JOURNAL OF COMPUTATIONAL ANALYSIS AND APPLICATIONS VOLUME 24, 2018

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Volume 24, Number 1 ISSN:1521-1398 PRINT,1572-9206 ONLINE

January 2018

Journal of Computational Analysis and Applications EUDOXUS PRESS,LLC

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

Journal of Computational Analysis and Applications(JoCAAA) is published by EUDOXUS PRESS,LLC,1424 Beaver Trail Drive,Cordova,TN38016,USA,[email protected] http://www.eudoxuspress.com. Annual Subscription Prices:For USA and Canada,Institutional:Print $800, Electronic OPEN ACCESS. Individual:Print $400. For any other part of the world add $150 more(handling and postages) to the above prices for Print. No credit card payments. Copyright©2018 by Eudoxus Press,LLC,all rights reserved.JoCAAA is printed in USA. JoCAAA is reviewed and abstracted by AMS Mathematical Reviews,MATHSCI,and Zentralblaat MATH. It is strictly prohibited the reproduction and transmission of any part of JoCAAA and in any form and by any means without the written permission of the publisher.It is only allowed to educators to Xerox articles for educational purposes.The publisher assumes no responsibility for the content of published papers.

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

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

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

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

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

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

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

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

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

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

011-49-203-379-3542 e-mail: [email protected] Approximation Theory, Computer Aided Geometric Design John R. Graef Department of Mathematics University of Tennessee at Chattanooga Chattanooga, TN 37304 USA [email protected] Ordinary and functional differential equations, difference equations, impulsive systems, differential inclusions, dynamic equations on time scales, control theory and their applications

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

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

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

Xing-Biao Hu Institute of Computational Mathematics AMSS, Chinese Academy of Sciences Beijing, 100190, CHINA [email protected]

H. H. Gonska Department of Mathematics University of Duisburg Duisburg, D-47048 Germany

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

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

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

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

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

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

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

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

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

Anfithea - Paleon Faliron GR-175 64 Athens, Greece [email protected] Numerical Analysis H. M. Srivastava Department of Mathematics and Statistics University of Victoria Victoria, British Columbia V8W 3R4 Canada tel.250-472-5313; office,250-4776960 home, fax 250-721-8962 [email protected] Real and Complex Analysis, Fractional Calculus and Appl., Integral Equations and Transforms, Higher Transcendental Functions and Appl.,q-Series and q-Polynomials, Analytic Number Th.

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

I. P. Stavroulakis Department of Mathematics University of Ioannina 451-10 Ioannina, Greece [email protected] Differential Equations Phone +3-065-109-8283

Tomasz Rychlik Polish Academy of Sciences Instytut Matematyczny PAN 00-956 Warszawa, skr. poczt. 21 ul. Śniadeckich 8 Poland [email protected] Mathematical Statistics, Probabilistic Inequalities

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

Boris Shekhtman Department of Mathematics University of South Florida Tampa, FL 33620, USA Tel 813-974-9710 [email protected] Approximation Theory, Banach spaces, Classical Analysis

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

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.

Juan J. Trujillo University of La Laguna Departamento de Analisis Matematico C/Astr.Fco.Sanchez s/n 38271. LaLaguna. Tenerife. SPAIN Tel/Fax 34-922-318209 [email protected]

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Fractional: Differential EquationsOperators-Fourier Transforms, Special functions, Approximations, and Applications

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

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

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

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

Richard A. Zalik Department of Mathematics Auburn University Auburn University, AL 36849-5310 USA. Tel 334-844-6557 office 678-642-8703 home Fax 334-844-6555 [email protected] Approximation Theory, Chebychev Systems, Wavelet Theory Ahmed I. Zayed Department of Mathematical Sciences DePaul University

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J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 24, NO.1, 2018, COPYRIGHT 2018 EUDOXUS PRESS, LLC

A novel approach for solving fully fuzzy linear programming problem with LR flat fuzzy numbers† Zeng-Tai Gonga,∗ , Wen-Cui Zhaoa,b a College of Mathematics and Statistics, Northwest Normal University, Lanzhou 730070, China b Group of Mathematics, YiFu Experiment Middle School of Tianshui, Tianshui 741200, China

Abstract: The fuzzy linear programming problem with triangular fuzzy numbers in its objective functions or constraints has been discussed by many scholars based on using Zadeh’s decomposition theorem of fuzzy numbers and transforming it into some crisp linear programming problems. However, existed methods and results are limited if the fuzzy linear programming problem with generalized fuzzy numbers in its objective functions and constraints. In this paper, we discuss fully fuzzy linear programming (FFLP) problems of which all parameters are LR flat fuzzy numbers and a simple but practical method is developed to solve it. In this method, the approximate representation of fully fuzzy constraints is investigated by means of the arithmetic operations on LR flat fuzzy numbers space firstly. Meanwhile, we constructed a auxiliary multi-objective programming to solve the FFLP problems. After that, three approaches are proposed to solve the constructed auxiliary multi-objective programming, including optimistic approach, pessimistic approach and linear sum approach based on membership function. Finally, the obtained results are compared with the existing works and numerical example is given to illustrate the effectiveness of the proposed method. Keywords: LR flat fuzzy number; multi-objective linear programming; fully fuzzy linear programming 1. Introduction Linear programming (LP) is an essential mathematical tool in science and technology. Although, it has been investigated and expanded for more than six decades by many researchers from various point of views, it is still useful to develop new approaches in order to better fit the real world problems within the framework of linear programming. In conventional approach, parameters of linear programming models must be well defined and precise. However, in real world environment, this is not a realistic assumption. Usually, most of information is not deterministic and in this situation human has a capability to make a rational decision based on this uncertainty. This is hard challenge for decision makers to design an intelligent system which make a decision the same as the human. In fact, some of parameters of the system may be represented by fuzzy quantities rather than crisp ones in practice. Hence, it is necessary to develop mathematical theory and numerical schemes to handle fuzzy linear programming (FLP) problems. Bellman and Zadeh [1] proposed the concept of decision making in fuzzy environments. Since then, a number of researchers have exhibited their interest to various types of the FLP problems and proposed several approaches for solving these problems [2-11]. FLP model with triangular fuzzy numbers (TFNs) [6]. Lai and Hwang [6] developed a new approach to some possibilistic linear programming problems with TFNs, they transformed the fuzzy linear programming into a multi-objective linear programming model, involving three objective functions: minimizing the low loss, maximizing the most possible value and maximizing the upper the profit. FLP model with trapezoidal fuzzy numbers (TrFNs) [7õ11]. For example, Ganesan [7] and Ebrahimnejad [8] studied the fuzzy linear programs with TrFNs, but the constructed fuzzy linear programming models are only suitable for the symmetrical TrFNs. MahadaviAmiri [9] and Campos [10] utilized the ranking function to solve the fuzzy linear programming models with TrFNs. Wan [11] developed a method which taken advantage of multi-objective linear programming model to solve FLP problems with TrFNs. However, in all of the above mentioned works, those cases of † ∗

This work is supported by the National Natural Science Foundations of China (11461062, 61262022). Corresponding Author: Zeng-Tai Gong. Tel.:+86 09317971430. E-mail addresses: [email protected], [email protected](Zeng-Tai Gong). 11

Zeng-Tai Gong et al 11-22

J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 24, NO.1, 2018, COPYRIGHT 2018 EUDOXUS PRESS, LLC

Zeng-Tai Gong and Wen-Cui Zhao: A novel approach for solving fully fuzzy linear programming...

FLP problems have been studied in which not all parts of the problem were assumed to be fuzzy, e.g., only the right hand side or the objective function coefficients were fuzzy but the variables were not fuzzy. The FLP problems in which all the parameters as well as the variables are represented by fuzzy numbers are known as fully fuzzy linear programming (FFLP) problems. Many authors [12-14] have proposed different methods for solving FFLP problems. Lotfi [12] proposed a novel method to obtain the approximate solution of FFLP problems by using the concept of the symmetric triangular fuzzy numbers and introduce an approach to defuzzify a general fuzzy quantity. In Kumar’s article [13], an exact optimal solution is achieved using a linear ranking function. In this method, the linear ranking function has been used to convert the fuzzy objective function to a crisp objective function. The shortcoming exists of it is that the fuzziness of objective function has been neglected by the linear ranking function. Ezzati [14] proposed a new algorithm to solve FFLP problems with TFNs. To the best of our knowledge, till now there is no method in the literature to obtain the exact solution of FFLP problems in which all the parameters as well as the variables are represented by LR flat fuzzy numbers. The LR flat fuzzy number and its operations were first introduced by Dubois [15]. We know that triangular fuzzy numbers are just specious cases of LR flat fuzzy numbers. In 2006, Dehgham [16] discussed the computational methods for fully fuzzy linear systems whose coefficient matrix and the right-hand side vector are denoted by LR fuzzy numbers. In this paper, the approximate representation e x=e e e e > 0, where e of fully fuzzy constraints (max(min) e cT ⊗e x, s.t. A⊗e b, x cT , A, b and e c are fuzzy matrixs which consist of LR flat fuzzy numbers) is investigated by means of the arithmetic operations on LR flat fuzzy numbers space firstly. Meanwhile, we constructed a auxiliary multi-objective programming to solve the FFLP problems. After that, three approaches are proposed to solve the constructed auxiliary multiobjective programming, including optimistic approach, pessimistic approach and linear sum approach based on membership function. Finally, the results obtained are compared with the existing works and numerical example is given to illustrate the effectiveness of the proposed method. The structure of this paper is organized as follows. In Section 2, we review some basic concepts and introduce the interval objective programming. In Section 3, the approximate representation of fully fuzzy constraints is given. The method for solving the FFLP problem is discussed in Section 4. In Section 5, the numerical example is solved and the obtained results are compared with the existing works. Conclusion is drawn in Section 6. 2. Preliminaries 2.1 Basic definitions and arithmetic operations In this section, some basic definitions and arithmetic operations of LR flat fuzzy numbers are presented. e defined on universal set of real numbers R, denoted as A e= Definition 2.1.[17] A fuzzy number A, (m, n, α, β)LR , is said to be an LR flat fuzzy number if its membership function µAe(x) is given by  m−x  L( α ) R( x−n uAe(x) = β )  1

x < m, α > 0, x > n, β > 0, m 6 x 6 n,

e α and β are the left and right spreads, respectively. where the closed interval [m, n] is the mode of A, The function L(·), which is called the left shape function satisfying: (1) L(x) = L(−x); (2) L(0) = 1 and L(1) = 0; (3) L(x) is non-increasing on [0, +∞). The definition of the right shape function R(·) is usually similar to that of L(·). Usually, we could define the predetermined left spreads shape functions L(·) and right spreads shape functions R(·) as follows according to the need of mathematical modeling: (1) L(x) = max{0, 1 − |x|p }(p > 0); p 1 (e1−|x| − 1)}(p > 0); (2) L(x) = max{0, e−1 2 (3) L(x) = max{0, 1+|x| p − 1}(p > 0), and similarly to R(·). 12

Zeng-Tai Gong et al 11-22

J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 24, NO.1, 2018, COPYRIGHT 2018 EUDOXUS PRESS, LLC

Zeng-Tai Gong and Wen-Cui Zhao: A novel approach for solving fully fuzzy linear programming...

e = (m, n, α, β)LR is said to be non-negative LR flat Definition 2.2.[18] An LR flat fuzzy number A fuzzy number if m − α > 0 and is said to be non-positive LR flat fuzzy number if n + β 6 0. f1 = (m1 , n1 , α1 , β1 )LR and A f2 = (m2 , n2 , α2 , β2 )LR are Definition 2.3.[17] LR flat fuzzy numbers A f1 = A f2 if and only if m1 = m2 , n1 = n2 , α1 = α2 , β1 = β2 . said to be equal, i.e., A Remark 2.1. If m = n, then an LR flat fuzzy number (m, n, α, β)LR is said to be an LR fuzzy number and is denoted as (m, α, β)LR or (n, α, β)LR . Remark 2.2. If m 6= n and L(x) = R(x) = max{0, 1 − |x|}, then an LR flat fuzzy number (m, n, α, β)LR is said to be a trapezoidal fuzzy number and is denoted as (m, n, α, β). Remark 2.3. If m = n and L(x) = R(x) = max{0, 1 − |x|}, then an LR flat fuzzy number (m, n, α, β)LR is said to be a triangular fuzzy number and is denoted as (m, α, β). The arithmetic operations between two LR flat fuzzy numbers are defined by the extension principle as follows[17] : f1 = (m1 , n1 , α1 , β1 )LR , A f2 = (m2 , n2 , α2 , β2 )LR be any LR flat fuzzy numbers and A f3 = Let A (m3 , n3 , α3 , β3 )RL be any RL flat fuzzy number. Then, f1 ⊕ A f2 = (m1 + m2 , n1 + n2 , α1 + α2 , β1 + β2 )LR . (i) A f f3 = (m1 − n3 , n1 − m3 , α1 + β3 , α3 + β1 )LR . (ii) A1 A f1 and A f2 are non-negative, then (iii) If A f1 ⊗ A f2 ∼ A = (m1 m2 , n1 n2 , m1 α2 + m2 α1 , n1 β2 + n2 β1 )LR . f f2 is non-negative, then (iv) If A1 is non-positive and A f1 ⊗ A f2 ∼ A = (m( 1 n2 , n1 m2 , α1 n2 − m1 β2 , m2 β1 − n1 α2 )LR . (λm1 , λn1 , λα1 , λβ1 )LR , λ > 0, ∼ f1 = (v) λ ⊗ A (λn1 , λm1 , −λβ1 , −λα1 )RL , λ < 0. e = (e Definition 2.4. A matrix A aij )m×n (i = 1, 2, · · · m, j = 1, 2, · · · n) is said to be a fuzzy matrix if e e is said to be a each element of A is a fuzzy number. If for every element e aij > 0 (or e aij 6 0), then A e > 0 (or A e 6 0). non-negative (or non-positive) fuzzy matrix, denoted by A e = (e Let all element of A aij )m×n (i = 1, 2, · · · m, j = 1, 2, · · · n) are LR flat fuzzy numbers, i.e., e aij = e e (mij , nij , αij , βij )LR . Then we can represent m × n fuzzy matrix A with new notation A = (A, B, M, N), where A = (mij ), B = (nij ), M = (αij ) and N = (βij ) are four m × n matrices. For brevity, a crisp matrix consisting of real numbers is written as a matrix directly throughout this paper, similar to linear systems, linear equations, matrix equations, and so on. e = (e e = (ebij )n×p . Then Definition 2.5.[16,19] Let A aij )m×n and B e ⊗B e =D e = (deij )m×p , A where

L

deij =

X

e aik ⊗ ebkj .

k=1,··· ,n

2.2 Interval objective programming Ishibuchi and Tanaka [20] gave the definitions of the maximization and minimization problems with the interval objective functions, which are introduced in Definitions 2.6 and 2.7 as follows. Definition 2.6.[20] Let e a = [al , au ] be an interval. The maximization problem with the interval objective function is described as follow: max {e a}, s.t. e a ∈ Ω, which is equivalent to the following bi-objective mathematical programming problem: 1 max {al , (al + au )}, 2 s.t. e a ∈ Ω, 13

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Zeng-Tai Gong and Wen-Cui Zhao: A novel approach for solving fully fuzzy linear programming...

where Ω is a set of constraints in which the variable e a should satisfy according to requirements in real situations. Definition 2.7.[20] Let e a = [al , au ] be an interval. The minimization problem with the interval objective function is described as follow: min {e a}, s.t. e a ∈ Ω, which is equivalent to the following bi-objective mathematical programming problem: 1 min {au , (al + au )}, 2 s.t. e a ∈ Ω. 3. Fully fuzzy linear programming problem Linear programming is one of the most frequently applied operations research technique. In the conventional approach, the value of the parameters and variables of linear programming models must be well defined and precise. However, this is not a realistic assumption in the real world environment. There may exists uncertainty about the parameters and variables in the real life problems. So, the fuzzy numbers and fuzzy variables should be used in the LP problem. Therefore, we encounter with the FFLP problems. Consider the standard form of FFLP problem which with m constraints and n variables as follows: e, max(min)Ze = e cT ⊗ x e e e = b, s.t. A⊗x

(0.1)

e is non-negative fuzzy number vector, x e = (e e = (e where A aij )m×n , e cT = (e cj )1×n , x xj )n×1 , e b = (ebi )m×1 , and e aij > 0 (or e aij < 0), e cj > 0 (or e cj < 0), ebi , x ej are LR flat fuzzy numbers. e ⊗x e ⊗x e 6 e e > e It should be noted that A b and A b can be transformed to the standard form by e = (te1 , te2 , · · · , tf introducing a vector variable T ), where tej (j = 1, 2, · · · , m) are LR flat fuzzy numbers, m e e e e e e e ⊕ T = b and A ⊗ x e T = b, respectively. as A ⊗ x Definition 3.1. The fuzzy exact optimal solution of FFLP problem (1) will be a fuzzy number vector e∗ if it satisfies the following characteristics: x e∗ = (xej ∗ )n×1 > 0, where xej ∗ (j = 1, 2, · · · , n) are LR flat fuzzy numbers; (1) x e ⊗x e∗ = e (2) A b; e ={ ex|A e ⊗x e=e e∈S (3) For any x b, ex = (xej )n×1 > 0, where xej are LR flat fuzzy numbers}, we have T ∗ T e e >e e (in case of maximization problem), e e∗ 6 e e (in case of minimization problem). c ⊗x c ⊗x cT ⊗ x cT ⊗ x e e∗ be an exact optimal solution of FFLP problem (1). If there exists an x e0 ∈ S Remark 3.1. Let x e∗ = e e0 , then x e0 is also an exact optimal solution of FFLP problem (1) and is called such that e cT ⊗ x cT ⊗ x an alternative exact optimal solution. e of the FFLP problem (1) have two forms, Note that the elements ( e aij ) in coefficient matrix A e as follows: i.e., (1) e aij > 0, (2) e aij < 0. So we define the coefficient matrix A ( e aij , e aij > 0, e 1 )ij = (A e 0, e aij < 0; ( e 2 )ij = (A

e aij , e aij 6 0, e 0, e aij > 0,

where 1 6 i 6 m, 1 6 j 6 n. Obviously, e =A e1 ⊕ A e 2, A 14

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e ⊗x e1 ⊗ x e2 ⊗ x e=A e⊕A e. A Similarly, we define the coefficient matrix (e cT ) of objective function of FFLP problem (1) as follows: ( e cj , e cj > 0, T (e c1 ) j = e 0, e cj < 0; ( (e cT2 )j

=

e cj , e 0,

e cj 6 0, e cj > 0,

where 1 6 j 6 n. Obviously, e cT = e cT1 ⊕ e cT2 , T T e e=e e⊕e e. c ⊗x c1 ⊗ x cT2 ⊗ x In the FFLP problems (1), let e c = (c, d, p, q)LR = ce1 ⊕ ce2 (where ce1 = (c1 , d1 , p1 , q1 )LR > 0, ce2 = e = (x, y, s, t)LR . Then, according to the operations of the LR flat fuzzy numbers, (c2 , d2 , p2 , q2 )LR 6 0), x we have e e=e e⊕e e = (c1 , d1 , p1 , q1 )LR ⊗ (x, y, s, t)LR ⊕ (c2 , d2 , p2 , q2 )LR ⊗ (x, y, s, t)LR cT ⊗ x cT1 ⊗ x cT2 ⊗ x ∼ = (c1 x, d1 y, c1 s + p1 x, d1 t + q1 y)LR ⊕ (c2 y, d2 x, p2 y − c2 t, q2 x − d2 s)LR = (c1 x + c2 y, d1 y + d2 x, c1 s − c2 t + p1 x + p2 y, q2 x + q1 y + d1 t − d2 s)LR . For brevity, we substitute (Zm , Zn , Zα , Zβ )LR for (c1 x + c2 y, d1 y + d2 x, c1 s − c2 t + p1 x + p2 y, q2 x + q1 y + d1 t − d2 s)LR throughout this paper. e = (A, B, M, N)LR = Theorem 3.1. (Approximate representation of fully fuzzy constraints) Let A f1 = (A1 , B1 , M1 , N1 )LR > 0, A f2 = f1 ⊕ A f2 , e e = (x, y, s, t)LR > 0, where A A b = (b, g, h, f)LR , x e ⊗x e=e (A2 , B2 , M2 , N2 )LR 6 0. Then A b can be represented approximately as follows:   A1 x + A2 y = b,  B y + B x = g, 1 2 (0.2)  M x + M y + A1 s − A2 t = h, 1 2    N2 x + N1 y + B1 t − B2 s = f. e = (A, B, M, N)LR = A e1 ⊕ A e 2, A e 1 = (A1 , B1 , M1 , N1 )LR > 0, A e 2 = (A2 , B2 , M2 , Proof. Since A e = (x, y, s, t)LR > 0, we have N2 )LR 6 0, x e ⊗x e1 ⊕ A e 2) ⊗ x e1 ⊗ x e2 ⊗ x e e = (A e=A e⊕A A = (A1 , B1 , M1 , N1 )LR ⊗ (x, y, s, t)LR ⊕ (A2 , B2 , M2 , N2 )LR ⊗ (x, y, s, t)LR ∼ = (A1 x, B1 y, A1 s + M1 x, B1 t + N1 y)LR ⊕ (A2 y, B2 x, M2 y − A2 t, N2 x − B2 s)LR = (A1 x + A2 y, B1 y + B2 x, A1 s − A2 t + M1 x + M2 y, N2 x + N1 y + B1 t − B2 s)LR =e b = (b, g, h, f)LR . According to Definition 2.3, we have A1 x + A2 y = b,

B1 y + B2 x = g,

M1 x + M2 y + A1 s − A2 t = h,

N2 x + N1 y + B1 t − B2 s = f.

e ⊗x e=e Hence, A b can be represented approximately as (2). 4. Proposed method to find the fuzzy exact optimal solution of FFLP problem In this section, in order to find an effective fuzzy solution of the type of Eq. (1) problem, we are going to introduce a method based on the definition of the interval objective programming and the arithmetic operations of LR flat fuzzy numbers. 15

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Zeng-Tai Gong and Wen-Cui Zhao: A novel approach for solving fully fuzzy linear programming...

We consider the case of maximizing fuzzy objective function at first. e = (c1 x + c2 y, d1 y + d2 x, c1 s − c2 t + p1 x + p2 y, q2 x + q1 y + In Eq. (1), the fuzzy objective Ze = e cT ⊗ x d1 t − d2 s)LR = (Zm , Zn , Zα , Zβ )LR is an LR flat fuzzy number. This fuzzy objective is fully defined by four corner points (Zm , 1), (Zn , 1), (Zm − Zα , 0) and (Zn + Zβ , 0), geometrically. Thus, maximizing the fuzzy objective can be obtained by pushing these four critical points in the direction of the right-hand side. Fortunately, the vertical coordinates of the critical points are fixed at either 1 or 0. The only considerations then are the four horizontal coordinates. Therefore, our problem is to solve {Zm − Zα , Zm , Zn , Zn + Zβ }, e ⊗x e=e s.t. A b, e > 0. x

max

(0.3)

However, the above four objectives (Zm − Zα ), Zm , Zn and (Zn + Zβ ) should always preserve the form of the LR flat fuzzy number (Zm , Zn , Zα , Zβ )LR during the optimization process. Thus, in order to keep the LR flat fuzzy number shape (normal and convex) of the possibility distribution, it is necessary to make a little change. For the mode of LR flat fuzzy number (Zm , Zn , Zα , Zβ )LR , since the objective function of Eq. (1) is e it is natural to maximize the interval [Zm , Zn ] for this objective function. According to to maximize Z, Definition 2.6, in order to maximize the interval [Zm , Zn ], we need maximize the left endpoint Zm and maximize the middle point 21 (Zm + Zn ) of this interval simultaneously. For the lower and upper limits of LR flat fuzzy number (Zm , Zn , Zα , Zβ )LR , we minimize Zα and maximize Zβ instead of maximizing the lower Zm − Zα and the upper Zn + Zβ , respectively. Therefore, combining Theorem 3.1, Eq. (3) can be transformed into the following multi-objective programming model: min max

Z1 = Zα , Z2 = Zm , 1 max Z3 = (Zm + Zn ), 2 max Z4 = Zβ , s.t. A1 x + A2 y = b, B1 y + B2 x = g, M1 x + M2 y + A1 s − A2 t = h, N2 x + N1 y + B1 t − B2 s = f, y − x > 0, x − s > 0, s > 0, t > 0.

(0.4)

Although Eq. (4) is also a multi-objective linear programming model, it can effectively keep the LR flat e To solve Eq. (4), we may use any MOLP technique [21] fuzzy number shape of objective function Z. such as utility theory, goal programming fuzzy programming or interactive approaches. In this paper, we propose three kinds of approaches to solve this multi-objective linear programming model. Since the e = (x, y, s, t)LR , simply denote by objective function Zi is the function of the decision variable vector x Zi = Zi (e x) (i = 1, 2, 3, 4). Let Z1min , Zimax (i = 2, 3, 4) and xei 0 (i = 1, 2, 3, 4) respectively be the minimum objective, maximum objective value and the optimal solution for the following single objective linear programming model: min Z1 = Z1 (e x), s.t. A1 x + A2 y = b, B1 y + B2 x = g, M1 x + M2 y + A1 s − A2 t = h, N2 x + N1 y + B1 t − B2 s = f, y − x > 0, x − s > 0, s > 0, t > 0, 16

(0.5)

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and max s.t.

Zi = Zi (e x), (i = 2, 3, 4) A1 x + A2 y = b, B1 y + B2 x = g, M1 x + M2 y + A1 s − A2 t = h, N2 x + N1 y + B1 t − B2 s = f, y − x > 0, x − s > 0, s > 0, t > 0.

(0.6)

Then, set Z1max = max{Z1 (xe1 0 ), Z1 (xe2 0 ), Z1 (xe3 0 ), Z1 (xe4 0 )}, Zimin = min{Zi (xe1 0 ), Zi (xe2 0 ), Zi (xe3 0 ), Zi (xe4 0 )} (i = 2, 3, 4). The linear membership function of the objective function Z1 can be calculated as follows:   1 if Z1 < Z1min ,  Z1max −Z1 if Z1min 6 Z1 6 Z1max , x) = µz1 (e Z1max −Z1min   0 if Z1 > Z1max . The linear membership function of the objective function Zi (i = 2, 3, 4) can be calculated as follows:  0 if Zi < Zimin ,   Zi −Zimin µzi (e x) = if Zimin 6 Zi 6 Zimax , Zimax −Zimin   1 if Zi > Zimax . Thus, Eq. (4) can be solved by the following linear programming model: max s.t.

µ, 4µzi (e x) +

4 X

µzi (e x) > 8µ (i = 1, 2, 3, 4),

i=1

A1 x + A2 y = b, B1 y + B2 x = g, M1 x + M2 y + A1 s − A2 t = h, N2 x + N1 y + B1 t − B2 s = f, y − x > 0, x − s > 0, s > 0, t > 0,

(0.7)

or max s.t.

µ, 4µzi (e x) +

4 X

µzi (e x) 6 8µ (i = 1, 2, 3, 4),

i=1

A1 x + A2 y = b, B1 y + B2 x = g, M1 x + M2 y + A1 s − A2 t = h, N2 x + N1 y + B1 t − B2 s = f, y − x > 0, x − s > 0, s > 0, t > 0,

(0.8)

or max w1 µz1 (e x) + w2 µz2 (e x) + w3 µz3 (e x) + w4 µz4 (e x), s.t. A1 x + A2 y = b, B1 y + B2 x = g, M1 x + M2 y + A1 s − A2 t = h, N2 x + N1 y + B1 t − B2 s = f, y − x > 0, x − s > 0, s > 0, t > 0, 17

(0.9)

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where w = (w1 , w2 , w3 , w4 )T is the weight vector of objective Zi (i = 1, 2, 3, 4), satisfies that wi > 0 (i = P 1, 2, 3, 4) and 4i=1 wi = 1. Eq. (7) is a kind of pessimistic approach which shows that the decision maker (DM) is very conservative, whereas Eq. (8) is a kind of optimistic approach which shows that the (DM) is very aggressive. Eq. (9) is the linear sum approach based on membership function. Analogously, for the case of minimization fuzzy objective function programming, using Definition 2.7, it can be transformed into the following multi-objective programming model: max min

Z1 = Zα , Z2 = Zn , 1 min Z3 = (Zm + Zn ), 2 min Z4 = Zβ , s.t. A1 x + A2 y = b, B1 y + B2 x = g, M1 x + M2 y + A1 s − A2 t = h, N2 x + N1 y + B1 t − B2 s = f, y − x > 0, x − s > 0, s > 0, t > 0,

(0.10)

which can be solved by using a similar approach (as previously described). 5. Examples In this section, we will demonstrate efficiency and superiority of the proposed method using numerical examples. At the same time, the shortcomings of the existing methods [11-14] for solving FFLP problems with equality constraints are pointed out. 2 1 (e1−x − 1)}. Consider the following Example 5.1. Let L(x) = max{0, 1 − |x|}, R(x) = max{0, e−1 FFLP: max Z = (2, 3, 1, 3) ⊗ x e1 ⊕ (−3, −2, 2, 1) ⊗ x e2 ⊕ (0, 1, 0, 0) ⊗ x e3 , s.t. (2, 2, 1, 0) ⊗ x e1 ⊕ (−2, −1, 2, 1) ⊗ x e2 ⊕ (−2, −1, 1, 0) ⊗ x e3 = (2, 9, 16, 4), (−3, −1, 1, 1) ⊗ x e1 ⊕ (2, 2, 1, 1) ⊗ x e2 ⊕ (−2, −1, 0, 1) ⊗ x e3 = (−13, 2, 11, 12), x e1 > 0, x e2 > 0, x e3 > 0.

(0.11)

Let x e1 = (x1 , y1 , s1 , t1 )LR , x e2 = (x2 , y2 , s2 , t2 )LR , x e3 = (x3 , y3 , s3 , t3 )LR . Then T e e cT = ((2, 3, 1, 3), (−3, −2, 2, 1), (0, 1, 0, 0)), b = ((2, 9, 16, 4), (−13, 2, 11, 12)),   (2, 2, 1, 0) (−2, −1, 2, 1) (−2, −1, 1, 0) e = A , (−3, −1, 1, 1) (2, 2, 1, 1) (−2, −1, 0, 1)         2, −2, −2 2, −1, −1 1, 2, 1 0, 1, 0 A= , B= , M= , N= , −3, 2, −2 −1, 2, −1 1, 1, 0 1, 1, 1         2, 0, 0 0, −2, −2 2, 0, 0 0, −1, −1 A1 = , A2 = , B1 = , B2 = , 0, 2, 0 −3, 0, −2 0, 2, 0 −1, 0, −1         1, 0, 0 0, 2, 1 0, 0, 0 0, 1, 0 M1 = , M2 = , N1 = , N2 = . 0, 1, 0 1, 0, 0 0, 1, 0 1, 0, 1

cT = (2, −3, 0), cT1 = (2, 0, 0), pT1 = (1, 0, 0), bT = (2, −13),

dT = (3, −2, 1),

pT = (1, 2, 0),

qT = (3, 1, 0),

cT2 = (0, −3, 0),

dT1 = (3, 0, 1),

dT2 = (0, −2, 0),

pT2 = (0, 2, 0),

qT1 = (3, 0, 0),

qT2 = (0, 1, 0),

gT = (9, 2),

hT = (16, 11),

fT = (4, 12).

18

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Zeng-Tai Gong and Wen-Cui Zhao: A novel approach for solving fully fuzzy linear programming...

By Eq.(4), Eq. (11) can be solved by the auxiliary MOLP model as follows: min max max max s.t.

Z1 = x1 + 2y2 + 2s1 + 3t2 , Z2 = 2x1 − 3y2 , Z3 = x1 − x2 + 1.5y1 − 1.5y2 + 0.5y3 , Z4 = x2 + 3y1 + 3t1 + t3 + 2s2 , 2x1 − 2y2 − 2y3 = 2, 2x2 − 3y1 − 2y3 = −13, 2y1 − x2 − x3 = 9, 2y2 − x1 − x3 = 2, x1 + 2y2 + y3 + 2s1 + 2t2 + 2t3 = 16, x2 + y1 + 2s2 + 3t1 + 2t3 = 11, x2 + 2t1 + s2 + s3 = 4, x1 + x3 + y2 + s1 + s3 + 2t2 = 12, yi − xi > 0, xi − si > 0, si > 0, ti > 0 (i = 1, 2, 3, 4).

(0.12)

Using Eq. (5,6), it follows that Z1max = 17.20,

Z2max = −0.40,

Z3max = 6.10,

Z4max = 23.86,

Z1min = 16.35,

Z2min = −0.85,

Z3min = 5.93,

Z4min = 22.60.

Then, we can get µz1 (e x) =

µz2 (e x) =

  1

17.20−Z1 0.85

 0   1

Z2 +0.85 0.45

 µz3 (e x) =

µz4 (e x) =

0   1

Z3 −5.93 0.17

 0   1

Z4 −22.60 1.26



0

if Z1 < 16.35, if 16.35 6 Z1 6 17.20, if Z1 > 17.20; if Z2 > −0.40, if − 0.85 6 Z2 6 −0.40, if Z2 < −0.85; if Z3 > 6.10, if 5.93 6 Z3 6 6.10, if Z3 < 5.93; if Z4 > 23.86, if 22.60 6 Z4 6 23.86, if Z4 < 22.60.

Next, we use three approach (i.e., Eqs. (7)-(9)) to solving MOLP (12), respectively. First, according to Eq. (7) (i.e., pessimistic approach), we can obtain the optimal solution of the MOLP (12) as follows: x∗1 = 5.28, x∗2 = 2.96, x∗3 = 0.15, y1∗ = 6.06, y2∗ = 3.72, y3∗ = 0.38, s∗1 = 0.00, s∗2 = 0.88, s∗3 = 0.15, t∗1 = 0.00, t∗2 = 1.35, t∗3 = 0.11. Therefore, the optimal solution of the FFLP (11) is:   (5.28, 6.06, 0.00, 0.00)LR e∗ =  (2.96, 3.72, 0.88, 1.35)LR  . x (0.15, 0.38, 0.15, 0.11)LR Substituted the above optimal solution into the objective function of FFLP (11), the optimal objective value is obtained as (−0.60, 12.64, 16.77, 23.01)LR . Namely, the most likely value of the objective is between -0.60 and 12.64, the upper and lower limits of the objective value are -17.37 and 35.65, respectively. Its membership function is given by   L( −0.60−t 16.77 ) t < −0.60, 1 −0.60 6 t 6 12.64, uZe (t) =  R( t−12.64 ) t > 12.64. 23.01 19

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Zeng-Tai Gong and Wen-Cui Zhao: A novel approach for solving fully fuzzy linear programming...

That is uZe (t) =

 1+     1    

0.60+t 16.77

1− 1 e−1 (e

(t−12.64)2 23.012

0,

−17.37 < t < −0.60, −0.60 6 t 6 12.64, − 1) 12.64 < t < 35.65, others.

Second, solving the MOLP (12) by using Eq. (8) (i.e., optimistic approach), we obtain the optimal solution as follows: x∗1 = 4.09, x∗2 = 1.14, x∗3 = 0.01, y1∗ = 5.08, y2∗ = 3.05, y3∗ = 0.02, s∗1 = 0.72, s∗2 = 0.54, s∗3 = 0.00, t∗1 = 1.16, t∗2 = 2.07, t∗3 = 0.11. Therefore, the optimal solution of the FFLP (11) is:   (4.09, 5.08, 0.72, 1.16)LR e∗ =  (1.14, 3.05, 0.54, 2.07)LR  . x (0.01, 0.02, 0.00, 0.11)LR Substituted the above optimal solution into the objective function of FFLP (11), the optimal objective value is obtained as (−0.97, 12.98, 17.84, 21.05)LR . Namely, the most likely value of the objective is between -0.97 and 12.98, the upper and lower limits of the objective value are -18.81 and 34.03, respectively. Its membership function is given by   L( −0.97−t 17.84 ) t < −0.97, 1 −0.97 6 t 6 12.98, uZe (t) =  R( t−12.98 ) t > 12.98, 21.05 That is uZe (t) =

 1+     1    

0.97+t 17.84

1− 1 e−1 (e

(t−12.98)2 21.052

0,

−18.81 < t < −0.97, −0.97 6 t 6 12.98, − 1) 12.98 < t < 34.03, others.

Finally, solving the MOLP (12) by using Eq. (9) (i.e., linear sum approach based on membership function), we obtain the optimal solution for w = ( 41 , 14 , 14 , 14 )T as follows: x∗1 = 5.32, x∗2 = 2.84, x∗3 = 0.05, y1∗ = 5.95, y2∗ = 3.68, y3∗ = 0.42, s∗1 = 0.00, s∗2 = 1.11, s∗3 = 0.05, t∗1 = 0.00, t∗2 = 1.45, t∗3 = 0.00. Therefore, the optimal solution of the FFLP (11) is:   (5.32, 5.95, 0.00, 0.00)LR e∗ =  (2.84, 3.68, 1.11, 1.45)LR  . x (0.05, 0.42, 0.05, 0.00)LR Substituted the above optimal solution into the objective function of FFLP (11), the optimal objective value is obtained as (−0.40, 12.59, 17.03, 22.91)LR . Namely, the most likely value of the objective is between -0.40 and 12.59, the upper and lower limits of the objective value are -17.43 and 35.50, respectively. Its membership function is given by   L( −0.40−t 17.03 ) t < −0.40, 1 −0.40 6 t 6 12.59, uZe (t) =  R( t−12.59 ) t > 12.59, 22.91 That is uZe (t) =

 1+     1    

0.40+t 17.03

1− 1 e−1 (e

0,

(t−12.59)2 22.912

−17.43 < t < −0.40, −0.40 6 t 6 12.59, − 1) 12.59 < t < 35.50, others.

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Zeng-Tai Gong and Wen-Cui Zhao: A novel approach for solving fully fuzzy linear programming...

To illustrate the influence of the weight vector w on the optimal value in this example, we use a different weight vector w to solve the MOLP (12) according to Eq. (9). Generally speaking, we need to consider some special cases. One is an average weight, i.e, w = ( 14 , 14 , 14 , 14 )T . Secondly, the objective functions Z2 and Z3 are relative to the mode of the flat fuzzy number Z, which are the most possible values of Z. Thus, more weights should be assigned to Z2 and Z3 , i.e., w = ( 16 , 62 , 26 , 16 )T . Thirdly, similar to the Olympic games, which discarded a maximum point and a minimum point, we can set w = (0, 12 , 12 , 0)T . Finally, w = ( 62 , 61 , 61 , 26 )T emphasizes both ends and reduces the middle. All the computation results are shown in Table 1. Table 1: The optimal solution and optimal objective value for FFLP (11) with different approaches. variable by Eq.(6) by Eq.(7) by Eq.(8) w = ( 14 , 41 , 14 , 14 )T w = ( 16 , 62 , 26 , 16 )T w = (0, 21 , 12 , 0)T w = ( 26 , 61 , 16 , 26 )T

x1 x2 x3 y1 y2 y3 s1 s2 s3 t1 t2 t3 optimal valueZ 5.28 2.96 0.15 6.06 3.72 0.38 0.00 0.88 0.15 0.00 1.35 0.11 (−0.60, 12.64, 16.77, 23.01)LR 4.09 1.14 0.01 5.08 3.05 0.02 0.72 0.54 0.00 1.16 2.07 0.11 (−0.97, 12.98, 17.84, 21.05)LR 5.32 2.84 0.05 5.95 3.68 0.42 0.00 1.11 0.05 0.00 1.45 0.00 (−0.40, 12.59, 17.03, 22.91)LR 5.32 2.84 0.05 5.95 3.68 0.42 0.00 1.11 0.05 0.00 1.45 0.00 (−0.40, 12.59, 17.03, 22.91)LR 5.20 2.60 0.00 5.80 3.60 0.40 0.00 1.00 0.00 0.20 1.60 0.00 (−0.40, 12.60, 17.20, 22.60)LR 5.45 3.55 0.36 6.45 3.91 0.36 0.00 0.45 0.00 0.00 1.14 0.05 (−0.83, 12.61, 16.69, 23.85)LR

It can be seen from Table 1 that, the optimal solutions and optimal objective value of Z for w = w = ( 16 , 62 , 26 , 61 )T and w = (0, 12 , 12 , 0)T are completely identical. The optimal solution and optimal objective value of Z for w = ( 62 , 61 , 16 , 62 )T are not the same as that for the other weight vectors. This signifies that the weight vector w does affect the optimal solutions and optimal objective value. In addition, Table 1 shows that applying different approaches to solving the multi-objective programming may result in different optimal solutions and optimal objective values. The DM can choose the proper approach to solving the multi-objective programming according to his/her risk preference and actual requirements. e∗ is more general Note that for a fully fuzzy linear programming problem, the fuzzy optimal solution x than the result which are calculated in Lotfi [12], Kumar [13] and Ezzati [14] when the left spreads shape functions L(·) and right spreads shape functions R(·) are linear functions. Furthermore, even the uncertain elements in a fuzzy linear programming problem were extended fuzzy numbers, we can construct a corresponding auxiliary multi-objective programming problem and solve it by use of three approaches (i.e., optimistic approach, pessimistic approach and linear sum approach based on membership function). In Wan [11], the FLP problems with trapezoidal fuzzy numbers have been studied in which not all parts of the problem were assumed to be fuzzy (the variables were not fuzzy). Lotfi [12] proposed a novel method to find the optimal solution of FFLP problems, this method was obtained by using the concept of the symmetric triangular fuzzy numbers and introduce an approach to defuzzify a general fuzzy quantity. However, it can be applied only if the elements of the coefficient matrix are symmetric fuzzy numbers. In Kumar’s article [13], the fuzzy optimal solution of FFLP problem can be obtained by using the arithmetic operations of triangular fuzzy numbers and linear ranking function which is used to convert the fuzzy objective function to the real objective function. Although the ranking function is convenient for the specific numerical computation, the fuzziness of the objective function is neglected by it. Ezzati [14] proposed a new algorithm to solve FFLP problems with TFNs. It can be applied only if all the parameters as well as the variables are represented by TFNs which is the specious case of LR flat fuzzy number, for the generalized fuzzy numbers, it is unsuitable. ( 14 , 14 , 14 , 14 )T ,

6. Conclusion In this paper, we propose a simple and practical method to solve a full fuzzy linear programming problem. The corresponding auxiliary MOLP problem is constructed and we solve it by use of three approaches. By numerical example, the obtained results of proposed algorithm with Wan [11], Lotfi [12], Kumar [13] and R. Ezzati [14] have been compared and shown the reliability and applicability of our 21

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Zeng-Tai Gong and Wen-Cui Zhao: A novel approach for solving fully fuzzy linear programming...

algorithm.

References [1] R. E. Bellman, L. A. Zadeh, Decision making in a fuzzy environment, Management Sci. 17 (1970) 141õ164. [2] H. Tanaka, T. Okuda, K. Asai, On fuzzy mathematical programming, Cybernetics Syst. 3(4) (1973) 37õ46. [3] H. J. Zimmerman, Fuzzy programming and linear programming with several objective functions, Fuzzy Sets and Systems 1(1) (1978) 45õ55. [4] H. R. Maleki, M. Tata, M. Mashinchi, Linear programming with fuzzy variables, Fuzzy Sets and Systems 109(1) (2000) 21-33. [5] H. R. Maleki, Ranking functions and their applications to fuzzy linear programming, Far East Journal of Mathematical Sciences 4(3) (2002) 283-302. [6] Y. J. Lai, C. L. Hwang, A new approach to some possibilistic linear programming problems, Fuzzy Sets and Systems 49 (1992) 121õ133. [7] K. Ganesan, P. Veeramani, Fuzzy linear programs with trapezoidal fuzzy numbers, Annals of Operations Research 143(1) (2006) 305-315. [8] A. Ebrahimnejad, Some new results in linear programs with trapezoidal fuzzy numbers: finite convergence of the Ganesan and Veeramani’s method and a fuzzy revised simplex method, Applied Mathematical Modelling 35 (2011) 4526õ4540. [9] N. Mahadavi-Amiri, S. H. Nasseri, Duality in fuzzy number linear programming by the use of a certain linear ranking function, Applied Mathematics and Computation 180 (2006) 206õ216. [10] L. Campos, J. L. Verdegay, Linear programming problems and ranking of fuzzy numbers, Fuzzy Sets and Systems 32(1) (1989) 1-11. [11] S. P. Wan, J. Y. Dong. Possibility linear programming with trapezoidal fuzzy numbers, Applied Mathematical Modelling 38(5õ6) (2014) 1660-1672. [12] F. H. Lotfi, T. Allahviranloo, M. A. Jondabeha, L. Alizadeh, Solving a fully fuzzy linear programming using lexicography method and fuzzy approximate solution, Applied Mathematical Modelling 33 (2009) 3151õ3156. [13] A. Kumar, J. Kaur, P. Singh, A new method for solving fully fuzzy linear programming problems, Applied Mathematical Modelling 35 (2011) 817õ823. [14] R. Ezzati, E. Khorram, R. Enayati, A new algorithm to solve fully fuzzy linear programming problems using the MOLP problem, Applied Mathematical Modelling 39(12) (2015) 3183-3193. [15] D. Dubois, H. Prade, Operations on fuzzy numbers, International Journal of Systems Science 9(6) (1978) 613-626. [16] M. Dehghan, B. Hashemi, M, Ghatee, Computational methods for solving fully fuzzy linear systems, Applied Mathematics and Computation 179(1) (2006) 328-343. [17] D. Dubois, H. Prade, Fuzzy Sets and Systems: Theory and Applications, Academic Press, New York, 1980. [18] A. Kauffmann, M. M. Gupta, Introduction to fuzzy Arithmetic: Theory and Applications, Van Nostrand Reinhold, New York, 1991. [19] Z. T. Gong, X. B. Guo, Inconsistent fuzzy matrix equations and its fuzzy least squares solutions, Applied Mathematical Modelling 35 (2011) 1456õ1469. [20] H. Ishibuchi, H. Tanaka, Multiobjective programming in optimization of the interval objective function, Eur. J. Oper. Res. 48 (1990) 219õ225. [21] E. Yazdany Peraei, H. R. Maleki, M. Mashinchi, A method for solving a fuzzy linear programming, J. Appl. Math. 8 (2) (2001) 347õ356. 22

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Generalized Bateman’s G−function and its bounds Mansour Mahmoud1 and Hanan Almuashi2 1 Mansoura 2 King

University, Faculty of Science, Mathematics Department, Mansoura 35516, Egypt.

Abdulaziz University, Faculty of Science, Mathematics Department, Jeddah 21589, Saudi Arabia. 1 2

[email protected]

beautyrose− 12− [email protected]

. Abstract In this paper, we presented some functional equations of the generalized Bateman’s G−function Gh (x) and its relation with the hypergeometric series 3 F2 . We deduced an asymptotic expansion of the function Gh (x) and studied the completely monotonic property of some functions involving it. Also, we presented some new bounds of the function Gh (x) and the double inequality ) ( ) ( h h 2h < Gh (x) − < ln 1 + , x > 0; 0 < h < 2 ln 1 + x+β x(x + h) x+α where the constants α = 1 and β =

h γ+ 2 +ψ h 2 e h

( ) −1

are the best possible.

2010 Mathematics Subject Classification: 33B15, 26D15, 41A60, 65Q20. Key Words: Psi function, Bateman’s G-function, functional equation, asymptotic formula, Laplace transform, inequality, monotonicity, best possible bound.

1

Introduction.

The ordinary gamma function Γ(x) is defined by [3] ∫ ∞ Γ(x) = tx−1 e−t dt,

x>0

0

and the Psi or digamma function ψ(x) is given by ψ(x) =

d log Γ(x). dx

The gamma function and its logarithmic derivatives ψ (n) (x) are of the most widely used special functions encountered in advanced mathematics . For more details about the properties of 1

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these functions and their bounds, please refer to [2], [3], [8], [9], [15]-[18], [23]-[27] and plenty of references therein. The Bateman’s G−function is defined by [7] ( ) (x) x+1 , x= ̸ 0, −1, −2, ... . G(x) = ψ −ψ (1) 2 2 The function G(x) is very useful in estimating and summing certain numerical and algebraic series. For more details about the properties, bounds and applications of the G(x), please refer to [7], [12]-[14], [15], [19], [28] and the references therein. The function G(x) satisfies the following relations [7] G(x) = 2

∞ ∑ (−1)k k=0

k+x

,

(2)

G(x + 1) + G(x) = 2x−1 , ( ) n−1 ∑ k k+1 −1 (−1) ψ x + G(nx) = 2n , n k=0 ( ) k (−1) G x + G(nx) = n , n k=0 ∫ ∞ e−xt G(x) = 2 dt, 1 + e−t 0 −1

n−1 ∑

k

(3) n = 2, 4, 6, ...

n = 1, 3, 5, ...

(4)

(5)

x>0

(6)

G(x) = 2x−1 2 F1 (1, x; x + 1; −1),

(7)

where r Fs (α1 , ..., αr ; β1 , ..., βs ; x)

=

∞ ∑ (α1 )k ...(αr )k xk k=0

(β1 )k ...(βs )k k!

is the generalized hypergeometric series [3] defined for r, s ∈ N, αj , βj ∈ C, βj ̸= 0, −1, −2, ... and the Pochhammer or shifted symbol (α)n is given by (α)0 = 1

and

(α)m =

Γ(α + m) , Γ(α)

m ≥ 1.

Qiu and Vuorinen [28] presented the double inequality 4(3/2 − ln 4) 1 < G(x) − x−1 < 2 , 2 x 2x

x > 0.5 .

(8)

Mahmoud and Agarwal [12] deduced the following asymptotic formula for Bateman’s G-function G(x) ∼ x−1 +

∞ ∑ (22k − 1)B2k

k

k=1

x−2k ,

x→∞

2

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(9)

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and they presented the inequality 2x2

1 1 < G(x) − x−1 < 2 , + 1.5 2x

x>0

(10)

√ ln 4 which improved the lower bound of the inequality (8) for x > 89−6 . Also, Mahmoud and ln 4−11 Almuashi [13] proved the following double inequality of the Bateman’s G−function 2m ∑ (2n − 1)B2n n=1

n

x−2n < G(x) − x−1
0, r = 0, 1, 2, ... , 0 < λ < 1, [

∑ 1 1 Tr (λ; x) = (1 − λ) + λ + r + 1 i=0 (x + i + 1)(x + i + λ)

and ωr (λ; x) =

r−1

]

1 (x + r)(x+r)(1−λ) (x + r + 1)(x+r+1)λ log . x+r+λ (x + r + λ)x+r+λ

Mahmoud, Talat and Moustafa [14] presented the following family of approximations of the function G(x) ( ) 1 2 M (µ, x) = ln 1 + + , x > 0; 1 ≤ µ ≤ 2 x+µ x(x + 1) ( ) ( ) (x+2)[(e2 −4)x+4] which is of an order of convergence of O ln (x+1)[(e for x > 2 and µ ∈ 1, e24−4 and is 2 −4)x+e2 ] asymptotically equivalent to G(x) as x → ∞. In this paper, we presented some functional equations of the generalized Bateman’s G−function ) ( (x) x+h −ψ , 0 < h < 2; x ̸= −2m, −2m − h for m = 0, 1, 2, ... (13) Gh (x) = ψ 2 2 and its relation with the hypergeometric function 3 F2 . We deduced an asymptotic expansion of the function Gh (x) and studied the completely monotonic property of the function Gh (x) − xsr for different values of the parameter s, r and h for x > 0. Also, some new bounds and best possible bounds of the generalized Bateman’s G−function are given. 3

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2

Some relations of the function Gh(x).

Lemma 2.1. The function Gh (x) satisfies the functional equation Gh (x + 1) + Gh (x) = 2 (ψ(x + h) − ψ(x)) , Proof. Using the integral representation [3] ∫ ∞ −t e − e−tz dt, ψ(z) = −γ + 1 − e−t 0 we get

∫

∞

Gh (x) = 2 0

1 − e−ht −xt e dt, 1 − e−2t

x > 0.

(14)

R(z) > 0

(15)

x > 0.

(16)

Also, ∫ ψ(x + h) − ψ(x) =

∞

∫0 ∞ = 0

1 − e−ht −xt e dt 1 − e−t ∫ ∞ 1 − e−ht −(x+1)t 1 − e−ht −xt e dt + e dt 1 − e−2t 1 − e−2t 0

1 = [Gh (x + 1) + Gh (x)] . 2

In case of h = 1 and using the functional equation ψ(x + 1) = (3).

1 x

+ ψ(x), we get the relation

Lemma 2.2. The function Gh (x) satisfies the functional equation ( ) m−1 1 ∑ 2r Gh (mx) = Gh x+ , m r=0 m m

x > 0; m ∈ N.

(17)

Proof. m−1 ∑ r=0

) ( ) ∫ ∞ (m−1 ∑ −2rt 1 − e −ht m 2r Gh x+ = e m e−xt dt −2t m m 1 − e 0 r=0 ) ∫ ∞( −ht 1 − e−2t 1 − e m −xt = e dt −2t 1 − e−2t 1−e m 0 ∫ ∞ −ht 1 − e m −xt dt = −2t e 1−e m 0 = m Gh (mx).

Remark 1. The following new functional equation of the ordinary function G(x) in terms of the generalized function Gh (x) can be obtained in case of h = 1. 4

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Corollary 2.3. The function G(x) satisfies the functional equation ( ) m−1 1 ∑ 2r G(mx) = G1 x+ , x > 0; m ∈ N. m r=0 m m

(18)

The following result relates the function Gh (x) and the hypergeometric function 3 F2 . Lemma 2.4. The function Gh (x) satisfies ) ( x+h+2 h h+2 ; 2, ;1 , Gh (x) = 3 F2 1, 1, x+h 2 2 Proof. Using the integral representation [3] ∫ 1 1 − tz−1 ψ(z) = −γ + dt, 1−t 0 we get

∫

1

Gh (x) =

t

x−2 2

0

x+h−2 2

−t 1−t

and then Gh (x) =

∫

1

(

dt =

t

x−2 2

−t

n=0

(19)

R(z) > 0

x+h−2 2

(∞ ) ) ∑ tn dt,

0 ∞ ∑

x > 0.

x>0

n=0

2h , (x + 2n)(x + h + 2n)

x > 0.

(20)

Using the relation x+n= we obtain

x(x + 1)n , (x)n

( x+h ) ( x ) ∞ ∑ 2h 2 2 n ( x+h+2 )n ( x+2 ) Gh (x) = x(x + h) n=0 2 2 n n ( ) 2h x x+h x+2 x+h+2 = ; , ;1 , 3 F2 1, , x(x + h) 2 2 2 2

x > 0.

Now using the two-term Thomae transformation formula [30], [21] 3 F2

(α, β, σ; δ, η; 1) =

with α= we have

Γ(δ)Γ(θ − σ) 3 F2 (η − α, η − β, σ; θ, η; 1) , Γ(θ)Γ(δ − σ)

x+h+2 x x+h , β= , σ = 1, η = , 2 2 2

δ=

θ =δ+η−α−β x+2 2

) ( ) ( h+2 x+h+2 x x+h x+2 x+h+2 x ; , ; 1 = 3 F2 1, 1, ; 2, ;1 , 3 F2 1, , 2 2 2 2 2 2 2

which complete the proof. Remark 2. From the formulas (7) and (19) for h = 1, we can conclude that ( ) x+3 2(x + 1) ;1 = 3 F2 1, 1, 3/2; 2, 2 F1 (1, x; x + 1; −1) , 2 x

x > 0.

5

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3

An asymptotic expansion of the function Gh(x).

Ii is well known that the Psi function has the asymptotic expansion [6] ∞ ∑ (−1)k Bk 1 ψ(z) ∼ ln z − k zk k=1

and its generalization is given by ψ(z + l) ∼ ln z −

∞ ∑ (−1)k Bk (l) 1 , k k z k=1

where Bk (l) are the Bernoulli polynomials defined by the generating function [11] ∑ Bk (l) zezt = zk ez − 1 k=0 k! ∞

and the Bernoulli constants Bk = Bk (0). Using the operations of the asymptotic expansions [5]; [20], we obtain ∞ ∑ (−1)k+1 1 ψ(z + l) − ψ(z) ∼ [Bk (l) − Bk ] k . k z k=1 For more details about the general theory of the asymptotic expansion of the function f (z + t) by the asymptotic expansion of the function f (z) using Appell polynomials, we refer to [4]. Now, using the identity [11] k ∑ (k ) k−r Bk (l) = , r Br l r=0

we get

[ k−1 ] ∞ ∑ (−1)k+1 ∑ (k ) 1 k−r ψ(z + l) − ψ(z) ∼ . r Br l k k z r=0 k=1

Then we obtain the following result. Lemma 3.1. The following asymptotic series holds: [ ( ) ] ∞ ∑ (−1)n+1 2n h 1 Gh (x) ∼ Br − Br n , n 2 x n=1 or Gh (x) ∼

[ n−1 ∞ ∑ (−1)n+1 ∑ n=1

n

] (nr ) 2r Br hn−r

r=0

1 , xn

Remark 3. As a special case at h = 1, we obtain [ ( ) ] ∞ 1 ∑ (−1)n+1 2n 1 1 G(x) ∼ + Br − Br n , x n=2 n 2 x

x → ∞.

(22)

x → ∞.

(23)

x→∞

6

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and using the identities [1]

( ) ( ) 1 Bn = 21−n − 1 Bn , 2

n = 0, 1, 2, ...

and B2n+1 = 0,

n = 1, 2, ...

then we get the asymptotic series (9). Now we will study the completely monotonic property of the function Gh (x) − xsr for different values of the parameters s, h and r for positive values of x. Lemma 3.2. The functions s x

χs,1 (x, h) = Gh (x) − and χs,r (x, h) = Gh (x) −

s xr

s ≤ h; x > 0; 0 < h < 2,

(24)

s < 0; x > 0; 0 < h < 2; r = 2, 3, 4, ...

(25)

are strictly completely monotonic. Proof. Using the relation (16) and the known formula ∫ ∞ −m (r − 1)! x = v m−1 e−xv dv,

m∈N

(26)

0

we get

∫ (−1)n χ(n) s,r (x, h)

∞

=

ϕh,s (r, t) 0

where

tn e−xt dt, e2t − 1

n = 0, 1, 2, 3, ...

(27)

( ) ) s tr−1 ( 2t ϕh,s (r, t) = 2 e2t − e(2−h)t − e −1 . (r − 1)!

Then ϕh,s (r, t) =

∞ ∑ 2k+1 tk

k!

k=1

Ph,s (r, t),

where

( )k h s Ph,s,k (r, t) = 1 − 1 − − tr−1 . 2 2(r − 1)! Firstly, if r = 1, we obtain ( )k ( )k h s s h Ph,s,k (1, t) = 1 − 1 − − > 0 iff ≤1− 1− 2 2 2 2

k = 1, 2, 3, ... .

But

( )k h h ≤1− 1− 0 < h < 2; k = 1, 2, 3, ... 2 2 and thus, ϕh,s (1, t) > 0 for all t ≥ 0 iff s ≤ h. Secondly, when r = 2, 3, 4, ..., then Ph,s,k (r, t) ( )k is increasing as a function of t if s < 0 with Ph,s,k (r, 0) = 1 − 1 − h2 > 0 for 0 < h < 2 and k = 1, 2, 3, ... . Thus ϕh,s (r, t) > 0 for all t ≥ 0, r = 2, 3, ... iff s < 0. 7

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As a result of the strictly completely monotonicity of the function χs,1 (x, h) and the relation (23), we obtain χs,1 (x, h) > lim (χs,1 (x, h)) = 0, s ≤ h. x→∞

Hence, we have the following result: Corollary 3.3. The following inequality holds Gh (x) >

4

h , x

x > 0; 0 < h < 2.

(28)

Some Bounds of the function Gh(x).

Lemma 4.1. Gh (x)
0; 0 < h < 2.

(29)

Proof. By using the formulas (16) and (26), we get for x > 0 that ) −xt ∫ ∞( ( 2t ) ( 2t ) h(2 − h) ( 2t ) e 2 h(2 − h) (2−h)t −2 e −1 − e − 1 t 2t Gh (x) − − = 2 e −e dt 2 x 2x 2 e −1 0 ) −xt ∫ ∞( ) ( ) h(2 − h) ( 2t e (2−h)t = e − 1 t 2t dt 2 1−e − 2 e −1 0 < 0 for 0 < h < 2.

Theorem 1. Gh (x)
0; 1 ≤ h < 2.

(30)

Proof. Using the two formulas (16) and (26), we have ∫ ∞ h h(2 − h) e−xt Gh (x) − − = dt, ρ (t) h x 2x2 e2t − 1 0 where

Then

( ) ( ) h(2 − h) ( 2t ) ρh (t) = 2 e2t − e(2−h)t − h e2t − 1 − e −1 t 2

t > 0.

ρ′′h (t) = 2(h − 2)e(2−h)t Qh (t)

with Qh (t) = 2 − h + eht (h − 2 + ht). , which is non positive for The function Qh (t) is convex function with minimum value at t0 = 1−h h 1 ≤ h < 2 and Qh (0) = 0. Hence Qh (t) > 0 for 1 ≤ h < 2. Hence ρh (t) is concave for 1 ≤ h < 2 and its has maximum value at t = 0. Then 1 ≤ h < 2; t > 0.

ρh (t) < 0,

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Then the function Gh (x) − hx − h(2−h) is strictly increasing function for 1 ≤ h < 2 and x > 0 and 2x2 using the asymptotic expansion (23), we get ) ( h h(2 − h) lim Gh (x) − − = 0, x→∞ x 2x2 which complete the proof. Remark 4. In case of h = 1, the inequality (30) will prove the right-hand side of the inequality (8). To obtain our next result, we will apply the following monotone form of L’Hˆopital’s rule [10] (see also [22] and [29] ). Theorem 2. Let −∞ < α < β < ∞ and L, U : [α, β] → R be continuous on [α, β] and differentiable on (α, β). Let U ′ (x) ̸= 0 on (α, β). If L′ (x)/U ′ (x) is increasing (decreasing) on (α, β), then so are L(x) − L(α) L(x) − L(β) and . (31) U (x) − U (α) U (x) − U (β) If L′ (x)/U ′ (x) is strictly monotone, then the monotonicity in the conclusion is also strict. Theorem 3. Gh (x) >

h h(2 − h) + , x 2 (x2 + 3h2 )

x > 0; 0 < h < 2.

(32)

Proof. Using the two formulas (16) and (26) and the Laplace transformation of the sine function, we get ∫ ∞ h h(2 − h) e−xt Gh (x) − − = dt, ξ (t) h x 2 (x2 + 3h2 ) 6 (e2t − 1) 0 where (

ξh (t) = 6 −2e − e 2t

(2+h)t

ht

(−2 + h) + he

)

+

√

ht

3e

(

−1 + e

Now consider the function √ ( ) 2 3e−ht −2e2t − e(2+h)t (−2 + h) + heht τh (t) = (−1 + e2t ) (2 − h)

2t

)

(√ ) (−2 + h) sin 3ht .

t > 0; 0 < h < 2.

The function √ ( )) ( √ −ht 2 3 dtd e−ht −2e2t − e(2+h)t (−2 + h) + heht 3e (−1 + eht ) = 2 d 2t ) (2 − h)) ((−1 + e dt is increasing function for t > 0. Using the monotone form of L’Hˆopital’s rule, we get that the function τh (t) is increasing. Similarly, the function τh (t) , Hh (t) = √ 3ht

t > 0; 0 < h < 2.

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is increasing function and lim Hh (t) = 1.

t→∞

Then √ ) ( ) ( 2 3e−ht −2e2t − e(2+h)t (−2 + h) + heht > ht −1 + e2t (2 − h),

t > 0; 0 < h < 2

and using Jordan’s inequality 2z ≤ sin z ≤ z, π

x ∈ [0, π/2]

we have (√ ) √ ( ) ( ) 2 3e−ht −2e2t − e(2+h)t (−2 + h) + heht > ht −1 + e2t (2−h) sin 3ht ,

t > 0; 0 < h < 2.

Hence ξh (t) > 0,

t > 0; 0 < h < 2.

h(2−h) Then the function Gh (x) − hx − 2(x 2 +3h2 ) is strictly decreasing function for 0 ≤ h < 2 and x > 0. Also, using the asymptotic expansion (23), we get ( ) h h(2 − h) lim Gh (x) − − = 0, x→∞ x 2 (x2 + 3h2 )

which complete the proof. Remark 5. In case of h = 1, the inequality (32) will prove the left-hand side of the inequality (10). Remark 6. Using the inequalities (29), (30) and (32) with the relation (20), we get the following estimations ∑ 1 2−h 1 1 2−h + < < + , 2 2 2x 4 (x + 3h ) n=0 (x + 2n)(x + h + 2n) 2x 4x2

x > 0; 1 ≤ h < 2

∑ 1 1 2−h 1 2−h + < < + , 2 2 2x 4 (x + 3h ) n=0 (x + 2n)(x + h + 2n) hx 4x2

x > 0; 0 < h < 2.

∞

and ∞

5

Sharp double inequality of the function Gh(x).

Theorem 4. For x > 0, we have ( ) ( ) h 2h h 2h ln 1 + + ≤ Gh (x) ≤ ln 1 + + . x+2 x(x + h) x x(x + h)

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Proof. Consider the following function for x > 0 and 0 < h < 2 Ax,h (t) =

2h , (x + h + 2t)(x + 2t)

t≥0

which is strictly decreasing since 4h(2x + h + 4t) d Ax,h (t) = − . dt (x + 2t)2 (x + h + 2t)2 The function Ax,h (t) is continuous for t ≥ 0, then its lower Riemann sum L(Ax,h (t), P ) with respect to the partition P = {0, 1, ..., n} of the interval [0, n], n ∈ N , satisfies ∫ n L(Ax,h (t), P ) < Ax,h (t)dt. 0

Hence

n ∑

( Ax,h (i) < ln

i=1

or

n ∑ i=0

2h < ln (x + h + 2i)(x + 2i)

Then as n → ∞, we get

(x + h)(x + 2n) x(x + h + 2n)

(

)

(x + h)(x + 2n) x(x + h + 2n)

) +

2h . x(x + h)

( ) h 2h Gh (x) ≤ ln 1 + + . x x(x + h)

Also, the function Ax,h (t + 1) is continuous for t ≥ −1, then its upper Riemann sum U (Ax,h (t + 1), P ) with respect to the partition P of the interval [0, n], n ∈ N , satisfies ∫ n Ax,h (t + 1)dt < U (Ax,h (t + 1), P ). 0

Hence

( ln

or

( ln

(x + h + 2)(x + 2 + 2n) (x + 2)(x + h + 2 + 2n)

(x + h + 2)(x + 2 + 2n) (x + 2)(x + h + 2 + 2n)

Then as n → ∞, we get

( ln 1 +

)

)
0; 0 < h < 2

) h − x = 1, lim x→∞ fh (x) ( ) h h lim −x = , 2 h x→0 fh (x) eγ+ h +ψ( 2 ) ( ) d h lim = 1, x→∞ dx fh (x) (

and

d lim x→0 dx

where

(

h fh (x)

(34) (35) (36)

) = δ(h),

(37)

( )) 2 h ( eγ+ h +ψ( 2 ) 24 + h2 π 2 − 6h2 ψ ′ h2 . δ(h) = ( )2 2 h 12h eγ+ h +ψ( 2 ) − 1

Proof. Using the expansion fh (x) =

h h − 2 + O(x−3 ), x x

(38)

we get ( lim

x→∞

(

) h lim h −x h −3 ) x→∞ − + O(x 2 x x ) ( h −2 − O(x ) = 1. = lim h x h −3 ) x→∞ − + O(x 2 x x

) h −x = fh (x)

The expansion 2h 2 Gh (x) − =γ+ +ψ x(x + h) h

( ) ( ( )) h 2 π2 1 ′ h + − 2− + ψ x + O(x2 ) 2 h 12 2 2

gives us that ( lim

x→0

h −x fh (x)

(

) = lim

x→0

( 2 γ+ h +ψ ( h + − 2)

e h = . 2 h eγ+ h +ψ( 2 ) − 1

h 2 2 − π12 + 12 ψ ′ h2

) )

( h2 )

x+O(x2 )

−1

−x

Now using the expansion (38) and the expansion G′h (x) +

2h(h + 2x) h = − + O(x−3 ), x2 (x + h)2 x2 12

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we have d lim x→∞ dx

(

h fh (x)

(

) =

lim

[h

[

−h −

+ O(x−3 ) ( (1)) h +O x ( ) = lim x→∞ h2 − O x1 = 1. x→∞

x 2

h x2

]2

][ ]) h h h −3 −3 1 + − 2 + O(x ) − 2 + O(x ) x x x

Also, using the expansions 2h

2

h

eGh (x)− x(x+h) = eγ+ h +ψ( 2 ) + O(x) and G′h (x)

2h(h + 2x) −2 π 2 1 + 2 = 2 − + + ψ′ 2 x (x + h) h 12 2

we obtain d x→0 dx lim

(

h fh (x)

)

( ) h + O(x), 2



] [ 2 −h  +ψ ( h γ+ h 2 ) + O(x) = lim  [ e ] 2 h 2 x→0 eγ+ h +ψ( 2 ) − 1 + O(x) [ ( ) ]) −2 π 2 1 h ′ − + +ψ + O(x) h2 12 2 2 = δ(h).

Lemma 5.2. ( )) 2 h ( eγ+ h +ψ( 2 ) 24 + h2 π 2 − 6h2 ψ ′ h2 δ(h) = < 1, )2 ( h 2 12h eγ+ h +ψ( 2 ) − 1

0 < h < 2.

Proof. Consider the two functions 2 P1 (h) = γ + + ψ h and

( ) h 2

( ) h P2 (h) = 24 + h π − 6h ψ . 2 2 2

2

′

Using the integral representation (15), we have ( ) ∫ ∞ −2 1 ′ h 2t ′ P1 (h) = 2 + ψ = e−xt dt > 0 2t h 2 2 e −1 0 and then the function P1 (h) is increasing and positive since lim P1 (h) = 0,

h→0

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( ) h 2 π2 ψ = −γ − + h + O(h2 ). 2 h 12

where

Also,

P2′ (h) = hP3 (h), ( ) ( ) h h ′′ P3 (h) = 2π − 12ψ − 3hψ . 2 2

where

′

2

Using the integral representation (15), we obtain ∫ ∞ 24e2t (e2t − 1 + t) −xt 2 e dt P3 (h) = 2π − (e2t − 1)2 0 ∫

and then P3′ (h)

∞

= 0

24e2t (e2t − 1 + t) −xt te dt > 0. (e2t − 1)2

Hence the function P3 (h) is strictly increasing and using the expansions ( ) h π2 4 ′ ψ + O(h) = 2+ 2 h 6 ( ) h −16 ψ = 3 + ψ ′ (1) + O(h), 2 h

and

′′

we get lim P3 (h) = 0.

h→0

Then P3 (h) > 0 and hence the function P2 (h) is strictly increasing and positive since ( ( )) 4 π2 2 2 2 lim P2 (h) = lim 24 + h π − 6h + + O(h) = 0. h→0 h→0 h2 6 Now consider the two functions P4 (h) = 24(e − 1)2 eP1 (h) P2 (h) and P5 (h) = 288eh(eP1 (h) − 1)2 . Using the properties of the two functions P1 (h) and P2 (h), we conclude that P4 (h) and P5 (h) are strictly increasing positive functions and P4 (0) = P5 (0),

where ψ

(1) 2

P4 (2) = P5 (2) = 576e(e − 1)2 ,

P4 (1) = 12(e − 1)2 (π 2 − 12)e2 < P5 (1) = 18e(e2 − 4)2 , = −γ − ln 4 (see [3]). Then we get P4 (h) < P5 (h),

00

2e−ht (eht − 1)t −xt e dt < 0 e2t − 1

then fh (x) is strictly decreasing positive function. Hence fhh(x) is strictly increasing positive function. Also, fh (x) is strictly convex function since [( ] )2 ∫ ∞ −ht ht 2 2h 2h(h + 2x) 2e (e − 1)t fh′′ (x) = eGh (x)− x(x+h) G′h (x) + 2 + e−xt dt > 0. 2 2t x (x + h) e −1 0 From the relations (36) and (37) with the inequality (40), we conclude that the function ( ) d h is increasing function. Thus we get convex and dx fh (x) ( ) ( ) d h d h < lim = 1, x > 0. x→∞ dx dx fh (x) fh (x)

h fh (x)

is

Then Mh (x) is strictly decreasing function for all x > 0, where d d h Mh (x) = − 1 < 0. dx dx fh (x) Hence lim Mh (x) < Mh (x) < lim+ Mh (x) and using the limits (34) and (35), we have x→∞

x→0

1 < Mh (x)
0; 0 < h < 2 ln 1 + + h x+α x or ( ) ∞ ∑ h 1 1 1 ln 1 + < + 2h x+β x(x + h) k=0 (x + 2k)(x + h + 2k) ( ) h 1 1 ln 1 + , x > 0; 0 < h < 2 < + 2h x+α x(x + h) with sharp bounds α = 1 and β =

h

( ) −1

γ+ 2 +ψ h 2 e h

.

References [1] M. Abramowitz, I. A. Stegun (Eds.), Handbook of Mathematical Functions with Formulas, Graphs and Mathematical Tables, Dover, New York, 1965. [2] H. Alzer, On some inequalities for the gamma and psi function, Math. Comput., 66, 217, 373-389, 1997. [3] G. E. Andrews, R.Askey and R.Roy, Special Functions, Cambridge Univ. Press, 1999. [4] T. Buri´c , N. Elezovi´c and L. Vukˇsi´c, Appell polynomials and asymptotic expansions, Mediterranean J. Math., Vol. 13, Issue 3, 899-912, 2016. [5] E. T. Copson, Asymptotic expansions, Cambridge University Press, 1965. [6] N. Elezovi´c, Asymptotic expansions of gamma and related functions, binomial coefficients, inequalities and means, Journal of Mathematical Inequalities, Vol. 9, N. 4, 1001-1054, 2015. [7] A. Erd´ elyi et al., Higher Transcendental Functions Vol. I-III, California Institute of Technology - Bateman Manuscript Project, 1953-1955 McGraw-Hill Inc., reprinted by Krieger Inc. 1981. [8] B.-N. Guo and F. Qi, A class of completely monotonic functions involving divided differences of the psi and tri-gamma functions and some applications, J. Korean Math. Soc. 48, 655-667, 2011. [9] B.-N. Guo and F. Qi, Sharp inequalities for the psi function and harmonic numbers, Analysis–International mathematical journal of analysis and its applications 34 , no. 2, 201-208, 2014. [10] G. H. Hardy, J. E. Littlewood and G. P´olya, Inequalities, Cambridge Univ. Press, 1959. 16

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[11] V. Kac and Pokman Cheung, Quantum Calculus, Springer-Verlag, 2002. [12] M. Mahmoud and R. P. Agarwal, Bounds for Bateman’s G−function and its applications, Georgian Mathematical Journal, Vol. 23 (4), 579-586, 2016. [13] M. Mahmoud and H. Almuashi, On some inequalities of the Bateman’s G−function, J. Comput. Anal. Appl., Vol. 22, No.4, , 672-683, 2017. [14] M. Mahmoud, A. Talat and H. Moustafa, Some approximations of the Bateman’s G−function, J. Comput. Anal. Appl., Vol. 23, No.6, 1165-1178, 2017. [15] C. Mortici, A sharp inequality involving the psi function, Acta Universitatis Apulensis, 41-45, 2010. [16] C. Mortici, Estimating gamma function in terms of digamma function, Math. Comput. Model., 52, no. 5-6, 942-946, 2010. [17] C. Mortici, New approximation formulas for evaluating the ratio of gamma functions. Math. Comp. Modell. 52(1-2), 425-433, 2010. [18] C. Mortici, Accurate estimates of the gamma function involving the psi function, Numer. Funct. Anal. Optim., 32, no. 4, 469-476, 2011. [19] K. Oldham, J. Myland and J. Spanier, An Atlas of Functions, 2nd edition. Springer, 2008. [20] F. W. J. Olver (ed.), D. W. Lozier (ed.), R. F. Boisvert (ed.) and C. W. Clark (ed.), NIST Handbook of Mathematical Functions, Cambridge University Press, Cambridge, 2010. [21] A. P. Prudnikov, Yu. A. Brychkov and O. I. Marichev, Integrals and Series: More Special Functions, Vol. 3, Gordon and Breach Science Publishers, New York, 1990. [22] I. Pinelis, L’Hospital rules for monotonicity and the Wilker-Anglesio inequality, Amer. Math. Monthly 111(10), 905-909, 2004. [23] F. Qi, R.-Q. Cui, C.-P. Chen and B.-N. Guo, Some completely monotonic functions involving polygamma functions and an application, Journal of Mathematical Analysis and Applications 310, no. 1, 303-308, 2005. [24] F. Qi and B.-N. Guo, Completely monotonic functions involving divided differences of the diand tri-gamma functions and some applications, Commun. Pure Appl. Anal. 8, 1975-1989, 2009. [25] F. Qi and B.-N. Guo, Necessary and sufficient conditions for functions involving the tri- and tetra-gamma functions to be completely monotonic, Adv. in Appl. Math. 44, 71-83, 2010. [26] F. Qi and Q.-M. Luo, Bounds for the ratio of two gamma functions: from Wendel’s asymptotic relation to Elezovi´c-Giordano-Peˇcari´c’s theorem, Journal of Inequalities and Applications 2013, 2013:542, 20 pages. [27] F. Qi , Bounds for the ratio of two gamma functions, J. Inequal. Appl. 2010 (2010), Article ID 493058, 84 pages. 17

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[28] S.-L. Qiu and M. Vuorinen, Some properties of the gamma and psi functions with applications, Math. Comp., 74, no. 250, 723-742, 2004. [29] J. S´andor, Sharp Cusa-Huygens and related inequalities, Notes on Number Theory and Discrete Mathematics Vol. 19, No. 1, 50-54, 2013. [30] M. A. Shpota and H. M. Srivastava, The Clausenian hypergeometric function 3 F2 with unit argument and negative integral parameter differences, Applied Mathematics and Computation Vol. 259, 819-827, 2015.

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Interval-valued fuzzy quasi-metric spaces Hanchuan Lua ,b ,?

∗

Shenggang Lic

a School of Mathematical Sciences, Nanjing Normal University,210023 Nanjing , P. R. China b Guizhou Provincial Key Laboratory of Public Big Data, Guizhou University,550025, Guiyang, P. R. China c College of Mathematics and Information Science, Shaanxi Normal University, 710062, Xi’an, P. R. China

Abstract — Under the context of quasi-metric, promoted the concept of intervalvalued fuzzy metric space, the main results are as follows :(1) topology induced by quasi-metric is consistent with which induced via a standard interval-valued fuzzy quasimetric; (2) proved that every quasi-metrizable topological space admits a compatible interval-valued fuzzy quasi-metric. On the contrary, topology generated by intervalvalued fuzzy quasi-metric is quasi-metrizable; (3) discussed some properties of intervalvalued fuzzy quasi-metric space which is bicompletion. proved that if an interval-valued fuzzy quasi-metric space has bicompletion, then it is unique up to isometry. In addition, we define a fuzzy contraction mapping of interval-valued fuzzy metric space, promote the Banach and Edelstein fixed point theorem to interval fuzzy metric space. Keywords —Interval-valued fuzzy quasi-metric spaces; Quasi-metric; Quasi-uniformity; Bicompletion; Isometry AMS classification 54A40

1

Introduction In 1975, Kramosil and Michalek introduced the concept of fuzzy metric spaces which

is closely related with probabilistic metric spaces in [1], it is also known as generalized Menger space. Currently, many mathematical workers use the concept of fuzzy sets giving different fuzzy metric space ideas [1-5]. Among them, A.George and P.Veeramani improved the concept of fuzzy metric space defined by Kramosil and Michalek in [2, 6], proposed the concept of continuous t-norm, and they used fuzzy sets to represent the uncertainty of the distance between two points in a fuzzy metric space, thus obtained a stronger form of fuzzy metric which has a Hausdorff topology. It is clearly that ∗

This work was supported by the International Science and Technology Cooperation Foundation of China (Grant No. 2012DFA11270), and the National Natural Science Foundation of China (Grant No. 11071151). ? Corresponding author. E-mail: [email protected] (H.-C. Lu) 41

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the concept of fuzzy metric space is different with the paper [1, 3, 4, 5] definates. In particular, every metric induces a fuzzy metric space which is under A.George and P.Veeramani significance. Instead, each fuzzy metric space in A.George and P.Veeramani significance generate a metrizable topology [7, 8]. As early as in 1975, Zadah has given the concept of interval-valued fuzzy sets [9]. It characterizes fuzzy set by interval-valued membership functions, which Is another generalize of fuzzy set. In 2012, based on the concept of A.George and P.Veeramani’s fuzzy metric space, Shen [10] given a conception of continuous interval value t- norm and interval-valued fuzzy metric space, and also discussed some topological properties of such metric spaces. In addition, it is known that quasi-metric space constitutes a very effective tool in discuss and resolve topology algebraic, approximation theory, theory computational science and other aspects [11,18]. In the context of quasi-metric, this article generalized interval-valued fuzzy metric that the paper [10] defined, proposed the concept of interval-valued fuzzy quasi-metric, as a basis for the study. Structure of the article is divided into five parts. The first part is to introduce the background knowledge; the second part is prior knowledge; the third section discuss the quasi-metrizable of interval-valued fuzzy metric space; part IV discuss the bicompletion of interval-valued fuzzy metric space; section V discuss the fixed point theorem of interval-valued fuzzy metric space.

2

Preliminaries Interval analysis (see http://www.cs.utep.edu/interval-comp/main.html) leaded by

interval numbers is an area of active research in mathematics, numerical analysis and computer science began in the late 1950s. An interval number is a point ha− , a+ i in the 2-dimensional Euclidean space R2 which satisfies a− ≤ a+ . The set of all interval numbers is denoted by I(R) 1 . For any ha− , a+ i, hb− , b+ i ∈ I(R) and each nonnegative real number r, define ha− , a+ i ⊕ hb− , b+ i = ha− + b− , a+ + b+ i, ha− , a+ i hb− , b+ i = hmin{a− − b− , a+ − b+ }, max{a− − b− , a+ − b+ }i, and rha− , a+ i = hra− , ra+ i. We write ha− , a+ i = a when a− = a+ = a, and I(I) = {ha− , a+ i ∈ I(R) | a− , a+ ∈ I} (where I = [0, 1] is the ordinary closed unit interval). In this paper we only involve the notion of interval-valued fuzzy set used generally to dispose uncertainness (for an overview of 1 Unless confusion arise, we identify a real number a ∈ R with an interval number a , and identify a closed interval [a, b] of R and a point ha, bi in R2 since there exists a nature one-to-one correspondence between the set of all closed intervals of R and I(R); we will also use a, b, c, · · · to denote interval numbers.

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interval analysis, its relationship to fuzzy set theory, and possible areas of further fruitful research, please see [10,12,13,14,15,17,19]). Definition 2.1 An interval-valued fuzzy set (resp., fuzzy set) on a set X is exactly a mapping A : X −→ I(I) (resp., A : X −→ I). The set of all interval-valued fuzzy sets on X is denoted by IVF(X). Each A ∈ IVF(X) induces two fuzzy sets A− : X −→ I and A+ : X −→ I whose values are determined by A(x) = hA− (x), A+ (x)i (∀x ∈ X). For A, B ∈ IVF(X), the infimum A ∧ B ∈ IVF(X) of A and B in IVF(X) is given by (A ∧ B)(x) = hmin{A− (x), B − (x)}, min{A+ (x), B + (x)}i (∀x ∈ X), the suprmum A ∨ B ∈ IVF(X) of A and B in IVF(X) is given by (A ∨ B)(x) = hmax{A− (x), B − (x)}, max{A+ (x), B + (x)}i (∀x ∈ X), and the complement A0 ∈ IVF(X) of A in IVF(X) is given by A0 (x) = h1 − A+ (x), 1 − A− (x)i (∀x ∈ X). Definition 2.2[10] An interval-valued t-norm is a binary operation ∗ : I(I)×I(I) −→ I(I) which satisfies the following conditions: (i) a ∗ b = b ∗ a (∀a, b ∈ I(I)). (ii) a ∗ (b ∗ c) = (a ∗ b) ∗ c (∀a, b, c ∈ I(I)). (iii) a ∗ b ≤ a ∗ c whenever b ≤ c. (iv) a ∗ 1 = a (∀a ∈ I(I)); ha− , a+ i ∗ h0, 1i = h0, a+ i (∀ha− , a+ i ∈ I(I)). Proposition 2.3[10]

An interval-valued t-norm ∗ has the following properties:

(i) 0 ∗ a = a ∗ 0 = 0 ; h0, 1i ∗ ha− , a+ i = h0, a+ i; 1 ∗ a = a. (ii) a ∗ b ≤ c ∗ d whenever a ≤ c and b ≤ d. (iii) 0 = a∗00 ≤ a∗b ≤ a∗11 = a, 0 = a∗00 ≤ 0 ∗b ≤ 1 ∗b = b, 0 = a∗00 ≤ a∗b ≤ a∧b. Definition 2.4[10]

+ (1) Let a = ha− , a+ i and an = ha− n , an i are in I(I) for each

n ∈ N (the natural number set). {an }n∈N (briefly, {an }) is said to be convergent to a − + + (write as lim an = a) if lim a− n = a and lim an = a . n→∞

n→∞

n→∞

(2) An interval-valued t-norm ∗ is said to be continuous in its first component if lim (an ∗ b) = ( lim an ) ∗ b = a ∗ b for all b ∈ I(I) whenever lim an = a ({an }n∈N ⊆

n→∞

I(I), a ∈ I(I)).

n→∞

n→∞

Proposition 2.5[10] The followings hold for a continuous interval-valued t-norm ∗: 43

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(i) For any a1 , a2 ∈ I(I) with a1 > a2 , there exists a a3 ∈ I(I) such that a1 ∗ a3 ≥ a2 . (ii) For any a4 ∈ I(I), there exists a a5 ∈ I(I) such that a5 ∗ a5 ≥ a4 . Definition 2.6[10]

An interval-valued fuzzy metric space is a triple (X, M, ∗),

where ∗ is a continuous t-norm and M (called an interval-valued fuzzy metric on X) is an interval-valued fuzzy set on X 2 ×(0, ∞) having the following properties (x, y, z ∈ X): (IV1) M (x, y, t) > 0 . (IV2) M (x, y, t) = 1 if and only if x = y. (IV3) M (x, y, t) = M (y, x, t). (IV4) M (x, y, t) ∗ M (y, z, s) ≤ M (x, z, t + s) for all t, s > 0. (IV5) M (x, y, ·) : (0, ∞) −→ I(I) − {00} is continuous, i.e. both M − (x, y, ·) = p1 ◦ M (x, y, ·) : (0, ∞) −→ (0, 1] and M + (x, y, ·) = p2 ◦ M (x, y, ·) : (0, ∞) −→ (0, 1] are continuous, where pi : I 2 −→ I is the projective mapping (i = 1, 2), and M − (x, y, t) and M + (x, y, t) denote the lower nearness and upper nearness degree between x and y with respect to t, respectively. (IV6) lim M (x, y, t) = 1 . t→∞

Remark 2.7[10] (1) An interval-valued fuzzy metric space (X, M, ∗) will degenerate into an ordinary fuzzy metric space if M − (x, y, t) = M + (x, y, t) for all t > 0. (2) Let (X, M, ∗) be an interval-valued fuzzy metric space. (i) The set B(x, r, t) = {y ∈ X | M (x, y, t) > 1 r} is called a open ball with center x ∈ X, where t > 0 and r ∈ I(I) − {00, 1 }. (ii) JM = {A ⊆ X | ∀x ∈ A, ∃tx > 0, ∃rx ∈ I(I) − {00, 1 }, B(x, rx , tx ) ⊆ A} is a topology on X, and {B(x, r, t) | x ∈ X, r ∈ I(I) − {00, 1}, t > 0} is a base of JM . (iii) For each (x, y, t, r) ∈ X 2 × (0, +∞) × (I(I) − {00, 1 }), there exists a t0 ∈ (0, t) such that M (x, y, t0 ) > 1 r whenever M (x, y, t) > 1 r. (iv) For a sequence {xn } ⊆ X, lim xn = x in (X, JM ) if and only if lim M (x, xn , t) = n→∞

n→∞

1. (v) A sequence {xn } in (X, M, ∗) is called a Cauchy sequence if for all ε > 0 and t > 0, there exists an n0 ∈ N such that M (xn , xm , t) > 1 ε for all n, m ≥ n0 . (vi) (X, M, ∗) is said to be complete if every Cauchy sequence in it is convergent. Remark 2.8[16] (1) A quasi-pseudo-metric space is a pair (X, d), where X is a set and d (called a quasi-pseudo-metric on X) is a mapping d : X −→ [0, +∞) satisfying the following conditions: (i) d(x, x) = 0 (∀x ∈ X); 44

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(ii) d(x, z) ≤ d(x, y) + d(y, z) (∀x, y, z ∈ X). (2) A quasi-metric space is a pair (X, d), where X is a set and d (called a quasi-metric on X) is a quasi-pseudo-metric on X satisfying the following condition: (iii) x = y ⇐⇒ d(x, y) = d(y, x) = 0. Each quasi-metric d on X induces a T0 topology Jd on X which has as a base the family of open balls {Bd (x, r) | x ∈ X, r > 0}, where Bd (x, r) = {y ∈ X | d(x, y) < r} (x ∈ X, r > 0). (3) A topological space (X, J ) is said to be quasi-metrizable if there is a quasi-metric d on X such that Jd = J (in this case, we say that d is compatible with J , and that J is a quasi-metrizable topology on X), where Jd = hh{Bd (x, r) | x ∈ X, r > 0}ii (i.e. Jd is the smallest topology on X containing {Bd (x, r) | x ∈ X, r > 0}). (4) If d is a quasi-pseudo-metric on X, then d−1 (called the conjugate of d which is defined by d−1 (x, y) = d(y, x) (∀x, y ∈ X)) and ds (defined by ds (x, y) = max{d(x, y), d−1 (x, y)} (∀x, y ∈ X)) are also quasi-pseudo-metrics on X. A quasi-metric space (X, d) is said to be bicomplete if (X, ds ) is a complete metric space. In this case we say that d is a bicomplete quasi-metric on X.

3

Quasi-metrizable of interval-valued fuzzy quasimetric space Definition 3.1

Let X be a set, ∗ a continuous t-norm, and M an interval-valued

fuzzy set on X 2 × (0, ∞) having the following properties: (IV1) M (x, y, t) > 0 (x, y ∈ X, t ∈ (0, ∞)). (IV2) M (x, x, t) = 1 (x ∈ X, t ∈ (0, ∞)). (IV3) M (x, y, t) ∗ M (y, z, s) ≤ M (x, z, t + s) (x, y, z ∈ X, s, t ∈ (0, ∞)). (IV4) M (x, y, ·) : (0, ∞) −→ I(I) − {00} is continuous (x, y ∈ X). (IV5) lim M (x, y, t) = 1 (x, y ∈ X). t→∞

(1) Such a triple (X, M, ∗) is called an interval-valued fuzzy quasi-pseudo-metric space, and (M, ∗) is called an interval-valued fuzzy quasi-pseudo-metric on X. (2) This triple (X, M, ∗) is called an interval-valued fuzzy quasi-metric space (and (M, ∗) is called an interval-valued fuzzy quasi-metric on X) if M satisfies the following condition: (IV10 ) x = y if and only if M (x, y, t) = M (y, x, t) = 1 for all t > 0. 45

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(3) This triple (X, M, ∗) is called a T1 interval-valued fuzzy quasi-metric space (and (M, ∗) is called a T1 interval-valued fuzzy quasi-metric on X) if M satisfies the following condition: (IV20 ) x = y if and only if M (x, y, t) = 1 for all t > 0, (4) This triple (X, M, ∗) is called an interval-valued fuzzy pseudo-metric space (and (M, ∗) is called an interval-valued fuzzy pseudo-metric on X) if M satisfies the following condition: (IV6) M (x, y, t) = M (y, x, t)for all t > 0. Definition 3.1’ An interval-valued fuzzy (pseudo-) metric on X is a interval-valued fuzzy quasi-(pseudo-)metric (M, ∗) on X such that for each x, y ∈ X: (IV6) M (x, y, t) = M (y, x, t)for all t > 0. Remark 3.2

(1) There are many interval-valued fuzzy quasi-metric spaces which

are not interval-valued fuzzy metric space. (2) There are many interval-valued fuzzy pseudo-metric spaces which are not intervalvalued fuzzy metric space. Interval-valued fuzzy quasi-pseudo metric space is a weakened form of the interval value fuzzy metric Spaces, which is a generalization of the interval-valued fuzzy metric Spaces, will have a greater scope in the application. Remark 3.4 It is clear that every interval-valued fuzzy metric is a T1 interval-valued fuzzy quasi-metric; every T1 interval-valued fuzzy quasi-metric is an interval-valued fuzzy quasi-metric, and every interval-valued fuzzy quasi-metric is an interval-valued fuzzy quasi-pseudo-metric. Definition 3.5

An interval-valued fuzzy quasi-(pseudo-)metric space is a triple

(X, M, ∗) such that X is a (nonempty) set and (M, ∗) is an interval-valued fuzzy quasi(pseudo-)metric on X. The notions of a T1 interval-valued fuzzy quasi-metric space and of a interval-valued fuzzy metric space are defined in the obvious manner. Remark 3.6

If (M, ∗) is an interval-valued fuzzy quasi-(pseudo-)metric on a set

X, it is immediate to show that (M −1 , ∗) is also an interval-valued fuzzy quasi-(pseudo)metric on X, where M −1 is the interval-valued fuzzy set in X × X × (0, +∞) defined by M −1 (x, y, t) = M (y, x, t). Moreover, if we denote M i the interval-valued fuzzy set in X × X × (0, +∞) given by M i (x, y, t) = min{M (x, y, t), M −1 (x, y, t)}, then (M i , ∗) is, clearly, an interval-valued fuzzy (pseudo-)metric on X. 46

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Thus, conditions (IV1’) of the definition 3.2 above is equivalent to the following: M (x, x, t) = 1 for all x ∈ X and t > 0,and M i (x, y, t) < 1 for all x 6= y and t > 0. Definition 3.7

Let (X, M, ∗) be an interval-valued fuzzy quasi-pseudo-metric

space. We define open ball BM (x, r, t) = {y ∈ X|M (x, y, t) > 1 − r} for t > 0 with center x ∈ X and the interval number r, 0 < r < 1 , t > 0. Similar to the [10], we get the following conclusion: Proposition 3.8

Let (X, M, ∗) be an interval-valued fuzzy quasi-pseudo-metric

space. Then every open ball BM (x, r, t) is an open set. Proof. Let BM (x, r, t) is an open ball. For any y ∈ BM (x, r, t), we know that M (x, y, t) > 1 − r. Therefore, there exists a t0 ∈ (0, t) such that M (x, y, t0 ) > 1 − r. Set

r0 = M (x, y, t0 ). Since r0 > 1 − r, there exists a s ∈ I(I) such that r0 > 1 − s > 1 − r. Now for given r0 and s with r0 > 1 − s, there exists a r1 ∈ I(I)such that r0 ∗I r1 ≥ 1 − s. Consider the open ball BM (y, 1 − r1 , t − t0 ), we will obtain that BM (y, 1 − r1 , t − t0 ) ⊂ BM (x, r, t). In fact, for every z ∈ BM (y, 1 − r1 , t − t0 ), we have M (y, z, t − t0 ) > r1 . Therefore, M (x, z, t) ≥ M (x, y, t0 ) ∗I M (y, z, t − t0 ) ≥ r0 ∗I r1 ≥ 1 − s > 1 − r Thus x ∈ BM (x, r, t), and hence BM (y, 1 − r1 , t − t0 ) ⊂ BM (x, r, t). Similar to the [10], we get the following conclusion: Theorem 3.9 Let (X, M, ∗) be an interval-valued fuzzy quasi-pseudo-metric space. Define τM = {A ⊂ X | ∀x ∈ A, ∃r ∈ I(I) − {00, 1 }, and t > 0 such that BM (x, r, t) ⊂ A} Then τM is a topology on X. Proposition 3.10

A sequence {xn }n in an interval-valued fuzzy quasi-pseudo-

metric space (X, M, ∗) is said to be a Cauchy sequence if and only if limn M (xn+p , xn , t) = 1 , for all p > 0, t > 0. A sequence {xn }n in an interval-valued fuzzy quasi-pseudo-metric space (X, M, ∗) is converging to x in X,denoted by xn −→ x, if and only if limn M (x, xn , t) = 1 for all t > 0. A interval-valued fuzzy quasi-pseudo-metric space (X, M, ∗) is said to be complete if and only if every Cauchy sequence is convergent. Proposition 3.11

Let (X, M, ∗) be an interval-valued fuzzy quasi-pseudo-metric

space. Then, for each x, y ∈ X the function M (x, y, −) is nondecreasing. 47

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Proof. Let x, y ∈ X and 0 ≤ t < s. Then M (x, y, s) ≥ M (x, x, s − t) ∗ M (x, y, t) = 1 ∗ M (x, y, t) = M (x, y, t). Proposition 3.12

Let (X, M, ∗) be an interval-valued fuzzy quasi-metric space.

Then, for each t > 0 the function M (−, −, t) : (X × X, τM i × τM i ) −→ I(I) − {00} is continuous. Proof. Fit t > 0. Let x, y ∈ X and let (x0n , yn0 )n be a sequence in X × X that converges to (x, y) with respect to τM i × τM i . Then, it will be sufficient to show that M (x, y, t) = limn M (xn , yn , t) for some subsequence (xn , yn )n of (x0n , yn0 )n . Indeed, since (M (x0n , yn0 , t))n is sequence in I(I)−{00}, there is a subsequence (xn , yn )n of (x0n , yn0 )n such that the sequence (M (xn , yn , t))n converges to some ε of I(I). Fix δ > 0 such that 2δ < t. Then M (xn , yn , t) ≥ M (xn , x, δ) ∗ M (x, y, t − 2δ) ∗ M (y, yn , δ), and M (x, y, t + 2δ) ≥ M (x, xn , δ) ∗ M (xn , yn , t) ∗ M (yn , y, δ). Since limn M i (x, xn , δ) = limn M i (y, yn , δ) = 1 , we deduce that M (x, y, t + 2δ) ≥ lim M (xn , yn , t) ≥ M (x, y, t − 2δ). n

Finally, it follows from the continuity of M (x, y, −) that M (x, y, t) = limn M (xn , yn , t). This completes the proof. Definition 3.13

Let (X, d) be a quasi-metric space. Define the interval-valued

t-norm

a ∗ b = ha− ∧ b− , a+ ∧ b+ i and interval-valued fuzzy quasi-metric Md (x, y, t) = hM − (x, y, t), M + (x, y, t)i = h

t t , i t + ld(x, y) t + md(x, y) (∀x, y ∈ X, ∀t, l, m ∈ R+ ).

Then (X, Md , ∗) is an interval-valued fuzzy quasi-metric space called the standard intervalvalued fuzzy quasi-metric space and (Md , ∗) is the interval-valued fuzzy quasi-metric induced by d. 48

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Furthermore, it is easy to check that (Md )−1 = Md−1 and (Md )i = Mds , where d−1 (x, y) = d(y, x), ds (x, y) = max{d(x, y), d−1 (x, y)}. In the paper [10], it is proved that every metric can induce an interval-valued fuzzy metric. Moreover, if (X, d) is a metric space ,then the topology generated by d coincides with the topology τMd generated by the interval-valued fuzzy metric. Finally, from proposition 3.8, 3.10, 3.11 and definition 3.13, it follows that the topology τd , generated by d, coincides with the topology τMd generated by the induced standard interval-valued fuzzy quasi-metric (Md , ∗). Definition 3.14

We say that a topological space (X, τ ) admits a compatible

interval-valued fuzzy quasi-metric if there is an interval-valued fuzzy quasi-metric (M, ∗) on X such that τ = τM . It follows from definition 3.11 that every quasi-metrizable topological space admits a compatible interval-valued fuzzy quasi-metric. Then, conversely, the topology generated by an interval-valued fuzzy quasi-metric space is quasi-metrizable. Lemma 3.15 Let (X, Md , ∗) be an interval-valued fuzzy quasi-metric space. Then {Un : n = 2, 3, · · ·} is a base for a quasi-uniformity UM on X compatible with τM , where Un = {(x, y) ∈ X × X : M (x, y, 1/n) > h1 − 1/n, 1i}, for n = 2, 3, · · ·. Moreover the conjugate quasi-uniformity (uM )−1 coincides with uM −1 and it is compatible with τM −1 . Form definition 3.11, Lemma 3.13 and the well-known result that the topology generated by a quasi-uniformity with a countable base is quasi-pseudometrizable, we immediately deduce the following. Theorem 3.16 For a topological space (X, τ ) the following are equivalent . (i) (X, τ ) is quasi-metrizable. (ii) (X, τ ) admits a compatible interval-valued fuzzy quasi-metric.

4

Bicomplete interval-valued fuzzy quasi-metric space Definition 4.1

An interval-valued fuzzy quasi-metric space (X, M, ∗) is called

bicomplete if (X, M i , ∗) is a complete interval-valued fuzzy metric space. In this case, we say that (M, ∗) is a bicomplete interval-valued fuzzy quasi-metric on X. Proposition 4.2 (1) Let (X, M, ∗) be a bicomplete interval-valued fuzzy quasi-metric space.Then (X, τM ) admits a compatible bicomplete quasi-metric space. 49

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(2)) Let (X, d) be a bicomplete quasi-metric space. Then (X, Md , ∗) is a bicomplete interval-valued fuzzy quasi-metric space. Proof. (1) Let d be a quasi-metric on X inducing the quasi-uniformity UM . Then d is compatible with τM . Now let (xn )n be a Cauchy sequence in (X, ds ). Clearly (xn )n is a Cauchy sequence in the interval-valued fuzzy metric space (X, M i , ∗). So it converges to a point y ∈ X with respect to τM i . Hence (xn )n converges to y with respect to τds . Consequently d is bicomplete. (2) This part is almost obvious because (Md )i = Mds (see definition 3.11) and thus each Cauchy sequence in (X, (Md )i , ∗) is clearly a Cauchy sequence in (X, ds ). Definition 4.3

Let (X, M, ∗) and (Y, N, ?I ) be two interval-valued fuzzy quasi-

metric space.Then (i) A mapping f from X to Y is called an isometry if for each x, y ∈ X and each t > 0, M (x, y, t) = N (f (x), f (y), t). (ii) (X, M, ∗) and (Y, N, ?I ) are called isometric if there is an isometry from X onto Y. Definition 4.4

Let (X, M, ∗) be an interval-valued fuzzy quasi-metric space. An

interval-valued fuzzy quasi-metric bicompletion of (X, M, ∗) is a bicomplete intervalvalued fuzzy quasi-metric space (Y, N, ?I ) such that (X, M, ∗) is isometric to a τN i −dense subspace of Y . Lemma 4.5

Let (X, M, ∗) be an interval-valued fuzzy quasi-metric space and

(Y, N, ?I ) a bicomplete interval-valued fuzzy quasi-metric space.If there is a τM i −dense subset A of X and an isometry f : (A, M, ∗) −→ (Y, N, ?I ), then there exists a unique isometry F : (X, M, ∗) −→ (Y, N, ?I ) such that F |A = f . Proof. It is clear that f is a quasi-uniformly continuous mapping from the quasiuniform space (A, UM |A × A) to the quasi-uniform space (Y, UN ). By Theorem 3.29 of [16], f has a unique quasi-uniformly continuous extension F : (X, UM ) −→ (Y, UN ). We shall show that actually F is an isometry from (X, M, ∗) to (Y, N, ?I ). Indeed, let x, y ∈ X and t > 0. Then, there exist two sequences (xn )n and (yn )n in A such that (xn )n −→ x and (yn )n −→ y with respect to τM i . Thus F (xn ) −→ F (x) and F (yn ) −→ F (y) with respect to τN i .Choose ε ∈ I(I) − {00, 1 } with 0 < ε < t. Therefore, there is nε such that for n > nε , M (x, xn , ε/2) > 1 − ε, M (yn , y, ε/2) > 1 − ε, N (F (xn ), F (x), ε/2) > 1 − ε, N (F (y), F (yn ), ε/2) > 1 − ε. 50

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Thus M (x, y, t) ≥ M (x, xn , ε/2) ∗ M (xn , yn , t − ε) ∗ M (yn , y, ε/2) ≥ (11 − ε) ∗ N (F (xn ), F (yn ), t − ε) ∗ (11 − ε) ≥ (11 − ε) ∗ h(11 − ε) ?I N (F (x), F (y), t − 2ε) ?I (11 − ε)i ∗ (11 − ε) By continuity of ∗and ?I and by continuity of N (F (x), F (y), −), it follows that M (x, y, t) ≥ N (F (x), F (y), t). Similarly we can show that N (F (x), F (y), t) ≥ M (x, y, t). Consequently F is an isometry from (X, M, ∗) to (Y, N, ?I ). Theorem 4.6

Suppose that (Y1 , N1 , ?I1 ) and (Y2 , N2 , ?I2 ) are two interval-valued

fuzzy quasi-metric bicompletions of (X, M, ∗). Then (Y1 , N1 , ?I1 ) and (Y2 , N2 , ?I2 ) are isometric. Thus, if an interval-valued fuzzy quasi-metric space has an interval-valued fuzzy quasi-metric bicompletion, it is unique in the mean of isometry. Proof. Since (Y2 , N2 , ?I2 ) is an interval-valued fuzzy quasi-metric bicompletion of (X, M, ∗), there is an isometry f from (X, M, ∗) onto a dense subspace of (Y2 , N2 , ?I2 ). By Lemma 4.5, f admits a (unique) extension F to (Y1 , N1 , ?I1 ) which is also an isometry. So, it remains to see that F is onto. But this fact follows from standard arguments. Indeed, given y2 ∈ Y2 , there is a sequence (xn )n in X such that F (xn ) −→ y2 . Since F is an isometry, (xn )n is a Cauchy sequence, so it converges to some point y1 ∈ Y1 . Consequently F (y1 ) = y2 . The proof is complete.

5

Fixed point theorems in interval-valued fuzzy metric spaces We will discuss the interval-valued fuzzy metric space fixed point problem in this

section. Firstly, the definition of interval-valued fuzzy contraction mapping in intervalvalued fuzzy metric space, then a generalization of Banach and Edelstein fixed point theorems is given. In an interval-valued fuzzy metric space (X, M, ∗), whenever M (x, y, t) > 1 − r for all x, y ∈ X, t > 0, r ∈ I(I) − {00, 1 }, there exists a t0 with 0 < t0 < t such that M (x, y, t0 ) > 1 − r. Definition 5.1

Let (X, M, ∗) be an interval-valued fuzzy metric space. We will

say the mapping f : X → X is t-uniformly continuous if for each ε ∈ I(I), there exists a

r ∈ (I) such that M (x, y, t) > 1 − r implies M (f (x), f (y), t) ≥ 1 − ε, for each x, y ∈ X and t > 0. Clearly if f is t-uniformly continuous it is uniformly continuous for the uniformity generated by M , and then continuous for the topology deduced from M . 51

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Definition 5.2

Let (X, M, ∗) be an interval-valued fuzzy metric space. We will

say the mapping f : X → X is fuzzy contractive if there exists a k ∈ (0, 1) such that for each x, y ∈ X, and t > 0, M (x, y, t) ∗ (11 − M (f (x), f (y), t) ≤ k(11 − M (x, y, t)) ∗ M (f (x), f (y), t), where k is called the contractive constant of f . The above definition is justified by the next Proposition 5.4. Proposition 5.3

Let (X, M, ∗) be an interval-valued fuzzy metric space. If f :

X → X is fuzzy contractive then f is t-uniformly continuous. Proposition 5.4

Let (X, d) be a metric space. The mapping f : X → X is

contractive (a contraction) on the metric space (X, d) with contractive constant k if and only if f is fuzzy contractive, with contractive constant k, on the standard intervalvalued fuzzy metric space (X, Md , ∗), induced by d. Recall that a sequence (xn ) in a metric space (X, d) is said to be contractive if there exists a k ∈ (0, 1) such that d(xn+1 , xn+2 ) ≤ kd(xn , xn+1 ), for all n ∈ N. Now, we give the following definition (compare with Definition 5.2). Definition 5.5

Let (X, M, ∗) be an interval-valued fuzzy metric space. We will

say that the sequence (xn ) in X is fuzzy contractive if there exists k ∈ (0, 1) such that for all t > 0, n ∈ N, M (xn , xn+1 , t) ∗ (11 − M (xn+1 , xn+2 , t)) ≤ k(11 − M (xn , xn+1 , t)) ∗ M (xn+1 , xn+2 , t). Proposition 5.6 Let (X, Md , ∗) be the standard interval-valued fuzzy metric space induced by the metric d on X. The sequence (xn ) in X is contractive in (X, d) if and only if (xn ) is fuzzy contractive in (X, Md , ∗). Research of the fixed point theorem in fuzzy metric space has attracted the attention of many scholars [20,23-29], below we will discuss the fuzzy metric space fixed point theorems in interval-valued fuzzy metric space. Definition 5.7 A sequence {xn } in an interval-valued fuzzy metric space (X, M, ∗) is a Cauchy sequence if and only if for each ε > 0 and each t > 0 there exists a n0 ∈ N such that M (xn , xm , t) > 1 − ε for all n, m ≥ n0 . Definition 5.8

An interval-valued fuzzy metric space (X, M, ∗) in which every

Cauchy sequence is convergent is called a complete fuzzy metric space. Theorem 5.9 A sequence {xn } in an interval-valued fuzzy metric space (X, M, ∗) converges to x if and only if M (x, xn , t) → 1 as n → ∞. 52

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Result 5.10

The metric space (X, d) is complete if and only if the standard

interval-valued fuzzy metric space (X, Md , ∗) is complete. Proof may refer to A. George and P. Veeramani literature [5]. Next, we extend the Banach fixed point theorem to fuzzy contractive mappings of interval-valued fuzzy metric spaces. Theorem 5.11 (fuzzy Banach contraction theorem) Let (X, M, ∗) be an intervalvalued fuzzy metric space in which fuzzy contractive sequences are Cauchy. Let T : X → X be a fuzzy contractive mapping being k the contractive constant. Then T has a unique fixed point. Proof. Fix x ∈ X. Let xn = T n (x), n ∈ N. We have for t > 0, M (x, x1 , t) ∗ (11 − M (T (x), T 2 (x), t)) ≤ k(11 − M (x, x1 , t)) ∗ M (T (x), T 2 (x), t), and by induction, for any n ∈ N, M (xn , xn+1 , t) ∗ (11 − M (xn+1 , xn+2 , t)) ≤ k(11 − M (xn , xn+1 , t)) ∗ M (xn+1 , xn+2 , t), Then (xn ) is a fuzzy contractive sequence, so it is a Cauchy sequence and, hence, (xn ) converges to y, for some y ∈ X . We will see y is a fixed point for T . By Theorem 5.9, we have M (y, xn , t) ∗ (11 − M (T (y), T (xn ), t)) ≤ k(11 − M (y, xn , t)) ∗ M (T (y), T (xn ), t)) −→ 0 as n −→ ∞, then limn M (T (y), T (xn ), t)) = 1 for each t > 0, and, therefore, limn T (xn ) = T (y), i.e., limn xn+1 = T (y) and then T (y) = y. To show uniqueness, assume T (z) = z for some z ∈ X. Then for t > 0 we have = ≤ = ≤ ≤ ≤

M (y, z, t) ∗ (11 − M (y, z, t)) M (y, z, t) ∗ (11 − M (T (y), T (z), t)) k[M (T (y), T (z), t)) ∗ (11 − M (y, z, t))] k[M (y, z, t) ∗ (11 − M (T (y), T (z), t))] k 2 [M (T (y), T (z), t)) ∗ (11 − M (y, z, t))] ··· k n [M (T (y), T (z), t)) ∗ (11 − M (y, z, t))]

then M (y, z, t) ∗ (11 − M (y, z, t)) → 0 as n → ∞. Hence, M (y, z, t) = 1 and theny = z. Now suppose (X, Md , ∗) is a complete standard interval-valued fuzzy metric space induced by the metric d on X . From Result 5.10 (X, d) is complete, then if (xn ) is a fuzzy contractive sequence, by Proposition 5.6 it is contractive in (X, d), hence convergent. 53

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So, from Theorem 5.11 we have the following corollary, which can be considered as the fuzzy version of the classic Banach contraction theorem on complete interval-valued metric spaces. Corollary 5.12 Let (X, Md , ∗) be a complete standard interval-valued fuzzy metric space and let T : X → X a fuzzy contractive mapping. Then T has a unique fixed point. Definition 5.13[10]

An interval-valued fuzzy metric space (X, M, ∗) is compact

space if X is a compact set. Lemma 5.14 Let (X, Md , ∗) be an interval-valued fuzzy metric space. If limn∈N xn = x and limn∈N yn = y, then for all t > 0 and 0 < ε < 2t , M (x, y, t − ε) ≤ lim M (xn , yn , t) ≤ M (x, y, t + ε). n∈N

Proof. By Definition 3.1(IV3), ε ε M (xn , yn , t) ≥ M (xn , x, ) ∗ M (x, y, t − ε) ∗ M (y, yn , ). 2 2 Thus, lim M (xn , yn , t) ≥ 1 ∗ M (x, y, t − ε) ∗ 1 = M (x, y, t − ε).

n∈N

On the other hand, ε M (x, y, t + ε) ≥ M (x, y, t − ε) ∗ M (xn , yn , t) ∗ M (yn , y, ), 2 hence M (x, y, t + ε) ≥ lim M (xn , yn , t). n∈N

Corollary 5.15 Let (X, M, ∗) be an interval-valued fuzzy metric space. If limn∈N xn = x and limn∈N yn = y, then limn∈N M (xn , yn , t) = M (x, y, t) for all t > 0. Theorem 5.16 (fuzzy Edelstein contraction theorem). Let (X, Md , ∗) be a compact interval-valued fuzzy metric space. Let T : X → X be a mapping satisfying for all x 6= y and t > 0, M (T x, T y, t) > M (x, y, t), then T has a unique fixed point. Proof. Let x ∈ X and xn = T n x(n ∈ N). Assume xn 6= xn+1 for each n (if not, T (xn ) = xn ). Now, assume xn 6= xm (n 6= m). Otherwise we get M (xn , xn+1 , t) = M (xm , xm+1 , t) > M (xm−1 , xm+1 , t) > · · · > M (xn , xn+1 , t), 54

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where m > n, a contradiction. Since X is compact, {xn } has a convergent subsequence {xni }. Let y = limi∈N xni . We also assume that y, T y ∈ / {xn : n ∈ N} (if not, choose a subsequence with such a property). According to the above assumptions we may now write M (T xni , T y, t) > M (xni , y, t) for all i ∈ N and t > 0. Since M (x, y, ·) is continuous for all x, y ∈ X,by Corollary 5.15 we obtain lim M (T xni , T y, t) ≥ lim M (xni , y, t) = 1 i∈N

i∈N

for each t > 0, hence lim T xni = T y, · · · · · · · · · · · · (1). i∈N

Similarly, we obtain lim T 2 xni = T 2 y, · · · · · · · · · · · · · · · (2), i∈N

since T y 6= T xni for all i. Now, observe that M (xn1 , T xn1 , t) < M (T xn1 , T x2n1 , t) < · · · < M (xni , T xni , t) < M (T xni , T 2 xni , t) < · · · < M (xni+1 , T xni+1 , t) < M (T xni+1 , T 2 xni+1 , t) < · · · < 1 for all t > 0. Thus {M (xni , T xni , t)} and {M (T xni , T 2 xni , t)} (t > 0) are convergent to a common limit [cf.[29]). So, by (1),(2) and Corollary 5.15 we get M (y, T y, t) = = = = =

M (limi xni , T (limi xni ), t) limi M (xni , T xni , t) limi M (T xni , T 2 xni , t) M (limi T xni , limi T 2 xni , t) M (T y, T 2 y, t)

for all t > 0. Suppose y 6= T y. Then, by M (T x, T y, t) > M (x, y, t), M (y, T y, t) > M (T y, T 2 y, t),(t > 0), a contradiction. Hence y = T y, a fixed point. Uniqueness follows at once from M (T x, T y, t) > M (x, y, t).

6

Concluding remarks In this paper, the concept of interval-valued fuzzy metric space is defined, and it

is proved that the topology induced by quasi-metric is consistent with which induced via a standard interval-valued fuzzy quasi-metric, every quasi-metrizable topological space admits a compatible interval-valued fuzzy quasi-metric. On the contrary, topology generated by interval-valued fuzzy quasi-metric is quasi-metrizable. Furthermore, 55

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some properties of interval-valued fuzzy quasi-metric space which is bicompletion are discussed, and it is proved that if an interval-valued fuzzy quasi-metric space has bicompletion, then it is unique in the mean of isometry. Finally, a fuzzy contraction mapping of interval-valued fuzzy metric space is defined, and the Banach and Edelstein fixed point theorem to interval fuzzy metric space are promoted.

References [1] I.Kramosil, Michalek. Fuzzy metric and statistical metric spaces[J]. Kybernetika, 1975,11:326-334. [2] A. George, P. Veeramani. On some results in fuzzy metric spaces[J]. Fuzzy Sets and Systems 1994, 64(3): 395-399. [3] Z.K.DENG. Fuzzy pseudo metric spaces[J]. Journal Of Mathematical Analysis And Applications, 1982, 86(1): 74-95. [4] M.A.Erceg. Metric spaces in fuzzy set theory[J]. Journal Of Mathematical Analysis And Applications, 1979, 69(1): 205-230. [5] O. Kaleva, S. Seikkala. On fuzzy metric spaces[J]. Fuzzy Sets and Systems ,1984 ,12(3): 215-229. [6] A.George, P. Veeramani. On some results of analysis for fuzzy metric spaces[J]. Fuzzy Sets and Systems, 1997, 90(3): 365-368. [7] A.George, P. Veeramani. Some theorems in fuzzy metric spaces[J]. Journal of Fuzzy Mathematics, 1995, 3: 933-940. [8] V.Gregori, S.Romaguera. Some properties of fuzzy metric spaces[J]. Fuzzy Sets and Systems, 2000, 115(3): 485-489. [9] L.A.Zadeh. The concept of a linguistic variable and its application to approximation reasoning I[J]. Information Sciences, 1975, 8(3): 199-249. [10] Y.H.Shen, H.F. Li, F. X. Wang . On Interval-Valued Fuzzy Metric Spaces[J]. International Journal of Fuzzy Systems, 2012, 14(1): 35-44. [11] H.P.A. K u ¨nzi. Nonsymmetric distances and their associated topologies:About the origins of basic ideas in the area of asymmetric topology,in: Handbook of the history of general topology[M]. Dordrecht: Kluwer,2001,3: 853-968. [12] V. Gregori, S.Romaguera, Veeramani P. A note on intuitionistic fuzzy metric spaces[J]. Chaos, Solitons and Fractals, 2006, 28(4): 902-905. [13] A. Mohamad. Fixed-point theorems in intuitionistic fuzzy metric spaces[J]. Chaos, Solitons and Fractals, 2007, 34(5): 1689-1695 . [14] D.H. Hong, S. Lee. Some algebraic properties and a distance measure for interval-valued fuzzy numbers, Information Sciences[J]. 2002, 148(1-4): 1-10. [15] G.J.Wang, X.P.Li. Correlation and information energy of interval-valued fuzzy numbers[J]. Fuzzy Sets and Systems, 1999,103(1): 169-175. [16] P. Fletcher, W.F. Lindgren. Quasi-Uniform spaces[M].New York:Marcel Dekker, 1982. [17] L.A.Zadeh, Fuzzy sets[J]. Inform. and Control, 1965, 8: 338-353.

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[18] Z.K.Deng. Fuzzy pseudo metric spaces[J]. Journal Of Mathematical Analysis And Applications, 1982, 86(1): 74-95. [19] C. Li. Distances between interval-valued fuzzy sets. Proceedings of The 28th North American Fuzzy Information Processing Society Annual Conference (NAFIPS2009), Cincinnati, USA, DOI: 10.1109/NAFIPS. 2009. 5156438, Jun. 2009. [20] M. Grabiec, Fixed points in fuzzy metric spaces[J]. Fuzzy Sets and Systems, 1989, 27: 385-389. [21] R.E.Moore, Interval Analysis[M]. Prentice Hall, New Jersey, 1996. [22] C.Chakraborty, D. Chakraborty, Y.Y.Li, A theoretical development on a fuzzy distance measure for fuzzy numbers[J]. Mathematical and Computer Modelling, 2006, 43: 254-261. [23] G.Yun , S. Hwang, J. Chang. Fuzzy Lipschitz maps and fixed point theorems in fuzzy metric spaces[J]. Fuzzy Sets and Systems, 2010, 161: 1117-1130. [24] R.Vasuki. A common fixed point theorem in a fuzzy metric space. Fuzzy Sets and Systems 97 (1998) 395-397. [25] S.Sharma. Common fixed point theorems in fuzzy metric spaces[J]. Fuzzy Sets and Systems, 2002, 127: 345-352. [26] S.Sedghi, I. Altun, N. Shobe, Coupled fixed point theorems for contractions in fuzzy metric spaces[J]. Nonlinear Analysis,2010, 72: 1298-1304. [27] D. Mihe. A class of contractions in fuzzy metric spaces[J]. Fuzzy Sets and Systems, 2010, 161: 1131-1137. [28] M.Imdad, J. Ali. Some common fixed point theorems in fuzzy metric spaces[J]. Mathematical Communications, 2006, 11: 153-163. [29] B.K.Ray, H.Chatterjee. Some results on fixed points in meteic and Banach spaces[J]. Bull. Acad. Polon. Math, 1977, 25: 1243-1247.

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BI-UNIVALENT FUNCTIONS OF COMPLEX ORDER BASED ON QUASI-SUBORDINATE CONDITIONS INVOLVING WRIGHT HYPERGEOMETRIC FUNCTIONS N.E.CHO1,∗ G. MURUGUSUNDARAMOORTHY2 AND K.VIJAYA3

1

Department of Applied Mathematics Pukyong National University Busan 608-737, Korea E-mail: [email protected]. 2,3 School of Advanced Sciences, VIT University, Vellore 632014, Tamilnadu, India E-mail: [email protected]; [email protected] ∗ Corresponding Author Abstract. In the present paper, we introduce and investigate a new subclass of biunivalent functions of complex order defined in the open unit disk, which are associated with Wright hypergeometric functions and satisfying quasi-subordinate conditions. Furthermore, we find estimates on the second and the third coefficients of the TaylorMaclaurin series for functions in the new subclass. Several special consequences of the results are also pointed out. 2010 Mathematics Subject Classification. Primary 30C45. Key Words and Phrases. Analytic functions; Univalent functions; Bi-univalent functions; Bi-starlike and bi-convex functions; Gaussian hypergeometric function; DziokSrivastava operator.

1. Introduction, Definitions and Preliminaries Let A denote the class of functions of the form: ∞ X f (z) = z + an z n ,

(1.1)

n=2

which are analytic in the open unit disk U = {z : z ∈ C and |z| < 1}. Further, by S we shall denote the class of all functions in A which are univalent in U. Some of the important and well-investigated subclasses of the univalent function class S include (for example) the class S∗ (α) of starlike functions of order α in U and the class K(α) of convex functions of order α in U. 1 58

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2

The study of operators plays an important role in the geometric function theory and its related fields. Many differential and integral operators can be written in terms of convolution of certain analytic functions. It is observed that this formalism brings an ease in further mathematical exploration and also helps to understand the geometric properties of such operators better. The convolution or Hadamard product of two functions f, h ∈ A is denoted by f ∗ h and is defined as ∞ X (f ∗ h)(z) = z + an b n z n , (1.2) n=2

where f (z) is given by (1.1) and h(z) = z +

∞ P

bn z n .

n=2

Now we briefly recall the definitions of the special functions and operators used in this paper. β α For complex parameters α1 , . . . , αl ( Ajj 6= 0, −1, . . . ; j = 1, 2, . . . l) and β1 , . . . , βm ( Bjj 6= 0, −1, . . . ; j = 1, 2, . . . m), Fox’s H-function ( for details, see [23]) we mean the Wright’s generalized hypergeometric functions l Ψm with Aj , Bj > 0, give (rather general and typical examples of H−functions, not reducible to G−functions):   X ∞ Γ(α1 + nA1 ) . . . Γ(αl + nAl ) z n (α1 , A1 ), . . . , (αm , Al ) Ψ : z = l m (β1 , B1 ), . . . , (βm , Bm ) Γ(β1 + nB1 ) . . . Γ(βm + nBm ) n! n=0   (1 − α1 , A1 ), . . . , (1 − αl , Al ) 1,l = Hl,m+1 −z| , (1.3) (0, 1), (1 − β1 , B1 ), . . . , (1 − βm , Bm ) where 1 +

m P n=1

Bn −

l P

An ≥ 0, (l, m ∈ N = {1, 2, ...}) and for suitably bounded values of

n=1

|z|. We note that when A1 = · · · = Al = B1 = · · · = Bm = 1, they turn into the generalized hypergeometric functions  Ql    Γ(αj ) (α1 , 1), . . . , (αl , 1)  l Fm (α1 , . . . , αl : β1 , . . . , βm : z) (1.4) : z =  j=1 m l Ψm Q (β1 , 1), . . . , (βm , 1) Γ(βj ) i=1

(l ≤ m + 1; l, m ∈ N0 = N ∪ {0}; z ∈ U). Now we state the linear operator due to Srivastava [23] (see [6]) and Wright [24] in terms of the Hadamard product (or convolution) involving the generalized hypergeometric function. Let l, m ∈ N and suppose that the parameters α1 , A1 . . . , αl , Al and β1 , B1 . . . , βm , Bm are also positive real numbers. Then, corresponding to a function l Φm [(αj , Aj )1,l ; (βj , Bj )1,m ; z]

defined by l Φm [(αj , Aj )1,l ; (βj , Bj )1,m ; z]

= Ωzl Ψm [(αj , Aj )1,l (βj , Bj )1,m ; z]

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(1.5)

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where Ω =

l Q

!−1 Γ(αj )

j=1

m Q

3

! Γ(βj ) , we consider a linear operator

j=1

W[(αj , Aj )1,l ; (βj , Bj )1,m ] : A → A defined by the following Hadamard product (or convolution) W[(αj , Aj )1,l ; (βj , Bj )1,m ]f (z) := z l Φm [(αj , Aj )1,l ; (βj , Bj )1,m ; z] ∗ f (z). We observe that , for f (z) of the form(1.1),we have W[(αj , Aj )1,l ; (βj , Bj )1,m ]f (z) = z +

∞ X

ϕ n an z n

(1.6)

n=2

where Ω Γ(α1 + A1 (n − 1)) . . . Γ(αl + Al (n − 1)) (n − 1)!Γ(β1 + B1 (n − 1)) . . . Γ(βm + Bm (n − 1)) If, for convenience, we write

(1.7)

ϕn =

Wlm f (z) = W[(α1 , A1 ), . . . , (αl , Al ); (β1 , B1 ), . . . , (βm , Bm )]f (z).

(1.8)

We state the following remark due to Srivastava [23] (see [6]) and Wright [24]. Remark 1.1. Other interesting and useful special cases of the Fox-Wright generalized hypergeometric function l Ψm defined by (1.3) include (for example) the generalized Bessel function ∞ X (−z)n ν . Ψ (−; (ν + 1, µ); −z) ≡ J = 0 1 µ n! Γ(nµ + ν + 1) n=0 For µ = 1, corresponds essentially to the classical Bessel function Jν (z), and the generalized Mittag-Leffler function ∞ X (z)n λ . 1 Ψ1 ((1, 1); (µ, λ); z) ≡ Eµ = Γ(nλ + µ) n=0 Remark 1.2. By setting Aj = 1(j = 1, ..., l) and Bj = 1(j = 1, ..., m) in (1.5), we are led immediately to the generalized hypergeometric function l Fm (z) is defined by ∞ X (α1 )n . . . (αl )n z n Ωl Fm (z) ≡ Ωl Fm (α1 , . . . αl ; β1 , . . . , βm ; z) := (1.9) (β ) . . . (β ) n! 1 n m n n=0 (l ≤ m + 1; l, m ∈ N0 := N ∪ {0}; z ∈ U), where (α)n is the Pochhammer symbol. In view of the relationship (1.9), the linear operator(1.6) includes the Dziok-Srivastava operator (see [5]), so that it includes (as its special cases) various other linear operators introduced and studied by Bernardi [3], Carlson and Shaffer [4], Libera [11], Livingston [12], Ruscheweyh [19] and Srivastava and Owa [22]. It is well known that every function f ∈ S has an inverse f −1 , defined by f −1 (f (z)) = z

60

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and f (f

−1

 (w)) = w

1 |w| < r0 (f ); r0 (f ) ≥ 4

4

 ,

where g(w) = f −1 (w) = w − a2 w2 + (2a22 − a3 )w3 − (5a32 − 5a2 a3 + a4 )w4 + · · · .

(1.10)

A function f ∈ A is said to be bi-univalent in U if both f (z) and f −1 (z) are univalent in U. Let Σ denote the class of bi-univalent functions in U given by (1.1). For some intriguing examples of functions and characterization of the class Σ, one could refer Srivastava et al., [21] and the references stated therein (see also, [9]). Recently there has been triggering interest to study the bi-univalent function class Σ (see [8, 9, 13, 15, 17, 21]) and obtain non-sharp estimates on the first two Taylor-Maclaurin coefficients |a2 | and |a3 |. The coefficient estimate problem for each of the following Taylor-Maclaurin coefficients |an | for n ∈ N \ {1, 2} is presumably still an open problem. In 1970 Robertson[18] introduced the concept of quasi-subordination. An analytic function f (z) is quasi-subordinate to an analytic function φ(z), in the open unit disk if there exist analytic functions h(z) and w, with w(0) = 0 such that |h(z)| ≤ 1, |w(z)| < 1 and f (z) = h(z)φ[w(z)]. Then we write f (z) ≺qe φ(z). If h(z) = 1, then the quasisubordination reduces to the subordination. Also,if w(z) = z then f (z) = h(z)φ(z) and in this case we say that f (z) is majorized by φ(z) and it is written as f (z) 0).

(1.11)

also let ψ(z) = D0 + D1 z + D2 z 2 + D3 z 3 + · · · (|ψ(z) ≤ 1|; z ∈ U). (1.12) Motivated by the earlier work of Deniz [7] (see [1, 17, 20]) in the present paper, we introduce new subclass of the function class Σ of complex order γ ∈ C\{0}, involving Wright hypergeometric functions Wlm , and find estimates on the coefficients |a2 | and |a3 | for functions in the new subclasses of function class Σ. Several related classes are also considered , and connection to earlier known results are made. Definition 1.3. A function f ∈ Σ given by (1.1) is said to be in the class Gl,m Σ (γ, λ, φ) if the following conditions are satisfied:   z(Wlm f (z))0 1 − 1 ≺qe (φ(z) − 1) (1.13) γ (1 − λ)Wlm f (z) + λz(Wlm f (z))0 and

  1 w(Wlm g(w))0 − 1 ≺qe (φ(w) − 1), γ (1 − λ)Wlm g(w) + λz(Wlm g(w))0 where γ ∈ C\{0}, 0 ≤ λ < 1, z ∈ U and the function g is given by (1.10).

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Example 1. For λ = 0 and γ ∈ C\{0}, a function f ∈ Σ, given by (1.1) is said to be in the class Sl,m Σ (γ, φ) if the following conditions are satisfied:   1 z(Wlm f (z))0 − 1 ≺qe (φ(z) − 1) (1.15) γ Wlm f (z) and

  1 w(Wlm g(w))0 − 1 ≺qe (φ(w) − 1), γ Wlm g(w) where z, w ∈ U and the function g is given by (1.10).

(1.16)

On specializing the parameters l, m one can state the various new subclasses of Σ (or ), as illustrations, we present some examples for the case with Aj = 1 (j = 1, 2, ..., l); Bj = 1 (j = 1, 2, ..., m). Gl,m Σ (γ, λ, φ)

Example 2. If l ≤ m + 1, l, m ∈ N0 := N ∪ {0}, and γ ∈ C\{0}, then a function f ∈ Σ, given by (1.1) is said to be in the class Sl,m Σ (γ, λ, φ) if the following conditions are satisfied:   l z(Hm 1 f (z))0 − 1 ≺qe (φ(z) − 1) l f (z)) + λ(Hl f (z))0 γ (1 − λ)(Hm m and

 l g(w))0 w(Hm − 1 ≺qe (φ(w) − 1), (0 ≤ λ < 1), l g(w)) + λ(Hl g(w))0 (1 − λ)(Hm m   ∞ P (α1 )n ...(αl )n (z)n l where Hm f (z) := z + ∗ f (z) ≡ W[(αt , 1)1,l ; (βt , 1)1,m ]f (z) is a well(β1 )n ...(βm )n n! 1 γ



n=2

known Dziok-Srivastava operator [5], the function g is given by (1.10) and z, w ∈ U. Example 3. If l = 2 and m = 1 with α1 = a (a > 0), α2 = b, (b > 0) β1 = c (c > 0), and γ ∈ C\{0}, a function f ∈ Σ, given by (1.1) is said to be in the class Sa,c,b Σ (γ, λ, φ), if the following conditions are satisfied:   0 1 z(Ia,b c f (z)) − 1 ≺qe (φ(z) − 1) a,b 0 γ (1 − λ)(Ia,b c f (z)) + λ(Ic f (z)) and 1 γ



0 w(Ia,b c g(w))

 − 1 ≺qe (φ(w) − 1),

(0 ≤ λ < 1; z, w ∈ U), a,b 0 (1 − λ)(Ia,b c g(w)) + λ(Ic g(w))   ∞ P (a)n−1 (b)n−1 n a,b where Ic f (z) := z + z ∗ f (z) ≡ H12 (a, b; c)f (z), is a well-known Hohlov (c)n−1 (n−1)! n=2

operator [10] and the function g is given by (1.10). Example 4. If l = 2 and m = 1 with α1 = a (a > 0), α2 = 1, β1 = c (c > 0), and γ ∈ C\{0}, a function f ∈ Σ, given by (1.1) is said to be in the class Sa,c Σ (γ, λ, φ), if the following conditions are satisfied:   z(L(a, c)f (z))0 1 − 1 ≺qe (φ(z) − 1) γ (1 − λ)(L(a, c)f (z)) + λ(L(a, c)f (z))0

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6

and   1 w(L(a, c)g(w))0 − 1 ≺qe (φ(w) − 1), (0 ≤ λ < 1; z, w ∈ U), γ (1 − λ)(L(a, c)g(w)) + λ(L(a, c)g(w))0   ∞ P (a)n n z ∗ f (z) ≡ H12 (a, 1; c)f (z), is a well-known Carlsonwhere L(a, c)f (z) := z + (c)n n=2

Shaffer operator [4] and the function g is given by (1.10). Example 5. If l = 2 and m = 1 with α1 = δ + 1 (δ ≥ −1), α2 = 1, β1 = 1, and γ ∈ C\{0}, a function f ∈ Σ, given by (1.1) is said to be in the class SδΣ (γ, λ, φ) if the following conditions are satisfied:   z(Dδ f (z))0 1 − 1 ≺qe (φ(z) − 1) γ (1 − λ)(Dδ f (z)) + λ(Dδ f (z))0 and 1 γ



 w(Dδ g(w))0 − 1 ≺qe (φ(w) − 1), (1 − λ)(Dδ g(w)) + λ(Dδ g(w))0

(0 ≤ λ < 1; z, w ∈ U),

where Dδ is called Ruscheweyh derivative[19] of order δ (δ ≥ −1) and Dδ f (z) := f (z) ≡ H12 (δ + 1, 1; 1)f (z) and the function g is given by (1.10).

z (1−z)δ+1

∗

Example 6. If l = 2 and m = 1 with α1 = 1, α2 = 1, β1 = 1, and γ ∈ C\{0}, a function f ∈ Σ, given by (1.1) is said to be in the class S∗Σ (γ, λ, φ) if the following conditions are satisfied:   zf 0 (z) 1 − 1 ≺qe (φ(z) − 1) γ (1 − λ)f (z) + λf 0 (z) and 1 γ



 wg 0 (w) − 1 ≺qe (φ(w) − 1), (1 − λ)g(w) + λg 0 (w)

where 0 ≤ λ < 1, z, w ∈ U and the function g is given by (1.10). Example 7. If l = 2 and m = 1 with α1 = 1, α2 = 1, β1 = 1, and λ = 0; γ ∈ C\{0}, a function f ∈ Σ, given by (1.1) is said to be in the class S∗Σ (γ, φ) if the following conditions are satisfied:   1 zf 0 (z) − 1 ≺qe (φ(z) − 1) γ f (z) and 1 γ



 wg 0 (w) − 1 ≺qe (φ(w) − 1), g(w)

where 0 ≤ λ < 1, z, w ∈ U and the function g is given by (1.10). Remark 1.4. For λ = 0 and γ ∈ C\{0}, a function f ∈ Σ, given by (1.1), as in Example 1, one can state various analogous subclasses defined in Examples 2 to 4.

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Definition 1.5. A function f ∈ Σ given by (1.1) is said to be in the class Bl,m Σ (γ, λ, φ) if the following conditions are satisfied:   1 z 1−λ (Wlm f (z))0 − 1 ≺qe (φ(z) − 1) (1.17) γ [Wlm f (z)]1−λ and 1 γ



 w1−λ (Wlm g(w))0 − 1 ≺qe (φ(w) − 1), [Wlm g(w)]1−λ

(1.18)

where γ ∈ C \ {0}, λ ≥ 0, z, w ∈ U and the function g is given by(1.10). On specializing the parameters λ one can define the various new subclasses of Σ associated with Wright hypergeometric functions Wlm , as illustrated in the following examples. l,m Example 8. For λ = 0 and a function f ∈ Σ, given by (1.1), Bl,m Σ (γ, 0, φ) ≡ SΣ (γ, φ).

Example 9. For λ = 1, a function f ∈ Σ, given by (1.1) is said to be in the class HΣl,m (γ, φ) if the following conditions are satisfied:
1 Wlm f (z))0 − 1 ≺qe (φ(z) − 1) (1.19) γ and
1 (Wlm g(w))0 − 1 ≺qe (φ(w) − 1), γ

(1.20)

where γ ∈ C \ {0}; z, w ∈ U and the function g is given by(1.10). It is of interest to note that for γ = 1 the class Bl,m Σ (γ, λ, φ) reduces to the following l,m new subclass BΣ (λ, φ). Example 10. A function f ∈ Σ given by (1.1) is said to be in the class Bl,m Σ (λ, φ) if the following conditions are satisfied:   1−λ l z (Wm f (z))0 − 1 ≺qe (φ(z) − 1) (1.21) [Wlm f (z)]1−λ and 

 w1−λ (Wlm g(w))0 − 1 ≺qe (φ(w) − 1), [Wlm g(w)]1−λ

(1.22)

where λ ≥ 0, z, w ∈ U and the function g is given by (1.10). Remark 1.6. On specializing the parameters l, m one can state the various new subclasses of Bl,m Σ (γ, λ, φ) , as illustrated in Examples 1 to Examples 7, with Aj = 1 (j = 1, 2, ..., l); Bj = 1 (j = 1, 2, ..., m). In the following section, we find estimates on the coefficients |a2 | and |a3 | for functions l,m in the above-defined subclasses Gl,m Σ (γ, λ, φ) and BΣ (γ, λ, φ) of the function class Σ.

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2. Coefficient Bounds for the Function Class Gl,m Σ (γ, λ, φ) In order to derive our main results, we shall need the following lemma. Lemma 2.1. (see [16]) If h ∈ P, then |ck | ≤ 2 for each k, where P is the family of all functions h, analytic in U, for which 0

(z ∈ U),

where h(z) = 1 + c1 z + c2 z 2 + · · ·

(z ∈ U).

We begin by finding the estimates on the coefficients |a2 | and |a3 | for functions in the class Gl,m Σ (γ, λ, φ). Define the functions p(z) and q(z) by p(z) :=

1 + u(z) = 1 + p1 z + p2 z 2 + · · · 1 − u(z)

q(z) :=

1 + v(z) = 1 + q1 z + q2 z 2 + · · · 1 − v(z)

and

or, equivalently,     p(z) − 1 1 p21 2 u(z) := = p1 z + p2 − z + ··· p(z) + 1 2 2 and     q(z) − 1 1 q12 2 v(z) := = q1 z + q2 − z + ··· . q(z) + 1 2 2 Then p(z) and q(z) are analytic in U with p(0) = 1 = q(0). Since u, v : U → U, the functions p(z) and q(z) have a positive real part in U, and |pi | ≤ 2 and |qi | ≤ 2. Theorem 2.2. Let the function f (z) given by (1.1) be in the class Gl,m Σ (γ, λ, φ). Then √ |γ| |D0 |C1 C1 |a2 | ≤ p (2.1) |[γ D0 (λ2 − 1)C12 + (1 − λ)2 (C1 − C2 )]ϕ22 + 2γ(1 − λ)D0 C12 ϕ3 | and |a3 | ≤

|γD0 |2 C12 |γD0 | C1 |γD1 |C1 + + . 2 2 (1 − λ) ϕ2 2(1 − λ)ϕ3 2(1 − λ)ϕ3

Proof. It follows from (1.17) and (1.18) that   1 z(Wlm f (z))0 − 1 = ψ(z)[φ(u(z)) − 1] γ (1 − λ)Wlm f (z) + λz(Wlm f (z))0

(2.2)

(2.3)

and 1 γ



 w(Wlm g(w))0 − 1 = ψ(w)[φ(v(w)) − 1], (1 − λ)Wlm g(w) + λz(Wlm g(w))0

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(2.4)

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where p(z) and q(w) in P and have the following forms:     1 1 1 p21 D0 C2 2 2 ψ(z)[φ(u(z)) − 1] = D0 C1 p1 z + D1 C1 p1 + D0 C1 p2 − + p z + ··· 2 2 2 2 4 1 (2.5) and     1 1 q12 D0 C2 2 2 1 + q w + ··· ψ(w)[φ(v(w)) − 1] = D0 C1 q1 w + D1 C1 q1 + D0 C1 q2 − 2 2 2 2 4 1 (2.6) respectively. Now, equating the coefficients in (2.3) and (2.4), we get 1 (1 − λ) ϕ2 a2 = D0 C1 p1 , γ 2

(2.7)

  (λ2 − 1) 2 2 2(1 − λ) 1 1 p21 D0 C2 2 ϕ 2 a2 + ϕ3 a3 = D1 C1 p1 + D0 C1 p2 − + p, γ γ 2 2 2 4 1 −

1 (1 − λ) ϕ2 a2 = D0 C1 q1 γ 2

(2.8) (2.9)

and   1 (λ2 − 1) 2 2 2(1 − λ) q12 D0 C2 2 1 2 ϕ 2 a2 + ϕ3 (2a2 − a3 ) = D1 C1 q1 + D0 C1 q2 − q . (2.10) + γ γ 2 2 2 4 1 From (2.7) and (2.9), we find that a2 =

−γD0 C1 q1 γD0 C1 p1 = , 2(1 − λ)ϕ2 2(1 − λ)ϕ2

(2.11)

which implies p1 = −q1 .

(2.12)

8(1 − λ)2 ϕ22 a22 = γ 2 D02 C12 (p21 + q12 ).

(2.13)

and Adding (2.8) and (2.10), by using(2.12)and(2.13) ,we obtain
4 [γD0 (λ2 − 1)C12 + (1 − λ)2 (C1 − C2 )]ϕ22 + 2γD0 (1 − λ)C12 ϕ3 a22 = γ 2 D02 C13 (p2 + q2 ). (2.14) Thus, a22 =

γ 2 D02 C13 (p2 + q2 ) . 4 ([γD0 (λ2 − 1)C12 + (1 − λ)2 (C1 − C2 )]ϕ22 + 2γD0 (1 − λ)C12 ϕ3 )

(2.15)

Applying Lemma 2.1 for the coefficients p2 and q2 , we immediately have |a2 |2 ≤

|γ|2 |D0 |2 C13 . |[γD0 (λ2 − 1)C12 + (1 − λ)2 (C1 − C2 )]ϕ22 + 2γD0 (1 − λ)C12 ϕ3 |

(2.16)

Since C1 > 0,the last inequality gives the desired estimate on |a2 | given in (2.1).

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Next, in order to find the bound on |a3 |, by subtracting (2.10) from (2.8), we get 4(1 − λ) 4(1 − λ) ϕ 3 a3 − ϕ3 a22 γ γ D1 C1 (p1 − q1 ) D0 C1 (p2 − q2 ) D0 (C2 − C1 )(p21 − q12 ) = + + . (2.17) 2 2 4 It follows from (2.11), (2.12) and (2.17) that a3 =

γ 2 D02 C12 (p21 + q12 ) γD1 C1 (p1 − q1 ) γC1 D0 (p2 − q2 ) + + . 8(1 − λ)2 ϕ22 8(1 − λ)ϕ3 8(1 − λ)ϕ3

Applying Lemma 2.1 once again for the coefficients p2 and q2 , we readily get |a3 | ≤

|γD0 |2 C12 |γD1 |C1 |γD0 | C1 + . + 2 2 (1 − λ) ϕ2 2(1 − λ)ϕ3 2(1 − λ)ϕ3

This completes the proof of Theorem 2.2.



Putting λ = 0 in Theorem 2.2, we have the following corollary. Corollary 2.3. Let the function f (z) given by (1.1) be in the class Sl,m Σ (γ, φ). Then √ |γD0 |C1 C1 |a2 | ≤ p (2.18) |[(C1 − C2 ) − γD0 C12 ]ϕ22 + 2γD0 C12 ϕ3 | and |a3 | ≤

|γD1 |C1 |γD0 |2 C12 |γD0 |C1 + ++ . 2 ϕ2 2ϕ3 2ϕ3

(2.19)

Now we state the following corollaries for the function classes S∗Σ (γ, λ, φ) and S∗Σ (γ, λ) defined in Example 5 and Example 6, respectively. Corollary 2.4. Let the function f (z) given by (1.1) be in the class S∗Σ (γ, λ, φ). Then √ |γD0 |C1 C1 p |a2 | ≤ (2.20) (1 − λ) |(C1 − C2 ) + γD0 C12 | and |a3 | ≤

|γD0 |C1 |γD1 |C1 |γD0 |2 C12 + + . 2 (1 − λ) 2(1 − λ) 2(1 − λ)

(2.21)

Corollary 2.5. Let the function f (z) given by (1.1) be in the class S∗Σ (γ, φ). Then √ |γD0 |C1 C1 |a2 | ≤ p (2.22) |(C1 − C2 ) + γD0 C12 | and |a3 | ≤ |γD0 |2 C12 +

|γD0 |C1 |γD1 |C1 + . 2 2

67

(2.23)

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3. Coefficient Bounds for the Function Class Bl,m Σ (γ, λ, φ) Theorem 3.1. Let the function f (z) given by (1.1) be in the class Bµ,b Σ (γ, λ, φ). Then √ |γ| |D0 |C1 2C1 (3.1) |a2 | ≤ p |γD0 C12 [(λ − 1)(λ + 2)ϕ22 + 2(λ + 2)ϕ3 ] − 2(C2 − C1 )(1 + λ)2 ϕ22 | and  |a3 | ≤

|γD0 |C1 (1 + λ)ϕ2

2 +

|γD0 |C1 |γD1 |C1 + . (λ + 2)ϕ3 2(1 + λ)ϕ3

(3.2)

−1 Proof. Let f ∈ Bl,m . Then there are analytic functions u, v : 4 −→ 4 Σ (γ, λ, φ) and g = f with u(0) = 0 = v(0), satisfying   1 z 1−λ (Wlm f (z))0 − 1 = ψ(z)[φ(u(z)) − 1] (3.3) γ [Wlm f (z)]1−λ

and   1 w1−λ (Wlm g(w))0 − 1 = ψ(w)[φ(u(w)) − 1]. γ [Wlm g(w)]1−λ In light of (1.1) - (1.11), from (2.5) and (2.6), it is evident that   1 (λ − 1)(λ + 2) 2 2 2 (λ + 1) ϕ 2 a2 z + ϕ 2 a2 z + · · · (λ + 2)ϕ3 a3 + γ γ 2     1 1 p21 D0 C2 2 2 1 p z + ··· + = D0 C1 p1 z + D1 C1 p1 + D0 C1 p2 − 2 2 2 2 4 1

(3.4)

and     1 (λ − 1)(λ + 2) 2 (λ + 1) ϕ 2 a2 w + ϕ2 + 2(λ + 2)ϕ3 a22 w2 + · · · −(λ + 2)ϕ3 a3 + − γ γ 2     1 1 q12 D0 C2 2 2 1 q w + ··· + = D0 C1 q1 w + D1 C1 q1 + D0 C1 q2 − 2 2 2 2 4 1 Now proceeding on lines similar to Theorem 2.2 we get the desired results.



Choosing λ = 0 and λ = 1 we state the initial Taylor coefficients for the function classes and HΣl,m (γ, φ).

Sl,m Σ (γ, φ)

4. Concluding remarks For the class of strongly starlike functions, the function φ is given by  α 1+z φ(z) = = 1 + 2αz + 2α2 z 2 + · · · (0 < α ≤ 1), 1−z

(4.1)

which gives C1 = 2α and C2 = 2α2 .

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On the other hand, for −1 ≤ B ≤ A < 1 if we take φ(z) =

1 + Az = 1 + (A − B)z − B(A − B)z 2 + B 2 (A − B)z 3 + · · · , 1 + Bz

(4.2)

then we have C1 = (A − B), C2 = −B(A − B). By taking, A = (1 − 2β) where 0 ≤ β < 1 and B = −1 in (4.2), we get 1 + (1 − 2β)z 1−z = 1 + 2(1 − β)z + 2(1 − β)z 2 + 2(1 − β)z 3 + · · · .

φ(z) =

(4.3)

Hence, we have C1 = C2 = 2(1 − β). Further, by taking β = 0, in (4.3), we get φ(z) =

1+z = 1 + 2z + 2z 2 + 2z 3 + · · · 1−z

,

(4.4)

Hence, C1 = C2 = 2. Various Choices of φ as mentioned above and suitably choosing the values of C1 and C2 , we state some interesting results analogous to Theorem 2.2 , Theorem 3.1 and the Corollaries 2.3 to 2.5 for various new subclasses of Σ. Remark 4.1. Setting Aj = 1 (j = 1, ..., l) and Bj = 1 (j = 1, ..., m) and if l = 2 and m = 1 with α1 = µ + 1(µ > −1), α2 = 1, β1 = µ + 2, where Jµ is a Bernardi operator [3] defined by Z µ + 1 z µ−1 Jµ f (z) := t f (t)dt ≡ H12 (µ + 1, 1; µ + 2)f (z). zµ 0 Note that the operator J1 was studied earlier by Libera [11] and Livingston [12] and various other interesting corollaries and consequences of our main results (which are asserted by Theorem 2.2 and Theorem 3.1 above) can be derived similarly. Further, by setting Aj = 1 (j = 1, ..., l) and Bj = 1 (j = 1, ..., m) and suitably choosing l, m, λ, we can state the various results for the new classes defined in Examples 1 to 10, but the details involved may be left as an exercise for the interested reader. Acknowledgement This research was supported by the Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education, Science and Technology (No. 2016R1D1A1A09916450).

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References [1] R. M. Ali, S. K. Lee, V. Ravichandran, S. Supramaniam, Coefficient estimates for bi-univalent MaMinda star-like and convex functions, Appl. Math. Lett. 25 (2012) 344 -351. [2] D. A. Brannan and T. S. Taha, On some classes of bi-univalent functions, Studia Univ. Babe¸s-Bolyai Math. 31 (2) (1986), 70–77. [3] S. D. Bernardi, Convex and starlike univalent functions, Trans. Amer. Math. Soc. 135 (1969), 429– 446. [4] B. C. Carlson and S. B. Shaffer, Starlike and prestarlike hypergeometric functions, SIAM J. Math. Anal. 15 (1984), 737–745. [5] J. Dziok and H. M. Srivastava, Certain subclasses of analytic functions associated with the generalized hypergeometric function, Intergral Transform Spec. Funct. 14 (2003), 7–18. [6] J. Dziok and R. K. Raina, Families of analytic functions associated with the Wright generalized hypergeometric function, Demonstratio Math. 37 (3) (2004), 533–542. [7] E. Deniz, Certain subclasses of bi-univalent functions satisfying subordinate conditions, J. Classical Anal. 2(1) (2013), 49–60. [8] B. A. Frasin and M. K. Aouf, New subclasses of bi-univalent functions, Appl. Math. Lett. 24 (2011), 1569–1573. [9] T. Hayami and S. Owa, Coefficient bounds for bi-univalent functions, Pan Amer. Math. J. 22 (4) (2012), 15–26. [10] Yu. E. Hohlov, Hadamard convolutions, hypergeometric functions and linear operators in the class of univalent functions, Dokl. Akad. Nauk Ukrain. SSR Ser. A 1984 (7) (1984), 25–27. [11] R. J. Libera, Some classes of regular univalent functions, Proc. Amer. Math. Soc. 16 (1965), 755– 758. [12] A. E. Livingston, On the radius of univalence of certain analytic functions, Proc. Amer. Math. Soc. 17 (1966), 352–357. [13] X.-F. Li and A.-P. Wang, Two new subclasses of bi-univalent functions, Internat. Math. Forum 7 (2012), 1495–1504. [14] W. C. Ma, D. Minda, A unified treatment of some special classes of functions, in: Proceedings of the Conference on Complex Analysis, Tianjin, 1992, 157 - 169, Conf. Proc. Lecture Notes Anal. 1. Int. Press, Cambridge, MA, 1994. [15] G. Murugusundaramoorthy and T. Janani, Bi-starlike function of complex order associated with hypergeometric functions, Miskolc Math. Notes. 16 (1) (2015), 305–319 . [16] C. Pommerenke, Univalent Functions, Vandenhoeck & Ruprecht, G¨ottingen, 1975. [17] T. Panigarhi and G. Murugusundaramoorthy, Coefficient bounds for Bi-univalent functions analytic functions associated with Hohlov operator, Proc. Jangjeon Math. Soc. 16 (1) (2013), 91-100. [18] M. S. Robertson, Quasi-subordination and coefficient conjectures, Bulletin of the American Mathematical Soceity, 76 (1970), 1–9. [19] St. Ruscheweyh, New criteria for univalent functions, Proc. Amer. Math. Soc. 49 (1975), 109–115. [20] H. M. Srivastava, G. Murugusundaramoorthy and N. Magesh, Certain subclasses of bi-univalent functions associated with the Hohlov operator, Global J. Math. Anal. 2 (1) (2013), 67–73. [21] H. M. Srivastava, A. K. Mishra and P. Gochhayat, Certain subclasses of analytic and bi-univalent functions, Appl. Math. Lett. 23 (2010), 1188–1192. [22] H. M. Srivastava and S. Owa, Some characterization and distortion theorems involving fractional calculus, generalized hypergeometric functions, Hadamard products, linear operators and certain subclasses of analytic functions, Nagoya Math. J. 106 (1987), 1–28. [23] H. M. Srivastava, Some Fox’s-Wright generalized hypergeometric functions and associated families of convolution operators, Appl. Anal. Discrete Math. 1 (2007), 56?71. [24] E. M. Wright, The asymptotic expansion of the generalized hypergeometric function, Proc. London. Math. Soc. 46 (1946), 389–408.

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On uni-soft mighty filters of BE-algebras Jeong Soon Han1 and Sun Shin Ahn2,∗ 1

Department of Applied Mathematics, Hanyang University, Ahnsan, 15588, Korea

2

Department of Mathematics Education, Dongguk University, Seoul 04620, Korea

Abstract. The notion of uni-soft mighty filters of a BE-algebra is introduced, and related properties are investigated. The problem of classifying uni-soft mighty by their γ-exclusive filter is solved. We construct a new quotient structure of a transitive BE-algebra using a unit soft filter and consider some properties of it.

1. Introduction Kim and Kim [7] introduced the notion of a BE-algebra, and investigated several properties. In [2], Ahn and So introduced the notion of ideals in BE-algebras. They gave several descriptions of ideals in BE-algebras. Various problems in system identification involve characteristics which are essentially nonprobabilistic in nature [12]. In response to this situation Zadeh [13] 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 [14]. 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 [10]. Maji et al. [9] and Molodtsov [10] 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 [10] 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. At present, works on the soft set theory are progressing rapidly. Maji et al. [9] described the application of soft set 0

2010 Mathematics Subject Classification: 06F35; 03G25; 06D72. Keywords: BE-algebra; (mighty) filter; uni-soft (mighty) filter. The corresponding author. Tel: +82 2 2260 3410, Fax: +82 2 2266 3409 0 E-mail: [email protected] (J. S. Han); [email protected] (S. S. Ahn) 0 This work was supported by the research fund of Hanyang University(HY-2016-G). 0

∗

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theory to a decision making problem. Maji et al. [8] also studied several operations on the theory of soft sets. H. R. Lee and S. S. Ahn [6] introduced the notion of a mighty filter in BE-algebras, and investigated some properties of it. Jun et al. [5] introduced the notion of a uni-soft filter of a BE-algebra, and studied their properties. In this paper, we introduce the notion of a uni-soft mighty filter of a BE-algebra, and investigate their properties. We solve the problem of classifying uni-soft mighty by their γ-exclusive filter. Also we construct a new quotient structure of a transitive BE-algebra using a unit soft filter and study some properties of it. 2. Preliminaries We recall some definitions and results discussed in [7]. An algebra (X; ∗, 1) of type (2, 0) is called a BE-algebra if (BE1) (BE2) (BE3) (BE4)

x ∗ x = 1 for all x ∈ X; x ∗ 1 = 1 for all x ∈ X; 1 ∗ x = x for all x ∈ X; x ∗ (y ∗ z) = y ∗ (x ∗ z) for all x, y, z ∈ X (exchange)

We introduce a relation “≤” on a BE-algebra X by x ≤ y if and only if x ∗ y = 1. A non-empty subset S of a BE-algebra X is said to be a subalgebra of X if it is closed under the operation “ ∗ ”. Noticing that x ∗ x = 1 for all x ∈ X, it is clear that 1 ∈ S. A BE-algebra (X; ∗, 1) is said to be self distributive if x ∗ (y ∗ z) = (x ∗ y) ∗ (x ∗ z) for all x, y, z ∈ X. Definition 2.1. Let (X; ∗, 1) be a BE-algebra and let F be a non-empty subset of X. Then F is called a filter of X if (F1) 1 ∈ F ; (F2) x ∗ y ∈ F and x ∈ F imply y ∈ F for all x, y ∈ X. F is a mighty filter [6] of X if it satisfies (F1) and (F3) z ∗ (y ∗ x) ∈ F and z ∈ F imply ((x ∗ y) ∗ y) ∗ x ∈ F for all x, y, z ∈ X. Proposition 2.2. Let (X; ∗, 1) be a BE-algebra and let F be a filter of X. If x ≤ y and x ∈ F for any y ∈ X, then y ∈ F . Proposition 2.3. Let (X; ∗, 1) be a self distributive BE-algebra. Then following hold: for any x, y, z ∈ X, (i) if x ≤ y, then z ∗ x ≤ z ∗ y and y ∗ z ≤ x ∗ z. (ii) y ∗ z ≤ (z ∗ x) ∗ (y ∗ z). (iii) y ∗ z ≤ (x ∗ y) ∗ (x ∗ z).

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J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 24, NO.1, 2018, COPYRIGHT 2018 EUDOXUS PRESS, LLC

On uni-soft mighty filters of BE-algebras

A BE-algebra (X; ∗, 1) is said to be transitive if it satisfies Proposition 2.3(iii). Theorem 2.4.([6]) A filter F of a BE-algebra X is mighty if and only if (2.1) (∀x, y ∈ X)(y ∗ x ∈ F ⇒ ((x ∗ y) ∗ y) ∗ x ∈ F ). A soft set theory is introduced by Molodtsov [10]. In what follows, let U be an initial universe set and X be a set of parameters. Let P(U ) denote the power set of U and A, B, C, · · · ⊆ X. Definition 2.5. A soft set (f, A) of X over U is defined to be the set of ordered pairs (f, A) := {(x, f (x)) : x ∈ X, f (x) ∈ P(U )} , where f : X → P(U ) such that f (x) = ∅ if x ∈ / A. For a soft set (f, A) of X and a subset γ of U, the γ-exclusive set of (f, A) , denoted by eA (f ; γ) , is defined to be the set eA (f ; γ) := {x ∈ A | f (x) ⊆ γ} . For any soft sets (f, X) and (g, X) of X, we call (f, X) a soft subset of (g, X) , denoted by ˜ (g, X) , if f (x) ⊆ g(x) for all x ∈ X. The soft union of (f, X) and (g, X), denoted by (f, X) ⊆ ˜ (g, X) , is defined to be the soft set (f ∪ ˜ g, X) of X over U in which f ∪ ˜ g is defined by (f, X) ∪ ˜ g) (x) = f (x) ∪ g(x) for all x ∈ X. (f ∪ ˜ (g, X) , is defined to be the soft The soft intersection of (f, X) and (g, X) , denoted by (f, X) ∩ ˜ g, M ) of X over U in which f ∩ ˜ g is defined by set (f ∩ ˜ g) (x) = f (x) ∩ g(x) for all x ∈ M. (f ∩ 3. Uni-soft mighty filters In what follows, we take a BE-algebra X, as a set of parameters unless specified. Definition 3.1.([5]) A soft set (f, X) of X over U is called a union-soft filter (briefly, uni-soft filter) of X if it satisfies: (US1) (∀x ∈ X) (f (1) ⊆ f (x)) , (US2) (∀x, y ∈ X) (f (y) ⊆ f (x ∗ y) ∪ f (x)). Proposition 3.2.([5]) Every uni-soft filter (f, X) of X over U satisfies the following properties: (i) (∀x, y ∈ X) (x ≤ y ⇒ f (y) ⊆ f (x)). (ii) (∀x, y, z ∈ X) (z ≤ x ∗ y ⇒ f (y) ⊆ f (x) ∪ f (z)). Definition 3.3. A soft set (f, X) of X over U is called a union-soft mighty filter (briefly, uni-soft mighty filter) of X if it satisfies (US1) and (US3) (∀x, y, z ∈ X) (f (((x ∗ y) ∗ y) ∗ x) ⊆ f (z ∗ (y ∗ x)) ∪ f (z))).

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J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 24, NO.1, 2018, COPYRIGHT 2018 EUDOXUS PRESS, LLC

Jeong Soon Han and Sun Shin Ahn

Example 3.4. Let E = X be the set of parameters where X := {1, a, b, c, d, 0} is a BE-algebra [7] with the following Cayley table: ∗ 1 a b c d 0

1 1 1 1 1 1 1

a a 1 1 a 1 1

b b a 1 b a 1

c c c c 1 1 1

d d c c a 1 1

0 0 d c b a 1

Let (f, X) be a soft set of X over U := Z defined as follows: { N if x ∈ {1, a, b} f : X → P(U ), x 7→ Z if x ∈ {c, d, 0}. It is easy to check that (f, X) is a uni-soft mighty filter of X. Proposition 3.5. Every uni-soft mighty filter of a BE-algebra X is a uni-soft filter of X. Proof. Let (f, X) be a uni-soft mighty filter of X. Setting y := 1 in (US3), we have f (x) = (((x ∗ 1) ∗ 1) ∗ x) ⊆ f (z ∗ (1 ∗ x)) ∪ f (z) = f (z ∗ x) ∪ (z). Hence (US2) holds. Therefore (f, X) is a uni-soft filter of X. □ The converse of Proposition 3.5 is not true in general as seen in the following example. Example 3.6. Let E = X be the set of parameters where X := {1, a, b, c, d} is a BE-algebra [7] with the following Cayley table: ∗ 1 a b c d 1 1 a b c d a 1 1 b c d b 1 a 1 c c c 1 1 b 1 b d 1 1 1 1 1 Let (f, X) be a soft set of X over U := Z defined as   3N f : X → P(U ), x 7→ 4N  3Z

follows: if x ∈ {1} if x ∈ {a} if x ∈ {b, c, d}.

It is easy to check that (f, X) is a uni-soft filter of X. But it is not a uni-soft mighty filter of X, since f (((a ∗ c) ∗ c) ∗ a) = f (a) = 4N ⊈ f (1 ∗ (c ∗ a)) ∪ f (1) = f (1) = 3N. We provide conditions for a uni-soft filter to be a uni-soft mighty filter. Theorem 3.7. Any uni-soft filter of a BE-algebra X is mighty if and only if it satisfies

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J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 24, NO.1, 2018, COPYRIGHT 2018 EUDOXUS PRESS, LLC

On uni-soft mighty filters of BE-algebras

(3.1) (∀x, y ∈ X)(f (((x ∗ y) ∗ y) ∗ x)) ⊆ f (y ∗ x). Proof. Assume that a uni-soft filter (f, X) of X is mighty. Putting z := 1 in (US3), we have f (((x ∗ y) ∗ y) ∗ x) ⊆ f (1 ∗ (y ∗ x)) ∪ f (1) = f (y ∗ x). Hence (3.1) holds. Conversely, suppose that the uni-soft filter (f, X) of X satisfies the condition (3.1). Using (3.1) and (US2), we have f (((x ∗ y) ∗ y) ∗ x) ⊆ f (y ∗ x) ⊆ f (z ∗ (y ∗ x)) ∪ f (z) for any x, y ∈ X. Hence (f, X) is an uni-soft mighty filter of X. □ Proposition 3.8. Let (f, X) be a uni-soft mighty filter of a BE-algebra X. Then Xf := {x ∈ X|f (x) = f (1)} is a mighty filter of X. Proof. Clearly, 1 ∈ Xf . Let z ∗ (y ∗ x), z ∈ Xf . Then f (z ∗ (y ∗ x)) = f (1) and f (z) = f (1). It follows from (US3) that f (((x ∗ y) ∗ y) ∗ x) ⊆ f (z ∗ (y ∗ x)) ∪ f (z) = f (1). By (US1), we get f (((x ∗ y) ∗ y) ∗ x) = f (1). Hence ((x ∗ y) ∗ y) ∗ x ∈ Xf . Therefore Xf is a mighty filter of X. □ Theorem 3.9. Let (f, X) and (g, X) be uni-soft filters of a transitive BE-algebra such that ˜ X) and f (1) = g(1). If (g, X) is mighty, then so is (f, X). (f, X)⊆(g, Proof. Let x, y ∈ X. Note that y ∗ ((y ∗ x) ∗ x) = (y ∗ x) ∗ (y ∗ x) = 1. Since (g, X) is a uni-soft ˜ X) we have g(1) = g(y ∗ ((y ∗ x) ∗ x)) ⊇ mighty filter of a BE-algebra X, by (3.1) and (f, X)⊆(g, g(((((y ∗ x) ∗ x) ∗ y) ∗ y) ∗ ((y ∗ x) ∗ x)) ⊇ f (((((y ∗ x) ∗ x) ∗ y) ∗ y) ∗ ((y ∗ x) ∗ x)). Since f (1) = g(1), we get f ((y ∗ x) ∗ ((((y ∗ x) ∗ x) ∗ y) ∗ y) ∗ x)) = f (((((y ∗ x) ∗ x) ∗ y) ∗ y) ∗ ((y ∗ x) ∗ x)) = f (1). It follows from (US1) and (US2) that f (y ∗ x) =f (1) ∪ f (y ∗ x) =f ((y ∗ x) ∗ (((((y ∗ x) ∗ x) ∗ y) ∗ y) ∗ x)) ∪ f (y ∗ x)

(3.2)

⊇f (((((y ∗ x) ∗ x) ∗ y) ∗ y) ∗ x). Since X is transitive, we get [((((y ∗ x) ∗ x) ∗ y) ∗ y) ∗ x]∗[((x ∗ y) ∗ y) ∗ x] ≥ ((x ∗ y) ∗ y) ∗ ((((y ∗ x) ∗ x) ∗ y) ∗ y) ≥ (((y ∗ x) ∗ x) ∗ y) ∗ (x ∗ y) ≥ x ∗ ((y ∗ x) ∗ x) = (y ∗ x) ∗ (x ∗ x) = (y ∗ x) ∗ 1 = 1. It follows from Proposition 3.2 that f (((((y∗x)∗x)∗y)∗y)∗x)∪f (1) = f (((((y∗x)∗x)∗y)∗y)∗x) ⊇ f (((x ∗ y) ∗ y) ∗ x). Using (3.2), we have f (y ∗ x) ⊇ f (((((y ∗ x) ∗ x) ∗ y) ∗ y) ∗ x) ⊇ f (((x ∗ y) ∗ y) ∗ x). Therefore f (y ∗ x) ⊇ f (((x ∗ y) ∗ y) ∗ x). By Theorem 3.7, (f, X) is a uni-soft mighty filter of X. □

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J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 24, NO.1, 2018, COPYRIGHT 2018 EUDOXUS PRESS, LLC

Jeong Soon Han and Sun Shin Ahn

Let (f, X) be a uni-soft filter of a transitive BE-algebra X. Define a binary relation “ ∼f ” on X by putting x ∼f y if and only if f (x ∗ y) = f (y ∗ x) = f (1) for any x, y ∈ X. Lemma 3.10. The relation “ ∼f ” is an equivalence relation on a transitive BE-algebra X. Proof. For any x ∈ X, x ∗ x = 1 by (BE1). So f (x ∗ x) = f (1), hence x ∼f x, which ∼f is reflexive. Suppose that x ∼f y for any x, y ∈ X. Then f (x ∗ y) = f (y ∗ x) = f (1). Hence ∼f is symmetric. Assume that x ∼f y and y ∼f z for any x, y, z ∈ X. Then f (x ∗ y) = f (y ∗ x) = f (1) and f (y ∗ z) = f (z ∗ y) = f (1). By transitivity, (x ∗ y) ∗ [(y ∗ z) ∗ (x ∗ z)] = 1 and (z ∗ y) ∗ [(y ∗ x) ∗ (z ∗ x)] = 1. By Proposition 3.2, we have f (x ∗ z) ⊆ f (y ∗ z) ∪ f (x ∗ y) = f (1) and f (z ∗ x) ⊆ f (y ∗ x) ∪ f (z ∗ y) = f (1). Hence f (z ∗ x) = f (z ∗ x) = f (1), i.e., x ∼f z. Thus ∼f is an equivalence relation on X. □ Lemma 3.11. The relation “ ∼f ” is a congruence relation on a transitive BE-algebra X. Proof. If x ∼f y and u ∼f v for any x, y, u, v ∈ X, then f (x ∗ y) = f (y ∗ x) = f (1) and f (u∗v) = f (v∗u) = f (1). By transitivity, (u∗v)∗[(x∗u)∗(x∗v)] = 1 and (v∗u)∗[(x∗v)∗(x∗u)] = 1, it follows from Proposition 3.2 that f (1) = f (u ∗ v) ⊇ f ((x ∗ u) ∗ ((x ∗ v)) and f (1) = f (v ∗ u) ⊇ f ((x ∗ v) ∗ (x ∗ u)). Hence f ((x ∗ u) ∗ (x ∗ v)) = f (1) and f ((x ∗ v) ∗ (x ∗ u)) = f (1). Therefore x ∗ u ∼f x ∗ v. By a similar way, we can prove that x ∗ v ∼f y ∗ v. Hence x ∗ u ∼f y ∗ v. Therefore ∼f is a congruence relation on X. □ X is decomposed by the congruence relation ∼f . The class containing x is denoted by fx . Denote X/f := {fx |x ∈ X}. We define a binary relation • on X/f by fx • fy := fx∗y . This definition is well defined since ∼f is a congruence relation on X Lemma 3.12. f1 = Xf . Proof. f1 = {x ∈ X|1 ∼f x} = {x ∈ X|f (1 ∗ x) = f (x ∗ 1) = f (1)} = {x ∈ X|f (x) = f (1)} = Xf . □ Theorem 3.13. Let (X, f ) be a uni-soft filter of a transitive BE-algebra X. Then (X/f ; •, f1 ) is a transitive BE-algebra. □

Proof. It is straightforward.

Theorem 3.14. Let X be a transitive BE-algebra. If every filter of the quotient algebra X/f is mighty, then a uni-soft filter of X is mighty. Proof. Suppose that every filter of the quotient algebra X/f is mighty and let x, y ∈ X be such that y ∗ x ∈ f1 . Then f (y ∗ x) = f (1) and so fy • fx ∈ f1 . Since {f1 } is a mighty filter of X/f , it follows from Theorem 2.4 that f((x∗y)∗y)∗x = ((fx • fy ) • fy ) • fx ∈ f1 . Hence f ((((x ∗ y) ∗ y)∗) ∗ x) = f (1). Therefore f (y ∗ x) = f (((x ∗ y) ∗ y)) ∗ x). Thus (f, X) is a uni-soft mighty filter by Theorem 3.7. □

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J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 24, NO.1, 2018, COPYRIGHT 2018 EUDOXUS PRESS, LLC

On uni-soft mighty filters of BE-algebras

Theorem 3.15. A uni-soft set (X, f ) of a BE-algebra X is a uni-soft mighty filter of X if and only if the set eX (f ; γ) := {x ∈ X|f (x) ⊆ γ} is a mighty filter of X for all γ ∈ P([0, 1]) whenever it is nonempty. Proof. Suppose that (f, X) is a uni-soft mighty filter of X. Let x, y, z ∈ X and γ ∈ P([0, 1]) be such that z ∗ (y ∗ x) ∈ eX (f ; γ) and z ∈ eX (f ; γ). Then f (z ∗ (y ∗ x)) ⊆ γ and f (z) ⊆ γ. It follows from (US1) and (US3) that f (1) ⊆ fX (((x ∗ y) ∗ y) ∗ x) ⊆ f (z ∗ (y ∗ x)) ∪ f (z) ⊆ γ. Hence 1 ∈ eX (f ; γ) and ((x ∗ y) ∗ y) ∗ x ∈ eX (f ; γ), and therefore eX (f ; γ) is a mighty filter of X. Conversely, assume that eX (f ; γ) is a mighty filter of X for all γ ∈ P([0, 1]) with eX (f ; γ) ̸= ∅. For any x ∈ X, let f (x) = γ. Then x ∈ eX (f ; γ). Since eX (f ; γ) is a mighty filter of X, we have 1 ∈ eX (f ; γ) and so f (x) = γ ⊇ f (1). For any x, y, z ∈ X, let f (z ∗ (y ∗ x)) = γz∗(y∗x) and f (z) = γz . Let γ := γz∗(y∗x) ∪ γz . Then z ∗ (y ∗ x) ∈ eX (f ; γ) and z ∈ eX (f ; γ) which imply that ((x ∗ y) ∗ y) ∗ x ∈ eX (f ; γ). Hence f (((x ∗ y) ∗ y) ∗ x) ⊆ γ = γz∗(y∗x) ∪ γz = f (z ∗ (y ∗ x)) ∪ f (z). Thus (f, X) is a uni-soft mighty filter of X. □ We call eX (f ; γ) a γ-exclusive filter of a BE-algebra X. Theorem 3.16. Every filter of a BE-algebra can be represented as a γ-exclusive set of a uni-soft mighty filter, i.e., given a filter F a BE-algebra X, there exists a uni-soft mighty filter (f, X) of X over U such that F is the γ-exclusive set of (f, X) for a non-empty subset γ of U . Proof. Let F be a filter of a BE-algebra X. For a subset γ of U, define a soft set (f, X) over U by { γ if x ∈ F, f : X → P(U ), x 7→ U if x ∈ / F. Obviously, F = eX (f ; γ). We now prove that (f, X) is a uni-soft mighty filter of X. Since 1 ∈ F = eX (f ; γ), we have f (1) = γ ⊆ f (x) for all x ∈ X. Let x, y, z ∈ X. If z ∗ (y ∗ x), z ∈ F, then ((x∗y)∗y)∗x ∈ F because F is a mighty filter of X. Hence f (z∗(y∗x)) = f (z) = f (((x∗y)∗y)∗x) = γ, and so f (z ∗(y ∗x))∪f (z) ⊇ f (((x∗y)∗y)∗x). If z ∗(y ∗x) ∈ F and z ∈ / F, then f (z ∗(y ∗x)) = γ and f (z) = U which imply that f (z ∗ (y ∗ x))) ∪ f (z) = γ ∪ U = U ⊇ f (((x ∗ y) ∗ y) ∗ x). Similarly, if z ∗ (y ∗ x) ∈ / F and z ∈ F, then f (z ∗ (y ∗ x))) ∪ f (z) ⊇ f (((x ∗ y) ∗ y) ∗ x). Obviously, if z ∗ (y ∗ x) ∈ / F and z ∈ / F, then f (z ∗ (y ∗ x)) ∪ f (z) ⊇ f (((x ∗ y) ∗ y) ∗ x). Therefore (f, X) is a uni-soft mighty filter of X. □ Any filter of a BE-algebra X cannot be represented as a γ-exclusive set of a uni-soft mighty filter (f, X) of X over U (see Example 3.17). Example 3.17. Let X = {1, a, b, c, d, 0} be the BE-algebra as in Example 3.4. Given U = X, consider a soft set (f, X) over U which is defined by { {c} if x ∈ {1, a, b}, f : X → P(U ), x 7→ {1, c} if x ∈ {c, d, 0}.

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J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 24, NO.1, 2018, COPYRIGHT 2018 EUDOXUS PRESS, LLC

Jeong Soon Han and Sun Shin Ahn

Then (f, X) is a uni-soft mighty filter of X. The γ-exclusive sets of (f, X) are described as follows:   if γ ⊇ {1, c},  X eX (f ; γ) = {1, a, b} if {c} ⊆ γ ⊊ {1, c}   ∅ otherwise. The filter {1} cannot be a γ-exclusive set eX (f ; γ), since there is no γ ⊆ U such that eX (f ; γ) = {1}. Proposition 3.18. Let (f, X) be a uni-soft filter of a transitive BE-algebra X. The mapping γ : X → X/f , given by γ(x) := fx , is a surjective homomorphism, and Kerγ = {x ∈ X|γ(x) = f1 } = Xf . Proof. Let fx ∈ X/f . Then there exists an element x ∈ X such that γ(x) = fx . Hence γ is surjective. For any x, y ∈ X, we have γ(x ∗ y) = fx∗y = fx • fy = γ(x) • γ(y). Thus γ is a homomorphism. Moreover, Ker γ = {x ∈ X|γ(x) = f1 } = {x ∈ X|x ∼f 1} = {x ∈ X|f (x) = f (1)} = Xf . □ Example 3.19. Let E = X be the set of parameters where X := {1, a, b, c, d, 0} is a transitive BE-algebra [6] with the following Cayley table: ∗ 1 a b c d 0

1 1 1 1 1 1 1

a a 1 a a 1 1

b b b 1 1 1 1

c c c b 1 b 1

d d b a a 1 1

0 0 c d a b 1

Let (f, X) be a soft set of X over U defined as follows: { γ1 if x ∈ {1, b, c} f : X → P(U ), x 7→ γ2 if x ∈ {a, d, 0}, where γ1 and γ2 are subsets of U with γ1 ⊊ γ2 . Then (f, X) is a uni-soft mighty filter over U . By Proposition 3.5, it is a uni-soft filter over U . Then Xf = {x ∈ X|f (x) = f (1)} = {1, b, c}. Define x ∼f y if and only if f (x ∗ y) = f (y ∗ x) = f (1). Then f1 = {x ∈ X|x ∼f 1} = {x ∈ X|f (x ∗ 1) = f (1 ∗ x) = f (1)} = {1, b, c}, fa = {x ∈ X|x ∼f a} = {x ∈ X|f (x ∗ a) = f (a ∗ x) = f (1)} = {a}, fd = {x ∈ X|x ∼f d} = {x ∈ X|f (x ∗ d) = f (d ∗ x) = f (1)} = {d}, and f0 = {x ∈ X|x ∼f 0} = {x ∈ X|f (x ∗ 0) = f (0 ∗ x) = f (1)} = {0}. Hence X/f = {f1 , fa , fd , f0 }. Let φ : X → X/f be a map defined by φ(1) = φ(b) = φ(c) = f1 , φ(a) = fa , φ(d) = fd , φ(0) = f0 . It is easy to check

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J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 24, NO.1, 2018, COPYRIGHT 2018 EUDOXUS PRESS, LLC

On uni-soft mighty filters of BE-algebras

that φ is a homomorphism and Kerφ = {x ∈ X|φ(x) = f1 } = {x ∈ X|x ∼f 1} = {x ∈ X|f (x) = f (1)} = Xf . Proposition 3.20. Let µ : X → Y be an epimorphism of BE-algebras. If (f, Y ) is a uni-soft filter of Y , then (f ◦ µ, X) is a uni-soft filter of X. Proof. For any x ∈ X, we have (f ◦ µ)(x) = f (µ(x)) ⊇ f (1Y ) = f (µ(1X )) = (f ◦ µ)(1X ) and (f ◦ µ)(y) = f (µ(y)) ⊆ f (a ∗Y µ(y)) ∪ f (a) for any a ∈ Y . Let x be any preimage of a under µ. Then (f ◦ µ)(y) ⊆f (a ∗Y µ(y)) ∪ f (a) =f (µ(x) ∗Y µ(y)) ∪ f (µ(x)) =f (µ(x ∗X y)) ∪ f (µ(x)) =(f ◦ µ)(x ∗X y) ∪ (f ◦ µ)(x). Therefore (f ◦ µ, X) is a uni-soft filter of X.

□

Proposition 3.21. Let (f, X) be a uni-soft filter of a transitive BE-algebra X. If J is a filter of X, then J/f is a filter of X/f . Proof. Let (f, X) be a uni-soft filter of X and let J be a filter of X. For any x ∗ y, x ∈ J, we obtain y ∈ J. Hence for any fx • fy = fx∗y , fx ∈ J/f , we have fy ∈ J/f . Thus J/f is a filter of X/f . □ Theorem 3.22. Let (f, X) be a uni-soft filter of a transitive BE-algebra X. If J ∗ is a filter of a transitive BE-algebra X/f , then there exists a filter J = {x ∈ X|fx ∈ J ∗ } in X such that J/f = J ∗ . Proof. Since J ∗ is a filter of X/f , so fx • fy = fx∗y , fx ∈ J ∗ imply fy ∈ J ∗ for any fx , fy ∈ J ∗ . Therefore x ∗ y, x ∈ J imply y ∈ J for any x, y ∈ J. Therefore J is a filter of X. By Proposition 3.18, we have J/f ={fj |j ∈ J} ={fj |∃fx ∈ J ∗ such that j ∼f x} ={fj |∃fx ∈ J ∗ such that fx = fj } ={fj |fj ∈ J ∗ } = J ∗ . Theorem 3.23. Let (f, X) be a uni-soft filter of a transitive BE-algebra X. If J is a filter of X/f ∼ X, then = X/J. J/f X/f X/f = {[fx ]J/f |fx ∈ X/f }. If we define φ : → X/J by φ([fx ]J/f ) = J/f J/f [x]J = {y ∈ X|x ∼J y}, then it is well defined. In fact, suppose that [fx ]J/f = [fy ]J/f . Then Proof. Note that

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Jeong Soon Han and Sun Shin Ahn

fx ∼J/f fy and so fx∗y = fx • fy ∈ J/f . Hence x ∗ y ∈ J. Therefore x ∼J y, i.e., [x]J = [y]J . X/f Given [fx ]J/f , [fy ]J/f ∈ , we have J/f φ([fx ]J/f • [fy ]J/f ) =φ([fx • fy ]J/f ) =[x ∗ y]J = [x]J ∗ [y]J =φ([fx ]J/f ) ∗ φ([fy ]J/f ). Hence φ is a homomorphism. Obviously, φ is onto. Finally, we show that φ is one-to-one. If φ([fx ]J/f ) = φ([fy ]J/f ), then [x]J = [y]J , i.e., x ∼J y. If fa ∈ [fx ]J/f , then fa ∼J/f fx and hence fa∗x ∈ J/f . It follows that a ∗ x ∈ J, i.e., a ∼J x. Since ∼J is an equivalence relation, a ∼J y and so Ja = Jy . Hence a ∗ y ∈ J and so fa∗y ∈ J/f . Therefore fa ∼J/f fy . Hence fa ∈ [fy ]J/f . Thus [fx ]J/f ⊆ [fy ]J/f . Similarly, we obtain [fy ]J/f ⊆ [fx ]J/f . Therefore [fx ]J/f = [fy ]J/f . It is completes the proof. □ References [1] S. S. Ahn, N. O. Alshehri and Y. B. Jun, Int-soft filters of BE-algebras, Disctete Dynamics in Nature and Society, accepted. [2] S. S. Ahn and K. S. So, On ideals and upper sets in BE-algerbas, Sci. Math. Jpn. 68 (2008), 279–285. [3] Y. B. Jun and S. S. Ahn, Applications of soft sets in BE-algebras, Algebra, Volume 2013, Article ID 368962, 8 pages. [4] Y. B. Jun, N. O. Alshehri and S. S. Ahn, Int-soft implicative filters in BE-algebras, J. Comput. Anal. Appl. 18(3) (2015), 559–571. [5] Y. B. Jun and N. O. Alshehri, Uni-soft filters of BE-algebras, J. Comput. Anal. Appl. 21(3) (2016), 552–563. [6] H. R. Lee and S. S. Ahn, Mighty filters in BE-algebras, Honam Mathematical J. 37(2) (2015), 221–233. [7] H. S. Kim and Y. H. Kim, On BE-algerbas, Sci. Math. Jpn. 66 (1) (2007), 113–116. [8] P. K. Maji, R. Biswas and A. R. Roy, Soft set theory, Comput. Math. Appl. 45 (2003) 555–562. [9] 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. [10] D. Molodtsov, Soft set theory - First results, Comput. Math. Appl. 37 (1999) 19–31. [11] S. T. Park and S. S. Ahn, On n-fold implicative vague filters in BE-algebras, J. Korean Soc. Math. Educ. Ser. B: Pure Appl. Math. 19(2012), 127–136. [12] L. A. Zadeh, From circuit theory to system theory, Proc. Inst. Radio Eng. 50 (1962) 856–865. [13] L. A. Zadeh, Fuzzy sets, Inform. Control 8 (1965) 338–353. [14] L. A. Zadeh, Toward a generalized theory of uncertainty (GTU) - an outline, Inform. Sci. 172 (2005) 1–40.

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INCLUSION RELATIONSHIPS FOR SOME SUBCLASSES OF ANALYTIC FUNCTIONS ASSOCIATED WITH GENERALIZED BESSEL FUNCTIONS K.A. SELVAKUMARAN, H. A. AL-KHARSANI, D. BALEANU, S.D. PUROHIT, AND K.S. NISAR Abstract. This paper introduces new subclasses of analytic functions and investigate the inclusion properties of these subclasses using the generalized Bessel functions of the first kind. We also derive a variety of special cases and corollaries of the main results.

1. Introduction Let U = {z : z ∈ C and |z| < 1} be an open disk and let A be the class of functions f of the form ∞ X f (z) = z + an+1 z n+1 , (1.1) n=1

which are analytic in U and satisfy the following normalization condition: f (0) = f 0 (0) − 1 = 0. Let S subclasses of A containing all functions which are univalent, C is the close-to-convex, S ∗ (α) starlike of order α and K(α) is the convex of order α in U. For functions fj ∈ A given by ∞ X fj (z) = z + an+1,j z n+1 , (j = 1, 2) n=1

we state the Hadamard product of f1 and f2 by (f1 ∗ f2 )(z) := z +

∞ X

an+1,1 an+1,2 z n+1 ,

(z ∈ U).

n=1

f (z) is named subordinate to g (z) if ∃ w(z) analytic in U in such a way that w(0) = 0,

|w(z)| < 1 (z ∈ U) and f (z) = g(w(z))

(z ∈ U).

and we write f (z) ≺ g (z) . Let M be the class of analytic functions ϕ(z) in U with ϕ(0) = 1. We consider that S denote the subclasses of A containing all functions which are univalent, C is the close-to-convex, S ∗ (α) starlike of order α and K(α) denote the convex of order α in U. Using the subordination between analytic functions, now we 2000 Mathematics Subject Classification. 30C45, 33C10, 30C80. Key words and phrases. Analytic functions; Starlike functions; Convex functions; Close-to-convex functions; Generalized Bessel functions.

1

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define the subclasses of A for 0 ≤ α, β < 1 and ϕ, ψ ∈ N (cf., [4, 5, 9, 11]):   0   1 zf (z) ∗ S (α; ϕ) := f ∈ A : − α ≺ ϕ(z), z ∈ U , 1 − α f (z)     1 zf 00 (z) K(α; ϕ) := f ∈ A : 1+ 0 − α ≺ ϕ(z), z ∈ U 1−α f (z) and   0   1 zf (z) ∗ C(α, β; ϕ, ψ) := f ∈ A : for some g ∈ S (α; ϕ) s.t. − β ≺ ψ(z), z ∈ U . 1−β g(z) clearly f (z) ∈ K(α; ϕ) ⇐⇒ zf 0 (z) ∈ S ∗ (α; ϕ). The particular choices of ϕ and ψ yields the familier subclasses of A as:       1+z 1+z 1+z 1+z ∗ ∗ = S (α), K α; = K(α) and C 0, 0; , = C. S α; 1−z 1−z 1−z 1−z Recently, Deniz et al. [7] gave the transformation φl,b,c (z) of generalized Bessel function of first kind of order l (cf. [1]):  2k+l ∞ X (−c)k z Wl,b,c (z) = (1.2) (z ∈ C, l, b, c ∈ R). b+1 k!Γ(l + k + 2 ) 2 k=0 by  1 b+1 φl,b,c (z) = 2 Γ l + z 1−l/2 Wl,b,c (z 2 ) 2   ∞ X (−c)k z k+1 b+1 − =z+ ν =l+ 6∈ Z0 := {0, −1, −2, · · · } , 4k · (ν)k k! 2 k=1 l



where (λ)k represents the Pochhammer symbol given by (λ)k = λ(λ + 1)(λ + 2) · · · (λ + k − 1) (k ∈ N := {1, 2, 3, · · · )}

and

(λ)0 = 1

Subsequently, by using φl,b,c (z), Deniz [6] developed the operator Bνc as follows: Bνc f (z)

= φl,b,c (z) ∗ f (z) = z +

∞ X (−c)k ak+1 z k+1 k=1

4k · (ν)k

(z ∈ C).

k!

(1.3)

clearly from (1.3),
0 c c z Bν+1 f (z) = νBνc f (z) − (ν − 1)Bν+1 f (z),

(1.4)

where

b+1 6∈ Z− 0. 2 Indeed, the operator Bνc given by (1.3) provides an elementary transform of the generalized hypergeometric function,  c  Bνc f (z) = z 0 F1 ; ν; − z ∗ f (z) 4 and  c  φν,c − z = z 0 F1 ( ; ν; z). 4 ν =l+

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3

In this article, we investigate several inclusion relationships for each of the following subclasses of A, associated with Wl,b,c (z) (see also [15], [16] and [17] for inclusion relationships of various other function classes). Indeed, for c ∈ C, ν ∈ R \ Z− 0 , 0 ≤ α, β < 1 and ϕ, ψ ∈ N we define Sνc (α; ϕ) := {f ∈ A : Bνc f (z) ∈ S ∗ (α; ϕ), z ∈ U} , Kνc (α; ϕ) := {f ∈ A : Bνc f (z) ∈ K(α; ϕ), z ∈ U} Cνc (α, β; ϕ, ψ) := {f ∈ A : Bνc f (z) ∈ C(α, β; ϕ, ψ), z ∈ U} . Also, f (z) ∈ Kνc (α; ϕ) ⇐⇒ zf 0 (z) ∈ Sνc (α; ϕ). Particularly, we set   δ  1 + Az c Sν α; = Sνc (α; A, B; δ), 1 + Bz

(1.5)

(0 < δ ≤ 1; −1 ≤ B < A ≤ 1)

and Kνc



 α;

1 + Az 1 + Bz

δ 

= Kνc (α; A, B; δ),

(0 < δ ≤ 1; −1 ≤ B < A ≤ 1).

We need the following results for the investigation of our inclusion properties. Lemma 1. [8] Let φ(z) be analytic and convex univalent in U with φ(0) = 1 and Re{ηφ(z) + σ} > 0 (η, σ ∈ C). If p(z) is analytic in U with p(0) = 1, then the subordination zp0 (z) p(z) + ≺ φ(z) (z ∈ U) ηp(z) + σ implies that p(z) ≺ φ(z) (z ∈ U). Lemma 2. [12] Let h(z) be convex in U with h(0) = 1. Let Q(z) be analytic in U with Re{Q(z)} ≥ 0 (z ∈ U). If q(z) is analytic in U such that q(0) = h(0), then we have q(z) + Q(z)zq 0 (z) ≺ h(z)

(z ∈ U)

which implies that q(z) ≺ h(z)

(z ∈ U).

2. The Main Inclusion Relationships Theorem 3. Let f ∈ A, c ∈ C, ν ∈ R \ Z− 0 and α + ν > 1

(0 ≤ α < 1). Then

c (α; ϕ) f ∈ Sνc (α; ϕ) =⇒ f ∈ Sν+1

or equivalently, c Sνc (α; ϕ) ⊂ Sν+1 (α; ϕ)

(ϕ ∈ N ).

Proof. Let f ∈ Sνc (α; ϕ) and set 1 q(z) = 1−α

!
0 c z Bν+1 f (z) −α , c Bν+1 f (z)

83

(2.1)

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where q(z) is analytic in U with q(0) = 1. From (1.4) we get,
0 c f (z) z Bν+1 Bνc f (z) = + (ν − 1). ν c c f (z) Bν+1 f (z) Bν+1

(2.2)

By using (2.1) and (2.2), we have  1 Bνc f (z) = (1 − α)q(z) + α + ν − 1 . c Bν+1 f (z) ν

(2.3)

Now, by applying logarithmic differentiation on (2.3), we get
0
0 c z Bνc f (z) z Bν+1 f (z) (1 − α)zq 0 (z) = + c Bνc f (z) Bν+1 f (z) (1 − α)q(z) + α + ν − 1 in view of (2.1), yields 1 1−α

!
0 z Bνc f (z) zq 0 (z) − α = q(z) + Bνc f (z) (1 − α)q(z) + α + ν − 1

(z ∈ U).

(2.4)

c Finally, by applying Lemma 1 and (2.4), we have q(z) ≺ ϕ(z), hence f ∈ Sν+1 (α; ϕ). 

Theorem 4. Let f ∈ A, c ∈ C, ν ∈ R \ Z− 0 and α + ν > 1

(0 ≤ α < 1). Then

c f ∈ Kνc (α; ϕ) =⇒ f ∈ Kν+1 (α; ϕ)

or equivalently, c Kνc (α; ϕ) ⊂ Kν+1 (α; ϕ)

(ϕ ∈ N ).

Proof. Using (1.5) and Theorem 3, we have f (z) ∈ Kνc (α; ϕ) ⇐⇒ Bνc f (z) ∈ K(α; ϕ) ⇐⇒ z (Bνc f (z))0 ∈ S ∗ (α; ϕ) ⇐⇒ Bνc (zf 0 (z)) ∈ S ∗ (α; ϕ) ⇐⇒ zf 0 (z) ∈ Sνc (α; ϕ) c =⇒ zf 0 (z) ∈ Sν+1 (α; ϕ) c ⇐⇒ Bν+1 (zf 0 (z)) ∈ S ∗ (α; ϕ)
0 c ⇐⇒ z Bν+1 f (z) ∈ S ∗ (α; ϕ) c ⇐⇒ Bν+1 f (z) ∈ K(α; ϕ) c ⇐⇒ f (z) ∈ Kν+1 (α; ϕ).

 If we take  ϕ(z) =

1 + Az 1 + Bz

δ (−1 ≤ B < A ≤ 1; 0 < δ ≤ 1; z ∈ U)

in Theorems 3 and 4,then we obtain the corollaries as:

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Corollary 5. Let f ∈ A, c ∈ C, ν ∈ R \ Z− 0 and α + ν > 1

5

(0 ≤ α < 1). Then

c Sνc (α; A, B; δ) ⊂ Sν+1 (α; A, B; δ)

(−1 ≤ B < A ≤ 1; 0 < δ ≤ 1)

c Kνc (α; A, B; δ) ⊂ Kν+1 (α; A, B; δ)

(−1 ≤ B < A ≤ 1; 0 < δ ≤ 1).

and Theorem 6. Let f ∈ A, c ∈ C, ν ∈ R \ Z− 0 and α + ν > 1

(0 ≤ α < 1). Then

c f ∈ Cνc (α, β; ϕ, ψ) =⇒ f ∈ Cν+1 (α, β; ϕ, ψ)

or equivalently, c Cνc (α, β; ϕ, ψ) ⊂ Cν+1 (α, β; ϕ, ψ)

(0 ≤ α, β < 1; ϕ, ψ ∈ N ).

Proof. Let f ∈ Cνc (α, β; ϕ, ψ). Then ∃ a function g ∈ Sνc (α; ϕ) 3 !
0 z Bνc f (z) 1 − β ≺ ψ(z), (0 ≤ β < 1, z ∈ U). 1−β Bνc g(z) Now let 1 ω(z) = 1−β

!
0 c f (z) z Bν+1 −β , c Bν+1 g(z)

(2.5)

where ω(z) is analytic in U with ω(0) = 1. Making use of (1.4) we also have

0 Bνc zf 0 (z) z Bνc f (z) = Bνc g(z) Bνc g(z)
0 c c z Bν+1 (zf 0 (z)) + (ν − 1)Bν+1 (zf 0 (z)) =
0 c c z Bν+1 g(z) + (ν − 1)Bν+1 g(z) #−1 # " "
0
0 c c c g(z) z Bν+1 z Bν+1 (zf 0 (z)) Bν+1 (zf 0 (z)) + (ν − 1) +ν−1 = · c c c Bν+1 g(z) Bν+1 g(z) Bν+1 g(z) (2.6) By Theorem 3, c g ∈ Sνc (α; ϕ) =⇒ g ∈ Sν+1 (α; ϕ),

therefore, we set 1 ϑ(z) = 1−α

!
0 c z Bν+1 g(z) −α , c Bν+1 g(z)

(2.7)


where < ϑ(z) > 0 (z ∈ U). Applying (2.5) and (2.7) into (2.6) we have h
0 i  c −1
0 c 0 c z Bν+1 (zf (z)) · Bν+1 g(z) + (ν − 1)[(1 − β)ω(z) + β] z Bν f (z) = Bνc g(z) (1 − α)ϑ(z) + α + ν − 1

(2.8)

The logarithmic differentiation of (2.5) gives
0 c z Bν+1 (zf 0 (z)) = (1 − β)zω 0 (z) + [(1 − α)ϑ(z) + α] · [(1 − β)ω(z) + β], c Bν+1 g(z)

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which, in conjunction with (2.8), yields !
0 c z B f (z) 1 zω 0 (z) ν − β = ω(z) + . 1−β Bνc g(z) (1 − α)ϑ(z) + α + ν − 1 Since α + ν > 1 and ϑ(z) ≺ ϕ(z) in U,  < (1 − α)ϑ(z) + α + ν − 1 > 0

(z ∈ U).

c Finally, by applying Lemma 2, we have ω(z) ≺ ψ(z), so that f ∈ Cν+1 (α, β; ϕ, ψ).



3. Inclusion Relationships Involving the Integral Operator Fσ The generalized Bernardi-Libera-Livingston integral operator Fσ (σ > −1) (cf. [3, 10, 13]) considered here and is defined by Z σ + 1 z σ−1 Fσ (f ) := Fσ (f )(z) = t f (t)dt (f ∈ A; σ > −1). (3.1) zσ 0 c Theorem 7. Let f (z) ∈ A, c ∈ C, ν ∈ R \ Z− 0 and σ ≥ 0. If f ∈ Sν (α; ϕ) c 1; ϕ ∈ N ), then Fσ (f ) ∈ Sν (α; ϕ) (0 ≤ α < 1; ϕ ∈ N ).

(0 ≤ α
0

(0 ≤ α < 1). Then

1 1 f ∈ Sp+1 (α; ϕ) =⇒ f ∈ Sp+2 (α; ϕ)

(ϕ ∈ N )

or equivalently, 1 Jp f (z) ∈ S ∗ (α; ϕ) then f (z) ∈ Sp+n (α; ϕ)

Corollary 13. Let f ∈ A, p ∈ R \ Z− and α + p > 0

(n ∈ N \ {1}; ϕ ∈ N ). (0 ≤ α < 1). Then

1 1 (α; ϕ) f ∈ Kp+1 (α; ϕ) =⇒ f ∈ Kp+2

(ϕ ∈ N )

or equivalently, 1 (α; ϕ) Jp f (z) ∈ K(α; ϕ) then f (z) ∈ Kp+n

88

(n ∈ N \ {1}; ϕ ∈ N ).

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Corollary 14. Let f ∈ A, p ∈ R \ Z− and α + p > 0

9

(0 ≤ α < 1). Then

1 1 f ∈ Cp+1 (α, β; ϕ, ψ) =⇒ f ∈ Cp+2 (α, β; ϕ, ψ)

(0 ≤ β < 1; ϕ, ψ ∈ N )

or equivalently, 1 Jp f (z) ∈ C(α, β; ϕ, ψ) then f (z) ∈ Cp+n (α, β; ϕ, ψ)

(n ∈ N \{1}; 0 ≤ β < 1; ϕ, ψ ∈ N ).

Finally, we remark that similar results can be obtained involving the operators Ip and Qp by specializing the parameter in Theorems 3-6. We also remark that several other applications and corollaries of our main results (Theorems 3-6) can indeed be derived similarly. Acknowledgment The research of H.A. Al-kharsani, and K.S. Nisar was supported by Deanship of Scientific Research in University of Dammam, project ID#2014219. References ´ Baricz, Generalized Bessel Functions of the First Kind, Lecture Notes in Mathematics, Vol. 1994, [1] A. Springer-Verlag, Berlin, Heidelberg and New York, 2010. ´ Baricz E. Deniz, Murat Caglar, Halit Orhan, Differential subordinations involving generalized [2] A. Bessel functions, Bull. Malays. Math. Sci. Soc. 38 (2015), no. 3, 1255–1280. [3] S. D. Bernardi, Convex and starlike univalent functions, Trans. Amer. Math. Soc. 135 (1969), 429– 446. [4] N. E. Cho and J. A. Kim, Inclusion properties of certain subclasses of analytic functions defined by a multiplier transformation, Comput. Math. Appl. 52 (2006), no. 3-4, 323–330. [5] J. H. Choi, M. Saigo and H. M. Srivastava, Some inclusion properties of a certain family of integral operators, J. Math. Anal. Appl. 276 (2002), no. 1, 432–445. [6] E. Deniz, Differential subordination and superordination results for an operator associated with the generalized Bessel function [arXiv:1204.0698v1 [math.CV]]. [7] E. Deniz, H. Orhan and H. M. Srivastava, Some sufficient conditions for univalence of certain families of integral operators involving generalized Bessel functions, Taiwanese J. Math. 15 (2011), 883–917. [8] P. Enigenberg, S.S. Miller, P.T. Mocanu and M.O. Reade, On a Briot-Bouquet differential subordination, In General Inequalities, Volume 3, pp. 339-348, Birkhiiuser Verlag-Basel, (1983). [9] Y. C. Kim, J. H. Choi and T. Sugawa, Coefficient bounds and convolution properties for certain classes of close-to-convex functions, Proc. Japan Acad. Ser. A Math. Sci. 76 (2000), no. 6, 95–98. [10] R. J. Libera, Some classes of regular univalent functions, Proc. Amer. Math. Soc. 16 (1965), 755–758. [11] W. Ma and D. Minda, An internal geometric characterization of strongly starlike functions, Ann. Univ. Mariae Curie-Sklodowska Sect. A 45 (1991), 89–97 (1992). [12] S. S. Miller and P. T. Mocanu, Differential subordinations and univalent functions, Michigan Math. J. 28 (1981), no. 2, 157–172. [13] S. Owa and H. M. Srivastava, Some applications of the generalized Libera integral operator, Proc. Japan Acad. Ser. A Math. Sci. 62 (1986), no. 4, 125–128. [14] J. K. Prajapat, Certain geometric properties of normalized Bessel functions, Appl. Math. Lett. 24 (2011), 2133–2139. [15] C. Selvaraj and K. A. Selvakumaran, On certain classes of multivalent functions involving a generalized differential operator, Bull. Korean Math. Soc. 46 (2009), no. 5, 905–915. [16] H. M. Srivastava, M. K. Aouf and R. M. El-Ashwah, Some inclusion relationships associated with a certain class of integral operators, Asian-European J. Math. 3 (2010), 667-684. [17] H. M. Srivastava, S. M. Khairnar and M. More, Inclusion properties of a subclass of analytic functions defined by an integral operator involving the Gauss hypergeometric function, Appl. Math. Comput. 218 (2011), 3810-3821. [18] G. N. Watson, A Treatise on the Theory of Bessel Functions, Second edition, Cambridge University Press, Cambridge, London and New York, 1944.

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K.A. Selvakumaran: Department of Mathematics, R.M.K College of Engg. and Technology, Puduvoyal - 601206, Tamil Nadu, India. E-mail address: [email protected] Huda A. Al-Kharsani: Department of mathematics, Girls College, 838, Dammam University, Saudi Arabia. E-mail address: [email protected] D. Baleanu: Department of Mathematics and Computer Sciences, Faculty of Arts and Sciences, Cankaya University-06530, Ankara, Turkey. Institute of Space Sciences, Magurele-Bucharest, Romania. E-mail address: [email protected] S.D. Purohit: Department of HEAS (Mathematics), Rajasthan Technical University, Kota-324010, Rajasthan, India E-mail address: sunil a [email protected] K.S. Nisar: Department of Mathematics, College of Arts and Science, Prince Sattam bin Abdulaziz University, Wadi Al-Dawaser, Saudi Arabia. E-mail address: [email protected]

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Fixed point properties of Suzuki generalized nonexpansive set-valued mappings in complete CAT(0) spaces I Jing Zhou∗,a , Yunan Cuib a

Department of Mathematics, Harbin Institute of Technology, Harbin 150080, P.R. China b Department of Mathematics, Harbin University of Science and Technology, Harbin 150080, P.R. China

Abstract In this paper, we introduce the notion of Suzuki generalized nonexpansive setvalued mappings from traditional Banach spaces to CAT(0) spaces. Moreover we discuss fixed point properties including the existence, 4−convergence and strong convergence for this kind of mappings in complete CAT(0) spaces. Our results extend the results of B. Nanjaras [12] for the Suzuki generalized nonexpansive single-valued mappings. Key words: Suzuki generalized nonexpansive set-valued mapping, CAT(0) space, Existence theorem, Convergence theorem, Fixed point 2010 MSC: 47H09, 47H10, 54E40

I

NFS of Hei Longjiang Provience (A2015018) Corresponding author Email addresses: [email protected] (Jing Zhou), [email protected] (Yunan Cui) ∗

Preprint submitted to Journal of Computational Analysis and ApplicationsApril 25, 2016

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1. Introduction In recent years, more and more classical fixed point theorems for singlevalued nonexpansive mappings are extended by set-valued nonexpansive mappings. As a result, fixed point theory for set-valued nonexpansive mappings get rapid development[see [6–10]]. In 2008, Suzuki [11] brought in a new kind of mappings which was called satisfying the condition (C) in Banach spaces. In 2010, B. Nanjaras, B. Panyanak and W. Phuengrattana [12] established fixed point theorems for the mappings satisfying the condition (C) in CAT(0) spaces. In 2011, Abkar and Eslamian [14] gave the definition of set-valued mappings for the condition (C) and then they proved the existence of fixed point in uniformly convex Banach spaces [13]. Let (X, d) be a metric space. A geodesic path joining x ∈ X to x0 ∈ X is a map c from a closed interval [0, h] ⊂ R to X such that c(0) = x, c(h) = x0 and d(c(l), c(l0 )) = |l − l0 |, for all l, l0 ∈ [0, h]. In particular, c is an isometry and d(x, x0 ) = h. The image γ of c is called a geodesic (or metric) segment joining x and x0 denoted by [x, x0 ] whenever it is unique. The space (X, d) is said to be a geodesic space if every two points of X are joined by a geodesic, and X is said to be uniquely geodesic if there is exactly one geodesic joining x and x0 for each x, x0 ∈ X. A geodesic space is said to be a CAT(0) space if the following CAT(0) inequality 2  1 1 1 x1 ⊕ x2 ≤ d(z, x1 )2 + d(z, x2 )2 − d(x1 , x2 )2 d z, 2 2 2 4 satisfies for all x1 , x2 , z ∈ X. This is the (CN) inequality of Bruhat and Tits [1](More details spaces see [2]). Lemma 1.1. Let (X, d) be a CAT(0) space. (i)[3, Lemma 2.1(iv)] For each x1 , x2 ∈ X and α ∈ [0, 1], there exists a unique point y ∈ [x1 , x2 ] such that d(x1 , y) = αd(x1 , x2 ),

d(x2 , y) = (1 − α)d(x1 , x2 ).

Denote y = (1 − α)x1 ⊕ αx2 in the above equations conveniently. (ii)[3, Lemma 2.4] For each x1 , x2 , y ∈ X and α ∈ [0, 1]. We have d((1 − α)x1 ⊕ αx2 , y) ≤ (1 − α)d(x1 , y) + αd(x2 , y).

(1.1)

2

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Obviously, Lemma 1.1. shows the convexness of CAT(0) spaces. Lemma 1.2. ([4, Lemma 2.5]) Let (X, d) be a CAT(0) space. Then
2 d (1 − t)x ⊕ ty, z ≤ (1 − t)d(x, z)2 + t d(y, z)2 − t(1 − t)d(x, y)2 for all t ∈ [0, 1] and x, y, z ∈ X. Let {xn } be a bounded sequence in a CAT(0) space X. For z ∈ X, we set r(z, {xn }) = lim sup d(z, xn ). n→∞

The asymptotic radius r({xn }) of {xn } is given by r({xn }) = inf{r(z, {xn }) : z ∈ X}. The asymptotic radius rD ({xn }) of {xn } with respect to D ⊂ X is given by rD ({xn }) = inf {r (z, {xn }) : z ∈ D} . The asymptotic center A({xn }) of {xn } is the set A({xn }) = {z ∈ X : r(z, {xn }) = r({xn })}. And the asymptotic center AD ({xn }) of {xn } with respect to D ⊂ X is the set AD ({xn }) = {z ∈ D : r(z, {xn }) = r({xn })}. It follows from [5, Proposition 7]) that A({xn }) consists of exactly one point in a CAT(0) space. In 1976, Lim [1] introduced the concept of ∆− convergence in a general metric space. In 2008, Kirk and Panyanak [4] brought in ∆−convergence to CAT(0) spaces and proved that there is an analogy between ∆−convergence and weak convergence. Definition 1.3. ([2]) A sequence {xn } in a CAT(0) space X is said to ∆converge to x ∈ X if x is the unique asymptotic center of {un } for every subsequence {un } of {xn }. In this case, we write ∆ − limn→∞ xn = x and call x the ∆-limit of {xn }. Lemma 1.4. ([2]) If D is a closed convex subset of a complete CAT(0) space and if {xn } is a bounded sequence in D, then the asymptotic centerasymptotic center of {xn } is in D. 3

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Lemma 1.5. ([2]) Every bounded sequence in a complete CAT(0) space always has a ∆-convergent subsequence. Lemma 1.6. ([4]) If {xn } is a bounded sequence in a complete CAT(0) space with A({xn }) = {p}, {un } is a subsequence of {xn } with A({un }) = {u}, and the sequence {d(xn , u)} converges, then p = u. The purpose of this paper is to bring in the concept of Suzuki generalized nonexpansive set-valued mappings in CAT(0) spaces. Then we shall prove a common fixed point theorem for commuting pairs consisting of a singlevalued and a set-valued mapping both satisfying the condition (C) which is analogous to the results in Banach spaces [13]. Furthermore, we also establish 4−convergence and strong convergence of Mann iteration in CAT(0) spaces. 2. Preliminaries Let D be a nonempty subset of a CAT(0) space X. We denote by B(D) the collection of all nonempty bounded closed subsets of D and C(D) the collection of all nonempty compact subsets of D. Suppose H is the Hausdorff metric with respect to d, that is,   H (U, V ) := max sup dist (u, V ) , sup dist (v, U ) , U, V ∈ B (X) u∈U

v∈V

where dist (u, V ) = inf v∈V d (u, v) is the distance from the point u to the set V. Let T : X → 2X be a set-valued mapping. If an element x ∈ X satisfies x ∈ T x, then x is called a fixed point of T . The set of fixed points of T is denoted by F ix(T ). Definition 2.1. A set-valued mapping T : X → B (X) is (i) nonexpansive provided H (T x, T y) ≤ d (x, y) ,

x, y ∈ X;

(ii) quasi nonexpansive if F ix (T ) 6= ∅ and H (T x, p) ≤ d (x, p) ,

x ∈ X,

for all p ∈ F ix (T ). 4

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Definition 2.2. A set-valued mapping T : X → B (X) is called to satisfy the condition (C) if 1 dist (x, T x) ≤ d (x, y) 2

implies H (T x, T y) ≤ d (x, y)

for all x, y ∈ X. This kind of set-valued mappings satisfying the condition (C) can be called Suzuki generalized nonexpansive as well. Proposition 2.3. Suppose a set-valued mapping T : X → B (X) satisfies the condition (C) with the nonempty fixed point set. Then T is a quasinonexpansive set-valued mapping. Theorem 2.4. ([12, Theorem 4.1]) Suppose D is a nonempty bounded closed convex subset of a complete CAT(0) space X and t : D → D is a mapping which satisfies the condition (C). Then F (t) is nonempty in D. Corollary 2.5. ([12, Corollary 4.2]) Suppose D is a nonempty bounded closed convex subset of a complete CAT(0) space X and t : D → D is a mapping which satisfies the condition (C). Then F (t) is nonempty closed, convex and hence contractible. Definition 2.6. Suppose D is a nonempty bounded closed convex subset of a CAT(0) space X. Let t : D → D and T : D → B(D) be a single-valued mapping and a set-valued mapping respectively. Then t and T are said to be commuting mappings if for every x, y ∈ D such that x ∈ T y and ty ∈ D, we have tx ∈ T ty. Definition 2.7. A set-valued mapping T : X → B (X) is said to satisfy the condition (Eµ ) provided that dist (x, T y) ≤ µdist (x, T x) + d (x, y) ,

x, y ∈ X.

We say that T satisfies the condition (E) whenever T satisfies (Eµ ) for some µ ≥ 1. Proposition 2.8. Let T : X → B (X) be a set-valued mapping which satisfies the condition (C). Then T satisfies the condition (E3 ). 5

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proof. Given x ∈ X. Since for any z ∈ T x, dist(x, T x) ≤ d(x, z)

(2.1)

then

1 1 dist(x, T x) ≤ d(x, z) ≤ d(x, z). 2 2 From our assumption, we have H(T x, T z) ≤ d(x, z)

(2.2)

for any x ∈ X and z ∈ T x. Then for x, y ∈ X and z ∈ T x either 1 dist(x, T x) ≤ d(x, y) or 21 H(T x, T z) ≤ d(y, z) holds. On the contrary, 2 together with (2.1) and (2.2), we get d(x, z) ≤ d(x, y) + d(y, z) 1 1 dist(x, T x) + H(T x, T z) < 2 2 1 1 d(x, z) + d(x, z) ≤ 2 2 = d(x, z) which is a contradiction. Case I. 12 dist(x, T x) ≤ d(x, y)holds. From the above hypothesis, we can obtain H(T x, T y) ≤ d(x, y) by the condition (C). Hence, dist(x, T y) ≤ dist(x, T x) + H(T x, T y) ≤ dist(x, T x) + d(x, y) ≤ 3dist(x, T x) + d(x, y). Case II.

1 H(T x, T z) 2

≤ d(y, z)holds. Then,

1 1 dist(z, T z) ≤ sup dist (u, T z) 2 2 u∈T x 1 ≤ max{ sup dist (u, T z) , sup dist (v, T x)} 2 u∈T x v∈T z 1 = H(T x, T z) 2 ≤ d(y, z). 6

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From the assumption, we have H(T y, T z) ≤ d(y, z).

(2.3)

Together with (2.1), (2.2) and (2.3), we can obtain dist(x, T y) ≤ ≤ ≤ ≤ =

dist(x, T x) + H(T x, T y) dist(x, T x) + H(T x, T z) + H(T z, T y) d(x, z) + d(x, z) + d(y, z) 2d(x, z) + d(x, y) + d(x, z) 3d(x, z) + d(x, y).

Because this is applied for any z ∈ T x, we get dist(x, T y) ≤ 3dist(x, T x) + d(x, y). By overall consideration, T satisfies the condition (E3 ).



Lemma 2.9. ([12, Lemma 2.5]) Let {xn }, {ynP } be bounded sequences in a CAT(0) space X and let {αn } ∈ [0, 1) such that ∞ n=1 αn = ∞ and lim supn αn < 1. Suppose that xn+1 = αn yn ⊕ (1 − αn ) xn and d (yn+1 , yn ) ≤ d (xn+1 , xn ) for all n ∈ N . Then lim d(xn , yn ) = 0. n→∞

3. Fixed point properties of Suzuki generalized nonexpansive setvalued mappings In this section, we denote by D a nonempty bounded closed convex subset of a complete CAT(0) space X. Firstly, we shall discuss the existence of a common fixed point. Theorem 3.1. Let t : D → D and T : D → C(D) be a single-valued mapping and a set-valued mapping respectively. If both t and T satisfy the condition (C) and in the meantime, they are commuting, then they have a common fixed point, that is, there exists a point z ∈ D such that z = tz ∈ T z. proof. By Theorem 2.4 and Corollary 2.5, we know that the mapping t has a fixed point set F ix(t) which is a nonempty closed convex subset of X. Let p ∈ F ix(t). As t and T are commuting, we have tq ∈ T tp = T p for each 7

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q ∈ T p. Therefore, T p is invariant under t for each p ∈ F ix(t). Since T p is a bounded closed T convex subset of X, we can obtain that t has a fixed point in T p. Hence, T p F ix(t) 6= ∅ for p ∈TF ix(t). Take T x1 ∈ F ix(t). Because T x1 F ix(t) 6= ∅, we are able to select y1 ∈ T x1 F ix(t). Define x2 = 12 (x1 + y1 ). Due to the fact that F ix(t) is a convex set, we have x2 ∈ F ix(t). Let y2 ∈ T x2 be selected as follows d(y1 , y2 ) = dist(y1 , T x2 ). Since 21 d(y1 , ty1 ) = 0 ≤ d(y1 , y2 ), from the assumption, we know d(y1 , ty2 ) = d(ty1 , ty2 ) ≤ d(y1 , y2 ). Note that ty2 ∈ T x2 , hence, there comes a contradiction. Therefore, y2 ∈ F ix(t). In such a way, we can find a sequence {xn } in F ix(t) such that 1 xn+1 = (xn + yn ), 2

n≥1

T where yn ∈ T xn F ix(t) and d(yn−1 , yn ) = dist(yn−1 , T xn ). Therefore, by Lemma 1.1.(i) we get 1 d(xn , yn ) = d(xn , xn+1 ) 2 from which we can conclude 1 1 dist(xn , T xn ) ≤ d(xn , yn ) = d(xn , xn+1 ), 2 2

n ≥ 1.

Hence, H(T xn , T xn+1 ) ≤ d(xn , xn+1 ),

n≥1

which implies d(yn , yn+1 ) = dist(yn , T xn+1 ) ≤ H(T xn , T xn+1 ) ≤ d(xn , xn+1 ). We now apply Lemma 2.9. to obtain that lim d(xn , yn ) = 0

n→∞

where yn ∈ T xn , i.e., we find an approximate fixed point sequence for T in F ix(t). 8

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Assume that z = A({xn }). By Lemma 1.4. we know that z ∈ F ix(t). For each n ≥ 1, we choose zn ∈ T z such that d(yn , zn ) = dist(yn , T z). Because limn→∞ d(xn , yn ) = 0, there exists n0 such that d(xn , yn ) ≤ d(xn , z),

n ≥ n0 .

This implies that 1 dist(xn , T xn ) ≤ d(xn , z), 2 and hence, H(T xn , T z) ≤ d(xn , z),

n ≥ n0 .

Therefore, d(yn , zn ) ≤ H(T xn , T z) ≤ d(xn , z),

n ≥ n0 .

As 12 d(yn , T yn ) = 0 ≤ d(yn , zn ) for each n ≥ 1, we get d(yn , tzn ) = d(tyn , tzn ) ≤ d(yn , zn ). Since z ∈ F ix(t) and zn ∈ T z, by the fact that the mappings t and T are commuting, we can obtain that tzn ∈ T tzn = T z. Now by the uniqueness of zn as the nearest point to yn , we get tzn = zn ∈ F ix(t). As T z is compact, the sequence {zn } has a convergent subsequence {znk } with lim znk = z ∗ ∈ T z. Because znk ∈ F ix(t) for all n, and F ix(t) is closed, k→∞

we can obtain that z ∗ ∈ F ix(t). Take the subsequence {xnk } of {xn } and {ynk } of {yn }, we have d(xnk , z ∗ ) ≤ d(xnk , ynk ) + d(ynk , znk ) + d(znk , z ∗ )

(3.1)

and for all nk ≥ n0 , d(ynk , znk ) ≤ d(xnk , z). Letting k → ∞ and taking superior limit on the both sides of (3.1), we have lim sup d(xnk , z ∗ ) ≤ lim sup d(xnk , z).

k→∞

k→∞

By the uniqueness of asymptotic centers, this shows z = z ∗ ∈ T z and hence z = tz ∈ T z.  9

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Corollary 3.2. Let t : D → D and T : D → C(D) be a single-valued mapping and a set-valued mapping respectively. Assume that t and T are commuting mappings. Then there exists a point z ∈ D such that z = tz ∈ T z. Corollary 3.3. Suppose T : D → C(D) is set-valued mapping which satisfies the condition (C). Then T has a fixed point. We will prove ∆−convergence and strong convergence theorems in the following. Before that, we discuss some Lemmas which will be used in the main proofs. Lemma 3.4. Suppose T : D → C (D) is a set-valued mapping which satisfies the condition (C). Define a sequence {xn } by x1 ∈ D and   1 xn+1 = (1 − αn ) xn ⊕ αn yn , αn ⊂ , 1 , n ≥ 1 2 where yn ∈ T xn such that d (yn−1 , yn ) = dist (yn−1 , T xn ). Then lim dist (T xn , xn ) = 0.

n→∞

proof. It follows from Lemma 1.1.(i) that for every natural number n ≥ 1 1 dist (xn , T xn ) ≤ αn dist (xn , T xn ) ≤ αn d (xn , yn ) = d (xn , xn+1 ) 2 From our assumption, we get H (T xn , T xn+1 ) ≤ d (xn , xn+1 ) ,

n ≥ 1.

Hence d (yn , yn+1 ) = dist (yn , T xn+1 ) ≤ H (T xn , T xn+1 ) ≤ d (xn , xn+1 ) ,

n ≥ 1.

We can conclude that limn→∞ d (xn , yn ) = 0 where yn ∈ T xn by Lemma 2.9., i.e., limn→∞ dist (T xn , xn ) ≤ limn→∞ d (xn , yn ) = 0  Lemma 3.5. Suppose T : D → C (D) is a set-valued mapping which satisfies the condition (C). Define a sequence {xn } as in Lemma 3.4. Then limn→∞ d (xn , p) exists for all p ∈ F ix (T ). 10

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proof. By Theorem 3.1. F (T ) is nonempty. Take p ∈ F ix(t), by Lemma 1.2. and Proposition 2.3. we have d2 (xn+1 , p) ≤ ≤ ≤ =

(1 − αn )d2 (xn , p) + αn d2 (yn , p) − αn (1 − αn )d2 (xn , yn ) (1 − αn )d2 (xn , p) + αn H 2 (T xn , T p) − αn (1 − αn )d2 (xn , yn ) (1 − αn )d2 (xn , p) + αn d2 (xn , p) − αn (1 − αn )d2 (xn , yn ) d2 (xn , p) − αn (1 − αn )d2 (xn , yn ).

This entails d2 (xn+1 , p) ≤ d2 (xn , p) . Therefore, d(xn+1 , p) ≤ d(xn , p) for all n ≥ 1 which implies {d(xn , p)}∞ n=1 is bounded and decreasing. Hence, limn→∞ d (xn , p) exists for each p ∈ F ix(T ).  Lemma 3.6. Let T : D → C(D) be a set-valued mapping which satisfies the condition (C). If {xn } is a sequence in D such that dist (T xn , xn ) → 0 and ∆− converges to some ω ∈ X. Then ω ∈ D and ω ∈ T ω. proof. We first note that ω ∈ D by Lemma 1.4. For each n ≥ 1, we select ωn ∈ T ω such that d(xn , ωn ) = dist(xn , T ω). By the compactness of T ω, there exists a subsequence {ωnk } of {ωn } such that {ωnk } → ω ∗ ∈ T ω. Taking the subsequence {xnk } of {xn }, it follows from Proposition 2.8. that dist(xnk , T ω) ≤ 3dist(xnk , T xnk ) + d(xnk , ω). Note that d(xnk , ω ∗ ) ≤ d(xnk , ωnk ) + d(ωnk , ω ∗ ) ≤ 3dist(xnk , T xnk ) + d(xnk , ω) + d(ωnk , ω ∗ ). Letting k → ∞ and taking superior limit on the both sides of the above inequation, we have lim sup d(xnk , ω ∗ ) ≤ lim sup d(xnk , ω).

k→∞

k→∞

By the uniqueness of asymptotic centers, we have ω = ω ∗ ∈ T ω.



Theorem 3.7. Let T : D → C(D) be a set-valued mapping which satisfies the condition (C). Define a sequence {xn } as in Lemma 3.4. Then {xn } ∆-converges to a fixed point of T . 11

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proof. By Lemma 3.4., limn→∞ dist (T xn , xn ) = 0. Now we prove that Wω (xn ) :=

∪

{µn }⊂{xn }

A({µn }) ⊂ F ix (T )

(3.2)

and Wω (xn ) consists of exactly one point. In fact, let µ ∈ Wω (xn ), then there exists a sequence {µn } of {xn } such that A({µn }) = {µ}. By Lemma 1.4. and 1.5., there exists a subsequence {νn } of {µn } such that ∆−limn→∞ νn = ν ∈ D. Since limn→∞ dist (T νn , νn ) = 0, then ν ∈ F ix (T ) by Lemma 3.6. and limn→∞ d (xn , ν) exists by Lemma 3.5. By Lemma 1.6. µ = ν. This implies that Wω (xn ) ⊂ F ix (T ). Next we prove that Wω (xn ) consists of exactly one point. Let {µn } be a subsequence of {xn } with A({µn }) = {µ} and let A({xn }) = {x}. Since µ ∈ Wω (xn ) ⊂ F ix (T ), from Lemma 3.5. we know that {d (xn , µ)} is convergent. In view of Lemma 1.6, x = µ.  Finally, we shall give the strong convergence for Suzuki generalized nonexpansive set-valued mappings in complete CAT(0) spaces. Theorem 3.8. Suppose D is a nonempty compact convex subset of a complete CAT(0) space and T : D → C(D) is a set-valued mapping which satisfies the condition (C). Define a sequence {xn } as in Lemma 3.4. Then {xn } converges strongly to a fixed point of T . proof. By Lemma 3.4., limn→∞ dist (T xn , xn ) = 0. Since D is compact, there exists a subsequence {xnk } of {xn } such that xnk → z for some z ∈ D. By Proposition 2.8., we have dist(z, T z) ≤ d(z, xnk ) + dist(xnk , T z) ≤ d(z, xnk ) + 3dist(xnk , T xnk ) + d(xnk , z) = 2d(z, xnk ) + 3dist(xnk , T xnk ). Letting k → ∞, we have z ∈ F ix(T ). Since {xnk } converges strongly to z and limn→∞ d (xn , z) exists, we can conclude that {xnk } converges strongly to z.  Acknowledgment The paper is supported by NFS of HeiLongjiang Provience(A2015018). 12

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References [1] T. C. Lim, Remarks on some fixed point theorems, Proc. Amer. Math. Soc. (1976) 60: 179–182 [2] W. A. Kirk and B. Panyanak, A Concept of Convergence in Geodesic Spaces, Nonlinear Anal. (2008) 68: 3689–3696 [3] F. Bruhat and J. Tits, Groupes r´eductifs sur un corps local, I. Dom´ees ´ radicielles valu´ees. Inst. Hautes Etudes Sci. Publ. Math. (1972) 41: 5251 [4] S. Dhompongsa and B. Panyanak, On ∆-convergence Theorems in CAT(0) Spaces, Comput. Math. Appl. (2008) 56: 2572–2579 [5] S. Dhompongsa, W. A. Kirk and B. Sims, Fixed Points of Uniformly Lipschitzian Mappings, Nonlinear Anal. (2006) 65(4): 762–772 [6] S. Dhompongsa, W. A. Kirk and B. Sims, Some Fixed Point Property for Multivalued Nonexpansive Mappings in Banach Spaces, J. Math. Inequal. (2013) 7(1): 129-137 [7] S. Dhompongsa, W. A. Kirk and B. Sims, Common Fixed Points of a Finite Family of Multivalued Quasi-Nonexpansive Mappings in Uniformly Convex Banach Spaces, Bull. Iran Math. Soc. (2013) 39(6): 1125-1135 [8] S. H. Khan and M. Abbas, Common Fixed Points of Two Multivalued Nonexpansive Maps in Kohlenbach Hyperbolic Spaces, Fixed Point Theory Appl. (2014) (DOI: 10.1186/1687-1812-2014-181) [9] J. Markin and N. Shahzad, Common Fixed Points for Commuting Mappings in Hyperconvex Spaces, Topology Appl. (2015) 180: 181-185 [10] L. H. Peng, C. Li and J. C. Yao, Porosity Results on Fixed Points for Nonexpansive Set-valued Maps in Hyperbolic Spaces, J. Math. Anal. Appl. (2015) 428(2): 989–1004 [11] T. Suzuki, Fixed point theorems and convergence theorems for some generalized nonexpansive mapping, Math. Anal. Appl. (2008) 340: 1088– 1095

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[12] B. Nanjaras, B. Panyanak and W. Phuengrattana, Fixed point theorems and convergence theorems for Suzuki-generalized nonexpansive mappings in CAT(0) spaces, Nonlinear Anal.-Hybrid Syst. (2010) 4: 25–31 [13] A. Abkar and M. Eslamian, Fixed Point Theorems for Suzuki Generalized Nonexpansive Multivalued Mappings in Banach Spaces, Fixed Point Theory Appl. (2010) (Article ID: 457935) [14] A. Abkar and M. Eslamian, A fixed point theorem for generalized nonexpansive multivalued mapping, Fixed Point Theory (2011) 12(2): 241– 246

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WEIGHTED COMPOSITION FOLLOWED AND PROCEEDED BY DIFFERENTIATION OPERATORS FROM ZYGMUND SPACES TO BERS-TYPE SPACES JIANREN LONG AND CONGLI YANG Abstract. In this paper, we investigate boundedness and compactness of the weighted composition followed and proceeded by differentiation operators from Zygmund spaces to Bers-type spaces and little Bers-type spaces. Some sufficient and necessary conditions for the boundedness and compactness of these operators are obtained.

1. Introduction Let ∆ = {z : |z| < 1} be the open unit disc in the complex plane C, and let H(∆) be the class of all analytic functions on ∆. Assume that µ is a positive continuous function on [0, 1), having the property that there exist positive numbers s and t, 0 < s < t, and δ ∈ [0, 1), such that µ(r) µ(r) is decreasing on [δ, 1), lim = 0, s r→1 (1 − r)s (1 − r) µ(r) µ(r) is increasing on [δ, 1), lim = ∞. t r→1 (1 − r)t (1 − r) Then µ is called a normal function (see [6], [18]). An analytic function f on ∆ is said to belong to the Bers-type space, denoted by Hµ∞ , if ∥f ∥Hµ∞ = sup µ(|z|)|f (z)| < ∞, z∈∆

∞ and it is said to belong to the little Bers-type space Hµ,0 if

lim µ(|z|)|f (z)| = 0.

|z|→1

∞ are Banach spaces with the norm It is clear that both Hµ∞ and Hµ,0 ∞ ∥ · ∥Hµ∞ , and Hµ,0 is a closed subspace of Hµ∞ . When µ ≡ 1, the space Hµ∞ is just H ∞ , which is defined by

H ∞ = {f ∈ H(∆) : ∥f ∥∞ = sup |f (z)| < ∞}. z∈∆

2010 Mathematics Subject Classification. Primary 47B38; Secondary 30H05. Key words and phrases. Zygmund spaces, Bers-type spaces, Weighted composition followed and proceeded by differentiation operators, Boundedness, Compactness. 1

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See [25] for more about Bers-type space. An f in H(∆) is said to belong to the Zygmund space, denoted by Z, if |f (ei(θ+h) ) + f (ei(θ−h) ) − 2f (eiθ )| < ∞, h where the supremum is taken over all eiθ ∈ ∂∆ and h > 0. By Theorem 5.3 in [2], we see that f ∈ Z if and only if sup

∥f ∥Z = |f (0)| + |f ′ (0)| + sup(1 − |z|2 )|f ′′ (z)| < ∞. z∈∆

It is easy to check that Z is a Banach space under the above norm. For every f ∈ Z, by using a result in [9], we have that e |f ′ (z)| ≤ C∥f ∥Z ln . 1 − |z|2 Let Z0 denote the subspace of Z consisting of those f ∈ Z for which lim (1 − |z|2 )|f ′′ (z)| = 0.

|z|→1

The space Z0 is called the little Zygmund space. Let φ be a nonconstant analytic self-map of ∆, and let ϕ be an analytic function in ∆. For f ∈ H(∆), we define the linear operators ϕCφ Df = ϕ(f ′ oφ) = ϕf ′ (φ) and ϕDCφ f = ϕ(f oφ)′ = ϕf ′ (φ)φ′ . They are called weighted composition followed and proceeded by differentiation operators respectively, where Cφ and D are composition and differentiation operators respectively. Associated with φ is the composition operator Cφ f = f ◦ φ and weighted composition operator ϕCφ f = ϕf ◦ φ for ϕ ∈ H(∆) and f ∈ H(∆). It is interesting to provide a function theoretic characterization for φ inducing a bounded or compact composition operator, weighted composition operator and related ones on various spaces (see, e.g., [1, 3, 10, 15, 17, 19-21, 23-24, 26]). For example, it is well known that Cφ is bounded on the classical Hardy, Bloch and Bergman spaces. Operators DCφ and Cφ D as well as some other products of linear operators were studied, for example, in [5, 7-8, 11, 13, 16, 22] (see also the references therein). There has been some considerable recent interest in investigation various type of operators from or to Zygmund type spaces (see, [4, 9-12, 27]). In this paper, we investigate the operators ϕDCφ and ϕCφ D from Zygmund spaces to Bers-type spaces and little Bers-type spaces by using the

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similar ways in [14]. Some sufficient and necessary conditions for the boundedness and compactness of these operators are given. 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 B ≤A≤CB. C 2. Main results and proofs In this section, we state and prove our main results. In order to formulate our main results, we quote several lemmas which will be used in the proofs of the main results in this paper. The following lemma can be proved in a standard way (see,e.g., Proposition 3.11 in [1]). Hence we omit the details. Lemma 2.1. Let φ be an analytic self-map of ∆, ϕ be an analytic function in ∆. Suppose that µ is normal. Then ϕDCφ (or ϕCφ D):Z (or Z0 )→ Hµ∞ is compact if and only if ϕDCφ (or ϕCφ D):Z (or Z0 )→Hµ∞ is bounded and for any bounded sequence {fn }n∈N in Z (or Z0 ) which converges to zero uniformly on compact subsets of ∆ as n → ∞, and ∥ϕDCφ fn ∥Hµ∞ → 0 (or ∥ϕCφ Dfn ∥Hµ∞ → 0) as n → ∞. ∞ Lemma 2.2. A closed set K of Hµ,0 is compact if and only if it is bounded and satisfies

(2.1)

lim sup µ(|z|)|f (z)| = 0.

|z|→1 f ∈K

Proof. First of all, we suppose that K is compact and let ε > 0. By the ∞ definition of Hµ,0 , we can choose an 2ε -net which center at f1 , f2 , · · · , fn in K respectively, and a positive number r ( 0 < r < 1), such that µ(|z|)|fi (z)| < ε , for 1 ≤ i ≤ n and |z| > r. If f ∈ K, ∥f − fi ∥Hµ∞ < 2ε for some fi , so we 2 have µ(|z|)|f (z)| ≤ ∥f − fi ∥Hµ∞ + µ(|z|)|fi (z)| < ε, for |z| > r. This establishes (2.1). On the other hand, if K is a closed bounded set which satisfies (2.1) and {fn } is a sequence in K, then by the Montel, s theorem, there is a subsequence {fnk } which converges uniformly on compact subsets of ∆ to some analytic function f . According to (2.1), for every ε > 0, there is an r, 0 < r < 1, such that for all g ∈ K, µ(|z|)|g(z)| < 2ε , if |z| > r. It follows that µ(|z|)|f (z)| < 2ε , if |z| > r. Since {fnk } converges uniformly to f on |z| ≤ r, it follows that limk→∞ sup ∥fnk − f ∥Hµ∞ ≤ ε, i.e limk→∞ ∥fnk − f ∥Hµ∞ = 0, so that K is compact. 

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Theorem 2.3. Let φ be an analytic self-map of ∆, and ϕ be an analytic function in ∆. Suppose that µ is normal. Then the following statements are equivalent. (i) ϕDCφ : Z → Hµ∞ is bounded; (ii) ϕDCφ : Z0 → Hµ∞ is bounded; (iii) (2.2)

sup µ(|z|)|ϕ(z)φ′ (z)| ln z∈∆

e < ∞. 1 − |φ(z)|2

Proof. (i)⇒(ii). This implication is obvious. (ii)⇒(iii). Assume that ϕDCφ :Z0 → Hµ∞ is bounded, i.e., there exists a constant C such that ∥ϕDCφ f ∥Hµ∞ ≤ C∥f ∥Z for all f ∈ Z0 . Taking the function f (z) = z ∈ Z0 , we get sup µ(|z|)|ϕ(z)φ′ (z)| < ∞.

(2.3)

z∈∆

Set h(z) = (z − 1)[(1 + ln

1 2 ) + 1] 1−z

and ha (z) =

(2.4)

h(¯ az) 1 (ln )−1 a ¯ 1 − |a|2

for a ∈ ∆ \ {0}. It is known that ha ∈ Z0 (see [9]). Since (2.5)

h′a (z) = (ln

1 1 )2 (ln )−1 , 1−a ¯z 1 − |a|2

for |φ(λ)| > 12 , we have C∥ϕDCφ ∥Z0 →Hµ∞ ≥ ∥ϕDCφ hφ(λ) ∥Hµ∞ ≥ µ(|λ|)|ϕ(λ)φ′ (λ)| ln

1 . 1 − |φ(λ)|2

Hence, we have that (2.6)

sup µ(|λ|)|ϕ(λ)φ′ (λ)| ln |φ(λ)|> 21

1 < ∞. 1 − |φ(λ)|2

On the other hand, from the inequality (2.3) we have that (2.7) sup µ(|λ|)|ϕ(λ)φ′ (λ)| ln |φ(λ)|≤ 12

1 4 ≤ sup µ(|λ|)|ϕ(λ)φ′ (λ)| ln < ∞. 2 1 − |φ(λ)| 3 λ∈∆

Hence, from (2.3), (2.6) and (2.7), we obtain (2.2).

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(iii)⇒(i). Assume that (2.2) holds. Then, for every f ∈ Z, we have µ(|z|)|(ϕDCφ f )(z)| = µ(|z|)|ϕ(z)φ′ (z)f ′ (φ(z))| ≤ Cµ(|z|)|ϕ(z)φ′ (z)| ln

(2.8)

e ∥f ∥Z . 1 − |φ(z)|2

Taking the supremum in (2.8) for z ∈ ∆, and employing (2.2), we deduce that ϕDCφ : Z → Hµ∞ is bounded. The proof of Theorem 2.3 is completed. Theorem 2.4. Let φ be an analytic self-map of ∆, and ϕ be an analytic function in ∆. Suppose that µ is normal. Then the following statements are equivalent. (i) ϕDCφ : Z → Hµ∞ is compact; (ii) ϕDCφ : Z0 → Hµ∞ is compact; (iii) ϕDCφ : Z → Hµ∞ is bounded, and (2.9)

lim µ(|z|)|ϕ(z)φ′ (z)| ln

|φ(z)|→1

e = 0. 1 − |φ(z)|2

Proof. (i)⇒(ii). This implication is clear. (ii)⇒(iii). Assume that ϕDCφ :Z0 → Hµ∞ is compact. Then it is clear that ϕDCφ :Z0 → Hµ∞ is bounded. By Theorem 2.3 we know that ϕDCφ :Z → Hµ∞ is bounded. Let (zn )n∈N be a sequence in ∆ such that |φ(zn )| → 1 as n → ∞ and φ(zn ) ̸= 0, n ∈ N (if such a sequence does not exist then (2.9) is vacuously satisfied). Set (2.10)

hn (z) =

h(φ(zn )z)

(ln

1 )−1 , n ∈ N. 1 − |φ(zn )|2

φ(zn ) Then from the proof of Theorem 2.3, we see that hn ∈ Z0 for each n ∈ N . Moreover hn → 0 uniformly on compact subsets of ∆ as n → ∞ and 1 h′n (φ(zn )) = ln . 1 − |φ(zn )|2 Since ϕDCφ :Z0 → Hµ∞ is compact, by Lemma 2.1, we have lim ∥ϕDCφ hn ∥Hµ∞ = 0.

n→∞

Hence, (2.11)

lim µ(|zn |)|ϕ(zn )φ′ (zn )| ln

n→∞

1 = 0. 1 − |φ(zn )|2

From (2.11) easily follows that limn→∞ µ(|zn |)|ϕ(zn )φ′ (zn )| = 0, which altogether imply (2.9). (iii)⇒(i). Suppose that ϕDCφ :Z → Hµ∞ is bounded and that conditions (2.9) holds. From Theorem 2.3, we know that (2.12)

C = sup µ(|z|)|ϕ(z)φ′ (z)| < ∞. z∈∆

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By the assumption, for every ε > 0, there is a δ ∈ (0, 1), such that e (2.13) µ(|z|)|ϕ(z)φ′ (z)| ln < ε, 1 − |φ(z)|2 whenever δ < |φ(z)| < 1. Assume that (fk )k∈N is a sequence in Z such that supk∈N ∥fk ∥Z ≤ L and fk converges to 0 uniformly on compact subsets of ∆ as k → ∞. Let K = {z ∈ ∆ : |φ(z)| ≤ δ}. Then by (2.12) and (2.13), we have that sup µ(|z|)|(ϕDCφ fk )(z)| = sup µ(|z|)|ϕ(z)φ′ (z)fk′ (φ(z))| z∈∆

≤ sup µ(|z|)|ϕ(z)φ

′

z∈∆ ′ (z)fk (φ(z))|

′

(z)fk′ (φ(z))|

z∈K

+ sup µ(|z|)|ϕ(z)φ′ (z)fk′ (φ(z))| z∈∆\K

≤ sup µ(|z|)|ϕ(z)φ z∈K

+ C sup µ(|z|)|ϕ(z)φ′ (z)| ln z∈∆\K

e ∥fk ∥Z 1 − |φ(z)|2

≤ C sup |fk′ (ω)| + Cε∥fk ∥Z , |ω|≤δ

i.e., we obtain (2.14) ∥ϕDCφ fk ∥Hµ∞ ≤ C sup |fk′ (ω)| + Cε∥fk ∥Z + |ϕ(0)||fk′ (φ(0))||φ′ (0)|. |ω|≤δ

Since fk converges to 0 uniformly on compact subsets of ∆ as k → ∞, Cauchy′ s estimate gives that fk′ → 0 as k → ∞ on compact subsets of ∆. Hence, letting k → ∞ in (2.14), and using the fact that ε is an arbitrary positive number, we obtain lim ∥ϕDCφ fk ∥Hµ∞ = 0.

k→∞

Combining this with Lemma 2.1 the result easily follows. The proof of Theorem 2.4 is completed.  Theorem 2.5. Let φ be an analytic self-map of ∆, and ϕ be an analytic ∞ function in ∆. Suppose that µ is normal. Then ϕDCφ : Z0 → Hµ,0 is ∞ bounded if and only if ϕDCφ : Z0 → Hµ is bounded and (2.15)

lim µ(|z|)|ϕ(z)φ′ (z)| = 0.

|z|→1

∞ is bounded. Then, it is clear that Proof. Assume that ϕDCφ :Z0 → Hµ,0 ϕDCφ :Z0 → Hµ∞ is bounded. Taking the test function f (z) = z, we obtain (2.15). Conversely, assume that ϕDCφ :Z0 → Hµ∞ is bounded and (2.15) holds. Then for each polynomial p, we have that

(2.16)

µ(|z|)|(ϕDCφ p)(z)| ≤ µ(|z|)|ϕ(z)φ′ (z)p′ (φ(z))|.

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In view of the facts that sup |p′ (ω)| < ∞, ω∈∆ ∞ from (2.15) and (2.16), it follows that ϕDCφ p ∈ Hµ,0 . Since the set of all polynomials is dense in Z0 (see [11]), we have that for every f ∈ Z0 , there is a sequence of polynomials (pn )n∈N such that ∥f − pn ∥Z → 0 as n → ∞. Hence

∥ϕDCφ f − ϕDCφ pn ∥Hµ∞ ≤ ∥ϕDCφ ∥Z0 →Hµ∞ ∥f − pn ∥Z → 0 as n → ∞. Since the operator ϕDCφ :Z0 → Hµ∞ is bounded, so ϕDCφ (Z0 ) ⊆ ∞ ∞ Hµ,0 , which implies the boundedness of ϕDCφ :Z0 → Hµ,0 . Theorem 2.6. Let φ be an analytic self-map of ∆, and ϕ be an analytic function in ∆. Suppose that µ is normal. Then the following statements are equivalent. ∞ (i) ϕDCφ : Z → Hµ,0 is compact; ∞ (ii) ϕDCφ : Z0 → Hµ,0 is compact; (iii)

(2.17)

lim µ(|z|)|ϕ(z)φ′ (z)| ln

|z|→1

e = 0. 1 − |φ(z)|2

Proof. (i)⇒(ii). This implication is trivial. ∞ (ii)⇒(iii). Assume that ϕDCφ : Z0 → Hµ,0 is compact. Then ϕDCφ : ∞ Z0 → Hµ,0 is bounded. From the proof of Theorem 2.5, we know that (2.18)

lim µ(|z|)|ϕ(z)φ′ (z)| = 0.

|z|→1

Hence, if ∥φ∥∞ < 1, from (2.18), we obtain that e e ≤ ln lim µ(|z|)|ϕ(z)φ′ (z)| ln lim µ(|z|)|ϕ(z)φ′ (z)| = 0, 2 |z|→1 1 − |φ(z)| 1 − ∥φ∥2∞ |z|→1 from which the result follows in this case. Now assume that ∥φ∥∞ = 1. Let (zk )k∈N be a sequence such that |φ(zk )| → 1 as k → ∞. Since ϕDCφ : Z0 → Hµ∞ is compact, by Theorem 2.4, e (2.19) lim µ(|z|)|ϕ(z)φ′ (z)| ln = 0. |φ(z)|→1 1 − |φ(z)|2 From (2.18) and (2.19), we have that for every ε > 0, there exists an r ∈ (0, 1) such that e µ(|z|)|ϕ(z)φ′ (z)| ln < ε, 1 − |φ(z)|2

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when r < |φ(z)| < 1, and there exists a σ ∈ (0, 1) such that ε µ(|z|)|ϕ(z)φ′ (z)| ≤ e , ln 1−r 2 when σ < |z| < 1. Therefore, when σ < |z| < 1 and r < |φ(z)| < 1, we have e (2.20) µ(|z|)|ϕ(z)φ′ (z)| ln < ε. 1 − |φ(z)|2 On the other hand, if σ < |z| < 1 and |φ(z)| ≤ r, we obtain e e (2.21) µ(|z|)|ϕ(z)φ′ (z)| ln < ε. < µ(|z|)|ϕ(z)φ′ (z)| ln 2 1 − |φ(z)| 1 − r2 Inequality (2.20) together with (2.21) gives the (2.17). (iii)⇒(i). Let f ∈ Z. we have µ(|z|)|(ϕDCφ f )(z)| ≤ Cµ(|z|)|ϕ(z)φ′ (z)| ln

e ∥f ∥Z . 1 − |φ(z)|2

Taking the supremum in this inequality over all f ∈ Z such that ∥f ∥Z ≤ 1, then letting |z| → 1, and using (2.17), we obtain that lim sup µ(|z|)|(ϕDCφ f )(z)| = 0.

|z|→1 ∥f ∥Z ≤1

∞ Using Lemma 2.2 we obtain that the operator ϕDCφ :Z → Hµ,0 is compact. Similarly to the proofs of Theorems 2.3-2.6, we can get the following results, we omit the proof.

Theorem 2.7. Let φ be an analytic self-map of ∆, and ϕ be an analytic function in ∆. Suppose that µ is normal. Then the following statements are equivalent. (i) ϕCφ D : Z → Hµ∞ is bounded; (ii) ϕCφ D : Z0 → Hµ∞ is bounded; (iii) sup µ(|z|)|ϕ(z)| ln z∈∆

e < ∞. 1 − |φ(z)|2

Theorem 2.8. Let φ be an analytic self-map of ∆, and ϕ be an analytic function in ∆. Suppose that µ is normal. Then the following statements are equivalent. (i) ϕCφ D : Z → Hµ∞ is compact; (ii) ϕCφ D : Z0 → Hµ∞ is compact; (iii) ϕCφ D : Z → Hµ∞ is bounded, lim µ(|z|)|ϕ(z)| ln

|φ(z)|→1

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Theorem 2.9. Let φ be an analytic self-map of ∆, and ϕ be an analytic ∞ function in ∆. Suppose that µ is normal. Then ϕCφ D : Z0 → Hµ,0 is ∞ ∞ bounded if and only if ϕCφ D : Z0 → Hµ is bounded and ϕ(z) ∈ Hµ,0 . Theorem 2.10. Let φ be an analytic self-map of ∆, and ϕ be an analytic function in ∆. Suppose that µ is normal. Then the following statements are equivalent. ∞ (i) ϕCφ D : Z → Hµ,0 is compact; ∞ (ii) ϕCφ D : Z0 → Hµ,0 is compact; (iii)

lim µ(|z|)|ϕ(z)| ln

|z|→1

e = 0. 1 − |φ(z)|2

Acknowledgements. This research is partly supported by the United Technology Foundation of Science and Technology Department of Guizhou Province and Guizhou Normal University (Grant No.LKS[2012]12), the National Natural Science Foundation of China (Grant No. 11501142, 11101099), and the Foundation of Science and Technology of Guizhou Province of China (Grant No. [2015]2112, [2012]2273). References [1] C. C. Cowen and B. D. Maccluer, Composition operators on spaces of analytic functions, Studies in Advanced Mathematics, CRC Press, Florida, 1995. [2] P. L. Duren, Theory of H p Spaces, Academic press, New York, 1970. [3] X. Fu and X. Zhu, Weighted composition operators on some weighted spaces in the unit ball, Abstract and applied analysis, 8(2008), Article ID 605807. [4] X. Fu and S. Li, Composition operators from Zygmund spaces into Qk spaces, J. Inequal. Appl. 2013(2013), no. 175, 9 pages. [5] R. A. Hibschweiler and N. Portnoy, Composition followed by differentiation between Bergman and Hardy spaces, The Rocky Mountain J. Math. 35(2005), no. 3, 843-855. [6] Z. Hu and S. Wang, Composition operators on Bloch-type spaces, Proc. Royal Society of Edinburgh A 135(2005), no. 6, 1229-1239. [7] S. Li and S. Stevi´ c, Composition followed by differentiation between Bloch type spaces, J. Comput. Anal. Appl. 9(2007), no. 2, 195-205. [8] S. Li and S. Stevi´ c, Composition followed by differentiation between ∞ H and α-Bloch spaces, Houston J. Math. 35(2009), no. 1, 327-340.

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[9] S. Li and S. Stevi´ c, Volterra type operators on Zygmund spaces, J. Inequ. Appl. 2007(2007), Article ID 32124, 10 pages. [10] S. Li and S. Stevi´ c, Generalized composition operators on Zygmund spaces and Bloch type spaces, J. Math. Anal. Appl. 338(2008), no. 2, 1282-1295. [11] S. Li and S. Stevi´ c, Products of composition and differentiation operators from Zygmund spaces to Bloch spaces and Bers spaces, Appl. Math. Comput. 217(2010), 3144-3154. [12] Y. Liu and Y. Yu, Riemann-Stieltjes operator from mixed norm spaces to Zygmund- type spaces on the unit ball, Taiwan. J. Math. 17(2013), no. 5, 1751-1764. [13] J. R. Long and P. C. Wu, Weighted composition followed and proceeded by differentiation operators from Qk (p, q) spaces to Bloch-type spaces, J. Inequal. Appl. 2012(2012), no. 160, 12 pages. [14] J. R. Long, C. H. Qiu and P. C. Wu, Weighted composition followed and proceeded by differentiation operators from Zygmund spaces to Blochtype spaces, J. Inequal. Appl. 2014(2014), no. 152, 12 pages. [15] K. Madigan and A. Matheson, Compact composition operators on the Bloch space, Trans. Amer. Math. Soc. 347(1995), 2679-2687. [16] S. Ohno, Products of composition and differentiation between Hardy spaces, Bull. Austral. Math. Soc. 73(2006), no. 2, 235-243. [17] J. H. Shapiro, Composition Operators and Classical Function Theory, Springer, New York, 1993. [18] A. L. Shields and D. L. Williams, Bonded projections, duality, and multipliers in spaces of analytic functions, Trans. Amer. Math. Soc. 162(1971), 287-302. [19] S. Stevi´ c, Generalized composition operators between mixed norm space and some weighted spaces, Numer. Funct. Anal. Opt. 29(2009), 426434. [20] S. Stevi´ c, Norm of weighted composition operators from Bloch space to ∞ Hµ on the unit ball, Ars. Combin. 88(2008), 125-127. [21] S. Stevi´ c, Norm of weighted composition operators from α−Bloch spaces to weighted-type spaces, Appl. Math. Comput. 215(2009), 818820. [22] S. Stevi´ c, Weighted differentiation composition operators from mixednorm spaces to weighted-type spaces, Appl. Math. Comput. 211(2009), 222-233.

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[23] S. I. Ueki, Composition operators on the Privalov spaces of the unit ball of Cn , J. Korean Math. Soc. 42(2005), no. 1, 111-127. [24] S. I. Ueki, Weighted composition operators on the Bargmann-Fock space, Int. J. Mod. Math. 3(2008), no. 3, 231-243. [25] S. Yamashita, Schilicht holomorphic functions and the Riccati differential equation, Math. Z. 157(1997), 19-22. [26] X. Zhu, Generalized weighted composition operators from Bloch-type spaces to weighted Bergman spaces, Indian J. Math. 49(2007), no. 2, 139-149. [27] X. Zhu, Extended Cesa´ ro operators from mixed norm spaces to Zygmund type spaces, Tamsui Oxf. J. Math. Sci. 26(2010), no. 4, 411-422. Jianren Long Department of Mathematical Science, Guizhou Normal University, 550001, Guiyang, P.R. China. E-mail address: [email protected] Congli Yang Department of Mathematical Science, Guizhou Normal University, 550001, Guiyang, P.R. China. E-mail address: [email protected]

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Global stability in n-dimensional stochastic difference equations for predator-prey models Sang-Mok Chooa , Young-Hee Kim∗,b a

b

Department of Mathematics, University of Ulsan, Ulsan 44610, Korea. Ingenium College of Liberal Arts-Mathematics, Kwangwoon University, Seoul 01897, Korea

Abstract There are relatively few theoretical papers to consider the positivity of solutions of discrete time stochastic difference equations (DSDEs), compared to many publications on theoretical analysis of solutions of deterministic difference equations and stochastic differential equations. Additionally, no papers theoretically investigate the global stability of nontrivial solutions of n-dimensional DSDEs. In this paper, we consider the Euler-Maruyama scheme for n-dimensional stochastic difference equations that are a generalization of a two-dimensional model of stochastic predator-prey interactions, and show the positivity and the global stability of nontrivial solutions of the scheme by applying a new discretized version of the Itˆo formula. Numerical simulations are introduced to support the results. Key words: Euler-Maruyama scheme, Positivity, Global stability, Stochastic difference equations.

1. Introduction Stochastic differential equation (SDE) models have been increasingly used in a range of application areas, including biology, chemistry, mechanics, economics, and finance. In general, the exact solutions of SDEs are not known, so one has to numerically solve these SDEs. This leads us to consider and analyze discrete time stochastic difference equations (DSDEs), which can be also viewed as stochastically perturbed versions of deterministic difference equations (DDEs). There are many publications on estimations of the difference between solutions of SDEs and DSDEs. The global asymptotic stability of the trivial solution of DSDEs has been also widely addressed (see [1], [2], [3] and references therein). However, relatively few studies theoretically consider the positivity of solutions of DSDEs that are scalar equations on a finite time interval (see [4] references therein). In particular, to the best of our knowledge, there is no paper that theoretically deals with the global stability of nontrivial solutions of DSDEs, except [5], in which two-dimensional DSDEs are treated with a new discretized version of the Itˆo formula. Therefore, the aim of this paper is to extend the method used in [5] for investigating the positivity and the global stability of nontrivial solutions of n-dimensional DSDEs on an infinite time interval with stochastic predator-prey models. ∗

Corresponding author Email addresses: [email protected] (Sang-Mok Choo), [email protected] (Young-Hee Kim)

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We generalize the two dimensional predator-prey model in [6] dx(t) = x(t){r1 − a11 x(t) − a12 y(t)}dt + σ1 x(t)dW1 (t), dy(t) = y(t){−r2 + a21 x(t) − a22 y(t)}dt + σ2 y(t)dW2 (t),

(1)

to the n-dimensional stochastic differential equations dxi (t) = xi (t) ri +

i−1 X

aij xj (t) −

n X

j=1

! aij xj (t) dt + σi xi (t)dWi (t),

(2)

j=i

where Wi are independent and real valued Wiener processes on a complete probability space (Ω, F, P). Although all the parameters ri , aij and σi in (1) are positive, we weaken the conditions on the signs in (2) such that ri ∈ R, aii > 0, aij ≥ 0, σi > 0 (1 ≤ i, j ≤ n, i 6= j). Consider the Euler-Maruyama scheme for (2) ! ( ) n i−1 X X √ i aij xjk + ∆tσi ξk+1 xik+1 = xik 1 + ∆t ri + aij xjk − ,

(3)

j=i

j=1

where 1 ≤ i ≤ n, k ≥ 0, xi0 > 0, ∆t = N1 for N ∈ N, tk = k∆t, and discrete √ i with a mutually independent and idenWiener processes Wi (tk+1 ) − Wi (tk ) are ∆tξk+1 1 n ∞ tically distributed sequence (ξk , · · · , ξk )k=1 of the standard normal random variables. The solutions of (3) are defined with respect to a complete, filtered probability space ∞ (ΩN , FN , {Fk }∞ k=1 , PN ), where {Fk }k=1 is the natural filtration generated by the stochas1 n ∞ tic sequence (ξk , · · · , ξk )k=1 . The positivity of the solutions of continuous time SDEs (1) is obtained in the infinite time interval [0, ∞) without the assumption of boundedness of the noises Wi (t) by using the concept of explosion time (see [7] and [6]). However, for obtaining the positivity of the solutions of discrete time DSDEs (3) in the infinite time interval, we restrict noises to bounded noises, which means that ξki are assumed to be truncated standard normal random variables with support [−ς, ς] for a positive constant ς that satisfies   E(ξki ) = 0, E (ξki )2 = 1 − ης . (4) 2ςφ(ς) can be assumed to be sufficiently close to 0, where φ and The positive value ης = Φ(ς)−Φ(−ς) Φ are the probability density and the cumulative distribution functions of the standard normal random variable, respectively. For example, when ς = 20, we have 0 < ης < 10−85 . The paper is organized as follows. Section 2 gives the positivity and the boundedness of the solutions of (3). In Section 3, we introduce the new discrete Itˆo formula developed in [5] by using a known discrete Itˆo formula for stochastic difference equations (see [8] [9] and [10]), which is the main tool for finding conditions for the global stability of the solutions of (3). Section 4 introduces auxiliary equations, the solutions of which are used for upper bounds of the solutions of (3). We show the global stability of the solutions of the two-dimensional model (3) in Section 5. The properties of the solutions of the auxiliary equations are used in Section 6 to find conditions for at least one of the solutions of (3) to converge to zero. In addition, the approaches in Section 5 are extended into the ndimensional model (3) for finding conditions for the global stability the solutions of (3). Section 7 gives simulation results to confirm the results obtained in this paper.

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2. Positivity and boundedness of solutions of DSDEs In this section, we show the positivity and boundedness of solutions of the n-dimensional model (3) by applying the approach used in the DDE model (3) with σ1 = σ2 = 0 (see [11]). For simplicity, we use the notations for every constant a, √ a ˆ = a · ∆t, a ˜ = a · ∆t and i+1 n xik = (x1k , · · · , xi−1 k , xk , · · · , xk ).

Then the n-dimensional discrete model (3) can be written as xik+1 = Fxik (xik ), where Fxik (xik )

=

xik



1 + rˆi +

X

a ˆij xjk 1≤j≤i−1

−

X

a ˆij xjk i≤j≤n



.

(5)

For a vector ζ ik = (ζk1 , · · · , ζki−1 , ζki+1 , · · · , ζkn ) of real numbers, define   Xi−1 Xn j j i −1 i a ˆij ζk − a ˆij ζk + σ V (ζ k ) = (2ˆ aii ) 1 + rˆi + ˜i ξk+1 .

(6)

j=1

+

i σ ˜i ξk+1

j=i+1

If V (ζ ik ) > 0, then (5) gives Fζ ik (x) is increasing on 0 ≤ x < V (ζ ik ).

(7)



(8)

For 1 ≤ i ≤ n, denote  X χi = a ˜−1 r ˜ + i ii

1≤j≤i−1

a ˜ij χj + σi ς

and assume that   X χi ≤ (2ˆ aii )−1 1 + rˆi − a ˆij χj − σ ˜i ς , i+1≤j≤n X rˆi + a ˆij χj + σ ˜i ς ≤ 1. 1≤j≤i−1

The model (3) is also assumed to satisfy the initial condition Y (x10 , · · · , xn0 ) ∈ (0, χi ).

(9) (10)

(11)

1≤i≤n

Theorem 1. Let xik be the solutions of (3) and χi be defined in (8). Then Y (x1k , · · · , xnk ) ∈ (0, χi ), k ≥ 0. 1≤i≤n

Proof. It follows from (11), (9) and (6) that for 1 ≤ i ≤ n   P 0 < xi0 < χi < (2ˆ aii )−1 1 + rˆi − i+1≤j≤n a ˆij χj − σ ˜i ς < V (xi0 ). Then letting ζ i0 = xi0 in (7) gives the positivity xi1 = Fxi0 (xi0 ) > Fxi0 (0) = 0, 1 ≤ i ≤ n. 3 118

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Let ω ∈ ΩN . If rˆi +

P

a ˆij xj0 (ω) −

1≤j≤i−1

a ˆij xj0 (ω) + σ ˜i ξ1i (ω) ≤ 0, then

P i≤j≤n

xi1 (ω) = Fxi0 (xi0 )(ω) ≤ xi0 (ω) < χi , and otherwise, we have 0 < xi0 (ω) < f (xi0 )(ω) with  X X j r ˜ + ˜−1 a ˜ x − f (xi0 ) = a i ij 0 ii 1≤j≤i−1

i+1≤j≤n

 a ˜ij xj0 + σi ξ1i .

Since 0 < f (xi0 ) < V (xi0 ) by (10), using both (7) with ζ ik = xi0 and (8) with (11), we have xi1 (ω) = Fxi0 (xi0 )(ω) < Fxi0 (f (xi0 ))(ω) = f (xi0 )(ω) < χi , 1 ≤ i ≤ n. Therefore if (x10 , · · · , xn0 ) ∈

Q

1≤i≤n (0, χi ),

then Q (x11 , · · · , xn1 ) ∈ 1≤i≤n (0, χi ),

and hence applying mathematical induction, we can complete the proof. Remark 1. Since (8) and (9) can be written as   X a ˜ij χj + σi ς , χi = (∆t)−0.5 aii −1 r˜i + 1≤j≤i−1   X −1 −1 a ˆij χj − σ ˜i ς , χi ≤ (∆t) (2aii ) 1 + rˆi − i+1≤j≤n

the conditions (9) and (10) can be satisfied when taking small values of ∆t. For example, take n = 3 in (3). The definition (8) gives χ1 = a ˜−1 r1 + σ1 ς), χ2 = a ˜−1 r2 + a ˜21 χ1 + σ2 ς) 11 (˜ 22 (˜ −1 and χ3 = a ˜33 (˜ r3 + a ˜31 χ1 + a ˜32 χ2 + σ3 ς). Let ∆t = 0.0001, ς = 20, ri = aij = σi = 1 for 1 ≤ i, j ≤ 3. Then the conditions (9) and (10) are satisfied.

3. A new discretized version of the Itˆ o formula In order to find conditions for the stability of the solutions of (3), we need a discretized form of the Itˆo formula. Although there are discretized versions of the Itˆo formula (see [8], [9] and [10]), we developed a variant which is suitable for our purpose in [5]. For the completeness of this paper, we include the proof of the new discretized version of the Itˆo formula in Appendix below. We write q1 (h) = O(q2 (h)) (or q1 (h) = O(q2 (h)) for h → 0 to be more precise) if there exist positive constants C and h0 such that |q1 (h)| ≤ C|q2 (h)| for all h with 0 < h ≤ h0 . We make the following assumptions about the noise ξ: (a) The noise ξ satisfies that for some C and µ with 0 < µ < 1  
E(ξ) = 0, E ξ 2 = 1 − µ, E |ξ|` ≤ C (` = 1, 3). (b) The probability density function p exists and satisfies that for some C and all sufficiently large |x| C . |x|3 p(x) ≤ |x| 4 119

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The truncated Normal random variables satisfy both (a) and (b) with µ = ης in (4). By C 3 (R), we denote the set of all functions defined on R that are continuously differentiable up to the order 3. Lemma 1. Consider functions φ, ϕ : R → R satisfying for some δ > 0, (i) ϕ = φ on [1 − δ, 1 + δ]. (ii) ϕ ∈ C 3 (R) and |ϕ000 (x)| ≤ C for some C and all x ∈ R. R (iii) R |ϕ(x) − φ(x)|dx < ∞. Let ξ be an F-independent random variable satisfying (a) and (b). Let f and g be Fmeasurable random variables satisfying that for some positive constants ε and C, √ (12) max{h|f |, h|g|} ≤ Chε Then     √ E φ 1 + hf + hgξ F 1 = φ(1) + φ0 (1)hf + φ00 (1)hg 2 · (1 − µ) + hf O (hε ) + hg 2 O (hε ) . 2 Pn P j j Remark 2. For the solutions xik of (3), let f = ri + i−1 j=i aij xk and g = σi j=1 aij xk − Then f and g satisfy (12) with ε = 0.5 due to 0 < xik ≤ χi = O(h−0.5 ) for 1 ≤ i ≤ n and k ≥ 0. Remark 3. In order to construct ϕ corresponding to the function  ln |x| (|x| > 0) φ (x) = , 0 (x = 0) we modify the function ϕ in [2]. Define the function ϕ as follows:  ln |x| (|x| ≥ e−1 ) ϕ (x) = . 6 3 3 − 41 e4 x4 + e2 x2 − 74 + e6 (x − e−1 ) (x + e−1 ) (|x| ≤ e−1 ) Then ϕ satisfies all the conditions in Lemma 1. Notation 1. For simplicity, we use the notations for 1 ≤ i ≤ n and k > 0, E(xik ) =


1 Xk−1 E xis , s=0 k

and for every constant a and ης in (4)  2 ˚ a = a · 1 + O(h0.5 ) , aη = a · (1 − ης ), riσ = ri − 0.5σiη . Remark 4. Since the solutions xik+1 of (3) are positive, we can take logarithm of (3). Then applying Lemma 1 with (φ, ϕ) in Remark 3, the Fk -independent and normally truncated random variable ξk+1 and (f, g) in Remark 2, we have     √     i i E ln xk+1 Fk = E ln xk Fk + E φ 1 + hf + hgξk+1 Fk , 5 120

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which gives

1 E(ln xik+1 ) = ln xik + hf + hg 2 · (1 − ης u) + hf O h0.5 + hg 2 O h0.5 2 ! n i−1 X X 1 j j 2 = ln xik + ˚ h ri − σiη aij xk . + aij xk − 2 j=i j=1

(13)

Taking expectation of (13) and adding the result, we can obtain h riσ + E(ln xik ) = E(ln xi0 ) + k˚

i−1 X

aij E(xjk )

j=1

−

n X

! aij E(xjk )

.

(14)

j=i

4. Auxiliary equations In order to find upper bounds of xik , we consider the auxiliary equations for 1 ≤ i ≤ n and k ≥ 0   Xi−1 j i i i i zk+1 = zk 1 + rˆi + a ˆij zk − a ˆii zk + σ ˜i ξk+1 , z0i = xi0 . (15) j=1

Since (15) is the system (3) with aij = 0 for 1 ≤ i < j ≤ n, Theorem 1 gives Y (zk1 , · · · , zkn ) ∈ (0, χi ), k ≥ 0.

(16)

Let βi be the solutions of the equations X riσ + aij βj − aii βi = 0, 1 ≤ i ≤ n.

(17)

1≤i≤n

1≤j≤i−1

Note that (13) and (14) with a1j = 0 (2 ≤ j ≤ n) become
˚ r1σ − a11 z 1 , = ln zk1 + ∆t k


˚ r1σ − a11 E zs1 E ln zk1 = E ln z01 + k ∆t n Xk−1

o 1 −1 1 ˚ = E ln z0 + k ∆ta11 β1 − k E zs ,

1 E ln zk+1




s=0

(18)

(19)

due to β1 = r1σ a−1 11 in (17). The proofs of Lemma 2 and 3 were given in [5]. For the completeness of this paper, we write the proofs again. Lemma 2. Let zk1 and β1 be the solutions of (15) and (17), respectively. If β1 > 0, then for every  > 0 and some integer N > 0 k −1

Xk−1 s=0


E zs1 ≤ β1 + , k ≥ N .

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Proof. Suppose that the theorem is false, which means that there exist a constant ε0 > 0 and an infinite increasing sequence {km } satisfying both for all km −1 km

Xkm −1


E zs1 ≥β1 + ε0 ,

(20)


E zs1 β1 + ε0 .

(26)

Hence there exist finitely many k satisfying (21) due to (24) and (26). Claim 2: As (20) implies (24), the equation (25) implies lim E(zk1 ) = 0,

k→∞

which is contradictive to (25) due to β1 + ε0 > 0. This contradiction completes the proof. Lemma 3. Let (zk1 , zk2 ) and (β1 , β2 ) be the solutions of (15) and (17), respectively. (a) If riσ < 0 (i = 1, 2), then lim zki = 0 (i = 1, 2), a.s. k→∞ P 1 (b) Assume r1σ > 0. Then limk→∞ k −1 k−1 s=0 E (zs ) = β1 . (i) If r2σ + a21 β1 < 0, then limk→∞ zk2 = 0, a.s. 7 122

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(ii) If r2σ + a21 β1 > 0, then limk→∞ k −1

Pk−1 s=0

E (zs2 ) = β2 .

Proof. (a) Since r1σ < 0 is equivalent to β1 < 0, the equation (19) with the positivity of zk1 gives that if r1σ < 0, then limk→∞ E (ln zk1 ) = −∞ and so we have lim zk1 = 0, a.s. k→∞

Hence the dominated convergence theorem yields lim E(zk1 ) = 0, which implies k→∞

lim E(zk1 ) = 0.

(27)

k→∞

It remains to show that limk→∞ zk2 = 0, a.s. Using (13) and (14) with a2j = 0 (3 ≤ j ≤ n), we have

2 ˚ r2σ + a21 z 1 − a22 z 2 , E ln zk+1 = ln zk2 + ∆t (28) k k

˚ r2σ + a21 E(zk1 ) − a22 E(zk2 ) E ln zk2 = E ln z02 + k ∆t
   1 2 ˚ 22 a−1 = E ln z02 + k ∆ta (29) 22 r2σ + a21 E(zk ) − E(zk ) Combining (27) and (29) with r2σ < 0 and the positivity of zk2 , we have
limk→∞ E ln zk2 = −∞ Therefore, as (22) implies (23), we can obtain that if r2σ < 0, then limk→∞ zk2 = 0, a.s. (b) Assume r1σ > 0, which gives β1 = r1σ a−1 11 > 0. Due to Lemma 2, it is enough to prove that for all  > 0 there exists an integer N satisfying Xk−1
β1 −  ≤ k −1 E zs1 , k ≥ N . (30) s=0

Suppose that (30) is false, which means that there exist a constant ε0 > 0 and an infinite increasing sequence {km } such that −1 β1 − ε0 > km

Xkm −1 s=0


E zs1 .

Then the boundedness of zk1 and (19) imply that for all km

˚ 11 ε0 , ∞ > E ln zk1m > E ln z01 + km ∆ta which is a contradiction. Therefore (30) is true and then Lemma 2 gives limk→∞ k −1

k−1 X


E zs1 = β1 .

(31)

s=0

(b)-(i) Assume that r1σ > 0 and r2σ + a21 β1 < 0. Applying (31) to (29) with r2σ + a21 β1 < 0 and the positivity of zk2 , we have lim E(ln zk2 ) = −∞.

k→∞

Therefore, as (22) implies (23), we can obtain limk→∞ zk2 = 0, a.s.. (b)-(ii) Assume that r1σ > 0 and r2σ + a21 β1 > 0 and then β2 = a−1 22 (r2σ + a21 β1 ) > 0. 8 123

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Following the proof of Lemma 2, we can obtain that for every  > 0 and some integer N > 0 Xk−1
E zs2 ≤ β2 + , k ≥ N , (32) k −1 s=0

zk1 ,

by replacing β1 , (18) and (19) in Lemma 2 with zk2 , β2 , (28) and (29), respectively, and using (31). On the other hand, following the proof of (30) with (29) instead of (19), we can obtain that for all  > 0 there exists an integer N satisfying β2 −  ≤ k −1

Xk−1 s=0


E zs2 , k ≥ N .

Combining (32) and (33), we obtain limk→∞ k −1

Pk−1 s=0

(33)

E (zs2 ) = β2 .

The proofs of Remark 5 and 6 are given in Appendix below. Remark 5. Using the idea in Lemma 3, we can find conditions under which the solution zk3 of (15) converges. Let β˜3 = a−1 33 (r3σ + a31 β1 ), which is equal to β3 when β2 = 0. (a) Assume that r1σ > 0 and r2σ + a21 β1 < 0. (i) If r3σ + a31 β1 < 0, then lim zk3 = 0, a.s. k→∞ P 3 ˜ (ii) If r3σ + a31 β1 > 0, then lim k −1 k−1 s=0 E (zs ) = β3 . k→∞

(b) Assume that r1σ > 0 and r2σ + a21 β1 > 0. P (i) If r3σ + 2j=1 a3j βj < 0, then lim zk3 = 0, a.s. k→∞ P P 3 (ii) If r3σ + 2j=1 a3j βj > 0, then lim k −1 k−1 s=0 E (zs ) = β3 . k→∞

Remark 6. Replacing (3) with (15), the equation (14) becomes n o Xi−1 ˚ riσ + aij E(zkj ) − aii E(zki ) . E(ln zki ) = E(ln z0i ) + k ∆t j=1

(34)

Substituting (17) to (34) yields E(ln zki )

=

E(ln z0i )

˚ + k ∆t

hXi−1 j=1

  i j i aij E(zk ) − βj − aii E(zk ) − βi ,

(35)

so that we can extend (b)-(ii) in both Lemma 3 and Remark 5 to the n-dimensional case: Pi−1 If riσ + j=1 aij βj > 0 (1 ≤ i ≤ n), then limk→∞ E(zki ) = βi (1 ≤ i ≤ n) and, as a result, (35) yields 1 lim E(ln zki ) = 0, 1 ≤ i ≤ n. (36) k→∞ k Lemma 4. Let xik and zki be the solutions of (3) and (15), respectively. Then 0 < xik ≤ zki , 1 ≤ i ≤ n, k ≥ 0. Proof. Let 0k0 be the zero vector of n − 1 entries for k ≥ 0. Since x10 > 0 and Fζ 10 (x) in (5) is decreasing in ζ 10 , we have x11 = Fx10 (x10 ) ≤ F010 (x10 ).

(37)

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Using (6) and (9), we have 0 < x10 ≤ z01 ≤ χ1 ≤ V (010 ) and then (7) yields F010 (x10 ) ≤ F010 (z01 ) = z11 .

(38)

Hence combining (37) and (38) gives x11 ≤ z11 .

(39)

x1k > 0, zk1 ≤ χ1 ≤ V (0k0 )

(40)

Note that for k ≥ 1 by Theorem 1, (16), (6), (9). Following the proof of (39), we can obtain x1k ≤ zk1 (k ≥ 0) by using mathematical induction with (40) and 0k0 . Similarly, letting z 20 = (z01 , 0, · · · , 0) and using 0 < x20 ≤ z02 ≤ χ2 ≤ V (z 20 ), we have x21 = Fx20 (x20 ) ≤ F020 (x20 ) ≤ F020 (z02 ) = z12 .

(41)

1 Hence mathematical induction with (41) and z 2k−1 = (zk−1 , 0, · · · , 0) gives

x2k ≤ zk2 (k ≥ 0). Therefore, using mathematical induction with z ik = (zk1 , · · · , zki−1 , 0, · · · , 0), we can complete the proof. P Remark 7. If riσ + i−1 j=1 aij βj > 0 (1 ≤ i ≤ n), then both Lemma 4 and (36) imply that for every  > 0 there exists an integer N such that 1 E(ln xik ) ≤ , k ≥ N , k

(42)

which will be used in Section 5 and 6 to show that all the solutions of (3) converge to positive values.

5. Global stability of the n-dimensional DSDEs In this section, we first find conditions under which at least one of the solutions of (3) converges to zero, a.s. and then another conditions under which all E(xik ) (1 ≤ i ≤ n) converge to positive values. The extension of the approach used for the global stability of the two-dimensional DSDEs can lead us to the global stability of the n-dimensional DSDEs, so that we again treat the two-dimensional DSDEs in [5].

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5.1. Global stability of the two-dimensional DSDEs In this section, we consider the two-dimensional model (3) with n = 2 1 x1k+1 = x1k (1 + rˆ1 − a ˆ11 x1k − a ˆ12 x2k + σ ˜1 ξk+1 ), 2 x2k+1 = x2k (1 + rˆ2 + a ˆ21 x1k − a ˆ22 x2k + σ ˜2 ξk+1 ).

(43)

Define A2 and D2i (i = 1, 2) by      1 a11 a12 r1σ D2 A2 = and = A2 . −a21 a22 r2σ D22

(44)

Theorem 2. Let xik and βi be the solutions of (43) and (17), respectively. (a) If riσ < 0 (i = 1, 2), then limk→∞ xik = 0 (i = 1, 2), a.s. (b) If r1σ > 0, r2σ + a21 β1 < 0, then lim E(x1k ) = β1 and lim x2k = 0, a.s. k→∞

(c)

If r1σ > a−1 22 a12 r2σ and where D2i is defined in

r2σ + a21 β1 > 0, then (44).

k→∞ limk→∞ E(xik )

= D2i (i = 1, 2),

Proof. (a) The proof is followed by applying Lemma 3-(a) and Lemma 4. (b) From Lemma 3-(b)-(i) and Lemma 4, we have limk→∞ x2k = 0, a.s. which gives that for every ε > 0, there exists an integer Nε satisfying 0 < x2k ≤ ε, k ≥ Nε .

(45)

Consider the system 1 uLk+1 = uLk (1 + rˆ1 − a ˆ11 uLk − a ˆ12 ε + σ ˜1 ξk+1 ), uL0 = x10 , 1 uUk+1 = uUk (1 + rˆ1 − a ˆ11 uUk + a ˆ12 ε + σ ˜1 ξk+1 ), uU0 = x10 .

Following the proofs of both Lemma 4 and (45), we can have 0 < uLk ≤ x1k ≤ uUk for k ≥ Nε , and then due to (31), we can obtain k−1

k−1


r1σ − a12 ε
r1σ + a12 ε 1X 1X E uLs = , lim E uUs = , lim k→∞ k k→∞ k a a 11 11 s=0 s=0 which gives the desired result. (c) Let |A2 | be the determinant of A2 in (44), which is positive. Then  1     D2 a22 r1σ − a12 r2σ −1 −1 r1σ = A2 = |A2 | D22 r2σ a11 (a21 β1 + r2σ ) and hence the conditions in (c) imply D2i > 0 for i = 1, 2. Applying (44) to (14) with n = 2 gives      1  E (ln x1k ) E (ln x10 ) D2 − E (x1k ) ˚ , = + k ∆tA2 E (ln x2k ) E (ln x20 ) D22 − E (x2k )

(46)

(47)

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and multiplying the matrix |A2 |A−1 2 to (47), we have
˚ 2 | D21 − E (x1 ) , a22 E(ln x1k ) − a12 E(ln x2k ) = C1 + k ∆t|A k
˚ 2 | D2 − E (x2 ) , a21 E(ln x1k ) + a11 E(ln x2k ) = C2 + k ∆t|A 2 k

(48) (49)

where C1 = a22 E(ln x10 ) − a12 E(ln x20 ) and C2 = a21 E(ln x10 ) + a11 E(ln x20 ). Since |A2 | > 0, D21 > 0 in (48), and −a12 E(ln x2k ) > −∞ from Theorem 1, we can follow the proof of Lemma 2 with (48) instead (19) and, as a result, obtain that for every 12 > 0 there exists an integer N12 such that E(x1k ) ≤ D21 + 12 , k ≥ N12 .

(50)

Substituting (42) into (49) gives that for every 22 > 0 there exists an integer N22 satisfying E(x2k ) ≥ D22 − 22 , k ≥ N22 .

(51)

Applying (50) to the second equation in (47), we have for k ≥ N12  2 1 2 ˚ 22 a−1 E(ln x2k ) ≤ E(ln x20 ) + k ∆ta 22 a21 2 + D2 − E(xk ) .

(52)

Instead of (19) and (18), using (52) and (14) with n = i = 2 and D22 > 0, and following the proof of Lemma 2, we can obtain that for every 02 2 > 0 there exists an integer N02 2 such that E(x2k ) ≤ D22 + 02 . (53) 2 , k ≥ N02 2 Hence (51) and (53) imply lim E(x2k ) = D22 .

(54)

k→∞

It remains to show lim E(x1k ) = D21 . Applying (54) to (49) with (42) yields k→∞

1 1 E(ln x1k ) = lim E(ln x2k ) = 0, k→∞ k k→∞ k lim

with which (48) gives the desired result. Remark 8. Assume r1σ < 0. Then it follows from Lemma 3-(a) and Lemma 4 that limk→∞ x1k = 0, a.s. In addition, if r2σ > 0, then we can have limk→∞ E(x2k ) = a−1 22 r2σ by following the proof of Theorem 2-(b) with limk→∞ x1k = 0, a.s., instead of limk→∞ x2k = 0, a.s. Remark 9. Since |A2 | > 0, the identity (46) gives that the two conditions in Theorem 2-(c) are equivalent to D21 > 0 and D22 > 0. This equivalence with |A2 | > 0 is used to find conditions for the global stability of solutions of the n-dimensional model (3) in the next section.

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5.2. Model reduction Using Lemma 3, Remark 5 and Lemma 4, we can find conditions under which at least one of the solutions of (3) converges to zero, a.s. For example, the three-dimensional model (3) with n = 3 can be reduced to the two-dimensional model (3) and hence we can apply Theorem 2 as follows. Theorem 3. Let xik and βi be the solutions of (3) and (17) with n = 3. (a) If r1σ < 0, then limk→∞ x1k = 0, a.s. (b) Assume r1σ > 0. (i) Let riσ +ai1 βi < 0 (i = 2, 3). Then lim xik = 0, a.s. (i = 2, 3) and limk→∞ E(x1k ) = k→∞

β1 . (ii) Let r2σ + a21 β1 < 0 and r3σ + a31 β1 > 0. Then lim x2k = 0, a.s. k→∞
−1 ˇ 23 ). ˇ 21 , D Furthermore, if r1σ > a33 a13 r3σ , then lim E(x1k ), E(x3k ) = (D k→∞

a3i βi < 0. Then lim x3k = 0, a.s. k→∞ i=1
−1 1 > a22 a12 r2σ , then lim E(xk ), E(x2k ) = (D21 , D22 ).

(iii) Let r2σ + a21 β1 > 0 and r3σ + Furthermore, if r1σ

2 P

k→∞

ˇ 21 , D ˇ 23 ) is equal to (D1 , D2 ) in (44) when a12 , a21 and a22 are replaced with a13 , a31 Here (D 2 2 and a33 , respectively. Remark 10. Assume r1σ < 0, which gives limk→∞ x1k = 0, a.s. by Lemma 3-(a) and Lemma 4. Then the three-dimensional model (3) can be considered as the two-dimensional model (3) with two state variables x2k , x3k and a21 = a31 = 0. Therefore, we can apply Theorem 2 and Remark 8. Remark 11. Assume that r3 < 0 and a31 > 0 in Theorem 3-(ii). Then r3σ < 0 and hence −1 r3σ + a31 β1 = r3σ + a31 a−1 11 r1σ > 0 gives r1σ > 0 ≥ a33 a13 r3σ . Therefore Theorem 3-(ii) is −1 satisfied without using the condition r1σ > a33 a13 r3σ if r3 < 0. Similarly, if r2 < 0 and a21 > 0, then Theorem 3-(iii) is satisfied without the condition r1σ > a−1 22 a12 r2σ . 5.3. Convergence of all state variables to nonzero values Now, we find conditions under which E(xik ) (1 ≤ i ≤ n) converge to positive values Dni . The conditions are extensions of the following three conditions used in the proof of Theorem 2-(c): a11 a12 = a11 a22 + a12 a21 > 0. (A1) |A2 | = −a21 a22 (A2) D2i > 0, i = 1, 2 (see Remark 9). (A3) The system of equations (48) and (49) has the sign-pattern matrix     a22 −a12 + −0 sgn = , a21 a11 +0 + which means that the signs of a22 , −a12 , a21 , and a11 are positive, non-positive, nonnegative, and positive, respectively.

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Define An and Dni , extensions of (44), by       a11 a12 · · · a1(n−1) a1n r1σ Dn1  −a21 a22 · · · a2(n−1) a2n   r2σ  D 2       n An =  .. .. .. ..  ,  ..  = An  ..  , . .  .  .  . . . .   .  −an1 −an2 · · · −an(n−1) ann rnσ Dnn

(55)

with the assumptions corresponding to (A1) and (A2): (A10 ) |An | > 0. (A20 ) Dni > 0 for 1 ≤ i ≤ n. Using (55), we can write (14) as ( i−1 ) n X X

˚ − aij Dnj − E(xjk ) + E(ln xik ) = E(ln xi0 ) + k ∆t aij Dnj − E(xjk ) , j=1

j=i

which is corresponding to (47). Hence (48) and (49) are extended as X X  ˚ n | Di − E(xi ) , Cji E(ln xjk ) = Cji E(ln xj0 ) + k ∆t|A k n 1≤j≤n

1≤j≤n

where cofactors Cij of An satisfy the    + C11 · · · Cn1  ..  ..  =  . .. (A30 ) sgn  ... . .   −0 C1n · · · Cnn +0

(56)

assumption corresponding to (A3):  −0 · · · −0 −0 .. . . . ..  . .. . .  . −0 · · · + −0  +0 · · · +0 +

Remark 12. It follows from (17) and (55) that (A20 ) gives X a11 β1 = r1σ = a1j Dnj ≥ a11 Dn1 > 0, 1≤j≤n X a2j Dnj ≥ a22 Dn2 . a22 β2 = r2σ + a21 β1 ≥ r2σ + a21 Dn1 = 2≤j≤n

P Repeating this process, we can conclude that (A20 ) implies riσ + i−1 j=1 aij βj > 0 (1 ≤ i ≤ n), and then we can use (42) in the proof of the following theorem. Now we can extend the proof of Theorem 2-(c) to the n-dimensional model (3). Theorem 4. Let xik be the solutions of (3) and Dni be defined in (55). Assume that (A10 )–(A30 ) are satisfied. Then
1 Xk−1 E xis = Dni , 1 ≤ i ≤ n. s=0 k→∞ k lim

Proof. Instead of (48), using (56) for 1 ≤ i ≤ n − 1 and following the proof of (50) with (A10 ) and (A20 ), we can obtain that for every in > 0 there exists an integer Nin such that E(xik ) ≤ Dni + in , 1 ≤ i ≤ n − 1, k ≥ Nin .

(57)

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Substituting (42) into (56) with i = n yields that for every nn > 0 there exists an integer Nnn such that E(xnk ) ≥ Dnn − nn , k ≥ Nnn . (58) As in the proof of (53), we can have that for every 0n n > 0 there exists an integer N0n n such that E(xnk ) ≤ Dnn + 0n (59) n , k ≥ N0n n by replacing both (50) and the second equation in (47) with (57) and ( n−1 ) X

˚ E(ln xn ) = E(ln xn ) + k ∆t anj E(xj ) − Dj − ann E(xn ) − Dn , k

0

k

n

k

n

j=1

respectively. Hence (58) and (59) give lim E(xnk ) = Dnn .

(60)

k→∞

Now it remains to show limk→∞ E(xik ) = Dni (1 ≤ i ≤ n − 1). Substituting (60) to (56) with i = n and using both Cin ≥ 0 (1 ≤ i ≤ n) in (A30 ) and (42), we can obtain 1 E(ln xik ) = 0, 1 ≤ i ≤ n − 1, k→∞ k lim

with which (56) gives the desired result. Remark 13. Following the proof of Lemma 1 under the assumptions about the noise ξ, we can obtain the new discretized Itˆo formula for the Milstein method     √ 1 2 2 E φ 1 + hf + hgξ + hg · (ξ − 1) F 2   1 2 1 0 = φ(1) + φ (1) hf − hg µ + φ00 (1)hg 2 · (1 − µ) + h(f + g 2 )O (hε ) , 2 2 which gives     √ 2 2 E ln 1 + hf + hgξ + 0.5hg · (ξ − 1) F 
= h f − 0.5g 2 + (f + g 2 )O (hε ) . √ √ 2 Therefore, replacing riσ = ri − 0.5σiη and ∆tσi ξki with riσ = ri − 0.5σi2 and ∆tσi ξki + 0.5∆tσi2 {(ξki )2 − 1} for 1 ≤ i ≤ n, respectively, we can conclude that the solutions of the Milstein scheme for (2) satisfy all the results in this paper.

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6. Numerical examples In this section, we provide some simulations that illustrate our results in Theorem 3 and 4 for the three-dimensional model (3) with ∆t = 0.001 and ς = 20 in (4). Let r1 = 1.1, a11 = 2.1, a12 = 0.2, a13 = 0.1, a21 = 0.5, a22 = 1.1, a23 = 0.1, a31 = 3.1, a32 = 5.1, a33 = 0.5 and σ2 = σ3 = 0.1 in the model (3) with n = 3. The model is simulated 1000 and 5000 times in Figure 1 and 2, respectively, for calculating the values of expectations. Example 1. Figure 1-(a) denotes three plots in the first column of Figure 1. Each figure in the ith row of Figure 1 is the sequence of the 1000 realizations of xik (i = 1, 2, 3). In Figure 1-(a), we let (−r2 , −r3 ) = (0.1, 2.1) and σ1 = 2. The values (−r2 , −r3 ) in Figure 1-(b), (c) and (d) are (5.1, 5.1), (5.1, 1.1), (0.1, 5.1), respectively, with σ1 = 0.1. Then the values of parameters in Figure 1-(a), (b), (c) and (d) satisfy the conditions in Theorems 3(a), (b)-(i), (ii) and (iii), respectively, and Figure 1 demonstrates Theorem 3: For example, all xik (i = 1, 2, 3) in Figure 1-(a) converge to zero. At k = 2 · 106 , which means time 2000, i ˇi the errors |E(x1k ) − a−1 11 r1σ | in Figure 1-(b), |E(xk ) − D2 | (i = 1, 3) in Figure 1-(c), and |E(xik )−D2i | (i = 1, 2) in Figure 1-(d) are 0.00012, 0.000075, 0.0019, 0.000038 and 0.00068, respectively. −4 (a) ×10

(b)

(c)

(d)

0.5

0.5

0.5

0.3

0.3

0.3

−8 1.0 ×10

−8 1.0 ×10

0.4

1

x

1.0 0

0 50 −4 1.0 ×10

100

x2 0.2 0 −4 1.0 ×10

0

0

−8 1.0 ×10

1.5

x3

0 −8 1.0 ×10

1 0

0

0.2

0

Figure 1: All the x-axes denote time k∆t from 0 to 100. The curves in each figure are 1000 realizations of one of the solutions xik (1 ≤ i ≤ 3) of DSDEs. The solutions xik (1 ≤ i ≤ 3) in (a), xik (i = 2, 3) in (b), x2k in (c), x3k in (d) are all convergent to zero.

Example 2. Let r2 = −0.1, r3 = −2.1 and σ1 = 0.1. Then all the conditions in Theorem 4 are satisfied. Consequently, the solutions xik (1 ≤ i ≤ 3) are positive, which is demonstrated in Figure 2-(a), and E(xik ) are convergent to positive values D3i , which is demonstrated in Figure 2-(b). At k = 5 · 106 , which means time 5000, the error vector (|E(x1k ) − D31 |, |E(x2k ) − D32 |, |E(x3k ) − D33 |) is marked with the larger star in Figure 2-(b) and equal to (0.000028, 0.00011, 0.0018).

7. Conclusion In this paper, dealing with the n-dimensional stochastic difference model, we have extended the new approach to obtain the global stability of the fixed point of the twodimensional stochastic and discrete predator-prey system. As in the two-dimensional model, we have found the conditions under which at least one discrete solution converges to 16 131

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(b)

x3

|E(x3k ) − D33 |

(a)

2 0.14

2.0

2.0 1.0 2 0.12

1 0.5

1.0 0. 1.5

x1

|E(x2k ) − D32 |

x

1.5 0

0

|E(x1k ) − D31 |

Figure 2: The start and the final points of each curve are denoted by the circle and the star symbols, respectively. Two thicker curves in (a) and (b) show one (x1k , x2k , x3k ) (0 ≤ k ≤ 106 ) and
realization of 1 1 2 2 3 3 6 error vectors |E(xk ) − D3 |, |E(xk ) − D3 |, |E(xk ) − D3 | (0 ≤ k ≤ 5·10 ), respectively. Each of the other curves is the projection of one of the thicker curves onto one plane: The two planes in Figure 2-(a) are x1 = 2 and x2 = 2.

zero by using a model reduction method. Additionally, we have also found the conditions under which all expectations of solutions globally converge to non-zero fixed values. While proving the global stability of solutions of the two- and n-dimensional stochastic difference models, we have used our new discretized form of the Itˆo formula and therefore found the possibility that the new discrete Itˆo formula will be applied to other stochastic difference models.

Appendix A.1. The proof of Lemma 1 By Taylor expansion, ϕ00 (1) 2 ϕ000 (θ) 3 x + x 2 6 √ with θ lying between 1 and x. We substitute x = hf + hgξ and take expectations. Since ξ is F-independent with E(ξ) = 0 and E(ξ 2 ) = 1 − µ, and f, g are F-measurable, we have
E (x| F) = hf, E x2 F = (hf )2 + hg 2 · (1 − µ). ϕ(1 + x) = ϕ(1) + ϕ0 (1)x +

Note that   000 C
ϕ (θ) 3 E x F 6 E x3 F = hf O (hε ) + hg 2 O (hε ) 6 6 by expanding x3 and using E (|ξ i |) < ∞ (i = 1, 2, 3) and (12). Therefore E (ϕ(1 + x) |F ) = ϕ(1) + ϕ0 (1)hf +

(61) 00

ϕ (1) 2 hg · (1 − µ) + hf O (hε ) + hg 2 O (hε ) . 2

(62)

Now it is enough to show       √ √ E φ 1 + hf + hgξ − ϕ 1 + hf + hgξ F = hg 2 O (hε ) . 17 132

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Since f, g are F-measurable and ξ is F -independent, letting c1 = 1 + hf, c2 = Uδ = [1 − δ, 1 + δ] gives       √ √ E φ 1 + hf + hgξ − ϕ 1 + hf + hgξ F Z = {φ (c1 + c2 x) − ϕ (c1 + c2 x)} p(x) dx   ZR s − c1 ds {φ (s) − ϕ (s)} p = c2 |c2 | R−Uδ

√ hg and

(63)

where p is the probabilty density function of ξ. Note that Z   s − c ds 1 {φ (s) − ϕ (s)} p c2 |c2 | R−Uδ      Z s − c1 1 ds sup p 6 |φ(s) − ϕ(s)| |c2 | s∈U c2 |c2 | / δ R−Uδ     s − c1 1 6 C · |c2 |2 sup p c2 |c2 |3 s∈U / δ       s − 1 − hf  1  2 √ = C · hg sup p √ 3 .  hg s∈U / δ  hg  Here letting y =

s−1−hf √ , hg

we have

    s − 1 − hf  √ sup p hg s∈U / δ 

  

1

(

√ 3 = sup  / δ hg  s∈U

p (y) |y|3 |s − 1 − hf |3

)

For all s ∈ / Uδ , there exists some δ0 such that for all sufficiently small h > 0 |s − 1 − hf | > |s − 1| − h|f | > δ − Ch > δ0 > 0 and then we have |y| ≡

δ0 |s − 1 − hf | √ > . O (hε ) h |g|

(64)

(65)

Hence it follows from (64), (65) and the assumption (b) that ( ) p (y) |y|3 sup = O (hε ) , 3 |s − 1 − hf | s∈U / δ which gives Z

R−Uδ

 {φ (s) − ϕ (s)} p

s − c1 c2



ds = hg 2 O (hε ) . |c2 |

(66)

Therefore using (61), (63) and (66), we obtain the desired result.

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A.2. The proof of Remark 5 Using (13) and (14) with a3j = 0 (4 ≤ j ≤ n), we have

3 ˚ r3σ + a31 zk1 + a32 zk2 − a33 zk3 , E ln zk+1 = ln zk3 + ∆t (67) 

  ˚ 33 a−1 r3σ + a31 E(z 1 ) + a32 E(z 2 ) − E(z 3 ) . (68) E ln z 3 = E ln z 3 + k ∆ta k

0

33

k

k

k

A.2.1. The proof of Remark 5-(a)-(i) Assume that r1σ > 0, r2σ + a21 β1 < 0, r3σ + a31 β1 < 0. Lemma 3-(b) and (b)-(i) give that limk→∞ E(zk1 ) = β1

(69)

limk→∞ zk2 = 0,

(70)

and respectively. Applying the dominated convergence theorem with (70), we have lim E(zk2 ) = 0

k→∞

and then limk→∞ E(zk2 ) = 0. Substituting (69) and (71) into (68) and using the positivity of
limk→∞ E ln zk3 = 0.

(71) zk3 ,

we obtain (72)

Therefore combining (72) and (67) with the boundedness of zki (i = 1, 2, 3) in (16), we can obtain the desired result: lim zk3 = 0, a.s. k→∞

A.2.2. The proof of Remark 5-(a)-(ii) Assume that r1σ > 0, r2σ + a21 β1 < 0, r3σ + a31 β1 > 0 and then limk→∞ E(zk1 ) = β1 , limk→∞ E(zk2 ) = 0, β˜3 = a−1 33 (r3σ + a31 β1 ) > 0.

(73)

Following the proof of Lemma 2, we can obtain that for every  > 0 and some N > 0 Xk−1
k −1 E zs3 ≤ β˜3 + , k ≥ N , (74) s=0

by replacing zk1 , β1 ,(18) and (19) in Lemma 2 with zk3 , β˜3 , (67) and (68), respectively and using (73). Hence it remains to show that for every  > 0 and some N > 0 Xk−1
β˜3 −  ≤ k −1 E zs3 , k ≥ N . (75) s=0

Following the proof of (30) instead of (31), we can obtain (75), which gives the P with (68) 3 desired result : lim k −1 k−1 E (z ) = β˜3 . s s=0 k→∞

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A.2.3. The proof of Remark 5-(b)-(i) Assume that X2 r1σ > 0, r2σ + a21 β1 > 0, r3σ +

j=1

and then

a3j βj < 0

 X2 r3σ + lim E(zki ) = βi (i = 1, 2), β3 = a−1 33

j=1

k→∞

 a3j βj < 0,

(76)

due to Lemma 3-(b) and (b)-(ii). Combining (76) and (68) with the positivity of zk3 , we can have
limk→∞ E ln zk3 = −∞. Therefore, as (22) implies (23), we can obtain the desired result: lim zk3 = 0, a.s. k→∞

A.3. The proofs of Remark 5-(b)-(ii) and Remark 6 Using mathematical induction, we prove that P i if riσ + i−1 j=1 aij βj > 0 (1 ≤ i ≤ n), then limk→∞ E(zk ) = βi (1 ≤ i ≤ n). Lemma 3-(a) gives that if r1σ > 0, then limk→∞ E(zk1 ) = β1 . Assume that for some ` ∈ {2, · · · , n} riσ +

i−1 X

aij βj > 0 (1 ≤ i ≤ `) and lim E(zki ) = βi (1 ≤ i ≤ ` − 1). k→∞

j=1

Using (15) with i = ` instead of (3), the equations (13) and (14) become   X`−1 ` ˚ r`σ + E(ln zk+1 ) = ln zk` + ∆t a`j zkj − a`` zk` , j=1 n o X`−1 j ` ˚ `` a−1 E(ln zk` ) = E(ln z0` ) + k ∆ta a E(z E(z ) . ) − `j k `` k j=1

(77)

(78) (79)

Following the proof of Lemma 2, we can obtain that for every  > 0 and some N > 0 lim E(zk` ) ≤ β` + , k ≥ N ,

k→∞

(80)

by replacing zk1 , β1 ,(18) and (19) in Lemma 2 with zk` , β` , (78) and (79), respectively and using (77). Hence it remains to show that for every  > 0 and some N > 0 β` −  ≤ lim E(zk` ), k ≥ N . k→∞

(81)

Following the proof of (30) with (79) instead of (31), we can obtain (81), which gives lim E(zk` ) = β` .

k→∞

Therefore the desired result is obtained. 20 135

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Conflict of Interests The authors declare that there is no conflict of interests regarding the publication of this paper. Acknowledgments This work was supported by the 2016 Research Fund of University of Ulsan. References [1] Berkolaiko, G., Kelly, C., Rodkina, A.: Sharp pathwise asymptotic stability criteria for planar systems of linear stochastic difference equations. Discrete Contin. Dyn. Syst. (Dynamical systems, differential equations and applications. 8th AIMS Conference. Suppl. Vol. I), 163–173 (2011) [2] Berkolaiko, G., Buckwar, E., Kelly, C., Rodkina, A.: Almost sure asymptotic stability analysis of the θ-Maruyama method applied to a test system with stabilising and destabilising stochastic perturbations. LMS J. Comput. Math. 15, 71–83 (2012). doi:10.1112/S1461157012000010 [3] Paternoster, B., Shaikhet, L.: Application of the general method of Lyapunov functionals construction for difference Volterra equations. Comput. Math. Appl. 47(8-9), 1165–1176 (2004). doi:10.1016/S0898-1221(04)90111-3 [4] Appleby, J.A.D., Guzowska, M., Kelly, C., Rodkina, A.: Preserving positivity in solutions of discretised stochastic differential equations. Appl. Math. Comput. 217(2), 763–774 (2010). doi:10.1016/j.amc.2010.06.015 [5] Choo, S.M., Kim, Y.H.: Global stability in stochastic difference equations for predator-prey models. J. Comput. Anal. Appl. 23(3), 462–486 (2017) [6] Bahar, A., Mao, X.: Stochastic delay population dynamics. Int. J. Pure Appl. Math. 11(4), 377–400 (2004) [7] Liu, M., Qiu, H., Wang, K.: A remark on a stochastic predator-prey system with time delays. Appl. Math. Lett. 26(3), 318–323 (2013). doi:10.1016/j.aml.2012.08.015 [8] Appleby, J.A.D., Berkolaiko, G., Rodkina, A.: Non-exponential stability and decay rates in nonlinear stochastic difference equations with unbounded noise. Stochastics 81(2), 99–127 (2009). doi:10.1080/17442500802088541 [9] Kelly, C., Palmer, P., Rodkina, A.: Almost sure instability of the equilibrium solution of a Milstein-type stochastic difference equation. Comput. Math. Appl. 66(11), 2220– 2230 (2013). doi:10.1016/j.camwa.2013.06.020 [10] Palmer, P.: Application of a discrete Itˆo formula to determine stability (instability) of the equilibrium of a scalar linear stochastic difference equation. Comput. Math. Appl. 64(7), 2302–2311 (2012). doi:10.1016/j.camwa.2012.03.012 [11] Choo, S.M.: Global stability in n-dimensional discrete lotka-volterra predator-prey models. Adv. Difference Equ., 2014–1117 (2014) 21 136

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Entire solutions of certain type of nonlinear differential equations and differential-difference equations

∗

Min Feng Chen†and Zong Sheng Gao LMIB and School of Mathematics and Systems Science, Beihang University, Beijing, 100191, P. R. China

Abstract. In this paper, we investigate the transcendental entire solutions of finite order of the differential equations or differential-difference equations f 2 (z) + P (f ) = p1 eα1 z + p2 eα2 z , where p1 , p2 , α1 , α2 are nonzero constants with α1 6= α2 , and P (f ) denotes a differential polynomial or differential-difference polynomial in f (z) with degree 1. And we partially answer a question proposed by Li [10] (P. Li, Entire solutions of certain type of differential equations II, J. Math. Anal. Appl. 375(2011), 310 − 319). Mathematics Subject Classification (2010). 39B32, 34M05, 30D35. Keywords. Entire solutions, Differential equations, Differential-difference equations, finite order.

1

Introduction and Results

In this paper, we assume that the reader is familiar with standard symbols and fundamental results of Nevanlinna theory [1, 2]. In addition, we denote by S(r, f ) any quantify satisfying S(r, f ) = o(T (r, f )), as r → ∞, outside of a possible exceptional set of finite logarithmic measure. Recently, a number of papers (including [3 − 12]) have focused on solvability and existence of meromorphic solutions of differential equations, difference equations or differentialdifference equations in complex plane. Specifically, it shows in [4] that the equation 4f 3 + 00 3f = − sin 3z has exactly three nonconstant entire solutions, namely f1 (z) = sin z, f2 (z) = √ √ 3 3 1 1 cos z − sin z and f (z) = − cos z − 3 2 2 2 2 sin z. The following two Theorems obtained more general results. Theorem A. (See [5]) Let n ≥ 2 be an integer, P (f ) be a differential polynomial in f (z) of degree at most n − 2, and λ, p1 , p2 be three nonzero constants. If f (z) is an entire solution of the following equation f n (z) + P (f ) = p1 eλz + p2 e−λz , (1.1) ∗ This research was supported by the National Natural Science Foundation of China (No: 11171013, 11371225). † Corresponding author. E-mail: [email protected].

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then f (z) = c1 eλz/n + c2 e−λz/n , where c1 and c2 are constants and c2i = pi , i = 1, 2. Theorem B. (See [10]) Let n ≥ 2 be an integer, P (f ) be a differential polynomial in f (z) of degree at most n − 2, and p1 , p2 , α1 , α2 be nonzero constants and α1 6= α2 . If f (z) is a transcendental meromorphic solution of the following equation f n (z) + P (f ) = p1 eα1 z + p2 eα2 z ,

(1.2)

and satisfying N (r, f ) = S(r, f ), then one of the following holds: (i) f (z) = c0 (z) + c1 eα1 z/n ; (ii) f (z) = c0 (z) + c2 eα2 z/n ; (iii) f (z) = c1 eα1 z/n + c2 eα2 z/n and α1 + α2 = 0, where c0 (z) is a small function of f (z) and c1 , c2 are constants satisfying c2i = pi , i = 1, 2. For further study, Li [10] proposed the following question. Question. How to find the solutions of (1.2) under the condition deg P (f ) = n − 1? In this paper, we study this question and partially answer this question, and obtain the following result. Theorem 1.1. Let p1 , p2 , α1 , α2 be nonzero constants such that α1 6= α2 , a(z) be a nonzero polynomial. If f (z) is a transcendental entire solution of finite order of the differential equation f 2 (z) + a(z)f 0 (z) = p1 eα1 z + p2 eα2 z , (1.3)   and satisfying N r, f1 = S(r, f ), then a(z) must be a constant, and one of the following holds: (i) f (z) = c1 eα1 z/2 , ac1 α1 = 2p2 , α1 = 2α2 ; (ii) f (z) = c2 eα2 z/2 , ac2 α2 = 2p1 , α2 = 2α1 ; where c1 , c2 are constants satisfying c2i = pi , i = 1, 2.   Remark 1.1. We conjecture that the condition N r, f1 = S(r, f ) in Theorem 1.1 can be omitted using another method although we have not found a suitable one yet. In [11], Wang and Li investigated the existence of entire solutions of differential-difference equation f n (z) + q(z)f (k) (z + c) = aeibz + de−ibz , (1.4) and obtained the following Theorem. Theorem C. (See [11]) For two integers n ≥ 3, k ≥ 0 and a nonlinear differentialdifference equation (1.4), where q(z) is a polynomial and a, b, c, d are constants such that |a| + |d| = 6 0, bc 6= 0. (i) Let n = 3. If q(z) is nonconstant, then the equation (1.4) does not admit entire solutions of finite order. If q := q(z) is a constant, then equation (1.4) admits three distinct transcendental entire solutions of finite order, provided that  3k 3i 3 m+1 27ad, bc = 3mπ (m 6= 0, if q 6= 0), q = (−1) b

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when k is an even, or bc =

3π + 3mπ (if q 6= 0), q 3 = i(−1)m 2



3i b

3k 27ad,

when k is an odd, for an integer m. (ii) Let n ≥ 3. If ad 6= 0, then the equation (1.4) does not admit entire solutions of finite order. If ad = 0, then equation (1.4) admits n distinct transcendental entire solutions of finite order, provided that q := q(z) ≡ 0. In this paper, we shall tackle differential-difference equations in the form (1.4) with n = 2, and obtain the following result. Theorem 1.2. Let p1 , p2 , λ, c be nonzero constants, k ≥ 0 be an integer and a(z) be a nonzero polynomial. If f (z) is a transcendental entire solution of finite order of the differential-difference equation f 2 (z) + a(z)f (k) (z + c) = p1 eλz + p2 e−λz ,

(1.5)

then a(z) must be a constant, and satisfying one of the following relations: (i) f (z) = ± 2i a( λ2 )k + c1 eλz/2 + c2 e−λz/2 , eλc = −1, when k is an odd; (ii) f (z) = ± 21 a( λ2 )k + c1 eλz/2 + c2 e−λz/2 , eλc = 1, when k is an even and k > 0, 1 4 λ 4k where a, c1 , c2 are constants with 64 a ( 2 ) = p1 p2 and c2i = pi , i = 1, 2; 1 λz/2 −λz/2 + c2 e , eλc = 1, when k = 0, (iii) f (z) = ± 2 a + c1 e 9 4 1 4 a = p1 p2 or 64 a = p1 p2 and c2i = pi , i = 1, 2. where a, c1 , c2 are constants with 64 Example 1.1. f (z) = ez is a transcendental entire solution of finite order of the differential equation f 2 (z) + f 0 (z) = e2z + ez . Example 1.2. The differential-difference equation 1 4z 9 e + e−4z . 4 16

f 2 (z) + f (z + πi/2) =

+ 21 e2z − 34 e−2z and f2 (z) =

1 2

has exactly two entire solutions, namely f1 (z) = 3 −2z . 4e

1 2

− 12 e2z +

Example 1.3. The differential-difference equation f 2 (z) − f 0 (z + πi/2) = has exactly two entire solutions, namely f1 (z) =

i 2

1 2z 1 e + e−2z . 4 16 + 12 ez + 14 e−z and f2 (z) =

i 2

− 21 ez − 14 e−z .

Example 1.4. The differential-difference equation f 2 (z) − 2f 00 (z + πi) =

1 2z e + e−2z . 4

has exactly two entire solutions, namely f1 (z) = −1 + 21 ez − e−z and f2 (z) = −1 − 21 ez + e−z .

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2

Lemmas for the Proof of Theorems

Lemma 2.1. (Clunie’s Theorem)(See [2, 13]) Let f (z) be a transcendental meromorphic solution of f n (z)P (z, f ) = Q(z, f ), where P (z, f ) and Q(z, f ) are polynomials in f (z) and its derivatives with meromorphic coefficients, say {aλ |λ ∈ I}, such that m(r, aλ ) = S(r, f ) for all λ ∈ I. If the total degree of Q(z, f ) as a polynomial in f (z) and its derivatives is ≤ n, then m(r, P (z, f )) = S(r, f ). Lemma 2.2. (See [14])

(2.1)

Let f (z) be a nonconstant finite order meromorphic solution of f n (z)P (z, f ) = Q(z, f ),

where P (z, f ) and Q(z, f ) are polynomials in f (z) and its shifts with small meromorphic coefficients, and let c ∈ C, δ < 1. If the total degree of Q(z, f ) as a polynomial in f (z) and its shifts is ≤ n, then   T (r + |c|, f ) + o(T (r, f )) (2.2) m(r, P (z, f )) = o rδ for all r outside of a possible exceptional set with finite logarithmic measure. Remark 2.1. In Lemma 2.2, if f (z) is transcendental with finite order, and P (z, f ), Q(z, f ) are differential-difference polynomials in f (z), then by using a similar method as in the proof of Lemma 2.1, we see that a similar conclusion of Lemma 2.2 holds. Moreover, we know that if the coefficients of P (z, f ) and Q(z, f ) are polynomials aj (z), j = 1, . . . , k, then (2.2) can be replaced by   k X m(r, P (z, f )) = S(r, f ) + O  m(r, aj (z)) , (2.3) j=1

where r is sufficiently large. Lemma 2.3. (See [1, 2]) an integer. Then

Let f (z) be a transcendental meromorphic function and k ≥ 1 be f (k) (z) m r, f (z) 

 = S(r, f ).

(2.4)

Lemma 2.4. Let α be a nonzero constant, and H(z) be a nonvanishing polynomial. Then the differential equation αf (z) − 2f 0 (z) = H(z) (2.5) has a special solution c0 (z) which is a nonzero polynomial. Proof. Similarly to the proof of [Lemma 2.3, 8]. If H(z) is a nonzero constant, then c0 (z) = H(z) is a special solution of (2.5). Now suppose that α H(z) = an z n + an−1 z n−1 + · · · + a1 z + a0 ,

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where n ≥ 1 is an integer, an 6= 0, an−1 , . . . , a1 , a0 are constants. We use the method of undetermined coefficients to derive the polynomial solution c0 (z) satisfying (2.5) by α, an , an−1 , . . . , a1 , a0 . By (2.5), we see that deg c0 (z) = deg H(z). For n = 1, equation (2.5) has a polynomial solution c0 (z) = α1 a1 z + α1 (a0 + α2 a1 ). In a general case, for n ≥ 2, (2.5) has a polynomial solution c0 (z) = bn z n + · · · + bj z j + · · · + b1 z + b0 , where

1 1 2(j + 1) an , bj = aj + bj+1 , j = n − 1, . . . , 0. α α α Therefore, (2.5) has a nonzero polynomial solution c0 (z). bn =

Lemma 2.5. (See [8]) Let λ be a nonzero constant, and H(z) be a nonvanishing polynomial. Then the differential equation 4f 00 (z) − λ2 f (z) = H(z)

(2.6)

has a special solution c0 (z) which is a nonzero polynomial. Lemma 2.6. (See [15]) Suppose that f1 (z), f2 (z), . . . , fn (z)(n ≥ 2) are meromorphic functions and g1 (z), g2 (z), . . . , gn (z) are entire functions satisfying the following conditions, Pn (1) j=1 fj (z)egj (z) ≡ 0; (2) gj (z) − gk (z) are not constants for 1 ≤ j < k ≤ n; (3) For 1 ≤ j ≤ n, 1 ≤ h < k ≤ n, T (r, fj (z)) = o(T (r, egh (z)−gk (z) ))(r → ∞, r 6∈ E), where E ⊂ [1, ∞) is finite linear measure or finite logarithmic measure. Then fj (z) ≡ 0 (j = 1, . . . , n). Lemma 2.7. Let P (z) be a nonzero polynomial, A, B, c be nonzero constants and A 6= 1. If P 2 (z) ≡ AP (z)P (z + c) + B,

(2.7)

then P (z) must be a constant, P (z) := p(constant). Proof. If P (z) is a nonconstant polynomial, then deg P (z) ≥ 1. Now suppose that P (z) = an z n + an−1 z n−1 + · · · + a0 , where n ≥ 1 is an integer, an 6= 0, an−1 , . . . , a0 are constants, then P (z + c) = an (z + c)n + an−1 (z + c)n−1 + · · · + a0 , P 2 (z) = a2n z 2n + 2an an−1 z 2n−1 + · · · , P (z)P (z + c) = a2n z 2n + (na2n c + 2an an−1 )z 2n−1 + · · · , P 2 (z) − AP (z)P (z + c) = (1 − A)a2n z 2n + · · · . From A 6= 1, we see that deg[P 2 (z) − AP (z)P (z + c)] = 2n ≥ 2, it contradicts with (2.7). Thus P (z) must be a nonzero constant, set P (z) := p(6= 0).

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3

Proof of Theorems

Proof of Theorem 1.1 Denote P = a(z)f 0 (z). Suppose that f (z) is a transcendental entire solution of finite order of equation (1.3). By differentiating (1.3), we have 2f f 0 + P 0 = α1 p1 eα1 z + α2 p2 eα2 z .

(3.1)

Eliminating eα1 z , eα2 z from (1.3) and (3.1), respectively, we have α1 f 2 − 2f f 0 + α1 P − P 0 = (α1 − α2 )p2 eα2 z ,

(3.2)

α2 f 2 − 2f f 0 + α2 P − P 0 = (α2 − α1 )p1 eα1 z .

(3.3)

Differentiating (3.3) yields 2α2 f f 0 − 2(f 0 )2 − 2f f 00 + α2 P 0 − P 00 = α1 (α2 − α1 )p1 eα1 z .

(3.4)

It follows from (3.3) and (3.4) that ϕ = −Q,

(3.5)

ϕ = α1 α2 f 2 − 2(α1 + α2 )f f 0 + 2(f 0 )2 + 2f f 00 ,

(3.6)

Q = α1 α2 P − (α1 + α2 )P 0 + P 00 .

(3.7)

where

and

If ϕ 6≡ 0, by (3.5) − (3.7) and Lemma 2.3, we see that     ϕ ϕ m r, = S(r, f ) and m r, 2 = S(r, f ). f f   From N r, f1 = S(r, f ) and (3.8), we see that

(3.8)

      ϕ 1 1 ≤ m r, + m r, T (r, f ) + S(r, f ) = m r, f f ϕ ≤ T (r, ϕ) + S(r, f ) = m(r, ϕ) + S(r, f )   ϕ ≤ m r, + m(r, f ) + S(r, f ) f = T (r, f ) + S(r, f ), that is, T (r, ϕ) = T (r, f ) + S(r, f ).

(3.9)

From (3.8) and (3.9), we see that     1 1 2T (r, f ) + S(r, f ) = 2m r, = m r, 2 f f     ϕ 1 ≤ m r, 2 + m r, f ϕ ≤ T (r, ϕ) + S(r, f ) = T (r, f ) + S(r, f ),

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that is T (r, f ) ≤ S(r, f ), a contradiction. Then we have ϕ ≡ 0, that is Q ≡ 0. By (3.7), we have α1 α2 P − (α1 + α2 )P 0 + P 00 ≡ 0. (3.10) Since P = a(z)f 0 (z), it is obvious that P 6≡ 0. Since α1 6= α2 , we see that α1 P − P 0 ≡ 0 and α2 P − P 0 ≡ 0 cannot hold simultaneously. Suppose α1 P − P 0 6≡ 0. By (3.10), we get α1 P − P 0 = Aeα2 z ,

(3.11)

where A is a nonzero constant. Substituting (3.11) into (3.2), we have f (α1 f − 2f 0 ) =

[(α1 − α2 )p2 − A]α1 (α1 − α2 )p2 − A 0 P− P . A A

(3.12)

Since the right-hand side of (3.12) is a differential polynomial in f (z) of degree ≤ 1. By Lemma 2.1, we have m(r, α1 f − 2f 0 ) = S(r, f ). (3.13) If α1 f − 2f 0 6≡ 0, since f (z) is a transcendental entire function of finite order, by (3.13), we see that m(r, α1 f − 2f 0 ) = S(r, f ) = O(log r), which implies that α1 f − 2f 0 is a nonzero polynomial. Then we have α1 f − 2f 0 = H(z),

(3.14)

where H(z) is a nonvanishing polynomial, but H(z) may be a nonzero constant. By Lemma 2.4, we know that (3.14) has a nonzero polynomial solution, say, c0 (z). Since the differential equation α1 f − 2f 0 = 0 (3.15) has a fundamental solution f (z) = eα1 z/2 . Then the general entire solution f (z) of (3.14) can be expressed as f (z) = c0 (z) + c1 eα1 z/2 , (3.16) where c1 is a constant, c0 (z) is a nonzero polynomial. Substituting (3.16) into (1.3), we have   α1 (c21 − p1 )eα1 z − p2 eα2 z + 2c0 (z) + a(z) c1 eα1 z/2 + c20 (z) + a(z)c00 (z) = 0. 2 By α1 6= α2 , if α2 6= α21 , by Lemma 2.6, we see that p2 = 0, a contradiction. If α2 = α21 , then (3.17) can be rewritten as h  i α1 (c21 − p1 )eα1 z + 2c0 (z) + a(z) c1 − p2 eα2 z + c20 (z) + a(z)c00 (z) = 0. 2 By α1 6= α2 and Lemma 2.6, we have   α1 a(z) c1 − p2 ≡ 0, c20 (z) + a(z)c00 (z) ≡ 0, 2c0 (z) + 2

(3.17)

(3.18)

(3.19)

then c0 (z) ≡ 0, a contradiction. Therefore, we have α1 f − 2f 0 ≡ 0, which yields f 2 = Beα1 z ,

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(3.20)

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where B is a nonzero constant. By (1.3), (3.11) and (3.20), we have 

1−

p1  2 α1 p2 − A p2 f = P − P 0. B A A

(3.21)

If p1 6= B, then by (3.21) and Lemma 2.1, we get T (r, f ) = S(r, f ), which is impossible. Therefore, p1 = B and f (z) = c1 eα1 z/2 , where c1 is a nonzero constant satisfying c21 = p1 . Substituting f (z) = c1 eα1 z/2 and c21 = p1 into (1.3), we have c1 α1 a(z)eα1 z/2 − p2 eα2 z = 0. 2

(3.22)

By α1 6= α2 and Lemma 2.6, we see that α1 = 2α2 and c1 α1 a(z) ≡ 2p2 , then a(z) must be a constant, set a(z) := a. If α2 P − P 0 6≡ 0, then by a similar method, we can deduce that f (z) = c2 eα2 z/2 , α2 = 2α1 , c2 α2 a = 2p1 , c22 = p2 . This completes the proof of Theorem 1.1. Proof of Theorem 1.2 Denote P1 (f ) = a(z)f (k) (z + c). Suppose that f (z) be a transcendental entire solution of finite order of equation (1.5). By differentiating both sides of equation (1.5), we have 2f f 0 + P10 (f ) = λ(p1 eλz − p2 e−λz ).

(3.23)

Differentiating (3.23), we obtain 2(f 0 )2 + 2f f 00 + P100 (f ) = λ2 (p1 eλz + p2 e−λz ).

(3.24)

Combining (1.5) with (3.24), we have (f 0 )2 =

1 2 2 λ f − f f 00 + Q1 (f ), 2

(3.25)

where Q1 (f ) = 21 (λ2 P1 (f ) − P100 (f )). Eliminating eλz , e−λz from (1.5) and (3.23), we obtain λ2 (f 2 + P1 (f ))2 − (2f f 0 + P10 (f ))2 = 4p1 p2 λ2 , which implies that λ2 f 4 − 4f 2 (f 0 )2 = R3 (f ),

(3.26)

where R3 (f ) = −[2λ2 f 2 P1 (f ) + λ2 (P1 (f ))2 − 4f f 0 P10 (f ) − (P10 (f ))2 − 4p1 p2 λ2 ]. Substituting (3.25) into (3.26), we see that f 3 (4f 00 − λ2 f ) = T3 (f ),

(3.27)

where T3 (f ) = 4f 2 Q1 (f ) + R3 (f ). Now, we consider two cases, case 1: T3 (f ) ≡ 0 and case 2: T3 (f ) 6≡ 0. Case 1: T3 (f ) ≡ 0. By (3.27), we have 4f 00 − λ2 f ≡ 0.

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(3.28)

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Every entire solution f (z)(6≡ 0) of (3.28) can be expressed as f (z) = c1 eλz/2 + c2 e−λz/2 ,

(3.29)

where c1 , c2 are constants, and at least one of them being not equal to zero. Then  k  k λ λ f (k) (z + c) = c1 eλc/2 eλz/2 + (−1)k+1 c2 e−λc/2 e−λz/2 . 2 2

(3.30)

Substituting (3.29) and (3.30) into (1.5), we obtain λ λ (c21 −p1 )eλz +(c22 −p2 )e−λz +a(z)c1 ( )k eλc/2 eλz/2 +(−1)k+1 a(z)c2 ( )k e−λc/2 e−λz/2 +2c1 c2 = 0. 2 2 (3.31) By Lemma 2.6, we see that  k  k λ λ λc/2 k+1 a(z)c1 e ≡ 0, (−1) a(z)c2 e−λc/2 ≡ 0. (3.32) 2 2 From λ 6= 0 and a(z) 6≡ 0, then c1 = c2 = 0, a contradiction. Case 2: T3 (f ) 6≡ 0. Since f (z) is a transcendental entire function of finite order, we see that (3.27) satisfies the conditions of Lemma 2.2 and Remark 2.1. Thus we have m(r, 4f 00 − λ2 f ) = S(r, f ) + O(m(r, a(z))) = O(log r), which implies that 4f 00 − λ2 f is a polynomial. Thus, from (3.27) and T3 (f ) 6≡ 0, we have 4f 00 − λ2 f = H(z),

(3.33)

where H(z) is a nonvanishing polynomial. By Lemma 2.5, we see that (3.33) must have a nonzero polynomial solution, say, c0 (z). Since the differential equation 4f 00 − λ2 f = 0, (3.34) has two fundamental solutions λ

f1 (z) = e 2 z ,

λ

f2 (z) = e− 2 z .

Then the general entire solution f (z)(6≡ 0) of (3.33) can be expressed as λ

λ

f (z) = c0 (z) + c1 e 2 z + c2 e− 2 z ,

(3.35)

where c1 , c2 are constants, c0 (z) is a nonzero polynomial. If k is an odd, then  k  k λ λ (k) eλc/2 eλz/2 − c2 e−λc/2 e−λz/2 . f (k) (z + c) = c0 (z + c) + c1 2 2

(3.36)

Substituting (3.35) and (3.36) into (1.5), we obtain (c21

− p1 )e

λz

+

(c22

− p2 )e

−λz

+

!  k λ λc/2 2c1 c0 (z) + a(z)c1 e eλz/2 2

!  k λ (k) −λc/2 + 2c2 c0 (z) − a(z)c2 e e−λz/2 + c20 (z) + a(z)c0 (z + c) + 2c1 c2 = 0. (3.37) 2

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By Lemma 2.6, we see that  2 c1 = p1 , c22 = p2 ;     2c c (z) + a(z)c ( λ )k eλc/2 ≡ 0; 1 0 1 2  2c2 c0 (z) − a(z)c2 ( λ2 )k e−λc/2 ≡ 0;    2 (k) c0 (z) + a(z)c0 (z + c) + 2c1 c2 ≡ 0.

(3.38)

From (3.38), we know that a(z) ≡ a(constant),

c0 (z) ≡ c0 (constant),

and then i c0 = ± a 2

 k  4k λ 1 4 λ , a = p1 p2 , eλc = −1, c2i = pi , i = 1, 2. 2 64 2

If k is an even, then (k)

f (k) (z + c) = c0 (z + c) + c1

 k  k λ λ eλc/2 eλz/2 + c2 e−λc/2 e−λz/2 . 2 2

(3.39)

Substituting (3.35) and (3.39) into (1.5), we obtain (c21

− p1 )e

λz

+

(c22

− p2 )e

−λz

+

!  k λ λc/2 2c1 c0 (z) + a(z)c1 e eλz/2 2

!  k λ (k) −λc/2 e e−λz/2 + c20 (z) + a(z)c0 (z + c) + 2c1 c2 = 0. (3.40) + 2c2 c0 (z) + a(z)c2 2 By Lemma 2.6, we see that  2 c1 = p1 , c22 = p2 ;     2c c (z) + a(z)c ( λ )k eλc/2 ≡ 0; 1 0 1 2 λ k −λc/2  2c c (z) + a(z)c ≡ 0; 2 0 2( 2 ) e    2 (k) c0 (z) + a(z)c0 (z + c) + 2c1 c2 ≡ 0.

(3.41)

If k is even and k > 0, from (3.41), we know that a(z) ≡ a(constant),

c0 (z) ≡ c0 (constant),

and then 1 c0 = ± a 2

 k  4k λ 1 4 λ , a = p1 p2 , eλc = 1, c2i = pi , i = 1, 2. 2 64 2

If k = 0, from (3.41), we see that c20 (z) ≡ ±2c0 (z)c0 (z + c) − 2c1 c2 .

(3.42)

By (3.42) and Lemma 2.7, we know that c0 (z) ≡ c0 (constant). By (3.41), we have a(z) ≡ a(constant), and 1 1 4 9 4 c0 = ± a, a = p1 p2 or a = p1 p2 , eλc = 1, c2i = pi , i = 1, 2. 2 64 64 This completes the proof of Theorem 1.2. Acknowledgements. The authors would like to thank the referee for his/her thorough reviewing with constructive suggestions and comments to the paper.

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References [1] W. K. Hayman., Meromorphic Function, Clarendon Press, Oxford, 1964. [2] I. Laine., Nevanlinna Theory and Complex Differential Equations, W. de Gruyter, Berlin, 1993. [3] P. Li and C. C. Yang., On the nonexistence of entire solutions of certain type of nonlinear differential equations, J. Math. Anal. Appl. 320(2006), 827-835. [4] C. C. Yang and P. Li., On the transcendental solutions of a certain type of nonlinear differential equations, Arch. Math. 82(2004), 442-448. [5] P. Li., Entire solutions of certain type of differential equations, J. Math. Anal. Appl. 344(2008), 253-259. [6] J. F. Tang and L. W. Liao., The transcendental meromorphic solutions of a certain type of nonlinear differential equations, J. Math. Anal. Appl. 334(2007), 517-527. [7] B. Q. Li., On certain non-linear differential equations in complex domains, Arch. Math. 91(2008), 344-353. [8] Z. X. Chen and C. C. Yang., On entire solutions of certain type of differential-difference equations, Taiwanese. J. Math. 18(2014), 677-685. [9] J. Zhang and L. W. Liao., On entire solutions of a certain type of nonlinear differential and difference equations, Taiwanese. J. Math. 15(2011), 2145-2157. [10] P. Li., Entire solutions of certain type of differential equations II, J. Math. Anal. Appl. 375(2011), 310-319. [11] S. M. Wang and S. Li., On entire solutions of nonlinear difference-differential equations, Bull. Korean. Math. Soc. 50(2013), 1471-1479. [12] Li. S and Gao Z. S., Finite order meromorphic solutions of linear difference equations, Proc. Japan. Acad. Ser. A. Math. Sci. 87(2011), 73-76. [13] J. Clunie., On integral and meromorphic functions, J. London. Math. Soc. 37(1962), 17-27. [14] R. G. Halburd and R. J. Korhonen., Difference analogue of the lemma on the logarithmic derivative with applications to difference equations, J. Math. Anal. Appl. 314(2006), 477-487. [15] C. C. Yang and H. X. Yi., Uniqueness theory of meromorphic functions, Science Press, Beijing, 1995. Kluwer Academic, Dordrecht, 2003. Min-Feng Chen LMIB & School of Mathematics and Systems Science Beihang University Beijing 100191 P. R. China Email: [email protected] Zong-Sheng Gao LMIB & School of Mathematics and Systems Science Beihang University Beijing 100191 P. R. China E-mail: [email protected]

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APPROXIMATE GENERALIZED QUADRATIC MAPPINGS IN (β, p)-BANACH SPACES HARK-MAHN KIM AND HONG-MEI LIANG

Abstract. In this article, we present general solution of a generalized quadratic functional equation with several variables, and then obtain its generalized Hyers–Ulam stability results for approximate generalized quadratic mappings in (β, p)-Banach spaces.

1. Introduction In 1940, S. M. Ulam [30] gave the following question associated with the stability of group homomorphisms: Let G be a group and let G0 be a metric group with the metric d(·, ·). Given ε > 0, does there exist a δ > 0 such that if a mapping f : G → G0 satisfies the inequality d(f (xy), f (x)f (y)) < δ for all x, y ∈ G, then there exists a homomorphism F : G → G0 with d(f (x), F (x)) < ε for all x ∈ G? The question of Ulam was first solved by D. H. Hyers [12] for approximate Cauchy additive mappings on Banach spaces. Th. M. Rassias [22] provided a generalized Hyers–Ulam stability for the unbounded Cauchy difference controlled by a sum of unbounded function ε(kxkp + kykp ) for case 0 ≤ p < 1. And then Z. Gajda [8] provided a generalized Hyers–Ulam stability for the Cauchy difference controlled by the same unbounded function ε(kxkp + kykp ) for case p > 1. In 1984, J. M. Rassias [23] gave a similar stability for the unbounded Cauchy difference controlled by a product of unbounded function εkxkp kykq , p+q 6= 1. More generally, Gˇavruta [10] established a generalized Hyers–Ulam stability under replacing the bound of Cauchy difference controlled by an integrated general control function with regular condition. The following functional equation f (x + y) + f (x − y) = 2[f (x) + f (y)] 2010 Mathematics Subject Classification. 39B52, 39B82. Key words and phrases. generalized Hyers–Ulam stability; quasi-β-normed spaces; (β, p)-Banach spaces. † Corresponding author. [email protected]. 1

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is called a quadratic functional equation since it may be originated from the important parallelogram equality kx + yk2 + kx − yk2 = 2[kxk2 + kyk2 ] in inner product spaces. In particular, every solution of the quadratic functional equation is said to be a quadratic mapping. The Hyers–Ulam stability problem for the quadratic functional equation was first proved by F. Skof [29] for a mapping f : X → Y , where X is a normed space and Y is a Banach space. In 1992, S. Czerwik [6] demonstrated the Hyers– Ulam stability of the quadratic functional equation with the sum of powers of norms in the sense of Th. M. Rassias approach using the direct method. In the same year, J. M. Rassias [24] certified the Hyers–Ulam stability of the quadratic functional equation with the product of powers of norms using the direct method. In 1995, C. Borelli and G. L. Forti [2] have verified the generalized Hyers–Ulam stability theorem of the quadratic functional equation. V. Radu [21], L. C˘adariu and V. Radu [3, 4] have proposed to investigate the stability of functional equations using the fixed point method which is based on the alternative fixed point theorem. Since then, the stability of several functional equations using the fixed point method has been extensively investigated by several mathematicians [9, 11, 14, 16, 17]. Recently, A. Zivari-Kazempour and M. Eshaghi Gordji [32] have determined the general solution of the quadratic functional equation   f (x + 2y) + f (y + 2z) + f (z + 2x) = 2f (x + y + z) + 3 f (x) + f (y) + f (z) , and then have investigated its generalized Hyers–Ulam stability. Motivated from this quadratic functional equation, we now consider a generalized functional equation (1.1)

n X



X

f kxi +

i=1 1≤i1 ≤···≤in ≤n

= (n + k − 1)

n−k X

x ij



j=1,ij 6=i

n−2 n−k−1

! f

n X i=1

xi



nk(k − 1) + n−k

n−2 n−k−1

!

n X

f (xi ),

i=1

where n, k are fixed integers with n ≥ 3 and 2 ≤ k ≤ n − 1. Kim and Liang [15] have presented the classical stability results of quadratic functional equation (1.1) by using the fixed point approach in normed spaces. In this article, we give the general solution of the functional equation (1.1), and then investigate generalized Hyers–Ulam stability results for approximate generalized quadratic mappings in (β, p)-Banach spaces. 2. The general solution of Equation (1.1) First of all, we introduce basic definitions, notations and preliminary theorems in the sequel. Let N be the set of all natural numbers, n ∈ N and let X and Y be vector spaces. A

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3

mapping An : X n → Y is called n-additive if it is additive in each of its variables. A mapping An is called symmetric if An (x1 , · · · , xn ) = An (xπ(1) , · · · , xπ(n) ) for every permutation {π(1), · · · , π(n)} of {1, · · · , n} and all x1 , · · · , xn ∈ X. If An (x1 , · · · , xn ) is an n-additive symmetric map, then An (x) will denote the diagonal An (x, · · · , x) for all x ∈ X, which will be called a monomial mapping of degree n. Note that An (rx) = rn An (x) for all x ∈ X and all rational number r ∈ Q. A mapping p : X → Y is called a generalized polynomial of degree n ∈ N provided that there exist an i-additive symmetric mappings Ai : X i → Y (1 ≤ i ≤ n) P such that p(x) = ni=0 Ai (x) for all x ∈ X, where An 6≡ 0 and A0 (x) = A0 ∈ Y is a constant. For f : X → Y , let ∆h be the difference operator defined as follows: ∆h f (x) = f (x + h) − f (x) for all x, h ∈ X. We notice that these difference operators satisfy commutativity: ∆h1 ◦∆h2 f (x) = ∆h2 ◦∆h1 f (x) for all x, h1 , h2 ∈ X, where ∆h1 ◦∆h2 denotes the composition of the operators ∆h1 and ∆h2 . Furthermore, let ∆0h f (x) = f (x), ∆1h f (x) = ∆h f (x) and n ∆n+1 h f (x) = ∆h ◦ ∆h f (x) for all n ∈ N and all x, h ∈ X. In explicit form, the functional n+1 equation ∆h f (x) = 0 can be written as ! n+1 X n+1 n+1 n+1−j ∆h f (x) = (−1) f (x + jh) = 0 j j=0 for all x, h ∈ X. For any given n ∈ N, the following is well-known Fr´echet functional equation ∆hn+1 ◦ · · · ◦ ∆h1 f (x) = 0 for all x, h1 , · · · , hn+1 ∈ X. The following theorem was proved by Mazur and Orlicz [19, 20] and by Djokovi´c [7] in greater generality. Theorem 2.1. Let X and Y be vector spaces, n ∈ N and f : X → Y , then the followings are equivalent. (1) ∆n+1 h f (x) = 0 for all x, h ∈ X. (2) ∆hn+1 ◦ · · · ◦ ∆h1 f (x) = 0 for all x, h1 , · · · , hn+1 ∈ X. (3) f (x) = An (x) + · · · + A0 (x) for all x ∈ X, where A0 (x) = A0 is an arbitrary element of Y and Ai (x) is the diagonal of an i-additive symmetric mapping Ai : X i → Y (i = 1, · · · , n). The following two theorems ([27]) are need for us to establish general solution of the functional equation (1.1) (see also [28, 31] for details). Theorem 2.2. Let G be a commutative semigroup with identity, S a commutative group and n a nonnegative integer. Let the multiplication by n! be bijective in S. The function f : G → S is a solution of Fr´echet functional equation ∆hn+1 ◦ · · · ◦ ∆h1 f (x) = 0 for all x, h1 , · · · , hn+1 ∈ G if and only if f is a generalized polynomial of degree at most n.

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Theorem 2.3. Let G and S be commutative groups, n a nonnegative integer, ϕi , ψi additive functions from G into G and ϕi (G) ⊆ ψi (G) (i = 1, · · · , n + 1). If the functions f, fi : G → S (i = 1, · · · , n + 1) satisfy f (x) +

n+1 X

fi (ϕi (x) + ψi (y)) = 0

i=1

for all x, y ∈ G, then f satisfies Fr´echet functional equation ∆hn+1 ◦ · · · ◦ ∆h1 f (x) = 0 for all x, h1 , · · · , hn+1 ∈ G. Before taking up our subject, we note that a mapping f is quadratic if and only if f satisfies the functional equation f (2x + y) + f (x + 2y) = 4f (x + y) + f (x) + f (y), which is equivalent to the functional equation f (3x + y) + f (x + 3y) = 6f (x + y) + 4f (x) + 4f (y) for all x, y ∈ X [5]. Moreover, it is very meaningful and elementary to see that the functional equation (2.1)

f (kx + y) + f (x + ky) = 2kf (x + y) + (k − 1)2 [f (x) + f (y)]

between vector spaces, where k is a fixed positive integer with k ≥ 2, is equivalent to the quadratic functional equation (2.2)

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

for all x, y ∈ X using Theorem 2.2 and Theorem 2.3 together with f (kx) = k 2 f (x) for all x ∈ X, and so the mapping f satisfying the equation (2.1) is quadratic. Now, we are ready to present the general solution of the functional equation (1.1). Theorem 2.4. Let X and Y be vector spaces. If a mapping f : X → Y is solution of the functional equation (1.1), then f satisfies the functional equation (2.2) and so it is quadratic. Proof. Assume that a mapping f : X → Y satisfies the functional equation (1.1). Letting x1 = · · · = xn := 0 in (1.1), we obtain ! ! ! n−1 n−2 n−2 n2 k(k − 1) n f (0) = (n + k − 1) f (0) + f (0), n−k n−k n−k−1 n−k−1 which yields f (0) = 0.

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5

On the other hand, one has the following identity by setting x1 := x, x2 = · · · = xn := 0 in (1.1) ! ! n−1 n−2 f (kx) + (n − 1) f (x) n−k n−k−1 ! ! n−2 n−2 nk(k − 1) f (x) = (n + k − 1) f (x) + n−k n−k−1 n−k−1 for all x ∈ X. Thus f (kx) = k 2 f (x) for all x ∈ X. Putting x1 := x, x2 := y, x3 = · · · = xn := 0 in (1.1), we get ! ! n−2 n−2 [f (kx + y) + f (x + ky)] + [f (kx) + f (ky)] n−k−1 n−k ! ! n−3 n−3 +(n − 2) f (x + y) + (n − 2) [f (x) + f (y)] n−k−2 n−k−1 ! ! n−2 n−2 nk(k − 1) f (x + y) + = (n + k − 1) [f (x) + f (y)], n−k n−k−1 n−k−1 from which it follows that f (kx + y) + f (x + ky) = 2kf (x + y) + (k − 1)2 [f (x) + f (y)] for all x, y ∈ X. Hence f is a quadratic mapping.



3. The generalized Hyers–Ulam stability of Equation (1.1) We recall some basic facts concerning the (β, p)-normed spaces [25]. Let β be a fixed real number with 0 < β ≤ 1 and X a linear space over K, where K denote either R or C. A quasi-β-norm is a real-valued function on X satisfying the following: (1) kxk ≥ 0 for all x ∈ X and kxk = 0 if and only if x = 0; (2) kλxk = |λ|β kxk for all λ ∈ K and all x ∈ X; (3) There is a constant M ≥ 1 such that kx + yk ≤ M (kxk + kyk) for all x, y ∈ X. In this case, the pair (X, k · k) is called a quasi-β-normed space. A quasi-β-Banach space is a complete quasi-β-normed space. Let p be a real number with 0 < p ≤ 1. The quasi-β-norm k · k on X is called a (β, p)-norm if, moreover, k · kp satisfies the following triangle inequality kx + ykp ≤ kxkp + kykp for all x, y ∈ X. In this case, a quasi-β-Banach space is called a (β, p)-Banach space. We notice that quasi-1-normed spaces are equivalent to quasi-normed spaces and that (1, p)Banach spaces with (1, p)-norm are equivalent to p-Banach spaces with p-norm. We may

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refer to [1, 26] for the concept of quasi-normed spaces and p-Banach spaces. Given a pnorm, the formula d(x, y) := kx − ykp gives us a translation invariant metric on X. By the Aoki–Rolewicz theorem [26], each quasi-norm is equivalent to some p-norm [1]. Now, we study the generalized Hyers–Ulam stability of the functional equation (1.1) by using the direct method. For convenience, we use the following abbreviations: Df (x1 , · · · , xn ) :=

n X

f kxi +

i=1 1≤i1 ≤···≤in ≤n

−

n−k n−2 n−k−1

k(k − 1)



xij − (n + k − 1)

j=1,ij 6=i

n−2 n−k−1

nk(k − 1) n−k

M :=

n−k X



X

!

n X

f (xi ),

n−2 n−k−1

! f

n X

xi



i=1

(x1 , · · · , xn ) ∈ X n := X × · · · × X;

i=1

!.

Theorem 3.1. Let X be a vector space and Y a (β, p)-Banach space. Assume that ϕ : X n → [0, ∞) is a function satisfying
p ∞ X ϕ nj x1 , · · · , nj xn (3.1) 0; N (x, t) = 1 if and and only if x = 0; t (N3) ∀t ∈ R, t > 0; N (x, t) = N (x, |c| ) if c 6= 0; (N4) ∀s, t ∈ R, x, u ∈ U ; N (x + u, s + t) ≥ min {N (x, s), N (u, t)} (N5) N (x, ·) is a non-decreasing function of R and lim N (x, t) = 1. t→∞

The pair (U, N ) will be referred to as a fuzzy normed linear space. Theorem 1.2. Let (U, N ) be a fuzzy normed linear space. Assume further that, (N6) ∀t > 0, N (x, t) > 0 implies x = 0. Define kxkα = ∧ {t > 0 : N (x, t) ≥ α} , α ∈ (0, 1). Then {k.kα : α ∈ (0, 1)}is an ascending family of norms on U and they are called α-norms on U corresponding to the fuzzy norm N on U . 2010 Mathematics Subject Classification. 47H10; 46S40; 47S40. Key words and phrases. hybrid mapping; fuzzy Hilbert spaces; fixed point theorem; nonexpansive mappings; nonspreading mappings. ∗ Corresponding authors.

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A. BATOOL, T. KAMRAN, C. PARK, AND J.R. LEE

Theorem 1.3. Let (U, N ) be a fuzzy normed linear space satisfying (N6). Assume further that (N7) for x 6= 0, N (x, ·) is a continuous function of R. Let kxkα = ∧ {t > 0 : N (x, t) ≥ α} , α ∈ (0, 1) and N 0 : U × R → [0, 1] be a function defined by  ∧ {α ∈ (0, 1) : kxkα ≤ t} , if (x, t) 6= (0, 0) 0 N (x, t) = 0 if (x, t) = (0, 0). Then (i) {k.kα : α ∈ (0, 1)} is an ascending family of norms on U. (ii) N 0 is a fuzzy norm on U. (iii) N 0 = N In a Hilbert space with inner product h·, ·i and norm k·k, respectively, it is known that kαx + (1 − α)yk2 = α kxk2 + (1 − α) kyk2 − α(1 − α) kx − yk2

(1.1)

for all x, y ∈ H and α ∈ R (see [31]). Furthermore, in a Hilbert space, we have that 2 hx − y, z − wi = kx − wk2 + ky − zk2 − kx − zk2 − ky − wk2

(1.2)

for all x, y, z, w ∈ H. Using means and the Riesz theorem, we can obtain the following result (see [22, 25, 26, 27]). Lemma 1.4. Let H be a Hilbert space, {xn } a bounded sequence in H and let µ be a mean on l∞ . Then there exists a unique point z0 ∈ co{xn |n ∈ N } such that µn hxn , yi = hz0 , yi , ∀y ∈ H. We can define the following nonlinear mappings (see [9, 16, 18, 19, 21, 28, 29]) in fuzzy Hilbert spaces. Let H be a fuzzy Hilbert space with inner product {h., .iα : α ∈ (0, 1)} and norm {k.kα : α ∈ (0, 1)}. Let C be a nonempty subset of H. A mapping T : C → H is said to be nonexpansive, nonspreading, and hybrid if kT x − T ykα ≤ kx − ykα , 2 kT x − T yk2α ≤ kT x − yk2α + kT y − xk2α and 3 kT x − T yk2α ≤ kx − yk2α + kT x − yk2α + kT y − xk2α for all x, y ∈ C, respectively. A mapping F : C → H is said to be firmly nonexpansive if kF x − F yk2α ≤ hx − y, F x − F yiα for all x, y ∈ C. A mapping T from C into H is said to be widely generalized hybrid if there exist α, β, γ, δ, ε, ζ ∈ R such that α kT x − T yk2α + β kx − T yk2α + γ kT x − yk2α + δ kx − yk2α n o +max ε kx − T xk2α , ζ ky − T yk2α ≤ 0

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FIXED POINT THEOREMS FOR GENERALIZED HYBRID MAPPINGS

for all x, y ∈ C and T is called symmetric generalized hybrid if there exist α, β, γ, δ ∈ R such that α kT x − T yk2α + β(kx − T yk2α + kT x − yk2α ) +γ kx − yk2α + δ(kx − T xk2α + ky − T yk2α ) ≤ 0

(1.3)

for all x, y ∈ C. Such a mapping T is also called (α, β, γ, δ)-symmetric generalized hybrid. If α = 1, β = δ = 0 and γ = −1 in (1.3), then the mapping T is nonexpansive. If α = 2, β = −1 and γ = δ = 0 in (1.3), then the mapping T is nonspreading. Furthermore, if α = 3, β = γ = −1 and δ = 0 in (1.3), then the mapping T is hybrid. Let H be a fuzzy Hilbert space and let C be a nonempty subset of H. Then T : C → H is called a widely strict pseudo-contraction if there exists r ∈ R with r < 1 such that kT x − T yk2α ≤ kx − yk2α + r k(I − T )x − (I − T )yk2α ,

∀ x, y ∈ C.

We call such T a widely r-strict pseudo-contraction. If 0 ≤ r < 1, then T is a strict pseudocontraction (see [10, 23, 24]). Furthermore, if r = 0, then T is nonexpansive. Conversely, let 1 n S : C → H be a nonexpansive mapping and define T : C → H by T = 1+n S + 1+n I for all x ∈ C and n ∈ N. Then T is a widely (-n)-strict pseudocontraction. In fact, from the definition of T , it follows that S = (1 + n)T − nI. Since S is nonexpansive, we have that for any x, y ∈ C, k(1 + n)T x − nx − ((1 + n) T y − ny)k2α ≤ kx − yk2α and hence kT x − T yk2α ≤ kx − yk2α + n k(I − T )x − (I − T )yk2α . We denote the strong convergence and the weak convergence of {xn } to x ∈ H by xn → x and xn * x, respectively. Let A be a nonempty subset of H. We denote by coA the closure of the convex hull of A. Let T be a mapping from C into H. We denote by F (T ) the set of fixed points of T . 2. Fixed point theorems Theorem 2.1. Let H be a fuzzy Hilbert space, C a nonempty closed convex subset of H and let T be an (α, β, γ, δ)-symmetric generalized hybrid mapping from C into itself such that the conditions (1) α + 2β + γ ≥ 0, (2) α + β + δ > 0 and (3) δ ≥ 0 hold. Then T has a fixed point if and only if there exists z ∈ C such that {T n z : n = 0, 1, ...} is bounded. In particular, a fixed point of T is unique in the case of α + 2β + γ > 0 on the condition (1). Proof. Suppose that T has a fixed point z. Then {T n z : n = 0, 1, ...} = {z} and hence {T n z : n = 0, 1, ...} is bounded. Conversely, suppose that there exists z ∈ C such that {T n z : n = 0, 1, ...} is bounded. Since T is an (α, β, γ, δ)-symmetric generalized hybrid mapping of C into itself, we have that

2

2 α T x − T n+1 z α + β( x − T n+1 z α + kT x − T n zk2α )

2 +γ kx − T n zk2α + δ(kx − T xk2α + T n z − T n+1 z α ) ≤ 0

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for all n ∈ N ∪ {0} and x ∈ C. Since {T n z} is bounded, we can apply a Banach limit µ to both

2

2 sides of the inequality. Since µn kT x − T n zk2α = µn T x − T n+1 z α and µn x − T n+1 z α = µn kx − T n zk2α , we have that (α + β)µn kT x − T n zk2α + (β + γ)µn kx − T n zk2α

2 +δ(kx − T xk2α + µn T n z − T n+1 z α ) ≤ 0. Furthermore, since µn kT x − T n zk2α = kT x − xk2α + 2µn hT x − x, x − T n ziα + µn kx − T n zk2α , we have that (α + β + δ) kT x − xk2α + 2(α + β)µn hT x − x, x − T n ziα

2

+(α + 2β + γ)µn kx − T n zk2α + δµn T n z − T n+1 z α ≤ 0. From (1) α + 2β + γ ≥ 0 and (3) δ ≥ 0, we have that (α + β + δ) kT x − xk2α + 2(α + β)µn hT x − x, x − T n ziα ≤ 0.

(2.1)

Since there exists p ∈ H from Lemma 1.4 such that µn hy, T n ziα = hy, piα for all y ∈ H, we have from (2.1) that (α + β + δ) kT x − xk2α + 2(α + β) hT x − x, x − piα ≤ 0.

(2.2)

Since C is closed and convex, we have that p ∈ co{T n x : n ∈ N } ⊂ C. Putting x = p, we obtain from (2.2) that (α + β + δ) kT p − pk2α ≤ 0.

(2.3)

We have from (2) α + β + δ > 0 that kT p − pk2α ≤ 0. This implies that p is a fixed point in T . Next suppose that α + 2β + γ > 0. Let p1 and p2 be fixed points of T . Then we have that α kT p1 − T p2 k2α + β(kp1 − T p2 k2α + kT p1 − p2 k2α ) +γ kp1 − p2 k2α + δ(kp1 − T p1 k2α + kp2 − T p2 k2α ) ≤ 0 and hence(α + 2β + γ) kp1 − p2 k2α ≤ 0. We have from α + 2β + γ > 0 that p1 = p2 . Therefore a fixed point of T is unique. This completes the proof.  We can derive the following theorem from Theorem 2.1. Theorem 2.2. Let H be a fuzzy Hilbert space, C a nonempty bounded closed convex subset of H and let T be an (α, β, γ, δ)-symmetric generalized hybrid mapping from C into itself such that the conditions (1) α + 2β + γ ≥ 0, (2) α + β + δ > 0 and (3) δ ≥ 0 hold. Then T has a fixed point. In particular, a fixed point of T is unique in the case of α + 2β + γ > 0 on the condition (1).

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A mapping T from C into H is called (α, β, γ, δ, ζ)-symmetric more generalized hybrid if there exist α, β, γ, δ, ζ ∈ R such that α kT x − T yk2α + β(kx − T yk2α + kT x − yk2α ) + γ kx − yk2α +δ(kx − T xk2α + ky − T yk2α ) + ζ kx − y − (T x − T y)k2α ≤ 0

(2.4)

for all x, y ∈ C. Theorem 2.3. Let H be a fuzzy Hilbert space, C a nonempty closed convex subset of H and let T be an (α, β, γ, δ, ζ)-symmetric more generalized hybrid mapping from C into itself such that the conditions (1) α + 2β + γ ≥ 0, (2) α + β + δ + ζ > 0 and (3) δ + ζ ≥ 0 hold. Then T has a fixed point if and only if there exists z ∈ C such that {T n z : n = 0, 1, ...} is bounded. In particular, a fixed point of T is unique in the case of α + 2β + γ > 0 on the condition (1). Proof. Since T : C → C is an (α, β, γ, δ, ζ)-symmetric more generalized hybrid mapping, there exist α, β, γ, δ, ζ ∈ R satisfying (2.4). We also have that kx − y − (T x − T y)k2α = kx − T xk2α + ky − T yk2α − kx − T yk2α − ky − T xk2α + kx − yk2α + kT x − T yk2α

(2.5)

for all x, y ∈ C. Thus we obtain from (2.4) that (α + ζ) kT x − T yk2α + (β − ζ)(kx − T yk2α + kT x − yk2α ) +(γ + ζ) kx − yk2α + (δ + ζ)(kx − T xk2α + ky − T yk2α ) ≤ 0.

(2.6)

The conditions (1) α + 2β + γ ≥ 0 and (2) α + β + δ + ζ > 0 are equivalent to (α + ζ) + 2(β − ζ) + (γ + ζ) ≥ 0 and (α + ζ) + (β − ζ) + (δ + ζ) > 0, respectively. Furthermore, since (3) δ + ζ ≥ 0 holds, we have the desired result from Theorem 2.1.  As a direct consequence of Theorem 2.3, we obtain the following. Theorem 2.4. Let H be a fuzzy Hilbert space, C a nonempty bounded closed convex subset of H and let T be an (α, β, γ, δ, ζ)-symmetric more generalized hybrid mapping from C into itself such that the conditions (1) α + 2β + γ ≥ 0, (2) α + β + δ + ζ > 0 and (3) δ + ζ ≥ 0 hold. Then T has a fixed point if and only if there exists z ∈ C such that {T n z : n = 0, 1, ...} is bounded. In particular, a fixed point of T is unique in the case of α + 2β + γ > 0 on the condition (1). We can extend the above theorem as follows. Theorem 2.5. Let H be a fuzzy Hilbert space, C a nonempty bounded closed convex subset of H and let T be an (α, β, γ, δ, ζ)-symmetric more generalized hybrid mapping from C into itself which satisfies the conditions (1) α + 2β + γ ≥ 0, (2) α + β + δ + ζ > 0 and (3) there exists λ ∈ [0, 1) such that (α + β) λ + δ + ζ ≥ 0. Then T has a fixed point. In particular, a fixed point of T is unique in the case of α + 2β + γ > 0 on the condition (1).

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Proof. Since T : C → C is an (α, β, γ, δ, ζ)-symmetric more generalized hybrid mapping, we obtain that

2

2 α T x − T n+1 z α + β( x − T n+1 z α + kT x − T n zk2α ) + γ kx − T n zk2α

2


2 +δ(kx − T xk2α + T n z − T n+1 z α ) + ζ (x − T x) − T n z − T n+1 z α ≤ 0 for all n ∈ N ∪ {0} and all x ∈ C. Let λ ∈ [0, 1) ∩ {λ : (α + β)λ + ζ + η ≥ 0} and define S = (1 − λ)T + λI. Since C is convex, S is a mapping from C into itself. Since C is bounded, {S n z : n = 0, 1, ...} is bounded for all 1 λ z ∈ C. Since λ 6= 1, we obtain that F (S) = F (T ). Moreover, from T = 1−λ S − 1−λ I and (2.1), we have that

    2

1 λ 1 λ

α Sx − x − Sy − y 1−λ 1−λ 1−λ 1 − λ α

2

  2  



λ λ 1 1 2

+β

x − 1 − λ Sy − 1 − λ y + β 1 − λ Sx − 1 − λ x − y + γ kx − ykα α α

  2   2



1 λ 1 λ

+δ

x − 1 − λ Sx − 1 − λ x + δ y − 1 − λ Sy − 1 − λ y α α

     2 

λ 1 λ 1

Sx − x − y − Sy − y +ζ x −

1−λ 1−λ 1−λ 1−λ α

  2

1 λ

= α

1 − λ (Sx − Sy) − 1 − λ (x − y) α

 2 

λ 1

+β

x − 1 − λ (x − Sy) − 1 − λ (x − y) α

 2 

λ 1 2

+β

x − 1 − λ (Sx − y) − 1 − λ (x − y) + γ kx − ykα α

2

2

1

1



+δ

1 − λ (x − Sx) + δ 1 − λ (y − Sy) α α

2

1 1

+ζ

1 − λ (x − Sx) − 1 − λ (y − Sy) α α β 2 2 = kSx − Sykα + kx − Sykα 1−λ 1−λ   β λ 2 + kSx − ykα + − (α + 2β) + γ kx − yk2α 1−λ 1−λ δ + βλ δ + βλ 2 + ky − Syk2α 2 kx − Sxkα + (1 − λ) (1 − λ)2

+

ζ + αλ k(x − Sx) − (y − Sy)k2α ≤ 0 (1 − λ)2

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  ζ+αλ β δ+βλ α λ , -symmetric more generalized , 1−λ , − 1−λ (α + 2β) + γ, (1−λ) Therefore, S is an 1−λ 2 (1−λ)2 hybrid mapping. Furthermore, we obtain that α 2β γ λ + + − (α + 2β) + γ = α + 2β + γ ≥ 0, 1−λ 1−λ 1−λ 1−λ α ζ + αλ α+β+δ+ζ β δ + βλ + = > 0, + + 1 − λ 1 − λ (1 − λ)2 (1 − λ)2 (1 − λ)2 δ + βλ ζ + αλ (α + β) λ + δ + ζ ≥ 0. 2 + 2 = (1 − λ) (1 − λ) (1 − λ)2 Therefore, by Theorem 2.4, we obtain F (S) 6= φ. Next, suppose that α + 2β + γ > 0. Let p1 and p2 be fixed points of T . Then α kT p1 − T p2 k2α + β(kp1 − T p2 k2α + kT p1 − p2 k2α ) + γ kp1 − p2 k2α δ(kp1 − T p1 k2α + kp2 − T p2 k2α ) + ζ k(p1 − T p1 ) + (p2 − T p2 )k2α = (α + 2β + γ) kp1 − p2 k2α ≤ 0 and hence p1 = p2 . Therefore a fixed point of T is unique.



For the case β + δ = 0 in Theorem 2.5, we have the following theorem. Theorem 2.6. Let H be a fuzzy Hilbert space, C a nonempty bounded closed convex subset of H and let T be an (α, −β, γ, β, ζ)-symmetric more generalized hybrid mapping from C into itself, i.e., there exist α, β, γ, ζ ∈ R such that α kT x − T yk2α + β(kx − T yk2α + kT x − yk2α ) + γ kx − yk2α −β(kx − T xk2α + ky − T yk2α ) + ζ kx − y − (T x − T y)k2α ≤ 0

(2.7)

for all x, y ∈ C. Furthermore, suppose that T satisfies the following conditions (1) α+2β +γ ≥ 0, (2) α + ζ > 0 and (3) there exists λ ∈ [0, 1) such that (α + β) λ + δ + ζ ≥ 0. Then T has a fixed point. In particular, a fixed point of T is unique in the case of α + 2β + γ > 0 on the condition (1). Using Theorem 2.3, we prove the following fixed point theorem. Theorem 2.7. Let H be a fuzzy Hilbert space, C a nonempty bounded closed convex subset of H and let T be a widely strict pseudo-contraction from C into itself, i.e., there exists r ∈ R with r < 1 such that kT x − T yk2α ≤ kx − yk2α + r k(I − T )x − (I − T )yk2α , ∀x, y ∈ C.

(2.8)

Then T has a fixed point in C. Proof. We first assume that r ≤ 0. We have from (2.8) that for all x, y ∈ C, kT x − T yk2α − kx − yk2α − r k(I − T )x − (I − T )yk2α ≤ 0

(2.9)

Then T is a (1, 0, −1, 0, −r)-symmetric more generalized hybrid mapping. Furthermore, (1) α + 2β + γ = 1 − 1 ≥ 0, (2) α + β + δ + ζ = 1 − r > 0 and (3) δ + ζ = −r ≥ 0 in Theorem 1.4

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are satisfied. Thus T has a fixed point from Theorem 2.3. Assume that 0 ≤ r < 1 and define a mapping T as follows: Sx = λx + (1 − λ)T x, ∀x ∈ C, where r ≤ λ < 1. Then S is a mapping from C into itself and F (S) = F (T ). From Sx = λx + (1 − λ)T x, we also have that Tx =

λ 1 Sx − x. 1−λ 1−λ

Thus we have

  2

1 λ 1 λ

Sx − x− Sy − y 0 ≥ 1−λ 1−λ 1−λ 1 − λ α

   2

1 λ 1 λ 2

− kx − ykα − r x − y − Sx − x− Sy − y

1−λ 1−λ 1−λ 1−λ α

2

1

λ

=

1 − λ (Sx − Sy) − 1 − λ (x − y) α

2

1 1 2

− kx − ykα − r

1 − λ (x − y) − 1 − λ (Sx − Sy) α

1 λ = kSx − Syk2α − kx − yk2α 1−λ 1−λ λ 1 . kx − y − (Sx − Sy)k2α − kx − yk2α + 1−λ 1−λ r − kx − y − (Sx − Sy)k2α (1 − λ)2 1 λ−r 1 2 kSx − Syk2α − kx − yk2α + = 2 kx − y − (Sx − Sy)kα . 1−λ 1−λ (1 − λ)   1 λ−r 1 , 0, − 1−λ , 0, (1−λ) Then S is a 1−λ 2 )-symmetric more generalized hybrid. From 1 1 + 2.0 − 1−λ 1−λ 1 λ−r + 1 − λ (1 − λ)2 λ−r (1 − λ)2

= 0, > 0

and

≥ 0,

(1) α + 2β + γ ≥ 0, (2) α + β + δ + ζ > 0 and (3) δ + ζ ≥ 0 in Theorem 1.4 are satisfied.Thus S has a fixed point in C from Theorem 2.3 and hence T has a fixed point. This completes the proof.  Let H be a fuzzy Hilbert space and let C be a nonempty subset of H. Let T be a mapping of C into H. For u ∈ H and s, t ∈ (0, 1), we define the following mapping: Sx = tx + (1 − t) (su + (1 − s)T x)

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for all x ∈ C. We call such S a TWY mapping generated by u, T, s, t. Since Sx = tx + s(1 − t)u + (1 − t)(1 − s)T x, we have that for all x, y ∈ C, kSx − Syk2α = kt (x − y) + (1 − t)(1 − s) (T x − T y)k2α 2

= t kx −

yk2α

2

2

+ (1 − t) (1 − s) kT x −

(2.10)

T yk2α

+2t(1 − t)(1 − s) hx − y, T x − T yiα = t2 kx − yk2α + (1 − t)2 (1 − s)2 kT x − T yk2α +t(1 − t)(1 − s)(kx − T yk2α + ky − T xk2α − kx − T xk2α − ky − T yk2α ) = t2 kx − yk2α + (1 − t)2 (1 − s)2 kT x − T yk2α +t(1 − t)(1 − s)(kx − T yk2α + ky − T xk2α ) −t(1 − t)(1 − s)(kx − T xk2α − ky − T yk2α ). Similarly, we have that kx − Syk2α + ky − Sxk2α   = s (1 − t)2 ku − xk2α + ku − yk2α

(2.11)

−s(1 − s)(1 − t)2 (ku − T xk2α + ku − T yk2α ) −t(1 − t)(1 − s)(kx − T xk2α + ky − T yk2α ) +(1 − t)(1 − s)(kx − T xk2α + ky − T yk2α ) + 2t kx − yk2α , and kx − Sxk2α + ky − Syk2α   = s (1 − t)2 ku − xk2α + ku − yk2α −s(1 − s)(1 − t)2 (ku − T xk2α + ku − T yk2α ) +s(1 − s)(1 − t)2 (kx − T xk2α + ky − T yk2α ). We also have that kx − y − Sx − Syk2α = (1 − s)(1 − t)2 (kx −

(2.12) T xk2α

−(1 − s)(1 − t)2 (kx −

+ ky − T yk2α ) T yk2α + ky − T xk2α ) 2 2

+(1 − t)2 kx − yk2α + (1 − t) (1 − s) kT x − T yk2α . Using (2.11) and (2.12), we have that kx − Sxk2α + ky − Syk2α − kx − Syk2α − ky − Sxk2α =

(2.13)

(1 − s)(1 − t)(kx − T xk2α + ky − T yk2α − kx − T yk2α + ky − T xk2α − 2t kx − yk2α ).

Using (2.10) and (2.13), we have the following theorem.

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Theorem 2.8. Let H be a fuzzy Hilbert space, C a nonempty bounded closed convex subset of H and let T be a widely strict pseudo-contraction from C into itself, i.e., there exists r ∈ R with r < 1 such that kT x − T yk2α ≤ kx − yk2α + r k(I − T )x − (I − T )yk2α ∀x, y ∈ C.

(2.14)

Let u ∈ C and s ∈ (0, 1). Define a mapping U : C → C as follows: U x = su + (1 − s)T x, ∀x ∈ C. Then U has a unique fixed point in C. Proof. Since T is a widely r-strict pseudo-contraction from C into itself, we have that for all x, y ∈ C, kT x − T yk2α − kx − yk2α − r k(I − T )x − (I − T )yk2α ≤ 0. If r ≤ 0, then T is a nonexpansive mapping. Therefore U is a contractive mapping. Using the fixed point theorem for contractive mappings, we have that U has a unique fixed point in C. Let 0 < r < 1. Since kx − y − (T x − T y)k2α = kx − T xk2α + ky − T yk2α − kx − T yk2α − ky − T xk2α + kx − yk2α + kT x − T yk2α , we have that (1 − r) kT x − T yk2α − (1 + r) kx − yk2α −r kx − T xk2α + ky − T yk2α − kx − T yk2α − ky − T xk2α ≤ 0. For u, T and s, r ∈ (0, 1), define a TWY mapping S as follows: Sx = rx + (1 − r) (su + (1 − s)T x) , ∀x ∈ C. Then we have from (2.10) that 1 r2 2 kSx − Syk − kx − yk2α α (1 − r)(1 − s)2 (1 − r)(1 − s)2   r + kx − T xk2α + ky − T yk2α − kx − T yk2α − ky − T xk2α (1 − s) − (1 + r) kx − yk2α −r(kx − T xk2α + ky − T yk2α − kx − T yk2α − ky − T xk2α ) ≤ 0. We have from (2.13) that 1 r2 2 kSx − Syk − kx − yk2α α (1 − r)(1 − s)2 (1 − r)(1 − s)2   r 2 2 2 2 kx − Sxk + ky − Syk − kx − Syk − ky − Sxk + α α α α (1 − r)(1 − s)2 2r2 + kx − yk2α − − (1 + r) kx − yk2α 2 (1 − r)(1 − s)   r − kx − Sxk2α + ky − Syk2α − kx − Syk2α − ky − Sxk2α (1 − r)(1 − s)

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−

2r2 kx − yk2α ≤ 0 (1 − r)(1 − s)

and hence 1 kSx − Syk2α (1 − r)(1 − s)2   rs 2 2 − + ky − Sxk kx − Syk α α (1 − r)(1 − s)2   r2 1 − s + r2 (1 + s) + kx − yk2α − 2 (1 − r)(1 − s) (1 − r)(1 − s)   rs 2 2 + kx − Sxk + ky − Syk α α ≤ 0. (1 − r)(1 − s)2 For this inequality, we apply Theorem 2.2. We first obtain that

=

1 2rs r2 1 − s + r2 (1 + s) − + − (1 − r)(1 − s)2 (1 − r)(1 − s)2 (1 − r)(1 − s)2 (1 − r)(1 − s) s (1 + r) (2 − s (1 − r)) > 0. (1 − r)(1 − s)2

Furthermore, we have that

=

1 rs rs − + (1 − r)(1 − s)2 (1 − r)(1 − s)2 (1 − r)(1 − s)2 1 > 0, (1 − r)(1 − s)2 rs ≥ 0. (1 − r)(1 − s)2

Thus S has a unique fixed point z in C from Theorem 1.3. Since z is a fixed point of S, we have z = rz + (1 − r)(su + (1 − s)T z. From 1 − r 6= 0, we have that z = su + (1 − s)T z. This completes the proof.

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A. BATOOL, T. KAMRAN, C. PARK, AND J.R. LEE

[10] F. E. Browder and W. V. Petryshyn, Construction of fixed points of nonlinear mappings in Hilbert spaces, J. Math. Anal. Appl. 20 (1967), 197-228. [11] S. C. Cheng and J. N. Mordeson, Fuzzy linear operators and fuzzy normed linear spaces, Bull. Calcutta Math. Soc. 86 (1994), 429-436. [12] A. M. El-Abyad and H. M. El-Hamouly, Fuzzy inner product spaces, Fuzzy Sets and Systems 44 (1991), 309-326. [13] C. Felbin, Finite dimensional fuzzy normed linear spaces, Fuzzy Sets and Systems 48 (1992), 239-248. [14] C. Felbin, The completion of a fuzzy normed linear space, J. Math. Anal. Appl. 174 (1993), 428-440. [15] C. Felbin, Finite dimensional normed linear space II, J. Anal. 7 (1999), 117-131. [16] K. Goebel and W. A. Kirk, Topics in Metric Fixed Point Theory, Cambridge University Press, Cambridge, 1990. [17] A. K. Katsaras, Fuzzy topological vector spaces II, Fuzzy Sets and Systems 12 (1984), 143-154. [18] T. Kawasaki and W. Takahashi, Fixed point and nonlinear ergodic theorems for new nonlinear mappings in Hilbert spaces, J. Nonlinear Convex Anal. 13 (2012), 529-540. [19] P. Kocourek, W. Takahashi and J.-C. Yao, Fixed point theorems and weak convergence theorems for generalized hybrid mappings in Hilbert spaces, Taiwanese J. Math. 14 (2010), 2497-2511. [20] J. K. Kohli and R. Kumar, On fuzzy inner product spaces and fuzzy co-inner product spaces, Fuzzy Sets and Systems 53 (1993), 227-232. [21] F. Kohsaka and W. Takahashi, Existence and approximation of fixed points of firmly nonexpansive-type mappings in Banach spaces, SIAM J. Optim. 19 (2008), 824-835. [22] L.-J. Lin and W. Takahashi, Attractive point theorems and ergodic theorems for nonlinear mappings in Hilbert spaces, Taiwanese J. Math. 16 (2012), 1763-1779. [23] Y. Shehu, Iterative approximation for split equality fixed point problem for family of multivalued mappings, Rev. R. Acad. Cienc. Exactas Fis. Nat. Ser. A Math. RACSAM 109 (2015), 627-643. [24] G. Soleimani Rad, S. Shukla and H. Rahimi, Some relations between n-tuple fixed point and fixed point results, Rev. R. Acad. Cienc. Exactas Fis. Nat. Ser. A Math. RACSAM 109 (2015), 471-481. [25] W. Takahashi, A nonlinear ergodic theorem for an amenable semigroup of nonexpansive mappings in a Hilbert space, Proc. Amer. Math. Soc. 81 (1981), 253-256. [26] W. Takahashi, A nonlinear ergodic theorem for a reversible semigroup of nonexpansive mappings in a Hilbert space, Proc. Amer. Math. Soc. 97 (1986), 55-58. [27] W. Takahashi, Nonlinear Functional Analysis, Fixed Points Theory and its Applications, Yokohama Publishers, Yokohama, 2000. [28] W. Takahashi, Introduction to Nonlinear and Convex Analysis, Yokohama Publishers, Yokohama, 2009. [29] W. Takahashi, Fixed point theorems for new nonlinear mappings in a Hilbert space, J. Nonlinear Convex Anal. 11 (2010), 79-88. Afshan Batool Department of Mathematics, Quaid-i-Azam University, Islamabad, Pakistan E-mail address: [email protected] Tayyab Kamran Department of Mathematics, Quaid-i-Azam University, Islamabad, Pakistan E-mail address: [email protected] Choonkil Park Research Institute for Natural Sciences Hanyang University, Seoul 04763, Korea E-mail address: [email protected] Jung Rye Lee Department of Mathematics, Daejin University, Kyunggi 11159, Korea E-mail address: [email protected]

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Some properties on Dirichlet-Hadamard product of Dirichlet series ∗ Yong-Qin Cuia 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 constructing a new form Dirichlet-Hadamard product of Dirichlet series, we investigate the relation about the growth of Dirichlet series and obtain some estimates on the upper and the lower bounds of the (lower) q-order and the (lower) q-type of Dirichlet-Hadamard product of Dirichlet series. We also study the growth on scalar multiplication and shift of Dirichlet series. Our results of this paper are improvements of the previous theorems given by Kong and Deng. Key words: Dirichlet-Hadamard product, growth, scalar multiplication, Dirichlet series. 2010 Mathematics Subject Classification: 30B50, 30D15, 11F66.

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 → ∞; s = σ + it (σ, t are real variables); an are nonzero complex numbers. Let f (s) satisfy log n = 0, λn

(2)

log |an | = −∞, λn

(3)

lim sup n→∞

lim sup n→∞

then we have the abscissas of convergence and absolute convergence are +∞ by applying the Valion’s formula (see [4]), that is, f (s) is an analytic function in the whole plane C. We denote D to be the class of all functions f (s) satisfying (2),(3). ∗ The authors were supported by the NSF of China(11561033), the Natural Science Foundation of Jiangxi Province in China 20151BAB201008), and the Foundation of Education Department of Jiangxi (GJJ150902) of China. † Corresponding author.

1

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Definition 1.1 (see [16]). Let f (s) ∈ D, the order of f (s) is defined by ρ = lim sup σ→+∞

log log M (σ, f ) , σ

where M (σ, f ) =

{|f (σ + it)|}, f or

sup

−∞ N = max{N1 , N2 }, we have γn log[q−1] γn < ρ1 + ε, log |an |−1

ξn log[q−1] ξn < ρ2 + ε. log |bn |−1

(10)

νξn log[q−1] ξn µγn log[q−1] γn + , ρ1 + ε ρ2 + ε

(11)

Since cn = aµn bνn , it follows from (10) that log |cn |−1 = µ log |an |−1 + ν log |bn |−1 > then from (11) we have λn log[q−1] λn < log |cn |−1 Since q = 2, 3, · · · and ξn =

λn log[q−1] λn [q−1] µ γn + ρ2ν+ε ξn ρ1 +ε γn log

1 αη+β λn , γn

=

1 α+ β η

log[q−1] ξn

.

(12)

λn , it follows

log[q−1] γn ∼ log[q−1] ξn ∼ log[q−1] λn .

(13)

Since ε is arbitrary, it follows from (12) and (13) that ρ = lim sup n→∞

λn log[q−1] λn ≤ log |cn |−1

µ 1 ρ1 α+ β η

1 +

ν 1 ρ2 αη+β

=

(αη + β)ρ1 ρ2 . µηρ2 + νρ1 2

Theorem 2.2 Let f1 (s), f2 (s)(∈ D) be of lower q-order χ1 , χ2 , respectively. If f1 (s), f2 (s) satisfy the conditions of Lemma 2.2, then the lower q-order χ of F (s) satisfies χ≥

(αη + β)χ1 χ2 . ηµχ2 + νχ1

Proof: Suppose that χ1 , χ2 > 0. From Theorem 1.2, for any ε > 0, there exists a positive number N ∈ N+ such that n > N , we have γn log[q−1] γn−1 ξn log[q−1] ξn−1 > χ1 − ε, > χ2 − ε. −1 log |an | log |bn |−1

(14)

Since cn = aµn bνn , it follows from (14) that log |cn |−1 = µ log |an |−1 + ν log |bn |−1 µ ν < (γn log[q−1] γn−1 ) + (ξn log[q−1] ξn−1 ). χ1 − ε χ2 − ε

(15)

Thus, from (13) and (15) we have λn log[q−1] λn−1 > log |cn |−1

λn log[q−1] λn−1 µ χ1 −ε (γn

log

[q−1]

γn−1 ) +

ν χ2 −ε (ξn

log[q−1] ξn−1 )

.

(16)

By Lemma 2.2, we have ϕ(n) is a non-decreasing function. And since ε is arbitrary, it follows from (16) that !−1 λn log[q−1] λn−1 1 µ 1 ν (αη + β)χ1 χ2 χ = lim inf ≥ + = . β χ n→∞ log |cn |−1 αη + β χ µηχ2 + νχ1 α+ η 1 2 2

This completes the proof of Theorem 2.2. 5

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Theorem 2.3 Suppose that f1 (s), f2 (s) are two ρ[q] -regular growth functions and of qorder ρ1 , ρ2 , respectively, and if f1 (s), f2 (s) satisfy the conditions of Lemma 2.2. (i) Then F (s) is also ρ[q] -regular growth function, and of q-order ρ satisfying (α + β)ρ1 ρ2 , µηρ2 + νρ1

ρ=

ρ1 , ρ2 ∈ [0, +∞).

(ii) If ρ1 , ρ2 ∈ (0, +∞) and f1 (s), f2 (s) are of q-type T1 , T2 , respectively, then q-type T of F (s) satisfy  µρη vρ (αη+β)ρ1  T2(αη+β)ρ2 ,  T1 T ≤ µρη vρ αη + β  (ρ1 T1 ) ρ1 (αη+β) (ρ2 T2 ) ρ2 (αη+β) ,  µρη ρη (αη+β)ρ1

q = 3, 4, 5, . . . , q = 2.

Proof: (i) Since f1 (s), f2 (s) are two ρ[q] -regular growth functions, thus χ1 = ρ1 , χ2 = ρ2 , where χ1 , χ2 are the lower q-order of f1 (s), f2 (s), respectively. Thus, it follows by Theorem 2.1 and Theorem 2.2 that ρ=

(αη + β)ρ1 ρ2 , µηρ2 + νρ1

ρ1 , ρ2 ∈ [0, +∞).

This proves (i). (ii) From Theorem 1.1, we have ρ1

T1 = lim sup |an | γn log[q−2] ( n→∞

ρ2 γn ξn ), T2 = lim sup |bn | ξn log[q−2] ( ). eρ1 eρ2 n→∞

So for any ε > 0, there exists a positive number N ∈ N+ such that n > N , we have ρ1

|an | γn ≤

ρ2 T1 + ε , |bn | ξn [q−2] γn log ( eρ1 )

≤

T2 + ε . [q−2] ξn log ( eρ2 )

If q = 3, 4, 5, · · · , we have log[q−2] (

γn ξn ) ∼ log[q−2] γn , log[q−2] ( ) ∼ log[q−2] ξn . eρ1 eρ2

(17)

And cn = aµn bνn , then it follows from (17) that ρ

|cn | λn log[q−2] (

ρ λn λn ) = (|an |µ |bn |ν ) λn log[q−2] ( ) eρ eρ ρ1

µργn

ρ2

νρξn

= (|an | γn ) ρ1 λn (|bn | ξn ) ρ2 λn log[q−2] ( ≤( =(

µργn T1 + ε ) ρ 1 λn [q−2] γn log ( eρ1 )

(

λn ) eρ

νρξn T2 + ε ) ρ 2 λn [q−2] ξn log ( eρ2 )

µηρ T1 + ε ) (αη+β)ρ1 [q−2] γn log ( eρ1 )

log[q−2] (

log[q−2] (

λn ). (18) eρ

µηρ νρ λn ) ≤ (T1 + ε) (αη+β)ρ1 (T2 + ε) (αη+β)ρ2 . eρ

(19)

(

νρ T2 + ε ) (αη+β)ρ2 [q−2] ξn log ( eρ2 )

λn ) eρ

Thus, from (18) we have ρ

T = lim sup |cn | λn log[q−2] ( n→∞

6

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If q = 2, from (5), we have for any ε > 0, there exists a positive number N ∈ N+ such that n > N ρ1 ρ2 T1 + ε T2 + ε (20) |an | γn ≤ γn , |bn | ξn ≤ ξn , eρ1

Since cn =

aµn bνn ,

eρ2

it follows from (20) that ρ

|cn | λn

µργn νρξn λ ρ2 ρ1 λn n = (|an | γn ) ρ1 λn (|bn | ξn ) ρ2 λn eρ eρ νρ µηρ T2 + ε λn T1 + ε ≤ ( γn ) (αη+β)ρ1 ( ξn ) (αη+β)ρ2 ( ). eρ eρ1 eρ

(21)

2

Since ε is arbitrary, it follows from (19) and (21) that  µρη vρ (αη+β)ρ1  T2(αη+β)ρ2 ,  T1 T ≤ µρη vρ αη + β  (ρ1 T1 ) ρ1 (αη+β) (ρ2 T2 ) ρ2 (αη+β) ,  µρη ρη (αη+β)ρ1

q = 3, 4, 5, . . . , q = 2. 2

Thus, this completes the proof of Theorem 2.3.

Theorem 2.4 Let f1 (s), f2 (s) be two ρ[q] -perfectly regular growth functions, and satisfy (8),the condition of lemma2.2 and log[q−2] γn−1 ∼ log[q−2] γn ,

log[q−2] ξn−1 ∼ log[q−2] ξn , n → ∞.

(22)

Set ρ1 , ρ2 , T1 , T2 , τ1 and τ2 be the q-order,q-type and lower q-type of f1 (s), f2 (s), then F (s) is of ρ[q] -perfectly regular growth ρ and its q-type T satisfies  µρη vρ (αη+β)ρ1  T2(αη+β)ρ2 ,  T1 T = µρη vρ αη + β  (ρ1 T1 ) ρ1 (αη+β) (ρ2 T2 ) ρ2 (αη+β) ,  µρη ρη (αη+β)ρ1

q = 3, 4, 5, . . . , q = 2.

Proof: Suppose that ρ1 , ρ2 ∈ (0, +∞),τ1 , τ2 < +∞. From Theorem 1.3, for any ε > 0, there exists a positive number N ∈ N+ such that n > N , we have ρ1

|an | γn log[q−2] (

γn−1 ) ≥ τ1 − ε, eρ1

ρ2

|bn | ξn log[q−2] (

ξn−1 ) ≥ τ2 − ε. eρ2

If q ≥ 3, it follows ρ

τ = lim inf |cn | λn log[q−2] ( n→∞

λn−1 ) eρ

ρ

= lim inf (|an |µ |bn |ν ) λn log[q−2] ( n→∞

≥ lim inf {[ n→∞

µγn τ1 − ε ] ρ1 [q−2] γn−1 log ( eρ1 ) µηρ

[

λn−1 ) eρ τ2 − ε

n−1 log[q−2] ( ξeρ ) 2

]

νξn ρ2

ρ

} λn log[q−2] (

λn−1 ) eρ

νρ

≥ (τ1 − ε) (αη+β)ρ1 (τ2 − ε) (αη+β)ρ2 .

(23)

From Theorem 2.3 and since ε > is arbitrary, it follows from (23) that µρη

vρ

µηρ

νρ

T1(αη+β)ρ1 T2(αη+β)ρ2 ≥ T ≥ τ ≥ (τ1 ) (αη+β)ρ1 (τ2 ) (αη+β)ρ2 .

(24)

7

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If q = 2, from (22) we have λn−1 eρ τ1 − ε ρµργλn τ2 − ε ρνρξλn ≥ lim inf ( γn−1 ) 1 n ( ξn−1 ) 2 n ρ

τ = lim inf |cn | λn n→∞

n→∞

eρ1

eρ2

νρ µηρ 1 (αη + β)ρ1 τ1 (αη+β)ρ 1 [(αη + β)ρ τ ] (αη+β)ρ2 ≥ [ ] 2 2 ρ η µρη vρ αη + β = (ρ1 τ1 ) ρ1 (αη+β) (ρ2 τ2 ) ρ2 (αη+β) µρη ρη (αη+β)ρ1

(25)

From Theorem 2.3 and since ε > is arbitrary, it follows from (25) that αη + β ρη

µρη (αη+β)ρ1

vρ

µρη

(ρ1 T1 ) ρ1 (αη+β) (ρ2 T2 ) ρ2 (αη+β) ≥ T ≥ τ αη + β

≥

ρη

µρη

µρη (αη+β)ρ1

vρ

(ρ1 τ1 ) ρ1 (αη+β) (ρ2 τ2 ) ρ2 (αη+β)

(26)

Since f1 (s), f2 (s) are ρ[q] -perfectly regular growth and τj = Tj , j = 1, 2, from (24) and (26), it is easy to get the conclusions of Theorem 2.4. Thus, we complete the proof of Theorem 2.4. 2

3

The linear substitution of Dirichlet series

Next, we define the scalar multiplication of Dirichlet series as follows Definition 3.1 Let k be a positive number, we define the scalar multiplication of Dirichlet series as follows H(s) = f (ks) =

∞ X

an eλn (ks) =

∞ X

an eζn s , ζn = kλn .

n=1

n=1

Theorem 3.1 Let f (s) ∈ D, then H(s) ∈ D. Furthermore, if ϕ(n) is a non-decreasing function with n, then the q-order ρ∗ and the lower q-order χ∗ of H(s) satisfy ρ∗ = kρ and χ∗ = kχ. Proof: Since lim sup n→∞

log |an | log |an | = lim sup = −∞, ζn kλn n→∞

thus, we have H(s) ∈ D. Furthermore, we have ρ∗ = lim sup n→∞

kλn log[q−1] kλn ζn log[q−1] ζn = lim sup log |an |−1 log |an |−1 n→∞

= lim sup k n→∞

λn log[q−1] λn = kρ, log |an |−1

and χ∗ = lim inf n→∞

ζn log[q−1] ζn−1 kλn log[q−1] kλn−1 = lim inf n→∞ log |an |−1 log |an |−1

= lim inf k n→∞

λn log[q−1] λn−1 = kχ. log |an |−1 8

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2

This completes the proof of Theorem 3.1. From Theorem 3.1, we can obtain the following result easily.

Theorem 3.2 Let f1 (s), f2 (s) ∈ D and satisfy the conditions of Lemma 2.2. Set ρ, χ be the q-order and the lower q-order of F (s), then the q-order ρ∗ and the lower q-order χ∗ of H ∗ (s) = F (ks) satisfy ρ∗ = kρ, χ∗ = kχ. Let f1 (s), f2 (s) ∈ D and k, m be positive numbers. Set H1 (s) = f1 (ks), H2 (s) = f2 (ms) and H ∗∗ (s) = (H1 4H2 )(µ, ν, s) =

∞ X

cn eλn s , cn = aµn bνn , λn = αkγn + βmξn ,

n=1

where α, β, µ, ν are positive numbers, an , bn are nonzero complex numbers, 0 < γn , ξn ↑ +∞. The following result is about the growth of H ∗∗ (s). Theorem 3.3 Let f1 (s), f2 (s) ∈ D satisfy (8) and the conditions of Lemma 2.2. Let ρ, χ be the q-order and lower q-order of F (s), then the q-order ρ∗∗ and the lower q-order χ∗∗ of H ∗∗ (s) satisfy ρ∗∗ =

kαη + mβ kαη + mβ ρ, χ∗∗ = χ. αη + β αη + β

Proof: Since ρ∗∗ = lim sup n→∞

= lim sup n→∞

λn log[q−1] λn log |cn |−1 (kαγn + mβξn ) log[q−1] (kαγn + mβξn ) log |cn |−1

kαη + mβ λn log[q−1] (kαγn + mβξn ) αη + β log |cn |−1 n→∞ kαη + mβ = ρ, αη + β = lim sup

(27)

and χ∗∗ = lim inf n→∞

= lim inf n→∞

λn log[q−1] λn−1 log |cn |−1 (kαγn + mβξn ) log[q−1] (kαγn−1 + mβξn−1 ) log |cn |−1

kαη + mβ λn log[q−1] (kαγn−1 + mβξn−1 ) n→∞ αη + β log |cn |−1 kαη + mβ = χ, αη + β = lim inf

thus from (27) and (28) we can prove the conclusions of Theorem 3.3. ∗

∗∗

∗

∗∗

Remark 3.1 From Theorem 3.3, we can get that ρ = ρ , χ = χ

(28) 2

if k = m.

In 2008, Kong defined the shift of Dirichlet series (see [6]). Definition 3.2 (see [6]). Let α be a real number, the shift of Dirichlet series can be defined as follows G(s) = f (s + α) =

∞ X

an eλn (s+α) =

n=1

∞ X

a0n eλn s ,

n=1

0

where an = an eλn α . 9

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Let f1 (s), f2 (s) ∈ D, α1 , α2 be two real numbers and k be a positive number. Set G1 (s) = f1 (s + α1 ), G2 (s) = f2 (s + α2 ), H1 (s) = f1 (ks), and G∗ (s) = (G1 4G2 )(µ, ν, s) =

∞ X

cn eλn s , cn = (a0n )µ (b0n )ν , λn = αγn + βξn ,

n=1

and G∗∗ (s) = (H1 4G2 )(µ, ν, s) =

∞ X

cn eλn s , cn = aµn (b0n )ν , λn = αkγn + βξn ,

n=1

where α, β, µ, ν are positive numbers, an , bn are nonzero complex numbers, 0 < γn , ξn ↑ +∞. We investigate the growth of G∗ (s), G∗∗ (s) and obtain the following results Theorem 3.4 Let f1 (s), f2 (s) ∈ D satisfy (8) and the conditions of Lemma 2.2. Let ρ, χ be the q-order and lower q-order of F (s), then the q-order ρ∗∗ 1 and the lower q-order ∗ ∗∗ ∗∗ χ∗∗ of G (s) satisfy ρ = ρ, χ = χ. 1 1 1 Proof: Since ρ∗∗ 1 = lim sup n→∞

= lim sup n→∞

= lim sup n→∞

λn log[q−1] λn log |cn |−1

(29)

λn log[q−1] λn −1 log |cn | − (µγn α1 + vξn α2 ) λn log[q−1] λn α1 +vξn α2 log |cn |−1 (1 + µγnlog ) |cn |

(30) (31)

= ρ,

(32)

that is, ρ∗∗ 1 = ρ. Similarly, we have χ∗∗ 1 = χ.

2

Theorem 3.5 Let f1 (s), f2 (s) ∈ D satisfy (8) and the conditions of Lemma 2.2. Let ρ, χ be the q-order and lower q-order of F (s), then the q-order ρ∗∗ 2 and the lower q-order ∗∗ χ∗∗ of G (s) satisfy 2 kαη + β kαη + β ρ, χ∗∗ χ. ρ∗∗ 2 = 2 = αη + β αη + β Proof: Since ρ∗∗ 2 = lim sup n→∞

= lim sup n→∞

= lim sup

λn log[q−1] λn log |cn |−1

(33)

(kαγn + βξn ) log[n−1] (kαγn + βξn ) log |cn |−1 − (vξn α2 )

(34)

kαη+β αη+β λn

n→∞

= that is, ρ∗∗ 2 =

log[n−1] (αkγn + βξn )

log |cn |−1 (1 +

vξn α2 log |cn | )

kαη + β ρ. αη + β

(35) (36)

kαη+β αη+β ρ.

Similarly, we have χ∗∗ 2 =

2

kαη+β αη+β χ.

∗∗ ∗∗ ∗∗ Remark 3.2 From Theorem 3.5, we can get that ρ∗∗ 2 = ρ1 , χ2 = χ1 if k = 1.

10

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References [1] P. V. Filevich and M. N. Sheremeta, Regularly increasing entire Dirichlet series, Math. 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 and D. C. Sun, The reuglar growth of Dirichlet series on the whole plane, Acta Mathematica Scientia 31 (2011), 991-997(in Chinese). [4] G. H. Hardy and M. Riesz, The general theory of Dirichlet series, New York: Stechert-Hafner, Inc, 1964. [5] Y. Y. Kong, On some q-orders and q-types of Dirichlet-Hadamard product function, Acta Mathematica Sinica 52 (6) (2009), 1165-1172 (in Chinese Series). [6] Y. Y. Kong, G. T. Deng, The Dirichlet-Hadamard product of Dirichlet series, Chinese Annals of Mathematics, 35A(2) (2014), 145-152. [7] Y. Y. Kong and H. L. Gan, On orders and types of Dirichlet series of slow growth, Turk J. Math. 34 (2010), 1-11. [8] L. Ya. Mykytyuk and M. M. Sheremeta, On the asymptotic behavior of the remainer of a Dirichlet series absolutely convergent in a half-plane, Ukrainian Mathematical Journal, 55 (3) (2003), 456-467. [9] A. Nautiyal, On the coefficients of analytic Dirichlet series of fast growth, Indian J. Pure Appl. Math. 15(10) (1984), 1102-1114. [10] A. Nautiyal and D. P. Shukla, On the approximation of an analytic function by exponential polynomials, Indian J. Pure appl. Math. 14 (6) (1983), 722-727. [11] K. A. M. Sayyed, M. S. Metwally and M. T. Mohamed, Some orders and types of generalized Hadamard Product of entire functions, Southeast Asian Bull. Math. 26 (2002), 121-132. [12] L. N. Shang and Z. S. Gao, Entire functions defined by Dirichlet series, J. Math. Anal. Appl. 339 (2008), 853-862. [13] H. Wang and H. Y. Xu, The approximation and growth problem of Dirichlet series of infinite order, J. Comput. Anal. Appl. 16 (2) (2014), 251-263. [14] H. Y. Xu and 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). ´ [15] J. R. Yu, Sur les droites de Borel de certaines fonction enti`eres, Annales Ecole Norm. Sup. 68(3) (1951), 65-104. [16] J. R. Yu, X. Q. Ding, and F. J. Tian, On The Distribution of Values of Dirichlet Series And Random Dirichlet Series, Wuhan: Press in Wuhan University, 2004(in Chinese).

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The differential and subdifferential for fuzzy mappings based on the generalized difference of n-cell fuzzy-numbers

b

Shexiang Haia , Zengtai Gongb∗ a School of Science, Lanzhou University of Technology, Lanzhou, 730050, P.R. China College of Mathematics and Statistics, Northwest Normal University, Lanzhou 730070, China

Abstract We use the concept of generalized difference for fuzzy n-cell numbers which is presented in this paper to introduce and study the differential and gradient of fuzzy n-cell mappings. At the same time, some connections between gradient, boundary-f unction-wise gradient and level-wise gradient of fuzzy n-cell mappings are established. Furthermore, the subdifferential for fuzzy n-cell mappings based on the ordering c is discussed. Keywords: Fuzzy numbers; fuzzy n-cell mappings; gradient; subdifferential. 1. Introduction Since the concept and operations of fuzzy set were introduced by Zadeh [1], enormous researchers have been dedicated on development of various aspects of the theory and applications of fuzzy sets. Soon after, Zadeh proposed the notion of fuzzy numbers in [2,3,4]. Since then, fuzzy numbers have been extensively investigated by many authors. The importance of the derivative of a function in the study of mathematical programming and fuzzy differential equations is well-known. It is necessary to introduce a concept of differentiability for fuzzy mappings. Toward this end, in fuzzy analysis, there are a variety of notions of derivative for fuzzy mappings. The concept of fuzzy derivative first introduced by Chang and Zadeh [5] in 1972. Since then, numerous definitions of the differentiability of fuzzy mappings have been presented. In 1983, Puri and Ralescu [6] defined the derivative and G-derivative of fuzzy mappings from an open subset of a normed space into n-dimension fuzzy number space E n by using embedding theorem (which shows how to isometrically embed E n into a Banach space as a closed convex cone of vertex zero) and Hukuhara difference. In 1987, Kaleva [7] discussed the G-derivative, obtained a sufficient condition of the H-differentiability of the fuzzy mappings from [a, b] into E n and a necessary condition for the H-differentiability of fuzzy mapping from [a, b] into E 1 . In 2003, Wang and Wu [8] put forward the concepts of directional derivative, differential and sub-differential of fuzzy mappings from Rn into E 1 by using Hukuhara difference. However, the usual Hukuhara difference between two fuzzy numbers exists only under very restrictive conditions [7] and the H-difference of two fuzzy numbers does not always exist [9]. The g-difference proposed in [9] overcomes these shortcomings of the above discussed concepts and the g-difference of two fuzzy numbers always exists. Based on the novel generalizations of the Hukuhara difference for fuzzy sets, Bede [9] introduced and studied new generalized differentiability concepts for fuzzy valued functions in 2013, in particular, a new very general fuzzy differentiability concept was defined and studied, the so-called g-derivative, and it was shown that the g-derivative is the most general among all similar definitions. Motivated both by [9] and the importance of the concept of differential for fuzzy analysis, the concept of differential and gradient for fuzzy n-cell mappings is introduced, which is based on the novel generalizations difference of fuzzy n-cell numbers presented in this paper. The remainder of the paper is organised as follows: First of all, we give the preliminary terminology used in the present paper. And then, in Section 3, we present the concept of generalized difference for fuzzy n-cell numbers and discuss several properties for it. We use the generalized difference for fuzzy n-cell numbers to introduce and study differential and gradient for fuzzy n-cell mappings in Section 4. At † ∗

Supported by the Natural Scientific Fund of China (11461062, 61262022). Corresponding Author:Zeng-Tai Gong. Tel.: +869317971430. E-mail addresses: [email protected] 184

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Shexiang Hai and Zengtai Gong: The differential and subdifferential for fuzzy mappings based on ...

last, using the concept of the ordering c for fuzzy n-cell numbers, section 5 deals with the subdifferential for fuzzy n-cell mappings. 2. Preliminaries Throughout this paper, Rn denote the n-dimensional Euclidean space and F (Rn ) denote the set of all fuzzy subsets on Rn . A fuzzy subset of (in short, a fuzzy set) Rn is a function u : Rn → [0, 1]. For each fuzzy sets u, we denote by [u]r = {x ∈ Rn : u(x) ≥ r}, for any r ∈ (0, 1], its r-level set. By suppu = {x ∈ Rn : u(x) > 0} we represent the support of u. Suppose u ∈ F (Rn ), satisfies the following conditions: (1) u is a normal fuzzy set, i.e., there exists an x0 ∈ Rn such that u(x0 ) = 1, (2) u is a convex fuzzy set, i.e., u(λx + (1 − λ)y) ≥ min{u(x), u(y)} for any x, y ∈ Rn and λ ∈ [0, 1], (3) u is upper semicontinuous , S (4) [u]0 = {x ∈ Rn : u(x) > 0} = r∈(0,1] [u]r is compact, here A denotes the closure of A. Then u is called a fuzzy number. We use E n to denote the fuzzy number space [10,11,12,13]. It is clear that each u ∈ Rn can be considered as a fuzzy number u defined by  1, x = u, u e(x) = 0, otherwise. In particular, the fuzzy number e 0 is defined as e 0(x) = 1 if x = 0, and e 0(x) = 0 otherwise. n r Definition 2.1. [14] If u ∈ E , and [u] is a cell, i.e., for any r ∈ [0, 1], [u]r =

n Y

+ − + + − − + [u− i (r), ui (r)] = [u1 (r), u1 (r)] × [u2 (r), u2 (r)] × · · · × [un (r), un (r)],

i=1 + − + where u− i (r), ui (r) ∈ R with ui (r) ≤ ui (r) (i = 1, 2, · · · , n), then we call u a fuzzy n-cell number. Denote the collection of all fuzzy n-cell numbers by L(E n ). − For any r ∈ [0, 1], li [u]r = u+ i (r) − ui (r) (i = 1, 2, · · · , n) is called the r-level length of a fuzzy n-cell number u with respect to the ith component. + Theorem 2.1. [14] (Representation theorem). If u ∈ L(E n ), then for i = 1, 2, · · · , n, u− i (r), ui (r) are real-valued functions on [0, 1], and satisfy (1) u− i (r) are non-decreasing, left continuous at r ∈ (0, 1] and right continuous at r = 0, (2) u+ i (r) are non-increasing, left continuous at r ∈ (0, 1] and right continuous at r = 0, + − + (3) u− i (r) ≤ ui (r) (it is equivalent to ui (1) ≤ ui (1)). Conversely if ai (r), bi (r) (i = 1, 2, · · · , n) are real-valued functions on [0, 1] which satisfy conditions Qn n r (1)-(3), then there exists a unique u ∈ L(E ) such that [u] = i=1 [ai (r), bi (r)] for any r ∈ [0, 1]. Theorem 2.2. [14] Let u, v ∈ Q L(E n ) and k ∈ R. Then for any r ∈ [0, 1], − + + r r r (1) [u + v] = [u] +[v] = ni=1 [u− i (r) + vi (r), ui (r) + vi (r)], Qn − + [kui (r), kui (r)], k ≥ 0, (2) [ku]r = k[u]r = Qi=1 n + − i=1 [kui (r), kui (r)], k < 0, Qn − − − + + − + + r (3) [uv] = i=1 [min{ui (r)vi (r), ui (r)vi (r), ui (r)vi (r), ui (r)vi (r)}, − − + + − + + max{u− i (r)vi (r), ui (r)vi (r), ui (r)vi (r), ui (r)vi (r)}]. Given u, v ∈ L(E n ), the distance D : L(E n ) × L(E n ) → [0, +∞) between u and v is defined by the equation D(u, v) = supr∈[0,1] d([u]r , [v]r ) − + + = supr∈[0,1] max1≤i≤n {| u− i (r) − vi (r) |, | ui (r) − vi (r) |}.

Then (L(E n ), D) is a complete metric space, and satisfies D(u + w, v + w) = D(u, v), D(ku, kv) = |k|D(u, v) for any u, v, w ∈ L(E n ), k ∈ R. In recent years, several authors have discussed different ordering relation of fuzzy numbers [15]. To the best of our knowledge, very few investigations have been appeared to study ordering relation of fuzzy n-cell numbers. For this reason, an ordering c of fuzzy n-cell numbers will be introduced and be applied to solve fuzzy constrained minimization problem. 185

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Definition 2.2. Let τ : L(E n ) → Rn be a vector-valued function defined by R 1 R ··· R r x1 dx1 dx2 ···dxn R 1 R ··· R r x2 dx1 dx2 ···dxn R 1 R ··· R r xn dx1 dx2 ···dxn τ (u) = (2 0 r R ··· R[u] 1dx dx ···dxn dr, 2 0 r R ··· R[u] 1dx dx ···dxn dr, · · · , 2 0 r R ··· R[u] 1dx dx ···dxn dr) 1 2 1 2 1 2 [u]r [u]r [u]r R1 R1 R1 + − + − + − = ( 0 r(u1 (r) + u1 (r))dr, 0 r(u2 (r) + u2 (r))dr, · · · , 0 r(un (r) + un (r))dr), where

R1 0

R

R ··· r xi dx1 dx2 ···dxn R R[u] dr ··· [u]r 1dx1 dx2 ···dxn

r

R

(i = 1, 2, · · · , n) is the Lebesque integral of r

R ··· r xi dx1 dx2 ···dxn R R[u] ··· [u]r 1dx1 dx2 ···dxn

(i =

1, 2, · · · , n) on [0, 1]. The vector-valued function τ is called a ranking value function defined on L(E n ). In this case τ (u) represents a centroid of the fuzzy n-cell number u. From the ranking value function τ (u), we consider the following ordering relation c on L(E n ). Definition 2.3. Let u, v ∈ L(E n ), C ⊆ Rn be a closed convex cone with 0 ∈ C and C 6= Rn . We say that u c v (u precedes v) if τ (v) ∈ τ (u) + C (τ (v) − τ (u) ∈ C). Obviously the order relation c is reflexive and transitive, and c is a partial order relation on L(E n ). If u, v ∈ E 1 , C = [0, +∞) ⊆ R, then Definition 2.3 coincides with Definition 2.5 of reference [15]. We say that u ≺c v if u c v and τ (u) 6= τ (v). Sometimes we may write v c u (resp. v c u) instead of u c v (resp. u ≺c v). Remark 2.1. Let u, v ∈ L(E n ), k1 , k2 ∈ R. According to Theorem 2.2 and Definition 2.2, it is easy to verify that τ (k1 u + k2 v) = k1 τ (u) + k2 τ (v). Theorem 2.3. Let u1 , u2 , v1 , v2 , ∈ L(E n ), k1 , k2 ∈ [0, +∞], C ⊆ Rn be a closed convex cone with 0 ∈ C and C 6= Rn . If u1 c v1 and u2 c v2 , then k1 u1 + k2 u2 c k1 v1 + k2 v2 . Proof. It is follows from Definition 2.3 that τ (v1 ) − τ (u1 ) ∈ C and τ (v2 ) − τ (u2 ) ∈ C. On the other hand, closed convex cone C is closed under addition and positive scalar multiplication, thus k1 (τ (v1 ) − τ (u1 )) + k2 (τ (v2 ) − τ (u2 )) ∈ C, which implies that k1 τ (v1 ) + k2 τ (v2 ) ∈ k1 τ (u1 ) + k2 τ (u2 ) + C. It is obvious from Remark 2.1 that τ (k1 v1 + k2 v2 ) ∈ τ (k1 u1 + k2 u2 ) + C, then k1 u1 + k2 u2 c k1 v1 + k2 v2 . 3. Generalized difference for fuzzy n-cell numbers n the family of all nonempty compact convex subsets of Rn , that is {Kn ⊂ Rn : A 6= We denote by KC C n ∅ is compact and convex}. Stefanini [16] defined the generalized Hukuhara difference of two sets A ∈ KC n and B ∈ KC as follows:  (1) A = B + C, A gH B = C ⇐⇒ or (2) B = A + (−1)C. n and B ∈ Kn , if the generalized Hukuhara difference C = A For any A ∈ KC gH B exists, it is unique. C Q Q + + , a ] and Bi = [b− Lemma 3.1. [16] Q Let A = ni=1 Ai , B = ni=1 Bi , where Ai = [a− i i i , bi ] are real n compact intervals ( i=1 denotes the cartesian product). If A gH B exists, then

A gH

n n Y Y − + + − − + + B= (Ai gH Bi ) = [min{a− i − bi , ai − bi }, max{ai − bi , ai − bi }]. i=1

i=1

Lemma 3.2. [16] The gH-difference A gH B exists if and only if one of the two conditions is satisfied: − + + (i) a− i − bi ≤ ai − bi , i = 1, 2, · · · , n or − + + (ii)a− i − bi ≥ ai − bi , i = 1, 2, · · · , n. According to Lemma 3.2, the definition of generalized Hukuhara difference for real compact intervals is extended to the fuzzy case. Definition 3.1. Let u, v ∈ L(E n ). If li [u]r ≤ li [v]r or li [u]r ≥ li [v]r for any r ∈ [0, 1] and i = 1, 2, · · · , n, then the generalized difference (g-difference for short) is given by its level sets as n Y − + + − − + + [u g v] = [ inf min{u− i (β) − vi (β), ui (β) − vi (β)}, sup max{ui (β) − vi (β), ui (β) − vi (β)}], r

i=1

β≥r

β≥r

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where β ∈ [r, 1]. Remark 3.1. If u, v ∈ E 1 , we have [u g v]r = [ inf min{u− (β) − v − (β), u+ (β) − v + (β)}, sup max{u− (β) − v − (β), u+ (β) − v + (β)}], β≥r

β≥r

which coincides with Definition 7 of reference [9]. Theorem 3.1. Let u, v ∈ L(E n ). If li [u]r ≤ li [v]r or li [u]r ≥ li [v]r for any r ∈ [0, 1] and i = 1, 2, · · · , n, then the g-difference u g v exists and u g v ∈ L(E n ). Proof. Assume that [w]r = [u g v]r Qn + + − − = i=1 [inf β≥r min{ui (β) − vi (β), ui (β) − vi (β)}, − + + supβ≥r max{u− i (β) − vi (β), ui (β) − vi (β)}],

for any r ∈ [0, 1]. We can prove that the class of sets [w]r determines a fuzzy n-cell number. For any r ∈ [0, 1], we have + + − wi− (r) = inf β≥r min{u− i (β) − vi (β), ui (β) − vi (β)} − + + ≤ supβ≥r max{u− i (β) − vi (β), ui (β) − vi (β)}

= wi+ (r). It can be easily seen that wi− (r) are non-decreasing while wi+ (r) are non-increasing, wi− (r) and wi+ (r) are left continuous on (0, 1] and right continuous at 0. It follows from Theorem 2.1 that the g-difference u g v exists and u g v = w ∈ L(E n ). From now on, throughout this paper, we will assume that the g-difference u g v for any fuzzy n-cell numbers u and v exists. Theorem 3.2. For any u, v, w ∈ L(E n ), we have (1) u g u = e 0, u g e 0 = u, e 0 g u = −u, (2) u g v = −(v g u), (3) (u + v) g (u + w) = v g w, (4) k(u g v) = ku g kv, k ∈ R, (5) (u + v) g v = u, (6) e 0 g (u g v) = v g u = (−u) g (−v), (7) u g v = v g u = w if and only if w = −w, furthermore, w = e 0 if and only if u = v. Proof. The proof of (1), (3) and (4) are immediate. (2) According to Definition 3.1, we have −[v g u]r Q + + = − ni=1 [inf β≥r min{vi− (β) − u− i (β), vi (β) − ui (β)}, + + supβ≥r max{vi− (β) − u− i (β), vi (β) − ui (β)}]

=

Qn

i=1 [− supβ≥r

+ + max{vi− (β) − u− i (β), vi (β) − ui (β)},

+ + − inf β≥r min{vi− (β) − u− i (β), vi (β) − ui (β)}]

=

− i=1 [− supβ≥r (− min{ui (β)

Qn

+ − vi− (β), u+ i (β) − vi (β)}),

− + + − inf β≥r (− max{u− i (β) − vi (β), ui (β) − vi (β)})]

=

Qn

i=1 [inf β≥r

− + + min{u− i (β) − vi (β), ui (β) − vi (β)},

− + + supβ≥r max{u− i (β) − vi (β), ui (β) − vi (β)}]

= [u g v]r , 187

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for any r ∈ [0, 1]. It follows from Theorem 2.2 that u g v = −(v g u). (5) We have from Theorem 2.2 that [(u + v) g v]r Qn − + + + − − = i=1 [inf β≥r min{(ui (β) + vi (β)) − vi (β), (ui (β) + vi (β)) − vi (β)}, − − + + + supβ≥r max{(u− i (β) + vi (β)) − vi (β), (ui (β) + vi (β)) − vi (β)}]

=

Qn

+ − + min{u− i (β), ui (β)}, supβ≥r max{ui (β), ui (β)}]

=

Qn

+ u− i (β), supβ≥r ui (β)]

=

Qn

i=1 [inf β≥r

i=1 [inf β≥r

− i=1 [ui (r),

u+ i (r)]

= [u]r , for any r ∈ [0, 1]. Then (u + v) g v = u. (6) It follows from (1), (2) and (3) that the proof of (6) is immediate. (7) We have from (2) that the proof of (7) is immediate. 4. The differential and gradient for fuzzy n-cell mappings In this work, let M be a convex set of m-dimensional Euclidean space Rm . We consider mappings F from M into L(E n ). Such a mapping is called a fuzzy n-cell mapping. For the sake of brevity, F is called Q a fuzzy mapping. Let Fe : M → L(E n ), for any r ∈ [0, 1], we denote Fr (t) = ni=1 [Fi− (r, t), Fi+ (r, t)]. Definition 4.1. Let Fe : M → L(E n ), t0 = (t01 , t02 , · · · , t0m ) ∈ intM, t = (t1 , t2 , · · · , tm ) ∈ intM. If g-difference Fe(t) g Fe(t0 ) exists and there exist uj ∈ L(E n ) (j = 1, 2, · · · , m), such that P 0 D(Fe(t) g Fe(t0 ), m j=1 uj (tj − tj )) = 0, lim t→t0 d(t, t0 ) then we say that Fe is differentiable at t0 and (u1 , u2 , · · · , um ) is the gradient of Fe at t0 , denoted by ∇Fe(t0 ), i.e., ∇Fe(t0 ) = (u1 , u2 , · · · , um ). Remark 4.1. Let Fe : M → L(E n ), t0 ∈ M. Then the gradient ∇Fe(t0 ) exists at t0 if and only if Fe(t) g Fe(t0 ) exists and there are uj ∈ L(E n ) (j = 1, 2, · · · , m), such that Fe(t01 , · · · , t0j + h, · · · , t0m ) g Fe(t01 , · · · , t0j , · · · , t0m ) , h→0 h

uj = lim

where h ∈ R and t = (t01 , · · · , t0j + h, · · · , t0m ) ∈ intM. Here the limit is taken in the metric space (L(E n ), D). Definition 4.2. Let Fe : M → L(E n ), t0 = (t01 , t02 , · · · , t0m ) ∈ intM, t = (t1 , t2 , · · · , tm ) ∈ intM. If there + exists u− ij (r), uij (r) ∈ R (i = 1, 2, · · · , n, j = 1, 2, · · · , m), such that P − 0 |Fi− (r, t) − Fi− (r, t0 ) − m j=1 uij (r)(tj − tj )| = 0 (i = 1, 2, · · · , n), lim t→t0 d(t, t0 ) and lim

|Fi+ (r, t) − Fi+ (r, t0 ) −

Pm

+ j=1 uij (r)(tj

d(t, t0 )

t→t0

− t0j )|

= 0 (i = 1, 2, · · · , n),

uniformly for r ∈ [0, 1], then we say that Fe is boundary-f unction-wise differentiable (b-differentiable for short) at t0 . Theorem 4.1. Let Fe : M → L(E n ) be a fuzzy mapping. If Fe is b-differentiable at t0 = (t01 , t02 , · · · , t0m ) ∈ intM, then there exist uj ∈ L(E n ), such that for any r ∈ [0, 1], r

[uj ] =

n Y i=1

+ − + [ inf min{u− ij (β), uij (β)}, sup max{uij (β), uij (β)}] (j = 1, 2, · · · , m). β≥r

β≥r

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Proof. For any r ∈ [0, 1], we can show that the class of sets n Y + − + [ inf min{u− ij (β), uij (β)}, sup max{uij (β), uij (β)}] (j = 1, 2, · · · , m) i=1

β≥r

β≥r

satisfies the conditions of Theorem 2.1. According to Definition 4.2, if there exists δ > 0, such that for any |h| < δ with t = (t01 , · · · , t0j + h, · · · , t0m ) ∈ intM, we have u− ij (r)

Fi− (r, t01 , · · · , t0j + h, · · · , t0m ) − Fi− (r, t01 , · · · , t0j , · · · , t0m ) , = lim h→0 h Fi+ (r, t01 , · · · , t0j + h, · · · , t0m ) − Fi+ (r, t01 , · · · , t0j , · · · , t0m ) , h→0 h

u+ ij (r) = lim

for all i = 1, 2, · · · , n and j = 1, 2, · · · , m. Since Fi− (r, t) and Fi+ (r, t) are left continuous with respect r ∈ (0, 1] and right continuous at r = 0, Fi− (r, t01 , · · · , t0j + h, · · · , t0m ) − Fi− (r, t01 , · · · , t0j , · · · , t0m ) h and

Fi+ (r, t01 , · · · , t0j + h, · · · , t0m ) − Fi+ (r, t01 , · · · , t0j , · · · , t0m ) h are left continuous at r ∈ (0, 1] and right continuous at r = 0. Thus for any i = 1, 2, · · · , n and j = 1, 2, · · · , m, inf β≥r min{

Fi− (r,t01 ,··· ,t0j +h,··· ,t0m )−Fi− (r,t01 ,··· ,t0j ,··· ,t0m ) , h Fi+ (r,t01 ,··· ,t0j +h,··· ,t0m )−Fi+ (r,t01 ,··· ,t0j ,··· ,t0m ) }, h

supβ≥r max{

Fi− (r,t01 ,··· ,t0j +h,··· ,t0m )−Fi− (r,t01 ,··· ,t0j ,··· ,t0m ) , h Fi+ (r,t01 ,··· ,t0j +h,··· ,t0m )−Fi+ (r,t01 ,··· ,t0j ,··· ,t0m ) } h

are left continuous at r ∈ (0, 1] and right continuous at r = 0. Therefore, for any r ∈ [0, 1], i = 1, 2, · · · , n and j = 1, 2, · · · , m, we have + (1) inf β≥r min{u− ij (β), uij (β)} are non-decreasing and left continuous at r ∈ (0, 1] and right continuous at r = 0, + (2) supβ≥r max{u− ij (β), uij (β)} are non-increasing and left continuous at r ∈ (0, 1] and right continuous at r = 0, + − + (3) inf β≥r min{u− ij (β), uij (β)} ≤ supβ≥r max{uij (β), uij (β)}. n Consequently, there exist uj ∈ L(E ) (j = 1, 2, · · · , m), such that for any r ∈ [0, 1], r

[uj ] =

n Y i=1

+ − + [ inf min{u− ij (β), uij (β)}, sup max{uij (β), uij (β)}] (j = 1, 2, · · · , m). β≥r

β≥r

Definition 4.3. Let Fe : M → L(E n ) is b-differentiable at t0 . For any r ∈ [0, 1], we denote [uj ]r =

n Y i=1

+ − + [ inf min{u− ij (β), uij (β)}, sup max{uij (β), uij (β)}] (j = 1, 2, · · · , m), β≥r

β≥r

then we say that (u1 , u2 , · · · , um ) is the boundary-f unction-wise gradient (b-gradient for short) of Fe at t0 , denoted by ∇b Fe(t0 ), i.e., ∇b Fe(t0 ) = (u1 , u2 , · · · , um ). 189

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Remark 4.2. Let Fe : M → L(E n ), t0 ∈ M. Then the b-gradient ∇b Fe(t0 ) exists at t0 if and only if there + are u− ij (r), uij (r) ∈ R (i = 1, 2, · · · , n, j = 1, 2, · · · , m), such that Fi− (r, t01 , · · · , t0j + h, · · · , t0m ) − Fi− (r, t01 , · · · , t0j , · · · , t0m ) h→0 h

u− ij (r) = lim and

Fi+ (r, t01 , · · · , t0j + h, · · · , t0m ) − Fi+ (r, t01 , · · · , t0j , · · · , t0m ) = lim h→0 h

u+ ij (r)

uniformly for r ∈ [0, 1], where h ∈ R with t = (t01 , · · · , t0j + h, · · · , t0m ) ∈ intM and r

[uj ] =

n Y

+ − + [ inf min{u− ij (β), uij (β)}, sup max{uij (β), uij (β)}] (j = 1, 2, · · · , m). β≥r

i=1

β≥r

Theorem 4.2. Let the b-gradient ∇b Fe(t0 ) of fuzzy mapping Fe : M → L(E n ) be exist at t0 ∈ intM. If Fe(t) g Fe(t0 ) exists, then the gradient ∇Fe(t0 ) of Fe at t0 exists and we have uj = vj (j = 1, 2, · · · , m), where ∇Fe(t0 ) = (u1 , u2 , · · · , um ), ∇b Fe(t0 ) = (v1 , v2 , · · · , vm ). Proof. Let t0 = (t01 , · · · , t0j , · · · , t0m ) ∈ intM, h ∈ R and t = (t01 , · · · , t0j + h, · · · , t0m ) ∈ intM. According to Theorem 2.2 and Definition 3.1, for any r ∈ [0, 1], we have [

Fe(t) g Fe(t0 ) r ] h

=

1 e h [F (t)

=

1 h

g Fe(t0 )]r

Qn

i=1 [inf β≥r

min{Fi− (β, t) − Fi− (β, t0 ), Fi+ (β, t) − Fi+ (β, t0 )},

supβ≥r max{Fi− (β, t) − Fi− (β, t0 ), Fi+ (β, t) − Fi+ (β, t0 )}] =

Qn

i=1 [inf β≥r

min{

Fi− (β,t)−Fi− (β,t0 ) Fi+ (β,t)−Fi+ (β,t0 ) , }, h h

supβ≥r max{

Fi− (β,t)−Fi− (β,t0 ) Fi+ (β,t)−Fi+ (β,t0 ) , }]. h h

Because the b-gradient ∇b Fe(t0 ) of Fe be exist at t0 ∈ intM, for any r ∈ [0, 1], we have limh→0 [ = limh→0

Fe(t01 ,··· ,t0j +h,··· ,t0m ) g Fe(t01 ,··· ,t0j ,··· ,t0m ) r ] h

Qn

i=1 [inf β≥r

min{

Fi− (β,t01 ,··· ,t0j +h,··· ,t0m )−Fi− (β,t01 ,··· ,t0j ,··· ,t0m ) , h Fi+ (β,t01 ,··· ,t0j +h,··· ,t0m )−Fi+ (β,t01 ,··· ,t0j ,··· ,t0m ) }, h

supβ≥r max{

Fi− (β,t01 ,··· ,t0j +h,··· ,t0m )−Fi− (β,t01 ,··· ,t0j ,··· ,t0m ) , h Fi+ (β,t01 ,··· ,t0j +h,··· ,t0m )−Fi+ (β,t01 ,··· ,t0j ,··· ,t0m ) }] h

=

Qn

i=1 [inf β≥r

min{limh→0 limh→0

Fi− (r,t01 ,··· ,t0j +h,··· ,t0m )−Fi− (r,t01 ,··· ,t0j ,··· ,t0m ) , h Fi+ (r,t01 ,··· ,t0j +h,··· ,t0m )−Fi+ (r,t01 ,··· ,t0j ,··· ,t0m ) }, h

supβ≥r max{limh→0 limh→0 =

Qn

i=1 [inf β≥r

Fi− (r,t01 ,··· ,t0j +h,··· ,t0m )−Fi− (r,t01 ,··· ,t0j ,··· ,t0m ) , h

Fi+ (r,t01 ,··· ,t0j +h,··· ,t0m )−Fi+ (r,t01 ,··· ,t0j ,··· ,t0m ) }] h

+ − + min{u− ij (β), uij (β)}, supβ≥r max{uij (β), uij (β)}]

= [vj ]r . 190

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Therefore, there exist uj = vj ∈ L(E n ), such that Fe(t01 , · · · , t0j + h, · · · , t0m ) g Fe(t01 , · · · , t0j , · · · , t0m ) = uj (j = 1, 2, · · · , m), lim h→0 h which implies that the gradient ∇Fe(t0 ) of Fe at t0 exists and uj = vj (j = 1, 2, · · · , m). Definition 4.4. Let Fe : M → L(E n ), t0 = (t01 , t02 , · · · , t0m ) ∈ intM and t = (t1 , tQ 2 , · · · , tm ) ∈ intM. If + the gH-difference Fr (t) gH Fr (t0 ) exist for all r ∈ [0, 1], and there exist [uj ]r = ni=1 [u− ij (r), uij (r)] ⊆ Rn (j = 1, 2, · · · , m), such that P r 0 d(Fr (t) gH Fr (t0 ), m j=1 [uj ] (tj − tj )) lim =0 t→t0 d(t, t0 ) uniformly for r ∈ [0, 1], then we say that Fe is level-wise differentiable (l-differentiable for short) at t0 . Theorem 4.3. Let Fe : M → L(E n ) be a fuzzy mapping. If Fe is l-differentiable at t0 = (t01 , t02 , · · · , t0m ) ∈ intM, then there exist uj ∈ L(E n ), such that for any r ∈ [0, 1], r

[vj ] =

n Y

+ [ inf u− ij (β), sup uij (β)] (j = 1, 2, · · · , m). β≥r

i=1

β≥r

Proof. For any r ∈ [0, 1], We can show that the class of sets n Y + [ inf u− ij (β), sup uij (β)] (j = 1, 2, · · · , m) i=1

β≥r

β≥r

satisfies the conditions of Theorem 2.1. According to Lemma 3.1 and Definition 4.4, if there exists δ > 0, such that for any |h| < δ with t = (t01 , · · · , t0j + h, · · · , t0m ) ∈ intM, we have u− ij (r) = min{limh→0

Fi− (r,t01 ,··· ,t0j +h,··· ,t0m )−Fi− (r,t01 ,··· , t0j ,··· , t0m ) , h

limh→0 u+ ij (r) = max{limh→0 limh→0

Fi+ (r,t01 ,··· ,t0j +h,··· ,t0m )−Fi+ (r,t01 ,··· , t0j ,··· , t0m ) }, h Fi− (r,t01 ,··· ,t0j +h,··· ,t0m )−Fi− (r,t01 ,··· , t0j ,··· , t0m ) , h Fi+ (r,t01 ,··· ,t0j +h,··· ,t0m )−Fi+ (r,t01 ,··· , t0j ,··· , t0m ) }, h

for all i = 1, 2, · · · , n and j = 1, 2, · · · , m. Since Fi− (r, t) and Fi+ (r, t) are left continuous with respect r ∈ (0, 1] and right continuous at r = 0, Fi− (r, t01 , · · · , t0j + h, · · · , t0m ) − Fi− (r, t01 , · · · , t0j , · · · , t0m ) h and

Fi+ (r, t01 , · · · , t0j + h, · · · , t0m ) − Fi+ (r, t01 , · · · , t0j , · · · , t0m ) h are left continuous at r ∈ (0, 1] and right continuous at r = 0. Thus, for any i = 1, 2, · · · , n and j = 1, 2, · · · , m, inf β≥r min{

Fi− (r,t01 ,··· ,t0j +h,··· ,t0m )−Fi− (r,t01 ,··· ,t0j ,··· ,t0m ) , h Fi+ (r,t01 ,··· ,t0j +h,··· ,t0m )−Fi+ (r,t01 ,··· ,t0j ,··· ,t0m ) }, h

and supβ≥r max{

Fi− (r,t01 ,··· ,t0j +h,··· ,t0m )−Fi− (r,t01 ,··· ,t0j ,··· ,t0m ) , h Fi+ (r,t01 ,··· ,t0j +h,··· ,t0m )−Fi+ (r,t01 ,··· ,t0j ,··· ,t0m ) } h 191 Shexiang Hai et al 184-195

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Shexiang Hai and Zengtai Gong: The differential and subdifferential for fuzzy mappings based on ...

are left continuous at r ∈ (0, 1] and right continuous at r = 0. Therefore, for any r ∈ [0, 1], i = 1, 2, · · · , n and j = 1, 2, · · · , m, we have (1) inf β≥r min{limh→0

Fi− (β,t01 ,··· ,t0j +h,··· ,t0m )−Fi− (β,t01 ,··· , t0j ,··· , t0m ) , h Fi+ (β,t01 ,··· ,t0j +h,··· ,t0m )−Fi+ (β,t01 ,··· , t0j ,··· , t0m ) } h

limh→0

are non-decreasing and left continuous at r ∈ (0, 1] and right continuous at r = 0, (2) Fi− (β,t01 ,··· ,t0j +h,··· ,t0m )−Fi− (β,t01 ,··· , t0j ,··· , t0m ) , h

supβ≥r max{limh→0 limh→0

Fi+ (β,t01 ,··· ,t0j +h,··· ,t0m )−Fi+ (β,t01 ,··· , t0j ,··· , t0m ) } h

are non-increasing and left continuous at r ∈ (0, 1] and right continuous at r = 0, (3) inf β≥r min{limh→0

Fi− (β,t01 ,··· ,t0j +h,··· ,t0m )−Fi− (β,t01 ,··· , t0j ,··· , t0m ) , h Fi+ (β,t01 ,··· ,t0j +h,··· ,t0m )−Fi+ (β,t01 ,··· , t0j ,··· , t0m ) } h

limh→0

≤ supβ≥r max{limh→0

Fi− (β,t01 ,··· ,t0j +h,··· ,t0m )−Fi− (β,t01 ,··· , t0j ,··· , t0m ) , h

Fi+ (β,t01 ,··· ,t0j +h,··· ,t0m )−Fi+ (β,t01 ,··· , t0j ,··· , t0m ) }. h

limh→0

Consequently, there exist vj ∈ L(E n ) (j = 1, 2, · · · , m), such that for any r ∈ [0, 1], r

[vj ] =

n Y i=1

+ [ inf u− ij (β), sup uij (β)] (j = 1, 2, · · · , m). β≥r

β≥r

Definition 4.5. Let Fe : M → L(E n ) is l-differentiable at t0 , for any r ∈ [0, 1], we denote [vj ]r =

n Y i=1

+ [ inf u− ij (β), sup uij (β)] (j = 1, 2, · · · , m), β≥r

β≥r

then we say that (v1 , v2 , · · · , vm ) is the level-wise gradient (l-gradient for short) of Fe at t0 , denoted by ∇l Fe(t0 ), i.e., ∇l Fe(t0 ) = (v1 , v2 , · · · , vm ). Remark 4.3. Let Fe : M → L(E n ), t0 ∈ Q M. Then the l-gradient ∇l Fe(t0 ) exists at t0 if and only if + n r Fr (t) gH Fr (t0 ) exist and there are [uj ] = ni=1 [u− ij (r), uij (r)] ⊆ R (j = 1, 2, · · · , m), such that u− ij (r) = min{limh→0

Fi− (r,t01 ,··· ,t0j +h,··· ,t0m )−Fi− (r,t01 ,··· , t0j ,··· , t0m ) , h

limh→0 and u+ ij (r) = max{limh→0 limh→0

Fi+ (r,t01 ,··· ,t0j +h,··· ,t0m )−Fi+ (r,t01 ,··· , t0j ,··· , t0m ) } h Fi− (r,t01 ,··· ,t0j +h,··· ,t0m )−Fi− (r,t01 ,··· , t0j ,··· , t0m ) , h Fi+ (r,t01 ,··· ,t0j +h,··· ,t0m )−Fi+ (r,t01 ,··· , t0j ,··· , t0m ) } h

uniformly for r ∈ [0, 1], where h ∈ R with t = (t01 , · · · , t0j + h, · · · , t0m ) ∈ intM and [uj ]r =

n Y i=1

+ − + [ inf min{u− ij (β), uij (β)}, sup max{uij (β), uij (β)}] (j = 1, 2, · · · , m). β≥r

β≥r

192

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Theorem 4.4. Let the l-gradient ∇l Fe(t0 ) of fuzzy mapping Fe : M → L(E n ) be exist at t0 ∈ intM. If Fe(t) g Fe(t0 ) exists, then the gradient ∇Fe(t0 ) of Fe at t0 exists and we have uj = vj (j = 1, 2, · · · , m), where ∇Fe(t0 ) = (u1 , u2 , · · · , um ), ∇l Fe(t0 ) = (v1 , v2 , · · · , vm ). Proof. Let t0 = (t01 , · · · , t0j , · · · , t0m ) ∈ intM, h ∈ R and t = (t01 , · · · , t0j + h, · · · , t0m ) ∈ intM. We denote Q + n [uj ]r = ni=1 [inf β≥r u− ij (β), supβ≥r uij (β)], then uj ∈ L(E ) and u− ij (r) = min{limh→0

Fi− (r,t01 ,··· ,t0j +h,··· ,t0m )−Fi− (r,t01 ,··· , t0j ,··· , t0m ) , h

limh→0 u+ ij (r) = max{limh→0 limh→0

Fi+ (r,t01 ,··· ,t0j +h,··· ,t0m )−Fi+ (r,t01 ,··· , t0j ,··· , t0m ) }, h Fi− (r,t01 ,··· ,t0j +h,··· ,t0m )−Fi− (r,t01 ,··· , t0j ,··· , t0m ) , h Fi+ (r,t01 ,··· ,t0j +h,··· ,t0m )−Fi+ (r,t01 ,··· , t0j ,··· , t0m ) }, h

for all i = 1, 2, · · · , n and j = 1, 2, · · · , m. It follows from Theorem 2.2 and Lemma 3.1 that D(

Fe(t01 ,··· ,t0j +h,··· ,t0m ) g Fe(t01 ,··· , t0j ,··· , t0m ) , uj ) h

= supr∈[0,1] d( h1

Qn

i=1 [inf β≥r

min{Fi− (β, t01 , · · · , t0j + h, · · · , t0m ) − Fi− (β, t01 , · · · , t0j , · · · , t0m ), Fi+ (β, t01 , · · · , t0j + h, · · · , t0m ) − Fi+ (β, t01 , · · · , t0j , · · · , t0m )},

supβ≥r max{Fi− (β, t01 , · · · , t0j + h, · · · , t0m ) − Fi− (β, t01 , · · · , t0j , · · · , t0m ), Fi+ (β, t01 , · · · , t0j + h, · · · , t0m ) − Fi+ (β, t01 , · · · , t0j , · · · , t0m )}], Qn

i=1 [inf β≥r

≤ supr∈[0,1] d( h1

+ u− ij (β), supβ≥r uij (β)])

− 0 i=1 [min{Fi (r, t1 , · · ·

Qn

, t0j + h, · · · , t0m ) − Fi− (r, t01 , · · · , t0j , · · · , t0m ),

Fi+ (r, t01 , · · · , t0j + h, · · · , t0m ) − Fi+ (r, t01 , · · · , t0j , · · · , t0m )}, max{Fi− (r, t01 , · · · , t0j + h, · · · , t0m ) − Fi− (r, t01 , · · · , t0j , · · · , t0m ), Fi+ (r, t01 , · · · , t0j + h, · · · , t0m ) − Fi+ (r, t01 , · · · , t0j , · · · , t0m )}], Qn

i=1 [inf β≥r

= supr∈[0,1] d(

+ u− ij (β), supβ≥r uij (β)])

Fr (t01 ,··· ,t0j +h,··· ,t0m ) gH Fr (t01 ,··· , t0j ,··· , t0m ) , [uj ]r ). h

Because

Fr (t01 , · · · , t0j + h, · · · , t0m ) gH Fr (t01 , · · · , t0j , · · · , t0m ) = [uj ]r h→0 h uniformly for r ∈ [0, 1], for any ε > 0, there exists δ > 0, when |h| < δ, we have lim

D(

Fe(t01 , · · · , t0j + h, · · · , t0m ) g Fe(t01 , · · · , t0j , · · · , t0m ) Fr (t0 + h) gH Fr (t0 ) , uj ) ≤ sup d( , [uj ]r ) < ε. h h r∈[0,1]

Therefore, the gradient ∇Fe(t0 ) of Fe at t0 exists and ∇Fe(t0 ) = (u1 , u2 , · · · , um ) = ∇l Fe(t0 ). 5. The subdifferential for fuzzy n-cell mappings In recent years, nonsmooth analysis has increasingly come to play a role in functional analysis, optimization, optimal design, differential equations and control theory. The subdifferential is an important tool, used widely in nonsmooth analysis and optimization, thus we will discuss subdifferential concept for fuzzy n-cell mappings based on the ordering c . Definition 5.1. [17] Let Fe : M → L(E n ) be a fuzzy n-cell mapping. Fe is said to be convex (c.) on M if Fe(λt + (1 − λ)t0 ) c λFe(t) + (1 − λ)Fe(t0 ) 193

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for any t, t ∈ M and λ ∈ [0, 1]. The convex fuzzy n-cell mappings in the following arguments are assumed to be comparable. Definition 5.2. Let Fe : M → L(E n ) be a convex fuzzy n-cell mapping on M, t0 = (t01 , t02 , · · · , t0m ) ∈ intM, t = (t01 , · · · , t0j + h, · · · , t0m ) ∈ intM. if there exist uj ∈ L(E n ) (j = 1, 2, · · · , m), such that Fe(t01 , · · · , t0j + h, · · · , t0m ) g Fe(t01 , · · · , t0j , · · · , t0m ) C huj , then we call (u1 , u2 , · · · , um ) a subgradient of Fe at t0 , and say the set of all subgradients of Fe at t0 to be subdifferential of Fe at t0 , denoted by ∂ Fe(t0 ), i.e., ∂ Fe(t0 ) = {(u1 , u2 , · · · , um ) : Fe(t01 , · · · , t0j + h, · · · , t0m ) g Fe(t01 , · · · , t0j , · · · , t0m ) C huj , uj ∈ L(E n )}. According to Theorem 2.3, it is easy to verify the following conclusion. Theorem 5.1. Let Fe : M → L(E n ) be a convex fuzzy n-cell mapping on M. Then, we have e e ∂(αFe(t) + β G(t)) = α∂ Fe(t) + β∂ G(t), for any t ∈ intM and α, β ≥ 0. Theorem 5.2. Let Fe : M → L(E n ) be a convex fuzzy n-cell mapping on M. Then the subdifferential ∂ Fe(t) is a convex set in L(E n ). Proof. For an empty subdifferential, the assertion is trivial. Take two arbitrary subgradients (u1 , u2 , · · · , um ), (v1 , v2 , · · · , vm ) ∈ ∂ Fe(t1 , · · · , tj , · · · , tm ). When (t1 , · · · , tj + h, · · · , tm ) ∈ intM, we have λ(Fe(t1 , · · · , tj + h, · · · , tm ) g Fe(t1 , · · · , tj , · · · , tm )) C λhuj , (1 − λ)(Fe(t1 , · · · , tj + h, · · · , tm ) g Fe(t1 , · · · , tj , · · · , tm )) C (1 − λ)hvj , for any λ ∈ [0, 1]. It follows from Theorem 2.3 that Fe(t1 , · · · , tj + h, · · · , tm ) g Fe(t1 , · · · , tj , · · · , tm ) C h(λuj + (1 − λ)vj ) for any j = 1, 2, · · · , m. Therefore, λ(u1 , u2 , · · · , um ) + (1 − λ)(v1 , v2 , · · · , vm ) ∈ ∂ Fe(t), which implies that the subdifferential ∂ Fe(t) is a convex set in E n . Next, We study the problems of minimizing and maximizing a convex fuzzy n-cell mapping and discuss the necessary and sufficient conditions for optimality. Let Fe : M → L(E n ) be a fuzzy n-cell mapping. We consider an unconstrained fuzzy minimization problem (FMP): Minimize Fe(t), Subject to t ∈ intM. A point t0 ∈ intM is called a feasible solution to the problem, if for no t ∈ intM such that Fe(t) c Fe(t0 ), then t0 is called an optimal solution, or a global minimum point. With the aid of definitions for subdifferential and optimal solution we can immediately present a necessary and sufficient optimality condition. This theorem is formulated without proof because it is an obvious consequence of the definition of the subdifferential. Theorem 5.3. Let Fe : M → L(E n ) be a convex fuzzy n-cell mapping on M. Then t0 is a global minimum point if and only if (e 0, e 0, · · · , e 0) ∈ ∂ Fe(t0 ).

194

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Shexiang Hai and Zengtai Gong: The differential and subdifferential for fuzzy mappings based on ...

6. Conclusion In this paper, we introduce the concept of generalized difference of n-cell fuzzy-numbers and an ordering relation on the fuzzy n-cell number space is considered. Using the generalized difference of n-cell fuzzy-numbers the generalized differential and gradient concepts for fuzzy n-cell mappings are discussed. Furthermore, we have used the ordering relation c to obtain the subdifferential for fuzzy n-cell mappings based on the generalized difference of n-cell fuzzy-numbers. Future research includes studying optimality conditions for fuzzy constrained minimization problem. One alternative is to define the concept of invex function using g-differentiability and the ordering relation c for fuzzy n-cell mappings.

References [1] L.A. Zadeh, Fuzzy sets, Inform Control. 8 (1965) 338-353. [2] L.A. Zadeh, The concept of a linguistic variable and its application to approximate reasoning-I, Inform. Sci. 8 (1975) 199-249. [3] L.A. Zadeh, The concept of a linguistic variable and its application to approximate reasoning-II, Inform. Sci. 8 (1975) 301-357. [4] L.A. Zadeh, The concept of a linguistic variable and its application to approximate reasoning-III, Inform. Sci. 9 (1975) 43-80. [5] S.L. Chang, L.A. Zadeh, On fuzzy mapping and control, IEEE T Syst Man Cy-S. 2 (1972) 30-34. [6] M.L. Puri, D.A. Ralescu, Differentials of fuzzy functions, J. Math. Anal. Appl. 91 (1983) 552-558. [7] O. Kaleva, Fuzzy differential equations, Fuzzy Sets Syst. 24 (1987) 301-317. [8] G.X. Wang, C.X. Wu, Directional derivatives and subdifferential of convex fuzzy mappings and application in convex fuzzy programming, Fuzzy Sets Syst. 138 (2003) 559-591. [9] B. Bede, L. Stefanini, Generalized differentiability of fuzzy-valued functions, Fuzzy Sets Syst. 230 (2013) 119-141. [10] P. Diamond, P. Kloeden, Characterization of compact subsets of fuzzy sets, Fuzzy Sets Syst. 29 (1989) 341-348. [10] [11] M. Ma, On embedding problems of fuzzy number spaces: Part 5, Fuzzy Sets Syst. 55 (1993) 313-318. [12] C.V. Negoita, D.A. Ralescu, Application of fuzzy sets to systems analysis, Wiley, NewYork, 1975. [13] C.X. Wu, M. Ma, J.X. Fang, Structure theory of fuzzy analysis, Guizhou Scientific Publication, 1994 (in Chinese). [14] G.X. Wang,, C.X. Wu, Fuzzy n-cell numbers and the differential of fuzzy n-cell mappings, Fuzzy Sets Syst. 130 (2002) 367-381. [15] R. Goetschel, W. Voxman, Elementary fuzzy calculus, Fuzzy Sets Syst. 18 (1986) 31-43. [16] L. Stefanini, A generalization of Hukuhara difference and division for interval and fuzzy arithmetic, Fuzzy Sets Syst. 161(2010)1564-1584. [17] Z.T. Gong, S.X. Hai, The convexity of n-dimensional fuzzy-number-valued functions and its applications [J]. Fuzzy Sets Syst. 295 (2016) 19-36.

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TABLE OF CONTENTS, JOURNAL OF COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 24, NO. 1, 2018

A novel approach for solving fully fuzzy linear programming problem with LR at fuzzy numbers, Zeng-Tai Gong and Wen-Cui Zhao,……………………………………………11 Generalized Bateman's G-function and its bounds, Mansour Mahmoud and Hanan Almuashi,23 Interval-valued fuzzy quasi-metric spaces, Hanchuan Lu and Shenggang Li,…………………41 Bi-univalent functions of complex order based on quasi-subordinate conditions involving wright hypergeometric functions, N.E.Cho, G. Murugusundaramoorthy, and K.Vijaya,………… 58 On uni-soft mighty filters of BE-algebras, Jeong Soon Han and Sun Shin Ahn,…………….71 Inclusion relationships for some subclasses of analytic functions associated with generalized Bessel functions, K.A. Selvakumaran, H. A. Al-Kharsani, D. Baleanu, S.D. Purohit, and K.S. Nisar,……………………………………………………………………………………………81 Fixed point properties of Suzuki generalized nonexpansive set-valued mappings in complete CAT(0) spaces, Jing Zhou and Yunan Cui,……………………………………………………91 Weighted composition followed and proceeded by differentiation operators from Zygmund spaces to Bers-type spaces, Jianren Long and Congli Yang,…………………………………105 Global stability in n-dimensional stochastic difference equations for predator-prey models, Sang-Mok Chooa and Young-Hee Kim,………………………………………………………116 Entire solutions of certain type of nonlinear differential equations and differential-difference equations, Min Feng Chen and Zong Sheng Gao,……………………………………………..137 Approximate generalized quadratic mappings in (𝛽,p)-Banach spaces, Hark-Mahn Kim and Hong-Mei Liang,………………………………………………………………………………148 Fixed point theorems for generalized hybrid mappings in fuzzy Hilbert spaces, Afshan Batool, Tayyab Kamran, Choonkil Park, and Jung Rye Lee,………………………………………161 Some properties on Dirichlet-Hadamard product of Dirichlet series, Yong-Qin Cui and HongYan Xu,………………………………………………………………………………………173 The differential and subdifferential for fuzzy mappings based on the generalized difference of ncell fuzzy-numbers, Shexiang Hai and Zengtai Gong,………………………………………184

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J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 24, NO.2, 2018, COPYRIGHT 2018 EUDOXUS PRESS, LLC

THE EXTENSION OF A MODIFIED INTEGRAL OPERATOR TO A CLASS OF GENERALIZED FUNCTIONS S. K. Q. Al-Omari1 and Dumitru Baleanu2 Abstract. In this paper, we investigate a class of modi…ed G-transforms having G-functions as kernels on a generalized space of sequences. We derive certain spaces of generalized functions named as Boehmians to legitimate the existence of the described integral. The modi…ed G-transform is partially sharing the classical transform with some general properties. An inversion formula is also discussed on the generalized sense.

1. Introduction H-functions being related to most of known special functions are de…ned by integrals of the Mellin-Barnes type with integrands involving products of Euler gamma functions. Being an intemperate generalization of the generalized hypergeometric functions p Fq ; H-functions are utilized for applications in a large variety of problems connected with statistical distributions, versatile integrals, reaction, di¤usion, reaction di¤usion, engineering, communications, fractional di¤erential and integral equations and many areas of theoretical physics and statistical distribution theory as well. Through a special case of H-integral transforms, the G-integral transform enfolds various integrals related to Laplace, Hankel, Hilbert and Riemann-Liouville fractional integral transforms and, that integrals of Gauss hypergeometric function kernel type. However, despite a variety of integral transforms may not be reduced to G-transform integral type they are indeed given in the form of H-transform integral type. With the interest to study integral, dual and tripple equations, integral transforms of special kernel functions were motivated to include many mathematical problems and engineering applications. Integral transforms having kernels of H-function type were frequently presented as [4] " # Z 1 (ai ; i )1;p m;n (H') ( ) = Hp;q ' ( ) d ; > 0; (1) bj ; j 1;q 0 1991 Mathematics Subject Classi…cation. Primary 54C40, 14E20; Secondary 46E25, 20C20. Key words and phrases. Fox’s H-function; modi…ed G-transform; generalized function; Boehmians. 1

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S. K. Q. Al-Omari et al 209-218

J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 24, NO.2, 2018, COPYRIGHT 2018 EUDOXUS PRESS, LLC

2

S. K. Q. AL-OM ARI 1 AND DUM ITRU BALEANU 2

m;n where Hp;q are functions given in terms of the Mellin-Barnes type contour integral [6] Z 1 m;n Hp;q (w) = X ( )w d ; (2) 2 i L where Ym Yn bj ; j (1 aj ; j ) 1 1 : (3) X ( ) = Yq Yp 1 bj ; j (aj ; j ) m+1

n+1

The particular case of H-transforms where gives the G-transform integral (G') ( ) =

Z

1

Gm;n p;q

= ::: =

(ai )1;p (bj )1;q

Gm;n p;q

0

and that the amendment

1

p

= 1 and

1

= ::: =

'( )d ;

q

= 1;

(4)

(ai )1;p (bj )1;q

(5)

of the H-function is the so-called G-function. For a somehow much more detailed account of G and H-functions we refer to [1; 8]. The numbers a ; ; a1 ; a2 ; and when they appear are given as follows [4; (6:1:5) (6:1:11)] 9 a = 2m + 2n p q; > > > > = q p; > > > > a1 = (m + n) p; > > > > a2 =((m + n) q; > > > min Re (bj ) ; m > 0 = 1 j m = : (6) ;m = 0 > > ( 1 > > > 1 max Re (ai ) ; n > 0 > > 1 i n > = > > > 1 ;n = 0 > > > Pq Pp p q > ; = j=1 bj a + i i=1 2 By lv;r ; v 2 R; 1 r < 1; we denote the summable space of those Lebesgue measurable complex valued functions such that 9 1 r1 01 Z > > > r d A > k'kv;r = @ j v ' ( )j

= : (7) 0 > > and > > ; k'kv;1 = ess sup ( v j' ( )j) ; v 2 R > >0

The modi…ed G-transform we consider in this note is given by the integral equation [4; (6:2:4)] Z 1 (ai )1;p d G1 ; ' ( ) = Gm;n '( ) : (8) p;q (b ) j 1;q 0

It is associated with the the radical integral transform (4) by the equation G1 ; ' ( ) = M (G (RM ')) ( ) ;

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S. K. Q. Al-Omari et al 209-218

J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 24, NO.2, 2018, COPYRIGHT 2018 EUDOXUS PRESS, LLC

THE EXTENSION OF A M ODIFIED INTEGRAL OPERATOR TO ....

3

where R and M are operators de…ned , respectively, by [4; (3:3:13) and (3:3:11)] (R') ( ) =

1

'

1

and (M ') ( ) =

'( );

2 C:

Parseval Formula 1 The Parseval’s formula for the modi…ed G-transform is derived as Z 1 Z 1 1 G2 ; ' ( ) g ( ) d ; '( ) G ; g ( )d = 0

0

where G2 ; ' is the modi…ed G-transform Z 1 Gm;n G2 ; ' ( ) = p;q 0

(ai )1;p (bj )1;q

'( )

d

:

The following remark is of great importance to our investigation .

Remark 2 ([4; Theorem 6.50 (i)]) Let and be real numbers and that numbers a ; ; , be de…ned as in (6). Suppose the following are satis…ed : (i) < v < ; (ii) Either of (a) a > 0 or (b) a = 0 and [v ] + Re 5 0 holds. Then the transform G1 ; ' is a one-one mapping from lv;2 into lv ;2 : For a somehow much more detailed account of several signi…cant results on the modi…ed G-transforms, we refer the reader to [4]. Boehmians are motivations of regular operators with algebraic character of Mikusinski operators and do not have restriction on the support. With di¤erent function spaces various spaces of Boehmians can be obtained. Distributions, ultradistributions, regular operators are indeed contained in some well established spaces of Boehmians. In a Boehmian context, various generalizations of various integral transforms were given once the topic was started. A complete account of the theory of Boehmian spaces was given in [2; 3; 5; 7] ; [10]-[17]. However, the existed results in this theory are classical and none were discussed in the space of Boehmians. In this article, we develope the classical theory of the modi…ed G1 ; transform to the theory of Boehmians. In the following section we discuss the construction of the spaces of Boehmians. In Section 3, we give the representative of the modi…ed G1 ; transform and its inverse in the de…ned spaces of Boehmians. We further discuss certain results related to the proposed integrals. 2. Construction of Spaces of Boehmians Let us …rst agree for the products we demand for our investigation. The …rst product we should use here is the so-called Mellin type convolution product of …rst kind de…ned as [9] Z 1 (' g g) ( ) = y 1 ' y 1 g (y) dy; (9) 0

provided the integral exists. A number of the properties of this integral that we …nd it worthwhile to be described here : (i) g1 g g2 = g2 g g1 ; (ii) (g1 g g2 ) g g3 = g1 g (g2 g g3 ) ; (iii) ( g1 ) g g2 = (g1 g g2 ) ; (iv) g1 g (g2 + g3 ) = g1 g g2 + g1 g g3 :

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4

It is of great importance to introduce the following convolution product that will be worthy of attention Z 1 (' g) ( ) = ' y 1 y 1 g (y) dy; (10) 0

where

and

are real numbers.

Properties of this integral are to be provided in the text of the paper. The relation between the convolution products are given by the following theorem. Theorem 3 Let ' and g be integrable functions on (0; 1). Then, we have G1 ; (' g g) ( ) = G1 ; ' g ( ) : Proof Under the hypothesis of the theorem and by using (8) for (9) we get Z 1 Z 1 (ai )1;p y 1 ' y 1 g (y) G1 ; (' g g) ( ) = Gm;n p;q (bj )1;q 0 0 dy =

Z

1

d

g (y) y

Z

1

0

1

(ai )1;p (bj )1;q

Gm;n p;q

0

d

'

y

1

(11)

dy:

On setting variables and using Fubini’s theorem, (11) produce Z 1 Z 1 yw (ai )1;p 1 1 G ; (' g g) ( ) = g (y) y Gm;n (yw) ' (w) p;q (bj )1;q 0 0 ydw dy yw Z 1 Z 1 (ai )1;p w = g (y) y 1 y w ' (w) Gm;n p;q 1 (bj )1;q y 0 0 dw dy w Z 1 Z 1 (ai )1;p w 1 1 = g (y) y y Gm;n p;q 1 (bj )1;q y 0 0 dw w ' (w) dy w Z 1

=

G1 ; '

y

1

y

1

g (y) dy

0

The proof of the theorem is completely …nished. Lemma 4 Let '; g and get

be integrable functions on the open interval (0; 1). We ' (g g ) = (' g)

:

Proof On account of (10) we write

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' (g g ) ( )

=

Z

1

'

y

1

0

=

Z

1

( )

Z

1

Z

1

y

1

g y

1

1

g y

( )d

dy

0

1

'

1

y

1

( w)

y

1

dy d : (12)

0

0

By change of variables, (12) yields Z 1 Z 1 ' (g g ) ( ) = ( ) ' ( w) 0 0 Z 1 Z 1 1 ' ( ) = Z

1

1

g (w) dwd

1

w

0

0

=

1

5

w

1

1

g (w) dwd 1

( )

1

(' g)

d :

0

The proof is completely …nished. Let D denote the standard notation of the space of test functions of compact supports in (0; 1) : Then we have the following results: Theorem 5 Let ' 2 lv

;2

and g 2 D be given. Then, we have ' g 2 lv

;2 :

Proof By appealing to (7) and the integral equation (10) ; we get Z 1 Z 1 1 2 v ' y 1 g (y) dy = k' gkv ;2

2

d

:

0

0

Applying Jensen’s inquality yields Z 1 2 v k' gkv ;2

Z

0

Using the Fubini’s theorem implies Z 1 2 g (y) y k' gkv ;2

1

'

y

1

2

y

1

g (y) dy

d

:

0

Z

1

0

1

v

'

y

1

2

d

dy:

0

Now, let [a; b] ; 0 < a < b; be an interval containing the support of g: Then, the hypothesis of the theorem ' 2 lv ;2 ; reveals Z b 2 2 k' gkv M k'k y 1 g (y) dy; v ;2 ;2 Rb

a

1

where M = a y g (y) dy: Thus, the above equation further reveals k' gkv

;2

< 1:

The proof is completely …nished. Theorem 6 There hold the following identities. (i) Let f'n g ; ' 2 lv ;2 be such that 'n ! ' as n ! 1: We have 'n g ! ' g as n ! 1 for every g 2 D:

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6

(ii) Let '1 ; '2 2 lv ;2 and g 2 D: Then we have the following identities satisfy ('1 + '2 ) g = '1 g + '2 g and ('1 g) = ( '1 ) g; for arbitrary complex number : The proof of this theorem can be followed by using simple integral calculus. Hence, we avoid adding more details. Definition Z 1 7 Let f n g 2 D such that (i) n ( ) d = 1 (n 2 N) : Z0 1 (ii) j n ( )j d < A (A 2 R being positive) : 0

(iii) supp n = f : n ( ) 6= 0g ! 0 as n ! 1: The set of all sequences f n g are denoted by . Every f n g in delta sequence which corresponds to the delta distribution. Theorem 8 Let f

ng

2

'

and ' 2 lv n

;2 :

! ' in lv

is said to be a

Then, we have ;2

as n ! 1:

(14)

Proof By the …rst part of De…nition 7 and Jensen’s inequality we have k('

n) (

)

'(

2 )kv

;2

Z

1

2

v

0

j

Z

1

1

y

'

y

1

'( )

2

0

n

(y)j dy

d

:

(15)

Therefore, by making use of Fubini’s theorem, (15) gives Z bn 2 k(' n ) ( ) ' ( )kv j n (y)j ;2 a Z n1 v

y

( )

'( )

2

d

dy; (16)

0

where supp n (y) [an ; bn ] ; 0 < an < bn ; 8n 2 N: Taking into account the fact that ' ( ) ; y ( ) = ' follows from (16) that Z 2 k(' n ) ( ) ' ( )kv M ;2

y

1

y

1

2 lv

;2 ;

it

bn

an

j

n

(y)j dy:

for some positive constant M : Therefore, 2 k(' n ) ( ) ' ( )kv M M1 (an ; bn ) ; ;2 M1 > 0: The last inequality follows from the identity (iii) of De…nition 7: The proof of the theorem is completely …nished. The space B (lv of Boehmians.

;2 ; (D; g) ;

; ) is therefore generated and regarded as a space

Construction of the space B (lv;2 ; (D; g) ; g; ) can be obtained by that technique similar to that of B (lv ;2 ; (D; g) ; ; ) and the properties of g we have already cited above.

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7

'n gn Sum of two Boehmians and in B (lv;2 ; (D; g) ; g; ) can be expressed "n n as 'n gn 'n g n + gn g n + = : "n n n g "n 'n 'n = = Multiplication in B (lv;2 ; (D; g) ; g; ) by 2 C is de…ned as n

'n

n

:

n

The extensions of g and D to B (lv;2 ; (D; g) ; g; ) are introduced as 'n n

Let

'n

g

gn 'n g gn = "n n g "n

and D

'n

=

n

D 'n n

;

2 R:

belong to B (lv;2 ; (D; g) ; g; ) and ! be in lv;2 . The operation g can be

n

extended to B (lv;2 ; (D; g) ; g; )

'n n

lv;2 by 'n g !

g! =

:

n

Let the sequence f n g be in B (lv;2 ; (D; g) ; g; ) : Then n ! in B (lv;2 ; (D; g) ; g; ) ; if there can be found a delta sequence f n g such that for ( n g k ) and ( g k ) 2 lv;2 , n; k 2 N; we have lim

n!1

n

g

k

!

g

k

in lv;2 for every k 2 N:

This can be expressed in B (lv;2 ; (D; g) ; g; ) as :

! ( as n ! 1) if and only if there are 'n;k ; 'k 2 lv;2 and f k g 2 , 'n;k 'k = ; = and to every k 2 N we have limn!1 'n;k = 'k in lv;2 : n

n

k

k

( as n ! 1) if there can be found a f n g 2 such that ( n )g n 2 n ! lv;2 (8n 2 N) and limn!1 ( n ) g n = 0 in lv;2 : On the other hand, addition of two Boehmians in B (lv ;2 ; (D; g) ; ; ) is de…ned as gn 'n n + gn n 'n + = : "n n n g "n Multiplication and convergence in B (lv ;2 ; (D; g) ; ; ) can be de…ned similarly as in B (lv ; (D; g) ; ; ). ;2 3. G1 ; of Boehmians In view of Remark 2 and Theorem 3, we extend the transform G1 ; to the space B (lv;2 ; (D; g) ; g; ) as " # z }| { ' 1 G ' n ; n G1 ; = (17) n

in B (lv

;2 ; (D; g) ;

n

; ):

z }| { We recite some properties of the transform G1 ; in the course of the following theorems.

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8

z }| { Theorem 9 (i) The operator G1 ; is well - de…ned and linear . z }| { (ii) The operator G1 ; is an isomorphism from B (lv;2 ; (D; g) ; g; ) onto the space B (lv ;2 ; (D; g) ; ; ) : z }| { (iii) The operator G1 ; is continuous with respect to and - convergence. z }| { (iv) The operator G1 ; : B (lv;2 ; (D; g) ; g; ) ! B (lv ;2 ; (D; g) ; ; ) transform is compatible with G1 ; : lv;2 ! lv ;2 : Proof We prove Part (iv) since similar proofs for Part(i) - Part(iii) are available in many cited papers of the same author and of Roopkumar in [20]. g n be its representative To prove the last part of the theorem, let 2 lv;2 and n

in B (lv;2 ; (D; g) ; g; ) where f n g 2 (8n 2 N) : Clearly, for all n 2 N; f n g is independent from the representative. Hence, from (17) and Theorem 3, we get z }| { G1 ;

Thus

"

g

z }| { = G1 ;

n

n

G1 ;

n n

#

g

n

=

n

"

G1 ; ( g

n)

n

is the representative of G1 ;

#

=

"

in the space lv

G1 ;

n n

#

:

;2 :

The proof is therefore …nished. z }| { In view of Theorem 9, we introduce the inverse transform of G1 ; as follows. Definition 10 Let integral operator of

"

"

G1 ; 'n n

G

1

'n

;

n

#

#

z }| { 2 B (lv;2 ; (D; g) ; g; ) : We de…ne the inverse G1 ; as

! z }| { 1 G ; for each f

ng

2

Theorem 11 Let

:"

G 1 ; 'n n

z }| { G1 ;

#

!

"

1

G1 ; 'n n

"

G 1 ; 'n n

and z }| { G1 ;

'n n

g'

=

=

"

#

'

!

=

G 1 ; 'n

216

n

'n

;

n

;2 ; (D; g) ;

2 B (lv 1

#!

; ) and ' 2 D: We have 'n n

#

g'

':

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THE EXTENSION OF A M ODIFIED INTEGRAL OPERATOR TO ....

Proof Assume

"

! z }| { G1 ;

G 1 ; 'n n 1

"

#

;2 ; (D; g) ;

2 B (lv

G 1 ; 'n n

#

'

!

i.e.

=

=

"

; ). For every

! z }| { G1 ; G1 ;

1

1

"

G 1 ; 'n ' n

Using Theorem 3 and De…nition 10 we obtain ! " # ! z }| { 1 G 1 ; 'n 1 G ; ' =

z }| { Proof of the part G1 ;

'n n

g'

=

"

G 1 ; 'n n

#

#!

#

:

'n g '

n

=

2 D, we have

G 1 ; 'n ' n

9

n

'n n

g ':

' is almost similar. We prefer

to omit the details of the proof. This completely …nishes the proof of the theorem. Con‡ict of Interests : The authors declare that they have no competing interests. Authors’Contribution : The authors read and approved the …nal version of the paper. References [1] C. Fox, "The G- and H-functions as symmetrical Fourier kernel", Trans. Amer. Math. Soc., 98, 395-429, 1961. [2] D. Nemzer, "The Laplace transform on a class of Boehmians", Bull. Austral. Math. Soc. 46, 347-352, 1992. [3] P. Mikusinski, "Fourier transform for integrable Boehmians", Rocky Moun. J. Math., 17(3), 577-582, 1987. [4] A. Kilbas and A. M. Saigo, H-transform, theory and applications, CRC press LLC, Boca Raton, London , NY, Washinton, 2004. [5] J. J. Betancor, M. Linares, and J. R. Méndez, "The Hankel transform of integrable Boehmians", Appl. Anal. 58, 367-382, 1995. [6] B. L. J. Braaksma, "Asymptotic expansion and analytic continuation for the class of Barnes integral", Comp. Math., 15, 239-341, 1964. [7] S. K. Q. Al-Omari and D. Baleanu, "On generalized space of quaternions and its application to a class of Mellin transforms", J. Nonlinear Sci. Applic., To appear, 2006. [8] A. M. Mathai , R. Kishore and S. J. Haubold, The H-Function : Theory and Applications, Springer Science+Business Media, LLC 2010, 2009. [9] A. H. Zemanian, Generalized integral transformation, Dover Publications, Inc.,New York, 1987 . [10] S. K. Q. Al-Omari and J. F. Al-Omari, "Some extensions of a certain integral transform to a quotient space of generalized functions", Open Math., 13, 816-825, 2015. [11] S. K. Q. Al-Omari and D. Baleanu, "On the generalized Stieltjes transform of Fox’s kernel function and its properties in the space of generalized functions", J. Comput. Anal. Applicat., To appear, 2016. [12] D. Atanasiu and P. Mikusinski, "On the Fourier transform, Boehmians, and distributions", Colloq. Math. 108, 263-276, 2007 .

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10

S. K. Q. AL-OM ARI 1 AND DUM ITRU BALEANU 2

[13] S. K. Q. Al-Omari, "Some characteristics of S transforms in a class of rapidly decreasing Boehmians", J. Pseudo-Di¤er. Oper. Applic. 01/2014; 5(4):527-537. DOI:10.1007/s11868014-0102-8, 2014. [14] A. Zayed, "Fractional fourier transforms of generalized functions", Integ. Trans. Spec. Funct. 7, 299-312, 1998. [15] A. Zayed and P. Mikusinski, "On the extension of the Zak transform", Methods Appl. Anal. 2, 160-172, 1995. [16] E. R. Dill and P. Mikusinski, "Strong Boehmians", Proc. Amer. Math. Soc. 119, 885-888, 1993. [17] S. B. Gaikwad and M. S. Chaudhary, "Fractional Fourier transforms of ultra Boehmians", J. Indian Math. Soc., 73, 53-64, 2006. Current address : 1 Department of Applied Sciences, Faculty of Engineering Technology, AlBalqa Applied University, Amman 11134, Jordan., 2 Department of Mathematics and Computer Sciences, Cankaya University, Eskisehir Yolu 29.km, 06810 Ankara, Turkey and,, Institute of Space Sciences, Magurele-Bucharest, Romania., E-mail address : [email protected],[email protected], Correspondence should be addressed to the first author; [email protected]

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HARMONIC QUASICONFORMAL MAPPINGS OF THE UNIT DISK ONTO THE HORIZONTAL STRIP AND HALF PLANE JIAN-FENG ZHU Abstract. In this paper we consider two types of harmonic mappings w(z) ∈ SH (D, Ω1 ) and w(z) ∈ SH (D, R), where D is the unit disk and Ω1 , R are the domain defined by (2) and (3). Using the representation of harmonic mappings, we find the sufficient and necessary conditions to make w(z) be a harmonic quasiconformal mapping. Furthermore, we obtain some estimates of w(z).

1. Introduction A real-valued function u(x, y) on an open set D ⊆ C is harmonic if it is C 2 on D and satisfies Laplace’s equation: △u = uxx + uyy = 0. Assume that z = x + iy, w(z) = u(x, y) + iv(x, y). Then a complex-valued function w(z) is harmonic if and only if u(x, y) and v(x, y) are both harmonic. This has an equivalent form wz z¯ = 0. Let w(z) be a harmonic mapping defined in the unit disk D = {z : |z| < 1}. Then there exist two analytic functions h(z) and g(z) such that w(z) = h(z) + g(z). For z = reiϕ ∈ D, denote by P (r, t − ϕ) =

1 − r2 1 2π (1 − 2r cos(t − ϕ) + r 2 )

the Poisson kernel. Then every bounded harmonic mapping w(z) defined on D has the following representation (1)

w(z) = P [f ](z) =

Z2π 0

P (r, t − ϕ)f (eit ) dt,

where z = reiϕ ∈ D and f is a bounded integrable function defined on the unit circle S 1 = ∂D. For z ∈ D, let Λw (z) = max |wz (z) + e−2iα wz¯(z)| = |wz (z)| + |wz¯(z)| 0≤α≤2π

2000 Mathematics Subject Classification. Primary: 30C62; Secondary: 30C20, 30F15. Key words and phrases. Harmonic mappings; harmonic quasiconformal mappings; harmonic kernels; unbounded convex domain. File: 15-2018*JOCAAA.tex, printed: 17-1-2017, 21.49. The author of this work was supported by NNSF of China Grant Nos. 11501220, 11471128, NNSF of Fujian Province Grant No. 2016J01020, and the Promotion Program for Young and Middle-aged Teacher in Science and Technology Research of Huaqiao University (ZQN-PY402).

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and λw (z) = min |wz (z) + e−2iα wz¯(z)| = ||wz (z)| − |wz¯(z)||. 0≤α≤2π

According to Lewy’s Theorem we know that w(z) is locally univalent and sensepreserving in D if and only if its Jacobian satisfies Jw (z) = |wz (z)|2 − |wz¯(z)|2 > 0 for every z ∈ D.

Suppose that w(z) is a sense-preserving univalent harmonic mapping of D onto a domain Ω ⊆ C. Then w(z) is a harmonic K-quasiconformal mapping if and only if |wz (z)| + |wz¯(z)| ≤ K. z∈U |wz (z)| − |wz¯(z)|

K(w) := sup

It is interesting to consider such a question: under what conditions on f is w = P [f ](z) a harmonic quasiconformal mapping? Several authors have studied such a problem (see [5], [6], [8],[10], [11], [12], [13]). However, a univalent sense-preserving harmonic mapping defined on D doesn’t determined by its image domain. In this paper we consider two types of harmonic mappings of D onto a unbounded convex domain. One maps the unit disk onto the horizontal strip and the other maps the unit disk onto the half plane. Let SH denote the class of all complex valued, sense-preserving univalent harmonic 0 mappings w(z) in D, with the normalization w(0) = wz (0) − 1 = 0. Let SH be the subclass of SH with wz¯(0) = 0 and wz (0) > 0. For a domain Ω ⊆ C containing the origin, SH (D, Ω) will denote the class of all sense-preserving univalent harmonic mappings of D onto Ω normalized by w(0) = wz¯(0) = 0 and wz (0) > 0. Considering the following domains: the horizontal strip (2)

Ω1 = {w : −1 < Imw < 1},

and the right half plane

−1 }. 2 A conformal mapping ϕ from D onto Ω1 normalized by ϕ(0) = 0 < ϕ′ (0) has the form 2 1+z (4) ϕ(z) = ln . π 1−z For z ∈ D and |η| = 1, define the kernel Zz Zz 1 + η¯ζ 4 1 + η¯ζ ′ (5) k(z, η) = ϕ (ζ) dζ = dζ, 1 − η¯ζ π (1 − η¯ζ)(1 − ζ 2 )

(3)

R = {w : Rew >

0

0

and the family (6)

̥ = {w : w(z) = Re

Z

|η|=1

k(z, η)dµ(η) + iImϕ(z), µ ∈ P},

where P is the set of probability measures on the Borel sets of the unit circle S 1 . According to [4] we have the following theorem.

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Harmonic quasiconformal mappings of the unit disk onto the horizontal strip and half plane

3

Theorem A. SH (D, Ω1 ) = ̥. Here SH (D, Ω1 ) is the closure of SH (D, Ω1 ). Similarly, define the kernel  z z Re 1−z + iIm (1−z)  2  F (z, η) =   2η 1−z  Re z + iIm + ln 1−ηz 1−z (1−η)2

if η = 1 1+η z 1−η 1−z



if η 6= 1.

According to [1], we have the following theorem.

Theorem B. Each harmonic mapping w(z) ∈ SH (D, R) if and only if there is a probability measure µ on the unit circle such that Z (7) w(z) = F (z, η)dµ(η). |η|=1

In this paper, we find the sufficient and necessary conditions on the kernel k(z, η) and F (z, η) which make harmonic mappings w(z) of SH (D, Ω1 ) and SH (D, R) to be quasiconformal. 2. Necessary and sufficient conditions Theorem 1. Suppose that w ∈ SH (D, Ω1 ), which has the representation (6). Then w is a harmonic quasiconformal mapping if and only if it’s kernel satisfies the following conditions: R η+z (i) c := ess inf Re |η|=1 η−z dµ(η) > 0, z∈D R η+z (ii) M1 := ess sup |η|=1 η−z dµ(η) < ∞, z∈D

where c and M1 are positive constant.

Proof. Let w(z) = h(z) + g(z) be a sense-preserving univalent harmonic mapping of D onto Ω1 . According to Theorem A we have Z w(z) = Re k(z, η)dµ(η) + iImϕ(z), |η|=1

where k(z, η) and µ are defined by (5) and (6). This implies that Z  Z  1 1 w(z) = h(z)+g(z) = k(z, η)dµ(η) + ϕ(z) + k(z, η)dµ(η) − ϕ(z) . 2 2 |η|=1 |η|=1 Then (8)

ϕ′ (z) h (z) = 2

Z

ϕ′ (z) g (z) = 2

Z

′

|η|=1

and (9)

′

|η|=1

η+z dµ(η) + 1 η−z



 η+z dµ(η) − 1 . η−z

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J. -F. Zhu

It follows from (8) and (9) that ′ R dµ(η) − 1 g (z) |η|=1 η+z η−z = R . η+z h′ (z) dµ(η) + 1 |η|=1 η−z

R R dµ(η) and A2 = Im |η|=1 η+z dµ(η). Let A1 = Re |η|=1 η+z η−z η−z The proof of ’if’ part: Since A1 ≥ c > 0 applying condition (ii) we see that ′ 2   g (z) (A1 − 1)2 + A22 4A1 < 1. ess sup ′ = ess sup = ess sup 1 − h (z) (A1 + 1)2 + A22 (A1 + 1)2 + A22 z∈D z∈D z∈D

This shows that w(z) is a harmonic quasiconformal mapping. The proof of ’only if’ part: Assume that w(z) ∈ SH (D, Ω1 ) is a harmonic quasiconformal mapping. Then the following inequality R ′ η+z dµ(η) − 1 g (z) |η|=1 η−z ess sup ′ = ess sup R ≤k η+z h (z) dµ(η) + 1 z∈D z∈D |η|=1 η−z

R holds for some constant k < 1. Hence |η|=1

η+z dµ(η) < ∞ and (A1 − 1)2 + A22 ≤ η−z

k 2 (A1 + 1)2 + k 2 A22 . This implies that 2A1 (1 + k 2 ) ≥ 1 − k 2 . Thus A1 ≥ The proof is completed.

1−k 2 2(1+k 2 )

> 0. 

2

1−r Remark: For any z = reiθ ∈ D and η = eit ∈ ∂D, we have Re η+z = 1−2r cos(θ−t)+r 2 > η−z R η+z ′ it 0. If we additional assume that µ (e ) > 0, then Re |η|=1 η−z dµ(η) > 0. According to (8) and (9) we see that ′ g (z) ess sup ′ < 1, h (z) z∈D

which implies that w(z) is a harmonic quasiconformal mapping.

Theorem 2. Let w ∈ SH (D, R) be a sense-preserving harmonic mapping which has the representation (7). Then w is a quasiconformal mapping if and only if its kernel satisfies the following conditions:   R ηz (i) d := ess inf 1 + 2Re |η|=1 1−ηz dµ(η) > 0, z∈D R ηz (ii) M2 := ess sup |η|=1 1−ηz dµ(η) < ∞, z∈D

where d and M2 are positive constant.

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5

Proof. According to [4] we know that w(z) has the following representation: Z w(z) = F (z, η)dµ(η) |η|=1   Z z 1 2η 1−z 1+η z dµ(η) = + ln + 2 |η|=1 1 − z (1 − η)2 1 − ηz 1 − η 1 − z   Z 1 z 2η 1−z 1+η z + dµ(η) − ln − 2 |η|=1 1 − z (1 − η)2 1 − ηz 1 − η 1 − z = h(z) + g(z).

Then (10) and (11) This implies that ′ g (z) h′ (z) =

1 h (z) = (1 − z)2

Z

1 dµ(η) 1 − ηz

1 g (z) = (1 − z)2

Z

−ηz dµ(η). 1 − ηz

′

′

|η|=1

|η|=1

R R ηz ηz dµ(η) dµ(η) |η|=1 1−ηz |η|=1 1−ηz R = .   R ηz ηz dµ(η) dµ(η) |η|=1 1 + 1−ηz 1 + |η|=1 1−ηz

The proof of ’if’ part: Let

(12) Then

B=

Z

ηz dµ(η). |η|=1 1 − ηz

′ 2 2 g (z) |B|2 = |B| = . h′ (z) |1 + B|2 1 + |B|2 + 2ReB Applying condition(i) and (ii) we have ′ 2 g (z) |B|2 ess sup ′ ≤ < 1. h (z) |B|2 + d z∈D

The proof of ’only if’ part: Assume that w(z) is a harmonic quasiconformal mapping of D onto R. Then the following inequality ′ 2 g (z) 2 h′ (z) ≤ k

holds for some constant k < 1. This is equivalent to |B| < ∞ and |B|2 ≤ k 2 (|B|2 + 1 + 2ReB). Hence Z ηz dµ(η) < ∞, |η|=1 1 − ηz and (1 − k 2 )|B|2 1 + 2ReB ≥ , k2

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where B is defined by (12). This completes the proof.



Remark: According to (6) we know R that µ ∈ P is a probability measure on the 1 Borel set of S . This implies that |η|=1 dµ(η) = 1.

Theorem 3. Suppose that w(z) = h(z) + g(z) ∈ SH (D, Ω1 ) is a harmonic Kquasiconformal mapping. Then its Jacobian satisfies c Jw ≥ 2 , π where c is a positive constant depends on K. ′

ϕ′ (z) 2

R

η+z dµ(η) |η|=1 η−z



+ 1 and Proof. According to (8) and (9) we have h (z) = R  R ′ η+z g ′ (z) = ϕ 2(z) |η|=1 η−z dµ(η) − 1 , where ϕ(z) = π2 ln 1+z . Let A1 = Re |η|=1 η+z dµ(η) 1−z η−z R η+z and A2 = Im |η|=1 η−z dµ(η). Since w(z) is a harmonic quasiconformal mapping, by using condition (i) in Theorem 1 we see that Jw (z) = |h′ (z)|2 − |g ′(z)|2 = A1 |ϕ′ (z)|2 ≥ where c is a positive constant depends on K. This completes the proof.

c 16c ≥ 2, 4 + |z|) π

π 2 (1



Theorem 4. Suppose that w(z) = h(z) + g(z) ∈ SH (D, R) is a harmonic Kquasiconformal mapping. Then its Jacobian satisfies d , 16 where d is is a positive constant depends on K. Jw (z) ≥

Proof. According to (10) and (11) we have   Z Z 1 1 1 ηz ′ h (z) = dµ(η) = 1+ dµ(η) (1 − z)2 |η|=1 1 − ηz (1 − z)2 |η|=1 1 − ηz R −ηz 1 dµ(η). Using (12) and condition (i) of Theorem2 we and g ′ (z) = (1−z) 2 |η|=1 1−ηz obtain that 1 d d Jw = |h′ |2 − |g ′|2 = (1 + 2ReB) ≥ ≥ , 4 4 |1 − z| (1 + |z|) 16 where d is a positive constant depends on K. This completes the proof.

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7

3. co-Lipschitz condition of w(z) A complex-valued mapping w(z) in D is said to be a co-Lipschitz (Lipschitz) mapping if there exists a constant L > 0 such that the following inequality |w(z1 ) − w(z2 )| ≥

|z1 − z2 | L

(|w(z1 ) − w(z2 )| ≤ L|z1 − z2 |) holds for any z1 , z2 ∈ D. Suppose that w(z) is a harmonic quasiconformal mapping of D onto a bounded convex domain. Many mathematicians have discussed about the bi-Lipschitz property of w(z) (cf.[6],[9] and [11]). We point out that if w(z) is a harmonic quasiconformal mapping defined by (6) and (7), then it would be a co-Lipschitz mapping. Theorem 5. Given K ≥ 1. Suppose that w(z) = h(z) + g(z) ∈ SH (D, Ω1 ) is a harmonic K-quasiconformal mapping. Then the following inequalities 1+c 2(M1 + 1) ≤ |h′ (z)| ≤ 2π π(1 − |z|)2 hold for every z ∈ D, where c and M1 are positive constant depend on K. Furthermore, the inequality |w(z1 ) − w(z2 )| ≥ holds for any z1 , z2 ∈ D, where k =

(1 − k)(1 + c) |z1 − z2 |, 2π

K−1 . K+1

Proof. According to (8), we have ′ Z ϕ (z) η+z 2(1 + M1 ) ′ |h (z)| = dµ(η) + 1 ≤ 2 π(1 − |z|2 ) |η|=1 η − z and

′ Z ϕ (z) |h (z)| = 2 ′

|η|=1

η+z 2(1 + A1 ) (1 + c) dµ(η) + 1 ≥ ≥ , η−z π(1 + |z|)2 2π

where c and M1 are positive constant depend on K. Since w(z) = h(z) + g(z) is ′ a harmonic K-quasiconformal mapping, we see that ess sup hg ′(z) ≤ k < 1, where (z) z∈D

k=

K−1 . K+1

Then

Λw (z) = |h′ (z)| + |g ′(z)| ≤ |h′ (z)|(1 + k) ≤

2(1 + k)(1 + M1 ) π(1 − |z|2 )

and λw (z) = ||h′ (z)| − |g ′(z)|| ≥

2(1 − k)(1 + c) (1 − k)(1 + c) ≥ . π(1 + |z|)2 2π

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Take ζ1 , ζ2 ∈ Ω1 satisfying z1 = w −1 (ζ1 ), z2 = w −1 (ζ2 ). Let ϕ(t) = w −1 (ζ1 + t(ζ2 − ζ1 )), t ∈ [0, 1]. Then dtd w(ϕ(t)) = ζ2 − ζ1 . Since Ω1 is a convex domain, we see that Z1 Z1 d ′ ′ |ζ1 − ζ2 | = dt w(ϕ(t)) dt = wz (ϕ(t))ϕ (t) + wz¯(ϕ(t))ϕ (t) dt 0

≥ (13)

Z1 0

0

  ′ ′ |wz (ϕ(t))ϕ (t)| − |wz¯(ϕ(t))ϕ (t)| dt

≥ inf (|wz (u)| − |wz¯(u)|) u∈D

Z1 0

|ϕ′ (t)| dt

(1 − k)(1 + c) |z1 − z2 |. 2π This completes the proof. ≥



Theorem 6. Given K ≥ 1. Suppose that w(z) = h(z) + g(z) ∈ SH (D, R) is a harmonic K-quasiconformal mapping. Then the following inequalities (M2 + 1) d ≤ |h′ (z)| ≤ , 4 (1 − |z|)2 hold for any z ∈ D, where d and M2 are positive constant depend on K. Furthermore, the following inequality (1 − k)d |z1 − z2 |, |w(z1 ) − w(z2 )| ≥ 4 K−1 holds for any z1 , z2 ∈ D, where k = K+1 . Proof. According to (10) and (12), we have Z √ 1 ηz 1 ′ 1 + = 1 + 2ReB + B 2 ≥ d , |h (z)| = dµ(η) (1 − z)2 (1 − z)2 4 |η|=1 1 − ηz and

Z 1 1+ |h (z)| = (1 − z)2 ′

|η|=1

ηz 1 dµ(η) ≤ (M2 + 1), 1 − ηz (1 − |z|)2

where d and M2 are positive constant depend on K. Since w(z) = h(z) + g(z) is g′ (z) a harmonic K-quasiconformal mapping, we see that k = ess sup h′ (z) < 1, where z∈D

k=

K−1 . K+1

Hence

Λw (z) = |h′ (z)| + |g ′ (z)| ≤ |h′ (z)|(1 + k) ≤ and

(1 + k)(1 + M2 ) , (1 − |z|)2

λw (z) = ||h′ (z)| − |g ′(z)|| ≥ |h′ (z)|(1 − k) ≥

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For each z1 , z2 ∈ D, using (13) we have Z (1 − k)d |w(z1 ) − w(z2 )| = wz dz + wz¯d¯ z ≥ inf λw (z)|z1 − z2 | ≥ |z1 − z2 |. z∈D 4 [z1 ,z2 ]

This completes the proof.

9



References 1. M.J.Dorff, harmonic univalent mappings onto asymmetric vertical strips, Computational Mthods and Function Theory 1997, N.Papamichael, St.Ruscheweyh and E.B.Saff(Eds.), World Scientific (1997), 171–175. 2. P. Duren, Theory of H p Spaces, Academic Press, New York, 1970. 3. P. Duren, Harmonic mappings in the plane, Cambridge University Press, New York, 2004. 4. A.Grigoryan and M. Nowak, Estimate of integral means of harmonic mappings, Complex Variables 42(2000), 151–161. 5. D.Kalaj and M.Pavlovic, Boundary correspondence under quasiconformal harmonic diffeomorphisms of a half-plane, Ann. Acad. Sci. Fenn. 30(2005), 159–165. 6. D. Kalaj, Quasiconformal and harmonic mappings between Jordan domains, Math. Z. 260(2)(2008), 237–252. 7. D. Kalaj, S. Ponnusamy and M. Vuorinen, Radius of close-to-convexity of harmonic functions, Complex Var. Elliptic Equ., 59(4)(2014), 539-552. 8. D. Partyka and K. Sakan, On bi-Lipschitz type inequalitites for quasiconformal harmonic mappings, Ann. Acad. Sci. Fenn. Ser. A I Math. 32(2007), 579–594. 9. M. Pavlovic, Boundary correspondence under harmonic quasiconformal homeomorphisma of the unit disk, Ann. Acad.Sci.Fenn.Math. 27(2002), 365–372. 10. J.F. Zhu and X.M. Zeng, Estimate for Heinz inequality in the small dilatation of harmonic quasiconformal mappings, J. Compu. Analy. Appl. 13(2011), 1081-1087. 11. J.F. Zhu, Some estimates for harmonic mappings with given boundary function, J. Math. Anal. Appl. 411(2014), 631–638. 12. J.F. Zhu, Landau theorem for planar harmonic mappings, Complex Anal. Oper. Theory, 9(2015), 1819–1826. DOI: 10.1007/s11785-015-0449-8. 13. J.F. Zhu, Coefficients estimate for harmonic v- Bloch mappings and harmonic K- quasiconformal mappings, Bull. Malaysian Math. Sci. Soc. July, 2015, DOI: 10.1007/s40840-015-0175-4. Jian-Feng Zhu, School of Mathematical Sciences, Huaqiao University, Quanzhou 362021, China. E-mail address: [email protected]

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Langevin fractional differential inclusions with nonlocal Katugampola fractional integral boundary conditions Sotiris K. Ntouyas

a,b

, Jessada Tariboon

c

a

Department of Mathematics, University of Ioannina, 451 10 Ioannina, Greece e-mail: [email protected] b Nonlinear Analysis and Applied Mathematics (NAAM)-Research Group, Department of Mathematics, Faculty of Science, King Abdulaziz University, P.O. Box 80203, Jeddah 21589, Saudi Arabia c

Nonlinear Dynamic Analysis Research Center, Department of Mathematics, Faculty of Applied Science, King Mongkut’s University of Technology North Bangkok, Bangkok, 10800 Thailand e-mail: [email protected] Abstract In this paper, we study a boundary value problem consisting from a fractional differential inclusion of Riemann-Liouville Langevin type subject to Katugampola fractional integral conditions. Some new existence results for convex as well as non-convex multivalued maps are obtained by using standard fixed point theorems. Enlighten examples illustrating the obtained results are also presented.

Key words and phrases: Fractional differential inclusions; generalized fractional integral; Katugampola fractional integral; nonlocal boundary conditions; fixed point theorems AMS (MOS) Subject Classifications: 34A60; 26A33; 34A08

1

Introduction

In this manuscript, we investigate the sufficient conditions of existence of solutions for the following fractional Langevin inclusion subject to the generalized nonlocal fractional integral conditions of the form  Dp1 (Dp2 + λ)x(t) ∈ F (t, x(t)), 0 < t < T,      x(0) = 0, (1) ∫ ξi n n i  ∑ ∑  sρi −1 x(s) ρ1−q  ρ q i i i  ds := αi I x(ξi ), αi  x(η) = Γ(qi ) 0 (tρi − sρi )1−qi i=1 i=1 where Dpi denote the Riemann-Liouville fractional derivative of order pi , i = 1, 2, 0 < p1 , p2 ≤ 1, 1 < p1 + p2 ≤ 2, λ is a given constant, ρi I qi are the generalized fractional integral of orders qi > 0, ρi > 0, η, ξi arbitrary, with η, ξi ∈ (0, T ), αi ∈ R for all i = 1, 2, . . . , n and F : [0, T ] × R → P(R) is a multivalued map, P(R) is the family of all nonempty subsets of R. The search for the existence of solutions to nonlinear fractional boundary value problems has expanded greatly over the past years. For examples and recent development of the topic, see [1]-[11] and the references cited therein. In fractional calculus, the fractional derivatives are defined via fractional integrals. There are several known forms of the fractional integrals which have been studied extensively for their applications. The most known fractional integrals are the Caputo, Riemann-Liouville and the Hadamard fractional integral. A new fractional integral, called generalized Riemann-Liouville fractional integral, which generalizes the Riemann-Liouville and the Hadamard integrals into a single form, was introduced in [12], [13]. See Definition 2.5 below. This integral is now known as ”Katugampola fractional integral” see for example 1 228

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[14, pp 15, 123]. The existence and uniqueness results for the Caputo-Katugampola derivative is given in [15]. For some recent work with this new operator and similar operators, for example, see [16]-[18] and the references cited therein. The Langevin equation (first formulated by Langevin in 1908 to give an elaborate description of Brownian motion) is found to be an effective tool to describe the evolution of physical phenomena in fluctuating environments [19]. For some new developments on the fractional Langevin equation, see, for example, [20]-[24]. The present paper is motivated by a recent paper [25], where it is considered problem (1) with F single valued. Existence and uniqueness results were proved in [25] by using a variety of fixed point theorems, such as Banach contraction principle, Krasnoselskii fixed point theorem, Leray-Schauder nonlinear alternative and Leray-Schauder degree theory. Here, we cover the multi-valued case. We establish some existence results for the problem (1), when the right hand side is convex as well as non-convex valued. In the first result, we use the nonlinear alternative of Leray-Schauder type while in the second result, we shall combine the nonlinear alternative of Leray-Schauder type for single-valued maps with a selection theorem due to Bressan and Colombo for lower semicontinuous multivalued maps with nonempty closed and decomposable values. The third result relies on the fixed point theorem for contraction multivalued maps due to Covitz and Nadler. Examples illustrating the obtained results are also presented. The methods used are well known, however their exposition in the framework of problem (1) is new.

2 2.1

Preliminaries Basic material of fractional calculus

In this section, we introduce some notations and definitions of fractional calculus [1, 2] and present preliminary results needed in our proofs later. Definition 2.1 [2] The Riemann-Liouville fractional integral of order p > 0 of a continuous function f : (0, ∞) → R is defined by I p f (t) =

1 Γ(p)

∫

t

(t − s)p−1 v(s)ds, 0

provided∫ the right-hand side is point-wise defined on (0, ∞), where Γ is the gamma function defined by ∞ Γ(p) = 0 e−s sp−1 ds. Definition 2.2 [2] The Riemann-Liouville fractional derivative of order p > 0 of a continuous function f : (0, ∞) → R is defined by Dp f (t) =

1 Γ(n − p)

(

d dt

)n ∫

t

(t − s)n−p−1 v(s)ds,

n − 1 ≤ p < n,

0

where n = [p]+1, [p] denotes the integer part of a real number p, provided the right-hand side is point-wise defined on (0, ∞). Lemma 2.3 [2] Let p > 0 and x ∈∑C(0, T )∩L(0, T ). Then the fractional differential equation Dp x(t) = n p−i 0∑has a unique solution x(t) = , and the following formula holds: I p Dp x(t) = x(t) + i=1 ci t n p−i , where ci ∈ R, i = 1, 2, . . . , n, and n − 1 ≤ p < n. i=1 ci t Lemma 2.4 ([2], page 71) Let α > 0 and β > 0. Then the following properties hold: I α (x − a)β−1 (t)

=

229

Γ(β) (t − a)β+α−1 . Γ(β + α)

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3

Definition 2.5 ([13]) The Katugampola fractional integral of order q > 0 and ρ > 0, of a function f (t), for all 0 < t < ∞, is defined as ρ1−q I f (t) = Γ(q)

∫

t

ρ q

0

sρ−1 v(s) ds, (tρ − sρ )1−q

provided the right-hand side is point-wise defined on (0, ∞). Lemma 2.6 ([25]) Let constants q > 0 and p > 0. Then the following formula holds ) ( Γ p+ρ ρ tp+ρq ρ q p ) q . I t = ( ρ Γ p+ρq+ρ ρ

(2)

Lemma 2.7 ([25]) Let 0 < p1 , p2 ≤ 1, 1 < p1 + p2 ≤ 2, qi , ρi > 0, η, ξi ∈ (0, T ), αi ∈ R for all i = 1, 2, . . . , n and h ∈ C([0, T ], R). Then x is a solution of the problem Dp1 (Dp2 + λ)x(t) = h(t), n ∑ x(0) = 0, x(η) = αi

0 < t < T,

(3)

ρi qi

I x(ξi ),

(4)

i=1

if and only if x(t) =

[ Γ(p1 ) tp1 +p2 −1 p1 +p2 I h(η) − λ I p2 x(η) Γ(p1 + p2 ) Ω ] n ∑ ( p1 +p2 ) ρi qi p2 − αi I I h(s) − λ I x(s) (ξi ) + I p1 +p2 h(t) − λ I p2 x(t),

(5)

i=1

where ) p1 + p2 + ρi − 1 Γ n ∑ ξ p1 +p2 +ρi qi +ρi −1 Γ(p1 ) αi Γ(p1 ) ρi ( ) i − Ω= η p1 +p2 −1 6= 0. qi p1 + p2 + ρi qi + ρi − 1 Γ(p + p ) ρ Γ(p + p ) 1 2 1 2 i i=1 Γ ρi (

2.2

(6)

Basic material for multivalued maps

Here, we outline some basic concepts of multivalued analysis [26, 27]. Let C([0, T ], R) denote the Banach space of all continuous functions from [0, T ] into R with the norm kxk = sup{|x(t)|, t ∈ [0, T ]}. Also by L1 ([0, T ], R), we denote the space of functions x : [0, T ] → R such ∫T that kxkL1 = 0 |x(t)|dt. For a normed space (X, k · k), let Pcl (X) = {Y ∈ P(X) : Y is closed}, Pb (X) = {Y ∈ P(X) : Y is bounded}, Pcl,b (X) = {Y ∈ P(X) : Y is closed and bounded}, Pcp (X) = {Y ∈ P(X) : Y is compact}, and Pcp,c (X) = {Y ∈ P(X) : Y is compact and convex}. A multi-valued map G : X → P(X) : (i) is convex (closed) valued if G(x) is convex (closed) for all x ∈ X. (ii) is bounded on bounded sets if G(Y ) = ∪x∈Y G(x) is bounded in X for all Y ∈ Pb (X) (i.e. supx∈Y {sup{|y| : y ∈ G(x)}} < ∞). (iii) is called upper semi-continuous (u.s.c.) on X if for each x0 ∈ X, the set G(x0 ) is a nonempty closed subset of X, and if for each open set N of X containing G(x0 ), there exists an open neighborhood N0 of x0 such that G(N0 ) ⊆ N.

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(iv) G is lower semi-continuous (l.s.c.) if the set {y ∈ X : G(y) ∩ Y 6= ∅} is open for any open set Y in X. (v) is said to be completely continuous if G(B) is relatively compact for every B ∈ Pb (X); If the multi-valued map G is completely continuous with nonempty compact values, then G is u.s.c. if and only if G has a closed graph, i.e., xn → x∗ , yn → y∗ , yn ∈ G(xn ) imply y∗ ∈ G(x∗ ). (vi) is said to be measurable if for every y ∈ X, the function t 7−→ d(y, G(t)) = inf{|y − z| : z ∈ G(t)} is measurable. (vii) has a fixed point if there is x ∈ X such that x ∈ G(x). The fixed point set of the multivalued operator G will be denoted by Fix G.

3

Existence results

Let C = C([0, T ], R) denotes the Banach space of all continuous functions from [0, T ] to R endowed with the norm defined by kxk = supt∈[0,T ] |x(t)|. Throughout of this paper, for convenience of proving, we let the notations I z v(s)(y) and ρ I z v(s)(y) defined by 1 I v(s)(y) = Γ(z)

∫

y

(y − s)

z

z−1

v(s)ds and

0

ρ1−z I v(s)(y) = Γ(z)

∫

y

ρ z

0

sρ−1 v(s) ds, (y ρ − sρ )1−z

where z > 0 and y ∈ [0, T ]. To simplify the notations, we use the following constants: Λ1

=

Λ2

=

Γ(p1 ) T p1 +p2 −1 , Γ(p1 + p2 ) |Ω| ( ) ( p2 +ρi [ ]) n p2 +ρi qi p2 Γ p2 ∑ ρi T 1 ξi η ( ) . + Λ1 + |αi | Γ(1 + p2 ) Γ(1 + p2 ) i=1 Γ(1 + p2 ) ρqi i Γ p2 +ρi qi +ρi ρi

(7)

(8)

Definition 3.1 A function x ∈ AC 2 ([0, T ], R) is a solution of the problem (1) if x(0) = 0, x(η) = n ∑ αi ρi I qi x(ξi ), and there exists a function v ∈ L1 ([0, T ], R) such that f (t) ∈ F (t, x(t)) a.e. on [0, T ] i=1

and x(t) =

[ n ∑ Γ(p1 ) tp1 +p2 −1 p1 +p2 I v(η) − λ I p2 x(η) − αi Γ(p1 + p2 ) Ω i=1

ρi qi

I

(

I

p1 +p2

) v(s) − λ I x(s) (ξi )

]

p2

+I p1 +p2 v(t) − λ I p2 x(t).

3.1

The Carath´ eodory case

In this subsection, we consider the case when F has convex values and prove an existence result based on nonlinear alternative of Leray-Schauder type, assuming that F is Carath´eodory. Definition 3.2 A multivalued map F : [0, T ] × R → P(R) is said to be Carath´eodory if (i) t 7−→ F (t, x) is measurable for each x ∈ R; (ii) x 7−→ F (t, x) is upper semicontinuous for almost all t ∈ [0, T ]; Further a Carath´eodory function F is called L1 −Carath´eodory if

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(iii) for each ρ > 0, there exists ϕρ ∈ L1 ([0, T ], R+ ) such that kF (t, x)k = sup{|v| : v ∈ F (t, x)} ≤ ϕρ (t) for all kxk ≤ ρ and for a.e. t ∈ [0, T ]. For each y ∈ C, define the set of selections of F by SF,y := {v ∈ L1 ([0, T ], R) : v(t) ∈ F (t, y(t)) on [0, T ]}. We define the graph of G to be the set Gr (G) = {(x, y) ∈ X × Y, y ∈ G(x)} and recall a result for closed graphs and upper-semicontinuity. Lemma 3.3 ([26, Proposition 1.2]) If G : X → Pcl (Y ) is u.s.c., then Gr (G) is a closed subset of X × Y ; i.e., for every sequence {xn }n∈N ⊂ X and {yn }n∈N ⊂ Y , if when n → ∞, xn → x∗ , yn → y∗ and yn ∈ G(xn ), then y∗ ∈ G(x∗ ). Conversely, if G is completely continuous and has a closed graph, then it is upper semi-continuous. The following lemma will be used in the sequel. Lemma 3.4 ([28]) Let X be a Banach space. Let F : J × R → Pcp,c (X) be an L1 − Carath´eodory multivalued map and let Θ be a linear continuous mapping from L1 (J, X) to C(J, X). Then the operator Θ ◦ SF : C(J, X) → Pcp,c (C(J, X)), x 7→ (Θ ◦ SF )(x) = Θ(SF,x ) is a closed graph operator in C(J, X) × C(J, X). We recall the well-known nonlinear alternative of Leray-Schauder for multivalued maps. Lemma 3.5 (Nonlinear alternative for Kakutani maps)[29]. Let E be a Banach space, C a closed convex subset of E, U an open subset of C and 0 ∈ U. Suppose that F : U → Pcp,c (C) is a upper semicontinuous compact map. Then either (i) F has a fixed point in U , or (ii) there is a u ∈ ∂U and ν ∈ (0, 1) with u ∈ νF (u). Theorem 3.6 Assume that: (H1 ) F : [0, T ] × R → Pcp,c (R) is L1 -Carath´eodory; (H2 ) there exists a continuous nondecreasing function ψ : [0, ∞) → (0, ∞) and a function p ∈ L1 ([0, T ], R+ ) such that kF (t, x)kP := sup{|y| : y ∈ F (t, x)} ≤ p(t)ψ(kxk) for each (t, x) ∈ [0, T ] × R; (H3 ) there exists a constant M > 0 such that {[ n ∑ Λ1 M > I p1 +p2 p(s)(η) + |αi | ψ(M ) (1 − |λ|Λ2 ) i=1 } +I p1 +p2 p(s)(T )

:= Ω1 ,

ρi qi

I

(

I

p1 +p2

)

]

p(s)(τ ) (ξi )

|λ|Λ2 < 1,

where Λ1 and Λ2 are defined by (7) and (8) respectively. Then the boundary value problem (1) has at least one solution on [0, T ].

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Proof. Define the operator F : C → P(C) by

F (x)

          =

        

h ∈ C : [  Γ(p1 ) tp1 +p2 −1 p1 +p2   I v(s)(η) − λ I p2 x(s)(η)   Ω   Γ(p1 + p2 ) ] n ( ) ∑ h(t) = ρi qi p1 +p2 p2  − αi I I v(s)(τ ) − λ I x(s)(τ ) (ξi )     i=1   p1 +p2 p2 +I v(s)(t) − λ I x(s)(t)

                  

(9)

for v ∈ SF,x . It is obvious that the fixed points of F are solutions of the boundary value problem (1). We will show that F satisfies the assumptions of Leray-Schauder Nonlinear alternative (Lemma 3.5). The proof consists of several steps. Step 1. F(x) is convex for each x ∈ C. This step is obvious since SF,x is convex (F has convex values), and therefore, we omit the proof. Step 2. F maps bounded sets (balls) into bounded sets in C. For a positive number ρ, let Bρ = {x ∈ C : kxk ≤ ρ} be a bounded ball in C. Then, for each h ∈ F(x), x ∈ Bρ , there exists v ∈ SF,x such that [ Γ(p1 ) tp1 +p2 −1 p1 +p2 h(t) = I v(s)(η) − λ I p2 x(s)(η) Γ(p1 + p2 ) Ω ] n ( ) ∑ ρi qi p1 +p2 p2 − αi I I v(s)(τ ) − λ I x(s)(τ ) (ξi ) + I p1 +p2 v(s)(t) − λ I p2 x(s)(t). i=1

Then, we have ¯ [ ¯ Γ(p ) tp1 +p2 −1 ¯ 1 I p1 +p2 v(s)(η) − λ I p2 x(s)(η) |h(x)| ≤ ¯ ¯ Γ(p1 + p2 ) Ω ¯ ] n ¯ ( ) ∑ ¯ ρi qi p1 +p2 p2 p1 +p2 p2 − αi I I v(s)(τ ) − λ I x(s)(τ ) (ξi ) + I v(s)(t) − λ I x(s)(t)¯ ¯ i=1 [ Γ(p1 ) tp1 +p2 −1 p1 +p2 ≤ I p(s)ψ(kxk)(η) + |λ| I p2 kxk(η) Γ(p1 + p2 ) |Ω| ] n ( ) ∑ ρi qi p1 +p2 p2 + |αi | I I p(s)ψ(kxk)(τ ) + |λ| I kxk(τ ) (ξi ) i=1 p1 +p2

p(s)ψ(kxk)(t) + |λ| I p2 kxk(t) [ Γ(p1 ) T p1 +p2 −1 ≤ ψ(kxk) I p1 +p2 p(s)(η) Γ(p1 + p2 ) |Ω| ] n ( ) ∑ ρi qi p1 +p2 + |αi | I I p(s)(τ ) (ξi ) + ψ(kxk)I p1 +p2 p(s)(T ) +I

i=1

[ ] n ( ) ∑ Γ(p1 ) T p1 +p2 −1 |λ|kxk I p2 (η) + + |αi | ρi I qi I p2 kxk(τ ) (ξi ) + |λ|kxkI p2 (T ) Γ(p1 + p2 ) |Ω| i=1 [ { p1 +p2 −1 Γ(p1 ) T ≤ ψ(kxk) I p1 +p2 p(s)(η) Γ(p1 + p2 ) |Ω| ] } n ( ) ∑ + |αi | ρi I qi I p1 +p2 p(s)(τ ) (ξi ) + I p1 +p2 p(s)(T ) i=1

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( Γ(p1 ) T p1 +p2 −1 η p2 T p2 + +|λ|kxk Γ(1 + p2 ) Γ(p1 + p2 ) |Ω| Γ(1 + p2 ) ( ) )} p2 +ρi [ ] n ∑ ρi 1 ξip2 +ρi qi Γ ( ) + |αi | Γ(1 + p2 ) ρiqi Γ p2 +ρi qi +ρi i=1 ρi [ ] n ( ) ∑ p1 +p2 p1 +p2 ρi qi p(s)(τ ) (ξi ) + |λ|kxkΛ2 , = ψ(kxk)Λ1 I p(s)(η) + |αi | I I {

i=1

and consequently, [ khk ≤

ψ(ρ)Λ1 I

p1 +p2

p(s)(η) +

n ∑

|αi |

ρi qi

I

(

I

p1 +p2

] ) p(s)(τ ) (ξi ) + |λ|ρΛ2 .

i=1

Step 3. F maps bounded sets into equicontinuous sets of C. Let τ1 , τ2 ∈ [0, T ] with τ1 < τ2 and x ∈ Bρ . For each h ∈ F(x), we obtain ¯ ¯ Γ(p ) tp1 +p2 −1 − tp1 +p2 −1 [ ¯ 1 1 2 I p1 +p2 v(s)(η) − λ I p2 x(s)(η) |h(τ2 ) − h(τ1 )| ≤ ¯ ¯ Γ(p1 + p2 ) Ω ]¯¯ n ( ) ∑ ¯ − αi ρi I qi I p1 +p2 v(s)(τ ) − λ I p2 x(s)(τ ) (ξi ) ¯ ¯ ¯ ¯i=1 ¯ ¯ +¯I p1 +p2 v(s)(t2 ) − I p1 +p2 v(s)(t1 )¯ + ¯λ I p2 x(s)(t2 ) − λ I p2 x(s)(t1 )¯ [ Γ(p1 ) |tp21 +p2 −1 − tp11 +p2 −1 | p1 +p2 I p(s)ψ(kxk)(η) + |λ| I p2 kxk(η) ≤ Γ(p1 + p2 ) |Ω| ] n ( ) ∑ + αi ρi I qi I p1 +p2 p(s)ψ(kxk)(τ ) + |λ| I p2 kxk(τ ) (ξi ) ¯i=1 ¯ +¯I p1 +p2 p(s)ψ(kxk)(t2 ) − I p1 +p2 p(s)ψ(kxk)(t1 )¯ ¯ ¯ +¯λ I p2 x(s)(t2 ) − λ I p2 x(s)(t1 )¯ [ Γ(p1 ) |tp21 +p2 −1 − tp11 +p2 −1 | p1 +p2 I p(s)ψ(ρ)(η) + |λ| ρI p2 (η) ≤ Γ(p1 + p2 ) |Ω| ] n ( ) ∑ + αi ρi I qi I p1 +p2 p(s)ψ(ρ)(τ ) + |λ| ρI p2 (τ ) (ξi ) ¯i=1 ¯ ( ) ( ) +¯ I p1 +p2 p(s)ψ(ρ) (t2 ) − I p1 +p2 p(s)ψ(ρ) (t1 )¯ ¯ ¯ +|λ|ρ¯ (I p2 ) (t2 ) − (I p2 ) (t1 )¯. Obviously the right hand side of the above inequality tends to zero independently of x ∈ Bρ as τ2 − τ1 → 0. As F satisfies the above three assumptions, therefore it follows by the Ascoli-Arzel´a theorem that F : C → P(C) is completely continuous. Since F is completely continuous, in order to prove that it is u.s.c. it is enough to prove that it has a closed graph (Lemma 3.3). Thus, in our next step, we show that Step 4. F has a closed graph. Let xn → x∗ , hn ∈ F(xn ) and hn → h∗ . Then, we need to show that h∗ ∈ F(x∗ ). Associated with hn ∈ F(xn ), there exists vn ∈ SF,xn such that for each t ∈ [0, T ], [ Γ(p1 ) tp1 +p2 −1 p1 +p2 I vn (s)(η) − λ I p2 x(s)(η) hn (t) = Γ(p1 + p2 ) Ω

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−

n ∑

αi

ρi qi

I

(

I

p1 +p2

] ) vn (s)(τ ) − λ I x(s)(τ ) (ξi ) + I p1 +p2 vn (s)(t) − λ I p2 x(s)(t). p2

i=1

Thus it suffices to show that there exists v∗ ∈ SF,x∗ such that for each t ∈ [0, T ], [ Γ(p1 ) tp1 +p2 −1 p1 +p2 I v∗ (s)(η) − λ I p2 x(s)(η) h∗ (t) = Γ(p1 + p2 ) Ω ] n ( ) ∑ p1 +p2 p2 ρi qi v∗ (s)(τ ) − λ I x(s)(τ ) (ξi ) + I p1 +p2 v∗ (s)(t) − λ I p2 x(s)(t). − αi I I i=1

Let us consider the linear operator Θ : L1 ([0, T ], R) → C given by [ Γ(p1 ) tp1 +p2 −1 p1 +p2 I v(s)(η) − λ I p2 x(s)(η) v 7→ Θ(v)(t) = Γ(p1 + p2 ) Ω ] n ( ) ∑ − αi ρi I qi I p1 +p2 v(s)(τ ) − λ I p2 x(s)(τ ) (ξi ) + I p1 +p2 v(s)(t) − λ I p2 x(s)(t). i=1

Observe that khn (t) − h∗ (t)k

° [ ° Γ(p ) tp1 +p2 −1 ° 1 I p1 +p2 (vn (s) − v∗ (s))(η) = ° ° Γ(p1 + p2 ) Ω ° ] n ° ( ) ∑ ° ρi qi p1 +p2 p1 +p2 − αi I I (vn (s) − v∗ (s))(τ ) (ξi ) + I (vn (s) − v∗ (s))(t)° → 0, ° i=1

as n → ∞. Thus, it follows by Lemma 3.4 that Θ ◦ SF is a closed graph operator. Further, we have hn (t) ∈ Θ(SF,xn ). Since xn → x∗ , therefore, we have [ Γ(p1 ) tp1 +p2 −1 p1 +p2 h∗ (t) = I v∗ (s)(η) − λ I p2 x(s)(η) Γ(p1 + p2 ) Ω ] n ( ) ∑ ρi qi p1 +p2 p2 − αi I I v∗ (s)(τ ) − λ I x(s)(τ ) (ξi ) + I p1 +p2 v∗ (s)(t) − λ I p2 x(s)(t) i=1

for some v∗ ∈ SF,x∗ . Step 5. We show there exists an open set U ⊆ C with x ∈ / θF(x) for any θ ∈ (0, 1) and all x ∈ ∂U. Let θ ∈ (0, 1) and x ∈ θF(x). Then there exists v ∈ L1 ([0, T ], R) with v ∈ SF,x such that, for t ∈ [0, T ], we have [ Γ(p1 ) tp1 +p2 −1 p1 +p2 I v(s)(η) − λ I p2 x(s)(η) x(t) = θ Γ(p1 + p2 ) Ω ] n ( ) ∑ ρi qi p1 +p2 p2 − αi I I v(s)(τ ) − λ I x(s)(τ ) (ξi ) + θI p1 +p2 v(s)(t) − θλ I p2 x(s)(t). i=1

Using the computations of the second step above, we have [ Γ(p1 ) tp1 +p2 −1 p1 +p2 I p(s)ψ(kxk)(η) + |λ| I p2 kxk(η) kxk ≤ Γ(p1 + p2 ) |Ω|

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+

n ∑

|αi | i=1 p1 +p2

ρi qi

(

I

I

p1 +p2

9

) p(s)ψ(kxk)(τ ) + |λ| I kxk(τ ) (ξi )

]

p2

p(s)ψ(kxk)(T ) + |λ| I p2 kxk(T ) [ { Γ(p1 ) T p1 +p2 −1 p1 +p2 I p(s)(η) ψ(kxk) Γ(p1 + p2 ) |Ω| ] } n ( ) ∑ p1 +p2 p1 +p2 ρi qi p(s)(τ ) (ξi ) + I p(s)(T ) + |αi | I I

+I ≤

i=1

=

{

( T p2 Γ(p1 ) T p1 +p2 −1 η p2 +|λ|kxk + Γ(1 + p2 ) Γ(p1 + p2 ) |Ω| Γ(1 + p2 ) ) ( p2 +ρi [ ])} n ∑ ρi 1 ξip2 +ρi qi Γ ) ( + |αi | Γ(1 + p2 ) ρqi i Γ p2 +ρi qi +ρi i=1 ρi { [ ] n ( ) ∑ p1 +p2 p1 +p2 ρi qi ψ(kxk) Λ1 I p(s)(η) + p(s)(τ ) (ξi ) |αi | I I i=1

} +I p1 +p2 p(s)(T ) which implies that kxk ψ(kxk)

≤

Λ1 (1 − |λ|Λ2 )

+ |λ|kxkΛ2 ,

{[ I p1 +p2 p(s)(η) +

n ∑

|αi |

ρi qi

I

(

)

]

}

I p1 +p2 p(s)(τ ) (ξi ) + I p1 +p2 p(s)(T ) .

i=1

In view of (H3 ), there exists M such that kxk 6= M . Let us set U = {x ∈ C : kxk < M }. Note that the operator F : U → P(C) is a compact multi-valued map, u.s.c. with convex closed values. From the choice of U , there is no x ∈ ∂U such that x ∈ θF(x) for some θ ∈ (0, 1). Consequently, by the nonlinear alternative of Leray-Schauder type (Lemma 3.5), we deduce that F has a fixed point x ∈ U which is a solution of the problem (1). This completes the proof. ¤

3.2

The lower semicontinuous case

In the next result, F is not necessarily convex valued. Our strategy to deal with this problem is based on the nonlinear alternative of Leray Schauder type together with the selection theorem of Bressan and Colombo [30] for lower semi-continuous maps with decomposable values. Let X be a nonempty closed subset of a Banach space E and G : X → P(E) be a multivalued operator with nonempty closed values. G is lower semi-continuous (l.s.c.) if the set {y ∈ X : G(y) ∩ B 6= ∅} is open for any open set B in E. Let A be a subset of [0, T ] × R. A is L ⊗ B measurable if A belongs to the σ−algebra generated by all sets of the form J × D, where J is Lebesgue measurable in [0, T ] and D is Borel measurable in R. A subset A of L1 ([0, T ], R) is decomposable if for all u, v ∈ A and measurable J ⊂ [0, T ] = J, the function uχJ + vχJ−J ∈ A, where χJ stands for the characteristic function of J . Definition 3.7 Let Y be a separable metric space and let N : Y → P(L1 ([0, T ], R)) be a multivalued operator. We say N has a property (BC) if N is lower semi-continuous (l.s.c.) and has nonempty closed and decomposable values. Let F : [0, T ]×R → P(R) be a multivalued map with nonempty compact values. Define a multivalued operator F : C([0, T ] × R) → P(L1 ([0, T ], R)) associated with F as F(x) = {w ∈ L1 ([0, T ], R) : w(t) ∈ F (t, x(t)) for a.e. t ∈ [0, T ]},

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which is called the Nemytskii operator associated with F. Definition 3.8 Let F : [0, T ] × R → P(R) be a multivalued function with nonempty compact values. We say F is of lower semi-continuous type (l.s.c. type) if its associated Nemytskii operator F is lower semi-continuous and has nonempty closed and decomposable values. Lemma 3.9 ([31]) Let Y be a separable metric space and let N : Y → P(L1 ([0, T ], R)) be a multivalued operator satisfying the property (BC). Then N has a continuous selection, that is, there exists a continuous function (single-valued) g : Y → L1 ([0, T ], R) such that g(x) ∈ N (x) for every x ∈ Y . Theorem 3.10 Assume that (H2 ), (H3 ) and the following condition holds: (H4 ) F : [0, T ] × R → P(R) is a nonempty compact-valued multivalued map such that (a) (t, x) 7−→ F (t, x) is L ⊗ B measurable, (b) x 7−→ F (t, x) is lower semicontinuous for each t ∈ [0, T ]; Then the boundary value problem (1) has at least one solution on [0, T ]. Proof. It follows from (H2 ) and (H4 ) that F is of l.s.c. type. Then from Lemma 3.9, there exists a continuous function f : AC 2 ([0, T ], R) → L1 ([0, T ], R) such that f (x) ∈ F(x) for all x ∈ C([0, T ], R). Consider the problem  q RL D x(t) = f (x(t)), 0 < t < T, 1 < α ≤ 2,    n (10) ∑   αi ρi I qi x(ξi ).  x(0) = 0, x(η) = i=1

Observe that if x ∈ AC ([0, T ], R) is a solution of (10), then x is a solution to the problem (1). In order to transform the problem (10) into a fixed point problem, we define the operator F as [ Γ(p1 ) tp1 +p2 −1 p1 +p2 Fx(t) = I f (x(s))(η) − λ I p2 x(s)(η) Γ(p1 + p2 ) Ω ] n ( ) ∑ ρi qi p1 +p2 p2 − αi I I f (x(s))(τ ) − λ I x(s)(τ ) (ξi ) + I p1 +p2 f (x(s))(t) − λ I p2 x(s)(t). 2

i=1

It can easily be shown that F is continuous and completely continuous. The remaining part of the proof is similar to that of Theorem 3.6. So, we omit it. This completes the proof. ¤

3.3

The Lipschitz case

In this subsection, we prove the existence of solutions for the problem (1) with a not necessary nonconvex valued right hand side, by applying a fixed point theorem for multivalued maps due to Covitz and Nadler [32]. Let (X, d) be a metric space induced from the normed space (X; k·k). Consider Hd : P(X)×P(X) → R ∪ {∞} given by Hd (A, B) = max{sup d(a, B), sup d(A, b)}, a∈A

b∈B

where d(A, b) = inf a∈A d(a; b) and d(a, B) = inf b∈B d(a; b). Then (Pcl,b (X), Hd ) is a metric space (see [33]). Definition 3.11 A multivalued operator N : X → Pcl (X) is called (a) γ−Lipschitz if and only if there exists γ > 0 such that Hd (N (x), N (y)) ≤ γd(x, y) for each x, y ∈ X;

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(b) a contraction if and only if it is γ−Lipschitz with γ < 1. Lemma 3.12 ([32]) Let (X, d) be a complete metric space. If N : X → Pcl (X) is a contraction, then F ixN 6= ∅. Theorem 3.13 Assume that: (A1 ) F : [0, T ] × R → Pcp (R) is such that F (·, x) : [0, T ] → Pcp (R) is measurable for each x ∈ R. (A2 ) Hd (F (t, x), F (t, x ¯)) ≤ m(t)|x − x ¯| for almost all t ∈ [0, T ] and x, x ¯ ∈ R with m ∈ L1 ([0, T ], R+ ) and d(0, F (t, 0)) ≤ m(t) for almost all t ∈ [0, T ]. Then the boundary value problem (1) has at least one solution on [0, T ] if [ ] n ( ) ∑ p1 +p2 ρi qi p1 +p2 Ω2 := Λ1 I m(s)(η) + |αi | I I m(s)(τ ) + I p1 +p2 m(s)(T ) < 1. i=1

Proof. Consider the operator F defined by (9). Observe that the set SF,x is nonempty for each x ∈ C by the assumption (A1 ), so F has a measurable selection (see Theorem III.6 [34]). Now, we show that the operator F satisfies the assumptions of Lemma 3.12. We show that F(x) ∈ Pcl (C) for each x ∈ C. Let {un }n≥0 ∈ F(x) be such that un → u (n → ∞) in C. Then u ∈ C and there exists vn ∈ SF,xn such that, for each t ∈ [0, T ], [ Γ(p1 ) tp1 +p2 −1 p1 +p2 un (t) = I vn (s)(η) − λ I p2 x(s)(η) Γ(p1 + p2 ) Ω ] n ( ) ∑ − αi ρi I qi I p1 +p2 vn (s)(τ ) − λ I p2 x(s)(τ ) (ξi ) + I p1 +p2 vn (s)(t) − λ I p2 x(s)(t). i=1

As F has compact values, we pass onto a subsequence (if necessary) to obtain that vn converges to v in L1 ([0, T ], R). Thus, v ∈ SF,x and for each t ∈ [0, T ], we have [ Γ(p1 ) tp1 +p2 −1 p1 +p2 I v(s)(η) − λ I p2 x(s)(η) un (t) → v(t) = Γ(p1 + p2 ) Ω ] n ( ) ∑ ρi qi p1 +p2 p2 − αi I I v(s)(τ ) − λ I x(s)(τ ) (ξi ) + I p1 +p2 v(s)(t) − λ I p2 x(s)(t). i=1

Hence, u ∈ F(x). Next, we show that there exists δ < 1 such that Hd (F(x), F(¯ x)) ≤ δkx − x ¯k for each x, x ¯ ∈ AC 2 ([0, T ], R). Let x, x ¯ ∈ AC 2 ([0, T ], R) and h1 ∈ F(x). Then there exists v1 (t) ∈ F (t, x(t)) such that, for each t ∈ [0, T ], [ Γ(p1 ) tp1 +p2 −1 p1 +p2 I v1 (s)(η) − λ I p2 x(s)(η) h1 (t) = Γ(p1 + p2 ) Ω ] n ( ) ∑ − αi ρi I qi I p1 +p2 v1 (s)(τ ) − λ I p2 x(s)(τ ) (ξi ) + I p1 +p2 v1 (s)(t) − λ I p2 x(s)(t). i=1

By (A2 ), we have

Hd (F (t, x), F (t, x ¯)) ≤ m(t)|x(t) − x ¯(t)|.

So, there exists w ∈ F (t, x ¯(t)) such that |v1 (t) − w| ≤ m(t)|x(t) − x ¯(t)|, t ∈ [0, T ].

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12

S.K. Ntouyas and J. Tariboon Define U : [0, T ] → P(R) by U (t) = {w ∈ R : |v1 (t) − w| ≤ m(t)|x(t) − x ¯(t)|}.

Since the multivalued operator U (t) ∩ F (t, x ¯(t)) is measurable (Proposition III.4 [34]), there exists a function v2 (t) which is a measurable selection for U . So v2 (t) ∈ F (t, x ¯(t)) and for each t ∈ [0, T ], we have |v1 (t) − v2 (t)| ≤ m(t)|x(t) − x ¯(t)|. For each t ∈ [0, T ], let us define h2 (t) =

[ Γ(p1 ) tp1 +p2 −1 p1 +p2 I v2 (s)(η) − λ I p2 x(s)(η) Γ(p1 + p2 ) Ω ] n ( ) ∑ p1 +p2 p2 ρi qi v2 (s)(τ ) − λ I x(s)(τ ) (ξi ) + I p1 +p2 v2 (s)(t) − λ I p2 x(s)(t). − αi I I i=1

Thus, |h1 (t) − h2 (t)| ≤

[ Γ(p1 ) T p1 +p2 −1 p1 +p2 I |v1 (s) − v2 (s)|(η) Γ(p1 + p2 ) |Ω| ] n ( ∑ + |αi | ρi I qi I p1 +p2 |v1 (s) − v2 (s)|(τ ) + I p1 +p2 |v1 (s) − v2 (s)|(t) i=1

[ Γ(p1 ) T p1 +p2 −1 p1 +p2 I m(s)kx − x ¯k(η) ≤ Γ(p1 + p2 ) |Ω| ] n ( ) ∑ ρi qi p1 +p2 + |αi | I I m(s)kx − x ¯k(τ ) + I p1 +p2 m(s)kx − x ¯k(T ) { =

i=1

[

Λ1 I

p1 +p2

n ∑

m(s)(η) +

+I

I

(

I

p1 +p2

)

]

m(s)(τ )

i=1

} p1 +p2

|αi |

ρi qi

m(s)(T ) kx − x ¯k.

Hence, { kh1 − h2 k

≤

[

Λ1 I p1 +p2 m(s)(η) +

n ∑

|αi |

ρi qi

(

I

I p1 +p2 m(s)(τ )

)

]

} + I p1 +p2 m(s)(T ) kx − x ¯k.

i=1

Analogously, interchanging the roles of x and x, we obtain { [ n ∑ Hd (F(x), F(¯ x)) ≤ Λ1 I p1 +p2 m(s)(η) + |αi | } +I

p1 +p2

ρi qi

I

(

) I p1 +p2 m(s)(τ )

]

i=1

m(s)(T ) kx − x ¯k.

So F is a contraction. Therefore, it follows by Lemma 3.12 that F has a fixed point x which is a solution of (1). This completes the proof. ¤

3.4

Examples

In this section, we will illustrate our main results with the help of some examples.

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Nonlocal Fractional Katugampola Langevin Inclusions

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Example 3.14 Let us consider the following Langevin fractional differential inclusions with nonlocal Katugampola fractional integral boundary conditions ( ) ( )  1 2 2/3 4/5   x(t) ∈ F (t, x(t)), , D D + t ∈ 0, 1   8 3      x(0) = 0,   ( ) ( ) √ ( ) ( ) (11) 3 √2 1 1 1 √3 1 1 4 2 2 42 2  5 I3x 8 Iπx  e I3x √ = + + x   9 4 9 3 5 9   ( ) 7   2  6 11 5 √  5 I 13 x  , + 15 9 where

[

)] x2 e−t + . (12) 3(1 + |x|) 2 √ Here p1 = 2/3, p2√= 4/5, λ√= 1/8, T = 2/3, η = 2/9, n =√4, α1 = 3/4, ρ1 = 2/5, q1 = 1/3, ξ1 = 1/9, α2 = 1/ √ 7, ρ2 = 3/8, q2 = 1/π, ξ2 = 1/3, α3 = 2/5, ρ3 = 4/e2 , q3 = 2/3, ξ3 = 4/9, α4 = 11/15, ρ4 = 2/ 5, q4 = 6/13, ξ4 = 5/9. From these constants, we can find that Ω = 0.2660602470, Λ1 = 4.756155970, Λ2 = 5.624515148 and also |λ|Λ2 = 0.7030643935 < 1. It is obvious that the condition (H1 ) is satisfied. For f ∈ F1 , we have ( ( ) ( )) 1 (t1/2 + 1) |x| sin |x| e−2t (t 2 + 1) x2 e−t |f | ≤ max + , + 20 5(3 + 2|x|) 3 15 3(1 + |x|) 2 ( ) (t1/2 + 1) 1 1 ≤ |x| + , t ∈ (0, 2/3), x ∈ R. 15 3 2 (t1/2 + 1) F1 (t, x) = 20

(

|x| sin |x| e−2t + 5(3 + 2|x|) 3

)

(t1/2 + 1) , 15

(

Therefore, we have kF1 (t, x)kP = sup{|y| : y ∈ F1 (t, x)} ≤ p(t)ψ(|x|),

t ∈ (0, 2/3) x ∈ R,

where p(t) = (t1/2 + 1)/15, ψ(|x|) = (1/3)|x| + (1/2). This means that the condition (H2 ) is fulfilled. By direct computation, we have Ω1 = 1.123809144 and also there exists a constant M > 0.8984766718 satisfying condition (H3 ). Therefore, all the conditions of Theorem 3.6 are satisfied. So, the problem (11) with F1 (t, x) given by (12) has at least one solution on [0, 2/3]. Example 3.15 Let us consider the following Langevin fractional differential inclusions with nonlocal Katugampola fractional integral boundary conditions ) ) ( (  3 1 8/9 6/7   D x(t) ∈ F2 (t, x(t)), t ∈ 0, , D +   12 2      x(0) = 0,   ( ) ( ) ( ) ( ) (13) 2 2 3 5 1 4 √3 √1 1 π 2 √3 5  4 8 5 13 6 8 √  x = I x + I x + 2 I x   3 3 7 2 e 6 11   ( ) √ √ ( )    52 3 7 3 11 13 4  15 I 17 x  + 9 I π2 x + , 4 6 9 3 where

[ ( ) ] (t1/3 + 1) x2 + 2|x| t F2 (t, x) = 0, + . 24 (1 + |x|) 2

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√ Here p1 = 6/7, p2 = 8/9, √ λ = 1/12, T = √ 3/2, η = 2/3, n = 5,2 α1 = 2/ 11, ρ1 = 3/4, √ q1 = 5/8, = π/e , ρ3 = 2/13, q3 = 3/ 8, ξ3 = 5/6, ξ1 = 1/3, α2 = 4/7, ρ2 = 3/5, q2 = 1/ 6, ξ2√= 1/2, α3 √ α4 = 5/4, ρ4 = 2/9, q4 = 3/π 2 , ξ4 = 7/6, α5 = 3/9, ρ5 = 11/15, q5 = 13/17, ξ5 = 4/3. From given constants, we can find that Ω = 2.105868955, Λ1 = 0.7738927855. In addition, we have ( ) t (t1/3 + 1) x2 + 2|x| + , sup{|x| : x ∈ F2 (t, x)} ≤ 24 (1 + |x|) 2 which yields Hd (F2 (t, x), F2 (t, y)) ≤

(t1/3 + 1) |x − y|. 12

Choosing m(t) = (t1/3 + 1)/12, we obtain Hd (F2 (t, x), F2 (t, y)) ≤ m(t)|x − y| such that d(0, F2 (t, 0)) ≤ m(t). By the previous setting, we find that Ω2 = 0.3397697571 < 1. Thus all assumptions of Theorem 3.13 are fulfilled. Therefore, by the conclusion of Theorem 3.13, we deduce that the problem (13) with F2 (t, x) given by (14) has at least one solution on [0, 3/2].

References [1] Podlubny, I: Fractional Differential Equations. Academic Press, San Diego, 1999. [2] A. A. Kilbas, H. M. Srivastava, J. J. Trujillo, Theory and Applications of Fractional Differential Equations, North-Holland Mathematics Studies, 204, Elsevier Science B.V., Amsterdam, 2006. [3] R. P. Agarwal, Y. Zhou, Y. He, Existence of fractional neutral functional differential equations, Comput. Math. Appl. 59 (2010), 1095-1100. [4] B. Ahmad, S. K. Ntouyas, A. Alsaedi, New existence results for nonlinear fractional differential equations with three-point integral boundary conditions, Adv. Differ. Equ. (2011), Art. ID 107384, 11 pp. [5] B. Ahmad, S. K. Ntouyas, A. Alsaedi, A study of nonlinear fractional differential equations of arbitrary order with Riemann-Liouville type multistrip boundary conditions, Math. Probl. Eng. Art. (2013), ID 320415, 9 pp. [6] B. Ahmad, J. J. Nieto, Boundary value problems for a class of sequential integrodifferential equations of fractional order, J. Funct. Spaces Appl. (2013), Art. ID 149659, 8 pp. [7] L. Zhang, B. Ahmad, G. Wang, R. P. Agarwal, Nonlinear fractional integro-differential equations on unbounded domains in a Banach space, J. Comput. Appl. Math. 249 (2013), 51-56. [8] X. Liu, M. Jia, W. Ge, Multiple solutions of a p-Laplacian model involving a fractional derivative, Adv. Differ. Equ. (2013) 2013:126. [9] S. K. Ntouyas, S. Etemad, On the existence of solutions for fractional differential inclusions with sum and integral boundary conditions, Appl. Math. Comp. 266 (2015), 235-243. [10] S. K. Ntouyas, S. Etemad, J. Tariboon, Existence of solutions for fractional differential inclusions with integral boundary conditions, Bound. Value Prob. (2015) 2015:92. [11] S. K. Ntouyas, S. Etemad, J. Tariboon, Existence results for multi-term fractional differential inclusions, Adv. Differ. Equ. (2015) 2015:140. [12] U. N. Katugampola, New Approach to a generalized fractional integral, Appl. Math. Comput. 218 (3) (2015), 860-865. [13] U. N. Katugampola, A new approach to generalized fractional derivatives, Bull. Math. Anal. Appl. 6 (4) (2014), 1-15.

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[14] A. B. Malinowska, T. Odzijewicz, D. F. M. Torres, Advanced Methods in the Fractional Calculus of Variations, Springer, 2015. [15] U. N. Katugampola, Existence and uniqueness results for a class of generalized fractional differential equations, preprint arXiv: 1411.5229. [16] S. Gaboury, R. Tremblay, B. J. Fugere, Some relations involving a generalized fractional derivative operator, J. Ineq. Appl. (2013) 2013:167. [17] M. A. Noor, M. U. Awan, Kl. Noor, On some inequalities for relative semi-convex functions, J. Ineq. Appl. (2013) 2013:332. [18] S. Pooseh, R. Almeida, D. F. M. Torres, Expansion formulas in terms of integer-order derivatives for the Hadamard fractional integral and derivative, Numer. Funct. Anal. Optim. 33 (2012), 301-319. [19] W. T. Coffey, Yu. Kalmykov, J. T. Waldron, The Langevin Equation, second ed., World Scientific, Singapore, 2004. [20] S. C. Lim, M. Li, L. P. Teo, Langevin equation with two fractional orders. Phys. Lett. A 372 (2008), 6309-6320. [21] A. Alsaedi, S. K. Ntouyas, B. Ahmad, Existence results for Langevin fractional differential inclusions involving two fractional orders with four-point multi-term fractional integral boundary conditions, Abstract Appl. Anal. Vol. 2013, Article ID 869837, 17 pages. [22] J. Tariboon, S. K. Ntouyas, Nonlinear second-order impulsive q-difference Langevin equation with boundary conditions, Bound. Value Probl. (2014) 2014: 85. [23] J. Tariboon, S. K. Ntouyas, Ch. Thaiprayoon, Nonlinear Langevin equation of Hadamard-Caputo type fractional derivatives with nonlocal fractional integral conditions, Adv. Math. Phys. Vol.2014, Article ID 372749, 15 pages. [24] W. Yukunthorn, S. K. Ntouyas, J. Tariboon, Nonlinear fractional Caputo-Langevin equation with nonlocal Riemann-Liouville fractional integral conditions, Adv. Differ. Equ. (2014) 2014:315. [25] Ch. Thaiprayoon, S. K. Ntouyas, J. Tariboon, On the nonlocal Katugampola fractional integral conditions for fractional Langevin equation, Adv. Differ. Equ. (2015) 2015:374. [26] K. Deimling, Multivalued Differential Equations, Walter De Gruyter, Berlin-New York, 1992. [27] Sh. Hu, N. Papageorgiou, Handbook of Multivalued Analysis, Theory I, Kluwer, Dordrecht, 1997. [28] A. Lasota, Z. Opial, An application of the Kakutani-Ky Fan theorem in the theory of ordinary differential equations, Bull. Acad. Polon. Sci. Ser. Sci. Math. Astronom. Phys. 13 (1965), 781-786. [29] A. Granas, J. Dugundji, Fixed Point Theory, Springer-Verlag, New York, 2005. [30] A. Bressan, G. Colombo, Extensions and selections of maps with decomposable values, Studia Math. 90 (1988), 69-86. [31] M. Frigon, Th´eor`emes d’existence de solutions d’inclusions diff´erentielles, Topological Methods in Differential Equations and Inclusions (edited by A. Granas and M. Frigon), NATO ASI Series C, Vol. 472, Kluwer Acad. Publ., Dordrecht, (1995), 51-87. [32] H. Covitz, S.B. Nadler Jr., Multivalued contraction mappings in generalized metric spaces, Israel J. Math. 8 (1970), 5-11. [33] M. Kisielewicz, Differential Inclusions and Optimal Control, Kluwer, Dordrecht, The Netherlands, 1991. [34] C. Castaing, M. Valadier, Convex Analysis and Measurable Multifunctions, Lecture Notes in Mathematics 580, Springer-Verlag, Berlin-Heidelberg-New York, 1977.

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On the Riemann-Liouville fractional Hermite-Hadamard-type inequalities for differentiable α-preinvex mappings Lianzi Chen Tingsong Du∗ Shasha Zhao and Sheng Zheng Department of Mathematics, College of Science, China Three Gorges University, Yichang, 443002, P. R. China. E-mail: [email protected], [email protected] [email protected], [email protected] ∗ Corresponding author

Received on May 11, 2016; Accepted on January 16, 2017 Abstract In the paper, by discovering a Riemann-Liouville fractional integral identity involving twice differentiable preinvex mappings, the authors establish the rightsided new Hermite-Hadamard type inequalities via Riemann-Liouville fractional integrals for α-preinvex functions. The new fractional integral inequalities are then applied to some special means. 2010 Mathematics Subject Classification: Primary 26A33; Secondary 26D15, 26E60, 41A55. Key words and phrases: Hermite-Hadamard’s inequality; α-preinvex functions ; Riemann-Liouville fractional integrals.

1

Introduction

Let f : I ⊂ R → R be a convex mapping defined on the interval I of real numbers and a, b ∈ I with a < b. The following inequality   Z b a+b 1 f (a) + f (b) (1.1) f ≤ f (x)dx ≤ 2 b−a a 2 referred to as Hermite-Hadamard inequality, is one of the most famous results for convex functions. A number of papers have been written on this inequality providing new proofs, noteworthy extensions, generalizations, refinements and

1

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new inequalities connected with the Hermite-Hadamard inequality. The reader may refer to [7, 12, 14, 16, 20, 21, 26, 27, 34] and the references therein. Let us recall some necessary definitions and preliminary results which are used for further discussion. Definition 1.1 ([3, 32]) A set S ⊆ Rn is said to be invex set with respect to the mapping η : S × S → Rn if x + tη(y, x) ∈ S for every x, y ∈ S and t ∈ [0, 1]. The invex set S is also called an η-connected set. Notice that every convex set is invex with respect to the mapping η(y, x) = y − x, but the converse is not necessarily true. See [3, 33], for example. Definition 1.2 ([3]) Let S ⊆ Rn be an invex set with respect to η : S ×S → Rn . For every x, y ∈ S, the η-path Pxv joining the points x and v = x + η(y, x) is defined by Pxv = {z|z = x + tη(y, x), t ∈ [0, 1]}. A significant generalization of convex mappings is that of preinvex mappings introduced by Weir and Mond in [32]. Definition 1.3 ([32]) The function f defined on the invex set K ⊆ Rn is said to be preinvex with respect to η if for every x, y ∈ K and t ∈ [0, 1] we have
f x + tη(y, x) ≤ (1 − t)f (x) + tf (y). The function f is said to be preincave if and only if −f is preinvex. The concept of preinvexity is more general than convexity since every convex function is preinvex with respect to the mapping η(y, x) = y − x. Further, there exist preinvex functions which are not convex. Moreover, Wang et al. gave the so-called α-preinvex function in [29] as follows. Definition 1.4 ([29]) Let S ⊆ Rn be an invex set with respect to η : S × S → Rn . A function f : S → R is said to be α-preinvex with respect to η for α ∈ (0, 1], if every x, y ∈ S and t ∈ [0, 1],
f x + tη(y, x) ≤ (1 − tα )f (x) + tα f (y). Certainly, α-preinvex mapping means just preinvex mapping when α = 1. For recent results on some new generalizations, refinements of integral inequalities involved with the preinvex functions, one can see [4, 13, 17–19, 23] and the references therein. We also need the following fractional calculus background. Definition 1.5 ([25]) Let f ∈ L1 [a, b]. The left-sided and right-sided RiemannLiouville fractional integrals of order α > 0 with a ≥ 0 are defined by Z x 1 α (x − t)α−1 f (t)dt, a < x Ja+ f (x) = Γ(α) a 2

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and Jbα− f (x) =

1 Γ(α)

Z

b

(t − x)α−1 f (t)dt, x < b

x

respectively, R ∞ where Γ(.) is Gamma function and its definition is Γ(α) = 0 e−u uα−1 du. It is to be noted that Ja0+ f (x) = Jb0− f (x) = f (x). In the case α = 1, the Riemann-Liouville fractional integral reduces to the classical and usual integral. In [28], Sarikaya et al. established the following interesting inequalities of Hermite-Hadamard type involving Riemann-Liouville fractional integrals. Theorem 1.1 Let f : [a, b] → R be a positive function with 0 ≤ a < b and f ∈ L1 [a, b]. If f is convex function on [a, b], then the following inequalities for fractional integrals hold: a + b Γ(α + 1) α f (a) + f (b) ≤ [Ja+ f (b) + Jbα− f (a)] ≤ f (1.2) α 2 2(b − a) 2 Observe that for α = 1, the inequalities (1.2) reduces to the classical HermiteHadamard inequality (1.1). For some recent results associated with the fractional integral inequalities, one can consult [1, 2, 5, 8–10, 15, 28, 30]. In the recently published article [25] by Qaisar et al., they obtained RiemannLiouville fractional Hadamard-type integral inequalities for mappings whose absolute value of first derivatives are preinvex, and in the paper [11] Dragomir et al. also found some Hadamard-type fractional integral inequalities for differentiable mappings whose absolute value of second derivatives are convex. Motivated and inspired by this idea, in the present paper, by discovering a Riemann-Liouville fractional integral identity involving twice differentiable preinvex mappings, the authors establish the right-sided new Hermite-Hadamard type inequalities via Riemann-Liouville fractional integrals for α-preinvex functions. The new fractional integral inequalities are then applied to some special means.

2

Main Results

To derive main results in this section, we prove the following Lemma. Lemma 2.1 Let A ⊆ R be an open invex subset with respect to η : A × A → R and Let a, b ∈ A with a < a + η(b, a). Assume that f : A → R be a twice differentiable mapping. If f 00 is preinvex on A and f 00 is integrable on the ηpath Pac : c = a + η(b, a), then the following identity for Riemann-Liouville fractional integral with α > 0 holds:

 f (a) + f a + η(b, a) Γ(α + 1)  α α − α Ja+ f a + η(b, a) + J(a+η(b,a)) − f (a) 2 2η (b, a) (2.1) Z 1 2 α+1
η (b, a) 1−t − (1 − t)α+1 00 f a + tη(b, a) dt, = 2 α+1 0 3

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where Γ(α) =

R∞ 0

e−t uα−1 du.

Proof. Set η 2 (b, a) 2

I=

Z 0

1


1 − tα+1 − (1 − t)α+1 00 f a + tη(b, a) dt. α+1

Since a, b ∈ A and A is an invex set with respect to η, for every t ∈ [0, 1], we have a + tη(b, a) ∈ A. Integrating by part yields that 
1 η 2 (b, a) 1 − tα+1 − (1 − t)α+1 0 I= f a + tη(b, a) 2 (α + 1)η(b, a) 0  Z 1 α α
−(α + 1)t + (α + 1)(1 − t) 0 − f a + tη(b, a) dt (α + 1)η(b, a) 0  α 2
1 η (b, a) t − (1 − t)α = f a + tη(b, a) 2 η 2 (b, a) 0  Z 1 α−1 α−1
αt + α(1 − t) f a + tη(b, a) dt − η 2 (b, a) 0
Z 1   
f (a) + f a + η(b, a) α = − tα−1 + (1 − t)α−1 f a + tη(b, a) dt . 2 2 0 Let u = a + tη(b, a), then du = η(b, a)dt, and using the reduction formula Γ(α + 1) = αΓ(α)(α > 0) for Euler gamma function, we have α 2

Z 0

1


Γ(α + 1) α tα−1 f a + tη(b, a) dt = α J − f (a) 2η (b, a) (a+η(b,a))

and similarly we get Z

α 1 Γ(α + 1) α Ja+ f a + η(b, a) . (1 − t)α−1 f a + tη(b, a) dt = α 2 0 2η (b, a) Thus, we have conclusion (2.1). Remark 2.1 Applying Lemma 2.1 for η(b, a) = b−a, we can obtain the Lemma 2.1 in [31], which may be discovered also in [22]. Furthermore, let α = 1, we can get lemma 1 in [24]. With the help of Lemma 2.1, new upper bound for the right-hand side of (1.2) for α-preinvex functions via the Riemann-Liouville fractional integral is presented in the following theorem. Theorem 2.1 Let A ⊆ R be an open invex subset with respect to η : A × A → R and a, b ∈ A with a < a + η(b, a). Suppose that f : A → R be a twice differentiable mapping and f 00 is integrable on the η-path Pac : c = a + η(b, a).

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If |f 00 | is α-preinvex on A then the following inequality for fractional integrals with 0 < α ≤ 1 holds:
f (a) + f a + η(b, a)
 Γ(α + 1)  α α f a + η(b, a) + J f (a) − J + − (a+η(b,a)) 2 2η α (b, a) a   η 2 (b, a)  2α2 + α − 2 ≤ + β(α + 1, α + 2) |f 00 (a)| 2(α + 1) (α + 2)(2α + 2)   1  + − β(α + 1, α + 2) |f 00 (b)| . 2α + 2 (2.2) Proof. Since a + tη(b, a) ∈ A for every t ∈ [0, 1], by using the properties of modulus on Lemma 2.1, we can obtain that
f (a) + f a + η(b, a)
 Γ(α + 1)  α α − α J + f a + η(b, a) + J(a+η(b,a))− f (a) 2 2η (b, a) a Z
η 2 (b, a) 1 1 − tα+1 − (1 − t)α+1 00 ≤ |f a + tη(b, a) |dt. 2 α+1 0 Using the α-preinvexity of |f 00 |, we have 1

1 − tα+1 − (1 − t)α+1
00 |f a + tη(b, a) |dt α+1 0 Z 1   1 ≤ 1 − tα+1 − (1 − t)α+1 (1 − tα )|f 00 (a)| + tα |f 00 (b)| dt α+1 0   2α2 + α − 2 1 + β(α + 1, α + 2) |f 00 (a)| ≤ α + 1 (α + 2)(2α + 2)   1  + − β(α + 1, α + 2) |f 00 (b)| . 2α + 2 Z

To prove the second inequality, we used the following fact that Z 1
2α2 + α − 2 1 − tα+1 − (1 − t)α+1 − tα + t2α+1 dt = , (α + 2)(2α + 2) 0 Z 1 1 , (tα − t2α+1 )dt = 2α +2 0 and Z

1

tα (1 − t)α+1 dt = β(α + 1, α + 2),

0

where the Beta function, Z β(x, y) =

1

tx−1 (1 − t)y−1 dt, ∀x, y > 0,

0

5

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which completes the proof. Another Riemann-Liouville fractional Hermit-Hadamard-type inequality for powers in terms of the second derivatives is obtained below. Theorem 2.2 Let A ⊆ R be an open invex subset with respect to η : A×A → R and a, b ∈ A with a < a+η(b, a). Suppose that f : A → R be a twice differentiable mapping and f 00 is integrable on the η-path Pac : c = a + η(b, a). Assume that q ∈ R, q ≥ 1 such that |f 00 |q is α-preinvex on A, then the following inequality for fractional integrals with 0 < α ≤ 1 holds:
f (a) + f a + η(b, a) 
Γ(α + 1)  α α − α J + f a + η(b, a) + J(a+η(b,a))− f (a) 2 2η (b, a) a  q1 η 2 (b, a)  1  q1  α 1 ≤ 1 − qα |f 00 (a)|q + |f 00 (b)|q . 2(α + 1) 2 α+1 α+1 (2.3) Proof. Since a + tη(b, a) ∈ A for every t ∈ [0, 1], by using the properties of modulus on Lemma 2.1 and using the well-known power-mean integral inequality for q ≥ 1, we can obtain that
f (a) + f a + η(b, a)
 Γ(α + 1)  α α − J f a + η(b, a) + J f (a) + − (a+η(b,a)) 2 2η α (b, a) a Z
η 2 (b, a) 1 1 − tα+1 − (1 − t)α+1 00 ≤ |f a + tη(b, a) |dt 2 α+1 0 1Z    q1 Z 1− 1 1 q
q
q η 2 (b, a) ≤ 1dt 1 − tα+1 − (1 − t)α+1 f 00 a + tη(b, a) dt 2(α + 1) 0 0 Z 1   q1  2
q η (b, a) 1 00 ≤ 1 − qα |f a + tη(b, a) dt . 2(α + 1) 0 2 To prove the third inequality above, we used the following inequality
q 1 − (1 − t)α+1 − tα+1 ≤ 1 − [(1 − t)α+1 + tα+1 ]q ≤ 1 − (2−α )q 1 ≤ 1 − qα 2 for any t ∈ [0, 1] with q ≥ 1, and also using the α-preinvexity of |f 00 |q , that is Z 0

1


q |f 00 a + tη(b, a) dt ≤

Z

1

  (1 − tα )|f 00 (a)|q + tα |f 00 (b)|q dt

0

α 1 = |f 00 (a)|q + |f 00 (b)|q . α+1 α+1

(2.4)

Therefore, we can get the required results (2.3).

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Corollary 2.1 With the same assumptions given in Theorem 2.2, if |f 00 (x)| ≤ M on [a, a + η(b, a)], we can deduce that
f (a) + f a + η(b, a)
 Γ(α + 1)  α α − α J + f a + η(b, a) + J(a+η(b,a))− f (a) 2 2η (b, a) a M η 2 (b, a)  1  q1 ≤ 1 − qα . 2(α + 1) 2 Another similar result may be presented in the following theorem. Theorem 2.3 Let A ⊆ R be an open invex subset with respect to η : A×A → R and a, b ∈ A with a < a+η(b, a). Suppose that f : A → R be a twice differentiable mapping and f 00 is integrable on the η-path Pac : c = a + η(b, a). Assume that p such that |f 00 |q is α-preinvex on A, then the following p ∈ R, p > 1 with q = p−1 inequality for fractional integrals with 0 < α ≤ 1 holds:
f (a) + f a + η(b, a) 
Γ(α + 1)  α α − α J + f a + η(b, a) + J(a+η(b,a))− f (a) 2 2η (b, a) a  q1 η 2 (b, a)  pα + p − 1  p1  α 1 ≤ |f 00 (a)|q + |f 00 (b)|q . 2(α + 1) pα + p + 1 α+1 α+1 (2.5) Proof. Since a + tη(b, a) ∈ A for every t ∈ [0, 1], by using the properties of modulus on Lemma 2.1 and making use of the well-known H¨older’s integral inequality for q > 1, we can obtain that
f (a) + f a + η(b, a)
 Γ(α + 1)  α α − J f a + η(b, a) + J f (a) + (a+η(b,a))− 2 2η α (b, a) a Z
η 2 (b, a) 1 1 − tα+1 − (1 − t)α+1 00 ≤ |f a + tη(b, a) |dt 2 α + 1 0 Z  p1  Z 1  q1
η 2 (b, a)  1 α+1 α+1 p ≤ |1 − t − (1 − t) | dt |f 00 a + tη(b, a) |q dt 2(α + 1) 0 0 Z
q  q1 η 2 (b, a)  pα + p − 1  p1  1 00 |f a + tη(b, a) | dt , ≤ 2(α + 1) pα + p + 1 0 where we use the following inequality 1 − (1 − t)α+1 − tα+1


p

≤ 1 − (1 − t)p(α+1) − tp(α+1)

(2.6)

for any t ∈ [0, 1], which follows from (A − B)p ≤ Ap − B p

(2.7)

for any A > B ≥ 0 and p > 1. By applying (2.4) and (2.6), we can get (2.5). Hence the proof is completed.

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Corollary 2.2 With the same assumptions given in Theorem 2.3, if |f 00 (x)| ≤ M on [a, a + η(b, a)], we obtain that
f (a) + f a + η(b, a)
 Γ(α + 1)  α α − α J + f a + η(b, a) + J(a+η(b,a))− f (a) 2 2η (b, a) a M η 2 (b, a)  pα + p − 1  p1 ≤ . 2(α + 1) pα + p + 1 A different approach leads to the following result. Theorem 2.4 Let A ⊆ R be an open invex subset with respect to η : A×A → R and a, b ∈ A with a < a+η(b, a). Suppose that f : A → R be a twice differentiable mapping and f 00 is integrable on the η-path Pac : c = a + η(b, a). Assume that p such that |f 00 |q is α-preinvex on A, then the following p ∈ R, p > 1 with q = p−1 inequality for fractional integrals with 0 < α ≤ 1 holds:
f (a) + f a + η(b, a) 
Γ(α + 1)  α α − α J + f a + η(b, a) + J(a+η(b,a))− f (a) 2 2η (b, a) a   η 2 (b, a)  α 1− q1  2α2 + α − 2 ≤ + β(α + 1, α + 2) |f 00 (a)|q 2(α + 1) α + 2 (α + 2)(2α + 2)  q1  1  00 q + − β(α + 1, α + 2) |f (b)| . 2α + 2 (2.8) Proof. Since a + tη(b, a) ∈ A for every t ∈ [0, 1], by utilizing the properties of modulus on Lemma 2.1 and using the H¨older’s integral inequality for q > 1 , we can obtain that
f (a) + f a + η(b, a) 
Γ(α + 1)  α α f (a) f a + η(b, a) + J − J + (a+η(b,a))− 2 2η α (b, a) a Z
η 2 (b, a) 1 1 − tα+1 − (1 − t)α+1 00 ≤ |f a + tη(b, a) |dt 2 α + 1 0 Z 1  1 2
1− q η (b, a) 1 − tα+1 − (1 − t)α+1 dt ≤ 2(α + 1) 0 Z 1  q1
00
q α+1 α+1 × 1−t − (1 − t) |f a + tη(b, a) | dt 0 2

η (b, a)  α 1− q1 = 2(α + 1) α + 2

Z

1 α+1

1−t

α+1

− (1 − t)




|f

00

 q1
q a + tη(b, a) | dt .

0

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Using the α-preinvexity of |f 00 |q , i.e. the inequality (2.4), we have Z

1



1 − tα+1 − (1 − t)α+1 |f 00 a + tη(b, a) |q dt

0

Z ≤

1

1 − tα+1 − (1 − t)α+1




 (1 − tα )|f 00 (a)|q + tα |f 00 (b)|q dt

0

 2α2 + α − 2  = + β(α + 1, α + 2) |f 00 (a)|q (α + 2)(2α + 2)  1  + − β(α + 1, α + 2) |f 00 (b)|q . 2α + 2 Thus, we get the desired inequality (2.8). Corollary 2.3 With the same assumptions given in Theorem 2.4, if |f 00 (x)| ≤ M on [a, a + η(b, a)], we obtain that
f (a) + f a + η(b, a) 
Γ(α + 1)  α α − α J + f a + η(b, a) + J(a+η(b,a))− f (a) 2 2η (b, a) a M αη 2 (b, a) ≤ . 2(α + 1)(α + 2) Finally we shall prove the following result. Theorem 2.5 Suppose that all the assumptions of Theorem 2.4 are satisfied. Then the following inequalities hold:
f (a) + f a + η(b, a)
 Γ(α + 1)  α α − α J + f a + η(b, a) + J(a+η(b,a))− f (a) 2 2η (b, a) a q−1 η 2 (b, a) h (q − p)α − p + 1 i q ≤ 2(α + 1) (q − p)α + 2q − p − 1   αp + α + 1 2 × − + β(α + 1, p(α + 1) + 1) |f 00 (a)|q (α + 1)(p + 1) p(α + 1) + 1  i h p − β(α + 1, p(α + 1) + 1) |f 00 (b)|q . + (α + 1)(p + 1) (2.9) Proof. Since a + tη(b, a) ∈ A for every t ∈ [0, 1], by using the properties of modulus on Lemma 2.1 and making use of the well-known H¨older’s integral inequality for q > 1, we can obtain that

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f (a) + f a + η(b, a) 
Γ(α + 1)  α α − α J + f a + η(b, a) + J(a+η(b,a))− f (a) 2 2η (b, a) a Z
η 2 (b, a) 1 1 − tα+1 − (1 − t)α+1 00 ≤ |f a + tη(b, a) |dt 2 α+1 0 Z 1  q−p h i q−1 2 η (b, a) q−1 q ≤ 1 − tα+1 − (1 − t)α+1 dt 2(α + 1) 0 p hZ 1 i q1
× 1 − tα+1 − (1 − t)α+1 |f 00 a + tη(b, a) |q dt 0 q−1 η 2 (b, a) h (q − p)α − p + 1 i q ≤ 2(α + 1) (q − p)α + 2q − p − 1 hZ 1 p i q1
× 1 − tα+1 − (1 − t)α+1 |f 00 a + tη(b, a) |q dt

0

Using the α-preinvexity of |f 00 |q , we have Z

1



1 − tα+1 − (1 − t)α+1

p



1 − tα+1 − (1 − t)α+1

p 



1 − t(α+1)p − (1 − t)(α+1)p



 1 − (1 − t)(α+1)p − t(α+1)p − tα + tα (1 − t)(α+1)p + tα+(α+1)p |f 00 (a)|q dt


|f 00 a + tη(b, a) |q dt

0

Z

1

≤

 (1 − tα )|f 00 (a)|q + tα |f 00 (b)|q dt

0

Z

1

≤

  (1 − tα )|f 00 (a)|q + tα |f 00 (b)|q dt

0

Z ≤

1

0

Z 1  + tα − tα (1 − t)(α+1)p − tα+(α+1)p |f 00 (b)|q dt   0 2 αp + α + 1 − + β(α + 1, p(α + 1) + 1) |f 00 (a)|q = (α + 1)(p + 1) p(α + 1) + 1 h i p − β(α + 1, p(α + 1) + 1) |f 00 (b)|q . + (α + 1)(p + 1) Here, we use (1 − (1 − t)α+1 − tα+1 )q ≤ 1 − (1 − t)q(α+1) − tq(α+1)

(2.10)

for any t ∈ [0, 1], which follows from (A − B)q ≤ Aq − B q

(2.11)

for any A > B ≥ 0 and q ≥ 1. Thus, we get the desired inequality (2.9). 10

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Corollary 2.4 From Theorems (2.2),(2.3),(2.4) and (2.5), we have
f (a) + f a + η(b, a)
 Γ(α + 1)  α α J + f a + η(b, a) + J(a+η(b,a))− f (a) − α 2 2η (b, a) a ≤ min{K1 , K2 , K3 , K4 }, where  q1 1  q1  α η 2 (b, a)  1 1 − qα |f 00 (a)|q + |f 00 (b)|q , 2(α + 1) 2 α+1 α+1 1    q1 2 α η (b, a) pα + p − 1 p 1 K2 = |f 00 (a)|q + |f 00 (b)|q , 2(α + 1) pα + p + 1 α+1 α+1   η 2 (b, a)  α 1− q1  2α2 + α − 2 K3 = + β(α + 1, α + 2) |f 00 (a)|q 2(α + 1) α + 2 (α + 2)(2α + 2)  q1  1  00 q , + − β(α + 1, α + 2) |f (b)| 2α + 2 K1 =

q−1

K4 =

3

η 2 (b, a) h (q − p)α − p + 1 i q 2(α + 1) (q − p)α + 2q − p − 1   αp + α + 1 2 × − + β(α + 1, p(α + 1) + 1) |f 00 (a)|q (α + 1)(p + 1) p(α + 1) + 1  h i p 00 q + − β(α + 1, p(α + 1) + 1) |f (b)| . (α + 1)(p + 1)

Applications to special means

In the following we give certain generalizations of some notions for a positive valued function of a positive variable. Definition 3.1 [6] A function M : R2+ → R+ , is called a Mean function if it has the following properties: (1) Homogeneity: M (ax, ay)=aM (x, y), for all a>0, (2) Symmetry: M (x, y) = M (y, x), (3) Reflexivity: M (x, x) = x, (4) Monotonicity: If x ≤ x0 and y ≤ y 0 , then M (x, y) ≤ M (x0 , y 0 ), (5) Internality: min{x, y} ≤ M (x, y) ≤ max{x, y}. We consider some means for arbitrary positive real numbers a > 0 and b > 0, √ 2ab define A := A(a, b) = a+b , G := G(a, b) = ab, H := H(a, b) = a+b , 2  r r  r1 ,r ≥ 1 Pr := Pr (a, b) = a +b 2

I := I(a, b) =

 

1 e



 a,

bb aa

1  b−a



, a 6= b, a = b,

L := L(a, b) =

b−a ln b−ln a ,

a,

a 6= b, a = b,

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and  h i1  bp+1 −ap+1 p    (p+1)(b−a) , p 6= 0, −1, and a 6= b, L(a, b), p = −1 and a 6= b, Lp := Lp (a, b) =   I(a, b), p = 0 and a 6= b,   a, a = b. It is well known that Lp is monotonic nondecreasing over p ∈ R, with L−1 := L and L0 := I. In particular, we have the following inequality H ≤ G ≤ L ≤ I ≤ A. Now, let a and b be positive real numbers such that a < b. Consider the function M := M (a, b) : [a + η(b, a)] × [a, a + η(b, a)] → R+ , which is one of the above mentioned means, therefore one can obtain various inequalities for these means below: Setting η(b, a) = M (b, a) in (2.2), (2.3), (2.5), (2.8) and (2.9), one can derive the following interesting inequalities concerning means:
f (a) + f a + M (b, a) 
Γ(α + 1)  α α − J + f a + M (b, a) + J(a+M (b,a))− f (a) (1) 2 2M α (b, a) a   M 2 (b, a)  2α2 + α − 2 ≤ + β(α + 1, α + 2) |f 00 (a)| 2(α + 1) (α + 2)(2α + 2)   1  00 + − β(α + 1, α + 2) |f (b)| , 2α + 2
f (a) + f a + M (b, a)
 Γ(α + 1)  α α (2) − J + f a + M (b, a) + J(a+M (b,a))− f (a) 2 2M α (b, a) a  q1 1 M 2 (b, a)  1  q1  α 1 − qα |f 00 (a)|q + |f 00 (b)|q , ≤ 2(α + 1) 2 α+1 α+1
f (a) + f a + M (b, a)
 Γ(α + 1)  α α (3) − J + f a + M (b, a) + J(a+M (b,a))− f (a) 2 2M α (b, a) a   q1 M 2 (b, a)  pα + p − 1  p1 α 1 ≤ |f 00 (a)|q + |f 00 (b)|q dt , 2(α + 1) pα + p + 1 α+1 α+1
f (a) + f a + M (b, a)
 Γ(α + 1)  α α (4) − J f a + M (b, a) + J f (a) + − (a+M (b,a)) 2 2M α (b, a) a  1     1− q M 2 (b, a) α 2α2 + α − 2 ≤ + β(α + 1, α + 2) |f 00 (a)|q 2(α + 1) α + 2 (α + 2)(2α + 2)  q1  1  00 q + − β(α + 1, α + 2) |f (b)| , 2α + 2

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and
f (a) + f a + M (b, a) 
Γ(α + 1)  α α (5) − J + f a + M (b, a) + J(a+M (b,a))− f (a) 2 2M α (b, a) a q−1 M 2 (b, a) h (q − p)α − p + 1 i q ≤ 2(α + 1) (q − p)α + 2q − p − 1   αp + α + 1 2 × − + β(α + 1, p(α + 1) + 1) |f 00 (a)|q (α + 1)(p + 1) p(α + 1) + 1  h i p + − β(α + 1, p(α + 1) + 1) |f 00 (b)|q . (α + 1)(p + 1) Letting M = A, G, H, Pr , I, L, Lp in (1), (2), (3), (4) and (5), we get the inequalities involving means for a particular choice of a twice differentiable α-preinvex function f , and the details are left to the interested reader.

Acknowledgments This work was partially supported by the National Natural Science foundation of China under Grant No. 61374028.

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[22] Y. M. Liao, J. H. Deng, and J. R. Wang, Riemann-Liouville fractional Hermite-Hadamard inequalities. Part II: for twice differentiable geometricarithmetically s-convex functions, J. Inequal. Appl., 2013, Article Number 517, 13 pages, (2013). [23] M. A. Noor, K. I. Noor, M. U. Awan, and S. Khan, Hermite-Hadamard type inequalities for differentiable hϕ -preinvex functions, Arab. J. Math., 4, 63-76, (2015). ¨ [24] M. E. Ozdemir, M. Avci, and E. Set, On some inequalities of HermiteHadamard type via m-convexity, Appl. Math. Lett., 23, 1065-1070, (2010). [25] S. Qaisar, M. Iqbal, and M. Muddassar, New Hermite-Hadamard’s inequalities for preinvex functions via fractional integrals, J. Comput. Anal. Appl., 20, no. 7, 1318-1328, (2016). [26] M.-H. Qu, W.-J. Liu, and J. Park, Some new Hermite-Hadamard-type inequalities for geometric-arithmetically s-convex functions, WSEAS Trans. Math., 13, 452-461, (2014). [27] M. Z. Sarikaya and M. E. Kiris, Some new inequalities of HermiteHadamard type for s-convex functions, Miskolc Math. Notes, 16, no. 1, 491-501, (2015). [28] M. Z. Sarikaya, E. Set, H. Yaldiz, and N. Ba¸sak, Hermite-Hadamard’s inequalities for fractional integrals and related fractional inequalities, Math. Comput. Modelling, 57, no. 9-10, 2403-2407, (2013). [29] Y. Wang, M. M. Zheng, and F. Qi, Integral inequalities of HermiteHadamard type for functions whose derivatives are α-preinvex, J. Inequal. Appl., 2014, Article Number 97, 10 pages, (2014). [30] S. -H. Wang and F. Qi, Hermite-Hadamard type inequalities for s-convex functions via Riemann-Liouville fractional integrals, J. Comput. Anal. Appl., 22, no. 6, 1124-1134, (2017). [31] J. R. Wang, X. Z. Li, M. Feˇckan, and Y. Zhou, Hermite-Hadamard-type inequalities for Riemann-Liouville fractional integrals via two kinds of convexity, Appl. Anal., 92, no. 11, 2241-2253, (2013). [32] T. Weir and B. Mond, Pre-invex functions in multiple objective optimization, J. Math. Anal. Appl., 136, 29-38, (1998). [33] X. M. Yang, X. Q. Yang, and K. L. Teo, Generalized invexity and generalized invariant monotonicity, J. Optim. Theory Appl., 117, no. 3, 607-625, (2003). [34] Z. Q. Yang, Y. J. Li, and T. S. Du, A generalization of Simpson type inequality via differentiable functions using (s, m)-convex functions, Ital. J. Pure Appl. Math., 35, 327-338, (2015). 15

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CONVOLUTION PROPERTIES FOR CERTAIN SUBCLASSES OF MEROMORPHIC BOUNDED FUNCTIONS HANAN DARWISH, ABD EL-MONEIM LASHIN, AND SULIMAN SOWILEH Abstract. By making use of the Hadamard product, we derive necessary and su¢ cient conditions for certain meromorphic function to be in the class S ( ; ; M )( 2 C = Cnf0g; M 1; 2 C) which uni…es the classes of bounded starlike and convex functions of complex order. By using Al-Oboudi operator a more general class S (n; ; ; M ) related to S ( ; ; M ) is also considered. Several properties of the class S (n; ; ; M ) are also obtained.

AMS (2010) Subject Classi…cation: 30C45, 30C50. Key Words. Univalent meromorphic functions, bounded starlike functions of complex order, bounded convex functions of complex order, -starlike functions, Hadamard product, subordination. 1. Introduction Let C be the complex plane and let functions having the form: (1.1)

f (z) = z

1

denote the class of all meromorphic

+

1 X

ak z k ;

k=0

which are analytic in the punctured unit disc E = fz : z 2 C;

0 < jzj < 1g =: Enf0g:

The familiar Hadamard product (or convolution) of two functions f (z) given by (1.1 and g(z) is given by (1.2)

g(z) = z

1

+

1 X

bk z k ;

k=0

is de…ned by (1.3)

(f

g)(z) = z

1

+

1 X

ak bk z k = (g f )(z):

k=0

An analytic function f is said to be subordinate to another analytic function g; written symbolically as follows: f (z)

g(z) (z 2 E);

if there exists a function !(z), analytic in E with such that

!(0) = 0 and j!(z)j < 1 (z 2 E); f (z) = g(w(z)) (z 2 E): 1

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Furthermore, if the function g(z) is univalent in E; then we have the following equivalence, (cf., e.g., [5], [9], [10] ): f (z)

g(z) , f (0) = g(0) and f (E)

g(E):

Making use of the principal of subordination between analytic functions, Aouf [2] de…ned the subclasses S ( ; M ) and C( ; M ) of the class as follows: (1.4) [ (1 + m) m] z + 1 zf 0 (z) 1 S ( ; M) = f 2 : 2C ; m=1 ; M f (z) 1 mz M

1; z 2 E

or, equivalently, zf 0 (z) f (z)

1 (1.5)

M :

0

z zf (z) :

2 C;

0

0

+ (1 + )zf (z)

zf 0 (z) + (1 + )f (z) 2C ; m=1

259

1 ; M M

9 > m] z + 1 =

[ (1 + m) 1 mz

1; z 2 E

> ;

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C O N VO LUTIO N FO R C ERTAIN SU BC LASSES O F M ERO M O RPH IC FU N C TIO N S 3

One can easily show that f 2 S ( ; ; M ) if and only if there is a function g 2 S (1; M ) such that (zg(z)) z It was shown in [8] g 2 S (1; M ) if and only if for z 2 E 0

(1.11)

zf (z) + (1 + )f (z) =

0

(1.12)

zg (z) 1 + !(z) = ; g(z) 1 m!(z)

!(0) = 0; j!(z)j < 1 and m = 1

1 : M

Thus from (1.11) and (1.12) follows that f 2 S ( ; ; M ) if and only if for M 1; 2 C and z 2 E 0

0

z zf (z) (1.13)

0

+ (1 + )zf (z) =

0

zf (z) + (1 + )f (z)

By specializing ; authors:

[ (1 + m) m] !(z) + 1 : 1 m!(z)

and M , we get the following subclasses studied by earlier

Remark 1. (i) S (0; ; 1) =: S ( ); with 2 C ; (see Aouf [2]); (ii) S ( 1; ; 1) =: C( ); with 2 C ; (see Aouf [2]); (iii) S (0; 1 a; M ) =: SM (a); with 0 a < 1; (see Kaczmarski [8]); (iv)S ( 1; 1 a; M ) =: CM (a); with 0 a < 1; (see Aouf [2]); (v) S (0; 1; 1) =: S (1); with 0 a < 1; (see Clunie [7]); (vi) S ( 1; 1; 1) =: C(1); with 0 a < 1; (see Aouf [2]); (vii) S (0; 1 a; 1) =: S (1 a); with 0 a < 1; (see Kaczmarski [8] and Pommerenke [11]); (viii) S ( 1; 1 a; 1) =: C(1 a);with 0 a < 1; (see Aouf [2]); a < 1; j j (ix) S (0; (1 a)e i cos ; M ) =: SM (a; ); with 0 2 ); (see Kaczmarski [8]); (x) S ( 1; (1 a)e i cos ; M ) =: CM (a; );with 0 a < 1; j j 2 ); (see Aouf [2]); (xi) S (0; (1 a)e i cos ; 1) =: S (a; ); with 0 a < 1; j j 2 ); (see Kaczmarski [8]); (xii) S ( 1; (1 a)e i cos ; 1) =: C(a; );with 0 a < 1; j j 2 ); (see Aouf [2]). For f (z) 2 , Al-Oboudi and Al-Zkeri [1] de…ned the following operator Dn f (n 2 N0 = N [ f0g = f0; 1; 2; 3; : : :g) which is called the Al-Oboudi operator: D0 f (z)

= f (z);

D1 f (z)

= =

(1.14)

(z 2 f (z))0 ; 0 z (1 + )f (z) + zf 0 (z) = D f (z); (1

)f (z) +

D2 f (z)

= D D1 f (z):

Dn f (z)

= D

Dn

1

f (z) ; n 2 N

From (1.1) and (1.14) we get (1.15)

Dn f (z) = z

1

+

1 X

k=0

n

[ (k + 1) + 1] ak z k (z 2 E ) :

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H AN AN DARW ISH , ABD EL-M O N EIM LASH IN , AND SU LIM AN SOW ILEH

With the aid of Al-Oboudi operator, we introduce the class S (n; ; ; M ) as follows: De…nition 2. Let the function f (z) de…ned by (1.1) be in the class S (n; ; ; M ) if and only if for …xed M; ! 0 0 0 z z (Dn f (z)) + (1 + )z (Dn f (z)) 1 1 1+ M < M (z 2 E) 0 z (Dn f (z)) + (1 + )Dn f (z) where, M

1;

2C ;

2 C and n 2 N0 :

We note that S (0; ; ; M ) S ( ; ; M ). In this paper we will investigate some convolution properties of the class S ( ; ; M ). Using these properties, we …nd the necessary and su¢ cient condition, and containment property for the subclass S (n; ; ; M ). The results obtained here extend some known results in [3], [4] and [6]. 2. Convolution properties

1;

Unless otherwise mentioned, we assume throughout this article that 2 C and n 2 N0 :

2 C ;M

Theorem 1. The function f (z) de…ned by (1.1) be in the class S ( ; ; M ) if and

only if (2.1)

z f (z)

where C = C =

(1 + ) e

i m (1+m) ;

2(C 1)z 2 + 3z (C 1) z + 1 + z(1 z)2 z(1 z)3

1

6= 0 (z 2 E)

2 [0; 2 ):

Proof. First suppose f (z) de…ned by (1.1) is in the class S ( ; ; M ); we have 0

0

(2.2)

0

+ (1 + )zf (z)

z zf (z)

[ (1 + m) m] z + 1 (z 2 E), 1 mz

zf 0 (z) + (1 + )f (z)

0

since the left-hand side of (2.2) is analytic in E, it follows zf (z) + (1 + )f (z) 6= 0 0 for all z 2 E ; i.e. z 2 f (z) + (1 + )zf (z) 6= 0, z 2 E, so (2.1) holds for C = 0. By using the principle of subordination, we can write (2.2) as 0

z zf (z)

0

0

+ (1 + )zf (z)

zf 0 (z) + (1 + )f (z) which is equivalent to h i0 0 z zf (z) + (1 + )f (z) 0

zf (z) + (1 + )f (z)

or (2.3) h z

Since (2.4)

6=

i0 0 zf (z) + (1 + )f (z) 1 f (z) = f (z)

1 z(1

z)

and

=

[ (1 + m) m]!(z) + 1 (z 2 E), 1 m!(z)

[ (1 + m) m]ei + 1 ; 1 mei

mei

h

2 [0; 2 )):

i 0 zf (z) + (1 + )f (z) [ (1 + m)

zf 0 (z) = f (z)

261

( z 2 E;

1 z(1

2 z)2

(1

z)2

m] ei + 1 6= 0:

:

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Applying (2.4) it is not di¢ cult to verify that 0

(2.5)

zf (z) + (1 + )f (z) = f (z)

Since z(f

0

z

h

0

zf (z) + (1 + )f (z)

z[f (z) f (1 + )(1

z) (1 + m)ei

+ (1

1)z + 1 : z)2

zg ; we can write

Using (2.5) and (2.6) in (2.3), we get (2.7)

z(1

0

g) = f

(2.6)

(

2 (1

i0

1)z 2 + 3z z(1 z)3

2(

= f (z)

z)[ (1 + m)ei + [ (1 + m)

mei )z 2 + 2(1

1

:

m]ei + 1 z]

z) (1 + m)ei zg=z(1

z)3 ] 6= 0:

The left hand side of (2.7) may be written as

z[f (z) f (1 + )(1 i

+

(1 + m)e

3

z)[ (1 + m)ei + 1 i

(1 + m)e z

2

1

mei i

(1 + m)ei 2

me

z +2

z]

(1 + m)ei z 2 ]g=z(1

z)3 ]:

Equation (2.7) can be rewritten in the form 2

z 4f (z)

8 < :

e i m (1+m)

(1 + )

z(1

i

m 2( e (1+m)

1 z+1 +

z)2

z(1

1)z 2 + 3z z)3

93 1= 5 6= 0 ;

where z 2 E; 2 [0; 2 ): Thus we have the …rst part of the proof. 0 (ii) Conversely, since (2.1) holds for C = 0, then z 2 f (z) + (1 + )zf (z) 6= 0 for z

h

i0 0 zf (z)+(1+ )f (z)

all z 2 E, hence the function '(z) = is analytic in E (i.e. it zf 0 (z)+(1+ )f (z) is regular at z0 = 0; with '(0) = 1). Since (2.7) is equivalent to (2.1), we have h i0 0 z zf (z) + (1 + )f (z) [ (1 + m) m]ei + 1 6= (z 2 E; 2 [0; 2 )): (2.8) 0 zf (z) + (1 + )f (z) 1 mei Assume that z '(z) =

h

0

zf (z) + (1 + )f (z) 0

zf (z) + (1 + )f (z)

i0

;

(z) =

[ (1 + m) m]ei + 1 : 1 mei

The relation (2.8) means that '(E) \ (@E) = ;: Thus, the simply connected domain '(E) is included in a connected component of Cn (@E): From this, using the fact that '(0) = (0) and the univalence of the function , it follows that '(z) (z); this implies that f (z) 2 S ( ; ; M ): Thus the proof of Theorem 1 is completed. Remark 2. (i) Taking = 0 in Theorem 1, we obtain the result obtained by Aouf [4, Theorem 2.1]. (ii) Taking = 1 in Theorem 1, we obtain the result obtained by Aouf [4, Theorem 2.3]. (iii) Taking = 0 and m = 1 in Theorem 1, we obtain the result obtained by Bulboac¼ a et al. [6, Theorem 1, with A = 1 and B = 1] and Aouf et al. [3, Theorem 4, with = 0; A = 1 and B = 1].

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H AN AN DARW ISH , ABD EL-M O N EIM LASH IN , AND SU LIM AN SOW ILEH

(iv) Taking = 0; = m = 1 and ei = x in Theorem 1, we obtain the result obtained by Ponnusamy [12, Theorem 4, with = 0; A = 1 and B = 1]. (iiv) Taking = 0; m = 1; = (1 )e i cos ( 2 R; j j < 1) and 2; 0 i e = x in Theorem 1, we obtain the result obtained by Ravichandran et al. [13, Theorem 1.2 with p = 1]. Theorem 2. A necessary and su¢ cient condition for the function f (z) de…ned by (1.1) to be in the class S (n; ; ; M ) is that (2.9)

1+

1 X

k=0

n

[ (k + 1) + 1] [ (k + 1) + 1]ak z k+1 6= 0

and (2.10)

1+

1 X

[ (k + 1) + 1]n

k=0

for all

h

i

(k+1)[e i m]+ (1+m) (1+m)

+ k]ak z k+1 6= 0

[(1 +

2 [0; 2 ) and z 2 E:

Proof. From Theorem 1, we have f (z) 2S (n; ; ; M ) if and only if (2.11) 2(C 1)z 2 + 3z 1 (C 1) z + 1 + z Dn f (z) (1 + ) 6= 0 (z 2 E) z(1 z)2 z(1 z)3 for all C = C = equations

e

i m (1+m) ;

1

(2.12)

z (1

z)

(0

1

=z

< 2 ) ; and also for C = 0. From (1.15) and the

+

1 X

1

zk ;

z(1

k=0

z)2

=z

1

+

1 X

(k + 2) z k ;

k=0

it is not di¢ cult to show that (2.10) holds for C = 0 i¤ (2.9) satis…ed. The left hand side of (2.11) may be written as (2.13) C 4C 1 2(C 1) 1 C 2C + + z Dn f (z) (1 + ) + 2 3 2 z (1 z) z(1 z) z(1 z) z(1 z) z (1 z)

:

Using (1.15), (2.12) and the formula 1 z(1

z)3

=z

1

+

1 X (k + 2) (k + 3)

k=0

2

zk

Equation (2.13) can be written as 1+

1 X

n

[ (k + 1) + 1]

k=0

h

(k+1)[e

i

m]+ (1+m) (1+m)

i

[1 +

+ k]ak z k+1 :

Thus, the proof of Theorem 2 is completed. Theorem 3. If the function f (z) given by (1.1) and satisfy the inequality (2.14)

1 X

(k + 1 + j j)[ (k + 1) + 1][ (k + 1) + 1]n jak j

k=0

j j

then f (z) 2 S (n; ; ; M ):

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Proof. Since (k+1)[e

i

(k+1+j j) j j

m]+ (1+m) (1+m)

then 1+

1 X

[ (k + 1) + 1]n

k=0

1 1

1 X

k=0 1 X

k=0

h

(k+1)[e i m]+ (1+m) (1+m)

[ (k + 1) + 1]n (k+1+j j) [ j j

(k+1)[e i m]+ (1+m) (1+m)

i

[ (k + 1) + 1]ak z k+1

[ (k + 1) + 1] jak j jzjk+1

(k + 1) + 1] [ (k + 1) + 1]n jak j > 0

(z 2 E) :

Which implies that inequality (2.14). Thus this completes the proof of Theorem 3.

Theorem 4. For

2 C; we have S (n + 1; ; ; M )

S (n; ; ; M ):

Proof. If f (z) 2 S (n + 1; ; ; M ); then Theorem 2 gives 1+

1 X

n+1

[ (k + 1) + 1]

k=0

and (2.15)

1+

1 X

[ (k + 1) + 1]n+1

k=0

h

(k+1)[e i m]+ (1+m) (1+m)

we can write (2.15) as (2.16) " # " 1 1 X X k+1 1+ [ (k + 1) + 1] z 1+ k=0

But

(2.17)

[ (k + 1) + 1]ak z k+1 6= 0 i

[ (k + 1) + 1] ak z k+1 6= 0

(k+1)[e i m]+ (1+m) (1+m)

n

[ (k + 1) + 1] [ (k + 1) + 1] ak z

k=0

"

1+

1 X

k=0

[ (k + 1) + 1] z

k+1

# "

1+

1 X

k=0

k+1

#

6= 0:

# 1 X 1 k+1 =1+ z k+1 : z [ (k + 1) + 1] k=0

By using the property, if f 6= 0 and g h 6= 0; then f (g h) 6= 0; (2.16) can be written as 1 h i X i m]+ (1+m) (2.18) 1+ [ (k + 1) + 1]n (k+1)[e (1+m) [ (k + 1) + 1] ak z k+1 6= 0: k=0

In view of Theorem 2, we conclude that f (z) 2 S (n; ; ; M ):

References [1] F. M. Al-Oboudi and H. A. Al-Zkeri, On some classes of meromorphic starlike functions de…ned by a di¤erential operator, Global J. Pure Appl. Math., 3(1)(2007), 1–11. [2] M. K. Aouf, Coe¢ cient results for some classes of meromorphic functions, J. Natural Sci. Math., 27(1987), 81-97. [3] M. K. Aouf, A.O. Mostafa and H. M. Zayed, Convolution properties for some subclasses of meromorphic functions of complex order, Abstr. Appl. Anal., 2015(2015), 1-6. [4] M. K. Aouf, A. O. Mostafa and H. M. Zayed, Convolution conditions for some subclasses of meromorphic bounded functions of complex order, Thai J. Math., (2015), 1-10. [5] T. Bulboac¼ a , Di¤erential subordinations and superordinations, Recent Results, House of Scienti…c Book Publ., Cluj-Napoca, 2005. [6] T. Bulboac¼ a , M. K. Aouf, and R. M. El-Ashwah, Convolution properties for subclasses of meromorphic univalent functions of complex order, Filomat, 26(1)(2012), 153–163. [7] J. Clunie, On meromorphic schlicht functions, J. London Math. Soc., 34(1959), 215-216.

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H AN AN DARW ISH , ABD EL-M O N EIM LASH IN , AND SU LIM AN SOW ILEH

[8] J. Kaczmarski, On the coe¢ cients of some classes of starlike functions, Bull. Acad. Polon. Sci. Ser. Math. Astronom. Phy., 17(1969), 495-501. [9] S. S. Miller and P. T. Mocanu, Di¤erential subordination: Theory and Applications, Series on Monographs and Textbooks in Pure and Applied Mathematics, Vol. 225, Marcel Dekker Inc., New York and Basel, 2000. [10] S. S. Miller and P. T. Mocanu, Subordinates of di¤erential superordinations, Complex Variables, 48(10)(2003), 815–826. [11] Ch. Pommerenke, On meromorphic starlike functions, Paci…c J. Math., 13(1963), 221-235. [12] S. Ponnusamy, Convolution properties of some classes of meromorphic univalent functions, Proc. Indian Acad. Sci. (Math. Sci.), 103(1993), 73-89. [13] V. Ravichandran, S. S. Kumar and K. G. Subramanian, Convolution conditions for spirallikeness and convex spirallikenesss of certain p valent meromorphic functions, J. Ineq. Pure Appl. Math., 5(1)(11)(2004), 1-7. Department of Mathematics Faculty of Science, Mansoura University Mansoura, 35516, EGYPT. E-mail address : [email protected] Department of Mathematics Faculty of Science, Mansoura University Mansoura, 35516, EGYPT. E-mail address : [email protected] Department of Mathematics Faculty of Science, Mansoura University Mansoura, 35516, EGYPT. E-mail address : [email protected]

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ON A GENERALIZED DEGENERATE λ-q-DAEHEE NUMBERS AND POLYNOMIALS JIN-WOO PARK

Abstract. In [4], Daehee numbers and polynomials are introduced by T. Kim et al. In this paper, we consider the generalized λ-q-Daehee polynomials by using the bosonic p-adic q-integral and give some relations between the generalized λ-q-Daehee polynomials and special polynomials.

1. Introduction Let d be fixed positive integer and let p be a fixed odd prime number. Throughout this paper, Zp , Qp , and Cp will respectively denote the ring of p-adic rational integers, the field of p-adic rational numbers and the completions of algebraic closure of Qp . The p-adic norm is defined |p|p = p1 . We set [ X = Xd = lim ZdpN Z, X ∗ = (a + dpZp ) , ← − N

0 0 and fp,m (x + k) < 0. Therefore, fp,m (x + 2pk)

= mfp,m (x + (2p − 1)k) + fp,m (x + (p − 1)k) = m2 fp,m (x + (2p − 2)k) + fp,m (x + (p − 1)k) +mfp,m (x + (p − 2)k) = m3 fp,m (x + (2p − 3)k) + fp,m (x + (p − 1)k) +mfp,m (x + (p − 2)k) + m2 fp,m (x + (p − 3)k) .. . = mp fp,m (x + pk) + fp,m (x + (p − 1)k) + · · · − mp−2 fp,m (x + k) + mp−1 fp,m (x) = Fp,m (p + 1)fp,m (x + pk) + Fp,m (1)fp,m (x + (p − 1)k) + · · · − Fp,m (p − 1)fp,m (x + k) + Fp,m (p)fp,m (x),

fp,m (x + (2p + 1)k)

= mfp,m (x + 2pk) + fp,m (x + pk) h = m mp fp,m (x + pk) + fp,m (x + (p − 1)k) i + · · · − mp−2 fp,m (x + k) + mp−1 fp,m (x) + f (x + pk) =

(mp+1 + 1)fp,m (x + pk) + mfp,m (x + (p − 1)k) + · · · − mp−1 fp,m (x + k) + mp fp,m (x)

= Fp,m (p + 2)fp,m (x + pk) + Fp,m (2)fp,m (x + (p − 1)k) + · · · − Fp,m (p)fp,m (x + k) + Fp,m (p + 1)fp,m (x).

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fp,m (x + (2p + 2)k)

= mfp,m (x + (2p + 1)k) + fp,m (x + (p + 1)k) h = m (mp+1 + 1)fp,m (x + pk) + mfp,m (x + (p − 1)k) i + · · · − mp−1 fp,m (x + k) + mp fp,m (x) +mf (x + pk) + f (x) =

(mp+2 + 2m)fp,m (x + pk) + m2 fp,m (x + (p − 1)k) + · · · − mp fp,m (x + k) + (mp+1 + 1)fp,m (x)

= Fp,m (p + 3)fp,m (x + pk) + Fp,m (3)fp,m (x + (p − 1)k) + · · · − Fp,m (p + 1)fp,m (x + k) + Fp,m (p + 2)fp,m (x). Continuing this process, we have

fp,m (x + nk)

= Fp,m (n − p + 1)fp,m (x + pk) =

+Fp,m (n − 2p + 1)fp,m (x + (p − 1)k)

=

+ · · · − Fp,m (n − p − 1)fp,m (x + k) + Fp,m (n − p)fp,m (x)

and fp,m (x + (n + 1)k)

= Fp,m (n − p + 2)fp,m (x + pk) =

+Fp,m (n − 2p + 2)fp,m (x + (p − 1)k)

=

+ · · · − Fp,m (n − p)fp,m (x + k) + Fp,m (n − p + 1)fp,m (x),

where Fp,m is an m−extension of Fibonacci p−sequence with the initial conditions, Fp,m (0) = 0, Fp,m (1) = 1, Fp,m (2) = m, . . ., Fp,m (p) = mp−1 . Given x0 ∈ R, there exists x ∈ R such that x0 = x + nk . Therefore, fp,m (x0 + k) fp,m (x0 )

fp,m (x + (n + 1)k) fp,m (x + nk)  Fp,m (n − p + 2)fp,m (x + pk) + · · · − Fp,m (n − p)fp,m (x + k)+    Fp,m (n − p + 1)fp,m (x)  =   F (n − p + 1)f (x + pk) + · · · − F (n − p − 1)f (x + k)+  p,m p,m p,m p,m  Fp,m (n − p)fp,m (x)   Fp,m (n − p) fp,m (x + k)+ F (n − p + 2) fp,m (x + pk) + · · · −  p,m F (n − p + 2) p,m     Fp,m (n − p + 1)  fp,m (x)  Fp,m (n − p + 2)  =    F (n − p + 1) f (x + pk) + · · · − Fp,m (n − p − 1) f (x + k)+ p,m p,m  p,m Fp,m (n − p + 1)     Fp,m (n − p) fp,m (x) Fp,m (n − p + 1) =

                .      

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

fp,m (x + pk) + · · · −

Fp,m (n − p) fp,m (x + k)+ Fp,m (n − p + 2)



     Fp,m (n − p + 1)        f (x) p,m   fp,m (x0 + k) Fp,m (n − p + 2)  Fp,m (n − p + 2)  lim = lim   n→∞ n→∞ Fp,m (n − p + 1)  Fp,m (n − p − 1)  fp,m (x0 )  fp,m (x + pk) + · · · − fp,m (x + k)+    F (n − p + 1) p,m    Fp,m (n − p)  fp,m (x) Fp,m (n − p + 1)  Fp,m (n − p) f (x + pk) + · · · − lim fp,m (x + k)+ n→∞ Fp,m (n − p + 2)  p,m   Fp,m (n − p + 1)   lim fp,m (x) Fp,m (n − p + 2)   n→∞ Fp,m (n − p + 2) = lim  n→∞ Fp,m (n − p + 1)  F (n − p − 1)  fp,m (x + pk) + · · · − p,m fp,m (x + k)+  Fp,m (n − p + 1)   Fp,m (n − p) fp,m (x) Fp,m (n − p + 1) Let N = n + 1. If n → ∞ then N → ∞. So, we can write the above expression as  Fp,m (N − p − 1) f (x + pk) + · · · − lim fp,m (x + k)+  p,m N →∞ Fp,m (N − p + 1)   Fp,m (N − p)  lim     fp,m (x)  0 N →∞ fp,m (x + k) Fp,m (n − p + 2)  Fp,m (N − p + 1) lim = lim  n→∞ n→∞ Fp,m (n − p + 1)  F (n − p − 1) fp,m (x0 )  fp,m (x + pk) + · · · − lim p,m fp,m (x + k)+  n→∞ Fp,m (n − p + 1)   Fp,m (n − p) lim fp,m (x) n→∞ Fp,m (n − p + 1)   Fp,m (N − p − 1) fp,m (x + pk) + · · · − lim fp,m (x + k)+   N →∞ Fp,m (N − p + 1)     Fp,m (N − p)  lim   N →∞ F (N − p + 1) fp,m (x)    p,m = αm   = αm . Fp,m (n − p − 1)    fp,m (x + pk) + · · · − lim fp,m (x + k)+    n→∞ Fp,m (n − p + 1)     Fp,m (n − p) lim fp,m (x) n→∞ Fp,m (n − p + 1) Here αm is the unique positive real root of the characteristic equation of m−extension of Fibonacci p−sequence. Next, suppose that Q(x) > 0, without loss of generality we assume fp,m (x) > 0, fp,m (x + k) > 0. Identically,   f (x+(n+1)k) we can easily obtain that limn→∞ p,m = αm . Hence we omit the proof. fp,m (x+nk) 6. Acknowledgments This study is a part of the second author’s PhD Thesis. This research is supported by TUBITAK. References [1] T. Koshy, Fibonacci and Lucas numbers with applications, Vol. 51, John Wiley & Sons, 2011.

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       .     

            

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[2] S. Falcon, A. Plaza, On the Fibonacci k-numbers, Chaos, Solitons & Fractals 32 (5) 1615–1624 (2007). [3] S. Falcon, A. Plaza, On k-Fibonacci sequences and polynomials and their derivatives, Chaos, Solitons & Fractals 39 (3) 1005–1019 (2009). [4] K. Uslu, N. Taskara, H. Kose, The generalized k-fibonacci and k-lucas numbers, Ars Combinatoria 99 25–32 (2011). [5] A. Horadam, Basic properties of a certain generalized sequence of numbers, The Fibonacci Quarterly 3 (3) 161–176 (1965). [6] O. Yayenie, A note on generalized Fibonacci sequences, Applied Mathematics and Computation 217 (12) 5603–5611 (2011). [7] Y. Yazlik, N. Taskara, A note on generalized k-horadam sequence, Computers & Mathematics with Applications 63 (1) 36–41 (2012). [8] E. Kilic, The binet formula, sums and representations of generalized Fibonacci p-numbers, European Journal of combinatorics 29 (3) 701–711 (2008). [9] E. G. Kocer, N. Tuglu, A. Stakhov, On the m−extension of the Fibonacci and Lucas p−numbers, Chaos, Solitons & Fractals 40 (4) 1890–1906 (2009). [10] A. Stakhov, B. Rozin, Theory of binet formulas for Fibonacci and Lucas p−numbers, Chaos, Solitons & Fractals 27 (5) 1162–1177 (2006). [11] D. Tollu, Y. Yazlik, N. Taskara, On fourteen solvable systems of difference equations, Applied Mathematics and Computation 233 310–319 (2014). [12] D. Tasci, M. C. Firengiz, Incomplete Fibonacci and Lucas p−numbers, Mathematical and Computer Modelling 52 (9) 1763–1770 (2010). [13] D. T. Tollu, Y. Yazlik, N. Taskara, On the solutions of two special types of riccati difference equation via Fibonacci numbers, Advances in Difference Equations 2013 (1) 174 (2013). [14] A. Stakhov, B. Rozin, The continuous functions for the fibonacci and lucas p−numbers, Chaos, Solitons & Fractals 28 (4) 1014–1025 (2006). [15] M. Basu, B. Prasad, Coding theory on the m−extension of the fibonacci p−numbers, Chaos, Solitons & Fractals 42 (4) 2522–2530 (2009). [16] J. S. Han, H. S. Kim, J. Neggers, On Fibonacci functions with Fibonacci numbers, Advances in Difference Equations 2012 (1) 126 (2012). [17] B. Sroysang, On Fibonacci functions with period k, Discrete Dynamics in Nature and Society 2013. [18] J. F. T. Rabago, On second-order linear recurrent functions with period k and proofs to two conjectures of sroysang, Hacet. J. Math. Stat 45(2) 429–446 (2016).

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FOURIER SPECTRAL METHODS FOR STOCHASTIC SPACE FRACTIONAL PARTIAL DIFFERENTIAL EQUATIONS DRIVEN BY SPECIAL ADDITIVE NOISES FANG LIU, MONZORUL KHAN AND YUBIN YAN

∗

Abstract. Fourier spectral methods for solving stochastic space fractional partial differential equations driven by special additive noises in one-dimensional case are introduced and analyzed. The space fractional derivative is defined by using the eigenvalues and eigenfunctions of Laplacian subject to some boundary conditions. The space-time noise is approximated by the piecewise constant functions in the time direction and by some appropriate approximations in the space direction. The approximated stochastic space fractional partial differential equations are then solved by using Fourier spectral methods. For the linear problem, we obtain the precise error estimates in the L2 norm and find the relations between the error bounds and the fractional powers. For the nonlinear problem, we introduce the numerical algorithms and MATLAB codes based on the FFT transforms. Our numerical algorithms can be adapted easily to solve other stochastic space fractional partial differential equations with multiplicative noises. Numerical examples for the semilinear stochastic space fractional partial differential equations are given. Key words. Space fractional partial differential equations, stochastic partial differential equations, Fourier spectral method, error estimates AMS subject classifications. 65M12; 65M06; Secondary 65M70;35S10

1. Introduction. Fourier spectral methods for solving the following stochastic space fractional partial differential equation are considered in this work, with 1/2 < α ≤ 1, (1.1) (1.2)

dW (t) du(t) + Aα u(t) = f (u(t)) + , dt dt u(0) = u0 .

0 < t < T,

Here A is an unbounded positive self-adjoint operator, u0 is an initial value and f (u) is a nonlinear term. The space-time white noise W (t) will be defined below. Let H be a separable Hilbert space and ∥ · ∥, (·, ·) denote the norm and inner product in H. Let A : D(A) ⊂ H → H be a positive definite self-adjoint operator such that A−1 is compact on H. From this we infer the existence of a complete orthonormal basis {ek }k≥0 for H of eigenfunctions of A such that the associated sequence of eigenvalues {λk } form an increasing unbounded sequence. Using the basis {ek } we may also define the fractional powers of A. Given 1/2 < α ≤ 1 define ∑ 2 H 2α := D(Aα ) = {v ∈ H : λ2α k |(v, ek )| < ∞}, k

and (1.3)

Aα v :=

∑

α λα k (v, ek )ek , v ∈ D(A ),

k ∗ Fang Liu: Department of Mathematics, Lvliang University, Lishi, P.R. China, 033000, ([email protected]), Monzorul Khan: Department of Mathematics, University of Chester, Thornton Science Park, Pool Lane, Ince, CH2 4NU, UK, (sohel [email protected]), Yubin Yan: Department of Mathematics, University of Chester, Thornton Science Park, Pool Lane, Ince, CH2 4NU, UK, ([email protected]). Dr. Yubin Yan is the corresponding author.

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FOURIER SPECTRAL METHODS FOR SPDEs

with the associated Hilbert norm defined by ∑ 2 ∥Aα v∥2 = λ2α k |(v, ek )| . k

The special space-time noise considered in this work is ∞

dW (t) ∑ = σk (t)β˙ k (t)ek , dt

(1.4)

k=1

k (t) where β˙ k (t) = dβdt , k = 1, 2, . . . is the derivative of the standard Brownian motions βk (t), k = 1, 2, . . . and σk (t), k = 1, 2, . . . are some appropriate functions of t. In 1/2 particular, when σk (t) = γ¯k , γ¯k > 0, the noise (1.4) reduces to

∞

dW (t) ∑ 1/2 ˙ γ¯k βk (t)ek , = dt k=1

which is a so-called H-valued Wiener process with the covariance operator ∑∞Q and the linear operator Q : H → H is a trace class operator, that is Tr(Q) = k=1 γ¯k < ∞ where Qek = γ¯k ek , k = 1, 2, . . . . Let us here give two possible operators in (1.1)-(1.2). One is A = −∆ with the homogeneous Dirichlet boundary condition, D(A) = H01 (0, 1) ∩ H 2 (0, 1), where ∆ = ∂ 2 /∂x2 denotes the√Laplacian. In this case, A has the eigenvalues λk = k 2 π 2 and eigenfunctions ek = 2 sin kπx, k = 1, 2, . . . . Our error estimates in this work are based on these eigenvalues and eigenfunctions. Another one is A = I −∆ with periodic 2 2 boundary conditions, D(A) = Hper (−π, π). Here Hper (−π, π) denotes the completion 2 with respect to the H (−π, π) norm of the set of u ∈ C ∞ ([−π, π]) such that the pth derivative u(p) (−π) = u(p) (π) for p = 0, 1, . . . . It is a Hilbert space with the H 2 (−π, π) inner product, see [24, Definition 1.47]. In this case, A has the eigenvalues λ1 = 1, λ2k = 1 + k 2 , λ2k+1 = 1 + k 2 and eigenfunctions e1 (x) = √12π , e2k (x) = √1 sin kx, e2k+1 (x) = √1 cos kx, k = 1, 2, . . . , see [24, Example 1.84]. π π We obtain the detailed error estimates, i.e., Theorems 2.1, 3.1, 3.3 below for the linear stochastic space fractional partial differential equation subject to the Dirichlet boundary conditions. More precisely, we shall consider the error estimates for the following linear problem, with 1/2 < α ≤ 1,

(1.6)

∂u(t, x) ∂ 2 W (t, x) + (−∆)α u(t, x) = , ∂t ∂t∂x u(t, 0) = u(t, 1) = 0, 0 < t < T,

(1.7)

u(0, x) = u0 (x),

(1.5)

0 < t < T, 0 < x < 1,

0 < x < 1. 2

W (t,x) Here the space-time noise ∂ ∂t∂x = dWdt(t) is define by (1.4). For the linear stochastic space fractional partial differential equation subject to the periodic boundary conditions, we may obtain the similar error estimates as in Theorems 2.1, 3.1, 3.3. For the length of the paper, we will not give the detailed proofs for the error estimates in this case. However, in the numerical examples in Section 4, we shall consider the spectral method for the semilinear stochastic space fractional partial differential equations subject to the periodic boundary conditions to illustrate the experimentally determined convergence orders.

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FANG LIU,, MONZORUL KHAN, AND YUBIN YAN

3

The stochastic partial differential equations driven by the white noise ( the covariance operator Q = I) often have poor regularity estimates. In the physical world, to take into account the short and long range correlations of the stochastic effects, both white noise and colored noises may be considered. There are many situations where colored noises model the reality more closely, and there are also instances where the important stochastic effects are the noises acting on a few selected frequencies. For t example one may choose σk (t) = cos k3 . [12] Space-fractional partial differential equations are widely used to model complex phenomena, for example, quasi-geostrophic flows, fast rotating fluids, dynamic of the frontogenesis in meteorology, diffusion in fractal or disordered medium, pollution problems, mathematical finance and the transport problems, see, e.g., [3], [7], [21], [36], [2]. Let us here consider two examples which apply the fractional Laplacian in the physical models. The first example is about the surface quasi-geostrophic (SQG) equation, ∂t θ + ⃗u · ∇θ + κ(−∆)α θ = 0, where κ ≥ 0 and α > 0, θ = θ(x1 , x2 , t) denotes the potential temperature, ⃗u = (u1 , u2 ) is the velocity field determined by θ. When κ > 0, the SQG equation takes into account the dissipation generated by a fractional Laplacian. The SQG equation with κ > 0 and α = 1/2 arises in geophysical studies of strongly rotating fluids. For the dissipative SQG equation, α = 1/2 appears to be a critical index. In the subcritical case when α > 1/2, the dissipation is sufficient to control the nonlinearity and the global regularity is a consequence of global a priori bound. In the critical case when α = 1/2, the global regularity issue is more delicate. The mystery in the supercritical case α < 1/2 is only partially uncovered at the moment. [9] The second example is about the wave propagation in complex solids, especially viscoelastic materials (for example Polymers).[4]. In this case, the relaxation function has the form k(t) = ct−ν , 0 < ν < 1, c ∈ R, instead of the exponential form known in the standard models. This polynomial relaxation is due to the non uniformity of the material. The far field is then described by a Burgers equation with the leading 1+ν operator (−∆) 2 instead of the Laplacian ∂t u = −(−∆)

1+ν 2

u + ∂x (u2 ).

This equation also describes the far-field evolution of acoustic waves propagating in a gas-filled tube with a boundary layer. Frequently, the initial value or the coefficients of the equation are random, therefore it is natural to consider the stochastic space-fractional partial differential equations. The existence, uniqueness and regularities of the solutions of stochastic spacefractional partial differential equations have been extensively studied, see, e.g., [3], [7], [10], [28]. In this work, we will focus on the case 1/2 < α ≤ 1 since the existence and uniqueness and regularity of the solution in this case is well understood in literature, see [11, Theorem 1.3]. However the numerical methods for solving spacefractional stochastic partial differential equations are quite restricted even for the case 1/2 < α ≤ 1. Debbi and Dozzi [11] introduced a discretization of the fractional Laplacian and used it to elaborate an approximation scheme for fractional heat equation perturbed by a multiplicative cylindrical white noise. As far as we know [11] is the only existing paper in the literature of dealing with the numerical approach of this

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kind of problems. In this work, we will use the ideas developed in [1] and [12], to consider the numerical methods for solving stochastic space fractional partial differential equations, see also [19], [8], [20]. We first approximate the noise by using piecewise constant functions and then obtain the approximate solution u ˆ(t) of the exact solution u(t). Finally we provide error estimates in L2 -norm for u(t) − u ˆ(t). For the deterministic space fractional partial differential equations, many numerical methods are available in literature. There are two approaches to define the fractional Laplacian. One approach is by using the eigenvalues and eigenfunctions of the Laplacian −∆ subject to the boundary conditions as in (1.3). Another approach is by using the left-handed and right-handed Riemann-Liouville fractional derivatives. For the deterministic space fractional partial differential equations defined by the Riemann-Liouville fractional derivatives, many numerical methods are available, e.g., finite difference methods [14]-[15], [26], [31]-[32], finite element methods [13], [18] and the spectral methods [22]-[23]. For the deterministic space fractional partial differential equations defined by (1.3), some numerical methods are also available, see, e.g., matric transfer technique (MTT) [14], [15], [6], Fourier spectral method [5]. In this work, we will use Fourier spectral method to solve the stochastic space fractional partial differential equations. The main advantage of this approach is that it gives a full diagonal representation of the fractional operator, being able to achieve spectral convergence regardless of the fractional power in the problem. Let Nt ∈ N and let 0 = t0 < t1 < t2 < · · · < tNt = T be the time partition of [0, T ] and ∆t the time step size. To find the approximate solution of (1.5)-(1.7), 2 W (t,x) we approximate the noise ∂ ∂t∂x by the piecewise constant functions in the time direction defined by, with l = 1, 2, ..., Nt , [12] t (∑ ) c (t, x) ∑ ∂2W 1 √ ηk,l χl (t) , = σkM (t)ek (x) ∂t∂x ∆t

(1.8)

∞

N

k=1

l=1

where (1.9)

1 ηk,l = √ ∆t

∫

) 1 ( βk (tl ) − βk (tl−1 ) ∈ N (0, 1), d βk (t) = √ ∆t tl−1 tl

and { χl (t) =

1, 0,

t ∈ [tl−1 , tl ], l = 1, 2, . . . , Nt , otherwise.

Here σkM (t) is the approximation of σk (t) in the space direction. For example, we can choose, with some positive integer M > 0, cos t σk (t) = 3 , k

{ σkM (t)

=

σk (t), k ≤ M, 0, k > M.

More precisely, replacing σk (t) by σkM (t), we get the noise approximation in space, ∑Nt 1 √ and replacing β˙ k (t) by j=1 η χ (t), we get the noise approximation in time. ∆t k,j j Substituting

∂ 2 W (t,x) ∂t∂x

with

c (t,x) ∂2W ∂t∂x

in (1.5)-(1.7), we get

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FANG LIU,, MONZORUL KHAN, AND YUBIN YAN

c (t, x) ∂u ˆ(t, x) ∂2W + (−∆)α u ˆ(t, x) = , ∂t ∂t∂x u ˆ(t, 0) = u ˆ(t, 1) = 0, 0 < t < T, u ˆ(0, x) = u0 (x), 0 < x < 1.

(1.10) (1.11) (1.12)

0 < t < T, 0 < x < 1,

2c

W (t,x) now is a function in L2 ((0, T ) × (0, 1)) and therefore we can Note that ∂ ∂t∂x solve (1.10)-(1.12) by using any numerical methods for deterministic space fractional partial differential equations. Assume that {σk (t)} and its derivative are uniformly bounded, [12]

|σk′ (t)| ≤ γk ,

|σk (t)| ≤ βk ,

(1.13)

∀ t ∈ [0, T ],

and the coefficients {σkM } are constructed such that (1.14)

|σk (t) − σkM (t)| ≤ αkM ,

|σkM (t)| ≤ βkM ,

|(σkM )′ (t)| ≤ γkM ,

∀ t ∈ [0, T ],

with positive sequences {αkM } being arbitrarily chosen, {βkM } and {γkM } being related to {βk } and {γk }. Further we assume that βkM ≤ k −α˜ ,

(1.15)

for some 0 ≤ α ˜ < 1/2.

Let E denote the expectation, in Theorem 2.1, we prove that, with 1/2 < α ≤ 1 and 0≤α ˜ < 1/2, ∫ T∫ 1 ( )2 (1.16) E u(t, x) − u ˆ(t, x) dxdt 0

≤C

0

∞ (∑ (αM )2 k

k=1

2λα k

+ ∆t2

∞ ( ∑

M M λα k β k + γk

)2

) α ˜ 1 + ∆t1+ α − 2α .

k=1

Let J ∈ N, we denote SJ = span{e1 , e2 , . . . , eJ }, and define by PJ : H → SJ the projection from H to SJ , J ∑ PJ v = (v, ej )ej .

(1.17)

j=1

The Fourier spectral method of (1.10)-(1.12) is to find u ˆJ (t) ∈ SJ such that, with gˆ(t, x) :=

c (t,x) ∂2W ∂t∂x .

(1.19)

∂u ˆJ (t, x) + (−∆)α u ˆJ (t, x) = PJ gˆ(t, x), ∂t u ˆJ (t, 0) = u ˆJ (t, 1) = 0, 0 < t < T,

(1.20)

u ˆJ (0, x) = PJ u0 (x),

(1.18)

0 < t < T, 0 < x < 1,

0 < x < 1,

In Theorem 3.1, we prove that, with 1/2 < α ≤ 1 and 0 ≤ α ˜ < 1/2, ∫ t 1 ∥ˆ g (s)∥2 ds. (1.21) ∥ˆ u(t) − u ˆJ (t)∥2 ≤ C∥u0 − PJ u0 ∥2 + C (J + 1)2α 0

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Combining Theorem 2.1 with Theorem 3.1, we have, with u0 ∈ D(Aα ), 1/2 < α ≤ 1, ∫

T

∫

1

E 0

≤C

( )2 u(t, x) − u ˆJ (t, x) dxdt

0

∞ (∑ (αM )2 k

k=1

2λα k

∞ ( ∑

+ ∆t2

M M λα k βk + γk

)2

+ ∆t1+ α − 2α α ˜

1

)

k=1

+ CE∥u0 − PJ u0 ∥2 + C

( ) ∑ ∑ 1 α 2 α M 2 M 2 ∆tE∥A u ∥ + ∆t λ (β ) + (β ) . 0 k k k (J + 1)2α ∞

∞

k=1

k=1

The paper is organized as follows. In Section 2, we consider the approximation of noise. In Section 3, we introduce the Fourier spectral methods for solving the approximated space fractional partial differential equations and the error estimates for the linear stochastic space fractional partial differential equations are proved. In Section 4, we consider the numerical examples for solving the semilinear stochastic space fractional partial differential equations subject to the periodic boundary conditions. From now on we denote by C a generic constant, which may not be the same at different occurrences. 2. Approximate the noise and regularity. It is well known that the mild solution of (1.5)-(1.7) has the following form ∫ 1 ∫ t∫ 1 (2.1) u(t, x) = Gα (t, x, y) u0 (y) dy + Gα (t − s, x, y) dW (s, y), 0

0

0

where Gα (t, x, y) =

∞ ∑

e−λj t ej (x)ej (y), α

j=1

∫t∫1 and the stochastic integral 0 0 Gα (t−s, x, y) dW (s, y) is well-defined. The existence and uniqueness of the solutions of (1.5)-(1.7) are discussed in, e.g., [10], [11], [28] and the references cited therein. Similarly the mild solution of (1.10)-(1.12) has the form of, see, e.g., [12] ∫ 1 ∫ t∫ 1 c (s, y), (2.2) u ˆ(t, x) = Gα (t, x, y) u0 (y) dy + Gα (t − s, x, y) dW 0

0

0

Theorem 2.1. Let u and u ˆ be the solutions of (1.5)-(1.7) and (1.10)-(1.12), respectively. Assume that the assumptions (1.13)-(1.15) hold. Then we have ∫ (2.3)

T

E 0

≤C

∫

1

(

0 ∞ (∑ k=1

u(t, x) − u ˆ(t, x)

)2

dxdt

)2 ) ∑( α ˜ 1 (αkM )2 M M + ∆t2 λα + ∆t1+ α − 2α . k β k + γk α 2λk ∞

k=1

Proof. See the Appendix. Remark 2.2. When α = 1, Theorem 2.1 should reduce to the Theorem 3.3 in [12]. However one term ∆t1/2+α˜ , 0 ≤ α ˜ < 1/2 of the bounds in (3.20) in Theorem 3.3 [12] is missing. The term ∆t1/2+α˜ , 0 ≤ α ˜ < 1/2 comes from the estimates II1 and

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∫T ∫1 II3 of the estimate for II = E 0 0 F2 (t, x) dxdt in (4.12). The authors in [12] only considered the estimate II2 and neglected the terms II1 and II3 which would produce the term ∆t1/2+α˜ , 0 ≤ α ˜ < 1/2. (See the estimates for the term II in [12, p.1441]). α ˜ 1 In Theorem 2.1, we include the terms O(∆t1+ α − 2α ). Theorem 2.3. Let u ˆ be the solution of (1.10)-(1.12). Assume that the assumptions (1.13)-(1.15) hold. Further assume that u0 ∈ D(Aα ), 1/2 < α ≤ 1 and E∥Aα u0 ∥2 < ∞. Then ∫ (2.4) E

∫

tj+1 tj

1 0

∞ ∞ ∂u ) ( ∑ ∑ ˆ(t, x) 2 λkα (βkM )2 + (βkM )2 , dxdt ≤ C ∆tE∥Aα u0 ∥2 +∆t ∂t k=1

k=1

and ∫ E

(2.5)

tj+1

tj

∫

1

∞ ( ) ∑ α 2 α 2 M 2 A u ˆ(t, x) dxdt ≤ C ∆tE∥A u0 ∥ + ∆t λα k (βk ) .

0

k=1

Proof. Assume that, with 0 < t ≤ tj+1 , (2.6)

u ˆ(t, x) =

∞ ∑

u ˆk (t)ek (x),

k=1

and, with u ˆk (0) = (u0 , ek ), k = 1, 2, . . . , u ˆ(0, x) = u0 (x) =

∞ ∑

u ˆk (0)ek (x).

k=1

Substituting (2.6) into (1.10), we get, with 0 < t ≤ tj+1 , j+1 (∑ ) dˆ uk (t) 1 M √ + λα u ˆ (t) = σ (t) η χ (t) , k,l l k k k dt ∆t

(2.7)

l=1

which implies that, with 0 < t ≤ tj+1 , u ˆk (t) = e−λk t u ˆk (0) + α

(2.8)

∫

t

e−λk (t−s) σkM (s) α

0

j+1 (∑ l=1

) 1 √ ηk,l χl (s) ds. ∆t

Let us first show (2.4). Note that {ek } is an orthonormal basis in H = L2 (0, 1), we have, by (2.7), ∫ E

tj+1

tj

∫

1 0

∞ ∫ tj+1 ∂u ∑ uk (t) 2 ˆ(t, x) 2 dˆ dxdt = E dt ∂t dt tj

∞ ( ∫ tj+1 ∑

k=1

∫

j+1 2 ) σ M (t) ∑ k √ ≤ 2E dt + ηk,l χl (t) dt ∆t l=1 tj tj k=1 ∫ tj+1 ∞ ∞ ∫ tj+1 M 2 ∑ ∑ σk (t) = 2E λ2α |ˆ uk (t)|2 dt + 2E ηk,j+1 χj+1 (t) dt √ k ∆t tj k=1 k=1 tj

|λα ˆk (t)|2 ku

tj+1

= 2(I + II).

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For I, we have, by (2.8), with t∗l = tl , 1 ≤ l ≤ j and t∗l = t, l = j + 1, (2.9)

∫ t∗l j+1 2 ∑ α ηk,l √ e−λk (t−s) σkM (s) ds dt ∆t tl−1 tj tj k=1 k=1 l=1 ∫ ∫ ∫ ∗ j+1 ∞ ∞ tj+1 ∑ )2 ∑ tj+1 ∑ )2 α ( 1 ( tl −λαk (t−s) M = 2E e−2λk t Aα u0 , ek dt + 2 λ2α e σk (s) ds dt k ∆t tl−1 tj tj

I ≤ 2E

∞ ∑

∫

tj+1

λ2α k

∫ ∞ 2 ∑ −λαk t 2α u ˆ (0) dt + 2E λ e k k

k=1

≤ 2E ≤ 2E ≤ 2E

∞ ∑ k=1 ∞ ∑ k=1 ∞ ∑

( ( (

)2 Aα u0 , ek ∆t + 2 )2 Aα u0 , ek ∆t + 2 )2 Aα u0 , ek ∆t + 2

k=1

∞ ∑ k=1 ∞ ∑ k=1 ∞ ∑

∫ λ2α k ∫

k=1

l=1

(∫

tj+1

λ2α k tj

( M )2 λ2α k βk

∫

≤ 2E∥Aα u0 ∥2 ∆t + ∆t

∫ ∗ ∫ ∗ 1 ( tl −2λαk (t−s) ( M )2 )( tl 2 ) e σk (s) ds 1 ds dt ∆t tl−1 tl−1 t

( )2 ) α e−2λk (t−s) σkM (s) ds dt

0 tj+1

tj

k=1 ∞ ∑

l=1

j+1 tj+1 ∑ tj

tj+1

1 − e−2λk t dt 2λα k α

( M )2 λα , k βk

k=1

where in the last inequality, we use the fact 1 − e−2λk t ≤ 1. For II, we have α

II = E

∞ ∫ ∑ k=1

∞ ∞ ∫ tj+1 ( M σ M (t) 2 ∑ ∑ ( M )2 σk (t) )2 k √ √ η χ (t) dt ≤ βk . dt = k,j+1 j+1 ∆t ∆t k=1 k=1 tj

tj+1

tj

Combining I with II we get (2.4). Similarly we have, ∫ tj+1 ∫ 1 ∫ tj+1 E |Aα u ˆ(t)|2 dxdt = E ∥Aα u ˆ(t, x)∥2 dt tj

=E

∫

0 tj+1 tj

∞ (∑

)

tj

ˆ2k (t) dt = E λ2α k u

∞ ∑

∫

|ˆ uk (t)|2 dt = I,

tj

k=1

k=1

tj+1

λ2α k

which implies (2.5) also holds. Together these estimates complete the proof of Theorem 2.3. 3. Fourier spectral method. Denote Eα (t) = e−tA , 1/2 < α ≤ 1, where is defined by (1.3). The mild solution of (1.10)-(1.12) has the form of, with α

Aα

gˆ(t) =

c (t,x) ∂2W ∂t∂x ,

∫ (3.1)

t

Eα (t − s)ˆ g (s) ds,

u ˆ(t) = Eα (t)ˆ u0 +

u ˆ(0) = u0 .

0

Similarly the solution of (1.18)-(1.20) has the form of ∫ (3.2)

t

Eα (t − s)PJ gˆ(s) ds,

u ˆJ (t) = Eα (t)PJ u ˆ0 +

u ˆ(0) = PJ u0 .

0

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9

Theorem 3.1. Assume that u ˆ and u ˆJ are the solutions of (1.10)-(1.12) and (1.18)-(1.20), respectively. Let 0 ≤ r < 1/2 and let u0 ∈ H. Then we have ∫ t

r/2 (

2 ) 2 1



∥ˆ g (s)∥2 ds. (3.3) A u ˆ(t) − u ˆJ (t) ≤ C u0 − PJ u0 + C (J + 1)2α(1−r/α) 0 In particular, with r = 0, (3.4)

1 ∥ˆ u(t) − u ˆJ (t)∥ ≤ C∥u0 − PJ u0 ∥ + C (J + 1)2α 2

∫

t

∥ˆ g (s)∥2 ds.

2

0

To prove Theorem 3.1, we need the following smoothing property for the solution operator Eα (t). Lemma 3.2. 1. Let s > 0. We have, with 1/2 < α ≤ 1, ∥As Eα (t)∥ ≤ Ct− α e−δt , t > 0, s

for some constants C = C(s, α) > 0 and δ = δ(α) > 0. 2. Let PJ : H → SJ be defined by (1.17). We have ∥Eα (t)(I − PJ )v∥ ≤ e−tλJ+1 ∥v∥, t > 0. α

Proof. Recall that A is positive definite and A has the eigenvalues 0 < λ1 < λ2 < λ3 < . . . . For any function h(·), we have ∥h(A)∥ = sup |h(λ)|, λ∈σ(A)

where σ(A) denotes the set of eigenvalues of A. Thus, with δ = 21 λα 1, ∥As Eα (t)∥ = ∥As Eα (t/2)Eα (t/2)∥ ≤ ∥As Eα (t/2)∥∥Eα (t/2)∥ ( ( t λα )s/α ( t )−s/α ) t α ( s − t λα ) ( − t λα ) 2 = sup λ e 2 · sup e 2 = sup e− 2 λ1 t α 2 e2λ λ∈σ(A) λ∈σ(A) λ∈σ(A) ≤ C(t/2)−s/α e−δt ≤ Ct−s/α e−δt , which shows (1). Further (2) follows from ∥Eα (t)(I − PJ )v∥ =

∞ ( ∑

e−2tλj (v, ej )2 α

)1/2

≤ e−tλJ+1 ∥v∥. α

j=J+1

Together these estimates complete the proof of Lemma 3.2. Proof. [Proof of Theorem 3.1] Subtracting (3.2) from (3.1), we get ∫ t ( ) (3.5) u ˆ(t) − u ˆJ (t) = Eα (t)(u0 − PJ u0 ) + Eα (t − s) gˆ(s) − PJ gˆ(s) ds = I + II. 0

For I, we have, with 0 ≤ r < 1/2, ( ) r ∥Ar/2 I∥ = ∥A 2 Eα (t) u0 − PJ u0 ∥ ∞ ( ∑ )1/2 α α = e−2tλj λrj (u0 , ej )2 ≤ e−tλJ+1 ∥Ar/2 (u0 − PJ u0 )∥. j=J+1

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For II, we have, by Lemma 3.2, for some γ ∈ (0, 1),

∫ t

( )

∥Ar/2 II∥ = Ar/2 Eα (t − s) I − PJ gˆ(s) ds 0

∫ t[

] ( )][ ( )

= Ar/2 Eα (1 − γ)(t − s) Eα γ(t − s) (I − PJ ) gˆ(s) ds 0 ∫ t r ≤C (t − s)− 2α e−κα (t−s) ∥ˆ g (s)∥ ds, 0

where κα = δ(1 − γ) + λα J+1 γ. By Cauchy- Schwarz inequality, we have ∥A

r/2

II∥ ≤ C

(∫ t (

) r − 2α −κα (t−s) 2

(t − s)

e

)1/2 ( ∫

t

∥ˆ g (s)∥2 ds

ds

0

)1/2 .

0

Note that r < α, we have, with λJ+1 = (J + 1)2 π 2 , ∫ ∞ −r/α −2s ∫ ∞ −2κα s ∫ t −2κα s s e ds 1 e e ≤ C 1−r/α ds ≤ ds ≤ 0 r/α r/α 1−r/α s s 0 0 κα κα 1 1 ≤ C α 1−r/α ≤ C . (λJ+1 ) (J + 1)2α(1−r/α) Thus ∥Ar/2 II∥ ≤ C

(∫

1 (J + 1)2α(1−r/α)

t

)1/2 ∥ˆ g (s)∥2 ds .

0

Together these estimates complete the proof of Theorem 3.1. Combining Theorem 2.1 with Theorem 3.1, we have Theorem 3.3. Let u and u ˆJ be the solutions of (1.5)-(1.7) and (1.18)-(1.20), respectively. Assume that the assumptions (1.13)-(1.15) hold. Further assume that u0 ∈ D(Aα ), 1/2 < α ≤ 1 and E∥Aα u0 ∥2 < ∞. Then we have ∫

T

∫

1

E 0

≤C

( )2 u(t, x) − u ˆJ (t, x) dxdt

0

∞ (∑ (αM )2 k

2λα k

k=1

+ ∆t2

∞ ( ∑

M M λα k βk + γk

)2

+ ∆t1+ α − 2α α ˜

1

)

k=1

+ CE∥u0 − PJ u0 ∥2 + C

) ( ∑ ∑ 1 α 2 α M 2 M 2 ∆tE∥A u ∥ + ∆t λ (β ) + (β ) . 0 k k k (J + 1)2α ∞

∞

k=1

k=1

Proof. Note that ∫

T

∫

1

E 0

0

∫

T

(

u(t, x) − u ˆJ (t, x)

∫

≤ 2E 0

0

1

)2

dxdt

( )2 ˆ(t, x) dxdt + 2E u(t, x) − u

∫

T 0

∫

1

(

)2 u ˆ(t, x) − u ˆJ (t, x) dxdt

0

= 2I + 2II.

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For I, we have, by Theorem 2.1, I≤C

∞ (∑ (αM )2 k

k=1

2λα k

∞ ( ∑

2

+ ∆t

M M λα k β k + γk

)2

) 1 α ˜ + ∆t1+ α − 2α .

k=1

For II, we have ∫ II = E

T

1 E ∥ˆ u(t) − u ˆJ (t)∥ dt ≤ CE∥u0 − PJ u0 ∥ + C (J + 1)2α 2

0

Note that gˆ(s) = ∫

T

∫

t

∥ˆ g (s)∥2 dsdt. 0

0

+ (−∆)α u ˆ(s), we have, by Theorem 2.3,

∫ t

2 u(s)

dˆ

+ (−∆)α u ˆ(s) dsdt

ds 0 0 0 0 ∫ T ∫ T ∫ 1 ( ) ˆ(s, x) 2 ∂u ≤ CE ˆ(s, x)|2 dxdsdt + |(−∆)α u ∂s 0 0 0 ∞ ∞ ( ) ∑ ∑ M 2 ≤ C ∆tE∥Aα u0 ∥2 + ∆t λα (βkM )2 . k (βk ) +

E

T

∫

dˆ u(s) ds

∫

2

∫

t

T

∥ˆ g (s)∥2 dsdt ≤ E

k=1

k=1

Together these estimates complete the proof of Theorem 3.3. 4. Numerical simulations. In this section, we will consider the numerical simulation of the Fourier spectral methods for solving the following semilinear stochastic space fractional partial differential equations subject to the periodic boundary conditions, with 1/2 < α ≤ 1, 0 < x < 1, 0 < t ≤ T , (4.1) (4.2) (4.3)

∂u(t, x) ∂ 2 W (t, x) + ϵ(−∆)α u(t, x) = f (u(t, x)) + , ∂t ∂t∂x u(t, 0) = u(t, 1), u′x (t, 0) = u′x (t, 1), u(0, x) = u0 (x),

where (−∆)α is the fractional Laplacian defined by using the eigenvalues and eigenfunctions of the Laplacian −∆ subject to the periodic boundary conditions. Here f : R → R is a smooth function and ϵ > 0 denotes the diffusion coefficient. Here we consider the problems with the periodic boundary conditions because we want to compare our numerical results with the results in [24, Example 10.39] where the algorithms of the spectral methods for stochastic semilinear parabolic equation subject to the periodic boundary conditions are given and discussed. One may also consider the algorithms and MATLAB codes for stochastic space fractional partial differential equations with the homogeneous boundary conditions following the approaches in, e.g., [16], [17]. Although the Laplacian is singular in (4.1)-(4.2) due to the periodic boundary conditions, we expect the errors to behave as in Theorem 3.3, see the comments in [24, Corollary 10.38]. ∂2 2 2 Denote A = − ∂x 2 with D(A) = Hper (0, 1), where D(A) = Hper (0, 1) is defined in the Introduction section. Then the eigenvalues and eigenfunctions of A can also be expressed by λk = (2πk)2 ,

ek = ei2πkx , k ∈ Z.

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The noise has the form of ∂ 2 W (t, x) ∑ = σk (t)β˙ k (t)ek (x), ∂t∂x

(4.4)

k∈Z

k (t) where β˙ k (t) = dβdt , k ∈ Z are the derivatives of the standard Brownian motions βk (t), k ∈ Z and σk (t), k ∈ Z are some appropriate functions of t. Here k ∈ Z since 1/2 we consider the periodic boundary conditions. When σk (t) = γ¯k , γ¯k > 0, k ∈ Z, the noise (4.4) reduces to

∂ 2 W (t, x) ∑ 1/2 ˙ = γ¯k βk (t)ek (x). ∂t∂x

(4.5)

k∈Z

The approximate noise

c (t,x) ∂2W ∂t∂x

c (t, x) ∂2W = ∂t∂x

(4.6)

is, with some positive integer M > 0, ∑ k∈Z,|k|≤M

1/2

γ¯k ek (x)

Nt ∑ ηk,l l=1

∆t

χl (t).

In our numerical example below, we assume that, [24, Example 10.8], (4.7)

γ¯0 = 0,

γ¯k = |k|−(2r1 +1+˜ϵ) , k ∈ Z, k ̸= 0.

where ϵ˜ > 0 is a very small positive number. When r1 = −1/2, we obtain so-called space-time white noise. When r1 = 1, we obtain the smooth noise. Let SJ := span{e0 , e1 , . . . , eJ/2 , e−J/2+1 . . . , e−1 }. We assume J ≤ M where M is determined in (4.5). Here the ordering 0, 1, 2, . . . , J/2, −J/2 + 1, . . . , −1 is consistent with the ordering in the MATLAB functions fft and ifft [33]. Let 0 = t0 < t1 < t2 < · · · < tNt = T, Nt ∈ N be the time partition of [0, T ] and ∆t the time step size with T = Nt ∆t. We use the semi-implicit Euler method to consider the time discretization. We will consider the convergence rate against the different time steps. Choose J = 64. The reference solution is obtained by using the time step size ∆tref = T /N ref with N ref = 104 . Let kappa = [5, 10, 20, 50, 100, 200, 500], we will consider the approximate solutions with the different time step sizes ∆ti = ∆tref ∗ kappa(i), i = 1, 2, . . . , 7. By Theorem 2.1, we have ∫ T ∫ 1( )2 E (4.8) u(t, x) − u ˆ(t, x) dxdt 0

≤C

0

( ∑ (αM )2 k

k∈Z

2λα k

+ ∆t2

∑(

M M λα k β k + γk

)2

) α ˜ 1 + ∆t1+ α − 2α .

k∈Z

We remark that here we choose k ∈ Z since we consider the periodic boundary conditions. In our numerical example, we will choose, with γ¯k given by (4.7), 1/2

σk (t) = γ¯k , γ¯k > 0, k ∈ Z, { 1/2 σk (t) = γ¯k , |k| ≤ M, σkM (t) = 0, |k| > M, which implies that 1/2

|σkM (t)| ≤ βkM , where βkM = γ¯k , |k| ≤ M,

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13

and 1/2

|σk (t) − σkM (t)| ≤ αkM , where αkM = γ¯k , |k| > M. We first observe that for sufficiently large the convergence order of the L2 ( M 1 ˜ 1 ) (1+ α − 2α ) 2 α norm of the error in (4.8) is dominated by O ∆t . In fact, we will choose M = J where J is sufficiently large. Then the first term of the right side of (4.8) satisfies, with λk = (2πk)2 , k ∈ Z, ∑ (αM )2 k

k∈Z

2λα k

( 1 ) ∑ (αM )2 1 k ≤C α + α + ... α 2λk λM +1 λM +2 |k|>M ) ( 1 1 + + . . . ≤C (M + 1)2α (M + 2)2α ( ) 1 1 =C + + ... . 2α 2α (J + 1) (J + 2) =

The second term of the right side of the error in (4.8) is O(∆t2 ). Hence for sufficiently ( 1 α ˜ 1 ) large J, the convergence order of the L2 norm of the error in (4.8) is O ∆t 2 (1+ α − 2α ) . We now consider two cases r1 = −1/2 and r1 = 1 in (4.7). For r1 = −1/2, we may choose α ˜ = 0 which implies that the convergence order of the L2 norm in (4.8) ( 1 (1+ α˜ − 1 ) ) ( 1 1 ) 2 α 2α is O ∆t = O ∆t 2 (1− 2α ) . Indeed, α ˜ = 0 satisfies (1.15), that is, 1/2

βkM = γ¯k

= |k|−

2r1 +1+˜ ϵ 2

= |k|−˜ϵ/2 ≤ |k|−α˜ .

For r1 = 1, we may choose α ˜ = 1/2−¯ ϵ ( since 0 ≤ α ˜ < 1/2 ) with arbitrarily small positive number ϵ¯ which implies that the convergence order of the L2 norm in (4.8) ( 1 ( 1 α ˜ 1 ) ϵ ¯ ) is O ∆t 2 (1+ α − 2α ) = O ∆t 2 (1− α ) ≈ O(∆t1/2 ). Indeed, in this case, α ˜ = 1/2 − ϵ¯ satisfies (1.15), that is, 1/2

βkM = γ¯k

= |k|−

2r1 +1+˜ ϵ 2

= |k|−

3+˜ ϵ 2

≤ |k|−α˜ .

Thus we have, by Theorem 2.1, the following error estimates, with 1/2 < α ≤ 1 and r1 = −1/2, (4.9)

1

1

∥ˆ u − u∥L2 (Ω,L2 ((0,T ),H)) ≤ C(∆t 2 (1− 2α ) ),

and, with 1/2 < α ≤ 1 and r1 = 1 (4.10)

∥ˆ u − u∥L2 (Ω,L2 ((0,T ),H)) ≤ C(∆t1/2 ),

where the norm is measured in L2 both for time and space. In particular, when α = 1, r1 = −1/2, we have ∥ˆ u − u∥L2 (Ω,L2 ((0,T ),H)) ≤ C(∆t1/4 ), which is consistent with the standard time discretization error for the stochastic heat equation driven by space-time white noise, see, e.g., [35]. In our numerical experiment below, we choose f (u) = u − u3 , u0 (x) = sin(2πx), and ϵ = 1. See the simulation of this problem for α = 1 in [30]. We will consider the error estimates ∥ˆ u(tn ) − u(tn )∥L2 (Ω,H) at time tn . We hope to observe the same convergence order as in (4.9) and (4.10).

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14

FOURIER SPECTRAL METHODS FOR SPDEs A plot of the error at T=1 against log2 (∆ t), with r=−1/2

log2(error)

0.5

0

−0.5

−1 −4.5

← reference line of slope 3/16

−4

−3.5

−3

−2.5

−2

log2(∆ t)

Fig. 1. A plot of the error at T = 1 against log2(∆t) with α = 0.8, r1 = −1/2

To do this, we consider M = 100 simulations. For each simulation ωm , m = 1, 2, . . . , M , we generate J independent Brownian motions βl , l = 0, 1, . . . , J/2, −J/2+ 1, . . . , −1 and compute u ˆJ (tn ) ≈ u ˆ(tn ) at time tn = 1 by using the different time step sizes. We then compute the following L2 norm of the error at tn = 1 for the simulation ωm , m = 1, 2, . . . , M , uJ (tn , ωm ) − uref(tn , ωm )∥2 , ϵ(∆ti , ωm ) = ϵ(∆ti , ωm , tn ) = ∥ˆ where the reference (“true ”) solution uref(tn , ωm ) is approximated by using the time step ∆tref = T /N ref and Jref = J. We then average ϵ(∆ti , ωm ) with respect to ωm to obtain the following approximation of ∥ˆ uJ (tn ) − uref(tn )∥L2 (Ω,H) for the different time step size ∆ti , S(∆ti ) =

M M ( 1 ∑ )1/2 ( 1 ∑ )1/2 . ϵ(∆ti , ωm ) = ∥ˆ uJ (tn , ωm ) − uref(tn , ωm )∥2 M m=1 M m=1

For example, in the case α = 0.8, r1 = −1/2, the convergence rate against the time 1 1 step size is O(∆t 2 (1− 2α ) ) = O(∆t3/16 ), i.e., with some positive constant C, 3/16

S(∆ti ) ≈ C∆ti

,

which implies that log(S(∆ti )) ≈ log(C) +

3 log(∆ti ), i = 1, 2, . . . , 7. 16

In Figure 1, we )consider the case α = 0.8, r1 = −1/2 and plot the points log(∆ti ), log(S(∆ti )) , i = 1, 2, . . . , 7 and we observe that the experimentally determined convergence order is higher than the theoretical order in this case. Here the 3 reference line has the slope 16 . In Figure 2, we consider the case α = 0.8, r1 = 1 and in this case the theoretical 1/2 (convergence order with ) respect to the time step size is O(∆t ). We plot the points log(∆ti ), log(S(∆ti )) , i = 1, 2, . . . , 7 and we observe that the experimentally determined convergence order is also higher than the theoretical order in this case. Here the reference line has the slope 1/2 . In Figure 3, we consider the convergence rate against the different J. Choose fixed time step ∆t = T /Nt with Nt = 104 . We then consider the different J = Jref ∗ ( 211 , 212 , 213 , . . . , 218 ) where Jref = 210 . We will first generate the reference Brownian motions

(

(4.11)

βj (t), j = 0, 1, 2, . . . , Jref /2, −Jref /2 + 1, · · · − 1

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15

FANG LIU,, MONZORUL KHAN, AND YUBIN YAN A plot of the error at T=1 against log2 (∆ t), with r=1

log2(error)

−1 ← reference line of slope 1/2 −2

−3

−4 −4.5

−4

−3.5

−3

−2.5

−2

log2(∆ t)

Fig. 2. A plot of the error at T = 1 against log2(∆t) with α = 0.8, r1 = 1 A plot of the error at T=1 against J 4 2

log2(error)

0 −2 −4 −6 −8

1

2

3

4

5 log2(J)

6

7

8

9

Fig. 3. A plot of the error at T = 1 against the J with α = 0.8, r1 = 1

for computing the reference (“true”) solution uref. When we consider the approximate solution u with J truncated terms, we will use the Brownian motions βj (t), j = 0, 1, 2, . . . , J/2, −J/2 + 1, · · · − 1 from (4.11). In Figure 3, we consider the case α = 0.8, r1 = 1 and plot the L2 norm error against the different J where the L2 norm error are approximated by using M = 100 simulations. We indeed observe the spectral convergence with respect to the different J. Appendix In the Appendix, we shall provide the proof of Theorem 2.1. To do this, we need the following lemma. Lemma 4.1. Let 1/2 < α ≤ 1 and 0 ≤ α ˜ < 1/2. We have ∫ ∞ ( ) 2α α ˜ 1 ˜ x−2(α+α) 1 − e−x ∆t dx ≤ C∆t1+ α − 2α . 0

Proof. With the variable change y = x2α ∆t, we have ∫ ∞ (∫ 1 ∫ ( ) 2α α ˜ 1 ˜ x−2(α+α) 1 − e−x ∆t dx = C∆t1+ α − 2α + 0

0

It is easy to see that, with α ∈ (1/2, 1], ∫ ∞ 1 − e−y y 2+ α − 2α α ˜

1

1

1

∞

) 1 − e−y y 2+ α − 2α α ˜

1

dy

dy ≤ C.

Further, we have ∫ 1 ∫ 1 ∫ 1 1 − e−y y 1 dy ≤ C dy ≤ C α ˜ 1 α ˜ 1 α ˜ 1 dy < ∞. 2+ − 2+ − 1+ − 2α α 2α α 2α α 0 y 0 y 0 y

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FOURIER SPECTRAL METHODS FOR SPDEs

˜ 1 if 1 + α ˜ < 1/2. α − 2α < 1, i.e., 0 ≤ α Together these estimates complete the proof of Lemma 4.1.

Proof. [Proof of Theorem 2.1] Subtracting (2.2) from (2.1), we have

u(t, x) − u ˆ(t, x) ∫ t∫ 1 ∫ t∫ = Gα (t − s, x, y) dW (s, y) − 0

0

[∫ t∫

0

0

1

Gα (t − s, x, y) dW (s, y) −

0 1

Gα (t − s, x, y) dW (s, y) 0

∫ t∫

0 1

Gα (t − s, x, y) dW (s, y) −

+ 0

c (s, y) Gα (t − s, x, y) dW

0

∫ t∫

1

= [∫ t ∫

1

0

0

c (s, y) Gα (t − s, x, y) dW

]

]

0

= F1 (t, x) + F2 (t, x), where, with ηk,l and χl (t) defined as in (1.9), [∑ ] ∂ 2 W (s, y) dsdy = σk (s)ek (y) dβk (s)dy, ∂s∂y ∞

d W (s, y) =

k=1

2

∞ [∑

2c

∞ [∑

∂ W (s, y) dsdy = ∂s∂y

d W (s, y) =

∂ W (s, y) dsdy = ∂s∂y

c (s, y) = dW

] σkM (s)ek (y) dβk (s)dy,

k=1 Nt (∑ ) ] η √k,l χl (s) ek (y) dsdy. σkM (s) ∆t k=1 l=1

Thus ∫

T

∫

∫

1

E

T

∫

1

ˆ(t, x)|2 dxdt ≤ CE |u(t, x) − u 0

0

∫

T

F12 (t, x) dxdt 0

∫

0

1

F22 (t, x) dxdt = C(I + II).

+ CE 0

0

I, we αhave, by using isometry property and (1.14), with Gα (t − s, x, y) = ∑∞For−(t−s)λ j e (x)e (y), e j j j=1 ∫

T

∫

I=E

1

[∫ t∫

0 0 T ∫ 1∫ t

∫

0

0

0

T

= 0

0

∫ t∑ ∞

Gα (t − s, x, y)

0 k=1

0 −2(t−s)λα k

e

1

Gα (t − s, x, y) dW (s, y) −

0 1

[∫

= ∫

∫ t∫

1

∞ (∑ ( k=1

( M )2 αk dsdt =

0

]2 Gα (t − s, x, y) dW (s, y) dxdt

0

) ]2 ) σk (s) − σkM (s) ek (y) dy dsdxdt.

∫

∞ T ∑ 0

k=1

∞

∑ 1 ( ) 1 − e−2tλk ( M )2 M 2 α dt ≤ C . k α αk 2λα 2λ k k α

k=1

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FANG LIU,, MONZORUL KHAN, AND YUBIN YAN

For II, we have ∫

∫

T

1

II = E

∫ t∫

1

tj

j=0

[∫

tj

∫

tj

∫

0

0

∫

0

0

∫

tj

c (s, y) − Gα (tj − s, x, y) dW

∫ t∫

0

0

∫

0

+

∫

0

0 1

0

Gα (t − s, x, y) dW (s, y) −

Gα (tj − s, x, y) dW (s, y) −

0

]2 } c (s, y) Gα (t − s, x, y) dW dxdt

1

1

+

1

Gα (t − s, x, y) dW (s, y) −

0 0 0 0 N t −1 ∫ tj+1 ∫ 1 {[ ∫ t ∑

≤ 3E

[∫

{[ ∫ t ∫

0

1

0 1

tj

∫

1

]2 Gα (tj − s, x, y) dW (s, y)

0

]2 c (s, y) Gα (tj − s, x, y) dW

c (s, y) Gα (tj − s, x, y) dW

]2 } dxdt

0

(4.12) ( ) ≤ 3 II1 + II2 + II3 . For II2 , we have, by isometry property,

II2 = E

N t −1 ∫ tj+1 ∑ tj

j=0

−

j−1 ∫ tl+1 ∑

= −

=

∫

1 ∆t

−

1

1

l=0

{∫

tl

k=1

tj

( 1 ∫ tl+1 )]2 s)ek (y) dyd˜ s σkM (˜ dβk (s) dxdt ∆t tl

∞ ) (∑ Gα (tj − s, x, y) σkM (s)ek (y) dy

0

0 tj+1

k=1

∞ ∑

1

) σkM (s)ek (y) dy dβk (s)

k=1

j−1 ∫ tl+1 1∑

∫ 0

l=0

tl

k=1

∞ { 1 ∫ tl+1 [ ∫ 1 (∑ ) Gα (tj − s, x, y) σkM (s)ek (y) dy ∆t tl 0 k=1

∞ (∑ ) ] }2 Gα (tj − s˜, x, y) σkM (˜ s)ek (y) dy d˜ s dsdxdt k=1

j−1 ∫ tl+1 ∑ N ∞ { t −1 ∫ tj+1 ∑ ∑ j=0

=

0

∞ (∑

∞ (∑ ) }2 Gα (tj − s˜, x, y) σkM (˜ s)ek (y) dyd˜ s dsdxdt

0

=

tl

j−1 ∫ tl+1 1∑ 0

tl+1 ∫

tl

j=0

∫

1

Gα (tj − s, x, y)

0

N t −1 ∫ ∑

∫

l=0

∫

tl+1

1

tj

∫

j−1 ∫ [∑

Gα (tj − s˜, x, y)

N t −1 ∫ tj+1 ∑ j=0

1 0

tl

l=0

∫

tj

l=0

tl

k=1

j−1 ∫ tl+1 ∑ N ∞ t −1 ∫ tj+1 ∑ ∑ j=0

tj

l=0

tl

k=1

1 ∆t

∫

[ ] }2 α α e−λk (tj −s) σkM (s) − e−λk (tj −˜s) σkM (˜ s) d˜ s dsdt

tl+1

tl

e−2λk tj { ∆t2 α

∫

tl+1

[ α ] }2 α eλk s σkM (s) − eλk s˜σkM (˜ s) d˜ s dsdt.

tl

By (1.14), we have, with some ξl1 , ξl2 which lie between s and s˜, α ( α ) α α α λk s M s) s) = eλk s − eλk s˜ σkM (s) + eλk s˜(σkM (s) − σkM (˜ e σk (s) − eλk s˜σkM (˜ ( ( ) ) 1 α M ′ 2 λα ˜ k ξl ∆t σ M (s) + eλk s (σ ) (ξ ) ∆t ≤ λα e k l k k ( ) α α λα k tl+1 β M ∆t + eλk tl+1 γ M ∆t ≤ eλk tl+1 λα β M + γ M ∆t. ≤ λα k k k k k ke

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FOURIER SPECTRAL METHODS FOR SPDEs

Hence j−1 ∫ tl+1 ∑ N ∞ t −1 ∫ tj+1 ∑ ∑

II2 ≤

tj

j=0

tl

l=0

N t −1 ∫ tj+1 ∑

≤ ∆t2

∫

tj

j=0

)2 ] e−2λk tj [ 2λαk tl+1 ( α M M 4 λ β + γ dsdt e ∆t k k k ∆t2 α

k=1 ∞ ( tj ∑

0

M M λα k βk + γk

)2

dsdt ≤ C∆t2

k=1

∞ ( ∑

M M λα k βk + γk

)2 ,

k=1

where we use the inequality e−2λk (tj −tl+1 ) ≤ 1 for l = 0, 1, 2, . . . , j − 1. For II1 , we have α

II1 = E

N t −1 ∫ tj+1 ∑

0

N t −1 ∫ tj+1 ∑

∫

tj

j=0

+ 2E

1

tj

j=0

≤ 2E

∫

N t −1 ∫ ∑ j=0

[∫ t ∫ 0 1

tj

tj

0

∫

0

1

(

∫

1

Gα (t − s, x, y) dW (s, y)

]2 dxdt

0

) ]2 Gα (t − s, x, y) − Gα (tj − s, x, y) dW (s, y) dxdt

0

1 [∫ t ∫

0

tj

Gα (t − s, x, y) dW (s, y) − 0

[∫

0

tj+1 ∫

∫

1

tj

1

]2 Gα (t − s, x, y) dW (s, y) dxdt = 2(II11 + II12 ).

0

For II11 , we have, by the isometry property and (1.14), II11 =

N t −1 ∫ tj+1 ∑ j=0

≤

∫

tj

∞ ( tj ∑

0

tj

α

α

∫

tj

(βkM )2

(

α

α

0

tj

α

dsdt

( )2 α α e−2λk (t−s) 1 − e−λk (tj −t) ds

0

= 1 − e−λk (tj −t)

)2

0

k=1

)2 e

(σkM (s))2 dsdt

e−λk (t−s) − e−λk (tj −s)

Note that ∫ tj ( ∫ )2 α α e−λk (t−s) − e−λk (tj −s) ds = (

)2

k=1

N ∞ t −1 ∫ tj+1 ∑ ∑ j=0

e−λk (t−s) − e−λk (tj −s)

−2λα k (t−tj )

−e 2λα k

−2λα kt

( ≤

1 − e−λk (t−tj ) α

)2 .

2λα k

Hence, we have II11 ≤

N t −1 ∫ tj+1 ∑ j=0

∞ (∑

tj

(βkM )2

( )2 α ) 1 − e−λk (t−tj ) 2λα k

k=1

dt ≤ C

∞ ∑

( (βkM )2

k=1

1 − e−λk ∆t α

2λα k

)2 .

By (1.15) and Lemma 4.1, we have

II11 ≤ C

∞ ∑ k=1

( k −2α˜

1 − e−λk ∆t α

2λα k

)2

∫ ≤C

∞

( ) 2α α ˜ 1 ˜ x−2(α+α) 1 − e−x ∆t dx ≤ C∆t1+ α − 2α .

1

For II12 , we have, by isometry property and (1.14) and (1.15),

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FANG LIU,, MONZORUL KHAN, AND YUBIN YAN

II12

=E

N t −1 ∫ tj+1 ∑ tj

j=0

=

N t −1 ∫ tj+1 ∑ tj

j=0

≤C

∫

1

[∫ t ∫

0

tj

∫ t∑ ∞

e

≤C 0

∞

tj

dxdt

N t −1 ∑ ( M )2 σk (s) dsdt ≤

∫

j=0

( 1 − e−2λαk ∆t )] λα k

k=1

1 − e−2x ∆t dx ≤ C x2α+2α˜ 2α

]2

0

−2λα k (t−s)

k −2α˜

∫

Gα (t − s, x, y) dW (s, y)

tj k=1

N ∞ [ t −1 ∫ tj+1 ∑ ∑ j=0

1

∫

∞

dt = C

∞ [ ∑

tj+1

tj

k −2α˜

∫ t∑ ∞ (

) α k −2α˜ e−2λk (t−s) dsdt

tj k=1

( 1 − e−2λαk ∆t )]

k=1

λα k

( ) 2α ˜ x−2(α+α) 1 − e−x ∆t dx.

0

By Lemma 4.1, we have (4.13)

II12 ≤ C∆t1+ α − 2α . α ˜

1

Similarly we may show, with 0 ≤ α ˜ < 1/2, II3 ≤ C∆t1+ α − 2α . α ˜

1

Together these estimates complete the proof of Theorem 2.1. Acknowledgements. We thank Prof. Neville Ford for his consistent support and encouragements for this research. We would also like to thank Dr. Dimitra Antonopoulou and Dr. Nikos Kavallaris for their fruitful discussions about this research topic. REFERENCES [1] E. J. Allen, S. J. Novosel and Z. Zhang, Finite element and difference approximation of some linear stochastic partial differential equations, Stochastics Stochastics Rep., 64(1998), pp. 117-142. [2] D. A. Benson, S. W. Wheatcraft, and M. M. Meerschaert, The fractional order governing equations of L´ evy motion, Water Resour. Res., 36 (2000), pp. 1413-1423. [3] P. Biler, T. Funaki, , and W. A. Woyczynski, Fractal Burgers’ equations, J. Differential Equations, 148 (1998), pp. 9-46. [4] Z. Brze´ zniak, L. Debbi and B. Goldys, Ergodic properties of fractional stochastic Burgers Equations, Global and Stochastic Analysis, 1(2011), 149-174. [5] A. Bueno-Orovio, D. Kay and K. Burrage, Fourier spectral methods for fractional-in-space reaction-diffusion equations, BIT Numer. Math., 2014, DOI 10.1007/s10543-014-0484-2. [6] K. Burrage, N. Hale, and D. Kay, An efficient implicit FEM scheme for fractional-in-space reaction-diffusion equations, SIAM J. Sci. Comput., 34 (2012), pp. A2145-A2172. [7] L. A. Caffarelli and A. Vasseur, Drift diffusion equations with fractional diffusion and the quasi-geostrophic equation, Ann. of Math., 171 (2010), pp. 1903-1930. [8] Y. Cao, H. Yang and H. Yin, Finite element methods for semilinear elliptic stochastic partial differential equations, Numer. Math. 106(2007), 181-198. [9] P. Constantin, M. C. Lai, R. Sharma, Y. H. Tseng and J. Wu, New numerical results for the surface quasi-geostrophic equation, J. Sci. Comput., 50 (2012), 1-28. [10] L. Debbi and M. Dozzi, On the solutions of nonlinear stochastic fractional partial differential equations in one spatial dimension, Stoc. Proc. Appl., 115(2005), pp. 1764-1781. [11] L. Debbi and M. Dozzi, On a space discretization scheme for the fractional stochastic heat equations, arXiv: 1102.4689v1, 2011 [12] Q. Du and T. Zhang, Numerical approximation of some linear stochastic partial differential equations driven by special additive noises, SIAM J. Numer. Anal., 40(2002), pp. 14211445.

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FOURIER SPECTRAL METHODS FOR SPDEs

[13] V. J. Ervin and J. P. Roop, Variational solution of fractional advection dispersion equations on bounded domains in Rd , Numer. Methods Partial Differential Equations, 23(2007), pp. 256-281. [14] M. Ilic, F. liu, I. Turner and V. Anh, Numerical approximation of a fractional-in-space diffusion equation I, Frac. Calc. Appl. Anal., 8(2005), pp. 323-341. [15] M. Ilic, F. liu, I. Turner and V. Anh, Numerical approximation of a fractional-in-space diffusion equation II: with nonhomogeneous boundary conditions, Frac. Calc. Appl. Anal., 9(2006), pp. 333-349. [16] A. Jentzen and P. E. Kloeden, The numerical approximation of stochastic partial differential equations, Milan J. Math., 77(2009), pp. 205-244. ¨ ckner, A break of the complexity of the numerical approximation of [17] A. Jentzen and M. Ro nonlinear SPDEs with multiplicative noise, Preprint, arXiv:1001.2751v2, 2010. [18] B. Jin, R. Lazarov, J. Pasciak and Z. Zhou, Error analysis of a finite element method for the space-fractional parabolic equation, SIAM J. Numer. Anal. 52(2014), pp. 2272-2294. [19] M. A. Katsoulakis, G. T. Kossioris and O. Lakkis, Noise regularization and computations for the 1-dimensional stochastic Allen-Cahn problem, Interfaces Free Bound. 9 (2007), pp. 1-30. [20] G. T. Kossioris and G. E. Zouraris, Fully-discrete finite element approximations for a fourthorder linear stochastic parabolic equation with additive space-time white noise, M2AN Math. Model. Numer. Anal., 44 (2010), pp. 289-322. [21] X. Leoncini and G. M. Zaslavsky, Jets, stickiness and anomalous transport, Phys. Rev. E, 65(046216), 2002. [22] X. J. Li and C. J. Xu, A space-time spectral method for the time fractional diffusion equation, SIAM J. Numer. Anal., 47(2009), pp. 2108-2131 [23] X. J. Li and C. J. Xu , Existence and uniqueness of the weak solution of the space-time fractional diffusion equation and a spectral method approximation, Commun. Comput. Phys., 8(2010), pp. 1016-1051 [24] G. J. Lord, C. E. Powell, and T. Shardlow, An Introduction to Computational Stochastic PDEs, Cambridge University Press, Cambridge, UK, 2014. [25] S. N. Majumdar and A. J. Bray, Spatial persistence of fluctuating interfaces, Phys. Rev. Lett., 86(2011), pp. 3700-3703. [26] M. M. Meerschaert and C. Tadjeran, Finite difference approximations for fractional advection-dispersion flow equations, J. Comput. Appl. Math., 172 (2004), pp. 65-77. [27] J. P. Roop, Computational aspects of FEM approximation of fractional advection dispersion equations on bouinded domains in R2 , J. Comput. Appl. Math., 193(2006), pp. 243-268. ¨ ckner, R. C. Zhu and X. C. Zhu, Local existence and non-explosion of solutions for [28] M. Ro stochastic fractional partial differential equations driven by multiplicative noise, Stochastic Processes and their Applications 124(2014), pp. 1974 -2002. [29] R. Servadei and E. Valdinoci, On the spectrum of two different fractional operators, Proc. Royal Soc. Edinburgh Sec. A., 144(2014), pp. 831-855. [30] T. Shardlow, Stochastic perturbations of the Allen-Cahn equation, Electron. J. Differential Equations, 2000 (2000), pp. 1-19. [31] C. Tadjeran, M. M. Meerschaert and H. Scheffler, A second-order accurate numerical approximation for the fractional diffusion equation, J. Comput. Phys., 213(2006), pp. 205213. [32] C. Tadjeran, M. M. Meerschaert, A second-order accurate numerical method for the twodimensional fractional diffusion equation, J. Comput. Phys., 220(2007), pp. 813-823. [33] L. N. Trefethen, Spectral Methods in MATLAB, SIAM, Philadelphia, 2000. ´ [34] J. B. Walsh, An introduction to stochastic partial differential equations. In Ecole d’´ et´ e de probabilit´ es de Saint-Flour, XI-1984, volume 1180 of Lecture Notes in Math., 265-439. Springer, Berlin, 1986. [35] Y. Yan, Galerkin finite element methods for stochastic parabolic partial differential equations, SIAM J. Numer. Anal. 43 (2005), pp. 13631384. [36] G. M. Zaslavsky and S. S. Abdullaev, Scaling property and anomalous transport of particals inside the stochastic layer, Phys. Rev. E, (1995), 3901-3910.

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J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 24, NO.2, 2018, COPYRIGHT 2018 EUDOXUS PRESS, LLC

On anti-periodic type boundary value problems of sequential fractional dierential equations of order q 2 (2; 3] Ahmed Alsaedi, Mohammed H. Aqlan, Bashir Ahmad

Nonlinear Analysis and Applied Mathematics (NAAM)-Research Group, Department of Mathematics, Faculty of Science, King Abdulaziz University, P.O. Box 80203, Jeddah 21589, Saudi Arabia e-mail: [email protected], [email protected], bashirahmad [email protected] Abstract

We investigate a new kind of anti-periodic type boundary value problems of sequential fractional dierential equation of order q 2 (2; 3]. We make use of Banach's contraction mapping principle to obtain the uniqueness result while the existence of solutions is established via Krasnoselskii's xed point theorem and Leray-Schauder nonlinear alternative. The paper concludes with some illustrative examples.

Key words and phrases: Fractional dierential equations; sequential; antiperiodic; existence; xed point AMS (MOS) Subject Classi cations: 34A08; 34A12; 34A37

1

Introduction

Boundary value problems constitute an important eld of research and arise in several disciplines such as applied mathematics, control theory, mechanical structures and physics. The literature on the topic ranges from theoretical aspects of existence and uniqueness of solutions to analytic and numerical methods for nding solutions of the problems. Linear and nonlinear, singular and nonsingular, wellposed and ill-posed, local and nonlocal, free and xed problems are well known types of boundary value problems. In relation to the boundary conditions, considerable attention has been given to two-point, multi-point, periodic/anti-periodic and integral boundary value problems. In particular, anti-periodic boundary conditions are found to be quite signi cant and important in the mathematical modeling of certain physical processes and phenomena, for example, wavelets, physics, trigonometric polynomials in the study of interpolation problems, etc., for example, see [1] and the references cited therein. Dierential and integral operators of fractional-order appear in the mathematical modelling of several phenomena occurring in engineering and scienti c disciplines such as biological sciences, ecology, control theory, aerodynamics, uid dynamics, polymer rheology, regular variation in thermodynamics, etc. For more details and explanation, for instance, see [2, 3, 4]. The interest in the study of fractional-order operators is mainly due to nonlocal nature of such operators which takes into account memory and hereditary properties of some important and useful materials and processes. In recent years, fractional-order boundary value problems involving a variety of boundary conditions have been studied by several researchers. For details and examples, we refer the reader to a series of papers ([5]-[10]). For some works on sequential fractional dierential equations, for example, see ([11][15]). Anti-periodic boundary value problems of fractional-order have also been investigated in the literature ([16]-[19]). However, the study of sequential fractional dierential equations equipped with anti-periodic boundary conditions has not been investigated yet. In this paper, we consider a nonlinear anti-periodic boundary value problem of sequential fractional 310

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J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 24, NO.2, 2018, COPYRIGHT 2018 EUDOXUS PRESS, LLC

A. Alsaedi, M.H. Aqlan, B. Ahmad

dierential equations given by (

(c Dq + k c Dq 1 )u(t) = f (t; u(t)); 2 < q  3; 0 < t < T; 1 u(0) + 1 u(T ) = a; 2 u0 (0) + 2 u0 (T ) = b; 3 u00 (0) + 3 u00 (T ) = c;

where c Dq denotes the Caputo fractional derivative of order q, i ; i ; (i = 1; 2; 3); a; b; c 2 R, k and f is a given continuous function.

2

(1)

2 R+

Preliminaries and an auxiliary lemma

First of all, let us recall some basic de nitions [2, 3].

De nition 2.1 The fractional integral of order r with the lower limit zero for a function f is de ned as

1 (r)

I r f (t) =

t

Z

0

f (s) ds; t > 0; r > 0; (t s)1 r

provided the right Rhand-side is point-wise de ned on [0; 1), where (
) is the gamma function, which is de ned by (r) = 01 tr 1 e t dt.

De nition 2.2 The Riemann-Liouville fractional derivative of order r > 0; n 1 < r < n; n 2 N , is de ned as

  Z 1 d n t (t s)n r 1 f (s)ds; 0+ f (t) = (n r) dt 0

Dr

where the function f (t) has absolutely continuous derivative up to order (n

1).

De nition 2.3 The Caputo derivative of order r for a function f : [0; 1) ! R can be written as c Dr f (t) = Dr

f (t)

0+

nX1 k t k=0

k!

!

(k)

f (0) ; t > 0; n 1 < r < n:

Remark 2.4 If f (t) 2 C n [0; 1); then c Dr f (t) =

1 (n r)

t

Z

0

f (n) (s) ds = I n r f (n) (t); t > 0; n 1 < q < n: (t s)r+1 n

The following lemma plays a pivotal role in de ning the solution for problem (1).

Lemma 2.5 Let h 2 AC ([0; T ]); R). Then the following linear boundary value problem (

(c Dq + k c Dq 1 )u(t) = h(t); 2 < q  3; 0 < t < T; 1 u(0) + 1 u(T ) = a; 2 u0 (0) + 2 u0 (T ) = b; 3 u00 (0) + 3 u00 (T ) = c

(2)

is equivalent to the fractional integral equation Z T  Z s  (s x)q 2 (s x)q 2 h(x)dx ds + 2 (t) e k(T s) h(x)dx ds (q 1) (q 1) 0 0 0 0 Z T Z T q 2 q 3 (T s) (T s) +3 (t) h(s)ds + 4 (t) h(s)ds; ( q 1) (q 2) 0 0 (3)

u(t) = 1 (t) +

Z

t

e k(t s)

Z

s

311

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On anti-periodic type boundary value problems

where

 a 1 T b c  ce kt t  c2    + k

T + + b+ ; 2 1 1 1 2 k2 1 2 3 2 1 k 2  3 2 k3   k 1 T    3 e kt kt   



2 2 3

2 2 3 ; 2 (t) = 3 1 3 1 +  3 1

3 1 2 3 3 2 3   1 3 1 T  2 3

3 e kt t  2 3 3 (t) =

2 + +

2 k1 3 1 2 3 k3 2  3  

3 e kt 3 2 t

2 1 + 2 k 1 T 4 (t) = 2 3 ; k 3 1 2 k 2 3 k3 2 i = i + i e kT ; i = 1; 2; 3; 3 6= 0; 1 = 1 + 1 6= 0; 2 = 2 + 2 6= 0:

1 (t) =

(4)

(5)

Proof. Rewrite the equation (c Dq + k c Dq 1 )u(t) = h(t) as c Dq

Applying the operator I q

1

1

(D + k)u(t) = h(t):

(6)

on both sides of (6), and solving the resulting equation, we get

u(t) = A0 e kt + A1 + A2 t +

t

Z

0

e k(t s) I q 1 h(s)ds;

(7)

where A0 ; A1 and A2 are arbitrary constants and

Iq

1

h(t) =

t

Z

0

(t x)q 2 h(x)dx: (q 1)

Dierentiating (7) with respect to t; we obtain

u0 (t) = kA0 e kt + A2 k u00 (t) = k2 A0 e kt + k2

t

Z

0

t

Z

0

e k(t s) I q 1 h(s)ds + I q 1 h(t);

e k(t s) I q 1 h(s)ds kI q 1 h(t) + I q 2 h(t):

(8) (9)

Using the boundary conditions of (2) in (7)-(9), we get

1 A0 + 1 A1 + A2 1 T + 1 k2 A0 + 2 A2 + 2 

A0 k2 3 + 3 k2

T

Z

0



k

T

Z

0

Z

0

T

e k(T s) I q 1 h(s)ds = a;

(10) 

e k(T s) I q 1 h(s)ds + I q 1 h(T ) = b; 

e k(T s) I q 1 h(s)ds kI q 1 h(T ) + I q 2 h(T ) = c:

(11) (12)

Solving the system (10)-(12) for A0 ; A1 and A2 ; we nd that

A0 =

Z T  o 1 n 2 k(T s) I q 1 h(s)ds kI q 1 h(T ) + I q 2 h(T ) ; c

k e 3 k 2 3 0

A1 =

a 1



1 T b  1 2





1 3  3 1



1 3 1 T  2 3 + k3 1 1 3 2

+

1 T  2 k 1 2

1 2 1 T )  + c k2 3 1 k3 1 2

k2 3   3 2

Z

1  T k(T s) q 1 e I h(s)ds 1 0



2  q 1   T I h(T ) + 21 3 + 2 3 1 I q 2 h(T ); 2 k 3 1 k3 1 2

312

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A. Alsaedi, M.H. Aqlan, B. Ahmad

Z k2 3  T k(T s) q 1 e I h(s)ds  3 2 0

b  c  k A2 = + 2 + 2 2 k3 2 2 

+

2 3  3 2

2 3 q 2 I h(T ); k3 2

2  q 1 I h(T ) 2

where we have used (5). Substituting the values of A0 ; A1 and A2 in (7) and using the notations (4), we obtain the solution (3). By direct computation, it is easy to show that (3) satis es the problem (2). This completes the proof. 

3

Main Result

Let C = C ([0; T ]; R) denotes the Banach space of all continuous functions from [0; T ] with the norm de ned by kuk = supfju(t)j; t 2 [0; T ]g. Via Lemma 2.5, we transform the problem (1) to an equivalent xed point problem as

! R endowed

u = Hu;

(13)

where H : C ! C is de ned by t

s

 (s x)q 2 f (x; u(x))dx ds (q 1) 0 0 Z T Z s  (s x)q 2 +2 (t) e k(T s) f (x; u(x))dx ds (q 1) 0 0 Z T Z T q 2 (T s)q 3 (T s) f (s; u(s))ds + 4 (t) f (s; u(s))ds: +3 (t) (q 1) (q 2) 0 0

(Hu)(t) = 1 (t) +

Z

e k(t s)

Z

(14)

Notice that the problem (1) has solutions if the operator equation (13) has xed points. For computational convenience, we set

Q = sup t2[0;T ]

n

tq 1 (1 e kt ) j2 (t)jT q 1 (1 e kT ) j3 (t)jT q + + k (q) k (q) (q)

1

+

j4 (t)jT q (q

2o

1)

:

(15)

Now we are in a position to present our rst result which deals with the existence of a unique solution of the problem (1) and is based on Banach's contraction mapping principle.

Theorem 3.1 Assume that f : [0; T ]  R condition:

!R

is a continuous functions satisfying the Lipschitz

(A1 ) there exists a positive number ` such that jf (t; u) f (t; v)j  `ju vj; 8t 2 [0; T ]; u; v 2 R: Then the problem (1) has a unique solution on [0; T ] if ` < 1=Q; where Q is given by (15).

QM + k1 k Proof. Let us x r  ; where supt2[0;T ] jf (t; 0)j = M and Q is given by (15), and de ne 1 `Q a set Br = fu 2 C : kuk  rg: In the rst step, we show that HBr  Br ; where the operator H is de ned by (14). For any u 2 Br ; t 2 [0; T ]; we have

jf (t; u(t))j = jf (t; u(t)) f (t; 0) + f (t; 0)j  jf (t; u(t)) f (t; 0)j + jf (t; 0)j  `kuk + M  `r + M: Then, for u 2 Br ; we obtain n

k(Hu)k  sup j1 (t)j + t2[0;T ]

Z

0

t

e k(t s)

s

Z

313

0

 (s x) 2 j f (x; u(x))jdx ds ( 1)

Ahmed Alsaedi et al 310-317

J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 24, NO.2, 2018, COPYRIGHT 2018 EUDOXUS PRESS, LLC

On anti-periodic type boundary value problems

+

j2 (t)j

+

j3 (t)j

Z

T

0 Z

T

k(T s)

e

0

(T

0

Z

T

0

0

e k(T s)

e k(t s) s

Z

0

s

Z

0

(s x) 2  dx ds ( 1)

(s x)q 2  dx ds (q 1)

T (T s)q 3 o s) 2 + j3 (t)j ds + j4 (t)j ds: + k1 k ( 1) (q 2) 0 0  (`r + M )Q + k1 k  r: This shows that HBr  Br : Next we show that the operator H is a contraction. Let u; v 2 C : Then nZ t Z s  (s x) 2 k (t s) kHu Hvk  sup e j f (x; u(x)) f (x; v(x))jdx ds ( 1) t2[0;T ] 0 0 Z

T

t

nZ

t2[0;T ]

j2 (t)j

 (s x)q 2 j f (x; u(x))jdx ds (q 1)

Z T o (T s)q 3 s) 2 j f (s; u(s))jds + j4 (t)j j f (s; u(s))jds: ( 1) (q 2) 0

 (`r + M ) sup +

s

Z

+

j2 (t)j

+

j3 (t)j

+

Z

(T

j4 (t)j

T

Z

e

0 Z

k(T s)

0

 (s x)q 2 f (x; v(x)) dx ds f (x; u(x)) (q 1)

T

(T

s)q 2 f (s; v(s)) ds f (s; u(s)) (q 1)

T

(T

o s)q 3 f (s; v(s)) ds f (s; u(s)) (q 2)

0 Z

s

Z

0

 `Q k u v k;

where we have used (15). By the given assumption: ` < 1=Q; it follows that the operator H is a contraction. Thus, by Banach's contraction mapping principle, we deduce that the operator H has a xed point, which equivalently means that the problem (1) has a unique solution on [0; T ]:  Now we show the existence of solutions for the problem (1) by means of Krasnoselskiis xed point theorem, which is stated below for the reader's convenience.

Lemma 3.2 (Krasnoselskii's xed point theorem [20]) Let Y be a closed bounded, convex and nonempty subset of a Banach space X : Let '1 ; '2 be the operators such that (i) '1 y1 + '2 y2 2 Y whenever y1 ; y2 2 Y ; (ii) '1 is compact and continuous and (iii) '2 is a contraction mapping. Then there exists y 2 Y such that y = '1 y + '2 y: Theorem 3.3 Let f : [0; T ]  R ! R be a continuous function such that jf (t; x)j  g(t); 8(t; x) 2 [0; T ]  R; where g 2 C ([0; T ]; R+ ); with supt2[0;T ] jg(t)j = kgk: In addition, it is assumed that `Q1 < 1; where

Q1 = sup t2[0;T ]

n

j2 (t)jT q 1 (1 e

kT )

k (q)

+

j3 (t)jT q 1 + j4 (t)jT q 2 o: (q) (q 1)

(16)

Then the problem (1) has at least one solution on [0; T ]:

Proof. With r  Qkgk + k1 k (Q is given by (15)), we de ne operators H1 and H2 on Br = fu 2 C : kuk  rg as follows (H1 u)(t) =

t

Z

0

e

k(t s)

s

Z

0

314

 (s x) 2 f (x; u(x))dx ds; ( 1)

Ahmed Alsaedi et al 310-317

J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 24, NO.2, 2018, COPYRIGHT 2018 EUDOXUS PRESS, LLC

A. Alsaedi, M.H. Aqlan, B. Ahmad

For u; v

2

T

s

 (s x)q 2 f (x; u(x))dx ds (q 1) 0 0 Z T Z T q 2 (T s) (T s)q 3 +3 (t) f (s; u(s))ds + 4 (t) f (s; u(s))ds: (q 1) (q 2) 0 0 Br; it is easy to verify that kH1 u + H2 vk  Qkgk + k1 k: Thus, H1 u + H2 v 2 Br: Using the

(H2 u)(t) = 1 (t) + 2 (t)

Z

e

k(T s)

Z

assumption (A1 ) and (3.3),one can get kH2 u H2 vk  `Q1 k u v k; which implies that the operator H2 is a contraction in view of the given condition: `Q1 < 1: Continuity of f implies that the operator H1 is continuous. Also, H1 is uniformly bounded on Br as kT q 1 kH1 uk  (1 e k ()qT) kgk : Finally, we establish that the operator H1 is compact. Letting sup(t;u)2[0;T ]Br jf (t; u)j = fr, for t1 ; t2 2 [0; T ];

we have

k(H1 u)(t2 ) (H1 u)(t1 )k  Z t Z t Z s Z s 1 2 (s x)q 2  (s x)q 2  ks kt kt k (t2 s) 1 2 e e j  fr je dx ds + e dx ds (q 1) (q 1) 0 0 t1 0 ! 0 as t2 t1 ! 0; independent of u: Thus the operator H1 is relatively compact on Br: Hence, by the Arzela-Ascoli Theorem, the operator H1 is compact on Br: Thus all the assumptions of Lemma 3.2 are satis ed. In consequence, by the conclusion of Lemma 3.2, the problem (1) has at least one solution on [0; T ]:  In our last result, we prove the existence of solutions the problem(1) by applying Leray-Schauder nonlinear alternative.

Lemma 3.4 (Nonlinear alternative for single valued maps [20]). Let S be a closed, convex subset of a Banach space E , and V be an open subset of S with 0 2 V : Suppose that A : V ! S is continuous and compact (that is, A(V ) is a relatively compact subset of S ) map. Then either (i) A has a xed point in V ; or (ii) there is a v 2 @ V (the boundary of V in S ) and  2 (0; 1) with v = A(v):

Let f : [0; T ]  R ! R be a continuous function. Assume that (A3 ) there exist a function p 2 C ([0; T ]; R+ ); and a nondecreasing function : R+ ! R+ such that jf (t; x)j  p(t) (kuk); 8(t; u) 2 [0; T ]  R; (A4 ) there exists a constant M > 0 such that M=Q > 1; where

Theorem 3.5

Q = k1 k + kpk (kM k)Q:

(17)

Then the boundary value problem (1) has at least one solution on [0; T ]: Proof. We complete the proof in several steps. Firstly we show that the operator H : C ! C de ned by (14) maps bounded sets into bounded sets in C . For the positive number r, let Br = fu 2 C : kuk  rg be a bounded set in C : Then, for u 2 Br ; we have

j(Hu)(t)j  j1 (t)j + + +

j2 (t)j j3 (t)j

0

T

Z

0

t

Z

Z s  (s x)q 2 e k(t s) j f (x; u(x))jdx ds (q 1) 0

Z s  (s x)q 2 e k(T s) j f (x; u(x))jdx ds (q 1) 0

T (T

Z

0

Z s) 2 jf (s; u(s))jds + j (t)j T (T s)q 3 jf (s; u(s))jds: 4 ( 1) (q 2) 0

n q 1 e kt ) + j (t)j T q 1 (1 e  j1 (t)j + kpk (kuk) t (1 2 k (q) k (q)

315

kT )

+ j3 (t)j

T q 1 + j (t)j T q 2 o; 4 k (q) k (q 1)

Ahmed Alsaedi et al 310-317

J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 24, NO.2, 2018, COPYRIGHT 2018 EUDOXUS PRESS, LLC

On anti-periodic type boundary value problems

which implies that k(Hu)k  k1 (t)k + kpk (r)Q; where Q is given by (15). Next we show that H maps bounded sets into equicontinuous sets of C . Let t1 ; t2 u 2 Br , where Br is a bounded set in C . Then we obtain

2 [0; T ] with t1 < t2 and

j(Hu)(t2 ) (Hu)(t1 )j  j1 (t2 ) 1 (t1 )j t2

Z

+

0

Z s  (s x)q 2 e k(t2 s) f (x; u(x))dx ds (q 1) 0

+j2 (t2 ) 2 (t1 )j

T

Z

+j3 (t2 ) 3 (t1 ))j

0

0

Z s) 2 jf (s; u(s))jds + j (t )  (t )j T (T s)q 3 jf (s; u(s))jds 4 2 4 1 ( 1) (q 2) 0

 ( b2 + k22c3 )(t2 t1 ) + k2c2 (e

+

0

Z s  (s x)q 2 e k(t1 s) f (x; u(x))dx ds (q 1) 0

 Z s (s x)q 2 j f (x; u(x))jdx ds e k(T s) (q 1) 0

T (T

Z

t1

Z

kt2

h k(t2 t1 ) )  e kt1 ) + kpk (r) (1 e tq1 1 (1 e kt1 ) + tq2 k (q)

1



T q 2 e kt1 3 n T 2 e kT  + q 1 o 1 e k(t2 t1 )  (q) 3 k k2

T q 2 n ( 2 3 3 2 )T 2 e kT  + q 1 2 3 o(t t )i: (q) 2 3 k 3 2 2 1 Obviously the right hand side of the above inequality tends to zero independently of u 2 Br as t2 t1 ! 0: As H satis es the above assumptions, therefore it follows by the Arzela-Ascoli theorem that H : C ! C is completely continuous. The conclusion will follow form the Leray-Schauder nonlinear alternative (Lemma 3.4) once we have proved the boundedness of the set of all solutions to equations u = Hu for  2 [0; 1]: Let u be a solution. Then, for t 2 [0; T ]; and using the computations in proving that H is bounded, we have +

ju(t)j = j(Hu)(t)j  j1 (t)j + +

j2 (t)j

+

j3 (t)j

T

Z

0

0

0

 Z s (s x)q 2 e k(t s) j f (x; u(x))jdx ds (q 1) 0

 Z s (s x)q 2 j f (x; u(x))jdx ds e k(T s) (q 1) 0

T (T

Z

t

Z

Z s) 2 jf (s; u(s))jds + j (t)j T (T s)q 3 jf (s; u(s))jds 4 ( 1) (q 2) 0

n q 1 e kt ) + j (t)j T q 1 (1 e  j1 (t)j + kpk (kuk) t (1 2 k (q) k (q)

kT )

+ j3 (t)j

T q 1 + j (t)j T q 2 o: 4 k (q) k (q 1)

In consequence, we get h

i

kuk= k1 k + kpk (kuk)Q  1: In view of (A4 ), there exists M such that kuk 6= M . Let us set U = fu 2 C : kuk < M g: Note that the operator H : U ! C ([0; T ]; R) is continuous and completely continuous. From the choice of U , there is no u 2 @U such that u = H(u) for some  2 (0; 1). Consequently, by the nonlinear alternative of Leray-Schauder type (Lemma 3.4), we deduce that H has a xed point u 2 U which is a solution of the problem (1).  Example 3.6

Consider the following anti-periodic fractional boundary value problem: 8 sin u < (c D5=2 + 2c D3=2 )u(t) = + e t cos t; t 2 [0; 2]; 25 : u(0) + u(2) = 1; u0 (0) (1=2)u0 (2) = 2; u00 (0) + (1=4)u00 (2) = 1;

(18)

where f (t; u(t)) = sin25u + e t cos t; T = 2; k = 2; 1 = 1; 1 = 1; 2 = 1; 2 = 1=2; 3 = 1; 3 = 1=4; a = 1; b = 2; c = 1: With the given data, we nd that the values of Q and Q1 respectively given by (15) and (16) are Q ' 7:557935, Q1 ' 6:513574: 316

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(a)

(b)

(c)

For the applicability of Theorem 3.1, we have that `and = 1=25 as jf (t; u) f (t; v)j  251 ju vj and `Q ' 0:302317 < 1: Thus all the conditions of Theorem 3.1 are satis ed. Hence the conclusion of Theorem 3.1 implies that there exists a unique solution for problem (18)on [0; 2]: 26 Observe that jf (t; u)j  g(t) = 251 + e t cos t with kgk = 25 and `Q1 ' 0:260543 < 1: Thus all the conditions of Theorem (3.3) are satis ed. Hence, by the conclusion of Theorem (3.3), the problem (18) has at least one solution on [0; 2]: Obviously jf (t; u)j  1=25 + e t cos t: Taking (kuk) = 1; p(t) = 1=25 + e t cos t; we have Q = k1 k + kpk (kM k)Q ' 13:224428 (Q is given by (17)) so that M > 13:224428. Thus all the conditions of Theorem 3.5 are satis ed. Hence it follows by the conclusion of Theorem 3.5 that the problem (18) has at least one solution on [0; 2]:

References [1] A. Alsaedi, S. Sivasundaram, B. Ahmad, On the generalization of second order nonlinear anti-periodic boundary value problems, Nonlinear Stud. 16 (2009), 415-420. [2] I. Podlubny, Fractional Dierential Equations, Academic Press, San Diego, 1999. [3] A.A. Kilbas, H.M. Srivastava, J.J. Trujillo, Theory and Applications of Fractional Dierential Equations, North-Holland Mathematics Studies, 204. Elsevier Science B.V., Amsterdam, 2006. [4] J. Klafter, S. C Lim, R. Metzler (Editors), Fractional Dynamics in Physics, World Scienti c, Singapore, 2011. [5] D. O'Regan, S. Stanek, Fractional boundary value problems with singularities in space variables, Nonlinear Dynam. 71 (2013), 641-652. [6] J. Henderson, R. Luca, Positive solutions for a system of nonlocal fractional boundary value problems. Fract. Calc. Appl. Anal. 16 (2013), 985-1008. [7] X. Liu, Z. Liu, X. Fu, Relaxation in nonconvex optimal control problems described by fractional dierential equations, J. Math. Anal. Appl. 409 (2014), 446-458. [8] F. Punzo, G. Terrone, On the Cauchy problem for a general fractional porous medium equation with variable density, Nonlinear Anal. 98 (2014), 27-47. [9] J.R. Wang, Y. Zhou, M. Feckan, On the nonlocal Cauchy problem for semilinear fractional order evolution equations, Cent. Eur. J. Math. 12 (2014), 911-922. [10] I. Area, J. Losada, J.J. Nieto, A note on the fractional logistic equation, Phys. A 444 (2016), 182-187. [11] B. Ahmad, J.J. Nieto, Sequential fractional dierential equations with three-point boundary conditions, Comput. Math. Appl. 64 (2012), no. 10, 3046-3052. [12] B. Ahmad, J.J. Nieto, Boundary value problems for a class of sequential integrodierential equations of fractional order, J. Funct. Spaces Appl. 2013(2013), Article ID 149659, 1-8. [13] B. Ahmad, S.K. Ntouyas, Existence results for a coupled system of Caputo type sequential fractional dierential equations with nonlocal integral boundary conditions, Appl. Math. Comput. 266 (2015), 615622. [14] M. Klimek, Sequential fractional dierential equations with Hadamard derivative, Commun. Nonlinear Sci. Numer. Simul. 16 (2011), 4689-4697. [15] H. Ye, R. Huang, On the nonlinear fractional dierential equations with Caputo sequential fractional derivative, Adv. Math. Phys. (2015), Art. ID 174156, 9 pp. [16] J. Cao, Q. Yang, Z. Huang, Existence of anti-periodic mild solutions for a class of semilinear fractional dierential equations, Commun. Nonlinear Sci. Numer. Simul. 17 (2012), no. 1, 277-283. [17] B. Ahmad, S.K. Ntouyas, A boundary value problem of fractional dierential equations with anti-periodic type integral boundary conditions, J. Comput. Anal. Appl. 15 (2013), 1372-1380. [18] B. Ahmad, J. Losada, J.J. Nieto, On antiperiodic nonlocal three-point boundary value problems for nonlinear fractional dierential equations, Discrete Dyn. Nat. Soc. 2015(2015), Art. ID 973783, 7 pp. [19] J. Jiang, Solvability of anti-periodic boundary value problem for coupled system of fractional p-Laplacian equation, Adv. Dierence Equ. 2015:305 (2015), 11 pp. [20] D.R. Smart, Fixed Point Theorems, Cambridge University Press, 1980.

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ADDITIVE-QUADRATIC ρ-FUNCTIONAL INEQUALITIES IN FUZZY NORMED SPACES JUNG RYE LEE1 , CHOONKIL PARK2∗ , DONG YUN SHIN3∗ , AND SUNGSIK YUN4 Abstract. Let 1 1 1 3 f (x + y) − f (−x − y) + f (x − y) + f (y − x) − f (x) − f (y), 4  4 4      4 x+y x−y y−x M2 f (x, y) : = 2f +f +f − f (x) − f (y). 2 2 2 Using the fixed point method, we prove the Hyers-Ulam stability of the additive-quadratic ρ-functional inequalities M1 f (x, y) :

=

N (M1 f (x, y), t) ≥ N (ρM2 f (x, y), t)

(0.1)

where ρ is a fixed real number with |ρ| < 1, and N (M2 f (x, y), t) ≥ N (ρM1 f (x, y), t)

(0.2)

where ρ is a fixed real number with |ρ| < 12 .

1. Introduction and preliminaries Katsaras [19] defined a fuzzy norm on a vector space to construct a fuzzy vector topological structure on the space. Some mathematicians have defined fuzzy norms on a vector space from various points of view [15, 21, 48]. In particular, Bag and Samanta [3], following Cheng and Mordeson [11], 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 [4]. We use the definition of fuzzy normed spaces given in [3, 25, 26] to investigate the Hyers-Ulam stability of additive ρ-functional inequalities in fuzzy Banach spaces. Definition 1.1. [3, 25, 26, 27] Let X be a real vector space. A function N : X × R → [0, 1] is called a fuzzy norm on X if for all x, y ∈ X and all s, t ∈ R, (N1 ) N (x, t) = 0 for t ≤ 0; (N2 ) x = 0 if and only if N (x, t) = 1 for all t > 0; t ) if c 6= 0; (N3 ) N (cx, t) = N (x, |c| (N4 ) N (x + y, s + t) ≥ min{N (x, s), N (y, t)}; (N5 ) N (x, ·) is a non-decreasing function of R and limt→∞ N (x, t) = 1. (N6 ) for x 6= 0, N (x, ·) is continuous on R. The pair (X, N ) is called a fuzzy normed vector space. The properties of fuzzy normed vector spaces and examples of fuzzy norms are given in [25, 28]. Definition 1.2. [3, 28, 26, 27] Let (X, N ) be a fuzzy normed vector space. A sequence {xn } in X is said to be convergent or converge if there exists an x ∈ X such that limn→∞ N (xn −x, t) = 1 for all t > 0. In this case, x is called the limit of the sequence {xn } and we denote it by N limn→∞ xn = x. 2010 Mathematics Subject Classification. Primary 46S40, 39B52, 47H10, 39B62, 26E50, 47S40. Key words and phrases. fuzzy Banach space; fixed point method; additive-quadratic ρ-functional inequality; Hyers-Ulam stability. ∗ Corresponding authors.

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Definition 1.3. [3, 28, 26, 27] Let (X, N ) be a fuzzy normed vector space. A sequence {xn } in X is called Cauchy if for each ε > 0 and each t > 0 there exists an n0 ∈ N such that for all n ≥ n0 and all p > 0, we have N (xn+p − xn , t) > 1 − ε. It is well-known that every convergent sequence in a fuzzy normed vector space is Cauchy. If each Cauchy sequence is convergent, then the fuzzy norm is said to be complete and the fuzzy normed vector space is called a fuzzy Banach space. We say that a mapping f : X → Y between fuzzy normed vector spaces X and Y is continuous at a point x0 ∈ X if for each sequence {xn } converging to x0 in X, then the sequence {f (xn )} converges to f (x0 ). If f : X → Y is continuous at each x ∈ X, then f : X → Y is said to be continuous on X (see [4]). The stability problem of functional equations originated from a question of Ulam [47] concerning the stability of group homomorphisms. The functional equation f (x + y) = f (x) + f (y) is called the Cauchy equation. In particular, every solution of the Cauchy equation is said to be an additive mapping. Hyers [17] gave a first affirmative partial answer to the question of Ulam for Banach spaces. Hyers’ Theorem was generalized by Aoki [2] for additive mappings and by Rassias [39] for linear mappings by considering an unbounded Cauchy difference. A generalization of the Rassias theorem was obtained by G˘avruta [16] by replacing the unbounded Cauchy difference by a general control function in the spirit of Rassias’ approach. The functional equation f (x+y)+f (x−y) = 2f (x)+2f (y) is called the quadratic functional equation. In particular, every solution of the quadratic functional equation is said to be a quadratic mapping. The stability of quadratic functional equation was proved by Skof [46] for mappings f : E1 → E2 , where E1 is a normed space and E2 is a Banach space. Cholewa [12] noticed that the theorem of Skof is still true if the relevant domain E1 is replaced by an Abelian group. The stability problems of various functional equations have been extensively investigated by a number of authors (see [1, 5, 9, 10, 14, 22, 24, 29, 34, 35, 36, 40, 41, 42, 43, 44, 45, 49, 50]). We recall a fundamental result in fixed point theory. Theorem 1.4. [6, 13] Let (X, d) be a complete generalized metric space and let J : X → X be a strictly contractive mapping with Lipschitz constant α < 1. Then for each given element x ∈ X, either d(J n x, J n+1 x) = ∞ for all nonnegative integers n or there exists a positive integer n0 such that (1) d(J n x, J n+1 x) < ∞, ∀n ≥ n0 ; (2) the sequence {J n x} converges to a fixed point y ∗ of J; (3) y ∗ is the unique fixed point of J in the set Y = {y ∈ X | d(J n0 x, y) < ∞}; 1 d(y, Jy) for all y ∈ Y . (4) d(y, y ∗ ) ≤ 1−α In 1996, G. Isac and Th.M. Rassias [18] 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 [7, 8, 30, 31, 38]). Park [32, 33] defined additive ρ-functional inequalities and proved the Hyers-Ulam stability of the additive ρ-functional inequalities in Banach spaces and non-Archimedean Banach spaces. In Section 2, we prove the Hyers-Ulam stability of the additive-quadratic ρ-functional inequality (0.1) in fuzzy Banach spaces by using the fixed point method. In Section 3, we prove the Hyers-Ulam stability of the additive-quadratic ρ-functional inequality (0.2) in fuzzy Banach spaces by using the fixed point method. 2. Additive-quadratic ρ-functional inequality (0.1) In this section, we prove the Hyers-Ulam stability of the additive-quadratic ρ-functional inequality (0.1) in fuzzy Banach spaces. Let ρ be a real number with |ρ| < 1. We need the following lemma to prove the main results.

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Lemma 2.1. (i) If an odd mapping f : X → Y satisfies N (M1 f (x, y), t) ≥ N (ρM2 f (x, y), t)

(2.1)

for all x, y ∈ X and all t > 0, then f is the Cauchy additive mapping. (ii) If an even mapping f : X → Y satisfies f (0) = 0 and (2.1), then f is the quadratic mapping. Proof. (i) Letting y = x in (2.1), we get N (f (2x) − 2f (x), t) = 1 for all t > 0 and so f (2x) = 2f (x) for all x ∈ X. Thus x 2

 

f

1 = f (x) 2

(2.2)

for all x ∈ X. It follows from (2.1) and (2.2) that x+y − f (x) − f (y) , t) N (f (x + y) − f (x) − f (y), t) = N (ρ 2f 2 = N (ρ(f (x + y) − f (x) − f (y)), t) 







for all t > 0 and so f (x + y) = f (x) + f (y) for all x, y ∈ X by (N5 ).   (ii) Letting y = x in (2.1), we get N 12 f (2x) − 2f (x), t = 1 for all t > 0 and so f (2x) = 4f (x) for all x ∈ X. Thus x 2

 

f

1 = f (x) 4

(2.3)

for all x ∈ X. It follows from (2.1) and (2.3) that 

N

1 1 f (x + y) + f (x − y) − f (x) − f (y), t 2 2         x+y x−y + 2f − f (x) − f (y) , t = N ρ 2f 2 2     1 1 =N ρ f (x + y) + f (x − y) − f (x) − f (y) , t 2 2 

for all t > 0 and so f (x + y) + f (x − y) = 2f (x) + 2f (y) for all x, y ∈ X by (N5 ).



Using the fixed point method, we prove the Hyers-Ulam stability of the additive-quadratic ρ-functional inequality (0.1) in fuzzy Banach spaces. Theorem 2.2. Let ϕ : X 2 → [0, ∞) be a function such that there exists an L < 1 with ϕ(x, y) ≤

L L ϕ (2x, 2y) ≤ ϕ (2x, 2y) 4 2

(2.4)

for all x, y ∈ X. (i) Let f : X → Y be an odd mapping satisfying t N (M1 f (x, y), t) ≥ min N (ρM2 f (x, y), t) , t + ϕ(x, y) 

320



(2.5)

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for all x, y ∈ X and all t > 0. Then A(x) := N -limn→∞ 2n f defines an additive mapping A : X → Y such that




exists for each x ∈ X and

(2 − 2L)t (2 − 2L)t + Lϕ(x, x)

N (f (x) − A(x), t) ≥

(2.6)

for all x ∈ X and all t > 0. (ii) Let f : X →
Y be an even mapping satisfying f (0) = 0 and (2.5). Then Q(x) := N limn→∞ 4n f 2xn exists for each x ∈ X and defines a quadratic mapping Q : X → Y such that (2 − 2L)t N (f (x) − Q(x), t) ≥ (2.7) (2 − 2L)t + Lϕ(x, x) for all x ∈ X and all t > 0. Proof. (i) Letting y = x in (2.5), we get t t + ϕ(x, x)

N (f (2x) − 2f (x), t) ≥

(2.8)

and so 

x t
,t ≥ 2 t + ϕ x2 , x2

 

N f (x) − 2f



(2.9)

for all x ∈ X. Consider the set S := {g : X → Y } and introduce the generalized metric on S: t , ∀x ∈ X, ∀t > 0 , t + ϕ(x, x) 



d(g, h) = inf µ ∈ R+ : N (g(x) − h(x), µt) ≥

where, as usual, inf φ = +∞. It is easy to show that (S, d) is complete (see [23, Lemma 2.1]). Now we consider the linear mapping J : S → S such that x 2

 

Jg(x) := 2g for all x ∈ X. Let g, h ∈ S be given such that d(g, h) = ε. Then N (g(x) − h(x), εt) ≥

t t + ϕ(x, x)

for all x ∈ X and all t > 0. Hence x x x L , Lεt = N g −h , εt 2 2 2 2 Lt Lt t 2 2 =
≥ Lt x x L t + ϕ(x, x) + ϕ 2, 2 2 + 2 ϕ(x, x)



≥

Lt 2

x 2

 

N (Jg(x) − Jh(x), Lεt) = N 2g

 



  

 



− 2h

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 (2.9) that N f (x) − 2f

x 2






, L2 t ≥

t t+ϕ(x,x)

for all x ∈ X and all t > 0. So

L 2.

d(f, Jf ) ≤ By Theorem 1.4, there exists a mapping A : X → Y satisfying the following:

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(1) A is a fixed point of J, i.e., x 1 = A(x) (2.10) 2 2 for all x ∈ X. Since f : X → Y is odd, A : X → Y is an odd mapping. The mapping A is a unique fixed point of J in the set  

A

M = {g ∈ S : d(f, g) < ∞}. This implies that A is a unique mapping satisfying (2.10) such that there exists a µ ∈ (0, ∞) satisfying t N (f (x) − A(x), µt) ≥ t + ϕ(x, x) for all x ∈ X; (2) d(J n f, A) → 0 as n → ∞. This implies the equality   x n = A(x) N - lim 2 f n→∞ 2n for all x ∈ X; 1 (3) d(f, A) ≤ 1−L d(f, Jf ), which implies the inequality d(f, A) ≤

L . 2 − 2L

This implies that the inequality (2.6) holds. By (2.5), 



n

N 2 M1 f

(

x y , , 2n t ≥ min N 2n M2 f 2n 2n 







x y t
, n , 2n t , n 2 2 t + ϕ 2xn , 2yn 



)

and so 

n

N 2 M1 f



(

x y , , t ≥ min N 2n M2 f 2n 2n 







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

x y , ,t , 2n 2n 



t 2n t Ln + 2n ϕ(x,y) 2n

t 2n

+

t 2n Ln 2n ϕ (x, y)

)

= 1 for all x, y ∈ X and all

N (M1 A(x, y), t) ≥ N (ρM2 A(x, y), t) for all x, y ∈ X and all t > 0. By Lemma 2.1, the mapping A : X → Y is Cauchy additive. (ii) Letting y = x in (2.5), we get   t 1 f (2x) − 2f (x), t ≥ (2.11) N 2 t + ϕ(x, x) and so     t x t 2

N f (x) − 4f ,t ≥ t (2.12) x x = t + 2ϕ x , x 2 + ϕ , 2 2 2 2 2 for all x ∈ X. Now we consider the linear mapping J : S → S such that   x Jg(x) := 4g 2 for all x ∈ X. Let g, h ∈ S be given such that d(g, h) = ε. Then t N (g(x) − h(x), εt) ≥ t + ϕ(x, x)

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for all x ∈ X and all t > 0. Hence x x x L − 4h , Lεt = N g −h , εt 2 2 2 4 Lt Lt t 4 4 =
≥ Lt x x L t + ϕ(x, x) + ϕ 2, 2 4 + 4 ϕ(x, x)

x N (Jg(x) − Jh(x), Lεt) = N 4g 2 

≥

Lt 4

 

 



  

 



for all x ∈ X and all t > 0. So d(g, h) = ε implies that d(Jg, Jh) ≤ Lε. This means that d(Jg, Jh) ≤ Ld(g, h) for all g, h ∈ S.  It follows from (2.12) that N f (x) − 4f

x 2






, L2 t ≥

t t+ϕ(x,x)

for all x ∈ X and all t > 0. So

L 2.

d(f, Jf ) ≤ By Theorem 1.4, there exists a mapping Q : X → Y satisfying the following: (1) Q is a fixed point of J, i.e.,   x 1 = Q(x) (2.13) Q 2 4 for all x ∈ X. Since f : X → Y is even, Q : X → Y is a even mapping. The mapping Q is a unique fixed point of J in the set M = {g ∈ S : d(f, g) < ∞}. This implies that Q is a unique mapping satisfying (2.13) such that there exists a µ ∈ (0, ∞) satisfying t N (f (x) − Q(x), µt) ≥ t + ϕ(x, x) for all x ∈ X; (2) d(J n f, Q) → 0 as n → ∞. This implies the equality   x n N - lim 4 f = Q(x) n→∞ 2n for all x ∈ X; 1 (3) d(f, Q) ≤ 1−L d(f, Jf ), which implies the inequality d(f, Q) ≤

L . 2 − 2L

This implies that the inequality (2.7) holds. By (2.5), 



n

N 4 M1 f

(

x y , , 4n t ≥ min N 4n M2 f 2n 2n 







x y t
, , 4n t , 2n 2n t + ϕ 2xn , 2yn 



)

and so 

n

N 4 M1 f



(

x y , , t ≥ min N 4n M2 f 2n 2n 







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

x y , ,t , 2n 2n 



t 4n t Ln + 4n ϕ(x,y) 4n

t 4n

+

t 4n n L 4n ϕ (x, y)

)

= 1 for all x, y ∈ X and all

N (M1 Q(x, y), t) ≥ N (ρM2 Q(x, y), t) for all x, y ∈ X and all t > 0. By Lemma 2.1, the mapping Q : X → Y is quadratic.



Corollary 2.3. Let θ ≥ 0 and let p be a real number with p > 2. Let X be a normed vector space with norm k · k.

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(i) Let f : X → Y be an odd mapping satisfying 

N (M1 f (x, y), t) ≥ min N (ρM2 f (x, y), t) ,

t t + θ(kxkp + kykp )



(2.14)

for all x, y ∈ X and all t > 0. Then A(x) := N -limn→∞ 2n f ( 2xn ) exists for each x ∈ X and defines an additive mapping A : X → Y such that N (f (x) − A(x), t) ≥

(2p

(2p − 2)t − 2)t + 2θkxkp

for all x ∈ X and all t > 0. (ii) Let f : X → Y be an even mapping satisfying f (0) = 0 and (2.14). Then Q(x) := N limn→∞ 4n f ( 2xn ) exists for each x ∈ X and defines a quadratic mapping Q : X → Y such that (2p − 4)t N (f (x) − Q(x), t) ≥ p (2 − 4)t + 4θkxkp for all x ∈ X and all t > 0. Proof. The proof follows from Theorem 2.2 by taking ϕ(x, y) := θ(kxkp + kykp ) for all x, y ∈ X. Choosing L = 21−p for an odd mapping case and L = 22−p for an even mapping case, then we obtain the desired results.  Theorem 2.4. Let ϕ : X 2 → [0, ∞) be a function such that there exists an L < 1 with     x y x y ϕ (x, y) ≤ 2Lϕ , ≤ 4Lϕ , (2.15) 2 2 2 2 for all x, y ∈ X.. (i) Let f : X → Y be an odd mapping satisfying (2.5). Then A(x) := N -limn→∞ 21n f (2n x) exists for each x ∈ X and defines an additive mapping A : X → Y such that N (f (x) − A(x), t) ≥

(2 − 2L)t (2 − 2L)t + ϕ(x, x)

for all x ∈ X and all t > 0. (ii) Let f : X → Y be an even mapping satisfying f (0) = 0 and (2.5). Then Q(x) := N limn→∞ 41n f (2n x) exists for each x ∈ X and defines a quadratic mapping Q : X → Y such that (2 − 2L)t N (f (x) − Q(x), t) ≥ (2 − 2L)t + ϕ(x, x) for all x ∈ X and all t > 0. Proof. Let (S, d) be the generalized metric space defined in the proof of Theorem 2.2. (i) It follows from (2.8) that   1 1 t N f (x) − f (2x), t ≥ 2 2 t + ϕ(x, x) for all x ∈ X and all t > 0. (ii) It follows from (2.11) that 1 1 t N f (x) − f (2x), t ≥ 4 2 t + ϕ(x, x) 



for all x ∈ X and all t > 0. The rest of the proof is similar to the proof of Theorem 2.2.



Corollary 2.5. Let θ ≥ 0 and let p be a real number with 0 < p < 1. Let X be a normed vector space with norm k · k.

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(i) Let f : X → Y be an odd mapping satisfying (2.14). Then A(x) := N -limn→∞ 21n f (2n x) exists for each x ∈ X and defines an additive mapping A : X → Y such that (2 − 2p )t N (f (x) − A(x), t) ≥ (2 − 2p )t + 2θkxkp for all x ∈ X and all t > 0. (ii) Let f : X → Y be an even mapping satisfying f (0) = 0 and (2.14). Then Q(x) := N limn→∞ 41n f (2n x) exists for each x ∈ X and defines a quadratic mapping Q : X → Y such that (4 − 2p )t N (f (x) − Q(x), t) ≥ (4 − 2p )t + 4θkxkp for all x ∈ X and all t > 0. Proof. The proof follows from Theorem 2.4 by taking ϕ(x, y) := θ(kxkp + kykp ) for all x, y ∈ X. Choosing L = 2p−1 for an odd mapping case and L = 2p−2 for an even mapping case, then we obtain the desired results.  3. Additive-quadratic ρ-functional inequality (0.2) In this section, we prove the Hyers-Ulam stability of the additive-quadratic ρ-functional inequality (0.2) in fuzzy Banach spaces. Let ρ be a real number with |ρ| < 21 . Lemma 3.1. (i) If an odd mapping f : X → Y satisfies N (M2 f (x, y), t) ≥ N (ρM1 f (x, y), t)

(3.1)

for all x, y ∈ X and all t > 0, then f is the Cauchy additive mapping. (ii) If an even mapping f : X → Y satisfies f (0) = 0 and (3.1), then f is the quadratic mapping.

Proof. (i) Letting y = 0 in (3.1), we get N 2f x2 − f (x), t = 1 for all t > 0. So x 2

 

f

1 = f (x) 2

(3.2)

for all x ∈ X. It follows from (3.1) and (3.2) that x+y − f (x) − f (y), t 2 = N (ρ(f (x + y) − f (x) − f (y)), t) 







N (f (x + y) − f (x) − f (y), t) ≥ N 2f

for all t > 0 and so f (x + y) = f (x) + f (y) for all x, y ∈ X by (N5 ).

(ii) Letting y = 0 in (3.1), we get N 4f x2 − f (x), t for all t > 0. So x 2

 

f

1 = f (x) 4

(3.3)

for all x ∈ X. It follows from (3.1) and (3.3) that   1 1 f (x + y) + f (x − y) − f (x) − f (y), t N 2 2       x+y x−y ≥ N 2f + 2f − f (x) − f (y), t 2 2     1 1 =N ρ f (x + y) + f (x − y) − f (x) − f (y) , t 2 2

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for all t > 0 and so f (x + y) + f (x − y) = 2f (x) + 2f (y) for all x, y ∈ X by (N5 ).



Using the fixed point method, we prove the Hyers-Ulam stability of the additive-quadratic ρ-functional inequality (0.2) in fuzzy Banach spaces. Theorem 3.2. Let ϕ : X 2 → [0, ∞) be a function satisfying (2.4). (i) Let f : X → Y be an odd mapping satisfying   t N (M2 f (x, y), t) ≥ min N (ρM1 f (x, y), t) , t + ϕ(x, y)

(3.4)

for all x, y ∈ X and all t > 0. Then A(x) := N -limn→∞ 2n f 2xn exists for each x ∈ X and defines an additive mapping A : X → Y such that (1 − L)t N (f (x) − A(x), t) ≥ (1 − L)t + ϕ(x, x) for all x ∈ X and all t > 0. (ii) Let f : X →
Y be an even mapping satisfying f (0) = 0 and (3.4). Then Q(x) := N limn→∞ 4n f 2xn exists for each x ∈ X and defines a quadratic mapping Q : X → Y such that (1 − L)t N (f (x) − Q(x), t) ≥ (1 − L)t + ϕ(x, x) for all x ∈ X and all t > 0.


Proof. (i) Letting y = 0 in (3.4), we get         x t x , t = N 2f − f (x), t ≥ N f (x) − 2f 2 2 t + ϕ(x, 0) for all x ∈ X. Consider the set S := {g : X → Y } and introduce the generalized metric on S:   t d(g, h) = inf µ ∈ R+ : N (g(x) − h(x), µt) ≥ , ∀x ∈ X, ∀t > 0 , t + ϕ(x, 0)

(3.5)

where, as usual, inf φ = +∞. It is easy to show that (S, d) is complete (see [23, Lemma 2.1]). The rest of the proof is similar to the proof of Theorem 2.2 (i). (ii) Letting y = 0 in (3.4), we get         x x t , t = N 4f − f (x), t ≥ N f (x) − 4f (3.6) 2 2 t + ϕ(x, 0) for all x ∈ X. The rest of the proof is similar to the proof of Theorem 2.2 (ii).  Corollary 3.3. Let θ ≥ 0 and let p be a real number with p > 2. Let X be a normed vector space with norm k · k. (i) Let f : X → Y be an odd mapping satisfying t N (M2 f (x, y) − ρM1 f (x, y), t) ≥ (3.7) t + θ(kxkp + kykp ) for all x, y ∈ X and all t > 0. Then A(x) := N -limn→∞ 2n f ( 2xn ) exists for each x ∈ X and defines an additive mapping A : X → Y such that (2p − 2)t N (f (x) − A(x), t) ≥ p (2 − 2)t + 2p θkxkp

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for all x ∈ X and all t > 0. (ii) Let f : X → Y be an even mapping satisfying f (0) = 0 and (3.7). Then Q(x) := N limn→∞ 4n f ( 2xn ) exists for each x ∈ X and defines a quadratic mapping Q : X → Y such that (2p − 4)t N (f (x) − Q(x), t) ≥ p (2 − 4)t + 2p θkxkp for all x ∈ X and all t > 0. Proof. The proof follows from Theorem 3.2 by taking ϕ(x, y) := θ(kxkp + kykp ) for all x, y ∈ X. Choosing L = 21−p for an odd mapping case and L = 22−p for an even mapping case, then we obtain the desired results.  Theorem 3.4. Let ϕ : X 2 → [0, ∞) be a function satisfying (2.15). (i) Let f : X → Y be an odd mapping satisfying (3.4). Then A(x) := N -limn→∞ exists for each x ∈ X and defines an additive mapping A : X → Y such that N (f (x) − A(x), t) ≥

1 2n f

(2n x)

(1 − L)t (1 − L)t + Lϕ(x, x)

for all x ∈ X and all t > 0. (ii) Let f : X → Y be an even mapping satisfying f (0) = 0 and (3.4). Then Q(x) := N limn→∞ 41n f (2n x) exists for each x ∈ X and defines a quadratic mapping Q : X → Y such that (1 − L)t N (f (x) − Q(x), t) ≥ (1 − L)t + Lϕ(x, x) for all x ∈ X and all t > 0. Proof. Let (S, d) be the generalized metric space defined in the proof of Theorem 3.2. (i) It follows from (3.5) that 1 t N f (x) − f (2x), 2 2 



≥

t t + ϕ(2x, 0)

and so

1 2Lt t N f (x) − f (2x), Lt ≥ = 2 2Lt + ϕ(2x, 0) t + ϕ(x, 0) for all x ∈ X and all t > 0. (ii) It follows from (3.6) that 



1 t N f (x) − f (2x), 4 4 



≥

t t + ϕ(2x, 0)

and so

4Lt t 1 N f (x) − f (2x), Lt ≥ = 4 4Lt + ϕ(2x, 0) t + ϕ(x, 0) for all x ∈ X and all t > 0. The rest of the proof is similar to the proof of Theorem 2.2. 





Corollary 3.5. Let θ ≥ 0 and let p be a real number with 0 < p < 1. Let X be a normed vector space with norm k · k. (i) Let f : X → Y be an odd mapping satisfying (3.7). Then A(x) := N -limn→∞ 21n f (2n x) exists for each x ∈ X and defines an additive mapping A : X → Y such that N (f (x) − A(x), t) ≥

(2 − 2p )t (2 − 2p )t + 2p θkxkp

for all x ∈ X.

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(ii) Let f : X → Y be an even mapping satisfying f (0) = 0 and (3.7). Then Q(x) := N limn→∞ 41n f (2n x) exists for each x ∈ X and defines a quadratic mapping Q : X → Y such that (4 − 2p )t N (f (x) − Q(x), t) ≥ (4 − 2p )t + 2p θkxkp for all x ∈ X. Proof. The proof follows from Theorem 3.4 by taking ϕ(x, y) := θ(kxkp + kykp ) for all x, y ∈ X. Choosing L = 2p−1 for an odd mapping case and L = 2p−2 for an even mapping case, then we obtain the desired results.  References [1] M. Adam, On the stability of some quadratic functional equation, J. Nonlinear Sci. Appl. 4 (2011), 50–59. [2] T. Aoki, On the stability of the linear transformation in Banach spaces, J. Math. Soc. Japan 2 (1950), 64–66. [3] T. Bag, S. K. Samanta, Finite dimensional fuzzy normed linear spaces, J. Fuzzy Math. 11 (2003), 687–705. [4] T. Bag, S. K. Samanta, Fuzzy bounded linear operators, Fuzzy Sets and Systems 151 (2005), 513–547. [5] L. C˘ adariu, L. G˘ avruta, P. G˘ avruta, On the stability of an affine functional equation, J. Nonlinear Sci. Appl. 6 (2013), 60–67. [6] L. C˘ adariu, V. Radu, Fixed points and the stability of Jensen’s functional equation, J. Inequal. Pure Appl. Math. 4, no. 1, Art. ID 4 (2003). [7] L. C˘ adariu, V. Radu, On the stability of the Cauchy functional equation: a fixed point approach, Grazer Math. Ber. 346 (2004), 43–52. [8] L. C˘ adariu, V. Radu, Fixed point methods for the generalized stability of functional equations in a single variable, Fixed Point Theory Appl. 2008, Art. ID 749392 (2008). [9] A. Chahbi, N. Bounader, On the generalized stability of d’Alembert functional equation, J. Nonlinear Sci. Appl. 6 (2013), 198–204. [10] I. Chang, Y. Lee, Additive and quadratic type functional equation and its fuzzy stability, Results Math. 63 (2013), 717–730. [11] S. C. Cheng, J. M. Mordeson, Fuzzy linear operators and fuzzy normed linear spaces, Bull. Calcutta Math. Soc. 86 (1994), 429–436. [12] P. W. Cholewa, Remarks on the stability of functional equations, Aequationes Math. 27 (1984), 76–86. [13] J. Diaz, B. Margolis, A fixed point theorem of the alternative for contractions on a generalized complete metric space, Bull. Amer. Math. Soc. 74 (1968), 305–309. [14] G. Z. Eskandani, P. Gˇ avruta, Hyers-Ulam-Rassias stability of pexiderized Cauchy functional equation in 2-Banach spaces, J. Nonlinear Sci. Appl. 5 (2012), 459–465. [15] C. Felbin, Finite dimensional fuzzy normed linear spaces, Fuzzy Sets and Systems 48 (1992), 239–248. [16] P. G˘ avruta, A generalization of the Hyers-Ulam-Rassias stability of approximately additive mappings, J. Math. Anal. Appl. 184 (1994), 431–436. [17] D.H. Hyers, On the stability of the linear functional equation, Proc. Nat. Acad. Sci. U.S.A. 27 (1941), 222–224. [18] G. Isac, Th. M. Rassias, Stability of ψ-additive mappings: Appications to nonlinear analysis, Internat. J. Math. Math. Sci. 19 (1996), 219–228. [19] A. K. Katsaras, Fuzzy topological vector spaces II, Fuzzy Sets and Systems 12 (1984), 143–154. [20] I. Kramosil, J. Michalek, Fuzzy metric and statistical metric spaces, Kybernetica 11 (1975), 326–334. [21] S. V. Krishna, K. K. M. Sarma, Separation of fuzzy normed linear spaces, Fuzzy Sets and Systems 63 (1994), 207–217. [22] G. Lu, Y. Wang, P. Ye n-Jordan ∗-derivations on induced fuzzy C ∗ -algebras, J. Comput. Anal. Appl. 20 (2016), 266–276. [23] D. Mihet¸, V. Radu, On the stability of the additive Cauchy functional equation in random normed spaces, J. Math. Anal. Appl. 343 (2008), 567–572. [24] D. Mihet, R. Saadati, On the stability of the additive Cauchy functional equation in random normed spaces, Appl. Math. Lett. 24 (2011), 2005–2009. [25] A. K. Mirmostafaee, M. Mirzavaziri, M. S. Moslehian, Fuzzy stability of the Jensen functional equation, Fuzzy Sets and Systems 159 (2008), 730–738. [26] A. K. Mirmostafaee, M. S. Moslehian, Fuzzy versions of Hyers-Ulam-Rassias theorem, Fuzzy Sets and Systems 159 (2008), 720–729.

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[27] A. K. Mirmostafaee, M. S. Moslehian, Fuzzy approximately cubic mappings, Inform. Sci. 178 (2008), 3791– 3798. [28] M. Mirzavaziri and M.S. Moslehian, A fixed point approach to stability of a quadratic equation, Bull. Braz. Math. Soc. 37 (2006), 361–376. [29] E. Movahednia, S. M. S. M. Mosadegh, C. Park, D. Shin, Stability of a lattice preserving functional equation on Riesz space: fixed point alternative, J. Comput. Anal. Appl. 21 (2016), 83–89. [30] C. Park, Fixed points and Hyers-Ulam-Rassias stability of Cauchy-Jensen functional equations in Banach algebras, Fixed Point Theory Appl. 2007, Art. ID 50175 (2007). [31] C. Park, Generalized Hyers-Ulam-Rassias stability of quadratic functional equations: a fixed point approach, Fixed Point Theory Appl. 2008, Art. ID 493751 (2008). [32] C. Park, Additive ρ-functional inequalities and equations, J. Math. Inequal. 9 (2015), 17–26. [33] C. Park, Additive ρ-functional inequalities in non-Archimedean normed spaces, J. Math. Inequal. 9 (2015), 397–407. [34] C. Park, Stability of ternary quadratic derivation on ternary Banach algebras: revisited, J. Comput. Anal. Appl. 20 (2016), 21–23. [35] C. Park, K. Ghasemi, S. G. Ghaleh, S. Jang, Approximate n-Jordan ∗-homomorphisms in C ∗ -algebras, J. Comput. Anal. Appl. 15 (2013), 365–368. [36] C. Park, A. Najati, S. Jang, Fixed points and fuzzy stability of an additive-quadratic functional equation, J. Comput. Anal. Appl. 15 (2013), 452–462. [37] W. Park, J. Bae, Approximate quadratic forms on restricted domains, J. Comput. Anal. Appl. 20 (2016), 388–410. [38] V. Radu, The fixed point alternative and the stability of functional equations, Fixed Point Theory 4 (2003), 91–96. [39] Th.M. Rassias, On the stability of the linear mapping in Banach spaces, Proc. Amer. Math. Soc. 72 (1978), 297–300. [40] K. Ravi, E. Thandapani, B. V. Senthil Kumar, Solution and stability of a reciprocal type functional equation in several variables, J. Nonlinear Sci. Appl. 7 (2014), 18–27. [41] S. Schin, D. Ki, J. Chang, M. Kim, Random stability of quadratic functional equations: a fixed point approach, J. Nonlinear Sci. Appl. 4 (2011), 37–49. [42] S. Shagholi, M. Bavand Savadkouhi, M. Eshaghi Gordji, Nearly ternary cubic homomorphism in ternary Fr´echet algebras, J. Comput. Anal. Appl. 13 (2011), 1106–1114. [43] S. Shagholi, M. Eshaghi Gordji, M. Bavand Savadkouhi, Stability of ternary quadratic derivation on ternary Banach algebras, J. Comput. Anal. Appl. 13 (2011), 1097–1105. [44] D. Shin, C. Park, S. Farhadabadi, On the superstability of ternary Jordan C ∗ -homomorphisms, J. Comput. Anal. Appl. 16 (2014), 964–973. [45] D. Shin, C. Park, S. Farhadabadi, Stability and superstability of J ∗ -homomorphisms and J ∗ -derivations for a generalized Cauchy-Jensen equation, J. Comput. Anal. Appl. 17 (2014), 125–134. [46] F. Skof, Propriet locali e approssimazione di operatori, Rend. Sem. Mat. Fis. Milano 53 (1983), 113–129. [47] S. M. Ulam, A Collection of the Mathematical Problems, Interscience Publ. New York, 1960. [48] J. Z. Xiao, X. H. Zhu, Fuzzy normed spaces of operators and its completeness, Fuzzy Sets and Systems 133 (2003), 389–399. [49] C. Zaharia, On the probabilistic stability of the monomial functional equation, J. Nonlinear Sci. Appl. 6 (2013), 51–59. [50] S. Zolfaghari, Approximation of mixed type functional equations in p-Banach spaces, J. Nonlinear Sci. Appl. 3 (2010), 110–122. 1

Department of Mathematics, Daejin University, Kyunggi 11159, Korea E-mail address: [email protected] 2

Research Institute for Natural Sciences, Hanyang University, Seoul 04763, Korea E-mail address: [email protected] 3

Department of Mathematics, t University of Seoul, Seoul 02504, Korea E-mail address: [email protected] 4

Department of Financial Mathematics, Hanshin University, Gyeonggi-do 18101, Korea E-mail address: [email protected]

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Hyers-Ulam stability of set-valued functional equations: a fixed point approach Sungsik Yun, Choonkil Park∗ and Hassan Azadi Kenary∗

Abstract. In [36], Park proved the Hyers-Ulam stability of set-valued functional equations by using the direct method. In this paper, we prove the Hyers-Ulam stability of set-valued functional equations by using the fixed point method.

1. Introduction and preliminaries Set-valued functions in Banach spaces have been developed in the last decades. The pioneering paper by Aumann [5] and Debreu [14] were inspired by problems arising in Control Theory and Mathematical Economics. We can refer to the papers by Arrow and Debreu [3], McKenzie [29], the momographs by Hindenbrand [20], Aubin and Frankowska [4], Castaing and Valadier [8], Klein and Thompson [26] and the survey by Hess [19]. The stability problem of functional equations originated from a question of Ulam [50] concerning the stability of group homomorphisms. Hyers [21] gave a first affirmative partial answer to the question of Ulam for Banach spaces. Hyers’ Theorem was generalized by Aoki [2] for additive mappings and by Rassias [40] for linear mappings by considering an unbounded Cauchy difference. A generalization of the Rassias theorem was obtained by G˘ avruta [18] by replacing the unbounded Cauchy difference by a general control function in the spirit of Rassias’ approach The functional equation f (x + y) + f (x − y) = 2f (x) + 2f (y) is called a quadratic functional equation. In particular, every solution of the quadratic functional equation is said to be a quadratic mapping. A Hyers-Ulam stability problem for the quadratic functional equation was proved by Skof [49] for mappings f : X → Y , where X is a normed space and Y is a Banach space. Cholewa [12] noticed that the theorem of Skof is still true if the relevant domain X is replaced by an Abelian group. Czerwik [13] proved the Hyers-Ulam stability of the quadratic 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, 17, 18, 22, 23], [41]–[48]). In [25], Jun and Kim considered the following cubic functional equation f (2x + y) + f (2x − y) = 2f (x + y) + 2f (x − y) + 12f (x).

(1.1)

It is easy to show that the function f (x) = x3 satisfies the functional equation (1.1), which is called a cubic functional equation and every solution of the cubic functional equation is said to be a cubic mapping. In [28], Lee et al. considered the following quartic functional equation f (2x + y) + f (2x − y) = 4f (x + y) + 4f (x − y) + 24f (x) − 6f (y).

(1.2)

4

It is easy to show that the function f (x) = x satisfies the functional equation (1.2), which is called a quartic functional equation and every solution of the quartic functional equation is said to be a quartic mapping. 02010 Mathematics Subject Classification: 47H10, 54C60, 39B52, 47H04, 91B44. 0Keywords: Hyers-Ulam stability, set-valued functional equation, fixed point. ∗

Corresponding authors.

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Hyers-Ulam stability of set-valued functional equations 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. Let (X, d) be a generalized metric space. An operator T : X → X satisfies a Lipschitz condition with Lipschitz constant L if there exists a constant L ≥ 0 such that d(T x, T y) ≤ Ld(x, y) for all x, y ∈ X. If the Lipschitz constant L is less than 1, then the operator T is called a strictly contractive operator. Note that the distinction between the generalized metric and the usual metric is that the range of the former is permitted to include the infinity. We recall the following theorem by Margolis and Diaz. Theorem 1.1. [9, 15] Let (X, d) be a complete generalized metric space and let J : X → X be a strictly contractive mapping with Lipschitz constant L < 1. Then for each given element x ∈ X, either d(J n x, J n+1 x) = ∞ for all nonnegative integers n or there exists a positive integer n0 such that (1) d(J n x, J n+1 x) < ∞, ∀n ≥ n0 ; (2) the sequence {J n x} converges to a fixed point y ∗ of J; (3) y ∗ is the unique fixed point of J in the set Y = {y ∈ X | d(J n0 x, y) < ∞}; 1 d(y, Jy) for all y ∈ Y . (4) d(y, y ∗ ) ≤ 1−L In 1996, Isac and Rassias [24] 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 [10, 11, 31, 39]). Let Y be a Banach space. We define the following: 2Y : the set of all subsets of Y ; Cb (Y ) : the set of all closed bounded subsets of Y ; Cc (Y ) : the set of all closed convex subsets of Y ; Ccb (Y ) : the set of all closed convex bounded subsets of Y . On 2Y we consider the addition and the scalar multiplication as follows: C + C 0 = {x + x0 : x ∈ C, x0 ∈ C 0 }, 0

λC = {λx : x ∈ C},

0

where C, C ∈ 2 and λ ∈ R. Further, if C, C ∈ Cc (Y ), then we denote by C ⊕ C 0 = C + C 0 . It is easy to check that Y

λC + λC 0 = λ(C + C 0 ),

(λ + µ)C ⊆ λC + µC.

Furthermore, when C is convex, we obtain (λ + µ)C = λC + µC for all λ, µ ∈ R+ . For a given set C ∈ 2Y , the distance function d(·, C) and the support function s(·, C) are respectively defined by d(x, C) ∗

s(x , C)

= =

inf{kx − yk : y ∈ C}, ∗

sup{hx , xi : x ∈ C},

x ∈ Y, x∗ ∈ Y ∗ .

For every pair C, C 0 ∈ Cb (Y ), we define the Hausdorff distance between C and C 0 by h(C, C 0 ) = inf{λ > 0 : C ⊆ C 0 + λBY ,

C 0 ⊆ C + λBY },

where BY is the closed unit ball in Y . The following proposition reveals some properties of the Hausdorff distance.

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S. Yun, C. Park, H. Azadi Kenary Proposition 1.2. For every C, C 0 , K, K 0 ∈ Ccb (Y ) and λ > 0, the following properties hold (a) h(C ⊕ C 0 , K ⊕ K 0 ) ≤ h(C, K) + h(C 0 , K 0 ); (b) h(λC, λK) = λh(C, K). Let (Ccb (Y ), ⊕, h) be endowed with the Hausdorff distance h. Since Y is a Banach space, (Ccb (Y ), ⊕, h) is a complete metric semigroup (see [8]). Debreu [14] proved that (Ccb (Y ), ⊕, h) is isometrically embedded in a Banach space as follows. Lemma 1.3. [14] Let C(BY ∗ ) be the Banach space of continuous real-valued functions on BY ∗ endowed with the uniform norm k · ku . Then the mapping j : (Ccb (Y ), ⊕, h) → C(BY ∗ ), given by j(A) = s(·, A), satisfies the following properties: (a) j(A ⊕ B) = j(A) + j(B); (b) j(λA) = λj(A); (c) h(A, B) = kj(A) − j(B)ku ; (d) j(Ccb (Y )) is closed in C(BY ∗ ) for all A, B ∈ Ccb (Y ) and all λ ≥ 0. Let f : Ω → (Ccb (Y ), h) be a set-valued function from a complete finite measure space (Ω, Σ, ν) into Ccb (Y ). Then f is Debreu integrable if the composition j ◦ f is Bochner integrable (see [7]). In this case, the Debreu R R integral of f in Ω is the unique element (D) Ω f dν ∈ Ccb (Y ) such tha j((D) Ω f dν) is the Bochner integral of j ◦ f . The set of Debreu integrable functions from Ω to Ccb (Y ) will be denoted by D(Ω, Ccb (Y )). Furthermore, on D(Ω, Ccb (Y )), we define (f + g)(ω) = f (ω) ⊕ g(ω) for all f, g ∈ D(Ω, Ccb (Y )). Then we obtain that ((Ω, Ccb (Y )), +) is an abelian semigroup. Set-valued functional equations have been extensively investigated by a number of authors and there are many interesting results concerning this problem (see [6], [32]–[35], [37, 38]). Using the fixed point method, we prove the additive set-valued functional equation, the quadratic set-valued functional equation, the cubic set-valued functional equation and the quartic set-valued functional equation. Throughout this paper, let X be a real vector space and Y a Banach space. 2. Stability of the additive set-valued functional equation Using the fixed point method, we prove the Hyers-Ulam stability of the additive set-valued functional equation. Definition 2.1. [27] Let f : X → Ccb (Y ). The additive set-valued functional equation is defined by f (x + y) = f (x) ⊕ f (y) for all x, y ∈ X. Every solution of the additive set-valued functional equation is called an additive set-valued mapping. Note that there are some examples in [27]. Theorem 2.2. Let ϕ : X 2 → [0, ∞) be a function such that there exists an L < 1 with ϕ(x, y) ≤

L ϕ (2x, 2y) 2

for all x, y ∈ X. Suppose that f : X → (Ccb (Y ), h) is a mapping satisfying h(f (x + y), f (x) ⊕ f (y)) ≤ ϕ(x, y)

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Hyers-Ulam stability of set-valued functional equations for all x, y ∈ X. Then there exists a unique additive set-valued mapping A : X → (Ccb (Y ), h) such that h(f (x), A(x)) ≤

L ϕ(x, x) 2 − 2L

(2.2)

for all x ∈ X. Proof. Let y = x in (2.1). Since f (x) is convex, we get h(f (2x), 2f (x)) ≤ ϕ(x, x)

(2.3)

  x  x x L h f (x), 2f ≤ϕ , ≤ ϕ (x, x) 2 2 2 2

(2.4)

and so

for all x ∈ X. Consider S := {g : g : X → Ccb (Y ), g(0) = {0}} and introduce the generalized metric on X, d(g, f ) = inf {µ ∈ (0, ∞) : h(g(x), f (x)) ≤ µϕ(x, x), x ∈ X}, where, as usual, inf φ = +∞. It is easy to show that (S, d) is complete (see [16, Theorem 2.4] and [30, 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, f ∈ S be given such that d(g, f ) = ε. Then h(g(x), f (x)) ≤ εϕ(x, x) for all x ∈ X. Hence  x    x   x   x , 2f = 2h g ,f ≤ Lϕ(x, x) h 2g 2 2 2 2 for all x ∈ X. So d(g, f ) = ε implies that d(Jg, Jf ) ≤ Lε. This means that h(Jg(x), Jf (x))

=

d(Jg, Jf ) ≤ Ld(g, f ) for all g, f ∈ S. It follows from (2.4) that d(f, Jf ) ≤ L2 . By Theorem 1.1, there exists a mapping A : X → Y satisfying the following: (1) A is a fixed point of J, i.e., x 1 A = A(x) 2 2 for all x ∈ X. The mapping A is a unique fixed point of J in the set

(2.5)

M = {g ∈ S : d(f, g) < ∞}. This implies that A is a unique mapping satisfying (2.5) such that there exists a µ ∈ (0, ∞) satisfying h(f (x), A(x)) ≤ µϕ(x, x) for all x ∈ X; (2) d(J n f, A) → 0 as n → ∞. This implies the equality

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lim 2n f

n→∞

for all x ∈ X; (3) d(f, A) ≤

1 d(f, Jf ), 1−L

x = A(x) 2n

which implies the inequality d(f, A) ≤

L . 2 − 2L

This implies that the inequality (2.2) holds. By (2.1),  x + y x  y  x y  n n n h 2n f , 2 f ⊕ 2 f ≤ 2 ϕ , ≤ Ln ϕ(x, y), 2n 2n 2n 2n 2n which tends to zero as n → ∞ for all x, y ∈ X. Thus A(x + y) = A(x) ⊕ A(y), as desired.



Corollary 2.3. Let p > 1 and θ ≥ 0 be real numbers, and let X be a real normed space. Suppose that f : X → (Ccb (Y ), h) is a mapping satisfying h(f (x + y), f (x) ⊕ f (y)) ≤ θ(||x||p + ||y||p )

(2.6)

for all x, y ∈ X. Then there exists a unique additive set-valued mapping A : X → Y satisfying h(f (x), A(x)) ≤

2θ ||x||p 2p − 2

for all x ∈ X. Proof. The proof follows from Theorem 2.2 by taking ϕ(x, y) := θ(kxkp + kykp ) for all x, y ∈ X. Then we can choose L = 21−p and we get the desired result.



Theorem 2.4. Let ϕ : X 2 → [0, ∞) be a function such that there exists an L < 1 with x y ϕ(x, y) ≤ 2Lϕ , 2 2 for all x, y ∈ X. Suppose that f : X → (Ccb (Y ), h) is a mapping satisfying (2.1). Then there exists a unique additive set-valued mapping A : X → (Ccb (Y ), h) such that h(f (x), A(x)) ≤

1 ϕ(x, x) 2 − 2L

for all x ∈ X. Proof. It follows from (2.3) that   1 1 h f (x), f (2x) ≤ ϕ (x, x) 2 2 for all x ∈ X. The rest of the proof is similar to the proof of Theorem 2.2.



Corollary 2.5. Let 1 > p > 0 and θ ≥ 0 be real numbers, and let X be a real normed space. Suppose that f : X → (Ccb (Y ), h) is a mapping satisfying (2.6). Then there exists a unique additive set-valued mapping A : X → Y satisfying h(f (x), A(x)) ≤

2θ ||x||p 2 − 2p

for all x ∈ X.

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Hyers-Ulam stability of set-valued functional equations Proof. The proof follows from Theorem 2.4 by taking ϕ(x, y) := θ(kxkp + kykp ) for all x, y ∈ X. Then we can choose L = 2p−1 and we get the desired result.



3. Stability of the quadratic set-valued functional equation Using the fixed point method, we prove the Hyers-Ulam stability of the quadratic set-valued functional equation. Definition 3.1. [27] Let f : X → Ccb (Y ). The quadratic set-valued functional equation is defined by 2f (x + y) ⊕ 2f (x − y) = f (2x) ⊕ f (2y) for all x, y ∈ X. Every solution of the quadratic set-valued functional equation is called a quadratic set-valued mapping. Note that there are some examples in [27]. Theorem 3.2. Let ϕ : X 2 → [0, ∞) be a function such that there exists an L < 1 with L ϕ (2x, 2y) 4 for all x, y ∈ X. Suppose that f : X → (Ccb (Y ), h) is a mapping satisfying f (0) = {0} and ϕ(x, y) ≤

h(2f (x + y) ⊕ 2f (x − y), f (2x) ⊕ f (2y)) ≤ ϕ(x, y)

(3.1)

for all x, y ∈ X. Then there exists a unique quadratic set-valued mapping Q : X → (Ccb (Y ), h) such that h(f (x), Q(x)) ≤

L ϕ(x, 0) 4 − 4L

for all x ∈ X. Proof. Let y = 0 in (3.1). Since f (x) is convex, we get h(f (2x), 4f (x)) ≤ ϕ(x, 0)

(3.2)

x    x  L ≤ϕ , 0 ≤ ϕ(x, 0) h f (x), 4f 2 2 4

(3.3)

and

for all x ∈ X. Consider S := {g : g : X → Ccb (Y ), g(0) = {0}} and introduce the generalized metric on X, d(g, f ) = inf {µ ∈ (0, ∞) : h(g(x), f (x)) ≤ µϕ(x, 0), x ∈ X}, where, as usual, inf φ = +∞. It is easy to show that (S, d) is complete (see [16, Theorem 2.4] and [30, Lemma 2.1]). Now we consider the linear mapping J : S → S such that x Jg(x) := 4g 2 for all x ∈ X. By the same reasoning as in the proof of Theorem 2.2, one can show that d(Jg, Jf ) ≤ Ld(g, f )

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S. Yun, C. Park, H. Azadi Kenary for all g, f ∈ S. It follows from (3.3) that d(f, Jf ) ≤ L4 . The rest of the proof is similar to the proof of Theorem 2.2.



Corollary 3.3. Let p > 2 and θ ≥ 0 be real numbers, and let X be a real normed space. Suppose that f : X → (Ccb (Y ), h) is a mapping satisfying f (0) = {0} and h(2f (x + y) ⊕ 2f (x − y), f (2x) ⊕ f (2y)) ≤ θ(||x||p + ||y||p )

(3.4)

for all x, y ∈ X. Then there exists a unique quadratic set-valued mapping Q : X → Y satisfying h(f (x), Q(x)) ≤

θ ||x||p 2p − 4

for all x ∈ X. Proof. The proof follows from Theorem 3.2 by taking ϕ(x, y) := θ(kxkp + kykp ) for all x, y ∈ X. Then we can choose L = 22−p and we get the desired result.



Theorem 3.4. Let ϕ : X 2 → [0, ∞) be a function such that there exists an L < 1 with x y , ϕ(x, y) ≤ 4Lϕ 2 2 for all x, y ∈ X. Suppose that f : X → (Ccb (Y ), h) is a mapping satisfying f (0) = {0} and (3.1). Then there exists a unique quadratic set-valued mapping Q : X → (Ccb (Y ), h) such that 1 ϕ(x, 0) 4 − 4L

h(f (x), Q(x)) ≤ for all x ∈ X. Proof. It follows from (3.2) that 

1 h f (x), f (2x) 4

 ≤

1 ϕ (x, 0) 4

for all x ∈ X. The rest of the proof is similar to the proofs of Theorems 2.2 and 3.2.



Corollary 3.5. Let 0 < p < 2 and θ ≥ 0 be real numbers, and let X be a real normed space. Suppose that f : X → (Ccb (Y ), h) is a mapping satisfying f (0) = {0} and (3.4). Then there exists a unique quadratic set-valued mapping Q : X → Y satisfying h(f (x), Q(x)) ≤

θ ||x||p 4 − 2p

for all x ∈ X. Proof. The proof follows from Theorem 3.4 by taking ϕ(x, y) := θ(kxkp + kykp ) for all x, y ∈ X. Then we can choose L = 2p−2 and we get the desired result.

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Hyers-Ulam stability of set-valued functional equations 4. Stability of the cubic set-valued functional equation Using the fixed point method, we define a cubic set-valued functional equation and prove the Hyers-Ulam stability of the cubic set-valued functional equation. Definition 4.1. [36] Let f : X → Ccb (Y ). The cubic set-valued functional equation is defined by f (2x + y) ⊕ f (2x − y) = 2f (x + y) ⊕ 2f (x − y) ⊕ 12f (x) for all x, y ∈ X. Every solution of the cubic set-valued functional equation is called a cubic set-valued mapping. Theorem 4.2. Let ϕ : X 2 → [0, ∞) be a function such that there exists an L < 1 with ϕ(x, y) ≤

L ϕ (2x, 2y) 8

for all x, y ∈ X. Suppose that f : X → (Ccb (Y ), h) is a mapping satisfying f (0) = {0} and h(f (2x + y) ⊕ f (2x − y), 2f (x + y) ⊕ 2f (x − y) ⊕ 12f (x)) ≤ ϕ(x, y)

(4.1)

for all x, y ∈ X. Then there exists a unique cubic set-valued mapping C : X → (Ccb (Y ), h) such that h(f (x), C(x)) ≤

L ϕ(x, 0) 16 − 16L

for all x ∈ X. Proof. Let y = 0 in (4.1). Since f (x) is convex, we get h(2f (2x), 16f (x)) ≤ ϕ(x, 0)

(4.2)

  x  1 x  L h f (x), 8f ≤ ϕ ,0 ≤ ϕ(x, 0) 2 2 2 16

(4.3)

and

for all x ∈ X. Consider S := {g : g : X → Ccb (Y ), g(0) = {0}} and introduce the generalized metric on X, d(g, f ) = inf {µ ∈ (0, ∞) : h(g(x), f (x)) ≤ µϕ(x, 0), x ∈ X}, where, as usual, inf φ = +∞. It is easy to show that (S, d) is complete (see [16, Theorem 2.4] and [30, Lemma 2.1]). Now we consider the linear mapping J : S → S such that x Jg(x) := 8g 2 for all x ∈ X. By the same reasoning as in the proof of Theorem 2.2, one can show that d(Jg, Jf ) ≤ Ld(g, f ) for all g, f ∈ S. L It follows from (4.3) that d(f, Jf ) ≤ 16 . The rest of the proof is similar to the proof of Theorem 2.2.

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S. Yun, C. Park, H. Azadi Kenary Corollary 4.3. Let p > 3 and θ ≥ 0 be real numbers, and let X be a real normed space. Suppose that f : X → (Ccb (Y ), h) is a mapping satisfying h(f (2x + y) ⊕ f (2x − y), 2f (x + y) ⊕ 2f (x − y) ⊕ 12f (x)) ≤ θ(||x||p + ||y||p )

(4.4)

for all x, y ∈ X. Then there exists a unique cubic set-valued mapping C : X → Y satisfying θ h(f (x), C(x)) ≤ ||x||p 2(2p − 8) for all x ∈ X. Proof. The proof follows from Theorem 4.2 by taking ϕ(x, y) := θ(kxkp + kykp ) for all x, y ∈ X. Then we can choose L = 23−p and we get the desired result.



Theorem 4.4. Let ϕ : X 2 → [0, ∞) be a function such that there exists an L < 1 with x y ϕ(x, y) ≤ 8Lϕ , 2 2 for all x, y ∈ X. Suppose that f : X → (Ccb (Y ), h) is a mapping satisfying (4.1). Then there exists a unique cubic set-valued mapping C : X → (Ccb (Y ), h) such that 1 h(f (x), C(x)) ≤ ϕ(x, 0) 16 − 16L for all x ∈ X. Proof. It follows from (4.2) that   1 1 h f (x), f (2x) ≤ ϕ (x, 0) 8 16 for all x ∈ X. The rest of the proof is similar to the proofs of Theorems 2.2 and 4.2.



Corollary 4.5. Let 3 > p > 0 and θ ≥ 0 be real numbers, and let X be a real normed space. Suppose that f : X → (Ccb (Y ), h) is a mapping satisfying (4.4). Then there exists a unique cubic set-valued mapping C : X → Y satisfying θ ||x||p h(f (x), C(x)) ≤ 2(8 − 2p ) for all x ∈ X. Proof. The proof follows from Theorem 4.4 by taking ϕ(x, y) := θ(kxkp + kykp ) for all x, y ∈ X. Then we can choose L = 2p−3 and we get the desired result.



5. Stability of the quartic set-valued functional equation Using the fixed point method, we define a quartic set-valued functional equation and prove the Hyers-Ulam stability of the quartic set-valued functional equation. Definition 5.1. [36] Let f : X → Ccb (Y ). The quartic set-valued functional equation is defined by f (2x + y) ⊕ f (2x − y) ⊕ 6f (y) = 4f (x + y) ⊕ 4f (x − y) ⊕ 24f (x) for all x, y ∈ X. Every solution of the quartic set-valued functional equation is called a quartic set-valued mapping.

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Hyers-Ulam stability of set-valued functional equations Theorem 5.2. Let ϕ : X 2 → [0, ∞) be a function such that there exists an L < 1 with L ϕ (2x, 2y) 16 for all x, y ∈ X. Suppose that f : X → (Ccb (Y ), h) is a mapping satisfying f (0) = {0} and ϕ(x, y) ≤

h(f (2x + y) ⊕ f (2x − y) ⊕ 6f (y), 4f (x + y) ⊕ 4f (x − y) ⊕ 24f (x)) ≤ ϕ(x, y)

(5.1)

for all x, y ∈ X. Then there exists a unique quartic set-valued mapping T : X → (Ccb (Y ), h) such that h(f (x), T (x)) ≤

L ϕ(x, 0) 32 − 32L

for all x ∈ X. Proof. Let y = 0 in (5.1). Since f (x) is convex, we get h(2f (2x), 32f (x)) ≤ ϕ(x, 0)

(5.2)

  x  1 x  L h f (x), 16f ≤ ϕ ,0 ≤ ϕ(x, 0) 2 2 2 32

(5.3)

and

for all x ∈ X. Consider S := {g : g : X → Ccb (Y ), g(0) = {0}} and introduce the generalized metric on X, d(g, f ) = inf {µ ∈ (0, ∞) : h(g(x), f (x)) ≤ µϕ(x, 0), x ∈ X}, where, as usual, inf φ = +∞. It is easy to show that (S, d) is complete (see [16, Theorem 2.4] and [30, Lemma 2.1]). Now we consider the linear mapping J : S → S such that x Jg(x) := 16g 2 for all x ∈ X. By the same reasoning as in the proof of Theorem 2.2, one can show that d(Jg, Jf ) ≤ Ld(g, f ) for all g, f ∈ S. L . It follows from (5.3) that d(f, Jf ) ≤ 32 The rest of the proof is similar to the proof of Theorem 2.2.



Corollary 5.3. Let p > 4 and θ ≥ 0 be real numbers, and let X be a real normed space. Suppose that f : X → (Ccb (Y ), h) is a mapping satisfying h(f (2x + y) ⊕ f (2x − y) ⊕ 6f (y), 4f (x + y) ⊕ 4f (x − y) ⊕ 24f (x)) ≤ θ(||x||p + ||y||p )

(5.4)

for all x, y ∈ X. Then there exists a unique quartic set-valued mapping T : X → Y satisfying θ h(f (x), T (x)) ≤ ||x||p 2(2p − 16) for all x ∈ X. Proof. The proof follows from Theorem 5.2 by taking ϕ(x, y) := θ(kxkp + kykp ) for all x, y ∈ X. Then we can choose L = 24−p and we get the desired result.

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S. Yun, C. Park, H. Azadi Kenary Theorem 5.4. Let ϕ : X 2 → [0, ∞) be a function such that there exists an L < 1 with x y ϕ(x, y) ≤ 16Lϕ , 2 2 for all x, y ∈ X. Suppose that f : X → (Ccb (Y ), h) is a mapping satisfying (5.1). Then there exists a unique quartic set-valued mapping T : X → (Ccb (Y ), h) such that h(f (x), T (x)) ≤

1 ϕ(x, 0) 32 − 32L

for all x ∈ X. Proof. It follows from (5.2) that   1 1 h f (x), f (2x) ≤ ϕ (x, 0) 16 32 for all x ∈ X. The rest of the proof is similar to the proofs of Theorems 2.2 and 5.2.



Corollary 5.5. Let 4 > p > 0 and θ ≥ 0 be real numbers, and let X be a real normed space. Suppose that f : X → (Ccb (Y ), h) is a mapping satisfying (5.4). Then there exists a unique quartic set-valued mapping T : X → Y satisfying h(f (x), T (x)) ≤

θ ||x||p 2(2p − 16)

for all x ∈ X. Proof. The proof follows from Theorem 5.4 by taking ϕ(x, y) := θ(kxkp + kykp ) for all x, y ∈ X. Then we can choose L = 2p−4 and we get the desired result.



Acknowledgments S. Yun was supported by Hanshin University Research Grant.

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Hyers-Ulam stability of set-valued functional equations [10] L. C˘ adariu and V. Radu, On the stability of the Cauchy functional equation: a fixed point approach, Grazer Math. Ber. 346 (2004), 43–52. [11] L. C˘ adariu and V. Radu, Fixed point methods for the generalized stability of functional equations in a single variable, Fixed Point Theory Appl. 2008, Art. ID 749392 (2008). [12] P.W. Cholewa, Remarks on the stability of functional equations, Aequationes Math. 27 (1984), 76–86. [13] S. Czerwik, On the stability of the quadratic mapping in normed spaces, Abh. Math. Sem. Univ. Hamburg 62 (1992), 59–64. [14] G. Debreu, Integration of correspondences, Proceedings of Fifth Berkeley Symposium on Mathematical Statistics and Probability, Vol. II, Part I (1966), 351–372. [15] J. Diaz and B. Margolis, A fixed point theorem of the alternative for contractions on a generalized complete metric space, Bull. Amer. Math. Soc. 74 (1968), 305–309. [16] M. Eshaghi Gordji, C. Park and M.B. Savadkouhi, The stability of a quartic type functional equation with the fixed point alternative, Fixed Point Theory 11 (2010), 265–272. [17] 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. [18] P. Gˇ avruta, A generalization of the Hyers-Ulam-Rassias stability of approximately additive mappings, J. Math. Anal. Appl. 184 (1994), 431–436. [19] C. Hess, Set-valued integration and set-valued probability theory: an overview, in Handbook of Measure Theory, Vols. I, II, North-Holland, Amsterdam, 2002. [20] W. Hindenbrand, Core and Equilibria of a Large Economy, Princeton Univ. Press, Princeton, 1974. [21] D.H. Hyers, On the stability of the linear functional equation, Proc. Nat. Acad. Sci. USA 27 (1941), 222–224. [22] D.H. Hyers, G. Isac and Th.M. Rassias, Stability of Functional Equations in Several Variables, Birkh¨ auser, Basel, 1998. [23] G. Isac and Th.M. Rassias, On the Hyers-Ulam stability of ψ-additive mappings, J. Approx. Theory 72 (1993), 131–137. [24] G. Isac and Th.M. Rassias, Stability of ψ-additive mappings: Applications to nonlinear analysis, Internat. J. Math. Math. Sci. 19 (1996), 219–228. [25] K. Jun and H. Kim, The generalized Hyers-Ulam-Rassias stability of a cubic functional equation, J. Math. Anal. Appl. 274 (2002), 867–878. [26] E. Klein and A. Thompson, Theory of Correspondence, Wiley, New York, 1984. [27] K. Lee, Stability of functional equations related to set-valued functions (preprint). [28] S. Lee, S. Im and I. Hwang, Quartic functional equations, J. Math. Anal. Appl. 307 (2005), 387–394. [29] L.W. McKenzie, On the existence of general equilibrium for a competitive market, Econometrica 27 (1959), 54–71. [30] 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. [31] M. Mirzavaziri and M.S. Moslehian, A fixed point approach to stability of a quadratic equation, Bull. Braz. Math. Soc. 37 (2006), 361–376. [32] K. Nikodem, On quadratic set-valued functions, Publ. Math. Debrecen 30 (1984), 297–301. [33] K. Nikodem, On Jensen’s functional equation for set-valued functions, Radovi Mat. 3 (1987), 23–33. [34] K. Nikodem, Set-valued solutions of the Pexider functional equation, Funkcialaj Ekvacioj 31 (1988), 227– 231. [35] K. Nikodem, K-Convex and K-Concave Set-Valued Functions, Zeszyty Naukowe Nr. 559, Lodz, 1989. [36] C. Park, Approximation of set-valued functional equations (preprint). [37] Y.J. Piao, The existence and uniqueness of additive selection for (α, β)-(β, α) type subadditive set-valued maps, J. Northeast Normal University 41 (2009), 38–40. [38] D. Popa, Additive selections of (α, β)-subadditive set-valued maps, Glas. Mat. Ser. III, 36 (56) (2001), 11–16.

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S. Yun, C. Park, H. Azadi Kenary [39] V. Radu, The fixed point alternative and the stability of functional equations, Fixed Point Theory 4 (2003), 91–96. [40] Th.M. Rassias, On the stability of the linear mapping in Banach spaces, Proc. Amer. Math. Soc. 72 (1978), 297–300. [41] Th.M. Rassias (Ed.), Functional Equations and Inequalities, Kluwer Academic, Dordrecht, 2000. [42] Th.M. Rassias, On the stability of functional equations in Banach spaces, J. Math. Anal. Appl. 251 (2000), 264–284. [43] Th.M. Rassias, On the stability of functional equations and a problem of Ulam, Acta Math. Appl. 62 (2000), 23–130. [44] S. Schin, D. Ki, J. Chang and M. Kim, Random stability of quadratic functional equations: a fixed point approach, J. Nonlinear Sci. Appl. 4 (2011), 37–49. [45] S. Shagholi, M. Bavand Savadkouhi and M. Eshaghi Gordji, Nearly ternary cubic homomorphism in ternary Fr´echet algebras, J. Comput. Anal. Appl. 13 (2011), 1106–1114. [46] S. Shagholi, M. Eshaghi Gordji and M. Bavand Savadkouhi, Stability of ternary quadratic derivation on ternary Banach algebras, J. Comput. Anal. Appl. 13 (2011), 1097–1105. [47] D. Shin, C. Park and Sh. Farhadabadi, On the superstability of ternary Jordan C ∗ -homomorphisms, J. Comput. Anal. Appl. 16 (2014), 964–973. [48] D. Shin, C. Park and Sh. Farhadabadi, Stability and superstability of J ∗ -homomorphisms and J ∗ -derivations for a generalized Cauchy-Jensen equation, J. Comput. Anal. Appl. 17 (2014), 125–134. [49] F. Skof, Propriet` a locali e approssimazione di operatori, Rend. Sem. Mat. Fis. Milano 53 (1983), 113–129. [50] S.M. Ulam, Problems in Modern Mathematics, Chapter VI, Science ed., Wiley, New York, 1940. Sungsik Yun Department of Financial Mathematics, Hanshin University, Gyeonggi-do 18101, Korea E-mail address: [email protected] Choonkil Park Research Institute for Natural Sciences, Hanyang University, Seoul 04763, Korea E-mail address: [email protected] Hassan Azadi Kenary Department of Mathematics, College of Science, Yasouj University, Yasouj 75914-353, Iran E-mail address: [email protected]

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On precompactness of the Hausdorff fuzzy metric on closed sets Chang-qing Lia, ∗ , Yan-lan Zhangb a School of Mathematics and Statistics, Minnan Normal University, Zhangzhou, Fujian 363000, China b College of Computer, Minnan Normal University, Zhangzhou, Fujian 363000, China Email: helen [email protected], zyl 1983 [email protected] ¯ ¯ ¯ August 24, 2016 In the paper, we construct a Hausdorff fuzzy metric on the family of nonempty closed subsets of a stationary and F-bounded fuzzy metric space. Using the construction of the Hausdorff fuzzy metric, we prove three equivalent characterizations for the given fuzzy metric space to be precompact. Furthermore, several examples are given. Keywords: Fuzzy metric, Continuous t-norm, The Hausdorff fuzzy metric, Closed subset, Precompact. AMS Subject Classifications: 54A40, 54B20, 54E35

1

Introduction

Fuzzy metric is an important notion in Fuzzy Topology. Many authors have introduced the concept of fuzzy metric from different points of view [2, 3, 12, 13]. In particular, George and Veeramani [3] obtained the concept of fuzzy metric with the help of continuous t-norms in 1994. Later, it was proved that the topological space induced by the fuzzy metric space is metrizable in [8]. This version of fuzzy metric determines the class of spaces that are tightly connected with the class of metrizable topological spaces. Hence it is interesting to study the version of fuzzy metric. Some contributions to the study of fuzzy metric spaces can be found in [4, 5, 6, 14, 15, 16, 19, 20]. In order to study the hyperspaces in a fuzzy metric space, Rodr´ıguez-L´opez and Romaguera [17] gave a definition of Hausdorff fuzzy metric on the family ∗ Corresponding author. This work was supported by Grants from the National Natural Science Foundation of China (Nos. 11526109, 11471153, 11571158, 61379021), Natural Science Foundation of Fujian (Nos. 2016J01671, 2015J05011, JK2014028), and the outstanding youth foundation of the Education Department of Fujiang Province. .

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of nonempty compact sets. Unfortunately, the Hausdorff fuzzy metric defined by the authors does not provide a fuzzy metric when one consider the family of nonempty closed and F-bounded subsets of a given fuzzy metric space. In [17], Rodr´ıguez-L´ opez and Romaguera illustrated the result above with the help of a example. It is a nature problem to explore under what condition the Hausdorff fuzzy metric defined by Rodr´ıguez-L´opez and Romaguera on the family of nonempty closed and F-bounded subsets of a given fuzzy metric space can provide a fuzzy metric. This is done in the present paper. We construct a Hausdorff fuzzy metric on the family of nonempty closed subsets of a stationary and F-bounded fuzzy metric space. Also, we prove three necessary and sufficient conditions for the given fuzzy metric space to be precompact. Moreover, we give some illustrative examples.

2

Preliminaries

Throughout the paper the letter N shall denote the set of all nature numbers. Our basic reference for general topology is [1]. Definition 2.1 [3] A binary operation ∗ : [0, 1] × [0, 1] → [0, 1] is a continuous t-norm if it satisfies the following conditions: (i) ∗ is associative and commutative; (ii) ∗ is continuous; (iii) a ∗ 1 = a for all a ∈ [0, 1]; (iv) a ∗ b ≤ c ∗ d whenever a ≤ c and b ≤ d, and a, b, c, d ∈ [0, 1]. The following are examples of t-norms: a ∗ b = min{a, b}; a ∗ b = a · b; a ∗ b = max{a + b − 1, 0}. Definition 2.2 [3] A 3-tuple (X, M, ∗) is said to be a fuzzy metric space if X is an arbitrary set, ∗ is a continuous t-norm and M is a fuzzy set on X ×X ×(0, ∞) satisfying the following conditions for all x, y, z ∈ X and s, t ∈ (0, ∞): (i) M (x, y, t) > 0; (ii) M (x, y, t) = 1 if and only if x = y; (iii) M (x, y, t) = M (y, x, t); (iv) M (x, y, t) ∗ M (y, z, s) ≤ M (x, z, t + s); (v) the function M (x, y, ·) : (0, ∞) → [0, 1] is continuous. If (X, M, ∗) is a fuzzy metric space, we will say that (M, ∗) is a fuzzy metric on X. Definition 2.3 [3] Let (X, M, ∗) be a fuzzy metric space and let r ∈ (0, 1), t > 0 and x ∈ X. The set BM (x, r, t) = {y ∈ X|M (x, y, t) > 1 − r} is called the open ball with center x and radius r with respect to t. Obviously, {BM (x, r, t)|x ∈ X, t > 0, r ∈ (0, 1)} forms a base of a topology in X. The topology is denoted by τM and is known to be metrizable (see [8]). 2

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Lemma 2.4 [3] Let (X, M, ∗) be a fuzzy metric space. Then, for each x ∈ X, {BM (x, n1 , n1 )|n ∈ N} is a neighborhood base at x for the topology τM . Definition 2.5 [3] Let (X, d) be a metric space. Define a ∗ b = ab for all a, b ∈ [0, 1], and let Md be the function on X × X × (0, ∞) defined by Md (x, y, t) =

t . t + d(x, y)

Then (X, Md , ·) is a fuzzy metric space and (Md , ·) is called the standard fuzzy metric induced by d. Definition 2.6 [3] Let (X, M, ∗) be a fuzzy metric space. A subset A of X is said to be F-bounded if there exist t > 0 and 0 < r < 1 such that M (x, y, t) > 1 − r for all x, y ∈ A. We call (X, M, ∗) a F-bounded fuzzy metric space provided that X is Fbounded. Clearly, a subset of an F-bounded fuzzy metric space is F-bounded. Definition 2.7 [10] A fuzzy metric space (X, M, ∗) is said to be stationary if M does not depend on t, i.e. if for each x, y ∈ X, the function M (x, y, ·) is constant. In the case we write M (x, y) and BM (x, r) instead of M (x, y, t) and BM (x, r, t), respectively. Lemma 2.8 [17] Let (X, M, ∗) be a fuzzy metric space. Then M is a continuous function on X × X × (0, ∞).

3

The Hausdorff fuzzy metric on Cld(X)

Given a fuzzy metric space (X, M, ∗), we will denote by P(X), Cld(X) and Fin(X), the set of nonempty subsets, the set of nonempty closed subsets and the set of nonempty finite subsets of (X, τM ), respectively. For every C ∈ P(X), a ∈ X and t > 0, let M (a, C, t) := sup M (a, c, t), M (C, a, t) := sup M (c, a, t) c∈C

c∈C

(see Definition 2.4 of [20]). It is clear that M (a, C, t) = M (C, a, t). Lemma 3.1 Let (X, M, ∗) be a fuzzy metric space, a, c ∈ X, D ∈ P(X) and t, s ∈ (0, ∞). Then M (a, D, t + s) ≥ M (a, c, t) ∗ M (c, D, s). Proof Note that, for each d ∈ D, M (a, D, t + s) ≥ M (a, d, t + s) ≥ M (a, c, t) ∗ M (c, d, s). It follows from continuity of ∗ that M (a, D, t + s) ≥ M (a, c, t) ∗ M (c, D, s). Let (X, M, ∗) be a fuzzy metric space, A, C ∈ Cld(X) and t > 0, define HM : Cld(X)× Cld(X) × (0, ∞) → [0, 1] by HM (A, C, t) = min{ inf M (a, C, t), inf M (A, c, t)}. a∈A

c∈C

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If (X, M, ∗) is a stationary fuzzy metric space, then we write HM (A, C), M (a, C) and M (A, c) instead of HM (A, C, t), M (a, C, t) and M (A, c, t), respectively. Theorem 3.2 Let (X, M, ∗) be a stationary and F-bounded fuzzy metric space. Then (Cld(X),HM , ∗) is a fuzzy metric space. Proof Let A, C, D ∈ Cld(X). Obviously, (i), (ii), (iii) and (v) in Definition 2.2 hold. Now, we are going to prove that (iv) in Definition 2.2 is satisfied, i.e., HM (A, D) ≥ HM (A, C) ∗ HM (C, D). Let a ∈ A. Then we can choose a sequence {can }n∈N in C such that lim M (a, can ) = M (a, C).

n→+∞

Since {M (can , D)}n∈N is a sequence in [0,1], there is a subsequence {canl }l∈N of {can }n∈N such that the sequence {M (canl , D)}l∈N converges to some point of [0,1]. It follows from Lemma 3.1 that M (a, D) ≥ M (a, cank ) ∗ M (cank , D) for every cank ∈ {canl |l ∈ N}. Therefore, by continuity of ∗, we have M (a, D) ≥ lim M (a, cank ) ∗ lim M (cank , D) k→+∞

k→+∞

= M (a, C) ∗ lim M (cank , D). k→+∞

According to continuity of ∗, we deduce that inf M (a, D) ≥ inf M (a, C) ∗ inf a∈A

a∈A

lim M (cank , D) a∈A k→+∞ inf {M (cank , D) | k ∈ N} a∈A

a∈A

c∈C

a∈A

≥ inf M (a, C) ∗

≥ inf M (a, C) ∗ inf M (c, D) ≥ HM (A, C) ∗ HM (C, D). Analogously, we get

inf M (A, d) ≥ HM (A, C) ∗ HM (C, D).

d∈D

Hence HM (A, D) ≥ HM (A, C) ∗ HM (C, D). (HM , ∗) will be called the Hausdorff fuzzy metric on Cld(X). Next we will give two examples. Example 3.3 Let X = (1, 10]. Denote a ∗ b = a · b for all a, b ∈ [0, 1]. Define M by min{x, y} M (x, y, t) = max{x, y} for all x, y ∈ X and t > 0. Then (X, M, ∗) is a stationary and F-bounded fuzzy metric space (see [7]). So, by Theorem 3.2, (Cld(X),HM , ∗) is a fuzzy metric space. 4

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Example 3.4 Let X = [3, 5]. Denote a∗b = max{a+b−1, 0} for all a, b ∈ [0, 1] and let M be a fuzzy set on X × X × (0, ∞) defined as follows: { 1 x = y, M (x, y, t) = 1 1 + x ̸= y, x y for all x, y ∈ X and t > 0. Then (X, M, ∗) is a stationary and F-bounded fuzzy metric space (see [18]). Thus, according to Theorem 3.2, (Cld(X),HM , ∗) is a fuzzy metric space. More examples of stationary and F-bounded fuzzy metric spaces may be found in [7, 11, 18].

4

Precompactness of the Hausdorff fuzzy metric We start this section by recalling the concept of precompact.

Definition 4.1 [8] A fuzzy metric space (X, M, ∗) is called precompact if for each ∪ r ∈ (0, 1) and t > 0, there is a finite subset A of X such that X = BM (a, r, t). a∈A

Theorem 4.2 Let Y be a dense subspace of a stationary and F-bounded fuzzy metric space (X, M, ∗). Then Fin(Y ) is dense in (Cld(X), HM , ∗) if and only if (X, M, ∗) is precompact. Proof Assume that Fin(Y ) is dense in (Cld(X), HM , ∗). Let r ∈ (0, 1). Since Fin(Y ) ⊆ Fin(X), we get Fin(X) ∩ BHM (X, r) ̸= Ø. Take A ∈ Fin(X) ∩ BHM (X, r). Then A ∈ Fin(X) and HM (X, A) > 1 − r. Hence inf M (x, A) > 1 − r. x∈X

Let x ∈ X. Then M (x, A) > 1 − r. Since A ∈ Fin(X), there exists an a ∈ A such that M (x, a) = M (x, A) > 1 − r. We have x ∈ BM (a, r). So X⊂

∪

BM (a, r).

a∈A

It follows that (X, M, ∗) is precompact. Conversely, assume that (X, M, ∗) is precompact. Let D ∈ Cld(X) and ε ∈ (0, 1). Then, by the continuity of ∗, there exists a δ ∈ (0, ε) such that (1 − δ) ∗ (1 − δ) > 1 − ε. 5

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We only need to verify that Fin(Y ) ∩ BHM (D, ε) ̸= Ø. Since (X, M, ∗) is precompact, there exists a C = {c1 , c2 , · · · , cn } ∈ Fin(X) such that X=

n ∪

BM (ci , δ).

i=1

Now, we can find C ′ = {cn1 , cn2 , · · · , cnk } ⊂ C such that D⊂

k ∪

BM (cni , δ)

i=1

and D ∩ BM (cni , δ) ̸= Ø(i = 1, 2, · · · , k). Take ani ∈ D ∩ BM (cni , δ) (i = 1, 2, · · · , k). Denote A = {an1 , an2 , · · · , ank }. Since Y is a dense subspace of X, we can find an eni ∈ BM (cni , δ) ∩ Y for every i ∈ {1, 2, · · · , k}. So E = {en1 , en2 , · · · , enk } ∈ Fin(Y ). For eni ∈ E (i = 1, 2, · · · , k), we have M (D, eni ) ≥ M (ani , eni ) ≥ M (ani , cni ) ∗ M (cni , eni ) ≥ (1 − δ) ∗ (1 − δ) > 1 − ε. Hence inf M (D, e) = min{M (D, eni )|i = 1, 2, · · · , k} > 1 − ε.

e∈E

On the other hand, let d ∈ D. Then there exists a cni ∈ C ′ such that d ∈ BM (cni , δ). Hence M (d, E) ≥ M (d, eni ) ≥ M (d, cni ) ∗ M (cni , eni ) ≥ (1 − δ) ∗ (1 − δ). So inf M (d, E) ≥ (1 − δ) ∗ (1 − δ) > 1 − ε.

d∈D

Hence HM (D, E) = min{ inf M (d, E), inf M (D, e)} > 1 − ε, e∈E

d∈D

that is, E ∈ BHM (D, ε). Consequently, E ∈ Fin(Y ) ∩ BHM (D, ε). We complete the proof. Let (X, M, ∗) be a fuzzy metric space and A ⊂ X. We will denote by M |A×A×(0,∞) the restriction of M on A × A × (0, ∞). It is easy to see that M |A×A×(0,∞) is a fuzzy metric on A. We will simply write M |A instead of M |A×A×(0,∞) when confusion is not possible.

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Lemma 4.3 Let (X, M, ∗) be a precompact fuzzy metric space and A ⊂ X. Then (A, M |A , ∗) is precompact. Proof Let ε ∈ (0, 1) and t > 0. Then there exists a δ ∈ (0, ε) such that (1 − δ) ∗ (1 − δ) > 1 − ε. Because of precompactness of (X, M, ∗) , we can find finite points x1 , x2 , · · · , xn in X such that X=

n ∪

t BM (xi , δ, ). 2 i=1

Since A ⊂ X, there exists {xn1 , xn2 , · · · , xnk } ⊂ {x1 , x2 , · · · , xn } such that A⊂ and

Choose ynj we have

k ∪

t BM (xnj , δ, ) 2 j=1

t A ∩ BM (xnj , δ, ) ̸= Ø(j = 1, 2, · · · , k). 2 ∈ A∩BM (xnj , δ, 2t ) (j = 1, 2, · · · , k). Then, for each z ∈ BM (xnj , δ, 2t ),

t t M (z, ynj , t) ≥ M (z, xnj , ) ∗ M (xnj , ynj , ) ≥ (1 − δ) ∗ (1 − δ) > 1 − ε. 2 2 So

t BM (xnj , δ, ) ⊂ BM (ynj , ε, t). 2

Hence A⊂

k ∪ t BM (ynj , ε, t). BM (xnj , δ, ) ⊂ 2 j=1 j=1 k ∪

Whence A=(

k ∪

BM (ynj , ε, t)) ∩ A =

j=1

k ∪

(BM (ynj , ε, t) ∩ A) =

j=1

k ∪

BM |A (ynj , ε, t).

j=1

We are done. Theorem 4.4 Let (X, M, ∗) be a stationary and F-bounded fuzzy metric space. Then (Cld(X), HM , ∗) is precompact if and only if (X, M, ∗) is precompact. Proof Suppose that (Cld(X), HM , ∗) is precompact. For each x, y ∈ X, we have HM ({x}, {y}) = M (x, y). So we can regard X as a subset of Cld(X) and M as HM |{{x}|x∈X} . It follows from Lemma 4.3 that (X, M, ∗) is precompact. Conversely, suppose that (X, M, ∗) is precompact. Let ε ∈ (0, 1) and D ∈ Cld(X). Then, by precompactness of (X, M, ∗), we can find an F ∈ Fin(X) such that ∪ ε X= BM (x, ). 2 x∈F

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Therefore, there exists an FD ⊂ F such that ∪ ε D⊂ BM (x, ). 2 x∈FD

Also, we can find xD ∈ BM (x, 2ε ) ∩ D for every x ∈ FD . Note that there exists an xy ∈ FD such that ε y ∈ BM (xy , ) 2 for every y ∈ D. It follows that ε M (y, FD ) ≥ M (y, xy ) > 1 − . 2 Hence inf M (y, FD ) ≥ 1 −

y∈D

ε 2

> 1 − ε.

On the other hand, for each x ∈ FD , we get ε M (D, x) ≥ M (xD , x) > 1 − . 2 Hence inf M (D, x) ≥ 1 −

x∈FD

ε > 1 − ε. 2

So HM (D, FD ) > 1 − ε, i.e., D ∈ BHM (FD , ε). Since F is a finite set, we see that F = {FD |D ∈ Cld(X)} is a finite family. Observe that ∪ BHM (FD , ε). Cld(X)= FD ∈F

It follows that (Cld(X), HM , ∗) is precompact. Definition 4.5 Let (X, M, ∗) be a fuzzy metric space, Y ⊂ X, r ∈ (0, 1) and t > 0. Y is said to be fuzzy r discrete with respect to t if M (x, y, t) < 1 − r whenever x, y ∈ Y and x ̸= y. Definition 4.6 Let (X, M, ∗) be a fuzzy metric space and Y ⊂ X. Y is called a fuzzy uniformly discrete set provided that there exist r ∈ (0, 1) and t > 0 such that Y is fuzzy r discrete with respect to t. According to Zorn’s lemma, it is straightforward to show that, by the inclusion relationship of the sets, X has a maximal subset which is fuzzy r discrete with respect to t for all r ∈ (0, 1) and t > 0. Lemma 4.7 Let (X, M, ∗) be a fuzzy metric space and Y be a fuzzy uniformly discrete subset of X. Then Y is a closed set in (X, τM ).

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Proof By assumption, we can find some r0 ∈ (0, 1) and t0 > 0 such that Y is fuzzy r0 discrete with respect to t0 . According to the continuity of ∗, there exists a ε ∈ (0, r0 ) such that (1 − ε) ∗ (1 − ε) > 1 − r0 . Let x ∈ / Y . To complete our proof, it suffices to prove that there exists a open set U of x in X such that U ∩ Y = Ø. For every y, z ∈ BM (x, ε, t20 ), we have M (y, z, t0 ) ≥ M (x, y,

t0 t0 ) ∗ M (x, z, ) ≥ (1 − ε) ∗ (1 − ε) > 1 − r0 . 2 2

So BM (x, ε, t20 ) contains at most one point of Y . If BM (x, ε, t20 ) ∩ Y = Ø, then BM (x, ε, t20 ) is the required open set. If BM (x, ε, t20 ) ∩ Y = {a}, then, by the Hausdorffness of (X, M, ∗), we can choose an n ∈ N such that a∈ / BM (x,

1 1 , ). n n

So we get x ∈ BM (x, ε, with BM (x, ε,

t0 1 1 ) ∩ BM (x, , ), 2 n n

t0 1 1 ) ∩ BM (x, , ) ∩ Y = Ø, 2 n n

which implies that BM (x, ε, t20 ) ∩ BM (x, n1 , n1 ) is the required open set. We are done. Lemma 4.8 Let (X, M, ∗) be a fuzzy metric space and Y be an uncountable fuzzy uniformly discrete subset of X. Then X is not separable. Proof Suppose that X is separable. Then we take a countable and dense subset A in X. According to assumption, we can find some r0 ∈ (0, 1) and t0 > 0 such that M (x, y, t0 ) < 1 − r0 for all x, y ∈ Y and x ̸= y. According to continuity of ∗, there exists a ε ∈ (0, r0 ) such that (1 − ε) ∗ (1 − ε) > 1 − r0 . Pick a, b ∈ Y , with a ̸= b. We conclude that BM (a, ε,

t0 t0 ) ∩ BM (b, ε, ) = Ø. 2 2

Indeed, otherwise, we can take a c ∈ BM (a, ε, t20 ) ∩ BM (b, ε, t20 ). Then M (a, b, t0 ) ≥ M (a, c,

t0 t0 ) ∗ M (c, b, ) ≥ (1 − ε) ∗ (1 − ε) > 1 − r0 , 2 2

which contradicts M (a, b, t0 ) < 1 − r0 . So {BM (y, ε, t20 )|y ∈ Y } is an uncountable and pair-wise disjoint open family. Since A is dense in X, we see that 9

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BM (y, ε, t20 ) ∩ A ̸= Ø for all y ∈ Y . This shows that A is uncountable, which is a contradiction. We complete the proof. Lemma 4.9 [8] Let (X, M, ∗) be a precompact fuzzy metric space. Then it is separable. Theorem 4.10 Let (X, M, ∗) be a stationary and F-bounded fuzzy metric space. Then (X, M, ∗) is precompact if and only if (Cld(X),HM , ∗) is separable. Proof Suppose that (X, M, ∗) is precompact. then, according to Theorem 4.4, we deduce that (Cld(X),HM , ∗) is precompact. So, by Lemma 4.9, we see that (Cld(X),HM , ∗) is separable. Conversely, Let (Cld(X),HM , ∗) be separable. Suppose that (X, M, ∗) fails to be precompact. Then there exists an infinite fuzzy uniformly discrete subset Y of X. Observe that P(Y ) is the set of nonempty subsets of Y . Then P(Y ) is uncountable. Take A, C ∈ P(Y ). Then, by Lemma 4.7, we see that A, C ∈ Cld(X). Now, for each a ∈ A and c ∈ C, we can find r0 ∈ (0, 1) such that M (a, c) < 1 − r0 . So M (a, C) ≤ 1 − r0 . It follows that inf M (a, C) ≤ 1 − r0 .

a∈A

Similarly, we obtain inf M (A, c) ≤ 1 − r0 .

c∈C

Hence

r0 , 2 which implies that P(Y ) is a fuzzy uniformly discrete subset of Cld(X). Thus, by Lemma 4.8, Cld(X) fails to be separable, which is a contradiction. Consequently, (X, M, ∗) is precompact. HM (A, C) ≤ 1 − r0 < 1 −

From Theorem 4.2, Theorem 4.4 and Theorem 4.10 we immediately deduce the next corollary. Corollary 4.11 Let Y be a dense subspace of a stationary and F-bounded fuzzy metric space (X, M, ∗). Then the following are equivalent. (i) (X, M, ∗) is precompact. (ii) Fin(Y ) is dense in (Cld(X), HM , ∗). (iii) (Cld(X), HM , ∗) is precompact. (iv) (Cld(X),HM , ∗) is separable.

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References [1] R. Engelking, General Topology, PWN-Polish Science Publishers, warsaw, 1977. [2] M. A. Erceg, Metric spaces in fuzzy set theory, J. Math. Anal. Appl. 69 (1) (1979) 205–230. [3] A. George, P. Veeramani, On some resules in fuzzy metric spaces, Fuzzy Sets and Systems 64 (3) (1994) 395–399. [4] A. George, P. Veeramani, Some theorems in fuzzy metric spaces, J. Fuzzy Math. 3 (1995) 933–940. [5] A. George, P. Veeramani, On some resules of analysis for fuzzy metric spaces, Fuzzy Sets and Systems 90 (3) (1997) 365–368. [6] V. Gregori, A. L´opez-Crevill´en, S. Morillas, A. Sapena, On convergence in fuzzy metric spaces, Topology and its Application 156 (18) (2009) 3002– 3006. [7] V. Gregori, S. Morillas, A. Sapena, Example of fuzzy metrics and applications, Fuzzy Sets and Systems 170 (1) (2011) 95–111. [8] V. Gregori, S. Romaguera, Some properties of fuzzy metric spaces, Fuzzy Sets and Systems 115 (3) (2000) 485–489. [9] V. Gregori, S. Romaguera, On completion of fuzzy metric spaces, Fuzzy Sets and Systems 130 (3) (2002) 399–404. [10] V. Gregori, S. Romaguera, Characterizing completable fuzzy metric spaces, Fuzzy Sets and Systems 144 (3) (2004) 411–420. [11] V. Gregori, S. Romaguera, A. Sapena, Uniform continuity in fuzzy metric spaces, Rend. Istit. Mat. Univ. Trieste, 32 suppl.2 (2001), 81–88. [12] O. Kaleva, S. Seikkala, On fuzzy metric spaces, Fuzzy Sets and Systems 12 (3) (1984) 215–229. [13] I. Kramosil, J. Michalek, Fuzzy metric and statistical metric spaces, Kybernetika 11 (1975) 336–344. [14] C. Q. Li, On some results of metrics induced by a fuzzy ultrametric, Filomat, 27 (6) (2013) 1133–1140. [15] C. Q. Li, Some properties of intuitionstic fuzzy metric spaces, Journal of Computational Analysis and Applications, 16 (4) (2014) 670–677. [16] C. Q. Li, Some properties of the Hausdorff fuzzy metric on finite sets, Journal of Computational Analysis and Applications, 19 (2) (2015) 359– 364. [17] J. Rodr´ıguez-L´ opez, S. Romaguera, The Hausdorff fuzzy metric on compact sets, Fuzzy Sets and Systems 147 (2) (2004) 273–283. [18] A.Sapena, A contribution to the study of fuzzy metric spaces, Applied General Topology 2 (1) (2001) 63–75. [19] A.Savchenko, M.Zarichnyi, Fuzzy ultrametrics on the set of probability measures, Topology 48 (2–4) (2009) 130–136. [20] P.Veeramani, Best approximation in fuzzy metric spaces, J. Fuzzy Math. 9 (2001) 75–80.

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Semilocal convergence of a modified Chebyshev-like’s method for solving nonlinear equations under generalized weak condition ∗ Lin Zheng

† 1, 2

1. School of Statistics and Applied Mathematics, Anhui University of Finance and Economics, Bengbu 233030, China; 2. Department of Mathematics, Shanghai University, Shanghai 200444, China

Abstract: In this paper, the semilocal convergence of the modified Chebyshev-like’s method is established by using recurrence relations under generalized weak condition. We prove an existence-uniqueness theorem and give a priori error bound which demonstrates the R-order convergence of the method. Moreover, the dynamical behavior of this method is also studied. Finally, numerical examples are presented to demonstrate our approach. Keywords: Nonlinear equations; Chebyshev-like’s method; Recurrence relations; Semilocal convergence; A priori error bounds; Dynamics.

1

Introduction

A number of problems arisen from scientific and engineering areas often needs to find the solution of nonlinear equations in Banach spaces F (x) = 0,

(1.1)

where F is a third-order Fr´echet-differentiable operator defined on a convex subset Ω of a Banach space X with values in a Banach space Y . Generally, iterative methods are often used to solve this problem [1]. Newton’s method being a second-order method is one of best known of these methods. The convergence of Newton’s method in Banach spaces was established by Kantorovich in [2]. The convergence of the sequence obtained by the iterative expression is derived from the convergence of majorizing sequences. This technique has been used by many authors in order to establish the order of convergence of the variants of Newton’s methods [3-9]. An alternative approach is developed to establish this convergence by using recurrence relations. The approach is also a very popular technique to ∗

The work are supported by National Natural Science Foundation of China (11371243, 11301001, 61300048), Natural Science Foundation of Universities of Anhui Province (KJ2014A003, KJ2014A223). † Corresponding author. E-mail addresses: [email protected].

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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 [10-23]. In [9], we introduce a modified Chebyshev-like’s method given by h i 1 1 xn+1 = xn − I + KF (xn ) + KF2 (xn ) Γn F (xn ), (1.2) 2 2 where Γn = [F 0 (xn )]−1 , KF (xn ) = Γn F 00 (un )Γn F (xn ) and un = xn − 31 Γn F (xn ). By assuming that (A1) Γ0 exist and kΓ0 F (x0 )k ≤ η, (A2) kΓ0 k ≤ β, 00 (A3) kF (x)k ≤ M, x ∈ Ω, (A4) kF 000 (x)k ≤ N, x ∈ Ω, 000 000 (A5) kF (x) − F (y)k ≤ Lkx − yk, ∀ x, y ∈ Ω, we have analyzed the semilocal convergence of the method given by (1.2) by majorizing sequences and proved the R-order is improved to four, the computation efficiency and error estimate were also given. Numerical applications shows this method can solve some equations successfully. But under assumptions (A1)-(A5), we can not study the solution of some equations. Such as the nonlinear integral equation of mixed Hammerstein type, which is given by m Z b X x(s) + Gi (s, t)Hi (x(t))dt = u(s), s ∈ [a, b], (1.3) i=1

a

where −∞ < a < b < ∞, u, Hi and Gi , for i = 1, 2, · · · , m, are known functions and x is a solution to be determined. The problem is from the dynamic model of a chemical reactor [24]. On the condition that Hi000 (x(t)) is (Li , qi )-H¨older continuous in Ω, i = 1, 2, . . . , m, then corresponding operator F : Ω ⊆ C[0, 1] → C[0, 1], m Z b X [F (x)](s) = x(s) + Gi (s, t)Hi (x(t))dt − u(s), s ∈ [a, b], (1.4) i=1

a

is such that its third Fr´echet derivative is neither Lipschitz continuous nor H¨older continuous in Ω while, for an example, we consider the max-norm. For this case, kF 000 (x) − F 000 (y)k ≤

m X

Li kx − ykqi , Li ≥ 0, qi ∈ [0, 1], x, y ∈ Ω.

(1.5)

i=1

Because of the importance of nonlinear integral equation of mixed Hammerstein type, several authors [18-20] have considered a mild condition kF 000 (x) − F 000 k ≤ ω(kx − yk), x, y ∈ Ω,

(1.6)

where ω(z) is nondecreasing continuous real valued function for z > 0, such that ω(0) ≥ 0, on F 000 to study the semilocal convergence of some iterative methods. In the paper, the semilocal convergence of the modified Chebyshev-like’s method is established by using recurrence relations under the assumption that F 000 satisfies the ω-continuity condition (1.6), An existence-uniqueness theorem is also established for the solution along with its a priori error bounds. Moreover, the dynamical behavior of modified Chebyshev-like’s method is also studied. Finally numerical examples are presented to demonstrate our approach. 2 355

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2

Recurrence relations

Let x0 ∈ Ω and the nonlinear operator F : Ω ⊂ X → Y be continuously third-order Fr´echet differentiable where Ω is an open set and X and Y are Banach spaces. We assume that (C1) Γ0 exist and kΓ0 F (x0 )k ≤ η, (C2) kΓ0 k ≤ β, 00 (C3) kF (x)k ≤ M, x ∈ Ω, (C4) kF 000 (x)k ≤ N, x ∈ Ω, 000 000 (C5) kF (x) − F (y)k ≤ ω(kx − yk), ∀ x, y ∈ Ω, where ω(z) is non-decreasing continuous real function for z > 0 and satisfy ω(0) ≥ 0, (C6) there exists a non-negative real function ϕ(t) ≤ 1, such that ω(tz) ≤ ϕ(t)ω(z), for t ∈ [0, 1], z ∈ (0, +∞). Notice that condition (C6) is not restrictive, since we can always consider ϕ(t) = 1, as a consequence of ω is non-decreasing function, but its interest is to sharp the priori error bounds. Firstly we give the following lemma to show an approximation of operator F , which will be used in the latter developments. Lemma 1 Assume that the nonlinear operator F : Ω ⊂ X → Y is continuously third-order Fr´echet differentiable where Ω is an open set and X and Y are Banach spaces. Then we have F (xn+1 ) = −

¤ 1 £ 00 F (un ) − F 00 (xn ) (yn − xn )KF (xn )(yn − xn ) 2 1 + F 00 (xn )(yn − xn )KF2 (xn )(yn − xn ) 2 Z 1 ¡ ¢ + F 00 yn + t(xn+1 − yn ) (1 − t)dt(xn+1 − yn )2 0 Z ´ i 1 1 1 h 000 ³ − F xn + t(yn − xn ) − F 000 (xn ) dt(yn − xn )3 6 0 3 Z 1h i ¡ ¢ 1 F 000 xn + t(yn − xn ) − F 000 (xn ) (1 − t)2 dt(yn − xn )3 + 2 0 Z 1 ¡ ¢ + F 000 xn + t(yn − xn ) (1 − t)dt(yn − xn )2 (xn+1 − yn ).

(2.1)

0

Now we consider the following scalar functions 1 1 g(t) = 1 + t + t2 , 2 2 1 h(t) = , 1 − tg(t) 1 3 1 I1 + 3I2 `(t, u, v) = t (t2 + 2t + 5) + (3t + 5)tu + v, 8 12 6 R1 R1 where I1 = 0 ϕ( 13 t)dt and I2 = 0 ϕ(t)(1 − t)2 dt.

(2.2) (2.3) (2.4)

Let Φ(t) = g(t)t − 1. Since Φ(0) = −1 and Φ(1) = 1 > 0, then Φ(t) has at least a zero in (0, 1). Let s is the smallest positive zero of the scalar function g(t)t − 1. Denote a0 = M βη, b0 = N βη 2 , c0 = w(η)βη 2 and d0 = h(a0 )`(a0 , b0 , c0 ). Next, some properties of the functions g, h, ` defined in (2.2)-(2.4) are given in the following lemma. 3 356

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Lemma 2 Let the real functions g, h and ` be given in (2.2)-(2.4). Then (i) g(t) and h(t) are increasing and g(t) > 1, h(t) > 1 for t ∈ (0, s), (ii) `(t, u, v) is increasing for t ∈ (0, s), u > 0, v > 0. For n = 0, the existence of Γ0 implies the existence of y0 , u0 . This gives us ky0 − x0 k = kΓ0 F (x0 )k ≤ η, and

° 1 1° ° ° ku0 − x0 k = °Γ0 F (x0 )° ≤ η. 3 3 ¡ ¢ This means that y0 , u0 ∈ B(x0 , Rη), where R = g(a0 )/ 1 − d0 . Furthermore, we have kKF (x0 )k = kΓ0 F 00 (u0 )Γ0 F (x0 )k ≤ kΓ0 kkF 00 (u0 )kkΓ0 F (x0 )k ≤ M βη = a0 . We can obtain ° ° 1 1 2 ° ° kx1 − x0 k = °I + KF (x0 ) + KF (x0 )°kΓ0 F (x0 )k 2 2 h 1 1 i ≤ 1 + a0 + a20 ky0 − x0 k = g(a0 )ky0 − x0 k. 2 2

(2.5)

From the assumption d0 < 1/h(a0 ) < 1, it follows that x1 ∈ B(x0 , Rη). We also have °1 ° 1 ° ° kx1 − y0 k ≤ ° KF (x0 ) + KF2 (x0 )°kΓ0 F (x0 )k 2 2 a0 (1 + a0 ) ky0 − x0 k. ≤ 2

(2.6)

By a0 < s and g(a0 ) < g(s), we have kI − Γ0 F 0 (x1 )k ≤ kΓ0 kkF 0 (x0 ) − F 0 (x1 )k ≤ M βkx1 − x0 k ≤ a0 g(a0 ) < 1. It follows by the Banach lemma that Γ1 = [F 0 (x1 )]−1 exists and kΓ1 k ≤

1 kΓ0 k = h(a0 )kΓ0 k. 1 − a0 g(a0 )

(2.7)

By Lemma 1, we can get kF (x1 )k ≤

a20 a0 1 M η 2 + N η 3 + M kx1 − y0 k2 2 6 2 1 1 1 + I1 w(η)η 3 + I2 w(η)η 3 + N η 2 kx1 − y0 k. 6 2 2

(2.8)

Then from (2.7) and (2.8), we have ky1 − x1 k = kΓ1 F (x1 )k ≤ kΓ1 kkF (x1 )k ≤ d0 ky0 − x0 k.

(2.9)

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Because of g(a0 ) > 1, we obtain ky1 − x0 k ≤ ky1 − x1 k + kx1 − x0 k ³ ´ ≤ g(a0 ) + d0 η ³ ´ < g(a0 ) 1 + d0 η < Rη,

(2.10)

M kΓ1 kkΓ1 F (x1 )k ≤ M h(a0 )kΓ0 kd0 ky0 − x0 k = a0 h(a0 )d0 ,

(2.11)

N kΓ1 kkΓ1 F (x1 )k2 ≤ N h(a0 )kΓ0 kd20 ky0 − x0 k2 = b0 h(a0 )d20 ,

(2.12)

which shows y1 ∈ B(x0 , Rη). In addition, we have

ω(ky1 − x1 k)kΓ1 kky1 − x1 k2 ≤ h(a0 )kΓ0 kω(d0 ky0 − x0 k)d20 ky0 − x0 k2 ≤ c0 h(a0 )d20 ϕ(d0 ),

(2.13)

kx2 − x0 k ≤ kx2 − x1 k + kx1 − x0 k ≤ g(a1 )ky1 − x1 k + g(a0 )ky0 − x0 k

(2.14)

≤ (1 + d0 )g(a0 )ky0 − x0 k < Rη. Since

kI − Γ1 F 0 (x2 )k ≤ kΓ1 kkF 0 (x1 ) − F 0 (x2 )k ≤ M kΓ1 kkx2 − x1 k ≤ a0 h(a0 )d0 g(a0 h(a0 )d0 ) < 1,

and by the Banach lemma, Γ2 = [F 0 (x2 )]−1 exists and kΓ2 k ≤ h(a0 h(a0 )d0 )kΓ1 k.

(2.15)

Hence x2 is well defined. We now write a0 h(a0 )d0 = a1 , b0 h(a0 )d20 = b1 , c0 h(a0 )d20 ϕ(d0 ) = c1 and define for n ≥ 0 an+1 = an h(an )dn , bn+1 = bn h(an )d2n , cn+1 = cn h(an )d2n ϕ(dn ),

(2.16)

dn+1 = h(an+1 )`(an+1 , bn+1 , cn+1 ). Later developments will require the following lemma, where some properties of the previous scalar sequences are given. Lemma 3 Let the real functions g, h and ` be given in (2.2)-(2.4). If 0 < a0 < s and h(a0 )d0 < 1,

(2.17)

then we have (i) h(an ) > 1 and h(an )dn < 1 for n ≥ 0, (ii) the sequences {an }, {bn }, {cn } and {dn } are decreasing, (iii) g(an )an < 1 and h(an )dn < 1 for n ≥ 0. 5 358

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Our next goal is to guarantee that (1.2) is well defined. To do this, the system of recurrence relations given in the next lemma must be satisfied. The proof follows by using a similar way that the above-mentioned and invoking the induction hypothesis. Lemma 4 Let the assumptions of Lemma 3 and the conditions (C1)-(C6) hold. Then the following items are true for all n ≥ 1: (i) There exists Γn = [F 0 (xn )]−1 and kΓn k ≤ h(an−1 )kΓn−1 k, (ii) kyn − xn k = kΓn F (xn )k ≤ dn−1 kyn−1 − xn−1 k ≤ dn0 ky0 − x0 k < η, (iii) M kΓn kkΓn F (xn )k ≤ an , (iv) N kΓn kkΓn F (xn )k2 ≤ bn , (v) ω(kyn − xn k)kΓn kkyn − xn k2 ≤ cn , (vi) kxn+1 − xn k ≤ g(an )kyn − xn k, (vii) kyn − x0 k ≤ Rη and kxn+1 − x0 k ≤ g(a0 )

3

1−dn+1 0 1−d0 ky0

− x0 k < Rη, where R =

g(a0 ) 1−d0 .

Semilocal convergence

We are now interested in proving that sequence (1.2) is convergent. To do this, we will see that (1.2) is a Cauchy sequence. We will give some properties of the scalar sequence {an }, {bn }, {cn } and {dn } in the following lemma. Lemma 5 Let the real functions g, h and ` be given in (2.2)-(2.4). Let τ ∈ (0, 1), then g(τ t) < g(t), h(τ t) < h(t) and `(τ t, τ 2 u, τ 2 v) < τ 2 `(t, u, v) for t ∈ (0, s). Lemma 6 Under the assumptions of Lemma 3. Let γ = h(a0 )d0 , δ = 1/h(a0 ). For n ≥ 0, we have 3n+1 −1 n an+1 ≤ γ 3 an ≤ γ 2 a0 , (3.1) ³ n ´2 n+1 bn+1 < γ 3 bn < γ 3 −1 b0 , (3.2) ³ n ´2 n+1 cn+1 < γ 3 cn < γ 3 −1 c0 , dn+1 = h(an+1 )`(an+1 , bn+1 , cn+1 ) ≤ δγ 3

(3.3) n+1

.

(3.4)

Proof The Lemma will be proved by induction. Since a1 = γa0 , by the above-mentioned assumption, we get γ < 1. We also get b1 = b0 h(a0 )d20 < γ 2 b0 , and c1 = c0 h(a0 )d20 ϕ(d0 ) < γ 2 c0 , as ϕ(d0 ) ≤ 1. Suppose (3.1)-(3.3) hold for n = k, then ak+1 = ak h(ak )dk = ak h2 (ak )`(ak , bk , ck ) ¡ k−1 ¢ ³ k−1 ¡ k−1 ¢2 ¡ k−1 ¢2 ´ k−1 ≤ γ 3 ak−1 h2 γ 3 ak−1 ` γ 3 ak−1 , γ 3 bk−1 , γ 3 ck−1 ¡ k−1 ¢2 k−1 k ≤ γ 3 ak−1 h2 (ak−1 ) γ 3 `(ak−1 , bk−1 , ck−1 ) = γ 3 ak .

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We also have bk+1 = bk h(ak )d2k < and ck+1 = ck h(ak )d2k ϕ(dk )
−1 and every integer k ≥ 0, we have (1 + x)k − 1 ≥ kx. Thus, kxm+n − xn k ≤ g(a0 )δ n γ

3n −1 2

n

1 − γ m·3 δ m η. 1 − γ 3n δ

(3.6)

It follows that {xn } is a Cauchy sequence. So there exists a x∗ such that lim xn = x∗ . n→∞ For m ≥ 1 and n = 0, we get kxm − x0 k ≤ g(a0 )

1 − γ mδm η < Rη. 1 − γδ

Hence, xm ∈ B(x0 , Rη), for all m ≥ 0. By letting n = 0, m → ∞ in (3.6), we obtain kx∗ − x0 k ≤ Rη.

(3.7)

This shows x∗ ∈ B(x0 , Rη). Now we prove that x∗ is a solution of F (x) = 0. Since kF 0 (xn )k ≤ kF 0 (x0 )k + kF 0 (xn ) − F 0 (x0 )k ≤ kF 0 (x0 )k + M kxn − x0 k ≤ kF 0 (x0 )k + M Rη, we can obtain kF (xn )k ≤ kF 0 (xn )kkΓn F (xn )k ≤ (kF 0 (x0 )k + M Rη)kΓn F (xn )k. Since

(3.8)

³ n−1 Y ´ 3n −1 kΓn F (xn )k ≤ dn−1 kyn−1 − xn−1 k = · · · = η di ≤ ηδ n γ 2 , i=0

by letting n → ∞, we obtain kΓn F (xn )k → 0, and kF (xn )k → 0 in (3.8). Hence, by the continuity of F in Ω, we obtain F (x∗ ) = 0. Now we prove the uniqueness of x∗ in B(x0 , M2β − Rη) ∩ Ω. Firstly we can obtain x∗ ∈ ¡ ¢ B(x0 , M2β − Rη) ∩ Ω. Since R = g(a0 )/ 1 − d0 < 1/a0 , then we have ³2 ´ 2 1 − Rη = − R η > η > Rη, Mβ a0 a0 and then B(x0 , Rη) ⊆ B(x0 , M2β − Rη) ∩ Ω. 8 361

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Let x∗∗ be another zero of F (x) in B(x0 , M2β − Rη) ∩ Ω. By Taylor theorem, we have Z 1 ¢ 0¡ ∗∗ ∗ 0 = F (x ) − F (x ) = F (1 − t)x∗ + tx∗∗ dt(x∗∗ − x∗ ). (3.9) 0

Since

°Z 1h i ° ° ° ¡ ¢ 0 ∗ ∗∗ 0 ° kΓ0 k° F (1 − t)x + tx − F (x0 ) dt° ° 0 Z 1 £ ¤ ≤ Mβ (1 − t)kx∗ − x0 k + tkx∗∗ − x0 k dt 0

i 2 Mβ h Rη + − Rη = 1, < 2 Mβ ¢ R1 ¡ it follows by the Banach lemma that 0 F 0 (1 − t)x∗ + tx∗∗ dt is invertible and hence x∗∗ = x∗ . By letting m → ∞ in (3.6), we obtain (3.5) and furthermore kxn − x∗ k ≤

g(a0 )η 1/3 γ (1 − d

¡ 0)

γ 1/3

¢3n

.

(3.10)

This means that the modified Chebyshev-like’s method given by (1.2) is of R-order of convergence at least three.

3.2

R-order of convergence

Now we consider the following mixed condition 000

000

kF (x) − F (y)k ≤

m X

Li kx − ykqi , Li ≥ 0, qi ∈ [0, 1], x, y ∈ Ω.

i=1

P qi qi By choosing w(µ) = i=1 (Li µqi ), we have w(tµ) = m i=1 (Li t µ ), since t ∈ [0, 1], qi ∈ [0, 1], p then ϕ(t) = t , where p = min{q1 , q2 , · · · , qm }. Now we consider that ϕ(t) = tp , where p ∈ (0, 1]. In this situation Z 1 ¡1 ¢ 1 1 I1 = ϕ t dt = p · , (3.11) 3 3 1+p 0 Pm

and

Z I2 =

1

ϕ(t)(1 − t)2 dt =

0

1 . (1 + p)(2 + p)(3 + p)

(3.12)

The sequence {cn } is reduced to cn+1 = cn h(an )d2+p n , n ≥ 1. Moveover `(τ t, τ 2 u, τ 2+p v) < τ 3+p `(t, u, v), for τ ∈ (0, 1), p ∈ [0, 1]. Then n

an+1 ≤ γ (3+p) an ≤ γ

(3+p)n+1 −1 2+p

n

bn+1 < γ 2(3+p) bn < γ cn+1 < γ

(2+p)(3+p)n

a0 , n ≥ 0,

2[(3+p)n+1 −1] 2+p

cn < γ

b0 , (3+p)n+1 −1

n ≥ 0,

c0 , n ≥ 0.

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Consequently, the new a prior error estimates for iteration (1.2) are: kx∗ − xn k ≤ g(a0 )ηγ so that kx∗ − xn k ≤

(3+p)n −1 2+p

δn , n ≥ 0, 1 − γ (3+p)n δ

³

g(a0 )η 1

γ 2+p (1 − γδ)

1

γ 2+p

´(3+p)n

, n ≥ 0,

and the R-order of convergence is then at least 3 + p. Remark Notice that w(z) = Lz, L ≥ 0, if F 000 is Lipschitz continuous in Ω and the R-order of convergence of iteration (1.2) is now at least four. And if F 000 is (L, p)-H¨older continuous in Ω, then w(z) = Lz p , L ≥ 0, so that (1.2) is of R-order of convergence at least 3 + p.

4

Application

We illustrate the previous study with an application to the following nonlinear integral equations. Example 1. Consider the mixed Hammerstein type integral equation [24]: x(s) = 1 +

1 3

Z

1

³ ´ G(s, t) x(t)10/3 + x(t)4 dt,

s, t ∈ [0, 1],

(4.1)

0

where x ∈ X. Here X = C[0, 1] is the space of continuous functions on [0, 1] with the max-norm kxk = max |x(s)| . s∈[0,1]

And the kernel G is the Green function ( G(s, t) =

(1 − s)t, t ≤ s, s(1 − t), s ≤ t.

The analysis and computation of these types of equations are justified by the dynamic model of a chemical reactor. Solving (4.1) is equivalent to solve F (x) = 0, where F : Ω ⊆ C[0, 1] → C[0, 1], 1 [F (x)](s) = x(s) − 1 − 3

Z

1

³ ´ G(s, t) x(t)10/3 + x(t)4 dt, s ∈ [0, 1],

(4.2)

0

and Ω is a suitable non-empty open convex domain. Note that the first, second and third Fr´echet derivatives of the operator F are given by Z ³ 10 ´ 1 1 0 G(s, t) x(t)7/3 + 4x(t)3 y(t)dt, (4.3) [F (x)y](s) = y(s) − 3 0 3 Z ³ 70 ´ 1 1 00 [F (x)yz](s) = − G(s, t) x(t)4/3 + 12x(t)2 z(t)y(t)dt, (4.4) 3 0 9 Z ³ 280 ´ 1 1 G(s, t) x(t)1/3 + 24x(t) u(t)z(t)y(t)dt. (4.5) [F 000 (x)yzu](s) = − 3 0 27

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Observe that F 000 is neither Lipschitz continuous nor H¨older continuous in Ω, but the operator F satisfies the assumptions of Theorem 1, so that a solution of (4.1) can be approximated by (1.2). Now we consider Ω = B(0, 3/2) ⊆ X as an open convex nonempty domain and choose x0 (s) = 1 as an initial approximation solution. One can easily obtain kΓ0 k ≤ 36/25 = β, kΓ0 F (x0 )k ≤ 3/25 = η, kF 00 (x)k ≤ 1.6815 = M, kF 000 (x)k ≤ 1.9946 = N, √ 35 √ 3 3 ω(z) = z + z, ϕ(t) = t, I1 = 0.5200, I2 = 0.0964. 81 Hence, a0 = M βη = 0.2906, b0 = N βη 2 = 0.0414 and c0 = ω(η)βη 2 = 0.0069, so that Φ(a0 ) = a0 g(a0 ) − 1 ' −0.6550 < 0, and h(a0 )d0 ' 0.0564 < 1. Besides, the solution x∗ belongs to B(x0 , Rη) = B(1, 0.1480 · · · ) ⊆ Ω and it is unique in B(1, 0.6780 · · · ) ∩ Ω. Finally, we discretize (4.1) to transform it into a finite dimensional problem and we apply (1.2) to approximate a solution. This procedure consists of approximating the integral appearing in (4.1) by a numerical quadrature formula. We consider the following Gauss-Legendre formula Z

1

v(t)dt ' 0

m X

wi v(ti ),

i=1

where the nodes ti and the weights wi can be easily computed. If we denote xj = x(tj ), (j = 1, 2, · · · , m), (4.1) becomes the following nonlinear system of equations m 1X 10/3 αjk (xk + x4k ), j = 1, 2, · · · , m, (4.6) xj = 1 + 3 k=1

where

( αjk =

wk tk (1 − tj ) if k ≤ j, wk tj (1 − tk ) if k < j.

Now we apply the method given by (1.2) to compute (4.6) and compare it with the Chebyshevlike’s method in [30]. Taking into account that we have previously considered the starting function x0 (s) = 1, we now choose the vector x0 = (1, 1, · · · , 1)T as the initial iterate. All computations are carried out with double arithmetic precision. In the tests, we take m = 10, 20 in (4.6) respectively. Displayed in Tables 1 and 2 is the max-norm of vector functions at each iterative step. The stopping criterion that we consider is kF (xn )k∞ ≤ 10−15 . From the numerical results, we can see that the performance of the method (1.2) is better. This means that our method can be of practical interest.

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Table 1: Results of system (4.6) with m = 10 n 1 2 3

Chebyshev-like method 3.098738e-004 1.645750e-011 6.661338e-016

the method (1.2) 6.359533e-005 6.661338e-016

Table 2: Results of system (4.6) with m = 20 n 1 2 3 4

5

Chebyshev-like method 2.956476e-004 1.313716e-011 1.443290e-015 4.440892e-016

the method (1.2) 5.838933e-005 9.992007e-016

Dynamics

The dynamical analysis of a method is becoming a trend in recent publications [25-28] on iterative methods because it allows us to classify the various iterative formulas, not only from the point of view of its order of convergence, but also analyzing how these formulas behave as function of the initial estimate that is taken. Let us first recall some dynamical concepts. Consider a Fr´echet differential function G : Rn −→ Rn . The orbit of x ∈ Rn is defined as: x, G(x), G2 (x), · · · , Gp (x), · · · . A point xf is a fixed point of G if G(xf ) = xf . The basin of attraction of xf is the set of points whose orbits tend to this fixed point A(xf ) = {x ∈ Rn : Gp (x) −→ xf for p → ∞}. In this section we study the dynamics of the method (1.2) when applied to the solution of a 2 × 2 nonlinear system and compare it with the dynamics of Chebyshev-like’s method in [30]. We show that the method is generally convergent and depict the attraction basins. Example 2. Consider the following system [29]  1 1 1   f1 (x) = sin(x1 x2 ) − x2 − x1 = 0 2 4π 2 ¡ ¢¡ 2x1 ¢ e 1   f2 (x) = 1 − e − e + x2 − 2ex1 = 0 4π π where 0.25 ≤ x1 ≤ 1 and 1.5 ≤ x2 ≤ 2π. The exact solutions are x∗ = ( 21 , π)T and x∗ = (0.29945, 2.83693)T . For the comparisons, we have run the methods with tolerance 10−5 , performing a maximum of 20 iterations. The starting points form a uniform grid of size 600 × 600 in a rectangle of the real plane. The attraction basins have been colored according to the corresponding fixed point. 12 365

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Figure 1: Attraction basins for Chebyshev-like’s method Figures 1 and 2 show the attraction basins of the method (1.2) and Chebyshev-like’s method in [30], respectively. According to the figure, for any starting point that arise from the red or blue regions, the methods are converge to the solution in that region, while starting points from other region failed to convergence. The presented basin of attraction show the good performance of the method (1.2) as compared to Chebyshev-like’s method in [30].

6

Conclusion

In this paper, the semilocal convergence of the modified Chebyshev-like’s method for solving nonlinear equations in Banach spaces is established by using recurrence relations under the assumption that F 000 satisfies ω-continuity condition. An existence-uniqueness theorem is given to show the R-order convergence of the method. Also a priori error bounds is given. From the numerical results, we can observe that the performance of our method in this paper is better. The dynamical behavior of the method (1.2) has been compared with that of Chebyshevlike’s method in [30]. The presented basin of attraction have also confirmed better performance of the method (1.2).

References [1] J. M. Ortega, W. C. Rheinboldt, Iterative Solution of Nonlinear Equation in Several Variables, Academic Press, New York, 1970. [2] L. V. Kantorovich, G.P. Akilov, Functional analysis, Pergamon Press, Oxford, 1982. [3] J. M. Guti´errez, M.A. Hern´andez, A family of Chebshev-Halley type methods in Banach spaces, Bull. Austral. Math. Soc., 1997, 55, 113-130.

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Figure 2: Attraction basins for Modified Chebyshev-like’s method [4] I. K. Argyros, S. Hilout, Majorizing sequences for iterative methods, J. Comput. Appl. Math. 2012, 236, 1947-1960. [5] I. K. Argyros, S. Hilout, The majorant method in the theory of Newton-Kantorovich approximations and generalized Lipschitz conditions J. Comput. Appl. Math. 2016, 291, 332-347. [6] I. K. Argyros, Y. J. Cho, S. Hilout, On the semilocal convergence of the Halley method using recurrent functions, J. Appl. Math. Comput., 2011, 37, 221-246. [7] Q. B. Wu, Y. Q. Zhao, Newton-Kantorovich type convergence theorem for a family of new deformed-Chebyshev method, Appl. Math. Comput., 2007, 192, 405-412. [8] Lin Zheng, Chuanqing Gu, Fourth-order convergence theorem by using majorizing functions for super-Halley method in Banach spaces, International Journal of Computer Mathematics, 2013, 90, 423-434. [9] Lin Zheng, Ke Zhang, Liang Chen, On the convergence of a modified Chebyshev-like’s method for solving nonlinear equations, Taiwanese Journal of mathematics, 2015, 19, 193209. [10] V. Candela, A. Marquina, Recurrence relations for rational cubic methods I: the Halley method, Computing, 1990, 44, 169-184. [11] V. Candela, A. Marquina, Recurrence relations for rational cubic methods II: the Chebyshev method, Computing, 1990, 45, 355-367. [12] S. Amat, S. Busquier, J. M. Guti´errez, Third-order iterative methods with applications to Hammerstein equations: A unified approach, J. Comput. Appl. Math., 2011, 235, 2936-2943.

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[13] S. Amat, C. Berm´ udez, S. Busquier and S. Plaza, On a third-order Newton-type method free of bilinear operators. Numer. Linear Algebra Appl., 2010, 17, 639-653. [14] P. K. Parida, D. K. Gupta, Recurrence relations for semilocal convergence of a Newton-like method in Banach spaces, J. Math. Anal. Appl., 2008, 345, 350-361. [15] Lin Zheng, Chuanqing Gu, Semilocal convergence of a sixth-order method in Banach spaces, Numer. Algor., 2012, 61, 413-427. [16] Lin Zheng, Chuanqing Gu, Recurrence relations for semilocal convergence of a fifth-order method in Banach spaces, Numer. Algor., 2012, 59, 623-638. [17] M. A. Hern´andez, E. Mart´ınez, On the semilocal convergence of a three steps Newton-type iterative process under mild convergence conditions, Numer. Algor., 2015, 70, 377-392. [18] Xiuhua Wang, Jisheng Kou, Semilocal convergence of a modified multi-point Jarratt method in Banach spaces under general continuity condition, Numer. Algor., 2012, 60, 369-390. [19] Xiuhua Wang, Jisheng Kou, Convergence for a class of multi-point modified ChebyshevHalley methods under the relaxed conditions, Numer. Algor., 2015, 68, 569-583. [20] J. A. Ezquerro, M. A. Hern´andez, New iterations of R-order four with reduced computational cost, BIT. Numer Math., 2009, 49, 325-342. [21] J. L. Hueso, E. Mart´ınez, Semilocal convergence of a family of iterative methods in Banach spaces, Numer. Algor., 2014, 67, 365-384. [22] A. Corderoa, M.A. Hern´andez, N. Romerob, J.R. Torregrosaa, Semilocal convergence by using recurrence relations for a fifth-order method in Banach spaces, J. Comput. Appl. Math., 2015, 273 (1), 205-213. [23] S. Amat, C. Berm´ udez, M.A. Hern´andez, E. Mart´ınez, On an efficient k-step iterative method for nonlinear equations, J. Comput. Appl. Math., 2016, 302, 258-271. [24] D.D. Bruns, J.E. Bailey, Nonlinear feedback control for operating a nonisothermal CSTR near an unstable steady state, Chem. Eng.Sci., 1977, 32, 257-264. [25] J. L. Hueso, E. Mart´ınez, C. Teruel, Convergence efficiency and dynamics of new fourht and sixth order families of iterative methods for nonlinear systems, J. Comput. Appl. Math., 2015, 275, 412-420. [26] H. Esmaeili, M. Ahmadi, An efficient three-step method to solve system of nonlinear equations, Appl. Math. Comput., 2015, 266, 1093-1101. [27] Beny Neta, Melvin Scott, Changbum Chun, Basins of attraction for several methods to simple roots of nonlinear equations, Appl.Math. Comput., 2012, 218, 10548-10556. [28] S. Amat, S. Busquier, C. Berm´ udez, A.A. Magre˜ n´an, On the election of the dampled parameter of a two-step relaxed Newton-type method, Nonlinear Dyn., 2016, 84, 9-18.

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[29] J. Alikhani Koupaei, S.M.M.Hosseini, A new hybrid algorithm based on chaotic maps for solving systems of nonlinear equations, Chaos, Solitons and Fractals, 2015, 81, 233-245. [30] J. A. Ezquerro, J. M. Guti´errez, M. A. Hern´andez, M. A. Salanova, Chebyshev-like methods and quadratic equations, Rev. Anal. Num´er. Th´eor. Approx., 1999, 28, 23-25.

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Generalized Hyers–Ulam stability of sextic functional equation in random normed spaces Shaymaa Alshybania , S. Mansour Vaezpoura , Reza Saadatib,∗ a Department

of Mathematics and Computer Science, Amirkabir University of Technology, Tehran, Iran of Mathematics, Iran University of Science and Technology, Tehran, Iran

b Department

Abstract By using the direct and fixed point methods, we establish the general solution and generalized Hyers–Ulam stability of the following sextic functional equation f (nx + y) + f (nx − y) + f (x + ny) + f (x − ny) = (n4 + n2 )[f (x + y) + f (x − y)] + 2(n6 − n4 − n2 + 1)[f (x) + f (y)] in random normed spaces. Also, we present an illustrative example with the Lukasiewicz t-norm that can be a suitable approximation using this sextic function.

1. Introduction If the values of norms are probability distribution functions, then we have a generalized notion of normed space named random normed space, that was introduced by Sherstnev in [31] and extended by Alsina, Schweizer and Sklar in [1]. The theory of random normed spaces have significant applications in quantum particle physics (see [20]). Also, it has very useful applications in many fields like population dynamics, chaos control, computer programming, nonlinear dynamical system, nonlinear operators, statistical convergence, etc. On the other hand, one of the most important issues in the theory of functional equations, concerning the famous Ulam stability problem is: when a mapping satisfying a functional equation approximately, must be close to an exact solution of a given functional equation? Ulam [35] in 1940 raised the first stability problem concerning group homomorphisms. Hyers [12] was the first mathematician to present an affirmative partial answer to the question of Ulam for Banach spaces. Subsequently, Hyers’ theorem was generalized by Aoki [2] for additive mappings and by Rassias [28] for linear mappings by considering an unbounded Cauchy difference. Gavruta [10] obtained a generalization of Rassias’ theorem, which allows the Cauchy difference to be controlled by a general unbounded function. ∗ Corresponding

author Email addresses: [email protected] (Shaymaa Alshybani), [email protected] (S. Mansour Vaezpour), [email protected] (Reza Saadati)

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The stability problems of a wide class of functional equations have been investigated by a number of authors, and there are many interesting results concerning those problem (see, e.g., [3, 11, 13, 15, 25, 28, 32–34, 36]). Also by using fixed point methods, the stability problems of several functional equations have been extensively investigated by a number of authors (see, e.g., [5–7, 19, 26]). The generalized Hyers–Ulam stability of different functional equations in random normed spaces, paranormed spaces, quasi-normed spaces and quasi-β-normed spaces has been studied by many authors (see , e.g., [8, 14, 18, 22–24]). Park and Lee [23] proved the Hyers–Ulam stability of the following additivequadratic-cubic-quatric functional equation f (x + 2y) + f (x − 2y) = 4f (x + y) + 4f (x − y) − 6f (x) + f (2y) + f (−2y) − 4f (y) − 4f (−y)

(1.1)

in paranormed spaces. The general solution of quintic and sextic functional equations f (x + 3y) − 5f (x + 2y) + 10f (x + y) − 10f (x) + 5f (x − y) − f (x − 2y) = 120y

(1.2)

and f (x + 3y) − 6f (x + 2y) + 15f (x + y) − 20f (x) + 15f (x − y) − 6f (x − 2y) + f (x − 3y) = 720f (y) (1.3) was introduced and investigated on the generalized Hyers–Ulam stability in quasi-β-normed spaces via fixed point method by Xu et al. [37]. The general solution and the generalized Hyers–Ulam stability of the sextic functional equation f (nx + y) + f (nx − y) + f (x + ny) + f (x − ny) = (n4 + n2 )[f (x + y) + f (x − y)] + 2(n6 − n4 − n2 + 1)[f (x) + f (y)]

(1.4)

in paranormed spaces was discussed by Ravi and Sabarinathan [29]. In this paper, we present the general solution and generalized Hyers–Ulam stability of the sextic functional equation (1.4) under arbitrary t-norms by direct method and under Min t-norm by fixed point method in random normed spaces and provide an example for random normed spaces with the Lukasiewicz t-norm, by direct method. 2. Preliminaries Before giving the main result, we present some basic facts related to random normed spaces and some preliminary results. We say that f : R −→ [0, 1] is a distribution function if and only if it is a monotone, nondecreasing, left continuous, inff (x) = 0 and supf (x) = 1. By 4+ we denote a collection of all distribution functions and by D+ the set of all distribution functions such that f (x) = 0. If a ∈ R0 , then Ha ∈ D+ where Ha (t) :=

  0

if t ≤ a,

 1

if t > a.

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It is obvious that H0 ≥ f for all f ∈ D+ . Definition 2.1 ([8, 30]). A mapping T : [0, 1] × [0, 1] → [0, 1] is a continuous triangular norm (briefly a t-norm) if T satisfies the following conditions: 1. T is commutative and associative; 2. T is continuous; 3. T (a, 1) = a for all a ∈ [0, 1]; 4. T (a, b) ≤ T (c, d) whenever a ≤ c and b ≤ d. 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). n xi is Recall (see [8, 11]) that if T is a t-norm and xn is a given sequence of numbers in [0, 1], Ti=1 n−1 1 n defined recurrently by Ti=1 xi = x1 and Ti=1 xi = T (Ti=1 xi , xn ) for n ≥ 2.

It is known [11] that for the Lukasiewicz t-norm the following implication holds: lim (TL )∞ i=1 xn+i = 1 ⇐⇒

n→∞

∞ X

(1 − xn ) < ∞.

n=1

Definition 2.2 ([31]). A random normed space (briefly RN-space) is a triple (X, µ, T ) where X is a vector space, T is a continuous t-norm, and µ is a mapping from X into D+ such that the following conditions hold: 1. µx (t) = H0 (t) for all t > 0 iff x = 0; t 2. µαx (t) = µx ( |α| ) for all x ∈ X, t > 0 and α 6= 0; 3. µx+y (t + s) ≥ T (µx (t), µy (s)) for all x, y, z ∈ X and t, s ≥ 0. Definition 2.3 ([18]). Let (X, µ, T ) be an RN-space. Then 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 2.4 ([30]). 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 2.5 ([16]). Let X be a set. A function d : X × X −→ [0, ∞] is called a generalized metric on X if it 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 2.6 ([4, 9]). Let (X, d) be a complete generalized metric space and let J : X −→ X be a strictly contractive mapping with Lipschitz constant α < 1. Then for each given element x ∈ X, either d(J n x, J n+1 x) = ∞ for all nonnegative integers n, or there exists a positive integer n0 such that 1. d(J n x, J n+1 x) < ∞, ∀n ≥ n0 ; 2. the sequence {J n x} converges to a fixed point y ∗ of J; 3. y ∗ is the unique fixed point of J in the set Y = {y ∈ X|d(J n0 x, y) < ∞}; 1 4. d(y, y ∗ ) 6 1−α d(y, Jy) for all y ∈ Y . 3 372

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3. Hyers–Ulam stability of the sextic functional equation (1.4) by direct method In this section, using the direct method, we prove the generalized Hyers–Ulam stability of the sextic functional equation (1.4) in complete RN-spaces. Also, we present an illustrative example with the Lukasiewicz t-norm that can be suitable approximation using this sextic function. Theorem 3.1. Let X be a real liner space, (Y, µ, T ) a complete RN-space and f : X −→ Y be a mapping with f (0) = 0 for which there is φ : X 2 −→ D+ (φ(x, y) is denoted by φx,y ) such that µDs f (x,y) (t) > φx,y (t),

(3.1)

where Ds f (x, y) := f (nx + y) + f (nx − y) + f (x + ny) + f (x − ny) − (n4 + n2 )[f (x + y) + f (x − y)] − 2(n6 − n4 − n2 + 1)[f (x) + f (y)] for all x, y ∈ X and t > 0. If ∞ lim Ti=1 (φni+m−1 x,0 (n6m+5i t)) = 1,

(3.2)

lim φnm x,nm y (n6m t) = 1

(3.3)

m→∞

and m→∞

for all x, y ∈ X and t > 0, then there exists a unique sextic mapping S : X −→ Y satisfying (1.4) and the inequality ∞ µf (x)−s(x) (t) ≥ Ti=1 (φni−1 x,0 (n5i t)

(3.4)

for all x ∈ X and t > 0. Proof. Letting y = 0 in (3.1), we get µf (nx)−n6 f (x) (t) ≥ φx,0 (2t) ≥ φx,0 (t)

(3.5)

µ f (nx) −f (x) (t) ≥ φx,0 (n6 t),

(3.6)

for all x ∈ X. Then we get n6

therefore, µ f (nk+1 x) n6k+6

−

f (nk x) n6k

(t) ≥ φnk x,0 (n6k+6 t),

(3.7)

that is, t ) ≥ φnk x,0 (n5(k+1) t) nk+1 for every k ∈ N , t > 0, n positive integer, n > 1. As µ f (nk+1 x) n6k+6

−

1>

f (nk x) n6 k

(

(3.8)

1 1 1 1 + + 3 + ... + k , n n2 n n

by the triangle inequality it follows: m−1 X

µ f (nm x) −f (x) (t) ≥ µ f (nm x) −f (x) ( n6m

n6m

≥

m−1 Tk=0

k=0



1 nk+1

t)

 µ f (nk+1 x) f (nk x) ( k+1 t) − n n6k+6 n6k 1

(3.9)

m−1 > Tk=0 (φnk x,0 (n5k+5 t)
m = Ti=1 φni−1 x,0 (n5i t) ,

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j

x) x ∈ X, t > 0, and n > 1. In order to prove the convergence of the sequence { f (n n6j }, we replace x by 6j nj x, and multiplying the left-hand side of (3.9) by nn6j , we get
m µ f (nm+j x) − f (nj x) (t) ≥ Ti=1 φnj+i−1 x,0 (n6j+5i t) . (3.10) n6m+6j

n6j

Since the right-hand side of the inequality (3.10) tends to 1 as m and j tend to infinity, the sequence is a Cauchy sequence. Therefore, we may define

j x) { f (n n6j }

f (nj x) j−→∞ n6j

S(x) = lim

for all x ∈ X. Replacing x, y by nm x and nm y, respectively, in (3.1), then multiplying the right hand-side by it follows that 6m 1 t) µ 6m Ds f (nm x,nm y) (t) ≥ φnm x,nm y (n

n6m n6m ,

n

for all x, y ∈ X, and positive integer n, n > 1. Taking the limit as m → ∞ we find that S satisfies (1.4), that is, S is a sextic map. To prove (3.4) take the limit as m → ∞ in (3.9). Finally, to prove the uniqueness of the sextic function S, let us assume that there exists a sextic function r which satisfies (3.4) and equation (1.4). Therefore µr(x)−s(x) (t) = µr(x)− f (nj x) + f (nj ) −s(x) (t) n6j

n6j

t t ≥ T (µr(x)− f (nj x) ( ), µ f (nj x) −s(x) ( )). 2 2 6j 6j n n Taking the limit as j → ∞, we find µr(x)−s(x) (t) = 1. Therefore r = s. Corollary 3.2. Let X be a real liner space and (Y, µ, T ) a complete RN-space such that (T = TM , Tp or TL ) and f : X −→ Y be a mapping satisfying µDs f (x,y) (t) > 1 −

kxk t + kxk

(3.11)

for all x ∈ X, t > 0. Then there exists a unique sextic mapping S : X −→ Y satisfying (1.4) and ∞ µf (x)−s(x) (t) > Ti=1 (1 −

kxk ) + kxk

n4i+1 t

for every x ∈ X, and t > 0. Proof. It is enough to put, φx,y (t) = 1 −

kxk t + kxk

for all x, y ∈ X and t > 0, in Theorem 3.1. Corollary 3.3. Let X be a real liner space and (Y, µ, T ) a complete RN-space such that (T = TM , Tp or TL ) and f : X −→ Y be a mapping satisfying µDs f (x,y) (t) >

t , t + εkx0 k

x0 ∈ X, t > 0, and ε > 0. Then there exists a unique sextic mapping S : X −→ Y satisfying (1.4) and ∞ µf (x)−s(x) (t) > Ti=1 (

n5i t ). n5i t + εkx0 k

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Proof. It is enough to put, t t + εkx0 k

φx,y (t) = for all x, y ∈ X and t > 0, in Theorem 3.1.

Corollary 3.4. Let X be a real linear space and (Y, µ, T ) a complete RN-space such that (T = TM , Tp or TL ) and let L ≥ 0 and p be a real number with 0 < p < 5 and f : X −→ Y be a mapping satisfying µDs f (x,y) (t) >

t t + L(k x kp + k y kp )

for all x, y ∈ X and t > 0. Then there exists a unique sextic mapping S : X −→ Y satisfying (1.4) and ∞ µf (x)−s(x) (t) ≥ Ti=1 (

t t+

Lni(p−5)−p

k x kp

)

for every x ∈ X and t > 0. Proof. It is enough to put φx,y (t) =

t t + L(k x kp + k y kp )

for all x, y ∈ X and t > 0, in Theorem 3.1. Example 3.5. Let (X, k.k) be a Banach algebra and ( max{1 − kxk t , 0} if t > 0, µx (t) = 0 if t ≤ 0, for all x, y ∈ X and t > 0. Let ( max{1 − ϕx,y (t) = 0

(8n6 )(kxk+kyk) , 0} t

if t > 0, if t ≤ 0.

We note that ϕx,y (t) is a distribution function and limj→∞ ϕnj x,nj y (n6j t) = 1 for all x, y ∈ X and t > 0. It is easy to show that (X, µ, TL ) is an RN-space (this was essentially proved by Mushtari in [21], see also [27]). Indeed, µx (t) = 1, ∀t > 0 implies kxk t = 0 and hence x = 0 for all x ∈ X and t > 0. Obviously, µλx (t) = µx ( λt ) for all x ∈ X and t > 0. Next, for all x, y ∈ X and t, s > 0, we have kx + yk , 0} t+s x+y = max{1 − k k, 0} t+s x y = max{1 − k + k, 0} t+s t+s x y ≥ max{1 − k k − k k, 0} t s = TL (µx (t), µy (s)).

µx+y (t + s) = max{1 −

It is easy to see that (X, µ, TL ) is complete, for µx−y (t) ≥ 1 −

kx − yk , t

∀x, y ∈ X,

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and t > 0 and (X, k.k) is complete. Define a mapping f : X −→ X by f (x) = x6 + kxkx0 for all x ∈ X, where x0 is a unit vector in X. A simple computation shows that kf (nx + y) + f (nx − y) + f (x + ny) + f (x − ny) − (n4 + n2 )[f (x + y) + f (x − y)] − 2(n6 − n4 − n2 + 1)[f (x) + f (y)]k = | k nx + y k + k nx − y k + k x + ny k + k x − ny k − (n2 + n4 )[k x + y k + k x − y k] − 2(n6 − n4 − n2 + 1)[k x k + k y k]| ≤ 2(n6 + n + 2)(k x k + k y k) ≤ 8n6 (k x k + k y k) for all x, y ∈ X. Hence µDs f (x,y) (t) ≥ φx,y (t) for all x, y ∈ X and t > 0. Fix x ∈ X and t > 0. Then it follows that, (∞ )    X ∞ 6j+5i) 6j+5i) φni+j−1 x,0 (n (TL )i=1 φni+j−1 x,0 (n t) = max t) − 1 + 1, 0 i=1

 = max 1 −

 8n5 kxk , 0 n5j (n4 − 1)t

for all x ∈ X, n ∈ N and t > 0. Hence   6j+5i) lim (TL )∞ t) = 1 i=1 ϕni+j−1 x,0 (n

j→∞

for all x ∈ X and t > 0. Thus, all the conditions of Theorem 3.1 hold. Since
8n5 kxk 5i (TL )∞ , 0} i=1 φni−1 x,0 (n t) = max{1 − (n4 − 1)t for all x ∈ X and t > 0, we can deduce that S(x) = x6 is the unique sextic mapping S : X −→ X such that µf (x)−s(x) (t) ≥ max{1 −

8n5 kxk , 0} (n4 − 1)t

for all x ∈ X and t > 0. 4. Hyers–Ulam stability of the sextic functional equation (1.4) by fixed point method In this section, using the fixed point method, we prove the generalized Hyers–Ulam stability of the sextic functional equation (1.4) in complete RN-spaces. Theorem 4.1. Let X be a real liner space and (Y, µ, TM ) be a complete RN-space and f : X −→ Y be a mapping with f (0) = 0 for which there is φ : X 2 −→ D+ (φ(x, y) is denoted by φx,y ) such that φnx,ny (αt) ≥ φx,y (t),

0 < α < n6 ,

and µDs f (x,y) (t) > φx,y (t)

(4.1)

for all x, y ∈ X, and t > 0, where Ds f (x, y) := f (nx + y) + f (nx − y) + f (x + ny) + f (x − ny) 7 376

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− (n4 + n2 )[f (x + y) + f (x − y)] − 2(n6 − n4 − n2 + 1)[f (x) + f (y)] for all x, y ∈ X and t > 0. Then there exists a unique sextic mapping g : X −→ Y such that µf (x)−g(x) (t) > φx,0 (2(n6 − α)t)

(4.2)

for all x ∈ X and t > 0. Moreover, we have g(x) = limm−→∞

f (nm x) . n6m

Proof. Let y = 0 in (4.1); we get µ2f (nx)−2n6 f (x) (t) ≥ φx,0 (t)

(4.3)

µ f (nx) −f (x) (t) ≥ φx,0 (2n6 t).

(4.4)

for all x ∈ X and t > 0 and hence n6

Consider the set E := {g : X → Y : g(0) = 0}, and the mapping dG defined on E × E by dG (g, h) = inf{ > 0 : µg(x)−h(x) (t) ≥ φx,0 (2n6 t)}, for all x ∈ X, t > 0. Then (E, dG ) is a complete generalized metric space (see the proof of [17, Lemma 2.1]). Now, let us consider the linear mapping J : E → E defined by Jg(x) =

g(nx) . n6

Now, we show that J is a strictly contractive self-mapping of E with the Lipschitz constant k = Indeed, let g, h ∈ E be the mappings such that dG (g, h) < . Then we have

α n6 .

µg(x)−h(x) (t) ≥ φx,0 (2n6 t) for all x ∈ X and t > 0 and hence µJg(x)−Jh(x) (

αt αt ) = µ g(nx) − h(nx) ( 6 ) 6 6 6 n n n n = µg(nx)−h(nx) (αεt) ≥ φnx,0 (2αn6 t)

for all x ∈ X and t > 0. Since φnx,ny (αt) ≥ φx,y (t), we have µJg(x)−Jh(x) (

0 < α < n6 ,

αt ) ≥ φx,0 (2n6 t), n6

that is, dG (g, h) <  =⇒ dG (Jg, Jh) < This means that dG (Jg, Jh)
0, 2n6 t dG (u, v) <  =⇒ µu(x)−v(x) (t) ≥ φx,0 ( ),  m x) it follows from dG (J n f, g) −→ 0 that limm−→∞ f (n = g(x) for all x ∈ X. Also from n6m dG (f, g) ≤ for all g, h ∈ E, we have dG (f, g) ≤

1 1− nα6

1 d(f, Jf ) 1−L

, and it immediately follows that

µg(x)−f (x) (

n6 t) > φx,0 (2n6 t) n6 − α

for all x ∈ X and t > 0. This means that µg(x)−f (x) (t) > φx,0 (2(n6 − α)t) for all x ∈ X and t > 0. Finally, the uniqueness of g follows from the fact that g is the unique fixed point of J such that there exists C ∈ (0, ∞) satisfying µg(x)−f (x) (Ct) > φx,0 (2n6 t) for all x ∈ X and t > 0. This completes the proof. Corollary 4.2. Let X be a real liner space, (Y, µ, TM ) a complete RN-space, and f : X −→ Y a mapping satisfying µDs f (x,y) (t) > 1 −

kxk t + kxk

(4.5)

for all x ∈ X, t > 0. Then there exists a unique sextic mapping s : X −→ Y satisfying (1.4) and µf (x)−s(x) (t) > 1 −

2(n6

kxk − α)t + kxk

for every x ∈ X, t > 0, and n positive integer. Moreover, we have f (nm x) . m−→∞ n6m

s(x) = lim Proof. It is enough to put,

φx,y (t) = 1 −

kxk t + kxk

for all x ∈ X and t > 0 in Theorem 4.1. Then we can choose n < α < n6 and so we get the desired result. Corollary 4.3. Let X be a real liner space, (Y, µ, TM ) a complete RN-space and f : X −→ Y a mapping satisfying t µDs f (x,y) (t) > , t + εkx0 k x0 ∈ X, t > 0, and ε > 0. Then there exists a unique sextic mapping s : X −→ Y satisfying (1.4) and µf (x)−s(x) (t) >

2(n6 − α)t 2(n6 − α)t + εkx0 k

for every x ∈ X, t > 0, and n positive integer. Moreover, we have f (nm x) . m−→∞ n6m

s(x) = lim

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Proof. It is enough to put φx,y (t) =

t t + εkx0 k

for all x ∈ X, and t > 0 in Theorem 4.1. Then we can choose n < α < n6 and so we get the desired result. Corollary 4.4. Let X be a real liner space, (Y, µ, TM ) a complete RN-space and f : X −→ Y a mapping satisfying µDs f (x,y) (t) >

t t + θ(k x kp + k y kp )

for all x, y ∈ X, t > 0, θ > 0, and 0 < p < 6. Then there exists a unique sextic mapping s : X −→ Y satisfying (1.4) and 2(n6 − α)t µf (x)−s(x) (t) ≥ 2(n6 − α)t + θ k x kp for every x ∈ X and t > 0. Moreover, we have s(x) = limm−→∞

f (nm x) . n6m

Proof. It is enough to put φx,y (t) =

t t + θ(k x kp + k y kp )

for all x, y ∈ X and t > 0 in Theorem 4.1. Then we can choose np < α < n6 and so we get the desired result. References [1] C. Alsina, B. Schweizer, A. Sklar, On the definition of a probabilistic normed space, Aequationes Math., 46 (1993), 91–98. 1 [2] T. Aoki, On the stability of the linear transformation in Banach spaces, J. Math. Soc. Japan, 2 (1950), 64–66. 1 [3] J.-H. Bae, W.-G. Park, On the Ulam stability of the Cauchy-Jensen equation and the additive-quadratic equation, J. Nonlinear Sci. Appl., 8 (2015), 710–718. 1 [4] L. Cadariul, V. Radu, Fixed points and the stability of Jensen’s functional equation, JIPAM. J. Inequal. Pure Appl. Math., 4 (2003), 7 pages. 2.6 [5] L. Cadariul, V. Radu, On the stability of the Cauchy functional equation:a fixed point approach, Grazer Math. Ber., 346 (2004), 43–52. 1 [6] L. Cadariul, V. Radu, Fixed points and generalized stability for functional equations in abstract spaces, J. Math. Inequal., 3 (2009), 463–473. [7] Y. J. Cho, C. Park, Y.-O. Yang, Stability of derivations in fuzzy normed algebras, J. Nonlinear Sci. Appl., 8 (2015), 1–7. 1 [8] Y. J. Cho, T. M. Rassias, R. Saadati, Stability of functional equations in random normed spaces, Springer, New York, (2013). 1, 2.1, 2 [9] J. Diaz, B. Margolis, A fixed point theorem of the alternative for contractions on a generalized complete metric space, Bull. Amer. Math. Soc., 74 (1968), 305–309. 2.6 [10] P. Gavruta, A generalization of the Hyers–Ulam-Rassias stability of approximately additive mappings, J. Math. Anal. Appl., 184 (1994), 431–436 . 1 [11] O. Hadzic, E. Pap, Fixed point theory in PM spaces, Kluwer Academic Publishers, Dordrecht, (2001). 1, 2

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[12] D. H. Hyers, On the stability of the linear functional equation, Proc. Nat. Acad. Sci. U. S. A., 27 (1941), 222–224. 1 [13] Y. Lan, Y. Shen, The general solution of a quadratic functional equation and Ulam stability, J. Nonlinear Sci. Appl., 8 (2015), 640–649. 1 [14] R. Lather, M. Kumar, Hyers–Ulam Stability of quartic functional equation in paranormed spaces, Internat. J. Eng. Sci., 5 (2015), 23–26. 1 [15] T. Li, A. Zada, S. Faisal, Hyers-Ulam stability of nth order linear differential equations, J. Nonlinear Sci. Appl., 9 (2016), 2070–2075. 1 [16] W. A. J. Luxemburg, on the convergence of successive approximations in theory of ordinary differential equations, Canad. Math. Bull., 1 (1958), 9–20. 2.5 [17] D. Mihet, V. Radu, on the stability of the additive cauchy functional equation in random normed spaces, J. Math. Anal. Appl., 343 (2008), 567–572. 4 [18] D. Mihet, R. Saadati, S. M. Vaezpour, The stability of the quartic functional equation in random normed spaces, Acta. Appl. Math., 110 (2010), 797–803. 1, 2.3 [19] D. Mihet, C. Zaharia, probabilistic (Quasi) metric versions for a stability result of Baker, Abstr. Appl. Anal., 2012 (2012), 10 pages. 1 [20] M. Mohamadi, Y. J. Cho, C. Park, F. Vetro, R. Saadati, Random stability on an additive-quadratic-quartic functional equation, J. Inequal. Appl., 2010 (2010), 18 pages. 1 [21] D. H. Mushtari, The linearity of isometric mappings of random normed spaces, Kazan. Gos. Univ. Ucen. Zap., 128 (1968), 86–90. 3.5 [22] A. Najati, G. Z. Eskandani, Stability of a mixed additive and cubic functional equation in quasi-Banach spaces, J. Math. Anal. Appl., 342 (2008), 1318–1331. 1 [23] C. Park, J. R. Lee, An AQCQ-functional equation in paranormed spaces, Adv. Difference Equ., 2012 (2012), 9 pages. 1 [24] C. Park, D. Y. Shin, Functional equations in paranormed spaces, Adv. Difference Equ., 2012 (2012), 14 pages. 1 [25] C. Park, S. Yun, Stability of cubic and quartic ρ-functional inequalities in fuzzy normed spaces, J. Nonlinear Sci. Appl., 9 (2016), 1693–1701. 1 [26] V. Radu, The fixed point alternative and the stability of functional equations, Fixed Point Theory, 4 (2003), 91–96. 1 [27] V. Radu, Some remarks on quasi-normed and random normed structures, Automat. Comput. Appl. Math., 13 (2004), 169–177. 3.5 [28] T. M. Rassias, On the stability of the linear mapping in Banach spaces, Proc. Amer. Math. Soc., 72 (1978), 297–300. 1 [29] K. Ravi, S. Sabarinathan, Generalized Hyers–Ulam stability of a sextic functional equation in paranormed spaces, Internat. J. Manag. Inform. Technol., 9 (2014), 61–69. 1 [30] B. Schweizer, A. Sklar, Probabilistic metric spaces, North-Holland Publishing Co., New York, (1983). 2.1, 2.4 ˇ [31] Serstnev AN, On the notion of a random normed space, Dokl. Akad. Nauk. SSSR, 149 (1963), 280–283. (in Russian) 1, 2.2 [32] W. Shatanawi, M. Postolache, Mazur-Ulam theorem for probabilistic 2-normed spaces, J. Nonlinear Sci. Appl., 8 (2015), 1228–1233. 1 [33] Y. Shen, An integrating factor approach to the Hyers–Ulam stability of a class of exact differential equations of second order, J. Nonlinear Sci. Appl., 9 (2016), 2520–2526. [34] Y. Shen, W. Chen, On the Ulam stability of an n-dimensional quadratic functional equation, J. Nonlinear Sci. Appl., 9 (2016), 332–341. 1 [35] S. M. Ulam, Problems in modern mathematics, Science Editions Wiley, New York, (1964). 1

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[36] C. Urs, Ulam-Hyers stability for coupled fixed points of contractive type operators, J. Nonlinear Sci. Appl., 6 (2013), 124–136. 1 [37] T. Z. Xu, J. M. Rassias, M. J. Rassias, W. X. Xu, A fixed point approach to the stability of quintic and sextic functional equations in quasi-β-normed spaces, J. Inequal Appl., 2010 (2010), 23 pages. 1

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On convergence theorem of a finite family of nonlinear mappings in uniformly convex metric spaces Atid Kangtunyakarn∗ Department of Mathematics, Faculty of Science, King Mongkut, s Institute of Technology Ladkrabang, Bangkok 10520, Thailand

Abstract In this paper, we introduce the S−mapping generated by a finite family of nonexpansive mapping and real numbers in convex metric space by using concept of the S−mapping defined by Kangtunyakarn and Suantai [1]. Then, we prove convergence of Ishikawa iteration generated by the S−mapping to a common fixed point of a finite family of nonexpansive mappings in uniformly convex metric space. Keywords: Convex metric space; Nonexpansive mapping; S−mapping.

2010 Mathematics Subject Classification 47H09, 47H10, 54E50.

1

Introduction

Throughout this paper, we assume that (X, d) is a complete metric space and C is a nonempty closed convex subset of (X, d). A point x is called a fixed point of T if T x = x. We use F (T ) to denote the set of fixed point of T . Recalled the following definitions; Definition 1.1. The mapping T : C → C is said to be nonexpansive if d(T x, T y) ≤ d(x, y), ∀x, y ∈ C. In 1970, Takahashi [9] introduce the following definition as follows: Definition 1.2. Let (X, d) be a metric space. A mapping W : X × X × [0, 1] → X is said to be a convex structure on X if for each (x, y, λ) ∈ X × X × [0, 1] and for all u ∈ X,
d u, W (x, y, λ) ≤ λd(u, x) + (1 − λ)d(u, y). ∗

E-mail: [email protected]

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2

Ishikawa iteration for a common fixed point of a finite...

We observe that W (x, y, λ) = λx + (1 − λ)y is a convex structure on a normed linear space. A metric space (X, d) together with a convex structure W is called a convex metric space denoted by (X, d, W ). A nonempty subset C of X is said to be convex if W (x, y, λ) ∈ C for all x, y ∈ C and λ ∈ [0, 1]. Two classical iteration processes are often used to approximate a fixed point of a nonexpansive mapping. The first one is introduced by Mann [10] and is defined as follows: xn+1 = αn xn + (1 − αn )T xn ,

∀n ≥ 1,

where x1 ∈ C, {αn } ⊆ [0, 1]. The second iteration process is referred to as Ishikawa, s iteration process [5] which is defined recursively by (

xn+1 = αn xn + (1 − αn )T yn yn = βn xn + (1 − βn )T xn , ∀n ≥ 1,

(1.1)

where x1 ∈ C, {αn }, {βn } ⊆ [0, 1]. In 2009, Kangtunyakarn and Suantai [1] introduced the mapping generated by a finite family of nonexpansive mapping and family of real numbers as follows: Definition 1.3. Let C be a nonempty convex subset of real Banach space. Let {Ti }N i=1 be a finite family of nonexpanxive mappings of C into itself. For each j = 1, 2, ..., N, let αj = (α1j , α2j , α3j ) ∈ I × I × I where I ∈ [0, 1] and α1j + α2j + α3j = 1. They define the mapping S : C → C as follows: U0 = I, U1 = α11 T1 U0 + α21 U0 + α31 I, U2 = α12 T2 U1 + α22 U1 + α32 I, U3 = α13 T3 U2 + α23 U2 + α33 I, . . . UN −1 = α1N −1 TN −1 UN −2 + α2N −1 UN −2 + α3N −1 I, S = UN = α1N TN UN −1 + α2N UN −1 + α3N I.

(1.2)

This mapping is called S-mapping generated by T1 , ...., TN and α1 , α2 , ..., αN . In this paper, by using the concept of the S−mapping in Definition 1.3, we define the S−mapping generated by a finite family of nonexpansive mappings and real numbers in convex metric space. Then, we prove convergence of Ishikawa iteration generated by the S−mapping to a common fixed point of a finite family of nonexpansive mappings in uniformly convex metric space. 383

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A. Kangtunyakarn

2

3

Preliminaries

In this section, we recall some lemmas and definitions to prove our main result as follows: Definition 2.1. (See [7]) A convex metric space (X, d, W ) is said to be uniformly convex if for any  > 0, there exists δ = δ() > 0 such that for all r > 0 and x, y, z ∈ X with d(z, x) < r, d(z, y) < r and d(x, y) ≥ r, 1
d z, W (x, y, ) ≤ (1 − δ)r. 2 Lemma 2.1. (See [11], [3]) Let (X, d, W ) be a convex metric space. For each x, y ∈ X and λ, λ1 , λ2 ∈ [0, 1], we have the following. (i) W (x, x, λ) = x,
W (x, y, 0) = y and W (x, y, 1) = x.
(ii) d x, W (x, y, λ) = (1 − λ)d(x, y) and d y, W
(x, y, λ) = λd(x, y).
(iii) d(x, y) = d x, W (x, y, λ) + d W (x, y, λ), y
. (iv) |λ1 − λ2 |d(x, y) ≤ d W (x, y, λ1 ), W (x, y, λ2 ) . We say that a convex metric space (X, d, W ) has the property: (C) if W (x, y, λ) = W (y, x, 1 −
λ) for all x, y ∈ X and λ ∈ [0, 1], (I) if d W (x, y, λ1 ), W (x, y, λ2 ) ≤ |λ1 − λ2 |d(x, y) for all x, y ∈ X and λ1 , λ2 ∈ [0, 1],
(H) if d W (x, y, λ), W (x, z, λ)
≤ (1−λ)d(y, z) for all x, y, z ∈ X and λ ∈ [0, 1], (S) if d W (x, y, λ), W (z, w, λ) ≤ λd(x, z) + (1 − λ)d(y, w) for all x, y, z, w ∈ X and λ ∈ [0, 1]. Remark 2.2. It is easy to see that the property (C) and (H) imply continuity of a convex structure W : X × X × [0, 1] → X and the property (S) implies the property (H). In 2005, Aoyama et al. [3] proved that a convex metric space with property (C) and (H) has the property (S). In 2011, Phuengrattana and Suantai [8] proved the following lemma as follows; Lemma 2.3. (See [8]) Property (C) holds in uniformly convex metric space. Remark 2.4. (See [8]) From Lemma 2.3, a uniformly convex metric space (X, d, W ) with the property (H) has the property S and the convex structure W is also continuous. Lemma 2.5. (See [6]) Let (X, d, W ) be a uniformly convex metric space with continuous convex structure. Then for arbitrary positive number , there exists η = η() > 0 such that
d z, W (x, y, λ) ≤ (1 − 2 min{λ, 1 − λ}η)r for all r > 0 and x, y, z ∈ X, d(z, x) ≤ r, d(z, y) ≤ r, d(x, y) ≥ rε and λ ∈ [0, 1]. 384

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Ishikawa iteration for a common fixed point of a finite...

Lemma 2.6. (See [2],[4]) Let {an }, {bn } and {δn } be sequences of nonnegative real numbers satisfying the inequality an+1 ≤ (1 + δn )an + bn , n ≥ 1. If

∞ X

δn < ∞, and

n=1

∞ X

bn < ∞,

n=1

then limn→∞ an exists. We introduce the following definition to use in the next section. Definition 2.2. Let (X, d, W ) be a complete convex metric space and C be a nonempty closed convex subset of (X, d, W ). Let {Ti }N i=1 be a finite family of mappings of C into C. For each j = 1, 2, · · · , N , let αj = (α1j , α2j , α3j ) where α1j , α2j , α3j ∈ [0, 1] and α1j + α2j + α3j = 1. For every x ∈ C, we define the mapping S : C × C × [0, 1] → C as follows: U0 x = x,
α21 ), α11 , 1 1 − α1
α22 ), α12 , U2 x = W T2 U1 x, W (U1 x, x, 2 1 − α1 .. .
α2N −1 N −1 ), α , UN −1 x = W TN −1 UN −2 x, W (UN −2 x, x, 1 1 − α1N −1
α2N Sx = UN x = W TN UN −1 x, W (UN −1 x, x, ), α1N . N 1 − α1 U1 x = W T1 U0 x, W (U0 x, x,

This mapping is called S−mapping generated by T1 , T2 , . . . , TN and α1 , α2 , . . . , αN . Lemma 2.7. Let C be a nonempty closed convex subset of a complete uniformly convex metric space (X, d, W ) with property (H). Let {Ti }N i=1 be a finite family of nonTN expanxive mappings of C into itself with i=1 F (Ti ) 6= ∅ and let αj = (α1j , α2j , α3j ) ∈ I ×I ×I, j = 1, 2, 3, ..., N , where I = [0, 1] , α1j +α2j +α3j = 1, α1j ∈ (0, 1) for all j = 1, 2, ..., N −1, α1N ∈ (0, 1] α2j , α3j ∈ [0, 1) for all j = 1, 2, ..., N. Let S be the mapping T generated by T1 , ...., TN and α1 , α2 , ..., αN . Then F (S) = N i=1 F (Ti ). T Proof. From Lemma 2.1 and definition of S−mapping, it is easy to see that N i=1 F (Ti ) ⊆ TN F (S). Next, we show that F (S) ⊆ i=1 F (Ti ). To show this let x0 ∈ F (S) and T q∈ N i=1 F (Ti ), we have  
α2N N ), α1 d(q, Sx0 ) = d q, W TN UN −1 x0 , W (UN −1 x0 , x0 , 1 − α1N   α2N N N ≤ α1 d(q, TN UN −1 x0 ) + (1 − α1 )d q, W (UN −1 x0 , x0 , ) 1 − α1N 385

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≤

α1N d(q, TN UN −1 x0 )



+ (1 −

α2N )d(q, x0 ) + (1 − 1 − α1N

α1N )

5

α2N d(q, UN −1 x0 ) 1 − α1N



= α1N d(q, TN UN −1 x0 ) + α2N d(q, UN −1 x0 ) + α3N d(q, x0 ) ≤ (1 − α3N )d(q, UN −1 x0 ) + α3N d(q, x0 )   N −1 N −1 N ≤ (1 − α3 ) (1 − α3 )d(q, UN −2 x0 ) + α3 d(q, x0 ) +α3N d(q, x0 ) = (1 − α3N )(1 − α3N −1 )d(q, UN −2 x0 ) + α3N −1 (1 − α3N )d(q, x0 ) +α3N d(q, x0 ) j
j N = ΠN j=N −1 (1 − α3 )d(q, UN −2 x0 ) + 1 − Πj=N −1 (1 − α3 ) d(q, x0 )

≤ .. . j j
N ≤ ΠN j=3 (1 − α3 )d(q, U2 x0 ) + 1 − Πj=3 (1 − α3 ) d(q, x0 ) j = ΠN j=3 (1 − α3 )d q, W T2 U1 x0 , W (U1 x0 , x0 ,



α22 ), α12 2 1 − α1

j
+ 1 − ΠN j=3 (1 − α3 ) d(q, x0 )  j N ≤ Πj=3 (1 − α3 ) α12 d(q, T2 U1 x0 ) + (1 − α12 )d q, W (U1 x0 , x0 ,

≤

=

≤

= =

α22
) 1 − α12



j
+ 1 − ΠN j=3 (1 − α3 ) d(q, x0 )  α22 j N Πj=3 (1 − α3 ) α12 d(q, T2 U1 x0 ) + (1 − α12 ) d(q, U1 x0 ) 1 − α12 
α22 j
+(1 − )d(q, x0 ) + 1 − ΠN j=3 (1 − α3 ) d(q, x0 ) 2 1 − α1   j N 2 2 2 Πj=3 (1 − α3 ) α1 d(q, T2 U1 x0 ) + α2 d(q, U1 x0 ) + α3 d(q, x0 ) j
+ 1 − ΠN j=3 (1 − α3 ) d(q, x0 )  
j N 2 2 Πj=3 (1 − α3 ) (1 − α3 )d(q, U1 x0 ) + α3 d(q, x0 ) j
+ 1 − ΠN j=3 (1 − α3 ) d(q, x0 ) j j
N ΠN j=2 (1 − α3 )d(q, U1 x0 ) + 1 − Πj=2 (1 − α3 ) d(q, x0 )

α21 j ), α11 ΠN (1 − α )d q, W T U x , W (U x , x , 1 0 0 0 0 0 j=2 3 1 1 − α1 j
N + 1 − Πj=2 (1 − α3 ) d(q, x0 )

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Ishikawa iteration for a common fixed point of a finite...

j 1 = ΠN j=2 (1 − α3 )d q, W T1 x0 , x0 , α1

j
+ 1 − ΠN j=2 (1 − α3 ) d(q, x0 )
j j
1 1 N ≤ ΠN j=2 (1 − α3 ) α1 d(q, T1 x0 ) + (1 − α1 )d(q, x0 ) + 1 − Πj=2 (1 − α3 ) d(q, x0 ) j j
N ≤ ΠN j=2 (1 − α3 )d(q, x0 ) + 1 − Πj=2 (1 − α3 ) d(q, x0 )





= d(q, x0 ).

(2.1)

From (2.1), we have
d(q, U1 x0 ) = d q, W (T1 x0 , x0 , α11 ) = d(q, x0 ) and d(q, T1 x0 ) = d(q, x0 ). Suppose x0 6= T1 x0 , then we have d(x0 , T1 x0 ) > 0. Choose r = d(q, x0 ) > 0 and  = d(x0 , T1 x0 ) , we have d(q, T1 x0 ) ≤ d(q, x0 ) = r, d(q, x0 ) ≤ r and d(x0 , T1 x0 ) ≥ r. r From Lemma 2.5, we have
d q, W (T1 x0 , x0 , α11 ) < d(q, x0 ) for α11 ∈ (0, 1). This is a contradiction, we have x0 ∈ T1 x0 , that is, x0 ∈ F (T1 ). Since x0 = T1 x0 definition of U1 and Lemma 2.1, we have U1 x0 = x0 , that is, x0 ∈ F (U1 ). From (2.1) and x0 = U1 x0 , we have d(q, U2 x0 ) = d q, W T2 x0 , x0 , α12





= d(q, x0 ) and d(q, T2 x0 ) = d(q, x0 ).

Suppose x0 6= T2 x0 , then we have d(x0 , T2 x0 ) > 0. Choose r1 = d(q, x0 ) > 0 and d(x0 , T2 x0 ) = , we have d(q, T2 x0 ) ≤ d(q, x0 ) = r1 , d(q, x0 ) ≤ r1 and d(x0 , T2 x0 ) ≥ r1 r1 . From Lemma 2.5, we have
d q, W (T2 x0 , x0 , α12 ) < d(q, x0 ) for α12 ∈ (0, 1). This is a contradiction, we have x0 = T2 x0 , that is, x0 ∈ F (T2 ). Since x0 = T2 x0 definition of U2 and Lemma 2.1, we have U2 x0 = x0 , that is, x0 ∈ F (U2 ). By continuing on this way we can conclude that x0 ∈ F (Ti ) and x0 ∈ F (Ui ) for all i = 1, 2, . . . , N − 1. Finally, we show that x0 ∈ F (TN ). From definition of S and Lemma 2.1, we have Sx0 = W TN UN −1 x0 , W (UN −1 x0 , x0 ,


α2N N ), α = W (TN x0 , x0 , α1N ). 1 1 − α1N

Since 0 = d(x0 , Sx0 ) = d(x0 , W (TN x0 , x0 , α1N )) = α1N d(TN x0 , x0 ), T we have x0 = TN x0 , that is, x0 ∈ F (TN ). Hence F (S) ⊆ N i=1 F (Ti ). 387

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7

Remark 2.8. From Theorem 2.7, we have the mapping S is nonexpansive. To show this, let x, y ∈ C. By remark 2.4, we have
α2N ), α1N , N 1 − α1 
α2N N W TN UN −1 y, W (UN −1 y, y, ), α1 1 − α1N

 d(Sx, Sy) = d W TN UN −1 x, W (UN −1 x, x,

≤ α1N d(TN UN −1 x, TN UN −1 y) +(1 − α1N )d W (UN −1 x, x,

α2N α2N
), W (U y, y, ) N −1 1 − α1N 1 − α1N

≤ α1N d(TN UN −1 x, TN UN −1 y)   α2N α2N
N +(1 − α1 ) d(UN −1 x, UN −1 y) + 1 − d(x, y) 1 − α1N 1 − α1N ≤ α1N d(UN −1 x, UN −1 y) + α2N d(UN −1 x, UN −1 y) + α3N d(x, y) = (1 − α3N )d(UN −1 x, UN −1 y) + α3N d(x, y)
≤ (1 − α3N ) (1 − α3N −1 )d(UN −2 x, UN −2 y) + α3N −1 d(x, y) + α3N d(x, y) j
j N = ΠN j=n−1 (1 − α3 )d(UN −2 x, UN −2 y) + 1 − Πj=N −1 (1 − α3 ) d(x, y) ≤ .. . j j
N = ΠN j=1 (1 − α3 )d(U0 x, U0 y) + 1 − Πj=1 (1 − α3 ) d(x, y)

= d(x, y).

3

Main results

Theorem 3.1. Let C be a nonempty compact closed convex subset of a complete uniformly convex metric space (X, d, W ) with propertyT(H). Let {Ti }N i=1 be a finite N family of nonexpansive mappings of C into itself with i=1 F (Ti ) 6= ∅ and let αj = (α1j , α2j , α3j ) ∈ I × I × I, j = 1, 2, 3, ..., N , where I = [0, 1] , α1j + α2j + α3j = 1, α1j ∈ (0, 1) for all j = 1, 2, ..., N − 1, α1N ∈ (0, 1] α2j , α3j ∈ [0, 1) for all j = 1, 2, ..., N. Let S be the mapping generated by T1 , ...., TN and α1 , α2 , ..., αN . Let x1 ∈ C and let {xn }, {yn } be sequences generated by (

xn+1 = W (xn , Syn , γn ), yn = W (xn , Sxn , βn )

(3.1)

for all n ≥ 1 where {γn }, {βn } are sequences in [0, 1] satisfying 0 < a ≤ P∞ TNγn ≤ b < 1 and n=1 βn (1 − βn ) = ∞. Then the sequence {xn } converges to z ∈ i=1 F (Ti ). Proof. First, we show that inf n∈N d(xn , Sxn ) = 0. Assume that inf n∈N d(xn , Sxn ) = 388

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Ishikawa iteration for a common fixed point of a finite...

r0 > 0. Let p ∈

TN

i=1 F (Ti ),

by nonexpansiveness of S−mapping, we have


d(p, xn+1 ) = d p, W (xn , Syn , γn ) ≤ γn d(p, xn ) + (1 − γn )d(p, Syn ) ≤ γn d(p, xn ) + (1 − γn )d(p, yn )  
= γn d(p, xn ) + (1 − γn ) d p, W (xn , Sxn , βn )   ≤ γn d(p, xn ) + (1 − γn ) βn d(p, xn ) + (1 − βn )d(p, Sxn ) ≤ d(p, xn ). It implies by Lemma 2.6 that limn→∞ d(p, xn ) exists. Then, we have limn→∞ d(p, xn ) = r00 > 0. By nonexpansiveness of S, we have d(p, Sxn ) ≤ d(p, xn ). Since {d(p, xn )} is a nonincreasing and inf n∈N d(xn , Sxn ) = r0 > 0, we have d(xn , Sxn ) ≥ r0 r0 d(p, xn ) d(p, xn ) r0 ≥ d(p, xn ) d(p, x1 ) > 0, ∀n ∈ N. ≥

By Lemma 2.5, there exists η = η

r0 d(p,x1 )




> 0 such that

d(p, xn+1 ) ≤ γn d(p, xn ) + (1 − γn )d p, W (xn , Sxn , βn )  
≤ γn d(p, xn ) + (1 − γn ) 1 − 2 min{βn , 1 − βn }η d(p, xn ) = γn d(p, xn ) + (1 − γn )d(p, xn ) − 2(1 − γn ) min{βn , 1 − βn }ηd(p, xn ) ≤ γn d(p, xn ) + (1 − γn )d(p, xn ) − 2(1 − γn )βn (1 − βn )ηd(p, xn ) = d(p, xn ) − 2(1 − γn )βn (1 − βn )ηd(p, xn ), which follows that 2(1 − b)βn (1 − βn )ηr00 ≤ 2(1 − γn )βn (1 − βn )ηd(p, xn ) ≤ d(p, xn ) − d(p, xn+1 ). (3.2) From (3.2), it implies that 2(1 − b)η

k X

βn (1 − βn )r00 ≤ d(p, x1 ) − d(p, xk+1 )

(3.3)

n=1

for all k ≥ 1. Letting k → ∞ in (3.3) and , we have ∞ ≤ d(p, x1 ) − r00 < ∞. This is a contradiction, then we have inf n∈N d(xn , Sxn ) = 0. Then, there exists a subsequence {xnj } of {xn } such that limj→∞ d(xnj , Sxnj ) = 0. Since C is compact, 389

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9

then there exists a subsequence {xnjl } of {xnj } and p such that liml→∞ xnjl = p. From nonexpansiveness of S, we have d(p, Sp) ≤ d(p, xnjl ) + d(xnjl , Sxnjl ) + d(Sxnjl , Sp) ≤ 2d(p, xnjl ) + d(xnjl , Sxnjl ). T Taking l → ∞, it implies that p ∈ F (S). From Lemma 2.7, we have p ∈ N F (Ti ). Ti=1 N Since limn→∞ d(p, xn ) exists, we can conclude that {xn } converges to p ∈ i=1 F (Ti ).

We can prove the following results by using Theorem 3.1. Corollary 3.2. Let C be a nonempty compact closed convex subset of a complete uniformly convex metric space (X, d, W ) with property (H). Let T : C → C be a nonexpansive mappings with F (T ) 6= ∅. Let x1 ∈ C and let {xn }, {yn } be sequences generated by ( xn+1 = W (xn , T yn , γn ), (3.4) yn = W (xn , T xn , βn ) for all P n ≥ 1 where {γn }, {βn } are sequences in [0, 1] satisfying 0 < a ≤ γn ≤ b < 1 and ∞ n=1 βn (1 − βn ) = ∞. Then the sequence {xn } converges to z ∈ F (T ). Proof. Put N = 1 in Theorem 3.1, we obtain the desired result.

Acknowledgements This research was supported by the Research Administration Division of King Mongkut’s Institute of Technology Ladkrabang.

References [1] A. Kangtunyakarn, S. Suantai, Hybrid iterative scheme for generalized equilibrium problems and fixed point problems of finite family of nonexpansive mappings, Nonlinear Analysis: Hybrid Systems 3 (2009) 296-309. [2] Browder, F.E., Petryshyn,W.V.: Construction of fixed points of nonlinear mappings in Hilbert spaces. J. Math. Anal. Appl. 20, 197-228 (1967). [3] K. Aoyama, K. Eshita, and W. Takahashi, Iteration processes for nonexpansive mappings in convex metric spaces, in Proceedings of the International Conference on Nonlinear and Convex Analysis, pp. 31-39, 2005. 390

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[4] Osilike, M.O.: Implicit iteration process for common fixed points of a finite family of strictly pseudocontractive maps. J. Math. Anal. Appl. 294, 73-84 (2004). [5] S. ISHIKAWA, Fixed points by a new iteration method, Proc. Am. Math. Soc. 44 (1974) 147-150. [6] T. Shimizu, A convergence theorem to common fixed points of families of nonexpansive mappings in convex metric spaces, in: Proceedings of the International Conference on Nonlinear and Convex Analysis, 2005, pp. 575-585. [7] T. Shimizu and W. Takahashi, Fixed points of multivalued mappings in certain convex metric spaces, Topological Methods in Nonlinear Analysis, vol. 8, no. 1, pp. 197-203, 1996. [8] W. Phuengrattana, S. Suantai, Strong Convergence Theorems for a Countable Family of Nonexpansive Mappings in Convex Metric Spaces, Hindawi Publishing Corporation Abstract and Applied Analysis Volume 2011, Article ID 929037, 18 pages doi:10.1155/2011/929037. [9] W. Takahashi, A convexity in metric space and nonexpansive mappings. I, Kodai Mathematical Seminar Reports, vol. 22, pp. 142149, 1970. [10] W. R. MANN, Mean value methods in iteration, Proc. Amer. Math. Soc. 4 (1953) 506-510. [11] W. Takahashi, A convexity in metric space and nonexpansive mappings. I, Kodai Mathematical Seminar Reports, vol. 22, pp. 142-149, 1970.

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TABLE OF CONTENTS, JOURNAL OF COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 24, NO. 2, 2018

The extension of a modified integral operator to a class of generalized functions, S. K. Q. AlOmari and Dumitru Baleanu,…………………………………………………………………209 Harmonic quasiconformal mappings of the unit disk onto the horizontal strip and half plane, Jian-Feng Zhu,…………………………………………………………………………………219 Langevin fractional differential inclusions with nonlocal Katugampola fractional integral boundary conditions, Sotiris K. Ntouyas and Jessada Tariboon,……………………………228 On the Riemann-Liouville fractional Hermite-Hadamard-type inequalities for differentiable 𝛼preinvex mappings, Lianzi Chen, Tingsong Du, Shasha Zhao, and Sheng Zheng,…………243 Convolution properties for certain subclasses of meromorphic bounded functions, Hanan Darwish, Abd El-Moneim Lashin, and Suliman Sowileh,………………………………….258 On a generalized degenerate 𝜆-q-Daehee numbers and polynomials, Jin-Woo Park,……….266 On the m-extension of Fibonacci p-functions with period k, Yasin Yazlik and Cahit Kome,274

Fourier spectral methods for stochastic space fractional partial differential equations driven by special additive noises, Fang Liu, Monzorul Khan, and Yubin Yan,………………………290 On anti-periodic type boundary value problems of sequential fractional differential equations of order q ∈ (2, 3], Ahmed Alsaedi, Mohammed H. Aqlan, and Bashir Ahmad,……………….310

Additive-quadratic 𝜌-functional inequalities in fuzzy normed spaces, Jung Rye Lee, Choonkil Park, Dong Yun Shin, and Sungsik Yun,…………………………………………………….318 Hyers-Ulam stability of set-valued functional equations: a fixed point approach, Sungsik Yun, Choonkil Park, and Hassan Azadi Kenary,…………………………………………………330 On precompactness of the Hausdorff fuzzy metric on closed sets, Chang-qing Li and Yan-lan Zhang,……………………………………………………………………………………….343 Semilocal convergence of a modified Chebyshev-like's method for solving nonlinear equations under generalized weak condition, Lin Zheng,…………………………………………….354 Generalized Hyers-Ulam stability of sextic functional equation in random normed spaces, Shaymaa Alshybani, S. Mansour Vaezpour, and Reza Saadati,…………………………370

TABLE OF CONTENTS, JOURNAL OF COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 24, NO. 2, 2018 (continued) On convergence theorem of a finite family of nonlinear mappings in uniformly convex metric spaces, Atid Kangtunyakarn,…………………………………………………………………382

Volume 24, Number 3 ISSN:1521-1398 PRINT,1572-9206 ONLINE

March 2018

Journal of Computational Analysis and Applications EUDOXUS PRESS,LLC

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

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Implicit Midpoint Type Picard Iterations for Strongly Accretive and Strongly Pseudocontractive Mappings Shin Min Kang1,2 , Arif Rafiq3 , Young Chel Kwun4,∗ and Faisal Ali5

1

Department of Mathematics and RINS, Gyeongsang National University, Jinju 52828, Korea e-mail: [email protected] 2

3

Center for General Education, China Medical University, Taichung 40402, Taiwan e-mail: [email protected]

Department of Mathematics and Statistics, Virtual University of Pakistan, Lahore 54000, Pakistan e-mail: [email protected] 4

5

Department of Mathematics, Dong-A University, Busan 49315, Korea e-mail: [email protected]

Center for Advanced Studies in Pure and Applied Mathematics, Bahauddin Zakariya University, Multan 60800, Pakistan e-mail: [email protected] Abstract We study the convergence of implicit midpoint type Picard sequence for strongly accretive and strongly pseudocontractive mappings. We have also improved the results of some authors. 2010 Mathematics Subject Classification: 47H06, 47J05, 47J25 Key words and phrases: Banach space, Lipschitzian mapping, strongly pseudocontractive mapping, strongly accretive mapping, implicit midpoint type Picard iteration

1

Introduction and Preliminaries

Let E be a real Banach space with dual E ∗ . A mapping T with domain D(T ) and range R(T ) in E is called strongly pseudocontractive if and only if for all x, y ∈ D(T ), the following inequality is satisfied: kx − yk ≤ k(1 + r)(x − y) − rt(T x − T y)k

(1.1)

for r > 0 and some t > 1. If t = 1 in inequality (1.1), then T is called pseudocontractive. ∗

Corresponding author

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∗

For E, we will denote by J the normalized duality mapping from E to 2E defined by J(x) = {f ∗ ∈ E ∗ : hx, f ∗i = kxk2 = kf ∗ k2 }, where h·, ·i denotes the generalized duality pairing. As a consequence of a result of Kato [15], it follows from inequality (1.1) that T is strongly pseudocontractive if and only if h(I − T )x − (I − T )y, j(x − y)i ≥ kkx − yk2

(1.2)

holds for all x, y ∈ D(T ) and for some j(x − y) ∈ J(x − y), where k = t−1 t ∈ (0, 1). Consequently, it follows easily (again from Kato [15] and inequality (1.2) that T is strongly pseudocontractive if and only if the following inequality holds: kx − yk ≤ kx − y + s[(I − T − kI)x − (I − T − kI)yk

(1.3)

for all x, y ∈ D(T ) and s > 0. Closely related to the class of pseudocontractive mappings is the class of accretive operators. A mapping A with domain D(A) and range R(A) in E is called accretive if the following inequality holds: kx − yk ≤ kx − y + s(Ax − Ay)k for all x, y ∈ D(A) and s > 0. Also, as a consequence of Kato [15], this accretive condition can be expressed in terms of the duality map as follows: For each x, y ∈ D(A), there exists j(x − y) ∈ J(x − y) such that hAx − Ay, j(x − y)i ≥ 0.

(1.4)

Consequently, inequality (1.1) with t = 1 yields that A is accretive if and only if T := (I − A) is pseudocontractive. Furthermore, setting A := (I − T ), it follows from inequality (1.3) that T is strongly pseudocontractive if and only if (A − kI) is accretive, and using (1.4), this implies that T (= I − A) is strongly pseudocontractive if and only if the following inequality holds hAx − Ay, j(x − y)i ≥ kkx − yk2

(1.5)

for all x, y ∈ D(A) and some k ∈ (0, 1). Operators A satisfying inequality (1.5) for all x, y ∈ D(A) and some k ∈ (0, 1) are called strongly accretive. It is then clear that A is strongly accretive if and only if T := (I − A) is strongly pseudocontractive. Thus, the mapping theory for strongly accretive operators is closely related to the fixed point theory of strongly pseudocontractive maps. We shall exploit this connection in the sequel. The notion of accretive operators was introduced independently in 1967 by Browder [2] and Kato [15]. An early fundamental result in the theory of accretive operators, due to Browder, states that the initial value problem du + Au = 0, dt

u(0) = u0

(1.6)

is solvable if A is locally Lipschitzian and accretive on E. If u is independent of t, then Au = 0 and the solution of this equation corresponds to the equilibrium points of the 2 406

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system (1.6). Consequently, considerable research efforts have been devoted, especially within the past 15 years or so, to developing constructive techniques for the determination of the kernels of accretive operators in Banach spaces (see [3, 4, 8–12, 14, 16, 17, 19,20, 22]). Two well known iterative schemes, the Mann iterative method (see [18]) and the Ishikawa iteration scheme (see [13]) have successfully been employed. In [16], Liu obtained a fixed point of the strictly pseudocontractive mapping as the limit of an iteratively constructed sequence with error estimation in general Banach spaces. Theorem 1.1. Let E be a Banach space, and let K be a nonempty closed convex and bounded subset of E. Let T : K → K be a Lipschitzian strictly pseudocontractive mapping. If F ix(T ) 6= ∅, where F ix(T ) is the fixed point set of T, then {xn } is a sequence in K generated by x1 ∈ K, xn+1 = (1 − αn )xn + αn T xn , where {αn } is a seqnence in (0, 1] satisfying ∞ X

αn = ∞,

αn → 0

n→∞

n=1

strongly converges to q ∈ F ix(T ) and F ix(T ) is a single set. In [21], Sastry and Babu showed that any fixed point of a Lipschitzian, strictly pseudocontractive mapping T on a closed convex subset K of a Banach space E is necessarily unique, and may be norm approximated by an iterative procedure. They also provided a convergence rate estimate and removes the boundedness assumption on K, generalizing Theorems of Liu. Theorem 1.2. Let (E, k · k), K, T, L and k be as described above. Let q ∈ K be a fixed point of T . Suppose that {αn } is a sequence in (0, 1] such that for some η ∈ (0, k), for all n ∈ N, ∞ X k−η αn ≤ , while αn = ∞. (L + 1)(L + 2 − k) n=1 Fix x1 ∈ K. Define for all n ∈ N, xn+1 := (1 − αn )xn + αn T xn . Then there exists a sequence {βn } in (0, 1) with each βn ≥ kxn+1 − qk ≤

n Y

η 1+k αn

such that for all n ∈ N,

(1 − βj )kx1 − qk.

j=1

In particular, {xn } converges strongly to q, and q is the unique fixed point of T . The Mann and Ishikawa iteration schemes are global and their rate of convergence is 1 generally of the order O(n− 2 ). It is clear that if, for an operator U , the classical iteration sequence of the form, xn+1 = U xn , x0 ∈ D(U ) (the so-called Picard sequence) converges, then it is certainly superior and preferred to either the Mann or the Ishikawa sequence since it requires less computations and moreover, its rate of convergence is always at least as fast as that of a geometric progression. In [5, 6], Chidume proved the following results. 3 407

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Theorem 1.3. Let E be an arbitrary real Banach space and A : E → E be a Lipschitz (with constant L > 0) and strongly accretive mapping with strong accretivity constant
k k ∈ (0, 1). Let x∗ denote a solution of the equation Ax = 0. Set  := 21 1+L(3+L−k) and define A : E → E by A x := x − Ax for each x ∈ E. For arbitrary x0 ∈ E, define the sequence {xn } in E by xn+1 = A xn , n ≥ 0. (1.7) ∗ Then {xn }∞ n=0 converges strongly to x with

kxn+1 − x∗ k ≤ δ n kx0 − x∗ k, where δ = 1 − 21 k ∈ (0, 1) is the Lipschitz constant of the operator A. Moreover, x∗ is unique. Corollary 1.4. Let E be an arbitrary real Banach space and K be nonempty convex subset of E. Let T : K → K be Lipschitz (with constant L > 0) and strongly pseudocontractive (i.e., T satisfies inequality (1.3) for all x, y ∈ K). Assume that T has a fixed point x∗ ∈ K.
k Set 0 := 21 1+L(3+L−k) and define T0 : K → K by T0 x = (1 − 0 )x + 0 T x for each x ∈ K. For arbitrary x0 ∈ K, define the sequence {xn }∞ n=0 in K by xn+1 = T0 xn ,

n ≥ 0.

(1.8)

Then {xn } converges strongly to x∗ with kxn+1 − x∗ k ≤ δ n kx0 − x∗ k, where δ := 1 − 21 k0 ∈ (0, 1). Moreover, x∗ is unique. ´ c et al. [7] presented the following results. Recently Ciri´ Theorem 1.5. Let E be an arbitrary real Banach space, A : E → E be a Lipschitz (with constant L > 0) and strongly accretive mapping with strong accretivity constant k ∈ (0, 1). k−η Let x∗ denote a solution of the equation Ax = 0. Set  := L(2+L) , η ∈ (0, k) and define A : E → E by A x := x − Ax for each x ∈ E. For arbitrary x0 ∈ E, define the sequence {xn } in E by xn+1 = A xn , n ≥ 0. Then {xn } converges strongly to x∗ with kxn+1 − x∗ k ≤ θn kx0 − x∗ k, k−η where θ = 1 − k−η+L(2+L) η ∈ (0, 1). Thus the choice η = ∗ Moreover, x is unique.

k 2

yields θ = 1 −

k2 . 2[k+2L(2+L)]

Corollary 1.6. Let E be an arbitrary real Banach space, K be a nonempty convex subset of E. Let T : K → K be Lipschitz (with constant L > 0) and strongly pseudocontractive (i.e., T satisfies inequality (1.3) for all x, y ∈ K). Assume that T has a fixed point x∗ ∈ K. k−η Set 0 := L(2+L) , η ∈ (0, k) and define T0 : K → K by T0 x = (1 − 0 )x + 0 T x for each x ∈ K. For arbitrary x0 ∈ K, define the sequence {xn } in K by xn+1 = T0 xn , n ≥ 0. 4 408

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Then {xn } converges strongly to x∗ with kxn+1 − x∗ k ≤ θn kx0 − x∗ k, where θ := 1 −

k−η η k−η+L(2+L)

∈ (0, 1). Moreover, x∗ is unique.

However Kang et al. [14] established the following results. Theorem 1.7. Let E be an arbitrary real Banach space, A : E → E be a Lipschitz (with constant L > 1) and strongly accretive mapping with strong accretivity constant k ∈ (0, 1). k−η , η ∈ (0, k) and Let x∗ denote a solution of the equation Ax = 0. Set  := L+(1+L)(k−η) define A : E → E by A xn := (1 − ) xn−1 + xn − Axn for each xn ∈ E. For arbitrary x0 ∈ E, define the sequence {xn } in E by xn = A xn ,

n ≥ 1.

Then {xn } converges strongly to x∗ with kxn+1 − x∗ k ≤ λn kx0 − x∗ k, where λ = 1 − k2

2[2L+k(1+L+k)] .

k−η L+(k−η)(1+L+k) η ∗

∈ (0, 1). Thus the choice η =

k 2

yields λ = 1 −

Moreover, x is unique.

Corollary 1.8. Let E be an arbitrary real Banach space ans K be a nonempty closed convex subset of E. Let T : K → K be Lipschitz (with constant L > 0) and strongly pseudocontractive (i.e., T satisfies inequality (1.3) for all x, y ∈ K). Assume that T has k−η a fixed point x∗ ∈ K. Set 0 := L+(1+L)(k−η) , η ∈ (0, k) and define A0 : K → K by A0 xn := (1 − 0 ) xn−1 + 0 xn − 0 Axn for each xn ∈ K. For arbitrary x0 ∈ K, define the sequence {xn } in K by xn = A0 xn , n ≥ 1. Then {xn } converges strongly to x∗ with kxn+1 − x∗ k ≤ λn0 kx0 − x∗ k, where λ0 = 1 − k2

2[2L+k(1+L+k)] .

k−η L+(k−η)(1+L+k) η ∗

∈ (0, 1). Thus the choice η =

k 2

yields λ0 = 1 −

Moreover, x is unique.

Let H be the Hilbert space. Recently Alghamdi et al. [1] defined the following algorithm. Algorithm 1.9. Initialize xn ∈ H arbitrarily and iterate   xn + xn+1 xn+1 = (1 − tn )xn + tn T , 2

n ≥ 0,

where tn ∈ (0, 1) for all n. For the approximation of fixed points of nonexpansive mappings under the setting of Hilbert spaces, they provide the following results. 5 409

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Lemma 1.10. Let {xn } be the sequence generated by Algorithm 1.9. Then (i) kxn+1 − pk ≤ kxn − pk for all n ≥ 0 and p ∈ F ix(T ), P (ii) ∞ tn kxn − xn+1 k2 < ∞, Pn=1 ∞ n+1 )k2 < ∞. (iii) n=1 tn (1 − tn )kxn − T ( xn +x 2

Lemma 1.11. Let {xn } be the sequence generated by Algorithm 1.9. Suppose that t2n+1 ≤ atn for all n ≥ 0 and a > 0. Then lim kxn+1 − xn k = 0.

n→∞

Lemma 1.12. Assume that (i) t2n+1 ≤ atn for all n ≥ 0 and a > 0, (ii) lim supn→∞ tn > 0. Then the sequence {xn } generated by Algorithm 1.9 satisfies the property lim kxn − T xn k = 0.

n→∞

Theorem 1.13. Let H be a Hilbert space and T : H → H be a nonexpansive mapping with F ix(T ) 6= ∅. Assume that {xn } is generated by Algorithm 1.9, where the sequence {tn } of parameters satisfies the conditions: (i) t2n+1 ≤ atn for all n ≥ 0 and a > 0, (ii) lim supn→∞ tn > 0. Then {xn } converges weakly to a fixed point of T . In this paper, we study the convergence of implicit Picard sequence for strongly accretive and strongly pseudocontractive mappings. We have also improved the results of [5–7, 14, 16, 19–21].

2

Main results

In the following theorems, L > 1 will denote the Lipschitz constant of the operator A and k > 0 will denote the strong accretivity constant of A (as in inequality (1.5)). Furthermore,  > 0 is defined by k−η  := , η ∈ (0, k). 1 L + 2 (1 + L) (k − η) With these notations, we prove the following theorem. Theorem 2.1. Let E be an arbitrary real Banach space, A : E → E be a Lipschitz and strongly accretive mapping with strong accretivity constant k ∈ (0, 1). Let x∗ denote a x +x solution of the equation Ax = 0. Define A : E → E by A xn := (1 − ) xn−1 +  n−12 n − x +x A n−12 n for each xn ∈ E. For arbitrary x0 ∈ E, define the sequence {xn }∞ n=0 in E by xn = A xn ,

n ≥ 1.

∗ Then {xn }∞ n=0 converges strongly to x with

kxn+1 − x∗ k ≤ ρn kx0 − x∗ k, 2(k−η) where ρ = 1− 2L+(k−η)(2+L+k) η ∈ (0, 1). Thus the choice η = ∗ Moreover, x is unique.

k 2

2

k . yields ρ = 1− 4L+k(2+L+k)

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Proof. Existence of x∗ follows from Theorem 13.1 of [8]. Define T := (I − A) where I denotes the identity mapping on E. Observe that Ax∗ = 0 if and only if x∗ is a fixed point of T . Moreover, T is strongly pseudocontractive (satisfies inequality (1.2) since A satisfies (1.5), and so T also satisfies inequality (1.3) for all x, y ∈ E and all s > 0. Furthermore, the recursion formula xn = A xn becomes   xn−1 + xn , n ≥ 1. xn = (1 − )xn−1 + T (2.1) 2 Observe that x∗ = (1 + )x∗ + (I − T − kI)x∗ − (1 − k)x∗ , and from the recursion formula (2.1) that xn−1 + xn xn−1 + xn xn−1 = (1 + )xn + (I − T − kI) − (1 − k) 2 2   x + x n−1 n , + 2 xn−1 − T 2

(2.2)

so that 

 xn−1 + xn ∗ xn−1 − x = (1 + )(xn − x ) +  (I − T − kI) − (I − T − kI)x 2      xn−1 + xn xn−1 + xn ∗ 2 − (1 − k) − x +  xn−1 − T . 2 2 ∗

∗

Assume that xn ' xn−1 , which yields that the second term of right hand side, we get

xn−1 +xn 2

' xn . Replace

xn−1 +xn 2

by xn in

xn−1 − x∗ = (1 + )(xn − x∗ ) +  [(I − T − kI)xn − (I − T − kI)x∗ ]      xn−1 + xn xn−1 + xn ∗ 2 − (1 − k) − x +  xn−1 − T . 2 2 This implies, using inequality (1.3) with s =

 1+

and y = x∗ that

kxn−1 − x∗ k

 

 ∗ ∗

≥ (1 + )

(xn − x ) + 1 +  [(I − T − kI)xn − (I − T − kI)x ]

 

xn−1 + xn

xn−1 + xn ∗ 2

− (1 − k) − x −  xn−1 − T

2 2  ≥ (1 + )kxn − x∗ k − (1 − k) kxn−1 − x∗ k

2  

 xn−1 + xn ∗ 2

− (1 − k) kxn − x k −  xn−1 − T

2 2     = −(1 − k) kxn−1 − x∗ k + 1 + (1 + k) kxn − x∗ k 2 2

 

xn−1 + xn 2

. −  xn−1 − T

2

(2.3)

7

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Observe that

and so

 

xn−1 − T xn−1 + xn

2

 

xn−1 + xn

≤ kxn−1 + T xn−1 k + T xn−1 + T

2

 

xn−1 + xn 1

− A Ax ≤ kAxn−1 k + kxn−1 − xn k +

n−1

2 2

xn−1 + xn 1 ∗

≤ L kxn−1 − x k + kxn−1 − xn k + L xn−1 −

2 2 1 = L kxn−1 − x∗ k + (1 + L) kxn−1 − xn k 2

 

1 xn−1 + xn ∗

, = L kxn−1 − x k + (1 + L)  xn−1 − T

2 2 kxn−1 − T



xn−1 + xn 2



k≤

1−

1 2

L kxn−1 − x∗ k , (1 + L) 

(2.4)

so that from (2.3) we obtain   1 + (1 − k) kxn−1 − x∗ k 2   L2 kxn−1 − x∗ k . ≥ 1 + (1 + k) kxn − x∗ k − 2 1 − 12 (1 + L)  Therefore kxn − x∗ k ≤ and consider ρ=

1 + (1 − k) 2 +

L2 1 1− 2 (1+L)

1 + (1 + k) 2

kxn−1 − x∗ k,

(2.5)

L2 1− 12 (1+L) + k) 2

1 + (1 − k) 2 +

1 + (1    L k− =1− 1 + (1 + k) 2 1 − 12 (1 + L)   =1− η 1 + (1 + k) 2 2 (k − η) η. =1− 2L + (k − η) (2 + L + k)

(2.6)

From (2.5) and (2.6), we get kxn − x∗ k ≤ ρkxn−1 − x∗ k ≤ · · · ≤ ρn kx0 − x∗ k →0 as n → ∞. Hence xn → x∗ as n → ∞. Uniqueness follows from the strong accretivity property of A. This completes the proof. 8 412

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The following is an immediate corollary of Theorem 2.1. Corollary 2.2. Let E be an arbitrary real Banach space and K be anonempty closed convex subset of E. Let T : K → K be Lipschitz (with constant L > 1) and strongly pseudocontractive (i.e., T satisfies inequality (1.3) for all x, y ∈ K). Assume that T has k−η a fixed point x∗ ∈ K. Set 0 := L+ 1 (1+L)(k−η) ; η ∈ (0, k) and Define A0 : K → K by 2

+x

x

A0 xn := (1 − 0 ) xn−1 + 0 n−12 n − 0 A define the sequence {xn } in K by

xn−1 +xn 2

xn = A0 xn ,

for each xn ∈ K. For arbitrary x0 ∈ K, n ≥ 1.

(2.7)

Then {xn } converges strongly to x∗ with kxn+1 − x∗ k ≤ ρn0 kx0 − x∗ k, 2(k−η) 2L+(k−η)(2+L+k) η ∈ Moreover, x∗ is unique.

where ρ0 = 1 − k2 4L+k(2+L+k)

.

(0, 1). Thus the choice η =

k 2

yields ρ0 = 1 −

Proof. Observe that x∗ is a fixed point of T if and only if it is a fixed point of T0 . Furthermore, the recursion formula (2.7) simplifies to the formula xn = (1 − 0 )xn−1 + 0 T xn , which is similar to (2.1). Following the method of computations as in the proof of the Theorem 2.1, we obtain kxn − x∗ k ≤

L20 1− 12 (1+L)0 + k) 20

1 + (1 − k) 20 +

kxn−1 − x∗ k   2 (k − η) η kxn−1 − x∗ k. = 1− 2L + (k − η) (2 + L + k)

Set ρ0 = 1 −

1 + (1

2(k−η) 2L+(k−η)(2+L+k) η.

(2.8)

Then from (2.8) we obtain

kxn − x∗ k ≤ ρ0 kxn−1 − x∗ k ≤ · · · ≤ ρn0 kx0 − x∗ k →0 as n → ∞. This completes the proof. Remark 2.3. Since L > 1, consider 2 (k − η) η 2L + (k − η) (2 + L + k) k−η =1− L + (k − η) (1 + L + k)   2 1 − − (k − η) η 2L + (k − η) (2 + L + k) L + (k − η) (1 + L + k) k−η 0 and p < 1. Then the limit f (2n x) L(x) = lim n→∞ 2n 0 exists for all x ∈ E and L : E → E is the unique additive mapping which satisfies 2 kf (x) − L(x)k ≤ kxkp 2 − 2p for all x ∈ E. Also, if for each x ∈ E the function f (tx) is continuous in t ∈ R, then L is R-linear. Th.M. Rassias [26] during the 27th International Symposium on Functional Equations asked the question whether such a theorem can also be proved for p ≥ 1. Gajda [9] following the same approach as in Th.M. Rassias [25], gave an affirmative solution to this question for p > 1. For further research developments in stability of functional equations the readers are referred to the ˇ works of G˘avruta [10], Jung [16], Park [23], Th.M. Rassias [27]–[30], Th.M. Rassias and Semrl [31], F. Skof [38] and the references cited therein. See also [32, 33, 34, 35, 36, 37] for functional equations. In an inner product space, the equality

1 x + y

2 kz − xk2 + kz − yk2 = kx − yk2 + 2 z −

2 2 holds, and is called the Apollonius’ identity. The following functional equation, which was motivated by this equation,  1 x + y Q(z − x) + Q(z − y) = Q(x − y) + 2Q z − , (1.1) 2 2 is quadratic. For this reason, the function equation (1.1) is called a quadratic functional equation of Apollonius type, and each solution of the functional equation (1.1) is said to be a quadratic mapping of Apollonius type. Jun and Kim [15] investigated the quadratic functional equation of Apollonius type. In this paper, employing the above equality (1.1), we introduce a new functional equation, which is called the Apollonius type additive functional equation and whose solution of the functional equation is said to be the Apollonius type additive mapping:  1 x + y . L(z − x) + L(z − y) = − L(x + y) + 2L z − 2 4

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M. RAGHEBI MOGHADAM, TH. M. RASSIAS, V. KESHAVARZ, C. PARK, Y. S. PARK

In this paper, we investigate Jordan homomorphisms and Jordan derivations in C ∗ -ternary algebras, and Jordan homomorphisms and Jordan derivations in JB ∗ -triples. 2. Jordan homomorphisms between C ∗ -ternary algebras Throughout this section, assume that A is a C ∗ -ternary algebra with norm k · kA and that B is a C ∗ -ternary algebra with norm k · kB . In this section, we investigate Jordan homomorphisms between C ∗ -ternary algebras. The following lemma was proved in [24]. Lemma 2.1. Let f : A → B be a mapping such that

1 x+y



)

f (z − x) + f (z − y) + f (x + y) ≤ 2f (z − 2 4 B B for all x, y, z ∈ A. Then f is additive. The following lemma was proved in [8]. Lemma 2.2. Let f : A → B be an additive mapping. Then the following assertions are equivalent   f [x, x, x] = [f (x), f (x), f (x)] for all x ∈ A, and   f [x, y, z] + [y, z, x] + [z, x, y] = [f (x), f (y), f (z)] + [f (y), f (z), f (x)] + [f (z), f (x), f (y)] for all x, y, z ∈ A. The following lemma was proved in [6]. Lemma 2.3. Let f : A → A be an additive mapping. Then the following assertions are equivalent.   f [x, x, x] = [f (x), x, x] + [x, f (x), x] + [x, x, f (x)] for all x ∈ A, and   f [xyz] + [yzx] + [zxy] = [f (x), b, c] + [x, f (y), z] + [x, y, f (z)] + [f (y), z, x] + [y, f (z), x] + [y, z, f (x)] + [f (z), x, y] + [z, f (x), y] + [z, x, f (y)], for all x, y, z ∈ A. Theorem 2.4. Let r 6= 1 and θ be nonnegative real numbers, and let f : A → B be a mapping such that

1 x+y



) , (2.1)

f (z − µx) + µf (z − y) + f (x + y) ≤ 2f (z − 2 4 B B





f [x, y, z] + [y, z, x] + [z, x, y]) − [f (x), f (y), f (z)] − [f (y), f (z), f (x)] − [f (z), f (x), f (y)] B   3r 3r 3r ≤ θ kxkA + kykA + kzkA (2.2)

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JORDAN HOMOMORPHISMS IN C ∗ -TERNARY ALGEBRAS AND JB ∗ -TRIPLES

for all µ ∈ T1 := {λ ∈ C | |λ| = 1} and all x, y, z ∈ A. Then the mapping f : A → B is a C ∗ -ternary algebra Jordan homomorphism. Proof. Assume r > 1. Let µ = 1 in (2.1). By Lemma 2.1, the mapping f : A → B is additive. Letting y = −x and z = 0, we get kf (−µx) + µf (x)kB ≤ k2f (0)kB = 0 for all x ∈ A and all µ ∈ T1 . So −f (µx) + µf (x) = f (−µx) + µf (x) = 0 for all x ∈ A and all µ ∈ T1 . Hence f (µx) = µf (x) for all x ∈ A and all µ ∈ T1 . By the same reasoning as in the proof of [21, Theorem 2.1], the mapping f : A → B is C-linear. It follows from (2.2) that





f [x, y, z] + [y, z, x] + [z, x, y] − [f (x), f (y), f (z)] − [f (y), f (z), f (x)] − [f (z), f (x), f (y)] B

 [x, y, z] [y, z, x] [z, x, y] 

+ + = lim 8n f n→∞ 8ni h 8n 8n h x y z y z x i h z x y i

− f ( n ), f ( n ), f ( n ) − f ( n ), f ( n ), f ( n ) − f ( n ), f ( n ), f ( n ) 2 2 2 2 2 2 2 2 2 B  8n θ  3r 3r =0 ≤ lim nr kxk3r A + kykA + kzkA n→∞ 8 for all x, y, z ∈ A. Thus   f [x, y, z] + [y, z, x] + [z, x, y] = [f (x), f (y), f (z)] + [f (y), f (z), f (x)] + [f (z), f (x), f (y)] for all x, y, z ∈ A. Hence the mapping f : A → B is a C ∗ -ternary algebra Jordan homomorphism. Similarly, one obtains the result for the case r < 1.  3. Jordan derivations on C ∗ -ternary algebras Throughout this section, assume that A is a C ∗ -ternary algebra with norm k · kA . In this section, we investigate Jordan derivations on C ∗ -ternary algebras. Theorem 3.1. Let r 6= 1 and θ be nonnegative real numbers, and let f : A → A be a mapping satisfying (2.1) such that

 

f [x, y, z] + [y, z, x] + [z, x, y] − [f (x), b, c] − [x, f (y), z] − [x, y, f (z)] − [f (y), z, x]

− [y, f (z), x] − [y, z, f (x)] − [f (z), x, y] − [z, f (x), y] − [z, x, f (y)] (3.1) A   3r 3r ≤ θ kxk3r A + kykA + kzkA for all x, y, z ∈ A. Then the mapping f : A → A is a C ∗ -ternary Jordan derivation.

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M. RAGHEBI MOGHADAM, TH. M. RASSIAS, V. KESHAVARZ, C. PARK, Y. S. PARK

Proof. Assume r > 1. By the same reasoning as in the proof of Theorem 2.4, the mapping f : A → A is C-linear. It follows from (3.1) that

 

f [x, y, z] + [y, z, x] + [z, x, y] − [f (x), y, z] − [x, f (y), z] − [x, y, f (z)] − [f (y), z, x]

− [y, f (z), x] − [y, z, f (x)] − [f (z), x, y] − [z, f (x), y] − [z, x, f (y)]kA

 x, y, z y, z, x z, x, y  h x y z i

= lim 8n f [ n ] + [ n ] + [ n ] − f ( n ), n , n n→∞ 8 8 8 2 2 2 hx y z i hx y z i h y z xi hy z xi − n , f ( n ), n − n , n , f ( n ) − f ( n ), n , n − n , f ( n ), n h 2y z 2 x2 i h 2 z2 x 2 y i h z 2 x2 2y i h 2z x 2 y2 i

− n , n , f ( n ) − f ( n ), n , n − n , f ( n ), n − n , n , f ( n ) 2 2 2 2 2 2 2 2 2 2 2 2 A  8n θ  3r 3r ≤ lim nr kxk3r =0 A + kykA + kzkA n→∞ 8 for all x, y, z ∈ A. So   f [x, y, z] + [y, z, x] + [z, x, y] = [f (x), y, z] + [x, f (y), z] + [x, y, f (z)] + [f (y), z, x] +[y, f (z), x] + [y, z, f (x)] + [f (z), x, y] + [z, f (x), y] + [z, x, f (y)] for all x, y, z ∈ A. Thus the mapping f : A → A is a C ∗ -ternary Jordan derivation. Similarly, one obtains the result for the case r < 1.



4. Jordan homomorphisms between JB ∗ -triples Throughout this paper, assume that J is a JB ∗ -triple with norm k · kJ and that L is a JB ∗ -triple with norm k · kL . In this section, we investigate Jordan homomorphisms between JB ∗ -triples. Theorem 4.1. Let r 6= 1 and θ be nonnegative real numbers, and let f : J → L be a mapping such that 1 x+y kf (z − µx) + µf (z − y) + f (x + y)kL ≤ k2f (z − )kL , 2 4





f {xyz} + {yzx} + {zxy} − {f (x)f (y)f (z)} − {f (y)f (z)f (x)} − {f (z)f (x)f (y)} L   3r 3r 3r ≤ θ kxkJ + kykJ + kzkJ

(4.1)

(4.2)

for all µ ∈ T1 and all x, y, z ∈ J . Then the mapping f : J → L is a JB ∗ -triple Jordan homomorphism. Proof. Assume r > 1. By the same reasoning as in the proof of Theorem 2.4, the mapping f : J → L is C-linear.

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JORDAN HOMOMORPHISMS IN C ∗ -TERNARY ALGEBRAS AND JB ∗ -TRIPLES

It follows from (4.2) that





f {xyz} + {yzx} + {zxy} − {f (x)f (y)f (z)} − {f (y)f (z)f (x)} − {f (z)f (x)f (y)} L

 {xyz} {yzx} {zxy}  n x y z o n = lim 8 f n n n + n n n + n n n − f ( n )f ( n )f ( n ) n→∞ 2 .2o .2 n 2 .2 .2 2 .2o .2 2 2 2 n y z x z x y − f ( n )f ( n )f ( n ) − f ( n )f ( n )f ( n ) 2 2 2 2 2 2 L  8n θ  3r 3r 3r ≤ lim nr kxkJ + kykJ + kzkJ = 0 n→∞ 8 for all x, y, z ∈ J . Thus   f {xyz} + {yzx} + {zxy} = {f (x)f (y)f (z)} + {f (y)f (z)f (x)} + {f (z)f (x)f (y)} for all x, y, z ∈ J . Hence the mapping f : J → L is a JB ∗ -triple Jordan homomorphism. Similarly, one obtains the result for the case r < 1.



5. Jordan derivations on JB ∗ -triples Throughout this paper, assume that J is a JB ∗ -triple with norm k · kJ . In this section, we investigate Jordan derivations on JB ∗ -triples. Theorem 5.1. Let r 6= 1 and θ be nonnegative real numbers, and let f : J → J be a mapping satisfying (4.1) such that

f ({xyz} + {yzx} + {zxy}) − {f (x)yz} − {xf (y)z} − {xyf (z)} − {f (y)zx}

− {yf (z)x} − {yzf (x)} − {f (z)xy} − {zf (x)y} − {zxf (y)} (5.1) A   3r 3r ≤ θ kxk3r A + kykA + kzkA for all x, y, z ∈ J . Then the mapping f : J → J is a JB ∗ -triple Jordan derivation. Proof. Assume r > 1. By the same reasoning as in the proof of Theorem 2.4, the mapping f : J → J is C-linear. It follows from (5.1) that

 

f {xyz} + {yzx} + {zxy} − {f (x)yz} − {xf (y)z} − {xyf (z)} − {f (y)zx}

− {yf (z)x} − {yzf (x)} − {f (z)xy} − {zf (x)y} − {zxf (y)} J

 {xyz}  n x y z o {yzx} {zxy}

= lim 8n f n n n + n n n + n n n − f ( n ) n n n→∞ 2 .2 .2 2 .2 .2 o 2n .2 .2 2 2 2 nx y z o nx y z y z xo ny z xo − n f( n ) n − n n f( n ) − f( n ) n n − n f( n ) n n 2y z 2 x2 o n 2 2z x 2 y o n z 2 x2 2y o n 2z x 2 y2 o

− n n f( n ) − f( n ) n n − n f( n ) n − n n f( n ) 2 2 2 2 2 2 2 2 2 2 2 2 J  8n θ  3r 3r ≤ lim nr kxk3r =0 J + kykJ + kzkJ n→∞ 8

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for all x, y, z ∈ J . So f ({xyz} + {yzx} + {zxy}) = {f (x)yz} + {xf (y)z} + {xyf (z)} + {f (y)zx} + {yf (z)x} + {yzf (x)} + {f (z)xy} + {zf (x)y} + {zxf (y)} for all x, y, z ∈ J . Thus the mapping f : J → J is a JB ∗ -triple Jordan derivation. Similarly, one obtains the result for the case r < 1.  References [1] V. Abramov, R. Kerner and B. Le Roy, Hypersymmetry: A Z3 graded generalization of supersymmetry, J. Math. Phys. 38 (1997), 1650–1669. [2] F. Bagarello and G. Morchio, Dynamics of mean-field spin models from basic results in abstract differential equations, J. Stat. Phys. 66 (1992), 849–866. [3] A. Cayley, On the 34concomitants of the ternary cubic, Am. J. Math 4 (1881). 1–15. [4] P. Czerwik, Functional Equations and Inequalities in Several Variables, World Scientific Publishing Company, New Jersey, Hong Kong, Singapore and London, 2002. [5] M. Eshaghi Gordji, Nearly involutions on Banach algebras: A fixed point approach, Fixed Point Theory 14 (2013), 117–124. [6] M. Eshaghi Gordji, Sh. Bazeghi, C. Park and S. Jang, Ternary Jordan ring derivations on Banach ternary algebras: A fixed point approach, J. Comput. Anal. Appl. 21 (2016). 829–834. [7] M. Eshaghi Gordji and N. Ghobadipour, Stability of (α, β, γ)-derivations on lie C ∗ -algebras, Int. J. Geom. Methods Modern Phys. 7 (2010), 1093–1102. [8] M. Eshaghi Gordji and V. Keshavarz, Ternary Jordan homomorphisms between unital ternary C ∗ -algebras (preprint). [9] Z. Gajda, On stability of additive mappings, Int. J. Math. Math. Sci. 14 (1991), 431–434. [10] P. Gˇ avruta, 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. Nat. 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] D. H. Hyers, G. Isac and Th. M. Rassias, On the asymptoticity aspect of Hyers-Ulam stability of mappings, Proc. Am. Math. Soc. 126 (1998), 425–430. [14] G. Isac and Th.M. Rassias, Stability of ψ-additive mappings: Applications to nonlinear analysis, Int. J. Math. Math. Sci. 19 (1996), 219–228. [15] K. Jun and H. Kim, On the stability of Appolonius’ equation, Bull. Belg. Math. Soc. - Simon Stevin 11 (2004), 615–624. [16] S. Jung, On the Hyers-Ulam-Rassias stability of approximately additive mappings, J. Math. Anal. Appl. 204 (1996), 221–226. [17] M. Kapranov, I. M. Gelfand and A. Zelevinskii, Discriminants, Resultants and Multidimensional Determinants, Birkh¨ auser, Berlin, 1994. [18] R. Kerner, The cubic chessboard: Geometry and physics, Classical Quantum Gravity 14 (1997), A203–A225. [19] J. Nambu, Physical Review D7 (1973), p. 2405. [20] C. Park, Approximate homomorphisms on JB ∗ -triples, J. Math. Anal. Appl. 306 (2005), 375–381. [21] C. Park, Homomorphisms between Poisson JC ∗ -algebras, Bull. Braz. Math. Soc. 36 (2005), 79–97. [22] C. Park, Isomorphisms between C ∗ -ternary algebras, J. Math. Anal. Appl. 327 (2007), 101–115. [23] C. Park, Hyers-Ulam-Rassias stability of a generalized Apollonius-Jensen type additive mapping and isomorphisms between C ∗ -algebras, Math. Nachr. 281 (2008), 402–411. [24] C. Park and Th. M. Rassias, Homomorphisms in C ∗ -ternary algebras and JB ∗ -triples, J. Math. Anal. Appl. 337 (2008), 13–20.

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[25] Th. M. Rassias, On the stability of the linear mapping in Banach spaces, Proc. Am. Math. Soc. 72 (1978), 297–300. [26] Th. M. Rassias, Problem 16; 2, Report of the 27th International Symp. on Functional Equations, Aequationes Math. 39 (1990), 292–293; 309. [27] Th. M. Rassias, The problem of S.M. Ulam for approximately multiplicative mappings, J. Math. Anal. Appl. 246 (2000), 352–378. [28] Th. M. Rassias, On the stability of functional equations in Banach spaces, J. Math. Anal. Appl. 251 (2000), 264–284. [29] Th. M. Rassias, On the stability of functional equations and a problem of Ulam, Acta Appl. Math. 62 (2000), 23–130. [30] Th. M. Rassias, Functional Equations, Inequalities and Applications, Kluwer Academic Publishers, Dordrecht, Boston and London, 2003. ˇ [31] Th. M. Rassias and P. Semrl, On the behaviour of mappings which do not satisfy Hyers-Ulam stability, Proc. Am. Math. Soc. 114 (1992), 989–993. [32] K. Ravi, E. Thandapani and B. V. Senthil Kumar, Solution and stability of a reciprocal type functional equation in several variables, J. Nonlinear Sci. Appl. 7 (2014), 18–27. [33] S. Schin, D. Ki, J. Chang and M. Kim, Random stability of quadratic functional equations: a fixed point approach, J. Nonlinear Sci. Appl. 4 (2011), 37–49. [34] S. Shagholi, M. Bavand Savadkouhi and M. Eshaghi Gordji, Nearly ternary cubic homomorphism in ternary Fr´echet algebras, J. Comput. Anal. Appl. 13 (2011), 1106–1114. [35] S. Shagholi, M. Eshaghi Gordji and M. Bavand Savadkouhi, Stability of ternary quadratic derivation on ternary Banach algebras, J. Comput. Anal. Appl. 13 (2011), 1097–1105. [36] D. Shin, C. Park and Sh. Farhadabadi, On the superstability of ternary Jordan C ∗ -homomorphisms, J. Comput. Anal. Appl. 16 (2014), 964–973. [37] D. Shin, C. Park and Sh. Farhadabadi, Stability and superstability of J ∗ -homomorphisms and J ∗ -derivations for a generalized Cauchy-Jensen equation, J. Comput. Anal. Appl. 17 (2014), 125–134. [38] F. Skof, Propriet` a locali e approssimazione di operatori, Rend. Sem. Mat. Fis. Milano 53 (1983), 113–129. [39] S. M. Ulam, A Collection of the Mathematical Problems, Interscience Publ. New York, 1960. [40] L. Vainerman and R. Kerner, On special classes of n-algebras, J. Math. Phys. 37 (1996), 25–53. [41] H. Zettl, A characterization of ternary rings of operators, Adv. Math. 48 (1983), 117–143. Mohammad Raghebi Moghadam, Department of Mathematics, Tabiat Modares University, P. O. Box 14115-134, Tehran, Iran E-mail address: [email protected] Themistocles M. Rassias, Department of Mathematics, National Technical University of Athens, Zografou Campus, 15780 Athens, Greece E-mail address: [email protected] Vahid Keshavaz, Department of Mathematics, Shiraz University of Technology, P. O. Box 71555313, Shiraz, Iran E-mail address: [email protected] Choonkil Park, Research Institute for Natural Sciences, Hanyang University, Seoul 04763, Republic of Korea E-mail address: [email protected] Young Sun Park, Research Institute for Natural Sciences, Hanyang University, Seoul 04763, Republic of Korea E-mail address: [email protected]

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THE GENERIC STABILITY OF KKM POINTS IN PMT SPACES M. TATARI, S. M. VAEZPOUR, AND REZA SAADATI*

Abstract. In this paper, we consider the generic stability of generalized KKM points and present a result concerning the generic continuity of set-valued mappings in PMT spaces. Then we prove that almost all of generalized KKM points of probabilistic upper semicontinuous set-valued mappings defined on compact subsets of such spaces are stable in the sense of Baire category theory. Also, we discuss on existence of the essential component of generalized KKM points.

Keywords: KKM point, PMT space, Hausdorff distance, Generic stability, Essential component, Generic continuity 1. Introduction In 2003, Yu et.al. [1], introduced the concept of KKM points of a KKM mapping G : X → K(X), from a bounded complete convex subset X of a normed linear space E into nonempty compact subsets of X. By Fort theorem, they prove that if M be the collection of all KKM mappings G, then there exists a dense residual subset Q of M such that for each G ∈ Q, G is essential. They also proved there exists at least one essential component of KKM points for each G ∈ M ; (see also [2, 3]). In this paper, we present a result concerning generic continuity of set-valued mappings based upon extensions of Fort’s theorems in probabilistic metric type spaces. 2. Preliminaries First, let us give the background and auxiliary results which will be needed. For more details see [4, 5, 6, 7, 8]. Definition 2.1. ([9, 10]) mapping F : (−∞, ∞) → [0, 1] is called a distribution function if it is nondecreasing and left-continuous with inf x∈R F (x) = 0 and supx∈R F (x) = 1. If in addition F (0) = 0, then F is called a distance distribution function. The set of all distance distribution functions (d.d.f )is denoted by ∆+ . The maximal element for ∆+ in this order is the d.d.f , 0 , given by  0 if t ≤ 0 , 0 (t) = 1 if t > 0 . Definition 2.2. ([9]) A triangular norm (shorter t-norm) is a binary operation T on [0, 1], i.e., a function 2 T : [0, 1] → [0, 1] which satisfies the following conditions: (1) T is associative and commutative; (2) T (a, 1) = a for all a ∈ [0, 1]; (3) T (a, b) ≤ T (c, d) whenever a ≤ c and b ≤ d, for each a, b, c, d ∈ [0, 1]. 2

In particular, a t−norm T is said to be continuous if it is a continuous function in [0, 1] . A t−norm is called sup-continuous if supλ∈Λ T (aλ , b) = T (supλ∈Λ aλ , b) for any family {aλ : λ ∈ Λ} ⊂ [0, 1] and b ∈ [0, 1]. The operatins TL (a, b) = max(a + b − 1, 0), TM (a, b) = min{a, b} and Tp (a, b) = ab on [0, 1] are T norms. Lemma 2.3. ([11]) Let T be a t−norm. (1) If T is left-continuous, then T satisfies sup0 1 − r} for all r ∈ (0, 1) and t > 0, and (X, F, T ) is a Hausdorff topological space. In virtue of this topology τ, a sequence {xn } in (X, F, T ) is said to be convergent to x (we write xn → x or limn→∞ xn = x) if limn→∞ Fxn ,x (t) = 1 for all t > 0; {xn } is called a Cauchy sequence in (X, F, T ) if for any given t > 0 and r ∈ (0, 1), there exists N = N (ε, λ) ∈ Z + such that Fxn ,xm (t) > 1 − r, whenever n, m ≥ N . S Let t > 0 and r ∈ (0, 1], A is said to have a finite (r, t)-net if there exists a finite set S ⊂ A such that A ⊂ x∈S Bx (r, t), i.e. , for each y ∈ A there is x ∈ S such that Fx,y (t) > 1 − r. A is said to be totally bounded if for each t > 0 and r ∈ (0, 1], A has a finite (r, t)-net. A is said to be probabilistically bounded( P −bounded ) if supt>0 inf x,y∈A Fx,y (t) = 1. Let P (X) denote the class of all nonempty subsets of X. We use the notions: (1) Pcl (X) = {Y ∈ P (X) : Y is closed}; (2) Pbd (X) = {Y ∈ P (X) : Y is probabilistic bounded}; (3) Pcp (X) = {Y ∈ P (X) T : Y is compact}; (4) Pcl,bd (X) = Pcl (X) Pbd (X) . Let Ψ : X → X be a mapping. Ψ is said to be closed if ΨA ∈ Pcl (X) for each A ∈ Pcl (X). It is said to be bounded if ΨA ∈ Pbd (X) for each A ∈ Pbd (X). Lemma 2.6. ([12]) Let (X, F, T ) be a PMT space. Let A ⊂ X. (1) A is compact if and only if A is sequentially compact; (2) If A is compact, then A is closed and totally bounded; (3) If A is totally bounded, then A ∈ Pbd (X) and A¯ is also totally bounded. 3. Probabilistic Hausdorff distance type Given x ∈ X, B ∈ P (X), the ”probabilistic distance type” from x to B is defined as  0 if t = 0 , Fx,B (t) = FB,x (t) = sups 1 − µ, and moreover, Fu,w (M + 2) ≥ T (Fu,y ((M + 1)/K), Fy,w (1/K)) ≥ T (1 − µ, 1 − ν) ≥ T (1 − µ, 1 − µ) > 1 − λ. Hence supt>0 inf u,w∈A Fu,w (t) ≥ inf u,w∈A Fu,w (M + 2) ≥ 1 − λ. By the arbitrariness of λ, we have supt>0 inf u,w∈A Fu,w (t) = 1, i.e. , A is probabilistically bounded. Also similar to proof of Theorem 2.2, [13] we can prove that A is closed. From Theorem (3.3) we see that A is closed. Therefore A ∈ Pcl,bd (X), and so the proof is complete.  Theorem 3.6. Let (X, F, T ) be a complete PMT space. Then (Pcp (X), H, T ) is complete PMT space. Proof. By Theorem (3.3), (Pcl,bd (X), H, T ) is a complete PMT space. By Corollary (3.4) we see that Pcp (X) is PMT space. Since Pcp (X) ⊂ Pcl,bd (X) , it is enough that Pcp (X) is closed with respect to H. ∞ Suppose that {An }n=1 ⊂ Pcp (X) , An → A with respect to H. We shall prove that A ∈ Pcp (X). Choose any ε > 0 and λ ∈ (0, 1]. By the left-continuity of T and lemma (2.3), we have sup0 1 − µ , f or all n ≥ N. 2 Since by Lemma (2.6), AN is compact, AN is also totally bounded. Thus, AN has a finite ( 2ε , µ)-net SN . From this we infer that SN is a finite (, λ)-net of A. In fact, for each x ∈ A, it follows the existence of 428

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ε ε y ∈ AN such that Fx,y ( 2K ) > 1 − µ. For such y we can select z ∈ SN with Fy,z ( 2K ) > 1 − µ. Hence, from (5) we have ε ε Fx,z (ε) ≥ T (Fx,y ( ), Fy,z ( )) ≥ T (1 − µ, 1 − µ) > 1 − λ. 2K 2K This shows that SN is a finite (ε, λ)-net of A, and so A is totally bounded. From the completeness of (X, F, T ) it follows that A is compact, i.e. , A ∈ Pcp (X). 

4. Stability of KKM points Stability of solution maps has been intensively investigated recently [15, 16, 17]. In this section, we first give some Lemmas and concepts, then we investigate on exictence of essential components and the stability of the set of KKM points in PMT space. For a set A, we denote the set of all nonempty finite subsets of A by hAi. Let A be a nonempty p-bounded subset of PMT space (X, F, T ). Then: T (1) co (A) = {B ⊂ X, B is a closed ball in X such that A ⊂ B}; (2) A(X) = {A ⊂ X, A = co (A)}, i.e. A ∈ A(X) if and only if A is an intersection of all closed balls containing A. In this case, we say that A is an admissible set in X; (3) A is called subadmissible, if for each D ⊂< A >, co(D) ⊂ A. Obviously, if A is an admissible subset of X, then A must be subadmissible. Recall that closed and open balls of X are defined as Bx [r, t] = {y ∈ X, Fx,y (t) ≥ 1 − r},

Bx (r, t) = {y ∈ X, Fx,y (t) > 1 − r} ,

for any x ∈ X and 0 < r < 1 and t > 0. Let (X, F, T ) be a PMT space and A a subadmissible subset of X and Pcp (X) the set of all nonempty compact subsets of X. G : X → Pcp (X) is called a KKM mapping, if for each A ∈< X >, we have co(A) ⊂ G(A). More generally, if G : X → Pcp (X), S : X → Pcp (X) are two set-valued functions such that for any A ∈< X >, S(co(A)) ⊆ G(A), then G is called a generalized KKM mapping with respect to S. If the set-valued function S : X → Pcp (X) satisfies the requirement that for any generalized KKM mapping G : X → Pcp (X) with respect to S the family {G(x) : x ∈ X} has the finite intersection property, then S is said to have the KKM property. We define KKM (X, Pcp (X)) := {S : X → Pcp (X) : S has the KKM property } . Thus if S ∈ KKM (X, Pcp (X)), then for any generalized TKKM mapping G : X → Pcp (X) with respect T to S we have x∈X G(x) 6= ∅. then such a point x∗ ∈ x∈X G(x), is called the KKM point of G and denote by K(G) the set of all generalized KKM points of G. Let M be the collection of all KKM mappings G : X → Pcp (X) with respect to S. For each G1 , G2 ∈ M define ˜ G ,G (t) = inf HG (x),G (x) (t) , H 1 2 1 2 x∈X

˜ T) where H is the probabilistic Hausdorff distance type defined on all compact subsets of X. Clearly (M, H, is a PMT space. ˜ T ) is a complete PMT space. Lemma 4.1. (M, H, ∞

Proof. Let {Gn }n=1 be any Cauchy sequence in M , then for any t > 0 and r ∈ (0, 1), there exists a positive ˜ G ,G (t) > 1 − r whenever n, m ≥ k, i.e. integer k such that H n m inf HGn (x),Gm (x) (t) > 1 − r

x∈X

,

∞

for any n, m ≥ k. It follows that for each x ∈ X, {Gn }n=1 is a Cauchy sequence in (Pcp (X), H, T ). By Theorem (3.6), there is G : X → Pcp (X) such that HGn (x),G(x) (t) > 1 − r for each x ∈ X. And it is easy to prove that inf x∈X HGn (x),G(x) (t) > 1 − r. Suppose that G were not generalized KKM mapping with Sm 0 0 respect to S, then there exist {x1 , ..., xm } ⊂ X and xS∈ S(co{x1 , ..., S xm }) such that x ∈ i=1 G(xi ). Since m m supx∈X HGn (x),G(x) (t) > 1 − r, there is n2 such that i=1 Gn (xi ) ⊂ i=1 BG(xi ) (r, t) for any n ≥ n2 . Thus Sm 0 x ∈ / i=1 Gn (xi ) for any n ≥ n2 which contradicts the assumption that Gn is generalized KKM mapping 429

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with respect to S for all n = 1, 2, ... . Hence G must be generalized KKM mapping with respect to S, and ˜ T ) is complete. (M, H,  Now we state some definitions. A set-valued mapping S from PMT space (X, F, T ), into nonempty subsets of a PMT space (Y, F ∗ , T ∗ ) is said to be probabilistic upper (lower) semicontinuous at x0 ∈ X, 0 0 if for any 0 < r < 1, there exists 0 < r < 1 such that S(x ) ⊂ BS(x0 ) (r, t) (S(x0 ) ⊂ BS(x0 ) (r, t)) for 0 0 each x ∈ X with Fx0 ,x0 (t) > 1 − r , for t > 0. S is probabilistic continuous at x0 ∈ X if S is both probabilistic upper semicontinuous and probabilistic lower semicontinuous at x0 . Also S is said probabilistic metric upper semicontinuous at x0 ∈ X if, for any 0 < r < 1, there exists a neighborhood U of x0 such that S(U ) ⊂ BS(x0 ) (r, t) for t > 0. It is easily verified that if S(x0 ) is compact, then S is probabilistic metric upper semicontinuous at x0 if and only if S is probabilistic upper semicontinuous at x0 . In general probabilistic metric upper semicontinuity is a weaker notion than probabilistic upper semicontinuity. On the other hand, the set-valued mapping S is said to be probabilistic metric lower semicontinuous at x0 if for any 0 < r < 1 there exists a neighborhood U of x0 such that S(x0 ) ⊂ BS(x) (r, t) for every x ∈ U and t > 0. It is easy to see that if S(x0 ) is totally bounded, then S is probabilistic lower semicontinuous at x0 if and only if S is probabilistic metric lower semicontinuous at x0 . However, we can also show that in general probabilistic lower semicontinuity is a weaker notion than probabilistic metric lower semicontinuity. Also a subset Q in X is called a residual set if it contains a countable intersection of open dense subsets of X. A set Q is called nowhere dense in X if int(Q) = ∅. If there exists a dense residual set Q of X such that S is continuous at each point of Q then we say that S is continuous at most point of X. In this case we shall also say that S is generically continuous on X. Result concerning generic continuity of set-valued mappings were first considered by Fort in [18]. After Fort’s theorems were published there have been several extensions of his original results; see [19, 20]. In the following we will extend Fort’s theorem in PMT space. Theorem 4.2. Let (X, F, T ) be a complete PMT space, (Y, F ∗ , T ∗ ) be a PMT space and S : X → 2Y be a probabilistic metric upper semicontinuous. Then there exists a dense residual set Q ⊂ X such that S is probabilistic metric lower semicontinuous at each x ∈ Q. Proof. For each 0 < r < 1 let 0

0

C(r) = {x ∈ X : ∀ 0 < r0 < r and 0 < r < 1, ∃ y ∈ Bx (r , t), such that BS(y) (r0 , t) + S(x) f or t > 0} 0

First, we prove that C(r) is a closed set. For any 0 < k < r and 0 < r < 1, let r0 < r00 < r and r00 − r0 = η. Due to the probabilistic metric upper semicontinuity of S, for each z ∈ C(r), there exists 00 0 00 00 0 < r < r such that S(x) ⊂ BS(z) (η, t) for all x ∈ Bz (r , t). Then there exists x ∈ C(r)∩Bz (r , t) such that 00 000 00 000 00 S(x) ⊂ BS(z) (η, t). From x ∈ Bz (r , t), Choose 0 < r < r such that Bx (r , t) ⊂ Bz (r , t). As x ∈ C(r), 000 00 0 it is easy to see that there exists y ∈ Bx (r , t) ⊂ Bz (r , t) ⊂ Bz (r , t) such that BS(y) (r00 , t) + S(x). Thus, it follows that BS(y) (r0 , t) ⊃ S(z). In fact, if BS(y) (r0 , t) ⊃ S(z), then BBS(y) (r0 ,t) (η, t) ⊃ BS(y) (η, t), so that BS(y) (r00 , t) + S(x). Thus, it follows that BS(y) (r0 , t) + S(z). In fact, if BS(q) (r0 , t) ⊃ S(z), then BBS(y) (r0 ,t) (η, t) ⊃ BS(z) (η, t), so that BS(y) (r00 , t) ⊃ S(x) which contradicts x ∈ C(r). Thus, it is proved that z ∈ C(r) and C(r) is a closed set. Next, we will prove that C(r) is a nowhere dense set, that is, to prove that C(r) contains no interior point. If not, let x1 ∈ C(r) be a interior point of C(r). For any 0 < r0 < r, choose r0 < r00 < r and set r00 − r0 = η. Then there exists 0 < r1 < 1 such that 0 0 Bx1 (r1 , t) ⊂ C(r) and S(x ) ⊂ BS(x1 ) (η, t), ∀ x ∈ Bx1 (r1 , t). From x1 ∈ C(r), it is known that there exists x2 ∈ Bx1 (r1 , t) such that BS(x2 ) (r00 , t) + S(x1 ). For x2 ∈ Bx1 (r1 , t) ⊂ C(r), choose 0 < r2 < r21 such that 0 0 Bx2 (r2 , t) ⊂ Bx1 (r1 , t) ⊂ C(r) and S(x ) ⊂ BS(x2 ) (η, t) for all x ∈ Bx2 (r2 , t). From x2 ∈ C(r), there exists x3 ∈ Bx2 (r2 , t) such that BS(x3 ) (r00 , t) + S(x2 ). The rest may be deduced by analogy, thus there exists rn−1 r1 , r2 , ..., rn−1 , rn , ... such that 0 < rn < , 2 Bxn (rn , t) ⊂ Bxn−1 (rn−1 , t) ⊂ ... ⊂ Bx2 (r2 , t) ⊂ Bx1 (r1 , t) ⊂ C(ε) and

0

0

S(x ) ⊂ BS(xn ) (η, t) , ∀x ∈ Bxn (rn , t) . 430

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We also have BS(xn+1 ) (r00 , t) + S(xn ) , n = 1, 2, ... . From the completeness of (X, F, T ) and the closedness of C(r), it is known that there exists x∗ ∈ C(r) and xn → x∗ . As x∗ ∈ Bxn (rn , t) for each n = 1, 2, ..., we have S(x∗ ) ⊂ BS(xn ) (η, t). Therefore,BS(p∗ ) (r0 , t) ⊂ BBS(xn ) (η,t) (r0 , t) = BS(xn ) (r00 , t). It follows from BS(xn ) (r00 , t) + S(xn−1 ) that BS(x∗ ) (r0 , t) + S(xn−1 ). In addition, from xn → x∗ and the upper semicontinuity of S at x∗ , for given r0 > 0, it is known that S(xn−1 ) ⊂ BS(x∗ ) (r0 , t) when n is sufficiently large, which is a contradiction. Thus, we can prove that C(r) is nowhere dense. S Let (0, 1)R be the rational number set in [0, 1], C = r∈(0,1) C(r) and Q = X \C. From the completeness R of X and nowhere density of C(r), it is easy to see that C is of first category. Hence, Q is a dense residual 0 set and of second category for any 0 < r < 1, choose r ∈ (0, 1)R such that 0 < r0 < r. For each x ∈ Q, by 0 the definition of Q we have x ∈ / C(r0 ). Also by the definition of C(r0 ), there exists 0 < r∗ < r0 and r > 0 0 such that BS(y) (r∗ , t) ⊃ S(x) for all y ∈ Bx (r , t), and hence BS(y) (r, t) ⊃ S(x). From the arbitrariness of 0 < r < 1, it is known that S is probabilistic metric lower semicotinuous at x. Therefore, S is probabilistic metric lower semicontinuous at each p ∈ Q.  Because in general probabilistic metric upper semi-continuity is a weaker notion than probabilistic upper semicontinuity, the following corollary is obvious. Corollary 4.3. Let (X, F, T ) be a complete PMT space, (Y, F ∗ , T ∗ ) be a PMT space and S : X → 2Y be probabilistic upper semicontinuous. Then there exists a dense residual set Q ⊂ X such that S is probabilistic metric lower semicontinuous at each x ∈ Q, and hence S is also probabilistic lower semi-continuity at each x ∈ Q. For each G ∈ M , K(G) is the set of all KKM points of G, G → K(G) indeed defines a set-valued mapping K : M → 2X Lemma 4.4. K : M → 2X is a probabilistic upper semicontinuous and compact-valued (pusco) mapping. ∞

Proof. For any G ∈ M , for any sequence {xn }m=1 in K(G) with xnT→ x∗ , then xn ∈ G(x) for each x ∈ X. Since G(x) is compact, then x∗ ∈ G(x) for each x ∈ X and x∗ ∈ x∈X G(x), x∗ ∈ K(G). Hence K(G) is closed, K(G) ⊆ G(x) must be compact. Fix t > 0, suppose that K were not probabilitic upper semicontinuous ∞ at G ∈ M , then there exist 0 < r0 < 1 and a sequence {Gn }n=1 in M with Gn → G such that T for each n = 1, 2, ..., there is xn ∈ K(Gn ) with xn ∈ / BK(G) (r0 , t). Since xn ∈ K(Gn ), we have xn ∈ x∈X Gn (x). T∞ S For any x ∈ X, since Gn (x) → G(x), Gn (x) (n = 1, 2, ...) and G(x) is compact, thus n=1TGn (x) G(x) is compact. xn ∈ Gn (x), we may assume that xn → x∗ , we obtain x∗ ∈ G(x). Thus x∗ ∈ x∈X Gn (x) and x∗ ∈ K(G) ⊂ BK(G) (r0 , t) which contradicts the assumption that xn → x∗ and xn ∈ / BK(G) (r0 , t) for each n = 1, 2, ... . Therefore, K must be probabilistic upper semicontinuous on M .  0

Definition 4.5. G ∈ M , (1) x ∈ K(G) is essential if for any 0 < r < 1, there exists 0 < r < 1 such that for 0 0 0 0 ˜ each G ∈ M with H G,G0 (t) > 1 − r , there exists x ∈ K(G ), with Fx,x0 (t) > 1 − r, (2) G is weakly essential if there exists x ∈ K(G) which is essential and (3) G is essential if every x ∈ K(G) is essential. Theorem 4.6. K : M → Pcp (X) is probabilistic lower semicontinuous at G ∈ M if and only if G is essential. 0

Proof. If K is probabilistic lower semicontinuous at G ∈ M , then for any 0 < r < 1, there exists 0 < r < 1 0 ˜ such that K(G) ⊂ BK(G0 ) (r, t) for each G ∈ M with H G,G0 (t) > 1 − δ. For each x ∈ K(G) there exists 0 0 x ∈ K(G ) with Fx,x0 (t) > 1 − r, x is essential and G is essential. Conversely, suppose that G is essential. ∞ If K were not probabilistic lower semicontinuous at G, then there exist 0 < r0 < 1 and a sequence {Gn }n=1 in M with Gn → G such that for each n = 1, 2, ..., there is an xn ∈ K(G) with xn ∈ BK(Gn ) (r0 , t). Since K(G) is compact, we map assume that xn → x ∈ K(G). Since x is essential, Gn → G, xn → x, there is an N such that Fxn ,x (t) > 1 − r21 and BK(Gn ) ( r21 , t) for all n ≥ N . Hence xn ∈ BK(Gn ) (r1 , t) for all n ≥ N which 431

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contracts the assumption that xn ∈ / BK(Gn ) (r0 , t) for all n = 1, 2, ... . Hence K must be probabilistic lower semicontinuous at G.  Theorem 4.7. There exists a dense residual subset Q of M such that for each G ∈ Q, G is essential. Proof. by Lemma (4.4), K : M → Pcp (X) is an pusco mapping. By Corollary (4.3), there exists a dense residual subset Q of M such that for each G ∈ Q, K is probabilistic lower semicontinuous at G. By Theorem (4.6), for each G ∈ Q, G is essential.  Remark 4.8. If G ∈ Q, by Lemma (4.4) and Theorem (4.6), K is probabilistic continuous, then for any 0 0 0 ˜ 0 < r < 1, there exists 0 < r < 1 such that for any G ∈ M with H G,G0 (t) > 1 − r , HK(G),K(G0 ) (t) > 1 − r, G is stable. Now we shall introduce some definitions. For each G ∈ M , the component of a point x ∈ K(G) is the union of all connected subsets of K(G) which contain the point x. Note that components are connected closed subsets of K(G) and are also connected compact. It is easy to see that the components of two distinct points K(G) either coincide or are disjoint, so that all components constitute a decomposition of K(G) into connected pairwise disjoint compact subsets, i.e. , [ K(G) = Cα (G) α∈Λ

where ΛTis an index set; for any α ∈ Λ, Cα (G) is a nonempty connected compact and for any α, β ∈ Λ(α 6= β), Cα (G) Cβ (G) = ∅. Definition 4.9. For each G ∈ M , let e(G) be a nonempty closed subset of K(G). Fix t > 0, e(G) is called 0 0 an essential set of K(G) if for eny 0 < r < 1, there exists 0 < r < 1 such that for any G ∈ M with 0 0 T ˜ H Be(G) (r, t) 6= ∅. If Cα (G), the component of K(G) is essential, then Cα (G) is G,G0 (t) > 1 − r , K(G ) called an essential component of K(G). Following theorem is the main result . Theorem 4.10. For each G ∈ M , there exists at least one essential component of K(G). Proof. By Lemma (4.4), K : M → Pcp (X) is probabilistic upper semicontinuous, that is for any 0 < r < 1, 0 0 0 0 ˜ there exists 0 < r < 1 such that for any G ∈ M with H G,G0 (t) > 1 − r , K(G ) ⊂ BK(G) (r, t). Hence 0 T K(G ) BK(G) (r, t) 6= ∅, K(G) is essential set of itself. Let Φ denote the family of all essential sets of K(G) ordered by set inclusion. Thus Φ is nonempty and every decreasing chain of elements in Φ has a lower bound (because by the compactness the intersection is in Φ); therefore by Zorn’s Lemma, Φ has a minimal element m(G) and m(G) is essential. Suppose that m(G) were not connected. Then S there exist two nonempty closed setsTc1 (G), c2 (G) and two open sets V1 and V2 such that m(G) = c1 (G) c2 (G), c1 (G) ⊂ V1 , c2 (G) ⊂ V2 , V1 V2 = ∅. Since m(G) is minimal, neither c1 (G) nor c2 (G) is essential, there exist 0 < r1 < 1, 0 < r2 < 1 0 ˜ G ,G (t) > 1 − r0 , H ˜ G,G (t) > 1 − such that for any 0 < r < 1, there exists G1 , G2 ∈ M such that H 1 2 2 T T T 0 r T with K(G1 ) [Bc1 (G) (r1 , t)] = ∅, K(G2 ) [Bc2 (G) (r2 , t)] = ∅. Denote W1 = V1 [Bc1 (G) (r1 , t)], W2 = V2 [Bc2 (G) (r2 , t)], then W1 , W2 are open. Since c1 (G) ⊂ W1 , c2 (G) ⊂ W2 , there exists 0 < r0 < 1 such 0 0 0 0 that [Bc1 (G) (r1 , t)] ⊂ W1 , [Bc2 (G) (r2 , t)] ⊂ W2 . Denote W1 = Bc1 (G) (r , t), W2 = Bc2 (G) (r , t), we know that 0 0 0 S 0 0 0 W1 , W2 are open. Since W1 W2 = Bc1 (G)∪c2 (G) (r , t) = Bm(G) (r , t) ⊃ m(G), then there exists 0 < r∗ < 1 S 0 0 T ∗ ˜ such that for any G ∈ M with H W2 ) 6= ∅. We may suppose that r∗ > a, G,G0 (t) > 1 − r , K(G ) (W1 ∗ † ∗ where a = FW 0 ,W 0 (t) > 0. For this r we can find an r > r such that T (1 − r† , 1 − r† ) ≥ 1 − r∗ . Thus for 1 2 T 0 0 0 0 t t † ˜ † ˜ 0 0 0 < r† < 1 there exist G1 , G2 ∈ M such that H W1 = ∅, G,G1 ( 2K ) > 1 − r , HG,G2 ( 2K ) > 1 − r , K(G1 ) T 0 0 K(G2 ) W2 = ∅. Note that t t ˜ 0 0 (t) > T (H ˜ ˜ 0 0 H )) H ) > T (1 − r† , 1 − r† ) ≥ 1 − r∗ > a . G1 ,G2 G,G1 ( G,G1 ( 2K 2K 432

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0

Now define G : X → Pcp (X) as follows: 0

0

0

G (x) = [G1 \ W2 ]

\

0

0

[G2 \ W1 ],

x∈X

0

Suppose that G were not a generalized KKM mapping with respect to S, then there exist {x1 , ..., xm } ⊂ X Sm 0 0 T 0 0 0 0 0 0 0 and x ∈ S(co{x1 , ..., xm }) such that x ∈ / i=1 G (xi ). Since W1 W2 = ∅, then x ∈ / W1 or x ∈ / W2 . 0 0 0 0 0 Without loss of generality, we may assume that x ∈ / W1 . Since x ∈ / G2 (xi ) \ W1 , then x ∈ / G2 (xi ) for each Sm 0 0 i = 1, ..., m, x ∈ / i=1 G2 (xi ) which contradicts that G2 ∈ M . Thus G ∈ M . ˜ 0 0 (t) ≥ H ˜ 0 0 (t). Note that Next we are going to prove that H G ,G G ,G 1

1

2

˜ 0 0 (t) = inf H 0 0 H G ,G G (x),G (x) (t) = inf {sup min{ inf 0

Fy,G0 (x) (s),

˜ 0 0 (t) = inf H 0 0 H G (x),G (x) (t) = inf {sup min{ inf G ,G 0

Fy,G0 (x) (s),

1

x∈X

x∈X s and T ( ) (y) tp > ; where ts; tp 2 ( ; 1] : It T ( ) : Then T ( ) (x) follows that ^ ^ = (z) _ = (z) _ T ( ) ((xa) y) _ z2T ((xa)y)

z2T (xa)T (y)

^

=

((bc) d) _

(bc)d2T ((xa)y)

^

=

bc2T (xa);d2T (y)

=

((bc) d) _

^

b2T (x);c2T (a);d2T (y)

^

b2T (x);d2T (y)

=

^

b2T (x)

(where z = (bc) d) =

^

bc2T (x)T (a);d2T (y)

((bc) d) _

((bc) d) _

( (b) ^ (d) ^ )

(b) ^

^

d2T (y)

(d) ^ = T ( ) (x) ^ T ( ) (y) ^ :

This implies that T ( ) ((xa) y) _ min ft1 ; t2 ; g. Hence by Theorem 4, T ( ) is an (2 ; 2 _q )-fuzzy bi-ideal of S: Conclusion; Associative algebras are being studied all over the globe, in particular semigroups have attracted many authors and researchers. The (2 ; 2 _q )fuzzy algebraic substructures are generalizations of fuzzy algebraic substructures and (2; 2 _qk )-fuzzy algebraic substructures. In this paper, generalized roughness have been studied for (2 ; 2 _q )-fuzzy algebraic substructures of semigroups. It is seen that, in order to preserve a particular algebraic substructure in case of its approximations, many types of set valued homomorphisms are required. This aspect of roughness study in semigroups makes this study more interesting. References [1] M. Banerjee, S. K. Pal, Roughness of a fuzzy set, Inform. Sci., 93 (1996), 235-246. [2] S. K. Bhakat, P. Das, On the de…nition of a fuzzy subgroup, Fuzzy Sets & Systems, 51 (1992), 235-241. [3] S. K. Bakat, P. Das, (2; 2 _q)-fuzzy subgroups, Fuzzy Sets & Systems, 80 (1996), 359-368. [4] S. K. Bakat, P. Das, Fuzzy subrings and ideals rede…ned, Fuzzy Sets & Systems, 81 (1996), 383-393. [5] R. Biswas, S. Nanda, Rough groups and Rough subgroups, Bull. polish Acad. of Sciences, 42 (1994), No 3.1. [6] K. Chakrabarty, R. Biswas, S. Nanda, Fuzziness in rough sets, Fuzzy Sets & Systems, 110 (2000), 247-251. [7] I. Couois, D. Dubois, Rough set, Coverings and incomplete information, Fundamenta informatica, XXI (2001) 1001-1025. [8] B. Davvaz, Roughness in rings, Inform. Sci., 164-(2004), 147-163.

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[9] D. Dubois, H. Prade, Rough fuzzy sets and fuzzy rough sets, Int. Journal of General System, 17 (1990), 191-208 [10] W. A. Dudek, M. Shabir, M. I. Ali, ( ; )-fuzzy ideals of hemirings, Comput. Math. Appl., 58 (2009), 310-32.5. [11] Y. B. Jun, Generalization of (2; 2 _q)-fuzzy subalgebras in BCK /BCI, Comput. Math. Appl., 58 (7) (2009), 1383-1390. [12] Y. B. Jun, Roughness of ideals in BCK-algebra, Scientiae Math. Japonica, 57 (1) (2008), 165-169. [13] S. B. Hosseini, T -rough semiprime ideals on commutative rings, The Journal of Nonlinear Science and Application, 4 (2011), 270-280. [14] S. B. Hosseini, N. Jafarzadeh, A. Gholami, T -rough Ideal and T -rough Fuzzy Ideal in a semigroup, Advanced Materials Research, Vol 433-440 (2012), 4915-4919. [15] N. Kuroki, Rough ideals in semigroups, Inform. Sci., 100 (1997), 139-163. [16] Z. Pawlak, Rough set , Int. J. Computer Science, 11 (1982), 341-356. [17] P. P. Ming, L. Y. Ming, Fuzzy topology I:Neighborhood structure of a fuzzy point and MoorSmith convergence, J. Math. Anal. Appl., 76 (1980), 571-59. [18] N. Rehman, M. Shabir, Characterization of ternary semigroup by ( ; )-fuzzy ideals,World Applied Sciences Journal, 18 (11) (2012), 1556-1570. [19] N. Rehman, M. Shabir, Some characterization of ternary semigroups by the properties of their (2 ; 2 _q )-fuzzy ideals, Journal of Intelligent & Fuzzy Systems, 26 (2014), 2107-2117. [20] M. Shabir, Y. B. Jun, Y. Nawaz, Characterization of Regular semigroup by ( ; )-fuzzy ideals, Comput. Math. Appl., 59 (2010), 161-17. [21] M. Shabir, Y. B. Jun, Y. Nawaz, Semigroup Characterized by (2; 2 _qk )-fuzzy ideals, Comput. Math. Appl., 60 (2010), 1473-1493. [22] M. Shabir, N. Rehman, Characterization of ternary semigroup by (2; 2 _qk )-fuzzy ideals, Iranian Journal of Science & Technology, Trans. A-Science, 36 (A3) (2012) 395-410. [23] M. Shabir, M. Ali, Characterization of semigroup by the properties of their (2 ; 2 _q )-fuzzy ideals, Iranian Journal of Science & Technology, Trans. A-Science 37 A2 (2013), 117-131. [24] L. A Zadeh, Fuzzy Set, Inf. control, 8 (1965), 338-353. Corresponding author: Department of Mathematics and Statistics, Riphah International University Islamabad Pakistan E-mail address : [email protected] Department of Mathematics, Islamia College University Peshawar KPK Pakistan Department of Mathematics and Statistics, Riphah International University Islamabad Pakistan E-mail address : [email protected] Department of Mathematics and Statistics, Riphah International University Islamabad Pakistan

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Equivalence between some iterations in CAT (0) spaces Kyung Soo Kim Graduate School of Education, Mathematics Education Kyungnam University, Changwon, Gyeongnam, 51767, Republic of Korea e-mail: [email protected] Abstract. We obtain some equivalence conditions for the convergence of iterative sequences for set-valued contraction mapping in CAT (0) spaces.

1. Introduction Let (X, d) be a metric space. One of the most interesting aspects of metric fixed point theory is to extend a linear version of known result to the nonlinear case in metric spaces. To achieve this, Takahashi [32] introduced a convex structure in a metric space (X, d). A mapping W : X × X × [0, 1] → X is a convex structure in X if d(u, W (x, y, λ)) ≤ λd(u, x) + (1 − λ)d(u, y) for all x, y ∈ X and λ ∈ [0, 1]. A metric space together with a convex structure W is known as a convex metric space. A nonempty subset K of a convex metric space is said to be convex if W (x, y, λ) ∈ K for all x, y ∈ K and λ ∈ [0, 1]. In fact, every normed space and its convex subsets are convex metric spaces but the converse is not true, in general (see, [32]). Example 1.1. ([15, 16]) Let X = {(x1 , x2 ) ∈ R2 : x1 > 0 , x2 > 0}. For all x = (x1 , x2 ), y = (y1 , y2 ) ∈ X and λ ∈ [0, 1]. We define a mapping W : X × X × [0, 1] → X by   λx1 x2 + (1 − λ)y1 y2 W (x, y, λ) = λx1 + (1 − λ)y1 , λx1 + (1 − λ)y1 and define a metric d : X × X → [0, ∞) by d(x, y) = |x1 − y1 | + |x1 x2 − y1 y2 |. Then we can show that (X, d, W ) is a convex metric space but not a normed linear space. 0

2010 Mathematics Subject Classification: 40G05, 41A60, 41A65, 51K05. Keywords: CAT(0) space, geodesic, Hausdorff metric, contraction, fixed point, multivalued mapping, iteration. 0

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2

A metric space X is a CAT (0) space. This term is due to M. Gromov [10] and it is an acronym for E. Cartan, A.D. Aleksandrov and V.A. Toponogov. If it is geodesically connected, and if every geodesic triangle in X is at least as ‘thin’ as its comparison triangle in the Euclidean plane(see, e.g., [2], p.159). It is well known that any complete, simply connected Riemannian manifold nonpositive sectional curvature is a CAT (0) space. The precise definition is given below. For a thorough discussion of these spaces and of the fundamental role they play in various branches of mathematics, see Bridson and Haefliger [2] or Burago et al. [1]. Let (X, d) be a metric space. A geodesic path joining x ∈ X to y ∈ X (or, more briefly, a geodesic from x to y) is a mapping c from a closed interval [0, l] ⊂ R to X such that c(0) = x, c(l) = y, and d(c(t), c(t0 )) = |t − t0 | for all t, t0 ∈ [0, l]. In particular, c is an isometry and d(x, y) = l. The image α of c is called a geodesic (or, metric) segment joining x and y. When it is unique, this geodesic is denoted by [x, y]. The space (X, d) is said to be a geodesic space if every two points of X are joined by a geodesic, and X is said to be uniquely geodesic if there is exactly one geodesic joining x and y for each x, y ∈ X. A subset Y ⊆ X is said to be convex if Y includes every geodesic segment joining any two of its points. A geodesic triangle 4(x1 , x2 , x3 ) is a geodesic metric space (X, d) consists of three points x1 , x2 , x3 ∈ X (the vertices of 4) and a geodesic segment between each pair of vertices (the edges of 4). A comparison triangle for the geodesic ¯ 1 , x2 , x3 ) = 4(x¯1 , x¯2 , x¯3 ) in triangle 4(x1 , x2 , x3 ) in (X, d) is a triangle 4(x 2 R such that dR2 (x¯i , x¯j ) = d(xi , xj ) for i, j ∈ {1, 2, 3}. Such a triangle always exists(see, [2]). A geodesic metric space is said to be a CAT (0) space if all geodesic triangles of appropriate size satisfy the following CAT (0) comparison axiom. ¯ ⊂ R2 be a comparison Let 4 be a geodesic triangle in X and let 4 triangle for 4. Then 4 is said to satisfy the CAT (0) inequality if for ¯ all x, y ∈ 4 and all comparison points x ¯, y¯ ∈ 4, d(x, y) ≤ d(¯ x, y¯). Complete CAT (0) spaces are often called Hadamard spaces(see [22]). If x, y1 , y2 are points of a CAT (0) space and if y0 is the midpoint of the segment [y1 , y2 ], 2 which we will denote by y1 ⊕y 2 , then the CAT (0) inequality implies   y1 ⊕ y2 1 1 1 ≤ d2 (x, y1 ) + d2 (x, y2 ) − d2 (y1 , y2 ). d x, 2 2 2 4 2

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3

This inequality is the (CN) inequality of Bruhat and Tits [3]. In fact, a geodesic space is a CAT (0) space if and only if satisfies the (CN) inequality (cf. [2], p.163). The above inequality has been extended by Khamsi and Kirk [12] as d2 (z, αx ⊕ (1 − α)y) ≤ αd2 (z, x) + (1 − α)d2 (z, y) − α(1 − α)d2 (x, y),

(CN∗ )

for any α ∈ [0, 1] and x, y, z ∈ X. The inequality (CN∗ ) also appeared in [5]. Let us recall that a geodesic metric space is a CAT (0) space if and only if it satisfies the (CN) inequality(see, [2], p.163). Moreover, if X is a CAT (0) metric space and x, y ∈ X, then for any α ∈ [0, 1], there exists a unique point αx ⊕ (1 − α)y ∈ [x, y] such that d(z, αx ⊕ (1 − α)y) ≤ αd(z, x) + (1 − α)d(z, y) for any z ∈ X and [x, y] = {αx ⊕ (1 − α)y : α ∈ [0, 1]}. In view of the above inequality, CAT (0) space have Takahashi’s convex structure W (x, y, α) = αx⊕ (1 − α)y. It is easy to see that for any x, y ∈ X and λ ∈ [0, 1], d(x, (1 − λ)x ⊕ λy) = λd(x, y), d(y, (1 − λ)x ⊕ λy) = (1 − λ)d(x, y). As a consequence, 1 · x ⊕ 0 · y = x, (1 − λ)x ⊕ λx = λx ⊕ (1 − λ)x = x.

(1.1)

Moreover, a subset K of CAT (0) space X is convex if for any x, y ∈ K, we have [x, y] ⊂ K. 2. Preliminaries Let D be a nonempty subset of a CAT (0) space X. We shall denote by CB(D) the family of nonempty bounded closed subset of D. Let H(·, ·) be the Hausdorff metric on CB(D), i.e.,   H(A, B) = max sup dist(a, B), sup dist(b, A) , A, B ∈ CB(D), a∈A

b∈B

where dist(a, B) = inf {d(a, b) : b ∈ B} is the distance from the point a to the set B. A multivalued mapping T : D → CB(D) is said to be a contraction if there exists a constant k ∈ (0, 1) such that H(T x, T y) ≤ k · d(x, y),

∀ x, y ∈ D.

A point x is called a fixed point of any mapping T if x ∈ T x. We denote by F (T ) the set of all fixed points of T .

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4

Let X be a CAT (0) space, and let {xn } be a bounded sequence in X, for x ∈ X we let r(x, {xn }) = lim sup d(x, xn ). n→∞

The asymptotic radius r({xn }) of {xn } is given by r({xn }) = inf {r(x, {xn }) : x ∈ X} , and the asymptotic center A({xn }) of {xn } is the set A({xn }) = {x ∈ X : r(x, {xn }) = r({xn })} . It is known that in a CAT (0) space asymptotic center A({xn }) consists of exactly one point(see, e.g., [6], Proposition 7). Definition 2.1. ([23]) A sequence {xn } in a CAT (0) space X is said to 4converge to x ∈ X if x is the unique asymptotic center of {un } for every subsequence {un } of {xn }. In this case one must write 4

xn −→ x

or

4 − lim xn = x n→∞

and call x the 4-limit of {xn }. Remark 2.1. In a CAT (0) space X, strong convergence implies 4-convergence. Lemma 2.1. ([28]) Let (X, d) be a complete metric space and T : X → CB(X) be a multivalued mapping. Then for any given ε > 0 and for any given x, y ∈ X, u ∈ T x, there exists v ∈ T y such that d(u, v) ≤ (1 + ε)H(T x, T y) where H(·, ·) is the Hausdorff metric on CB(X). Definition 2.2. Let D be a nonempty convex subset of a CAT (0) space X, T : D → CB(D) be a multivalued mapping. Let {αn }, {βn } and {γn } are three sequences in [0, 1] satisfying some conditions. (1) The sequence of Picard iterates (cf., [30]) is defined by w0 ∈ D, wn+1 = νn ,

(P)

where νn ∈ T wn such that d(νn+1 , νn ) ≤ (1 + ε)H(T wn+1 , T wn ). (2) The sequence of Mann iterates (cf., [27]) is defined by u0 ∈ D, un+1 = (1 − αn )un ⊕ αn θn ,

(M)

where θn ∈ T un such that d(θn+1 , θn ) ≤ (1 + ε)H(T un+1 , T un ).

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(3) The sequence of Ishikawa iterates (cf., [11]) is defined by r0 ∈ D, sn = (1 − βn )rn ⊕ βn δn ,

(I)

rn+1 = (1 − αn )rn ⊕ αn σn , where δn ∈ T rn and σn ∈ T sn such that d(δn+1 , δn ) ≤ (1 + ε)H(T rn+1 , T rn ), d(σn+1 , σn ) ≤ (1 + ε)H(T sn+1 , T sn ). (4) The sequence of three-step iterates (cf., [13, 14]) is defined by x0 ∈ D, zn = (1 − γn )xn ⊕ γn µn , yn = (1 − βn )xn ⊕ βn ξn ,

(TH)

xn+1 = (1 − αn )xn ⊕ αn ηn , where µn ∈ T xn , ξn ∈ T zn and ηn ∈ T yn such that d(µn+1 , µn ) ≤ (1 + ε)H(T xn+1 , T xn ), d(ξn+1 , ξn ) ≤ (1 + ε)H(T zn+1 , T zn ), d(ηn+1 , ηn ) ≤ (1 + ε)H(T yn+1 , T yn ). Another iteration processes and other some results in CAT (0) space have been studied extensively by various authors(see e.g. [4, 9, 17, 24, 26, 31]). Lemma 2.2. ([7]) Let {an } be recursively generated by an+1 = (1 − tn )an + b2n with n ≥ 1, an ≥ 0, {tn } ⊆ [0, 1] and ∞ X b2n < ∞, n=1

∞ X

tn = ∞.

n=1

Then lim an = 0.

n→∞

3. Main theorems Theorem 3.1. Let (X, d) be a CAT (0) space and D be a nonempty convex subset of X. Let T : D → CB(D) be a multivalued contraction mapping with 1 k < 1+ε and F (T ) 6= ∅ satisfying T p = {p} for any fixed point p ∈ F (T ). Let a constant L satisfying supw∈T x,x∈D d(p, w) ≤ L, for all x ∈ D. Let {wn } and {xn } be the Picard and three step iterative sequence defined by (P) and (TH) respectively and satisfying the following conditions: (i) αn , βn , γn ∈ [0, 1], ∀ n ≥ 0;

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(ii) P limn→∞ βn = 0; P∞ ∞ (iii) n=0 αn βn < ∞, n=0 (1 − αn ) = ∞. If w0 = x0 , then the following statements are equivalent: (1) the Picard iterative sequence {wn } 4-converegs to x∗ ∈ F (T ); (2) the three step iterative sequence {xn } 4-converegs to x∗ ∈ F (T ). Furthermore, x∗ is the unique fixed point of T . Proof. From Nadler [28], there exists a fixed point x∗ ∈ F (T ). Put M 0 = L + d(p, x0 ). From the contractive of T , we have d2 (xn+1 , p) = d2 ((1 − αn )xn ⊕ αn ηn , p) ≤ (1 − αn )d2 (xn , p) + αn d2 (ηn , p) − (1 − αn )αn d2 (n , ηn ) ≤ (1 − αn )d2 (xn , p) + αn (H(T yn , T p))2 ≤ (1 − αn )d2 (xn , p) + αn k 2 d2 (yn , p) ≤ (1 − αn )d2 (xn , p) + αn k 2 (d2 ((1 − βn )xn ⊕ βn ξn , p)) ≤ (1 − αn )d2 (xn , p) + αn k 2 ((1 − βn )d2 (xn , p) + βn d2 (ξn , p) − βn (1 − βn )d2 (xn , ξn )) ≤ (1 − αn )d2 (xn , p) + αn · k 2 (1 − βn )d2 (xn , p) + αn βn · k 4 · d2 (zn , p) − αn βn (1 − βn )k 2 · d2 (xn , ξn ) ≤ (1 − αn )d2 (xn , p) + αn (1 − βn ) · k 2 d2 (xn , p) + αn βn · k 4 ((1 − γn )d2 (xn , p) + γn d2 (µn , p) − (1 − γn )γn d2 (xn , µn )) − αn βn (1 − βn )k 2 · d2 (xn , ξn ) ≤ d2 (xn , p) − αn βn γn (1 − γn ) · k 4 · d2 (xn , µn ), for p ∈ F (T ). This implies 0 ≤ αn βn γn (1 − γn )k 4 d2 (xn , µn ) ≤ d2 (xn , p) − d2 (xn+1 , p). Therefore, we have d(xn+1 , p) ≤ d(xn , p). By induction, it is easy to see that sup{d(p, µn ), d(p, ηn ), d(p, ξn ), d(p, xn ), d(p, yn ), d(p, zn )} ≤ M 0 , n≥0

for µn ∈ T xn , ηn ∈ T yn and ξn ∈ T zn , n ≥ 0. By hypothesis, let M 00 = d(p, w0 ) + d(p, w1 ) < ∞,

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Put M = max{M 0 , M 00 }. From {wn } be the Picard iterative sequence defined by (P), we have d(νn , νn+1 ) ≤ (1 + ε)H(T wn , T wn−1 ) ≤ (1 + ε)k · d(wn , wn−1 ) = (1 + ε)k · d(νn−1 , νn−2 ) ≤ (1 + ε)k(1 + ε)H(T wn−1 , T wn−2 ) (3.1)

≤ ((1 + ε)k)2 d(wn−1 , wn−2 ) .. . ≤ ((1 + ε)k)n d(w1 , w0 ) ≤ ((1 + ε)k)n M

for any given ε > 0. From {xn } be the three step iterative sequence defined by (TH) and (3.1), for each n ≥ 0 d(xn+1 , wn+1 ) = d((1 − αn )xn ⊕ αn ηn , νn ) ≤ (1 − αn )d(xn , νn ) + αn · d(ηn , νn ) ≤ (1 − αn ){d(xn , wn ) + d(wn , νn )}

(3.2)

+ αn k · d(yn , wn ) ≤ (1 − αn ){d(xn , wn ) + ((1 + ε)k)n M } + αn k · d(yn , wn ), d(yn , wn ) = d((1 − βn )xn ⊕ βn ξn , wn ) ≤ (1 − βn )d(xn , wn ) + βn · d(ξn , νn−1 ) ≤ (1 − βn )d(xn , wn )

(3.3)

+ βn (d(ξn , , νn ) + d(νn , νn−1 )) ≤ (1 − βn )d(xn , wn ) + βn k · d(zn , wn ) + βn ((1 + ε)k)n M and d(zn , wn ) = d((1 − γn )xn ⊕ γn µn , wn ) ≤ (1 − γn )d(xn , wn ) + γn · d(µn , νn−1 ) ≤ (1 − γn )d(xn , wn )

(3.4)

+ γn k{d(xn , wn ) + d(wn , wn−1 )} ≤ (1 − γn )d(xn , wn ) + γn k{d(xn , wn ) + ((1 + ε)k)n−1 M }.

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Substituting (3.4) into (3.3), we get d(yn , wn ) ≤ (1 − βn )d(xn , wn ) h i + βn k (1 − γn )d(xn , wn ) + γn k{d(xn , wn ) + ((1 + ε)k)n−1 M } + βn ((1 + ε)k)n M

(3.5)

≤ (1 − βn )d(xn , wn ) + βn ((1 + ε)k)n M + βn k{(1 − γn (1 − k))d(xn , wn ) + γn ((1 + ε)k)n M }. Combining (3.5) and (3.2), we can obtain d(xn+1 , wn+1 ) ≤ (1 − αn ){d(xn , wn ) + ((1 + ε)k)n M } h + αn k (1 − βn )d(xn , wn ) + βn ((1 + ε)k)n M n oi + βn k (1 − γn (1 − k))d(xn , wn ) + γn ((1 + ε)k)n M = (1 − αn )d(xn , wn ) + (1 − αn )((1 + ε)k)n M + αn k(1 − βn )d(xn , wn ) + αn βn k((1 + ε)k)n M n o + αn βn k 2 (1 − γn (1 − k))d(xn , wn ) + γn ((1 + ε)k)n M h i = 1 − αn + αn (1 − βn )k + αn βn k 2 (1 − γn (1 − k)) d(xn , wn )

(3.6)

+ (1 − αn )((1 + ε)k)n M + αn βn k(1 + kγn )((1 + ε)k)n M ≤ (1 − αn (1 − k))d(xn , wn ) + {(1 − αn ) + αn βn (1 + kγn )}((1 + ε)k)n M. Take an = d(xn , wn ),

tn = αn (1 − k)

and b2n = {(1 − αn ) + αn βn (1 + kγn )}((1 + ε)k)n M P P∞ in (3.6). Since (1 + ε)k < 1, ∞ n=0 αn βn < ∞ and n=0 (1 − αn ) < ∞, we have ∞ ∞ X X tn = ∞, b2n < ∞. n=0

n=0

By Lemma 2.2, we know that 4 − lim d(xn , wn ) = 0. n→∞

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4

If wn −→ x∗ ∈ F (T ) as n → ∞, by Definition 2.1, we have 4

d(xnk , x∗ ) ≤ d(xnk , wnk ) + d(wnk , x∗ ) −→ 0 4

as n → ∞. If xn −→ x∗ ∈ F (T ) as n → ∞, we have 4

d(wnk , x∗ ) ≤ d(wnk , xnk ) + d(xnk , x∗ ) −→ 0 as n → ∞. Therefore, the equivalence between the statement (1) and (2) was proved. Finally, we prove that x∗ ∈ X is the unique fixed point of T. In fact, let x∗ , y ∗ ∈ X be two fixed points of T . Since T is a multivalued contraction with constant 0 < k < 1, we have d(x∗ , y ∗ ) ≤ (1 + ε)H(T x∗ , T y ∗ ) ≤ (1 + ε)k · d(x∗ , y ∗ ). Since ε is arbitrary, this implies that d(x∗ , y ∗ ) = 0, i.e., x∗ = y ∗ . This completes the proof.



If γn = 0 in (TH), then it reduces to (I). So we can easily prove the following corollary. Corollary 3.1. Let (X, d) be a CAT (0) space and D be a nonempty convex subset of X. Let T : D → CB(D) be a multivalued contraction mapping with 1 k < 1+ε and F (T ) 6= ∅ satisfying T p = {p} for any fixed point p ∈ F (T ). Let a constant L satisfying supw∈T x,x∈D d(p, w) ≤ L, for all x ∈ D. Let {wn } and {rn } be the Picard and Ishikawa iterative sequence defined by (P) and (I) respectively and satisfying the following conditions: (i) αn , βn ∈ [0, 1], ∀ n ≥ 0; (ii) P limn→∞ βn = 0; P∞ ∞ (iii) n=0 (1 − αn ) = ∞. n=0 αn βn < ∞, If w0 = r0 , then the following statements are equivalent: (1) the Picard iterative sequence {wn } 4-converegs to x∗ ∈ F (T ); (2) the Ishikawa iterative sequence {xn } 4-converegs to x∗ ∈ F (T ). Furthermore, x∗ is the unique fixed point of T . If βn = 0 in (I), then it reduces to (M). So we can easily prove the following corollary.

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Corollary 3.2. Let (X, d) be a CAT (0) space and D be a nonempty convex subset of X. Let T : D → CB(D) be a multivalued contraction mapping with 1 k < 1+ε and F (T ) 6= ∅ satisfying T p = {p} for any fixed point p ∈ F (T ). Let a constant L satisfying supw∈T x,x∈D d(p, w) ≤ L, for all x ∈ D. Let {wn } and {rn } be the Picard and Mann iterative sequence defined by (P) and (M) respectively and satisfying the following conditions: (i) α Pn∞∈ [0, 1], ∀ n ≥ 0; (ii) n=0 (1 − αn ) = ∞. If w0 = u0 , then the following statements are equivalent: (1) the Picard iterative sequence {wn } 4-converegs to x∗ ∈ F (T ); (2) the Mann iterative sequence {xn } 4-converegs to x∗ ∈ F (T ). Furthermore, x∗ is the unique fixed point of T .

4. Some remarks and open problem For a real number κ, a CAT (κ) space is defined by a geodesic metric space whose geodesic triangle is sufficiently thinner than the corresponding triangle in a model space with curvature κ. For κ = 0, the 2-dimensional model space Mκ2 = M02 is the Euclidean space R2 with the metric induced from the Euclidean norm. For κ > 0, Mκ2 is the 2dimensional sphere √1κ S2 whose metric is length of a minimal great arc joining 1 each two points. For κ < 0, Mκ2 is the 2-dimensional hyperbolic space √−κ H2 with the metric defined by a usual hyperbolic distance. For more details about the properties of CAT (κ) spaces, see [2], [8], [20], [21], [29].

Open Problem 1. It will be interesting to obtain a generalization of both Theorem 3.1 and Theorem 3.2 to CAT (κ) space. Open Problem 2. Can Theorem 3.1 be generalized to more than one contractive, or a commutative or left amenable semigroup S of mappings for which the sequence is defined by a strongly left invariant sequence (or net) of finite means on S(see [18], [19], [25])? Competing interests The author declares to have no competing interests. Acknowledgments This work was supported by Kyungnam University Foundation Grant, 2016.

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References [1] D. Burago, Y. Burago and S. Ivanov, A course in metric Geometry, in:Graduate studies in Math., 33, Amer. Math. Soc., Providence, Rhode Island, 2001. [2] M. Bridson and A. Haefliger, Metric spaces of Non-Positive Curvature, Springer-Verlag, Berlin, Heidelberg, 1999. [3] F. Bruhat and J. Tits, Groups r´ eductifss sur un corps local. I. Donn´ ees radicielles ´ valu´ ees, Publ. Math. Inst. Hautes Etudes Sci., 41 (1972), 5–251. [4] P. Chaoha and A. Phon-on, A note on fixed point sets in CAT (0) spaces, J. Math. Anal. Appl., 320 (2006), 983–987. [5] S. Dhompongsa and B. Panyanak, On triangle-convergence theorems in CAT (0) spaces, Comput. Math. Anal., 56 (2008), 2572–2579. [6] S. Dhompongsa, W.A. Kirk and B. Sims, Fixed point of uniformly Lipschitzian mappings, Nonlinear Anal., 65(4) (2006), 762–772. [7] J.C. Dunn, Iterative construction of fixed points for multivalued operators of the monotone type, J. Funct. Anal., 27(1) (1978), 38–50. [8] R. Esp´inola and A. Fern´ andez-Le´ on, CAT (κ)-spaces, weak convergence and fixed points, J. Math. Anal. Appl., 353 (2009), 410–427. [9] R. Esp´ınola and B. Pi¸atek, The fixed point property and unbounded sets in CAT (0) spaces, J. Math. Anal. Appl., 408 (2013), 638–654. [10] M. Gromov, Hyperbolic groups, Essays in group theory, Math. Sci. Res. Inst. Publ. 8. Springer, New York, 1987. [11] S. Ishikawa, Fixed point by a new iteration, Proc. Amer. Math. Soc., 44 (1974), 147–150. [12] M.A. Khamsi and W.A. Kirk, On uniformly Lipschitzian multivalued mappings in Banach and metric spaces, Nonlinear Anal., 72 (2010), 2080–2085. [13] J.K. Kim, K.H. Kim and K.S. Kim, Three-step iterative sequences with errors for asymptotically quasi-nonexpansive mappings in convex metric spaces, Proc. of RIMS Kokyuroku, Kyoto Univ., 1365 (2004), 156–165. [14] J.K. Kim, K.H. Kim and K.S. Kim, Convergence theorems of modified three-step iterative sequences with mixed errors for asymptotically quasi-nonexpansive mappings in Banach spaces, PanAmerican Math. Jour., 14(1) (2004), 45–54. [15] J.K. Kim, K.S. Kim and S.M. Kim, Convergence theorems of implicit iteration process for a finite family of asymptotically quasi-nonexpansive mappings in convex metric spaces, Proc. of RIMS Kokyuroku, Kyoto Univ., 1484 (2006), 40–51. [16] J.K. Kim, K.S. Kim and Y.M. Nam, Convergence and stability of iterative processes for a pair of simultaneously asymptotically quasi-nonexpansive type mappings in convex metric spaces, J. of Compu. Anal. Appl., 9(2) (2007), 159–172. [17] K.S. Kim, Some convergence theorems for contractive type mappings in CAT (0) spaces, Abst. Appl. Anal., Vol. 2013, Article ID 381715, 9 pages. [18] K.S. Kim, Invariant means and reversible semigroup of relatively nonexpansive mappings in Banach spaces, Abst. Appl. Anal., Vol. 2014, Article ID 694783, 9 pages. [19] K.S. Kim, Convergence of a hybrid algorithm for a reversible semigroup of nonlinear operators in Banach spaces, Nonlinear Analysis, 73 (2010), 3413–3419. [20] Y. Kimura and K. Satˆ o, Convergence of subsets of a complete geodesic space with curvature bounded above, Nonlinear Anal., 75 (2012), 5079–5085. [21] Y. Kimura and K. Satˆ o, Halpern iteration for strongly quasinonexpansive mappings on a geodesic space with curvature bounded above by one, Fixed Point Theory Appl., 2013(7) (2013).

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[22] W.A. Kirk, A fixed point theorem in CAT (0) spaces and R-trees, Fixed Point Theory Appl., 2004(4) (2004), 309–316. [23] W.A. Kirk and B. Panyanak, A concept of convergence in geodesic spaces, Nonlinear Anal., 68(12) (2008), 3689–3696. [24] U. Kohlenbach and L. Leustean, Mann iterates of directionally nonexpansive mappings in hyperbolic spaces, Abst. Appl. Anal., 2003(8) (2003), 449–477. [25] A.T-M. Lau, Semigroup of nonexpansive mappings on a Hilbert space, J. Math. Anal. Appl., 105(2) (1985), 514-522. [26] L. Leustean, A quadratic rate of asymptotic regularity for CAT (0)-spaces, J. Math. Anal. Appl., 325 (2007), 386–399. [27] W.R. Mann, Mean value methods in iteration, Proc. Amer. Math. Soc., 4 (1953), 506– 510. [28] S.B. Nadler, Multi-valued contraction mappings, Pacific J. Math., 30 (1969), 475–488. [29] B. Panyanak, On an open problem of Kyung Soo Kim, Fixed Point Theory and Appl., 2015:186 DOI:10.1186/s13663-015-0438-7. [30] E. Picard, Sur les groupes de transformation des e´quations diff´ erentielles lin´ eaires, Comptes Rendus Acad. Sci. Paris, 96 (1883), 1131–1134. [31] S. Saejung, Halpern’s iteration in CAT (0) spaces, Fixed Point Theory Appl., 2010, Article ID 471781, 13 pages. [32] W. Takahashi, A convexity in metric spaces and nonexpansive mappings, Kodai Math. Sem. Rep., 22 (1970), 142–149.

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Strong convergence theorems for the generalized viscosity implicit rules of asymptotically nonexpansive mappings in Hilbert spaces Qian Yan School of Mathematics Science, Chongqing Normal University, Chongqing 401331, China

Shaotao Hu∗ School of Marxism, Chongqing Normal University, Chongqing 401331, China

Abstract The purpose of this paper is to introduce the generalized viscosity implicit rules of one asymptotically nonexpansive mapping in Hilbert spaces. We obtain some strong convergence theorems under certain assumptions imposed on the parameters. We also apply our main results to solve mixed equilibrium problem in Hilbert spaces. A numerical example is also given to support our main results. The results obtained in this paper improve and extend many recent ones in this field. Keywords: Fixed point; Generalized implicit rule; Asymptotically nonexpansive mapping; Hilbert spaces. 2000 AMS Subject Classification: 47H09; 47H10.

1. Introduction Let C be a subset of real Hilbert space H. Let F (T ) be the set of fixed points of mapping T . We recall some basic definitions. A mapping f : C → C is called a strict contraction, if there exists a constant α ∈ (0, 1) such that kf (x) − f (y)k ≤ α kx − yk , ∀ x, y ∈ C.

(1.1)

A mapping T : C → C is called nonexpansive if kT x − T yk ≤ kx − yk , ∀ x, y ∈ C.

(1.2)

A mapping T : C → C is called asymptotically nonexpansive if there exists a sequence {θn } ⊂ [0, +∞) with limn→∞ θn = 0 such that kT n x − T n yk ≤ (1 + θn ) kx − yk , ∀ n ≥ 0, x, y ∈ C.

(1.3)

It is easy to see that asymptotically nonexpansive mapping contains strict contraction, nonexpansive mapping as a special case. A mapping A : C → H is called monotone if hAx − Ay, x − yi ≥ 0, ∀x, y ∈ C.

(1.4)

∗ Corresponding

author. Email addresses: [email protected] (Qian Yan), [email protected] (Shaotao Hu∗ ) 1 This work was supported by the NSF of China (No. 11401063), the Natural Science Foundation of Chongqing (cstc2014jcyjA00016), Science and Technology Project of Chongqing Education Committee (Grant No. KJ1500314) and the graduate students’ innovative research project of Chongqing normal University (YKC16001)

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A mapping A : C → H is called α-inverse strongly monotone if there exists a positive real number α such that 2 hAx − Ay, x − yi ≥ α kAx − Ayk , ∀x, y ∈ C. (1.5) Recently, viscosity iterative algorithms for finding a common element of the set of fixed point of nonexpansive mappings, the set of solution of variational inequality problem and mixed equilibrium problems have been investigated extensively by many authors, see [1-15] and the references therein. For instance, Moudafi[1] introduced the viscosity technique for nonexpansive mappings in Hilbert spaces. Xu [2] refined the main results of [1] in Hilbert spaces and extended them to more general uniformly smooth spaces. Precisely, he proved that the suggested viscosity iterative sequence converges strongly to a fixed point of one nonexpansive mapping, which also solves some variational inequality. Very recently, the implicit midpoint rule has become a powerful methods for solving ordinary differential equations; see [16-22] and the references therein. Xu et al. [20] considered the following viscosity implicit midpoint rule: xn + xn+1 ), n ≥ 0. (1.6) xn+1 = αn f (xn ) + (1 − αn )T ( 2 By using contractions to regularize the implicit midpoint rule for nonexpansive mappings, they proved that the iterative sequence defined by (1.6) converges in norm to a fixed point of T , which also solves the variational inequality: h(I − f )q, x − qi ≥ 0, x ∈ F (T ). (1.7) On the other hand, many authors studied the Mann and Ishikawa iterations processes for asymptotically nonexpansive mapping in Hilbert spaces or Banach spaces, see [23-30] and the references therein. For example, Lou et al.[24] investigated some iterative algorithms for asymptotically nonexpansive mapping on a uniformly convex Banach space with uniformly Gˆ ateaux differentiable norm. In this paper, we introduce a viscosity implicit rules for an asymptotically nonexpansive mapping in Hilbert spaces. Under suitable assumptions imposed on the parameters, we obtain some strong convergence theorems for finding a fixed point of an asymptotically nonexpansive mapping. We also apply our main results to solve mixed equilibrium problem in Hilbert spaces. 2. Preliminaries Let C be a nonempty closed convex subset of H. For all x ∈ H, there exists a unique nearest point in C, denoted by PC x, such that kx − PC xk ≤ kx − yk for all y ∈ C. (2.1) In this case, P is called a metric projection of H onto C. It is well known that PC is a nonexpansive mapping of H onto C and satisfies 2

hx − y, PC x − PC yi ≥ kPC x − PC yk , ∀ x, y ∈ H.

(2.2)

Furthermore, PC x is characterized by the following properties: PC x ∈ C and hx − PC x, y − PC xi ≤ 0, 2

2

2

kx − yk ≥ kx − PC xk + ky − PC xk , ∀ x ∈ H, y ∈ C.

(2.3) (2.4)

We need the following lemmas for proving our main results. Lemma 2.1 ([2]). Assume {an } is a sequence of nonnegative real numbers such that an+1 ≤ (1 − αn )an + δn , n ≥ 0, where {α Pn∞} is a sequence in (0, 1) and {δn } is a sequence in R such that (i) n=0 αn = ∞, P∞ (ii) either lim supn→∞ αδnn ≤ 0 or n=1 |δn | < ∞. Then limn→∞ an = 0. Lemma 2.2 ([21]). Let T be an asymptotically nonexpansive mapping defined on a nonempty bounded closed convex subset C of a Hilbert space H. If {xn } is a sequence in C such that xn * z and T xn − xn → 0, then z ∈ F (T ).

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3. Main results Theorem 3.1. Let H be a Hilbert space, C a nonempty closed convex subset of H. Let T : C → C be an asymptotically nonexpansive mapping with a sequence {θn } such that F (T ) 6= ∅ and f : C → C a strict contraction with coefficient α ∈ [0, 1). Pick any x0 ∈ C. Let {xn } be a sequence generated by xn+1 = αn f (xn ) + (1 − αn )T n (

xn + xn+1 ), 2

(3.1)

where {αn } is a real sequence P∞ in [0, 1] satisfying the following conditions: (i) limn→∞ αn = 0, n=0 αn = ∞; (ii) limn→∞ αθnn = 0; P∞ (iii) n=1 |αn+1 − αn | < ∞;

P∞ 0 (iv) n=0 supx∈C 0 T n+1 x − T n x < ∞, where C is a closed convex subset of C that contains sequence {xn }. Then {xn } converges strongly to a fixed point q of the asymptotically nonexpansive mapping T , which is also the solution of the variational inequality h(I − f )q, y − qi ≥ 0, for all y ∈ F (T ). Proof. First, we show that {xn } is bounded. Indeed, take p ∈ F (T ) arbitrarily, since limn→∞ αθnn = 0, then there exists N ∈ N such that for all n ≥ N, αθnn ≤ 1−α 2 . Choose a constant M1 > 0 sufficiently large such that kxN − pk ≤ M1 , kf (p) − pk ≤

1−α M1 . 2

We proceed by induction to show that kxn − pk ≤ M1 , ∀n ≥ 1. Assume kxn − pk ≤ M1 , for some n ≥ N . We show that kxn+1 − pk ≤ M1 . We observe kxn+1 − pk xn + xn+1 ) − pk 2 xn + xn+1 = kαn (f (xn ) − p) + (1 − αn )[T n ( ) − p]k 2

= kαn f (xn ) + (1 − αn )T n (

xn + xn+1 ) − pk 2 xn + xn+1 ≤ ααn kxn − pk + αn kf (p) − pk + (1 − αn )(θn + 1)k − pk 2 (1 − αn )(θn + 1) (1 − αn )(θn + 1) ≤ ααn kxn − pk + αn kf (p) − pk + kxn − pk + kxn+1 − pk 2 2 ≤ αn kf (xn ) − f (p)k + αn kf (p) − pk + (1 − αn )kT n (

It follows that 1 − αn + 2αn α + (1 − αn )θn αn kxn − pk + kf (p) − pk 1 + αn − (1 − αn )θn 1 + αn − (1 − αn )θn 2αn (1 − α) − 2(1 − αn )θn αn = [1 − ]kxn − pk + kf (p) − pk 1 + αn − (1 − αn )θn 1 + αn − (1 − αn )θn αn (1 − α) αn (1 − α) kf (p) − pk ≤ [1 − ]kxn − pk + 1 + αn − (1 − αn )θn 1 + αn − (1 − αn )θn 1 − α kf (p) − pk ≤ max{kxn − pk, } 1−α ≤ M1 .

kxn+1 − pk ≤

(3.2)

This implies that {xn } is bounded. 488

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Next, we prove that limn→∞ kxn+1 − xn k = 0. It follows from (3.1) that kxn+1 − xn k xn−1 + xn xn + xn+1 ) − αn−1 f (xn−1 ) − (1 − αn−1 )T n−1 ( )k 2 2 xn + xn+1 xn−1 + xn = kαn (f (xn ) − f (xn−1 )) + (αn − αn−1 )f (xn−1 ) + (1 − αn )[T n ( ) − T n( )] 2 2 xn−1 + xn xn−1 + xn + (1 − αn )T n ( ) − (1 − αn−1 )T n ( ) 2 2 xn−1 + xn xn−1 + xn ) − T n−1 ( )]k + (1 − αn−1 )[T n ( 2 2 xn + xn+1 xn−1 + xn = kαn (f (xn ) − f (xn−1 )) + (1 − αn )[T n ( ) − T n( )] + (αn − αn−1 ) 2 2 xn−1 + xn xn−1 + xn xn−1 + xn · [f (xn−1 ) − T n ( )] + (1 − αn−1 )[T n ( ) − T n−1 ( )]k 2 2 2 kxn+1 − xn k kxn − xn−1 k + ) ≤ ααn kxn − xn−1 k + (1 − αn )(θn + 1)( 2 2 xn−1 + xn + |αn − αn−1 | kf (xn−1 ) − T n ( )k + sup kT n x − T n−1 xk 2 x∈C 0 = kαn f (xn ) + (1 − αn )T n (

=

2ααn + (1 − αn )(θn + 1) (1 − αn )(θn + 1) kxn − xn−1 k + kxn+1 − xn k 2 2 + |αn − αn−1 | M2 + sup kT n x − T n−1 xk, x∈C 0

where M2 is a constant such that M2 = sup kf (xn−1 ) − T n ( n≥0

xn−1 + xn )k. 2

It follows that 2ααn + (1 − αn )(θn + 1) 2 − (1 − αn )(θn + 1) kxn+1 − xn k ≤ kxn − xn−1 k + |αn − αn−1 | M2 2 2 + sup kT n x − T n−1 xk. x∈C 0

This implies kxn+1 − xn k 2ααn + (1 − αn )(θn + 1) 2M2 kxn − xn−1 k + · 2 − (1 − αn )(θn + 1) 2 − (1 − αn )(θn + 1) 2 sup kT n x − T n−1 xk |αn − αn−1 | + 2 − (1 − αn )(θn + 1) x∈C 0   2[1 − ααn − (1 − αn )(θn + 1)] 2M1 = 1− kxn − xn−1 k + · 2 − (1 − αn )(θn + 1) 2 − (1 − αn )(θn + 1) 2 |αn − αn−1 | + sup kT n x − T n−1 xk. 2 − (1 − αn )(θn + 1) x∈C 0 ≤

(3.3)

Let γn =

2[1 − ααn − (1 − αn )(θn + 1)] . 2 − (1 − αn )(θn + 1)

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We note 2[αn (1 − α) + θn (αn − 1)] 1 − θn + αn (θn + 1) 2[αn (1 − α) + θn (αn − 1)] ≥ 1 − θn + (θn + 1) = αn (1 − α) + θn (αn − 1) 1−α ≥ αn (1 − α) − θn ≥ αn . 2 P∞ By condition (i), we have n=0 γn = ∞. Apply Lemma 2.1 to (3.3), we get γn =

lim kxn+1 − xn k = 0.

(3.4)

n→∞

Next, we prove that limn→∞ kxn − T xn k = 0. In fact, we have kxn+1 − T n (

xn + xn+1 xn + xn+1 )k = αn kf (xn ) − T n ( )k 2 2 → 0 as n → ∞.

(3.5)

Moreover, we get kxn − T n xn k xn + xn+1 xn + xn+1 ) + T n( ) − T n xn k 2 2 xn + xn+1 xn + xn+1 ≤ kxn+1 − xn k + kxn+1 − T n ( )k + kT n ( ) − T n xn k 2 2 xn + xn+1 θn + 1 ≤ kxn+1 − xn k + kxn+1 − T n ( )k + kxn+1 − xn k 2 2 xn + xn+1 θn + 3 kxn+1 − xn k + kxn+1 − T n ( )k. = 2 2 = kxn − xn+1 + xn+1 − T n (

Combining (3.4) and (3.5), we can obtain lim kxn − T n xn k = 0.

(3.6)

n→∞

We notice kxn − T xn k = kxn − T n xn + T n xn − T n+1 xn + T n+1 xn − T xn k ≤ kxn − T n xn k + kT n xn − T n+1 xn k + (1 + θ1 )kT n xn − xn k ≤ kxn − T n xn k + sup kT n x − T n+1 xk + (1 + θ1 )kT n xn − xn k. x∈C 0

By condition (iv) and (3.6), we have lim kxn − T xn k = 0.

(3.7)

n→∞

Next, we claim that lim sup hq − f (q), q − xn i ≤ 0,

(3.8)

n→∞

where q = PF (T ) f (q). Indeed, there exists a subsequence {xni } of {xn } such that lim sup hq − f (q), q − xn i = lim hq − f (q), q − xni i . n→∞

i→∞

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Since {xn } is bounded, there exists a subsequence of {xn } which converges weakly to p. Without loss of generality, we may assume that xni * p. From (3.7) and Lemma 2.2, we have p ∈ F (T ). This together with the property of the metric projection implies that lim sup hq − f (q), q − xn i = lim hq − f (q), q − xni i = hq − f (q), q − pi ≤ 0. n→∞

i→∞

Then (3.8) holds. Finally, we show that xn → q as n → ∞. In fact, we have kxn+1 − qk2 xn + xn+1 ) − q, xn+1 − qi 2 xn + xn+1 = hαn (f (xn ) − q) + (1 − αn )(T n ( ) − q), xn+1 − qi 2

= hαn f (xn ) + (1 − αn )T n (

= αn hf (xn ) − f (q), xn+1 − qi + αn hf (q) − q, xn+1 − qi + (1 − αn )hT n ( ≤ ααn kxn − qk · kxn+1 − qk + (1 − αn )(θn + 1)k

xn + xn+1 ) − q, xn+1 − qi 2

xn − q xn+1 − q + k · kxn+1 − qk 2 2

+ αn hf (q) − q, xn+1 − qi ≤

ααn ααn (1 − αn )(θn + 1) kxn − qk2 + kxn+1 − qk2 + kxn − qk2 2 2 4 (1 − αn )(θn + 1) (1 − αn )(θn + 1) + kxn+1 − qk2 + kxn+1 − qk2 + αn hf (q) − q, xn+1 − qi, 4 2

which implies 4 − 2ααn − 3(1 − αn )(θn + 1) kxn+1 − qk2 4 2ααn + (1 − αn )(θn + 1) kxn − qk2 + αn hf (q) − q, xn+1 − qi. ≤ 4 That is kxn+1 − qk2 2ααn + (1 − αn )(θn + 1) 4αn kxn − qk2 + hf (q) − q, xn+1 − qi 4 − 2ααn − 3(1 − αn )(θn + 1) 4 − 2ααn − 3(1 − αn )(θn + 1) 4(αn θn + αn − ααn − θn ) 4αn = [1 − ]kxn − qk2 + · 4 − 2ααn − 3(1 − αn )(θn + 1) 4 − 2ααn − 3(1 − αn )(θn + 1) hf (q) − q, xn+1 − qi. (3.9)

≤

Put γn =

4(αn θn + αn − ααn − θn ) . 4 − 2ααn − 3(1 − αn )(θn + 1)

We have 4[θn (αn − 1) + αn (1 − α)] 1 − 2ααn + 3θn (αn − 1) + 3αn 4[θn (αn − 1) + αn (1 − α)] ≥ 1 + 3αn ≥ θn (αn − 1) + αn (1 − α) 1−α ≥ αn (1 − α) − θn ≥ αn . 2

γn =

Apply Lemma 2.1 to (3.9), we obtain xn → q as n → ∞. This completes the proof. 491

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Since nonexpansive mapping is asymptotically nonexpansive, so we obtain the main results of [20]. Theorem 3.2. Let H be a Hilbert space, C a nonempty closed convex subset of H. Let T : C → C be a nonexpansive mapping with such that F (T ) 6= ∅ and f : C → C a strict contraction with coefficient α ∈ [0, 1). Pick any x0 ∈ C. Let {xn } be a sequence generated by xn+1 = αn f (xn ) + (1 − αn )T (

xn + xn+1 ), 2

(3.10)

where {αn } is a real sequence P∞ in [0, 1] satisfying the following conditions: (i) lim α = 0, n→∞ n n=0 αn = ∞; P∞ n = 0. (iii) n=1 |αn+1 − αn | < ∞ or limn→∞ ααn−1 Then {xn } converges strongly to a fixed point q of nonexpansive mapping T , which is also the solution of the variational inequality h(I − f )q, y − qi ≥ 0, for all y ∈ F (T ). Now we give an example that one mapping satisfies condition (iv) in Theorem 3.1. Example 3.3. Let T : C → C be a strict contraction with a constant β ∈ (0, 1) and let C 0 be a bounded subset of C. Then

n+1

T x − T n x ≤ β n kT x − xk ≤ β n K1 , ∀ x ∈ C 0 , where K1 is a constant such that K1 = supx∈C 0 (kxk + kT xk). It follows that ∞

X

βK1 β n K1 = < ∞. sup T n+1 x − T n x ≤ 0 1 −β n=1 n=1 x∈C ∞ X

Example 3.4. Let C be a nonempty closed convex subset of a Banach space. Define mapping T : C → C as T n x = (1 + n1 )x for any x ∈ C. It is easy to see that T is asymptotically nonexpansive mapping in the intermediate sense. Let {xn } be a bounded sequence in C, we observe

n+1

T xn − T n xn =

1 1 1 kxn k ≤ 2 kxn k ≤ 2 K2 , n(n + 1) n n

where K2 is a constant such that K2 = supn≥1 kxn k. Hence we obtain ∞ ∞ X

n+1

X 1

T xn − T n xn ≤ K < ∞. 2 2 n n=1 n=1

4. Applications In this section, we apply our main results to solve mixed equilibrium problems. Let ϕ : C → R be a real-valued function and F : C × C → R be a bifunction. The mixed equilibrium problem is to find x ∈ C such that F (x, y) + ϕ(y) − ϕ(x) ≥ 0, ∀ y ∈ C.

(4.1)

The set of solutions of (1.1) is denoted by M EP (F, ϕ). If ϕ = 0, then problem (4.1) reduces to equilibrium problem which is to find x ∈ C such that F (x, y) ≥ 0, ∀ y ∈ C.

(4.2)

We denote the set of solutions of (4.2) by EP (F ). For solving the mixed equilibrium problem, let us give the following assumptions for the bifunction F, ϕ and the set C([13]): (A1) F (x, x) = 0 for all x ∈ C; 492

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(A2) F is monotone, i.e., F (x, y) + F (y, x) ≤ 0 for any x, y ∈ C; (A3) for each y ∈ C, x 7→ F (x, y) is weakly upper semicontinuous; (A4) for each x ∈ C, y 7→ F (x, y) is convex; (A5) for each x ∈ C, y 7→ F (x, y) is lower semicontinuous; (B1) for each x ∈ H and r > 0, there exists a bounded subset Dx ⊆ C and yx ∈ C such that for any z ∈ C\Dx , F (z, yx ) + ϕ(yx ) +

1 hyx − z, z − xi < ϕ(z); r

(B2) C is bounded set. Lemma 4.1 ([13]). Let C be a nonempty closed convex subset of H. Let F be a bifunction from C × C to R satisfying (A1)-(A5) and let ϕ : C → R ∪ {+∞} be a proper lower semicontinuous and convex function. (F,ϕ) Assume that either (B1) or (B2) holds. For r > 0 and x ∈ H, define a mapping Tr : H → C as follows.   1 Tr(F,ϕ) (x) = z ∈ C : F (z, y) + ϕ(y) + hy − z, z − xi ≥ ϕ(z), ∀ y ∈ C r for all x ∈ H. Then the following conclusions hold: (F,ϕ) (x) 6= ∅; (1) for each x ∈ H, Tr (F,ϕ) is single-valued; (2) Tr (F,ϕ) is firmly nonexpansive, i.e., for any x, y ∈ H, (3) Tr

2 D E

(F,ϕ)

(x) − TrF,ϕ (y) ≤ Tr(F,ϕ) (x) − Tr(F,ϕ) (y), x − y ;

Tr (F,ϕ)

) = MEP(F, ϕ); (4) F (Tr (5) MEP(F, ϕ) is closed and convex. Theorem 4.1. Let H be a Hilbert space, C a nonempty closed convex subset of H. Let F be a bifunction from C × C to R satisfying (A1)-(A5), ϕ : C → R ∪ {+∞} be a proper lower semicontinuous and convex function. Assume that either (B1) or (B2) holds. Let f : C → C be a strict contraction with coefficient α ∈ [0, 1). Pick any x0 ∈ C. Let {xn } be a sequence generated by   xn+1 = αn f (xn ) + (1 − αn )zn , zn ∈ C such that F (zn , y) + ϕ(y) + 1r hy − zn , zn − un i ≥ ϕ(zn ), ∀ r > 0, y ∈ C, (4.3)  n+1 un = xn +x , 2 where {αn } is a real sequence P∞ in [0, 1] satisfying the following conditions: (i) lim α = 0, n n=0 αn = ∞; Pn→∞ ∞ n (iii) n=1 |αn+1 − αn | < ∞ or limn→∞ ααn−1 = 0. Then {xn } converges strongly to an element of mixed equilibrium problem (4.1), which is also the solution of the variational inequality h(I − f )q, y − qi ≥ 0, for all y ∈ MEP(F, ϕ). Proof. We can rewrite (4.3) as follows: xn+1 = αn f (xn ) + (1 − αn )Tr(F,ϕ) (

xn + xn+1 ). 2

Then we obtain the desired results by Theorem 3.2 easily.

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5. Numerical Examples Example 5.1. Let inner product < ·, · >: R3 × R3 → R be defined by hx, yi = x · y = x1 · y1 + x2 · y2 + x3 · y3 and the usual norm k·k : R3 → R be defined by q kxk = x21 + y12 + z12 , ∀ x = (x1 , x2 , x3 ), y = (y1 , y2 , y3 ) ∈ R3 . Let T, f : R3 → R3 be defined by Tx =

1 1 (x), f (x) = x, ∀ x ∈ R. 3 4

1 Let αn = 5n , ∀ n ∈ N and let {xn } be a sequence generated by (3.10). It is easy to see that F (T ) = {0}. Then {xn } converges strongly to 0 by Theorem 3.2. We can rewrite (3.10) as follows: 10n + 1 xn . (5.1) xn+1 = 50n + 2

Choosing x1 = (1, 3, 5) in (5.1), we have the following numerical results in Figure 1 and Figure 2.

Figure 1:

Figure 2:

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HERMITE HADAMARD TYPE INEQUALITIES FOR m-CONVEX AND (α, m)-CONVEX FUNCTIONS FOR FUZZY INTEGRALS M. A. LATIF 1 , WAJEEHA IRSHAD 2 , AND M. MUSHTAQ

3

Abstract. In this paper we prove Hermite–Hadamard type inequalities for m-convex and (α, m)-convex functions for fuzzy integrals. Some examples are also given to illustrate the results.

1. Main Results Let [0, b], where b > 0, be an interval of the real line R. A function f is said to be convex on [0, b] if f (tx + (1 − t)y) ≤ tf (x) + (1 − t)f (y), holds for all x, y ∈ [0, b] and t ∈ [0, 1] and a function f is starshaped with respect to the origin on [0, b] if f (tx) ≤ tf (x), holds for all x ∈ [0, b] and t ∈ [0, 1] . In [26] G. Toader, (see also [1, 2, 4]) defined m-convexity: another intermediate between the usual convexity and starshaped convexity as follow: Definition 1. The function f : [0, b] → R, b > 0, is said to be m-convex, where m ∈ [0, 1] , if we have f (tx + m(1 − t)y) ≤ tf (x) + m(1 − t)f (y) for all x, y ∈ [0, b] and t ∈ [0, 1] . We say that f is m-concave if −f is m-convex. The class of all m-convex functions on [0, b] for which f (0) ≤ 0 is denoted by Km (b). Obviously, for m = 1, m-convexity is the standard convexity of functions on [0, b] , and for m = 0 the concept of starshaped functions. The following lemmas hold (see [26] see also [1, Lemma A & Lemma B, Page 2]). Lemma 1. [1, Lemma A, Page 2] If f is in the class Km (b) , then it is starshaped. Lemma 2. [1, Lemma B, Page 2] If f is in the class Km (b) and 0 < n < m ≤ 1, then f is in the class Kn (b). From Lemma 2 and Lemma 3 it follows that K1 (b) ⊂ Km (b) ⊂ K0 (b) whenever m ∈ (0, 1) . Note that in the class K1 (b) are only convex functions f : [0, b] → R for which f (0) ≤ 0, that is, K1 (b) is a proper subclass of the class of Date: June, 23, 2016. Key words and phrases. Hermite–Hadamard inequality, Sugeno integral, m-convex function, (α, m)-convex function. This paper is in final form and no version of it will be submitted for publication elsewhere. 1

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convex functions on [0, b]. It is interesting to point out that for any m ∈ (0, 1) there are continuous and differentiable functions which are m-convex, but which are not convex in the standard sense (see [27]). The notion of m-convexity was further generalized in the following definition (see also [1, 2]). Definition 2. [11] The function f : [0, b] → R, b > 0, is said to be (α, m)-convex, where (α, m) ∈ [0, 1]2 , if we have f (tx + m(1 − t)y) ≤ tα f (x) + m(1 − tα )f (y) for all x, y ∈ [0, b] and t ∈ [0, 1]. The class of all (α, m)-convex functions on [0, b] for which f (0) ≤ 0 is denoted α by Km (b). If we take (α, m) = (1, m) , it can be easily seen that (α, m)-convexity reduces to m-convexity and for (α, m) = (1, 1) , (α, m)-convexity reduces to the concept of usual convexity defined on [0, b], b > 0. For further results on inequalities related to m-convex and (α, m)-convex functions we refer the readers to [1, 2, 4]. In [4], S.S. Dragomir and G. Toader proved the following Hadamard type inequality for m-convex functions: Theorem 1. Let f : [0, ∞) → R be an m-convex function with m ∈ (0, 1] . If 0 ≤ a < b < ∞ and f ∈ L1 ([a, b]) then (

) Z b a b f (b) + mf m f (a) + mf m 1 , f (x)dx ≤ min (2.1) b−a a 2 2 We will see that this inequalitiy does not valid for fuzzy integrals in general. To prove our assertion we consider the function f : [0, ∞) → [0, ∞), f (x) = axn , n ∈ N, n ≥ 2, a ≥ 0, then f is m-convex on [0, ∞), m ∈ (0, 1]. Example 1. Take X = [0, 1] and let µ be the usual Lebesgue measure on X. Let 2 9 f : [0, ∞) → [0, ∞) be defined as f (x) = x3 with m = 10 . Now to calculate the R 1 x2 Sugeno integral 0 3 dµ, consider the distribution function F associated to f on [0, 1] then   2  x F (α) = µ ([0, 1] ∩ {f ≥ α}) = µ [0, 1] ∩ ≥α 3  n o √ √ = µ [0, 1] ∩ x ≥ 3α = 1 − 3α √ and we solve the equation 1 − 3α = α. It can be easily seen that the solution of √ this equation is 52 − 12 21, therefore by Remark 1, we have that Z 1 2 x 5 1√ dµ = − 21 ≈ 0.208 71. 2 2 0 3 Now f (a) + mf 2

b m


=

498

5 ≈ 0.1851852 27

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and on the other hand f (b) + mf 2

a m


=

1 ≈ 0.16666667. 6

Therefore ( min

f (a) + mf 2

b m




f (b) + mf , 2

a m


) =

1 ≈ 0.16666667. 6

Which follows that (2.1) is not satisfied in the fuzzy context. Now we prove Hdadamard type inequalites like (2.1) but for Sugeno Integral or fuzzy integral. Theorem 2. Let g : [0, ∞) → [0, ∞) be an
m-convex function with m ∈ (0, 1] 1 . Let µ be the Lebesgue measure on such that mg(0) < g(1) and g(0) < mg m [0, 1] ⊂ [0, ∞), then ) (
Z 1 1 mg m g (1)
, . (2.2) gdµ ≤ min 1, 1 1 + mg m − g(0) 1 + g (1) − mg(0) 0 Proof. Since g is an m-convex function, therefore for x ∈ [0, 1] and m ∈ (0, 1], we have   1 g(x) = g ((1 − x) 0 + 1 · x) ≤ (1 − x) g(0) + mxg = h(x) m and hence by (3) of Proposition 1,   Z 1 Z 1 Z 1 1 gdµ ≤ (1 − x) g(0) + mxg dµ = h(x)dµ. m 0 0 0 Let F be the distribution function associated to h on [0, 1], then      1 ≥α F (α) = µ ([0, 1] ∩ {h ≥ α}) = µ [0, 1] ∩ (1 − x) g(0) + mxg m )! ( α − g(0)
= µ [0, 1] ∩ x ≥ 1 mg m − g(0) =1−

mg

α − g(0)
1 m − g(0)

and as a solution of the equation α = 1 −

α−g(0) , 1 mg ( m )−g(0)

we get


1 mg m
. α= 1 1 + mg m − g(0)

(2.3)

Analogously by the m-convexity of g, we also have g(x) = g ((1 − x) 0 + 1 · x) ≤ m (1 − x) g(0) + xg(1) = h1 (x). Arguing smilarly, let F1 be the distribution function accociated to h1 on [0, 1], then (2.4)

α=

g (1) . 1 + g (1) − mg(0)

By (1) of Proposition 1, we have that Z 1 Z 1 (2.5) h(x)dµ = h1 (x)dµ ≤ µ ([0, 1]) = 1. 0

0

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The equations (2.3), (2.4), (2.5) and the definition of Sugeno integral give us the desired inequality.  A similar results may be stated as follow, however, we leave the details for the intrested readers. Proposition 1. Let g : [0, ∞) → [0, ∞) be
an m-convex function with m ∈ (0, 1] 1 . Let µ be the Lebesgue measure on such that mg(0) > g(1) and g(0) > mg m [0, 1] ⊂ [0, ∞), then ( ) Z 1 g (0) mg (0)
gdµ ≤ min 1, (2.6) , . 1 1 − mg m + g(0) 1 − g (1) + mg(0) 0 Remark 1. If m = 1, then the inequalities (2.2) and (2.6) become those inequalities proved in Theroem 1 and Theorem 2 from [3, p. 3]. Now we give general cases of Theroem 2 and Theorem 3. Theorem 3. Let f : [0, ∞) → [0, ∞) be
an m-convex function with m ∈ (0, 1] such b a ) < g(b) and g(a) < mg m . Let µ be the Lebesgue measure on [a, b] that mg( m and 0 ≤ a < b < ∞. Then ) (
Z b b (b − a) mg m (b − a) g (b)
.
, (2.7) gdµ ≤ min 1, a b b − a + mg m − g(a) b − a + g(b) − mg m a Proof. Since g is an m-convex function m ∈ (0, 1], therefore for x ∈ [a, b], 0 ≤ a < b < ∞, we have    x−a x−a g(x) = g 1− ·b a+ b−a b−a       b−x x−a b ≤ g(a) + m g = h(x) b−a b−a m By (3) of Proposition 1, we have      Z b Z b  Z b b−x x−a b gdµ ≤ g(a) + m g dµ = h(x)dµ. b−a b−a m a a a Let us consider the distribution function F given by F (α) = µ ([a, b] ∩ {h ≥ α})         b−x x−a b = µ [a, b] ∩ g(a) + m g ≥α b−a b−a m ( )!
b α (b − a) + mag m − bg(a)
= µ [a, b] ∩ x ≥ b mg m − g(a)
b α (b − a) + mag m − bg(a)
=b− b mg m − g(a) and as solution of the equation b − (2.8)

α=

b α(b−a)+mag ( m )−bg(a) b mg ( m )−g(a)

= α, we get that


b

(b − a)
. b b − a + mg m − g(a) mg

m

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Analogously by the m-convexity of g, we also have    x−a x−a g(x) = g 1− a+ ·b b−a b−a     b−x a x−a ≤m g( ) + g (b) = h1 (x). b−a m b−a Arguing smilarly, let F1 be the distribution function accociated to h1 on [a, b], then (2.9)

α=

(b − a) g (b) b − a + g(b) − mg

a m


.

Moreover by (1) of Proposition 1, we have Z b Z b (2.10) h(x)dµ = h1 (x)dµ ≤ µ ([a, b]) = b − a. a

a

From (2.8), (2.9), (2.10) and by the definition of fuzzy integral, we obtain (2.7). This completes the proof of the Teorem.  Again, we state smilar results like the one proved in Theorem 4, however, the details are left to the intrested readers. Proposition 2. Let f : [0, ∞) → [0, ∞) be an
m-convex function with m ∈ (0, 1] a b such that mg( m ) > g(b) and g(a) > mg m . Let µ be the Lebesgue measure on [a, b] and 0 ≤ a < b < ∞. Then ( )
Z b a m (b − a) g m (b − a) g (a)

, (2.11) gdµ ≤ min 1, . a b b − a + mg m − g(b) b − a + g(a) − mg m a Remark 2. If m = 1, then the inequalities (2.7) and (2.11) become those inequalities proved in Theorem 3 from [3, p. 4]. Example 2. Take X = [0, 1] and let µ be the usual Lebesgue measure on X. Let 2 f : [0, ∞) → [0, ∞) be defined as f (x) = x
, then f is an m-convex function on 1 [0, 1] with mg(0) < g(1) and g(0) < mg m , m ∈ (0, 1]. Now
1 mg m 1
= 1 m+1 1 + mg m − g(0) and g (1) 1 = 1 + g (1) − mg(0) 2 Therefore by Theorem 2, we have Z 1 1 x2 dµ ≤ . 2 0 Now we give our results for (α, m)-convex functions Theorem 4. Let g : [0, ∞) → [0, ∞) be an (α,
m)-convex function with α, m ∈ 1 (0, 1]2 such that mg(0) < g(1) and g(0) < mg m . Let µ be the Lebesgue measure on [0, 1], then Z 1 (2.12) gdµ ≤ min {1, α1 , α2 } , 0

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  α1 0 α −g(0) where α1 and α2 are positive real solutions of the equations α = 1− mg 1 −g(0) (m)  0  α1 0 α −g(0) and α = 1 − g(1)−mg(0) respectively. 0

Proof. Since g is an (α, m)-convex function, therefore for x ∈ [a, b] and α, m ∈ (0, 1]2 , we have   1 α α g(x) = g ((1 − x) 0 + 1 · x) ≤ (1 − x ) g(0) + mx g = h(x) m and hence by (3) of Proposition 1,   Z 1 Z 1 Z 1 1 α α gdµ ≤ (1 − x ) g(0) + mx g dµ = h(x)dµ. m 0 0 0 Let F be the distribution function associated to h on [0, 1], then       n o 0 0 0 1 ≥α F (α ) = µ [0, 1] ∩ h ≥ α = µ [0, 1] ∩ (1 − xα ) g(0) + mxα g m    1 ! 0 α  α − g(0) 
= µ [0, 1] ∩ x ≥ 1   − g(0) mg m 0

=1−

α − g(0)
1 mg m − g(0)

! α1

and hence we get the equation 0

α − g(0)
1 mg m − g(0)

0

α =1−

(2.13)

! α1 .

Analogously by the (α, m)-convexity of g, we also have g(x) = g ((1 − x) 0 + 1 · x) ≤ m (1 − xα ) g(0) + xα g(1) = h1 (x). Arguing smilarly, let F1 be the distribution function accociated to h1 on [0, 1], then we have that the following equation: ! α1 0 0 α − g(0) (2.14) α =1− . g (1) − mg(0) By (1) of Proposition 1, we have that Z 1 Z 1 (2.15) h(x)dµ = h1 (x)dµ ≤ µ ([0, 1]) = 1. 0

0

The equations (2.13), (2.14), (2.15) and the definition of Sugeno integral give us the required inequality.  A similar result can be stated as follow, however, the details are left to the intrested reasers:

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Proposition 3. Let g : [0, ∞) → [0, ∞) be an (α, m)-convex function with α,
1 m ∈ (0, 1]2 such that mg(0) > g(1) and g(0) > mg m . Let µ be the Lebesgue measure on [0, 1] ⊂ [0, ∞), then Z

1

gdµ ≤ min {1, α1 , α2 } ,

(2.16) 0

0

where α1 and α2 are positive real solutions of the equations α =   α1 0 0 g(0)−α and α = mg(0)−g(1) respectively.



0

g(0)−α 1 g(0)−mg ( m )

 α1

Remark 3. If (α, m) = (1, 1), then the inequalities (2.12) and (2.16) become those inequalities proved in Theroem 1 and Theorem 2 from [3, p.4]. Now in following results we give the general case of the last two results. Theorem 5. Let g : [0, ∞) → [0, ∞) be an (α, m)-convex function with α, m ∈
a b (0, 1]2 such that mg( m ) < g(b) and g(a) < mg m . Let µ be the Lebesgue measure on [a, b], 0 ≤ a < b < ∞, then Z

1

gdµ ≤ min {1, α1 , α2 } ,

(2.17) 0

" 0



where α1 and α2 are positive real solutions of the equations α = (b − a) 1 − "  0  α1 # a 0 α −g ( m ) and α = (b − a) 1 − g(b)−mg a respectively. (m)

0

α −g(a) b mg ( m )−g(a)

 α1 #

Proof. Since g is an (α, m)-convex function, therefore for x ∈ [0, 1] and α, m ∈ (0, 1]2 , we have    x−a x−a g(x) = g 1− a+ ·b b−a b−a   α   α   x−a x−a b ≤ 1− g(a) + m g = h(x) b−a b−a m and hence by (3) of Proposition 1, Z

1

Z gdµ ≤

0

0

1

  α   α   Z 1 x−a x−a b 1− g(a) + m g dµ = h(x)dµ. b−a b−a m 0

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Let F be the distribution function associated to h on [0, 1], then  n o 0 0 F (α ) = µ [a, b] ∩ h ≥ α    α   α    0 b x−a x−a g g(a) + m ≥α = µ [a, b] ∩ 1− b−a b−a m    1 ! 0 α  α − g(a) 
= µ [a, b] ∩ x ≥ a + (b − a) b   − g(a) mg m  ! α1  0 α − g(a) 
= (b − a) 1 − b mg m − g(a) and hence we get the equation  (2.18)

0

α = (b − a) 1 −

0

α − g(a)
b mg m − g(a)

! α1  .

Analogously by the (α, m)-convexity of g, we also have    x−a x−a g(x) = g 1− ·b a+ b−a b−a   α     α x−a a x−a ≤m 1− g g(b) = h1 (x). + b−a m b−a Arguing smilarly, let F1 be the distribution function accociated to h1 on [0, 1], then we have that the following equation: 
! α1  0 a α −g m 0 .
(2.19) α = (b − a) 1 − a g (b) − mg m By (1) of Proposition 1, we have that Z 1 Z 1 (2.20) h(x)dµ = h1 (x)dµ ≤ µ ([a, b]) = b − a. 0

0

The equations (2.18), (2.19), (2.20) and the definition of Sugeno integral give us the required inequality.  A similar resutl is stated below, however, the details are left: Theorem 6. Let g : [0, ∞) → [0, ∞) be an (α, m)-convex function with α, m ∈
b a ) > g(b) and g(a) > mg m . Let µ be the Lebesgue measure (0, 1]2 such that mg( m on [a, b], 0 ≤ a < b < ∞, then Z 1 (2.21) gdµ ≤ min {1, α1 , α2 } , 0 0

where α1 and α2 are positive real solutions of the equations α = (b − a) "  α1 # 0 a 0 g( m )−α and α = (b − a) respectively. a mg ( m )−g(b)

504

"

0

g(a)−α b g(a)−mg ( m )

 α1 #

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Remark 4. If (α, m) = (1, 1), then the inequalities (2.17) and (2.21) become those inequalities proved in Theroem 3 from [3, p.4].

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[29] M. Ying, Linguistic quantifiers modeled by Sugeno integrals, Artificial Intelligence 170 (2006) 581–606. College of Science, Department of Mathematics, University of Hail, Hail 2440, Saudi Arabia. E-mail address: m amer [email protected] E-mail address: [email protected] E-mail address: [email protected] 2,3 Department of Mathematics, University of Engineering and Technology, Lahore, Pakistan

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Very true operators on equality algebras Jun Tao Wanga , Xiao Long Xina,∗ ,Young Bae Junb a

b

School of Mathematics, Northwest University, Xi’an, 710127, China Department of Mathematics Education, Gyeongsang National University, Jinju 660-701, Korea

Abstract In this paper, we introduce the concept of very true operators on equality algebras and investigate some related properties of such operators. As an application of properties of very true operators on equality algebras, we give a characterization of prelinear equality algebras, and discuss the relation between very true operator and internal state operator on equality algebras. Moreover, we put forth the notion of very true filter on equality algebras and obtain some important results. In particular, using very true filter, we characterize the simple very true equality algebras and establish the uniform structures on very true equality algebras. Keywords: fuzzy higher logic; equality algebras; very true operators; very true filters; uniform structures. 1. Introduction Fuzzy type theory [13, 14, 15, 16], whose basic connective is a fuzzy equality ∼, was developed as a fuzzy counterpart of the classical higher-order logic (type theory in which identity is a basic connective, see [9]). Since the truth values for algebra of fuzzy type theory is no longer a residuated lattice, a specific algebra called an EQ-algebra [17] was proposed. Viewing the axioms of EQ-algebras with a purely algebraic eye it appears that unlike in the case of residuated lattices where the adjointness condition ties product with implication. By contrast, the product in EQ-algebras is quite loosely related to the other connectives, which lead to the product in EQ-algebra may be replaced by other similar connectives. Furthermore, the freedom in choosing the product might prohibit to find deep related algebraic results. For this reason, a new algebraic structure was introduced in [11], called equality algebra, which consisting of two binary operations-meet and equivalence, and constant 1. It was proved in [12] that any equality algebra has a corresponding BCK-meet-semilattice and any BCK-meet-semilattice with (D) has a corresponding equality algebras. Apart from their algebraic interest, the general motivation for equality algebras from the side of logic was to define an algebraic structure which (with appropriate extensions) is suitable to axiomatize a large class of substructural logics based on an equivalence connective rather than implication. The very first step toward this aim has been done in [11]. Indeed, the equality algebras could also be candidates for a possible algebraic semantics for fuzzy type theory, which lead to the study of equality algebra is highly motivated. ∗ Corresponding author. Email addresses: [email protected](J.T. Wang), [email protected] (X.L. Xin), [email protected](Y.B. Jun).

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Jun Tao Wang et al 507-521

J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 24, NO.3, 2018, COPYRIGHT 2018 EUDOXUS PRESS, LLC

In the sense of Zadeh [5]( 1975), fuzzy logic distinguish in broad and narrow sense. In narrow sense, fuzzy logic deals with some many valued logic but asks questions different from those asked by logicians, who devote to completeness and soundness, etc. Compare to narrow sense, Zadeh stress the importance of fuzzy truth values as very true, quite true and so on, that themselves are fuzzy subsets of all truth degrees. He always gives some examples of handling these fuzzy truth values but seems uninterested in any sort of axiomatization. In order to answer the question “ if any natural axiomatization is possible and how far can even this sort of fuzzy logic be captured by standard methods of mathematical logic“, H´ajek[8] introduced the concept of very true operator on BLalgebras as a tool for reducing the number of possible logical values in many valued fuzzy logic. Consequently, the notion of very true operator has been extended to other logical algebras such as MV-algebra [2], R`-monoid[3], residuated lattices[10, 20] and provided an algebraic foundation for reasoning about fuzzy truth valued of events in many valued logic system, which belong to a subclass of substructural logic based on an fuzzy implication. As for BL-algebras, MV-algebras R`-monoid and residuated lattices, we observed that although they are different algebras they all are essentially particular types of equality algebras. Thus, it is natural to generalized the concept of very true operators to equality algebras for studying the most general results regarding very true operators in the abovementioned algebras. On the other hand, BL, Lukasiewicz, ML, they are all many valued logic system belong to a subclass of substructural logic based on an fuzzy implication. However, the logic system corresponding to equality algebra is a fuzzy higher logic, which belong to subclass of substructural logic based on fuzzy equality and is different from above common many valued logic system that based on fuzzy implication. Moreover, all results of this paper may be considered providing an algebraic foundation for reasoning about probabilities of fuzzy events for higher fuzzy logic. This is the motivation for us to investigate very true operators on equality algebras. Based on above consideration, we enrich the language of equality algebras by adding a very true operator to get algebras named very true equality algebras. This paper is structured in five sections. In order to make the paper as self-contained as possible, we recapitulate in Section 2 the definition of equality algebras, and review their basic properties that will be used in the remainder of the paper. In Section 3, we introduce very true operator on equality algebras and study some properties of them. Also, we give a characterizations of a prelinear equality algebras and discuss the relation between very true operator and internal state on equality algebras. In Section 4, we investigate very true filter of very true equality algebras. In particular, by using very true filter, simple very true equality algebras are characterized and the uniform structures on very true equality algebras are established. 2. Preliminaries In this section, we summarize some definitions and results about equality algebras which will be used in the following and we shall not cite them every time they are used. Definition 2.1. [11] An algebra (E, ∼, ∧, 1) of type (2, 2, 0) is called an equality algebra if it satisfies the following conditions: (E1) (E, ∧, 1) is a commutative idempotent integral monoid (i.e., ∧-semilattice with top element 1), 508

Jun Tao Wang et al 507-521

J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 24, NO.3, 2018, COPYRIGHT 2018 EUDOXUS PRESS, LLC

(E2) (E3) (E4) (E5) (E6) (E7)

x ∼ y = y ∼ x, x ∼ x = 1, x ∼ 1 = x, x ≤ y ≤ z implies x ∼ z ≤ y ∼ z and x ∼ z ≤ x ∼ y, x ∼ y ≤ (x ∧ z) ∼ (y ∧ z), x ∼ y ≤ (x ∼ z) ∼ (y ∼ z),

for all x, y, z ∈ E. In what follows, by E we denote the universe of (E, ∼, ∧, 1). For any x, y ∈ E, we define fuzzy implication as x → y = x ∼ (x ∧ y) and agree that ∼ and → have higher priority than ∧. On an equality algebra (E, ∼, ∧, 1) we define x ≤ y iff x ∧ y = x. It is easy to check that ≤ is a partial order relation on E and for all x ∈ E, x ≤ 1. Definition 2.2. [6, 7] Let (E, ∼, ∧, 1) be an equality algebra. Then E is called: (1) bounded if there exists an element 0 ∈ E such that 0 ≤ x for all x ∈ E, (2) prelinear if for all x, y ∈ E, 1 is the unique upper bounded in E of the set {(y → x), (x → y)}. Proposition 2.3. [6, 7] If (E, ∼, ∧, 1) is a prelinear equality algebra, then (E, ≤) is a lattice, where the join operation is given by x ∨ y = ((x → y) → y) ∧ (y → x) → x), for any x, y ∈ E. Proposition 2.4. [6, 7] An equality algebra (E, ∼, ∧, 1) is prelinear if and only if (x → y) → z ≤ ((y → x) → z) → z, for all x, y, z ∈ E. Proposition 2.5. [11, 12] In every equality algebra (E, ∼, ∧, 1) the following properties hold for all x, y, z ∈ E: (1) (2) (3) (4) (5) (6) (7) (8) (9) (10) (11) (12) (13)

x ∼ y ≤ x ↔ y ≤ x → y, x ∼ y = 1 iff x = y, x → y = 1 iff x ≤ y, x → y = 1 and y → x = 1 imply x = y, 1 → x = x, x → 1 = 1, x → x = 1, x ≤ y → x, x ≤ (x → y) → y, x → y ≤ (y → z) → (x → z), x ≤ y → z iff y ≤ x → z, if x ≤ y, then x ≤ x ∼ y, x ≤ y imply y → x = y ∼ x, x ≤ y imply y → z ≤ x → z, z → x ≤ z → y, V W If E is a prelinear equality algebra, then i∈I (xi → y) = i∈I xi → y, provided that both infimum as well as supremum exist.

Definition 2.6. [11] Let (E, ∼, ∧, 1) and (E 0 , , u, 10 ) be two equality algebras and f : E −→ E 0 be a mapping. We call f a homomorphism if the following conditions hold for all x, y ∈ E: 509

Jun Tao Wang et al 507-521

J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 24, NO.3, 2018, COPYRIGHT 2018 EUDOXUS PRESS, LLC

(1) f (x ∧ y) = f (x) u f (y), (2) f (x ∼ y) = f (x) f (y). The following theorem provides a connection of equality algebras with a special class of BCK-algebras with meet. Theorem 2.7. [4, 11, 12] The following two statements hold: (1) For any equality algebra (E, ∼, ∧, 1), the structure Ψ(E) = {E, ∧, →, 1} is a BCKmeet-semilattice, where → denotes the implication of E, (2) For any BCK(D)-meet-semilattice (B, ∧, →, 1), the structure Φ(B) = {B, ∧, ↔, 1) is an equality algebra, where ↔ denotes the equivalence operation of B. Moreover, the implication of Φ(B) coincide with →, that is, x → y = x ↔ (x ∧ y). Let (E, ∼, ∧, 1) be an equality algebra. A nonempty set F is called a filter of E if it satisfies: (1) x ∈ F , x ≤ y implies y ∈ F , (2) x ∈ F , x ∼ y ∈ F implies y ∈ F . One can prove that the set of filters of an equality algebra coincide with the set of filter of its underlying BCK-algebra. A filter F of an equality algebra E is proper if F 6= E. A proper filter is called maximal if it is not strictly contained in any other proper filter of E. We will denote by F (E) the set of all filter of E. Clearly, {1}, E ⊆ F (E) and F (E) is closed under arbitrary intersections. As a consequence, (F (E), ⊆) forms a complete lattice. An equality algebra (E, ∼, ∧, 1) is calle a simple if F (E) = {{1}, E}. (see [4, 12]) Definition 2.8. [4, 11, 12] Let (E, ∼, ∧, 1) be an equality algebra. A subset θ ⊆ E × E is called a congruence of E if it is an equivalence relation on E and for all x1 , y1 , x2 , y2 ∈ E such that (x1 , y1 ), (x2 , y2 ) ∈ θ the following hold: (1) (x1 ∧ x2 , y1 ∧ y2 ) ∈ θ, (2) (x1 ∼ x2 , y1 ∼ y2 ) ∈ θ. Let F be a filter of E. Define the congruence relation ≡F on E by x ≡F y if and only if x ∼ y ∈ F . The set of all congruence class is denote by E/F , i.e. E/F = {[x]|x ∈ E}, where [x] = {x ∈ E|x ≡F y}. Define •, * on E/F as follows: [x] • [y] = [x ∧ y], [x] * [y] = [x ∼ y]. Therefore, (E/F, •, *, [1]) is an equality algebra which is called a quotient equality algebra of E with respect to F . (see [4]) In what follows, we review some notions about uniformity which will be necessary in the following section. Let X be a nonempty set and U, V be any subset of X × X. Defined U ◦ V = {(x, y) ∈ X × X| for some z ∈ X, (z, y) ∈ U and (x, z) ∈ V }, U −1 = {(x, y) ∈ X × X|(y, x) ∈ U }, 4 = {(x, x) ∈ X × X|x ∈ X}. Definition 2.9. [1, 18, 19] By an uniformity on X we shall mean a nonempty collection K of subsets of X × X which satisfies the following conditions: (U1) (U2) (U3) (U4) (U5)

4 ⊆ U for any U ∈ K, If U ∈ K, then U −1 ∈ K, If U ∈ K, then there exists a V ∈ K such that V ◦ V ⊆ U , If U, V ∈ K, then U ∩ V ∈ K, If U ∈ K and U ⊆ V ⊆ X × X then V ∈ K,

The pair (X, K) is called an uniform structure (uniform space). 510

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3. Very true operators on equality algebras In this section, we introduce the notion of very true operators in an equality algebras and investigate some related properties of such operators. Also, we give characterizations of prelinea equality algebras and discuss relations between very true operators and internal state operators on equality algebras. Definition 3.1. Let E be an equality algebra. The mapping τ : E −→ E is called a very true operator if it satisfies the following conditions: (VE1) (VE2) (VE3) (VE4) (VE5)

τ (1) = 1, τ (x) ≤ x, τ (x) ≤ τ τ (x), τ (x ∼ y) ≤ τ (x) ∼ τ (y), τ (x ∧ y) = τ (x) ∧ τ (y).

The pair (E, τ ) is said to be very true equality algebra. Note. (1) Such a proliferation of logics deserves some explanation. Since 1 is considered as the logical value absolutely true. First note that (VE1) means that absolutely true is very true, which is the standard axiom obtain the classical logic. (VE2) means that if ϕ is very true then it is true. (VE3) says that very true of very true is very true, which is a kind of necessitation with respect to very true connective. (VE4) means that if both ϕ and ϕ ∼ ψ are very true then so is ψ, that means the connective τ preserve modus ponens based on fuzzy equality ∼. (VE5) says that if both ϕ and ψ are very true then so is conjunction ϕ ∧ ψ, one can easily to check that (VE5) is sound for each natural interpretation in equality algebra. Indeed, the order in equality algebra is lattice order, that is the conjunction ∧ is interpreted as the lattice meet ∧. (2) Although equality algebras belong to some subclasses of substructural logics based on fuzzy equality rather than on fuzzy implication, we define very true operators on them in the way which is in accordance with traditional definitions in residuated lattices. In fact, a very true operator on residuated lattice was introduced by Vychodil [10] in 2005 as a mapping τ : E → E satisfying conditions (VE1)-(VE3) in Definition 3.1 and (VE4’) τ (x → y) ≤ τ (x) → τ (y). We know that residuated lattices are special cases of equality algebras satisfying the residuated law. In residuated lattice, fuzzy equality ∼ can be defined by x ∼ y = (x → y) ∧ (y → x). From (VE4) and (VE5), one can obtain that the connective τ always preserve modus ponens based on fuzzy implication. Based on (VE4’) and the isotonicity of very true connective τ , one can easily obtain τ (x ∼ y) = τ ((x → y) ∧ (y → x)) ≤ τ (x → y) ∧ τ (y → x) ≤ (τ (x) → τ (y)) ∧ (τ (y) → τ (x)) = τ (x) ∼ τ (y), thus the (VE5) hold. From this point of view, the very true equality algebra essentially generalize residuated lattice with very true operator. Thus, it is the most general logic algebras with very true in the existing ones founded in the literature, at least to the authors knowledge. Now, we will give some important examples to illustrate above definition is existing and meaningful. Example 3.2. For every equality algebra E there exist at least two very true operator. One is the identical mapping τ0 (x) = x for any x ∈ E, and the other is defined by τ1 (1) = 1 and τ1 (x) = 0 for any x < 1. It is evident that if τ is a very true operator on E, we have τ1 (x) ≤ τ (x) ≤ τ0 (x). Thus these τ0 ,τ1 are extremal. 511

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J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 24, NO.3, 2018, COPYRIGHT 2018 EUDOXUS PRESS, LLC

Example 3.3. Let E = {0, a, b, 1} with 0 ≤ a ≤ b ≤ 1. Consider the operation ∼ and → given by the following tables: ∼ 0 a b 1

0 1 a 0 0

a a 1 a a

b 0 a 1 b

→ 0 a b 1

1 0 a b 1

0 1 a 0 0

a 1 1 a a

b 1 1 1 b

1 1 1 1 1

Then (E, ∼, ∧, 1) is an equality algebra in [4]. Now,we define τ (0) = 0, τ (a) = a, τ (b) = a, τ (1) = 1. One can easily check that τ is a very true operator on E. However, τ is not a endhomomorphism on E since τ (a ∼ b) = a 6= 1 = τ (a) ∼ τ (b). Next, we present some useful properties of very true operator on equality algebras. Proposition 3.4. Let (E, τ ) be a very true equality algebra, then for any x, y, z ∈ E we have, (1) (2) (3) (4) (5) (6) (7) (8) (9) (10) (11) (12) (13) (14) (15) (16)

If E is a bounded equality algebra, then τ (0) = 0, τ (x) = 1 if and only if x = 1, τ τ (x) = τ (x), τ (x → y) ≤ τ (x) → τ (y), x ≤ y implies τ (x) ≤ τ (y), τ (x) ≤ y if and only if τ (x) ≤ τ (y), τ (E) = F ixτ ,where F ixτ = {x ∈ E|τ (x) = x}, F ixτ is closed under ∧, If y ≤ x, then τ (x) → τ (y) = τ (x) ∼ τ (y), τ (x ∼ y) ≤ τ (x) → τ (y), τ (x ∼ y) ≤ (x ∧ z) ∼ (y ∧ z), If τ (E) = E, then τ = idE , Ker(τ ) = {1}, where Ker(τ ) = {x ∈ E|τ (x) = 1}, Ker(τ ) is a filter of E, τ (x) = x or τ (x) and x are not comparable, If E is linearly order, then τ = idE .

Proof. (1) Applying (VE2), we have τ (0) ≤ 0 and hence τ (0) = 0. (2) If τ (x) = 1 for some x ∈ E then by (VE2), 1 = τ (x) ≤ x giving x = 1. The converse follows by (VE1). (3) Applying (VE2) and (VE3), we have τ τ (x) = τ (x). (4) By (VE4) and (VE5) we have τ (x → y) = τ (x ∧ y ∼ x) = τ (x ∧ y) ∼ τ (x) ≤ τ (x) ∧ τ (y) ∼ τ (x) = τ (x) → τ (y). (5) If x ≤ y, then x → y = 1. It follows from (VE1) and (4) that τ (x) → τ (y) = 1. Therefore, τ (x) ≤ τ (y). (6) Assume that τ (x) ≤ y, we have τ τ (x) ≤ τ (y). By (3), we get τ τ (x) = τ (x). Thus τ (x) ≤ τ (y). Conversely, suppose that τ (x) ≤ τ (y), we have τ (x) ≤ τ (y) ≤ y. (7) Let y ∈ τ (E), so there exists x ∈ E such that y = τ (x). Hence τ (y) = τ τ (x) = τ (x) = y. It follows that y ∈ F ixτ . Conversely, if y ∈ F ixτ , we have y ∈ τ (E). Therefore, τ (E) = F ixτ . (8) By (VE5), we obtain that F ixτ is closed under ∧. 512

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(9) Since y ≤ x, we have τ (y) ≤ τ (x) and τ (x) → τ (y) = τ (x) ∼ τ (x) ∧ τ (y) = τ (x) ∼ τ (y). (10) By Proposition 2.5(1) and (4),(5), we have τ (x ∼ y) ≤ τ (x → y) ≤ τ (x) → τ (y). (11) By (E6) and (VE2), one can obtain it very easy and hence we omit the process of this proof. (12) For any x ∈ E, we have x = τ (x0 ) for some x0 ∈ E. By (3), we have τ (x) = τ (τ (x0 )) = τ (x0 ) = x. Therefore, τ = idE . (13) Assume that x ∈ E but x 6= 1 such that τ (x) = 1. Applying (VE2), we have 1 = τ (x) ≤ x and hence x = 1, which is a contradiction. Therefore, Ker(τ ) = {1}. (14) It is easy to check it and hence we omit the process. (15) Assume x ∈ E such that τ (x) 6= x and τ (x) and x are comparable. Then τ (x) < x or x < τ (x), from (3),(5), we have τ (x) < τ (x), which is a contradiction. (16) It follows from (15) directly. From the above Proposition 3.4, one can see that τ (E) is closed under the operation ∧. However, the following example shows that τ (E) is not a subalgebra of E since it is not closed under ∼ in general. Example 3.5. Let E = {0, a, b, c, 1} with 0 ≤ a ≤ b, c ≤ 1. Consider the operation ∼ and → given by the following tables: ∼ 0 a b c 1

0 1 0 0 0 0

a 0 1 c b a

b 0 c 1 0 b

c 0 b 0 1 1

→ 0 a b c 1

1 0 a b c b

0 1 0 0 0 0

a 1 1 c b a

b 1 1 1 b b

c 1 1 c 1 c

1 1 1 1 1 1

Then (E, ∼, ∧, 1) is an equality algebra. Now,we define τ (0) = 0, τ (a) = a, τ (b) = a, τ (c) = c, τ (1) = 1. One can easily check that τ is a very true operator on E. However, τ (E) is not a subalgebra of E since a ∼ c = b ∈ / τ (E). Although the τ (E) is not necessary a subalgebra of an equality algebra in general, while it forms an equality algebra after redefined its fuzzy equality, which reveals the essence of the fixed point set. Theorem 3.6. Let (E, τ ) be a very true equality algebra. Then (τ (E), ∧, an equality algebra, where x y = τ (x ∼ y) for all x, y ∈ τ (E).

, 1) is also

Proof. Now, we will show that (τ (E), ∧, , 1) is an equality algebra. For (E1), we show that (τ (E), ∧, 1) is a semilattice with 1 as the greatest element. From Definition 3.1(5), we have that τ (E) is closed under ∧. Therefore (τ (E), ∧) is a semilattice. For all x ∈ τ (E), one can easily check that x ∧ 1 = x. Thus, 1 is the greatest element in τ (E). For (E2), we will show that x y=y x. It is easy to prove. For (E3), we will show that x x = 1. Applying (VE1), we have τ (1) = 1 and hence x x = τ (x ∼ x) = τ (1) = 1. For (E4), we will show that x 1 = x. Since x ∈ τ (E), we have τ (x) = x and hence x 1 = τ (x ∼ 1) = τ (x) = x. In the similarly way, we can show that (E5)-(E7) hold. Combine them, we obtain that (τ (E), ∧, , 1) is an equality algebra. 513

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Note. In fact, it is easily checked that (τ (E), ∧, , 1) is an equality algebra, where τ (x) τ (y) = τ (x ∼ y). Furthermore, we also obtain that if τ (x) τ (y) = 1, then τ (x) = τ (y). Indeed, if τ (x), τ (y) ∈ F ixτ , by above Theorem 3.6, τ (x) τ (y) = τ (τ (x) ∼ τ (y)) = τ (x) ∼ τ (y). Thus τ (x) ∼ τ (y) = 1, and we have τ (x) = τ (y) by Proposition 2.5(3). In what following, we will give an analogy first isomorphism theorem related to very true operator on equality algebras, which will be used in the next section. Theorem 3.7. Let (E, τ ) be a very true equality algebra and (τ (E), ∧, equality algebra. Then the following properties hold:

, 1) be an

(1) τ : E −→ τ (E) is a homomorphism, (2) The mapping τ0 : E/Ker(τ ) −→ τ (E) defined by τ0 ([x]) = τ (x) is a isomorphism. Proof. (1) It follows from (VE1) and (VE5) in Definition 3.1 that τ (1) = 1 and τ (x∧y) = τ (x) ∧ τ (y). Moreover, from the above Note, we have τ (x) τ (y) = τ (x ∼ y). Therefore τ : E −→ τ (E) is a homomorphism. (2) Assume that [x] = [y] and hence (x, y) ∈ Ker(τ ). Then x ∼ y ∈ Ker(τ ), that is, τ (x ∼ y) = 1. From the above Note, we have τ (x) τ (y) = 1 and (τ (E), ∧, , 1) is an equality algebras and so τ (x) = τ (y). Therefore, τ0 is well defined. Now, we will show that τ0 is a isomorphism. First, we will show that τ0 is a homomorphism. From (1), we have τ0 ([x] * [y]) = τ0 ([x ∼ y]) = τ (x ∼ y) = τ (x) τ (y) = τ0 ([x]) τ0 ([y]). Moreover, we have τ0 ([x] • [y]) = τ0 ([x ∧ y]) = τ (x ∧ y) = τ (x) ∧ τ (y) = τ0 ([x]) ∧ τ0 ([y]). Clearly, τ0 ([1]) = 1. Hence τ0 is a homomorphism. Next, we will show that τ0 is one to one. From the above Note, if τ0 ([x]) = τ0 ([y]) then τ (x) = τ (y) and hence τ (x) τ (y) = τ (x ∼ y) = 1, that is means (x, y) ∈ Ker(τ ) and hence [x] = [y] and so τ0 is one to one. Furthermore, since τ is onto, τ0 is onto. Combine them, we obtain that τ0 is a isomorphism. As an application of the properties respect to very true operators, we give a characterization of prelinear equality algebras. For obtaining this important result, we need the following theorem. Theorem 3.8. The following conditions are equivalent in each very true equality algebra (E, τ ): (1) τ (x → y) ≤ (x ∧ z) → (y ∧ z), (2) τ (y) ≤ z → (y ∧ z), (3) τ (y) ≤ u → (u ∧ (z → (y ∧ z)). Proof. (1) ⇒ (2) Assume (1) holds. From Proposition 2.5 (1) and Proposition 3.5 (11), we have τ (x → y) ≤ ((1 ∧ z) ∼ (y ∧ z)) ≤ ((1 ∧ z) → (y ∧ z)) = z → (y ∧ z). (2) ⇒ (3) Assume (2) holds. By (VE3), we get τ (y) ≤ τ τ (y) ≤ τ (z → (y ∧ z)), which implies, by (2) again, τ (y) ≤ u → (u ∧ (z → (z ∧ y))). (3) ⇒ (2) Taking u = 1, we obtain that (2) holds. (2) ⇒ (1) Assume (2) holds. By (3), we have τ (x → y) ≤ (x∧z) → ((x → y)∧(x∧z)). Thus, τ (x → y) ≤ (x ∧ z) → ((x → y) ∧ x ∧ z). Furthermore, from Proposition 2.5(12), we have τ (x → y) ≤ (x ∧ z) → ((x → y) ∧ x ∧ z) ≤ (x ∧ z) → (y ∧ z). 514

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Conditions for an equality algebra to be a prelinear equality algebra is gave via very true operator on equality algebra. Theorem 3.9. The following conditions are equivalent in each very true equality algebras: (1) (2) (3) (4)

E is prelinear, τ (x ∨ y) = τ (x) ∨ τ (y), τ (x → y) ∨ τ (y → x) = 1, τ ((x → y) → z) ≤ τ (y → x) → z) → z.

Proof. (1) ⇒ (2) Assume that E is prelinear equality algebra. By(VE5) and Proposition 2.3, we obtain, for all x, y ∈ L, τ (x ∨ y) = τ ((x → y) → y) ∧ τ ((y → x) → x). By Proposition 3.4(5) and (VE3), we get τ (x ∨ y) ≤ τ ((x → y) → (τ (x) ∨ τ (y))) ∧ (τ (y → x) → (τ (x) ∨ τ (y))). Hence by Proposition 2.5(13), we obtain τ (x ∨ y) ≤ (τ (x → y)) ∨ (τ (y → x)) → (τ (x) ∨ τ (y)) and hence τ (x ∨ y) ≤ τ (x) ∨ τ (y). The other inequality easily by Proposition 3.4(5). (2) ⇒ (3) straightforward. (3) ⇒ (4) This follows exactly by a proof similar to the proof of the equivalence between prelinearity and (x → y) → z ≤ ((y → x) → z) → z) in Proposition 2.4. (4) ⇒ (1) Taking very true operator τ = idE and direct from Proposition 2.4. Note. We know a BL-algebras is an prelinear quality algebra satisfies the divisibility. From the above theorem, one can check that the very true prelinear equality algebra essentially generalize BL-algebras with very true operator, which introduced by H´ajek [8] in 2001 as a mapping τ : E → E satisfying conditions (VE1)-(VE3), (4) in Proposition 3.5 and (2) in Theorem 3.9. In what follows, we will give a relationship between very true equality algebras and sate (morphism) equality algebras, which was introduced by L.C. Ciungu [4] to providing an algebraic foundation for reasoning about probabilities of fuzzy events of a large class of substructural logics based on an fuzzy equality. Definition 3.10. [4] An equality algebra with internal state or state equality algebra is a structure (E, σ) = (E, ∧, ∼, σ, 1), where (E, ∧, ∼, 1) is an equality algebra and σ : E → E is a unary operator on E, called internal state or state operator, such that for all x, y ∈ E the following conditions are satisfied: (1) (2) (3) (4)

σ(x) ≤ σ(y), whenever x ≤ y, σ(x ∼ x ∧ y) = σ((x ∼ x ∧ y) ∼ y) ∼ σ(y), σ(σ(x) ∼ σ(y)) = σ(x) ∼ σ(y), σ(σ(x) ∧ σ(y)) = σ(x) ∧ σ(y).

Definition 3.11. [4] Let (E, ∧ ∼, 1) be an equality algebra. A state morphism operator on E is a map σ : E → E satisfying the following condition for all x, y ∈ E: (1) σ(x ∼ y) = σ(x) ∼ σ(y), (2) σ(x ∧ y) = σ(x) ∧ σ(y), (3) σ(σ(x)) = σ(x). The pair (E, σ) is called a state morphism equality algebra. 515

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Theorem 3.12. [4] Any state morphism on a linearly ordered equality algebra E is an internal state on E. Theorem 3.13. Let (E, τ ) be a very true equality algebra. Then the following conditions are equivalent: (1) (E, τ ) is a state morphism equality algebras, (2) τ = idE . Proof. (1) ⇒ (2) We note that (E, τ ) is not only a state morphism equality algebras but also a very true equality algebras. In order to prove this important result, we need some useful result. First, we will prove that x < y implies τ (x) < τ (y). Assume that x < y, then τ (x) ≤ τ (y). If τ (x) = τ (y), then τ (x → y) = τ (x) → τ (y) = 1. By Proposition 3.4(13), we have y → x = 1 and hence y ≤ x, which is a contraction to x < y. Now, we will show that τ (x) = x for all x ∈ E. By (VE2), we have τ (x) ≤ x. Assume that τ (x) 6= x, then τ (x) < x and hence τ τ (x) < τ (x). Thus τ (x) < τ (x), which is a contradiction. Therefore, τ = idE . (2) ⇒ (1) straightforward. Theorem 3.14. Let (E, τ ) be a very true linearly order equality algebra. Then the following conditions are equivalent: (1) (E, τ ) is a state equality algebras, (2) τ = idE . Proof. It follows from Proposition 3.4(16), we can easily to obtain this result. 4. Very true filters in very true equality algebras In this section, we introduce very true filters of very true equality algebras and obtain some important result of them. In particular, using very true filter, we give a characterization of simple very true equality algebras and construct the uniform structures on very true equality algebras. Definition 4.1. Let (E, τ ) be a very true equality algebra. A nonempty subset F of E is called a very true filter of (E, τ ) if F is a filter of E such that if x ∈ F , then τ (x) ∈ F for all x ∈ E. Example 4.2. Let (E, τ ) be a very true equality algebra. Then Ker(τ ) = {x ∈ X|τ (x) = 1} is a very true filter of (E, τ ). Example 4.3. Consider the Example 3.3, one can easily check that the very true filter of (E, τ ) are {1}, {a, b, 1} and E. On the other hand, one can see that {b, 1} is not a very true filter of (E, τ ), but it is a filter of E, that is to say, not every filter is very true filter. As an application of very true filters, we give a characterizations of simple very true equality algebras. For obtaining this important result, we need the following proposition. Proposition 4.4. Let (E, τ ) be a very true equality algebra. (1) If F is a filter of τ (E), then τ −1 (F ) is a very true filter of (E, τ ), 516

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(2) If F is a very true filter of (E, τ ), then τ (F ) is a filter of τ (E). Proof. (1) Suppose that F is a filter of τ (E). If x, x ∼ y ∈ τ −1 (F ). Then τ (x), τ (x ∼ y) ∈ F . Since τ (x ∼ y) ≤ τ (x) ∼ τ (y) and τ (x) ∼ τ (y) ∈ F , thus τ (y) ∈ F , that is y ∈ τ −1 (F ). Let x, y ∈ E such that x ∈ τ −1 (F ) and x ≤ y. Then τ (x) ≤ τ (y). Since τ (x) ∈ F and τ (y) ∈ τ (E), we can obtain that τ (y) ∈ F , that is y ∈ τ −1 (F ). Obviously, 1 ∈ τ − (F ). Therefore τ −1 (F ) is a filter of E. If x ∈ τ −1 (F ), then τ (x) ∈ F , so τ τ (x) = τ (x) ∈ F , that is, τ (x) ∈ τ −1 (F ). Therefore, τ −1 (F ) is a very true filter of (E, τ ). (2) First, we have τ (F ) = F ∩τ (E). Indeed, if x ∈ F ∩τ (E), then x = τ (x) as x ∈ τ (E), and τ (x) ∈ τ (F ) as x ∈ F . Thus, we have x ∈ τ (F ). It follows that F ∩ τ (E) ⊆ τ (F ). Conversely, if y ∈ τ (F ), then there exists x ∈ F such that y = τ (x). Since F is an very true filter of (E, τ ), we obtain y = τ (x) ∈ F . Therefore, τ (F ) = F ∩ τ (E). If x, x y ∈ τ (F ) = F ∩ τ (E), then τ (y) ∈ F ∩ τ (E) = τ (F ) and hence y ∈ τ (F ). On the other hand, if x ∈ τ (F ) = F ∩ τ (E), y ∈ τ (E) such that x ≤ y, then y ∈ F ∩ τ (E) = τ (F ). Therefore, τ (F ) is a filter of τ (E). Now, we introduce simple very true equality algebras and give a characterizations of it via very true filter. Definition 4.5. A very true equality algebra (E, τ ) is called simple very true if it has exactly two very true filter: {1} and E. Example 4.6. For each equality algebra E, (E, τ1 ) is a simple very true equality algebra, where τ1 (1) = 1 and τ1 (0) = 0 if x < 1. Theorem 4.7. Let (E, τ ) be a very true equality algebra. Then the following are equivalent: (1) (E, τ ) is a simple very true equality algebra, (2) τ (E) is a simple equality algebra. Proof. (1) ⇒ (2) Let F be a filter of τ (E) and F 6= {1}. It follows from Proposition 4.4(1) that τ −1 (F ) is a very true filter of (E, τ ). Since (E, τ ) is very true simple, we have that τ −1 (F ) = {1} or τ −1 (F ) = E. Notice that F ⊆ τ −1 (F ) (if x ∈ F , then τ (x) = x, that is, x ∈ τ −1 (F ), we obtain that τ −1 (F ) 6= {1}. Thus, τ −1 (F ) = E. Then 0 ∈ τ −1 (F ), that is, 0 = τ (0) ∈ F . So we obtain that F = τ (E). Therefore, τ (E) is simple equality algebra. (2) ⇒ (1) Let F be a very true filter of (E, τ ) and F 6= {1}. By Proposition 4.4(2), we obtain that τ (F ) is a filter of τ (E). Since τ (E) is simple equality algebra, we obtain that τ (F ) = {1} or τ (F ) = E. Since Ker(τ ) = {1}, we have F 6= {1}. Thus, τ (F ) = E. Then 0 ∈ τ (F ), that is, 0 ∈ F . It follows that F = E. Therefore (E, τ ) is a simple very true equality algebra. Note. The above theorem brings a method of how to check a very true equality algebra is simple very true. As an application of above theorem, one can easily check that the very true equality algebra in example 4.6 is simple very true since τ (E) = {0, 1} is a simple equality algebra. For any very true filter F of (E, τ ). Defined by τF : E/F −→ E/F as a mapping τF ([x]) = [τ (x)]. 517

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Proposition 4.8. Let (E, τ ) be a very true equality algebra and F be a very true filter of very true equality algebra (E, τ ). Then τF is a very true operator on E/F . Proof. First, we will prove that τF is well defined. Indeed, assume that [x] = [y] for x, y ∈ E. Then (x, y) ∈≡F , i.e., x ∼ y ∈ F . Since F is a very true filter and hence τ (x ∼ y) ∈ F . Now, applying (VE4) of Definition 3.1, τ (x ∼ y) ≤ τ (x) ∼ τ (y) ∈ F . Thus (τ (x), τ (y)) ∈ F and [τ (x)] = [τ (y)]. The rest of the proof is easy: (VE1) (VE2) (VE3) (VE4) (VE5)

τF ([1]) = [τ (1)] = [1], τF ([x]) = [τ (x)] ≤ [x], τF ([x]) = [τ (x)] ≤ [τ τ (x)] ≤ τF τF ([x]), τF ([x] ∼ [y]) = [τ (x ∼ y)] ≤ [τ (x) ∼ τ (y)] = τF ([x]) ∼ τF ([y]), τF ([x] ∧ [y]) = [τ (x ∧ y)] = [τ (x) ∧ τ (y)] = τF ([x]) ∧ τF ([y]).

Combine them, one can obtain that (E/F, τF ) is a very true equality algebra. Proposition 4.9. In the very true equality algebra (E/Ker(τ ), τKer(τ ) ) we have: (1) [x] ≤ [y] iff τ (x ∼ x ∧ y) = 1 iff τ (x → y) = 1, (2) [x] = [y] iff τ (x ∼ y) = 1. Proof. (1) Applying the definition of E/Ker(τ ) we get: [x] ≤ [y] iff [x] • [y] = [x] iff [x] = [x ∧ y] iff [x] * [x ∧ y] = [1] iff [x ∼ x ∧ y] = [1] iff x ∼ x ∧ y ∈ Ker(τ ) iff τ (x ∼ x ∧ y) = 1 iff τ (x → y) = 1. (2) We have [x] = [y] iff [x] * [y] = [1] iff [x ∼ y] = [1] iff x ∼ y ∈ Ker(τ ) iff τ (x ∼ y) = 1. Definition 4.10. Let (E, τ ) be a very true equality algebra and θ be a congruence on E. Then θ is called a very true congruence on (E, τ ) if (x, y) ∈ θ implies (τ (x), τ (y)) ∈ θ for each x, y ∈ E. Example 4.11. Consider the Example 3.4, one can see that R = {{0, 0}, {a, a}, {b, b}, {1, 1}, {a, b}, {b, a}, {a, 1}, {1, a}, {1, b}, {b, 1}} is a very true congruence on a very true equality algebra (E, τ ). Theorem 4.12. Let (E, τ ) be a very true equality algebra. Then there is a one to one correspondence between its very true filters and its very true congruences. Proof. Suppose that θ is a very true congruence relation on (E, τ ). Clearly Fθ = {x ∈ E|(x, 1) ∈ θ} is a very true filter of (E, τ ). Now given x ∈ Fθ , we have (x, 1) ∈ η and hence (τ (x), 1) = (τ (x), τ (1)) ∈ θ and therefore τ (x) ∈ Fθ . This proves that Fθ is a very true filter on (E, τ ). Conversely, let F be a very true filter. Then θF is a very true congruence on very true equality algebra, since for each (x, y) ∈ θF , we have x ∼ y ∈ F . Since F is a very true filter and hence τ (x ∼ y) ∈ F . By (VE4), we have τ (x) ∼ τ (y) ∈ θF , thus θF is an very true congruence of (E, τ ). It can be easily shown that gh(θ) = θ and hg(F ) = F , for all very true congruence θ and very true filter F of (E, τ ). As another applications of very true filter on very true equality algebra, we consider the uniformity structure on a very true equality algebra. 518

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Theorem 4.13. Let (E, τ ) be a very true equality algebra and F be a very true filter of (E, τ ). Define UF = {(x, y) ∈ E × E|τ (x) ∼ τ (y) ∈ F } and K ∗ = {UF |F is a very true filter of (E, τ )}. Then K ∗ satisfies the conditions (U1 )-(U4 ). Proof. Now, we will show that K ∗ satisfies the conditions (U1 )-(U4 ). (U1 ) Let UF ∈ K ∗ and (x, x) ∈ 4. Since x ∼ x = 1 ∈ F , we have (x, x) ∈ UF . Therefore (U1 ) holds. (U2 ) Note that (x, y) ∈ UF if and only if τ (x) ∼ τ (y) ∈ F if and only if τ (y) ∼ τ (x) ∈ F if and only if (y, x) ∈ UF if and only if (x, y) ∈ UF−1 . Thus UF−1 = UF ∈ K ∗ . Therefore (U2 ) is true. (U3 ) Let Σ(F ) = {Fa |Fa ⊆ F } be the collection of very true filters contained in F . Clearly, Σ(F ) is not empty. Let G be the very true filter generated by ∪a Fa . Then UG ⊆ K ∗ . It is sufficient to show that UG ◦ UG ⊆ UF . If (x, y) ∈ UG ◦ UG , then there exists z ∈ E such that (x, z) ∈ UG and (z, y) ∈ UG . It follows from (E7) in Definition 2.1 that (x, y) ∈ UG , that is, τ (x) ∼ τ (y) ∈ G. Since G is the minimal very true filter containing ∪a Fa and ∪a Fa ⊆ F , it follows that G ⊆ F . Thus τ (x) ∼ τ (y) ∈ F and hence (x, y) ∈ UF . Therefore UG ◦ UG ⊆ UF is true. (U4 ) This will from the observation that UG ∩ UF = UG∩F for all UG , UF ∈ K ∗ . Indeed, if (x, y) ∈ UG ∩ UF . Then (x, y) ∈ UG and (x, y) ∈ UF , which implies τ (x) ∼ τ (y) ∈ G and τ (x) ∼ τ (y) ∈ F . Thus τ (x) ∼ τ (y) ∈ G ∩ F and hence (x, y) ∈ UG∩F . Similary, we can show that UF ∩G ⊆ UF ∩UG , whence UF ∩G = UF ∩UG . This completeness the proof. Theorem 4.14. Let (E, τ ) be a very true equality algebra. Define K = {U ∈ E ×E|UF ⊆ U , for some UF ∈ K ∗ }. Then K satisfies a uniformity on (E, τ ) and hence the pair ((E, τ ), K) is a uniform space. Proof. Using Theorem 4.13, we can show that K satisfies the conditions (U1 ) − (U4 ). Now, we will show that it satisfies (U5 ). If U ∈ K and U ⊆ V ⊆ E × E. Then there exists a UF ∈ K ∗ such that UF ⊆ U ⊆ V , which implies that V ∈ K. Therefore,((E, τ ), K) is a uniform space. For x ∈ E and U ∈ K, we define U [x] = {y ∈ E|(x, y) ∈ U }. Theorem 4.15. Let (E, τ ) be a very true equality algebra. Define T = {G ⊆ E|τ x ∈ G, ∃U ∈ K, U [x] ⊆ G}. Then T is a topology on (E, τ ). Proof. It is clear ∅ ∈ T and E ∈ T . Also from the nature of that definition, it is clear that T is closed under arbitrary union. Finally to show that T is closed under finite intersection, let G, W ∈ T and suppose x ∈ G ∩ W . Then there exist U and V ∈ K such that U [x] ⊆ G and V [x] ⊆ W . Let N = U ∩ V , then N ∈ K. Also N [x] ⊆ U [x] ∩ V [x] and hence N [x] ⊆ G ∩ W , thus G ∩ W ∈ T . Therefore T is a topology on (E, τ ). Note We denote the uniform topology obtained by an arbitrary family Λ, by TΛ and If Λ = {F }, we denote it by TF . Theorem 4.16. Let (E, τ ) be a very true equality algebra. For each x ∈ E, the collection Ux = {U [x]|U ∈ K} forms a neighborhood base at x, making (E, τ ) a topological space. 519

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Proof. We note that x ∈ U [x] for each x. Since U1 [x] ∩ U2 [x] = (U1 ∩ U2 )[x], which means that the intersection of neighborhoods is a neighborhood. Finally, if U [x] ∈ Ux then there exists a W ∈ K such that W ◦ W ⊆ U by (U3 ). Then for any y ∈ W [x], W [y], so this property of neighborhoods is satisfied. Theorem 4.17. Let (E, τ ) be a very true equality algebra and (E,TKer(τ ) ) be a compact topological space. Then τ (E) is finite. Proof. Since E = ∪{UF [x]|x ∈ E} and (E, TF ) is a compact topological space. Thus there exist x1 , x2 · · · , xn ∈ E such that E = ∪UF [xi ]. Now, let x be a arbitrary element of E. By Theorem 3.7, τ : E −→ τ (E) is a homomorphism and by the Note after the Theorem 3.6, (τ (E), , 1) is an equality algebra, hence we have τ (UF [x]) = {τ (y)|y ∈ UF [x]} = {τ (y)|(τ (x), τ (y) ∈ θF } = {τ (y)|τ (x) ∼ τ (y) ∈ F } = {τ (y)|τ (x) = τ (y)} = {τ (x)}. Thus τ (UF [x]) = {τ (x)} and so F ixτ = ∪UF [xi ] = {τ (x1 ), · · · , τ (xn )}, that is, τ (E) is finite. Corollary 4.18. Let (E, τ ) be a very true equality algebra and (E,TKer(τ ) ) be a compact topological space. Then τ (E) is a compact subset of E. Proof. It follows from Theorem 4.17 directly and hence we omit it. 5. Conclusion As we mentioned in the introduction, the study of equality algebras is motivated by the goal to develop appropriate algebraic semantics for fuzzy type theory, so a concept of fuzzy type theory should be introduced based on these algebras. In this paper, motivating by the previous research on very true residuated lattice, very true MV-algebras and very true BL-algebras, we extended the concept of very true operators to the more general fuzzy structures, namely equality algebras. We introduce and study very true equality algebras and prove some new results regarding these structures. The aim of this paper is to provide an algebraic foundation for fuzzy type theory. Since the above topics are of current interest we suggest further directions of research: 1. Characterize very true filter generated by a subset of an very true equality algebra in terms of fuzzy equality operation. 2. Define and characterize subdirectly irreducible very true equality algebras. 3. Establish the logic system corresponding to very true equality algebra and prove the soundness and completeness theorem of this logic. Acknowledgments This research is partially supported by a grant of National Natural Science Foundation of China (11571281). References [1] A.B.Saeid, H.Babaei, M. Haveshki, Uniform Topology on Hilbert algebras, Kyungpook Math. J, 45(2005),405-411. [2] I. Leu¸stean, Non-commutative Lukasiewicz propositional logic, Arch. Math. Logic, 45 (2006),191-213. 520

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ˆ [3] J. Rachunek, D. Salounov´ a. Truth values on generalizations of some commutative fuzzy structures. Fuzzy Sets Syst, 157(2006),3159-3168. [4] L.C.Ciungu. Internal states on equality algebras, Soft Comput,19(2014), 939-953. [5] L.Zadeh. Fuzzy logic and approximate reasoning, Synthese, 30 (1975), 407-428. [6] M. El-Zekey, V. Nov´ ak, R. Mesiar, On good EQ-algebras, Fuzzy Sets and Systems 178 (2011), 1-23. [7] M. El-Zekey, Representable good EQ-algebras, Soft Comput, 14 (2010) 1011-1023. [8] P. H´ajek. On very true, Fuzzy Sets Syst, 124 (2001),329-333. [9] P. Andrews. An Itroduction to Mathematical Logic and Type Theory: To Truth Through Proof, Kluwer, Dordrecht, 2002. [10] R.B˘ elohl´ avek, V. Vyhodil, Reducing the size of fuzzy concept lattices by hedges. In: FUZZ-IEEE 2005, the IEEE internationnal conference on fuzzy systems, Reno, NV, USA, 65-74. [11] S. Jenei, Equality algebras, Studia Logica, 100 (2012),1201-1209. [12] S. Jenei, L. Korodi, Pseudo equality algebras, Arch. Math. Logic, 52(2013)469-481. [13] V. Nov´ ak, From fuzzy type theory to fuzzy intensional logic, in: Proceedings of the 3rd Conference of the European Society for Fuzzy Logic and Technology, Zittau, Germany, 2003. [14] V. Nov´ ak, Fuzzy type theory as higher order fuzzy logic, in: Proceedings of the 6th International Conference on Inteligent Techologies, Bangkok, Thailand, 2005. [15] V. Nov´ ak, On fuzzy type theory, Fuzzy Sets Syst, 149 (2005), 235-273. [16] V. Nov´ ak, EQ-algebra based fuzzy type theory and its exetensions, Log, J.IGPL, 19 (2011), 512-542. [17] V. Nov´ ak, B.De Bates, EQ-algebras, Fuzzy Sets Syst, 160 (2009), 2956-2978. [18] Y.B.Jun, E.H.Ron. On uniformities of BCK-algebras. Comm.Korean Math. Soc, 10(1995), 11-14. [19] Y.B.Jun, H.S.Kim. Uniformity structures in positive implicative algebras, International Mathematical Journal, 2(2002),215-218. [20] Z.D.Wang, J.X.Fang, On v-filters and normal v-filters of a residuated lattice with a weak vt-operator, Information Science, 178(2008), 3465-3473.

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A Splitting Iterative Method for a System of Accretive Inclusions in Banach Spaces Birendra Kumar Sharma1 , Niyati Gurudwan2 , Avantika Awadhiya3 and Shin Min Kang4,5,∗ 1

2

School of Studies in Mathematics, Pt. Ravishankar Shukla University, Raipur 492010, India e-mail: [email protected]

Department of Mathematics, Govt. Rajeev Lochan College, Rajim, Gariaband 493885, India e-mail: [email protected]

3

School of studies in Mathematics, Pt. Ravishankar Shukla University, Raipur 492010, India e-mail: [email protected] 4

5

Center for General Education, China Medical University, Taichung 40402, Taiwan e-mail: [email protected]

Department of Mathematics and RINS, Gyeongsang National University, Jinju 52828, Korea e-mail: [email protected] Abstract In this paper, a system of accretive inclusions is proposed and a splitting iterative method is investigated for solutions of proposed system of operator inclusion problems. Under suitable conditions on the parameters, strong convergence of our splitting iterative method is established in a reflexive Banach space. 2010 Mathematics Subject Classification: 47H06, 47H09 Key words and phrases: Accretive operator, reflexive Banach space, splitting iteration, resolvent

1

Introduction

In the area of nonlinear analysis, the theory of accretive operators is an important and developing field [3, 4]. The class of accretive operators is firmly connected with equations of evolutions found in the heat, wave, Schr¨odinger and similar other equations [5]. Many problems in operations research and mathematical physics can be written as variational inequalities, equilibrium problems or operator inclusions with accretive operators [2, 10, 17]. Let H be a real Hilbert space whose inner product and norm are denoted by h·, ·i and k · k, respectively. One popular method for solving the following inclusion problem: find z ∈ H such that 0 ∈ Az, ∗

(1.1)

Corresponding author

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where A : H → 2H is an m-accretive operator, is the proximal point algorithm, which was proposed by Martinet [15, 16] and generalized by Rockafellar [20, 21]. Rockafellar [20] proved the weak convergence of the sequence {xn } defined by xn+1 = JrAn xn ,

n∈N

(1.2)

to an element of solution set of problem (1.1). The weak and strong convergence of the sequence {xn } defined by (1.2) have been extensively discussed in Hilbert and Banach spaces (see [7, 28, 29, 30, 31] and the references therein). The proximal point like methods for finding solutions of problem (1.1) have been studied by Lehdili and Moudafi [12] and Tossings [26] in Hilbert spaces and by Sahu and Yao [23] in Banach spaces. Many nonlinear problems arising in applied areas such as image recovery, signal processing, and machine learning can be mathematically modeled in form of inclusion problem: to find z ∈ C such that 0 ∈ (A + B)z, (1.3) where C is a nonempty closed convex subset of H, A : H → 2H and B : C → H are monotone operators. For instance, a stationary solution to the initial value problem of the evolution equation ∂u 0∈ + F u, u0 = u(0) ∂t can be recast as (1.3) when the governing maximal monotone F is of the form F = A + B, see, for example, [13]. The central problem is to iteratively find the solution of the inclusion problem (1.3) when A and B are two monotone operators in a Hilbert space H. One method for finding solutions of problem (1.3) is splitting method. A splitting method for (1.3) means an iterative method for which each iteration involves only with the individual operators A and B, but not the sum A + B. Splitting methods for linear equations were introduced by Douglas and Rachford [8] and Peaceman and Rachford [18]. Extensions to nonlinear equations in Hilbert spaces were carried out by Kellogg [9] and Lions and Mercier [13] (see also [22, 27]). In this paper, we are interested in the following system of operator inclusion problems: find z ∈ C such that 0 ∈ (Ai + Bi )z,

i ∈ ∆N := {1, 2, · · · , N },

(1.4)

in the framework of a Banach space X, where N ≥ 1 is a positive integer, C is a nonempty closed convex subset of X, Ai : X → 2X is an m-accretive operator such T that i∈∆N D(Ai ) ⊆ C and Bi : C → X an operator. The inclusion problem (1.4) is more general in nature. For instance, if Bi is the operator constantly zero for all i ∈ ∆N , the problem (1.4) reduces find z ∈ C such that 0 ∈ Ai z,

i ∈ ∆N .

(1.5)

The purpose of this paper is to introduce a forward-backward splitting method to solve the system of operator inclusion problem (1.4) in a Banach space. We prove strong convergence of iterative sequences generated by our algorithm. In Section 2, we give duality mappings, nonexpansive mappings and their properties and accretive operators and their properties. In Section 3, we introduce a forward-backward splitting method and state main theoretical result of the paper. Our iterative method improves and generalizes the corresponding results of inclusion problem (1.5). 2 523

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2

Preliminaries

2.1

Duality mappings

A continuous strictly increasing function ϕ : R+ → R+ is said to be a gauge if ϕ(0) = 0 ∗ and limt→∞ ϕ(t) = ∞. The mapping Jϕ : X → 2X defined by Jϕ (x) = {j ∈ X ∗ : hx, ji = kxkkjk∗, kjk∗ = ϕ(kxk)},

x ∈ X,

is called the duality mapping with gauge ϕ. In the special case where ϕ(t) = t, the duality mapping Jϕ =: J is the classical normalized duality mapping. In the case ϕ(t) = tp−1 , p > 1, the duality mapping Jϕ =: Jp is called the generalized duality mapping and it is given by Jp (x) := {j ∈ X ∗ : hx, ji = kxkkjk∗, kjk∗ = kxkp−1 }, x ∈ X. Note that if p = 2, then J2 = J is the normalized duality mapping. By ϕ we always mean a gauge and by Φ the corresponding function defined by Z t Φ(t) = ϕ(s)ds. 0

For a smooth Banach space X, we have Φ(kx + yk) ≤ Φ(kxk) + hy, Jϕ(x + y)i for all x, y ∈ X;

(2.1)

or considering the normalized duality mapping J, we have kx + yk2 ≤ kxk2 + 2hy, J(x + y)i for all x, y ∈ X.

2.2

Nonexpansive mappings

Let C be a nonempty subset of a Banach space X and T : C → X a mapping. T is said to be nonexpansive if kT x − T yk ≤ kx − yk for all x, y ∈ C. The set of fixed point of T is denoted by F ix(T ). The following result was proved by Kirk [11]. Lemma 2.1. (Kirk [11]) Let X be a reflexive Banach space and let C be a nonempty closed convex bounded subset of X which has normal structure. Let T be a nonexpansive mapping of C into itself. Then T has a fixed point. A subset C of a Banach space X is called a retract of X if there exists a continuous mapping P from X onto C such that P x = x for all x in C. We call such P a retraction of X onto C. It follows that if a mapping P is a retraction, then P y = y for all y in the range of P . A retraction P is said to be sunny if P (P x + t(x − P x)) = P x for each x in X and t ≥ 0. If a sunny retraction P is also nonexpansive, then C is said to be a sunny nonexpansive retract of X. The following lemmas will be useful for our main result. 3 524

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Lemma 2.2. ([1, Corollary 5.6.4]) Let X be a Banach space with a weakly continuous duality mapping Jϕ : X → X ∗ with gauge function ϕ. Let C be a nonempty closed convex subset of X and T : C → C an nonexpansive mapping. Then I − T is demiclosed at zero, that is, if {xn } is a sequence in C which converges weakly to x and if the sequence {xn − T xn } converges strongly to zero, then x − T x = 0. Lemma 2.3. ([6]) Let X be a strictly convex Banach space. Let C be a nonempty closed convex subset of X and let N ≥ 1 be a positive integer. For each i ∈ ∆N , let Ti : C → C T be a nonexpansive mapping such that i∈∆N F ix(Ti) 6= ∅. Let {δi }i∈∆N ⊂ (0, 1) such P PN PN that N i=1 δi = 1. Then the mapping i=1 δi Ti is nonexpansive with F ix( i=1 δi Ti) = T i∈∆N F ix(Ti ).

Lemma 2.4. Let C be a convex subset of a smooth Banach space X, D a nonempty subset of C and P a retraction from C onto D. Then the following are equivalent: (a) P is a sunny and nonexpansive. (b) hx − P x, J(z − P x)i ≤ 0 for all x ∈ C, z ∈ D. (c) hx − y, J(P x − P y)i ≥ kP x − P yk2 for all x, y ∈ C. Lemma 2.5. Let X be a reflexive Banach space which has a weakly continuous duality map Jϕ . Let C be a nonempty closed convex subset of X and T : C → C a nonexpansive mapping such that F ix(T ) 6= ∅. Then F ix(T ) is the sunny nonexpansive retract of C. The property (N ) alludes to the fact that in order to solve the system of operator inclusions (1.4). Definition 2.6. ([22]) Let C be a nonempty closed convex subset of a Banach space X. An operator B : C → X is said to satisfy the property (N ) on (0, γX,B ) if there exists γX,B ∈ (0, ∞], depends on X and B, such that I − ξB : C → C is nonexpansive for each ξ ∈ (0, γX,B ). Remark 2.7. For a nonexpansive mapping T : C → C with B = I − T , the average mapping Tw = I − wB is always nonexpansive for each w ∈ (0, γX,B ), where γX,B = 1.

2.3

Accretive operators

Let X be a real Banach space. For an operator A : X → 2X , we define its domain, range and graph as follows: D(A) = {x ∈ X : Ax 6= ∅}, [ R(A) = {Az : z ∈ D(A)}, and

G(A) = {(x, y) ∈ X × X : x ∈ D(A), y ∈ Ax}, respectively. Thus, we write A : X → 2X as follows: A ⊆ X × X. The inverse A−1 of A is defined by x ∈ A−1 y if and only if y ∈ Ax. The operator A is said to be accretive if for each xi ∈ D(A) and yi ∈ Axi (i = 1, 2), there exists j ∈ J(x1 − x2 ) such that hy1 − y2 , ji ≥ 0. An accretive operator A is said 4 525

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to be maximal accretive if there is no proper accretive extension of A and m-accretive if R(I +A) = X, where I stands for the identity operator on X (It follows that R(I +rA) = X for all r > 0). If A is m-accretive, then it is maximal accretive, but the converse is not true in general. If A is accretive, then we can define, for each λ > 0, a nonexpansive single-valued mapping JλA : R(1 + λA) → D(A) by JλA = (I + λA)−1 . It is called the resolvent of A. It is well known that if A is an m-accretive operator on a Banach space X, then for each λ > 0, the resolvent JλA = (I + λA)−1 is a single-valued nonexpansive mapping whose domain is entire space X. An accretive operator A defined on a Banach space X is said to satisfy the range condition if D(A) ⊂ R(1 + λA) for all λ > 0, where D(A) denotes the closure of the domain of A. It is well known that for an accretive operator A which satisfies the range condition, A−1 (0) = F ix(JλA ) for all λ > 0. We also define the Yosida approximation Ar by Ar = (I − JrA )/r. We know that Ar x ∈ AJrA x for all x ∈ R(I + rA) and kAr xk ≤ |Ax| = inf{kyk : y ∈ Ax} for all x ∈ D(A) ∩ R(I + rA). We also know the following [25]: For each λ, µ > 0 and x ∈ R(I + λA) ∩ R(I + µA), it holds that

A

J x − JµA x ≤ |λ − µ| kx − J A xk. λ λ λ

Lemma 2.8. ([22]) Let C be a nonempty closed convex subset of a Banach space X, T A ⊆ X × X an accretive operator such that D(A) ⊆ C ⊆ t>0 R(I + tA) and B : C → X an operator such that Zer(A+B) 6= ∅ and B has the property (N ) on (0, γX,B ), where γX,B is a constant depends on X and B. For r ∈ (0, γX,B ), define an operator JrA,B : C → C by JrA,B x = JrA (I − rB)x, x ∈ C. Then the following statements hold. (a) JrA,B is nonexpansive. (b) F ix(JrA,B ) = Zer(A + B).

Lemma 2.9. ([24]) Let {xn } and {yn } be bounded sequences in a Banach space X and let {βn } be a sequence in [0, 1] with 0 < lim inf n→∞ βn ≤ lim supn→∞ βn < 1. Suppose that xn+1 = (1 − βn )xn + βn yn

for all n ∈ N

and lim sup(kyn+1 − yn k − kxn+1 − xn k) ≤ 0. n→∞

Then limn→∞ kyn − xn k = 0. Lemma 2.10. ([14]) Let {an } and {cn } be two sequences of nonnegative real numbers and let {bn} be a sequence in R satisfying the following condition: an+1 ≤ (1 − αn )an + bn + cn for all n ∈ N, P where {αn } is a sequence in (0, 1]. Assume that ∞ n=1 cn < ∞. Then the following statements hold: 5 526

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(a) If bn ≤ Kαn for all n ∈ N and for some K ≥ 0, then an+1 ≤ δn a1 + (1 − δn )K +

n X

cj

for all n ∈ N,

j=1

Q where δn = nj=1 (1 − αj ) and hence {an } is bounded. P bn ∞ (b) If ∞ n=1 αn = ∞ and lim supn→∞ αn ≤ 0, then {an }n=1 converges to zero.

3

Main results

Now we are ready to prove our main result for solving the system of operator inclusions (1.4) in the framework of Banach space. Theorem 3.1. Let X be a strictly convex and reflexive Banach space which has a weakly continuous duality map Jϕ . Let C be a nonempty closed convex subset of X and let N ≥ 1 be a positive integer. Let f : C → C be a contraction mapping with Lipschitz constant kf . T For each i ∈ ∆N , let Ai : X → 2X be an m-accretive operator such that i∈∆N D(Ai) ⊆ C T and Bi : C → X an operator such that S := i∈∆N Zer(Ai + Bi ) 6= ∅. For each i ∈ ∆N , let Bi has the property (N ) on (0, γX,Bi ), where γX,Bi is a constant depends on X and Bi . Let {αn }, {βn} and {γn } be real number sequences in (0, 1) and let {δn,i } be a real number sequence in (0, 1) for each i ∈ ∆N satisfying the following conditions: P (C1) αn + βn + γn = N δ = 1 for all n∈ N, i=1 P n,i (C2) limn→∞ αn = 0 and ∞ n=1 αn = ∞, (C3) 0 < lim inf n→∞ βn ≤ lim supn→∞ βn < 1, (C4) limn→∞ δn,i = δi ∈ (0, 1) for all i ∈ ∆N . Let {xn } be a sequence in C generated by the following splitting iterative method: xn+1 = αn f (xn ) + βn xn + γn

N X

δn,i JrAi i (I − ri B)xn

for all n ∈ N,

(3.1)

n=1

where {ri}i∈∆N is a set of positive real numbers. Then {xn } converges strongly to x∗ ∈ C, which is the unique solution to the following variational inequality: \ to find z ∈ Zer(Ai + Bi ) such that h(I − f )z, x − zi ≥ 0 (3.2) i∈∆N

for all x ∈

T

i∈∆N

Zer(Ai + Bi ).

P Proof. (a) Define T = N JrAi i (I − riB). From Lemmas 2.3 and 2.8, we see that T is i=1 δiT nonexpansive with F ix(T ) = i∈∆N Zer(Ai + Bi ). From Lemma 2.5 shows that S is the Sunny nonexpansive retract of C. Let QS be the sunny nonexpansive retraction of C onto S. It follows that QS f is a contraction. Hence there exists a unique fixed point x∗ ∈ C of QS f. From Lemma 2.4 that the variational inequality problem (3.2) has a unique solution x∗ ∈ C. (b) We proceed with the following steps: Step I: {xn } is bounded. 6 527

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From (3.1), we have

N

X

Ai ∗ kxn+1 − x k ≤ αn kf (xn ) − x k + βn kxn − x k + γn δn,i Jri (I − ri B)xn − x

∗

∗

∗

i=1

≤ kf αn kxn − x∗ k + αn kf (x∗ ) − x∗ k + βn kxn − x∗ k + γn

N X

δn,i kJrAi i (I − ri B)xn − x∗ k

i=1

≤ kf αn kxn − x∗ k + αn kf (x∗ ) − x∗ k + (1 − αn )kxn − x∗ k = (1 − (1 − kf )αn )kxn − x∗ k + αn kf (x∗ ) − x∗ k   ∗ ∗ ∗ kf (x ) − x k ≤ max kxn − x k, 1 − kf   ∗ ) − x∗ k kf (x ≤ max kx1 − x∗ k, . 1 − kf Thus, {xn } is bounded. Step II: kxn+1 − xn k → 0 and kxn − T xn k → 0 as n → ∞. P Ai Set yn = N n=1 δn,i Jri (I − riB)xn . Note

N

N

X

X

kyn+1 − yn k = δn+1,i JrAi i (I − ri B)xn+1 − δn,i JrAi i (I − ri B)xn

i=1 i=1

N N

X X

Ai Ai δn+1,i Jri (I − ri B)xn δn+1,i Jri (I − ri B)xn+1 − ≤

i=1 i=1

N N

X X

δn,i JrAi i (I − riB)xn δn+1,i JrAi i (I − riB)xn − +

i=1

≤

N X

i=1

δn+1,i kJrAi i (I − ri B)xn+1 − JrAi i (I − riB)xn k

i=1

N

X

+ (δn+1,i − δn,i )JrAi i (I − riB)xn

i=1

≤ kxn+1 − xn k +

N X

|δn+1,i − δn,i | kJrAi i (I − ri B)xn k.

i=1

From (3.1), we have xn+1 = αn f (xn ) + βn xn + γn yn = βn xn + (1 − βn )zn , where zn =

1 [αn f (xn ) + γn yn ] . 1 − βn 7 528

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Hence 1 1 [αn+1 f (xn+1 ) + γn+1 yn+1 ] − [αn f (xn ) + γn yn ] 1 − βn+1 1 − βn 1 = [αn+1 f (xn+1 ) + (1 − αn+1 − βn+1 )yn+1 ] 1 − βn+1 1 − [αn f (xn ) + (1 − αn − βn )yn ] 1 − βn αn+1 αn [f (xn ) − yn ] + yn+1 − yn . = [f (xn+1 ) − yn+1 ] − 1 − βn+1 1 − βn

zn+1 − zn =

Note αn+1 αn kf (xn+1 ) − yn+1 k + kf (xn ) − yn k + kyn+1 − γn k 1 − βn+1 1 − βn αn αn+1 ≤ kf (xn+1 ) − yn+1 k + kf (xn ) − yn k 1 − βn+1 1 − βn N X |δn+1,i − δn,i | kJrAi i (I − riB)xn k. + kxn+1 − xn k +

kzn+1 − zn k ≤

i=1

Thus, we have kzn+1 − zn k − kxn+1 − xn k ≤

αn+1 αn kf (xn+1 ) − yn+1 k + kf (xn ) − yn k 1 − βn+1 1 − βn N X + |δn+1,i − δn,i |kJrAi i (I − ri B)xn k. i=1

From the conditions (C1)-(C4), we get lim sup(kzn+1 − zn k − kxn+1 − xn k) ≤ 0. n→∞

From Lemma 2.9, we obtain that lim kzn − xn k = 0.

n→∞

Since xn+1 − xn = (1 − βn )(zn − xn ), we have kxn+1 − xn k = (1 − βn )kzn − xn k ≤ kzn − xn k → 0 as n → ∞. Observe that kxn − T xn k ≤ kxn − xn+1 k + kxn+1 − T xn k ≤ kxn − xn+1 k + αn kf (xn ) − T xn k + βn kxn − T xn k

N

X

+ γn (δn,i − δi )JrAi i (I − riB)xn

i=1

≤ kxn − xn+1 k + αn kf (xn ) − T xn k + βn kxn − T xn k + γn

N X

|δn,i − δi | kJrAi i (I − riB)xn k,

i=1

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which implies that (1 − βn )kxn − T xn k ≤ kxn − xn+1 k + αn kf (xn ) − T xn k + γn

N X

|δn,i − δi | kJrAi i (I − ri B)xn k.

i=1

Thus, we have lim kxn − T xn k = 0.

n→∞

Step III: {xn } converges strongly to x∗ . Take a subsequence {xni } of {xn } such that lim suphf (x∗ ) − x∗ , Jϕ (xn − x∗ )i = lim hf (x∗ ) − x∗ , Jϕ(xni − x∗ )i. i→∞

n→∞

Since X is reflexive, we may further assume that xni * z for some z ∈ C. It follows from Lemma 2.2 that z ∈ F ix(T ). From the weak continuity of the duality mapping Jϕ and (3.2) we obtain that lim suphf (x∗ ) − x∗ , Jϕ(xn − x∗ )i = lim hf (x∗ ) − x∗ , Jϕ (xni − x∗ )i i→∞

n→∞

= hf (x∗ ) − x∗ , Jϕ(z − x∗ )i ≤ 0. From (2.1) and (3.1), we have Φ(kxn+1 − x∗ k) = Φ(kαnf (xn ) + βn xn + γn yn − x∗ k) ≤ Φ(kαn(f (xn ) − f (x∗ ) + βn (xn − x∗ ) + γn (yn − x∗ ) + αn (f (x∗ ) − x∗ )k) ≤ Φ(kαn(f (xn ) − f (x∗ ) + βn (xn − x∗ ) + γn (yn − x∗ )k) + αn hf (x∗ ) − x∗ , Jϕ(xn+1 − x∗ )i ≤ Φ(αn kf (xn ) − f (x∗ )k + βn kxn − x∗ k + γn kyn − x∗ k) + αn hf (x∗ ) − x∗ , Jϕ(xn+1 − x∗ )i ≤ Φ((1 − (1 − kf )αn )kxn − x∗ k) + αn hf (x∗ ) − x∗ , Jϕ (xn+1 − x∗ )i ≤ (1 − (1 − kf )αn )Φ(kxn − x∗ k) + αn hf (x∗ ) − x∗ , Jϕ (xn+1 − x∗ )i. P Noticing that lim supn→∞ hf (x∗ ) − x∗ , Jϕ(xn+1 − x∗ )i ≤ 0 and ∞ n=1 αn = ∞. Therefore, we conclude from Lemma 2.10 that Φ(kxn − x∗ k) → 0, that is, {xn } converges strongly to x∗ . Theorem 3.1 is more general in nature due to the property (N ) of operators Bi , therefore, we are able to derive the some new and known results from it. To demonstrate the wide range of applicability of our convergence theory, a few examples are detailed below. In particular for Bi = 0, we immediately obtain an improvement upon Qing and Lv [19, Theorem 2.1] as follows: 9 530

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Corollary 3.2. Let X be a strictly convex and reflexive Banach space which has a weakly continuous duality map Jϕ . Let C be a nonempty closed convex subset of X and let N ≥ 1 be a positive integer. Let f : C → C be a contraction mapping with Lipschitz constant kf . T For each i ∈ ∆N , let Ai : X → 2X be an m-accretive operator such that i∈∆N D(Ai) ⊆ C T such that i∈∆N A−1 i (0) 6= ∅. Let {αn }, {βn} and {γn} be real number sequences in (0, 1) and let {δn,i } be a real number sequence in (0, 1) for each i ∈ ∆N satisfying the conditions (C1)-(C4). Let {xn } be a sequence in C generated in the following iterative process: xn+1 = αn f (xn ) + βn xn + γn

N X

δn,i JrAi i xn

for all n ∈ N,

n=1

where {ri}i∈∆N is a set of positive real numbers. Then {xn } converges strongly to x∗ ∈ C, which is the unique solution to the following variational inequality: \ to find z ∈ A−1 i (0) such that h(I − f )z, x − zi ≥ 0 i∈∆N

for all x ∈

T

i∈∆N

A−1 i (0).

Theorem 3.3. Let X be a strictly convex and reflexive Banach space which has a weakly continuous duality map Jϕ . Let C be a nonempty closed convex subset of X and let N ≥ 1 be a positive integer. Let f : C → C be a contraction mapping with Lipschitz constant kf . T For each i ∈ ∆N , let Ai : X → 2X be an m-accretive operator such that i∈∆N D(Ai) ⊆ C T and Ti : C → C a nonexpansive with Bi = I − Ti such that S := i∈∆N Zer(Ai + Bi ) 6= ∅. Let {αn }, {βn} and {γn } be real number sequences in (0, 1) and let {δn,i } be a real number sequence in (0, 1) for each i ∈ ∆N satisfying the conditions (C1)-(C4). Let {xn } be a sequence in C generated in the following iterative process: xn+1 = αn f (xn ) + βn xn + γn

N X

δn,i JrAi i (I − ri Bi )xn

for all n ∈ N,

n=1

where {ri}i∈∆N is a set of positive real numbers. Then {xn } converges strongly to x∗ ∈ C, which is the unique solution to the following variational inequality: \ to find z ∈ Zer(Ai + Bi ) such that h(I − f )z, x − zi ≥ 0 i∈∆N

for all x ∈

T

i∈∆N

Zer(Ai + Bi ).

Proof. Note each Ti is nonexpansive with Bi = I − Ti . It follows from Remark 2.7 that (i) the average mapping Tw = I − wBi is always nonexpansive for each w ∈ (0, γX,Bi), where γX,Bi = 1. Therefore, Theorem 3.3 follows from Theorem 3.1.

Acknowledgment The author are thankful to CSIR,New Delhi for the financial support under the research project No.25(0234)/14/EMR-II. Dated April 16, 2014. 10 531

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[17] A. Nagurney, Network Economics: a Variational Inequality Approach, Advances in Computational Economics, 1, Kluwer Academic Publishers Group, Dordrecht, 1993. [18] D. H. Peaceman and H. H. Rachford, The numerical solution of parabolic and elliptic differential equations, J. Soc. Industrial Appl. Math., 3 (1955), 28–41. [19] Y. Qing and S. Lv, Strong convergence of a parallel iterative algorithm in a reflexive Banach space, Fixed Point Theory Appl., 2014, 125 (2014), 9 pages. [20] R. T. Rockafellar, Monotone operators and the proximal point algorithm, SIAM J. Control Optim., 14 (1976), 877–898. [21] R. T. Rockafellar, Augmented Lagrangians and applications of the proximal point algorithm in convex programming, Math. Oper. Res., 1 (1976), 97–116. [22] D. R. Sahu, Q. H. Ansari and J. C. Yao, The prox-Tikhonov-like forward-backward method and applications, Taiwanese J. Math., 19 (2015), 481–503. [23] D. R. Sahu and J. C. Yao, The prox-Tikhonov regularization method for the proximal point algorithm in Banach spaces, J. Global Optim., 51 (2011), 641–655. [24] T. Suzuki, Strong convergence theorems for infinite families of nonexpansive mappings in general Banach spaces, Fixed Point Theory Appl., 2005 (2005), 103–123. [25] W. Takahashi, Nonlinear Functional Analysis, Fixed Point Theory and Its Applications, Yokohama Publishers, Yokohama, 2000. [26] P. Tossings, The perturbed proximal point algorithm and some of its applications, Appl. Math. Optim., 29 (1994), 125–159. [27] P. Tseng, Applications of a splitting algorithm to decomposition in convex programming and variational inequalities, SIAM J. Control Optim., 29 (1991), 119–138. [28] H. K. Xu, Strong convergence of an iterative method for nonexpansive and accretive operators, J. Math. Anal. Appl., 314 (2006), 631–643. [29] H. K. Xu, Iterative algorithms for nonlinear operators, J. London Math. Soc., 66 (2002), 240–256. [30] H. Zegeye and N. Shahzad, Strong convergence theorems for a common zero of a finite family of m-accretive mappings, Nonlinear Anal., 66 (2007), 1161–1169. [31] Q. N. Zhang and Y. S. Song, Halpern type proximal point algorithm of accretive operators, Nonlinear Anal., 75 (2012), 1859–1868.

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Sidi-Israeli Quadrature Method for Steady-State Anisotropic Field Problems by Direct Domain Mapping∗ Xin Luo

†

Jin Huang‡

Tai-Song Xiong§

Abstract In this paper, the two-dimensional steady-state anisotropic field problems are transformed into the Laplace equation by direct domain mapping, and then the Sidi-Israeli quadrature method is applied to solve the weakly singular boundary integral equation of the Laplace equation. Especially, the kress’s variable transformation is used for the polygon case in order to improve the accuracy by smoothing the singularities of the exact solution at the corner points of the boundary. The convergence and error analysis of numerical solutions are given by use of collective compact theory. At last, numerical examples are tested and results verify the theoretical analysis. Keyword : Boundary integral equation, singularity, variable transformation, convergence

1

Introduction

Consider an anisotropic medium in domain Ω ∈ R2 bounded by its boundary Γ = ∪m j=1 Γj (m ≥ 1) which may consist of m segments each being sufficiently smooth ( in the sense of Liapunov ). In the absence of heat sources, the equation governing steady-state heat conduction with Dirichlet condition can be described as (see as [1, 2, 5])  2  κ ∂ u(x) = 0, (i, j = 1, 2), ij ∂xi ∂xj (1.1)  u(x) = g, x ∈ Γ, ∗

This work is supported by Project (NO. KYTZ201505) Supported by the Scientific Research Foundation of CUIT † College of Applied Mathematics, Chengdu University of Information Technology, Chengdu 610225, P.R. China, corresponding author: [email protected] ‡ School of Mathematical Sciences, University of Electronic Science and Technology of China, Chengdu 611731, P.R. China § College of Applied Mathematics, Chengdu University of Information Technology, Chengdu 610225, P.R. China

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where u represents the temperature, and κ = {κij }1≤i,j≤2 denotes the thermal conductivity matrix which satisfies the symmetry (κ12 = κ21 ) and positive-definite (|κ| = κ11 κ22 − κ212 > 0) conditions. Using the following coordinate transformation [2] p |κ| κ12 x1 , xˆ2 = x2 − x1 , (1.2) xˆ1 = κ11 κ11 Eq. (1.1) can be written as the ’isotropic’ Laplacian form in a mapped plane in the transformed xˆi −system x) ∂ 2 u(ˆ x) ∂ 2 u(ˆ + = 0, 2 2 ∂ xˆ1 ∂ xˆ2

xˆ = (ˆ x1 , xˆ2 ) ∈ Ω′ .

(1.3)

Then by single-layer potential theory [8], Eq. (1.3) can be converted into the following weakly singular boundary integral equation (BIE): Z 1 v(ˆ x) ln |ˆ x − yˆ|dsxˆ = g(ˆ y ), yˆ ∈ Γ′ , (1.4) − 2π Γ′ where Γ′ is the boundary of the transformed domain Ω′ . The solution of Eq. (1.4) exists and is unique as long as CΓ′ 6= 1, where CΓ′ is the logarithmic capacity [11, 12]. As soon as v(ˆ x) is solved from (1.4), the solution u(ˆ x) of the problem (1.3) can be calculated by the following Z 1 v(ˆ x) ln |ˆ u(ˆ y) = − x − yˆ|dsxˆ , yˆ ∈ Ω′ . (1.5) 2π Γ′ Finally, using the inverse transformation of (1.2) κ11 x1 = p xˆ1 , |κ|

κ12 x2 = xˆ2 + p xˆ1 , |κ|

(1.6)

we can obtain the solution u(x) of the problem (1.1). As far as we know, the most popular numerical methods for engineering problems include, for example, the finite element method (FEM) [17], the finite difference method (FDM) [15], and the boundary element method (BEM) [11, 12]. The former two, used frequently in numerical modeling, are referred to as domain solution techniques and require full discretization of the whole domain and are often computationally costly and mathematically tricky in the volume mesh generation [1]. The BEM has been recognized as an efficient computational method that only the boundary needs to be modeled and owing the high approximation. For the application of BEM for the steady-state anisotropic heat conduction problems (1.1), various types of elements, namely constant, continuous and discontinuous linear elements and continuous and discontinuous quadratic elements has been investigated in the literature [5]. In this article, the Sidi-Israeli quadrature formula [3] is applied to calculate weakly singular integrals. Especially for the case of closed curved polygons Γ′ , we use the 2

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Kress’s variable transformation [4] to overcome the corner singularities and improve the accuracy at the boundary corners. This paper is organized as follows: in Section 2, the convergence and error analysis are carried out based on the theory of collectively compact operators [6, 7] for closed smooth boundaries. The Kress’s variable transformation is introduced to overcome the corner singularities for curved polygons in Section 3. Numerical examples are provided to verify the theoretical results in Section 4, and some useful conclusions are made in Section 5.

2

Sidi-Israeli quadrature method for boundary integral equation on smooth domain

Suppose that the boundary Γ′ (= ∂Ω′ ) to be a smooth closed curve and assume that the curve Γ′ can be parameterized by yˆ(t) = ϕ(t) = (ϕ1 (t), ϕ2 (t)) : [0, 2π) → ∂Ω′ . Then Eq. (1.4) can be written as Z 2π 1 ln ϕ(s) − ϕ(t) v(t)dt, (2.1) g(s) = − 2π 0

where v(t) = |ϕ′ (t)|v(ϕ(t)) and g(t) = g(ϕ(t)) are periodic functions with period 2π. In order to achieve high accuracy for numerical computation of finite-range integrals with weakly singular kernels, the following lemma about Sidi’s quadrature formula is introduced. Lemma 2.1. [3] Assume that the functions H1 (t, τ ) and H2 (t, τ ) are 2ℓ times differentiable on [0, 2π]. Assume also that the functions H(t, τ ) are periodic with period e = (−∞, ∞)\{τ + kT }∞ T = 2π, and that they are 2ℓ times differentiable on R k=−∞ . If a(t, τ ) = H1 (t, τ )ln|t − τ | + H2 (t, τ ), then Qn [a(t, τ )] = h

n X

a(tj , τ ) + H2 (tj , τ )h + ln(

j=0 tj 6=τ

2π h )H1 (tj , τ )h, h = , tj = jh, 2π n

and ℓ−1 ′ X
ζ (−2µ) ∂ 2µ H1 (tj , τ ) h2µ+1 + O (h2ℓ ), as h → 0, En [a(t, τ )] = 2 2µ (2µ)! ∂τ µ=1 R 2π where ζ(z) is a Riemann function [9, 13] and En [a(t, τ )] = 0 a(t, τ )dt − Qn [a(t, τ )]. Define the integral operator Z 2π (Lv)(s) = l(s, t)v(t)dt, (2.2) 0

with the kernel (l(s, t) = −

1 ln ϕ(s) − ϕ(t) , 2π 3

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and construct the Nystr¨om operator by the Sidi’s quadrature formula n−1 X  h h 2π ′ (L v)(s) = − ln ) + ln |ϕ (s)| v(s) + h . l(s, tj )v(tj ), tj = jh, h = 2π 2π n j=0 h

tj 6=s

Let V h = span{e0 (s), e1 (s), · · · , en (s)} ⊂ C[0, 2π) be a piecewise linear function subspace with nodes {sj }nj=0 , where ei (s) is the basis function satisfying ei (sj ) = δij . Define a prolongation operator P h : Rn → V h and a restricted operator Qh satisfying  h P v = v · e, v = (v0 , · · · , vn ), e = (e0 , · · · , en ) ∈ Rn , Qh v = (v(s0 ), · · · , v(sn )) ∈ Rn , v ∈ C[0.2π). Then Eq. (2.1) and its approximation equation are  Lv = g Lh v h = Qh g where Lh = [lij ]n−1 i,j=0 and the entries are lij =



hl(si , tj ), i 6= j,
′ h ln h|ϕ2π(ti )| , i = j. − 2π

Define the following integral operator Z 2π (A0 v)(s) = a0 (s, t)v(t)dt,

(2.3)

0

with the kernel a0 (s, t) = −

s − t 1 . ln 2e−1/2 sin 2π 2

Let L − A0 = A1 , then the integral equation (1.4) can be split into a singularity part and a compact perturbation part A0 v + A1 v = g,

(2.4)

where (A1 v)(s) = (v(.), a1 (s, .))L2 with the kernel  1/2 ϕ(s)−ϕ(t) 1 , s − t 6= 2πZ,   − 2π ln e 2sin s−t 2 a1 (s, t) =   − 1 ln e1/2 |ϕ′ (s)|
, s − t = 2πZ. 2π

Now we construct the approximations of A0 and A1 . For the logarithmically singular operators A0 , by the Sidi’s quadrature formula [3], we can construct the Fredholm approximation (Ah0 v)(s)

=h

n−1 X

a0 (s, tj )v(tj ),

(2.5)

j=0

4

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where

 1 s − tj
 ln 2e−1/2 sin s 6= tj , − 2π 2 a0 (s, tj ) = −1/2
 h  − 1 ln e s = tj , 2π 2π which has the following error bounds: (Ah0 v)(si )

2l−1 ′ −2 X ζ (−2µ) (2µ) [v(s)] − (A0 v)(si ) = h2µ+1 + O(h2l ). π µ=1 (2µ)! s=si

(2.6)

(2.7)

For the integral operators A1 with periodic kernels, we can construct the Nystr¨om approximation by the trapezoidal rule [8], (Ah1 v)(s)

=h

n−1 X

a1 (s, tj )v(tj ) j = 0, 1, · · · , n − 1,

i=0

which has the error bounds O (h2l ), l ∈ N. Consider the discrete approximation of (2.4) (Ah0 + Ah1 )v h = g h ,

(2.8)

n−1 h h h where v h = (v0h , v1h , · · · , vn−1 )T , Ah0 = [a0 (si , tj )]n−1 i,j=0 , A1 = [a1 (si , tj ]i,j=0 , and g = T (g(ϕ(s0)), · · · , g(ϕ(sn−1))) . Obviously, (2.8) is a linear equation system with n unknowns. Once v h is solved from (2.8), the solution of (1.5) u(ˆ y ) (ˆ y ∈ Ω′ ) can be computed by n h X h h v (si ) xˆ′ (si ) ln xˆ(si ) − yˆ . (2.9) u (ˆ y) = − 2π i=0

From (2.6), we have −1 h − 1 (n−1)h 
1 h 2e 2 sin 2e 2 sin ln e− 2 2π ln · · · ln 2 2
1 1 1 (n−2)h  ln 2e− 2 sin h −2 h −2 sin ln e · · · ln 2e h  2 2π 2 Ah0 = −  .. .. .. .. 2π  . − 1 . (n−1)h − 1 . (n−2)h .
1 h − ln 2e 2 sin 2 ln 2e 2 sin 2 ··· ln e 2 2π

By [10], we know that k(Ah0 )−1 k ≤ cn. Hence the Eq. (2.8) is equivalent to (E h + P h (Ah0 )−1 Qh Ah1 )v h = P h (Ah0 )−1 Qh g h ,



  .  (2.10)

where E h is the identity operator. n o Lemma 2.2. The operator sequence P h (Ah0 )−1 Qh Ah0 : C 3 [0, 2π) → C[0, 2π) is uniformly bounded and p P h (Ah0 )−1 Qh A0 → I.

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where → denotes the pointwisely convergence and I is the embedding operator . Proof. Let v ∈ C 3 [0, 2π) and v h be the solutions of the auxiliary equations A0 v = g and Ah0 v h = Qh g respectively, where Ah0 Qh v = Ah0 v(ti ) = h

n−1 X

a0 (si , tj )v(tj ) −

j=0 j6=i

and h

Q g = g(si) =

Z

he−1/2
1 ln v(ti ), 2π 2π

2π

a0 (si , t)v(t)dt.

0

From (2.7), we obtain that Qh g − Ah0 Qh v = O (h3 ), and

1

5

kQh g − Ah0 Qh vk2 = (n(O (h3 ))2 ) 2 = O (h 2 ).

(2.11)

By (2.11), the following holds kQh v − (Ah0 )−1 Qh A0 vk2 = kQh v − (Ah0 )−1 Qh gk2 = kQh v − v h k2 = k(Ah0 )−1 Ah0 (Qh v − v h )k2 1 ≤ kAh0 (Qh v − v h )k2 h 1 = kAh0 Qh v − Ah0 v h k2 h 3 1 = kQh g − Ah0 Qh vk2 = O (h 2 ), h the proof of Lemma 2.2 is completed.  Theorem 2.3. Assume that ∂Ω′ is a simply smooth and closed curve, Qh is a restricted operator and P h is a prolongation operator with nodes {si }ni=0 , then the operator sequence {P h (Ah0 )−1 Qh Ah1 } is collectively compactly convergent to A−1 0 A1 in C[0, 2π), that is, c.c P h (Ah0 )−1 Qh Ah1 → A−1 0 A1 . Proof. Since (P h (Ah0 )−1 Qh )(P h Ah1 Qh ) = (P h (Ah0 )−1 Qh A0 )((A0 )−1 P h Ah1 Qh ), we get k(P h (Ah0 )−1 Qh )(P h Ah1 Qh )k ≤ kP h (Ah0 )−1 Qh A0 k0,3 k(A0 )−1 P h Ah1 Qh k3,0 . c.c

p

h h −1 h From [10] and by Lemma 2.2, we have (A0 )−1 P h Ah1 Qh → A−1 0 A1 and P (A0 ) Q A0 → I. The proof of Theorem 2.3 is completed. 

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J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 24, NO.3, 2018, COPYRIGHT 2018 EUDOXUS PRESS, LLC h h Replacing (Ah0 )−1 , Ah0 , Ah1 , v h and g h by (Aˆ0 )−1 = P h (Ah0 )−1 Qh , Aˆ0 = P h Ah0 Qh h Aˆ1 = P h Ah1 Qh , vˆh = P h v h and gˆh = P h (Ah0 )−1 Qh g h , respectively. Then (2.10) can

, be written as

h h h (E h + (Aˆ0 )−1 Aˆ1 )ˆ v = gˆh .

(2.12)

Theorem 2.4. Assume that ∂Ω′ is a simply smooth and closed curve, v and vˆh are the solutions of (2.4) and (2.12), respectively, xi ∈ C 6 [0, 2π) and g(s) ∈ C 5 [0, 2π), then the following holds (ˆ v h − v) s=si = O (h3 ). (2.13) Proof. By the trapezoidal rule, the asymptotic expansion holds [10] (g − g h ) s=si = h3 P h Qh ϕ1 s=si + O (h5 ),

(2.14)

′

with ϕ1 (s) = −ζ (−2)g ′′(s)/π . Using (2.7) and (2.14), we can obtain P h (Ah0 + Ah1 )Qh (v h − v) s=si = g h − P h (Ah0 + Ah1 )Qh v s=si = g h − [((A0 + A1 )v − h3 P h Qh ϕ2 ) s=si + O (h5 )] = (g h − g) s=si + h3 P h Qh ϕ2 s=si + O (h5 ) = h3 P h Qh ϕ s=si + O (h5 ), ′

where ϕ2 (s) = −ξ (−2)v ′′ (s)/π, and ϕ(s) = ϕ1 (s) + ϕ2 (s). From Theorem 2.3, we have h h (E h + (Aˆ0 )−1 Ah1 )(v − vˆh ) s=si = h3 (Aˆ0 )−1 P h Qh ϕ(s) s=si + O (h5 ). (2.15) h Since (E h + (Aˆ0 )−1 Ah1 )−1 is uniformly bounded, we immediately get (ˆ v h − v) s=si = O (h3 ).

(2.16)



3

Corner singularity and convergence analysis

Definition 3.1. [14] A real-valued function γ is said to be a sigmoidal transformation if the following conditions are satisfied: (i) γ ∈ C 1 [0, 1] ∪ C ∞ (0, 1) with γ(0) = 0; (ii) γ(x) + γ(1 − x) = 1, 0 ≤ x ≤ 1; (iii) γ is strictly increasing on [0, 1] and its derivative γ ′ is strictly increasing on [0, 1/2] with γ ′ (0) = 0.

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0.9 0.8 0.7

γr (t)

0.6 0.5 0.4 0.3 0.2 0.1 0

0

0.2

0.4 0.6 Kress’s variable transformation

0.8

1

Figure 1: γr transformation

The Kress’s variable transformation is an example of ”algebraic” sigmoidal transformations, which was first proposed by Kress [4]. The function γ, defined on [0, 1] by γ(x) = f (x)/(f (x) + f (1 − x)), where f (x) = (x + cB3 (x))r , here r > 1, c is a constant to be determined and B3 is the Bernoulli polynomial of degree 3 defined by B3 (x) := x(x − 1/2)(x − 1). If we choose c = −8(1/r − 1/2), then we have γr (t) =

(θ(t))r Θ1 (t) = : [0, 1] → [0, 1], r ≥ 1, Θ2 (t) (θ(t))r + (θ(2π − 2πt))r

(3.1)

where θ(t) = ( 1r − 12 )(1 − 2t)3 + 1r (2t − 1) + 12 . The plots for γr is shown in Figure 1. ′ Assume that Γ′ = ∪m of a polygonal domain Ω′ in q=1 Γq (q > 1) be the boundary R2 , Γ′q ∈ C 2ℓ+1 (q = 1, ..., m, ℓ ∈ N), and let gq = g Γ′ . Define the boundary integral q operators on Γ′q , 1 (Lpq vq )(ˆ y) = − 2π

Z

Γ′q

vq (ˆ x) ln xˆ − yˆ dsxˆ , yˆ = (ˆ y1 , yˆ2 ) ∈ Γ′p (p, q = 1, ..., m). (3.2)

Then Eq. (1.4) can be converted into a matrix operator equation Lv = G,

(3.3)

where L = [lpq ]m y ), ..., gm (ˆ y ))T and v = (v1 (ˆ x), ..., vm (ˆ x))T . Assume p,q=1 , G = (g1 (ˆ ′ that Γq can be described by the parameter mapping: xˆq (s) = ϕ(s) = (ϕq1 (s), ϕq2 (s)) 8

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J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 24, NO.3, 2018, COPYRIGHT 2018 EUDOXUS PRESS, LLC ′ ′ ′ [0, 1] → Γ′q with |ϕq (s)| = [|ϕq1 (s)|2 + |ϕq2(s)|2 ]1/2 > 0. In order to degrade the

: singularities at corners, we apply the Kress’s variable transformation to (3.3) and give the following decomposition of Lpq , Lpq = A0(pp) + A1(pq) . The operators A0(pp) and A1(pq) are singular and compact operators respectively, and their Nystr¨om approximation are  R  (A0(pp) wp )(t) = 1 a0(pq) (t, τ )wp (τ )dτ, t ∈ [0, 1], 0 P −1/2 hp nm (A w )(t) = h hp wp (t), hp = 1/np , p j=1 a0(pp) (t, τj )wp (τj ) − hp ln e  0(pp) p t6=τj

and  R1  A1(pq) (wq )(t) = 0 a1(pq) (t, τ )wq (τ )dτ, t ∈ [0, 1], nq P hq a1(pq) (t, τj )wq (τj ), t ∈ [0, 1], τj = jh p, q = 1, ..., m,  (A1(pq) wq )(t) = hq j=1

where

a0(pp) (t, τ ) = − and

here

1 ln 2e−1/2 sin π(t − τ ) , 2π

 |ˆ xp (t) − yˆq (τ )| 1   ln −1/2 as p = q, − 2π |2e sinπ(t − τ )| a1(pq) (t, τ ) =  1  − ln |ˆ xp (t) − yˆq (τ )| as p 6= q, 2π xˆq (t) = (ϕq1 (γr (t)), ϕq2 (γr (t))), wq (t) = vq (ϕq (γr (t)))|ϕ′q (γr (t))|γr′ (t).

Then Eq. (3.3) and its discrete equations are  (A0 + A1 )W = G, (Ah0 + Ah1 )W h = Gh .

(3.4)

where A0 = diag(A0(11) , · · · , A0(mm) ),

A1 = [A1(pq) ]m p,q=1 ,

W = (w1 , · · · , wm )T , G = (g1 , · · · , gm )T , gq (t) = gq (ϕq (t)), hm hm W h = (w1h1 (t1 ), ..., w1h1 (tn1 ), ..., wm (t1 ), ..., wm (tnm ))T , np 1 m Ah0 = diag(Ah0(11) , ..., Ah0(mm) ), Ah0(pp) = [a0(pp) (tj , τi )]j,i=1 , h

q Ah1 = [A1(pq) ]m p,q=1 ,

n ,n

p q Ah1(pq) = [a1(pq) (tj , τi )]j,i=1 ,

Gh = (g1 (t1 ), ..., g1 (tn1 ), ..., gm (t1 ), ..., gm (tnm ))T . 9

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J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 24, NO.3, 2018, COPYRIGHT 2018 EUDOXUS PRESS, LLC The operator A0(pp) is an isometry operator from (H r [0, 1])m to (H r+1[0, 1])m [11, 12]. hq hq In addition, from [10], we know that A0(qq) and Ah0 are invertible, and ||(A0(qq) )−1 || = O(nq ) and ||(Ah0 )−1 || = O(h−1 q ), where k · k denotes the spectral norm. Hence Eq.

(3.4) is equivalent to 

−1 ˜ (E + A−1 0 A1 )W = A0 G = G ˜ h. (E h + (Ah0 )−1 Ah1 )W h = G

(3.5)

Obviously, the second equation of (3.5) is a system of linear equations with n h (= Σm y ) (ˆ y ∈ Ω′ ) can be j=1 nj ) unknowns. Once W is solved by (3.5), the solution u(ˆ computed by m np 1 XX hp ln |ˆ xpq (τq ) − yˆ||ˆ x′p (τq )|wqh (τq ). u (ˆ y) = − 2π p=1 q=1 h

Let the function vq (t) = tαq φq (t) (0 > αq ≥ −1/2), where φq (t) is differentiable enough on [0, 1] with φq (0) 6= 0. From Taylor’s formula we have vq (t) =

l (j) X φq (0) j=0

and γr′ (t)

∼

∞ X

j!

tj+αq + O(tl+αq +1 ) as t → 0+

δj tr−1+j as t → 0+ , and δ0 > 0.

(3.6)

(3.7)

j=0

By substituting (3.6) and (3.7) into the expression of wq (t), then the function wq (t) can be expressed by wq (t) = c1 φq (0)tr(αq +1)−1 (1 + O(t)) as t → 0+ ,

(3.8)

where c1 is a constant independent of t. Similarly, let the function vq (t) = (1 − t)αq φ˜q (t) (0 > αq ≥ −1/2), where φ˜q (t) is differentiable enough on [0, 1] with φ˜q (1) 6= 0. Then the function wq (t) can be expressed by wq (t) = c2 φ˜q (1)(1 − t)r(αq +1)−1 (1 + O(1 − t))) as t → 1− ,

(3.9)

where c2 is a constant independent of t. Remark 1 The function vq (t) has singularities at endpoints t = 0 and t = 1 [16], but wq (t) has no singularities by Kress transformation at t = 0 and t = 1. Lemma 3.2. Let a ˜1(pq) (t, τ ) = a1(pq) (t, τ )γr′ (t),

γ ≥ 1,

Γp ∩ Γq 6= ∅,

(3.10)

then a ˜1(pq) (t, τ ) is smooth on [0, 1]2 . Proof. By using the continuity of a˜1(pp) (t, τ ) and the boundness of γr′ (t), we can 10

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immediately complete the proof for the case p = q. Let Γp−1 ∩ Γp = Pp = (0, 0) and βp ∈ (0, 2π) be the corresponding interior angle. Since ϕp−1(1) = ϕp−1 (0), the kernel a1(p−1,p) (t, τ ) have singularities at the points (t, τ ) = (0, 1) and (t, τ ) = (1, 0). For convenience of analysis, we only discuss the case in that (t, τ ) = (1, 0). If (t, τ ) 6= (1, 0), we write 1 a ˜1(p−1,p) (t, τ ) = − (S1 (t, τ ) + S2 (t, τ )), (3.11) 4π where S1 (t, τ ) = γ ′ (t) ln(|ϕp−1(t)|2 + |ϕp (τ )|2 ) and S2 (t, τ ) = γ ′ (t) ln[1 − 2|ϕp−1(t)||ϕp (τ )| cos βp−1 /(|ϕp−1(t)|2 + |ϕp (τ )|2 )]. Since

2 2 2|ϕp−1(t)||ϕp (τ )| cos βp−1 /(|ϕp−1(t)| + |ϕp (τ )| ) < 1,

the function S2 (t, τ ) and its first derivative are bounded. Noting that γr(k) (0) = γr(k) (1) = 0, we have

(k)

(k)

|ϕp¯ (0)| = |ϕp¯ (1)| = 0,

k = 0, · · · , γ,

p¯ = p − 1 or p, k = 1, · · · , γ.

Let (t, τ ) ∈ [ε/2, ε] × [1 − ε, 1 − ε/2] for all ε > 0, we have |S1 (t, τ )| = O(εr−1| ln ε|), so S1 (t, τ ) is also bounded. In addition, from ′ ∂ 2|ϕp (τ )||ˆ x′p−1 (γr (τ ))||γr (τ )| S1 (t, τ ) ≤ O(tr−1) ∂τ |ϕp−1(t)|2 + |ϕp (τ )|2

= O(εr−1)O(ε2r−2)/O(ε2r−2) = O(εr−1),

∂˜ a

(t,τ )

we know 1(pq) ∂τ pleted.  Suppose that

is also continuous in (C[0, 1])2. The proof of Lemma 3.2 is com-

tν = (ν + 1)/2 for − 1 < ν ≤ 1, [ν,r]

so that −1 < ν ≤ 1 with t0 = 1/2 and t1 = 1. The offset trapezoidal rule Qn f with

11

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transformation γr , n ∈ N, −1 < ν ≤ 1 is defined by  n−1  1X   f (γr ((j + tν )/n))γr′ ((j + tν )/n),   n j=0 Q[ν,r] n f = n−1   1X   f (γr (j/n))γr′ (j/n), ν = 1.  n j=1

−1 < ν < 1,

[ν,r]

At the same time, we define the truncation error En f by En[ν,r] f := If − Q[ν,r] n f.

(3.12)

Let us assume that near x = 0 we can write r

γr (x) = c0 (r)x (1 +

∞ X

dk (r)xk ).

(3.13)

k=1

Theorem 3.3. [14] Assume that f is holomorphic at both 0 and 1, and fr (τ ) := f (γr (τ ))γr′ (τ ), 0 ≤ τ ≤ 1 can be continuous into the strip S such that (i) fr (τ ) is continuous in S and holomorphic in int(S); (ii) fr (τ ) = o(exp(2πn|Rz|)) as Rz → ∞ in S, uniformly with respect to Rz. For any sigmoidal transformation γr of order r > 1 and for −1 < ν ≤ 1 then for n >> 1 (2πn)r En[0,r] f ∼ 2c0 (r)Γ(r + 1)(f (0) + f (1)) × {cos(rπ/2)(1 − 21−r )ζ(r) − sin(rπ/2)(r + 1)d1(r)(1 − 2−r )ζ(r + 1)/(2πn)} + O(1/nmin(2,r) ), (3.14) and (2πn)r En[1,r] f ∼ 2c0 (r)Γ(r + 1)(f (0) + f (1)) × {cos(rπ/2)ζ(r) − sin(rπ/2)(r + 1)d1 (r)ζ(r + 1)/(2πn)} + O(1/nmin(2,r) ).

(3.15)

Theorem 3.4. [14] Suppose f is defined on S by f (z) = z α (1 − z)β g(z) for α, β > 1,

(3.16)

where g is holomorphic on S, real on [0, 1] and such that g(0) 6= 0, g(1) 6= 0. Let γr be a sigmoidal transformation of order r, r > 1. Then for n >> 1 nr En[ν,r] f ∼ Jν (α, r, n)g(0) + J−ν (β, r, n)g(1), where the strip S of the complex z− plane defined by S := {z : 0 ≤ x = Rz ≤ 1, −∞ < y = Iz < ∞},

12

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and

Jν (α, r, n) : = −r(c0 (r))1+α {ζ(1 − r(α + 1), tν ) + (α + 1 + 1/r)d1 (r)ζ(−r(α + 1), tν )/n}/nrα .

(3.17)

and t−ν = 1 − tν , for − 1 < ν ≤ 1. Remark 2 If we choose r = 3 for γr , then γ3 ∼ O (t3 ) and d1 (r) = 0. In addition, if β = 0, we have En[ν,r] f ∼ n−ω , ω = min{r, (α + 1)r}. (3.18) h

q For the Nystr¨om approximation operator A1(pq) of the integral operator A1(pq) , we have the error bounds [8]

h

q (A1(pq) wq )(t) − (A1(pq) wq )(t) = O (hr ), for Γp = Γq or Γp ∩ Γq = ∅, r ∈ N, (3.19)

and

h

q A1(pq) (wq )(t) − (A1(pq) wq )(t) = O (hω ), for Γp ∩ Γq ∈ {Pq },

(3.20)

where ω = min{r, (α + 1)r}. h

p of the logarithmically singular operator A0(pp) , For the approximate operator A0(pp)      nq   hl X −1/2 hp ln 2e sin π(t − τj ) wp (τj ) (A0(pp) wp )(t) = −  2π    j=1 (3.21)  

t6=τj

−

hp  −1/2 ln 2e hp wp (t) 2π

(i = 1, ..., np ),

which have the error bounds [3] hp wp )(t) (A0(pq)

2ℓ−1 2 X ζ ′ (−2µ) [wp (t)](2µ) h2µ+1 − (A0(pp) wp )(t) = − + O(h2ℓ p p ), π µ=1 (2µ)!

t ∈ {ti },

where ζ ′(t) is the derivative of the Riemann zeta function. From (3.21), we can obtain  p p) p −1)πhp ) ln( eh1/2 ) ln( sin(πh ) · · · ln( sin((ne1/2 ) e1/2 /2 /2  hp sin((np −2)πhp ) sin(πhm ) ln( e1/2 ) · · · ln( ) hp  ln( e−1/2 /2 ) hp e1/2 /2 A0(pp) =−  . . . .  .. .. .. .. 2π  p −1)πhp ) p −2)πhp ) p ) ln( sin((ne1/2 ) ··· ) ln( sin((ne1/2 ln( eh1/2 /2 /2



  ,  

Define the subspace C0 [0, 1] = {v(t) ∈ C[0, 1] : v(t)/γ3 (πt) ∈ C[0, 1]} of the space C[0, 1] with the norm ||v||∗ = max0≤t≤1 |v(t)/γ3(πt)|. Let S hp = span{ej (t), j = 1, ..., np } ⊂ C0 [0, 1] be a piecewise linear function subspace with the basis nodes 13

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satisfying 

P np P hp v = j=1 vpj epj (τ ), v = (vp1 , · · · , vpnp ) ∈ Rn , Qhp v = (v(τp1 ), · · · , v(τpnp )) ∈ Rn , v ∈ C[0, 2π). h

p For the properties of A0(pp) , we have the following lemma from [10].

h

p Lemma 3.5. The operator sequence {P hp (A0(pp) )−1 Qhp A0(pp) : C 2 [0, 1] → C[0, 1]} is uniformly bounded and

p

h

p P hp (A0(pp) )−1 Qhp A0(pp) → I.

(3.22)

If |p − q| 6= 1 or m − 1, by the definition of A1(pq) we know the kernel a1(pq) (t, τ ) of the operator A1(pq) and its derivatives of higher order are continuous . ′ Lemma 3.6. Let Γ′ = ∪m q=1 Γq satisfy CΓ′ 6= 1, and also let hq A¯1(pq)

=

(

h

q A1(pq) , Γ′p = Γ′q or Γ′p ∩ Γ′q = ∅,

hq A˜1(pq) , Γ′p ∩ Γ′q ∈ {Pq },

where the kernel a ˜1(pq) (t, τ ) of A˜1(pq) is defined by (3.10). Then under the transformation (3.1), we have hq k(A0(pp) )−1 A¯1(pq) k2,0 ≤ M (3.23a) and c.c hq hp → (A0(pp) )−1 A1(pq) , in C[0, 1] → C[0, 1], )−1 Qhp A¯1(pq) P hp (A0(pp)

(3.23b)

where M is a constant. proof. From [10] and by Lemma 3.2, a1(pq) (t, τ ) and a˜1(pq) (t, τ ) are continuous on 2 (C [0, 1])2 , and then we have (3.23a). Using the results of Lemma 3.5, and by hq hp hq hp )k0,0 )−1 Qhp A0(pp) )((A0(pp) )−1 A¯1(pq) k0,0 = k(P hp (A0(pp) )−1 Qhp A¯1(pq) kP hp (A0(pp) q p k2,0 )−1 Qhp A0(pp) k0,2 k(A0(pp) )−1 A¯1(pq) ≤ kP hp (A0(pp)

h

h

≤ C, where C is a constant. Thus, we complete the proof of Lemma 3.6.  Consider the following discrete equation (E h + P h (Ah0 )−1 Qh Ah1 )W h = P h (Ah0 )−1 Qh Gh ,

(3.24)

where P h =diag(P h1 , ..., P hm ) and Qh =diag(Qh1 , ..., Qhm ). ′ ′ Theorem 3.7. Assume Γ′ = ∪m q=1 Γq satisfy CΓ′ 6= 1, and Γq (q = 1, ..., m) are smooth curves. Then we have c.c

P h (Ah0 )−1 Qh Ah1 → A−1 0 A1 .

(3.25)

14

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Proof. Let B = {z : ||z|| ≤ 1, z ∈ (C[0, 1]) } be the unit ball in the space V = (C[0, 1])m, and (n)

H = {H (1) , H (2) , · · · },

H (n) = {h1 , · · · , h(n) m } (n)

are multi-parameter sequences. Also let max1≤q≤m hq the sequence {Zh , h ∈ H} ⊂ Θ and Zh = (Z1h , · · · , Zmh ),

→ 0 as nq → ∞. Choosing

Zqh = (zq1 , · · · , zqnq ), q = 1, · · · , m,

satisfying h max max max |zpq (t)/γ3 (πt)| ≤ 1.

(3.26)

1≤p≤m 0≤q≤np 0≤t≤1

From

P

h

(Ah0 )−1 Qh



1 P h1 (Ah0(11) )−1 Qh1

  = 

2 P h2 (Ah0(22) )−1 Qh2

..

 . m P hm (Ah0(mm) )−1 Qhm

   

and

P

h

Ah1 Qh



  = 

h1 P h1 A1(11) Qh1 1 P h1 Ah1(21) Qh1 .. .

2 P h2 Ah1(12) Qh2 2 P h2 Ah1(22) Qh2 .. .

m . . . P hm Ah1(1m) Qhm m Qhm . . . P hm Ah1(2m) .. .

m 1 2 Qhm P h1 Ah1(m1) Qh1 P h2 Ah1(m2) Qh2 . . . P hm Ah1(mm)



  , 

we have

P

h

(Ah0 )−1 Qh Ah1 Qh Zh



  =  

Pm h1 h1 −1 h1 hq hq q=1 P (A0(11) ) Q A1(1q) Q Zqh Pm h2 hq h2 −1 h2 hq q=1 P (A0(22) ) Q A1(2q) Q Zqh .. . Pm hm hm −1 hm hq (A0(mm) ) Q A1(mq) Qhq Zqh q=1 P

If Γ′p ∩ Γ′q = ∅, from Lemma 3.6 we obtain h

h

cc

p q P hp (A0(pp) )−1 Qhp A1(pq) → A−1 0(pp) A1(pq) .



  .   (3.27)

If Γ′p ∩ Γ′q 6= ∅, using Lemma 3.5 and Lemma 3.6, and by hq hq ||P hp (Ah0(pp) )−1 Qhp A1(pq) Qhq Zqh ||0 = ||P hp (Ah0(pp) )−1 Qhp A˜1(pq) Qhq Zqh /γ3 (πt)||0

h

∗ q ˜hq

, ≤ P hpp (Ah0(pp) )−1 Qhp A0(pp) 0,3 ||A−1 0(pp) A1(pq) ||3,0 ||Q Zqh (3.28)

15

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J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 24, NO.3, 2018, COPYRIGHT 2018 EUDOXUS PRESS, LLC hp hq can find an infinite subsequence in {P hp (A0(pp) )−1 Qhp A1(pq) Qhq Zqh } which con-

we verges as h → 0. Hence, there exists an infinite subsequence Hl ⊂ H such that (3.28) converges, As above, there exists an infinite subsequence {Hl , l = 1, · · · , m} such that {P h (Ah0 )−1 Qh Ah1 , h ∈ Hm } is a convergent sequence in the space V = (C0 [0, 1])m . This shows that {P h (Ah0 )−1 Qh Ah1 } is a collectively compact sequence, and P h (Ah0 )−1 Qh Ah1 is pointwisely convergent to A−1 0 A1 . The proof of Theorem 3.7 is completed.  Similar to Theorem 2.4, we have the following theorem. ′ 6 ′ Theorem 3.8. Assume Γ′ = ∪m q=1 Γq satisfy CΓ′ 6= 1, gq = g|Γ′q ∈ C (Γq ), then when we choose an appropriate number r in (3.18) such that ω > 3, there exists a vector function Φ = (Φ1, ..., Φm )T ∈ (C0 [0, 1])m independent of h = (h1 , ..., hm )T such that the following multi-parameter asymptotic expansions hold at nodes w − wˆ h = diag(h31 , ..., h3m )Φ + O(h5max )e,

(3.29)

where hmax = max1≤q≤m hq , and e = (1, 1, · · · , 1)T is a m dimensional vector.

4

Numerical experiments

In this section, two numerical examples are presented to verify the efficiency of the Sidi-Israeli quadrature method for anisotropic heat conduction problems. Suppose that en = u − un be the errors by Sidi-Israeli quadrature method using n boundary nodes, and let rn = log2 (en /en/2 ) be the error ratio. Example 1. [5] Consider the steady state heat conduction in an anisotropic material in the two dimensional disc Ω of radius unity. The thermal conductivity tensor is chosen to be κ11 = 5.0, κ12 = κ21 = 2.0, and κ22 = 1.0. Dirichlet boundary conditions corresponding to the analytical solution u(x1 , x2 ) = x31 /5 − x21 x2 + x1 x22 + x32 /3 are applied to the whole boundary Γ = {(x1 , x2 )|x21 + x22 = 1}. Under the transformation (1.2), the physical domain is distorted into an oblique ellipse Ω′ with the boundary Γ′ = {(ˆ x1 , xˆ2 )|(5ˆ x1 )2 + (ˆ x2 + 2ˆ x1 )2 = 1} on the mapped plane, as shown in Fig. 2. The computed values at the interior points P1 = (0.2, 0.2), P2 = (0.4, 0.4) and P3 = (0.6, 0.6) using different boundary nodes are listed in Table 1, from the numerical results we can see that rn ≈ 3. In addition, the numerical √ solution u of the interior points along the line x2 = x1 are computed, where x1 = 2/2ρ and ρ = −0.9 : 0.05 : 0.9. The plots of computed errors are shown in Figure 3 to Figure 5. P Example 2. Consider a square domain Ω with the boundary Γ = 4q=1 Γq , where Γ1 = {(x1 , 0) : 0 ≤ x1 ≤ 1}, Γ2 = {(1, x2 ) : 0 ≤ x2 ≤ 1}, Γ3 = {(x1 , 1) : 0 ≤ x1 ≤ 1}, and Γ4 = {(0, x2 ) : 0 ≤ x2 ≤ 1}. The invariant coefficients are chosen to be κ11 = 1, κ12 = 0.5, and κ22 = 1. The Dirichlet condition are applied to the Γ is u(x1 , x2 ) = 12 x21 + x1 x2 − x22 + 2. Let each boundary Γq (q = 1, · · · , 4) be divided into 2k (k = 4, · · · , 10) segments. The physical domain and the ’isotropic’ mapped domain (parallelogram) are shown in Fig. 6. In order to overcome the singularities at the corners, we use Sidi-Israeli quadrature method with the Kress’s 16

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Table 1: The errors for u at the points P1 = (0.2, 0.2), P2 = (0.4, 0.4) and P3 = (0.6, 0.6) 25 26 27 28 29 210 211 8.633E-4 2.420E-5 3.672E-6 4.589E-7 5.735E-8 7.169E-9 8.961E-10 5.517 2.720 3.000 3.000 3.000 3.000 1.533E-3 9.791E-6 6.180E-6 7.621E-7 9.525E-8 1.191E-8 1.488E-9 7.291 0.644 3.020 3.000 3.000 3.000 2.377E-3 1.532E-3 7.495E-5 3.283E-7 9.447E-8 1.178E-8 1.472E-9 0.6324 4.355 7.835 1.797 3.004 3.000

n en (P1 ) rn (P1 ) en (P2 ) rn (P2 ) en (P3 ) rn (P3 )

x ˆ2

x2

x ˆ1

O

x

O

1

Figure 2: Left: The physical domain Ω; Right: The mapped domain Ω′ .

−3

9

−3

x 10

2

x 10

1.8

8

1.6

7

1.4

6 Errors

Errors

1.2 5 4

1 0.8

3

0.6

2

0.4

1 0 −0.8

0.2 −0.6

−0.4 −0.2 0 0.2 0.4 x coordinate of the points (n = 25 )

0.6

0 −0.8

0.8

−0.6

−0.4 −0.2 0 0.2 0.4 x coordinate of the points (n = 26 )

0.6

0.8

Figure 3: Left: Errors for u by 25 boundary nodes; Right: Errors for u by 26 boundary nodes. 17

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−5

−5

x 10

1.8

8

1.6

7

1.4

6

1.2

5

Errors

Errors

9

4

1 0.8

3

0.6

2

0.4

1

0.2

0 −0.8

−0.6

−0.4 −0.2 0 0.2 0.4 x coordinate of the points (n = 27 )

0.6

x 10

0 −0.8

0.8

−0.6

−0.4 −0.2 0 0.2 0.4 x coordinate of the points (n = 28 )

0.6

0.8

Figure 4: Left: Errors for u by 27 boundary nodes; Right: Errors for u by 28 boundary nodes.

−7

−8

x 10

1.4

1.2

1.2

1

1

0.8

0.8

Errors

Errors

1.4

0.6

0.6

0.4

0.4

0.2

0.2

0 −0.8

−0.6

−0.4 −0.2 0 0.2 0.4 x coordinate of the points (n = 29 )

0.6

x 10

0 −0.8

0.8

−0.6

−0.4 −0.2 0 0.2 0.4 x coordinate of the points (n = 210 )

0.6

0.8

Figure 5: Left: Errors for u by 29 boundary nodes; Right: Errors for u by 210 boundary nodes.

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x ˆ2

x

2

O

O

x ˆ1

x

1

Figure 6: Left: The physical domain Ω; Right: The mapped domain Ω′ . variable transformation γ3 for this problem. The computed values at the interior points P1 = (0.1, 0.1), P2 = (0.3, 0.3) and P3 = (0.5, 0.5) using n (= 4 × 2k , k = 4, · · · , 10 nodes are listed in Table 2, from the numerical results we can also see that rn ≈ 3. In addition, the numerical solution u of the interior points along the line x2 = x1 are computed, where x1 = ρ and ρ = 0.1 : 0.02 : 0.9. The plots of computed errors are shown in Figure 7 to Figure 9. Table 2: The errors for u at the points P1 = (0.1, 0.1), P2 = (0.3, 0.3) and P3 = (0.5, 0.5) n en (P1 ) rn (P1 ) en (P2 ) rn (P2 ) en (P3 ) rn (P3 )

5

24 25 26 27 28 29 210 1.154E-4 1.254E-5 1.565E-6 1.956E-7 2.444E-8 3.055E-9 3.819E-10 3.202 3.002 3.000 3.000 3.000 3.000 1.158E-4 1.445E-5 1.805E-6 2.256E-7 2.820E-8 3.525E-9 4.406E-10 3.003 3.001 3.000 3.000 3.000 3.000 1.198E-4 1.495E-5 1.869E-6 2.336E-7 2.919E-8 3.649E-9 4.561E-10 3.002 3.001 3.000 3.000 3.000 3.000

Conclusions

In this paper, the Sidi-Israeli quadrature method is used to solve the boundary integral equations of steady state anisotropic heat conduction problems on the twodimensional domain with smooth boundaries and polygons respectively. Especially, in order to provide a good accuracy in the solution near the singular points, the Kress’s variable transformation is used for the weakly singular integral equations of 19

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−5

2

x 10

2.4

x 10

2.3

1.9

2.2

1.8

2.1 Errors

Errors

1.7 1.6

2 1.9

1.5 1.8 1.4

1.7

1.3

1.6

1.2 0.1

0.2

0.3 0.4 0.5 0.6 0.7 x coordinate of the points (n = 4 × 25 )

0.8

0.9

0.1

0.2

0.3 0.4 0.5 0.6 0.7 x coordinate of the points (n = 4 × 26 )

0.8

0.9

Figure 7: Left: Errors for u by 4 × 25 boundary nodes; Right: Errors for u by 4 × 26 boundary nodes. −7

3

−8

x 10

3.8

x 10

3.6

2.8

3.4

Errors

Errors

2.6

2.4

3.2

3 2.2 2.8 2

2.6

1.8 0.1

0.2

0.3 0.4 0.5 0.6 0.7 x coordinate of the points (n = 4 × 27 )

0.8

2.4 0.1

0.9

0.2

0.3 0.4 0.5 0.6 0.7 x coordinate of the points (n = 4 × 28 )

0.8

0.9

Figure 8: Left: Errors for u by 4 × 27 boundary nodes; Right: Errors for u by 4 × 28 boundary nodes. −10

−9

4.6

x 10

5.8

x 10

5.6

4.4

5.4 4.2 5.2 5 Errors

Errors

4 3.8

4.8 4.6

3.6

4.4 3.4 4.2 3.2 3 0.1

4 0.2

0.3 0.4 0.5 0.6 0.7 x coordinate of the points (n = 4 × 29 )

0.8

3.8 0.1

0.9

0.2

0.3 0.4 0.5 0.6 0.7 x coordinate of the points (n = 4 × 210 )

0.8

0.9

Figure 9: Left: Errors for u by 4 × 29 boundary nodes; Right: Errors for u by 4 × 210 boundary nodes. 20

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problems (1.1). The numerical results show that the presented algorithm has a high accuracy of O (n−3 ), which coincides with our theoretical analysis.

References [1] Yan Gu, Wen Chen and Xiao-Qiao He, Singular boundary method for steadystate heat conduction in three dimensional general anisotropic media, International Journal of Heat and Mass Transfer. 55 ( 2012) 4837-4848. [2] Y.C. Shiah and C.L. Tan, BEM treatment of two-dimensional anisotropic field problems by direct domain mapping, Eng. Anal. Boundary Elem. 20 (1997) 347351. [3] A. Sidi and M. Israeli,Quadrature methods for periodic singular and weakly singular Fredholm integral equation, J. Sci. Comput. 3 (1988) 201-231. [4] R. Kress, A Nystr¨om method for boundary integral equations in domains with corners. Numer. Math. 58 (1990) 145-161. [5] N.S. Mera, L. Elliott, D.B. Ingham, and D. Lesnic, A comparison of boundary element method formulations for steady state anisotropic heat conduction problems, Eng. Anal. Boundary Elem. 25 (2001) 115-128. [6] P.M. Anselone, Collectively Compact Operator Approximation Theory, PrenticeHall, Englewood Cliffs, NJ. 1971. [7] P.M. Anselone, Singularity subtraction in numerical solution of integral equations, J. Austral Math. Soc. 22 (1981) 408-418. [8] P. Davis, Methods of Numerical Integration, Second edition, Academic Press, New York, 1984. [9] Abramowitz M, Stegun I., Handbook of Mathematical Functions. New York: Dover Publications, 1965. [10] J. Huang and Z. Wang, Extrapolation algorithms for solving mixed boundary integral equations of the Helmholtz equation by mechanical quadrature methods, SIAM J. Sci. Comput. 31 (2009) 4115-4129. [11] I.H. Sloan and A. Spence, The Galerkin method for integral equations of firstkind with logarithmic kernel: theory, IMA J. Numer. Anal. 8 (1988) 105-122. [12] Y. Yan and I. Sloan, On integral equations of the first kind with logarithmic kernels, J. Integral Equations Appl. 1 (1988) 517-548. [13] George E. Andrews et al., Special Functions, Cambridge University Press, 2001

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[14] David Elliott, Sigmoidal Transformations and the Trapezoidal Rule, J. Austral. Math. Soc. B 40 (1998) 77-137. [15] Zl´aml, M., Asymptotic error estimates in solving elliptic equations of the fourth order by the method of finite differences, SIAM J. Numer. Anal. 2 (1965) 337-344. [16] Pan Cheng, Zhi lin, Wenzhong Zhang, Five-Order Algorithms for Solving Laplace’s Steklov Eigenvalue on Polygon by Mechanical Quadrature Methods, J. Comput. Analysis and Application. 18 (2015) 138-148. [17] O.C. Zienkiewicz, The finite element method, McGraw Hill, Maidenhead, 1977.

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Hyers-Ulam stability of an additive set-valued functional equation Gang Lu, Jun Xie, Choonkil Park∗ and Yuanfeng Jin∗

Abstract. In this paper, we define the following additive set-valued functional equation f (2x + 3y − z) + f (2y + 3z − x) + f (3x + 2z − y) (1)

= f (x + y) + f (y + z) + f (x + z) + f (2x) + f (2y) + f (2z) and prove the Hyers-Ulam stability of the above additive set-valued functional equation.

1. Introduction and preliminaries The stability problem of functional equations was originated from a question of Ulam [28] concerning the stability of group homomorphisms. Hyers [6] gave a first affirmative partial answer to the question of Ulam for Banach spaces. Hyers’ Theorem was generalized by Aoki [2] for additive mappings and by Rassias [19] for linear mappings by considering an unbounded Cauchy difference. A generalization of the Rassias theorem was obtained by G˘ avruta [5] by replacing the unbounded Cauchy difference by a general control function in the spirit of Rassias’ approach (see [1, 4, 5, 7, 8, 14, 15, 20, 21, 22, 23, 24, 25, 26, 27]). It is easy to show that if f : R → R is a solution of the inequality |f (x + y) − f (x) − f (y)| < ε

(1.1)

for some ε > 0 then there exists a linear function g(x) = mx, m ∈ R, such that |f (x) − g(x)| < ε for all x ∈ R. The inequality (1.1) can be written as the form f (x + y) − f (x) − f (y) ∈ B(0, ε), where B(0, ε) := (−ε, ε). Hence we have f (x + y) + B(0, ε) ⊆ f (x) + B(0, ε) + f (y) + B(0, ε) and denoting by F (x) = f (x) + B(0, ε), x ∈ R, we get F (x + y) ⊆ F (x) + F (y), x, y ∈ R and g(x) ∈ F (x). Let Y be a real normed space. The family of all closed and convex subsets, containing 0, of Y will be denoted by ccz(Y ). 0

2010 Mathematics Subject Classification: 54C60, 39B52, 47H04, 49J54. Keywords: Hyers-Ulam stability, additive set-valued functional equation, closed and convex subset, cone ∗ Corresponding authors. 0

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Hyers-Ulam stability of an additive set-valued functional equation Let A, B be nonempty subsets of a real vector space X and λ a real number. We define A+B

= {x ∈ X : x = a + b,

λA = {x ∈ X : x = λa,

a ∈ A, b ∈ B}, a ∈ A}.

Lemma 1.1. ([13]) Let λ and µ be real numbers. If A and B are nonempty subsets of a real vector space X, then λ(A + B) = λA + λB, (λ + µ)A ⊆ λA + µB. Moreover, if A is a convex set and λµ ≥ 0, then we have (λ + µ)A = λA + µA. A subset A ⊆ X is said to be a cone if A + A ⊆ A and λA ⊆ A for all λ > 0. If the zero vector in X belongs to A, then we say that A is a cone with zero. Set-valued functional equations have been extensively investigated by a number of authors and there are many interesting results concerning this problem (see [3, 10, 11, 12]). 2. Stability of the set-valued functional equation (1) In this section, let X be a real vector space, A ⊆ X a cone with zero and Y a Banach space. The following theorem is similar to the results of [16] and [18] Theorem 2.1. If F : A → ccz(Y ) is a set-valued map satisfying F (2x + 3y − z) + F (2y + 3z − x) + F (3x + 2z − y) (2.1)

⊆ F (x + y) + F (y + z) + F (x + z) + F (2x) + F (2y) + F (2z) and sup{diam(F (x)) : x ∈ A} < +∞ for all x, y, z ∈ A, then there exists a unique additive mapping g : A → Y such that g(x) ∈ F (x). Proof. Take an element x ∈ A. Letting y = z = x in (2.1) and using Lemma 1.1, we get 3F (4x) ⊆ 6F (2x).

(2.2)

Replacing 2x by 2n x in (2.2), we obtain
F 2n+1 x ⊆ 2F (2n x) and F (2n+1 x) F (2n x) ⊆ . 2n+1 2n n Denoting by Fn (x) = F (22n x) , x ∈ A, n ∈ N, we obtain that (Fn (x))n≥0 is a decreasing sequence of closed subsets of the Banach space Y . We have also diam(Fn (x)) =

1 diam (F (2n x)) . n 2 557

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G. Lu, J. Xie, C. Park, Y. Jin Taking account of sup{diam(F (x)) : x ∈ A} < +∞, we get lim diam(Fn (x)) = 0.

n→∞

Using the Cantor theorem for the sequence (Fn (x))n≥0 , we obtain that the intersection ∩n≥0 Fn (x) is a singleton set and we denote this intersection by g(x) for all x ∈ A. Thus we get a mapping g : A → Y and g(x) ∈ F0 (x) = F (x) for all x ∈ A. We now show that g is additive. For all xi ∈ A, i = 1, 2, · · · , N and n ∈ N, Fn (2x + 3y − z) + Fn (2y + 3z − x) + Fn (3x + 2z − y) F (2n (2x + 3y − z)) F (2n (2y + 3z − x)) F (2n (3x + 2z − y)) = + + 2n 2n 2n n n n n F (2 (x + y)) + F (2 (y + z)) + F (2 (x + z)) + F (2 (2x)) + F (2n (2y)) + F (2n (2z)) ⊆ . 2n = Fn (x + y) + Fn (y + z) + Fn (x + z) + Fn (2x) + Fn (2y) + Fn (2z). By the definition of g, we obtain g(2x + 3y − z) + g(2y + 3z − x) + g(3x + 2z − y) = ⊆ ⊆

∞ \

Fn (2x + 3y − z) +

∞ \

Fn (2y + 3z − x) +

n=0

n=0 ∞ \

∞ \

Fn (3x + 2z − y)

n=0

{Fn (2x + 3y − z) + Fn (2y + 3z − x) + Fn (3x + 2z − y)}

n=0 ∞ \

{Fn (x + y) + Fn (y + z) + Fn (x + z) + Fn (2x) + Fn (2y) + Fn (2z)}

n=0

and g(xi ) ∈ Fn (xi ). Thus we get kg(2x + 3y − z) + g(2y + 3z − x) + g(3x + 2z − y) −g(x + y) − g(y + z) − g(z + x) − g(2x) − g(2y) − g(2z)k ≤ diam (Fn (x + y)) + diam (Fn (y + z)) + diam (Fn (x + z)) + diam (Fn (2x)) +diam (Fn (2y)) + diam (Fn (2z)) which tends to zero as n tends to ∞. Thus g(2x + 3y − z) + g(2y + 3z − x) + g(3x + 2z − y) = g(x + y) + g(y + z) + g(x + z) + g(2x) + g(2y) + g(2z)

(2.3)

for all x, y, z ∈ A. Letting x = y = z = 0 in (2.3), we have 3g(0) = 6g(0). Thus g(0) = 0. Letting x = y = z in (2.3), we get g(2x) = 2g(x) for all x ∈ A. And letting y = z = 0 in (2.3), we have g(−x) + g(3x) = 2g(x) = g(2x)

(2.4)

for all x ∈ A. Letting z = −x, y = 2x in (2.4), we get g(z) + g(y − z) = g(y)

(2.5)

for all y, z ∈ A. Letting y = 0 in (2.5), we have g(−z) = −g(z) for all z ∈ A. Hence g(y − z) = g(y) + g(−z) 558

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Hyers-Ulam stability of an additive set-valued functional equation for all y, z ∈ A. That is, g is an additive mapping.



Acknowledgement G. Lu was supported by Doctoral Science Foundation of Shengyang University of Technology (No.521101302) and the Project Sponsored by the Scientific Research Foundation for the Returned Overseas Chinese Scholars, State Education Ministry (No.2014-62009). Y. Jin was supported by National Natural Science Foundation of China (11361066) and the study of high-precision algorithm for high dimensional delay partial differential equations 2014-2017.

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] T. Cardinali, K. Nikodem and F. Papalini, Some results on stability and characterization of Kconvexity of set-valued functions, Ann. Polon. Math. 58 (1993), 185–192. [4] P.W. Cholewa, Remarks on the stability of functional equations, Aequationes Math. 27 (1984), 76–86. [5] P. Gˇavruta, A generalization of the Hyers-Ulam-Rassias stability of approximately additive mappings, J. Math. Anal. Appl. 184 (1994), 431–436. [6] D.H. Hyers, On the stability of the linear functional equation, Proc. Nat. Acad. Sci. USA 27 (1941), 222–224. [7] D.H. Hyers, G. Isac and Th.M. Rassias, Stability of Functional Equations in Several Variables, Birkh¨auser, Basel, 1998. [8] G. Isac and Th.M. Rassias, On the Hyers-Ulam stability of ψ-additive mappings, J. Approx. Theory 72 (1993), 131–137. [9] G. Lu and C. Park, Hyers-Ulam stability of additive set-valued functional equations, Appl. Math. Lett. 24 (2011), 1312–1316. [10] K. Nikodem, On quadratic set-valued functions, Publ. Math. Debrecen 30 (1984), 297–301. [11] K. Nikodem, On Jensen’s functional equation for set-valued functions, Radovi Mat. 3 (1987), 23–33. [12] K. Nikodem, Set-valued solutions of the Pexider functional equation, Funkcialaj Ekvacioj 31 (1988), 227–231. [13] K. Nikodem, K-Convex and K-Concave Set-Valued Functions, Zeszyty Naukowe Nr. 559, Lodz, 1989. [14] C. Park, K. Ghasemi, S. G. Ghaleh and S. Jang, Approximate n-Jordan ∗-homomorphisms in C ∗ algebras, J. Comput. Anal. Appl. 15 (2013), 365–368. [15] C. Park, A. Najati and S. Jang, Fixed points and fuzzy stability of an additive-quadratic functional equation, J. Comput. Anal. Appl. 15 (2013), 452–462. [16] Y.J. Piao, The existence and uniqueness of additive selection for (α, β)-(β, α) type subadditive setvalued maps, J. Northeast Normal Univ. (Natural Science) 41 (2009), 32–34 (In Chinese). [17] Y.J. Piao, The unique existence of additive selection maps for a waek subadditive set-valued maps, Chin. Quart. J. Math. 28 (2013), 53–59. [18] D. Popa, Additive selections of (α, β)-subadditive set-valued maps, Glas. Mat. Ser. III 36 (56) (2001), 11–16. [19] Th.M. Rassias, On the stability of the linear mapping in Banach spaces, Proc. Amer. Math. Soc. 72 (1978), 297–300. 559

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G. Lu, J. Xie, C. Park, Y. Jin [20] Th.M. Rassias (Ed.), Functional Equations and Inequalities, Kluwer Academic, Dordrecht, 2000. [21] Th.M. Rassias, On the stability of functional equations in Banach spaces, J. Math. Anal. Appl. 251 (2000), 264–284. [22] Th.M. Rassias, On the stability of functional equations and a problem of Ulam, Acta Math. Appl. 62 (2000), 23–130. [23] S. Schin, D. Ki, J. Chang and M. Kim, Random stability of quadratic functional equations: a fixed point approach, J. Nonlinear Sci. Appl. 4 (2011), 37–49. [24] S. Shagholi, M. Bavand Savadkouhi and M. Eshaghi Gordji, Nearly ternary cubic homomorphism in ternary Fr´echet algebras, J. Comput. Anal. Appl. 13 (2011), 1106–1114. [25] S. Shagholi, M. Eshaghi Gordji and M. Bavand Savadkouhi, Stability of ternary quadratic derivation on ternary Banach algebras, J. Comput. Anal. Appl. 13 (2011), 1097–1105. [26] D. Shin, C. Park and Sh. Farhadabadi, On the superstability of ternary Jordan C ∗ -homomorphisms, J. Comput. Anal. Appl. 16 (2014), 964–973. [27] D. Shin, C. Park and Sh. Farhadabadi, Stability and superstability of J ∗ -homomorphisms and J ∗ derivations for a generalized Cauchy-Jensen equation, J. Comput. Anal. Appl. 17 (2014), 125–134. [28] S.M. Ulam, Problems in Modern Mathematics, Chapter VI, Science ed., Wiley, New York, 1940. Gang Lu Department of Mathematics, School of Science, ShenYang University of Technology, Shenyang 110178, P. R. China; Department of Mathematics, Zhejiang University, Hangzhou 310027, P. R. China E-mail address: [email protected] Jun Xie Department of Mathematics, School of Science, ShenYang University of Technology, Shenyang 110178, P. R. China E-mail address: [email protected] Choonkil Park Department of Mathematics, Research Institute for Natural Sciences,, Hanyang University, Seoul 04763, Republic of Korea E-mail address: [email protected] Yuanfeng Jin Department of Mathematics, Yanbian University, Yanji 133001, P. R. China E-mail address: [email protected] E-mail address: [email protected]

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Topological spaces induced by fuzzy prime ideals in BCC-algebras Sun Shin Ahn1 and Keum Sook So2,∗ 1

Department of Mathematics Education, Dongguk University, Seoul 04620, Korea 2 Department of Mathematics, Hallym University, Chuncheon, 24252, Korea

Abstract. In this paper, we construct a topological space on the set of all fuzzy prime ideals of a commutative BCC-algebra X, and we discuss fuzzy prime ideals in commutative BCC-algebras. 1. Introduction In 1966, Imai and Is´eki ([7]) defined a class of algebras of type (2,0), called BCK-algebras which generalized the notion of an algebra of sets with the set subtraction as the only fundamental non-nullary operation, and also the notion of implication algebras ([8]). The class of all BCK-algebras is a quasi-variety. Is´eki posed an interesting problem whether the class of BCK-algebras is a variety. That problem was solved by Wro´ nski ([11]), who proved that BCK-algebra do not form a variety. In connection with this problem, Komori ([9]) introduced the notion of BCC-algebras, and Dudeck ([1, 2]) redefined the notion of BCC-algebras by using a dual form of the original definition in the sense of Komori. In [5], Dudek and Zhang introduced a new notion of ideals in BCC-algebras and described some connections between such ideals and congruences. Dudek and Jun ([3]) considered the fuzzification of ideals in BCC-algebras. Dudek, Jun and Stojakovic ([4]) described fuzzy BCC-ideals and its image. In this paper, we define a topology on the set of all fuzzy prime ideals of a commutative BCC-algebra X and the resulting space, denoted by F -spec(X), and obtain some related properties. 2. Preliminaries By a BCC-algebra ([6]) we mean a nonempty set X with a constant 0 and a binary operation “ ∗ ” satisfying axioms: for all x, y, z ∈ X, (I) ((x ∗ y) ∗ (z ∗ y)) ∗ (x ∗ z) = 0, (II) 0 ∗ x = 0, (III) x ∗ 0 = x, (IV) x ∗ y = 0 and y ∗ x = 0 imply x = y. For brevity, we also call X a BCC-algebra. In X we can define a binary relation “ ≤ ” by x ≤ y if and only if x ∗ y = 0. Then ≤ is a partial ordering on X. The relation “ ≤ ” is called a BCC-order on X. A non-empty subset S of a BCC-algebra X is called a subalgebra of X if x ∗ y ∈ S for all x, y ∈ S. In a BCC-algebra X, the following hold: for any x, y, z ∈ X, (2.1) x ∗ x = 0, (2.2) (x ∗ y) ∗ x = 0, (2.3) x ≤ y ⇒ x ∗ z ≤ y ∗ z, 0

*Correspondence: +82 33 248 2011(phone), +82 33 256 2011(fax) (K. S. So). 561

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Sun Shin Ahn and Keum Sook So∗ (2.4) x ≤ y ⇒ z ∗ y ≤ z ∗ x. Any BCK-algebra is a BCC-algebra, but there exist BCC-algebras which are not BCK-algebras (cf. [2]). Note that a BCC-algebra is a BCK-algebra if and only if it satisfies: (2.5) (x ∗ y) ∗ z = (x ∗ z) ∗ y, for all x, y, z ∈ X. Definition 2.1 ([5]). Let X be a BCC-algebra and ∅ 6= I ⊆ X. I is called an ideal (or a BCK-ideal) of X if it satisfies the following conditions: (i) 0 ∈ I, (ii) x ∗ y, y ∈ I imply x ∈ I for all x, y ∈ X. Theorem 2.2 ([5]). In a BCC-algebra X, every ideal of X is a subalgebra of X. Definition 2.3 ([5]). Let X be a BCC-algebra and ∅ 6= I ⊆ X. I is called a BCC-ideal of X if it satisfies the following conditions: (i) 0 ∈ I, (ii) (x ∗ y) ∗ z ∈ I and y ∈ I imply x ∗ z ∈ I, for all x, y, z ∈ X. Lemma 2.4 ([5]). In a BCC-algebra X, any BCC-ideal of X is an ideal of X. Corollary 2.5 ([5]). Any BCC-ideal X of a BCC-algebra X is a subalgebra of X. Remark. In a BCC-algebra, a subalgebra need not be an ideal, and an ideal need not be a BCC-ideal in general (see [2, 4]). We now review some fuzzy logic concept. Let X be a BCC-algebra. A fuzzy set µ in X is a function µ : X → [0, 1]. The set µt := {x ∈ X|µ(x) ≥ t}, where t ∈ [0, 1] is fixed, is called a level set of X. By Im(µ) we denote the image set of µ. A fuzzy set µ : X → [0, 1] is called a fuzzy subalgebra ([3]) of X if µ(x ∗ y) ≥ min{µ(x), µ(y)} for all x, y ∈ X. Definition 2.6. For t ∈ [0, 1], fuzzy point xt is a fuzzy subset of X such that ( t if y = x, xt (y) := 0 if y 6= x. Definition 2.7 ([3]). A fuzzy set µ in a BCC-algebra X is called a fuzzy BCK-ideal if (i) µ(0) ≥ µ(x) for all x ∈ X, (ii) µ(x) ≥ min{µ(x ∗ y), µ(y)} for all x, y ∈ X. Lemma 2.8 ([3]). Let X be a BCC-algebra and µ be a fuzzy BCK-ideal of X. (i) if x ∗ y = 0, then µ(x) ≥ µ(y) for any x, y ∈ X, (ii) µ(x ∗ y) ≥ min{µ(x ∗ z), µ(z ∗ y)} for all x, y, z ∈ X. Definition 2.9 ([3]). A fuzzy set µ in a BCC-algebra X is called a fuzzy BCC-ideal if (i) µ(0) ≥ µ(x) for all x ∈ X, (ii) µ(x ∗ z) ≥ min{µ((x ∗ y) ∗ z), µ(y)} for all x, y, z ∈ X. 562

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On fuzzy spectrum of a BCC-algebra Any fuzzy BCC-ideal is a fuzzy BCK-ideal in BCC-algebras. Lemma 2.10 ([4]). If µ is a fuzzy BCC-ideal of a BCC-algebra X, then, for any x, y, z ∈ X, (i) x ≤ y implies µ(y) ≤ µ(x), (ii) µ(x ∗ y) = µ(0), then µ(x) ≥ µ(y), (iii) µ(x ∗ y) ≥ µ(x), (iv) µ(x ∗ y) ≥ min{µ(x), µ(y)}, (v) µ(x ∗ (y ∗ z)) ≥ min{µ(x), µ(y), µ(z)}, (i) µ((x ∗ y) ∗ (x ∗ z)) ≥ µ(z ∗ y). Proposition 2.11 ([3]). A fuzzy set µ in a BCC-algebra X is a fuzzy BCK(BCC, resp.)-ideal (subalgebra, resp.) if and only if for every t ∈ [0, 1], the level subset µt is either empty or a BCK(BCC, resp.)-ideal (subalgebra, resp.) of X. Theorem 2.12. If I is an ideal of a BCC-algebra, then the characteristic function χI : X → [0, 1] of I, is a fuzzy ideal of X with I = XχI , where XχI =: {x ∈ X|χI (x) = χI (0). Proof. It is easily checked that χI is a fuzzy ideal of X. Given an ideal I, we have XχI ={x ∈ X|χI (x) = χI (0)} ={x ∈ X|χI (x) = 1} =I.  3. Toplogical Spaces by fuzzy prime ideals Theorem 3.1. Let X be a BCC-algebra and let {ηi }{i∈Λ} be a family of fuzzy BCK(BCC, resp.)-ideals of X. Then ∩i∈Λ ηi is a fuzzy BCK(BCC, resp.)-ideal of X. Proof. Straighforward.



If µ is a fuzzy subset of a BCC-algebra X, then the ideal generated by µ which is denoted by hµi is defined as follows: hµi = ∩{η|µ ⊆ η, η is a fuzzy BCC(BCK, resp.)-ideal of X}. For all x, y in a BCC-algebra X, y ∗ (y ∗ x) is denoted by x ∧ y. A BCC-algebra X is said to be commutative ([2]) if x ∗ (x ∗ y) = y ∗ (y ∗ x), for all x, y ∈ X, i.e., x ∧ y = y ∧ x. If X is a commutative BCC-algebra, then it is easy to check that x ∧ y ≤ x and x ∧ y ≤ y.

(∗)

A proper ideal P of a BCC-algebra X is said to be prime if for all ideals A, B of X such that AB ⊆ P , either A ⊆ P or B ⊆ P , where AB = {a ∧ b|a ∈ A, b ∈ B}.

In what follows, let X be a BCC-algebra, unless otherwise specified. 563

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Sun Shin Ahn and Keum Sook So∗ Definition 3.2. Let µ and η be two fuzzy subsets of X. Then the fuzzy set µη is defined by µη(x) = sup{min{µ(y), η(z)}|x = y ∧ z}.

Clearly, since x ∧ x = x, each x ∈ X is expressible as x = y ∧ z for some y, z ∈ X. Definition 3.3. A non-constant fuzzy ideal µ of X is said to be fuzzy prime if for all fuzzy ideals θ, σ such that θσ ⊆ µ, either θ ⊆ µ or σ ⊆ µ. Lemma 3.4. Let µ and η be two fuzzy BCK(BCC, resp.)-ideals of X. Then µη ⊆ µ ∩ η. Proof. Let x ∈ X such that x = a ∧ b for some a, b ∈ X and let µ, η be fuzzy BCK(BCC, resp.)-ideals of X. Then µ(a) ≤ µ(a ∧ b) = µ(x) and η(b) ≤ η(a ∧ b) = η(x) from Lemma 2.10-(i). Hence min{µ(a), η(b)} ≤ (µ ∩ η)(x). Thus µη(x) ≤ (µ ∩ η)(x).



Let Y be the set of all fuzzy prime ideals of X. Let V (θ) := {µ ∈ Y |θ ⊆ µ}, where θ is any fuzzy subset of X. Put Y (θ) = Y \ V (θ), the complement of V (θ) in Y . Lemma 3.5. If θ is a fuzzy subset of X, then V (hσi) = V (σ). In particular, V (hxβ i) = V (xβ ) for any fuzzy point xβ of X. Proof. Clearly, V (σ) ⊆ V (hσi). Now, let µ ∈ V (hσi) = {µ ∈ Y |hσi ⊆ µ}. Then we have hσi ⊆ µ. Since σ ⊆ hσi, we have σ ⊆ µ which implies that µ ∈ V (σ). Thus V (hσi) ⊆ V (σ).



Theorem 3.6. Let τ = {Y (θ)|θ is a fuzzy BCK(BCC)-ideal of X}. Then the pair (Y, τ ) is a topological space. Proof. Consider the fuzzy ideals θ and σ of Y defined by θ(x) := 0 and σ(x) := 1 for all x ∈ X. Then V (θ) = Y and V (σ) = ∅ so that ∅, Y ∈ τ . Now let θ1 and θ2 be two fuzzy BCK(BCC)-ideals of X. We show that V (θ1 ) ∪ V (θ2 ) = V (θ1 ∩ θ2 ). To do this, if µ ∈ V (θ1 ) ∪ V (θ2 ), then µ ∈ V (θ1 ) or µ ∈ V (θ2 ), i.e., θ1 ⊆ µ or θ2 ⊆ µ and hence θ1 ∩ θ2 ⊆ µ. Therefore µ ∈ V (θ1 ∩ θ2 ). Thus V (θ1 ) ∪ V (θ2 ) ⊆ V (θ1 ∩ θ2 ). On the other hand, if µ ∈ V (θ1 ∩ θ2 ), then θ1 ∩ θ2 ⊆ µ. By Lemma 3.4, θ1 θ2 ⊆ θ1 ∩ θ2 ⊆ µ and θ1 θ2 ⊆ µ. Since µ is a fuzzy prime ideal of X, θ1 ⊆ µ or θ2 ⊆ µ. Hence µ ∈ V (θ1 ) ∪ V (θ2 ). Therefore V (θ1 ∩ θ2 ) ⊆ V (θ1 ) ∪ V (θ2 ) and hence V (θ1 ) ∪ V (θ2 ) = V (θ1 ∩ θ2 ). Thus Y (θ1 ) ∩ Y (θ2 ) = Y (θ1 ∩ θ2 ), i.e., τ is closed under finite intersection. Now we will prove that if {θi }i∈Λ is a family of fuzzy BCK(BCC)-ideal of X, then ∩i∈Λ V (θi ) = V (h∪i∈Λ θi i).

(∗∗)

Let µ ∈ Y . Then we have µ ∈ V (θi ), ∀i ∈ Λ ⇔ θi ⊆ µ, ∀i ∈ Λ ⇔ ∪i∈Λ θi ⊆ µ ⇔ h∪i∈Λ θi i ⊆ µ ⇔ µ ∈ V (h∪i∈Λ θi i). Therefore (∗∗) holds. Thus ∩i∈Λ Y (θi ) = Y (h∪i∈Λ θi i). This proves that τ is closed under arbitrary union. Thus (Y, τ ) is a topological space.

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On fuzzy spectrum of a BCC-algebra The topological space (Y, τ ) described in Theorem 3.6 is called a fuzzy spectrum of Y or F -spectrum of Y and is denoted by F -spec (Y ). Theorem 3.7. Let (Y, τ ) be a topological space. Then the subfamily B = {Y (xβ )|x ∈ X, β ∈ (0, 1]} of τ is a base for τ . Proof. It is enough to show that for all Y (θ) ∈ τ and µ ∈ Y (θ) there exists Y (xβ ) ∈ B such that µ ∈ Y (xβ ) and Y (xβ ) ⊆ Y (θ). To do this, if Y (θ) ∈ τ and µ ∈ Y (θ), then θ * µ. Hence there exists x ∈ X such that θ(x) > µ(x). If θ(x) = β, then µ ∈ Y (xβ ).

(1)

If σ ∈ V (θ) is an arbitrary element, then σ(x) ≥ θ(x) = β = xβ (x), which implies that xβ ⊆ σ. Therefore σ ∈ V (xβ ) and hence V (θ) ⊆ V (xβ ). Thus we have Y (xβ ) ⊆ Y (θ).

(2)

By (1) and (2), the proof is complete.



4. Fuzzy prime ideals of commutative BCC-algebras Proposition 4.1. Let µ be a fuzzy BCK(BCC)-ideal of a BCC-algebra X. Then Xµ := {x ∈ X|µ(x) = µ(0)} is a BCK(BCC)-ideal of X. Proof. Straightforward.



If µ is not a fuzzy BCK(BCC)-ideal of a BCC-algebra X, then Proposition 4.1 need not be true as shown in the following example. Example 4.2. Let X be a BCC-algebra with the following table: ∗ 0 1 2 3 4 5

0 0 1 2 3 4 5

1 0 0 2 2 4 5

2 0 0 0 1 4 5

3 0 0 0 0 4 5

4 0 0 1 1 0 5

5 0 1 1 1 1 0

Then (X; ∗, 0) is not a BCK-algebra, since (2 ∗ 1) ∗ 4 = 1 6= 0 = (2 ∗ 4) ∗ 1. Let S := {0, 1, 2, 3, 4} and T := {0, 1, 2}. Then S is a BCC-ideal of X and T is a BCC-subalgebra of X, but not a BCK-ideal of X, since 3 ∗ 2 = 0 ∈ T and 3 ∈ / T ([5]). Let µ : X → [0, 1] be a map defined by µ(0) = µ(1) = µ(2) = 1 and µ(3) = µ(4) = µ(5) = 21 . Then µ is not a BCK-ideal, since

1 2

= µ(3) < min{µ(3 ∗ 2), µ(2)} = 1. Xµ = {0, 1, 2}

is not a BCK-ideal of X, since 1 = 3 ∗ 2 ∈ Xµ and 2 ∈ Xµ , but 3 ∈ / Xµ . Define a fuzzy subset ν in X by ν(0) = ν(1) = ν(5) = 0.9 and ν(2) = ν(3) = ν(4) = 0.3. Then ν is not a fuzzy BCC-ideal of X, since ν(4 ∗ 2) = ν(4) = 0.3 < min{ν((4 ∗ 5) ∗ 2) = ν(1 ∗ 2) = ν(0) = 0.9, ν(5) = 0.9} = 0.9. But Xν = {0, 1, 5} is not a BCC-ideal of X, since (4 ∗ 5) ∗ 0 = 1, 5 ∈ Xν but 4 ∗ 0 = 4 ∈ / Xν . A proper ideal P of a BCC-algebra X is said to be s-prime if x ∧ y ∈ P ⇔ x ∈ P or y ∈ P, for all x, y ∈ X. 565

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Sun Shin Ahn and Keum Sook So∗ Definition 4.3. Let X be a commutative BCC-algebra. A non-constant fuzzy BCK-ideal (or fuzzy ideal) µ of X is said to be s-prime if for all x, y ∈ X, either µ(x ∧ y) = µ(x) or µ(x ∧ y) = µ(y). Lemma 4.4. A non-constant fuzzy set µ of a commutative X is a fuzzy s-prime ideal of X if and only if for each t ∈ [0, 1], µt is either empty or an s-prime ideal of X if it is proper. Proof. Suppose that µ is a fuzzy s-prime ideal of X. For each t ∈ [0, 1], assume that µt 6= ∅ and x ∧ y ∈ µt , where x, y ∈ X. Then µ(x ∧ y) ≥ t. Since µ is a fuzzy s-prime ideal of X, we obtain µ(x ∧ y) = µ(x) ≥ t or µ(x ∧ y) = µ(y) ≥ t. Hence either x ∈ µt or y ∈ µt . Thus µt is an s-prime ideal of X. Conversely, if µ is not an s-prime ideal of X, then µ(x ∧ y) 6= µ(x) and µ(x ∧ y) 6= µ(y) for all x, y ∈ X. Let x ∧ y ∈ µt for all x, y ∈ X. Then µ(x ∧ y) ≥ t. Since µ is not an s-prime ideal of X, we obtain µ(x) < t and µ(y) < t. Hence x ∈ / µt and y ∈ / µt , which is a contradiction.



Lemma 4.5. Let µ be a fuzzy prime ideal of a BCC-algebra X. Then for any t ∈ [0, 1], µt is either empty or a prime ideal of X if it is proper. Proof. Let t ∈ [0, 1] and µt 6= ∅. By Proposition 2.11, µt is a BCK-ideal of X. Now let A, B be two ideals of X such that AB ⊆ µt = {x ∈ X|µ(x) ≥ t}. If we define the fuzzy subsets θ := χA and σ := χB , then it is easy to show that θσ ⊆ µ, which implies θ ⊆ µ or σ ⊆ µ, since µ is a fuzzy prime ideal of X. It follows that A ⊆ µt or B ⊆ µt .



Lemma 4.6. Let X be a commutative BCC-algebra X. If z ≤ x and z ≤ y for all x, y, z ∈ X, then z ≤ x ∧ y. Proof. Since z ≤ x and z ≤ y, we have z ∗ x = 0 and z ∗ y = 0. Then z = z ∗ 0 = z ∗ (z ∗ x) = x ∗ (x ∗ z) and z = z ∗ 0 = z ∗ (z ∗ y) = y ∗ (y ∗ z), since X is commutative. Hence z = x ∗ (x ∗ z) = x ∗ (x ∗ (y ∗ (y ∗ z))) ≤ x ∗ (x ∗ y) = y ∧ x = x ∧ y. This competes the proof.



A BCC-algebra X is said to be positive implicative ([2]) if for any x, y, z ∈ X, (x ∗ y) ∗ z = (x ∗ z) ∗ (y ∗ z). Lemma 4.7. Let X be a positive implicative BCC-algebra which is commutative. Then a proper ideal P of X is an s-prime ideal of X if and only if P is a prime ideal of X. Proof. Suppose that P is an s-prime ideal such that AB ⊆ P for some ideals A, B of X. In order to prove that A ⊆ P or B ⊆ P , let us assume that neither A ⊆ P nor B ⊆ P . Then there exist a ∈ A, b ∈ B such that a ∈ /P and b ∈ / P . Since a ∧ b ∈ AB and AB ⊆ P , we have a ∧ b ∈ P . Since P is an s-prime ideal of X, a ∈ P or b ∈ P , which is a contradiction. Thus AB ⊆ P implies A ⊆ P or B ⊆ P . Conversely, suppose that for any ideals A, B of X, AB ⊆ P implies A ⊆ P or B ⊆ P . We prove that P is an s-prime ideal of X. Let a ∧ b ∈ P , where a, b ∈ X. Put A(a) := {x ∈ X|x ≤ a} and A(b) := {y ∈ X|y ≤ b}. Clearly, 0, a ∈ A(a). Let x ∗ y, y ∈ A(a). Then x ∗ y ≤ a and y ≤ a. Since X is positive implicative, we have (x ∗ y) ∗ a = (x ∗ a) ∗ (y ∗ a) = (x ∗ a) ∗ 0 = x ∗ a = 0. Hence x ∈ A(a). Therefore A(a) is a BCK-ideal of X. Similarly, A(b) is a BCK-ideal of X. We claim that A(a)A(b) ⊆ P . Let x ∈ A(a) and y ∈ A(b). Then x ≤ a and y ≤ b. Since X is commutative, we obtain x ∧ y ≤ x. Since (X, ≤) is a partially ordered set, we have x ∧ y ≤ a. Similarly, x ∧ y ≤ b. By Lemma 4.6, we obtain x ∧ y ≤ a ∧ b and a ∧ b ∈ P . Since P is a BCK-ideal, x ∧ y ∈ P . Hence A(a)B(b) ⊆ P . By hypothesis, A(a) ⊆ P or A(b) ⊆ P . Since a ∈ A(a), b ∈ A(b), we have a ∈ P or b ∈ P . Thus P is an s-prime ideal of X.

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On fuzzy spectrum of a BCC-algebra Theorem 4.8. Let µ be a fuzzy prime ideal of a BCC-algebra X. Then Xµ is a prime ideal of X. Proof. Clearly, Xµ = µµ(0) . By Lemma 4.5, Xµ is a prime ideal of X.



Theorem 4.9. Let µ be a fuzzy prime ideal of a positive implicative BCC-algebra X which is commutative. Then µ is a fuzzy s-prime ideal of X. Proof. Let µ be a fuzzy prime ideal of X. By Lemma 4.5, µt is a prime ideal of X. Using Lemma 4.7, µt is an s-prime ideal of X. It follows from Lemma 4.4 that µ is a fuzzy s-prime ideal of X.



The following example shows that the converse of Theorem 4.9 does not hold. Example 4.10. Let X := {0, e} be a set with the following table: ∗ 0 e 0 0 0 e e 0 Then (X; ∗, 0) is a positive implicative BCC-algebra which is commutative. Define the fuzzy subset µ of X by µ(0) = 0.7, µ(e) = 0. Clearly µ is a fuzzy s-prime ideal of X. Now consider the fuzzy ideals σ and θ of X which are defined by σ(x) =

1 2

for all x ∈ X and θ(e) = 0, θ(0) = 1. Then we have σθ ⊆ µ but σ * µ and θ * µ. Thus

µ is not a fuzzy prime ideal of X. Theorem 4.11. Let X be a commutative BCC-algebra. Then I is an s-prime ideal of X if and only if χI is a fuzzy s-prime ideal of X. Proof. Suppose that I is an s-prime ideal of X. By Theorem 2.12, χI is a fuzzy ideal of X. Since I is proper, χI is a non-constant function. Let x, y ∈ X. If x ∈ I or y ∈ I, then x ∧ y ∈ I. Hence χI (x ∧ y) = 1 = χI (x) ∨ χI (y). If x ∈ / I and y ∈ / I, then x ∧ y ∈ / I. Hence χI (x ∧ y) = 0 = χ(x) ∨ χI (y). Thus χI is a fuzzy s-prime ideal of X. Conversely, since I = XχI , if χI is a fuzzy s-prime ideal of X, it follows by Lemma 4.4 that I is an s-prime ideal of X.



Corollary 4.12. Let X be a positive implicative BCC-algebra which is commutative. Then P is a prime ideal of X if and only if χP is a fuzzy prime ideal of X. Proof. Let P be a prime ideal of X. Then χP is a fuzzy ideal of X. Now let θ, σ be two fuzzy ideals such that θσ ⊆ χP . We shall show that θ ⊆ χP or σ ⊆ χP .

(∗ ∗ ∗)

If (∗ ∗ ∗) does not hold, then there exist x, y ∈ X \ P such that θ(x) > 0 and σ(y) > 0. By Lemma 4.7, we have x∧y ∈ / P . Since θσ ⊆ χP , we have 0 < min{θ(x), σ(y)} ≤ θσ(x ∧ y) ≤ χP (x ∧ y). In other words, x ∧ y ∈ P , which is a contradiction. Hence (∗ ∗ ∗) holds. Conversely, let χP be a fuzzy prime ideal. By Theorem 4.9, χP is a fuzzy s-prime ideal. By Theorem 4.11, P is an s-prime ideal of X. By Lemma 4.7, P is a prime ideal of X. 567

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Sun Shin Ahn and Keum Sook So∗ References [1] W. A. Dudek, The number of subalgebras of finite BCC-algebras, Bull. Inst. Math. Academia Sinica 20 (1992), 129-136. [2] W. A. Dudek, On proper BCC-algebras, Bull. Inst. Math. Academia Sinica 20 (1992), 137-150. [3] W. A. Dudek and Y. B. Jun, Fuzzy BCC-algebras in BCC-algebras, Math. Montisnigri 10 (1999), 21-30. [4] W. A. Dudek, Y. B. Jun and Z. Stojakovic, On fuzzy ideald in BCC-algebras, Fuzzy Sets and Systems 123 (2001), 251-258. [5] W. A. Dudek and X. H. Zhang, On ideals and congruences in BCC-algebras, Czech. Math. Journal 48(123) (1998), 21-29. [6] J. Hao, Ideal Theory of BCC-algebras, Sci. Math. Japon. 1 (1998), 373-381. [7] Y. Imai and K. Is´eki, On axiom system of propositional calsuli XIV, Proc. Japan Academy 42 (1966), 19-22. [8] K. Is´eki and S. Tanaka, An introduction to the theory of BCK-algebras, Math. Japon. 23 (1978), 1-26. [9] Y. Komori, The class of BCC-algebras is not a variety, Math. Japon. 29 (1984), 391-394. [10] J. Meng and Y. B. Jun, BCK-algebras, Kyungmoonsa, Seoul, Korea, 1994. [11] A. Wronski, BCK-algebra do not form a variety, Math. Japon. 28 (1983), 211-213. [12] J. R. Munkres, Topology, Prentice Hall, New York, 1975.

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RIESZ FUZZY NORMED SPACES AND STABILITY OF A LATTICE PRESERVING FUNCTIONAL EQUATION CHOONKIL PARK, EHSAN MOVAHEDNIA∗ , SEYED MOHAMMAD SADEGH MODARRES MOSADEGH, AND MOHAMMAD MURSALEEN

Abstract. The main objective of this paper is to introduce and to study fuzzy normed Riesz spaces. By the direct method, we prove the Hyers-Ulam stability of the following lattice preserving functional equation in fuzzy Banach Riesz space N2 (f (τ x ∨ ηy) − τ f (x) ∨ ηf (y), t) ≥ N1 (ϕ(τ x ∨ ηy, τ x ∧ ηy), t) where (X , N1 ), (Y, N2 ) are fuzzy normed Riesz space and fuzzy Banach Riesz space, respectively; and ϕ : X × X → X is a mapping such that   α x y ϕ(x, y) ≤ (τ η) 2 ϕ , τ η for all τ, η ≥ 1 and α ∈ [0, 1).

1. Introduction Riesz spaces are named after Frigyes Riesz who first defined them in [1] . Riesz spaces are real vector spaces equipped with a partial order. Under this partial order the Riesz space must satisfy some axioms, including the axiom that it is a lattice. For the basic theory of vector lattices (Riesz spaces) and Banach lattices and for unexplained terminology we refer to [2, 3, 4]. In 1984, Katrasas [5] defined a fuzzy norm on a linear space to construct a fuzzy vector topological structure on the space. Later, some mathematicians have defined fuzzy norms on a linear space from various points of view [6, 7]. In particular, Bag and Samanta [8], following Cheng and Mordeson [9], gave an idea of a fuzzy norm in such a manner that the corresponding fuzzy metric is of Kramosil and Michalek type [10]. They also established a decomposition theorem of a fuzzy norm into a family of crisp norms and investigated some properties of fuzzy normed spaces. A classical question in the theory of functional equations is the following: When is it true that a function which approximately satisfies a functional equation D must be close to an exact solution of D? If the problem accepts a solution, we say that the equation D is stable. The first stability problem concerning group homomorphisms was raised by Ulam [11] in 1940. In 1941, Hyers [12] solved this stability problem for additive mappings subject to the Hyers condition on approximately additive mappings. The result of Hyers was generalized by Rassias [13] for linear mapping by considering an unbounded Cauchy difference. 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 ([14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25]). Recently, considerable attention has been increasing to the problem of fuzzy stability of functional equations. Several fuzzy stability results concerning Cauchy, Jensen, simple quadratic, and cubic functional equations have been investigated [26]–[31]. In this paper, Riesz fuzzy normed spaces are defined and the stability condition are verified. 2010 Mathematics Subject Classification. 54A40, 46S40, 39B52. Key words and phrases. fuzzy normed Riesz space; fixed point; Hyers-Ulam stability; lattice preserving functional equation. ∗ Corresponding author.

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C. PARK, E. MOVAHEDNIA, S.M.S.M. MOSADEGH, AND M. MURSALEEN

2. Preliminary estimates A non empty set M with a relation “≤” is said to be an ordered set whenever the following conditions are satisfied : 1. x ≤ x for every x ∈ M. 2. x ≤ y and y ≤ x implies that x = y. 3. x ≤ y and y ≤ z implies that x ≤ z. If, in addition, for all x, y ∈ M either x ≤ y or y ≤ x, then M is called a totally ordered set. Let A be subset of an ordered set M. x ∈ M is called an upper bound of A if y ≤ x for every y ∈ A. z ∈ M is called a lower bound of A if y ≥ z for all y ∈ A. Moreover, if there is an upper bound of A, then A is said to be bounded from above. If there is a lower bound of A, then A is said to be bounded from below. If A is bounded from above and from below, then we will briefly say that A is order bounded. An order set (M, ≤) is called a lattice if any two elements x, y ∈ M have a least upper bound denoted by x ∨ y = sup(x, y) and a greatest lower bound denoted by x ∧ y = inf(x, y). A real vector space E which is also an ordered set is an ordered vector space if the order and the vector space structure are compatible in the following sense: 1. If x, y ∈ E such that x ≤ y, then x + z ≤ y + z for all z ∈ E. 2. If x, y ∈ E such that x ≤ y, then αx ≤ αy for all α ≥ 0. (E, ≤) is called a Riesz space if (E, ≤) is a lattice and ordered vector space. A norm k · k on Riesz space E, is called a lattice norm if kxk ≤ kyk whenever |x| ≤ |y|. In the latter case (E, k · k) is called a normed Riesz space. (E, k · k) is called a Banach lattice if for all x, y ∈ E 1. (E, k · k) is a Banach space. 2. E is a Riesz space. 3. k · k is a lattice norm. Example 2.1. Suppose that X is compact Hausdorff space. We denote by C(K) the Banach space of all real continuous functions on X . Let “≤” be a point-wise order on C(K), f ≤ g if and only if f (t) ≤ g(t) for all t ∈ K. It is easy to see that (C(K), ≤) is a Banach lattice. Let E be a Riesz space and let the positive cone E + of E consist of all x ∈ E such that x ≥ 0. For every x ∈ E let x+ = x ∨ 0 x− = −x ∨ 0 |x| = x ∨ −x. Let E be a Riesz space. For all x, y, z ∈ E and a ∈ R the following assertions hold 1. x + y = x ∨ y + x ∧ y , −(x ∨ y) = −x ∧ y. 2. x + (y ∨ z) = (x + y) ∨ (x + z) , x + (y ∧ z) = (x + y) ∧ (x + z). 3. |x| = x+ + x− , |x + y| ≤ |x| + |y|. 4. x ≤ y is equivalent to x+ ≤ y + and y − ≤ x− . 5. (x ∨ y) ∧ z = (x ∧ y) ∨ (y ∧ z) , (x ∧ y) ∨ z = (x ∨ y) ∧ (y ∨ z). A Riesz space E is Archimedean if x ≤ 0 holds whenever the set {nx : n ∈ N} is bounded from above. Definition 2.1. [2] Let X and Y be Banach lattices. A mapping T : X → Y is called positive if T (X + ) = {T (|x|) : x ∈ X } ⊂ Y + . Theorem 2.1. [3] For an operator T : X → Y between two Riesz spaces the following statements are equivalent: 1. T is a lattice homomorphism. 2. T (x+ ) = T (x)+ for all x ∈ X . 3. T (x ∧ y) = T (x) ∧ T (y). 4. If x ∧ y = 0 in X , then T (x) ∧ T (y) = 0 holds in Y.

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RIESZ FUZZY NORMED SPACES

5. T (|x|) = |T (x)|. Definition 2.2. [4] Let X and Y be Banach lattices and let T : X → Y be a positive mapping. We define (P1 ) lattice homomorphism functional equation: T (|x| ∨ |y|) = T (|x|) ∨ T (|y|); (P2 ) semi-homogeneity: for all x ∈ X and every number α ∈ R+ T (α|x|) = αT (|x|). Remark 2.1. [4] Given two Banach lattices X and Y, let a positive mapping f : X → Y satisfy the property (P1 ). Then the following statements are valid. 1. f (|x ∨ y|) ≤ f (|x|) ∨ f (|y|) for all x, y ∈ X . 2. The semi-homogeneity implies that f (0) = 0. 3. f is an increasing operator, in the sense that if x, y ∈ X are such that |x| ≤ |y|, then f (|x|) ≤ f (|y|). 3. Main results Definition 3.1. Let (X , ≤) be a Riesz space. A function N : X × R → [0, 1] is called a Riesz fuzzy norm on X if for all x, y ∈ X and all s, t ∈ R (N1 ) (N2 ) (N3 ) (N4 ) (N5 )

N (x, t) = 0 for t ≤ 0 ; x = 0 if and only if N (x, t) = 1 for all t> 0 ; N (cx, t) = N (x, t/|c|) if c6= 0 ; N (x + y, t + s) ≥ min {N (x, t), N (y, s)} ; N (x, .) is a non-decreasing function of R and limt→∞ N (x, t) = 1 ;

(N6 ) for x 6= 0, N (x, .) is continuous on R ; (N7 ) N (x, t) ≥ N (y, t) whenever |x| ≤ |y|. Then (X , ≤, N ) is called a Riesz fuzzy normed space. Example 3.1. Let (X , ≤, k.k) be a normed Riesz space. One can easily verify that for each k > 0 ,  t    t + kkxk if t > 0 Nk (x, t) =    0 if t ≤ 0 defines a Riesz fuzzy norm on X . Note that (N1 )−(N6 ) have been checked in [8]. We show that (N7 ) is satisfied. Suppose that |x| ≤ |y| for all x, y ∈ X . Then ||x|| ≤ ||y|| since (X , ≤, k · k) is a normed Riesz space. So t t ≥ t + kkxk t + kkyk and so N (x, t) ≥ N (y, t) for all t > 0 and k > 0. Therefore, (X , ≤, N ) is a Riesz fuzzy normed space. Example 3.2. Let (X , ≤, k.k ) be a normed Riesz space. We define   0 if t ≤ kxk N (x, t) =  1 if t > kxk.

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J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 24, NO.3, 2018, COPYRIGHT 2018 EUDOXUS PRESS, LLC

C. PARK, E. MOVAHEDNIA, S.M.S.M. MOSADEGH, AND M. MURSALEEN

It is very easy to show that (X , ≤, N ) is a Riesz fuzzy normed space. Remark 3.1. Convergent and Cauchy sequences in Riesz fuzzy normed space are same as in fuzzy normed space. Definition 3.2. Let (X , ≤, N ) be a fuzzy normed Riesz space. A sequence {xn } in X is said to be convergent if there exists an x ∈ X such that limn→∞ N (xn − x, t) = 1 for all t >0. In this case, x is called the limit of the sequence {xn } and we denote it by N − limn→∞ N (xn − x, t) = x. Definition 3.3. Let (X , ≤, N ) be a fuzzy normed Riesz space. A sequence {xn } in X is said to be Cauchy if for each  > 0 and each δ > 0 there exists an n0 ∈ N such that N (xm − xn , δ) > 1 −  (m, n ≥ n0 ). Definition 3.4. Let (X , ≤, N ) be a fuzzy normed Riesz space. A sequence yn in X is called order fuzzy convergent to y as n → ∞ if there exists a sequence xn ↓ 0 in X as n → ∞ and N (yn − y, t) ≥ N (xn , t) for all n ∈ N. We write y = of − limn→∞ yn . It is well-known that every convergent sequence in a fuzzy normed Riesz space is Cauchy. If each Cauchy sequence is convergent, then the Riesz fuzzy norm is said to be complete and the fuzzy normed Riesz space is called a fuzzy Banach Riesz space. Theorem 3.1. Let (X , ≤, N ) be a fuzzy normed Riesz space and let {xn }, {yn } be sequences in X such that x = of − lim xn

and

n→∞

y = of − lim yn . n→∞

Then xn + yn

= of − lim x + y,

xn ∨ yn

= of − lim x ∨ y,

xn ∧ yn

= of − lim x ∧ y.

n→∞

n→∞

n→∞

Theorem 3.2. Let (X , ≤, N ) be a fuzzy normed Riesz space. Then lattice operators are continuous. Proof. Assume that lim N (xn − x, s) = 1

n→∞

limn→∞ N (yn − y, t) = 1

(3.1)

for all t, s > 0. Therefore, N (xn ∧ yn − x ∧ y, t + s)

= N (xn ∧ yn − xn ∧ y + xn ∧ y − x ∧ y, t + s) ≥ min{N (xn ∧ yn − xn ∧ y, t), N (xn ∧ y − x ∧ y, s)} ≥ min{N (yn − y, t), N (xn − x, s)}.

So lim N (xn ∧ yn − x ∧ y, t + s) = 1.

n→∞

It is easy to see that the other lattice operations are continuous.



Theorem 3.3. Every fuzzy normed Riesz space is Archimedean.

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Proof. Let (X , ≤, N ) be fuzzy normed Riesz space. We show that X has Archimedean properties. Suppose that x, y ∈ X+ and nx ≤ y for all n ∈ N. N (nx, t) ≥ N (y, t)

t>0

and so  N

x,

t n

 ≥ N (y, t)

t > 0.

Therefore, N (x, t) ≥ N (y, nt)

t>0

and N (x, t) ≥ N (n−1 y, t)

t > 0.

Since n ∈ N is arbitrary as n → ∞, we have x = 0. Hence X has Archimedean properties.



Theorem 3.4. Let (E, ≤, N ) be a fuzzy normed Riesz space. Then the positive cone E+ is closed. Proof. Assume that xn ∈ E+  lim N

n→∞

t xn − x, 2

 for all t > 0, x ∈ E.

=1

By Theorem 3.2, we have  lim N

n→∞

xn ∨ 0 − x ∨ 0,

t 2

 =1

So xn ∨ 0 = xn since xn ∈ E+ . Therefore,   t lim N xn − x ∨ 0, =1 n→∞ 2

for all t > 0, x ∈ E.

for all t > 0, x ∈ E

and hence by (N4 ), we get      t t , N xn − x, N (x − x ∨ 0, t) ≥ min N xn − x ∨ 0, 2 2 for all t > 0 and x ∈ E. Two terms on the right hand side of the above inequality tend to 1 as n → ∞, and so x = x ∨ 0. Hence x ∈ E+ . Thus the proof is complete.  Theorem 3.5. Let (E, ≤, N ) be a fuzzy normed Riesz space. For every increasing convergent sequence {xn } ⊂ E lim N (xn − u, t) = 1 for all t > 0,

n→∞

where u = sup{xn : n ∈ N}. Proof. Suppose that {xn } is an increasing convergent sequence and lim N (xn − x, t) = 1

n→∞

for all t > 0 and all n ∈ N.

(3.2)

For every m ≥ n, we have xm − xn ∈ E+ . It follows from Theorem 3.4 that x − xn ≥ 0 and xn ≤ u ≤ x for all n ∈ N. So by (N7 ) N (xn − x, t) ≤ N (xn − u, t) for all t > 0. Therefore, as n → ∞, we have limn→∞ N (xn − u, t) = 1 and hence u = x. This completes the proof.  Definition 3.5. The sequence {xn } is called uniformly bounded if there exist e ∈ E+ and an ∈ l1 such that xn ≤ an · e.

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J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 24, NO.3, 2018, COPYRIGHT 2018 EUDOXUS PRESS, LLC

C. PARK, E. MOVAHEDNIA, S.M.S.M. MOSADEGH, AND M. MURSALEEN

Definition 3.6. Let (E, ≤, N ) be a fuzzy normed Riesz space. Then E is called uniformly complete if Pn sup{ i=1 xi : n ∈ N} exists for every uniformly bounded sequence xn ⊂ E+ . Theorem 3.6. Every fuzzy Banach Riesz space is uniformly complete. Proof. Let (E, ≤, N ) be a fuzzy Banach Riesz space and {xn } ⊂ E+ be a sequence such that xn ≤ an e Pn for a suitable sequence {an } ∈ l1 and some e ∈ E+ . We show that sup{ i=1 xi : n ∈ N} exists. We set yn = x1 + x2 + ... + xn

∞ X

and bn =

aj .

(3.3)

j=n+1

By (3.3) and (N7 ) we have N (yn+p − yn , t)

= N (xn+1 + · · · + xn+p , t) ∞ X ≥ N( an+j · e, t) j=1

= N (bn · e, t) for all t > 0. As n → ∞, we get lim N (yn+p − yn , t) = 1.

n→∞

So (yn ) is a Cauchy sequence in fuzzy Banach Riesz space and therefore there exists y ∈ E such that yn −→N y. Since yn is increasing and convergence sequence, by Theorem 3.5, we have lim N (yn − ∨yn , t) = 1

n→∞

P∞ that is, yn −→N sup{ i=1 xi : n ∈ N}. Using a unique limit we have y = sup{

∞ X

xi : n ∈ N}.

i=1

Thus the proof is complete.



Definition 3.7. Let (E, ≤, N ) be a fuzzy normed Riesz space. A ⊂ E is solid if (1) f ∈ A if and only if |f | ∈ A; (2) if 0 ≤ f ∈ A and g ∈ E+ then f ∧ g ∈ A. Definition 3.8. Every solid subset I of E is called an ideal in E. Definition 3.9. An ordered closed ideal is referred to as a band. Theorem 3.7. Let (E, ≤, N ) be a fuzzy normed Riesz space. Then the closure of every solid subset of E is solid. Proof. Let A ⊂ E be solid and f ∈ A. We show that |f | ∈ A. There exists {fn } ∈ A such that fn −→N f . It follows from ||fn | − |f || ≤ |fn − f | and (N7 ) that N (|fn | − |f |, t) ≥

N (|fn − f |, t)

= N (fn − f, t) for all n ∈ N and t > 0. Therefore, |fn | →N |f | as n tends to infinity. Hence |f | ∈ A, since A is solid and fn ∈ A. Conversely, assume that |f | ∈ A. Then there exists fn ∈ A+ such that fn −→N |f |. By (3.2) we have fn ∧ f −→N f ∧ |f | = f.

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Therefore, f ∈ A, since fn ∧ f ∈ A. Now, assume that 0 ≤ f ∈ A and g ∈ E+ . There exists a sequence {fn } ∈ A+ such that fn −→N f. Hence by (3.2) fn ∧ g →N f ∧ g. So fn ∧ g ∈ A. Since fn ∈ A and A is solid, f ∧ g ∈ A.



Theorem 3.8. Let (E, ≤, N ) be a fuzzy normed Riesz space. Then every band in E is closed. Proof. Suppose that B is a band and assume that {fn } ⊂ B is a sequence such that fn −→N f for some f ∈ E. It follows from (3.2) that |fn | ∧ |f | →N |f | as n → ∞. For every n ∈ N, let gn = (|fn | ∨ · · · ∨ |f1 |) ∧ |f |. Then {gn } is an increasing sequence and gn = (|fn | ∧ |f |) ∨ ... ∨ (|f1 | ∧ |f |). So |fn | ∧ |f | ≤ gn ≤ |f |. Therefore, by (N7 ) we have N (|f | − gn , t) ≥ N (|f | − |fn | ∧ |f |, t) for all t > 0. Hence gn −→N |f | as n → ∞.



4. Hyers-Ulam stability of lattice homomorphisms in fuzzy normed Riesz spaces Using the direct method, we prove the Hyers-Ulam stability of lattice homomorphisms in fuzzy Banach Riesz space as below. Theorem 4.1. Let f be a positive operator from a fuzzy normed Riesz space (X , N1 ) to a fuzzy Banach Riesz space (Y, N2 ) such that N2 (f (τ x ∨ ηy) − τ f (x) ∨ ηf (y), t) ≥ N1 (ϕ(τ x ∨ ηy, τ x ∧ ηy), t)

(4.4)

for all x, y ∈ X and t > 0. Here ϕ : X × X → X is a mapping such that   α x y , ϕ(x, y) ≤ (τ η) 2 ϕ τ η for all τ, η ≥ 1 and for which there are a number α ∈ [0, 1) and a unique positive operator T : X → Y satisfying the properties (P1 ) and (P2 ) for x ∈ X+ and the inequality N2 (T (x) − f (x), t) ≥ N1 (ϕ(x, x),

τ − τα , t). τα

Proof. Putting y = x and τ = η in (??), we have N2 (f (τ x) − τ f (x), t) ≥

N1 (ϕ(τ x, τ x), t)

N1 (τ α ϕ(x, x), t)   t = N1 ϕ(x, x), α . τ ≥

Therefore,  N2

1 f (τ x) − f (x), τ α−1 t τ

575

 ≥

N1 (ϕ(x, x), t).

(4.5)

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J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 24, NO.3, 2018, COPYRIGHT 2018 EUDOXUS PRESS, LLC

C. PARK, E. MOVAHEDNIA, S.M.S.M. MOSADEGH, AND M. MURSALEEN

Now, replacing x by τ x in (4.5) and using the assumption that ϕ(τ x, τ x) ≤ τ α ϕ(x, x) and the property (N3 ) and (N7 ) of Definition 3.1, we obtain   1 N2 f (τ 2 x) − f (τ x), τ α−1 t ≥ N1 (ϕ(τ x, τ x), t) τ ≥ N1 (τ α ϕ(x, x), t)   t = N1 ϕ(x, x), α . τ Hence  N2

 1 1 2 2α−2 f (τ x) − f (τ x), τ t τ2 τ

≥

N1 (ϕ(x, x), t).

By comparing (4.5) and (4.6) and using property (N4 ), we obtain     1 2 α−1 2(α−1) f (τ x) − f (x), τ t ≥ N2 +τ τ2 Again, replacing x by τ x in (4.7), we have     1 1 3 2(α−1) 3(α−1) N2 f (τ x) − f (τ x), τ +τ t τ3 τ

N1 (ϕ(x, x), t).

≥

N1 (ϕ(x, x), t).

By comparing (4.5) and (4.8) and the property (N4 ), we obtain     1 3 (α−1) 2(α−1) 3(α−1) N2 f (τ x) − f (x), τ +τ +τ t ≥ τ3

(4.6)

(4.7)

(4.8)

N1 (ϕ(x, x), t).

With this process, we obtain n

N2

X 1 f (τ n x) − f (x), τ k(α−1) t n τ

! ≥

N1 (ϕ(x, x), t)

(4.9)

k=1

for all n ∈ N. If m ∈ N and n > m > 0, then n − m ∈ N. Replacing n by n − m in (4.9), we get ! n−m X 1 f (τ n−m x) − f (x), τ k(α−1) t ≥ N1 (ϕ(x, x), t). (4.10) N2 τ n−m k=1

m

By replacing x by τ x and using (N7 ), we obtain N2

n−m 1 1 1 X k(α−1) n m τ t f (τ x) − f (τ x), τn τm τm

! ≥

N1 (ϕ(τ m x, τ m x), t)

k=1

N1 (τ mα ϕ(x, x), t)   t = N1 ϕ(x, x), mα . τ ≥

It follows that N2

1 1 1 f (τ n x) − m f (τ m x), m τn τ τ

n X

! τ k(α−1) t

≥

N1 (ϕ(x, x), t) .

(4.11)

k=m+1

Let c > 0, and let  be given. Since limt→∞ N1 (ϕ(x, x), t) = 1, there is some t0 > 0 such that N1 (ϕ(x, x), t) ≥ 1 − . P∞ k(α−1) Fix t > t0 . The convergence of series k=1 τ guarantees that there exists some n0 ≥ 0 such that, for each n > m > n0 , the inequality n X τ k(α−1) < c k=m+1

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RIESZ FUZZY NORMED SPACES

holds. It follows that   1 1 n m N2 f (τ f (τ x) − x), c ≥ τn τm

n X 1 1 n m f (τ f (τ x) − x), τ k(α−1) t τn τm

N2

!

k=m+1

≥

N1 (ϕ(x, x), t) ≥ 1 − .

 f (τ n x) is a Cauchy sequence in fuzzy Banach Riesz space (Y, N2 ) and thus this sequence τn converges to some T (x) ∈ Y. It means that 

Hence

f (τ n x) n→∞ τn

T (x) = N2 − lim Furthermore, by putting m = 0 in (4.11), we have n

N2

X 1 n f (τ x) − f (x), τ k(α−1) t τn

! ≥

N1 (ϕ(x, x), t).

k=1

So  N2

 1 n f (τ x) − f (x), t τn

≥

 N1 ϕ(x, x), Pn

t k(α−1) τ k=1

 .

As n → ∞, we have   τ − τα N1 ϕ(x, x), t . τα

N2 (T (x) − f (x), t) ≥

Next, we show that T satisfies (P1 ). Putting τ = η = τ n in (4.4), we get N2 (f (τ n x ∨ τ n y) − τ n f (x) ∨ τ n f (y), t) ≥

N1 (ϕ(τ n x ∨ τ n y, τ n x ∧ τ n y, t)   t ≥ N1 ϕ(x ∨ y, x ∧ y), nα . τ

Substituting x with τ n x and y with τ n y in this last inequality, one can get N2 (f (τ n (τ n x ∨ τ n y)) − τ n f (τ n x) ∨ τ n f (τ n y), t)   t n n n n ≥ N1 ϕ(τ x ∨ τ y, τ x ∧ τ y, nα τ   t ≥ N1 ϕ(x ∨ y, x ∧ y), 2nα , τ which yields N2

!
f τ 2n (x ∨ y) f (τ n x) f (τ n y) t − ∨ , 2n ≥ τ 2n τn τn τ

  N1 ϕ(x ∨ y, x ∧ y), τ 2(1−α) t .

The term on the right-hand side of the above inequality tends to 1 as n → ∞. By Theorem 3.2, we obtain N2 (T (x ∨ y) − T x ∨ T y, t) ≥ 1. This means that T (x ∨ y)

= T x ∨ T y,

consequently, the property (P1 ) holds. Next, we show that T (τ x) = τ T (x) for all x ∈ X+ and τ ≥ 1. In fact, in the inequality (4.4), choose η = τ , y = 0 and substitute 2n τ for τ and consider Remark 2.1. N2 (f (2n τ x ∨ 0) − 2n τ f (x) ∨ 0, t) ≥

577

N1 (ϕ (2n τ x, 0) , t)

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C. PARK, E. MOVAHEDNIA, S.M.S.M. MOSADEGH, AND M. MURSALEEN

for all x ∈ X. Now we replace x with 2n x. Consequently, by (2.1)   f (4n τ x) τ f (2n x) t N2 − , n ≥ N1 (ϕ (4n τ x, 0) , t) 4n 2n 4 ≥ N1 (4nα τ α ϕ (x, 0) , t) . Therefore,  N2

 f (4n τ x) τ f (2n x) − , t ≥ 4n 2n

  N1 4n(α−1) τ α ϕ (x, 0) , t .

The term on the right-hand side of the above inequality tends to 1 as n → ∞. Thus T (τ x) = τ (T x) for all x ∈ X+ .



Theorem 4.2. Let X , Y be Banach lattices and p : [0, ∞) → [0, ∞) be a continuous function. Consider a positive map f : X → Y for which there are numbers ν ∈ R and 0 ≤ r < 1 such that   αp(α)f (|x|) ∨ βp(β)f (|y|) N2 f (α|x| ∨ β|y|) − , t ≥ N1 (ν (xr ∨ y r ) , t) (4.12) p(α) ∨ p(β) for all x, y ∈ X and α, β ∈ R+ . Then there exists a unique positive mapping T : X → Y which satisfies the properties (P1 ), (P2 ) and the inequality   2νx N2 (F (|x|) − T (|x|), t) ≥ N1 ,t . 2 − 2r Proof. Putting α = β = 2 and x = y in (4.12), we get   2p(2)f (|x|) ∨ 2p(2)f (|x|) N2 f (2|x|) − , t ≥ N1 (νxr , t) p(2) ∨ p(2) for all x ∈ X and r ∈ [0, 1). Therefore, N2 (f (2|x|) − 2f (|x|), t) ≥ N1 (νxr , t) ,   1 N2 f (2|x|) − f (|x|), t ≥ N1 (νxr , 2t) . 2 The rest of the proof is similar to the previous one.



5. Conclusion In the classical Riesz space theory, Banach lattice requires more attention. In the present research work, we briefly introduced and studied the fuzzy normed Riesz spaces. Thus we think that there are many open problems and applications in this new research area. For example we will introduce M-space, L-space and order unit in fuzzy normed Riesz spaces in our future research work. References [1] F, Riesz, Sur la decomposition des oprations fonctionnelles linaires, Atti Congr. Internaz. Mat., Bologna 3 (1930), 143–148. [2] P. M. Nieberg, Banach Lattice, Springer-Verlag, Berlin Heidelberg, 1991. [3] C. D. Aliprantis and O. Burkinshaw, Positive Operators, Springer Science and Business Media, 2006. [4] C. Zaanen, Introduction to Operator Theory in Riesz Spaces, Springer Science and Business Media, 2012. [5] A. K. Katsaras, Fuzzy topological vector spaces, Fuzzy Sets Syst. 12 (1984), 143-154. [6] C. Felbin, Finite dimensional fuzzy normed linear space, Fuzzy Sets Syst. 48 (1992), 239-248. [7] J. Z. Xiao and X. H. Zhu, Fuzzy normed space of operators and its completeness, Fuzzy Sets Syst. 133 (2003), 389–399. [8] T. Bag and S. K. Samanta, Finite dimensional fuzzy normed linear spaces, J. Fuzzy Math. 11 (2003), 687-705.

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[9] S. C. Cheng and J. N. Mordeson, Fuzzy linear operators and fuzzy normed linear spaces, Bull. Calcutta Math. Soc.86 (1994), 429-436. [10] I. Kramosil and J. Michalek, Fuzzy metric and statistical metric spaces, Kybernetika 11 (1975), 326-334. [11] S. M. Ulam, Problems in Modern Mathematics, Wiley, New York, 1964. [12] D. H. Hyers, On the stability of the linear functional equation, Proc. Nat. Acad. Sci. USA 27 (1941), 222–224. [13] T. M. Rassias, On the stability of the linear mapping in Banach spaces, Proc. Amer. Math. Soc. 72 (1978), 297–300. [14] S. Jung, Hyers-Ulam-Rassias stability of Jensen’s equation and its application, Proc. Amer. Math. Soc. 126 (1998), 3137–3143. [15] M. Mursaleen and K. J. Ansari, On the stability of some positive linear operators from approximation theory, Bull. Math. Sci. 5 (2015), 147–157. [16] M. Mursaleen and K. J. Ansari, Stability results in intuitionistic fuzzy normed spaces for a cubic functional equation, Appl. Math. Inf. Sci. 7 (2013), 1685–1692. [17] N. K. Agbeko, Stability of maximum preserving functional equation on Banach lattice. Miskols Math. Notes 13 (2012), 187–196. [18] A. Hazy and Z. P ales, On a certain stability of the Hermite-Hadamard inequality, Proc. R. Soc. Lond, Ser. A, Math. Phys. Eng. Sci. 465 (2009), 571-583. [19] C. Park, K. Ghasemi, S. G. Ghaleh and S. Jang, Approximate n-Jordan ∗-homomorphisms in C ∗ -algebras, J. Comput. Anal. Appl. 15 (2013), 365–368. [20] C. Park, A. Najati and S. Jang, Fixed points and fuzzy stability of an additive-quadratic functional equation, J. Comput. Anal. Appl. 15 (2013), 452–462. [21] S. Schin, D. Ki, J. Chang and M. Kim, Random stability of quadratic functional equations: a fixed point approach, J. Nonlinear Sci. Appl. 4 (2011), 37–49. [22] S. Shagholi, M. Bavand Savadkouhi and M. Eshaghi Gordji, Nearly ternary cubic homomorphism in ternary Fr´ echet algebras, J. Comput. Anal. Appl. 13 (2011), 1106–1114. [23] S. Shagholi, M. Eshaghi Gordji and M. Bavand Savadkouhi, Stability of ternary quadratic derivation on ternary Banach algebras, J. Comput. Anal. Appl. 13 (2011), 1097–1105. [24] D. Shin, C. Park and Sh. Farhadabadi, On the superstability of ternary Jordan C ∗ -homomorphisms, J. Comput. Anal. Appl. 16 (2014), 964–973. [25] D. Shin, C. Park and Sh. Farhadabadi, Stability and superstability of J ∗ -homomorphisms and J ∗ -derivations for a generalized Cauchy-Jensen equation, J. Comput. Anal. Appl. 17 (2014), 125–134. [26] E. Movahednia, Fuzzy stability of quadratic functional equations in general cases, ISRN Math. Anal. 2012, Art. ID 503164 (2011). [27] E. Movahednia, S. Eshtehar and Y. Son, Stability of quadratic functional equations in fuzzy normed spaces, Int. J. Math. Anal. 6 (2012), 2405–2412. [28] A. K. Mirmostafaee and M. S. Moslehian, Fuzzy versions of Hyers-Ulam-Rassias theorem, Fuzzy Sets Syst. 159 (2006), 720–729. [29] A. K. Mirmostafaee, M. Mirzavaziri and M. S. Moslehian, Fuzzy stability of the Jensen functional equation, Fuzzy Sets Syst. 159 (2008), 730–738. [30] A. K. Mirmostafaee and M. S. Moslehian, Fuzzy almost quadratic functions, Results Math. 52 (2008), 161–177. [31] A. K. Mirmostafaee and M. S. Moslehian, Fuzzy approximately cubic mappings, Inform. Sci.178 (2008), 3791–3798. Choonkil Park Research Institute for Natural Sciences, Hanyang University, Seoul 04763, Republic of Korea E-mail address: [email protected] Ehsan Movahednia Department of Mathematics, Behbahan Khatam Al-Anbia University of Technology, Behbahan, Iran E-mail address: [email protected] Seyed Mohammad Sadegh Modarres Mosadegh Department of mathematics, University of Yazd, P. O. Box 89195-741, Yazd, Iran E-mail address: [email protected] Mohammad Mursaleen Department of Mathematics, Aligarh Muslim University, Aligarh 202002, India E-mail address: [email protected]

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BIPOLAR FUZZY SETS OF BCK-MODULES M. A. ALGHAMDI1 , N. O. ALSHEHRI2 , AND N. M. MUTHANA 3 Abstract. The notion of bipolar fuzzy BCK-submodules are introduced, and some characterizations of bipolar fuzzy BCK-submodules are given. The conceopt of homomorphic images and preimages of bipolar fuzzy BCK-submodules are investigated. Normality and completely normality of bipolar fuzzy BCKsubmodules are discussed.

keywords: bipolar fuzzy BCK-submodule, normal (completely normal) bipolar fuzzy BCK-submodule, maximal bipolar fuzzy BCK-submodule.

1. Introduction In the traditional fuzzy sets, which presented by Zadeh [8] in 1965, the membership of elements are expressed in degrees ranging from 0 to 1. The membership degree 0 is assigned to elements which do not satisfy a corresponding property to the concerned fuzzy set. It is of interest to know whether these elements are satisfying a counter-property of our fuzzy set, but the restriction of the membership degrees to the interval [0,1] led to a great di¢ culty in doing this. For this reason, Lee [7] introduced the concept of the bipolar-valued fuzzy sets as an extension of the fuzzy sets. In the case of bipolar-valued fuzzy sets, the membership degrees range is increased from the interval [0,1] to the interval [-1,1]. The representation of bipolar-valued fuzzy sets express the di¤erence of contrary elements from irrelevant elements. The notion of bipolar-valued fuzzy subalgebra and bipolar-valued fuzzy ideal was introduced by Lee [6]. H. A. S. Abujabal, M. Aslam and A. B. Thaheem [1], introduced the notion of BCK-modules as an action of BCK-algebra over a commutative group. The concept of fuzzy BCK-submodules was introduced by M. Bakhshi [2], where he characterized the fuzzy BCK-submodules and provided some operations of it. In this paper, we apply the notion of bipolar-valued fuzzy set on BCK-modules and introduce the notion of bipolar-valued fuzzy BCK-submodules. Then we present some characterization of bipolar-valued fuzzy BCK-submodules by means of positive t-level cut and negative s-level cut. Moreover, a certain form of bipolar fuzzy BCK-submodules is derived from a given BCK-submodule. We investigate the homomorphic image and preimage of the bipolar-valued fuzzy BCK-submodules under some conditions. The later work is devoted to discuss the normality and completely normality of bipolar fuzzy BCK-submodules. A maximal bipolar fuzzy BCK-submodule is de…ned and its range is speci…ed quit so. Many examples are given to illustrate our concepts and results.

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M . A. ALGHAM DI 1 , N. O. ALSHEHRI 2 , AND N. M . M UTHANA 3

2. Preliminaries In this section we review some de…nitions and results regarding BCK-algebras, BCK-modules and bipolar fuzzy sets. By a BCK-algebra, we mean an algebra (X; ; 0) of type (2; 0) satisfying the following axioms: (I) ((x y) (x z))(z y) = 0; (II) (x (x y)) y = 0; (III) x x = 0; (IV) 0 x = 0; (V) x y = 0 and y x = 0 implies x = y, for all x; y; z 2 X. Let (X; ; 0) be a BCK-algebra. Then X is a partially ordered set with the partial ordering de…ned on X by: x y if and only if x y = 0. X is said to be bounded if there is an element 1 2 X such that x 1 for all x 2 X. X is said to be commutative (implicative) if x ^ y = y ^ x (x (y x) = x) for all x; y 2 X where x ^ y = y (y x). De…nition 2.1[1] Let X be a BCK-algebra. Then by a left X-module (abbreviated X-module), we mean an abelian group M with an operation X M ! M with (x; m) 7! xm satis…es the following axioms for all x; y 2 X and m; n 2 M; (1) (x ^ y)m = x(ym); (2) x(m + n) = xm + xn; (3) 0m = 0: Moreover, if X is bounded and M satis…es 1m = m, for all m 2 M , then M is said to be unitary. Example 2.2 Any bounded implicative BCK-algebra X forms an X-module, where "+ "is de…ned as x + y = (x y) _ (y x) and xy = x ^ y. A subgroup N of an X-module M is called submodule of M if N is also an X-module. Theorem 2.3 [2] A subset N of a BCK-module M is a BCK-submodule of M if and only if n1 n2 ; xn 2 N for all n1 ; n2 ; n 2 N and x 2 X. De…nition 2.4 [1] Let M; N be modules over a BCK-algebra X. A mapping f : M ! N is called an X-homomorphism if (1) f (m1 + m2 ) = f (m1 ) + f (m2 ) (2) f (xm) = xf (m) for all m1 ; m2 ; m 2 M , x 2 X. A BCK-module homomorphism is said to be monomorphism (epimorphism) if it is one to one (onto). If it is both one to one and onto, then we say that it is an isomorphism. Let X be the universe of discourse. A bipolar valued fuzzy set object having the form

of X is an

= f(x; + (x); (x)) j x 2 Xg + where : X ! [0; 1] and : X ! [ 1; 0] are mappings. The positive membership degree + (x) denotes the satisfaction degree of an element x to the

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property corresponding to a bipolar -valued fuzzy set = f(x; + (x); (x)) j x 2 Xg, and the negative membership degree (x) denotes the satisfaction degree of x to some implicit counter-property of = f(x; + (x); (x)) j x 2 Xg. For the sake of simplicity, we shall use the symbol = (X; + ; ) for the + bipolar-valued fuzzy set = f(x; (x); (x)) j x 2 Xg, and use the notion of bipolar fuzzy sets instead of the notion of bipolar-valued fuzzy sets. For a bipolar fuzzy set = (X; + ; ) and (t; s) 2 [0; 1] [ 1; 0], we de…ne P ( ; t) = fx 2 X j N ( ; s) = fx 2 X j

+

(x) (x)

tg; sg

which are called the positive t-level cut of = (X; + ; ) and the negative + s-level cut of = (X; ; ), respectively. For k 2 [0; 1], the set L( ; k) = P ( ; k) \ N ( ; k) is called the k-level cut of = (X; + ; ) (see [6]). If = (X; + ; ) and = (X; + ; ) are bipolar fuzzy sets de…ned on X, then the union and the intersection of and are bipolar fuzzy sets of X de…ned as follows: + + [ = (X; ( [ ) ; ( [ ) ) and \ = (X; ( \ ) ; ( \ ) ); respectively, where +

+

(x) ;

+

(x) ; ( [ ) (x) = min

(x) ;

(x) ;

+

+

(x) ;

+

(x) ; ( \ ) (x) = max

(x) ;

(x) ;

( [ ) (x) = max and ( \ ) (x) = min for all x 2 X.

De…nition 2.5 [3] Let = (X; + ; ) and = (X; + ; ) be bipolar fuzzy + + sets of X. If (x) (x) and (x) (x) for all x 2 X, then we say that = (X; + ; ) is a bipolar fuzzy extension of = (X; + ; ) (simply is subset of ) and we write .

In what follows, X will denote a bounded BCK-algebra and M; N are X-modules unless otherwise speci…ed. 3. Bipolar Fuzzy BCK-Submodules In this section applying bipolar fuzzy sets theory to BCK-modules, we introduce the notion of bipolar fuzzy BCK-submodules and discuss their properties. De…nition 3.1 A bipolar fuzzy set = (M ; + ; ) of a BCK-module M is said to be a bipolar fuzzy BCK-submodule if it satis…es: (BFS1) + (m1 + m2 ) minf + (m1 ); + (m2 )g and (m1 + m2 ) maxf (m1 ); (m2 )g; (BFS2) + ( m) = + (m) and ( m) = (m); + (BFS3) + (xm) (m) and (xm) (m):

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For the sake of simplicity, we shall use the symbols BF(M ) and BFS(M ) for the set of all bipolar fuzzy sets of M , and the set of all bipolar fuzzy BCK-submodules of M , respectively. Example 3.2 Let X = P (Z) with de…ned by A B = A \ B c and 0 is the empty set ?, then X is a BCK-algebra. M = ZZ = ff j f : Z ! Zg, considered with the traditional addition of maps and 0 is the zero map, is an abelian group. If we de…ne an action of X on M by Af = A f , then M forms an X-module. De…ne a bipolar fuzzy set = (M ; + ; ) on M by +

1 if f = 0; 0 otherwise,

(f ) =

and 1 if f = 0; 0 otherwise. is a bipolar fuzzy BCK-submodule of M. (f ) =

Then

Example 3.3 Let X = f0; a; b; c; dg and consider the following table: 0 a b 0 0 0 a 0 a b b 0 c b a d d d

0 a b c d

c d 0 0 0 a 0 b 0 d d 0

Tab. 3.1

Then (X; ; 0) is a commutative BCK-algebra which is not bounded. The subset M = f0; a; b; cg of X along with the operation + de…ned by Table 3.2 is an abelian group. Table 3.3 shows the action of X on M (xm = x ^ m). Consequently, M forms an X-module. + 0 a b c

0 a b 0 a b a 0 c b c 0 c b a

^ 0 a b c d

0 0 0 0 0 0

c c b a 0

Tab. 3.2

a 0 a b a 0

b 0 0 b b 0

c 0 a b c 0

Tab. 3.3

Now let

= (M ;

+

;

) be a bipolar fuzzy set on M de…ned as follows: M +

0 1

a b c 0:7 0:7 0:7 0:8 0:6 0:6 0:6 Tab. 3.4

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Then

= (M ;

+

;

5

) is a bipolar fuzzy BCK-submodule of M .

Theorem 3.4 A bipolar fuzzy set = (M ; + ; ) 2 BFS(M ) if and only if + (i) + (xm) (m) and (xm) (m); (ii) + (m1 m2 ) minf + (m1 ); + (m2 )g and (m1 m2 ) maxf (m1 ); (m2 )g for all m; m1 ; m2 2 M and x 2 X: Proof Let = (M ; + ; ) 2 BFS(M ), then +

(m1

m2 )

minf = minf m2 ) maxf = maxf

(m1

+

(m1 ); + (m1 ); (m1 ); (m1 );

+ +

( m2 )g (m2 )g; ( m2 )g (m2 )g:

Conversely, assume that (i) and (ii) are satis…ed. Put x = 0 in (i), then + (m); and (0) (m) for all m 2 M . So using (ii), we have +

+

( m) = + (m) =

(0 + (0

+

(0)

+ m) minf + (0); + (m)g (m); + + + ( m)) minf (0); ( m)g ( m);

which implies that +

( m) =

+

(m):

Moreover, +

(m1 + m2 )

We can verify that ( m) = by similar argument.

=

+

(m1 ( m2 )) minf + (m1 ); + ( m2 )g = minf + (m1 ); + (m2 )g: (m) and

(m1 +m2 )

maxf

(m1 );

(m2 )g

Theorem 3.5 A bipolar fuzzy set = (M ; + ; ) 2 BFS(M ) if and only if + (i) + (0) (m) and (0) (m); (ii) + (x1 m1 x2 m2 ) minf + (m1 ); + (m2 )g and (x1 m1 x2 m2 ) maxf (m1 ); (m2 )g for all m; m1 ; m2 2 M and x1 ; x2 2 X: Proof Let = (M ; + ; ) 2 BFS(M ) and let m; m1 ; m2 2 M and x1 ; x2 2 X: (i) is already shown in the proof of Theorem 3.4. Moreover, +

(x1 m1

x2 m2 )

minf minf

(x1 m1

x2 m2 )

maxf maxf

+

(x1 m1 ); + (x2 m2 )g + (m1 ); + (m2 )g;

and (x1 m1 ); (x2 m2 )g (m1 ); (m2 )g:

Now, let = (M ; + ; ) be a bipolar fuzzy set of M . Assume that (i) and (ii) hold. Let m; m1 ; m2 2 M and x 2 X, then + (xm) = + (xm 0) = + (xm 0m) minf + (m)); + (m)g = + (m); and (xm) = (xm 0) = (xm 0m) maxf (m)); (m)g = (m): Now

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6 +

(m1 m2 ) = + (1m1 1m2 ) minf + (m1 ); + (m2 )g; and (m1 (1m1 1m2 ) maxf (m1 ); (m2 )g: Using Theorem 3.4, = (M ; is a bipolar fuzzy BCK-submodule of M .

m2 ) = ; )

+

Theorem 3.6 Let = (M ; + ; ) 2 BF(M ). Then 2 BF S(M ) if and only if ? 6= P ( ; t) and ? 6= N ( ; s) are submodules of M for all (t; s) 2 [0; 1] [ 1; 0]. Proof Assume that = (M ; + ; ) 2 BFS(M ) and (t; s) 2 [0; 1] [ 1; 0] be such that ? 6= P ( ; t); ? 6= N ( ; s): Let m; m1 ; m2 2 P ( ; t) and m0 ; m01 ; m02 2 N ( ; s) and x 2 X. Then +

(m1 (m01

m2 ) m02 )

i.e. m1 m2 2 P ( ; t) and m01 Further,

minf maxf

+

(m1 ); (m01 );

+

(m2 )g (m02 )g

t; s:

m02 2 N ( ; s):

+

+

(xm) (xm0 )

(m) (m0 )

t; s:

Thus we have xm 2 P ( ; t) and xm0 2 N ( ; s): Hence P ( ; t) and N ( ; s) are submodules of M . Conversely, assume that ? 6= P ( ; t) and ? 6= N ( ; s) are submodules of M for all (t; s) 2 [0; 1] [ 1; 0]. For m; m0 2 M; let t0 = minf + (m); + (m0 )g and s0 = maxf (m); (m0 )g: Then m; m0 2 P ( ; t0 ) and m; m0 2 N ( ; s0 ): Since P ( ; t0 ) and N ( ; s0 ) are submodules of M; then m m0 2 P ( ; t0 ) and m m0 2 N ( ; s0 ), which means that (m

m0 )

t0 = minf

(m

m0 )

s0 = maxf

+

+

(m);

+

(m0 )g,

and (m);

(m0 )g:

Now, let + (m) = t1 , (m0 ) = s1 and x 2 X: Then m 2 P ( ; t1 ) and m0 2 N ( ; s1 ) which implies that xm 2 P ( ; t1 ) and xm0 2 N ( ; s1 ): i.e. + (xm) t1 = + (m); and (xm0 ) s1 = (m0 ): Thus by Theorem 3.4, 2 BFS(M ): Corollary 3.7 If = (M ; + ; ) 2 BFS(M ), then the intersection of a non) empty positive t-level cut and a non-empty negative s-level cut of = (M ; + ; is a submodule of M for all (t; s) 2 [0; 1] [ 1; 0]: In particular, the non-empty k-level cut of = (M ; + ; ) is a submodule of M for all k 2 [0; 1]: The union of a non-empty positive t-level cut and a non-empty negative s-level cut of = (M ; + ; ) 2 BFS(M ) is not a submodule of M in general as seen in the following example. Example 3.8 Let X = f0; a; b; c; d; 1g with a binary operation de…ned on Table 3.5. For the subset M = f0; a; c; dg of X, de…ne an operation + as x + y = (x y) _ (y x). It follows by Table 3.6 that (M; +) is an abelian group. M is an X-module according to Table 3.7.

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BIPOLAR FUZZY SETS OF BCK-M ODULES

0 a b c 0 0 0 0 a 0 0 a b a 0 b c c c 0 d c c a 1 d c b

0 a b c d 1

d 0 0 a 0 0 a

7

1 0 0 0 0 0 0

Tab. 3.5

+ 0 a c d

0 0 a c d

a c a c 0 d d 0 c a

d d c a 0

Tab. 3.6

^ 0 a b c d 1

0 0 0 0 0 0 0

a 0 a a 0 a a

c 0 0 0 c c c

d 0 a a c d d

Tab. 3.7

De…ne a bipolar fuzzy set on M by the following table: M +

0 a c d 0:9 0:6 0:4 0:4 1 0:5 0:7 0:5 Tab. 3.8

We can easily check that = (X; + ; ) is a bipolar fuzzy BCK-submodule of M . The positive 0:5-level cut is P ( ; 0:5) = f0; ag; and the negative 0:7-level cut is N ( ; 0:7) = f0; cg: It is clear that P ( ; 0:5) [ N ( ; 0:7) = f0; a; cg is not a submodule of M . Furthermore, P ( ; 0:6) [ N ( ; 0:6) = f0; a; cg is not a submodule of M . A su¢ cient condition for P ( ; k) [ N ( ; k) to be a submodule of M is given in the next theorem. without this condition, P ( ; k) [ N ( ; k) need not be a submodule of M as we have already seen. Theorem 3.9 If (*)

= (M ;

+

;

) 2 BFS(M ) such that

+

(m) +

(m)

0;

+

(m) +

(m)

0;

or (**)

for all m 2 M , then the union of a non-empty positive k-level cut and a non-empty negative k-level cut of = (M ; + ; ) is a submodule of M for all k 2 [0; 1]:

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Proof Let k 2 [0; 1] such that P ( ; k) 6= ? and N ( ; k) 6= ?. Then they are submodules of M by Theorem 3.6. Let x 2 X and m; m1 ; m2 2 P ( ; k)[N ( ; k). If m 2 P ( ; k); then xm 2 P ( ; k) If m 2 N (

P ( ; k) [ N ( ; k):

; k); then xm 2 N ( ; k)

P ( ; k) [ N ( ; k): +

Now we shall prove that m1 m2 2 P ( three cases: i. m1 ; m2 2 P ( ; k); ii. m1 ; m2 2 N ( ; k); iii. m1 2 P ( ; k); m2 2 N ( ; k): Case i. implies that m1

m2 2 P ( ; k)

; k) [ N (

; k): We have the following

P ( ; k) [ N ( ; k):

Case ii. gives m1 +

m2 2 N ( ; k)

In case iii. (m1 ) k and (m2 ) 0 which means that m1 If we consider (**), then m1

+

(m2 ) (m2 )

m2 2 P ( ; k) +

(m1 ) +

P ( ; k) [ N ( ; k): k: If we consider (*), then k and so

+

(m2 ) +

P ( ; k) [ N ( ; k):

(m1 )

m2 2 N ( ; k)

0 implies that

(m1 )

k. Hence

P ( ; k) [ N ( ; k):

Therefore P ( ; k) [ N ( ; k) is a submodule of M . For a bipolar fuzzy set = (M ; (m) = ( ; ) in the meaning of

+ +

; ) and an element m 2 M , we shall write (m) = and (m) = .

Theorem 3.10 Let M be a module over a BCK-algebra X and ? 6= N M: Suppose that = (M ; + ; ) is a bipolar fuzzy set on M de…ned as follows: (m) =

( ; ) if m 2 N; ( ; ) otherwise,

where ( ; ); ( ; ) 2 [0; 1] [ 1; 0] with > and < : Then = (M ; + ; ) is a bipolar fuzzy BCK-submodule of M if and only if N is a submodule of M . Proof Assume that = (M ; + ; ) 2 BFS(M ) and we shall prove that N is + (n) = > which a submodule of M . Let n 2 N and x 2 X. Then + (xn) + implies that (xn) = i.e. xn 2 N: Now let n1 ; n2 2 N , then + (n1 ) = + (n2 ) = and + (n1 n2 ) minf + (n1 ); + (n2 )g = > and this gives + (n1 n2 ) = i.e. n1 n2 2 N: Hence N is a submodule of M: Conversely, let N be a submodule of M and let m; m1 ; m2 2 M and x 2 X: We + shall prove that + (xm) (m) and (xm) (m): If m 2 N;then xm 2 N and we obtain + (xm) = = + (m); and (xm) = = (m):

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BIPOLAR FUZZY SETS OF BCK-M ODULES

If m 2 = N; then

+

(m) =

and

9

(m) = . So we have

+

(xm)

=

(xm)

=

+

(m)

and (m):

To show that +

(m1 m2 )

+

minf

(m1 );

+

(m2 )g and

(m1 m2 )

we consider the following cases: i. m1 ; m2 2 N; ii. m1 ; m2 2 = N; iii. m1 2 N; m2 2 = N: For case i. we have m1 m2 2 N and so +

(m1

m2 ) =

= minf

(m1

m2 ) =

= maxf

(m1

m2 )

= minf

(m1

m2 )

= maxf

+

maxf

+

(m1 );

(m1 );

(m2 )g;

(m2 )g

and (m1 );

(m2 )g:

For case ii. and iii. we have +

+

+

(m1 );

(m2 )g

and Therefore

= (M ;

+

;

(m1 );

(m2 )g:

) 2 BFS(M ):

For a submodule N of M , denote by N = (M; + N ; N ) the bipolar fuzzy set de…ned in the theorem above with ( ; ) = (1; 1) and ( ; ) = (0; 0). Example 3.11 Let f : M ! N be a homomorphism of BCK-modules. We know that ker f and Im f are submodules of M and N respectively. So ker f = + (M ; + ker f ; ker f ) and Im f = (N ; Im f ; Im f ) are bipolar fuzzy BCK-submodules. De…nition 3.12 Let f : M ! N be a BCK-module homomorphism and let 2 BFS(M ): Then the homomorphic image f ( ) = (N ; f ( + ); f ( )) of under f de…ned as follows: f(

+

)(n) =

(

+

sup m2f

(m) if f

1

(n) 6= ?;

if f

1

(n) = ?;

(m) if f

1

if f

1

(n) 6= ?;

1 (n)

0

and f(

)(n) =

(

inf1

m2f

(n)

0

(n) = ?:

Theorem 3.13 Let f : M ! N be a BCK-module epimorphism. If = (M ; + ; ) 2 BFS(M ), then the homomorphic image f ( ) 2 BFS(N ): Proof According to Theorem 3.6, it is su¢ cient to prove that P (f ( ); t) and N (f ( ); s) are submodules of N for all (t; s) 2 [0; 1] [ 1; 0] satisfying P (f ( ); t) 6=

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?; N (f ( ); s) 6= ?. Let n0 ; n1 ; n2 2 P (f ( ); t) and x 2 X: Since f is an epimorphism, then there exist m0 2 f 1 (n0 ); m1 2 f 1 (n1 ); m2 2 f 1 (n2 ) such that + (mi ) t, i = 0; 1; 2. Now f(

+

)(xn0 ) =

+

sup m2f

+

(m)

+

(xm0 )

(m0 )

t;

1 (xn)

and f(

+

)(n1 n2 ) =

+

sup 1 (n

m2f

+

(m)

(m1 m2 )

minf

+

(m1 );

+

(m2 )g

t:

n2 )

1

Which implies that xn0 ; n1 n2 2 P (f ( ); t). Therefore P (f ( ); t) is a submodule of N for all t 2 [0; 1]. Analogously, we can verify that N (f ( ); s) is a submodules of N for all s 2 [ 1; 0]. This completes the proof. De…nition 3.14 Let f : M ! N be a homomorphism of BCK-modules, and = (N ; + ; ) be a bipolar fuzzy set of N . Then the preimage of , f 1 ( ) = 1 + (M ; f ( ); f 1 ( )), is the bipolar fuzzy set on M given by f 1 ( + )(m) = + (f (m)), f 1 ( )(m) = (f (m)) for all m 2 M . Theorem 3.15 Let f : M ! N be a homomorphism of BCK-modules, and = (N ; + ; ) 2 BFS(N ), then the preimage f 1 ( ) = (M ; f 1 ( + ); f 1 ( )) 2 BFS(M ). Proof Suppose that = (N ; + ; ) 2 BFS(N ) and f is a homomorphism of BCK-modules from M to N . Then for all m1 ; m2 2 M , we have f

1

+

(

)(m1

m2 )

=

+

(f (m1 m2 )) = + (f (m1 ) minf + (f (m1 )); + (f (m2 ))g

=

minff

1

+

(

1

)(m1 ); f

+

(

f (m2 ))

)(m2 )g

Moreover, let x 2 X and m 2 M: then f

1

+

(

)(xm)

+

=

+

(f (xm)) =

+

(f (m)) = f

1

(xf (m))

(

+

)(m1 ); f

1

)(m):

Analogously, we have f

1

(

)(m1

m2 )

maxff

1

(

(

)(m2 )g

and f Hence, f M.

1

( ) = (M ; f

1

(

)(xm)

1

+

(

); f

1

max f (

1

(

)(m):

)) is a bipolar fuzzy BCK-submodule of

Theorem 3.16 Let f : M ! N be an epimorphism of BCK-modules. If = (N ; + ; ) is a bipolar fuzzy set on N such that the preimage f 1 ( ) = 1 + (M ; f ( ); f 1 ( )) 2 BFS(M ), then 2 BFS(N ): Proof Let f 1 ( ) = (M ; f 1 ( + ); f 1 ( )) be a bipolar fuzzy BCK-submodule on M and let f : M ! N be an epimorphism. For n0 ; n1 ; n2 2 N; there exist m0 ; m1 ; m2 2 M such that f (m0 ) = n0 ; f (m1 ) = n1 ; and f (m2 ) = n2 .

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11

Now +

(n1

n2 )

+

=

(f (m1

1

m2 )) = f

1

+

(

1

+

)(m1

m2 )

+

minff ( )(m1 ); f ( )(m2 )g = minf + (f (m1 ); + (f (m2 ))g minf + (n1 ); + (n2 )g and +

+

(xn0 ) =

(f (xm0 )) = f

1

+

(

)(xm0 )

1

f

+

(

+

)(m0 ) =

(f (m0 )) =

+

(n0 );

for all x 2 X. By similar argument, we have (n1

n2 )

maxf

(n1 );

(n2 )g;

and (xn0 )

(n0 ):

This …nishes the proof. For a bipolar fuzzy set = (M ; elements m 2 M with (m) = (0).

+

;

), we de…ne M

to be the set of all

Proposition 3.17 If = (M ; + ; ) 2 BFS(M ), then M is a submodule of M. Proof Clearly M 6= ?, since 0 2 M . Let m; m1 ; m2 2 M and x 2 X. Then +

i.e. + (xm) = more, +

+

(0)

+

(0)

(0). Similarly, +

(m1

+

min

+

(m) =

(xm) =

m2 )

+

+

(xm)

(0)

(0) and so xm 2 M . Further+

(m1 ) ;

(m2 ) =

+

+

which means (m1 m2 ) = (0). Analogously, (m1 Hence m1 m2 2 M . Therefore M is a submodule of M . De…nition 3.18 If said to be normal.

(0) = (1; 1) for

= (M ;

+

;

(0) ;

m2 ) =

) 2 BFS(M ), then

(0).

is

Theorem 3.19 Let = (M ; + ; ) 2 BFS(M ). The normalization = + + (M ; ; ) of de…ned by + (m) = + (m)+1 (0) and (m) = (m) 1 (0), for all m 2 M , is normal bipolar fuzzy BCK-submodule of M containing . + Proof Clearly, + (m) (m) and (m) (m) for all m 2 M . i.e. . Now let m; m1 ; m2 2 M and x 2 X. Then +

(xm) =

+

+

(xm) + 1

+

(0)

+

(m) + 1

(0) =

+

(m);

and +

(m1

m2 )

= = =

+

+ (m1 m2 ) + 1 (0) + + min (m1 ) ; (m2 ) + 1

min

+

(m1 ) + 1

min

+

+

(m1 );

590

+

(0) ;

+ +

(0)

(m2 ) + 1

+

(0)

(m2 ) :

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12

Analogously, (xm) (m) and Hence 2 BFS(M ). Moreover, +

(0) =

+

(m1

m2 ) +

(0) + 1

max

(m1 );

(m2 ) .

(0) = 1;

and (0) = Which means that

(0)

1

(0) =

1:

is normal.

Let S(M ) (respectively N (M )) denote the set of all submodules (respectively, normal bipolar fuzzy BCK-submodules) of M . We de…ne functions F : S(M ) ! N (M ) and G : N (M ) ! S(M ) by F (N ) = N and G ( ) = M . Then GF = 1S(M ) and F G ( ) = F (M ) = M . Note that S(M ) (respectively N (M )) is a poset under the set inclusion (respectively, bipolar fuzzy set inclusion). Theorem 3.20 If N; K 2 S(M ), then N \K = N \ K , that is, F (N \ K) = F (N ) \ F (K). If ; 2 N (M ), then M \ = M \ M , that is, G ( \ ) = G ( ) \ G( ). Proof Let m 2 M . If m 2 N \ K, then N \K (m) = (1; 1). From m 2 N and m 2 K it follows that N (m) = K (m) = (1; 1). Hence (

N

+ K)

\

+ N

(m) = minf

+ K

(m) ;

+ N \K

(m)g = 1 =

(m) ;

and (

N

\

K)

(m) = maxf

N

(m) ;

K

If m 2 = N \ K, then m 2 = N or m 2 = K. Thus

(m)g =

1=

(

N

\

+ K)

(m) = minf

+ N

(m) ;

+ K

(m)g = 0 =

(

N

\

K)

(m) = maxf

N

(m) ;

K

(m)g = 0 =

N \K + N \K

(m) :

(m) ;

and Therefore N \K = N (M ). Then M

\

= = = =

N

\

K,

m2M j

m2M j

(m) :

and so F (N \ K) = F (N ) \ F (K). Now let

m 2 M j ( \ )+ (m) = 1; ( \ ) (m) =

m 2 M j minf

N \K

+

(m) ;

+

+

(m) =

+

(m) = 1;

+

(m)g = 1; max

(m) = 1; (m) =

+

(m) =

;

2

1 (m) ;

(m) =

(m) =

1

1

1 \

m2M j (m) = 1; (m) = 1 = fm 2 M j (m) = (0)g \ fm 2 M j = M \M ,

(m) =

(0)g

that is, G ( \ ) = G ( ) \ G( ). This completes the proof. Proposition 3.21 Let = (M ; + ; ) be a non-constant normal bipolar fuzzy BCK-submodule which is maximal in (N (M ); ), then takes only a value among (0; 0) ; (1; 0) ; (0; 1) and (1; 1). Proof Let = (M ; + ; ) be a non-constant maximal element in (N (M ); ). Since is normal, then (0) = (1; 1). Let m0 2 M be such that + (m0 ) 6= 1.

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Then + (m0 ) = 0. Otherwise, 0 < de…ned by

+

(m0 ) < 1. Consider

13

= (M;

+

;

)

1 + 1 ( (m) + + (m0 )), (m) = ( (m) + (m0 )); 2 2 for all m 2 M . Clearly is well de…ned. Let m; m1 ; m2 2 M , then +

+

(m) =

(m1

1 + ( (m1 m2 ) + + (m0 )) 2 1 (minf + m1 ); + (m2 g + + (m0 )) 2 1 1 = minf ( + (m1 ) + + (m0 )); ( + (m2 ) + 2 2 + = min (m1 ) ; + (m2 ) ,

m2 )

=

+

(m0 ))g

and +

(xm)

=

1 ( 2 1 ( 2 +

=

+

(xm) +

+

(m) +

+

+

(m0 ))

(m0 ))

(m) :

By similar argument, we can show (m1

m2 )

maxf

(m1 ) ;

(m2 )g;

and (xm) Hence, M; + ;

(m) :

2 BFS(M ). Clearly, is non-constant and by Theorem 3.19, 2 N (M ). Now for all m 2 M , we have +

(m)

= = =

+

=

+

(m) + 1

(0) 1 + ( (m) + + (m0 )) + 1 2 1 + (1 + + (m)) (m) ; 2

1 ( 2

+

(0) +

+

(m0 ))

and (m)

= = =

(m) 1 ( 2 1 ( 2

1

(0)

(m) + (m)

(m0 ))

1

1 ( 2

(0) +

(m0 ))

1)

(m) : Furthermore, 1 (1 + + (m0 )) > + (m0 ) : 2 This means that is a proper subset of , which contradicts the maximality of in N (M ). Thus the possible values of + are only 0 and 1. Likewise, we can show that 0 and 1 are the only possible values of . Therefore, takes only a value among (0; 0) ; (1; 0) ; (0; 1) and (1; 1). This …nishes the proof. +

(m0 ) =

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14

For

+

= (M ;

M M M

(1;0)

(0; 1)

;

(0;0)

) 2 BFS(M ), consider the following sets: +

= fm 2 M j

+

= fm 2 M j M

+

= fm 2 M j

(1; 1)

(m)

(m)

(m)

1;

0;

(m)

0;

(m)

= fm 2 M j = fm 2 M j

(m)

0g = M;

0g = fm 2 M j

+

(m) = 1g;

1g = fm 2 M j

+

(m)

+

(m) = 1;

1;

(m) =

(m)

1g

(m) =

1g:

1g;

Clearly, we have the relations M

(1; 1)

M

(1;0)

M

(0;0)

;M

(1; 1)

M

(0; 1)

M

(0;0)

;M

(1;0)

De…nition 3.22 A normal bipolar fuzzy BCK-submodule said to be completely normal if there exists m 2 M such that Example 3.23 Let N be a proper submodule of M . Then

\M

(0; 1)

=M

(1; 1)

:

= (M ; + ; ) is (m) = (0; 0). N

= (M ;

+ N;

N)

is completely normal bipolar fuzzy BCK-submodule. Denote by C(M ) the set of all completely normal bipolar fuzzy BCK-submodules of M . Note that C(M ) N (M ) and the restriction of the partial ordering " " of N (M ) gives a partial ordering of C(M ). Theorem 3.24 A non-constant maximal element of (N (M ); ) is also a maximal element of ( C(M ); ). Proof First, we show that if = (M ; + ; ) is a non-constant maximal element of (N (M ); ),then 2 C(M ). Suppose that there is no m 2 M with (0;0) (1;0) (0; 1) (m) = (0; 0) i.e. M M [M = ?. Since is non-constant normal, then assumes the value (1; 0) or (and) (0; 1) at some points in M and so we have the following cases: (1;0) (1; 1) (0; 1) (1; 1) i. M M 6= ?; M M = ?: (0; 1) (1; 1) (1;0) (1; 1) ii. M M 6= ?; M M = ?: (1;0) (1; 1) (0; 1) (1; 1) iii. M M 6= ?; M M 6= ?: For case i. let ( (1; 1) (1; 1) if m 2 M ; (m) = (1;0) (1; 1) M : 1; 21 if m 2 M For case ii. let (m) =

(

(m) =

8 >
:

(1;0)

(1; 1)

(1; 1) if m 2 M ; (0; 1) (1; 1) 1 M : 2 ; 1 if m 2 M (1; 1)

(1; 1) if m 2 M ; (1;0) (1; 1) 1 1; 2 if m 2 M M ; (0; 1) (1; 1) 1 ; 1 if m 2 M M : 2 (0; 1)

, and M are submodules of M , it is not di¢ cult to ) 2 BFS(M ) in each case. obviously, is non-constant

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15

normal and . But this contradicts the fact that is non-constant maximal in (N (M ); ). Thus should has the value (0; 0) at some points m 2 M and 0 0 so 2 C(M ). Now let 0 2 C(M ) such that . It follows that 0 in N (M ). Since is maximal in (N (M ) ; ) and since is non-constant, then = 0 . Therefore is maximal in (C (M ) ; ). De…nition 3.25 A bipolar fuzzy set = (M ; maximal if it satis…es: (i) is non-constant (ii) is maximal element in (N (M ); ).

+

;

) 2 BFS(M ) is said to be

Theorem 3.26 A maximal bipolar fuzzy BCK-submodule is completely normal and equivalent to its normalization. Proof If = (M ; + ; ) is a maximal bipolar fuzzy BCK-submodule of M , then is non-constant maximal in (N (M ); ) and so it is maximal in (C (M ) ; ). So for some m 2 M , +

0 = 0 =

(m) = (m) =

+

(m) + 1 (m) 1

+

(0) ; (0) :

Which implies that +

Since + (m) = and so

+

(m) = (m) =

0 and (m) 0, then is completely normal.

(0) 1 (0) + 1 +

0; 0.

(0) = 1 and

(0) =

1. Therefore

Now we arrive at one of our main theorems Theorem 3.27 A maximal bipolar fuzzy BCK-submodule takes exactly the values (1; 1); (0; 0). Proof Assume that = (M ; + ; ) is a maximal bipolar fuzzy BCK-submodule. Then takes a value among (0; 0) ; (1; 0) ; (0; 1) and (1; 1) and it is completely (0;0) (1;0) (0; 1) (1;0) (1; 1) normal. So M M [M 6= ?. The subsets M M and M

(0; 1)

M

(1; 1)

of M are empty. If not, then we have the following cases: (0; 1) (1; 1) i. M M 6= ?; M M =? (0; 1) (1; 1) (1;0) (1; 1) ii. M M 6= ?; M M =? (1;0) (1; 1) (0; 1) (1; 1) iii. M M 6= ?; M M 6= ? For case i. let 8 (1; 1) > if m 2 M ; < (1; 1) (1;0) (1; 1) 1 (m) = M ; 1; 2 if m 2 M > : (0;0) (1;0) (0; 0) if m 2 M M : (1;0)

(1; 1)

For case ii. let

(m) =

8 > < > :

(1; 1)

(1; 1) if m 2 M ; (0; 1) (1; 1) 1 M ; : 2 ; 1 if m 2 M (0;0) (0; 1) (0; 0) if m 2 M M :

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16

For case iii. let

(m) =

8 > > > >
> > > : (0; 0) if m 2 M (0;0)

M

(1;0)

[M

(0; 1)

:

In each case, is non-constant completely normal bipolar fuzzy BCK-submodules (1;0) (1; 1) (0; 1) containing . A contradiction. Hence M M = ? and M (1; 1) M = ?, i.e. does not assume the values (1; 0) and (0; 1) at any point in M . Therefore takes exactly the values (0; 0) and (1; 1). Clearly M

(1; 1)

= M , for

Corollary 3.28 If then M = .

= (M ;

2 N (M ). We are thus led to the following result. +

;

) is a maximal bipolar fuzzy BCK-submodule,

Theorem 3.29 For a maximal bipolar fuzzy BCK-submodule = (M ; + ; ), M is a maximal submodule of M . Proof Let = (M ; + ; ) be a maximal bipolar fuzzy BCK-submodule and suppose that N is a proper submodule of M such that M N . Consider the normal bipolar fuzzy BCK-submodule N = (M ; + ; ). If M is a proper N N submodule of N then is proper subset of N . A contradiction. Hence M is a maximal submodule of M .

4. Conclusion The notion of bipolar-valued fuzzy set was introduced by K. M. Lee in 2000. Since then, bipolarity has been applied to various algebraic structures by many researchers. In this paper, we have applied the notion of bipolar-valued fuzzy sets to BCK-modules. We have characterized our new concept "bipolar fuzzy BCKsubmodule " in several ways. Next, the homomorphic images and pre images of bipolar fuzzy BCK-submodules were discussed. The remainder of the paper was focused on the normal and completely normal bipolar fuzzy BCK-submodules which guided …nally to the concept of maximality. References [1] H. A. S. Abujabal, M. Aslam and A. B. Thaheem, "On actions of BCK-algebras on groups", Panamerican Mathematical Journal, 4 (3)(1994), 43-48. [2] M. Bakhshi, "Fuzzy set theory applied to BCK-modules", Advances in Fuzzy Sets and Systems, 2 (8)(2011), 61-87. [3] Y. B. Jun, H. S. Kim and K. J. Lee, "Bipolar fuzzy translations in BCK/BCI-algebras", Journal of Chungcheong Mathematical Society, 3 (22)(2009), 399-408. ½ u rk, "Fuzzy maximal ideals of Gamma near-rings", [4] Y. B. Jun, K. H. Kim and M. A. Ozt½ Turkish Journal of Mathematics, 25 (2001), 457-463. [5] A. Kashif and M. Aslam, "Homology theory of BCK-modules", Southeast Asian Bulletin of Mathematics, 38 (2014), 61-72. [6] K. J. Lee, "Bipolar fuzzy subalgebras and bipolar fuzzy ideals of BCK/BCI-algebras", Bulletin of the Malaysian Mathematical Sciences Society, 32 (3)(2009), 361-373.

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17

[7] K. M. Lee, "Bipolar-valued fuzzy sets and their operations" Proc. Int. Conf. on Intelligent Technologies, Bangkok, Thailand, 2000, 307-312. [8] L. A. Zadeh, "Fuzzy sets", Information and Control, 8 (1965), 338-353. [9] M. Zhou and S. Li, "Application of bipolar fuzzy sets in semirings", Journal of Mathematical Research with Applications, 1 (34)(2014), 61-72. [10] M. Zhou and S. Li, "Applications of bipolar fuzzy theory to hemirings", International Journal of Innovative Computing, Information and Control, 2 (10)(2014), 767-781. 1 Department of Mathematics, Faculty of Sciences, King Abdulaziz University, Jeddah, Saudi Arabia. E-mail address : [email protected] 2 Department of Mathematics, Faculty of Sciences, King Abdulaziz University, Jeddah, Saudi Arabia. E-mail address : [email protected] 3 Department

of Mathematics, Faculty of Sciences, King Abdulaziz University, Jeddah, Saudi Arabia. E-mail address : [email protected]

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TABLE OF CONTENTS, JOURNAL OF COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 24, NO. 3, 2018

Implicit Midpoint Type Picard Iterations for Strongly Accretive and Strongly Pseudocontractive Mappings, Shin Min Kang, Arif Rafiq, Young Chel Kwun, and Faisal Ali,…………………405 Jordan homomorphisms in C*-ternary algebras and JB*-triples, Mohammad Raghebi Moghadam, Themistocles M. Rassias, Vahid Keshavarz, Choonkil Park, Young Sun Park,416 The generic stability of KKM points in PMT spaces, M. Tatari, S. M. Vaezpour, and Reza Saadati,……………………………………………………………………………………425 The essential norm of the generalized integration operator, Yongmin Liu and Yanyan Yu,435 Solution of a third order fractional system of difference equations, Asim Asiri, M. M. ElDessoky, and E. M. Elsayed,…………………………………………………………………444 Inequalities for Orlicz mixed Harmonic Quermassintegrals, Lewen Ji and Zhenbing Zeng,454 Roughness in (∈𝛾 , ∈𝛾 , ⋁ 𝑞𝑞 )-fuzzy substructures of semigroups based on set valued mapping, Noor Rehman, Syed Inayat Ali Shah, Abbas Ali, and Rabia Aslam,………………………463 Equivalence between some iterations in CAT(0) spaces, Kyung Soo Kim,………………474 Strong convergence theorems for the generalized viscosity implicit rules of asymptotically nonexpansive mappings in Hilbert spaces, Qian Yan and Shaotao Hu,…………………486 Hermite Hadamard type inequalities for m-convex and (𝛼,m)-convex functions for fuzzy integrals, M. A. Latif, Wajeeha Irshad, and M. Mushtaq,…………………………………497 Very true operators on equality algebras, Jun Tao Wang, Xiao Long Xin,Young Bae Jun,507 A Splitting Iterative Method for a System of Accretive Inclusions in Banach Spaces, Birendra Kumar Sharma, Niyati Gurudwan, Avantika Awadhiya, and Shin Min Kang,……………522 Sidi-Israeli Quadrature Method for Steady-State Anisotropic Field Problems by Direct Domain Mapping, Xin Luo, Jin Huang, and Tai-Song Xiong,………………………………………534 Hyers-Ulam stability of an additive set-valued functional Equation, Gang Lu, Jun Xie, Choonkil Park, and Yuanfeng Jin, ……………………………………………………………………556

TABLE OF CONTENTS, JOURNAL OF COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 24, NO. 3, 2018 (continued) Topological spaces induced by fuzzy prime ideals in BCC-algebras, Sun Shin Ahn and Keum Sook So,………………………………………………………………………………………561 Riesz fuzzy normed spaces and stability of a lattice preserving functional equation, Choonkil Park, Ehsan Movahednia, Seyed Mohammad Sadegh Modarres Mosadegh, and Mohammad Mursaleen,……………………………………………………………………………………569 Bipolar fuzzy sets of BCK-modules, M. A. Alghamdi, N. O. Alshehri, and N. M. Muthana,580

Volume 24, Number 4 ISSN:1521-1398 PRINT,1572-9206 ONLINE

April 2018

Journal of Computational Analysis and Applications EUDOXUS PRESS,LLC

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

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

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

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

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

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

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

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

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

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

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Zeros of functions in weighted Dirichlet spaces and Carleson type measure Ruishen Qian, Songxiao Li∗ and Yanhua Zhang Abstract. In this paper, we study the zeros of functions in weighted Dirichlet space and a class of Carleson type measure. MSC 2000: 30J99, 30H99 Keywords: Zero, weighted Dirichlet space, Carleson measure.

1 Introduction Let D denote the open unit disk in the complex plane C, ∂D its boundary and H(D) the space of all analytic functions in D. For a ∈ D, let σa be the automorphism of a−z ∞ D exchanging 0 for a, namely σa (z) = 1−¯ denote the space of az , z ∈ D. Let H bounded analytic function. Throughout this paper, we assume that K : [0, ∞) → [0, ∞) is a right-continuous and nondecreasing function. An f ∈ H(D) is said to belong to the weighted Dirichlet space, denoted by DK , if (see, e.g., [24]) ∫ ( ) ∥f ∥2DK = |f (0)|2 + |f ′ (z)|2 K 1 − |z|2 dA(z) < ∞, D

where dA(z) is the normalized Lebesgue measure on D. Clearly, DK is a Hilbert space. When K(t) = ts , 0 ≤ s < ∞, the space DK gives the usual Dirichlet type space Ds . In particular, if s = 0, this gives the classical Dirichlet space D. We refer to [16, 19, 20] for the space Ds . The space DK has been extensively studied. For example, under some conditions on K, Kerman and Sawyer [11] characterized Carleson measures and multipliers of DK in terms of a maximal operator. Aleman [1] proved that each element of the space DK can be written as a quotient of two bounded functions in the same space. See [2, 3, 8, 14, 18, 24] for more results on weighted Dirichlet spaces. We say that Z = {zn } ⊂ D is a zero set of an analytic function space X defined on D if there is a f ∈ X that vanishes on Z and nowhere else. Describing the zero sets for an analytic function space is a difficult problem. See [4, 5, 6, 13, 17, 21] for more information about this topic. Let µ denote a positive Borel measure on D. For a subarc I ⊆ ∂D, let S(I) be the Carleson box based on I with z ∈ I}. S(I) = {z ∈ D : 1 − |I| ≤ |z| < 1 and |z| 1

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If I = ∂D, let S(I) = D. Let 0 ≤ s < ∞. We say that µ is a (K, s)-Carleson measure on D if µ(S(I)) sup < ∞. s I⊆∂D K(|I|)|I| Here and henceforth supI⊆∂D indicates the supremum taken over all subarcs I of ∂D. We say that µ is a vanishing (K, s)-Carleson measure, if µ(S(I)) = 0. |I|→0 K(|I|)|I|s lim

If K(t)ts = t, then we get the classical Carleson measure and vanishing Carleson measure, respectively. Carleson measure was firstly introduced in [4] by Carleson and it has many applications, such as in the interpolating sequence, ∂-equations, composition operators and integral operators. Hence Carleson measure is a very important tool for the function theory, harmonic analysis and operator theory. For more results on Carleson measure and its’ generalization, we refer to [7, 10, 12, 22, 23, 25]. Recently, Pau and Pel´aez [13, Theorem 1] gave a nice characterization of zero sets of Dirichlet spaces Ds (0 < s < 1). Motivated by [13, Theorem 7], in this paper we study the zero sets of the space DK . Moreover, we will characterize (K, s)-Carleson measure and vanishing (K, s)-Carleson measure. In particular, we will characterize vanishing (K, s)-Carleson measure by functions in the space DK . Throughout this paper, we assume that K(0) = 0 such that ∫

1 0

φK (s) ds < ∞ s

(1)

φK (s) ds < ∞, s2

(2)

and ∫

∞ 1

where φK (s) = sup K(st)/K(t),

0 < s < ∞.

0≤t≤1

In this paper, the symbol f ≈ g means that f . g . f . We say that f . g if there exists a constant C such that f ≤ Cg.

2 Zero sets In order to study the zero sets of the DK space, we define the space SK , which consists of those f ∈ H(D) such that ( ) ∫ K 1 − |z|2 ) ∥f ∥2SK = |f (0)|2 + sup |f ′ (z)|2 ( dA(z) < ∞. 2 a∈D D K |1−az| 2 1−|a|

2

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We say that a positive Borel measure µ on D is a Carleson measure for DK if there is a positive constant C such that ∫ |f (z)|2 dµ(z) ≤ C∥f ∥2DK D

for all f ∈ DK . We also need the following results. Lemma 1. [11] Let K be increasing, concave and limx→0 x/K(x) = 0. Then, µ is a Carleson measure for DK if and only if there exists a constant C > 0 such that ∫ µ(S(J))2 sup dθ ≤ Cµ(S(I)), I θ∈J⊂I K(|J|)|J| for all arcs I ⊂ ∂D. Here the supremum is taken over all closed arcs J ⊂ I. Remark 1. From [9, Lemma 2.3], we known that if K satisfy (1) and (2), there exists K3 , such that K3 is increasing, concave, limx→0 x/K3 (x) = 0 and K3 (t) ≈ t1−c K(t), 0 < t < ∞. As in [9], let c ∈ (0, 1) be a small constant such that K(t) tc and K(t) are nondecreasing functions when 0 < t < 1, moreover φK (s) . s1−c , s ≥ 1

(3)

φK (s) . sc , s ≤ 1.

(4)

and

Lemma 2. Suppose that K satisfy (1) and (2). Then f ∈ SK if and only if ∫ K(1 − |z|2 ) sup |f ′ (z)|2 dA(z) < ∞. K(|I|) I⊆∂D S(I)

(5)

Proof. Assume that f ∈ SK . For any I ⊆ ∂D, let b = (1 − |I|)η ∈ D, where η is the center of I. Then 1 − |b| ≈ |1 − bz| ≈ |I|, z ∈ S(I). Thus,

( K

|1 − bz|2 1 − |b|2

) ≈ K(|I|), z ∈ S(I).

Therefore, ( ) ∫ ∫ 2 2 ′ 2 K(1 − |z| ) ′ 2 K (1 − |z| ) |f (z)| |f (z)| dA(z) . dA(z) 2 K(|I|) S(I) D K |1−bz| 1−|b|2 ( ) ∫ 2 ′ 2 K (1 − |z| ) . sup |f (z)| dA(z) < ∞, 2 a∈D D K |1−az| 1−|a|2 3

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which implies the desired result. Conversely, assume that (5) holds. Without loss of generality, we can assume |a| > 1/2. Let I be a subarc of ∂D such that |In | = 2n |I|, n = 0, 1, 2..., N −1 and IN = ∂D. Then we have |1 − aη|2 ≈ |I|, η ∈ I, 1 − |a|2 and |1 − aη|2 ≈ 22n |I|, η ∈ In+1 /In , n = 0, 1, 2..., N − 1. 1 − |a|2 Since K satisfy (1) and (2), by Remark 1 we obtain ( ) ∫ 2 ′ 2 K (1 − |z| ) |f (z)| dA(z) 2 D K |1−az| 1−|a|2 ∫ ∞ ∑ 1 |f ′ (z)|2 K(1 − |z|2 )dA(z) . 2n |I|) K(2 n+1 I)\S(2n I) S(2 n=1 ∫ 1 + |f ′ (z)|2 K(1 − |z|2 )dA(z) K(|I|) S(2I) ∫ ∞ ∑ 1 K(2|I|) . |f ′ (z)|2 K(1 − |z|2 )dA(z) + 2n |I|) K(2 K(|I|) n+1 S(2 I) n=1 . .

∞ ∑ K(2n+1 |I|) + φK (2) K(22n |I|) n=1 ∞ ∑

2(1−n)c + 21−c < ∞.

n=1

Here c is defined in Remark 1. Hence f ∈ SK . From Lemma 2, we can easily obtain the following corollary. Corollary 1. Suppose that K satisfy the conditions (1) and (2). Then f ∈ SK if and only if |f ′ (z)|2 K(1 − |z|2 )dA(z) is (K, 0)-Carleson measure. Let M (DK ) denote the space of multipliers of DK , that is, M (DK ) = {g ∈ H(D) : gf ∈ DK for all f ∈ DK }. Theorem 1. Suppose that K satisfy the conditions (1) and (2). Then M (DK ), SK ∩ H ∞ , SK and DK have the same zero sets. Proof. Since M (DK ) ⊆ DK , we have any zero sets in M (DK ) is zero sets in DK . Next we prove that any zero set in DK is zero set in M (DK ). Suppose that ∥f ∥DK = 1. Let {zk } be the zeros of f . Fix z0 ∈ D such that f (z0 ) ̸= 0. Set w0 = ∥Rfz(z∥0D) and wj = 0 (j ≥ 1), where o

K

Rzj (z) = 1 +

∞ ∑

1 n n 1 zj z , z ∈ D, j ≥ 0. nK( ) n n=1 4

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From [3, Lemma 3.1], we know that Rzj (z) is the reproducing kernel of DK space at zj . Then, for each n ≥ 1 and a0 , a1 , ..., an ∈ C, we have n ∑ n ∑

ai aj (1 − wi wj )⟨Rzi , Rzj ⟩

i=0 j=0

2

∑

n

= aj Rzj − |a0 |2 |f (z0 )|2

j=0

DK

2 2

n

n

∑

∑

= aj Rzj − ⟨f, aj Rzj ⟩ ≥ 0.

j=0

j=0 DK

Combine with Lemma 3.2 of [3], we know that DK has Pick property. Thus, there exists Fn ∈ M (DK ) with ∥Fn ∥M (DK ) ≤ 1 such that Fn (z0 ) =

f (z0 ) ∥Rz0 ∥DK

Fn (zj ) = 0 (j = 1, ..., n).

and

Then, for all n, we have ∥Fn ∥H ∞ ≤ ∥Fn ∥M (DK ) ≤ 1. So {Fn }n≥1 is a normal family, the limit function F is also a multiplier with ∥F ∥M (DK ) ≤ 1, F (z0 ) =

f (z0 ) ̸= 0 ∥Rz0 ∥DK

and

F (zj ) = 0 (j = 1, ..., n).

By f -property of DK space (see [15]), we have every function f ∈ DK , there exist F ∈ M (DK ) with the same zero set. That is, DK and M (DK ) have the same zero sets. Note that SK ∩ H ∞ ⊆ SK ⊆ DK . We only need to prove that M (DK ) ⊆ SK ∩ H ∞ . Suppose that f ∈ M (DK ). From [3, Theorem 4.6], we known that |f ′ (z)|2 K(1 − |z|2 )dA(z) is a Carleson measure for DK . Let |I| = |J| in Lemma 1, that is, if µ is a Carleson measure for DK , we can deduce that µ(S(I)) . K3 (|I|) ≈ K(|I|). Thus, ∫ |f ′ (z)|2 K(1 − |z|2 )dA(z) . K(|I|).

S(I)

Combine with Lemma 2, we deduce that f ∈ SK . Notice that M (DK ) ⊆ H ∞ (see [3, Theorem 4.6]). That is, M (DK ) ⊆ SK ∩ H ∞ .

3 Carleson type measure In this section, we give a characterization for (K, s)-Carleson measure and vanishing (K, s)-Carleson measure. Theorem 2. Suppose that K satisfy the conditions (1) and (2). Let µ be a positive Borel measure on D, 0 ≤ s < ∞ such that s + c > 1. Then µ is a (K, s)-Carleson 5

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measure if and only if 1 K(1 − |a|2 )

sup a∈D

∫ D

(1 − |a|2 )s dµ(z) < ∞. |1 − az|2s

(6)

Proof. Suppose that (6) holds. For any I ⊆ ∂D, let b = (1 − |I|)ζ ∈ D, where ζ is the center of I. Then, for any z ∈ S(I), we have 1 − |b| ≈ |1 − bz| ≈ |I| and K(1 − |b|2 ) ≈ K(|I|). Therefore, 1 µ(S(I)) . K(|I|)|I|s K(1 − |b|2 )

∫ S(I)

∫

(1 − |b|2 )s dµ(z) |1 − bz|2s

(1 − |b|2 )s dµ(z) 2s D |1 − bz| ∫ (1 − |a|2 )s 1 ≤ sup dµ(z) < ∞, 2 2s a∈D K(1 − |a| ) D |1 − az|

1 ≤ K(1 − |b|2 )

which implies that µ is a (K, s)-Carleson measure by the arbitrary of I. Conversely, assume that µ is a (K, s)-Carleson measure. Without loss of generality, a we assume |a| > 45 . Let In be the arc on ∂D such that |a| is the center of In and 1

|In | = A(n−1) (1 − |a|), where 1 < A < 2 s , n = 1, 2, ..., N , where N is the smallest integer such that A(N −1) (1 − |a|) ≥ 1. Since (1 − |a|2 )s 1 1 . 2(n−1)s . 2ns , z ∈ S(In ) \ S(In−1 ), 2s s |1 − az| A (1 − |a|)s A (1 − |a|) and

( ) K A(n−1) (1 − |a|) . A(n−1)(1−c) , K(1 − |a|2 )

|In |s = A(n−1)s−ns < 1, ns A (1 − |a|)s we obtain 1 K(1 − |a|2 )

∫ D

N ∑ (1 − |a|2 )s |In |s K(|In |) dµ(z) ≤ |1 − az|2s A2ns (1 − |a|)s K(1 − |a|2 ) n=1 ( ) N ∑ K A(n−1) (1 − |a|) . Ans K(1 − |a|2 ) n=1

.

∞ N ∑ ∑ A(n−1)(1−c) An(1−c) < ns A Ans n=1 n=1

1. 6

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Theorem 3. Suppose that K satisfy the conditions (1) and (2). Let µ be a positive Borel measure on D, 0 ≤ s < ∞ such that s + c > 1. Then µ is a vanishing (K, s)Carleson measure if and only if ∫ (1 − |a|2 )s 1 lim dµ(z) = 0. (7) 2 |a|→1 K(1 − |a| ) D |1 − az|2s Proof. First we assume that µ is a vanishing (K, s)-Carleson measure. For any ϵ > 0, there is a η > 0 such that for all arcs I ⊆ ∂D with |I| ≤ η, such that µ(S(I)) < ϵ. K(|I|)|I|s Assume a = reiθ and r > 1 − η. Let Iη ⊆ ∂D such that eiθ is the center of Iη and |Iη | = η. Then ∫ (1 − |a|2 )s dµ(z) =: M1 + M2 , 2s D |1 − az| ∫

where M1 =:

D\S(Iη )

∫

and

(1 − |a|2 )s dµ(z) |1 − az|2s

M2 =: S(Iη )

(1 − |a|2 )s dµ(z). |1 − az|2s

Suppose that e is also the center of {In }, |In | = An−1 (1 − |a|), A > 1, n = 1, 2, ..., N − 1 and N is the smallest integer such that |IN | > η. Let I0 = ϕ. Note that iθ

(1 − |a|2 )s 1 . 2ns , 2s |1 − az| A (1 − |a|)s

z ∈ S(In ) \ S(In−1 ).

We have M2 ≤

N ∫ ∑ n=1

.

S(In )\S(In−1 )

(1 − |a|2 )s dµ(z) |1 − az|2s

N −1 ∑ 1 µ(S(In )\S(In−1 )) µ(S(Iη ) \ S(IN −1 )) + s (1 − |a|) n=1 A2ns A2N s (1 − |a|)s

N ∑ 1 µ(S(In )) . . (1 − |a|)s n=1 A2ns

Note that

µ(S(I)) K(|I|)|I|s

< ϵ and |In | = An−1 (1 − |a|), using the fact that φK (t) .

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t1−c , t ≥ 1, we deduce that M2

.

N ∑ 1 ϵK(|In |)|In |s (1 − |a|)s n=1 A2ns

.

N ∑ ϵK(An−1 (1 − |a|)) K(1 − |a|) K(1 − |a|)Ans+s n=1

.

N ∑

φK (An−1 )

n=1

. ϵ

N ∑

ϵK(1 − |a|) Ans+s

A(n−1)(1−c)−ns−s K(1 − |a|) . ϵK(1 − |a|).

(8)

n=1

Now, we estimate M1 . Since |1 − az| ≥ η, z ∈ D\S(Iη ), and notice the fact that is a nondecreasing function when 0 < t < 1, we obtain M1

. .

µ(D) (1 − |a|)s K(1 − |a|) η 2s K(1 − |a|) (1 − |a|)1−c+s−1+c K(1 − |a|) . (1 − |a|)s+c−1 K(1 − |a|). K(1 − |a|)

t1−c K(t)

(9)

From (8) and (9) we see that (7) holds. The proof for another side is similar to Theorem 2. Thus, we omit the details. Finally, we give another characterization of vanishing (K, s)-Carleson measure by using functions in DK . Theorem 4. Suppose that K satisfy the conditions (1) and (2). Let µ be a positive Borel measure on D, 0 ≤ s < ∞ such that s + c > 1. Let {gn } be a bounded sequence in DK such that gn → 0 uniformly on compact subset of D as n → ∞. Then µ is a vanishing (K, s)-Carleson measure if and only if ∫ (1 − |a|2 )s lim sup |gn (a)|2 dµ(z) = 0. (10) n→∞ a∈D D |1 − az|2s Proof. First we assume that µ is a vanishing (K, s)-Carleson measure. Following the proof of Theorem 3, for any given ϵ > 0, we may find κ > 0 such that ∫ 1 (1 − |a|2 )s sup dµ(z) < ϵ, 2s a∈D\Dκ K(1 − |a|) D |1 − az| where Dκ = {z ∈ D : |z| < κ}. Since ∥g∥DK |g(z)| . √ , g ∈ DK , K(1 − |z|) 8

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we obtain ∫

(1 − |a|2 )s dµ(z) |1 − az|2s a∈D\Dκ D ∫ (1 − |a|2 )s 1 dµ(z) < ϵ. . sup 2s a∈D\Dκ K(1 − |a|) D |1 − az| |gn (a)|2

sup

(11)

Also, since gn → 0 uniformly on compact subsets of D, we see that for n sufficiently large, ∫ ∫ (1 − |a|2 )s (1 − |a|2 )s sup dµ(z) ≤ ϵ sup dµ(z) . ϵ. (12) |gn (a)|2 2s 2s |1 − az| a∈Dκ D a∈Dκ D |1 − az| From (11) and (12) we see that (10) holds. Conversely, assume that (10) holds. For a ∈ D, it is easy to check that ga (z) =

1 − |a| √ ∈ DK . (1 − az) K(1 − |a|)

For any In ⊆ ∂D such that |In |1 as n → ∞, let an = (1 − |In |)eiθn ∈ D, where eiθn is the center of In . It is easy to check that {gan } is a bounded sequence in DK and gan → 0 uniformly on compact subsets of D as n → ∞. By (10), we have ∫ (1 − |a|2 )s lim sup |gan (z)|2 dµ(z) = 0. n→∞ a∈D D |1 − az|2s ∫

Thus, lim

n→∞

1 |In |s

Notice the fact that |gan (z)|2 &

|gan (z)|2 dµ(z) = 0. S(In )

1 K(|In |) ,

we get

µ(S(In )) → 0, as n → ∞, K(|In |)|In |s as desired.

References [1] A. Aleman, Hilbert spaces of analytic functions between the Hardy space and the Dirichlet space, Proc. Amer. Math. Soc. 115 (1992), 97–104. [2] N. Arcozzi, R. Rochberg and E. Sawyer, Carleson measures for analytic Besov spaces, Rev. Mat. Iberoamericana. 18 (2002), 443–510. [3] G. Bao, Z. Lou, R. Qian and H. Wulan, On multipliers of Dirichlet type spaces, Complex Anal. Oper. Theory 9 (2015), 1701–1732. [4] L. Carleson, An interpolation problem for bounded analytic functions, Amer. J. Math. 80 (1985), 921–930.

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[5] L. Carleson, Interpolation by bounded analytic functions and the Corona problem, Ann. Math. 76 (1962), 547–559. [6] J. Caughran, Two results concerning the zeros of functions with finite Dirichlet integral, Canad. J. Math. 21 (1969), 312–316. [7] P. Duren, Theory of H p Spaces, Academic Press, New York, 1970. [8] O. El-Fallah, K. Kellay, J. Mashreghi and T. Ransford, A Primer on the Dirichlet Space, Cambridge Tracts in Mathematics 203, Cambridge University Press, 2014. [9] M. Essen, H. Wulan and J. Xiao, Several function-theoretic characterizations of M¨obius invariant QK spaces, J. Funct. Anal. 230 (2006), 78–115. [10] J. Garnett, Bounded Analytic Functions, Springer, New York, 2007. [11] R. Kerman and E. Sawyer, Carleson measures and multipliers of Dirichlet-type spaces, Trans. Amer. Math. Soc. 309 (1988), 87–98. [12] B. MacCluer and R. Zhao, Vanishing logarithmic Carleson measures, Illinois J. Math. 46 (2002), 507–518. [13] J. Pau and J. Pel´aez, On the zeros of functions in Dirichlet-type spaces, Trans. Amer. Math. Soc. 363 (2011), 1981–2002. [14] R. Qian and Y. Shi, Inner function in Dirichlet type spaces, J. Math. Anal. Appl. 421 (2015), 1844– 1854. [15] M. Rabindranathan, Toeplitz operators and division by inner function, Indiana Univ. J. Math. 22 (1972), 523–529. [16] R. Rochberg and Z. Wu, A new characterization of Dirichlet type spaces and applications, Illinois J. Math. 37 (1993), 101–122. [17] H. Shapiro and A. Shields, On the zeros of functions with finite Dirichlet integral and some related function spaces, Math. Z. 80 (1962), 217–229. [18] S. Shimorin, Complete Navanlinna-Pick property of Dirichlet-type spaces, J. Funct. Anal. 191 (2002), 272–296. [19] D. Stegenga, Multipliers of the Dirichlet space, Illinois J. Math. 24 (1980), 113–139. [20] G. Taylor, Multipliers on Dα , Trans. Amer. Math. Soc. 123 (1966), 229–240. [21] B. Taylor and D. Williams, Zeros of Lipschitz functions analytic in the unit disc, Michigan Math. J. 18 (1971), 129–139. [22] J. Xiao, Holomorphic Q Classes, Springer, LNM 1767, Berlin, 2001. [23] R. Zhao, On logarithmic Carleson measures, Acta Sci. Math. (Szeged) 69 (2003), 605–618. [24] J. Zhou and Y. Wu, Decomposition theorems and conjugate pair in DK spaces, Acta Math. Sinica 30 (2014), 1513–1525. [25] K. Zhu, Operator Theory in Function Spaces, American Mathematical Society, Providence, RI, 2007. Ruishen Qian: School of Mathematics and Statistics, Lingnan normal University, 524048, Zhanjiang, Guangdong, P. R. China. Email: [email protected] Songxiao Li: Institute of System Engineering, Macau University of Science and Technology, Avenida Wai Long, Taipa, Macau. Email: [email protected] Yanhua Zhang: Department of Mathematics, Qufu Normal University, 273165, Qufu, ShanDong, P. R. China. Email: [email protected] ⋆ Corresponding author: Songxiao Li

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Smarandache fuzzy BCI-algebras Sun Shin Ahn1 and Young Joo Seo∗,2 1

2

Department of Mathematics Education, Dongguk University, Seoul 04620, Korea Research Institute for Natural Sciences, Department of Mathematics, Hanyang University, Seoul, 04763, Korea

Abstract. The notions of a Smarandache fuzzy subalgebra (ideal) of a Smarandache BCI-algebra, a Smarandache fuzzy clean(fresh) ideal of a Smarandache BCI-algebra are introduced. Examples are given, and several related properties are investigated. 1. Introduction Generally, in any human field, a Smarandache structure on a set A means a weak structure W on A such that there exists a proper subset B of A with a strong structure S which is embedded in A. In [4], R. Padilla showed that Smarandache semigroups are very important for the study of congruences. Y. B. Jun ([1,2]) introduced the notion of Smarandache BCI-algebras, Smarandache fresh and clean ideals of Smarandache BCI-algebras, and obtained many interesting results about them. In this paper, we discuss a Smarandache fuzzy structure on BCI-algebras and introduce the notions of a Smarandache fuzzy subalgebra (ideal) of a Smarandache BCI-algebra, a Smarandache fuzzy clean (fresh) ideal of a Smarandache BCI-algebra are introduced, and we investigate their properties. 2. Preliminaries An algebra (X; ∗, 0) of type (2,0) is called a BCI-algebra if it satisfies the following conditions: (I) (∀x, y, z ∈ X)(((x ∗ y) ∗ (x ∗ z)) ∗ (z ∗ y) = 0), (II) (∀x, y ∈ X)((x ∗ (x ∗ (x ∗ y)) ∗ y = 0), (III) (∀x ∈ X)((x ∗ x = 0), (IV) (∀x, y ∈ X)(x ∗ y = 0 and y ∗ x = 0 imply x = y). If a BCI-algebra X satisfies the following identity; (V) (∀x ∈ X)(0 ∗ x = 0), then X is said to be a BCK-algebra. We can define a partial order “ ≤ ” on X by x ≤ y if and only if x ∗ y = 0. Every BCI-algebra X has the following properties: (a1 ) (∀x ∈ X)(x ∗ 0 = x), (a1 ) (∀x, y, z ∈ X)(x ≤ y implies x ∗ z ≤ y ∗ z, z ∗ y ≤ z ∗ x). A non-empty subset I of a BCI-algebra X is called an ideal of X if it satisfies the following conditions: (i) 0 ∈ I, (ii) (∀x ∈ X)(∀y ∈ I)(x ∗ y ∈ I implies x ∈ I). 0∗

Correspondence: Tel: +82 10 9247 6575 (Y. J. Seo). 619

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Sun Shin Ahn1 and Young Joo Seo2 Definition 2.1. ([1]) A Smarandache BCI-algebra is defined to be a BCI-algebra X in which there exists a proper subset Q of X such that (i) 0 ∈ Q and |Q| ≥ 2, (ii) Q is a BCK-algebra under the same operation of X. By a Smarandache positive implicative (resp. commutative and implicative) BCI-algebra, we mean a BCIalgebra X which has a proper subset Q of X such that (i) 0 ∈ Q and |Q| ≥ 2, (ii) Q is a positive implicative (resp. commutative and implicative) BCK-algebra under the same operation of X. Let (X; ∗, 0) be a Smarandache BCI-algebra and H be a subset of X such that 0 ∈ H and |H| ≥ 2. Then H is called a Smarandache subalgebra of X if (H; ∗, 0) is a Smarandache BCI-algebra. A non-empty subset I of X is called a Smarandache ideal of X related to Q if it satisfies: (i) 0 ∈ I, (ii) (∀x ∈ Q)(∀y ∈ I)(x ∗ y ∈ I implies x ∈ I), where Q is a BCK-algebra contained in X. If I is a Smarandache ideal of X related to every BCK-algebra contained in X, we simply say that I is a Smarandache ideal of X. In what follows, let X and Q denote a Smarandache BCI-algebra and a BCK-algebra which is properly contained in X, respectively. Definition 2.2. ([2]) A non-empty subset I of X is called a Smarandache ideal of X related to Q (or briefly, a Q-Smarandache ideal) of X if it satisfies: (c1 ) 0 ∈ I, (c2 ) (∀x ∈ Q)(∀y ∈ I)(x ∗ y ∈ I implies x ∈ I). If I is a Smarandache ideal of X related to every BCK-algebra contained in X, we simply say that I is a Smarandache ideal of X. Definition 2.3. ([2]) A non-empty subset I of X is called a Smarandache fresh ideal of X related to Q (or briefly, a Q-Smarandache fresh ideal of X) if it satisfies the conditions (c1 ) and (c3 ) (∀x, y, z ∈ Q)(((x ∗ y) ∗ z) ∈ I and y ∗ z ∈ I imply x ∗ z ∈ I). Theorem 2.4. ([2]) Every Q-Smarandache fresh ideal which is contained in Q is a Q-Smarandache ideal. The converse of Theorem 2.4 need not be true in general. Theorem 2.5. ([2]) Let I and J be Q-Smarandache ideals of X and I ⊂ J. If I is a Q-Smarandache fresh ideal of X, then so is J. Definition 2.6. ([2]) A non-empty subset I of X is called a Smarandache clean ideal of X related to Q (or briefly, a Q-Smarandache clean ideal of X) if it satisfies the conditions (c1 ) and (c4 ) (∀x, y ∈ Q)(z ∈ I)((x ∗ (y ∗ x)) ∗ z ∈ I implies x ∈ I). 620

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Smarandache fuzzy BCI-algebras Theorem 2.7. ([2]) Every Q-Smarandache clean ideal of X is a Q-Smarandache ideal. The converse of Theorem 2.7 need not be true in general. Theorem 2.8. ([2]) Every Q-Smarandache clean ideal of X is a Q-Smarandache fresh ideal. Theorem 2.9. ([2]) Let I and J be Q-Smarandache ideals of X and I ⊂ J. If I is a Q-Smarandache clean ideal of X, then so is J. A fuzzy set µ in X is called a fuzzy subalgebra of a BCI-algebra X if µ(x ∗ y) ≥ min{µ(x), µ(y)} for all x, y ∈ X. A fuzzy set µ in X is called a fuzzy ideal of X if (F1 ) µ(0) ≥ µ(x) for all x ∈ X, (F2 ) µ(x) ≥ min{µ(x ∗ y), µ(y)} for all x, y ∈ X. Let µ be a fuzzy set in a set X. For t ∈ [0, 1], the set µt := {x ∈ X|µ(x) ≥ t} is called a level subset of µ. 3. Smarandache fuzzy ideals Definition 3.1. Let X be a Smarandache BCI-algebra. A map µ : X → [0, 1] is called a Smarandache fuzzy subalgebra of X if it satisfies (SF1 ) µ(0) ≥ µ(x) for all x ∈ P , (SF2 ) µ(x ∗ y) ≥ min{µ(x), µ(y)} for all x, y ∈ P , where P ( X, P is a BCK-algebra with |P | ≥ 2. A map µ : X → [0, 1] is called a Smarandache fuzzy ideal of X if it satisfies (SF1 ) and (SF2 ) µ(x) ≥ min{µ(x ∗ y), µ(y)} for all x, y ∈ P , where P ( X, P is a BCK-algebra with |P | ≥ 2. This Smarandache fuzzy subalgebra (ideal) is denoted by µP , i.e., µP : P → [0, 1] is a fuzzy subalgebra(ideal) of X. Example 3.2. Let X := {0, 1, 2, 3, 4, 5} be a Smarandache BCI-algebra ([1]) with the following Cayley table: ∗ 0 1 2 3 4 5

0 0 1 2 3 4 5

1 0 0 2 3 3 3

2 0 1 0 3 4 5

3 3 3 3 0 1 1

4 3 3 3 0 0 1

5 3 3 3 0 0 0

Define a map µ : X → [0, 1] by µ(x) :=

( 0.5 0.7

if x ∈ {0, 1, 2, 3}, otherwise

Clearly µ is a Samrandache fuzzy subalgebra of X. It is verified that µ restricted to a subset {0, 1, 2, 3} which is a subalgebra of X is a fuzzy subalgebra of X, i.e., µ{0,1,2,3} : {0, 1, 2, 3} → [0, 1] is a fuzzy subalgebra of X. Thus µ : X → [0, 1] is a Smarandache fuzzy subalgebra of X. Note that µ : X → [0, 1] is not a fuzzy subalgebra of X, since µ(5 ∗ 4) = µ(0) = 0.5 ≯ min{µ(5), µ(4)} = 0.7. 621

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Sun Shin Ahn1 and Young Joo Seo2 Example 3.3. Let X := {0, 1, 2, 3, 4, 5} be a Smarandache BCI-algebra ([1]) with the following Cayley table: ∗ 0 1 2 3 4 5

0 0 1 2 3 4 5

1 0 0 2 3 44 4

2 0 0 0 3 4 4

3 0 1 2 0 0 5

4 4 4 4 4 0 1

5 4 4 4 4 0

Define a map µ : X → [0, 1] by µ(x) :=

( 0.5 0.7

if x ∈ {0, 1, 2} otherwise

Clearly µ is a Samrandache fuzzy ideal of X. It is verified that µ restricted to a subset {0, 1, 2} which is an ideal of X is a fuzzy ideal of X, i.e., µ{0,1,2} : {0, 1, 2} → [0, 1] is a fuzzy ideal of X. Thus µ : X → [0, 1] is a Smarandache fuzzy ideal of X. Note that µ : X → [0, 1] is not a fuzzy ideal of X, since µ(2) = 0.5 ≯ min{µ(2∗4) = µ(4), µ(4)} = µ(4) = 0.7. Lemma 3.4. Every Smarandache fuzzy ideal µP of a Smarandache BCI-algebra X is order reversing. Proof. Let P be a BCK-algebra with P ( X and |P | ≥ 2. If x, y ∈ P with x ≤ y, then x ∗ y = 0. Hence we have µ(x) ≥ min{µ(x ∗ y), µ(y)} = min{µ(0), µ(y)} = µ(y).



Theorem 3.5. Any Smarandache fuzzy ideal µP of a Smarandache BCI-algebra X must be a Smarandache fuzzy subalgebra of X. Proof. Let P be a BCK-algebra with P ( X and |X| ≥ 2. Since x ∗ y ≤ x for any x, y ∈ P , it follows from Lemma 3.4 that µ(x) ≤ µ(x ∗ y), so by (SF2 ) we obtain µ(x ∗ y) ≥ µ(x) ≥ min{µ(x ∗ y), µ(y)} ≥ min{µ(x), µ(y)}. This shows that µ is a Smarandache fuzzy subalgebra of X, proving the theorem.



Proposition 3.6. Let µP be a Smarandache fuzzy ideal of a Smarandache BCI-algebra X. If the inequality x ∗ y ≤ z holds in P , then µ(x) ≥ min{µ(x), µ(z)} for all x, y, z ∈ P. Proof. Let P be a BCK-algebra with P ( X and |P | ≥ 2. If x ∗ y ≤ z in P , then (x ∗ y) ∗ z = 0. Hence we have µ(x ∗ y) ≥ min{µ((x ∗ y) ∗ z), µ(z)} = min{µ(0), µ(z)} = µ(z). It follows that µ(x) ≥ min{µ(x ∗ y), µ(y)} ≥ min{µ(y), µ(z)}.



Theorem 3.7. Let X be a Smarandache BCI-algebra. A Smarandache fuzzy subalgebra µP of X is a Smarandache fuzzy ideal of X if and only if for all x, y ∈ P , the inequality x ∗ y ≤ z implies µ(x) ≥ min{µ(y), µ(z)}. Proof. Suppose that µP is a Smarandache fuzzy subalgebra of X satisfying the condition x ∗ y ≤ z implies µ(x) ≥ min{µ(y), µ(z)}. Since x ∗ (x ∗ y) ≤ y for all x, y ∈ P , it follows that µ(x) ≥ min{µ(x ∗ y), µ(y)}. Hence µP is a Smarandache fuzzy ideal of X. The converse follows from Proposition 3.6.



Definition 3.8. Let X be a Smarandache BCI-algebra. A map µ : X → [0, 1] is called a Smarandache fuzzy clean ideal of X if it satisfies (SF1 ) and (SF3 ) µ(x) ≥ min{µ(x ∗ (y ∗ x)) ∗ z), µ(z)} for all x, y, z ∈ P , 622

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Smarandache fuzzy BCI-algebras where P ( X and P is a BCK-algebra with |P | ≥ 2. This Smarandache fuzzy clean ideal is denoted by µP , i.e., µP : P → [0, 1] is a Smarandache fuzzy clean ideal of X. Example 3.9. Let X := {0, 1, 2, 3, 4, 5} be a Smarandache BCI-algebra ([2]) with the following Cayley table: ∗ 0 1 2 3 4 5

0 0 1 2 3 4 5

1 0 0 1 4 4 5

2 0 0 0 4 4 5

3 0 0 1 4 4 5

4 0 0 0 0 0 5

5 5 5 5 5 5 0

Define a map µ : X → [0, 1] by µ(x) :=

( 0.4 0.8

if x ∈ {0, 1, 2, 3} otherwise

Clearly µ is a Samrandache fuzzy clean ideal of X, but µ is not a fuzzy clean ideal of X, since µ(3) = 0.4 ≯ min{µ((3 ∗ (0 ∗ 3)) ∗ 5), µ(5)} = min{µ(5), µ(5)} = µ(5) = 0.8. Theorem 3.10. Let X be a Smarandache BCI-algebra. Any Smarandache fuzzy clean ideal µP of X must be a Smarandache fuzzy ideal of X. Proof. Let X be a BCK-algebra with P ( X and |P | ≥ 2. Let µP : P → [0, 1] be a Smarndache fuzzy clean ideal of X. If we let y := x in (SF3 ), then µ(x) ≥ min{µ((x ∗ (x ∗ x)) ∗ z), µ(z)} = min{µ((x ∗ 0) ∗ z), µ(z)} = min{µ(x ∗ z), µ(z)}, for all x, y, z ∈ P . This shows that µ satisfies (SF2 ). Combining (SF1 ), µP is a Smarandache fuzzy ideal of X, proving the theorem.



Corollary 3.11. Every Smarandache fuzzy clean ideal µP of a Smarndache BCI-algebra X must be a Smarandache fuzzy subalgebra of X. Proof. It follows from Theorem 3.5 and Theorem 3.10.



The converse of Theorem 3.10 may not be true as shown in the following example. Example 3.12. Let X := {0, 1, 2, 3, 4, 5} be a Smarandache BCI-algebra with the following Cayley table: ∗ 0 1 2 3 4 5

0 0 1 2 3 4 5

1 0 0 2 3 3 5

2 0 1 0 3 4 5

3 0 0 0 0 1 5

4 0 0 0 0 0 5

5 5 5 5 5 5 0

Let µP be a fuzzy set in P = {0, 1, 2, 3, 4} defined by µ(0) = µ(2) = 0.8 and µ(1) = µ(3) = µ(4) = 0.3. It is easy to check that µP is a fuzzy ideal of X. Hence µ : X → [0, 1] is a Smarandache fuzzy ideal of X. But it is not a Smarandache fuzzy clean ideal of X since µ(1) = 0.3 ≯ min{µ((1 ∗ (3 ∗ 1)) ∗ 2), µ(2)} = min{µ(0), µ(2)} = 0.8. Theorem 3.13. Let X be a Smarandache implicative BCI-algebra. Every Smarandache fuzzy ideal µP of X is a Smarandache fuzzy clean ideal of X. 623

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Sun Shin Ahn1 and Young Joo Seo2 Proof. Let P be a BCK-algebra with P ( X and |P | ≥ 2. Since X is a Smarandache implicative BCI-algebra, we have x = x ∗ (y ∗ x) for all x, y ∈ P . Let µP be a Smarandache fuzzy ideal of X. It follows from (SF2 ) that µ(x) ≥ min{µ(x ∗ z), µ(z)} ≥ min{µ((x ∗ (y ∗ x)) ∗ z), µ(z)}, for all x, y, z ∈ P . Hence µP is a Smarandache clean ideal of X. The proof is complete.



In what follows, we give characterizations of fuzzy implicative ideals. Theorem 3.14. Let X be a Smarandache BCI-algebra. Suppose that µP is a Smarandache fuzzy ideal of X. Then the following equivalent: (i) µP is Smarandache fuzzy clean, (ii) µ(x) ≥ µ(x ∗ (y ∗ x)) for all x, y ∈ P , (iii) µ(x) = µ(x ∗ (y ∗ x)) for all x, y ∈ P . Proof. (i) ⇒ (ii): Let µP be a Smarandache fuzzy clean ideal of X. It follows from (SF3 ) that µ(x) ≥ min{µ((x ∗ (y ∗ x)) ∗ 0), µ(0)} = min{µ(x ∗ (y ∗ x)), µ(0)} = µ(x ∗ (y ∗ x)), ∀x, y ∈ P. Hence the condition (ii) holds. (ii) ⇒ (iii): Since X is a Smarnadache BCI-algebra, we have x ∗ (y ∗ x) ≤ x for all x, y ∈ P . It follows from Lemma 3.4 that µ(x) ≤ µ(x ∗ (y ∗ x)). By (ii), µ(x) ≥ µ(x ∗ (y ∗ x)). Thus the condition (iii) holds. (iii) ⇒ (i): Suppose that the condition (iii) holds. Since µP is a Smarandache fuzzy ideal, by (SF2 ), we have µ(x ∗ (y ∗ x)) ≥ min{µ((x ∗ (y ∗ x)) ∗ z), µ(z)}. Combining (iii), we obtain µ(x) ≥ min{µ((x ∗ (y ∗ x)) ∗ z), µ(z)}. Hence µ satisfies the condition (SF3 ). Obviously, µ satisfies (SF1 ). Therefore µ is a fuzzy clean ideal of X. Hence the condition (i) holds. The proof is complete.



For any fuzzy sets µ and ν in X, we write µ ≤ ν if and only if µ(x) ≤ ν(x) for any x ∈ X. Definition 3.15. Let X be a Smarandache BCI-algebra and let µP : P → [0, 1] be a Smarandache fuzzy BCI-algebra of X. For t ≤ µ(0), the set µt := {x ∈ P |µ(x) ≥ t} is called a level subset of µP . Theorem 3.16. A fuzzy set µ in P is a Smarandache fuzzy clean ideal of X if and only if, for all t ∈ [0, 1], µt is either empty or a Smarandache clean ideal of X. Proof. Suppose that µP is a Smarandache fuzzy clean ideal of X and µt 6= ∅ for any t ∈ [0, 1]. It is clear that 0 ∈ µt since µ(0) ≥ t. Let µ((x ∗ (y ∗ x)) ∗ z) ≥ t and µ(z) ≥ t. It follows from (SF3 ) that µ(x) ≥ min{µ((x ∗ (y ∗ x)) ∗ z), µ(z)} ≥ t, namely, x ∈ µt . This shows that µt is a Smarandache clean ideal of X. Conversely, assume that for each t ∈ [0, 1], µt is either empty or a Smaranadche clean ideal of X. For any x ∈ P , let µ(x) = t. Then x ∈ µt . Since µt (6= ∅) is a Smarandache clean ideal of X, therefore 0 ∈ µt and hence µ(0) ≥ µ(x) = t. Thus µ(0) ≥ µ(x) for all x ∈ P . Now we show that µ satisfies (SF3 ). If not, then there exist x0 , y 0 , z 0 ∈ P such that µ(x0 ) < min{µ((x0 ∗ (y 0 ∗ z 0 )) ∗ z 0 ), µ(z 0 )}. Taking t0 := 12 {µ(x0 ) + min{µ((x0 ∗ (y 0 ∗ z 0 )) ∗ z 0 ), µ(z 0 )}}, we have µ(x0 ) < t0 < min{µ((x0 ∗ (y 0 ∗ z 0 )) ∗ z 0 ), µ(z 0 )}. Hence x0 ∈ / µt0 , (x0 ∗ (y 0 ∗ x0 )) ∗ z ∈ µt0 , and z 0 ∈ µt0 , i.e., µt0 is not a Smaraqndache clean of X, which is a contradiction. Therefore, µP is a Smarnadche fuzzy clean ideal, completing the proof. Theorem 3.17.



([2]) (Extension Property) Let X be a Smarandache BCI-algebra.

Let I and J be Q-

Smarandache ideals of X and I ⊆ J ⊆ Q. If I is a Q-Smarandache clean ideal of X, then so is J. Next we give the extension theorem of Smarandache fuzzy clean ideals. 624

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Smarandache fuzzy BCI-algebras Theorem 3.18. Let X be a Smarandache BCI-algebra. Let µ and ν be Smarandache fuzzy ideals of X such that µ ≤ ν and µ(0) = ν(0). If µ is a Smarndache fuzzy clean ideal of X, then so is ν. Proof. It suffices to show that for any t ∈ [0, 1], νt is either empty or a Smarandache clean ideal of X. If the level subset νt is non-empty, then µt 6= ∅ and µt ⊆ νt . In fact, if x ∈ µt , then t ≤ µ(x); hence t ≤ ν(x), i.e, x ∈ νt . So µt ⊆ νt . By the hypothesis, since µ is a Smarandache fuzzy clean ideal of X, µt is a Smarandache clean of X by Theorem 3.16. It follows from Theorem 3.17 that νt is a Smarandache clean ideal of X. Hence ν is a Smarandache fuzzy clean of X. The proof is complete.



Definition 3.19. Let X be a Smarandache BCI-algebra. A map µ : X → [0, 1] is called a Smarandache fuzzy fresh ideal of X if it satisfies (SF1 ) and (SF4 ) µ(x ∗ z) ≥ min{µ((x ∗ y) ∗ z), µ(y ∗ z)} for all x, y, z ∈ P , where P is a BCK-algebra with P ( X and |P | ≥ 2. This Smarandache fuzzy ideal is denoted by µP , i.e., µP : P → [0, 1] is a Smarandache fuzzy fresh ideal of X. Example 3.20. Let X := {0, 1, 2, 3, 4, 5} be a Smarandache BCI-algebra ([2]) with the following Cayley table: ∗ 0 1 2 3 4 5

0 0 1 2 3 4 5

1 0 0 2 1 4 5

2 0 1 0 3 4 5

3 0 0 2 0 4 5

4 0 1 0 3 0 5

5 5 5 5 5 5 0

Define a map µ : X → [0, 1] by µ(x) :=

( 0.5 0.9

if x ∈ {0, 1, 3}, otherwise

Clearly µ is a Samrandache fuzzy fresh ideal of X. But it is not a fuzzy fresh ideal of X, since µ(2 ∗ 4) = µ(0) = 0.5 ≯ min{µ((2 ∗ 5) ∗ 4), µ(5 ∗ 4)} = µ(5) = 0.9. Theorem 3.21. Any Smarandache fuzzy fresh ideal of a Smarandache BCI-algebra X must be a Smarandache fuzzy ideal of X. Proof. Taking z := 0 in (SF4 ) and x ∗ 0 = x, we have µ(x ∗ 0) ≥ min{µ((x ∗ y) ∗ 0), µ(y ∗ 0)}. Hence µ(x) ≥ min{µ(x ∗ y), µ(y)}. Thus (SF2 ) holds.



The converse of Theorem 3.21 may not be true as show in the following example. Example 3.22. Let X := {0, 1, 2, 3, 4, 5} be a Smarandache BCI-algebra ([2]) with the following Cayley table: ∗ 0 1 2 3 4 5

0 0 1 2 3 4 5

1 0 0 1 1 4 5

2 0 0 0 1 4 5 625

3 0 0 1 0 4 5

4 0 1 2 3 0 5

5 5 5 5 5 5 0 Sun Shin Ahn et al 619-627

J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 24, NO.4, 2018, COPYRIGHT 2018 EUDOXUS PRESS, LLC

Sun Shin Ahn1 and Young Joo Seo2 Define a map µ : X → [0, 1] by µ(x) :=

( 0.5 0.4

if x ∈ {0, 4}, otherwise

Clearly µ(x) is a Samrandache fuzzy ideal of X. But µ(x) is not a Samrandache fuzzy fresh ideal of X, since µ(2 ∗ 3) = µ(1) = 0.4 ≯ min{µ((2 ∗ 1) ∗ 3), µ(1 ∗ 3)} = min{µ(1 ∗ 3), µ(0)} = µ(0) = 0.5. Proposition 3.23. Let X be a Smarandache BCI-algebra. A Smarandache fuzzy ideal µP of X is a Smarandache fuzzy fresh ideal of X if and only if it satisfies the condition µ(x ∗ y) ≥ µ((x ∗ y) ∗ y) for all x, y ∈ P . Proof. Assume that µP is a Smarandache fuzzy fresh ideal of X. Putting z := y in (SF4 ), we have µ(x ∗ y) ≥ min{µ((x ∗ y) ∗ y), µ(y ∗ y)} = min{µ((x ∗ y) ∗ y), µ(0)} = µ((x ∗ y) ∗ y), ∀x, y ∈ P. Conversely, let µP be Smarandache fuzzy ideal of X such that µ(x ∗ y) ≥ µ((x ∗ y) ∗ y). Since, for all x, y, z ∈ P , ((x ∗ z) ∗ z) ∗ (y ∗ z) ≤ (x ∗ z) ∗ y = (x ∗ y) ∗ z, we have µ((x ∗ y) ∗ z) ≤ µ(((x ∗ z) ∗ z) ∗ (y ∗ z)). Hence µ(x ∗ z) ≥ µ((x ∗ z) ∗ z) ≥ min{µ(((x ∗ z) ∗ z) ∗ (y ∗ z)), µ(y ∗ z)} ≥ min{µ((x ∗ y) ∗ z), µ(y ∗ z)}. This completes the proof.



Since (x ∗ y) ∗ y ≤ x ∗ y, it follows from Lemma 3.4 that µ(x ∗ y) ≤ µ((x ∗ y) ∗ y). Thus we have the following theorem. Theorem 3.24. Let X be a Smarandache BCI-algebra. A Smarandache fuzzy ideal µP of X is a Smarandache fuzzy fresh if and only if it satisfies the identity µ(x ∗ y) = µ((x ∗ y) ∗ y), textf or all x, y ∈ X.

We give an equivalent condition for which a Smarandache fuzzy subalgebra of a Smarandache BCI-algebra to be a Smarandache fuzzy clean ideal of X. Theorem 3.25. A Smarandache fuzzy subalghebra µP of X is a Smarandache fuzzy clean ideal of X if and only if it satisfies (x ∗ (y ∗ x)) ∗ z ≤ u implies µ(x) ≥ min{µ(z), µ(u)} for allx, y, z, u ∈ P.

(∗)

Proof. Assume that µP is a Smarandache fuzzy clean ideal of X. Let x, y, z, u ∈ P be such that (x ∗ (y ∗ x)) ∗ z ≤ u. Since µ is a Smarandache fuzzy ideal of X, we have µ(x ∗ (y ∗ x)) ≥ min{µ(z), µ(u)} by Theorem 3.7. By Theorem 3.14-(iii), we obtain µ(x) ≥ min{µ(z), µ(u)}. Conversely, suppose that µP satisfies (∗). Obviously, µP satisfies (SF1 ), since (x ∗ (y ∗ x)) ∗ ((x ∗ (y ∗ x)) ∗ z) ≤ z, by (∗), we obtain µ(x) ≥ min{µ((x ∗ (y ∗ x)) ∗ z), µ(z)}, which shows that µP satisfies (SF3 ). Hence µP is a Smarandache fuzzy clean ideal of X. The proof is complete.



References [1] Y. B. Jun, Smarandache BCI-algebras, Sci. Math. Japo. 62(1) (2005), 137–142; e2005, 271–276. [2] Y. B. Jun, Smarandache fresh and clean ideals of Smarandache BCI-algebras, Kyungpook Math. J. 46 (2006), 409–416. [3] J. Meng and Y. B. Jun, BCK-algebras, Kyung Moon Sa, Seoul, 1994. 626

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Smarandache fuzzy BCI-algebras [4] R. Padilla, Smarandache algebraic structures, Bull. Pure Sci. Delhi, 121;http://www.gallup.unm.edu/smarandache/alg-s-tx.txt.

627

17E (1998),

no.1,

119–

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On Fibonacci derivative equations Hee Sik Kim1 , J. Neggers2 and Keum Sook So3,∗ 1

2

Department of Mathematics, Hanyang University, Seoul, 133-791, Korea Department of Mathematics, University of Alabama, Tuscaloosa, AL 35487-0350, U. S. A 3,∗ Department of Mathematics, Hallym University, Chuncheon 200-702, Korea

Abstract. In this paper, we introduce the notions of Fibonacci (co-)derivative of real-valued functions. We find general solutions of the equations 4(f (x)) = g(x) and (4 + I)(f (x)) = g(x). 1. Introduction The theory of Fibonacci-numbers has been studied in many different forms for centuries and the literature on the subject is consequently incredibly vast. The most amazing qualities of these numbers is the variety of mathematical models where they play some sort of role and where their properties are of importance in elucidating the ability of the model under discussion to explain whatever implications are inherent in it. Atanassov et al. [1] and Dunlap [2] provided general and fundamental surveys on the theory of Fibonacci numbers. Hyers-Ulam studied the stability of Fibonacci functional equations [5]. Han et al. [3] discussed Fibonacci sequences in both several groupoids and groups. The present authors [6] introduced the notion of generalized Fibonacci sequences over a groupoid, and investigated these in particular for the case of a groupoid containing idempotents and pre-idempotents. Han et al. [4] studied Fibonacci functions on the real numbers R, i.e., functions f : R → R such that for all x ∈ R, f (x + 2) = f (x + 1) + f (x), and they developed the notion of Fibonacci functions using the concept of f -even and f -odd functions. The present authors [7] studied Fibonacci functions using the (ultimately) periodicity and also discussed the exponential Fibonacci functions. Especially, given a non-negative real-valued function, the present authors obtained several exponential Fibonacci functions. In this paper, we introduce the notions of Fibonacci (co-)derivative of real-valued functions. We find general solutions of the equations 4(f (x)) = g(x) and (4 + I)(f (x)) = g(x). 2. Preliminaries A function f defined on the real numbers is said to be a Fibonacci function ([4]) if it satisfies the formula f (x + 2) = f (x + 1) + f (x) for any x ∈ R, where R (as usual) is the set of real numbers. Example 2.1. ([4]) Let f (x) := ax be a Fibonacci function on R where a > 0. Then ax a2 = f (x + 2) = f (x + 1) + f (x) = ax (a + 1). Since a > 0, we have a2 = a + 1 and a =

√ 1+ 5 2 .

√

Hence f (x) = ( 1+2 5 )x is a Fibonacci

function, and the unique Fibonacci function of this type on R. If we let u0 = 0, u1 = 1, then we consider the full Fibonacci sequence: · · · , 5, −3, 2, −1, 1, 0, 1, 1, 2, 3, 5, · · · , i.e., u−n = (−1)n un for n > 0, and un = Fn , the nth Fibonacci number. 0∗

Correspondence: Tel.: +82 33 248 2011, Fax: +82 33 256 2011 (K. S. So). 628

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Hee Sik Kim, J. Neggers and Keum Sook So∗ ∞ Example 2.2. ([4]) Let {un }∞ n=−∞ and {vn }n=−∞ be full Fibonacci sequences. We define a function f (x)

by f (x) := ubxc + vbxc t, where t = x − bxc ∈ (0, 1). Then f (x + 2) = ubx+2c + vbx+2c t = u(bxc+2) + v(bxc+2) t = (u(bxc+1) + ubxc ) + (v(bxc+1) + vbxc )t = f (x + 1) + f (x) for any x ∈ R. This proves that f is a Fibonacci function. Note that if a Fibonacci function is differentiable on R, then its derivative is also a Fibonacci function. Proposition 2.3. ([4]) Let f be a Fibonacci function. If we define g(x) := f (x + t) where t ∈ R for any x ∈ R, then g is also a Fibonacci function. √

√

√

For example, since f (x) = ( 1+2 5 )x is a Fibonacci function, g(x) = ( 1+2 5 )x+t = ( 1+2 5 )t f (x) is also a Fibonacci function where t ∈ R.

3. Fibonacci derivatives Let f : R → R be a real-valued function. We shall consider the expression (4f )(x) := f (x + 2) − f (x + 1) − f (x) to be the Fibonacci derivative of f (x). For example, if Φ :=

√ 1+ 5 2 ,

then f (x) = Φx yields (4f )(x) = Φx+2 −

Φx+1 − Φx = Φx (Φ2 − Φ − 1) = 0. If f is any Fibonacci function, then (4f )(x) = 0 for all x ∈ R and conversely. Note that if 4f = 4g, then f − g is a Fibonacci function. Example 3.1. If f (x) := ax + b, then 4(ax + b)

=

[a(x + 2) + b] − [a(x + 1) + b] − [ax + b]

= −ax + (a − b) and 4(b) = −b, 4(x) = −x + 1. Simultaneously we shall also consider the Fibonacci co-derivative of f , denoted (4 + I)(f ), by the formula (4 + I)(f )(x) = 4(f ) + f (x) = f (x + 2) − f (x + 1) Thus for example, if f (x) = ax + b, then (4 + I)(ax + b) = [a(x + 2) + b] − [a(x + 1) + b] − [ax + b] = a, which coincides with

d dx (ax

+ b).

We pose a question: what is the “anti-derivative” of a function f : R → R, i.e., given f : R → R, find g : R → R such that 4g = f . For example, 4(−x − 1) = [−(x + 2) − 1] − [−(x + 1) − 1] − [−x − 1] = x. Hence the Fibonacci anti-derivative of x is −x − 1 + ϕ where ϕ is a Fibonacci function. Proposition 3.2. Fibonacci functions are fixed points for Fibonacci co-derivative operator 4 + I. Proof. Let f (x) be a Fibonacci function. Then (4f )(x) = 0 and hence (4 + I)(f )(x) = (4f )(x) + f (x) = f (x).



Proposition 3.3. If (4 + I)(f )(x) = 0, then (42 f )(x) = f (x). Proof. If (4 + I)(f )(x) = 0, then (4f )(x) = −f (x). It follows that (42 f )(x) = 4(−f (x)) = (−f )(x + 2) − (−f )(x + 1) − (−f )(x) = −(4f )(x) = −(−f (x)) = f (x). 629

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On Fibonacci derivative equations Example 3.4. Suppose that 4f = x2 and f (x) = ax2 +bx+c. Since 4(x2 ) = −x2 +2x+3 and 4(x) = −x+1, we obtain x2

= 4(ax2 + bx + c) = a 4 (x2 ) + b 4 (x) + 4(c) = −ax2 + (2a − b)x + (3a + b − c)

It follows that a = −1, b = −2 and c = −5, i.e., 4(−x2 − 2x − 5) = x2 . Thus, the general Fibonacci anti-derivative of x2 is −x2 − 2x − 5 + ϕ where ϕ is a Fibonacci function. Example 3.5. Suppose that 4f = x3 and f (x) = ax3 + bx2 + cx + d. Since 4(x3 ) = −x3 + 3x2 + 9x + 7, we obtain 4(−x3 − 3x2 − 15x − 31) = x3 as in Example 3.4. Theorem 3.6. Let 4fn = xn and let f0 , f1 , · · · , fn−1 be determined to yield particular solutions for 4fk = xk (k = 0, 1, · · · , n − 1). Then n

fn = −x +

n−1 X k=0

n k



[2n−k − 1]fk + ϕ

where ϕ is a Fibonacci function. Proof. Let 4fn = xn and let fn = −xn + Qn (x) where Qn (x) is a polynomial of x of degree n − 1. Then xn

= 4(fn ) = 4(−xn + Qn (x)) = − 4 (xn ) + 4(Qn (x)) = −[(x + 2)n − (x + 1)n − xn ] + 4(Qn (x))

 n [2n−k −1]xk . Assume f0 , f1 , · · · , fn−1 are determined k=0 k to have a particular solutions for 4fk = xk (k = 0, 1, · · · , n − 1). Then

It follows that 4(Qn (x)) = (x+2)n −(x+1)n =

4(Qn (x))

Pn−1



n−1 X

 n [2n−k − 1] 4 fk k k=0 n−1 X n  = 4( [2n−k − 1]fk ) k =

k=0

Pn−1



It follows that Qn (x) = k=0   Pn−1 n [2n−k − 1]fk + ϕ. k=0 k

n k



[2n−k − 1]fk + ϕ for some Fibonacci function ϕ. Hence fn = −xn + Qn (x) =  630

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Hee Sik Kim, J. Neggers and Keum Sook So∗ Example 3.7. In the above examples it was known that f0 = −1 + ϕ, f1 = −x − 1 + ϕ, f2 = −x2 − 2x − 5 + ϕ and f3 = −x3 − 3x2 − 15x − 31 + ϕ where ϕ is a Fibonacci function. We compute Q4 (x) as follows: Q4 (x)

=

 3  X 4 [24−k − 1]fk + ϕ k

k=0

=

15f0 + 28f1 + 18f2 + 4f3 + ϕ

=

15(−1) + 28(−x − 1) + 18(−x2 − 2x − 5) +4(−x3 − 3x2 − 15x − 31)

= −4x3 − 30x2 − 124x − 257 + ϕ This shows that f4 = −x4 + Q4 (x) + ϕ = −x4 − 4x3 − 30x2 − 124x − 257 + ϕ where ϕ is a Fibonacci function. Theorem 3.8. Given a polynomial g(x) := a0 + a1 x + · · · + an xn , we have a particular solution for 4(f (x)) = g(x) as f (x) = a0 f0 + a1 f1 + · · · + an fn , where 4fk = xk (k = 0, 1, · · · , n) and a general solution f (x) + ϕ(x) where 4(ϕ(x)) = 0. Proof. It follows immediately from Theorem 3.6.



4. Fibonacci co-derivatives Let us consider the problem (4 + I)k (f (x)) = xn . We have (4 + I)(1) = 4(1) + I(1) = 0, (4 + I)(x) = 4(x) + I(x)(−x + 1) + x = 1 and (4 + I)(x2 ) = 4(x2 ) + I(x2 ) = 2x + 3. Using Theorem 3.6, we obtain the following proposition. Proposition 4.1. The Fibonacci co-derivative of xn is n

(4 + I)(x ) =

n−1 X k=0

n k



[2n−k − 1]xk

Proof. Using Theorem 3.6, we obtain (4 + I)(xn )

= 4(xn ) + I(xn ) =

(x + 2)n − (x + 1)n

= 4(Qn (x)) n−1 X n  = [2n−k − 1]xk , k k=0

proving the proposition.

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On Fibonacci derivative equations Example 4.2. If we let n := 4 in Proposition 4.1, then  3  X 4 (4 + I)(x ) = [24−k − 1]xk k k=0       4 4 4 = (24 − 1)x0 + (23 − 1)x1 + (22 − 1)x2 0 1 2   4 + (23 − 1)x3 3 4

=

4x3 + 18x2 + 28x + 15

It follows that 4(x4 ) = −x4 + 4x3 + 18x2 + 28x + 15. Consider now (4 + I)2 (1) = (4 + I)[(4 + I)(1)] = (4 + I)(0) = 0 and (4 + I)2 (x) = (4 + I)[(4 + I)(x2 )] = (4 + I)(2x + 3) = 2. Similarly we obtain (4 + I)2 (x3 ) = (4 + I)[3x2 + 9x + 7] = 6x + 18. Using Proposition 4.1, we obtain the following formula. Proposition 4.3. For any natural number n, we have (4 + I)2 (xn ) =

n−1 X X k−1



n k

k=0 j=0

k j



(2n−k − 1)(2k−j − 1)xj

Proof. Using Proposition 4.1, we obtain the following. (4 + I)2 (xn )

=

n−1 X

(4 + I)[

k=0

=

n−1 X k=0

=

n k



n−1 X k−1 X k=0 j=0

n k



(2n−k − 1)xk ]

(2n−k − 1)

k−1 X j=0



n k

k j



k j



(2k−j − 1)xj

(2n−k − 1)(2k−j − 1)xj

 Example 4.4. We compute (4 + I)2 (x4 ) as follows. (4 + I)2 (x4 )

=

(4 + I)[4x3 + 18x2 + 28x + 15]

=

4(4 + I)(x3 ) + 18(4 + I)(x2 ) + 28(4 + I)(x) +15(4 + I)(1)

=

12x2 + 72x + 110 632

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Hee Sik Kim, J. Neggers and Keum Sook So∗ Upon checking Proposition 4.3 when n = 4, we find that 3 k−1 X X 4  k  (24−k − 1)(2k−j − 1)xj (4 + I)2 (x4 ) = j k k=0 j=0

=

  0  X 4 1 (23 − 1)(21−j − 1)xj 1 j j=0

+

  1  X 4 2 (22 − 1)(22−j − 1)xj 2 j j=0

+

  2  X 3 4 (21 − 1)(23−j − 1)xj j 3 j=0

=

28x0 + 54x0 + 36x + 28x0 + 36x + 12x2

=

12x2 + 72x + 110

Next, we want to obtain an exact analog of Theorem 3.6 for the Fibonacci co-derivative 4 + I. Example 4.5. Let f1 (x) := ax2 + bx + c be a polynomial satisfying (4 + I)(f1 (x)) = x. Then x = (4 + I)(ax2 + bx + c) = 2ax + 3a + b. It follows that 2a = 1, 3a + b = 0, i.e., a =

1 2, b

= − 32 and c is arbitrary.

Hence (4 + I)( 12 x2 − 32 x + c) = x where c is a constant. Similarly, we may find a polynomial f2 (x) satisfying (4 + I)(f2 (x)) = x2 , i.e., (4 + I)( 13 x3 − 32 x2 + polynomial fn (x) =

1 n+1 n+1 x

13 6 x

+ d) = x2 where d is a constant. In this fashion, we obtain a

+ qn+1 (x) which can be determined so that (4 + I)(fn (x)) = xn where qn+1 (x) is

a polynomial of degree n. Theorem 4.6. Given a polynomial g(x) := a0 +a1 x+· · ·+an xn , a particular solution for (4+I)(f (x)) = g(x) as f (x) = a0 f0 + a1 f1 + · · · + an fn is obtained, where fn (x) =

1 n+1 n+1 x

+ qn+1 (x) where qn+1 (x) is a polynomial

of degree n. Proof. The proof is similar to the proof of Theorem 3.8.



5. Solving the equation (4 + I)n (f (x)) = q(x) Consider (4 + I)(f (x)) = (4 + I)(g(x)). It means that (4 + I)(f (x) − g(x)) = 0, i.e., (f − g)(x + 2) − (f − g)(x + 1) = 0 for all x ∈ R. This shows that there exists a map ψ : R → R with ψ(x + 2) = ψ(x + 1) for all x ∈ R such that f = g + ψ. If we let B1 := {ψ|(4 + I)(ψ(x)) = 0, ∀x ∈ R}, then B1 consists of all functions ϕ : R → R such that ϕ is periodic of period 1. This means that ϕ ∈ B1 ⇐⇒ ϕ(x + 1) − ϕ(x) = 0, ∀x ∈ R Hence general solution of (4 + I)n (f (x)) = q(x) is {p(x) + ψ(x) | 4 (p(x)) = q(x), ψ(x) ∈ B1 } = {p(x) + ψ(x) | 4 (p(x)) = q(x), ψ(x + 1) = ψ(x), ∀x ∈ R}. Consider (4 + I)2 (f (x)) = q(x). Let p(x) be a polynomial in R[x] such that (4 + I)2 (p(x)) = q(x). Then (4 + I)2 (p(x)) = (4 + I)2 (f (x)). It follows that (4 + I)2 (f (x) − p(x)) = 0, i.e., 633

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On Fibonacci derivative equations there exists a polynomial ψ(x) ∈ R[x] such that f (x) − p(x) = ψ(x) and (4 + I)2 (ψ(x)) = 0. This means that (4 + I)[(4 + I)(ψ(x))] = 0, i.e., (4 + I)(ψ(x)) ∈ B1 . If we let B2 := {ϕ(x) ∈ R[x] | (4 + I)(ϕ(x)) ∈ B1 }, then ϕ(x) ∈ B2

⇐⇒

(4 + I)(ϕ(x)) ∈ B1

⇐⇒

∃h(x) ∈ B1 such that ϕ(x + 2) − ϕ(x + 1) = h(x)

⇐⇒

h(x + 1) − h(x) = 0, ϕ(x + 2) − ϕ(x + 1) = h(x)

⇐⇒

ϕ(x + 2) − 2ϕ(x + 1) + ϕ(x) = 0

Hence the set of all general solutions of (4 + I)2 (f (x)) = q(x) is {p(x) + ψ(x) | (4 + I)2 (p(x)) = q(x), ψ(x) ∈ B2 } = {p(x) + ψ(x) | (4 + I)2 (p(x)) = q(x), ψ(x + 2) − 2ψ(x + 1) + ψ(x) = 0, ∀x ∈ R}. Similarly, if we let B3 := {ϕ(x) | (4 + I)(ϕ(x)) ∈ B2 }, then B3 = {ϕ ∈ R[x] | ϕ(x + 3) − 3ϕ(x + 2) + 3ϕ(x + 1) − ϕ(x) = 0}. We generalize this fact as follows: Lemma 5.1. If we let Bn := {ϕ(x) | (4+I)(ϕ(x)) ∈ Bn−1 }, then Bn = {ϕ ∈ R[x] |

Pn



r=0

n r

 ϕ(x+n−r) =

0, ∀x ∈ R}. Theorem 5.2. Given a polynomial p(x) ∈ R[x], there exists a polynomial qn (x) ∈ R[x] such that (4 + n

I) (qn (x)) = p(x), and its general solution f (x) is of the form qn (x) + ϕ(x) where ϕ(x) ∈ Bn . Proof. It follows from Theorem 4.6 and Lemma 5.1.



6. Concluding remark Given Theorem 3.6 and the fact the ϕ(x) ≡ 0 is a Fibonacci function, a particular solution to the Fibonacci derivative equation 4fn = xn , is given iteratively by the formula: fn = −xn +

n−1 X k=0

n k



[2n−k − 1]fk

where if we set f0 = 1, we obtain a sequence of polynomials of degree n for fn , n = 0, 1, 2, · · · . From the structure of the formula we may surmise the existence of many combinatorial properties of the sequence. Also upon rewriting: fn =

n X

Aln xl , Ann = −1,

l=0

the coefficients Aln , thought of as analogs of binomial numbers, should illustrate a great number of combinatorial relations among themselves as well as with other families, including the binomial numbers (coefficients). Since   Pn−1 n n (4 + I)(x ) = k=0 [2n−k − 1]xk exhibits a “similar” form, we expect there to be confirmation of the k claim made above in a multitude of ways, above and beyond what has already been illustrated.

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Hee Sik Kim, J. Neggers and Keum Sook So∗ 7. Future works Given what has been done in this paper, it is clear that very much remains to be done. Thus, a much more detailed study of functions of the F (x) type, as described above, remains to be done. Furthermore, as pointed out in the concluding remarks, there is much to be done still in completing the combinatorial grammar which is associated with the solution of a particular kind to the equation 4m fmn = xn , of which some cases have been looked at above, but for which very significant gaps still remain to be explored. Also, as usual in this type of research, the law of natural growth of problems prevails, i.e., as one problem is successfully resolved, novel gaps noted present themselves for consideration and no finality is in sight (nor expected) for the area of study touched upon in this case as well, to the benefit of those engaged in furthering knowledge of this (as well as any other) subject. Acknowledgement This work was supported by Hallym University Research Fund HRF-201604-004.

References [1] K. Atanassov et al., New Visual Perspectives on Fibonacci numbers, World Scientific, New Jersey, 2002. [2] R. A. Dunlap, The Golden Ratio and Fibonacci Numbers, World Scientific, New Jersey, 1997. [3] J. S. Han, H. S. Kim and J. Neggers, J., Fibonacci sequences in groupoids, Advances in Difference Equations 2012 2012:19 (doi:10.1186/1687-1847-2012-19). [4] S. M. Jung, Hyers-Ulam stability of Fibonacci functional equation, Bull. Iranian Math. Soc. 35 (2009), 217227. [5] H. S. Kim and J. Neggers, Fibonacci Means and Golden Section Mean, Computers and Mathematics with Applications 56 (2008), 228-232. [6] H. S. Kim, J. Neggers and K. S. So, Generalized Fibonacci sequences in groupoids, Advances in Difference Equations 2013 2013:26 (doi:10.1186/1687-1847-2013-26). [7] H. S. Kim, J. Neggers and K. S. So, On Fibonacci functions with periodicity, Advances in Difference Equations 2014 2014:293 (doi:10.1186/1687-1847-2014-293).

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A NOTE ON SYMMETRIC IDENTITIES FOR TWISTED DAEHEE POLYNOMIALS JONGKYUM KWON1 AND JIN-WOO PARK2,∗

Abstract. In this paper, we consider the twisted Daehee numbers and polynomials. We investigate some new and explicit symmetric identities for the twisted Daehee polynomials arising from p-adic invariant integral on Zp .

1. Introduction Throughout this paper, Zp , Qp , and Cp will respectively denote the ring of padic rational integers, the field of p-adic rational numbers and the completions of algebraic closure of Qp . The p-adic norm is defined |p|p = p1 . Let f (x) be a uniformly differentiable function on Zp . Then the p-adic invariant integral on Zp is defined by N pX −1

Z f (x)dµ0 (x) = lim

N →∞

Zp

f (x)dµ0 x + pN Zp




x=0

(1.1)

N

p −1 1 X = lim N f (x). N →∞ p n=0

Thus, by (1.1), we get Z Z f1 (x)du0 (x) − Zp

f (x)du0 (x) = f 0 (0),

(1.2)

Zp

where f1 (x) = f (x + 1) (see [1, 4, 9]). From (1.2), we can derive Z Z n−1 X fn (x)du0 (x) − f (x)du0 (x) = f 0 (l), (n ∈ N), Zp

Zp

(1.3)

l=0

where fn (x) = f (x + n) (see [1, 5, 6]). As is well known, the Bernoulli polynomials are defined by the generating function to be   ∞ X t tn xt e = Bn (x) , (see [1, 2, 4]). (1.4) t e −1 n! n=0 When x = 0, Bn = Bn (0) are called the Bernoulli numbers. 2010 Mathematics Subject Classification. 11B68, 11S40, 11S80. Key words and phrases. Daehee polynomials, twisted Daehee polynomials, twisted λ-Daehee polynomials. ∗ corresponding author. 1

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2

For n ∈ N, let Tp be the p-adic locally constant space defined by Tp = ∪ Cpn = lim Cpn , n→∞

n≥1

 n where Cpn = ω|ω p = 1 is the cyclic group of order pn . It is well known that for ξ ∈ Tp , the twisted Bernoulli polynomials are defined as   ∞ X t tn xt e = (1.5) Bn,ξ (x) , (see [1, 2]). t ξe − 1 n! n=0 When x = 0, Bn,ξ = Bn,ξ (0) are called the twisted Bernoulli numbers. 1 For t ∈ Cp with |t|p < p− p−1 , the Daehee polynomials are defined by the generating function to be ∞ X log(1 + t) tn (1 + t)x = (1.6) Dn (x) , (see [1, 2, 4 - 14, 16,17]). t n! n=0 When x = 0, Dn = Dn (0) are called the Daehee numbers. From (1.4) and (1.6), we can derive the following equation: ∞ X 1 tn Dn (x) (et − 1)m Bn (x) = n! m! m=0 n=0 ∞ X

=

∞ X

n X

n=0

m=0

! Dm (x)S2 (n, m)

tn , n!

(1.7)

where S2 (n, m) is the Stirling number of the second kind which is given by the generating function to be ∞ X 1 t tn (e − 1)m = S2 (n, m) , (see [3, 15]). m! n! n=m

By (1.7), we get Bn (x) =

n X

Dm (x)S2 (n, m), (n ≥ 0).

(1.8)

m=0

From (1.4), we have ∞ X

Dn (x)

n=0

∞ X tn 1 = Bm (x) (log(1 + t))m n! m=0 m! ! ∞ n X X tn = Bm (x)S1 (n, m) , n! n=0 m=0

(1.9)

where S1 (n, m) is the Stirling number of the first kind which is defined by falling factorials as follows: n X (x)0 = 1, (x)n = x(x − 1) · · · (x − n + 1) = S1 (n, l)xl , (n ∈ N), l=0

Thus, by (1.9), we get Dn (x) =

n X

Bm (x)S1 (n, m), (n ≥ 0).

(1.10)

m=0

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From (1.2), we derive Witt’s formula for Daehee polynomials as follows:

Z

(1 + t)x+y du0 (y) =

Zp

=

log(1 + t) (1 + t)x t ∞ X n=0

Dn (x)

(1.11)

tn . n!

Thus, by (1.11), we get

Z (x + y)n du0 (y) = Dn (x), (n ≥ 0), (see [8]).

(1.12)

Zp

Now, we consider the twisted Daehee polynomials defined by the generating function to be

∞ X tn log(1 + ξt) Dn,ξ (x) , (see [5, 13, 14]). (1 + ξt)x = ξt n! n=0

(1.13)

When x = 0, Dn,ξ = Dn,ξ (0) are called the twisted Daehee numbers. In [5], authers define twisted λ-Daehee polynomials which are given by the p-adic invariant integral on Zp to be

Z

(1 + ξt)λ(x+y) dµ0 (y) =

Zp

λ log(1 + ξt) (1 + ξt)λx (1 + ξt)λ − 1 ∞ X

tn = Dn,λ,ξ (x) . n! n=0

(1.14)

In the special case, λ = 1, ξ = 1, we note that Dn,1,1 (x) = Dn (x). When x = 0, then Dn,λ,ξ = Dn,λ,ξ (0) are called twisted λ - Daehee numbers. Recently, several authors have researched twisted Daehee polynomials in the several areas (see [5, 13, 14]). In this paper, we investigate some explicit and new symmetric identities for the twisted Daehee polynomials which are derived from the p-adic invariant integral on Zp .

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4

2. Symmetric identities for the twisted Daehee polynomials 1

Let t ∈ Cp with |t|p < p− p−1 . Now, we consider the following p-adic integral on Zp . From (1.3), we easily get ! Z Z 1 n+x x (1 + ξt) du0 (x) − (1 + ξt) du0 (x) log(1 + ξt) Zp Zp =

n−1 X 1 (1 + ξt)l log(1 + ξt) log(1 + ξt) l=0

=

n−1 X

(1 + ξt)l

(2.1)

l=0

=

=

n−1 ∞ XX

n−1 X

i=0 n=0

m=0

∞ X

n X

n=0

m=0

! m

i S1 (n, m)

S1 (n, m)

n−1 X

ξ n tn n!

! i

m

i=0

ξ n tn . n!

Then, by (2.1), we get 1 log(1 + ξt) =

∞ X

n X

n=0

m=0

Z

n+x

(1 + ξt)

Z du0 (x) −

Zp

! x

(1 + ξt) du0 (x) Zp

! S1 (n, m)Sm (n − 1)

(2.2)

ξ n tn , n!

where for given positive integer k, Sk (n) = 0k + 1k + 2k + · · · + nk . From (1.2) and (1.3), we have ! Z Z 1 n+x x (1 + ξt) du0 (x) − (1 + ξt) du0 (x) log(1 + ξt) Zp Zp R n Zp (1 + ξt)x du0 (x) =R (1 + ξt)n+x du0 (x) Zp ! ∞ k X X ξ n tk S1 (k, m)Sk (n − 1) . = k! m=0

(2.3)

k=0

We recall that Cauchy numbers are defined by the generating function to be ∞ X t tn = Cn . log(1 + t) n=0 n!

(2.4)

 ! k k t (Dm,ξ (k) − Dm,ξ ) Ck−m m k! k=0 m=0 ! ∞ k−1 X X tk = S1 (k − 1, m)Sk−1 (n − 1) . k! m=0

(2.5)

By (2.3) and (2.4), we get ∞ X

k X

k=0

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5

From (2.5), we have 1 k =

 ! k (Dm,ξ (k) − Dm,ξ )Ck−m m m=0 k X

k−1 X

(2.6)

S1 (k − 1, m)Sk−1 (n − 1), (k ∈ N, n ∈ N).

m=0

Therefore, by (2.6), we obtain the following theorem. Theorem 2.1. For k, n ∈ N, we have  ! k−1 X k = (Dm,ξ (k) − Dm,ξ )Ck−m S1 (k − 1, m)Sk−1 (n − 1). m m=0 m=0 k X

1 k

(2.7)

Now, we consider symmetric identities for the twisted Daehee polynomials. Let w1 , w2 ∈ N. Then, we easily see that R R w1 log(1+ξt) w2 log(1+ξt) (1 + ξt)w1 x1 +w2 x2 du0 (x1 )du0 (x2 ) Zp Zp ((1+ξt)w1 −1) ((1+ξt)w2 −1) R = w1 w2 log(1+ξt) (1 + ξt)w1 w2 x du0 (x) Zp ((1+ξt)w1 w2 −1) (2.8) log(1 + ξt)((1 + ξt)w1 w2 − 1) = . ((1 + ξt)w1 − 1)((1 + ξt)w2 − 1 We consider the following double p-adic invariant integral on Zp as follows: R R (1 + ξt)w1 x1 +w2 x2 +w1 w2 x dµ0 (x1 )dµ0 (x2 ) Z Z R I= p p (1 + ξt)w1 w2 x dµ0 (x) Zp (2.9) log(1 + ξt)(1 + ξt)w1 w2 x ((1 + ξt)w1 w2 − 1) = . ((1 + ξt)w1 − 1)((1 + ξt)w2 − 1) From (1.2) and (1.3), we have R 1 −1 w1 Zp (1 + ξt)x du0 (x) wX R = (1 + ξt)k (1 + ξt)w1 x du0 (x) Zp k=0

=

∞ X

l X

l=0

m=0

! S1 (l, m)Sm (w1 − 1)

(2.10) ξ l tl l!

From (2.10), we get I=

1 w1

=

1 w1

=

1 w1

Z

! w1 (x1 +w2 x)

(1 + ξt)

dµ0 (x1 )

w1 R

R Zp

(1 + ξt)w2 x2 dµ0 (x2 )

!

(1 + ξt)w1 w2 x dµ0 (x) ! ∞ ! ! ∞ k X X X ti ξ k tk m Di,w1 ,ξ (w2 x) w2 S1 (k, m)Sm (w1 − 1) i! k! i=0 k=0 m=0 !   ∞ n n−i X X X n ξ n−i tn Di,w1 ,ξ (w2 x) w2m S1 (n − i, m)Sm (w1 − 1) . i n! n=0 i=0 m=0 (2.11) Zp

Zp

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On the other hand, by (2.10), we get I=

1 w2

!

Z

w2 (x2 +w1 x)

(1 + ξt)

dµ0 (x2 )

w2 R

Zp

R Zp

Zp

(1 + ξt)w1 x1 dµ0 (x1 )

!

(1 + ξt)w1 w2 x dµ0 (x)

! ∞ k ! X X ti ξ k tk m Di,w2 ,ξ (w1 x) ( w1 S1 (k, m)Sm (w2 − 1)) i! k! i=0 k=0 m=0 !   ∞ n n−i X 1 X X n ξ n−i tn = Di,w2 ,ξ (w1 x) . w1m S1 (n − i, m)Sm (w2 − 1) w2 n=0 i=0 i n! m=0 (2.12)

1 = w2

∞ X

Therefore, by (2.9), (2.11) and (2.12), we obtain the following theorem. Theorem 2.2. For w1 , w2 ∈ N and n ∈ N ∪ {0}, we have n   n−i X 1 X n Di,w1 ,ξ (w2 x) ξ n−i w2m S1 (n − i, m)Sm (w1 − 1) w1 i=0 i m=0   n n−i X 1 X n Di,w2 ,ξ (w1 x) ξ n−i w1m S1 (n − i, m)Sm (w2 − 1). = w2 i=0 i m=0

Remark. By replacing t by ∞ X

1 ξ

(et − 1) in (1.14), we get

∞

Bn (x)λn

n=0

tn X 1 = Dn,λ,ξ (x) n! n=0 n! =

∞ X

n X

n=0

m=0




n 1 t e −1 ξ !

ξ −n S2 (n, m)Dm,λ,ξ (x)

tn . n!

(2.13)

Thus, by (2.13), we have λn Bn (x) =

n X

ξ −n S2 (n, m)Dm,λ,ξ (x), (n ≥ 0).

(2.14)

m=0

  By replacing t by log t + 1ξ in (1.5), we have ∞

∞ X

1 tn X Bn (x) Dn,ξ (x) = n! n=0 n! n=0 =

=

∞ X

Bn (x)

n=0 ∞ X

ξn

n=0



∞ X

l=n n X



1 log t + ξ

S1 (l, n)

n

(ξt)n n!

(2.15) !

Bm (x)S1 (n, m)

m=0

tn . n!

Thus, by (2.15), we get ξ −n Dn,ξ (x) =

n X

Bm (x)S1 (n, m), (n ≥ 0).

(2.16)

m=0

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From (2.9), we note that R ! ! Z w1 Zp (1 + ξt)w2 x2 dµ0 (x2 ) (1 + ξt)w1 w2 x w 1 x1 R I= (1 + ξt) dµ0 (x1 ) w1 (1 + ξt)w1 w2 x dµ0 (x) Zp Zp ! w −1 ! Z 1 X (1 + ξt)w1 w2 x w1 x w2 l = (1 + ξt) dµ0 (x1 ) (1 + ξt) w1 Zp l=0 ! w −1 ! Z 1 X w2 1 w (w x+ l = (1 + ξt)w1 x dµ0 (x1 ) (1 + ξt) 1 2 w1 w1 Zp l=0 wX 1 −1 Z w 2 1 = (1 + ξt)w1 (x1 +w2 x+ w1 l) dµ0 (x1 ) w1 Zp l=0 ! w ∞ 1 −1 X 1 X w2 ξ n tn Dn,w1 ,ξ (w2 x + l) . = w1 w1 n! n=0 l=0 (2.17) On the other hand, we obtain the following equation by the symmetric property of p-adic invariant integral on Zp as follows: R ! ! Z w2 Zp (1 + ξt)w1 x1 dµ0 (x1 ) (1 + ξt)w1 w2 x w 2 x2 R I= (1 + ξt) dµ0 (x2 ) w2 (1 + ξt)w1 w2 x dµ0 (x) Zp Zp ! ! Z wX 2 −1 1 w2 x2 = (1 + ξt) dµ0 (x2 ) (1 + ξt)w1 l (1 + ξt)w1 w2 x w2 Zp l=0 ! w −1 ! Z 2 X w 1 w2 (w1 x+ w1 l) w2 x2 2 (1 + ξt) = (1 + ξt) dµ0 (x2 ) w2 Zp l=0 w2 −1 Z w1 1 X (1 + ξt)w2 (x2 +w1 x+ w2 l) dµ0 (x2 ) = w2 Zp l=0 ! w2 −1 ∞ X 1 X w1 ξ n tn . = Dn,w2 ,ξ (w1 x + l) w2 w2 n! n=0 l=0 (2.18) Therefore, by comparing the coefficients on the both sides of (2.17) and (2.18), we obtain the following theorem. Theorem 2.3. For w1 , w2 ∈ N and n ≥ 0, we have     w1 −1 w2 −1 1 X 1 X w2 w1 Dn,w1 ,ξ w2 x + l = Dn,w2 ,ξ w1 x + l . w1 w1 w2 w2 l=0

l=0

Corollary 2.4. For w1 , w2 ∈ N and n ≥ 0, we have   wX n 1 −1 X w2 m−1 w1 Bm w2 x + l S1 (n, m) w1 l=0 m=0   wX n 2 −1 X w1 m−1 = w2 Bm w1 x + l S1 (n, m). w2 m=0 l=0

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JONGKYUM KWON1 AND JIN-WOO PARK2,∗

8

References [1] Y.-H. Cho, T. Kim, T. Mansour, S.-H. Rim, Higher-order q-Daehee polynomials, J. Comput. Anal. Appl. 19 (2015), no. 1, 167-173. [2] Y.-H. Cho, T. Kim, T. Mansour, S.-H. Rim, On a (r,s)-analogue of Changhee and Daehee numbers and polynomials, Kyungpook Math. J. 55 (2015), 225-232. [3] L. Comtet, Advanced Combinatorics, Reidel, Dordrecht, 1974. [4] D. S. Kim, T. Kim, Daehee numbers and polynomials, Appl. Math. Sci. 7(120),5969 5976(2013). [5] D. S. Kim, T. Kim, S. -H. Lee, J.-J. Seo, A note on the twisted λ- Daehee polynomials, Appl. Math. Sci. 7(2013). [6] D. S. Kim, T. Kim, Identites arsing from higher-order Daehee polynomial bases, Open Math.13 2015, 196-208. [7] D. S. Kim, T. Kim, S.-H. Lee, J.-J. Seo, A note on the λ-Daehee polynomials, Int. J. Math. Anal.(Ruse) 7 (2013), no. 61-64, 3069-3080. [8] D. S. Kim, T. Kim, J.-J. Seo, Higher-order Daehee polynomials of the first kind with umbral calculus, Adv. Stud. Contemp. Math.(Kyungshang) 24 (2014), no. 1, 5-18. [9] T. Kim, An invariant p-adic integral associated with Daehee numbers, Integral Transforms Spec. Funct. 13(1),65-69(2002). [10] E.-J. Moon, J.-W. Park, S.-H. Rim, A note on the generalized q-Daehee numbers of higher order, Proc. Jangjeon Math. Soc. 17 (2014), no. 4, 557-565. [11] H. Ozden, I.N. Cangul, Y. Simsek, Remarks on the q− Bernoulli numbers associated with Daehee numbers, Adv. Stud. Contemp. Math.(Kyungshang) 18 (2009), no. 1, 41-48. [12] J.-W. Park, On the q-analogue of λ-Daehee polynomials, J. Comput. Anal. Appl. 19 (2015), no. 6, 966-974. [13] J.-W. Park, S.-H. Rim, J. Kwon, The twisted Daehee numbers and polynomials, Adv. Difference Equ. 2014 (2014:1), 9pp. [14] J.-W. Park, On the twisted Daehee polynomials with q-parameter, Adv. Difference Equ. 2014 (2014:304), 10pp. [15] S. Roman, The umbral calculus, Dover Publ. Inc. New York, 2005. [16] J.-J.Seo, T. Kim, Some identites of symmetry for Daehee polynomials arising from p-adic invariant integral on Zp , Proc. Jangjeon Math. Soc. 19 (2016), no. 2, 285 - 292. [17] J.-J.Seo, S.-H. Rim, T. Kim, S.H. Lee, Sums products of generalized Daehee numbers, Proc. Jangjeon Math. Soc. 17 (2014), no. 1, 1-9. 1 Department of Mathematics Education and RINS, Gyeongsang National University, Jinju, Gyeongsangnamdo, 52828, Republic of Korea E-mail address: [email protected] 2,∗

Department of Mathematics Education, Daegu University, Gyeongsan-si, Kyungsangbukdo, 38453, Republic of Korea E-mail address: [email protected]

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Dynamics and Behavior of

+1 =  + ¡1 +

 + ¡2  + ¡2

M. M. El-Dessoky12 and Aatef Hobiny13 1 King Abdulaziz University, Faculty of Science, Mathematics Department, P. O. Box 80203, Jeddah 21589, Saudi Arabia. 2 Mansoura University, Faculty of Science, Department of Mathematics, Mansoura 35516, Egypt. 3 Nonlinear Analysis and Applied Mathematics (NAAM)-Research Group, Department of Mathematics, Faculty of Science, King Abdulaziz University, P.O. Box 80203, Jeddah 21589, Saudi Arabia. E-mail: [email protected]; [email protected] ABSTRACT The main objective of this paper is to study the local and the global stability of the solutions, the periodic character and the boundedness of the di¤erence equation +1 =  + ¡1 +

 + ¡2   + ¡2

where the parameters      and  are positive real numbers and the initial conditions ¡2  ¡1 and 0 are positive real numbers. Some numerical examples will be given to illustrate our results. Keywords: Di¤erence equations, Stability, Boundedness, Periodic solutions. Mathematics Subject Classi…cation: 39A10 —————————————————

1. INTRODUCTION Di¤erence equations or discrete dynamical systems are diverse …eld which impact almost every branch of pure and applied mathematics. Every dynamical system +1 =  ( ) determines a di¤erence equation and vice versa. Recently, there has been great interest in studying di¤erence equations systems. One of the reasons for this is a necessity for some techniques which can be used in investigating equations arising in mathematical models describing real life situations in many applied sciences. The theory of discrete dynamical systems and di¤erence equations developed greatly during the last twenty-…ve years of the twentieth century. Applications of discrete dynamical systems and di¤erence equations have appeared recently in many areas. The theory of di¤erence equations occupies a central position in applicable analysis. There is no doubt that the theory of di¤erence equations will continue to play an important role in mathematics as a whole. Nonlinear di¤erence equations of order greater than one are of paramount importance in applications. Such equations also appear naturally as discrete analogues and as numerical solutions of di¤erential and delay di¤erential equations which model various diverse phenomena in biology, physiology, ecology, engineering, physics, economics, genetics, probability theory, psychology and resource management. It is very interesting to investigate the behavior of solutions of a system of higher-order rational di¤erence equations and to discuss the local asymptotic stability of their equilibrium points. Systems of rational di¤erence equations have been studied by several authors. Especially there has been a great interest in the study of the attractivity of the solutions of such systems [1-33]. Many research have been done to study the global attractivity, boundedness character, periodicity and the solution form of nonlinear di¤erence equations. For example, Agarwal et al. [2] looked at the global stability, periodicity character and found the solution form of some special cases of the di¤erence equation +1 =  +

¡ ¡  ¡¡

644

 = 0 1 

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where     and the initial conditions ¡  ¡+1   ¡1  0 are positive real numbers, while  6=  for  = ¡ ¡  + 1  0 where  = maxf  g. Hamza and Morsy in [3] investigated the global behavior of the di¤erence equation +1 =  +

¡1   

 = 0 1 

where the parameters   2 (0 1) and the initial values ¡1 and 0 are arbitrary positive real numbers.

Elsayed et al. [4] studied the global stability character and the periodicity of solutions of the di¤erence equation +¡1 +1 =  + +   = 0 1  ¡1 where the parameters     and  are positive real numbers and the initial conditions ¡1 and 0 are positive real numbers. Zayed et al. [5] studied the behavior of the following rational recursive sequence 

+1 =

  +



 X =0

 X

  ¡ 



 = 0 1 2 

  ¡

=0

where the coe¢cients ,  ,   and the initial conditions ¡  ¡+1   ¡1  0 are positive real numbers, while  is a positive integer number. Also, in [6] Zayed et al. obtained the global behavior of the di¤erence equation  X

 ¡

+1 =

=0 

X

+



 = 0 1 2 

  ¡

=0

where the coe¢cients ,  ,   and the initial conditions ¡  ¡+1   ¡1  0 are arbitrary positive real numbers, while  is a positive integer number. In [7] El-Moneam investigated the periodicity, the boundedness and the global stability of the positive solutions of nonlinear di¤erence equation +1 =  + ¡ + ¡ + ¡ +

¡ ¡ ¡¡ 

 = 0 1 2 

where the coe¢cients        2 (0 1), while   and  are positive integers and the initial conditions ¡   ¡   ¡   ¡1  0 are arbitrary positive real numbers such that     . Yalç¬nkaya [8] investigated the global behaviour of the di¤erence equation +1 =  +

¡   

 = 0 1 

where the parametere   2 (0 1) and the initial values are arbitrary positive real numbers.

Elabbasy et al. [9] studied the dynamics, the global stability, periodicity character and the solution of special case of the recursive sequence  +1 =  ¡  ¡   = 0 1  ¡1 where the initial conditions ¡1 , 0 are arbitrary real numbers and     are positive constants.

El-Owaidy et al. [10] investigated local stability, oscillation and boundedness character of the di¤erence equation  +1 =  + ¡1  = 0 1    

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under speci…ed conditions. Elsayed [11] studied some qualitative behavior of the solutions of the di¤erence equation +1 =  +

   ¡¡1

 = 0 1 

where the initial conditions ¡1 , 0 are arbitrary real numbers and     are positive constants with 0 ¡ ¡1 6= 0.

Elsayed and El-Dessoky [12] investigated the global convergence, boundedness, and periodicity of solutions of the di¤erence equation ¡ +¡ +1 = ¡ +    = 0 1  ¡ +¡

where the parameters     and  are positive real numbers and the initial conditions ¡  ¡+1   ¡1 , 0 are positive real numbers where  = f  g. This paper aims to study the global stability character and the periodicity of solutions of the di¤erence equation  + ¡2 +1 =  + ¡1 +   = 0 1  (1)  + ¡2 where the parameters      and  are positive real numbers and the initial conditions ¡2  ¡1 and 0 are positive real numbers.

2. SOME BASIC PROPERTIES AND DEFINITIONS In this section, we state some basic de…nitions and theorems that we need in this paper. Let  be some interval of real numbers and let  :  3 !  be a continuously di¤erentiable function. Then for every set of initial conditions ¡2  ¡1  0 2  the di¤erence equation +1 =  (  ¡1   ¡2 ) (2) has a unique solution f g1 =¡2 .

De…nition 1. (Equilibrium Point) A point  2  is called an equilibrium point of Eq.(2) if  =  (   ).

That is,  =  for  ¸ 0 is a solution of Eq.(2), or equivalently,  is a …xed point of 

Definition 2.1. (Stability) (i) The equilibrium point  of Eq.(2) is locally stable if for every   0 there exists   0 such that for all ¡2  ¡1  0 2  with j¡2 ¡ j + j¡1 ¡ j + j0 ¡ j   we have

j ¡ j  

for all  ¸ ¡

(ii) The equilibrium point  of Eq.(2) is locally asymptotically stable if  is locally stable solution of Eq.(2) and there exists   0 such that for all ¡2  ¡1  0 2  with j¡2 ¡ j + j¡1 ¡ j + j0 ¡ j   we have lim  = 

!1

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(iii) The equilibrium point  of Eq.(2) is global attractor if for all ¡2  ¡1  0 2  we have lim  = 

!1

(iv) The equilibrium point  of Eq.(2) is globally asymptotically stable if  is locally stable, and  is also a global attractor of Eq.(2). (v) The equilibrium point  of Eq.(2) is unstable if  is not locally stable. Definition 2.2. (Boundedness) A sequence f g1 =¡2 is said to be bounded and persists if there exist posiyive constants  and  such that  ·  · 

for all  ¸ ¡2

Definition 2.3. (Periodicity) 1 A sequence f g1 =¡2 is said to be periodic with period  if + =  for all  ¸ ¡1 A sequence f g=¡2 is said to be periodic with prime period  if  is the smallest positive integer having this property. The linearized equation of Eq.(2) about the equilibrium  is the linear di¤erence equation +1 =

 (   )  (   )  (   )  + ¡1 + ¡2   ¡1 ¡2

(3)

Now, assume that the characteristic equation associated with (3) is () = 0 2 + 1  + 2 = 0 where 0 =

(4)

 (   )  (   )  (   )  1 = and 2 =   ¡1 ¡2

Theorem A [18]: Assume that  2   = 1 2 3. Then j1 j + j2 j + j3 j  1 is a su¢cient condition for the asymptotic stability of the di¤erence equation +3 + 1 +2 + 2 +1 + 3  = 0 Theorem B [19]: Let  : [ ]3 ! [ ] be a continuous function, where 3 is a positive integer, and [ ] is an interval of real numbers and consider the di¤erence equation +1 = (  ¡1  ¡2 )

(5)

Suppose that  satis…es the following conditions: (i) For every integer  with 1 ·  · 3, the function (1  2  3 ) is weakly monotonic in  , for …xed 1  2  3 .

(ii) If   is a solution of the system

 = (1  2  3 ) and  = (1  2  3 ) then  =  , where for each  = 1 2 3, we set ½  if  is non-decreasing in   =  if  is non-increasing in 

and

 =

½

 

if  is non-decreasing in   if  is non-increasing in 

Then, there exists exactly one equilibrium point  of the di¤erence equation (5), and every solution of (5) converges to .

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3. LOCAL STABILITY OF THE EQUILIBRIUM POINT OF EQ.(1) In his section, we study the local stability character of the equilibrium point of Eq.(1). Eq.(1) has equilibrium point and is given by  =  +  + or

 +    + 

(1 ¡  ¡ )2 + ( ¡  ¡  ¡ ) ¡  = 0

Then if  +   1 the only positive equilibrium point of Eq.(1) is given by p ( +  +  ¡ ) + ( +  +  ¡ )2 + 4(1 ¡  ¡ )  = 2(1 ¡  ¡ ) Theorem 3.1. The equilibrium  of Eq. (1) is locally asymptotically stable if and only if ( + )2 

j ¡ j  (1 ¡  ¡ )

(6)

Proof: Let  : (0 1)3 ¡! (0 1) be a continuous function de…ned by  (  ) =  +  + Therefore,

 +    + 

(7)

 (  )  (  ) ( ¡ )  (  ) =  =  = .    ( + )2

So, we can write  (  ) (  )  (  ) ( ¡ ) = 3  =  = 1  =  = 2  =    ( + )2 Then the linearized equation of Eq.(1) about  is +1 ¡ 1 ¡1 ¡ 2  ¡ 3 ¡2 = 0

(8)

It follows by Theorem A that, Eq.(1) is asymptotically stable if and only if j1 j + j2 j + j3 j  1 Thus,

and so

¯ ¯ ¯ ( ¡ ) ¯ ¯ + jj  1 jj + ¯¯ ( + )2 ¯

¯ ¯ ¯ ( ¡ ) ¯ ¯ ¯ ¯ ( + )2 ¯  1 ¡  ¡ 

j ¡ j  ( + )2 (1 ¡  ¡ )

or

The proof is complete.

j ¡ j  ( + )2  (1 ¡  ¡ )

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Example 1. The solution of the di¤erence equation (1) is global stability if  = 055  = 03  = 08  = 05  = 7 and  = 2 and the initial conditions ¡2 = 4 ¡1 = 9 and 0 = 03 (See Fig. 1). x(n+1)=ax(n)+bx(n-1)+(alfa+cx(n-2))/(beta+dx(n-2)) 14

12

10

x(n)

8

6

4

2

0

0

10

20

30

40 n

50

60

70

80

Figure 1. Plot the behavior of the solution of equation (1).

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  +   1 Proof: Let f g1 =¡2 be a solution of Eq.(1). It follows from Eq.(1) that +1 =

+1 =  + ¡1 +

 + ¡2  ¡2 =  + ¡1 + +   + ¡2  + ¡2  + ¡2

+1   + ¡1 +

 ¡2   + =  + ¡1 + +  ¡2  

Then

for all

 ¸ 0.

By using a comparison, the right hand side can be written as follows +1 =  + ¡1 +

  +   

and this equation is locally asymptotically stable if  +   1 and converges to the equilibrium point  =  +  . (1 ¡  ¡ ) Therefore  +  lim sup · . (1 ¡  ¡ ) !1 Hence, the solution is bounded.

Theorem 4.2. Every solution of Eq.(1) is unbounded if   1 or   1 Proof: Let f g1 =¡2 be a solution of Eq.(1). Then from Eq.(1) we see that +1 =  + ¡1 +

 + ¡2    + ¡2

for all

 ¸ 0.

The right hand side can be written as follows +1 = 

)  =  0 

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and this equation is unbounded because   1 and lim  = 1. Then by using ratio test f g1 =¡2 is !1 unbounded from above. Similarly from Equation (1) we see that +1 =  + ¡1 +

 + ¡2  ¡1  + ¡2

for all

 ¸ 0.

We see that the right hand side can be written as follows ) 2¡1 =  ¡1 and 2 =  0 

+1 = ¡1

and this equation is unbounded because   1 and lim 2¡1 = lim 2 = 1. Then by using ratio test !1

f g1 =¡2 is unbounded from above.

!1

Example 2. Figure (2) shows that behavior of the solution of the di¤erence equation (1) is boundedness if we take  = 03  = 01  = 08  = 05  = 7 and  = 2 and the initial conditions ¡2 = 4 ¡1 = 9 and 0 = 03 x(n+1)=ax(n)+bx(n-1)+(alfa+cx(n-2))/(beta+dx(n-2)) 9 8 7 6

x(n)

5 4 3 2 1 0

0

5

10

15

20

25 n

30

35

40

45

50

Figure 2. Show the boundedness of the solution of equation (1). Example 3. Figure (3) shows the behavior of the solution of the di¤erence equation (1) is undounded when we put  = 15  = 08  = 2  = 3  = 6 and  = 5 and the initial conditions ¡2 = 04 ¡1 = 09 and 0 = 03 3

x 10

13

x(n+1)=ax(n)+bx(n-1)+(alfa+cx(n-2))/(beta+dx(n-2))

2.5

x(n)

2

1.5

1

0.5

0

0

5

10

15

20

25 n

30

35

40

45

50

Figure 3. Show the unboundedness of the solution of equation (1).

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5. GLOBAL ATTRACTIVITY OF THE EQUILIBRIUM POINT OF EQ.(1) In this section, the global asymptotic stability of Eq.(1) is studied. Theorem 5.1. The equilibrium point  is a global attractor of Eq.(1) if  +   1. Proof: Suppose that  and  are real numbers and assume that  : [ ]3 ¡! [ ] is a function de…ned by (  ) =  +  + Then

 +    + 

(  ) (  ) ( ¡ ) (  ) . =  =  =    ( + )2

Now, two cases must be considered : Case (1): Let  ¡   0, then we can easily see that the function (  ) increasing in   and decreasing in  Let (  ) be a solution of the system  = (  ) and  = (   ). Then from Eq.(1), we see that  +   +    =  +  +   =  +  +  +   +  or  +   +   (1 ¡  ¡ ) =  (1 ¡  ¡ ) =   +   +  then (1 ¡  ¡ ) + (1 ¡  ¡ )  =  +  (1 ¡  ¡ ) + (1 ¡  ¡ ) =  +  Subtracting we obtain

( ¡ )f(1 ¡  ¡ ) + g = 0

under the condition  +   1 we see that

 =  It follows by Theorem B that  is a global attractor of Eq.(1). This completes the proof of the theorem. Case (2): Assume that  ¡   0 is true, then we can easily see that the function (  ) increasing in    and decreasing in  Let (  ) be a solution of the system  = (   ) and  = (  ). Then from Eq.(1), we see that  +   +   =  +  +   =  +  +   +   +  or  +   +   (1 ¡  ¡ ) =  (1 ¡  ¡ ) =   +   +  then (1 ¡  ¡ ) + (1 ¡  ¡ ) 2 (1 ¡  ¡ ) + (1 ¡  ¡ )2

=  +  =  + 

Subtracting we obtain (1 ¡  ¡ )( ¡ ) + (1 ¡  ¡ )( 2 ¡ 2 ) = ( ¡ ) ( ¡ )f(1 ¡  ¡ )( + ) + (1 ¡  ¡ ) ¡ g = 0 under the condition  +   1 and (1 ¡  ¡ )   we see that  =  It follows by Theorem B that  is a global attractor of Eq.(1).

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6. EXISTENCE OF PERIODIC SOLUTIONS In this section we investigate the existence of periodic solutions of Eq.(1). The following theorem states the necessary and su¢cient conditions that this equation has periodic solutions of prime period two. Theorem 6.1. Eq.(1) has positive prime period two solutions if and only if () [( ¡ 1) ¡ ( + )]2 ( ¡  ¡ 1) + 4f(1 ¡ )[( ¡ 1) ¡ ( + )] + g  0 Proof: Firstly, suppose that there exists a prime period two solution       of Eq.(1). We will show that Condition (i) holds. From Eq.(1), we get  =  +  +

 +    + 

 =  +  +

 +    + 

and Therefore,

 +  =  +  2 +  +  +  + 

(9)

 +  =  + 2 +  +  +  + 

(10)

and Subtracting (10) from (9) gives ( ¡ ) + (2 ¡  2 ) = ( ¡ ) ¡ ( ¡ ) ¡ ( ¡ ). Since  6=  it follows that + =

( ¡  ¡ 1) ¡   

(11)

Again adding (9) and (10) yields ( + ) + 2 = ( + )( + ) + (2 +  2 ) + 2 + 2 + ( + ) (2 +  2 ) = ( + ) + 2 ¡ ( + )( + ) ¡ 2 ¡ 2 ¡ ( + ) (2 +  2 ) = ( ¡  ¡  ¡ )( + ) + 2 ¡ 2 ¡ 2

(12)

By using (11) (12) and the relation

2 +  2 = ( + )2 ¡ 2 for all   2  we obtain (( + )2 ¡ 2) = ( ¡  ¡  ¡ )( + ) + 2 ¡ 2 ¡ 2

(¡¡¡)2 

¡ 2

Then,  = ¡

=

(¡¡¡)(¡¡¡) 

+ 2 ¡ 2 ¡ 2

( ¡ 1)[( ¡ 1) ¡ ( + )] +   2 ( ¡  ¡ 1)

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Now it is obvious from Eq.(11) and Eq.(13) that  and  are the two distinct roots of the quadratic equation ( ¡ 1)[( ¡ 1) ¡ ( + )] +  ( ¡  ¡ 1) ¡  ¡ = 0  2 ( ¡  ¡ 1) ( ¡ 1)[( ¡ 1) ¡ ( + )] +   2 ¡ ( ¡  ¡  ¡ ) ¡ = 0 ( ¡  ¡ 1) 2 ¡

and so 2

[( ¡ 1) ¡ ( + )] + or

(14)

4f( ¡ 1)[( ¡ 1) ¡ ( + )] + g  0 ( ¡  ¡ 1)

[( ¡ 1) ¡ ( + )]2 ( ¡  ¡ 1) + 4f( ¡ 1)[( ¡ 1) ¡ ( + )] + g  0

for  + 1   then the inequalities (i) holds.

Conversely, suppose that inequality (i) is true. We will prove that Eq.(1) has a prime period two solution. Suppose that =

+  2

=

¡  2

and

where  =

q

2 +

4[(¡1)+] and (¡¡1)

 = ( ¡ 1) ¡ ( + )

We see from the inequality (i) that 2 ( ¡  ¡ 1) + 4 [( ¡ 1) + ]  0 which equivalents to 2 + Therefore  and  are distinct real numbers.

4 [( ¡ 1) + ]  0 ( ¡  ¡ 1)

Set ¡2 =  ¡1 =  and 0 =  We would like to show that 1 = ¡2 = 

and 2 = ¡1 = 

It follows from Eq.(1) that 1 =  +  +

+ +

=

µ

+ 2

¶

+

µ

¡ 2

¶

+ +  +

³

³

Dividing the denominator and numerator by 2 we get µ ¶ µ ¶ + ¡ 2 +  ( + ) 1 =  + +  2 2 2 +  ( + )

+ 2 + 2

´

´

Multiplying the denominator and numerator of the right side by 2 +  ( ¡ ) and by computation we obtain 1 =  Similarly as before, it is easy to show that 2 = 

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Then by induction we get 2 = 

and

2+1 = 

for all

Thus Eq.(1) has the prime period two solution

 ¸ ¡2.

 where  and  are the distinct roots of the quadratic equation (14) and the proof is complete. Example 4. Figure (4) shows the period two solution of equation (1) when  = 02  = 5  = 04  = 5  = 07 and  = 02 and the initial conditions ¡2 =  ¡1 =  and 0 =  since  and  as in the previous theorem. x(n+1)=ax(n)+bx(n-1)+(alfa+cx(n-2))/(beta+dx(n-2)) 0.5

0.4

0.3

x(n)

0.2

0.1

0

-0.1

-0.2

0

5

10

15 n

20

25

30

Figure 4. Plot the periodicity of the solution of equation (1).

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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J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 24, NO.4, 2018, COPYRIGHT 2018 EUDOXUS PRESS, LLC

Algebraic and Order Properties of Tracy-Singh Products for Operator Matrices Arnon Ploymukda1 , Pattrawut Chansangiam1∗, Wicharn Lewkeeratiyutkul2 1 Department of Mathematics, Faculty of Science, King Mongkut’s Institute of Technology Ladkrabang, Bangkok 10520, Thailand. 2 Department of Mathematics and Computer Science, Faculty of Science, Chulalongkorn University, Bangkok 10330, Thailand.

Abstract We generalize the tensor product for operators to the Tracy-Singh product for operator matrices acting on the direct sum of Hilbert spaces. This kind of operator product is compatible with algebraic operations and order relations for operators. It follows that this product preserves many structure properties of operators.

Keywords: tensor product, Tracy-Singh product, operator matrix, MoorePenrose inverse Mathematics Subject Classifications 2010: 15A69, 47A05, 47A80.

1

Introduction

In scientific computing, we consider a matrix to be a two-dimensional array for stacking data. A processing of such data can be performed using matrix products. One of extremely useful matrix products is the Kronecker product. For any complex matrices A ∈ Mm,n (C) and B ∈ Mp,q (C), the Kronecker product of A and B is given by the block matrix ˆ B = [aij B]ij ∈ Mmp,nq (C). A⊗ ˆ B is the unique complex matrix of order mp × nq satisfying Equivalently, A ⊗ ˆ B)(x ⊗ ˆ y) = Ax ⊗ ˆ By (A ⊗

(1)

for all x ∈ Cn and y ∈ Cq . This matrix product has wide applications in mathematics, computer science, statistics, physics, system theory, signal processing, and related fields. See [2, 5, 6, 12] for more information. ∗ Corresponding

author. Email: [email protected]

1

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Tracy-Singh Products for Operator Matrices

Kronecker product was generalized to the Tracy-Singh product of partitioned matrices by Tracy and Singh [10]. Let A = [Aij ] ∈ Mm,n (C) be a ∑ partitioned matrix with A of order m × n as the (i, j)th submatrix where ij i j i mi = m ∑ and j nj = n. Let B = [Bkl ] ∈ Mp,q (C) be a∑partitioned matrix with Bkl of ∑ order pk × ql as the (k, l)th submatrix where k pk = p and l ql = q. The Tracy-Singh product of A and B is defined by ˆB = A

[[ ] ] ˆ Bkl Aij ⊗ ∈ Mmp,nq (C), kl ij

ˆ Bkl is of order mi pk × nj ql . This kind of matrix prodwhere each block Aij ⊗ uct has several attractive properties in algebraic, order, and analytic points of views; see, e.g., [3, 8, 9, 10]. The Tracy-Singh product can be applied widely in statistics, econometrics and related fields; see, e.g., [9, 10]. As a natural generalization of a complex matrix, we consider a bounded linear operator between complex Hilbert spaces. The tensor product of Hilbert space operators can be viewed as an extension of the Kronecker product of complex matrices. Using the universal mapping property in the monoidal category of Hilbert spaces, the tensor product of A ∈ B(H, H′ ) and B ∈ B(K, K′ ) is the unique bounded linear operator from H ⊗ K into H′ ⊗ K′ such that for all x ∈ H and y ∈ K, (A ⊗ B)(x ⊗ y) = Ax ⊗ By.

(2)

A fundamental property of tensor product is the mixed product property: (A ⊗ B)(C ⊗ D) = AC ⊗ BD.

(3)

The theory of tensor product of operators has been continuously developed in the literature; see, e.g., [4, 11]. From the previous discussion, it is natural to extend the notion of tensor product for operators to the “Tracy-Singh product”of operators. We shall propose a natural definition of such operator product. It turns out that this product is compatible with algebraic operations and order relations for operators. One of the most attractive properties, the mixed product property, also holds for Tracy-Singh products. It follows that this product preserves attractive properties of operators, such as being invertible, Hermitian, unitary, positive, and normal. Our results generalize the results known so far in the literature for both Tracy-Singh products of matrices and tensor products of operators. This paper is organized as follows. In section 2, we introduce the TracySingh product for operator matrices and deduce its algebraic properties. In section 3, we show that the Tracy-Singh product is compatible with various kinds of operator inverses. We investigate the relationship between Tracy-Singh products and operator orderings in Section 4.

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A. Ploymukda, P. Chansangiam, W. Lewkeeratiyutkul

2

Tracy-Singh products and algebraic operations for operators

In this section, we introduce the Tracy-Singh product of operators on a Hilbert space. Then we will show that this product is compatible with addition, scalar multiplication, adjoint operation, usual multiplication, power, and direct sum of operator inverses. Throughout this paper, let H, H′ , K and K′ be complex Hilbert spaces. When X and Y are Hilbert spaces, denote by B(X , Y) the Banach space of bounded linear operators from X into Y, and abbreviate B(X , X ) to B(X ). The projection theorem for Hilbert spaces allows us to decompose H =

n ⊕

Hj ,

′

H =

j=1

m ⊕

Hi′ ,

K =

q ⊕

i=1

Kl ,

′

K =

l=1

p ⊕

Kk′

k=1

Hj , Hi′ , Kl , Kk′

where each are Hilbert spaces. Such decompositions are fixed throughout the paper. For each j = 1, . . . , n, let Ej be the canonical embedding from Hj into H, defined by xj 7→ (0, . . . , 0, xj , 0, . . . , 0). Similarly, let Fl be the canonical embedding from Kl into K for each l = 1, . . . , q. For each i = 1, . . . , m and k = 1, . . . , p, let Pi′ : H′ → Hi′ and Q′k : K′ → Kk′ be the orthogonal projections. Thus, each operator A ∈ B(H, H′ ) and B ∈ B(K, K′ ) can be expressed uniquely as operator matrices m,n

A = [Aij ]i,j=1

p,q

and B = [Bkl ]k,l=1

where Aij = Pi′ AEj and Bkl = Q′k BFl for each i, j, k, l. p,q ′ ′ Definition 1. Let A = [Aij ]m,n i,j=1 ∈ B(H, H ) and B = [Bkl ]k,l=1 ∈ B(K, K ) be operator matrices defined as above. We define the Tracy-Singh product of A and B to be the operator matrix [ ] A  B = [Aij ⊗ Bkl ]kl ij (4)

which is a bounded linear operator from

q n ⊕ ⊕ j=1 l=1

Hj ⊗ Kl to

p m ⊕ ⊕

Hi′ ⊗ Kk′ .

i=1 k=1

Note that if both A and B are 1 × 1 block operator matrices i.e. m = n = p = q = 1, then their Tracy-Singh product A  B is just the tensor product A ⊗ B. Next, we shall show that the Tracy-Singh product of two linear maps induced by two matrices is just the linear map induced by the Tracy-Singh product of these matrices. Recall that for each A ∈ Mm,n (C) and B ∈ Mp,q (C), the induced maps LA : Cn → Cm , x 7→ Ax and LB : Cq → Cp , y 7→ By

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Tracy-Singh Products for Operator Matrices

are bounded linear operators. Using the universal mapping property, we identify Cn ⊗ Cq with Cnq ∼ = Mn,q (C) together with the canonical bilinear map (x, y) 7→ ˆ y for each (x, y) ∈ Cn × Cq . It is similar for Cm ⊗ Cp . x⊗ Lemma 2. For each A ∈ Mm,n (C) and B ∈ Mp,q (C), we have LA ⊗ LB = LA ⊗ ˆ B.

(5)

Proof. For any x ⊗ y ∈ Cn ⊗ Cq , we obtain from the mixed product property of the Kronecker product (1) that ˆ LB (y) (LA ⊗ LB )(x ⊗ y) = LA (x) ⊗ LB (y) = LA (x) ⊗ ˆ By = (A ⊗ ˆ B)(x ⊗ ˆ y) = Ax ⊗ ˆ B)(x ⊗ y) = LA ⊗ = (A ⊗ ˆ B (x ⊗ y). Thus, by the uniqueness of tensor product, LA ⊗ LB = LA ⊗ ˆ B. Proposition 3. For any complex matrices A = [Aij ] and B = [Bkl ] partitioned in block-matrix forms, we have LA  LB = LA  ˆ B.

(6)

Proof. Recall that the (i, j)th block of the matrix representation of LA is the matrix Aij . It follows from Lemma 2 that [[ ] ] [[ ] ] LA  LB = LAij ⊗ LBkl kl ij = LAij ⊗ = LA  ˆ B. ˆ Bkl kl ij

The next proposition shows that the Tracy-Singh product is compatible with the addition, the scalar multiplication and the adjoint operation of operators. Proposition 4. Let A ∈ B(H, H′ ) and B, C ∈ B(K, K′ ) be operator matrices, and let α ∈ C. Then (αA)  B = α(A  B) = A  (αB), ∗

∗

∗

(A  B) = A  B , A  (B + C) = A  B + A  C, (B + C)  A = B  A + C  A.

(7) (8) (9) (10)

Proof. Since each (i, j)th block of αA is given by (αA)ij = αAij , we get [ ] [ ] (αA)  B = [(αAij ) ⊗ Bkl ]kl ij = [α(Aij ⊗ Bkl )]kl ij = α(A  B). ∗ Similarly, A  (αB) = α(A  B). Since A∗ = [A∗ji ]ij and B ∗ = [Blk ]kl for all i, j, k, l, we obtain [[ [ ] ] ∗ ] ∗ (A  B)∗ = [Aji ⊗ Bkl ]kl ij = A∗ji ⊗ Blk = A∗  B ∗ . kl ij

The proofs of (9) and (10) are done by using the fact that (B + C)kl = Bkl + Ckl for all k, l together with the left/right distributivity of the tensor product over the addition.

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A. Ploymukda, P. Chansangiam, W. Lewkeeratiyutkul

Properties (7), (9) and (10) say that the map (A, B) 7→ A  B is bilinear. Proposition 5. Let A = [Aij ] ∈ B(H, H′ ) and let B ∈ B(K, K′ ) be operator matrices. Then   A11  B · · · A1n  B   .. .. .. A  B = [Aij  B]ij =  . . . . Am1  B · · · Amn  B That is, the (i, j)th block of A  B is just Aij  B, regardless of how to partition B. Proof. It follows directly from the definition of the Tracy-Singh product. Remark 6. It is not true in general that the (k, l)th block of A  B is A  Bkl . When H = H1 ⊕ H2 and K = K1 ⊕ K2 , the direct sum of A1 ∈ B(H1 , K1 ) and A2 ∈ B(H2 , K2 ) is defined to be the operator [ ] A1 0 A1 ⊕ A2 = ∈ B(H, K). 0 A2 The next result gives a relation between the direct sum and the Tracy-Singh product. Proposition 7. The Tracy-Singh product is right distributive over the direct sum of operators. That is, for any operator matrices A, B and C, we have (A ⊕ B)  C = (A  C) ⊕ (B  C). Proof. It follows from Proposition 5 that [ ] [ AC 0C AC (A ⊕ B)  C = = 0C BC 0

(11)

0 BC

]

= (A  C) ⊕ (B  C). It is not true in general that the Tracy-Singh product is left distributive over the direct sum of operators. The next theorem shows that the Tracy-Singh product is compatible with the ordinary product of operators. This fundamental property, called the mixed product property, will be used many times in later discussions. Theorem 8. Let H, H′ , H′′ , K, K′ and K′′ be complex Hilbert spaces. Let A = n,r p,q ′ ′′ ′ ′ ′′ [Aij ]m,n i,j=1 ∈ B(H , H ), C = [Cij ]i,j=1 ∈ B(H, H ), B = [Bkl ]k,l=1 ∈ B(K , K ) q,s ′ and D = [Dkl ]k,l=1 ∈ B(K, K ) be operator matrices partitioned so that they are compatible with the decompositions of the corresponding Hilbert spaces. Then (A  B)(C  D) = AC  BD.

660

(12)

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J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 24, NO.4, 2018, COPYRIGHT 2018 EUDOXUS PRESS, LLC

Tracy-Singh Products for Operator Matrices

Proof. Using block multiplication of operators and the mixed product property of the tensor product (3), we have [ ] [ ] (A  B)(C  D) = [Aij ⊗ Bkl ]kl ij [Cij ⊗ Dkl ]kl ij    q n ∑ ∑ =  (Aiα ⊗ Bkβ )(Cαj ⊗ Dβl )   = 

α=1 β=1 n ∑

q ∑

 

=

n ∑

ij

(Aiα Cαj ⊗ Bkβ Dβl ) 

α=1 β=1

[

kl



]



Aiα Cαj

α=1

ij

kl q ∑

β=1



ij

Bkβ Dβl  kl

= AC  BD. Corollary 9. For any operator matrices A ∈ B(H) and B ∈ B(K), we have (A  B)r = Ar  B r

(13)

for any r ∈ N. In the rest of section, we investigate structure properties of operators under taking Tracy-Singh products. Recall that an operator T ∈ B(H) is said to be involutary if T 2 = I, idempotent if T 2 = T , an isometry if T ∗ T = I, a partial isometry if the restriction of T to a closed subspace is an isometry, or equivalently, T T ∗ T = T . Corollary 10. Let A ∈ B(H) and B ∈ B(K). If both A and B satisfy one of the following properties, then the same property holds for AB: Hermitian, unitary, isometry, co-isometry, partial isometry, idempotent, involutary, projection. Proof. Applying Theorem 8 and Proposition 4, we get the results. If A and B are skew-Hermitian operators, then A  B is Hermitian. Recall that an operator T ∈ B(H) is said to be nilpotent if there is a positive integer k such that T k = 0. The smallest such integer k is called the degree of nilpotency of T . If A ∈ B(H) and B ∈ B(K) are nilpotent operators with degrees of nilpotency r and s, respectively, then A  B is also nilpotent with degree of nilpotency not exceed min{r, s}.

3

Tracy-Singh products and operator inverses

Next, we discuss the invertibility of the Tracy-Singh product of operators. Recall that an operator A ∈ B(H, K) is said to be regular if there is an operator A− ∈ B(K, H) such that AA− A = A. The operator A− is called an inner inverse of A. An operator X ∈ B(K, H) is said to be an outer inverse of A if XAX = X.

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A. Ploymukda, P. Chansangiam, W. Lewkeeratiyutkul

Proposition 11. Let A ∈ B(H, H′ ) and B ∈ B(K, K′ ). ˆ respectively, then (i) If A and B are left invertible with left inverses Aˆ and B ˆ ˆ A  B is left invertible and A  B is its left inverse. ˆ respectively, (ii) If A and B are right invertible with right inverses Aˆ and B ˆ ˆ then A  B is right invertible and A  B is its right inverse. (iii) If A and B are regular with inner inverses A− and B − respectively, then A  B is regular with A−  B − as its inner inverse. (iv) If A and B have A− and B − as their outer inverses respectively, then A  B has A−  B − as its outer inverse. Proof. It follows from Theorem 8 and the facts that IX  IY = IX ⊗Y for any Hilbert spaces X and Y. As a consequence of (i) and (ii) in Proposition 11, we obtain the following result. Corollary 12. Let A ∈ B(H) and B ∈ B(K). If A and B are invertible, then A  B is invertible and (A  B)−1 = A−1  B −1 .

(14)

Next, we consider a kind of operator inverse, called Moore-Penrose inverse. Recall that a Moore-Penrose inverse of A ∈ B(H, K) is an operator A† ∈ B(K, H) satisfying the following Penrose conditions ([7]) (i) A† is an inner inverse of A ; (ii) A† is an outer inverse of A ; (iii) AA† is Hermitian ; (iv) A† A is Hermitian. It is well known that the following statements are equivalent for A ∈ B(H, K) (see e.g. [1]): (i) a Moore-Penrose inverse of A exists ; (ii) a Moore-Penrose inverse of A is unique ; (iii) the range of A is closed. Theorem 13. Let A ∈ B(H, H′ ) and B ∈ B(K, K′ ). If A and B have closed ranges, then 1. the range of A  B is closed ; 2. (A  B)† = A†  B † .

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Tracy-Singh Products for Operator Matrices

Proof. Since the ranges of A and B are closed, the Moore-Penrose inverses A† and B † exist and are unique. Making use of Theorem 8 and Proposition 4, we can verify that A†  B † satisfies the following Penrose equations: (i) (A  B)(A†  B † )(A  B) = A  B (ii) (A†  B † )(A  B)(A†  B † ) = A†  B † ( )∗ (iii) (A  B)(A†  B † ) = (A  B)(A†  B † ) ( )∗ (iv) (A†  B † )(A  B) = (A†  B † )(A  B). Hence, a Moore-Penrose inverse of A  B exists and it is uniquely determined by A†  B † . It follows that A  B has a closed range. The results in this section indicate that the Tracy-Singh product is compatible with various kinds of operator inverses.

4

Tracy-Singh products and operator orderings

Now, we focus on order properties of Tracy-Singh products related to algebraic properties. Theorem 14. Let A ∈ B(H) and B ∈ B(K). (i) If A, B > 0, then A  B > 0. (ii) If A, B > 0, then A  B > 0. Proof. Assume A, B > 0. Using Theorem 8 and property (8), we obtain ( 1 )( 1 ) 1 1 1 1 1 1 A  B = A2 A2  B 2 B 2 = A2  B 2 A2  B 2 ( 1 )∗ ( 1 ) 1 1 = A2  B 2 A 2  B 2 > 0. Consider the case A, B > 0. We have immediately by (i) that A  B > 0. By Corollary 12, A  B is invertible. This implies that A  B > 0. The next result provides the monotonicity of Tracy-Singh product. Corollary 15. Let A1 , A2 ∈ B(H) and B1 , B2 ∈ B(K). (i) If A1 > A2 > 0 and B1 > B2 > 0, then A1  B1 > A2  B2 . (ii) If A1 > A2 > 0 and B1 > B2 > 0, then A1  B1 > A2  B2 . Proof. Suppose that A1 > A2 > 0 and B1 > B2 > 0. Applying Proposition 4 and Theorem 14 yields A1  B1 − A2  B2 = A1  B1 − A2  B1 + A2  B1 − A2  B2 = (A1 − A2 )  B1 + A2  (B1 − B2 ) > 0. The proof of (ii) is similar to that of (i).

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Acknowledgement. This research was supported by Thailand Research Fund grant no. MRG6080102.

References [1] S. R. Caradus, Generalized Inverses and Operator Theory, Queen’s Papers in Pure and Applied Mathematics no. 50, Queen’s University, Kingston, Ontario, 1978. [2] R. A. Horn, C. R. Johnson, Topics in Matrix Analysis, Cambridge University Press, Cambridge, 1999. [3] R. H. Koning, H. Neudecker, T. Wansbeek, Block Kronecker products and the vecb operator. Linear Algebra Appl., 149, 165184 (1991). [4] C. S. Kubrusly, P. C.M. Vieira, Convergence and decomposition for tensor products of Hilbert space operators. Oper. Matrices, 2, 407–416 (2008). [5] J. R. Magnus, H. Neudecker, Matrix differential calculus with applications to simple, Hadamard, and Kronecker products., J. Math. Psych., 29, 474-492 (1985). [6] H. Neudecker, A. Satorra, G. Trenkler, S. Liu, A Kronecker matrix inequality with a statistical application. Econometric Theory, 11, 654-655 (1985). [7] R. Penrose, A generalized inverse for matrices. Proc. Cambridge Philos. Soc., 51, 406-413 (1955). [8] L. Shuangzhe, Matrix results on the Khatri-Rao and Tracy-Singh products. Linear Algebra Appl., 289, 267-277 (1999). [9] D. S. Tracy, K. G. Jinadasa, Partitioned Kronecker products of matrices and applications. Canad. J. Statist., 17, 107-120 (1989). [10] D. S. Tracy, R. P. Singh, A new matrix product and its applications in partitioned matrix differentiation. Stat. Neerl., 26, 143157 (1972). [11] J. Zanni, C. S. Kubrsly, A note on compactness of tensor products., Acta Math. Univ. Comenian. (N.S.), 84, 59-62 (2015). [12] H. Zhang, F. Ding, On the Kronecker products and their applications. J. Appl. Math., 2013, 8 pages (2013).

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Analytic Properties of Tracy-Singh Products for Operator Matrices Arnon Ploymukda1 , Pattrawut Chansangiam1∗, Wicharn Lewkeeratiyutkul2 1 Department of Mathematics, Faculty of Science, King Mongkut’s Institute of Technology Ladkrabang, Bangkok 10520, Thailand. 2 Department of Mathematics and Computer Science, Faculty of Science, Chulalongkorn University, Bangkok 10330, Thailand.

Abstract We show that the Tracy-Singh product of Hilbert space operators is continuous with respect to the operator-norm topology. The Tracy-Singh product of two nonzero operators is compact if and only if both factors are compact. We provide upper and lower bounds for certain Schatten p-norms of the Tracy-Singh product of operators. It turns out that this product is continuous with respect to the topologies on norm ideals of compact operators, trace class operators, and Hilbert-Schmidt class operators. Thus the Tracy-Singh product preserves such classes of operators.

Keywords: tensor product, Tracy-Singh product, operator matrix, compact operator, Schatten p-class operator Mathematics Subject Classifications 2010: 47A80, 47A30, 47B10.

1

Introduction

In matrix theory, one of useful matrix products is the Kronecker product. Recall that the Kronecker product of two complex matrices A ∈ Mm,n (C) and B ∈ Mp,q (C) is given by the block matrix ˆ = [aij B]ij ∈ Mmp,nq (C). A ⊗B This matrix product was generalized to the Tracy-Singh product by Tracy and Singh [3]. Let A = [Aij ] ∈ Mm,n (C) be a partitioned matrix with Aij as the (i, j)th submatrix. Let B = [Bkl ] ∈ Mp,q (C) be a partitioned matrix with Bkl as the (k, l)th submatrix. The Tracy-Singh product of A and B is defined by [[ ] ] ˆ B = Aij ⊗ ˆ Bkl A ∈ Mmp,nq (C). kl ij ∗ Corresponding

author. Email: [email protected]

1

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Analytic Properties of Tracy-Singh Products for Operator Matrices

This kind of matrix product has several attractive properties and can be applied widely in statistics, econometrics and related fields; see e.g., [3, 5, 7, 8, 9]. The tensor product of Hilbert space operators is a natural extension of the Kronecker product to infinite-dimensional setting. Theory of Hilbert tensor product has been continuously investigated in the literature; see, e.g., [2, 4, 10]. It is well known that the tensor product is continuous with respect to the operator-norm topology. Moreover, on the norm ideals of compact operators generated by Schatten p-norm for p = 1, 2, ∞, the tensor product are also continuous. Recently, the tensor product for operators was generalized to the Tracy-Singh product for operator matrices acting on the direct sum of Hilbert spaces in [6]. This kind of operator product satisfies certain pleasing algebraic and order properties. In this paper, we discuss continuity, convergence, and compactness of the Tracy-Singh product for operators in the operator-norm topology. Then we obtain relations between Tracy-Singh product and certain analytic functions. We also investigate the Tracy-Singh product on norm ideals of compact operators generated by certain Schatten p-norms. In fact, this product is continuous with respect to the Schatten p-norm for p = 1, 2, ∞. Estimations by such norms for Tracy-Singh products are provided. It follows that trace class operators and Hilbert-Schmidt class operators are preserved under this product. This paper is organized as follows. In section 2, we give preliminaries on Tracy-Singh products for operators on a Hilbert space. In section 3, we establish analytic properties of the Tracy-Singh product in the operator-norm topology. We investigate the Tracy-Singh product on the norm ideals of compact operators generated by certain Schatten p-norms in Section 4.

2

Preliminaries on Tracy-Singh products for operator matrices

Throughout, let H, H′ , K and K′ be complex Hilbert spaces . When X and Y are Hilbert spaces, denote by B(X, Y ) the Banach space of bounded linear operators from X into Y , and abbreviate B(X, X) to B(X). In order to define the Tracy-Singh product, we have to fix the decompositions of Hilbert spaces, namely, H =

n ⊕ j=1

Hj ,

H′ =

m ⊕ i=1

Hi′ ,

K =

q ⊕

K′ =

Kl ,

p ⊕

Kk′

k=1

l=1

where each Hj , Hi′ , Kl , Kk′ are Hilbert spaces. For each j = 1, . . . , n and l = 1, . . . , q, let Ej : Hj → H and Fl : Kl → K be the canonical embeddings. For each i = 1, . . . , m and k = 1, . . . , p, let Pi′ and Q′k be the orthogonal projections. Thus, each operator A ∈ B(H, H′ ) and B in B(K, K′ ) can be expressed uniquely as operator matrices m,n

A = [Aij ]i,j=1

666

p,q

and B = [Bkl ]k,l=1

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A. Ploymukda, P. Chansangiam, Wicharn Lewkeeratiyutkul

where Aij = Pi′ AEj : Hj → Hi′ and Bkl = Q′k BFl : Kl → Kk′ for each i, j, k, l. We define of A and B to be a bounded linear operator ⊕n,qthe Tracy-Singh⊕product m,p ′ ′ from j,l=1 Hj ⊗ Kl to i,k=1 Hi ⊗ Kk represented in the block-matrix form as follows: [ ] A  B = [Aij ⊗ Bkl ]kl ij . When m = n = p = q = 1, the Tracy-Singh product A  B becomes the tensor product A ⊗ B. Lemma 1 ([6]). Fundamental properties of the Tracy-Singh product for operators are listed below (provided that each term is well-defined): 1. The map (A, B) 7→ A  B is bilinear. 2. Compatibility with adjoints: (A  B)∗ = A∗  B ∗ . 3. Mixed-product property: (A  B)(C  D) = AC  BD. 4. Compatibility with powers: (A  B)r = Ar  B r for any r ∈ N. 5. Compatibility with inverses: if A and B are invertible, then A  B is invertible with (A  B)−1 = A−1  B −1 . 6. Positivity: if A > 0 and B > 0, then A  B > 0. 7. Strictly positivity: if A > 0 and B > 0, then A  B > 0. 8. If A and B are partial isometries, then so is AB. Recall that an operator T is a partial isometry if and only if the restriction of T to a closed subspace is an isometry.

3

Analytic properties of the Tracy-Singh product

In this section, we establish some analytic properties of the Tracy-Singh product involving operator norms. These properties involve continuity, convergence, norm estimates, and certain analytic functions. We denote the operator norm by ∥ · ∥∞ . In order to discuss the continuity of the Tracy-Singh product, recall the following bounds for the operator norm of operator matrices. n,n

Lemma 2 ([1]). Let A = [Aij ]i,j=1 ∈ B(H) be an operator matrix. Then n−2

n ∑

∥Aij ∥2∞ 6 ∥A∥2∞ 6

i,j=1

n ∑

∥Aij ∥2∞ .

(1)

i,j=1

∞ Lemma 3. Let A = [Aij ]n,n i,j=1 ∈ B(H) be an operator matrix and let (Ar )r=1 (r)

be a sequence in B(H) where Ar = [Aij ]n,n i,j=1 for each r ∈ N. Then Ar → A if (r)

and only if Aij → Aij for all i, j = 1, . . . , n.

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Analytic Properties of Tracy-Singh Products for Operator Matrices

Proof. It is a direct consequence of Lemma 2. The next theorem explains that the Tracy-Singh product is (jointly) continuous with respect to the topology induced by the operator norm. Theorem 4. Let A = [Aij ] ∈ B(H) and B = [Bkl ] ∈ B(K) be operator matrices, ∞ ∞ and let (Ar )r=1 and (Br )r=1 be sequences in B(H) and B(K), respectively. If Ar → A and Br → B, then Ar  Br → A  B. (r)

Proof. Suppose that Ar → A and Br → B. By Lemma 3, we have Aij → Aij (r)

and Bkl → Bkl for each i, j, k, l. Since the tensor product is continuous, we have (r) (r) Aij ⊗ Bkl → Aij ⊗ Bkl for each i, j, k, l. It follows that Ar  Br → A  B by Lemma 3. The next theorem provides upper/lower bounds for the operator norm of the Tracy-Singh product. Theorem 5. For any operator matrices A = [Aij ]n,n i,j=1 ∈ B(H) and B = [Akl ]q,q ∈ B(K), we have k,l=1 1 ∥A∥∞ ∥B∥∞ 6 ∥A  B∥∞ 6 nq∥A∥∞ ∥B∥∞ . nq

(2)

Proof. It follows from Lemma 2 that ∑∑ ∑∑ ∥A  B∥2∞ 6 ∥Aij ⊗ Bkl ∥2∞ = ∥Aij ∥2∞ ∥Bkl ∥2∞ i,j

k,l

k,l

i,j

) (∑ )( ∑ ∥Bkl ∥2∞ 6 (nq)2 ∥A∥2∞ ∥B∥2∞ . = ∥Aij ∥2∞ i,j

We also have ∥A  B∥2∞ > (nq)−2

∑∑ ∑∑ ∥Aij ⊗ Bkl ∥2∞ = (nq)−2 ∥Aij ∥2∞ ∥Bkl ∥2∞ k,l

−2

= (nq)

k,l

i,j

k,l

i,j

) (∑ )( ∑ ∥Aij ∥2∞ ∥Bkl ∥2∞ > (nq)−2 ∥A∥2∞ ∥B∥2∞ . i,j

k,l

Hence, we obtain the bound (2). Theorem 6. Let A ∈ B(H). (i) If f is an analytic function on a region containing the spectra of A and I  A, then f (I  A) = I  f (A). (3) (ii) If f is an analytic function on a region containing the spectra of A and A  I, then f (A  I) = f (A)  I. (4)

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A. Ploymukda, P. Chansangiam, Wicharn Lewkeeratiyutkul

Proof. (i) Since f is analytic on spectra of A and I  A, we have the Taylor series expansion f (z) =

∞ ∑

αr z r .

r=0

It follows that f (A) =

∞ ∑

αr Ar

f (I  A) =

and

r=0

∞ ∑

αr (I  A)r .

r=0

Making use of the bilinearity of Tracy-Singh product and Theorem 4 yields f (I  A) =

∞ ∑

αr (I  Ar ) =

r=0

= I

∞ ∑

(I  αr Ar )

r=0 ∞ ∑

αr Ar = I  f (A).

r=0

Similarly, we obtain the assertion (ii). Theorem 7. Let A ∈ B(H) and B ∈ B(K) be positive operators. For any α > 0, we have (A  B)α = Aα  B α . (5) Proof. First, note that A  B is positive by property (6) of Lemma 1. It follows from the property (4) in Lemma 1 that for any r, s ∈ N, ( r r )s As  B s = Ar  B r = (A  B)r , r

r

r

and thus (A  B) s = A s  B s . Now, for α > 0, there is a sequence (qn ) of positive rational numbers such that qn → α. It follows from the previous claim and the continuity of Tracy-Singh product (Theorem 4) that (A  B)α = lim (A  B)qn = lim Aqn  B qn n→∞

n→∞

= lim Aqn  lim B qn = Aα  B α . n→∞

n→∞

Corollary 8. Let A ∈ B(H) and B ∈ B(K) be strictly positive operators. For any real number α, we have (A  B)α = Aα  B α .

(6)

Proof. Note that A  B is strictly positive by property (7) of Lemma 1. For α < 0, it follows from Theorem 7 and the property (5) in Lemma 1 that (A  B)α = [(A  B)−1 ]−α = (A−1  B −1 )−α = (A−1 )−α  (B −1 )−α = Aα  B α .

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Analytic Properties of Tracy-Singh Products for Operator Matrices

Corollary 9. Let A ∈ B(H, H′ ) and B ∈ B(K, K′ ). Then |A  B| = |A|  |B|.

(7)

Proof. Applying Lemma 1 and property (5), we get |A  B| = [(A  B)∗ (A  B)] 2 = [(A∗  B ∗ )(A  B)] 2 1

1

= (A∗ A  B ∗ B) 2 = (A∗ A) 2  (B ∗ B) 2 = |A|  |B|. 1

1

1

Recall the polar decomposition theorem: for any A ∈ B(H, K), there exists a partial isometry U such that A = U |A|. The next result is a polar decomposition for the Tracy-Singh product of operators. Corollary 10. Let A ∈ B(H, H′ ) and B ∈ B(K, K′ ). If A = U |A| and B = V |B| are polar decompositions of A and B, respectively, then a polar decomposition of A  B is given by A  B = (U  V )|A  B|. (8) Proof. Let U and V be partial isometries such that A = U |A| and B = V |B|. It follows from Lemma 1(3) and Corollary 9 that A  B = U |A|  V |B| = (U  V )(|A|  |B|) = (U  V )|A  B|. Note that U  V is also a partial isometry, according to property (8) in Lemma 1. Hence, the decomposition (8) is a polar one.

4

Tracy-Singh products on norm ideals of compact operators

In this section, we investigate the Tracy-Singh product on norm ideals of B(H). Recall that any proper ideal of B(H) is contained in the ideal S∞ of compact operators. For any compact operator A ∈ B(H), let (si (A))∞ i=1 be the sequence of decreasingly-ordered singular values of A (i.e. eigenvalues of |A|). For each 1 6 p < ∞, the Schatten p-norm of A is defined by ( ∥A∥p =

∞ ∑

)1/p spi (A)

.

i=1

If ∥A∥p is finite, we say that A is a Schatten p-class operator. The Schatten ∞-norm is just the operator norm. For each 1 6 p 6 ∞, let Sp be the Schatten p-class operators . In particular, S1 and S2 are the trace class and the HilbertSchmidt class, respectively. Each Schatten p-norm induces a norm ideal of B(H) and this ideal is closed under the topology generated by this norm. Lemma 11. Let A = [Aij ] ∈ B(H) be an operator matrix. Then A is compact if and only if Aij is compact for all i, j.

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A. Ploymukda, P. Chansangiam, Wicharn Lewkeeratiyutkul

Proof. If A is compact, then Aij = Pi′ AEj is also compact for each i, j due to the fact that S∞ is an ideal of B(H). Conversely, suppose that Aij is compact for all i, j. Recall that a bounded linear operator is compact if and only if it maps a bounded sequence into a sequence having a convergent subsequence. Let (xr )∞ r=1 ⊕n (1) (2) (n) T be a bounded sequence in H = H . Write x = [x x . . . x ] ∈ r r r i r i=1 ⊕n H for each r ∈ N. Consider i i=1      (1)  (1) (n) A11 · · · A1n xr A11 xr + · · · + A1n xr     ..   .. ..  .   . Axr =  ... . .   ..  =  .  (n) (1) (n) An1 · · · Ann xr An1 xr + · · · + Ann xr ∞ For each l = 1, 2, . . . , n, since (xr )∞ r=1 is bounded, the sequence (Aij xr )r=1 (l) ∞ has a convergent subsequence, namely, (Aij xrk )k=1 . Hence,   (n) (1) A11 xrk + · · · + A1n xrk   ..   .   (n) (1) An1 xrk + · · · + Ann xrk (l)

(l)

is a desired convergent subsequence of (Axr )∞ r=1 . n,n

Lemma 12 ([1]). Let A = [Aij ]i,j=1 be an operator matrix in the Schatten p-class. (i) For 1 6 p 6 2, we have n ∑

∥Aij ∥2p 6 ∥A∥2p 6 n4/p−2

i,j=1

n ∑

∥Aij ∥2p .

(9)

∥Aij ∥2p .

(10)

i,j=1

(ii) For 2 6 p < ∞, we have n4/p−2

n ∑

∥Aij ∥2p 6 ∥A∥2p 6

i,j=1

n ∑ i,j=1

Lemma 13. Let 1 6 p < ∞. An operator matrix A = [Aij ] ∈ B(H) is a Schatten p-class operator if and only if Aij is a Schatten p-class operator for all i, j. Proof. This is a direct consequence of the norm estimations in Lemma 12. Lemma 14. Let 1 6 p 6 ∞. Let A = [Aij ]n,n matrix in i,j=1 be an operator [ ]n,n (r) ∞ the class Sp and let (Ar )r=1 be a sequence in Sp where Ar = Aij for i,j=1

(r)

each r ∈ N. Then Ar → A in Sp if and only if Aij → Aij in Sp for all i, j = 1, . . . , n.

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Analytic Properties of Tracy-Singh Products for Operator Matrices

(r)

Proof. Lemma 13 assures that Aij and Aij belong to Sp for any i, j = 1, . . . , n and r ∈ N. Consider the case 1 6 p 6 2. Suppose that Ar → A in Sp . For any fixed i, j ∈ {1, ..., n}, we have from the estimation (9) that (r)

∥Aij − Aij ∥2p 6

n ∑

(r)

∥Aij − Aij ∥2p 6 ∥Ar − A∥2p .

i,j=1 (r)

(r)

Hence, Aij → Aij in Sp . Conversely, suppose Aij → Aij in Sp for each i, j. Lemma 12 implies that n ∑

∥Ar − A∥2p 6 n4/p−2

(r)

∥Aij − Aij ∥2p .

i,j=1

Hence, Ar → A in Sp . The case 2 < p < ∞ and the case p = ∞ are done by using the norm estimations (10) and (1), respectively. Next, we discuss compactness of Tracy-Singh product of operators. Lemma 15 ([10]). Let A ∈ B(H) and B ∈ B(K) be nonzero operators. Then A ⊗ B is compact if and only if both A and B are compact. Theorem 16. Let A ∈ B(H) and B ∈ B(K) be nonzero operator matrices. Then A  B is compact if and only if both A and B are compact. Proof. Write A = [Aij ] and B = [Bkl ]. For sufficiency, suppose that A and B are compact. By Lemma 11, we deduce that Aij and Bkl are compact for all i, j, k, l. It follows from Lemma 15 that Aij ⊗ Bkl is compact for all i, j, k, l. Lemma 11 ensures the compactness of A  B. For necessity part, reverse the previous procedure. The following theorem supplies bounds for Schatten 1-norm of the TracySingh product of operators. Theorem 17. For any nonzero compact operator A = [Aij ]n,n i,j=1 ∈ B(H) and B = [Akl ]q,q ∈ B(K), we have k,l=1 1 ∥A∥1 ∥B∥1 6 ∥A  B∥1 6 nq∥A∥1 ∥B∥1 . nq

(11)

Hence, A  B is trace-class if and only if both A and B are trace-class. Proof. Suppose that both A and B are nonzero and compact. Then the operator A  B is compact by Theorem 16. It follows from the norm bound (9) that ∑∑ ∑∑ ∥A  B∥21 6 (nq)2 ∥Aij ⊗ Bkl ∥21 = (nq)2 ∥Aij ∥21 ∥Bkl ∥21 k,l

i,j

k,l

i,j

(∑ )( ∑ ) = (nq)2 ∥Aij ∥21 ∥Bkl ∥21 6 (nq)2 ∥A∥21 ∥B∥21 . i,j

k,l

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A. Ploymukda, P. Chansangiam, Wicharn Lewkeeratiyutkul

We also have ∥A  B∥21 >

∑∑ ∑∑ ∥Aij ⊗ Bkl ∥21 = ∥Aij ∥21 ∥Bkl ∥21 i,j

k,l

i,j

k,l

(∑ )( ∑ ) = ∥Aij ∥21 ∥Bkl ∥21 > (nq)−2 ∥A∥21 ∥B∥21 . i,j

k,l

Hence, we obtain the bound (11). Theorem 18. For any nonzero compact operator matrices A ∈ B(H) and B ∈ B(K), we have ∥A  B∥2 = ∥A∥2 ∥B∥2 .

(12)

Hence, A  B is a Hilbert-Schmidt operator if and only if both A and B are Hilbert-Schmidt operators. Proof. Since both A and B are nonzero and compact, the operator A  B is compact by Theorem 16. Write A = [Aij ] and B = [Bkl ]. Then by Lemma 12(ii), we have ∑∑ ∑∑ ∥Aij ⊗ Bkl ∥22 = ∥Aij ∥22 ∥Bkl ∥22 ∥A  B∥22 = k,l

=

i,j

(∑

∥Aij ∥22

)( ∑

i,j

k,l

∥Bkl ∥22

)

i,j

= ∥A∥22 ∥B∥22 .

k,l

Hence, we get the multiplicative property (12). The final result asserts that the Tracy-Singh product is continuous with respect to the topology induced by the Schatten p-norm for each p ∈ {1, 2, ∞}. Theorem 19. Let p ∈ {1, 2, ∞}. If a sequence (Ar )∞ r=1 converges to A and a sequence (Br )∞ r=1 converges to B in the norm ideal Sp , then Ar  Br converges to A  B in Sp . Proof. Write A = [Aij ] and B = [Bkl ]. In the viewpoint of Lemma 14, it suffices (r) (r) to show that Aij ⊗ Bkl → Aij ⊗ Bkl in Sp for all i, j, k, l. Since Ar → A and (r)

(r)

Br → B in Sp , we have by Lemma 14 that Aij → Aij and Bkl → Bkl for all i, j, k, l. It follows that (r)

(r)

(r)

(r)

(r)

(r)

∥Aij ⊗ Bkl − Aij ⊗ Bkl ∥p = ∥Aij ⊗ Bkl − Aij ⊗ Bkl + Aij ⊗ Bkl − Aij ⊗ Bkl ∥p (r)

(r)

(r)

(r)

(r)

6 ∥Aij ⊗ (Bkl − Bkl )∥p + ∥(Aij − Aij ) ⊗ Bkl ∥p (r)

= ∥Aij ∥p ∥Bkl − Bkl ∥p + ∥Aij − Aij ∥p ∥Bkl ∥p → ∥Aij ∥p · 0 + 0 · ∥Bkl ∥p = 0. (r)

(r)

Hence, Aij ⊗ Bkl → Aij ⊗ Bkl in Sp for all i, j, k, l. Acknowledgement. This research was supported by Thailand Research Fund (grant no. MRG6080102).

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Analytic Properties of Tracy-Singh Products for Operator Matrices

References [1] R. Bhatia, F. Kittaneh, Norm inequalities for partitioned operators and an application. Math. Ann., 287, 719-726 (1990). [2] A. Brown, C. Pearcy, Spectra of tensor products of operators. Proc. Amer. Math. Soc., 17, 162-166 (1966). [3] R. H. Koning, H. Neudecker, T. Wansbeek, Block Kronecker products and the vecb operator. Linear Algebra Appl., 149, 165-184 (1991). [4] C. S. Kubrusly, P. C.M. Vieira, Convergence and decomposition for tensor products of Hilbert space operators. Oper. Matrices, 2, 407–416 (2008). [5] J.R. Magnus, H. Neudecker, Matrix differential calculus with applications to simple, Hadamard, and Kronecker products. J. Math. Psych., 29, 474492 (1985). [6] A. Ploymukda, P. Chansangiam, W. Lewkeeratiyutkul, Algebraic and order properties of Tracy-Singh products for operator matrices. J. Comput. Anal. Appl., 9 pages, to appear in 2018. [7] L. Shuangzhe, Matrix results on the Khatri-Rao and Tracy-Singh products. Linear Algebra Appl. 289, 267-277 (1999). [8] D. S. Tracy, K. G. Jinadasa, Partitioned Kronecker products of matrices and applications. Canad. J. Statist., 17, 107-120 (1989). [9] D. S. Tracy, R. P. Singh, A new matrix product and its applications in partitioned matrix differentiation. Stat. Neerl., 26, 143-157 (1972). [10] J. Zanni, C. S. Kubrsly, A note on compactness of tensor products. Acta Math. Univ. Comenian. (N.S.), 84, 59-62 (2015).

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ON THE RADIAL DISTRIBUTION OF JULIA SET OF SOLUTIONS OF f ′′ + Af ′ + Bf = 0 JIANREN LONG Abstract. The paper is devoted to study the dynamical properties of solutions of f ′′ + A(z)f ′ + B(z)f = 0, where A(z) is nontrivial solution of w′′ + P (z)w = 0, P (z) is a polynomial, B(z) is a transcendental entire function of lower order less than 21 . The lower bound of the size of the radial distribution of Julia sets of infinite order solutions of the equation are obtained. Another proof of the result in [9] is discussed in which the modified Phragm´en-Lindel¨of principle is needed.

1. Introduction and main results For a function meromorphic f in the complex plane C, the order of growth, lower order of growth and the convergence exponent of zero-sequence of f are given respectively by log+ T (r, f ) ρ(f ) = lim sup , log r r→∞ log+ T (r, f ) log r

µ(f ) = lim inf r→∞

and λ(f ) = lim sup

log+ N (r, f1 )

. log r In what follows, we assume that the reader is familiar with standard notation and basic results in Nevanlinna theory of meromorphic functions, such as T (r, f ), m(r, f ) and N (r, f ), see [10, 12, 23] for more details. We define the nth iterate of meromorphic function f as follows: r→∞

f 0 (z) = z, f 1 (z) = f (z), . . . , f n (z) = f (f n−1 (z)),

n ∈ N,

n ≥ 2.

The Fatou set F (f ) of transcendental meromorphic function f is the subset of C where the iterates {f n (z)}∞ n=1 of f form a normal family, and its complement J(f ) = C\F (f ) is called the Julia set of f . It is well known that F (f ) is open and completely invariant under f , and J(f ) is closed and 2010 Mathematics Subject Classification. 34M10; 37F10; 30D35. Key words and phrases. Complex differential equation, Entire function, Infinite order, Radial distribution, Julia set. 1

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2

J. R. LONG

non-empty, see [4]. We also need the following notations and definitions. For α < β and r ∈ (0, ∞), set S(α, β) = {z : α < arg z < β}, S(α, β, r) = {z : |z| < r, α < arg z < β}, S(r, α, β) = {z : |z| > r, α < arg z < β}. Let F denotes the closure of F ⊂ C. Given θ ∈ [0, 2π), if S(θ−ε, θ+ε)∩J(f ) is unbounded for any small ε > 0, then the ray arg z = θ from the origin is called the radial distribution of J(f ). Define ∆(f ) = {θ ∈ [0, 2π) : arg z = θ is the radial distribution of J(f )}. Obviously, ∆(f ) is closed and so measurable. Let m(∆(f )) denotes the linear measure of ∆(f ). What can we say according to m(∆(f )) of any meromorphic function f in C? It is interesting topic, many results have been obtained by several authors. Baker [1] considered the radial distribution of the Julia set and constructed an entire function with infinite lower order whose Julia set lies in a horizontal trip. Qiao [16] proved that if f is a transcendental entire function of finite lower order, then { = 2π, µ(f ) < 1/2, m(∆(f )) ≥ π/µ(f ), µ(f ) ≥ 1/2. Later, some observations on radial distribution of the Julia sets of transcendental meromorphic functions with finite lower order were made; see, for example, [25] and [17]. It seems that there are few work done on the case of meromorphic functions of infinite order. Recently, Huang-Wang [8, 9] studied the radial distribution of the Julia sets of entire functions of infinite lower order by using the tool of differential equations, i.e., for any nontrivial solutions f of (1.2) below, a lower bound of m(∆(f )) is obtained. Zhang-Wang-Yang [24] also studied the radial distribution of the Julia sets of entire soutions f of (1.2), a lower bound of m(∆(f )) is obtained when the coefficient A(z) and B(z) satisfy different conditions with [8, 9]. Our idea of this paper comes from [24], a new lower bound of m(∆(f )) is found when A(z) and B(z) satisfy new conditions which are different with the conditions of [24]. Our starting point is a result which is related to the growth of solutions of (1.2). Theorem 1.1 ([14]). Let A(z) be a nontrivial solution of the equation (1.1)

w′′ + P (z)w = 0,

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RADIAL DISTRIBUTION SOLUTIONS OF DIFFERENTIAL EQUATIONS

3

where P (z) = an z n + · · · + a0 , an = ̸ 0, and let λ(A) < ρ(A). Let B(z) be a transcendental entire function with µ(B) < 21 . Then every nontrivial solution of the equation f ′′ + A(z)f ′ + B(z)f = 0

(1.2) is of infinite order.

The following result shows that m(∆(f )) has a lower bound when A(z) and B(z) satisfy the conditions of Theorem 1.1. Theorem 1.2. Let A(z) and B(z) be given as in Theorem 1.1. Then every 2π nontrivial solution f of (1.2) satisfies m(∆(f )) ≥ n+2 . To state the following results, the definition of accumulation lines of zero-sequence is needed, which can be found in [13, 18, 21, 22]. Definition 1.3. Let f be a meromorphic function in C, and let arg z = θ ∈ [0, 2π) be a ray from the origin. We denote, for each ε > 0, the convergence exponent of zero-sequence of f in the region S(θ − ε, θ + ε, r) by λθ,ε (f ) and by λθ (f ) = lim+ λθ,ε (f ). That is, ε→0

log+ nθ−ε,θ+ε (r, 0, f ) , ε→0 log r r→∞ where nθ−ε,θ+ε (r, 0, f ) is the number of zeros of f , counting multiplicity in S(θ − ε, θ + ε, r). λθ (f ) = lim+ lim sup

The ray arg z = θ is called an accumulation line of the zero-sequence of f if λθ (f ) = ρ(f ). By Lemma 2.1 below, we know that the number of accumulation lines of zero-sequence of nontrivial solutions of (1.1) less than or equal to n + 2 and the set of the accumulation lines of zero-sequence of nontrivial solutions of (1.1) is the subset of {arg z = θj , 0 ≤ j ≤ n + 1}, n) . Let w be a nontrivial solution of (1.1), where P (z) = where θj = 2jπ−arg(a n+2 n an z + · · · + a0 is a polynomial of degree n ≥ 1, let p(w) denotes the number of the rays arg z = θj , j = 0, 1, . . . , n + 1, which are not accumulation lines of zero-sequence of w. Remark 1.4. It follows from Lemma 2.1 that p(w) must be an even number for every nontrivial solution w of (1.1). Theorem 1.5. Let A(z) be a nontrivial solution of (1.1), and the number of accumulation lines of zero-sequence of A(z) strictly less than n+2. Let B(z) be a transcendental entire function with µ(B) < 12 . Then every nontrivial 2π solution f of (1.2) satisfies ρ(f ) = ∞ and m(∆(f )) ≥ n+2 .

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Furthermore, we study the radial distribution of Julia set of the derivatives of nontrivial solutions of (1.2). Theorem 1.6. Let A(z) and B(z) be given as in Theorem 1.1. Then every ( ) 2π nontrivial solution f of (1.2) satisfies m ∆(f ) ∩ ∆(f (k) ) ≥ n+2 , where k ≥ 1 is an integer. By using similar reasoning in proving Theorems 1.5 and 1.6, we have the following result. Theorem 1.7. Let A(z) and B(z) be given as in Theorem 1.5. Then every ( ) 2π nontrivial solution f of (1.2) satisfies m ∆(f ) ∩ ∆(f (k) ) ≥ n+2 , where k ≥ 1 is an integer. Applying Theorems 1.6 and 1.7, we immediately obtain the following corollaries. Corollary 1.8. Let A(z) and B(z) be given as in Theorem 1.6. Then ( ) 2π for every nontrivial solution f of (1.2), where k ≥ 1 m ∆(f (k) ) ≥ n+2 is an integer. Corollary 1.9. Let A(z) and B(z) be given as in Theorem 1.7. Then ( ) 2π m ∆(f (k) ) ≥ n+2 for every nontrivial solution f of (1.2), where k ≥ 1 is an integer. Obviously, we can obtain Theorems 1.2 and 1.5 from Theorems 1.6 and 1.7, however, we need the results of Theorems 1.2 and 1.5 in proving Theorems 1.6 and 1.7. So we will give the proofs of Theorems 1.2 and 1.5 in Sections 3 and 4, respectively. 2. Auxiliary results In this section, we will give some auxiliary results for proving our theorems. To this end, we introduce following notations. Let f be an entire function of order ρ(f ) ∈ (0, ∞). For simplicity, set ρ(f ) = ρ and S = S(α, β). If for any θ ∈ (α, β), log log |f (reiθ )| = ρ, r→∞ log r lim

then we say that f blows up exponentially in S. If for any θ ∈ (α, β), log log |f (reiθ )|−1 = ρ, r→∞ log r lim

then we say that f decays to zero exponentially in S. The following lemma, originally due to Hille [11, Chapter 7.4], which also be found in [6, 19], plays an important role in proving our results.

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Lemma 2.1. Let w be a nontrivial solution of (1.1), where P (z) = an z n + n) · · · + a0 , an ̸= 0. Set θj = 2jπ−arg(a and Sj = S(θj , θj+1 ), where j = n+2 0, 1, 2, . . . , n + 1 and θn+2 = θ0 + 2π. Then w has the following properties. (i) In each sector Sj , w either blows up or decays to zero exponentially. (ii) If, for some j, w decays to zero in Sj , then it must blow up in Sj−1 and Sj+1 . However, it is possible for w to blow up in many adjacent sectors. (iii) If w decays to zero in Sj , then w has at most finitely many zeros in any closed subsector within Sj−1 ∪ Sj ∪ Sj+1 . (iv) If w blows up in Sj−1 and Sj , then for each ε > 0, w has infinitely many zeros in each sector S(θj − ε, θj + ε), and furthermore, as r → ∞, √ ( ) 2 |an | n+2 n S(θj − ε, θj + ε, r), 0, w = (1 + o(1)) r 2 , π(n + 2) where n(S(θj −ε, θj +ε, r), 0, w) is the number of zeros of w, counting multiplicity in S(θj − ε, θj + ε, r). Before stating the next lemma, for E ⊂ [0, ∞), we define the Lebesgue ∫ linear measure of E by m(E) = E dt, and the logarithmic measure of ∫ F ⊂ [1, ∞) is ml (F ) = F dtt . The upper and lower logarithmic density of F ⊂ [1, ∞) are given by log dens(F ) = lim sup r→∞

ml (F ∩ [1, r]) log r

and log dens(F ) = lim inf r→∞

ml (F ∩ [1, r]) , log r

respectively. The following result is due to Barry [3]. Lemma 2.2. Let f be an entire function with 0 ≤ µ(f ) < 1, and denote m(r) = inf log |f (z)| and M (r) = sup log |f (z)|. Then, for every α ∈ |z|=r

|z|=r

(µ(f ), 1), log dens ({r ∈ [1, ∞) : m(r) > M (r) cos πα}) ≥ 1 −

µ(f ) . α

We say that an open set is hyperbolic if it has at least three boundary points in C = C ∪ {∞}. Let W be a hyperbolic open set in C. For an a ∈ C\W , define CW (a) = inf{λW (z)|z − a| : z ∈ W },

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where λW (z) is the hyperbolic density on W . We know that if every component of W is simply connected, then CW (a) ≥ 12 . The following result was proved in [25, Lemma 2.2]. Lemma 2.3. Let f be an analytic in S(r0 , θ1 , θ2 ), let U be a hyperbolic domain and f : S(r0 , θ1 , θ2 ) → U . If there exists a point a ∈ ∂U \{∞} such that CU (a) > 0, then there exists a constant l > 0 such that, for sufficiently small ε > 0, one has |f (z)| = O(|z|l ),

z ∈ S(r0 , θ1 + ε, θ2 − ε),

|z| → ∞.

The following lemma is related to the Nevanlinna theory in an angular domain. To the end, we recall some notations and properties of Nevanlinna theory in an angular domain, see [5] for more details. Let f be meromorphic in S(α, β), where 0 < α < β ≤ 2π. Then we have ∫ ω r 1 tω dt Aα,β (r, f ) = ( ω − 2ω ){log+ |f (teiα )| + log+ |f (teiβ )|} , π 1 t r t ∫ β 2ω Bα,β (r, f ) = ω log+ |f (teiφ )| sin ω(φ − α)dφ, πr α ∑ 1 |bn |ω Cα,β (r, f ) = 2 ( − ) sin ω(θn − α), |bn |ω r2ω 1 0 only depending on f, ε1 , . . . , εn−1 and S(αn−1 , βn−1 ), and not depending on z, such that ′ f (z) −2 M f (z) ≤ Kr (sin k(ψ − α)) and

( )−2 (n) n−1 ∏ f (z) M sin k(ψ − α) sin kεj (ψ − αj ) f (z) ≤ Kr j=1

for all z ∈ S(αn−1 , βn−1 ) outside an R−set H, where k = π , j = 1, 2, . . . , n − 1. βj −αj

π β−α

and kεj =

3. Proof of Theorem 1.2 2π Set d = n+2 . Suppose on the contrary to the assertion that there exists a nontrivial solution f of (1.2) with m(∆(f )) < d. We aim for a contradiction. Set η = d − m(∆(f )). Since ∆(f ) is closed, then S = [0, 2π)\∆(f ) consists of at most countable many open intervals. Therefore, we choose finite many open intervals Ii = (αi , βi ), i = 1, 2, . . . , m, which satisfy [αi , βi ] ⊂ S and η m(S\ ∪m i=1 Ii ) < 4 . For the sector domain S(αi , βi ), and for sufficiently large ri , we have

(αi , βi ) ∩ ∆(f ) = ∅,

S(ri , αi , βi ) ∩ J(f ) = ∅.

This shows that, for each i = 1, 2, · · · , m, there exist the corresponding ri and unbounded Fatou component Ui of F (f ), such that S(ri , αi , βi ) ⊂ Ui (see [2]). In boundary of Ui , we take an unbounded and connected section γi ⊂ ∂Ui , then the mapping f : S(ri , αi , βi ) → C\γi is analytic. According to the choose of γi , we know that C\γi is simply connected, thus for any a ∈ γi \{∞}, CC\γi (a) ≥ 21 . In every S(ri , αi , βi ), applying Lemma 2.3 to f , there exists a positive constant l1 such that |f (z)| = O(|z|l1 ),

z ∈ ∪m i=1 S(ri , αi + ε, βi − ε),

|z| → ∞,

η i where 0 < ε < min{ 16m , βi −α }, i = 1, 2, . . . , m. Hence we immediately get 8

Sαi +ε,βi −ε (r, f ) = O(1),

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and then σαi +ε,βi −ε (f ) is finite. Applying Lemma 2.4, there exist two constants M > 0 and K > 0 such that (k) f (z) M (3.1) f (z) ≤ Kr , k = 1, 2, for all z ∈ ∪m i=1 S(ri , αi + 2ε, βi − 2ε), outside a R−set H. n) and Sj = {z : θj < arg z < θj+1 }, j = 0, 1, 2, . . . , n + Set θj = 2jπ−arg(a n+2 1, if j = n + 1, set θj+1 = θ0 + 2π. Since λ(A) < ρ(A), by Lemma 2.1, there exists at least one sector of the n + 2 sectors, such that A(z) decays to zero exponentially, say Sj0 = {z : θj0 < arg z < θj0 +1 }, 0 ≤ j0 ≤ n + 1. This implies that for any θ ∈ (θj0 + ε, θj0 +1 − ε), (3.2)

lim

r→∞

log log |A(re1 iθ )| log r

=

n+2 . 2

Set Sj′ 0 = {θ ∈ [0, 2π) : reiθ ∈ Sj0 (ε)}, where Sj0 (ε) = {z : θj0 + ε < arg z < θj0 +1 − ε}, then m(Sj′ 0 ) = θj0 +1 − ε − (θj0 + ε) ≥ d − η4 . Thus ( ) 3η m(Sj′ 0 ∩ S) = m Sj′ 0 \(Sj′ 0 ∩ ∆(f )) ≥ m(Sj′ 0 ) − m(∆(f ))) > > 0. 4 So, ( ) ′ ′ m m Sj′ 0 ∩ (∪m i=1 Ii ) = m(Sj0 ∩ S) − m(Sj0 ∩ (S\ ∪i=1 Ii )) 3η > − m(S\ ∪m i=1 Ii ) 4 η > > 0. 2 m ∪ Thus, there exists an open interval Ii0 = (α, β) ⊂ Ii ⊂ S, such that i=1

(3.3)

( ) η m Sj′ 0 ∩ Ii0 > > 0. 2m

By equation (1.2), we get (3.4)

′′ ′ f (z) + |A(z)| f (z) . |B(z)| ≤ f (z) f (z)

We divide into two cases to B(z) for finishing the proof. Case 1. 0 < µ(B) < 21 . By Lemma 2.2, there exists a set E1∗ ⊂ [1, ∞) µ(B)+ 1

2 with log dens(E1∗ ) ≥ 1− µ(B) , where α0 = , E1∗ = {r ∈ [1, ∞) : m(r) > α0 2 M (r) cos πα0 }, m(r) = inf log |B(z)|, M (r) = sup log |B(z)|. Hence there

|z|=r

|z|=r

exists a constant R0 > 1, such that (3.5)

|B(z)| > exp(rµ(B)−ε )

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for all |z| = r ∈ E1 = E1∗ \[0, R0 ). Then there exists a sequence of points {rs eiθ } outside H, θ ∈ Ii0 , rs ∈ E1 satisfying rs → ∞ as s → ∞, such that (3.1), (3.5) hold for z = rs eiθ , and (3.6)

lim

log log |A(rs1eiθ )|

s→∞

log rs

=

n+2 . 2

It follows from (3.1), (3.4), (3.5) and (3.6) that exp(rsµ(B)−ε ) < KrsM (1 + o(1)) for sufficiently large s. Obviously, this is a contradiction. Case 2. µ(B) = 0. By Lemma 2.2, there exists a set E2 ⊂ [1, ∞) with log dens(E2 ) = 1, such that for all z satisfying |z| = r ∈ E2 , we have √ 2 (3.7) log M (r, B), log |B(z)| > 2 where M (r, B) = max |B(z)|. It follows from (3.3) and (3.7), there exists a |z|=r

sequence of points {rs eiθ } outside H, θ ∈ Ii0 , rs ∈ E2 satisfying rs → ∞ as s → ∞, such that (3.1), (3.6) and (3.7) hold for z = rs eiθ . We deduce from (3.1), (3.4), (3.6) and (3.7) that √

(3.8)

M (rs , B)

2 2

≤ KrsM (1 + o(1))

for sufficiently large s. However B(z) is a transcendental entire function, we have log M (r, B) lim inf (3.9) = ∞. r→∞ log r We get a contradiction from (3.8) and (3.9). This completes the proof. 4. Proof of Theorem 1.5 We begin by recalling a lemma on logarithmic derivative due to Gundersen plays an important role in proving Theorem 1.5, it can be found in [7]. Lemma 4.1. Let f be a transcendental meromorphic function of finite order ρ(f ). Let ε > 0 be a given real constant, and let k and j be integers such that k > j ≥ 0. Then there exists a set E1 ⊂ [0, 2π) of linear measure zero, such that if ψ0 ∈ [0, 2π)\E1 , then there is a constant R0 = R0 (ψ0 ) > 1 such that for all z satisfying arg z = ψ0 and |z| ≥ R0 , we have (k) f (z) (k−j)(ρ(f )−1+ε) . f (j) (z) ≤ |z|

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Firstly, we prove that every nontrivial solution of (1.2) is of infinite order. To the end, suppose on the contrary to the assertion that there exists a nontrivial solution f of (1.2) with ρ(f ) < ∞. We aim for a contradiction. n) Set θj = 2jπ−arg(a and Sj = {z : θj < arg z < θj+1 }, j = 0, 1, 2, . . . , n + 1, n+2 if j = n + 1, set θj+1 = θ0 + 2π. Since p(A) ≥ 2, by Lemma 2.1, there exists at least one sector of the n + 2 sectors, such that A(z) decays to zero exponentially, say Sj0 = {z : θj0 < arg z < θj0 +1 }, 0 ≤ j0 ≤ n + 1. That is, for any θ ∈ (θj0 , θj0 +1 ), we have (3.2) holds. In the following, we divide into two cases to B(z). Case 1. 0 < µ(B) < 21 . By using the similar reasoning as in the case 1 of proof of Theorem 1.2, we have (3.5) holds for all |z| = r ∈ E1 , where µ(B)+ 12 , α = . log dens(E1 ) ≥ 1 − µ(B) 0 α0 2 Applying Lemma 4.1, there exists a set E2 ⊂ [0, 2π) of linear measure zero, such that if ψ0 ∈ [0, 2π)\E2 , then there is a constant R0 = R0 (ψ0 ) > 1 such that for all z satisfying arg z = ψ0 and |z| ≥ R0 , (k) f (z) 2ρ(f ) (4.1) , k = 1, 2. f (z) ≤ |z| Thus, there exists a sequence of points zs = rs eiθ with rs → ∞ as s → ∞, rs ∈ E1 and θ ∈ (θj0 , θj0 +1 )\E2 , such that (3.2), (3.5) and (4.1) hold. It follows from (1.2), (3.2), (3.5) and (4.1) that exp(rsµ(B)−ε ) ≤ |B(rs eiθ )| ′′ ′ f (rs eiθ ) f (rs eiθ ) iθ + |A(rs e )| ≤ f (rs eiθ ) f (rs eiθ ) (4.2)

≤ rs2ρ(f ) (1 + o(1))

for sufficiently large s. Obviously, this is a contradiction for arbitrary small ε. Hence we have ρ(f ) = ∞ for every nontrivial solutions f of (1.2). Case 2. µ(B) = 0. By Lemma 2.2, there exists a set E3 ⊂ [0, ∞) with log dens(E3 ) = 1, such that for all z satisfying |z| = r ∈ E3 , we have (3.7) holds. Thus, there exists a sequence of points zs = rs eiθ with rs → ∞ as s → ∞, rs ∈ E3 and θ ∈ (θj0 , θj0 +1 )\E2 , such that (3.2), (3.7) and (4.1) hold. Therefore, we deduce from (1.2), (3.2), (3.7) and (4.1) that √

(4.3)

M (rs , B)

2 2

≤ rs2ρ(f ) (1 + o(1))

for sufficiently large s. But B(z) is a transcendental entire function, we have (3.9) holds. We obtain a contradiction from (3.9) and (4.3). So, ρ(f ) = ∞ for every nontrivial solutions f of (1.2). 2π Secondly, we can prove m(∆(f )) ≥ n+2 by using the similar reasoning in proving Theorem 1.2, we omit the details. This completes the proof.

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5. Proof of Theorem 1.6 2π Set d = n+2 . Suppose on the contrary to the assertion that there exists a nontrivial solution f of (1.2) with m(∆(f ) ∩ ∆(f (k) )) < d. We aim for a contradiction. Set η = d − m(∆(f ) ∩ ∆(f (k) )). By using the idea as in proving Theorem 1.2, in order to finish the proof, we need find an open interval I = (α, β) ⊂ ∆(f (k) )c , 0 < β − α < d, such that ( ) (5.1) m ∆(f ) ∩ Sj′ 0 ∩ I > 0,

where ∆(f (k) )c = [0, 2π)\∆(f (k) ), Sj′ 0 is defined as in the proof of Theorem 1.2. First, we claim that ( ) (5.2) m Sj′ 0 \∆(f ) = 0. If it is not true, then there exist ϕ0 ∈ ∆(f )c and ζ > 0 satisfying ( ) (5.3) m (ϕ0 − ζ, ϕ0 + ζ) ∩ (Sj′ 0 \∆(f )) > 0. Since arg z = ϕ0 is not the radial distribution of J(f ), there exists a constant r0 > 0 such that S(r0 , ϕ0 − ζ, ϕ0 + ζ) ∩ J(f ) = ∅. It follows that there exists an unbounded component U of Fatou set F (f ), such that S(r0 , ϕ0 −ζ, ϕ0 +ζ) ⊂ U . In boundary of U , we take an unbounded and connected set γ ⊂ ∂U , then the mapping f : S(r0 , ϕ0 − ζ, ϕ0 + ζ) → C\γ is analytic. Since C\γ is simply connected, then, for any a ∈ γ\{∞}, we get CC\γ (a) ≥ 21 . For any 0 < ξ < ζ4 , applying Lemma 2.3 to f in S(r0 , ϕ0 − ζ, ϕ0 + ζ), we get (5.4)

|f (z)| = O(|z|l1 ),

z ∈ S(r0 , ϕ0 − ζ + ξ, ϕ0 + ζ − ξ),

|z| → ∞,

where l1 is a positive constant. Hence we get Sϕ0 −ζ+ξ,ϕ0 +ζ−ξ (r, f ) = O(1), and then σϕ0 −ζ+ξ,ϕ0 +ζ−ξ (f ) is finite. Applying Lemma 2.4, there exist two constants M > 0 and K > 0 such that (3.1) holds for all z ∈ S(r0 , ϕ0 − ζ + 2ξ, ϕ0 + ζ − 2ξ), outside a R−set H. Since ξ is arbitrary small, from (5.3), we have ( ) m (ϕ0 − ζ + 2ξ, ϕ0 + ζ − 2ξ) ∩ Sj′ 0 > 0. By the similar reasoning as in the cases 1 and 2 of the proof of Theorem 1.2, then there exists a sequence of points {rs eiϕ }, where ϕ ∈ (ϕ0 − ζ + 2ξ, ϕ0 + ζ − 2ξ) and rs → ∞ as s → ∞, such that (3.5), (3.6) and (3.7) hold for z = rs eiϕ . Combining (1.2), (3.1), (3.5), (3.6) and (3.7), we get a contradiction. Therefore, (5.2) is valid.

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We know that m(∆(f )) ≥ d from Theorem 1.2. It follows from the definition of Sj′ 0 and Lemma 2.1 that m(Sj′ 0 ) ≥ d − 2ε for any small ε > 0. From this and (5.2), we have ( ) η (5.5) m ∆(f ) ∩ Sj′ 0 ≥ d − . 4 Since ∆(f (k) ) is closed, then ∆(f (k) )c consists of at most countable many open intervals. We can choose finite many open intervals Ii such that ( ) m ∪ η Ii ⊂ ∆(f (k) )c , m ∆(f (k) )c \ Ii < , i = 1, 2, . . . , m. 4 i=1 Since m(∆(f ) ∩

Sj′ 0

∩(

m ∪

Ii )) + m(∆(f ) ∩ Sj′ 0 ∩ ∆(f (k) ))

i=1

(

= m ∆(f ) ∩ Sj′ 0 ∩ (∆(f (k) ) ∪ (

m ∪

i=1

then

( m ∆(f ) ∩ Sj′ 0 ∩ (

m ∪

) Ii ))

η ≥d− , 2

) Ii )

i=1

η − m(∆(f ) ∩ Sj′ 0 ∩ ∆(f (k) )) 2 η η ≥ d − − m(∆(f ) ∩ ∆(f (k) )) = > 0. 2 2 m ∪ Thus, there exists an open interval Ij0 = (α, β) ⊂ Ii ⊂ ∆(f (k) )c , such ≥d−

i=1

that ( ) η m ∆(f ) ∩ Sj′ 0 ∩ Ij0 ≥ > 0. 2m Thus (5.1) is valid. From (5.1), we know that there are ϕe0 and ζe > 0, such e ϕe0 + ζ) e ⊂ Ij , and m(∆(f ) ∩ S ′ ∩ (ϕe0 − ζ, e ϕe0 + ζ)) e > 0. Then that (ϕe0 − ζ, 0 j0 e ϕe0 + ζ) e ∩ J(f (k) ) = ∅. By using there exists re0 > 0, such that S(re0 , ϕe0 − ζ, e similar reasoning as in proving (5.4), for any 0 < ξe < ζ6 , we have (5.6) |f (k) (z)| = O(|z|l2 ),

e ϕe0 + ζe − ξ), e z ∈ S(re0 , ϕe0 − ζe + ξ,

|z| → ∞,

where l2 is a positive constant. e ϕe0 + ζe − ξ), e and for any Fix r∗ eiϕ∗ , where r∗ > re0 and ϕ∗ ∈ (ϕe0 − ζe + ξ, e ϕe0 + ζe − ξ). e Take a simple Jordan arc γz in z = reiϕ ∈ S(re0 , ϕe0 − ζe + ξ, e ϕe0 + ζe − ξ) e which connects r∗ eiϕ∗ to r∗ eiϕ along |z| = r∗ S(re0 , ϕe0 − ζe + ξ, and connects r∗ eiϕ to reiϕ along arg z = ϕ. It follows from (5.6) and Cauchy

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integral formula that |f

(k−1)

13

∫

(z)| ≤

|f (k) (z)||dz| + ck ≤ O(|z|l2 +1 ),

|z| → ∞.

γz

Similarly, |f

(k−2)

∫ (z)| ≤

|f (k−1) (z)||dz| + ck−1 ≤ O(|z|l2 +2 ),

|z| → ∞.

γz

By induction, we have ∫ |f (z)| ≤ |f ′ (z)||dz| + c1 ≤ O(|z|l2 +k ),

|z| → ∞,

γz

where ci , i = 1, 2, . . . , k, are positive constants. Therefore, Sϕf0 −ζ+ e ξ, eϕ f0 +ζ− e ξe(r, f ) = O(1), and then σϕf0 −ζ+ e ξ, eϕ f0 +ζ− e ξe(f ) < ∞. Applying Lemma 2.3, we know that (3.1) e ξ, e ϕe0 +ζ−2 e ξ), e outside a R−set H. By applying holds for all z ∈ S(re0 , ϕe0 −ζ+2 similar reasoning as in the cases 1 and 2 of the proof of Theorem 1.2, we can get a contradiction. Therefore, we have ( ) m ∆(f ) ∩ ∆(f (k) ) ≥ d. This completes the proof. 6. Annex remarks In [9], Huang and Wang proved the following result by using the spread relation and P´olya peaks of meromorphic functions. Theorem 6.1. Suppose that Ai (z), i = 0, 1, . . . , k − 1, are entire functions satisfying ρ(Aj ) < µ(A0 ) < ∞, j = 1, 2, . . . , k − 1. Then every nontrivial solution f of the equation (6.1)

f (k) + Ak−1 (z)f (k−1) + · · · + A0 (z)f = 0

π satisfies m(∆(f )) ≥ min{2π, µ(A }. 0)

In this section, we will give a simple proof of Theorem 6.1 than original way which is given in [9], which rely heavily on the following modified Phragm´en-Lindel¨of principle. Lemma 6.2 ([20]). Let f be an entire function of lower order µ(f ) ∈ [ 21 , ∞). Then there exists a sector domain S(α, β) = {z : α < arg z < β} with π , where 0 ≤ α < β ≤ 2π, such that β − α ≥ µ(f ) log log |f (reiθ )| ≥ µ(f ) log r r→∞ for all the ray arg z = θ ∈ (α, β). lim sup

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Another proof of Theorem 6.1. We divide into two cases to finish our proof. π for every Case 1. µ(A0 ) ≥ 12 . We need prove that m(∆(f )) ≥ µ(A 0) nontrivial solution f of (6.1). To the end, suppose on the contrary to the assertion that there exists a nontrivial solution f of (6.1) with m(∆(f )) < π . Set S = (0, 2π)\∆(f ). By Lemma 6.2 to A0 (z), there exists a sector µ(A0 ) π domain S(α, β) with β − α ≥ µ(A , such that 0) (6.2)

lim sup r→∞

log log |A0 (reiθ )| ≥ µ(A0 ) log r

for any θ ∈ (α, β). π π and β − α ≥ µ(A , then there exists a sector Since m(∆(f )) < µ(A 0) 0) ′ ′ ′ ′ domain S(α , β ), such that α < α < β < β and (α′ , β ′ ) ⊂ S. Then for any α′ < θ < β ′ , we have (6.2) holds. For the sector domain S(α′ , β ′ ), it is easy to see that (α′ , β ′ ) ∩ ∆(f ) = ∅,

S(r, α′ , β ′ ) ∩ J(f ) = ∅

for sufficiently large r. This implies that there exists r0 > 0 and unbounded Fatou component U of F (f ) such that S(r0 , α′ , β ′ ) ⊂ U . We take a unbounded and connected section γ of ∂U , then the mapping f : S(r0 , α′ , β ′ ) → C\γ is analytic. Since we have chosen γ such that C\γ is simply connected, for any a ∈ γ\{∞}, we have CC\γ (a) ≥ 12 . Applying Lemma 2.3 to f , there exists a positive constant l such that (6.3)

|f (z)| = O(|z|l ), ′

z ∈ S(r0 , α′ + ε, β ′ − ε),

|z| → ∞,

′

where 0 < ε < β −α . Thus we immediately obtain Sα′ +ε,β ′ −ε (r, f ) = O(1), 8 and then σα′ +ε,β ′ −ε (f ) < ∞. Applying Lemma 2.4, there exist two constants M > 0 and K > 0, such that (j) f (z) M (6.4) f (z) ≤ Kr , j = 1, 2, . . . , k, for all z ∈ S(α′ + 2ε, β ′ − 2ε), outside a R−set H. Let max {ρ(Aj )} = η, and δ ∈ (η + ε, µ(A0 ) − 2ε) be a constant. Since 1≤j≤k−1

ρ(Aj ) < µ(A0 ), j = 1, 2, . . . , k − 1, then there exists a constant r1 > r0 , such that for any |z| = r > r1 , we have (6.5)

|Aj (z)| ≤ exp(rδ ),

j = 1, 2, . . . , k − 1.

Thus there exists a sequence of points zn = rn eiθ outside H, rn → ∞ as n → ∞, α′ < θ < β ′ , such that (6.2), (6.4) and (6.5) hold. It follows from

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(6.1), (6.2), (6.4) and (6.5) that exp(rnµ(A0 )−ε ) ≤ |A0 (rn eiθ )| (k) ′ f (rn eiθ ) f (rn eiθ ) iθ + · · · + |A1 (rn e )| ≤ f (rn eiθ ) f (rn eiθ ) (6.6)

≤ KrnM (k − 1) exp(rnδ ).

Obviously, this is a contradiction for sufficiently large n. Case 2. µ(A0 ) < 12 . We need prove that m(∆(f )) = 2π for every nontrivial solution f of (6.1). Suppose on the contrary to the assertion that there exists a nontrivial solution f of (6.1) with m(∆(f )) < 2π. Then, by similar reasoning as in case 1 above, there exist (α, β) ⊂ [0, 2π)\∆(f ) and constant r0 > 1 such that S(r0 , α, β) ⊂ F (f ), and (6.3) holds for z ∈ S(r, α+ε, β −ε). Hence we have Sα+ε,β−ε (r, f ) = O(1), and then σα+ε,β−ε (f ) < ∞. Applying Lemma 2.4, we see that (6.4) holds for all z ∈ S(r, α + 2ε, β − 2ε) outside a R−set H. Since µ(A0 ) < 21 , applying Lemma 2.2 to A0 (z), there exists a set E1∗ ⊂ µ(A )+ 1

0 0) 2 , where α0 = , E1∗ = {r ∈ [1, ∞) : [1, ∞) with log dens(E1∗ ) ≥ 1− µ(A α0 2 m(r) > M (r) cos πα0 }, m(r) = inf log |A0 (z)|, and M (r) = sup log |A0 (z)|.

|z|=r

|z|=r

Thus, there exists a constant R0 > 1 such that (6.7)

|A0 (z)| > exp(rµ(A0 )−ε )

for all |z| = r ∈ E1 = E1∗ \[0, R0 ]. Thus, there exists a sequence of points zn = rn eiθ outside H, rn → ∞ as n → ∞, α < θ < β, such that (6.4), (6.5) and (6.7) hold. It follows from (6.1), (6.4), (6.5) and (6.7) that exp(rnµ(A0 )−ε ) ≤ KrnM (k − 1) exp(rnδ ). Obviously, this is a contradiction for sufficiently large n. This completes the proof. Acknowledgements. This research is supported by the National Natural Science Foundation of China (Grant No. 11501142), and the Foundation of Science and Technology of Guizhou Province of China (Grant No. [2015] 2112). References [1] I. N. Baker, Sets of non-normality in iteration theory, J. London Math. Soc. 40 (1965), 499-502. [2] I. N. Baker, The domains of normality of an entire function, Ann. Acad. Sci. Fenn. Math. 1 (1975), no. 2, 277-283.

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[3] P. D. Barry, Some theorems related to the cos πρ theorem, Proc. London. Math. Soc. 21 (1970), no. 3, 334-360. [4] W. Bergweiler, Iteration of meromorphic functions, Bull. Amer. Math. Soc. 29 (1993), 151-188. [5] A. A. Gol’dberg and I. V. Ostrovskii, Value Distribution of Meromorphic Functions, Translations of Mathematical Monographs, vol. 236, American Mathematical Society, Providence, RI, 2008. [6] G. G. Gundersen, On the real zeros of solutions of f ′′ + A(z)f = 0 where A(z) is entire, Ann. Acad. Sci. Fenn. Math. 11 (1986), 275-294. [7] G. G. Gundersen, Estimates for the logarithmic derivative of a meromorphic function, plus similar estimates, J. London Math. Soc. 37 (1988), 88-104. [8] Z. G. Huang and J. Wang, On the radial distribution of Julia sets of entire solutions of f (n) + A(z)f = 0, J. Math. Anal. Appl. 409 (2012), no. 2, 1106-1113. [9] Z. G. Huang and J. Wang, On limit directions of Julia sets of entire solutions of linear differential equations, J. Math. Anal. Appl. 387 (2014), no. 1, 478-484. [10] W. Hayman, Meromorphic Functions, Clarendon Press, Oxford, 1964. [11] E. Hille, Lectures on Ordinary Differntial Equations, Addison–Wesley Publishing Company, Reading, Massachusetts-Menlo Park, California– London–Don Mills, Ontario, 1969. [12] I. Laine, Nevanlinna Theory and Complex Differential Equations, Walter de Gruyter Berlin, New York, 1993. [13] J. R. Long, P. C. Wu and X. B. Wu, On the zero distribution of solutions of second order linear differential equations in the complex plane, New Zealand J. Math. 42 (2012), 9-16. [14] J. R. Long and K. E. Qiu, Growth of solutions to a second-order complex linear differential equation, Math. Pract. Theory 45 (2015), no. 2, 243-247. [15] A. Z. Mokhon’ko, An estimate of the modules of the logarithmic derivative of a function which is meromorphic in an angular region, and its application, Ukrain Mat. Zh. 41 (1989), no. 6, 839-843. [16] J. Y. Qiao, Stable sets for iterations of entire functions, Acta Math. Sinica 37 (1994), no. 5, 702-708. [17] L. Qiu and S. J. Wu, Radial distributions of Julia sets of meromorphic functions, J. Austra. Math. Soc. 81 (2006), no. 3, 363-368.

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[18] J. Rossi and S. P. Wang, The radial osillation of solutions to ODE’s in the complex domain, Proc. Edinburgh Math. Soc. 39 (1996), 473-483. [19] K. C. Shin, New polynomials P for which f ′′ +P (z)f = 0 has a solution with almost all real zeros, Ann. Acad. Sci. Fenn. Math. 27 (2002), 491498. [20] X. B. Wu, J. R. Long, J. Heittokangas and K. E. Qiu, Secondorder complex linear differential equations with special functions or extremal functions as coefficients, Electron. J. Differential Equations 2015 (2015), No. 143, 1-15. [21] S. J. Wu, On the location of zeros of solution of f ′′ + Af = 0 where A(z) is entire, Math. Scand. 74 (1994), 293-312. [22] S. J. Wu, Angular distribution in the complex oscilation theory, Sci. China Ser. A 48 (2005), no. 1, 107-114. [23] L. Yang, Value Distribution Theory, Springer-Verlag, Berlin, 1993. [24] G. W. Zhang, J. Wang and L.Z. Yang, On radial distribution of Julia sets of solutions to certain second order complex linear differential equations, Abstr. Appl. Anal. 2014 (2014), Article ID 842693, 1-6. [25] J. H. Zheng, S. Wang and Z. G. Huang, Some properties of Fatou and Julia sets of transcendental meromorphic functions, Bull. Austra. Math. Soc. 66 (2002), no. 1, 1-8. Jianren Long School of Mathematicial Sciences, Guizhou Normal University, Guiyang, 550001, P.R. China. E-mail address: [email protected], [email protected]

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Common Fixed Point Results for the Family of Multivalued Mappings Satisfying Contractions on a Sequence in Hausdor¤ Fuzzy Metric Space Abdullah Shoaib1 , Akbar Azam2 , and Aqeel Shahzad3 Abstract: The aim of this paper is to establish common …xed point results on a sequence contained in a closed ball for family of multivalued mapping in complete fuzzy metric space. Simple and di¤erent technique has been used. Example has been constructed to demonstrate the novelty of our results. Our results unify, extend and generalize several results in the existing literature. _____________________ 2010 Mathematics Subject Classi…cation: 46S40; 47H10; 54H25. Keywords and Phrases: common …xed point; complete fuzzy metric space; closed ball; family of multivalued mappings; Hausdor¤ fuzzy metric space. _____________________

1

Introduction and Preliminaries

The notion of fuzzy sets was …rst introduced by Zadeh [5]. Kramosil et al. [10] introduced the concept of fuzzy metric space and obtained many …xed point results. Later on many authors [7, 8, 9, 11] used this concept and prove many …xed point results using the di¤erent contractive conditions. Lopez et al. [11] discuss the method for constructing a Hausdor¤ fuzzy metric on nonempty compact subsets of a given fuzzy metric space. Sometimes, it happens that the …xed point of a mapping exists, but the contraction does not hold. Recently, Shoaib et al. [1, 2, 3, 4, 6, 13] obtained the necessary and su¢ cient conditions for the existence of a …xed point of such self mapping. In this paper, we prove the existence of a common …xed point of a family of such multivalued mappings which are contractive on a sequence contained in a closed ball instead of the whole space, by using the concept of Hausdor¤ fuzzy metric space. We also present an example to support our results. De…nition 1.1 [7] A binary operation : [0; 1] [0; 1] ! [0; 1] is said to be a continuous t-norm if it is satis…es the following conditions: i) is associative and commutative; ii) is continuous; iii) a 1 = a for all a 2 [0; 1]; iv) a b c d whenever a c and b d for each a; b; c; d 2 [0; 1]. De…nition 1.2 [10] The 3-tuple (X; M; ) is said to be a fuzzy metric space if X is an arbitrary set, is a continuous t-norm, and F is a fuzzy set on X 2 [0; 1), satisfying the following conditions for all x; y; z 2 X and t; s > 0: 1

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F1) F (x; y; 0) = 0; F2) F (x; y; t) = 1 if and only if x = y; F3) F (x; y; t) = F (y; x; t); F4) F (x; z; t + s) F (x; y; t) F (y; z; s); F5) F (x; y; :) : (0; 1) ! [0; 1] is left-continuous. Example 1.3 [7] Let (X; d) be a metric space. De…ne a b = ab and F (x; y; t) =

ktn ; ktn + md(x; y)

for all x; y 2 X and k; m; n 2 R+ . Then (X; F; ) is a fuzzy metric space. De…nition 1.4 [9] Let (X; F; ) be a fuzzy metric space. Then, we have i) A sequence fxn g in X is said to be convergent to a point x 2 X denoted xn ! x; if lim F (xn ; x; t) = 1 for each t > 0. n!1

ii) A sequence fxn g in X is said to be a Cauchy sequence, if lim F (xn ; xn+p ; t) = n!1 1 for each t > 0, p > 0. iii) A fuzzy metric space (X; F; ) in which every Cauchy sequence is convergent is called a complete fuzzy metric space. De…nition 1.5 [11] Let (X; F; ) be a fuzzy metric space. De…ne a function HF M on C^0 (X) C^0 (X) (0; 1) by HF M (A; B; t) = min

inf F (a; B; t); inf F (A; b; t) ;

a2A

b2B

for all A; B 2 C^0 (X) and t > 0, where C^0 (X) is the collection of all nonempty compact subsets of X. De…nition 1.6 [7] Let (X; F; ) be a fuzzy metric space. Then, BF (x; r; t) = fy 2 X : F (x; y; t) > 1

rg

BF (x; r; t) = fy 2 X : F (x; y; t)

rg

and 1

are called open and closed balls respectively, with centre x 2 X and radius r for 0 < r < 1, t > 0. Lemma 1.7 [11] Let (X; F; ) be a complete fuzzy metric space. Then, for each a 2 X, B 2 C^0 (X) and for all t > 0 there is bo 2 B such that F (a; bo ; t) = F (a; B; t): Lemma 1.8 [12] Let (X; F; ) be a complete fuzzy metric space. (C^0 (X); HF M ; ) is a hausdor¤ fuzzy metric space on C^0 (X). Then, for all A; B 2 C^0 (X), for each a 2 A and for all t > 0 there exists ba 2 B; satis…es F (a; B; t) = F (a; ba ; t); then HF M (A; B; t) F (a; ba ; t):

2

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2

Main Results

Let (X; F; ) be a fuzzy metric space, x0 2 X and let fS : 2 g be a family of multivalued mappings from X to C^0 (X). Then, there exists x1 2 Sa x0 for some a 2 , such that F (x0 ; Sa x0 ; t) = F (x0 ; x1 ; t); for all t > 0: Let x2 2 Sb x1 be such that F (x1 ; Sb x1 ; t) = F (x1 ; x2 ; t): Continuing this process, we construct a sequence xn of points in X such that xn+1 2 S xn , F (xn ; S xn ; t) = F (xn ; xn+1 ; t); for all t > 0: We denote this iterative sequence fXS (xn ) : 2 g and say that fXS (xn ) : 2 g is a sequence in X generated by x0 . Theorem 2.1 Let (X; F; ) be a complete fuzzy metric space, where be a continuous t-norm, de…ned as a a a or a b = minfa; bg. Let (C^0 (X); HF M ; ) be a Hausdor¤ fuzzy metric space on C^0 (X), fS : 2 g be a family of multivalued mappings from X to C^0 (X) and fXS (xn ) : 2 g be a sequence in X generated by x0 . Assume that, for some 0 < i;j k < 1; for all t > 0, x0 2 X, for all x; y 2 BF (x0 ; r; t) \ fXS (xn ) : 2 g; with x 6= y and for all i; j 2 with i 6= j; we have HF M (Si x; Sj y;

i;j t)

F (x; y; t)

(2.1)

and, for some t > 0 F (x0 ; x1 ; (1

k)t))

1

r:

(2.2)

Then, fXS (xn ) : 2 g is a sequence in BF (x0 ; r; t) and fXS (xn ) : 2 g ! z 2 BF (x0 ; r; t): Also, if (2.1) holds for z; then there exists a common …xed point for the family of multivalued mappings fS : 2 g in BF (x0 ; r; t). Proof: Let fXS (xn ) : 2 g be a sequence in X generated by x0 . If x0 = x1 , then x0 is a common …xed point of Sa for all a 2 . Let x0 6= x1 and by Lemma 1:8; we have F (x1 ; x2 ; t) HF M (Sa x0 ; Sb x1 ; t): By induction, we have by Lemma 1:8; we have F (xn ; xn+1 ; t)

HF M (Si xn

1; S

xn ; t):

(2.3)

First, we will show that xn 2 BF (x0 ; r; t). By (2:2); we get F (x0 ; x1 ; t) = F (x0 ; Sa x0 ; t) > F (x0 ; x1 ; (1 F (x0 ; x1 ; t) > 1 r:

k)t)

1

r

3

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This shows that x1 2 BF (x0 ; r; t): Let x2 ; F (xj ; xj+1 ; t)

HF M (S xj

1; S

; xj 2 BF (x0 ; r; t): Now, we have xj ; t)

F (xj

1 ; xj ;

t

)

;

HF M (S xj

2; S

xj

t

1;

)

(by Lemma 1:8)

;

F (xj F (xj F (xj ; xj+1 ; t)

t ;m ;

2 ; xj 1 ; 2 ; xj 1 ;

F (x0 ; x1 ;

t ) k2

) ;

::::

F (x0 ; x1 ;

t ) kj

t ) kj

(2.4)

Now, F (x0 ; xj+1 ; t)

F (x0 ; xj+1 ; t)

F (x0 ; xj+1 ; (1 k j+1 )t) F (x0 ; x1 ; (1 k)t) F (x1 ; x2 ; (1 k)kt) :::: F (xj ; xj+1 ; (1 k)k j t) F (x0 ; x1 ; (1 k)t) F (x0 ; x1 ; (1 k)t) :::: F (x0 ; x1 ; (1 k)t) (by (2.4)) 1 r 1 r :: 1 r = 1 r 1 r:

This implies that xj+1 2 BF (x0 ; r; t): Now, inequality (2.4) can be written as F (xn ; xn+1 ; t)

F (x0 ; x1 ;

t ): kn

(2.5)

Let n; m 2 N with m > n: Assume that m = n + p; we have F (xn ; xn+p ; t)

F (xn ; xn+1 ; (1 F (xn ; xn+1 ; (1 F (xn ; xn+1 ; (1

k)t) F (xn+1 ; xn+p ; kt) k)t) HF M (Sj xn ; Sk xn+p 1 ; kt) kt k)t) F (xn ; xn+p 1 ; ) j;k

F (xn ; xn+1 ; (1 k)t) F (xn ; xn+1 ; (1 k)t) F (xn+1 ; xn+p 1 ; (1 F (xn ; xn+1 ; (1 k)t) HF M (Sj xn ; Sl xn+p F (xn ; xn+1 ; (1 k)t) kt F (xn ; xn+p 2 ; )

F (xn ; xn+p 1 ; t) F (xn ; xn+1 ; (1 k)t) k)t) F (xn ; xn+1 ; (1 k)t) 2 ; kt) F (xn ; xn+1 ; (1 k)t)

j;l

F (xn ; xn+p ; t)

F (xn ; xn+1 ; (1 k)t) F (xn ; xn+1 ; (1 F (xn ; xn+p 2 ; t):

k)t)

4

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Using the above, we have F (xn ; xn+p ; t)

F (xn ; xn+1 ; (1 k)t) F (xn ; xn+1 ; (1 k)t) ::: F (xn ; xn+1 ; t) F (xn ; xn+1 ; (1 k)t) F (xn ; xn+1 ; (1 k)t) ::: F (xn ; xn+1 ; (1 k)t) (1 k)t (1 k)t F (x0 ; x1 ; ) F (x0 ; x1 ; ) ::: n k kn (1 k)t F (x0 ; x1 ; ) (by (2.5)) kn (1 k)t F (x0 ; x1 ; ): kn

F (xn ; xn+p ; t)

F (xn ; xn+p ; t) As, we have

lim F (x; y; t) = 1 for all x; y 2 X:

t!1

In particular F (x0 ; x1 ;

(1

k)t ) = 1 as n ! 1: kn

By using above, we get F (xn ; xm ; t) = 1 as n ! 1: Hence, fXS (xn )g is a Cauchy sequence in BF (x0 ; r; t): As every closed ball in a complete fuzzy metric space is complete. So, BF (x0 ; r; t) is complete. Then, there exists z 2 BF (x0 ; r; t), such that xn ! z as n ! 1: Now, for some q 2 ; we have F (z; Sq z; t) F (z; xn ; (1 k)t) F (xn ; Sq z; kt): By Lemma 1:8, we have F (z; Sq z; t)

F (z; xn ; (1

k)t) HF M (Sr xn

F (z; xn ; (1

k)t) F (xn

1 ; Sq z; kt) kt ) 1 ; z;

F (z; xn ; (1

k)t) F (xn

1 ; z; t):

F (z; Sq z; t)

1 1 = 1:

r;q

Letting n ! 1; we have

This implies that z 2 Sq z: Hence, z 2 \ Sq z. This completes the proof. q2

Let (X; F; ) be a fuzzy metric space, x0 2 X and let S be a multivalued mapping from X to C^0 (X). Then, there exists x1 2 Sx0 , such that F (x0 ; Sx0 ; t) = F (x0 ; x1 ; t); for all t > 0: Let x2 2 Sx1 be such that F (x1 ; Sx1 ; t) = F (x1 ; x2 ; t): 5

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Continuing this process, we construct a sequence xn of points in X such that xn+1 2 Sxn , F (xn ; Sxn ; t) = F (xn ; xn+1 ; t); for all t > 0: We denote this iterative sequence fXS(xn )g and say that fXS(xn )g is a sequence in X generated by x0 . Corollary 2.2 Let (X; F; ) be a complete fuzzy metric space, where be a continuous t-norm, de…ned as a a a or a a = minfa; bg. Let (C^0 (X); HF M ; ) is Hausdor¤ fuzzy metric space on C^0 (X), x0 2 X, S : X ! C^0 (X) be a multivalued mapping and fXS(xn )g be a sequence in X generated by xo . Assume that for some k 2 (0; 1) t > 0, and xo 2 X, we have HF M (Sx; Sy; kt)

F (x; y; t) for all x; y 2 BF (x0 ; r; t) \ fXS(xn )g

(2.6)

and F (x0 ; Sx0 ; (1

k)t))

1

r

Then, fXS(xn )g is a sequence in BF (x0 ; r; t) and fXS(xn )g ! z 2 BF (x0 ; r; t): Also, if (2.6) holds for z; then there exists a …xed point for S in BF (x0 ; r; t). Proof: By using the similar steps as we have used in Theorem 2.1, it can be proved easily. Corollary 2.3 Let (X; F; ) be a complete fuzzy metric space, where be a continuous t-norm, de…ned as a a a or a a = minfa; bg. Let x0 2 X and S : X ! X be a self mapping. Assume that for some k 2 (0; 1), t > 0 and xo 2 X, we have F (Sx; Sy; kt)

F (x; y; t) for all x; y 2 BF (x0 ; r; t)

and F (x0 ; Sx0 ; (1

k)t))

1

r:

Then S has a …xed point in BF (x0 ; r; t). Example 2.4 Let X = [0; 5] and d : X X ! R be a complete metric space de…ned by, d(x; y) = jx yj for all x; y 2 X t Denote a b = ab or a b = minfa; bg for all a; b 2 [0; 1] and F (x; y; t) = t+d(x;y) for all x; y 2 X and t > 0. Then, we can …nd that (X; F; ) is a complete fuzzy metric space. Consider the multivalued mappings S : X ! C^0 (X) where = a; 1; 2; 3; : de…ned as, 8 x x ; 2n ] if x 2 [0; 72 ] < [ 3n ; where n = 1; 2; ; Sn x = : [2nx; 3nx] if x 2 ( 72 ; 5]

and

Sa x =

8 x 5x < [ 3 ; 12 ] if x 2 [0; 27 ] :

:

[2x; 3x] if x 2 ( 72 ; 5]

6

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Considering, x0 =

1 2

and r = 34 . then, BF (x0 ; r; t) = [0; 72 ]. Now,

F (x0 ; Sa x0 ; t) F (x1 ; S1 x1 ; t)

1 1 1 5 = F ( ; Sa ; t) = F ( ; ; t) 2 2 2 24 5 5 5 5 = F ( ; S1 ; t) = F ( ; ; t) 24 24 24 48

5 5 5 ; 48 ; 192 ; :::g in X generated by So, we obtain a sequence fXS (xn )g = f 21 ; 24 5 x0 : Now, for x = 4, y = 5, k = 1;a = 6 and t = 1; we have

5 HF M (S1 4; Sa 5; ) 6

=

F (4; 5; 1)

=

5 5 inf F (a; Sa 5; ); inf F (S1 4; b; ) a2S1 4 6 b2Sa 5 6 1 1 = = 0:5 1 + j4 5j 2

min

= 0:22

So, we have 5 HF M (S1 4; Sa 5; ) 6

F (4; 5; 1)

So, the contractive condition does not hold on X. Now, for all x; y 2 BF (x0 ; r; t)\ fXS (xn )g, we have HF M (Sn x; Sa y; kt)

=

min

inf F (a; Sa y; kt); inf F (Sn x; b; kt)

a2Sn x

b2Sa y

y 5y 5 x x 5 inf F (a; [ ; ]; t); inf F ([ ; ]; b; t) 3 12 6 b2Sa y 3n 2n 6 x 5y 5 x y 5 = min F ( ; ; t); F ( ; ; t) 2n 12 6 3n 3 6 (5=6)t (5=6)t = min ; (5=6)t + jx=3 y=3j (5=6)t + jy=3 x=3j t (5=6)t = F (x; y; t) HF M (Sx; Sy; kt) = (5=6)t + jx=3 y=3j t + jx yj =

min

a2Sn x

So, the contractive condition holds on BF (x0 ; r; t) \ fXS (xn )g. Also, for t = 1; we have F (x0 ; x1 ; (1

1 5 1 = F( ; ; ) 2 24 6 4 1 = > =1 11 4

k)t))

r

Hence, all the conditions of above theorem are satis…ed. Now, we have fXS (xn )g is a sequence in BF (x0 ; r; t), and fXS (xn )g ! 0 2 BF (x0 ; r; t). Moreover, fS : = a; 1; 2 g has a common …xed point 0. Competing interests The authors declare that they have no competing interests. 7

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References [1] M. Arshad, A. Shoaib, I. Beg, Fixed point of a pair of contractive dominated mappings on a closed ball in an ordered complete dislocated metric space, Fixed Point Theory and Appl. (2013), 2013:115. [2] M. Arshad, A. Shoaib, and P. Vetro, Common Fixed Points of a Pair of Hardy Rogers Type Mappings on a Closed Ball in Ordered Dislocated Metric Spaces, Journal of Function Spaces, 2013 (2013), Article ID 63818. [3] M. Arshad , A. Shoaib, M. Abbas and A. Azam, Fixed Points of a pair of Kannan Type Mappings on a Closed Ball in Ordered Partial Metric Spaces, Miskolc Mathematical Notes, 14(3), 2013. [4] M. Arshad, A. Azam, M. Abbas and A. Shoaib, Fixed point results of dominated mappings on a closed ball in ordered partial metric spaces without continuity, U.P.B. Sci. Bull., Series A, 76(2), 2014. [5] L. A. Zadeh, Fuzzy Sets. Information and Control, 8(3), 1965. [6] I. Beg, M. Arshad , A. Shoaib, Fixed Point on a Closed Ball in ordered dislocated Metric Space, Fixed Point Theory, 16(2), 2015. [7] A. George and P. Veeramani, On some results in fuzzy metric spaces, Fuzzy Sets and Systems 64(3), 1994. [8] A. George and P. Veeramani, On some results of analysis for fuzzy metric spaces, Fuzzy Sets and Systems, 90(3), 1997. [9] D. Gopal, C. Vetro, Some New Fixed Point Theoerms In Fuzzy Metric Spaces, Iranian Journal of Fuzzy Systems, 11(3), 2014. [10] I. Kramosil, J. Michalek, Fuzzy metric and statistical metric spaces, Kybernetika 11(5) (1975). [11] J. Rodriguez-Lopez, S. Romaguera, The Hausdor¤ fuzzy metric on compact sets, Fuzzy Sets and Systems 147(2), 2004. [12] A. Shoaib, Ph.D thesis, Fixed points theorems for locally and globally contractive mappings in ordered spaces, 2016. [13] A. Shoaib, M. Arshad and M. A. Kutbi, Common …xed points of a pair of Hardy Rogers Type Mappings on a Closed Ball in Ordered Partial Metric Spaces, J. Comput. Anal. Appl., 17(2014), 255-264. 1;3

Department of Mathematics and Statistics, Riphah International University, Islamabad - 44000, Pakistan. 1 E-mail address: [email protected]. 3 E-mail address: [email protected]. 2 Department of Mathematics, COMSATS Institute of Information Technology, Chack Shahzad, Islamabad - 44000, Pakistan. E-mail address:[email protected]. 8

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ADDITIVE-QUADRATIC ρ-FUNCTIONAL INEQUALITIES IN NON-ARCHIMEDEAN BANACH SPACES CHOONKIL PARK1 , JUNG RYE LEE2 , AND DONG YUN SHIN3∗ Abstract. Let 3 1 f (x + y) − f (−x − y) 4 4 1 1 + f (x − y) + f (y − x) − f (x) − f (y), 4 4      x+y x−y y−x M2 f (x, y) : = 2f +f +f − f (x) − f (y). 2 2 2 We solve the additive-quadratic ρ-functional inequalities M1 f (x, y) :

=

kM1 f (x, y)k ≤ kρM2 f (x, y)k,

(0.1)

where ρ is a fixed non-Archimedean number with |ρ| < 1, and kM2 f (x, y)k ≤ kρM1 f (x, y)k,

(0.2)

where ρ is a fixed non-Archimedean number with |ρ| < |2|. Furthermore, we prove the Hyers-Ulam stability of the additive-quadratic ρ-functional inequalities (0.1) and (0.2) in non-Archimedean Banach spaces.

1. Introduction and preliminaries A valuation is a function | · | from a field K into [0, ∞) such that 0 is the unique element having the 0 valuation, |rs| = |r| · |s| and the triangle inequality holds, i.e., |r + s| ≤ |r| + |s|,

∀r, s ∈ K.

A field K is called a valued field if K carries a valuation. The usual absolute values of R and C are examples of valuations. Let us consider a valuation which satisfies a stronger condition than the triangle inequality. If the triangle inequality is replaced by |r + s| ≤ max{|r|, |s|},

∀r, s ∈ K,

then the function | · | is called a non-Archimedean valuation, and the field is called a nonArchimedean field. Clearly |1| = | − 1| = 1 and |n| ≤ 1 for all n ∈ N. A trivial example of a non-Archimedean valuation is the function | · | taking everything except for 0 into 1 and |0| = 0. Throughout this paper, we assume that the base field is a non-Archimedean field, hence call it simply a field. Definition 1.1. ([8]) Let X be a vector space over a field K with a non-Archimedean valuation | · |. A function k · k : X → [0, ∞) is said to be a non-Archimedean norm if it satisfies the following conditions: 2010 Mathematics Subject Classification. Primary 46S10, 39B62, 39B52, 47S10, 12J25. Key words and phrases. Hyers-Ulam stability; non-Archimedean normed space; additive-quadratic ρ-functional inequality. ∗ Corresponding author: Dong Yun Shin (email: [email protected]).

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(i) kxk = 0 if and only if x = 0; (ii) krxk = |r|kxk (r ∈ K, x ∈ X); (iii) the strong triangle inequality kx + yk ≤ max{kxk, kyk},

∀x, y ∈ X

holds. Then (X, k · k) is called a non-Archimedean normed space. Definition 1.2. (i) Let {xn } be a sequence in a non-Archimedean normed space X. Then the sequence {xn } is called Cauchy if for a given ε > 0 there is a positive integer N such that kxn − xm k ≤ ε for all n, m ≥ N . (ii) Let {xn } be a sequence in a non-Archimedean normed space X. Then the sequence {xn } is called convergent if for a given ε > 0 there are a positive integer N and an x ∈ X such that kxn − xk ≤ ε for all n ≥ N . Then we call x ∈ X a limit of the sequence {xn }, and denote by limn→∞ xn = x. (iii) If every Cauchy sequence in X converges, then the non-Archimedean normed space X is called a non-Archimedean Banach space. The stability problem of functional equations originated from a question of Ulam [19] concerning the stability of group homomorphisms. The functional equation f (x + y) = f (x) + f (y) is called the Cauchy equation. In particular, every solution of the Cauchy equation is said to be an additive mapping. Hyers [7] gave a first affirmative partial answer to the question of Ulam for Banach spaces. Hyers’ Theorem was generalized by Aoki [2] for additive mappings and by Rassias [12] for linear mappings by considering an unbounded Cauchy difference. A generalization of the Rassias theorem was obtained by G˘avruta [6] by replacing the unbounded Cauchy difference  by a general control  x+y = 21 f (x) + 21 f (y) is function in the spirit of Rassias’ approach. The functional equation f 2 called the Jensen equation. The functional equation f (x + y) + f (x − y) = 2f (x) + 2f (y) is called the quadratic functional equation. In particular, every solution of the quadratic functional equation is said to be a quadratic mapping. The stability of quadratic functional equation was proved by Skof [18] for mappings f : E1 → E2 , where E1 is a normed space and E2 is a Banach space. Cholewa [5] noticed that the theorem of Skof isstill true if the relevant domain E1 is replaced by an Abelian group. The functional equation 2f x+y +2 x−y = f (x)+f (y) is called a Jensen type quadratic equation. The 2 2 stability problems of various functional equations have been extensively investigated by a number of authors (see [1, 3, 4, 10, 11, 13, 14, 15, 16, 17, 20, 21]). In Section 2, we solve the additive-quadratic ρ-functional inequality (0.1) and prove the HyersUlam stability of the additive-quadratic ρ-functional inequality (0.1) in non-Archimedean Banach spaces. In Section 3, we solve the additive-quadratic ρ-functional inequality (0.2) and prove the HyersUlam stability of the additive-quadratic ρ-functional inequality (0.2) in non-Archimedean Banach spaces. Throughout this paper, assume that X is a non-Archimedean normed space and that Y is a non-Archimedean Banach space. Let |2| 6= 1.

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2. Additive-quadratic ρ-functional inequality (0.1) in non-Archimedean normed spaces Throughout this section, assume that ρ is a fixed non-Archimedean number with |ρ| < 1. In this section, we solve the additive-quadratic ρ-functional inequality (0.1) in non-Archimedean normed spaces. Lemma 2.1. (i) If an odd mapping f : X → Y satisfies kM1 f (x, y)k ≤ kρM2 f (x, y)k

(2.1)

for all x, y ∈ X, then f : X → Y is additive. (ii) If an even mapping f : X → Y satisfies (2.1), then f : X → Y is quadratic. Proof. (i) Assume that f : X → Y satisfies (2.1). Since f is an odd mapping, f (0) = 0. Letting y = x in (2.1), we get kf (2x) − 2f (x)k ≤ 0 and so f (2x) = 2f (x) for all x ∈ X. Thus x 2

 

f

1 = f (x) 2

(2.2)

for all x ∈ X. It follows from (2.1) and (2.2) that



x+y − f (x) − f (y)

2 = |ρ|kf (x + y) − f (x) − f (y)k 





kf (x + y) − f (x) − f (y)k ≤

ρ 2f and so

f (x + y) = f (x) + f (y) for all x, y ∈ X. (ii) Assume that f : X → Y satisfies (2.1). Letting x = y = 0 in (2.1), we get kf (0)k ≤ k2ρf (0)k = |2| · |ρ| · kf (0)k. So f (0) = 0. Letting y = x in (2.1), we get

1

f (2x) − 2f (x) ≤ 0

2

and so f (2x) = 4f (x) for all x ∈ X. Thus x 2

 

f

1 = f (x) 4

(2.3)

for all x ∈ X.

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It follows from (2.1) and (2.3) that



1

f (x + y) + 1 f (x − y) − f (x) − f (y)

2 2

     

x+y x−y

≤ ρ 2f + 2f − f (x) − f (y)

2 2

1

1

= |ρ|

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

and so f (x + y) + f (x − y) = 2f (x) + 2f (y) for all x, y ∈ X.



We prove the Hyers-Ulam stability of the additive-quadratic ρ-functional inequality (2.1) in nonArchimedean Banach spaces for an odd mapping case. Theorem 2.2. Let r < 1 and θ be nonnegative real numbers and let f : X → Y be an odd mapping such that kM1 f (x, y)k ≤ kρM2 f (x, y)k + θ(kxkr + kykr )

(2.4)

for all x, y ∈ X. Then there exists a unique additive mapping A : X → Y such that 2θ kxkr |2|r

(2.5)

kf (2x) − 2f (x)k ≤ 2θkxkr

(2.6)

kf (x) − A(x)k ≤ for all x ∈ X. Proof. Since f is an odd mapping, f (0) = 0. Letting y = x in (2.4), we get




≤ 2r θkxkr for all x ∈ X. Hence |2|

   

l x

2 f x − 2m f

l 2 2m

         

l

m−1 x x x x l+1 m

2

≤ max 2 f − 2 f , · · · , f − 2 f

2l 2l+1 2m−1 2m

 

       

x x x x l m−1

= max |2| f − 2f , · · · , |2| f − 2f

2l 2l+1 2m−1 2m ( )

for all x ∈ X. So f (x) − 2f

≤ max

x 2

|2|l |2|m−1 , · · · , |2|rl+r |2|r(m−1)+r

2θkxkr =

2θ

|2|(r−1)l+r

(2.7)

kxkr

for all nonnegative integers m and l with m > l and all x ∈ X. It follows from (2.7) that the sequence {2n f ( 2xn )} is a Cauchy sequence for all x ∈ X. Since Y is complete, the sequence {2n f ( 2xn )} converges. So one can define the mapping A : X → Y by x A(x) := lim 2n f ( n ) n→∞ 2 for all x ∈ X. Moreover, letting l = 0 and passing the limit m → ∞ in (2.7), we get (2.5).

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It follows from (2.4) that

 

x y

kM1 A(x, y)k = lim |2| M1 f , n→∞ 2n 2n

 

x y |2|n θ

M f + lim ≤ lim |2|n |ρ| , (kxkr + kykr )

2 n→∞ 2n 2n n→∞ |2|nr n

= |ρ| kM2 A(x, y)k = kρM2 A(x, y)k for all x, y ∈ X. So kM1 A(x, y)k ≤ kρM2 A(x, y)k for all x, y ∈ X. By Lemma 2.1, the mapping A : X → Y is additive . Now, let T : X → Y be another additive mapping satisfying (2.5). Then we have

   

q x x q

2 A kA(x) − T (x)k = − 2 T

q 2 2q          

q

x 2θ x q

, 2q T x − 2q f x ≤ 2 A ≤ max kxkr , − 2 f



q q q q (r−1)q+r 2 2 2 2 |2|

which tends to zero as q → ∞ for all x ∈ X. So we can conclude that A(x) = T (x) for all x ∈ X. This proves the uniqueness of h. Thus the mapping A : X → Y is a unique additive mapping satisfying (2.5).  Theorem 2.3. Let r > 1 and θ be nonnegative real numbers and let f : X → Y be an odd mapping satisfying (2.4). Then there exists a unique additive mapping A : X → Y such that kf (x) − A(x)k ≤

2θ kxkr |2|

(2.8)

for all x ∈ X. Proof. It follows from (2.6) that



f (x) − 1 f (2x) ≤ 2 θkxkr

2 |2|

for all x ∈ X. Hence

1  l 

f 2 x − 1 f (2m x)

2l

m 2

       

1

1

1 1 l l+1 m−1 m

≤ max l f 2 x − l+1 f 2 x , · · · , m−1 f 2 x − m f (2 x) 2 2 2 2

 



     1

1 1 1 l l+1 m−1 m

= max f 2 x − f 2 x , · · · , m−1 f 2 x − f (2 x)

l |2| 2 |2| 2 (

≤ max

|2|lr |2|r(m−1) , · · · , |2|l+1 |2|(m−1)+1

)

2θkxkr =

2θ |2|(1−r)l+1

for all nonnegative integers m and l with m > l and all x ∈ X. The rest of the proof is similar to the proof of Theorem 2.2.

kxkr



Now, we prove the Hyers-Ulam stability of the additive-quadratic ρ-functional inequality (2.1) in non-Archimedean Banach spaces for an even mapping case.

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Theorem 2.4. Let r < 2 and θ be nonnegative real numbers and let f : X → Y be an even mapping satisfying (2.4). Then there exists a unique quadratic mapping Q : X → Y such that kf (x) − Q(x)k ≤

|2| 2θkxkr |2|r

(2.9)

for all x ∈ X. Proof. Letting x = y = 0 in (2.4), we get kf (0)k ≤ |ρ|k2f (0)k. So f (0) = 0. Letting y = x in (2.4), we get

1

f (2x) − 2f (x) ≤ 2θkxkr

2

(2.10)




≤ |2|r 2θkxkr for all x ∈ X. Hence |2|

   

l x

4 f x − 4m f (2.11)

2l 2m

         

l

x x x x l+1 m

, · · · , 4m−1 f

4 f ≤ max − 4 f − 4 f

2l 2l+1 2m−1 2m

 

       

x x x x l m−1

= max |4| f − 4f , · · · , |4|

f 2m−1 − 4f 2m 2l 2l+1 ( )

for all x ∈ X. So f (x) − 4f

≤ max

x 2

|4|l |4|m−1 , · · · , |2|rl |2|r(m−1)

|2| 2θ |2| 2θkxkr = (r−2)l r kxkr r |2| |2| |2|

for all nonnegative integers m and l with m > l and all x ∈ X. It follows from (2.11) that the sequence {4n f ( 2xn )} is a Cauchy sequence for all x ∈ X. Since Y is complete, the sequence {4n f ( 2xn )} converges. So one can define the mapping Q : X → Y by x Q(x) := lim 4n f ( n ) n→∞ 2 for all x ∈ X. Moreover, letting l = 0 and passing the limit m → ∞ in (2.11), we get (2.9). It follows from (2.4) that

 

x y

kM1 Q(x, y)k = lim |4|n M f ,

1 n→∞ 2n 2n

  n

x y n

+ lim |4| θ (kxkr + kykr ) ≤ lim |4| |ρ| M2 f ,

n→∞ n→∞ |2|nr 2n 2n = |ρ| kM2 Q(x, y)k for all x, y ∈ X. So kM1 Q(x, y)k ≤ kρM2 Q(x, y)k for all x, y ∈ X. By Lemma 2.1, the mapping h : X → Y is quadratic. Now, let T : X → Y be another quadratic mapping satisfying (2.9). Then we have

   

q x x q

kQ(x) − T (x)k = 4 Q q − 4 T 2 2q          

q

x |2| x q

, 4q T x − 4q f x ≤ ≤ max 4 Q q − 4 f 2θkxkr ,

q q q (r−2)q+r 2 2 2 2 |2| which tends to zero as q → ∞ for all x ∈ X. So we can conclude that Q(x) = T (x) for all x ∈ X. This proves the uniqueness of Q. Thus the mapping Q : X → Y is a unique quadratic mapping satisfying (2.9). 

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Theorem 2.5. Let r > 2 and θ be positive real numbers, and let f : X → Y be an even mapping satisfying (2.4). Then there exists a unique quadratic mapping Q : X → Y such that kf (x) − Q(x)k ≤

2θ kxkr |2|

for all x ∈ X. Proof. It follows from (2.10) that



f (x) − 1 f (2x) ≤ 2θ kxkr

4 |2|

for all x ∈ X. The rest of the proof is similar to the proof of Theorem 2.4.



3. Additive-quadratic ρ-functional inequality (0.2) Throughout this section, assume that ρ is a fixed non-Archimedean number with |ρ| < |2|. In this section, we solve the additive-quadratic ρ-functional inequality (0.2) in non-Archimedean normed spaces. Lemma 3.1. (i) If an odd mapping f : X → Y satisfies kM2 f (x, y)k ≤ kρM1 f (x, y)k

(3.1)

for all x, y ∈ X, then f : X → Y is additive. (ii) If an even mapping f : X → Y satisfies f (0) = 0 and (3.1), then f : X → Y is quadratic. Proof. (i) Assume that f : X → Y satisfies (3.1). Letting y = 0 in (3.1), we get

 



2f x − f (x) ≤ 0

2

(3.2)

and so f x2 = 12 f (x) for all x ∈ X. It follows from (3.1) and (3.2) that






x+y − f (x) − f (y)

2 ≤ |ρ|kf (x + y) − f (x) − f (y)k 



kf (x + y) − f (x) − f (y)k =

2f and so

f (x + y) = f (x) + f (y) for all x, y ∈ X. (ii) Assume that f : X → Y satisfies (3.1). Letting y = 0 in (3.1), we get

 



4f x − f (x) ≤ 0

2

and so f

x 2




(3.3)

= 14 f (x) for all x ∈ X.

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It follows from (3.1) and (3.3) that

1

f (x + y) + 1 f (x − y) − f (x) − f (y)

2

2



  

x+y x−y

= 2f + 2f − f (x) − f (y)

2 2

1

1

≤ |ρ|

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

and so f (x + y) + f (x − y) = 2f (x) + 2f (y) for all x, y ∈ X.



We prove the Hyers-Ulam stability of the additive-quadratic ρ-functional inequality (3.1) in nonArchimedean Banach spaces for an odd mapping case. Theorem 3.2. Let r < 1 and θ be nonnegative real numbers, and let f : X → Y be an odd mapping such that kM2 f (x, y)k ≤ kρM1 f (x, y)k + θ(kxkr + kykr )

(3.4)

for all x, y ∈ X. Then there exists a unique additive mapping A : X → Y such that kf (x) − A(x)k ≤ θkxkr

(3.5)

for all x ∈ X. Proof. Since f is an odd mapping, f (0) = 0. Letting y = 0 in (3.4), we get

 



2f x − f (x) ≤ θkxkr

2

(3.6)

for all x ∈ X. So

   

l x

2 f x − 2m f

2l 2m

         

l

m−1 x x x x l+1 m



≤ max 2 f −2 f , · · · , 2 f −2 f 2l 2l+1 2m−1 2m



        

x x x x l m−1

− 2f , · · · , |2| = max |2| f

f 2m−1 − 2f 2m 2l 2l+1 ( )

≤ max

|2|l |2|m−1 , · · · , r(m−1) rl |2| |2|

θkxkr =

(3.7)

θ kxkr |2|(r−1)l

for all nonnegative integers m and l with m > l and all x ∈ X. It follows from (3.7) that the sequence {2n f ( 2xn )} is a Cauchy sequence for all x ∈ X. Since Y is complete, the sequence {2n f ( 2xn )} converges. So one can define the mapping A : X → Y by x A(x) := lim 2n f ( n ) n→∞ 2 for all x ∈ X. Moreover, letting l = 0 and passing the limit m → ∞ in (3.7), we get (3.5). The rest of the proof is similar to the proof of Theorem 2.2. 

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Theorem 3.3. Let r > 1 and θ be positive real numbers, and let f : X → Y be an odd mapping satisfying (3.4). Then there exists a unique additive mapping A : X → Y such that kf (x) − A(x)k ≤

|2|r θ kxkr |2|

(3.8)

for all x ∈ X. Proof. It follows from (3.6) that

r

f (x) − 1 f (2x) ≤ |2| θ kxkr

2 |2|

for all x ∈ X. Hence



1

f (2l x) − 1 f (2m x)

2l m 2     

1

1 l l+1

,··· ≤ max f 2 x − f 2 x

2l

l+1 2

 

  

1

f 2l x − 1 f 2l+1 x , · · · , = max

l |2| 2 ( )

≤ max

|2|rl |2|r(m−1) , · · · , |2|l+1 |2|(m−1)+1

(3.9)





1

 

1 m−1 f 2 x − m f (2m x)

m−1 2 2



 

1

f 2m−1 x − 1 f (2m x)

m−1 |2| 2

,

|2|r θkxkr =

|2|r θ |2|(1−r)l+1

kxkr

for all nonnegative integers m and l with m > l and all x ∈ X. It follows from (3.9) that the sequence { 21n f (2n x)} is a Cauchy sequence for all x ∈ X. Since Y is complete, the sequence { 21n f (2n x)} converges. So one can define the mapping A : X → Y by A(x) := lim

1

n→∞ n

f (2n x)

for all x ∈ X. Moreover, letting l = 0 and passing the limit m → ∞ in (3.9), we get (3.8). The rest of the proof is similar to the proofs of Theorems 2.2 and 3.2.



Now, we prove the Hyers-Ulam stability of the additive-quadratic ρ-functional inequality (3.1) in non-Archimedean Banach spaces for an even mapping case. Theorem 3.4. Let r < 2 and θ be nonnegative real numbers, and let f : X → Y be an even mapping satisfying (3.4). Then there exists a unique quadratic mapping Q : X → Y such that kf (x) − Q(x)k ≤ θkxkr

(3.10)

for all x ∈ X. Proof. Letting x = y = 0 in (3.4), we get k2f (0)k ≤ |ρ|kf (0)k. So f (0) = 0. Letting y = 0 in (3.4), we get

 



4f x − f (x) ≤ θkxkr

2

708

(3.11)

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

for all x ∈ X. So

   

l x

4 f x − 4m f (3.12)

2l 2m

         

l

x x x x l+1 m

, · · · , 4m−1 f

4 f ≤ max − 4 f − 4 f

2l 2l+1 2m−1 2m

 

       

x x x x l m−1

f = max |4| f , · · · , |4| − 4f − 4f

2l 2l+1 2m−1 2m ( )

≤ max

|4|l |4|m−1 , · · · , |2|rl |2|r(m−1)

θkxkr =

θ

|2|(r−2)l

kxkr

for all nonnegative integers m and l with m > l and all x ∈ X. It follows from (3.12) that the sequence {4n f ( 2xn )} is a Cauchy sequence for all x ∈ X. Since Y is complete, the sequence {4n f ( 2xn )} converges. So one can define the mapping Q : X → Y by x Q(x) := lim 4n f ( n ) n→∞ 2 for all x ∈ X. Moreover, letting l = 0 and passing the limit m → ∞ in (3.12), we get (3.10). The rest of the proof is similar to the proof of Theorem 2.2.  Theorem 3.5. Let r > 2 and θ be positive real numbers, and let f : X → Y be an even mapping satisfying (3.4). Then there exists a unique quadratic mapping Q : X → Y such that kf (x) − Q(x)k ≤

|2|r θ kxkr |4|

(3.13)

for all x ∈ X. Proof. It follows from (3.11) that

r

f (x) − 1 f (2x) ≤ |2| θ kxkr

4 |4|

for all x ∈ X. Hence



1

f (2l x) − 1 f (2m x) (3.14)

4l m 4

     

1  l 

1 l+1

, · · · , 1 f 2m−1 x − 1 f (2m x) ≤ max f 2 x − f 2 x

4l

l+1 m−1 m 4 4 4

 



    

1 1

f 2l x − 1 f 2l+1 x , · · · ,

f 2m−1 x − 1 f (2m x) = max



l m−1 |4| 4 |4| 4 (

≤ max

|2|rl |2|r(m−1) , · · · , |4|l+1 |4|(m−1)+1

)

|2|r θkxkr =

|2|r θ |2|(2−r)l+2

kxkr

for all nonnegative integers m and l with m > l and all x ∈ X. It follows from (3.14) that the sequence { 41n f (2n x)} is a Cauchy sequence for all x ∈ X. Since Y is complete, the sequence { 41n f (2n x)} converges. So one can define the mapping Q : X → Y by 1 f (2n x) n→∞ 4n for all x ∈ X. Moreover, letting l = 0 and passing the limit m → ∞ in (3.14), we get (3.13). The rest of the proof is similar to the proofs of Theorems 2.2 and 3.4. Q(x) := lim

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References [1] M. Adam, On the stability of some quadratic functional equation, J. Nonlinear Sci. Appl. 4 (2011), 50–59. [2] T. Aoki, On the stability of the linear transformation in Banach spaces, J. Math. Soc. Japan 2 (1950), 64–66. [3] L. C˘ adariu, L. G˘ avruta and P. G˘ avruta, On the stability of an affine functional equation, J. Nonlinear Sci. Appl. 6 (2013), 60–67. [4] A. Chahbi and N. Bounader, On the generalized stability of d’Alembert functional equation, J. Nonlinear Sci. Appl. 6 (2013), 198–204. [5] P. W. Cholewa, Remarks on the stability of functional equations, Aequationes Math. 27 (1984), 76–86. [6] P. Gˇ avruta, A generalization of the Hyers-Ulam-Rassias stability of approximately additive mappings, J. Math. Anal. Appl. 184 (1994), 431–43. [7] D. H. Hyers, On the stability of the linear functional equation, Proc. Natl. Acad. Sci. U.S.A. 27 (1941), 222–224. [8] M.S. Moslehian and Gh. Sadeghi, A Mazur-Ulam theorem in non-Archimedean normed spaces, Nonlinear Anal.– TMA 69 (2008), 3405–3408. [9] 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. [10] C. Park, K. Ghasemi, S. G. Ghaleh and S. Jang, Approximate n-Jordan ∗-homomorphisms in C ∗ -algebras, J. Comput. Anal. Appl. 15 (2013), 365-368. [11] C. Park, A. Najati and S. Jang, Fixed points and fuzzy stability of an additive-quadratic functional equation, J. Comput. Anal. Appl. 15 (2013), 452–462. [12] Th. M. Rassias, On the stability of the linear mapping in Banach spaces, Proc. Amer. Math. Soc. 72 (1978), 297–300. [13] S. Schin, D. Ki, J. Chang and M. Kim, Random stability of quadratic functional equations: a fixed point approach, J. Nonlinear Sci. Appl. 4 (2011), 37–49. [14] S. Shagholi, M. Bavand Savadkouhi and M. Eshaghi Gordji, Nearly ternary cubic homomorphism in ternary Fr´echet algebras, J. Comput. Anal. Appl. 13 (2011), 1106–1114. [15] S. Shagholi, M. Eshaghi Gordji and M. Bavand Savadkouhi, Stability of ternary quadratic derivation on ternary Banach algebras, J. Comput. Anal. Appl. 13 (2011), 1097–1105. [16] D. Shin, C. Park and Sh. Farhadabadi, On the superstability of ternary Jordan C ∗ -homomorphisms, J. Comput. Anal. Appl. 16 (2014), 964–973. [17] D. Shin, C. Park and Sh. Farhadabadi, Stability and superstability of J ∗ -homomorphisms and J ∗ -derivations for a generalized Cauchy-Jensen equation, J. Comput. Anal. Appl. 17 (2014), 125–134. [18] F. Skof, Propriet locali e approssimazione di operatori, Rend. Sem. Mat. Fis. Milano 53 (1983), 113–129. [19] S. M. Ulam, A Collection of the Mathematical Problems, Interscience Publ. New York, 1960. [20] C. Zaharia, On the probabilistic stability of the monomial functional equation, J. Nonlinear Sci. Appl. 6 (2013), 51–59. [21] S. Zolfaghari, Approximation of mixed type functional equations in p-Banach spaces, J. Nonlinear Sci. Appl. 3 (2010), 110–122. 1

Research Institute for Natural Sciences, Hanyang University, Seoul 04763, Republic of Korea E-mail address: [email protected] 2

Department of Mathematics, Daejin University, Kyunggi 11159, Republic of Korea E-mail address: [email protected] 3

Department of Mathematics, University of Seoul, Seoul 02504, Republic of Korea E-mail address: [email protected]

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Differential equations associated with the generalized Euler polynomials of the second kind C. S. Ryoo Department of Mathematics, Hannam University, Daejeon 306-791, Korea

Abstract : In this paper, we study linear differential equations arising from the generating functions of the generalized Euler polynomials of the second kind. We give explicit identities for the second kind Euler polynomials. Key words : linear differential equations, the second kind Euler numbers, generalized Euler polynomials of the second kind. 2000 Mathematics Subject Classification : 11B68, 11S40, 11S80. 1. Introduction Recently, many mathematicians have studied in the area of the Bernoulli numbers, Euler numbers, Genocchi numbers, and the second kind Euler numbers(see [1, 2, 3, 4, 6, 7]). The generalized Euler polynomials En (x)(n ≥ 0) of the second kind, were introduced by Ryoo(see [5, 6]). The generalized Euler polynomials En (x) of the second kind are defined by the generating function: ( F = F (t, x) =

2et 2t e +1

)x =

∞ ∑

En (x)

n=0

tn . n!

(1.1)

We recall that the classical Stirling numbers of the first kind S1 (n, k) and S2 (n, k) are defined by the relations(see [8]) n n ∑ ∑ (x)n = S1 (n, k)xk and xn = S2 (n, k)(x)k , (1.2) k=0

k=0

respectively. Here (x)n = x(x − 1) · · · (x − n + 1) denotes the falling factorial polynomial of order n. The numbers S2 (n, m) also admit a representation in terms of a generating function ∞ ∑

tn (et − 1)m = . n! m!

(1.3)

tn (log(1 + t))m = . n! m!

(1.4)

S2 (n, m)

n=m

We also have

∞ ∑

S1 (n, m)

n=m

If x is a variable, we use the following notation: < x >k = x(x + 1) · · · (x + k − 1),

( ) x (x)k = , k k!

x

(1 + t) =

∞ ( ) ∑ x k=0

k

tk .

(1.5)

Nonlinear differential equations arising from the generating functions of special polynomials are studied by many authors in order to give explicit identities for special polynomials(see [3, 4]). In this paper, we study linear differential equations arising from the generating functions of the generalized Euler polynomials of the second kind. We give explicit identities for the generalized Euler polynomials of the second kind.

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2. Differential equations associated with the generalized Euler polynomials of the second kind In this section, we study linear differential equations arising from the generating functions of the generalized Euler polynomials En (x) of the second kind. Let ( )x 2et F = F (t, x) = . e2t + 1 Then, by (2.1), we have ( )x ( )x−1 ( ( )2 ) d d 2et 2et 2et 2et (1) F = F (t, x) = = x 2t − et dt dt e2t + 1 e +1 e2t + 1 e2t + 1

(2.1)

= xF (t, x) − xF (t, x + 1)et , d (1) F = xF (t, x)(1) − xF (1) (t, x + 1)et − xF (t, x + 1)et dt ( ) = x xF (t, x) − xF (t, x + 1)et ( ) − x (x + 1)F (t, x + 1) − (x + 1)F (t, x + 2)et et − xF (t, x + 1)et ,

F (2) =

(2.2)

= x2 F (t, x) − (2x2 + 2x)F (t, x + 1)et + x(x + 1)F (t, x + 2)e2t , and F (3) =

d (2) F = x2 F (1) (t, x) − (2x2 + 2x)F (1) (t, x + 1)et − (2x2 + 2x)(t, x + 1)et dt + x(x + 1)F (1) (t, x + 2)e2t + 2x(x + 1)F (t, x + 2)e2t

(2.3)

= x3 F (t, x) − (3x3 + 6x2 + 4x)F (t, x + 1)et − (3x3 + 9x2 + 6x)F (t, x + 2)e2t − (x3 + 3x2 + 2x)F (t, x + 3)e3t . Continuing this process, we can guess that ( )N d F (N ) = F (t, x) dt =

N ∑

(2.4) it

ai (N, x)F (t, x + i)e ,

(N = 0, 1, 2, . . .).

i=0

Taking the derivative with respect to t in (2.4), we have F (N +1) = =

N ∑

dF (N ) dt

ai (N, x)ieit F (t, x + i) +

i=0

=

N ∑

N ∑

ai (N, x)eit F (1) (t, x + i)

(2.5)

i=0

ai (N, x)(x + 2i)eit F (t, x + i) −

i=0

N +1 ∑

ai−1 (N, x)(x + i − 1)eit F (t, x + i).

i=1

On the other hand, by replacing N by N + 1 in (2.4), we get F (N +1) =

N +1 ∑

ai (N + 1, x)eit F (t, x + i)

(2.6)

i=0

By (2.5) and (2.6), we have N ∑

ai (N, x)(x + 2i)eit F (t, x + i) −

i=0

=

N +1 ∑ i=1

N +1 ∑

ai−1 (N, x)(x + i − 1)eit F (t, x + i) (2.7)

ai (N + 1, x)eit F (t, x + i).

i=0

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J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 24, NO.4, 2018, COPYRIGHT 2018 EUDOXUS PRESS, LLC

Comparing the coefficients on both sides of (2.7), we obtain aN +1 (N + 1, x) = −(x + N )aN (N, x),

a0 (N + 1, x) = xa0 (N, x),

(2.8)

and ai (N + 1, x) = (x + 2i)ai (N, x) − (x + i − 1)ai−1 (N, x), (1 ≤ i ≤ N ).

(2.9)

In addition, by (2.2) and (2.4), we get F (0) = F (0) (t, x) = a0 (0, x) = F (t, x).

(2.10)

a0 (0, x) = 1.

(2.11)

Thus, by (2.10), we obtain

It is not difficult to show that xF (t, x) − xF (t, x + 1)e = t

1 ∑

ai (1, x)eit F (t, x + i)

(2.12)

i=0 t

= a0 (1, x)F (t, x) + a1 (1, x)F (t, x + 1)e . Thus, by (2.12), we also get a1 (1, x) = −x.

a0 (1, x) = x,

(2.13)

From (2.8), we note that a0 (N + 1, x) = xa0 (N, x) = · · · = xN +1 a0 (0, x) = xN +1 , and aN +1 (N + 1, x) = −(x + N )aN (N, x) = · · ·

(2.14)

= (−1)N +1 (x + N )(x + N − 1) · · · (x + 1)x. For i = 1, 2, 3 in (2.9), we get a1 (N + 1, x) = −x

N ∑

(x + 2 · 1)k a0 (N − k, x),

k=0

a2 (N + 1, x) = −(x + 1)

N −1 ∑

(x + 2 · 2)k a1 (N − k, x),

k=0

and a3 (N + 1, x) = −(x + 2)

N −2 ∑

(x + 2 · 3)k a2 (N − k, x).

k=0

Continuing this process, we can deduce that, for 1 ≤ i ≤ N, ai (N + 1, x) = −(x + i − 1)

N∑ +1−i

(x + 2 · i)k ai−1 (N − k, x).

(2.15)

k=0

Here, we note that the matrix ai (j, x)0≤i,j≤N is given by 

1 x  0 (−1) < x >1  0 0   0 0 . .. . . . 0

0



x2 ·

x3 ·

··· ···

xN ·

(−1)2 < x >2 0 .. .

· (−1) < x >3 .. .

··· ··· .. .

· · .. .

0

0

···

(−1)N < x >N

713

3

         

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J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 24, NO.4, 2018, COPYRIGHT 2018 EUDOXUS PRESS, LLC

Now, we give explicit expressions for ai (N + 1, x). By (2.14) and (2.15), we get a1 (N + 1, x) = −x

N ∑

(x + 2)k1 a0 (N − k1 , x)

k1 =0

= −x

N −1 ∑

(x + 2)k1 xN −k1

k1 =0 N −1 ∑

a2 (N + 1, x) = −(x + 1)

(x + 2 · 2)k2 a1 (N − k2 , x)

k2 =0

= (−1)2 x(x + 1)

N −1 ∑

(x + 2 · 2)k2 (x + 2)k1 xN −k2 −k1 −1 ,

k2 =0

and a3 (N + 1) = −(x + 2)

N −2 ∑

(x + 4)k3 a2 (N − k3 , x)

k3 =0 3

= (−1) x(x + 1)(x + 2) ×

N −2 N −k 3 −2 N −k∑ 3 −k2 −2 ∑ ∑ k3 =0

k2 =0

(x + 2 · 3)k3 (x + 2 · 2)k2 (x + 2)k1 xN −k3 −k2 −k1 −2 .

k1 =0

Continuing this process, we have ai (N + 1) = (−1)i < x >i

N∑ −i+1 N −k i −i+1 ∑ ki =0

···

N −ki −···−k i ∑ 2 −i+1 ∏

(x + 2l)kl xN −ki −ki−1 −···−k2 −k1 −i+1 .

ki−1 =0

k1 =0

(2.16)

l=1

Therefore, by (2.16), we obtain the following theorem. Theorem 1. For N = 0, 1, 2, . . . , the linear functional equations (N ( )i ) ∑ 2et (N ) eit F F = ai (N, x) 2t e + 1 i=0 (

have a solution F = F (t, x) =

2et e2t + 1

)x ,

where a0 (N, x) = xN , aN (N, x) = (−1)N < x >N , ai (N ) = (−1)i < x >i

N −i N −k ∑ ∑i −i

···

N −ki −···−k i ∑ 2 −i ∏

ki =0 ki−1 =0

(x + 2l)kl xN −ki −ki−1 −···−k2 −k1 −i ,

k1 =0

l=1

(1 ≤ i ≤ N − 1).

From (1.1), we note that ( F (N ) =

d dt

)N F (t, x) =

∞ ∑ k=0

714

Ek+N (x)

tk . k!

(2.17)

C. S. Ryoo 711-716

J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 24, NO.4, 2018, COPYRIGHT 2018 EUDOXUS PRESS, LLC

From Theorem 1 and (2.17), we can derive the following equation: ∞ ∑

Ek+N (x)

k=0

tk = F (N ) k! )x+i 2et e2t + 1 i=0 (∞ )( ∞ ) N ∑ ∑ tk ∑ tk k = ai (N, x) i Ek (x + i) k! k! i=0 k=0 k=0 ( ) ( ) ∞ N ∑ k ∑ ∑ k tk ai (N, x)ik−l El (x + i) = . l k! i=0 =

N ∑

(

ai (N, x)eit

k=0

(2.18)

l=0

By comparing the coefficients on both sides of (2.18), we obtain the following theorem. Theorem 2. For k = 0, 1, . . . , and N = 0, 1, 2, . . . , we have N ∑ k ( ) ∑ k Ek+N (x) = ai (N, x)ik−l El (x + i), l i=0

(1.19)

l=0

a0 (N, x) = xN , aN (N, x) = (−1)N < x >N , ai (N ) = (−1)i < x >i

N −i N −k ∑ ∑i −i

···

N −ki −···−k i ∑ 2 −i ∏

(x + 2l)kl xN −ki −ki−1 −···−k2 −k1 −i ,

ki =0 ki−1 =0

k1 =0

l=1

(1 ≤ i ≤ N − 1). Let us take k = 0 in (2.19). Then, we have the following corollary. Corollary 3. For N = 0, 1, 2, . . . , we have EN (x) =

N ∑

ai (N, x).

i=0

The first few of them are E0 (x) = 1, (x)

E1 (x) = 0,

E2

E3 (x) = 0,

E4 (x) = 2x + 3x2 ,

E5 (x) = 0,

E6 (x) = −16x − 30x2 − 15x3 ,

E7 (x) = 0,

E8 (x) = 272x + 588x2 + 420x3 + 105x4 ,

E9 (x) = 0,

E10 (x) = −7936x − 18960x2 − 16380x3 − 6300x4 − 945x5 .

= −x,

For N = 0, 1, 2, . . . , the linear functional equations (N ( )i ) ∑ 2et (N ) F = ai (N, x) 2t eit F e + 1 i=0 (

have a solution F = F (t, x) =

715

2et 2t e +1

)x .

C. S. Ryoo 711-716

J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 24, NO.4, 2018, COPYRIGHT 2018 EUDOXUS PRESS, LLC

FHt, xL 3

2

FHt,xL

20

1

4

10

t

2

0 0

-2

0

x -1

-2

0 t

2

-2

-4

-3 -4

-2

0 x

2

4

Figure 1: The surface for the solution F (t, x)

Here is a plot of the surface for this solution. In Figure 1(left), we plot of the surface for this solution. In Figure 1(right), we shows a higher-resolution density plot of the solution. The author has no doubt that investigations along this line will lead to a new approach employing numerical method in the research field of the generalized Euler polynomials of the second kind to appear in mathematics and physics(see [5, 6, 7]).

REFERENCES 1. A. Bayad, T. Kim, Higher recurrences for Apostal-Bernoulli-Euler numbers, Russ. J. Math. Phys. 19(1), (2012), 1-10. 2. A. Erdelyi, W. Magnus, F. Oberhettinger, F.G. Tricomi, Higher Transcendental Functions, Vol 3. New York: Krieger, 1981. 3. D.S. Kim, T. Kim, Some identities of Bell polynomials, Sci. China Math. 58(10)(2015), 2095-2104. 4. T. Kim, D.S. Kim, Identities involving degenerate Euler numbers and polynomials arising from non-linear differential equations, J. Nonlinear Sci. Appl., 9(2016), 2086-2098. 5. Y.H. Kim, H.Y. Jung, C.S. Ryoo, On the generalized Euler polynomials of the second kind, J. Appl. Math. & Informatics, 31(2013), 623 - 630 6. C. S. Ryoo, Y.H. Kim, H.Y. Jung, Distribution of the zeros of the generalized Euler polynomials of second kind, Int. Journal of Math. Analysis, 7(43)(2013), 2129 - 2136. 7. C.S. Ryoo, Calculating zeros of the second kind Euler polynomials, Journal of Computational Analysis and Applications, 12 (2010), 828-833. 8. P. T. Young, Degenerate Bernoulli polynomials, generalized factorial sums, and their applications, Journal of Number Theorey., 128(2008), 738-758.

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J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 24, NO.4, 2018, COPYRIGHT 2018 EUDOXUS PRESS, LLC

Chlodowsky Variant of Bernstein-Schurer Operators Based on (p,q)-Integers Eser Gemikonakli · Tuba Vedi-Dilek

Abstract In this paper, we introduce the Chlodowsky variant of Bernstein-Schurer operators based on (p, q)-integers. By obtaining first few moments of these operators, we prove well-known Korovkin-type approximation theorems in different function spaces. We also compute the error of approximation by using modulus of continuity and Lipschitz-type functionals. Moreover, we study the generalization of the Chlodowsky variant of BernsteinSchurer operators based on (p, q)-integers and investigate their approximations. Finally, numerical results are presented in detail. Keywords (p,q)-integers, q-Bernstein operators, q-Bernstein-Schurer operators.

1 Introduction

The classical Bernstein-Chlodowsky operators were defined by Chlodowsky [4] as   r  n  x r  n x n−r Cn ( f ; x) = ∑ f bn 1− , r n bn bn r=0 where the function f is defined on [0, ∞) and (bn ) is a positive increasing sequence with bn → 0 as n → ∞. bn → ∞ and n In 2008, Karsh and Gupta [9] defined q-analogue of Chlodowsky operators as follows: !      n [k]q x k n−k−1 n s x Cn ( f ; q; x) = ∑ f bn 1 − q , 0 ≤ x ≤ bn ∏ k q bn [n]q bn s=0 k=0 where (bn ) has the same properties of Bernstein-Chlodowsky operators. Eser Gemikonakli Institute of Applied Sciences, University of Kyrenia, Girne, Mersin 10, Turkey E-mail: [email protected] Tuba Vedi-Dilek Institute of Applied Sciences, University of Kyrenia, Girne, Mersin 10, Turkey E-mail: [email protected]

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In 1987, Lupas¸ [11] defined the q-based Bernstein operators and obtained the Korovkintype approximation theorem. Over the past several years, there has been a considerable amount of research on the q-based operators (see [2], [3], [7], [12], [13],[16],[17], [20], [21], [22], [24]). To date, the focus of published work has been largely on (p,q) based operators. In 2015, (p, q)-analogue of Bernstein operators were introduced by Mursaleen et al [14] as ! n   n−k−1 [k] p,q k(k−1) 1 n k s s , x ∈ [0, 1] . Bn,p,q ( f ; x) = n(n−1) ∑ p 2 x ∏ (p − q x) f pk−n [n] p,q s=0 p 2 k=0 k p,q (1.1) For 0 < q ≤ p < 1, the (p, q)-numbers are given as [6] [k] p,q =

pk − qk . p−q

For each k ∈ N0 the (p, q)-factorial is represented by  [k] [k − 1] ... [1] , k = 1, 2, 3, ..., [k] p,q ! = 1 , k=0 and (p, q)-binomial coefficients are defined as   [n] p,q ! n = k p,q [n − k] p,q ! [k] p,q ! where n ≥ k ≥ 0. Note that, as it is introduced in [18], the operators are reduced to the q-Bernstein operators for p = 1 in Eq.(1.1). Recently, the (p, q)-analogue of Berntein-Schurer operators have been introduced by Sidharth and Agrawal [1] as !  n+s k(k−1) (n+s)(n+s−1)  [k] p,q n + s n+s−k − k n−k 2 B¯ n,s ( f ; p, q; x) = ∑ p 2 x (1 − x) p,q f p k p,q [n] p,q k=0 where f ∈ C [0, 1 + s] , s ∈ N0 and n ∈ N. Corollary 1 If we use the properties of (p, q) integers, we have [n] p,q = pn−1 [n]q/p and

    n n = pk(n−k) . k p,q k q/p

This paper is structured in the following way; The next section introduces the Chlodowsky variant of Bernstein-Schurer operators based on (p, q)-integers and investigate the moments of the operator. Section 3 discusses several Korovkin-type theorems in different function spaces. In section 4, we obtain the order of convergence of the Chlodowsky variant of Bernstein-Schurer operators based on (p, q)integers by means of Lipschitz class functions and the first modulus of continuity. Section 5 provides the generalization of the Chlodowsky variant of Bernstein-Schurer operators based on (p, q)-integers and investigate their approximations. Finally, in section 6, numerical results to illustrate the contribution of the Chlodowsky Variant of Bernstein-Schurer Operators based on (p; q)-integers are presented. 2

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2 Construction of the Operators We construct the Chlodowsky variant of Bernstein-Schurer operators based on (p, q)-integers as C¯n,s ( f ; p, q; x) n+s

:=

∑f

pn−k

k=0

[k] p,q [n] p,q

! p

bn

k(k−1) (n+s−1)(n+s) 2 − 2



n+s k



 p,q

x bn

k n+s−k−1 

∏

j=0

(2.1)  x pj −qj , bn

where n, s ∈ N, 0 ≤ x ≤ bn and 0 < q < p ≤ 1. Note that, in case p=1 in Eq.(2.1), Chlodowsky variant of Bernstein-Schurer reduces to the Chlodowsky variant of q-Bernstein-Schurer operators. First of all, we obtained the following lemma and used it throughout the paper. Lemma 1 Let C¯n,s ( f ; p, q; x) be given in Eq.(2.1). Then we get, (i) C¯n,s (1; p, q; x) = 1, (ii) C¯n,s (t; p, q; x) =

[n + s] p,q ps [n] p,q

p1−2s q[n+s−1] p,q [n+s]p,q 2 pn−s−1 [n+s] p,q x + bn x, 2 [n] p,q [n]2p,q


(iii) C¯n,s t 2 ; p, q; x =

(iv)C¯n,s ((t − x) ; p, q; x) =

(v) C¯n,s

x,

(t − x)2 ; p, q; x




=

[n + s] p,q ps [n] p,q

! − 1 x,

p1−2s q [n + s − 1] p,q [n + s] p,q [n]2p,q

−2

[n + s] p,q [n] p,q

! + 1 x2 +

pn−s−1 [n + s] p,q [n]2p,q

bn x.

Proof Applying the Corollary 1, we have n+s

C¯n,s (1; p, q; x) =

∑p

k(k−1) (n+s−1)(n+s) 2 − 2

k=0 n+s

=

∑

p

k(k−1) (n+s−1)(n+s) 2 − 2



n+s k



 p,q

pk(n+s−k)

k=0



x bn

n+s k

k

(1 − x)n+s−k p,q

 q/p



x bn

k   x n+s−k 1− bn q/p

=1 q ≤ 1. p Using the linearity of the operators and Corollary 1 with some basic calculations, we can obtain the assertions (ii), (iii). Then, from (i) and (ii), we have when 0
0, we have ! !2 [k] p,q [k] p,q 2M n−k n−k bn − f (x) < ε + 2 p bn − x , f p [n] p,q δ [n] p,q where x ∈ [0, bn ] and δ = δ (ε) are independent of n. With the help of the following equality !2    k n+s−k−1   n+s [k] p,q k(k−1) (n+s−1)(n+s) x n+s n−k j j x 2 − 2 p b − x p p − q n ∑ ∏ k p,q bn [n] p,q bn j=0 k=0 = C¯n,s ( f ; p, q; x) , 5

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we get by Theorem 3 and Lemma 1 that sup C¯n,s ( f ; p, q; x) − f (x) 0≤x≤bn

2M ≤ ε+ 2 δ Since

"

p1−2s q [n + s − 1] p,q [n + s] p,q [n]2p,q

−2

[n + s] p,q [n] p,q

! +1

b2n +

pn−s−1 [n + s] p,q [n]2p,q

# b2n

.

b2n → 0 as n → ∞, we have the desired result. [n]

4 Order of Convergence In this section, we compute the rate of convergence of the operators in terms of the elements of Lipschitz classes and the modulus of continuity of the function. Now, we give the rate of convergence of the operators C¯n,s in terms of the Lipschitz class LipM (γ) , for 0 < γ ≤ 1. Let CB [0, ∞) denotes the space of bounded continuous functions on [0, ∞). A function f ∈ CB [0, ∞) belongs to LipM (γ) if | f (t) − f (x)| ≤ M |t − x|γ

(t, x ∈ [0, ∞))

is satisfied. Theorem 4 Let f ∈ LipM (γ) |C¯n,s ( f ; p, q; x) − f (x)| ≤ M (λn (x))γ/2 where λn,q (x) =

p1−2s q [n + s − 1] p,q [n + s] p,q [n]2p,q

−2

[n + s] p,q [n] p,q

! + 1 x2 +

pn−s−1 [n + s] p,q [n]2p,q

bn x.

Proof Considering the monotonicity and the linearity of the operators, and taking into account that f ∈ LipM (γ) |C¯n,s ( f ; p, q; x) − f (x)| n+s

= | ∑ ( f (pn−k k=0

[k] p,q [n] p,q

bn ) − f (x)p

k(k−1) (n+s−1)(n+s) 2 − 2



n+s k



 p,q

x bn

k n+s−k−1 

∏

j=0

pj −qj

x bn

 |

  k n+s−k−1   k(k−1) (n+s−1)(n+s)  [k] p,q x n+s n−k j j x 2 ≤ ∑ f (p bn ) − f (x) p 2 − p − q ∏ k p,q bn [n] p,q bn j=0 k=0       k n+s n+s−k−1 [k] p,q k(k−1) (n+s−1)(n+s) x x n+s 2 ≤ M ∑ |pn−k bn − x|γ p 2 − ∏ p j − q j bn . k [n] b n p,q p,q j=0 k=0 n+s

Using H¨older’s inequality with p =

2 2 and q = , we get by the statement (Lemma 2) γ 2−γ

|C¯n,s ( f ; p, q; x) − f (x)| (    k n+s−k−1   n+s [k] p,q k(k−1) (n+s−1)(n+s) γ x x n+s 2 ≤ M ∑ [(pn−k bn − x)2 p 2 − pj −qj ]2 ∏ k [n] b b n n p,q p,q j=0 k=0 )     k n+s−k−1  k(k−1) (n+s−1)(n+s) 2−γ x x n+s 2 × [p 2 − ∏ p j − q j bn ] 2 k p,q bn j=0 6

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"

   k n+s−k−1  γ x n+s j j x p − q ]} 2 ∏ k [n] p,q bn p,q bn j=0 k=0 #       k n+s k(k−1) (n+s−1)(n+s) n+s−k−1 2−γ x x n+s 2 pj −qj ]} γ × { ∑ [p 2 − ∏ k p,q bn bn j=0 k=0 n+s

≤ M { ∑ [(pn−k

[k] p,q

bn − x)2 p

k(k−1) (n+s−1)(n+s) 2 − 2



  γ = M[C¯n,s (t − x)2 ; p, q; x ] 2 γ

≤ M(λn,q (x)) 2 . Now we give the rate of convergence of the operators by means of the modulus of continuity which is denoted by ω( f ; δ ). Let f ∈ CB [0, ∞) and x ≥ 0. Then the definition of the modulus of continuity of f is given by ω( f ; δ ) = max | f (t) − f (x)|.

(4.1)

|t−x|≤δ t,x∈[0,∞)

It is following that for any δ > 0 the following inequality   |x − y| | f (x) − f (y)| ≤ ω ( f ; δ ) +1 δ

(4.2)

is satisfied ([5]). Theorem 5 If f ∈ CB [0, ∞), we have C¯n,s ( f ; p, q; x) − f (x) ≤ 2ω

  q f ; λn,p,q (x)

where ω ( f ; ·) is modulus of continuity of f which is defined in Eq.(4.1) and λn,q (x) be the same as in Theorem 4. Proof By triangular inequality, we get C¯n,s ( f ; p, q; x) − f (x)    k n+s−k−1   n+s [k] p,q k(k−1) (n+s−1)(n+s) x x n + s n−k − j j 2 = ∑ f (p bn )p 2 p − q − f (x) ∏ k p,q bn k=0 [n] p,q bn j=0       k n+s n+s−k−1 [k] p,q x n+s k(k−1) (n+s−1)(n+s) j j x 2 ≤ ∑ ( f (pn−k bn ) − f (x) p 2 − p − q . ∏ k p,q bn [n] p,q bn j=0 k=0 Now, using Eq.(4.2) and H¨older inequality, we can write C¯n,s ( f ; p, q; x) − f (x)   n−k [k] p,q    k n+s−k−1   b − x n+s p n k(k−1) (n+s−1)(n+s) [n] p,q x n+s   − j j x 2 2 ≤ ∑ + 1 ω ( f ; λ ) p ∏ p − q bn k p,q bn λ j=0 k=0 n+s

= ω ( f ;λ) ∑ p k=0

k(k−1) (n+s−1)(n+s) 2 − 2



n+s k



 p,q

x bn

k n+s−k−1 

∏

j=0

  k(k−1) (n+s−1)(n+s)  ω ( f ;λ) x n+s n−k [k] p,q 2 − 2 + p b − x p n ∑ k λ [n] p,q p,q bn k=0 n+s

 x bn k n+s−k−1 

pj −qj

∏

j=0

pj −qj

x bn



7

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= ω ( f ;λ)   !2    k n+s−k−1  1/2 [k] k(k−1) (n+s−1)(n+s) x ω ( f ; λ ) n+s x n + s p,q + ∑ pn−k [n] bn − x p 2 − 2 ∏ p j − q j bn  k p,q bn k=0 λ p,q j=0 = ω ( f ;λ)+

 o1/2 ω ( f ;λ) n ¯  Cn,s (t − x)2 ; p, q; x . λ

Now, choosing λn,q (x) same as in Theorem 4, then we have  q  C¯n,s ( f ; p, q; x) − f (x) ≤ 2ω f ; λn,p,q (x) .

5 Generalization of the Chlodowsky Variant of q-Bernstein-Schurer-Stancu Operators In this section, we introduce generalization of Chlodowsky variant of Bernstein-Schurer operators based on (p, q)-integers and this provides us to obtain approximate continuous functions on more general weighted spaces. For x ≥ 0, consider any continuous function ω (x) ≥ 1 and define 1 + t2 G f (t) = f (t) . ω (t) Let us consider the generalization of the C¯n,s ( f ; p, q; x) as follows Ln ( f ; p, q; x) ω (x) n+s = ∑ Gf 1 + x2 k=0

[k] p,q n−k

p

[n] p,q

! bn

p

k(k−1) (n+s−1)(n+s) 2 − 2



n+s k



 p,q

x bn

k n+s−k−1 

∏

pj −qj

j=0

x bn

 ,

where 0 ≤ x ≤ bn and (bn ) has the same properties of Chlodowsky variant of q-BernsteinSchurer-Stancu operators. Theorem 6 For the continuous functions satisfying lim

x→∞

we have lim sup

f (x) = K f < ∞, ω (x)

|Ln ( f ; p, q; x) − f (x)| = 0. ω (x) n

n→∞ 0≤x≤b

Proof Clearly, Ln ( f ; p, q; x) − f (x) ω (x) = 1 + x2

n+p

∑ Gf

∏

j=0

pj −qj

x bn

[k] p,q [n] p,q

k=0

n+s−k−1 

×

p

n−k



! bn

p

k(k−1) (n+s−1)(n+s) 2 − 2



n+s k



 p,q

x bn

k

! − G f (x)

8

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thus


C¯n,s G f ; p, q; x − G f (x) |Ln ( f ; p, q; x) − f (x)| sup = sup . ω (x) 1 + x2 0≤x≤bn 0≤x≤bn
By using | f (x)| ≤ M f ω (x) and continuity of the function f , we get that G f (x) ≤ M f 1 + x2 for x ≥ 0 and G f (x) is continuous function on [0, ∞). Thus, from the Theorem 1 we get the result.
Finally note that, the operators Ln ( f ; p, q; x) reduces to C¯n,s G f ; p, q; x by taking ω(x) = 1 + x2 .

6 Numerical Results and Discussions In order to show the effectiveness and accuracy of C¯n,s ( f ; p, q; x) to f(x) with different values of parameters, numerical results are presented in this section. Sensitivity analysis is carried out to minimise the error of approximation of C¯n,s ( f ; p, q; x) to the function f (x) = 1 − cos (4ex ) for minimum n and s values by taken into account different p and q values. In Figure 1, C¯n,s ( f ; p, q; x) results are given as a function of x for different n values. To minimise the error of the approximation of C¯n,s ( f ; p, q; x) to f (x), two different sequences n n2 + 4 ) and bn = 2 are considered respectively. It is evident that the bn = 1 + log( n + 12 n + 18n C¯n,s ( f ; p; q; x) converges to f (x) for both sequences as the value of q and p approaches ton ) for C¯n,s ( f ; p, q; x) rather than wards 1, while s = 2. However, using bn = 1 + log( n + 12 n2 + 4 bn = 2 results better approximation results. Therefore, the effect of increasing n n + 18n further than 20 is less evident for x < 0.5 for the convergence of C¯n,s ( f ; p, q; x) to the function f (x). On the other hand, it is required to increase the value of n further than 50 for x > 0.5 in order to have more accurate results for each sequences. Comparative results are given in Table 1 and Table 2, for the errors of the approximation of C¯n,s ( f ; p, q; x) to f (x), considering each sequences for different n values. f (x) − C¯30,2 ( f ; p, q; x) f (x) − C¯50,2 ( f ; p, q; x) 0.0310 0.0190 0.0432 0.0207 0.0200 0.0049 0.0411 0.0544 0.1224 0.1050 0.1998 0.1280 0.2723 0.1210 0.3627 0.1242 0.3925 0.1406   2 n +4 Table 1 Errors of approximation C¯n,s ( f ; p, q; x) to f (x) s = 2, bn = 2 , p = 1, q = 0.98 n + 18n x 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

f (x) 1.2876 0.8276 0.3657 0.0494 0.0482 0.4642 1.1997 1.8665 1.9157

f (x) − C¯20,2 ( f ; p, q; x) 0.0460 0.0720 0.0543 0.0180 0.1369 0.2913 0.4942 0.7628 0.9275

f (x) − C¯80,2 ( f ; p, q; x) 0.0125 0.0090 0.0172 0.0595 0.0933 0.0898 0.0489 0.0224 0.0521

Figure 2 demonstrates the convergence of C¯n,s ( f ; p, q; x) to f (x) but this time considn ering different p and q values, when n = 50 for bn = 1 + log( ). In Figures 2(a), and n + 12 2(b), as q values are increased, the errors of the approximation of C¯n,s ( f ; p, q; x) to f (x) is 9

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(a) n = 20

(b) n = 30

(c) n = 50

(d) n = 80

Fig. 1 Convergence of C¯n,s ( f ; p, q; x)for different n values f (x) − C¯30,2 ( f ; p, q; x) f (x) − C¯50,2 ( f ; p, q; x) f (x) − C¯80,2 ( f ; p, q; x) 0.0317 0.0196 0.0130 0.0427 0.0204 0.0089 0.0156 0.0088 0.0198 0.0520 0.0644 0.0663 0.1390 0.1203 0.1040 0.2187 0.1412 0.0993 0.2650 0.1157 0.0461 0.3110 0.0825 0.0045 0.2897 0.0640 0.0038   n ), p = 1, q = 0.98 Table 2 Errors of approximation C¯n,s ( f ; p, q; x) to f (x) s = 2, bn = 1 + log( n + 12 x 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

f (x) 1.2876 0.8276 0.3657 0.0494 0.0482 0.4642 1.1997 1.8665 1.9157

f (x) − C¯20,2 ( f ; p, q; x) 0.0459 0.0721 0.0546 0.0173 0.1358 0.2903 0.4947 0.7665 0.9361

minimised for x < 0.5 and x > 0.8 for any given p values. On the other hand, the results show that decreasing q values has an effect on the convergence of C¯n,s ( f ; p, q; x) which provide better approximate results for x > 0.6 and x < 0.8 (See Table 2).

(a) p = 1

(b) p = 0.8

Fig. 2 Convergence of C¯n,s ( f ; p, q; x) for different p and q values

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Dynamical behavior of a general HIV-1 infection model with HAART and cellular reservoirs Ahmed M. Elaiw, Abdullah M. Althiabi, Mohammed A. Alghamdi and Nicola Bellomo Department of Mathematics, Faculty of Science, King Abdulaziz University, P.O. Box 80203, Jeddah 21589, Saudi Arabia. Email: a m [email protected] (A. Elaiw) Abstract This paper study the dynamical behavior of a general HIV-1 infection model under the effect of Highly active antiretroviral therapies (HAART). The model includes three types of infected cells (i) long-lived productively infected cells which live for long time and creat small amount of HIV-1 particles, (ii) latently infected cells which do not creat HIV-1 until they have been activated (iii) short-lived productively infected cells which live for long time and creat large amount of HIV-1. The model incorporates humoral immune response and general nonlinear forms for the incidence rate of infection, the generation and removal rates of all compartments. The nonnegativity and boundedness of the solutions of the model as well as global stability of the steady states are studied. The global stability are established using Lyapunov method. Using MATLAB we conduct some numerical simulations to confirm our results.

Keywords: HIV-1 infection; HAART; global stability; humoral immune response; latency; viral reservoirs

1

Introduction

Human immunodeficiency virus type 1 (HIV-1) infects the CD4+ T cells which play the central role in the immune system of the human body. HIV-1 causes gradual depletion in the concentration of the uninfected CD4+ T cells which decreases the efficiency of the immune system against other infections. During the last decades, substantial efforts have been paid to propose treatment strategies for HIV-1 [1], [2]. Highly active antiretroviral therapies (HAART) which combines two classes of antiviral drugs, reverse transcriptase inhibitor (RTI) and protease inhibitor (PI), can rapidly decrease the concentration of the HIV-1 and increase the concentration of the healthy CD4+ T cells in the plasma. However, HAART can not eradicate the HIV-1 completely due to the presence of viral reservoirs such as latently infected cells. Mathematical modeling and analysis of the dynamics of HIV-1 are helpful in understanding the virus dynamics and improving diagnosis and treatment strategies [3]-[23]. Modeling the HIV-1 dynamics with latent infection has been studied by several researchers [24]-[29]. The HIV-1 dynamics model with latently infected cells consists of four compartments: uninfected CD4+ T cells, latently infected cells, actively infected cells and free HIV-1 particles [24]. x˙ = ρ − dx − (1 − εr )βxv,

(1)

w˙ = h(1 − εr )βxv − (a1 + δ1 )w,

(2)

y˙ = (1 − h)(1 − εr )βxv + a1 w − δ2 y,

(3)

v˙ = N δ2 y − δ4 v,

(4)

where x, w, y, v represent the concentrations of the uninfected CD4+ T cells, latently infected cells, actively infected cells and free HIV-1 particles, respectively. ρ > 0 is the replenished rate of uninfected CD4+ T cells from

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body’s sources such as bone marrow and thymus. The parameters d, δ1 , δ2 and δ4 are the death rate constants of the uninfected CD4+ T cells, latently infected cells, actively infected cells and HIV-1 particles, respectively. The uninfected CD4+ T cells become infected by viruses with infectivity β. The efficacy of the RTI drugs is given by εr , where εr ∈ [0, 1]. The latently infected cells are activated at rate a1 w. The parameter N is the average number of HIV-1 particles generated in the lifetime of the actively infected. A fraction h ∈ (0, 1) of infection events result in latent infection. The global stability analysis of model (1)-(4) has been studied by Wang et al. in [26]. As reported in [24], [2] and [30] three are two types of productively infected cells, the first is short-lived productively infected cells which live for short time and produce hight numbers of HIV-1 particles, and the second is the long-lived productively infected cells which live for long time and produce small numbers of HIV1 particles. Long-lived productively infected cells can be seen as another reservoirs which a major obstacle to eliminate the HIV-1 completely by HAART. Model (1)-(4) has been modified by including: (i) mitotic proliferation of the uninfected CD4+ T cells, (ii) three types of infected cells, latently infected cells (w), shortlived productively infected cells (y), and long-lived productively infected cells (u) [30].   x x˙ = ρ − dx + px 1 − − (1 − εr )(β 1 + β 2 + β 3 )xv, (5) xmax w˙ = (1 − εr )β 1 xv − (a1 + δ1 )w,

(6)

y˙ = (1 − εr )β 2 xv + a1 w − δ2 y,

(7)

u˙ = (1 − εr )β 3 xv − δ3 u,

(8)

v˙ = (1 − εp )N δ2 y + (1 − εp )M δ3 u − δ4 v.

(9)

Uninfected CD4+ T cells can be produced by proliferation of existing healthy cells in the body. The parameter p > 0 is the maximum proliferation rate of uninfected cells. The parameter xmax > 0 is the maximum level of uninfected cell concentration in the body. If the concentration arrives at xmax , it should decreases. The parameters δ2 and δ3 are the death rate constants of the short-lived productively infected cells and long-lived productively infected cells, respectively. The uninfected CD4+ T cells become infected by viruses with infectivity β 1 + β 2 + β 3 . The efficacy of the PI drugs is given by εp , where εp ∈ [0, 1]. The parameter M is the average number of HIV-1 particles generated in the lifetime of the long-lived productively infected cells. In model (5)-(9) we note the following (i) the infection rate is given by bilinear incidence which may not describe the virus dynamics accurately, (ii) the death rate of all compartments, the production rate of viruses and the latent-to-active transmission rate are given by linear functions, however, these rates are generally not known, (iii) the effect of immune response has been neglecting. The aim of this paper is to propose an HIV1 infection model which improves model (5)-(9) by taking into account the humoral immune response and by assuming that the intrinsic growth rate of uninfected CD4+ T cells as well as the death rate of HIV-1 and infected cells are given by general nonlinear functions. We study the qualitative behavior of the proposed model. The existence and global stability of all the steady states of the model is established. Lyapunov functionals and LaSalle’s invariance principle are used to prove the global stability of the model.

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2

Mathematical HIV-1 dynamics model

Based on the above discussion we formulate a general nonlinear HIV dynamics model with humoral immunity. The model can be considered as a generalization of several existing HIV-1 models. x˙ = π(x) − (1 − εr )(β 1 + β 2 + β 3 )ξ(x, v),

(10)

w˙ = (1 − εr )β 1 ξ(x, v) − (a1 + δ1 )g1 (w),

(11)

y˙ = (1 − εr )β 2 ξ(x, v) + a1 g1 (w) − δ2 g2 (y),

(12)

u˙ = (1 − εr )β 3 ξ(x, v) − δ3 g3 (u),

(13)

v˙ = (1 − εp )N δ2 g2 (y) + (1 − εp )M δ3 g3 (u) − δ4 g4 (v) − qg4 (v)g5 (z),

(14)

z˙ = rg4 (v)g5 (z) − δ5 g5 (z),

(15)

where z represents the concentration of the B cells. Function π(x) represents the intrinsic growth rate of uninfected CD4+ T cells accounting for both production and natular mortality. The viruses are neutralized at rate qg5 (z)g4 (v) and die at rate δ4 g4 (v), where q and δ4 are positive constants. The B cells are activated at rate rg5 (z)g4 (v) and die at rate δ5 g5 (z). All the parameters are positive. Let us define β i = (1 − εr )β i , i = 1, 2, 3, N = (1 − εp )N and M = (1 − εp )M . Functions π, ξ, gi , i = 1, ..., 5 are continuously differentiable, moreover, they satisfy some hypotheses: (H1). (i) there exists x0 such that π(x0 ) = 0, π(x) > 0 for x ∈ [0, x0 ), (ii) π 0 (x) < 0 for x ∈ (0, ∞), (iii) there are b > 0 and b > 0 such that π(x) ≤ b − bx for x ∈ [0, ∞). (H2). (i) ξ(x, v) > 0 and ξ(0, v) = ξ(x, 0) = 0 for x, v ∈ (0, ∞), (ii) ∂ξ(x,v) > 0, ∂ξ(x,v) > 0 and ∂ξ(x,0) > 0 for all x, v ∈ (0, ∞), ∂v ∂x 0 ∂v ∂ξ(x, 0) (iii) > 0 for x ∈ (0, ∞). ∂v (H3). (i) gj (η) > 0 for η ∈ (0, ∞), gj (0) = 0, j = 1, ..., 5 (ii) gj0 (η) > 0 for η ∈ (0, ∞), j = 1, 2, 3, 5, g40 (η) > 0, for η ∈ [0, ∞), (iii) there  are αj >0, j = 1, ..., 5 such that gj (η) ≥ αj η for η ∈ [0, ∞). ξ(x, v) ∂ ≤ 0 for v ∈ (0, ∞). (H4). ∂v g4 (v)

3

Basic properties

In this section we study the basic properties of model (10)-(15). The non-negativity and boundedness of the solutions of the model will be established in the next theorem: Theorem 1. Let Hypotheses (H1)-(H3) be hold true, then there exist a set  ∆ = (x, w, y, u, v, z) ∈ R6≥0 : 0 ≤ x, w, y, u ≤ κ1 , 0 ≤ v ≤ κ2 , 0 ≤ z ≤ κ3 which is positively invariant with respect to system (10)-(15), where κ1 , κ2 and κ3 are positive numbers. Proof. First, we show that R6≥0 is positively invariant for system (10)-(15) as: x˙ |x=0 = π(0) > 0, w˙ |w=0 = β 1 ξ(x, v) ≥ 0

for x, v ∈ [0, ∞),

y˙ |y=0 = β 2 ξ(x, v) + a1 g1 (w) ≥ 0

for x, w, v ∈ [0, ∞),

u˙ |u=0 = β 3 ξ(x, v) ≥ 0

for x, v ∈ [0, ∞),

v˙ |v=0 = N δ2 g2 (y) + M δ3 g3 (u) ≥ 0

for y, u ∈ [0, ∞),

z˙ |z=0 = 0.

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Hence, all the state variables of system (10)-(15) are non-negative. Let T1 (t) = x(t) + w(t) + y(t) + u(t), then T˙1 = π(x) − δ1 g1 (w) − δ2 g2 (y) − δ3 g3 (u) ≤ b − bx − δ1 α1 w − δ2 α2 y − δ3 α3 u. ≤ b − σ1 (x + w + y + u) = b − σ1 T1 , b . The non-negativity of σ1 x(t), w(t), y(t) and u(t) implies 0 ≤ x(t), w(t), y(t), u(t) ≤ κ1 , if 0 ≤ x(0) + w(0) + y(0) + u(0) ≤ κ1 . Moreover, we let T2 (t) = v(t) + rq z(t). Then where σ1 = min{b, δ1 α1 , δ2 α2 , δ3 α3 }. Hence, T1 (t) ≤ κ1 , if T1 (0) ≤ κ1 , where κ1 =

qδ5 T˙2 = N δ2 g2 (y) + M δ3 g3 (u) − δ4 g4 (v) − g5 (z) r qδ5 ≤ N δ2 g2 (κ1 ) + M δ3 g3 (κ1 ) − δ4 α4 v − α5 z r ≤ N δ2 g2 (κ1 ) + M δ3 g3 (κ1 ) − σ2 T2 , N δ2 g2 (κ1 ) + M δ3 g3 (κ1 ) . The σ2 q non-negativity of v(t) and z(t) implies 0 ≤ v(t) ≤ κ2 and 0 ≤ z(t) ≤ κ3 if 0 ≤ v(0) + r z(0) ≤ κ2 , where rκ2 . κ3 = q Theorem 2. Suppose that Hypotheses (H1)-(H4) are valid, then there exist two bifurcation parameters R0 and R1 with R0 > R1 > 0 such that (i) if R0 ≤ 1, then the system has only one positive steady state S0 ∈ ∆, (ii) if R1 ≤ 1 < R0 , then the system has only two positive steady states S0 ∈ ∆ and S1 ∈ ∆,

where σ2 = min{δ4 α4 , δ5 α5 }. Hence, T2 (t) ≤ κ2 if T2 (0) ≤ κ2 , where κ2 =

◦

(iii) if R1 > 1, then the system has three positive steady states S0 ∈ ∆, S1 ∈ ∆ and S2 ∈ ∆. Proof. Let S(x, w, y, u, v, z) be any steady state of (10)-(15) satisfying the following equations: 0 = π(x) − (β 1 + β 2 + β 3 )ξ(x, v),

(16)

0 = β 1 ξ(x, v) − (a1 + δ1 )g1 (w),

(17)

0 = β 2 ξ(x, v) + a1 g1 (w) − δ2 g2 (y),

(18)

0 = β 3 ξ(x, v) − δ3 g3 (u),

(19)

0 = N δ2 g2 (y) + M δ3 g3 (u) − δ4 g4 (v) − qg4 (v)g5 (z),

(20)

0 = rg4 (v)g5 (z) − δ5 g5 (z).

(21)

From Eq. (21) we have two possible solutions, g5 (z) = 0 and g4 (v) = δ5 /r. Let us consider the case g5 (z) = 0, then from Hypothese (H3) we get z = 0. Hypothese (H3) implies that gi−1 , i = 1, ..., 5 exist, strictly increasing and gi−1 (0) = 0. Let us define     β1 a1 β 1 + (a1 + δ1 )β 2 θ(x) = g1−1 π(x) , ψ(x) = g2−1 π(x) , β(a1 + δ1 ) δ2 β(a1 + δ1 )     β3 γ µ(x) = g3−1 π(x) , `(x) = g4−1 π(x) , (22) δ3 β β where β = β 1 + β 2 + β 3 and γ =

N (a1 β 1 +(a1 +δ1 )β 2 )+M β 3 (a1 +δ1 ) . δ4 (a1 +δ1 )

w = θ(x),

y = ψ(x),

It follows from Eqs. (16)-(21) that:

u = µ(x),

v = `(x).

(23)

Obviousely, θ(x), ψ(x), µ(x), `(x) > 0 for x ∈ [0, x0 ) and θ(x0 ) = ψ(x0 ) = µ(x0 ) = `(x0 ) = 0. From Eqs. (16), (22) and (23) we obtain γξ(x, `(x)) − g4 (`(x)) = 0. (24)

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Eq. (24) admits a solution x = x0 which gives the infection-free steady state S0 (x0 , 0, 0, 0, 0, 0). Let Ψ1 (x) = γξ(x, `(x)) − g4 (`(x)) = 0. It is clear from Hypotheses (H1) and (H2) that, Ψ1 (0) = −g4 (`(0)) < 0 and Ψ1 (x0 ) = 0. Moreover,   ∂ξ(x0 , 0) ∂ξ(x0 , 0) 0 0 Ψ1 (x0 ) = γ + ` (x0 ) − g40 (0)`0 (x0 ). ∂x ∂v We note from Hypothese (H2) that

∂ξ(x0 ,0) ∂x

= 0. Then,   γ ∂ξ(x0 , 0) Ψ01 (x0 ) = `0 (x0 )g40 (0) − 1 . g40 (0) ∂v

From Eq. (22), we get Ψ01

γ (x0 ) = π 0 (x0 ) β



 γ ∂ξ(x0 , 0) −1 . g40 (0) ∂v

0 ,0) > 1, then Ψ01 (x0 ) < 0 and there Therefore, from Hypothese (H1), we have π 0 (x0 ) < 0. Therefore, if g0 γ(0) ∂ξ(x ∂v 4 exists x1 ∈ (0, x0 ) such that Ψ1 (x1 ) = 0. Hypotheses (H1)-(H3) imply that

w1 = θ(x1 ) > 0, y1 = ψ(x1 ) > 0, u1 = µ(x1 ) > 0, v1 = `(x1 ) > 0.

(25)

0 ,0) It means that, a humoral-inactivated infection steady state S1 (x1 , w1 , y1 , u1 , v1 , 0) exists when g0 γ(0) ∂ξ(x > 1. ∂v 4 Let us define γ ∂ξ(x0 , 0) R0 = 0 , g4 (0) ∂v   δ5 δ5 which yields v2 = g4−1 > 0. Substituting v = v2 in Eq. (16) The other solution of Eq. (21) is g4 (v2 ) = r r and letting Ψ2 (x) = π(x) − βξ(x, v2 ) = 0. According to Hypotheses (H1) and (H2), Ψ2 is strictly decreasing, Ψ2 (0) = π(0) > 0 and Ψ2 (x0 ) = −βξ(x0 , v2 ) < 0. Thus, there exists a unique x2 ∈ (0, x0 ) such that Ψ2 (x2 ) = 0. It follows from Eqs. (20) and (23) that,   δ5 −1 w2 = θ(x2 ) > 0, y2 = ψ(x2 ) > 0, u2 = µ(x2 ) > 0, v2 = g4 > 0, r    δ4 ξ(x2 , v2 ) z2 = g5−1 −1 . γ q g4 (v2 ) 2 ,v2 ) Thus, z2 > 0 when γ ξ(x g4 (v2 ) > 1. Now we define the paramater R1 as:

R1 = γ If R1 > 1, then z2 = g5−1 S2 (x2 , w2 , y2 , u2 , v2 , z2 ).



δ4 q (R1

ξ(x2 , v2 ) . g4 (v2 )

 − 1) > 0 and exists a humoral-activated infection steady state ◦

Now we show that S0 ∈ ∆, S1 ∈ ∆ and S2 ∈ ∆. Clearly, S0 ∈ ∆. Now we show that S1 ∈ ∆. We have x1 ∈ (0, x0 ), then from Hypothese (H1) we obtain 0 = π(x0 ) < π(x1 ) ≤ b − bx1 . It follows that 0 < x1
1 we get δ4 g4 (v2 ) < N δ2 g2 (y2 ) + M δ3 g3 (u2 ) ⇒ δ4 α4 v2 < N δ2 g2 (κ1 ) + M δ3 g3 (κ1 ) ⇒ 0 < v2
1 and Hypotheses (H1)-(H4) are satisfied, then

sgn(R1 − 1) = sgn(v1 − v2 ) = sgn(x2 − x1 ). Proof. Using Hypotheses (H1) and (H2), that for x1 , x2 , v1 , v2 > 0, we get (x1 − x2 ) (π(x2 ) − π(x1 )) > 0,

(31)

(x2 − x1 )(ξ(x2 , v2 ) − ξ(x1 , v2 )) > 0,

(32)

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(v2 − v1 ) (ξ(x1 , v2 ) − ξ(x1 , v1 )) > 0,

(33)

and from Hypothese (H4), we obtain  (v1 − v2 )

ξ(x1 , v2 ) ξ(x1 , v1 ) − g2 (v2 ) g2 (v1 )

 > 0.

(34)

First, we show that sgn(v1 − v2 ) = sgn(x2 − x1 ). Suppose that sgn(v2 − v1 ) = sgn(x2 − x1 ). Using the steady states conditions of S1 and S2 we obtain π(x2 ) − π(x1 ) = β [ξ(x2 , v2 ) − ξ(x1 , v1 )] = β [(ξ(x2 , v2 ) − ξ(x1 , v2 )) + (ξ(x1 , v2 ) − ξ(x1 , v1 ))] . Therefore, from inequalities (31)-(33) we obtain sgn (x2 − x1 ) = sgn (x1 − x2 ), which is a contradiction, hence, sgn (v1 − v2 ) = sgn (x2 − x1 ) . Using Eq. (25) and the definition of R1 we get     ξ(x2 , v2 ) ξ(x1 , v1 ) 1 ξ(x1 , v2 ) ξ(x1 , v1 ) R1 − 1 = γ − =γ (ξ(x2 , v2 ) − ξ(x1 , v2 )) + − . g4 (v2 ) g4 (v1 ) g4 (v2 ) g4 (v2 ) g4 (v1 ) Thus, from Eqs. (32) and (34) we obtain sgn(R1 − 1) = sgn(v1 − v2 ). Theorem 4. Suppose that Hypotheses (H1)-(H4) are satisfied and R1 ≤ 1 < R0 , then S1 is globally asymptotically stable in ∆. Proof. We introduce Lyapunov function     Zw Zy Zx g (w ) g (y ) ξ(x1 , v1 ) 1 1 2 1 dη + k1 w − w1 − dη  + k2 y − y1 − dη  W1 = x − x1 − ξ(η, v1 ) g1 (η) g2 (η) w1 y1 x1     u v Z Z g3 (u1 )  g4 (v1 )  + k3 u − u1 − dη + k4 v − v1 − dη + k5 z, g3 (η) g4 (η) u1

and evaluate

v1

dW1 dt along

the trajectories of (10)-(15):     ξ(x1 , v1 ) dW1 g1 (w1 ) = 1− (π(x) − βξ(x, v)) + k1 1 − (β 1 ξ(x, v) − (a1 + δ1 )g1 (w)) dt ξ(x, v1 ) g1 (w)     g2 (y1 ) g3 (u1 ) + k2 1 − (β 2 ξ(x, v) + a1 g1 (w) − δ2 g2 (y)) + k3 1 − (β 3 ξ(x, v) − δ3 g3 (u)) g2 (y) g3 (u)   g4 (v1 ) (N δ2 g2 (y) + M δ3 g3 (u) − δ4 g4 (v) − qg4 (v)g5 (z)) + k4 1 − g4 (v) + k5 (rg4 (v)g5 (z) − δ5 g5 (z)) .

(35)

Collecting terms of Eq. (35) and applying π(x1 ) = βξ(x1 , v1 ) we get     dW1 ξ(x1 , v1 ) ξ(x1 , v1 ) = (π(x) − π(x1 )) 1 − + βξ(x1 , v1 ) 1 − dt ξ(x, v1 ) ξ(x, v1 ) ξ(x1 , v1 ) g1 (w1 ) g2 (y1 ) + βξ(x, v) − k1 β1 ξ(x, v) + k1 (a1 + δ1 )g1 (w1 ) − k2 β2 ξ(x, v) ξ(x, v1 ) g1 (w) g2 (y) g2 (y1 )g1 (w) g3 (u1 ) g4 (v1 ) − k2 a1 + k2 δ2 g2 (y1 ) − k3 β 3 ξ(x, v) + k3 δ3 g3 (u1 ) − k4 N δ2 g2 (y) g2 (y) g3 (u) g4 (v) g4 (v1 ) − k4 M δ3 g3 (u) − k4 δ4 g4 (v) + k4 δ4 g4 (v1 ) + k4 qg4 (v1 )g5 (z) − k5 δ5 g5 (z). g4 (v) Utilizing conditions of the steady state S1 , we obtain (a1 + δ1 )g1 (w1 ) = β1 ξ(x1 , v1 ), δ3 g3 (u1 ) = β 3 ξ(x1 , v1 ),

735

k2 δ2 g2 (y1 ) = (k1 β 1 + k2 β 2 )ξ(x1 , v1 ), k4 δ4 g4 (v1 ) = βξ(x1 , v1 ),

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J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 24, NO.4, 2018, COPYRIGHT 2018 EUDOXUS PRESS, LLC

then, we have     dW1 ξ(x1 , v1 ) ξ(x1 , v1 ) + βξ(x1 , v1 ) 1 − = (π(x) − π(x1 )) 1 − dt ξ(x, v1 ) ξ(x, v1 )   ξ(x, v) g4 (v) ξ(x, v)g1 (w1 ) + βξ(x1 , v1 ) − − k1 β1 ξ(x1 , v1 ) + k1 β1 ξ(x1 , v1 ) ξ(x, v1 ) g4 (v1 ) ξ(x1 , v1 )g1 (w) g2 (y1 )g1 (w) ξ(x, v)g2 (y1 ) − k1 β1 ξ(x1 , v1 ) + (k1 β 1 + k2 β 2 )ξ(x1 , v1 ) − k2 β2 ξ(x1 , v1 ) ξ(x1 , v1 )g2 (y) g2 (y)g1 (w1 ) ξ(x, v)g3 (u1 ) g2 (y)g4 (v1 ) − k3 β 3 ξ(x1 , v1 ) + k3 β 3 ξ(x1 , v1 ) − (k1 β 1 + k2 β 2 )ξ(x1 , v1 ) ξ(x1 , v1 )g3 (u) g2 (y1 )g4 (v)   g3 (u)g4 (v1 ) δ5 − k3 β 3 ξ(x1 , v1 ) + βξ(x1 , v1 ) + k5 r g4 (v1 ) − g5 (z). g3 (u1 )g4 (v) r

(37)

Equation (37) can be simplified as:     dW1 ξ(x1 , v1 ) βξ(x1 , v1 )g4 (v) ξ(x, v) ξ(x, v1 ) = (π(x) − π(x1 )) 1 − + − (ξ(x, v) − ξ(x, v1 )) dt ξ(x, v1 ) ξ(x, v)ξ(x, v1 )) g4 (v) g4 (v1 )   ξ(x1 , v1 ) ξ(x, v)g1 (w1 ) g2 (y1 )g1 (w) g2 (y)g4 (v1 ) g4 (v)ξ(x, v1 ) + k1 β1 ξ(x1 , v1 ) 5 − − − − − ξ(x, v1 ) ξ(x1 , v1 )g1 (w) g2 (y)g1 (w1 ) g2 (y1 )g4 (v) g4 (v1 )ξ(x, v)   ξ(x1 , v1 ) ξ(x, v)g2 (y1 ) g2 (y)g4 (v1 ) g4 (v)ξ(x, v1 ) + k2 β2 ξ(x1 , v1 ) 4 − − − − ξ(x, v1 ) ξ(x1 , v1 )g2 (y) g2 (y1 )g4 (v) g4 (v1 )ξ(x, v)   ξ(x, v)g3 (u1 ) g3 (u)g4 (v1 ) g4 (v)ξ(x, v1 ) ξ(x1 , v1 ) − − − + k3 β3 ξ(x1 , v1 ) 4 − ξ(x, v1 ) ξ(x1 , v1 )g3 (u) g3 (u1 )g4 (v) g4 (v1 )ξ(x, v) + k5 r (g4 (v1 ) − g4 (v2 )) g5 (z).

(38)

Hypotheses (H1), (H2), (H4), Lemma 1 and the condition R1 ≤ 1 imply that   ξ(x1 , v1 ) (π(x) − π(x1 )) 1 − ≤ 0, ξ(x, v1 )   ξ(x, v) ξ(x, v1 ) − (ξ(x, v) − ξ(x, v1 )) ≤ 0, g4 (v) g4 (v1 ) g4 (v1 ) − g4 (v2 ) ≤ 0. It is known that the arithmetical mean is greater than or equal to the geometrical mean. It follows that for all  dW1 1 x, y, v, z > 0 we have dW dt ≤ 0 . Clearly, the largest invariant set Γ0 ⊆ Γ = (x, w, y, u, v, z) : dt = 0 is the singlton {S1 }. By LaSalle’s invariance principle, S1 is globally asymptotically stable.  Theorem 5. Let R1 > 1 and Hypotheses (H1)-(H4) are satisfied, then S2 is globally asymptotically stable ◦

in ∆. Proof. Define a Lyapunov functional     Zw Zy ξ(x2 , v2 ) g (w ) g (y ) 1 2 2 2 W 2 = x − x2 − dη + k1 w − w2 − dη  + k2 y − y2 − dη  ξ(η, v2 ) g1 (η) g2 (η) x2 w2 y2       Zu Zv Zz g (v ) g (z ) g (u ) 4 2 5 2 3 2 dη  + k4 v − v2 − dη  + k5 z − z2 − dη  . + k3 u − u2 − g3 (η) g4 (η) g5 (η) Zx

u2

v2

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Calculating

dW2 dt

along the solutions of model (10)-(15), we get     ξ(x2 , v2 ) g1 (w2 ) dW2 = 1− (π(x) − βξ(x, v)) + k1 1 − (β 1 ξ(x, v) − (a1 + δ1 )g1 (w)) dt ξ(x, v2 ) g1 (w)     g2 (y2 ) g3 (u2 ) + k2 1 − (β 2 ξ(x, v) + a1 g1 (w) − δ2 g2 (y)) + k3 1 − (β 3 ξ(x, v) − δ3 g3 (u)) g2 (y) g3 (u)   g4 (v2 ) + k4 1 − (N δ2 g2 (y) + M δ3 g3 (u) − δ4 g4 (v) − qg4 (v)g5 (z)) g4 (v)   g5 (z2 ) + k5 1 − (rg4 (v)g5 (z) − δ5 g5 (z)) . g5 (z)

(39)

Collecting terms of Eq. (39) and applying π(x2 ) = βξ(x2 , v2 ) we get     dW1 ξ(x2 , v2 ) ξ(x2 , v2 ) = (π(x) − π(x2 )) 1 − + βξ(x2 , v2 ) 1 − dt ξ(x, v2 ) ξ(x, v2 ) ξ(x2 , v2 ) g1 (w2 ) + βξ(x, v) − k1 β1 ξ(x, v) + k1 (a1 + δ1 )g1 (w2 ) ξ(x, v2 ) g1 (w) g2 (y2 ) g2 (y2 )g1 (w) g3 (u2 ) − k2 β2 ξ(x, v) − k2 a1 + k2 δ2 g2 (y2 ) − k3 β 3 ξ(x, v) g2 (y) g2 (y) g3 (u) g4 (v2 ) g4 (v2 ) + k3 δ3 g3 (u2 ) − k4 N δ2 g2 (y) − k4 M δ3 g3 (u) − k4 δ4 g4 (v) + k4 δ4 g4 (v2 ) g4 (v) g4 (v) + k4 qg4 (v2 )g5 (z) − k5 δ5 g5 (z) − k5 rg5 (z2 )g4 (v) + k5 δ5 g5 (z2 ) Using the following steady state conditions for S1 : (a1 + δ1 )g1 (w2 ) = β1 ξ(x2 , v2 ), δ3 g3 (u2 ) = β 3 ξ(x2 , v2 ),

k2 δ2 g2 (y2 ) = (k1 β 1 + k2 β 2 )ξ(x2 , v2 ), k4 δ4 g4 (v2 ) = βξ(x2 , v2 ) − k4 qg5 (z2 )g4 (v2 ),

we obtain     dW2 ξ(x2 , v2 ) ξ(x2 , v2 ) = (π(x) − π(x2 )) 1 − + βξ(x2 , v2 ) 1 − dt ξ(x, v2 ) ξ(x, v2 )   ξ(x, v) g4 (v) ξ(x, v)g1 (w2 ) + βξ(x2 , v2 ) − + k1 β1 ξ(x2 , v2 ) − k1 β1 ξ(x2 , v2 ) ξ(x, v2 ) g4 (v2 ) ξ(x2 , v2 )g1 (w) g2 (y2 )g1 (w) ξ(x, v)g2 (y2 ) − k2 β2 ξ(x2 , v2 ) − k1 β1 ξ(x2 , v2 ) + (k1 β 1 + k2 β 2 )ξ(x2 , v2 ) ξ(x2 , v2 )g2 (y) g2 (y)g1 (w2 ) ξ(x, v)g3 (u2 ) − k3 β 3 ξ(x2 , v2 ) + k3 β 3 ξ(x2 , v2 ) ξ(x2 , v2 )g3 (u) g3 (u)g4 (v2 ) g2 (y)g4 (v2 ) − k3 β 3 ξ(x2 , v2 ) + βξ(x2 , v2 ). − (k1 β 1 + k2 β 2 )ξ(x2 , v2 ) g2 (y2 )g4 (v) g3 (u2 )g4 (v)

(40)

Equation (40) can be simplified as:     dW2 ξ(x2 , v2 ) βξ(x2 , v2 )g4 (v) ξ(x, v) ξ(x, v2 ) = (π(x) − π(x2 )) 1 − + − (ξ(x, v) − ξ(x, v2 )) dt ξ(x, v2 ) ξ(x, v)ξ(x, v2 )) g4 (v) g4 (v2 )   ξ(x2 , v2 ) ξ(x, v)g1 (w2 ) g2 (y2 )g1 (w) g2 (y)g4 (v2 ) g4 (v)ξ(x, v2 ) + k1 β1 ξ(x2 , v2 ) 5 − − − − − ξ(x, v2 ) ξ(x2 , v2 )g1 (w) g2 (y)g1 (w2 ) g2 (y2 )g4 (v) g4 (v2 )ξ(x, v)   ξ(x2 , v2 ) ξ(x, v)g2 (y2 ) g2 (y)g4 (v2 ) g4 (v)ξ(x, v2 ) + k2 β2 ξ(x2 , v2 ) 4 − − − − ξ(x, v2 ) ξ(x2 , v2 )g2 (y) g2 (y2 )g4 (v) g4 (v2 )ξ(x, v)   ξ(x2 , v2 ) ξ(x, v)g3 (u2 ) g3 (u)g4 (v2 ) g4 (v)ξ(x, v2 ) + k3 β3 ξ(x2 , v2 ) 4 − − − − . (41) ξ(x, v2 ) ξ(x2 , v2 )g3 (u) g3 (u2 )g4 (v) g4 (v2 )ξ(x, v) According to Hypotheses (H1), (H2) and (H4) and the relation between the geometrical and arithmetical means  dW2 2 we get dW dt ≤ 0. Clearly, the largest invariant set Γ0 ⊆ Γ = (x, w, y, u, v, z) : dt = 0 is the singlton {S2 }. By LaSalle’s invariance principle S2 is globally asymptotically stable. 

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5

Numerical simulations

We now perform some computer simulations on the following application:   (1 − εr )βxv x − , x˙ = ρ − dx + px 1 − xmax 1 + η1 x + η2 v

(42)

(1 − εr )β 1 xv − (a1 + δ1 )w, 1 + η1 x + η2 v (1 − εr )β 2 xv y˙ = + a1 w − δ2 y, 1 + η1 x + η2 v (1 − εr )β 3 xv u˙ = − δ3 u, 1 + η1 x + η2 v

w˙ =

(43) (44) (45)

v˙ = (1 − εp )N δ2 y + (1 − εp )M δ3 u − δ4 v − qvz,

(46)

z˙ = rvz − δ5 z.

(47)

We assume that p < d. In this application, we consider the following specific forms of the general functions:   x xv π(x) = ρ − dx + px 1 − , ξ(x, v) = , gi (θ) = θ, i = 1, ..., 5. xmax 1 + η1 x + η2 v First we verify Hypotheses (H1)-(H4) for the chosen forms, then we solve the system using MATLAB. Clearly, π(0) = ρ > 0 and π(x0 ) = 0, where r   4ρp xmax p − d + (p − d)2 + . x0 = 2p xmax We have π 0 (x) = −d + p −

2px < 0. xmax

(48)

Clearly, π(x) > 0, for x ∈ [0, x0 ) and π(x) = ρ − (d − p)x − p

x2 ≤ ρ − (d − p)x xmax

Then Hypothese (H1) is satisfied. We also have ξ(x, v) > 0, ξ(0, v) = ξ(x, 0) = 0 for x,v ∈ (0, ∞), and ∂ξ(x, v) v (1 + δv) = , ∂x (1 + η1 x + η2 v)2 Then, ∂ξ(x,v) > 0, ∂x addition

∂ξ(x,v) ∂v

> 0 and

∂ξ(x,0) ∂v

∂ξ(x, v) x(1 + η1 x) = 2, ∂v (1 + η1 x + η2 v)

∂ξ(x, 0) x = . ∂v 1 + η1 x

> 0 for x, v ∈ (0, ∞). Therefore, Hypothese (H1) is satisfied. In

xv ∂ξ(x, 0) xv ≤ =v , 1 + η1 x + η2 v 1 + η1 x ∂v 0  1 ∂ξ(x, 0) = > 0 for all x > 0. ∂v (1 + η1 x)2

ξ(x, v) =

It follows that, (H2) is satisfied. Clearly Hypothese (H3) holds true. Moreover,   ∂ ξ(x, v) −η2 x = < 0. ∂v g4 (v) (1 + η1 x + η2 v) Therefore, Hypothese (H4) hold true and Theorems 3-5 are applicable. The parameters R0 and R1 for this application are given by:  (1 − εr )(1 − εp ) N (a1 β 1 + (a1 + δ1 )β 2 ) + M β 3 (a1 + δ1 ) x0 R0 = , δ4 (a1 + δ1 ) 1 + η 1 x0  (1 − εr )(1 − εp ) N (a1 β 1 + (a1 + δ1 )β 2 ) + M β 3 (a1 + δ1 ) x2 R1 = . δ4 (a1 + δ1 ) 1 + η1 x2 + η2 v2

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Now we are ready to perform some numerical simulations for system (42)-(47). The data of system (42)-(47) are provided in Table 1. • Effect of the drug efficacy on the stability of the steady states Now we verify our theoretical results given in Theorems 3-5 by numerical simulation. To discuss our global results we choose three different initial conditions: IC1: (x(0), w(0), y(0), u(0), v(0), z(0)) = (900, 10, 12, 60, 40, 1.6). IC2: (x(0), w(0), y(0), u(0), v(0), z(0)) = (700, 7, 8, 30, 25, 1.0). IC3: (x(0), w(0), y(0), u(0), v(0), z(0)) = (500, 4, 5, 10, 15, 0.6). Let us address three scenarios for three different groups of the parameters εr , εp and r. Scenario (I): In this case we choose εr = 0.6, εp = 0.6 and r = 0.001 which gives R0 = 0.4941 < 1 and R1 = 0.4430 < 1. Therefore, based on Theorems 2 and 3, the system has unique steady state, that is S0 and it is globally asymptotically stable. As we can see from Figures 1-6 that the concentration of the uninfected CD4+ T cells is increased and approached its normal value before infection that is x0 = 1083.9, while concentrations of the other compartments converge to zero for all the three initial conditions. This case corresponds to the uninfected state where the HIV-1 is removed from the plasma. Scenario (II): By taking εr = 0.2, εp = 0.5 and r = 0.001. With such choice we get, R1 = 0.9351 < 1 < R0 = 1.2352. Consequently, based on Theorems 2 and 4, the humoral-inactivated infection steady state S1 is positive and is globally asymptotically stable. Figures 1-6 confirm that the numerical results support the theoretical results presented in Theorem 4. It can be observed that, the variables of the model eventually converge to S1 = (309.165, 13.2492, 15.4574, 94.0263, 72.0754, 0.0) for all the three initial conditions. This case corresponds to a chronic HIV-1 infection in the absense of immune response. Scenario (III): εr = 0.2, εp = 0.5 and r = 0.003. Then, we calculate R0 = 1.2352 > 1 and R1 = 1.19604 > 1. According to Lemma 1 and Theorem 3, the humoral-activated infection steady state S2 is positive and is globally asymptotically stable. We can see from Figures 1-6 that, there is a consistency between the numerical results and theoretical results of Theorem 5. The states of the system converge to S2 = {820.603, 5.8629, 6.8401, 41.6079, 26.6667, 1.1762) for all the three initial conditions, in the same time frame. In this case the humoral immune response is activated and can control the disease. • Effect of the HAART on the basic reproduction number: Let us define the overall HAART effect as εe = εr + εp − εr εp [9]. If εe = 0, then the HAART has no effect, if εe = 1, the HIV-1 growth is completely halted. Consequently, the parameter R0 is given by  (1 − εe ) N (a1 β 1 + (a1 + δ1 )β 2 ) + M β 3 (a1 + δ1 ) x0 . R0 (εe ) = δ4 (a1 + δ1 ) 1 + η 1 x0 We note that, the value of R0 (εe ) does not depend on the values of the parameters q, r and δ5 . This means that, humoral immune response can play a significant role in reducing the infection progress but do not play a role in clearing the HIV-1 from the body. Since the goal is to clear the HIV-1 from the body, then we have to determine the drug efficacies that make R0 (εe ) ≤ 1 for system (42)-(47). Now, we calculate the critical overall treatment effect εcrit (i.e, the minimum overall treatment effect required to stabilize the system around the e infection-free steady state). Let R0 (εe ) ≤ 1 , then   R0 (0) − 1 crit crit εe ≤ εe < 1, εe = max 0, , R0 (0) Figure 15 shows the effect of the HAART on the basic reproduction number R0 (ε). We note that, if εcrit ≤ εe < e 1, then R0 (εe ) ≤ 1 and S0 is globally asymptotically stable. Moreover, if 0 ≤ εe < εcrit , then S is unstable. 0 e

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Parameter ρ d p xmax q

Table 1: The values Value Parameter 10 δ1 0.01 δ2 0.008 δ3 1200 δ4 0.5 β1

of the parameters of example (42)-(47). Value Parameter Value Parameter 0.02 β2 0.0625 µ 0.36 β3 0.0625 N 0.031 a1 0.2 M 3.0 η1 1 εr , εp 0.0625 η2 1 r

Value 0.08 62 30 Varied Varied

20

1100 Scenario (I)

18

1000

16 Scenario (III)

Latently infected cells

Uninfected CD4+ T cells

900 800 700 600 500 400

Scenario (II)

14 12 10 8

Scenario (III) 6 4

Scenario (II)

300

2 Scenario (I)

200 0

100

200

300 Time

400

500

0 0

600

Figure 1: The concentration of uninfected CD4+ T cells for system (42)-(47).

300 Time

400

500

600

120

18

Long−lived productively infected cells

Short−lived productively infected cells

200

Figure 2: The concentration of latently infected cells for system (42)-(47).

20

Scenario (II)

16 14 12 10 8

Scenario (III)

6 4 2 0 0

100

Scenario (II)

100

80

60 Scenario (III) 40

20

Scenario (I) 100

200

300 Time

400

500

Scenario (I) 0 0

600

Figure 3: The concentration of short-lived productively infected cells for system (42)-(47).

740

100

200

300 Time

400

500

600

Figure 4: The concentration of long-lived productively infected cells for system (42)-(47).

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90

2.5

80

Scenario (II) 2

60 1.5 50

B cells

Free virus particles

70

40

1

Scenario (III)

30

Scenario (III)

20

Scenario (I)

0.5

Scenario (II)

10 Scenario (I) 0 0

100

200

300 Time

400

500

0 0

600

Figure 5: The concentration of free virus particles for system (42)-(47).

100

200

300 Time

400

500

600

Figure 6: The concentration of B cells for system (42)-(47).

The basic reproduction number R0

3.5 3 2.5 S0 unstable 2 1.5 R0=1 1 0.5

S stable 0

0 0

0.2

0.4 0.6 The overall treatment effect ε

0.8

1

e

Figure 7: The basic reproduction number as a function of the overall treatment effect εe of system (42)-(47).

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6

Acknowledgment

This research is supported by the Deanship of Scientific Research (DSR), King Abdulaziz University, Jeddah, under Grant No. (29-130-36-HiCi). The authors, therefore, acknowledge with thanks DSR technical and financial support.

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[15] G. Huang, Y. Takeuchi and W. Ma, Lyapunov functionals for delay differential equations model of viral infections, SIAM J. Appl. Math., 70(7) (2010), 2693-2708. [16] 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. [17] A.M. Elaiw, Global properties of a class of HIV models, Nonlinear Analysis: Real World Applications, 11 (2010), 2253-2263. [18] D. Huang, X. Zhang, Y. Guo, and H. Wang, Analysis of an HIV infection model with treatments and delayed immune response, Applied Mathematical Modelling, 40(4) (2016), 3081-3089. [19] C. Monica and M. Pitchaimani, Analysis of stability and Hopf bifurcation for HIV-1 dynamics with PI and three intracellular delays, Nonlinear Analysis: Real World Applications, 27 (2016), 55-69. [20] M. Y. Li and L. Wang, Backward bifurcation in a mathematical model for HIV infection in vivo with anti-retroviral treatment, Nonlinear Analysis: Real World Applications, 17 (2014), 147-160. [21] S. Liu, and L. Wang, Global stability of an HIV-1 model with distributed intracellular delays and a combination therapy, Mathematical Biosciences and Engineering, 7(3) (2010), 675-685. [22] Elaiw AM, Almuallem NA. Global dynamics of delay-distributed HIV infection models with differential drug efficacy in cocirculating target cells. Mathematical Methods in the Applied Sciences 2016; 39:4-31. [23] X. Wang, A. M. Elaiw, X. Song, Global properties of a delayed HIV infection model with CTL immune response, Applied Mathematics and Computation, 218 (2012), 9405-9414. [24] D.S. Callaway, and A.S. Perelson, HIV-1 infection and low steady state viral loads, Bull. Math. Biol., 64 (2002), 29-64. [25] B. Buonomo, and C. Vargas-De-Le, Global stability for an HIV-1 infection model including an eclipse stage of infected cells, Journal of Mathematical Analysis and Applications, 385 (2012), 709-720. [26] H. Wang, R. Xu, Z. Wang, H. Chen, Global dynamics of a class of HIV-1 infection models with latently infected cells, Nonlinear Analysis: Modelling and Control, 20 (1) (2012), 21-37. [27] A. Korobeinikov, Global properties of basic virus dynamics models, Bull. Math. Biol. 66, (2004), 879-883. [28] S. Pankavich, The effects of latent infection on the dynamics of HIV, Differential Equations and Dynamical Systems (2015), doi: 10.1007/s12591-014-0234-6. [29] A. M. Elaiw and N. H. AlShamrani, Global stability of humoral immunity virus dynamics models with nonlinear infection rate and removal, Nonlinear Analysis: Real World Applications, 26, (2015), 161-190. [30] W. S. Hlavacek, N. I. Stilianakis, A. S. Perelson, Influence of follicular dendritic cells on HIV dynamics, Philos.Trans. R. Soc. London B Biol. Sci. 355 (2000) 1051-1058. [31] J. K. Hale, and S. M. V. Lunel, Introduction to functional differential equations, Springer-Verlag, New York, (1993).

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Ideal theory of pre-logics based on the theory of falling shadows Young Bae Jun1 and Sun Shin Ahn2,∗ 1

Department of Mathematics Education, Gyeongsang National University, Jinju, 52828, Korea 2

Department of Mathematics Education, Dongguk University, Seoul 04620, Korea

Abstract. Based on the theory of a falling shadow which was first formulated by Wang [8], a theoretical approach of the ideal structure in pre-logics is established. The notions of a falling subalgebra, a falling and a positive implicative falling ideal of a pre-logic are introduced. Some fundamental properties are investigated. Relations among a falling subalgebra, a falling ideal and a positive implicative falling ideal are stated. Characterizations of falling deals and positive implicative falling ideals are discussed.

1. Introduction In the study of a unified treatment of uncertainty modelled by means of combining probability and fuzzy set theory, Goodman [3] pointed out the equivalence of a fuzzy set and a class of random sets. Wang and Sanchez [7] introduced the theory of falling shadows which directly relates probability concepts to the membership function of fuzzy sets. Falling shadow representation theory shows us a method of selection relied on the joint degree distributions. It is a reasonable and convenient approach for the theoretical development and the practical applications of fuzzy sets and fuzzy logics. The mathematical structure of the theory of falling shadows is formulated in [8]. Y. B. Jun and C. H. Park [5] discussed the notion of a falling fuzzy subalgebra/ideal of a BCK/BCI-algebra. Y. B. Jun and M. S. Kang [4] established a theoretical approach for defining a fuzzy positive implicative ideal in a BCK-algebra based on the theory of falling shadows. I. Chajda and R. Halas [2] introduced the concept of a pre-logic which is an algebra weaker than a Hilbert algebra (an algebraic counterpart of intuitionistic logic) but strong enough to have deductive systems. Y. B. Jun and S. S. Ahn [1] defined the notion of pseudo-valuations (valuation) on pre-logics and induced a pseudo-metric by using a pseudo-valuation on pre-logics. In this paper, we introduce the notions of a falling subalgebra, a falling ideal and a positive implicative falling ideal of a pre-logic. We investigate some fundamental properties. Also we give relations among a falling subalgebra, a falling ideal and a positive implicative falling ideal. We establish characterizations of falling ideals and positive implicative falling ideals. 2. Preliminaries 0

2010 Mathematics Subject Classification: 06F35; 03G25; 08A72. Keywords: falling shadow; falling subalgebra; falling ideal; positive implicative falling 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. S. Ahn) 0

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Young Bae Jun and Sun Shin Ahn

We recall some definitions and results (see [1, 2, 6]). Definition 2.1. ([2]) By a pre-logic, we mean a triple (X; ∗, 1) where X is a non-empty set, ∗ is a binary operation on X and 1 ∈ X is a constant such that the following identities hold: (P1) (P2) (P3) (P4)

(∀x ∈ X) (x ∗ x = 1), (∀x ∈ X) (1 ∗ x = x), (∀x ∈ X) (x ∗ (y ∗ z) = (x ∗ y) ∗ (x ∗ z)), (∀x, y, z ∈ X) (x ∗ (y ∗ z) = y ∗ (x ∗ z)).

In what follows, let X denote a pre-logic unless otherwise specified. Lemma 2.2. ([2]) Let X be a pre-logic. Then the following hold: (a) (∀x ∈ X) (x ∗ 1 = 1); (b) (∀x, y ∈ X) (x ∗ (y ∗ x) = 1); (c) an order relation ≤ on X defined by (∀x, y ∈ X) (x ≤ y if and only if x ∗ y = 1) is a quasiorder on X (i.e., a reflexive and transitive order relation on X); (d) 1 ≤ x for all x ∈ X implies x = 1. Remark 2.3. ([2]) The quasiorder ≤ of Lemma 2.2(c) is called the induced quasiorder of a pre-logic X. Lemma 2.4. ([2]) Let ≤ be the induced quasiorder of a pre-logic X and let x, y, z ∈ X. If x ≤ y, then z ∗ x ≤ z ∗ y and y ∗ z ≤ x ∗ z. Definition 2.5. ([2]) Let X = (X; ∗, 1) be a pre-logic. A non-empty subset D of X is called a deductive system of X if the following conditions hold: (d1) 1 ∈ D, (d2) if x ∈ D and x ∗ y ∈ D, then y ∈ D. Definition 2.6. ([2]) Let X be a pre-logic. A non-empty subset I of X is called an ideal of X if the following conditions are satisfied: (I1) x ∈ X and y ∈ I imply x ∗ y ∈ I; (I2) x ∈ X and y1 , y2 ∈ I imply (y2 ∗ (y1 ∗ x)) ∗ x ∈ I. Lemma 2.7. ([2]) Let X be a pre-logic and ≤ its induced quasiorder. The the following hold: (a) (∀x, y ∈ X) (x ∗ ((x ∗ y) ∗ y) = 1), (b) (∀x, y, z ∈ X) ((y ∗ z) ∗ ((x ∗ y) ∗ (x ∗ z)) = 1), (c) if D is a deductive system of X, a ∈ D, and a ≤ b, then b ∈ D.

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J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 24, NO.4, 2018, COPYRIGHT 2018 EUDOXUS PRESS, LLC

Ideal theory of pre-logics based on the theory of falling shadows

Theorem 2.8. ([1]) A non-empty subset I of a pre-logic X is an ideal of X if and only if it satisfies the following two conditions: (I1′ ) (1 ∈ I); (I2′ ) (∀x, z ∈ X)(∀y ∈ I) (x ∗ (y ∗ z) ∈ I ⇒ x ∗ z ∈ I). Definition 2.9. ([6]) A non-empty subset I of a pre-logic X is a positive implicative ideal of X if it satisfies (I1′ ) and (I3) (∀y, z ∈ X)(∀x ∈ I) (x ∗ ((y ∗ z) ∗ y) ∈ I ⇒ y ∈ I). Theorem 2.10. ([6]) Every positive implicative ideal of a pre-logic X is an ideal of X. We now display the basic theory on falling shadows. We refer the reader to the papers [3, 4, 5, 7, 8] for further information regarding the theory of falling shadows. Given a universe of discourse U, let P(U ) denote the power set of U. For each u ∈ U, let u˙ := {E | u ∈ E and E ⊆ U },

(2.1)

E˙ := {u˙ | u ∈ E}.

(2.2)

and for each E ∈ P(U ), let An ordered pair (P(U ), B) is said to be a hyper-measurable structure on U if B is a σ-field in P(U ) and U˙ ⊆ B. Given a probability space (Ω, A , P ) and a hyper-measurable structure (P(U ), B) on U, a random set on U is defined to be a mapping ξ : Ω → P(U ) which is A -B measurable, that is, (∀C ∈ B) (ξ −1 (C) = {ω | ω ∈ Ω and ξ(ω) ∈ C} ∈ A ).

(2.3)

˜ ˜ is a Suppose that ξ is a random set on U. Let H(u) := P (ω | u ∈ ξ(ω)) for each u ∈ U. Then H ˜ a falling shadow of the random set ξ, and ξ is called a cloud of kind of fuzzy set in U. We call H ˜ H. For example, (Ω, A , P ) = ([0, 1], A , m), where A is a Borel field on [0, 1] and m is the usual ˜ be a fuzzy set in U and H ˜ t := {u ∈ U | H(u) ˜ ˜ Lebesgue measure. Let H ≥ t} be a t-cut of H. ˜ t is a random set and ξ is a cloud of H. ˜ We shall call ξ defined Then ξ : [0, 1] → P(U ), t 7→ H ˜ (see [3]). above as the cut-cloud of H 3. Falling subalgebras and falling ideals Definition 3.1. Let (Ω, A , P ) be a probability space, and let ξ : Ω → P(X) be a random set, where X is a pre-logic. If ξ(ω) is a subalgebra (resp. ideal) of X for any ω ∈ Ω with ξ(ω) ̸= ∅, ˜ of the random set ξ, i.e., H(x) ˜ then the falling shadow H = P (ω | x ∈ ξ(ω)) is called a falling subalgebra (resp. falling ideal) of X. ˜ denote a falling shadow of the random set ξ : Ω → P(X). In what follows, let H

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Young Bae Jun and Sun Shin Ahn

Example 3.2. (1) Let X := {1, a, b, c, d} be a set with the following Cayley table: ∗

1

a

b

c

d

1 a b c d

1 1 1 1 1

a 1 a 1 1

b b 1 b 1

c c c 1 1

d d c b 1

Then (X; ∗, 1) is a pre-logic (see [6]). Let (Ω, A , P ) = ([0, 1], A , m) and define a random set ξ : Ω → P(X) as follows:    {1, a, b} if ω ∈ [0, 0.6) ξ(ω) := ∅ if ω ∈ [0.6, 0.7),   X if ω ∈ [0.7, 1]. ˜ of ξ is both a falling subalgebra of X and a falling ideal of X. Then the falling shadow H Define a random set η : Ω → P(X) as follows:   if ω ∈ [0, 0.3),  ∅ η(ω) := {1, b, c} if ω ∈ [0.3, 0.8),   X if ω ∈ [0.8, 1]. Then η(ω) is a subalgebra of X for all ω ∈ Ω with η(ω) ̸= ∅, but not an ideal of X, since ˜ of ξ is a falling (b ∗ (a ∗ a)) ∗ a = (b ∗ 1) ∗ a = 1 ∗ a = a ∈ / {1, b, c}. Hence the falling shadow H subalgebra of X, but not a falling ideal of X. For a probability space (Ω, A , P ) and any element x of a BCC-algebra X, let Ω(x; ξ) := {ω ∈ Ω | x ∈ ξ(ω)}.

(3.1)

Then Ω(x; ξ) ∈ A . ˜ is a falling subalgebra of a pre-logic X, then (∀x ∈ X) (Ω(x; ξ) ⊆ Ω(1; ξ)) . Lemma 3.3. If H Proof. If Ω(x; ξ) = ∅, then it is clear. Assume that Ω(x; ξ) ̸= ∅ and let ω ∈ Ω be such that ω ∈ Ω(x; ξ). Then x ∈ ξ(ω), and so 1 = x ∗ x ∈ ξ(ω) since ξ(ω) is a subalgebra of X. Hence ω ∈ Ω(1; ξ), and therefore Ω(x; ξ) ⊆ Ω(1; ξ) for all x ∈ X. □ Proposition 3.4. Every falling ideal of a pre-logic X is a falling subalgebra of X. ˜ be a falling ideal of X. Then ξ(ω) is an ideal of X for any ω ∈ Ω with ξ(ω) ̸= ∅. Proof. Let H ˜ is a Let x, y ∈ ξ(ω). Using (I1), we have x ∗ y ∈ ξ(ω). Hence ξ(ω) is a subalgebra of X. Thus H falling subalgebra of X. □

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Ideal theory of pre-logics based on the theory of falling shadows

The converse of Proposition 3.4 is not true in general (see Example 3.2). We provide a characterization of a falling ideal. ˜ is a falling ideal of X if and only if the following Theorem 3.5. Let X be a pre-logic. Then H conditions are valid: (i) (∀x, y ∈ X) (Ω(x ∗ (y ∗ z); ξ) ∩ Ω(y; ξ) ⊆ Ω(x ∗ z; ξ)) , (ii) (∀x ∈ X) (Ω(x; ξ) ⊆ Ω(1; ξ)) . ˜ is a falling ideal of X. For any x, y, z ∈ X, if ω ∈ Ω(x ∗ (y ∗ z); ξ) ∩ Ω(y; ξ), Proof. Assume that H then x ∗ (y ∗ z) ∈ ξ(ω) and y ∈ ξ(ω). It follows from (I2′ ) that x ∗ z ∈ ξ(ω) since ξ(ω) is an ideal of X. Hence ω ∈ Ω(x ∗ z; ξ). Therefore Ω(x ∗ (y ∗ z); ξ) ∩ Ω(y; ξ) ⊆ Ω(x ∗ z; ξ), for any x, y, z ∈ X. Thus (i) is valid. The second condition (ii) follows from Lemma 3.3 and Proposition 3.4. Conversely, suppose that two conditions (i) and (ii) are valid. Let x, y, z ∈ X and ω ∈ Ω be such that x ∗ (y ∗ z) ∈ ξ(ω) and y ∈ ξ(ω). Then ω ∈ Ω(x ∗ (y ∗ z); ξ) and ω ∈ Ω(y; ξ). If follows from (i) that ω ∈ Ω(x ∗ z; ξ). Hence x ∗ z ∈ ξ(ω). Now, assume that x ∈ ξ(ω) for every x ∈ X and for all ω ∈ Ω. Then ω ∈ Ω(x; ξ) ⊆ Ω(1; ξ) and so 1 ∈ ξ(ω) for all ω ∈ Ω. Therefore ξ(ω) is ˜ is a falling ideal of X. an ideal of X for all ω ∈ Ω with ξ(ω) ̸= ∅. Hence H □ ˜ is a falling ideal of X if and only if the following Theorem 3.6. Let X be a pre-logic. Then H conditions are valid: (i) (∀x, y ∈ X) (Ω(y; ξ) ⊆ Ω(x ∗ y; ξ) , (ii) (∀x, y, z ∈ X) (Ω(x; ξ) ∩ Ω(y; ξ) ⊆ Ω((x ∗ (y ∗ z)) ∗ z; ξ)) . ˜ satisfies two conditions (i) and (ii). Let x, y ∈ X and ω ∈ Ω such that y ∈ ξ(ω). Proof. Assume H Then ω ∈ Ω(y; ξ). Using (i), we have ω ∈ Ω(x ∗ y; ξ). Hence x ∗ y ∈ ξ(ω). Now, let x, y, z ∈ X and ω ∈ Ω such that x, y ∈ ξ(ω). Then ω ∈ Ω(x; ξ) and ω ∈ Ω(y; ξ) and so ω ∈ Ω(x; ξ) ∩ Ω(y; ξ). It follows from (ii) that ω ∈ Ω((x ∗ (y ∗ z)) ∗ z; ξ). Hence (x ∗ (y ∗ z)) ∗ z ∈ ξ(ω) and so ξ(ω) is ˜ is a falling ideal of X. an ideal of X. Therefore H ˜ is a falling ideal of X. Let x, y ∈ X and ω ∈ Ω be such that Conversely, suppose that H ω ∈ Ω(y; ξ). Then y ∈ ξ(ω). Since ξ(ω) is an ideal of X, we have x ∗ y ∈ ξ(ω). Hence ω ∈ Ω(x ∗ y; ξ). Therefore (i) is valid. For any x, y, z ∈ X, if ω ∈ Ω(x; ξ) ∩ Ω(y; ξ), then x ∈ ξ(ω) and y ∈ ξ(ω). Since ξ(ω) is an ideal of X, we get (x ∗ (y ∗ z)) ∗ z ∈ ξ(ω). Therefore ω ∈ Ω((x ∗ (y ∗ z)) ∗ z; ξ). Thus (ii) is true. □ Proposition 3.7. Every falling ideal of a pre-logic satisfies the following assertions: (i) (∀x ∈ X) (Ω(x; ξ) ⊆ Ω(1; ξ)), (ii) (∀x, y ∈ X) (Ω(x; ξ) ⊆ Ω((x ∗ y) ∗ y; ξ)) , (iii) (∀x, y ∈ X) (x ≤ y ⇒ Ω(x; ξ) ⊆ Ω(y; ξ)) . Proof. (i) Using (P1) and Theorem 3.6(i), we have Ω(x; ξ) ⊆ Ω(1; ξ).

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(ii) Taking x := x, y := 1, and z := y in Theorem 3.6(ii) and using (P2) and (i), we get Ω(x; ξ) = Ω(x; ξ) ∩ Ω(1; ξ) ⊆ Ω((x ∗ (1 ∗ y)) ∗ y; ξ) = Ω((x ∗ y) ∗ y; ξ). (iii) Let x, y ∈ X be such that x ≤ y. Then x ∗ y = 1. Using (P2), we have Ω(x; ξ) ⊆ Ω((x ∗ y) ∗ y; ξ) = Ω(1 ∗ y; ξ) = Ω(y; ξ). □ ˜ of a pre-logic X satisfies the following property: Lemma 3.8. Every falling ideal of H (∀x, y ∈ X)(Ω(x ∗ y; ξ) ∩ Ω(x; ξ) ⊆ Ω(y; ξ)).

(3.2)

Proof. Using (P1), (P2), and Theorem 3.6(ii), we have Ω(x ∗ y; ξ) ∩ Ω(x; ξ) ⊆ Ω(((x ∗ y) ∗ (x ∗ y)) ∗ y; ξ) = Ω(1 ∗ y; ξ) = Ω(y; ξ) for all x, y ∈ X. □ ˜ is a falling ideal of X if and only if it satisfies the Corollary 3.9. Let X be a pre-logic. Then H condition (3.2) and (i) (∀x ∈ X) (Ω(x; ξ) ⊆ Ω(1; ξ)). ˜ is a falling ideal of X. Using Proposition 3.7, (i) holds. By Lemma 3.8, Proof. Assume that H the condition (3.2) holds. ˜ satisfies two conditions (3.2) and (i). Using (3.2), we have Ω(y ∗ Conversely, suppose that H (x ∗ z); ξ) ∩ Ω(y; ξ) ⊆ Ω(x ∗ z; ξ). Using (P4), we have Ω(x ∗ (y ∗ z); ξ) ∩ Ω(y; ξ) ⊆ Ω(x ∗ z; ξ). By ˜ is a falling ideal of X. Theorem 3.5, H □ ˜ of a pre-logic X, the following are equivalent: Lemma 3.10. For any falling ideal H (i) (∀x, y ∈ X) (Ω(x ∗ y; ξ) ∩ Ω(x; ξ) ⊆ Ω(y; ξ)) . (ii) (∀x, y, z ∈ X) (Ω(x ∗ (y ∗ z); ξ) ∩ Ω(x ∗ y; ξ) ⊆ Ω(x ∗ z; ξ)) . ˜ satisfies (i). For any x, y, z ∈ X, using (P3), we have Ω(x ∗ (y ∗ z); ξ) ∩ Proof. Assume that H Ω(x ∗ y; ξ) = Ω((x ∗ y) ∗ (x ∗ z); ξ) ∩ Ω(x ∗ y; ξ)) ⊆ Ω(x ∗ z; ξ). Thus (ii) is valid. ˜ satisfies (ii). Putting x := 1 in (ii) and using (P2), we have Conversely, suppose that H Ω(y ∗ z; ξ) ∩ Ω(y; ξ) = Ω(1 ∗ (y ∗ z)); ξ) ∩ Ω(1 ∗ y; ξ)) ⊆ Ω(1 ∗ z; ξ) = Ω(z; ξ). Thus (i) is true. □ ˜ is a falling ideal of X if and only if the Proposition 3.11. Let X be a pre-logic. Then H following conditions are valid: (i) (∀x ∈ X) (Ω(x; ξ) ⊆ Ω(1; ξ)). (ii) (∀x, y, z ∈ X) (Ω(x ∗ (y ∗ z); ξ) ∩ Ω(x ∗ y; ξ) ⊆ Ω(x ∗ z; ξ)) . □

Proof. It follows from Corollary 3.9 and Lemma 3.10. ˜ of a pre-logic X satisfies the following property: Corollary 3.12. Every falling ideal H (∀x, y ∈ X)(Ω(x ∗ (x ∗ y); ξ) ⊆ Ω(x ∗ y; ξ)).

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Ideal theory of pre-logics based on the theory of falling shadows

Proof. Putting x := x, z := y and y := x in Proposition 3.11(ii), we have Ω(x ∗ (x ∗ y); ξ) = Ω(x ∗ (x ∗ y); ξ) ∩ Ω(1; ξ) = Ω(x ∗ (x ∗ y); ξ) ∩ Ω(x ∗ x; ξ) ⊆ Ω(x ∗ y; ξ), for all x, y ∈ X. □ 4. Positive implicative falling ideals Definition 4.1. Let (Ω, A , P ) be a probability space, and let ξ : Ω → P(X) be a random set, where X is a pre-logic. If ξ(ω) is a positive implicative ideal of X for any ω ∈ Ω with ξ(ω) ̸= ∅, ˜ of the random set ξ, i.e., H(x) ˜ then the falling shadow H = P (ω | x ∈ ξ(ω)) is called a positive implicative falling ideal of X. Example 4.2. Let X = {1, a, b, c, d} be a pre-logic as in Example 3.2. ˜ of ξ is a positive (1) Consider a random set ξ as in Example 3.2. Then the falling shadow H implicative falling ideal of X, since {1, a, b} is a positive implicative ideal of X. (2) Define a random set η : Ω → P(X) as follows:   if ω ∈ [0, 0.3),  ∅ η(ω) := {1, b} if ω ∈ [0.3, 0.7),   X if ω ∈ [0.7, 1]. Note that J := {1, b} is an ideal of X but not a positive implicative ideal of X since b∗((a∗d)∗a) = ˜ is a falling ideal of X, but not a positive b ∗ (d ∗ a) = b ∗ 1 = 1 ∈ J and b ∈ J but a ∈ / J. Hence H implicative falling ideal of X. Proposition 4.3. Every positive implicative falling ideal of a pre-logic X is a falling ideal of X. □

Proof. Straightforward by Definition 4.1 and Theorem 2.10. The converse of Proposition 4.3 is not true in general (see Example 4.2(2)).

˜ is a positive implicative falling ideal of X if and Theorem 4.4. Let X be a pre-logic. Then H ˜ satisfies the following two conditions: only if H (i) (∀x ∈ X)(Ω(x; ξ) ⊆ Ω(1; ξ)), (ii) (∀x, y, z ∈ X)(Ω(x ∗ ((y ∗ z) ∗ y); ξ) ∩ Ω(x; ξ) ⊆ Ω(y; ξ)). ˜ satisfies two conditions (i) and (ii). Let x ∈ ξ(ω) for every x ∈ X and for Proof. Assume that H all ω ∈ Ω. Then ω ∈ Ω(x; ξ) ⊆ Ω(1; ξ) and so 1 ∈ ξ(ω). Let x, y, z ∈ X be such that x ∈ ξ(ω) and x ∗ ((y ∗ z) ∗ y) ∈ ξ(ω). Then ω ∈ Ω(x; ξ) and ω ∈ Ω(x ∗ ((y ∗ z) ∗ y); ξ). Using (ii), we have ˜ is a ω ∈ Ω(y; ξ). Hence y ∈ ξ(ω) and so ξ(ω) is a positive implicative ideal of X. Therefore H positive implicative falling ideal of X. ˜ is a positive implicative falling ideal of X. The first condition (i) Conversely, suppose that H follows from Proposition 3.7(i) and Proposition 4.3. For any x, y, z ∈ X, if ω ∈ Ω(x ∗ ((y ∗ z) ∗ y); ξ) ∩ Ω(x; ξ), then x ∗ ((y ∗ z) ∗ y) ∈ ξ(ω) and x ∈ ξ(ω). Since ξ(ω) is a positive implicative ideal

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of X, we have ω ∈ Ω(y; ξ). Therefore Ω(x ∗ ((y ∗ z) ∗ y); ξ) ∩ Ω(x; ξ) ⊆ Ω(y; ξ) for any x, y, z ∈ X. Thus (ii) holds. □ ˜ be a falling ideal of a pre-logic X. Then the following are equivalent: Theorem 4.5. Let H ˜ is a positive implicative falling ideal of X. (i) H (ii) (∀x, y ∈ X)(Ω((x ∗ y) ∗ x; ξ) ⊆ Ω(x; ξ)). ˜ is a positive implicative ideal of X. Putting x := 1, y := x, and z := y in Proof. Assume that H Theorem 4.4(ii), we have Ω(1∗((x∗y)∗x); ξ)∩Ω(1; ξ) = Ω((x∗y)∗x; ξ)∩Ω(1; ξ) = Ω((x∗y)∗x; ξ) ⊆ Ω(x; ξ). Hence (ii) holds. ˜ satisfies (ii). By Lemma 3.8, for any x, y, z ∈ X, we Conversely, suppose that a falling ideal H ˜ is a positive have Ω(x ∗ ((y ∗ z) ∗ y); ξ) ∩ Ω(x; ξ) ⊆ Ω((y ∗ z) ∗ y; ξ) ⊆ Ω(y; ξ). By Theorem 4.4, H implicative falling ideal of X. Thus (i) is true. □ Corollary 4.6. Any positive implicative falling ideal of a pre-logic X satisfies the following property: (∀x, y ∈ X)(Ω((x ∗ y) ∗ y; ξ) ⊆ Ω((y ∗ x) ∗ x; ξ)). Proof. Since x ≤ (y ∗ x) ∗ x for all x, y ∈ X, it follows from Lemma 2.4 that ((y ∗ x) ∗ x) ∗ y ≤ x ∗ y. Then (x ∗ y) ∗ y ≤ (y ∗ x) ∗ ((x ∗ y) ∗ x) = (x ∗ y) ∗ ((y ∗ x) ∗ x) ≤ (((y ∗ x) ∗ x) ∗ y) ∗ ((y ∗ x) ∗ x). By Proposition 3.7(iii) and Proposition 4.5, we have Ω((x ∗ y) ∗ y; ξ) ⊆ Ω((((y ∗ x) ∗ x) ∗ y) ∗ ((y ∗ x) ∗ x); ξ) ⊆ Ω((y ∗ x) ∗ x; ξ), for any x, y ∈ X. This completes the proof. □ References [1] S. S. Ahn and Y. B. Jun, Pseudo-valuations on pre-logics, Acta Math. Hungar., 134(2012), 499-510. [2] I. Chajada and R. Halas, Algebraic properties of pre-logics, Math. Slovaca, 52(2002), 157-751. [3] I. R. Goodman, Fuzzy sets as equivalence classes of random sets, in: R. Yager(Ed.), Recent Ddevelopments in Fuzzy Sets and Possibility Theory, Pergamon, New York, 1982. [4] Y. B. Jun and M. S. Kang, Fuzzy positive implicative ideals of BCK-algebras based on the theory of falling shadows, Comput. Math. Appl., 61 (2011), 62–67. [5] Y. B. Jun and C. H. Park, Falling shadows applied to subalgebras and ideals of BCK/BCI-algebras, Chaos Solutions Fractals, (submitted for publication). [6] Y. H. Kim and S. S. Ahn, Ideal theory of pre-logics based on N -structures, Honam Math. J., 33 (2011), 535-546. [7] P. Z. Wang and E. Sanchez, Treating a fuzzy subset as a projectable random set, in: M. M. Gupta, E. Sanchez, Eds., “Fuzzy Information and Decision” (Pergamon, New York, 1982), 212–219. [8] P. Z. Wang, Fuzzy Sets and Falling Shadows of Random Sets, Beijing Normal Univ. Press, People’s Republic of China, 1985 [In Chinese].

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Young Bae Jun et al 744-751

J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 24, NO.4, 2018, COPYRIGHT 2018 EUDOXUS PRESS, LLC

QUADRATIC ρ-FUNCTIONAL EQUATIONS IN NON-ARCHIMEDEAN BANACH SPACES CHOONKIL PARK, GANG LU, YINHUA CUI, AND MING FANG∗ Abstract. In this paper, we solve the quadratic ρ-functional equations     x+y f (x + y) + f (x − y) − 2f (x) − 2f (y) = ρ 4f + f (x − y) − 2f (x) − 2f (y) ,(0.1) 2 where ρ is a fixed non-Archimedean number with |ρ| < |2|, and   x+y 4f + f (x − y) − 2f (x) − 2f (y) = ρ(f (x + y) + f (x − y) − 2f (x) − 2f (y)),(0.2) 2 where ρ is a fixed non-Archimedean number with |ρ| < |2|. Furthermore, we prove the Hyers-Ulam stability of the quadratic ρ-functional equations (0.1) and (0.2) in non-Archimedean Banach spaces.

1. Introduction and preliminaries A valuation is a function | · | from a field K into [0, ∞) such that 0 is the unique element having the 0 valuation, |rs| = |r| · |s| and the triangle inequality holds, i.e., |r + s| ≤ |r| + |s|,

∀r, s ∈ K.

A field K is called a valued field if K carries a valuation. The usual absolute values of R and C are examples of valuations. Let us consider a valuation which satisfies a stronger condition than the triangle inequality. If the triangle inequality is replaced by |r + s| ≤ max{|r|, |s|},

∀r, s ∈ K,

then the function | · | is called a non-Archimedean valuation, and the field is called a non-Archimedean field. Clearly |1| = | − 1| = 1 and |n| ≤ 1 for all n ∈ N. A trivial example of a non-Archimedean valuation is the function | · | taking everything except for 0 into 1 and |0| = 0. Throughout this paper, we assume that the base field is a non-Archimedean field, hence call it simply a field. Definition 1.1. ([8]) Let X be a vector space over a field K with a non-Archimedean valuation | · |. A function k · k : X → [0, ∞) is said to be a non-Archimedean norm if it satisfies the following conditions: 2010 Mathematics Subject Classification. Primary 46S10, 39B62, 39B52, 47S10, 12J25. Key words and phrases. Hyers-Ulam stability; non-Archimedean normed space; quadratic ρfunctional equation. ∗ Corresponding author.

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J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 24, NO.4, 2018, COPYRIGHT 2018 EUDOXUS PRESS, LLC

C. PARK, G. LU, Y. CUI, AND M. FANG

(i) kxk = 0 if and only if x = 0; (ii) krxk = |r|kxk (r ∈ K, x ∈ X); (iii) the strong triangle inequality kx + yk ≤ max{kxk, kyk},

∀x, y ∈ X

holds. Then (X, k · k) is called a non-Archimedean normed space. Definition 1.2. (i) Let {xn } be a sequence in a non-Archimedean normed space X. Then the sequence {xn } is called Cauchy if for a given ε > 0 there is a positive integer N such that kxn − xm k ≤ ε for all n, m ≥ N . (ii) Let {xn } be a sequence in a non-Archimedean normed space X. Then the sequence {xn } is called convergent if for a given ε > 0 there are a positive integer N and an x ∈ X such that kxn − xk ≤ ε for all n ≥ N . Then we call x ∈ X a limit of the sequence {xn }, and denote by limn→∞ xn = x. (iii) If every Cauchy sequence in X converges, then the non-Archimedean normed space X is called a non-Archimedean Banach space. The stability problem of functional equations originated from a question of Ulam [18] concerning the stability of group homomorphisms. The functional equation f (x + y) = f (x) + f (y) is called the Cauchy equation. In particular, every solution of the Cauchy equation is said to be an additive mapping. Hyers [7] gave a first affirmative partial answer to the question of Ulam for Banach spaces. Hyers’ Theorem was generalized by Aoki [2] for additive mappings and by Rassias [11] for linear mappings by considering an unbounded Cauchy difference. A generalization of the Rassias theorem was obtained by G˘avruta [6] by replacing the unbounded Cauchy difference by a general control function in the spirit of Rassias’ approach. The functional equation f (x + y) + f (x − y) = 2f (x) + 2f (y) is called the quadratic functional equation. In particular, every solution of the quadratic functional equation is said to be a quadratic mapping. The stability of quadratic functional equation was proved by Skof [17] for mappings f : E1 → E2 , where E1 is a normed space and E2 is a Banach space. Cholewa [5] noticed that the theorem of Skof is still true if the relevant domain E1 is replaced by an Abelian group. The functional equation  x+y 4f 2 + (x − y) = f (x) + f (y) is called a Jensen type quadratic equation. The stability problems of various functional equations have been extensively investigated by a number of authors (see [1, 3, 4, 9, 10, 12, 13, 14, 15, 16, 19, 20]). In Section 2, we solve the quadratic ρ-functional equation (0.1) and prove the HyersUlam stability of the quadratic ρ-functional equation (0.1) in non-Archimedean Banach spaces. In Section 3, we solve the quadratic ρ-functional equation (0.2) and prove the HyersUlam stability of the quadratic ρ-functional equation (0.2) in non-Archimedean Banach spaces.

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CHOONKIL PARK et al 752-759

J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 24, NO.4, 2018, COPYRIGHT 2018 EUDOXUS PRESS, LLC

QUADRATIC ρ-FUNCTIONAL EQUATIONS

Throughout this paper, assume that X is a non-Archimedean normed space and that Y is a non-Archimedean Banach space. Let |2| 6= 1. 2. Quadratic ρ-functional equation (0.1) in non-Archimedean normed spaces Throughout this section, assume that ρ is a fixed non-Archimedean number with |ρ| < |2|. In this section, we solve the quadratic ρ-functional equation (0.1) in non-Archimedean normed spaces. Lemma 2.1. If a mapping f : G → Y satisfies f (0) = 0 and f (x + y) + f (x − y) − 2f (x) − 2f (y)     x+y + f (x − y) − 2f (x) − 2f (y) = ρ 4f 2 for all x, y ∈ G, then f : G → Y is quadratic. Proof. Assume that f : G → Y satisfies (2.1). Letting y = x in (2.1), we get f (2x) − 4f (x) = 0 for all x ∈ G. Thus   x 1 f = f (x) 2 4 for all x ∈ G. It follows from (2.1) and (2.2) that

(2.1)

(2.2)

f (x + y) + f (x − y) − 2f (x) − 2f (y)     x+y = ρ 4f + f (x − y) − 2f (x) − 2f (y) 2 = ρ(f (x + y) + f (x − y) − 2f (x) − 2f (y)) and so f (x + y) + f (x − y) = 2f (x) + 2f (y) for all x, y ∈ G.



Now, we prove the Hyers-Ulam stability of the quadratic ρ-functional equation (2.1) in non-Archimedean Banach spaces. Theorem 2.2. Let r < 2 and θ be nonnegative real numbers and let f : X → Y be a mapping satisfying

   

x+y

f (x + y) + f (x − y) − 2f (x) − 2f (y) − ρ 4f + f (x − y) − 2f (x) − 2f (y)

2 ≤ θ(kxkr + kykr ) (2.3) for all x, y ∈ X. Then there exists a unique quadratic mapping Q : X → Y such that 2 kf (x) − Q(x)k ≤ r θkxkr (2.4) |2| for all x ∈ X.

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J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 24, NO.4, 2018, COPYRIGHT 2018 EUDOXUS PRESS, LLC

C. PARK, G. LU, Y. CUI, AND M. FANG

Proof. Letting x = y = 0 in (2.3), we get −2f (0) = ρf (0). Since |ρ| < |2|, f (0) = 0. Letting y = x in (2.3), we get kf (2x) − 4f (x)k ≤ 2θkxkr

(2.5)

 

for all x ∈ X. So f (x) − 4f x2 ≤ |2|2r θkxkr for all x ∈ X. Hence

   

l x x

m

4 f (2.6) − 4 f

m 2l 2

         

x x x x ≤ max

4l f l − 4l+1 f l+1

, · · · ,

4m−1 f m−1 − 4m f m

2 2  2 2  

 

    

x x x

x

l m−1 = max |4| f l − 4f l+1 , · · · , |4| − 4f m

f 2 2 2m−1 2 ( )

≤ max

|4|l |4|m−1 , · · · , |2|rl |2|r(m−1)

2 2θ 1 r θkxk = kxkr |2|r |2|(r−2)l |2|r

for all nonnegative integers m and l with m > l and all x ∈ X. It follows from (2.6) that the sequence {4n f ( 2xn )} is a Cauchy sequence for all x ∈ X. Since Y is complete, the sequence {4n f ( 2xn )} converges. So one can define the mapping Q : X → Y by x Q(x) := lim 4n f ( n ) n→∞ 2 for all x ∈ X. Moreover, letting l = 0 and passing the limit m → ∞ in (2.6), we get (2.4). It follows from (2.3) that

   

x+y

Q(x + y) + Q(x − y) − 2Q(x) − 2Q(y) − ρ 4Q + Q (x − y) − 2Q(x) − 2Q(y)

2 

     

x + y x − y x y = lim |4|n

f +f − 2f n − 2f n n n n→∞ 2 2 2 2         

x+y x−y x y |4|n θ

−ρ 4f +f − 2f n − 2f n ≤ lim (kxkr + kykr ) = 0 n+1 n nr n→∞ 2 2 2 2 |2| for all x, y ∈ X. So x+y + Q (x − y) − 2Q(x) − 2Q(y) Q(x + y) + Q(x − y) − 2Q(x) − 2Q(y) = ρ 4Q 2 for all x, y ∈ X. By Lemma 2.1, the mapping h : X → Y is quadratic. Now, let T : X → Y be another quadratic mapping satisfying (2.4). Then we have

   

q x x

q

kQ(x) − T (x)k = 4 Q q − 4 T 2 2q          

x x

x x 2 q ≤ max

4q Q q − 4q f q

,

4q T − 4 f θkxkr , ≤

2 2 2q 2q |2|(r−2)q+r 







which tends to zero as q → ∞ for all x ∈ X. So we can conclude that Q(x) = T (x) for all x ∈ X. This proves the uniqueness of Q. Thus the mapping Q : X → Y is a unique quadratic mapping satisfying (2.4). 

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CHOONKIL PARK et al 752-759

J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 24, NO.4, 2018, COPYRIGHT 2018 EUDOXUS PRESS, LLC

QUADRATIC ρ-FUNCTIONAL EQUATIONS

Theorem 2.3. Let r > 2 and θ be positive real numbers, and let f : X → Y be a mapping satisfying (2.3). Then there exists a unique quadratic mapping Q : X → Y such that 2θ kxkr kf (x) − Q(x)k ≤ |4| for all x ∈ X. Proof. It follows from (2.5) that



1 2θ

f (x) − f (2x) ≤ kxkr

|4|

4

for all x ∈ X. The rest of the proof is similar to the proof of Theorem 2.2.



3. Quadratic ρ-functional equation (0.2) Throughout this section, assume that ρ is a fixed non-Archimedean number with |ρ| < |2|. In this section, we solve the quadratic ρ-functional equation (0.2) in non-Archimedean normed spaces. Lemma 3.1. If a mapping f : G → Y satisfies f (0) = 0 and   x+y 4f + f (x − y) − 2f (x) − 2f (y) 2 = ρ(f (x + y) + f (x − y) − 2f (x) − 2f (y))

(3.1)

for all x, y ∈ G, then f : G → Y is quadratic. Proof. Assume that f : G → Y satisfies (3.1). Letting y = 0 in (3.1), we get   x 4f = f (x) 2 for all x ∈ G. It follows from (3.1) and (3.2) that

(3.2)

f (x + y) + f (x − y) − 2f (x) − 2f (y)   x+y + f (x − y) − 2f (x) − 2f (y) = 4f 2 = ρ(f (x + y) + f (x − y) − 2f (x) − 2f (y)) and so f (x + y) + f (x − y) = 2f (x) + 2f (y) for all x, y ∈ G.



Now, we prove the Hyers-Ulam stability of the quadratic ρ-functional equation (3.1) in non-Archimedean Banach spaces.

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CHOONKIL PARK et al 752-759

J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 24, NO.4, 2018, COPYRIGHT 2018 EUDOXUS PRESS, LLC

C. PARK, G. LU, Y. CUI, AND M. FANG

Theorem 3.2. Let r < 2 and θ be nonnegative real numbers, and let f : X → Y be a mapping satisfying





x+y

4f + f (x − y) − 2f (x) − 2f (y) − ρ(f (x + y) + f (x − y) − 2f (x) − 2f (y))

2 ≤ θ(kxkr + kykr ) (3.3) for all x, y ∈ X. Then there exists a unique quadratic mapping Q : X → Y such that kf (x) − Q(x)k ≤ θkxkr

(3.4)

for all x ∈ X. Proof. Letting x = y = 0 in (3.3), we get f (0) = 2ρf (0). Since |ρ| < |2|, f (0) = 0. Letting y = 0 in (3.3), we get

 

x

4f (3.5) − f (x)

≤ θkxkr

2 for all x ∈ X. So

   

l x x

m

4 f − 4 f (3.6)

m 2l 2

         

x x x x ≤ max

4l f l − 4l+1 f l+1

, · · · ,

4m−1 f m−1 − 4m f m

2 2  2 2  

 

    



x x x x = max |4|l

f l − 4f l+1

, · · · , |4|m−1

f m−1 − 4f m

2 2 2 2 ) ( |4|m−1 θ |4|l , · · · , r(m−1) θkxkr = (r−2)l kxkr ≤ max |2|rl |2| |2| for all nonnegative integers m and l with m > l and all x ∈ X. It follows from (3.6) that the sequence {4n f ( 2xn )} is a Cauchy sequence for all x ∈ X. Since Y is complete, the sequence {4n f ( 2xn )} converges. So one can define the mapping Q : X → Y by x Q(x) := lim 4n f ( n ) n→∞ 2 for all x ∈ X. Moreover, letting l = 0 and passing the limit m → ∞ in (3.6), we get (3.4). The rest of the proof is similar to the proof of Theorem 2.2.  Theorem 3.3. Let r > 2 and θ be positive real numbers, and let f : X → Y be an even mapping satisfying (3.3). Then there exists a unique quadratic mapping Q : X → Y such that |2|r θ kf (x) − Q(x)k ≤ kxkr (3.7) |4| for all x ∈ X. Proof. It follows from (3.5) that



1 |2|r θ

f (x) − f (2x) ≤ kxkr

|4|

4

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J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 24, NO.4, 2018, COPYRIGHT 2018 EUDOXUS PRESS, LLC

QUADRATIC ρ-FUNCTIONAL EQUATIONS

for all x ∈ X. Hence

1

1 m

f (2l x) − (3.8) f (2 x)

l

4 4m

   

1  l 

1

1  l+1 

1 m−1 m

≤ max l f 2 x − l+1 f 2 x , · · · , m−1 f 2 x − m f (2 x) 4 4 4 4 (





)    



1 1 1 1 l l+1

,··· ,

f 2m−1 x − f (2m x) f 2 x − f 2 x = max



|4|l 4 |4|m−1 4 |2|rl |2|r θ |2|r(m−1) r r ≤ max |2| θkxk = , · · · , kxkr l+1 (m−1)+1 (2−r)l+2 |4| |4| |2| (

)

for all nonnegative integers m and l with m > l and all x ∈ X. It follows from (3.8) that the sequence { 41n f (2n x)} is a Cauchy sequence for all x ∈ X. Since Y is complete, the sequence { 41n f (2n x)} converges. So one can define the mapping Q : X → Y by 1 f (2n x) 4n for all x ∈ X. Moreover, letting l = 0 and passing the limit m → ∞ in (3.8), we get (3.7). The rest of the proof is similar to the proofs of Theorems 2.2 and 3.2.  Q(x) := lim

n→∞

References [1] M. Adam, On the stability of some quadratic functional equation, J. Nonlinear Sci. Appl. 4 (2011), 50–59. [2] T. Aoki, On the stability of the linear transformation in Banach spaces, J. Math. Soc. Japan 2 (1950), 64–66. [3] L. C˘adariu, L. G˘ avruta and P. G˘ avruta, On the stability of an affine functional equation, J. Nonlinear Sci. Appl. 6 (2013), 60–67. [4] A. Chahbi and N. Bounader, On the generalized stability of d’Alembert functional equation, J. Nonlinear Sci. Appl. 6 (2013), 198–204. [5] P. W. Cholewa, Remarks on the stability of functional equations, Aequationes Math. 27 (1984), 76–86. [6] P. Gˇavruta, A generalization of the Hyers-Ulam-Rassias stability of approximately additive mappings, J. Math. Anal. Appl. 184 (1994), 431–43. [7] D. H. Hyers, On the stability of the linear functional equation, Proc. Natl. Acad. Sci. U.S.A. 27 (1941), 222–224. [8] M.S. Moslehian and Gh. Sadeghi, A Mazur-Ulam theorem in non-Archimedean normed spaces, Nonlinear Anal.–TMA 69 (2008), 3405–3408. [9] C. Park, K. Ghasemi, S. G. Ghaleh and S. Jang, Approximate n-Jordan ∗-homomorphisms in C ∗ -algebras, J. Comput. Anal. Appl. 15 (2013), 365-368. [10] C. Park, A. Najati and S. Jang, Fixed points and fuzzy stability of an additive-quadratic functional equation, J. Comput. Anal. Appl. 15 (2013), 452–462. [11] Th. M. Rassias, On the stability of the linear mapping in Banach spaces, Proc. Amer. Math. Soc. 72 (1978), 297–300. [12] S. Schin, D. Ki, J. Chang and M. Kim, Random stability of quadratic functional equations: a fixed point approach, J. Nonlinear Sci. Appl. 4 (2011), 37–49. [13] S. Shagholi, M. Bavand Savadkouhi and M. Eshaghi Gordji, Nearly ternary cubic homomorphism in ternary Fr´echet algebras, J. Comput. Anal. Appl. 13 (2011), 1106–1114.

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[14] S. Shagholi, M. Eshaghi Gordji and M. Bavand Savadkouhi, Stability of ternary quadratic derivation on ternary Banach algebras, J. Comput. Anal. Appl. 13 (2011), 1097–1105. [15] D. Shin, C. Park and Sh. Farhadabadi, On the superstability of ternary Jordan C ∗ homomorphisms, J. Comput. Anal. Appl. 16 (2014), 964–973. [16] D. Shin, C. Park and Sh. Farhadabadi, Stability and superstability of J ∗ -homomorphisms and J ∗ -derivations for a generalized Cauchy-Jensen equation, J. Comput. Anal. Appl. 17 (2014), 125–134. [17] F. Skof, Propriet locali e approssimazione di operatori, Rend. Sem. Mat. Fis. Milano 53 (1983), 113–129. [18] S. M. Ulam, A Collection of the Mathematical Problems, Interscience Publ. New York, 1960. [19] C. Zaharia, On the probabilistic stability of the monomial functional equation, J. Nonlinear Sci. Appl. 6 (2013), 51–59. [20] S. Zolfaghari, Approximation of mixed type functional equations in p-Banach spaces, J. Nonlinear Sci. Appl. 3 (2010), 110–122. 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 110870, P.R. China; Department of Mathematics, Zhejiang University, Hangzhou 310027, P.R. China E-mail address: [email protected] Yinhua Cui Department of Mathematics, Yanbian University, Yanji 133001, P.R. China E-mail address: [email protected] Ming Fang Department of Mathematics, Yanbian University, Yanji 133001, P.R. China E-mail address: [email protected]

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The Incomplete Global GMERR Algorithm to Solve AX = B ∗ Yu-Hui Zhenga†, Jian-Lei Lib , Dong-Xu Chenga , Ling-Ling Lvb a

College of Science, Zhongyuan University of Technology, Zhengzhou, Henan, 450007, PR China.

b

College of Mathematics and Information Science, North China University of,

Water Resources and Electric Power, Zhengzhou, Henan, 450011, PR China.

Abstract To reduce the computational cost and storage requirement of global generalized minimal error(GLGMERR) method, in this paper, we propose a truncated version of GLGMERR method, which is termed as incomplete global generalized minimal error method. The proposed approach uses only a few rather than all of the prior computed matrices in recurrences to generate the next matrix. Moreover a quasi-minimum error solution is obtained as well. Finally, we present the numerical results by comparing with the traditional global GMERR method in CPU time and storage requirements to show the effectiveness and advantages of our method. Key words:matrix equation; incomplete global generalized minimal error. AMSC(2000): 65F10, 49M15, 65H10, 15A24

1

Introduction

Consider the following problem: Ax(i) = b(i) , i = 1, 2, · · · s,

(1.1)

∗

This research was supported by National Natural Science Foundation of China(11501200,11601152,11501525) and National Natural Science Tianyuan Foundation of China(11626238), the Natural Science Foundation of Henan Province(16A110018,17A110037,15A110050), the Key Scientific Research Project of colleges and universities in Henan Province(No.15A110045), Growth Funds for Scientific Research team of NCWU(320009-00200), Youth Science and Technology Innovation talents of NCWU(70491), Doctoral Research Project of NCWU(201119). † Corresponding author. E-mail:[email protected].

1

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J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 24, NO.4, 2018, COPYRIGHT 2018 EUDOXUS PRESS, LLC

Yu-Hui Zheng etal: The incomplete global GMERR algorithm to Solve AX = B where A is a n×n unsymmetric matrix, x(i) , b(i) are all n×1 real vectors, s ≤ n. In our daily life, sometimes we have to solve this problem. Therefore, it is of importance that researchers are interested in the study of the numerical solutions, algorithms design and software development for solving problem (1.1). Krylov subspace method, as one of the effective method for solving Ax = b, can be used to solve s linear systems one by one. However, when the order of A is large, it is not enough to use this method to solve this problem. Therefore, we have to find the other new method to solve it. It should be noticed that when all b(i) do impact on the whole system, the problem (1.1) can be rewritten as AX = B,

(1.2)

where X = [x(1) , x(2) , · · · x(s) ]T , B = [b(1) , b(2) , · · · b(s) ]T . In the past decades, some related works have been achieved to solve the problem (1.2). In 1999, Jbilou [1] et al proposed the global Arnoldi method and moreover, they proposed global FOM and GMRES methods based on global Arnoldi method, which extended the Krylov subspace method. Among all Krylov subspace methods, GMERR method is one of the most effective methods, because it can minimize the error norm of this method on Krylov subspace. The literature [2] presented the global GMRES method for solving unsymmetric linear systems, which maps the initial residual matrix to the Krylov subspace. In some sense, global GMERR and global GMERS [1] methods have similar structures. The global generalized minimal error(GMERR) algorithm is an effective Krylov subspace method to solve the linear equations with multiple right-hand sides. As the global GMERR method and the GMERR method have the long recurrence, which result in the dramatic increase of the calculation and storage along with the increase of the step numbers. At present, there are many truncation strategies. For example, Young [3] presented the truncated forms of the orthogonal direction method and the orthogonal residual method. In [4, 5], a truncated forms of FOM method has been given. The truncated forms of IGMRES method or QGMRES method are presented in [6–10]. In this paper, we use the truncation strategy to improve the global GMERR algorithm, and propose a incomplete global GMERR algorithm, which use a few of the previously generated matrix to construct the new basis matrix, and we also give the quasi global minimum error solution on the Krylov subspace. The remainder of this paper is organized as follows. In Section 2, we present the incomplete global GMERR algorithm. Section 3 and 4 give some numerical experiments to test the effectiveness of the incomplete global GMERR algorithm and conclusions, respectively

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J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 24, NO.4, 2018, COPYRIGHT 2018 EUDOXUS PRESS, LLC

Yu-Hui Zheng etal: The incomplete global GMERR algorithm to Solve AX = B

2

Incomplete global GMERR algorithm

The incomplete global GMERR algorithm, which is based on the incomplete orthogonality of the Krylov subspace matrix, is to seek the quasi global minimum error solution. The basis matrices {Vi }(i = 1, 2, ..., m) of the Krylov subspace Km (AT , R0 ) can be obtained through the incomplete orthogonal process. AT Vi (i = 1, 2, ..., m) is carried out in the orthogonal process with the first q(q < m) matrices Vi0 , ...Vi (i0 = max{1, i−q+1}), The incomplete global GMERR algorithm is to seek the approximate solution Xm = X0 + Zm , Zm ∈ AT Km (AT , R0 ). Moreover Rm = B − AXm ⊥ Km (AT , R0 ), i.e., R0 − AZm ⊥ Km (AT , R0 ).

(2.1)

Note that Um = [V1 , V2 , ..., Vm ]. let Zm = AT Um ∗ ym , then we can have Xm = X0 + AT Um ∗ ym , Rm = R0 − AAT Um ∗ ym . Since R0 − AZm is orthogonal to Km (AT , R0 ) from equation (2.1). Therefore, for i = 1, 2, ..., m, we can obtain < Vi , R0 >=< Vi , AAT Um ∗ ym > .

(2.2)

Let V1 = R0 / ∥ R0 ∥F , then for the formula (2.2), when i = 1, it means tr(V1T R0 ) = tr(V1T AAT Um ∗ ym ), i.e., ∥ R0 ∥F = (tr(V1T AAT V1 ), tr(V1T AAT V2 ), ...tr(V1T AAT Vm ))ym ; when i = q + 2, ..., m, (tr(ViT AAT V1 ), tr(ViT AAT V2 ), ...tr(ViT AAT Vm ))ym = 0; when i = 2, ..., q + 1, tr(ViT R0 ) = (tr(ViT AAT V1 ), tr(ViT AAT V2 ), ...tr(ViT AAT Vm ))ym . Therefore, through the above discussion, we can obtain ym by solving the following linear system:   ∥ R0 ∥F     0   tr(V1T AAT V1 ) tr(V1T AAT V2 ) · · · tr(V1T AAT V2 ) ..   .   tr(V T AAT V1 ) tr(V T AAT V2 ) · · · tr(V T AAT V2 )   2 2 2     0    ym =  .. .. .. . .     . . . . T R ) tr(V  q+2 0  T T T T T T   tr(Vm AA V1 ) tr(Vm AA V2 ) · · · tr(Vm AA V2 ) ..   . tr(VmT R0 ) (2.3) To sum up, we obtain the following restarting incomplete GMERR Algorithm. Algorithm 1 (The restarting in complete GMERR Algorithm) Step 1. Choose the restarting step number m, let 2 ≤ q ≤ m, set the precision tol and initial estimation n × s moment X0 . Then calculate R0 = B − AX0 , Let V1 = R0 / ∥ R0 ∥F ;

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J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 24, NO.4, 2018, COPYRIGHT 2018 EUDOXUS PRESS, LLC

Yu-Hui Zheng etal: The incomplete global GMERR algorithm to Solve AX = B

Step 2. For i = 1, 2, ..., m , do the following incomplete orthogonal process 2.1 W = AT Vi , 2.2 For j = max(1, i − q + 1), ..., i, calculate hj,i = tr(VjT AT Vi ), W = W − hj,i Vj , 2.3 hi+1,i =∥ W ∥F , Vi+1 = W/hi+1,i ; Step 3. Solve the linear system (2.3) to get ym ; Step 4. Calculate Xm = X0 + AT Um ∗ ym ; Step 5. If ∥ Rm ∥F =∥ B − AX0 ∥F ≤ tol, stop; otherwise, let Xm = X0 , calculate R0 = B − AX0 , V1 = R0 / ∥ R0 ∥F , go to step 2. It is not difficult to find that the matrices and Hessenberg matrix Hm produced by the above incomplete orthogonal process satisfy the following theorem. Theorem 1 If the incomplete global GMERR algorithm doesnt interrupt before the mth step, i.e., hi+1,i ̸= 0, (i = 1, 2, · · · , m), then {Vi }(i = 1, 2, · · · , m), which are produced by the incomplete orthogonal process, constitute a basis of the Krylov subspace. In addition, we have AT Um = Um ∗ Hm + Sm+1 , tr(ViT Vj ) = 0(i ̸= j, | i − j |≤ q),

tr(ViT Vi ) = 1, (i, j = 1, 2, · · · , m),

where Sm+1 = hm+1,m [0n×s , 0n×s ...Vk+1 ]. By analyzing the above theorem, we can achieve Hm and Bm = (tr(ViT Vj ))m×m , (i, j = 1, 2, · · · , m) in detail 

tr(V1T AV1 ) · · · tr(V1T AVq )   tr(V T AV1 ) . . . 2   Hm =     0         Bm =       

1 0 .. . 0 ∗

 0 ... ...

    , T tr(Vm−q+1 AVm )   ..  .

tr(VmT AVm−1 ) tr(VmT AVm )  0 ··· 0  ∗   ... ...  1  .. .. ..  . . . 0  , ..  .. .. .. . . . .    ... ... ... 0    0 ··· 0 1

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Yu-Hui Zheng etal: The incomplete global GMERR algorithm to Solve AX = B

Now, let us state some results which are indispensable for our subsequent discussions. Lemma 1 For inner product < X, Y >, we have < X, Y >≤∥ X ∥F ∥ Y ∥F , where X, Y ∈ Zn×s , Zn×s represents the n × s matrix space over R Proof. Obviously, < X, Y >= tr(X T Y ) =

n ∑ s ∑

xij yij , By Cauchy-Schwarz in-

i=1 j=1

equality, we have n ∑ s ∑

xij yij ≤

i=1 j=1

n ∑

( ) 12 ( s ) 21  s ∑ ∑   x2ij · yij2

i=1

≤

j=1

( n s ∑∑

j=1

)

(

1 2

·

x2ij

i=1 j=1

n ∑ s ∑

) 12 yij2

i=1 j=1

=∥ X ∥F ∥ Y ∥F . Hence, the proof of the theorem is completed.  Lemma 2 If j + 2 ≤ i ≤ m + 1, then tr(ViT AT Vj ) =

min{i−q−1,j+1} ∑

hk,j tr(ViT Vk ).

k=j−q+1

Proof. By analyzing the algorithm and components of Hm , we have AT Vj =

m+1 ∑

hk,j Vk , where j = 1, 2, · · · , m. Multiplying ViT left to the two sides of the above

k=1

formula, we have ViT AT Vj =

min{i−q−1,j+1} ∑

hk,j ViT Vk .

k=1

Taking the trace, we can have ∑

i−q−1

tr(ViT AT Vj )

=

hk,j tr(ViT Vk )

+ hi,j +

k=1

∑

hk,j tr(ViT Vk )

k=i+q+1

i−q−1

=

m+1 ∑

hk,j tr(ViT Vk ) + hi,j +

k=1

j+1 ∑

hk,j tr(ViT Vk ) (k > 1, hk,j = 0)

k=i+q+1

∑

min{i−q−1,j+1}

=

tr(hk,j ViT Vk )

k=1

∑

min{i−q−1,j+1}

=

hk,j tr(ViT Vk )

k=j−q+1

If k ≤ 0, let tr(ViT Vk ), we have hk,j = 0.  Theorem 2 Suppose q ≥ 2, i ≤ m + 1 and i − j ≥ q + 1, if the incomplete global

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Yu-Hui Zheng etal: The incomplete global GMERR algorithm to Solve AX = B

GMERR algorithm doesnt interrupt before the step, we have | tr(VjT Vi ) |≤ ci ∥ AT − A ∥F /hi,i−1 ,

(2.4)

where ci+1 = max γi+1,j , 1≤j≤i−q



∑



min{i−q−1,j+1}

γi+1,j = 1 + ci 

(

| hk,j | /hi,i−1 +

k=j−q+1

i ∑

) ck | hk,i | /hk,k−1

k=j+q+1

and if k ≤ 0, then γi+1,j = 1. Proof. Let Um+1 = [V1 , V2 , . . . , Vm+1 ], Bm+1 = (tr(ViT Vj ))(m+1)×(m+1) . From Algorithm 1 and Lemma 2, when i + 1 ≤ m + 1 and i + 1 − j ≥ q + 1, we obtain that i ∑ T hi+1,i Vi+1 = A Vi − hk,i Vk , (2.5) k=i0

Left-multiplying VjT to both sides of equation (2.5) and taking trace, we can get ( ( )) i ∑ tr(VjT Vi+1 hi+1,i ) = tr VjT AT Vi − hk,i Vk k=i0

= tr

(

VjT

(

)

∑

min{i−q−1,j+1}

)

A − A Vi + T

hk,j tr(VkT Vi )

−

i ∑

hk,i tr(VjT Vk ),

k=i0

k=j−q+1

(2.6) where i0 = max{1, i − q + 1}. In the following part, the inductive method is used to prove our theorem. When i + 1 = q + 2 ≤ m + 1, j = 1, from equation (2.2) and Lemma 2, we can obtain ( ( ) ) hq+2,q+1 tr(V1T Vq+2 ) = tr V1T AT − A Vq+1 . Assume that the incomplete global GMERR algorithm doesnt interrupt, and then we can have ( ) ( ) tr(V1T Vq+2 ) = tr V1T AT − A Vq+1 /hq+2,q+1 ( ) = < V1T , AT − A Vq+1 > /hq+2,q+1 ≤∥ V1 ∥F ∥ AT − A ∥F ∥ Vq+1 ∥F /hq+2,q+1 =∥ AT − A ∥F /hq+2,q+1 , which shows that it satisfies formula (2.5), where cq+2 = 1.

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J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 24, NO.4, 2018, COPYRIGHT 2018 EUDOXUS PRESS, LLC

Yu-Hui Zheng etal: The incomplete global GMERR algorithm to Solve AX = B

Assume the first i columns in the upper right corner of matrix Bm+1 satisfy equation (2.5). In the following we process the (i+1)th column, where 1 ≤ j ≤ j−q, i.e. i−j ≥ q. For formula (2.6), we can separately get the following inequations min{i−q−1,j+1} min{i−q−1,j+1} ∑ ∑ T ≤ hk,j tr(VkT Vi ) h tr(V V ) k,j i k k=j−q+1 k=j−q+1 ∑

min{i−q−1,j+1}

≤ ci

|hk,j | ∥ AT − A ∥F /hi,i−1 ,

k=j−q+1

and

i i ∑ ∑ T hk,i tr(VjT Vk ) = hk,i tr(Vj Vk ) ≤ k=i0

k=i0

≤

i ∑

i ∑ hk,i tr(VjT Vk ) k=j+q+1

ck |hk,i | ∥ AT − A ∥F /hk,k−1 .

k=j+q+1

Then, we can conclude that (

) min{i−q−1,j+1} ∑ i ∑ ∥ Vj ∥F ∥ AT − A ∥F ∥ Vi ∥F + hk,j tr(VkT Vi ) + hk,i tr(VjT Vk ) k=j−q+1 k=i0

tr(VjT Vi+1 ) ≤ [

( 1+

≤

ci

min{i−q−1,j+1} ∑

) |hk,j | /hi,i−1 +

k=j−q+1

hi+1,i ( i ∑

)] ck |hk,i | /hk,k−1

∥ AT − A ∥F

k=j+q+1

hi+1,i

= γi+1,j ∥ AT − A ∥F /hi+1,i , where ci+1 = max γi+1,j . The proof the theorem is completed.  1≤j≤i−q ( ) Theorem 3 Any singular value σ(Bm ) of Bm = tr(VIT Vj ) m×m satisfies } { max 0, 1 − (m − q − 1)c ∥ AT − A ∥F ≤ σ(Bm ) ≤ 1 + (m − q − 1)c ∥ AT − A ∥F where c is a function generated by Hm . Proof. By Gerschgorin Circular disc Theorem and Theorem 1, for any singular value σ(Bm ) of Bm , there must exist i, satisfying ∑ ∑ |tr(ViT Vj )|. |σ(Bm ) − 1| ≤ |tr(ViT Vj )| = |i−j|≥q+1

j̸=i

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Yu-Hui Zheng etal: The incomplete global GMERR algorithm to Solve AX = B

Therefore, we can get that ∑ 1− |tr(ViT Vj )| ≤ σ(Bm ) ≤ 1 + |i−j|≥q+1

∑

|tr(ViT Vj )|.

|i−j|≥q+1

By analying the algorithm and the structure of Bm , we have ∑ |i−j|≥q+1

If i ≤ q + 1,

i−q−1 ∑

∑

m ∑

i−q−1

|tr(ViT Vj )|

=

|tr(ViT Vj )|

+

j=1

|tr(ViT Vj )|

j=i+q+1

|tr(ViT Vj )| = 0, then

j=1

∑

|tr(ViT Vj )| ≤ (m − i − q)

|i−j|≥q+1

max

i+q+1≤j≤m

|tr(ViT Vj )|

≤ (m − q − 1) max |tr(ViT Vj )| q+2≤j≤m

≤ (m − q − 1)c ∥ AT − A ∥F , where c =

max cj /hj,j−1 .

q+2≤j≤m

If i ≥ m − q,

m ∑

|tr(ViT Vj )| = 0, then

j=i+q+1

∑

|tr(ViT Vj )| ≤ (i − q − 1)

|i−j|≥q+1

max

1≤j≤i−q−1

≤ (m − q − 1)

|tr(ViT Vj )|

max

1≤j≤m−q−1

|tr(VjT Vi )|

≤ (m − q − 1)c ∥ AT − A ∥F , where c =

max ci /hi,i−1 .

m−q≤i≤m

If q + 2 ≤ i ≤ m − q − 1, based on Lemma 2, we have ∑ |tr(ViT Vj )| ≤ (i − q − 1) max |tr(ViT Vj )| + (m − i − q) 1≤j≤i−q−1

|i−j|≥q+1

≤ (i − q − 1) ∥ AT − A ∥F

max

q+2≤i≤m−q−1

+ (m − i − q) ∥ AT − A ∥F

max

max

i+q+1≤j≤m

|tr(ViT Vj )|

ci /hi,i−1

i+q+1≤j≤m

cj /hj,j−1

≤ (m − 2q − 1)c ∥ AT − A ∥F ≤ (m − q − 1)c ∥ AT − A ∥F , { } where c = max max ci /hi,i−1 , max cj /hj,j−1 . Based the above disscusq+2≤i≤m−q−1

i+q+1≤j≤m

sion, the proof the theorem is completed. 

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Yu-Hui Zheng etal: The incomplete global GMERR algorithm to Solve AX = B

Theorem 3 presents that the orthogonal degree of basis matrices is determined by the symmetry degree of the coefficient matrix. By the above theorems, we can obtain the conclusion on the algorithm convergence. If the algorithm is interrupted, which means h1+i,i = 0(1 ≤ i ≤ m) at ith step, the invariant subspace of AT could be generated, moreover the approximate solution Xi generated by incomplete global GMERR algorithm is the exact solution of AX = B. Meanwhile, the error Ri = 0. During the incomplete orthogonal process, the generated basis matrix may lose the orthogonality to some extent. From theorem 2 and theorem 3, we can find that the incomplete global GMERR algorithm can not control the orthogonal degree of the generated basis matrix when the coefficient matrix is far away from the symmetric property. And so, the algorithm may not converge.

3

Numerical experiments

In this section, we give numerical experiments to test the effectiveness of the incomplete global GMERR algorithm. Moreover we compare it with the global GMERR algorithm and find that our proposed method is more effective than the traditional method when they are set in the same accuracy. Example Consider the two-dimension Convection-Diffusion Equation which is defined on the domain Ω = [0, 1] × [0, 1]   − ∆u(x, y) + α ∂ u(x, y) = f (x, y), ∂x  u(x, y) = 0. In this paper, we use the central difference method with grid length h = 1/(l + 1) to discrete the above equation, and then we obtain the n = l2 order non-symmetric matrix A(α)   B(α) −I .. ..   . .  −I  A(α) =  , . . . .  . . −I  −I B(α)   4 a . .    b .. ..  α α , b = −1 − 2(l+1) , where B(α) =   is l order matrix, a = −1 + 2(l+1) .. ..   . . a b 4 I is l order identity matrix, parameter α control the A(α) and deviation of symmetry.

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Yu-Hui Zheng etal: The incomplete global GMERR algorithm to Solve AX = B

Table 1: The incomplete global GMERR(m) algorithm with n = 2500, α = 0.25, ∥ A(0.25)T − A(0.25) ∥F = 0.3431 m

40

50

q 2 5 10 20 30 40 2 5 10 20 30 40 50

CP U 43.8590 45.7030 50.7810 55.6400 57.4530 60.7350 34.4850 37.0470 39.9530 43.9370 47.7030 49.4220 50.1560

ratio 0.7221 0.7725 0.8361 0.9161 0.9459 1 0.6876 0.7386 0.7966 0.8760 0.9511 0.9854 1

IT 28 28 28 28 28 28 15 15 15 15 15 15 15

∥ R ∥F 9.2951e-7 9.2951e-7 9.2951e-7 9.2951e-7 9.2951e-7 9.2951e-7 7.3342e-7 7.3342e-7 7.3342e-7 7.3342e-7 7.3342e-7 7.3342e-7 7.3342e-7

Table 2: The incomplete global GMERR(m) algorithm with n = 2500, α = 2.5, ∥ A(2.5)T − A(2.5) ∥F = 3.4314 m

40

50

q 2 5 10 20 30 40 2 5 10 20 30 40 50

CP U 44.6410 47.2960 48.1100 58.7340 59.6250 60.0780 34.4690 33.4690 38.4060 39.2660 42.0780 44.0630 47.8440

ratio 0.7431 0.7872 0.8008 0.9778 0.9812 1 0.7204 0.6995 0.8027 0.8207 0.8795 0.9209 1

769

IT 24 24 24 24 24 24 13 13 13 13 13 13 13

∥ R ∥F 9.1938e-7 9.1938e-7 9.1938e-7 9.1938e-7 9.1938e-7 9.1938e-7 6.5037e-7 6.5037e-7 6.5037e-7 6.5037e-7 6.5037e-7 6.5037e-7 6.5037e-7

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Yu-Hui Zheng etal: The incomplete global GMERR algorithm to Solve AX = B

Table 3: The incomplete global GMERR(m) algorithm with n = 2500, α = 25, ∥ A(25)T − A(25) ∥F = 34.3137 m

40

50

q 2 5 10 20 30 40 2 5 10 20 30 40 50

CP U 393.7810 381.4370 420.1400 461.5940 489.7580 520.3590 431.6090 324.9540 352.1880 392.1410 427.0160 442.0160 486.5620

ratio 0.7567 0.7330 0.8074 0.8871 0.9412 1 0.8871 0.6679 0.7238 0.8059 0.8776 0.9084 1

IT 208 208 208 208 208 209 192 137 137 137 137 136 136

∥ R ∥F 8.2432e-7 9.6001e-7 8.4919e-7 9.3441e-7 9.6380e-7 8.2174e-7 7.7878e-7 8.8358e-7 8.4919e-7 9.1390e-7 9.0901e-7 9.7113e-7 9.1955e-7

Table 4: The incomplete global GMERR(m) algorithm with n = 2500, α = 2500, ∥ A(α)T − A(α) ∥F = 3.4314e + 3 m

40

50

q 2 5 10 20 30 40 2 5 10 20 30 40 50

CP U 153.0620 145.2810 162.4370 180.1560 191.8900 196.8130 196.8280 183.8900 205.8750 228.0320 256.4370 244.1870 255.7810

ratio 0.7777 0.7382 0.8253 0.9154 0.9750 1 0.7695 0.7189 0.8049 0.8915 1.0026 0.9547 1

770

IT 90 92 92 92 92 92 71 71 71 71 71 71 71

∥ R ∥F 9.6087e-7 9.8770e-7 9.8773e-7 9.8773e-7 9.8773e-7 9.8773e-7 9.5397e-7 9.3412e-7 9.5319e-7 9.5319e-7 9.5319e-7 9.5319e-7 9.5319e-7

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Yu-Hui Zheng etal: The incomplete global GMERR algorithm to Solve AX = B

The purpose of this section is to demonstrate that, with the same accuracy, the incomplete global GMERR algorithm is more effective than the global GMERR algorithm when we solve the large linear equations with multiple right-hand sides. Without loss of generality, we set s = 2, which means the two right-hand sides. Assuming B = rand(n, s), X0 = 0, l = 50, n = 2500, tol = 10−6 , we test the incomplete global GMERR algorithm for numerical analysis for m = 40, 50, Moreover, we find that the incomplete global GMERR algorithm degenerate to global GMERR algorithm when q = m, Table 1-4 show us the numerical results, where CPU is denoted as the algorithm running time (in seconds), IT represents the iterate times, Ratio means the running time ratio of the incomplete global GMERR algorithm to the GMERR algorithm with the same accuracy requirements. For α = 0.25, A(α) is approximatly symmetric. Thus, the loss of the orthogonality of the basis matrices is not serious. The CPU time of incomplete global GMERR algorithm is shorter than the global GMERR algorithm, which shows effective of our proposed method. For α = 2.5, we can see the incomplete global GMERR algorithm is more effective than the global GMERR algorithm from table 2 and table 3. For α = 2500, although A(α) is far away from the symmetric property and the loss of the orthogonality of the basis matrices is serious, we can find that the incomplete global GMERR algorithm is still effective than the global GMERR algorithm from table 4. The experimental results show that, with the same accuracy, the incomplete global GMERR algorithm is more effective than the global GMERR algorithm. With the same computational cost, operation time and storage of our method is less than these of traditional method.

4

Conclusion

The incomplete global GMERR algorithm can overcome the long recurrence of the global GMERR algorithm by truncation strategy, which can save the computation and storage requirements effectively. In this paper, we present the incomplete global GMERR algorithm theoretically. Finally, the experimental results show effectiveness of the incomplete global GMERR algorithm by comparing with the traditional global GMERR algorithm.

References [1] Jbilou. K, Messaoudi. A, Sadok. H. Global FOM and GMRES algorithms for matrix equations. Appl Numer Math., 1999, 31, pp. 49–63.

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Yu-Hui Zheng etal: The incomplete global GMERR algorithm to Solve AX = B

[2] Y. H. Zheng, D. X. Cheng, X. H. Qian. Global GMERR algorithm for linear systems with multiple right-hand sides. Journal of Northeast Normal University(Natural Science Edition), 2012, 44(3), pp. 41–45. [3] D. M. Young, K. C. Jea. Generalized conjugate-gradient acceleration of nonsymmetrizable iterative methods. Linear Algebra Appl., 1980, 34, pp. 159–194. [4] Y. Saad. Krylov subspace methods for solving large unsymmetric linear systems. Math Comput., 1981, 37, pp. 105–126. [5] Z. Jia. On IOM(q): the incomplete orthogonalization method for large unsymmetric linear system. Numer Linear Alg Appl., 1996, 3(6), pp. 491–512. [6] P. N. Brown, A. C. Hindmarsh. Reduced storage methods in stiff ODE systems. Appl Math Comput., 1989, 31, pp. 40–91. [7] P. N. Brown, Y. Saad. Hybrid Krylov subspace methods for nonlinear systems of equations. SIAM J Sci Stat Comput., 1990, 11, pp. 450–481. [8] P. N. Brown. A theoretical comparison of the Arnoldi and GMRES algorithms. SIAM J Sci Stat Comput., 1991, 12, pp. 58–78. [9] Y. Saad K. Wu. DQGMRES: a direct qusi-minimal residual algorithms based on incomplete orthogonalization. Numer Linear Alg Appl., 1996, 3, pp. 329–344. [10] Z. Jia. On IGMRES: an incomplete generalized minimal residual method for large unsymmetric linear systems. Science in China(Series A)., 1998, 41(12), pp. 1278–1288.

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DOUBLE DIFFERENCE SPACES OF ALMOST NULL AND ALMOST CONVERGENT SEQUENCES FOR ORLICZ FUNCTION KULDIP RAJ AND RENU ANAND

Abstract. The objective of this paper is to introduce and study some double difference spaces of almost null and almost convergent sequences defined by a MusielakOrlicz function. We prove that these spaces are Banach, Barreled and Bornological spaces. An attempt is also made to prove that these spaces are BDK spaces and prove some interrelationship between these spaces.

1. Introduction and Preliminaries The initial work on double sequences is found in Bromwich [4]. Later on, it was studied by Hardy [10], M´ oricz [15], M´ oricz and Rhoades [16], Tripathy ([27],[28]), Ba¸sarır and Sonalcan [2] and many others. Hardy [10] introduced the notion of regular convergence for double sequences. Quite recently, Zeltser [30] in her Ph.D thesis has essentially studied both the theory of topological double sequence spaces and the theory of summability of double sequences. Mursaleen and Edely [17] have recently introduced the statistical convergence and Cauchy convergence for double sequences and given the relation between statistical convergent and strongly Ces`aro summable double sequences. Next, Mursaleen [21] and Mursaleen and Edely [18] have defined the almost strong regularity of matrices for double sequences and applied these matrices to establish a core theorem and introduced the M -core for double sequences and determined those four dimensional matrices transforming every bounded double sequences x = (xkl ) into one whose core is a subset of the M -core of x. The set of all complex valued double sequences is a vector space with coordinatewise addition and scalar multiplication which is denoted by Ω. By the convergence of a double sequence we mean the convergence in the Pringsheim sense i.e. a double sequence x = (xkl ) has Pringsheim limit L (denoted by P − lim x = L) provided that given  > 0 there exists n0 ∈ N such that |xkl − L| <  whenever k, l > n0 . We shall write more briefly as P -convergent. The space of all convergent double sequences in Pringsheim’s sense is denoted by Cp . A double sequence x = (xkl ) of complex numbers is said to be bounded if ||x||∞ = sup |xkl | < ∞, where N = {0, 1, 2, ...}. The space of all bounded double sequences is k,l∈N

denoted by Mu , which is a Banach space with the norm ||.||∞ . It is well known that there are such sequences in the space Cp but not in the space Mu . Indeed, if we define the sequence x = (xkl ) by   k, k ∈ N l, l ∈ N xkl =  0, k, l ∈ N \ {0}, 2010 Mathematics Subject Classification. 46A45, 40C05. Key words and phrases. Orlicz function, Musielak-Orlicz function, sequence space, double sequence, P-convergent, almost convergence. 1

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KULDIP RAJ AND RENU ANAND

for all k, l ∈ N, then, it is trivial that x ∈ Cp\Mu , since P − lim xkl = 0 but ||x||∞ = ∞. k,l→∞

Therefore, we can consider the space Cbp of the double sequences that are both convergent in Pringsheim’s sense and bounded which we write Cbp = Cp ∩ Mu . A sequence in the space Cp is said to be regularly convergent if it is a single convergent sequence with respect to each index and denote the space of all such sequences by Cr . Also by Cbp0 and Cr0 , we denote the spaces of all double sequences converging to 0 contained in the sequence spaces Cbp and Cr , respectively. Moricz [15] proved that Cbp , Cbp0 , Cr and Cr0 are Banach spaces with the norm ||.||∞ . The concept of almost convergence for single sequences was introduced by Lorentz [13] and for double sequences by M´ oricz and Rhoades [16]. A double sequence x = (xkl ) of complex numbers is said to be almost convergent to a generalized limit α if P − lim sup q,r→∞ s,t>0

s+q X t+r X 1 (xkl − α) = 0. (q + 1)(r + 1)

(see [29])

k=s l=t

Here, α is called the f2 - limit of x. The space of all almost convergent double sequences is denoted by Cf . A P − convergent double sequence need not to be almost convergent. However, every bounded convergent double sequence is almost convergent and every almost convergent double sequence is also bounded. Definition 1.1. [6] A bounded double sequence x = (xkl ) of real numbers is said to σ− convergent to a limit L if P − lim τqrst (x) = L uniformly in s, t ∈ N, q,r

where q

τqrst (x) =

r

XX 1 xσk (s),σl (t) . (q + 1)(r + 1) k=0 l=0

In this case, we write σ2 − lim x = L. The set of all bounded σ− convergent double sequences is denoted by Vσ2 . Clearly, Cbp ⊂ Vσ2 . Definition 1.2. [26] A topological vector space λ over R or C is called locally convex if it is a Hausdorff space such that every neighbourhood of any x ∈ λ contains a convex neighbourhood of x. Definition 1.3. [30] A locally convex double sequence space λ is called a DK− space if all of the seminorms rkl : λ → R, x = (xkl ) → |xkl | for all k, l ∈ N are continuous. A DK− space with a Frechet topology is called an F DK− space. A normed F DK− space is called a BDK− space. Definition 1.4. [26] Let λ be a vector space over the field C and let A, B be subsets of λ. Then A absorbs B if there exists α0 ∈ C such that B ⊂ αA whenever |α| > |α0 |. A subset C of λ is circled if αC ⊂ C whenever |α| ≤ 1. Definition 1.5. [26] A locally convex space λ is bornological if every circled, convex subset A ⊂ λ that absorbs every bounded set in λ is a neighbourhood of 0 in λ. Definition 1.6. [5] Let λ be a locally convex space. Then a subset is called barrel if it is absolutely convex, absorbing and closed in λ. Moreover, λ is called a barreled space if each barrel is a neighbourhood of zero. Lemma 1.7. [26] Every Banach space and every F r´ echet space is a barreled space.

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DOUBLE DIFFERENCE SPACES OF ALMOST NULL AND ALMOST CONVERGENT SEQUENCES 3

Lemma 1.8. [26] Every F r´ echet space and hence every Banach space is a bornological. Lemma 1.9. [5] Let (X, p) be a seminormed space and q be a seminorm on X. Then the following are equivalent: (a) q is continuous. (b) q is continuous at zero. (c) There exists M > 0 such that q(x) ≤ M p(x) for all x ∈ X. Altay and Ba¸sar [1] introduced the space BS of bounded series as follows: m,n X o n xkl < ∞ . BS = x = (xkl ) ∈ Ω : ||x||BS = sup m,n∈N

k,l=0

The space is also a Banach space with the norm k.kBS . One can refer to Mursaleen and Mohiuddine [19] for relevant terminology and required details on the spaces of double sequences and related topics. The notion of difference sequence spaces was introduced by Kızmaz [11], who studied the difference sequence spaces l∞ (∆), c(∆) and c0 (∆). The notion was further generalized by Et and C ¸ olak [7] by introducing the spaces l∞ (∆m ), c(∆m ) and c0 (∆m ). Later the concept have been studied by Bekta¸s et al. [3] and Et et al. [8]. Another type of generalization of the difference sequence spaces is due to Tripathy and Esi [27] who studied the spaces l∞ (∆v ), c(∆v ) and c0 (∆v ) where m, v are non-negative integers. Now, for Z = c, c0 and l∞ , we have sequence spaces Z(∆m ) = {x = (xk ) ∈ Ω : (∆m xk ) ∈ Z}, where ∆m x = (∆m xk ) = (∆m−1 xk − ∆m−1 xk+1 ) and ∆0 xk = xk for all k ∈ N, which is equivalent to the following binomial representation   m X m ∆m xk = (−1)v xk+v . v v=0

Taking m = 1, we get the spaces studied by Et and C ¸ olak [7]. An Orlicz function M is a function, which is continuous, non-decreasing and convex with M (0) = 0, M (x) > 0 for x > 0 and M (x) −→ ∞ as x −→ ∞. Lindenstrauss and Tzafriri [12] used the idea of Orlicz function to define the following sequence space. Let w be the space of all real or complex sequences x = (xk ), then ∞ n  |x |  o X k `M = x ∈ w : M < ∞, for some ρ > 0 ρ k=1

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

It is shown in [12] that every Orlicz sequence space `M contains a subspace isomorphic to `p (p ≥ 1). The ∆2 −condition is equivalent to M (Lx) ≤ kLM (x) for all values of x ≥ 0 and for L > 1. For more details about sequence spaces (see [9], [20], [23], [24], [25]) and references therein . A sequence M = (Mk ) of Orlicz functions is called a Musielak-Orlicz function (see [14],[22]). A sequence N = (Nk ) is defined by Nk (v) = sup{|v|u − (Mk ) : u ≥ 0}, k = 1, 2, · · ·

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is called the complementary function of a Musielak-Orlicz function M. For a given Musielak-Orlicz function M, the Musielak-Orlicz sequence space tM and its subspace hM are defined as follows: n o tM = x ∈ w : IM (cx) < ∞ for some c > 0 , n o hM = x ∈ w : IM (cx) < ∞ for all c > 0 , where IM is a convex modular defined by ∞ X IM (x) = Mk (xk ), x = (xk ) ∈ tM . k=1

We consider tM equipped with the Luxemburg norm o n x ≤1 ||x|| = inf k > 0 : IM k or equipped with the Orlicz norm n1  o ||x||0 = inf 1 + IM (kx) : k > 0 . k Let M = (Mkl ) be Musielak-Orlicz function, p = (pkl ) be a bounded sequence of positive real numbers and u = (ukl ) be a double sequence of strictly positive real numbers. In the present paper we define the following classes of sequences: Cf (M, u, ∆m , p) = s+q X t+r n  u ∆m x − α  pkl X 1 kl kl x = (xkl ) ∈ Ω : P − lim sup Mkl = 0, q,r→∞ s,t>0 (q + 1)(r + 1) % k=s l=t o for some % > 0 and Cf0 (M, u, ∆m , p) = q X r  u ∆m x  p n X 1 kl k+s,l+t kl Mkl = x = (xkl ) ∈ Ω : P − lim sup q,r→∞ s,t>0 (q + 1)(r + 1) % k=0 l=0 o 0, for some % > 0 . Remark 1.10. Let us consider a few special cases of the above sequence spaces: (i) If we take M(x) = x, then the above classes of sequences reduces to following sequence spaces: Cf (u, ∆m , p) = s+q X t+r  n X 1 ukl ∆m xkl − α  pkl x = (xkl ) ∈ Ω : P − lim sup = 0, q,r→∞ s,t>0 (q + 1)(r + 1) % k=s l=t o for some % > 0 and Cf0 (u, ∆m , p) = q X r  n X 1 ukl ∆m xk+s,l+t  pkl x = (xkl ) ∈ Ω : P − lim sup = 0, q,r→∞ s,t>0 (q + 1)(r + 1) % k=0 l=0 o for some % > 0 . (ii) If we take p = (pkl ) = 1, then the above sequence space becomes

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DOUBLE DIFFERENCE SPACES OF ALMOST NULL AND ALMOST CONVERGENT SEQUENCES 5

Cf (M, u, ∆m ) = s+q X t+r n  u ∆m x − α  X 1 kl kl x = (xkl ) ∈ Ω : P − lim sup Mkl = 0, q,r→∞ s,t>0 (q + 1)(r + 1) % k=s l=t o for some % > 0 and Cf0 (M, u, ∆m ) = q X r n   u ∆m x X 1 kl k+s,l+t x = (xkl ) ∈ Ω : P − lim sup Mkl = 0, q,r→∞ s,t>0 (q + 1)(r + 1) % k=0 l=0 o for some % > 0 . (iii) If we take M(x) = x, p = (pkl ) = 1, u = (ukl ) = 1 m = 0 and % = 1, then we have s+q X t+r  n  X 1 Cf = x = (xkl ) ∈ Ω : P − lim sup xkl − α = 0 and q,r→∞ s,t>0 (q + 1)(r + 1) k=s l=t q X r  n o  X 1 Cf0 = x = (xkl ) ∈ Ω : P − lim sup xk+s,l+t = 0 . q,r→∞ s,t>0 (q + 1)(r + 1) k=0 l=0 which were introduced and studied by Ye¸silkayagiil and Ba¸sar [29]. The main purpose of the present paper is to show that the sequence spaces Cf (M, u, ∆m , p) and Cf0 (M, u, ∆m , p) are BDK-spaces, Barreled and Bornological. Furthermore, we also studied some inclusion relations between these spaces. 2. Main Results Theorem 2.1. Let M = (Mkl ) be Musielak-Orlicz function, p = (pkl ) be a bounded sequence of positive real numbers and u = (ukl ) be a double sequence of strictly positive real numbers. Then the sequence spaces Cf (M, u, ∆m , p) and Cf0 (M, u, ∆m , p) are Banach spaces with the supremum norm. Proof. We are going to prove this for the space Cf (M, u, ∆m , p) and the other can be proved in the similar way. Define norm ||.|| on Cf (M, u, ∆m , p) as: m ,p ||x||M,u,∆ Cf

=

sup q,r,s,t∈N



q

r

 u ∆m x  p XX 1 kl k+s,l+t kl Mkl . (q + 1)(r + 1) % k=0 l=0

m

Clearly, Cf (M, u, ∆ , p) is a normed linear space by the above defined norm. Now, we have to prove that Cf (M, u, ∆m , p) is complete. For this, let (∆m x(b) ) be a Cauchy (b) sequence in Cf (M, u, ∆m , p). Then (∆m xkl ) is a Cauchy sequence in C, for each k, l. (b) Therefore, ∆m xkl → ∆m xkl (say). Put ∆m x = (∆m xkl ). For given , there exists an integer N () = N (say) such that for each b, n > N  ||∆m x(b) − ∆m x(n) || < . 2 Hence, sup |τqrst (∆m x(b) − ∆m x(n) )| < q,r,s,t

 . 2

Then, for each q, r, s, t and b, n > N, we have  . 2 Now, for fixed b, the above inequality holds. Since for fixed b, ∆m x(b) ∈ Cf (M, u, ∆m , p), we get (2.1)

|τqrst (∆m x(b) − ∆m x(n) )|
0, there exists positive integers q0 , r0 such that  (2.2) |τqrst (∆m x(b) ) − L| < , 2 for q ≥ q0 , r ≥ r0 and for all s, t. Here q0 , r0 are independent of s, t but depend upon . Now, by using (2.1) and (2.2), we obtain |τqrst (∆m x) − L| = |τqrst (∆m x) − τqrst (∆m x(b) ) + τqrst (∆m x(b) ) − L| ≤
0 for all x ∈ Cf (M, u, ∆m , p) such that m

rkl (x) = |xkl | ≤ T ||x||M,u,∆ Cf

778

,p

∀ k, l ∈ N.

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DOUBLE DIFFERENCE SPACES OF ALMOST NULL AND ALMOST CONVERGENT SEQUENCES 7

So, the seminorm is continuous. Therefore, Cf (M, u, ∆m , p) is a DK− space and so is Banach space, it follows that Cf (M, u, ∆m , p) has Frechet topology. Thus, it is BDK−space with the above given norm. Hence, the proof is complete.  Theorem 2.5. Let M = (Mkl ) be Musielak-Orlicz function, p = (pkl ) be a bounded sequence of positive real numbers and u = (ukl ) be a double sequence of strictly positive real Mkl(w) numbers. Suppose that β = lim < ∞. Then, Cf0 (u, ∆m , p) = Cf0 (M, u, ∆m , p). w→∞ w Proof. In order to prove that Cf0 (u, ∆m , p) = Cf0 (M, u, ∆m , p), it is sufficient to show that Cf0 (M, u, ∆m , p) ⊂ Cf0 (u, ∆m , p). Now, let β > 0. By definition of β, we have w ≤ β1 Mkl (w), ∀ w ≥ 0. Let x = (xkl ) ∈ Cf0 (M, u, ∆m , p). Thus, we have q X r  X 1 ukl ∆m xk+s,l+t  pkl lim sup q,r→∞ s,t>0 (q + 1)(r + 1) % k=0 l=0

q

≤

r

 u ∆m x  p XX 1 1 kl k+s,l+t kl lim sup Mkl β q,r→∞ s,t>0 (q + 1)(r + 1) % k=0 l=0

which implies that x = (xkl ) ∈ Cf0 (u, ∆m , p).



Theorem 2.6. Let M = (Mkl ) be Musielak-Orlicz function and u = (ukl ) be a double sequence of strictly positive real numbers. If p = (pkl ) and v = (vkl ) are bounded sequences of positive real numbers with 0 ≤ pkl ≤ vkl < ∞ ∀ k, l; then Cf0 (M, u, ∆m , p) ⊂ Cf0 (M, u, ∆m , v). Proof. Let x = (xkl ) ∈ Cf0 (M, u, ∆m , p). Then q X r  u ∆m x  p X 1 kl k+s,l+t kl sup Mkl → 0 as q, r → ∞. % s,t>0 (q + 1)(r + 1) k=0 l=0 This implies that   pkl u ∆m x ≤ 1, for sufficiently large values of k and l. Since (Mkl ) is in Mkl kl %k+s,l+t creasing and pkl ≤ vkl , we have q X r  u ∆m x  v X 1 kl k+s,l+t kl sup Mkl % s,t>0 (q + 1)(r + 1) k=0 l=0

≤

sup

s,t>0

q

r

 u ∆m x  p XX 1 kl k+s,l+t kl Mkl . (q + 1)(r + 1) % k=0 l=0

m

Thus, x = (xkl ) ∈ Cf0 (M, u, ∆ , v). This completes the proof.



Theorem 2.7. Let M = (Mkl ) be Musielak-Orlicz function, p = (pkl ) be a bounded sequence of positive real numbers and u = (ukl ) be a double sequence of strictly positive real numbers. Then the following inclusions hold: (i) If 0 < inf pkl < pkl ≤ 1 then Cf0 (M, u, ∆m , p) ⊂ Cf0 (M, u, ∆m ); (ii) If 1 ≤ pkl ≤ sup pkl < ∞ then Cf0 (M, u, ∆m ) ⊂ Cf0 (M, u, ∆m , p). Proof. (i) Let x = (xkl ) ∈ Cf0 (M, u, ∆m , p). Then, since 0 < inf pkl < pkl ≤ 1, we obtain the following: q X r  u ∆m x  X 1 kl k+s,l+t Mkl lim sup q,r→∞ s,t>0 (q + 1)(r + 1) % k=0 l=0

≤

lim sup

q,r→∞ s,t>0

q

r

 u ∆m x  p XX 1 kl k+s,l+t kl Mkl . (q + 1)(r + 1) % k=0 l=0

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Thus, x = (xkl ) ∈ Cf0 (M, u, ∆m ). (ii) Let p = (pkl ) ≥ 1 for each k and l and sup pkl < ∞. Let x = (xkl ) ∈ Cf0 (M, u, ∆m ). Then for each 0 <  < 1, there exists a positive integer N such that q X r  u ∆m x  X 1 kl k+s,l+t Mkl lim sup ≤  < 1 ∀ q, r ≥ N. q,r→∞ s,t>0 (q + 1)(r + 1) % k=0 l=0 This implies that q X r  u ∆m x  p X 1 kl k+s,l+t kl lim sup Mkl q,r→∞ s,t>0 (q + 1)(r + 1) % k=0 l=0

≤

lim sup

q,r→∞ s,t>0

q

r

  u ∆m x XX 1 kl k+s,l+t Mkl . (q + 1)(r + 1) % k=0 l=0

m

Therefore, x = (xkl ) ∈ Cf0 (M, u, ∆ , p). This concludes the proof.



0 Theorem 2.8. Let M = (Mkl ) and M0 = (Mkl ) be two Musielak-Orlicz functions, p = (pkl ) be a bounded sequence of positive real numbers and u = (ukl ) be a double sequence of strictly positive real numbers. Then,

Cf0 (M, u, ∆m , p) ∩ Cf0 (M0 , u, ∆m , p) ⊂ Cf0 (M + M0 , u, ∆m , p). Proof. Let x = (xkl ) ∈ Cf0 (M, u, ∆m , p) ∩ Cf0 (M0 , u, ∆m , p). Then q X r  u ∆m x  p X 1 kl k+s,l+t kl sup Mkl → 0 as q, r → ∞ % s,t>0 (q + 1)(r + 1) k=0 l=0

and q X r  u ∆m x  p X 1 kl k+s,l+t kl 0 Mkl → 0 as q, r → ∞. % s,t>0 (q + 1)(r + 1) k=0 l=0 Then, we have q X r  u ∆m x  p X 1 kl k+s,l+t kl 0 (Mkl + Mkl ) sup % s,t>0 (q + 1)(r + 1)

sup

k=0 l=0

q

r

≤ K sup

 u ∆m x  p i XX 1 kl k+s,l+t kl Mkl (q + 1)(r + 1) %

+ K sup

 u ∆m x  p i XX 1 kl k+s,l+t kl 0 Mkl → 0 as q, r → ∞. (q + 1)(r + 1) %

h

s,t>0

h

s,t>0

Thus, sup s,t>0

k=0 l=0 q r

k=0 l=0

1 (q + 1)(r + 1)

q X r X

0 (Mkl + Mkl )

k=0 l=0

 u ∆m x  p kl k+s,l+t kl → 0 as q, r → ∞. %

Therefore, x = (xkl ) ∈ Cf0 (M + M0 , u, ∆m , p). This completes the proof.



0 Theorem 2.9. Let M = (Mkl ) and M0 = (Mkl ) be two Musielak-Orlicz functions, p = (pkl ) be a bounded sequence of positive real numbers and u = (ukl ) be a double sequence of strictly positive real numbers. Then,

Cf0 (M0 , u, ∆m , p) ⊂ Cf0 (M ◦ M0 , u, ∆m , p). Proof. Let x = (xkl ) ∈ Cf0 (M0 , u, ∆m , p). Then, we have q X r  u ∆m x  p X 1 kl k+s,l+t kl 0 Mkl lim sup = 0. q,r→∞ s,t>0 (q + 1)(r + 1) % k=0 l=0 Let  > 0 and choose δ > 0 with 0 < δ < 1 such that Mkl (n) <  for 0 ≤ n ≤ δ.

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DOUBLE DIFFERENCE SPACES OF ALMOST NULL AND ALMOST CONVERGENT SEQUENCES 9

i h  ukl ∆m xk+s,l+t 0 and consider Write ykl = Mkl % q,r pkl X 1 sup Mkl (ykl ) s,t>0 (q + 1)(r + 1) k,l=0,0

=

sup

s,t>0

pkl pkl X X 1 1 Mkl (ykl ) + sup Mkl (ykl ) (q + 1)(r + 1) 1 s,t>0 (q + 1)(r + 1) 2

where the first summation is over ykl ≤ δ and second summation is over ykl > δ. Since Mkl is continuous, we have pkl X 1 (2.3) Mkl (ykl ) < H sup s,t>0 (q + 1)(r + 1) 1 where H = sup pkl and for ykl > δ, we use the fact that ykl ykl δ, Mkl (ykl ) < 2Mkl (1) yδkl . Hence, ykl
0 (q + 1)(r + 1) 2 (2.4)

  ≤ max 1, (2Fk (1)δ −1 )H sup s,t>0

q,r pkl X 1 ykl (q + 1)(r + 1) k,l=0,0

Therefore, from equations (2.3) and (2.4), we have Cf0 (M0 , u, ∆m , p) ⊂ Cf0 (M ◦ M0 , u, ∆m , p). This completes the proof.



Theorem 2.10. Let BS be the space of bounded series of double sequences and Cf0 (M, u, ∆m , p) be the space of all almost null double sequences. Then the inclusion relation BS ⊂ Cf0 (M, u, ∆m , p) holds. s,t X xkl < ∞. Proof. Let x = (xkl ) ∈ BS. Then T = sup s,t∈N

k,l=0,0

Therefore, for all q, r, s, t ∈ N, we have s+q X s+q X s+q X t+r s−1 X t+r t−1 t+r X X X X xkl − xkl − xkl xkl = k=s l=t

k=0 l=0

k=0 l=0

k=s l=0

s+q X s+q X s+q X t+r t+r t−1 X X X ≤ xkl + xkl + xkl k=0 l=0 s+q X t−1 X

≤ 2T +

k=s l=0

k=s l=0

xkl

k=s l=0

=

s+q X t−1 s X t−1 X X 2T + xkl − xkl k=0 l=0

k=0 l=0

s+q X t−1 s X t−1 X X ≤ 2T + xkl + xkl ≤ 4T, k=0 l=0

k=0 l=0

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which implies that

s+q X s+q X t+r t+r  u ∆m x  pkl  u ∆m x  pkl X X 1 4T kl kl kl kl Mkl Mkl ≤ (q + 1)(r + 1) % (q + 1)(r + 1) % k=s l=t

k=s l=t

Further, if we take supremum over s, t ∈ N in the above relation and also apply the P-limit as q, r → ∞, then, we have P − lim sup q,r→∞ s,t>0

s+q t+r

 u ∆m x  pkl XX 1 kl kl Mkl = 0, (q + 1)(r + 1) % k=s l=t

m

therefore, x ∈ Cf0 (M, u, ∆ , p). Hence, the result holds.



References [1] A. Altay and F. Ba¸sar, Some new spaces of double sequences, J Math Anal Appl, 309 (2005), 70-90. [2] M. Ba¸sarır and O. Sonalcan, On some double sequence spaces, J. Indian Acad. Math., 21 (1999), 193-200. [3] C ¸ . A. Bekta¸s, M. Et and R. C ¸ olak, Generalized difference sequence spaces and their dual spaces, J. Math. Anal. Appl., 292 (2004), 423-432. [4] T. J. Bromwich, An introduction to the theory of infinite series, Macmillan and co. Ltd., New York (1965). [5] J. Boos, Classical and Modern Methods in Summability, NewYork, NY, USA: Oxford University Press Inc., 2000. [6] C. C ¸ akan, B. Altay and M. Mursaleen, The σ-convergence and σ-core of double sequences, Appl. Math. Letters, 19 (2006), 1122-1128. [7] M. Et and R. C ¸ olak, On generalized difference sequence spaces, Soochow J. Math., 21 (1995), 377-386. [8] M. Et and A. Esi, On K¨ othe-Toeplitz duals of generalized difference sequence spaces, Bull. Malays. Math. Sci. Soc., 23 (2000), 25-32. [9] A. Esi, B. C. Tripathy and B. Sarma, On some new type generalized difference sequence spaces, Math. Slovaca., 57 (2007), 475-482. [10] G. H. Hardy, On the convergence of certain multiple series, Proc. Camb. Phil., Soc., 19 (1917), 86-95. [11] H. Kizmaz, On certain sequence spaces, Canad. Math. Bull., 24 (1981), 169-176. [12] J. Lindenstrauss and L. Tzafriri, On Orlicz sequence spaces, Israel J. Math., 10 (1971), 379-390. [13] G.G. Lorentz, A contribution to the theory of divergent sequences, Acta Math., 80 (1948), 167-190. [14] L. Maligranda, Orlicz spaces and interpolation, Seminars in Mathematics 5, Polish Academy of Science, 1989. [15] F. M´ oricz, Statistical convergence of multiple sequences, Arch. Math. (Basel), 81 (2003), 82-89. [16] F. M´ oricz and B. E. Rhoades, Almost convergence of double sequences and strong reqularity of summability matrices, Math. Proc. Camb. Phil. Soc., 104 (1988), 283-294. [17] M. Mursaleen and O. H. H. Edely, Statistical convergence of double sequences, J. Math. Anal. Appl., 288(1), (2003), 223-231. [18] M. Mursaleen and O. H. H. Edely, Almost convergence and a core theorem for double sequences, J. Math. Anal. Appl., 293, (2004), 532-540. [19] M. Mursaleen and S.A. Mohiuddine, Invariant mean and some core theorems for double sequences, Taiwanese J Math, 14 (2010), 21-33. [20] M. Mursaleen and S.A. Mohiuddine, Convergence Methods For Double Sequences and Applications, New Delhi, India: Springer, 2014. [21] M. Mursaleen, Almost strongly regular matrices and a core theorem for double sequences, J Math Anal Appl., 293 (2004), 523-531. [22] J. Musielak, Orlicz spaces and modular spaces, Lecture notes in Mathematics, 1034 (1983). [23] K. Raj, A. Azimhan and K. Ashirbayev, Some generalized difference sequence spaces of ideal convergence and Orlicz functions, J. Comput. Anal. Appl., 22 (2017), 52-63. [24] K. Raj and S. K. Sharma, Applications of double lacunary sequences to n-norm, Acta Univ. Sapientiae Mathematica, 7 (2015), 67-88. [25] K. Raj, S. Jamwal and S. K. Sharma, New classes of generalized sequence spaces defined by an Orlicz function, J. Comput. Anal. Appl., 15 (2013), 730-737. [26] H. H. Schaefer, Topological vector spaces, Graduate Texts in Mathematics, Vol. 3, 5th priniting, 1986.

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[27] B. C. Tripathy and A. Esi, A new type of difference sequence spaces, Int. J. Sci. Tech., 1 (2006), 11-14. [28] B. C. Tripathy, A. Esi and B. Tripathy, On a new type of generalized difference Ces` aro sequence spaces, Soochow J. Math., 31 (2005), 333-340. [29] M. Ye¸silkayagiil and Feyzi Ba¸sar, Some topological properties of the spaces of almost null and almost convergent double sequences Turk. J. Math. 40 (2016), 624-630. [30] M. Zeltser, Investigation of double sequence spaces by soft and hard analytical methods, Diss. Math. Univ. Tartu., 25, Tartu University Press, Univ. of Tartu, Faculty of Mathematics and Computer Science, Tartu (2001). School of Mathematics Shri Mata Vaishno Devi University, Katra-182320, J & K (India) E-mail address: [email protected] E-mail address: [email protected]

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ADDITIVE ρ-FUNCTIONAL INEQUALITIES HARIN LEE, JAE YOUNG CHA, MIN WOO CHO, MYUNGJUN KWON AND CHOONKIL PARK∗ Abstract. In this paper, we solve the additive ρ-functional inequalities kf (2x − y) + f (y − x) − f (x)k ≤ kρ (f (x + y) − f (x) − f (y))k ,

(0.1)

where ρ is a fixed complex number with |ρ| < 1, and kf (x + y) − f (x) − f (y)k ≤ kρ(f (2x − y) + f (y − x) − f (x))k,

(0.2)

1 2.

where ρ is a fixed complex number with |ρ| < Furthermore, we prove the Hyers-Ulam stability of the additive ρ-functional inequalities (0.1) and (0.2) in complex Banach spaces.

1. Introduction and preliminaries The stability problem of functional equations originated from a question of Ulam [17] concerning the stability of group homomorphisms. The functional equation f (x + y) = f (x) + f (y) is called the Cauchy equation. In particular, every solution of the Cauchy equation is said to be an additive mapping. Hyers [6] gave a first affirmative partial answer to the question of Ulam for Banach spaces. Hyers’ Theorem was generalized by Aoki [1] for additive mappings and by Rassias [10] for linear mappings by considering an unbounded Cauchy difference. A generalization of the Rassias theorem was obtained by G˘avruta [5] by replacing the unbounded Cauchy difference by a general control function in the spirit of Rassias’ approach. The stability of quadratic functional equation was proved by Skof [16] for mappings f : E1 → E2 , where E1 is a normed space and E2 is a Banach space. Cholewa [3] noticed that the theorem of Skof is still true if the relevant domain E1 is replaced by an Abelian group. See [2, 4, 7, 8, 9, 11, 12, 13, 14, 15, 18] for more information on the stability problems of functional equations. In Section 2, we solve the additive ρ-functional inequality (0.1) and prove the HyersUlam stability of the additive ρ-functional inequality (0.1) in complex Banach spaces. In Section 3, we solve the additive ρ-functional inequality (0.2) and prove the HyersUlam stability of the additive ρ-functional inequality (0.2) in complex Banach spaces. 2010 Mathematics Subject Classification. Primary 39B62, 39B72, 39B52. Key words and phrases. Hyers-Ulam stability; additive ρ-functional inequality. ∗ Corresponding author.

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Throughout this paper, let G be a 2-divisible abelian group. Assume that X is a real or complex normed space with norm k · k and that Y is a complex Banach space with norm k · k. 2. Additive ρ-functional inequality (0.1) Throughout this section, assume that ρ is a fixed complex number with |ρ| < 1. In this section, we solve and investigate the additive ρ-functional inequality (0.1) in complex Banach spaces. Lemma 2.1. If a mapping f : G → Y satisfies kf (2x − y) + f (y − x) − f (x)k ≤ kρ (f (x + y) − f (x) − f (y))k

(2.1)

for all x, y ∈ G, then f : G → Y is additive. Proof. Assume that f : G → Y satisfies (2.1). Letting x = 0 and y = 0 in (2.1), we get kf (0)k ≤ kρ (f (0))k and so f (0) = 0 with |ρ| < 1. Letting x = 0 in (2.1), we get kf (−y) + f (y)k ≤ 0 and so f is an odd mapping. Letting x = z and y = z − w in (2.1), we get kf (z + w) − f (z) − f (w)k ≤ kρ (f (2z − w) + f (w − z) − f (z))k

(2.2)

for all z, w ∈ G. It follows from (2.1) and (2.2) that kf (2x − y) + f (y − x) − f (x)k ≤ kρ (f (x + y) − f (x) − f (y))k ≤ |ρ|2 kf (2x − y) + f (y − x) − f (x)k and so f (2x − y) + f (y − x) = f (x) for all x, y ∈ G. It is easy to show that f is additive.  We prove the Hyers-Ulam stability of the additive ρ-functional inequality (2.1) in complex Banach spaces. Theorem 2.2. Let r > 1 and θ be nonnegative real numbers, and let f : X → Y be a mapping such that kf (2x − y) + f (y − x) − f (x)k

(2.3) r

r

≤ kρ (f (x + y) − f (x) − f (y))k + θ(kxk + kyk ) for all x, y ∈ X. Then there exists a unique additive mapping h : X → Y such that kf (x) − h(x)k ≤

2θ kxkr r 2 −2

(2.4)

for all x ∈ X.

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Proof. Letting x = y = 0, in (2.3), we get kf (0)k ≤ 0. So f (0) = 0. Letting y = 0 in (2.3), we get kf (2x) + f (−x) − f (x)k ≤ θkxkr

(2.5)

for all x ∈ X. Letting x = 0 in (2.3), we get kf (y) + f (−y)k ≤ θkykr

(2.6)

for all y ∈ X. From (2.5) and (2.6), we get kf (2x) − 2f (x)k ≤ kf (2x) + f (−x) − f (x)k + kf (x) + f (−x)k ≤ 2θkxkr

(2.7)

for all x ∈ X. So,

 

x

2 r

f (x) − 2f

≤ r θkxk

2

2

for all x ∈ X. Hence

   

l x x

m

2 f −2 f m ≤

l

2

2

m−1 X

j=l

2j f



x 2j



j+1

−2



f

x



j+1

2

X 2j 2 m−1 ≤ r θkxkr rj 2 j=l 2

(2.8)

for all nonnegative integers m and l with m > l and all x ∈ X. It follows from (2.8) that the sequence {2n f ( 2xn )} is a Cauchy sequence for all x ∈ X. Since Y is complete, the sequence {2n f ( 2xn )} converges. So one can define the mapping h : X → Y by x h(x) := lim 2n f ( n ) n→∞ 2 for all x ∈ X. Moreover, letting l = 0 and passing the limit m → ∞ in (2.8), we get (2.4). It follows from (2.3) that kh(2x − y) + h(y − x) − h(x)k

     

2x − y y−x x n = lim 2 f +f − f n

n n n→∞ 2 2 2

     

x y 2n θ x + y

+ lim − f − f (kxkr + kykr ) ≤ n→∞ lim 2n |ρ|

f 2n 2n 2n n→∞ 2nr = |ρ|kh(x + y) − h(x) − h(y)k for all x, y ∈ X. So kh(2x − y) + h(y − x) − h(x)k ≤ |ρ|kh(x + y) − h(x) − h(y)k for all x, y ∈ X. By Lemma 2.1, the mapping h : X → Y is additive.

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Now, let T : X → Y be another additive mapping satisfying (2.4). Then we have

x x

kh(x) − T (x)k = 2n

h n − T 2 2n          

x x x x

≤ 2n

h n − f n

+

T − f 2 2 2n 2n 4 · 2n ≤ θkxkr , r nr (2 − 2)2









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 (2.4).  Theorem 2.3. Let r < 1 and θ be positive real numbers, and let f : X → Y be a mapping satisfying f (0) = 0 and (2.3). Then there exists a unique additive mapping h : X → Y such that kf (x) − h(x)k ≤

2θ kxkr r 2−2

(2.9)

for all x ∈ X. Proof. It follows from (2.7) that



1

f (x) − f (2x) ≤ θkxkr

2

for all x ∈ X. Hence

1

1 m

f (2l x) − f (2 x) ≤

l m

2

2

≤

m−1 X

j=l m−1 X j=l



1 1 f (2j x) − j+1 f (2j+1 x)

j 2 2

2rj θkxkr 2j

(2.10)

for all nonnegative integers m and l with m > l and all x ∈ X. It follows from (2.10) that the sequence { 21n f (2n x)} is a Cauchy sequence for all x ∈ X. Since Y is complete, the sequence { 21n f (2n x)} converges. So one can define the mapping h : X → Y by h(x) := n→∞ lim

1 f (2n x) 2n

for all x ∈ X. Moreover, letting l = 0 and passing the limit m → ∞ in (2.10), we get (2.9). The rest of the proof is similar to the proof of Theorem 2.2.  Remark 2.4. If ρ is a real number such that −1 < ρ < 1 and Y is a real Banach space, then all the assertions in this section remain valid.

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ADDITIVE ρ-FUNCTIONAL INEQUALITIES

3. Additive ρ-functional inequality (0.2) Throughout this section, assume that ρ is a fixed complex number with |ρ| < 21 . In this section, we solve and investigate the additive ρ-functional inequality (0.2) in complex Banach spaces. Lemma 3.1. If a mapping f : G → Y satisfies kf (x + y) − f (x) − f (y)k ≤ kρ(f (2x − y) + f (y − x) − f (x))k

(3.1)

for all x, y ∈ G, then f : G → Y is additive. Proof. Assume that f : G → Y satisfies (3.1). Letting x = y = 0 in (3.1), we get kf (0)k ≤ 0. So f (0) = 0. Letting y = x in (3.1), we get kf (2x) − 2f (x)k ≤ 0 and so 2f (x) = f (2x)

(3.2)

for all x ∈ G. Letting y = 2x in (3.1), we get kf (3x) − f (x) − f (2x)k ≤ 0 and from (3.2), 3f (x) = f (3x)

(3.3)

for all x ∈ G. Letting y = −x in (3.1), we get kf (x) + f (−x)k ≤ kρ(f (3x)+f (−2x)−f (x))k. From (3.2) and (3.3), f (3x) + f (−2x) − f (x) = 2f (x) + 2f (−x), so kf (x) + f (−x)k ≤ 0, and we get f (x) + f (−x) = 0

(3.4)

for all x ∈ G. So f is an odd mapping. Letting x = z, y = z − w in (3.1), we get kf (2z − w) − f (z) − f (z − w)k ≤ kρ(f (z + w) + f (−w) − f (z))k and from (3.4), kf (2z − w) + f (w − z) − f (z)k ≤ kρ(f (z + w) − f (z) − f (w))k

(3.5)

for all z, w ∈ G. It follows from (3.1) and (3.5) that kf (x + y) − f (x) − (y)k ≤ kρ(f (2x − y) + f (y − x) − f (x))k ≤ |ρ|2 kf (x + y) − f (x) − f (y)k and so f (x + y) = f (x) + f (y) for all x, y ∈ G. So f is additive.



We prove the Hyers-Ulam stability of the additive ρ-functional inequality (3.1) in complex Banach spaces. Theorem 3.2. Let r > 1 and θ be positive real numbers, and let f : X → Y be a mapping such that kf (x + y) − f (x) − f (y) k

(3.6)

≤ kρ(f (2x − y) + f (y − x) − f (x))k + θ(kxkr + kykr )

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

2θ kxkr −2

(3.7)

2r

for all x ∈ X. Proof. Letting x = y = 0 in (3.4), we get kf (0)k ≤ 0. So f (0) = 0. Letting y = x in (3.4), we get kf (2x) − 2f (x)k ≤ 2θkxkr

(3.8)

for all x ∈ X. So

   

l x x

m

2 f ≤ − 2 f

l m

2

2

≤

m−1 X

j=l

2j f



x 2j



− 2j+1 f



x



j+1

2

X 2j 2 m−1 θkxkr 2r j=l 2rj

(3.9)

for all nonnegative integers m and l with m > l and all x ∈ X. It follows from (3.9) that the sequence {2n f ( 2xn )} is a Cauchy sequence for all x ∈ X. Since Y is complete, the sequence {2n f ( 2xn )} converges. So one can define the mapping h : X → Y by x h(x) := lim 2n f ( n ) n→∞ 2 for all x ∈ X. Moreover, letting l = 0 and passing the limit m → ∞ in (3.9), we get (3.7). It follows from (3.6) that kh (x + y) − h (x) − h(y)k

     

x+y x y n = lim 2 f − f n − f n

n n→∞ 2 2 2

      

y−x x 2n θ 2x − y n

≤ lim 2 ρ f +f − f n + lim nr (kxkr + kykr ) n n n→∞ n→∞ 2 2 2 2 = kρ(h(2x − y) + h(y − x) − h(x))k for all x, y ∈ X. So kh (x + y) − h (x) − h(y)k ≤ kρ(h(2x − y) + h(y − x) − h(x))k for all x, y ∈ X. By Lemma 3.1, the mapping h : X → Y is additive. Now, let T : X → Y be another additive mapping satisfying (3.7). Then we have

   

x x

kh(x) − T (x)k = 2 h n − T 2 2n          

x x



x x n ≤ 2 h n − f n + T − f n

n n

2 2 2 · 2n · 2 ≤ θkxkr , (2r − 2)2nr

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ADDITIVE ρ-FUNCTIONAL INEQUALITIES

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 (3.7).  Theorem 3.3. Let r < 1 and θ be positive real numbers, and let f : X → Y be a mapping satisfying (3.4). Then there exists a unique additive mapping h : X → Y such that 2θ kf (x) − h(x)k ≤ kxkr (3.10) r 2−2 for all x ∈ X. Proof. It follows from (3.8) that



1

f (x) − f (2x) ≤ θkxkr

2

for all x ∈ X. Hence

m−1 X

1

1

1 1 m j+1

f (2l x) −

f (2j x) − f (2 x) ≤ f (2 x)

l

j 2 2m 2j+1 j=l 2 ≤

m−1 X j=l

2rj kxkr 2j

(3.11)

for all nonnegative integers m and l with m > l and all x ∈ X. It follows from (3.11) that the sequence { 21n f (2n x)} is a Cauchy sequence for all x ∈ X. Since Y is complete, the sequence { 21n f (2n x)} converges. So one can define the mapping h : X → Y by 1 f (2n x) n→∞ 2n for all x ∈ X. Moreover, letting l = 0 and passing the limit m → ∞ in (3.11), we get (3.10). The rest of the proof is similar to the proof of Theorem 3.2.  h(x) := lim

Remark 3.4. If ρ is a real number such that − 12 < ρ < space, then all the assertions in this section remain valid.

1 2

and Y is a real Banach

References [1] T. Aoki, On the stability of the linear transformation in Banach spaces, J. Math. Soc. Japan 2 (1950), 64–66. [2] A. Chahbi and N. Bounader, On the generalized stability of d’Alembert functional equation, J. Nonlinear Sci. Appl. 6 (2013), 198–204. [3] P. W. Cholewa, Remarks on the stability of functional equations, Aequationes Math. 27 (1984), 76–86. [4] G. Z. Eskandani and P. Gˇ avruta, Hyers-Ulam-Rassias stability of pexiderized Cauchy functional equation in 2-Banach spaces, J. Nonlinear Sci. Appl. 5 (2012), 459–465. [5] P. Gˇavruta, A generalization of the Hyers-Ulam-Rassias stability of approximately additive mappings, J. Math. Anal. Appl. 184 (1994), 431–436. [6] D. H. Hyers, On the stability of the linear functional equation, Proc. Natl. Acad. Sci. U.S.A. 27 (1941), 222–224.

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[7] C. Park, Orthogonal stability of a cubic-quartic functional equation, J. Nonlinear Sci. Appl. 5 (2012), 28–36. [8] C. Park, K. Ghasemi, S. G. Ghaleh and S. Jang, Approximate n-Jordan ∗-homomorphisms in C ∗ -algebras, J. Comput. Anal. Appl. 15 (2013), 365-368. [9] C. Park, A. Najati and S. Jang, Fixed points and fuzzy stability of an additive-quadratic functional equation, J. Comput. Anal. Appl. 15 (2013), 452–462. [10] Th. M. Rassias, On the stability of the linear mapping in Banach spaces, Proc. Amer. Math. Soc. 72 (1978), 297–300. [11] K. Ravi, E. Thandapani and B. V. Senthil Kumar, Solution and stability of a reciprocal type functional equation in several variables, J. Nonlinear Sci. Appl. 7 (2014), 18–27. [12] S. Shagholi, M. Bavand Savadkouhi and M. Eshaghi Gordji, Nearly ternary cubic homomorphism in ternary Fr´echet algebras, J. Comput. Anal. Appl. 13 (2011), 1106–1114. [13] S. Shagholi, M. Eshaghi Gordji and M. Bavand Savadkouhi, Stability of ternary quadratic derivation on ternary Banach algebras, J. Comput. Anal. Appl. 13 (2011), 1097–1105. [14] D. Shin, C. Park and Sh. Farhadabadi, On the superstability of ternary Jordan C ∗ homomorphisms, J. Comput. Anal. Appl. 16 (2014), 964–973. [15] D. Shin, C. Park and Sh. Farhadabadi, Stability and superstability of J ∗ -homomorphisms and J ∗ -derivations for a generalized Cauchy-Jensen equation, J. Comput. Anal. Appl. 17 (2014), 125– 134. [16] F. Skof, Propriet locali e approssimazione di operatori, Rend. Sem. Mat. Fis. Milano 53 (1983), 113–129. [17] S. M. Ulam, A Collection of the Mathematical Problems, Interscience Publ. New York, 1960. [18] C. Zaharia, On the probabilistic stability of the monomial functional equation, J. Nonlinear Sci. Appl. 6 (2013), 51–59. Harin Lee, Jae Young Cha, Min Woo Cho, Myungjun Kwon Mathematics Branch, Seoul Science High School, Seoul 03066, Republic of Korea E-mail address: [email protected]; [email protected]; [email protected] Choonkil Park Research Institute for Natural Sciences, Hanyang University, Seoul 04763, Republic of Korea E-mail address: [email protected]

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TABLE OF CONTENTS, JOURNAL OF COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 24, NO. 4, 2018

Zeros of functions in weighted Dirichlet spaces and Carleson type measure, Ruishen Qian, Songxiao Li, and Yanhua Zhang,……………………………………………………………609 Smarandache fuzzy BCI-algebras, Sun Shin Ahn and Young Joo Seo,……………………619 On Fibonacci derivative equations, Hee Sik Kim, J. Neggers, and Keum Sook So,………628 A note on symmetric identities for twisted Daehee polynomials, Jongkyum Kwon and Jin-Woo Park,…………………………………………………………………………………………636 𝛼+𝑐𝑥

Dynamics and Behavior of 𝑥𝑛+1 = 𝑎𝑥𝑛 + 𝑏𝑥𝑛−1 + 𝛽+𝑑𝑥𝑛−2 , M. M. El-Dessoky and Aatef 𝑛−2

Hobiny,………………………………………………………………………………………644

Algebraic and Order Properties of Tracy-Singh Products for Operator Matrices, Arnon Ploymukda, Pattrawut Chansangiam, and Wicharn Lewkeeratiyutkul,……………………656 Analytic Properties of Tracy-Singh Products for Operator Matrices, Arnon Ploymukda, Pattrawut Chansangiam, and Wicharn Lewkeeratiyutkul,………………………………………………665 On the radial distribution of Julia set of solutions of 𝑓′′ + 𝐴𝐴′ + 𝐵𝐵 = 0, Jianren Long,.675

Common Fixed Point Results for the Family of Multivalued Mappings Satisfying Contractions on a Sequence in Hausdorff Fuzzy Metric Space, Abdullah Shoaib, Akbar Azam, and Aqeel Shahzad,………………………………………………………………………………………692 Additive-quadratic 𝜌-functional inequalities in non-Archimedean Banach spaces, Choonkil Park, Jung Rye Lee, and Dong Yun Shin,………………………………………………………….700 Differential equations associated with the generalized Euler polynomials of the second kind, C. S. Ryoo,………………………………………………………………………………………711 Chlodowsky Variant of Bernstein-Schurer Operators Based on (p,q)-Integers, Eser Gemikonakli and Tuba Vedi-Dilek,……………………………………………………………………….717 Dynamical behavior of a general HIV-1 infection model with HAART and cellular reservoirs, Ahmed M. Elaiw, Abdullah M. Althiabi, Mohammed A. Alghamdi and Nicola Bellomo,…728

TABLE OF CONTENTS, JOURNAL OF COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 24, NO. 4, 2018 (continued) Ideal theory of pre-logics based on the theory of falling shadows, Young Bae Jun and Sun Shin Ahn,…………………………………………………………………………………………744 Quadratic 𝜌-functional equations in non-Archimedean Banach spaces, Choonkil Park, Gang Lu, Yinhua Cui, and Ming Fang,………………………………………………………………752 The Incomplete Global GMERR Algorithm to Solve AX = B*, Yu-Hui Zheng, Jian-Lei Li, Dong-Xu Cheng, and Ling-Ling Lv,………………………………………………………760 Double difference spaces of almost null and almost convergent sequences for Orlicz function, Kuldip Raj and Renu Anand,………………………………………………………………773 Additive 𝜌-functional inequalities, Harin Lee, Jae Young Cha, Min Woo Cho, Myungjun Kwon and Choonkil Park,…………………………………………………………………………784

Volume 24, Number 5 ISSN:1521-1398 PRINT,1572-9206 ONLINE

May 2018

Journal of Computational Analysis and Applications EUDOXUS PRESS,LLC

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On generalized Fibonacci k-sequences and Fibonacci k-numbers Hee Sik Kim, J. Neggers and Choonkil Park∗ Abstract. In this paper analogs of Fibonacci sequences and Fibonacci numbers as well as Fibonacci functions (the case n = 2) for cases n = 3, 4, · · · are introduced. It is shown that these analogs are related to each other in a regular manner and that if limn→∞ then ϕ1 = 1, ϕ2 =

√ 1+ 5 2

Fn+1 Fn

= ϕk for a Fibonacci k-sequence,

< ϕ3 < · · · < ϕn < · · · < limn→∞ ϕn = 2. Many

identities of types similar to those which hold for the case n = 2 (i.e., the Fibonacci case) are also established, indicating the existence of a larger theory of which the Fibonacci case is an integrated part.

1. Introduction Fibonacci-numbers have been studied in many different forms for centuries and the literature on the subject is consequently incredibly vast (see [1, 3, 4, 9]). Han et al. [5] considered several properties of Fibonacci sequences in arbitrary groupoids. Kim et al. [7] introduced the notion of generalized Fibonacci sequences over a groupoid and discussed these in particular for the case where the groupoid contains idempotents and pre-idempotents. In [6], Han et al. discussed Fibonacci functions on the real numbers R, i.e., functions f : R → R such that for all x ∈ R, f (x + 2) = f (x + 1) + f (x), and developed the notion of Fibonacci functions using the concept of f -even and f -odd√functions. Moreover, they showed 1+ 5 that if f is a Fibonacci function then limx→∞ f (x+1) 2 . f (x) = Kim et al. [7] discussed Fibonacci functions using the (ultimately) periodicity and they also discussed the exponential Fibonacci functions. Especially, given a non-negative real-valued function, they obtained examples of several classes of exponential Fibonacci functions. In this paper we introduce the family of Fibonacci k-sequences, where {Fn }ωn=0 is a Fibonacci k-sequence provided F (n + k) = F (n + k − 1) + · · · + F (n) for all n ∈ N = {1, 2, · · · }. Thus, if k = 2, then F (n + 2) = F (n + 1) + F (n) and {F (n)}ωn=0 is an ordinary Fibonacci sequence. Similarly, for k = 3, we obtain the formula F (n+3) = F (n+2)+F (n+1)+F (n), where F (0), F (1), F (2) may be taken as arbitrary elements of an abelian group A, usually taken to be the group of integers, rationals, real or complex numbers. If F (0) = F (1) = F (2) = 1, then we will consider this special case, i.e., {1, 1, 1, 3, 5, 9, 17, 31, · · · } the sequence of Fibonacci 3-numbers. The properties of this sequence can be expected to be analogous to those of the sequence {1, 1, 2, 3, 5, 8, · · · } of Fibonacci (i.e.,√Fibonacci 2-) numbers. Thus, e.g., = ϕ3 (+ 1.839) is an analogue of ϕ2 = 1+2 5 , the golden section. What this limn→∞ FFn+1 n number ϕ3 may mean in other settings is itself a question of interest as is the question of 2010 Mathematics Subject Classification: 11B39, 39A10. Key words and phrases: ((α, β, γ)-)Fibonacci (k-)function, ((α, β, γ)-)Fibonacci k-sequence. ∗ Correspondence: +82 2 2220 0892(phone), +82 2 2290 0019(fax) (C. Park).

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H. S. Kim, J. Neggers and C. Park

whether and where this number √ can be observed in nature. It turns out that the correspond1+ 5 ing sequence ϕ1 = 1, ϕ2 = 2 < ϕ3 < · · · < ϕn < · · · < limn→∞ ϕn = 2 is itself a sequence of interest as we hope to show below as well. From Fibonacci sequences to go to Fibonacci functions in a natural way has proven to be interesting and it has yields a theory of such functions [6, 8]. The second part of this paper is concerned with the introduction of Fibonacci k-functions, where a real-valued function f : R → R is a Fibonacci k-function if f (x + k) = f (x + k − 1) + f (x + k − 2) + · · · + f (x). Again, in analogy with the class of Fibonacci 2-functions (i.e., Fibonacci functions), we are able to construct many examples of such functions for k = 3, 4, 5, · · · etc., indicating that there is a great deal of material which has yet to be uncovered in this area. Nevertheless, it is clear that there is much work to be done, some of which we will discuss below in further detail.

2. Preliminaries A function f defined on the real numbers is said to be a Fibonacci function ([5]) if it satisfies the formula f (x + 2) = f (x + 1) + f (x)

(1)

for any x ∈ R, where R (as usual) is the set of real numbers. Example 2.1. ([6]) Let f (x) := ax be a Fibonacci function on R where a > 0. Then √ ax a2 = f (x + 2) = f (x + 1) + f (x) = ax (a + 1). Since a > 0, we have a2 = a + 1 and a = 1+2 5 . √

Hence f (x) = ( 1+2 5 )x is a Fibonacci function, and the unique Fibonacci function of this type on R. √

Example 2.2. ([8]) (a). A function f (x) = (x − bxc)( 1+2 5 )x is a Fibonacci function. (b). A function f (x) defined by ( √ (x − bxc)( 1+2 5√)x if x ∈ Q f (x) = 1+ 5 x −(x − bxc)( 2 ) otherwise is a Fibonacci function. √ x (c). A function f (x) = sin(πx)( 5−1 2 ) is a Fibonacci function. Proposition 2.3. ([5]) If f (x) is a Fibonacci function, then √ f (x + 1) 1+ 5 lim = x→∞ f (x) 2

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Fibonacci k-sequences and Fibonacci k-numbers

3. Higher Fibonacci k-sequences Let α, β, γ be non-zero integers. A sequence {Fn } is said to be an (α, β)-Fibonacci 2sequence if F1 = α, F2 = β and Fn+2 = Fn+1 + Fn for n = 1, 2, 3, · · · . A sequence {Fn } is said to be an (α, β, γ)-Fibonacci 3-sequence if F1 = α, F2 = β, F3 = γ and Fn+3 = Fn+2 +Fn+1 +Fn for n = 1, 2, 3, · · · . Especially, if α = β = 1 or α = β = γ = 1, then we say {Fn } a Fibonacci 2-sequence or a Fibonacci 3-sequence respectively. Example 3.1. (a). It is well known that {1, 1, 2, 3, 5, 8, 13, · · · } is the Fibonacci 2-sequence. (b). {1, 1, 1, 3, 5, 9, 17, 31, 57, 105, 193, 355, 653, · · · } is the Fibonacci 3-sequence. We give some formulas for the Fibonacci 3-sequence as follows. Proposition 3.2. If {Fn } is the Fibonacci 3-sequence, then 1 (1) Fn = [(Fn+3 − Fn+1 ) − (Fn+2 − Fn )]. 2 Proof. Since Fn+3 = Fn+2 + Fn+1 + Fn , we obtain Fn+3 − Fn+1 = Fn+2 + Fn . It follows that 2Fn = (Fn+2 + Fn ) − (Fn+2 − Fn ) = (Fn+3 − Fn+1 ) − (Fn+2 − Fn ) so that we obtain the equality (1).



Proposition 3.3. If {Fn } is the Fibonacci 3-sequence, then 1 (2) Fn+2 = [(Fn+3 − Fn+1 ) + (Fn+2 − Fn )]. 2 Proof. Since Fn+3 = Fn+2 + Fn+1 + Fn , we obtain Fn+3 − Fn+1 = Fn+2 + Fn . It follows that (Fn+3 − Fn+1 ) + (Fn+2 − Fn ) = (Fn+2 + Fn ) + (Fn+2 − Fn ) = 2Fn+2 so that we obtain the equality (2).



Proposition 3.4. If {Fn } is the Fibonacci 3-sequence, then 1 (3) Fn Fn+2 = [(Fn+3 − Fn+1 )2 − (Fn+2 − Fn )2 ]. 4 Proof. It follows that 2 2 4Fn Fn+2 = Fn+2 + 2Fn Fn+2 + Fn2 − Fn+2 + 2Fn Fn+2 − Fn2

= (Fn+2 + Fn )2 − (Fn+2 − Fn )2 = (Fn+3 − Fn+1 )2 − (Fn+2 − Fn )2 so that we obtain the equality (3).



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Proposition 3.5. If {Fn } is the Fibonacci 3-sequence, then Fn+3 = 2(F1 + F2 + · · · + Fn ) + Fn+1 where F0 = −1. Proof. Since Fn+3 = Fn+2 + Fn+1 + Fn , we obtain F3 − F2 = F1 + F0 , F4 − F3 = F2 + F1 , · · · , Fn+3 − Fn+2 = Fn+1 + Fn , which proves the proposition.  4. Solutions of a Fibonacci polynomial ξn (x). Given a natural number n, we define a polynomial ξn (x) by ξn (x) := xn − xn−1 − xn−2 − · · · − x − 1. We call such a polynomial ξn (x) a Fibonacci polynomial. Let ϕn be the largest real root of the equation ξn (x) = 0. Then ξn (2) = 2n −(2n−1 +2n−2 +· · ·+2+1) = 1 and ξn (1) = 1−n < 0 when n > 1. Let x ≥ 2. Then xn − 1 = (x − 1)(xn−1 + xn−2 + · · · + x + 1) ≥ xn−1 + xn−2 + · · · + x + 1. It follows that ξn (x) = xn − (xn−1 + xn−2 + · · · + x + 1) ≥ 1 = ξn (2). Hence we obtain 1 ≤ ϕn ≤ 2 for all n ∈ N. If n = 1, then ξ1 (x) = x − 1 and ϕ1 = 1, and if n = 2, then √ 1+ 5 2 ξ2 (x) = x − x − 1 and ϕ2 = 2 . Proposition 4.1. Let {ϕn } be the sequence of the largest roots of ξn (x) = 0. Then it is increasing, i.e., ϕn < ϕn+1 for all n ∈ N and limn→∞ ϕn ≤ 2. Proof. Since ϕn is the largest real root of ξn (x) = 0 for all n ∈ N, ξn−1 (ϕn−1 ) = 0. It follows that n−3 n−2 ϕn−1 n−1 = ϕn−1 + ϕn−1 + · · · + ϕn−1 + 1.

(4)

If we let x := ϕn−1 in ξn (x) = 0, then by (4) we have n−2 ξn (ϕn−1 ) = ϕkn−1 − (ϕn−1 n−1 + ϕn−1 + · · · + ϕn−1 + 1

= ϕnn−1 − 2ϕn−1 n−1 = ϕn−1 n−1 (ϕn−1 − 2) < 0. Since ϕn is the largest real root of ξn (x) = 0, we obtain ϕn−1 < ϕn for all n ∈ Z with n ≥ 2. Since {ϕn } is an increasing sequence and it is bounded above, we have limn→∞ ϕn ≤ 2.  Given a Fibonacci 3-sequence {Fn }, we consider the following sequence { FFn+1 }: n {

Fn+1 } = {1, 1, 3, 1.667, 1.8, 1.889, 1.832, 1.839, 1.842, 1.838, 1.839, 1.839, · · · } Fn

Thus we expect that the limit limn→∞ Note that

limn→∞ FFn+1 n

√

=

1+ 5 2

Fn+1 Fn

is approximately 1.839.

for the Fibonacci 2-sequence.

A quick computation yields ξ3 (1.839) = (1.839)3 − (1.839)2 − 1.839 − 1 = 0.002, i.e., F3 (1.839) = 0.002 + 0.

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Since ϕ3 is the largest real root of ξ3 (x) = 0, we obtain ξ3 (x) = (x − ϕ3 )(x2 + αx + β) for some α, β ∈ R. It follows that ξ3 (x) = x3 + (α − ϕ3 )x2 + (β − αϕ3 )x − ϕ3 β It follows that α − ϕ3 = −1, β − αϕ3 = −1 and −ϕ3 β = −1. Hence β = αϕ3 − 1 = (ϕ3 − 1)ϕ3 − 1 = ϕ23 − ϕ3 − 1 and α2 − 4β = −3ϕ23 + 2ϕ3 + 5 + −1.468 < 0 if we let ϕ3 = 1.839. This means that x2q + αx + β = 0 has imaginary roots. Let ρ1 , ρ2 be roots of x2 + q αx + β = 0. Then ρi = − α2 ± i

then ρ1 =

√1 ϕ3

β − 14 α2 and kρi k =

expiτ and ρ2 =

√1 ϕ3

√1 ϕ3

< 1. If we let τ := arg(− α2 ± i β − 41 α2 ),

exp−iτ . If we let ϕ3 + 1.839, then ϕ23 = 3.382 and hence

β = ϕ23 −ϕ3 −1 + 0.543 and α + 0.839. Hence x3 −x2 −x−1 + (x−1.839)(x2 +0.839x+0.543) and ρi + −0.420 ± 0.606i. Now, we may assume the polynomial x3 − x2 − x − 1 = 0 is the characteristic equation of the linear operator ξn , the Fibonacci 3-sequence (as in a linear differential equation: y 000 − y 00 − y 0 − y = 0) which means that the roots ϕ3 , √1ϕ3 eτ i , √1ϕ3 e−τ i provide the expression: 1 1 Fn = Aϕn3 + B( √ )n enτ i + C( √ )n e−nτ i ϕ3 ϕ3

(5) for some A, B, C ∈ R.

Theorem 4.2. If Fn is of the form (5), then A= where M =

1 (e2τ i ϕ23

1 1 1 1 2τ i [ 2 (e − e−2τ i ) + )(e−τ i − eτ i ) ], √ (1 + M ϕ3 ϕ3 ϕ3 ϕ3

− e−2τ i ) +

√1 (1 ϕ3

+

1 )(e−τ i ϕ33

− eτ i ).

Proof. If we take n = 1, 2, 3 in (5) respectively, then we have 1 1 1 = F1 = Aϕ3 + B √ eτ i + C √ e−τ i , ϕ3 ϕ3 1 1 1 = F2 = Aϕ23 + B( √ )2 e2τ i + C( √ )2 e−2τ i , ϕ3 ϕ3 1 1 1 = F3 = Aϕ33 + B( √ )3 e3τ i + C( √ )3 e−3τ i . ϕ3 ϕ3 Using Cramer’s rule we obtain ϕ 3 √1 eτ i √1 e−τ i ϕ3 ϕ3 1 1 M= ϕ23 ( √ϕ3 )2 e2τ i ( √ϕ3 )2 e−2τ i 3 ϕ3 ( √1ϕ )3 e3τ i ( √1ϕ )3 e−3τ i 3 3 which becomes M=

1 2τ i 1 1 (e − e−2τ i ) + √ (1 + 3 )(e−τ i − eτ i ) 2 ϕ3 ϕ3 ϕ3

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by simple computations. To obtain A, we change the first column into 1’s in M. 1 √1 eτ i √1 e−τ i ϕ ϕ 3 3 1 1 1 A = 1 ( √ϕ3 )2 e2τ i ( √ϕ3 )2 e−2τ i M 1 ( √1ϕ )3 e3τ i ( √1ϕ )3 e−3τ i 3 3 which becomes A=

1 1 1 1 2τ i )(e−τ i − eτ i ), ] [ (e − e−2τ i ) + √ (1 + M ϕ23 ϕ3 ϕ3 ϕ3

by simple computations. This proves the theorem.



Note that we can obtain the coefficients B, C, but it is not necessary to find those. Since 1 = √1+α , we obtain that limn→∞ ( √1ϕ3 )n = 0 and limn→∞ ( √1ϕ3 )n [Benτ i + Ce−nτ i ] =

√1 ϕ3

0. Hence limn→∞ limn→∞ FFn+1 n

Fn ϕn 3

= A. Thus limn→∞

Fn+1 ϕ3 Fn

= limn→∞

Fn+1 /ϕn+1 3 Fn /ϕn 3

= 1. It follows that

Fn ϕn 3

for a “sufficiently large

= ϕ3 .

Example 4.3. To approximate A we may take an expression A +

n and a sufficiently accurate value of ϕ3 ”. Thus, if n = 5, then F5 = 9 and (1.839)5 = 21.033 and hence A + 0.428. If n = 7, then F7 = 31, ϕ7 = 71.132 and hence A + ϕF77 + 0.436. If n = 12, then F12 = 653, (1.836)12 = 1496.145 and hence A +

F12 (1.839)12

3

= 0.436.

5. Higher Fibonacci k-functions Let k ≥ 2 be an integer. A function f defined on the real numbers is said to be a Fibonacci k-function if it satisfies the formula (2)

f (x + k) = f (x + k − 1) + f (x + k − 2) + · · · + f (x + 1) + f (x)

for any x ∈ R, where R (as usual) is the set of real numbers. = ϕx3 ϕ33 and f (x + 2) + f (x + 1) + Example 5.1. Let f (x) := ϕx3 . Then f (x + 3) = ϕx+3 3 x+2 x+1 x x 2 2 3 2 f (x) = ϕ3 + ϕ3 + ϕ3 = ϕ3 [ϕ3 + ϕ3 + 1]. Since ϕ3 = ϕ3 + ϕ23 + 1, f (x) = ϕx3 is a Fibonacci 3-function. In a similar manner, we know that g(x) = ϕx2 is a Fibonacci 2-function where ϕ2 = Similarly, h(x) = ϕxk defines a Fibonacci k-function.

√ 1+ 5 2 .

Example 5.2. Let ϕk := ωθ and f (x) := Aω bxc θx where A ∈ R. Then we have f (x + k) = Aω bx+kc θx+k = Aω bxc θx ω k θk = Aω bxc θx (1 + ωθ + · · · + ω k−1 θk−1 ) = Aω bxc θx + Aω bx+1c θx+1 + · · · + Aω bx+k−1c θx+k−1 = f (x) + f (x + 1) + · · · + f (x + k − 1).

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Hence f (x) = Aω bxc θx is a Fibonacci k-function. Note that the collection of all Fibonacci k-functions over the real numbers forms a vector space over R. Proposition 5.3. Let f (x) be a Fibonacci k-function. If we define g(x) := f (x + t) where t ∈ R for any x ∈ R, then g(x) is also a Fibonacci k-function. Proof. Since f (x) is a Fibonacci k-function, we have g(x + k) = f (x + k + t) = f (x + t + k − 1) + f (x + t + k − 2) + · · · + f (x + t) = g(x + k − 1) + g(x + k − 2) + · · · + g(x + 1) + g(x), proving that g(x) is also a Fibonacci k-function.



For example, a function g(x) := (ϕ3 )x+t is a Fibonacci 3-function. Theorem 5.4. Let ϕk be the largest real root of Fk (x) = 0 and let ϕk = ω1 θ1 = ω2 θ2 = P bxc x · · · = ωN θN , where ωi , θi ∈ R. If we define F (x) := N i=1 Ai ωi θi , then F (x) is a Fibonacci k-function, where Ai ∈ R. Proof. Given x ∈ R, we have F (x + k) =

=

N X i=1 N X

bx+kc x+k θi

Ai ωi

bxc

Ai ωi θix ωik θik

i=1

=

=

=

N X i=1 N X i=1 N X i=1

bxc

Ai ωi θix ϕkk bxc

Ai ωi θix (1 + ωi θi + · · · + ωik−1 θik−1 ) bxc

Ai ωi θix +

N X

bx+1c x+1 θi

A i ωi

+ ··· +

i=1

N X

bx+k−1c x+k−1 θi

Ai ωi

i=1

= F (x) + F (x + 1) + · · · + F (x + k − 1), proving that F (x) is a Fibonacci k-function.



Example 5.5. In Example 5.2, if we let A := 1 and ω := n a natural number, then ϕk = nθ. Assume f (x) := nbxc θx . Then f (x) = nbxc θx = nbxc ( ϕnk )x = nbxc−x (ϕk )x . If we let √ x := 2.5, then b2.5c = 2. Hence f (2.5) = nb2.5c−2.5 (ϕk )2.5 = √1n (ϕk )2 ϕk . If we let n := 8, √ since {ϕk } goes to 2, we obtain f (2.5) + √18 22 2 = 2.

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Proposition 5.6. Let f (x) be a Fibonacci k-function and differentiable (integrable, resp.). Then its derivative(integration, resp.) is also a Fibonacci k-function. Example 5.7. Consider a Fibonacci k-function f (x) := ϕxk . Then f (x) = ex ln ϕk and hence f 0 (x) = (ex ln ϕk )0 = ln ϕk ex ln ϕk = Rϕxk ln ϕk = f (x) ln ϕk . Hence f 0 (x) is a Fibonacci x k-function. Similarly, if we define g(x) := 0 eu ln ϕk du, then g(x) = ln1ϕk [ϕxk − 1] and g 0 (x) = 1 x x ln ϕk [ϕk ln ϕk − 0] = ϕk . Hence g(x) is also a Fibonacci k-function. Example 5.8. In Example 5.2, if ϕk = ωθ, then f (x) := Aω bxc θx is a Fibonacci k-function. f (x+1) bx+c θ x+1 )/(Aω bxc θ x ) = ωθ = ϕ , we obtain lim Since f (x+1) x→∞ f (x) = ϕk . k f (x) = (Aω Conjecture. If f (x) is a Fibonacci k-function, then lim

x→∞

f (x + 1) = ϕk f (x)

It is known that if f (x) is a Fibonacci 2-function, then limx→∞

f (x+1) f (x)

= ϕ2 [5].

6. Fibonacci k-sequences and Fibonacci k-numbers Proposition 6.1. Let f (x) be a Fibonacci 3-function, then f (x + n + 2) = αn f (x + 2) + βn f (x + 1) + γn f (x) for all natural numbers n, where {αn } is an (1, 2, 4)-Fibonacci 3-sequence, {βn } is an (1, 2, 3)Fibonacci 3-sequence and {γn } is an (1, 1, 2)-Fibonacci 3-sequence. Proof. Let f (x) be a Fibonacci 3-function. Then f (x + 3) = f (x + 2) + f (x + 1) + f (x), and f (x + 4) = 2f (x + 2) + 2f (x + 1) + f (x) for all x ∈ R. In this fashion we obtain f (x + 5) = 4f (x + 2) + 2f (x + 1) + 2f (x), f (x + 6) = 7f (x + 2) + 6f (x + 1) + 4f (x), f (x + 7) = 13f (x + 2) + 11f (x + 1) + 7f (x), f (x + 8) = 24f (x + 2) + 20f (x + 1) + 13f (x). The sequence {αn } of coefficients of f (x + 2) is 1, 2, 4, 7, 13, 24, · · · , the sequence {βn } of the coefficients of f (x + 1) is 1, 2, 3, 6, 11, 20, · · · and the sequence {γn } of coefficients of f (x) is 1, 1, 2, 4, 7, 13, · · · . This shows that {αn } is the (1, 2, 4)-Fibonacci 3-sequence, {βn } is the (1, 2, 3)-Fibonacci 3-sequence and {γn } is the (1, 1, 2)-Fibonacci 3-sequence. This shows that f (x + n + 2) = αn f (x + 2) + βn f (x + 1) + γn f (x), proving the theorem.  Corollary 6.2. Given a natural number n, we have ϕn+2 = αn ϕ23 + βn ϕ3 + γn 3 where {αn } is the (1, 2, 4)-Fibonacci 3-sequence, {βn } is the (1, 2, 3)-Fibonacci 3-sequence and {γn } is the (1, 1, 2)-Fibonacci 3-sequence. Proof. f (x) = ϕx3 is a Fibonacci 3-function.

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Theorem 6.3. Let a, b, c be non-zero integers. If we define f (x + 3) := af (x + 2) + bf (x + 1) + cf (x) for all x ∈ R, then f (x + n + 2) = αn f (x + 2) + βn f (x + 1) + γn f (x) for all x ∈ R and n ≥ 4, where {αn } is a (a, a2 + b, a3 + 2ab + c)-Fibonacci 3-sequence, {βn } is a (b, ab + c, a2 b + ac + b2 )-Fibonacci 3-sequence and {γn } is a (c, ac, a2 c + bc)-Fibonacci 3-sequence. Proof. Since f (x + 3) := af (x + 2) + bf (x + 1) + cf (x), we let (α1 , β1 , γ1 ) := (a, b, c). f (x + 4) = af (x + 3) + bf (x + 2) + cf (x + 1) = a[af (x + 2) + bf (x + 1) + cf (x)] = (a2 + b)f (x + 2)+(ab+c)f (x+1)+acf (x) leads to (α2 , β2 , γ2 ) = (a2 +b, ab+c, ac). By simple computations, we obtain f (x + 5) = (a3 + 2ab + c)f (x + 2) + (a2 b + ac + b2 )f (x + 1) + (a2 c + bc)f (x). Let (α3 , β3 , γ3 ) := (a3 + 2ab + c, a2 b + ac + b2 , a2 c + bc). We compute f (x + 6) as follows: f (x + 6) = = + + = +

af (x + 5) + bf (x + 4) + cf (x + 3) a[α3 f (x + 2) + β3 f (x + 1) + γ3 f (x)] b[α2 f (x + 2) + β2 f (x + 1) + γ2 f (x)] c[α1 f (x + 2) + β1 f (x + 1) + γ1 f (x)] (aα3 + bα2 + cα1 )f (x + 2) + (aβ3 + bβ2 + cβ1 )f (x + 1) (aγ3 + bγ2 + cγ1 )f (x),

i.e., α4 = aα3 + bα2 + cα1 , β4 = aβ3 + bβ2 + cβ1 and γ4 = aγ3 + bγ2 + cγ1 . If we let αn = aαn−1 + bαn−2 + cαn−3 , βn = aβn−1 + bβn−2 + cβn−3 and γn = aγn−1 + bγn−2 + cγn−3 for n ≥ 4, then {αn } is an (a, a2 + b, a3 + 2ab + c)-Fibonacci 3-sequence, {βn } is a (b, ab + c, a2 b + ac + b2 )-Fibonacci 3-sequence and {γn } is a (c, ac, a2 c + bc)-Fibonacci 3-sequence. Hence f (x + n + 2) = αn f (x + 2) + βn f (x + 1) + γn f (x) for all n ≥ 4. This proves the theorem.  Note that Proposition 6.1 is a special case of Theorem 6.3 if we let a = b = c = 1.

References [1] K. Atanassove et al, New Visual Perspectives on Fibonacci Numbers, World Scientific, New Jersey, 2002. [2] M. Bidkham, M. Hosseini, C. Park and M. Eshaghi Gordji, Nearly (k, s)-Fibonacci functional equations in β-normed spaces, Aequationes Math., 83 (2012), 131–141. [3] R. A. Dunlap, The Golden Ratio and Fibonacci Numbers, World Scientific, New Jersey, 1997. [4] E. Erkus-Duman and N. Tuglu, Generating functions for the generalized bivariate Fibonacci and Lucas polynomials, J. Comput. Anal. Appl. 18 (2015), 815–821. [5] J. S. Han, H. S. Kim and J. Neggers, Fibonacci sequences in groupoids, Adv. Difference Equ. 2012 , 2012:19. [6] J. S. Han, H. S. Kim and J. Neggers, On Fibonacci functions with Fibonacci numbers, Adv. Difference Equ. 2012, 2012:126. [7] H. S. Kim and J. Neggers and K. S. So, Generalized Fibonacci sequences in groupoids, Adv. Difference Equ. 2013, 2013:26.

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H. S. Kim, J. Neggers and C. Park [8] H. S. Kim and J. Neggers and K. S. So, On Fibonacci functions with periodicity, Adv. Difference Equ. 2014, 2014:293. [9] B. Peker and H. Senay, Solutions of the Pell equation x2 − (a2 + 2a)y 2 = N via generalized Fibonacci and Lucas numbers, J. Comput. Anal. Appl. 18 (2015), 721–726. Hee Sik Kim, Department of Mathematics, Research Institute for Natural Sciences, Hanyang University, Seoul 04763, Republic of Korea E-mail address: [email protected] J. Neggers, Department of Mathematics, University of Alabama, Tuscaloosa, AL 35487-0350, U. S. A E-mail address: [email protected] Choonkil Park, Department of Mathematics, Research Institute for Natural Sciences, Hanyang University, Seoul 04763, Republic of Korea E-mail address: [email protected]

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CONVERGENCE OF SP ITERATIVE SCHEME FOR THREE MULTIVALUED MAPPINGS IN HYPERBOLIC SPACE BIROL GUNDUZ AND IBRAHIM KARAHAN Abstract. The present paper aims to deal with multivalued version of SP iterative scheme to approximate a common …xed point of three multivalued nonexpansive mappings in a uniformly convex hyperbolic space and obtain strong and -convergence theorems for the SP process. Our results extend some existing results in the contemporary literature.

1. Introduction Fixed point theory is one of the most important area of nonlinear analysis and it has applications in di¤erent disciplines of science such as in economics, biology, chemistry, engineering and technology, game theory and physics. It has become attractive to many scientists because it directly a¤ects our daily lives. Iterative methods play an important role in calculating the …xed point of nonlinear mappings (see [16, 17, 18, 19]). The oldest known iterative method is Picard iteration which is the pioneer of iterative approximation of …xed point of di¤erent class of nonlinear mappings. W.R. Mann [1], S. Ishikawa [2], M. A. Noor [3] introduced the Mann, Ishikawa, Noor iteration process respectively for a single valued map T de…ned on nonempty subset of a normed space. Metric space versions of these iterations are following: (Mann)

un+1 = W (un ; T un ;

n) ;

(Ishikawa)

un+1 = (un ; T vn ; n ) ; vn = (un ; T un n )

(Noor)

un+1 = (un ; T vn ; vn = (un ; T wn ; wn = (un ; T un ;

n) n)

;

n)

where f n g ; f n g and f n g are sequences of real numbers in [0; 1]. In 2008, Thianwan [4] introduced a new two steps iteration. Gunduz et al. [20] modi…ed this iteration process as following and used for computing …xed point of nonexpansive mappings in hyperbolic spaces. un+1 = (vn ; T vn ; vn = (un ; T un ;

(Thianwan) where f

ng

and f

ng

n) n)

;

are sequences in [0; 1].

2000 Mathematics Subject Classi…cation. 47H10, 54H25. Key words and phrases. …xed point; hyperbolic space; multivalued map;

-convergence.

1

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2

BIROL GUNDUZ AND IBRAHIM KARAHAN

In 2011, Phuengrattana and Suantai [5] gave the SP iteration which de…ned in metric spaces as follows: (SP)

un+1 = (vn ; T vn ; n ) vn = (wn ; T wn ; n ) ; wn = (un ; T un ; n )

where f n g ; f n g and f n g are sequences of real numbers in [0; 1]. They also showed that SP-iteration is a generalized version of Mann, Ishikawa, Noor iterations and converges faster than all of others for the class of non-decreasing and continuous functions and they supported their analytical results by numerical examples. Chugh and Kumar [6] showed that the SP iterative scheme converges faster than the new two step iterative schemes of Thianwan [4] for increasing functions and decreasing functions. On the other hand, it is well known that the theory of multivalued maps is more complex than according to the theory of single valued maps. Now we discourse on multivalued maps. Let E be a metric space and D be a nonempty subset of E: If there exists an element k 2 D such that d(x; k) = inffd(x; y) : y 2 Dg = d(x; D) for each x 2 E, then the set D is called proximinal. We shall denote the compact subsets, proximinal bounded subsets, and closed and bounded subsets of K by C(D), P (D), and CB(D); respectively. A Hausdor¤ metric H induced by the metric d of E is de…ned by H(A; B) = maxfsup d(x; B); sup d(y; A)g x2A

y2B

for every A; B 2 CB(E): A multivalued mapping T : D ! P (D) is said to be a contraction if there exists a constant k 2 [0; 1) such that for any x; y 2 D; H(T x; T y)

kd(x; y);

and T is said to be nonexpansive if H(T x; T y)

d(x; y)

for all x; y 2 D: A point p 2 D is called a …xed point of T if p 2 T p: Denote the set of all …xed points of T by F (T ). By using the Hausdor¤ metric, Markin [7, 8] studied the …xed points of multivalued nonexpansive mappings and contractions. Moreover, Lim [33] proved the existence of …xed points for multivalued nonexpansive mappings under suitable conditions in uniformly convex Banach spaces. Later on, since the …xed point theory for this kind of mappings has a lot of application areas such as convex optimization problem, control problem and di¤erential inclusion problem (see, [9] and references cited therein), an interesting …xed point theory was developed. From then on di¤erent authors have studied on the convergence of …xed points for this class of mappings in convex metric spaces. For instance, Shimizu and Takahashi [25] generalized the results of Lim [33] to the convex metric spaces. The study of multivalued maps is a rapidly growing area of research (see, [10, 11, 12, 13, 14, 15]). Shahzad and Zegeye [13] studied strong convergenge of Ishikawa iterative process for quasi-nonexpansive multivalued maps in uniformly convex Banach spaces. They de…ned Ishikawa iteration as follows:

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Let T : D ! P (D) and PT (x) = fy 2 T x : d(x; y) = d(x; T x)g. The sequence of Ishikawa iteration is de…ned by x0 2 D, ( yn = (1 bn )xn + bn zn ; 0 xn+1 = (1 an )xn + an zn ;

0

where zn 2 T xn ; zn 2 T yn ; and fan g ; fbn g are sequences of real numbers in [0; 1]. In this paper, we …rst give multivalued version of the SP iteration scheme in Kohlenbach hyperbolic spaces and then use this scheme to approximate a common …xed point of three multivalued nonexpansive mappings. Our results improve and extend current results in recently literature by via faster, more general iteration and more general space. 2. Hyperbolic Spaces and Lemmas There are di¤erent de…nitions for hyperbolic space in mathematics. We will study in the hyperbolic space de…ned by Kohlenbach [22]. We review hyperbolic space and it’s features in this section. De…nition 1. [22] A metric space (E; d) is said to be hyperbolic space if there exists a map W : E 2 [0; 1] ! E satisfying: W1. W2. W3. W4.

d (u; W (x; y; )) (1 ) d (u; x) + d (u; y) d (W (x; y; ) ; W (x; y; )) = j j d (x; y) W (x; y; ) = W (y; x; (1 )) d (W (x; z; ) ; W (y; w; )) (1 ) d (x; y) + d (z; w)

for all x; y; z; w 2 E and ; 2 [0; 1]. A metric space (E; d) is called a convex metric space introduced by Takahashi [23] if it satis…es only W1. Every normed space (and Banach space) is a special convex metric space, but the converse of this statement is not true, in general (see [16]). The class of hyperbolic spaces includes normed spaces, the Hilbert ball (see [24] for a book treatment) and CAT (0)-spaces. The readers can found detailed discussion in [21]. De…nition 2. [25] A hyperbolic space (E; d; W ) is said to be uniformly convex if for all u; x; y 2 E, r > 0 and " 2 (0; 2], there exists a 2 (0; 1] such that 9 d (x; u) r = 1 d (y; u) r ;u (1 ) r: ) d W x; y; ; 2 d (x; y) "r

A map : (0; 1) (0; 2] ! (0; 1] which provides such a = (r; ") for given r > 0 and " 2 (0; 2], is called modulus of uniform convexity. We call monotone if it decreases with r (for a …xed "). A subset D of a hyperbolic space E is convex if W (x; y; ) 2 D for all x; y 2 D and 2 [0; 1]: Now, we give de…nition of -convergence which coined by Lim [26] in general metric spaces. To give the de…nition of -convergence, we …rst recall the notions of asymptotic radius and asymptotic center. Let fxn g be a bounded sequence in a metric space E. For x 2 E, de…ne a continuous functional r (:; fxn g) : E ! [0; 1) by r (x; fxn g) = lim supn!1 d (x; xn ) : Then the asymptotic radius = r (fxn g)

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of fxn g is given by = inf fr (x; fxn g) : x 2 Eg and the asymptotic center of a bounded sequence fxn g with respect to a subset D of E is de…ned as follows: AD (fxn g) = fx 2 E : r (x; fxn g)

r (y; fxn g) for any y 2 Dg :

The set of all asymptotic centers of fxn g is denoted by A(fxn g). It has been shown in [29] that bounded sequences have unique asymptotic center with respect to closed convex subsets in a complete and uniformly convex hyperbolic space with monotone modulus of uniform convexity. A sequence fxn g in E is said to -converge to x 2 E if x is the unique asymptotic center of fun g for every subsequence fun g of fxn g [27]. In this case, we write -limn xn = x. We want to point out that convergence coincides with weak convergence in Banach spaces with Opial’s property [30]. Kirk and Panyanak [27] specialized this concept to CAT(0) spaces and showed that many Banach space results involving weak convergence have precise analogs in this setting. Dhompongsa and Panyanak [28] continued to work in this direction and studied the -convergence of Picard, Mann and Ishikawa iterates in CAT (0) spaces. Khan et al. [31] was studied this concept in hyperbolic spaces and they gave a couple of helpful lemma as follows. Lemma 1. [31] Let (E; d; W ) be a uniformly convex hyperbolic space with monotone modulus of uniform convexity . Let x 2 E and f n g be a sequence in [b; c] for some b; c 2 (0; 1). If fxn g and fyn g are sequences in E such that lim supn!1 d (xn ; x) r; lim supn!1 d (yn ; x) r and limn!1 d (W (xn ; yn ; n ) ; x) = r for some r 0, then limn!1 d (xn ; yn ) = 0: Lemma 2. [31] Let D be a nonempty closed convex subset of a uniformly convex hyperbolic space and fxn g be a bounded sequence in D such that A (fxn g) = fyg and r (fxn g) = . If fym g is another sequence in D such that limm!1 r (ym ; fxn g) = , then limm!1 ym = y. The following useful …rst lemma can be found in [14] gives some properties of PT in metric (and hence hyperbolic) spaces and second can be found in [8]. Lemma 3. [14] Let T : D ! P (D) be a multivalued mapping and PT (x) = fy 2 T x : d(x; y) = d(x; T x)g :Then the following are equivalent. (1) x 2 F (T ); that is, x 2 T x, (2) PT (x) = fxg, that is, x = y for each y 2 PT (x), (3) x 2 F (PT ), that is, x 2 PT (x): Moreover, F (T ) = F (PT ): Lemma 4. Let A; B 2 CB (E) and a 2 A. If that d (a; b) H (A; B) + .

3. Strong and

> 0, then there exists b 2 B such

-Convergence Theorems

Before giving our main results, we give the multivalued version of the SP iteration scheme (SP) in hyperbolic spaces. Let E be a hyperbolic space and D be a nonempty convex subset of E. Let T; S; R : D ! P (D) be three multivalued maps

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5

and PT (x) = fy 2 T x : d(x; y) = d(x; T x)g. Choose x0 2 D and de…ne fxn g as 8 >

: zn = W (wn ; xn ; n )

where un 2 PT (yn ); vn 2 PS (zn ); wn 2 PR (xn ) and f n g; f n g ; f n g are real sequences such that 0 < a b < 1 for all n 2 N: The iterative sequence n; n; n (3.1) is called the modi…ed SP iterative process for a multivalued nonexpansive mapping in a Kohlenbach hyperbolic space. Lemma 5. Let D be a nonempty closed convex subset of a hyperbolic space X and let S; T; R : D ! P (D) be three multivalued mappings such that PT ; PS and PR are nonexpansive mappings with a nonempty common …xed point set F . Then for the modi…ed SP iterative process fxn g in (3.1), limn!1 d (xn ; p) exists for each p 2 F . Proof. Let p 2 F . Then p 2 PT (p) = fpg, p 2 PS (p) = fpg and p 2 PR (p) = fpg. Using (3.1), we have d (xn+1 ; p)

(3.2)

= d (W (un ; yn ; n ) ; p) (1 n ) d (un ; p) + n d (yn ; p) (1 n ) d (un ; PT (p)) + n d (yn ; p) (1 n ) H (PT (yn ) ; PT (p)) + n d (yn ; p) (1 n ) d (yn ; p) + n d (yn ; p) = d (yn ; p) = d (W (vn ; zn ; n ) ; p) (1 n ) d (vn ; p) + n d (zn ; p) (1 n ) d (vn ; PS (p)) + n d (zn ; p) (1 n ) H (PS (zn ) ; PS (p)) + n d (zn ; p) (1 n ) d (zn ; p) + n d (zn ; p) = d (zn ; p) = d (W (wn ; xn ; n ) ; p) (1 n ) d (wn ; p) + n d (xn ; p) (1 n ) d (wn ; PR (p)) + n d (xn ; p) (1 n ) H (PR (xn ) ; PR (p)) + n d (xn ; p) (1 n ) d (xn ; p) + n d (xn ; p) = d (xn ; p)

That is, d (xn+1 ; p)

d (xn ; p) :

Hence limn!1 d (xn ; p) exists. Lemma 6. Let D be a nonempty closed convex subset of a uniformly convex hyperbolic space X and let S; T; R : D ! P (D) be three multivalued mappings such that PT ; PS and PR are nonexpansive mappings with a nonempty common …xed point set F . Then for modi…ed SP iterative process fxn g in (3.1), we have limn!1 d (xn ; PT (yn )) = limn!1 d (xn ; PS (zn )) = limn!1 d (xn ; PR (xn )) = 0.

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Proof. By Lemma 5, limn!1 d (xn ; p) exists for each p 2 F . Assume that lim d (xn ; p) = n!1 c for some c 0. For c = 0, the result is trivial. Suppose c > 0. Now limn!1 d (xn+1 ; p) = c can be rewritten as (3.3)

lim d (W (un ; yn ;

n!1

n ) ; p)

= c:

Since PT is nonexpansive, we have d (un ; p)

= d (un ; PT (p)) H (PT (yn ) ; PT (p)) d (yn ; p) = d (W (vn ; zn ; n ) ; p) (1 n ) d (vn ; p) + n d (zn ; p) (1 n ) H (PS (zn ) ; PS (p)) + n d (zn ; p) (1 n ) d (zn ; p) + n d (zn ; p) = d (zn ; p) = d (W (wn ; xn ; n ) ; p) (1 n ) d (wn ; p) + n d (xn ; p) (1 n ) H (PR (xn ) ; PR (p)) + n d (xn ; p) (1 n ) d (xn ; p) + n d (xn ; p) d (xn ; p) :

Hence (3.4)

lim sup d (un ; p)

c; lim sup d (zn ; p)

n!1

c; lim sup d (yn ; p)

n!1

c:

n!1

Next d (vn ; p)

= d (vn ; PS (p)) H (PS (zn ) ; PS (p)) d (zn ; p)

and so (3.5)

lim sup d (vn ; p)

c:

n!1

Further d (wn ; p)

= d (wn ; PR (p)) H (PR (xn ) ; PR (p)) d (xn ; p)

and so (3.6)

lim sup d (wn ; p)

c:

n!1

On the other hand, since d (W (wn ; xn ;

n ) ; p)

= d (zn ; p)

d (xn ; p) ;

we have (3.7)

lim sup d (W (wn ; xn ; n!1

n ) ; p)

c:

From (3.2), we have (3.8)

c

lim inf d (W (wn ; xn ; n!1

820

n ) ; p)

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7

So, it follows from (3.7) and (3.8) that (3.9)

lim d (W (wn ; xn ;

n!1

n ) ; p)

= lim d (zn ; p) = c: n!1

Since we assume that limn!1 d (xn ; p) = c, we can write (3.10)

lim sup d (xn ; p)

c:

n!1

By using the inequalities (3.6), (3.9) and (3.10) and Lemma 1, we obtain that (3.11)

lim d (xn ; wn ) = 0:

n!1

On the other hand, it follows from (3.3), (3.4) and Lemma 1 that lim d (un ; yn ) = 0:

n!1

Nothing that d (xn+1 ; p)

= d (W (un ; yn ; n ) ; p) (1 n ) d (un ; p) + n d (yn ; p) (1 n ) d (yn ; un ) + d (un ; p)

which yields that (3.12)

c

lim inf d (un ; p) : n!1

Then from (3.4) and (3.12), we have c = lim d (un ; p) : n!1

Since d (xn+1 ; p) c

d (yn ; p), this implies that lim d (yn ; p) = lim d (W (vn ; zn ;

n!1

lim

n!1

n!1

(1

lim [(1

n!1

n ) lim sup d (vn ; p) n!1 n) c +

+

n ) ; p)

n

lim sup d (zn ; p) n!1

n c]

= c Hence, we get (3.13)

c = lim d (yn ; p) = lim d (W (vn ; zn ; n!1

n!1

n ) ; p) :

So, it follows from Lemma 1, (3.4), (3.5) and (3.13) that lim d (zn ; vn ) = 0

n!1

and d (yn ; p)

= d (W (vn ; zn ; n ) ; p) (1 n ) d (vn ; p) + n d (zn ; p) n d (zn ; vn ) + d (vn ; p)

this yields that c

lim inf d (vn ; p) ; n!1

so (3.5) gives that c = lim d (vn ; p) : n!1

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On the other hand, since lim d (zn ; xn )

n!1

=

lim d (W (wn ; xn ;

n ) ; xn )

n!1

lim

n!1

(1

n ) lim sup d (xn ; xn ) n!1

+

n

lim sup d (wn ; xn ) ; n!1

and lim d (yn ; zn )

n!1

=

lim d (W (vn ; zn ;

n ) ; zn )

n!1

lim

n!1

(1

n ) lim sup d (zn ; zn ) n!1

+

n

lim sup d (vn ; zn ) n!1

we have lim d (zn ; xn ) = 0

n!1

and lim d (yn ; zn ) = 0:

n!1

Also, d (un ; xn )

d (un ; yn ) + d (yn ; zn ) + d (zn ; xn ) ;

then lim d (un ; xn ) = 0:

n!1

Also, d (vn ; xn )

d (vn ; zn ) + d (zn ; xn ) ;

that is, lim d (vn ; xn ) = 0:

n!1

Since d (x; PR (x)) =

inf

z2PR (x)

d (x; z) ;

therefore d (xn ; PR (xn ))

d (xn ; wn ) ! 0 as n ! 1:

Similarly d (xn ; PT (yn ))

d (xn ; un ) ! 0

d (xn ; PS (zn ))

d (xn ; vn ) ! 0

and as n ! 1: Firstly, we prove that modi…ed SP iterative process de…ned in (3:1) a common …xed point of S; T and R.

-converges

Theorem 1. Let D be a nonempty closed and convex subset of a complete uniformly convex hyperbolic space E with monotone modulus of uniform convexity . Let S; T; R; PS ; PT ; PR and F be as in Lemma 6. Then the modi…ed SP iterative process fxn g -converges to a p in F .

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9

Proof. Let p 2 F (T ) = F (PT ): By the Lemma 5, fxn g is bounded and so lim d (xn ; p) n!1

exists. This gives that fxn g has a unique asymptotic center. In other words, we have A(fxn g) = fxg. Let fvn g be any subsequence of fxn g such that A(fvn g) = fvg. From Lemma 6, we have limn!1 (xn ; PS (xn )) = 0. We claim that v is a …xed point of PS . To prove this, take another sequence fzm g in PS (v). Then r(zm ; fvn g)

=

lim sup d(zm ; vn )

n!1

lim sup fd(zm ; PS (vn )) + d(PS (vn ) ; vn )g

n!1

lim sup H(PS (v); PS (vn ))

n!1

lim sup d (v; vn )

n!1

= r (v; fvn g) : This gives jr(zm ; fvn g r (v; fvn g)j ! 0 (m ! 1). From Lemma 2; we have limm!1 zm = v. Note that Sv 2 P (K) being proximinal is closed, hence PS (v) is closed. Moreover, PS (v) is bounded. Consequently limm!1 zm = v 2 PS (v). Similarly, v 2 PT (v) and v 2 PR (v). Hence v 2 F . From the uniqueness of asymptotic center, we have lim sup d (vn ; v)
n: Then it follows (along the lines similar to Lemma 5) that d(xm ; p)

823

d (xn ; p)

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BIROL GUNDUZ AND IBRAHIM KARAHAN

for all p 2 F . Thus we have d (xm ; xn )

d (xm ; p) + d (p; xn )

2d (xn ; p) :

Taking inf on the set F , we have d (xm ; xn ) d (xn ; F ). We show that fxn g is a Cauchy sequence in D by taking limit as m ! 1; n ! 1 in the inequality d (xm ; xn ) d (xn ; F ) : Thus, it converges to a q 2 D. Now it is left to show that q 2 F (S). Indeed, by d (xn ; F (PS )) = inf y2F (PS ) d (xn ; y) : So for each " > 0, there (") exists pn 2 F (PS ) such that, " < d (xn ; F (PS )) + : d xn ; p(") n 2 (")

" 2:

This implies limn!1 d xn ; pn follows that

(")

From d pn ; q

(")

d xn ; pn

+ d (xn ; q) it

" : 2

lim d p(") n ;q

n!1

Finally, d (PS (q) ; q)

d q; p(") + d p(") n n ; PS (q) d q; p(") + H PS p(") ; PS (q) n n 2d p(") n ;q

shows d (PS (q) ; q) < ". Therefore d (PS (q) ; q) = 0. In a similar way, we get d (PT (q) ; q) = 0 and d (PR (q) ; q) = 0. Since F is closed, therefore q 2 F . As appropriate our aim, we give de…nition of multivalued version of condition (I) of Senter and Dotson [32] for three maps and de…nition of semi-compact map. De…nition 3. The multivalued nonexpansive mappings S; T; R : D ! CB(D) where D a subset of E; are said to satisfy condition (I) if there exists a nondecreasing function f : [0; 1) ! [0; 1) with f (0) = 0; f (r) > 0 for all r 2 (0; 1) such that 13 [d(x; Sx) + d(x; T x) + d(x; Rx)] f (d(x; F )) for all x 2 D: De…nition 4. A map T : D ! P (D) is called semi-compact if any bounded sequence fxn g satisfying d(xn ; T xn ) ! 0 as n ! 1 has a convergent subsequence. We now give applications of above theorem. Theorem 3. Let D be a nonempty closed convex subset of a complete uniformly convex hyperbolic space E with monotone modulus of uniform convexity and, S; T; R; PS ; PT ; PR and F be as in Lemma 6. Suppose that PS ; PT and PR satisfy condition (I), then the modi…ed SP iterative process fxn g de…ned in (3:1) converges strongly to p 2 F . Proof. For all p 2 F; limn!1 d (xn ; p) exists from Lemma 5. We call it c for some c 0: If c = 0; there is nothing to prove. Assume c > 0: Now d (xn+1 ; p) d (xn ; p) gives that inf

p2F (T )

d (xn+1 ; p)

824

inf

p2F (T )

d (xn ; p) ;

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which means that d(xn+1 ; F )

11

d(xn ; F ):Hence lim d(xn ; F ) exists. By using n!1

condition (I) and Lemma 6;we get lim f (d(xn ; F ))

n!1

1 [d(x; Sx) + d(x; T x) + d(x; Rx)] = 0: n!1 3 lim

and so lim f (d(xn ; F )) = 0:

n!1

By the properties of f; we obtain that lim d(xn ; F ) = 0: Finally applying Theorem n!1 2, we get the result. Since the proof of following theorem is similar to proof of theorem proved in Banach spaces by various authors, we omit. Theorem 4. Let D be a nonempty closed convex subset of a complete uniformly convex hyperbolic space E with monotone modulus of uniform convexity and, S; T; R; PS ; PT ; PR and F be as in Lemma 6. Suppose that one of PS ; PT ; PR is semi-compact, then the modi…ed SP iterative process fxn g de…ned in (3:1) converges strongly to p 2 F . As a corollary of Theorem 2, we have the following theorem which is new in the literature. Theorem 5. Let D be a nonempty closed convex subset of a complete uniformly convex hyperbolic space X and let T : D ! P (D) be a multivalued mapping such that PT is nonexpansive mapping with a nonempty …xed point set F . Let fxn g de…ned as 8 >xn+1 = W (un ; yn ; n ) < ; yn = W (vn ; zn ; n ) > : zn = W (wn ; xn ; n ) where un 2 PT (yn ); vn 2 PT (zn ); wn 2 PT (xn ). Let f n g; f n g ; f n g be real sequences such that 0 < a b < 1 for all n 2 N. Then fxn g n; n; n converges to a …xed point of T . Remark 1. (1) Since SP iterative process converges faster than Mann and Ishikawa iterative processes, our theorems are better than results of Fukhar-ud-din et al. [34]. (2) Since CAT(0)-spaces are uniformly convex hyperbolic spaces with a ’nice’ 2 monotone modulus of uniform convexity (r; ") := "8 ; then our results valid in CAT(0) spaces besides Banach spaces. References [1] W.R. Mann, Mean value methods in iteration, Proc. Amer. Math. Soc. 4 (1953), 506-510. [2] S. Ishikawa, Fixed points by a new iteration method, Proc. Amer. Math. Soc. 44(1974), 147150. [3] M. A. Noor, New approximation schemes for general variational inequalities, J. Math. Anal. Appl. 251(2000), 217-229. [4] S. Thiainwan, Common …xed points of new iterations for two asymptotically nonexpansive nonself mappings in Banach spaces, J. Comput. Appl. Math., 224 (2) (2009), 688-695. [5] W. Phuengrattana, S. Suantai, On the rate of convergence of Mann, Ishikawa, Noor and SPiterations for continuous functions on an arbitrary interval, J. Comput. Appl. Math. 235, 3006-3014 (2011). [6] R. Chugh and V Kumar, Strong Convergence of SP Iterative Scheme for Quasi-Contractive Operators, International Journal of Computer Applications, 31, No.5, 2011.

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[7] J. T. Markin, Continuous dependence of …xed point sets, Proc. Amer. Math. Soc., 38 (1973), 545-547. [8] S. B. Nadler, Jr., Multivalued contraction mappings , Paci…c J. Math., 30 (1969), 475-488. [9] L. Gorniewicz, Topological …xed point theory of multivalued mappings, Kluwer Academic Pub., Dordrecht, Netherlands, 1999. [10] K. P. R. Sastry and G. V. R. Babu, Convergence of Ishikawa iterates for a multivalued mapping with a …xed point, Czechoslovak Math. J., 55 (2005), 817-826. [11] B. Panyanak, Mann and Ishikawa iterative processes for multivalued mappings in Banach spaces , Comp. Math. Appl., 54 (2007), 872-877. [12] Y. Song and H. Wang, Erratum to "Mann and Ishikawa iterative processes for multivalued mappings in Banach spaces" [Comp. Math. Appl., 54(2007), 872-877]. Comp. Math. Appl., 55(2008), 2999-3002. [13] N. Shahzad and H. Zegeye, On Mann and Ishikawa iteration schemes for multi-valued maps in Banach spaces, Nonlinear Anal. 71 (2009), no. 3-4, 838-844. [14] Y. Song and Y.J. Cho, Some notes on Ishikawa iteration for multivalued mappings,Bull. Korean. Math. Soc., 48 (2011), No. 3, pp. 575-584. DOI 10.4134/BKMS.2011.48.3.575 [15] S. H. Khan, M. Abbas and B.E. Rhoades, A new one-step iterative scheme for approximating common …xed points of two multivalued nonexpansive mappings , Rend. del Circ. Mat., 59 (2010), 149-157. [16] B. Gunduz and S. Akbulut, Strong convergence of an explicit iteration process for a …nite family of asymptotically quasi-nonexpansive mappings in convex metric spaces, Miskolc Mathematical Notes, 14 (3), 905-913, (2013). [17] B. Gunduz, S. H. Khan, S. Akbulut, On convergence of an implicit iterative algorithm for non self asymptotically non expansive mappings, Hacettepe Journal of Mathematics and Statistics, 43 (3), 399-411, (2014). [18] B. Gunduz and S. Akbulut, Strong and -convergence theorems in hyperbolic spaces, Miskolc Mathematical Notes, 14 (3), 915-925, (2013). [19] S. H. Khan, B. Gunduz, S. Akbulut, Solving nonlinear -strongly accretive operator equations by a one-step-two-mappings iterative scheme, J. Nonlinear Sci. Appl. 8 (2015), 837–846. [20] B. Gunduz, S. H. Khan, S. Akbulut, Common …xed points of two …nite families of nonexpansive mappings in kohlenbach hyperbolic spaces, J. Nonlinear Funct. Anal. 2015, 2015:15. [21] U. Kohlenbach and L. Leustean, Applied Proof Theory: Proof Interpretations and Their Use in Mathematics. Springer Monographs in Mathematics. Springer, Berlin (2008). [22] U. Kohlenbach, Some logical metatheorems with applications in functional analysis, Trans. Amer. Math. Soc. 357 (2005) 89-128. [23] W. Takahashi, A convexity in metric spaces and nonexpansive mappings. Kodai Math Sem Rep. 22, 142-149 (1970). [24] K. Goebel, S. Reich, Uniform convexity, hyperbolic geometry, and nonexpansive mappings, Monographs and Textbooks in Pure and Applied Mathematics, vol. 83, Marcel Dekker Inc., 1984. [25] T. Shimizu and W. Takahashi, Fixed points of multivalued mappings in certain convex metric spaces. Topol Methods Nonlinear Anal. 8, 197–203, (1996). [26] T.C. Lim, Remarks on some …xed point theorems, Proc. Amer. Math. Soc. 60, 179–182, 1976. [27] W.A. Kirk and B. Panyanak, A concept of convergence in geodesic spaces, Nonlinear Anal. 68, 3689–3696, 2008. [28] S. Dhompongsa and B. Panyanak, On -convergence theorems in CAT(0) spaces, Comput. Math. Appl. 56, 2572–2579, 2008. [29] L. Leu¸stean, Nonexpansive iterations in uniformly convex W-hyperbolic spaces, Contemp. Math., 513(2010), 193-210. [30] T. Kuczumow, An almost convergence and its applications, Ann. Univ. Mariae CurieSklodowska, Sect. A, 32 (1978), 79-88. [31] A.R. Khan, H. Fukhar-ud-din and M.A.A. Khan, An implicit algorithm for two …nite families of nonexpansive maps in hyperbolic spaces. Fixed Point Theory and Applications, 2012, 2012:54. [32] H.F. Senter and W.G. Dotson, Approximatig …xed points of nonexpansive mappings, Proc. Amer. Math. Soc., 44(2) (1974), 375–380. [33] T.C. Lim, A …xed point theorem for multivalued nonexpansive mappings in a uniformly convex Banach spaces, Bull. Amer. Math. Soc., 80 (1974), 1123-1126.

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13

[34] H. Fukhar-ud-din, A.R. Khan and M. Ubaid-ur-rehman, Ishikawa type algorithm of two multivalued quasi-nonexpansive maps on nonlinear domains, Ann. Funct. Anal. 4 (2) (2013),97-109. Department of Mathematics, Faculty of Science and Art, Erzincan University, Erzincan, 24000, Turkey. E-mail address : [email protected] Department of Mathematics, Faculty of Science, Erzurum Technical University, Erzurum, 25700, Turkey E-mail address : [email protected]

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ADDITIVE-QUADRATIC ρ-FUNCTIONAL EQUATIONS IN NON-ARCHIMEDEAN BANACH SPACES CHOONKIL PARK1 , JUNG RYE LEE2 , AND DONG YUN SHIN3∗ Abstract. Let 3 1 f (x + y) − f (−x − y) 4 4 1 1 + f (x − y) + f (y − x) − f (x) − f (y), 4( 4 ) ( ) ( ) x+y x−y y−x M2 f (x, y) : = 2f +f +f − f (x) − f (y). 2 2 2 We solve the additive-quadratic ρ-functional equations M1 f (x, y) :

=

M1 f (x, y) = ρM2 f (x, y),

(0.1)

where ρ is a fixed non-Archimedean number with |ρ| < 1, and M2 f (x, y) = ρM1 f (x, y),

(0.2)

where ρ is a fixed non-Archimedean number with |ρ| < |2|. Furthermore, we prove the Hyers-Ulam stability of the additive-quadratic ρ-functional equations (0.1) and (0.2) in non-Archimedean Banach spaces.

1. Introduction and preliminaries A valuation is a function | · | from a field K into [0, ∞) such that 0 is the unique element having the 0 valuation, |rs| = |r| · |s| and the triangle inequality holds, i.e., |r + s| ≤ |r| + |s|,

∀r, s ∈ K.

A field K is called a valued field if K carries a valuation. The usual absolute values of R and C are examples of valuations. Let us consider a valuation which satisfies a stronger condition than the triangle inequality. If the triangle inequality is replaced by |r + s| ≤ max{|r|, |s|},

∀r, s ∈ K,

then the function | · | is called a non-Archimedean valuation, and the field is called a nonArchimedean field. Clearly |1| = | − 1| = 1 and |n| ≤ 1 for all n ∈ N. A trivial example of a non-Archimedean valuation is the function | · | taking everything except for 0 into 1 and |0| = 0. Throughout this paper, we assume that the base field is a non-Archimedean field, hence call it simply a field. 2010 Mathematics Subject Classification. Primary 46S10, 39B62, 39B52, 47S10, 12J25. Key words and phrases. Hyers-Ulam stability; non-Archimedean normed space; additive-quadratic ρ-functional equation. ∗ Corresponding author: Dong Yun Shin (email: [email protected]). Dong Yun Shin was supported by the 2017 Research Fund of the University of Seoul.

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Definition 1.1. ([8]) Let X be a vector space over a field K with a non-Archimedean valuation | · |. A function ∥ · ∥ : X → [0, ∞) is said to be a non-Archimedean norm if it satisfies the following conditions: (i) ∥x∥ = 0 if and only if x = 0; (ii) ∥rx∥ = |r|∥x∥ (r ∈ K, x ∈ X); (iii) the strong triangle inequality ∥x + y∥ ≤ max{∥x∥, ∥y∥},

∀x, y ∈ X

holds. Then (X, ∥ · ∥) is called a non-Archimedean normed space. Definition 1.2. (i) Let {xn } be a sequence in a non-Archimedean normed space X. Then the sequence {xn } is called Cauchy if for a given ε > 0 there is a positive integer N such that ∥xn − xm ∥ ≤ ε for all n, m ≥ N . (ii) Let {xn } be a sequence in a non-Archimedean normed space X. Then the sequence {xn } is called convergent if for a given ε > 0 there are a positive integer N and an x ∈ X such that ∥xn − x∥ ≤ ε for all n ≥ N . Then we call x ∈ X a limit of the sequence {xn }, and denote by limn→∞ xn = x. (iii) If every Cauchy sequence in X converges, then the non-Archimedean normed space X is called a non-Archimedean Banach space. The stability problem of functional equations originated from a question of Ulam [19] concerning the stability of group homomorphisms. The functional equation f (x + y) = f (x) + f (y) is called the Cauchy equation. In particular, every solution of the Cauchy equation is said to be an additive mapping. Hyers [7] gave a first affirmative partial answer to the question of Ulam for Banach spaces. Hyers’ Theorem was generalized by Aoki [2] for additive mappings and by Rassias [12] for linear mappings by considering an unbounded Cauchy difference. A generalization of the Rassias theorem was obtained by G˘avruta [6] by replacing the unbounded Cauchy difference ( ) by a general control x+y function in the spirit of Rassias’ approach. The functional equation f 2 = 12 f (x) + 12 f (y) is called the Jensen equation. The functional equation f (x + y) + f (x − y) = 2f (x) + 2f (y) is called the quadratic functional equation. In particular, every solution of the quadratic functional equation is said to be a quadratic mapping. The stability of quadratic functional equation was proved by Skof [18] for mappings f : E1 → E2 , where E1 is a normed space and E2 is a Banach space. Cholewa [5] noticed that the theorem of Skof is(still )true (if the) relevant domain E1 is replaced by an Abelian group. The functional equation 2f x+y +2 x−y = f (x)+f (y) is called a Jensen type quadratic equation. The 2 2 stability problems of various functional equations have been extensively investigated by a number of authors (see [1, 3, 4, 10, 11, 13, 14, 15, 16, 17, 20, 21]). In Section 2, we solve the additive-quadratic ρ-functional equation (0.1) and prove the HyersUlam stability of the additive-quadratic ρ-functional equation (0.1) in non-Archimedean Banach spaces. In Section 3, we solve the additive-quadratic ρ-functional equation (0.2) and prove the HyersUlam stability of the additive-quadratic ρ-functional equation (0.2) in non-Archimedean Banach spaces.

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Throughout this paper, assume that X is a non-Archimedean normed space and that Y is a non-Archimedean Banach space. Let |2| ̸= 1. 2. Additive-quadratic ρ-functional equation (0.1) in non-Archimedean normed spaces Throughout this section, assume that ρ is a fixed non-Archimedean number with |ρ| < 1. In this section, we solve the additive-quadratic ρ-functional equation (0.1) in non-Archimedean normed spaces. Lemma 2.1. (i) If an odd mapping f : X → Y satisfies M1 f (x, y) = ρM2 f (x, y)

(2.1)

for all x, y ∈ X, then f : X → Y is additive. (ii) If an even mapping f : X → Y satisfies (2.1), then f : X → Y is quadratic. Proof. (i) Assume that f : X → Y satisfies (2.1). Since f is an odd mapping, f (0) = 0. Letting y = x in (2.1), we get f (2x) − 2f (x) = 0 and so f (2x) = 2f (x) for all x ∈ X. Thus

( )

f

x 2

for all x ∈ X. It follows from (2.1) and (2.2) that

1 = f (x) 2

(

(

(2.2)

)

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

)

and so f (x + y) = f (x) + f (y) for all x, y ∈ X. (ii) Assume that f : X → Y satisfies (2.1). Letting x = y = 0 in (2.1), we get −f (0) = 2ρf (0). So f (0) = 0. Letting y = x in (2.1), we get 1 f (2x) − 2f (x) = 0 2 and so f (2x) = 4f (x) for all x ∈ X. Thus ( )

f

x 2

1 = f (x) 4

(2.3)

for all x ∈ X.

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It follows from (2.1) and (2.3) that 1 1 f (x + y) + f (x − y) − f (x) − f (y) 2 2 ( ( ) ( ) ) x+y x−y = ρ 2f + 2f − f (x) − f (y) 2 2 ) ( 1 1 f (x + y) + f (x − y) − f (x) − f (y) =ρ 2 2 and so f (x + y) + f (x − y) = 2f (x) + 2f (y) for all x, y ∈ X.



We prove the Hyers-Ulam stability of the additive-quadratic ρ-functional equation (2.1) in nonArchimedean Banach spaces for an odd mapping case. Theorem 2.2. Let r < 1 and θ be nonnegative real numbers and let f : X → Y be an odd mapping such that ∥M1 f (x, y) − ρM2 f (x, y)∥ ≤ θ(∥x∥r + ∥y∥r )

(2.4)

for all x, y ∈ X. Then there exists a unique additive mapping A : X → Y such that 2θ ∥x∥r |2|r

(2.5)

∥f (2x) − 2f (x)∥ ≤ 2θ∥x∥r

(2.6)

∥f (x) − A(x)∥ ≤ for all x ∈ X. Proof. Since f is an odd mapping, f (0) = 0. Letting y = x in (2.4), we get

( ) for all x ∈ X. So f (x) − 2f x2 ≤ |2|2r θ∥x∥r for all x ∈ X. Hence

( ) ( )

l x

2 f x − 2m f

2l 2m

{ ( ) ( ) ) ( ) } (

l

m−1 x x x x l+1 m



≤ max 2 f −2 f −2 f , · · · , 2 f 2l 2l+1 2m−1 2m

( )

( ( ) ) ( ) } {

x x x x m−1 l

− 2f = max |2| f , · · · , |2|

f 2m−1 − 2f 2m 2l 2l+1 { }

≤ max

|2|m−1 |2|l , · · · , |2|rl+r |2|r(m−1)+r

2θ∥x∥r =

2θ

|2|(r−1)l+r

(2.7)

∥x∥r

for all nonnegative integers m and l with m > l and all x ∈ X. It follows from (2.7) that the sequence {2n f ( 2xn )} is a Cauchy sequence for all x ∈ X. Since Y is complete, the sequence {2n f ( 2xn )} converges. So one can define the mapping A : X → Y by x A(x) := lim 2n f ( n ) n→∞ 2 for all x ∈ X. Moreover, letting l = 0 and passing the limit m → ∞ in (2.7), we get (2.5).

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It follows from (2.4) that ∥M1 A(x, y) − ρM2 A(x, y)∥ = ≤



lim |2|n

M1 f n→∞

(

x y , 2n 2n

)

(

− ρM2 f

)

x y

, 2n 2n

|2|n θ (∥x∥r + ∥y∥r ) = 0 n→∞ |2|nr lim

for all x, y ∈ X. So M1 A(x, y) = ρM2 A(x, y) for all x, y ∈ X. By Lemma 2.1, the mapping A : X → Y is additive . Now, let T : X → Y be another additive mapping satisfying (2.5). Then we have

( ) ( )

q x x q

∥A(x) − T (x)∥ = 2 A q − 2 T 2 2q { ( ) ( ) } ( ) ( )

q

x x 2θ q

, 2q T x − 2q f x ≤ ≤ max 2 A q − 2 f ∥x∥r ,

q q q (r−1)q+r 2 2 2 2 |2| which tends to zero as q → ∞ for all x ∈ X. So we can conclude that A(x) = T (x) for all x ∈ X. This proves the uniqueness of h. Thus the mapping A : X → Y is a unique additive mapping satisfying (2.5).  Theorem 2.3. Let r > 1 and θ be nonnegative real numbers and let f : X → Y be an odd mapping satisfying (2.4). Then there exists a unique additive mapping A : X → Y such that 2θ ∥x∥r (2.8) ∥f (x) − A(x)∥ ≤ |2| for all x ∈ X. Proof. It follows from (2.6) that



f (x) − 1 f (2x) ≤ 2 θ∥x∥r

2 |2|

for all x ∈ X. Hence

1 ( l )

f 2 x − 1 f (2m x)

2l

2m

} { ( ) ( )

1 ( l )

1

1 1 l+1 m−1 m

≤ max l f 2 x − l+1 f 2 x , · · · , m−1 f 2 x − m f (2 x) 2 2 2 2

( )

(

} { ( ) ) 1

1 1 1 l l+1 m−1 m

= max f 2 x − f 2 x , · · · , m−1 f 2 x − f (2 x)

l |2| 2 |2| 2 {

≤ max

|2|r(m−1) |2|lr , · · · , |2|l+1 |2|(m−1)+1

}

2θ∥x∥r =

2θ |2|(1−r)l+1

for all nonnegative integers m and l with m > l and all x ∈ X. The rest of the proof is similar to the proof of Theorem 2.2.

∥x∥r 

Now, we prove the Hyers-Ulam stability of the additive-quadratic ρ-functional equation (2.1) in non-Archimedean Banach spaces for an even mapping case. Theorem 2.4. Let r < 2 and θ be nonnegative real numbers and let f : X → Y be an even mapping satisfying (2.4). Then there exists a unique quadratic mapping Q : X → Y such that ∥f (x) − Q(x)∥ ≤

|2| 2θ∥x∥r |2|r

832

(2.9)

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for all x ∈ X. Proof. Letting x = y = 0 in (2.4), we get −f (0) = 2ρf (0). So f (0) = 0. Letting y = x in (2.4), we get

1

f (2x) − 2f (x) ≤ 2θ∥x∥r

2

(2.10)

( x )

≤ |2|r 2θ∥x∥r for all x ∈ X. Hence 2 |2|

( ) ( )

l

x x

4 f

− 4m f (2.11)

2l 2m

{ ( ) ) ( ) } ( ) (

l

x x x x l+1 m

, · · · , 4m−1 f

≤ max − 4 f − 4 f 4 f

2l 2l+1 2m−1 2m

(

( ) ( ) ) ( ) } {

x x x x

, · · · , |4|m−1 f

− 4f − 4f = max |4|l f

2l 2l+1 2m−1 2m { }

for all x ∈ X. So f (x) − 4f

≤ max

|4|m−1 |4|l , · · · , |2|rl |2|r(m−1)

2θ |2| |2| 2θ∥x∥r = (r−2)l r ∥x∥r r |2| |2| |2|

for all nonnegative integers m and l with m > l and all x ∈ X. It follows from (2.11) that the sequence {4n f ( 2xn )} is a Cauchy sequence for all x ∈ X. Since Y is complete, the sequence {4n f ( 2xn )} converges. So one can define the mapping Q : X → Y by x Q(x) := lim 4n f ( n ) n→∞ 2 for all x ∈ X. Moreover, letting l = 0 and passing the limit m → ∞ in (2.11), we get (2.9). It follows from (2.4) that

( ) ( )

x y x y n

∥M1 Q(x, y) − ρM2 Q(x, y)∥ = lim |4| M1 f , − ρM2 f , n→∞ 2n 2n 2n 2n |4|n θ (∥x∥r + ∥y∥r ) = 0 ≤ lim n→∞ |2|nr for all x, y ∈ X. So M1 Q(x, y) = ρM2 Q(x, y) for all x, y ∈ X. By Lemma 2.1, the mapping h : X → Y is quadratic. Now, let T : X → Y be another quadratic mapping satisfying (2.9). Then we have

( ) ( )

q x x q

∥Q(x) − T (x)∥ = 4 Q − 4 T

q 2 2q { ( ) ( ) ( ) ( ) }

q |2| x x q

, 4q T x − 4q f x ≤ − 4 f ≤ max 4 Q 2θ∥x∥r ,



q q q q (r−2)q+r 2 2 2 2 |2| which tends to zero as q → ∞ for all x ∈ X. So we can conclude that Q(x) = T (x) for all x ∈ X. This proves the uniqueness of Q. Thus the mapping Q : X → Y is a unique quadratic mapping satisfying (2.9).  Theorem 2.5. Let r > 2 and θ be positive real numbers, and let f : X → Y be an even mapping satisfying (2.4). Then there exists a unique quadratic mapping Q : X → Y such that 2θ ∥x∥r ∥f (x) − Q(x)∥ ≤ |2| for all x ∈ X.

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Proof. It follows from (2.10) that



f (x) − 1 f (2x) ≤ 2θ ∥x∥r

4 |2|

for all x ∈ X. The rest of the proof is similar to the proof of Theorem 2.4.



3. Additive-quadratic ρ-functional equation (0.2) Throughout this section, assume that ρ is a fixed non-Archimedean number with |ρ| < |2|. In this section, we solve the additive-quadratic ρ-functional equation (0.2) in non-Archimedean normed spaces. Lemma 3.1. (i) If an odd mapping f : X → Y satisfies M2 f (x, y) = ρM1 f (x, y)

(3.1)

for all x, y ∈ X, then f : X → Y is additive. (ii) If an even mapping f : X → Y satisfies f (0) = 0 and (3.1), then f : X → Y is quadratic. Proof. (i) Assume that f : X → Y satisfies (3.1). Letting y = 0 in (3.1), we get ( ) x 2f − f (x) = 0 2

(3.2)

( )

and so f x2 = 12 f (x) for all x ∈ X. It follows from (3.1) and (3.2) that

(

)

x+y f (x + y) − f (x) − f (y) = 2f − f (x) − f (y) 2 = ρ(f (x + y) − f (x) − f (y)) and so f (x + y) = f (x) + f (y) for all x, y ∈ X. (ii) Assume that f : X → Y satisfies (3.1). Letting y = 0 in (3.1), we get ( ) x 4f − f (x) = 0 2

(3.3)

( )

and so f x2 = 14 f (x) for all x ∈ X. It follows from (3.1) and (3.3) that 1 1 f (x + y) + f (x − y) − f (x) − f (y) 2 2 ( ) ( ) x+y x−y = 2f + 2f − f (x) − f (y) 2 2 1 1 = ρ( f (x + y) + f (x − y) − f (x) − f (y)) 2 2 and so f (x + y) + f (x − y) = 2f (x) + 2f (y)

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for all x, y ∈ X.



We prove the Hyers-Ulam stability of the additive-quadratic ρ-functional equation (3.1) in nonArchimedean Banach spaces for an odd mapping case. Theorem 3.2. Let r < 1 and θ be nonnegative real numbers, and let f : X → Y be an odd mapping such that ∥M2 f (x, y) − ρM1 f (x, y)∥ ≤ θ(∥x∥r + ∥y∥r )

(3.4)

for all x, y ∈ X. Then there exists a unique additive mapping A : X → Y such that ∥f (x) − A(x)∥ ≤ θ∥x∥r

(3.5)

for all x ∈ X. Proof. Since f is an odd mapping, f (0) = 0. Letting y = 0 in (3.4), we get

( )



2f x − f (x) ≤ θ∥x∥r

2

(3.6)

for all x ∈ X. So

( ) ( )

l x

2 f x − 2m f

2l 2m

( ) ) ( ) } { ( ) (

l

x x x x l+1 m

, · · · , 2m−1 f

− 2 f − 2 f ≤ max 2 f

2l 2l+1 2m−1 2m

(

( ) ( ) ) ( ) } {

x x x x

, · · · , |2|m−1 f

− 2f − 2f = max |2|l f

2l 2l+1 2m−1 2m } {

≤ max

|2|m−1 |2|l , · · · , |2|rl |2|r(m−1)

θ∥x∥r =

θ

|2|(r−1)l

(3.7)

∥x∥r

for all nonnegative integers m and l with m > l and all x ∈ X. It follows from (3.7) that the sequence {2n f ( 2xn )} is a Cauchy sequence for all x ∈ X. Since Y is complete, the sequence {2n f ( 2xn )} converges. So one can define the mapping A : X → Y by x A(x) := lim 2n f ( n ) n→∞ 2 for all x ∈ X. Moreover, letting l = 0 and passing the limit m → ∞ in (3.7), we get (3.5). The rest of the proof is similar to the proof of Theorem 2.2.  Theorem 3.3. Let r > 1 and θ be positive real numbers, and let f : X → Y be an odd mapping satisfying (3.4). Then there exists a unique additive mapping A : X → Y such that ∥f (x) − A(x)∥ ≤

|2|r θ ∥x∥r |2|

(3.8)

for all x ∈ X. Proof. It follows from (3.6) that

r

f (x) − 1 f (2x) ≤ |2| θ ∥x∥r

2 |2|

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ADDITIVE-QUADRATIC ρ-FUNCTIONAL EQUATIONS

for all x ∈ X. Hence

1

f (2l x) − 1 f (2m x)

2l

2m

}

{ ( ) ( )

1 ( l )

1 1 1 l+1 m−1 m

x − m f (2 x) ≤ max l f 2 x − l+1 f 2 x , · · · , m−1 f 2 2 2 2 2

}

( )

( { ( ) ) 1

1 1 1 l+1 m l m−1

= max f 2 x − f 2 x , · · · , m−1 f 2 x − f (2 x)

l |2| 2 |2| 2 {

≤ max

|2|rl |2|r(m−1) , · · · , |2|l+1 |2|(m−1)+1

}

|2|r θ∥x∥r =

|2|r θ |2|(1−r)l+1

(3.9)

∥x∥r

for all nonnegative integers m and l with m > l and all x ∈ X. It follows from (3.9) that the sequence { 21n f (2n x)} is a Cauchy sequence for all x ∈ X. Since Y is complete, the sequence { 21n f (2n x)} converges. So one can define the mapping A : X → Y by A(x) := lim

1

n→∞ n

f (2n x)

for all x ∈ X. Moreover, letting l = 0 and passing the limit m → ∞ in (3.9), we get (3.8). The rest of the proof is similar to the proofs of Theorems 2.2 and 3.2.



Now, we prove the Hyers-Ulam stability of the additive-quadratic ρ-functional equation (3.1) in non-Archimedean Banach spaces for an even mapping case. Theorem 3.4. Let r < 2 and θ be nonnegative real numbers, and let f : X → Y be an even mapping satisfying (3.4). Then there exists a unique quadratic mapping Q : X → Y such that ∥f (x) − Q(x)∥ ≤ θ∥x∥r

(3.10)

for all x ∈ X. Proof. Letting x = y = 0 in (3.4), we get 2f (0) = ρf (0). So f (0) = 0. Letting y = 0 in (3.4), we get

( )

4f x − f (x) ≤ θ∥x∥r (3.11)

2 for all x ∈ X. So

( ) ( )

l x

4 f x − 4m f

(3.12)

2l 2m

( ) ) ( ) } { ( ) (

l

x x x x l+1 m

, · · · , 4m−1 f

− 4 f − 4 f ≤ max 4 f

2l 2l+1 2m−1 2m

(

( ) { ( ) ) ( ) }

x x x x

, · · · , |4|m−1 f

= max |4|l − 4f − 4f f

2l 2l+1 2m−1 2m { } |4|l |4|m−1 θ ≤ max , · · · , r(m−1) θ∥x∥r = (r−2)l ∥x∥r rl |2| |2| |2| for all nonnegative integers m and l with m > l and all x ∈ X. It follows from (3.12) that the sequence {4n f ( 2xn )} is a Cauchy sequence for all x ∈ X. Since Y is complete, the sequence {4n f ( 2xn )} converges. So one can define the mapping Q : X → Y by x Q(x) := lim 4n f ( n ) n→∞ 2 for all x ∈ X. Moreover, letting l = 0 and passing the limit m → ∞ in (3.12), we get (3.10).

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The rest of the proof is similar to the proof of Theorem 2.2.

Theorem 3.5. Let r > 2 and θ be positive real numbers, and let f : X → Y be an even mapping satisfying (3.4). Then there exists a unique quadratic mapping Q : X → Y such that ∥f (x) − Q(x)∥ ≤

|2|r θ ∥x∥r |4|

(3.13)

for all x ∈ X. Proof. It follows from (3.11) that

r

f (x) − 1 f (2x) ≤ |2| θ ∥x∥r

4 |4|

for all x ∈ X. Hence

1

f (2l x) − 1 f (2m x) (3.14)

4l

4m

}

{ ( ) ( )

1 ( l )

1 l+1

, · · · , 1 f 2m−1 x − 1 f (2m x) ≤ max f 2 x − f 2 x

4l

4l+1 4m−1 4m

( )

} { ( ) ( )

1 1

f 2l x − 1 f 2l+1 x , · · · ,

f 2m−1 x − 1 f (2m x) = max



|4|l 4 |4|m−1 4 {

≤ max

|2|rl |2|r(m−1) , · · · , |4|l+1 |4|(m−1)+1

}

|2|r θ∥x∥r =

|2|r θ |2|(2−r)l+2

∥x∥r

for all nonnegative integers m and l with m > l and all x ∈ X. It follows from (3.14) that the sequence { 41n f (2n x)} is a Cauchy sequence for all x ∈ X. Since Y is complete, the sequence { 41n f (2n x)} converges. So one can define the mapping Q : X → Y by 1 f (2n x) 4n for all x ∈ X. Moreover, letting l = 0 and passing the limit m → ∞ in (3.14), we get (3.13). The rest of the proof is similar to the proofs of Theorems 2.2 and 3.4. Q(x) := lim

n→∞



References [1] M. Adam, On the stability of some quadratic functional equation, J. Nonlinear Sci. Appl. 4 (2011), 50–59. [2] T. Aoki, On the stability of the linear transformation in Banach spaces, J. Math. Soc. Japan 2 (1950), 64–66. [3] L. C˘ adariu, L. G˘ avruta and P. G˘ avruta, On the stability of an affine functional equation, J. Nonlinear Sci. Appl. 6 (2013), 60–67. [4] A. Chahbi and N. Bounader, On the generalized stability of d’Alembert functional equation, J. Nonlinear Sci. Appl. 6 (2013), 198–204. [5] P. W. Cholewa, Remarks on the stability of functional equations, Aequationes Math. 27 (1984), 76–86. [6] P. Gˇ avruta, A generalization of the Hyers-Ulam-Rassias stability of approximately additive mappings, J. Math. Anal. Appl. 184 (1994), 431–43. [7] D. H. Hyers, On the stability of the linear functional equation, Proc. Natl. Acad. Sci. U.S.A. 27 (1941), 222–224. [8] M.S. Moslehian and Gh. Sadeghi, A Mazur-Ulam theorem in non-Archimedean normed spaces, Nonlinear Anal.– TMA 69 (2008), 3405–3408. [9] 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. [10] C. Park, K. Ghasemi, S. G. Ghaleh and S. Jang, Approximate n-Jordan ∗-homomorphisms in C ∗ -algebras, J. Comput. Anal. Appl. 15 (2013), 365-368. [11] C. Park, A. Najati and S. Jang, Fixed points and fuzzy stability of an additive-quadratic functional equation, J. Comput. Anal. Appl. 15 (2013), 452–462.

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[12] Th. M. Rassias, On the stability of the linear mapping in Banach spaces, Proc. Amer. Math. Soc. 72 (1978), 297–300. [13] S. Schin, D. Ki, J. Chang and M. Kim, Random stability of quadratic functional equations: a fixed point approach, J. Nonlinear Sci. Appl. 4 (2011), 37–49. [14] S. Shagholi, M. Bavand Savadkouhi and M. Eshaghi Gordji, Nearly ternary cubic homomorphism in ternary Fr´echet algebras, J. Comput. Anal. Appl. 13 (2011), 1106–1114. [15] S. Shagholi, M. Eshaghi Gordji and M. Bavand Savadkouhi, Stability of ternary quadratic derivation on ternary Banach algebras, J. Comput. Anal. Appl. 13 (2011), 1097–1105. [16] D. Shin, C. Park and Sh. Farhadabadi, On the superstability of ternary Jordan C ∗ -homomorphisms, J. Comput. Anal. Appl. 16 (2014), 964–973. [17] D. Shin, C. Park and Sh. Farhadabadi, Stability and superstability of J ∗ -homomorphisms and J ∗ -derivations for a generalized Cauchy-Jensen equation, J. Comput. Anal. Appl. 17 (2014), 125–134. [18] F. Skof, Propriet locali e approssimazione di operatori, Rend. Sem. Mat. Fis. Milano 53 (1983), 113–129. [19] S. M. Ulam, A Collection of the Mathematical Problems, Interscience Publ. New York, 1960. [20] C. Zaharia, On the probabilistic stability of the monomial functional equation, J. Nonlinear Sci. Appl. 6 (2013), 51–59. [21] S. Zolfaghari, Approximation of mixed type functional equations in p-Banach spaces, J. Nonlinear Sci. Appl. 3 (2010), 110–122. 1

Research Institute for Natural Sciences, Hanyang University, Seoul 04763, Republic of Korea E-mail address: [email protected] 2

Department of Mathematics, Daejin University, Kyunggi 11159, Republic of Korea E-mail address: [email protected] 3

Department of Mathematics, University of Seoul, Seoul 02504, Republic of Korea E-mail address: [email protected]

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Complete asymptotic expansions for the genuiune Bernstein-Durrmeyer operator 1 Chungou Zhang School of Mathematical Sciences, Capital Normal University, Beijing 100048, People’s Republic of China E-mail: [email protected]

Abstract In this paper, we discuss properties of asymptotic approximation of the genuiune Bernstein-Durrmeyer operator, for which establish a complete asymptotic expansion formula of approximation and present the saturation theorems as an application. Key words The genuiune Bernstein-Durrmeyer operator, Jacobi weight, Complete asymptotic expansion, Approximation, Saturation.

1. INTRODUCTION The Bernstein-Durrmeyer operator with weights, which is one of the objects of interest in approximation theory of operators and play as an important role in learning theory, is defined as follows Mnω (f ; x) =

n X (f, bn,k )ω

(e0 , bn,k )ω

k=0

bn,k (x)

where ek = ek (x) = xk , k = 0, 1, · · ·, Bernstein basis functions à !

bn,k (x) =

n k x (1 − x)n−k k

x ∈ [0, 1], k = 0, 1, · · · , n

and inner product weighted ω defined by Z

(f, g)ω = 1

Foundation item:

0

1

f (t)g(t)ω(t)dt.

Supported by the Natural Science Foundation of China ( 11371253,

11671271 ) and by Beijing Municipal Education Commission science and technology plan projects ( KM201510028003 )

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A popular case is that the weight ω is taken as the Jacobi weight ω(t) = tα (1 − (α,β)

t)β , α, β > −1, at this time we denote this operator Mnω (f ) by Mn

(f ). Because

when α = −1, β = −1, the inner products (e0 , bn,k )ω are no definition at k = 0 and n, W. Chen ( [1] 1987), T.N.T. Goodman and A. Sharma ([2] 1987) independently modified the operator into Mn∗ (f ; x) = Lf + (n − 1)

n−1 X

(f − Lf, bn−2,k−1 )bk,n (x)

k=1

where Lf (x) = (1 − x)f (0) + xf (1) and

Z

(f, g) =

1

0

f (x)g(x)dx.

The operator Mn∗ f is also called as the genuiune Bernstein-Durrmeyer operator(see [3] or [4]) and can be rewritten into the following form Mn∗ (f ; x) = f (0)(1 − x)n + f (1)xn + (n − 1)

n−1 X

(f, bn−2,k−1 )bn,k (x),

k=1

which is a particular case of those operators introduced by D. C´ardenas-Morales and V. Gupta in [5] Mn,α,β (f ; x) =

X

k bn,k (x)f ( ) + (n − α + 1) n

k∈ln n−α+β X Z 1

·

k=1

0

f (t) bn−α,k−β (t)dt bn,k (x)

where ln ⊂ {0, 1, · · · , n}. Obviously when α = 2, β = 1 and ln = {0, n}, Mn,α,β (f ) is Mn∗ (f ), for which there has been extensive research (see [3,4] and [6-10]). In the next sect, we will establish a complete asymptotic expansion formula for the operator Mn∗ (f ), and in the section 3, we will present the saturation theorems of approximation as a application of the asymptotic expansion formula.

2. Complete Asymptotic expansions From [5], one can find that Mn∗ (e0 ; x) = Mn∗ (1, x) = 1; Mn∗ (e1 ; x) = Mn∗ (t; x) = x; 2

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2x(1 − x) . n+1 For higher order moments of the operator Mn∗ (f ), we have the following result. Mn∗ (e2 ; x) = Mn∗ (t2 , x) = x2 +

Lemma 2.1 For any natural numbers n, m, there holds that Mn∗ (em ; x) = x(1 − x)

m X k=0

à !

i(k) m 1 h m−1 , x (1 − x)k−1 k nk

where the up factorial is defined by ¯

¯

nk = n(n + 1) · · · (n + k − 1), n0 = 1 and the fall factorial, which to be used in the following proof, done by nk = n(n − 1) . . . (n − k + 1), n0 = 1. Proof Since when n, m ≤ 1, the conclusion is true obviously, we only need to consider the case n, m ≥ 2. At this moment, we have Mn∗ (em ; x) = (n − 1)

n−1 X

(tm , bn−2,i−1 )bn,i (x) + xn

i=1

à !

X (m + i − 1)! n (n − 1)! n−1 = xi (1 − x)n−i + xn (m + n − 1)! i=1 (i − 1)! i à !

= = = =

i X n (n − 1)! h dm n−1 m+i−1 n−i x y + xn y=1−x (m + n − 1)! dxm i=1 i i (n − 1)! dm h m−1 x (x + y)n m y=1−x (m + n − 1)! dx à ! m h i X m (m − 1)! v−1 n! (n − 1)! x (x + y)n−v y=1−x (m + n − 1)! v=1 v (v − 1)! (n − v)! à ! m 1 X m (m − 1)m−v nv xv . nm v=1 v

Using the Vandermonde formula we get v

n =

v X k=0

à ! v ³

k

´k

n − (1 − m)

(1 − m)v−k ,

and notice that when v = 0, (m − 1)m−v = 0, therefore it follows that Mn∗ (em ; x) 3

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à !

à !

v m ´k X 1 X v ³ m = m n − (1 − m) (1 − m)v−k (m − 1)m−v xv n v=0 v k k=0

=

m X

k 1 X 1 m! (k + m − v − 2)k−v (m − 1)v xm−v . (−1)k−v k (m − k)! v! (k − v)! n v=0 k=0

Again using the Vandermonde formula we have Ã

!

k X (k + m − v − 2)k−v k−v (m − 1)v = (m − 1)v (m − 1 − v)i (k − 1)k−i , (k − v)! i − v i=v

hence the second sum in the previous equation can be turned into k X

(−1)k−v

v=0

= m! = m!

k m! m−v X (m − 1)i (k − 1)k−i x · v! (i − v)! (k − i)! i=v

k X (m − 1)i (k − 1)k−i i=0 k X i=0

i!

(k − i)! 1)i

(m − i!

·

i X

à !

k−v

(−1)

v=0

i m−v x v

(k − 1)k−i (−1)k−i xm−i (1 − x)i (k − i)!

à !

k i m! X k di h m−1 i dk−i h = x(1 − x) x (1 − x)k−1 i k−i k! i=0 i dx dx h i (k) m! m−1 . x (1 − x)k−1 = x(1 − x) k! That completes the proof of Lemma 2.1.

If denoting ψxs (t) = (t − x)s , then we have the following assertion. Lemma 2.2 For arbitrary natural number s and x ∈ [0, 1], there holds Mn∗ (ψxs ; x)

Ã

s X

== x(1 − x)

k=[ s+1 ] 2

Proof On account of Mn∗ (ψxs ; x)

=

s X m=0

!

h i(2k−s) s! 1 k xk−1 (1 − x)k−1 k! nk s − k

à !

s (−x)s−m Mn∗ (em ; x), m

by Lemma 2.1 we see easily à !

à !

m i(k) X s m 1 h m−1 x (1 − x)k−1 = x(1 − x) (−x)s−m · m k nk k=0 Ã ! ¡ s ¢ k m=0 s s−k h X s−k i¯ X s−k−m m−1+k k−1 ¯ k d (−x) y (1 − y) = x(1 − x) ¯ k dy k y=x m m=0 k=0 ¡n ¢ s s k h i¯ X ¯ k d = x(1 − x) y k−1 (1 − y)k−1 (y − x)s−k ¯ , k dy k y=x n k=0

Mn∗ (ψxs ; x)

s X

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and recall that i dk h k−1 k−1 s−k y (1 − y) (y − x) k y=x dy Ã ! 2k−s i h  k d  k−1 k−1  (s − k)! , s ≤ 2k y (1 − y) y=x s − k dy 2k−s =    0, s > 2k.

The proof is completed. Remark By Lemma 2.2, we immediately obtain Mn∗ (ψxs ; x) = O(n−[

s+1 ] 2

).

For arbitrary natural number q and x ∈ I, let f ∈ K[q; x] denote the class of functions f ∈ B(I) ( space of bounded functions on I) which q times differentiable at x, then we have the following theorems of approximation. Lemma 2.3

[11]

Let q be arbitrary natural number, x ∈ I and An : B(I) →

C(I)(space of continuous functions on I) be a sequence of positive linear operators such that An (ψxs ; x) = O(n−[

s+1 ] 2

) (n → ∞)

(s = 0, 1, · · · , 2q + 2),

then for arbitrary f ∈ K[2q; x], there holds An (f ; x) =

2q X f (s) (x) s=0

s!

An (ψxs ; x) + o(n−q ) (n → ∞).

In particular, if f (2q+2) (x) exists, then the o(n−q ) can be replaced by O(n−q−1 ) Theorem 2.1 For arbitrary natural number q, x ∈ [0.1] and f ∈ K[2q; x], there holds that Mn∗ (f ; x) =

q X

1

k k=0 k!n

h

i(k)

x(1 − x) xk−1 (1 − x)k−1 f (k) (x)

+ o(n(−q)

Proof By Lemma 2.2 and Lemma 2.3, we have Mn∗ (f ; x) =

2q X f (s) (x) s=0

·

s!

s X

k=[ s+1 ] 2

x(1 − x) Ã

!

h i(2k−s) s! 1 k xk−1 (1 − x)k−1 + o(n−q ). k! nk s − k

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Interchanging the two sums and substituting s + k into s, it is followed by (Mn∗ (f ; x) = x(1 − x) ·

k X

q X

à !k=0

1 k!nk

i(k−s) k (s+k) h k−1 + o(n−q ). f (x) x (1 − x)k−1 s

s=0

With the Leibniz formula, it becomes Mn∗ (f ; x)

= x(1 − x)

q X

1

k k=0 k!n

· [xk−1 (1 − x)k−1 f (k) (x)](k) + o(n−q ).

That is the proof of Theorem 2.1. Remark 1 If taking q = 1, then we can get so-call the Vonorovskaja type asymptotic expansion formula by Theorem 2.1 as below lim n(Mn∗ (f ; x) − f (x)) = x(1 − x)f 00 (x)

n−→∞

Remark 2 Theorem 2.1 shows that the complete asymptotic expansion formula (α,β)

of the operator Mn∗ (f ) coincides with which of the operator Mn −1. This seems to be one of reasons why

Mn∗ (f )

(f ) at α = −1, β =

is called as the genuiune Bernstein-

Durrmeyer operator.

3. Saturations of Approximation As an application of the asymptotic formula, in this section we will present the saturation theorems of approximation for the operator Mn∗ f . Along with those denotations and signs in [12], denote the space of all continuous functions on [0, 1] by C[0, 1], for v(x), ω(x) ∈ C[0, 1] and strictly positive, that is v(x), ω(x) > 0, x ∈ (0, 1), let

Z

ϕ(x) =

0

x

Z

v(t)dt,

φ(x) =

Z

ψ(x) =

x

0

0

x

ω(t)dt

ψ(t)v(t)dt.

For a function f ∈ C[0, 1], we define that Df (x) = Dψ Dϕ f (x) =

1 h f 0 (x) i0 ω(x) v(x)

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where Dϕ f (t) =

f 0 (t) . v(t)

Suppose that Ln : C[0, 1] → C[0, 1] is a sequence of positive linear operators, {λn } is a sequence of positive numbers such as λn →∝ (n →∝) and ρ(x) ∈ C[0, 1] is strictly positive. We say that {Ln } satisfy with the Voronovskaya condition if and only if for Df (x), {λn } and ρ(x) as above, there holds lim λn {Ln (f ; x) − f (x)} = ρ(x)Df (x),

n−→∝

x ∈ [0, 1]

Lemma 3.1[12] Suppose that Ln : C[0.1] → C[0.1] is a sequence of positive linear operators and satisfy the Voronovskaya condition and G ∈ C[0, 1]. If for all x ∈ (0, 1)

¯ ¯

¯ ¯

λn ¯Ln (G; x) − G(x)¯ ≤ M ρ(x) + ox (1), for some positive constant M , than there exists Dϕ G(x) ∈ C[0, 1] and ¯ ¯ ¯ ¯ ¯ ¯ ¯ ¯ ¯Dϕ G(y) − Dϕ G(x)¯ ≤ M ¯ψ(y) − ψ(x)¯ x, y ∈ [0, 1],

and vice versa. Remark 1 of Theorem 2.1 shows that the operator Mn∗ f satisfies the Voronovskaya condition and v(x) = ω(x) = 1, ρ(x) = x(1 − x), λn = n Df (x) = Dψ Dϕ f (x) = f 00 (x), thus from Lemma 3.1 we can get the following pointwise saturation theorem without any difficult. Theorem 3.1 (1) Let f ∈ C[0, 1], then for all x ∈ [0, 1], lim n{Mn∗ (f ; x) − n−→∝

f (x)} = 0 if and only if f (x) = A + Bx; (2) Let f ∈ C[0, 1], than for all x ∈ [0, 1] ¯ ¯

¯ ¯

n¯Mn∗ (f ; x) − f (x)¯ ≤ M ρ(x) + ox (1) if and only if for any x, y ∈ [0, 1], ¯ ¯ ¯ ¯ ¯ 0 ¯ ¯ ¯ ¯f (y) − f 0 (x)¯ ≤ M ¯y − x¯.

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For the operator Mn∗ (f ), we also establish the following saturation theorem in LP . Since it is the same as the approach taken for the Kantorovich operator in [13] and [14], here the process is omitted. Let f ∈ Lp [0, 1], (p ≥ 1) denote |f |p Lebesgue integrable on [0, 1],  n  h|h ∈ Lp [0, 1], h(0) = h(1) = 0 & h0 ∈ Lp [0, 1]}, p > 1 Up =  {h|h ∈ L [0, 1], h(0) = h(1) = 0 & h ∈ BV [0, 1]}, p = 1. p

and

n

Sp = f | f ∈ Lp [0, 1], ∃h ∈ Up , ξ ∈ (0, 1)and constants c, d such as f (x) = c + dx +

Rx Ru ξ

(

h(t) ξ t(1−t) dt)du

o

.

Theorem 3.2 Suppose f ∈ Lp [0, 1](p ≥ 1), then (i) k Mn∗ (f ) − f kp = O( n1 ), if and only if f ∈ Sp (p ≥ 1) (ii) k Mn∗ (f ) − f kp = o( n1 ), if and only if f (x) is a linear function. References [1] W. Chen, On the modified Bernstein-Durrmeyer operators, Report of the Fifth Chinese Conference on Approximation Theory, Zhen Zhou, China 1987. [2] T. N. T. Goodman, A. Sharma, A modified Bernstein-Schoenberg operator, in: Proc. of the Conference on Constructive Theory of Functions, Varna 1987 (ed. by Sendov et al.), Sofia: Publ. House Bulg. Acad. of Sci. 1988, 166-173. [3] Heiner Gonska, Daniela Kacso and Ioan Rasa, on genuine Bernstein-Durrmeyer operators, Result Math. 50(2007), 213-225. [4] Heiner Gonska, Ioan Rasa and Elena-Dorina Stanila, The Eigenstructure of Operators Linking the Bernstein and the Genuine Bernstein-Durrmeyer operators, Mediterr. J. Math. 11 (2014), 561-576. [5] Daniel Crdenas-Morales, Vijay Gupta, Two families of Bernstein-Durrmeyer type operators, Applied Mathematics and Computation 248(2014), 342-353. [6] K. G. Ivanov, P. E. Parvanov, Weighted Approximation by the GoodmanSharma Operators, East J. Approx. 15(4)(2009), 473-486. [7] J. A. Adell, J. de la Cal, Bernstein-Durrmeyer operators, Comput. Math. Appl. 30(1995), 1-14. [8] Z. Finta, Direct and Converse Theorems for Generalized Bernstein-type Operators, Serdica Math. J. 30(2004), 33-42. 8

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[9] D. Crdenas-Moralesa, P. Garrancho, I. Rasa, Approximation properties of Bernstein-Durrmeyer type operators, Applied Mathematics and Computation 232 (2014), 1-8. [10] Germain E. Randriambelosoa, On a Modified Durrmeyer-Bernstein Operator and Applications, Applied Mathematics Research express 4(2005), 169-182. [11] P. C. Sikkema, On some linear positive operators, Indag. Math. 32(1970), 327-337. [12] H. Berens, Pointwise Saturation of Positive Operators, J. A. T. 6(1972), 135-146. [13] V. Maier, The L1 saturation class of the Kantorovich operator, J. A. T. 22(3)(1978), 223-232. [14] S. D. Riemenschneider, The Lp -saturation of the Bernstein-Kantorovich polynomials, J. A. T. 23(2)(1978), 158-162.

9

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J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 24, NO.5, 2018, COPYRIGHT 2018 EUDOXUS PRESS, LLC

IMPULSIVE PERIODIC SOLUTIONS FOR A SINGULAR DAMPED DIFFERENTIAL EQUATION VIA VARIATIONAL METHODS SHENGJUN LI

1,2 ,XIANHUA

TANG 2 , HUXIAO LUO

2

Abstract. In this paper, we study impulsive periodic solutions for second order non-autonomous singular damped differential equations. The proof of the main result relies on a variational approach on mountain pass theorem, together with a truncation technique.

1. Introduction In this work, we are concerned with the existence of periodic solutions for the following second order non-autonomous singular damped problems  u00 + a(t)u0 − b(t) uα = g(t), a.e. t ∈ (0, T ), (1.1) u(0) = u(T ), u0 (0) = u0 (T ), under the impulse conditions 0 − ∆u0 (tj ) = u0 (t+ j ) − u (tj ) = Ij (u(tj )), j = 1, 2, . . . , p − 1, RT 0 where u0 (t± j ) = lim± u (t), α > 1, a, g ∈ C(R/T Z, R) with 0 a(t)dt = 0, and

(1.2)

t→tj

b ∈ C(R/T Z, (0, ∞)), tj for j = 1, 2, . . . , p − 1, are the instants when the impulses occur and 0 = t0 < t1 < t2 < . . . < tp−1 < tp = T, Ij : R → R(j = 1, 2, . . . , p − 1) are continuous. Impulsive effects occur widely in many evolution processes in which their states are changed abruptly at certain moments of time. In recent years, second-order differential boundary value problems with impulses have been studied extensively in the literature [1, 3, 11, 12, 14, 15, 16, 17, 18]. In [18], Tian and Ge studied the existence of solutions for impulsive differential equations:   −(ρ(t)φp (u0 (t)))0 + s(t)φp (u(t)) = f (t, u(t)), t 6= tj a.e. t ∈ [a, b], −∆(ρ(tj )φp (u0 (tj )))0 = Ij (u(tj )), j = 1, 2, . . . , l,  αu0 (a) + βu(a) = A, γu(b) + σu0 (b) = B, by using a variational method. Later, Nieto and O’Regan [12] further developed the variational framework for impulsive problems and established existence results for the following impulsive differential equations with Dirichlet boundary conditions:   −u00 + λu(t) = f (t, u(t)), t 6= tj a.e. t ∈ (0, T ), ∆(u0 (tj )) = Ij (u(tj )), j = 1, 2, . . . , l,  u(0) = u(T ) = 0, 2010 Mathematics Subject Classification. 34C25. Key words and phrases. Periodic solution; Impulse; Singular differential equation; Mountain pass theorem. 1

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J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 24, NO.5, 2018, COPYRIGHT 2018 EUDOXUS PRESS, LLC

2

SHENGJUN LI,

XIANHUA TANG

AND

HUXIAO LUO

From then on, the variational method has been a powerful tool in the study of impulsive differential equations. On the other hand, singular problems without impulse effects have been investigated extensively in the literature. Usually, in the literature, the proof is based on variational methods [10], or topological methods[2, 4, 6, 7, 8, 9], which were started with the pioneering paper of Lazer and Solimini [5]. In 1987, Lazer and Solimini [5] considered a the second order singular problem u00 (t) −

(1.3)

1 = e(t), uα (t)

t ∈ (0, T ).

By using the method of upper and lower solutions, they obtained a famous sufficient and necessary condition on positive T -periodic solution for Problem (1.3) as follows Theorem 1.1[5] Assume that e ∈ L1 ([0, T ], R) is T -periodic. Then Problem RT (1.3) has a positive T -periodic weak solution if and only if 0 e(t)dt < 0. Motivated by the above fact, in the present paper we shall consider Problem (1.1) with impulsive effects, In general cases, it is impossible to apply variational RT methods to Eq.(1.1) when 0 a(t)dt > 0. In this paper, using a variant of the RT mountain pass theorem, we consider the case 0 a(t)dt = 0, on an appropriate Sobolev space, we establish the corresponding variational framework of periodic solutions to guarantee the existence of at least one nontrivial solution of Eq.(1.1). In order to state our main result, we need the following assumptions: RT (H1 ) a ∈ C(R/T Z) with 0 a(t)dt = 0; (H2 ) b ∈ C(R/T Z, (0, ∞)) is T −periodic and b0 (t) ≥ 0 for all t ∈ [0, T ]; RT (H3 ) g ∈ L2 ([0, T ], R) is T −periodic and 0 g(t)dt < 0; (H4 ) There exist two constants m, M such that for any t ∈ R, m ≤ Ij (t) ≤ M, j = 1, 2, . . . , p − 1, RT 1 g(t)dt; where m < 0 and 0 ≤ M < − p−1 0 (H5 ) For any t ∈ R, Z t Ij (s)ds ≥ 0, j = 1, 2, . . . , p − 1. 0

Theorem 1.1 Assume that (H1 ) − (H5 ) are satisfied. Then problem (1.1)-(1.2) has at least one solution. The rest of this paper is organized as follows. In Section 2, some preliminary results will be given. In Section 3, by the use of variational method, we will state and prove the main results. 2. preliminaries In this section, we present some results which will be applied in Sections 3. Let HT1 = {u : [0, T ] → R | u is absolutely continuous, u(0) = u(T ) and u0 ∈ L2 ([0, T ], R)} with the inner product Z T Z (u, v) = u(t)v(t)dt + 0

T

u0 (t)v 0 (t)dt,

∀u, v ∈ HT1 .

0

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IMPULSIVE PERIODIC SOLUTIONS FOR A SINGULAR DAMPED DIFFERENTIAL EQUATION 3

The corresponding norm is defined by Z kukHT1 =

T

|u(t)|2 dt +

Z

! 21

T

|u0 (t)|2 dt)

, ∀u ∈ HT1 .

0

0

Then H1T is a Banach space. If u ∈ HT1 , then u is absolutely continuous and u0 ∈ L2 ([0, T ], R). In this case, ∆u0 (t) = u0 (t+ ) − u0 (t− ) = 0 is not necessarily valid for every t ∈ (0, T ) and the derivative u0 may exist some discontinuities. It may lead to impulse effects. From (1.1), we get   h i0 A(t) 0 A(t) b(t) (2.1) − e + g(t) = 0, u (t) + e uα Rt where A(t) = 0 a(s)ds. Following the ideas of [12], take v ∈ HT1 and multiply the two sides of (2.1) by v and integrate from 0 to T , so we have   Z T h Z T i0 b(t) (2.2) − eA(t) u0 (t) v(t)dt + + g(t) v(t)dt = 0 eA(t) uα 0 0 Note that, since u0 (0) = u0 (T ), one has Z Th i0 eA(t) u0 (t) v(t)dt =

0 p−1 X Z tj+1 j=0

=

p−1 X

h i0 eA(t) u0 (t) v(t)dt

tj p−1   X − + 0 + eA(t) u0 (t− )v(t ) − u (t )v(t ) − j+1 j+1 j+1 j+1

j=0

j=0

= eA(T ) u0 (T )v(T ) − eA(0) u0 (0)v(0) −

p−1 X

= −eA(t)

Z Ij (u(tj ))v(tj ) −

j=1

tj+1

eA(t) u0 (t)v 0 (t)dt

tj

eA(t) ∆u0 (tj )v(tj ) −

j=0 p−1 X

Z

Z

T

eA(t) u0 (t)v 0 (t)dt

0

T

eA(t) u0 (t)v 0 (t)dt.

0

Combining with (2.2), we get   Z T p−1 X b(t) A(t) 0 0 e u (t)v (t) + α v(t)dt + g(t)v(t) dt + eA(t) Ij (u(tj ))v(tj ) = 0. u 0 j=1 As a result we introduce the following concept of a weak solution for problem (1.1)(1.2). Definition 2.1 We say that a function u ∈ HT1 is a weak solution of problem (1.1)-(1.2) if   Z T p−1 X b(t) eA(t) u0 (t)v 0 (t) + α v(t)dt + g(t)v(t) dt + eA(t) Ij (u(tj ))v(tj ) = 0 u 0 j=1 holds for any v ∈ HT1 .

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J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 24, NO.5, 2018, COPYRIGHT 2018 EUDOXUS PRESS, LLC

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SHENGJUN LI,

XIANHUA TANG

AND

HUXIAO LUO

Define the functional Φ : HT1 → R by " # Z T Z u(t) p−1 Z u(tj ) X 1 A(t) 1 0 A(t) 2 Φ(u) := e Ij (s)ds ds + g(t)u(t) dt + e |u (t)| + b(t) 2 sα 0 1 j=1 0 for every u ∈ HT1 . Under the conditions of Theorem 1.2, it is easy to verify that Φ is well defined on HT1 , continuously differentiable and weakly lower semi-continuous. Moreover, the critical points of Φ are the weak solutions of problem (1.1)-(1.2). In next section, the following version of the mountain pass theorem will be used in our argument. Theorem 2.2 [13] Let X be a Banach space and let ϕ ∈ C(X, R). Assume that there exist x0 , x1 ∈ X and a bounded open neighborhood Ω of x0 such that ¯ and x1 ∈ X/Ω max{ϕ(x0 ), ϕ(x1 )} < inf ϕ(x). x∈∂Ω

Let Γ = {h ∈ C([0, 1], X) : h(0) = x0 , h(1) = x1 } and c = inf max ϕ(h(s)). h∈Γ s∈[0,1]

If ϕ satisfies the (P S)-condition, i.e., a sequence {un } in X satisfying ϕ(un ) is bounded and ϕ0 (un ) → 0 as n → ∞ has a convergent subsequence, then c is a critical value of ϕ and c > max{ϕ(x0 ), ϕ(x1 )}. 3. Main results In order to study problem (1.1)-(1.2), for any λ ∈ (0, 1) we consider the following modified problem  00  u + a(t)u0 + b(t)fλ (u(t)) = g(t), a.e. t ∈ (0, 1), ∆u0 (tj ) = Ij (u(tj )), j = 1, 2, . . . , p − 1, (3.1)  u(0) = u(T ), u0 (0) = u0 (T ), where fλ : [0, T ] × R → R is defined by  − u1α , u ≥ λ, fλ (u) = − λ1α , u < λ. Ru Let Fλ (u) = 1 fλ (s)ds Φλ : HT1 → R defined by Z Φλ (u) := 0

T

e

A(t)



 p−1 Z u(tj ) X 1 0 2 |u (t)| − b(t)Fλ (u(t))dt + g(t)u(t) dt + eA(t) Ij (s)ds. 2 j=1 0

Clearly, Φλ is well defined on HT1 , continuously differentiable and weakly lower semicontinuous. Moreover, the critical points of Φλ are the weak solutions of problem (3.1). Proof. The proof will be divided into four steps. Step 1. Φλ satisfies the Palais-Smale condition.

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IMPULSIVE PERIODIC SOLUTIONS FOR A SINGULAR DAMPED DIFFERENTIAL EQUATION 5

Let a sequence {un } in HT1 satisfy Φλ (un ) is bounded and Φ0λ (un ) → 0 as n → +∞. That is, there exist a constant c1 > 0 and a sequence {n }n∈N ⊂ R+ with n → 0 as n → +∞ such that, for all n,   Z T A(t) 1 0 2 | e |u (t)| − b(t)Fλ (un (t))dt + g(t)un (t) dt (3.2) 2 n 0 p−1 Z un (tj ) X A(t) Ij (s)ds |≤ c1 , +e j=1

0

and for every v ∈ HT1 , Z T (3.3) eA(t) [u0n (t)v 0 (t) − b(t)fλ (un (t))v(t) + g(t)v(t)] dt | 0

+eA(t)

p−1 X

Ij (un (tj ))v(tj ) |≤ n kvkHT1 .

j=1

Using a standard argument, it is suffices to show that {un } is bounded when verifying the (PS)-condition. Taking v(t) ≡ −1 in (3.3), one has Z T p−1 X √ A(t) A(t) e [b(t)fλ (un (t)) − g(t)] dt − e Ij (un (tj )) ≤ n T for all n. 0 j=1 By (H3 ), we have Z Z p−1 T T X √ |Ij (un (tj ))| eA(t) b(t)fλ (un (t)) ≤ n T + eA(t) g(t)dt+ + eA(t) 0 0 j=1 Z T √ ≤ n T + ekakL1 g(t) + ekakL1 (p − 1)M := c2 . 0 Note that for any t ∈ [0, T ], b(t)fλ (un (t)) < 0. Thus Z Z T T A(t) b(t)fλ (un (t)) dt = eA(t) b(t)fλ (un (t)) ≤ c2 . e 0 0 On the other hand, if we take, in (3.3), v(t) ≡ wn (t) := un (t) − u ¯n , where u ¯n is the average of un over the interval [0, T ], we have Z T p−1 X   c3 kwkHT1 ≥ eA(t) wn0 (t)2 − b(t)fλ (un (t))wn (t) + g(t)wn (t) dt + eA(t) Ij (un (tj ))wn (tj ) 0 j=1 ≥ e−kakL1 kwn0 kL2 − (c2 + ekakL1 kgkL1 )kwn kL∞ + e−kakL1 (p − 1)mkwn kL∞ ≥ e−kakL1 kwn0 kL2 − (c2 + ekakL1 kgkL1 − e−kakL1 (p − 1)m)kwn kL∞ ≥ e−kakL1 kwn0 kL2 − c4 kwn kHT1 , where c3 and c4 are two positive constants. Consequently, using the Wirtinger inequality for zero mean functions in the Sobolev space HT1 , there exists c5 > 0 such that (3.4)

ku0n k2L2 ≤ kwn kHT1 ≤ c5 .

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SHENGJUN LI,

XIANHUA TANG

AND

HUXIAO LUO

Now, suppose that kun kHT1 → +∞ as n → +∞. Since (3.4) holds, we have, passing to subsequence if necessary, that either Mn := max un → +∞ as n → +∞, or mn := min un → −∞ as n → +∞, (i) Assume that the first possibility occurs. By (H3 ) and the fact that fλ < 0, one has Z T p−1 Z T X A(t) A(t) Ij (s)ds e [b(t)Fλ (un (t)) − g(t)un (t)] dt − e 0

Z

T

eA(t)

≥

un (t)

# b(t)fλ (s)ds − g(t)un (t) dt − ekakL1 (p − 1)M Mn

1

0

Z

"Z

T A(t)

=

Mn (t)

1

b(t)fλ (s)ds − g(t)un (t) dt un (t)

−ekakL1 (p − 1)M Mn Z T Z = eA(t) b(t)Fλ (Mn )dt − 0

#

Mn

Z b(t)fλ (s)ds −

e 0

T

e

A(t)

Z

"Z

#

Mn

e 0

0

≥

T

Mn g(t)dt −

0

−ekakL1 (p − 1)M Mn Z T Z A(t) ≥ e [b(t)Fλ (Mn ) − Mn g(t)] dt + Z

0

j=1

"Z

A(t)

(b(t)fλ (s) − g(t)) ds dt

un (t)

T

eA(t) (Mn − un (t))g(t)dt − ekakL1 (p − 1)M Mn

0 T

eA(t) [b(t)Fλ (Mn ) − Mn g(t)] dt − ekakL1 kMn − un kkgkL1 − ekakL1 (p − 1)M Mn .

0

Thus, using Sobolev and Poincare’s inequalities, one has ! Z T kakL1 A(t) −e (p − 1)M + e g(t)dt Mn 0

Z

T

eA(t) [b(t)Fλ (un (t)) − g(t)un (t)]dt − eA(t)

≤

p−1 Z X

0

j=1

Z √ + T ekakL1 kgkL1 ku0n kL2 − Fλ (Mn )ekakL1

un (tj )

Ij (s)ds

0

T

b(t)dt

0

Z ≤

T

eA(t) [b(t)Fλ (un (t)) − g(t)un (t)]dt − eA(t)

0

p−1 Z X j=1

√ + T ekakL1 kgkL1 ku0n kL2 −

RT

b(t)dt α−1

0



1 Mnα−1

un (tj )

Ij (s)ds

0

 − 1 ekakL1 ,

1 From (3.2),(3.4) and the fact that M α−1 → 0 as n → +∞, we see that the right n hand side of the above inequality is bounded, which is contradiction. (ii) Assume the second possibility occurs, i.e., mn → −∞ as n → +∞. We replace Mn by mn in the preceding arguments, and we also get a contradictions.

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IMPULSIVE PERIODIC SOLUTIONS FOR A SINGULAR DAMPED DIFFERENTIAL EQUATION 7

Therefore, Φλ satisfies the Palais-Smale condition. This completes the proof of the claim. Step 2. Let   Ω = u ∈ HT1 | min u(t) > 1 , t∈[0,T ]

and ∂Ω =

  u ∈ HT1 | min u(t) ≥ 1 for all t ∈ (0, T ), ∃tu ∈ (0, T ) such that u(tn ) = 1 . t∈[0,T ]

We show that there exists d > 0 such that inf u∈∂Ω Φλ (u) ≥ −d whenever λ ∈ (0, 1). For any u ∈ ∂Ω, there exists some tu ∈ (0, T ) such that mint∈[0,T ] u(t) = u(tn ) = 1. By (H4 ) and extending the functions by T -periodicity, we obtain that   Z tu +T p−1 Z u(tj ) X 1 eA(t) |u0 (t)|2 − b(t)Fλ (u(t))dt + g(t)u(t) dt + eA(t) Φλ (u) = Ij (s)ds 2 tu j=1 0   Z Z tu +T 1 1 1 tu +T A(t) 0 2 A(t) e u (t) dt + e b(t) 1 − dt ≥ 2 tu α − 1 tu u(t)α−1 Z tu +T Z tu +T + eA(t) g(t)(u(t) − 1)dt + eA(t) g(t)dt 1 ≥ 2

Z

tu tu +T

tu

e

A(t) 0

2

Z

tu +T

u (t) dt +

tu

e

A(t)

Z

tu +T

g(t)(u(t) − 1)dt +

tu

eA(t) g(t)dt.

tu

By the Schwarz inequality and the fact that u0 (t) = (u(·) − 1)0 (t), one has e−kakL1 k(u(·) − 1)0 kL2 − ekakL1 kgkL2 ku(·) − 1kL2 − ekakL1 kgkL1 . 2 Applying Poincare’s inequality to u(·) − 1, we get Φλ (u) ≥

e−kakL1 k(u(·) − 1)0 kL2 − ekakL1 γkgkL2 ku0 kL2 − ekakL1 kgkL1 , 2 where γ = γ(tu ). The above inequality shows that Φλ (u) ≥

Φλ (u) → +∞ as ku0 kL2 → +∞. Since mint∈[0,T ] u(t) = 1, we have that ku(·) − 1kHT1 → +∞ is equivalent to ku0 kL2 → +∞. Hence Φλ (u) → +∞ as kukHT1 → +∞, ∀u ∈ ∂Ω, which shows that Φλ is coercive. Thus it has a minimizing sequence. The weak lower semi-continuity of Φλ yields inf Φλ (u) > −∞.

u∈∂Ω

It follows that there exists d > 0 such that inf Φλ (u) > −d for all λ ∈ (0, 1). u∈∂Ω

Step 3. We show that there exists λ0 ∈ (0, 1) with the property that, for every λ ∈ (0, λ0 ), any solution u of problem (3.1) satisfying Φλ (u) > −d such that minu∈[0,T ] u(t) ≥ λ0 , and hence u is a solution of problem (1.1)-(1.2). Assume on the contrary that there are sequence {λn }n∈N and {un }n∈N such that (i) λn ≤ n1 ; (ii) un is a solution of (3.1) with λ = λn ;

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SHENGJUN LI,

XIANHUA TANG

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HUXIAO LUO

(iii) Φλn (un ) ≥ −d; (iv) mint∈[0,T] un (t) < n1 . Since fλn < 0 and Z T eA(t) [b(t)fλn (un (t)) − g(t)]dt = 0, (3.5) 0

one has keA(t) b(·)fλn (un (·))kL1 ≤ c6 ,

for some constant c6 > 0.

On the other hand, since un (0) = un (T ), there exists τn ∈ (0, T ) such that u0n (τn ) = 0. Therefore, we obtain that eA(t) u0n (t) − eA(τn ) u0n (τn ) =

Z

t

eA(s) [fλ (un (s)) − g(s)]ds,

τn

which, from (3.5), yields that ku0n kL∞ ≤ c7

(3.6)

for some constant c7 > 0.

inf Φλ (u) > −∞.

u∈∂Ω

From Φλn (un ) ≥ −d, it follows that there must exist two constants R1 and R2 , with 0 < R1 < R2 such that max{un (t); t ∈ [0, T ]} ⊂ [R1 , R2 ]. If not, un would tend uniformly to 0 or +∞. In both cases, by (H2 ) − (H3 ) and (3.6), we have Φλn (un ) → −∞ as n → +∞, which contradicts Φλn (un ) ≥ −d. Let τn1 , τn2 be such that, for n large enough 1 un (τn1 ) = < R1 = un (τn2 ). n Multiplying the differential equation in (3.1) by u0n and integrating the equation on [τn1 , τn2 ], (or [τn2 , τn1 ]), we get Z τn2 Z τn2 Z τn2 Ψ := u00n (t)u0n (t)dt + a(t)u0n (t)2 (t)dt + b(t)fλn (un (t))u0n (t)dt 1 τn

Z

1 τn

2 τn

= 1 τn

1 τn

g(t)u0n (t)dt.

It is easy to verify that Ψ = Ψ1 +

 1 0 2 un (τn ) − u0n (τn1 ) + 2

Z

1 τn

2 τn

a(t)u02 n (t)dt,

where Z

2 τn

Ψ1 = 1 τn

b(t)fλn (un (t))u0n (t)dt.

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IMPULSIVE PERIODIC SOLUTIONS FOR A SINGULAR DAMPED DIFFERENTIAL EQUATION 9

From (H2 ) and (3.6), it follows that Ψ is bounded, and consequently Ψ1 is bounded. On the other hand, it is easy to see that b(t)fλn (un (t))u0n (t) =

d [b(t)Fλn (un (t))] − b0 (t)Fλn (un (t)). dt

Thus, by (H1 ) we have   Z τn2 1 − b0 (t)Fλn (un (t))dt n 1 τn     Z τn2 1 1 1 − b0 (t) − 1 dt. ≤ b(τn2 )Fλn (R1 ) − b(τn1 )Fλn n α − 1 τn1 R2α−1

Ψ1 = b(τn2 )Fλn (R1 ) − b(τn1 )Fλn

From the fact that Fλn ( n1 ) → +∞ as n → +∞, we obtain Ψ1 → −∞, i.e., Ψ1 is unbounded. This is a contradiction. Step 4. We prove that Φλ has a mountain-pass geometry for λ ≤ λ0 . Fix λ ∈ (0, λ0 ], one has 0

Z

fλ (s)ds = −

Fλ (0) =

fλ (s)ds

1

0

Z

λ

=−

Z

1

fλ (s)ds − 0

=

1

Z

1 λα−1

fλ (s)ds λ

1

Z −

fλ (s)ds. λ

This implies that Z

1

Fλ (0) > −

Z

λ

fλ (s)ds = λ

fλ (s)ds = Fλ (λ). 1

Hence Z Φλ (0) = −Fλ (0)

(3.7)

T

e

A(t)

Z b(t)dt < −Fλ (λ)

0

RT ≤−

0

T

eA(t) b(t)dt

0

eA(t) b(t)dt α−1



1 λα−1

 −1 .

Consider λ ∈ (0, λ0 ] such that 1 d(α − 1) > 1+ RT . λα−1 eA(t) b(t)dt 0 Thus it follows from (3.7) that Φλ (0) < −d. Also, using (H3 ) we can choose R > 1 enough large such that −e

kakL1

Z M (p − 1) + 0

T

RT   ekakL1 0 b(t)dt 1 g(t)dt R − 1 − α−1 > d. α−1 R !

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SHENGJUN LI,

XIANHUA TANG

AND

HUXIAO LUO

Thus, Φλ (R) = e

A(t)

p−1 Z X j=1

R

Z Ij (s)ds − Fλ (R)

T A(t)

e

Z 0

0

0

T

g(t)dt

b(t)dt + R

Z T  1 1 kakL1 ≤ e M (p − 1)R + b(t)dt e 1 − α−1 α−1 R 0 Z T +RekakL1 g(t)dt 0 ! RT   Z T 1 kakL1 kakL1 0 b(t)dt = e M (p − 1) + g(t)dt R + e 1 − α−1 α−1 R 0 kakL1

< −d. Since Ω is a neighborhood of R, 0 ∈ / Ω and max{Φλ (0), Φλ (R)} < inf Φλ (u). x∈∂Ω

Step 1 and Step 2 imply that Φλ has a critical point uλ such taht Φλ (uλ ) = inf max Φλ (h(s)) ≥ inf Φλ (u), h∈Γ s∈[0,1]

x∈∂Ω

where Γ = {h ∈ C([0, 1], HT1 ) : h(0) = 0, h(1) = R}. Since inf u∈∂Ω Φλ (uλ ) ≥ −d, it follows from Step 3 that uλ is a solution of problem (1.1)-(1.2). The proof of the main result is complete.  Acknowledgment This work is supported by the National Natural Science Foundation of China (Grant No.11461016), Hainan Natural Science Foundation(Grant No.20167246), CSU Postdoctoral Science Foundtion funded Project (Grant No. 169731). References 1. D. Chen, B. Dai, Periodic solution of second order impulsive delay differential systems via variational method, Appl. Math. Lett. 38 (2014), 61-66. 2. J. Chu, S. Li, H. Zhu, Nontrivial periodic solutions of second order singular damped dynamical systems, Rocky Mountain J. Math., 45 (2015), 457C474. 3. B. Dai, D. Zhang, The existence and multiplicity of solutions for second-order impulsive differential equations on the half-line, Results Math. 63 (2013), 135-149. 4. R. Hakl, P. J. Torres, On periodic solutions of second-order differential equations with attractive-repulsive singularities. J. Differential Equations 248 (2010), 111-126. 5. A.C. Lazer, S. Solimini, On periodic solutions of nonlinear differential equations with singularities. Proc. Amer. Math. Soc. 99 (1987), 109-114. 6. S. Li, F. Liao, H. Zhu, Periodic solutions of second order non-autonomous differential systems, Fixed Point Theory, 15 (2014), 487-494. 7. S. Li, F. Liao, W. Xing, Periodic solutions of Li´ enard differential equations with singularity, Electron. J. Differential Equations, 151 (2015), 1-12. 8. S. Li, W. Li, Y. Fu, Periodic orbits of singular radially symmetric systems, J. Comput. Anal. Appl. 22 (2017), 393-401. 9. S. Li, Y. Zhu, Periodic orbits of radially symmetric Keplerian-like systems with a singularity, J. Funct. Spaces, 2016, ID 7134135. 10. J. Li, S. Li, Z. Zhang, Periodic solutions for a singular damped differential equation, Bound. Value Probl. 5 (2015).

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IMPULSIVE PERIODIC SOLUTIONS FOR A SINGULAR DAMPED DIFFERENTIAL EQUATION 11

11. R. Liang, Z. Liu, Nagumo type existence results of Sturm-Liouville BVP for impulsive differential equations, Nonlinear Anal. 74 (2011), 6676-6685. 12. J.J. Nieto, D. O’Regan, Variational approach to impulsive differential equations. Nonlinear Anal. Real World Appl. 10 (2009) , 680-690. 13. J. Mawhin, M. Willem, Critical Point Theory and Hamiltonian Systems. Springer, 1989. 14. H. Shi, H. Chen, Multiplicity results for a class of boundary value problems with impulsive effects, Math. Nachr. 289 (2016), 718-726. 15. J. Sun, D. O’Regan, Impulsive periodic solutions for singular problems via variational methods. Bull. Aust. Math. Soc.86 (2012), 193-204. 16. J. Sun, H. Chen, J. J. Nieto, M. Otero-Novoa, Multiplicity of solutions for perturbed secondorder Hamiltonian systems with impulsive effects. Nonlinear Anal. 72 (2010), 4575-4586. 17. J. Sun, H. Chen, J. J. Nieto, Infinitely many solutions for second-order Hamiltonian system with impulsive effects. Math. Comput. Modelling. 54 (2011), 544-555. 18. Y. Tian, W. Ge, Applications of variational methods to boundary value problem for impulsive differential equation. Proc. Edin. Math. Soc. 51 (2008), 509-527. 1 College of Information Sciences and Technology, Hainan University, Haikou, 570228, China 2 School of Mathematics and Statistics, Central South University, Changsha, Hunan, 410083, China E-mail address: [email protected] (S. Li) E-mail address: [email protected] (X. Tang) E-mail address: [email protected] (H. Luo)

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Accelerated SNS and accelerated SSS iteration methods for non-Hermitian linear systems Min-Li Zeng and Guo-Feng Zhang Abstract. Recently, Bai proposed the skew-normal splitting (SNS) and skew-scaling splitting (SSS) iteration methods for large sparse nonHermitian positive definite systems. Compared with the Hermitian and skew-Hermitian splitting (HSS) iteration method, both of the SNS and SSS methods are making more use of the skew-Hermitian parts than the HSS method. In this paper, we introduce an accelerated skew-normal splitting (ASNS) iteration method and an accelerated skew-scaling splitting (ASSS) iteration method for solving large sparse non-Hermitian positive definite system of linear equations. We study the convergence properties of the the new iteration methods and the quasi-optimal parameters. Moreover, the inexact forms of the new methods are proposed by employing some subspace methods as the inner iteration processes at each step of the outer iterations. Numerical experiments are given to verify the correctness of the theoretical results and the effectiveness of the new methods. Mathematics Subject Classification (2010). 65F10; 65F50. Keywords. HSS iteration method, skew-normal splitting, skew-scaling splitting, quasi-optimal parameters, non-Hermitian positive definite system.

1. Introduction Consider the numerical solution of the large sparse system of linear equations of the form Ax = b, A ∈ Cn×n and x, b ∈ Cn , (1.1) This work was supported by the National Natural Science Foundation of China (11271174, 11511130015), the Natural Science Foundation of Fujian Province (2016J05016) and the Scientific Research Project of Putian University (2015061, 2016021, 2016075).

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where A is a non-Hermitian positive definite matrix. The linear system (1.1) arises from many scientific computing areas, such as diffuse optical tomography [1], lattice quantum chromo dynamics [16], structural dynamics [15], eddy current problems [11] and so on. See [2] and references therein for more applications of the linear system of the form (1.1). Based on the Hermitian and skew-Hermitian (HS) splitting of the coefficient matrix A: A = H + S, with 1 1 H = (A + A∗ ), S = (A − A∗ ), 2 2 Bai, Golub and Ng [2] present and studied an efficient Hermitian and skewHermitian splitting (HSS) iteration method. Because of the unconditionally convergent property and effectiveness, the HSS iteration method has captured a lot of researchers’ attention. A multitude of researchers focused on the HSS method and proposed varieties of variants based on the Hermitian and skew-Hermitian splitting, such as the HSS-like method [9], the modified HSS method [10], the accelerated HSS method [5] and the preconditioned HSS method [6, 3] and so on, see [12, 20, 22, 17, 4, 18]. As is shown in [7] that the HSS method is more effective when S dominates H than vice versa. Furthermore, when S is very small compared to H, the subsystem 1

(αI + S)x(k+1) = (αI − H)x(k+ 2 ) + b in the HSS method contributes little to convergence and the inner iterations must be designed to terminate properly since 1 1 1 (I − S)−1 = I + S + 2 S 2 + · · · . α α α As is known that the iterative matrix of HSS (αI + S)−1 (αI − H)(αI + H)−1 (αI − S) is similar to W (α)Q(α), where 1 1 H)(I + H)−1 α α is Hermitian, so forth ∥W (α)∥2 < 1, and 1 1 Q(α) = (I − S)(I + S)−1 α α is unitary for all α > 0. Therefore, according to [7], when S is small compared to H, different iterations combing H and S ∗ S are more naturally. Suppose that S is invertible, then we can multiply (1.1) on the left by −S to obtain the following equivalent form: W (α) = (I −

−SHx − S 2 x = −Sb,

(1.2)

i.e., −SHx = S 2 x − Sb or − S 2 x = SHx − Sb. Adding αH to each side will lead to the following two fixed-point equations (αH − SH)x ≡ (αI − S)Hx = (αH + S 2 )x − Sb, 860

(1.3)

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ASNS and ASSS iteration methods for non-Hermitian linear systems (αH − S 2 )x = (αH + SH)x − Sb ≡ (αI + S)Hx − Sb.

3

(1.4)

Based on the above fixed-point equations, Bai [7] established the skew-normal splitting (SNS) methods as the following algorithm. Algorithm 1.1. (SNS method) Given an initial approximate solution x(0) , for k = 0, 1, 2, · · · until convergence, solve { 1 (αI − S)x(k+ 2 ) = (αH + S 2 )x(k) − Sb, 1

(αH − S 2 )x(k+1) = (αI + S)x(k+ 2 ) − Sb, where α is a given positive constant. Another way to use the skew-Hermitian matrix S is to employ it to scale the linear system (1.1). By first adding and then subtracting α1 S 2 x to the fixed-point equations −Sx = Hx − b

and Hx = −Sx + b,

respectively, it follows, 1 1 1 1 (I − S)(−Sx) = (H + S 2 )x − b and (H − S 2 )x = (I + S)(−Sx) + b. α α α α In analogy to the SNS method, Bai further present the skew-scaling splitting (SSS) method in [7] as the following algorithm. Algorithm 1.2. (SSS method) Given an initial approximate solution x(0) , for k = 0, 1, 2, · · · , until convergence, solve { 1 (αI − S)x(k+ 2 ) = (αH + S 2 )x(k) − αb, 1

(αH − S 2 )x(k+1) = (αI + S)x(k+ 2 ) + αb, where α is a given positive constant. For the SNS and SSS methods, it has been shown in [7] that when S is small compared to H, the Corollary 2.3 in [2] makes clear how to choose a good iterative parameter. When S dominates H, it is not clear. However, the SNS and SSS methods give an exact way to choose the optimal iterative parameter no matter S dominates H or not. In this paper, we will first accelerate the SNS and SSS methods by adding another parameter in the second iterate step of each iterations and then we obtain two new methods, which are named as the ASNS method and the ASSS method. The new methods can be seen as generalized forms of the original SNS and SSS methods. Futher, the iterative parameters can be chosen in a more extensive range. The outline of this paper is arranged as follows. In Section 2, we present the accelerated SNS (ASNS) and the accelerated SSS (ASSS) methods. Then we analyze the convergence properties of both methods. In Section 3, we determine the quasi-optimal parameters by minimizing the upper bound of the spectral radius of the iteration matrix and then show the case about the new methods superiority to the SNS and SSS methods. Section 4 is devoted to 861

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the inexact variant form of the new methods and the asymptotically convergent rate property of the new methods. Numerical experiments are given in Section 5 to illustrate the correctness of the theoretical results obtained in this paper. Section 6 draws some conclusions and remarks to end this paper. Throughout this paper, the skew-Hermitian part S of the coefficient matrix A is assumed to be invertible and the Hermitian part H to be positive definite. We use A ∼ B to denote that the matrix A is similar to the matrix B.

2. The ASNS and ASSS methods In this section, we will propose the accelerated SNS (ASNS) and the accelerated SSS (ASSS) methods. Firstly, we add αH to each side of the equation −SHx = S 2 x − Sb and then add βH to each side of the equation −S 2 x = SHx − Sb. Then we obtain the fixed-point equations (αH − SH)x ≡ (αI − S)Hx = (αH + S 2 )x − Sb

(2.1)

and (βH − S 2 )x = (βH + SH)x − Sb ≡ (βI + S)Hx − Sb. Subsequent algorithm is the ASNS iteration method.

(2.2)

Algorithm 2.1. (ASNS method) Given an initial approximate solution x(0) , for k = 0, 1, 2, · · · until convergence, solve { 1 (αI − S)x(k+ 2 ) = (αH + S 2 )x(k) − Sb, 1

(βH − S 2 )x(k+1) = (βI + S)x(k+ 2 ) − Sb, where α and β are given positive constants. For the fixed-point equations −Sx = Hx − b we first add

1 2 αS x

and Hx = −Sx + b,

and then subtract (I −

1 2 βS x

on both sides to obtain

1 1 S)(−Sx) = (H + S 2 )x − b α α

and

1 2 1 S )x = (I + S)(−Sx) + b. β β After rearranging these equations and choosing x(0) wisely, we can straightforwardly get the accelerated skew-scaling splitting (ASSS) method as the next algorithm. (H −

Algorithm 2.2. (ASSS method) Given an initial approximate solution x(0) , for k = 0, 1, 2, · · · , until convergence, solve { 1 (αI − S)x(k+ 2 ) = (αH + S 2 )x(k) − αb, 1

(βH − S 2 )x(k+1) = (βI + S)x(k+ 2 ) + βb, where α and β are given positive constants. 862

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Obviously, when α = β, the ASNS method and the ASSS method reduce to the SNS method and the SSS method, respectively. Comparing the ASSS method with the ASNS method, we find that the coefficient matrices of the ASNS and ASSS method are exactly the same. Therefore, they have the same iteration matrix M (α, β), where M (α, β) := (βH − S 2 )−1 (βI + S)(αI − S)−1 (αH + S 2 ). It can be seen that the ASSS method is much cheaper than the ASNS method, because of the constant vector terms αb, βb instead of Sb. It is seen that S −1 is skew-Hermitian, then −S −1 HS −1 = (S −1 )∗ HS −1 is Hermitian positive definite. Denote ξmax = max{|ξj | iξj ∈ σ(S)}, ξmin = min{|ξj | iξj ∈ σ(S)}, λmax = max{λj λj ∈ σ((S −1 )∗ HS −1 )}, λmin = min{λj λj ∈ σ((S −1 )∗ HS −1 )}, the following theorem concentrates on the convergence property of the ASNS method. Theorem 2.1. Given a non-Hermitian matrix A ∈ Cn×n . Let H = 21 (A + A∗ ) and S = 12 (A−A∗ ). If H is positive definite , S is invertible, then the spectral radius ρ(M (α, β)) of the ASNS and ASSS iteration matrix is bounded by δ(α, β), where √ αλk − 1 β 2 + ξj2 . · max δ(α, β) = max 2 2 −1 ∗ −1 α + ξj λk ∈σ((S ) HS ) βλk + 1 iξj ∈σ(S) Further, if α and β satisfy δ(α, β) < 1, then the ASNS method and the ASSS method are convergent to the unique solution of the linear system (1.1). Proof. As M (α, β) = (βH − S 2 )−1 (βI + S)(αI − S)−1 (αH + S 2 ) ∼ S −1 (βI + S)(αI − S)−1 (αH + S 2 )(βH − S 2 )−1 S = (βI + S)S −1 (αI − S)−1 S(−S −1 )(αH − S ∗ S)S −1 S(βH + S ∗ S)−1 (−S) = (βI + S)(αI − S)−1 (α(S −1 )∗ HS −1 − I)(β(S −1 )∗ HS −1 + I)−1 := M (α, β), (2.3) then it follows ρ(M (α, β)) ≤ ∥(βI + S)(αI − S)−1 ∥ · ∥(α(S −1 )∗ HS −1 − I)(β(S −1 )∗ HS −1 + I)−1 ∥ √ αλk − 1 β 2 + ξj2 ≤ max−1 · max α2 + ξj2 λk ∈σ((S))∗ HS −1 ) βλk + 1 iξj ∈σ(S ) = δ(α, β) If δ(α, β) < 1, then ρ(M (α, β)) ≤ δ(α, β) < 1, i.e., the ASNS and the ASSS methods are convergent.  863

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Theorem 2.2. The ASNS and ASSS iteration methods are convergent if either of the following conditions holds: 2 2 2 (1) α ≥ β, (λ2min ξmax − 1)α − 2ξmax λmin < β(λ2min ξmax − 1 + 2αλmin ) and 2 2 2 2 2 (λmax ξmax − 1)α − 2ξmax λmax < β(λmax ξmax − 1 + 2αλmax ); 2 2 2 (2) α < β, (λ2min ξmin − 1)α − 2ξmin λmin < β(λ2min ξmin − 1 + 2αλmin ) and 2 2 2 2 2 (λmax ξmin − 1)α − 2ξmin λmax < β(λmax ξmin − 1 + 2αλmax ). Proof. δ(α, β) < 1 leads to √ αλk − 1 β 2 + ξj2 < 1. max · max 2 2 α + ξj λk ∈σ((S −1 )∗ HS −1 ) βλk + 1 iξj ∈σ(S) Or equivalently, αλk − 1 < min max −1 ∗ −1 iξj ∈σ(S) λk ∈σ((S ) HS ) βλk + 1 (1) If α ≥ β, then

√ min

iξj ∈σ(S)

Since max −1 ∗

λk ∈σ((S

)

HS −1 )

α2 + ξj2 = β 2 + ξj2

√

√

α2 + ξj2 . β 2 + ξj2

(2.4)

2 α2 + ξmax . 2 2 β + ξmax

αλk − 1 { } = max | αλmin − 1 |, | αλmax − 1 | , βλk + 1 βλmin + 1 βλmax + 1

then (2.4) is equivalent to  2 α2 + ξmax αλmin − 1 2   , ) < 2  ( βλ 2 β + ξmax min + 1 2  αλmax − 1 2 α2 + ξmax  ( . ) < 2 2 βλmax + 1 β + ξmax After some simple computations, it follows { 2 2 (αλmin − 1)2 (β 2 + ξmax ) < (α2 + ξmax )(βλmin + 1)2 , 2 2 )(βλmax + 1)2 . ) < (α2 + ξmax (αλmax − 1)2 (β 2 + ξmax

(2.5)

Because α and β are positive constants, the first equation of (2.5) leads to 2 2 2 (λ2min ξmax − 1)α − 2ξmax λmin < β(λ2min ξmax − 1 + 2αλmin ).

The second equation of (2.5) leads to 2 2 2 − 1 + 2αλmax ). − 1)α − 2ξmax λmax < β(λ2max ξmax (λ2max ξmax

(2) If α < β, then

√ min

iξj ∈σ(S)

α2 + ξj2 = β 2 + ξj2

√

2 α2 + ξmin . 2 β 2 + ξmin

Using the same strategy as α ≥ β, we can easily obtain 2 2 2 (λ2min ξmin − 1)α − 2ξmin λmin < β(λ2min ξmin − 1 + 2αλmin )

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and 2 2 2 (λ2max ξmin − 1)α − 2ξmin λmax < β(λ2max ξmin − 1 + 2αλmax ).

 Remark 2.3. If α = β, from the result (1) of Theorem 2.2, we obtain that when 2 α2 +ξmin > 0, the ASNS and ASSS iteration methods are convergent. Because 2 2 α + ξmin > 0 invariably holds, then we know the ASNS and ASSS iteration methods are convergent unconditionally, which agrees with the results in [7].

3. The quasi-optimal iterative parameters In this section, we will give the quasi-optimal parameters of the ASNS iteration method and the ASSS iteration method. Then we will analyze the optimal parameters by minimizing the upper bound δ(α, β). According to the analysis in the previous section, we have √ 2 { αλmin − 1 αλmax − 1 }  β 2 + ξmin     α2 + ξ 2 · max | βλmin + 1 |, | βλmax + 1 | , α ≥ β; min δ(α, β) = √  2 2  { αλmin − 1 αλmax − 1 } β + ξmax   · max | |, | | , α < β.  2 α2 + ξmax βλmin + 1 βλmax + 1 We rewrite the matrix M (α, β) in (2.3) as M (α, β) 1 1 1 1 = ( I + S −1 )( I − S −1 )−1 ( I − (S −1 )∗ HS −1 )( I + (S −1 )∗ HS −1 )−1 β α α β 1 1 1 1 ∼ ( I + (S −1 )∗ HS −1 )−1 ( I − (S −1 )∗ HS −1 )( I − S −1 )−1 ( I + S −1 ). β α α β By making use of the same strategy of Theorem 4.2 in [19], we can obtain the optimal iterative parameters and the corresponding upper bound in the following theorem. Theorem 3.1. Let A ∈ Cn×n be a non-Hermitian matrix. If the ASNS and ASSS iteration methods are convergent, then √ √ 1 1 (1) when λmin λmax > ξmin or λmin λmax < ξmax , the quasi-optimal parameters are given by √ (c2 + λ2min )(c2 + λ2max ) + c2 − λmin λmax , α∗ = (λmax + λmin )c2 √ (c2 + λ2min )(c2 + λ2max ) − c2 + λmin λmax β∗ = , (λmax + λmin )c2 and the corresponding optimal upper bound δ(α∗ , β ∗ ) of ρ(M (α, β)) is √ (c2 + λ2min )(c2 + λ2max ) − (c2 + λmin λmax ) ∗ ∗ δ(α , β ) = , (λmax − λmin )c 865

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where the constant c is given by  √ 1 1   , if ; λmax λmin >  ξmax ξmin c= √ 1 1    , if . λmax λmin < ξmin ξmax √ 1 1 (2) when ξmax ≤ λmin λmax ≤ ξmin , the quasi-optimal parameters α∗ and β∗ are 1 α∗ = β∗ = √ λmin λmax and the corresponding optimal upper bound is √ √ λmax − λmin √ δ(α∗ , β∗ ) = √ . λmax + λmin √ Remark 3.2. When emin ≤ λmin λmax ≤ emax , the optimal upper bound δ(α, β) of the ASNS and GSSS iteration methods reduces to the optimal bound of the SNS and SSS iteration methods, respectively, i.e., 1 α∗ = β∗ = √ λmin λmax and the corresponding optimal bound is √ √ λmax − λmin √ ν∗ = √ . λmax + λmin √ √ 1 1 or λmin λmax < ξmax , then we have Theorem 3.3. When λmin λmax > ξmin δ(α∗ , β∗ ) < ν∗ , where δ(α∗ , β∗ ) and ν∗ are defined in Theorem 3.1 and Remark 3.2, respectively. Proof. We rewrite ν∗ as ν∗ =

√ √ ( λmax − λmin )2 . λmax − λmin

Then δ(α∗ , β∗ ) < ν∗ if and only if √ √ (λ2min + c2 )(λ2max + c2 ) < [c( λmin − λmin )2 + (λmin λmin + c2 )]2 . Denote k1 = λmax + λmin and k2 = λmax λmin , then the above inequality can be simplified as √ √ √ 4c2 k2 − 2c2 k1 k2 + ck1 k2 − 2c2 c3 k2 − 2ck2 k2 + c3 k1 > 0. Or equivalently,

√ √ (k1 − 2 k2 )( k2 − c)2 > 0.

It follows

√ √ √ λmax − λmin )2 ( λmin λmax − c)2 > 0, √ √ √ 1 i.e., ( λmin λmax − c)2 > 0. That is, λmin λmax > ξmin or λmin λmax < (

866

1 ξmax .

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ASNS and ASSS iteration methods for non-Hermitian linear systems

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4. The inexact ASNS and ASSS methods The two half-steps at each step of the ASNS and ASSS methods require finding solutions with the coefficient matrices αI − S and βH − S 2 . Because βH − S 2 is Hermitian positive definite, then we may use the CG method to solve the linear system with coefficient matrix βH −S 2 . For solving the linear system with coefficient matrix αI − S, we may use some Krylov subspace method [21]. The inexact ASNS (IASNS) method can be described as follows. Algorithm 4.1. (IASNS method) Given an initial guess x ¯(0) , for k = 0, 1, 2, · · · , (k) (k+ 21 ) until {¯ x } converges, solve {¯ x } approximately from 1

(αI − S)¯ x(k+ 2 ) = (αH + S 2 )¯ x(k) − Sb by employing an inner iteration (e.g., some Krylov subspace method) with x ¯(k) as the initial guess, then solve x ¯(k+1) approximately from 1

(βH − S 2 )¯ x(k+1) = (βI + S)¯ x(k+ 2 ) − Sb 1

¯(k+ 2 ) as the by employing an inner iteration (e.g., the CG method) with x initial guess, where α and β are given positive constants. The inexact ASSS (IASSS) method can be described as follows. Algorithm 4.2. (IASSS method) Given an initial guess x ¯(0) , for k = 0, 1, 2, · · · , 1 until {¯ x(k) } converges, solve {¯ x(k+ 2 ) } approximately from 1

(αI − S)¯ x(k+ 2 ) = (αH + S 2 )¯ x(k) − αb by employing an inner iteration (e.g., some Krylov subspace method) with x ¯(k) as the initial guess, then solve x ¯(k+1) approximately from 1

(βH − S 2 )¯ x(k+1) = (βI + S)¯ x(k+ 2 ) + βb 1

by employing an inner iteration (e.g., the CG method) with x ¯(k+ 2 ) as the initial guess, where α and β are given positive constants. Subsequently, we will concentrate on the IASNS iteration method. The results about the IASSS iteration method can be obtained in the similar way. To simplify numerical implementation and convergence analysis, the IASNS iteration method can be rewritten as the following equivalent scheme. Given an initial guess x ¯(0) , for k = 0, 1, 2, · · · , compute Step 1 and Step (k) 2 until x ¯ converges: Step 1. approximate the solution of (αI − S)¯ z (k) = −S r¯(k) , (¯ r(k) = b − A¯ x(k) ) by iterating until z¯(k) is such that the residual p¯(k) = r¯(k) − (αI − S)¯ z (k) satisfies ∥¯ p(k) ∥ ≤ εk ∥¯ r(k) ∥, 1

and then compute x ¯(k+ 2 ) = x ¯(k) + z¯(k) ; 867

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Step 2. approximate the solution of 1

1

1

1

(βH − S 2 )¯ z (k+ 2 ) = −S r¯(k+ 2 ) , (¯ r(k+ 2 ) = b − A¯ x(k+ 2 ) ) 1

by iterating until z¯(k+ 2 ) is such that the residual 1

1

1

q¯(k+ 2 ) = r¯(k+ 2 ) − (βH − S 2 )¯ z (k+ 2 ) satisfies 1

1

∥¯ p(k+ 2 ) ∥ ≤ ηk ∥¯ r(k+ 2 ) ∥, 1

1

¯(k+ 2 ) + z¯(k+ 2 ) . Here ∥ · ∥ is a norm of a vector. and then compute x ¯(k+1) = x If the two inner systems are solved inexactly with corresponding quantities {εk } and {ηk }. Denote εmax = maxk {εk } and ηmax = maxk {ηk }. Let ∥| · |∥M denote ∥|X|∥M = ∥M XM −1 ∥ for all X ∈ Cn×n . Then the following theorem concentrates on the convergent results of the IASNS and IASSS iteration methods. According to Theorem 3.1 in [8] and by specializing the splitting as (−S)A = M1 − N1 := (αH − SH) − (αH + S 2 ) = M2 − N2 := (βH − S 2 ) − (βH + SH), we can immediately obtain the following theorem. Theorem 4.1. Let A ∈ Cn×n be a non-Hermitian positive definite matrix. H = 21 (A + A∗ ) and S = 12 (A − A∗ ) be its Hermitian and skew-Hermitian parts, and let α and β be positive constants. If S is invertible, {¯ x(k) } is an iterative sequence generated by the IASNS iteration method and x∗ ∈ Cn is the exact solution of the linear system (1.1), then it holds that ∥|¯ x(k+1) − x∗ |∥M2 ≤ (σ(α, β) + µ(α, β)θ(β)εk + θ(β)(ρ(α, β) + θ(β)ν(α, β)εk )ηk ) · ∥|¯ x(k) − x∗ |∥M2 , where σ(α, β) = ∥(βI + S)(αI − S)−1 (αH + S 2 )(βH − S 2 )−1 ∥, ρ(α, β) = ∥(βH − S 2 )H −1 (αI − S)−1 (αH + S 2 )(βH − S 2 )−1 ∥, µ(α, β) = ∥(βI + S)(αI − S)−1 ∥, θ(β) = ∥(−SA)(βH − S 2 )−1 ∥, ν(α, β) = ∥(βH − S 2 )H −1 (αI − S)−1 ∥. In particularly, if σ(α, β) + µ(α, β)θ(β)εmax + θ(β)(ρ(α, β) + θ(β)ν(α, β)εmax )ηmax < 1, then the iterative sequence {¯ x(k) } converges to x∗ ∈ Cn , where εmax = max{εk } k

and

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Theorem 4.2. Let the assumption in Theorem 4.1 be satisfied. Suppose that both {τ1 (k)} and {τ2 (k)} are nondecreasing and positive sequences satisfying τ1 (k) ≥ 1 and τ2 (k) ≥ 1, and lim sup τ1 (k) = lim sup τ2 (k) = +∞,

k→∞

k→∞

and that both δ1 and δ2 are real constants in the interval [0, 1] satisfying τ (k)

εk ≤ c1 δ11

and

τ (k)

ηk ≤ c2 δ22

,

for k = 0, 1, 2, · · · , where c1 and c2 are nonnegative constants. Then we have √ ∥|¯ x(k+1) − x∗ |∥M2 ≤ ( σ(α, β) + ω(α, β)θ(β)δ τ (k) )2 · ∥|¯ x(k) − x∗ |∥M2 , where k = 0, 1, 2, · · · , τ (k) = min{τ1 (k), τ2 (k)}, δ = max{δ1 , δ2 } and √ 1 ω = max{ c1 c2 ν(α, β), √ (c1 µ(α, β) + c2 ρ(α, β))}. 2 σ(α, β) In particular, we have ∥|¯ x(k+1) − x∗ |∥M2 = σ(α, β), k→∞ ∥|¯ x(k) − x∗ |∥M2 i.e., the convergence rate of the IASNS method is asymptotically the same as that of the exact two-step iterative scheme ASNS. lim sup

5. Numerical results In this section, we will consider the three-dimensional convection-diffusion equation −(uxx + uyy + uzz ) + q(ux + uy + uz ) = f (x, y, z) (5.1) on the unit cube Ω = [0, 1] × [0, 1] × [0, 1], with constant coefficient q and subject to Dirichlet-type boundary conditions. When the seven-point finite difference discretization, for example, the centered differences to diffusive terms, and the centered differences or the first order upwind approximations to the convective terms are applied to the above model convection-diffusion equation, we get the system of linear equations (1.1) with the coefficient matrix A = Tx ⊗ I ⊗ I + I ⊗ Ty ⊗ I + I ⊗ I ⊗ Tz , 1 where the equidistant step-size h = n+1 is used in the distretization on all of the three directions and the natural lexicographic ordering is employed to the unknowns. In addition, ⊗ denotes the Kronecker product, and Tx , Ty , and Tz are tri-diagonal matrices given by Tx = tridiag(t2 , t1 , t3 ),

Ty = tridiag(t2 , 0, t3 ),

Tz = tridiag(t2 , 0, t3 ),

with t1 = 6, t2 = −1 − r, t3 = −1 + r if the first order derivatives are approximated by the centered difference scheme and with t1 = 6 + 6r, t2 = −1 − 2r, t3 = −1 869

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if the first order derivatives are approximated by the upwind difference scheme. Here r = qh 2 is the mesh Reynolds number. For details, we refer to [13, 14, 2]. Figure 1 plots ρ(M (α, β)) and δ(α, β) for the centered difference scheme with n = 8 and q = 1, 10 when α varies in [0, 1]. When α is fixed, according to [19], we can compute β as αλmax − 1 1 − αλmin = . βλmin + 1 βλmax + 1 That is, β=

α(λmax + λmin ) − 2 . λmax + λmin − 2αλmax λmin

(5.2)

Besides, Figure 2 plots ρ(M (α, β)) and δ(α, β) for the upwind difference scheme with n = 8 and q = 1, 10 when α varies in [0, 1]. β is also computed according to (5.2). From Figures 1-2, we find that, when β is chosen according to (5.2) and let α vary in [0, 1], the point such that the value of ρ(M (α, β)) reaches the minimum is extremely close to the point such that the value of δ(α, β) reaches the minimum. Therefore, the theoretical optimal parameters in Theorem 3.1 would be intensely close to the real optimal parameters of the methods. In Figure 3 and Figure 4, we plot the distributions of the eigenvalues of the iterative matrices. Here, we choose the experimental optimal parameters α = β = qh 2 in the SNS method and we plot the eigenvalues distribution in Figure 3. We replace α in the ASNS method by h2 and we plot the eigenvalues distributions in Figure 4. It can be seen from Figures 3-4 that the eigenvalues of the ASNS iterative matrix are more cluster around 0 than the eigenvalues of the SNS iterative matrix when the iterative parameters are chosen appropriately. Further, if we can find a cheaper way to choose the optimal parameters, the accelerated methods would be more efficient, which will be our next work.

1.04

1.01

ρ(M(α,β)) δ(α,β)

1.02

1

1 ρ(M(α,β)) and δ(α,β)

ρ(M(α,β)) and δ(α,β)

0.99

0.98

0.97

0.98 0.96 0.94 0.92

0.96 0.9 0.95

0.94

0.88

ρ(M(α,β)) δ(α,β) 0

0.2

0.4

α

0.6

0.8

0.86

1

0

0.2

0.4

α

0.6

0.8

1

Figure 1. The comparison between ρ(M (α, β)) and δ(α, β) when q = 1 (left) and q = 10 (right) (centered).

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ASNS and ASSS iteration methods for non-Hermitian linear systems 13 1.01

1.04 1.02

1

ρ(M(α,β)) and δ(α,β)

ρ(M(α,β)) and δ(α,β)

1 0.99

0.98

0.97

0.98 0.96 0.94 0.92 0.9

0.96

0.95

0.88

ρ(M(α,β)) δ(α,β) 0

0.2

0.4

α

0.6

0.8

0.86

1

ρ(M(α,β)) δ(α,β) 0

0.2

0.4

α

0.6

0.8

1

Figure 2. The comparison between ρ(M (α, β)) and δ(α, β) when q = 1 (left) and q = 10 (right) (upwind). The ASNS(ASSS) method 0.6

0.4

0.4

0.2

0.2

imag part

imag part

The SNS(SSS) method 0.6

0

−0.2

0

−0.2

−0.4

−0.4

−0.6

−0.6

−0.8 −0.2

0

0.2

0.4 0.6 real part

0.8

1

−0.8

1.2

0

0.1

0.2

0.3

0.4 real part

0.5

0.6

0.7

0.8

Figure 3. The distribution of the eigenvalues of the iterative matrix for the SNS method (left) and the ASNS method (right). The ASNS(ASSS) method

The SNS(SSS) method 1

0.4

0.8

0.3

0.6

0.2

0.4 imag part

imag part

0.1 0.2 0 −0.2

0 −0.1

−0.4 −0.2 −0.6 −0.3

−0.8 −1 −0.4

−0.2

0

0.2

0.4 real part

0.6

0.8

1

−0.4 −0.1

1.2

0

0.1

0.2

0.3

0.4 0.5 real part

0.6

0.7

0.8

Figure 4. The distribution of the eigenvalues of the iterative matrix for the SNS method (left) and the ASNS method (right).

6. Conclusions For solving the non-Hermitian positive definite with invertible skew-Hermitian parts, we proposed an accelerated SNS and an accelerated SSS iteration method in this paper. Comparied with the HSS iteration method, the new 871

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methods not only concentrate more on the balance between the Hermitian parts and the skew-Hermitian parts, but also accelerate the original SNS and SSS methods. The convergence properties and the quasi-optimal parameters are analyzed. In actual implementations, we give the inexact forms of the new methods. Numerical results demonstrate that the point such that δ(α, β) reaches the minimum is exact the same as the point such that ρ(M (α, β)) reaches the minimum. Therefore, the theoretical results obtained in this paper is correct. Meanwhile, by choosing appropriate parameters, the new methods are more efficient than the SNS and SSS methods.

References [1] S.R. Arridge, Optical tomography in medical imaging, Inverse Probl. 15, R41 (1999). [2] Z.-Z. Bai, G.-H. Golub, and M.K. Ng, Hermitian and skew-Hermitian splitting methods for non-Hermitian positive definite linear systems, SIAM J. Matrix Anal. Appl. 24, 603–626 (2003). [3] Z.-Z. Bai, G.H. Golub, and J.-Y. Pan, Preconditioned Hermitian and skewHermitian splitting methods for non-Hermitian positive semidefinite linear systems, Numer. Math. 98, 1–32 (2004). [4] Z.-Z. Bai, G.H. Golub, and L.-Z. Lu, Block triangular and skew-Hermitian splitting methods for positive-definite linear systems, SIAM J. Sci. Comput. 26, 844–863 (2005). [5] Z.-Z. Bai, and G.H. Golub, Accelerated Hermitian and skew-Hermitian splitting iteration methods for saddle-point problems, IMA J. Numer. Anal. 27, 1–23 (2007). [6] Z.-Z. Bai, G.H. Golub, and C.-K. Li, Convergence properties of preconditioned Hermitian and skew-Hermitian splitting methods for non-Hermitian positive semidefinite matrices, Math. Comp. 76, 287–298 (2007). [7] Z.-Z. Bai, Several splittings for non-Hermitian linear systems, Sci. China Ser. A 51, 1339–1348 (2008). [8] Z.-Z. Bai, G.H. Golub, and M.K. Ng, On inexact Hermitian and skewHermitian splitting methods for non-Hermitian positive definite linear systems, Linear Algebra Appl. 428, 413–440 (2008). [9] Z.-Z. Bai, Optimal parameters in the HSS-like methods for saddle-point problems. Numer. Linear Algebra Appl. 16, 447–479 (2009). [10] Z.-Z. Bai, M. Benzi, and F. Chen, Modified HSS iteration methods for a class of complex symmetric linear systems, Computing 87, 93–111 (2010). [11] Z.-Z. Bai, Block alternating splitting implicit iteration methods for saddle-point problems from time-harmonic eddy current models, Numer. Linear Algebra Appl. 19, 914–936 (2012). [12] Z.-Z. Bai, and M. Rozlozn´Ik, On the numerical behavior of matrix splitting iteration methods for solving linear systems, SIAM J. Numer. Anal. 53, 1716– 1737 (2015). 872

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[13] C. Greif, and J. Varah, Iterative solution of cyclically reduced systems arising from discretization of the three-dimensional convection-diffusion equation, SIAM J. Sci. Comput. 19, 1918–1940 (1998). [14] C. Greif, and J. Varah, Block stationary methods for nonsymmetric cyclically reduced systems arising from three-dimensional elliptic equations, SIAM J. Matrix Anal. Appl. 20, 1038–1059 (1999). [15] A. Feriani, F. Perotti, and V. Simoncini, Iterative system solvers for the frequency analysis of linear mechanical systems, Comput. Methods Appl. Mech. Engrg. 190, 1719–1739 (2000). [16] A. Frommer, T. Lippert, B. Medeke, and K. Schilling, Numerical challenges in lattice quantum chromodynamics, Lecture notes in computational science and engineering, Springer, Heidelberg, 15 (2000). [17] C.-X. Li, and S.-L. Wu, A modified GHSS method for non-Hermitian positive definite linear systems, Jpn. J. Ind. Appl. Math. 29, 253–268 (2012). [18] C.-X. Li, and S.-L. Wu, A single-step HSS method for non-Hermitian positive definite linear systems, Appl. Math. Lett. 44, 26–29 (2015). [19] W. Li, Y.-P. Liu, and X.-F. Peng, The generalized HSS method for solving singular linear systems, J. Comput. Appl. Math. 236, 2338–2353 (2012). [20] H.N. Pour, and H.S. Goughery, New Hermitian and skew-Hermitian splitting methods for non-Hermitian positive-definite linear systems, Numer. Algorithms 69, 207–225 (2015). [21] Y. Saad, Iterative Methods for Sparse Linear Systems, SIAM, 2003. [22] Y.-J. Wu, X. Li, and J.-Y. Yuan, A non-alternating preconditioned HSS iteration method for non-Hermitian positive definite linear systems, Comput. Appl. Math. DOI: 10.1007/s40314-015-0231-6 (2015). Min-Li Zeng School of Mathematics, Putian University, Putian 351100, P.R. China e-mail: [email protected] Guo-Feng Zhang School of Mathematics and Statistics, Lanzhou University, Lanzhou 730000, P.R. China

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Aggregating of Interval-valued Intuitionistic Uncertain Linguistic Variables based on Archimedean t-norm and It Applications in Group Decision Makings† a

Juan Lia,∗ , Xiao-Lei Zhangb , Zeng-Tai Gongc Department of Mathematics, Baoji University of Arts and Sciences, Baoji, Shaanxi, 721013, P.R. China b Group of Mathematics, Dongfanghong Middle School of Dingxi, Dingxi, Gansu 743000, China b College of Mathematics and Statistics, Northwest Normal University, Lanzhou, Gansu 730070 China

Abstract With respect to multiple attribute group decision making (MAGDM) problems in which the attribute weights and the expert weights take the form of real numbers and the attribute values take the form of interval-valued intuitionistic uncertain linguistic variable, we propose group decision making methods of interval-valued intuitionistic uncertain linguistic variable based on Archimedean t-norm and Choquet integral. First, we introduce some concepts of fuzzy measure and interval-valued intuitionistic uncertain lingistic variables based on Archimedean t-norm. Then, interval-valued intuitionistic uncertain linguistic weighted average(geometric) and interval-valued intuitionistic uncertain lingistic ordered weighted average operator based on Archimedean t-norm are developed. Furthermore, some desirable properties of these operators, such as commutativity, idempotency and monotonicity have been studied, and interval-valued intuitionistic uncertain linguistic hybrid average operator based on Archimedean tnorm are developed. Based on these operators, two methods for multiple attribute group decision making problems with intuitionistic uncertain linguistic information have been proposed. Finally, an illustrative example is given to verify the developed approaches and demonstrate their practicality and effectiveness. Keywords: Interval-valued intuitionistic fuzzy sets; aggregation operators; Archimedean t-norm; group decision making. 1. Introduction Multiple attribute decision making (MADM) problems are an important research topic in decision theory. Because the objects are fuzzy and uncertain, the attributes involved in decision problems are not always expressed as real numbers, and some better suited to be denoted by fuzzy numbers, such as interval numbers, triangular fuzzy numbers, trapezoidal fuzzy numbers, linguistic numbers on uncertain linguistic variables, and intuitionistic fuzzy numbers. Because Zadeh initially proposed the basic model of fuzzy decision making based on the theory of fuzzy mathematics, fuzzy MADM has been receiving more and more attention. The fuzzy set (FS) theory proposed by Zadeh [1] was a very good tool to research the fuzzy MADM problems, the fuzzy set is used to character the fuzziness just by membership degree. Different from fuzzy set,there is another parameter: non-membership degree in intuitionistic fuzzy set (IFS) which is proposed by Atanassov [2,3]. Clearly, the IFS can describe and character the fuzzy essence of the objective world more accurately [2] than the fuzzy set, and has received more and more attention since its appearance. Later, Atanassov and Gargov [4,5] further introduced the interval-valued intuitionistic fuzzy set (IVIFS), which is a generalization of the IFS. The fundamental characteristic of the IVIFS is that the values of its membership function and non-membership function are interval numbers rather real numbers. On the other hand, in the real decision-making, there are many qualitative attributes which are difficult to give attribute values by quantitative measurement. While, they are easy to give linguistic assessment values. However, for a linguistic assessment value, it is usually implied that the membership †

Supported by the National Natural Scientific Foundation of China (11461062) and the Scientific Research Project of Education Department of Shaanxi Gansu Provincial Government (16JK1047). ∗ Corresponding Author: Juan Li. Tel.: +8613992795738. Email Address: [email protected] 874

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degree is one, and the non-membership degree and hesitation degree of decision makers cannot be expressed. On the basis of the intuitionistic fuzzy set and the linguistic assessment set, Wang and Li [6] proposed the concept of intuitionistic linguistic set, the intuitionistic linguistic number, the intuitionistic two-semantic and the Hamming distance between two intuitionistic two-semantics and ranked the alternatives by calculating the comprehensive membership degree to the ideal solution for each alternative. As an aggregation function, the Choquet integral [7] with respect to fuzzy measures has performed successfully in multicriteria decision making (MCDM). There are many works on the Choquet integral of single-valued functions. It is of interest to combine the Choquet integral and the IFS theory or MCDM under intuitionistic fuzzy environment, because, by doing this, we cannot only deals with the imprecise and uncertain decision information but also efficiently take into account the various interactions among the decision criteria. Base on Archimedean t-conorm and t-norm [8-11], and the aggregation functions for the classical fuzzy sets (FSs), Beliakov et al. gave some operations about intuitionistic fuzzy sets, proposed two general concepts for constructing other types of aggregation operators for intuitionistic fuzzy sets (IFSs) extending the existing methods and showed that the operators obtained by using the Lukasiewicz t-norm are consistent with the ones on ordinary FSs. We can find above aggregation operators are all based on different relationships of the aggregated arguments, which can provide more choices for the decision makers. In summary, based on intuitionistic linguistic set proposed by Wang and Li [6] , combining intervalvalued uncertain linguistic variables, Archimedean t-norm and Choquet integral, in this paper, we propose the interval-valued uncertain linguistic variables based on Archimedean t-norm and investigate the MAGDM problems. First, we introduced some concepts of fuzzy measure and interval-valued intuitionistic uncertain linguistic variables based on Archimedean t-norm. Then, interval-valued intuitionistic uncertain linguistic weighted average(geometric) operator based on Archimedean t-norm ,interval-valued intuitionistic uncertain linguistic ordered weighted average(geometric) operator based on Archimedean t-norm are developed. Furthermore, some desirable properties of these operators, such as commutativity, idempotency and monotonicity have been studied, and an intuitionistic uncertain linguistic hybrid average(geometric) operator based on Archimedean t-norm was developed. Based on these operators, two methods for multiple attribute group decision making problems with intuitionistic uncertain linguistic information have been proposed. 1. Preliminaries A function T : [0, 1] × [0, 1] → [0, 1] is called a t-norm if it satisfies the following four conditions (ref. to [8,9]): 1) T (1, x) = x, for all x. 2) T (x, y) = T (y, x), for all x and y. 3) T (x, T (y, z)) = T (T (x, y), z), for all x, y and z. 0 0 0 0 0 0 4) x 6 x , y 6 y implies T (x, y) 6 T (x , y ), x, y, x , y ∈ [0, 1]. A function S : [0, 1] × [0, 1] → [0, 1] is called a t-conorm if it satisfies the following four conditions (ref. to [8,9]): 1) S(0, x) = x, for all x. 2) S(x, y) = S(y, x), for all x and y. 3) S(x, S(y, z)) = S(S(x, y), z), for all x, y and z. 0 0 0 0 0 0 4) x 6 x , y 6 y implies S(x, y) 6 S(x , y ), x, y, x , y ∈ [0, 1]. A t-norm function T (x, y) is called Archimedean t-norm if it is continuous and T (x, x) < x for all x ∈ [0, 1]. An Archimedean t-norm is called strictly Archimedean t-norm if it is strictly increasing in each variable for x, y ∈ (0, 1), (ref. to [8,9]). A t-conorm function S(x, y) is called Archimedean t-conorm if it is continuous and S(x, x) > x for all x ∈ [0, 1]. An Archimedean t-conorm is called strictly Archimedean t-conorm if it is strictly increasing in each variable for x, y ∈ (0, 1), (ref. to [8,9]). A mapping N : [0, 1] → [0, 1] is called negation operator§if N is decreasing and N (0) = 1 , N (1) = 0. Suppose that S = (s0 , s1 , . . . , sl−1 ) is a finite and fully ordered discrete term set, where l is an odd number. In real situations, l would be equal to 3,5,7,9,etc. For example, when l = 7, a set S can be given 875

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Juan Li, Xiao-Lei Zhang and Zeng-Tai Gong: Aggregating of Interval-valued Intuitionistic Uncertain Linguistic...

as follows: S = (s0 , s1 , s2 , s3 , s4 , s5 , s6 ) = {very poor, poor, slightly poor, f air, slightly good, good, very good}. For any linguistic set S = (s0 , s1 , . . . , sl−1 ), the relationship between the element si and its subscript i is strictly monotonically increasing [12,13,14], so the function can be defined as follows: f : si = f (i). Clearly, the function f (i) is a strictly monotonically increasing function about a subscript i. To preserve all of the given information, the discrete linguistic label S = (s0 , s1 , . . . , sl−1 ) is extended to a continuous linguistic label S = {sα | α ∈ R}, which satisfies the above characteristics. Suppose se = [sa , sb ], sa , sb ∈ S and a 6 b, sa and sb are the lower limit and the upper limit of se, respectively. Then, se is called an uncertain linguistic variable [15]. For each x, µA (x) and νA (x) are closed intervals and their lower and upper end points are, respectively, U L U denoted by µL A (x), µA (x),νA (x),νA (x). We can denote by U L U A = {hx[[sθ(x) , sτ (x) ], [µL A (x), µA (x)], [νA (x), νA (x)]]i|x ∈ X}, U L L where sθ(x) , sτ (x) ∈ S, 0 6 µU A (x) + νA (x) 6 1, x ∈ X, µA (x) > 0 and νA (x) > 0. For each element x, we can compute its hesitation interval of x to uncertain linguistic variable [sθ(x) , sτ (x) ] as: L U U L L πA (x) = [πA (x), πA (x)] = [1 − νA (x) − µU A (x), 1 − νA (x) − µA (x)]. U L U Definition 1.1. Let A = {hx[[sθ(x) , sτ (x) ], [µL A (x), µA (x)], [νA (x), νA (x)]]i|x ∈ X} be IV IU LS, 6U L U Tuple < [sθ(x) , sτ (x) ], [µL A (x), µA (x)], [νA (x), νA (x)] > is called an interval-valued intuitionistic uncertain linguistic number (IV IU LN ), and A can also be viewed as a collection of the interval-valued intuitionistic U L uncertain linguistic variables. So it can also be expressed as A = {h[sθ(x) , sτ (x) ], [µL A (x), µA (x)], [νA (x), U (x)]i|x ∈ X}. νA

2. Interval-valued intuitionistic uncertain linguistic variables based on Archimedean t-norm Definition 2.1. Let α ei = h[sθ(αi ) , sτ (αi ) ], [µL (αi ), µU (αi )], [ν L (αi ), ν U (αi )]i (i = 1, 2) be two intervalvalued intuitionistic uncertain linguistic variables based on Archimedean t-norm and λ > 0, we can define the operational rules about α e1 and α e2 as follows (1) α e1 ⊕ α e2 = h[sθ(α1 )+θ(α2 ) , sτ (α1 )+τ (α2 ) ], [S(µL (α1 ), µL (α2 )), S(µU (α1 ), µU (α2 ))], [T (ν L (α1 ), ν L (α2 )), T (ν U (α1 ), ν U (α2 ))]i = h[sθ(α1 )+θ(α2 ) , sτ (α1 )+τ (α2 ) ], [h−1 (h(µL (α1 )) + h(µL (α2 ))), h−1 (h(µU (α1 )) + h(µU (α2 )))], [g −1 (g(ν L (α1 )) + g(ν L (α2 ))), g −1 (g(ν U (α1 )) + g(ν U (α2 )))]i; (2) α e1 ⊗ α e2 = h[sθ(α1 )×θ(α2 ) , sτ (α1 )×τ (α2 ) ], [T (µL (α1 ), µL (α2 )), T (µU (α1 ), µU (α2 ))], L [S(ν (α1 ), ν L (α2 )), S(ν U (α1 ), ν U (α2 ))]i = h[sθ(α1 )×θ(α2 ) , sτ (α1 )×τ (α2 ) ], [g −1 (g(µL (α1 )) + g(µL (α2 ))), g −1 (g(µU (α1 )) + g(µU (α2 )))], [h−1 (h(ν L (α1 )) + h(ν L (α2 ))), h−1 (h(ν U (α1 )) + h(ν U (α2 )))]i; (3)λe α1 = h[sλ×θ(α1 ) , sλ×τ (α1 ) ], [h−1 (λh(µL (α1 ))), h−1 (λh(µU (α1 )))], [g −1 (λg(ν L (α1 ))), g −1 (λg(ν U (α1 )))]i; (4) α e1λ = h[s(θ(α1 ))λ , s(τ (α1 ))λ ], [g −1 (λg(µL (α1 ))), g −1 (λg(µU (α1 )))], [h−1 (λh(ν L (α1 ))), h−1 (λh(ν U (α1 )))]i. Theorem 2.1. For any two interval-valued intuitionistic uncertain linguistic numbers based on Archimedean t-norm α ei = h[sθ(αi ) , sτ (αi ) ], [µL (αi ), µU (αi )], [ν L (αi ), ν U (αi )]i (i = 1, 2), it can be proved the calculation rules shown as follows: (a) α e1 ⊕ α e2 = α e2 ⊕ α e1 ; (b) α e1 ⊗ α e2 = α e2 ⊗ α e1 ; (c) λ(e α1 ⊕ α e2 ) = λe α1 ⊕ λe α2 , λ > 0; (d) λ1 α e1 ⊕ λ2 α e1 = (λ1 + λ2 )e α1 , λ1 , λ2 > 0; (e) α e1λ1 ⊗ α e1λ2 = (e α1 )λ1 +λ2 , λ1 , λ2 > 0; (f ) α e1λ ⊗ α e2λ = (e α1 ⊗ α e2 )λ , λ > 0. Definition 2.2 [16]. Let α e1 = h[sθ(α1 ) , sτ (α1 ) ], [µL (α1 ), µU (α1 )], [ν L (α1 ), ν U (α1 )]i be an intervalvalued intuitionistic uncertain linguistic number based on Archimedean t-norm, an expected value E(e α1 ) of α e1 can be represented as follows 876

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Juan Li, Xiao-Lei Zhang and Zeng-Tai Gong: Aggregating of Interval-valued Intuitionistic Uncertain Linguistic...

1 µL (α1 ) + µU (α1 ) ν L (α1 ) + ν U (α1 ) ×( +1− ) × s(θ(α1 )+τ (α1 ))/2 2 2 2 = s((θ(α1 )+τ (α1 ))×(µL (α1 )+µU (α1 )+2−ν L (α1 )−ν U (α1 ))/8 . Definition 2.3 [16]. Let α e1 = h[sθ(α1 ) , sτ (α1 ) ], [µL (α1 ), µU (α1 )], [ν L (α1 ), ν U (α1 )]i be an intervalvalued intuitionistic uncertain linguistic number based on Archimedean t-norm, an accuracy function H(e α1 ) can be represented as follows E(e α1 ) =

µL (α1 ) + µU (α1 ) ν L (α1 ) + ν U (α1 ) + ) × s(θ(α1 )+τ (α1 ))/2 2 2 = s((θ(α1 )+τ (α1 ))×(µL (α1 )+µU (α1 )+ν L (α1 )+ν U (α1 ))/4 . Definition 2.4 [16]. If α ei = h[sθ(αi ) , sτ (αi ) ], [µL (αi ), µU (αi )], [ν L (αi ), ν U (αi )]i (i = 1, 2) are any two interval-valued intuitionistic uncertain linguistic numbers based on Archimedean t-norm, then (1) If E(e α1 ) > E(e α2 ), then α e1  α e2 . (2) If E(e α1 ) = E(e α2 ), then: If H(e α1 ) > H(e α2 ), then α e1  α e2 . If H(e α1 ) = H(e α2 ), then α e1 = α e2 . H(e α1 ) = (

3. Aggregating of the interval-valued intuitionistic uncertain linguistic variables based on Archimedean t-norm aggregating operator and fuzzy measure A fuzzy measure on X is a set function µ : P (X) → [0, 1] such that (i) µ(∅) = 0, µ(X) = 1; (ii) A, B ⊆ X, A ⊆ B implies µ(A) 6 µ(B). Let A, B ∈ P (X), A ∩ B = ∅. If fuzzy measure g satisfies the following conditions: g(A ∪ B) = g(A) + g(B) + λg(A)g(B) and λ ∈ (−1, ∞). If λ = 0, then g is an additive measure, which means there is no interaction between coalitions A and B . If λ > 0, then g is called a supperadditive measure, which reflects there exists complementary interaction between coalitions A and B. If −1 < λ < 0, then g is said to be a subadditive measure, which shows there exists redundancy interaction between coalitions A and B. S Let X = {x1 , x2 , ...xn } be a attribute index set, if i, j = 1, 2, ..., n and i 6= j, xi ∩ xj = ∅, ni=1 xi = X, then  1 Qn λ 6= 0, λ ( i=1 [1 + λg(xi )] − 1) g(X) = P n g(x ) λ = 0, i i=1 where xi , g(xi ) is called a fuzzy measure function, and it indicates the importance degree of xi . Qn From g(X) = 1, we know λ is determined by λ + 1 = i=1 (1 + λg(xi )). Based on the the above operational rules, we propose weighted average (geometric) operator, ordered weighted average (geometric) operator and hybrid average (geometric) operator for interval-valued intuitionistic uncertain linguistic variables based on Archimedean t-norm in this part. Definition 3.1. Let α ei = h[sθ(αi ) , sτ (αi ) ], [µL (αi ), µU (αi )], [ν L (αi ), ν U (αi )]i (i = 1, 2, . . . , n) be a collection of interval-valued intuitionistic uncertain linguistic variables based on Archimedean t-norm, and AT S − IV IU LW A(or AT S − IV IU LW GA) : Ωn → Ω, if AT S − IV IU LW Aµ (e α1 , α e2 , . . . , α en ) =

n X

µj α ej ,

j=1

(or AT S − IV IU LW GAµ (e α1 , α e2 , . . . , α en ) =

n Y

(e αj )µj , )

j=1

877

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Juan Li, Xiao-Lei Zhang and Zeng-Tai Gong: Aggregating of Interval-valued Intuitionistic Uncertain Linguistic...

where Ω is the set of all interval-valued intuitionistic uncertain linguistic variables based on Archimedean t-norm, and µ = (µ1 , µ2 , . . . , µn )T is the weightedP vector of α ej (j = 1, 2, . . . , n), µ is a fuzzy measure on X n with µj ∈ [0, 1], µj = µ(A(j) ) − µ(A(j+1) ), and j=1 µj = 1, A(j) = (j, . . . , n) with A(n+1) = ∅, then AT S − IV IU LW A (AT S − IV IU LW GA) is called the interval-valued intuitionistic uncertain linguistic weighted average (weighted geometric average) operator based on Archimedean t-norm. Specifically, if µ = ( n1 , n1 , . . . , n1 ), then AT S − IV IU LW A operator is interval-valued intuitionistic uncertain linguistic arithmetic average operator based on Archimedean t-norm (AT S − IV IU LAA)µ AT S − IV IU LAA(e α1 , α e2 , . . . , α en ) =

1 (e α1 ⊕ α e2 ⊕ . . . ⊕ α en ). n

Theorem 3.1. Let α ei = h[sθ(αi ) , sτ (αi ) ], [µL (αi ), µU (αi )], [ν L (αi ), ν U (αi )]i (i = 1, 2, . . . , n) be a collection of interval-valued intuitionistic uncertain linguistic variables based on Archimedean t-norm, then, the result aggregated by Definition 3.1 is still an interval-valued intuitionistic uncertain linguistic variable based on Archimedean t-norm, and AT S − IV IU LW Aµ (e α1 , α e2 , . . . , α en ) = h[sPnj=1 (µj ×θ(αj )) , sPnj=1 (µj ×τ (αj )) ], P P P P [h−1 ( nj=1 µj h(µL (αj ))), h−1 ( nj=1 µj h(µU (αj )))], [g −1 ( nj=1 µj g(ν L (αj ))), g −1 ( nj=1 µj g(ν U (αj )))]i. (or AT S − IV IU LW GAµ (e α1 , α e2 , . . . , α en ) = h[sQnj=1 ((θ(αj ))µj ) , sQnj=1 ((τ (αj ))µj ) ], P P P P [g −1 ( nj=1 µj g(µL (αj ))), g −1 ( nj=1 µj g(µU (αj )))], [h−1 ( nj=1 µj h(ν L (αj ))), h−1 ( nj=1 µj h(ν U (αj )))]i, ) where Pn µ = (µ1 , µ2 , . . . , µn ) is a fuzzy measure on X with µj ∈ [0, 1], µj = µ(A(j) ) − µ(A(j+1) ), and e1 6 α e2 6 · · · 6 j=1 µj = 1, the parentheses used for indices represent a permutation on X such that α α en , A(j) = (j, ..., n), A(n+1) = ∅. Theorem 3.1 can be proven by mathematical induction. The steps in the proof are shown as follows: Proof. We only prove the case of AT S − IV IU LW Aµ (e α1 , α e2 , . . . , α en ). (1) When n = 1, obviously, it is right. (2) When n = 2, µ1 α e1 = h[sµ1 ×θ(α1 ) , sµ1 ×τ (α1 ) ], [h−1 (µ1 h(µL (α1 ))), h−1 (µ1 h(µU (α1 )))], [g −1 (µ1 g(ν L (α1 ))), g −1 (ν1 g(ν U (α1 )))]i. µ2 α e2 = h[sµ2 ×θ(α2 ) , sµ2 ×τ (α2 ) ], [h−1 (µ2 h(µL (α2 ))), h−1 (µ2 h(µU (α2 )))], −1 [g (µ2 g(ν L (α2 ))), g −1 (ν2 g(ν U (α2 )))]i. AT S − IV IU LW Aµ (e α1 , α e 2 ) = µ1 α e 1 ⊕ µ2 α e2 −1 L = h[sµ1 ×θ(α1 ) , sµ1 ×τ (α1 ) ], [h (µ1 h(µ (α1 ))), h−1 (µ1 h(µU (α1 )))], [g −1 (µ1 g(ν L (α1 ))), g −1 (ν1 g(ν U (α1 )))]i ⊕h[sµ2 ×θ(α2 ) , sµ2 ×τ (α2 ) ], [h−1 (µ2 h(µL (α2 ))), h−1 (µ2 h(µU (α2 )))], [g −1 (µ2 g(ν L (α2 ))), g −1 (ν2 g(ν U (α2 )))]i = h[sµ1 ×θ(α1 )+µ2 ×θ(α2 ) , sµ1 ×τ (α1 )+sµ ×τ (α ) ], 2 2 [h−1 (h(h−1 (µ1 h(µL (α1 )))) + h(h−1 (µ2 h(µL (α2 ))))), h−1 (h(h−1 (µ1 h(µU (α1 )))) + h(h−1 (µ2 h(µU (α2 )))))], [g −1 (g(g −1 (µ1 g(ν L (α1 )))) + g(g −1 (µ2 g(ν L (α2 ))))), g −1 (g(g −1 (µ1 g(ν U (α1 )))) + g(g −1 (µ2 g(ν U (α2 )))))]i P P = h[sP2 (µj ×θ(αj )) , sP2 (µj ×τ (αj )) ], [h−1 ( 2j=1 µj h(µL (αj ))), h−1 ( 2j=1 µj h(µU (αj )))], j=1 j=1 P P [g −1 ( 2j=1 µj g(ν L (αj ))), g −1 ( 2j=1 µj g(ν U (αj )))]i. Therefore, when n = 2, the conclusion is right. (3) Suppose when n = k, the conclusion is right, i.e. P AT S − IV IU LW Aµ (e α1 , α e2 , . . . , α ek ) = h[sPk (µj ×θ(αj )) , sPk (µj ×τ (αj )) ], [h−1 ( kj=1 µj h(µL (αj ))), j=1 j=1 P P P h−1 ( kj=1 µj h(µU (αj )))], [g −1 ( kj=1 µj g(ν L (αj ))), g −1 ( kj=1 µj g(ν U (αj )))]i. Then, when n = k + 1, P AT S − IV IU LW Aµ (e α1 , α e2 , . . . , α ek , α ek+1 ) = h[sPk (µj ×θ(αj )) , sPk (µj ×τ (αj )) ], [h−1 ( kj=1 µj h(µL (αj ))), j=1 j=1 P P P h−1 ( kj=1 µj h(µU (αj )))], [g −1 ( kj=1 µj g(ν L (αj ))), g −1 ( kj=1 µj g(ν U (αj )))]i ⊕h[sµk+1 ×θ(αk+1 ) , sµk+1 ×τ (αk+1 ) ], [h−1 (µk+1 h(µL (αk+1 ))), h−1 (µk+1 h(µU (αk+1 )))], [g −1 (µk+1 g(ν L (αk+1 ))), g −1 (νk+1 g(ν U (αk+1 )))]i = h[sPk (µj ×θ(αj ))+µk+1 ×θ(αk+1 ) , sPk (µj ×τ (αj ))+µk+1 ×τ (αk+1 ) ], j=1

j=1

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Juan Li, Xiao-Lei Zhang and Zeng-Tai Gong: Aggregating of Interval-valued Intuitionistic Uncertain Linguistic...

P [h−1 (h(h−1 ( kj=1 µj h(µL (αj )))) + h(h−1 (µk+1 h(µL (αk+1 ))))), P h−1 (h(h−1 ( kj=1 µj h(µU (αj )))) + h(h−1 (µk+1 h(µU (αk+1 )))))], P [g −1 (g(g −1 ( kj=1 µj g(ν L (αj )))) + g(g −1 (µk+1 g(ν L (αk+1 ))))), P g −1 (g(g −1 ( kj=1 µj g(ν U (αj )))) + g(g −1 (µk+1 g(ν U (αk+1 )))))]i P P U = h[sPk+1 (µj ×θ(αj )) , sPk+1 (µj ×τ (αj )) ], [h−1 ( k+1 µj h(µL (αj ))), h−1 ( k+1 j=1 j=1 µj h(µ (αj )))], j=1 j=1 P Pk+1 L −1 U [g −1 ( k+1 j=1 µj g(ν (αj ))), g ( j=1 µj g(ν (αj )))]i. So, when n = k + 1, the conclusion is right, too. According to steps (1), (2) and (3), we can conclude the conclusion is right for all n. Example 3.1. If N (x) = 1 − x, g(t) = − log t, and Algebraic t-conorm and t-norm [11] defined by T A (x, y) = x · y, S A (x, y) = x + y − xy, then IV IU LW Aµ (e α1 , α e2 , . . . , α en ) = h[sPnj=1 (µj ×θ(αj )) , sPnj=1 (µj ×τ (αj )) ], Qn Q [1 − j=1 (1 − µL (αj ))µj , 1 − nj=1 (1 − µU (αj ))µj ], Qn Q [ j=1 (ν L (αj ))µj , nj=1 (ν U (αj ))µj ]i, which is the interval-valued intuitionistic uncertain linguistic weighted average(IVIULWA) operator. Example 3.2. If N (x) = 1 − x, g(t) = log( 2−t t ), and Einstein t-conorm and t-norm [11] defined by x+y xy E E T (x, y) = 1+(1−x)(1−y) , S (x, y) = 1+xy , then EIV IU LW Aµ (e α1 , α e2 , . . . , α en ) = h[sPnj=1 (µj ×θ(αj )) , sPnj=1 (µj ×τ (αj )) ], Q µj Qn µj µ j Qn U U L (1+µL (αj ))µj − n j=1 (1+µ (αj )) −Qj=1 (1−µ (αj )) j=1 (1−µ (αj )) µ j Qn µ j , Qn µj µj n L U U j )) + j=1 (1−µ (αj )) j=1 j=1 (1+µ (αj )) + j=1 (1−µ (αj )) Qn Qn µj µj L U 2 2 (ν (αj )) (ν (αj )) [ Qn (2−ν L (αj=1))µj +Qn (ν L (α ))µj , Qn (2−ν U (αj=1))µj +Qn (ν U (α ))µj ]i, j j j j j=1 j=1 j=1 j=1 Qn

[ Qj=1 n (1+µL (α

],

which is called the Einstein interval-valued intuitionistic uncertain linguistic weighted average(EIVIUL WA) operator. Example 3.3. If N (x) = 1 − x, g(t) = log( γ+(1−γ)t ) , γ > 0, and Hamacher t-conorm and t-norm t xy H H [11] defined by Tγ (x, y) = γ+(1−γ)(x+y−xy) , γ > 0, Sγ (x, y) = x+y−xy−(1−γ)xy , γ > 0, then 1−(1−γ)xy P P n n HIV IU LW Aµ (e α1 , α e2 , . . . , α en ) = h[s j=1 (µj ×θ(αj )) , s j=1 (µj ×τ (αj )) ], µj Qn µj L L j=1 (1+(γ−1)µ (αj )) − j=1 (1−µ (αj )) Qn µj µj L L j=1 (1+(γ−1)µ (αj )) +(γ−1) j=1 (1−µ (αj )) Qn µj L γ j=1 (ν (αj )) Qn Qn µj µj L L j=1 (1+(γ−1)(1−ν (αj ))) +(γ−1) j=1 (ν (αj ))

Qn

[ Qn [

µj Qn U (1−µU (αj ))µj j=1 (1+(γ−1)µ (αj )) − j=1 Qn µj µj U U j=1 (1+(γ−1)µ (αj )) +(γ−1) j=1 (1−µ (αj )) Qn µj U γ j=1 (ν (αj )) Qn Qn µj µj U U j=1 (1+(γ−1)(1−ν (αj ))) +(γ−1) j=1 (ν (αj ))

Qn

, Qn ,

],

]i,

which is called the Hammer interval-valued intuitionistic uncertain linguistic weighted average(HIVIUL WA) operator. Especially, if γ = 1, then the HIV IU LW A operator reduces to the IV IU LW A operator; if γ = 2, then the HIV IU LW A operator reduces to the EIV IU LW A operator. Example 3.4. If N (x) = 1 − x, g(t) = log( γγ−1 t −1 ) , γ > 1, and Frank t-conorm and t-norm [11] x

y

1−x

1−y

−1) −1) defined by TγF (x, y) = logγ (1 + (γ −1)(γ ) , γ > 1, SγF (x, y) = 1 − logγ (1 + (γ −1)(γ ) , γ > 1, γ−1 γ−1 then F IV IU LW Aµ (e α1 , α e2 , . . . , α en ) = h[sPnj=1 (µj ×θ(αj )) , sPnj=1 (µj ×τ (αj )) ], Q Q L U [1 − logγ (1 + nj=1 (γ 1−µ (αj ) − 1)µj ), 1 − logγ (1 + nj=1 (γ 1−µ (αj ) − 1)µj )], Q Q U L [logγ (1 + nj=1 (γ ν (αj ) − 1)µj ), logγ (1 + nj=1 (γ ν (αj ) − 1)µj )]i, which is called the Frank interval-valued intuitionistic uncertain linguistic weighted average(FIVIULWA) operator. Especially, if γ → 1, then the F IV IU LW A operator reduces to the IV IU LW A operator. Example 3.5. If N (x) p = 1 − x2 , g(t) = − log t, and Algebraic t-conorm and t-norm [11] defined by A A T2 (x, y) = xy , S2 (x, y) = 1 − (1 − x2 )(1 − y 2 ), then IV IU LW Aµ (e α1 , α e2 , . . . , α en ) = h[sPnj=1 (µj ×θ(αj )) , sPnj=1 (µj ×τ (αj )) ], q q Qn Q [ 1 − j=1 (1 − (µL (αj ))2 )µj , 1 − nj=1 (1 − (µU (αj ))2 )µj ], Q Q [ nj=1 (ν L (αj ))µj , nj=1 (ν U (αj ))µj ]i, which is the interval-valued intuitionistic uncertain linguistic weighted average(IVIULWA) operator.

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J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 24, NO.5, 2018, COPYRIGHT 2018 EUDOXUS PRESS, LLC

Juan Li, Xiao-Lei Zhang and Zeng-Tai Gong: Aggregating of Interval-valued Intuitionistic Uncertain Linguistic...

Example 3.6. If N (x) = 1 − x2 , g(t) = log( 2−t t ), and Einstein t-conorm and t-norm [11] defined by q 2 2 xy x +y T2E (x, y) = 1+(1−x)(1−y) , S2E (x, y) = 1+x2 y2 , then EIV IU LW Aµ (e α1 , α e2 , . . . , α en ) = h[sPnj=1 (µj ×θ(αj )) , sPnj=1 (µj ×τ (αj )) ], r Qn r Qn Q Qn n L 2 µj L 2 µj U 2 µj U 2 µj j=1 (1+(µ (αj )) ) −Qj=1 (1−(µ (αj )) ) j=1 (1+(µ (αj )) ) −Qj=1 (1−(µ (αj )) ) Q Q ], [ µj , µj µj n n n n L 2 U L 2 U 2 (1−(µ (α )) ) (1−(µ (α ))2 )µj (1+(µ (α )) ) + (1+(µ (α )) ) + j=1

j

j

j=1

Q 2 n (ν L (αj ))µj Q [ n (2−ν L (αj=1))µj +Qn (ν L (α ))µj j j j=1 j=1

,

j

j=1

Q µj U 2 n j=1 (ν (αj )) Qn µj Qn µj U U j=1 (2−ν (αj )) + j=1 (ν (αj ))

j

j=1

]i,

which is called the Einstein interval-valued intuitionistic uncertain linguistic weighted average(EIVIUL WA) operator. Example 3.7. If N (x) = 1 − x2 , g(t) = log( γ+(1−γ)t ) , γ > 0, and Hamacher t-conorm and t-norm t q 2 2 2 2 −x y −(1−γ)x2 y 2 xy H H defined by T2γ (x, y) = γ+(1−γ)(x+y−xy) , γ > 0, S2γ (x, y) = x +y 1−(1−γ)x , γ > 0, then 2 y2 P P n n HIV IU LW Aµ (e α1 , α e2 , . . . , α en ) = h[s j=1 (µj ×θ(αj )) , s j=1 (µj ×τ (αj )) ], r Qn r Qn Q µj Qn L 2 (1−(µL (αj ))2 )µj (1+(γ−1)(µU (αj ))2 )µj − n (1−(µU (αj ))2 )µj j=1 (1+(γ−1)(µ (αj )) ) − j=1 j=1 Qn ], [ Qn (1+(γ−1)(µL (α ))2 )µj +(γ−1) Qn (1−(µL (α ))2 )µj , Qn j=1 µj U 2 (1−(µU (α ))2 )µj (1+(γ−1)(µ (α )) ) +(γ−1) j=1

j

j=1

j

Q µj L γ n j=1 (ν (αj )) Qn Q [ n (1+(γ−1)(1−ν L (α µj µj L j ))) +(γ−1) j=1 (ν (αj )) j=1

j

j=1

,

j=1

Q µj U γ n j=1 (ν (αj )) Qn Qn µj µj U U j=1 (ν (αj )) j=1 (1+(γ−1)(1−ν (αj ))) +(γ−1)

j

]i,

which is called the Hammer interval-valued intuitionistic uncertain linguistic weighted average(HIVIUL WA) operator. Especially, if γ = 1, then the HIV IU LW A operator reduces to the IV IU LW A operator; if γ = 2, then the HIV IU LW A operator reduces to the EIV IU LW A operator. Example 3.8. If N (x) = 1 − x2 , g(t) = log( γγ−1 t −1 ) , γ > 1, and Frank t-conorm and t-norm defined q x y 1−x2 −1)(γ 1−y 2 −1) F (x, y) = log (1 + (γ −1)(γ −1) ) , γ > 1, S F (x, y) = by T2γ 1 − logγ (1 + (γ ) , γ > 1, then γ 2γ γ−1 γ−1 P P n n F IV IU LW Aµ (e α1 , α e2 , . . . , α en ) = h[s j=1 (µj ×θ(αj )) , s j=1 (µj ×τ (αj )) ], q q Qn Q L 2 U 2 [ 1 − logγ (1 + j=1 (γ 1−(µ (αj )) − 1)µj ), 1 − logγ (1 + nj=1 (γ 1−(µ (αj )) − 1)µj )], Q Q U L [logγ (1 + nj=1 (γ ν (αj ) − 1)µj ), logγ (1 + nj=1 (γ ν (αj ) − 1)µj )]i, which is called the Frank interval-valued intuitionistic uncertain linguistic weighted average(FIVIULWA) operator. Especially, if γ → 1, then the F IV IU LW A operator reduces to the IV IU LW A operator. The AT S − IV IU LW A operator has the following properties, such as idempotency, monotonicity, bounded, and so on. In the following section, we propose the interval-valued intuitionistic uncertain linguistic ordered weighted average operator based on Archimedean t-norm. Definition 3.2. Let α ei = h[sθ(αi ) , sτ (αi ) ], [µL (αi ), µU (αi )], [ν L (αi ), ν U (αi )]i (i = 1, 2, . . . , n) be a collection of interval-valued intuitionistic uncertain linguistic variables based on Archimedean t-norm, and AT S − IV IU LOW A (AT S − IV IU LOW G) : Ωn → Ω, if AT S − IV IU LOW Aw (e α1 , α e2 , . . . , α en ) =

n X

wj α eσj ,

j=1

(or AT S − IV IU LOW Gw (e α1 , α e2 , . . . , α en ) =

n Y

w

α eσjj , )

j=1

where Ω is the set of all interval-valued intuitionistic uncertain linguistic variables based on Archimedean T t-norm, and w = (w1 , w2 , . . . , wP n ) is an associated weighted vector with AT S − IV IU LOW A (AT S − IV IU LOW G) and wj ∈ [0, 1], nj=1 wj = 1. If (σ1 , σ2 , . . . , σn ) is any permutation of (1, 2, . . . , n), such that α eσj−1  α eσj for all j = 1, 2, . . . , n, then AT S − IV IU LOW A (AT S − IV IU LOW G) operator is called the interval-valued intuitionistic uncertain linguistic ordered weighted average (weighted geometric) operator based on Archimedean t-norm. wj is decided only by the jth position in the aggregation process. Therefore, w can also be called the position-weighted vector. The position-weighted vector w can be determined according to actual needs, or it can be determined based on the fuzzy semantic quantitative operator[17] or the combination number[18]. 880

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Juan Li, Xiao-Lei Zhang and Zeng-Tai Gong: Aggregating of Interval-valued Intuitionistic Uncertain Linguistic...

Theorem 3.2. Let α ei = h[sθ(αi ) , sτ (αi ) ], [µL (αi ), µU (αi )], [ν L (αi ), ν U (αi )]i (i = 1, 2, . . . , n) be a collection of interval-valued intuitionistic uncertain linguistic variables based on Archimedean t-norm, the result aggregated from the Definition 3.2 is still an interval-valued intuitionistic uncertain linguistic variable based on Archimedean t-norm, and P AT S − IV IU LOW Aw (e α1 , α e2 , . . . , α en ) = h[sPnj=1 (wj ×θ(ασj )) , sPnj=1 (µj ×τ (ασj )) ], [h−1 ( nj=1 wj h(µL P P P (ασj ))), h−1 ( nj=1 wj h(µU (ασj )))], [g −1 ( nj=1 wj g(ν L (ασj ))), g −1 ( nj=1 wj g(ν U (ασj )))]i. P (or AT S−IV IU LOW Gw (e α1 , α e2 , . . . , α en ) = h[sQnj=1 ((θ(ασ ))wj ) , sQnj=1 ((τ (ασ ))wj ) ], [g −1 ( nj=1 wj g(µL j j P P P (ασj ))), g −1 ( nj=1 wj g(µU (ασj )))], [h−1 ( nj=1 wj h(ν L (ασj ))), h−1 ( nj=1 wj h(ν U (ασj )))]i, ) where w = (w1 , w2 , . . . , wn )T is Pan associated weighted vector with AT S − IV IU LOW A (AT S − IV IU LOW G) and wj ∈ [0, 1], nj=1 wj = 1. (σ1 , σ2 , . . . , σn ) is any permutation of (1, 2, . . . , n), such that α eσj−1  α eσj for all j = 1, 2, . . . , n. In a similar way to the proof of Theorem 3.1, Theorem 3.2 can be proven by mathematical induction, and the proof steps are therefore omitted. The AT S − IV IU LOW A operator has some similar properties to the AT S − IV IU LW A operator, such as idempotency, monotonicity, bounded, commutativity and so on.). Definition 3.3. Let α ei = h[sθ(αi ) , sτ (αi ) ], [µL (αi ), µU (αi )], [ν L (αi ), ν U (αi )]i (i = 1, 2, . . . , n) be a collection of interval-valued intuitionistic uncertain linguistic variables based on Archimedean t-norm, and AT S − IV IU LHA : Ωn → Ω, if AT S − IV IU LHAµ,w (e α1 , α e2 , . . . , α en ) =

n X

wj βeσj ,

j=1

(or AT S − IV IU LHGµ,w (e α1 , α e2 , . . . , α en ) =

n Y

(βeσj )wj , )

j=1

where Ω is the set of all interval-valued intuitionistic uncertain linguistic variables based on Archimedean t-norm, and w = (w1 , w2 , . . . , wn )T is an associated weighted vector with AT S − IV IU LHA (AT S − P IV IU LHG) and wj ∈ [0, 1], nj=1 wj = 1. If βeσj is the jth the largest of the interval-valued intuitionistic uncertain linguistic weighted argument βek (βek = nµk α ek , k = 1, 2, . . . , n), µ = (µ1 , µ2 , . . . , µn )T is the weighted vector ej (j = 1, 2, . . . , n), µ is a fuzzy measure on X with µj ∈ [0, 1], µj = µ(A(j) ) − Pn of α µ(A(j+1) ), j=1 µj = 1, A(j) = (j, . . . , n) with A(n+1) = ∅ and n is the balancing coefficient, then AT S − IV IU LHA (AT S − IV IU LHG) is called the interval-valued intuitionistic uncertain linguistic hybrid average (hybrid geometric) operator based on Archimedean t-norm. Theorem 3.3. Let α ei = h[sθ(αi ) , sτ (αi ) ], [µL (αi ), µU (αi )], [ν L (αi ), ν U (αi )]i (i = 1, 2, . . . , n) be a collection of interval-valued intuitionistic uncertain linguistic variables based on Archimedean t-norm, then, the result aggregated from Definition 3.3 is still an interval-valued intuitionistic uncertain linguistic variable based on Archimedean t-norm, and P AT S − IV IU LHAµ,w (e α1 , α e2 , . . . , α en ) = h[sPnj=1 (wj ×θ(βσj )) , sPnj=1 (wj ×τ (βσj )) ], [h−1 ( nj=1 wj h(µL (βσj ))), P P P h−1 ( nj=1 wj h(µU (βσj )))], [g −1 ( nj=1 wj g(ν L (βσj ))), g −1 ( nj=1 wj g(ν U (βσj )))]i. P (or AT S − IV IU LHGµ,w (e α1 , α e2 , . . . , α en ) = h[sQnj=1 ((θ(βσ ))wj ) , sQnj=1 ((τ (βσ ))wj ) ], [g −1 ( nj=1 wj g(µL (βσj j j P P P ))), g −1 ( nj=1 wj g(µU (βσj )))], [h−1 ( nj=1 wj h(ν L (βσj ))), h−1 ( nj=1 wj h(ν U (βσj )))]i), where w = (w1 , w2 , . . . , wn )T is an associated weighted vector with AT S−IV IU LHA (AT S−IV IU LHG) P and wj ∈ [0, 1], nj=1 wj = 1. If βeσj is the jth the largest of the interval-valued intuitionistic uncertain linguistic weighted argument βek (βek = nµk α ek , k = 1, 2, . . . , n), µ = (µ1 , µ2 , . . . , µn )T is the weighted vector of α ej (j = 1, 2, . . . , n), µ is a fuzzy measure on X with µj ∈ [0, 1], µj = µ(A(j) ) − µ(A(j+1) ), P n j=1 µj = 1, A(j) = (j, . . . , n) with A(n+1) = ∅ and n is the balancing coefficient. 4. Aggregating of the interval-valued intuitionistic uncertain linguistic variables in group decision making based on Archimedean t-norm Consider a multiple attribute group decision making problem with intuitionistic linguistic information. Let A = {A1 , A2 , . . . , Am } be a discrete set of alternatives and C = {C1 , C2 , . . . , Cn } be the 881

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set of attributes. µ = (µ1 , µ2 , . P . . , µn )T is the weighting vector of the attribute Cj (j = 1, 2, . . . , n), n where µj > 0 (j = 1, 2, . . . .n), j=1 µj = 1. Let D = {D1 , D2 , . . . , Dp } be the set of decision makT ers and λ = (λ1 , λ2 , · · · , λP ) be the weighted vector of the decision makers, where λk > 0 (k = Pp k] ek 1, 2, . . . , p), rij m×n is the decision matrix, where k=1 λk = 1. Suppose that R = [e k L U L U L U reij = h[αijk , αijk ], [µijk (αi ), µijk (αi )], [νijk (αi ), νijk (αi )]i takes the form of the intuitionistic uncertain linguistic variable given by the decision maker Dk for an alternative Ai with respect to an attribute Cj , L L U L U U U L U and 0 6 µL ijk 6 1, 0 6 νijk 6 1, µijk 6 µijk , νijk 6 νijk , µijk + νijk 6 1, αijk , αijk ∈ S. Then, the ranking of alternatives is required. In the following section, we will apply the above operators to solve multiple group decision making problems based on intuitionistic uncertain linguistic information. There are two methods, which are as follows: 4.1 The method of aggregating the attribute values first Step 1. We can utilize the AT S − IV IU LW A operator( or AT S − IV IU LW GA operator) k ,r k ,...,r k ) given by each decision maker with respect to each to aggregate the attribute values (e ri1 ei2 ein alternative into a comprehensive attribute value reik , reik

= AT S − IV

k k k IU LW Aµ (e ri1 , rei2 , . . . , rein )

=

n X

k µj reij

j=1

where i = 1, 2, . . . , m; k = 1, 2, . . . , p, µ = (µ1 , µ2 , . . . , µn )T , µj = µ(A(j) ) − µ(A(j+1) ) ∈ [0, 1] is the attribute weight vector, A(j) = (j, . . . , n) with A(n+1) = ∅. Step 2. We can utilize the AT S − IV IU LHA operator( or AT S − IV IU LHG operator) operator to aggregate the information (e ri1 , rei2 , . . . , reip ) of each decision maker into a collective value rei for each alternative, p X rei = AT S − IV IU LHAλ,w (e ri1 , rei2 , . . . , reip ) = wkebσi k k=1

i = 1, 2, . . . , m, where w = (w1 , w2 , . . . , wp )T is an associated weighted vector with AT S − IV IU LHA P (or AT S − IV IU LHG) and wk ∈ [0, 1], pk=1 wk = 1. ebσi k is the kth the largest of the interval-valued intuitionistic uncertain linguistic weighted argument ebli (ebli = pλl reil or ebli = P reilpλl , l = 1, 2, . . . , p), λ = (λ1 , λ2 , . . . , λp )T is the weighted vector of the decision makers, λk ∈ [0, 1], nk=1 λk = 1 and p is the balancing coefficient. Step 3. By Definition 2.2, we can calculate the expected value E(e ri )(i = 1, 2, . . . , m) of the collective value rei (i = 1, 2, . . . , m), rank all of the alternatives Ai (i = 1, 2, . . . , m) and then select the best one(s). If there is no difference between two expected values E(e ri ) and E(e rj ), then by Definition 2.3, we must calculate the accuracy degrees H(e ri ) and H(e rj ) of the collective overall preference values rei and rej , respectively, and then rank the alternatives Ai and Aj in accordance with the accuracy function values H(e ri ) and H(e rj ). Step 4. Rank all of the alternatives Ai (i = 1, 2, . . . , m) and select the best one(s) in accordance with E(e ri ) and H(e ri ) (i = 1, 2, . . . , m). 4.2 The method of first aggregating the information from different decision makers Step 1. We can utilize the AT S − IV IU LW A operator( or AT S − IV IU LW GA operator) to ek (k = 1, 2, . . . , p) into a collective decision matrix R e = [e aggregate all of the decision matrices R rij ]m×n , reij = AT S − IV

p 1 2 IU LW Aλ (e rij , reij , . . . , reij )

=

p X

k λk reij

k=1

i = 1, 2, . . . , m; j = 1, 2, . . . , n where λ = (λ1 , λ2 , . . . , λp )T is the weighted vector of the decision makers.

882

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Juan Li, Xiao-Lei Zhang and Zeng-Tai Gong: Aggregating of Interval-valued Intuitionistic Uncertain Linguistic...

Step 2. We can utilize the AT S − IV IU LHA operator( or AT S − IV IU LHG operator) to derive the collective overall preference value rei (i = 1, 2, . . . , m) of the alternative Ai , rei = AT S − IV IU LHAµ,w = (e ri1 , rei2 , . . . , rein ) =

n X

wjebiσj

j=1

i = 1, 2, . . . , m, where w = (w1 , w2 , . . . , wn )T is an associated weighted vector with AT S − IV IU LHA P (or AT S − IV IU LHG) and wj ∈ [0, 1], nj=1 wj = 1. ebiσj is the jth the largest of the interval-valued intuitionistic uncertain linguistic weighted argument ebik (ebik = nµk reik or ebik = rik )nµk , k = 1, 2, . . . , n), P(e n T µ = (µ1 , µ2 , . . . , µn ) is the weight vector of reij (j = 1, 2, . . . , n), µj ∈ [0, 1], j=1 µj = 1, and n is the balancing coefficient. Step 3. By Definition 2.2, we can calculate the expected value E(e ri )(i = 1, 2, . . . , m) of the collective value rei (i = 1, 2, . . . , m), rank all of the alternatives Ai (i = 1, 2, . . . , m) and then select the best one(s). If there is no difference between two expected values E(e ri ) and E(e rj ), then by Definition 2.3, we must calculate the accuracy degrees H(e ri ) and H(e rj ) of the collective overall preference values rei and rej , respectively, and then rank the alternatives Ai and Aj in accordance with the accuracy function values H(e ri ) and H(e rj ). Step 4. Rank all of the alternatives Ai (i = 1, 2, . . . , m) and select the best one(s) in accordance with E(e ri ) and H(e ri ) (i = 1, 2, . . . , m). 5. An application example Let us consider an investment company that wants to invest a sum of money in the best option. There is a panel with four possible alternatives in which to invest the money: (1) A1 is a car company. (2) A2 is a computer company. (3) A3 is a TV company. (4) A4 is a food company. The investment company must make a decision according to the following four attributes: (1) C1 is the risk index. (2) C2 is the growth index. (3) C3 is the social-political impact index. (4) C4 is the environment impact index. The fuzzy measure of each attributes: g(x1 ) = 0.4, g(x2 ) = 0.25, g(x3 ) = 0.37, g(x4 ) = 0.2. Since λ+1=

n Y (1 + λg(xi )), i = 1, 2, 3, 4, i=1

we have λ = −0.44, then g(x1 , x2 ) = 0.60, g(x1 , x3 ) = 0.70, g(x1 , x4 ) = 0.56, g(x2 , x3 ) = 0.68, g(x2 , x4 ) = 0.43, g(x3 , x4 ) = 0.54, g(x1 , x2 , x4 ) = 0.88, g(x2 , x3 , x4 ) = 0.75, g(x1 , x3 , x4 ) = 0.73, g(x1 , x2 , x3 , x4 ) = 1. The four possible alternatives A1 , A2 , A3 , A4 are evaluated by three decision makers Dk (k = 1, 2, 3) (whose weight vector is λ = (0.4, 0.32, 0.28)T ) using the linguistic term set S = (s0 , s1 , s2 , s3 , s4 , s5 , s6 ) k = about the interval-valued intuitionistic uncertain linguistic variables based on Archimedean t-norm reij L , αU ], [µL , µU ], [ν L , ν U ])i under the above four attributes. The decision matrices R k] ek = [e h[αijk rij 4×4 ijk ijk ijk ijk ijk (k = 1, 2, 3)) are listed in Tables 1-3. To obtain the best alternative(s), we can use the two methods introduced to obtain the selection results. Let N (x) = 1 − x, g(t) = log( γ+(1−γ)t ) , γ = 2, i.e. g(t) = log( 2−t t t ). 5.1 The method of aggregating the attribute values first Step 1. k ,r k ,r k ,r k) (e ri1 ei2 ei3 ei4

Table 4.

We can utilize the AT S − IV IU LW GA operator to aggregate the attribute values into a comprehensive attribute value reik , we can obtain the aggregating results shown in 883

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Juan Li, Xiao-Lei Zhang and Zeng-Tai Gong: Aggregating of Interval-valued Intuitionistic Uncertain Linguistic...

Step 2. We can utilize the AT S − IV IU LHG operator to aggregate the information (e ri1 , rei2 , rei3 ) of each decision maker into a collective value rei for each alternative. Suppose the position weight is w = (0.25, 0.5, 0.25)T . We can obtain re1 = h[s3.97 , s4.85 ], [0.73, 0.78], [0.12, 0.16]i, re2 = h[s4.12 , s5.03 ], [0.61, 0.75], [0.17, 0.14]i, re3 = h[s4.28 , s4.40 ], [0.72, 0.75], [0.10, 0.18]i, re4 = h[s3.28 , s3.97 ], [0.63, 0.70], [0.16, 0.22]i. Step 3. We can calculate the expected value E(e ri )(i = 1, 2, 3, 4) of the collective interval-valued intuitionistic uncertain linguistic variables values based on Archimedean t-norm rei (i = 1, 2, 3, 4) as E(e r1 ) = s3.976 , E(e r2 ) = s3.701 , E(e r3 ) = s3.499 , E(e r4 ) = s2.984 . Step 4. By definition 4.2, we can rank all of the alternatives (A1 , A2 , A3 , A4 ) in accordance with the expected values (E(e r1 ), E(e r2 ), E(e r3 ), E(e r4 )) of the collective interval-valued intuitionistic uncertain linguistic variables values based on Archimedean t-norm (e r1 , re2 , re3 , re4 ). We can obtain A1  A2  A3  A4 , and thus, the most desirable alternatives is A1 . 5.2 The method of first aggregating the information from different decision makers Step 1. We can utilize the AT S − IV IU LW GA operator to aggregate all of the decision matrie e = [e ces Rk (k = 1, 2, 3) into a collective decision matrix R rij ]4×4 . We can obtain the aggregation result shown in Table 5. Step 2. We can utilize rei = AT S − IV IU LHGµ,w = (e ri1 , rei2 , . . . , rei4 ) =

4 Y

(ebiσj )wj (i = 1, 2, 3, 4)

j=1

to derive the collective overall values rei of alterative Ai , where ebiσj is the jth the largest of the intervalvalued intuitionistic uncertain linguistic weighted argument ebik (ebik = (e rik )4µk , k = 1, 2, 3, 4). Suppose the position weight is w = (0.15, 022., 0.35, 0.28)T . We can obtain re1 = h[s4.65 , s5.11 ], [(0.72, 0.78], [0.10, 0.15]i, re2 = h[s4.74 , s5.17 ], [(0.63, 0.70], [0.15, 0.20]i, re3 = h[s4.34 , s4.45 ], [(0.70, 0.76], [0.11, 0.17]i, re4 = h[s3.71 , s4.08 ], [(0.66, 0.73], [0.13, 0.23]i. Step 3. We can calculate the expected value E(e ri )(i = 1, 2, 3, 4) of the collective interval-valued intuitionistic uncertain linguistic variables values based on Archimedean t-norm rei (i = 1, 2, 3, 4) as E(e r1 ) = s3.971 , E(e r2 ) = s3.694 , E(e r3 ) = s3.494 , E(e r4 ) = s2.952 . Step 4. By definition 4.2, we can rank all of the alternatives (A1 , A2 , A3 , A4 ) in accordance with the expected values (E(e r1 ), E(e r2 ), E(e r3 ), E(e r4 )) of the collective interval-valued intuitionistic uncertain linguistic variables values based on Archimedean t-norm (e r1 , re2 , re3 , re4 ). We can obtain A1  A2  A3  A4 , and thus, the most desirable alternatives is A1 . e1 Table 1: Decision matrix R A1 A2 A3 A4

C1 h[s4 , s5 ], [0.7, 0.8], [0.1, 0.2]i h[s5 , s5 ], [0.6, 0.6], [0.1, 0.2]i h[s4 , s4 ], [0.7, 0.7], [0.2, 0.2]i h[s3 , s4 ], [0.6, 0.7], [0.2, 0.3]i

C2 h[s5 , s5 ], [0.6, 0.6], [0.1, 0.3]i h[s5 , s6 ], [0.7, 0.7], [0.2, 0.2]i h[s4 , s4 ], [0.7, 0.8], [0.1, 0.2]i h[s3 , s3 ], [0.5, 0.6], [0.2, 0.3]i

C3 h[s5 , s6 ], [0.8, 0.8], [0.1, 0.1]i h[s4 , s5 ], [0.5, 0.6], [0.2, 0.3]i h[s5 , s5 ], [0.7, 0.7], [0.1, 0.2]i h[s4 , s4 ], [0.6, 0.7], [0.2, 0.3]i

C4 h[s4 , s4 ], [0.8, 0.8], [0.1, 0.1]i h[s4 , s5 ], [0.5, 0.6], [0.1, 0.3]i h[s5 , s5 ], [0.7, 0.8], [0.1, 0.3]i h[s3 , s4 ], [0.7, 0.7], [0.2, 0.2]i

e2 Table 2: Decision matrix R A1 A2 A3 A4

C1 h[s5 , s6 ], [0.6, 0.7], [0.1, 0.1]i h[s5 , s5 ], [0.5, 0.7], [0.2, 0.2]i h[s5 , s5 ], [0.6, 0.7], [0.0, 0.2]i h[s5 , s5 ], [0.7, 0.8], [0.1, 0.2]i

C2 h[s5 , s5 ], [0.7, 0.7], [0.1, 0.1]i h[s4 , s5 ], [0.6, 0.7], [0.2, 0.2]i h[s4 , s5 ], [0.8, 0.9], [0.1, 0.1]i h[s4 , s4 ], [0.5, 0.6], [0.2, 0.3]i

C3 h[s4 , s5 ], [0.7, 0.9], [0.2, 0.1]i h[s5 , s4 ], [0.7, 0.7], [0.1, 0.2]i h[s4 , s4 ], [0.6, 0.6], [0.2, 0.2]i h[s3 , s3 ], [0.9, 0.9], [0.0, 0.1]i

C4 h[s5 , s5 ], [0.7, 0.8], [0.1, 0.2]i h[s6 , s6 ], [0.6, 0.7], [0.1, 0.1]i h[s4 , s4 ], [0.7, 0.7], [0.2, 0.2]i h[s3 , s4 ], [0.8, 0.8], [0.1, 0.2]i

e3 Table 3: Decision matrix R A1 A2 A3 A4

C1 h[s5 , s5 ], [0.7, 0.8], [0.1, 0.1]i h[s5 , s6 ], [0.6, 0.7], [0.1, 0.2]i h[s5 , s5 ], [0.8, 0.8], [0.0, 0.1]i h[s4 , s5 ], [0.8, 0.9], [0.1, 0.1]i

C2 h[s5 , s5 ], [0.8, 0.9], [0.1, 0.1]i h[s5 , s6 ], [0.7, 0.7], [0.1, 0.2]i h[s5 , s5 ], [0.7, 0.8], [0.1, 0.2]i h[s4 , s4 ], [0.8, 0.8], [0.0, 0.2]i 884

C3 h[s5 , s5 ], [0.8, 0.9], [0.1, 0.1]i h[s5 , s5 ], [0.8, 0.8], [0.1, 0.1]i h[s4 , s4 ], [0.7, 0.8], [0.1, 0.2]i h[s4 , s5 ], [0.8, 0.8], [0.0, 0.1]i

C4 h[s5 , s6 ], [0.7, 0.8], [0.2, 0.2]i h[s5 , s5 ], [0.9, 0.9], [0.1, 0.1]i h[s4 , s4 ], [0.7, 0.8], [0.1, 0.1]i h[s4 , s5 ], [0.7, 0.7], [0.1, 0.2]i

Juan Li et al 874-885

J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 24, NO.5, 2018, COPYRIGHT 2018 EUDOXUS PRESS, LLC

Juan Li, Xiao-Lei Zhang and Zeng-Tai Gong: Aggregating of Interval-valued Intuitionistic Uncertain Linguistic...

Table 4: The comprehensive attribute value reik A1 A2 A3 A4

D1 h[s4.07 , s4.89 ], [0.58, 0.67], [0.12, 0.16]i h[s4.46 , s5.10 ], [0.61, 0.75], [0.1, 0.2]i h[s3.82 , s4.19 ], [0.68, 0.79], [0.07, 0.10]i h[s4.37 , s4.94 ], [0.72, 0.80], [0.10, 0.16]i

D2 h[s4.21 , s4.97 ], [0.64, 0.80], [0.11, 0.18]i h[s4.21 , s4.86 ], [0.67, 0.78], [0.13, 0.17]i h[s4.30 , s4.81 ], [0.61, 0.74], [0.12, 0.22]i h[s4.79 , s5.04 ], [0.61, 0.74], [0.08, 0.20]i

D3 h[s5.21 , s5.43 ], [0.56, 0.73], [0.14, 0.19]i h[s4.60 , s4.96 ], [0.68, 0.80], [0.13, 0.20]i h[s4.33 , s4.62 ], [0.71, 0.84], [0.11, 0.23]i h[s4.57 , s5.08 ], [0.64, 0.76], [0.12, 0.19]i

e = (C1 , C2 , C3 , C4 ). Table 5: The collective decision matrix R A1 A2 A3 A4

C1 h[s4.57 , s5.28 ], [0.67, 0.77], [0.10, 0.14]i h[s5.00 , s5.28 ], [0.57, 0.66], [0.13, 0.20]i h[s4.57 , s4.57 ], [0.70, 0.73], [0.09, 0.17]i h[s3.81 , s4.57 ], [0.69, 0.79], [0.14, 0.21]i

C2 h[s5.00 , s5.00 ], [0.69, 0.71], [0.10, 0.19]i h[s4.68 , s5.68 ], [0.67, 0.70], [0.17, 0.20]i h[s4.28 , s4.57 ], [0.73, 0.83], [0.10, 0.17]i h[s3.57 , s3.57 ], [0.58, 0.65], [0.14, 0.27]i

A1 A2 A3 A4

C3 h[s4.68 , s5.38 ], [0.83, 0.86], [0.07, 0.10]i h[s4.57 , s4.68 ], [0.64, 0.69], [0.14, 0.21]i h[s4.37 , s4.37 ], [0.67, 0.70], [0.13, 0.20]i h[s3.67 , s3.92 ], [0.74, 0.79], [0.09, 0.19]i

C4 h[s4.57 , s4.83 ], [0.74, 0.80], [0.13, 0.16]i h[s4.83 , s5.28 ], [0.63, 0.71], [0.19, 0.19]i h[s4.37 , s4.37 ], [0.70, 0.77], [0.13, 0.17]i h[s3.50 , s4.28 ], [0.73, 0.73], [0.14, 0.20]i

References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18]

L.A. Zadeh, Fuzzy sets, Information and Control 8 (1965) 338-356. K.T. Atanassov, Intuitionistic fuzzy sets, Fuzzy Sets and Systems 20 (1986) 87-96. K.T. Atanassov, More on intuitionistic fuzzy sets, Fuzzy Sets and Systems 33 (1989) 37-46. K.T. Atanassov, G.Gargov, Interval-valued intuitionistic fuzzy sets, Fuzzy Sets and Systems 3 (1989) 343-349. K.T. Atanassov, Operators over interval-valued intuitionistic fuzzy sets, Fuzzy Sets and System 64 (1994) 159-174. J.Q. Wang, J.J. Li, The multi-cricteria group decision making method based on multi-granularity intuitionistic two semantics, Science and Technology Information 33 (2009) 8-9. G. Choquet, Theory of capacities, Annales de l’institut Fourier 5 (1953) 131-295. G. Klir, B. Yuan, Fuzzy sets and fuzzy logic: theory and applications. NJ: Prentice Hall, Upper Saddle River, 1995. H.T. Nguyen, E.A. Walker, A first course in fuzzy logic. Boca Raton, Florida: CRC Press, 1997. E.P. Klement, R. Mesiar(Eds.), Logical, algebraic, analytic, and probabilistic aspects of triangular norms. New York: Elsevier, 2005. G. Beliakov, H. Bustince, D.P. Goswami, U.K. Calvo, Aggregation Functions: A guide for Practitioners. Springer, Heidelberg Berlin, New York, 2007. M.M. Xia, Z.S. Xu, B. Zhu, Some issues on intuitionistic fuzzy aggregation operators based on Archimedean t-conorm and t-norm, Knowledge-Based Systems 31 (2012) 78-88 F. Herrera, E. Herrera-Vieddma, J.L. Verdegay, A model of consensus in group decision making under linguistic assessments, Fuzzy Sets and Systems 79 (1996) 73-87. Z.S. Xu, A note on linguistic hybrid arithmetic averaging operator in multiple attribute group decision making with linguistic information, Group Decision and Negotiation 15 (2006) 593-604. Z.S. Xu, Induced uncertain linguistic OWA operators applied to group decision making,Information Fusion Fusion 7 (2006) 231-238. Peide Liu, Fang Jin, Methods for aggregating intuitionistic uncertain linguistic variables and their application to group decision making, Information Sciences 205 (2012) 58-71. F. Herrera, E. Herrera-Vieddma, F. Chiclana, Multiperson decision-making based on mulitiplicative preference relations, European Journal of Operational Research 129 (2001) 3606-3618. Yu Wang, Zeshui Xu, A new method of giving OWA weights, Mathematics in Practice and Theory 38 (2008) 51-61.

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Non-integer variable order dynamic equations on time scales involving Caputo-Fabrizio type differential operator Dumitru Baleanu1∗, Mehdi Nategh2†,

Abstract This work deals with the conecept of a Caputo-Fabrizio type non-integer variable order differential opertor on time scales that involves a non-singular kernel. A measure theoretic discussion on the limit cases for the order of differentiation is presented. Then, corresponding to the fractional derivative, we discuss on an integral for constant and variable orders. Beside the obtaining solutions to some dynamic problems on time scales involving the proposed derivative, a fractional folrmulation for the viscoelastic oscillation problem is studied and its conversion into a third order dynamic equation is presented. keywords: Time scales, Fractional calculus, Caputo-Fabrizio derivative, Non-integer variable order derivative and integral , Dirac delta functional, Viscoelastic Oscillation. MSC 2010: 34N05, 26A33.

1

Introduction

This work deals with two theories, namely non-integer order calculus (or as what it is called, fractional calculus) and ∆-calculus (calculus on time scales). The first one, originally is as old as the classical calculus in the sense of Leibnitz and the latter which was started by an effort in 80’s, was aimed to unify the difference and the differential. For an overview on the trends and achievements in ∆-calculus, see [5]. In the recent years, to propose a non-integer order counterpart of ∆-calculus, a number of efforts have been made [1, 3]. One of the main challenges for such proposals was to overcome the limitations caused by the nature of the time scales, since a typical cluster of points may appear in a variety of scattered or dense patterns. In view of the real world applications of the non-local fractional calculus, one presumption is to find a suitable kernel for the purpose of the better description to a class of phenomena. ∗

1. Corresponding author. Department of Mathematics and Computer Sciences, C ¸ ankaya University, Ankara, Turkey. E-mail: [email protected], Tel: +903122331420, Fax: +903122331025. † 2. Department of Mathematics, University of Mazandran, Babolsar, Iran. E-mail: [email protected], Tel: +989216246609, Fax: +981135302460.

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1

INTRODUCTION

2

Recently, a Caputo-like non-integer order derivative with non-singular kernel is proposed by Caputo and Fabrizio [7], " # Z α(t − τ ) M (α) t 0 α f (τ ) exp − dτ, (1.1) Dt f (t) = 1−α a 1−α in which, M (α) is called, the normalization function and it satisfies M (0) = M (1) = 1. As is is graphically illustrated in [7], this new definition of the non-integer order derivative seems to be more appropriate in describing a process which is affected by its past. Indeed, compare to Caputo derivative, the new derivative with an exponential kernel, shows rapid stabilization with respect to the memory effect. A substitution of the exponential kernel with Mittag-Leffler type, to define another nonlocal derivative, is suggested by Atangana and Baleanu with an application to the non-integer order heat transfer model [2]. Fractional integral of variable order with singular kernel first was introduced in 1993 by Samko and Ross in [14] using direct and Fourier based approaches. The direct appraoch formulation for the derivative reads Z 1 d t f (τ ) α(t) Da+ = . (1.2) Γ(1 − α(t)) dt a (t − τ )−α(τ ) with the assumtion that 0 ⇒ t + F (d(T (x), T (y))) ≤ F (d(x, y))

(1.1)

for all x, y ∈ X and some t > 0. Here F : R+ → R is a mapping satisfying the following properties: 1∗ Corresponding

authors.

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M. NAZAM, M. ARSHAD, C. PARK, A. HUSSAIN, AND D. SHIN

(F1 ) F is strictly increasing. (F2 ) For each sequence {an } of positive numbers limn→∞ an = 0 if and only if limn→∞ F (an ) = −∞. (F3 ) There exists θ ∈ (0, 1) such that limα→0+ (α)θ F (α) = 0. Wardowski [22] established the following result using F -contraction: Theorem 1. [22] Let (X, d) be a complete metric space and T : X → X be an F contraction. Then T has a unique fixed point υ ∈ X and for every X0 ∈ X a sequence {T n (X0 )} is convergent to υ. We denote by ∆F the set of all functions satisfying the conditions (F1 ) − (F3 ). Example 1. [22] Let F : R+ → R be given by the formula F (α) = ln α. It is clear that F satisfies (F1 ) − (F3 ) for any κ ∈ (0, 1). Each mapping T : X → X satisfying (1.1) is an F -contraction such that d(T (x), T (y)) ≤ e−τ d(x, y) for all x, y ∈ X with T (x) 6= T (y). Obviously, for all x, y ∈ X such that T (x) = T (y), the inequality d(T (x), T (y)) ≤ e−τ d(x, y) holds, that is, T is a Banach contraction. Remark 1. From (F1 ) and (1.1) it is easy to conclude that every F -contraction is necessarily continuous. 2. Main result We begin with the following definitions. Definition 2. Let (X, d) be a metric space. A mapping T : X → X is called a rational type F -contraction if, for all x, y ∈ X, we have τ + F (d(T (x), T (y))) ≤ F (N (x, y)) ,

(2.1)

where F ∈ ∆F and τ > 0, and   d(x, T (x))d(y, T (y)) d(x, T (x))d(y, T (y)) , . N (x, y) = max d(x, y), 1 + d(x, y) 1 + d(T (x), T (y)) Definition 3. Let (X, d) be a metric space. Mappings S, T : X → X are called a pair of rational type F -contractions if for all x, y ∈ X, we have τ + F (d(S(x), T (y))) ≤ F (M (x, y)) ,

(2.2)

where F ∈ ∆F and τ > 0, and   d(x, S(x))d(y, T (y)) d(x, S(x))d(y, T (y)) M (x, y) = max d(x, y), , . 1 + d(x, y) 1 + d(S(x), T (y)) The following theorem is one of our main results. Theorem 2. Let (X, d) be a complete metric space and S, T : X → X be a pair of mappings such that (1) (S, T ) is a pair of continuous mappings,

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SYSTEM OF INTEGRAL EQUATIONS VIA FIXED POINT METHOD

(2) (S, T ) is a pair of rational type F -contractions. Then there exists a common fixed point υ of a pair (S, T ) in X. Proof. We begin with the following observation: M (x, y) = 0 if and only if x = y is a common fixed point of (S, T ). Indeed, if x = y is a common fixed point of (S, T ), then T (y) = T (x) = x = y = S(y) = S(x) and   d(x, S(x))d(y, T (y)) d(x, S(x))d(y, T (y)) M (x, y) = max d(x, x), , = 0. 1 + d(x, y) 1 + d(S(x), T (y)) Conversely, if M (x, y) = 0, it is easy to check that x = y is a fixed point of S and T . In order to find common fixed points of S and T for the situation when M (x, y) > 0 for all x, y ∈ X with x 6= y, we construct an iterative sequence {xn } of points in X such a way that x2i+1 = S(x2i ) and x2i+2 = T (x2i+1 ) where i = 0, 1, 2, . . . . If xn 6= xn+1 for all n ≥ 0, then from contractive condition (2.2), we get F (d(x2i+1 , x2i+2 )) = F (d(S(x2i ), T (x2i+1 ))) ≤ F (M (x2i , x2i+1 )) − τ for all i ∈ N ∪ {0}, where   d(x2i , S(x2i ))d(x2i+1 , T (x2i+1 ))    ,   d(x2i , x2i+1 ),  1 + d(x 2i , x2i+1 ) M (x2i , x2i+1 ) = max d(x2i , S(x2i ))d(x2i+1 , T (x2i+1 ))       1 + d(S(x2i ), T (x2i+1 ))   d(x2i , x2i+1 )d(x2i+1 , x2i+2 ) d(x2i , x2i+1 )d(x2i+1 , x2i+2 ) = max d(x2i , x2i+1 ), , 1 + d(x2i , x2i+1 ) 1 + d(x2i+1 , x2i+2 ) ≤ max {d(x2i , x2i+1 ), d(x2i+1 , x2i+2 )} . If M (x2i , x2i+1 ) = d(x2i+1 , x2i+2 ), then F (d(x2i+1 , x2i+2 )) ≤ F (d(x2i+1 , x2i+2 )) − τ, which is a contradiction due to F1 . Therefore, F (d(x2i+1 , x2i+2 )) ≤ F (d(x2i , x2i+1 )) − τ, for all i ∈ N ∪ {0}. Hence F (d(xn+1 , xn+2 )) ≤ F (d(xn , xn+1 )) − τ,

(2.3)

for all n ∈ N ∪ {0}. By (2.3), we obtain F (d(xn , xn+1 )) ≤ F (d(xn−2 , xn−1 )) − 2τ. Repeating these steps, we get F (d(xn , xn+1 )) ≤ F (d(x0 , x1 )) − nτ.

(2.4)

From (2.4), we obtain limn→∞ F (d(xn , xn+1 )) = −∞. Since F ∈ ∆F , lim d(xn , xn+1 ) = 0.

n→∞

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(2.5)

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M. NAZAM, M. ARSHAD, C. PARK, A. HUSSAIN, AND D. SHIN

From the property (F3 ) of F -contraction, there exists κ ∈ (0, 1) such that lim ((d(xn , xn+1 ))κ F (d(xn , xn+1 ))) = 0.

(2.6)

n→∞

By (2.4), for all n ∈ N, we obtain (d(xn , xn+1 ))κ (F (d(xn , xn+1 )) − F (d(x0 , x1 ))) ≤ − (d(xn , xn+1 ))κ nτ ≤ 0.

(2.7)

Considering (2.5), (2.6) and letting n → ∞ in (2.7), we have lim (n (d(xn , xn+1 ))κ ) = 0.

(2.8)

n→∞

Since (2.8) holds, there exists n1 ∈ N, such that n (d(xn , xn+1 ))κ ≤ 1 for all n ≥ n1 or, 1 d(xn , xn+1 ) ≤ 1 for all n ≥ n1 . (2.9) nκ Using (2.9), we get for m > n ≥ n1 , d(xn , xm ) ≤ d(xn , xn+1 ) + d(xn+1 , xn+2 ) + d(xn+2 , xn+3 ) + · · · + d(xm−1 , xm ); m−1 ∞ X X = d(xi , xi+1 ) ≤ d(xi , xi+1 ) ≤

i=n ∞ X

i=n

1

i=n

ik

1

. P∞

1

1 entails limn,m→∞ d(xn , xm ) = 0. Hence {xn } is a iκ Cauchy sequence in (X, d). Since (X, d) is a complete metric space, there exists υ ∈ X such that xn → υ as n → ∞, moreover, x2n+1 → υ and x2n+2 → υ. Now the continuity of T implies

The convergence of the series

i=n

υ = lim xn = lim x2n+1 = lim x2n+2 = lim T (x2n+1 ) = T ( lim x2n+1 ) = T (υ). n→∞

n→∞

n→∞

n→∞

n→∞

Analogously, υ = S(υ). Thus we have S(υ) = T (υ) = υ. Hence (S, T ) has a common fixed point. Now we show that υ is the unique common fixed point of S and T . Assume the contrary, that is, there exists ω ∈ X such that ω = T (ω). From the contractive condition (2.2), we have τ + F (d(S(υ), T (ω))) ≤ F (M (υ, ω)) ,

(2.10)

where   d(υ, S(υ))d(ω, T (ω)) d(υ, S(υ))d(ω, T (ω)) M (υ, ω) = max d(υ, ω), , . 1 + d(υ, y) 1 + d(S(υ), T (ω)) From (2.10), we have τ + F (d(υ, ω)) ≤ F (d(υ, ω)) , ,

(2.11)

which implies d(υ, ω) < d(υ, ω), which is a contradiction. Hence υ = ω and υ is a unique common fixed point of a pair (S, T ).  1477

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SYSTEM OF INTEGRAL EQUATIONS VIA FIXED POINT METHOD

Let us consider an example to illustrate Theorem 2. Example 2. Let X = [1, ∞] and d (x, y) = |x − y| . Then (X, d) is a complete metric space. Define the mappings S, T : X → X as follows: S(x) = x2 and T (x) = x + 3 for all x ∈ X. Define the function F : R+ → R by F (x) = ln(x) for all x ∈ R+ > 0 and τ > 0. Then the contractive condition (2.2) is satisfied. Indeed, for all x, y ∈ X, the following inequality τ + ln (d(S(x), T (y))) ≤ ln (M (x, y)) holds. Particularly, for x = 2 and y = 3, we have   d(2, S(2))d(3, T (3)) d(2, S(2))d(3, T (3)) M (2, 3) = max d(2, 3), , 1 + d(2, 3) 1 + d(S2, T 3) = max {1, 3, 2} = 3 and d(S2, T 3) = d(4, 6) = 2. Thus τ + ln (d(S(2), T (3))) = τ + ln 2 ≤ ln (M (2, 3)) = ln 3, which implies τ + F (d(S(x), T (y))) ≤ F (M (x, y)) . Hence all the hypotheses of Theorem 2 are satisfied and so (S, T ) have a common fixed point. By setting S = T , we obtain the following result. Corollary 1. Let (X, d) be a complete metric space and T : X → X be a mapping such that (1) T is a continuous mapping, (2) T is a rational type F -contraction. Then T has a unique fixed point υ in X. Remark 2. If we set N (x, y) = max {d(x, y), d(x, T (x)), d(y, T (y))} in (2.1), then Corollary 1 remains true. Similarly, if we set M (x, y) = max {d(x, y), d(x, T (x)), d(y, S(y))} in (2.2), then Theorem 2 remains true. 3. Application to system of integral equations Now we discuss an application of fixed point theorem, proved in the previous section, in solving the system of Volterra type integral equations. Such a system is given by the

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following equations Zt u(t) = f (t) +

K1 (t, s, u(s))ds.

(3.1)

K2 (t, s, w(s))ds.

(3.2)

0

Zt w(t) = f (t) + 0

for all t ∈ [0, a], and a > 0. We shall show, by using Theorem 2, that the solution of integral equations (3.1) and (3.2) exists. Let C([0, a], R) be the space of all continuous functions defined on [0, a]. For u ∈ C([0, a], R), define supremum norm as: kukτ = sup {u(t)e−τ t }, where τ > 0. Let C([0, a], R) be endowed with the metric t∈[0,a]

dτ (u, v) = sup k |u(t) − v(t)| e−τ t kτ

(3.3)

t∈[0,a]

for all u, v ∈ C([0, a], R). Obviously, C([0, a], R, k · kτ ) is a Banach space. Now we prove the following theorem to ensure the existence of solution of system of integral equations. Theorem 3. Assume the following conditions are satisfied: (i) K1 , K2 : [0, a] × [0, a] × R → R and f, g : [0, a] → R are continuous; (ii) Define the operators Zt Su(t) = f (t) +

K1 (t, s, u(s))ds, 0

Zt T u(t) = f (t) +

K2 (t, s, u(s))ds, 0

and there exists τ ≥ 1 such that |K1 (t, s, u) − K2 (t, s, v)| ≤ τ e−τ [M (u, v)] for all t, s ∈ [0, a] and u, v ∈ C([0, a], R), where M (u, v) = max{|u(t) − v(t)| ,

|u(t) − Su(t)| |v(t) − T v(t)| |u(t) − Su(t)| |v(t) − T v(t)| , }. 1 + |u(t) − v(t)| 1 + |Su(t) − T v(t)|

Then the system of integral equations given in (3.1) and (3.2) has a unique solution.

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SYSTEM OF INTEGRAL EQUATIONS VIA FIXED POINT METHOD

Proof. By assumption (ii), we have Zt |T u(t) − Sv(t)| =

|K1 (t, s, u(s) − K2 (t, s, v(s)))| ds 0

Zt ≤

τ e−τ ([M (u, v)]e−τ s )eτ s ds

0

Zt ≤

τ e−τ kM (u, v)kτ eτ s ds

0

Zt

−τ

≤ τ e kM (u, v)kτ

eτ s ds

0

1 ≤ τ e−2τ kM (u, v)kτ eτ t τ −τ τt ≤ e kM (u, v)kτ e . This implies |T u(t) − Sv(t)| e−τ t ≤ e−τ kM (u, v)kτ , that is, kT u(t) − Sv(t)kτ ≤ e−τ kM (u, v)kτ . So we have τ + ln kT u(t) − Sv(t)kτ ≤ ln kM (u, v)kτ . Thus all the conditions of Theorem 2 are satisfied. Hence the system of integral equations given in (3.1) and (3.2) has a unique common solution.  References [1] M. Abbas, B. Ali, S. Romaguera, Fixed and periodic points of generalized contractions in metric spaces, Fixed Point Theory Appl. 2013, 2013:243 (2013). [2] M. Arshad , A. Shoaib, I. Beg, Fixed point of a pair of contractive dominated mappings on a closed ball in an ordered complete dislocated metric space, Fixed Point Theory Appl. 2013, 2013:115 (2013). [3] R. P. Agarwal, D. ORegan, N. Shahzad, Fixed point theorems for generalized contractive maps of Mei-Keeler type, Mathematische Nachrichten 276 (2004) 3-12. [4] M. Arshad, A. Shoaib, P. Vetro, Common fixed points of a pair of Hardy Rogers type mappings on a closed ball in ordered dislocated metric spaces, J. Function Spaces Appl. 2013 (2013), Article ID 638181. [5] R. Batra, S. Vashistha, Fixed points of an F -contraction on metric spaces with a graph. Int. J. Comput. Math. 91 (2014) 1–8. [6] R. Batra, S. Vashistha, R. Kumar, A coincidence point theorem for F -contractions on metric spaces equipped with an altered distance, J. Math. Comput. Sci. 4 (2014) 826-833. [7] V. Berinde, On the approximation of fixed points of weak contractive mappings, Carpathian J. Math. 19 (2003) 7-22. [8] V. Berinde, Iterative Approximation of Fixed Points, Springer-Verlag, Berlin Heidelberg, 2007.

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M. NAZAM, M. ARSHAD, C. PARK, A. HUSSAIN, AND D. SHIN

[9] D. W. Boyd, J. S. W. Wong, On nonlinear contractions, Proc. Am. Math. Soc. 20 (1969) 458-464. [10] Lj. B. Ciric, A generalization of Banachs contraction principle, Proc. Am. Math. Soc. 45 (1974) 267-273. [11] M. Cosentino, P. Vetro, Fixed point results for F -contractive mappings of Hardy-Rogers-type, Filomat 28 (2014) 715–722. [12] N. Hussain, E. Karapınar, P. Salimi, F. Akbar, α-Admissible mappings and related fixed point theorems, J. Inequal. Appl. 2013, 2013:114 (2013). [13] N. Hussain, P. Salimi, Suzuki-Wardowski type fixed point theorems for α-GF -contractions, Taiwanese J. Math. 18 (2014) 1879–1895. [14] R. Kannan, Some results on fixed points, Bull. Calcutta Math. Soc. 60 (1968) 71–76. [15] E. Kryeyszig, Introductory Functional Analysis with Applications, John Wiley & Sons, New York, 1989. ´ c type generalized F -contractions on complete metric spaces [16] G. Minak, A. Halvaci, I. Altun, Ciri´ and fixed point results, Filomat 28 (2014) 1143–1151. [17] H. Piri, P. Kumam, Some fixed point theorems concerning F -contraction in complete metric spaces, Fixed Point Theory Appl. 2014, 2014:210 (2014). [18] M. Sgroi, C. Vetro, Multi-valued F -contractions and the solution of certain functional and integral equations, Filomat 27 (2013) 1259–1268. [19] P. Salimi, A. Latif, N. Hussain, Modified α-ψ-contractive mappings with applications, Fixed Point Theory Appl. 2013, 2013:151 (2013). [20] N. A. Secelean, Iterated function systems consisting of F -contractions, Fixed Point Theory Appl. 2013, 2013:277 (2013). [21] B. Samet, C. Vetro and P. Vetro, Fixed point theorems for α-ψ-contractive type mappings, Nonlinear Anal. 75 (2012) 2154–2165. [22] D. Wardowski, Fixed point theory of a new type of contractive mappings in complete metric spaces, Fixed Point Theory Appl. 2012, 2012:94 (2012). [23] D. Wardowski and N. Van Dung, Fixed points of F -weak contractions on complete metric space, Demonstr. Math. XLVII (2014) 146–155. Muhammad Nazam Department of Mathematics and Statistics, International Islamic University, H-10, Islamabad, Pakistan E-mail address: [email protected] Muhammad Arshad Department of Mathematics and Statistics, International Islamic University, H-10, Islamabad, Pakistan E-mail address: [email protected] Choonkil Park Research Institute for Natural Sciences, Hanyang University, Seoul 04763, Korea E-mail address: [email protected] Aftab Hussain International Islamic University, Department of Mathematics and Statistics, H-10 Islamabad, Pakistan; Lahore Leads University, Department of Mathematical Sciences, Lahore - 33000, Lahore, Pakistan E-mail address: [email protected] Dong Yun Shin Department of Mathematics, University of Seoul, Seoul 02504, Korea E-mail address: [email protected]

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On continuous Fibonacci functions Hee Sik Kim1 , J. Neggers2 and Keum Sook So3,∗ 1

2

Department of Mathematics, Hanyang University, Seoul, 133-791, Korea Department of Mathematics, University of Alabama, Tuscaloosa, AL 35487-0350, U. S. A 3,∗ Department of Mathematics, Hallym University, Chuncheon 200-702, Korea

Abstract. In this paper, we define and study a function F : [0, ∞) → R and extensions F : R → C, Fe : C → C which are continuous and such that if n ∈ Z, the set of all integers, then F (n) = Fn , the nth Fibonacci number based on F0 = F1 = 1. If x is not an integer and x < 0, then F (x) may be a complex number, e.g., F (−1.5) = 12 +i. If z = a + bi, then Fe(z) = F (a) + iF (b − 1) defines complex Fibonacci numbers. In connection with this function (and in general) we define a Fibonacci derivative of f : R → R as (4f )(x) = f (x + 2) − f (x + 1) − f (x) so that e is given as if (4f )(x) ≡ 0 for all x ∈ R, then f is a (real) Fibonacci function. A complex Fibonacci derivative 4 e 4f (a + bi) = 4f (a) + i 4 f (b − 1) and its properties are discussed in same detail.

1. Introduction Fibonacci-numbers have been studied in many different forms for centuries and the literature on the subject is consequently incredibly vast. One of the amazing qualities of these numbers is the variety of mathematical models where they play some sort of role and where their properties are of importance in elucidating the ability of the model under discussion to explain whatever implications are inherent in it. The fact that the ratio of successive Fibonacci numbers approaches the Golden ratio (section) rather quickly as they go to infinity probably has a good deal to do with the observation made in the previous sentence. Surveys and connections of the type just mentioned are provided in [1] and [2] for a very minimal set of examples of such texts, while in [7] Kim and Neggers showed that there is a mapping D : M → DM on means such that if M is a Fibonacci mean so is DM , that if M is the harmonic mean, then DM is the arithmetic mean, and if M is a Fibonacci mean, then limn→∞ Dn M is the golden section mean. Hyers-Ulam stability of Fibonacci functional equation was studied in [6]. Surprisingly novel perspectives are still available and will presumably continue to be so for the future as long as mathematical investigations continue to be made. In the following the authors of the present paper are making another small offering at the same spot many previous contributors have visited in both recent and more distant pasts. Han et al. [4] considered several properties of Fibonacci sequences in arbitrary groupoids. They discussed Fibonacci sequences in both several groupoids and groups. The present authors [8] introduced the notion of generalized Fibonacci sequences over a groupoid and discussed these in particular for the case where the groupoid contains idempotents and pre-idempotents. Using the notion of Smarandache-type P -algebras they obtained several relations on groupoids which are derived from generalized Fibonacci sequences. In [5] Han et al. discussed Fibonacci functions on the real numbers R, i.e., functions f : R → R such that for all x ∈ R, f (x + 2) = f (x + 1) + f (x), and developed the notion of Fibonacci functions using the concept of f -even and f -odd functions. Moreover, they showed that if f is a Fibonacci function then limx→∞

f (x+1) f (x)

=

√ 1+ 5 2 .

KNS[4445] discussed Fibonacci functions using the (ultimately) periodicity and we also discuss the exponential 0∗

Correspondence: Tel.: +82 33 248 2011, Fax: +82 33 256 2011 (K. S. So). 1482

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Hee Sik Kim, J. Neggers and Keum Sook So∗ Fibonacci functions. Especially, given a non-negative real-valued function, we obtain several exponential Fibonacci functions. In this paper we are interested in describing properties of a function F : R → C, the complex numbers, where F (x) := (Fbx−1c )x−bxc + Fbxc + (x − bxc − 1) and bxc is the greatest integer function. It follows that if x = n ∈ Z, then bnc = n and F (n) = (Fn−1 )0 + Fn + (n − n − 1), where (Fn−1 )0 = 1 implies F (n) = Fn with F0 = F1 = 1 and F−n = (−1)n Fn−2 so that F−1 = −F−1 = 0 for example. If one computes F1 directly, then F1 = F0 + F−1 and 1 = 1 + 0 yields F0 = 1 as well. It also follows that F (x) is not itself a Fibonacci function in the sense that F (x + 2) 6= F (x + 1) + F (x) if x − bxc 6= 0, i.e., if x 6∈ Z. Nevertheless it is a very interesting function which allows one to define continuous Fibonacci numbers in an interesting manner. If one computes F (−1.5) for example, then one finds that F (−1.5) = 1 2 +i,

which suggests that the function F (x) may deserve looking at in the context of the study of the zeta-function.

Given the fact that F : R → C and that F (R) ∩ C 6= ∅, it also becomes a question of interest to study possible complex extensions of F to Fe : C → C where z = a + bi means Fe(z) = F (a) + iF (b − 1), so that if z = a, then Fe(z) = F (a) + iF (−1), where F (−1) = 0 implies Fe(a) = F (a), i.e., Fe is an extension of F . According to this construction we find that Fe(1 + i) = F (1) + iF (0) = 1 + i as one would hope. A second component of the paper is a study of properties of the Fibonacci derivative 4f of a function f : R → R, given by the formula (4f )(x) = f (x + 2) − f (x + 1) − f (x), so that (4f )(x) ≡ 0 means that f is then a Fibonacci function. If one notes that (4f ) exists for any function f : R → R, then a variety of questions may be asked about properties of this operator. For example (4f )(x) ≡ f (x) is a simple type of Fibonacci derivative equation with many types of solutions. Other analogs of standard differential equations may also be addressed. Given that F : R → C is itself a function of interest in this context, 4F : R → C is looked at below. Finally, e : C → C defined by 4F e (z) = 4F e (a + bi) = 4F (a) + i 4 F (b − 1), reduces for b = 0 to a complex version 4F e (a) = 4F (a), i.e., 4 e extends the operator 4, and thus again it is a matter of interest to study the behavior 4F e Fe(z) for complex numbers. of the function 4 Note that because of the very rich structure of relations among the coefficients Fn , we may expect there to eventually be development of an equally rich structure of relations among the various values of F (x) (and Fe(z)) extending the ones already known.

2. Preliminaries A function f defined on the real numbers is said to be a Fibonacci function ([5]) if it satisfies the formula f (x + 2) = f (x + 1) + f (x) for any x ∈ R, where R (as usual) is the set of real numbers. Example 2.1. ([5]) Let f (x) := ax be a Fibonacci function on R where a > 0. Then ax a2 = f (x + 2) = f (x + 1) + f (x) = ax (a + 1). Since a > 0, we have a2 = a + 1 and a =

√ 1+ 5 2 .

√

Hence f (x) = ( 1+2 5 )x is a Fibonacci

function, and the unique Fibonacci function of this type on R. If we let u0 = 0, u1 = 1, then we consider the full Fibonacci sequence: · · · , 5, −3, 2, −1, 1, 0, 1, 1, 2, 3, 5, · · · , i.e., u−n = (−1)n un for n > 0, and un = Fn , the nth Fibonacci number. 1483

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On continuous Fibonacci functions ∞ Example 2.2. ([5]) Let {un }∞ n=−∞ and {vn }n=−∞ be full Fibonacci sequences. We define a function f (x)

by f (x) := ubxc + vbxc t, where t = x − bxc ∈ (0, 1). Then f (x + 2) = ubx+2c + vbx+2c t = u(bxc+2) + v(bxc+2) t = (u(bxc+1) + ubxc ) + (v(bxc+1) + vbxc )t = f (x + 1) + f (x) for any x ∈ R. This proves that f is a Fibonacci function. Note that if a Fibonacci function is differentiable on R, then its derivative is also a Fibonacci function. Proposition 2.3. ([5]) Let f be a Fibonacci function. If we define g(x) := f (x + t) where t ∈ R for any x ∈ R, then g is also a Fibonacci function. √

√

√

For example, since f (x) = ( 1+2 5 )x is a Fibonacci function, g(x) = ( 1+2 5 )x+t = ( 1+2 5 )t f (x) is also a Fibonacci function where t ∈ R. Theorem 2.4. ([5]) If f (x) is a Fibonacci function, then the limit of the quotient

f (x+1) f (x)

exists.

Corollary 2.5. ([5]) If f (x) is a Fibonacci function, then

√ f (x + 1) 1+ 5 lim = x→∞ f (x) 2

3. Continuous Fibonacci functions Given a real number x 6∈ Z, we define a map F (x) by F (x) := (Fbx−1c )x−bxc + Fbxc + (x − bxc − 1)

(1)

where {Fn } is the sequence of Fibonacci numbers with F0 = F1 = 1. Example 3.1. We compute some F (x) as follows: F (1.5) = (Fb1.5−1c )1.5−b1.5c +Fb1.5c + (1.5 − b1.5c − 1) = (F0 )0.5 + F1 + (1.5 − 1 − 1) = 1.5 and F (1.75) = (F0 )0.75 + F1 + (1.75 − b1.75c − 1) = 1.75. Moreover, √ F (3.25) = (Fb3.25−1c )0.25 + Fb3.25c + (3.25 − b3.25c − 1) = (F2 )0.25 + F3 + (3.25 − 3 − 1) = 4 2 + 2.25. Theorem 3.2. If we define F (n) := Fn , the nth Fibonacci function, then F (x) is continuous for all x ∈ R.  Proof. Let x := n +  where n ∈ Z and 0 <  < 1. Then F (x) = Fn−1 + Fn + ( − 1). It follows that  lim→0+ F (x) = lim→0+ (Fn−1 + Fn + ( − 1)) = Fn . Let x := n −  where n ∈ Z and 0 <  < 1. Then

F (x)

=

(Fbn−−1c )n−−bn−c + Fbn−c + (n −  − bn − c − 1)

=

(Fn−2 )n−−(n−1) + Fn−1 + (n −  − (n − 1) − 1)

=

(Fn−2 )1− + Fn−1 − 

It follows that lim→0+ F (x) = lim→0+ [(Fn−2 )1− + Fn−1 − ] = Fn−2 + Fn−1 = Fn .



In Theorem 3.2, we call the real number F (x) the (continuous) Fibonacci function at x. Let f : R → R be a real-valued function. We shall consider the expression (4f )(x) := f (x + 2) − f (x + 1) − f (x) to be the Fibonacci derivative of f (x). For example, if Φ :=

√ 1+ 5 2 ,

then f (x) = Φx yields (4f )(x) = Φx+2 −

Φx+1 − Φx = Φx (Φ2 − Φ − 1) = 0 and similarly, if f is any Fibonacci function, then (4f )(x) = 0 for all x ∈ R. 1484

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Hee Sik Kim, J. Neggers and Keum Sook So∗ We are next concerned with determining the Fibonacci derivative of F (x) as we have defined above. Theorem 3.3. If F (x) is a continuous Fibonacci function, then its Fibonacci derivative is (4F )(x) = (Fn+1 ) − (Fn ) − (Fn−1 ) − ( − 1)

(2)

where x = n + , n ∈ Z, 0 <  < 1 Proof. Given x = n + , n ∈ Z, 0 <  < 1, by using the formula (1), we obtain (4F )(x)

= F (n + 2 + ) − F (n + 1 + ) − F (n + ) =

(Fbn+1+c ) + Fn+2 + ( − 1) −(Fbn+c ) − Fn+1 − ( − 1) −(Fbn−1+c ) − Fn − ( − 1)

=

(Fn+1 ) − (Fn ) − (Fn−1 ) − ( − 1) 

Note that the map F (x) in Theorem 3.3 is not necessarily a Fibonacci function. The formula (2) is a function depending on , and so we need to know the value of d [(4F )(x)] d

= =

We denote

d d [4F (x)]

d d [4F (x)].

d [(Fn+1 ) − (Fn ) − (Fn−1 ) − ( − 1)] d ln(Fn+1 )(Fn+1 ) − ln(Fn )(Fn ) − ln(Fn−1 )(Fn−1 ) − 1

by (4F )0 (x).

Proposition 3.4. If F (x) is a continuous Fibonacci function, then (3)

(4(4F ))(x) = (Fn+3 ) − 2(Fn+2 ) − (Fn+1 ) + 2(Fn ) + (Fn−1 ) + ( − 1)

Proof. It follows from the formula (2) that (4(4F ))(x)

= 4F (x + 2) − 4F (x + 1) − 4F (x) =

(Fn+3 ) − (Fn+2 ) − (Fn+1 ) − ( − 1) −(Fn+2 ) + (Fn+1 ) − (Fn ) + ( − 1) −(Fn+1 ) + (Fn ) − (Fn−1 ) + ( − 1)

=

(Fn+3 ) − 2(Fn+2 ) − (Fn+1 ) + 2(Fn ) + (Fn−1 ) + ( − 1),

proving the proposition.



Proposition 3.5. (4F )(x) is a continuous function and (4F )(n) = 0 for all n ∈ Z. Proof. It follows from Theorem 3.3 that (4F )(x) is a continuous function. Since lim→0 (4F )(x) = lim→0 [(Fn+1 ) − (Fn ) −(Fn−1 ) −(−1)] = 0 and lim→1 (4F )(x) = lim→1 [(Fn+1 ) −(Fn ) −(Fn−1 ) −(−1)] = Fn+1 −Fn −Fn−1 = 0 for any n ∈ Z.



Theorem 3.6. If F (x) is a continuous Fibonacci function, then there exists a γn ∈ (n, n + 1) such that (4F )0 (γn ) = 0 for all n ∈ Z. 1485

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On continuous Fibonacci functions Proof. Since (4F )(x) is a continuous function and (4F )(n) = (4F )(n + 1) = 0, by Rolle’s Theorem, there exists a γn ∈ (n, n + 1) such that (4F )0 (γn ) =

d d (4F )(γn )

= 0.



Theorem 3.7. If F (x) is a continuous Fibonacci function, then (4F )(x) is concave down. d2 d2 [(4F )(x)],

Proof. If we let T (x) :=

then

d d [ ((4F )(x))] d d d = [ln(Fn+1 )(Fn+1 ) − ln(Fn )(Fn ) − ln(Fn−1 )(Fn−1 ) − 1] d = {ln(Fn+1 )}2 (Fn+1 ) − {ln(Fn )}2 (Fn ) − {ln(Fn−1 )}2 (Fn−1 ) .

T (x)

=

Let n be very large so that

Fk+1 Fk

=Φ=

√ 1+ 5 2 .

It follows that

ln(Fn+1 ) 2 Fn+1  ln(Fn ) 2 Fn  } [ } [ ] −{ ] −1 ln(Fn−1 ) Fn−1 ln(Fn−1 ) Fn−1 2 ln Φ + ln(Fn−1 ) 2 = { } (Φ)2 ln(Fn−1 ) ln Φ + ln(Fn−1 ) 2 } (Φ) − 1. −{ ln(Fn−1 )

T (x) {ln(Fn−1 )}2 (Fn−1 )

= {

If we let n → ∞, then T (x) = Φ2 − Φ − 1 {ln(Fn−1 )}2 (Fn−1 ) q √ √ √ √ √ 2 1+ 5 2 If we let  := 21 , then Φ2 − Φ − 1 = Φ− Φ − 1 and Φ − 1 = 5−1 Φ = , 2 2 , so that (Φ − 1) − ( Φ) =

(4)

√ 4−4 5 4

lim

n→∞

< 0, proving that T (x) < 0. This shows that (4F )(x) is concave down.



We discuss a Fibonacci derivative of a function which is not a Fibonacci function as below. Proposition 3.8. Let f (x) := ax + b for some a, b ∈ R. Then 4k+1 (f )(x) = (−1)k+1 ax + (−1)k [(k + 1)a − b]

Proof. The Fibonacci derivative (4f )(x) of f (x) = ax + b is f (x + 2) − f (x + 1) − f (x) = [a(x + 2) + b] − [a(x + 1) + b] − [ax + b] = −ax + a − b. Similarly, we obtain [42 (f )](x) = ax − 2a + b, [43 (f )](x) = −ax + 3a − b and [44 (f )](x) = ax − 4a + b. Assume 4k (f )(x) = (−1)k ax + (−1)k−1 (ka − b). Then 4k+1 (f )(x)

= 4[4k (f )(x)] = 4k (f )(x + 2) − 4k (f )(x + 1) − 4k (f )(x) =

(−1)k a(x + 2) + (−1)k−1 (ka − b) −[(−1)k a(x + 1) + (−1)k−1 (ka − b)] −[(−1)k ax + (−1)k−1 (ka − b)]

=

(−1)k+1 ax + (−1)k [(k + 1)a − b]. 

Note that [(4k+3 + 4k+2 ) − (4k+1 + 4k )](f )(x) = (−1)k+2 2a − (−1)k 2a = 0. 1486

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Hee Sik Kim, J. Neggers and Keum Sook So∗ We need to find some conditions for a map f : R → R satisfying (4f )(x) = f (x). Proposition 3.9. Let f : R → R be a map. If it satisfies the condition either (i) 2f (x) = f (x + 2) − f (x + 1) or (ii) f (x + 1) = −f (x) for all x ∈ R, then (4f )(x) = f (x). Proof. Straightforward.



Example 3.10. If f (x) := 2x , then 21 (2x+2 − 2x+1 ) = 2x and hence 4(2x )(x) = 2x . If we let f (x) := sin(πx), then f (x + 1) = sin π(x + 1) = − sin πx = −f (x) and hence 4(sin πx)(x) = sin πx. Now, we define a function f : R → R satisfying the condition: 2f (x) = f (x + 2) − f (x + 1). If we make such a function, then it satisfies the condition (4f )(x) = f (x). Suppose that one defines f (x) for 0 ≤ x < 2 at will. Then for x ∈ [2, 3) one defines f (2 + θ) := 2f (θ) + f (1 + θ) where 0 ≤ θ < 1. If f (x) has been defined for x ∈ [m−1, m), then we define f (m+θ) := 2f ((m−2)+θ)+f (m−1+θ), where 0 ≤ θ < 1. Then f (x) is uniquely determined on [0, ∞). To define f (x) for [−1, 0), we have f (1 + θ) = 2f (−1 + θ) + f (θ) or f (−1 + θ) = (f (1 + θ) − f (θ))/2, and thus f (−m + θ) = [f (−m + 2 + θ) − f (−m + 1 + θ)]/2 inductively as well to obtain f (x) defined on the entire real line. Example 3.11. if f (x) := 1 on [0, 2), then f (2+θ) = 2f (θ)+f (1+θ) = 2+1 = 3, f (3+θ) = 2f (1+θ)+f (2+θ) = 5 and f (4 + θ) = 2f (2 + θ) + f (3 + θ) = 2 · 3 + 5 = 11. If we take F1∗ = 1, F2∗ = 3, F3∗ = 5, F4∗ = 11, · · · , then ∗ ∗ = 2Fn−1 + Fn∗ . We have f (4 + θ) = 2f (2 + θ) + f (3 + θ) = {Fn∗ } is a Fibonacci sequence type satisfying Fn+1 ∗ ∗ 5f (1 + θ) + 6f (θ) = F3∗ f (1 + θ) + 2F2∗ f (θ). Assume that f (n + θ) = Fn−1 f (1 + θ) + 2Fn−2 f (θ). Then

f (n + 1 + θ)

=

2f (n − 1 + θ) + 2f (n + θ)

=

∗ ∗ ∗ ∗ 2[Fn−2 f (1 + θ) + 2Fn−3 f (θ)] + [Fn−1 f (1 + θ) + 2Fn−2 f (θ)]

=

∗ ∗ ∗ ∗ (2Fn−2 + Fn−1 )f (1 + θ) + 2(2Fn−3 + Fn−2 )f (θ)

∗ = Fn∗ f (1 + θ) + 2Fn−1 f (θ)

4. Complex Fibonacci functions Given the Fibonacci sequence F0 = 1, F1 = 1, F2 = 2, F3 = 3, · · · , we may compute F−n , n = 1, 2, · · · via the equation (5)

F−n+2 = F−n+1 + F−n

so that F−1 = F1 −F0 = 0, F−2 = F0 −F−1 = 1−0 = 1, F−3 = F−1 −F−2 = 0−1 = −1, F4 = F−2 −F−3 = 1−(−1) = 2, F−5 = F−3 − F−4 = −1 − 2 = (−1)5 F3 and F−6 = F−4 − F−5 = (−1)6 F4 . Assume F−n = (−1)n Fn−2 (n ≥ 5). Then F−(n+1) = F−(n+1)+2 − F−(n+1)+1 = (−1)n−1 Fn−1 − (−1)n Fn = (−1)n+1 (Fn−3 + Fn−2 ) = (−1)n Fn−1 so that F−2 = (−1)2 F0 = 1, F−3 = (−1)1 F1 = −1. For F−1 , the formula would yield F−1 = (−1)1 F−1 , i.e., F−1 = −F−1 which would imply F−1 = 0 as well. Hence we have the result: for n ≥ 1, (6)

F−n = (−1)n Fn−2

Thus, we may apply the formula (1) for x < 0 as well. For example, F (−1.5) = (Fb−1.5−1c )(−1.5−b−1.5c) +Fb−1.5c + √ (−1.5 − b−1.5c − 1) = (F−3 )0.5 + F−2 + (−1.5 − (−2) − 1) = −1 + 1 − 21 = 12 + i, i.e., F (−1.5) = 21 + i, the complex number. 1487

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On continuous Fibonacci functions Example 4.1. We compute F (−4 + θ), (0 ≤ θ < 1) as follows: F (−4 + θ)

=

(Fb−4+θ−1c )−4+θ−b−4+θc + Fb−4+θc + (−4 + θ − b−4 + θc − 1)

=

(F−5 )θ + F−4 + (θ − 1)

=

(−3)θ + 2 + (θ − 1)

=

3θ (−1)θ + θ + 1

where (−1)θ = expθ ln(−1) , so that ln(−1) = Log(−1), where Log(−1) is a “suitable branch of the Log-function”. Note that i2 = −1, so Log(−1) = Log(i2 ) = 2Log(i) = 2 ln(i). If we set ln i := a + bi, then expln i = i = expa expb i and a = 0, b =

π 2

yields ln i =

π 2 i.

Thus 2 ln i = πi = Log(−1) and (−1)θ = expπθi , e.g., θ = 1 yields

(−1)1 = expπi = −1 as required. Thus we set (−3)θ = 3θ expπθi = 3θ (cos πθ + i sin πθ). Hence F (−4 + θ) = (3θ cos πθ + 1 + θ) + (3θ sin πθ)i Hence the evaluation of F (x) for x < 0 may involve complex numbers. Definition 4.2. Given a complex number z := a + bi ∈ C, we define a map Fe : C → C by Fe(z) := F (a) + iF (b − 1) where F (x) is the continuous Fibonacci function on R. We call such a map Fe a complex Fibonacci function. Given a real number a ∈ R, we have Fe(a) = Fe(a + i0) = F (a) + iF (−1) = F (a), so that Fe extends the function F already defined on R to the complex numbers C. Proposition 4.3. Given a Gaussian integer z = m + in (m, n ∈ Z), we have Fe((m + 2) + (n + 3)i) = Fe((m + 1) + (n + 2)i) + Fe(m + (n + 1)i))

Proof. Since Fe(m + ni) = F (m) + F (n − 1)i = Fm + Fn−1 i, we obtain Fe((m + 2) + (n + 3)i) = Fm+2 + Fn+2 i = (Fm+1 + Fn+1 i) + (Fm + Fn i) = Fe((m + 1) + (n + 2)i) + Fe(m + (n + 1)i)).  Using the fact that the Fibonacci derivative is a linear mapping, we define the Fibonacci derivative for complex Fibonacci numbers as follows: Given z = a + bi (a, b ∈ R), e (z) := 4F (a) + i 4 F (b − 1) 4F

Proposition 4.4. Given a complex Fibonacci function Fe(z), we have e Fe(z) = Fe(z + 2(1 + i)) − Fe(z + 1 + i) − Fe(z) 4

Proof. Given z := a + bi ∈ R, we have e Fe(z) 4

= 4F (a) + i 4 F (b − 1) =

[F (a + 2) − F (a + 1) − F (a)] + i[F (b + 1) − F (b) − F (b − 1)]

= Fe((a + 2) + i(b + 2)) − Fe((a + 1) + i(b + 1)) − Fe(a + ib) = Fe(z + 2(1 + i)) − Fe(z + 1 + i) − Fe(z), 1488

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Hee Sik Kim, J. Neggers and Keum Sook So∗ e is the complex Fibonacci derivative. i.e., 4

 5. Concluding remark

As was already been mentioned in the introduction and has been demonstrated in the paper, the extensions of {Fn }n∈Z to F (x) and Fe(z) show themselves to be rather remarkable functions. We should note that F (0) = F (1) = 1, only selects one among a family of functions of this type. Considering the usual property limx→∞

f (x+1) f (x)

=

√ 1+ 5 2

for Fibonacci function f : R → R, it is naturally of interest to check on a variety of limit problems of this type and discover properties and solutions to these problems among others. 6. Future works One area which needs further investigation is the adapting of the theory developed above to general groupoids. In order to do this we will need to reduce formulas such as given above, which involve two (closely related) binary operations to those using only one such binary operation. Thus, consider an arbitrary groupoid (X, ∗) and an element a ∈ X. We consider functions f : X → X such that (5ca f )(x) = (f (x) ∗ f (x ∗ a)) ∗ f ((x ∗ a) ∗ a) ≡ c and let these be (a, c)-(X, ∗)-Fibonacci-functions. That this is a true generalization can be seen as follows. Let (X, ∗) = (R, −) and let a = 1, c = 0. Then (501 f )(x) = f (x) − f (x − 1) − f (x − 2) = 0 means f (x) = f (x − 1) + f (x − 2), i.e., an (1, 0)-(R, −)-Fibonacci-function as defined above. At the same time, if (4f )(x) = f (x + 2) − f (x + 1) − f (x), then (4f )(x − 2) = f (x) − f (x − 1) − f (x − 2) = (501 f )(x), so that 4f is a translation of 501 f on (R, −). Using these approach one hopes to develop very general Fibonacci properties which may be directly applied to a great variety of situations and thus also with an improved chance for possible applications, due to a much larger range of possible models which may be available. Therefore it is among our plans to follow through with this approach as well as what has been mentioned in the concluding remark section also. Acknowledgement This work was supported by Hallym University Research Fund HRF-201604-004.

References [1] Atanassove, K. et al, New Visual Perspectives on Fibonacci numbers, World Scientific, New Jersey, 2002. [2] Dunlap, R. A., The Golden Ratio and Fibonacci Numbers, World Scientific, New Jersey, 1997. [3] Han, JS, Kim, HS, Neggers, J., The Fibonacci norm of a positive integer n- observations and conjectures -, Int. J. Number Th. 6 (2010), 371-385. [4] Han, JS, Kim, HS, Neggers, J., Fibonacci sequences in groupoids, Advances in Difference Equations 2012 2012:19 (doi:10.1186/1687-1847-2012-19). [5] Han, JS, Kim, HS, Neggers, J., On Fibonacci functions with Fibonacci numbers, Advances in Difference Equations 2012 2012:126 (doi:10.1186/1687-1847-2012-126). [6] Jung, SM, Hyers-Ulam stability of Fibonacci functional equation, Bull. Iranian Math. Soc. 35 (2009), 217-227. [7] Kim, HS, Neggers, J., Fibonacci Means and Golden Section Mean, Computers and Mathematics with Applications 56 (2008), 228-232. 1489

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On continuous Fibonacci functions [8] Kim, HS, Neggers, J., So, KS, Generalized Fibonacci sequences in groupoids, Advances in Difference Equations 2013 2013:26 (doi:10.1186/1687-1847-2013-26). [9] Kim, HS, Neggers, J., So, KS, On Fibonacci functions with periodicity, Advances in Difference Equations 2014 2014:293. (doi:10.1186/1687-1847-2014-293).

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Decomposition and improved hyperbolic cross approximation of bivariate functions on [0, 1]2 ∗ Zhihua Zhang College of Global Change and Earth System Science, Beijing Normal University, Beijing, China Joint Centre for Global Change Studies, Beijing, China E-mail: [email protected]

Abstract. For a bivariate function on the unit square, if we extend it robustly into a periodic function on the plane, then its Fourier coefficients decay very slowly due to the discontinuity on the boundary of the unit square, therefore, we need a lot of Fourier coefficients to reconstruct this bivariate function. In order to solve this problem, for any bivariate smooth function on the unit square, we introduce a Fourier expansion with a polynomial term and several polynomial factors such that the corresponding Fourier coefficients decay fast. Using this expansion, we can construct a good approximation tool for any bivariate function on the unit square. 1. Introduction It is well-known that smooth period functions can be approximated well by Fourier series. But, for a bivariate function f on the unit square [0, 1]2 , if we extend it into a periodic function on the plane, then its Fourier coefficients decay very slowly due to the discontinuity on the boundary of [0, 1]2 . So we need a lot of Fourier coefficients to reconstruct this bivariate function. In order to reconstruct f by fewest Fourier coefficients, we will develop a new approximation tool in this paper. We first construct four simple univariate polynomial ϕi (i = 1, ..., 4) of degree 3 which is independent of f . With the help of these polynomials, we express f into a sum: f = f1 + f2 + f3 . In this decomposition formula, f1 is a linear combination of ϕi (x)ϕj (y) (i, j = 1, ..., 4), f2 is a sum of products of a polynomial ϕi and a univariate function, and f3 is a bivariate function whoes partial derivatives vanish on the boundary of [0, 1]2 . Then we expand these univariate functions and this bivariate function into Fourier series, where the corresponding Fourier coefficients will decay fast. We call this process a Fourier expansion of f with a polynomial term and several polynomial factors. Based on this expansion, we can develop a good approximation tool of f by using the partial sums of these univariate Fourier series and the hyperbolic cross truncation of bivariate Fourier series. Precisely say, for a function f satisfying

∂4f ∂x2 ∂y 2

∈ C([0, 1]2 ), if we use our

∗ This research is supported by National Key Science Programme for Global Change Research 2015CB953602, Beijing Higher Education Young Elite Teacher Project, and Scientific Research Foundation for the Returned Overseas Chinese Scholars, State Education Ministry.

1

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approximation tool, then the approximation error is equivalent to

log4 Nd , Nd3

however, if we directly expand f into

Fourier series, the approximation errors of its partial sums and hyperbolic cross truncation are equivalent to 2 √1 , log Nd , Nd Nd

respectively, where Nd is the number of Fourier coefficients used. It is clear that our approximation

tool is much better than traditional Fourier approximation. At the end of this paper, we will extend these results to the case of random processes. Throughout this paper we always assume bivariate functions on [0, 1]2 are real-valued. Denote by {0, 1}2 vertexes of the unit square [0, 1]2 and by ∂([0, 1]2 ) the boundary of [0, 1]2 . We say f ∈ C (2,2) ([0, 1]2 ) if

∂4f ∂x2 ∂y 2

is a continuous function on [0, 1]2 . Denote by sN (f ; x, y) the Fourier series partial sum of f on [0, 1]2 , i.e., sN (f ; x, y) =

X

X

cmn (f ) e2πimx e2πiny .

|m|≤N |n|≤N (h)

Denote by sn (f ; x, y) the Fourier series hyperbolic cross truncation of f on [0, 1]2 , i.e., N P

(h)

sN (f ; x, y) =

cm0 (f ) e2πimx

|m|=0

(1, 1) +

N P |n|=1

P

c0n e2πiny +

cmn (f ) e2πi(mx+ny) .

1≤|mn|≤N

For a random variable ξ, denote by E[ξ] and Var(ξ) its expectation and variance, respectively. For two random variables ξ, η, denote by Cov(ξ, η) their covariance. We also always assume that ξ is real-valued. The concept of the random calculus may refer the reference [2,6]. This paper is organized as follows: In section 2 we give a decomposition formula of a bivariate function on [0, 1]2 . In section 3 we discuss Fourier expansions with polynomial term and polynomial factors and estimate Fourier coefficients. In Section 4 we present a new approximation tool and estimate the corresponding approximation error. In Section 5 we generalize these results to random processes on [0, 1]2 . 2. Decomposition of bivariate functions on the unit square Suppose that f (x, y) is a real-valued function on [0, 1]2 and f ∈ C (2,2) ([0, 1]2 ). First we introduce four fundamental polynomials:

ϕ1 (x) = (1 + 2x)(x − 1)2 = −(3 − 2x)x2 + 1, ϕ2 (x) = (3 − 2x)x2 = −ϕ1 (x) + 1, (2.1) ϕ3 (x) = x(x − 1)2 , ϕ4 (x) = x2 (x − 1)

2

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satisfying the following conditions ϕ1 (0) = 1,

ϕ1 (1) = ϕ01 (0) = ϕ01 (1) = 0,

ϕ2 (1) = 1,

ϕ2 (0) = ϕ02 (0) = ϕ02 (1) = 0,

ϕ03 (0) = 1,

ϕ3 (0) = ϕ3 (1) = ϕ03 (1) = 0,

ϕ04 (1) = 1,

ϕ4 (0) = ϕ4 (1) = ϕ04 (0) = 0.

(2.2)

Define a bivariate polynomial: f1 (x, y) =

+

1 P ν=0

1 ³ P ∂f ν=0

+

(f (0, ν)ϕ1 (x) + f (1, ν)ϕ2 (x)) ϕ1+ν (y) ´

∂x (0, ν)ϕ3 (x) +

ϕ1+ν (y)

(2.3)

´

1 ³ P ∂f ν=0

∂f ∂x (1, ν)ϕ4 (x)

∂y (0, ν)ϕ1 (x)

+

∂f ∂y (1, ν)ϕ2 (x)

ϕ3+ν (y).

This is a linear combination of {ϕi (x)ϕj (y)}i,j=1,...,4 whose coefficients depend only on values of f and partial derivatives of f at vertexes {0, 1}2 = {(0, 0), (0, 1), (1, 0), (1, 1)}. Denote g(x, y) = f (x, y) − f1 (x, y). We construct a bivariate function f2 which depends only on values of g and partial derivatives of g at the boundary of [0, 1]2 . Define f2 (x, y) = g(x, 0)ϕ1 (y) + g(x, 1)ϕ2 (y) + g(0, y)ϕ1 (x) + g(1, y)ϕ2 (x) ∂g + ∂x (0, y)ϕ3 (x) +

∂g ∂x (1, y)ϕ4 (x)

+

∂g ∂y (x, 0)ϕ3 (y)

+

∂g ∂y (x, 1)ϕ4 (y).

(2.4)

This is a sum of products of univariate functions and fundamental polynomials ϕi . Finally, we let f3 (x, y) = f (x, y) − f1 (x, y) − f2 (x, y). Then the following decomposition formula holds. Theorem 2.1. Let f ∈ C (2,2) ([0, 1]2 ). Then f (x, y) = f1 (x, y) + f2 (x, y) + f3 (x, y),

(2.5)

where f1 , f2 , and f3 are stated as above and satisfy (i) f1 (x, y) is a bivariate polynomial and for (x, y) ∈ {0, 1}2 , f1 (x, y) = f (x, y),

∂f1 ∂x (x, y)

∂f1 ∂y (x, y)

∂ 2 f1 ∂x∂y (x, y)

=

∂f ∂y (x, y),

=

∂f ∂x (x, y),

=

∂2f ∂x∂y (x, y),

3

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i.e., g = f − f1 satisfies g(x, y) =

∂g ∂g ∂2g (x, y) = (x, y) = (x, y) = 0, ∂x ∂y ∂x∂y

(x, y) ∈ {0, 1}2 ;

(ii) the remainder f3 ∈ C (2,2) ([0, 1]2 ) and for (x, y) ∈ ∂([0, 1]2 ), f3 (x, y) =

∂f3 ∂f3 ∂ 2 f3 (x, y) = (x, y) = (x, y) = 0. ∂x ∂y ∂x∂y

From the definitions of f1 , f2 , f3 , and (2.2), we can directly check (i) and (ii). With the help of this decomposition formula, we give a Fourier expansion with polynomial factors and a new approximation tool such that we can reconstruct functions on [0, 1]2 by fewest Fourier coefficients. 3. A kind of new Fourier expansions In this section we give a Fourier expansion of the function on [0, 1]2 with a polynomial term and several polynomial factors. Suppose that f ∈ C (2,2) ([0, 1]2 ). By Theorem 2.1, f (x, y) = f1 (x, y) + f2 (x, y) + f3 (x, y), where f1 is a polynomial which is stated in (2.3) and f2 is stated in (2.4). We expand the first factor of each term in (2.4) into univariate Fourier series, such as we expand g(x, ν) (ν = 0, 1) into the Fourier series: g(x, ν) =

X

2πimx a(ν) , m e

m (ν)

where am =

R1 0

g(x, ν) e−2πimx dx and

P m

=

P∞

m=−∞ .

Using the integration by parts, by Theorem 2.1 (i) and

Riemann-Lebesgue lemma, the Fourier coefficients satisfy (ν)

am

=

R1 0

g(x, ν) e−2πimx dx =

= − 4π21m2 Similarly, we expand g(ν, y),

1 2πim

R1

∂g (x, ν) e−2πimx dx 0 ∂x

R1

∂2g (x, ν) e−2πimx dx 0 ∂x2

∂g ∂g ∂x (ν, y), ∂y (x, ν) (ν

g(ν, y) =

n

=

∂g ∂y (x, ν)

=

¡

1 m2

¢

.

= 0, 1) into Fourier series:

P

∂g ∂x (ν, y)

=o

(ν)

bn e2πiny ,

P n

P m

(ν)

αn e2πiny , (ν)

βm e2πimx .

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From this, we see that f2 (x, y) can be expanded into the following Fourier series with polynomial factors ϕi : P (0) P (1) f2 (x, y) = ϕ1 (y) am e2πimx + ϕ2 (y) am e2πimx m

m

+ ϕ1 (x) + ϕ3 (x)

n

P n

P

+ ϕ3 (y) and

P

m

(ν)

¡

(ν)

¡

am = o αn = o

(0)

bn e2πiny + ϕ2 (x)

P

(0)

αn e2πiny + ϕ4 (x) (0)

βm e2πimx + ϕ4 (y)

1 m2 1 n2

¢ ¢

(ν)

¡

(ν)

¡

,

bn = o

,

βm = o

1 n2

n

(1)

bn e2πiny (3.1)

P

(1)

αn e2πiny

n

P m

¢

1 m2

(1)

βm e2πimx

,

¢

.

Finally, expand f3 into a bivariate Fourier series: X f3 (x, y) = cmn (f3 ) e2πi(mx+ny) , m,n

where cmn (f3 ) =

R1R1 0

0

P

f3 (x, y) e−2πi(mx+ny) dxdy and

interior integral is equal to Z 1 f3 (x, y) e−2πi(mx+ny) dx = 0

So the Fourier coefficients: cmn (f3 ) =

1 (2πim)2

Z

1

1 (2πim)2 µZ

e−2πimx

0

=

1 (2πim)2

R1 0

1 16π 4 m2 n2

1

0

Again, by Theorem 2.1 (ii), cmn (f3 ) =

m,n

e−2πimx

³

1 2πin

= Z

P∞

1

0

P∞

n=−∞ .

m=−∞

By Theorem 2.1 (ii), the

∂ 2 f3 (x, y) e−2πimx dx. ∂x2

¶ ∂ 2 f3 −2πiny (x, y) e dy dx. ∂x2 R1

∂ 3 f3 (x, y) e−2πiny dy 0 ∂x2 ∂y

´ dx (3.2)

R1R1 0

∂ 4 f3 (x, y) e−2πi(mx+ny) dxdy. 0 ∂x2 ∂y 2

Summarizing up the above results, we get a Fourier expansion with polynomial term and polynomial factors, where Fourier coefficients decay fast. Theorem 3.1. Let f ∈ C (2,2) ([0, 1]2 ) and f1 , f2 , f3 be stated as in (2.3)-(2.5), and ϕi (i = 1, ..., 4) be stated as in (2.1). Then f can be expanded into Fourier series with a polynomial term and several polynomial factors as follows: f (x, y) = f1 (x, y) +

+

1 P ν=0

+

P m,n

1 P ν=0

µ ϕ1+ν (y)

µ ϕ3+ν (x)

P n

P m

(ν) am e2πimx

(ν) αn e2πiny

+ ϕ1+ν (x)

+ ϕ3+ν (y)

P m

P n

¶ (ν) bn e2πiny

¶ (ν) βm e2πimx

cmn (f3 ) e2πi(mx+ny) , 5

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where f1 (x, y) is a polynomial which is stated in (2.3), both the second term and the third term are a combination of four univariate Fourier expansions and four fundamental polynomials, and for ν = 0, 1, ¡ ¢ R1 (ν) am = 0 g(x, ν) e−2πimx dx = o m12 , R1

(ν)

bn =

g(ν, y) e−2πiny dx = o

0

(ν)

R1

(ν)

R1

∂g (ν, y) e−2πiny dy 0 ∂x

αn = βm =

¡

1 n2

=o

∂g (x, ν) e−2πimx dx 0 ∂y

¡

=o

¢

,

1 n2

¡

¢

1 m2

(3.4) , ¢

and g(x, y) = f (x, y) − f1 (x, y), and the last term is a bivariate Fourier series of f3 whose coefficients satisfy ¡ ¢ cmn (f3 ) = o m21n2 (m → ∞ or n → ∞), c0n (f3 ) = o

¡

cm0 (f3 ) = o

1 n2

¡

¢

1 m2

(n → ∞),

¢

(3.5)

(m → ∞).

4. A new approximation tool We want to reconstruct the bivariate function f (x, y) by the fewest Fourier coefficients. For this purpose, we take partial sums of univariate Fourier series and hyperbolic cross truncation of the bivariate Fourier series of P cmn (f3 ) e2πi(mx+ny) in (3.3), we get a hyperbolic cross truncation of Fourier expansion of f with a polynomial m,n

term and several polynomial factors. For an appropriate N ∈ Z+ , we define such a combination of polynomials and trigonometric polynomials: (h) TN (x, y)

+

1 P ν=0

= f1 (x, y) + Ã

ν=0

P

ϕ3+ν (x)

Ã

1 P

P

ϕ1+ν (y)

(ν) an e2πinx

!

P

+ ϕ1+ν (x)

(ν) bn e2πiny

|n|≤N

|n|≤N

P

(ν)

αn e2πiny + ϕ3+ν (y)

(ν)

βn e2πinx

|n|≤N

|n|≤N

(4.1)

! (h)

+ sN (f3 ; x, y),

where the last term is the hyperbolic cross truncation of f3 which is stated in (1.1). From this and (3.3), it follows that (h)

(1)

(2)

f (x, y) − TN (x, y) = sN (x, y) + sN (x, y), where (1) sN (x, y)

=

1 P ν=0

+

1 P ν=0

(2) sN (x, y)

à ϕ1+ν (y)

P

(ν) an e2πinx

+ ϕ1+ν (x)

|n|>N

à ϕ3+ν (x)

= f3 (x, y) −

P

P

(4.2) ! (ν) bn e2πiny

|n|>N

(ν)

αn e2πiny + ϕ3+ν (y)

|n|>N

P

! (ν)

βn e2πinx

|n|>N

(h) sN (f3 , x, y).

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(4.3)

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Consider the square error: Z

1

(h)

k f − TN k22 =

Z

0

1

0

(h)

|f (x, y) − TN (f ; x, y)|2 dxdy.

By (4.2), we have (h)

(1)

(2)

k f − TN k22 ≤ 2 k sN k22 +2 k sN k22 . R1 By (4.3) and k h1 (x)h2 (y) k22 =k h1 (x) k22 k h2 (y) k22 , where k h(t) k22 = 0 |h(t)|2 dt, we have ! Ã 1 P (ν) 2πiny 2 P P (ν) 2πinx 2 (1) 2 2 2 k ϕ1+ν k2 k an e k2 + k ϕ1+ν k2 k bn e k2 k sN k2 ≤ 64 ν=0

+

64

1 P

|n|>N

|n|>N

à k

ν=0

ϕ3+ν k22 k

P

(4.4)

(ν) αn e2πiny

k22

|n|>N

+k

!

P

ϕ3+ν k22 k

(ν) βn e2πinx

|n|>N

k22

.

By the Parseval identity of the univariate Fourier series, we get   1 X X X (1) 2 (ν) 2 2 k ϕ1+ν k22 k sN k22 ≤ 64 (|a(ν) (|αn(ν) | + |βn(ν) |2 ) . n | + |bn | )+ k ϕ3+ν k2 ν=0

|n|>N

Again, by (3.4),

|n>N |



(1)

k sN

 µ ¶ X 1 =o 1 . k22 = o  n4 N3

(4.5)

|n|>N

For

(2) sN ,

by (1.1), we have P

(2)

sN (f3 ; x, y) =

P

cm0 (f3 ) e2πimx +

|m|>N

+

∞ P

cmn (f3 ) e2πi(mx+ny)

|n|>N |m|=0

N P

P

cmn (f3 ) e2πi(mx+ny) .

N |n|=1 |m|> |n|

(2)

(1)

(2)

(3)

By the Parseval identity and (3.5), we have k sN k22 = JN + JN + JN , where (1)

JN =

(2) JN

=

P

|cm0 (f3 )|2 = o

¡

|m|>N

P

1 m4

¢

=o

Ã

∞ P

1 N3

|cmn (f3 )| = o

¢

, !Ã

P

2

|n|>N |m|=1

¡

|n|>N

1 n4

!

∞ P

|m|=1

1 m4

 (3)

JN =

N P

P

|n|=1 |m|>[

]

 = O

 N P

|n|=1

1 n4

P N |m|> |n|

1  m4

P

|n|=1

N |m|> |n|

à =O

¡

1 N3

¢

,

 N P

|cmn (f3 )|2 = O 

N |n|

=o

P

|n|>N

1  m4 n4

! 3 1 |n| n4 N 3

=O

¡

1 N3

¢ log N .

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Therefore,

µ k

(2) sN

k22 =

O

1 log N N3

¶ .

From this and (4.4), and (4.5), it follows that µ (h)

k f − TN k22 = O Since tool

P

1≤|mn|≤N 1 ∼ N log N , by (4.1), we see (h) TN (x, y) is equivalent to N log N , i.e.,

log N N3

¶ .

that the number Nd of Fourier coefficients in the approximation

Nd ∼ N log N. This implies the following: (h)

Theorem 4.1. Let f ∈ C (2,2) ([0, 1]2 ) and TN be the hyperbolic cross truncation of its Fourier expansion with a polynomial term and several polynomial factors which are stated in (4.1). Then µ 4 ¶ log Nd (h) , k f − TN k22 = O Nd3

(4.6)

(h)

where Nd is the number of Fourier coefficients in TN . For f ∈ C (2,2) ([0, 1]2 ), consider the partial sums of the Fourier series of f : X

sN (x, y) =

X

cmn (f ) e2πi(mx+ny) .

|m|≤N |n|≤N

By the Parseval identity, k f − sN k22 =

∞ X X

|cmn (f )|2 +

|n|>N |m|=0

From this and cm0 (f ) = O

¡1¢ m

N X X

|cmn (f )|2 .

|n|=0 |m|>N

¢ 1 , cmn (f ) = O mn , it follows that µ ¶ 1 k f − sN k22 = O . N

, c0n (f ) = O

¡1¢

¡

n

Note that the number Nd of Fourier coefficients in the partial sum is equivalent to N 2 , i.e., Nd ∼ N 2 . So ¶ µ 1 k f − sN k22 = O √ . (4.7) Nd Consider the hyperbolic cross truncation of the Fourier series of f . By (1.1) and the Parseval identity, (h)

k f − sN k22 =

X

|cm0 (f )|2 +

|m|>N

∞ X X

|cmn (f )|2 +

From this and Nd ∼ N log N ,

µ kf−

X

N |n|=1 |m|> |n|

|n|>N |m|=1

(h) sN

N X

k22 =

o

log2 Nd Nd

µ |cmn (f )|2 = o

log N N

¶ .

¶ .

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(h)

Comparing (4.7), (4.8) with (4.6), we see that the approximation tool TN is such that we reconstruct f by the fewest Fourier coefficients. 5. Uncertainty analysis Now we extend the above results to the case of random processes. Suppose that f is a real-valued random process on [0, 1]2 and f ∈ C (2,2) ([0, 1]2 ) (Refer to [6] for random calculus). Then the decomposition formula (2.5) is still valid: f = f1 + f2 + f3 , where f1 and f2 are stated as in (2.3) and (2.4), respectively, and f3 is the residual. However, now f1 is a random polynomial, f2 is a sum of products of univariate random processes and fundamental polynomials ϕi . Theorem (ν)

(ν)

(ν)

(ν)

2.1 and the expansion (3.3) are still valid. However, Fourier coefficients in (3.3): am , bn , αn , βm (ν = 0, 1), and cmn (f3 ) are now all random variables. Consider their expectations and variances. Note that Z a(ν) m =

1

g(x, ν) e−2πimx dx

(ν = 0, 1).

(5.1)

0

Since the expectation and the integral can be exchanged, Z 1 (ν) E[g(x, ν)] e−2πimx dx E[am ] =

(ν = 0, 1).

(5.2)

0

Since the random process g = f − f1 belongs to C (2,2) ([0, 1]2 ) and the expectation and the partial derivatives can be exchanged, the deterministic function E[g(x, y)] ∈ C (2,2) ([0, 1]2 ). Noticing that Theorem 2.1 is still valid, for (x, y) ∈ {0, 1}2 , ¸ · ¸ · 2 ¸ ∂g ∂ g ∂g (x, y) = E (x, y) = E (x, y) = 0. E[g(x, y)] = E ∂x ∂y ∂x∂y ·

Exchanging the expectation and the partial derivatives, for (x, y) ∈ {0, 1}2 , we get E[g(x, y)] =

∂ ∂ ∂2 E[g(x, y)] = E[g(x, y)] = E[g(x, y)] = 0. ∂x ∂y ∂x∂y

Therefore, (ν)

E[am ] =

1

1 − 2πim E[g(x, ν)]|x=0 +

¯1 ∂ ¯ ∂x E[g(x, ν)] x=0

1 = − (2πim) 2

=

1 (2πim)2

R1

+

R1

∂ (E[g(x, ν)]) e−2πimx dx 0 ∂x

1 (2πim)2

∂2 (E[g(x, ν)]) e−2πimx dx 0 ∂x2

This implies that E[a(ν) m ]≤ (ν)

1 2πim

1

max |

4π 2 m2 0≤x≤1 (ν)

R1

∂2 (E[g(x, ν)]) e−2πimx dx 0 ∂x2

=o

∂2 E[g(x, ν)]| ∂x2

¡

1 m2

¢

(ν = 0, 1).

(ν = 0, 1).

(ν)

Similarly, we compute E[bn ], E[αn ], E[βm ], and E[cmn (f3 )]. So we have the following: 9

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Theorem 5.1. Let f be a random process on [0, 1]2 and f ∈ C (2,2) ([0, 1]2 ). Then, in the Fourier expansion (3.3) with a random polynomial and several random polynomial factors, Fourier coefficients satisfy (ν)

¡

(ν)

¡

E[am ] = o E[bn ] = o

1 n2

(ν)

¡

(ν)

¡

E[αn ] = o E[βm ] = o

¢

1 m2

¢

1 n2

E[cmn (f3 )] = o

,

¢

1 m2

,

(ν)

1 ∂2 max | ∂x 2 E[g(x, ν)]|, 4π 2 m2 0≤x≤1

(ν)

∂2 1 max | ∂y 2 E[g(ν, y)]|, 4π 2 n2 0≤y≤1

E[bn ] ≤ (ν)

,

¢ ¡

E[am ] ≤

E[αn ] ≤ (ν)

,

E[βm ] ≤

1 m2 n2

¢

,

∂3 1 max | ∂x∂y 2 E[g(ν, y)]|, 4π 2 n2 0≤y≤1 3 1 max | ∂x∂2 ∂y E[g(x, ν)]| 4π 2 m2 0≤x≤1

(ν = 0, 1),

4 1 max | ∂x∂2 ∂y2 E[f3 (x, y)]|, 16π 4 m2 n2 0≤x,y≤1

E[cmn (f3 )] ≤

where g = f − f1 and f1 , f2 are stated as above. Now we consider the variances of Fourier coefficients in (3.3). Since g is a real-valued, by (5.1), we deduce that for ν = 0, 1, R1R1

(ν)

|am |2 =

0

g(x, ν)g(t, ν) e−2πim(x−t) dxdt,

0

(ν)

E[ |am |2 ] =

R1R1 0

0

E[ g(x, ν)g(t, ν) ] e−2πim(x−t) dxdt,

Since E[g(x, ν)g(t, ν)] ∈ C (2,2) ([0, 1]2 ), by Theorem 2.1 (i) and using integration by parts, it follows that 2 E[ |a(ν) m | ]=

Z

1 16π 4 m4

1

Z

0

1

0

∂4 E[g(x, ν)g(t, ν)] e−2πim(x−t) dxdt ∂x2 ∂t2

and so

µ 2 E[ |a(ν) m | ]=o (ν)

1 m4

(ν = 0, 1),

(5.3)

¶ (ν = 0, 1).

(ν)

(ν)

Noticing that Var(am ) = E[ |am |2 ] − ( E[am ] )2 , we get µ (ν) 2 Var(a(ν) m ) ≤ E[(am ) ] = o

1 m4

¶ (ν = 0, 1).

(5.4)

From (5.2), it follows that 2 (E[ a(ν) m ]) =

1 16π 4 m4

Z

1

Z

0

1

0

∂4 (E[g(x, ν)]E[g(t, ν)]) e−2πim(x−t) dxdt ∂x2 ∂t2

(ν = 0, 1).

Again, by the covariance formula: Cov(g(x, ν), g(t, ν)) = E[g(x, ν)g(t, ν)] − E[g(x, ν)]E[g(t, ν)], we have Var(a(ν) m )=

1 16π 4 m4

Z 0 (ν)

1

Z 0

1

∂4 Cov(g(x, ν), g(t, ν)) e−2πim(x−t) dxdt ∂x2 ∂t2 (ν)

(ν = 0, 1).

(ν)

Similarly, we compute Var(bn ), Var(αn ), Var(βm ), and Var(cmn (f3 )). So we have

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J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 24, NO.8, 2018, COPYRIGHT 2018 EUDOXUS PRESS, LLC

Theorem 5.2. Under conditions of Theorem 5.1, for ν = 0, 1, we have (ν)

4 1 max | ∂x∂2 ∂t2 Cov(g(x, ν), 16π 4 m4 0≤x,t≤1

(ν)

4 1 max | ∂y∂2 ∂t2 Cov(g(ν, y), 16π 4 n4 0≤y,t≤1

Var(am ) ≤ Var(bn ) ≤

³

(ν)

4 1 max | ∂y∂2 ∂t2 Cov 16π 4 n4 0≤y,t≤1

(ν)

4 1 max | ∂x∂2 ∂t2 Cov 16π 4 m4 0≤x,t≤1

Var(αn ) ≤ Var(βm ) ≤

Var(cmn (f3 )) ≤

g(t, ν))|, g(ν, t))|,

∂g ∂g ∂x (ν, y), ∂x (ν, t)

³

´ |, ´

∂g ∂g ∂y (x, ν), ∂y (t, ν)

1 max |Cov 256π 8 m4 n4 0≤x,y,t,s≤1

³

|, ´

∂ 4 f3 ∂ 4 f3 ∂x2 ∂y 2 (x, y), ∂t2 ∂s2 (t, s)

|,

Similar to (5.5), we can obtain that the second-order moments are as follows. For ν = 0, 1, (ν)

¡

(ν)

¡

E[|am |2 ] = o E[|αn |2 ] = o

1 m4 1 n4

E[|cmn (f3 )|2 ] = o

¢ ¢ ¡

(ν)

¡

(ν)

¡

,

E[|bn |2 ] = o

,

E[|βm |2 ] = o

1 m4 n4

¢

1 n4

¢

1 m4

,

¢

(5.6)

,

. (h)

(h)

Finally, for a random process, we still define an approximation tool TN (x, y) as in (4.1). Now TN (x, y) is a combination of random polynomials of degree 3 and random trigonometric polynomials of degree N . Using the Parseval identity, by (5.6), we have the following: (h)

Theorem 5.3. Let f be a random process on [0, 1]2 and f ∈ C (2,2) ([0, 1]2 ), and TN Then the mean square error of approximation by

(h) TN

be stated as above.

satisfies µ

(h)

E[ k f − TN k22 ] = o

log4 Nd Nd3

¶ ,

(h)

where Nd is the number of Fourier coefficients in TN .

References [1] B. Boashash, Time-frequency signal analysis and processing, Second edition, Academic press, 2016. [2] B. Hajek, An exploration of random processes for engineer, 2002, see: jek/Papers/randomprocDec 11. pdf

http://www.ifp.illinois.edu/ ha-

[3] J.-P. Kahane, Some random series of functions, Cambridge Studies in Advanced Mathematics, Vol. 5, 2rd. Cambridge University Press, Cambridge, 1994. [4] D. W. Kammler, A first course in Fourier analysis, Cambridge Univ. Press, 2008. [5] Z. Zhang, Approximation of bivariate functions via smooth extensions, The Scientific World Journal, vol. 2014, Article ID 102062, 2014. doi:10.1155/2014/102062.

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[6] F. C. Klebaner, Introduction to stochastic calculus with application, World Scientific Publishing, 2012. [7] G. G. Lorentz, Approximation, Holt, Rinebart and Winston, Inc. 1966. [8] E. M. Stein, and G. Weiss, Introduction to Fourier analysis on Euclidean space, Princeton Univ. Press, 1971. [9] P. Stoica and R. Moses, Spectral analysis of signals, Prentice hall, 2005. [10] Z. Zhang, Environmental Data Analysis, DeGruyter, December 2016. [11] Z. Zhang, P. Jorgensen, Modulated Haar wavelet analysis of climatic background noise, Acta Appl Math, 140, 71-93, 2015

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Gronwall-Bellman type inequalities for the distributional Henstock-Kurzweil integral and applications

∗

Wei Liu, Guoju Ye, Dafang Zhao College of Science, Hohai University, Nanjing 210098, P.R. China E-mails: [email protected], [email protected], [email protected]

October 14, 2016

Abstract: This paper is devoted to studying the Gronwall-Bellman type inequalities involving the distributional Henstock-Kurzweil integral. Moreover, an linear differential equation with distributional coefficients is considered as an application. Keywords: distributional derivative, distributional Henstock-Kurzweil integral, Gronwall-Bellman inequality. MSC 2010: 26D15, 26A39, 46G12.

1

Introduction

It is well-known that the Gronwall-Bellman inequality (also called the Gronwall’s lemma or the Gronwall’s inequality) has played a fundamental role in the study of the qualitative behaviour of solutions of differential and integral equations. In 1919, T. H. Gronwall [1] firstly established the following integral inequality. Lemma 1.1. Let f (t) be a continuous function defined on [a, a + δ] ⊂ R, for which the inequality Z 0 ≤ f (t) ≤

t

(αf (s) + β)ds,

t ∈ [a, a + δ]

(1.1)

a ∗

Supported by the Program of High-end Foreign Experts of the SAFEA (No.

GDW20163200216).

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holds, where α, β, δ are nonnegative constants. Then 0 ≤ f (t) ≤ βδ exp(αδ),

t ∈ [a, a + δ].

(1.2)

In 1943, R. Bellman [2] generalized Lemma 1.1 to the following result. Lemma 1.2. Let f (t) and g(t) be nonnegative, continuous functions on [a, b] ⊂ R, for which the inequality Z t 0 ≤ f (t) ≤ η + g(s)f (s)ds,

t ∈ [a, b]

(1.3)

t ∈ [a, b].

(1.4)

a

holds, where η is a nonnegative constant. Then Z t  0 ≤ f (t) ≤ η exp g(s)ds , a

Because of the importance of this inequality, over the years investigators have discovered many useful generalizations in order to achieve a diversity of desired goals in various branches of differential and integral equations [3–18]. However, almost all the generalizations are based on continuous functions under Riemann and Lebesgue integrals, they are not applicable to generalized ordinary differential equations [19, 20]. This did not really ˇ Schwabik [11] presented the Gronwall-Bellman inequality change until S. for the Henstock-Kurzweil integral in 1985, while K. Ostaszewski and J. Sochacki [12] gave a simpler and significant proof in 1987.

As for the

Gronwall-Bellman type inequalities for the Stieltjes integrals, we refer the reader to [14–18]. In this paper, we study the Gronwall-Bellman type inequalities for the distributional Henstock-Kurzweil integral, which defined by using Schwartz distributional derivative. It is a very wide integral form including the Riemann integral, the Lebesgue integral, and the Henstock-Kurzweil integral (see [19–22, 27–30] for details). The space of such integrable distributions, denoted by DHK , is a completion of the space of Henstock-Kurzweil integrable functions (shortly, HK). This paper is organized as follows. Section 2 is devoted to the basic notations of the distributional Henstock-Kurzweil integral. Section 3 contains our main results on the Gronwall-Bellman type inequalities involving the distributional Henstock-Kurzweil integral, while Section 4 sets forth an application to an linear differential equation with distributional coefficients. 2

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2

The Distributional Henstock-Kurzweil Integral

Let (a, b) be an open interval in R, we define D((a, b)) = {φ : (a, b) → R | φ ∈ Cc∞ and φ has a compact support in (a, b)}. The distributions on (a, b) are defined to be the continuous linear functionals on D((a, b)). The dual space of D((a, b)) is denoted by D0 ((a, b)). For all f ∈ D0 ((a, b)), we define the distributional derivative f 0 of f to be a distribution satisfying hf 0 , φi = −hf, φ0 i, where φ ∈ D((a, b)) is a test function. Further, we write distributional derivative as f 0 and its pointwise derivative as f 0 (t) where t ∈ R. From now on, all derivative in this paper will be distributional derivatives unless stated otherwise. Denote the space of continuous functions on [a, b] by C[a, b]. Let C0 = {F ∈ C[a, b] : F (a) = 0}.

(2.1)

Then C0 is an Banach space under the norm kF k∞ = sup |F (t)| = max |F (t)|. t∈[a,b]

t∈[a,b]

Definition 2.1 ( [29, Definition 1]). A distribution f ∈ D0 ((a, b)) is said to be Henstock–Kurzweil integrable (shortly DHK ) on an interval [a, b] if there exists a continuous function F ∈ C0 such that F 0 = f , i.e., the distributional derivative of F is f . The distributional Henstock–Kurzweil integral of f on Rb [a, b] is denoted by a f (t)dt = F (b) − F (a). The function F is called the primitive of f . For every f ∈ DHK , φ ∈ D((a, b)), we write Z hf, φi =

b

Z f (t)φ(t)dt = −

a

b

F (t)φ0 (t)dt.

a

The distributional Henstock–Kurzweil integral is very wide and it includes Riemann integral, Lebesgue integral, Henstock–Kurzweil integral, restricted and wide Denjoy integral (see [21, 22, 27–29]). From now on, we Rb Rb write “ a f (t)dt ” as “ a f ” for short. For f ∈ DHK , define the Alexiewicz norm in DHK as kf k = kF k∞ = sup |F (t)| = max |F (t)|. t∈[a,b]

t∈[a,b]

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Under the Alexiewicz norm, DHK is a Banach space, see [28, Theorem 2]. For F ∈ C0 , the positive part F + = maxt∈[a,b] {F (t), 0}, the negative part F − = maxt∈[a,b] {−F (t), 0}, and hence F = F + − F − and the absolute value |F | = F + + F − . Moreover, F + , F − , |F | all belong to C0 . Let f ∈ DHK with the primitive F ∈ C0 , as in [28], define f + = (F + )0 ,

f − = (F − )0 ,

|f | = |F |0 .

(2.2)

Then, f = f + − f −,

|f | = f + + f − .

(2.3)

In C0 there exists a pointwise order: for F, G ∈ C0 , F ≤ G if and only if F (t) ≤ G(t) for all t ∈ [a, b]. For f, g ∈ DHK with primitives F, G ∈ C0 , respectively, we say f

(p)



g (or g

(p)



f)

if and only if F (t) ≤ G(t), ∀t ∈ [a, b],

and f

(m)



g (or g

(m)



Z f)

(2.4)

Z f≤

if and only if

g,

I

(2.5)

I

where I is arbitrary subinterval of [a, b]. Obviously, f

(m)



g

⇒

f

(p)



g,

(2.6)

but the converse is not true. Particularly, if f, g are functions, then f (t) ≤ g(t) (∀t ∈ [a, b])

⇔

f

(m)



⇒

g

f

(p)



g.

(2.7)

Lemma 2.2. Let f, g ∈ DHK . Then Rt Rt (I) |f | ∈ DHK and | a f | ≤ a |f | for all t ∈ [a, b]; (II) k |f | k = k |F 0 | k = k |F | k∞ = kf k; (III) |f + g|

(p)



|f | + |g|.

Proof. (I) and (II) see [28, Theorem 24]. Since |F + G| ≤ |F | + |G| in C0 , (III) follows immediately from (2.3) and (2.4). If g : [a, b] → R, its variation is V g = sup

P

n |g(tn )

− g(sn )|, where

the supremum is taken over every sequence {(tn , sn )} of disjoint intervals in [a, b]. If V g < ∞ then g is called a function with bounded variation. Denote the set of functions with bounded variation by BV (see [21–23]). 4

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Lemma 2.3 ( [23, Theorem 2.2]). Let g, h ∈ BV. Then (i) g ± h ∈ BV and V (g ± h) ≤ V g + V h; (ii) gh ∈ BV; (iii) gh−1 =

g h

∈ BV if there exists constant δ > 0 such that |h| ≥ δ.

Moreover, we have the following result. Lemma 2.4 ( [25, Lemma 1.5]). Let F ∈ C[a, b] and g ∈ BV. Then F 0g =

Z

0

t

gdF

,

(2.8)

.

(2.9)

a

and 0

Z

0

t

Fg =

F dg a

Lemma 2.5 ( [29, Lemma 2, Integration by parts]). Let f ∈ DHK , and g ∈ BV. Then f g ∈ DHK and Z

b

Z

b

f g = F (b)g(b) − a

F dg. a

By Lemmas 2.4 and 2.5, it is easy to see that Lemma 2.6. Let f ∈ DHK be the distributional derivative of F ∈ C[a, b], and g ∈ BV. Then (F g)0 = f g + F g 0 . From (2.5) and Lemma 2.5, the following lemma holds. Lemma 2.7. Let f ∈ DHK and let g be a nonnegative function on [a, b]. (I) If f

(m)



0 and g is monotone on [a, b], then fg

(II) If f

(p)



(m)



0.

0 and g is nonincreasing on [a, b], then fg

(p)



0.

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Proof. (I) Let F (t) = because f

(m)

Rt a

f, t ∈ [a, b]. Then, F ∈ C[a, b], and F ≥ 0 on [a, b],

0. Since g ≥ 0 is monotone, then g ∈ BV. By the first mean



value theorem for Riemann integrals, there exists ξ ∈ [c, d] ⊂ [a, b] such that Z d F dg = F (ξ)(g(d) − g(c)), ξ ∈ [c, d]. c

In view of Lemma 2.5 and (2.5), one has f g ∈ DHK , and Z d Z d d f g = F g |c − F dg c

c

= F (d)g(d) − F (c)g(c) − F (ξ)(g(d) − g(c)) Z ξ Z d = g(d) f ≥ 0, ∀ [c, d] ⊂ [a, b]. f + g(c) c

ξ

Hence, by (2.5), (I) follows. Rt (II) Let F (t) = a f, t ∈ [a, b]. Then, F ∈ C[a, b], and F ≥ 0 on [a, b], because f

(p)



0. Since g ≥ 0 is nonincreasing on [a, b], then g ∈ BV and

f g ∈ DHK . Moreover, Z t Z t t f g = F g|a + F d(−g) a

a

≥ F (t)g(t) − F (a)g(a) + F (η)(g(a) − g(t)) ≥ 0,

t ∈ [a, b],

where F (η) = mins∈[a,t] F (s). Thus, by(2.4), (II) holds. The proof is therefore complete. Remark 2.8. In Lemma 2.7, (II) is not true if g is nondecreasing on [a, b]. For example, let (   0, 5π f = sin t, t ∈ 0, , and g(t) = 4 1,

t ∈ [0, π),   t ∈ π, 5π 4 .

(p)

It is easy to see that f  0 and g is nonnegative and nondecreasing on √  5π  R 5π R 5π 0, 4 . However, 0 4 f g = π 4 sin t = 22 − 1 < 0. This implies by (2.4) that (II) is not true.

3

Main Results

In this section, we shall prove that the Gronwall-Bellman type inequalities involving the distributional Henstock-Kurzweil integral remain valid. 6

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Theorem 3.1. Let f ∈ DHK , g : [a, b] → R be a nonnegative nonincreasing function. If there is a constant η such that Z t (p) (p) f g, t ∈ [a, b]. 0  f  η+

(3.1)

a

Then 0

(p)



(p)

f



Z

t

η exp

 g ,

t ∈ [a, b].

(3.2)

a

Proof. Since g is a nonnegative nonincreasing function on [a, b], then Rt g exp − a g is also nonnegative and nonincreasing on [a, b]. This together with Lemma 2.5 implies that  Z t  g ∈ DHK . f g ∈ DHK , f g exp − a

Let x(t) =

Rt a

f g, then x(t) ∈ C0 , x0 = f g, and (3.1) can be transformed

into 0

(p)



f

(p)



η + x,

on [a, b].

(3.3)

Furthermore,  R  by Lemma 2.7, 0 ≤ x(t), t ∈ [a, b]. t g exp − a g on both sides of (3.3), one has  Z t  (f g − xg) exp − g

(p)



 Z t  ηg exp − g ,

a

Multiplying by

t ∈ [a, b].

(3.4)

a

By Lemma 2.6, 

 Z t 0 x exp − g

(p)





 Z t 0 η − η exp − g ,

a

t ∈ [a, b].

(3.5)

a

Taking in account (2.4), we get  Z t   0 ≤ x(t) ≤ η exp g −1 ,

∀t ∈ [a, b].

(3.6)

a

It follows from (2.7), (3.3) and (3.6) that  Z t   Z t  (p) (p) 0  f  η + η exp g − 1 = η exp g , a

t ∈ [a, b].

a

This completes the proof. As Lemma 1.1, we have the following consequence.

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Corollary 3.2. Let f ∈ DHK . If there exist positive constants K and η such that 0

(p)

t

Z

(p)

f



t ∈ [a, b].

(Kf (s) + η)ds,

 a

Then 0

(p)



(p)

f



η(b − a) exp (K(b − a)) .

Remark 3.3. If f (t) is nonnegative and Henstock-Kurzweil integrable on [a, b], then Theorem 3.1 and Corollary 3.2 are still valid. Therefore, Lemma 1.1 and the corresponding result in [12] are only special cases of our results. For the ordering (2.5), we have the following result. Theorem 3.4. Let f ∈ DHK , g : [a, b] → R be a positive monotone function. If there is a constant η such that 0

(m)



f

(m)



t

Z η+

t ∈ [a, b].

f g,

(3.7)

a

Then 0

(m)



f

t

Z

(m)

η exp



 g ,

t ∈ [a, b].

(3.8)

a

Proof. The proof is similar to Theorem 3.1, so we omit it. Remark 3.5. Assume that f, g are nonnegative continuous functions. Since the ordering (2.5) equals to the pointwise ordering (see (2.7)), it is easy to see that Theorem 3.4 is a generalization of Lemma 1.2. Next we give a more general version of the Gronwall-Bellman type inequality due to H. E. Gollwitzer [13]. Theorem 3.6. Let f ∈ DHK , g : [a, b] → R be a nonnegative nonincreasing (p)

function. If there exist l ∈ DHK , l

0 on [a, b], and h ∈ HK, h ≥ 0 on



[a, b], such that 0

(p)



f

Z

(p)

t

l+h



t ∈ [a, b].

f g,

(3.9)

a

Then 0

(p)



f

(p)



Z l+h

t

Z lg exp

t

 gh ,

a

t ∈ [a, b].

(3.10)

a

 R  t Proof. Since g(t) ∈ BV and f, l ∈ DHK . Then exp − a g ∈ BV, f g ∈ Rt DHK , and lg ∈ DHK . Suppose that x(t) = a f g, one has x(a) = 0 and x0 = f g,

on [a, b].

(3.11)

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According to (3.9), 0

(p)



f

(p)



l + hx,

on [a, b].

(3.12)

It turns out from (3.11), (3.12) and Lemma 2.7 that (p)

x0 − ghx  lg, on [a, b].  R  t Multiplying exp − a gh on both sides of (3.13), we have 

 Z t  0 exp − gh x

(p)



 Z t  lg exp − gh ≤ lg,

a

(3.13)

t ∈ [a, b].

(3.14)

a

Applying (2.5) yields that t

Z

t

Z

0 ≤ x(t) ≤ exp

gh

t ∈ [a, b],

gl,

a

(3.15)

a

and hence, by (2.7) and (3.12), 0

(p)



(p)

f



t

Z l + h exp

t

Z gh

gl,

a

t ∈ [a, b],

a

which completes the proof. Corollary 3.7. Let f ∈ DHK , K be a positive constant. If there exist l ∈ DHK , l

(p)



0 on [a, b], and h ∈ HK, h ≥ 0 on [a, b], such that Z t (p) (p) 0  f  l+h f K, t ∈ [a, b].

(3.16)

a

Then 0

(p)



f

(p)



t

Z klkKh exp

 Kh ,

t ∈ [a, b].

(3.17)

a

Proof. Let F (t) =

Rt a

f and L(t) =

Rt a

l. By Theorem 3.6 and Lemma 2.7,

we get 0

(p)



f

(p)



Z

t

l + Kh exp

Z

t

Kh a

l,

t ∈ [a, b].

(3.18)

a

Moreover, in view of (2.4), Z t

Z

s

Z

s



0 ≤ F (t) ≤ L(t) + Kh exp Kh l a a a  Z s  Z t ≤ klk 1 + Kh exp Kh a a Z t  = klk exp Kh , t ∈ [a, b].

(3.19)

a

Therefore, by (2.4) and Lemma 2.4, the assertion follows. 9

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Remark 3.8. In Corollary 3.7, without loss of generality, let h ∈ C[a, b]. Obviously, the inequality (3.16) implies by (2.5) that t

Z

t

Z KhF ≤ klk +

0 ≤ F (t) ≤ L(t) +

KhF,

t ∈ [a, b].

(3.20)

a

a

Since F, h ∈ C[a, b] are nonnegative, K, klk are positive constants, then by Lemma 1.2, t

Z 0 ≤ F (t) ≤ klk exp

 t ∈ [a, b],

Kh , a

which is the same result as in (3.19). Hence, our results are extensions of Lemma 1.2. Furthermore, we have another result for the ordering (2.5). Theorem 3.9. Let f ∈ DHK , g : [a, b] → R be a nonnegative nonincreasing function. If there exist l ∈ DHK , l

(m)

0 on [a, b], and h ∈ HK, h ≥ 0 on



[a, b], such that (m)

0



f

Z

(m)

t

l+h



t ∈ [a, b].

f g,

(3.21)

a

Then 0

(m)



f

(m)



Z

t

Z

 gh ,

lg exp

l+h

t

t ∈ [a, b].

(3.22)

a

a

Remark 3.10. If f, g, h, l are nonnegative continuous functions, the inequalities in Theorem 3.6 also hold for the pointwise order in C[a, b], because of (2.7). Therefore, Theorem 3.6 extends the corresponding result in [13].

4

Application

In this section, we will give an application concerned about the GronwallBellman type inequalities. We consider the system A1 (A0 x)0 − A02 x = F 0 ,

(4.1)

where the derivatives, products and equality are understood in the sense of distributions, see [26]. Assumptions 4.1. The function A0 ∈ C[0, T ], A0 6= 0 on [0, T ], A−1 1 ∈ BV with |A−1 1 | ≥ δ1 > 0, and A2 ∈ C[0, T ] ∩ BV. Furthermore, F ∈ C[0, T ]. 10

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Let us notice that under Assumptions 4.1 the products A1 (A0 x)0 and A02 x, by Lemma 2.4, are well defined for any x ∈ C[0, T ]. Definition 4.2. A function x(t) is called a solution to the equation (4.1) on the interval [0, T ] if x ∈ C[0, T ] and A1 (A0 x)0 − A02 x − F 0 is the zero distribution. Firstly, we show the estimate of solutions to the equation (4.1). Theorem 4.3. Let the assumptions 4.1 be satisfied. If x(t) ∈ C[0, T ] is a solution to the equation (4.1) on [0, T ], then Z t  −1 |x(t)| ≤ |A0 (t)|klk exp h ,

t ∈ [0, T ],

(4.2)

0

where 0 l = |A−1 1 F |,

0 −1 h = |A−1 1 A2 A0 |.

(4.3)

Proof. By Assumptions 4.1 and (4.1), −1 0 −1 0 (A0 x)0 = A−1 1 F + A1 A2 A0 (A0 x).

(4.4)

Hence, by Lemma 2.2, 0

|(A0 x) |

(p)



−1 0 A F + |A−1 A02 A−1 | 0

1

1

t

Z

|(A0 x)0 |.

(4.5)

0

Let 0 l = A−1 1 F ,

0 −1 h = |A−1 1 A2 A0 |.

g = 1,

(4.6)

It is easy to see that l ∈ DHK , h ∈ HK. Therefore, by Corollary 3.7, Z t  0 (p) |(A0 x) |  klkh exp h , t ∈ [0, T ], (4.7) 0

which yields by (2.4) and Lemma 2.2 that Z t  |x(t)| ≤ |A−1 (t)|klk exp h , 0

t ∈ [0, T ].

(4.8)

0

The proof is therefore complete. Finally, we give an existence and uniqueness result.

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Theorem 4.4. Let the Assumptions 4.1 be satisfied. A−1 0

∈ BV with

|A−1 0 |

Moreover, either

≥ δ0 > 0 or A2 ∈ AC holds. Then (4.1) has a

unique solution x(t) ∈ C[0, T ] satisfying Z t −1 −1 kA−1 x(t) = A0 (t)k (t) 1 dF,

t ∈ [0, T ],

(4.9)

t ∈ [0, T ].

(4.10)

0

where

  Z t −1 k(t) = exp − A−1 A dA 2 , 0 1 0

Proof. We only prove the necessity, the sufficiency is easy to prove. By Assumptions 4.1 and (4.1), −1 0 0 −1 (A0 x)0 − A−1 1 A2 A0 (A0 x) = A1 F .

Let

  Z t −1 −1 k(t) = exp − A0 A1 dA2 ,

t ∈ [0, T ].

(4.11)

(4.12)

0

It is easy to see that k(t) ∈ C[0, T ] ∩ BV. Multiplying both side of (4.11) by k(t), we get −1 0 0 −1 k(A0 x)0 − kA−1 1 A2 A0 (A0 x) = kA1 F .

(4.13)

0 (kA0 x)0 = kA−1 1 F .

(4.14)

By Lemma 2.6,

Therefore, x(t) =

−1 A−1 0 (t)k (t)

Z

t

0

kA−1 1 dF,

t ∈ [0, T ].

Let y(t) be another solution to (4.1). Then, x(t) − y(t) = 0,

t ∈ [0, T ].

Therefore, x(t) satisfying (4.9) is a unique solution of (4.1).

References [1] T. H. Gronwall, Note on the derivatives with respect to a parameter of the solutions of a system of differential equations, Ann. of Math, 20 (4) (1919), 292–296. [2] R. Bellman, The stability of solutions of linear differential equations, Duke Math. J. 10 (4) (1943), 643–647. 12

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[3] W.-S. Cheung, On some new integrodifferential inequalities of the Gronwall and Wendroff type, J. Math. Anal. Appl. 178 (2) (1993), 438–449. [4] W.-S. Cheung, Some discrete nonlinear inequalities and applications to boundary value problems for difference equations, J. Differ. Equ. Appl. 10 (2) (2004), 213–223. [5] W.-S. Cheung, Some new nonlinear inequalities and applications to boundary value problems, Nonlinear Anal-Theor. 64 (9) (2006), 2112– 2128. [6] W.-S. Cheung, J. Ren, Discrete non-linear inequalities and applications to boundary value problems, J. Math. Anal. Appl. 319 (2) (2006), 708– 724. [7] W.-S. Cheung, Q.-H. Ma, J. Peˇcari´c, Some discrete nonlinear inequalities and applications to difference equations, Acta Mathematica Scientia, 28 (2) (2008), 417–430. [8] A. Younus, M. Asif, K. Farhad, On Gronwall type inequalities for interval-valued functions on time scales, J. Inequal. Appl. 2015, 2015: 271, 18 pp. [9] T. Abdeljawad, On conformable fractional calculus, J. Comput. Appl. Math. 279 (1) 2015, 57–66. [10] S. S. Dragomir, Some Gronwall type inequalities and applications. Nova Science Publishers, Inc., NY, 2003. ˇ Schwabik, Generalized differential equations. Fundamental results, [11] S. ˇ ˇ Vˇed Rada Mat. Pˇr´ırod. Vˇed, Rozpravy Ceskoslovensk´ e Akad. 95 (6) (1985), 103 pp. [12] K. Ostaszewski, J. Sochacki, Gronwall’s inequality and Henstock integral, J. Math. Anal. Appl. 128 (2) (1987), 370–374. [13] H. E. Gollwitzer, A note on a functional inequality. Proc. Amer. Math. Soc. 23 (3) (1969) 642–647. [14] W. W. Schmaedeke, G. R. Sell, The Gronwall inequality for modified Stieltjes integrals. Proc. Amer. Math. Soc. 19 (5) (1968), 1217–1222. [15] B. W. Helton, A special integral and a Gronwall inequality. Trans. Amer. Math. Soc. 217 (1976), 163–181. [16] V. Sree Hari Rao, Integral inequalities of Gronwall type for distributions. J. Math. Anal. Appl. 72 (2) (1979), 545–550. 13

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[17] P. C. Das, R. R. Sharma, Some Stieltjes integral inequalities. J. Math. Anal. Appl. 73 (2) (1980), 423–433. [18] X. Ding, G. Ye, Generalized Gronwall-Bellman inequalities using the Henstock-Kurzweil integral. Southeast Asian Bull. Math. 33 (4) (2009), 703–713. [19] J. Kurzweil, Generalized ordinary differential equations. Not absolutely continuous solutions, World Scientific, Singapore, 2012. ˇ Schwabik, Generalized ordinary differential equations, World Scien[20] S. tific, Singapore, 1992. ˇ Schwabik, G. Ye, Topics in Banach space integration, World Scien[21] S. tific, Singapore, 2005. [22] P. Y. Lee, Lanzhou Lecture on Henstock Integration, World Scientific, Singapore, 1989. [23] C. Wu, L. Zhao, T. Liu, Bounded variation functions and applications, Heilongjiang Science Press, Harbin, 1988. (In Chinese) [24] M. Tvrd´ y, Differential and integral equations in the space of regulated functions. Mem. Differential Equations Math. Phys., 25 (2002), 1–104. [25] M. Tvrd´ y, Linear distributional differential equations of the second order. Math. Bohem. 119 (4) (1994), 415–436. [26] M. Pelant, M. Tvrd´ y, Linear distributional differential equations in the space of regulated functions. Math. Bohem. 118 (4) (1993), 379–400. [27] D. D. Ang, K. Schmitt, L. K. Vy, A multidimensional analogue of the Denjoy-Perron-Henstock-Kurzweil integral, Bull. Belg. Math. Soc. 4 (3) (1997), 355–371. [28] E. Talvila, The distributional Denjoy integral, Real. Anal. Exchang. 33 (1) (2007/2008), 51–82. [29] G. Ye, W. Liu, The distributional Henstock-Kurzweil integral and applications. Monatsh. Math., 2015, in press. Doi: 10.1007/s00605-0150853-1. [30] W. Liu, G. Ye, Y. Wang, X. Zhou, On Periodic Solutions for First order differential equations involving the distributional Henstock-Kurzweil integral, Bull. Aust. Math. Soc. 86 (2) (2012), 327–338.

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Formulas and properties of some class of nonlinear difference equations E. M. Elsayed1,2 , Faris Alzahrani1 , and H. S. Alayachi1 1 Mathematics Department, Faculty of Science, King Abdulaziz University, P. O. Box 80203, Jeddah 21589, Saudi Arabia. 2 Department of Mathematics, Faculty of Science, Mansoura University, Mansoura 35516, Egypt. E-mails: [email protected], [email protected], [email protected]. ABSTRACT We obtain the formulas of the solutions of the recursive sequences xn+1 =

xn xn−5 , n = 0, 1, ..., xn−4 (±1 ± xn xn−5 )

where the initial conditions are arbitrary non zero real numbers. Also, we discuss and illustrate the stability of the solutions in the neighborhood of the critical points and the periodicity of the considered equations. Keywords: equilibrium point, recursive sequences, periodicity. Mathematics Subject Classification: 39A10. ––––––––––––––––––––––

1. INTRODUCTION In recent years, the qualitative study of difference equations has become an active research area among a considerable number of mathematicians. Some economical and biological examples can be seen in [9,36,40,47,48,54]. It is commonly known that nonlinear difference equations are able to produce and present sophisticated behaviors regardless their orders. Some articles show that a great effort has been done to demonstrate and explore the dynamics of nonlinear difference equations (see [40]-[61]). In fact, investigating these equations is a challenge and still new in the mathematical world. It is strongly believed that the rational difference equations are significant in their own right. Abo-Zeid and Cinar [1] illustrated the global stability, cyclical behavior, oscillation of all acceptable solutions of the equation Axn−1 . xn+1 = B−Cx n xn−2 In [7], [8] Cinar considered the solutions of the equations yn+1 =

yn−1 1+ayn yn−1 ,

yn+1 =

yn−1 −1+ayn yn−1 .

A. El-Moneam, and Alamoudy [16] examined the positive solutions of the equation in terms of its periodicity, boundedness and the global stability. The considered difference equation is given by xn+1 = axn +

bxn−1 +cxn−2 +f xn−3 +rxn−4 dxn−1 +exn−2 +gxn−3 +sxn−4 .

Khatibzadeh and Ibrahim [42] studied the boundedness, asymptotic stability, oscillatory behavior and discovered the closed form of solutions of the equation

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xn+1 = axn +

bxn xn−1 cxn +dxn−1 .

Simsek et al. [49] has found and explored solutions for the recursive formula yn+1 =

yn−3 1+yn−1 .

For other related papers, see [25—46]. We analyze and explore the solutions of the following nonlinear recursive equation xn+1 =

xn xn−5 , xn−4 (±1 ± xn xn−5 )

n = 0, 1, ...,

(1)

with conditions posed on the initial values are arbitrary non zero real numbers. Also, we will survey some dynamic behaviors of its solutions. The linearized equation of equation xn+1 = f (xn , xn−1 , ..., xn−k ), n = 0, 1, ...,

(2)

about the equilibrium x is the linear difference equation yn+1 =

k P

i=0

∂f (x,x,...,x) yn−i . ∂xn−i

Theorem A [43]: Assume that pi ∈ R, i = 1, 2, ..., k and k ∈ {0, 1, 2, ...}. Then condition for the asymptotic stability of the difference equation

Pk

i=1

|pi | < 1, is a sufficient

xn+k + p1 xn+k−1 + ... + pk xn = 0, n = 0, 1, ... .

2. THE FIRST EQUATION XN+1 =

XN XN −5 XN −4 (1+XN XN −5 )

This section is devoted to give a specific solution of the first difference equation which is xn+1 =

xn xn−5 . xn−4 (1 + xn xn−5 )

(3)

Theorem 2.1. Let {xn }∞ n=−5 be a solution of Eq.(3). Then x10n−5 x10n−3 x10n−1 x10n+1

x10n+3

= f

= d

n−1 Yµ

i=0 n−1 Yµ

i=0 n−1 Yµ

1 + (10i)af 1 + (10i + 5)af 1 + (10i + 2)af 1 + (10i + 7)af

¶

¶

,

,

x10n−4 = e x10n−2 = c

n−1 Yµ

i=0 n−1 Yµ

i=0 n−1 Yµ

1 + (10i + 1)af 1 + (10i + 6)af

1 + (10i + 3)af 1 + (10i + 8)af

¶

¶

,

,

¶ ¶ 1 + (10i + 4)af 1 + (10i + 5)af , x10n = a , = b 1 + (10i + 9)af 1 + (10i + 10)af i=0 i=0 n−1 n−1 Y µ 1 + (10i + 6)af ¶ Y µ 1 + (10i + 7)af ¶ af af , x10n+2 = , = e(1 + af ) i=0 1 + (10i + 11)af d(1 + 2af ) i=0 1 + (10i + 12)af

n−1 Y µ 1 + (10i + 8)af ¶ af , = c(1 + 3af ) i=0 1 + (10i + 13)af

x10n+4

n−1 Y µ 1 + (10i + 9)af ¶ af , = b(1 + 4af ) i=0 1 + (10i + 14)af

where we put x−5 = f, x−4 = e, x−3 = d, x−2 = c, x−1 = b, x0 = a.

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Proof: The result holds for n = 0. Assume that n > 0 and our assumption true for n − 1. Then; x10n−15 x10n−13 x10n−11

= f

= d

= b

n−2 Yµ

i=0 n−2 Yµ

i=0 n−2 Yµ i=0

x10n−9

=

x10n−7

=

1 + (10i)af 1 + (10i + 5)af 1 + (10i + 2)af 1 + (10i + 7)af

1 + (10i + 4)af 1 + (10i + 9)af

¶

¶

¶

¶ 1 + (10i + 1)af x10n−14 = e , 1 + (10i + 6)af i=0 n−2 Y µ 1 + (10i + 3)af ¶ x10n−12 = c , 1 + (10i + 8)af i=0 n−2 Y µ 1 + (10i + 5)af ¶ x10n−10 = a , 1 + (10i + 10)af i=0 n−2 Yµ

,

,

,

n−2 Y µ 1 + (10i + 6)af ¶ af , e(1 + af ) i=0 1 + (10i + 11)af n−2 Y µ 1 + (10i + 8)af ¶ af , c(1 + 3af ) i=0 1 + (10i + 13)af

n−2 Y µ 1 + (10i + 7)af ¶ af , d(1 + 2af ) i=0 1 + (10i + 12)af n−2 Y µ 1 + (10i + 9)af ¶ af . = b(1 + 4af ) i=0 1 + (10i + 14)af

x10n−8 = x10n−6

Now, it follows from Eq.(3) that x10n−5

=

x10n−6 x10n−11 x10n−10 (1 + x10n−6 x10n−11 ) n−2 Y³ af b(1+4af )

=

"

a

=

=

=

=

n−2 Y³ i=0

# "i=0 ´ 1+(10i+5)af 1+ 1+(10i+10)af

1+(10i+9)af 1+(10i+14)af

´ n−2 Y ³ 1+(10i+4)af ´ b 1+(10i+9)af i=0

af b(1+4af )

n−2 Y³

1+(10i+9)af 1+(10i+14)af

i=0

# ´ n−2 Y ³ 1+(10i+4)af ´ b 1+(10i+9)af i=0

n−2 Y µ 1 + (10i + 4)af ¶ f (1 + 4af ) i=0 1 + (10i + 14)af "n−2 µ #" # n−2 Y 1 + (10i + 5)af ¶ Y µ 1 + (10i + 4)af ¶ af 1+ 1 + (10i + 10)af (1 + 4af ) i=0 1 + (10i + 14)af i=0 ¶ µ f 1 + (10n − 6)af "n−2 µ # ¸ Y 1 + (10i + 5)af ¶ ∙ af 1+ 1 + (10i + 10)af 1 + (10n − 6)af i=0

f "n−2 µ # Y 1 + (10i + 5)af ¶ [1 + (10n − 6)af + af ] 1 + (10i + 10)af i=0 n−2 n−1 Y µ 1 + (10i + 10)af ¶ Y µ 1 + (10i)af ¶ f =f . [1 + (10n − 5)af ] i=0 1 + (10i + 5)af 1 + (10i + 5)f i=0

Also, we have x10n−4 =

x10n−5 x10n−10 x10n−9 (1 + x10n−5 x10n−10 )

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f ="

n−1 Y³

1+(10i)af 1+(10i+5)af

i=0

af e(1+af )

n−2 Y³

1+(10i+6)af 1+(10i+11)af

i=0

´

#"

1+f

1+(10i)af 1+(10i+5)af

i=0

# ´ n−2 Y ³ 1+(10i+5)af ´ a 1+(10i+10)af i=0

¶ 1 af 1 + (10n − 5)af # " ¸ n−2 Y ³ 1+(10i+6)af ´ ∙ af af 1 + e(1+af ) 1+(10i+11)af 1 + (10n − 5)af i=0 "

af

¶# 1 + (10i + 6)af [1 + (10n − 5)af + af ] 1 + (10i + 11)af i=0 n−2 n−1 Y µ 1 + (10i + 11)af ¶ Y µ 1 + (10i + 1)af ¶ e(1 + af ) =e . [1 + (10n − 4)af ] i=0 1 + (10i + 6)af 1 + (10i + 6)af i=0

=

=

af e(1 + af )

n−2 Yµ

Similarly x10n−3 =

x10n−4 x10n−9 x10n−8 (1 + x10n−4 x10n−9 )

¶ n−2 Y µ 1 + (10i + 6)af ¶ 1 + (10i + 1)af af 1 + (10i + 6)af e(1+af ) i=0 1 + (10i + 11)af i=0 #" # " n−2 n−2 n−1 ³ ´ ³ ´ ³ ´ Y Y Y 1+(10i+7)af 1+(10i+1)af 1+(10i+6)af af af 1+e d(1+2af ) 1+(10i+12)af 1+(10i+6)af e(1+af ) 1+(10i+11)af e

n−1 Yµ

i=0

=

i=0 n−1 Y³

µ

=

=

´ n−2 Y ³ 1+(10i+5)af ´ a 1+(10i+10)af

i=0

¶ n−2 Y

n−1 Yµ

af 1 + (10i + 6)af i=0 # " " n−2 n−1 Y ³ 1+(10i+7)af ´ Yµ af 1+ d(1+2af ) 1+(10i+12)af i=0

=

=

=

i=0

i=0

(1 + (10i + 6)af )

i=0

af 1 + (10i + 6)af

¶ n−2 Y

#

(1 + (10i + 6)af )

i=0

µ

¶ af 1 + (10n − 4)af # " µ ¶¸ n−2 ³ ´ ∙ Y af 1+(10i+7)af af 1+ d(1+2af ) 1+(10i+12)af 1 + (10n − 4)af i=0 " "

1 (1+2af )

n−2 Y³

d # ´

1+(10i+7)af 1+(10i+12)af

i=0

d 1 (1+2af )

n−2 Y³ i=0

1+(10i+7)af 1+(10i+12)af

# ´

[1 + (10n − 4)af + af ] =d [1 + (10n − 3)af ]

n−1 Yµ i=0

1 + (10i + 2)af 1 + (10i + 7)af

¶

.

Similarly, one can simply find the other relations. Thus, the proof is done. Theorem 2.2. The unique equilibrium point of Eq.(3) is the number zero which is not locally asymptotically stable.

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Proof: The equilibrium points of Eq.(3) obtained by x2 ¢. ¡ x 1 + x3

x=

Arranging the previous equation gives x4 = 0. Thus x = 0. Let f : (0, ∞)3 −→ (0, ∞) be a function takes the form f (u, v, w) = Therefore fu (u, v, w) =

w

2,

v (1 + uw)

uw . v (1 + uw)

fv (u, v, w) = −

uw u , fw (u, v, w) = 2. v 2 (1 + uw) v (1 + uw)

So fu (x, x, x) = 1,

fv (x, x, x) = 1,

fw (x, x, x) = 1.

Then by using Theorem A the proof follows. Example 1. We assume x−5 = 6, x−4 = 11, x−3 = 3, x−2 = 2, x−1 = 1.8, x0 = −7. See Fig. 1. plot of x(n+1)=x(n)x(n−5)/(x(n−4)(1+x(n)x(n−5)) 12 10 8 6

x(n)

4 2 0 −2 −4 −6 −8

0

10

20

30

40

50 n

60

70

80

90

100

Figure 1. Example 2. See Fig. 2, since x−5 = 1.6, x−4 = 1.2, x−3 = −3, x−2 = .7, x−1 = 1.8, x0 = 3.

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plot of x(n+1)=x(n)x(n−5)/(x(n−4)(1+x(n)x(n−5)) 3

2

x(n)

1

0

−1

−2

−3

0

10

20

30

40

50 n

60

70

80

90

100

Figure 2.

3. THE SECOND EQUATION XN +1 =

XN XN −5 XN −4 (−1+XN XN −5 )

This section is devoted to obtain the solution of the difference equation which is xn+1 =

xn xn−5 , xn−4 (−1 + xn xn−5 )

(4)

where x0 x−5 6= 1.

Theorem 3.1. Let {xn }∞ n=−5 be a solution of Eq.(4). Then for x10n−5

=

x10n−3

=

x10−1

=

x10n+1

=

x10n+3

=

f , (−1 + af )n d , (−1 + af )n b , (−1 + af )n af , e(−1 + af )n+1 af , c(−1 + af )n+1

x10n−4 = e(−1 + af )n , x10n−2 = c(−1 + af )n , x10n = a(−1 + af )n , af (−1 + af )n , d af (−1 + af )n . = b

x10n+2 = x10n+4

Proof: The result holds for n = 0. Assume that n > 0 and that our assumption true for n − 1. Then;

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x10n−15

=

x10n−13

=

x10−11

=

x10n−9

=

x10n−7

=

f , (−1 + af )n−1 d , (−1 + af )n−1 b , (−1 + af )n−1 af , e(−1 + af )n af , c(−1 + af )n

x10n−14 = e(−1 + af )n−1 , x10n−12 = c(−1 + af )n−1 , x10n−10 = a(−1 + af )n−1 , af (−1 + af )n−1 , d af (−1 + af )n−1 . = b

x10n−8 = x10n−6

It follows from (4) that x10n−5 =

af af x10n−6 x10n−11 = = , n−1 x10n−10 (−1 + x10n−6 x10n−11 ) a(−1 + af ) [−1 + af ] a(−1 + af )n

x10n−4

=

=

=

x10n−5 x10n−10 x10n−9 (−1 + x10n−5 x10n−10 ) af a(−1 + af )n−1 a(−1 + af )n ∙ ¸∙ ¸ af af n−1 −1 + a(−1 + af ) e(−1 + af )n a(−1 + af )n e(−1 + af )n e(−1 + af )n ¸= ∙ = e(−1 + af )n . af [1 − af + af ] (−1 + af ) −1 + (−1 + af )

Similarly one can simply prove the other relations. Theorem 3.2. Eq.(4) has a period ten solution iff af = 2 and will be in the following form ¾ ½ af af af af , , , , f, e, d, ... . f, e, d, c, b, a, e d c b Proof: Firstly, assume that there exists a period ten solution ¾ ½ af af af af , , , , f, e, d, ... , f, e, d, c, b, a, e d c b of Eq.(4). Then, we can notice from the solution of Eq.(4) that

f

=

b = af d or,

=

d f , e = e(−1 + af )n , d = , c = c(−1 + af )n , (−1 + af )n (−1 + af )n af af b = , a = a(−1 + af )n , , n (−1 + af ) e e(−1 + af )n+1 af (−1 + af )n af af af (−1 + af )n af , = = . , d c c(−1 + af )n+1 b b (−1 + af )n = 1.

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Then af = 2. Secondly, suppose that af = 2. Then, it is easily seen from the solution of Eq.(4) that

x10n−5

= f,

x10n

= a,

x10n−4 = e, x10n−3 = d, x10n−2 = c, x10−1 = b, af af af , x10n+2 = , x10n+3 = , x10n+1 = e d c

x10n+4 =

af . b

Thus, the periodic solution of period ten is obtained and this proves the theorem. √ Theorem 3.3. Eq.(4) has two equilibrium points which are 0, 3 2 and these equilibrium points are not locally asymptotically stable. Proof: The equilibrium points of Eq.(4) can be written in the following form x2 ¢. ¡ x −1 + x2

x= Arranging this gives ¡ ¢ x2 −1 + x2 √ Therefore, the fixed points are 0, ± 2.

x2

= ⇒

¡ ¢ x2 x2 − 2 = 0.

Let f : (0, ∞)3 −→ (0, ∞) be a function defined by f (u, v, w) =

uw . v (−1 + uw)

Then it follows that fu (u, v, w) = −

w

2, v (−1 + uw) uw , fv (u, v, w) = − 2 v (−1 + uw) u fw (u, v, w) = − , v (−1 + uw)2

It can be seen that fu (x, x, x) = −1,

fv (x, x, x) = ±1,

fw (x, x, x) = −1.

Then by using Theorem A the proof follows. Example 3. We consider x−5 = .8, x−4 = 1.7, x−3 = .3, x−2 = 2, x−1 = 1.8, x0 = .7. See Fig. 3.

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plot of x(n+1)=x(n)x(n−5)/(x(n−4)(−1+x(n)x(n−5)) 50

40

30

x(n)

20

10

0

−10

−20

−30

0

5

10

15

20

25 n

30

35

40

45

50

Figure 3. Example 4. See Fig. 4, since x−5 = 8, x−4 = 1.7, x−3 = .3, x−2 = 2, x−1 = 1.8, x0 = 1/4. plot of x(n+1)=x(n)x(n−5)/(x(n−4)(−1+x(n)x(n−5)) 8

7

6

x(n)

5

4

3

2

1

0

0

5

10

15

20

25 n

30

35

40

45

50

Figure 4.

4. THE THIRD EQUATION XN+1 =

XN XN −5 XN −4 (1−XN XN −5 )

In this section we will obtain and present the solution of the third difference equation which is xn xn−5 . xn+1 = xn−4 (1 − xn xn−5 )

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J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 24, NO.8, 2018, COPYRIGHT 2018 EUDOXUS PRESS, LLC

Theorem 4.1. Let {xn }∞ n=−5 be a solution of Eq.(5). Then for n = 0, 1, ... x10n−5 x10n−3

= f

= d

n−1 Yµ

i=0 n−1 Yµ

i=0 n−1 Yµ

1 − (10i)af 1 − (10i + 5)af 1 − (10i + 2)af 1 − (10i + 7)af

¶

¶

,

,

x10n−4 = e x10n−2 = c

n−1 Yµ

i=0 n−1 Yµ

i=0 n−1 Yµ

¶ 1 − (10i + 4)af , x10n = a 1 − (10i + 9)af i=0 i=0 ¶ µ n−1 Y 1 − (10i + 6)af af , e(1 − af ) i=0 1 − (10i + 11)af n−1 Y µ 1 − (10i + 7)af ¶ af , d(1 − 2af ) i=0 1 − (10i + 12)af n−1 Y µ 1 − (10i + 8)af ¶ af , c(1 − 3af ) i=0 1 − (10i + 13)af n−1 Y µ 1 − (10i + 9)af ¶ af . b(1 − 4af ) i=0 1 − (10i + 14)af

x10n−1

= b

x10n+1

=

x10n+2

=

x10n+3

=

x10n+4

=

1 − (10i + 1)af 1 − (10i + 6)af

1 − (10i + 3)af 1 − (10i + 8)af

¶

¶

1 − (10i + 5)af 1 − (10i + 10)af

,

,

¶

,

Theorem 4.2. The unique critical point of Eq.(5) is the number zero which is not locally asymptotically stable. Example 5. Suppose that x−5 = 8, x−4 = 1.7, x−3 = .3, x−2 = 2, x−1 = 1.8, x0 = 1/4 see Fig. 5. plot of x(n+1)=x(n)x(n−5)/(x(n−4)(1−x(n)x(n−5)) 8

6

x(n)

4

2

0

−2

−4

0

5

10

15

20

25 n

30

35

40

45

50

Figure 5. Example 6. See Fig. 6 since x−5 = −7, x−4 = 1.5, x−3 = −3, x−2 = 2, x−1 = 12, x0 = 4.

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plot of x(n+1)=x(n)x(n−5)/(x(n−4)(1−x(n)x(n−5)) 12 10 8 6

x(n)

4 2 0 −2 −4 −6 −8

0

5

10

15

20

25 n

30

35

40

45

50

Figure 6.

5. THE FOURTH EQUATION XN+1 =

XN XN −5 XN −4 (−1−XN XN −5 )

Now, we will explore and discover the solution of the following difference equation xn+1 =

xn xn−5 , n = 0, 1, ..., xn−4 (−1 − xn xn−5 )

(6)

where x−5 x0 6= −1.

Theorem 5.1. Let {xn }∞ n=−5 be a solution of Eq.(6). Then Eq.(6) has unboundedness solution (except in the case if af = −2) and for n = 0, 1, ... x10n−5

=

x10n−3

=

x10−1

=

x10n+1

=

x10n+3

=

f , (−1 − af )n d , (−1 − af )n b , (−1 − af )n af , e(−1 − af )n+1 af , c(−1 − af )n+1

x10n−4 = e(−1 − af )n , x10n−2 = c(−1 − af )n , x10n = a(−1 − af )n , af (−1 − af )n , d af (−1 − af )n . = b

x10n+2 = x10n+4

Theorem 5.2. Eq.(6) has a periodic ½ ¾ solution of period ten iff af = −2 and written in the following form af af af f, e, d, c, b, a, , , , f, e, d, c, ... . e d c

Theorem 5.3. The unique equilibrium of Eq.(6) is the number zero which is not locally asymptotically stable. Example 7. Consider x−5 = −7, x−4 = 1.5, x−3 = −3, x−2 = 2, x−1 = 12, x0 = 4 see Fig. 7.

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6

5

plot of x(n+1)=x(n)x(n−5)/(x(n−4)(−1−x(n)x(n−5))

x 10

4

3

x(n)

2

1

0

−1

−2

0

5

10

15

20

25 n

30

35

40

45

50

Figure 7. Example 8. Fig. 8 illustrates the solutions when x−5 = −7, x−4 = 1.5, x−3 = −3, x−2 = 2, x−1 = 12, x0 = 2/7. plot of x(n+1)=x(n)x(n−5)/(x(n−4)(−1−x(n)x(n−5)) 12 10 8 6

x(n)

4 2 0 −2 −4 −6 −8

0

5

10

15

20

25 n

30

35

40

45

50

Figure 8.

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Axn−1 , Bol. Soc. B − Cxn xn−2

2. A. M. Ahmed, and N. A. Eshtewy, Basin of attraction of the recursive sequence xn+1 =

α + βxn + γxn−1 , A + Bxn + Cxn−1

Journal of Fractional Calculus and Applications, 5 (3S) (10) (2014), 1-8. 3. M. Aloqeili, Dynamics of a rational difference equation, Appl. Math. Comp., 176(2), (2006), 768-774. 4. A. Asiri, E. M. Elsayed, and M. M. El-Dessoky, On the solutions and periodic nature of some systems of difference equations, Journal of Computational and Theoretical Nanoscience, 12 (10) (2015), 3697-3704. 5. F. Bozkurt, Stability Analysis of a Nonlinear Difference Equation, International Journal of Modern Nonlinear Theory and Application, 2 (2013), 1-6. 6. D. Chen, and C. Wang, Boundedness of a Max-type Fourth Order Difference Equation with Periodic Coefficients, Journal of Informatics and Mathematical Sciences, 6 (1) (2014), 1—21. xn−1 7. C. Cinar, On the positive solutions of the difference equation xn+1 = , Appl. Math. Comp., 1 + axn xn−1 158(3) (2004), 809-812. xn−1 8. C. Cinar, On the positive solutions of the difference equation xn+1 = , Appl. Math. Comp., −1 + axn xn−1 158(3) (2004), 793-797. 9. Q. Din, Global Stability of Beddington Model, Qualitative Theory of Dynamical Systems, In press (2016), DOI 10.1007/s12346-016-0197-9. 10. E. M. Elabbasy, M. Y. Barsoum, H. S. Alshawee, Behavior of solutions of a class of nonlinear rational βxδ difference equation xn+1 = αxn−k + δn−l , Electronic Journal of Mathematical Analysis and Applications, γxn−s 4 (2) (2016), 78-87. 11. E. M. Elabbasy , H. El-Metwally and E. M. Elsayed, Global Behavior of the Solutions of Difference Equation, Advances in Difference Equations 2011, 2011: 28. 12. E. M. Elabbasy, A. A. Elsadany, and S. Ibrahim, Qualitative behavior of rational difference equations of higher order, Malaya J. Mat., 3 (4) (2015), 530—539. 13. H. El-Metwally and E. M. Elsayed, Qualitative Behavior of some Rational Difference Equations, Journal of Computational Analysis and Applications, 20 (2) (2016), 226-236. 14. H. El-Metwally and E. M. Elsayed, Form of solutions and periodicity for systems of difference equations, Journal of Computational Analysis and Applications, 15(5) (2013), 852-857. 15. H. El-Metwally and E. M. Elsayed, Solution and Behavior of a Third Rational Difference Equation, Utilitas Mathematica, 88 (2012), 27—42. 16. M. A. El-Moneam, and S. O. Alamoudy, On study of the asymptotic behavior of some rational difference equations, Dynamics of Continuous, Discrete and Impulsive Systems Series A: Mathematical Analysis, 22 (2015) 157-176. 17. M. M. El-Dessoky, and E. M. Elsayed, On the solutions and periodic nature of some systems of rational difference equations, Journal of Computational Analysis and Applications, 18 (2) (2015), 206-218. 18. E. M. Elsayed, Solution and attractivity for a rational recursive sequence, Discrete Dynamics in Nature and Society, Volume 2011, Article ID 982309, 17 pages. 19. E. M. Elsayed, Behavior and expression of the solutions of some rational difference equations, Journal of Computational Analysis and Applications 15 (1) (2013), 73-81. 20. E. M. Elsayed, Solution for systems of difference equations of rational form of order two, Computational and Applied Mathematics 33 (3) (2014), 751-765. 21. E. M. Elsayed, On the solutions and periodic nature of some systems of difference equations, International Journal of Biomathematics 7 (6) (2014), 1450067, (26 pages). 22. E. M. Elsayed, Dynamics and Behavior of a Higher Order Rational Difference Equation, The Journal of Nonlinear Science and Applications, 9 (4) (2016), 1463-1474.

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A rational bicubic spline for visualization of shaped data Xilian Fu College of Science, Shandong University of Technology, Zibo, 255049, China E-mail: [email protected]

Abstract The shaped data usually needs to be represented in such a way that its visual display looks smooth and pleasant, its shape is preserved everywhere and the computation cost is economical. This work contributes to the graphical display of positive or monotone data. For this purpose, a new bicubic rational interpolating spline with biquadratic denominator is developed based on function values and partial derivatives, and simple sufficient conditions are derived on the shape parameters in the description of the rational function to visualize the positive or monotone data in the view of positive or monotone surfaces. Keywords: Rational bicubic spline, shape parameter, positivity, monotonicity.

1

Introduction

The construction method of curve and surface and the mathematical description of them is a important issue in Computer-Aided Geometry Design (CAGD). Generally speaking, the interpolating data are often given as a set of values, in order to display these data, it is first necessary to construct an interpolant through those data; and then this interpolant is used in the subsequent contouring or curve and surface drawing. Thus, for the data obtained from some complex function or from some scientific phenomena, smooth curve or surface expression becomes crucial to incorporate the inherited features of the data. In many problems of industrial design and manufacturing, the given data often have some special shape properties, such as positivity, monotonicity and convexity, it is usually needed to generate a smooth function, which passes through the given set of data and preserves those certain shape properties of the data. In recent years, a good amount of work has been published that focuses on shape preserving curves and surfaces. Goodman and Ong [7] presented a local convexity preserving interpolation scheme using parametric C 2 cubic splines with uniform knots produced by a vector subdivision scheme. In [2, 4, 8, 9, 13, 14, 16, 18], several shape-preserving rational curves were shown for shaped data, such as positive data, monotonic data and convex data. Beatson and Ziegler [1] interpolated monotone data, given on a rectangular grid, with a C 1 monotone quadratic spline, and derived necessary and sufficient conditions to visualize monotone data. Floater and Pena [6] defined three kinds of monotonicity preservation of systems of bivariate functions on a triangle, and investigated some geometric applications. In [10], authors proposed a kind of monotonicitypreserving interpolating schemes for 2D/3D monotone data by constraints on shape parameters in the description of rational spline interpolants. In [12], Hussain et al. presented the C 1 rational bi-cubic local interpolation schemes for the shape preservation of convex, monotone and positive surface data. In [11], A bi-quadratic trigonometric interpolation scheme with four free parameters is developed for the positive and monotone 3D data. Piah et al. [15] discussed the problem of positivity preserving for scattered data interpolation.

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FU: VISUALIZATION OF SHAPED DATA In [5], Duan et al. developed a C 1 bivariate rational spline interpolant based on function values and partial derivatives under some suitable hypotheses. In [17], Sun et al. proposed a surface modeling method by using C 2 piecewise rational spline interpolation. This paper is concerned with the preservation of 3D positive data and monotone data. To solve the problem, motivated by [5, 17], we will first construct a new bicubic rational interpolating spline with biquadratic denominator based function values and derivative values of an original function. Further more, a positivitypreserving scheme and a monotonicity-preserving scheme are developed to visualize 3D positive data and monotone data in the view of positive surfaces and monotone surfaces, respectively. This paper is arranged as follows. Section 2 describes about the C 1 bicubic rational spline interpolant to be used in the surface schemes. In Section 3, the positivity problem is discussed for the generation of a smooth surface which can preserve the shape of positive data. In Section 4, a method is developed to preserve the shape of monotone data in the view of monotone surfaces by making constraints on shape parameters in the description of bicubic rational interpolant. Finally, Numerical examples are presented to discuss and demonstrate the performance of the method in Section 5.

2

Bicubic rational interpolant

Let Ω = [a, b; c, d] be the plane region, and {(xi , yj , fi,j ) : i = 1, 2, · · · , n; j = 1, 2, · · · , m} be a given set of data points, where a = x1 < x2 < ... < xn = b, c = y1 < y2 < · · · < ym = d are the knot spacings, fi,j represents fi,j (x, y) at the point (xi , yj ). Let d∗i,j and di,j be chosen partial (x,y) (x,y) and ∂f ∂y at the knots (xi , yj ), respectively. Denote hi = xi+1 − xi , derivative values ∂f∂x lj = yj+1 − yj , I = {1, 2, · · · , n}, J = {1, 2, · · · , m}, and for any point (x, y) ∈ [xi , xi+1 ; yj , yj+1 ], y−yj ∗ ∗ ∗ i θ := x−x hi , η := lj . Let αi,j , βi,j and γi,j be the positive parameters. First, we construct the ∗ x-direction interpolating curve Pi,j (x) in [xi , xi+1 ], this is given by ∗ Pi,j (x)

∗ f 2 ∗ 2 ∗ 3 ∗ (1 − θ)3 αi,j i,j + θ(1 − θ) Vi,j + θ (1 − θ)Wi,j + θ βi,j fi+1,j , = ∗ + θ(1 − θ)γ ∗ + θ 2 β ∗ (1 − θ)2 αi,j i,j i,j

(1)

where ∗ = (α∗ + γ ∗ )f ∗ ∗ Vi,j i,j i,j i,j + hi αi,j di,j , ∗ = (β ∗ + γ ∗ )f ∗ ∗ Wi,j i,j i,j i+1,j − hi βi,j di+1,j , ∗ (x) is called a rational cubic spline interpolation in [x , x ], and which satisfies: The interpolant Pi,j 1 n ∗ ∗ ∗ 0 ∗ 0 Pi,j (xi ) = fi,j , Pi,j (xi+1 ) = fi+1,j , Pi,j (xi ) = d∗i,j , Pi,j (xi+1 ) = d∗i+1,j . ∗ (x) defines the bivariate function on [x , x Using the x-direction Pi,j i i+1 ; yj , yj+1 ] as follow:

P (x, y) ≡ Pi,j (x, y) =

pi,j (x, y) , qi,j (y)

(2)

where ∗ (x) + η(1 − η)2 V 2 3 ∗ pi,j (x, y) = (1 − η)3 αi,j Pi,j i,j + η (1 − η)Wi,j + η βi,j Pi,j+1 (x),

qi,j (y) = (1 − η)2 αi,j + η(1 − η)γi,j + η 3 βi,j , with ∗ (x) + l α D (x), Vi,j = (αi,j + γi,j )Pi,j j i,j i,j ∗ Wi,j = (βi,j + γi,j )Pi,j+1 (x) − lj βi,j Di,j+1 (x),

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J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 24, NO.8, 2018, COPYRIGHT 2018 EUDOXUS PRESS, LLC

FU: VISUALIZATION OF SHAPED DATA

and Di,j (x) =

∗ + β ∗ )d ∗ + θγ ∗ )d θ2 ((1 − θ)γi,j (1 − θ)2 (αi,j i,j i,j i,j i+1,j + , ∗ + θ(1 − θ)γ ∗ + θ 2 β ∗ 2 α∗ + θ(1 − θ)γ ∗ + θ 2 β ∗ (1 − θ)2 αi,j (1 − θ) i,j i,j i,j i,j i,j

(3)

and αi,j > 0, βi,j > 0, γi,j > 0. The interpolant P (x, y) defined by (2) is called a bicubic rational interpolation in [x1 , xn ; y1 , ym ], and which satisfies P (xr , ys ) = f (xr , ys ),

∂P (xr , ys ) = d∗r,s , ∂x

∂P (xr , ys ) = dr,s , r = i, i + 1; s = j, j + 1. ∂y It is easy to test that the interpolant P (x, y) is C 1 in the interpolating region [a, b; c, d] if the shape parameters satisfy αi,j =constant, βi,j =constant and γi,j =constant, for each j ∈ J and all ∗ and γ ∗ might be. i ∈ I, no matter what the shape parameters a∗i,j , βi,j i,j

3

Positivity-preserving surface interpolating scheme

In engineering, industrial, and scientific problems, the construction of shape preserving interpolants is an everlasting demand and one of the major research areas of computer aided design. In this section, we identify suitable values for the shape parameters involved in P (x, y) defined by (2), which make the interpolating surface to preserve positive property of given data. Let {(xi , yj , fi,j ) : i = 1, 2, · · · , n; j = 1, 2, · · · , m} be a monotone data set defined over the rectangular grid [xi , xi+1 ; yj , yj+1 ] such that fi,j > 0 for all i, j. Here, the aim is to construct a piecewise rational bivariate function P (x, y) on Ω = [x1 , xn ; y1 , ym ] such that P (xi , yj ) = fi,j ,

i = 1, 2, · · · , n; j = 1, 2, · · · , m,

and P (x, y) > 0 for (x, y) ∈ Ω. From (2), qi,j (y) > 0 for the positive shape parameters, therefore, P (x, y) is positive if the following constraints hold: ∗ ∗ Pi,j (x) > 0, Vi,j > 0, Wi,j > 0, Pi,j+1 (x) > 0. ∗ (x) > 0 holds if the following equalities are satisfied: From (1), it is easy to see that Pi,j ∗ = (α∗ + γ ∗ )f ∗ ∗ Vi,j i,j i,j i,j + hi αi,j di,j > 0, ∗ = (β ∗ + γ ∗ )f ∗ ∗ Wi,j i,j i,j i+1,j − hi βi,j di+1,j > 0. ∗ (x) > 0 if Thus, Pi,j ∗ ∗ γi,j hi d∗i,j γi,j hi d∗i+1,j > − − 1, > − 1. ∗ ∗ αi,j fi,j βi,j fi+1,j

(4)

∗ ∗ hi d∗i,j+1 γi,j+1 hi d∗i+1,j+1 γi,j+1 > − − 1, > − 1. ∗ ∗ αi,j+1 fi,j+1 βi,j+1 fi+1,j+1

(5)

∗ Similarly, Pi,j+1 (x) > 0 if

Consider Vi,j which, after simplification, leads to Vi,j =

(1 − θ)3 κ1 + θ(1 − θ)2 κ2 + θ2 (1 − θ)κ3 + θ3 κ4 , ∗ + θ(1 − θ)γ ∗ + θβ ∗ (1 − θ)2 αi,j i,j i,j 1534

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J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 24, NO.8, 2018, COPYRIGHT 2018 EUDOXUS PRESS, LLC

FU: VISUALIZATION OF SHAPED DATA

where ∗ ((α κ1 = αi,j i,j + γi,j )fi,j + lj di,j αi,j ), ∗ ∗ ∗ ∗ ∗ + γ ∗ )f κ2 = (αi,j + γi,j )((αi,j i,j i,j + hi di,j αi,j ) + lj di,j αi,j (αi,j + γi,j ), ∗ + γ ∗ )f ∗ ∗ ∗ ∗ κ3 = (αi,j + γi,j )((βi,j i,j i+1,j − hi di+1,j βi,j ) + lj di+1,j αi,j (βi,j + γi,j ), ∗ ((α κ4 = βi,j i,j + γi,j )fi+1,j + lj di+1,j αi,j ).

Note that Vi,j > 0 if  ∗ 2hi d∗i,j γi,j   > − − 1,   ∗

αi,j

 γi,j   

αi,j

∗ γi,j 2hi d∗i+1,j > − 1, ∗ fi,j βi,j fi+1,j 2hi di,j 2hi di+1,j > max{0, − − 1, − − 1}. fi,j fi+1,j

(6)

Moreover, Wi,j can be rewritten as Wi,j =

(1 − θ)3 τ1 + θ(1 − θ)2 τ2 + θ2 (1 − θ)τ3 + θ3 τ4 , ∗ + θ(1 − θ)γ ∗ + θβ ∗ (1 − θ)2 αi,j i,j i,j

where ∗ τ1 = αi,j+1 ((βi,j + γi,j )fi,j+1 − lj di,j+1 βi,j ), ∗ ∗ ∗ ∗ ∗ τ2 = (βi,j + γi,j )((αi,j+1 + γi,j+1 )fi,j+1 + hi d∗i,j+1 αi,j+1 ) − lj di,j+1 βi,j (αi,j+1 + γi,j+1 ), ∗ ∗ ∗ ∗ ∗ τ3 = (βi,j + γi,j )((βi,j+1 + γi,j+1 )fi+1,j+1 − hi d∗i+1,j+1 βi,j+1 ) − lj di+1,j+1 βi,j (βi,j+1 + γi,j+1 ), ∗ τ4 = βi,j+1 ((βi,j + γi,j )fi+1,j+1 − lj di+1,j+1 βi,j ).

Thus, it is easy to derive that Vi,j > 0 if  ∗ γi,j+1 2hi d∗i,j+1   > − − 1,   ∗

αi,j+1

fi,j+1

 γi,j 2lj di,j+1   > max{0, 

βi,j

fi,j+1

∗ γi,j+1 2hi d∗i+1,j+1 > − 1, ∗ βi,j+1 fi+1,j+1 2lj di+1,j+1 − 1, − 1}. fi+1,j+1

(7)

Based on the analysis above, from (4)-(7), the following theorem can be obtained. Theorem 1. Let {(xi , yj , fi,j ) : i = 1, 2, · · · , n; j = 1, 2, · · · , m} be a positive data set defined over the plane region [x1 , xn ; y1 , ym ] such that fi,j > 0 for all i and j, where fi,j represents fi,j (x, y) at the point (xi , yj ). Let d∗i,j and di,j be the chosen partial derivatives. Then the bivariate rational spline interpolant P (x, y) defined in (2) visualize positive data in the view of positive surface if the positive shape parameters satisfy the following constraints: ∗ γi,j ∗ αi,j ∗ γi,j ∗ βi,j γi,j αi,j γi,j βi,j

2hi d∗i,j − 1}, fi,j 2hi d∗i+1,j > max{0, − 1}, fi+1,j 2hi di,j 2hi di+1,j > max{0, − − 1, − − 1}, fi,j fi+1,j 2lj di,j+1 2lj di+1,j+1 > max{0, − 1, − 1}. fi,j+1 fi+1,j+1 > max{0, −

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FU: VISUALIZATION OF SHAPED DATA

4

Monotonicity-preserving surface interpolating scheme

In this section, we will develop a monotonicity preserving surface interpolating scheme for the given monotone interpolating data. (x)

(y)

Denote ∆i,j = (fi+1,j − fi,j )/hi , ∆i,j = (fi,j+1 − fi,j )/lj . Let {(xi , yj , fi,j ) : i = 1, 2, · · · , n; j = 1, 2, · · · , m} be a monotone data set defined over the rectangular grid [xi , xi+1 ; yj , yj+1 ] such that (x) (y) fi+1,j > fi,j , fi,j+1 > fi,j for all i, j, or equivalently ∆i,j > 0, ∆i,j > 0. For a monotone surface P (x, y), it is necessary that the corresponding first partial derivatives d∗i,j and di,j should meet: d∗i,j > 0, di,j > 0, for all i = 1, 2, · · · , n; j = 1, 2, · · · , m. Using the result developed in [3]: Bicubic partially blended surface patch inherits all the properties of network of boundary curves, we just need to consider the monotonicity of the boundary curves in the interpolating surface. The function P (x, yj ) is monotonic increasing if and only if P 0 (x, yj ) > 0. From (2), we can derive that P 0 (x, yj ) =

(1 − θ)4 C1 + θ(1 − θ)3 C2 + θ2 (1 − θ)2 C3 + θ3 (1 − θ)C4 + θ4 C5 , ∗ + θ(1 − θ)γ ∗ + θ 2 β ∗ )2 ((1 − θ)2 αi,j i,j i,j

(8)

with ∗2 C1 = αi,j di,j , (x)

∗ ∗ ∗ ∗ C2 = 2αi,j [(βi,j + γi,j )∆i,j − βi,j di+1,j ], (x)

∗2 ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ C3 = (γi,j + γi,j (αi,j + βi,j ) + 4αi,j βi,j )∆i,j − αi,j (βi,j + γi,j )di,j − βi,j (αi,j + γi,j )di+1,j , (x)

∗ ∗ ∗ ∗ C4 = 2βi,j [(αi,j + γi,j )∆i,j − αi,j di,j ], ∗2 C5 = βi,j di+1,j .

It is evident that C1 and C5 are positive, and C2 > 0 if

∗ γi,j α∗i,j

>

Further, since

d∗i,j (x) ∆i,j

− 1, C4 > 0 if

∗ γi,j ∗ βi,j

>

d∗i+1,j (x)

∆i,j

− 1.

1 (x) ∗2 ∗ ∗ ∗ ∗ ∗ ∗ C3 = (C2 + C4 ) + (γi,j + 2αi,j βi,j )∆i,j − αi,j γi,j di,j − βi,j γi,j di+1,j 2 1 (x) (x) ∗ 1 ∗ ∗ ∗ 1 ∗ ∗ ≥ (C2 + C4 ) + γi,j ( γi,j ∆i,j − αi,j di,j ) + γi,j ( γi,j ∆i,j − βi,j di+1,j ), 2 2 2 it is easy to see that C3 > 0 if

∗ γi,j α∗i,j

>

2d∗i,j (x) ∆i,j

and

∗ γi,j ∗ βi,j

>

2d∗i+1,j (x)

∆i,j

.

Similarly, we have P 0 (xi , y) =

(1 − η)4 K1 + 2η(1 − η)3 K2 + η 2 (1 − η)2 K3 + η 3 (1 − η)K4 + η 4 K5 , ((1 − η)2 αi,j + η(1 − η)γi,j + η 2 βi,j )2

where 2 K1 = αi,j di,j , (y)

K2 = 2αi,j [(βi,j + γi,j )∆i,j − βi,j di,j+1 ], (y)

2 K3 = (γi,j + γi,j (αi,j + βi,j ) + 4αi,j βi,j )∆i,j − βi,j (αi,j + γi,j )di,j+1 − αi,j (βi,j + γi,j )di,j , (y)

K4 = 2βi,j [(αi,j + γi,j )∆i,j − αi,j di,j ], 2 K5 = βi,j di,j+1 .

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(9)

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FU: VISUALIZATION OF SHAPED DATA

Hence, Ki > 0 (k = 1, 2, · · · , 5) hold if

γi,j αi,j

>

2di,j (y)

∆i,j

and

γi,j βi,j

>

2di,j+1 (y)

∆i,j

.

Thus, the above discussion is epitomized in the form of following theorem. Theorem 2. Let {(xi , yj , fi,j ) : i = 1, 2, · · · , n; j = 1, 2, · · · , m} be a monotone data set defined over the plane region [x1 , xn ; y1 , ym ] such that fi+1,j > fi,j , fi,j+1 > fi,j for all i and j, where a = x1 < x2 < ... < xn = b, c = y1 < y2 < · · · < ym = d are the knot spacings, fi,j represents fi,j (x, y) at the point (xi , yj ). Let d∗i,j and di,j be the chosen partial derivatives, so that d∗i,j > 0 and di,j > 0. Then the bivariate rational spline interpolant P (x, y) defined in (2) visualize monotonic data in the view of monotone surface if the positive shape parameters satisfy the following constraints: ∗ ∗ 2d∗i,j γi,j 2d∗i+1,j γi,j 2di,j γi,j 2di,j+1 γi,j > (y) , > > , > , . ∗ ∗ (x) (x) (y) αi,j βi,j αi,j βi,j ∆i,j ∆i,j ∆i,j ∆i,j

5

Demonstration

In this section, we shall illustrate the positivity preserving scheme and the monotonicity preserving scheme developed in Sections 3 and 4 with some examples, respectively. Example 1. First of all, let us take the example of a positive data in Table 1. This data is generated approximately from the following smooth positive function by taking the values truncated to four decimal places: f (x, y) = exp(−x2 − 2y) + 0.01, 0 ≤ x, y ≤ 8.

(10)

The interpolant P (x, y) defined by (2) is identified uniquely by the given interpolating data and Table 1: A positive data set taken from function (10). y/x 0 2 4 6 8

0 1.0100 0.0283 0.0100 0.0100 0.0100

2 0.0283 0.0103 0.0100 0.0100 0.0100

4 0.0103 0.0100 0.0100 0.0100 0.0100

6 0.0100 0.0100 0.0100 0.0100 0.0100

8 0.0100 0.0100 0.0100 0.0100 0.0100

∗ = β∗ = α ∗ the values of shape parameters. We take αi,j i,j = βi,j = 1 and γi,j = γi,j = 2, the interi,j polant coincides with the bicubic Hermite interpolant. Figure 1 shows the graph of corresponding interpolating surface P1 (x, y), which loses the positivity. For the same data set in table 1, we employ Theorem 1 to compute the values of shape param∗ = β∗ = α ∗ eters, which are taken as: αi,j i,j = βi,j = 0.4 and γi,j = γi,j = 3. Figure 2 provides the i,j graph of the corresponding bicubic rational interpolant P2 (x, y). It is obvious to see from Figure 2 that the shape of the data has been preserved by the surface representation. Example 2. Here, we consider the example of a monotonic data in Table 2. This data has been generated approximately from the following smooth function:

f (x, y) = sin(x2 + y 2 + xy), 0 ≤ x, y ≤ 0.6.

(11)

This example will illustrate how visualization of 3D monotone data can be achieved in the view of monotone surfaces only by selecting suitable shape parameters for unchanged interpolating data in Table 2. For any values of the shape parameters, it cannot be guaranteed that the bicubic rational ∗ = α ∗ surface generated by (2) is monotone. For example, we take αi,j i,j = 1, βi,j = βi,j = 8 and

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FU: VISUALIZATION OF SHAPED DATA

1.2

1.2

1

1

0.8 0.8 0.6 0.6 0.4 0.4 0.2 0.2

0

−0.2 8

0 8 6

6

8 6

4

8

0

6

4

4

2

4

2

2

0

0

Figure 1: Non-positive Hermite surface P1 .

2 0

Figure 2: Positive rational surface P2 .

Table 2: A monotone data set taken from function (11). y/x 0 0.2 0.4 0.6

0 0 0.0400 0.1593 0.3523

0.2 0.0400 0.1197 0.2764 0.4969

0.4 0.1593 0.2764 0.4618 0.6889

0.6 0.3523 0.4969 0.6889 0.8820

∗ = γ γi,j i,j = 0.1. Figure 3 shows the graph of the corresponding rational surface P3 (x, y), which loses the monotonicity in its display. Figure 4 is a different view of Figure 3 obtained after making a rotation, it confirms quite clearly that the surface is not preserving monotonicity feature.

0.9

0.8 0.9

0.7

0.8

0.6

0.7 0.6

0.5

0.5 0.4

0.4

0.3 0.2

0.3

0.1

0.2

0 −0.1 0.6

0.1 0.6

0.4

0.5

0

0.4 0.3

0.2

0.2 0

0.1

−0.1

0

0.50

Figure 3: Non-monotonic rational surface P3 .

0.1

0.2

0.3

0.4

0.5

0.6

0.7

Figure 4: A different view of surface P3 .

Now, we employ Theorem 2 to compute the values of shape parameters, which are taken as: ∗ = β∗ = α ∗ αi,j i,j = βi,j = 0.6 and γi,j = γi,j = 1.2. Figure 5 provides the graph of the corresponding i,j bicubic rational interpolant P4 (x, y). It is obvious to see from Figure 5 that the shape of the monotone data in Table 2 has been preserved by the surface representation. Figure 6 is produced from this data set using a bicubic Hermite interpolant.

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FU: VISUALIZATION OF SHAPED DATA

0.9

0.9

0.8

0.8

0.7

0.7

0.6

0.6

0.5

0.5

0.4

0.4

0.3

0.3

0.2

0.2

0.1

0.1

0 0.6

0 0.6 0.6

0.4

0.6

0.4

0.5

0.5

0.4

0.4

0.3

0.2

0.2 0

0.1

0.2 0

0

Figure 5: Monotonic rational surface P4 .

6

0.3

0.2 0.1 0

Figure 6: Monotonic Hermite surface P5 .

Concluding remarks

In engineering, industrial and scientific problems, the construction of shape preserving interpolants is an everlasting demand and one of the major research areas of computer aided design. This work is a contribution towards the graphical display of data when it is positive or monotone. To overcome the problem, we present a new bicubic rational interpolating spline with the shape parameters based on function values and partial derivatives. Further, a positivity-preserving scheme and a monotonicity-preserving scheme are developed to visualize positive data and monotone data in the view of positive surface and monotone surface, respectively, and the simple sufficient conditions are derived on the shape parameters in the description of the rational function.

References [1] R.K. Beatson, Z. Ziegler, Monotonicity preserving surface interpolation, SIAM J. Numer. Anal. 22(1985)(2):401-411. [2] S. Butt, K.W. Brodlie, Preserving positivity using piecewise cubic interpolation, Comput. & Graph., 17(2003):55-64. [3] G. Casciola, L. Romani, Rational interpolants with tension parameters, in: Tom Lyche, MarieLaurence Mazure, Larry L. Schumaker (Eds.), Curve and Surface Design, Nashboro Press, Brentwood, TN, 2003, pp.41-50. [4] Q. Duan, L. Wang, E.H. Twizell, A novel approach to the convexity control of interpolant curves, Commun. Numer. Meth. En., 19(2003):833-845. [5] Q. Duan, S. Li, F. Bao, E.H. Twizell, Hermite interpolation by piecewise rational surface, Appl. Math. Comput., 198(2008)59-72. [6] M.S. Floater, J.M. Penna, Monotonicity preservation on triangles, Math. Comput. 69(2000):1505-1519. [7] T.N.T. Goodman, B.H. Ong, Shape-preserving interpolation by splines using vector subdivision, Adv. Comput. Math., 22(2005):49-77. [8] X. Han, Convexity-preserving piecewise rational quartic interpolation, SIAM J. Numer. Anal., 46(2008):920-929.

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[9] X. Han, Convexity-preserving approximation by univariate cubic splines, J. Comput. Appl. Math., 287(2015):196-206. [10] M.Z. Hussain, M. Hussain, Visualization of data preserving monotonicity, Appl. Math. Comput., 190(2007):1353-1364. [11] M. Hussain, M.Z. Hussain, A. Waseem and at al., Shape-preserving trigonometric surfaces, J. Math. Imaging Vis., 53(2015)1:21-41. [12] M.Z. Hussain, M. Hussain, B. Aqeel, Shape-preserving surfaces with constraints on tension parameters, Appl. Math. Comput., 247(2015):442-464. [13] B.I. Kvasov, Monotone and convex interpolation by weighted quadratic splines, Adv. Comput. Math., 40(2014)1:91-116. [14] H.T. Nguyen, A. Cuyt, O.S. Celis, Comonotone and coconvex rational interpolation and approximation, Numer. Algor., 58(2011):1-21. [15] A.R.M. Piah, T.N.T. Goodman, K. Unsworth, Positivity preserving scattered data interpolation, in: Proceedings of the 11th IMA Mathematics of Surfaces Conference, Loughborough, UK, September 5-7, 2005, pp. 336-349. [16] M. Sarfraz, M.Z. Hussain, Data visualization using spline interpolation, J. Comput. Appl. Math., 189(2006):513-525. [17] Q. Sun, F. Bao, Q. Duan, A Surface Modeling Method by Using C2 Piecewise Rational Spline Interpolation, J. Math. Imaging Vis., 53(2015)1:12-20. [18] Y. Zhu, X. Han, C 2 rational quartic interpolation spline with local shape preserving property, Appl. Math. Lett., 46(2015):57-63.

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J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 24, NO.8, 2018, COPYRIGHT 2018 EUDOXUS PRESS, LLC

Explicit Viscosity Rules and Applications of Nonexpansive Mappings Young Chel Kwun1 , Waqas Nazeer2,∗, Mobeen Munir3 and Shin Min Kang4,5,∗ 1

2

Division of Science and Technology, University of Education, Lahore 54000, Pakistan e-mail: [email protected]

3

Division of Science and Technology, University of Education, Lahore 54000, Pakistan e-mail: [email protected] 4

5

Department of Mathematics, Dong-A University, Busan 49315, Korea [email protected]

Center for General Education, China Medical University, Taichung 40402, Taiwan e-mail: [email protected]

Department of Mathematics and RINS, Gyeongsang National University, Jinju 52828, Korea e-mail: [email protected] Abstract This article presents a new explicit viscosity rule for nonexpansive mappings in Hilbert spaces. The strong convergence theorems of the rule is proved under certain assumptions imposed on the sequence of parameters. Moreover, we give applications to a more general system of variational inequalities, the constrained convex minimization problem and K-mapping. 2010 Mathematics Subject Classification: 47H05, 47H09, 47H10 Key words and phrases: Explicit viscosity rule, nonexpansive mappings, variational inequality, minimization problem, K-mapping

1

Introduction

In this paper, we will take H as a real Hilbert space with inner product h·, ·i and the induced norm k · k, and C as a nonempty closed subset of the Hilbert space H. A mapping T : H → H is called nonexpansive if kT (x) − T (y)k ≤ kx − yk,

∀x, y ∈ H.

A mapping f : H → H is called a contraction if there exists θ ∈ [0, 1) such that kf (x) − f (y)k ≤ θkx − yk,

∀x, y ∈ H.

Note that F (T ) is the set of fixed points of T . The following strong convergence theorem for nonexpansive mappings in real Hilbert spaces is given by Moudafi [8] in 2000. ∗

Corresponding author

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Theorem 1.1. Let C be a noneempty closed convex subset of the a Hilbert space H. Let T be a nonexpansive mapping of C into itself such that F (T ) is nonempty. Let f be a contraction of C into itself with coefficient θ ∈ [0, 1). Pick any x0 ∈ C, let {xn } be a sequence generated by xn+1 =

n 1 f (xn ) + T (xn ), 1 + n 1 + n

n ≥ 0,

where the sequence {n } in (0, 1) satisfies (i) limn→∞ n = 0, P (2) ∞ n=0 n = ∞, 1 (3) limn→∞ | n+1 − 1n | = 0. Then {xn }converges strongly to a fixed point x∗ of the mapping T, which is also the unique solution of the variational inequality h(I − f )x, y − xi ≤ 0,

∀ ∈ F (T ).

In other words, x∗ is the unique fixed point of the contraction PF (T )f, that is, PF (T ) f (x∗ ) = x∗ . This type of method for approximation of fixed points is called the viscosity approximation method. In 2015, Xu et al. [11] applied the viscosity method on the midpoint rule for nonexpansive mappings and give the following generalized viscosity implicit rule:   xn + xn+1 xn+1 = αn f (xn ) + (1 − αn )T , ∀n ≥ 0. 2 This use contraction to regularize the implicit midpoint rule for nonexpansive mappings. They also proved that the sequence generated by the generalized viscosity implicit rule converges strongly to a fixed point of T, which can also solved variational inequality. Ke and Ma [6] motivated and inspired by the idea of Xu et al. [11] and they proposed two generalized viscosity implicit rules: xn+1 = αn f (xn ) + (1 − αn )T (sn xn + (1 − sn )xn+1 ) and xn+1 = αn xn + βf (xn ) + γn T (sn xn + (1 − sn )xn+1 ) for n ≥ 0. In [3, 7], new viscosity rules and applications are developed. But they correspond to one step viscosity rule. In this paper, we give the following new two step explicit viscosity rule: ( xn+1 = (1 − αn )f (xn ) + αn T (yn ), yn = (1 − βn )xn + βn T (xn ).

We also give many applications of above rule. 2 1542

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2

Preliminaries

Now, we recall the properties of the metric projection. Definition 2.1. PC : H → C is called a metric projection if for every point x ∈ H, there exist a unique nearest point in C, denoted by PC x, such that kx − PC xk ≤ kx − yk,

∀y ∈ C.

The following lemma gives the condition for a projection mapping to be nonexpansive. Lemma 2.2. Let C be a nonempty closed convex subset of a real Hilbert space H and PC : H → H be a metric projection. Then (1) kPC x − PC yk2 ≤ hx − y, PC x − PC yi for all x, y ∈ H. (2) PC is a nonexpansive mapping, that is, kx − PC xk ≤ kx − yk for all y ∈ C. (3) hx − PC x, y − PC xi ≤ 0 for all x ∈ H and y ∈ C. In order to verify the weak convergence of an algorithm to a fixed point of a nonexpansive mapping we need the demiclosedness principle: Lemma 2.3. (The demiclosedness principle) ([2]) Let C be a nonempty closed convex subset of a real Hilbert space H and T : C → C such that xn * x ∗ ∈ C

and

(I − T )xn → 0.

Then x∗ = T x∗ , where → and * denote strong and weak convergence, respectively. In addition, we also need the following convergence lemma. Lemma 2.4. ([11]) Assume that {xn } is a sequence of non-negative real numbers such that an+1 ≤ (1 − γn )an + δn , ∀n ≥ 0, where {γn } is a sequence in (0, 1) and{δn } is a sequence such that P (1) ∞ n=0 γn = ∞, P (2) limn→∞ sup γδnn ≤ 0or ∞ n=0 |δn | < ∞. Then limn→∞ an = 0 as n → ∞.

3

Main results

Theorem 3.1. Let C be a nonempty closed convex subset of a real Hilbert space H. Let T : C → C be a nonexpansive mapping with F (T ) 6= ∅ and f : C → C be a contraction with coefficient θ ∈ [0, 1). Pick any x0 ∈ C, let {xn } be a sequence generated by ( xn+1 = (1 − αn )f (xn ) + αn T (yn ), (3.1) yn = (1 − βn )xn + βn T (xn ), where {αn } and {βn } are sequences in (0, 1) satisfying the following conditions: 3 1543

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(1) limn→∞ αn = 1 and limn→∞ βn = 1, P P (ii) ∞ αn = ∞ and ∞ = ∞, n=0 n=0 βn P P∞ (iii) n=0 |αn+1 − αn | < ∞ and ∞ n=0 |βn+1 − βn | < ∞, ∀n ≥ 0, (iv) limn→∞ kxn − T (xn )k = 0. Then {xn } converges strongly to a fixed point x∗ of the mapping T which is also the unique solution of the variational inequality h(I − f )x, y − xi ≥ 0,

∀y ∈ F (T ).

In other words, x∗ is the unique fixed point of the contraction PF (T )f, that is, PF (T ) f (x∗ ) = x∗ . Proof. We divide the proof into the following five steps. Step 1. Firstly, we show that xn is bounded. Indeed, take p ∈ F (T ) arbitrarily, we have kxn+1 − pk = k(1 − αn )f (xn ) + αn T (yn ) − pk = k(1 − αn )f (xn ) − (1 − αn )p + αn T (yn ) − αn pk ≤ (1 − αn )kf (xn) − pk + αn kT (yn ) − pk ≤ (1 − αn )kf (xn) − f (p)k + (1 − αn )kf (p) − pk

(3.2)

+ αn kyn − pk ≤ (1 − αn )θkxn − pk + (1 − αn )kf (p) − pk + αn kyn − pk. Now, consider kyn − pk = k(1 − βn )xn + βn T (xn ) − pk = k(1 − βn )xn − (1 − βn )p + βn T (xn ) − βn pk ≤ (1 − βn )kxn − pk + βn kT (xn ) − pk ≤ (1 − βn )kxn − pk + βn kxn − pk ≤ kxn − pk. Using this in (3.2) we have kxn+1 − pk ≤ (1 − αn )θkxn − pk + (1 − αn )kf (p) − pk + αn kxn − pk = [(1 − αn )θ + αn ]kxn − pk + (1 − αn )kf (p) − pk = [1 − 1 + α + (1 − αn )θ]kxn − pk + (1 − αn )kf (p) − pk = [1 − (1 − α) + (1 − αn )θ]kxn − pk + (1 − αn )kf (p) − pk = [1 − (1 − αn )(1 − θ)]kxn − pk   1 + (1 − αn )(1 − θ) kf (p) − pk . 1−θ Thus, we have    1 kxn+1 − pk ≤ max kxn − pk, kf (p) − pk . 1−θ 4 1544

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Similarly    1 kf (p) − pk kxn − pk ≤ max kxn−1 − pk, . 1−θ From this

   1 kf (p) − pk kxn+1 − pk ≤ max kxn − pk, 1−θ    1 kf (p) − pk kx − pk, ≤ max n−1 1−θ .. .    1 kf (p) − pk , ≤ max kx0 − pk, 1−θ

which shows that {xn } is bounded. From this we deduce immediately that {f (xn )}, {T (xn )} are bounded. Step 2. Next, we want to prove that limn→∞ kxn+1 − xn k = 0. For this consider kxn+1 − xn k = k(1 − αn )f (xn ) + αn T (yn ) − (1 − αn−1 )f (xn−1 ) − αn−1 T (yn−1 )k = k(1 − αn )(f (xn ) − f (xn−1 )) − (αn − αn−1 )f (xn−1 )

(3.3)

+ αn (T (yn ) − T (yn−1 )) + (αn − αn−1 )T (yn−1 )k ≤ (1 − αn )θkxn − xn−1 k + |αn − αn−1 |kT (yn−1 ) − f (xn−1 )k + αn kyn − yn−1 k. Now, consider kyn − yn−1 k = k(1 − βn )xn + βn T (xn ) − (1 − βn−1 )xn−1 − βn−1 T (xn−1 )k = k(1 − βn )(xn − xn−1 ) − (βn − βn−1 )xn−1 + βn (T (xn) − T (xn−1 )) + (βn − βn−1 )T (xn−1 )k ≤ (1 − βn )kxn − xn−1 k + |βn − βn−1 |kT (xn−1 ) − xn−1 k + βn kxn − xn−1 k ≤ kxn − xn−1 k + |βn − βn−1 |kT (xn−1 ) − xn−1 k. Using this in (3.3) we get kxn+1 − xn k ≤ (1 − αn )θkxn − xn−1 k + |αn − αn−1 |kT (yn−1 ) − f (xn−1 )k + αn kxn − xn−1 k + αn |βn − βn−1 |kT (xn−1 ) − xn−1 k = [(1 − αn )θ + αn ]kxn − xn−1 k + |αn − αn−1 |kT (yn−1 ) − f (xn−1 )k + αn |βn − βn−1 |kT (xn−1 ) − xn−1 k.

P P∞ Let λn = (1−αn ) so λn ∈ (0, 1) since αn ∈ (0, 1) ∞ n=0 λn = ∞, n=0 |αn −αn−1 | < ∞ P∞ and n=0 |βn − βn−1 | < ∞. By using Lemma 2.4, we get limn→∞ kxn+1 − xn k = 0. Step 3. Now we want to prove that limn→∞ kxn − T (yn )k = 0. 5 1545

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Now, consider kxn − T (yn )k ≤ kxn − T (xn )k + kT (xn ) − T (yn )k ≤ kxn − T (xn )k + kxn − yn k = kxn − T (xn )k + kxn − (1 − βn )xn − βn T (xn )k ≤ kxn − T (xn )k + βn kxn − T (xn )k ≤ (1 + βn )kxn − T (xn )k → 0 as n → ∞. Step 4. In this step, we claim that lim supn→∞ hx∗ − f (x∗ ), x∗ − xn i ≤ 0, where = PF (T )f (x∗ ). Indeed, we take a subsequence {xni } of {xn } which converges weakly to a fixed point p of T . Without loss of generality, we may assume that {xni } * p. From limn→∞ kxn − T xn k = 0 and Lemma 2.3 we have p = T p. This together with the property of the metric projection implies that

x∗

lim suphx∗ − f (x∗ ), x∗ − xn i = lim suphx∗ − f (x∗ ), x∗ − xni i n→∞

n→∞ ∗

= hx − f (x∗ ), x∗ − pi ≤ 0. Step 5. Finally, we show that xn → x∗ as n → ∞. Here again x∗ ∈ F (T ) is the unique fixed point of the contraction PF (T ) f . Consider kxn+1 − x∗ k2 = k(1 − αn )f (xn ) + αn T (yn ) − x∗ k2 = k(1 − αn )[f (xn) − x∗ ] + αn [T (yn ) − x∗ ]k2 = (1 − αn )2n kf (xn ) − x∗ k2 + (1 − αn )2 kT (xn ) − x∗ k2 + 2αn (1 − αn )hf (xn) − x∗ , T (yn ) − x∗ i ≤ α2n kyn − x∗ k2 + (1 − αn )2 kf (xn ) − x∗ k2 + 2αn (1 − αn )hf (xn) − f (x∗ ), T (yn) − x∗ i + 2αn (1 − αn )hf (x∗) − x∗ , T (yn ) − x∗ i ≤

α2n kyn

∗ 2

2

(3.4)

∗ 2

− x k + (1 − αn ) kf (xn ) − x k

+ 2αn (1 − αn )kf (xn) − f (x∗ )kkT (yn) − x∗ k + 2αn (1 − αn )hf (x∗) − x∗ , T (yn ) − x∗ i ≤ α2n kyn − x∗ k2 + 2αn (1 − αn )θkxn − x∗ kkyn − x∗ k + (1 − αn )2 kf (xn ) − x∗ k2 + 2αn (1 − αn )hf (x∗) − x∗ , T (yn ) − x∗ i. Now, consider kyn − x∗ k = k(1 − βn )xn + βn T (xn ) − x∗ k = k(1 − βn )xn − (1 − βn )x∗ + βn T (xn ) − βn x∗ k ≤ (1 − βn )kxn − x∗ k + βn kT (xn ) − x∗ k ≤ (1 − βn )kxn − x∗ k + βn kxn − x∗ k ≤ kxn − x∗ k. 6 1546

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Using (3.5) in (3.4) we get kxn+1 − x∗ k2 ≤ α2n kxn − x∗ k2 + 2αn (1 − αn )θkxn − x∗ kkxn − x∗ k + (1 − αn )2 kf (xn ) − x∗ k2 + 2αn (1 − αn )hf (x∗ ) − x∗ , T (yn) − x∗ i ≤ α2n kxn − x∗ k2 + 2αn (1 − αn )θkxn − x∗ kkxn − x∗ k + (1 − αn )2 kf (xn ) − x∗ k2 + 2αn (1 − αn )hf (x∗ ) − x∗ , T (yn) − x∗ i ≤ [α2n + 2αn (1 − αn )θ]kxn − x∗ k2 + (1 − αn )2 kf (xn ) − x∗ k2 + 2αn (1 − αn )hf (x∗ ) − x∗ , T (yn) − x∗ i. Note that αn θ < αn since αn ∈ (0, 1) and θ ∈ [0, 1). 2αn θ < 2αn implies 2αn θ(1 − αn ) < 2αn (1 − αn ) implies α2n + 2αn θ(1 − αn ) < α2n + 2αn (1 − αn ). So we have kxn+1 − x∗ k2 ≤ [α2n + 2αn (1 − αn )]kxn − x∗ k2 + (1 − αn )2 kf (xn ) − x∗ k2 + 2αn (1 − αn )hf (x∗ ) − x∗ , T (yn ) − x∗ i ≤ [2αn − α2n )]kxn − x∗ k2 + (1 − αn )2 kf (xn ) − x∗ k2 + 2αn (1 − αn )hf (x∗ ) − x∗ , T (yn ) − x∗ i

(3.6)

≤ 2αn kxn − x∗ k2 + (1 − αn )2 kf (xn ) − x∗ k2 + 2αn (1 − αn )hf (x∗ ) − x∗ , T (yn ) − x∗ i ≤ 2[1 − (1 − αn )]kxn − x∗ k2 + (1 − αn )2 kf (xn ) − x∗ k2 + 2αn (1 − αn )hf (x∗ ) − x∗ , T (yn ) − x∗ i. By limn→∞ αn = 1 we have (1 − αn )2 kf (xn ) − x∗ k2 + 2αn (1 − αn )hf (x∗ ) − x∗ , T (yn ) − x∗ i (1 − αn ) n→∞ ∗ 2 = lim sup[(1 − αn )kf (xn) − x k + 2αn hf (x∗ ) − x∗ , T (yn) − x∗ i] ≤ 0.

lim sup

(3.7)

n→∞

From (3.6), (3.7) and Lemma 2.4 we have lim kxn+1 − xn k2 = 0,

n→∞

which implies that xn → x∗ as n → ∞. This completes the proof. 7 1547

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4

Applications

4.1

A more general system of variational inequalities

Let Cbe a nonempty closed convex subset of a real Hilbert space H and {Ai }N i=1 : C → H be a family of mappings. In [1], Cai and Bu considered the problem of finding x∗1 , x∗2 , ..., x∗N ∈ C × C × · · · × C such that   hλN AN x∗N + x∗1 − x∗N , x − x∗1 i ≥ 0,     ∗ ∗ ∗ ∗   hλN −1 AN −1 xN −1 + xN − xN −1 , x − xN i ≥ 0, .. (4.1) .    ∗ ∗ ∗ ∗  hλ2 A2 x2 + x3 − x2 , x − x3 i ≥ 0,     hλ1 A1 x∗1 + x∗2 − x∗1 , x − x∗2 i ≥ 0, ∀x ∈ C. The equation (4.1) can be written as   hx∗1 − (I − λN AN )x∗N , x − x∗1 i ≥ 0,     ∗ ∗ ∗   hxN − (I − λN −1 AN −1 )xN −1 , x − xN i ≥ 0, .. .    ∗  hx3 − (I − λ2 A2 )x∗2 , x − x∗3 i ≥ 0,     ∗ hx2 − (I − λ1 A1 )x∗1 , x − x∗2 i ≥ 0, ∀x ∈ C,

which is a more general system of variational inequalities in a Hilbert space, where λi > 0 for all i ∈ {1, 2, 3, ..., N }. We also have following lemmas. Lemma 4.1. ([6]) Let C be a nonempty closed convex subset of a real Hilbert spaces H. For i ∈ {1, 2, 3, ..., N}, let Ai : C → H be δi -inverse-strongly monotone for some positive real number δi , namely, hAix − Ai y, x − yi ≥ δi kAi x − Ai yk2 ,

∀x, y ∈ C

Let G : C → C be a mapping defined by G(x) = PC (I − λN AN )PC (I − λN −1 AN −1 ) · · · PC (I − λ2 A2 )PC (I − λ1 A1 )x,

∀x ∈ C.

(4.2)

If 0 < λi ≤ 2δi for all i ∈ {1, 2, ..., N }, then G is nonexpansive. Lemma 4.2. ([5]) Let C be a nonempty closed convex subject of a real Hilbert space H. Let Ai : C → H be a nonlinear mapping, where i ∈ {1, 2, 3, ..., N }. For given x∗i ∈ C, i ∈ {1, 2, 3, ..., N}, (x∗1 , x∗2 , x∗3 , ..., x∗N ) is a solution of the problem (4.1) if and onli if x∗1 = PC (I − λN AN )x∗N , x∗i = PC (I − λi−1 Ai−1 )x∗i−1 ,

i = 2, 3, 4, ..., N,

(4.3)

that is, x∗1 = PC (I − λN AN )PC (I − λN −1 AN −1 ) · · · PC (I − λ2 A2 )PC (I − λ1 A1 )x∗1 ,

∀x ∈ C.

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From Lemma 4.2, we know that x∗1 = G(x∗1 ), that is, x∗1 is a fixed point of the mapping G, where G is defined by (4.2). Moreover, if we find the fixed point x∗1 , it is easy to get the other points by (4.3). Applying Theorem 3.1 we get the result Theorem 4.3. Let C be a nonempty closed convex subject of a real Hilbert space H. For i ∈ {1, 2, 3, ......, N }, let Ai : C → H be δi -inverse-strongly monotone for some positive real number δi with F (G) 6= ∅, where G : C → C is defined by G(x) = PC (I − λN AN )PC (I − λN −1 AN −1 ) · · · PC (I − λ2 A2 )PC (I − λ1 A1 )x,

∀x ∈ C.

Let f : C → C be a contraction with coefficient θ ∈ [0, 1). Pick any x0 ∈ C, let {xn } be a sequence generated by ( xn+1 = (1 − αn )f (xn ) + αn G(yn ), yn = (1 − βn )xn + βn G(xn ),

where λi ∈ (0, 2δi), i = 1, 2, 3, ..., N, {αn } and {βn } are sequences in (0, 1) satisfying the conditions (i)-(iv). Then {xn } converges strongly to a fixed point x∗ of the nonexpansive mapping G, which is also the unique solution of the variational inequality h(I − f )x, y − xi ≥ 0,

∀y ∈ F (T ).

In other words, x∗ is the unique fixed point of the contraction PF (G) f, that is, PF (G) f (x∗ ) = x∗ .

4.2

The constrained convex minimization problem

Now, we consider the following constrained convex minimization problem: min φ(x), x∈C

(4.4)

where φ : C → R is a real-valued convex function and assumes that the problem (4.4) is consistent. Let Ω denote its solution set. For the minimization problem (4.4), if φ is (Fr´echet)differentiable, then we have the following lemma. Lemma 4.4. (Optimality Condition) ([5]) A necessary condition of optimality for a point x∗ ∈ C to be a solution of the minimization problem (4.4) is that x∗ solves the variational inequality h∇φ(x∗ ), x − x∗ i ≥ 0, ∀x ∈ C. (4.4) Equivalently, x∗ ∈ C solves the fixed point equation x∗ = PC (x∗ − λ∇φ(x∗ )) for every constant λ > 0. If, in a addition φ is convex, then the optimality condition (4.5) is also sufficient. 9 1549

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It is well known that the mapping PC (I − λA) is nonexpansive when the mapping A is δ-inverse-strongly monotone and 0 < λ < 2δ. We therefore have the following result. Theorem 4.5. Let C be a nonempty closed convex subset of the real Hilbert Space H. For the minimization problem (4.4), assume that φ is (Fr´echet) differentiable and the gradient ∇φ is a δ-inverse-strongly monotone mapping for some positive real number δ. Let f : C → C be a contraction with coefficient θ ∈ [0, 1). Pick any x0 ∈ C, let {xn } be a sequence generated by ( xn+1 = (1 − αn )f (xn ) + αn PC (1 − λ∇φ)(yn), yn = (1 − βn )xn + βn PC (1 − λ∇φ)(xn),

where λi ∈ (0, 2δ), {αn } and {βn } are sequences in (0, 1) satisfying the conditions (i)-(iv). Then {xn } converges strongly to a solution x∗ of the minimization problem (4.4), which is also the unique solution of the variational inequality h(I − f )x, y − xi ≥ 0,

∀y ∈ Ω.

In other words, x∗ is the unique fixed point of the contraction PΩ f, that is, PΩ f (x∗ ) = x∗ .

4.3

The K-mapping

In 2009, Kangtunyakarn and Suantai [4] gave a K-mapping generated by T1 , T2, T3 , ..., TN and λ1 , λ2 , λ3, ..., λN as follows. Definition 4.6. Let C be a nonempty convex subset of a real Banach space. Let {Ti }N i=1 be a family of mappings of C into itself and let λ1 , λ2 , λ3, ..., λN be real numbers such that 0 ≤ λi ≤ 1 for every i = 1, 2, 3, ..., N. We define a mapping K : C → C as follows:   U1 = λ1 T1 + (1 − λ1 )I,       U2 = λ2 T2 U1 + (1 − λ2 )U1 , .. .     UN −1 = λN −1 TN −1 UN −2 + (1 − λN −1 )UN −2 ,     K = UN = λN TN UN −1 + (1 − λN )UN −1 . Such a mapping is called a K-mapping generated by T1 , T2 , T3, ..., TN and λ1 , λ2, λ3, ..., λN . In 2014, Suwannaut and Kangtunyakarn [10] established the following result for Kmapping generated by T1 , T2 , T3, ..., TN and λ1 , λ2 , λ3, ..., λN . Lemma 4.7. Let C be a nonempty closed convex subset of a real Hilbert space H. For i = 1, 2, 3, ..., N, let {Ti}N i=1 be aTfinite family of ki -strictly pseudo-contractive mapping of C into itself with ki ≤ ω1 and N i=1 F (Ti ) 6= ∅, namely, there exist constants ki ∈ [0, 1) such that kTi x − Ti yk2 ≤ kx − yk2 + ki k(I − Ti )x − (I − Ti )yk2 ,

∀x, y ∈ C.

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Let λ1 , λ2 , λ3, ..., λN be real numbers with 0 < λi < ω2 for all i = 1, 2, 3, ..., N and ω1 + ω2 < 1. Let K be the K-mapping generated by T1 , T2 , T3, ..., TN and λ1 , λ2 , λ3, ..., λN . Then the following properties hold: T (a) F (K) = N i=1 F (Ti ), (b) K is a nonexpansive mapping. On the bases of above lemma, we have the following results. Theorem 4.8. Let C be a nonempty closed convex subset of a real Hilbert space H. For i = 1, 2, 3, ..., N, let {Ti}N a finite family of ki -strictly pseudo-contractive mapping of i=1 be T C into itself with ki ≤ ω1 and N i=1 F (Ti ) 6= ∅. Let λ1 , λ2, λ3 , ..., λN be real numbers with 0 < λi < ω2 for all i = 1, 2, 3, ..., N and ω1 + ω2 < 1. Let K be the K-mapping generated by T1 , T2 , T3, ..., TN and λ1 , λ2, λ3 , ..., λN .Let f : C → C be a contraction with coefficient θ ∈ [0, 1). Pick any x0 ∈ C, let {xn } be sequence generated by ( xn+1 = (1 − αn )f (xn ) + αn K(yn ), , yn = (1 − βn )xn + βn K(xn ). where {αn } and {βn } are sequences in (0, 1) satisfying the conditions (i)-(iv). Then {xn } converges strongly to a fixed point x∗ of the mappings {Ti }N i=1 , which is also the unique solution of the variational inequality h(I − f )x, y − xi,

∀y ∈ F (K) =

N \

F (Ti).

i=1 ∗

In other words, x is the unique fixed point of the contraction PTN F (Ti ) f, that is, i=1 ∗ ∗ F (Ti ) f (x ) = x .

PTN

i=1

Acknowledgement This work was supported by the Dong-A University research fund.

References [1] G. Cai and S. Q. Bu, Hybrid algorithm for generalized mixed equilibrium problems and vari- ational inequality problems and fixed point problems, Comput. Math. Appl., 62 (2011), 4772–4782. [2] K. Goebel and W. Kirk, Topics in Metric Fixed Point Theory, Cambridge Studies in Ad- vanced Mathematics, vol. 28. Cambridge University Press, Cambridge, 1990. [3] C. Y. Jung, W. Nazeer, S. F. A. Naqvi and S. M. Kang, An implicit viscosity technique of nonexpansive mappings in Hilbert spaces, Int. J. Pure Appl. Math., 108 (2016), 635–650. [4] A. Kangtunyakarn and S. Suantai, A new mapping for finding common solutions of equilibrium problems and fixed point problems of finite family of nonexpansive mappings, Nonlinear Anal., 71 (2009), 4448–4460. 11 1551

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[5] Y. F. Ke and C. F. Ma, A new relaxed extragradient-like algorithm for approaching common solutions of generalized mixed equilibrium problems, a more general system of variational inequalities and a fixed point problem, Fixed Point Theory Appl., 126 (2013), 21 pages. [6] Y. F. Ke and C. F. Ma, The generalized viscosity implicit rules of nonexpansive mappings in Hilbert spaces, Fixed Point Theory Appl., 190 (2015), 21 pages. [7] Y. C. Kwun, W. Nazeer, S. F. A. Naqvi and S. M. Kang, Viscosity approximation methods of nonexpansive mappings in Hilbert spaces and applications, Int. J. Pure Appl. Math., 108 (2016), 929–944. [8] A. Moudafi, Viscosity approximation methods for fixed points problems, J. Math. Anal. Appl., 241 (2000), 46–55. [9] M. Su and H. K. Xu, Remarks on gradient-projection algorithm, J. Nonlinear Anal. Optim., 1 (2010), 35–43. [10] S. Suwannaut and A. Kangtunyakarn, Strong convergence theorem for the modified generalized equilibrium problem and fixed point problem of strictly pseudocontractive mappings, Fixed Point Theory Appl., 86 (2014), 31 pages. [11] H. K. Xu, M. A. Alghamdi and N. Shahzad, The viscosity technique for the implicit mid point rule of nonexpansive mappings in Hilbert spaces, Fixed Point Theory Appl., 41 (2015), 12 pages.

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Applications and Strong Convergence Theorems of Asymptotically Nonexpansive Non-self Mappings Young Chel Kwun1 , Waqas Nazeer2,∗, Mobeen Munir3 and Shin Min Kang4,5,∗

1

2

Division of Science and Technology, University of Education, Lahore 54000, Pakistan e-mail: [email protected]

3

Division of Science and Technology, University of Education, Lahore 54000, Pakistan e-mail: [email protected] 4

5

Department of Mathematics, Dong-A University, Busan 49315, Korea [email protected]

Center for General Education, China Medical University, Taichung 40402, Taiwan e-mail: [email protected]

Department of Mathematics and RINS, Gyeongsang National University, Jinju 52828, Korea e-mail: [email protected] Abstract We give some strong convergence theorems about asymptotically nonexpansive non-self mappings. In the end we give some applications of our results in the form of examples. Our results extend the results already proved. 2010 Mathematics Subject Classification: 47J25, 47N20, 34G20, 65J15 Key words and phrases: Strong convergence, asymptotically nonexpansive non-self mapping, fixed point, uniformly convex Banach space

1

Introduction

Fixed points of special mappings like asymptotically nonexpansive, nonexpansive, contractive and other mappings has become a field of interest on its on and has a variety of applications in related fields like image recovery, signal processing and geometry of objects. Almost in all branches of mathematics we see some versions of theorems relating to fixed points of functions of special nature. As a result we apply them in industry, toy making, finance, aircrafts and manufacturing of new model cars. Because of its vast range of applications almost in all directions, the research in it is moving rapidly and an immense literature is present now. ∗

Corresponding author

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Let C be a nonempty subset of a real normed space E of A mapping T : C → C is asymptotically nonexpansive if there exist a sequence {kn } ⊂ [0, ∞) with limn→∞ kn = 1 such that kT n x − T n yk ≤ kn kx − yk for all x, y ∈ C and n ≥ 1. This class was introduced by Goebel and Kirk in 1972, see [4]. Ever since it has occupied a central place in fixed point theory and other related fields concerning about special mappings. Theorem 1.1. If C is a nonempty closed convex subset of a real uniformly convex Banach space E and T : C → C is an asymptotically nonexpansive mapping, then T has a unique fixed point in C. However, in 1991, Schu [9] developed the modified Mann process for the approximation of fixed points of an asymptotically nonexpansive self mapping which is defined on a nonempty closed convex bounded subset of a Hilbert space given as follows: Theorem 1.2. Let C be a nonempty closed convex bounded subset of a Hilbert space H and T : C → C be a completely continuous and asymptotically nonexpansive mapping with P 2 sequence {kn } ⊂ [1, ∞] having limn→∞ kn = 1 and ∞ n=1 (kn − 1) ≤ ∞. Let {αn } be a real sequence in [0, 1] with condition  ≤ αn ≤ 1 −  for all n > 1 and for some  > 0. Then the sequence {xn } defined recursively by ( x1 ∈ C (arbitrarily), (1.1) xn+1 = (1 − αn )xn + αn T n xn , ∀ n ≥ 1 converges strongly to some fixed point of T. In the above theorem, T is a self mapping of C, where C is a nonempty closed convex subset of H. If, however, the domain D(T ) of T is a proper subset of H and T : D(T ) → H is a mapping the modified iteration {xn } may fail to be well defined. To overcome this, in 2003, Chidume et al. [3] introduce the concept of asymptotically nonexpansive non-self mappings. Let E be a real Banach space. A subset C of E is called a retract of E if there exist a continuous mapping P : E → E such that P x = x for all x ∈ C. Every closed convex subset of a uniform convex Banach space is a retract. A mapping P : E → E is called a retraction if P 2 = P. It is clear that if P is a retraction, then P y = y for all y in the the range of P (see [3]). Definition 1.3. Let C be a nonempty subset of a normed linear space E. Let P : E → C be a nonexpansive retraction of E onto C. A non-self mapping T : C → E is said to be asymptotically nonexpansive if there exist a sequence kn ⊂ [0, 1) with limn∞ kn = 1 such that kT (P T )n−1 (x) − T (P T )n−1 (y)k ≤ kn kx − yk (1.2) for all x, y ∈ C and n ≥ 1. Chidume et al. [3] introduced the following iterative scheme: ( x1 ∈ C,

xn+1 = P ((1 − αn )xn + αn T (P T )n−1 xn )

(1.3)

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for all n ≥ 1, where {αn } ⊂ (0, 1) and proved some results of strong convergence and weak convergence for asymptotic nonexpansive non-self mapping. Throughout this paper, we denote F ix(T ) = {x ∈ C : T x = x}. Remark 1.4. If T : C → C is a self mapping, then P becomes identity and we have 1. The non-self mapping with (1.2) coincides with an asymptotically nonexpansive self mapping in (1.1). 2. Both iterations (1.3) and (1.1) coincide. After Chidume et al. many authors proved weak and strong convergence theorems for asymptotically nonexpansive non-self mapping in Banach spaces [6–8, 10]. Guo and Guo [7] introduced following new iterative scheme which is given as: Let E be a real Banach space, C be a nonempty closed convex subset of E and P : C → E be a nonexpansive retraction of E onto C. Let T : C → E be an asymptotically nonexpansive non-self mapping defined by   x1 ∈ C, (1.4) yn = P ((1 − βn )xn + βn T (P T )n−1 xn ),   n−1 xn+1 = P ((1 − αn )xn + αn T (P T ) yn ) for all n ≥ 1, where {αn } ⊂ (0, 1) and {βn } ⊂ [0, 1]. They proved some theorems of strong convergence and weak convergence of the above iteration for an asymptotically nonexpansive non-self mapping T : C → E. In this paper, we first introduce a new iterative scheme {xn } defined as follows:   x1 ∈ C,    y = P ((1 − β )x + β T (P T )n−1 x ), n n n n n (1.5) n−1  zn = P ((1 − γn )yn + γn T (P T ) yn ),  

 x n−1 yn +zn n+1 = P (1 − αn )xn + αn T (P T ) 2

for all n ≥ 1, where {αn } ⊂ (0, 1) and {βn }, {γn} ⊂ [0, 1]. We first use the condition which is weaker than the completely continuous mappings, given in [5] named as the condition (BP). Secondly, we prove some strong convergence theorems for our iteration scheme for an asymptotically nonexpansive non-self mapping. It is important to remark that our results extend the results in [3, 5]. Finally, we give examples to explain the main results of this paper.

2

Some lemmas

In this section we give some key lemmas which will be used to prove the main results of this paper. Lemma 2.1. ([11]) Let p > 1 and R > 0 and E be a Banach space. Then E is uniformly convex if and only if there exist a continuous, strictly increasing and convex function g : [0, ∞) → [0, ∞) with g(0) = 0 such that kλx + (1 − λ)ykp ≤ λkxkp + (1 − λ)kykp − wp(λ)g(kx − yk) 3 1555

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for all x, y ∈ B(0, R) = {x ∈ E : |x| ≤ R} with λ ∈ [0, 1], where wp(λ) = λ(1 − λ)p + λp(1 − λ). Lemma 2.2. Let E be a real uniformly convex Banach space and C be a nonempty convex subset of E. Let T : C → E be an asymptotically nonexpansive non-self mapping with P {hn } ⊂ [1, ∞) such that ∞ n=1 (hn − 1) < ∞ and F ix(T ) 6= ∅. Let {xn } be the sequence defined by (1.5), where {αn } ⊂ [0, 1) and {βn }, {γn} ⊂ [0, 1]. Then (a) kxn+1 − pk ≤ h2n kxn − pk for all p ∈ F ix(T ). (b) limn→∞ kxn − pk exists for all p ∈ F ix(T ). Proof. Take T2 = T1 = T and S2 = S1 = I in [6], we obtain the required result (see [5]). Lemma 2.3. Let E be a real uniform convex Banach space and C be a nonempty closed convex subset of E. Let T : C → E be an asmyptotically nonexpansive non-self mapping P P∞ 0 with {hn }, {h0n} ⊂ [1, ∞) and h0n ≤ hn such that ∞ n=1 (hn − 1) < ∞, n=1 (hn − 1) < ∞ and F ix(T ) 6= ∅. Let {xn } a sequence defined in (1.5), where 0 < lim inf αn ,

lim sup αn < 1, lim sup βn < 1,

n→∞

n→∞

n→∞

lim sup γn < 1. n→∞

Then limn→∞ kxn − T xn k = 0. Proof. By Lemma 2.2, we know that limn→∞ kxn − pk exists for all p ∈ F ix(T ). So {xn − p}, {yn − p}, {zn − p}, {T (P T )n−1 xn − p}, {T (P T )n−1 yn − p} and {T (P T )n−1 zn − p} are all bounded so we have a real number R > 0 such that {xn − p, yn − p, zn − p, T (P T )n−1 xn − p, T (P T )n−1 yn − p T (P T )n−1 zn − p} ⊂ B(0, R), for all n ≥ 1. It follows from (1.5) and Lemma 2.1 that kyn − pk2 ≤ k(1 − βn )(xn − p) + βn (T (P T )n−1 xn − p)k2 ≤ (1 − βn )k(xn − p)k2 + βn k(T (P T )n−1 xn − p)k2 − βn (1 − βn )g(kxn − T (P T )n−1 xn k) ≤ (1 − βn )h2n k(xn − p)k2 + βn h2n kxn − pk2 = h2n k(xn − p)k2 , kzn − pk2 ≤ k(1 − γn )(yn − p) + γn (T (P T )n−1 yn − p)k2 ≤ (1 − γn )k(yn − p)k2 + γn k(T (P T )n−1 yn − p)k2 − γn (1 − γn )g(kyn − T (P T )n−1 yn k) 2

2

≤ (1 − γn )h0 n k(yn − p)k2 + γn h0 n kyn − pk2 2

= h0 n |(yn − p)k2 2

≤ h0 n h2n |(xn − p)k2 ≤ h4n k(xn − p)k2 . 4 1556

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Also

  2  

n−1 yn + zn

kxn+1 − pk ≤ (1 − αn )(xn − p) + αn T (P T ) −p

2

   2 

n−1 yn + zn −p ≤ (1 − αn )kxn − pk2 + αn

T (P T ) 2    

n−1 yn + zn − αn (1 − αn )g

xn − T (P T ) 2

2

8 2 0 2 2 yn + zn ≤ (1 − αn )hn kxn − pk + αn h n hn − p

2    

n−1 yn + zn − αn (1 − αn )g

xn − T (P T )

2 1 2 ≤ (1 − αn )h8n kxn − pk2 + αn h0 n h2n (kyn − pk2 + kzn − pk2 ) 2    

n−1 yn + zn

− αn (1 − αn )g xn − T (P T )

2 2

≤ (1 − αn )h8n kxn − pk2 + αn h8n kxn − pk2    

n−1 yn + zn

− αn (1 − αn )g xn − T (P T )

2    

8 2 n−1 yn + zn

= hn kxn − pk − αn (1 − αn )g xn − T (P T )

, 2

where g : [0, ∞) → [0, ∞) is a continuous, strictly increasing and convex function with the condition g(0) = 0. Since we have condition 0 < lim inf n→∞ αn and lim supn→∞ αn < 1. So there exist two real numbers a, b ∈ (0, 1) and a positive integer n0 such that a ≤ αn ≤ b for all n ≥ n0 , so a(1 − b)g(k xn − T (P T )n−1 yn k) ≤ h4 k xn − p k2 − k xn+1 − p k2 and

4

a(1 − b)g(k xn − T (P T )n−1 zn k) ≤ h0 k xn − p k2 − k xn+1 − p k2 . Since limn→∞ hn = 1 and limn→∞ h0 n = 1, there exist positive integer m0 , to and real number s, s0 ∈ (0, 1) such that βn hn ≤ s and γn h0 n ≤ s0 for all n > m0 and kyn − xn k ≤ βn kT (P T )n−1 xn − xn k ≤ βn kT (P T )n−1 xn − T (P T )n−1 yn k + βn kxn − T (P T )n−1 yn k ≤ βn hn kxn − yn k + kxn − T (P T )n−1 yn k. Similarly, kzn − xn k ≤ γn h0 n kxn − zn k + kxn − T (P T )n−1 zn k. Hence (1 − s)kyn − xn k ≤ (1 − βn hn )kyn − xn k ≤ kxn − T (P T )n−1 yn k and (1 − s0 )kzn − xn k ≤ (1 − γn h0 n )kzn − xn k ≤ kxn − T (P T )n−1 zn k. 5 1557

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So we have lim kyn − xn k = 0 and

n→∞

lim kzn − xn k = 0.

n→∞

Furthermore from kxn − T (P T )n−1 xn k ≤ kxn − T (P T )n−1 yn k + kT (P T )n−1 yn − T (P T )n−1 xn k + kxn − T (P T )n−1 yn k + hn kyn − xn k, it follows lim kxn − T (P T )n−1 xn k = 0.

n→∞

Using equations, we get lim kxn+1 − T (P T )n−1 yn k = 0.

n→∞

lim kxn+1 − T (P T )n−1 zn k = 0.

n→∞

lim kxn+1 − yn k = 0.

n→∞

lim kxn+1 − zn k = 0.

n→∞

Now kxn − T xn k ≤ kxn − T (P T )n−1 xn k + kT (P T )n−1 xn − T xn k ≤ kxn − T (P T )n−1 xn k + kT (P T )n−1 xn − T (P T )n−1 yn−1 k + kT (P T )n−1 yn−1 − T xn k + kT (P T )n−1 xn − T (P T )n−1 zn−1 k + kT (P T )n−1 zn−1 − T xn k, and using above condition we get required result limn→∞ kxn − T xn k = 0. The completes the proof.

3

The conditions (BP)

Here we recall the condition introduced in [5]. Let E be a Banach space and T : E → E be a bounded linear operator. In 1966, Browdin and Petryshyn [2] considered the existence of a solution of the equation f = u − T u by iteration of Picard-Poincor´e-Newmann, ( x0 ∈ E (3.1) xn+1 = T xn + f, or

( x0 ∈ E

xn+1 = T n x0 + (f + T f + · · · + T n−1 f ),

∀n ≥ 0, f ∈ E.

(3.2)

In fact, in 1958, Browder [1] proved the following. Theorem 3.1. Let E be a reflexive Banach space. Then a solution u of the equation u − T u = f exists for a given point f ∈ E and an operator T which is asmyptotic bounded if and only if the sequence {xn } defined by (3.1) is bounded for any fixed x0 ∈ E. 6 1558

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Relaxing the assumption of reflexivity on E, under a slight sharp condition on T, Browden and Petryshyn proved the following: Theorem 3.2. Let E be a Banach space, T : E → E be a bounded linear operator which is asymptotically convergent, that is, {T n x} converges in E for all x ∈ E. Then we have the following: (1) If f ∈ R(1 − T ), the sequence {xn } converges to a solution u of the equation u + T u = f. (2) If any subsequence {xnk } of the sequence {xn } converges to an element y ∈ E, then y is a solution of the equation y − T y = f. (3) If E is a reflexive Banach space and the sequence {xn } is bounded, then the sequence {xn } converges to a solution of the equation u + T u = f. Motivated by the above theorem, we have the concept of the condition (BP) as in [5] given by Let E be a real normed linear space, C be a nonempty subset of E and T : C → E be a mapping Definition 3.3. (Condition) The pair (T, C) is said to satisfy the condition (BP) if for any bounded closed subset G of C, {z : z = x − T x, x ∈ G} is a closed subset of E. Let E and F be Banach spaces. Recall that a mapping T : E → F is completely continuous if it is continuous and compact (that is, C is bounded implies that T (C) is compact) or a weakly convergent sequence (xn → x weakly) implies a strong convergent sequence (T xn → T x). Next we establish a relation between the condition (BP) and completely continuous mapping given as: Proposition 3.4. Let E be a real normed linear space, C be a nonempty subset of E and T : C → E be a completely continuous mapping. Then the pair (T, C) satisfy the condition (BP). Proof. It is similar as in [5]. Remark 3.5. ([5]) The converse of the above proposition does not holds in general.

4

Strong convergence theorems

Now we turn to strong convergence theorems for the asymptotically nonexpansive non-self mappings with condition in the real uniformly convex Banach spaces. First two results correspond to our new scheme where as remaining results are the same results in [3] and [5], which are the simple derivations of our result. Theorem 4.1. Let E be a real uniformly convex Banach space and C be a nonempty closed convex subset of E. Let T : C → E be an asymptotically nonexpansive non-self mapping 7 1559

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P P∞ 0 with {hn }, {h0n} ⊂ [1, ∞) and h0n ≤ hn such that ∞ n=1 (hn − 1) < ∞, n=1 (hn − 1) < ∞ and F ix(T ) 6= ∅. Let {xn } be a sequence defined by   x1 ∈ C,    y = P ((1 − β )x + β T (P T )n−1 x ), n n n n n (4.1) n−1  z = P ((1 − γ )y + γ T (P T ) y ),  n n n n n 

 x = P (1 − α )x + α T (P T )n−1 yn +zn n+1

n

n

n

2

for all n ≥ 1, where

0 ≤ lim inf αn , n→∞

lim sup αn < 1,

lim sup βn < 1,

n→∞

n→∞

lim sup γn < 1. n→∞

If the pair (T, C) satisfy the condition (BP), then the sequence {xn } converges strongly to a fixed point of T. Proof. Let G = {xn }, where A denotes the closure of A. Since the sequence {xn } is bounded in C by Lemma 2.2 and so G is a bounded closed subset of C. As pair (T, C) satisfy (BP) conditions, it follows that N = {z = x − T x : x ∈ G} is closed. Lemma 2.3 guarantees {xn − T xn } ⊂ N and xn − T xn → 0, {yn − T yn } ⊂ N and yn − T yn → 0 as n → ∞. Clearly the zero vector 0 ∈ N so there exist a q ∈ G such that q = T q so q ∈ F ix(T ). Since q ∈ G so there exists a positive integer n0 such that xn0 = q or there exists a subsequence {xnk } of the sequence {xn } such that xnk → q as k → ∞. If xnk = q, then it follows from Lemma 2.2 that xn = q for all n ≥ n0 and so xn → q as n → ∞. If xnk → q, then, since limn→∞ kxn − qk exists, we have xn → q as n → ∞. This completes the proof. Using Theorem 4.1 and Proposition 3.4, we have Corollary 4.2. Let E be a real uniformly convex Banach space and C be a nonempty closed convex subset of E. Let T : C → E be an asymptotically nonexpansive non-self mapping P P∞ 0 with {hn }, {h0n} ⊂ [1, ∞) and h0n ≤ hn such that ∞ n=1 (hn − 1) < ∞, n=1 (hn − 1) < ∞ and F ix(T ) 6= ∅. Let {xn } be a sequence defined by (4.1), where 0 ≤ lim inf αn , n→∞

lim sup αn < 1,

lim sup βn < 1,

n→∞

n→∞

lim sup γn < 1. n→∞

If T is completely continuous, then the sequence {xn } converges strongly to a fixed point of T. Theorem 4.3. Let E be a real uniformly convex Banach space and C be a nonempty closed convex subset of E. Let T : C → E be an asymptotically nonexpansive non-self P mapping with {hn } ⊂ [1, ∞) such that ∞ n=1 (hn − 1) < ∞ and F ix(T ) 6= ∅. Let {xn } be a sequence defined by (1.3), where 0 ≤ lim inf αn n→∞

and

lim sup αn < 1. n→∞

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If the pair (T, C) satisfy the condition (BP), then the sequence {xn } converges strongly to a fixed point of T. Proof. In Theorem 4.1, take βn = 0 and γn = 0 for all n ≥ 1, we arrive at the result. Using Theorem 4.3 and Proposition 3.4, we have Corollary 4.4. Let E be a real uniformly convex Banach space and C be a nonempty closed convex subset of E. Let T : C → E be an asymptotically nonexpansive non-self P mapping with {hn } ⊂ [1, ∞) such that ∞ n=1 (hn − 1) < ∞ and F ix(T ) 6= ∅. Let {xn } be a sequence defined by (1.3), where 0 ≤ lim inf αn n→∞

lim sup αn < 1.

and

n→∞

If T is completely continuous, then the sequence {xn } converges strongly to a fixed point of T. Remark 4.5. The results proved in [3] can also be obtained from our Theorem 4.1 under special assumptions of sequences on αn , βn and γn .

5

Examples

Here we focus on the families of examples to apply on our results. First example also extends the example presented in [5]. Example 5.1. Let X be a Hilbert space and C = {x ∈ X : kxk ≤ r, ∀r > 0}. Let P : C → C by Px =

(

x,

if x ∈ C,

rx , kxk

if x ∈ X − C.

Then P is a nonexpansive retraction of X onto C (see [5]). Take X = Rn with n X hx, yi = xi yi i=1

and

kxk =

n X i=1

Then X is a Hilbert space. Let C = {x ∈ X : kxk ≤ 1, }. Take P : C → C by Px =

(

!1 2

(xi )

2

.

x,

if x ∈ C,

x kxk ,

if x ∈ X − C. 9 1561

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Then P is a nonexpansive retraction of X onto C by [8], taking r = 1. Define Ti : C → X by Ti = (0, 0, ..., 1 − xi, 0, 0, ..., 0) for all X = (x1 , x2 , x3 , ..., xn) ∈ C. Then kT xi − T yi k = k(0, 0, ...1 − xi , 0, 0, ..., 0) − (0, 0, ..., 1 − yi , 0, 0, ..., 0)k = k(0, 0, ..., xi − yi , 0, 0, ..., 0)k ≤ kx − yk. So Ti x are a family of nonexpansive non-self mappings, so kTi (P Ti)2−1 x − Ti (P Ti )2−1 yk ≤ kP (Tix) − P (Ti y)k ≤ kTix − Tiyk ≤ kx − yk. Suppose that kTi(P Ti )k−1 x − Ti (P Ti )k−1 yk ≤ kx − yk,

∀n = k.

Taking n = k + 1, we have kTi (P Ti)(k+1)−1 x − Ti(P Ti )(k+1)−1 yk ≤ kP (Tix)(k+1)−1 − P (Tiy)(k+1)−1 k = kP (Ti)P (Ti x)k−1 − P (Ti)P (Ti y)k−1 k ≤ kTiP (Ti x)k−1 − TiP (Ti y)k−1 k ≤ kx − yk. It follws that from Mathematical Induction, Ti is a family of an asymptotically non expensive non-self mapping with sequence {hi,n } defined by hi,n = 1 for all n ≥ 1. Put n o 1 F ix(Ti) = (0, 0, ..., , 0, 0, ..., 0) . 2 Now, we prove that the pairs (Ti , C) for each i satisfy the condition. For any closed subset G of C, we denoted Ni = {z = x − Ti x : x ∈ G} Then Ni are closed. Reality is that, for any zn ∈ Ni with zn → z, there exist xn ∈ G such that zn = xn − Tixn . As G is bounded and closed in C so is compact. Therefore, there exists a convergent subsequence {xnk } of {xn }. Letting xnk → x0 as k → ∞, we have x0 ∈ G. Also as Ti are continuous so z = lim znk = lim (xnk − Ti xnk ) = x0 − Ti x0 ∈ Ni. k→∞

k→∞

For any given x1 ∈ C, take a sequence {xn }  n−1 x ),  n yn = P ((1 − β1 )xn + βn Ti (P Ti ) n−1 zn = P ((1 − γ1 )yn + γn Ti (P Ti) yn ), 

 yn +zn n−1 xn+1 = P (1 − α1 )xn + αn Ti(P Ti ) 2

for any n ≥ 1, where

4 1 + , n = 0 (mod 2), 5 6n 1 1 αn = + , n = 1 (mod 2), 10 2n 2n βn = γ n = , n ≥ 1. 3n + 2 αn =

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Clearly lim inf αn = n→∞

1 , 10

lim sup αn = n→∞

lim sup βn = lim sup γn = n→∞

n→∞

4 , 5

2 . 3

So all condition of Theorem 4.1 are satisfied and so {xn } converge strongly to a fixed point (0, 0, 0, ..., 21 , 0, 0, ..., 0) of Ti . Remark 5.2. If we take T1 (x1 , x2 ) = (1 − x1 , 0) with x1 = (0, 0), then {xn } converges strongly to F ix(Ti ) = ( 21 , 0) after single iteration. P P∞ 2
21 Example 5.3. Let X = l 2 with hx, yi = ∞ . Then X is a i=1 xi yi and kxk = i=1 xi real infinite dimensional Hilbert space. Let C = {x ∈ X : kxk ≤ 1}. Define a mapping P : X → C by ( x, if x ∈ X, Px = x , if x ∈ X − C. kxk Then P is nonexpansive retraction of X onto C. Define T : C → X by T x = (−x1 , −x2 , ..., −xi, ...), Then we have

∀x ∈ C.

kT x − T yk = k(y1 − x1 , y2 − x2 , ..., yi − xi , ...)k !1 ∞ 2 X 2 = (yi − xi ) = kx − yk i=1

for all x = (x1 , x2 , ..., xi, ...), y = (y1 , y2 , ..., yi, ...) ∈ C so T is an asymptotically nonexpansive non-self mapping with sequence {hn } defined by hn = 1 for all n ≥ 1 and F ix(T ) = {(0, 0, ..., 0)}. Now we prove that the pair (T, C) satisfy our condition and T is not completely continuous. In fact for any closed subset G of C, we denote N = {z = x − T x : c ∈ G}. For any zn ∈ N with zn → z as n → ∞, there exists zn = xn − T xn = 2xn . It follows from zn → z that xn → 21 z as n → ∞. Since G is closed in C, it follows that 21 z ∈ G. Since T is continuous, it follows that   1 1 z = lim zn = lim (xn − T xn ) = z − T z ∈ N. n→∞ n→∞ 2 2 This shows that the pair (T, C) satisfy the condition (BP). Since T is surjective, and unit ball C = {x ∈ X : kxk ≤ 1}, is not sequentially compact, so T is not completely continuous (see [5]). For x1 ∈ C, define sequence {xn } by  n−1  xn ), yn = P ((1 − βn )xn + βn T (P T ) n−1 zn = P ((1 − γn )yn + γn T (P T ) yn ), 

 n xn+1 = P (1 − αn )xn + αn T (P T )n−1 yn +z , 2 11

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1 1 where αn = 35 + 4n , n = 1 (mod 2), and αn = 71 + 2n , n = 0 (mod 2) for all c ≥ 1, 3n 3n βn = 4n+2 , γn = 4n+2 for all n ≥ 1. Clearly lim inf n→∞ αn = 71 and lim supn→∞ αn = 53 and lim supn→∞ βn = lim supn→∞ γn = 43 . So all conditions of Theorem 4.1 are satisfied. Hence the sequence {xn } converges strongly to fixed point (0, 0, ..., 0) of T.

Acknowledgement This work was supported by the Dong-A University research fund.

References [1] F. E. Browder, On the iteration of transformations in noncompact minimal dynamical systems, Proc. Amer. Math. Soc., 9 (1958), 773–780. [2] F. E. Browder and W. V. Petryshyn, The solution by iteration of linear functional equations in Banach spaces, Bull. Amer. Math. Soc., 72 (1966), 566–570. [3] C. E. Chidume, E. U. Ofoedu and H. Zegeye, Strong and weak convergence theorems for asymptotically nonexpansive mappings, J. Math. Anal. Appl., 280 (2003), 364– 374. [4] K. Goebel and W. A. Kirk, A fixed point theorem for asymptotically nonexpansive mapping, Proc. Amer. Math. Soc., 35 (1972), 171–174. [5] W. Guo, A. A. Abdou, L. A. Khan and Y. J. Cho, Strong convergence theorems for asymptoticlly nonexpansive nonself-mappings with applications, Fixed Point Theory Appl., 2015 (2015), Article ID 212, 12 pages. [6] W. Guo, Y. J. Cho and W. Guo, Convergence theorems for mixed type asyptotically nonexpansive mappings, Fixed Point Theory Appl., 2015 (2015), Article ID 224, 15 pages. [7] W. Guo and W. Guo, Weak convergence theorems for asyptotically nonexpansive nonself-mappings, Appl. Math. Lett., 24 (2011), 2181–2185. [8] Y. Q. Liu and W. Guo, Convergence theorems of composite implied iretation scheme for mixed type asyptotically nonexpansive mappings, Acta Math. Sci. Ser. A, 35 (2015), 422–440. [9] J. Schu, Iteration construction of fixed points of asymptotically nonexpansive mappings, J. Math. Anal. Appl., 158 (1991), 407–413. [10] L. Wang, Strong convergence theorems for common fixed points of nonself asymptotically nonexpansive mappings, J. Math. Anal. Appl., 323 (2006), 550–557. [11] H. K. Xu, Inequalities in Banach spaces with applications, Nonlinear Anal., 16 (1991), 1127–1138.

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On existence of nondecreasing solutions of q-quadratic integral equations February 16, 2017 Mohamed Abdalla Darwisha,b , Maryam Al-Yamib and Kishin Sadaranganic a

b

Department of Mathematics, Faculty of Science, Damanhour University Damanhour, Egypt e-mail: [email protected]

Department of Mathematics, Sciences Faculty for Girls, King Abdulaziz University Jeddah, Saudi Arabia e-mail: [email protected] c

Departamento de Matem´aticas, Universidad de Las Palmas de Gran Canaria Campus de Tafira Baja, 35017 Las Palmas de Gran Canaria, Spain e-mail: [email protected] Abstract We investigate a q-fractional integral equation with supremum and prove an existence theorem for it. We will prove that our q-integral equation has a solution in C[0, 1] which is monotonic on [0, 1]. The monotonicity measure of noncompactness due to Bana´s and Olszowy and Darbo’s theorem are the main tools used in the proof our main result.

MSC: 45G10, 47H09, 45M99. Keywords: q-fractional; integral equation; monotonic solutions; Darbo theorem; monotonicity measure of noncompactness.

1

Introduction

Jackson in [20, 21] introduced the concept of quantum calculus (q-calculus). This area of research has rich history and several applications, see [1, 3, 22, 23] and references therein. There are several developments and applications of the q-calculus in mathematical physics,

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2 especially concerning quantum mechanics, the theory of relativity and special functions [1, 3, 17, 23, 24]. Recently, several researchers attracted their attention by the concept of q-calculus, and we can find several new results in [2, 3, 18, 25] and the references therein. In several papers among them [4, 19], differential equations with supremum as well as integral equations with supremum have been studied. In [10, ?, 12, 13, 14, 15] Darwish et al. studied differential and integral equations of arbitrary orders with supremum. Also, Caballero et al. [8, 9] introduced and studied the quadratic Volterra equations with supremum. They showed that these equations have monotonic solutions in the space C[0, 1]. In [10], Darwish generalized and extend Caballero et al. [8] results to the quadratic integral equations of arbitrary orders with supremum. In this paper we will study the q-quadratic integral equation with supremum Z tβ−1 (T y)(t) t y(t) = f (t) + (qs/t; q)β−1 κ(t, s) max |y(τ )| dq s, t ∈ J = [0, 1], (1.1) Γq (β) [0,σ(s)] 0 where 0 < β, q ∈ (0, 1), f : J → R, T : C(J) → C(J), κ : J × J → R+ and σ : J → J. By using Darbo fixed point theorem and the monotonicity measure of noncompactness due to Bana´s and Olszowy [6] we prove the existence of monotonic solution to Eq.(1.1) in C[0, 1].

2

Fractional q-calculus

We collect basic definitions and results of the q-fractional integrals and q-derivatives, for more details, see [2, 3, 7, 17, 18, 24, 25] and references therein. First, for a real parameter q ∈ (0, 1), we define a q-real number [a]q by [a]q =

1 − qa , a ∈ R, 1−q

and a q-analog of the Pochhammer symbol (q-shifted factorial) is defined by  1, n = 0,    (a; q)n = n−1 Q   (1 − aq k ), n ∈ N.  k=0

Also, the q-analog of the power (a − b)n is given by  1, n = 0,    (a − b)(n) = n−1 Q   (a − bq k ), n ∈ N; a, b ∈ R.  k=0

Moreover, (a − b)(n) = an (b/a; q)n , a 6= 0.

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3 Notice that, lim (a; q)n exists and we will denote it by (a; q)∞ . n→∞

More generally, for β ∈ R, aq β 6= q −n (n ∈ N), we define (a; q)β =

(a; q)∞ (aq β ; q)∞

and (a − b)(β) = aβ

(b/a; q)∞ . (q β b/a; q)∞

Notice that (a − b)(β) = aβ (b/a; q)β . Therefore, if b = 0, then a(β) = aβ . Now, the q-gamma function is given by G(q x ) , x ∈ R\{0, −1, −2, · · · }, (1 − q)x−1 G(q)

Γq (x) = where G(q x ) =

1 (q x ;q)∞ .

Or, equivalently, Γq (x) =

(1−q)(x−1) (1−q)x−1

and satisfies Γq (x+1) = [x]q Γq (x).

Next, the q-derivative of a function f is given by (Dq f )(t) =

f (t) − f (qt) , (Dq f )(0) = lim(Dq f )(t), t→0 t − qt

and the q-derivative of higher order of a function f is defined by  n = 0,  f (t), (Dqn f )(t) =  Dq (Dqn−1 f )(t), n ∈ N. Let f be a function defined on [0, b]. The q-integral of f is defined as follows Z

t

f (s) dq s = t(1 − q)

(Iq f )(t) = 0

∞ X

q n f (tq n ), t ∈ [0, b].

(2.2)

n=0

If f is given on the interval [0, b] and a ∈ [0, b], then Z b Z b Z f (s) dq s = f (s) dq s − a

0

a

f (s) dq s.

0

The operator Iqn is defined by (Iqn f )(t)

=

  f (t), 

n = 0,

Iq (Iqn−1 f )(t), n ∈ N.

The fundamental theorem of calculus satisfies for Dq and Iq , i.e., (Dq Iq f )(t) = f (t), and if f is continuous at t = 0, then (Iq Dq f )(t) = f (t) − f (0).

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4 The following four formulas will be used later in this paper [a(t − s)](β) = aβ (t − s)(β) , t Dq (t s Dq (t

and Z

− s)(β) = [β]q (t − s)(β−1) ,

− s)(β) = −[β]q (t − qs)(β−1)

t

Z

t

f (t, s) dq s =

t Dq

t Dq f (t, s)

dq s + f (qt, t),

0

0

where t Dq denotes the derivative with respect to variable t. Notice that, if β > 0 and a ≤ b ≤ t, then (t − b)(β) ≤ (t − a)(β) . Definition 1. [1] Let f be a function defined on [0, 1]. The fractional q-integral of the Riemann-Liouville type of order β ≥ 0 is given by  β = 0,   f (t),  (Iqβ f )(t) = ∞ Rt β ;q) P  n  q n (q(q;q) f (tq n ), β > 0, t ∈ [0, 1].  Γq1(β) (t − qs)(β−1) f (s) dq s = tβ (1 − q)β n n=0

0

Notice that, for β = 1, the above q-integral reduces to (2.2). Definition 2. [1] The fractional q-derivative of the Riemann-Liouville type of order β ≥ 0 is given by  β = 0,  f (t), (Dqβ f )(t) =  [β] [β]−β (Dq Iq f )(t), β > 0, where [β] stands for the smallest integer equal or greater than β. In q−calculus, the derivative rule for the product of two functions and integration by parts formulas are (Dq f g)(t) = (Dq f )(t)g(t) + f (qt)(Dq g)(t), Z t Z t t f (s)(Dq g)(s) dq s = [f (s)g(s)]0 − (Dq f )(s)g(qs) dq s. 0

0

Lemma 1. Let β, γ ≥ 0. Then the following are verified for a function f defined on [0, 1]: (1) (Iqγ Iqβ f )(t) = (Iqβ+γ f )(t), (2) (Dqβ Iqβ f )(t) = f (t). Lemma 2. [24] For β > 0. Then q-integration by parts allows us to have (Iqβ 1)(t) = or Z

t(β) Γq (β + 1)

t

(t − qs)(β−1) dq s =

0

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5

3

Measure of noncompactness

We assume that (E, k.k) is a real Banach space with zero element θ and we denote by B(x, r) the closed ball with radius r and centre x, where Br ≡ B(θ, r). Now, let X ⊂ E and denote by X and ConvX the closure and convex closure of X, respectively. Also, the symbols X +Y and λY stands for the usual algebraic operators on sets. Moreover, the families ME and NE are defined by ME = {A ⊂ E : A 6= ∅, A is bounded} and NE = {B ⊂ ME : B is relatively compact}, respectively. Definition 3. [5] Let µ : ME → [0, +∞). If the following conditions 1◦ ∅ 6= {X ∈ ME : µ(X) = 0} = kerµ ⊂ NE , 2◦ if X ⊂ Y , then µ(X) ≤ µ(Y ), 3◦ µ(X) = µ(X) = µ(ConvX), 4◦ µ(λX + (1 − λ)Y ) ≤ λµ(X) + (1 − λ)µ(Y ), 0 ≤ λ ≤ 1 and 5◦ if (Xn ) is a sequence of closed subsets of ME with Xn+1 ⊂ Xn , n = 1, 2, 3, ..., and lim µ(Xn ) = 0 then X∞ = ∩∞ n=1 Xn 6= ∅ n→∞

hold. Then, the mapping µ is said to be a measure of noncompactness in E. Here, kerµ is the kernel of the measure of noncompactness µ. Our result will establish in C(J) the Banach space of all defined, continuous and real functions on J ≡ [0, 1] with kyk = max{|y(τ )| : τ ∈ J}. Next, we defined the measure of noncompactness related to monotonicity in C(J), see [5, 6]. We fix a bounded subset Y 6= ∅ of C(J). For ε ≥ 0 and y ∈ Y , ω(y, ε) denotes the modulus of continuity of the function y given by ω(y, ε) = sup{|y(t) − y(s)| : t, s ∈ J, |t − s| ≤ ε}. Moreover, we let ω(Y, ε) = sup{ω(y, ε) : y ∈ Y } and ω0 (Y ) = lim ω(Y, ε). ε→0

Define d(y) =

(|y(t) − y(s)| − [y(t) − y(s)])

sup t,s∈J, s≤t

and d(Y ) = sup d(y). y∈Y

Notice that all functions in Y are nondecreasing on J if and only if d(Y ) = 0.

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6 Now, we define the map µ on MC(J) as µ(Y ) = d(Y ) + ω0 (Y ). Clearly, µ verifies all conditions in Definition 3 and, therefore it is a measure of noncompactness in C(J) [6]. Definition 4. Let ∅ 6= M ⊂ E. Let P : M → E be a continuous operator. Suppose that P maps bounded sets onto bounded ones. If there exists a bounded Y ⊂ M with µ(PY ) ≤ αµ(Y ), α ≥ 0, then P is said to be satisfies the Darbo condition with respect to a measure of noncompactness µ. If α < 1, then P is called a contraction operator with respect to µ. Theorem 1. [16] Let Ω 6= ∅ be a bounded, convex and closed subset of E. If P : Ω → Ω is a contraction operator with respect to µ. Then P has at least one fixed point belongs to Ω. We will need the following two lemmas throughout our proof [8]. Lemma 3. Let r : J → J be a continuous function and y ∈ C(J). If, for t ∈ J, (F y)(t) = max |y(τ )|, [0,σ(t)]

then F y ∈ C(J). Lemma 4. Let (yn ) be a sequence in C(J) and y ∈ C(J). If (yn ) converges to y ∈ C(J), then (F yn ) converges uniformly to F y uniformly J.

4

Main Theorem

Let us consider the following hypotheses: (h1 ) f ∈ C(J). Moreover, f is nondecreasing and nonnegative on J. (h2 ) The operator T : C(J) → C(J) is continuous and satisfies the Darbo condition with a constant c for the measure of noncompactness µ . Moreover, T x ≥ 0 if x ≥ 0. (h3 ) ∃ a, b ≥ 0 s.t. |(T x)(t)| ≤ a + bkxk ∀x ∈ C(J), t ∈ J. (h4 ) The function κ : J × J → R+ is continuous on J × J and nondecreasing ∀t and s separately. Moreover, κ∗ = sup κ(t, s). (t,s)∈J×J

(h5 ) The function σ : J → J is nondecreasing and continuous on J. (h6 ) ∃ r0 > 0 such that kf k + and

ck∗ r0 Γq (β+1)

κ∗ r0 (a + br0 ) ≤ r0 Γq (β + 1)

(4.3)

< 1.

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7 Now, we rewrite Eq.(1.1) as x(t) = f (t) +

(T x)(t) Γq (β)

Z

t

κ(t, s) (t − qs)(β−1) max |x(τ )| dq s, 0 < β ≤ 1, t ∈ J,

(4.4)

[0,σ(s)]

0

and define the two operators K and F on C(J) as follows 1 (Ky)(t) = Γq (β)

Z

t

κ(t, s)(t − qs)(β−1) max |y(τ )| dq s

(4.5)

[0,σ(s)]

0

and (Fy)(t) = f (t) + (T y)(t) · (Ky)(t),

(4.6)

respectively. Finding a fixed point of the operator F is equivalent to solving Eq.(4.4). Under the above hypotheses, we state and prove our main theorem. Theorem 2. Assume the hypotheses (h1 ) − (h6 ) be verified. Then Eq.(4.4) has at least one solution x ∈ C(J) which is nondecreasing on J. Proof. First, we will show that the operator F maps C(J) into itself. For this, it is sufficient to show that Kx ∈ C(J) if x ∈ C(J). Fix ε > 0 and let x ∈ C(J) and t1 , t2 ∈ J (t1 ≤ t2 ) with |t2 − t1 | ≤ ε. We have Z t2 1 |(Kx)(t2 ) − (Kx)(t1 )| = κ(t2 , s)(t2 − qs)(β−1) max |x(τ )| dq s Γq (β) 0 [0,σ(s)] Z t1 1 (β−1) − κ(t1 , s)(t1 − qs) max |x(τ )| dq s Γq (β) 0 [0,σ(s)] Z t2 1 κ(t2 , s)(t2 − qs)(β−1) max |x(τ )| dq s ≤ Γq (β) 0 [0,σ(s)] Z t2 1 (β−1) − κ(t1 , s)(t2 − qs) max |x(τ )| dq s Γq (β) 0 [0,σ(s)] Z t2 1 + κ(t1 , s)(t2 − qs)(β−1) max |x(τ )| dq s Γq (β) 0 [0,σ(s)] Z t1 1 (β−1) κ(t1 , s)(t2 − qs) max |x(τ )| dq s − Γq (β) 0 [0,σ(s)] Z t1 1 + κ(t1 , s)(t2 − qs)(β−1) max |x(τ )| dq s Γq (β) 0 [0,σ(s)] Z t1 1 (β−1) κ(t1 , s)(t1 − qs) max |x(τ )| dq s − Γq (β) 0 [0,σ(s)] Z t2 1 ≤ |κ(t2 , s) − κ(t1 , s)|(t2 − qs)(β−1) max |x(τ )| dq s Γq (β) 0 [0,σ(s)] Z t2 1 + |κ(t1 , s)|(t2 − qs)(β−1) max |x(τ )| dq s Γq (β) t1 [0,σ(s)]

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8 1 + Γq (β)

t1

Z

|κ(t1 , s)| |(t2 − qs)(β−1) − (t1 − qs)(β−1) | max |x(τ )| dq s [0,σ(s)]

0 t2

kxk ωκ (ε, .) (t2 − qs)(β−1) dq s Γq (β) 0  Z t1 Z t2 κ∗ kxk (β−1) (β−1) (β−1) (t2 − qs) dq s + [(t1 − qs) − (t2 − qs) ]dq s + Γq (β) t1 0 kxk κ∗ kxk (β) (β) (β) = ωκ (ε, .) t2 + [t1 − t2 + 2(t2 − t1 )(β) ] Γq (β + 1) Γq (β + 1) 2κ∗ kxk kxk (β) ωκ (ε, .) t2 + (t2 − t1 )(β) ≤ Γq (β + 1) Γq (β + 1) kxk κ∗ kxk β ≤ ωκ (ε, .) tβ2 + ε , (4.7) Γq (β + 1) Γ(β + 1) Z

≤

where we used ωκ (ε, .) =

|κ(t, s) − κ(τ, s)|.

sup t, τ ∈J, |t−τ |≤ε

Notice that, since the function κ is uniformly continuous on J × J, then when ε → 0 we have that ωκ (ε, .) → 0. Therefore, Kx ∈ C(J) and consequently, Fx ∈ C(J). Now, ∀t ∈ J, we have Z (T x)(t) t (β−1) |(Fx)(t)| ≤ f (t) + κ(t, s)(t − qs) max |x(τ )| dq s Γq (β) 0 [0,σ(s)] Z t a + bkxk ≤ kf k + κ(t, s)(t − qs)(β−1) max |x(τ )| dq s Γq (β) 0 [0,σ(s)] a + bkxk ∗ ≤ kf k + κ kxk. Γq (β + 1) Hence kFxk ≤ kf k +

a + bkxk ∗ κ kxk. Γq (β + 1)

From hypothesis (h6 ), if kxk ≤ r0 , we get kFxk ≤ kf k +

a + br0 ∗ κ r0 Γq (β + 1)

≤ r0 . Therefore, F maps Br0 into itself. Next, we consider the operator F on the set Br+0 = {x ∈ Br0 : x(t) ≥ 0, ∀t ∈ J}. It is clear that Br+0 6= ∅ is closed, convex and bounded. By this facts and hypotheses (h1 ), (h3 ) and (h5 ), we obtain F transforms Br+0 into itself. In what follows, we will show that F is continuous on Br+0 . For, let (xn ) be us a sequence in Br+0 such that xn → x and we will show that Fxn → Fx. We have, ∀t ∈ J, Z (T xn )(t) t κ(t, s)(t − qs)(β−1) max |xn (τ )|dq s |(Fxn )(t) − (Fx)(t)| = Γq (β) 0 [0,σ(s)]

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9 Z (T x)(t) t (β−1) κ(t, s)(t − qs) max |x(τ )|dq s − Γq (β) 0 [0,σ(s)] Z (T xn )(t) t ≤ κ(t, s)(t − qs)(β−1) max |xn (τ )| dq s Γq (β) 0 [0,σ(s)] Z t (T x)(t) (β−1) − κ(t, s)(t − qs) max |xn (τ )| dq s Γq (β) 0 [0,σ(s)] Z t (T x)(t) κ(t, s)(t − qs)(β−1) max |xn (τ )| dq s + Γq (β) 0 [0,σ(s)] Z t (T x)(t) (β−1) − κ(t, s)(t − qs) max |x(τ )| dq s Γq (β) 0 [0,σ(s)] Z t |(T xn )(t) − (T x)(t)| ≤ |κ(t, s)|(t − qs)(β−1) max |xn (τ )| dq s Γq (β) [0,σ(s)] 0 Z t |(T x)(t)| (β−1) dq s. + |κ(t, s)|(t − qs) max |x (τ )| − max |x(τ )| n [0,σ(s)] Γq (β) 0 [0,σ(s)] Applying Lemma 4, we obtain kFxn − Fxk ≤

κ∗ r0 κ∗ (a + b r0 ) kT xn − T xk + kxn − xk. Γq (β + 1) Γq (β + 1)

(4.8)

By the continuity of T , ∃n1 ∈ N such that kT xn − T xk ≤ Also, ∃n2 ∈ N such that kxn − xk ≤

εΓq (β + 1) ∀n ≥ n1 . 2κ∗ r0

εΓq (β + 1) ∀n ≥ n2 . 2κ∗ (a + br0 )

Now, take max{n1 , n2 } ≤ n, then (4.8) gives us that kFxn − Fxk ≤ ε. This shows that F is continuous in Br+0 . Now, we take ∅ 6= X ⊂ Br+0 . Let us fix an arbitrarily number ε > 0 and choose x ∈ X and t1 , t2 ∈ J with |t2 − t1 | ≤ ε. We will assume that t1 ≤ t2 because no generality will be loss. Then, by using our hypotheses and inequality (4.7), we get |(Fx)(t2 ) − (Fx)(t1 )| ≤ |f (t2 ) − f (t1 )| + |(T x)(t2 ) (Kx)(t2 ) − (T x)(t2 ) (Kx)(t1 )| + |(T x)(t2 ) (Kx)(t1 ) − (T x)(t1 ) (Kx)(t1 )| ≤ ω(f, ε) + |(T x)(t2 )| |(Kx)(t2 ) − (Kx)(t1 )| + |(T x)(t2 ) − (T x)(t1 )| |(Kx)(t1 )| i (a + bkxk) h ω(T x, ε) ≤ ω(f, ε) + kxkωκ (ε, .) + 2κ∗ kxkεβ + kxkκ∗ Γq (β + 1) Γq (β + 1)

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10 ≤ ω(f, ε) +

r0 (a + br0 ) κ∗ r0 [ωκ (ε, .) + 2 κ∗ εβ ] + ω(T x, ε). Γq (β + 1) Γq (β + 1)

Hence, ω(Fx, ε) ≤ ω(f, ε) +

r0 (a + br0 ) κ∗ r0 [ωκ (ε, .) + 2κ∗ εβ ] + ω(T x, ε). Γq (β + 1) Γq (β + 1)

Consequently, ω(FX, ε) ≤ ω(f, ε) +

κ∗ r0 r0 (a + br0 ) [ωκ (ε, .) + 2κ∗ εβ ] + ω(T X, ε). Γq (β + 1) Γq (β + 1)

Since the function κ is uniformly continuous on J × J and the function f is continuous on J, then the last inequality gives us that ω0 (FX) ≤

κ∗ r0 ω0 (T X). Γq (β + 1)

(4.9)

Further, fix arbitrary x ∈ X and t1 , t2 ∈ J with t2 > t1 . Then, by our hypotheses, we have |(Fx)(t2 ) − (Fx)(t1 )| − [(Fx)(t2 ) − (Fx)(t1 )] Z (T x)(t2 ) t2 = f (t2 ) + κ(t2 , s)(t2 − qs)(β−1) max |x(τ )| dq s Γq (β) 0 [0,σ(s)] Z t1 (T x)(t1 ) (β−1) κ(t1 , s)(t1 − qs) max |x(τ )| dq s −f (t1 ) − Γq (β) 0 [0,σ(s)]  Z t2 (T x)(t2 ) − f (t2 ) + κ(t2 , s)(t2 − qs)(β−1) max |x(τ )| dq s Γ(q β) 0 [0,σ(s)]  Z t1 (T x)(t1 ) (β−1) −f (t1 ) − κ(t1 , s)(t1 − qs) max |x(τ )| dq s Γq (β) 0 [0,σ(s)] Z (T x)(t2 ) t2 ≤ {|f (t2 ) − f (t1 )| − [f (t2 ) − f (t1 )]} + κ(t2 , s)(t2 − qs)(β−1) max |x(τ )| dq s Γq (β) 0 [0,σ(s)] Z t2 (T x)(t1 ) − κ(t2 , s)(t2 − qs)(β−1) max |x(τ )| dq s Γq (β) 0 [0,σ(s)] Z (T x)(t1 ) t2 κ(t2 , s)(t2 − qs)(β−1) max |x(τ )| dq s + Γq (β) 0 [0,σ(s)] Z t1 (T x)(t1 ) (β−1) − κ(t1 , s)(t1 − qs) max |x(τ )| dq s Γq (β) 0 [0,σ(s)]  Z t2 (T x)(t2 ) − κ(t2 , s)(t2 − qs)(β−1) max |x(τ )| dq s Γq (β) 0 [0,σ(s)]  Z t2 (T x)(t1 ) (β−1) κ(t2 , s)(t2 − qs) max |x(τ )| dq s − Γq (β) 0 [0,σ(s)]  Z (T x)(t1 ) t2 + κ(t2 , s)(t2 − qs)(β−1) max |x(τ )| dq s Γq (β) 0 [0,σ(s)]

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11  Z (T x)(t1 ) t1 (β−1) κ(t1 , s)(t1 − qs) max |x(τ )| dq s − Γq (β) 0 [0,σ(s)] ≤ {|(T x)(t2 ) − (T x)(t1 )| − [(T x)(t2 ) − (T x)(t1 )]} Z t2 1 × κ(t2 , s)(t2 − qs)(β−1) max |x(τ )| dq s Γq (β) 0 [0,σ(s)]  Z t2 (T x)(t1 ) κ(t2 , s)(t2 − qs)(β−1) max |x(τ )| dq s + Γq (β) [0,σ(s)] 0 Z t1 (β−1) − κ(t1 , s)(t1 − qs) max |x(τ )| dq s [0,σ(s)] 0 Z t2 − κ(t2 , s)(t2 − qs)(β−1) max |x(τ )| dq s [0,σ(s)] 0  Z t1 (β−1) κ(t1 , s)(t1 − qs) max |x(τ )| dq s . − Now, we will prove that Z t2 Z (β−1) κ(t2 , s)(t2 − qs) max |x(τ )| dq s − [0,σ(s)]

0

t1

κ(t1 , s)(t1 − qs)(β−1) max |x(τ )| dq s ≥ 0. [0,σ(s)]

0

In fact, we have Z t2 Z (β−1) κ(t2 , s)(t2 − qs) max |x(τ )| dq s − [0,σ(s)]

0 t2

Z =

κ(t1 , s)(t1 − qs)(β−1) max |x(τ )| dq s [0,σ(s)]

0

+ 0 t2

=

[0,σ(s)]

κ(t1 , s)(t2 − qs)(β−1) max |x(τ )| dq s − κ(t1 , s)(t2 − qs)(β−1) max |x(τ )| dq s −

Z

[0,σ(s)]

t1

κ(t1 , s)(t2 − qs)(β−1) max |x(τ )| dq s

0

Z

0

Z

t2

Z

[0,σ(s)]

t2

+

Z

t1

κ(t2 , s)(t2 − qs)(β−1) max |x(τ )| dq s −

0

Z

(4.10)

[0,σ(s)]

0

t1

κ(t1 , s)(t2 − qs)(β−1) max |x(τ )| dq s [0,σ(s)]

0

[0,σ(s)]

t1

κ(t1 , s)(t1 − qs)(β−1) max |x(τ )| dq s [0,σ(s)]

0

(κ(t2 , s) − κ(t1 , s))(t2 − qs)(β−1) max |x(τ )| dq s [0,σ(s)]

0 t2

Z ++

κ(t1 , s)(t2 − qs)(β−1) max |x(τ )| dq s [0,σ(s)]

t1

Z +

t1

κ(t1 , s)[(t2 − qs)(β−1) − (t1 − qs)(β−1) ] max |x(τ )| dq s. [0,σ(s)]

0

But, κ(t1 , s) ≤ κ(t2 , s) because κ(t, s) is increasing with respect to t, then Z t2 (κ(t2 , s) − κ(t1 , s))(t2 − qs)(β−1) max |x(τ )| dq s ≥ 0

(4.11)

[0,σ(s)]

0

and, since (t2 − qs)(β−1) − (t1 − qs)(β−1) ≥ 0 for s ∈ [0, t1 ) then Z t1 κ(t1 , s)[(t2 − qs)(β−1) − (t1 − qs)(β−1) ] max |x(τ )| dq s [0,σ(s)]

0

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12 Z

t2

+ Z

κ(t1 , s)(t2 − qs)(β−1) max |x(τ )| dq s [0,σ(s)]

t1 t1

κ(t1 , t1 )[(t2 − qs)(β−1) − (t1 − qs)(β−1) ] max |x(τ )| dq s

≥

[0,σ(t1 )]

0

Z

t2

κ(t1 , t1 )(t2 − qs)(β−1) max |x(τ )| dq s [0,σ(t1 )] t1 Z t2 Z = κ(t1 , t1 ) max |x(τ )| (t2 − qs)(β−1) dq s − +

[0,σ(t1 )]

t1

(β−1)

(t1 − qs)

dq s

0

0

tβ2 − tβ1 = κ(t1 , t1 ) max |x(τ )| [β]q [0,σ(t1 )] ≥ 0.

(4.12)

Finally, (4.11) and (4.12) imply that Z Z t2 (β−1) κ(t2 , s)(t2 − qs) max |x(τ )| dq s − [0,σ(s)]

0



t1

κ(t1 , s)(t1 − qs)(β−1) max |x(τ )| dq s ≥ 0.

0

[0,σ(s)]

The above inequality and (4.10) leads us to |(Fx)(t2 ) − (Fx)(t1 )| − [(Fx)(t2 ) − (Fx)(t1 )] = {|(T x)(t2 ) − (T x)(t1 )| − [(T x)(t2 ) − (T x)(t1 )]} Z t2 1 × κ(t2 , s)(t2 − qs)(β−1) max |x(τ )| dq s Γq (β) 0 [0,σ(s)] κ∗ r0 ≤ d(T x). Γq (β + 1) Thus, d(Fx)Γq (β + 1) ≤ κ∗ r0 d(T x) and therefore, d(FX)Γq (β + 1) ≤ κ∗ r0 d(T X).

(4.13)

Finally, (4.9) and (4.13) gives us that ω0 (FX) + d(FX) ≤

κ∗ r0 (ω0 (FX) + d(T X)) Γq (β + 1)

or µ(FX) ≤ ≤ But

κ∗ r0 c Γq (β+1)

r0 κ∗ µ(T X) Γq (β + 1) κ∗ cr0 µ(X). Γq (β + 1)

< 1, then µ(FX) ≤ µ(X).

(4.14)

Inequality (4.14) enables us to use Theorem 1, then there are solutions to Eq.(1.1) in C(J). This finishes our proof.

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13 Acknowledgment: The third author was partially supported by the project MTM 201344357-P.

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TABLE OF CONTENTS, JOURNAL OF COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 24, NO. 8, 2018

Oscillation criteria for forced and damped nabla fractional difference equations, Jehad Alzabut, Thabet Abdeljawad, and Hussam Alrabaiah,……………………………………………1387 A type of 𝐶 2 smooth surfaces generated by bivariate rational spline interpolation, Jianxun Pan, Qinghua Sun, Ling Zhang, and Fangxun Bao,………………………………………….1395 Brunn-Minkowski type inequalities for width-integrals of index i, Yibin Feng,Shanhe Wu,1408 On fuzzy mighty filters in BE-algebras, Sun Shin Ahn and Jeong Soon Han,……………1419 Symmetric identities for (h,q)-extensions of the generalized higher order modified q-Euler polynomials, Jongkyum Kwon, Gyoyong Sohn, and Jin-Woo Park,……………………1431 Some properties of (p,q)-tangent polynomials, R. P. Agarwal, J. Y. Kang,C. S. Ryoo,1439 Strong Convergence Theorems for a Non-convex Hybrid Method for Quasi-Lipschitz Mappings and Applications, Waqas Nazeer, Shin Min Kang, Mobeen Munir, and Samina Kausar,1455 Certain subclasses of k-uniformly starlike functions associated with symmetric q-derivative operator, Nanjundan Magesh, Sahsene Altinkaya, and Sibel Yalcin,……………………1464 On solution of system of integral equations via fixed point method, Muhammad Nazam, Muhammad Arshad, Choonkil Park, Aftab Hussain, and Dong Yun Shin,…………….1474 On continuous Fibonacci functions, Hee Sik Kim, J. Neggers, and Keum Sook So,……1482 Decomposition and improved hyperbolic cross approximation of bivariate functions on [0,1]2, Zhihua Zhang,………………………………………………………………………………1491 Gronwall-Bellman type inequalities for the distributional Henstock-Kurzweil integral and applications, Wei Liu, Guoju Ye, and Dafang Zhao,………………………………………1503 Formulas and properties of some class of nonlinear difference Equations, E. M. Elsayed, Faris Alzahrani, and H. S. Alayachi,………………………………………………………………1517 A rational bicubic spline for visualization of shaped data, Xilian Fu,……………………..1532 Explicit Viscosity Rules and Applications of Nonexpansive Mappings, Young Chel Kwun, Waqas Nazeer, Mobeen Munir, and Shin Min Kang,……………………………………….1541

TABLE OF CONTENTS, JOURNAL OF COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 24, NO. 8, 2018 (continued) Applications and Strong Convergence Theorems of Asymptotically Nonexpansive Non-self Mappings, Young Chel Kwun, Waqas Nazeer, Mobeen Munir, and Shin Min Kang,……1553 On existence of nondecreasing solutions of q-quadratic integral equations, Mohamed Abdalla Darwish, Maryam Al-Yami, and Kishin Sadarangani,…………………………………….1565