∗ c) < {gC ((a ∗ b) ∗ c), gC (b)}. For the case fC (a ∗ c) > {fC ((a ∗ b) ∗ c), fC (b)} and V W gC (a ∗ c) ≥ {gC ((a ∗ b) ∗ c), gC (b)}, there exist s0 , t0 ∈ [−1, 0) such that fC (a ∗ c) > t0 > {fC ((a ∗ V b) ∗ c), fC (b)} and s0 = {gC ((a ∗ b) ∗ c), gC (b)}. It follows that (a ∗ b) ∗ c, b ∈ N {(fC , gC ); (t0 , s0 )}, W but a∗c ∈ / N {(fC , gC ); (t0 , s0 )}. This is impossible. For the case fC (a∗c) ≥ {fC ((a∗b)∗c), fC (b)} V and gC (a ∗ c) < {gC ((a ∗ b) ∗ c), gC (b)}, there exist s0 , t0 ∈ [−1, 0) such that t0 = fC (a ∗ b) and 744

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V gC (a ∗ c) < s0 < {gC ((a ∗ b) ∗ c), gC (b)}. Then (a ∗ b) ∗ c, b ∈ N {(fC , gC ); (t0 , s0 )}, but a ∗ c ∈ / W N {(fC , gC ); (t0 , s0 )}. This is a contradiction. If fC (a ∗ c) > {fC ((a ∗ b) ∗ c), fC (b)} and gC (a ∗ c) < V {gC ((a ∗ b) ∗ c), gC (b)}, then (a ∗ b) ∗ c, b ∈ N {(fC , gC ); (t0 , s0 )}, but a ∗ c ∈ / N {(fC , gC ); (t0 , s0 )}, W V 1 1 where t0 := (fC (a ∗ c) + {fC ((a ∗ b) ∗ c), fC (b)}) and s0 := (gC (a ∗ c) + {gC ((a ∗ b) ∗ c), gC (b)}). 2 2 This is a contradiction. Therefore C = (fC , gC ) is a coupled BCC-N -ideal of X.  Proposition 3.12. Every coupled BCK-N -ideal C = (fC , gC ) of a BCC-algebra X is a coupled N -subalgebra of X. Proof. Let a coupled N -structure C = (fC , gC ) be a coupled N -ideal of a BCC-algebra X and let x, y ∈ X. Then _ _ _ fC (x ∗ y) ≤ {fC ((x ∗ y) ∗ x), fC (x)} = {fC (0), fC (x)} ≤ {fC (x), fC (y)} and gC (x ∗ y) ≥

^

{gC ((x ∗ y) ∗ x), gC (x)} =

^

{gC (0), gC (x)} ≥

^

{gC (x), gC (y)}.

Hence C = (fC , gC ) is a coupled N -subalgebra of X.



The converse of Proposition 3.12 may not be true in general (see Example 3.6(1)) as seen in the following example. Proposition 3.13. Every coupled BCC-N -ideal C = (fC , gC ) of a BCC-algebra X is a coupled BCK-N -ideal of X. Proof. Put z := 0 in (c83).



Corollary 3.14. Every coupled BCC-N -ideal of a BCC-algebra X is a coupled N -subalgebra of X. Proof. It is easily verified from Proposition 3.12 and Proposition 3.13.



Proposition 3.15. Let C = (fC , gC ) be a coupled BCC- N -ideal of a BCC-algebra X. Then the following hold: (i) (ii) (iii) (iv) (v)

If x ≤ y for any x, y ∈ X, then fC (x) ≤ fC (y), gC (x) ≥ gC (y). If fC (x ∗ y) = fC (0) for any x, y ∈ X, then fC (x) ≤ fC (y). If gC (x ∗ y) = gC (0) for any x, y ∈ X, then gC (x) ≥ gC (y). (∀x, y ∈ X)(fC (x ∗ y) ≤ fC (x), gC (x ∗ y) ≥ gC (x)). W V (∀x, y, z ∈ X)(fC (x∗(y∗z)) ≤ {fC (x), fC (y), fC (z)}, gC (x∗(y∗z)) ≥ {gC (x), gC (y), gC (z)}).

Proof. (i) It follows from Proposition 3.7 and Proposition 3.13. (ii) For any x, y ∈ X, we have _ _ fC (x) = fC (x ∗ 0) ≤ {fC ((x ∗ y) ∗ 0), fC (y)} = {fC (x ∗ y), fC (y)} _ = {fC (0), fC (y)} = fC (y). 745

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(iii) For any x, y ∈ X, we have gC (x) = gC (x ∗ 0) ≥

^

{gC ((x ∗ y) ∗ 0), gC (y)} =

=

^

{gC (0), gC (y)} = gC (y).

^ {gC (x ∗ y), gC (y)}

(iv) By (b1), we have x ∗ y ≤ x for any x, y ∈ X. Using (i), we obtain fC (x ∗ y) ≤ fC (x) and gC (x ∗ y) ≥ gC (x) for any x, y ∈ X. (v) For any x, y, z ∈ X, using Corollary 3.14 we have _ fC (x ∗ (y ∗ z)) ≤ {fC (x), fC (y ∗ z)} _ ≤ {fC (x), fC (y), fC (z)} and gC (x ∗ (y ∗ z)) ≥

^

{gC (x), gC (y ∗ z)}

≥

^

{gC (x), gC (y), gC (z)}. 

Theorem 3.16. In a BCK-algebra, every coupled BCK-N -ideal is a coupled BCC-N -ideal of X. Proof. Let C = (fC , gC ) be a coupled BCK-N -ideal of a BCC-algebra X. Using (c82) and (a7), we have _ fC (x ∗ z) ≤ {fC ((x ∗ z) ∗ y), fC (y)} _ = {fC ((x ∗ y) ∗ z), fC (y)} and gC (x ∗ z) ≥

^

{gC ((x ∗ z) ∗ y), gC (y)}

=

^

{gC ((x ∗ y) ∗ z), gC (y)}

for all x, y, z ∈ X. Hence (c83) holds. This completes the proof.



Definition 3.17. A coupled N -structure C = (fC , gC ) in a BCC-algebra X is called a coupled strong BCC-N -ideal of X if it satisfies (c81) and W V (c84) fC (x) ≤ {fC ((x ∗ y) ∗ z), fC (y)} and gC (x) ≥ {gC ((x ∗ y) ∗ z), gC (y)} , for all x, y ∈ X. Theorem 3.18. For a coupled N -structure C = (fC , gC ) in a BCC-algebra X, the following are equivalent: (1) C = (fC , gC ) is a coupled strong BCC-N -ideal of X. (2) The nonempty N (t, s)-level set N {(fC , gC ); (t, s)} is a strong BCC-ideal of X for all t, s ∈ [−1, 0] with t + s ≥ −1. 746

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Proof. Straightforward.



Theorem 3.19. Every coupled strong BCC-N -ideal of a BCC-algebra X is a coupled BCCN -ideal of X. Proof. Let C = (fC , gC ) be a coupled strong BCC-N -ideal of a BCC-algebra X. Then the nonempty N (t, s)-level set N {(fC , gC ); (t, s)} is a strong BCC-ideal of X for all t, s ∈ [−1, 0] with t + s ≥ −1 by Theorem 3.18. Let x, y, z ∈ X be such that y, (x ∗ y) ∗ z ∈ N {(fC , gC ); (t, s)}. Then x ∈ N {(fC , gC ); (t, s)} by (2.4). It follows from (b1) that ((x ∗ z) ∗ x) ∗ 0 = 0 ∈ N {(fC , gC ); (t, s)} which implies from (2.4) that x ∗ z ∈ N {(fC , gC ); (t, s)}. Hence N {(fC , gC ); (t, s)} is a BCC-ideal of X. By Theorem 3.11, C = (fC , gC ) is a coupled BCC-N -ideal of X.  The converse of Theorem 3.19 is not true in general as seen in the following example. Example 3.20 Consider a BCC-algebra X = {0, 1, 2, 3, 4, 5} and a coupled N -structure C = (fC , gC ) as in Example 3.9 (2). Then C = (fC , gC ) is a coupled BCC-N -ideal of X(see Example 3.9), but not a coupled strong BCC-N -ideal of X since _ fC (5) = −0.2  −0.6 = {fC ((5 ∗ 2) ∗ 5) = fC (0), fC (2)} and/or gC (5) = −0.5  −0.3 =

^

{gC ((5 ∗ 2) ∗ 5) = gC (0), gC (2)} .

Lemma 3.21.([4]) If a and b are non-zero distinct atoms of a BCC-algebra X, then a ∗ b = a. Theorem 3.22. Let X be a BCC-algebra in which every non-zero element is an atom. Then every coupled BCC-N -ideal of X is a coupled strong BCC-N -ideal of X. Proof. Assume that every non-zero element is an atom in a BCC-algebra X. Let C = (fC , gC ) be a coupled BCC-N -ideal of X. Let x, y, z ∈ X be such that y, (x ∗ y) ∗ z ∈ N {(fC , gC ); (t, s)}. Then x ∗ z ∈ N {(fC , gC ); (t, s)}. It follows from Lemma 3.21 that x = x ∗ z ∈ N {(fC , gC ); (t, s)}. Hence C = (fC , gC ) be a coupled BCC-N -ideal of X.  For any element a of a BCC-algebra X, let Xa := {x ∈ X | fC (x) ≤ fC (a), gC (x) ≥ gC (a)} . Obviously, Xa is a non-empty subset of X. Theorem 3.23. Let a be any element of a BCC-algebra X. If C = (fC , gC ) is a coupled (strong) BCK(BCC)-N -ideal of X, then the set Xa is a (strong) BCK(BCC)-ideal of X. Proof. Since fC (0) ≤ fC (x) and gC (0) ≥ gC (x) for any x ∈ X, we have 0 ∈ Xa . Let x, y ∈ X be such that x ∗ y ∈ Xa and y ∈ Xa . Then fC (x ∗ y) ≤ fC (a), gC (x ∗ y) ≥ gC (a), fC (y) ≤ fC (a) W and gC (y) ≥ gC (a). It follows from (c82) that fC (x) ≤ {fC (x ∗ y), fC (y)} ≤ fC (a) and gC (x) ≥ V {gC (x ∗ y), gC (y)} ≥ gC (a) so that x ∈ Xa . Therefore Xa is a BCK-ideal of X. Let x, y, z ∈ X be such that (x ∗ y) ∗ z ∈ Xa and y ∈ Xa . Then fC ((x ∗ y) ∗ z) ≤ fC (a), gC ((x ∗ W y) ∗ z) ≥ gC (a), fC (y) ≤ fC (a) and gC (y) ≥ gC (a). It follows from (c83) that fC (x ∗ z) ≤ {fC ((x ∗ 747

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V y) ∗ z), fC (y)} ≤ fC (a) and gC (x ∗ z) ≥ {gC ((x ∗ y) ∗ z), gC (y)} ≥ gC (a) so that x ∗ z ∈ Xa . Therefore Xa is a BCC-ideal of X. Let x, y, z ∈ X be such that (x ∗ y) ∗ z ∈ Xa and y ∈ Xa . Then fC ((x ∗ y) ∗ z) ≤ fC (a), gC ((x ∗ W y) ∗ z) ≥ gC (a), fC (y) ≤ fC (a) and gC (y) ≥ gC (a). It follows from (c84) that fC (x) ≤ {fC ((x ∗ V y) ∗ z), fC (y)} ≤ fC (a) and gC (x) ≥ {gC ((x ∗ y) ∗ z), gC (y)} ≥ gC (a) so that x ∈ Xa . Therefore Xa is a strong BCC-ideal of X.  Proposition 3.24. Let a be any element of a BCC-algebra X and let C = (fC , gC ) be a coupled N -structure in X. Then (i) If Xa is an ideal of X, then C = (fC , gC ) satisfies the following assertion:   W fC (x) ≥ {fC (y ∗ z), fC (z)} ⇒ fC (x) ≥ fC (y) V (∀x, y, z ∈ X) . gC (x) ≤ {gC (y ∗ z), gC (z)} ⇒ gC (x) ≤ gC (y)

(3.2)

(ii) If C = (fC , gC ) satisfies (3.2) and (∀x ∈ X) (fC (0) ≤ fC (x), gC (0) ≥ gC (x)) ,

(3.3)

then Xa is an ideal of X. Proof. (i) Assume that Xa is an ideal of X for all a ∈ X. Let x, y, z ∈ X be such that fC (x) ≥ W V {fC (y ∗ z), fC (z)} and gC (x) ≤ {gC (y ∗ z), gC (z)} . Then y ∗ z ∈ Xx and z ∈ Xx . Since Xx is an ideal of X, it follows that y ∈ Xx so that fC (y) ≤ fC (x) and gC (y) ≥ gC (x). (ii) Suppose that C = (fC , gC ) satisfies two conditions (3.2) and (3.3). Let x, y ∈ X be such that x ∗ y ∈ Xa and y ∈ Xa . Then fC (x ∗ y) ≤ fC (a), gC (x ∗ y) ≥ gC (a), fC (y) ≤ fC (a) and W V gC (y) ≥ gC (a). Hence fC (a) ≥ {fC (x ∗ y), fC (y)} and gC (a) ≤ {gC (x ∗ y), gC (y)} , which imply from (3.2) that fC (a) ≥ fC (x) and gC (a) ≤ gC (x). Thus x ∈ Xa . Obviously, 0 ∈ Xa . Therefore Xa is an ideal of X.  Theorem 3.25. Let C = (fC , gC ) be a coupled N -structure in a BCC-algebra X. Then Xa is a BCK-ideal of X for any a ∈ X if and only if (i) fC (0) ≤ fC (a), gC (0) ≥ gC (a)). (ii) (∀x, y ∈ X)(fC (x ∗ y) ≤ fC (a) and fC (y) ≤ fC (a) imply fC (x) ≤ fC (a)). (iii) (∀x, y ∈ X)(gC (x ∗ y) ≥ gC (a) and gC (y) ≥ gC (a) imply gC (x) ≥ gC (a)). Proof. Assume that Xa is a BCK-ideal of X. Then 0 ∈ Xa and so fC (0) ≤ fC (a) and gC (0) ≥ gC (a). Let x, y, z ∈ X be such that fC (x ∗ y) ≤ fC (a), gC (x ∗ y) ≥ gC (a), fC (y) ≤ fC (a), and gC (y) ≥ gC (a). Then x ∗ y, y ∈ Xa . Since Xa is a BCK-ideal of X, we have x ∈ Xa . Hence fC (x) ≤ fC (a) and gC (x) ≥ gC (a). Conversely, consider Xa for any a ∈ X. Obviously, 0 ∈ Xa for any a ∈ X. Assume that x ∗ y, y ∈ Xa . Then fC (x ∗ y) ≤ fC (a), gC (x ∗ y) ≥ gC (a), fC (y) ≤ fC (a), and gC (y) ≥ gC (a). It follows from hypothesis that fC (x) ≤ fC (a) and gC (x) ≥ gC (a). Hence x ∈ Xa . Thus Xa is a BCK-ideal of X. 748

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Note that any coupled N -subalgebra of a BCC-algebra may not be a coupled BCK-N -ideal (See Example 3.6(1)). We now provide a condition for a coupled N -subalgebra to be a coupled BCK-N -ideal of X. Theorem 3.26. If every non-zero element of a BCC-algebra X is an atom, then every coupled N -subalgebra of X is a coupled BCK-N -ideal of X. Proof. Assume that every non-zero element is an atom in a BCC-algebra X. Let C = (fC , gC ) be a coupled BCC-N -ideal of X. It follows from Lemma 3.3 that (c81) holds. Let x, y ∈ X be such that x ∗ y, y ∈ N {(fC , gC ); (t, s)}. Since x is an atom, (b1) implies x ∗ y = 0 or x ∗ y = x ∈ N {(fC , gC ); (t, s)}. If x ∗ y = 0, then x ≤ y gives x = 0 or x = y. Hence x ∈ N {(fC , gC ); (t, s)}. Thus C = (fC , gC ) is a coupled BCK-N -ideal of X.  References [1] W. A. Dudek, On proper BCI-algebras, Bull. Inst. MAth. Academia Sinica 20(1992) 137-150. [2] W. A. Dudek and Y. B. Jun, Fuzzy BCC-ideals in BCC-algebras, Math. Montisnigri 10(1999) 21-30. [3] W. A. Dudek, Y. B. Jun and Z. Stojakovi´c, On fuzzy BCC-ideals in BCC-algebras, Fuzzy Sets and Systems 123(2001) 251-258. [4] W. A. Dudek and X. H. Zhang, On atoms in BCC-algeberas, Discuss. Math. Gen. Algebra Appl. 15(1995) 82-85. [5] W. A. Dudek and X. H. Zhang, On ideals and congruences in BCC-algebras, Czec. Math. J. 48(1998) 21-29. [6] W. A. Dudek and X. H. Zhang, On ideals and fuzzy BCC-ideals in BCC-algebras, Fuzzy Sets and Systems 123(2001) 251-258. [7] K. Is´eki, On BCI-algebras, Math. Seminar Notes 8(1980) 125–130. [8] K. Is´eki and S. Tanaka, An introduction to the theory of BCK-algebras, Math. Jpn. 23(1978) 1-26. [9] Y. B. Jun, S. S. Ahn and D. R. Prince Williams, Coupled N -structures and its applications in BCK/BCIalgebras, Iranian J. Sci. Tech. 37(2013), 133-140. [10] B. Karamdin and S. A. Bhatti, Ideals and branches of BCC-algebras, East. Asian Math. J. 23(2007) 247-255. [11] Y. Komori, The class of BCC-algebras is not a variety, Math. Jpn. 29(1984) 391-394. [12] A. Wro´ nski, A BCK-algebras do not form a variety, Math. Jpn. 28(1983) 211-213.

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Analytical solution of nonlinear second-order periodic boundary value problem using reproducing kernel method Nabil Shawagfeha,1 , Omar Abu Arquba , Shaher Momanib,* a b

Department of Mathematics, Al-Balqa Applied University, Salt 19117, Jordan Department of Mathematics, The University of Jordan, Amman 11942, Jordan *Corresponding author: e-mail: [email protected] (Shaher Momani)

||||||||||||||||||||||||||||||||||||||||||||{ Abstract This paper investigates the numerical solution of nonlinear second-order periodic boundary value problems by using reproducing kernel Hilbert space method. The solution was calculated in the form of a convergent series in the space W23 with easily computable components. In the proposed method, the n-term approximation is obtained and is proved to converge to the analytical solution. Meanwhile, the error of the approximate solution is monotone decreasing in the sense of the norm of W23 . The proposed technique is applied to several examples to illustrate the accuracy, e ciency, and applicability of the method. The results reveal that the method is very e ective, straightforward, and simple. Keywords: periodic boundary value problems; Reproducing kernel Hilbert space method AMS Subject Classi cation: 34K28; 47B32; 34B15 ||||||||||||||||||||||||||||||||||||||||||||||||{

1

Introduction

Second-order boundary value problems (BVPs) for ordinary di erential equations arise very frequently in many branches of applied mathematics and physics such as atomic calculations, gas dynamics, nuclear physics, atomic structures, deformation of beams and plate de ection theory, chemical reactions, and so on [1{4]. In recent years, the nonlinear second-order periodic BVPs which are a combination of second-order ordinary di erential equations and periodic boundary conditions have been widely studied by many authors [5{8], due to a wide range of applications in applied mathematics, physics, and engineering, particularly in the homogenization of composite materials with a periodic microstructure [9,10]. In most cases, nonlinear second-order periodic BVPs do not always have solutions which we can obtain using analytical methods. In fact, many of real physical phenomena encountered, are almost impossible to solve by this technique, these problems must be attacked by various approximate and numerical methods. This paper discusses and investigates the analytical approximate solution using reproducing kernel Hilbert space (RKHS) method for nonlinear second-order BVP with periodic boundary conditions which is as follows: u00 (x) = F (x; u (x) ; u0 (x)) , 0

x

1;

(1)

subject to the periodic boundary conditions u (0) = u (1) ;

(2)

u0 (0) = u0 (1) ;

where u 2 W23 [0; 1] is an unknown function to be determined, F (x; y; z) is continuous term in W21 [0; 1] as y = y (x) ; z = z (x) 2 W23 [0; 1], 0 x 1, 1 < y; z < 1, and is depending on the problem discussed, and 1 3 W2 [0; 1] ; W2 [0; 1] are two reproducing kernel spaces. |||||||||||| 1 On sabbatical leave from Department of Mathematics, Faculty of Science, The University of Jordan, AmmanJordan. 750

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2

Recently, many authors have discussed the numerical solvability of periodic BVPs. To mention a few, the existence and multiplicity of positive solutions have been discussed to rst-order periodic BVPs as described in [11]. In [12] the authors have discussed the existence of nontrivial periodic solutions for second-order periodic BVPs. In [13] also, the author has provided the existence and multiplicity of positive solutions to further investigation to second-order periodic BVPs. Furthermore, the existence of solutions is carried out in [14] for third-order periodic BVPs. The existence of positive solution has been investigated to solve fourth-order periodic BVPs as presented in [15]. However, we assume that Eq. (1) subject to the periodic boundary conditions (2) has a unique solution on [0; 1]. But on the other aspects as well, the numerical solvability of di erential and integro-di erential equations of di erent types and orders can be found in [16{20] and references therein. Also, for numerical solvability of di erent categories of second-order BVPs one can consult the references [21{24]. Investigation about second-order periodic BVPs numerically is scarce. In this work, we utilize an a methodical way to solve these type of di erential equations. The new method is accurate, need less e ort to achieve the results, and is developed especially for nonlinear case. Meanwhile, the proposed method has an advantage that it is possible to pick any point in the interval [0; 1] and as well the approximate solutions and all its derivatives up to order two will be applicable. Reproducing kernel theory has important application in numerical analysis, di erential equations, integral equations, probability and statistics, and so fourth [25{27]. In the last years, extensive work has been done using RKHS method, which provides numerical approximations for linear and nonlinear equations. This method has been implemented in several operator, di erential, integral, and integro-di erential equations, such as nonlinear operator equations [28], nonlinear system of second-order BVPs [29], linear initial-boundary-value problems, [30], nonlinear second-order singular BVPs [31, 32], nonlinear partial di erential equations [33], nonlinear FredholmVolterra integral equation [34], nonlinear fourth-order integro-di erential equations [35, 36], nonlinear FredholmVolterra integro-di erential equations [37], and others. The rest of the paper is organized as follows: in the next section, several reproducing kernel spaces are described. In section 3 a linear operator, a complete normal orthogonal system, and some essential results are introduced. Also, a method for the existence of solutions for Eqs. (1) and (2) based on reproducing kernel space is described. In section 4, we give an iterative method to solve Eqs. (1) and (2) numerically in the space W23 [0; 1]. A numerical examples are presented in section 5. Section 6 ends this paper with a brief conclusion.

2

Several reproducing kernel spaces

In this section, two reproducing kernels needed are constructed in order to solve Eqs. (1) and (2) using RKHS method. Before the construction, we utilize the reproducing kernel concept. Throughout this paperR C the set of b complex numbers, the superscript (i) in u(i) (x) denotes the i-th derivative of u (x), L2 [a; b] = fu j a u2 (x) dx < 1 P 2 1g, and l2 = fA j (Ai ) < 1g. i=1

An abstract set is supposed to have elements, each of which has no structure, and is itself supposed to have no internal structure, except that the elements can be distinguished as equal or unequal, and to have no external structure except for the number of elements. De nition .1 [31] Let E be a nonempty abstract set. A function K : E Hilbert space H if

E ! C is a reproducing kernel of the

1. for each x 2 E, K ( ; x) 2 H. 2. for each x 2 E and ' 2 H, h' ( ) ; K ( ; x)i = ' (x). The condition (2) is called "the reproducing property": the value of the function ' at the point x is reproducing by the inner product of ' ( ) with K ( ; x). A Hilbert space which possesses a reproducing kernel is called a RKHS [31]. Next, we rst utilize the reproducing kernel space W23 [0; 1] in which every function satis es the periodic boundary conditions (2) and then construct the space W21 [0; 1].

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3

De nition .2 [32] The inner product space W23 [0; 1] is de ned as W23 [0; 1] = fu (x) j u; u0 ; u00 are absolutely continuous real-valued functions on [0; 1], u; u0 ; u00 ; u000 2 L2 [0; 1], and u (0) = u (1), u0 (0) = u0 (1)g. On the other hand, the inner product and norm in W23 [0; 1] are de ned, respectively, by hu(x); v(x)iW 3 = 2

and jjujjW 3 = 2

2 P

u

(i)

(0) v

i=0

(i)

(0) +

Z

1

u000 (x)v 000 (x)dx;

(3)

0

q hu; uiW 3 , where u; v 2 W23 [0; 1]. 2

The Hilbert space W23 [0; 1] is called a reproducing kernel if for each xed x 2 [0; 1], there exist R (x; y) 2 W23 [0; 1] (simply Rx (y)) such that hu (y) ; Rx (y)iW 3 = u (x) for any u(y) 2 W23 [0; 1] and y 2 [0; 1]. Next theorem utilize the 2 reproducing kernel function Rx (y) on the space W23 [0; 1]. Theorem .1 The Hilbert space W23 [0; 1] is a complete reproducing kernel and its reproducing kernel function Rx (y) can be written as ( a1 (x) + a2 (x)y + a3 (x)y 2 + a4 (x)y 3 + a5 (x)y 4 + a6 (x)y 5 ; y x; Rx (y) = (4) b1 (x) + b2 (x)y + b3 (x)y 2 + b4 (x)y 3 + b5 (x)y 4 + b6 (x)y 5 ; y > x; where ai (x) and bi (x), i = 1; 2; :::; 6; are unknown coe cients of Rx (y). Proof. The proof of the completeness and reproducing property of W23 [0; 1] is similar to the proof in [30]. Now, let us nd out the expression form of the reproducing kernel function Rx (y) in the space W23 [0; 1]. Through several 2 R1 R1 P i integration by parts, we obtain 0 u000 (y)@y3 Rx (y)dy = ( 1) u(i) (y) @y5 i Rx (y) jy=1 u (y) @y6 Rx (y) dy. Thus, y=0 0 i=0

from Eq. (3) we can write hu (y) ; Rx (y)iW 3

2 P

=

2

i=0

u(i) (0) [@yi Rx (0) + ( 1)i+1 @y5 i Rx (0)] +

Z

1

u (y) @y6 Rx (y) dy:

2 P

i=0

( 1)i u(i) (1) @y5 i Rx (1)

0

Since Rx (y) 2 W23 [0; 1], it follows that Rx (0) = Rx (1) and @y1 Rx (0) = @y1 Rx (1). Again, since u(x) 2 W23 [0; 1], it yield that u(i) (a) = u(i) (b), i = 0; 1. Hence, hu (y) ; Rx (y)iW 3 = 2

2 P ( 1)i u(i) (1) @y5 i Rx (1) u(i) (0) [@yi Rx (0) + ( 1)i+1 @y5 i Rx (0)] + i=0 Z i=0 1 6 u (y) @y Rx (y) dy + c1 (u(0) u(1)) + c2 (u0 (0) u0 (1)): 2 P

(5)

0

On the other hand, if @y3 Rx (1) = 0, Rx (0) @y5 Rx (0) + c1 = 0, @y2 Rx (0) @y3 Rx (0) = 0, @y5 Rx (1) c1 = 0, @y1 Rx (0) + @y4 Rx (0) + c2 = 0, and @y4 Rx (1) + c2 = 0, then Eq. (5) implies that hu (y) ; Rx (y)iW 3 = 2 R1 6 u (y) ( @ R (y))dy. Now, for any x 2 [0; 1], if R (y) satis es x y x 0 @y6 Rx (y) =

(x

y) ,

dirac-delta function,

(6)

then hu (y) ; Rx (y)iW 3 = u (x). Obviously, Rx (y) is the reproducing kernel function of the space W23 [0; 1]. 2 Next, we give the expression of the reproducing kernel function Rx (y). The auxiliary equation of Eq. (6) is given by 6 = 0, and their auxiliary values are = 0 with multiplicity 6. So, let the expression of the reproducing kernel function Rx (y) be as de ned in Eq. (4). But on the other aspect as well, for Eq. (6) let Rx (y) satisfy the equation @ym Rx (x + 0) = @ym Rx (x 0), m = 0; 1; 2; 3; 4. Integrating @y6 Rx (y) = (x y) from x " to x + " with respect to y and let " ! 0, we have the jump degree of @y5 Rx (y) at y = x given by @y5 Rx (x + 0) @y5 Rx (x 0) = 1. Through the last descriptions the unknown coe cients ai (x) and bi (x), i = 1; 2; :::; 6 of Eq. (4) can be obtained. This completes the proof. 752

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4

Remark .1 By using Mathematica 7:0 software package, the coe cients ai (x) and bi (x), i = 1; 2; :::; 6 of the reproducing kernel function Rx (y) in the space W23 [0; 1] are obtained and are given as follows: a1 (x)

=

a2 (x)

=

a3 (x)

=

a4 (x)

=

a5 (x)

=

a6 (x)

=

b1 (x)

=

b2 (x)

=

b3 (x)

=

b4 (x)

=

b5 (x)

=

b6 (x)

=

1; 1 x(27 60x 20x2 + 85x3 32x4 ); 3867 1 x(240 963x + 968x2 247x3 + 2x4 ); 15468 1 x(240 963x + 968x2 247x3 + 2x4 ); 46404 1 x( 1827 + 1482x + 494x2 166x3 + 17x4 ); 9208 1 (3867 3840x 60x2 20x3 + 85x4 32x5 ); 464040 1+

1 5 x ; 120

1 x(216 480x 160x2 609x3 256x4 ); 30936 1 x(240 963x 321x2 247x3 + 2x4 ); 15468 1 x(240 + 2904x + 968x2 247x3 + 2x4 ); 46404 1 x(2040 + 1482x + 494x2 166x3 + 17x4 ); 92808 1 x(3840 + 60x + 20x2 85x3 + 32x4 ): 464040

The reproducing kernel function Rx (y) posses some important properties such as: Rx (y) is symmetric, unique, and nonnegative for any xed x 2 [0; 1]. De nition .3 [33] The inner product space W21 [0; 1] is de ned as W21 [0; 1] = fu (x) j u is absolutely continuous real-valued function on [0; 1] and u0 2 L2 [0; 1]g. On the other hand, the inner product and norm in W21 [0; 1] are de ned, respectively, by hu(x); v(x)iW 1 = u (0) v (0) + 2

and jjujjW 1 2

Z

1

u0 (x)v 0 (x)dx;

0

q = hu; uiW 1 , where u; v 2 W21 [0; 1]. 2

Remark .2 In [33], it has been proved that the Hilbert space W21 [0; 1] is a complete reproducing kernel and its 1 + y; y x; reproducing kernel function is given by Gx (y) = 1 + x; y > x:

3

Structure representation of solution

In this section, the representation of the analytical solution of Eqs. (1) and (2) and the implementation method are given in the reproducing kernel space W23 [0; 1]. After that, we construct an orthogonal function system of the space W23 [0; 1] based on the Gram-Schmidt orthogonalization process. To do this, we de ne a di erential operator L as L : W23 [0; 1] ! W21 [0; 1] ; such that Lu (x) = u00 (x) : 753

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5

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As a result, Eqs. (1) and (2) can be converted into the equivalent form as follows: Lu (x) = F (x; u (x) ; u0 (x)) , 0 u (0)

x

1; (7)

u (1) = 0;

0

u (0)

0

u (1) = 0;

where u (x) 2 W23 [0; 1] and F (x; y; z) 2 W21 [0; 1] for y = y (x) ; z = z (x) 2 W23 [0; 1], 1 < y; z < 1, and 0 x 1. It is easy to show that L is a bounded linear operator from the space W23 [0; 1] into space W21 [0; 1]. Initially, we construct an orthogonal function system of W23 [0; 1]. To do so, put 'i (x) = Rxi (x) and i (x) = 1 Li ' (x), where fxi gi=1 is dense on [0; 1] and L is the adjoint operator of L. In terms of the properties of reproducing kernel Rx (y), one obtains hu (x) ; i (x)iW 3 = hu (x) ; L 'i (x)iW 3 = hLu (x) ; 'i (x)iW 1 = Lu(xi ), i = 1; 2; :::. 2

2

1

2

3 For the orthonormal function system i (x) i=1 of the space W2 [0; 1], it can be derived from the Gram-Schmidt 1 orthogonalization process of f i (x)gi=1 as follows:

i

(x) =

i P

ik

k

(x) ;

(8)

k=1

where

ik

are orthogonalization coe cients and are given by the following subroutine:

ij

=

ij

=

ij

=

1 , for i = j = 1; k 1k 1 , for i = j 6= 1; dik 1 iP1 cik kj , for i > j; dik k=j

such that dik =

s

k

2 ik

iP1

k=1

c2ik , cik =

k W3, 2

i;

and f

i

1

(x)gi=1 is the orthonormal system in the space

W23 [0; 1]. It is easy to see that, i (x) = L 'i (x) = hL 'i (x) ; Kx (y)iW 3 = h'i (x) ; Ly Kx (y)iW 1 = Ly Kx (y)jy=xi 2 2 2 W23 [0; 1]. Thus, i (x) can be written in the form i (x) = Ly Kx (y)jy=xi , where Ly indicates that the operator L applies to the function of y. 1

Theorem .2 If fxi gi=1 is dense on [0; 1], then f

i

1

(x)gi=1 is a complete function system of the space W23 [0; 1].

Proof. For each xed u (x) 2 W23 [0; 1], let hu (x) ; i (x)iW 3 = 0, i = 1; 2; :::. In other word, hu (x) ; i (x)iW 3 = 2 2 1 hu (x) ; L 'i (x)iW 3 = hLu (x) ; 'i (x)iW 1 = Lu (xi ) = 0. Note that fxi gi=1 is dense on [0; 1], therefore Lu (x) = 0. 2 2 It follows that u (x) = 0 from the existence of L 1 . So, the proof of the theorem is complete. Lemma .1 If u (x) 2 W23 [0; 1], then there exists a positive constant M such that i = 0; 1; 2, where jju (x)jjC = max ju(x)j.

u(i) (x)

C

M jju(x)jjW 3 , 2

0 x 1

Proof. For any x; y 2 [0; 1], we have u(i) (x) = u(y); @xi Rx (y) follows that @xi Rx (y)

W23

Mi , i = 0; 1; 2. Thus, u(i) (x) =

Mi ku(x)kW 3 , i = 0; 1; 2. Hence, u 2

(i)

(x)

C

W23

, i = 0; 1; 2. By the expression of Rx (y), it

u (x) ; @xi Rx (x)

W23

@xi Rx (x)

max fMi g jju(x)jjW 3 , i = 0; 1; 2.

i=0;1;2

W23

ku(x)kW 3 2

2

The internal structure of the following theorem is as follows: rstly, we will give the representation of the exact solution of Eqs. (1) and (2) in the space W23 [0; 1]. After that, the convergence of approximate solution un (x) to the analytic solution u (x) will be proved. Theorem .3 For each u in the space W23 [0; 1], the series

norm of

W23

[0; 1]. On the other hand, if

1 fxi gi=1

1 P

i=1

u (x) ;

i

(x)

i

(x) is convergent in the sense of the

is dense on [0; 1], then the following are hold: 754

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(i) the exact solution of Eq. (7) could be represented by u (x) =

i 1 P P

(xk ; u (xk ) ; u0 (xk ))

ik F

i=1 k=1

i

(x) :

(9)

(ii) the approximate solution of Eq. (7) un (x) =

n P i P

i=1 k=1

ik F

(xk ; u (xk ) ; u0 (xk ))

i

(x) ;

(10)

(i)

and un (x), i = 0; 1; 2 are converging uniformly to the exact solution u (x) and all its derivative as n ! 1, respectively. rst part, let u (x) be solution of Eq. (7) in the space W23 [0; 1]. Since u (x) 2 W23 [0; 1],

Proof. For the 1 P u (x) ; i (x)

i

i=1

(x) is the Fourier series expansion about normal orthogonal system 1 P

is the Hilbert space, then the series

u (x) ;

i

i=1

using Eq. (8), it easy to see that u (x)

=

1 P

u (x) ;

i

i=1

=

1 P i P

i=1 k=1

=

1 P i P

i=1 k=1

=

1 P i P

i=1 k=1

=

1 P i P

i=1 k=1

=

1 P i P

i=1 k=1

(x)

i

W23

(x)

i

1 , i=1

and W23 [0; 1]

2

(x)

hu (x) ;

ik

hu (x) ; L 'k (x)iW 3

(x)iW 3

ik

hLu (x) ; 'k (x)iW 1

ik

hF (x; u (x) ; u0 (x)) ; 'k (x)iW 1

i

2

(x) i

2

2

(x)

(x)

i

2

ik F

(x)

(x) is convergent in the sense of k kW 3 . On the other hand,

ik

k

i

(xk ; u (xk ) ; u0 (xk ))

i

i

(x)

(x) :

Therefore, the form of Eq. (9) is the exact solution of Eq. (7). For the second part, it easy to see that by Lemma .1, for any x 2 [0; 1] jun (x)

u (x)j =

hun (x)

u (x) ; Rx (x)iW 3 2

kRx (x)kW 3 kun (x) 2

M0 kun (x)

u (x)kW 3 2

u (x)kW 3 : 2

On the other hand, u(i) n (x)

u(i) (x)

=

D

un (x)

@xi Rx (x)

Mi kun (x)

E u (x) ; Rx(i) (x) W23

kun (x)

W23

u (x)kW 3 2

u (x)kW 3 , i = 1; 2: 2

where Mi , i = 0; 1; 2 are positive constants. Hence, if kun (x) (i) un

u (x)kW 3 ! 0 as n ! 1, the approximate solution 2

un (x) and (x), i = 0; 1; 2 are converge uniformly to the exact solution u (x) and all its derivative, respectively. So, the proof of the theorem is complete. Remark .3 We mention here that, the approximate solution un (x) in Eq. (10) can be obtained directly by taking nitely many terms in the series representation for u (x) of Eq. (9). 755

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4

Procedure of constructing iterative method

In this section, an iterative method of obtaining the solution of Eq. (5) is represented in the reproducing kernel space W23 [0; 1] for linear and nonlinear case. Initially, we will mention the following remark about the exact and approximate solutions of Eqs. (1) and (2). Remark .4 In order to apply the RKHS technique for solve Eqs. (1) and (2), we have the following two cases based on the structure of the function F . Case 1: if Eq. (1) is linear, that is F (x; u (x) ; u0 (x)) = p(x)u0 (x)+q(x)u(x)+r(x), then the exact and approximate solutions can be obtained directly from Eqs. (9) and (10), respectively. Case 2: if Eq. (1) is nonlinear, that is F (x; u (x) ; u0 (x)) is not a linear combination of u0 (x) and u(x), then in this case the exact and approximate solutions can be obtained by using the following iterative algorithm: Algorithm 1 According to Eq. (9), the representation of the solution of Eq. (1) can be denoted by u (x) =

1 P

Bi

i=1

where Bi =

i P

(x) ;

i

ik F

xk ; uk

(11)

1

k=1

(xk ) ; u0k

1

(xk ) . In fact, Bi , i = 1; 2; ::: in Eq. (11) are unknown, we will approxi-

mate Bi using known Ai . For a numerical computations, we de ne initial function u0 (x1 ) = 0, put u0 (x1 ) = u (x1 ), and de ne the n-term approximation to u (x) by n P

un (x) =

Ai

i=1

i

(x) ;

where the coe cients Ai of A1 =

11 F

(12)

i

(x), i = 1; 2; :::; n are given as

(x1 ; u0 (x1 ) ; u00 (x1 )) ;

u1 (x) = A1 1 (x) ; P2 A2 = k=1 2k F xk ; uk 1 (xk ) ; u0k 1 (xk ) ; P2 u2 (x) = i=1 Ai i (x) ; .. . Pn 1 un 1 (x) = i=1 Ai i (x) ; Pn An = k=1 nk F xk ; uk 1 (xk ) ; u0k 1 (xk ) :

(13)

Here, we note that: in the iterative process of Eq. (12), we can guarantee that the approximation un (x) satis es the periodic boundary conditions (2). Now, the approximate solution uN n (x) can be obtained by taking nitely many terms in the series representation of un (x) and uN n (x) =

N P i P

i=1 k=1

ik F

xk ; un

1

(xk ) ; u0n

1

(xk )

i

(x) :

(14)

Now, we will proof that un (x) in the iterative formula (12) is converge to the exact solution u (x) of Eq. (1), in fact this result is a fundamental in the RKHS theory and its applications. The next two lemmas are collected in order to prove the recent theorem. Lemma .2 If kun (x) u (x)kW 3 ! 0, xn ! y as n ! 1, and F (x; v; w) is continuous in [0; 1] with respect to 2 x; v; w for x 2 [0; 1] and v; w 2 ( 1; 1), then F (xn ; un 1 (xn ); u0n 1 (xn )) ! F (y; u (y) ; u0 (y)) as n ! 1. Proof. Since kun (x) u (x)kW 3 ! 0 as n ! 1, by Theorem .3 and Lemma .1, we know that un 1 (xn ) and 2 u0n 1 (xn ) are convergent uniformly to u (x) and u0 (x), respectively, as xn ! y and n ! 1. Hence, the continuity of F gives the result. 756

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Lemma .3 Lun (xj ) = Lu (xj ) = F xj ; uj

(xj ) ; u0j

1

Proof. The proof of Lun (xj ) = F xj ; uj 1 (xj ) ; u0j n n P P Lun (xj ) = Ai L i (xj ) = Ai L i (x) ; 'j (x) i=1

i=1

Using the orthogonality of j P

jl Lun (xl )

i

n P

=

1 , i=1

(x)

Ai

i (x) ;

i=1

l=1

n P

=

j P

jl

l

Ai

j P

jl F

n.

(xj ) will be obtained by induction as follows: if j n n P P = A (x) ; L ' (x) = Ai i (x) ; i 1 3 i j W W

n, then

2

2

i=1

i=1

j

(x)

W23

.

(x) W23

(x) ;

j

(x)

xl ; ul

1

(xl ) ; u0l

i

(xj ) as j

1

l=1

i=1

=

yields that

1

W23

= Aj 1

(xl ) :

l=1

Now, if j = 1, then Lun (x1 ) = F (x1 ; u0 (x1 ) ; u00 (x1 )). Again, if j = 2, then 21 Lun (x1 ) + 22 Lun (x2 ) = 0 0 0 21 F (x1 ; u0 (x1 ) ; u0 (x1 )) + 22 F (x2 ; u1 (x2 ) ; u1 (x2 )). Thus, Lun (x2 ) = F (x2 ; u1 (x2 ) ; u1 (x2 )). It is easy to see 0 that Lun (xj ) = F xj ; uj 1 (xj ) ; uj 1 (xj ) by using mathematical induction. On the other hand, from Theorem .3, un (x) converge uniformly to u (x). It follows that, on taking limits in Eq. 1 P (12), u (x) = Ai i (x). Therefore, un (x) = Pn u (x), where Pn is an orthogonal projector from the space W23 [0; 1] i=1

to Spanf

u (x) ; Pn

1; j

2 ; :::;

(x)

W23

n g.

Thus, Lun (xj ) = Lun (x) ; 'j (x)

= u (x) ;

j

(x)

W23

= Lu (x) ; 'j (x)

W21

W21

= un (x) ; Lj ' (x)

W23

= Pn u (x) ;

j

(x)

W23

=

= Lu (xj ).

1

Theorem .4 If jjun jjW 3 is bounded and fxi gi=1 is dense on [0; 1], then the n-term approximate solution un (x) 2 in the iterative formula (12) converges to the exact solution u (x) of Eq. (7) in the space W23 [0; 1] and u (x) = 1 P Ai i (x), where Ai is given by Eq. (13).

i=1

1

Proof. The proof consists of the following three steps: rstly, we will prove that the sequence fun gn=1 in Eq. 1 (12) is monotone increasing in the sense of k kW 3 . By Theorem .2, i i=1 is the complete orthonormal system 2 n n n P P P 2 2 Ai i (x) ; in the space W23 [0; 1] : Hence, we have kun kW 3 = hun (x) ; un (x)iW 3 = Ai i (x) = (Ai ) . 2

2

i=1

i=1

W23

i=1

Therefore, kun kW 3 is monotone increasing. 2 Secondly, we will prove the convergence of un (x). From Eq. (12), we have un+1 (x) = un (x) + An+1 n+1 (x). 1 2 2 2 2 2 From the orthogonality of i (x) i=1 , it follows that jjun+1 jjW23 = jjun jjW23 + (An+1 ) = jjun 1 jjW23 + (An ) + n+1 P 2 2 (An+1 )2 = ::: = jju0 jjW 3 + (Ai )2 . Since, the sequence kun kW 3 is monotone increasing. Due to the condition 2

2

i=1

that jjun jjW 3 is bounded, jjun jjW 3 is convergent as n ! 1. Then, there exists a constant c such that 2

2

c: It implies that Ai =

i P

ik F

xk ; uk

k=1

um

1 )?(um 1

jjum (x)

um

2 )?:::?(un+1 2

un (x)jjW 3 2

1

(xk ) ; u0k

1

1 P

(Ai )2 =

i=1

(xk ) 2 l2 ; i = 1; 2; :::. On the other hand, since (um

un ) it follows for m > n that

= jjum (x) = jjum (x)

um

1 (x)

+ um

2 um 1 (x)jjW 3 2

1 (x)

::: + un+1 (x)

+ ::: + jjun+1 (x)

2

2

un (x)jjW 3

2 un (x)jjW 3 2

2

: 2

Furthermore, jjum (x) um 1 (x)jjW 3 = (Am )2 . Consequently, as n; m ! 1; we have jjum (x) un (x)jjW 3 = 2 2 m P 2 3 3 (Ai ) ! 0. Considering the completeness of W2 [0; 1], there exists a u (x) 2 W2 [0; 1] such that un (x) ! u(x) i=n+1

as n ! 1 in the sense of jj jjW 3 . 2 1 Thirdly, we will prove that u (x) is the solutions of Eq. (7). Since fxi gi=1 is dense on [0; 1], for any x 2 [0; 1], 1 there exists subsequence xnj j=1 , such that xnj ! x as j ! 1. From Lemma .3, It is clear that Lu xnj = 757

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F xnj ; unj

1

(xk ) ; u0nj

1

0

9

(xk ) . Hence, let j ! 1, by lemma .2 and the continuity of F , we have Lu (x) =

(x) 2 W23 [0; 1], clearly, u (x) satis es the periodic 1 P boundary conditions (2). In other words, u (x) is the solution of Eqs. (1) and (2), where u (x) = Ai i (x) and F (x; u (x) ; u (x)). That is, u (x) satis es Eq. (1). Also, since

i

i=1

Ai are given by Eq. (13). The proof is complete.

It obvious that, if we let u (x) denote the exact solution of Eq. (7), un (x) denote the approximate solution obtained by the RKHS method as given by Eq. (12), and rn (x) is the di erence between un (x) and u (x), where x 2 2 1 1 1 P P P 2 2 2 2 2 = (Ai ) and jjrn 1 (x)jjW 3 = (Ai ) Ai i (x) [0; 1], then jjrn (x)jjW 5 = jju (x) un (x)jjW 3 = 2

or jjrn (x)jjW 3 2 sense of k kW 3 .

jjrn

2

1

i=n+1

W23

2

i=n+1

i=n

(x)jjW 3 . Consequently, this show that the di erence rn (x) is monotone decreasing in the 2

2

5

Numerical outcomes

In this section, we propose few numerical simulations implemented by Mathematica 7:0 software package for solving some speci c examples of Eqs. (1) and (2). However, we apply the algorithm described in the previous sections to some linear and nonlinear test examples in order to demonstrate the e ciency, accuracy, and applicability of the proposed method. Results obtained by the method are compared with the analytical solution of each example by computing the exact and relative errors and are found to be in good agreement with each other. Example 1 Consider the following linear nonhomogeneous equation: u00 (x) + u (x) = f (x) , 0

x

1;

subject to the periodic boundary conditions u (0) = u (1) ; u0 (0) = u0 (1); where f (x) = (4x4

8x3 + 4x2 + 3)(1

2

2x)2 ex

(x 1)2

2

. The exact solution is u (x) = ex

(x 1)2

.

Using RKHS method, taking xi = ni 11 , i = 1; 2; :::; n with the reproducing kernel function Rx (y) on [0; 1], the approximate solution un (x) is calculated by Eq. (10). The numerical results at some selected grid points for n = 26 are given in Table 1. Table 1. Numerical results for Example 1. x Exact solution Approximate solution 0:16 1:0182274892397234 1:0182295775058678 0:32 1:0484886643504874 1:0484907460220763 0:48 1:0642817515996124 1:0642838267814154 0:64 1:0545183896526067 1:0545204690600505 0:80 1:0259304941903820 1:0259325816419569 0:96 1:0014756476981566 1:0014777346809174

Absolute 2:08827 2:08167 2:07518 2:07941 2:08745 2:08698

error 10 6 10 6 10 6 10 6 10 6 10 6

Relative 2:05088 1:98540 1:94984 1:97190 2:03469 2:08391

error 10 6 10 6 10 6 10 6 10 6 10 6

Example 2 Consider the following nonlinear nonhomogeneous equation: u00 (x) + 2u(x) +

1 = f (x) , 0 1 + (u(x))2

x

1;

subject to the periodic boundary conditions u (0) = u (1) ; u0 (0) = u0 (1); 758

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where f (x) =

x4 (x

1 + 8x3 1)4 + 1

8x + 2. The exact solution is u (x) = x4

10

2x3 + x2 .

Using RKHS method, taking xi = Ni 11 , i = 1; 2; :::; N with the reproducing kernel function Rx (y) on [0; 1], the approximate solution uN n (x) is calculated by Eq. (14). The numerical results at some selected grid points for N = 51 and n = 3 are given in Table 2. Table 2. Numerical results for Example 2. x Exact solution Approximate solution 0:16 0:01806336 0:01806360646957179 0:32 0:04734976 0:04734996786169693 0:48 0:06230016 0:06230018632258992 0:64 0:05308416 0:05308398923694464 0:80 0:02560000 0:02559974238385324 0:96 0:00147456 0:00147446292690514

Absolute 2:46470 2:07862 2:63226 1:70763 2:57616 9:70731

error 10 7 10 7 10 8 10 7 10 7 10 8

Relative 1:36447 4:38992 4:22512 3:21684 1:00631 6:58319

error 10 5 10 6 10 7 10 6 10 5 10 5

Example 3 Consider the following nonlinear nonhomogeneous equation: u00 (x) + u0 (x)

2

(2x

1) u (x) + cosh

1

(u (x)) = f (x) , 0

x

1;

subject to the periodic boundary conditions u (0) = u (1) ; u0 (0) = u0 (1); where f (x) = (1 + 2x) sinh (x (x

1)) + x

x2 . The exact solution is u (x) = cosh x2

x .

Using RKHS method, taking xi = Ni 11 , i = 1; 2; :::; N with the reproducing kernel function Rx (y) on [0; 1], the approximate solution uN n (x) is calculated by Eq. (14). The numerical results at some selected grid points for N = 51 and n = 3 are given in Table 3. Table 3. Numerical results for Example 3. x Exact solution Approximate solution 0:16 1:0090452833957488 1:0090454991505815 0:32 1:0237684442237780 1:0237671617164725 0:48 1:0313121374632044 1:0313104084921125 0:64 1:0266597016257117 1:0266584708541906 0:80 1:0128273299790107 1:0128274421174757 0:96 1:0007373706014195 1:0007395991224600

Absolute 2:15755 1:28251 1:72897 1:23077 1:12138 2:22852

error 10 7 10 6 10 6 10 6 10 7 10 6

Relative 2:13821 1:25273 1:67648 1:19881 1:10718 2:22688

error 10 7 10 6 10 6 10 6 10 7 10 6

Example 4 Consider the following nonlinear nonhomogeneous equation: 2

u00 (x) + (u0 (x)) + e

u(x)

= f (x) , 0

x

1;

subject to the periodic boundary conditions u (0) = u (1) ; u0 (0) = u0 (1); where f (x) =

3 (2x x2 (1

2

1) 2

x) + 1

. The exact solution is u (x) = ln x4

2x3 + x2 + 1 .

Using RKHS method, taking xi = Ni 11 , i = 1; 2; :::; N with the reproducing kernel function Rx (y) on [0; 1], the approximate solution uN n (x) is calculated by Eq. (14). The numerical results at some selected grid points for N = 51 and n = 3 are given in Table 4. 759

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Table 4. Numerical results for Example 4. x Exact solution Approximate solution 0:16 0:017902155877180255 0:017902119683171697 0:32 0:046262935319852880 0:046262940200774260 0:48 0:060436519420405940 0:060436573149274840 0:64 0:051723153984674340 0:051723175796916536 0:80 0:025277807184268607 0:025277772899866330 0:96 0:001473473903947873 0:001473467202387802

Absolute 3:61940 4:88092 5:37289 2:18122 3:42844 6:70156

error 10 8 10 9 10 7 10 8 10 8 10 9

Relative 2:02177 1:05504 8:89013 4:21711 1:35630 4:54814

11

error 10 6 10 7 10 7 10 7 10 6 10 6

As we mentioned earlier, it is possible to pick any point in [0; 1] and as well the approximate solutions and all its derivative up to order two will be applicable. Next, new numerical results for Example 4 which include the absolute error at some selected gird nodes in [0; 1] for u(i) (x), i = 0; 1; 2, where xi = Ni 11 , i = 1; 2; :::; N , N = 51, and n = 3 are given in Table 5. Table 5. Absolute error of u(i) (x), i = 0; 1; 2 for Example 4. i x = 0:16 x = 0:48 x = 0:64 0 3:61940 10 8 5:37289 10 7 2:18122 10 7 6 1 2:32325 10 1:95001 10 5:04347 10 2 9:12947 10 6 9:88580 10 6 5:81507 10

6

8 7 6

x = 0:96 6:70156 10 4:52410 10 3:72029 10

9 8 7

Conclusions

The main concern of this work has been to propose an e cient algorithm for the solutions of second-order periodic BVPs. The main goal has been achieved by introducing the RKHS method to solve this class of di erential equations. We can conclude that the RKHS method is powerful and e cient technique in nding approximate solution for linear and nonlinear second-order periodic BVPs. In the proposed algorithm, the solution u (x) and the approximate solution un (x) are represented in the form of series in W23 . Moreover, the approximate solution and all its derivatives converge uniformly to the exact solution and all its derivatives up to order two, respectively. There is an important point to make here, the results obtained by the RKHS method are very e ective and convenient in linear and nonlinear cases with less computational, iteration steps, work, and time. This con rms our belief that the e ciency of our technique gives it much wider applicability in the future for general classes of linear and nonlinear periodic problems. Acknowledgments The rst author would like to thank the University of Jordan for the nancial support during the preparation of this article.

References [1] S. Chandrasekhar, Hydrodynamic and Hydromagnetic Stability, Dover, New York, 1981. [2] W. Soedel, Vibrations of Shells and Plates, Dekker, New York, 1993. [3] T.Y. Na, Computational Methods in Engineering Boundary Value Problems, Academic Press, New York, 1979. [4] E. Dulacska, Soil Settlement E ects on Buildings (Developments in Geotechnical Engineering), Elsevier Science Ltd., Amsterdam, 1992. [5] F.M. Atici, G.S. Guseinov, On the existence of positive solutions for nonlinear di erential equations with periodic boundary conditions, Journal of Computational and Applied Mathematics 132 (2001) 341-356. [6] F. Li, Z. Liang, Existence of positive periodic solutions to nonlinear second-order di erential equations, Applied Mathematics Letters 18 (2005) 1256-1264. 760

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[7] R.P. Agarwal, M.E. Filippakis, D. O'Regan, N.S. Papageorgiou, Degree theoretic methods in the study of nonlinear periodic problems with nonsmooth potentials, Di erential and Integral Equations 19 (2006) 279-296. [8] V. Seda, J.J. Nieto, M. Gera, Periodic boundary value problem for nonlinear higher ordinary di erential equations, Applied Mathematics and Computation 48 (1992) 71-82. [9] A. Bensoussan, J.L. Lions, G. Papanicolaou, Asymptotic Analysis for Periodic Structures, North-Holland, Amsterdam, 1978. [10] E.S. Palencia, Non-Homogeneous Media and Vibration Theory (Lecture Notes in Physics), Springer-Verlag, Berlin, 1980. [11] S. Peng, Positive solutions for rst order periodic boundary value problem, Applied Mathematics and Computation 158 (2004) 345-351. [12] B. Liu, L. Liu, Y. Wuc, Existence of nontrivial periodic solutions for a nonlinear second order periodic boundary value problem, Nonlinear Analysis 72 (2010) 3337-3345. [13] Q. Yao, Positive solutions of nonlinear second-order periodic boundary value problems, Applied Mathematics Letters 20 (2007) 583-590. [14] H. Yu, M. Pei, Solvability of a nonlinear third-order periodic boundary value problem, Applied Mathematics Letters 23 (2010) 892-896. [15] Y. Lia, Positive solutions of fourth-order periodic boundary value problems, Nonlinear Analysis 54 (2003) 1069-1078. [16] O. Abu Arqub, Series solution of fuzzy di erential equations under strongly generalized di erentiability, Journal of Advanced Research in Applied Mathematics 5 (2013) 31-52. [17] O. Abu Arqub, A. El-Ajou, S. Momani, N. Shawagfeh, Analytical solutions of fuzzy initial value problems by HAM, Applied Mathematics and Information Sciences, in press. [18] O. Abu Arqub, A. El-Ajou, Solution of the fractional epidemic model by homotopy analysis method, Journal of King Saud University (Science) 25 (2013) 73-81. [19] A. El-Ajou, O .Abu Arqup, Solving fractional two-point boundary value problems using continuous analytic method, Ain Shams Engineering Journal, in press. [20] A. El-Ajou, O. Abu Arqub, S. Momani, Homotopy analysis method for second-order boundary value problems of integro-di erential equations, Discrete Dynamics in Nature and Society, vol. 2012, Article ID 365792, 18 pages, 2012. doi:10.1155/2012/365792. [21] O. Abu Arqub, Z. Abo-Hammour, S. Momani, Application of continuous genetic algorithm for nonlinear system of second-order boundary value problems, Applied Mathematics and Information Sciences. In press. [22] O. Abu Arqub, Z. Abo-Hammour, S. Momani, N. Shawagfeh, Solving singular two-point boundary value problems using continuous genetic algorithm, Abstract and Applied Analysis, vol. 2012, Article ID 205391, 25 page, 2012, doi.10.1155/2012/205391. [23] G. Adomian, R. Rach, N. Shawagfeh, On the analytic solution of Lane-Emden equation, Foundations of Physics Letters 8 (1995) 161-181. [24] N. Shawagfeh, Nonperturbative approximate solution for Lane-Emden equation, Journal of Mathematical Physics 34 (1993) 4364-4369. [25] A. Berlinet, C.T. Agnan, Reproducing Kernel Hilbert Space in Probability and Statistics, Kluwer Academic Publishers, 2004. 761

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[26] M. Cui, Y. Lin, Nonlinear Numercial Analysis in the Reproducing Kernel Space, Nova Science Publisher, New York, 2008. [27] A. Daniel, Reproducing Kernel Spaces and Applications, Springer, 2003. [28] C. Li, M. Cui, The exact solution for solving a class nonlinear operator equations in the reproducing kernel space, Applied Mathematics and Computation 143 (2003) 393-399. [29] F. Geng, M. Cui, Solving a nonlinear system of second order boundary value problems, Journal of Mathematical Analysis and Applications 327 (2007) 1167-1181. [30] L.H. Yang, Y. Lin, Reproducing kernel methods for solving linear initial-boundary-value problems, Electronic Journal of Di erential Equations 2008 (2008) 1-11. [31] F. Geng, Solving singular second order three-point boundary value problems using reproducing kernel Hilbert space method, Applied Mathematics and Computation 215 (2009) 2095-2102. [32] W. Wang, M. Cui, B. Han, A new method for solving a class of singular two-point boundary value problems, Applied Mathematics and Computation 206 (2008) 721-727. [33] Y. Lin, M. Cui, L. Yang, Representation of the exact solution for a kind of nonlinear partial di erential equations, Applied Mathematics Letters 19 (2006) 808-813. [34] M. Cui, H. Du, Representation of exact solution for the nonlinear Volterra-Fredholm integral equations, Applied Mathematics and Computation 182 (2006) 1795-1802. [35] M. Al-Smadi, O. Abu Arqub, N. Shawagfeh, Approximate solution of BVPs for 4th-order IDEs by using RKHS method, Applied Mathematical Sciences 6 (2012) 2453-2464. [36] M. Al-Smadi, O. Abu Arqub, S. Momani, A computational method for two-point boundary value problems o ourth-order mixed integro-di erential equations, Mathematical Problems in Engineering, In press [37] O. Abu Arqub, M. Al-Smadi, S. Momani, Application of reproducing kernel method for solving nonlinear Fredholm-Volterra integro-di erential equations, Abstract and Applied Analysis, vol. 2012, Article ID 839836, 16 pages, 2012. doi:10.1155/2012/839836.

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J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 16, NO.4, 2014, COPYRIGHT 2014 EUDOXUS PRESS, LLC

GENERALIZED CHEBYSHEV INEQUALITIES WITH APPLICATIONS

Marwan A Kutbi Department of Mathematics, King Abdulaziz University, P.O. Box 80203, Jeddah 21589, Saudi Arabia [email protected]

Nawab Hussain Department of Mathematics, King Abdulaziz University, P.O. Box 80203, Jeddah 21589, Saudi Arabia [email protected]

Arif Ra…q Department of Mathematics, Lahore Leads University, Lahore, Pakistan aara…[email protected]

Mohammad Masjed-Jamei Department of Mathematics, K. N. Toosi, University of Technology, P.O. Box: 16315-1618, Tehran, Iran [email protected]

Abstract In this paper, we generalize the Pecaric work on Montgomery’s identity via an arbitrary weight function, which no longer needs to be a probability density function, and apply it to derive some generalized Chebyshev type inequalities for any absolutely continuous function. Key words: Chebyshev inequality, Absolutely continuous functions, Peano kernel, Montgomery identity, Weight function, Optimal constants MSC (2010): 26D15; 26D10

Preprint submitted to Elsevier Science

763

5 September 2013

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J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 16, NO.4, 2014, COPYRIGHT 2014 EUDOXUS PRESS, LLC

Introduction Let Lp [a; b] (1 p 1) denote the space of p-power integrable functions on the interval [a; b] with the standard norm 1 p1 0 b Z kf kp = @ jf (t)jp dtA ; a

and L1 [a; b] the space of all essentially bounded functions on [a; b] with the norm kf k1 = sup jf (t)j : t2[a;b]

For two absolutely continuous functions f; g : [a; b] ! R and the positive function w : [a; b] ! R+ such that wf; wg; wf g 2 L1 [a; b], the weighted Chebyshev functional [10] is de…ned by

T (w; f; g) =

Zb

0 b 10 b 1 Z Z @ w(t)f (t)dtA @ w(t)g(t)dtA :

w(t)f (t)g(t)dt

a

a

(1.1)

a

If w(t) is uniformly distributed on [a; b] then (1.1) is reduced to the usual Chebyshev functional

b 1 Z f (t)g(t)dt T (f; g) = b aa

0

10

1

Zb Zb 1 1 @ f (t)dtA @ g(t)dtA : b aa b aa

(1.2)

To date, extensive research has been done on the bounds of Chebyshev functional, see e.g. [1, 3]. The …rst work dates back to 1882, when Chebyshev [2] proved that if f 0 ; g 0 2 L1 [a; b] then jT (f; g)j

1 (b 12

a)2 kf 0 k1 kg 0 k1 :

(1.3)

Later on, in 1934 Grüss [4] showed that 1 (M1 m1 )(M2 m2 ); (1.4) 4 where m1 ; m2 ; M1 and M2 and are real numbers satisfying the conditions jT (f; g)j

m1

f (t)

M1 and m2

g(t) 2

764

M2 for all t 2 [a; b]:

(1.5)

KUTBI ET AL 763-776

J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 16, NO.4, 2014, COPYRIGHT 2014 EUDOXUS PRESS, LLC

The optimal constant 41 is the best possible number in (1.4) in the sense that it cannot be replaced by a smaller quantity. A mixture type of inequalities (1.3) and (1.4) was introduced in [10] as 1 (b a)(M1 m1 ) kg 0 k1 ; (1.6) 8 in which f is a Lebesgue integrable function satisfying (1.5) and g is absolutely continuous so that g 0 2 L1 [a; b]. The optimal constant 81 is also the best possible number in (1.6). jT (f; g)j

Probably the most recent work about appropriate bounds of the usual Chebyshev functional is due to Niezgoda [8]: Let f; ; 2 Lp [a; b] and g 2 Lq [a; b] 1 + 1q = 1; 1 p 1 be functions such that (t) + (t) is a constant funcp tion and (t) f (t) (t) for all t 2 [a; b]. Then we have 1

jT (f; g)j

2(b

a)

k

b 1 Z g(t)dt b aa

kp g

:

(1.7)

q

For p = 2 = q, (1.7) leads to the well-known inequality [7]

such that m1

f (t)

q

1 (M1 2

jT (f; g)j

m1 ) T (g; g)

(1.8)

M1 for all t 2 [a; b]:

See [1, 3, 5-6] for further works on Chebyshev functional. In this paper, we …rst generalize the Pecaric work on Montgomery’s identity via an arbitrary weight function, which no longer needs to be a probability density function and then apply it to derive some generalized Chebyshev type inequalities for any absolutely continuous functions.

1

Main Results

Let f : [a; b] ! R be absolutely continuous function on [a; b] ; then the Montgomery type identity [7] reads as: f (x) =

1 b

a

Z

b

f (t) dt +

a

Z

b

K (x; t) f 0 (t) dt;

(2.1)

a

3

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J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 16, NO.4, 2014, COPYRIGHT 2014 EUDOXUS PRESS, LLC

where K (x; t) is the Peano kernel de…ned by: 8 >
:

t a ;a b a

t

x

t b ;x b a

t

b:

The weighted version of identity (2.1) given by Peµcari´c in [10] is in the form:

f (x) =

Z

b

r (t) f (t) dt +

a

Z

b

a

Kw (x; t) f 0 (t) dt;

(2.2)

where r (t) is a probability density function and the weighted Peano kernel is de…ned by:

Kw (x; t) =

8 >
: t r (s) ds a

t

1; x

x t

b:

We now introduce a further generalization of (2.2) by considering the positive function w : [a; b] ! [0; +1) which is not necessarily a probability density function, integrable and Zb

a

w (s) ds < 1: R

The domain of w may be …nite or in…nite. If m (a; b) = ab w (s) ds as total area of w such that m (a; x) = 0 for any x < a; then a generalized type of weighted Peano kernel can be de…ned as follows: 8 >
R :

where

2 [0; 1);

Rt

(1

t (1

)a+

)a+ b

w (s) ds; a R

b w (s) ds

6= 21 ; and x 2 [(1

a+(1 )b (1 )a+ b

t

x

(2.3)

w (s) ds; x

)a + b; a + (1

t

b;

)b] :

To simplify the details of presentation, let us de…ne 4

766

KUTBI ET AL 763-776

J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 16, NO.4, 2014, COPYRIGHT 2014 EUDOXUS PRESS, LLC

~

(2.4)

T (w; f; g; ) =

R (1

)a+ b

a

R (1

)a+ b

a

R (1

0

@1

R

0

@1

R

R

Z

R

a+(1 )b (1 )a+ b

a+(1 )b (1 )a+ b

a+(1 )b (1 )a+ b )a+ b

A

w (s) dx

)b

w (s) ds g(b)

a+(1

w (s) ds Rb

a+(1

Z

R

a+(1 )b (1 )a+ b Z b

1 A

w (s) ds

w (s) ds f (b)

w (x) g (x) dx

a

a+(1 )b (1 )a+ b

m(a; b)

)b

b

Rb

w (s) dx g(a) +

a+(1

)b

w (s) ds g(b)

w (s) ds m(a; b) Z

b

w (x) f (x) dx

a

w (x) f (x) g (x) dx

a

a+(1 )b (1 )a+ b

2

w (s) ds m (a; b)

w (x) f (x) dx

! Z

b

w (x) g (x) dx

a

a

Z

Rb

w (s) ds

w (s) ds m(a; b)

1

m(a; b)

b

+ R

a+(1 )b (1 )a+ b

w (s) ds f (a) + R

1 m (a; b)

w (s) ds f (b)

a+(1

w (s) ds g(a) +

a

+

+

R

)a+ b

R (1

)b

w (s) ds f (a) +

a

+

Rb

1 a+(1 )b (1 )a+ b

w (s) ds

b

w (x) f (x) dx

a

!

2

! Z

!

b

w (x) g (x) dx ;

a

(2.5)

Sw;f;g; (x) = f (x)g(x)

0 R (1

1 + @ 2 +

2 and

a

R (1 a

R

)a+ b

)a+ b

w (s) ds g(a) + R

a+(1 )b (1 )a+ b

w (s) ds f (a) +

1 a+(1 )b (1 )a+ b

Rb

R

a+(1 )b (1 )a+ b

w (s) ds

5

0

)b

w (s) ds g(b)

w (s) ds

Rb

a+(1

)b

w (s) ds f (b)

w (s) ds

@f (x)

767

a+(1

Zb a

w(x)g(x)dx + g(x)

1

g(x)A

Zb a

f (x)

1

w(x)f (x)dxA ;

KUTBI ET AL 763-776

J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 16, NO.4, 2014, COPYRIGHT 2014 EUDOXUS PRESS, LLC

~

T w;f;g;

Z b 1 w (x) f (x) g (x) dx = m(a; b) a

+

R (1

)a+ b

a

R

w (s) ds f (a) + R

a+(1 )b (1 )a+ b

(2.6) Rb

a+(1

w (s) ds m(a; b) Z

1 a+(1 )b (1 )a+ b

w (s) ds f (b) Z

)b

w (s) ds m(a; b)

b

w (x) g (x) dx

a

b

w (x) f (x) dx

a

! Z

!

b

w (x) g (x) dx :

a

It is easy to note that

Z 1 w(x)Sw;f;g; (x)dx = Tw;f;g; ; m(a; b) a b

(2.7)

where

Tw;f;g; =

Z b 1 w (x) f (x) g (x) dx m (a; b) a 0 R (1

1 @ + 2m (a; b) Z

)a+ b

a

w (s) ds f (a) + R

b

a

+

w (x) g (x) dx R (1

)a+ b

a

R

w (s) ds g(a) + 1

a+(1 )b (1 )a+ b

R

a+(1 )b (1 )a+ b

a+(1 )b (1 )a+ b

Rb

a+(1

w (s) ds

w (s) ds m (a; b)

It is worth to mention that for Chebyshev functional (1.2).

(2.8)

Z

)b

Rb

a+(1

)b

w (s) ds f (b)

w (s) ds

w (s) ds g(b) Z

b

w (x) f (x) dx

a

b

a

! Z

b

1

w (x) f (x) dxA !

w (x) g (x) dx :

a

= 0; (2.4), (2.6) and (2.8) reduce to the

We have given the generalized weighted Montgomery’s identity in the following result: Theorem 1 Let f : [a; b] ! R be absolutely continuous, then 6

768

KUTBI ET AL 763-776

J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 16, NO.4, 2014, COPYRIGHT 2014 EUDOXUS PRESS, LLC

f (x) =

R (1

)a+ b

a

+R

+R

Rb

w (s) ds f (a) + R

1 a+(1 )b (1 )a+ b

a+(1 )b (1 )a+ b Z b

a+(1 )b (1 )a+ b

w (s) ds

)b

w (s) ds f (b)

w (s) ds

w (t) f (t) dt

w (s) ds

1

a+(1

a

Z

b

a

Kw; (x; t) f 0 (t) dt;

for all x 2 [a; b] ; where Kw; (x; t) is de…ned in (2.3). PROOF. Consider the kernel de…ned in (2.3), we have Z

b

a

Kw; (x; t) f 0 (t) dt =

Z

Z

x

a b

x

=

Z

w (s) ds f 0 (t) dt

a

Z

t

w (s) ds )a+ b

(1

Z

b

!

)a+ b

(1

Z

+

t

t

(1

!

Z

a+(1

(1

!

w (s) ds f 0 (t) dt

)a+ b

Z

w (s) ds f 0 (t) dt )a+ b

)b

(1

a+(1

!Z

)b

b

w (s) ds )a+ b

f 0 (t) dt:

x

Integrating by parts and simplifying we obtain Z

b

a

0

Kw; (x; t) f (t) dt =

Z

!

)a+ b

(1

w (s) ds f (a)

a

+ Z

Z

b

!

w (s) ds f (b) + a+(1

)b

Z

(1

a+(1

)b

!

w (s) ds f (x) )a+ b

b

w (t) f (t) dt:

a

Hence proved.

2

Applications of generalized Montgomery identity

We now give a generalization of the Chebyshev inequality in the following result: Theorem 2 Let f; g : [a; b] ! R be absolutely continuous functions on [a; b]. Also let the function w satis…es the conditions given in Theorem 1. Suppose f 0 ; g 0 2 L1 [a; b] ; 7

769

KUTBI ET AL 763-776

J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 16, NO.4, 2014, COPYRIGHT 2014 EUDOXUS PRESS, LLC

then 1

~

T (w; f; g; )

R

a+(1 )b (1 )a+ b

w (s) ds

0

2

0

kf k1 kg k1

m (a; b)

Z

b

w (x)

a

2 ;a;b (x)dx;

(3.1)

for all x 2 [a; b] ; where

;a;b (x) =

Z

a

!

)a+ b

(1

w (s) ds + x

a

x

Z

a+(1

+

!

)a+ b

+

Z

sw (s) ds

a

Z

Z

w (s) ds + b

(1

a+(1

b

!

(3.2)

!

w (s) ds a+(1

)b

sw (s) ds

Z

sw (s) ds

x

w (s) ds )a+ b

x

(1

!

)b

x

(1

!

)b

x

Z

Z

)a+ b

b

! !

sw (s) ds : a+(1

)b

PROOF. Since the functions f and g are absolutely continuous, we have

f (x) +

=R

R

R (1

)a+ b

a

1 a+(1 )b (1 )a+ b

w (s) ds

1

a+(1 )b (1 )a+ b

w (s) ds

Rb

w (s) ds f (a) +

Z

Z

b

R

a+(1 )b (1 )a+ b

a+(1

)b

w (s) ds f (b)

)b

w (s) ds g(b)

w (s) ds

(3.3)

w (t) f (t) dt

a

b

Kw; (x; t) f 0 (t) dt;

a

and

g (x) +

=R

R

R (1

)a+ b

a

w (s) ds g(a) +

1 a+(1 )b (1 )a+ b

w (s) ds

1

a+(1 )b (1 )a+ b

w (s) ds

Z

Z

b

R

a+(1 )b (1 )a+ b

Rb

a+(1

w (s) ds

w (t) g (t) dt

a

b

a

Kw; (x; t) g 0 (t) dt:

Using (3.3) and (3.4) we have 8

770

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J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 16, NO.4, 2014, COPYRIGHT 2014 EUDOXUS PRESS, LLC

hR

0

1 a+(1 )b (1 )a+ b

= @f (x) + 0

R

R (1

+

1 w (s) ds

a+(1 )b (1 )a+ b

R (1

)a+ b

a

)a+ b

R (1

)a+ b

a

R (1

a+(1 )b (1 )a+ b

b

a

a+(1 )b (1 )a+ b

R

b

a+(1 )b (1 )a+ b

)a+ b

a

a+(1 )b (1 )a+ b

1 a+(1 )b (1 )a+ b

)b

w (s) ds g(b)

a+(1

w (s) ds

Rb

a+(1 )b (1 )a+ b

a+(1

)b

w (s) ds g(b)

w (s) ds

f (x)

g (x)

w (s) ds f (b)

Rb

)b

w (s) ds f (b)

a+(1

g(x)

Rb

a+(1

)b

w (s) ds g(b)

Rb

a+(1

w (s) ds

w (s) ds f (b) Z

)b 2

b

w (t) g (t) dt

a

b

w (t) f (t) dt

a

a+(1 )b (1 )a+ b

w (s) ds

)b

a+(1

w (s) ds

a+(1 )b (1 )a+ b

Z

Rb

w (s) ds

w (s) ds g(a) + R

Rb

w (s) ds

w (s) ds

w (s) ds f (a) +

w (s) ds

w (s) ds f (b)

w (t) g (t) dt

a+(1 )b (1 )a+ b

a+(1 )b (1 )a+ b

)b

w (s) ds g(a) +

w (s) ds f (a) +

1

Kw; (x; t) g (t) dt

1

w (s) ds g(a) +

)a+ b

!

a

R

R

0

Rb

w (t) g (t) dtA Z

a

a+(1

w (t) f (t) dtA R

b

1

w (s) ds f (a) +

R

+ R

R

f (x)

R

a

R (1

Z

)a+ b

w (s) ds

Kw; (x; t) f (t) dt

w (s) ds g(a) +

a

1

a

R

R (1

b

! Z

0

w (s) ds f (a) +

a

)a+ b

a

b

a

Z

w (s) ds

R (1

a+(1 )b (1 )a+ b

R (1

+

)a+ b

1

= f (x) g (x) + R

i2

a

a+(1 )b (1 )a+ b

@g (x) + R

w (s) ds

Z

2

Z

Rb

a+(1

w (s) ds

)b 2

w (s) ds g(b) Z b

w (t) f (t) dt

a

w (t) f (t) dt

a

! Z

b

b

a

!

w (t) g (t) dt :

w(x) Now …rst multiplying both sides by m(a;b) and then integrating both sides with respect to x over the interval [a; b] and simplifying by using (2.4), we get

9

771

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J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 16, NO.4, 2014, COPYRIGHT 2014 EUDOXUS PRESS, LLC

1

~

T (w; f; g; ) = R Z

a+(1 )b (1 )a+ b

Z

b

w (x)

a

2

w (s) ds

m(a; b)

b

! Z

=

Kw; (x; t) f (t) dt

a

b

!

0

Kw; (x; t) g (t) dt dx;

a

which implies that 1

~

T (w; f; g; )

R

a+(1 )b (1 )a+ b

0

2

w (s) ds

m(a; b)

0

kf k1 kg k1

Z

b

w (x)

a

Z

!2

b

jKw; (x; t)j dt

a

(3.5)

It can be easily seen that Z

b

a

jKw; (x; t)j dt = =

Z

x

jKw; (x; t)j dt +

a Z (1

Z

)a+ b

a

+

Z

x )a+ b

Z

+

jKw; (x; t)j dt

x )a+ b

(1

!

w (s) ds dt

a+(1

Z

w (s) ds dt

Z

)b

!

t

(1

)a+ b a+(1

)b

b a+(1

Z

= a

)b

Z

w (s) ds dt a+(1

)b

!

Z

a+(1

w (s) ds + b

+

(1

!

)a+ b

sw (s) ds

a

+

Z

a+(1

!

)b

sw (s) ds

x

x

w (s) ds )a+ b

(1

!

)b

x

Z

Z

w (s) ds + x

a

x

!

t

)a+ b

(1

!

w (s) ds dt

t

x

Z

b

t

(1

+

Z

Z

Z

w (s) ds a+(1

)b

x

!

!

sw (s) ds

(1

Z

b

!

)a+ b

b

!

sw (s) ds ; a+(1

)b

so that (3.5) turns in (3.1). Hence proved. Remark 3 If in (2.10) and (3.1), = 0 and w is the probability density function, then we recapture the results obtained in [10]. We now give another generalization of the Chebyshev inequality in the following result: Theorem 4 Let f; g : [a; b] ! R be absolutely continuous functions on [a; b]. 10

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Also let the function w satis…es the conditions given in Theorem 1. Suppose f 0 ; g 0 2 L1 [a; b] ; then jSw;f;g; (x)j

R

2

1 a+(1 )b (1 )a+ b

(jf (x)j kg 0 k1 + jg(x)j kf 0 k1 )

w (s) ds

;a;b (x);

(3.6)

and

jTw;f;g; j

2 Z

b

R

1 a+(1 )b (1 )a+ b

w (x)

a

w (s) ds m(a; b)

(kf k1 kg 0 k1 + kgk1 kf 0 k1 )

;a;b (x)dx

(3.7) R

2 8 > > > > > < > > > > > :

1 a+(1 )b (1 )a+ b

w (s) ds m(a; b) kwk1

kwkp

;a;b q

;a;b 1

if w 2 L1 [a; b] and

if w 2 Lp [a; b] and

kwk1

for all x 2 [a; b] ; where

(kf k1 kg 0 k1 + kgk1 kf 0 k1 )

;a;b (x)

;a;b 1

Rb

=

a

;a;b

;a;b

2 L1 [a; b] ;

2 Lq [a; b] ; where p > 1,

if w 2 L1 [a; b] and

;a;b

1 p

+

1 q

2 L1 [a; b] ;

jKw; (x; t)j dt is given by (3.2).

PROOF. Since the functions f and g are absolutely continuous, multiplying both sides of (3:3) and (3:4) by g(x) and f (x) respectively, adding the resulting identities and rewriting, we have f (x)g(x) + 0 R (1

1@ 2

)a+ b

a

R (1

)a+ b

a

+

2

R

1 a+(1 )b (1 )a+ b

1

w (s) ds g(a) + R

a+(1 )b (1 )a+ b

a+(1 )b (1 )a+ b

w (s) ds

0

@f (x)

= R a+(1 )b f (x) 2 (1 )a+ b w (s) ds

Z

a+(1

)b

a+(1

)b

w (s) ds f (b)

w (s) ds

Zb

w(t)g(t)dt + g(x)

a

Zb a

b

a

w (s) ds g(b)

w (s) ds Rb

w (s) ds f (a) + R

Rb

0

773

1

g(x)A

1

w(t)f (t)dtA

Kw; (x; t) g (t) dt + g(x)

11

f (x)

Z

b

a

0

!

Kw; (x; t) f (t) dt ;

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implies 1 Sw;f;g; (x) = R a+(1 )b 2 (1 )a+ b w (s) ds f (x)

Z

b

(3.8) 0

Kw; (x; t) g (t) dt + g(x)

a

Z

b

a

!

0

Kw; (x; t) f (t) dt ;

gives us

jSw;f;g; (x)j

2

R

1 a+(1 )b (1 )a+ b

Z

jf (x)j 2 for all x 2 [a; b] :

R

w (s) ds

b

0

jKw; (x; t)j jg (t)j dt + jg(x)j

a

1 a+(1 )b (1 )a+ b

Z

b

a

0

jKw; (x; t)j jf (t)j dt

0

0

(jf (x)j kg k1 + jg(x)j kf k1 )

w (s) ds

!

Z

b

jKw; (x; t)j dt;

a

This completes the proof of (3.6). w(x) Now multiplying both sides of (3.8) by m(a;b) and then integrating with respect to x from a to b and rewriting, we get

Tw;f;g; =

2 Z

b

R

1 a+(1 )b (1 )a+ b

w(x)

a

Z

w (s) ds m(a; b)

b

0

0

!

Kw; (x; t) (f (x)g (t) + g(x)f (t)) dt dx;

a

implies

jTw;f;g; j

2 Z

b

R

1 a+(1 )b (1 )a+ b

Z

w(x)

a

2

b

a

R

a+(1 )b (1 )a+ b 0

w (s) ds m(a; b) 0

0

!

jKw; (x; t)j (jf (x)j jg (t)j + jg(x)j jf (t)j) dt dx 1 w (s) ds m(a; b) 0

(kf k1 kg k1 + kgk1 kf k1 )

Z

b

a

w (x)

Z

b

a

!

jKw; (x; t)j dt dx:

Now (3.7) can be easily derived form the last inequality. 12

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Another variant of Chebyshev inequality is given in the form of the following result: Theorem 5 Let f : [a; b] ! R be absolutely continuous function on [a; b]. Also let the function w satis…es the conditions given in Theorem 1. Suppose f 0 ; g 2 L1 [a; b] ; then 1

~

T w;f;g;

R R

8 > > > > >
> > > > :

;a;b

a+(1 )b (1 )a+ b a+(1 )b (1 )a+ b

w (s) ds m(a; b) 1 w (s) ds m(a; b) kwk1

kwkp

;a;b q

;a;b 1

kf 0 k1 kgk1

;a;b 1

b

w(x)

a

(3.9)

;a;b (x)dx

kf 0 k1 kgk1 if w 2 L1 [a; b] and

if w 2 Lp [a; b] and

kwk1

Z

;a;b

;a;b

2 L1 [a; b] ;

2 Lq [a; b] ; where p > 1,

if w 2 L1 [a; b] and

;a;b

1 p

+

1 q

2 L1 [a; b] ;

is as de…ned in (3.2):

PROOF. Since the function f is absolutely continuous, multiplying both sides of (3.3) by w(x)g(x) and then integrating with respect to x from a to b m(a;b) and rewriting, we obtain T w;f;g; = R

Z

1

~ a+(1 )b (1 )a+ b

w (s) ds m(a; b)

Z

b

w(x)g(x)

b

a

a

!

0

Kw; (x; t) f (t)dt dx;

which implies, by taking the modulus on both sides, ~

T w;f;g; R R

1 a+(1 )b (1 )a+ b

w (s) ds m(a; b) 1

a+(1 )b (1 )a+ b

w (s) ds m(a; b)

Z

Z

b

w(x) jg(x)j

a 0

kf k1 kgk1

Z

b

a

!

jKw; (x; t)j jf (t)j dt dx

b

a

0

w(x)

Z

b

a

!

jKw; (x; t)j dt dx:

Hence proved. 13

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Acknowledgements This article was funded by the Deanship of Scienti…c Research (DSR), King Abdulaziz University, Jeddah. Therefore, the …rst and second authors acknowledges with thanks DSR, KAU for …nancial support. References (1) P. Cerone, S.S. Dragomir, New bounds for the Chebyshev functional, Appl. Math. Lett. 18 (2005) 603-611. (2) P. L.Chebyshev, Sur les expressions approximative des integrals par les auters prises entre les mêmes limites, Proc. Math. Soc. Charkov, 2 (1882) 93-98. (3) S. S. Dragomir, Bounds for some perturbed Chebyshev functionals, J. Inequal. Pure Appl. Math. 9 (3) (2008) Art. 64. (4) G. Grüss, Uber das Maximum des absoluten Betrages von , Math. Z., 39 (1935) 215-226. (5) M. Masjed-Jamei, A Main inequality for several special functions, Comput. Math. Appl., 60 (2010) 1280-1289. (6) M. Masjed-Jamei, Feng Qi, H.M Srivastava, Generalizations of some classical inequalities via a special functional property, Integral Transforms Spec. Funct, 21 (2010) 327-336. (7) D. S. Mitrinovic, J.E. Pecaric and A.M. Fink, Classical and New Inequalities in Analysis, Kluwer Academic Publishers, Dordrecht/ Boston/ London, 1993. (8) M. Niezgoda, A new inequality of Ostrowski-Gruss type and applications to some numerical quadrature rules, Comput. Math. Appl, 58 (2009) 589596. (9) A.M. Ostrowski, On an integral inequality, Aequat. Math. 4 (1970) 358373. (10) B. G. PACHPATTE, On Chebyshev-Grüss type inequalities via Peµcari´c’s extention of the Montgomery identity, J. Inequal. Pure and Appl. Math., 7(1) (2006), Art 108.

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Comment on “Generalized ternary bi-derivations on ternary Banach algebras: a fixed point approach” [Esgahi Gordji et al., J. Comput. Anal. Appl. 15 (2013) 45-54] Choonkil Park1 , Madjid Eshaghi2 , Gordji Jung Rye Lee3 and Dong Yun Shin4∗ 1

2

Research Institute for Natural Sciences, Hanyang University, Seoul 133-791, Korea Department of Mathematics, Semnan University, P.O. Box 35195-363, Semnan, Iran 3 Department of Mathematics, Daejin University, Kyeonggi 487-711, Korea 4

Department of Mathematics, University of Seoul, Seoul 130-743, Korea

Abstract. Eshaghi Gordji et al. [10] proved the Hyers-Ulam stability of generalized ternary bi-derivations on ternary Banach algebras. It is easy to show that the resulting generalized ternary bi-derivations of [10] are meaningless, since the definition of generalized ternary bi-derivation, given in [10], is meaningless. In this paper, we correct the definition of generalized ternary bi-derivation, and correct the statements of the results and prove the corrected results. 1. Introduction and preliminaries A C ∗ -ternary algebra is a complex Banach space A, equipped with a ternary product (x, y, z)  [xyz] of A3 into A, which is C-linear in the outer variables, conjugate C-linear in the middle variable, and associative in the sense that [xy[zvw]] = [x[wzy]v] = [[xyz]wv], and satisfies ∥[xyz]∥ ≤ ∥x∥∥y∥∥z∥ and ∥[xxx]∥ = ∥x∥3 (see [3]). Definition 1.1. ([2]) Let A be a C ∗ -ternary algebra. A C-bilinear mapping δ : A × A → A is called a ternary bi-derivation if it satisfies δ([abc], d) = [δ(a, d)bc] + [aδ(b, d)c] + [abδ(c, d)], δ(a, [bcd]) = [δ(a, b)cd] + [bδ(a, c)d] + [bcδ(a, d)] for all a, b, c, d ∈ A. Note that the d-variable in the left side of the first equality and the a-variable in the left side of the second equality are C-linear. But the d-variable in the right side of the first equality and the a-variable in the right side of the second equality are not C-linear. So we correct the definition of ternary bi-derivation as follows. Definition 1.2. Let A be a C ∗ -ternary algebra. A C-bilinear mapping δ : A × A → A is called a ternary bi-derivation if it satisfies δ([abc], d) = [δ(a, d)bc] + [aδ(b, d∗ )c] + [abδ(c, d)], δ(a, [bcd]) = [δ(a, b)cd] + [bδ(a∗ , c)d] + [bcδ(a, d)] for all a, b, c, d ∈ A. 0

2010 Mathematics Subject Classification: 39B52, 47B47, 47H10, 17A40, 39B82, 20N10. Keywords: Hyers-Ulam stability; generalized bi-derivation; C ∗ -ternary algebra; fixed point. ∗ Corresponding author. 0 E-mail: [email protected]; [email protected]; [email protected]; [email protected] 0

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C. Park, M. Eshaghi Gordji, J. Lee, D. Shin Definition 1.3. ([10]) Let A be a C ∗ -ternary algebra. A C-bilinear mapping D : A × A → A is called a generalized ternary bi-derivation if there exists a bi-derivation δ : A × A → A such that D([abc], d) = [D(a, d)bc] + [aδ(b, d)c] + [abδ(c, d)], D(a, [bcd]) = [D(a, b)cd] + [bδ(a, c)d] + [bcδ(a, d)] for all a, b, c, d ∈ A. Note that the d-variable in the left side of the first equality and the a-variable in the left side of the second equality are C-linear. But the d-variable in the right side of the first equality and the a-variable in the right side of the second equality are not C-linear. So we correct the definition of generalized ternary bi-derivation as follows. Definition 1.4. Let A be a C ∗ -ternary algebra. A C-bilinear mapping D : A × A → A is called a generalized ternary bi-derivation if there exists a bi-derivation δ : A × A → A such that D([abc], d) = [D(a, d)bc] + [aδ(b, d∗ )c] + [abδ(c, d)], D(a, [bcd]) = [D(a, b)cd] + [bδ(a∗ , c)d] + [bcδ(a, d)] for all a, b, c, d ∈ A. The stability problem of functional equations originated from a question of Ulam [14] concerning the stability of group homomorphisms. Hyers [12] gave a first affirmative partial answer to the question of Ulam for Banach spaces. Hyers’ Theorem was generalized by Aoki [1] for additive mappings and by Th.M. Rassias [13] for linear mappings by considering an unbounded Cauchy difference. Gˇavruta [11] obtained the generalized result of Rassias’s theorem which allows the Cauchy difference to be controlled by a general unbounded function. The stability problems of various functional equations have been extensively investigated by a number of authors (see [5, 6, 7, 8, 9]). We recall a fundamental result in fixed point theory. Theorem 1.5. ([4]) Suppose that a complete generalized metric space (X , d) and a strictly contractive mapping J : X → X with Lipschits constant 0 < L < 1 are given. Then, for a given element x ∈ X , exactly one of the following assertions is true: either (1) d(J n x, J n+1 x) = ∞ for all n ≥ 0 or (2) there exists n0 such that d(J n x, J n+1 x) < ∞ for all n ≥ n0 . Actually, if (2) holds, then the sequence J n x is convergent to a fixed point x∗ of J and (3) x∗ is the unique fixed point of J in Λ := {y ∈ X , d(J n0 x, y) < ∞}; (4) d(y, x∗ ) ≤ d(y,Jy) 1−L for all y ∈ Λ. In this paper, we correct the statements of the results and prove the corrected results. 2. Main results From now on, we assume that A is a C ∗ -ternary algebra. For a given mapping f : A × A → A, we define the difference operator Eλ,µ f : A4 → A by Eλ,µ f (a, b, c, d) = f (λa − λb, µc) + f (λa, µc − µd) − λµ(2f (a, c) − f (b, c) − f (a, d)) for all λ, µ ∈ T1 := {λ ∈ C : |λ| = 1} and all a, b, c, d ∈ A. We prove the Hyers-Ulam stability of generalized ternary bi-derivations. Theorem 2.1. Let f, g : A × A → A be uniformly continuous mappings such that g(0, 0) = f (0, 0) = 0. Let φ : A4 → [0, ∞) be a function such that max{∥Eλ,µ f (a, b, c, d)∥, ∥Eλ,µ g(a, b, c, d)∥} ≤ φ(a, b, c, d), max{∥f ([abc], d) − [f (a, d)bc] − [af (b, d∗ )c] − [abf (c, d)]∥, ∥f (a, [bcd]) − [f (a, b)cd] − [bf (a∗ , c)d] − [bcf (a, d)]∥} ≤ φ(a, b, c, d), 778

(2.1) (2.2) PARK ET AL 777-781

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Generalized ternary bi-derivations max{∥g([abc], d) − [g(a, d)bc] − [af (b, d∗ )c] − [abf (c, d)]∥,

(2.3) ∥g(a, [bcd]) − [g(a, b)cd] − [bf (a∗ , c)d] − [bcf (a, d)]∥} ≤ φ(a, b, c, d), 1 lim φ(2n a, 2n b, 2n c, 2n d) = 0 n→∞ 4n for all λ, µ ∈ T1 and all a, b, c, d ∈ A. If there exists an L < 1 such that Ψ(a, b) ≤ 4LΨ( a2 , 2b ) for all a, b ∈ A, where Ψ(a, b) := φ(0, a, 2b, 0) + φ(a, −a, 2b, b) + φ(0, 0, 2b, 0) + 3(φ(a, 0, b, −b) +φ(a, 0, 0, b) + φ(a, 0, 0, 0) + φ(0, 0, b, 0)), then there exist a unique ternary bi-derivation δ : A × A → A and a unique generalized ternary biderivation D : A × A → A (related to δ) such that (a c) L max{∥g(a, c) − D(a, c)∥, ∥f (a, c) − δ(a, c)∥} ≤ Ψ , (2.4) 1−L 2 2 for all a, c ∈ A. Proof. By the same reasoning as in the proof of [10, Theorem 2.1], there exist a unique C-bilinear mapping δ : A × A → A and a unique C-bilinear mapping D : A × A → A satisfying (2.4). The C-bilinear mapping δ : A × A → A and the C-bilinear mapping D : A × A → A are given by 1 δ(a, d) := lim n f (2n a, 2n d), n→∞ 4 1 D(a, d) := lim n g(2n a, 2n d) n→∞ 4 for all a, d ∈ A, respectively. It is easy to show that 1 1 δ(a, d) = lim f (8n a, 2n d) = lim f (2n a, 8n d) , n→∞ 16n n→∞ 16n 1 1 D(a, d) = lim g (8n a, 2n d) = lim g (2n a, 8n d) n→∞ 16n n→∞ 16n for all a, d ∈ A, since δ, D are bi-additive and f, g are uniformly continuous. It follows from (2.2) that ∥δ([abc], d) − [δ(a, d)bc] − [aδ(b, d∗ )c] − [abδ(c, d)]∥ 1 (∥f ([(2n a)(2n b)(2n c)], 2n d) − [f (2n a, 2n d)(2n b)(2n c)] 16n − [(2 a)f (2n b, 2n d)(2n c)] − [(2n a)(2n b)f (2n c, 2n d∗ )]∥) = lim

n→∞ n

1 1 φ(2n a, 2n b, 2n c, 2n d) ≤ lim n φ(2n a, 2n b, 2n c, 2n d) = 0 n n→∞ 16 n→∞ 4 for all a, b, c, d ∈ A. This means that ≤ lim

δ([abc], d) = [δ(a, d)bc] + [aδ(b, d∗ )c] + [abδ(c, d)] for all a, b, c, d ∈ A. Similarly, we can show that δ(a, [bcd]) = [δ(a, b)cd] + [bδ(a∗ , c)d] + [bcδ(a, d)] for all a, b, c, d ∈ A. Hence δ is a bi-derivation. On the other hand, by (2.3), we have ∥D([abc], d) − [D(a, d)bc] − [aδ(b, d∗ )c] − [abδ(c, d)]∥ 1 (∥g([(2n a)(2n b)(2n c)], 2n d) − [g(2n a, 2n d)(2n b)(2n c)] n→∞ 16n − [(2n a)f (2n b, 2n d∗ )(2n c)] − [(2n a)(2n b)f (2n c, 2n d)]∥) = lim

≤ lim

n→∞

1 1 φ(2n a, 2n b, 2n c, 2n d) ≤ lim n φ(2n a, 2n b, 2n c, 2n d) = 0 n→∞ 4 16n 779

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C. Park, M. Eshaghi Gordji, J. Lee, D. Shin for all a, b, c, d ∈ A. It follows that D([abc], d) = [D(a, d)bc] + [aδ(b, d∗ )c] + [abδ(c, d)] for all a, b, c, d ∈ A. Similarly, we can show that D(a, [bcd]) = [D(a, b)cd] + [bδ(a∗ , c)d] + [bcδ(a, d)] for all a, b, c, d ∈ A. This means that D is a generalized bi-derivation related to δ.



Corollary 2.2. Let p ∈ (0, 2) and q ∈ (0, ∞) be real numbers. Suppose that f, g : A × A → A are uniformly continuous mappings satisfying g(0, 0) = f (0, 0) = 0 and max{∥Eλ,µ f (a, b, c, d)∥, ∥Eλ,µ g(a, b, c, d)∥, ∥f ([abc], d) − [f (a, d)bc] − [af (b, d∗ )c] − [abf (c, d)]∥, ∥f (a, [bcd]) − [f (a, b)cd] − [bf (a∗ , c)d] − [bcf (a, d)]∥, ∥g([abc], d) − [g(a, d)bc] − [af (b, d∗ )c] − [abf (c, d)]∥, ∥g(a, [bcd]) − [g(a, b)cd] − [bf (a∗ , c)d] − [bcf (a, d)]} ≤ q(∥a∥p + ∥b∥p + ∥c∥p + ∥d∥p ) for all λ, µ ∈ T1 and all a, b, c, d ∈ A. Then there exist a unique ternary bi-derivation δ : A × A → A and a unique generalized ternary bi-derivation D : A × A → A (related to δ) such that max{∥g(a, c) − D(a, c)∥, ∥f (a, c) − δ(a, c)∥} ≤

5q (∥a∥p + ∥c∥p ) 4 − 2p

for all a, c ∈ A. Proof. The proof follows from Theorem 2.1 by putting φ(a, b, c, d) := q(∥a∥p + ∥b∥p + ∥c∥p + ∥c∥p ) for all a, b, c, d ∈ A and L = 2p−2 .



Theorem 2.3. Let f, g : A × A → A be uniformly continuous mappings satisfying g(0, 0) = f (0, 0) = 0. Let φ : A4 → [0, ∞) be a function satisfying (2.1), (2.2) and (2.3). Let lim 16n φ(2−n a, 2−n b, 2−n c, 2−n d) = 0

n→∞

for all a, b, c, d ∈ A. If there exists an L < 1 such that Ψ(a, b) ≤ L4 Ψ(2a, 2b) for all a, b ∈ A, where Ψ(a, b) is defined in Theorem 2.1, then there exist a unique ternary bi-derivation δ : A × A → A and a unique generalized ternary bi-derivation D : A × A → A (related to δ) such that max{∥g(a, c) − D(a, c)∥, ∥f (a, c) − δ(a, c)∥} ≤

L Ψ(a, c) 4 − 4L

for all a, c ∈ A. Proof. The proof is similar to the proof of Theorem 2.1.



Corollary 2.4. Let p ∈ (4, ∞) and q ∈ (0, ∞) be real numbers. Suppose that f, g : A × A → A are uniformly continuous mappings satisfying g(0, 0) = f (0, 0) = 0 and max{∥Eλ,µ f (a, b, c, d)∥, ∥Eλ,µ g(a, b, c, d)∥, ∥f ([abc], d) − [f (a, d)bc] − [af (b, d∗ )c] − [abf (c, d)]∥, ∥f (a, [bcd]) − [f (a, b)cd] − [bf (a∗ , c)d] − [bcf (a, d)]∥, ∥g([abc], d) − [g(a, d)bc] − [af (b, d∗ )c] − [abf (c, d)]∥, ∥g(a, [bcd]) − [g(a, b)cd] − [bf (a∗ , c)d] − [bcf (a, d)]} ≤ q(∥a∥p + ∥b∥p + ∥c∥p + ∥d∥p ) 780

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Generalized ternary bi-derivations for all λ, µ ∈ T1 and all a, b, c, d ∈ A. Then there exist a unique ternary bi-derivation δ : A × A → A and a unique generalized ternary bi-derivation D : A × A → A (related to δ) such that 5q max{∥g(a, c) − D(a, c)∥, ∥f (a, c) − δ(a, c)∥} ≤ p (∥a∥p + ∥c∥p ) 2 −4 for all a, c ∈ A. Proof. It follows from Theorem 2.3 by taking φ(a, b, c, d) := q(∥a∥p + ∥b∥p + ∥c∥p + ∥c∥p ) for all a, b, c, d ∈ A and L = 22−p .

 Acknowledgments

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

References [1] T. Aoki, On the stability of the linear transformation in Banach spaces, J. Math. Soc. Japan 2 (1950) 64–66. [2] J. Bae and W. Park, Approximate bi-homomorphisms and bi-derivations in C ∗ -ternary algebras, Bull. Korean Math. Soc. 47 (2010) 195–209. [3] A. Cayley, On the 34 concomitants of the ternary cubic, Amer. J. Math. 4 (1881) 1–15. [4] J.B. Diaz, B. Margolis, A fixed point theorem of the alternative for the contractions on generaliuzed complete metric space, Bull. Amer. Math. Soc. 74 (1968) 305–309. [5] A. Ebadian, N. Ghobadipour and H. Baghban, Stability of bi-θ-derivations on JB ∗ -triples, Int. J. Geom. Methods Mod. Phys. 9 (2012), No. 7, Art. ID 1250051, 12 pages. [6] A. Ebadian, I. Nikoufar and M. Eshaghi Gordji, Nearly (θ1 , θ2 , θ3 , ϕ)-derivations on C ∗ -modules, Int. J. Geom. Methods Mod. Phys. 9 (2012), No. 3, Art. ID 1250019, 12 pages. [7] M. Eshaghi Gordji, A. Fazeli and C. Park, 3-Lie multipliers on Banach 3-Lie algebras, Int. J. Geom. Methods Mod. Phys. 9 (2012), No. 7, Art. ID 1250052, 15 pages. [8] M. Eshaghi Gordji, M.B. Ghaemi and B. Alizadeh, A fixed point method for perturbation of higher ring derivationsin non-Archimedean Banach algebras, Int. J. Geom. Methods Mod. Phys. 8 (2011) 1611–1625. [9] M. Eshaghi Gordji and N. Ghobadipour, Stability of (α, β, γ)-derivations on Lie C ∗ -algebras, Int. J. Geom. Methods Mod. Phys. 7 (2010) 1097–1102. [10] M. Eshaghi Gordji, G.H. Kim, J.R. Lee and C. Park, Generalized ternary bi-derivations on ternary Banach algebras: a fixed point approach, J. Comput. Anal. Appl. 15 (2013) 45–54. [11] P. Gˇavruta, A generalization of the Hyers-Ulam-Rassias stability of approximately additive mappings, J. Math. Anal. Appl. 184 (1994) 431–436. [12] D.H. Hyers, On the stability of the linear functional equation, Proc. Natl. Acad. Sci. U.S.A. 27 (1941) 222–224. [13] Th.M. Rassias, On the stability of the linear mapping in Banach spaces, Proc. Amer. Math. Soc. 72 (1978) 297–300. [14] S.M. Ulam, Problems in Modern Mathematics, Chapter VI, Science ed., Wiley, New York, 1940.

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J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 16, NO.4, 2014, COPYRIGHT 2014 EUDOXUS PRESS, LLC

EXAMPLES OF UMBRAL CALCULUS D. S. KIM1 , T. KIM2 , W. J. KIM3 , AND DMITRY V. DOLGY4

Abstract. In this paper, we introduce some interesting Sheffer sequences of polynomials. From the properties of those sequences of polynomials, we derive some identities involving multiple power and alternating(power) sums.

1. INTRODUCTION For α ∈ R, the Frobenious-Euler polynomials are defined by the generating function to be )α ( ∞ ∑ tn 1−λ xt e = Hn(α) (x | λ) (1) t e −λ n! n=0 (α)

(α)

where λ ∈ C with λ ̸= 1. In the special case, x = 0, Hn (0 | λ) = Hn (λ) are called the n−th Frobenius-Euler numbers,(see [1-15]). As is well knowen, the Bernoulli polynomials of order α are also defined by the generating function to be ( )α ∞ ∑ t tn xt (α) e = , (2) B (x) n et − 1 n! n=0 (α)

(α)

(see[3,4,5,6]). In the special case, x = 0, Bn (0) = Bn are called the n−th Bernoulli numbers of order α. From (1) and (2), we have n ( ) n ( ) ∑ ∑ n n (α) (α) (α) n−l (α) Hn (x | λ) = Hl (λ)x , Bn (x) = Bl (λ)xn−l . (3) l l l=0

l=0

Let F = {f (t) =

∞ ∑

ak

k=0

tk | ak ∈ C}. k!

(4)

Let us assume that P is the algebra of polynomials in the single variable x over C and P∗ is the vector space of all linear functionals on P. The action of the linear functional on a polynomial p(x) is denoted by < L | p(x) >. We remind that the vector space structure on P∗ are defined by < L + M | p(x) >=< L | p(x) > + < M | p(x) >, < cL | p(x) >= c < L | p(x) >, where c is a complex constant (see[11,12,13,15]). For f (t) ∈ F, we define a linear functional on P by setting < f (t) | xn >= an ,

(5)

for all n ≥ 0, (see[11,12,15]). From (4) and (5), we note that < tk | xn >= n!δn,k ,

(6)

for all n, k ≥ 0, where δn,k is the Kronecker’s symbol (see[13,15]). 1

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2

∑∞ k t , we have < fL (t) | xn >=< L | xn >. So the For fL (t) = k=0 k! map L 7→ fL (t) is a vector space isomorphism from P∗ onto F. Henceforth, F is thought of as both a formal power series and a linear functional. We call F the umbral algebra. The umbral calculus is the study of umbral algebra(see[12,15]). The order O(f (t)) of the non-zero power series f (t) is the smallest integer k for which the coefficient of tk does not vanish(see[11,15]). If O(f (t)) = 1, then f (t) is called a delta series. If O(f (t)) = 0, then f (t) is called an invertible series. Let O(f (t)) = 1 and O(g(t)) = 0. Then there exists a unique sequence Sn (x) of polynomials such that < g(t)f (t)k | Sn (x) >= n!δn,k , where n, k ≥ 0. The sequence Sn (x) is called the Sheffer sequence for (g(t), f (t)), which is denoted by Sn (x) ∼ (g(t), f (t)) (see[13,15]). For f (t) ∈ F and p(x) ∈ P, we have f (t) =

∞ ∞ ∑ ∑ < f (t) | xk > k < tk | p(x) > k t , p(x) = x , (see[15]). k! k!

k=0

(7)

k=0

By (7), we easily get pk (0) =< tk | p(x) > ; < 1 | p(k) >= p(k) (0).

(8)

From (8), we have dk p(x) , dxk

tk p(x) = p(k) (x) =

k ≥ 0.

(9)

For Sn (x) ∼ (g(t), f (t)), the generating function of Sheffer sequence Sn (x) is given by ∞

∑ Sk (y) 1 ¯ eyf (t) = tk , for all y ∈ C, ¯ k! g(f (t) k=0

(10)

where f¯(t) is the compositional inverse of f (t), (see[12,15]). Let Sn (x) ∼ (1, g(t)) and tn (x) ∼ (1, f (t)). Then we have ( Sn (x) = x

f (t) g(t)

)n

x−1 tn (x), n ≥ 0,

(11)

(see[11,15]). This equation (11) is important in deriving our main results in this paper. The purpose of this paper is to introduce some interesting Sheffer sequences of polynomials and to investigate some properties of those sequences of polynomials.

2. Sheffer sequences and Applications Let us consider the following Sheffer sequences ( Sn (x) ∼

t 1, 1 − λet

)

( , tn (x) ∼

783

t 1, 1 − λm emt

) ,

(12)

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3

where λ ∈ C with λm ̸= 1. From xn ∼ (1, t), (11) and (12), we have  n ( )n t )  x−1 xn = x 1 − λet xn−1 Sn (x) = x  ( t 1−λet

= x(1 − λ − λ(et − 1))n xn−1 n ( ) ∑ n =x (1 − λ)n−l (−λ)l (et − 1)l xn−1 l l=0 n−1 n−l−1 ∑ ( n) ∑ l! =x (1 − λ)n−l (−λ)l S2 (l + m, l)tl+m xn−1 l (l + m)! m=0 l=0 ( )( ) n−1 n−l−1 ∑ ∑ n n−1 =x l!(1 − λ)n−l (−λ)l S2 (l + m, l)xn−l−m−1 l l + m l=0 m=0 n−1 ∑ n−l−1 ∑ (n − 1)(n) = l!(1 − λ)n−l (−λ)l S2 (n − 1 − r, l)xr+1 , (13) r l r=0 l=0

where S2 (n, k) is the Stirling number of the second kind. For n ≥ 1, by (11) and (12), we get  n ( )n t )  xn−1 = x 1 − λm emt xn−1 tn (x) = x  ( t 1−λm emt

= x(1 − λm − λm (emt − 1))n xn−1 n ( ) ∑ n =x (1 − λm )n−l (−λm )l (emt − 1)l xn−1 l l=0 n−1 ∑ n−l−1 ∑ (n)(n − 1) =x (1 − λm )n−l (−λm )l l!mk+l S2 (l + k, l)xn−k−l−1 l k+l l=0 k=0 n−1 ∑ n−l−1 ∑ (n)( n − 1 ) =x (1 − λm )n−l (−λm )l l!mn−1−r S2 (n − 1 − r, l)xr l n − 1 − r l=0 r=0 n−1 ∑ n−l−1 ∑ (n)(n − 1) = (1 − λm )n−l (−λm )l l!mn−1−r S2 (n − 1 − r, l)xr+1 . l r l=0 r=0 (14) Therefore, by (13) and (14), we obtain the following theorem. Theorem 1. For n ≥ 1, let ( Sn (x) ∼ 1,

t 1 − λet

) , tn (x) ∼

( 1,

t 1 − λm emt

)

where λ ∈ C with λm ̸= 1. Then we have n−1 ∑ n−l−1 ∑ (n − 1)(n) Sn (x) = l!(1 − λ)n−l (−λ)l S2 (n − 1 − r, l)xr+1 , r l r=0 l=0

and tn (x) =

n−1 ∑ n−l−1 ∑ ( l=0

r=0

n l

)(

) n−1 (1 − λm )n−l (−λm )l l!mn−1−r S2 (n − 1 − r, l)xr+1 . r

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4

By (11) and (12), we get ( )n t tn (x) = x ( =x

1−λet t 1−λm emt

m 1 ∑ k kt λ e λet

x

−1

( Sn (x) = x

)n

k=1

)n

x−1 Sn (x)

x−1 Sn (x) (

∑

= λ−n xe−nt

1 − λm emt 1 − λet

) n (λet )v1 +2v2 +···+mvm x−1 Sn (x) v1 , · · · , vm

0 ≤ v1 , · · · , vm ≤ n v1 + · · · + vm = n )( ∞ (∞ ( ) ∑ ∑ ∑ (−n)l n l −n t =λ x λv1 +2v2 +···+mvm l! v1 , · · · , vm k=0 l=0 0 ≤ v1 , · · · , v m ≤ n v1 + · · · + vm = n ) k k t × (v1 + 2v2 + · · · + mvm ) x−1 Sn (x) k! { s ( ) ( ) ∞ ∑ ∑ s ∑ n −n s−k =λ x (−n) λv1 +2v2 +···+mvm k v1 , · · · , v m s=0 k=0 0 ≤ v1 , · · · , vm ≤ n v1 + · · · + vm = n } ts −1 k × (v1 + 2v2 + · · · + mvm ) x Sn (x) s! { s ( ) ( ) n−1 ∑ ∑ ∑ n s −n λv1 +2v2 +···+mvm =λ x (−n)s−k v , · · · , v k 1 m s=0 k=0 0 ≤ v1 , · · · , v m ≤ n v1 + · · · + vm = n } ts −1 × (v1 + 2v2 + · · · + mvm )k x Sn (x). (15) s! Let (n)

Dk (m | λ) =

(

∑

0 ≤ v1 , · · · , v m ≤ n v1 + · · · + vm = n

) n λv1 +2v2 +···+mvm (v1 + 2v2 + · · · + mvm )k . v1 , · · · , vm (16)

By (13), (15) and (16), we get )( ) n−1 s ( ) n−1 n−l−1 ( ∑∑ s ts ∑ ∑ n − 1 n (n) tn (x) = λ−n x (−n)s−k Dk (x | λ) l!(1 − λ)n−l s! r l k s=0 r=0 k=0

l=0

× (−λ) S2 (n − 1 − r, l)x n−1 s n−1 ∑∑ ∑ n−1−l ∑ (n − 1)( s )(n)(r) (n) = λ−n x (−n)s−k Dk (x | λ)l!(1 − λ)n−l r k l s s=0 r=0 l

r

k=0 l=0

× (−λ) S2 (n − 1 − r, l)xr−s . l

(17)

Therefore, by Theorem1 and (17), we obtain the following theorem.

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Theorem 2. For n ≥ 1, λ ∈ C with λm ̸= 1, we have n−1 ∑ n−1−l ∑ (n)(n − 1) (1 − λm )n−l (−λm )l l!mn−1−r S2 (n − 1 − r, l)xr+1 l r l=0 r=0 )( )( )( ) n−1 s ( ∑ n−1−l ∑ n−1 ∑∑ n−1 s n r (n) −n =λ (−n)s−k Dk (x | λ)l!(1 − λ)n−l r k l s r=0 s=0 l=0

k=0

× (−λ)l S2 (n − 1 − r, l)xr+1−s . Let us consider the following Sheffer sequences Sn (x) ∼ (1, t(1 − λet )), tn (x) ∼ (1, t(1 − λm emt )),

(18)

where λm ̸= 1. For n ≥ 1, by (18) and xn ∼ (1, t), we have ( )n t Sn (x) = x x−1 xn = x(1 − λet )−n xn−1 t(1 − λet ) = x(1 − λ − λ(et − 1))−n xn−1 ( t )l n−1 ∑ (−n) x e −1 xn l l + (−1) λ xn−1 = (1 − λ)n (1 − λ)n l 1−λ l=1 ( )( )l ∑ n−1 l ( ) ∑ x n+l−1 λ l = (−1)l−r ert xn−1 (1 − λ)n l 1 − λ r=0 r l=0 )( ) ( )l n−1 l ( ∑∑ x n+l−1 l λ = (−1)l−r (x + r)n−1 . (1 − λ)n l r 1 − λ r=0

(19)

l=0

From the generating function of the Stirling number of the second kind, we can derive n−1 ∑ (n + l − 1) ( λ )l x t −n n−1 Sn (x) = x(1 − λe ) x = (et − 1)l xn−1 (1 − λ)n l 1−λ l=0 ( )( )l ( ) n−1 n−1−l ∑ ∑ x n+l−1 λ n − 1 n−m−l−1 = l!S2 (m + l, l) x (1 − λ)n l 1−λ m+l l=0 m=0 ( ) n−1 ∑ n−1−l ∑ (n + l − 1) ( λ )l x n−1 r = l!S (n − 1 − r, l) x 2 (1 − λ)n l 1−λ r l=0 r=0 n−1 ∑ n−1−r ∑ (n − 1)(n + l − 1) ( λ )l 1 l! = S2 (n − 1 − r, l)xr+1 . r l (1 − λ)n r=0 1−λ l=0 (20) For n ≥ 1, by (18) and xn ∼ (1, t), we get )n ( t xn−1 = x(1 − λm emt )−n xn−1 tn (x) = x t(1 − λm emt ) ( )l ∑ n−1 l ( ) ∑ (−n) λm l m −n l = x(1 − λ ) (−1) (−1)l−r ermt xn−1 m l 1−λ r r=0 l=0 )l ( ) )( n−1 l ( m ∑∑ n+l−1 λ l m −n (−1)l−r (x + rm)n−1 . = x(1 − λ ) m 1 − λ r l l=0 r=0 (21)

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6

By the same method as (20), we get tn (x) = x(1 − λm emt )−n xn−1 n−1 ∑ (−n) ( λm )l = x(1 − λm )−n (−1)l (emt − 1)l xn−1 l 1 − λm l=0 n−1−l n−1 ∑ l!S2 (r + l, l) ∑ (−n) ( λm )l x l = (−1) m n m (1 − λ ) 1−λ (r + l)! l r=0 l=0

× (n − 1)r+l mr+l xn−r−l−1 n−1 ∑ n−1−l ∑ (−n) ( λm )l l!S2 (n − r − 1, l) x (−1)l = m (1 − λm )n 1 − λ (n − r − 1)! l r=0 l=0

× (n − 1)n−r−1 mn−r−1 xr )l ( ) ( )( n−1 ∑ n−1−r ∑ 1 n−1 λm l −n = l!S2 (n − r − 1, l) (−1) (1 − λm )n r=0 1 − λm r l ×m

l=0 n−r−1 r+1

x

.

(22)

Therefore, by (20) and (22), we obtain the following theorem. Theorem 3. For n ≥ 1, let Sn (x) ∼ (1, t(1 − λet )), tn (x) ∼ (1, t(1 − λm emt )), where λ ∈ C with λm ̸= 1. Then we have n−1 ∑ n−1−r ∑ (n − 1)(n + l − 1) ( λ )l 1 Sn (x) = l! S2 (n − 1 − r, l)xr+1 , (1 − λ)n r=0 r l 1−λ l=0

and n−1 ∑ n−1−r ∑ (n + l − 1) ( λm )l (n − 1) 1 tn (x) = l!S2 (n − r − 1, l) (1 − λm )n r=0 l 1 − λm r l=0

× mn−r−1 xr+1 . By (11) and (18), we get )n m 1 ∑ l lt x−1 tn (x) Sn (x) = x x tn (x) = x λe λet l=1 ( ) ∑ x n = n (e−nt ) (λet )v1 +2v2 +···+mvm x−1 tn (x) λ v1 , · · · , v m 0 ≤ v1 , · · · , vm ≤ n v1 + · · · + vm = n ( ∞ ( ) ∑ x −nt ∑ n = n (e ) λv1 +2v2 +···+mvm λ v1 , · · · , v m k=0 0 ≤ v1 , · · · , vm ≤ n v1 + · · · + vm = n ) k k t × (v1 + 2v2 + · · · + mvm ) x−1 tn (x). (23) k! (

t(1 − λm emt ) t(1 − λet )

)n

(

−1

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From (16), (21) and (23), we have ( s ( ) ) ∞ ∑ ∑ s ts −1 −n s−k (n) Sn (x) = λ x (−n) Dk (m | λ) x tn (x) k s! s=0 k=0 ∞ ∑ s ( ) n−1 ∑ ∑ n−1−r ∑ (n − 1) s (n) = λ−n x (−n)s−k Dk (m | λ)(1 − λm )−n k r s=0 k=0 r=0 l=0 ) ( ) ( l ts n+l−1 λm mn−1−r S2 (n − 1 − r, l) xr × l! m 1−λ s! l ( )( )( )( ) s n−1 n−1−r r ∑ ∑ ∑∑ r s n−1 n+l−1 = (λ − λm+1 )−n s k r l r=0 l=0 s=0 k=0 ( ) l λm (n) × (−n)s−k mn−1−r l!Dk (m | λ)S2 (n − 1 − r, l)xr−s+1 . 1 − λm (24) Therefore, by Theorem 3 and (24), we obtain the following theorem. Theorem 4. For n ≥ 1, λ ∈ C with λm ̸= 1, we have n−1 ∑ n−1−r ∑ (n + l − 1) ( λ )l (n − 1) 1 l!S2 (n − r − 1, l)xr+1 n 1−λ r l (1 − λ) r=0 l=0 )( ) n−1 r ∑ s ( )( )( ∑ n−1−r ∑ ∑ r s n−1 n+l−1 m+1 −n = (λ − λ ) (−n)s−k mn−1−r s k r l r=0 l=0 s=0 k=0 )l ( m λ (n) l!Dk (m | λ)S2 (n − 1 − r, l)xr−s+1 . × 1 − λm Let Sn (x) ∼

( 1,

)

t 1 + λet

, tn (x) ∼

( 1,

)

t 1 + (−1)m+1 λm emt

,

(25)

where (−λ)m ̸= 1. From x ∼ (1, t) and (25), we have t n n−1

Sn (x) = x(1 + λe ) x

n ( ) ∑ n l =x λ (x + l)n−1 . l

(26)

l=0

By(1), we get ( Sn (x) = x(1 + λet )n xn−1 = x(1 + λ)n = (1 + λ)n

n−1 ∑( l=0

λ−1 + et λ−1 + 1

)n xn−1

) n−1 (−n) Hl (−λ−1 )xn−l , where λ ̸= 0. l

(27)

For n ≥ 1, by xn ∼ (1, t) and (25), we get tn (x) = x(1 + (−1)m+1 λm emt )n xn−1 = x

n ( ) ∑ n (−1)l (−λ)ml (x + lm)n−1 . (28) l l=0

From the generating function of the Stirling number of the second kind and (28), we have

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8

tn (x) = x(1 + (−1)m+1 λm emt )n xn−1 ( mt )l n ( ) ∑ n e −1 m n l ml = (1 − (−λ) ) x (−1) (−λ) xn−1 l 1 − (−λ)m l=0 ( ) ( ) n−1 ∑ n−l−1 ∑ (−λ)m+l n l n−1 m n l (−1) l!S2 (n − r − 1, l)mn−r−1 xr+1 . = (1 − (−λ) ) m )l (1 − (−λ) r l=0 r=0 (29) By (11) and (25), we get (

)n 1 + (−1)m+1 λm emt tn (x) = x x−1 Sn (x) 1 + λet (( )n )∑ m 1 =x (−λet )l x−1 Sn (x) −λet l=1 ( ) ∑ n −n −nt (−λet )v1 +2v2 +·+mvm x−1 Sn (x) = x(−λ) e v1 , · · · , v m 0 ≤ v1 , · · · , vm ≤ n v1 + · · · + vm = n )( ∞ (∞ ( ) ∑ ∑ ∑ (−n)l n l −n t = x(−λ) l! v1 , · · · , v m k=0 l=0 0 ≤ v1 , · · · , vm ≤ n v1 + · · · + vm = n ) k v1 +2v2 +·+mvm kt −1 × (−λ) (v1 + 2v2 + · + mvm ) x Sn (x) k! { s ( ) ( ) ∞ ∑ ∑ ∑ n −n s−k s = x(−λ) (−n) k v1 , · · · , vm s=0 k=0 0 ≤ v1 , · · · , v m ≤ n v1 + · · · + vm = n } s v1 +2v2 +·+mvm k t × (−λ) (v1 + 2v2 + · + mvm ) x−1 Sn (x). s! (30) (n)

Let us define Tk (m | λ) as follows: ∑ (n) Tk (m | λ) =

(

0 ≤ v1 , · · · , vm ≤ n v1 + · · · + vm = n

) n (−λ)v1 +2v2 +·+mvm v1 , · · · , v m

× (v1 + 2v2 + · + mvm )k .

(31)

From (26), (30) and (31), we have ( ) ∞ ∑ s n ( ) ∑ s ts ∑ n l (n) λ (x + l)n−1 tn (x) = x(−λ)−n (−n)s−k Tk (m | λ) s! l k s=0 k=0 l=0 ) n n−1 s ( )( )( ∑ ∑∑ n s n−1 (n) = x(−λ)−n (−n)s−k λl Tk (m | λ)(x + l)n−1−s . l k s l=0 s=0 k=0 (32)

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Therefore, by (28) and (32), we obtain the following theorem. Theorem 5. For n ≥ 1, λ ∈ C with (−λ)m ̸= 1, we have n ( ) ∑ n (−1)l (−λ)ml (x + lm)n−1 l l=0 ) n n−1 s ( )( )( ∑ ∑∑ n s n−1 (n) −n = (−λ) (−n)s−k λl Tk (m | λ)(x + l)n−1−s . l k s s=0 l=0

k=0

Remark. Let Sn (x) ∼ (1, t(1 + λet )), tn (x) ∼ (1, t(1 + (−1)m+1 λm emt )), (−λ)m ̸= 1.

(33)

For n ≥ 1, by xn ∼ (1, t) and (33), we get )( ) ( )l n−1 l ( ∑∑ x n+l−1 l λ Sn (x) = (−1)r (x + r)n−1 (1 + λ)n l r 1 + λ l=0 r=0 n−1 n−1−r ∑ ∑ (n + l − 1)(n − 1) ( −λ )l x l!S2 (n − r − 1, l)xr , = (1 + λ)n r=0 l r 1+λ l=0 (34) and tn (x) = (1 − (−λ)m )−n

n−1 ∑ n−1−l ∑ ( l=0

r=0

)l )( )( n+l−1 n−1 (−λ)m l! 1 − (−λ)m l r × S2 (n − 1 − r, l)mn−1−r xr+1 . (35)

From (11), (33), (34) and (35), we have m+1 −n

Sn (x) = (λ + (−λ)

)

n−1 l n−1 s ( ∑∑ ∑∑ l=0 r=0 s=0 k=0 )l m

( × (−1)l−r

(−λ) 1 − (−λ)m

)( )( )( ) n−1 s n+l−1 l (−n)s−k s k l r

(n)

Tk (m | λ)x(x + rm)n−1−s .

(36)

Let Sn (x) ∼ (1, ebt − 1), tn (x) ∼ (1,

t2 ), b ̸= 0. ebt − 1

From xn ∼ (1, t) and (37), we have ( )n (x) t x (n) Sn (x) = x bt xn−1 = n (bn−1 Bn−1 ) e −1 b b ) n−1 ( x ∑ n−1 (n) = n Bl bl xn−1−l , b l

(37)

(38)

l=0

and ( tn (x) = x

ebt − 1 t

)n n−1

x

n

=b x

n−1 ∑ l=0

(n−1) l b (n+l ) S2 (n + l, l)xn−1−l . l

(39)

l

Therefore, by (37), (38) and (39), we obtain the following theorem.

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D. S. KIM1 , T. KIM2 , W. J. KIM3 , AND DMITRY V. DOLGY4

10

Theorem 6. For n ≥ 1, b ̸= 0, let (x) t2 ∼ (1, ebt − 1), tn (x) ∼ (1, bt ). Sn (x) = b n e −1 Then we have ) n−1 ( (x) x ∑ l n−1 (n) Sn (x) = = n Bl xn−1−l , b b n b l l=0

and n

tn (x) = b x

n−1 ∑ l=0

(n−1) l b (n+l ) S2 (n + l, l)xn−1−l . l

l

From Theorem 6, we note that n−1 (x) ∑ (n − 1) (n) −n =m x ml Bl xn−1−l ∼ (1, emt − 1). m n l

(40)

l=0

Let b = 1. Then, by Theorem 6, we get n−1 ∑ (n − 1) (n) (x)n = x Bl xn−1−l ∼ (1, et − 1). l

(41)

l=0

From (11), (40) and (41), we have ( mt )n n−1 (x) ∑ (n − 1) (n) e −1 n−1−l −1 (x)n = x Bl x =x x l et − 1 m n l=0 ( ) n m (x) (x) ( t ) 1 ∑ lt −1 −nt 2t mt n −1 =x e x = xe e + e + · · · + e x et m n m n l=1 ( ) (x) ∑ n = xe−nt e(v1 +2v2 +···+mvm )t x−1 v1 , v 2 , · · · , v m m n 0 ≤ v1 , · · · , v m ≤ n v1 + · · · + vm = n )( ∞ ( (∞ ( ) ∑ ∑ ∑ (−n)l n l =x t l! v1 , v2 , · · · , vm k=0 l=0 0 ≤ v1 , · · · , vm ≤ n v1 + · · · + vm = n ) ) (x) k kt −1 × (v1 + 2v2 + · · · + mvm ) x k! m n { ( ) ( ) ∞ s ∑ ∑ ∑ n s =x (−n)s−k v , v , k 1 2 · · · , vm s=0 k=0 0 ≤ v1 , · · · , v m ≤ n v1 + · · · + vm = n } (x) s k t × (v1 + 2v2 + · · · + mvm ) x−1 . (42) s! m n (n)

Let us define Sk (m) as follows: ( ) ∑ n (n) Sk (m) = (v1 + 2v2 + · · · + mvm )k . (43) v1 , v 2 , · · · , v m 0 ≤ v1 , · · · , v m ≤ n v1 + · · · + vm = n

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11

Then, by (40), (42) and (43), we get { } ∞ ∑ s ( ) n−1 ∑ ∑ (n − 1) (n) ts s s−k (n) −n l n−1−l (−n) Sk (m) (x)n = x m m Bl x k s! l s=0 k=0 l=0 ( )( ) n−1 s s ∑ n−1−l ∑ ∑ n−1 (n) (n) t −n+l s (−n)s−k Sk (m)Bl m xn−1−l =x k l s! l=0 s=0 k=0 ( )( )( ) n−1 s ∑ n−1−l ∑ ∑ s n−1 n−1−l (n) (n) =x m−n+l (−n)s−k Sk (m)Bl xn−1−l−s k l s l=0 s=0 k=0 ( )( )( ) n−1 ∑ n−1−l ∑ n−1−l−r ∑ n−1−l−r n−1 n−1−l =x m−n+l k l r r=0 l=0

k=0

(n)

(n)

× (−n)n−1−l−r−k Sk (m)Bl xr .

(44)

Therefore, by (44), we obtain the following theorem. Theorem 7. For n ≥ 1, we have (x)n =

n−1 ∑ n−1−l ∑ n−1−l−r ∑ l=0

r=0

k=0

m−n+l

(

n−1−l−r k

)(

)( ) n−1 n−1−l l r (n)

(n)

× (−n)n−1−l−r−k Sk (m)Bl xr+1 .

References [1] S. Araci and M. Acikgoz, A note on the Frobenius-Euler numbers and polynomials associated with Bernstein polynomials, Adv. Stud. Contemp. Math . 22 (2012) no.3, 399-406. [2] S. Araci, M. Acikgoz, H. Jolany and J. J. Seo, A unied generating function of the q-Genocchi polynomials with their interpolation functions , Proc. Jangjeon Math. Soc., 15 (2012) no.2, 227-233. [3] M. Can, M. Cenkci, V. Kurt and Y. Simsek, Twisted Dedekind type sums associated with Barnes’ type multiple Frobenius-Eulerl-functions, Adv. Stud. Contemp. Math. 18 (2009), no. 2, 135-160. [4] L.Carlitz, The product of two Eulerian polynomials, Math. Mag. 368 (1963) 37-41. [5] R. Dere and Y. Simsek, Applications of umbral algebra to some special polynomials, Adv. Stud. Contemp. Math. 22 (2012) 433-438. [6] J. Y. Kang and C. S. Ryoo, n multiple interpolation functions of the q-Genocchi numbers and polynomials with weight and weak weight , Adv. Stud. Contemp. Math., 22 (2012) no. 3, 407-420. [7] D. S. Kim and T. Kim, Bernoulli basis and the product of several Bernoulli polynomials , Int. J. Math. Math. Sci. 2012, Art. ID 463659, 12 pp. [8] T. Kim, q-Bernoulli numbers and polynomials associated with Gaussian binomial coecients , Russ. J. Math. Phys. 15 (2008), no. 1, 51-57. [9] T. Kim, Identities involving Frobenius-Euler polynomials arising from non-linear dierential equations, J. Number Theory. 132 (2012), no. 1, 2854-2865. [10] T. Kim, An identity of the symmetry for the Frobenius-Euler polynomials associated with the fermionic p-adic invariant q-integrals on Zp , Rocky Mountain J. Math. 41 (2011), no. 1, 239-247. [11] T. Kim A note on q-Bernstein polynomials , Russ. J. Math. Phys. 18 (2011), no. 1, 73-82. [12] T. Kim Symmetry p-adic invariant integral on Zp for Bernoulli and Euler polynomials , J. Difference Equ. Appl. 14 (2008), no. 12, 1267-1277. [13] D. S. Kim, T. Kim, S. H. Lee and S. H. Rim ome identities of Bernoulli, Euler and Abel polynomials arising from umbral calculus , Adv. Difference Equ. (2013), 2013:15. [14] H. Ozden, I. N. Cangul and Y. Simsek, Remarks on q-Bernoulli numbers associated with Daehee numbers , Adv. Stud. Contemp. Math. 18 (2009), no. 1, 41-48. [15] S. Roman, The umbral calculus, Dover Publ. Inc. New York, (2005).

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D. S. KIM1 , T. KIM2 , W. J. KIM3 , AND DMITRY V. DOLGY4

12 1

Department of Mathematics, Sogang University, Seoul 121-742, Republic of Korea.

2 Department of Mathematics, Kwangwoon University, Seoul 139-701, Republic of Korea. E-mail address: [email protected] 3 Department of Mathematics, Kwangwoon University, Seoul 139-701, Republic of Korea. 4

Hanrimwon, Kwangwoon University, Seoul 139-701, Republic of Korea. E-mail address: [email protected]

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Dynamics and Behavior of a Second Order Rational Difference equation E. M. Elsayed1;2 , M. M. El-Dessoky1;2 , and Asim Asiri1 1 King Abdulaziz University, Faculty of Science, Mathematics Department, P. O. Box 80203, Jeddah 21589, Saudi Arabia. 2 Department of Mathematics, Faculty of Science, Mansoura University, Mansoura 35516, Egypt. E-mail: [email protected], [email protected], [email protected] ABSTRACT In this paper we investigate the global convergence result, boundedness, and periodicity of solutions of the difference equation xn+1 = axn +

b + cxn d + exn

1

;

n = 0; 1; :::;

1

where the parameters a; b; c; d and e are positive real numbers and the initial conditions x 1 and x0 are positive real numbers.

Keywords: stability, periodic solutions, boundedness, difference equations. Mathematics Subject Classi cation: 39A10 —————————————————

1

Introduction

Difference equations have been used to describe evolution phenomena since most measurements of time-evolving variables are discrete. More signi cantly, difference equations are used in the study of discretization methods for differential equations. The theory of difference equations has some results that have been acquired approximately as natural discrete analogues of corresponding results of differential equations [35]. The study of rational difference equations of order greater than one is quite ambitious and worthwhile since some paradigms for the development of the basic theory of the global behavior of nonlinear difference equations of order greater than one come from the results of rational difference equations. However, there have not been any useful general methods to study the global behavior of rational difference equations of order greater than one so far. Therefore, the study of rational difference equations of order greater than one deserves further consideration. Many research have been done to study the global attractivity, boundedness character, periodicity and the solution form of nonlinear difference equations. For example, Agarwal et al. [2]

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looked at the global stability, periodicity character and found the solution form of some special cases of the difference equation xn+1 = a +

dxn l xn b cxn

k

:

s

The form of the solutions of the difference equation xn 1 a xn xn

xn+1 =

; 1

was obtained by Aloqeili [4]. The dynamics, the global stability, periodicity character and the solution of special case of the recursive sequence bxn cxn dxn

xn+1 = axn

; 1

was investigated by Elabbasy et al in [8]. Elabbasy et al. [9] studied the behavior of the difference equation, especially global stability, boundedness, periodicity character and gave the solution of some special cases of the difference equation xn+1 =

xn Qk

k

i=0 xn

+

: i

Karatas et al. [31] researched the behavior of the solutions of the difference equation xn+1 =

axn (2k+2) Q a + 2k+2 i=0 xn

: i

In [36] Simsek et al. acquired the solution of the difference equation xn+1 =

xn 3 1 + xn

: 1

The dynamics of the difference equation xn+1 =

+

xn m ; xkn

was studied by Yalç nkaya et al. in [44]. Zayed et al. [46], [47] looked at the behavior of the following rational recursive sequences xn+1 = axn

cxn

bxn dxn

;

xn+1 = Axn + Bxn

k

k

+

pxn + xn k : q + xn k

Other related results on rational difference equations and systems can be found in refs. [1-45]. This paper aims to study the global stability character and the periodicity of solutions of the difference equation xn+1 = axn +

b + cxn d + exn

1

;

(1)

1

where the parameters a; b; c; d and e are positive real numbers and the initial conditions x

1

and x0 are positive real numbers.

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2

Some Basic Properties and De nitions

In this section, we state some basic de nitions and theorems that we need in this paper. Suppose that I is an interval of real numbers and let F :I

I ! I;

be a continuously differentiable function. Then for every set of initial conditions x

1 ; x0

difference equation xn+1 = F (xn ; xn has a unique solution fxn g1 n=

1 );

2 I; the (2)

n = 0; 1; :::;

1.

De nition 2.1. (Equilibrium Point) A point x 2 I is called an equilibrium point of Eq.(2) if x = F (x; x). That is, xn = x for n

0; is a solution of Eq.(2), or equivalently, x is a xed point of F .

De nition 2.2. (Periodicity) A sequence fxn g1 n=

1

is said to be periodic with period p if xn+p = xn for all n

1:

De nition 2.3. (Stability) (i) The equilibrium point x of Eq.(2) is locally stable if for every > 0; there exists that for all x

1 ; x0

2 I with

jx

1

xj + jx0

> 0 such

xj < ;

we have jxn

xj
0; such that for all x jx

1

xj + jx0

1;

x0 2 I with

xj < ;

we have lim xn = x:

n!1

(iii) The equilibrium point x of Eq.(2) is global attractor if for all x

1 ; x0

2 I; we have

lim xn = x:

n!1

(iv) The equilibrium point x of Eq.(2) is globally asymptotically stable if x is locally stable, and x is also a global attractor of Eq.(2). (v) The equilibrium point x of Eq.(2) is unstable if x is not locally stable. The linearized equation associated with Eq.(2) about the equilibrium pointx is the linear difference equations yn+1 = pyn + qyn 796

1:

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where p=

@F (x; x); @xn

q=

@F (x; x): @xn 1

Theorem A [34]: (Linearized Stability) (a) If both roots of the quadratic equation 2

p

(3)

q = 0:

lie in the open unit disk j j < 1, then the equilibrium x of Eq.(2) is locally asymptotically stable. (b) If at least one of the roots of Eq.(3) has absolute value greater than one, then the equilibrium

x of Eq.(2) is unstable. (c) A necessary and suf cient condition for both roots of Eq.(3) to lie in the open unit disk j j < 1, is

jpj < 1

(4)

q < 2:

In this case the locally asymptotically stable equilibrium x is also called a sink. Now, consider the following equation xn+1 = g(xn ; xn

(5)

1 ):

The following two theorems will be useful for the proof of our results in this paper. Theorem B [34]: Suppose that [ ; ] is an interval of real numbers and assume that g : [ ; ]2 ! [ ; ]; is a continuous function satisfying the following properties: (a) g(x; y) is non-decreasing in each of its arguments; (b) The equation g(x; x) = x; has a unique positive solution. Then Eq.(5) has a unique equilibrium point x 2 [ ; ] and every

solution of Eq.(5) converges to x:

Theorem C [34]: Suppose that [ ; ] is an interval of real numbers and let g : [ ; ]2 ! [ ; ]; be a continuous function that satis es the following properties : (a) g(x; y) is non-decreasing in x in [ ; ] for each y 2 [ ; ]; and is non-increasing in y 2 [ ; ]

for each x in [ ; ];

(b) If (m; M ) 2 [ ; ]

[ ; ] is a solution of the system M = g(M; m)

and

m = g(m; M );

then m = M: Then Eq.(5) has a unique equilibrium point x 2 [ ; ] and every solution of Eq.(5) converges to

x:

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3

Local Stability of the Equilibrium Point of Eq.(1)

In this section, we study the local stability character of the equilibrium point of Eq.(1). Eq.(1) has equilibrium point and is given by x = ax +

b + cx ; d + ex

or a)x2 + (d

e(1

da

c)x

b = 0:

Then the only positive equilibrium point of Eq.(1) is given by p (c d+da)+ (c d+da)2 +4be(1 x= 2e(1 a)

a)

:

Theorem 3.1. The equilibrium x of Eq. (1) is locally asymptotically stable if and only if (d + ex)2 >

jcd (1

bej ; a)

a < 1:

Proof: Let f : (0; 1)2 ! (0; 1) be a continuous function de ned by f (u; v) = au +

b + cv : d + ev

(6)

Therefore, @f (u; v) @f (u; v) (cd be) = a; = . @u @v (d + ev)2 So, we can write @f (x; x) @f (x; x) (cd be) = a = p; = = q: @u @v (d + ex)2 Then the linearized equation of Eq.(1) about x is yn+1

pyn

(7)

qyn = 0:

1

It follows by Theorem A that, Eq.(1) is asymptotically stable if and only if jpj < 1

q < 2:

Thus, jaj +

(cd be) < 1; (d + ex)2

and so (cd be) (d + ex)2 jcd or

< (1

a);

bej < (d + ex)2 (1 jcd (1

bej < (d + ex)2 ; a)

a < 1; a);

a < 1;

a < 1:

The proof is complete.

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4

Existence of Bounded and Unbounded Solutions of Eq.(1)

Here we look at the boundedness nature of solutions of Eq.(1). Theorem 4.1. Every solution of Eq.(1) is bounded if a < 1: Proof: Let fxn g1 n=

1

be a solution of Eq.(1). It follows from Eq.(1) that

xn+1 = axn + Then xn+1

axn +

b + cxn d + exn b cxn + d exn

1

= axn +

1

1

= axn +

1

b d + exn

+ 1

b c + d e

cxn 1 d + exn

for all

n

: 1

1.

By using a comparison, the right hand side can be written as follows yn+1 = ayn +

b c + : d e

So, we can write yn = an y0 + constant; and this equation is locally asymptotically stable because a < 1; and converges to the equilibbe + cd . rium point y = de(1 a) Therefore be + cd . lim supxn de(1 a) n!1 Hence, the solution is bounded. Theorem 4.2. Every solution of Eq.(1) is unbounded if a > 1: Proof: Let fxn g1 n=

1

be a solution of Eq.(1). Then from Eq.(1) we see that xn+1 = axn +

b + cxn d + exn

1

> axn

for all

n

1.

1

The right hand side can be written as follows yn+1 = ayn

) yn = an y0 ;

and this equation is unstable because a > 1; and lim yn = 1: Then by using ratio test

fxn g1 n=

5

1

n!1

is unbounded from above.

Existence of Periodic Solutions

In this section we investigate the existence of periodic solutions of Eq.(1). The following theorem states the necessary and suf cient conditions that this equation has periodic solutions of prime period two. Theorem 5.1. Eq.(1) has positive prime period two solutions if and only if (i) [c

ad

d]2 (1 + a) + 4 [be + (c

799

ad

d)ad] > 0:

(8)

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Proof: Firstly, suppose that there exists a prime period two solution :::; p; q; p; q; :::; of Eq.(1). We will show that Condition (i) holds. From Eq.(1), we get p = aq +

b + cp ; d + ep

q = ap +

b + cq : d + eq

and

Therefore, dp + ep2 = adq + aepq + b + cp;

(9)

dq + eq 2 = adp + aepq + b + cq:

(10)

and Subtracting (10) from (9) gives q) + e(p2

d(p Since p 6= q; it follows that

q2) =

ad(p

c

p+q =

ad e

q) + c(p d

q).

(11)

:

Again; adding (9) and (10) yields d(p + q) + e(p2 + q 2 ) = ad(p + q) + 2aepq + 2b + c(p + q); e(p2 + q 2 ) = (ad

d + c)(p + q) + 2aepq + 2b:

(12)

By using (11); (12) and the relation p2 + q 2 = (p + q)2

2pq for all p; q 2 R;

we obtain e((p + q)2

2pq) = (ad

2(1 + a)epq =

d + c)(p + q) + 2aepq + 2b

2ad(p + q)

2b:

ad d)ad (1 + a)e2

be

Then, pq =

(c

(13)

:

Now it is obvious from Eq.(11) and Eq.(13) that p and q are the two distinct roots of the quadratic equation t2 et2

c

ad e

d

t

(c

ad

d)t

[c

ad

d]2 +

be + (c ad d)ad (1 + a)e2 be + (c ad d)ad (1 + a)e

= 0; = 0;

(14)

and so 4[be+(c ad d)ad] (1+a)

800

> 0;

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or [c

d]2 (1 + a) + 4 [be + (c

ad

ad

d)ad] > 0:

Therefore inequality (i) holds. Conversely, suppose that inequality (i) is true. We will prove that Eq.(1) has a prime period two solution. Suppose that p= and q= where

q = [c

d]2 +

ad

c

ad d + ; 2e

c

ad d 2e

;

4[be+(c ad d)ad] : (1+a)

We see from the inequality (i) that [c

d]2 (1 + a) + 4 [be + (c

ad

ad

d)ad] > 0;

which equivalents to [c

d]2 +

ad

4[be+(c ad d)ad] (1+a)

> 0:

Therefore p and q are distinct real numbers. Set x

1

= p and x0 = q:

We would like to show that x1 = x

1

= p and x2 = x0 = q:

It follows from Eq.(1) that b + cp x1 = aq + =a d + ep

c ad d 2e

c ad d+ 2e c ad d+ d+e 2e b+c

+

:

Dividing the denominator and numerator by 2(d + ae) we get x1 = a

c ad d 2e

+

2eb+c(c ad d+ ) 2ed+e(c ad d+ ) :

Multiplying the denominator and numerator of the right side by 2ed + e (c

ad

d

) and by

computation we obtain x1 = p: Similarly as before, it is easy to show that x2 = q: Then by induction we get x2n = q

and

x2n+1 = p

for all

n

1.

Thus Eq.(1) has the prime period two solution :::;p;q;p;q;:::; where p and q are the distinct roots of the quadratic equation (14) and the proof is complete.

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6

Global Attractivity of the Equilibrium Point of Eq.(1)

In this section, the global asymptotic stability of Eq.(1) is studied. Lemma 6.1. For any values of the quotient

b d

and ec , the function f (u; v) de ned by Eq.(6) has

the monotonicity behavior in its two arguments. Proof: The proof follows by some computations and it will be omitted. Theorem 6.2. The equilibrium point x is a global attractor of Eq.(1) if one of the following statements holds be and c > d(1

(1) cd

(2) cd Proof: Suppose that

and

(15)

a); a < 1:

be and a < 1.

(16)

are real numbers and assume that g : [ ; ]2

function de ned by

! [ ; ] is a

b + cv : d + ev

g(u; v) = au + Then @g(u;v) @u

@g(u;v) @v

= a;

=

(cd be) . (d+ev)2

Now, two cases must be considered : Case (1): Suppose that (15) is true, then we can easily see that the function g(u; v) increasing in u; v: Let x be a solution of the equation x = g(x; x): Then from Eq.(1), we can write x = ax +

b+cx d+ex ;

or x(1

a) =

b+cx d+ex ;

then the equation a)x2 + fd(1

e(1

a)

has a unique positive solution when c > d(1 x=

(c d(1 a))+

cgx

b = 0;

a); a < 1 which is p

(c d(1 a))2 +4be(1 a) ; 2e(1 a)

By using Theorem B, it follows that x is a global attractor of Eq.(1) and then the proof is complete. Case (2): Suppose that (16) is true, let

and

be real numbers and assume that g : [ ; ]2 ! b + cv [ ; ] be a function de ned by g(u; v) = au + , then we can easily see that the function d + ev g(u; v) increasing in u and decreasing in v: Let (m; M ) be a solution of the system M = g(M; m) and m = g(m; M ). Then from Eq.(1), we see that M = aM +

b + cm ; d + em

m = am +

b + cM ; d + eM

M (1

b + cm ; d + em

m(1

b + cM ; d + eM

or a) =

802

a) =

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J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 16, NO.4, 2014, COPYRIGHT 2014 EUDOXUS PRESS, LLC

then d(1

a)M + e(1

a)M m = b + cm; d(1

a)m + e(1

a)mM = b + cM:

Subtracting we obtain (M

m)fd(1

a)(M + m) + cg = 0;

under the condition a < 1; we see that M = m: It follows by Theorem C that x is a global attractor of Eq.(1). This completes the proof of the theorem.

7

Numerical examples

To con rm the results of this paper, we consider numerical examples which represent different types of solutions to Eq. (1). Example 1. We assume that x

1

= 7; x0 = 11; a = 0:1; b = 2; c = 5; d = 3; e = 7. See Fig.

1. p l o t o f x( n + 1 ) = a x( n ) + ( b + c x( n - 1 ) ) /( d + e x( n - 1 ) ) 12

10

x(n)

8

6

4

2

0 0

5

10

15

n

Figure 1. Example 2. See Fig. 2, since x

1

= 13; x0 = 5; a = 0:8; b = 7; c = 2; d = 0:4; e = 2. p l o t o f x( n + 1 ) = a x( n ) + ( b + c x( n - 1 ) ) /( d + e x( n - 1 ) )

13 12 11

x(n)

10 9 8 7 6 5 0

5

10

15

20

25

30

n

Figure 2.

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Example 3. We consider x

1

= 2; x0 = 5; a = 1:2; b = 8; c = 5; d = 4; e = 1. See Fig. 3. p l o t o f x( n + 1 ) = a x( n ) + ( b + c x( n - 1 ) ) /( d + e x( n - 1 ) )

700 600 500

x(n)

400 300 200 100 0 0

2

4

6

8

10

12

14

16

18

20

n

Figure 3.

Example 0 4. Fig. 4. shows thersolutions when a = 2; b = 1 1; c = 11; d = 3; e = 4; x x0 = q: @Since p; q =

c ad d

4[be+(c ad d)ad] [c ad d] + (1+a) 2e

1

= p;

2

:A

p l o t o f x( n + 1 ) = a x( n ) + ( b + c x( n - 1 ) ) /( d + e x( n - 1 ) ) 1 0 .8 0 .6

x(n)

0 .4 0 .2 0 - 0 .2 - 0 .4 0

2

4

6

8

10

12

14

16

18

20

n

Figure 4.

Acknowledgements This article was funded by the Deanship of Scienti c Research (DSR), King Abdulaziz University, Jeddah. The authors, therefore, acknowledge with thanks DSR technical and nancial support.

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TABLE OF CONTENTS, JOURNAL OF COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 16, NO. 4, 2014 Disjoint Mixing Weighted Backward Shifts on the Space of All Complex Valued Square Summable Sequences, Liang Zhang, and Ze-Hua Zhou,………………………………………618 Some Identities Involving Associated Sequences of Special Polynomials, Taekyun Kim, and Dae San Kim,………………………………………………………………………………….626 Some New Integral Inequalities of the Type of Hermite-Hadamard's for the Mappings Whose Absolute Values of Their Derivative are Convex, Muhammad Iqbal, Muhammad Iqbal Bhatti, and Muhammad Muddassar,………………………………………………………………….643 A Generalized Additive Functional Inequality in Banach Spaces, Choonkil Park, Gang Lu, and Dong Yun Shin,……………………………………………………………………………….654 An Efficient Spectral Collocation Algorithm for Solving Neutral Functional-Differential Equations, L.M. Assas, A.H. Bhrawy, and M.A. Alghamdi,…………………………………661 Some Properties of Intuitionistic Fuzzy Metric Spaces, Chang-qing Li,………………………670 A Certain Class of Harmonic Mappings Related to Functions of Bounded Boundary Rotation, Yaşar Polatoğlu, Emel Yavuz Duman, and Melike Aydoğan,…………………………………678 Fuzzy Norms on BCK-Algebras and Non-Negativity of Norms in Algebras, Jeong Soon Han, and Keum Sook So,……………………………………………………………………………687 Extended Cesàro Operator from Hardy Space to Zygmund-Type Space on the Unit Ball, Yu-Xia Liang, and Hong-Gang Zeng,…………………………………………………………694 Some Special Polynomials and Sheffer Sequences, Dae San Kim, Taekyun Kim, Sang-Hun Lee, and Dmitry V. Dolgy,…………………………………………………………………………702 Pseudo-differentiability, Pseudo-integrability and Nonlinear Differential Equations, Zeng-tai Gong, and Ting Xie,……………………………………………………………………………713 A General Theorem Associated with the Briot-Bouquet Differential Subordination, Adel A. Attiya, and Teodor Bulboacă,…………………………………………………………………722 An SVD Free Construction of an Indicator Function as an Imaging Algorithm, K. Kim, K. H. Leem, and G. Pelekanos,………………………………………………………………………731

TABLE OF CONTENTS, JOURNAL OF COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 16, NO. 4, 2014 (continued)

Applications of Coupled N-structures in BCC-Algebras, Young Bae Jun, and Sun Shin Ahn,740 Analytical Solution of Nonlinear Second-Order Periodic Boundary Value Problem Using Reproducing Kernel Method, Nabil Shawagfeh, Omar Abu Arqub, and Shaher Momani,……750 Generalized Chebyshev Inequalities with Applications, Marwan A Kutbi, Nawab Hussain, Arif Rafiq, and Mohammad Masjed-Jamei,……………………………………………………763 Comment on "Generalized ternary bi-derivations on ternary Banach algebras: a fixed point approach" [Esgahi Gordji et al., J. Comput. Anal. Appl. 15 (2013) 45-54], Choonkil Park, Madjid Eshaghi, Gordji Jung Rye Lee, and Dong Yun Shin,…………………………………777 Examples of Umbral Calculus, D.S. Kim, T. Kim, W.J. Kim, and Dmitry V. Dolgy,………782 Dynamics and Behavior of a Second Order Rational Difference equation, E.M. Elsayed, M.M. El-Dessoky, and Asim Asiri,……………………………………………………..……794

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J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 16, NO.5, 2014, COPYRIGHT 2014 EUDOXUS PRESS, LLC

OSTROWSKI TYPE INEQUALITIES FOR m AND ( ; m) LOGARITMICALLY CONVEX FUNCTIONS HAVVA KAVURMACI| Abstract. In this paper, we give some informations about the Ostrowski type inequality and ( ; m) logaritmically convex functions. And then, we establish some Ostrowski type inequalities for this class of functions.

1. INTRODUCTION Let f : I [0; 1] ! R be a di¤erentiable mapping on I , the interior of the interval I, such that f 0 2 L [a; b] where a; b 2 I with a < b. If jf 0 (x)j M , then the following inequality holds (see [1]).

(1.1)

f (x)

1 b

a

Z

b

f (u)du

a

M b

a

"

(x

2

a) + (b 2

2

x)

#

:

This inequality is well known in the literature as Ostrowski inequality. For some results which generalize, improve and extend the inequality (1.1) see ([2]-[4]) and the references therein. Let us recall some known de…nitions and results which will be used in this paper. The function f : [a; b] ! R; is said to be convex, if we have f (tx + (1

t) y)

tf (x) + (1

t) f (y)

for all x; y 2 [a; b] and t 2 [0; 1] : In [13], Toader de…ned m convexity as following: De…nition 1. The function f : [0; b] ! R; b > 0, is said to be m convex where m 2 [0; 1]; if we have f (tx + m(1

t)y)

tf (x) + m(1

t)f (y)

for all x; y 2 [0; b] and t 2 [0; 1]: We say that f is m concave if

f is m convex.

For recent results related to above de…nitions we refer interest of readers to [5]-[7], [9], [15] and [12]. In [11], Mihe¸san de…ned ( ; m) convexity as following: De…nition 2. The function f : [0; b] ! R; b > 0 is said to be ( ; m) convex, where ( ; m) 2 [0; 1]2 ; if we have f (tx + m(1

t)y)

t f (x) + m(1

t )f (y)

Date : November 4, 2012. 2000 Mathematics Subject Classi…cation. Primary 26D15; Secondary 26A51. Key words and phrases. ( ; m) logaritmically convex functions, Ostrowski type inequalities, Hölder inequality. Corresponding Author. 1

820

KAVURMACI 820-826

J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 16, NO.5, 2014, COPYRIGHT 2014 EUDOXUS PRESS, LLC

HAVVA KAVURM ACI |

2

for all x; y 2 [0; b] and t 2 [0; 1]: Denote by Km (b) the class of all ( ; m) convex functions on [0; b] for which f (0) 0: If we choose ( ; m) = (1; m), it can be easily seen that ( ; m) convexity reduces to m convexity and for ( ; m) = (1; 1); we have ordinary convex functions on [0; b]: For the recent results based on the above de…nition see the papers [3], [5]-[6], [8] and [10]. The m and ( ; m) logaritmically convex functions were de…ned in [12] as following: De…nition 3. A function f : [0; b] ! (0; 1) ; is said to be m logaritmically convex if the inequality f (tx + m(1

f (x)t f (y)m(1

t)y)

t)

holds for all x; y 2 [0; b]; m 2 (0; 1] and t 2 [0; 1]: Obviously, if putting m = 1 in De…nition 3, then f is just the ordinary logaritmically convex function on [0; b]. De…nition 4. A function f : [0; b] ! (0; 1) ; is said to be ( ; m) logaritmically convex if f (tx + m(1

f (x)t f (y)m(1

t)y)

t )

2

holds for all x; y 2 [0; b]; m 2 (0; 1] and t 2 [0; 1]: Clearly, if taking = 1 in De…nition 4, then f becomes the standart m logaritmically convex function on [0; b]. We will use the following lemma, [2], in proofs of our main results: Lemma 1. Let f : I R ! R be a di¤ erentiable mapping on I where a; b 2 I with a < b: If f 0 2 L [a; b], then the following equality holds: (1.2)

f (x)

1 b

a

Z

b

f (u) du = (a

a

for each t 2 [0; 1] ; where p (t) =

for all x 2 [a; b].

b)

Z

1

p (t) f 0 (ta + (1

t) b) dt

0

8 > >
> : t

t

; 1 ;

h t 2 0; bb t2

x a

i

i

b x b a;1

2. Ostrowski Type Inequalities Theorem 1. Let I [0; 1) be an open real interval and let f : I ! (0; 1) be a q di¤ erentiable function on I such that f 0 2 L [a; b] for 0 a < b < 1. If jf 0 (x)j is

821

KAVURMACI 820-826

J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 16, NO.5, 2014, COPYRIGHT 2014 EUDOXUS PRESS, LLC

AND ( ; m)

m

LOGARITM ICALLY CONVEX

3

2

b ( ; m) logaritmically convex on 0; m for ( ; m) 2 (0; 1] and jf 0 (x)j

Z

1

f (x)

b

a

M;then

b

f (u) du

a

a) M m

(b

1

(p + 1) p 2 4 b x b a where

=

1 1+ p

q(b x) b a

1

q ln

jf 0 (a)j m and jf 0 ( mb )j

1 p

+

1 q

! q1

x b

+

1 1+ p

a a

! q1 3 5

q(b x) b a

q

q ln

= 1, p > 1:

Proof. By using Lemma, De…nition 4 and Hölder integral inequality, we have (2.1) f (x)

(b

b

a)

Z

a b b

x a

0

(b

+

(b

+

=

(b

a) Z

b b

p

t) dt

8 < Z :

b b

x a

(2.2)

tp dt

p

(1

t) dt

x a

a) f 0

b m

m

1

< 1, we have

qt

Z Z

! p1

Z

b b

b b

t) jf 0 (ta + (1

(1 b b

x a

q

jf 0 (ta + (1

t)b)j dt

b b

x a

0

qt

qt

x a

t)b)j dt

jf 0 (a)j

1 b b

q

jf 0 (ta + (1

x a

x a

qt

x a

1

1

Z Z

8 < b : b

b b

0

! p1

Z

t)b)j dt +

! p1

! p1

0

(p + 1) p For

tp dt

x a

1 b b

x a

0

(1 b b

f (u) du

t jf 0 (ta + (1

8 < Z :

b

a

1

a) Z

Z

1

jf 0 (a)j Z

1 1+ p

f0

b b

x a

qt

dt

! q1 9 = ;

dt

mq(1 t )

b m

0

! q1

mq(1 t )

b m

f0

t)b)j dt

dt ! q1

+

x b

! q1

! q1 9 =

a a

;

1 1+ p

Z

1 qt b b

x a

dt

! q1

:

, thereby

x a

qt

Z

dt

0

b b

x a

qt

dt

0 q(b x) b a

=

822

1

q ln

KAVURMACI 820-826

J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 16, NO.5, 2014, COPYRIGHT 2014 EUDOXUS PRESS, LLC

HAVVA KAVURM ACI |

4

and

Z

(2.3)

Z

1 qt b b

dt

x a

1 qt b b

dt

x a q(b x) b a

q

=

:

q ln If we write (2.2)-(2.3) in (2.1) and then use jf 0 (x)j

M; we get the desired result.

Corollary 1. Let I [0; 1) be an open real interval and let f : I ! (0; 1) be a q di¤ erentiable function on I such that f 0 2 L [a; b] for 0 a < b < 1. If jf 0 (x)j is b for m 2 (0; 1] and jf 0 (x)j M;then m logaritmically convex on 0; m Z b 1 f (x) f (u) du b a a a) M m

(b

1

where

(p + 1) p 2 4 b x b a

1 1+ p

q(b x) b a

1

q ln

! q1

x b

+

a a

1 1+ p

q(b x) b a

q

q ln

is in Theorem 1.

Corollary 2. If in Theorem 1, we choose x = f a) M m

(b

1 2

1

(p + 1) p

a+b 2 2 1 1+ p

which is an Ostrowski type inequality.

a+b 2 ,

1 b

a

q 2

4

Z

1

! q1 3 5

we get

b

f (u) du

a 1 q

+

q ln

q 2

q

q ln

1 q

3 5

Theorem 2. Let I [0; 1) be an open real interval and let f : I ! (0; 1) be a q di¤ erentiable function on I such that f 0 2 L [a; b] for 0 a < b < 1. If jf 0 (x)j is 2 b 0 1, jf (x)j M, ( ; m) logaritmically convex on 0; m for ( ; m) 2 (0; 1] , q then Z b 1 f (x) f (u) du b a a a) M m

(b

+ where

"

1 2

x b

8 " > < 1 > : 2 a a

is in Theorem 1.

# 2 1

b b 1 q

x a 0

# 2 1

1 q

2

q(b x) b a

4

2 q2

q(b x) b a

q

@

2 q2

823

q(b x) b a

1+ ln2

ln

1 +1

ln2

q(x a) ln b a

1 q1 9 > = A > ;

3 q1 5

KAVURMACI 820-826

J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 16, NO.5, 2014, COPYRIGHT 2014 EUDOXUS PRESS, LLC

AND ( ; m)

m

LOGARITM ICALLY CONVEX

5

Proof. By using Lemma, De…nition 4 and Power integral inequality, we have (2.4)

(b

b

a)

Z

a b b

a) Z

+

(b

a) "

1 + 2

=

"

1 + 2 For (2.5)

f (u) du

a

0

t jf (ta + (1

8 < Z :

b b

!1

x a

!1

1

(1 b b

8" < 1

b b

x b

# 2 1

: 2

x b

a a

Z

1 q

Z

1 q

1 q

qt

t

Z

qt

qt

dt

0

x a

Z

b b

x a

# 2 1

x a (1

b b

t)

, thereby Z bb xa t

(2.6)

b b

x a

t

qt

dt

qt

! q1 9 = ;

qt

;

mq(1 t )

dt

mq(1 t )

b m

f0 b b

b m

! q1 9 =

dt

dt

! q1

! q1

! q1 9 = ;

:

dt

0

t)

qt

dt

x a

q(b x) b a 2 q2

Z

1

(1

q

t)b)j dt

f0

0

x a

= Z

Z

1 q

1

q(b x) b a

and

t jf 0 (a)j

t) jf 0 (a)j

(1 x a

! q1

q

qt

qt

t)b)j dt

t)b)j dt

t) jf 0 (ta + (1

(1 b b

b b

Z

x a

1 b b

t) jf 0 (ta + (1

(1 b b

t jf 0 (ta + (1

0

: 2

# 2 1

a a

x a

1

1

1 q

8" < 1

m

b b

0

# 2 1

x a

b m

< 1, we have Z bb xa

1 q

t) dt

x a

Z

t)b)j dt +

tdt

0

a) f 0

(b

b

x a

0

(b

Z

1

f (x)

ln

1 +1

ln2

1

(1 b b

qt

t)

dt

x a

q

=

q(b x) b a

2 q2

If we write (2.5)-(2.6) in (2.4) and then use jf 0 (x)j

824

1+ ln2

q(x a) ln b a

:

M; we get the desired result.

KAVURMACI 820-826

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HAVVA KAVURM ACI |

6

Corollary 3. Let I [0; 1) be an open real interval and let f : I ! (0; 1) be a q di¤ erentiable function on I such that f 0 2 L [a; b] for 0 a < b < 1. If jf 0 (x)j is b 0 m logaritmically convex on 0; m for m 2 (0; 1], q 1, jf (x)j M , then Z b 1 f (x) f (u) du b a a 8 3 q1 " # 1 2 q(bb ax) q(b x) > < 1 b x 2 1 q ln 1 + 1 b a 5 4 (b a) M m > 2 b a q 2 ln2 : + where

"

1 2

x b

# 2 1

a a

1 q

0

q

q(b x) b a

@

is in Theorem 1.

(b

a) M 0 B @

"

q 2

1 8

1

q(x a) ln b a

q 2 ln2

Corollary 4. If in Thorem 2, we choose x = Z b a+b 1 f (u) du f 2 b a a m

1+

a+b 2 ,

1 q1 9 > = A > ;

we get

1 q

q 2 ln 2 q 2 ln2

1 +1

# q1

which is an Ostrowski type inequality.

0

+@

q

q 2

1+

2 q2

ln2

q ln 2

1 q1 1 A C A

References [1] A. Ostrowski, Über die Absolutabweichung einer di¤ erentierbaren Funktion von ihren Integralmittelwert, Comment. Math. Helv., 10, 226-227, (1938). [2] M. Alomari and M. Darus, Some Ostrowski Type Inequalities For Convex Functions With Applications, RGMIA Res. Rep. Coll. 13 (1) (2010), Article 3. [3] E. Set, M. Sardari, M.E. Özdemir and J. Rooin, On generalizations of the Hadamard inequality for ( ; m) convex functions, Kyungpook Math. J., 52 (3) (2012), 307— 317. [4] H. Kavurmac¬, M. Avc¬ and M.E. Özdemir, New Ostrowski type inequalities for m convex functions and applications, Hacettepe Journal of Mathematics and Statistics, Volume 40 (2) (2011), 135 – 145. [5] M.K. Bakula, M. E Özdemir, J. Peµcari´c, Hadamard type inequalities for m convex and ( ; m) convex functions, J. Inequal. Pure Appl. Math. 9 (2008), Article 96. [6] M.K. Bakula, J. Peµcari´c, M. Ribiµci´c, Companion inequalities to Jensen’s inequality for m convex and ( ; m) convex functions, J. Inequal. Pure Appl. Math. 7 (2006), Article 194. [7] S.S. Dragomir and G. Toader, Some inequalities for m convex functions, Studia Univ. Babesbolyai. Mathematica, 38(1993), 1, 21-28. [8] M.E. Özdemir, H. Kavurmaci, E. Set, Ostrowski’s type inequalities for ( ; m) convex functions, Kyungpook Math. J. 50 (2010) 371–378. [9] M.E. Özdemir, M. Avc¬ and E. Set, On some inequalities of Hermite-Hadamard type via m convexity, Applied Mathematics Letters, 23 (2010), 1065-1070.

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KAVURMACI 820-826

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m

AND ( ; m)

LOGARITM ICALLY CONVEX

7

[10] M.E. Özdemir, M. Avc¬ and H. Kavurmac¬, Hermite–Hadamard-type inequalities via ( ; m) convexity, Computers and Mathematics with Applications, 61 (2011), 2614–2620. [11] V.G. Mihe¸san, A generalization of the convexity, Seminar of Functional Equations, Approx. and Convex, Cluj-Napoca (Romania) (1993). [12] Rui-Fang Bai, Feng Qi, Bo-Yan Xi, Hermite-Hadamard type inequalities for the m and ( ; m) logarithmically convex functions, Filomat 27( 2013), 1-7; Available at: http://www.pmf.ni.ac.rs/…lomat. [13] G. Toader, Some generalization of the convexity, Proc. Colloq. Approx. Opt., Cluj-Napoca, (1984), 329-338. [14] G. Toader, On a generalization of the convexity, Mathematica, 30 (53) (1988), 83-87. [15] S.S. Dragomir, On some new inequalities of Hermite-Hadamard type for m convex functions, Tamkang Journal of Mathematics, 33 (1) (2002). AG¼ r¬ I·brahim Çeçen University, Faculty of Science and Letters, Department of Mathematics, 04100, AG¼ r¬, Turkey E-mail address : [email protected]

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A new version of Mazur-Ulam theorem under weaker conditions in linear n-normed spaces Choonkil Park1 , and Cihangir Alaca2,∗ 1

Department of Mathematics, Research Institute for Natural Sciences, Hanyang University, Seoul 133-791, South Korea 2 Department of Mathematics, Faculty of Science and Arts, Celal Bayar University, 45140 Manisa, Turkey

Abstract: The purpose of this paper is to prove a new result of Mazur-Ulam theorem for n-isometry without any other conditions in linear n-normed spaces. 2010 Mathematics Subject Classification: 46B20, 51M25, 46B04. Key words and phrases: n-norm, linear n-normed spaces, n-isometry, Mazur-Ulam theorem.

1

Introduction

Let X and Y be metric spaces. A mapping f : X → Y is called an isometry if f satisfies dY (f (x), f (y)) = dX (x, y) for all x, y ∈ X, where dX (·, ·) and dY (·, ·) denote the metrics in the spaces X and Y , respectively. Two metric spaces X and Y are defined to be isometric if there exists an isometry of X onto Y . In 1932, Mazur and Ulam [1] proved the following theorem. Mazur-Ulam Theorem. Every isometry of a real normed linear space onto a real normed linear space is a linear mapping up to translation. Baker [2] showed an isometry from a real normed linear space into a strictly convex real normed linear space is affine. Also, Jian [3] investigated the generalizations of the Mazur-Ulam theorem in F ∗ -spaces. Th.M. Rassias and Wagner [4] described all volume preserving mappings from a real finite dimensional vector space into itself and V¨ais¨al¨a [5] gave a short and simple proof of the Mazur-Ulam theorem. Chu [6] proved that the Mazur-Ulam theorem holds when X is a linear 2-normed space. Chu et al. [7] generalized the Mazur-Ulam theorem when X is a linear n-normed space, that is, the Mazur-Ulam theorem holds, when the n-isometry mapped to a linear n-normed space is affine. They also obtained extensions of the Th.M. Rassias and ˇ Semrl’s theorem [8]. The Mazur-Ulam theorem has been extensively studied by many authors (see [9, 10]). Recently, Moslehian and Sadeghi [11] investigated the Mazur-Ulam theorem in non-Archimedean spaces. Cho et al. [12] investigated the Mazur-Ulam theorem on probabilistic 2-normed spaces. Choy and Ku [13] proved that the barycenter of triangle carries the barycenter of corresponding triangle. They showed the Mazur-Ulam problem on non-Archimedean 2-normed spaces using ∗

Corresponding author. Tel.: +90 368 2715520; fax: +90 368 2715524. E-mail addresses: [email protected] (C. Park), [email protected] (C. Alaca).

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the above statement. Chen and Song [14] introduced the concept of weak n-isometry and then they got under some conditions, a weak n-isometry is also an n-isometry. Alaca [15] gave the concepts of 2-isometry, collinearity, 2-Lipschitz mapping in 2-fuzzy 2-normed linear spaces. Also, he gave a new generalization of the Mazur-Ulam theorem when X is a 2-fuzzy 2-normed linear space or ℑ(X) is a fuzzy 2-normed linear space. Choy et al. [16] proved the Mazur-Ulam theorem for the interior preserving mappings in linear 2-normed spaces and also proved the theorem on non-Archimedean 2-normed spaces over a linear ordered non-Archimedean field without the strict convexity assumption. Ren [17] showed that every generalized area n preserving mapping between real 2-normed linear spaces X and Y which is strictly convex is affine under some conditions. In the present paper, we show that every n-isometry without any other conditions from a linear n-normed space to another linear n-normed space is affine. A new version of Mazur-Ulam theorem is proved under much weaker conditions.

2

A new result for Mazur-Ulam theorem

Definition 2.1 ([18]) Let n ∈ N and let X be a real linear space of dimension d ≥ n. (Here we allow d to be infinite.) A real-valued function ∥•, ..., •∥ on X · · × X} satisfying the following | × ·{z n

properties (nN1 ) ∥x1 , x2 , ..., xn ∥ = 0 if and only if x1 , x2 , ..., xn are linearly dependent, (nN2 ) ∥x1 , x2 , ..., xn ∥ is invariant under any permutation, (nN3 ) ∥αx1 , x2 , ..., xn ∥ = |α| ∥x1 , x2 , ..., xn ∥ for any α ∈ R, (nN4 )∥x + y, x2 , ..., xn−1 , xn ∥ ≤ ∥x, x2 , ..., xn ∥ + ∥y, x2 , ..., xn ∥, is called an n-norm on X and the pair (X, ∥•, ..., •∥) is called an linear n-normed space. From now on, let X and Y be linear n-normed spaces and f : X → Y a mapping without special statements. Chu et al. [19] introduced the concept of n-isometry which is suitable to represent the notion of area preserving mappings in linear n-normed spaces as follows. For x0 , x1 , ..., xn ∈ X, ∥x1 − x0 , ..., xn − x0 ∥ is called an area of x0 , x1 , ..., xn . We call f an n-isometry if ∥x1 − x0 , ..., xn − x0 ∥ = ∥f (x1 ) − f (x0 ), ..., f (xn ) − f (x0 )∥ for all x0 , x1 , ..., xn ∈ X. A version of Mazur-Ulam theorem has been obtained in [7] as follows. Theorem 2.1 ([7]) Assume that X and Y are linear n-normed spaces. If f : X → Y is an n-isometry and satisfies f (x0 ), f (x1 ), ..., f (xn ) are n-collinear when x0 , x1 , ..., xn are n-collinear, then f is affine. A natural question is that whether the n-isometry in linear n-normed spaces is also affine without the condition of preserving n-collinearity. In this section, we will find a reply to this question in linear n-normed spaces. Lemma 2.1 ([19]) Let xi be an element of a linear n-normed spaces X for every i ∈ {1, ..., n} and γ ∈ R. Then ∥x1 , ..., xi , ..., xj , ..., xn ∥ = ∥x1 , ..., xi , ..., xj + γxi , ..., xn ∥ for all 1 ≤ i ̸= j ≤ n.

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Lemma 2.2 Let x0 , x1 be elements of X. Then u = satisfying

x0 +x1 2

is the unique element of X

∥x0 − u, x0 − xn , x2 − xn , ..., xn−1 − xn ∥ = ∥x1 − u, x1 − xn , x2 − xn , ..., xn−1 − xn ∥ 1 = ∥x0 − xn , x1 − xn , x2 − xn , ..., xn−1 − xn ∥ 2 with ∥x0 − xn , x1 − xn , x2 − xn , ..., xn−1 − xn ∥ ̸= 0 and u ∈ {tx0 + (1 − t)x1 : t ∈ R}. Proof. From Lemma 2.1, it is obvious that u =

x0 +x1 2

satisfies

∥x0 − u, x0 − xn , x2 − xn , ..., xn−1 − xn ∥ = ∥x1 − u, x1 − xn , x2 − xn , ..., xn−1 − xn ∥ 1 = ∥x0 − xn , x1 − xn , x2 − xn , ..., xn−1 − xn ∥ 2 with ∥x0 − xn , x1 − xn , x2 − xn , ..., xn−1 − xn ∥ ̸= 0 and u ∈ {tx0 + (1 − t)x1 : t ∈ R}. Now we prove the uniqueness of u. Assume that ν is an element of X satisfying the above properties. ∥x0 − ν, x0 − xn , x2 − xn , ..., xn−1 − xn ∥ = ∥x1 − ν, x1 − xn , x2 − xn , ..., xn−1 − xn ∥ 1 = ∥x0 − xn , x1 − xn , x2 − xn , ..., xn−1 − xn ∥ 2 with ∥x0 − xn , x1 − xn , x2 − xn , ..., xn−1 − xn ∥ ̸= 0 and ν ∈ {tx0 + (1 − t)x1 : t ∈ R}. Let ν = tx0 + (1 − t)x1 for some t ∈ R. From Lemma 2.1, we get ∥x0 − xn , x1 − xn , x2 − xn , ..., xn−1 − xn ∥ = 2 ∥x0 − ν, x0 − xn , x2 − xn , ..., xn−1 − xn ∥ = 2 ∥x0 − (tx0 + (1 − t)x1 ) , x0 − xn , x2 − xn , ..., xn−1 − xn ∥ = 2 |1 − t| ∥x0 − x1 , x0 − xn , x2 − xn , ..., xn−1 − xn ∥ = 2 |1 − t| ∥x1 − xn , x0 − xn , x2 − xn , ..., xn−1 − xn ∥ and ∥x0 − xn , x1 − xn , x2 − xn , ..., xn−1 − xn ∥ = 2 ∥x1 − ν, x1 − xn , x2 − xn , ..., xn−1 − xn ∥ = 2 ∥x1 − (tx0 + (1 − t)x1 ) , x1 − xn , x2 − xn , ..., xn−1 − xn ∥ = 2 |t| ∥x1 − x0 , x1 − xn , x2 − xn , ..., xn−1 − xn ∥ = 2 |t| ∥x1 − xn , x0 − xn , x2 − xn , ..., xn−1 − xn ∥ . Since ∥x0 − xn , x1 − xn , x2 − xn , ..., xn−1 − xn ∥ ̸= 0, we have 1 = 2 |1 − t| = 2 |t|. So t = 1 ν = u = x0 +x 2 .

1 2

and

Theorem 2.2 Let X and Y be linear n-normed spaces. If f : X → Y is an n-isometry, then f is affine. Proof. Let g(x) = f (x)−f (0). Then g is an n-isometry and g(0) = 0. For x0 , x1 , ..., xn ∈ X with ∥x0 − xn , x1 − xn , x2 − xn , ..., xn−1 − xn ∥ ̸= 0, ∥g(x0 ) − g(xn ), g(x1 ) − g(xn ), g(x2 ) − g(xn ), ..., g(xn−1 ) − g(xn )∥ ̸= 0.

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From Lemma 2.1, we get

( )

g(x0 ) − g x0 + x1 , g(x0 ) − g(xn ), g(x2 ) − g(xn ), ..., g(xn−1 ) − g(xn )

2

( )

x0 + x1

= x − , x − x , x − x , ..., x − x 0 0 n 2 n n−1 n

2

x0 − x1

=

2 , x0 − xn , x2 − xn , ..., xn−1 − xn 1 = ∥x0 − x1 , x0 − xn , x2 − xn , ..., xn−1 − xn ∥ 2 1 = ∥x1 − xn , x0 − xn , x2 − xn , ..., xn−1 − xn ∥ 2 1 = ∥g(x1 ) − g(xn ), g(x0 ) − g(xn ), g(x2 ) − g(xn ), ..., g(xn−1 ) − g(xn )∥ . 2 Similarly,

and

( )

x + x 0 1

g(x1 ) − g , g(x1 ) − g(xn ), g(x2 ) − g(xn ), ..., g(xn−1 ) − g(xn )

2 1 = ∥g(x1 ) − g (xn ) , g(x0 ) − g(xn ), g(x2 ) − g(xn ), ..., g(xn−1 ) − g(xn )∥ 2

(

)

g x0 + x1 − g(x1 ), g(x0 ) − g(x1 ), g(x2 ) − g(x1 ), ..., g(xn−1 ) − g(x1 )

2

( )

x0 + x1 − x1 , x0 − x1 , x2 − x1 , ..., xn−1 − x1 =

2 1 = ∥x0 − x1 , x0 − x1 , x2 − x1 , ..., xn−1 − x1 ∥ 2 = 0. (

So g

x0 + x1 2

) − g(x1 ) = t (g(x0 ) − g(x1 ))

for some t ∈ R by Definition 2.1. That is, ( ) x0 + x1 g = tg(x0 ) + (1 − t)g(x1 ). 2 By Lemma 2.2,

( g

x0 + x1 2

) =

g(x0 ) + g(x1 ) 2

for all x0 , x1 ∈ X. Since g(0) = 0, we have ( ) (x ) (x ) x0 + 0 g(x0 ) + g(0) 0 0 g =g = =g 2 2 2 2 and

( g (x0 + x1 ) = g

2x0 + 2x1 2

) =

g(2x0 ) + g(2x1 ) g(2x0 ) g(2x1 ) = + = g(x0 ) + g(x1 ). 2 2 2

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It follows that g is additive. Let r ∈ R+ and x0 ∈ X. Since g(0) = 0 and g is an n-isometry, we get ∥g(rx0 ) − 0, g(x0 ) − 0, g(x2 ) − 0, ..., g(xn−1 ) − 0∥ = ∥g(rx0 ) − g(0), g(x0 ) − g(0), g(x2 ) − g(0), ..., g(xn−1 ) − g(0)∥ = ∥rx0 − 0, x0 − 0, x2 − 0, ..., xn−1 − 0∥ = ∥rx0 , x0 , x2 , ..., xn−1 ∥ = 0. So g(rx0 ) = sg(x0 ) for some s ∈ R by Definition 2.1. Since dim X > 1, there exists an x1 ∈ X such that ∥x0 , x1 , x2 , ..., xn−1 ∥ ̸= 0. It is easy to see that r ∥x0 , x1 , x2 , ..., xn−1 ∥ = = = = =

∥rx0 , x1 , x2 , ..., xn−1 ∥ ∥g(rx0 ), g(x1 ), g(x2 ), ..., g(xn−1 )∥ ∥sg(x0 ), g(x1 ), g(x2 ), ..., g(xn−1 )∥ |s| ∥g(x0 ), g(x1 ), g(x2 ), ..., g(xn−1 )∥ |s| ∥x0 , x1 , x2 , ..., xn−1 ∥ .

So s = r or s = −r. If s = −r, then |r − 1| ∥x0 , x1 , x2 , ..., xn−1 ∥ = ∥(r − 1)x0 , x1 , x2 , ..., xn−1 ∥ = ∥rx0 − x0 , x1 − 0, x2 − 0, ..., xn−1 − 0∥ = ∥(g(rx0 ) − g(x0 ), g(x1 ) − g(0), g(x2 ) − g(0), ..., g(xn−1 ) − g(0)∥ = ∥−rg(x0 ) − g(x0 ), g(x1 ) − g(0), g(x2 ) − g(0), ..., g(xn−1 ) − g(0)∥ = (r + 1) ∥g(x0 ), g(x1 ), g(x2 ), ..., g(xn−1 )∥ = (r + 1) ∥x0 , x1 , x2 , ..., xn−1 ∥ . So |r − 1| = r + 1. This is a contradiction since r ∈ R+ . Thus, g(rx0 ) = rg(x0 ) for every r ∈ R+ and x0 ∈ X. Similarly, we can prove g(rx0 ) = rg(x0 ) for every r ∈ R− and x0 ∈ X. Hence, g is linear and f is affine. Remark 2.1 Theorem 2.1 has been substantially improved by Theorem 2.2. Remark 2.2 It is clear that the Mazur-Ulam theorem has been proved under much weaker conditions than the main result of Chu et al. [7] in the framework of 2-fuzzy 2-normed linear spaces.

Acknowledgments 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 (NRF2012R1A1A2004299).

References [1] S. Mazur, S. Ulam, Sur les transformationes isom´etriques d’espaces vectoriels norm´es, C.R. Acad. Sci. Paris 194 (1932), 946–948.

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[2] J.A. Baker, Isometries in normed spaces, Amer. Math. Monthly 78 (1971), 655–658. [3] W. Jian, On the generations of the Mazur-Ulam isometric theorem, J. Math. Anal. Appl. 263 (2001), 510–521. [4] Th.M. Rassias, P. Wagner, Volume preserving mappings in the spirit of the Mazur-Ulam theorem, Aequationes Math. 66 (2003), 85–89. [5] J. V¨ais¨al¨a, A proof of the Mazur-Ulam theorem, Amer. Math. Monthly 110 (2003), 633–635. [6] H. Chu, On the Mazur-Ulam problem in linear 2-normed spaces, J. Math. Anal. Appl. 327 (2007), 1041–1045. [7] H. Chu, S. Choi, D. Kang, Mappings of conservative distances in linear n-normed spaces, Nonlinear Analysis–TMA 70 (2009), 1168–1174. ˇ [8] Th.M. Rassias, P. Semrl, On the Mazur-Ulam problem and the Aleksandrov problem for unit distance preserving mappings, Proc. Amer. Math. Soc. 118 (1993), 919–925. [9] Th.M. Rassias, On the A.D. Aleksandrov problem of conservative distances and the MazurUlam theorem, Nonlinear Analysis–TMA 47 (2001), 2597–2608. [10] S. Xiang, Mappings of conservative distances and the Mazur-Ulam theorem, J. Math. Anal. Appl. 254 (2001), 262–274. [11] M.S. Moslehian, Gh. Sadeghi, A Mazur-Ulam theorem in non-Archimedean normed spaces, Nonlinear Analysis–TMA 69 (2008), 3405–3408. [12] Y. Cho, F. Rahbarnia, R. Saadati, Gh. Sadeghi, Isometries in probabilistic 2-normed spaces, J. Chungcheong Math. Soc. 22 (2009), 623–634. [13] J. Choy, S. Ku, Characterization on 2-isometries in non-Archimedean 2-normed spaces, J. Chungcheong Math. Soc. 22 (2009), 65–71. [14] X.Y. Chen, M.M. Song, Characterizations on isometries in linear n-normed spaces, Nonlinear Analysis–TMA 72 (2010), 1895–1901. [15] C. Alaca, A new perspective to the Mazur-Ulam problem in 2-fuzzy 2-normed linear spaces, Iranian J. Fuzzy Systems 7 (2010), 109–119. [16] J. Choy, H. Chu, S. Ku, Characterizations on Mazur-Ulam theorem, Nonlinear Analysis– TMA 72 (2010), 1291–1297. [17] W. Ren, On the generalized 2-isometry, Rev. Mat. Complut. 23 (2010), 97–104. [18] Y. Cho, P.C.S. Lin, S. Kim, A. Misiak, Theory of 2-Inner Product Spaces, Nova Science Publ., New York, 2001. [19] H. Chu, K. Lee, C. Park, On the Aleksandrov problem in linear n-normed spaces, Nonlinear Analysis–TMA 59 (2004), 1001–1011.

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ON UMBRAL CALCULUS INVOLVING SPECIAL POLYNOMIALS DAE SAN KIM, TAEKYUN KIM, SANG-HUN LEE AND SEOG-HOON RIM

Abstract. In this paper, we give some new and interesting identities involving special polynomials which are derived from the transfer formula for the associated sequences.

1. Introduction Let α ∈ R, the Bernoulli polynomials of order α are defined by the generating function to be )α ∑ ( ∞ t tn (α) = , (see [1,11,12,18]). B (x) (1) n et − 1 n! n=0 (α)

(α)

In the special case, x = 0, Bn (0) = Bn order α. From (1), we note that (2)

Bn(α) (x)

=

are called the n-th Bernoulli numbers of

n ( ) ∑ n

l

l=0

(α)

Bl xn−l .

As is well known, the Euler polynomials of of order α are also defined by the generating function to be ( )α ∑ ∞ 2 tn = En(α) (x) , (see [2,4,9,13,19,20]). (3) t e +1 n! n=0 (α)

(α)

In the special case, x = 0, En (0) = En are called the n-th Euler numbers of order α. By (3), we get n ( ) ∑ n (α) (α) (x) = En−l xl , (see [3,7,10,17]). E (4) n l l=0

Let F be the set of all formal power series in the variable t over C with { } ∞ ∑ ak k F = f (t) = t ak ∈ C , (see [5,8,16]). (5) k! k=0

Let us assume that P is the algebra of polynomials in the variable x over C and P∗ be the vector space of all linear functionals on P. ⟨L|p(x)⟩ denotes the action of the linear functional L on a polynomial p(x). For f (t) ∈ F , we define the linear functional f (t) on P by (6)

⟨f (t)|xn ⟩ = an ,

(n ≥ 0),

(see [6,10,14,15]).

1

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DAE SAN KIM, TAEKYUN KIM, SANG-HUN LEE AND SEOG-HOON RIM

Thus, by (5) and (6), we get ⟨tk |xn ⟩ = n!δn,k ,

(7)

(n, k ≥ 0),

(see [8,15]),

where δn,k is the Kronecker symbol. k ∑∞ ⟩ k Let fL (t) = k=0 ⟨L|x k! t . From (7), we note that ⟨fL (t)|xn ⟩ = ⟨L|xn ⟩.

(8)

Thus, by (8), we see that fL (t) = L. The map L 7→ fL (t) is a vector space isomorphism from P∗ onto F. Henceforth, F is thought of as both a formal power series and a linear functional (see [9,15,16]). We call F the umbral algebra. The umbral calculus is the study of umbral algebra. The order o(f (t)) of the non-zero power series f (t) is the smallest integer k for which the coefficient of tk does not vanish. If o(f (t)) = 1, then f (t) is called a delta series. If o(f (t)) = 0, then f (t) is called an invertible series (see [15,16]). Let o(f (t)) = 1 and o(g(t)) = 0. Then there exists a unique sequence Sn (x) of polynomials such that ⟨g(t)f (t)k |Sn (x)⟩ = n!δn,k where n, k ≥ 0. The sequence Sn (x) is called the Sheffer sequence for (g(t), f (t)), which is denoted by Sn (x) ∼ (g(t), f (t)). If Sn (x) ∼ (1, f (t)), then Sn (x) is called the associated sequence for f (t). By (7), we easily see that ⟨eyt |p(x)⟩ = p(y). For f (t) ∈ F and p(x) ∈ P, we have (9)

f (t) =

∞ ∑ ⟨f (t)|xk ⟩ k=0

k!

k

t ,

p(x) =

∞ ∑ ⟨tk |p(x)⟩ k=0

k!

xk ,

(see [10,15]).

Thus, by (9), we get p(k) (0) = ⟨tk |p(x)⟩,

(10)

⟨1|p(x)⟩ = p(k) (0).

From (10), we have (11)

tk p(x) = p(k) (x) =

dk p(x) , dxk

(k ≥ 0),

(see [9,15,16]).

By (11), we easily get eyt p(x) = p(x + y). Let Sn (x) ∼ (g(t), f (t)). Then we see that ∞

(12)

∑ Sk (y) 1 y f¯(t) e = tk , for all y ∈ C, k! g(f¯(t)) k=0

where f¯(t) is the compositional inverse of f (t). For pn (x) ∼ (1, f (t)), qn (x) ∼ (1, g(t)), we have ( )n f (t) (13) qn (x) = x x−1 pn (x), g(t)

(see [8,9,15,16]),

(see [8,15,16]).

Now, we introduce several important sequences which are used to derive our results in this paper (see [8,9,10,15,16]): (The Poisson-Charlier sequences) n ( ) ∑ t n Cn (x; a) = (−1)n−k a−k (x)k ∼ (ea(e −1) , a(et − 1)), (14) k k=0

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ON UMBRAL CALCULUS INVOLVING SPECIAL POLYNOMIALS

3

where a ̸= 0, (x)n = x(x − 1) · · · (x − n + 1), ( )n ∞ ∑ tk −t t−a Cn (k; a) e = , (a ̸= 0), n ∈ N, (15) k! a k=0

(The Abel sequences) An (x; b) = x(x − bn)n−1 ∼ (1, tebt ), (b ̸= 0),

(16)

(The Mittag-Leffler sequences) ( ) n ( ) ∑ n et − 1 k , Mn (x) = (n − 1)n−k 2 (x)k ∼ 1, t (17) k e +1 k=0

(The Laguerre sequences) Ln (x) =

(18)

) n ( ∑ n − 1 n! k=1

k − 1 k!

( (−x)k ∼

1,

) t . t−1

In this paper, we give some new and interesting identities involving special polynomials which are derived from the transfer formula for the associated sequences. 2. On Associated Sequences of Polynomials Let us consider the following associated sequences: ( ) ( ) t t (19) pn (x) ∼ 1, , qn (x) ∼ 1, , (a ̸= 0). (t − 1)a (t + 1)a By (13), we easily see that pn (x) = x(t − 1)an xn−1 = x

n−1 ∑ k=0

Can (k; 1)

1 k t (x − 1)n−1 k!

( ) n−1 =x Can (k; 1) (x − 1)n−1−k k k=0 ( ) n−1 ∑ n−1 =x Can (n − 1 − k; a) (x − 1)k , k n−1 ∑

(20)

k=0

and (21)

( qn (x) = x

)n

t t (t+1)a

= x(−1)an

n−1 ∑ k=0

x−1 xn = x(t + 1)an xn−1 = (−1)an x

∞ ∑ k=0

Can (k; −1)

tk −t n−1 e x k!

( ) n−1 Can (n − 1 − k; −1) (x − 1)k . k

From (12) and (17), we can derive the generating function of Mittag-Leffler sequences Mn (x) as follows: ( )x ∞ ∑ tk 1+t Mk (x) = . (22) k! 1−t k=0

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DAE SAN KIM, TAEKYUN KIM, SANG-HUN LEE AND SEOG-HOON RIM

By (13) and (19), we get (23)

( )an ( )an t+1 1+t qn (x) = x x−1 pn (x) = (−1)an x x−1 pn (x) t−1 1−t ) ( ) (n−1 ∑ ∑ Ml (an) )(n−1 n−1 Can (n − k − 1; 1) (x − 1)k = (−1)an x tl k l! k=0 l=0 ( )( ) n−1 ∑ n−1 ∑ n−1 k = (−1)an x Ml (an)Can (n − k − 1; 1)(x − 1)k−l k l l=0 k=l n−1 ∑ n−1−m ∑ ( n − 1 )(l + m) an = (−1) x Ml (an)Can (n − 1 − l − m; 1)(x − 1)m . l+m l m=0 l=0

Therefore, by (21) and (23), we obtain the following lemma. Lemma 2.1. For n ≥ 1, 0 ≤ m ≤ n − 1, and a ∈ Z+ = N ∪ {0}, we have ( ) n−1−m ∑ ( n − 1 )(l + m) n−1 Ml (an)Can (n−1−l −m; 1). Can (n−1−m; −1) = l+m l m l=0

It is easy to show that (24)

(an)

xBn−1 (x) ∼

( ( t )a ) e −1 1, t , t

(x)n ∼ (1, et − 1).

For n ≥ 1, by (13) and (24), we get ( t )(a−1)n e −1 (an) (x)n = x Bn−1 (x) t (25) n−1 ∑ ((a − 1)n)!S2 (l + (a − 1)n, (a − 1)n) (an) = x(n − 1)! Bn−1−l (x), (l + (a − 1)n)!(n − 1 − l)! l=0

where S2 (n, k) is the Stiring number of the second kind. It is known that (26)

(x)n =

n−1 ∑

S1 (n, k + 1)xk+1 , (n ∈ N),

k=0

where S1 (n, k) is the Stiring number of the first kind. Therefore, by (2), (25) and (26), we obtain the following theorem. Theorem 2.2. For n ≥ 1, 0 ≤ m ≤ n − 1, and a ∈ Z+ = N ∪ {0}, we have n−1−m ∑ ((a − 1)n)!S2 (l + (a − 1)n, (a − 1)n) (n − 1 − l) (an) S1 (n, m+1) = (n−1)! Bn−1−l−m . (l + (a − 1)n)!(n − 1 − l)! m l=0

Let us consider the following associated sequences: ( ( )a ) ( ) t pn (x) ∼ 1, t t , (x)k ∼ 1, et − 1 , a ∈ Z+ . e −1

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From (13) and (24), we can derive ( t )an n−1 ∑ (an)! e −1 S2 (l + an, an)(n − 1)l xn−1−l , pn (x) = x xn−1 = x t (l + an)!

(27)

l=0

and, for n ≥ 1, we have ( t )(a+1)n )(a+1)n ( t e −1 e −1 −1 (28) pn (x) = x x (x)n = x (x − 1)n−1 . t t By (27) and (28), we get (an)!(n − 1)! S2 ((a + 1)n − m − 1, an) ((a + 1)n − m − 1)!m! ( )( ) n−1−m ∑ n−1 ∑ k k−l k−l−m ((a + 1)n)!l! = (−1) (l + (a + 1)n)! l m

(29)

l=0

k=l+m

× S2 (l + (a + 1)n, (a + 1)n)S1 (n − 1, k), where n ≥ 1, 0 ≤ m ≤ n − 1, and a ∈ Z+ . Let us consider the following associated sequences: ( ( t )a ) e +1 (30) (x)n ∼ (1, et − 1), pn (x) ∼ 1, t , a ∈ Z+ . 2 From (13) and (30), we can prove the following Exercise. Exercise. (I) For n ≥ 1, we have (an)

En−1 (x + 1) =

n−1 ∑ n−1−m ∑ m=0

k=0

n!(k + m)! (an) (x). S1 (n − 1, k + m)S2 (k + n, n)Em (k + n)!m!

(II) For n ≥ 1, 0 ≤ j ≤ n − 1, we have ( ) an ( ) n − 1 ∑ an n−1−j k j k k=0 an n−1 ∑ ∑ m−j ∑ (an)(m − k ) n!m! = (l − 1)m−k−j S1 (n − 1, m)S2 (k + n, n). l j (k + n)!(m − k)! m=j l=0

k=0

(III) For n ≥ 1, a ∈ N, 0 ≤ j ≤ n − 1, we have ( ) an (a−1)n n−1 m−j ( ∑ ∑ ∑ (a − 1)n)(m − k ) n − 1 ∑ n−1−j k = (l − 1)m−k−j j l j m=j k=0

l=0

×

k=0

n!2n+k m! S1 (n − 1, m)S2 (k + n, n). (k + n)!(m − k)!

Let us consider the following associated sequences: ( ( t )a ) e −1 (an) t (31) (x)n ∼ (1, e − 1), xBn−1 (x) ∼ 1, t , (n, a ∈ N). t

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DAE SAN KIM, TAEKYUN KIM, SANG-HUN LEE AND SEOG-HOON RIM

By (13) and (31), we get (32)

( t )(a−1)n e −1 (an) (x)n = x Bn−1 (x) t =x

n−1 ∑ k=0

((a − 1)n)! (an) S2 (k + n(a − 1), (a − 1)n)tk Bn−1 (x) (k + (a − 1)n)!

n−1 ∑

(n − 1)!((a − 1)n)! (an) S2 (k + n(a − 1), (a − 1)n)Bn−1−k (x) (k + (a − 1)n)!(n − k − 1)! k=0 n−1 ∑ {n−1−l ∑ (n − 1 − k ) (an) (n − 1)!((a − 1)n)! Bn−1−k−l =x l (k + (a − 1)n)!(n − k − 1)! l=0 k=0 } × S2 (k + n(a − 1), (a − 1)n) xl , =x

and, for n ∈ N, we have (x)n = x

(33)

n−1 ∑

S1 (n, k + 1)xk .

k=0

Therefore, by (32) and (33), we obtain the following theorem. Theorem 2.3. For n ≥ 1, 0 ≤ l ≤ n − 1, we have S1 (n, l + 1) n−1−l ∑ (n − 1 − k ) (an) (n − 1)!((a − 1)n)! = S2 (k + (a − 1)n, (a − 1)n). Bn−1−k−l (k + (a − 1)n)!(n − k − 1)! l k=0

In particular, a = 1,

( S1 (n, l + 1) =

) n−1 (n) Bn−1−l . l

From (13) and (31), we can easily derive the following equation: (34)

(an)

Bn−1 (x + 1) =

n−1 ∑

n(a−1)

S1 (n − 1, k)Bk

(x).

k=0

Let us consider the following associated sequences: ( ( t )a ) e −1 (an) (35) xBn−1 (x) ∼ 1, t , An (x; b) = x(x − bn)n−1 ∼ (1, tebt ), t where n, a ∈ N and b ̸= 0. By (13) and (35), we get ( t )an e −1 (an) Bn−1 (x) An (x; b) = x(x − bn)n−1 = xe−bnt t (36)

= xe−bnt

n−1 ∑ l=0

=x

n−1 ∑ l=0

(an)! (an) S2 (l + an, an)(n − 1)l Bn−1−l (x) (l + an)!

(an)!(n − 1)! (an) S2 (l + an, an)Bn−1−l (x − nb) (l + an)!(n − 1 − l)!

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Thus, by (36), we get (x − bn)n−1 =

n−1 ∑ l=0

(an)!(n − 1)! (an) S2 (l + an, an)Bn−1−l (x − nb). (l + an)!(n − 1 − l)!

It is not difficult to show that (37)

(k ) } ∞ {∑ k ∑ S2 (j + n, n) k−j tk (et − 1)n k−j j ( ) = (−1) . x j+n etx tn k! n j=0 k=0

By (13), (35) and (37), we see that (38) (et − 1)an (an) Bn−1 (x) enbt tan ( ) } k k ∞ {∑ k ∑ t (an) j S2 (j + an, an) (j+an) B =x (nb)k−j (x) (−1)k−j k! n−1 j k=0 j=0 (k ) } n−1 k ∑ {∑ S2 (j + an, an) (n − 1)k (an) k−j j (j+an) (nb)k−j Bn−1−k (x) =x (−1) k! j k=0 j=0 (k)(n−1) n−1 k ∑∑ j k S2 (j + an, an) (an) (j+an) =x (−nb)k−j Bn−1−k (x).

An (x; b) = x(x − bn)n−1 = x

j

k=0 j=0

Therefore, by (38), we obtain the following theorem. Theorem 2.4. For n ≥ 1, a ∈ N and b ̸= 0, we have (k)(n−1) n−1 k ∑∑ j k (an) n−1 (j+an) S2 (j + an, an)(−nb)k−j Bn−1−k (x). (x − bn) = j

k=0 j=0

Let (39)

(an)

xEn−1 (x) ∼

( ( t )a ) e +1 1, t , 2

where n, a ∈ N and b ̸= 0. By (13) and (39), we get

)an et + 1 (an) En−1 (x) 2 ) an ( ∑ an (l−nb)t (an) −an =2 x e En−1 (x) l l=0 ) an ( ∑ an (an) = 2−an x En−1 (x + l − nb). l

An (x; b) = xe−bnt (40)

An (x; b) = x(x − bn)n−1 ∼ (1, tebt ),

(

l=0

Thus, from (40), we have (41)

(x − bn)

n−1

=2

−an

) an ( ∑ an (an) En−1 (x + l − nb). l l=0

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DAE SAN KIM, TAEKYUN KIM, SANG-HUN LEE AND SEOG-HOON RIM

Let us assume that (42)

(

(an) xBn−1 (x)

∼

(

et − 1 1, t t

)a ) ,

( Mn (x) ∼

) et − 1 1, t , e +1

where n, a ∈ N. By (13) and (42), we get (43)

(

)(a−1)n et − 1 (an) (et + 1)n Bn−1 (x) t )(a−1)n n ( )( t ∑ n e −1 (an) =x elt Bn−1 (x) l t l=0 n ( ) n−1 ∑ n ∑ ((a − 1)n)! (n − 1)! =x S2 (k + (a − 1)n, (a − 1)n) l (k + (a − 1)n)! (n − k − 1)!

Mn (x) = x

l=0 (an)

k=0

× Bn−1−k (x + l). Therefore, by (43), we obtain the following proposition. Proposition 2.5. For n, a ∈ N, we have n n−1 ∑ ∑ (n) ((a − 1)n)!S2 (k + (a − 1)n, (a − 1)n)(n − 1)! (an) Mn (x) = x Bn−1−k (x + l). (k + (a − 1)n)!(n − k − 1)! l l=0 k=0

From (13) and (42), we note that ( )(a−1)n ( )n t 1 (an) (44) xBn−1 (x) = x t x−1 Mn (x), e −1 et + 1 and, by (17), we get x

−1

−1

Mn (x) = x

n ( ) ∑ n k=1

k

(n − 1)n−k 2k (x)k

n ( ) ∑ n = (n − 1)n−k 2k (x − 1)k−1 k k=1 n ( ) k−1 ∑ ∑ n = (n − 1)n−k 2k S1 (k − 1, l)(x − 1)l . k

(45)

k=1

l=0

From (44), we have ( )(a−1)n )n ( t 2 e −1 (an) −n (46) Bn−1 (x) = 2 x−1 Mn (x). t et + 1 RHS of (46) =

n k−1 ∑ ∑ (n) k=1 l=0

(47) =

k

n k−1 ∑ ∑ (n)

k=1 l=0

k

(n − 1)n−k 2

k−n

( S1 (k − 1, l)

2 t e +1

)n (x − 1)l

(n)

(n − 1)n−k 2k−n S1 (k − 1, l)El (x − 1),

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and (48) LHS of (46) =

n−1 ∑ k=0

=

n−1 ∑ k=0

((a − 1)n)! (an) S2 (k + (a − 1)n, (a − 1)n)tk Bn−1 (x) (k + (a − 1)n)! ((a − 1)n)!(n − 1)! (an) S2 (k + (a − 1)n, (a − 1)n)Bn−1−k (x). (k + (a − 1)n)!(n − 1 − k)!

Therefore, by (46), (47) and (48), we obtain the following proposition. Proposition 2.6. For n ≥ 1, we have n−1 ∑

((a − 1)n)!(n − 1)! (an) S2 (k + (a − 1)n, (a − 1)n)Bn−1−k (x) (k + (a − 1)n)!(n − 1 − k)! k=0 n k−1 ∑ ∑ (n) (n) = (n − 1)n−k 2k−n S1 (k − 1, l)El (x − 1). k k=1 l=0

Let (49)

( ( pn (x) ∼ 1, t

t et − 1

)a )

( ,

Mn (x) ∼

) et − 1 1, t . e +1

For n ≥ 1, by (13) and (49), we get ( )(a+1)n t (50) Mn (x) = x t (et + 1)n x−1 pn (x), e −1 and

(51)

( )n ( t )an t e −1 pn (x) = x ( t )a x−1 xn = x xn−1 t t et −1 =x

n−1 ∑ l=0

(an)! S2 (l + an, an)(n − 1)l xn−1−l . (l + an)!

From (50) and (51), we can derive n n−1 ∑ ∑ ( n) (an)!(n − 1)! ((a+1)n) Mn (x) = S2 (l + an, an)xBn−1−l (x + k). k (l + an)!(n − 1 − l)! k=0 l=0

Let us assume that (52)

Mn (x) ∼

( 1,

) et − 1 , et + 1

pn (x) ∼

( ( 1, t

2 et + 1

)a ) , (a ∈ N).

It is easy to see that )n ( t pn (x) = x ( 2 )a x−1 xn = x2−an (et + 1)an xn−1 t et +1 (53) ) ) an ( an ( ∑ ∑ an lt n−1 an −an −an = x2 e x = x2 (x + l)n−1 . l l l=0

l=0

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DAE SAN KIM, TAEKYUN KIM, SANG-HUN LEE AND SEOG-HOON RIM

For n ≥ 1, by (13) and (52), we get ( t )n e −1 −an (54) pn (x) = 2 x (et + 1)(a−1)n x−1 Mn (x). t By (53) and (54), we get ) )n ( t an ( ∑ an e −1 (x + k)n−1 = (et + 1)(a−1)n x−1 Mn (x). (55) k t k=0

Thus, from (55), we have )n )( an ( ∑ an t (x + k)n−1 = (et + 1)(a−1)n x−1 Mn (x) et − 1 k k=0 n ( ) ∑ n t (a−1)n = (e + 1) (n − 1)n−k 2k x−1 (x)k (56) k k=1

=

(a−1)n n k−1 ( ∑ ∑∑ m=0 k=1 l=0

n k

)( ) (a − 1)n (n − 1)n−k 2k S1 (k − 1, l)emt (x − 1)l . m

Therefore, by (56), we obtain the following proposition. Proposition 2.7. For n ≥ 1, a ∈ N we have ) an ( ∑ an (n) Bn−1 (x + k) k k=0

=

n k−1 ∑ ∑ (a−1)n ∑ (n)((a − 1)n) (n − 1)n−k 2k S1 (k − 1, l)(x + m − 1)l . k m m=0

k=1 l=0

From (45) and (55), we can also derive the following identity: (57) ) an ( ∑ an k=0

(

=

k

( (x + k)

e −1 t t

)n

n−1

=

et − 1 t

)n t

(e + 1)

(a−1)n

n ( ) ∑ n k=1

k

(n − 1)n−k 2k (x − 1)k−1

n k−1 ∑ ∑ (n) t (a−1)n (e + 1) (n − 1)n−k 2k S1 (k − 1, l)(x − 1)l k k=1 l=0

) )n n k−1 (a−1)n ( )( e − 1 ∑ ∑ ∑ n (a − 1)n (n − 1)n−k 2k S1 (k − 1, l)(x + m − 1)l = t k m k=1 l=0 m=0 n k−1 l (n)((a−1)n) k ∑ ∑ (a−1)n ∑ ∑ 2 S1 (k − 1, l) k m = n!(n − 1)! S2 (r + n, n)l!(x + m − 1)l−r . (r + n)!(k − 1)!(l − r)! m=0 r=0 (

t

k=1 l=0

For n ≥ 1, by (13), (52) and (53), we get ( )n ( )(a−1)n t 2 n (58) Mn (x) = 2 x t x−1 pn (x). e −1 et + 1

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Thus, by (53) and (58), we see that ( t ) )n )(a−1)n ( an ( ∑ an 2 −n e − 1 −1 −an (x + k)n−1 . 2 x Mn (x) = 2 (59) k t et + 1 k=0

By (59), we get (60) ) ( t )n an ( ∑ an ((a−1)n) (a−1)n e − 1 En−1 (x + k) = 2 x−1 Mn (x) k t k=0 )n ( t n k−1 ∑ ∑ (n) e −1 k (a−1)n (x − 1)l =2 (n − 1)n−k 2 S1 (k − 1, l) t k k=1 l=0 (n) k n k−1 l ∑ ∑∑ (a−1)n k 2 l!S2 (m + n, n) = n!(n − 1)!2 S1 (k − 1, l)(x − 1)l−m . (k − 1)!(m + n)!(l − m)! m=0 k=1 l=0

Therefore, by (60), we obtain the following proposition. Proposition 2.8. For a, n ∈ N, we have ) an ( ∑ an ((a−1)n) (x + k) En−1 k k=0 (n) k n k−1 l ∑ ∑∑ (a−1)n k 2 l!S2 (m + n, n) = n!(n − 1)!2 S1 (k − 1, l)(x − 1)l−m . (k − 1)!(m + n)!(l − m)! m=0 k=1 l=0

For λ ̸= 0, let us consider the following associated sequences: ( ) t (61) pn (x) ∼ 1, , xn ∼ (1, t). 1 − λ(et − 1) From (61), we can derive (62)

pn (x) = x(1 − λ(et − 1))n xn−1 = x

n ( ) ∑ n (1 + λ)n−k (−λ)k (x + k)n−1 . k

k=0

By (13) and (61), we get (

1 x =x 1 − λ(et − 1)

)n

n

(63)

x−1 pn (x).

From Boyadzhiev, we get ∞

∑ tl 1 = Wl (λ) , t 1 − λ(e − 1) l!

(64)

l=0

∑n

where Wn (x) = k=0 k!S2 (n, k)xk , (see [15,16]). Thus, by (64), we get )n ∑ ( ( ) } l ∞ { ∑ 1 l t = W (λ) · · · W (λ) . (65) l1 ln 1 − λ(et − 1) l1 , · · · , ln l! l=0

l1 +···+ln =l

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DAE SAN KIM, TAEKYUN KIM, SANG-HUN LEE AND SEOG-HOON RIM

For n ≥ 1, by (63) and (65), we see that (66) xn−1 =

n−1 ∑{ l=0

=

∑

l1 +···+ln =l

n n−1 ∑ ∑

(

l l1 , · · · , ln (

∑

)(∏ n

) l }∑ n ( ) t n Wli (λ) (1 + λ)n−k (−λ)k (x + k)n−1 l! k i=1 k=0

)( )( ) (∏ ) n l n n−1 n−k k (1 + λ) (−λ) Wli (λ) l1 , · · · , ln k l i=1

k=0 l=0 l1 +···+ln =l

× (x + k)n−1−l . Therefore, by (60), we obtain the following theorem. Theorem 2.9. For n ≥ 1, we have xn−1 =

n n−1 ∑ ∑

∑

k=0 l=0 l1 +···+ln =l

(

l l1 , · · · , ln

)( )( ) (∏ ) n n n−1 (1 + λ)n−k (−λ)k Wli (λ) k l i=1

× (x + k)n−1−l . The Boole polynomials Bln (x; λ) are Sheffer sequences for (1 + eλt , et − 1). That is (67)

Bln (x; λ) ∼ (1 + eλt , et − 1),

(see [15,16]).

From (12) and (67), we can derive the generating function of Boole sequences as follows: ∞ ∑ Blk (x; λ)

(68)

k=0

k!

tk =

(1 + t)x . (1 + t)λ + 1

For λ = 0, we have Bln (x; 0) = (x)n . Let us consider the following associated sequences: ( ) t (69) Snµ (x) ∼ 1, . (1 + t)µ By (13) and (69), we get ( )n t Snµ (x) = x x−1 xn = x(1 + t)µn xn−1 t ( (1+t) ) µ (70) ( ) ) n−1 n ( ∑ µn ∑ µn (n − 1)! k n−k x . = (n − 1)k x = k n − k (k − 1)! k=0

k=1

For λ = 1, we note that Sn1 (x) = Sn (x) = Ln (−x). Let us assume that ( ) t (71) tn (x) ∼ 1, . 1 + (1 + t)λ

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By (13) and (71), we get (72)

(

)n

t

x−1 xn = x(1 + (1 + t)λ )n xn−1 t ( 1+(1+t) λ) ( ) n ( ) n ( ) n−1 ∑ ∑ n n ∑ λa b n−1 =x (1 + t)λa xn−1 = x t x a a b a=0 a=0 b=0 ) ) n ∑ n ( )( n ∑ n ( )( ∑ ∑ n λa n λa (n − 1)! b b x . = (n − 1)n−b x = a n−b a n − b (b − 1)! a=0 a=0

tn (x) = x

b=1

b=1

From (13), (69) and (71), we can derive (73)

(

Snµ (x) = x =x

∞ { ∑ l=0

=

=

(∑ )n ∞ Bll (µ; λ) l t x−1 tn (x) l! l=0 )(∏ )∑ ) } n n ∑ n ( )( l n λa (n − 1)! 1 l b−1 Blli (µ; λ) tx l1 , · · · , ln a n − b (b − 1)! l! a=0 i=1

(1 + t)µ 1 + (1 + t)λ ( ∑

l1 +···+ln =l

n {∑ n ∑ b−1 ∑

)n

∑

a=0 b=1 l=0 l1 +···+ln =l n {∑ n ∑ n ∑ ∑ k=1

x−1 tn (x) = x

b=1

)( )( )( ) } n λa b − 1 (n − 1)! b−l l Blli (µ; λ) x a n−b l (b − 1)! l1 , · · · , ln i=1 ( n )} ( )( )( )( ) n λa b−k b − 1 (n − 1)! ∏ Blli (µ; λ) xk a n − b l 1 , · · · , ln k − 1 (b − 1)! i=1

(

)(∏ n

a=0 b=k l1 +···+ln =b−k

Therefore, by (70) and (73), we obtain the following proposition. Proposition 2.10. For n ∈ N, 1 ≤ k ≤ n we have ( ) µn (n − 1)! n − k (k − 1)! )( ) ( n ) ( )( )( n ∑ n ∑ ∑ b − 1 (n − 1)! ∏ n λa b−k = Blli (µ; λ) . a n − b l1 , · · · , ln k − 1 (b − 1)! i=1 a=0 b=k l1 +···+ln =b−k

Remark Let (74)

x(n) = x(x + 1) · · · (x + n − 1) ∼ (1, 1 − e−t ).

Then the generating function of x(n) is given by ( )x ∞ ∑ 1 tk . x(k) = (75) k! 1−t k=0

The Stirling polynomials Stn (x) are Sheffer sequences for (e−t , f (t)), where f¯(t) = log( 1−et −t ). ( ) t −t ¯ (76) Stn (x) ∼ (e , f (t)), where f (t) = log . 1 − e−t

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DAE SAN KIM, TAEKYUN KIM, SANG-HUN LEE AND SEOG-HOON RIM

Thus, from (12) and (76), we have ∞ ∑

(77)

Stk (x)

k=0

tk = k!

(

t 1 − e−t

)x+1 .

Indeed, we note that (x+1)

(78)

Stk (x) = Bk

(x + 1),

for k ≥ 0.

For n ≥ 1, we have ( )n ( )n t t (n) −1 n x =x x x =x xn−1 1 − e−t 1 − e−t (79) ) n−1 n ( ∑ Stk (n − 1) ∑ n−1 n−1−k =x (n − 1)k x = Stn−k (n − 1)xk . k! k−1 k=0

As x(n) =

∑n k=1

k=1

|S1 (n, k)| xk , we also have : for 1 ≤ k ≤ n, ( ) n−1 |S1 (n, k)| = Stn−k (n − 1). k−1

Let us consider the following associated sequences: An (x; a) = x(x − an)n−1 ∼ (1, teat ), ( ) x ∼ (1, eat − 1), (a ̸= 0). a n

(80)

For n ≥ 1, by (13) and (80), we get (81)

( ) ( ) n−1 ∑ x k−n n − 1 = a Stk (n − 1)x(x − an)n−1−k . a n k k=0

Let (82)

( pn (x) ∼

( 1, t

t−b b

)) ,

x(n) ∼ (1, 1 − e−t ),

(b ̸= 0).

From (13) and (82), we have

(83)

( )n ( )−n b t pn (x) = x x−1 xn = (−1)n x 1 − xn−1 t−b b n−1 ∑ (−n) = (−1)n x (−1)l b−l tl xn−1 l l=0 n−1 ∑ (n + l − 1) (n − 1)! n−1−l n = (−1) x b−l x (n − l − 1)! l l=0 n−1 ∑ (2n − l − 2) (n − 1)! l n = (−1) x b−(n−1−l) x. n−1 l! l=0

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15

By (13) and (81), we get ( t−b )n t( b ) x(n) = x x−1 pn (x) 1 − e−t )n ( )n ( n−1 ∑ (2n − l − 2) (n − 1)! t t−b n (−1) xl =x (84) b 1 − e−t n−1 l!bn−1−l l=0 )( ) ( )n n−1 l ( ∑∑ 2n − l − 2 l (n − 1)! t−b n = (−1) Stk (n − 1)x xl−k . n−1 k l!bn−1−l b l=0 k=0

From (15) and (84), we can derive (85) x

(n)

=(−1)

n

n−1 l ∑ l−k ( ∑∑ l=0 k=0 r=0

)( )( ) 2n − l − 2 l l − k (n − 1)! Stk (n − 1)Cn (l − k − r; b) n−1 k r l!bn−1−l

× x(x − 1)r . Let us assume that (86)

qn (x) ∼

(

( 1, t

) ) t − b at e , b

( ) x ∼ (1, eat − 1), a n

(a, b ̸= 0).

We easily see that (87) ( )n ( )−n t −1 n −nat t − b qn (x) = x x x = xe xn−1 at b t( t−b )e b ( )−n n−1 ∑ (n + l − 1) t = xe−nat (−1)n 1 − xn−1 = xe−ant (−1)n b−l (n − 1)l xn−1−l b l l=0 n−1 ∑ (2n − l − 2) (n − 1)! = (−1)n x(x − an)l . n−1 bn−1−l l! l=0

From (13), (86) and (87), we can prove the following exercise. Exercise For n ≥ 1, a, b ̸= 0, we have ( ) )( )( ) n−1 l ∑ l−k ( ∑∑ x 2n − l − 2 l k + r (n − 1)! −n =(−a) Stl−k−r (n − 1) a n n−1 k+r k l!bn−1−l r=0 l=0 k=0

× Cn (k; b)x(x − na − 1)r .

References [1] S. Araci, M. Acikgoz, A. Esi, A note on the q-Dedekind-type DaeheeChanghee sums with weight alpha arising from modified q-Genocchi polynomials with weight alpha, arXiv:1211.2350. [2] S. Araci, M. Acikgoz, Extended q-Dedekind-type Daehee-Changhee sums associated with Extended q-Euler polynomials, arXiv:1211.1233. [3] S. Araci, E Sen, M. Acikgoz, A note on the modified q-Dedekind sums, arXiv:1212.5837.

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[4] L. Carlitz, Some generating functions for Laguerre polynomials, Duke Math. J. 35 (1968) 825-827. [5] D. Ding, J. Yang, Some identities related to the Apostol-Euler and ApostolBernoulli polynomials, Adv. Stud. Contemp. Math. 20 (2010), no. 1, 7-21. [6] Q. Fang, T. Wang, Umbral calculus and invariant sequences, Ars Combin. 101 (2011), 257–264. [7] T. Kim, On the von Staudt-Clausen’s theorem for the q-Euler numbers, Russ. J. Math. Phys. 20 (2013), no. 1, 34-39. [8] D. S. Kim, T.Kim, S.-H. Lee, S.-H. Rim, Frobenius-Euler polynomials and umbral calculus in the p-adic case, Adv. Stud. Contemp. Math. 2012, 2012:222. [9] D. S. Kim, T. Kim, Some new identities of Frobenius-Euler numbers and polynomials, Journal of Inequalities and Applications 2012, 2012:307. [10] D. S. Kim, T. Kim, Applications of Umbral Calculus Associated with pAdic Invariant Integrals on Zp , Abstract and Applied Analysis 2012 (2012), Article ID 865721, 12 pages. [11] D. S. Kim, T. Kim, Some identities of Frobenius-Euler polynomials arising from umbral calculus, Advances in Difference Equations 2012, 2012:196. [12] T. Kim, S.-H. Rim, D. V. Dolgy, S.-H. Lee, Some identities on Bernoulli and Euler polynomials arising from the orthogonality of Laguerre polynomials, Advances in Difference Equations 2012, 2012:201. [13] T. Kim, Some identities on the q-Euler polynomials of higher order and q-Stirling numbers by the fermionic p-adic integral on Zp , Russ. J. Math. Phys. 16 (2009), no. 4, 484-491. [14] T. J. Robinson, Formal calculus and umbral calculus, Electron. J. Combin. 17 (2010), no. 1 , Research Paper 95, 31 pp. [15] S. Roman, More on the umbral calculus, with emphasis on the q-umbral calculus, J. Math. Anal. Appl.107 (1985), 222-254. MR0786026 (86h:05024) [16] S. Roman, The umbral calculus, Dover Publ. Inc. New York, 2005. [17] C. S. Ryoo, Some relations between twisted q-Euler numbers and Bernstein polynomials, Adv. Stud. Contemp. Math. 21 (2011), no. 2, 217–223. [18] C. S. Ryoo, T. Kim, R. P. Agarwal, Exploring the multiple Changhee qBernoulli polynomials, Int. J. Comput. Math. 82 (2005), no. 4, 483-493. [19] J. Sandor, An additive analogue of the Euler minimum function, Adv. Stud. Contemp. Math. 10 (2005), no. 1, 53-62. [20] Y. Simsek, S.-H. Rim, L.-C. Jang, D.-J. Kang, J.-J. Seo, A note on q-Daehee sums, J. Anal. Comput. 1 (2005), no. 2, 151-160.

Dae San Kim Department of Mathematics, Sogang University, Seoul 121-742, Republic of Korea TaeKyun Kim

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17

Department of Mathematics, Kwangwoon University, Seoul 139-701, Republic of Korea E-mail: [email protected] Sang-Hun Lee Division of General Education, Kwangwoon University, Seoul 139-701, Republic of Korea E-mail: [email protected] Seog-Hoon Rim Department of Mathematics Education, Kyungpook National University, Daegu 702-701, Republic of Korea E-mail: [email protected]

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Semilocal convergence theorem by using majorizing functions for Harmonic mean Newton’s method in Banach spaces Hong-Xiu Zhong1 , Guo-Liang Chen2 , Xue-Ping Guo3

Abstract: In this paper, we study the semilocal convergence of the Harmonic mean Newton’s method for solving nonlinear equations in Banach spaces. We establish the Newton-Kantorovich-type convergence theorem for the method by using majorizing functions. We obtain an existence-uniqueness theorem and an error estimate. In comparison with the results obtained in Chen et al., we can provide a larger convergence radius. Finally, some numerical applications is presented to demonstrate our approach. Keywords: Nonlinear equation; Banach spaces; Majorizing functions; Semilocal convergence; Newton’s method; A priori error bounds AMS classifications: 65D10; 65D99; 47H17

1

Introduction

Solving nonlinear operator equation is an important issue in the engineering and technology field. In this study, we are concerned with the problem of approximating a locally unique solution x∗ of the equation F (x) = 0,

(1.1)

where F is a twice-order Fr´ echet differentiable operator defined on an open convex subset Ω of a Banach space X with values in a Banach space Y . This equation can represent differential equations, integral equations or a system of nonlinear equations in the simplest case. There are kinds of methods to find a solution of equation (1.1). Iterative methods are often used to solve this problem [10]. The Newton’s method which has quadratic convergence is the most well known iterative method. Recently, a lot of research has been carried out to provide improvements in these methods. Third-order iterative methods such as Halley’s method, Chebyshev’s method, super-Halley’s method and Newton-like methods [3, 4, 8, 12, 13] are used to solve equation (1.1). The convergence of these iterative methods in Banach spaces is established by using recurrence relations. An alternative approach is developed to establish the convergence by using majorizing functions. The approach is also a very popular technique to establish the convergence of iterative methods. For example, it has been successfully applied to the convergence analysis of Newton’s method and some high-order methods [1, 2, 5, 6, 7, 14, 15]. Our goals in this paper is to increase the speed of convergence of Newton’s method and not to increase its operational cost very much. Taking into account these goals, we consider a multipoint Newton-type ¨ method called the Harmonic mean Newton’s method studied by Ozban [11] and Homeier [9]. This method 1 Department

of Mathematics, East China Normal University, Shanghai 200241, P. R. China (zhonghongxiu1123@ yahoo.com.cn). 2 Corresponding author. Department of Mathematics, East China Normal University, Shanghai 200241, P. R. China ([email protected]). This author is supported by the National Natural Science Foundation of China (No.11071079). 3 Department of Mathematics, East China Normal University, Shanghai 200241, P. R. China ([email protected]). This author is supported by the National Natural Science Foundation of China (No.44102470).

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is defined for all n ≥ 0 by yn = xn − Γn F (xn ), 1 xn+1 = xn − [Γn + Γn ]F (xn ), 2 ′

(1.2)

′

where Γn = F (xn )−1 and Γn = F (yn )−1 . Recently, Chen et al. [6] use recurrence relations to establish the convergence of third-order Harmonic mean Newton’s method for solving nonlinear operator equations (1.1). In this paper, we apply majorizing functions to establish the semilacal convergence of the method to solve nonlinear equations in Banach spaces. We prove the Newton-Kantorovich-type convergence theorem, along with a priori error bounds, which demonstrates the R-order convergence of the method. In comparison with the results obtained in [6], we can provide a larger convergence radius. And in the process of the proving, we find that the R-order of Harmonic mean Newton’s method can be reached at four. It is an interesting discover. The paper is organized as follows. Section 1 is the introduction. The convergence analysis based on majorizing functions is given in Section 2. In Section 3, some numerical examples are worked out and some simple comparisons are made. Finally, conclusions form Section 4.

2

Analysis of convergence

Let X, Y be Banach spaces and F : Ω ⊆ X → Y be a nonlinear twice Fr´ echet differentiable operator, where Ω is an open convex domain. The Harmonic mean Newton’s method to solve the equation (1.1) given by (1.2) can be written in the following form: yn = xn − Γn F (xn ), ′

′

H(xn , yn ) = Γn [F (yn ) − F (xn )], 1 xn+1 = yn − H(xn , yn )(yn − xn ), 2 ′

(2.3) n = 0, 1, 2, · · · ,

′

where Γn = F (xn )−1 and Γn = F (yn )−1 . Let x0 ∈ Ω, we assume that (C1) ∥Γ0 ∥ ≤ β, (C2) ∥Γ0 F (x0 )∥ ≤ η, ′′ (C3) ∥F (x)∥ ≤ M, x ∈ Ω, (C4) there exists a positive real number N such that ′′

′′

∥F (x) − F (y)∥ ≤ N ∥x − y∥, We denote g(t) =

∀x, y ∈ Ω.

1 2 t η Kt − + , 2 β β

where K, β, η, M and N are positive real numbers and K≥M+

5N . 3M β

(2.4)

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Let h = Kβη. When h ≤ 21 , t∗ = hη (1 − g(t). Let

√

1 − 2h) and t∗∗ = hη (1 +

sn = tn −

g(tn ) , g ′ (tn )

′

√ 1 − 2h) are two positive roots of

t0 = 0,

′

′

hn = g (sn )−1 [g (sn ) − g (tn )],

(2.5)

1 g(tn ) tn+1 = sn + hn ′ . 2 g (tn ) First, we have the following lemmas. Lemma 2.1. Suppose the sequences {tn } and {sn } are generated by Equation (2.5). If h ≤ 21 , then the sequences {tn } and {sn } increase monotonically and converge to t∗ . Let an = t∗ − tn , bn = t∗∗ − tn . Then, for all natural numbers n, t∗ − tn+1 =

a4n , (an + bn )(a2n + b2n )

t∗∗ − tn+1 =

b4n , (an + bn )(a2n + b2n )

0 ≤ tn ≤ sn ≤ tn+1 < t∗ .

(2.6)

Proof. By direct calculating, we have g(tn ) = ′

g (tn ) = −

K ∗ K (t − tn )(t∗∗ − tn ) = an bn , 2 2

K K ∗ [(t − tn ) + (t∗∗ − tn )] = − (an + bn ). 2 2

′

We also know g (tn ) = Ktn − β1 , so it follows that −

K 1 (an + bn ) = Ktn − . 2 β

Therefore, sn − tn = − ′

hn =

g(tn ) an bn = , g ′ (tn ) an + bn

(2.7)

′

g (sn ) − g (tn ) K(sn − tn ) 2an bn , = =− 2 ′ 1 g (sn ) an + b2n Ktn − β + K(sn − tn ) tn+1 − sn =

1 g(tn ) an bn an bn hn = · , 2 g ′ (tn ) an + bn a2n + b2n

and tn+1 − tn = tn+1 − sn + sn − tn =

a2n + an bn + b2n an bn . · a2n + b2n an + bn

Thus

(2.8)

(2.9)

an+1 = t∗ − tn+1 = an − (tn+1 − tn ) =

a4n , (an + bn )(a2n + b2n )

(2.10)

bn+1 = t∗∗ − tn+1 = bn − (tn+1 − tn ) =

b4n . (an + bn )(a2n + b2n )

(2.11)

By equations (2.7)-(2.11), t0 = 0 < t∗ , and by induction, we know that equation (2.6) holds. So we get tn and sn increase and converge to t∗ (see the proof of Theorem 2.2). 3 852

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Lemma 2.2. Assume that the nonlinear operator F : Ω ⊂ X → Y is continuously second-order Fr´ echet differentiable, where Ω is an open set, X and Y are Banach spaces. The sequences {xn } and {yn } are generated by iterations (2.3), then we have ∫

1

′′

′′

[F (xn + t(yn − xn )) − F (xn )](1 − t)dt(yn − xn )2

F (xn+1 ) = 0

−

1 2 ∫

∫

1

′′

′′

[F (xn + t(yn − xn )) − F (xn )]dt(yn − xn )2

(2.12)

0 1

′′

F (yn + t(xn+1 − yn ))(1 − t)dt(xn+1 − yn )2 .

+ 0

Proof. By Taylor expansion, we have ′

F (xn+1 ) = F (yn ) + F (yn )(xn+1 − yn ) ∫ 1 ′′ + F (yn + t(xn+1 − yn ))(1 − t)dt(xn+1 − yn )2 .

(2.13)

0 ′

′

′′

F (yn ) − F (xn ) = F (xn )(yn − xn ) ∫ 1 ′′ ′′ + [F (xn + t(yn − xn )) − F (xn )]dt(yn − xn ),

(2.14)

0

From iterations (2.3), we note that ′ ′ 1 ′ F (yn )(xn+1 − yn ) = − [F (yn ) − F (xn )](yn − xn ), 2

(2.15)

′

F (xn ) + F (xn )(yn − xn ) = 0.

(2.16)

By Taylor expansion and equation (2.16), we obtain ′ 1 ′′ F (yn ) = F (xn ) + F (xn )(yn − xn ) + F (xn )(yn − xn )2 2 ∫ 1 ′′ ′′ + [F (xn + t(yn − xn )) − F (xn )](1 − t)dt(yn − xn )2

0

=

1 ′′ F (xn )(yn − xn )2 + 2

∫

1

′′

(2.17)

′′

[F (xn + t(yn − xn )) − F (xn )](1 − t)dt(yn − xn )2 , 0

Substituting equation (2.14) into equation (2.15), we have ′ 1 ′′ F (yn )(xn+1 − yn ) = − F (xn )(yn − xn )2 2 ∫ ′′ 1 1 ′′ − [F (xn + t(yn − xn )) − F (xn )]dt(yn − xn )2 , 2 0

(2.18)

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Then, substituting equations (2.17) and (2.18) into equation (2.13), we can obtain 1 ′′ F (xn+1 ) = F (xn )(yn − xn )2 + 2

∫

1

′′

′′

[F (xn + t(yn − xn )) − F (xn )](1 − t)dt(xn+1 − yn )2 0

∫ ′′ 1 ′′ 1 1 ′′ 2 − F (xn )(yn − xn ) − [F (xn + t(yn − xn )) − F (xn )]dt(yn − xn )2 2 2 0 ∫ 1 ′′ + F (yn + t(xn+1 − yn ))(1 − t)dt(xn+1 − yn )2 0

∫

1

′′

′′

[F (xn + t(yn − xn )) − F (xn )](1 − t)dt(yn − xn )2

= 0

−

1 2 ∫

∫

1

′′

′′

[F (xn + t(yn − xn )) − F (xn )]dt(yn − xn )2 0 1

′′

F (yn + t(xn+1 − yn ))(1 − t)dt(xn+1 − yn )2 .

+ 0

This completes the proof. Lemma 2.3. Under the conditions (C1)-(C4), equation (2.4), and h = Kβη ≤ 12 , considering the sequences {tn } and {sn } generated by iterations (2.5), the following items are verified for all n ≥ 0: (I) ∥xn − x0 ∥ ≤ tn , ′ ′ (II) ∥F (xn )−1 ∥ ≤ −g (tn )−1 , (III) ∥yn − xn ∥ ≤ sn − tn , (IV) ∥xn+1 − yn ∥ ≤ tn+1 − sn , (V) ∥xn+1 − xn ∥ ≤ tn+1 − tn . Proof. It can be easily proved that when n = 0 the above formula holds. Suppose that (I)-(V) are true for all integers k ≤ n. (I). From the above assumptions, we have ∥xn+1 − x0 ∥ ≤ ∥xn+1 − xn ∥ + ∥xn − x0 ∥ ≤ tn+1 − tn + tn = tn+1 . (II). By conditions (C3) and (2.4), we can obtain ′

′

∥F (xn+1 ) − F (x0 )∥ ≤ M ∥xn+1 − x0 ∥ ≤ M tn+1 < Kt∗ √ 1 − 1 − 2h 1 1 ≤ ≤ . = Kη ′ h β ∥F (x0 )−1 ∥ ′

By perturbation lemma see [10], page45, we get that F (xn+1 )−1 exists, and ′

′

β ∥F (x0 )−1 ∥ ≤ ′ ′ −1 1 − ∥F (x0 ) ∥∥F (xn+1 ) − F (x0 )∥ 1 − βM ∥xn+1 − x0 ∥ ′ β 1 ≤ = 1 = −g (tn+1 )−1 . 1 − βKtn+1 β − Ktn+1

∥F (xn+1 )−1 ∥ ≤

′

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(III). By lemma 2.2, and using induction hypothesis, one has ∫ 1 ′′ ′′ ∥F (xn+1 )∥ ≤ ∥[F (xn + t(yn − xn )) − F (xn )]∥(1 − t)dt∥yn − xn ∥2 0

+

1 2 ∫

∫

1

′′

′′

∥[F (xn + t(yn − xn )) − F (xn )]∥dt∥yn − xn ∥2 0 1

′′

∥F (yn + t(xn+1 − yn ))∥(1 − t)dt∥xn+1 − yn ∥2

+ 0

N N M ≤ ∥yn − xn ∥3 + ∥yn − xn ∥3 + ∥xn+1 − yn ∥2 6 4 2 M 5N (sn − tn )3 + (tn+1 − sn )2 ≤ 12 2 5N (sn − tn )2 M = (tn+1 − sn )(sn − tn ) + (tn+1 − sn )2 . 12 tn+1 − sn 2 Since

(sn − tn )2 (an bn )2 (an + bn )(a2n + b2n ) 2 2 = · ≤ an + bn = t∗ + t∗∗ = ≤ , 2 2 tn+1 − sn (an + bn ) (an bn ) Kβ Mβ using lemma 2.1, we know 5N 1 M (tn+1 − sn )(sn − tn ) + (tn+1 − sn )2 3M β 2 2 K ≤ (sn − tn+1 )(sn − tn+1 ) 2 K ∗ ≤ (t − tn+1 )(t∗ − tn+1 ) = g(tn+1 ). 2

∥F (xn+1 )∥ ≤

Hence, we get ′

′

∥yn+1 − xn+1 ∥ = ∥ − F (xn+1 )−1 F (xn+1 )∥ ≤ −g (tn+1 )−1 g(tn+1 ) = sn+1 − tn+1 .

(2.19)

(IV). By the assumption (C3), we get ′

′

∥F (yn+1 ) − F (x0 )∥ ≤ M ∥yn+1 − x0 ∥ ≤ M (∥yn+1 − xn+1 ∥ + ∥xn+1 − x0 ∥) ≤ M sn+1 1 < Kt∗ ≤ . ∥F ′ (x0 )−1 ∥ ′

So, F (yn+1 )−1 exists, and ′

′

∥F (x0 )−1 ∥ ′ −1 1 − ∥F (x0 ) ∥∥F ′ (yn+1 ) − F ′ (x0 )∥ ′ β 1 ≤ = 1 = −g (sn+1 )−1 . 1 − βM sn+1 β − Ksn+1

∥F (yn+1 )−1 ∥ ≤

Thus ∥xn+2 − yn+1 ∥ ≤

′ ′ ′ 1 ′ ∥F (yn+1 )−1 ∥∥F (yn+1 ) − F (xn+1 )∥∥F (xn+1 )−1 F (xn+1 )∥ 2 1 ′ ≤ − g (sn+1 )−1 · M (sn+1 − tn+1 )2 2 1 ′ ≤ − g (sn+1 )−1 · K(sn+1 − tn+1 )(sn+1 − tn+1 ) 2 ′ ′ ′ 1 ′ = g (sn+1 )−1 (g (sn+1 ) − g (tn+1 ))g (tn+1 )−1 g(tn+1 ) 2 = tn+2 − sn+1 .

(2.20)

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(V). From inequalities (2.19) and (2.20), we can obtain ∥xn+2 − xn+1 ∥ ≤ ∥xn+2 − yn+1 ∥ + ∥yn+1 − xn+1 ∥ ≤ tn+2 − tn+1 . This completes the proof. Theorem 2.1. Let X and Y be two Banach spaces and F : Ω ⊆ X → Y be a third-order Fr´ echet differentiable on a non-empty open convex subset Ω. Assume that all conditions (C1)-(C4) hold and x0 ∈ Ω, h = Kβη ≤ 12 , B(x0 , t∗ ) ⊂ Ω, where B(x0 , t) = {x|∥x − x0 ∥ < t, x ∈ Ω}, B(x0 , t) is the closed domain of B(x0 , t). Then, the sequence {xn } generated by iterations (2.3) is well defined, xn ∈ B(x0 , t∗ ) and ∥xn − x∗ ∥ ≤ t∗ − tn , and {xn } converges to the unique solution x∗ ∈ B(x0 , t∗∗ ) of (1.1). Proof. From Lemma 2.3, we can obtain that the sequence {xn } generated by iterations (2.3) is well defined, xn ∈ B(x0 , t∗ ) and converges to the unique solution x∗ ∈ B(x0 , t∗∗ ) of (1.1). Now, we prove the uniqueness. Also suppose y ∗ is the solution of (1.1) on B(x0 , t∗∗ ). Then, we have ′

∥F (x0 )−1

∫

1

′

F (x∗ + t(y ∗ − x∗ ))dt − I∥

0 ′

−1

≤ ∥F (x0 ) ∫

∫

1

∥·∥

′

′

[F (x∗ + t(y ∗ − x∗ )) − F (x0 )]dt∥

0 1

≤ Mβ

∥x∗ + t(y ∗ − x∗ ) − x0 ∥dt

0

∫ ≤ Mβ

1

[(1 − t)∥x∗ − x0 ∥ + t∥y ∗ − x0 ∥]dt

0

≤

Mβ ∗ (t + t∗∗ ) < 1. 2

By perturbation lemma, we get that the inverse of F (y ∗ ) − F (x∗ ) =

∫

1

∫1 0

′

F (x∗ + t(y ∗ − x∗ ))dt exists. Because

′

F (x∗ + t(y ∗ − x∗ ))dt(y ∗ − x∗ ),

0

y ∗ = x∗ . This completes the proof of the unique solution. Moreover, when m > n, ∥xm − xn ∥ ≤ ∥xm − xm−1 ∥ + ∥xm−1 − xm−2 ∥ + · · · + ∥xn+1 − xn ∥ ≤ tm − tn , and let m → ∞, we get ∥xn − x∗ ∥ ≤ t∗ − tn . This completes the proof. Theorem 2.2. Suppose F satisfies the conditions of Theorem 2.1. Then, (I). when h < 12 , (1 − θ2 )η 4n −1 ∥xn − x∗ ∥ ≤ t∗ − tn = θ , 1 − θ4n (II). when h = 21 , ∥xn − x∗ ∥ ≤ t∗ − tn = where θ =

t∗ t∗∗

=

√ 1−√1−2h . 1+ 1−2h

2η , 4n

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Proof. When h < 12 , by lemma 2.1, we get

Because t∗∗

n t∗ − tn t∗ − tn−1 4 t∗ − tn−2 42 t∗ − t0 4n = ( ∗∗ ) = ( ∗∗ ) = · · · ( ∗∗ ) = θ4 . ∗∗ t − tn t − tn−1 t − tn−2 t − t0 √ = hη (1 + 1 − 2h) = η( θ1 + 1) and t∗∗ − t∗ = η( θ1 + 1)(1 − θ) = ηθ (1 − θ2 ), we get

t∗ − tn =

(1 − θ2 )η 4n −1 θ . 1 − θ4n

When h = 21 , t∗ = t∗∗ = 2η. By lemma 2.1, we get t∗ − tn =

t∗ − t0 2η t∗ − tn+1 = ··· = = n. n 4 4 4

This completes the proof. Remark 1. From Theorem 2.2, we note that the R-order of convergence of method (2.3) is at least four when h < 12 , and for h = 12 , the order of convergence goes down to one. In [6], Chen obtain the R-order of convergence of the Harmonic mean Newton’s method is third by using recurrence relations, while, we obtain the fourth-order convergence in this paper, it’s the interesting thing.

3

Numerical examples

In this section, we illustrate the previous study with applications to the following two nonlinear equations. 1 3 Example 1. Consider the root of the equation F (x) = 10 x − 15 x − 21 = 0 on [1, 3]. If we select the initial point x0 = 2, then we easily get ′

∥F (x0 )−1 ∥ = 1 = β,

′

∥F (x0 )−1 F (x0 )∥ = 0.1 = η,

′′

∥F (x)∥ ≤

9 = M, 5

x ∈ Ω,

and the Lipschitz condition with N = 35 , ′′

′′

∥F (x) − F (y)∥ ≤

3 ∥x − y∥, 5

x, y ∈ Ω.

Note that K = 2.3556, and h = Kβη = 0.2356 ≤ 12 , therefor t∗ = 0.1158, t∗∗ = 0.7333. This means that the hypotheses of Theorem 2.1 are satisfied. Hence, the solution of (1.1) exists in B(2, 0.1158) ⊆ Ω, and the unique solution exists in the ball B(2, 0.7333) ∩ Ω. However, by the convergence method given in [6] or [14], the solution of F (x) exists in B(2, 0.1115) ⊆ Ω, which is inferior to our result. We apply the method given by the iterations (2.3) to compute the solution of example 1, and then compare it with Newton’s method. The initial values x0 = 2 and x0 = 2.5 are considered, respectively. In Table 1, the value of |xn − xn−1 | at each iterative step is displayed. The stopping criterion that we consider is |F (xn )| ≤ 1e − 15. Table 1. Example 1 step 1 2 3 4

Method (2.3) 0.0945 2.7884e-5 8.8818e-16

x0 = 2 Newton’s method 0.1 0.0054 1.6639e-5 1.5587e-10

x0 = 2.5 Method (2.3) Newton’s method 0.1944 0.2198 1.1035e-4 0.0249 5.9952e-14 3.5217e-4 6.9829e-8

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From the numerical results, we can see that the R-order of Harmonic mean Newton’s method achieves four, which is coincided with Theorem 2.2. Example 2. We now consider the nonlinear integral equation F (x) = 0, where 7 1 F (x)(s) = x(s) − + 5 2

∫

1

s · cos(x(t))dt, 0

where s ∈ [0, 1] and x ∈ Ω = B(0, 2) ⊂ X. Here, X = C[0, 1] is the space of continuous functions on [0, 1] with the max-norm ∥x∥ = max |x(s)|. s∈[0,1]

We can obtain the derivatives of F given by ′

F (x)y(s) = y(s) − ′′

F (x)yz(s) = − ′′

Furthermore, we have ∥F (x)∥ ≤ ′′

1 2

1 2

∫

∫

1 2

1

s · sin(x(t))y(t)dt,

y ∈ Ω,

0

1

s · cos(x(t))y(t)z(t)dt,

y, z ∈ Ω,

0

= M,

x ∈ Ω, and the Lipschitz condition with N = 12 ,

′′

∥F (x) − F (y)∥ ≤

1 ∥x − y∥, 2

x, y ∈ Ω.

A constant function, that is, x0 (t) = 75 , is chosen as the initial approximate solution. It follows that ∥F (x0 )∥ ≤

1 7 cos . 2 5

In this case, we have 1 7 sin , 2 5 and then by the perturbation lemma, we include that Γ0 exists and obtain ′

∥I − F (x0 )∥ ≤

∥Γ0 ∥ ≤

2 = β, 2 − sin(7/5)

∥Γ0 F (x0 )∥ ≤

cos(7/5) =η 2 − sin(7/5)

and

K = 1.34545855834295.

Note that h = 0.44434280650540 ≤ 12 , therefore t∗ = 0.25123688628137, t∗∗ = 0.50281849782578 and θ = 0.49965720705927. This means that the hypotheses of Theorem 2.1 are satisfied. Hence, the solution of F (x) exists in B( 75 , 0.25123688628137) ⊆ Ω, and the unique solution exists in the ball B( 75 , 0.50281849782578) ∩ Ω. However, by the convergence method given in [6], the solution of F (x) exists in B( 75 , 0.23567274887651) ⊆ Ω, which is inferior to our result.

4

Conclusions

This paper is devoted to a fourth-order variant of the Newton’s method for solving nonlinear equations in Banach spaces. We establish the Newton-Kantorovich-type convergence theorem for this method by using majorizing functions and get the error estimate. This approach is simple and efficient in comparison with the approach using recurrence relations in [6]. Numerical examples are worked out to demonstrate our approach. 9 858

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References [1] I. K. Argyros, On the Newton-Kantorovich hypothesis for solving equations, J. Comput. Appl. Math., 169: 315–332, 2004. [2] I. K. Argyros and S. Hilout, Majorizing sequences for iterative methods, J. Comput. Appl. Math., 236: 1947–1960, 2012. [3] V. Candela and A. Marquina, Recurrence relations for rational cubic methods I: The Halley method, Computing, 44: 169–184, 1990. [4] V. Candela and A. Marquina, Recurrence relations for rational cubic methods II: The Chebyshev method, Computing, 45: 355–367, 1990. [5] J.-H. Chen, I. K. Argyros, and R. P. Agarwal, Majorizing function and two-point Newton-type method, J. Comput. Appl. Math., 234: 1473–1484, 2010. [6] L. Chen, C.-Q. Gu, and Y.-F. Ma, Recurrence relations for the Harmonic mean Newton’s method in Banach Spaces, J. Comput. Anal. Appl., 14(6): 1154–1164, 2012. [7] X.-P. Guo, On semilocal convergence of inexact newton methods, J. Comput. Math., 25(2): 231–242, 2007. [8] J. M. Guti´ errez and M. A. Hernndez, Recurrence relations for the super-Halley method, Comput. Math. Appl., 36: 1–8, 1998. [9] H. Homeier, On Newton-type methods with cubic convergence, J. Comput. Appl. Math., 176: 425–432, 2005. [10] J. M. Ortega and W. C. Rheinbolt, Iterative solution of nonlinear equations in several variables, Academic Press, NewYork, 1970. ¨ [11] A. Y. Ozban, Some new variants of Newton’s method, Appl. Math. Lett., 17: 677–682, 2004. [12] P. K. Parida and D. K. Gupta, Recurrence relations for a Newton-like method in Banach spaces, J. Comput. Appl. Math., 206: 873–887, 2007. [13] P. K. Parida and D. K. Gupta, Semilocal convergence of a family of a third-order Chebyshev-type methods under a mild differentiability condition, Int. J. Comput. Math., 87: 3405–3419, 2010. [14] Q.-B. Wu and Y.-Q. Zhao, Third-order convergence theorem by using majorizing function for a modified Newton method in Banach space, Appl. Math. Comput., 175: 1515–1524, 2006. [15] L. Zheng and C.-Q. Gu, Fourth-order convergence theorem by using majorizing functions for superHalley method in Banach spaces, Int. J. Comput. Math., 1: 1–12, 2012.

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FERMIONIC p-ADIC INTEGRALS ON Zp AND UMBRAL CALCULUS DAE SAN KIM, TAEKYUN KIM, SANG-HUN LEE AND DMITRY V. DOLGY

Abstract. In this paper we study some properties of the fermionic p-adic integrals on Zp arising from the umbral calculus.

1. Introduction Let p be a fixed odd prime number. Throughout this paper Zp , Qp and Cp denote the ring of p-adic rational integers, the field of p-adic rational numbers and the completion of the algebraic closure of Qp , respectively. Let N be the set of natural numbers and Z+ = N∪{0}. Let C(Zp ) be the space of continuous functions on Zp . For f ∈ C(Zp ), the fermionic p-adic integral on Zp is defined by ∫ Zp

(1)

N p∑ −1

f (x)dµ−1 (x) = lim

N →∞

= lim

x=0 N p∑ −1

N →∞

f (x)(−1)x , (see [1,2,11]).

x=0

For n ∈ N, we have ∫ ∫ f (x + n)dµ−1 (x) + (−1)n−1 (2) Zp

f (x)µ−1 (x + pN Zp )

Zp

f (x)dµ−1 (x) = 2

n−1 ∑

(−1)n−1−l f (l).

l=0

In the special case, n = 1, we note that ∫ ∫ f (x + 1)dµ−1 (x) + f (x)dµ−1 (x) = 2f (0), (see [ 11]). (3) Zp

Zp

Let F be the set of all formal power series in the variable t over Cp with ∞ { } ∑ ak k F = f (t) = t ak ∈ Cp . k! k=0

Let P = Cp [x] and let P∗ denote the vector space of all linear functionals on P. The formal power series (4)

f (t) =

∞ ∑ ak k=0

k!

tk ∈ F.

defines a linear functional on P by setting (5)

⟨f (t)|xn ⟩ = an for all n ≥ 0, (see [ 7,14]). 1

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DAE SAN KIM, TAEKYUN KIM, SANG-HUN LEE AND DMITRY V. DOLGY

Thus, by (4) and (5), we get ⟨tk |xn ⟩ = n!δn,k , (n, k ≥ 0),

(6)

where δn,k is the Kronecker symbol (see [7,14]). Here, F denotes both the algebra of formal power series in t and the vector space of all linear functionals on P, and so an element f (t) of F will be thought of as both a formal power series and a linear functional. We shall call F the umbral algebra. The umbral calculus is the study of umbral algebra (see [7,14]). The order O(f (t)) of power series f (t)(̸= 0) is the smallest integer k for which ak does not vanish (see [7,4]). The series f (t) has a multiplicative inverse, denoted 1 by f (t)−1 or f (t) , if and only if O(f (t)) = 0. Such series is called an invertible series. A series f (t) for which O(f (t)) = 1 is called a delta series (see [7,14]). For f (t), g(t) ∈ F , we have ⟨f (t)g(t)|p(x)⟩ = ⟨f (t)|g(t)p(x)⟩ = ⟨g(t)|f (t)p(x)⟩. By (6), we get ⟨eyt |xn ⟩ = y n ,

(7)

⟨eyt |p(x)⟩ = p(y), (see [ 7,14]).

Let f (t) ∈ F . Then we note that f (t) =

(8)

∞ ∑ ⟨f (t)|xk ⟩ k=0

k!

tk ,

and p(x) =

(9)

∞ ∑ ⟨tk |p(x)⟩ k=0

k!

xk , for p(x) ∈ P, (see [14]).

Let f1 (t), f2 (t), · · · , fm (t) ∈ F . It is known in [7,14] that ) ∑( n n ⟨f1 (t) · · · fm (t)|x ⟩ = (10) ⟨f1 (t)|xi1 ⟩ · · · ⟨fm (t)|xim ⟩, i1 , · · · , im where the sum is over all nonnegative integers i1 , · · · , im such that i1 + i2 + · · · + im = n (see [7,14]). By (9), we get ∞

p (11)

(k)

dk p(x) ∑ ⟨tl |p(x)⟩ (x) = = l(l − 1) · · · (l − k + 1)xl−k dxk l! l=k ( ) ∞ ∑ l k! l−k l = ⟨t |p(x)⟩ x . k l! l=k

Thus, from (11), we have p(k) (0) = ⟨tk |p(x)⟩ = ⟨1|p(k) (x)⟩,

(12) and (13)

tk p(x) = p(k) (x) =

dk p(x) dxk

(see [6,7,14]).

From (13), we note that (14)

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

(see [7,14]).

In this paper, sn (x) denotes a polynomial of degree n. Let us assume that f (t), g(t) ∈ F with o(f (t)) = 1 and o(g(t)) = 1. Then there exists a unique sequence sn (x)

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3

of polynomials satisfying ⟨g(t)f (t)k |sn (x)⟩ = n!δn,k for all n, k ≥ 0. The sequence sn (x) is called the Sheffer sequence for (g(t), f (t)), which is denoted by sn (x) ∼ (g(t), f (t)). If sn (x) ∼ (g(t), t), then sn (x) is called the Appell sequence for g(t) (see [6,7,14]). Let p(x) ∈ P. Then we note that ⟨f (t)|xp(x)⟩ = ⟨∂t f (t)|p(x)⟩ = ⟨f ′ (t)|p(x)⟩,

(15) and

⟨eyt − 1|p(x)⟩ = p(y) − p(0),

(16)

(see [7,14]).

Let us assume that sn (x) ∼ (g(t), f (t)). Then we have h(t) =

(17)

∞ ∑ ⟨h(t)|sk (x)⟩ k=0

p(x) =

(18)

k!

g(t)f (t)k ,

∞ ∑ ⟨g(t)f (t)k |p(x)⟩ k=0

k!

sk (x),

h(t) ∈ F , p(x) ∈ P,

∞

∑ sk (y) 1 y f¯(t) tk , e = k! g(f¯(t)) k=0

(19)

for all y ∈ Cp ,

where f¯(t) is the compositional inverse of f (t), and f (t)sn (x) = nsn−1 (x),

(20)

(see [7,14]).

As is well known, the Euler polynomials are defined by the generating function to be ∞ ∑ 2 tn xt E(x)t e = e = , (see [1-19]), E (x) (21) n et + 1 n! n=0 with the usual convention about replacing E n (x) by En (x). In the special case, x = 0, En (0) = En are called the n-th Euler numbers Let sn (x) ∼ (g(t), t). Then Appell identity is known to be n ( ) n ( ) ∑ ∑ n n k sn (x + y) = sn−k (x)y = sk (x)y n−k . (22) k k k=0

k=0

From (21), we note that the recurrence relation of the Euler numbers is given by (23)

E0 = 1,

(E + 1)n + En = En (1) + En = 2δ0,n .

By (1) and (21), we get ∫ (x + y)n dµ−1 (y) = En (x), (24) Zp

∫ Zp

xn dµ−1 (y) = En ,

where n ≥ 0 (see [1,11,16]). Recently, D. S. Kim and T. Kim have studied applications of umbral calculus associated with p-adic invariant integrals on Zp (see [7]). In this paper we study some properties of the fermionic p-adic integrals on Zp arising from the umbral calculus.

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DAE SAN KIM, TAEKYUN KIM, SANG-HUN LEE AND DMITRY V. DOLGY

2. Umbral calculus and fermionic p-adic integrals on Zp Let sn (x) ∼ (g(t), t). Then, by (19), we get 1 n x = sn (x) if and only if (25) g(t) Let us assume that g(t) = By (21), we get

et +1 2 .

xn = g(t)sn (x).

Then we note that g(t) is an invertible functional. ∞

tk 1 xt ∑ Ek (x) . e = g(t) k!

(26)

k=0

Thus, from (26), we have 1 n x = En (x), (27) g(t)

tEn (x) =

n n−1 x = nEn−1 (x). g(t)

By (19), (20) and (27), we see that En (x) is an Appell sequence for g(t) = It is easy to show that ( g ′ (t) ) En+1 (x) = x − En (x), (n ≥ 0). (28) g(t) From (2), (21) and (24), we note that ∫ ∫ e(x+y+1)t dµ−1 (y) + (29) Zp

Zp

et +1 2 .

e(x+y)t dµ−1 (y) = 2ext .

Thus, by (29), we get ∫ ∫ n (x + y + 1) dµ (y) + (x + y)n dµ−1 (y) = 2xn . (30) −1 Zp

Zp

From (24) and (30), we have En (x + 1) + En (x) = 2xn ,

(31)

(n ≥ 0).

By (28), we see that g(t)En+1 (x) = g(t)xEn (x) − g ′ (t)En (x),

(32)

(n ≥ 0).

Thus, we have (et + 1)En+1 (x) = (et + 1)xEn (x) − et En (x).

(33) By (33), we get (34)

En+1 (x + 1) + En+1 (x) = (x + 1)En (x + 1) + xEn (x) − En (x + 1).

Thus, from (34) and (31), we have (35)

En+1 (x + 1) + En+1 (x) = x(En (x + 1) + En (x)).

By (35), we get En (x + 1) + En (x) = x(En−1 (x + 1) + En−1 (x)) = x2 (En−2 (x + 1) + En−2 (x)) = · · · = xn (E0 (x + 1) + E0 (x)) = 2xn . Let us consider the functional f (t) such that ∫ ⟨f (t)|p(x)⟩ = p(u)dµ−1 (u), (36) Zp

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for all polynomials p(x). It can be determined from (8) to be ∫ ∞ ∫ ∞ ∑ tk ⟨f (t)|xk ⟩ k ∑ k t = u dµ−1 (u) = eut dµ−1 (u). f (t) = (37) k! k! Zp Zp k=0

k=0

By (29) and (37), we get

∫ f (t) =

(38)

Zp

eut dµ−1 (u) =

et

2 . +1

Therefore, by (38), we obtain the following theorem. Theorem 2.1. For p(x) ∈ P, we have ∫ ∫ ⟨ eyt dµ−1 (y)|p(x)⟩ = Zp

Zp

That is, ⟨

2 |p(x)⟩ = et + 1

p(u)dµ−1 (u).

∫ Zp

p(u)dµ−1 (u).

Also, the n-th Euler number is given by ∫ En = ⟨ eyt dµ−1 (y)|xn ⟩. Zp

By (3) and (30), we get ∫ ∞ ∫ ∞ ∫ ∑ ∑ tn tn e(x+y)t dµ−1 (y) = eyt dµ−1 (y)xn . (x + y)n dµ−1 (y) = (39) n! n! Zp n=0 Zp n=0 Zp From (24) and (39), we have En (x) =

(40)

∫ Zp

eyt dµ−1 (y)xn =

2 xn , et + 1

where n ≥ 0. Therefore, by (40), we obtain the following theorem. Theorem 2.2. For p(x) ∈ P, we have ∫ ∫ p(x + y)dµ−1 (y) = eyt dµ−1 (y)p(x) = Zp

Zp

et

2 p(x). +1

From (22), we note that n ( ) ∑ n En (x + y) = Ek (x)y n−k . k k=0

The Euler polynomials of order r are defined by the generating function to be (

(41)

∞ ( 2 )r ( 2 ) ∑ 2 ) ( 2 ) tn xt xt × × · e = e = En(r) (x) . · · × t t t t e +1 e +1 e +1 e +1 n! n=0 | {z } r times

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DAE SAN KIM, TAEKYUN KIM, SANG-HUN LEE AND DMITRY V. DOLGY (r)

(r)

In the special case, x = 0, En (0) = En

are called the n-th Euler numbers of ( t ) r e +1 . Then we see that g r (t) order r (r ≥ 0), (see [1-19]). Let us take g (t) = 2 is an invertible functional in F. By (41), we get r

∞ 1 xt ∑ (r) tn e = En (x) . r g (t) n! n=0

(42) Thus, we have

1 n x = En(r) (x), g r (t)

n n−1 (r) x = nEn−1 (x). g r (t) ( t )r (r) So, by (42), we see that En (x) is the Appell sequence for e 2+1 . From (22), we have n ( ) ∑ n (r) (r) E (x + y) = En−k (x)y k . (44) n k (43)

tEn(r) (x) =

k=0

It is easy to show that (45) ∫ Zp

∫ ···

Zp

e(x1 +···xr +x)t dµ−1 (x1 ) · · · dµ−1 (xr ) =

(

∞ tn 2 )r xt ∑ (r) e = (x) . E n et + 1 n! n=0

By (6) and (45), we get ∫ ∫ (r) e(x1 +···xr )t dµ−1 (x1 ) · · · dµ−1 (xr )|xn ⟩, En = ⟨ ··· (46) Zp Zp | {z }

(n ≥ 0),

r−times

and, by (10), ∫ ∫ ⟨ ··· Zp

(47)

Zp

e(x1 +···xr )t dµ−1 (x1 ) · · · dµ−1 (xr )|xn ⟩

)∫ ∫ n ⟨ ex1 t dµ−1 (x1 )|xi1 ⟩ · · · ⟨ exr t dµ−1 (xr )|xir ⟩ i , · · · , i 1 r Z Z p p n=i1 +···+ir ( ) ∑ n = Ei1 Ei2 · · · Eir . i , · · · , ir 1 n=i +···+i =

(

∑

1

r

From (46) and (47), we have (48)

En(r) =

(

∑ n=i1 +···+ir

) n Ei1 · · · Eir . i1 , · · · , ir

(r)

By (44) and (48), we see that En (x) is a monic polynomial of degree n with coefficients in Q. Let r ∈ N. Then we note that ( e t + 1 )r 1 ∫ = . g r (t) = ∫ 2 (x1 +···xr )t dµ ) · · · e (x ) · · · dµ (x −1 1 −1 r (49) Zp Zp | {z } r−times

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FERMIONIC p-ADIC INTEGRALS ON Zp AND UMBRAL CALCULUS

7

By (49), we get (50)

∫

1 xt e = g r (t)

∫

Zp

From (50), we have

···

Zp

e(x1 +···xr +x)t dµ−1 (x1 ) · · · dµ−1 (xr ) =

∫

En(r) (x) =

∞ ∑

En(r) (x)

n=0

tn . n!

∫ ···

Zp

=∫ (51) |

Zp

(x1 + · · · xr + x)n dµ−1 (x1 ) · · · dµ−1 (xr ) 1

∫ ···

Zp

e(x1 +···xr )t dµ Zp

{z

xn −1 (x1 ) · · · dµ−1 (xr )

}

r−times

=

1 g r (t)

xn .

Therefore, by (51), we obtain the following theorem. Theorem 2.3. For p(x) ∈ P and r ∈ N. Then we have ∫ ∫ ( ··· p(x1 + · · · xr + x)dµ−1 (x1 ) · · · dµ−1 (xr ) = |

Zp

{z

Zp

}

2 )r p(x). et + 1

r−times

In particular, En(r) (x) = That is,

(

( En(r) (x) ∼ ∫

∫

∫

2 )r n x = et + 1

Zp

···

Zp

e(x1 +···xr )t dµ−1 (x1 ) · · · dµ−1 (xr )xn .

) 1 , t . · · · Zp e(x1 +···xr )t dµ−1 (x1 ) · · · dµ−1 (xr ) Zp ∫

Let us take the functional f r (t) such that ∫ ∫ r p(x1 + · · · xr )dµ−1 (x1 ) · · · dµ−1 (xr ), ⟨f (t)|p(x)⟩ = ··· (52) Zp Zp | {z } r−times

for all polynomials p(x). It can be determined from (8) to be f r (t) = =

∞ ∑ ⟨f r (t)|xk ⟩ k=0 ∞ ∫ ∑

∫

k=0 | Zp

(53)

··· {z

Zp

(x1 + · · · xr )k dµ−1 (x1 ) · · · dµ−1 (xr )

}

tk k!

r−times

∫

∫

= |

tk

k!

Zp

··· {z

Zp

e(x1 +···xr )t dµ−1 (x1 ) · · · dµ−1 (xr )

}

r−times

Therefore, by (52) and (53) , we obtain the following theorem.

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8

DAE SAN KIM, TAEKYUN KIM, SANG-HUN LEE AND DMITRY V. DOLGY

Theorem 2.4. For p(x) ∈ P, we have ∫ ∫ ∫ ⟨ ··· e(x1 +···xr )t dµ−1 (x1 ) · · · dµ−1 (xr )|p(x)⟩ = |

Zp

{z

Zp

}

|

r−times

Moveover, ( ⟨

∫

Zp

··· {z

Zp

p(x1 +· · · xr )dµ−1 (x1 ) · · · dµ−1 (xr ).

}

r−times

2 )r |p(x)⟩ = et + 1

∫

∫ |

Zp

··· {z

Zp

p(x1 + · · · xr )dµ−1 (x1 ) · · · dµ−1 (xr ).

}

r−times

ACKNOWLEDGEMENTS. 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 2012R1A1A2003786. The first author was supported in part by the Research Grant of Kwangwoon University in 2013.

References [1] S. Araci, M. Acikgoz, A note on the Frobenius-Euler numbers and polynomials associated with Bernstein polynomials, Adv. Stud. Contemp. Math. 22 (2012), no. 3, 399–406. [2] S. Araci, M. Acikg¨ oz, H. Jolany, J. J. Seo, A unified generating function of the q-Genocchi polynomials with their interpolation functions, Proc. Jangjeon Math. Soc. 15 (2012), no. 2, 227–233. [3] L. Carlitz, Some remarks on the multiplication theorems for the Bernoulli and Euler polynomials, Glas. Mat. Ser. III 16(36) (1981), no. 1, 3–23. [4] L. Carlitz, Some polynomials related to the Bernoulli and Euler polynomials, Utilitas Math. 19 (1981), 81–127. [5] M. Can, M. Cenkci, V. Kurt, Y. Simsek, On the higher-order w-q-Genocchi numbers, Adv. Stud. Contemp. Math. 19 (2009), no. 1, 39–57. [6] T. Kim, Symmetry p-adic invariant integral on Zp for Bernoulli and Euler polynomials, J. Difference Equ. Appl. 14 (2008), no. 12, 1267–1277. [7] T. Kim, Some identities for the Bernoulli, the Euler and the Genocchi numbers and polynomials, Adv. Stud. Contemp. Math. 20 (2010), no. 1, 23–28. [8] T. Kim, An identity of the symmetry for the Frobenius-Euler polynomials associated with the fermionic p-adic invariant q-integrals on Zp , Rocky Mountain J. Math. 41 (2011), no. 1, 239–247 [9] T. Kim, A note on q-Bernstein polynomials, Russ. J. Math. Phys. 18 (2011), no. 1, 73–82. [10] T. Kim, Identities involving Frobenius?Euler polynomials arising from nonlinear differential equations, J. Number Theory 132 (2012), no. 12, 2854– 2865. [11] T. Kim, Symmetry of power sum polynomials and multivariate fermionic p-adic invariant integral on Zp , Russ. J. Math. Phys. 16(2009), no. 1, 93-96 . [12] S.-H. Rim, J. Jeong, On the modified q-Euler numbers of higher order with weight, Adv. Stud. Contemp. Math. 22 (2012), no. 1, 93-98.

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9

[13] T, Kim, S.-H. Rim, D.V. Dolgy, S.-H. Lee, Some identities on Bernoulli and Euler polynomials arising from the orthogonality of Laguerre polynomials, Adv. Difference Equ. 2012, 2012:201. [14] S. Roman, The umbral calculus, Dover Publ. Inc. New York, 2005. [15] T. Kim, D. S. Kim, A. Bayad, S.-H. Rim Identities on the Bernoulli and the Euler numbers and polynomials, Ars Combin. 107 (2012), 455–463. [16] C. S. Ryoo, Some relations between twisted q-Euler numbers and Bernstein polynomials, Adv. Stud. Contemp. Math. 21 (2011), no. 2, 217–223. [17] T. Kim, J. Choi, Y. H. Kim, C. S. Ryoo, On the fermionic p-adic integral representation of Bernstein polynomials associated with Euler numbers and polynomials, J. Inequal. Appl. 2010, Art. ID 864247, 12 pp. [18] K. Shiratani, On Euler numbers, Mem. Fac. Sci., Kyushu Univ., Ser. A 27(1973), 1-5 . [19] K. Shiratani, S. Yamamoto, On a p-adic interpolation function for the Euler numbers and its derivatives, Mem. Fac. Sci. Kyushu Univ. Ser. A 39 (1985), no. 1, 113–125.

Dae San Kim Department of Mathematics, Sogang University, Seoul 121-742, Republic of Korea E-mail: [email protected]

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

Sang-Hun Lee Division of General Education, Kwangwoon University, Seoul 139-701 , Republic of Korea E-mail: [email protected]

Dmitry V. Dolgy Hanrimwon, Kwangwoon University, Seoul 139-701, Republic of Korea E-mail: [email protected]

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Soft set theory and N -structures applied to BCH-algebras Young Bae Juna , N. O. Alshehrib,∗ and Kyoung Ja Leec a

Department of Mathematics Education and (RINS), Gyeongsang National University, Chinju 660-701, Korea b

Department of Mathematics, King Abdulaziz University Faculty of science for girls, Jeddah, KSA c

Department of Mathematics Education Hannam University, Daejeon 306-791, Korea Abstract The notions of (closed) N -filters, regular N -subalgebras and N -soft BCH-algebras are introduced, and related properties are investigated. Conditions for an N -subalgebra (resp. N -closed ideal) to be an N -closed ideal (resp. closed N -filter) are provided. Characterizations of an N -structure with N -regularity are considered. A condition for an N -closed ideal to satisfy the N -structure is discussed. The union (resp. intersection) of two N -soft BCH-algebras are discussed. Keywords: N -closed ideal, N -subalgebra, (closed) N -filter, N -transfer principle, N regularity, regular subset, N -soft BCH-algebra, 2010 Mathematics Subject Classification. 06F35, 03G25, 06D72

1

Introduction

A (crisp) set A in a universe X can be defined in the form of its characteristic function µA : X → {0, 1} yielding the value 1 for elements belonging to the set A and the value 0 for elements excluded from the set A. So far most of the generalization of the crisp set have been conducted on the unit interval [0, 1] and they are consistent with the asymmetry observation. In other words, the generalization of the crisp set to fuzzy sets relied on *Corresponding author. e-mail: [email protected] (Y. B. Jun), [email protected] (N. O. Alshehri), [email protected] (K. J. Lee)

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spreading positive information that fit the crisp point {1} into the interval [0, 1]. Because no negative meaning of information is suggested, we now feel a need to deal with negative information. To do so, we also feel a need to supply mathematical tool. To attain such object, Jun et al. [7] introduced and used a new function which is called negative-valued function. They studied the ideal theory in BCK/BCI-algebras based on N -structures. The real world is inherently uncertain, imprecise and vague. Various problems in system identification involve characteristics which are essentially non-probabilistic in nature [13]. In response to this situation Zadeh [14] introduced fuzzy set theory as an alternative to probability theory. Uncertainty is an attribute of information. In order to suggest a more general framework, the approach to uncertainty is outlined by Zadeh [15]. To solve complicated problem in economics, engineering, and environment, we can not successfully use classical methods because of various uncertainties typical for those problems. There are three theories: theory of probability, theory of fuzzy sets, and the interval mathematics which we can consider as mathematical tools for dealing with uncertainties. But all these theories have their own difficulties. Uncertainties can not be handled using traditional mathematical tools but may be dealt with using a wide range of existing theories such as probability theory, theory of (intuitionistic) fuzzy sets, theory of vague sets, theory of interval mathematics, and theory of rough sets. However, all of these theories have their own difficulties which are pointed out in [12]. Maji et al. [11] and Molodtsov [12] suggested that one reason for these difficulties may be due to the inadequacy of the parametrization tool of the theory. To overcome these difficulties, Molodtsov [12] introduced the concept of soft set as a new mathematical tool for dealing with uncertainties that is free from the difficulties that have troubled the usual theoretical approaches. Molodtsov pointed out several directions for the applications of soft sets. Worldwide, there has been a rapid growth in interest in soft set theory and its applications in recent years. Evidence of this can be found in the increasing number of high-quality articles on soft sets and related topics that have been published in a variety of international journals, symposia, workshops, and international conferences in recent years. Maji et al. [11] described the application of soft set theory to a decision making problem. In [4, 5], Hu and Li introduced the notion of BCH–algebras which are a generalization of BCK/BCI-algebras. Ahmad [1] classified BCH–algebras, and decompositions of BCH– algebras are considered by Dudek and Thomys [3]. Chaudhry et al. studied closed ideals and filters in BCH–algebras. In this paper, we apply the N -structures and soft set theory to BCH–algebras. We introduce the notions of (closed) N -filters, regular N -subalgebras and N -soft BCH-algebras, and investigate related properties. We provide conditions for 2 870

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an N -subalgebra (resp. N -closed ideal) to be an N -closed ideal (resp. closed N -filter). We consider characterizations of an N -structure with N -regularity. We also discuss a condition for an N -closed ideal to satisfy the N -structure, and deal with the union (resp. intersection) of two N -soft BCH-algebras.

2

Preliminaries

By a BCH–algebra we mean an algebra (X, ∗, 0) of type (2, 0) satisfying the following axioms: (H1) x ∗ x = 0, (H2) x ∗ y = 0 and y ∗ x = 0 imply x = y, (H3) (x ∗ y) ∗ z = (x ∗ z) ∗ y for all x, y, z ∈ X. In a BCH–algebra X, the following conditions are valid (see [3, 4]). (a1) x ∗ 0 = x, (a2) x ∗ 0 = 0 implies x = 0, (a3) 0 ∗ (x ∗ y) = (0 ∗ x) ∗ (0 ∗ y), (a4) 0 ∗ (0 ∗ (0 ∗ x)) = 0 ∗ x. A BCH–algebra X is said to be medial if it satisfies: (∀x, y, a, b ∈ X)((x ∗ y) ∗ (a ∗ b) = (x ∗ a) ∗ (y ∗ b)).

(2.1)

A subset R of a BCH–algebra X is said to be regular if it satisfies: (∀x ∈ R)(∀y ∈ X)(x ∗ y ∈ R ⇒ y ∈ R).

(2.2)

A nonempty subset S of a BCH–algebra X is called a subalgebra of X if x ∗ y ∈ S for all x, y ∈ S. A nonempty subset A of a BCH–algebra X is called a closed ideal of X (see [2]) if it satisfies: ¡ ¢ (1) (∀x ∈ X) x ∈ A ⇒ 0 ∗ x ∈ A , ¡ ¢ (2) (∀y ∈ X) (∀x ∈ A) y ∗ x ∈ A ⇒ y ∈ A . 3 871

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Note that every closed ideal is a subalgebra, but the converse is not true (see [2]). Since every closed ideal is a subalgebra, we know that any closed ideal contains the element 0. A filter of a BCH–algebra X is a nonempty subset F of X satisfying the following conditions: (1) (∀x, y ∈ X) (x ∈ F, y ∈ F ⇒ x ∗ (x ∗ y) ∈ F, y ∗ (y ∗ x) ∈ F ), (2) (∀x, y ∈ X) (x ∈ F, x ≤ y ⇒ y ∈ F ). A filter F of a BCH–algebra X is said to be closed if 0 ∗ x ∈ F for all x ∈ F. For any family {ai | i ∈ Λ} of real numbers, we define ( _ max{ai | i ∈ Λ} if Λ is finite, {ai | i ∈ Λ} := sup{ai | i ∈ Λ} otherwise. ( ^ min{ai | i ∈ Λ} if Λ is finite, {ai | i ∈ Λ} := inf{ai | i ∈ Λ} otherwise. Denote by F(X, [−1, 0]) the collection of functions from a set X to [−1, 0]. We say that an element of F(X, [−1, 0]) is a negative-valued function from X to [−1, 0] (briefly, N -function on X). By an N -structure we mean an ordered pair (X, f ) of X and an N -function f on X. In what follows, let X denote a BCH–algebra and f an N -function on X unless otherwise specified. For any N -structure (X, f ) and α ∈ [−1, 0], the set C(f ; α) := {x ∈ X | f (x) ≤ α} is called a closed (f, α)-cut of (X, f ). Using the similar method to the transfer principle in fuzzy theory (see [6, 10]), we can consider transfer principle in N -structures. Let A be a subset of X and satisfy the following property P expressed by a first-order formula: P:

t1 (x,··· ,y)∈A,··· ,tn (x,··· ,y)∈A , t(x,··· ,y)∈A

where t1 (x, · · · , y), · · · , tn (x, · · · , y) and t(x, · · · , y) are terms of X constructed by variables x, · · · , y. We note that the subset A satisfies the property P if, for all elements a, · · · , b ∈ X, t(a, · · · , b) ∈ A whenever t1 (a, · · · , b), · · · , cn (a, · · · , b) ∈ A. For the subset A we define an N -structure (X, fA ) which satisfies the following property _ P¯ : fA (t(x, · · · , y) ≤ {fA (t1 (x, · · · , y)), · · · fA (tn (x, · · · , y))}. 4 872

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Theorem 2.1 ([8]). (N -transfer principle) An N -structure (X, f ) satisfies the property P¯ if and only if for all α ∈ [−1, 0], C(f ; α) 6= ∅ ⇒ C(f ; α) satisfies the property P. Definition 2.2 ([8]). By an N -subalgebra of X we mean an N -structure (X, f ) in which f satisfies: _ ¡ ¢ (∀x, y ∈ X) f (x ∗ y) ≤ {f (x), f (y)} .

(2.3)

Definition 2.3 ([8]). By an N -closed ideal of X we mean an N -structure (X, f ) in which f satisfies: _ ¡ ¢ (∀x, y ∈ X) f (0 ∗ x) ≤ f (x) ≤ {f (x ∗ y), f (y)} .

3

(2.4)

Closed N -filters of BCH–algebras

In what follows let X denote a BCH–algebra unless otherwise specified. For any N structure (X, f ), consider the following conditions: (b1) f is order reversing. ¢ (b2) (∀x ∈ X) (f (0 ∗ x) ≤ f (x) . ¡W ¢ W (b3) (∀x, y ∈ X) {f (x ∗ (x ∗ y)), f (y ∗ (y ∗ x))} ≤ {f (x), f (y)} . Definition 3.1. By an N -filter of X we mean an N -structure (X, f ) in which f satisfies (b1) and (b3). If an N -filter (X, f ) of X satisfies the condition (b2), then we say (X, f ) is closed. Theorem 3.2. For any N -structure (X, f ), thew following are equivalent: (1) (X, f ) is a (closed) N -filter of X. (2) (∀α ∈ [−1, 0]) (C(f ; α) 6= ∅ ⇒ C(f ; α) is a (closed) filter of X). Proof. It follows from the N -transfer principle.

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Table 1: ∗-operation ∗

0

1

2

3

0 1 2 3

0 1 2 3

0 0 2 3

3 3 0 2

2 2 3 0

Example 3.3. Let X = {0, 1, 2, 3} be a set with the ∗-operation given by Table 1. Then (X; ∗, 0) is a BCH–algebra. Define an N -function f on X by X f

0

1

2

3

−0.8 −0.8 −0.5 −0.2

Then (X, f ) is an N -filter of X. Since f (0 ∗ 2) = f (3) = −0.2 > −0.5 = f (2), (X, f ) is not closed. Proposition 3.4. Let (X, f ) be an N -structure satisfying conditions (b1) and (b2). Then (1) (∀x, y ∈ X) (f (y ∗ x) ≤ f (x ∗ y)). (2) (∀x, y ∈ X) (f (x ∗ (x ∗ y)) ≤ f (y)). (3) f is order preserving. Proof. (1) Using (a3), (H3) and (H1), we have (0 ∗ (x ∗ y)) ∗ (y ∗ x) = 0 for all x, y ∈ X. It follows from (b1) and (b2) that f (y ∗ x) ≤ f (0 ∗ (x ∗ y)) ≤ f (x ∗ y) for all x, y ∈ X. This proves (1). (2) For any x, y ∈ X, we get f (x ∗ (x ∗ y)) ≤ f ((x ∗ y) ∗ x) = f ((x ∗ x) ∗ y) = f (0 ∗ y) ≤ f (y). (3) Let x, y ∈ X be such that x ∗ y = 0. Then 0 ∗ x = (x ∗ y) ∗ x = (x ∗ x) ∗ y = 0 ∗ y, and so f (x) = f (x ∗ 0) ≤ f (0 ∗ x) = f (0 ∗ y) ≤ f (y ∗ 0) = f (y). Hence f is order preserving. 6 874

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We provide conditions for an N -subalgebra to be an N -closed ideal. Theorem 3.5. Let (X, f ) be an N -subalgebra of X. If (X, f ) satisfies two conditions (b1) and (b2), then (X, f ) is an N -closed ideal of X. W Proof. It is sufficient to show that f (y) ≤ {f (y ∗ x), f (x)} for all x, y ∈ X. Let x, y ∈ X. Then f (y) = f (y ∗ 0) ≤ f (0 ∗ y) = f ((x ∗ x) ∗ y) = f ((x ∗ y) ∗ x) _ ≤ {f (x ∗ y), f (x)} _ ≤ {f (y ∗ x), f (x)} . This completes the proof. We provide conditions for an N -closed ideal to be a closed N -filter. Theorem 3.6. Let (X, f ) be an N -closed ideal of X. If (X, f ) satisfies the condition (1) in Proposition 3.4, then (X, f ) is a closed N -filter of X. Proof. Let x, y ∈ X be such that x ∗ y = 0. Then 0 ∗ x = 0 ∗ y, and so f (y) = f (y ∗ 0) ≤ f (0 ∗ y) = f (0 ∗ x) ≤ f (x ∗ 0) = f (x), i.e., f is order reversing. Note that every N -closed ideal is an N -subalgebra (see [8, Theorem 3.5]). Hence _ f (x ∗ (x ∗ y)) ≤ {f (x), f (x ∗ y)} o _ _ _n f (x), {f (x ∗ y)} = {f (x), f (y)} . ≤ W Similarly, we have f (y ∗ (y ∗ x)) ≤ {f (x), f (y)} . Therefore _ _ {f (x ∗ (x ∗ y)), f (y ∗ (y ∗ x))} ≤ {f (x), f (y)} for all x, y ∈ X. Consequently, (X, f ) is a closed N -filter of X. Corollary 3.7. Let (X, f ) be an N -structure of X that satisfies (b1), (b2) and the condition (1) in Proposition 3.4. Then (X, f ) is a closed N -filter of X. Definition 3.8. An N -structure (X, f ) is said to satisfy the N -regularity if it satisfies: ³ ´ _ (∀x, y ∈ X) f (y) ≤ {f (x ∗ y), f (x)} . (3.1) 7 875

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Table 2: ∗-operation ∗

0

1

2

3

0 1 2 3

0 1 2 3

3 0 3 0

0 3 0 3

3 2 1 0

Table 3: ∗-operation ∗

0

1

2

3

4

0 1 2 3 4

0 1 2 3 4

0 0 2 3 4

4 4 0 2 3

3 3 4 0 2

2 2 3 4 0

An N -subalgebra (X, f ) satisfying the N -regularity is called a regular N -subalgebra of X. Example 3.9. Let X = {0, 1, 2, 3} be a set with the ∗-operation given by Table 2. Then (X; ∗, 0) is a BCH–algebra. Define an N -function f on X by X f

0

1

3

2

−0.8 −0.2 −0.8 −0.2

Then (X, f ) is a regular N -subalgebra of X. Example 3.10. Let X = {0, 1, 2, 3, 4} be a set with the ∗-operation given by Table 3. Then (X; ∗, 0) is a BCH–algebra. Define an N -function f on X by 1

2

3

4

X

0

f

α1 α2 α3 α2 α3 8 876

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where α1 < α2 < α3 in [−1, 0]. Then (X, f ) is an N -subalgebra of X. Since f (1) = α2 > W α1 = {f (0 ∗ 1), f (0)} , we know that (X, f ) does not satisfy the N -regularity. Lemma 3.11. If an N -structure (X, f ) of X satisfies the N -regularity, then f assigns 0 the least value of the image set of f. Proof. Taking y = 0 in (3.1) and using (a1) induce the desired result. Proposition 3.12. If an N -structure (X, f ) of X satisfies the N -regularity and the following inequality: (∀x, y ∈ X) (f (x ∗ y) ≤ f (y)) ,

(3.2)

then f is a constant mapping. Proof. Using (a1) and (3.2), we have f (x) = f (x ∗ 0) ≤ f (0) for all x ∈ X. It follows from Lemma 3.11 that f (x) = f (0) for all x ∈ X. Proposition 3.13. Every N -structure (X, f ) of X with the N -regularity satisfies: (∀x, y ∈ X) (x ≤ y ⇒ f (y) ≤ f (x)) .

(3.3)

Proof. Let x, y ∈ X be such that x ≤ y. Then x ∗ y = 0, and so _ _ f (y) ≤ {f (x ∗ y), f (x)} = {f (0), f (x)} = f (x) by (3.1) and Lemma 3.11. Proposition 3.14. Let (X, f ) be an N -structure of X satisfying the N -regularity. If X satisfies the following assertion: (∀x, y, z ∈ X) (z ≤ x ∗ y) , then f (y) ≤

W

(3.4)

{f (x), f (z)} for all x, y, z ∈ X.

Proof. Assume that (3.4) is valid. Then _ _ f (x ∗ y) ≤ {f (z ∗ (x ∗ y)), f (z)} = {f (0), f (z)} = f (z) for all x, y, z ∈ X. It follows that _ _ f (y) ≤ {f (x ∗ y), f (x)} ≤ {f (x), f (z)} for all x, y, z ∈ X. 9 877

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Theorem 3.15. For any N -structure (X, f ), the following are equivalent: (1) (X, f ) satisfies the N -regularity. (2) (∀α ∈ [−1, 0]) (C(f ; α) 6= ∅ ⇒ C(f ; α) is a regular subset of X) . Proof. Assume that (X, f ) satisfies the N -regularity. Let α ∈ [−1, 0] be such that C(f ; α) 6= ∅. Let x, y ∈ X be such that x ∈ C(f ; α) and x ∗ y ∈ C(f ; α). Then f (x) ≤ α and f (x ∗ y) ≤ α, which imply from (3.1) that f (y) ≤

_

{f (x ∗ y), f (x)} ≤ α.

Hence y ∈ C(f ; α), and therefore C(f ; α) is a regular subset of X. Conversely suppose that (2) is valid. Assume that there exist x, y ∈ X such that f (y) >

_

{f (x ∗ y), f (x)} = β.

Then x ∗ y ∈ C(f ; β) and x ∈ C(f ; β), but y ∈ / C(f ; β). This is a contradiction, and so f (y) ≤

_

{f (x ∗ y), f (x)}

for all x, y ∈ X. Therefore (X, f ) satisfies the N -regularity. Corollary 3.16. If an N -structure (X, f ) satisfies the N -regularity, then the set Xw := {x ∈ X | f (x) ≤ f (w)} is a regular subset of X for all w ∈ X. Proposition 3.17. If an N -structure (X, f ) satisfies the N -regularity, then the following implication is valid: ³ ´ _ (∀x, y, z ∈ X) f (x) ≥ {f (y ∗ z), f (y)} ⇒ f (z) ≤ f (x) . (3.5) W Proof. Let x, y, z ∈ X be such that f (x) ≥ {f (y ∗ z), f (y)} . Then y ∗ z ∈ Xx and y ∈ Xx . Since Xx is a regular subset of X by Corollary 3.16, it follows that z ∈ Xx , that is, f (z) ≤ f (x). Theorem 3.18. If an N -structure (X, f ) satisfies the condition (3.5), then the set Xw is a regular subset of X for all w ∈ X.

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Proof. Let x, y ∈ X be such that x ∈ Xw and x ∗ y ∈ Xw for all w ∈ X. Then f (x) ≤ f (w) and f (x ∗ y) ≤ f (w), which imply that f (w) ≥

_

{f (x ∗ y), f (x)} .

It follows from (3.5) that f (y) ≤ f (w). Hence y ∈ Xw , and so Xw is a regular subset of X for all w ∈ X. Corollary 3.19. If an N -structure (X, f ) satisfies the N -regularity, then the set Xw is a regular subset of X for all w ∈ X. Theorem 3.20. If an N -structure (X, f ) of X satisfies the N -regularity and the condition ³ ´ _ (∀x, y ∈ X) f (x) ≤ {f (x ∗ y), f (y)} , (3.6) then (X, f ) is an N -closed ideal of X. Proof. For any x ∈ X, we have _ f (0 ∗ x) ≤ {f (0 ∗ (0 ∗ x)), f (0)} = f (0 ∗ (0 ∗ x)) _ ≤ {f ((0 ∗ (0 ∗ x)) ∗ x), f (x)} _ = {f ((0 ∗ x) ∗ (0 ∗ x)), f (x)} _ = {f (0), f (x)} = f (x) by using (3.1), Lemma 3.11, (3.6), (a3) and (H1). Therefore (X, f ) is an N -closed ideal of X. We provide a condition for an N -closed ideal to satisfy the N -regularity. Proposition 3.21. If X is medial, then every N -closed ideal of X satisfies the N regularity. Proof. Let (X, f ) be an N -closed ideal of a medial BCH-algebra X. Then f (0 ∗ (x ∗ y)) ≤ f (x ∗ y) for all x, y ∈ X. Note from [3, Lemma 1] that a medial BCH-algebra X satisfies the equality x ∗ y = 0 ∗ (y ∗ x), It follows from (2.4) that f (x) ≤

_

{f (x ∗ y), f (y)} =

_

{f (0 ∗ (y ∗ x)), f (y)} ≤

_

{f (y ∗ x), f (y)} .

Therefore (X, f ) satisfies the N -regularity. 11 879

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4

N -soft BCH-algebras

Definition 4.1 ([9]). Let X be an initial universe set and E a set of attributes. By an N -soft set over X we mean a pair (f˜, A) where A ⊂ E and f˜ is a mapping from A to F(X, [−1, 0]), i.e., for each a ∈ A, f˜(a) := f˜a is an N -function on X. Denote by N (X, E) the collection of all N -soft sets over X with attributes from E and we call it an N -soft class. Definition 4.2 ([9]). Let (f˜, A) and (˜ g , B) be N -soft sets in N (X, E). Then (f˜, A) is e (˜ called an N -soft subset of (˜ g , B), denoted by (f˜, A) ⊆ g , B), if it satisfies: (i) A ⊆ B, ³ ´ (ii) (∀e ∈ A) f˜e ⊆ g˜e , i.e., f˜e (x) ≤ g˜e (x) for all x ∈ X . Definition 4.3. Let (f˜, A) be an N -soft set over a BCH-algebra X where A is a subset of E. If there exists an attribute u ∈ A for which the N -structure (X, f˜u ) is an N -subalgebra of X, then we say that (f˜, A) is an N -soft BCH-algebra related to the attribute u (briefly, Nu -soft BCH-algebra). If (f˜, A) is an Nu -soft BCH-algebra for all u ∈ A, we say that (f˜, A) is an N -soft BCH-algebra. Example 4.4. Let X := {apple, banana, carrot, peach, radish} be a universe, and consider a soft machine $ which makes X into a BCH-algebra as follows: x $ x = apple for all x ∈ X, x $ apple = x for all x ∈ X, x $ radish = radish for all x (6= radish) ∈ X, apple $ y = apple if y ∈ {banana, carrot, peach}, ( apple if y = carrot, banana $ y = banana if y = peach, ( carrot if y = banana, carrot $ y = apple if y = peach, peach $ y = peach if y ∈ {banana, carrot}, radish $ y = radish if y ∈ {banana, carrot, peach}. 12 880

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Table 4: Tabular representation of (f˜, A) (f˜, A)

apple

banana

carrot

peach radish

cat cow horse

−0.8 −0.7 −0.5

−0.3 −0.6 −0.6

−0.6 −0.5 −0.2

−0.3 −0.4 −0.1

−0.8 −0.3 −0.3

Consider a set of attributes A := {cat, cow, horse}, and let (f˜, A) be an N -soft set over X with the tabular representation which is given by Table 4. Then (f˜, A) is an N -soft BCH-algebra over X related to attributes “cat” and “cow”. But it is not an N -soft BCH-algebra over X related to the attribute “horse” since o _n f˜hourse (banana), f˜hourse (banana) . f˜hourse (apple) = −0.5 > −0.6 = Proposition 4.5. Every N -soft BCH-algebra (f˜, A) over a BCH-algebra X satisfies the following inequality: ³ ´ (∀x ∈ X)(∀u ∈ A) f˜u (0) ≤ f˜u (x) . (4.1) Proof. For any x ∈ X and u ∈ A, we have o _n f˜u (0) = f˜u (x ∗ x) ≤ f˜u (x), f˜u (x) = f˜u (x). This completes the proof. The problem we now discuss is: If (˜ g , B) is an N -soft BCH-algebra over a BCH-algebra X, then is every N -soft subset of (˜ g , B) an N -soft BCH-algebra over X? Unfortunately this is not true as seen in the following example. Example 4.6. Suppose there are four colors in the universe X, that is, X := {white, blackish, reddish, green} and E := {beautiful, fine, moderate, delicate, elegant, smart, chaste} be a set of attributes. Let ♥ be a soft machine to mix two colors according to order in such a way that we have the following results. 13 881

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Table 5: Tabular representation of (˜ g , B) (˜ g , B)

white blackish reddish

green

beautiful fine moderate smart

−0.9 −0.8 −0.7 −0.6

−0.4 −0.5 −0.5 −0.6

−0.4 −0.5 −0.3 −0.4

−0.7 −0.8 −0.3 −0.4

x ♥ white = x for all x ∈ X, y ♥ y = white for all y ∈ X, ( white if z = reddish, white ♥ z = green if z ∈ {blackish, green}, ( green if w = reddish, blackish ♥ w = redish if w = green, ( green if u = blackish, reddish ♥ u = blackish if u = green, ( green if v = reddish, green ♥ v = white if v = blackish. Then (X, ♥, white) is a BCH-algebra. Take B = {beautiful, fine, moderate, smart} and let (˜ g , B) be an N -soft set over X with the tabular representation which is given by Table 5. Then (˜ g , B) is an N -soft BCH-algebra over X. Now let (f˜, A) be an N -soft subset of (˜ g , B), where A = {beautiful, fine, smart} ⊂ B and the tabular representation of (f˜, A) is given by Table 6. Then f˜fine (redish ♥ blackish) = f˜fine (green) = −0.55 o _n > −0.65 = f˜fine (redish), f˜fine (blackish) , and so (f˜, A) is not an N -soft BCH-algebra over X related to the attribute “fine”. Hence (f˜, A) is not an N -soft BCH-algebra over X. 14 882

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Table 6: Tabular representation of (f˜, A) (f˜, A)

white

blackish reddish

green

beautiful fine smart

−0.99 −0.88 −0.66

−0.44 −0.65 −0.44

−0.44 −0.55 −0.66

−0.77 −0.88 −0.44

But, we have the following theorem. Theorem 4.7. For any subset A of E, let (f˜, A) be an N -soft BCH-algebra over a BCH-algebra X. If B is a subset of A, then (f˜|B , B) is an N -soft BCH-algebra over X. Proof. Straightforward. Definition 4.8 ([9]). For any (f˜, A), (˜ g , B) ∈ N (X, E), the union of (f˜, A) and (˜ g , B) is ˜ C) in (X, E) satisfying the following conditions: defined to be the N -soft set (h, (i) C = A ∪ B, (ii) for all x ∈ C,

 ˜  if x ∈ A \ B,  fx ˜hx = g˜x if x ∈ B \ A,   ˜ fx ∪ g˜x if x ∈ A ∩ B.

˜ C). e (˜ In this case, we write (f˜, A) ∪ g , B) = (h, Lemma 4.9. If (X, f˜) and (X, g˜) are N -subalgebras of a BCH-algebra X, then the union (X, f˜ ∪ g˜) of (X, f˜) and (X, g˜) is an N -subalgebra of X. Proof. Straightforward. Theorem 4.10. If (f˜, A) and (˜ g , B) are N -soft BCH-algebras over a BCH-algebra X, then the union of (f˜, A) and (˜ g , B) is an N -soft BCH-algebra over X. ˜ C) be the union of (f˜, A) and (˜ e (˜ Proof. Let (f˜, A) ∪ g , B) = (h, g , B). Then C = A∪B. For ˜ ˜ ˜ x ) = (X, g˜x )) any x ∈ C, if x ∈ A \ B (resp. x ∈ B \ A) then (X, hx ) = (X, fx ) (resp. (X, h ˜ x ) = (X, f˜x ∪ g˜x ) is an N -subalgebra is an N -subalgebra of X. If A ∩ B 6= ∅, then (X, h ˜ C) is an N -soft BCH-algebra over of X for all x ∈ A ∩ B by Lemma 4.9. Therefore (h, a BCH-algebra X. 15 883

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Table 7: Tabular representation of (˜ g , B) (f˜, A) moderate elegant

white blackish reddish −0.6 −0.1 −0.6 −0.4 −0.2 −0.2

green −0.1 −0.3

Definition 4.11 ([9]). Let (f˜, A) and (˜ g , B) be two N -soft sets in (X, E). The intersec˜ C) in (X, E) where C = A ∪ B and for every tion of (f˜, A) and (˜ g , B) is the N -soft set (h, x ∈ C,  ˜  if x ∈ A \ B,  fx ˜hx = g˜x if x ∈ B \ A,   ˜ fx ∩ g˜x if x ∈ A ∩ B. ˜ C). e (˜ In this case, we write (f˜, A) ∩ g , B) = (h, Theorem 4.12. Let (f˜, A) and (˜ g , B) be N -soft BCH-algebras over a BCH-algebra X. If A and B are disjoint, then the intersection of (f˜, A) and (˜ g , B) is an N -soft BCH-algebra over X. ˜ C) be the intersection of (f˜, A) and (˜ e (˜ Proof. Let (f˜, A) ∩ g , B) = (h, g , B). Then C = A ∪ B. Since A ∩ B = ∅, if x ∈ C then either x ∈ A \ B or x ∈ B \ A. If x ∈ A \ B, then ˜ x ) = (X, f˜x ) is an N -subalgebra of X. If x ∈ B \ A, then (X, h ˜ x ) = (X, g˜x ) is an (X, h ˜ C) is an N -soft BCH-algebra over a BCH-algebra X. N -subalgebra of X. Hence (h, The following example shows that Theorem 4.12 is not valid if A and B are not disjoint. Example 4.13. Let X and (˜ g , B) be the BCH-algebra and the N -soft BCH-algebra, respectively, in Example 4.6. Take A = {moderate, elegant} and let (f˜, A) be an N -soft set over X with the tabular representation which is given by Table 7. Then (f˜, A) is an N -soft BCH-algebra over X. Note that A and B are not disjoint. The intersection ˜ C) of (f˜, A) and (˜ e (˜ (f˜, A) ∩ g , B) = (h, g , B) is not an N -soft BCH-algebra over X since ³ ´ f˜moderate ∩ f˜moderate (blackish ♥ reddish) = f˜moderate (green) = −05 > −0.6 ´ ³ ´ o _ n³ = f˜moderate ∩ f˜moderate (blackish), f˜moderate ∩ f˜moderate (reddish) .

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References [1] B. Ahmad, On classification of BCH–algebras, Math. Japon. 35 (1990), no. 5, 801– 804. [2] M. A. Chaudhry, On BCH–algebras, Math. Japon. 36 (1991), no. 4, 665–676. [3] W. A. Dudek and J. Thomys, On decompositions of BCH–algebras, Math. Japon. 35 (1990), no. 6, 1131–1138. [4] Q. P. Hu and X. Li, On BCH–algebras, Math. Sem. Notes Kobe Univ. 11 (1983), no. 2, part 2, 313–320. [5] Q. P. Hu and X. Li, On proper BCH–algebras, Math. Japon. 30 (1985), no. 4, 659–661. [6] Y. B. Jun and M. Kondo, On transfer principle of fuzzy BCK/BCI-algebras, Sci. Math. Jpn. 59 (2004), no. 1, 35–40. [7] Y. B. Jun, K. J. Lee and S. Z. Song, N -ideals of BCK/BCI-algebras, J. Chungcheong Math. Soc. 22 (2009), 417–437. ¨ urk and E. H. Roh, N -structures applied to closed ideals in [8] Y. B. Jun, M. A. Ozt¨ BCH–algebras, Int. J. Math. Math. Sci. 2010 Article ID 943565, 9 pages. [9] Y. B. Jun, S. Z. Song and K. J. Lee, The combination of soft sets and N -structures with applications, Commun. Korean Math. Soc. (submitted). [10] M. Kondo and W. A. Dudek, On the transfer principle in fuzzy theory, Mathware Soft Comput. 12 (2005), no. 1, 41–55. [11] P. K. Maji, A. R. Roy and R. Biswas, An application of soft sets in a decision making problem, Comput. Math. Appl. 44 (2002) 1077–1083. [12] D. Molodtsov, Soft set theory - First results, Comput. Math. Appl. 37 (1999) 19–31. [13] L. A. Zadeh, From circuit theory to system theory, Proc. Inst. Radio Eng. 50 (1962) 856–865. [14] L. A. Zadeh, Fuzzy sets, Inform. Control 8 (1965) 338–353.

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[15] L. A. Zadeh, Toward a generalized theory of uncertainty (GTU) - an outline, Inform. Sci. 172 (2005) 1–40.

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Construction of Orthogonal Shearlet Tight Frames with Symmetry ∗

Yan Feng1,2 , Dehui Yuan 3 , Shouzhi Yang2,† Abstract Shearlet frames play an important role in describing the singularities of multidimensional data. In this paper, we present a simple but complete method for constructing symmetric orthogonal shearlet tight frames from any given shearlet tight frames. This includes a family of cone-adapted ones. Finally, two examples are given. Keywords: Shearlets, Tight frame, Orthogonality, Symmetry, 2000 MR Subject Classification: 42C15, 94A12

1

introduction

It is well known that the traditional theory of wavelet is based on the use of isotropic dilations. It can capture a point singularity of a function or distribution f ∈ R, effectively. However, it is unable to describe the geometric regularity along the singularities of surfaces and lacks directional sensitivity(see [1]). These limitations have led to several new schemes, such as curvelets, contourlets and shearlets. Comparing to this methods, the shearlets stands out since it is based on a simple and rigorous mathematical framework which not only provides a more flexible theoretical tool for the geometric representation of multidimensional data, but also is more natural for implementation. Moveover, it can provide optimally sparse representations(see [2]). As a consequence, it can be associated to a multiresolution analysis and then this leads to a unified treatment of both the continuous and discrete world, i.e., allowing a digital theory to be a natural digitization of the continuum theory. It therefore has become popular in many applications, such as image denoising, enhancement, edge analysis and detection and separation(see [1], [3-7]). In general, the construction of shearlet systems can be come in two classes today: One class is constructed through bandlimited functions on the space of R2 , this is generated by a unitary representation of the shearlet group and equipped with a particulary ‘nice’ mathematical structure. For more details, we refer the reader to [1] and references therein. The other is constructed based on the cone, referred as cone-adapted shearlet. It is restricted to a horizontal and vertical cone in frequency domain, thereby ensuring an equal treatment of all direction. The interest reader is referred to [19] and references therein for more details. Both have their particular advantages and disadvantages. Orthogonal frames, introduced by Weber in [8], is useful in multiple access communication systems, and has received much attention recently(see [9-12]). Just as the traditional theory of wavelets and frames, shearlet frames, as general frames, with symmetry is very desirable in various applications, since it can preserver linear phase properties and also allow symmetric boundary conditions ∗ This work was supported by the National Natural Science Foundation of China (Grant No.11071152), the Natural Science Foundation of Guangdong Province (Grant No. 10151503101000025) 1 School of Computer and information technology, Xinyang Normal University, Xinyang, 464000, China 2 Department of Mathematics, Shantou University, Shantou, Guangdong, 521041, China. 3 Department of Mathematics, Hanshan Normal University, Chaozhou, Guangdong, 521041, China. † Corresponding Author

1

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in shearlet algorithms. Wu et al.(see [13]) constructed (anti)symmetric composite dilation multiwavelet frames with any symmetric points. However, the shearlet is the special case of composite dilation wavelet. In [14], in order to provide the parametrization of orthogonal and symmetric multiwavelets, Li and Yang presented a algorithms for constructing paraunitary symmetric matrices. Inspired by [13] and [14], we give a simple but complete method for constructing symmetric shearlet tight frames for L2 (R2 ) and for the cone from any given shearlet tight frames, respectively.

2

Notations and lemmas

In this section, let us introduce some notations and lemmas. The Fourier transform of a function R f ∈ L1 (R2 ) is defined to be fˆ(ξ) := R2 f (x)e−iξ·x dx and can be extended to L2 (R2 ) functions and tempered distributions, naturally, where · denotes the standard inner product in R2 . An r × r integer matrix A is called a dilation matrix if limn→∞ A−n = 0, i.e., all eigenvalues of A are greater than one in modulus. For a matrix A(z), we denote A∗ (z) its transpose conjugate. We say that a matrix A(z) is paraunitary symmetric if its entries are all (anti)symmetric Laurent polynomials and A(z)A∗ (z) = Id . Let Aa , B ∈ GL2 (R), where a > 0, and GL2 (R) denotes the set of all 2 × 2 invertible matrices with real entries, which are defined by ¶ µ µ ¶ 1 1 a √0 and B = A= , (2.1) a 0 0 1 where A and B denote a parabolic scaling matrix and a shear matrix, respectively. A shearlet system for L2 (R2 ) is given by k m {DA DB Tn ψ` : k, m ∈ Z, n ∈ Z2 , ` = 1, ..., L}, where ψ ∈ L2 (R2 ), DA is the dilation operator, defined by DA f (x) = |detA|−1/2 f (A−1 x), Tn is the translation operator, defined by Tn f (x) = f (x − n). In this paper, we are interested in the special case where a = 4. It is interesting to observe that this choice gives a special case of the affine systems with composite dilations introduced in [15]. In this case, the shearlet system associated with shearlets ψ` ∈ L2 (R2 ) is given by the following expression k m {DA DB Tn ψ` (x) := 2−3k/2 ψ` (B−m A−k x − n), k, m ∈ Z, n ∈ Z2 , ` = 1, ..., L},

(2.2)

where k denotes the scale, and m, n denote the direction and position of singularities, respectively. Recall that a countable collection {ψ` }`∈Γ ∈ L2 (R2 ) is a tight frame for L2 (R2 ) if X |hf, ψ` i|2 = kf k2 , for all f ∈ L2 (R2 ). `∈Γ

P This is equivalent to the reproducing formula f = `∈Γ hf, ψ` iψ` , for all f ∈ L2 (R2 ), Γ is a countable index set. With respect to characterizations of shearlet tight frames, there is the following Lemma, which is adapted from Theorem 5.5 in [15]. 2 Lemma 2.1. Let A, B ⊂ GL2 (Z) be given by (2.1) and Ψ = {ψ` }L `=1 ⊂ L2 (R ). Suppose that L X X Z X `=1 k∈Z n∈Z2

supp fˆ

|fˆ(ξ + nBAk )|2 |ψˆ` (ξA−k B −1 )|2 dξ < ∞,

for all f ∈ D, where D is a dense subspace of L2 (R2 ) contained in the set {f ∈ L2 (R2 ) : fˆ ∈ L∞ (R2 ) and supp fˆ is compact}. k m Then shearlet system {DA DB Tn ψ` : k, m ∈ Z, n ∈ Z2 , ` = 1, ..., L} is a shearlet tight frame for 2 L2 (R ) if and only if

2

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 L P P    |ψˆ` (ξAk B)|2 = 1,    `=1 k∈Z  P L P ψˆ` (ξAk B)ψˆ` ((ξ + γ)Ak B) = 0, f or γ ∈ Z2 \(Z2 A),  `=1 k≥0    L P  P T   ψˆ` (ξAk B)ψˆ` ((ξ + γ)Ak B) = 0, f or γ ∈ k∈Z (Z2 Ak )\{0}. 

(2.3)

`=1 k∈Z

In [13], Wu et al. constructed the symmetric composite dilation multiwavelet frames from given 2 L 2 dilation multiwavelet frames. For some functions {ψ` }L `=1 ∈ L2 (R ) and points {x` }`=1 ∈ R , defining new functions through ( ` (x−x` ) , ψ`1 (x) = ψ` (x+x` )+ψ 2 (2.4) ψ` (x+x` )−ψ` (x−x` ) 2 ψ` (x) = . 2 It is obvious that functions defined by (2.4) are (anti)symmetric with respect to points {x` }L `=1 . As a special case of Theorem3.1 in [13], we have the following Lemma Lemma 2.2. Suppose that shearlet system k m {DA DB Tn ψ` : k, m ∈ Z, n ∈ Z2 , ` = 1, 2, ..., L}

defined by (2.2) is a shearlet tight frame for L2 (R2 ). Then shearlet system [ k m k m {DA DB Tn ψ`1 DA DB Tn ψ`2 : k, m ∈ Z, n ∈ Z2 , ` = 1, 2, ..., L} is a (anti)symmetric shearlet tight frame for L2 (R2 ) with symmetric points {x` }L `=1 , where functions ψ`1 (x) and ψ`2 (x) are defined by (2.4). Orthogonal frames plays an important role in multiple access communication systems and characterizations of superframes, the interested reader can find more details in [8], [9], [11]. Definition 2.3. Two Bessel sequence {ψ` }`∈Γ and {ψe` }`∈Γ in L2 (R2 ) are said to be orthogonal, if, for any function f ∈ L2 (R2 ), X hf, ψ` iψe` = 0. `∈Γ

Suppose that two sequence {ψ` }`∈Γ and {ψe` }`∈Γ are both tight frames and are orthogonal to each other. Then for any functions f, g ∈ L2 (R2 ), we have X X f= (hf, ψ` i + hg, ψ˜` i)ψ` and g = (hf, ψ` i + hg, ψ˜` i)ψ˜` . `∈Γ

`∈Γ

That is, the frames can be used to encode two signals f and g, which can be sent over a single com˜` }`∈Γ is a tight frame of the space L2 (R2 ) L L2 (R2 ), munications channel. Moreover, sequence {ψ , ψ ` L also it is referred as superframes, where denotes the direct sum of L2 (R2 ) 2 times. For more details see [8-11]. The following Lemma, which be trivial deduced from Theorem 1.5 in [8], describes the orthogonality of shearlet tight frames. k m k m f` : k, m ∈ Z, n ∈ Lemma 2.4. Let {DA DB Tn ψ` : k, m ∈ Z, n ∈ Z2 , ` = 1, 2, ...L} and {DA DB Tn ψ 2 2 Z , ` = 1, 2, ...L} be two shearlet tight frames for L2 (R ). Then they are a pair of orthogonal frames for L2 (R2 ) if and only if, for a.e. ξ ∈ R2 , the following equations hold  L P P  c  f` (ξAk B) = 0,  ψb` (ξAk B)ψ    `=1 k∈Z  P L P c f` ((ξ + γ)Ak B) = 0, f or γ ∈ Z2 \ (Z2 A), ψb` (ξAk B)ψ (2.5)  `=1 k≥0    L P  P c  f` ((ξ + γ)Ak B) = 0, f or γ ∈ T (Z2 Ak ) \ {0}.  ψb` (ξAk B)ψ  `=1 k∈Z

k∈Z

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3

Construction of symmetric orthogonal shearlet tight frames

In this section, we present a method for the construction of symmetric orthogonal shearlet tight frames of L2 (R2 ). This also includes a family of cone-adapted shearlet system. Theorem 3.1. Suppose that shearlet system k m {DA DB Tn ψ` : k, m ∈ Z, n ∈ Z2 , ` = 1, 2, ..., L}

is a shearlet tight frame for L2 (R2 ). Let M (z) be a Q × P matrix with π4 -periodic entries Mi,j (e−iξ ) and satisfy M (e−iξ ) × M ∗ (e−iξ ) = IQ , where ξ ∈ R2 . Construct LQP functions ψ`;i,j (x) through −iξ c [ ψ )ψ` (ξ), ` = 1, ..., L; i = 1, ..., Q; j = 1, ..., P. `;i,j (ξ) = Mi,j (e k m Then, for any integer i ∈ {1, ..., Q}, {DA DB Tn ψ`;i,j : k, m ∈ Z, j ∈ Z2 , ` = 1, ..., L} is also 2 a shearlet tight frames for L2 (R ). Moreover, for any two different integers i1 , i2 ∈ {1, ..., Q}, k m k m {DA DB Tn ψ`;i1 ,j : k, m ∈ Z, j ∈ Z2 , ` = 1, ..., L; j = 1, ..., P } and {DA DB Tn ψ`;i2 ,j : k, m ∈ Z, j ∈ 2 Z , ` = 1, ..., L; j = 1, ..., P } are orthogonal to each other.

Proof. P X L ¯ ¯2 XX ¯[ ¯ ¯ψ`;i,j (ξAk B)¯

=

k∈Z j=1 `=1

P X L ¯ ¯2 ¯ ¯2 XX k ¯ ¯ ¯c ¯ k ¯Mi,j (e−iξA B )¯ ¯ψ ` (ξA B)¯ k∈Z j=1 `=1

=

L X P ·¯ ¯2 ¯2 ¸¯ XX ¯ ¯ −iξAk B ¯ ¯c )¯ ¯ψ` (ξAk B)¯ ¯Mi,j (e k∈Z `=1 j=1

=

L ¯ ¯2 XX ¯c ¯ ¯ψ` (ξAk B)¯ = 1, k∈Z `=1

where we use the property of M (e−iξ )M ∗ (e−iξ ) = IQ in the last equality. For a.e. ξ ∈ R2 and γ ∈ Z2 \ (Z2 A), we obtain L XX P X

ψb`;i,j (ξAk B)ψb`;i,j ((ξ + γ)Ak B)

`=1 k≥0 j=1

=

¸ P · L XX ³ ´ X k Mi,j (e−iξAk B ) × Mi,j e−i(ξ+γ)A B ψˆ` (ξAk B)ψˆ` (ξ + γ)Ak B `=1 k≥0 j=1

=

L X X

ψˆ` (ξAk B)ψˆ` ((ξ + q)ak b) = 0,

`=1 k≥0

where in the above equality, we use the periods of the components of M (z) and the condition M (e−iξ )MT∗ (e−iξ ) = IQ . If γ ∈ k∈Z (Z2 Ak ) \ {0}, we also deduce that L XX P X

ψb`;i,j (ξAk B)ψb`;i,j ((ξ + γ)Ak B)

`=1 k∈Z j=1

¸ L XX P · X kB −i(ξ+γ)Ak B ˆ k −iξA k ˆ = Mi,j (e ) × Mi,j (e )ψ` (ξA B)ψ` (ξ + γ)A B `=1 k∈Z j=1

=

L X X

ψˆ` (ξAk B)ψˆ` ((ξ + γ)Ak B) = 0.

`=1 k∈Z

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k m Therefore, {DA DB Tn ψ`;i,j : ` = 1, ..., L; i = 1, ..., Q; j = 1, ..., P } is a shearlet tight frame from Lemma 2.1. In the following, we prove that they are orthogonal for any different integers i1 , i2 ∈ {1, ..., Q}. In fact, L XX P X

ψb`;i1 ,j (ξAk B)ψb`;i2 ,j (ξAk B)

`=1 k∈Z j=1

¸ L XX P · X kB ˆ −iξAk B ˆ k −iξA k Mi1 ,j (e )ψ`;i1 ,j (ξA B) × Mi2 ,j (e = )ψ`;i2 ,j (ξA B) `=1 k∈Z j=1

=

L X·X P µ X `=1 k∈Z

¶ k Mi1 ,j (e−iξAk B )Mi2 ,j (e−iξA B )

¸ k k ˆ ˆ × ψ`;i1 ,j (ξA B)ψ`;i2 ,j (ξA B) = 0.

j=1

where in the second equality we use the periods of Mi,j (e−iξ ) and the condition M (e−iξ )×M ∗ (e−iξ ) = IQ . If γ ∈ Z2 \ (Z2 A), we can deduce that L XX P X

ψb`;i1 ,j (ξAk B)ψb`;i2 ,j ((ξ + γ)Ak B)

`=1 k∈Z j=1

¸ P · L XX X kB ˆ −i(ξ+γ)Ak B ˆ k k −iξA )ψ`;i2 ,j ((ξ + γ)A B) = Mi1 ,j (e )ψ`;i1 ,j (ξA B) × Mi2 ,j (e `=1 k∈Z j=1

=

P µ L X·X X `=1 k∈Z

Mi1 ,j

(e−iξAk B )M

−i(ξ+γ)Ak B

i2 ,j (e

¶ ¸ k k ˆ ˆ ) × ψ`;i1 ,j (ξA B)ψ`;i2 ,j ((ξ + γ)A B) = 0.

j=1

Similarly, for γ ∈

T

2 k k∈Z (Z A )

\ {0}, we can have that

L XX P X

ψb`;i1 ,j (ξAk B)ψb`;i2 ,j ((ξ + γ)Ak B) = 0

`=1 k∈Z j=1

Thus we obtain the desired result. The following Corollary provides the construction of symmetric orthogonal shearlet tight frames from any symmetric shearlet tight frames. Corollary 3.2. Suppose that shearlet system k m {DA DB Tn ψ` : k, m ∈ Z, n ∈ Z2 , ` = 1, 2, ..., L}

is a symmetric shearlet tight frame for L2 (R2 ). Let M (e−iξ ) be a Q × P paraunitary symmetric matrix with π4 -periodic entries Mi,j (e−iξ ), where ξ ∈ R2 . Construct LQP functions ψ`;i,j through −iξ c [ ψ )ψ` (ξ), ` = 1, ..., L; i = 1, ..., Q; j = 1, ..., P. `;i,j (ξ) = Mi,j (e k m Then, for any integer i ∈ {1, ..., Q}, {DA DB Tn ψ`;i,j : k, m ∈ Z, n ∈ Z2 , ` = 1, 2, ..., L} is also 2 a symmetric shearlet tight frame for L2 (R ). Moreover, for any two different integers i1 , i2 ∈ k m k m {1, ..., Q}, {DA DB Tn ψ`;i1 ,j : k, m ∈ Z, j ∈ Z2 , ` = 1, ..., L; j = 1, ..., P } and {DA DB Tn ψ`;i2 ,j : 2 k, m ∈ Z, j ∈ Z , : ` = 1, ..., L; j = 1, ..., P } are orthogonal to each other. k m Note that if two orthogonal shearlet systems {DA DB Tn ψ` : k, m ∈ Z, n ∈ Z2 , ` = 1, 2, ...L} k m 2 and {DA DB Tn ψe` : k, m ∈ Z, n ∈ Z , ` = 1, 2, ...L} are shearlet tight frames for L2 (R2 ). Then we

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S k m e k m DB Tn ψ` DA DB Tn ψ` : k, m ∈ Z, n ∈ Z2 , ` = 1, 2, ...L} as super shearlet refer shearlet system {DA L 2 2 frames for L2 (R ) L2 (R ). Note that it is easy to see that this construction has a drawback that the directional parameter m runs over the non-compact set R. And then the distribution of directions becomes infinitely dense as m grows. In order to overcome this drawback, we can construct shearlet tight frames on the cone. First, we partition the frequency plane into the following four cones C1 − C4 :  {(ξ1 , ξ2 ) ∈ R2 : ξ1 ≥ 1, |ξ2 /ξ1 | ≤ 1} : κ = 1,    {(ξ , ξ ) ∈ R2 : ξ ≥ 1, |ξ /ξ | ≤ 1} : κ = 2, 1 2 1 1 2 Cκ = (3.1) 2  {(ξ1 , ξ2 ) ∈ R : ξ1 ≤ −1, |ξ2 /ξ1 | ≤ 1} : κ = 3,    {(ξ1 , ξ2 ) ∈ R2 : ξ1 ≤ −1, |ξ1 /ξ2 | ≤ 1} : κ = 4, and a centered rectangle R = {(ξ1 , ξ2 ) ∈ R2 : k(ξ1 , ξ2 )k∞ < 1}

(3.2)

Through adapting the construction of Corollary 3.2, we can obtain symmetric shearlet tight frames on the cone. Proposition 3.3. Let shearlet system k m {DA DB Tn ψ` : k, m ∈ Z, n ∈ Z2 , ` = 1, 2, ..., L}

be a shearlet tight frame for L2 (R2 ). Then the system k m {DA DB Tn ψ` : k > 0, −2k ≤ m ≤ 2k , n ∈ Z2 , ` = 1, ..., L} S is a shearlet tight frame for L2 (C1 C3 )∨ . Moreover, k m {DA DB Tn ψ`\ : k > 0, −2k ≤ m ≤ 2k , n ∈ Z2 , ` = 1, ..., L}

is also a shearlet tight frame for L2 (C2 by (3.1).

S

C4 )∨ , where ψ`\ (ξ1 , ξ2 ) = ψ` (ξ2 , ξ1 ), C1 − C4 are defined

Based on the shearlet tight frames on the cone, we give the following theorem for constructing symmetric orthogonal shearlet tight frames on the same cone, which can be proved easily. Theorem 3.4. Suppose that shearlet system k m {DA DB Tn ψ` : k, m ∈ Z, n ∈ Z2 , ` = 1, 2, ..., L} S is a symmetric shearlet tight frame for L2 (C1 C3 )∨ . Let M (e−iξ ) be a Q×P paraunitary symmetric S π −iξ matrix with 4 -periodic entries Mi,j (e ) and satisfy M (e−iξ ) × M ∗ (e−iξ ) = IQ , where ξ ∈ C1 C3 . Construct LQP functions ψ`;i,j (x) through −iξ c [ ψ )ψ` (ξ), ` = 1, ..., L; i = 1, ..., Q; j = 1, ..., P. `;i,j (ξ) = Mi,j (e k m 2 Then, for any integer i ∈ {1, ..., Q}, {DS A DB Tn ψ`;i,j : k, m ∈ Z, j ∈ Z , ` = 1, ..., L} is also a ∨ symmetric shearlet tight frame for L2 (C1 C3 ) . Moreover, for any two different integers i1 , i2 ∈ k m k m {1, ..., Q}, {DA DB Tn ψ`;i1 ,j : k, m ∈ Z, j ∈ Z2 , ` = 1, ..., L; j = 1, ..., P } and {DA DB Tn ψ`;i2 ,j : 2 k, m ∈ Z, j ∈ Z , : ` = 1, ..., L; j = 1, ..., P } are orthogonal to each other. S In the same way, we can construct symmetric shearlet tight frames for L2 (C2 C4 )∨ . Comparing to Corollary 3.2, the benefit of this construction is that the shear parameter k ranges over a finite set for each j. This is a obvious advantage for the numerical implementation.

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4

Examples

k m DB Tn ψ : k, m ∈ Z, n ∈ Z2 } is a shearlet tight frame It is well known that the shearlet system {DA for L2 (R2 )(see [1]), where function ψ ∈ L2 (R2 ) satisfies

ˆ ˆ 1 , ξ2 ) = ψˆ1 (ξ1 )ψˆ2 ( ξ2 ), ψ(ξ) = ψ(ξ ξ1

(4.1)

on condition given by which is called classical shearlet, ψ1 ∈ L2 (R) satisfies the discrete Calder´ P S 1 1 1 1 −j 2 ∞ ˆ ˆ ˆ j∈Z |ψ1 (2 ξ)| = 1 for a.e. ξ ∈ R, with ψ1 ∈ C (R) and supp ψ1 ⊆ [− 2 , − 16 ] [ 16 , 2 ], and P1 2 ˆ ψ2 ∈ L2 (R) is a ’bump’ function, i.e., for a.e. ξ ∈ [−1, 1], k=−1 |ψ2 (ξ + k)| = 1, satisfying ∞ ˆ ˆ ψ2 ∈ C (R) and supp ψ2 ⊆ [−1, 1]. There exist several choices of function ψ1 and ψ2 satisfying those conditions. One possible choice is to set ψ1 to be a Lemari` e-Meyer wavelet and ψ2 to be a spline, for more details see [16] and references therein. Note that this shearlet tight frame is welllocalized waveforms with frequency support increasing elongated at finer scale (k → −∞) and with the directions depending on m and n. Example 1. Let functions ψ 1 (x) and ψ 2 (x) be defined by (2.4) in the case of ` = 1, where ψ is classical shearlet given in (4.1). Assume that # " 4 −4 −4 4 M (z) = (Mi,j (z)) =

z +z 2 z 4 −z −4 2

z

−z 2 z 4 +z −4 2

,

γ where z = e−i(ξ1 /2+ξ2 ) , Construct functions {ψ`;i,j , γ, `, i, j = 1, 2} through  1 \ c1  ψ  1;1,j (ξ) = M1,j (z)ψ (ξ),   ψ 1 \ c2 2;1,j (ξ) = M1,j (z)ψ (ξ), 2 \ c1  ψ  1;2,j (ξ) = M2,j (z)ψ (ξ),   \ 2 c2 (ξ). ψ2;2,j (ξ) = M2,j (z)ψ γ k m Then, for every integer γ ∈ {1, 2}, {DA DB Tn ψ`;i,j : γ, `, i, j = 1, 2} is symmetric shearlet tight 2 k m 1 k m 2 frame for L2 (R ). Moreover, {DA DB Tn ψ`;i,j : `, i, j = 1, 2} and {DA DB Tn ψ`;i,j : `, i, j = 1, 2} are L γ m k orthogonal, namely, {DA DB Tn ψ`;i,j : γ, `, i, j = 1, 2} is a shearlet tight frame of L2 (R2 ) L2 (R2 ).

Example 2. Let ψ is also classical shearlet given by (4.1), then according to Theorem 3 in [17], the system k m {DA DB Tn ψ : k > 0, −2k ≤ m ≤ 2k , n ∈ Z2 } S S is a shearlet tight frame for L2 (C1 C3 )∨ . In the cone of (C1 C3 )∨ , constructing functions φ1 (x) and φ2 (x) by (2.4) with ` = 1, then they are (anti)symmetric. Let 2 × 2 matrix N (e−iξ ) be given by. " # −iξ

N (e where ξ ∈ C1

S

−iξ

) = (Ni,j (e

)) =

e−iξ +eiξ 2 e−iξ −eiξ 2

eiξ −e−iξ 2 e−iξ +eiξ 2

,

C3 , Construct functions {φγ`;i,j , γ, `, i, j = 1, 2} through  1 c1 (ξ),  (ξ) = N1,j (z)φ φ[   1;1,j  φ[ 1 c2 2;1,j (ξ) = N1,j (z)φ (ξ), 2 c1 φ[  1;2,j (ξ) = N2,j (z)φ (ξ),   [ 2 c2 (ξ). φ2;2,j (ξ) = N2,j (z)φ

k m Then, for every integer γ ∈ {1, 2}, system {DA DB Tn φγ`;i,j : k > 0, −2k ≤ m ≤ 2k , γ, `, i, j = 1, 2} is S e 1 , ξ2 ) = ψ(ξ2 , ξ1 ), where symmetric shearlet tight frame for L2 (C1 C3 )∨ . In the same way, let ψ(ξ S ψ(ξ1 , ξ2 ) is classical shearlet, we can obtain a shearlet tight frame for L2 (C2 C4 )∨ .

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References auser, Basel, [1] G. Kutyniok D. Labate, Shearlets: multiscale analysis for multivariate data, Birkh¨ 2012. [2] K. Guo, D. Labate, Optimally sparse multidimensional representation useing shearlets, SIAM J. Math. Anal., 39 (2007) 298-318. [3] G. R. Easley, D. Labate, and F. Colonna, Shearlet-based total variation for denoising, IEEE Trans. Image Processing, 18(2) (2009) 260-268. [4] K. Guo and D. Labate, Characterization and analysis of edges using the continuous shearlet transform, SIAM J. Imaging Sciences, 2 (2009) 959-986. [5] G. Kutyniok and W. Lim, Image separation using shearlets, in: Curves and Surfaces (Avignon, France, 2010), Lecture Notes in Computer Science 6920, Springer, 2012. [6] W. Q. Lim, The discrete shearlet transform: a new directional transform and compactly supported shearlet frames, IEEE Trans. Image Proc., 19(5) (2010) 1166-1180. [7] S. Yi, D. Labate, G. R. Easley, and H. Krim, A Shearlet approach to edge analysis and detection, IEEE Trans. Image Proc., 18(5) (2009) 929-941. [8] E. Weber, Orthogonal frames of translates, Appl. Comput. Harmon. Anal., 17 (2004) 69-90. [9] G. Bhatt, B. Johnson, E. Weber, Orthogonal wavelet frames and vector-valued wavelet transforms, Appl. Comput. Harmon. Anal., 23 (2) (2007) 215-234. [10] Y. Li, S. Yang, Explicit construction of symmetric orthogonal wavelet frames in L2 (Rs ), J. Approx. Theory, 162 (2010) 891-909. [11] H. Kim, R. Kim, J. Lim, Z. Shen, A pair of orthogonal frames, J. Approx. Theory, 147 (2007) 196-204. [12] Z. Liu, Y. Ren, G. Wu, Orthogonal frames and their dual frames in L2 (Rd ), Journal of Information Computational Science, 9(5) (2012) 1329-1336. [13] G. Wu, X. Tian, D. Li, The construction of symmetric or antisymmetric multiwavelets with composite dilations, Acta Math. Sinica(Chin. Ser.),54 (2011) 731-738. [14] Y. Li, S. Yang, Construction of paraunitary symmetric matrix and parametrization of symmetric and orthogonal multiwavelets filter banks, Acta Math. Sinica(Chin. Ser.), 53 (2010) 279-290. [15] K. Guo, D. Labate, W. Lim, G. Weiss, E. Wilson, Wavelets with composite dilations and their MRA properties, Appl. Comput. Harmon. Anal., 20 (2006) 202-236. [16] G. Easley, D. Labate, W. Lim, Sparse directional image representations using the discrete shearlet transform, Appl. Comput. Harmon. Anal., 25 (2008) 25-46. [17] K. Guo, G. Kutyniok, D. Labate, Sparse multidimensional representations using anisotropic dilation and shear operators, in Wavelets and Splines (Athens, GA, 2005), Nashboro Press, Nashville, TN, 2006, 189-201. [18] P. Grohs, Continuous shearlet tight frames, J. Fourier Anal. Appl., 17 (2011) 506-518. [19] P. Kittipoom, G. Kutyniok W. Lim, Construction of compactly supported shearlet frames, Constr. Approx., 35(2012) 21-72. DOI: 10.1007/s00365-011-9142-y.

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Exact orders in simultaneous approximation by complex q-Durrmeyer type operators Mei-Ying Ren1,∗ , Xiao-Ming Zeng2,∗, Liang Zeng3 1

Department of Mathematics and Computer Science, Wuyi University, Wuyishan 354300, China

2,3

Department of Mathematics, Xiamen University, Xiamen 361005, Chnia

E-mail: [email protected], [email protected], [email protected]

Abstract. In this paper we study the simultaneous approximation properties of the complex q-Durrmeyer type operators which were introduced by Agarwal and Gupta [3]. We obtain the exact orders in approximation by these operators and their derivatives on compact disks. Keywords: complex q-Durrmeyer type operators; simultaneous approximation; exact orders; q-calculus Mathematical subject classification: 30E10, 41A25

1. Introduction Let q > 0, for each nonnegative integer k, the q−integer [k]q and the q−factorial [k]q ! are defined by ½ (1 − q k )/(1 − q), q 6= 1 [k]q := k, q=1 and

½ [k]q ! :=

[k]q [k − 1]q · · · [1]q , k ≥ 1 1, k=0

respectively. Then for q > 0 and integers n, k, n ≥ k ≥ 0, we have [k + 1]q = 1 + q[k]q

and

[k]q + q k [n − k]q = [n]q .

For the integers n, k, n ≥ k ≥ 0, the q−binomial coefficients is defined by · ¸ [n]q ! n := . k q [k]q ![n − k]q ! ∗ Corresponding

authors: Mei-Ying Ren and Xiao-Ming Zeng.

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Let q > 0, q 6= 1, we can define the derivative Dq f of functions f in the q−calculus by ( f (qx)−f (x) x 6= 0, (q−1)x , Dq f (x) = f 0 (0), x = 0. Let a > 0, the q−Jackson integrals in the interval [0, a] is defined as Z

a

f (t)dq t = (1 − q)a 0

∞ X

f (aq j )q j , 0 < q < 1.

j=0

The q−analogue of Beta function is defined as Z 1 Bq (m, n) = tm−1 (1 − qt)n−1 dq t, q

m, n > 0,

0

where (a − b)nq =

n−1 Y

(a − q j b).

j=0

Also, it is known that Bq (m, n) =

[m − 1]q ![n − 1]q ! . [m + n − 1]q !

All of the previous concepts can be found in [1, 4, 11]. In 1986, the approximation properties of complex Bernstein polynomials in compact disks were initially studied by Lorentz [12]. Recently, the problem of the approximation of complex operators has been causing great concern, which is becoming a hot topic of research. (for instance, see [2, 5-10, 13-16]). In 2012, Agarwal and Gupta [3] introduced and studied the complex q-Durrmeyer type operators as follows: Mn,q (f ; z) = [n + 1]q

n X

Z

1

q 1−k pn,k (q; z)

pn,k−1 (q; qt)f (t)dq t 0

k=1

+ f (0)pn,0 (q; z),

(1.1)

where z ∈ C, n = 1, 2, . . . , 0 < q < 1 and · pn,k (q; z) :=

n k

¸ z q

k

n−k−1 Y

· s

(1 − q z) =

s=0

n k

¸ z k (1 − z)n−k . q q

In [3], the upper bound for the operators (1.1) was obtained as follows: ∞ P Theorem 1. Let f (z) = cm z m for all |z| < R and let 1 ≤ r ≤ R, then for m=0

all |z| ≤ r, 0 < q < 1 and n ∈ N, we have |Mn,q (f ; z) − f (z)| ≤

Kr (f ) , |n + 2|q

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where Kr (f ) = (1 + r)

∞ P m=1

|cm |m(m + 1)rm−1 < ∞.

The aim of the present article is to obtain the simultaneous approximation results for the complex q-Durrmeyer type operators (1.1) in the case 0 < q < 1.

2. Auxiliary results In the sequel, we shall need the following auxiliary results. Lemma 1.(see [3]) Let 0 < q < 1. Then, for all em (t) = tm , m ∈ N ∪ {0} and z ∈ C, we have the following recurrence relation: Mn,q (em+1 ; z) =

q m z(1 − z) [m]q + zq m [n]q Dq Mn,q (em ; z)+ Mn,q (em ; z). (2.1) [m + n + 2]q [m + n + 2]q

Lemma 2. If Pm (z) is a polynomial of degree m, for all |z| ≤ r, we have 0 |Dq Pm (z)| ≤ kPm kr ≤

m kPm kr , r

(2.2)

where kPm kr = max{|Pm (z)|; |z| ≤ r}. Proof. The proof is easy by using the Bernstein inequality and the complex mean value theorem, the proof is omitted in this. The following Voronovskaja-type result with a quantitative estimate holds. Lemma 3. Let 0 < q < 1, R > 1, DR = {z ∈ C : |z| < R}. Suppose ∞ P that f : DR → C is analytic in DR , i.e. f (z) = ck z k for all z ∈ DR . Then k=0

for any fixed r ∈ [1, R] and for all n ∈ N, |z| ≤ r, we have ¯ ¯ ∞ 0 00 X ¯ ¯ ¯Mn,q (f ; z) − f (z) + [2]q zf (z) − z(1 − z)f (z) ¯ ≤ Mr (f ) +2(1−q) |ck |krk , ¯ [n + 2]q [n + 2]q ¯ [n + 2]2q k=1

where Mr (f ) =

∞ P

|ck |kFk,r rk < ∞ with Fk,r = (k − 1)(k − 2)(2k − 3) + 4(k +

k=1

1)(k − 1)2 + 2(k − 1)(k + 1)2 + 4(k − 1)2 k(1 + r). Proof. Denoting ek (z) = z k , k = 0, 1, 2, . . ., by hypothesis that f (z) is an∞ P ck z k for all z ∈ DR , we can write Mn,q (f ; z) = alytic in DR , i.e. f (z) = ∞ P

k=0

ck Mn,q (ek ; z), thus, for all z ∈ DR and n ∈ N, we have

k=0

¯ ¯ 0 00 ¯ ¯ ¯Mn,q (f ; z) − f (z) + [2]q zf (z) − z(1 − z)f (z) ¯ ¯ [n + 2]q [n + 2]q ¯ 3

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≤

∞ X k=1

¯ ¯ ¯ k[2]q z k k(k − 1)(1 − z)z k−1 ¯¯ ¯ − |ck | ¯Mn,q (ek ; z) − ek (z) + ¯. [n + 2]q [n + 2]q

Denoting Ek,n (q; z) = Mn,q (ek ; z) − ek (z) +

k[2]q z k k(k − 1)(1 − z)z k−1 − , [n + 2]q [n + 2]q

it is obvious that Ek,n (q; z) is a polynomial of degree less than or equal to k and (1−q 2 )[n] z

that E1,n (q; z) = [n+2]q q . For k ≥ 2, by simple computation and the use of Lemma 1, we can get Ek,n (q; z) =

q k−1 z(1 − z) Dq Ek−1,n (q; z) [n + k + 1]q +

q k−1 [n]q z + [k − 1]q Ek−1,n (q; z) + Gk,n (q; z), [n + k + 1]q

(2.3)

where Gk,n (q; z) =

© 2 z k−2 z [k[n + k + 1]q ([2]q + k − 1) [n + 2]q [n + k + 1]q −[n + 2]q [n + k + 1]q − q k−1 (k − 1)([2]q + k − 2)[n]q +q k−1 [n]q [n + 2]q + q k−1 (k − 1)[k − 1]q ([2]q + k − 2) ¤ £ −q k−1 [k − 1]q [n + 2]q + z q k−1 [k − 1]q [n + 2]q −q k−1 (k − 1)[k − 1]q ([2]q + k − 2) − q k−1 (k − 1)(k − 2)[k − 2]q +[k − 1]q [n + 2]q − (k − 1)[k − 1]q ([2]q + k − 2) ¤ +q k−1 (k − 1)(k − 2)[n]q − k(k − 1)[n + k + 1]q £ ¤ + q k−1 (k − 1)(k − 2)[k − 2]q + (k − 1)(k − 2)[k − 1]q }

=:

z k−2 (z 2 Ak,n (q) + zBk,n (q) + Ck,n (q)). [n + 2]q [n + k + 1]q

For all k ≥ 2, we easily obtain |Ck,n (q)| ≤ (k − 1)(k − 2)(2k − 3). Since [n + k + 1]q = [k − 1]q + q k−1 [n + 2]q and [n]q = [n + 2]q − q n − q n+1 , so, for all k ≥ 2, we can get Bk,n (q) = [n + 2]q [q k−1 [k − 1]q + [k − 1]q − 2q k−1 (k − 1)] − q k−1 (k − 1)[k − 1]q ([2]q + k − 2) − q k−1 (k − 1)(k − 2)[k − 2]q − q n+k−1 (q + 1)(k − 1)(k − 2) − (k − 1)[k − 1]q ([2]q + k − 2) − k(k − 1)[k − 1]q . In view of [k − 1]q − (k − 1) = (q − 1)

k−2 P j=0

[j]q , [k − 1]q − q k−1 (k − 1) =

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(1 − q)

k−1 P j=1

[j]q q k−1−j and [n + 2]q =

q n+2 −1 q−1 ,

we have

[n + 2]q [q k−1 [k − 1]q + [k − 1]q − 2q k−1 (k − 1)] = q k−1 (q n+2 − 1)

k−2 X

[j]q + (1 − q n+2 )

j=0

k−1 X

[j]q q k−1−j ,

j=1

thus, by simple calculation, for all k ≥ 2, we get |Bk,n (q)| ≤ 4(k + 1)(k − 1)2 . Now we estimate Ak,n (q). Similar to the calculation of the Bk,n (q), for all k ≥ 2, we have Ak,n (q) = −q k−1 (q n+2 − 1)

k−2 X

[j]q − (1 − q n+2 )

j=0 n

k

+ (1 − q )(q + q +q

n+k−1

+q

k−1

k−1

k−1 X

[j]q q k−1−j

j=1

)[n + 2]q + k[k − 1]q ([2]q + k − 1)

(q + 1)(k − 1)([2]q + k − 2)

(k − 1)[k − 1]q ([2]q + k − 2),

by simple calculation, for all k ≥ 2, it follows that |Ak,n (q)| ≤ 2(k − 1)(k + 1)2 + 2(1 − q n )[n + 2]q . Thus, for all n ∈ N, k ≥ 2 and |z| ≤ r, we can obtain rk−2 [(k − 1)(k − 2)(2k − 3) + 4r(k + 1)(k − 1)2 [n + 2]2q

|Gk,n (q; z)| ≤

+ 2r2 (k − 1)(k + 1)2 ] + 2rk (1 − q). By formula (2.3), for all n ∈ N, k ≥ 2 and |z| ≤ r, we have |Ek,n (q; z)| ≤

r(1 + r) |Dq Ek−1,n (q; z)| [n + k + 1]q +

q k−1 [n]q r + [k − 1]q |Ek−1,n (q; z)| + |Gk,n (q; z)|, [n + k + 1]q

since q k−1 [n]q r + [k − 1]q ≤ [n + k + 1]q r, it follows |Ek,n (q; z)| ≤

r(1 + r) |Dq Ek−1,n (q; z)| + r|Ek−1,n (q; z)| + |Gk,n (q; z)|. [n + k + 1]q

Using the estimate in the proof of Theorem 1, we get |Mn,q (ek ; z) − ek (z)| ≤

(1 + r)k(k + 1)rk−1 , [n + 2]q

for all k, n ∈ N, |z| ≤ r, 1 ≤ r.

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Denote kf kr = max{|f (z)|; |z| ≤ r}, by Lemma 2 we have k−1 k Ek−1,n kr r k−1 ≤ [kMn,q (ek−1 ; ·) − ek−1 kr °r ° ¸ ° (k − 1)[2]q ek−1 (k − 1)(k − 2)(1 − e1 )ek−2 ° ° +° − ° ° [n + 2]q [n + 2]q r · ¸ k − 1 k(k − 1)(1 + r)rk−2 k(k − 1)(1 + r)rk−2 ≤ + r [n + 2]q [n + 2]q

|Dq Ek−1.n (q; z)| ≤

≤

4(k − 1)2 krk−1 , [n + 2]q

so, for all n ∈ N, k ≥ 2 and |z| ≤ r, we have |Ek,n (q; z)| ≤ where |Gk,n (q; z)| ≤

4(k − 1)2 k(1 + r)rk + r|Ek−1,n (q; z)| + |Gk,n (q; z)|, [n + 2]2q rk [n+2]2q Dk 2

+ 2rk (1 − q), Dk = (k − 1)(k − 2)(2k − 3) + 4(k +

1)(k − 1)2 + 2(k − 1)(k + 1) . (1−q 2 )[n] z On the other hand, for all n ∈ N and |z| ≤ r, |E1,n (q; z)| = | [n+2]q q | ≤ 2r(1 − q), therefore, for all k, n ∈ N and |z| ≤ r, we have |Ek,n (q; z)| ≤ rk k r|Ek−1,n (q; z)| + [n+2] 2 Fk,r + 2r (1 − q), where Fk,r is a polynomial of degree 3 q

in k defined as Fk,r = Dk + 4(k − 1)2 k(1 + r). Since E0.n (q; z) = 0 for any z ∈ C , therefore, by writing the last inequality for k = 1, 2, . . ., we easily obtain step by step the following |Ek,n (q; z)| ≤

k X rk krk k F + Fk,r + 2krk (1 − q). 2kr (1 − q) ≤ j,r [n + 2]2q j=1 [n + 2]2q

As a conclusion, we have ¯ X ¯ ∞ 0 00 ¯ ¯ ¯Mn,q (f ; z) − f (z) + [2]q zf (z) − z(1 − z)f (z) ¯ ≤ |ck ||Ek,n (q; z)| ¯ [n + 2]q [n + 2]q ¯ k=1

≤

∞ ∞ X X 1 k |c |kF r + 2(1 − q) |ck |krk . k k,r [n + 2]2q k=1

As f (4) (z) =

∞ P

k=1

ck k(k −1)(k −2)(k −3)z k−4 and the series is absolutely con-

k=4

∞ P

vergent in |z| ≤ r, it easily follows that which implies that

∞ P

|ck |k(k − 1)(k − 2)(k − 3)rk−4 < ∞,

k=4

|ck |kFk,r rk < ∞, this completes the proof of theorem.

k=1

Remark 1. Let 0 < q < 1 be fixed. Since we have

1 [n+2]q

→ 1 − q as n → ∞,

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by passing to limit with n → ∞ in the estimates in Lemma 2 we don’t obtain 1 convergence. But the situation can be improved by choosing 1− (n+2) 2 ≤ qn < 1 1 with qn → 1 as n → ∞. Indeed, since in this case [n+2]q → 0 as n → ∞ ( see n 1 1 Videnskii [17], formula (2.7) ) and 1 − qn ≤ (n+2) 2 ≤ [n+2]2 , from Lemma 2 we qn get ¯ ¯ 0 00 ¯ ¯ ¯Mn,qn (f ; z) − f (z) + [2]qn zf (z) − z(1 − z)f (z) ¯ ¯ [n + 2]qn [n + 2]qn ¯ ∞ X Mr (f ) 2 ≤ + |ck |krk , (2.4) [n + 2]2qn [n + 2]2qn k=1

that is the order of approximation

1 [n+2]2qn

.

3. Main results In the following theorem, we will obtain the simultaneous approximation properties of the operators (1.1). Theorem 2. Let f (z) =

∞ P m=0

cm z m for all |z| < R and let 1 ≤ r ≤ R, 0 < q < 1.

If r < r1 < R are arbitrary fired, then for all |z| ≤ r, n, p ∈ N, we have (p) |Mn,q (f ; z) − f (p) (z)| ≤

where Kr1 (f ) = (1 + r1 )

∞ P m=1

Kr1 (f )p!r1 , [n + 2]q (r1 − r)p+1

|cm |m(m + 1)r1m−1 < ∞.

Proof. Denoting by Γ the circle of radius r1 > r and center 0, since for any |z| ≤ r and v ∈ Γ, we have |v − z| ≥ r1 − r, by the Cauchy’s formulas and the Theorem 1, it follows that for all |z| ≤ r and n, p ∈ N, we have ¯ ¯Z p! ¯¯ Mn,q (f ; v) − f (v) ¯¯ (p) p dv ¯ |Mn,q (f ; z) − f (z)| = 2π ¯ Γ (v − z)p+1 Kr1 (f ) p! 2πr1 ≤ [n + 2]q 2π (r1 − r)p+1 Kr1 (f ) p!r1 , = · [n + 2]q (r1 − r)p+1 which proves the theorem. Theorem 3. Let f (z) =

∞ P m=0

cm z m for all |z| < R and let 1 −

1 (n+2)2

≤

qn < 1, n ∈ N. Suppose that 1 ≤ r < r1 < R and p ∈ N be fixed. If f is not a polynomial of degree ≤ p − 1, then we have (p) kMn,q (f ; ·) − f (p) kr ³ n

1 , n ∈ N, [n + 2]qn

where kf kr = max{|f (z)|; |z| ≤ r} and the constants in the equivalence depend only on f , r, r1 , p and on the sequence {qn }n∈N . 7

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Proof. Taking into account the upper estimate in Theorem 1, it remains to prove the lower estimate only. Denoting by Γ the circle of radius r1 > r and center 0 , by the Cauchy’s formula, it follows that for all |z| ≤ r and n ∈ N, we have Z p! Mn,qn (f ; v) − f (v) (p) (p) dv. Mn,q (f ; z) − f (z) = n 2πi Γ (v − z)p+1 Let Hn,qn (f ; z) = Mn,qn (f ; z) − f (z) +

[2]qn zf 0 (z) z(1 − z)f 00 (z) − . [n + 2]qn [n + 2]qn

For all n ∈ N, we have

=

1 [n + 2]qn

Mn,qn (f ; z) − f (z) ½ ¾ £ ¤ 1 00 2 0 [n + 2]qn Hn,qn (f ; z) , z(1 − z)f (z) − [2]qn zf (z) + [n + 2]qn

by using Cauchy’s formula, for all v ∈ Γ, we get n 1 (p) (p) [z(1 − z)f 00 (z) − [2]qn zf 0 (z)](p) Mn,q (f ; z) − f (z) = n [n + 2]qn ) Z [n + 2]2qn Hn,qn (f ; v) p! 1 dv , + [n + 2]qn 2πi Γ (v − z)p+1 passing now to k · kr and denoting e1 (z) = z, it follows ° ° h 1 ° (p) ° k[e1 (1 − e1 )f 00 − [2]qn e1 f 0 ](p) kr °Mn,qn (f ; ·) − f (p) ° ≥ [n + 2]qn r ° ° # ° p! Z [n + 2]2 H ° 1 ° ° qn n,qn (f ; v) − dv ° ° . ° [n + 2]qn ° 2πi Γ (v − ·)p+1 r

By hypothesis on f , we have k[e1 (1 − e1 )f 00 − [2]qn e1 f 0 ](p) kr > 0. Indeed, let p = 1, supposing the contrary, it follows that z(1 − z)f 00 (z) − [2]qn zf 0 (z) is a constant. Clearly, this is possible only if f is constant ( since contrariwise z(1 − z)f 00 (z) − [2]qn zf 0 (z) is a polynomial of degree at least 1, which cannot be equal to a constant ), which implies f is a polynomial of degree ≤ p − 1, a contradiction. let p = 2, supposing the contrary, it follows that z(1 − z)f 00 (z) − [2]qn zf 0 (z) is a polynomial of degree ≤ 1. Clearly, this is possible only if f is a polynomial of degree ≤ 1 (since contrariwise z(1 − z)f 00 (z) − [2]qn zf 0 (z) is a polynomial of degree at least 2, which cannot is a polynomial of degree ≤ 1), which implies f is a polynomial of degree ≤ p − 1, a contradiction. Now let p ≥ 3, suppose that k[e1 (1 − e1 )f 00 − [2]qn e1 f 0 ](p) kr = 0, it follows that z(1 − z)f 00 (z) − [2]qn zf 0 (z) is a polynomial of degree ≤ p − 1, that is z(1 − z)f 00 (z) − [2]qn zf 0 (z) = Qp−1 (z), for all |z| ≤ r, where Qp−1 (z) is an algebraic polynomial of degree ≤ p − 1, with complex coefficients. 8

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From above we also get Qp−1 (0) = 0, which means that Qp−1 (z) is necesp−1 P sarily of the form Qp−1 (z) = Ak z k and that we can simplify with z in the k=1

equation. Now, denoting f 0 = F , the above differential equation one reduces to (1 − z)F 0 (z) − [2]qn F (z) = Hp−2 (z), for all |z| ≤ r, where Hp−2 (z) =

p−2 P

Ak+1 z k is a polynomial of degree ≤ p − 2. In what follows,

k=0

(1)

(2)

(1)

(2)

denote F (x) = F1 (x) + iF2 (x), Ak+1 = Ak+1 + iAk+1 , where Ak+1 , Ak+1 ∈ p−2 P (j) k R, and Hp−2,j (x) = Ak+1 x , j = 1, 2. Evidently we have Hp−2 (x) = k=0

Hp−2,1 (x) + iHp−2,2 (x), for all x ∈ [−1, 1]. Note here that F1 (x), F2 (x), Hp−2,1 (x) and Hp−2,2 (x) are real functions of real variable. Also, recall that i2 = −1. Because r ≥ 1, it follows that taking z = x ∈ [−1, 1] in the equation in z for F , the functions Fj , j = 1, 2, necessarily verify the differential equations in x (1 − x)Fj0 (x) − [2]qn Fj (x) = Hp−2,j (x), for all x ∈ [−1, 1], j = 1, 2. The standard theory says that the general solution of a linear different equations of real functions of real variable is obtained by adding to the general solution of the homogenous equation, a particular solution of the inhomogenous equation. But reasoning exactly as in the proof of Lemma 3, the unique solutions of the homogenous equation are Fj (x) = 0, for all x ∈ [−1, 1], j = 1, 2. On the other hand, if we consider the differential equation of the form 0

(1 − x)G (x) − [2]qn G(x) =

p−2 X

dk+1 xk , for all x ∈ [−1, 1],

k=0

where G(x) is considered real-valued function and dk+1 ∈ R for all k, looking for p−2 P a particular solution of it, of the form G(x) = Ck xk , with Ck ∈ R, simply k=0

calculation show that the differential equation one reduces to p−3 X

(k + 1)Ck+1 xk − [2]qn

k=0

p−2 X

(k + 1)Ck xk =

k=0

p−2 X

dk+1 xk , for all x ∈ [−1, 1],

k=0

this immediately leads to the algebraic system Cp−2 = −

dk+1 dp−1 , Ck+1 − [2]qn Ck = , k ∈ {0, 1, . . . , p − 3}, [2]qn (p − 1) k+1

that evidently has unique solution for the unknowns Ck . Therefore, these considerations show that we can take F1 and F2 as polynomials of degree ≤ p − 2, solutions of the corresponding inhomogenous equations in x, which implies that necessarily these are the unique solutions of the above inhomogenous different equations in x. This also implies the uniqueness of F (x) 9

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too as polynomial of degree ≤ p − 2 in x, solution of the corresponding differential equation in x. Now, because F (z) is the analytic continuation of F (x), from the identity theorem on analytic function, it follows that F (z) as polynomial of degree ≤ p−2 in z, necessarily is the unique solution of the corresponding differential equation in z, for |z| ≤ r. This implies that f 0 (z) is a polynomial of degree ≤ p−2, which means that f (z) is a polynomial of degree ≤ p−1, a contradiction with the hypothesis. In conclusion, k[e1 (1 − e1 )f 00 − [2]qn e1 f 0 ](p) kr > 0. Since for any |z| ≤ r and υ ∈ Γ we have |υ − z| ≥ r1 − r, so, by the formula (2.4), we get ° ° ° p! Z [n + 2]2 H ° p! 2πr1 [n + 2]2qn kHn,qn (f ; ·)kr1 ° ° qn n,qn (f ; v) dv ≤ ° ° ° 2πi Γ ° (v − ·)p+1 2π (r1 − r)p+1 r

≤ where Nr1 (f ) = Mr1 (f ) + 2

∞ P k=1

Nr1 (f )p!r1 , (r1 − r)p+1

|ck |kr1k . Taking into account

1 [n+2]qn

→ 0 as

n → ∞, therefore, there exists an index n0 depending only on f , r and on sequence {qn }n∈N , such that for all n ≥ n0 we have ° ° ° ° p! Z [n + 2]2 H (f ; v) 1 n,q ° ° q n n ke1 (1 − e1 )f 00 − [2]qn e1 f 0 ](p) kr − dv ° ° ° [n + 2]qn ° 2πi Γ (v − ·)p+1 r

° 1° ° ° ≥ °[e1 (1 − e1 )f 00 − [2]qn e1 f 0 ](p)) ° , 2 r which implies (p) kMn,q (f ; ·) − f (p) kr ≥ n

1 k[e1 (1 − e1 )f 00 − [2]qn e1 f 0 ](p) kr , ∀n ≥ n0 . 2[n + 2]qn (p)

For n ∈ {1, 2, · · ·, n0 − 1}, we have kMn,qn (f ; ·) − f (p) kr ≥

Wr,n (f ) [n+2]qn ,

where

(p) Wr,n (f ) = [n + 2]qn · kMn,qn (f ; ·) − f (p) kr > 0. (p) Cr (f ) , for all n ∈ N, where As a conclusion, we have kMn,qn (f ; ·)−f (p) kr ≥ [n+2] qn 1 Cr (f ) = min{Wr,1 (f ), Wr,2 (f ) . . . , Wr,n0 −1 (f ), 2 k[e1 (1−e1 )f 00 −[2]qn e1 f 0 kr ](p) },

this complete the proof.

Acknowledgements This work is supported by the National Natural Science Foundation of China (Grant No. 61170324), the Natural Science Foundation of Fujian Province of China (Grant No. 2010J01012), and the Class A Science and Technology Project of Education Department of Fujian Province of China (Grant No. JA12324).

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J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 16, NO.5, 2014, COPYRIGHT 2014 EUDOXUS PRESS, LLC

ON POSITIVE SOLUTIONS OF A SYSTEM OF MAX-TYPE DIFFERENCE EQUATIONS ´ ∗ , ABDULLAH ALOTAIBI, NASEER SHAHZAD, STEVO STEVIC AND MOHAMMED A. ALGHAMDI Abstract. The boundedness character and global stability of positive solutions of the next system of difference equations with maximum     p  p  zn xpn yn , yn+1 = max c, p , zn+1 = max c, p , xn+1 = max c, p zn−1 xn−1 yn−1 n∈

N0 , where p, c ∈ (0, ∞), are studied in this paper.

1. Introduction There has been some recent interest in nonlinear systems of difference equations (see, e.g., [8, 9, 10, 11, 12, 13, 14, 15, 17, 18, 32, 34, 35, 36, 37, 38, 39, 40]), as well as in difference equations containing maximum operator, so called, max-type difference equations (see, e.g., [1, 6, 7, 15, 23, 24, 25, 26, 28, 29, 30, 31, 33, 36, 37, 41, 42, 43]). However, systems of difference equations with the maximum operator are barely touched (see [15, 36, 37]). Another interesting direction in theory of difference equations is investigation of those equations containing non-integer powers of their variables (see, e.g., [2, 3, 4, 5, 16, 19, 22, 23, 24, 25, 26, 27, 28, 29, 31, 43]). Some starting points and motivations for our investigations of difference equations containing non-integer powers of their variables were papers [20], [21] and [22]. These three papers along with some results on difference equations with maximum motivated S. Stevi´c to study in [23] the next difference equation ¾ ½ xp (1) xn+1 = max c, p n , n ∈ N0 , xn−1 where initial values x−1 , x0 , and parameters c and p are positive numbers. In view of all above mentioned investigations, it is a natural problem to study systems of max-type difference equations containing non-integer powers of their variables. One of the first papers in the area was [38] where S. Stevi´c studied solutions of the following max-type system of difference equations ¾ ¾ ½ ½ yp xp xn+1 = max c, p n , yn+1 = max c, p n , n ∈ N0 , xn−1 yn−1 with positive initial values x−1 , x0 , y−1 and y0 and parameters p and c, which is a natural extension of equation (1). 2000 Mathematics Subject Classification. Primary 39A11. Key words and phrases. System of difference equations, max-type system, positive solution, boundedness character, convergence. ∗ Corresponding author. 1

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´ ∗ , A. ALOTAIBI, N. SHAHZAD, AND M. A. ALGHAMDI STEVO STEVIC

2

Continuing this line of research, here we study long term behavior of positive solutions of the next system of max-type difference equations ½ ½ ½ ¾ ¾ ¾ yp zp xp xn+1 = max c, p n , yn+1 = max c, p n , zn+1 = max c, p n , (2) zn−1 xn−1 yn−1 n ∈ N0 , where parameters p and c are positive. System (2) is a natural threedimensional extension of scalar difference equation (1). Solution (xn , yn , zn )n≥−1 of system (2) is called positive if min{xn , yn , zn } > 0 for every n ≥ −1. We say that the system of difference equations xn+1 = f1 (xn , yn , zn , xn−1 , yn−1 , zn−1 ) yn+1 = f2 (xn , yn , zn , xn−1 , yn−1 , zn−1 ) zn+1 = f3 (xn , yn , zn , xn−1 , yn−1 , zn−1 ) is permanent with respect to a class of solutions F, if there are constants m and M such that for every solution (xn , yn , zn )n≥−1 ∈ F of the system the following inequalities hold m ≤ min{xn , yn , zn } ≤ max{xn , yn , zn } ≤ M, for sufficiently large n. Our focus in the study of system (2) will be on the permanence, the existence of unbounded solutions, and on the convergence in the class of positive solutions. Our results are presented in terms of parameters p and c. 2. Permanence and unbounded solutions of system (2) The permanence and the existence of unbounded positive solutions of system (2) are studied in this section. Theorem 1. Assume p ≥ 4 and c > 0. Then there are positive unbounded solutions of system (2). Proof. Assume (xn , yn , zn )n≥−1 is a positive solution of system (2). From the equations in (2) we obtain µ xn+1 ≥ yn+1 ≥ µ zn+1 ≥

¶p ,

n ∈ N0 ,

(3)

,

n ∈ N0 ,

(4)

,

n ∈ N0 .

(5)

zn−1 µ

and

yn zn xn−1 xn yn−1

¶p

¶p

From (3)-(5) we easily get ln xn+1 ≥p ln yn − p ln zn−1 ,

(6)

ln yn+1 ≥p ln zn − p ln xn−1

(7)

ln zn+1 ≥p ln xn − p ln yn−1 ,

(8)

for n ∈ N0 .

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SYSTEM OF MAX-TYPE DIFFERENCE EQUATIONS

3

Set vn = ln xn yn zn , n ≥ −1. Then (6)-(8) imply vn+1 − pvn + pvn−1 ≥ 0,

n ∈ N0 .

(9)

Note that when p ≥ 4 the polynomial P (r) = r2 − pr + p has two real roots r1 and r2 such that min{r1 , r2 } > 1. From (9) we have vn+1 − r1 vn − r2 (vn − r1 vn−1 ) ≥ 0, and consequently xn+1 yn+1 zn+1 ≥ (xn yn zn )r1

µ

xn yn zn (xn−1 yn−1 zn−1 )r1

n ∈ N0 ,

¶r2 ,

n ∈ N0 .

This implies xn yn zn ≥ (xn−1 yn−1 zn−1 )r1

µ

x0 y0 z0 (x−1 y−1 z−1 )r1

¶r2n ,

n ∈ N0 .

Now choose x−1 , y−1 , z−1 , x0 , y0 and z0 such that © ª x0 y0 z0 > max 1, (x−1 y−1 z−1 )r1 .

(10)

(11)

Then from (10) and (11) we get xn yn zn > (xn−1 yn−1 zn−1 )r1 ,

n ∈ N0 ,

from which along with (11) it follows that n

xn yn zn > (x0 y0 z0 )r1 → +∞,

as n → +∞.

(12)

The existence of an unbounded solution (xn , yn , zn )n≥−1 of system (2) follows from (12) and by the inequality p √ √ x2n + yn2 + zn2 ≥ 3 3 xn yn zn . ¤

Theorem 2. Let p ∈ (0, 4) and c > 0. Then system of difference equations (2) is permanent. Proof. First note that from equations in system (2) we immediately obtain that for every positive solution (xn , yn , zn )n≥−1 of the system c ≤ min{xn , yn , zn },

(13)

for every n ∈ N. Let (pk )k≥0 be defined by pk+1 = We have

p0 = 0.

½ µ ¶p ¾ yn = max c, = max c, p zn−1 zn−1 ( Ã ( )!p ) p−p c zn−11 = max c, max , . zn−1 xpn−2 ½

xn+1

p , p − pk ynp

(14)

¾

908

(15)

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´ ∗ , A. ALOTAIBI, N. SHAHZAD, AND M. A. ALGHAMDI STEVO STEVIC

Assume that p ∈ (0, 1] = (p0 , p1 ]. Then (13) and (15) imply )!p ) ( Ã ( ¾ ½ 1 1 c , p1 −p p ≤ max c, 1, p , xn+1 = max c, max zn−1 zn−1 xn−2 c

(16)

for n ≥ 3, which along with (13) imply

¾ ½ 1 c ≤ xn ≤ max c, 1, p , c

for n ≥ 4. Assume that p ∈ (p1 , p2 ]. By (2) and (15) we get !p−p1 )!p ) ( Ã ( Ã c zn−1 xn+1 = max c, max , p p−p1 zn−1 xn−2 ( Ã ( Ã ( )!p−p1 )!p ) 2 xp−p c c n−2 = max c, max , max , p . 2 zn−1 xpn−2 yn−3

(17)

(18)

Using (13) in (18), and the facts (p2 − 1)(p − p1 )p = p and p2 (p − p1 )p = p2 , we get à ( ( à ( )!p−p1 )!p ) c 1 c , max , p2 −p p xn+1 = max c, max 2 zn−1 xpn−2 xn−2 yn−3 ¾ ½ 1 1 ≤ max c, 1, p , p2 , c c for n ≥ 4, which along with (13) implies ¾ ½ 1 1 (19) c ≤ xn ≤ max c, 1, p , p2 , c c for n ≥ 5. Assume that p ∈ (p2 , p3 ]. By (2) and (18) we get à ( à !p−p2 )!p−p1 )!p ) ( à ( c xn−2 c , max , xn+1 = max c, max p3 2 zn−1 xpn−2 yn−3 à ( à ( )!p−p2 )!p−p1 )!p ) ( à ( p−p c c yn−33 c , max , max = max c, max p3 , p 2 zn−1 xpn−2 yn−3 zn−4 (20) for n ≥ 5. Using (13) in (20) we get à ( à ( ( à ( )!p−p2 )!p−p1 )!p ) c c 1 c , max , max xn+1 = max c, max p3 , p3 −p p 2 zn−1 xpn−2 yn−3 yn−3 zn−4 ½ ¾ 1 1 1 ≤ max c, 1, p , (p −1)(p−p )(p−p )p , p (p−p )(p−p )p , , 2 1 2 1 c c 3 c 3 for n ≥ 5, which along with (13) implies ¾ ½ 1 1 1 c ≤ xn ≤ max c, 1, p , (p −1)(p−p )(p−p )p , p (p−p )(p−p )p , , 2 1 2 1 c c 3 c 3 for n ≥ 6.

909

(21)

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SYSTEM OF MAX-TYPE DIFFERENCE EQUATIONS

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By induction we get that for each fixed l ∈ N0 Ã ( Ã ( )!p−p2 )!p−p1 )!p ) ( Ã ( p−p c c yn−33 c , max , max xn+1 = max c, max p3 , p 2 zn−1 xpn−2 yn−3 zn−4 =···

( Ã

= max c,

( max

c zn−1

à ,

Ã

(

··· ,

p−p

3l+1 zn−(3l+1)

c

)!p−p3l

, p p3l+1 xn−(3l+2) zn−(3l+1)

max

!p−p1 )!p ) ···

, (22)

for n ≥ 3l + 1 and p ∈ (p3l , p3l+1 ], ( Ã xn+1 = max c, =···

( max

( Ã

= max c,

c zn−1

( max

c zn−1

à ,

( max

à ,

Ã

c 2 xpn−2

à ··· ,

,

(

( max

c

p−p3 yn−3 , p3 p yn−3 zn−4

c

p−p

3l+2 xn−(3l+2)

)!p−p2 )!p−p1 )!p )

)!p−p3l+1

, p p3l+2 yn−(3l+3) xn−(3l+2)

max

!p−p1 )!p ) ···

,

(23) for n ≥ 3l + 2 and p ∈ (p3l+1 , p3l+2 ], and ( Ã ( Ã ( Ã ( )!p−p2 )!p−p1 )!p ) p−p c c c yn−33 xn+1 = max c, max , max , max p3 , p 2 zn−1 xpn−2 yn−3 zn−4 =···

( Ã

= max c,

( max

c zn−1

à ,

à ··· ,

( max

c

p−p

3l+3 yn−(3l+3)

, p p3l+3 zn−(3l+4) yn−(3l+3)

)!p−p3l+2

!p−p1 )!p ) ···

,

(24) for n ≥ 3l + 3 and p ∈ (p3l+2 , p3l+3 ]. If p = ps for some s ∈ N, difference equation (14) defines pi for i = 0, s, and the method described above is finished after s + 1 steps. The monotonicity of f (x) = p/(p−x) on the interval (0, p), and the fact 0 = p0 < p1 = 1, imply that pk−1 < pk as far as pk < p. The case pk ∈ (0, p) for every k ∈ N, is not possible. Namely, if it were then it would exist limk→∞ pk = p∗ ∈ (0, p], and it would be (p∗ )2 − pp∗ + p = 0, which for p ∈ (0, 4) is not possible. This implies that there is a k0 ∈ N such that p ∈ (pk0 −1 , pk0 ]. If k = k0 = 3l0 + 1, from (22) we get à à ( !p−p1 )!p ) ( à ( p−p3l0 +1 )!p−p3l0 zn−(3l c c 0 +1) , p , · · · , max ··· xn+1 = max c, max p3l0 +1 zn−1 xn−(3l0 +2) zn−(3l 0 +1) ( à ( à à ( )!p−p3l0 !p−p1 )!p ) 1 1 ≤ max c, max 1, · · · , max ··· , , cp3l0 +1 −1 cp3l0 +1 (25) for n ≥ 3l0 + 3.

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´ ∗ , A. ALOTAIBI, N. SHAHZAD, AND M. A. ALGHAMDI STEVO STEVIC

If k = k0 = 3l0 + 2, from (23) we get ( à ( à à ( !p−p1 )!p ) p−p3l0 +2 )!p−p3l0 +1 xn−(3l+2) c c xn+1 = max c, max , p , · · · , max ··· p3l0 +2 zn−1 yn−(3l0 +3) xn−(3l +2) 0 à ( ( à ( à )!p−p3l0 +1 !p−p1 )!p ) 1 1 , ≤ max c, max 1, · · · , max ··· , cp3l0 +2 −1 cp3l0 +2 (26) for n ≥ 3l0 + 4. If k = k0 = 3l0 + 3, from (24) we get à à ( !p−p1 )!p ) ( à ( p−p3l0 +3 )!p−p3l0 +2 yn−(3l c c +3) 0 , · · · , max ··· xn+1 = max c, max , p p3l0 +3 zn−1 zn−(3l0 +4) yn−(3l +3) 0 ( à ( à )!p−p3l0 +2 !p−p1 )!p ) à ( 1 1 ≤ max c, max 1, · · · , max ··· , , cp3l0 +3 −1 cp3l0 +3 (27) for n ≥ 3l0 + 5. From (13), (17), (19), (21), (25)-(27) and the method of induction it follows that for every p ∈ (0, 4) there is a k0 ∈ N0 such that p ∈ (pk0 −1 , pk0 ], and ( ) 1 c ≤ xn ≤ max c, , (28) Qk0 −1 cpk0 j=0 (p−pj ) for n ≥ k0 + 2. Bearing in mind that system (2) is invariant with respect to cyclic permutations of variables xn , yn and zn , from (28) it follows that ( ) 1 c ≤ min{yn , zn } ≤ max{yn , zn } ≤ max c, , (29) Qk0 −1 cpk0 j=0 (p−pj ) for n ≥ k0 + 2. From (28) and (29) the permanence of system (2) follows, as desired. ¤ The following corollary is a direct consequence of estimates (28) and (29) in Theorem 2. Corollary 1. Let p ∈ (0, 4) and c ≥ 1. Then every positive solution (xn , yn , zn )n≥−1 of system (2) is eventually equal to (c, c, c). 3. Convergence of positive solutions In this section we prove a result on the convergence of positive solutions of system of difference equations (2). Theorem 3. If p ∈ (0, 1) and c ∈ (0, 1), then every positive solution of system (2) converges to (1, 1, 1).

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Proof. Using (13) and (16) we get ¾ ½ 1 1 c ≤ xn+1 ≤ max c, 1, p = p , c c On the other hand, the invariance with respect to ables xn , yn and zn of system (2), implies that ¾ ½ 1 1 c ≤ yn+1 ≤ max c, 1, p = p , c c and ¾ ½ 1 1 c ≤ zn+1 ≤ max c, 1, p = p , c c

for

n ≥ 3.

zn+1 = max xn

½

for

n ≥ 3,

(31)

for

n ≥ 3.

(32)

¾ c 1 . , p xn x1−p n yn−1

Using (31) and (32) in (33), as well as the assumption p ∈ (0, 1), we get 1 xn+1 ≤ , for n ≥ 5; cp ≤ yn c using (30) and (32) in (34), and p ∈ (0, 1), we get 1 yn+1 ≤ , for n ≥ 5; cp ≤ zn c and finally, using (30) and (31) in (35), and p ∈ (0, 1), we get 1 zn+1 ≤ , for n ≥ 5. cp ≤ xn c Using p, c ∈ (0, 1) and (37) in the first equation in (2), we get 2 1 for n ≥ 6; cp ≤ xn+1 ≤ p , c using p, c ∈ (0, 1) and (38) in the second equation in (2), we get 2 1 for n ≥ 6; cp ≤ yn+1 ≤ p , c and finally using p, c ∈ (0, 1) and (36) in the third equation in (2), we get 2 1 cp ≤ zn+1 ≤ p , for n ≥ 6. c From (33)-(35), (39)-(41) it follows that 1 xn+1 ≤ p2 , cp ≤ for n ≥ 8, yn c cp ≤

yn+1 1 ≤ p2 , zn c

912

for

(30)

the cyclic permutations of vari-

Write the equations in (2) as follows ¾ ½ xn+1 c 1 , = max , p yn yn yn1−p zn−1 ½ ¾ c 1 yn+1 = max , , zn zn zn1−p xpn−1 and

7

n ≥ 8,

(33)

(34)

(35)

(36)

(37)

(38)

(39)

(40)

(41)

(42) (43)

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8

and 1 zn+1 ≤ p2 , for n ≥ 8. xn c Using (43) in the first equation in (2) it follows that 2 1 cp ≤ xn+1 ≤ p3 , for n ≥ 9, c Using (44) in the second equation in (2) it follows that 2 1 cp ≤ yn+1 ≤ p3 , for n ≥ 9. c Using (42) in the third equation in (2) it follows that 2 1 for n ≥ 9. cp ≤ zn+1 ≤ p3 , c A simple inductive argument shows that cp ≤

cp

2k

≤ min{xn+1 , yn+1 , zn+1 } ≤ max{xn+1 , yn+1 , zn+1 } ≤

(44)

1 , cp2k+1

(45)

for n ≥ 6k + 3, and 2k+2

cp

≤ min{xn+1 , yn+1 , zn+1 } ≤ max{xn+1 , yn+1 , zn+1 } ≤

1 cp2k+1

,

(46)

for n ≥ 6k + 6. From (45), (46) and the assumption p ∈ (0, 1) the result easily follows by letting k → ∞. ¤ Acknowledgements This work was funded by the Deanship of Scientific Research (DSR), King Abdulaziz University, under grant No. (11-130/1433 HiCi). The authors, therefore, acknowledge technical and financial support of KAU. References [1] C. C ¸ inar, S. Stevi´ c and I. Yal¸cinkaya, On positive solutions of a reciprocal difference equation with minimum, J. Appl. Math. Comput. 17 (1-2) (2005), 307-314. [2] B. Iriˇ canin, On a higher-order nonlinear difference equation, Abstr. Appl. Anal. Vol. 2010, Article ID 418273, (2010), 8 pages. [3] B. Iriˇ canin, The boundedness character of two Stevi´ c-type fourth-order difference equations, Appl. Math. Comput. 217 (5) (2010), 1857-1862. [4] B. Iriˇ canin and S. Stevi´ c, On a class of third-order nonlinear difference equations, Appl. Math. Comput. 213 (2009), 479-483. [5] G. Karakostas, Asymptotic behavior of the solutions of the difference equation xn+1 = x2n f (xn−1 ), J. Differ. Equations Appl. 9 (6) (2003), 599-602. [6] C. M. Kent, M. Kustesky M, A. Q. Nguyen and B. V. Nguyen, Eventually periodic solutions of xn+1 = max{An /xn , Bn /xn−1 } when the parameters are two cycles, Dyn. Contin. Discrete Impuls. Syst. Ser. A Math. Anal. 10 (1-3) (2003), 33-49. [7] C. M. Kent and M. A. Radin, On the boundedness nature of positive solutions of the difference equation xn+1 = max{An /xn , Bn /xn−1 }, with periodic parameters, Dyn. Contin. Discrete Impuls. Syst. Ser. B Appl. Algorithms 2003, suppl., 11-15. [8] G. Papaschinopoulos, M. Radin and C. J. Schinas, Study of the asymptotic behavior of the solutions of three systems of difference equations of exponential form, Appl. Math. Comput. 218 (2012), 5310-5318. [9] G. Papaschinopoulos and C. J. Schinas, On a system of two nonlinear difference equations, J. Math. Anal. Appl. 219 (2) (1998), 415-426. [10] G. Papaschinopoulos and C. J. Schinas, On the behavior of the solutions of a system of two nonlinear difference equations, Comm. Appl. Nonlinear Anal. 5 (2) (1998), 47-59.

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[11] G. Papaschinopoulos and C. J. Schinas, Invariants for systems of two nonlinear difference equations, Differential Equations Dynam. Systems 7 (2) (1999), 181-196. [12] G. Papaschinopoulos and C. J. Schinas, Invariants and oscillation for systems of two nonlinear difference equations, Nonlinear Anal. TMA 46 (7) (2001), 967-978. [13] G. Papaschinopoulos and C. J. Schinas, Oscillation and asymptotic stability of two systems of difference equations of rational form, J. Difference Equat. Appl. 7 (2001), 601-617. [14] G. Papaschinopoulos and C. J. Schinas, On the dynamics of two exponential type systems of difference equations, Comput. Math. Appl. 64 (7) (2012), 2326-2334. [15] G. Papaschinopoulos, C. Schinas and V. Hatzifilippidis, Global behavior of the solutions of a max-equation and of a system of two max-equations, J. Comput. Anal. Appl. 5 (2) (2003), 237-254. [16] G. Papaschinopoulos, C. J. Schinas and G. Stefanidou, On the nonautonomous difference equation xn+1 = An + (xpn−1 /xqn ), Appl. Math. Comput. 217 (2011), 5573-5580. [17] G. Stefanidou, G. Papaschinopoulos, and C. J. Schinas, On a system of two exponential type difference equations, Commun. Appl. Nonlinear Anal. 17 (2) (2010), 1-13. [18] S. Stevi´ c, A global convergence results with applications to periodic solutions, Indian J. Pure Appl. Math. 33 (1) (2002), 45-53. Pk pi [19] S. Stevi´ c, A note on the difference equation xn+1 = i=0 (αi /xn−i ), J. Differ. Equations Appl. 8 (7) (2002), 641-647. Q Q2(k+1) [20] S. Stevi´ c, On the recursive sequence xn+1 = A/ ki=0 xn−i + 1/ j=k+2 xn−j , Taiwanese J. Math. 7 (2) (2003), 249-259. [21] S. Stevi´ c, On the recursive sequence xn+1 = αn + (xn−1 /xn ) II, Dynam. Contin. Discrete Impuls. Systems 10a (6) (2003), 911-917. [22] S. Stevi´ c, On the recursive sequence xn+1 = α + (xpn−1 /xpn ), J. Appl. Math. & Computing 18 (1-2) (2005) 229-234. [23] S. Stevi´ c, On the recursive sequence xn+1 = max{c, xpn /xpn−1 }, Appl. Math. Lett. 21 (8) (2008), 791-796. [24] S. Stevi´ c, Boundedness character of a class of difference equations, Nonlinear Anal. TMA 70 (2009), 839-848. [25] S. Stevi´ c, Boundedness character of two classes of third-order difference equations, J. Differ. Equations Appl. 15 (11-12) (2009), 1193-1209. [26] S. Stevi´ c, Global stability of a difference equation with maximum, Appl. Math. Comput. 210 (2009), 525-529. [27] S. Stevi´ c, On a class of higher-order difference equations, Chaos Solitons Fractals 42 (2009), 138-145. [28] S. Stevi´ c, Global stability of a max-type equation, Appl. Math. Comput. 216 (2010), 354-356. [29] S. Stevi´ c, On a generalized max-type difference equation from automatic control theory, Nonlinear Anal. TMA 72 (2010), 1841-1849. [30] S. Stevi´ c, Periodicity of max difference equations, Util. Math. 83 (2010), 69-71. [31] S. Stevi´ c, On a nonlinear generalized max-type difference equation, J. Math. Anal. Appl. 376 (2011), 317-328. [32] S. Stevi´ c, On a system of difference equations, Appl. Math. Comput. 218 (2011), 3372-3378. [33] S. Stevi´ c, Periodicity of a class of nonautonomous max-type difference equations, Appl. Math. Comput. 217 (2011), 9562-9566. [34] S. Stevi´ c, On a third-order system of difference equations, Appl. Math. Comput. 218 (2012), 7649-7654. [35] S. Stevi´ c, On some solvable systems of difference equations, Appl. Math. Comput. 218 (2012), 5010-5018. [36] S. Stevi´ c, On some periodic systems of max-type difference equations, Appl. Math. Comput. 218 (2012), 11483-11487. [37] S. Stevi´ c, Solutions of a max-type system of difference equations, Appl. Math. Comput. 218 (2012), 9825-9830. [38] S. Stevi´ c, On a symmetric system of max-type difference equations, Appl. Math. Comput. 219 (2013) 8407-8412. [39] S. Stevi´ c, On the system of difference equations xn = cn yn−3 /(an + bn yn−1 xn−2 yn−3 ), yn = γn xn−3 /(αn + βn xn−1 yn−2 xn−3 ), Appl. Math. Comput. 219 (2013), 4755-4764.

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10

´ ∗ , A. ALOTAIBI, N. SHAHZAD, AND M. A. ALGHAMDI STEVO STEVIC

[40] S. Stevi´ c, On the system xn+1 = yn xn−k /(yn−k+1 (an + bn yn xn−k )), yn+1 = xn yn−k /(xn−k+1 (cn + dn xn yn−k )), Appl. Math. Comput. 219 (2013), 4526-4534. [41] H. D. Voulov, Periodic solutions to a difference equation with maximum, Proc. Amer. Math. Soc. 131 (7) (2003) 2155-2160. [42] H. D. Voulov, On the periodic nature of the solutions of the reciprocal difference equation with maximum, J. Math. Anal. Appl. 296 (1) (2004) 32-43. [43] X. Yang and X. Liao, On a difference equation with maximum, Appl. Math. Comput. 181 (2006) 1-5. ´, Mathematical Institute of the Serbian Academy of Sciences, Knez Stevo Stevic Mihailova 36/III, 11000 Beograd, Serbia Department of Mathematics, King Abdulaziz University, P.O. Box 80203, Jeddah 21589, Saudi Arabia E-mail address: [email protected] Abdullah Alotaibi, Department of Mathematics, King Abdulaziz University, P.O. Box 80203, Jeddah 21859, Saudi Arabia E-mail address: [email protected] Naseer Shahzad, Department of Mathematics, King Abdulaziz University, P.O. Box 80203, Jeddah 21859, Saudi Arabia E-mail address: [email protected] Mohammed A. Alghamdi, Department of Mathematics, King Abdulaziz University, P.O. Box 80203, Jeddah 21859, Saudi Arabia E-mail address: [email protected]

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J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 16, NO.5, 2014, COPYRIGHT 2014 EUDOXUS PRESS, LLC

SOLUTION AND STABILITY OF A MULTI-VARIABLE FUNCTIONAL EQUATION JAE-HYEONG BAE AND WON-GIL PARK* Abstract. We obtain some combinatorial identities and investigate the monomial functional equation ( n ) ( n ) ∑h k−1 [ h−k − d h−k−1 ][f (kx + y) + f (kx − y)] k=1 (−1) (n) h +n!(−1) (1 + d)f (x) − h (1 − d)f (y) = 0, {n { , n : even 0 , n : even 2 where h := n+1 and d := . , n : odd 1 , n : odd 2

1. Introduction (n)Throughout this paper, let X and(nY) be vector spaces and n a positive integer. For an integer r, r is the binomial coefficient. Here r := 0 for r < 0 or r > n. For a mapping f : X → Y , consider the monomial functional equation: [( ) ( )] h ∑ [ ] n n k−1 (1.1) (−1) −d f (kx + y) + f (kx − y) h−k h−k−1 k=1 ( ) n h + n! (−1) (1 + d)f (x) − (1 − d)f (y) = 0, h {n { 0 , n : even , n : even 2 where h = hn := n+1 and d = dn := 1 , n : odd. 2 , n : odd For X = Y = R, the monomial f (x) = cxn is a solution of (1.1) for each n ≥ 2. If n is even, then the monomial functional equation (1.1) can be rewritten as n ( ) ( ) 2 ∑ [ ] n n n (−1)k−1 n (1.2) f (kx + y) + f (kx − y) + n!(−1) 2 f (x) − n f (y) = 0. − k 2 2 k=1

The quadratic functional equation f (x + y) + f (x − y) = 2f (x) + 2f (y) is the functional equation (1.2) for n = 2. If n ≥ 3 is odd, then the monomial functional equation (1.1) can be rewritten as [(

n+1

(1.3)

2 ∑

k=1

(−1)

k−1

) ( )] [ ] n+1 n n − n−1 f (kx + y) + f (kx − y) + n!(−1) 2 2f (x) = 0. n+1 2 −k 2 −k

2000 Mathematics Subject Classification. 39B52, 05A19. Key words and phrases. Monomial functional equation, Combinatorial identity. * Corresponding author. 1

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2

In 2002, K.-W. Jun and H.-M. Kim [6] solved the cubic functional equation f (2x + y) + f (2x − y) = 2f (x + y) + 2f (x − y) + 12f (x) which is the functional equation (1.3) for n = 3. Note that a mapping f : X → Y satisfies the functional equation (1.3) for n = 1 if and only if it is a Jensen mapping. The authors [2, 9] investigated some functional equations in order to induce the monomial functional equation (1.1). Some books [5, 7, 8, 10] provide useful information on functional equations associated with monomials. In this paper, we obtain some combinatorial identities and investigate the monomial functional equation (1.1). 2. Combinatorial Identities In this section, we prove some combinatorial identities needed to investigate the monomial functional equation (1.1). Lemma 2.1. Assume that n ≥ 2 is an even integer. Then n ) ( 2 ∑ n n k−1 (a) For n ≥ 4 , k n−2j = 0 for all j = 1, 2, · · · , − 1. (−1) n 2 2 −k k=1

(

n 2

(b) 2

∑

(−1)

k−1 n 2

k=1

) n n k n = n! (−1) 2 −1 . −k

{ } Proof. (a) Let n ≥ 4 be even and j ∈ 1, 2, · · · , n2 − 1 . Note that (

) n (−1) (2.1) k n−2j n − k 2 k=1 ) ( ) ( ) ( ( )( ) n n n n n n n−2j n−2j n−2j n−2j −1 = n 1 − n 2 + n 3 − · · · + (−1) 2 0 2 2 −1 2 −2 2 −3 n −1 ( )( 2 )n−2j ∑ n n n = (−1) 2 −1−k −k . k 2 k=0 (n) ( n ) Since k = n−k for all k = 1, 2, · · · , n2 , by shifting of indices, we gain n

2 ∑

k−1

(

n

2 ∑

(2.2)

(−1)k−1

k=1

n 2

) ( )( n ∑ n n n n )n−2j k n−2j = (−1)k− 2 −1 k− . −k k 2 n k= 2 +1

By equalities (2.1) and (2.2), we get −1 2 ∑ n

(2.3)

(−1)

n −1−k 2

k=0

( )( ( )( n )n−2j ∑ n n n )n−2j k− n −1 n 2 −k (−1) . = k− k 2 k 2 n k= 2 +1

Since n is even, we have (−1) 2 −1−k = (−1) 2 −1−k (−1)−n+2k = (−1)k− 2 −1 n

n

n

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A MULTI-VARIABLE FUNCTIONAL EQUATION

and

(n 2

−k

)n−2j

3

( n )n−2j = k− 2

for all k = 0, 1, · · · , n. By equality (2.3) and the above equalities, we obtain n −1 ( )( ( )( n 2 )n−2j )n−2j ∑ ∑ n n n n n n −1−k −1−k (−1) 2 (−1) 2 −k =2 −k . k 2 k 2

k=0

k=0

By equality (2.1) and the above equality, we see that (

n

2 ∑

(−1)k−1

k=1

n 2

) ( )( n )n−2j n n 1∑ n n k n−2j = (−1) 2 −1−k −k . −k 2 k 2 k=0

Since n is even and (−1)−k = (−1)−k (−1)2k = (−1)k for all k = 0, 1, · · · , n, the above equality implies that n ( ) ( )( n 2 ∑ ∑ n n 1 n n )n−2j k−1 n−2j −1 k 2 (−1) k = (−1) (−1) k − . n 2 k 2 2 −k k=1

k=0

Note that N ∑

(2.4)

k=0

( ) N (−1) (α + k)m−1 = 0 k k

for N ≥ m ≥ 1, N, m ∈ N (see [4]). Replacing N , α and m by n, − n2 and n − 2j + 1 in the combi( )( n ∑ n )n−2j k n natorial identity (2.4), respectively, we get (−1) k− = 0 for all j = 1, 2, · · · , n2 − 1. 2 k k=0 Hence we obtain the desired combinatorial identity. ( )n ( )n (b) Since n is even, we gain − n2 + k = n2 − k for all k = 0, 1, · · · , n. Shifting of indices, we get n n −1 ( )( ( )( 2 2 )n ∑ )n ∑ n n n n k k−1 (−1) −k = (−1) −k+1 k 2 k−1 2 k=0

k=1

and n ∑ k= n +1 2

By letting k =

n 2

n ( )( ( ) 2 )n ∑ n n n n +k 2 (−1) − +k = (−1) kn . n k 2 + k 2

k

k=1

− j + 1 for k = 1, · · · , n2 , we have (

n

2 ∑

k=1

(−1)

k−1

n )( ( ) 2 )n ∑ n n n n −k+1 = jn. (−1) 2 −j n k−1 2 − j 2

j=1

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4

By the above equality and shifting of indices, we obtain ) ( )( ( 2 2 )n ∑ ∑ n n n n n k−1 −k (−1) 2 k = (−1) − k + 1 n k−1 2 2 −k n

n

k=1

k=1 n −1 2

=

∑

(−1)k

k=0

( )( )n n n −k . k 2

Since n is even, the above equality implies that n

2 ∑

(2.5)

(−1)

n −k 2

( n 2

k=1

n −1 ) ( )( 2 )n ∑ n n n k n k = (−1) − +k . 2 −k k

k=0

Shifting of indices, we obtain ) ( ( )( n 2 )n ∑ ∑ n n n n n +k k k = + k . (−1) 2 (−1) − n 2 k n 2 +k n

(2.6)

k=1

Since (−1) 2 −k = (−1) 2 +k and we see that n

(2.7)

n

k= 2 +1

(

n

)

n −k 2

=

(

n

n +k 2

)

for all k = 1, · · · , n2 , by the equalities (2.5) and (2.6),

( ) ( ) ( ) 2 2 2 ∑ ∑ ∑ n n n n n n +k n −k n +k 2 2 2 2 (−1) k = (−1) k + (−1) kn n n n − k − k + k 2 2 2 n

n

n

k=1

k=1 n −1 2

k=1

( )( ( )( n )n )n ∑ ∑ n n k n k n (−1) = −k + (−1) − +k k 2 k 2 n k=0

( )( n )n ∑ n k n = (−1) − +k . 2 k

k= 2 +1

k=0

Note that (2.8)

n ∑ k=0

( ) n (−1) (α + k)n = n! (−1)n k k

for n ≥ 0 (see [4]). Replacing α by − n2 in the combinatorial identity (2.8), we have n ∑ k=0

( )( )n n n (−1) − + k = n! (−1)n . 2 k k



By (2.7) and the above equality, the desired combinatorial identity holds.

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5

Lemma 2.2. Assume that n ≥ 3 is an odd integer. Then n+1

(a)

2 ∑

(−1)

[(

) ( )] ( ) ( ) n n n−1 n−1 − n−1 = n−1 − n−3 . n+1 2 −k 2 −k 2 2

[(

) ( )] n n n−1 − n−1 k n−2j = 0 for all j = 1, 2, · · · , . n+1 2 2 −k 2 −k

k−1

k=1 n+1

(b)

2 ∑

(−1)

k−1

k=1

[(

n+1 2

(c)

∑

k−1

(−1)

k=1

) ( )] n−1 n n − n−1 k n = (−1) 2 n!. n+1 2 −k 2 −k

Proof. (a) Let m be a nonnegative integer. Since (−1)k−1 = (−1)2k−2 (−1)−k+1 = (−1)−k+1 , we gain ) m+1 ( ) ∑ n n m−k+1 (−1) (−1) = (−1) m−k+1 m−k+1 k=1 k=1 ( ) ( ) ( ) ∑ ( ) m n n n n = − + − · · · + (−1)m = (−1)k . 0 1 m k m

(

m+1 ∑

k−1

k=0

Since

( ) ( ) m ∑ n n−1 (−1)k = (−1)m (see [4]), using the above equality, we get k m k=0

m+1 ∑

(2.9)

( (−1)

k−1

k=1

Replacing m by

n−1 2

n m−k+1

)

( =

) n−1 . m

in equality (2.9), we have (

n+1

2 ∑

(2.10)

(−1)

k−1

k=1

n n+1 2 −k

)

( =

) n−1 n−1 2

. (

n−1

And replacing m by (n)

n−3 2

in equality (2.9), we obtain (

n+1

2 ∑

n (−1) n−1 −1 = 0, we have 2 −k k=1 the desired combinatorial identity holds. k−1

)

( =

n−1 n−3 2

920

)

2 ∑

k=1

k−1

(−1)

n n−1 2 −k

)

( =

) n−1 n−3 2

. Since

. By equality (2.10) and the above equality,

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6

} { . Consider equality (b) Let n is odd and j ∈ 1, 2, · · · , n−1 2 [(

) ( )] n n (−1) − n−1 k n−2j n+1 − k − k 2 2 k=1 [( ) ( )] [( ) ( )] n n n n n−2j = n+1 − n−1 1 − − n−1 2n−2j n+1 − 1 − 1 − 2 − 2 2 2 2 )2 ( )] [( ) ]( [( n−1 n n n + 1 )n−2j n n−2j 2 − n−1 3 − · · · + (−1) −0 + n+1 0 2 2 −3 2 −3 n+1

2 ∑

(2.11)

k−1

n−1

= Since

(n) k

2 ∑

(−1)

k=0

=

(

n n−k

)

n−1 −k 2

and

(

[( ) ( )]( )n−2j n n n+1 − −k . k k−1 2

n k+1

)

=

(

n n−k−1

for all k = 1, 2, · · · , n+1 2 , by shifting of indices, we gain

[(

) ( )] n n (−1) − n−1 k n−2j n+1 − k − k 2 2 k=1 [( ) ( )]( n ∑ n n n − 1 )n−2j k− n+1 2 = (−1) . − k− 2 k k+1 n+1 n+1

(2.12)

)

2 ∑

k=

k−1

2

By the equalities (2.11) and (2.14), we get [( ) ( )]( )n−2j n n n+1 −k (−1) − 2 k k−1 k=0 [( ) ( )]( n ∑ n n n − 1 )n−2j k− n+1 2 = (−1) − k− . k k+1 2 n+1 n−1

2 ∑

k=

n−1 −k 2

2

By shifting of indices and the above equality, we have [( ) ( )]( n+1 ∑ n n n + 1 )n−2j k− n+1 2 (−1) − k− k k−1 2 n+3 k=

=

2

n ∑

=

(−1)

k− n+1 2

[( ) ( )]( n n n − 1 )n−2j − k− k k+1 2

(−1)

n−1 −k 2

[( ) ( )]( )n−2j n n n+1 − −k . k k−1 2

k= n+1 2 n−1

=

2 ∑

k=0

) ( )]( n n n − 1 )n−2j − k− k+1 k 2

(−1)

k= n+1 2 n ∑

[(

k− n−1 2

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A MULTI-VARIABLE FUNCTIONAL EQUATION

7

So we obtain that [( ) ( )]( n ∑ n+1 n n n + 1 )n−2j (2.13) (−1)k− 2 − k− k k−1 2 n+3 k=

2

n+1 ∑

=

(−1)

k− n+1 2

k= n+3 2 n−1

2 ∑

=

(−1)

n−1 −k 2

k=0

[( ) ( )]( ( n + 1 )n−2j n+1 n n n + 1 )n−2j − k− + (−1) 2 k k−1 2 2

[( ) ( )]( )n−2j ( n + 1 )n−2j n+1 n n n+1 − −k + (−1) 2 . k k−1 2 2

Since n is odd, we have (−1) and

n−1 −k 2

(

= −(−1)

n+1 −k 2

n−1 −k 2

)n−2j

(−1)−n+2k = −(−1)k−

n+1 2

) ( n + 1 n−2j =− k− 2

for all k = 0, 1, · · · , n. By the above equalities and using equality (2.15), we have [( ) ( )]( n ∑ n n n + 1 )n−2j k− n+1 2 (−1) − k− k k−1 2 k=0

[( ) ( )]( n n n + 1 )n−2j = − k− (−1) k k−1 2 k=0 [( ) ( )]( n ∑ n+1 n n n + 1 )n−2j (−1)k− 2 − k− + k k−1 2 n+3 n−1

2 ∑

k− n+1 2

k=

2

n−1 2

=

∑

(−1)

k− n+1 2

k=0

[( ) ( )]( n n n + 1 )n−2j − k− 2 k k−1

n−1

+

2 ∑

(−1)

n−1 −k 2

k=0 n−1

=2

2 ∑

(−1)

n−1 −k 2

k=0

[( ) ( )]( )n−2j ( n + 1 )n−2j n+1 n n n+1 − −k + (−1) 2 k k−1 2 2

[( ) ( )]( )n−2j ( n + 1 )n−2j n+1 n n n+1 − −k + (−1) 2 . k k−1 2 2

By equality (2.11) and the above equality, we gain [( ) ( )]( n ∑ n n n + 1 )n−2j k− n+1 2 (2.14) (−1) − k− k k−1 2 k=0

[(

n+1

=2

2 ∑

k=1

(−1)

k−1

) ( )] ( n + 1 )n−2j n+1 n n − n−1 k n−2j + (−1) 2 . n+1 2 2 −k 2 −k

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8

and n − 2j + 1 in the combinatorial identity (2.4), respectively, Replacing N , α and m by n, − n+1 (n)( ) 2 ∑n n+1 n−2j k we get k=0 (−1) k k − 2 = 0. By the above equality, we obtain n ∑

(2.15)

( )( n n + 1 )n−2j k− 2 k ( )( n ∑ n + 1 )n−2j k n = 0. (−1) k− 2 k

(−1)k−

k=0

=(−1)−

n+1 2

n+1 2

k=0

And replacing N , α and m by n, 1− n+1 and n−2j +1 in the combinatorial identity (2.4), respectively, (n)( )2 ∑n n+1 n−2j k we have k=0 (−1) k k + 1 − 2 = 0 and so we obtain ( )( ( )( n n + 1 )n−2j n + 1 )n−2j n n (−1) = −(−1) n+1− k+1− k 2 n 2 k=0 ( n + 1 )n−2j =(−1)n+1 . 2 (n) By shifting of indices, the fact that −1 = 0 and the above equality, we see that n−1 ∑

k

( )( n ∑ n n + 1 )n−2j k− n+1 2 (−1) − k− k−1 2 k=0 ( )( n ∑ n n + 1 )n−2j k − n+1 2 (−1) k− = − (−1) k−1 2 k=0 ( ) n ( n+1 ∑ n n + 1 )n−2j (−1)k k− = − (−1)− 2 k−1 2 k=1 ( )( n−1 n+1 ∑ n n + 1 )n−2j (−1)k+1 k+1− = − (−1)− 2 k 2 k=0 ( )( n−1 ∑ n n + 1 )n−2j k − n+1 (−1) k+1− =(−1) 2 k 2 k=0 ( ) n+1 n + 1 n−2j =(−1) 2 . 2 By the equalities (2.14) and (2.15) and the above equality, we obtain the desired combinatorial identity. (c) Shifting of indices, we gain

(2.16)

n+1 ( )( ( ) 2 )n ∑ n−1 n n−1 n (−1) +k = kn − (−1) 2 +k n−1 k 2 + k 2 n+1

n ∑ k=

k

k=1

2

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A MULTI-VARIABLE FUNCTIONAL EQUATION

9

and n−3

2 ∑

(2.17)

k=0

Setting j =

n−1 ( )( ( )( 2 )n ∑ )n n n − 1 n n+1 k k−1 (−1) − +k = (−1) − +k . k 2 k−1 2

k=1

− k for k = 1, · · · , n−1 2 in the right hand side of equality (2.17), we get

n+1 2

(

n−1

2 ∑

(−1)k−1

n−1 )( ( ) 2 )n ∑ n−1 n n+1 n −j +k = − (−1) 2 (−j)n n−1 2 k−1 − j 2

j=1

k=1

n−1

=(−1)

n−1

2 ∑

(−1)

n+1 −j 2

(

j=1

) n jn. n−1 − j 2

Since n is odd, we have (

n−1

2 ∑

(2.18)

(−1)k−1

n−1 )( ( ) 2 )n ∑ n+1 n n+1 n −j 2 − +k = (−1) jn n−1 k−1 2 2 −j

j=1

k=1

By the equalities (2.16), (2.17) and (2.18), we obtain n−1

2 ∑

(−1)

n+1 −j 2

j=1

∑

(−1)k

k=0

=

n ∑ k=0

n+1 ( ) ) 2 ∑ n−1 n n +k n 2 j + (−1) kn n−1 n−1 − j + k 2 2

k=1

n−3 2

=

(

( )( ( )( n )n )n ∑ n n−1 n n−1 +k + +k − (−1)k − 2 2 k k n+1 k=

( )( )n n−1 k n (−1) − +k . k 2

2

Replacing α by − n−1 2 in the combinatorial identity (2.8), we see that n−1

2 ∑

(−1)

n+1 −k 2

(

k=1

Since (−1)

n+1 −k 2

k=1

= (−1)

n+1 +k 2

n−1

2 ∑

(−1)

n+1 +k 2

and

(

k=1

Multiplying (−1)

n+1 ) ( ) 2 ∑ n−1 n n n +k k + k n = (−1)n n!. (−1) 2 n−1 n−1 − k + k 2 2

−n+1 2

k=1

n

n−1 +k 2

)

=

(

n

n+1 −k 2

)

for all k = 1, · · · , n+1 2 , we have

n+1 ) ( ) 2 ∑ n−1 n n n +k k + (−1) 2 k n = (−1)n n!. n−1 n+1 − k − k 2 2

k=1

by the both hand sides in the above equality, we obtain (

n−1

2 ∑

(

(−1)k+1

n+1 ) ( ) 2 ∑ n+1 n n n k k + (−1) n+1 k n = (−1) 2 n!. n−1 2 −k 2 −k

k=1

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10

k+1 = (−1)k−1 for all integers Dividing ( n−1 ) by the both hand sides in the above equality and using (−1) k and −1 = 0, we see that

[(

n+1

2 ∑

(−1)

k−1

k=1

) ( )] n−1 n n − n−1 k n = (−1) 2 n!. n+1 2 −k 2 −k 

3. Solution of Monomial Functional Equations The following lemma is needed to investigate the monomial functional equations (1.2) and (1.3). Lemma 3.1. Let n ≥ 2 and let f : X → Y be a mapping satisfying the monomial functional equation (1.1) for all x, y ∈ X. If n is even, then so is f . And, if n is odd, then so is f . Proof. Assume that n is even. Then (1.1) becomes (1.2). Thus, putting in (1.2) x = y = 0, we obtain ( ) ( ) 2 ∑ n n n k−1 2 (−1) f (0) + n!(−1) 2 f (0) − n f (0) = 0. n 2 −k 2 k=1 ( ) ( ) ( ) ∑n n for k = 0, · · · , n, we get Since k=0 (−1)k nk = 0 and nk = n−k n

(

n

2

(3.1)

2 ∑

(−1)k−1

n 2

k=1

n −k

) =

( ) n n 2

.

n 2

Thus we have n!(−1) f (0) = 0. Hence we obtain f (0) = 0. Putting in (1.2) x = 0, we get (

) ( ) n n [f (y) + f (−y)] − n n f (y) = 0 − k 2 2 k=1 ( ) ] n [ for all y ∈ X. By equality (3.1) and the above equation, we have n f (−y) − f (y) = 0 for all n

2 ∑

(−1)k−1

2

y ∈ X. Therefore f is even. Suppose that n is odd. Setting x = y = 0 in the functional equation (1.3), we gain [(

n+1

2 ∑

(−1)

k−1

k=1

By Lemma 2.2 (a), we get

[(

) ( )] n+1 n n − n−1 f (0) + n!(−1) 2 f (0) = 0. n+1 2 −k 2 −k

) n−1 n−1 2

( −

) n−1 n−3 2

+ n!(−1)

n+1 2

] f (0) = 0.

Thus we have f (0) = 0. Taking x = 0 in the functional equation (1.3), we obtain that [(

n+1

2 ∑

k=1

(−1)

k−1

) ( )] [ ] n n − n−1 f (y) + f (−y) = 0 n+1 2 −k 2 −k

925

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A MULTI-VARIABLE FUNCTIONAL EQUATION

11

for all y ∈ X. By Lemma 2.2 (a), we see that [(

) n−1 n−1 2

( −

)] n−1 [ n−3 2

] f (y) + f (−y) = 0

for all y ∈ X. Therefore f is odd.



In the following theorem, we investigate the solution of the monomial functional equation (1.2).

Theorem 3.2. Let n ≥ 2 be even. If a mapping f : X → Y satisfies the functional equation (1.2) for all x, y ∈ X, then there is a symmetric mapping Sn : X n → Y satisfying Sn (x, · · · , x) = 21n f (2x) for all x ∈ X. On the contrary, if Sn : X n → Y is a symmetric multi-additive mapping and a mapping f : X → Y satisfies f (x) = Sn (x, · · · , x) for all x ∈ X, then f satisfies the functional equation (1.2) for all x, y ∈ X. Proof. Suppose that a mapping f : X → Y satisfies equation (1.2) for all x, y ∈ X. By Lemma 3.1, f is even. Define the mapping Sn : X n → Y by Sn (x1 , · · · , xn ) :=

(3.2)

1 n! 2n−1

∑

σ2 · · · σn f (x1 + σ2 x2 + · · · + σn xn )

σ2 ,··· ,σn ∈{1,−1}

for all x1 , · · · , xn ∈ X. For 2 ≤ j ≤ n, we gain (3.3) Sn (xj , x2 , · · · , xj−1 , x1 , xj+1 , · · · , xn ) ∑ ) ( 1 = n−1 σ2 · · · σn f x1 + σ2 x2 + · · · + σn xn + (σj − 1)(x1 − xj ) n!2 σ2 ,··· ,σn ∈{1,−1} [ ∑ 1 = n−1 σ2 · · · σj−1 (1)σj+1 · · · σn n!2 σ2 ,··· ,σj−1 ,σj+1 ,··· ,σn ∈{1,−1} ) ( f x1 + σ2 x2 + · · · + σj−1 xj−1 + xj + σj+1 xj+1 + · · · + σn xn ∑ + σ2 · · · σj−1 (−1)σj+1 · · · σn σ2 ,··· ,σj−1 ,σj+1 ,··· ,σn ∈{1,−1}

(

)

]

f x1 + σ2 x2 + · · · + σj−1 xj−1 − xj + σj+1 xj+1 + · · · σn xn − 2x1 + 2xj [ ∑ ( ) 1 σ2 · · · σj−1 σj+1 · · · σn f x1 + σ2 x2 + · · · + σn xn + (1 − σj )xj = n−1 n!2 σ2 ,··· ,σj−1 ,σj+1 ,··· ,σn ∈{1,−1}

−

∑

(

σ2 · · · σj−1 σj+1 · · · σn f − x1 + σ2 x2 + · · · + σn xn + (1 − σj ) xj

σ2 ,··· ,σj−1 ,σj+1 ,··· ,σn ∈{1,−1}

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]

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JAE-HYEONG BAE AND WON-GIL PARK

12

for all x1 , · · · xn ∈ X. Since f is even, we get (3.4)

∑

( ) σ2 · · · σj−1 σj+1 · · · σn f − x1 + σ2 x2 + · · · + σn xn + (1 − σj ) xj

σ2 ,··· ,σj−1 ,σj+1 ,··· ,σn ∈{1,−1}

∑

=

( ) σ2 · · · σj−1 σj+1 · · · σn f x1 − σ2 x2 − · · · − σn xn − (1 − σj )xj

σ2 ,··· ,σj−1 ,σj+1 ,··· ,σn ∈{1,−1}

∑

=

( ) (−σ2 ) · · · (−σj−1 )(−σj+1 ) · · · (−σn )f x1 − σ2 x2 − · · · − σn xn − (1 − σj )xj

σ2 ,··· ,σj−1 ,σj+1 ,··· ,σn ∈{1,−1}

∑

=

( ) τ2 · · · τj−1 τj+1 · · · τn f x1 + τ2 x2 + · · · + τn xn − (1 + τj )xj

τ2 ,··· ,τj−1 ,τj+1 ,··· ,τn ∈{1,−1}

for all x1 , · · · , xn ∈ X. By the equalities (3.3) and (3.4), we have (3.5) Sn (xj , x2 · · · , xj−1 , x1 , xj+1 , · · · , xn ) [ ∑ ) ( 1 = σ2 · · · σj−1 σj+1 · · · σn f x1 + σ2 x2 + · · · + σn xn + (1 − σj )xj n−1 n!2 σ2 ,··· ,σj−1 ,σj+1 ,··· ,σn ∈{1,−1} ] ∑ ) ( − τ2 · · · τj−1 τj+1 · · · τn f x1 + τ2 x2 + · · · + τn xn − (1 + τj )xj τ2 ,··· ,τj−1 ,τj+1 ,··· ,τn ∈{1,−1}

1 = n! 2n−1

∑

σ2 · · · σn f (x1 + σ2 x2 + · · · + σn xn )

σ2 ,··· ,σn ∈{1,−1}

= Sn (x1 , · · · , xn ) for all x1 , · · · , xn ∈ X. For 2 ≤ j < k ≤ n, putting ϵi := σi for i ∈ {2, · · · , n} \ {j, k}, ϵj := σk and ϵk := σj , we obtain (3.6)

Sn (x1 , · · · , xj−1 , xk , xj+1 , · · · , xk−1 , xj , xk+1 , · · · , xn ) ∑ ( ) 1 σ2 · · · σn f x1 + σ2 x2 + · · · + σn xn + (σk − σj )xj + (σj − σk )xk = n−1 n! 2 σ2 ,··· ,σn ∈{1,−1}

1 = n! 2n−1

=

1 n! 2n−1

∑

( ϵ2 · · · ϵn f x1 + ϵ2 x2 + · · · + ϵj−1 xj−1 + ϵk xj + ϵj+1 xj+1 + · · · +

ϵ2 ,··· ,ϵn ∈{1,−1}

ϵk−1 xk−1 + ϵj xk + ϵk+1 xk+1 + · · · + ϵn xn + (ϵj − ϵk )xj + (ϵk − ϵj )xk ∑ ϵ2 · · · ϵn f (x1 + ϵ2 x2 + · · · + ϵn xn )

)

ϵ2 ,··· ,ϵn ∈{1,−1}

=Sn (x1 , · · · , xn ) for all x1 , · · · xn ∈ X. Hence Sn is symmetric. By the proof of Lemma 3.1, we get f (0) = 0. Letting y = 0 in (1.2), we have (

n

2 ∑

k=1

k−1

(−1)

n 2

) n n n! f (kx) = (−1) 2 +1 f (x) −k 2

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A MULTI-VARIABLE FUNCTIONAL EQUATION

13

for all x ∈ X. By the above equality, we have Sn (x, · · · , x) 1 = n! 2n−1

∑ σ2 ,··· ,σn ∈{1,−1}

( ) n−1 ∑ 1 k n−1 σ2 · · · σn f ((1 + σ2 + · · · + σn )x) = (−1) f ((n − 2k)x) n! 2n−1 k k=0

(

( ) n−1 ∑ ∑ 1 1 n−1 f (nx) k n−1 k = + (−1) f ((n − 2k)x) + (−1) f ((2k − n)x) n! 2n−1 n! 2n−1 k n! 2n−1 n n−k−1 n −1 2

)

k=1

k= 2 +1

(

( ) ∑ ∑ n n f (nx) 1 n−1 1 n−1 −k −k = + (−1) 2 f (2kx) + (−1) 2 f (2kx) n n n! 2n−1 n! 2n−1 n! 2n−1 2 −k 2 −k−1 n −1 2

)

n −1 2

k=1

n 2

= =

∑ n 1 (−1) 2 −k n! 2n−1

k=1

( n 2

k=1 n +1 2

( ) ∑ n n (−1) k−1 (−1) f (2kx) = f (2kx) n −k n! 2n−1 2 −k )

n +1 2

n 2

k=1

n 1 (−1) n! (−1) 2 +1 f (2x) = n f (2x) n! 2n−1 2 2

On the contrary, suppose that there exists a symmetric multi-additive mapping Sn : X n → Y such that f (x) = Sn (x, · · · , x) for all x ∈ X. By Section 11.1 in [1], it suffices to show for n ≥ 4. By equality (3.1) and Lemma 2.1 (b), we obtain that (

n

2 ∑

k−1

(−1)

n 2

k=1

( ) ) [ ] n n n 2 f (kx + y) + f (kx − y) + n!(−1) f (x) − n f (y) −k 2

(

) [ ] n (−1) = Sn (kx + y, · · · , kx + y) + Sn (kx − y, · · · , kx − y) n 2 −k k=1 ( ) n n 2 + n!(−1) Sn (x, · · · , x) − n Sn (y, · · · , y) n

2 ∑

k−1

=2

∑

(−1)k−1

n 2

k=1

−

( ) n n 2

=2

n )∑ ) 2 ( n n n Sn (kx, · · · , kx, y, · · · , y ) + n!(−1) 2 Sn (x, · · · , x) | {z } | {z } −k 2j

j=0

(

(−1)k−1

k=1

n 2

n −k

+2

k=1

2j

n −1 ( ) )∑ 2 n Sn (kx, · · · , kx, y, · · · , y ) | {z } | {z } 2j

(

n 2

∑

n−2j

Sn (y, · · · , y)

n

2 ∑

2

(

n 2

(−1)

k−1 n 2

j=0

n−2j

2j

) ( ) n n n 2 Sn (y, · · · , y) + n!(−1) Sn (x, · · · , x) − n Sn (y, · · · , y) −k 2

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14

(

n

=2

2 ∑

(−1)k−1

n 2

k=1

n −k

n −1 ( ) )∑ 2 n n−2j k Sn (x, · · · , x, y, · · · , y ) | {z } | {z } 2j

j=0

n−2j

2j

( ) ( )] [ ∑ n n n k−1 (−1) − n Sn (y, · · · , y) + n!(−1) 2 Sn (x, · · · , x) + 2 n − k 2 2 n 2

k=1

(

n 2

=2

∑

(−1)k−1

n 2

k=1

∑∑

n −1 ( ) )∑ 2 n n n−2j k Sn (x, · · · , x, y, · · · , y ) + n!(−1) 2 Sn (x, · · · , x) | {z } | {z } 2j

(

n −1 2

n 2

=2

n −k

(−1)k−1

n 2

k=1 j=1

j=0

n−2j

2j

)( ) n n n−2j k Sn (x, · · · , x, y, · · · , y ) | {z } | {z } −k 2j n−2j

2j

( ) ∑ n n k−1 +2 (−1) k n Sn (x, · · · , x) + n!(−1) 2 Sn (x, · · · , x) n 2 −k n 2

k=1

=2

(

n −1 2

n 2

∑∑

k−1

(−1)

n 2

k=1 j=1

n −k

)( ) n n−2j k Sn (x, · · · , x, y, · · · , y ) | {z } | {z } 2j n−2j

∑ (n)

∑

j=1

k=1

n −1 2

=2

(

n 2

Sn (x, · · · , x, y, · · · , y ) | {z } | {z } 2j n−2j

2j

(−1)k−1

n 2

2j

) n k n−2j −k

By Lemma 2.1 (a), we see that n ( ) ( ) 2 ∑ [ ] n n n k−1 (−1) f (kx + y) + f (kx − y) + n!(−1) 2 f (x) − n f (y) = 0. n 2 −k 2 k=1

 Theorem 3.3. Let n ≥ 3 be odd. If a mapping f : X → Y satisfies the functional equation (1.3) for all x, y ∈ X, then there is a symmetric mapping Sn : X n → Y satisfying Sn (x, · · · , x) = 21n f (2x) for all x ∈ X. On the contrary, if Sn : X n → Y is a symmetric multi-additive mapping and a mapping f : X → Y satisfies f (x) = Sn (x, · · · , x) for all x ∈ X, then f satisfies the functional equation (1.3) for all x, y ∈ X. Proof. Suppose that a mapping f : X → Y satisfies equation (1.3) for all x, y ∈ X. By Lemma 3.1, f is odd. Define the mapping Sn : X n → Y by equality (3.2) for all x1 , · · · , xn ∈ X. For 2 ≤ j ≤ n, we gain equality (3.3) for all x1 , · · · xn ∈ X. Since f is odd, we get ∑ ( ) σ2 · · · σj−1 σj+1 · · · σn f − x1 + σ2 x2 + · · · + σn xn + (1 − σj ) xj σ2 ,··· ,σj−1 ,σj+1 ,··· ,σn ∈{1,−1}

=−

∑

( ) σ2 · · · σj−1 σj+1 · · · σn f x1 − σ2 x2 − · · · − σn xn − (1 − σj )xj

σ2 ,··· ,σj−1 ,σj+1 ,··· ,σn ∈{1,−1}

=

∑

( ) (−σ2 ) · · · (−σj−1 )(−σj+1 ) · · · (−σn )f x1 − σ2 x2 − · · · − σn xn − (1 − σj )xj

σ2 ,··· ,σj−1 ,σj+1 ,··· ,σn ∈{1,−1}

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A MULTI-VARIABLE FUNCTIONAL EQUATION

∑

=

15

( ) τ2 · · · τj−1 τj+1 · · · τn f x1 + τ2 x2 + · · · + τn xn − (1 + τj )xj

τ2 ,··· ,τj−1 ,τj+1 ,··· ,τn ∈{1,−1}

for all x1 , · · · , xn ∈ X. By equality (3.3) and the above equality, we have equality (3.5) for all x1 , · · · , xn ∈ X. For 2 ≤ j < k ≤ n, putting ϵi := σi for i ∈ {2, · · · , n} \ {j, k}, ϵj := σk and ϵk := σj , we obtain equality (3.6) for all x1 , · · · xn ∈ X. Hence Sn is symmetric. By a similar method to the proof of Theorem 3.2, we obtain Sn (x, · · · , x) = 21n f (2x) for all x ∈ X. On the contrary, suppose that there exists a symmetric multi-additive mapping Sn : X n → Y such that f (x) = Sn (x, · · · , x) for all x ∈ X. By Lemma 2.2 (c), we obtain that [(

n+1

2 ∑

(−1)

k−1

k=1

) ( )] [ ] n+1 n n − n−1 f (kx + y) + f (kx − y) + n!(−1) 2 2f (x) n+1 2 −k 2 −k

[(

) ( )] [ n n = (−1) − n−1 Sn (kx + y, · · · , kx + y) n+1 2 −k 2 −k k=1 ] n+1 + Sn (kx − y, · · · , kx − y) + n!(−1) 2 2Sn (x, · · · , x) n+1

2 ∑

k−1

[(

n+1

=2

2 ∑

(−1)

k−1

j=0

k=1

+ n!(−1)

n+1 2

=2

(−1)k−1

+ n!(−1)

n+1 2

[( (−1)

k−1

k=1 j=1

+2

[( (−1)

k−1

k=1

+ n!(−1)

n+1 2

=2

) ( )] n n − n−1 k n Sn (x, · · · , x) n+1 − k − k 2 2

[( (−1)

k=1 j=1

∑(n)

k−1

) ( )]( ) n n n n−2j − n−1 k Sn (x, · · · , x, y, · · · , y ) n+1 | {z } | {z } 2j 2 −k 2 −k n−2j

n−1 2

=2

j=1

2j

2j

2Sn (x, · · · , x)

n+1 n−1

2 ∑ 2 ∑

2j

) ( )]( ) n n n n−2j − n−1 k Sn (x, · · · , x, y, · · · , y ) n+1 | {z } | {z } 2j 2 −k 2 −k n−2j

n+1 2

∑

n−2j

2Sn (x, · · · , x)

n+1 n−1 2 2

∑∑

2j

n−1 ) ( )] ∑ ) 2 ( n n n n−2j − n−1 k Sn (x, · · · , x, y, · · · , y ) n+1 | {z } | {z } 2j 2 −k 2 −k

j=0

k=1

=2

n−2j

2Sn (x, · · · , x)

[(

n+1 2

∑

n−1 ) ( )] ∑ ) 2 ( n n n − Sn (kx, · · · , kx, y, · · · , y ) n+1 n−1 | {z } | {z } 2j 2 −k 2 −k

[(

n+1 2

Sn (x, · · · , x, y, · · · , y ) | {z } | {z } n−2j

2j

∑

(−1)

k=1

930

k−1

2j

) ( )] n n − n−1 k n−2j n+1 − k − k 2 2

BAE ET AL 916-931

J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 16, NO.5, 2014, COPYRIGHT 2014 EUDOXUS PRESS, LLC

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16

By Lemma 2.2 (b), we see that [(

n+1

2 ∑

(−1)

k−1

k=1

) ( )] [ ] n+1 n n − n−1 f (kx + y) + f (kx − y) + n!(−1) 2 2f (x) = 0. n+1 2 −k 2 −k 

Acknowledgment This research was supported by Basic Science Research Program through the National Research Foundation of Korea(NRF) funded by the Ministry of Education, Science and Technology(grant number 2013026492) References [1] J. Aczel and J. Dhombres, Functional Equations in Several Variables, Cambridge Univ. Press, Cambridge, 1989. [2] J.-H. Bae and W.-G. Park, On stability of a functional equation with n variables, Nonlinear Anal. 64 (2006), 856–868. [3] J.K. Chung and P.K. Sahoo, On the general solution of a quartic functional equation, Bull. Korean Math. Soc. 40 (2003), 565–576. [4] I.S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series, and Products, Elsevier Inc., Amsterdam, 2007. [5] D.H. Hyers, G. Isac and Th.M. Rassias, Stability of Functional Equations in Several Variables, Birkhauser, Boston, 1998. [6] 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. [7] S.-M. Jung, Hyers-Ulam-Rassias Stability of Functional Equations in Nonlinear Analysis, Springer, New York, 2011. [8] Pl. Kannappan, Functional Equations and Inequalities with Applications, Springer, New York, 2009. [9] W.-G. Park and J.-H. Bae, On a bi-quadratic functional equation and its stability, Nonlinear Anal. 62 (2005), 643–654. [10] P.K. Sahoo and Pl. Kannappan, Introduction to Functional Equations, Chapman & Hall / CRC Press, Boca Raton, Florida, 2011. [11] L. Sz´ekelyhidi, Convolution Type Functional Equation on Topological Abelian Groups, World Scientific, Singapore, 1991. Jae-Hyeong Bae, Humanitas College, Kyung Hee University, Yongin 446-701, Republic of Korea E-mail address: [email protected] Won-Gil Park, Department of Mathematics Education, Mokwon University, Daejeon 302-729, Republic of Korea E-mail address: [email protected]

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On the Behaviour of the Solutions of Difference Equation Systems (1)

Y. Yazlik∗, (2,3) E. M. Elsayed†‡, (4) N. Taskara§¶

ABSTRACT In this paper, we investigate the behaviour of the solutions of difference equations systems yn−5 xn−5 xn+1 = , yn+1 = , ±1 + yn−1 xn−3 yn−5 ±1 + xn−1 yn−3 xn−5

where the initial values are arbitrary real numbers such that the denominator is always nonzero. Keywords: System of difference equation, Explicit solutions, Periodicity. AMS Classification: 39A10, 39A12.

1 Introduction Our aim in this study is to investigate the periodic character of all solutions of the following difference equations systems xn+1 =

yn−5 xn−5 , yn+1 = , n ∈ N0 , ±1 + yn−1 xn−3 yn−5 ±1 + xn−1 yn−3 xn−5

(1.1)

where the initial conditions are arbitrary real numbers. Throughout this paper, we will assume that our solutions are well-defined, that is, the denominator is always nonzero. Also, we take 1, n instead of 1, 2, . . . , n. Nonlinear difference equations have long interested mathematics as well as other sciences. They play a key concept in many applications such as the natural model of a discrete process. There have been many recent investigations and interest in the field of nonlinear difference equations by several authors [1–22]. For instance, in [16], Stevic obtained behaviour of the solutions of the following difference equation xn+1 =

xn−1 . 1 + xn xn−1

Karatas et al., in [12], gave that the solution of the difference equation xn+1 =

xn−5 . 1 + xn−2 xn−5

∗(1)

Nevsehir University, Faculty of Science and Art, Department of Mathematics, Nevsehir, Turkey. King Abdulaziz University, Faculty of Science, Mathematics Department, P. O. Box 80203, Jeddah 21589, Saudi Arabia. ‡(3) Department of Mathematics, Faculty of Science, Mansoura University, Mansoura 35516, Egypt. §(4) Selcuk University, Science Faculty, Department of Mathematics, Konya,TURKEY. ¶ e mail: [email protected], [email protected], [email protected]. †(2)

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In [6], Elsayed dealed with the dynamics and found the solution of the following rational recursive sequences xn−5 xn+1 = . ±1 + xn−1 xn−3 xn−5 Grove et al., in [11], have studied existence and behavior of solutions of the rational system xn+1 =

a b c d + , yn+1 = + . xn yn xn yn

Ozban [15], has investigated the positive solutions of the rational difference system xn+1 =

1 yn , yn+1 = . yn−k xn−m yn−m−k

In [13], Kurbanli et al. have studied the positive solutions of the system of difference equations xn+1 =

xn−1 yn−1 , yn+1 = . yn xn−1 + 1 xn yn−1 + 1

In [17], Touafek and Elsayed dealed with the periodic nature and the form of the solutions of the following systems of rational difference equations xn+1 =

xn−3 yn−3 , yn+1 = . ±1 ± xn−3 yn−1 ±1 ± yn−3 xn−1

Similar nonlinear system of rational difference equations were investigated; see [4,5,8,18-22].

2 On System xn+1 =

yn−5 , 1+yn−1 xn−3 yn−5

yn+1 =

xn−5 1+xn−1 yn−3 xn−5

In this section we study the solutions of the difference equation system xn+1 =

yn−5 xn−5 , yn+1 = , n ∈ N0 . 1 + yn−1 xn−3 yn−5 1 + xn−1 yn−3 xn−5

(2.1)

Theorem 2.1. Suppose that initial conditions are any positive real numbers. Let {xn , yn }∞ n=1 §n¨ be a solution of system (2.1). For k = 6 , all solutions of system (2.1) ; are given by: i) If k is odd and p is equal to 2 or 3,

xn = y−p

n−i bY 6 c

j=0

n−i ¢ ¢ ¡ § ¨ ¡ § ¨ bY 6 c 1 + 3j + 2i − 1 α 1 + 3j + 2i − 1 β ¡ § ¨¢ ¡ § i ¨¢ , yn = x−p . β 1 + 3j + 2i α 1 + 3j + 2 j=0

ii) If k is odd and p is equal to otherwise,

xn = y−p

n−i bY 6 c

j=0

n−i ¢ ¢ ¡ § ¨ ¡ § ¨ bY 6 c 1 + 3j + 2i − 1 β 1 + 3j + 2i − 1 α ¡ § ¨¢ ¡ § ¨¢ , yn = x−p . 1 + 3j + 2i β 1 + 3j + 2i α j=0

iii) If k is even and p is equal to 2 or 3

xn = x−p

n−i bY 6 c

j=0

n−i ¢ ¢ ¡ § ¨ ¡ § ¨ bY 6 c 1 + 3j + 2i − 1 β 1 + 3j + 2i − 1 β ¡ § ¨¢ ¡ § ¨¢ , yn = y−p . 1 + 3j + 2i β 1 + 3j + 2i β j=0

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iv) If k is even and p is equal to otherwise xn = x−p

n−i ¢ ¢ ¡ § ¨ ¡ § ¨ bY 6 c 1 + 3j + 2i − 1 α 1 + 3j + 2i − 1 α ¡ § ¨¢ ¡ § i ¨¢ , yn = y−p , 1 + 3j + 2i α 1 + 3j + α 2 j=0

n−i bY 6 c

j=0

¢ ¡ ¢ ¡ where for i = 1, 6 , n − i ≡ 0 (mod 6) and for p = 0, 5 , n + p ≡ 0 (mod 6) , r + n ≡ 0 (mod 2) , s = r + 2, t = r + 4, α = x−r y−s x−t , β = y−r x−s y−t . Proof. We will prove this theorem by mathematical induction on n. For n = 1, we obtain as k = 1, i = 1, p = 5, r = 1, s = 3, t = 5. Then, we get x1 =

y−5 x−5 , y1 = . 1 + x−1 y−3 x−5 1 + x−1 y−3 x−5

Now, suppose that our assumption holds as follows: x12n−5 = x−5 x12n−4 = x−4 x12n−3 = x−3 x12n−2 = x−2 x12n−1 = x−1 x12n = x0

2n−1 Y j=0

2n−1 Y j=0

2n−1 Y j=0

2n−1 Y j=0

2n−1 Y j=0

2n−1 Y j=0

2n−1 Y 1 + 3jα 1 + 3jβ , y12n−5 = y−5 , 1 + (3j + 1) α 1 + (3j + 1) β j=0

1 + 3jα , y12n−4 = y−4 1 + (3j + 1) α 1 + (3j + 1) β , y12n−3 = y−3 1 + (3j + 2) β 1 + (3j + 1) β , y12n−2 = y−2 1 + (3j + 2) β 1 + (3j + 2) α , y12n−1 = y−1 1 + (3j + 3) α

1 + (3j + 2) α , y12n = y0 1 + (3j + 3) α

2n−1 Y j=0

2n−1 Y j=0

2n−1 Y j=0

2n−1 Y j=0

2n−1 Y j=0

1 + 3jβ , 1 + (3j + 1) β 1 + (3j + 1) α , 1 + (3j + 2) α 1 + (3j + 1) α , 1 + (3j + 2) α 1 + (3j + 2) β , 1 + (3j + 3) β

1 + (3j + 2) β , 1 + (3j + 3) β

where k = 2n. To end up the proof, we have to show that the cases in {xm , ym } hold for m = 12n + 1, 12n + 6. For x12n+1 and y12n+1 , we obtain » ¼ ¹ º 12n + 1 12n i = 1, p = 5, k = = 2n + 1, = 2n, r = 1. 6 6 Firstly, we consider x12n+1 =

y12n−5 1+y12n−1 x12n−3 y12n−5 .Therefore,

y−5 x12n+1 = 1 + y−1 y−5 =

2n−1 Y

2n−1 Y

j=0

2n−1 Y

1+3jβ 1+(3j+1)β

j=0 1+(6n+1)β 1+6nβ

= y−5

934

1+3jβ 1+(3j+1)β

j=0 2n−1 Y

1+(3j+2)β 1+(3j+3)β x−3

j=0

2n Y i=0

we can write

1+(3j+1)β 1+(3j+2)β y−5

2n−1 Y

1+3jβ 1+(3j+1)β

j=0

1 + 3jβ . 1 + (3j + 1) β

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x12n−5 1+x12n−1 y12n−3 x12n−5 .

Secondly, we consider y12n+1 =

x−5 y12n+1 = 1 + x−1 y−5 =

2n−1 Y

2n−1 Y

j=0

2n−1 Y

1+3jα 1+(3j+1)α

j=0 2n−1 Y

1+(3j+2)α 1+(3j+3)α y−3

1+3jα 1+(3j+1)α

j=0 1+(6n+1)α 1+6nα

Then we can write

= y−5

j=0

2n Y i=0

1+(3j+1)α 1+(3j+2)α x−5

2n−1 Y

1+3jα 1+(3j+1)α

j=0

1 + 3jα . 1 + (3j + 1) α

Similarly one can prove the other relations. The proof is complete. Remark 2.1. If α = x−r y−s x−t 6= −1/n or β = y−r x−s y−t 6= −1/n for all n ∈ Z+ , then Theorem 2.1 also represents solutions of system (2.1) in the case where initial conditions are real numbers. Theorem 2.2. System (2.1) has one equilibrium point which is (0, 0). Proof. For the equilibrium points of system (2.1), we can write x=

y x and y = . 2 1+y x 1 + x2 y

Then we have x + y 2 x2 = y and y + x2 y 2 = x, or, y − x = x − y. Hence, we obtain x = y = 0, which is desired. Theorem 2.3. For all n ∈ Z+ , α = 0 and β = 0 iff the system (2.1) has periodic solutions of period 12. Proof. First, let α = 0 and β = 0. By considering Theorem 2.1, the solutions of system (2.1) is reduced as xn = y−p , yn = x−p ; k is odd, xn = x−p , yn = y−p ; k is even,

¡ ¢ where for p = 0, 5 , n + p ≡ 0 (mod 6) . Therefore, for n = 1, 2, . . . , we get

x12n−11 = y−5 , y12n−11 = x−5 ; x12n−10 = y−4 , y12n−10 = x−4 , x12n−9 = y−3 , y12n−9 = x−3 ; x12n−8 = y−2 , y12n−8 = x−2 , x12n−7 = y−1 , y12n−7 = x−1 ; x12n−6 = y0 , y12n−6 = x0 , x12n−5 = x−5 , y12n−5 = y−5 ; x12n−4 = x−4 , y12n−4 = y−4 , x12n−3 = x−3 , y12n−3 = y−3 ; x12n−2 = x−2 , y12n−2 = y−2 , x12n−1 = x−1 , y12n−1 = y−1 ; x12n = x0 , y12n = y0 , 935

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which is desired. Second assume that the system (2.1) has periodic solutions of period 12. Then, we have ¡ § ¨ ¢ 2n−2 Y 1 + 3j + i − 1 α 2 ¡ § i ¨¢ x12n−11 = y−5 1 + 3j + α 2 j=0 ¨ ¢ ¡ § 2n Y 1 + 3j + 2i − 1 α ¡ § i ¨¢ = x12n+1 , = y−5 α 1 + 3j + 2 j=0

and

y12n−11

¢ ¡ § ¨ 1 + 3j + 2i − 1 β ¡ § ¨¢ = x−5 1 + 3j + 2i β j=0 ¢ ¡ § ¨ 2n Y 1 + 3j + 2i − 1 β ¡ § ¨¢ = y12n+1 . = x−5 1 + 3j + 2i β j=0 2n−2 Y

In here, in order to ensure above equalities iff α, β = 0. Similarly one can prove the other conditions. The proof is complete. Theorem 2.4. Assume that α, β 6= 0. Then every solution of system (2.1) converges to (0, 0) . Proof. In here, there are 16 different states. We will present only the case α < 0 for xn in Theorem 2.1-(i). By considering Theorem 2.1, we obtain

xn = y−p

¢ ¡ § ¨ 1 + 3j + 2i − 1 α ¡ § ¨¢ 1 + 3j + 2i α

n−i bY 6 c

j=0

= y−p exp

n−i bY 6 c

j=0

¢ ¡ § ¨ 1 + 3j + 2i − 1 α ¡ § ¨¢ ln 1 + 3j + 2i α

⎞ n−i à ! bX 6 c α ⎟ ⎜ § i ¨¢ = y−p exp ⎝− ln 1 + ¡ ⎠ 3j + 2 α + 1 j=0 ⎞ ⎛ n−i à ! bX 6 c 1 1 ⎟ ⎜ ¡ § i ¨¢ = y−p c(n0 ) exp ⎝−α + O( 2 ) ⎠ → 0, n → ∞. i 3j + 2 α + 1 j=n ⎛

0

Here, c(n0 ) is a positive constant depending on n0 ∈ N. Similarly one can prove the other relations. The proof is complete.

3 On System xn+1 =

yn−5 , −1+yn−1 xn−3 yn−5

yn+1 =

xn−5 −1+xn−1 yn−3 xn−5

In here, we investigate on the solutions of the difference equation system xn+1 =

yn−5 xn−5 , yn+1 = , n ∈ N0 . −1 + yn−1 xn−3 yn−5 −1 + xn−1 yn−3 xn−5

(3.1)

Theorem 3.1. Let {xn , yn } be a solution of system (3.1) . For each n ∈ N, assume that α, β 6= 1. § ¨ For k = n6 , then all solutions of system (3.1) 936

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i) If k is even, xn = x−p , yn = y−p . ii) If k is odd and p is equal to 2 or 3 xn = y−p (−1 + α) , yn = x−p (−1 + α) . iii) If k is odd and p is equal to otherwise, xn =

y−p x−p , yn = , −1 + β −1 + α

where for (p = 0, 1, 2, 3, 4, 5) , n + p ≡ 0 (mod 6) , r + n ≡ 0 (mod 2) , s = r + 2, t = r + 4, α = x−r y−s x−t , β = y−r x−s y−t . Proof. By induction. For n = 1, we obtain as k = 1, p = 5, r = 1. Then we can write x1 = xn =

y−5 x−5 , yn = , −1 + β −1 + α

where α = x−1 y−3 x−5 , β = y−1 x−3 y−5 , and the relation holds. Suppose that our assumption holds as follows: x12n−5 = x−5 , y12n−5 = y−5 ; x12n−4 = x−4 , y12n−4 = y−4 , x12n−3 = x−3 , y12n−3 = y−3 ; x12n−2 = x−2 , y12n−2 = y−2 , x12n−1 = x−1 , y12n−1 = y−1 ; x12n = x0 , y12n = y0 . To end up the proof, we have to show that the cases in {xm , ym } hold for m = 12n + 1, 12n + 6. For x12n+1 and y12n+1 , we obtain » ¼ 12n + 1 k= = 2n + 1, p = 5, r = 1. 6 Firstly, we consider x12n+1 =

y12n−5 −1+y12n−1 x12n−3 y12n−5 .Therefore,

x12n+1 = = Secondly, we consider y12n+1 =

x−5 −1 + y−1 x−3 y−5 x−5 . −1 + β

x12n−5 −1+x12n−1 y12n−3 x12n−5 .

y12n+1 = =

we can write that

Then we obtain that

y−5 −1 + x−1 y−3 x−5 x−5 . −1 + α

Similarly one can prove the other relations. The proof is complete. Corollary 3.2. Let {xn , yn } be a solution of system (3.1) . Then all solutions of system (3.1) are periodic with period 12.

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4 On System xn+1 =

yn−5 , ±1+yn−1 xn−3 yn−5

yn+1 =

xn−5 ∓1+xn−1 yn−3 xn−5

In this section, firstly, we consider the solution of the following system yn−5 xn−5 , yn+1 = xn+1 = 1 + yn−1 xn−3 yn−5 −1 + xn−1 yn−3 xn−5

(4.1)

where the initial values are arbitrary real numbers such that the denominator is always nonzero. If we interchange xn and yn in the following theorem, then we obtain the solution of the other system. So, we can omit the solution of other system.

Theorem 4.1. Suppose that {xn , yn } are solutions of system (4.1). Then every solutions of system (4) are periodic with period 24 and given by the following formula for n = 0, 1, 2, ..., x24n−5 = x−5 , x24n−4 = x−4 , x24n−3 = x−3 , x24n−2 = x−2 , x24n−1 = x−1 , y−5 y−4 x24n = x0 , x24n+1 = , x24n+2 = , 1 + y−5 x−3 y−1 1 + y−4 x−2 y0 y−3 (−1 + x−5 y−3 x−1 ) y−2 (−1 + x−4 y−2 x0 ) , x24n+4 = , x24n+3 = (−1 + 2x−5 y−3 x−1 ) (−1 + 2x−4 y−2 x0 ) y−1 y0 x24n+5 = , x24n+6 = , 1 − y−5 x−3 y−1 1 − y−4 x−2 y0 x24n+7 = −x−5 , x24n+8 = −x−4 , x24n+9 = −x−3 , x24n+10 = −x−2 , −y−5 x24n+11 = −x−1 , x24n+12 = −x0 , x24n+13 = , 1 + y−5 x−3 y−1 −y−4 , x24n+14 = 1 + y−4 x−2 y0 −y−3 (−1 + x−5 y−3 x−1 ) −y−2 (−1 + x−4 y−2 x0 ) x24n+15 = , x24n+16 = , (−1 + 2x−5 y−3 x−1 ) (−1 + 2x−4 y−2 x0 ) −y−1 −y0 x24n+17 = , x24n+18 = , 1 − y−5 x−3 y−1 1 − y−4 x−2 y0 y24n−5 = y−5 , y24n−4 = y−4 , y24n−3 = y−3 , y24n−2 = y−2 , y24n−1 = y−1 , x−5 x−4 y24n = y0 , y24n+1 = , y24n+2 = , −1 + x−5 y−3 x−1 −1 + x−4 y−2 x0 y24n+3 = −x−3 (1 + y−5 x−3 y−1 ), y24n+4 = −x−2 (1 + y−4 x−2 y0 ), x−1 (1 − 2x−5 y−3 x−1 ) x0 (1 − 2x−4 y−2 x0 ) y24n+5 = , y24n+6 = , (−1 + x−5 y−3 x−1 ) (−1 + x−4 y−2 x0 ) y24n+7 = y24n+9 = y24n+10 = y24n+12 = y24n+14 = y24n+15 = y24n+17 =

y−5 (−1 + y−5 x−3 y−1 ) y−4 (−1 + y−4 x−2 y0 ) , y24n+8 = , (1 + y−5 x−3 y−1 ) (1 + y−4 x−2 y0 ) y−3 , (−1 + 2x−5 y−3 x−1 ) y−2 y−1 (1 + y−5 x−3 y−1 ) , y24n+11 = , (−1 + 2x−4 y−2 x0 ) (−1 + y−5 x−3 y−1 ) y0 (1 + y−4 x−2 y0 ) x−5 (−1 + 2x−5 y−3 x−1 ) , y24n+13 = , (−1 + y−4 x−2 y0 ) (−1 + x−5 y−3 x−1 ) x−4 (−1 + 2x−4 y−2 x0 ) , (−1 + x−4 y−2 x0 ) x−3 (1 − y−5 x−3 y−1 ), y24n+16 = x−2 (1 − y−4 x−2 y0 ), x−1 x0 , y24n+18 = . 1 − x−5 y−3 x−1 1 − x−4 y−2 x0 938

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5 Numerical Examples In order to illustrate the results of the previous sections and to support our theoretical discussions, we consider several interesting numerical examples in this section. These examples represent different types of qualitative behavior of solutions to nonlinear difference equations systems. Example 1. Consider the difference system equation (2.1) with the initial conditions x−5 = .5, x−4 = .2, x−3 = .11, x−2 = .7, x−1 = .4, x0 = −3, y−5 = .7, y−4 = .12, y−3 = −1.2, y−2 = .2, y−1 = −.11, and y0 = .13. (See Fig. 1). plot of X(n+1)=Y(n−5)/1+Y(n−1)X(n−3)Y(n−5),Y(n+1)=X(n−5)/1+X(n−1)Y(n−3)X(n−5) 6 x(n) y(n) 4

x(n),y(n)

2 0 −2 −4 −6

0

10

20

30 n

40

50

60

Figure 1. Example 2. For the initial conditions x−5 = 9, x−4 = 11, x−3 = 7, x−2 = 5, x−1 = 4, x0 = 3, y−5 = 8, y−4 = 3, y−3 = 1.2, y−2 = 3.4, y−1 = 1.9, and y0 = 13, when we take the system (2.1). (See Fig. 2). plot of X(n+1)=Y(n−5)/1+Y(n−1)X(n−3)Y(n−5),Y(n+1)=X(n−5)/1+X(n−1)Y(n−3)X(n−5) 14 x(n) y(n) 12

x(n),y(n)

10 8 6 4 2 0

0

10

20

30

40 n

50

60

70

80

Figure 2. Example 3. If we consider the difference equation system (3.1) with the initial conditions x−5 = .7, x−4 = −.16, x−3 = 1.5, x−2 = −.3, x−1 = .24, x0 = −.2, y−5 = .8, y−4 = 1.1, y−3 = −1.2, y−2 = .4, y−1 = 1.9, and y0 = −.13. (See Fig. 3).

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plot of X(n+1)=Y(n−5)/−1+Y(n−1)X(n−3)Y(n−5),Y(n+1)=X(n−5)/−1+X(n−1)Y(n−3)X(n−5) 2 x(n) y(n) 1.5 1 x(n),y(n)

0.5 0 −0.5 −1 −1.5

0

5

10

15 n

20

25

30

Figure 3. Example 4. See Figure 4, since we take the difference system equation (4.1) with the initial conditions x−5 = −.7, x−4 = .16, x−3 = −1.5, x−2 = .3, x−1 = −.24, x0 = .2, y−5 = −.8, y−4 = 1.1, y−3 = −1.2, y−2 = 1.4, y−1 = −1.9, and y0 = .13. plot of X(n+1)=Y(n−5)/1+Y(n−1)X(n−3)Y(n−5),Y(n+1)=X(n−5)/−1+X(n−1)Y(n−3)X(n−5) 2 x(n) y(n) 1

x(n),y(n)

0 −1 −2 −3 −4 −5

0

10

20

30

40

50

60

70

n

Figure 4.

Acknowledgements This article was funded by the Deanship of Scientific Research (DSR), King Abdulaziz University, Jeddah. The authors, therefore, acknowledge with thanks DSR technical and financial support.

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[5] D. Clark and M. Kulenovic, A coupled system of rational difference equations, Comput. Math. Appl., 43 (2002), 849–867. [6] E. M. Elsayed, Dynamics of a rational recursive sequences, Int. J. Differ. Equ., 4(2) (2009), 185–200. [7] E. M. Elsayed, Behavior and expression of the solutions of some rational difference equations, J. Comp. Anal. Appl., 15 (1) (2013), 73-81. [8] E. M. Elsayed, Solutions of rational difference systems of order two, Math. Comput. Mod., 55(3-4) (2012) 378–384. [9] E. M. Elsayed, Solution and attractivity for a rational recursive sequence, Disc. Dyn. Nat. Soc., Volume 2011, Article ID 982309, 17 pages. [10] E. M. Elsayed and H. A. El-Metwally, On the solutions of some nonlinear systems of difference equations, Adv. Differ. Equ., 2013, 2013:16, doi:10.1186/1687-1847-2013-161. [11] E.A. Grove, G. Ladas, L.C McGrath, C.T. Teixeira, Existence and behavior of solutions of a rational system, Commun. Appl. Nonlinear Anal. 8 (2001), 1–25. [12] R. Karatas, C. Cinar, D. Simsek, On positive solutions of the difference equation xn+1 = xn−5 /(1 + xn−2 xn−5 ), Int. J. Contemp. Math. Sci., 1(10) (2006) 495 500. [13] A. S. Kurbanli, C. Cinar, I. Yalcinkaya, On the behavior of positive solutions of the system of rational difference equations, Math. Comput. Mod., 53 (2011), 1261–1267. [14] R. E. Mickens, Difference equations, Theory and applications, Van Nostrand Reinhold Co., New York (1990). [15] A.Y. Ozban, On the positive solutions of the system of rational difference equations, xn+1 = 1/yn−k , yn+1 = yn /xn−m yn−m−k , J. Math. Anal. Appl., 323 (2006), 26–32. [16] S. Stevic, More on a rational recurrence relation, Appl. Math. E-notes 4 (2004), 80-84. [17] N. Touafek and E. M. Elsayed, On the solutions of systems of rational difference equations, Math. Comput. Mod., 55 (2012) 1987-1997. [18] N. Touafek and E. M. Elsayed, On the periodicity of some systems of nonlinear difference equations, Bull. Math. Soc. Sci. Math. Roumanie, 2 (2012), 217-224. [19] I. Yalcinkaya, On the global asymptotic stability of a second-order system of difference equations, Disc. Dyn. Nat. Soc., (2008), Article ID 860152. [20] I. Yalcinkaya, On the global asymptotic behavior of a system of two nonlinear difference equations, ARS Combinatoria, 95 (2010) 151–159. [21] I. Yalcinkaya, C. Cinar, M. Atalay, On the solutions of systems of difference equations, Adv. Differ. Equ., (2008) 9. Article ID 143943. [22] X. Yang, On the system of rational difference equations xn = A + yn−1 /xn−p yn−q , yn = A + xn−1 /xn−r yn−s , J. Math. Anal. Appl., 307 (2005) 305–311.

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The shared set of meromorphic functions and differential polynomials ∗ Hong-Yan XU Department of Informatics and Engineering, Jingdezhen Ceramic Institute, Jingdezhen, Jiangxi, 333403, China

Abstract The purpose of our paper is to deal with some uniqueness problem of meromorphic functions whose differential polynomials with more general form then the formers sharing a set with finite weight. These results in this paper complement some results given by Lin, Yi. Key words: Meromorphic function, small function, weighted sharing, differential polynomial. Mathematical Subject Classification (2010): 30D 30, 30D 35.

1

Introduction and Main Results

In this paper the term ”meromorphic” will always mean meromorphic in the complex plane C. We shall use the following standard notations of the value distribution theory(see Hayman [7], Yi and Yang [17]). Let f be a nonconstant meromorphic function and a ∈ C = C ∪ {∞} and S be a subset of C. Define [ E(S, f ) = {z : f (z) − a = 0, counting multiplicity}, a∈S

E(S, f ) =

[

{z : f (z) − a = 0, ignoring multiplicity}.

a∈S

If E(S, f ) = E(S, g) we say that f and g share the set S CM ; if E(S, f ) = E(S, g), we say that f and g share the set S IM . Especially, let S = {a}, we say that f and g share the value a CM if E(S, f ) = E(S, g); and we say that f and g share the value a IM if E(S, f ) = E(S, g)(see [6]). Let m be a nonnegative integer, we denote by Em (a; f ) the set of all a-points of f with multiplicities not exceeding m, where an a-point is counted according to its multiplicity. Also we denote by E m (a; f ) the set of distinct a-points of f with multiplicities not greater than m. If E∞ (a; f ) = E∞ (a; g) for some a ∈ C, we say that f, g share the value a CM . For any positive integer m, we define [ [ Em (S, f ) = Em (a; f ), and E m (S, f ) = E m (a; f ). a∈S

a∈S

In 1997, Yang and Hua [16] proved the following result. Theorem 1.1 (see [16]). Let f and g be two nonconstant meromorphic functions, n ≥ 11 an integer, and a ∈ C − {0}. If f n f 0 and g n g 0 share the value a CM , then either f = dg for some (n + 1)th root of unity d or g = c1 ecz and f = c2 e−cz where c, c1 , and c2 are constants satisfying (c1 c2 )n+1 c2 = −a2 . ∗ This work was supported by the NNSF of China(11301233, 61202313), the Natural Science Foundation of Jiang-Xi Province in China(Grant No. 2010GQS0119, No.20132BAB211001).

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In 2001, an idea of gradation of sharing of values was introduced in [8, 10] which measures how close a shared value is to being shared CM or to being shared IM . This notion is known as weighted sharing. The author studied some problem on the uniqueness of meromorphic function sharing some values and sets with finite weight (see [15, 14, 13]) In 2002, Fang and Fang [5] employed the idea of weighted sharing of values and obtained the following results: Theorem 1.2 (see [5]). Let f and g be two nonconstant entire functions, n be a positive integer. If Ek (1, f n (f − 1)f 0 ) = Ek (1, g n (g − 1)g 0 ) and one of the following conditions is satisfied: (a) k ≥ 3 and n ≥ 8, (b) k = 2 and n ≥ 9, (c) k = 1 and n ≥ 14, then f ≡ g. The following example shows that Theorem 1.2 is not valid when f and g are two meromorphic functions. Example 1.1 (see [11]). Let f=

(n + 2)(h − hn+2 ) , (n + 1)(1 − hn+2 )

(n + 2)(1 − hn+1 ) , (n + 1)(1 − hn+2 )

g=

where h = ez . Then f n (f − 1)f 0 and g n (g − 1)g 0 share 1 CM , but g 6≡ f . For meromorphic functions, Fang and Fang [5], Lin and Yi [11] obtained some unicity theorems corresponding to the above theorems. Theorem 1.3 (see [5]). Let f and g be two nonconstant meromorphic functions, n be a positive integer. If Ek (1, f n (f −1)2 f 0 ) = Ek (1, g n (g −1)2 g 0 ) and one of the following conditions is satisfied: (a) k ≥ 3 and n ≥ 13, (b) k = 2 and n ≥ 15, (c) k = 1 and n ≥ 23, then f ≡ g. Theorem 1.4 (see [11]). Let f and g be two nonconstant meromorphic functions satisfying 2 , n ≥ 12. If [f n (f − 1)]f 0 and [g n (g − 1)]g 0 share 1 CM , then f ≡ g. Θ(∞, f ) > n+1 In the mean time, Lahiri and Sarkar [9] also studied the uniqueness of meromorphic functions corresponding to nonlinear differential polynomials which are different from the forms previously mentioned, and proved the following result. Theorem 1.5 (see [9]). Let f and g be two nonconstant meromerophic functions, n(≥ 13) is an integer. If E2 (1, f n (f 2 − 1)f 0 ) = E2 (1, g n (g 2 − 1)g 0 ), then either f ≡ g or f ≡ −g. If n is an even integer then the possibility of f ≡ −g does not arise. In this paper, we will investigate the uniqueness of meromorphic functions when two nonlinear differential polynomials of more general form namely f n (f − a)(f − b)f 0 and g n (g − a)(g − b)g 0 2π where a 6= b and a, b 6= 0, share a set Sm = {1, ω, ω 2 , · · · , ω m−1 } where ω = e m i , m is a integer. Now, we state our main results of the paper as follows. Theorem 1.6 Let f and g be two nonconstant meromorphic functions, n and m(≥ 2) be two positive integers. Let Ek (Sm , f n (f − a)(f − b)f 0 ) = Ek (Sm , g n (g − a)(g − b)g 0 ), and f or g be Pn+1 a+b meromorphic function only having multiple poles, and let the two functions n+2 g s=0 ( fg )s − Pn+2 f s Pn f s ab s=0 ( g ) and s=0 ( g ) have no common simple zeros. If one of the following conditions is n+1 satisfied: (i) k ≥ 3: n > 4 +

8 m

(ii) k = 2: n > 4 + (iii) k = 1: n > 4 +

when 2 ≤ m ≤ 3 and n > 4 +

11 m 20 m

4 m

when 2 ≤ m ≤ 3 and n > 4 + when 2 ≤ m ≤ 3 and n > 4 +

when m ≥ 4;

4 m 4 m

when m ≥ 4; when m ≥ 4.

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Then f ≡ g. Remark 1.1 A. Banerjee [2] obtained some theorems when m = 1, that is. Sm = {1}. Theorem 1.7 Let f and g be two nonconstant meromorphic functions, n and m(≥ 2) be two positive integers. If Ek (Sm , f n (f − a)2 f 0 ) = Ek (Sm , g n (g − a)2 g 0 ), and one of the following conditions is satisfied: (i) k ≥ 3 and n > 4 +

8 m;

 (ii) k = 2 and n > max 4 + (iii) k = 1: n > 4 +

20 m

4 m, 2

+

10 m



;

when 2 ≤ m ≤ 3 and n > 4 +

4 m

when m ≥ 4.

Then f ≡ g. Next, some definitions and notations used in the paper are explained as follows. For a ∈ C and a positive integer k, we denote by N (r, a; f | = 1) the counting function of simple a-points of f , and denote by N (r, a; f | ≤ k) (N (r, a; f | ≥ k)) the counting functions of those a-points of f whose multiplicities are not greater (less) than k where each a-point is counted according to its multiplicity(see [7]). N (r, a; f | ≤ k)(N (r, a; f | ≥ k)) are defined similarly, where in counting the a-points of f we ignore the multiplicities. Set Nk (r, a; f ) = N (r, a; f ) + N (r, a; f | ≥ 2) + · · · + N (r, a; f | ≥ k). Definition 1.1 [1, 17] When f and g share 1 IM , We denote by N L (r, 1; f ) the counting function of the 1-points of f whose multiplicities are greater than 1-points of g, where each zero is counted 1) only once; Similarly, we have N L (r, 1; g). We also denote by NE (r, 1; f ) the counting function of (2

common simple 1-points of f and g; N E (r, 1; f ) denotes the counting function of those multiplicity 1-points of f and g,each point in these counting functions is counted only once. In the same way (2 1) ,one can define NE (r, 1; g), N E (r, 1; g).

2

Some Lemmas

To prove our results, we need the following Lemmas. Lemma 2.1 (see [12]). Let f be a nonconstant meromorphic function and let Pp ak f k R(f ) = Pk=0 q j j=0 bj f be an irreducible rational function in f with constant coefficient {ak } and {bj }, where ap 6= 0 and bq 6= 0. Then T (r, R(f )) = dT (r, f ) + S(r, f ), where d = max{p, q}. Lemma 2.2 (see [17]). Let f be a nonconstant meromorphic function, then   T r, f (k) ≤ T (r, f ) + kN (r, ∞; f ) + S(r, f ), and

  N r, 0; f (k) ≤ N (r, 0; f ) + kN (r, ∞; f ) + S(r, f ).

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Lemma 2.3 Let f and g be two nonconstant meromorphic functions and n, m be two positive 4 . If f or g is meromorphic function having only multiple poles and the integers such that n > 1 + m Pn f s P Pn+2 f s n+1 a+b ab two expressions n+2 g s=0 ( fg )s − n+1 s=0 ( g ) and s=0 ( g ) have no common simple zeros, and   n+3  m m g n+3 f (a + b)f n+2 abf n+1 (a + b)g n+2 abg n+1 ≡ , − + − + n+3 n+2 n+1 n+3 n+2 n+1 where a, b 6= C − {0}, then f ≡ g. Proof: We propose to follow the idea in the proof of [3, Lemma 2.11]. From the assumption of Lemma 2.3, we have  n+3  f n+3 (a + b)f n+2 abf n+1 g (a + b)g n+2 abg n+1 − + ≡t − + , (1) n+3 n+2 n+1 n+3 n+2 n+1 where tm = 1. From (1), we get that f and g share ∞ CM . Without loss of generality, from the assumption of Lemma 2.3, we may assume that g has some multiple poles. Let h = fg . From (1), we have Ag 2 (hn+3 − t) + Bg(hn+2 − t) + C(hn+1 − t) ≡ 0, i.e., Ag 2 = −Bg

hn+1 − t hn+2 − t − C , hn+3 − t hn+3 − t

(2)

a+b ab 1 , B = − n+2 and C = n+1 . where A = n+3 Let z0 be a pole of g with multiplicity p1 (≥ 2), which is not a root of h − uk = 0, where un+3 = t. From (2), we have 2p1 = p1 i.e., p1 = 0. Thus, we get a contradiction. k Therefore, we can see that the poles of g are precisely the roots of h − uk = 0. Let z1 be a zero of h − uk of multiplicity p2 which is a pole of g with multiplicity q2 , then from (2) we have 2q2 = p1 + q2 i.e., p2 = q2 . Since g has no simple pole, it follows that such points are multiple zeros of h − uk . From (2), we have Pn+1 Pn Bg j=0 hj + C j=0 hj 2 Ag = − . (3) Pn+2 j j=0 h

Suppose z2 be a simple zero of h − uk where k = 1, 2, . . . , n + 2, which is a zero of multiplicity q1 (≥ 2) of numerator of (3). Then from (3), z2 would be a zero of order q1 − 1 of g 2 . So it follows Pn Pn Pn+2 that z2 would be a zero of j=0 hj . Since j=0 hj and j=0 hj may have at most one common factor and a meromorphic function can not have more than two Picard exceptional values, we see that h − uk has multiple zeros for at least n − 1 values of k ∈ {1, 2, . . . , n + 2}. Therefore, we have 4 . Θ(uk ; h) ≥ 12 for at least n − 1 values of k, which implies a contradiction as n > 1 + m Thus, we complete the proof of Lemma 2.3. 2 Lemma 2.4 Let f and g be two nonconstant meromorphic functions, n(> 3), m be two positive integers. Then we have m

m

(f n (f − a)(f − b)f 0 ) (g n (g − a)(g − b)g 0 )

6≡ 1,

where a, b ∈ C − {0}. m

m

Proof: Suppose (f n (f − a)(f − b)f 0 ) (g n (g − a)(g − b)g 0 )

6≡ 1 , then we have

f n (f − a)(f − b)f 0 g n (g − a)(g − b)g 0 ≡ t,

(4)

where tm = 1. Without loss of generality , we suppose that there exists a set I with infinite measure such that T (r, g) ≤ T (r, f ), r ∈ I. Next, we consider the two following cases. 4

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Case 1: a = b. From (4), we have f n (f − a)2 f 0 g n (g − a)2 g 0 ≡ t. Using the similar method of [5, P.611], we can get that the equality is impossible. Case 2: a 6= b. Let z0 be a zero of f with multiplicity p1 , then from (4) we can see that z0 is a pole of g (say with multiplicity q1 ). Thus, we have np1 +p1 −1 = nq1 +2q1 +q1 +1 i.e., 2q1 +2 = (n+1)(p1 −q1 ) ≥ n+1, that is, q1 ≥ n−1 2 . Hence, from this we can deduce that (n + 1)p1 ≥

(n + 3)(n − 1) + 4 2

i.e.,

p1 ≥

n+1 . 2

(5)

Let z1 be a zero of f − a with multiplicity p2 , then from (4) we can see that z1 is a pole of g (say with multiplicity q2 ). Thus, we have 2p2 − 1 = (n + 3)q2 + 1

i.e.,

p2 =

n+5 (n + 3)q2 + 2 ≥ . 2 2

(6)

Let z2 be a zero of f − b with multiplicity p3 , then from (4) we can see that z2 is a pole of g (say with multiplicity q3 ). Similarly, we have p3 ≥

n+5 . 2

(7)

Let z3 be a zero of f 0 with multiplicity p4 which is not a zero of f (f − a)(f − b), then from (4) we get that z3 is a pole of g (say with multiplicity q4 ). Therefore, we have p4 = (n + 3)q4 + 1 ≥ n + 4.

(8)

Similarly, we have the same results for the zeros of g(g − a)(g − b)g 0 . Thus, we have N (r, ∞; f ) = N (r, ∞; f (f − a)(f − b)f 0 ) ≤ N (r, 0; g) + N (r, a; g) + N (r, b; g) + N 0 (r, 0; g 0 ) 2 2 1 2 N (r, 0; g) + N (r, a; g) + N (r, b; g) + N0 (r, 0; g 0 ) ≤ n+1 n+5 n+5 n+4   2 4 1 ≤ + + T (r, g) + S(r, g). n+1 n+5 n+4 By the second main theorem and from the above inequality, we have 2T (r, f ) ≤N (r, 0; f ) + N (r, a; f ) + N (r, b; f ) + N (r, ∞; f ) + S(r, f ) 2 2 2 ≤ N (r, 0; f ) + N (r, a; f ) + N (r, b; f ) n+1 n+5 n+5   4 1 2 + + T (r, g) + S(r, g) + S(r, f ) + n+1 n+5 n+4   4 8 2 ≤ + + T (r, f ) + S(r, f ). n+1 n+5 n+4 Since n > 3, we can get a contradiction. Thus, we complete the proof of Lemma 2.4.

2

Lemma 2.5 (see [5]). Let f and g be two meromorphic functions, and let k be a positive integer. If Ek (1, f ) = Ek (1, g), then one of the following cases must occur: 5

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(i) T (r, f ) + T (r, g) ≤N2 (r, ∞; f ) + N2 (r, 0; f ) + N2 (r, ∞; g + N2 (r, 0; g)) 1)

+ N (r, 1; f ) + N (r, 1; g) − NE (r, 1; f ) + N (r, 1; f | ≥ k + 1) + N (r, 1; g| ≥ k + 1) + S(r, f ) + S(r, g); (ii) f =

(B+1)g+(A−B−1) , Bg+(A−B)

where A(6= 0), B are two constants.

Lemma 2.6 Let f and g be two nonconstant meromorphic functions and n, m be two positive 4 integers such that n > 4 + m . Let F = f n (f − a)(f − b)f 0 and G = g n (g − a)(g − b)g 0 , where a 6= b and a, b ∈ C − {0}. If one of f and g is meromorphic function having and only having multiple Pn+1 Pn f s Pn+2 f s ab a+b g s=0 ( fg )s − n+1 poles and the two expressions n+2 s=0 ( g ) and s=0 ( g ) have no common simple zeros, and (B + 1)Gm + A − B − 1 , (9) Fm = BGm + A − B where A(6= 0) and B are constants, then f ≡ g. Proof: Let P (z) =

a + b n+2 ab n+1 1 z n+3 − z + z . n+3 n+2 n+1

(10)

Then we have F = (P (f ))0 = f n (f − a)(f − b)f 0 ,

G = (P (g))0 = g n (g − a)(g − b)g 0 .

(11)

By Lemmas 2.1 and 2.2, we have T (r, F ) ≤ T (r, f n (f − a)(f − b)) + T (r, f 0 ) ≤ (n + 2)T (r, f ) + 2T (r, f ) + S(r, f ) = (n + 4)T (r, f ) + S(r, f ). On the other hand, we have (n + 2)T (r, f ) =T (r, f n (f − a)(f − b)) ≤T (r, f n (f − a)(f − b)f 0 ) + T (r, f 0 ) + O(1) ≤T (r, F ) + 2T (r, f ) + S(r, f ). Therefore, we have nT (r, f ) + S(r, f ) ≤ T (r, F ) ≤ (n + 4)T (r, f ) + S(r, f ).

(12)

Thus, we get S(r, F ) = S(r, f ). Similarly, we can get S(r, G) = S(r, g). By Lemma 2.1, we have (n + 3)T (r, f ) = T (r, P (f )) ≤ T (r, (P (f ))0 ) + N (r, 0; P (f )) − N (r, 0; (P (f ))0 ) + S(r, f ) = T (r, F ) + N (r, 0; f ) + N (r, γ1 ; f ) + N (r, γ2 ; f ) − N (r, a; f ) − N (r, b; f ) − N (r, 0; f 0 ) + S(r, f ), where γ1 , γ2 are the two roots of the equation

1 2 n+3 z

−

a+b n+2 z

+

ab n+1

(13)

= 0.

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Similarly, we can get (n + 3)T (r, g) = T (r, P (g)) ≤ T (r, G) + nN (r, 0; g) + N (r, γ1 ; g) + N (r, γ2 ; g) − N (r, a; g) − N (r, b; g) − N (r, 0; g 0 ) + S(r, g).

(14)

Without loss of generality, we suppose that there exists a set I with infinite measure such that T (r, g) ≤ T (r, f ), r ∈ I. Next we consider three cases as follows. m m Case 1. Suppose B 6= 0, −1. From (9), we have N (r, B+1 B ; F ) = N (r, ∞; G ). By the second main theorem and S(r, F m ) = S(r, f ), we get mT (r, F ) = T (r, F m )

(15)

B+1 m ; F ) + S(r, f ) B = N (r, ∞; F m ) + N (r, 0; F m ) + N (r, ∞; Gm ) + S(r, f )

≤ N (r, ∞; F m ) + N (r, 0; F m ) + N (r,

≤ N (r, ∞; f ) + N (r, 0; f ) + N (r, a; f ) + N (r, b; f ) + N (r, 0; f 0 ) + N (r, ∞; g) + S(r, f ). From (13) and (15), we can get (n + 3)T (r, f ) =T (r, P (f )) 1 1 ≤ N (r, ∞; f ) + (1 + )N (r, 0; f ) + N (r, γ1 ; f ) m m 1 + N (r, γ2 ; f ) + N (r, ∞; g) + S(r, f ) m 1 2 ≤(3 + )T (r, f ) + T (r, g) + S(r, f ) m m 3 ≤(3 + )T (r, f ) + S(r, f ), m i.e., 3 )T (r, f ) ≤ S(r, f ). m 4 Since n > 4 + m , we get a contradiction. m m Case 2. Suppose B = 0. From (9), we have N (r, A−1 A ; F ) = N (r, 0; G ). We consider two subcases as follows. Subcase 2.1. A 6= 1. By the second main theorem and S(r, F m ) = S(r, f ), we get (n −

mT (r, F ) = T (r, F m )

(16)

≤ N (r, ∞; F m ) + N (r, 0; F m ) + N



A−1 m r, ;F A

 + S(r, f )

= N (r, ∞; F m ) + N (r, 0; F m ) + N (r, 0; Gm ) + S(r, f ) ≤ N (r, ∞; f ) + N (r, 0; f ) + N (r, a; f ) + N (r, b; f ) + N (r, 0; f 0 ) + N (r, 0; g) + N (r, a; g) + N (r, b; g) + N (r, 0; g 0 ) + S(r, f ). From (13) and (16), we can get (n + 3)T (r, f ) =T (r, P (f )) 5 2 ≤(3 + )T (r, f ) + T (r, g) + S(r, f ) m m 7 ≤(3 + )T (r, f ) + S(r, f ), m 7

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7 4 Thus, we get (n − m )T (r, f ) ≤ S(r, f ). Since n > 4 + m , we can deduce a contradiction. m m Subcase 2.2. A = 1. Then we have F = G , that is F = tG, where tm = 1. By integration we have P (f ) = tP (g) + s, where s is a constant. If s 6= 0, by the second main theorem we have

(n + 3)T (r, f ) =T (r, P (f )) ≤N (r, ∞; P (f )) + N (r, 0; P (f )) + N (r, s; P (f )) + S(r, f ) =N (r, ∞; P (f )) + N (r, 0; P (f )) + N (r, 0; tP (g)) + S(r, f ) ≤N (r, ∞; f ) + N (r, 0; f ) + N (r, γ1 ; f ) + N (r, γ2 ; f ) + N (r, 0; g) + N (r, γ1 ; g) + N (r, γ2 ; g) + S(r, f ) ≤4T (r, f ) + 3T (r, g) + S(r, f ) ≤7T (r, f ) + S(r, f ) 4 Since n > 4 + m , we get a contradiction. Hence s = 0, that is. P (f ) ≡ tP (g). By Lemma 2.3 we get f ≡ g. Case 3. B + 1 = 0. Proceeding as in the proof of Case 2, we can get F m Gm ≡ 1, that is. n f (f − a)(f − b)f 0 g n (g − a)(g − b)g 0 ≡ t, where tm = 1. By Lemma 2.4, we have f ≡ g. Therefore, we complete the proof of Lemma 2.6. 2

Lemma 2.7 (see [4]). Let Q(ω) = (n − 1)2 (ω n − 1)(ω n−2 − 1) − n(n − 2)(ω n−1 − 1)2 , then Q(ω) = (ω − 1)4 (ω − β1 )(ω − β2 ) · · · (ω − β2n−6 ), where βj ∈ C − {0, 1} (j = 1, 2, . . . , 2n − 6), which are distinct respectively.

3

Proofs of Theorems 1.7 and 1.8

3.1

Proof of Theorem 1.7

Proof: Let F and G be given by (11), and P (z) by (10). From the assumptions of Theorem 1.7, we have Ek (Sm , F ) = Ek (Sm , G) that is. Ek (1, F m ) = Ek (1, Gm ) and N2 (r, 0; F m ) + N2 (r, ∞; F m ) ≤2N (r, 0; f ) + 2N (r, a; f ) + 2N (r, b; f ) + 2N (r, 0; f 0 )

(17)

+ 2N (r, ∞; f ) + S(r, f ), N2 (r, 0; Gm ) + N2 (r, ∞; Gm ) ≤2N (r, 0; g) + 2N (r, a; g) + 2N (r, b; g) + 2N (r, 0; g 0 )

(18)

+ 2N (r, ∞; g) + S(r, g), (i) k ≥ 3. Since N (r, 1; F m ) + N (r, 1; Gm ) + N (r, 1; F m | ≥ k + 1)

(19)

1) NE (r, 1; F m )

m

+ N (r, 1; G | ≥ k + 1) − 1 1 ≤ N (r, 1; F m ) + N (r, 1; Gm ) + S(r, F m ) + S(r, Gm ) 2 2 m m ≤ T (r, F ) + T (r, G) + S(r, F ) + S(r, G). 2 2

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Suppose that F m , Gm satisfy (i) of Lemma 2.5, then from Lemma 2.1, we can get mT (r, F ) + mT (r, G) =T (r, F m ) + T (r, Gm ) m m ≤ T (r, F ) + T (r, G) + N2 (r, 0; F m ) + N2 (r, ∞; F m ) 2 2 + N2 (r, 0; Gm ) + N2 (r, ∞; Gm ) + S(r, F ) + S(r, G), i.e., T (r, F ) + T (r, G) ≤

2 2 2 N2 (r, 0; F m ) + N2 (r, ∞; F m ) + N2 (r, 0; Gm ) m m m 2 + N2 (r, ∞; Gm ) + S(r, F ) + S(r, G). m

(20)

(i1 ) 2 ≤ m ≤ 3. From (13),(14),(17),(18),(20) and the definitions of F, G, we have (n + 3)T (r, f ) + (n + 3)T (r, g) 4 ≤(1 + )N (r, 0; f ) + N (r, γ1 ; f ) + N (r, γ2 ; f ) + N (r, a; f ) m 4 4 + N (r, b; f ) + N (r, ∞; f ) + N (r, 0; f 0 ) + (1 + )N (r, 0; g) m m + N (r, γ1 ; g) + N (r, γ2 ; g) + N (r, a; g) + N (r, b; g) 4 + N (r, ∞; g) + N (r, 0; g 0 ) + S(r, f ) + S(r, g) m 8 8 ≤(7 + )T (r, f ) + (7 + )T (r, g) + S(r, f ) + S(r, g), m m i.e., 

8 n−4− m

 {T (r, f ) + T (r, g)} ≤ S(r, f ) + S(r, g).

(21)

8 Since n > 4 + m , we can get a contradiction. By Lemma 2.5, we have (B + 1)Gm + A − B − 1 Fm = , BGm + A − B

where A(6= 0) and B are constants. 8 From Lemma 2.6 and n > 4 + m , we can get f ≡ g. 4 (i2 ) If m ≥ 4, then m − 1 ≤ 0. Thus, from (13),(14),(17),(18) and (20), we can get (n + 3)T (r, f ) + (n + 3)T (r, g) 4 4 4 ≤(1 + )N (r, 0; f ) + N (r, γ1 ; f ) + N (r, γ2 ; f ) + N (r, ∞; f ) + (1 + )N (r, 0; g) m m m 4 + N (r, γ1 ; g) + N (r, γ2 ; g) + N (r, ∞; g) + S(r, f ) + S(r, g) m 8 8 ≤(3 + )T (r, f ) + (3 + )T (r, g) + S(r, f ) + S(r, g), m m i.e., 

8 n− m

 {T (r, f ) + T (r, g)} ≤ S(r, f ) + S(r, g).

4 Since n > 4 + m , we can get a contradiction. By Lemma 2.5, we have (B + 1)Gm + A − B − 1 , Fm = BGm + A − B

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where A(6= 0) and B are constants. 4 , we can get f ≡ g. From Lemma 2.6 and n > 4 + m (ii) k = 2. Since 1 N (r, 1; F m ) + N (r, 1; Gm ) + N (r, 1; F m | ≥ 3) 2 1 1) m + N (r, 1; G | ≥ 3) − N E (r, 1; F m ) 2 1 1 ≤ N (r, 1; F m ) + N (r, 1; Gm ) + S(r, F ) + S(r, G) 2 2 m m ≤ T (r, F ) + T (r, G) + S(r, F ) + S(r, G). 2 2

(22)

Suppose that F m , Gm satisfy (i) of Lemma 2.5, then from Lemma 2.1 and (22), we can get m m T (r, F ) + T (r, G) ≤N2 (r, 0; F m ) + N2 (r, ∞; F m ) + N2 (r, 0; Gm ) + N2 (r, ∞; Gm ) (23) 2 2 1 1 + N (r, 1; F m | ≥ 3) + N (r, 1; Gm | ≥ 3) + S(r, F ) + S(r, G). 2 2 Since     1 Fm (F m )0 1 N (r, 1; F m | ≥ 3) ≤ N r, ∞; m 0 = N r, ∞; + S(r, F ) 2 (F ) 2 Fm 1 1 ≤ N (r, ∞; F m ) + N (r, 0; F m ) + S(r, F ) 2 2 1 1 1 ≤ N (r, ∞; f ) + N (r, 0; f ) + N (r, a; f ) 2 2 2 1 1 0 + N (r, b; f ) + N (r, 0; f ) + S(r, f ), 2 2

(24)

and 1 1 1 N (r, 1; Gm | ≥ 3) ≤ N (r, ∞; g) + N (r, 0; g) + N (r, a; g) 2 2 2 1 1 0 + N (r, b; g) + N (r, 0; g ) + S(r, g). 2 2

(25)

(ii1 ) 2 ≤ m ≤ 3. From (13),(14), (17),(18) and (23)-(25), we can get (n + 3)T (r, f ) + (n + 3)T (r, g) ≤T (r, F ) + T (r, G)     1 9 N (r, 0; f ) + N (r, γ1 ; f ) + N (r, γ2 ; f ) + 1 + N (r, a; f ) ≤ 1+ 2m 2m     1 1 9 N (r, ∞; f ) + 1+ N (r, b; f ) + 1 + N (r, 0; f 0 ) + 2m 2m 2m     9 1 + 1+ N (r, 0; g) + N (r, γ1 ; g) + N (r, γ2 ; g) + 1 + N (r, a; g) 2m 2m     1 1 9 + 1+ N (r, b; g) + 1 + N (r, 0; g 0 ) + N (r, ∞; g) + S(r, f ) + S(r, g) 2m 2m 2m     11 11 ≤ 7+ T (r, f ) + 7 + T (r, g) + S(r, f ) + S(r, g), m m that is, 

11 n−4− m

 {T (r, f ) + T (r, g)} ≤ S(r, f ) + S(r, g).

(26)

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Since n > 4 + 11 m , we can get a contradiction. Thus, from Lemma 2.5, we can get that F m , Gm satisfy the equality Fm =

(B + 1)Gm + A − B − 1 , BGm + A − B

where A(6= 0) and B are constants. From Lemma 2.6 and n > 4 + 11 m , we can get f ≡ g. (ii2 ) m ≥ 4. Like (i2 ), we can get (n + 3)T (r, f ) + (n + 3)T (r, g) ≤T (r, F ) + T (r, G)   1 1 9 N (r, 0; f ) + N (r, γ1 ; f ) + N (r, γ2 ; f ) + N (r, a; f ) + N (r, b; f ) ≤ 1+ 2m 2m 2m   1 9 9 + N (r, 0; f 0 ) + N (r, ∞; f ) + 1 + N (r, 0; g) + N (r, γ1 ; g) + N (r, γ2 ; g) 2m 2m 2m 1 1 1 9 + N (r, a; g) + N (r, b; g) + N (r, 0; g 0 ) + N (r, ∞; g) + S(r, f ) + S(r, g), 2m 2m 2m 2m that is, 

11 n− m

 {T (r, f ) + T (r, g)} ≤ S(r, f ) + S(r, g).

4 7 Since n > 4 + m and m ≥ 4, then a contradiction n − 11 m > 4 − m > 0 exists. m m Thus, from Lemma 2.5, we can get that F , G satisfy the equality

Fm =

(B + 1)Gm + A − B − 1 , BGm + A − B

where A(6= 0) and B are constants. 4 From Lemma 2.6 and n > 4 + m , we can get f ≡ g. (iii) k = 1. Since 1)

N (r, 1; F m ) + N (r, 1; Gm ) − NE (r, 1; F m ) 1 1 ≤ N (r, 1; F m ) + N (r, 1; Gm ) + S(r, F ) + S(r, G) 2 2 m m ≤ T (r, F ) + T (r, G) + S(r, F ) + S(r, G), 2 2 m



N (r, 1; F | ≥ 2) ≤N

Fm r, ∞; m 0 (F )



 =N

(F m )0 r, ∞; Fm

(27)

 + S(r, F )

(28)

≤N (r, ∞; F m ) + N (r, 0; F m ) + S(r, F ) ≤N (r, 0; f ) + N (r, a; f ) + N (r, b; f ) + N (r, 0; f 0 ) + N (r, ∞; f ) + S(r, f ), and N (r, 1; Gm | ≥ 2) ≤N (r, 0; g) + N (r, a; g) + N (r, b; g) + N (r, 0; g 0 ) + N (r, ∞; g) + S(r, g).

(29)

Suppose that F m , Gm satisfy (i) of Lemma 2.5, then from Lemma 2.1 and (27), we can get m m T (r, F ) + T (r, G) ≤N2 (r, 0; F m ) + N2 (r, ∞; F m ) + N2 (r, 0; Gm ) + N2 (r, ∞; Gm ) 2 2 + N (r, 1; F m | ≥ 2) + N (r, 1; Gm | ≥ 2) + S(r, F ) + S(r, G).

(30)

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(iii1 ) 2 ≤ m ≤ 3. From (13),(14),(17),(18) and (28)-(30), we can get (n + 3)T (r, f ) + (n + 3)T (r, g) ≤T (r, F ) + T (r, G)     6 2 ≤ 1+ N (r, 0; f ) + N (r, γ1 ; f ) + N (r, γ2 ; f ) + 1 + N (r, a; f ) m m     2 2 6 + 1+ N (r, b; f ) + 1 + N (r, 0; f 0 ) + N (r, ∞; f ) m m m     2 2 + N (r, γ1 ; g) + N (r, γ2 ; g) + 1 + N (r, a; g) + 1 + N (r, b; g) m m   2 6 + 1+ N (r, 0; g 0 ) + N (r, ∞; g) + S(r, f ) + S(r, g), m m that is,  n−4−

20 m

 {T (r, f ) + T (r, g)} ≤ S(r, f ) + S(r, g).

(31)

Since n > 4 + 20 m , we get a contradiction. Thus, from Lemma 2.5, we can get that F m , Gm satisfy the equality Fm =

(B + 1)Gm + A − B − 1 , BGm + A − B

where A(6= 0) and B are constants. From Lemma 2.6 and n > 4 + 11 m , we can get f ≡ g. (iii2 ) m ≥ 4. Like (i2 ) and (iii1 ), we can get (n + 3)T (r, f ) + (n + 3)T (r, g) ≤T (r, F ) + T (r, G)   2 2 6 N (r, 0; f ) + N (r, γ1 ; f ) + N (r, γ2 ; f ) + N (r, a; f ) + N (r, b; f ) ≤ 1+ m m m   6 2 6 + N (r, 0; f 0 ) + N (r, ∞; f ) + 1 + N (r, 0; g) + N (r, γ1 ; g) + N (r, γ2 ; g) m m m 2 2 6 2 + N (r, a; g) + N (r, b; g) + N (r, 0; g 0 ) + N (r, ∞; g) + S(r, f ) + S(r, g), m m m m that is, 

20 n− m

 {T (r, f ) + T (r, g)} ≤ S(r, f ) + S(r, g).

4 16 Since n > 4 + m and m ≥ 4, then n − 20 m > 4 − m ≥ 0, a contradiction. m m Thus, from Lemma 2.5, we can get that F , G satisfy the equality

Fm =

(B + 1)Gm + A − B − 1 , BGm + A − B

where A(6= 0) and B are constants. 4 , we can get f ≡ g. From Lemma 2.6 and n > 4 + m Thus, the proof of Theorem 1.7 is completed.

3.2

2

Proof of Theorem 1.8

Proof: By Lemma 2.7 and using the same argument in Theorem 1.7, we can easily get the conclusions of Theorem 1.8. Here, the process of the proof of Theorem 1.8 is omitted. 2 12

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References [1] A. Banerjee, Weighted sharing of a small function by a meromorphic function and its derivative, Comput. Math. Appl. 53 (2007), 1750-1761. [2] A. Banerjee, S. Mukherjee, Nonlinear differential polynomials sharing a small function, Arch. Math. (Brno) 44 (2008), 41-56 [3] A. Banerjee, S. Mukherjee, Corrigendum to ”Nonlinear differential polynomials sharing a small function” [Arch. Math. (Brno) 44(2008): 41-56], Arch. Math. (Brno) 44 (2008), 335-337. [4] G. Frank, M. Reinders, A unique range set for meromorphic functions with 11 elements, Complex Var. Elliptic Equ. 37 (1998), no. 9, 185-193. [5] C.-Y. Fang, M.-L. Fang, Uniqueness of meromorphic functions and differential polynomials, Comput. Math. Appl. 44 (2002), 607-617. [6] G.G. Gundersen, Meromorphic functions that share three or four values, J. London Math. Soc. 20 (1979), 457-466. [7] W.K. Hayman, Meromorphic Functions, The Clarendon Press, Oxford, 1964. [8] I. Lahiri, Weighted sharing and uniqueness of meromorphic functions, Nagoya Math. J. 161 (2001), 193-206. [9] I. Lahiri, A. Sarkar, Nonlinear differential polynomials sharing 1-points with weight two, Chinese J. Contemp. Math. 25 (2004), no. 3, 325-334. [10] I. Lahiri, Weighted value sharing and uniqueness of meromorphic functions, Complex Var. Elliptic Equ. 46 (2001), 241-253. [11] W. C. Lin, H. X. Yi, Uniqueness theorems for meromorphic function, Indian J. Pure Appl. Math. 35 (2004), no. 2, 121-132. [12] A. Z. Mokhonko, The Navanlinna charateristic of some meromorphic functions, Funct. Anal. Appl. 14 (1971), 83-87. [13] H. Y. Xu, T. B. Cao, Uniqueness of entire ormeromorphic functions sharing one value or a function with finiteweight, J. Inequal. Pure and Appl. Math. 10(3) (2009), Art. 88, 14 pp. [14] H. Y. Xu, T. B. Cao, S. Liu, Uniquenessresults of meromorphic functions whose nonlinear differentialpolynomials have one nonzero pseudo value, Mat. Vesnik 64 (1) (2012), 1-16. [15] H. Y. Xu, C. F. Yi, T. B. Cao, Uniqueness ofmeromorphic functions and differential polynomials sharing onevalue with finite weight, Ann. Polon. Math. 95 (2009), 55-66. [16] C. C. Yang, X. H. Hua, Uniqueness and value-sharing of meromorphic functions, Ann. Acad. Sci. Fenn. Math. 22 (1997), 395-406. [17] H. X. Yi, C. C. Yang, Uniqueness theory of meromorphic functions, Kluwer Academic Publishers, Dordrecht, 2003; Chinese original: Science Press, Beijing, 1995.

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QUASICONFORMAL HARMONIC MAPPINGS RELATED TO STARLIKE FUNCTIONS Ya¸sar Polato˜glu1 , Emel Yavuz Duman2 , Yasemin Kahramaner3 and Melike Aydo˜gan 4 2013 Abstract Let f = h(z)+g(z) be a univalent sense-preserving harmonic mapping of the 0unit disc D = {z ∈ C||z| < 1}. If f satisfies the condition g (z) |w(z)| = h0 (z) < k, (0 ≤ k < 1), then f is called k−quasiconformal harmonic mapping in D. The aim of this paper is to investigate a subclass of k−quasiconformal harmonic mappings.

1

Introduction

Let Ω be the family of functions φ(z) regular in the open unit disc D and satisfying the conditions φ(0) = 0, |φ(z)| < 1 for every z ∈ D. Next, let S ∗ denote the family of functions h(z) = z + c2 z 2 + ... regular in D such that h(z) is in S ∗ if and only if z

1 + φ(z) h0 (z) = h(z) 1 − φ(z)

(1.1)

2010 Mathematics Subject Classification: 30C45, 30C55 Key words and phrases: k- quasiconformal harmonic mappings, distortion theorem, coefficient inequality

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for some function φ(z) ∈ Ω and every z ∈ D. Moreover, let s1 (z) = z + d2 z 2 + d3 z 3 + ... and s2 (z) = z + e2 z 2 + e3 z 3 + ... be analytic functions in the open unit disc D. If there exists a function φ(z) ∈ Ω such that s1 (z) = s2 (φ(z)) for all z ∈ D, then we say that s1 (z) is subordinate to s2 (z) and we write s1 (z) ≺ s2 (z). Specially if s2 (z) univalent in D, then s1 (z) ≺ s2 (z) if and only if s1 (D) ⊂ s2 (D) and s1 (0) ⊂ s2 (0) implies s1 (Dr ) ⊂ s2 (Dr ) where Dr = {z ∈ C||z| < r, 0 < r < 1} (Subordination and Lindelof Principle [3]). Finally, a planar harmonic mapping in the open unit disc D is a complex valued harmonic function f , which maps D onto the some planar domain f (D). Since D is a simply connected domain, the mapping f has a canonical decomposition f = h(z) + g(z), where h(z) and g(z) are analytic in D and have the following power series expansions h(z) =

∞ X

n

an z , g(z) =

n=0

∞ X

bn z n ,

n=0

Where an , bn ∈ C, n = 0, 1, 2, ... as usual, we call h(z) the analytic part of f and g(z) is co-analytic part of f . An elegant and complete account of the theory of harmonic mappings is given in Duren’s monograph [1]. Lewy [4] proved in 1936 that the harmonic function f is locally univalent in D if and only if its Jacobien 2

2

Jf = |h0 (z)| − |g 0 (z)|

is different from zero in D. In view of this result, locally univalent harmonic mappings in the open unit disc D are either sense-reversing if |g 0 (z)| > |h0 (z)| in D or sense-preserving if |g 0 (z)| < |h0 (z)| in D. Throughout this paper we will restrict ourselves to the study of sense-preserving harmonic mappings. We also note that f = h + g is sense-preserving in D if and only if h0 (z) 0 doesn’t vanish in D and the second dilatation w(z) = ( hg 0(z) ) has the property (z) |w(z)| < 1 for all z ∈ D. Therefore the class of all sense-preserving harmonic mappings in the open unit disc with a0 = b0 = 0 and a1 = 1 will be denoted by SH . Thus SH contains standard class S of univalent functions. The family of all mappings f ∈ SH with the additional property g 0 (0) = 0, i.e, b1 = 0 is 0 0 denoted by SH . Hence it is clear that S ⊂ SH ⊂ SH . For the aim of this paper we need the following lemma and theorem.

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Lemma 1.1. ([6])Let φ(z) be a non-constant and analytic function in the unit disc D with φ(0) = 0. If |φ(z)| attains its maximum value on the circle |z| = r at the point z0 , then z0 φ0 (z0 ) = kφ(z0 ), k ≥ 1. Theorem 1.2. ([3]) Let h(z) be an element of S ∗ , then r r ≤ |h(z)| ≤ 2 (1 + r) (1 − r)2 1−r 1+r ≤ |h0 (z)| ≤ 3 (1 + r) (1 − r)3 A univalent harmonic mapping is called k−quasiconformal (0 ≤ k < 1) if |w(z)| < k. For the general definition of quasiconformal mapping see [1], [5]. The main idea of this paper is to investigate the subclass of k−quasiconformal harmonic mappings o n (∗) SH(kq) = f = h(z) + g(z) ∈ SH | |w(z)| < k, 0 ≤ k < 1, h(z) ∈ S ∗ . (1.2)

2

Main Results (∗)

Theorem 2.1. Let f = (h(z) + g(z)) be an element of SH(kq) , then g(z) k 2 (b1 − z) ≺ 2 h(z) k − b1 z

(2.1)

Proof. We consider the linear transformation (

k 2 (b1 − z) ). k 2 − b1 z

The transfromation maps |z| < k onto itself. On the other hand we have, w(z) =

(b1 z + b2 z 2 + ...)0 b1 + 2b2 z + ... g 0 (z) = = ⇒ w(0) = b1 0 2 0 h (z) (z + a2 z + ...) 1 + 2a2 z + ...

Therefore the function φ(z) =

k 2 (b1 − w(z)) M 2 − b1 w(z) 3

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satisfies the conditions of Schwarz Lemma, then we have w(z) =

k 2 (b1 − z) g 0 (z) ≺ h0 (z) k 2 − b1 z

(2.2)

and the transformation

k 2 (b1 − z) ) k 2 − b1 z maps |z| = r onto the disc with the centre (

C(r) = ( and the radius

k 2 Reb1 (1 − r2 ) k 2 Imb1 (1 − r2 ) , ) k 2 − |b1 |2 r2 k 2 − |b1 |2 r2 k(k 2 − |b1 |2 )r ρ(r) = k 2 − |b1 |2 r2

then we can write n 2 2 k(k 2 − |b1 |2 )r o k (1 − r )b 1 ≤ W (Dr ) = z | w(z) − . k 2 − |b1 |2 r2 k 2 − |b1 |2 r2

(2.3)

Now we define the function φ(z) by k 2 (b1 − φ(z)) g(z) = 2 h(z) k − b1 φ(z)

(2.4)

Then φ(z) is analytic and φ(0) =

0 = 0. |b1 | − k 2 2

If we take the derivative of (2.4) and after the brief calculations we get k 2 (b1 − φ(z)) k 2 (|b1 |2 + k 2 − 2b1 φ(z))zφ(z) 1 − φ(z) g 0 (z) = 2 . w(z) = 0 + h (z) (k 2 − b1 φ(z))2 1 + φ(z) k − b1 φ(z) (2.5) Now it is easy to realize that the subordination (2.1) is equivalent to |φ(z)| < 1 for all z ∈ D. Indeed, we assume the contrary; then there is a z1 ∈ D such that |φ(z1 )| = 1. So by I. S. Jack’s Lemma z1 φ0 (z1 ) = mφ(z1 ) 4

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for some m ≥ 1 and for such z1 we have w(z1 ) =

k 2 (b1 − φ(z1 )) k 2 (|b1 |2 + k 2 − 2b1 φ(z1 ))zφ(z1 ) 1 − φ(z1 ) g 0 (z1 ) = + . h0 (z1 ) (k 2 − b1 φ(z))2 1 + φ(z1 ) k 2 − b1 φ(z1 ) = w(φ(z1 )) ∈ W (D)

but this contradicts to (2.1); so our assumption is wrong, i. e, |φ(z)| < 1 for every z ∈ D. (∗)

Corollary 2.2. Let f = (h(z) + g(z)) be an element of SH(kq) , then rF (k, |b1 | , −r) ≤ |g(z)| ≤ rF (k, |b1 | , r)

(2.6)

G(k, |b1 | , −r) ≤ |g 0 (z)| ≤ G(k, |b1 | , r)

(2.7)

where

1 k(|b1 | + kr) 2 (1 − r ) k + |b1 | r 1 + r k(|b1 | + kr) G(k, |b1 | , r) = (1 − r)3 k + |b1 | r F (k, |b1 | , r) =

Proof. These inequalities is a simple consequence of Theorem 2.1 and the (∗) defination of SH(kq) . (∗)

Corollary 2.3. Let f = (h(z) + g(z)) be an element of SH(kq) , then (1 − r)2 (1 + r)2 F (k, |b | , r) ≤ J ≤ F1 (k, |b1 | , r) 2 1 f (1 + r)6 (1 − r)6

(2.8)

where F1 (k, |b1 | , r) =

[(k + k |b1 |) − (|b1 | + k 2 )r][(k − k |b1 |) − (|b1 | − k 2 )r] (k − |b1 | r)2

F2 (k, |b1 | , r) =

[(k + k |b1 |) + (|b1 | + k 2 )r][(k − k |b1 |) + (|b1 | − k 2 )r] (k + |b1 | r)2

Proof. Using (2.3) we obtain F2 (k, |b1 | , r) ≤ (1 − |w(z)|2 ) ≤ F1 (k, |b1 | , r)

(2.9)

On the other hand we have 2

2

2

2

2

Jf = |h0 (z)| − |g 0 (z)| = |h0 (z)| − |h0 (z)| |w(z)|2 = |h0 (z)| (1 − |w(z)|2 ) (2.10) Considering (2.9) and (2.10) with the Theorem 2.1 we obtain (2.8). 5

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(∗)

Corollary 2.4. Let f = (h(z) + g(z)) be an element of SH(kq) , then r

Z

1 − ρ k(1 − |b1 |) + (|b1 | + k 2 )ρ dρ ≤ |f | ≤ (1 + ρ)3 k + |b1 | ρ

0

Z 0

r

1 + ρ k(1 + |b1 |) + (|b1 | + k 2 )ρ dρ (1 − ρ)3 k + |b1 | ρ (2.11)

Proof. Using (2.3) we get (k − k |b1 |) + (|b1 | − k 2 )r (k − k |b1 |) − (|b1 | − k 2 )r ≤ (1 − |w(z)|) ≤ (2.12) (k + |b1 | r) (k − |b1 | r) (k + k |b1 |) − (|b1 | + k 2 )r (k + k |b1 |) + (|b1 | + k 2 )r ≤ (1 + |w(z)|) ≤ (2.13) (k − |b1 | r) (k + |b1 | r) On the other hand (|h0 (z)| − |g 0 (z)|) |dz| ≤ |df | ≤ (|h0 (z)| + |g 0 (z)|) |dz| ⇒

|h0 (z)| (1 − |w(z)|) |dz| ≤ |df | ≤ |h0 (z)| (1 + |w(z)|) |dz|

(2.14)

Using (2.12), (2.13) and Theorem 1.2 in the inequality (2.14) and integrating we obtain (2.11). (∗)

Theorem 2.5. Let f = (h(z) + g(z)) be an element of SH(kq) , then n X

n X b1 b m − k 2 am 2 k |bm − b1 am | ≤ (|b1 | − k ) + 2

4

2

2 2

m=2

(2.15)

m=2

Proof. Using Theorem 2.1, then we can write k 2 (b1 − φ(z)) g(z) = 2 ⇔ (k 2 g(z) − k 2 b1 h(z)) = (b1 g(z) − k 2 h(z))φ(z) h(z) k − b1 φ(z) Therefore we have n X

2

2

m

(k bm −k b1 am )z +

m=2

∞ X

2

m

2

dm z = ((|b1 | −k )z+

m=n+1

∞ X

(b1 bm −k 2 am )z m )φ(z)

m=2

(2.16) 6

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Where the coefficients dm have been chosen suitably. The equality (2.16) can be written in the F (z) = G(z)φ(z), |φ(z)| < 1, then we have |F (z)|2 = |G(z)φ(z)|2 = |G(z)|2 |φ(z)|2 ⇒ |F (z)|2 ≤ |G(z)|2 ⇒ 2 2 n ∞ n X X X 2 2 m 2 2 m m 2 (k bm − k b1 am )z + dm z ≤ (|b1 | − k )z + (b1 bm − k am )z m=2

m=n+1

m=2

Assume z = reiθ , 0 < r < 1, 0 ≤ θ and integrate the resulting inequality in the interval [0, 2π]. Then we find the inequality n X

k 4 |bm − b1 am |2 r2k +

m=2

∞ X

|dm |2 r2k ≤ (|b1 |2 −k 2 )2 r2k +

m=n+1

n X b1 bm − k 2 am 2 r2k .

m=2

Hence we get n X

k 4 |bm − b1 am |2 r2k ≤ (|b1 |2 − k 2 )2 r2k +

n X b1 bm − k 2 am 2 r2k . m=2

m=2

passing to the limit as r → 1 we obtain (2.15). The proof of this method was introduced by J. Clunie [2].

References [1] L. Ahlfors, Lectures on quasiconformal mappings, Van Nastrand Princeton, N. J, (1966). [2] J. Clunie, On Meromorphic Schicht Functions, J. London Math. Soc. 34 (1959), 215-216. [3] P. Duren, Univalent Functions, Springer Verlag, (1983). [4] P. Duren, Harmonic Mappings in the Plane, Vol. 156 of Cambridge Tracts in Mathematics, Cambridge University Press, Cambridge UK, (2004).

7

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[5] D. Kolaj, Quasiconformal Harmonic Mappings and Close-to-Convex Domains, Filomat 24:1, (2010), 63-68. [6] I. S. Jack, Functions starlike and convex of order α, J. London Math. Soc. (2), 3, (1971), no. 2, 469 − 474.

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˜ lu Yas¸ar Polatog Department of Mathematics and Computer Sciences, ˙ ˙ Istanbul K¨ ult¨ ur University, Istanbul, Turkey e-mail: [email protected] Emel Yavuz Duman Department of Mathematics and Computer Sciences, ˙ ˙ Istanbul K¨ ult¨ ur University, Istanbul, Turkey e-mail: [email protected] Yasemin Kahramaner Department of Mathematics, ˙ ˙ Istanbul Ticaret University, Istanbul, Turkey e-mail: [email protected] ˜ an Melike Aydog Department of Mathematics, ˙ I¸sık University, Me¸srutiyet Koyu, S¸ile Istanbul, Turkey e-mail: [email protected]

9

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J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 16, NO.5, 2014, COPYRIGHT 2014 EUDOXUS PRESS, LLC

On the superstability of ternary Jordan C ∗ -homomorphisms

Dong Yun Shin1 , Choonkil Park2∗ and Shahrokh Farhadabadi3 1

Department of Mathematics, University of Seoul, Seoul 130-743, Korea 2

Department of Mathematics, Research Institute for Natural Sciences Hanyang University, Seoul, 133-791, Korea 3

Department of Mathematics, Urmia University, Urmia, Iran

E-mail: [email protected]; [email protected]; shahrokh [email protected] Abstract. In this paper, we prove the superstability of ternary Jordan C ∗ -homomorphisms associated with the following Cauchy-Jensen functional equation: (x + y ) (x + z ) (y + z ) f +z +f +y +f + x = 2 [f (x) + f (y) + f (z)] (1) 2 2 2 and the Hyers-Ulam stability of ternary Jordan C ∗ -homomorphisms associated with the following generalized Cauchy-Jensen functional equation:   p p p ∑ ∑ ∑  1  f xj + xi  = 2 f (xi ). p − 1 j=1 i=1 i=1

(2)

j̸=i

Keywords: Superstability; Hyers-Ulam stability; Cauchy-Jensen functional equation; Ternary Jordan C ∗ -homomorphism. 2010 MSC: 17C65, 39B52, 17A40, 20N10, 11E20. 1. Introduction and preliminaries It is clear that the functional equation (2) is a generalized form of the functional equation (1). In order to investigate of the functional equation (2), we will suppose that p ≥ 2, and so the simplest case of (2) is the Cauchy equation with p = 2. Definition 1.1. [25] Let A, B be C*-ternary algebras. A C-linear mapping H : A → B is called a C ∗ -ternary Jordan homomorphism if H ([x, x, x]) = [H(x), H(x), H(x)] for all x ∈ A. If, in addition, H (x∗ ) = H(x)∗ for all x ∈ A, then H is called a ternary Jordan C ∗ -homomorphism. Definition 1.2. [25] Let A be a C*-ternary algebra. A C-linear mapping δ : A → A is called a C ∗ -ternary Jordan derivation if δ ([x, x, x]) = [δ(x), x, x] + [x, δ(x), x] + [x, x, δ(x)] 0∗

Corresponding author: [email protected] (C. Park)

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Superstability and ternary Jordan C ∗ -homomorphisms for all x ∈ A. If, in addition, δ (x∗ ) = δ(x)∗ for all x ∈ A, then δ is called a ternary Jordan C ∗ -derivation. In this paper, we will just obtain our results for ternary Jordan C ∗ -homomorphisms, and the reader can also investigate and get the results for ternary Jordan C ∗ -derivations similarly. We say a functional equation (ξ) is stable if any function g satisfying the equation (ξ) approximately is near to true solution of (ξ). We say that a functional equation is superstable if every approximately solution is an exact solution of it [33]. The stability of functional equations originated from Ulam [37] in 1940. Ulam proposed the following question “When does an exact solution of functional equation (ξ), near an approximately solution of that exist?” In 1941, Hyers [18] affirmatively answered to this question of Ulam for Banach spaces. In 1950, Aoki [1] generalized the Hyers’ theorem for approximately additive mappings. In 1978, Rassias [32] generalized the theorem of Hyers by considering the stability problem with unbounded Cauchy differences. In 1994, a generalization of Rassias’ theorem was obtained by ˘ Gavrut ¸a [17]. He proved the following: Theorem 1.3. [17] Let G be an abelian group and E a Banach space. Denote by φ : G×G → [0, ∞) a mapping such that ∞ ∑ ϕ(x, y) := 2−(n+1) φ(2n x, 2n y) < ∞ n=0

for all x, y ∈ G. Suppose f : G → E is a mapping satisfying ∥ f (x + y) − f (x) − f (y) ∥ ≤ φ(x, y) for all x, y ∈ G. Then there exists a unique additive mapping A : G → E such that ∥ f (x) − A(x) ∥ ≤ ϕ(x, x) for all x ∈ G. The stability problems of several functional equations have been extensively investigated by a number of authors, and there are many interesting results concerning these problems. A list of references concerning these results can be found in [2, 3, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 19, 20, 24, 26, 27, 28, 29, 31, 33, 35, 36]. 2. Superstability of ternary Jordan C ∗ -homomorphisms Throughout this section, we prove the superstability of ternary Jordan C ∗ -homomorphisms associated with the functional equation (1). From now on, A and B are C ∗ -ternary algebras with norm ∥ · ∥A and ∥ · ∥B respectively. We will use the following lemmas in the proof of our theorems. Lemma 2.1. [22] Let X and Y be linear spaces and let f : X → Y be an additive mapping such that f (µx) = µf (x) for all µ ∈ T1 := {λ ∈ C : |λ| = 1} and all x ∈ X. Then the mapping f is C-linear. Lemma 2.2. [22] Let {xn }n , {yn }n and {zn }n be convergent sequences in A. Then the sequence {[xn , yn , zn ]}n is a convergent sequence in A.

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D.Y. Shin, C. Park, S. Farhadabadi Lemma 2.3. Let f : A → B be a mapping such that

(

) (x + z ) ) (x + y y+z



+z −f + y ≤ f +x

2[f (x) + f (y) + f (z)] − f 2 2 2 B B

(2.1)

for all x, y, z ∈ A. Then f is additive. Proof. Letting x = y = z = 0 in (2.1), we get ∥4f (0)∥B ≤ ∥f (0)∥B . So f (0) = 0. Letting y = z = −x in (2.1), we get ∥2f (x) + 2f (−x)∥B ≤ ∥f (0)∥B = 0 for all x ∈ A. Hence f (−x) = −f (x) for all x ∈ A. Letting z = −2x − y in (2.1), we obtain ) ( (y − x) 3x + y −f =0 2f (x) + 2f (y) − 2f (2x + y) + f 2 2

(2.2)

for all x, y ∈ A. Letting x = −y−z in (2.1), we get 2 ( ) ( ) (y + z ) 3z + y 3y + z −2f + 2f (y) + 2f (z) − f −f =0 2 4 4 and so − 2f

(x + y) 2

( + 2f (x) + 2f (y) − f

3y + x 4

)

( −f

3x + y 4

) =0

(2.3)

for all x, y ∈ A. Now by (2.2) and (2.3), we get ( ) ( ) ( ) (y − x) (x + y) 3x + y 3x + y 3y + x 2f (2x + y) + f − 2f −f −f −f =0 2 2 2 4 4 for all x, y ∈ A. Letting y = 0 in the top line, we obtain f (x) + 3f (2x) + f (3x) + f (6x) − 2f (8x) = 0

(2.4)

for all x ∈ A. Letting y = −3x, z = x in (2.1), and then y = 0, z = −2x and then y = 3x, z = −5x and then y = − 23 x, z = − 12 x and then y = 2x, z = −4x and then y = − 32 x, z = − 34 x, respectively, we get 4f (x) + f (2x) − 2f (3x) = 0, f (x) + 2f (2x) + f (3x) − 2f (4x) = 0, f (x) + 3f (3x) − 2f (5x) = 0, 2f (2x) − f (3x) − 2f (4x) − f (5x) + 2f (6x) = 0, f (x) − 2f (2x) − 2f (4x) − f (5x) + 2f (8x) = 0, 2f (4x) − f (5x) − 2f (6x) − f (7x) + 2f (8x) = 0 for all x ∈ A. By these equations and by (2.4), we obtain f (nx) = nf (x) with n = 1, · · · , 8. So by (2.2), we have 1 1 (2.5) f (2x + y) = f (x) + f (y) − f (y − x) + f (3x + y) 4 4 for all x, y ∈ A. Letting x = u + v, y = u − v, z = −3u − v in (2.1), we get f (v) + 2f (u + v) − 2f (v − u) + f (2u + v) − 2f (3u + v) = 0

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Superstability and ternary Jordan C ∗ -homomorphisms and so f (2x + y) = 2f (3x + y) + 2f (y − x) − f (y) − 2f (x + y) for all x, y ∈ A. By this equation and (2.5), we obtain that f (3x + y)

4 8 8 9 f (x) + f (y) + f (x + y) − f (y − x) 7 7 7 7

=

and so 4 8 8 9 f (y) + f (x) + f (x + y) + f (y − x) 7 7 7 7 for all x, y ∈ A. By (2.3), we can obtain the result. f (3y + x)

=



Theorem 2.4. Let φ : A3 → [0, ∞) be a function such that lim 23n φ(2−n x, 2−n x, 2−n x)

=

0

lim 2−n φ(2n x, 2n x, 2n x)

=

0

n→∞

or

n→∞

for all x ∈ A. Let f : A → B be a mapping satisfying

) ( x+z ) ( x+y

+ µz − f µ + µy

2µ[f (x) + f (y) + f (z)] − f µ 2 2 B

( ) y+z

≤ f µ + µx , 2 B ∥f ([x, x, x]) − [f (x), f (x), f (x)]∥B ≤ φ(x, x, x), ∗

∗

∗

∗

∗

∥f (x ) − f (x) ∥B ≤ φ(x , x , x )

(2.6) (2.7) (2.8)

for all µ ∈ T1 and all x, y, z ∈ A. Then the mapping f : A → B is a ternary Jordan C ∗ homomorphism. Proof. Assume that limn→∞ 23n φ(2−n x, 2−n x, 2−n x) = 0. Let µ = 1 in (2.6). By Lemma 2.3, the mapping f : A → B is additive. Letting y = z = −x in (2.6), we get ∥ − 2µf (x) + 2f (µx)∥B = ∥2µf (x) + 4µf (−x) − 2f (−µx)∥B ≤ ∥f (0)∥B = 0 for all µ ∈ T1 and all x ∈ A. Hence f (µx) = µf (x) for all µ ∈ T1 and all x ∈ A. By Lemma 2.1, the mapping f : A → B is C-linear. By (2.7) and Lemma 2.2, we have ∥f ([x, x, x]) − [f (x), f (x), f (x)]∥B

([ x x x ]) [ ( x ) ( x ) ( x )]

, , − f ,f ,f = lim 23n f

n→∞ 2n 2n 2n 2n 2n 2n B (x x x) 3n ≤ lim 2 φ n , n , n = 0 n→∞ 2 2 2 for all µ ∈ T1 and all x ∈ A. Hence f ([x, x, x]) = [f (x), f (x), f (x)] for all µ ∈ T1 and all x ∈ A. It follows from (2.8) that

( ∗) ( x )∗ x

∥f (x∗ ) − f (x)∗ ∥B = lim 2n f − f

n→∞ 2n 2n B ( ∗ ∗ ∗) ( ∗ ∗ ∗) x x x x x x 3n ≤ lim 2n φ , , ≤ lim 2 φ , , =0 n→∞ n→∞ 2n 2n 2n 2n 2n 2n for all µ ∈ T1 and all x ∈ A. Hence f (x∗ ) = f (x)∗ for all µ ∈ T1 and all x ∈ A. Therefore, the mapping f : A → B is a ternary Jordan C ∗ -homomorphism.

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D.Y. Shin, C. Park, S. Farhadabadi Assume that limn→∞ 2−n φ(2n x, 2n x, 2n x) = 0. By (2.7), (2.8) and Lemma 2.2, we have ∥f ([x, x, x]) − [f (x), f (x), f (x)]∥B 1 = lim 3n ∥f ([2n x, 2n x, 2n x]) − [f (2n x), f (2n x), f (2n x)]∥B n→∞ 2 1 1 ≤ lim 3n φ(2n x, 2n x, 2n x) ≤ lim n φ(2n x, 2n x, 2n x) = 0, n→∞ 2 n→∞ 2 1 ∥f (x∗ ) − f (x)∗ ∥B = lim n ∥f (2n x∗ ) − f (2n x)∗ ∥B n→∞ 2 1 ≤ lim n φ(2n x∗ , 2n x∗ , 2n x∗ ) = 0 n→∞ 2 for all x ∈ A. Therefore, the mapping f : A → B is a ternary Jordan C ∗ -homomorphism.



Corollary 2.5. Let θ be a nonnegative real number and q1 , q2 , q3 be positive real numbers such that q1 , q2 , q3 > 3 or q1 , q2 , q3 < 1. Let f : A → B be a mapping satisfying (2.6) and ( ) ∥f ([x, x, x]) − [f (x), f (x), f (x)]∥B ≤ θ ∥x∥qA1 + ∥x∥qA2 + ∥x∥qA3 , ( q ) q q ∥f (x∗ ) − f (x)∗ ∥B ≤ θ ∥x∗ ∥A1 + ∥x∗ ∥A2 + ∥x∗ ∥A3 for all x ∈ A. Then the mapping f : A → B is a ternary Jordan C ∗ -homomorphism. ( ) Proof. Defining φ(x, y, z) = θ ∥x∥qA1 + ∥y∥qA2 + ∥z∥qA3 and applying Theorem 2.4, we get the result.  Corollary 2.6. Let θ be a nonnegative real number and q1 , q2 , q3 be positive real numbers such that q1 + q2 + q3 > 3 or q1 + q2 + q3 < 1. Let f : A → B be a mapping satisfying (2.6) and ∥f ([x, x, x]) − [f (x), f (x), f (x)]∥B

≤

θ∥x∥qA1 +q2 +q3 ,

∥f (x∗ ) − f (x)∗ ∥B

≤

θ∥x∗ ∥A1

q +q2 +q3

for all x ∈ A. Then the mapping f : A → B is a ternary Jordan C ∗ -homomorphism. ( ) Proof. Defining φ(x, y, z) = θ ∥x∥qA1 ∥y∥qA2 ∥z∥qA3 and applying Theorem 2.4, we get the result.



We can also put q1 = q2 = q3 = q in Corollaries 2.5 and 2.6, and obtain better and simpler results. 3. Hyers-Ulam stability of ternary Jordan C ∗ -homomorphisms Throughout this section, We prove the Hyers-Ulam stability of ternary Jordan C ∗ -homomorphisms associated with the functional equation (2). In order to do that, for a given mapping f : A → B, we define   p p p ∑  1 ∑ ∑  f Γµ f (x1 , · · · , xp ) := µxj + µxi  − 2 µf (xi ) p − 1 j=1 i=1 i=1 j̸=i

for all µ ∈ T1 and all x1 , · · · , xp ∈ A.

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Superstability and ternary Jordan C ∗ -homomorphisms Lemma 3.1. [25] A mapping f : A → B is a C-linear mapping if and only if Γµ f (x1 , · · · , xp ) = 0 for all µ ∈ T1 and all x1 , · · · , xp ∈ A. Theorem 3.2. Let φ : Ap → [0, ∞) be a function. Denote by ϕ a function such that ϕ(x1 , · · · , xp ) :=

∞ ∑

( ) 2n φ 2−(n+1) x1 , · · · , 2−(n+1) xp < ∞,

(3.1)

n=0

( ) lim 23n φ 2−n x, · · · , 2−n x = 0,

(3.2)

n→∞

or

ϕ(x1 , · · · , xp ) :=

∞ ∑

2−(n+1) φ(2n x1 , · · · , 2n xp ) < ∞

(3.3)

n=0

for all x, x1 , · · · , xp ∈ A. Suppose that f : A → B is a mapping satisfying ∥ Γµ f (x1 , · · · , xp ) ∥B

≤

φ(x1 , · · · , xp ),

(3.4)

∥f ([x, x, x]) − [f (x), f (x), f (x)]∥B

≤

φ(x, · · · , x),

(3.5)

∗

∗

∥f (x ) − f (x) ∥B

≤

∗

∗

φ(x , · · · , x )

(3.6)

for all µ ∈ T1 and all x, x1 , · · · , xp ∈ A. Then there exists a unique ternary Jordan C ∗ -homomorphism H : A → B such that 1 (3.7) ∥f (x) − H(x)∥B ≤ ϕ(x, · · · , x) p for all x ∈ A. Proof. Firstly, assume that (3.1) and (3.2) hold. Putting µ = 1 and x1 = · · · = xp = x in (3.4), we get ∥f (2x) − 2f (x)∥B

≤

( x )

f (x) − 2f

2 B

≤

1 φ(x, · · · , x) p 1 (x x) φ ,··· , p 2 2

for all x ∈ A. Using the induction method, we obtain n−1

( x ) ) 1 ∑ s ( −(s+1)

n −(s+1) ≤ 2 φ 2 x, · · · , 2 x

f (x) − 2 f

2n B p s=0

(3.8)

for each n ≥ 1 and all x ∈ A. Now assume that m, l are positive integers, with m > l. By (3.8), we have

( )

( x ) ( x ) ( x )

1 x

m

l l m−l

−2 f = 2 2 f −f

2 f

2m 2l B 2m−l 2l 2 l B ≤

m−1 ) 1 ∑ s ( −(s+1) 2 φ 2 x, · · · , 2−(s+1) x p

≤

∞ ) 1 ∑ s ( −(s+1) 2 φ 2 x, · · · , 2−(s+1) x . p

s=l

s=l

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D.Y. Shin, C. Park, S. Farhadabadi Now, the relation (3.1) shows that the right side converges to 0 when l → ∞, and this clarifies that the sequence {2n f ( 2xn )} is a Cauchy sequence. Since A is a complete space, the sequence {2n f ( 2xn )} is a convergent sequence. Therefore, we can define, for all x ∈ A, the mapping H : A → B by H(x) := lim 2n f n→∞

(x) . 2n

Passing the limit n → ∞ in (3.8) and by (3.1), we obtain (3.7). It follows from (3.4) and (3.1) that

(x xp )

1 , · · · , ∥Γµ H(x1 , · · · , xp )∥B = lim 2n Γµ f

n→∞ 2n 2n B ) (x xp 1 ≤ lim 2n φ n , · · · , n = 0 n→∞ 2 2 for all µ ∈ T1 and all x1 , · · · , xp ∈ A. By Lemma 3.1, H is C-linear. x By Lemma 2.2 and replacing x by 2n in (3.5) and by (3.2), we obtain ∥H([x, x, x]) − [H(x), H(x), H(x)]∥B

([ x x x ]) [ ( x ) ( x ) ( x )]

= lim 23n f , , − f ,f ,f

n→∞ 2n 2n 2n 2n 2n 2n B (x ) x 3n ≤ lim 2 φ n , · · · , n = 0 n→∞ 2 2 for all x ∈ A. Thus H([x, x, x]) = [H(x), H(x), H(x)] for all x ∈ A. x By (3.1) and replacing x by 2n in (3.6), we get ∥H(x∗ ) − H(x)∗ ∥B

= ≤

( ∗) ( x )∗

x

lim 2n f − f

n n n→∞ 2 2 B ( ∗ ) ∗ x x lim 2n φ ,··· , n = 0 n→∞ 2n 2

for all x ∈ A. Thus H(x∗ ) = H(x)∗ for all x ∈ A. Therefore, H : A → B is a ternary Jordan C ∗ -homomorphism. Let T : A → B be another ternary Jordan C ∗ -homomorphism that satisfies (3.7). Then we have

( )

( ) ( x ) ( x ) x x

n − H + 2 − T ∥H(x) − T (x)∥B ≤ 2n f

f

2n 2n B 2n 2n B ( ) 2 (x x) ≤ 2n ϕ n,··· , n p 2 2 ∞ ( n+1 ∑ 2 x x) ≤ 2s φ 2−(s+1) n , · · · , 2−(s+1) n p s=0 2 2 =

∞ ) 2 ∑ s ( −(s+1) 2 φ 2 x, · · · , 2−(s+1) x p s=n

for all x ∈ A. Now if n → ∞, then (3.1) shows that the right side converges to 0. So H : A → B is unique. If we assume that (3.3) holds, then by the same method as in the proof of last part, one can obtain  a C-linear mapping H(x) := limn→∞ 21n f (2n x) satisfying (3.7), and get the desired result.

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Superstability and ternary Jordan C ∗ -homomorphisms Corollary 3.3. Let θ be a nonnegative real number and, for every 1 ≤ j ≤ p, qj be positive real numbers such that all qj > 3 or all qj < 1, and let f : A → B be a mapping satisfying q

∥Γµ f (x1 , · · · , xp )∥B

≤

θ(∥x1 ∥qA1 + · · · + ∥xp ∥Ap ),

∥f ([x, x, x]) − [f (x), f (x), f (x)]∥B

≤

θ(∥x∥qA1 + · · · + ∥x∥Ap ),

∥f (x∗ ) − f (x)∗ ∥B

≤

θ(∥x∗ ∥A1 + · · · + ∥x∗ ∥Ap )

q

q

q

for all µ ∈ T1 and all x, x1 , · · · , xp ∈ A. Then there exists a unique ternary Jordan C ∗ -homomorphism H : A → B such that p q ∑ θ∥x∥Aj ∥f (x) − H(x)∥B ≤ p |2qj − 2| j=1 for all x ∈ A. Proof. Defining φ(x1 , · · · , xp ) = θ

p ∑

q

∥xj ∥Aj

j=1



and applying Theorem 3.2, we get the result.

Corollary 3.4. Let θ be a nonnegative real number and, for every 1 ≤ j ≤ p, qj be positive real numbers such that q1 + · · · + qp > 3 or q1 + · · · + qp < 1, and let f : A → B be a mapping satisfying q

∥Γµ f (x1 , · · · , xp )∥B

≤

θ(∥x1 ∥qA1 · · · ∥xp ∥Ap ),

∥f ([x, x, x]) − [f (x), f (x), f (x)]∥B

≤

θ∥x∥A j=1

∥f (x∗ ) − f (x)∗ ∥B

≤

∑p

qj

,

∑p

θ∥x∗ ∥A

j=1 qj

for all µ ∈ T1 and all x, x1 , · · · , xp ∈ A. Then there exists a unique ternary Jordan C ∗ -homomorphism H : A → B such that ∑p

qj

θ∥x∥ j=1 ∥f (x) − H(x)∥B ≤ ∑p A p 2 j=1 qj − 2 for all x ∈ A. Proof. Defining φ(x1 , · · · , xp ) = θ

p ∏

q

∥xj ∥Aj

j=1



and applying Theorem 3.2, we get the result.

One can also put q1 = · · · = qp = q in these last corollaries, and obtain better and simpler results.

Acknowledgments Dong Yun Shin was supported by the 0000 Research Fund of the University of Seoul.

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D.Y. Shin, C. Park, S. Farhadabadi

References [1] T. Aoki, On the stability of the linear transformation in Banach space, J. Math. Soc. Japan 2 (1950), 64–66. [2] I. Chang, Stability of higher ring derivations in fuzzy Banach algebras, J. Computat. Anal. Appl. 14 (2012), 1059–1066. [3] I. Cho, D. Kang, H. Koh, Stability problems of cubic mappings with the fixed point alternative, J. Computat. Anal. Appl. 14 (2012), 132–142. [4] P. Czerwik, Functional Equations and Inequalities in Several Variables, Word Scientific Publishing Company, New Jersey, Hong Kong, Singapore and London, 2002. [5] A. Ebadian, N. Ghobadipour, M. Eshaghi Gordji, A fixed point method for perturbation of bimultipliers and Jordan bimultipliers in C ∗ -ternary algebras, J. Math. Phys. 51, No. 10, Art. ID 103508, 10 pages (2010). [6] A. Ebadian, I. Nikoufar, M. Eshaghi Gordji, Nearly (θ1 , θ2 , θ3 , ϕ)-derivations on C ∗ -modules, Int. J. Geom. Methods Mod. Phys. 9, No. 3, Art. ID 1250019, 12 pages (2012). [7] M. Eshaghi Gordji, M. Bavand Savadkouhi, M. Bidkham, Stability of a mixed type additive and quadratic functional equation in non-Archimedean spaces, J. Computat. Anal. Appl. 12 (2010), 454–462. [8] M. Eshaghi Gordji, A. Bodaghi, On the stability of quadratic double centralizers on Banach algebras, J. Computat. Anal. Appl. 13 (2011), 724–729. [9] M. Eshaghi Gordji, R. Farokhzad Rostami, S.A.R. Hosseinioun, Nearly higher derivations in unital C ∗ -algebras, J. Computat. Anal. Appl. 13 (2011), 734–742. [10] M. Eshaghi Gordji, A. Fazeli, C. Park, 3-Lie multipliers on Banach 3-Lie algebras, Int. J. Geom. Methods Mod. Phys. (to appear). [11] M. Eshaghi Gordji, M.B. Ghaemi, S. Kaboli Gharetapeh, S. Shams, A. Ebadian, On the stability of J ∗ -derivations, J. Geom. Phys. 60 (2010), 454–459. [12] M. Eshaghi Gordji, M.B. Ghaemi, B. Alizadeh, A fixed point method for perturbation of higher ring derivationsin non-Archimedean Banach algebras, Int. J. Geom. Methods Mod. Phys. 8 (2011), 1611–1625. [13] M. Eshaghi Gordji, N. Ghobadipour, Stability of (α, β, γ)-derivations on Lie C ∗ -algebras, Int. J. Geom. Methods Mod. Phys. 7 (2010), 1097–1102. [14] M. Eshaghi Gordji, S. Kaboli Gharetapeh, T. Karimi, E. Rashidi, M. Aghaei, Ternary Jordan derivations on C ∗ -ternary algebras, J. Computat. Anal. Appl. 12 (2010), 463–470. [15] M. Eshaghi Gordji, A. Najati, Approximately J ∗ -homomorphisms: a fixed point approach, J. Geom. Phys. 60 (2010), 809–814. [16] Z. Gajda, On stability of the additive mappings, Int. J. Math. Math. Sci. 14 (1991), 431–434. ˘ [17] P. Gavrut ¸a, A generalization of the Hyers-Ulam-Rassias stability of approximately additive maooings, J. Math. Anal. Appl. 184 (1994), 431–436. [18] D.H. Hyers, On the stability of the linear functional equation, Proc. Natl. Acad. Sci. U.S.A. 27 (1941), 222–224.

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Superstability and ternary Jordan C ∗ -homomorphisms [19] D.H. Hyers, G. Isac, Th.M. Rassias, Stability of Functional Equations in Several Variables, Birkh¨ auser, Basel, 1998. [20] H.A. Kenary, J. Lee, C. Park, Non-Archimedean stability of an AQQ-functional equation, J. Computat. Anal. Appl. 14 (2012), 211–227. [21] M.S. Moslehian, Almost derivations on C ∗ -ternary rings, Bull. Belgian Math. Soc–Simon Stevin 14 (2007), 135–142. [22] A. Najati, C. Park, J. Lee, Homomorphisms and derivations in C ∗ -ternary algebras, Abs. Appl. Anal. 2009, Art. ID 612392, 16 pages (2009). [23] C. Park, Isomorphisms between C ∗ -ternary algebras, J. Math. Phys. 47, Art. ID 103512, 12 pages (2006). [24] C. Park, Y. Cho, H.A. Kenary, Orthogonal stability of a generalized quadratic functional equation in non-Archimedean spaces, J. Computat. Anal. Appl. 14 (2012), 526–535. [25] C. Park, S. Farhadabadi Ternary Jordan C ∗ -homomorphisms and ternary Jordan C ∗ derivations for a generalized Cauchy-Jensen functional equation, (preprint). [26] C. Park, S. Jang, R. Saadati, Fuzzy approximate of homomorphisms, J. Computat. Anal. Appl. 14 (2012), 833–841. [27] C. Park, J. Lee, D. Shin, Stability of J ∗ -derivations, Int. J. Geom. Meth. Mod. Phys. 9, No. 5, Art. ID 1220009, 10 pages (2012). [28] C. Park, A. Najati, Homomorphisms and derivations in C ∗ -algebras, Abs. Appl. Anal. 2007, Art. ID 80630, 12 pages (2007). [29] C. Park, W. Park, On the Jensen’s equation in Banach modules, Taiwanese J. Math. 6 (2002), 523–531. [30] J.M. Rassias, On approximation of approximately linear mappings by linear mappings, J. Funct. Anal. 46 (1982), 126–130. [31] J.M. Rassias, Solution of the Ulam stability problem for the quartic mapping, Glasnik Math. 34(54) (1999), 243–252. [32] Th.M. Rassias, On the stability of the linear mapping in Banach spaces, Proc. Amer. Math. Soc. 72 (1978), 297–300. [33] Th.M. Rassias, On the stability of functional equations and a problem of Ulam, Acta Appl. Math. 62 (2000), 23–130. ˘ [34] Th.M. Rassias, P. Semrl, On the behavior of mappings which do not satisfy Hyers-Ulam stability, Proc. Amer. Math. Soc. 114 (1992), 989–993. [35] S. Shagholi, M. Eshaghi Gordji, M. Bavand Savadkouhi, Stability of ternary quadratic derivations on ternary Banach algebras, J. Computat. Anal. Appl. 13 (2011), 1097–1105. [36] S. Shagholi, M. Eshaghi Gordji, M. Bavand Savadkouhi, Nearly ternary cubic homomorphism in ternary Fr´echet algebras, J. Computat. Anal. Appl. 13 (2011), 1106–1114. [37] S.M. Ulam, A Collection of Mathematical Problems, Interscience Tracts in Pure and Applied Mathematics, no. 8, Interscience, New York, USA, 1960. [38] H. Zettl, A Chararcterization of ternary rings of operators, Adv. Math. 48 (1983), 117–143.

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Coupled fixed points for generalized weakly contractive mappings in partial metric spaces Rattanaporn Wangkeeree1 , Rabian Wangkeeree, and Nithirat Sisarat Department of Mathematics, Faculty of Science, Naresuan University, Phitsanulok 65000, Thailand

Abstract In this paper, we establish coupled fixed point results for generalized weakly contractive mappings having the mixed monotone property in ordered partial metric spaces. The results on fixed point theorems are generalizations of the recent results of Alsulami, Hussain and Alotaibi [S. Alsulami, N. Hussain and A. Alotaibi, Coupled Fixed and Coincidence Points for Monotone Operators in Partial Metric Spaces, Fixed Point Theory and Applications 2012, 2012:173]. Keywords: Coupled fixed point; Partial metric space; Generalized weakly contractive mapping; Coupled coincidence point 1. Introduction and Preliminaries The existence and uniqueness of fixed and common fixed point theorems of operators has been a subject of great interest since Banach [1] proved the Banach contraction principle in 1922. Many authors generalized the Banach contraction principle in various spaces such as quasi-metric spaces, generalized metric spaces, cone metric spaces and fuzzy metric spaces. Matthews [2] introduced the notion of partial metric spaces in such a way that each object does not necessarily have to have a zero distance from itself and proved a modified version of the Banach contraction principle. Afterwards, many authors proved many existing fixed point theorems in partial metric spaces (see [5]-[44] for examples). In recent times, fixed point theory has developed rapidly in partially ordered metric spaces, that is, metric spaces endowed with a partial ordering. Some of these works are noted in [11, 17, 18, 36]. Bhaskar and Lakshmikantham [18] introduced the concept of a coupled fixed point and the mixed monotone property. Furthermore, they proved some coupled fixed point theorems for mappings which satisfy the mixed monotone property and gave some applications in the existence and uniqueness of a solution for a periodic boundary value problem. After the publication of this work, several coupled fixed point and coincidence 1

Corresponding author. Email address: [email protected] (R. Wangkeeree), [email protected] (R. Wangkeeree), and [email protected] (N. Sisarat) Preprint submitted to ...

March 22, 2013

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2 point results have appeared in the recent literature. Works noted in [31, 38, 39] are some examples of these works. We recall below the definition of partial metric space and some of its properties. Definition 1.1. [2] A partial metric on a nonempty set X is a function p : X × X −→ R+ 0 such that for all x, y, z ∈ X: (p1) x = y ⇔ p(x, x) = p(x, y) = p(y, y), (p2) p(x, x) ≤ p(x, y), (p3) p(x, y) = p(y, x), (p4) p(x, y) ≤ p(x, z) + p(z, y) − p(z, z). A partial metric space is a pair (X, p) such that X is a nonempty set and p is a partial metric on X. It is clear that, if p(x, y) = 0, then from (p1) and (p2), x = y. But if x = y, p(x, y) may not be 0. The function p(x, y) = max{x, y} for all x, y ∈ R+ defines a partial metric on R+ . Each partial metric p on X generates a T0 topology τp on X which has as a base the family of open p-balls {Bp (x, ) : x ∈ X,  > 0}, where Bp (x, ) = {y ∈ X : p(x, y) < p(x, x) + } for all x ∈ X and  > 0. If p is a partial metric on X, then the function dp : X × X −→ R+ given by dp (x, y) = 2p(x, y) − p(x, x) − p(y, y) is a metric on X. Definition 1.2. Let (X, p) be a partial metric space. Then (i) A sequence {xn } in a partial metric space (X, p) converges to a point x ∈ X if and only if p(x, x) = limn−→∞ p(x, xn ). (ii) A sequence {xn } in a partial metric space (X, p) is called a Cauchy sequence iff limn,m−→∞ p(xn , xm ) exists (and is finite). (iii) A partial metric space (X, p) is said to be complete if every Cauchy sequence {xn } in X converges, with respect to τp , to a point x ∈ X such that p(x, x) = limn,m−→∞ p(xn , xm ). (iv) A subset A of a partial metric space (X, p) is closed if whenever {xn } is a sequence in A such that {xn } converges to some x ∈ X, then x ∈ A. Remark 1.3. The limit in a partial metric space is not unique. Theorem 1.4. Let (Y, d0 ) be a subspace of metric space (X, d). If (X, d) is a complete metric space and Y is a closed set in X, then (Y, d0 ) is a complete metric space. Lemma 1.5. [2, 33] Let (X, p) be a partial metric space. (a) {xn } is a Cauchy sequence in (X, p) if and only if it is a Cauchy sequence in the metric space (X, dp ).

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3 (b) A partial metric space (X, p) is complete if and only if the metric space (X, dp ) is complete. Furthermore, limn−→∞ dp (xn , x) = 0 if and only if p(x, x) = lim p(xn , x) = n−→∞

lim

n,m−→∞

p(xn , xm ).

Let (X, p) be a partial metric. We endow the product space X × X with the partial metric q defined as follows: for (x, y), (u, v) ∈ X × X,

q((x, y), (u, v)) = p(x, u) + p(y, v).

A mapping F : X × X → X is said to be continuous at (x, y) ∈ X × X is for each ε > 0, there exists δ > 0 such that F (Bq ((x, y), δ)) ⊂ Bp ((x, y), ε). The concept of a mixed monotone property has been introduced by Bhaskar and Lakshmikantham [18]. Definition 1.6 ([18]). Let (X, ) be a partial ordered set. A mapping F : X × X → X is said to be have mixed monotone property if F (x, y) is monotone non-decreasing in x and is monotone non-increasing in y, that is, for any x, y ∈ X x1 , x2 ∈ X, x1  x2

⇒

F (x1 , y)  F (x2 , y)

y1 , y2 ∈ X, y1  y2

⇒

F (x, y1 )  F (x, y2 ).

and

Definition 1.7 ([18]). Let F : X × X → X. An element (x, y) ∈ X × X is said to be a coupled fixed point of a mapping F if x = F (x, y) and y = F (y, x). Let Φ denote the set of all functions φ : [0, ∞) → [0, ∞) which satisfy (φ1) (φ2) (φ3) (φ4)

φ is continuous and non-decreasing, φ(t) = 0 if and only if t = 0, φ(t + s) ≤ φ(t) + φ(s) for all t, s ∈ [0, ∞), φ(αt) ≤ αφ(t) for all α ∈ (0, ∞).

and let Ψ denote the set of all functions ψ : [0, ∞) → [0, ∞) which satisfy lim ψ(t) > 0 for all r > 0 and lim+ ψ(t) = 0. t→r

t→0

Alsulami, Hussain and Alotaibi [4] proved some coupled fixed point results for (φ, ϕ)- weakly contractive mappings in ordered partial metric spaces. More precisely, they obtained the following results.

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4 Theorem 1.8. [4, Theorem 3.1] Let (X, ) be a partially ordered set and suppose there is a partial metric p on X such that (X, p) is a complete partial metric space. Let F : X ×X → X be a mapping having the mixed monotone property on X. Suppose that there exists φ ∈ Φ and ψ ∈ Ψ such that   1 p(x, u) + p(y, v) φ(p(F (x, y), F (u, v))) ≤ φ(p(x, u) + p(y, v)) − ψ (1.1) 2 2 for all x, y, u, v ∈ X for which x  u and y  v. If there exists x0 , y0 ∈ X such that x0  F (x0 , y0 ) and y0  F (y0 , x0 ), and also suppose either (a) F is continuous or (b) X has the following property: (i) if a non-decreasing sequence {xn } is such that xn → x,then xn  x for all n, (ii) if a non-increasing sequence {yn } is such that yn → y,then y  yn for all n, then F has a coupled fixed point in X, that is there exists (x, y) ∈ X×X such that x = F (x, y) and y = F (y, x). Starting from the results in Alsulami, Hussain and Alotaibi [4], our main aim in this paper is to obtain more general coupled fixed point theorems for mixed monotone operators F : X → X satisfying a contractive condition which is significantly more general that the corresponding conditions (1.1) in [4], thus extending many other related results in literature. We also provide illustrative example in support of our results. 2. Coupled fixed points for generalized weakly contractive mappings We start with an example which shows the weakness of Theorem 1.8. Example 2.1. Let X = R+ be a set endowed with order x  y ⇔ x ≤ y. Let p(x, y) = max{x, y}, then (X, p) is a partial metric space. Define the mapping F : X × X → X by F (x, y) =

5x − 2y for all x, y ∈ X. 8

Then the following properties hold: (1) F is mixed monotone; (2) the condition (1.1) does not hold. Indeed, we show that F does not satisfy condition (1.1). Assume on the contrary, that there exist φ and ψ, such that (1.1) holds. Therefore it implies that there exist φ ∈ Φ and ψ ∈ Ψ such that, for all x ≥ u and y ≤ v, we have      5x − 2y 5x − 2y 5u − 2v = φ max , φ 8 8 8 = φ(p(F (x, y), F (u, v)))

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5 1 φ(p(x, u) + p(y, v)) − ψ 2   1 x+v = φ(x + v) − ψ . 2 2

≤



p(x, u) + p(y, v) 2



Setting x = 5, y = 1/2 and v = 1 to the last inequality, we get, since ψ(3) > 0, that       1 1 1 φ(6) − ψ(3) < φ(6) ≤ 2φ(3) = φ(3) φ(3) ≤ 2 2 2 which gives a contradiction. Hence F does not satisfy condition (1.1). Notice, however, that (0, 0) ∈ X 2 is the coupled fixed point of F .  We now state and prove our first result which successively guarantees the existence of a coupled fixed point and generalizes Theorem 3.1 in [4]. Theorem 2.2. Let (X, ) be a partially ordered set and suppose there is a partial metric p on X such that (X, p) is a complete partial metric space. Let F : X × X → X be a mapping having the mixed monotone property on X. Suppose that there exist φ ∈ Φ and ψ ∈ Ψ such that   p(x, u) + p(y, v) φ,ψ MF (x, y, u, v) ≤ φ(p(x, u) + p(y, v)) − 2ψ (2.1) 2 for all x, y, u, v ∈ X for which x  u and y  v, where MFφ,ψ (x, y, u, v) = φ(p(F (x, y), F (u, v))) + φ(p(F (y, x), F (v, u))). If there exist two elements x0 , y0 ∈ X such that x0  F (x0 , y0 ) and y0  F (y0 , x0 ), and also suppose either (a) F is continuous or (b) X has the following property: (i) if a non-decreasing sequence {xn } is such that xn → x,then xn  x for all n, (ii) if a non-increasing sequence {yn } is such that yn → y,then y  yn for all n, then F has a coupled fixed point in X, that is there exists (x, y) ∈ X×X such that x = F (x, y) and y = F (y, x). Proof. Let x0 , y0 ∈ X such that x0  F (x0 , y0 ) and y0  F (y0 , x0 ). We construct sequence {xn } and {yn } in X as xn+1 = F (xn , yn ) and yn+1 = F (xn , yn ) for all n ≥ 0.

(2.2)

Next, we show that xn  xn+1 for all n ≥ 0

(2.3)

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6 and yn  yn+1 for all n ≥ 0.

(2.4)

For this we shall use mathematical induction. Let n = 0. Since x0  F (x0 , y0 ) and y0  F (y0 , x0 ) and as x1 = F (x0 , y0 ) and y1 = F (y0 , x0 ), we have x0  x1 and y0  y1 . Thus (2.3) and (2.4) hold for n = 0. Suppose now that (2.3) and (2.4) hold for some fixed n ≥ 0, then, since xn  xn+1 and yn  yn+1 , we have xn+2 = F (xn+1 , yn+1 )  F (xn , yn+1 )  F (xn , yn ) = xn+1

(2.5)

yn+2 = F (yn+1 , xn+1 )  F (yn+1 , xn )  F (yn , xn ) = yn+1 .

(2.6)

and

Using (2.5) and (2.6), we get xn+1  xn+2 and yn+1  yn+2 . Hence, by the induction method we conclude that (2.3) and (2.4) hold for all n ≥ 0. Therefore, x0  x1  x2  · · ·  xn  xn+1  · · ·

(2.7)

y0  y1  y2  · · ·  yn  yn+1  · · ·.

(2.8)

and

For each n ≥ 0, let ξn+1 = p(xn+1 , xn ) + p(yn , yn+1 ). Since xn  xn−1 and yn  yn−1 , using (2.1) and (2.2), we have φ(p(xn+1 , xn )) + φ(p(yn , yn+1 )) = φ(p(F (xn , yn ), F (xn−1 , yn−1 ))) + φ(p(F (yn−1 , xn−1 ), F (yn , xn ))) = MFφ,ψ (xn , yn , xn−1 , yn−1 ) ≤ φ(p(xn , xn−1 ) + p(yn , yn−1 ))  p(x , x ) + p(y , y )  n n−1 n n−1 . −2ψ 2

(2.9)

By property (φ3), we have φ(p(xn+1 , xn ) + p(yn , yn+1 )) ≤ φ(p(xn , xn−1 ) + p(yn , yn−1 ))  p(x , x ) + p(y , y )  n n−1 n n−1 −2ψ , 2

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7 which implies, since ψ is a non-negative function, φ(p(xn+1 , xn ) + p(yn , yn+1 )) ≤ φ(p(xn , xn−1 ) + p(yn , yn−1 ));

(2.11)

that is φ(ξn+1 ) ≤ φ(ξn ) for all n ≥ 0.

(2.12)

Using the fact that φ is a non-decreasing, we get ξn+1 ≤ ξn for all n ≥ 0. This shows that {ξn } is decreasing. Therefore is some ξ ≥ 0 such that lim ξn = lim [p(xn+1 , xn ) + p(yn+1 , yn )] = ξ.

n→∞

n→∞

(2.13)

We shall prove that ξ = 0. Suppose, to the contrary, that ξ > 0. Then taking the limit as n → ∞ (equivalently, ξn → ξ) of both sides of (2.10) and remembering limt→r ψ(t) > 0 for all r > 0 and φ is continuous, we have h  ξ i n−1 φ(ξ) = lim φ(ξn ) ≤ lim φ(ξn−1 ) − 2ψ n→∞ n→∞ 2 ξ  n−1 = φ(ξ) − 2 lim ψ < φ(ξ), ξn−1 →ξ 2 which gives a contradiction. Thus ξ = 0, that is, lim ξn = lim [p(xn+1 , xn ) + p(yn+1 , yn )] = 0.

n→∞

n→∞

(2.14)

Let ξnp = dp (xn+1 , xn ) + dp (yn , yn+1 ) for all n ∈ N. From the definition of dp , it is clear that ξnp ≤ 2ξn for all n ∈ N. Using (2.14), we get lim ξnp = lim [dp (xn+1 , xn ) + dp (yn+1 , yn )] = 0.

n→+∞

n→+∞

Now, we prove that {xn } and {yn } are Cauchy sequences in the partial metric space (X, p). From Lemma 1.5 (a), it is sufficient to prove that {xn } and {yn } are Cauchy sequences in the metric space (X, dp ). Suppose, to the contrary, that at least one of {xn } or {yn } is not a Cauchy sequence. Then there exists an ε > 0 for which we can find subsequences {xnk }, {xmk } of {xn } and {ynk }, {ymk } of {yn } with nk > mk ≥ k such that dp (xnk , xmk ) + dp (ynk , ymk ) ≥ ε.

(2.15)

Further, corresponding to mk , we can choose nk in such a way that it is the smallest integer with nk > mk and satisfying (2.15). Then dp (xnk −1 , xmk ) + dp (ynk −1 , ymk ) < ε.

(2.16)

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8 Using (2.15), (2.16) and the triangle inequality, we have ε ≤ rkp := dp (xnk , xmk ) + dp (ynk , ymk ) ≤ dp (xnk , xnk −1 ) + dp (xnk −1 , xmk ) + dp (ynk , ynk −1 ) + dp (ynk −1 , ymk ) ≤ dp (xnk , xnk −1 ) + dp (ynk , ynk −1 ) + ε. Letting k → ∞ and using (2.14), we get lim rkp = lim [dp (xnk , xmk ) + dp (ynk , ymk )] = ε.

k→∞

(2.17)

k→∞

By the triangle inequality, rkp = ≤ + =

dp (xnk , xmk ) + dp (ynk , ymk ) dp (xnk , xnk +1 ) + dp (xnk +1 , xmk +1 ) + dp (xmk +1 , xmk ) dp (ynk , ynk +1 ) + dp (ynk +1 , ymk +1 ) + dp (ymk +1 , ymk ) p ξnpk + ξm + dp (xnk +1 , xmk +1 ) + dp (ynk +1 , ymk +1 ). k

Using the properties of φ, we have p φ(rkp ) ≤ φ(ξnpk + ξm + dp (xnk +1 , xmk +1 ) + dp (ynk +1 , ymk +1 )) k p p ≤ φ(ξnk + ξmk ) + φ(dp (xnk +1 , xmk +1 )) + φ(dp (ynk +1 , ymk +1 )).

(2.18)

Now, let rk = p(xnk , xmk ) + p(ynk , ymk ). By the definition of rkp , we have rkp = = + =

dp (xnk , xmk ) + dp (ynk , ymk ) 2p(xnk , xmk ) − p(xnk , xnk ) − p(xmk , xmk ) 2p(ynk , ymk ) − p(ynk , ynk ) − p(ymk , ymk ) 2rk − p(xnk , xnk ) − p(xmk , xmk ) − p(ynk , ynk ) − p(ymk , ymk ).

(2.19)

In view of property (p2) and (2.14), we have lim p(xnk , xnk ) =

k→+∞

= =

lim p(xmk , xmk )

k→+∞

lim p(ynk , ynk )

k→+∞

lim p(ymk , ymk ) = 0.

k→+∞

Therefore, letting k → +∞ in (2.19) and using (2.17), we get ε lim rk = . k→+∞ 2 Since xnk  xmk and ynk  ymk , we have φ(dp (xnk +1 , xmk +1 )) + φ(dp (ynk +1 , ymk +1 ))

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9 ≤ ≤ = = ≤

φ(2p(xnk +1 , xmk +1 )) + φ(2p(ynk +1 , ymk +1 )) 2φ(p(xnk +1 , xmk +1 )) + 2φ(p(ynk +1 , ymk +1 )) 2φ(p(F (xnk , ynk )), p(F (xmk , ymk ))) + 2φ(p(F (ynk , xnk )), p(F (ymk , xmk ))) 2MFφ,ψ (xnk , ynk , xmk , ymk ) 2φ(p((xnk , xmk )) + p((ynk , ymk )))  p((x , x )) + p((y , y ))  nk mk nk mk −4ψ r  2 k = 2φ(rk ) − 4ψ . 2

(2.20)

Thus, from (2.18), we have p ) + 2φ(rk ) − 4ψ φ(rkp ) ≤ φ(ξnpk + ξm k

r  k

2

.

Letting k → +∞, and using the properties of φ and ψ together with the inequalities established above, we have rk ε rk φ(ε) ≤ φ(0) + 2φ( ) − 4 lim ψ( ) ≤ φ(ε) − 4 rlimε ψ( ) k→ k→+∞ 2 2 2 2 4 ≤ φ(ε) − 4 limε ψ(t) t→ 4

< φ(ε)

(2.21)

which is a contradiction. Therefore, {xn } and {yn } are Cauchy sequences in the complete metric space (X, dp ). Thus, there are x, y ∈ X such that lim dp (xn , x) = lim dp (yn , y) = 0,

n→+∞

(2.22)

n→+∞

which implies that lim F (xn , yn ) = lim xn = x

n→+∞

n→+∞

lim F (yn , xn ) = lim yn = y.

n→+∞

(2.23)

n→+∞

Therefore, from Lemma 1.5 (b), using (2.14) and the property (p2), we have p(x, x) = lim p(xn , x) = lim p(xn , xn ) = 0,

(2.24)

p(y, y) = lim p(yn , y) = lim p(yn , yn ) = 0.

(2.25)

n→+∞

n→+∞

n→+∞

n→+∞

On utilizing p(x, x) = p(y, y) = 0 in (2.1), we get φ(p(F (x, y), F (x, y))) + φ(p(F (y, x), F (y, x)))  p(x, x) + p(y, y)  ≤ φ(p(x, x) + p(y, y)) − 2ψ 2

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10 = φ(0) − 2ψ(0) = −2ψ(0) ≤ 0, which implies φ(p(F (x, y), F (x, y))) + φ(p(F (y, x), F (y, x))) = 0. Therefore, from the property (φ2), we get p(F (x, y), F (x, y)) = 0 = p(F (y, x), F (y, x)).

(2.26)

We now show that x = F (x, y) and y = F (y, x). Suppose that the assumption (a) holds. For any given ε > 0, the commutativity of F at a point (x, y) implies that there exists ξ > 0 such that if (u, v) ∈ X × X with q((x, y), (u, v)) < q((x, y), (x, y)) + ξ = ξ, meaning that p(x, u) + p(y, v) < p(x, x) + p(y, y) + ξ = ξ, because p(x, x) = p(y, y) = 0, then we have ε p(F (x, y), F (u, v)) < p(F (x, y), F (x, y)) + . 2

(2.27)

Since limn→+∞ p(xn , x) = limn→+∞ p(yn , y) = 0, and ξ = min{ 2ξ , 2ε } > 0, there exist n0 ∈ N such that for any n ≥ n0 , p(xn , x) < ξ and p(yn , y) < ξ, which gives that p(xn , x) + p(yn , y) < 2ξ < ξ. Using (2.27), we get that ε p(F (x, y), F (xn , yn )) < p(F (x, y), F (x, y)) + . 2

(2.28)

Then, for any n ≥ n0 , we have p(F (x, y), x) ≤ p(F (x, y), xn+1 ) + p(xn+1 , x) = p(F (x, y), F (xn , yn ) + p(xn+1 , x) ε ≤ p(F (x, y), F (x, y)) + + ξ 2 ≤ p(F (x, y), F (x, y)) + ε.

(2.29)

From (2.26), we have p(F (x, y), x) < ε. Since ε is arbitrary, we can conclude that p(F (x, y), x) = 0.

(2.30)

Similarly, we show that p(F (y, x), y) = 0. These together with (2.26) and (p1) imply that F (x, y) = x and F (y, x) = y. Hence, (x, y) is a coupled fixed point of F .

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11 Finally, suppose that (b) holds. By (2.3), (2.22) and (2.24), we have {xn } is a nondecreasing sequence, xn → x and {yn } is a non-increasing sequence, yn → y as n → ∞. Hence, by the assumption (b), we have for all n ≥ 0, xn  x and y  yn .

(2.31)

By property (p4), we have p(F (x, y), x) ≤ p(F (x, y), xn+1 ) + p(xn+1 , x) = p(F (x, y), F (xn , yn )) + p(xn+1 , x) and p(F (y, x), y) ≤ p(F (y, x), yn+1 ) + p(yn+1 , y) = p(F (y, x), F (yn , xn )) + p(yn+1 , y) Therefore, φ(p(F (x, y), x)) + φ(p(F (y, x), y)) ≤ φ(p(xn+1 , x)) + φ(p(yn+1 , y)) + φ(p(F (x, y), F (xn , yn ))) + φ(p(F (y, x), F (yn , xn )))  (p(x, x ) + p(y, y )  n n ≤ φ(p(xn+1 , x)) + φ(p(yn+1 , y)) + φ(p(x, xn ) + p(y, yn )) − 2ψ . 2 Taking limit as n → ∞ in the above inequality, using (2.24) and (2.25) and the properties of φ and ψ, we get φ(p(x, F (x, y))) = 0 = φ(p(y, F (y, x))), which implies p(x, F (x, y)) = 0 = p(y, F (y, x)). These together with (2.26), we have x = F (x, y) and y = F (y, x) Hence, (x, y) is a coupled coincidence point of F . This complete the proof.



Remark 2.3. Theorem 2.2 is more general than [4, Theorem 3.1], since the contractive condition (2.1) is weaker than (1.1), a fact which is clearly illustrated by the following example. Example 2.4. Let us recall Example 2.1. Define the mappings φ, ψ : [0, ∞) → [0, ∞) by φ(t) = 2t and ψ(t) =

t for all t ∈ [0, ∞), 2

We show that F, φ and ψ satisfy condition (2.1). For all x ≥ u and y ≤ v, we observe that φ(p(F (x, y), F (u, v))) + φ(p(F (y, x), F (v, u))) =

5x − 2y 5v − 2u + , 4 4

φ(p(x, u) + p(y, v)) = φ(x + v) = 2(x + v),

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12 and 

p(x, u) + p(y, v) 2ψ 2





x+v = 2ψ 2

 =

x+v . 2

Furthermore, we can have the following fact x + v ≥ −2u − 2y ⇔ 6x + 6v ≥ 5x − 2y + 5v − 2u 3 5x − 2y 5v − 2u ⇔ (x + v) ≥ + . 2 4 4 Therefore, we arrive that 5x − 2y 5v − 2u + 4 4 3 ≤ (x + v) 2 x+v = 2(x + v) − 2

φ(p(F (x, y), F (u, v))) + φ(p(F (y, x), F (v, u))) =



 p(x, u) + p(y, v) = φ(p(x, u) + p(y, v)) − 2ψ . 2 Hence F, φ and ψ satisfy (2.1). By Theorem 2.2, we conclude that F has a coupled fixed point in X. Moreover, (0, 0) ∈ X 2 is a coupled fixed point of F .  As an immediate consequence of the above theorem, by taking φ(t) = t, we have: Corollary 2.5. Let (X, ) be a partially ordered set and suppose there is a partial metric p on X such that (X, d) is a complete partial metric space. Let F : X × X → X be a mapping having the mixed monotone property on X. Assume that there exist two elements x0 , y0 ∈ X with x0  F (x0 , y0 ) and y0  F (y0 , x0 ). Suppose there exist φ ∈ Φ and ψ ∈ Ψ such that  p(x, u) + p(y, v)  1 p(F (x, y), F (u, v)) ≤ (p(x, u) + p(y, v)) − ψ 2 2 for all x, y, u, v ∈ X with x  u and y  v. Suppose either (a) F is continuous or (b) X has the following property: (i) if a non-decreasing sequence xn → x,then xn  x for all n, (ii) if a non-decreasing sequence yn → y,then y  yn for all n. Then there exist x, y, ∈ X such that x = F (x, y) and y = F (y, x), that is F has a coupled fixed point in X.

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13 Moreover, if we take ψ(t) =

1−k t 2

where k ∈ [0, 1) in Corollary 2.5, we get:

Corollary 2.6. Let (X, ) be a partially ordered set and suppose there is a partial metric p on X such that (X, d) is a complete partial metric space. Let F : X × X → X be a mapping having the mixed monotone property on X. Assume that there exist two elements x0 , y0 ∈ X with x0  F (x0 , y0 ) and y0  F (y0 , x0 ). Suppose there exist φ ∈ Φ and ψ ∈ Ψ such that 1 p(F (x, y), F (u, v)) ≤ (p(x, u) + p(y, v)) 2 for all x, y, u, v ∈ X with x  u and y  v. Suppose either (a) F is continuous or (b) X has the following property: (i) if a non-decreasing sequence xn → x,then xn  x for all n, (ii) if a non-decreasing sequence yn → y,then y  yn for all n. Then there exist x, y, ∈ X such that x = F (x, y) and y = F (y, x), that is F has a coupled fixed point in X. Acknowledgements. The authors would like to thank Naresuan University and the Thailand Research Fund for financial support. References [1] Banach, S: Surles operations dans les ensembles et leur application aux equation sitegrales, Fund. Math. 3, 133181 (1922) [2] Matthews, SG: Partial metric topology, Proc. 8th Summer Conference on General Topology and Applications, Ann. New York Acad. Sci. 728, 183197 (1994) [3] Abbas, M, Khan, SH, Nazir, T: Common fixed points of R-weakly commuting maps in generalized metric spaces. Fixed Point Theory Appl. 2011, 41 (2011) [4] Alsulami, S, Hussain, N, Alotaibi, A: Coupled Fixed and Coincidence Points for Monotone Operators in Partial Metric Spaces, Fixed Point Theory and Applications 2012, 2012:173 [5] Abdeljawad, TH, Karapinar, E, Tas, K: Existence and uniqueness of a common fixed point on partial metric spaces, Applied Mathematics Letters 24, 19001904 (2011) [6] Abdeljawad, TH, Karapinar, E, Tas, K: A generalized contraction principle with control functions on partial metric spaces, Computers and Mathematics with Applications 6, 716719 (2012) [7] Abdeljawad, TH: Fixed points for generalized weakly contractive mappings in partial metric spaces, Mathematical and Computer Modelling 54 29232927 (2011) [8] Altun, I, Erduran, A: Fixed point theorems for monotone mappings on partial metric spaces, Fixed Point Theory and Applications 2011, Article ID 508730, 10 pages (2011) [9] Altun, I, Sola, F, Simsek, H: Generalized contractions on partial metric spaces, Topology and Its Applications, 157, 27782785 (2010)

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14 [10] Agarwal, RP, Alghamdi, MA, Shahzad, N: Fixed point theory for cyclic generalized contractions in partial metric spaces, Fixed Point Theory and Applications 2012, 2012:40 [11] Agarwal, RP, El-Gebeily, MA, O’Regan, D: Generalized contractions in partially ordered metric spaces. Appl. Anal. 87, 1–8 (2008) [12] Aydi, H: Some fixed point results in ordered partial metric spaces, The J. Nonlinear Sci. Appl. 4, 112 (2011) [13] Aydi, H: Some coupled fixed point results on partial metric spaces, International Journal of Mathematical Sciences 2011, Article ID 647091, 11 pages (2011) [14] Aydi, H: Fixed point theorems for generalized weakly contractive condition in ordered partial metric spaces, Journal of Nonlinear Analysis and Optimization: Theory and Applications 2, 33.48 (2011) [15] Aydi, H, Karapinar, E, Shatanawi, W: Coupled fixed point results for (ψ, φ)-weakly contractive condition in ordered partial metric spaces, Computer and Mathematics with Applications 62, 4449.4460 (2011) [16] Aydi, H, Postolache, M, Shatanawi, W: Coupled fixed point results for (ψ, φ)-weakly contractive mappings in ordered G-metric spaces, Computers and Mathematics with Applications 63, 298.309 (2012) [17] Berinde, V: Coupled coincidencepointtheorems for mixedmonotone nonlinear operators, Computers and Mathematics with Applications (2012), doi:10.1016/j.camwa.2012.02.012 [18] Bhaskar, TG, Lakshmikantham, V: Fixed point theorems in partially ordered metric spaces and applications. Nonlinear Anal. 65, 1379–1393 (2006) [19] Cho, YJ, Rhoades, BE, Saadati, R, Samet, B, Shatanawi, W: Nonlinear coupled fixed point theorems in ordered generalized metric spaces with integral type, Fixed Point Theory and Applications 2012, 2012:8, doi:10.1186/1687-1812-2012-8 [20] GoluboviLc, Z, Kadelburg, Z, RadenoviLc, S: Coupled coincidence points of mappings in ordered partial metric spaces, Abstract and Applied Analysis 2012, Article ID 192581, 10 pages (2012). [21] Heckmann, R: Approximation of metric spaces by partial metric spaces, Appl. Categ. Structures 7, 71.83 (1999) [22] Karapinar, E: Generalizations of Caristi Kirkfs theorem on partial metric spaces, Fixed Point Theory and Appl. (in press) (2011). [23] Karapinar, E, Erhan, IM: Fixed point theorems for operators on partial metric spaces, Applied Mathematics Letters (2011), 10.1016/j.aml.2011.05.013. [24] Kirk, WA, Srinavasan PS, Veeramani P: Fixed points for mapping satisfying cyclical contractive conditions, Fixed Point Theory 4 79-89 (2003) [25] Karapinar, E, Erhan, IM: Best proximity point on different type contractions, Appl. Math. Inf. Sci. 5 342-353 (2011) [26] Karapinar, E, Erhan,IM, Ulus, AY: Fixed point theorem for cyclic maps on partial metric spaces, Appl. Math. Inf. Sci. 6, 239.244 (2012) [27] Karapinar, E, Erhan, IM: Cyclic contractions and fixed point theorems, Filomat 26, 777-7-82 (2012) [28] Khan, MS, Swaleh M, Sessa, S: Fixed point theorems by altering distances between the points, Bull. Aust. Math. Soc. 30, 1.9 (1984) ´ c, LJ., Coupled fixed point theorems for nonlinear contractions in partially [29] Lakshmikantham, V, Ciri´ ordered metric spaces, Nonlinear Anal. 70 (2009) 4341-4349. [30] Lakzian, H, Samet, B: Fixed points for (ψ, φ) -weakly contractive mappings in generalized metric spaces, Applied Mathematics Letters 25 , 902.906 (2012) [31] Nashine, HK, Kadelburg, Z, Radenovi´c, S: Coupled common fixed point theorems for w∗ -compatible mappings in ordered cone metric spaces. Appl. Math. Comput. 218, 5422–5432 (2012) [32] Nashine, HK, Kadelburg, Z, RadenoviLc, S: Common fixed point theorems for weakly isotone increasing mappings in ordered partial metric spaces, Mathematical and Computer Modelling (2012) Article in Press. [33] Oltra,S, Valero, O: Banachfs fixed point theorem for partial metric spaces, Rend. Istid. Math. Univ. Trieste 36, 17-26 (2004) [34] ] Rus, IA, Cyclic representations and fixed points, Ann. T. Popoviciu Seminar Funct. Eq. Approx.

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15 Convexity 3 (2005) 171-178 [35] Romaguera, S: A Kirk type characterization of completeness for partial metric spaces, Fixed Point Theory Appl. 2010, Article ID 493298, 6 pages (2010) [36] Radenovi´c, S, Kadelburg, Z: Generalized weak contractions in partially ordered metric spaces. Comput. Math. Appl. 60, 1776–1783 (2010) [37] Samet, B, RajoviLc, M, LazoviLc, R, StoiljkoviLc, R: Common fixed point results for nonlinear contractions in ordered partial metric spaces, Fixed Point Theory Appl., 2011:71 (2011) [38] Shatanawi, W, Abbas, M, Nazir, T: Common coupled fixed points results in two generalized metric spaces. Fixied Point Theory Appl. 2011, 80 (2011). doi:10.1186/1687-1812-2011-80 [39] Sintunavarat, W, Cho, YJ, Kumam, P: Coupled coincidence point theorems for contractions without commutative condition in intuitionistic fuzzy normed spaces. Fixed Point Theory Appl. 2011, 81 (2011) [40] Shatanawi, W, Al-Rawashdeh, A: Common fixed points of almost generalized (ψ, φ)-contractive mappings in ordered metric spaces, Accepted in Fixed Point Theory and Applications. [41] Shatanawi, W, Mustafa, Z, Tahat, N: Some coincidence point theorems for nonlinear contraction in ordered metric spaces, Fixed point Theory and Applications 2011, 2011:68. [42] Shatanawi, W, Samet, B: On (ψ, φ)-weakly contractive condition in partially ordered metric spaces, Computers and Mathematics with Applications 62, 3204.3214 (2011) [43] Shatanawi, W, Nashine, HK: A generalization of Banachfs contraction principle for nonlinear contraction in a partial metric space, J. Nonlinear Sci. Appl. 5, 37.43 (2012) [44] Valero, O: On Banach fixed point theorems for partial metric spaces, Appl. General Topology 6, 229-240 (2005)

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A note on structures of fuzzy approximation spaces ∗ Gangqiang Zhang† Yu Han‡ Zhaowen Li§ April 2, 2013 Abstract: In this paper, fuzzy rough approximation operators are further established. Topological structures of fuzzy approximation spaces are given. Keywords: Fuzzy set; Fuzzy relation; Fuzzy approximation space; Fuzzy topology.

1

Introduction

Rough set theory, proposed by Pawlak [9], is a mathematical tool for approximate reasoning about data. It may be seen as an extension of classical set theory and has been successfully applied to machine learning, intelligent systems, inductive reasoning, pattern recognition, mereology, image processing, signal analysis, knowledge discovery, decision analysis, expert systems and many other fields [10, 14, 15, 16]. The basic structure of rough set theory is an approximation space. Based on it, lower and upper approximations can be induced. Using these approximations, knowledge hidden in information systems may be revealed and expressed in the form of decision rules. Various fuzzy generalizations of rough approximations have been proposed [1, 8, 18, 20]. The most common fuzzy rough set is obtained by replacing the crisp relations with fuzzy relations on the universe and crisp subsets with fuzzy sets. An interesting and natural research topic in rough set theory is to study the relationship between rough sets or approximation spaces and topologies. Many authors studied topological properties of rough sets or approximation spaces [3, 6, 13, 22]. In the study of topological properties of fuzzy rough sets or fuzzy ∗ This work is supported by the National Natural Science Foundation of China (No.11061004). † Corresponding Author, College of Information Science and Engineering, Guangxi University for Nationalities, Nanning, Guangxi 530006, P.R.China. [email protected] ‡ College of Science, Guangxi University for Nationalities, Nanning, Guangxi 530006, P.R.China. [email protected] § College of Science, Guangxi University for Nationalities, Nanning, Guangxi 530006, P.R.China. [email protected]

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approximation spaces, Qin et al. [17] investigated the topological properties of fuzzy rough sets. The purpose of this paper is to investigate topological structures of fuzzy approximation spaces.

2

Preliminaries

Throughout this paper, X denotes a nonempty set. I denotes [0, 1]. F (X) denotes the set of all fuzzy sets in X. For a ∈ I, a ¯ denotes the constant fuzzy set in X. For each A ∈ F (X), we denote RA = {(x, y) ∈ X × X : A(x) > A(y)}. Obviously, RA = ∅ ⇐⇒ A = a ¯ for some a ∈ I. A fuzzy set is called a fuzzy point in X, if it takes the value 0 for each y ∈ X except one, say, x ∈ X. If its value at x is λ (0 < λ ≤ 1), we denote this fuzzy point by xλ , where the point x is called its support and λ is called its height (see [4, 12]). Denote P (X) = {xλ : x ∈ X, λ ∈ (0, 1]}. For a fuzzy point xλ and A ∈ F (X), we defined xλ ∈ A by xλ ⊆ A. Obviously, xλ ∈ A ⇐⇒ λ ≤ A(x). Definition 2.1 ([2]). σ ⊆ F (X) is called a fuzzy topology on X, if (i) For each a ∈ I, a ¯ ∈ σ. (ii) A, B ∈ σ =⇒ A ∩ B ∈Sσ, (iii) {Ai : i ∈ J} ⊆ σ =⇒ Ai ∈ σ. i∈J

In this case the pair (X, σ) is called a fuzzy topological space. Every member of σ is called a fuzzy open set in X. Its complement is called a fuzzy closed set in X. We denote σ c = {A ∈ F (X) : Ac ∈ σ}. Interior and closure of A denoted respectively by intσ (A) and clσ (A) for each A ∈ F (X), are defined as follows: [ \ intσ (A) = {B ∈ σ : B ⊆ A}, clσ (A) = {B ∈ σ c : B ⊇ A}. A fuzzy topology σ is called Alexandrov [5] if (ii) in Definition 2.2 is replaced by (ii)0 {Ai : i ∈ J} ⊆ σ =⇒

T

Ai ∈ σ.

i∈J

Definition 2.2 ([4]). Let (X, σ) be a fuzzy topological space and let xλ ∈ P (X) and A ∈ F (X). A is called a closed remote-neighborhood of xλ , if A ∈ σ c and xλ 6∈ A. 2

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Definition 2.3 ([4]). Let (X, σ) be a fuzzy topological space. (1) (X, σ) is called T−1 space, if for any xλ , xµ ∈ P (X) and µ < λ, there exists A such that xµ ∈ A and A is a closed remote-neighborhood of xλ , or, there exists B such that xλ ∈ B and B is a closed remote-neighborhood of xµ . (2) (X, σ) is called sub-T0 space, if for any x, y ∈ X and x 6= y, there exist λ ∈ (0, 1] and A such that yλ ∈ A and A is closed remote-neighborhood of xλ , or, there exist λ ∈ (0, 1] and B such that xλ ∈ B and B is a closed remoteneighborhood of yλ . Definition 2.4 ([21]). Let R be a crisp relation on X. For each x ∈ X, denote Rp (x) = {y ∈ X : (y, x) ∈ R} and Rs (x) = {y ∈ X : (x, y) ∈ R}. Rp (x) and Rs (x) are called the predecessor and successor neighborhood of x, respectively.

3

Fuzzy approximation spaces and fuzzy rough sets

Recall that R is called a fuzzy relation on X if R ∈ F (X × X). Definition 3.1. Let R be a fuzzy relation on X. Then R is called (1) reflexive, if R(x, x) = 1 for each x ∈ X. (2) symmetric, if R(x, y) = R(y, x) for any x, y ∈ X. (3) transitive, if R(x, z) ≥ R(x, y) ∧ R(y, z) for any x, y, z ∈ X. Let R be a fuzzy relation on X. R is called preorder (resp. equivalence) if R is reflexive and transitive (resp. reflexive, symmetric and transitive). Definition 3.2 ([11, 17]). Let R be a fuzzy relation on X. The pair (X, R) is called a fuzzy approximation space. For each A ∈ F (X), the fuzzy lower and the fuzzy upper approximation of A with respect to (X, R), denoted by R(A) and R(A) are respectively, defined as follows: ^ R(A)(x) = (A(y) ∨ (1 − R(x, y))) (x ∈ X) y∈X

and R(A)(x) =

_

(A(y) ∧ R(x, y))

(x ∈ X).

y∈X

The pair (R(A), R(A)) is called the fuzzy rough set of A with respect to (X, R). Remark 3.3. R(x1 )(y) = R(y, x) and R((x1 )c )(y) = 1 − R(y, x) (x, y ∈ X).

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Proposition 3.4 ([17, 19]). Let (X, R) be a fuzzy approximation space. Then for any A, B ∈ F (X), {Ai : i ∈ J} ⊆ F (X) and λ ∈ I, (1) R(¯ 1) = ¯ 1, R(¯ 0) = ¯ 0. (2) A ⊆ B =⇒ R(A) ⊆ R(B), R(A) ⊆ R(B). (3) R(Ac ) = (R(A))c , R(Ac ) = (R(A))c . ∩ R(B), R(A (4) R(A T∩ B) = R(A) T S ∪ B) = S R(A) ∪ R(B). (5) R( Ai ) = (R(Ai )), R( Ai ) = R(Ai ). i∈J

i∈J

i∈J

i∈J

Theorem 3.5 ([7, 11, 17]). Let (X, R) be a fuzzy approximation space. Then (1) R is ref lexive

⇐⇒ ⇐⇒ (2) R is symmetric ⇐⇒ ⇐⇒ (3) R is transitive ⇐⇒ ⇐⇒

(ILR) ∀A ∈ F (X), R(A) ⊆ A. (IU R) ∀A ∈ F (X), A ⊆ R(A). (ILS) ∀(x, y) ∈ X × X, R((x1 )c )(y) = R((y1 )c )(x). (IU S) ∀(x, y) ∈ X × X, R(x1 )(y) = R(y1 )(x). (ILT ) ∀A ∈ F (X), R(A) ⊆ R(R(A)). (IU T ) ∀A ∈ F (X), R(R(A)) ⊆ R(A).

Remark 3.6. (1) For each a ∈ I, R(a) ⊆ a ¯ ⊆ R(¯ a); (2) If R is reflexive, then for each a ∈ I, R(¯ a) = a ¯ = R(¯ a). Proposition 3.7. Let (X, R) be a fuzzy approximation space. Then for each A ∈ F (X) with RA 6= ∅, (1) a) R(A) ⊇ A ⇐⇒ (F LO)∀(x, y) ∈ RA , 1 − R(x, y) ≥ A(x) ∨ A(y). b) R(A) ⊆ A ⇐⇒ (F U O)∀(x, y) ∈ RA , R(y, x) ≤ A(x) ∧ A(y). (2) If R is reflexive, then a) R(A) = A ⇐⇒ (F LR)∀(x, y) ∈ RA , 1 − R(x, y) ≥ A(x) ∨ A(y). b) R(A) = A ⇐⇒ (F U R)∀(x, y) ∈ RA , R(y, x) ≤ A(x) ∧ A(y). Proof. (1) a) Necessity. Suppose that R(A) ⊇ A. Note that for each x ∈ X, ^ (A(y) ∨ (1 − R(x, y))) = (R(A))(y) ≥ A(x). y∈U

Then A(y) ∨ (1 − R(x, y)) ≥ A(x) for any x, y ∈ X. Since A(x) > A(y) for each (x, y) ∈ RA , we have 1 − R(x, y) ≥ A(x) = A(x) ∨ A(y)

((x, y) ∈ RA ).

Sufficiency. Suppose that (F LO) holds. Let x ∈ X. (i) If y ∈ (RA )s (x), then A(y) ∨ (1 − R(x, y)) ≥ A(y) ∨ (A(x) ∨ A(y)) ≥ A(x). (ii) If y 6∈ (RA )s (x), then A(y) ≥ A(x) and so A(y) ∨ (1 − R(x, y)) ≥ A(y) ≥ A(x).

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V

Hence R(A)(x) =

(A(y) ∨ (1 − R(x, y))) ≥ A(x).

y∈U

Thus R(A) ⊇ A. b) Necessity. Suppose that R(A) ⊆ A. Note that for each y ∈ X, _ (A(x) ∧ R(y, x)) = R(A)(y) ≤ A(y). x∈X

Then A(x) ∧ R(y, x) ≤ A(y) (x, y ∈ X) for any x, y ∈ X. Since A(x) > A(y) for each (x, y) ∈ RA , we have R(y, x) ≤ A(y) = A(x) ∧ A(y)

((x, y) ∈ RA ).

Sufficiency. Suppose that (F LO) holds. Let y ∈ X. (i) If x ∈ (RA )p (y), then (x, y) ∈ RA and so A(x) ∨ R(y, x) ≤ A(x) ∧ (A(x) ∧ A(y)) ≤ A(y). (ii) If x 6∈ (RA )p (y), then A(x) ≤ A(y) and so A(x) ∧ R(y, x) ≤ A(x) ≤ A(y). W (A(x) ∧ R(y, x)) ≤ A(y). Thus R(A) ⊆ A. Hence (R(A))(y) = x∈X

(2) This holds by (1) and Theorem 3.5(1).

4

Topological structures of fuzzy approximation spaces

Let (X, R) be a fuzzy approximation space. We denote σR = {A ∈ F (X) : A ⊆ R(A)}; sR =

^

R(x, y),

tR =

x,y∈X

_

R(x, y).

x,y∈X, x6=y

Theorem 4.1. Let (X, R) be a fuzzy approximation space. (1) σR is an Alexandrov fuzzy topology on X. (2) intσR (A) ⊆ R(A) and R(A) ⊆ clσR (A) (A ∈ F (X)). (3) A ∈ (σR )c ⇐⇒ A ⊇ R(A). (4) For each a ∈ I, a ¯ ∈ (σR )c . Proof. (1) (i) For each a ∈ I, by Remark 3.6(1), a ¯ ⊆ R(¯ a). Then a ¯ ∈ σR . (ii) Let {Ai : i ∈ J} ⊆ σR . Then Ai ⊆ R(Ai ) for each i ∈ J. By Proposition 3.4(5), \ \ \ Ai ⊆ R(Ai ) = R( Ai ). i∈J

i∈J

i∈J

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Hence

T

Ai ∈ σR .

i∈J

(iii) Let {Ai : i ∈ J} ⊆ σR . Then Ai ⊆ R(Ai ) for each i ∈ J. By Proposition 3.4(2), [ [ [ Ai ⊆ R(Ai ) ⊆ R( Ai ). Then

S

i∈J

i∈J

i∈J

Ai ∈ σR .

i∈J

Hence σR is an Alexandrov fuzzy topology on X. (2) For each A ∈ F (X), by Proposition 3.4(2), [ intσR (A) = {B ∈ σR : B ⊆ A} [ ⊆ {B ∈ σR : R(B) ⊆ R(A)} [ = {B ∈ F (X) : B ⊆ R(B) ⊆ R(A)} ⊆ R(A). By Proposition 3.4(3), clσR (A) = (intσR (Ac ))c ⊇ (R(Ac ))c = R(A) (A ∈ F (X)). (3) This holds by Proposition 3.4(3). (4) This holds by (3) and Remark 3.6(1). Theorem 4.2. Let (X, R) be a fuzzy approximation space. (1) (X, σR ) is not connected. (2) (X, σR ) is T−1 . (3) a) If tR < 1, then (X, σR ) is sub-T0 . b) If (X, σR ) is sub-T0 , then for any x, y ∈ U with x 6= y, R(x, y) ∧ R(y, x) < 1. (4) If R is preorder, then (X, σR ) is T1

⇐⇒ σR = F (X).

Proof. (1) This holds by Theorem 4.1(4). (2) For any xλ , xµ ∈ P (X) and µ < λ, put u ∈ (µ, λ), we have xλ 6∈ u ¯. By Theorem 4.1(4), u ¯ is a closed remote-neighborhood of xλ . Note that xµ ∈ u ¯. Then (X, σR ) is T−1 . (3) a) Let x, y ∈ X with x 6= y. By tR < 1, there exist λ ∈ (tR , 1]. Put A ∈ F (X) such that ( tR , t = x, A(t) = , λ, t 6= x then RA = {(t, x) : t ∈ X − {x}}. For each (t, x) ∈ RA , _ R(x, t) ≤ R(x, y) = tR = λ ∧ tR = A(t) ∧ A(x). x,y∈X,x6=y

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By Proposition 3.8(1), R(A) ⊆ A. By Theorem 4.1(3), A ∈ (σR )c . Note that A(x) = tR < λ. Then A is a closed remote-neighborhood of xλ . Obviously, yλ ∈ A. Hence (X, σR ) is sub-T0 . (4) b) For any x, y ∈ X and x 6= y, by (X, σR ) is sub-T0 , then there exist λ ∈ (0, 1] and A such that yλ ∈ A and A is a closed remote-neighborhood of xλ , or, there exist λ ∈ (0, 1] and B such that xλ ∈ B and B is a closed remote-neighborhood of yλ . (i) If there exist λ ∈ (0, 1] and A such that yλ ∈ A and A is a closed remoteneighborhood of xλ , we can obtain that A(y) ≥ λ and A(x) < λ. By Theorem 4.1(3) and A ∈ (σR )c , R(A) ⊆ A . By (y, x) ∈ RA and Proposition 3.8(1), R(x, y) ≤ A(x) ∧ A(y) = A(x) < λ ≤ 1. Then R(x, y) < 1. (ii) If there exist λ ∈ (0, 1] and B such that xλ ∈ B and B is a closed remote-neighborhood of yλ , similarly, we can prove that R(y, x) < 1. So R(x, y) ∧ R(y, x) < 1. Definition 4.3. Let R be a fuzzy relation on X. R is called pseudo-constant if there exists a ∈ I such that for any x, y ∈ X, ( 1, if x = y, R(x, y) = a, if x 6= y. We write R by a∗ . Obviously, every pseudo-constant fuzzy relation is an equivalence fuzzy relation. Remark 4.4. (1) For any a, b ∈ I, a ≤ b implies a∗ ⊆ b∗ . (2) For each a ∈ I, θa∗ = σa∗ . (3) σ0∗ = F (X), σ1∗ = {¯ a : a ∈ I}. Remark 4.5. Let R be a fuzzy relation on X. Then (1) R ⊆ t∗R . (2) R is reflexive ⇐⇒ s∗R ⊆ R. Lemma 4.6. Let R1 and R2 be two fuzzy relations on U . If R1 ⊆ R2 , then σR2 ⊆ σR1 . Proof. Let A ∈ σR2 . Then A ⊆ R2 (A). Note that R2 (A) ⊆ R1 (A) by R1 ⊆ R2 and Proposition 3.9(1). Then A ⊆ R1 (A) and so A ∈ σR1 . Thus σR2 ⊆ σR1 . Theorem 4.7. Let (X, R) be a fuzzy approximation space. Then (1) σR ⊇ σt∗R . (2) If R is reflexive, then σt∗R ⊆ τR ⊆ σs∗R . (3) σR = σ1∗ ∪ {A ∈ F (X) : ∀(x, y) ∈ RA , A(x) ∨ A(y) ≤ 1 − R(x, y)}. 7

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Proof. (1) This holds by Remark 4.5(1) and Lemma 4.6. (1) This holds by Remark 4.5 and Lemma 4.6. (3) This holds by Proposition 3.8(1), Theorem 4.1(1) and Remark 4.4(3).

References [1] D.Dubois, H.Prade, Rough fuzzy sets and fuzzy rough sets, International Journal of General Systems, 17(1990), 191-208. [2] R.Lowen, Fuzzy topological spaces and fuzzy compactness, Journal of Mathematical Analysis and Applications, 56(1976), 621-633. [3] E.F.Lashin, A.M.Kozae, A.A.Abo Khadra, T.Medhat, Rough set theory for topological spaces, International Journal of Approximate Reasoning, 40(2005), 35-43. [4] Y.Liu, M.Luo, Fuzzy Topology, World Scientific Publishing, Singapore, 1998. [5] H.Lai, D.Zhang, Fuzzy preorder and fuzzy topology, Fuzzy Sets and Systems, 157(2006), 1865-1885. [6] Z.Li, T.Xie, Q.Li, Topological structure of generalized rough sets, Computers and Mathematics with Applications, 63(2012), 1066-1071. [7] J.Mi, W.Wu, W.Zhang, Constructive and axiomatic approaches for the study of the theory of rough sets, Pattern Recognition Artificial Intelligence, 15(2002), 280-284. [8] S.Nanda, Fuzzy rough sets, Fuzzy Sets and Systems, 45(1992), 157-160. [9] Z.Pawlak, Rough sets, International Journal of Computer and Information Science, 11(1982), 341-356. [10] Z.Pawlak, Rough Sets: Theoretical Aspects of Reasoning about Data, Kluwer Academic Publishers, Dordrecht, 1991. [11] D.Pei, A generalized model of fuzzy rough sets, International Journal of General Systems, 34(5)(2005), 603-613. [12] B.Pu, Y.Liu, Fuzzy Topology I. Neighborhood strucure of a fuzzy point and Moore-Smiith convergence, Journal of Mathematical Analysis and Applications, 79(1980), 571-599. [13] Z.Pei, D.Pei, L.Zheng, Topology vs generalized rough sets, International Journal of Approximate Reasoning, 52(2011), 231-239. [14] Z.Pawlak, A.Skowron, Rudiments of rough sets, Information Sciences, 177(2007), 3-27.

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[15] Z.Pawlak, A.Skowron, Rough sets: some extensions, Information Sciences, 177(2007), 28-40. [16] Z.Pawlak, A.Skowron, Rough sets and Boolean reasoning, Information Sciences, 177(2007), 41-73. [17] K.Qin, Z.Pei, On the topological properties of fuzzy rough sets, Fuzzy Sets and Systems, 151(2005), 601-613. [18] A.M. Radzikowska, E.E.Kerre, A comparative study of fuzzy rough sets, Fuzzy Sets and Systems, 126(2002), 137-155. [19] H.Thiele, Fuzzy rough sets versus rough fuzzy setsAn interpretation and a comparative study using concepts of modal logics, in: Proc. 5th Europ. Congr. on Intelligent Technigues and Soft Computing, Aachen, Germany, September, 1997, pp.159-167. [20] W.Wu, J.Mi, W.Zhang, Generalized fuzzy rough sets, Information Sciences, 151(2003), 263-282. [21] Y.Y.Yao, Relational interpretations of neighborhood operators and rough set approximation operators, Information Sciences, 111(1998), 239-259. [22] L.Yang, L.Xu, Topological properties of generalized approximation spaces, Information Sciences, 181(2011), 3570-3580.

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SOME OSTROWSKI TYPE INEQUALITIES VIA RIEMANN-LIOUVILLE FRACTIONAL INTEGRALS FOR h−CONVEX FUNCTIONS WENJUN LIU Abstract. In this paper, some Ostrowski type inequalities via Riemann-Liouville fractional integrals for h−convex functions, which are super-multiplicative or super-additive, are given. These results not only generalize those of [24, 25], but also provide new estimates on these types of Ostrowski inequalities for fractional integrals.

1. Introduction Let f : I → R, where I ⊆ R is an interval, be a mapping differentiable in the interior I ◦ of I, and let a, b ∈ I ◦ with a < b. If |f ′ (x)| ≤ M for all x ∈ [a, b], then [ ( ) ] ∫ b a+b 2 x − 1 1 2 f (x) − + f (t)dt ≤ M (b − a) , ∀ x ∈ [a, b] . (1.1) b−a a 4 (b − a)2 This is the well-known Ostrowski inequality (see [19] or [18, page 468]), which gives an upper ∫b 1 bound for the approximation of the integral average (b−a) a f (t)dt by the value f (x) at point x ∈ [a, b] . In recent years, a number of authors have written about generalizations, extensions and variants of such inequalities (see [1, 7, 8, 14, 15, 16, 21]). Let us recall definitions of some kinds of convexity as follows. Definition A. [11] We say that f : I → R is a Godunova-Levin function or that f belongs to the class Q(I) if f is non-negative and for all x, y ∈ I and t ∈ (0, 1), one has f (tx + (1 − t)y) ≤

f (x) f (y) + . t 1−t

Definition B. [9] We say that f : I ⊆ R → R is a P −function or that f belongs to the class P (I) if f is non-negative and for all x, y ∈ I and t ∈ [0, 1], one has f (tx + (1 − t)y) ≤ f (x) + f (y). Definition C. [13] We say that f : (0, ∞] → [0, ∞] is s−convex in the second sense, or that f belongs to the class Ks2 , if for all x, y ∈ (0, b], t ∈ [0, 1] and for some fixed s ∈ (0, 1], one has f (tx + (1 − t)y) ≤ ts f (x) + (1 − t)s f (y). Definition D. [26] Let h : J ⊆ R → R be a positive function. We say that f : I ⊆ R → R is h−convex, or that f belongs to the class SX(h, I), if f is non-negative and for all x, y ∈ I and t ∈ [0, 1], one has f (tx + (1 − t)y) ≤ h(t)f (x) + h(1 − t)f (y). (1.2) If inequality (1.2) is reversed, then f is said to be h−concave, i.e. f ∈ SV (h, I). If h(t) = t, then all non-negative convex functions belong to SX(h, I) and all non-negative concave functions belong to SV (h, I); if h(t) = 1t , then SX(h, I) = Q(I); if h(t) = 1, then SX(h, I) ⊇ P (I); and if h(t) = ts for s ∈ (0, 1], then SX(h, I) ⊇ Ks2 . 2000 Mathematics Subject Classification. 26A33, 26A51, 26D07, 26D10, 26D15. Key words and phrases. Ostrowski type inequality, h−convex function, Riemann-Liouville fractional integral.

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Remark 1. [26] Let h be a non-negative function such that h(t) ≥ t for all t ∈ (0, 1). If f is a non-negative convex function on I, then for x, y ∈ I, t ∈ (0, 1), one has f (tx + (1 − t)y) ≤ tf (x) + (1 − t)f (y) ≤ h(t)f (x) + h(1 − t)f (y).

(1.3)

So, f ∈ SX(h, I). Similarly, if the function h has the property: h(t) ≤ t for all t ∈ (0, 1), then any non-negative concave function f belongs to the class SV (h, I). Definition E. [26] We say that h : J → R is a super-multiplicative function, if for all x, y ∈ J, one has h(xy) ≥ h(x)h(y). Definition F. [2] We say that h : J → R is a super-additive function, if for all x, y ∈ J, one has h(x + y) ≥ h(x) + h(y). For recent results concerning h−convex functions see [5, 23, 25, 26] and references therein. More recently, Tunc [25] established some new Ostrowski type inequalities for the class of h−convex functions which are super-multiplicative or super-additive. We then recall some definitions and mathematical preliminaries of fractional calculus theory which will be used throughout this paper. α f and J α f of order α > 0 Definition G. Let f ∈ L1 [a, b]. The Riemann-Liouville integrals Ja+ b− with a ≥ 0 are defined by ∫ x 1 α (x − t)α−1 f (t)dt, x > a Ja+ f (x) = Γ(α) a

and

∫ b 1 α (t − x)α−1 f (t)dt, x < b, Jb− f (x) = Γ(α) x ∫∞ 0 f (x) = J 0 f (x) = f (x). respectively, where Γ(α) = 0 e−u uα−1 du. Here, Ja+ b− In the case of α = 1, the fractional integral reduces to the classical integral. For some recent results connected with fractional integral inequalities we refer the reader to the papers [3, 4, 6, 10, 12, 17, 20, 22] and the reference cited therein. In [24], Set established some new Ostrowski type inequalities for s−convex functions in the second sense via Riemann-Liouville fractional integral. Motivated by these results, in the present paper, we establish some Ostrowski type inequalities via Riemann-Liouville fractional integrals for h−convex functions, which are super-multiplicative or super-additive. So, new estimates on these types of Ostrowski inequalities via fractional integrals are provided and the results of [24, 25] are generalized. 2. Ostrowski type fractional integral inequalities for h-convex functions To prove our main theorems, we need the following identity established by Set in [24]: Lemma 1. Let f : [a, b] → R be a differentiable mapping on (a, b) with a < b. If f ′ ∈ L1 [a, b] , then for all x ∈ [a, b] and α > 0, one has ] (x − a)α + (b − x)α Γ(α + 1) [ α α f (x) − Jx− f (a) + Jx+ f (b) b−a (b − a) ∫ α+1 ∫ 1 (x − a) (b − x)α+1 1 α ′ = tα f ′ (tx + (1 − t)a) dt − t f (tx + (1 − t)b) dt. b−a b−a 0 0

(2.1)

Using this lemma, we can obtain the following fractional integral inequalities for h−convex functions. 999

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OSTROWSKI TYPE FRACTIONAL INTEGRAL INEQUALITIES FOR h-CONVEX FUNCTIONS

Theorem 1. Let h : J ⊆ R → R ([0, 1] ⊆ J) be a non-negative and super-multiplicative function, h (t) ≥ t for 0 ≤ t ≤ 1, f : [a, b] ⊂ [0, ∞) → R be a differentiable mapping on (a, b) with a < b such that f ′ ∈ L1 [a, b] . If |f ′ | is h−convex on [a, b] and |f ′ (x)| ≤ M, x ∈ [a, b] , then the following inequalities for fractional integrals with α > 0 hold: (x − a)α + (b − x)α [ α ] Γ(α + 1) α f (x) − Jx− f (a) + Jx+ f (b) b−a (b − a) [ ] M (x − a)α+1 + (b − x)α+1 ∫ 1 ≤ [tα h (t) + tα h (1 − t)] dt (2.2) b−a 0 [ ] M (x − a)α+1 + (b − x)α+1 ∫ 1 [ ( ) ] ≤ h tα+1 + h (tα (1 − t)) dt. (2.3) b−a 0 Proof. From (2.1) and since |f ′ | is h−convex, we have (x − a)α + (b − x)α [ α ] Γ(α + 1) α f (x) − Jx− f (a) + Jx+ f (b) b−a (b − a) ∫ ∫ (x − a)α+1 1 α ′ (b − x)α+1 1 α ′ ≤ t f (tx + (1 − t)a) dt + t f (tx + (1 − t)b) dt b−a b−a 0 0 α+1 ∫ 1 [ ] (x − a) ≤ tα h(t) f ′ (x) + tα h(1 − t) f ′ (a) dt b−a 0 ∫ ] (b − x)α+1 1 [ α t h(t) f ′ (x) + tα h(1 − t) f ′ (b) dt + b−a 0 ∫ ∫ M (b − x)α+1 1 α M (x − a)α+1 1 α α [t h(t) + t h(1 − t)] dt + [t h(t) + tα h(1 − t)] dt, ≤ b−a b − a 0 0 which completes the proof of (2.2). By using the additional properties of h in the assumptions, we further have ∫ 1 ∫ 1 α α [t h(t) + t h(1 − t)] dt ≤ [h (tα ) h(t) + h (tα ) h(1 − t)] dt 0 0 ∫ 1 [ ( α+1 ) ] ≤ h t + h (tα (1 − t)) dt.

(2.4)

0



Hence, the proof of (2.3) is complete.

Remark 2. We note that in the proof of (2.2) we does not use the additional super-multiplicative property of h and the condition “h (t) ≥ t for 0 ≤ t ≤ 1”. In Theorem 1, if we choose α = 1, then (2.3) reduces the inequality [25, (2.1)], i.e., [ ] M (x − a)2 + (b − x)2 ∫ 1 ∫ b [ ( 2) ( )] f (x) − 1 f (u) du ≤ h t + h t − t2 dt, b−a a b−a 0 which can be better than the inequality (1.1) provide that h is chosen such that ∫ 1 [ ( 2) ( )] 1 h t + h t − t2 dt < . 2 0 In Theorem 1, if we choose h(t) = t, then (2.2) and (2.3) reduce the inequality in [24, Corollary 1]. In the next corollary, we will also make use of the Beta function of Euler type, which is defined as

∫ β (x, y) =

1

tx−1 (1 − t)y−1 dt =

0

1000

Γ (x) Γ (y) , Γ (x + y)

∀ x, y > 0.

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WENJUN LIU

Corollary 1. If we choose h (t) = ts , s ∈ (0, 1], in Theorem 1, then we have (x − a)α + (b − x)α ] Γ(α + 1) [ α α f (x) − Jx− f (a) + Jx+ f (b) b−a (b − a) [ ] M Γ (α + 1) Γ (s + 1) (x − a)α+1 + (b − x)α+1 ≤ 1+ b−a Γ (α + s + 1) α+s+1 [ ] M Γ (αs + 1) Γ (s + 1) (x − a)α+1 + (b − x)α+1 ≤ 1+ , b−a Γ (αs + s + 1) αs + s + 1 due to the fact that ∫ 1 ∫ [ ( α+1 ) ] α h t + h (t (1 − t)) dt =

∫

1

1

dt + tαs (1 − t)s dt 0 0 0 [ ] 1 Γ (αs + 1) Γ (s + 1) 1 Γ (αs + 1) Γ (s + 1) = + = 1+ . αs + s + 1 Γ (αs + s + 2) αs + s + 1 Γ (αs + s + 1) t

s(α+1)

The first inequality is the same as the one established in [24, Theorem 7]. Theorem 2. Let h : J ⊆ R → R ([0, 1] ⊆ J) be a non-negative and super-additive function, and f : [a, b] ⊂ [0, ∞) → R be a differentiable mapping on (a, b) with a < b such that f ′ ∈ L1 [a, b] . If |f ′ |q is h−convex on [a, b], p, q > 1, p1 + 1q = 1, and |f ′ (x)| ≤ M, x ∈ [a, b] , then the following inequality for fractional integrals with α > 0 holds: (x − a)α + (b − x)α [ α ] Γ(α + 1) α f (x) − Jx− f (a) + Jx+ f (b) b−a (b − a) [ ] ) 1q M (x − a)α+1 + (b − x)α+1 (∫ 1 [h (t) + h (1 − t)] dt ≤ (2.5) 1 0 (1 + pα) p (b − a) [ ] M (x − a)α+1 + (b − x)α+1 1 h q (1). (2.6) ≤ 1 (1 + pα) p (b − a) Proof. From Lemma 1 and using the well-known H¨older’s inequality, we have (x − a)α + (b − x)α [ α ] Γ(α + 1) α f (x) − Jx− f (a) + Jx+ f (b) b−a (b − a) ∫ ∫ (x − a)α+1 1 α ′ (b − x)α+1 1 α ′ t f (tx + (1 − t)a) dt + t f (tx + (1 − t)b) dt ≤ b−a b−a 0 0 1 (∫ ) ( ) 1q ∫ α+1 1 1 p (x − a) f ′ (tx + (1 − t)a) q dt tpα dt ≤ b−a 0 0 ) p1 (∫ 1 ) 1q α+1 (∫ 1 q ′ (b − x) pα + t dt f (tx + (1 − t)b) dt . b−a 0 0 Since |f ′ |q is h−convex and |f ′ (x)| ≤ M , we get ∫ 1 ∫ 1 ′ q q q ] [ f (tx + (1 − t) a) dt ≤ h (t) f ′ (x) + h (1 − t) f ′ (a) dt 0 0 ∫ 1 q ≤M [h (t) + h (1 − t)] dt 0

and similarly

∫

1

f ′ (tx + (1 − t) b) q dt ≤ M q

0

∫

1

[h (t) + h (1 − t)] dt.

0

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By simple computation, we have

∫

1

1 . pα + 1 0 Using these results, we complete the proof of (2.5). By using the super-additive property of h in the assumptions, we further have ∫ 1 ∫ 1 [h (t) + h (1 − t)] dt ≤ h(1)dt = h(1). tpα dt =

0

0



Hence, the proof of (2.6) is complete.

Remark 3. We note that in the proof of (2.5) we does not use the additional super-additive property of h. In Theorem 2, if we choose h(t) = t, then (2.5) reduces the inequality in [24, Corollary 2]; in Theorem 2, if we choose α = 1, then (2.5) becomes [ ] M (x − a)2 + (b − x)2 (∫ 1 ) 1q ∫ b 1 f (x) − f (t)dt ≤ , (2.7) [h (t) + h (1 − t)] dt 1 b−a a 0 (1 + p) p (b − a) which can be better than the inequality (1.1) provide that p, q and h are chosen such that ) 1q (∫ 1 1 1 [h (t) + h (1 − t)] dt < (1 + p) p . 2 0 Corollary 2. If we choose h (t) = ts , s ∈ (0, 1], in Theorem 2, then we have (x − a)α + (b − x)α ] Γ(α + 1) [ α α f (x) − Jx− f (a) + Jx+ f (b) b−a (b − a) ( )1 q (x − a)α+1 + (b − x)α+1 2 M , ≤ 1 b−a (1 + pα) p s + 1 due to the fact that

∫

1

[h (t) + h (1 − t)] dt =

0

2 . s+1

This is the inequality established in [24, Theorem 8]. Theorem 3. Let h : J ⊆ R → R ([0, 1] ⊆ J) be a non-negative and super-multiplicative function, h (t) ≥ t for 0 ≤ t ≤ 1, f : [a, b] ⊂ [0, ∞) → R be a differentiable mapping on (a, b) with a < b such that f ′ ∈ L1 [a, b] . If |f ′ |q is h−convex on [a, b], q ≥ 1 and |f ′ (x)| ≤ M, x ∈ [a, b] , then the following inequalities for fractional integrals with α > 0 hold: (x − a)α + (b − x)α [ α ] Γ(α + 1) α f (x) − Jx− f (a) + Jx+ f (b) b−a (b − a) (∫ 1 ) 1q M (x − a)α+1 + (b − x)α+1 α α ≤ [t h (t) + t h (1 − t)] dt (2.8) 1− 1 b−a 0 (1 + α) q (∫ 1 ) 1q [ ( α+1 ) ] M (x − a)α+1 + (b − x)α+1 α ≤ h t + h (t (1 − t)) dt . (2.9) 1− 1 b−a 0 (1 + α) q Proof. From Lemma 1 and using the well-known power mean inequality, we have (x − a)α + (b − x)α ] Γ(α + 1) [ α α f (x) − Jx− f (a) + Jx+ f (b) b−a (b − a) ∫ ∫ (x − a)α+1 1 α ′ (b − x)α+1 1 α ′ ≤ t f (tx + (1 − t)a) dt + t f (tx + (1 − t)b) dt b−a b−a 0 0 1002

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(x − a)α+1 ≤ b−a

(∫

Since

|f ′ |q

)1− 1q (∫ t dt α

0

α+1

(b − x) + b−a

1

(∫

1

q t f (tx + (1 − t)a) dt α ′

0

1 α

)1− 1q (∫

1

t dt 0

) 1q

) 1q q t f (tx + (1 − t)b) dt . α ′

0 ′ |f (x)|

is h−convex on [a, b] and ≤ M , we get ∫ 1 ∫ 1 q q q ] [α tα f ′ (tx + (1 − t)a) dt ≤ t h(t) f ′ (x) + tα h(1 − t) f ′ (a) dt 0 0 ∫ 1 ≤M q [tα h(t) + tα h(1 − t)] dt 0

and similarly

∫

1

q tα f ′ (tx + (1 − t)b) dt ≤ M q

0

∫

1

[tα h(t) + tα h(1 − t)] dt.

0

Using these inequalities, we complete the proof of (2.8). By using the additional properties of h in the assumptions, we further have (2.4). Hence, the proof of (2.9) is complete.  Remark 4. We note that in the proof of (2.8) we does not use the additional super-multiplicative property of h and the condition “h (t) ≥ t for 0 ≤ t ≤ 1”. In Theorem 3, if we choose α = 1, then (2.9) reduces the inequality [25, (2.4)]; in Theorem 3, if we choose h(t) = t, then (2.8) and (2.9) reduce the inequality in [24, Corollary 3]. Corollary 3. If we choose h (t) = ts , s ∈ (0, 1], in Theorem 3, then we have (x − a)α + (b − x)α [ α ] Γ(α + 1) α f (x) − Jx− f (a) + Jx+ f (b) b−a (b − a) [ ]1 1 Γ (α + 1) Γ (s + 1) q (x − a)α+1 + (b − x)α+1 M 1+ ≤ 1 1− 1 Γ (αs + s + 1) b−a (1 + α) q (α + s + 1) q [ ]1 1 Γ (αs + 1) Γ (s + 1) q (x − a)α+1 + (b − x)α+1 M 1 + ≤ . 1 1− 1 Γ (αs + s + 1) b−a (1 + α) q (α + s + 1) q The first inequality is the same as the one established in [24, Theorem 9]. Acknowledgements This work was partly supported by the National Natural Science Foundation of China (Grant No. 11301277), the Qing Lan Project of Jiangsu Province, and the Teaching Research Project of NUIST (Grant No. 12JY052, 13ZYKC01). References [1] M. Alomari et al., Ostrowski type inequalities for functions whose derivatives are s-convex in the second sense, Appl. Math. Lett. 23 (2010), no. 9, 1071–1076. [2] H. Alzer, A superadditive property of Hadamard’s gamma function, Abh. Math. Semin. Univ. Hambg. 79 (2009), no. 1, 11–23. [3] G. Anastassiou et al., Montgomery identities for fractional integrals and related fractional inequalities, JIPAM. J. Inequal. Pure Appl. Math. 10 (2009), no. 4, Article 97, 6 pp. [4] S. Belarbi and Z. Dahmani, On some new fractional integral inequalities, JIPAM. J. Inequal. Pure Appl. Math. 10 (2009), no. 3, Article 86, 5 pp. [5] M. Bombardelli and S. Varoˇsanec, Properties of h-convex functions related to the Hermite-Hadamard-Fej´er inequalities, Comput. Math. Appl. 58 (2009), no. 9, 1869–1877. [6] Z. Dahmani, New inequalities in fractional integrals, Int. J. Nonlinear Sci. 9 (2010), no. 4, 493–497. [7] S. S. Dragomir, The Ostrowski integral inequality for mappings of bounded variation, Bull. Austral. Math. Soc. 60 (1999), no. 3, 495–508.

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[8] S. S. Dragomir, The Ostrowski’s integral inequality for Lipschitzian mappings and applications, Comput. Math. Appl. 38 (1999), no. 11-12, 33–37. [9] S. S. Dragomir, J. Peˇcari´c and L. E. Persson, Some inequalities of Hadamard type, Soochow J. Math. 21 (1995), no. 3, 335–341. [10] B. Dyda, Fractional Hardy inequality with a remainder term, Colloq. Math. 122 (2011), no. 1, 59–67. [11] E. K. Godunova and V. I. Levin, Inequalities for functions of a broad class that contains convex, monotone and some other forms of functions, in Numerical mathematics and mathematical physics (Russian), 138–142, 166, Moskov. Gos. Ped. Inst., Moscow. [12] R. Gorenflo and F. Mainardi, Fractional calculus: integral and differential equations of fractional order, in Fractals and fractional calculus in continuum mechanics (Udine, 1996), 223–276, CISM Courses and Lectures, 378 Springer, Vienna. [13] H. Hudzik and L. Maligranda, Some remarks on s-convex functions, Aequationes Math. 48 (1994), no. 1, 100–111. [14] W. J. Liu and Q. -A. Ngˆ o, A generalization of Ostrowski inequality on time scales for k points, Appl. Math. Comput. 203 (2008), no. 2, 754–760. [15] Z. Liu, Some companions of an Ostrowski type inequality and applications, JIPAM. J. Inequal. Pure Appl. Math. 10 (2009), no. 2, Article 52, 12 pp. [16] Z. X. L¨ u, On sharp inequalities of Simpson type and Ostrowski type in two independent variables, Comput. Math. Appl. 56 (2008), no. 8, 2043–2047. [17] K. S. Miller and B. Ross, An introduction to the fractional calculus and fractional differential equations, A Wiley-Interscience Publication, Wiley, New York, 1993. [18] D. S. Mitrinovi´c, J. E. Peˇcari´c and A. M. Fink, Inequalities involving functions and their integrals and derivatives, Mathematics and its Applications (East European Series), 53, Kluwer Acad. Publ., Dordrecht, 1991. ¨ [19] A. Ostrowski, Uber die Absolutabweichung einer differentiierbaren Funktion von ihrem Integralmittelwert, Comment. Math. Helv. 10 (1937), no. 1, 226–227. [20] I. Podlubny, Fractional differential equations, Mathematics in Science and Engineering, 198, Academic Press, San Diego, CA, 1999. [21] M. Z. Sarikaya, On the Ostrowski type integral inequality, Acta Math. Univ. Comenian. (N.S.) 79 (2010), no. 1, 129–134. [22] M. Z. Sarikaya and H. Ogunmez, On new inequalities via Riemann-Liouville fractional integration, Abstr. Appl. Anal. 2012, Art. ID 428983, 1-10. [23] M. Z. Sarikaya, A. Saglam and H. Yildirim, On some Hadamard-type inequalities for h-convex functions, J. Math. Inequal. 2 (2008), no. 3, 335–341. [24] E. Set, New inequalities of Ostrowski type for mappings whose derivatives are s-convex in the second sense via fractional integrals, Comput. Math. Appl. 63 (2012), no. 7, 1147–1154. [25] M. Tunc, Ostrowski-type inequalities via h-convex functions with applications to special means, J. Inequal. Appl. 2013 (2013), no. 326, 1-10. [26] S. Varoˇsanec, On h-convexity, J. Math. Anal. Appl. 326 (2007), no. 1, 303–311. College of Mathematics and Statistics, Nanjing University of Information Science and Technology, Nanjing 210044, China E-mail address: [email protected]

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SOME SIMPSON TYPE INEQUALITIES FOR h−CONVEX AND (α, m)−CONVEX FUNCTIONS WENJUN LIU Abstract. In this paper, we establish some Simpson type inequalities for functions whose third derivatives in the absolute value are h−convex and (α, m)−convex, respectively.

1. Introduction The following inequality is well known in the literature as Simpson’s inequality: ∫ b [ ( )] 1 a + b b − a f (a) + f (b) (4) 5 + 2f f (t)dt − (1.1) ≤ 2880 ∥f ∥∞ (b − a) , 3 2 2 a

where the mapping f : [a, b] → R is supposed to be four time differentiable on the interval (a, b) and having the fourth derivative bounded on (a, b), that is ∥f (4) ∥∞ = supx∈(a,b) |f (4) (x)| < ∞. This inequality gives an error bound for the classical Simpson quadrature formula, which, actually, is one of the most used quadrature formulae in practical applications. In recent years, such inequalities were studied extensively by many researchers and numerious generalizations, extensions and variants of them appeared in a number of papers (see [1, 5, 6, 10, 11, 12, 13, 19, 21]). Let us recall definitions of some kinds of convexity as follows. Definition A. [8] We say that f : I → R is Godunova-Levin function or that f belongs to the class Q (I) if f is non-negative and for all x, y ∈ I and t ∈ (0, 1) we have f (x) f (y) + . t 1−t Definition B. [7] We say that f : I ⊆ R → R is a P −function or that f belongs to the class P (I) if f is non-negative and for all x, y ∈ I and t ∈ [0, 1] we have (1.2)

f (tx + (1 − t) y) ≤

(1.3)

f (tx + (1 − t) y) ≤ f (x) + f (y) .

Definition C. [9] Let s ∈ (0, 1] . A function f : (0, ∞] → [0, ∞] is said to be s−convex in the second sense if (1.4)

f (tx + (1 − t) y) ≤ ts f (x) + (1 − t)s f (y) ,

for all x, y ∈ (0, b] and t ∈ [0, 1]. This class of s−convex functions is usually denoted by Ks2 . Definition D. [22] Let h : J ⊆ R → R be a positive function. We say that f : I ⊆ R → R is h−convex function, or that f belongs to the class SX (h, I), if f is non-negative and for all x, y ∈ I and t ∈ [0, 1] we have (1.5)

f (tx + (1 − t) y) ≤ h (t) f (x) + h (1 − t) f (y) .

If inequality (1.5) is reversed, then f is said to be h−concave, i.e. f ∈ SV (h, I). Obviously, if h (t) = t, then all non-negative convex functions belong to SX (h, I) and all non-negative concave functions belong to SV (h, I); if h (t) = 1t , then SX (h, I) = Q (I); if h (t) = 1, then SX (h, I) ⊇ P (I); and if h (t) = ts , where s ∈ (0, 1], then SX (h, I) ⊇ Ks2 . For recent results concerning h−convex functions see [3, 4, 14, 18, 22] and references therein. 2000 Mathematics Subject Classification. 26A51, 26D07, 26D10, 26D15. Key words and phrases. Simpson type inequality, h−convex function, (α, m)−convex function.

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Definition E. [20] The function f : [0, b] → R is said to be m−convex, where m ∈ [0, 1], if for every x, y ∈ [0, b] and t ∈ [0, 1] we have f (tx + m(1 − t)y) ≤ tf (x) + m(1 − t)f (y). Denote by Km (b) the set of the m−convex functions on [0, b] for which f (0) ≤ 0. Definition F. [15] The function f : [0, b] → R, b > 0 is said to be (α, m)−convex, where (α, m) ∈ [0, 1]2 , if for all x, y ∈ [0, b] and t ∈ [0, 1] we have f (tx + m(1 − t)y) ≤ tα f (x) + m(1 − tα )f (y). α (b) the class of all (α, m)−convex functions on [0, b] for which f (0) ≤ 0. Denote by Km

If we choose (α, m) = (1, m), it can be easily seen that (α, m)−convexity reduces to m−convexity and for (α, m) = (1, 1), we have ordinary convex functions on [0, b]. ¨ Recently, Ozdemir et al. [16] established some Simpson type inequalities for functions whose ¨ third derivatives in the absolute value are m−convex. In [17], Ozdemir et al. established the following inequalities for functions whose third derivatives in the absolute value are s−convex in the second sense. Theorem A. Let f : I ⊂ [0, ∞) → R be a differentiable function on I ◦ such that f ′′′ ∈ L1 [a, b], where a, b ∈ I ◦ with a < b. If |f ′′′ | is s−convex in the second sense on [a, b] for some fixed s ∈ (0, 1], then ∫ b [ ( ) ] a+b b−a f (a) + 4f + f (b) f (x)dx − 6 2 a [ ( )] (b − a)4 2−4−s (1 + s)(2 + s) + 34 + 24+s (−2 + s) + 11s + s2 [ ′′′ ′′′ ] f (a) + f (b) . (1.6) ≤ 6 (1 + s)(2 + s)(3 + s)(4 + s) Theorem B. Let f : I ⊂ [0, ∞) → R be a differentiable function on I ◦ such that f ′′′ ∈ L1 [a, b], where a, b ∈ I ◦ with a < b. If |f ′′′ |q is s−convex in the second sense on [a, b] for some fixed s ∈ (0, 1] and q > 1 with p1 + 1q = 1, then ∫ b [ ( ) ] a+b b−a f (a) + 4f + f (b) f (x)dx − 6 2 a ( )1 ( ) 1 {[ ]1 s+1 − 1 ′′′ q q q 1 (b − a)4 1 p Γ(2p + 1)Γ(p + 1) p 2 ′′′ f (a) + f (b) ≤ 48 2 Γ(3p + 2) 2s+1 (s + 1) 2s+1 (s + 1) [ s+1 ]1 } ′′′ q q 2 − 1 ′′′ q 1 f (b) (1.7) + s+1 f (a) + s+1 . 2 (s + 1) 2 (s + 1) Theorem C. Suppose that all the assumptions of Theorem B are satisfied. Then ∫ b [ ( ) ] a+b b−a f (x)dx − f (a) + 4f + f (b) 6 2 a 1 ( ) 1 1− q (b − a)4 ≤ 6 192 ( )1 ( )  ′′′ q 2−4−s 34 + 24+s (−2 + s) + 11s + s2 ′′′ q q 2−4−s f (a) + f (b) ×  (3 + s)(4 + s) (1 + s)(2 + s)(3 + s)(4 + s) ( )1  ( ) q −4−s 4+s 2 −4−s 2 34 + 2 (−2 + s) + 11s + s ′′′ q 2 f ′′′ (b) q (1.8) + f (a) + .  (1 + s)(2 + s)(3 + s)(4 + s) (3 + s)(4 + s) The main purpose of this paper is to establish some new Simpson type inequalities for functions whose third derivatives in the absolute value are h−convex and (α, m)−convex, respectively.

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2. Simpson type inequalities for h-convex functions To prove our main theorems, we need the following identity established in [2]: Lemma 1. Let f : I → R be a function such that f ′′′ be absolutely continuous on I ◦ , the interior of I. Assume that a, b ∈ I ◦ , with a < b and f ′′′ ∈ L1 [a, b]. Then, the following equality holds: [ ( ) ] ∫ b ∫ 1 b−a a+b f (x)dx − f (a) + 4f + f (b) = (b − a)4 p(t)f ′′′ (ta + (1 − t)b)dt, 6 2 a 0 {

where p(t) =

1 2 6t 1 6 (t

(

t−

−

)

1 2 , ( ) 1)2 t − 21 ,

t ∈ [0, 21 ], t ∈ ( 12 , 1].

Using this lemma, we can obtain the following inequalities for h−convex functions. Theorem 1. Let h : J ⊆ R → R ([0, 1] ⊆ J) be a non-negative function, and f : I ⊂ [0, ∞) → R be a differentiable function on I ◦ such that f ′′′ ∈ L1 [a, b], where a, b ∈ I ◦ with a < b. If |f ′′′ | is h−convex on [a, b], then ∫ b [ ( ) ] b−a a+b f (x)dx − f (a) + 4f + f (b) 6 2 a [ ] ( ) ) ∫ 1 ( 4 ∫ 21 ] [ 2 (b − a) 2 1 2 1 ≤ t − t h(t)dt + t − t h(1 − t)dt f ′′′ (a) + f ′′′ (b) . (2.1) 6 2 2 0 0 Proof. From Lemma 1 and h−convexity of |f ′′′ |, we have ∫ b [ ( ) ] a + b b − a f (a) + 4f + f (b) f (x)dx − 6 2 a {∫ 1 ( ) 2 1 t2 t − 1 f ′′′ (ta + (1 − t)b) dt ≤ (b − a)4 6 2 0 } ( ) ∫ 1 1 1 f ′′′ (ta + (1 − t)b) dt (t − 1)2 t − + 1 6 2 2 {∫ 1 ( ) ) ( 2 (b − a)4 2 1 t − t h(t) f ′′′ (a) + h(1 − t) f ′′′ (b) dt ≤ 6 2 0 } ( ) ∫ 1 ′′′ ′′′ ) 1 ( 2 + (t − 1) t − h(t) f (a) + h(1 − t) f (b) dt 1 2 2 [∫ 1 ( ] ) ( ) ∫ 1 ] [ 2 1 1 (b − a)4 t2 − t h(t)dt + (t − 1)2 t − h(t)dt f ′′′ (a) + f ′′′ (b) , = 1 6 2 2 0 2

where we have used the fact that ( ) ) ∫ 1 ∫ 1 ( 2 1 2 1 2 t − t h(t)dt + (t − 1) t − h(t)dt 1 2 2 0 2 ) ( ) ∫ 1 ( ∫ 1 2 1 2 1 2 = t − t h(1 − t)dt + (t − 1) t − h(1 − t)dt 1 2 2 0 2 ) ) ∫ 1 ( ∫ 1 ( 2 2 2 1 2 1 = t − t h(t)dt + t − t h(1 − t)dt. 2 2 0 0 

Hence, the proof of (2.1) is complete. Remark 1. In Theorem 1, if we choose h(t) = ts , s ∈ (0, 1], then (2.1) reduces to (1.6).

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Theorem 2. Let h : J ⊆ R → R ([0, 1] ⊆ J) be a non-negative function, and f : I ⊂ [0, ∞) → R be a differentiable function on I ◦ such that f ′′′ ∈ L1 [a, b], where a, b ∈ I ◦ with a < b. If |f ′′′ |q is h−convex on [a, b] and q > 1 with p1 + 1q = 1, then ∫ b [ ( ) ] b−a a+b f (x)dx − f (a) + 4f + f (b) 6 2 a )1 4 ( ) p1 ( (b − a) 1 Γ(2p + 1)Γ(p + 1) p ≤ 48 2 Γ(3p + 2) [( ) (∫ 1 ) ]1  ∫ 12 ′′′ q ′′′ q q 2 × h(t)dt f (a) + h(1 − t)dt f (b)  0 0 [(∫ 1 ) (∫ 1 ) ]1  q q q  2 2 + h(1 − t)dt f ′′′ (a) + h(t)dt f ′′′ (b) (2.2) .  0 0 Proof. From Lemma 1, and using the h−convexity of |f ′′′ |q and the well-known H¨older’s inequality, we have ∫ b [ ( ) ] a+b b−a f (a) + 4f + f (b) f (x)dx − 6 2 a ( )1 ))p ) p1 (∫ 1 ∫ 1( ( q 2 2 1 (b − a)4  q f ′′′ (ta + (1 − t)b) dt −t t2 dt ≤  0 6 2 0 (∫ ( )1  ( ))p ) p1 (∫ 1 q 1 ′′′ 1 f (ta + (1 − t)b) q dt + (t − 1)2 t − dt 1 1  2 2 2 ( )1 ( )1 ∫ 1 q ′′′ q ′′′ q ] 2 [ (b − a)4 Γ(2p + 1)Γ(p + 1) p  f (a) + h(1 − t) f (b) dt h(t) ≤  0 6 23p+1 Γ(3p + 2) (∫ )1  q 1[ q q ] h(t) f ′′′ (a) + h(1 − t) f ′′′ (b) dt + 1  2

( )1 ( )1 (b − a)4 1 p Γ(2p + 1)Γ(p + 1) p ≤ 48 2 Γ(3p + 2) [( ) ) (∫ 1 ]1  ∫ 12 ′′′ q ′′′ q q 2 × h(t)dt f (a) + h(1 − t)dt f (b)  0 0 [(∫ ) (∫ ) ]1  1 1 ′′′ q ′′′ q q  + h(t)dt f (a) + h(1 − t)dt f (b) , 1 1  2

2

where we have used the fact that ))p ( ))p ∫ 1( ( ∫ 1( 2 1 Γ(2p + 1)Γ(p + 1) 2 2 1 (2.3) (t − 1) t − t −t dt = dt = 1 2 2 23p+1 Γ(3p + 2) 0 2



and Γ is the Gamma function. Hence, the proof of (2.2) is complete. Remark 2. In Theorem 2, if we choose h(t) = ts , s ∈ (0, 1], then (2.2) reduces to (1.7). A different approach leads to the following result.

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Theorem 3. Suppose that all the assumptions of Theorem 2 are satisfied. Then ∫ b [ ( ) ] a+b b−a f (a) + 4f + f (b) f (x)dx − 6 2 a 1 ( ) (b − a)4 1 1− q ≤ 6 192 [( ) (∫ 1 ( ) ]1 ) )  ∫ 21 ( 1 ′′′ q ′′′ q q 2 1 2 2 × t − t h(t)dt f (a) + t − t h(1 − t)dt f (b)  2 2 0 0 [(∫ 1 ( ) (∫ 1 ( ) ]1  ) ) q 2 2 q q 2 1 ′′′ 2 1 ′′′ (2.4) + t − t h(1 − t)dt f (a) + t − t h(t)dt f (b) .  2 2 0 0 Proof. From Lemma 1 and using the well-known power-mean inequality we have ∫ b [ ( ) ] a+b b−a f (a) + 4f + f (b) f (x)dx − 6 2 a ( 1 ( ) )1 1− q ) ) ∫ 1 ( ∫ 1 ( q 2 2 (b − a)4  1 1 q ≤ t2 t2 − t dt − t f ′′′ (ta + (1 − t)b) dt  0 6 2 2 0 (∫

1

+ 1 2

)1  ) )1− 1q (∫ 1 ) ( ( q q 1 1 ′′′ dt . (t − 1)2 t − (t − 1)2 t − f (ta + (1 − t)b) dt 1  2 2 2

Since |f ′′′ |q is h−convex, we have ) ∫ 1 ( q 2 1 t2 − t f ′′′ (ta + (1 − t)b) dt 2 0 ) ∫ 1 ( q q ) ( 2 2 1 t − t h(t) f ′′′ (a) + h(1 − t) f ′′′ (b) dt ≤ 2 ) ) (∫ 1 ( (0∫ 1 ( ) ) ′′′ q q 2 2 1 2 1 2 − t h(t)dt f (a) + − t h(1 − t)dt f ′′′ (b) t t = 2 2 0 0 and

( ) q 1 ′′′ (t − 1)2 t − f (ta + (1 − t)b) dt 1 2 2 ( ) ∫ 1 q q ) 1 ( 2 (t − 1) t − h(t) f ′′′ (a) + h(1 − t) f ′′′ (b) dt ≤ 1 2 (2∫ 1 ( ) (∫ 1 ( ) ) ) q 2 2 1 1 q = t2 − t h(1 − t)dt f ′′′ (a) + t2 − t h(t)dt f ′′′ (b) . 2 2 0 0 ∫

1



Hence, the proof of (2.4) is complete. Remark 3. In Theorem 3, if we choose h(t) = ts , s ∈ (0, 1], then (2.4) reduces to (1.8). 3. Simpson type inequalities for (α, m)−convex functions We use the following modified identity:

Lemma 2. [16, Lemma 2] Let f : I → R be a function such that f ′′′ be absolutely continuous on I ◦ , the interior of I. Assume that a, b ∈ I ◦ , with a < b, m ∈ (0, 1] and f ′′′ ∈ L1 [a, b]. Then, the

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following equality holds: ∫ mb

[ ( ) ] a + mb mb − a f (a) + 4f + f (mb) f (x)dx − 6 2 a ∫ 1 4 = (mb − a) p(t)f ′′′ (ta + m(1 − t)b)dt, 0

where

{

1 2 6t 1 6 (t

p(t) =

(

t−

−

)

1 2 , ( ) 1)2 t − 21 ,

t ∈ [0, 21 ], t ∈ ( 12 , 1].

Using this lemma, we can obtain the following inequalities for (α, m)−convex functions. Theorem 4. Let f : I ⊂ [0, b∗ ] → R, be a differentiable function on I ◦ such that f ′′′ ∈ L1 [a, b] where a, b ∈ I with a < b, b∗ > 0. If |f ′′′ |q is (α, m)−convex on [a, b] for (α, m) ∈ [0, 1]2 , q > 1 with p1 + 1q = 1, then ∫ mb [ ( ) ] mb − a a + mb f (x)dx − f (a) + 4f + f (mb) 6 2 a ) 1 {[ ′′′ ]1 4 ( (mb − a) Γ(2p + 1)Γ(p + 1) p |f (a)|q + m[2α (1 + α) − 1] |f ′′′ (b)|q q ≤ 96 Γ(3p + 2) 2α (1 + α) [ 1+α ]1 } (2 − 1) |f ′′′ (a)|q + m[2α (1 + α) − (21+α − 1)] |f ′′′ (b)|q q (3.1) . + 2α (1 + α) Proof. From Lemma 2 and using H¨older’s inequality we have ∫ mb [ ( ) ] mb − a a + mb f (x)dx − f (a) + 4f + f (mb) 6 2 a ( 1 ( ) )1 )) ( ( ∫ 1 ∫ 1 p q p 2 2 1 (mb − a)4  f ′′′ (ta + m(1 − t)b) q dt −t t2 dt ≤  0 6 2 0 (∫ ( )1  ))p ) p1 (∫ 1 ( q 1 1 f ′′′ (ta + m(1 − t)b) q dt + (t − 1)2 t − dt . 1 1  2 2

2

Due to the (α, m)−convexity of |f ′′′ |q , we have ∫ 1 ∫ 1 q q ] 2 2 [ f ′′′ (ta + m(1 − t)b) q dt ≤ tα f ′′′ (a) + m(1 − tα ) f ′′′ (b) dt 0

and

∫

0

|f ′′′ (a)|q + m[2α (1 + α) − 1] |f ′′′ (b)|q = 21+α (1 + α)

1 1 2

f ′′′ (ta + m(1 − t)b) q dt ≤ =

∫

q q ] tα f ′′′ (a) + m(1 − tα ) f ′′′ (b) dt

1[ 1 2

(21+α − 1) |f ′′′ (a)|q + m[2α (1 + α) − (21+α − 1)] |f ′′′ (b)|q . 2α (1 + α) 

The proof of (3.1) is complete by combining the above inequalities and (2.3). Remark 4. In Theorem 4, if we choose α = 1, we get the inequality in [16, Theorem 4].

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Theorem 5. Let the assumptions of Theorem 4 hold with q ≥ 1. Then ∫ mb [ ( ) ] mb − a a + mb f (x)dx − f (a) + 4f + f (mb) 6 2 a {( )1 (mb − a)4 12 |f ′′′ (a)|q + m[2α (3 + α)(4 + α) − 12] |f ′′′ (b)|q q ≤ 1152 2α (3 + α)(4 + α) ( 12[α2 + 11α + 34 − 24+α (2 − α)] ′′′ q f (a) + 2α (1 + α)(2 + α)(3 + α)(4 + α) [ ] )1 } 12[α2 + 11α + 34 − 24+α (2 − α)] ′′′ q q (3.2) +m 1 − α . f (b) 2 (1 + α)(2 + α)(3 + α)(4 + α) Proof. From Lemma 2, using the well known power-mean inequality and (α, m)−convexity of |f ′′′ |q , we have ∫ mb [ ( ) ] mb − a a + mb f (x)dx − f (a) + 4f + f (mb) 6 2 a ( )1 ) )1− 1q (∫ 1 ( ) ∫ 1 ( q 2 2 (b − a)4  1 1 q ≤ t2 t2 − t dt − t f ′′′ (ta + m(1 − t)b) dt  0 6 2 2 0 (∫ )1  ( ) )1− 1q (∫ 1 ( ) q 1 q 1 1 ′′′ (t − 1)2 t − dt (t − 1)2 t − f (ta + m(1 − t)b) dt + 1 1  2 2 2 2 ( )1 ) )1− 1q (∫ 1 ( ) ∫ 1 ( q [ ] 2 2 (b − a)4  1 1 q q ≤ t2 − t dt t2 − t tα f ′′′ (a) + m(1 − tα ) f ′′′ (b) dt  0 6 2 2 0 (∫

1

+ 1 2

)1  ) )1− 1q (∫ 1 ) ( ( q q ] 1 1 [ α ′′′ q dt t f (a) + m(1 − tα ) f ′′′ (b) dt (t − 1)2 t − (t − 1)2 t − . 1  2 2 2

By using the fact that

) 1 1 − t tα dt = , α (3 + α)(4 + α) 2 16 × 2 0 ) ∫ 1 ( 2 1 2α (3 + α)(4 + α) − 12 t2 − t (1 − tα )dt = , 2 192 × 2α (3 + α)(4 + α) 0 ( ) ∫ 1 1 α α2 + 11α + 34 − 24+α (2 − α) 2 (t − 1) t − t dt = 1 2 16 × 2α (1 + α)(2 + α)(3 + α)(4 + α) ∫

1 2

(

t2

2

and ( ) ∫ 1 1 2α (1 + α)(2 + α)(3 + α)(4 + α) − 12[α2 + 11α + 34 − 24+α (2 − α)] 2 (t−1) t − (1−tα )dt = , 1 2 192 × 2α (1 + α)(2 + α)(3 + α)(4 + α) 2

we obtain

∫

[ ( ) ] mb − a a + mb f (x)dx − f (a) + 4f + f (mb) 6 2 a )1− 1 {( )1 q 4 ( q (mb − a) 1 12 |f ′′′ (a)| + m[2α (3 + α)(4 + α) − 12] |f ′′′ (b)|q q ≤ 6 192 192 × 2α (3 + α)(4 + α) ( ′′′ q α2 + 11α + 34 − 24+α (2 − α) f (a) + α 16 × 2 (1 + α)(2 + α)(3 + α)(4 + α) mb

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] )1 } ′′′ q q 1 α2 + 11α + 34 − 24+α (2 − α) f (b) , +m − 192 16 × 2α (1 + α)(2 + α)(3 + α)(4 + α) [



which implies the desired result. Remark 5. In Theorem 5, if we choose α = 1, we have the inequality in [16, Theorem 5]. Acknowledgements

This work was partly supported by the National Natural Science Foundation of China (Grant No. 11301277), the Qing Lan Project of Jiangsu Province, and the Teaching Research Project of NUIST (Grant No. 12JY052, 13ZYKC01). References [1] M. Alomari and M. Darus, On some inequalities of Simpson-type via quasi-convex functions and applications, Transylv. J. Math. Mech. 2 (2010), no. 1, 15–24. [2] M. Alomari and S. Hussain, Two inequalities of Simpson type for quasi-convex functions and applications, Appl. Math. E-Notes 11 (2011), 110–117. [3] M. Bombardelli and S. Varoˇsanec, Properties of h-convex functions related to the Hermite-Hadamard-Fej´er inequalities, Comput. Math. Appl. 58 (2009), no. 9, 1869–1877. [4] P. Burai and A. H´ azy, On approximately h-convex functions, J. Convex Anal. 18 (2011), no. 2, 447–454. [5] S. S. Dragomir, On Simpson’s quadrature formula for mappings of bounded variation and applications, Tamkang J. Math. 30 (1999), no. 1, 53–58. [6] S. S. Dragomir, R. P. Agarwal and P. Cerone, On Simpson’s inequality and applications, J. Inequal. Appl. 5 (2000), no. 6, 533–579. [7] S. S. Dragomir, J. Peˇcari´c and L. E. Persson, Some inequalities of Hadamard type, Soochow J. Math. 21 (1995), no. 3, 335–341. [8] E. K. Godunova and V. I. Levin, Inequalities for functions of a broad class that contains convex, monotone and some other forms of functions, in Numerical mathematics and mathematical physics (Russian), 138–142, 166, Moskov. Gos. Ped. Inst., Moscow. [9] H. Hudzik and L. Maligranda, Some remarks on s-convex functions, Aequationes Math. 48 (1994), no. 1, 100–111. [10] V. N. Huy and Q. -A. Ngˆ o, New inequalities of Simpson-like type involving n knots and the mth derivative, Math. Comput. Modelling 52 (2010), no. 3-4, 522–528. [11] W. J. Liu, Y. N. Sun and Q. L. Zhang, Some new error inequalities for a generalized quadrature rule of open type, Comput. Math. Appl. 62 (2011), no. 5, 2218–2224. [12] Z. Liu, An inequality of Simpson type, Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci. 461 (2005), no. 2059, 2155–2158. [13] Z. Liu, Some sharp modified Simpson type inequalities and applications, Vietnam J. Math. 39 (2011), no. 2, 135–144. [14] Z. Liu, A note on Ostrowski type inequalities related to some s-convex functions in the second sense, Bull. Korean Math. Soc. 49 (2012), no. 4, 775–785. [15] V.G. Mihe¸san, A generalization of the convexity, Seminar of Functional Equations, Approx. and Convex, Cluj-Napoca (Romania) (1993). ¨ [16] M. E. Ozdemir, M. Avci and H. Kavurmaci, Simpson type inequalities for m−convex functions, arXiv: 1112.3559v1 [math.FA]. ¨ [17] M. E. Ozdemir, M. Avci and H. Kavurmaci, Simpson type inequalities for functions whose third derivatives in the absolute value are s−convex and s−concave functions, arXiv:1206.1193v1 [math.CA]. [18] M. Z. Sarikaya, A. Saglam and H. Yildirim, On some Hadamard-type inequalities for h-convex functions, J. Math. Inequal. 2 (2008), no. 3, 335–341. [19] M. Z. Sarikaya, E. Set and M. E. Ozdemir, On new inequalities of Simpson’s type for s-convex functions, Comput. Math. Appl. 60 (2010), no. 8, 2191–2199. [20] G. H. Toader, Some generalisations of the convexity, Proc. Colloq. Approx. Optim (1984), 329–338. [21] K.-L. Tseng, G.-S. Yang and S. S. Dragomir, On weighted Simpson type inequalities and applications, J. Math. Inequal. 1 (2007), no. 1, 13–22. [22] S. Varoˇsanec, On h-convexity, J. Math. Anal. Appl. 326 (2007), no. 1, 303–311. College of Mathematics and Statistics, Nanjing University of Information Science and Technology, Nanjing 210044, China E-mail address: [email protected]

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TABLE OF CONTENTS, JOURNAL OF COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 16, NO. 5, 2014 Ostrowski Type Inequalities for m- and (α,m)- Logaritmically Convex Functions, Havva Kavurmaci,……………………………………………………………………………………820 A New Version of Mazur-Ulam Theorem Under Weaker Conditions in Linear n-Normed Spaces, Choonkil Park, and Cihangir Alaca,…………………………………………………………827 On Umbral Calculus Involving Special Polynomials, Dae San Kim, Taekyun Kim, Sang-Hun Lee, and Seog-Hoon Rim,……………………………………………………………………833 Semilocal Convergence Theorem by Using Majorizing Functions for Harmonic Mean Newton's Method in Banach Spaces, Hong-Xiu Zhong, Guo-Liang Chen, and Xue-Ping Guo,………850 Fermionic p-Adic Integrals on Zp and Umbral Calculus, Dae San Kim, Taekyun Kim, Sang-Hun Lee, and Dmitry V. Dolgy,………………………………………………………………….860 Soft Set Theory and N-Structures Applied to BCH-algebras, Young Bae Jun, N. O. Alshehri, and Kyoung Ja Lee,………………………………………………………………………………869 Construction of Orthogonal Shearlet Tight Frames with Symmetry, Yan Feng, Dehui Yuan, and Shouzhi Yang,……………………………………………………………………………….887 Exact Orders in Simultaneous Approximation by Complex q-Durrmeyer type Operators, Mei-Ying Ren, Xiao-Ming Zeng, and Liang Zeng,…………………………………………895 On Positive Solutions of a System of Max-Type Difference Equations, Stevo Stević, Abdullah Alotaibi, Naseer Shahzad, and Mohammed A. Alghamdi,…………………………………906 Solution and Stability of a Multi-Variable Functional Equation, Jae-Hyeong Bae, and Won-Gil Park,…………………………………………………………………………………………916 On the Behaviour of the Solutions of Difference Equation Systems, Y. Yazlik, E. M. Elsayed, and N. Taskara,………………………………………………………………………………932 The Shared Set of Meromorphic Functions and Differential Polynomials, Hong-Yan Xu,…942 Quasiconformal Harmonic Mappings Related to Starlike Functions, Yaşar Polatoğlu, Emel Yavuz Duman, Yasemin Kahramaner, and Melike Aydoğan,……………………………….955 On the Superstability of Ternary Jordan C*-homomorphisms, Dong Yun Shin, Choonkil Park, and Shahrokh Farhadabadi,………………………………………………………………….964

TABLE OF CONTENTS, JOURNAL OF COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 16, NO. 5, 2014 (continued)

Coupled Fixed Points for Generalized Weakly Contractive Mappings in Partial Metric Spaces, Rattanaporn Wangkeeree, Rabian Wangkeeree, and Nithirat Sisarat,…………………………974 A Note on Structures of Fuzzy Approximation Spaces, Gangqiang Zhang, Yu Han, and Zhaowen Li,………………………………………………………………………………………………989 Some Ostrowski type Inequalities via Riemann-Liouville Fractional Integrals for h-Convex Functions, Wenjun Liu,………………………………………………………………………...998 Some Simpson type Inequalities for h-Convex and (α,m)−Convex Functions, Wenjun Liu,..1005