JOURNAL OF COMPUTATIONAL ANALYSIS AND APPLICATIONS VOLUME 18 (5-6), 2015 0651098283

Table of contents :
BLOCK-5-18
FACE-5-18.pdf
JCAAA-2015-V18-front-5
SCOPE--JOCAAA--2015
EB--JOCAAA--2015
Instructions--JOCAAA--2015
Binder5-vol-18.pdf
65-2015-KOMATSU--JOCAAA--1-5-2014.pdf
66-2015-BERINDE-JOCAAA--1-5-2014.pdf
67-2015-FNL-RIDVAN-AHMET-JoCAAA-25-02-2014.pdf
68-2015-erkus-duman-tuglu-JOCAAA--1-5-14.pdf
69-2015--FNL--Elsayed--JOCAAA--3-16-2014.pdf
70-2015-FNL-Mahmudov-Akkeles-Oneren-JoCAAA-4-14-2014.pdf
71-2015-FNL-SITTHIWIRATTHAM--JOCAAA--11-11-2014.pdf
72-2015-Omar Abu Arqub-JOCAAA--1-5-14.pdf
73-2015-FNL-Park-Jang-Saadati-Shin-JoCAAA-2-7-2014.pdf
74-2015-FNL-Kim-SoonLee-JoCAAA-2-11-2014.pdf
76-2015-FNL-Abdollahpour-Park-JoCAAA-2-7-2014.pdf
77-2015-Jong Soo Jung-JOCAAA--1-1-14.pdf
78-2015-FNL-Ahn-Kim-Park-Jocaaa-27-2-2014.pdf
79-2015-T-KIM-JOCAAA--1-8-2014.pdf
81-2015-T-KIM-JOCAAA--1-20-2014.pdf
BLANK-JoCAAA-2015
TOC-2015-JOCAAA-VOL-18-V
BLOCK-6-18
FACE-6-18.pdf
JCAAA-2015-V18-front-6
SCOPE--JOCAAA--2015
EB--JOCAAA--2015
Instructions--JOCAAA--2015
Binder6-vol-18.pdf
82-2015-FNL-Lu-Anastassiou-Park-Jin-JoCAAA-2-8-2014.pdf
83-2015-FNL-FNL-Jang-Park-Efteghar-Farhadabadi-2-7-2014.pdf
84-2015-fnl-Park-Lee-Shim-Shin-JoCAAA-2-7-2014.pdf
85-2015-FNL-El-Metwally-Elsayed and El-Morshedy Elsayed-jocaaa-10-2-2014.pdf
86-2015-Yongmin Liu-JOCAAA-1-18-2014.pdf
87-2015-fnl-QingboCai-JoCAAA-2-7-2014.pdf
88-2015-FNL-Abdul Qadeer Khan-JOCAAA--2-8-14.pdf
89-2015--Loriana Andrei--JOCAAA--1-26-2014.pdf
90-2015--Loriana Andrei--JOCAAA-1-28-2014.pdf
91-2015-FNL-Muhiuddin-Alroqi-JoCAAA-6-02-2014.pdf
92-2015-FNL-Zuo-Yang-JoCAAA-2-6-2014.pdf
93-2015-Taekyun Kim-jocaaa-2-6-2014.pdf
94-2015-CHOONKIL-PARK-JOCAAA--2-10-2014.pdf
95-2015-Tian Zhou Xu-JOCAAA--2-10-2014.pdf
96-2015-FNL-AYDOGAN-JOCAAA--9-6-2014.pdf
BLANK-JoCAAA-2015
TOC-2015-JOCAAA-VOL-18-VI

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Volume 18, Number 5 ISSN:1521-1398 PRINT,1572-9206 ONLINE

May 2015

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

28601, USA.

Journal of Computational Analysis and Applications(JoCAAA) is published by EUDOXUS PRESS,LLC,1424 Beaver Trail Drive,Cordova,TN38016,USA,[email protected] http//:www.eudoxuspress.com. Annual Subscription Prices:For USA and Canada,Institutional:Print $650, Electronic OPEN ACCESS. Individual:Print $300. For any other part of the world add $100 more(postages) to the above prices for Print. No credit card payments. Copyright©2015 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.

754

Editorial Board Associate Editors of Journal of Computational Analysis and Applications 1) George A. Anastassiou Department of Mathematical Sciences The University of Memphis Memphis,TN 38152,U.S.A Tel.901-678-3144 e-mail: [email protected] Approximation Theory,Real Analysis, Wavelets, Neural Networks,Probability, Inequalities. 2) J. Marshall Ash Department of Mathematics De Paul University 2219 North Kenmore Ave. Chicago,IL 60614-3504 773-325-4216 e-mail: [email protected] Real and Harmonic Analysis

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

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

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

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

5) Carlo Bardaro Dipartimento di Matematica e Informatica

755

23) Burkhard Lenze Fachbereich Informatik Fachhochschule Dortmund University of Applied Sciences Postfach 105018 D-44047 Dortmund, Germany e-mail: [email protected] Real Networks, Fourier Analysis,Approximation Theory 24) Hrushikesh N.Mhaskar Department Of Mathematics California State University Los Angeles,CA 90032 626-914-7002 e-mail: [email protected] Orthogonal Polynomials, Approximation Theory,Splines, Wavelets, Neural Networks

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. 6) Martin Bohner Department of Mathematics and Statistics Missouri S&T Rolla, MO 65409-0020, USA [email protected] web.mst.edu/~bohner Difference equations, differential equations, dynamic equations on time scale, applications in economics, finance, biology. 7) Jerry L.Bona Department of Mathematics The University of Illinois at Chicago 851 S. Morgan St. CS 249 Chicago, IL 60601 e-mail:[email protected] Partial Differential Equations, Fluid Dynamics

25) M.Zuhair Nashed Department Of Mathematics University of Central Florida PO Box 161364 Orlando, FL 32816-1364 e-mail: [email protected] Inverse and Ill-Posed problems, Numerical Functional Analysis, Integral Equations,Optimization, Signal Analysis 26) Mubenga N.Nkashama Department OF Mathematics University of Alabama at Birmingham Birmingham, AL 35294-1170 205-934-2154 e-mail: [email protected] Ordinary Differential Equations, Partial Differential Equations

27)Svetlozar (Zari) Rachev, Professor of Finance, College of Business,and Director of Quantitative Finance Program, Department of Applied Mathematics & Statistics Stonybrook University 312 Harriman Hall, Stony Brook, NY 117943775 Phone: +1-631-632-1998, Email : [email protected] 28) Alexander G. Ramm Mathematics Department Kansas State University Manhattan, KS 66506-2602 e-mail: [email protected] Inverse and Ill-posed Problems, Scattering Theory, Operator Theory, Theoretical Numerical Analysis, Wave Propagation, Signal Processing and Tomography

8) Luis A.Caffarelli Department of Mathematics The University of Texas at Austin Austin,Texas 78712-1082 512-471-3160 e-mail: [email protected] Partial Differential Equations 9) George Cybenko Thayer School of Engineering Dartmouth College 8000 Cummings Hall, Hanover,NH 03755-8000 603-646-3843 (X 3546 Secr.) e-mail: [email protected] Approximation Theory and Neural Networks

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

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

756

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

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

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

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

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

32) Manfred Tasche Department of Mathematics University of Rostock D-18051 Rostock,Germany [email protected] Numerical Fourier Analysis, Fourier Analysis,Harmonic Analysis, Signal Analysis, Spectral Methods, Wavelets, Splines, Approximation Theory

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

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

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

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

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

757

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

Lotharstr.65,D-47048 Duisburg,Germany e-mail:[email protected] Fourier Analysis,Computer-Aided Geometric Design, Computational Complexity, Multivariate Approximation Theory, Approximation and Interpolation Theory

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

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

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

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

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

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

19) Weimin Han Department of Mathematics University of Iowa Iowa City, IA 52242-1419 319-335-0770 e-mail: [email protected] Numerical analysis, Finite element method, Numerical PDE, Variational inequalities, Computational mechanics NEW MEMBERS 39)Xing-Biao Hu Institute of Computational Mathematics AMSS, Chinese Academy of Sciences Beijing, 100190, CHINA [email protected] Computational Mathematics

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

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J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 18, NO.5, 2015, COPYRIGHT 2015 EUDOXUS PRESS, LLC

An umbral calculus approach to poly-Cauchy polynomials with a q parameter Dae San Kim Department of Mathematics, Sogang University Seoul 121-741, Republic of Korea [email protected]

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

Takao Komatsu

∗

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

Jong-Jin Seo Department of Applied Mathematics, Pukyong National University Pusan 608-739, Republic of Korea [email protected] MR Subject Classifications: 05A15, 05A40, 11B68, 11B75, 65Q05

Abstract In this paper, we investigate the properties of the poly-Cauchy polynomials with a q parameter which were studied by the third named author, and give various identities with Bernoulli polynomials, Korobov polynomials, Stirling numbers, Frobenius-Euler polynomials, falling and rising factorials by an umbral calculus approach.

∗

The third author was supported in part by the Grant-in-Aid for Scientific research (C) (No.22540005), the Japan Society for the Promotion of Science.

1

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1

Introduction

Let n, k be integers with n ≥ 0, and let q be a real number with q 6= 0. The poly-Cauchy (k) (k) numbers with a q parameter of the first kind cn,q and of the second kind b cn,q are defined by ∫ 1 ∫ 1 (k) (x1 x2 . . . xk − z)(x1 x2 . . . xk − q − z) cn,q (z) = ... 0 0 | {z } k

. . . (x1 x2 . . . xk − (n − 1)q − z)dx1 dx2 . . . dxk and ∫ b c(k) n,q (z)

= |

0

∫

1

... {z

0

1

}

(−x1 x2 . . . xk + z)(−x1 x2 . . . xk − q + z)

k

. . . (−x1 x2 . . . xk − (n − 1)q + z)dx1 dx2 . . . dxk , respectively ([9]). The generating function of the poly-Cauchy polynomials with a q (k) (k) parameter of the first kind cn,q (z) and of the second kind cˆn,q (z) are given by ( ) ∑ ∞ ln(1 + qt) tn −z/q (k) (1 + qt) Lif k = cn,q (z) , q n! n=0 and

( z/q

(1 + qt)

Lif k

ln(1 + qt) − q

) =

∞ ∑

cˆ(k) n,q (z)

n=0

tn , n!

respectively ([9, Theorem 6]), where Lif k (z) :=

∞ ∑

zm m!(m + 1)k m=0

is the polylogarithm factorial function (or simply polyfactorial function), which is intro(k) (k) (k) (k) cn,1 (z) = b cn (z) are the poly-Cauchy duced in [8, 9]. If q = 1, then cn,1 (z) = cn (z) and b polynomials of the first kind and of the second kind, respectively ([2]). Notice that z is (k) (k) (k) (k) replaced by −z in [2]. If q = 1 and z = 0, then cn,1 (0) = cn and b cn,1 (0) = b cn are the poly-Cauchy numbers of the first kind and of the second kind, respectively ([8]). If (1) (1) q = k = 1 and z = 0, then cn,1 (0) = cn and b cn,1 (0) = b cn are the classical Cauchy numbers of the first kind and of the second kind, respectively (see e.g. [1, 15]). The concept about the poly-Cauchy numbers and polynomials have been introduced, and the characteristic and combinatorial properties have been investigated ([2, 8, 9, 10, 11, 12, 13]). The falling factorial is defined by (x)n = x(x − 1) · · · (x − n + 1) =

n ∑

s(n, l)xl ,

l=0

2

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where s(n, l) is the signed Stirling number of the first kind. The rising factorial is defined by n ∑ (n) (x) = x(x + 1) · · · (x + n − 1) = (−1)n−l s(n, l)xl . l=0

Recently, the method of umbral calculus has been introduced to yield various identities in the study of poly-Cauchy numbers ([6, ?]) as well as that of poly-Bernoulli polynomials ([5]). In this paper, we investigate the properties of the poly-Cauchy polynomials with a q parameter of the first kind and of the second kind with umbral calculus viewpoint, give various identities with Bernoulli polynomials, Korobov polynomials, Stirling numbers, Frobenius-Euler polynomials, falling and rising factorials.

2

Umbral calculus

Let C be the complex number field and let F be the set of all formal power series in the variable t: } { ∞ ∑ ak k (1) F = f (t) = t ak ∈ C . k! k=0

∗

Let P = C[x] and let P be the vector space of all linear functionals on P. hL|p(x)i is the action of the linear functional L on the polynomial p(x), and we recall that the vector space operations on P∗ are defined by hL + M |p(x)i = hL|p(x)i + hM |p(x)i, hcL|p(x)i = c hL|p(x)i, where c is a complex constant in C. For f (t) ∈ F, let us define the linear functional on P by setting hf (t)|xn i = an , (n ≥ 0). (2) In particular,

k n t |x = n!δn,k

(n, k ≥ 0),

(3)

where δn,k is the Kronecker’s symbol. ∑ hL|xk i k For fL (t) = ∞ t , we have hfL (t)|xn i = hL|xn i. That is, L = fL (t). The map k=0 k! L 7→ fL (t) is a vector space isomorphism from P∗ onto F. Henceforth, F denotes both the algebra of formal power series in t and the vector space of all linear functionals on P, and so an element f (t) of F will be thought of as both a formal power series and a linear functional. We call F the algebra and the umbral calculus is the study of ( umbral ) umbral algebra. The order O f (t) of a power series ( f (t)(6 ) = 0) is the smallest integer k k for which the of t does not vanish. If O f (t) = 1, then f (t) is called a delta ( coefficient ) series; f (t) is called an invertible series. For f (t), g(t) ∈ F with ( )if O f (t) =( 0, then ) O f (t) = 1 and O g(t) = 0, there exists a unique sequence sn (x) (deg sn (x) = n) such

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

(4)

3

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and

∞ ∑

f (t) =

k=0

 tk , f (t)|x k! k

∞ ∑

k  xk p(x) = t |p(x) k! k=0

(5)

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

(6)

([16, Theorem 2.2.5]). Thus, by (5), we get tk p(x) = p(k) (x) =

dk p(x) dxk

Sheffer sequences are characterized in the generating function ([16, Theorem 2.3.4]). ( ) Lemma 1 The sequence sn (x) is Sheffer for g(t), f (t) if and only if ∑ sk (y) 1 ¯ ) eyf (t) = tk ¯ k! g f (t) k=0 ∞

(

(y ∈ C) ,

where f¯(t) is the compositional inverse of f (t). ( ) For sn (x) ∼ g(t), f (t) , we have the following equations ([16, Theorem 2.3.7, Theorem 2.3.5, Theorem 2.3.9]): f (t)sn (x) = nsn−1 (x) (n ≥ 0), n ∑ 1 D ( ¯ )−1 ¯ j n E j g f (t) f (t) |x x , sn (x) = j! j=0 n ( ) ∑ n sn (x + y) = sj (x)pn−j (y) , j j=0

(7) (8) (9)

where pn (x) = g(t)sn (x). ( ) ( ) Assume that pn (x) ∼ 1, f (t) and qn (x) ∼ 1, g(t) . Then the transfer formula ([16, Corollary 3.8.2]) is given by ( )n f (t) x−1 pn (x) (n ≥ 1). qn (x) = x g(t) ( ) ( ) For sn (x) ∼ g(t), f (t) and rn (x) ∼ h(t), l(t) , assume that sn (x) =

n ∑

Cn,m rn (x) (n ≥ 0) ,

m=0

Then we have ([16, p.132]) Cn,m

1 = m!

+ * ( ) h f¯(t) ( ¯ )m n ( ) l f (t) x . g f¯(t)

(10)

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3

Main results (k)

It is immediate to see that cn,q (z) is the Sheffer sequence for the pair ( ) 1 e−qt − 1 , f (t) = g(t) = Lif k (−t) q (

or c(k) n,q (z)

∼

1 e−qt − 1 , Lif k (−t) q

) ,

(11)

(k)

and b cn,q (z) is that for the pair ( g(t) =

1 eqt − 1 , f (t) = Lif k (−t) q (

or b c(k) n,q (z)

∼

eqt − 1 1 , Lif k (−t) q

) ,

) (12)

because f¯(t) = − ln(1 + qt)/q and f¯(t) = ln(1 + qt)/q, respectively in Lemma 1. (k) (k) (k) (k) cn,q (0)) are called the poly-Cauchy cn,q = b When x = 0, cn,q = cn,q (0) (respectively, b numbers of the first kind (respectively, the poly-Cauchy numbers of the second kind).

3.1

Explicit expressions

It is known that ( )(n) ( ) ( ) x x x x = + 1 ··· + n − 1 ∼ (1, 1 − e−qt ) . q q q q So, ( ) + x (n) −qt k = (1 − e ) q *( )k ( )(n) + e−qt − 1 x = , (−q)n q q *

n!δn,k

yielding that

( )(n) ( −qt ) x e −1 (−q) ∼ 1, . q q n

Similarly, by

( ) ( ) ( ) x x x x = − 1 ··· − n + 1 ∼ (1, 1 − e−qt ) q n q q q

5

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we get

( ) + x = (eqt − 1)k q n *( )k ( ) + eqt − 1 n x = , q q q n *

n!δn,k

yielding that

( qt ) ( ) e −1 x q ∼ 1, . q n q First, we shall show the following results by the different methods derived from the umbral calculus, which have been already obtained in [9, Theorem 5]. n

Theorem 1 For integers n and k with n ≥ 0, we have n m ( ) ∑ ∑ m (−x)j (k) n−m , cn,q (x) = s(n, m)q j (m − j + 1)k m=0 j=0 n m ( ) ∑ ∑ m (−x)j (k) m n−m b cn,q (x) = (−1) s(n, m)q j (m − j + 1)k m=0 j=0 Proof. Since 1 n c(k) n,q (x) = (−q) Lif k (−t) we have c(k) n,q (x)

) ( )(n) ( −qt x e −1 ∼ 1, , q q

( )(n) x = (−q) Lif k (−t) q ( )m n ∑ x n n−m = (−q) Lif k (−t) (−1) s(n, m) q m=0 n

=q

n

n ∑ m=0 n ∑

(−q −1 )m s(n, m)Lif k (−t)xm m ∑

(−1)i i m tx k i!(i + 1) m=0 i=0 ( ) n m i m ∑ ∑ (−1) i xm−i = qn (−q −1 )m s(n, m) k (i + 1) m=0 i=0 ( ) n ∑ m m−j m ∑ (−1) j = qn (−q −1 )m s(n, m) xj k (m − j + 1) m=0 j=0 n m ( ) ∑ ∑ m (−x)j n−m = s(n, m)q . j (m − j + 1)k m=0 j=0

= qn

(−q −1 )m s(n, m)

6

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Since

1 b c(k) (x) = (−q)n Lif k (−t) n,q

we have b c(k) n,q (x)

( ) ( qt ) x e −1 ∼ 1, , q n q

( ) x = q Lif k (−t) q n ( )m n ∑ x n = q Lif k (−t) s(n, m) q m=0 n

= =

n ∑ m=0 n ∑ m=0 n ∑

s(n, m)q n−m Lif k (−t)xm s(n, m)q

n−m

i=0 m ∑

(i +

(m)

i 1)k

(−1)m−j

xm−i (m) j

xj k (m − j + 1) m=0 j=0 n m ( ) ∑ ∑ (−x)j m m n−m = (−1) s(n, m)q . k j (m − j + 1) m=0 j=0 =

s(n, m)q n−m

m ∑ (−1)i

Proof of Theorem 1 (A different version). As the second different proof, we use the conjugation formula (8). (k) Since g(t) = 1/Lif k (−t) and f¯(t) = − ln(1 + qt)/q for sn = cn,q (x), by (8) we get * )( )j + ( n ∑ 1 ln(1 + qt) ln(1 + qt) n c(k) Lif k − x xj . n,q (x) = j! q q j=0

Here, * )( )j + ( ln(1 + qt) n ln(1 + qt) − Lif k x q q D( )m+j n E (−1)j = ln(1 + qt) x m!(m + 1)k q m+j m=0 n−j ∑

n−j ∑

n−m−j ∑

 (−1)j (m + j)! l+m+j n s(l + m + j, m + j) (qt) = |x m!(m + 1)k q m+j l=0 (l + m + j)! m=0 n−m−j ∑ (m + j)! l+m+j (−1)j = q s(l + m + j, m + j)(l + m + j)!δl+m+j,n k m+j m!(m + 1) q (l + m + j)! m=0 l=0 n−j ∑

=

n−j ∑ (−1)j (m + j)!q n s(n, m + j) . k q m+j m!(m + 1) m=0

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Therefore,

( n−j ) ( ) ∑ (−1)j m+j q n m s(n, m + j) xj c(k) n,q (x) = k m+j (m + 1) q m=0 j=0 ( ) ( ) n n ∑ ∑ (−1)j mj q n s(n, m) xj = k qm (m − j + 1) j=0 m=j n m ( ) ∑ ∑ m (−x)j n−m = s(n, m)q . j (m − j + 1)k m=0 j=0 n ∑

(k) Since g(t) = 1/Lif k (−t) and f¯(t) = ln(1 + qt)/q for sn = b cn,q (x), by (8) we get * ( )( )j + n ∑ 1 − ln(1 + qt) ln(1 + qt) n b c(k) Lif k x xj . n,q (x) = j! q q j=0

Here, * ( )( )j + − ln(1 + qt) ln(1 + qt) n Lif k x q q =

D( )m+j n E (−1)m ln(1 + qt) x k q m+j m!(m + 1) m=0 n−j ∑

n−m−j ∑ (m + j)! l+m+j (−1)m q s(l + m + j, m + j)(l + m + j)!δl+m+j,n = k q m+j m!(m + 1) (l + m + j)! m=0 l=0 n−j ∑

n−j ∑ (−1)m (m + j)!q n s(n, m + j) . = m!(m + 1)k q m+j m=0

Therefore,

( n−j ) ( ) ∑ (−1)m m+j q n m b c(k) s(n, m + j) xj n,q (x) = k q m+j (m + 1) m=0 j=0 ( n ) ( ) n ∑ ∑ (−1)m−j mj q n = s(n, m) xj k qm (m − j + 1) j=0 m=j n m ( ) ∑ ∑ m (−x)j m n−m = . (−1) s(n, m)q j (m − j + 1)k m=0 j=0 n ∑

We shall show the formulae for the poly-Cauchy polynomials with a q parameter in (r) terms of Bernoulli polynomials Bn (x) of order r, defined by ( )r ∞ (r) ∑ t Bn (x) n xt e = t (13) et − 1 n! n=0 8

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(see e.g. [16, Section 2.2]). Observe that for any formal power series g(t) = (x) n n n (n ≥ 0, a 6= 0) p(x) = g(t)x , g(at)x = a p a because t as the differential operator, ∞ ∑

∑∞

m m=0 bm t /m!

tm n x m! m=0 ( ) ∞ ∑ n n−m = bm x m m=0

p(x) =

bm

and ∞ ∑

(at)m n x m! m=0 ( ) ∞ ∑ m n = bm a xn−m m m=0 ( )( ) ∞ ∑ n x n−m n =a bm m a m=0 (x) = an p . a Theorem 2 For integers n and k with n ≥ 1, we have (n−1)(n−l) n−j n ∑ ∑ l j (n) q l Bl xj , c(k) (−1)j n,q (x) = k (n − l − j + 1) j=0 l=0 ( )( ) n−j n l n−1 n−l ∑ ∑ (−1) l j (n) n b c(k) (−1)j q l B l xj . n,q (x) = (−1) k (n − l − j + 1) j=0 l=0 g(at)xn =

bm

Proof. Since

( −qt ) 1 e −1 (k) c (x) ∼ 1, Lif k (−t) n,q q n and x ∼ (1, t), for n ≥ 1 we have ( )n 1 t (k) c (x) = x x−1 xn Lif k (−t) n,q (e−qt − 1)/q ( )n −qt n = (−1) x −qt xn−1 e −1 ) n−1 ( ∑ n−1 (n) n = (−1) Bl (−q)l xn−l l l=0 ( ) n ∑ n−1 (n) n = (−1) Bl (−q)l xn−l . l l=0 9

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Thus, c(k) n,q (x)

n

= (−1)

) n ( ∑ n−1 l

l=0

(n)

Bl (−q)l Lif k (−t)xn−l

( )(n−l) n ∑ n−l ∑ (−1)m n−1 (n) n l m (−q)l Bl xn−l−m = (−1) k (m + 1) l=0 m=0 ( )(n−l) n ∑ n−l ∑ (−1)n−j n−1 l j (n) n q l Bl xj = (−1) k (n − l − j + 1) l=0 j=0 (n−1)(n−l) n−j n ∑ ∑ l j (n) = (−1)j q l Bl xj . k (n − l − j + 1) j=0 l=0 Since

1 b c(k) (x) ∼ Lif k (−t) n,q

(

eqt − 1 1, q

)

and xn ∼ (1, t), for n ≥ 1 we have )n t x−1 xn (eqt − 1)/q ( )n qt = x qt xn−1 e −1 ) n−1 ( ∑ n−1 (n) Bl q l xn−l = l l=0 ( ) n ∑ n−1 (n) = Bl q l xn−l . l l=0

1 b c(k) (x) = x Lif k (−t) n,q

(

Thus, b c(k) n,q (x)

=

) n ( ∑ n−1 l=0

l

(n)

Bl q l Lif k (−t)xn−l

( )(n−l) n−l ∑ (−1)m n−1 (n) l m = q l Bl xn−l−m k (m + 1) m=0 ( )( ) n−l n−l−j n−1 n−l ∑ (−1) l j (n) q l Bl xj = sumnl=0 k (n − l − j + 1) j=0 ( )( ) n−j n l n−1 n−l ∑ ∑ (−1) l j (n) = (−1)n q l B l xj . (−1)j k (n − l − j + 1) j=0 l=0 sumnl=0

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3.2

Sheffer identities

Theorem 3 n ( ) ∑ n

( )(n−j) y , + y) = j q j=0 ( ) n ( ) ∑ n n−j (k) y (k) b cn,q (x + y) = q b cj,q (x) . j q n−j j=0 c(k) n,q (x

(k) (−q)n−j cj,q (x)

(k)

Proof. Put sn = cn,q with (11) in (9). Since 1 c(k) (x) Lif k (−t) n,q ( )(n) ( −qt e − 1) x n ∼ 1, . = (−q) q q

pn (x) =

Thus, c(k) n,q (x

+ y) =

n ( ) ∑ n j=0

j

(k) cj,q (x)(−q)n−j

( )(n−j) y . q

(k)

Put sn = b cn,q with (12) in (9). Since 1 b c(k) (x) Lif k (−t) n,q ( ) ( eqt − 1 ) x n ∼ 1, . =q q n q

pn (x) =

Thus, b c(k) n,q (x

3.3

( ) n ( ) ∑ n (k) y n−j + y) = b cj,q (x)q . j q n−j j=0

Recurrence relations

Theorem 4 For integers n and k with n ≥ 0, we have (k)

(k) c(k) n,q (x − q) − cn,q (x) = nqcn−1,q (x) , (k)

cn−1,q (x) . c(k) b c(k) n,q (x) = nqb n,q (x + q) − b (k)

Proof. Put sn = cn,q in (7). Then ( −qt ) e − 1 (k) (k) cn,q (x) = ncn−1,q (x) . q 11

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So, we get the first relation. (k) Put sn = b cn,q in (7). Then (

) eqt − 1 (k) (k) cn−1,q (x) . b cn,q (x) = nb q

So, we get the second relation.

3.4

Differentiations

The following results ([9, Proposition 2]) can be also obtained by using the umbral calculus. Theorem 5 For integers n and k with n ≥ 0, we have ∑ (−q)n−l−1 (k) d (k) cn,q (x) = −n! cl,q (x) , dx (n − l)l! l=0 n−1

∑ (−q)n−l−1 (k) d (k) b c (x) = n! b c (x) . dx n,q (n − l)l! l,q l=0 n−1

We use a formula for sn (x) in terms of sl (x). ( ) Lemma 2 For sn (x) ∼ g(t), f (t) , n−1 ( ) ∑  d n ¯ sn (x) = f (t)|xn−l sl (x) . dx l l=0

(k) Proof of Theorem 5. Since f¯(t) = − ln(1 + qt)/q for sn = cn,q , by Lemma 2 * + n−1 ( ) ∑ d (k) n ln(1 + qt) n−l (k) c (x) = − cl,q (x) x dx n,q l q l=0 *∞ + n−1 ( ) ∑ (−1)j−1 q j tj n−l (k) 1∑ n cl,q (x) =− x q l=0 l j j=1 ( ) n−1 1 ∑ n (−1)n−l−1 q n−l (k) =− (n − l)!cl,q (x) q l=0 l n−l

= −n!

n−1 ∑ (−q)n−l−1 l=0

(n − l)l!

(k)

cl,q (x) .

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(k) Since f¯(t) = ln(1 + qt)/q for sn = b cn,q , by Lemma 2 * + n−1 ( ) ∑ n ln(1 + qt) n−l (k) d (k) b c (x) = b cl,q (x) x l dx n,q q l=0 + *∞ n−1 ( ) ∑ 1∑ n (−1)j−1 q j tj n−l (k) = b cl,q (x) x q l=0 l j j=1 n−1 ( ) 1 ∑ n (−1)n−l−1 q n−l (k) = (n − l)!b cl,q (x) q l=0 l n−l

= n!

n−1 ∑ (−q)n−l−1 l=0

3.5

(n − l)l!

(k)

b cl,q (x) .

Recurrence relations including Cauchy numbers

Theorem 6 n−1 ( ) ) 1 ∑ n l ( (k−1) (k) q cl cn−l (x + q) − cn−l (x + q) , = + q) + n l=0 l n−1 ( ) ) 1 ∑ n l ( (k−1) (k) (k) (k) b cn,q (x) = xb cn−1,q (x − q) + q cl b cn−l (x − q) − b cn−l (x − q) . n l=0 l

c(k) n,q (x)

(k) −xcn−1,q (x

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Proof. By (2), + *∞ ∑ (k) tl (k) (y) = cl,q (y) xn cn,q l! * l=0 (

) + n = (1 + qt)−y/q Lif k x * + ( ) ln(1 + qt) n−1 −y/q = (1 + qt) Lif k x · x q * ( + ( )) ln(1 + qt) n−1 −y/q = ∂t (1 + qt) Lif k x q * + ( ) ( ) ln(1 + qt) n−1 −y/q = ∂t (1 + qt) Lif k x q + * ( ( )) ln(1 + qt) n−1 + (1 + qt)−y/q ∂t Lif k x q * + ( ) ln(1 + qt) n−1 = −y (1 + qt)−(y+q)/q Lif k x q * + ( ln(1+qt) ) ( ln(1+qt) ) − Lif Lif k k−1 qt q q + q (1 + qt)−(y+q)/q xn−1 . ln(1 + qt) qt ln(1 + qt) q

Since the generating function of the classical Cauchy numbers of the first kind cn is given by ∞ ∑ t tn = cn ln(1 + t) n=0 n!

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(see e.g. [1, 8, 15]), we have (k)

c(k) n,q (y) = −ycn−1,q (y + q) *

Lif k−1 (1 + qt)−(y+q)/q

+q

=

(k) −ycn−1,q (y

+ q) + q

Lif k−1 (1 + qt)−(y+q)/q (k)

n−1 ( ∑

− Lif k

qt

l

l=0

= −ycn−1,q (y + q) +

q

) n−1 ( ∑ n−1

* ×

( ln(1+qt) )

q l cl

( ln(1+qt) ) q

+ ( ln(1+qt) ) n−1 ( ∑ n − 1) q q l cl xn−l−1 l l=0

− Lif k

qt )

( ln(1+qt) ) ( )+ n−l x q t n−l

n−1 1 l q cl n−l l

+ ) ( ) ( ) ln(1 + qt) ln(1 + qt) n−l −(y+q)/q × (1 + qt) Lif k−1 − Lif k x q q n−1 ( ) ) 1 ∑ n l ( (k−1) (k) (k) = −ycn−1,q (y + q) + q cl cn−l (y + q) − cn−l (y + q) . n l=0 l *

l=0

(

Thus, we get the first relation.

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Similarly, by (2), + *∞ ∑ (k) tl b c(k) b cl,q (y) xn n,q (y) = l! * l=0 (

) + n = (1 + qt)y/q Lif k x + * ( ) − ln(1 + qt) = (1 + qt)y/q Lif k x · xn−1 q * ( + ( )) − ln(1 + qt) = ∂t (1 + qt)y/q Lif k xn−1 q * + ) ( ( ) − ln(1 + qt) ∂t (1 + qt)y/q Lif k = xn−1 q * + ( ( )) − ln(1 + qt) + (1 + qt)y/q ∂t Lif k xn−1 q * + ) ( − ln(1 + qt) = y (1 + qt)(y−q)/q Lif k xn−1 q + * ( − ln(1+qt) ) ( − ln(1+qt) ) − Lif Lif k k−1 qt q q xn−1 . + q (1 + qt)(y−q)/q ln(1 + qt) qt ln(1 + qt) − q

Since

) n−1 ( ∑ n − 1 l n−l−1 qt n−1 x = q cl x , ln(1 + qt) l l=0

we have b c(k) n,q (y)

=

(k) yb cn−1,q (y

* ×

− q) + q

) n−1 ( ∑ n−1 l=0

Lif k−1 (1 + qt)(y−q)/q

l

q l cl

( − ln(1+qt) ) q

− Lif k

qt

( − ln(1+qt) ) ( )+ xn−l q t n−l

) n−1 ( ∑ n−1 1 l = − q) + q cl l n−l l=0 + * ) ( ( ) ( ) − ln(1 + qt) − ln(1 + qt) × (1 + qt)(y−q)/q Lif k−1 − Lif k xn−l q q n−1 ( ) ) 1 ∑ n l ( (k−1) (k) (k) = yb cn−1,q (y − q) + q cl b cn−l (y − q) − b cn−l (y − q) . n l=0 l (k) yb cn−1,q (y

Thus, we get the second relation. 16

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3.6

More recurrence relations

Theorem 7 For integers n and k with n ≥ 1, we have ( ) ( ) n ∑ n (k) n − 1 (k−1) m−1 (−q) (m − 1)! c = (−q) (m − 1)! c , m n−m,q m=1 m − 1 n−m,q m=1 ( ) ( ) n n ∑ ∑ n (k) n − 1 (k−1) m−1 m−1 (−q) (m − 1)! b cn−m,q = (−q) (m − 1)! b cn−m,q . m m − 1 m=1 m=1 n ∑

m−1

Proof. We shall compute *

ln(1 + qt) Lif k q

(

) + ln(1 + qt) n x q

in two different ways. On the one hand, * ( ) + ln(1 + qt) n ln(1 + qt) Lif k x q q * + ) ( 1 ln(1 + qt) = Lif k ln(1 + qt)xn q q * + ) ∞ ( ln(1 + qt) ∑ (−1)m−1 (qt)m n 1 Lif k x = q q m m=1 * + ) ∑ ( ( ) n 1 ln(1 + qt) m−1 m n n−m = Lif k (−1) (m − 1)!q x q q m m=1 + ) ( )* ( n ∑ ln(1 + qt) n−m m−1 m−1 n = (−1) (m − 1)!q Lif k x q m m=1 + * ( ) n ∞ i ∑ ∑ (k) t n−m m−1 m−1 n = (−1) (m − 1)!q ci,q x i! m m=1 i=0 ( ) n ∑ n (k) = (−1)m−1 (m − 1)!q m−1 cn−m,q . m m=1

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On the other hand, ) + ln(1 + qt) n ln(1 + qt) Lif k x q q *∫ ( ( ))0 + t ln(1 + qs) ln(1 + qs) = Lif k ds xn q q 0 *∫ ( ln(1+qs) ) + t Lif k−1 q = ds xn 1 + qs 0 *∫ ( ∞ )( ∞ ) + t ∑ ∑ (k−1) sj i (−qs) = cj,q ds xn j! 0 i=0 j=0 *∫ ( ∞ r ) + t r ∑∑ (k−1) s = (−q)r−j cj,q ds xn j! 0 r=0 j=0 + *∞ r ∑∑ tr+1 n (k−1) = (−q)r−j cj,q x j!(r + 1)

*

(

r=0 j=0

=

n−1 ∑

(k−1)

(−q)n−j−1 cj,q

j=0

= (n − 1)!

n! j!n

n−1 ∑ (−q)n−m−1 (k−1) cm,q . m! m=0

Thus, for n ≥ 1, we obtain n ∑

m−1

(−1)

(m − 1)!q

m=1

m−1

( ) n ∑ (−q)m−1 (k−1) n (k) cn−m,q = (n − 1)! cn−m,q . m (n − m)! m=1

Similarly, we shall compute *

( ) + ln(1 + qt) ln(1 + qt) n − Lif k − x q q

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in two different ways. On the one hand, it is equal to + * ( ) 1 ln(1 + qt) − Lif k − ln(1 + qt)xn q q + ( ) ( )* n ∑ n ln(1 + qt) n−m = Lif k − (−1)m (m − 1)!q m−1 x q m m=1 * + ( ) ∑ n ∞ i ∑ (k) t m m−1 n = (−1) (m − 1)!q b ci,q xn−m m i! m=1 i=0 ( ) n ∑ (k) m m−1 n = (−1) (m − 1)!q b cn−m,q . m m=1 On the other hand, it is equal to *∫ ( ( ))0 + t − ln(1 + qs) − ln(1 + qs) Lif k ds xn q q 0 *∫ ( − ln(1+qs) ) + t −Lif k−1 q = ds xn 1 + qs 0 (∞ )( ∞ ) + *∫ t ∑ ∑ (k−1) sj (−qs)i b cj,q ds xn = (−1) j! 0 i=0 j=0 + * ∞ r ∑∑ tr+1 n (k−1) = − (−q)r−j b cj,q x j!(r + 1) r=0 j=0

=−

n−1 ∑

(k−1)

(−q)n−j−1b cj,q

j=0

n! j!n

n ∑ (−q)m−1 (k−1) = −(n − 1)! b c . (n − m)! n−m,q m=1

Thus, for n ≥ 1, we obtain n ∑ m=1

3.7

m−1

(−1)

(m − 1)!q

m−1

( ) n ∑ (−q)m−1 (k−1) n (k) b cn−m,q = (n − 1)! b cn−m,q . m (n − m)! m=1

Some relations with Korobov polynomials

The Korobov polynomials of the first kind Kn,q (x) (q 6= 0) ([14]) are given by ∑ qt(1 + t)x tj = K (x) . j,q (1 + t)q − 1 j! j=0 ∞

(14)

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Theorem 8 For integers n and k with n ≥ 0, we have c(k) n,q (x) =

n ∑ n−i n−i−l ∑ ∑

n−i−l−m

( ( )( ) ) x n n − l n−l−m (n − i − l)! (k−1) cl,q cm,q Ki, 1 − q , q m!(n − i − l + 1) q l i

n−i−l−m

( ) ( )( ) x n n − l n−l−m (n − i − l)! (k−1) b cl,q b cm,q Ki,− 1 . q q m!(n − i − l + 1) q l i

(−1)

i=0 l=0 m=0

b c(k) n,q (x) =

n ∑ n−i n−i−l ∑ ∑ i=0 l=0 m=0

(−1)

(k)

Proof. By the definition of cn,q (x), * ( ) + ln(1 + qt) n c(k) (1 + qt)−y/q Lif k x n,q (y) = q * ) + ( ) ( (1 + qt)−y/q ln(1 + qt) ln(1 + qt) q (1 + qt)1/q − 1 n = Lif k x . (1 + qt)1/q − 1 q q ln(1 + qt) ( ) ∞ ∑ q (1 + qt)1/q − 1 tl cl,q , = ln(1 + qt) l! l=0

Since

we get + ) ∑ ( ∞ ( ) −y/q ln(1 ln(1 + qt) n (1 + qt) + qt) c(k) Lif k cl,q xn−l n,q (y) = (1 + qt)1/q − 1 q q l l=0 * + ( ) ))0 ( ( ∫ ∞ t ∑ n (1 + qt)−y/q ln(1 + qs) ln(1 + qs) n−l = cl,q Lif k ds x 1/q − 1 l (1 + qt) q q 0 l=0 * + ∞ ( ) ∞ r ∑ n (1 + qt)−y/q ∑ ∑ tr+1 n−l r−m (k−1) = cl,q (−q) c x m,q 1/q − 1 l (1 + qt) m!(r + 1) r=0 m=0 l=0 * + ∞ ( ) ∞ r r ∑ n t(1 + qt)−y/q ∑ ∑ t = cl,q (−q)r−m c(k−1) xn−l m,q 1/q − 1 (1 + qt) m!(r + 1) l r=0 m=0 l=0 * + (n−l) ( ) ∞ n−l r −y/q ∑ n ∑∑ r! t(1 + qt) n−l−r r = cl,q (−q)r−m c(k−1) . x m,q 1/q − 1 l m!(r + 1) (1 + qt) r=0 m=0 *

l=0

Replacing t by qt, q by 1/q and x by −y/q in (14), we have ∑ t(1 + qt)−y/q = Kj, 1 q (1 + qt)1/q − 1 j=0 ∞

(

y − q

)

(qt)j . j!

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Hence, c(k) n,q (y) =

∞ ( ) ∑ n l=0

Since

l

cl,q

*∞ ∑

+ *∞ ( ) j ∑ r! y (qt) n−l−r r−m (k−1) r . (−q) cm,q K 1 − x m!(r + 1) j=0 j, q q j! r=0 m=0 (

Kj, 1 q

j=0

(n−l)

n−l ∑ r ∑

−

y q

)

+ ( ) y (qt) n−l−r = Kn−l−r, 1 − q n−l−r , x q j! q j

we have ( ) ( )( ) y n n − l n−l−m r! (k−1) cl,q cm,q Kn−l−r, 1 − = (−1) q q m!(r + 1) q l r l=0 r=0 m=0 ( ) ( )( ) n ∑ n−l ∑ n−l ∑ y n − l n−l−m r! (k−1) r−m n cl,q cm,q Kn−l−r, 1 − = (−1) q q m!(r + 1) q l r l=0 m=0 r=m ( ) ( )( ) n ∑ n−l n−l−m ∑ ∑ y n − l n−l−m (n − l − i)! n−l−m−i n (k−1) = (−1) cl,q cm,q Ki, 1 − q q m!(n − l − i + 1) q l i l=0 m=0 i=0 ( )( ) ( ) n ∑ n−i n−i−l ∑ ∑ n − l n−l−m (n − i − l)! y n−i−l−m n (k−1) (−1) q = cl,q cm,q Ki, 1 − . q l i m!(n − i − l + 1) q i=0 l=0 m=0

c(k) n,q (y)

n ∑ n−l ∑ r ∑

r−m

Thus, we obtain the first relation. (k) Similarly, by the definition of cn,q (x), * ( ) + ln(1 + qt) n b c(k) (1 + qt)y/q Lif k − x n,q (y) = q * ) + ) ( ( − ln(1 + qt) q 1 − (1 + qt)−1/q n (1 + qt)y/q − ln(1 + qt) Lif k x . = (1 + qt)−1/q − 1 q q ln(1 + qt) Since

( ) ∞ ∑ q 1 − (1 + qt)−1/q tl b cl,q , = ln(1 + qt) l! l=0

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we get

+ * ( ) ∞ ( ) y/q ∑ − ln(1 + qt) n (1 + qt) − ln(1 + qt) b cl,q Lif k b c(k) xn−l n,q (y) = −1/q − 1 (1 + qt) q q l l=0 * + ( ))0 ∫ t( ∞ ( ) ∑ n (1 + qt)y/q − ln(1 + qs) − ln(1 + qs) n−l = b cl,q Lif k ds x l (1 + qt)−1/q − 1 0 q q l=0 * + ∞ ( ) ∞ r ∑ n −t(1 + qt)y/q ∑ ∑ tr r−m (k−1) = b cl,q xn−l (−q) b c m,q −1/q − 1 l (1 + qt) m!(r + 1) r=0 m=0 l=0 * + (n−l) ( ) ∞ n−l r y/q ∑ n ∑∑ r! −t(1 + qt) n−l−r r = b cl,q (−q)r−mb c(k−1) . x m,q l m!(r + 1) (1 + qt)−1/q − 1 l=0

r=0 m=0

Replacing t by qt, q by −1/q and x by y/q in (14), we have ( ) ∞ ∑ y (qt)j −t(1 + qt)y/q 1 = K . j,− q (1 + qt)−1/q − 1 q j! j=0 Hence,

(n−l) ( ) ∞ ( ) n−l ∑ r ∑ ∑ r! n y r−m (k−1) r = b cl,q (−q) b cm,q Kn−l−r,− 1 q n−l−r q l m!(r + 1) q r=0 m=0 l=0 ( )( ) ( ) n−l r n ∑∑∑ n − l n−l−m r! y r−m n (k−1) (−1) q = b cl,q b cm,q Kn−l−r,− 1 q l r m!(r + 1) q l=0 r=0 m=0 ( )( ) ( ) n−l ∑ n−l n ∑ ∑ n − l n−l−m r! y r−m n (k−1) (−1) q = b cl,q b cm,q Kn−l−r,− 1 q l r m!(r + 1) q l=0 m=0 r=m ( )( ) ( ) n−l n−l−m n ∑ ∑ ∑ n − l n−l−m (n − l − i)! y n−l−m−i n (k−1) (−1) q = b cl,q b cm,q Ki,− 1 q l i m!(n − l − i + 1) q l=0 m=0 i=0 ( )( ) ( ) n ∑ n−i n−i−l ∑ ∑ n − l n−l−m (n − i − l)! y n−i−l−m n (k−1) = (−1) q b cl,q b cm,q Ki,− 1 . q l i m!(n − i − l + 1) q i=0 l=0 m=0

c(k) n,q (y)

Thus, we obtain the second relation.

3.8

Some relations including Stiring numbers of the first kind

Theorem 9 For integers n and k with n ≥ 0, we have ( ) n ∑ n j −m m ∑ (−1) q j (k) n s(n, m)(x + q)j , cn+1,q (x) = −xc(k) n,q (x + q) + q k (m − j + 1) j=0 m=j ( ) n n m−j −m m ∑ ∑ (−1) q j (k) n c(k) b cn+1,q (x) = xb s(n, m)(x − q)j . n,q (x − q) − q k (m − j + 1) j=0 m=j 22

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We use the following recurrence formula for Sheffer sequences ([16, Corollary 3.7.2]). ( ) Lemma 3 If sn (x) ∼ g(t), f (t) , then ( ) g 0 (t) 1 sn (x) . sn+1 = x − 0 g(t) f (t) (k)

Proof of Theorem 9. Consider the Sheffer sequence sn = cn,q in Lemma 3. By f (t) = (e−qt − 1)/q and g(t) = 1/Lif k (−t), we get 1/f 0 (t) = −eqt and Lif 0k (−t) g 0 (t) = . g(t) Lif k (−t) Thus, (k)

cn+1,q (x) = eqt

Lif 0k (−t) (k) c (x) − xc(k) n,q (x + q) . Lif k (−t) n,q

We obtain ( )(n) x q ( )(n) n ∑ x 0 n n−m = (−q) Lif k (−t) (−1) s(n, m) q m=0

Lif 0k (−t) (k) c (x) = Lif 0k (−t)(−q)n Lif k (−t) n,q

=q

n

= qn

n ∑ m=0 n ∑

(−q −1 )m s(n, m)Lif 0k (−t)xm m ∑

(−1)r r m tx k r!(r + 1) r=0 ( ) −m m

(−q −1 )m s(n, m)

m=0 n ∑ m ∑

(−1)m+r q r s(n, m)xm−r k (r + 1) m=0 r=0 ( ) n ∑ m j −m m ∑ (−1) q j = qn s(n, m)xj . k (m − j + 1) m=0 j=0 = qn

Therefore, we have (k)

cn+1,q (x) = −xc(k) n,q (x + q)

+ qn

n ∑ n ∑ (−1)j q −m j=0 m=j

(m) j

(m − j + 1)k

s(n, m)(x + q)j .

(k)

Similarly, consider the Sheffer sequence sn = b cn,q in Lemma 3. By f (t) = (eqt − 1)/q and g(t) = 1/Lif k (−t), we get (k) b cn+1,q (x)

=

0 −qt Lif k (−t) (k) b c (x) −e Lif k (−t) n,q

+ xb c(k) n,q (x − q) .

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We obtain

( ) x q n ( )m n ∑ x 0 n = q Lif k (−t) s(n, m) q m=0

Lif 0k (−t) (k) b cn,q (x) = Lif 0k (−t)q n Lif k (−t)

=q

n

= qn =q

n

n ∑ m=0 n ∑ m=0 n ∑

q −m s(n, m)Lif 0k (−t)xm q −m s(n, m) q

−m

s(n, m)

m=0

=q

n

m ∑ (−1)r r=0 m ∑

m=0 j=0

(−1)m−j

(m − j +

j k 1)

n ∑ n ∑ (−1)m−j q −m j=0 m=j

3.9

(m)

(m − j + ( ) −m m

Therefore, we have (k)

xm−r

r

(r + 1)k

j=0

n ∑ m ∑ (−1)m−j q

n b cn+1,q (x) = xb c(k) n,q (x − q) − q

(m)

j

1)k

xj

s(n, m)xj .

(m) j

(m − j + 1)k

s(n, m)(x − q)j .

Some relations with Bernoulli polynomials

By applying Lemma 1 about (13), for nonnegative integer r, we have )r ) (( t e −1 (r) ,t . Bn ∼ t

(15)

Theorem 10 For integers n and k with n ≥ 0, we have ) n (n−m ∑ ∑ n−m−l ∑ (n)( n − l ) (s) (k) (k) m i (s) b c q s(i + m, m)C cn,q (x) = (−1) l,q n−m−l−i,q Bm (x) , l i+m m=0 l=0 i=0 ) n (n−m ∑ n−m−l ∑ (n)( n − l ) ∑ (s) (k) (k) i (s) b b cn,q (x) = q s(i + m, m)Cl,q b cn−m−l−i,q Bm (x) . l i + m m=0 l=0 i=0 ∑ (s) (k) Proof. Put cn,q (x) = nm=0 Cn,m Bm (x) for (11) and (15). Then by (10) we get * ( )( − ln(1+qt)/q )s ( )m + 1 ln(1 + qt) e −1 ln(1 + qt) n Cn,m = Lif k − x m! q − ln(1 + qt)/q q + * )) ( )( ( )m n q 1 − (1 + qt)−1/q s ( (−q −1 )m ln(1 + qt) . = Lif k ln(1 + qt) x m! q ln(1 + qt) 24

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b (r) (r > 0) by Define C l,q

( ( ) )r ∞ l ∑ q 1 − (1 + qt)−1/q b (r) t , = C l,q ln(1 + qt) l! l=0

(1) b (1) = b so that C cl,q = b cl,q . Then, l,q + * ( ) n l )m ∑ ln(1 + qt) ( (−q −1 )m t b (s) xn C Lif k ln(1 + qt) Cn,m = l,q m! q l! l=0 * + ( ( ) ) n )m n−l (−q −1 )m ∑ b (s) n ln(1 + qt) ( = Lif k ln(1 + qt) x Cl,q m! l q l=0 ( ) n−m (−q −1 )m ∑ b (s) n = Cl,q m! l l=0 * + ) n−m−l ( ∑ m! ln(1 + qt) s(i + m, m)(qt)i+m xn−l × Lif k q (i + m)! i=0 ( ) n−m−l n−m m! (−q −1 )m ∑ b (s) n ∑ Cl,q s(i + m, m)q i+m (n − l)i+m = l m! (i + m)! i=0 l=0 * + ) ( ln(1 + qt) n−m−l−i × Lif k . x q

*

Since

(

Lif k

ln(1 + qt) q

+ ) n−m−l−i (k) = cn−m−l−i,q , x

we have m

Cn,m = (−1)

n−m ∑ ( ∑ n−m−l l=0

i=0

n l

)(

) n−l i b (s) c(k) q s(i + m, m)C l,q n−m−l−i,q . i+m

Thus, c(k) n,q (x)

m

= (−1)

n (n−m ∑ ∑ n−m−l ∑ (n)( n − l ) m=0

l=0

i=0

l

i+m

) (s) (k) (s) b q s(i + m, m)Cl,q cn−m−l−i,q Bm (x) . i

∑ (s) (k) Similarly, put b cn,q (x) = nm=0 Cn,m Bm (x) for (12) and (15). Then by (10) we get * ( )( ln(1+qt)/q )s ( )m + 1 ln(1 + qt) e −1 ln(1 + qt) n Cn,m = Lif k − x m! q ln(1 + qt)/q q + * )) ( )( ( )m n q (1 + qt)1/q − 1 s ( 1 ln(1 + qt) . = Lif k − ln(1 + qt) x m!q m q ln(1 + qt) 25

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(r)

Define Cl,q (r > 0) by

(1)

( ( ) )r ∞ l ∑ q (1 + qt)1/q − 1 (r) t = Cl,q , ln(1 + qt) l! l=0

(1)

so that Cl,q = cl,q = cl,q . Then, * + ( ) n l )m ∑ ln(1 + qt) ( 1 t (s) Lif k − Cl,q xn ln(1 + qt) Cn,m = m m!q q l! l=0 * + ( ) ( ) n ∑ ( ) 1 ln(1 + qt) m (s) n = Cl,q Lif k − ln(1 + qt) xn−l m l m!q l=0 q ( ) n−m 1 ∑ (s) n C = m!q m l=0 l,q l * + ( ) n−m−l ln(1 + qt) ∑ m! × Lif k − s(i + m, m)(qt)i+m xn−l q (i + m)! i=0 ) ( n−m−l n−m m! 1 ∑ (s) n ∑ Cl,q s(i + m, m)q i+m (n − l)i+m = m l m!q l=0 (i + m)! i=0 * + ) ( ln(1 + qt) n−m−l−i × Lif k − . x q *

Since

Lif k we have Cn,m =

(

ln(1 + qt) − q

n−m ∑ ( ∑ n−m−l l=0

i=0

n l

)(

+ ) n−m−l−i (k) =b cn−m−l−i,q , x

) n−l i (s) (k) q s(i + m, m)Cl,q b cn−m−l−i,q . i+m

Thus, b c(k) n,q (x)

=

n (n−m ∑ (n)( n − l ) ∑ ∑ n−m−l m=0

3.10

l=0

i=0

l

i+m

i

q s(i +

(k) b (s)b m, m)C l,q cn−m−l−i,q

) (s) Bm (x) .

Some relations with Frobenius-Euler polynomials (r)

For λ ∈ C with λ 6= 1, the Frobenius-Euler polynomials of order r, Hn (x|λ) are defined by the generating function )r ( ∞ ∑ tn 1−λ xt (r) (x|λ) e = H n et − λ n! n=0 26

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(see e.g. [4]). Hence, by Lemma 1 we have (( t )r ) e −λ (r) ,t . Hn (x|λ) ∼ 1−λ

(16)

Theorem 11 For integers n and k with n ≥ 0, we have (

)r ∑ n λ = (−1)m λ − 1 m=0 ) (n−m ( )( ) r ∑∑ n (k) (r) l −1 i r (x|λ) , × q (−λ ) s(l + m, n)cn−l−m,q (i) Hm i l + m l=0 i=0 ( )r ∑ n λ (k) b cn,q (x) = λ − 1 m=0 (n−m ) ( )( ) r ∑∑ n (k) (r) l −1 i r (x|λ) . × q (−λ ) s(l + m, n)b cn−l−m,q (i) Hm i l+m l=0 i=0 c(k) n,q (x)

∑ (r) (k) Proof. Put cn,q (x) = nm=0 Cn,m Hm (x|λ) for (11) and (16). Then by (10) we get * )( − ln(1+qt)/q )r ( )m + ( 1 e −λ ln(1 + qt) ln(1 + qt) n Cn,m = Lif k − x m! q 1−λ q * + ) ( ( ) ) (−q −1 )m ln(1 + qt) ( r m = Lif k (1 + qt)−1/q − λ ln(1 + qt) xn . m!(1 − λ)r q Since

(

∑ )m n−m ln(1 + qt) = l=0

m! s(l + m, m)(qt)l+m , (l + m)!

(17)

we have Cn,m

n−m (−q −1 )m ∑ m! s(l + m, m)q l+m (n)l+m = m!(1 − λ)r l=0 (l + m)! + * ( ) r ( ) ln(1 + qt) ∑ r (−λ)r−i (1 + qt)−i/q xn−l−m × Lif k q i i=0 n−m (−q −1 )m ∑ m! = s(l + m, m)q l+m (n)l+m m!(1 − λ)r l=0 (l + m)! * + ( ) r ( ) ∑ r ln(1 + qt) × (−λ)r−i Lif k (1 + qt)−i/q xn−l−m . i q i=0

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Since

( Lif k

) ∞ ∑ ln(1 + qt) tj (k) −i/q (1 + qt) = cj,q (i) , q j! j=0

we have Cn,m

)( ) n−m r ( ∑ (−q −1 )m ∑ m! i=0 r (k) l+m = s(l + m, m)q (n)l+m (−λ)r−i cn−l−m,q (i) m!(1 − λ)r l=0 (l + m)! r i i=0 ( )r n−m ( )( ) r ∑ ∑ λ n (k) m+r l −1 i r = (−1) q (−λ ) s(l + m, n)cn−l−m,q (i) . 1−λ i l + m l=0 i=0

Thus, (

c(k) n,q (x)

)r ∑ n λ = (−1)m λ − 1 m=0 ) (n−m ( )( ) r ∑∑ n (k) l −1 i r (r) (x|λ) . × q (−λ ) s(l + m, n)cn−l−m,q (i) Hm i l + m l=0 i=0

∑ (r) (k) Similarly, put b cn,q (x) = nm=0 Cn,m Hm (x|λ) for (12) and (16). Then by (10) we get * )( ln(1+qt)/q )r ( )m + ( 1 e −λ ln(1 + qt) ln(1 + qt) n Cn,m = Lif k − x m! q 1−λ q * + ) ( ( ) ) ln(1 + qt) ( q −m r m Lif k − (1 + qt)1/q − λ ln(1 + qt) xn . = m!(1 − λ)r q By (17) we have Cn,m =

=

n−m ∑ m! q −m s(l + m, m)q l+m (n)l+m m!(1 − λ)r l=0 (l + m)! * + ) r ( ) ( ln(1 + qt) ∑ r (−λ)r−i (1 + qt)i/q xn−l−m × Lif k − i q i=0 n−m ∑ m! q −m s(l + m, m)q l+m (n)l+m m!(1 − λ)r l=0 (l + m)! * + ( ) r ( ) ∑ ln(1 + qt) r × (−λ)r−i Lif k − (1 + qt)i/q xn−l−m . i q i=0

Since

( Lif k

) ∞ ∑ tj ln(1 + qt) (k) (1 + qt)i/q = b cj,q (i) , − q j! j=0 28

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we have ( Cn,m =

λ λ−1

)r n−m r ∑∑ l=0 i=0

( )( ) r n (k) q (−λ ) s(l + m, n)b cn−l−m,q (i) . i l+m l

−1 i

Thus, (

b c(k) n,q (x)

3.11

)r ∑ n λ = (−1)m λ − 1 m=0 ) (n−m ( )( ) r ∑∑ n (k) (r) l −1 i r (x|λ) . × q (−λ ) s(l + m, n)b cn−l−m,q (i) Hm i l + m l=0 i=0

Some relations with falling and rising factorials

Theorem 12 For integers n and k with n ≥ 0, we have ( m ) ( ) n ∑ ∑ m 1 (k) c(k) (−1)i cn,i (−i) x(m) , n,q (x) = m! i m=0 ( i=0 ) ( ) m n ∑ 1 ∑ m (k) (−1)i b c (m − i) xm . b c(k) n,q (x) = m! i n,i m=0 i=0 (k)

Proof. Put cn,q (x) =

∑n

Cn,m = = = = Thus,

Cn,m x(m) for (11) and x(n) ∼ (1, 1 − e−t ). Then by (10) we get + * ) ( ln(1 + qt) 1 ln(1+qt)/q m n Lif k (1 − e ) x m! q + * ) ( ( ) ln(1 + qt) 1 1/q m n Lif k 1 − (1 + qt) x m! q + * ( ) ) ( m ∑ m ln(1 + qt) 1 i i/q n (−1) Lif k (1 + qt) x m! i=0 i q m ( ) 1 ∑ m (−1)i c(k) n,q (−i) . m! i=0 i m=0

n ∑ 1 c(k) (x) = n,q m! m=0

( m ∑

) ( ) m (k) (−1)i cn,i (−i) x(m) . i i=0

29

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∑ (k) Similarly, put b cn,q (x) = nm=0 Cn,m xm for (12) and xn ∼ (1, et − 1). Then by (10) we get + * ( ) ln(1 + qt) 1 ln(1+qt)/q m n Lif k − (e − 1) x Cn,m = m! q + * ( ) ) 1 ln(1 + qt) ( m 1/q = Lif k − (1 + qt) − 1 xn m! q + * ( ) m ( ) 1 ∑ m ln(1 + qt) m−i i/q n = Lif k − (−1) (1 + qt) x m! i=0 i q ( ) m 1 ∑ m = (−1)m−ib c(k) n,q (−i) m! i=0 i m ( ) 1 ∑ m = (−1)ib c(k) n,q (m − i) . m! i=0 i Thus,

n ∑ 1 (k) b cn,q (x) = m! m=0

(

) ( ) m (k) (−1)i b cn,i (m − i) xm . i i=0

m ∑

References [1] L. Comtet, Advanced Combinatorics, Reidel, Dordrecht, 1974. [2] K. Kamano and T. Komatsu, Poly-Cauchy polynomials, Mosc. J. Comb. Number Theory 3 (2013), 183–209. [3] M. Kaneko, Poly-Bernoulli numbers, J. Th. Nombres Bordeaux 9 (1997), 221–228. [4] D. S. Kim and T. Kim, Some identities of Frobenius-Euler polynomials arising from umbral calculus, Adv. Difference Equ. 2012 (2012), #196. [5] D. S. Kim, T. Kim, S. H. Lee, A note on poly-Bernoulli polynomials arising from umbral calculus , Adv. Studies Theor. Phys., 7 (2013), no. 15, 731-744. [6] D. S. Kim, T. Kim, S.-H. Lee, Poly-Cauchy numbers and polynomials with umbral calculus viewpoint , Int. Journal of Math. Analysis, 7 (2013), 2235-2253. [7] D. S. Kim, T. Kim, S.-H. Lee, Higher-order Cauchy of the first kind and Poly-Cauchy of the first kind mix-type polynomials, Adv. Stud. Contemp. Math. 23 (2013), 543– 554. [8] T. Komatsu, Poly-Cauchy numbers, Kyushu J. Math. 67 (2013), 143–153. 30

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[9] T. Komatsu, Poly-Cauchy numbers with a q parameter, Ramanujan J. 31 (2013), 353–371. [10] T. Komatsu, Sums of products of Cauchy numbers, including poly-Cauchy numbers, J. Discrete Math. 2013 (2013), Article ID 373927, 10 pages. [11] T. Komatsu, Hypergeometric Cauchy numbers, Int. J. Number Theory 9 (2013), 545– 560. [12] T. Komatsu, K. Liptai and L. Szalay, Some relationships between poly-Cauchy type numbers and poly-Bernoulli type numbers, East-West J. Math. 14 (2012), 114-120. [13] T. Komatsu and F. Luca, Some relationships between poly-Cauchy numbers and polyBernoulli numbers, Ann. Math. Inform. 41 (2013), 99-105. [14] N. M. Korobov, On some properties of special polynomials, Chebyshevski˘i Sb. 1 (2001), 40–49. [15] D. Merlini, R. Sprugnoli and M. C. Verri, The Cauchy numbers, Discrete Math. 306 (2006) 1906–1920. [16] S. Roman, The umbral Calculus, Dover, New York, 2005.

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TRIPLED FIXED POINT THEOREMS FOR MIXED MONOTONE CHATTERJEA TYPE CONTRACTIVE OPERATORS ˘ ALINA ˘ ˘ MARIN BORCUT, MAD PACURAR AND VASILE BERINDE

Abstract. Starting from the papers [Berinde, V., Borcut, M., Tripled fixed point theorems for contractive type mappings in partially ordered metric spaces, Nonlinear Anal., 74 (2011), 4889-4897], [ Borcut, M., Berinde, V., Tripled coincidence theorems for contractive type mappings in partially ordered metric spaces , Appl. Math. Comput., 218 (10) (2012), 5929-5936] and [Borcut, M., Tripled coincidente point theorems for contractive type mappings in partially ordered metric spaces, Appl. Math. Comput., 218 (2012), 7339-7346.], we present new results on the existence and uniqueness of tripled fixed points for nonlinear mappings in partially ordered complete metric spaces satisfying more general contractive conditions.

1. INTRODUCTION In some very recent papers, Berinde and Borcut [6], Borcut and Berinde [7], Borcut [8] have introduced and studied the concept of tripled fixed point, respectively tripled coincidence point for nonlinear contractive mappings F : X 3 → X, in partially ordered complete metric spaces and obtained existence as well as existence and uniqueness theorems of tripled fixed points, respectively of tripled coincidence points, for some classes of contractive type mappings. The presented theorems in [6], [7], [8], extend several existing results in the literature: [14], [18], [15]. For the sake of completeness, we recall the main concepts and results from [6] which are needed for the present paper. Let(X, ≤) be a partially ordered set and d be a metric on X such that (X, d) is a complete metric space. Consider on the product space X 3 the following partial order: for (x, y, z) , (u, v, w) ∈ X 3 , (u, v, w) ≤ (x, y, z) ⇔ x ≥ u, y ≤ v, z ≥ w. Definition 1. [6] Let (X, ≤) be a partially ordered set and F : X 3 → X a mapping. We say that F has the mixed monotone property if F (x, y, z) is nondecreasing in x and z, and is nonincreasing in y, that is, for any x, y, z ∈ X, x1 , x2 ∈ X, x1 ≤ x2 ⇒ F (x1 , y, z) ≤ F (x2 , y, z) , y1 , y2 ∈ X, y1 ≤ y2 ⇒ F (x, y1 , z) ≥ F (x, y2 , z) , and z1 , z2 ∈ X, z1 ≤ z2 ⇒ F (x, y, z1 ) ≤ F (x, y, z2 ) . Definition 2. [6] An element (x, y, z) ∈ X 3 is called a tripled fixed point of F : X 3 → X if F (x, y, z) = x, F (y, x, y) = y, and F (z, y, x) = z. 1

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Let (X, d) be a metric space. The mapping d : X × X × X → X, given by d [(x, y, z) , (u, v, w)] = d (x, u) + d (y, v) + d (z, w) , defines a metric on X × X × X, which will be denoted for convenience by d, too. Definition 3. Let X, Y, Z be nonempty sets and F : X × X × X → Y, G : Y × Y × Y → Z. We define the symmetric composition (or, the s-composition, for short) of F and G, G ∗ F : X × X × X → Z, by (G ∗ F ) (x, y, z) = G (F (x, y, z) , F (y, x, y) , F (z, y, x)) (x, y, z ∈ X). For each nonempty set X, denote by Px the projection mapping PX : X × X × X → X, P (x, y, z) = x for x, y, z ∈ X. The symmetric composition has the following properties: Proposition 1. (Associativity). If F : X × X × X → Y, G : Y × Y × Y → Z and H : Z × Z × Z → W, then (H ∗ G) ∗ F = H ∗ (G ∗ F ). Proposition 2. (Identity Element). If F : X × X × X → Y, then F ∗ PX = PY ∗ F = F. Proposition 3. (Mixed Monotonicity). If (X, ≤), (Y, ≤), (Z, ≤) are partially ordered sets and the mappings F : X × X × X → Y, G : Y × Y × Y → Z are mixed monotone, then G ∗ F is mixed monotone. Proposition 4. If (X, ≤) is a partially ordered set and F is mixed monotone, then F n = F ∗ F n−1 = F n−1 ∗ F is mixed monotone, for every n ≥ 1. The first main result in [6] is given by the following theorem. Theorem 1. [6] Let (X, ≤) be a partially ordered set and suppose there is a metric d on X such that (X, d) is a complete metric space. Let F : X × X × X → X be a continuous mapping having the mixed monotone property on X. Assume that there exist the constants j, k, l ∈ [0, 1) with j + k + l < 1 for which (1.1)

d (F (x, y, z) , F (u, v, w)) ≤ jd (x, u) + kd (y, v) + ld (z, w) ,

∀x ≥ u, y ≤ v, z ≥ w. If there exist x0 , y0 , z0 ∈ X such that x0 ≤ F (x0 , y0 , z0 ) , y0 ≥ F (y0 , x0 , y0 ) and z0 ≤ F (z0 , y0 , x0 ) , then there exist x, y, z ∈ X such that x = F (x, y, z) , y = F (y, x, y) and z = F (z, y, x) . Remark 1. If we take j = k = l = α3 in Theorem 1, then the contraction condition (1.1) can be written in a slightly simplified form α (1.2) d (F (x, y, z) , F (u, v, w)) ≤ [d (x, u) + d (y, v) + d (z, w)]. 3 Theorem 2. [6] By adding to the hypotheses of Theorem 1 the condition: for every (x, y, z) , (x1 , y1 , z1 ) ∈ X 3 , there exists a (u, v, w) ∈ X 3 that is comparable to (x, y, z) and (x1 , y1 , z1 ), then the tripled fixed point of F is unique. Theorem 3. [6] In addition to the hypotheses of Theorem 1, suppose that x0 , y0 , z0 ∈ X are comparable. Then x = y = z.

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3

2. Main results Based on the notions and results presented in the first section, we will prove new existence and uniqueness theorems for operators which verify a Chatterjea contraction type condition, adapted to the case X 3 . Theorem 4. Let (X, ≤) be a partially ordered set and suppose there is a metric d on X such that (X, d) is a complete metric space. Let F : X 3 → X be a mapping having the mixed monotone property on X. Assume that there exists a k ∈ [0, 1) such that k (2.1) d (F (x, y, z) , F (u, v, w)) ≤ [d (x, F (u, v, w)) + d (y, F (v, u, v)) + 8 +d (z, F (w, v, u)) + d (u, F (x, y, z)) + d (v, F (y, x, y)) + d (w, F (z, y, x))]. Also suppose either (a) F is continuous or (b) X has the following property: (i) if a nondecreasing sequence {xn } → x, then xn ≤ x for all n, (ii) if a nonincreasing sequence {yn } → y, then yn ≥ y for all n. If there exist x0 , y0 , z0 ∈ X such that, (2.2)

x0 ≤ F (x0 , y0 , z0 ) , y0 ≥ F (y0 , x0 , y0 ) and z0 ≤ F (z0 , y0 , x0 ) ,

then there exist x, y, z ∈ X such that, x = F (x, y, z) , y = F (y, x, y) and z = F (z, y, x) . Proof. Let the sequences {xn } , {yn } , {zn } ⊂ X be defined by xn+1 = F (xn , yn , zn ) = F n+1 (x0 , y0 , z0 ), yn+1 = F (yn , xn , yn ) = F n+1 (y0 , x0 , y0 ), zn+1 = F (zn , yn , xn ) = F n+1 (z0 , y0 , x0 ), (n = 0, 1, ...). Since F n is mixed monotone for every n (by Proposition 4), it follows by (2.2) that {xn } and {zn } are nondecreasing and {yn } is nonincreasing. Indeed, due to the mixed monotone property of F , it is easy to show that x2 = F (x1 , y1 , z1 ) ≥ F (x0 , y0 , z0 ) = x1 y2 = F (y1 , x1 , y1 ) ≤ F (y0 , x0 , y0 ) = y1 z2 = F (z1 , y1 , x1 ) ≥ F (z0 , y0 , x0 ) = z1 and thus we obtain three sequences satisfying the following conditions x0 ≤ x1 ≤ ... ≤ xn ≤ ..., y0 ≥ y1 ≥ ... ≥ yn ≥ ..., z0 ≤ z1 ≤ ... ≤ zn ≤ .... Now, for n ∈ N, denote Dxn+1 = d (xn+1 , xn ) , Dyn+1 = d (yn+1 , yn ) , Dzn+1 = d (zn+1 , zn ) and Dn+1 = Dxn+1 + Dyn+1 + Dzn+1 . Using (2.1), we get Dxn+1 = d (xn+1 , xn ) = d(F (xn , yn , zn ), F (xn−1 , yn−1 , zn−1 ))

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k d xn , Fxn−1 + d yn , Fyn−1 + d zn , Fzn−1 8 +d (xn−1 , Fxn ) + d (yn−1 , Fyn ) + d (zn−1 , Fzn )] k = [d (xn , xn ) + d (yn , yn ) + d (zn , zn ) 8 +d (xn−1 , xn+1 ) + d (yn−1 , yn+1 ) + d (zn−1 , zn+1 )] k = [d (xn−1 , xn+1 ) + d (yn−1 , yn+1 ) + d (zn−1 , zn+1 )] 8 k ≤ [d (xn−1 , xn ) + d (yn−1 , yn ) + d (zn−1 , zn ) 8 +d (xn , xn+1 ) + d (yn , yn+1 ) + d (zn , zn+1 )]  k = Dxn + Dyn + Dzn + Dxn+1 + Dyn+1 + Dzn+1 , 8 ≤

and therefore  k Dxn + Dyn + Dzn + Dxn+1 + Dyn+1 + Dzn+1 . 8   Similarly, we obtain for the sequences Dyn+1 , Dzn+1 (2.3)

Dxn+1 ≤

Dyn+1 = d (yn+1 , yn ) = d(F (yn , xn , yn ), F (yn−1 , xn−1 , yn−1 ))


k ≤ d yn , Fyn−1 + d xn , Fxn−1 + d yn , Fyn−1 8 +d (yn−1 , Fyn ) + d (xn−1 , Fx n ) + d (yn−1 , Fyn )] k = [d (yn , yn ) + d (xn , xn ) + d (yn , yn ) 8 +d (yn−1 , yn+1 ) + d (xn−1 , xn+1 ) + d (yn−1 , yn+1 )] k = [2d (yn−1 , yn+1 ) + d (xn−1 , xn+1 )] 8 k ≤ [2d (yn−1 , yn ) + d (xn−1 , xn ) + d (xn , xn+1 ) + 2d (yn , yn+1 )] 8  k = Dxn + 2Dyn + Dxn+1 + 2Dyn+1 , 8 and so (2.4)

Dyn+1 ≤

 k Dxn + 2Dyn + Dxn+1 + 2Dyn+1 8

and Dzn+1 = d (zn+1 , zn ) = d(F (zn , yn , xn ), F (zn−1 , yn−1 , xn−1 ))


k ≤ d zn , Fzn−1 + d yn , Fyn−1 + d xn , Fxn−1 8 +d (zn−1 , Fzn ) + d (yn−1 , Fyn ) + d (xn−1 , Fxn )] k = [d (xn , xn ) + d (yn , yn ) + d (zn , zn ) 8 +d (xn−1 , xn+1 ) + d (yn−1 , yn+1 ) + d (zn−1 , zn+1 )] k = [d (xn−1 , xn+1 ) + d (yn−1 , yn+1 ) + d (zn−1 , zn+1 )] 8 k ≤ [d (xn−1 , xn ) + d (yn−1 , yn ) + d (zn−1 , zn ) 8 +d (xn , xn+1 ) + d (yn , yn+1 ) + d (zn , zn+1 )]

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=

5

 k Dxn + Dyn + Dzn + Dxn+1 + Dyn+1 + Dzn+1 , 8

and therefore  k Dxn + Dyn + Dzn + Dxn+1 + Dyn+1 + Dzn+1 . 8 By using relations (2.3), (2.4) and (2.5), we get  k 3Dxn + 4Dyn + 2Dzn + 3Dxn+1 + 4Dyn+1 + 2Dzn+1 Dn+1 ≤ 8  k ≤ 4Dxn + 4Dyn + 4Dzn + 4Dxn+1 + 4Dyn+1 + 4Dzn+1 8 k ≤ [Dn + Dn+1 ] . 2 Therefore, for all n ≥ 1, we have k ∈ [0, 1), when k ∈ [0, 1). Dn+1 ≤ α · Dn ≤ ... ≤ αn · D1 , where α = 2−k Because Dxn+1 ≤ Dn+1 , Dyn+1 ≤ Dn+1 and Dzn+1 ≤ Dn+1 , then we have (2.5)

(2.6)

Dzn+1 ≤

Dxn+1 ≤ αn · D1 , Dyn+1 ≤ αn · D1 and Dzn+1 ≤ αn · D1

This implies that {xn } , {yn } , {zn } are Cauchy sequences in X. Indeed, let m ≥ n, then d (xm , xn ) ≤ Dxm + Dxm−1 + ... + Dxn+1 ≤ αn αn − αm · D1 < · D1 . ≤ [αm−1 + αm−2 + ... + αn ] · D1 = 1−α 1−α Similarly, we can verify that {yn } and {zn } are also Cauchy sequences. Since X is a complete metric space, there exist x, y, z ∈ X such that, lim xn = x, lim yn = y, lim zn = z.

x→∞

x→∞

x→∞

Finally, we claim that x = F (x, y, z) , y = F (y, x, y) and z = F (z, y, x) . Assume the first assumption (a) holds. This means F is continuous at (x, y, z) , and hence, for a given 2 > 0, there exists a δ > 0 such that, d ((x, y, z), (u, v, w)) = d (x, u) + d (y, v) + d (z, w) < δ  ⇒ d (F (x, y, z) , F (u, v, w)) < . 2 Since lim xn = x, lim yn = y, lim zn = z,

x→∞

for η = min

 2, 2


δ

x→∞

x→∞

, there exist n0 , m0 , p0 such that, for n ≥ n0 , m ≥ m0 , p ≥ p0 , d (xn , x) < η, d (yn , y) < η, d (zn , z) < η.

Now, for n ∈ N, n ≥ max {n0 , m0 , p0 } , we have d (F (x, y, z) , x) ≤ d (F (x, y, z) , xn+1 ) + d (xn+1 , x)  = d (F (x, y, z) , F (xn , yn , zn )) + d (xn+1 , x) < + η ≤ , 2 and this implies that x = F (x, y, z) . Similarly, we can show that y = F (y, x, y) and z = F (z, y, x) .

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Suppose now that (b) holds. Since {xn } , {zn } are non-decreasing and xn → x, zn → z, and also {yn } is non-increasing and yn → y, from (b) we have xn ≤ x, yn ≥ y and zn ≤ z, for all n. Then by triangle inequality and (2.1), we get d (x, F (x, y, z)) ≤ d (x, xn+1 ) + d (xn+1 , F (x, y, z))

(2.7)

= d (x, xn+1 ) + d (F (xn , yn , zn ), F (x, y, z)) k ≤ d (x, xn+1 ) + [d(xn , xn+1 ) + d(yn , yn+1 ) + d(zn , zn+1 ) 8 +d(x, F (x, y, z)) + d(y, F (y, x, y)) + d(z, F (z, y, x))], (2.8)

k [d(xn , xn+1 ) + 2d(yn , yn+1 ) 8 +d(x, F (x, y, z)) + 2d(y, F (y, x, y))],

d(y, F (y, x, y)) ≤ d (y, yn+1 ) +

and k [d(xn , xn+1 ) + d(yn , yn+1 ) + d(zn , zn+1 ) 8 +d(x, F (x, y, z)) + d(y, F (y, x, y)) + d(z, F (z, y, x))]. By summing (2.7), (2.8), (2.9) we obtain (2.9)

d(z, F (z, y, x)) ≤ d (z, zn+1 ) +

d(x, F (x, y, z)) + d(y, F (y, x, y)) + d(z, F (z, y, x)) ≤

2 [d (x, xn+1 ) + d (y, yn+1 ) + d (z, zn+1 )] 2−k

k [3d(xn , xn+1 ) + 4d(yn , yn+1 ) + 2d(zn , zn+1 )], 4(2 − k) and let n → ∞ one obtains +

d (x, F (x, y, z)) + d(y, F (y, x, y)) + d(z, F (z, y, x)) ≤ 0, that is, x = F (x, y, z), y = F (y, x, y), z = F (z, y, x).



3. Uniqueness of tripled fixed points In [6], [7] and [8] the authors also considered some additional conditions to ensure the uniqueness of the tripled fixed point and also appropriate conditions to ensure that for such a tripled fixed point (x, y, z) we have all components equal: x = y = z. Similarly, one can prove that the tripled fixed point ensured by Theorem 4 is in fact unique, provided that the product space X × X × X endowed with the partial order mentioned earlier possesses an additional property. Theorem 5. If, in addition to the hypotheses of Theorem 4, the condition: for every (x, y, z) , (x1 , y1 , z1 ) ∈ X × X × X, there exists a (u, v, w) ∈ X × X × X that is comparable to (x, y, z) and (x1 , y1 , z1 ), is satisfied, then the tripled fixed point of F is unique. Proof. If (x∗ , y ∗ , z ∗ ) ∈ X × X × X is another tripled fixed point of F, then we show that d ((x, y, z) , (x∗ , y ∗ , z ∗ )) = 0, where lim xn = x, lim yn = y, lim zn = z. x→∞

x→∞

x→∞

We consider two cases.

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Case 1. If (x, y, z) is comparable to (x∗ , y ∗ , z ∗ ) with respect to the ordering in X × X × X then, for every n = 0, 1, 2, . . . , the triple (F n (x, y, z) , F n (y, x, y) , F n (z, y, x)) = (x, y, z) is comparable to (F n (x∗ , y ∗ , z ∗ ) , F n (y ∗ , x∗ , y ∗ ) , F n (z ∗ , y ∗ , x∗ )) = (x∗ , y ∗ , z ∗ ) . Also, using the process for obtaining (2.6), we get d ((x, y, z) , (x∗ , y ∗ , z ∗ )) = d (x, x∗ ) + d (y, y ∗ ) + d (z, z ∗ ) = d (F n (x, y, z) , F n (x∗ , y ∗ , z ∗ )) + d (F n (y, x, y) , F n (y ∗ , x∗ , y ∗ )) +d (F n (z, y, x) , F n (z ∗ , y ∗ , x∗ )) ∗

≤ α [d (x, x ) + d (y, y ∗ ) + d (z, z ∗ )] = αn d ((x, y, z) , (y ∗ , x∗ , z ∗ )) , α ∈ [0, 1). By letting n → ∞, this implies that d ((x, y, z) , (y ∗ , x∗ , z ∗ )) = 0. Case 2 : If (x, y, z) are not comparable to (x∗ , y ∗ , z ∗ ) , then there exists an upper bound or a lower bound (u, v, w) ∈ X × X × X of (x, y, z) and (x∗ , y ∗ , z ∗ ) . Then, for all n = 1, 2, ..., n

(F n (u, v, w) , F n (v, u, v) , F n (w, v, u)) is comparable to (F n (x, y, z) , F n (y, x, y) , F n (z, y, x)) = (x, y, z) and to (F n (x∗ , y ∗ , z ∗ ) , F n (y ∗ , x∗ , y ∗ ) , F n (z ∗ , y ∗ , x∗ )) = (x∗ , y ∗ , z ∗ ) . We have, 

  n ∗ ∗ ∗   n   ∗  F (x , y , z ) F (x, y, z) x x d  y  ,  y ∗  = d  F n (y, x, y)  ,  F n (y ∗ , x∗ , y ∗ )  F n (z ∗ , y ∗ , x∗ ) F n (z, y, x) z∗ z  n   n  F (x, y, z) F (u, v, w) ≤ d  F n (y, x, y)  ,  F n (v, u, v)  F n (z, y, x) F n (w, v, u)  n   n ∗ ∗ ∗  F (u, v, w) F (x , y , z ) +d  F n (v, u, v)  ,  F n (y ∗ , x∗ , y ∗ )  F n (w, v, u) F n (z ∗ , y ∗ , x∗ )

≤ αn {[d (x, u) + d (y, v) + d (z, w)] + [d (u, x∗ ) + d (v, y ∗ ) + d (w, z ∗ )]} → 0   ∗   x x as n → ∞, and so d  y  ,  y ∗  = 0. z∗ z  Theorem 6. In addition to the hypotheses of Theorem 4, suppose that x0 , y0 , z0 ∈ X are comparable. Then x = y = z. Proof. Recall that x0 , y0 , z0 , ∈ X are such that x0 ≤ F (x0 , y0 , z0 ), y0 ≥ F (y0 , x0 , y0 ), z0 ≤ F (z0 , y0 , x0 ). Now, if x0 ≤ y0 , and z0 ≤ y0 we claim that, for all n ∈ N, xn ≤ yn and zn ≤ yn . Indeed, by the mixed monotone property of F, x1 = F (x0 , y0 , z0 ) ≤ F (y0 , x0 , y0 ) = y1 and z1 = F (z0 , y0 , x0 ) ≤ F (y0 , x0 , y0 ) = y1 .

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Assume that xn ≤ yn and zn ≤ yn for some n. Then xn+1 = F n+1 (x0 , y0 , z0 ) = F (F n (x0 , y0 , z0 ), F n (y0 , x0 , y0 ), F n (z0 , y0 , x0 )) = F (xn , yn , zn ) ≤ F (yn , xn , yn ) = yn+1 , and similarly for zn . Now, d(x, y) ≤ d(x, xn+1 ) + d(y, xn+1 ) ≤ d(x, xn+1 ) + d(xn+1 , yn+1 ) + d(y, yn+1 ) = d(x, F n+1 (x0 , y0 , z0 )) + d [F (F n (x0 , y0 , z0 ), F n (y0 , x0 , y0 ), F n (z0 , y0 , x0 )), , F (F n (y0 , x0 , y0 ), F n (x0 , y0 , x0 ), F n (y0 , x0 , y0 ))] + d(y, yn+1 ) → 0 as n → ∞. This implies that d(x, y) = 0 and hence we have x = y. Similarly, we obtain that d(x, z) = 0 and d(y, z) = 0. The other cases for x0 , y0 , z0 are similar.  4. Example and final remarks Let X = [0, 1] be endowed with the usual metric d (x, y) = |x  − y| and let 1 4 F : X 3 → X be given by F (x, y, z) = , for (x, y, z) ∈ 0, × [0, 1]2 and 20 5   4 11 , for (x, y, z) ∈ , 1 × [0, 1]2 . F (x, y, z) = 80 5 14 Then F satisfies Chatterjea’s contractive condition (2.1) with k = < 1 but 15 does not satisfy the Banach type contractive condition (1.1). Let us first prove the first part of the assertion above. It suffices to completely cover the following  case.   limit  4 4 , 1 , and u, y, z, v, w ∈ 0, Case 1. x ∈ 5 5 11 1 In this case F (x, y, z) = , F (u, v, w) = and so condition (2.1) reduces to 80 20 (4.1)   11 − 1 ≤ k x − 1 + y − 1 + z − 1 + u − 11 + v − 1 + w − 1 . 80 20 8 20 20 20 80 20 20   4 For x ∈ , 1 , we have 5 x − 1 ≥ 4 − 1 = 3 20 5 20 4 and hence the minimum value of the right hand side of (4.1) is greater or equal to k 3 · . 8 4   4 Therefore, in order to have (4.1) satisfied for all x ∈ , 1 and u, y, z, v, w ∈ 5   4 0, , with x ≥ u, y ≤ v, z ≥ w, i.e., 5 1 − 11 ≤ k · 3 , 20 80 8 4 it suffices to take k such that

14 ≤ k < 1. 15

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Note that for the remaining cases to be discussed, the right hand side of (2.1) will be greater than the value obtained in Case  1.   4 4 For example, in the Case 2. x, v ∈ , 1 and u, y, z, w ∈ 0, , the minimum 5 5 k 6 value of the right hand side of (2.1) will be greater or equal to · .     8 4 4 4 Note also that in the cases x, u ∈ , 1 or x, u ∈ 0, , the left hand side of 5 5 (2.1) is always zero and so (1.2) is satisfied for all values of y, z, v, w ∈ [0, 1]. 14 This proves that, indeed, F satisfies (2.1) with k = < 1. 15 F is not continuous but X satisfies property (b) in Theorem 4. Moreover, by 1 1 and z0 = , one can check that (2.2) is fulfilled. Thus, taking x0 = 0, y0 = 5 8 all assumptions in Theorem 4 are satisfied and hence F does admit tripled fixed points. By Theorem 5 we actually conclude that F has a unique tripled fixed point,   1 1 1 , , . 20 20 20 Now let us show that F does not satisfy (1.1). Assume the that is, that F does satisfy (1.1) and take  > 0 such that  contrary,  4 4 4 u = −  ∈ 0, , x = and y = z, v = w ∈ [0, 1] arbitrary in (1.1) to obtain 5 5 5 7 ≤ i · ,  > 0. 80 Now letting  → 0 in (4.2) we reach to a contradiction. This proves that, indeed, F does not satisfy (1.1).

(4.2)

Acknowledgements The research was supported by the Grant PN-II-RU-TE-2011-3-0239 of the Romanian Ministry of Education and Research. References [1] Abbas, M., Ali Khan, M., Radenovi´ c, S., Common coupled fixed point theorems in cone metric spaces for w-compatible mappings, Appl. Math. Comput. 217 (2010), no. 1, 195–202 [2] Altun, I., Damjanovi´ c, B. Djori´ c, D., Fixed point and common fixed point theorems on ordered cone metric spaces, Appl. Math. Lett. 23 (2010), no. 3, 310-316 [3] Altun, I., Rakocevi´ c, V., Ordered cone metric spaces and fixed point results, Comput. Math. Appl. 60 (2010), no. 5, 1145–1151 [4] Beg, I., Abbas, M., Fixed points and invariant approximation in random normed spaces, Carpathian J. Math. 26 (2010), no. 1, 36–40 [5] Berinde, V., Iterative approximation of fixed points. Second edition, Lecture Notes in Mathematics, 1912, Springer, Berlin, 2007 [6] Berinde, V.,Borcut, M., Tripled fixed point theorems for contractive type mappings in partially ordered metric spaces, Nonlinear Anal. , 74 (2011) 4889-4897. [7] Borcut, M., Berinde, V., Tripled coincidence theorems for contractive type mappings in partially ordered metric spaces , Aplied Mathematics and Computation, 218 (10) (2012) pp. 5929-5936 [8] Borcut, M., Tripled coincidente point theorems for contractive type mappings in partially ordered metric spaces, Aplied Mathematics and Computation, 218 (2012) pp. 7339-7346 [9] Borcut, M., Tripled fixed point theorems for monotone contractive type mappings in partially ordered metric spaces, Carpathian J. MAath., 28 (2012), No. 2, 207-214.

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[10] Borcut, M., Tripled coincidente point theorems for monotone contractive type mappings in partially ordered metric spaces, Creative Mathematics and Informatics, 21 (2012), No. 2, 135-142. [11] Borcut, M., Tripled fixed point theorems for operators which verify the contraction-type condition Kannan in partially ordered metric spaces, Applied Mathematical Sciences, (Submitted). [12] Borcut, M., Tripled fixed point theorems in partially ordered metric spaces, Hacettepe Journal of Mathematics and Statistics (Submited). [13] Borcut, M., Tripled coincidente point theorems for monotone φ-contractive type mappings in partially ordered metric spaces, Filomat J. (submitted). [14] Gnana Bhaskar, T., Lakshmikantham, V., Fixed point theorems in partially ordered metric spaces and applications, Nonlinear Anal. 65 (2006), no. 7, 1379–1393 ´ pez, B; Sadarangani, K. Fixed point theorems for mixed monotone oper[15] Harjani, J; Lo ators and applications to integral equations , Nonlinear Anal. 74 (2011), 1749–1760. [16] Kannan, R. Some results on fixed points,, Bull. Calcutta Math. Soc., 10 (1968), 71-76 [17] Karapinar, E., Coupled fixed point theorems for nonlinear contractions in cone metric spaces, Comput. Math. Appl., 59 (2010), no. 12, 3656–3668 ´ c, L., Coupled fixed point theorems for nonlinear contractions in [18] Lakshmikantham, V., Ciri´ partially ordered metric spaces, Nonlinear Anal. 70 (2009), 4341-4349 [19] Nguyen V. L., Nguyen X. T., Coupled fixed points in partially ordered metric spaces and application, Nonlinear Anal., 74 (2011), 983–992 [20] Nieto,Juan J.; Rodriguez-Lopez, Rosana., Existence and uniqueness of fixed point in partially ordered sets and applications to ordinary differential equations , Acta. Math. Sin. ,(Engl. Ser.) 23(2007), no. 12, 2205–2212 . [21] Ran, A. C. M., Reurings, M. C. B., A fixed point theorem in partially ordered sets and some applications to matrix equations, Proc. Amer. Math. Soc. 132 (2004), no. 5, 1435–1443 [22] Rus, I. A., Generalized Contractions and Applications, Cluj University Press, Cluj-Napoca, 2001 [23] Rus, I. A., Petru¸sel, A., Petru¸sel, G., Fixed Point Theory, Cluj University Press, Cluj-Napoca, 2008 [24] Rus, M-D., Fixed point theorems for generalized contractions in partially ordered metric spaces with semi-monotone metric, Nonlinear Anal. [25] Sabetghadam, F., Masiha, H.P., Sanatpour, A.H., Some coupled fixed point theorems in cone metric spaces, Fixed Point Theory. Appl., 2009, Art. ID 125426, 8 pp [26] Samet, B., Coupled fixed point theorems for a generalized Meir-Keeler contraction in partially ordered metric spaces, Nonlinear Anal., 72 (2010), no. 12, 4508–4517 [27] Sedghi, S., Altun, I., Shobe, N., Coupled fixed point theorems for contractions in fuzzy metric spaces, Nonlinear Anal., 72 (2010), no. 3-4, 1298–1304

Department of Mathematics and Computer Science North University of Baia Mare Victoriei 76, 430122 Baia Mare ROMANIA E-mail: [email protected] Department of Mathematics and Computer Science North University of Baia Mare Victorie1 76, 430072 Baia Mare ROMANIA E-mail: [email protected]; vasile [email protected]; Department of Statistics, Analysis, Forecast and Mathematics Faculty of Economics and Bussiness Administration Babe¸s-Bolyai University of Cluj-Napoca 56-60 T. Mihali St., 400591 Cluj-Napoca ROMANIA E-mail: [email protected]

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SOFT BOOLEAN ALGEBRA AND ITS PROPERTIES ¨ ¸ UK ¨ RIDVAN S ¸ AHIN AND AHMET KUC

Abstract. Molodtsov [21] introduced the concept of soft theory which can be used as a generic mathematical tool for dealing with uncertainty. In this paper, we apply the notion of the soft set theory of Molodtsov to the theory of Boolean algebras which is a well-known algebraic structure. We introduce the concepts of soft filter and soft ideal on the soft Boolean algebra as well as notions of a soft Boolean algebra and soft Boolean homomorphism, and investigate basic properties as intersection, union and product of the soft Boolean algebras. Also we give several illustrative examples.

1. Indroduction In 1999, Molodtsov [21] initiated the theory of soft sets as a new mathematical tool to deal with uncertainties while modelling the problems in engineering, physics, computer science, economics, social sciences, and medical science. Maji et al. [18] showed the applications of soft set theory in decision making problem by defining several operations on soft set. In theoretical aspects, Maji et al. [19] introduced several operators for soft set theory such as equality of two soft sets, subset and superset of a soft set, complement of a soft set, null soft set, and absolute soft set. Recently, some new operations in soft set theory has been given by Irfan Ali et al. in [2], also see [23]. Later, the properties and applications of soft set theory have been studied by many authors (e.g. [3, 5, 14, 15, 17, 20, 22, 25, 26] ). At present, studies on the soft set theory is progressing rapidly on algebraic structures. Aktas and Cagman [1] defined a basic version of soft group theory. Sezgin et al. [24] introduced the concepts of normalistic soft group and normalistic soft group homomorphism. Feng et al. [4] studied soft semi rings. Jun et al. [9, 11] applied soft sets in the theories of BCK/BCI-algebras. Kazancı et al. [13] introduced soft BCH-algebras and studied their basic properties. Several other studies on soft BCH-algebras have been discussed in [7, 8, 10]. Jun et al. [12] applied the notion of the soft sets to the theory of Hilbert algebras. In this paper, we apply the notion of the soft set theory of Molodtsov to the theory of Boolean algebras. We introduce the concepts of soft filter and soft ideal on the soft Boolean algebra as well as concept of a soft Boolean algebra. We also investigate basic properties as intersection, union and product of the soft Boolean algebras, and define soft Boolean homomorphism and obtain some properties. Also we give several illustrative examples. This paper is organized as follows. In the next two sections, we give some important concepts of Boolean algebra and basic definitions of soft set theory. In 1991 Mathematics Subject Classification. 2010 Primary 06D72; Secondary 54A40. Key words and phrases. Boolean algebra, soft set, soft Boolean algebra, atomic soft Boolean algebra, complete soft Boolean algebra, soft Boolean homomorphism. 1

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Section 4, we present the definition of soft Boolean algebra and some properties of soft Boolean algebra. Finally, we summary the paper in Section 5. 2. Basic Results on Boolean Algebras In the middle of the 19th century, George Boole introduced the concept of Boolean algebras by attempting to formalize propositional logic. Boolean algebra now plays a central role in mathematical logic, probability theory and computer design. In this section, we give some basic notions in Boolean algebra. For more details on Boolean algebras, we refer the reader to [6, 16]. Definition 1. A Boolean algebra is a tuple (K, ∧, ∨, ¬, ¯0, ¯1) (briefly K), where K is a set with two distinguished elements ¯0, ¯1 ∈ K, ¬ : K → K is a unary operation and ∧, ∨ : K × K → K are binary operations (called meet and join, respectively) such that (1) ∧, ∨ are associative, (2) ∧, ∨ are commutative, (3) ∧ and ∨ are distributive, i.e. ∀x, y, z ∈ B : x ∧ (y ∨ z) = (x ∧ y) ∨ (x ∧ y) and x ∨ (y ∧ z) = (x ∨ y) ∧ (x ∨ z), (4) ∀x ∈ K : x ∧ ¬x = ¯ 0 and x ∨ ¬x = ¯1. Let K be a Boolean algebra and x, y ∈ K. Then K carries a natural partial order. In other words, an relation ”≤ ” defined x ≤ y if x = x ∧ y or x ∨ y = y, is an ordering relation on K, where x ∨ y and x ∧ y are least upper bound and greatest lower bound of {x, y}, respectively. The element ¯0 in Boolean algebra K is called to be zero element if x ∨ ¯0 = ¯0 ∨ x = x and x ∧ ¯0 = ¯0 ∧ x = ¯0 for any x ∈ K. Similarly, the element ¯1 in Boolean algebra K is called to be unit element if x ∨ ¯ 1=¯ 1∨x=¯ 1 and x ∧ ¯ 1 = ¯1 ∧ x = x for any x ∈ K. Remark 1. Let K be a Boolean algebra, X be any set and P (X) its power set. Then (1) (P (X), ∩, ∪, −, ∅, X), where ”−” is the complement operation of sets, is a Boolean algebra of sets. (2) For x and y in the Boolean algebra K, x  y if x ≤ y does not hold. x < y (x is strictly smaller than y) if x ≤ y but x 6= y. Notation 1. Throughout this article, we assume that ”≤” is a natural partial order defined on natural integer N. Definition 2. Let K be a Boolean algebra. Then (1) a ∈ K is an atom of K, if 0¯ < a but there is no x in K satisfying ¯0 < x < a. K is atomless if it has no atoms and atomic if for each positive element x (i.e., x 6= ¯ 0) of K, there is some atom a such that a ≤ x. (2) For any S ⊆ K with S 6= ∅, K is complete iff both inf (S) and sup(S) exist for every nonempty subset S of K. Definition 3. Let K be a Boolean algebra. (1) A nonempty subset M of K is said to be a subalgebra of K if x, y ∈ M implies x ∨ y, x ∧ y and ¬x ∈ M . Moreover, note that every Boolean algebra is itself a subalgebra.

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(2) A filter on K is a subset F of K such that: (i) ¯ 0 ∈ / F, ¯ 1 ∈ F ; (ii) if x ∈ F and y ∈ F , then x ∧ y ∈ F ; (iii) if x, y ∈ K, x ∈ F and x ≤ y, then y ∈ F . (3) An ideal on K is a subset I of K such that: (i) ¯ 0 ∈ I, ¯ 1∈ / I; (ii) if x ∈ I and y ∈ I, then x ∨ y ∈ I; (iii) if x, y ∈ K, x ∈ I and y ≤ x, then y ∈ I. Definition 4. Let K and L be two Boolean algebras. A mapping φ : K → L is called a (Boolean) homomorphism if it preserves the operations:   φ(a ∧ b) = φ(a) ∧ φ(b) φ(a ∨ b) = φ(a) ∨ φ(b)  φ(¬a) = ¬φ(a) for all a, b ∈ K. If φ is bijective, then it is called a (Boolean) isomorphism. If there is a Boolean isomorphism φ : K → L, then K and L are said to be isomorphic, and denoted by K ' L. 3. Basic Results on Soft Sets In this paper, U is an initial universe set, P (U ) its power set and E is always the universe set of parameters with respect to U unless otherwise specified. Now, we recall some basic notions in soft set theory. Definition 5. [19, 20]. A pair (F, A) is called a soft set over U if A ⊆ E and F : A −→ P (U ), such that F (x) 6= ∅, if x ∈ A ⊆ E and F (x) = ∅ if x ∈ / A. Definition 6. [3]. Let U be an initial universe set and E be a universe set of parameters. Let (F, A) and (G, B) be soft sets over a common universe set U and A, B ⊆ E. Then ˜ (1) (F, A) is a subset of (G, B), denoted by (F, A)⊆(G, B), if (i) A ⊆ B; (ii) F (x) ⊆ G(x) for all x ∈ A, ˜ (2) (F, A) equals (G, B), denoted by (F, A) = (G, B), if (F, A)⊆(G, B) and ˜ (G, B)⊆(F, A). Definition 7. [19]. Let (F, A) and (G, B) be two soft sets over a common universe U. The union of (F, A) and (G, B) is defined to be a soft set (H, C), where C = A∪B and H is defined as follows:  if x ∈ A − B  F (x) G(x) if x ∈ B − A H(x) =  F (x) ∪ G(x) if x ∈ A ∩ B ˜ (G, B). We write (H, C) = (F, A)∪ Definition 8. [19]. Let (F, A) and (G, B) be two soft sets over a common universe U . The intersection of (F, A) and (G, B) is defined to be a soft set (H, C) satisfying the following conditions: (1) C = A ∩ B, (2) H(x) = F (x) or G(x) for each x ∈ C (as both are same set). ˜ (G, B). We write (H, C) = (F, A)∩ Definition 9. [19]. Let (F , A) and (G, B) be soft sets over a common universe set U.

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˜ (G, B) is a soft set defined by (F, A)∧ ˜ (G, B) = (H, A × B), where (1) (F, A)∧ H(x, y) = F (x)∩G(y) for any x ∈ A and y ∈ B, where ∩ is the intersection operation of sets. ˜ (G, B) is a soft set defined by (F, A)∨ ˜ (G, B) = (K, A × B), where (2) (F, A)∨ K(x, y) = F (x) ∪ G(y) for any x ∈ A and y ∈ B, where ∪ is the union operation of sets. 4. Soft Boolean Algebras Let K be a Boolean algebra and A be a nonempty set. R will refer to an arbitrary binary relation between an element of A and an element of K; that is, R is a subset of A × K unless otherwise specified. A set-valued function F : A → P (K) can be define as F (x) = {y ∈ K : (x, y) ∈ R} for all x ∈ A. Then the pair (F, A) is a soft set over K, which is derived from the relation R. Definition 10. Let (F, A) be a soft set over K. Then (F, A) is called a soft Boolean algebra over K if F (x) is a subalgebra of K for all x ∈ A. Example 1. Let K = {1, 2, 3, 6} be set of all divisors of 6. Consider x ∧ y = mcd(x, y) (2 ∧ 3 = 1, 2 ∧ 6 = 2), x ∨ y = mcm(x, y)(2 ∨ 3 = 6, 2 ∨ 6 = 6) and ¬x = x6 (¬2 = 3). Then the structure hK, ∧, ∨, ¬, 1, 6i is a Boolean algebra under the relation ”  ” which is given by x  y if x = x ∧ y or x ∨ y = y. Let (F, A) be a soft set over K, where A = K and F : A −→ P (K) is a set-valued function defined by    y ∈ {1, 6} if x ∈ / {2, 3} F (x) = y ∈ K : xRy ⇐⇒ y if x ∈ {2, 3} Then F (1) = F (6) = {1, 6} , F (2) = F (3) = {1, 2, 3, 6} . Therefore, F (x) is a subalgebra of K for all x ∈ A. Hence (F, A) is a soft Boolean algebra over K. Now, let (G, B) be a soft set over K, where B = K and G : B −→ P (K) is a set-valued function defined by G (x) = {y ∈ K : xR0 y ⇐⇒ mcd(x, y) ≤ mcm(x, y)} . Then G(1) = G(2) = G(3) = G(6) = {1, 2, 3, 6} = K. Hence G(x) is a subalgebra of K for all x ∈ B. Then (G, B) is a soft Boolean algebra over K. Example 2. Let M = {a, b, c} be a set and K = F U N (M, {0, 1}) be the set of all functions from M and to {0, 1}. Define the Boolean operations on K as follows for all f, g ∈ K: (f ∨ g)(x) = max {f (x), g(x)} , (f ∧ g)(x) = min {f (x), g(x)} and ¬f (x) = {0, 1} − {f (x)} for all x ∈ A. Then K together with these operations is a Boolean algebra and consists of elements {f1 , f2 , f3 , f4 , f5 , f6 , f7 , f8 } defined by f1 (a) = 0 f1 (b) = 0 f1 (c) = 0

f2 (a) = 1 f2 (b) = 1 f2 (c) = 1

f3 (a) = 1 f3 (b) = 0 f3 (c) = 0

f4 (a) = 1 f4 (b) = 1 f4 (c) = 0

f5 (a) = 1 f5 (b) = 0 f5 (c) = 1

f6 (a) = 0 f6 (b) = 1 f6 (c) = 1

f7 (a) = 0 f7 (b) = 0 f7 (c) = 1

f8 (a) = 0 f8 (b) = 1 f8 (c) = 0

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SOFT BOOLEAN ALGEBRA

5

Let (F, A) be a soft set over K, where A = M and F : A −→ P (K) is a set-valued function defined by    if x = a    f (a) = f (c) f (b) = f (a) if x = b F (x) = f ∈ K : xRf ⇐⇒    f (c) = f (b) if x = c Then F (a) = {f1 , f2 , f5 , f8 } , F (b) = {f1 , f2 , f4 , f7 } and F (c) = {f1 , f2 , f3 , f6 } . Therefore, F (x) is a subalgebra of K for all x ∈ A. Hence (F, A) is a soft Boolean algebra over K. Example 3. Every Boolean algebra can be considered as a soft Boolean algebra. Theorem 1. Let (F, A) and (G, B) be two soft Boolean algebras over K. If A∩B 6= ˜ (G, B) is a soft Boolean algebra over K. ∅, then (F, A)∩ ˜ (G, B) = (H, C), where C = A ∩ B Proof. By Definition 8, we can write (F, A)∩ and H(x) = F (x) or G(x) for all x ∈ C. For a mapping H : C −→ P (K), (H, C) is a soft set over K. Since (F, A) and (G, B) are soft Boolean algebras over K, there exists an equality such that H(x) = F (x) or H(x) = G(x) for all x ∈ C. But in ˜ (G, B) either case, H(x) is a subalgebra of K for all x ∈ C. Hence (H, C) = (F A)∪ is a soft Boolean algebra over K.  Theorem 2. Let (F, A) and (G, B) be two soft Boolean algebras over K. If A and ˜ (G, B) is a soft Boolean algebra over K. B are disjoint, then (F, A)∪ ˜ (G, B) = (H, C), where C = A ∪ B Proof. By Definition 7, we can write (F, A)∪ and for all x ∈ C,  if x ∈ A − B  F (x) G(x) if x ∈ B − A H(x) =  F (x) ∪ G(x) if x ∈ A ∩ B Since A ∩ B = ∅, either x ∈ A − B or x ∈ B − A. Since (F, A) is a soft Boolean algebra over K, then H(x) = F (x) is a subalgebra of K for x ∈ A − B. Similarty, since (G, B) is a soft Boolean algebra over K, then H(x) = G(x) is a subalgebra of K for x ∈ B − A. Hence (H, C) is a soft Boolean algebra over K and so ˜ (G, B) is a soft Boolean algebra over K. (H, C) = (F, A)∪  Remark 2. Let K be a Boolean algebra and S be a non-empty family of subalgebras of K. Note that intersection of members of S is again a subalgebra of K. But this is not correct for union. So Theorem 2 does not hold in general if A ∩ B 6= ∅. Theorem 3. Let (F, A) and (G, B) be two soft Boolean algebras over K, then ˜ (G, B) is a soft Boolean algebra over K. (F, A)∧ Proof. By Definition 9, we have ˜ (G, B) = (H, A × B) (F, A)∧ where H(x, y) = F (x) ∩ G(y) for all x ∈ A and y ∈ B. Since (F, A) and (G, B) are soft Boolean algebras over K, then F (x) and G(y) are subalgebras of K for all x ∈ A, y ∈ B and so the intersection F (x) ∩ G(y) is also a subalgebra of K. Hence H(x, y) ˜ (G, B) = (H, A × B) is is a subalgebra of K for all x ∈ A and y ∈ B. Then (F, A)∧ a soft Boolean algebra over K. 

807

RIDVAN SAHIN ET AL 803-814

J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 18, NO.5, 2015, COPYRIGHT 2015 EUDOXUS PRESS, LLC

6

¨ ¸ UK ¨ RIDVAN S ¸ AHIN AND AHMET KUC

Theorem 4. Let (F, A) be a soft Boolean algebra over K. If B is a subset of A, then (F |B , B) is a soft Boolean algebra over K. Proof. Since (F, A) is a soft Boolean algebra over K, then F (x) is a subalgebra of K for all x ∈ A. Therefore, F (x) is also a subalgebra of K for all x ∈ B ⊆ A. Hence (F |B , B) is a soft Boolean algebra over K.  Proposition 1. Let (G, B) be a soft set over K and (Fα , Aα ) be a soft Boolean algebra over K for α ∈ Λ, where Λ is an index set. Then h(G, B)i = ∩ {(Fα , Aα ) : (G, B) ⊆ (Fα , Aα ) } , where (Fα , Aα ) is a soft Boolean algebra over K for α ∈ Λ, is a soft Boolean algebra over K. We say that h(G, B)i is a soft Boolean algebra over K, which is generated by (G, B) . Definition 11. Let (F, A) and (G, B) be two soft Boolean algebras over K. Then ˜ (G, B) is a soft subalgebra of (F, A), denoted by (G, B) > pi (x)y i ; < > > :

i=0 9 P

qi (x)y i ;

y

x; (6)

y > x:

i=0

where pi (x) and qi (x), i = 0; 2; :::; 9 are unknown coe cients of Kx (y) and are given as p0 (x)

=

p3 (x)

=

p4 (x)

0, p1 (x) =

1 362884x 725764

725782x3 + 362903x4

12x7 + 9x8

2x9 , p2 (x) = 0;

1 ( 43895295360x 43894206720 +87800025610x3 43910173465x4 + 10160696x5 5806148x7 + 1088673x8 6x9 ); 1 x(43896746880 = 87788413440 87820346930x2 + 43950816165x3 30482088x4 + 4354668x6 1088709x7 + 14x8 );

p5 (x)

= p6 (x) = 0; 1 p7 (x) = x 362880 2177292x + 2903074x2 1088667x3 + 12x6 21947103360 1 p8 (x) = x 362884 + 725782x2 362903x3 + 12x6 9x7 + 2x8 ; 29262804480 1 p9 (x) = 362882 362880x 18x3 + 21x4 12x7 + 9x8 2x9 ; 131682620160 1 x9 ; 362880 1 q1 (x) = 7315701120 x 3657870720 + 7315882560x2 1 q2 (x) = x7 ; 10080 q0 (x)

9x7 + 2x8 ;

=

3658062240x3 + 120960x6 + 90721x7 + 20160x8 ;

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Omar Abu Arqub 857-874

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J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 18, NO.5, 2015, COPYRIGHT 2015 EUDOXUS PRESS, LLC

q3 (x)

=

q4 (x)

=

q5 (x)

=

q7 (x)

=

q8 (x)

=

q9 (x)

=

1 43894206720 x 43895295360 87800025610x2 + 43910173465x3 + 5806148x6 1088673x7 + 6x8 ; 1 87788413440 x 43896746880 87820346930x2 + 43950816165x3 + 4354668x6 1088709x7 + 14x8 ; 1 4 1 3 x , q6 (x) = x ; 2880 4320 1 x 362880 + 2903074x2 1088667x3 + 12x6 9x7 + 2x8 ; 21947103360 1 x 362880 + 725782x2 362903x3 + 12x6 9x7 + 2x8 ; 29262804480 1 x 362880 + 18x2 21x3 + 12x6 9x7 + 2x8 : 131682620160

Proof. The proof of the completeness and reproducing property of W25 [0; 1] is similar to the proof in [17]. Let us now nd out the expression form of the reproducing kernel function Kx (y) in the space W25 [0; 1]. Clearly, 4 R 1 (5) P (10) (5) (9 i) 4 i (i) 3R1 z (y) Kx (y) dy. Hence, hz (y) ; Kx (y)iW 5 = z (y) Kx (y) dy = ( 1) z (y) Kx (y) jy=1 y=0 + ( 1) 0 0 2

i=0

2 P

(i)

z (i) (0) Kx (0) +

1 P

(i)

z (i) (1) Kx (1) +

4 i

( 1)

(9 i)

z (i) (y) Kx

i=0

i=0

i=0 W25

4 P

(y) jy=1 y=0

R1 0

(10)

z (y) Kx

(y) dy. Since Kx (y) 2

[0; 1], it follows that Kx (0) = Kx00 (0) = Kx (1) = Kx00 (1) = 0. Further, since z (x) 2 W25 [0; 1], one obtains (8) (i) (i) z (0) = z 00 (0) = z (1) = z 00 (1) = 0. Thus, if Kx (0) = Kx (1) = 0, i = 5; 6, Kx0 (0) + Kx (0) = 0, and R 1 (8) (10) Kx (y) dy. Now, for each x 2 [0; 1], if Kx (y) also Kx0 (1) Kx (1) = 0, then hz (y) ; Kx (y)iW 5 = 0 z (y) 2

(10)

satis es Kx (y) = (x y), where is the dirac-delta function, then hz (y) ; Kx (y)iW 5 = z (x). Obviously, 2 Kx (y) is the reproducing kernel function of W25 [0; 1]. Let us now utilizing the expression form of the reproducing (10) kernel function Kx (y). The characteristic equation of Kx (y) = (y x) is 10 = 0, and their characteristic values are = 0 with 10 multiple roots. So, let the expression form of the reproducing kernel function Kx (y) be (m) (m) as de ned in Eq. (6). On the other aspect as well, let Kx (y) satis es Kx (x + 0) = Kx (x 0), m = 0; 1; :::; 8. (10) Integrating Kx (y) = (x y) from x " to x + " with respect to y and let " ! 0, we have the jump (9) (9) (9) degree of Kx (y) at y = x given by Kx (x 0) Kx (x + 0) = 1. Through the last descriptions and by using MATHEMATICA 7:0 software package, the unknown coe cients pi (x) and qi (x), i = 0; 2; :::; 9 of Eq. (6) can be obtained as given in the theorem. This completes the proof. De nition .4 [18] The inner product space W21 [0; 1] is de ned as W21 [0; 1] = fz (x) : z is absolutely continuous 1 real-valued function on [0; 1] and z 0 2 L2 [0; 1]g. The inner product q and the norm in W2 [0; 1] are de ned as R1 0 hz1 (x) ; z2 (x)iW 1 = 0 (z1 (x) z20 (x) + z1 (x) z2 (x)) dx and jjzjjW 1 = hz (x) ; z (x)iW 1 respectively, where z1 ; z2 2 2 2 2 n R o 1 W21 [0; 1] and L2 [0; 1] = z : 0 z 2 (x) dx < 1 . Theorem .2 [18] The Hilbert space W21 [0; 1] is a complete reproducing kernel and its reproducing kernel function Rx (y) can be written as ( p0 (x)ey + p1 (x)e y ; y x; Rx (y) = q0 (x)ey + q1 (x)e y ; y > x: where pi (x) and qi (x), i = 0; 1 are unknown coe cients of Rx (y) and are given as p0 (x)

=

p1 (x)

=

1 cosh (x 2 sinh (1) 1 cosh (x 2 sinh (1)

1) ; 1) ;

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Omar Abu Arqub 857-874

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J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 18, NO.5, 2015, COPYRIGHT 2015 EUDOXUS PRESS, LLC

q0 (x)

=

q1 (x)

=

1 ex 4 sinh (1) 1 e1 4 sinh (1)

1

+e

1 x

x

+ e1+x :

;

In fact, it easy to see that q0 (x)ey + q1 (x)e y = p0 (y) ex + p1 (y) e x . As a result, the reproducing kernel function posses some important properties such as: it is symmetric, unique, and nonnegative. The reader is asked to refer to [12{31] in order to know more details about reproducing kernel function including its mathematical properties, types and kinds, applications, method of calculations, and others.

5

Problem formulation in the space W25 [0; 1]

Problem formulation is normally the most important part of the process. It is the selection of linear operator, orthogonal basis, and orthonormal basis. In this section, Eqs. (3) and (4) are rst formulated as a di erential linear operator based on the spaces W25 [0; 1] and W21 [0; 1]. After that, the Gram-Schmidt orthogonalization process of 1 f i (x)gi=1 is presented. In order to apply the RKHS method, as in [12, 13, 17{31], we rs de ne a di erential linear operator L as L : W25 [0; 1] ! W21 [0; 1] such that Lv (x) = v (4) (x). Thus, discretized form of Eqs. (3) and (4) can be obtained as follows: 000

00

0

Lv (x) = F x; (v + ) (x) ; (v + ) (x) ; (v + ) (x) ; (v + ) (x) + [T (v + )] (x) ;

(7)

subject to the two-point boundary conditions v (0) = 0, v (1) = 0;

(8)

v 00 (0) = 0, v 00 (1) = 0; where v and

are as given in Algorithm 1.

Theorem .3 The operator L : W25 [0; 1] ! W21 [0; 1] is bounded and linear. 2

2

Proof. For boundedness, we need to prove kLv(x)kW 1

M kLv(x)kW 5 ; where M is a positive constant. From

2

2

2

the de nition of the inner product and the norm of W21 [0; 1], we have k(Lv) (x)kW 1 = h(Lv) (x) ; (Lv) (x)iW 1 = 2 2 o R1 n 2 0 2 (Lv) (x) + [(Lv) (x)] dx: By reproducing property of Kx (y), we have v(x) = hv (y) ; Kx (y)iW 5 , (Lv) (x) = 2

0

hv (y) ; (LKx ) (y)iW 5 , and (Lv)0 (x) = hv (y) ; (LKx )0 (y)iW 5 . Again, by Schwarz inequality, we get 2

2

j(Lv)(x)j = hv (x) ; (LKx ) (x)iW 5

kLKx (x)kW 5 kv (x)kW 5 = M1 kv (x)kW 5 ; M1 > 0;

2

2

j(Lv)0 (x)j = hv (x) ; (LKx )0 (x)iW 5 2

R1 n

2

0

2

2

2

k(LKx )0 (x)kW 5 kv (x)kW 5 = M2 kv (x)kW 5 ; M2 > 0: 2

2

2

2

o

2

(Lv) (x) + [(Lv) (x)] dx (M12 + M22 ) kv (x)kW 5 or k(Lv)(x)kW 1 M kv (x)kW 5 ; Thus, k(Lv)(x)kW 1 = 2 2 2 2 0 p where M = M12 + M22 . The linearity part is obvious. This complete the proof. After that, we construct an orthogonal function system of W25 [0; 1] as follows: put 'i (x) = Rxi (x) and 1 i (x) = Li ' (x), where fxi gi=1 is dense on [0; 1] and L is the adjoint operator of L. In terms of the properties of reproducing kernel function Kx (y), one can obtains hv (x) ; i (x)iW 5 = hv (x) ; L 'i (x)iW 5 = hLv (x) ; 'i (x)iW 1 = 2

Lv(xi ), i = 1; 2; :::. In fact, the orthonormal function system 1 the Gram-Schmidt orthogonalization process of f i (x)gi=1 as i

(x) =

i P

ik

k

i (x)

1 i=1

2

2

of the space W25 [0; 1] can be derived from

(x) ;

(9)

k=1

where

ik

are orthogonalization coe cients and are given as follows: 863

ij

=

1 k

1k

for i = j = 1,

ij

=

1 di

Omar Abu Arqub 857-874

for

8

J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 18, NO.5, 2015, COPYRIGHT 2015 EUDOXUS PRESS, LLC

i = j 6= 1, and

ij

1 di

=

iP1

cik

kj

for i > j such that di =

k=j

s

k

ik

iP1

2

k=1

c2ik , cik =

i;

k W5, 2

and f

i

1

(x)gi=1

is the orthonormal system in the space W25 [0; 1]. Through the next theorem the subscript y by the operator L (Ly ) indicates that the operator L applies to the function of y. 1

Theorem .4 If fxi gi=1 is dense on [0; 1], then f Ly Kx (y)jy=xi .

i

1

(x)gi=1 is a complete function system of W25 [0; 1] and

i

(x) =

Proof. Clearly, i (x) = Li ' (x) = hLi ' (y) ; Kx (y)iW 5 = h'i (y) ; Ly Kx (y)iW 1 = Ly Kx (y)jy=xi 2 W25 [0; 1]. 2 2 Now, for each xed v (x) 2 W25 [0; 1], let hv (x) ; i (x)iW 5 = 0, i = 1; 2; :::. In other word, hv (x) ; i (x)iW 5 = 2 2 1 hv (x) ; L 'i (x)iW 5 = hLv (x) ; 'i (x)iW 1 = Lv (xi ) = 0. Note that fxi gi=1 is dense on [0; 1], therefore Lv (x) = 0. 2 2 It follows that v (x) = 0 from the existence of L 1 . So, the proof of the theorem is complete. M jjv(x)jjW 5 , i = 0; 1; 2; 3; 4, Lemma .1 If v (x) 2 W25 [0; 1], then there exists M > 0 such that v (i) (x) C 2 where jjv (x)jjC = max jv(x)j. a x b D E (i) Proof. For any x; y 2 [0; 1], we have v (i) (x) = v(y); Kx (y) 5 , i = 0; 1; 2; 3; 4: By the expression of Kx (y), it W2 E D (i) (i) (i) Kx (x) 5 kv(x)kW 5 follows that Kx (y) 5 Mi , i = 0; 1; 2; 3; 4. Thus, v (i) (x) = v (x) ; Kx (x) 5

6

M jjv(x)jjW 5 , i = 0; 1; 2; 3; 4, where M =

C

2

2

W2

W2

W2

Mi kv(x)kW 5 , i = 0; 1; 2; 3; 4. Hence, v (i) (x)

2

max

i=0;1;2;3;4

fMi g.

Representation of exact and approximate solutions

In this section, we will give the representation form of exact and approximate solutions of Eqs. (3) and (4) in the space W25 [0; 1]. After that, an iterative formulas of obtaining approximate solution is represented for both linear and nonlinear case. Theorem .5 For each v (x) 2 W25 [0; 1], the series 1

1 P

v (x) ;

i

i=1

(x)

i

(x) is convergent in the sense of the norm

of W25 [0; 1]. On the other hand, if fxi gi=1 is dense on [0; 1], then the following are hold: (i) The exact solution of Eqs. (7) and (8) could be represented by v (x) =

1 P i P

i=1 k=1

ik

(10) 000

00

0

F xk ; (v + ) (xk ) ; (v + ) (xk ) ; (v + ) (xk ) ; (v + ) (xk ) + [T (v + )] (xk )

i

(x) :

(ii) The approximate solution of Eqs. (7) and (8) vn (x) =

n P i P

i=1 k=1

ik

(11) 000

00

0

F xk ; (v + ) (xk ) ; (v + ) (xk ) ; (v + ) (xk ) ; (v + ) (xk ) + [T (v + )] (xk )

i (x) ;

and its derivative up to order four are converging uniformly to the exact solution v (x) and all its derivative as n ! 1, respectively. Proof. For the rst part, let v (x) be solution of Eqs. (7) and (8) in the space W25 [0; 1]. Since v (x) 2 W25 [0; 1], 1 P 1 5 v (x) ; i (x) i (x) is the Fourier series about normal orthogonal system i (x) i=1 , and W2 [0; 1] is the i=1

Hilbert space, then the series

1 P

i=1

v (x) ;

i

(x)

i

(x) is convergent in the sense of k kW 5 . On the other aspect as 2

864

Omar Abu Arqub 857-874

J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 18, NO.5, 2015, COPYRIGHT 2015 EUDOXUS PRESS, LLC

9

well, using Eq. (9), we have v (x)

=

1 P

v (x) ;

i

i=1

=

1 P i P

i=1 k=1

=

1 P i P

i=1 k=1

=

1 P i P

i=1 k=1

=

i

W25

(x)

ik

hv (x) ;

ik

hv (x) ; L 'k (x)iW 5

ik

hLv (x) ; 'k (x)iW 1

1 P i P

ik

i 1 P P

ik

i=1 k=1

(x)

k

(x)iW 5

(x)

i

2

i

2

i

2

000

(x)

(x)

00

0

F x; (v + ) (x) ; (v + ) (x) ; (v + ) (x) ; (v + ) (x) + [T (v + )] (x) ; 'k (x)

=

i=1 k=1

000

00

i

W21

0

F xk ; (v + ) (xk ) ; (v + ) (xk ) ; (v + ) (xk ) ; (v + ) (xk ) + [T (v + )] (xk )

i

(x)

(x) :

Therefore, the form of Eq. (10) is the exact solution of Eqs. (7) and (8). For the second part, it easy to see that by Lemma .1, for any x 2 [0; 1] vn(i) (x)

v (i) (x)

D

=

E v (x) ; Kx(i) (x)

vn (x)

Kx(i) (x)

kvn (x)

W25

Mi kvn (x)

W25

v (x)kW 5 2

v (x)kW 5 , i = 0; 1; 2; 3; 4; 2

where Mi , i = 0; 1; 2; 3; 4 are positive constants. Hence, if kvn (x) (i) vn

v (x)kW 5 ! 0 as n ! 1, the approximate 2

(x), i = 0; 1; 2; 3; 4 are converge uniformly to the exact solution v (x) and all its derivative, solution vn (x) and respectively. So, the proof of the theorem is complete. Next, we will mention the following remark about the exact and approximate solutions of Eqs. (3) and (4). Remark .2 [12, 13, 17{31] In order to apply the RKHS technique for solve Eqs. (3) and (4), we de ne an initial guess approximation function as v0 (x1 ) = v (x1 ) = 0. On the other hand, we have the following two cases based on the form of Eq. (11) and the structure of the functions F , G1 , and G2 in Eq. (3). Case 1: If Eq. (3) is linear, then the approximate solution can be obtained directly as follows: vn (x) =

n P i P

i=1 k=1

ik fF

+ [T (vk

1

xk ; (vk

1

+ )] (xk )g

000

+ ) (xk ) ; (vk i

1

00

+ ) (xk ) ; (vk

1

0

+ ) (xk ) ; (vk

1

+ ) (xk )

(x) :

Case 2: If Eq. (3) is nonlinear, then the approximate solution can be obtained immediately as follows: vnN (x) =

N P i P

i=1 k=1

+ [T (vn

ik fF 1

xk ; (vn

+ )] (xk )g

1

000

+ ) (xk ) ; (vn i

1

00

+ ) (xk ) ; (vn

1

0

+ ) (xk ) ; (vn

1

+ ) (xk )

(x) :

The reader is asked to refer to [12,13,17{31] in order to know more details about these two case, including their derivation, their importance, and their relationship to the exact solution.

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7

Error estimation and error bound

When solving practical problems, it is necessary to take into account all the errors of the measurements. Moreover, in accordance with the technical progress and the degree of complexity of the problem, it becomes necessary to improve the technique of measurement of quantities. Considerable errors of measurement become inadmissible in solving complicated mathematical, physical, and engineering problems. The reliability of the numerical result will depend on an error estimate or bound, therefore the analysis of error and the sources of error in numerical methods is also a critically important part of the study of numerical technique. In this section, we derive an error bounds for the present method and problem in order to capture behavior of the solution. In the next theorem, we show that the error of approximate solution is monotone decreasing, while the next lemma is presented in order to prove the recent theorem. 2

Theorem .6 Let "2s;n = jjv (x) vn (x)jjW 3 , where v (x) and vn (x) are given by Eq. (10) and Eq. (11), respectively. 2 Then, the sequence of numbers f"n g are monotone decreasing in the sense of the norm of W25 [0; 1] and "n ! 0 as n ! 1. 1 P Proof. Since, v (x) = v (x) ; i (x) W 5 i (x) it obvious that 2

i=1

"2n = jjv (x) "2n

1

Clearly, "n

1 P

2

vn (x)jjW 5 = 2

= jjv (x)

vn

i=n+1

2

1

2

v (x) ;

(x)jjW 5 = 2

1 P

i

(x)

W25

i

=

(x) W25

1 P

i=n

i (x)

W25

=

i (x)

i

(x)

v (x) ;

i

(x)

i=n+1

2

v (x) ;

2

v (x) ;

1 P

i=n

W25

;

W25

2

:

W25

"n , and consequently f"n g is monotone decreasing in the sense of k kW 5 . On the other aspect as 2 1 P well, by Theorem .5, we know that v (x) ; i (x) W 5 i (x) is convergent in the sense of k kW 5 . Thus, we have "2n =

1

1 P

2

2

i=1

2

v (x) ;

i

i=n+1

(x)

W25

! 0 or "n ! 0. This complete the proof.

Lemma .2 Let v (x) is the exact solution of Eqs. (7) and (8), vn (x) is the approximate solution of v (x), and T = xk+1 = 2ki : k = 0; 1; :::; 2i . Then, Lv (xk ) = Lvn (xk ), for n = 2i + 1 and xk 2 T . Pn Proof. Set the projective operator Pn : W25 [0; 1] ! f m=1 cm m (x) ; cm 2 Rg, Then, we have Lvn (xk ) = hvn ( ) ; Lxk Fxk ( )iW 5 = hvn ( ) ; k ( )iW 5 = hPn v ( ) ; k ( )iW 5 = hv ( ) ; Pn k ( )iW 5 = hv ( ) ; k ( )iW 5 = 2 2 2 2 2 hv ( ) ; Lxk Fxk ( )iW 5 = Lxk hv ( ) ; Fxk ( )iW 5 = Lxk v (xk ) = Lv (xk ). 2

2

Theorem .7 Let v (x) is the exact solution of Eqs. (7) and (8), vn (x) is the approximate solution of v (x), and T = xk+1 = 2ki : k = 0; 1; :::; 2i . Then, jv (x) vn (x)j < M n , where M is the product of the sup of convergent basis 1 i P P 000 00 0 and the ik fF xk ; (v + ) ( k ) ; (v + ) ( k ) ; (v + ) ( k ) ; (v + ) ( k ) + [T (v + )] ( k )g i ( ) i=n+1 k=1

W25

maximum of determinate function

@ @

K ( )

W25

about the variable in [0; 1].

Proof. Since jv (x) vn (x)j = L 1 (Lv (x) Lvn (x)) and for every given x 2 [0; 1], there is always x0 2 T satisfying x0 < x and x x0 = n1 . On the other hand, Lemma .2 and x0 2 T implying that Lv (x0 ) = Lvn (x0 ). So, we obtain jLv (x)

Lvn (x)j = j(Lv (x)

Lv (x0 ))

(Lvn (x)

Lvn (x0 ))j :

(12)

By applying the reproducing kernel properties v (x) = hv ( ) ; Rx ( )iW 5 and Lv (x) = hv ( ) ; LKx ( )iW 5 to Eq. 2

866

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Omar Abu Arqub 857-874

J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 18, NO.5, 2015, COPYRIGHT 2015 EUDOXUS PRESS, LLC

11

(12), we conclude Lv (x)

Lvn (x)

=

(Lv (x)

Lv (x0 ))

= hv ( ) ; LKx ( ) = hv ( )

(Lvn (x)

Lvn (x0 ))

LKx0 ( )iW 5

hvn ( ) ; LKx ( )

2

vn ( ) ; LKx ( )

LRx0 ( )iW 5 2

LKx0 ( )iW 5 : 2

But on the other aspect as well, we have jv (x)

vn (x)j =

1

L

(Lv (x)

v( )

jjv ( )

1

vn ( ) ; L

hv ( )

=

Lvn (x)) LKx ( )

vn ( ) ; Kx ( )

1

L

LKx0 ( )

W25

Kx0 ( )iW 5 2

vn ( )jjW 5 jjKx ( ) 2

Kx0 ( )jjW 5 : 2

Here, we take the norm of jjKx ( ) Kx0 ( )jjW 5 for the variable and the function Kx ( ) is derived on x in [0; 1]. 2 So, we have Kx ( ) Kx0 ( ) = @@ K ( ) (x x0 ). Hence, we can write jv (x)

vn (x)j

jjv ( ) =

k

vn ( )jjW 5 2

1 P

i P

i=n+1 k=1

ik fF

+ [T (v + )] ( k )g =

@ @

K ( ) (x

x0 )

W25

000

00

0

xk ; (v + ) ( k ) ; (v + ) ( k ) ; (v + ) ( k ) ; (v + ) ( k ) i

( ) kW25 k

@ @

K ( ) kW25 (x

x0 )

M n :

So, the proof of the theorem is complete.

8

Software libraries and numerical outcomes

Software packages have great capabilities for solving mathematical, physical, and engineering problems. Sometimes, it is very di cult to solve these problems analytically, so it is required to obtain an e cient approximate solution. Thus, some software mathematical packages such as MATHEMATICA or MAPLE can be helpful in visualizing the behavior of the solutions of such problems. Indeed, throughout the whole paper we used MATHEMATICA 7:0 software package for numerical experiment. The object of the next algorithm is to implement a procedure to solve Eqs. (1) and (2) in numeric form in terms of their grid nodes based on the use of RKHS method. Algorithm 2 To approximate the solution of Eqs. (1) and (2), we do the following steps: Input: The endpoints of [0; 1]; the integers n and N ; the kernel functions Kx (y) and Rx (y); the di erential operator L; the function F ; the operator [T u]. Output: Approximate solution un (x) or uN n (x) of Eqs. (1) and (2). Step 1: Fixed x in [0; 1] and set y 2 [0; 1]; If y

x then set Kx (y) =

9 P

pi (x)y i ;

i=0

else set Kx (y) =

9 P

qi (x)y i ;

i=0

For i = 1; 2; :::; n do the following: Set xi =

i 1 n 1;

867

Omar Abu Arqub 857-874

J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 18, NO.5, 2015, COPYRIGHT 2015 EUDOXUS PRESS, LLC

Set

i (x)

= Ly [Kx (y)]y=xi ;

Output: the orthogonal function system

i (x).

Step 2: For i = 2; 3:::; n and j = 1; 2:::; i do the following: s iP1 2 Set di = k i k c2ik ; k=1

If j 6= i then set

ij

iP1

1 di

= 1 di ;

else set

ij

=

else set

11

=

1 k

1k

cik

kj ;

k=j

;

Output: the orthogonalization coe cients Step 3: For i = 2; 3:::; n and k = 1; 2:::; i Set

i

(x) =

i P

ik

k

ij .

1 do the following:

(x);

k=1

Set cik =

i;

k W5; 2

Output: the orthonormal function system

i

(x).

Step 4: Set v0 (x1 ) = v (x1 ) = 0; For i = 1; 2; :::n do the following: If F and [T u] are linear then set Bi =

i P

ik fF (xk ; (vk 1

k=1

(vk

set vi (x) =

0

+ ) (xk ) ; (vk

1 i P

Bi

i=1

i

000

+ ) (xk ) ; (vk

1

00

+ ) (xk ) ;

+ ) (xk )) + [T (vk

1

1

+ )] (xk )g;

(x);

else for i = 1; 2; :::N do the following: set xi = set Bi =

i 1 N 1; i P

ik f(F xk ; (vn 1

k=1

(vn

1

set vni (x) =

i P

i=1

0

1

1

+ )] (xk )g

+ ) (xk ) ; (vn

+ [T (vn Bi

000

+ ) (xk ) ; (vn

i

1

00

+ ) (xk ) ;

+ ) (xk )) i

(x);

(x);

Output: the approximate solution vn (x) or vnN (x) of Eqs. (3) and (4). N Step 5: Use the transformation un (x) = vn (x) + (x) or uN n (x) = vn (x) + (x);

Output: the approximate solution un (x) or uN n (x) of Eqs. (1) and (2). Step 6: Stop. 868

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12

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13

Next, we propose few numerical simulations for solving some speci c examples of Eqs. (1) and (2). However, we apply the techniques described in the previous sections to some linear and nonlinear test examples in order to demonstrate the e ciency, accuracy, and applicability of the proposed method. Results obtained by the method are compared with the analytical solution of each example by computing the exact and relative errors and are found to be in good agreement with each other. Problem .1 Consider the following linear equation: u(4) (x) =

u00 (x) +

4

u(x) + f (x) + [T u] (x) ;

in which the mixed operator is given as R1

[T u] (x) =

0

Rx x2 tu0 (t)dt + (x + 1)tu(t)dt; 0

and subject to the boundary conditions u (0) = 0; u (1) = 0; u00 (0) = 0; u00 (1) = 0; where 0

t l, all x ∈ X and all w ∈ Y . It follows from (2.6) that the sequence {2k g( 2ak )} is Cauchy for each x ∈ X. Since Y is 2-Banach space, the sequence {2k g( 2ak )} converges. So one can define the mapping A : X → Y by   x A(x) := lim 2j g j j→∞ 2 for all x ∈ X. That is,

 

x

j

lim 2 g j − A(x), w = 0

j→∞ 2 for all x ∈ X and all w ∈ Y . Moreover, letting 0 = 0 and passing the limit m → ∞ in (2.6), we get (2.2). By (2.1),

 

k

x y (2p + 8)2k

kDA(x, y), wk = lim 2 Dg k , k , w

≤ lim θkxkp kykq kwk = 0 (p+q)k k→∞ k→∞

2 2 2 and so kDA(x, y), wk = 0 for all x, y ∈ X and all w ∈ Y . Hence DA(x, y) = 0 for all x, y ∈ X. Since g : X → Y is odd, A : X → Y is odd. So the mapping A : X → Y is additive. Now, let T : X → Y be another additive mapping satisfying (2.2). Then we have

   

l

x x l

kA(x) − T (x), wk = 2 A l − 2 T , w

l 2 2

  

  

   

l

l

x x x x

≤ 2 A l − g l , w + 2 T −g l , w

l 2 2 2 2 l p 4+2 2 ≤ 2 p+q θkxkp+q kwk, (p+q)l 2 −22 which tends to zero as l → ∞ for all x ∈ X and all w ∈ Y . So we can conclude that A(a) = T (a) for all a ∈ X. This proves the uniqueness of A.

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J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 18, NO.5, 2015, COPYRIGHT 2015 EUDOXUS PRESS, LLC

AQCQ-FUNCTIONAL EQUATION IN NORMED 2-BANACH SPACES

Therefore, A : X → Y is a unique additive mapping satisfying (2.2), as desired.



Theorem 2.2. Let θ ∈ [0, ∞), p, q ∈ (0, ∞) with p + q < 1 and let f : X → Y be an odd mapping satisfying (2.1). Then there is a unique additive mapping A : X → Y such that kf (2x) − 8f (x) − A(x), wk ≤

4 + 2p θkxkp+q kwk p+q 2−2

for all x ∈ X and all w ∈ Y . Proof. Replacing y by x and letting g(x) := f (2x) − 8f (x) in (2.5), we get



1 4 + 2p

g(x) − g (2x) , w ≤ θkxkp+q kwk

2

2

for all x ∈ X and all w ∈ Y . The rest of the proof is similar to the proof of Theorem 2.1.



Theorem 2.3. Let θ ∈ [0, ∞), p, q ∈ (0, ∞) with p + q > 3 and let f : X → Y be an odd mapping satisfying (2.1). Then there is a unique cubic mapping C : X → Y such that 4 + 2p kf (2x) − 2f (x) − C(x), wk ≤ p+q θkxkp+q kwk 2 −8 for all x ∈ X and all w ∈ Y . Proof. Replacing y by

x 2

and letting g(x) := f (2x) − 2f (x) in (2.5), we get

 

4 + 2p x

≤

g(x) − 8g θkxkp+q kwk , w

p+q

2

2

for all x ∈ X and all w ∈ Y . The rest of the proof is similar to the proof of Theorem 2.1.



Theorem 2.4. Let θ ∈ [0, ∞), p, q ∈ (0, ∞) with p + q < 3 and let f : X → Y be an odd mapping satisfying (2.1). Then there is a unique cubic mapping C : X → Y such that kf (2x) − 2f (x) − C(x), wk ≤

4 + 2p θkxkp+q kwk 8 − 2p+q

for all x ∈ X and all w ∈ Y . Proof. Replacing y by x and letting g(x) := f (2x) − 2f (x) in (2.5), we get



1 4 + 2p

g(x) − g (2x) , w ≤ θkxkp+q kwk

8

8

for all x ∈ X and all w ∈ Y . The rest of the proof is similar to the proof of Theorem 2.1.



Now we prove the superstability of the AQCQ-functional equation (1.3) in normed 2-Banach spaces for an odd mapping case. Theorem 2.5. Let θ ∈ [0, ∞), p, q, r ∈ (0, ∞) with r 6= 1 and let f : X → Y be an odd mapping such that kDf (x, y), wk ≤ θkxkp kykq kwkr

(2.7)

for all x, y ∈ X and all w ∈ Y . Then f : X → Y is realized as the sum of an additive mapping and a cubic mapping.

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CHOONKIL PARK ET AL 875-884

J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 18, NO.5, 2015, COPYRIGHT 2015 EUDOXUS PRESS, LLC

C. PARK, S.Y. JANG, R. SAADATI, AND D.Y. SHIN

Proof. Replacing w by sw in (2.7) for s ∈ R\ {0}, we get kDf (x, y), swk ≤ θkxkp kykq kswkr and so p

q

kDf (x, y), wk ≤ θkxk kyk kwk

r |s|

r

(2.8)

|s|

for all x, y ∈ X, all w ∈ Y and all s ∈ R\ {0}. If r > 1, then the right side of (2.8) tends to kf (x + y + z), wk as s → 0. If r < 1, then the right side of (2.8) tends to kf (x + y + z), wk as s → +∞. Thus kDf (x, y), wk = 0 for all x, y ∈ X and all w ∈ Y . By [13, Lemma 2.2], f : X → Y is realized as the sum of an additive mapping and a cubic mapping.  3. Hyers-Ulam stability of the AQCQ-functional equation (1.3) in normed 2-Banach spaces: even mapping case In this section, we prove the Hyers-Ulam stability of the AQCQ-functional equation (1.3) in normed 2-Banach spaces for an even mapping case. Theorem 3.1. Let θ ∈ [0, ∞), p, q ∈ (0, ∞) with p + q > 2 and let f : X → Y be an even mapping satisfying f (0) = 0 and (2.1). Then there is a unique quadratic mapping Q : X → Y such that 4 + 2p θkxkp+q kwk kf (2x) − 16f (x) − Q(x), wk ≤ p+q 2 −4 for all x ∈ X and all w ∈ Y . Proof. Letting x = y in (2.1), we get kf (3y) − 6f (2y) + 15f (y), wk ≤ θkykp+q kwk for all y ∈ X and all w ∈ Y . Replacing x by 2y in (2.1), we get

(3.1)

kf (4y) − 4f (3y) + 4f (2y) + 4f (y), wk ≤ 2p θkykp+q kwk for all y ∈ X and all w ∈ Y . By (3.1) and (3.2),

(3.2)

kf (4y) − 20f (2y) + 64f (y), wk (3.3) ≤ k4(f (3y) − 6f (2y) + 15f (y)), wk + kf (4y) − 4f (3y) + 4f (2y) + 4f (y), wk = 4kf (3y) − 6f (2y) + 15f (y), wk + kf (4y) − 4f (3y) + 4f (2y) + 4f (y), wk ≤ (4 + 2p )θkykp+q kwk for all y ∈ X and all w ∈ Y . Replacing y by x2 and letting g(x) := f (2x) − 16f (x) in (3.3), we get

 

4 + 2p x

g(x) − 4g

, w ≤ p+q θkxkp+q kwk

2 2 for all x ∈ X and all w ∈ Y . The rest of the proof is similar to the proof of Theorem 2.1. 

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CHOONKIL PARK ET AL 875-884

J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 18, NO.5, 2015, COPYRIGHT 2015 EUDOXUS PRESS, LLC

AQCQ-FUNCTIONAL EQUATION IN NORMED 2-BANACH SPACES

Theorem 3.2. Let θ ∈ [0, ∞), p, q ∈ (0, ∞) with p + q < 2 and let f : X → Y be an even mapping satisfying f (0) = 0 and (2.1). Then there is a unique quadratic mapping Q : X → Y such that 4 + 2p kf (2x) − 16f (x) − Q(x), wk ≤ θkxkp+q kwk 4 − 2p+q for all x ∈ X and all w ∈ Y . Proof. Replacing y by x and letting g(x) := f (2x) − 16f (x) in (3.3), we get



1 4 + 2p

g(x) − g (2x) , w ≤ θkxkp+q kwk

4 4 for all x ∈ X and all w ∈ Y . The rest of the proof is similar to the proof of Theorem 2.1.  Theorem 3.3. Let θ ∈ [0, ∞), p, q ∈ (0, ∞) with p + q > 4 and let f : X → Y be an even mapping satisfying f (0) = 0 and (2.1). Then there is a unique quartic mapping R : X → Y such that 4 + 2p kf (2x) − 4f (x) − R(x), wk ≤ p+q θkxkp+q kwk 2 − 16 for all x ∈ X and all w ∈ Y . Proof. Replacing y by x2 and letting g(x) := f (2x) − 4f (x) in (3.3), we get

 

x 4 + 2p

g(x) − 16g

≤ θkxkp+q kwk , w

2 2p+q for all x ∈ X and all w ∈ Y . The rest of the proof is similar to the proof of Theorem 2.1.  Theorem 3.4. Let θ ∈ [0, ∞), p, q ∈ (0, ∞) with p + q < 4 and let f : X → Y be an even mapping satisfying f (0) = 0 and (2.1). Then there is a unique quartic mapping R : X → Y such that 4 + 2p kf (2x) − 4f (x) − R(x), wk ≤ θkxkp+q kwk p+q 16 − 2 for all x ∈ X and all w ∈ Y . Proof. Replacing y by x and letting g(x) := f (2x) − 16f (x) in (3.3), we get



1 4 + 2p

g(x) − g (2x) , w

≤ θkxkp+q kwk

16 16 for all x ∈ X and all w ∈ Y . The rest of the proof is similar to the proof of Theorem 2.1.  Now we prove the superstability of the AQCQ-functional equation (1.3) in normed 2-Banach spaces for an even mapping case. Theorem 3.5. Let θ ∈ [0, ∞), p, q, r ∈ (0, ∞) with r 6= 1 and let f : X → Y be an even mapping satisfying f (0) = 0 and (2.7). Then f : X → Y is realized as the sum of a quadratic mapping and a quartic mapping. Proof. By the same reasoning as in the proof of Theorem 2.5, one can obtain kDf (x, y), wk = 0 for all x, y ∈ X and all w ∈ Y . By [10, Lemma 2.1], f : X → Y is realized as the sum of a quadratic mapping and a quartic mapping. 

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CHOONKIL PARK ET AL 875-884

J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 18, NO.5, 2015, COPYRIGHT 2015 EUDOXUS PRESS, LLC

C. PARK, S.Y. JANG, R. SAADATI, AND D.Y. SHIN (−x) (−x) Let fo (x) := f (x)−f and fe (x) := f (x)+f . Then fo is odd and fe is even. fo , fe 2 2 satisfy the functional equation (1.3). Let go (x) := fo (2x) − 2fo (x) and ho (x) := fo (2x) − 8fo (x). Then fo (x) = 16 go (x) − 61 ho (x). Let ge (x) := fe (2x) − 4fe (x) and he (x) := 1 1 fe (2x) − 16fe (x). Then fe (x) = 12 ge (x) − 12 he (x). Thus

1 1 1 1 f (x) = go (x) − ho (x) + ge (x) − he (x). 6 6 12 12 We summarize the above results as follows. Theorem 3.6. Let θ ∈ [0, ∞) and p, q ∈ (0, ∞) with p + q > 4. Let f : X → Y be a mapping satisfying f (0) = 0 and (2.1). Then there exist an additive mapping A : X → Y , a quadratic mapping Q : X → Y , a cubic mapping C : X → Y and a quartic mapping R : X → Y such that



1 1 1 1

f (x) − A(x) − Q(x) − C(x) − R(x), w

6 12 6 12 ! p p 4+2 4+2 4 + 2p 4 + 2p ≤ + + + θkxkp+q kwk 6(2p+q − 2) 12(2p+q − 4) 6(2p+q − 8) 12(2p+q − 16) for all x ∈ X and all w ∈ Y . Theorem 3.7. Let θ ∈ [0, ∞) and p, q ∈ (0, ∞) with p + q < 1. Let f : X → Y be a mapping satisfying f (0) = 0 and (2.1). Then there exist an additive mapping A : X → Y , a quadratic mapping Q : X → Y , a cubic mapping C : X → Y and a quartic mapping R : X → Y such that



1 1 1 1

f (x) − A(x) − Q(x) − C(x) − R(x), w

6 12 6 12 ! p p 4+2 4+2 4 + 2p 4 + 2p ≤ + + + θkxkp+q kwk p+q p+q p+q p+q 6(2 − 2 ) 12(4 − 2 ) 6(8 − 2 ) 12(16 − 2 ) for all x ∈ X and all w ∈ Y . Theorem 3.8. Let θ ∈ [0, ∞), p, q, r ∈ (0, ∞) with r = 6 1 and let f : X → Y be a mapping satisfying f (0) = 0 and (2.7). Then f : X → Y is realized as the sum of an additive mapping, a quadratic mapping, a cubic mapping and a quartic mapping. Acknowledgments C. Park, S. Y. Jang, D. Y. Shin were supported by Basic Science Research Program through the National Research Foundation of Korea funded by the Ministry of Education, Science and Technology (NRF-2012R1A1A2004299), (NRF-2013-007226), and (NRF-2010-0021792), respectively. References [1] J. Aczel and J. Dhombres, Functional Equations in Several Variables, Cambridge Univ. Press, Cambridge, 1989. [2] M. Alimohammady and A. Sadeghi, Some new results on the superstability of the Cauchy equation on semigroups, Results Math. 63 (2013), 705–712. [3] J. M. Almira, A note on classical and p-adic Fr´echet functional equations with restrictions, Results Math. 63 (2013), 649–656. [4] T. Aoki, On the stability of the linear transformation in Banach spaces, J. Math. Soc. Japan 2 (1950), 64–66.

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CHOONKIL PARK ET AL 875-884

J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 18, NO.5, 2015, COPYRIGHT 2015 EUDOXUS PRESS, LLC

AQCQ-FUNCTIONAL EQUATION IN NORMED 2-BANACH SPACES

[5] I. Chang, M. Eshaghi Gordji, H. Khodaei and H. Kim, Nearly quartic mappings in β-homogeneous F -spaces, Results Math. 63 (2013), 529–541. [6] P. W. Cholewa, Remarks on the stability of functional equations, Aequationes Math. 27 (1984), 76–86. [7] S. Czerwik, On the stability of the quadratic mapping in normed spaces, Abh. Math. Sem. Univ. Hamburg 62 (1992), 59–64. [8] P. Czerwik, Functional Equations and Inequalities in Several Variables, World Scientific Publishing Company, New Jersey, Hong Kong, Singapore and London, 2002. [9] A. Ebadian and H. Ghobadipour, A fixed point approach to almost double derivations and Lie ∗double drivations, Results Math. 63 (2013), 409–423. [10] M. Eshaghi Gordji, S. Abbaszadeh and C. Park, On the stability of a generalized quadratic and quartic type functional equation in quasi-Banach spaces, J. Inequal. Appl. 2009, Article ID 153084, 26 pages (2009). [11] M. Eshaghi Gordji and A. Bodaghi, On the stability of quadratic double centralizers on Banach algebras, J. Comput. Anal. Appl. 13 (2011), 724–729. [12] M. Eshaghi Gordji, R. Farokhzad Rostami and S.A.R. Hosseinioun, Nearly higher derivations in unital C ∗ -algebras, J. Comput. Anal. Appl. 13 (2011), 734–742. [13] M. Eshaghi Gordji, S. Kaboli-Gharetapeh, C. Park and S. Zolfaghari, Stability of an additive-cubicquartic functional equation Adv. Difference Equat. 2009, Article ID 395693, 20 pages (2009). [14] M. Eshaghi Gordji, G. Kim, J. Lee and C. Park, Generalized ternary bi-derivations on ternary Banach algebras: a fixed point approach, J. Comput. Anal. Appl. 15 (2013), 45–54. [15] 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. [16] Z. Gajda, On stability of additive mappings, Internat. J. Math. Math. Sci. 14 (1991), 431–434. [17] P. Gˇavruta, A generalization of the Hyers-Ulam-Rassias stability of approximately additive mappings, J. Math. Anal. Appl. 184 (1994), 431–436. [18] S. G¨ahler, 2-metrische R¨ aume und ihre topologische Struktur, Math. Nachr. 26 (1963), 115–148. [19] S. G¨ahler, Lineare 2-normierte R¨ aumen, Math. Nachr. 28 (1964), 1–43. ¨ [20] S. G¨ahler, Uber 2-Banach-R¨ aume, Math. Nachr. 42 (1969), 335–347. [21] D. H. Hyers, On the stability of the linear functional equation, Proc. Natl. Acad. Sci. U.S.A. 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] K. Jun and H. Kim, The generalized Hyers-Ulam-Rassias stability of a cubic functional equation, J. Math. Anal. Appl. 274 (2002), 867–878. [25] K. Jun and Y. Lee, A generalization of the Hyers-Ulam-Rassias stability of the Pexiderized quadratic equations, J. Math. Anal. Appl. 297 (2004), 70–86. [26] S. Jung, Hyers-Ulam-Rassias Stability of Functional Equations in Mathematical Analysis, Hadronic Press lnc., Palm Harbor, Florida, 2001. [27] S. Lee, S. Im and I. Hwang, Quartic functional equations, J. Math. Anal. Appl. 307 (2005), 387–394. [28] J. Lee, S. Lee and C. Park, Fixed points and stability of the Cauchy-Jensen functional equation in fuzzy Banach algebras, J. Comput. Anal. Appl. 15 (2013), 692–698. [29] J. Lee, C. Park, Y. Cho and D. Shin, Orthogonal stability of a cubic-quartic functional equation in non-Archimedean spaces, J. Comput. Anal. Appl. 15 (2013), 572–583. [30] L. Li, G. Lu, C. Park and D. Shin, Additive functional inequalities in generalized quasi-Banach spaces, J. Comput. Anal. Appl. 15 (2013), 1165–1175. [31] G. Lu, Y. Jiang and C. Park, Additive functional equation in Fr´echet spaces, J. Comput. Anal. Appl. 15 (2013), 369–373. [32] C. Park, Homomorphisms between Poisson JC ∗ -algebras, Bull. Braz. Math. Soc. 36 (2005), 79–97.

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C. PARK, S.Y. JANG, R. SAADATI, AND D.Y. SHIN

[33] C. Park, K. Ghasemi, S. G. Ghaleh, S. Jang, Approximate n-Jordan ∗-homomorphisms in C ∗ algebras, J. Comput. Anal. Appl. 15 (2013), 365–368. [34] 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. [35] W. Park, Approximate additive mappings in 2-Banach spaces and related topics, J. Math. Anal. Appl. 376 (2011), 193–202. [36] J. M. Rassias, On approximation of approximately linear mappings by linear mappings, J. Funct. Anal. 46 (1982) 126–130. [37] Th. M. Rassias, On the stability of the linear mapping in Banach spaces, Proc. Amer. Math. Soc. 72 (1978), 297–300. [38] Th. M. Rassias, Problem 16; 2, Report of the 27th International Symp. on Functional Equations, Aequationes Math. 39 (1990), 292–293; 309. ˇ [39] Th. M. Rassias and P. Semrl, On the behaviour of mappings which do not satisfy Hyers-Ulam stability, Proc. Amer. Math. Soc. 114 (1992), 989–993. [40] L. Reich and J. Tomaschek, Some remarks to the formal and local theory of the generalized Dhombresw functional equation, Results Math. 63 (2013), 377–395. [41] S. Shagholi, M. Eshaghi Gordji and M. Bavand Savadkouhi, Stability of ternary quadratic derivations on ternary Banach algebras, J. Comput. Anal. Appl. 13 (2011), 1097–1105. [42] S. Shagholi, M. Eshaghi Gordji and M. Bavand Savadkouhi, Nearly ternary cubic homomorphisms in ternary Fr´echet algebras, J. Comput. Anal. Appl. 13 (2011), 1106–1114. [43] D. Shin, C. Park and Sh. Farhadabadi, On the superstability of ternary Jordan C ∗ -homomorphisms, J. Comput. Anal. Appl. 16 (2014), 964–973. [44] 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. [45] F. Skof, Propriet` a locali e approssimazione di operatori, Rend. Sem. Mat. Fis. Milano 53 (1983), 113–129. [46] S. M. Ulam, A Collection of the Mathematical Problems, Interscience Publ. New York, 1960. [47] A. White, 2-Banach spaces, Doctorial Diss., St. Louis Univ., 1968. [48] A. White, 2-Banach spaces, Math. Nachr. 42 (1969) 43–60. Choonkil Park Department of Mathematics, Research Institute for Natural Sciences, Hanyang University, Seoul 133-791, Korea E-mail address: [email protected] Sun Young Jang Department of Mathematics, University of Ulsan, Ulsan 680-749, Korea E-mail address: [email protected] Reza Saadati Department of Mathematics, Iran University of Science and Technology, Tehran, Iran E-mail address: [email protected] Dong Yun Shin Department of Mathematics, University of Seoul, Seoul 130-743, Korea E-mail address: [email protected]

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J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 18, NO.5, 2015, COPYRIGHT 2015 EUDOXUS PRESS, LLC

REFINED GENERAL QUADRATIC EQUATION WITH FOUR VARIABLES AND ITS STABILITY RESULTS HARK-MAHN KIM AND SOON LEE

Abstract. In this article, we establish the general solution of a functional equation x−y+z+w x+y−z+w x+y+z−w −x + y + z + w ) + rf ( ) + rf ( ) + rf ( ) rf ( s s s s = tf (x) + tf (y) + tf (z) + tf (w) and present the generalized Hyers–Ulam stability of the equation.

1. Introduction In 1940, S.M. Ulam [13] gave a wide ranging talk before the Mathematics Club of the University of Wisconsin in which he discussed a number of unsolved problems. Among these was the following question concerning the stability of homomorphisms: Let G1 be a group and G2 a metric group with metric ϕ(·, ·). Given ε > 0, does there exist a δ > 0 such that if f : G1 → G2 satisfies ϕ(f (xy), f (x)f (y)) < δ for all x, y ∈ G1 , then a homomorphism h : G1 → G2 exists with ϕ(f (x), h(x)) < ε for all x ∈ G1 ? Let X and Y be Banach spaces with norms k · k and k · k, respectively. D.H. Hyers [6] showed that if ε > 0 and f : X → Y such that kf (x + y) − f (x) − f (y)k ≤ ε for all x, y ∈ X, then there exists a unique additive mapping T : X → Y such that kf (x) − T (x)k ≤ ε for all x ∈ X. In 1950 T. Aoki [1] and in 1951 D.G. Bourgin [2] provided a generalized the Hyers theorem for additive mapping and in 1978 Th.M. Rassias [11] generalized the Hyers theorem for liner mapping by allowing the Cauchy difference to be unbounded. Let f : X → Y be a mapping such that f (tx) is continuous in t ∈ R for each fixed x ∈ X. Assume that there exist constants ε ≥ 0 and p ∈ [0, 1) such that kf (x + y) − f (x) − f (y)k ≤ ε(kxkp + kykp ) 2000 Mathematics Subject Classification: 39B82. 39B72 Key words and phrases: Drygas functional equation, general quadratic equation, generalized Hyers–Ulam stability. 1

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2

for all x, y ∈ X. Then Th.M. Rassias proved that there exists a unique R-linear mapping T : X → Y such that 2ε kxkp kf (x) − T (x)k ≤ 2 − 2p for all x ∈ X. And then, the result of Th.M. Rassias theorem has been generalized by P. Gˇavruta [5] by allowing the Cauchy difference to be a generalized control function. A square norm on an inner product space satisfies the important parallelogram equality kx + yk2 + kx − yk2 = 2kxk2 + 2kyk2 . The functional equation f (x + y) + f (x − y) = 2f (x) + 2f (y), which may be originated from this parallelogram equality, is called a quadratic functional equation. In particular, every solution of the quadratic functional equation is said to be a quadratic function. A Hyers–Ulam stability problem for the quadratic functional equation was proved by F. Skof [12] for mappings f : X → Y , where X is a normed space and Y is a Banach space. P.W. Cholewa [3] noticed that the theorem of Skof is still true if the relevant domain X is replaced by an Abelian group. In S. Czerwik [4] proved the Hyers–Ulam stability of the quadratic functional equation. In the last decade, S. Lee and K. Jun [8] and S. Lee and C. Park [9] have proved the Hyers–Ulam stability of quadratic type functional equation with three variables. C. Park [10] has investigated the Hyers–Ulam stability of the following functional equation, which has exactly quadratic mappings as solutions up to f (0), x+y+z+w x+y−z−w x−y+z−w ) + rf ( ) + rf ( ) s s s x−y−z+w ) = tf (x) + tf (y) + tf (z) + tf (w) +rf ( s for all x, y, z, w ∈ X under the assumption of an even mapping f : X → Y with f (0) = 0. rf (

Recently, the authors [7] have established the general solution of the above functional equation and then improved the Hyers–Ulam stability of the equation without the even condition and f (0) = 0. In this paper, we are going to establish the general solution of the following modified functional equation, which has exactly quadratic and additive mappings as solutions up to f (0), −x + y + z + w x−y+z+w x+y−z+w ) + rf ( ) + rf ( ) (1.1) s s s x+y+z−w ) = tf (x) + tf (y) + tf (z) + tf (w) +rf ( s for fixed nonzero real numbers r, s, t, and then investigate the Hyers–Ulam stability of the rf (

functional equation for mappings f : X → Y between normed spaces.

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J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 18, NO.5, 2015, COPYRIGHT 2015 EUDOXUS PRESS, LLC

REFINED GENERAL QUADRATIC EQUATION

3

2. General solution of the functional equation. First of all, we solve the general solution of the equation (1.1) in the class of mappings between linear spaces. Lemma 2.1. If a mapping f : X → Y satisfies the equation x−y+z+w x+y−z+w −x + y + z + w ) + rf ( ) + rf ( ) s s s x+y+z−w ) = tf (x) + tf (y) + tf (z) + tf (w) +rf ( s

rf (

for all x, y, z, w ∈ X, then f (x) = Q(x) + A(x) + f (0), where Q is quadratic and A is additive, and f (0) = 0 if r 6= t. Proof. Let f be a solution of the equation (1.1). Now, letting x = y = z = w := 0 in f (x) + f (−x) (1.1), one has f (0) = 0 if r 6= t. First, we prove the case r 6= t. Let fe (x) := 2 f (x) − f (−x) be an odd part of f . Then, we see that be an even part of f and fo (x) := 2 fe , fo are also solutions of the equation (1.1). Putting y = z = w := 0 in (1.1) for the even mapping fe , we have x 4rfe ( ) = tfe (x) s

(2.1)

for all x ∈ X, which yields fe (−x + y + z + w) + fe (x − y + z + w) + fe (x + y − z + w)

(2.2)

+fe (x + y + z − w) = 4fe (x) + 4fe (y) + 4fe (z) + 4fe (w) for all x, y, z, w ∈ X. Putting z = w := 0 in (2.2), we deduce fe (x + y) + fe (x − y) = 2fe (x) + 2fe (y) for all x, y ∈ X. So fe := Q is quadratic.

x Putting y = z = w := 0 in (1.1) for the odd mapping fo , we have the relation 2rfo ( ) = s tfo (x) for all x ∈ X. Thus, it follows that fo (−x + y + z + w) + fo (x − y + z + w) + fo (x + y − z + w)

(2.3)

+fo (x + y + z − w) = 2fo (x) + 2fo (y) + 2fo (z) + 2fo (w) for all x, y, z, w ∈ X. Putting z = w := 0 in (2.3), one conclude fo (x + y) = fo (x) + fo (y), and so, fo := A is additive. Therefore, f (x) = fe (x) + fo (x) = Q(x) + A(x), where Q is quadratic and A is additive.

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Next, we prove the case r = t. Let f (x) − f (0) = f˜(x), x ∈ X. In this case, f˜(0) = 0, and we see the functional equation x−y+z+w x+y−z+w −x + y + z + w ) + f˜( ) + f˜( ) (2.4) f˜( s s s x+y+z−w ) = f˜(x) + f˜(y) + f˜(z) + f˜(w) +f˜( s f˜(x) + f˜(−x) f˜(x) − f˜(−x) and f˜o (x) = for all x, y, z, w ∈ X. It follows that f˜e (x) := 2 2 also satisfy the equation (2.4). Putting y = z = w := 0 in (2.4) for the even mapping f˜e , x we have 4f˜e ( ) = f˜e (x), and so, s f˜e (−x + y + z + w) + f˜e (x − y + z + w) + f˜e (x + y − z + w) (2.5) +f˜e (x + y + z − w) = 4[f˜e (x) + f˜e (y) + f˜e (z) + f˜e (w)] for all x, y, z, w ∈ X. Thus, f˜e := Q is quadratic. Similarly, putting y = z = w := 0 in x (2.4) for the odd mapping f˜o , we have 2f˜o ( ) = f˜o (x), and so, we get s f˜o (−x + y + z + w) + f˜o (x − y + z + w) + f˜o (x + y − z + w) (2.6) +f˜o (x + y + z − w) = 2[f˜o (x) + f˜o (y) + f˜o (z) + f˜o (w)] for all x, y, z, w ∈ X. Thus, we conclude that f˜o (x + y) = f˜o (x) + f˜o (y), and hence, f˜o = A is additive. Therefore, f (x) − f (0) = f˜(x) = f˜e (x) + f˜o (x) = Q(x) + A(x), where Q is quadratic and A is additive. 2 Remark 2.2. If r = t and s 6= 2 is a rational number, and if f is a solution of the equation (1.1), then we note that f˜o (x) := A(x) ≡ 0 identically, and f˜e (x) := Q(x) ≡ 0 identically. Hence, f (x) = f (0) must be a constant solution. If r = t and s = 2, and if f is a solution of the equation (1.1) with f (0) = 0, then the equation f(

−x + y + z + w x−y+z+w x+y−z+w ) + f( ) + f( ) 2 2 2 x+y+z−w ) = f (x) + f (y) + f (z) + f (w), x, y, z, w ∈ X +f ( 2

yields 2f (

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

⇔ 2f (u) + f (v) + f (−v) = f (u + v) + f (u − v), which is well-known Drygas functional equation with general solution f (x) = Q(x) + A(x), x ∈ X. Therefore, if r = t and s = 2, and if f is a solution of the equation (1.1), then f (x) = Q(x) + A(x) + f (0) is a general solution of the equation.

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J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 18, NO.5, 2015, COPYRIGHT 2015 EUDOXUS PRESS, LLC

REFINED GENERAL QUADRATIC EQUATION

5

If r 6= t and s = 2, and if f is a solution of the equation (1.1), then we see that fe (x) = 0, x ∈ X identically and fo (x) = 0, x ∈ X identically. Thus, f (x) = f (0) = 0 must be a constant solution. We remark that if a mapping f : X → Y satisfies the equation f (−x + y + z + w) + f (x − y + z + w) + f (x + y − z + w) +f (x + y + z − w) = 4[f (x) + f (y) + f (z) + f (w)] for all x, y, z, w ∈ X, then (i) f (0) = 0; (ii) f (−x) = f (x); (iii) f (x + y) + f (x − y) = 2[f (x) + f (y)] for all x, y ∈ X, and thus, f is quadratic. 3. Stability of the functional equation for even mappings. We now prove the Hyers–Ulam stability of the functional equation for even mappings f : X → Y with some regularity conditions. Given a mapping f : X → Y and a function ϕ : X 4 → R+ := [0, ∞), we set for notational convenience kDf (x, y, z, w)k ≤ ϕ(x, y, z, w), (3.1) x−y+z+w x+y−z+w −x + y + z + w ) + rf ( ) + rf ( ) Df (x, y, z, w) := rf ( s s s x+y+z−w ) − [tf (x) + tf (y) + tf (z) + tf (w)] +rf ( s for all x, y, z, w ∈ X. From now on, we assume that that X and Y are a normed linear space with norm k · k and a Banach space with norm k · k, respectively. Theorem 3.1. Assume that an even mapping f : X → Y satisfies the functional inequality (3.1) and Φ1 (x, y, z, w) := Ã

∞ X ϕ(2i x, 2i y, 2i z, 2i w) i=0 ∞ X

4i

< ∞,

(3.2) !

x y z w Φ2 (x, y, z, w) := 4 ϕ( i , i , i , i ) < ∞, resp. 2 2 2 2 i=1 i

for all x, y, z, w ∈ X. Then, there exists a unique quadratic mapping Q1 : X → Y, f (2n x) , x ∈ X, (Q2 : X → Y, resp.) , defined as Q1 (x) = limn→∞ 4n ¶ µ x Q2 (x) = lim 4n [f ( n ) − f (0)], x ∈ X, resp. n→∞ 2 such that 1 [Φ1 (x, x, x, x) + Φ1 (2x, 0, 0, 0)], (3.3) kf (x) − f (0) − Q1 (x)k ≤ 4|t| Ã ! 1 [Φ2 (x, x, x, x) + Φ2 (2x, 0, 0, 0)], resp. kf (x) − f (0) − Q2 (x)k ≤ 4|t| for all x ∈ X.

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Proof. First, we observe that 4 | r − t | kf (0)k ≤ ϕ(0, 0, 0, 0), 2x k4rf ( ) − 4tf (x)k ≤ ϕ(x, x, x, x), s

k2rf (

2x ) − 2tf (x) + 2(r − t)f (0)k ≤ ϕ(x, x, 0, 0), s x k4rf ( ) − tf (x) − 3tf (0)k ≤ ϕ(x, 0, 0, 0) s

(3.4) (3.5) (3.6) (3.7)

for all x ∈ X. By using (3.5) and (3.6), we have k4(r − t)f (0)k ≤ ϕ(x, x, x, x) + 2ϕ(x, x, 0, 0)

(3.8)

for all x ∈ X. By using (3.5) and (3.7), one has kf (2x) − 4f (x) + 3f (0)k ≤

1 [ϕ(x, x, x, x) + ϕ(2x, 0, 0, 0)] |t|

(3.9)

for all x ∈ X. Let f (x) − f (0) := f˜(x), x ∈ X. Then one obtains from (3.9) kf˜(2x) − 4f˜(x)k ≤

1 [ϕ(x, x, x, x) + ϕ(2x, 0, 0, 0)] |t|

(3.10)

for all x ∈ X. Thus, we can prove by triangle inequality X 1 1 n−1 f˜(2n x) ˜ − f (x)k [ϕ(2i x, 2i x, 2i x, 2i x) + ϕ(2i+1 x, 0, 0, 0)] ≤ k n i 4 4|t| i=0 4

(3.11)

for all x ∈ X. Now, it follows from the last inequality that for all nonnegative integers n, m with n > m ≥ 0 f˜(2n x) f˜(2m x) − k 4n 4m n−1 X f˜(2i+1 x) f˜(2i x) − k ≤ k 4i+1 4i i=m

(3.12)

k

≤

X 1 1 n−1 [ϕ(2i x, 2i x, 2i x, 2i x) + ϕ(2i+1 x, 0, 0, 0)] 4|t| i=m 4i

for all x ∈ X, of which the right-hand side approaches 0 as m tends to infinity. This shows f˜(2n x) } is a Cauchy sequence for all x ∈ X. Since Y is complete, the that the sequence { 4n f˜(2n x) } converges in Y for all x ∈ X, and so one can define a mapping sequence { 4n Q1 : X → Y by f˜(2n x) f (2n x) = lim n→∞ n→∞ 4n 4n

Q1 (x) = lim for all x ∈ X.

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Now, it follows from (3.1) and the definition of Q1 that kDf (2n x, 2n y, 2n z, 2n w)k n→∞ 4n n n ϕ(2 x, 2 y, 2n z, 2n w) =0 ≤ lim n→∞ 4n for all x, y, z, w ∈ X. Thus, the mapping Q1 is quadratic by Lemma 2.1. Moreover, if we kDQ1 (x, y, z, w)k =

lim

let n → ∞ in (3.11), we get the desired approximation (3.3). 0

To prove the uniqueness, let Q : X → Y be another quadratic mapping satisfying (3.3). Then, we have 1 0 kQ1 (2n x) − Q (2n x)k n 4 µ ¶ 1 0 n n n n ≤ n kQ1 (2 x) − f (2 x) + f (0)k + kf (2 x) − f (0) − Q (2 x)k 4 µ ¶ 1 n n n n n+1 Φ1 (2 x, 2 x, 2 x, 2 x) + Φ1 (2 x, 0, 0, 0) , ≤ n 4 · 2|t|

0

k Q1 (x) − Q (x) k =

0

which tends to zero as n → ∞ for all x ∈ X. So one can conclude that Q1 (x) = Q (x) for all x ∈ X. This proves the uniqueness. 2 In the following, we consider another stability results of the functional equation (1.1) by using the similar manner to the reference [7]. Theorem 3.2. Assume that an even mapping f : X → Y satisfies the functional inequality (3.1) and the condition (3.2). Then, there exists a unique quadratic mapping f (2n x) , x∈X Q1 : X → Y, (Q2 : X → Y, resp.) , defined as Q1 (x) = limn→∞ 4n ! Ã (4r − t) x n f (0)], x ∈ X, resp.) lim 4 [f ( n ) − Q2 (x) = n→∞ 2 3t such that (4r − t) 1 f (0) − Q1 (x)k ≤ [2Φ1 (x, x, 0, 0) + Φ1 (2x, 0, 0, 0)], (3.13) 3t 4|t| ! Ã 1 (4r − t) f (0) − Q2 (x)k ≤ [2Φ2 (x, x, 0, 0) + Φ2 (2x, 0, 0, 0)], resp.) kf (x) − 3t 4|t|

kf (x) −

for all x ∈ X, where f (0) = 0 if r 6= t. Proof. Associating (3.6) with (3.7), one has kf (2x) − 4f (x) +

1 (4r − t) f (0)k ≤ [2ϕ(x, x, 0, 0) + ϕ(2x, 0, 0, 0)], t |t|

which yields k

˜ 1 f (2x) − f˜(x)k ≤ [2ϕ(x, x, 0, 0) + ϕ(2x, 0, 0, 0)], 4 4|t|

891

(3.14)

HARK-MAHN KIM ET AL 885-898

J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 18, NO.5, 2015, COPYRIGHT 2015 EUDOXUS PRESS, LLC

H. KIM AND S. LEE

8

(4r − t) f (0), x ∈ X. It follows from (3.14) that where f˜(x) := f (x) − 3t X 2ϕ(2i x, 2i x, 0, 0) + ϕ(2i+1 x, 0, 0, 0) 1 n−1 f˜(2n x) k [ ] ≤ kf˜(x) − 4n 4|t| i=0 4i for all x ∈ X. The rest of proof is similarly verified by the same argument as that of Theorem 3.1. 2 In Theorem 3.2, we remark that Q2 (0) = 0 by definition if r = t, and also, Q2 (0) = 0 if r 6= t because f (0) = 0 = ϕ(0, 0, 0, 0) by the convergence of Φ2 (0, 0, 0, 0). Corollary 3.3. Let δ, θ be nonnegative real numbers and p 6= 2 be a positive real number. Assume that an even mapping f : X → Y satisfies the functional inequality kDf (x, y, z, w)k ≤ δ + θ(kxkp + kykp + kzkp + kwkp ) for all x, y, z, w ∈ X, where δ = 0 when p > 2. Then, there exists a unique quadratic mapping Q : X → Y such that

·

4θkxkp 2p θkxkp 1 2δ + + kf (x) − f (0) − Q(x)k ≤ |t| 3 |4 − 2p | |4 − 2p |

¸

for all x ∈ X, where f (0) = 0 if r 6= t and p > 2. 4. Stability of the functional equation for odd mappings. We now prove the Hyers–Ulam stability of the functional equation for odd mappings f : X → Y with some regularity conditions. Theorem 4.1. Assume that an odd mapping f : X → Y satisfies the functional inequality (3.1) and Φ3 (x, y, z, w) = Ã

∞ X ϕ(2i x, 2i y, 2i z, 2i w) i=0 ∞ X

2i

1. Then, there exists a unique additive mapping A : X → Y such that ·

kf (x) − A(x)k ≤

2θkxkp 2p θkxkp 1 3δ + + |t| 2 |2 − 2p | |2 − 2p |

¸

for all x ∈ X. 5. Stability of the functional equation for general mappings. Finally, we now prove the Hyers–Ulam stability of the functional equation (1.1) for general mappings f : X → Y with some regularity conditions. Theorem 5.1. Assume that a mapping f : X → Y satisfies the functional inequality (3.1) and Φ3 (x, y, z, w) :=

∞ X ϕ(2i x, 2i y, 2i z, 2i w)

2i

i=0

1. Then, there exist a unique quadratic mapping Q : X → Y and a unique additive mapping A : X → Y such that ·

k

¸

4θkxkp 2p θkxkp f (x) + f (−x) 1 2δ − f (0) − Q(x)k ≤ + + , 2 |t| 3 |4 − 2p | |4 − 2p | · ¸ 2θkxkp 2p θkxkp f (x) − f (−x) 1 3δ − A(x)k ≤ + + , k 2 |t| 2 |2 − 2p | |2 − 2p | · ¸ 4θkxkp 2p θkxkp 2θkxkp 2p θkxkp 1 13δ + + + + kf (x) − f (0) − A(x) − Q(x)k ≤ |t| 6 |4 − 2p | |4 − 2p | |2 − 2p | |2 − 2p |

for all x ∈ X, where f (0) = 0 if r 6= t and p > 2. Acknowledgements This research was supported by Basic Research Program through the National Research Foundation of Korea(NRF) funded by the Ministry of Education(No. 2012R1A1A2008139). References [1] T. Aoki, On the stability of the linear transformation in Banach spaces, J. Math. Soc. Japan, 2 (1950), 64–66. [2] D.G. Bourgin, Classes of transformations and bordering transformations, Bull. Amer. Math. Soc. 57 (1951), 223–237. [3] P.W. Cholewa, Remarks on the shability of functional equations, Aequationes Math. 27(1984), 76–86.

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[4] S. Czerwik, On the stability of the quadratic mapping in normed spaces, Abh. Math. Sem. Univ. Hamburg, 62(1992), 59–64. [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. [7] H. Kim and S. Lee, Refined stability results of functional equation in four variables, to be appeared in Kyungpook Math. J. [8] S. Lee and K. Jun, Hyers–Ulam–Rassias stability of a quadratic type functional equation, Bull. Korean Math. Soc. 40(2003), 183–193. [9] S. Lee and C. Park, Hyers–Ulam–Rassias stability of a functional equation in three variables, J. Chungcheong Math. Soc. 16(2)(2003), 11–21. [10] C. Park, Hyers–Ulam–Rassias stability of an even functional equation in four variables, Kyungpook Math J. 44(2)(2004), 299–304. [11] Th.M. Rassias, On the stability of the linear mapping in Banach spaces, Proc. Amer. Math. Soc. 72 (1978), 297–300. [12] F. Skof, Propriet`a locali e approssimazione di operatori, Rend. Sem. Mat. Fis. Milano, 53(1983), 113–129. [13] S.M. Ulam, A Collection of the Mathematical Problems, Interscience Publ. New York, 1960. (Hark-Mahn Kim) Department of Mathematics, Chungnam National University, 79 Daehangno, Yuseong-gu, Daejeon 305-764, Korea E-mail address: [email protected] (Soon Lee) Department of Mathematics, Chungnam National University, 79 Daehangno, Yuseong-gu, Daejeon 305-764, Korea E-mail address: [email protected]

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HARK-MAHN KIM ET AL 885-898

J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 18, NO.5, 2015, COPYRIGHT 2015 EUDOXUS PRESS, LLC

HYERS-ULAM STABILITY OF A CLASS OF DIFFERENTIAL EQUATIONS OF SECOND ORDER MOHAMMAD REZA ABDOLLAHPOUR AND CHOONKIL PARK∗ Abstract. In this paper we prove the Hyers-Ulam stability of a class of differential equations of second order which includes Euler differential equation and second order linear differential equations with constant coefficients.

1. Introduction In 1940, Ulam [20] discussed the question concerning the stability of homomorphisms as follows: let G1 be a group and G2 be a metric group with a metric d(·, ·). For a given ε > 0, is there a δ > 0 such that if a function h : G1 → G2 satisfies the inequality d(h(xy), h(x)h(y)) < δ for all x, y ∈ G1 , then there exists a homomorphism H : G1 → G2 with d(h(x), H(x)) < δ for all x ∈ G1 ? The question of Ulam was answered by Hyers in [7] for the case both G1 and G2 are Banach spaces. See [3, 5, 6, 17, 18, 19] for the stability problems of functional equations. Definition 1.1. Let I ⊂ R be an open interval. We say that the differential equation an (t)y (n) (t) + an−1 (t)y (n−1) (t) + · · · + a1 (t)y 0 (t) + a0 y(t) + h(t) = 0

(1.1)

has the Hyers-Ulam stability, if for any function f : I → R satisfying the differential inequality |an (t)y (n) (t) + an−1 (t)y (n−1) (t) + · · · + a1 (t)y 0 (t) + a0 y(t) + h(t)| ≤ ε for all t ∈ I and for some ε > 0, there exists a solution g : I → R of (1.1) such that |f (t) − g(t)| ≤ K(ε) for any t ∈ I, where K(ε) is a constant depending only on ε. The first result concerning the Hyers-Ulam stability of ordinary differential equations was due to Alsina and Ger, see [2] (see also [15, 16]). In fact, their result dealt with the Hyers-Ulam stability of linear differential equations of first order. The result of Alsina and Ger has been generalized by many mathematicians (Ref. [8, 9, 10, 12, 13]). The Hyers-Ulam stability of second order linear differential equations has been investigated in [4] and [14]. Furthermore, Abdollahpour and Najati [1] proved the Hyers-Ulam stability of the third order differential equation y (3) (t)+αy 00 (t)+βy 0 (t)+ γy(t) = f (t). The aim of this paper is to investigate the Hyers-Ulam stability of the differential equation 2    α h00 (x) 1 00 y (x) + − y 0 (x) + βy(x) = f (x). (1.2) h0 (x) h0 (x) (h0 (x))3 MSC(2010): 34K20, 26D10, 39B52, 39B82 Keywords: Hyers-Ulam stability, Differential equation. ∗ Corresponding author: [email protected] (C. Park).

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CHOONKIL PARK ET AL 899-903

J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 18, NO.5, 2015, COPYRIGHT 2015 EUDOXUS PRESS, LLC

M. R. ABDOLLAHPOUR, C. PARK

More precisely, the problem, we will deal with the following. Let ε > 0 be fixed and [a, b] ⊂ R. h ∈ C 2 [a, b] be a function for which either h0 (x) > 0 or h0 (x) < 0 holds for all x ∈ [a, b] and let f ∈ C[a, b]. Assume that for the unknown function y ∈ C 2 [a, b]  2   00 (x) 1 α h 0 y (x) + βy(x) − f (x) y 00 (x) + − 0 0 (or h0 (x) < 0) for all x ∈ [a, b]. Proof. Suppose that λ, µ are the (real or complex) roots of m2 + αm + β = 0 with p = 0 then  1 − e−p[h(b)−h(a)]    ε if p 6= 0; p |z(x) − g(x)| ≤    (h(b) − h(a))ε if p = 0

(2.2)

for all x ∈ [a, b] and if h0 (x) < 0 then  e−p[h(b)−h(a)] − 1    ε if p 6= 0; p |z(x) − g(x)| ≤    (h(a) − h(b))ε if p = 0

(2.3)

for all x ∈ [a, b]. Now, we define u(x) = y(b)eλ[h(x)−h(b)] − eλh(x)

b

Z

h0 (t)z(t)e−λh(t) dt,

x ∈ [a, b].

x

Due to the definition of function u, we immediately obtain that u ∈ C 2 [a, b] and u0 (x) = λh0 (x)u(x) + z(x)h0 (x).

(2.4)

Then u00 (x)h0 (x) − h00 (x)u0 (x) = λu0 (x) + z 0 (x). [h0 (x)]2 It follows from (2.1) and (2.4) that 

1 h0 (x)

2

00

u (x) +



α h00 (x) − h0 (x) (h0 (x))3

901



u0 (x) + βu(x) = f (x),

x ∈ [a, b].

CHOONKIL PARK ET AL 899-903

J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 18, NO.5, 2015, COPYRIGHT 2015 EUDOXUS PRESS, LLC

M. R. ABDOLLAHPOUR, C. PARK

Furthermore, for the function u, the inequality Z b h0 (t)z(t)e−λh(t) dt |y(x) − u(x)| = y(x) − y(b)eλ[h(x)−h(b)] + eλh(x) x Z b λh(x) −λh(x) −λh(b) 0 −λh(t) =|e | y(x)e − y(b)e + h (t)z(t)e dt x Z b Z b −λh(t) 0 0 −λh(t) qh(x) [e y(t)] dt h (t)z(t)e dt − =e x x Z b 1 0 qh(x) 0 −λh(t) =e h (t)e [z(t) − 0 y (t) + λy(t)] dt h (t) x Z b |h0 (t)e−λh(t) ||z(t) − g(t)| dt ≤eqh(x) x

=eqh(x)

Z

b

e−qh(t) |h0 (t)||z(t) − g(t)| dt

x

is also valid for all x ∈ [a, b]. It follows from (2.2) that  [1 − e−p[h(b)−h(a)] ][1 − e−q[h(b)−h(a)] ]    ε   pq        [1 − e−p[h(b)−h(a)] ](h(b) − h(a))    ε p |y(x) − u(x)| ≤     [1 − e−q[h(b)−h(a)] ](h(b) − h(a))    ε   q       (h(b) − h(a))2 ε and (2.3) implies that  −p[h(b)−h(a)] [e − 1][e−q[h(b)−h(a)] − 1]    ε   pq        [e−p[h(b)−h(a)] − 1](h(a) − h(b))    ε p |y(x) − u(x)| ≤      [e−q[h(b)−h(a)] − 1](h(a) − h(b))   ε   q       (h(a) − h(b))2 ε for all x ∈ [a, b]. This completes the proof.

if p, q 6= 0;

if p 6= 0, q = 0;

if p = 0, q 6= 0; if p, q = 0

if p, q 6= 0;

if p 6= 0, q = 0;

if p = 0, q 6= 0; if p, q = 0 

In [11], Li and Shen proved that a second order differential equation with constant coefficients has the Hyers-Ulam stability, if its characteristic equation has two positive roots. It is necessary to mention that our next result is more general than their result, because in our result, there is no restriction on roots of characteristic equation. Corollary 2.2. The second order differential equation with constant coefficients y 00 (x) + αy 0 (x) + βy(x) = f (x) has the Hyers-Ulam stability.

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CHOONKIL PARK ET AL 899-903

J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 18, NO.5, 2015, COPYRIGHT 2015 EUDOXUS PRESS, LLC

HYERS-ULAM STABILITY OF A CLASS OF DIFFERENTIAL EQUATIONS

Proof. It is enough to take h(x) = x and replace α by α + 1 in Theorem 2.1.



Corollary 2.3. If 0 < a < b or a < b < 0 then the second order Euler differential equation x2 y 00 (x) + αxy 0 (x) + βy(x) = f (x) has the Hyers-Ulam stability. Proof. It is enough to take h(x) = Ln(|x|) and replace α by α + 1 in Theorem 2.1.  References [1] M. R. Abdollahpour and A. Najati, Stability of linear differential equations of third order, Appl. Math. Lett. 24 (2011), 1827–1830. [2] C. Alsina and R. Ger, On some inequalities and stability results related to the exponential function, J. Inequal. Appl. 2 (1998), 373–380. [3] M. Eshaghi Gordji and A. Bodaghi, On the stability of quadratic double centralizers on Banach algebras, J. Comput. Anal. Appl. 13 (2011), 724–729. [4] M. Eshaghi Gordji, Y. Cho, M. B. Ghaemi and B. Alizadeh, Stability of the exact second order partial differential equations, J. Inequal. Appl. 2011,Article ID 306275 (2011). [5] M. Eshaghi Gordji, R. Farokhzad Rostami and S.A.R. Hosseinioun, Nearly higher derivations in unital C ∗ -algebras, J. Comput. Anal. Appl. 13 (2011), 734–742. [6] M. Eshaghi Gordji, G. Kim, J. Lee and C. Park, Generalized ternary bi-derivations on ternary Banach algebras: a fixed point approach, J. Comput. Anal. Appl. 15 (2013), 45–54. [7] D.H. Hyers, On the stability of the linear functional equation, Proc. Natl. Acad. Sci. USA 27 (1941), 222–224. [8] S. Jung, Hyers-Ulam stability of linear differential equations of first order, Appl. Math. Lett. 17 (2004), 1135–1140. [9] S. Jung, Hyers-Ulam stability of linear differential equations of first order, III, J. Math. Anal. Appl. 311 (2005), 139–146. [10] S. Jung, Hyers-Ulam stability of linear differential equations of first order, II, Appl. Math. Lett. 19 (2006), 854–858. [11] Y. Li and Y. Shen, Hyers-Ulam stability of linear differential equations of second order, Appl. Math. Lett. 23 (2010), 306–309. [12] T. Miura, On the Hyers-Ulam stability of a differentiable map, Sci. Math. Japon. 55 (2002), 17–24. [13] T. Miura, S. Jung and S.-E. Takahasi, Hyers-Ulam-Rassias stability of the Banach space valued differential equations y 0 = λy, J. Korean Math. Soc. 41 (2004), 995–1005. [14] A. Najati, M. R. Abdollahpour and Y. Cho, Superstability of linear differential equations of second order, preprint. [15] M. Obloza, Hyers stability of the linear differential equation, Rocznik Nauk.-Dydakt. Prace Mat. 13 (1993), 259–270. [16] M. Obloza, Connections between Hyers and Lyapunov stability of the ordinary differential equations, Rocznik Nauk.-Dydakt. Prace Mat. 14 (1997), 141–146. [17] S. Shagholi, M. Eshaghi Gordji and M. Bavand Savadkouhi, Stability of ternary quadratic derivations on ternary Banach algebras, J. Comput. Anal. Appl. 13 (2011), 1097–1105. [18] S. Shagholi, M. Eshaghi Gordji and M. Bavand Savadkouhi, Nearly ternary cubic homomorphisms in ternary Fr´echet algebras, J. Comput. Anal. Appl. 13 (2011), 1106–1114. [19] D. Shin, C. Park and Sh. Farhadabadi, On the superstability of ternary Jordan C ∗ homomorphisms, J. Comput. Anal. Appl. 16 (2014), 964–973. [20] S.M. Ulam, A Collection of the Mathematical Problems, Interscience Publ. New York, 1960. Mohammad Reza Abdollahpour Department of Mathematics and Applications, Faculty of Mathematical Sciences, University of Mohaghegh Ardabili, Ardabil 56199-11367, Iran E-mail address: [email protected], [email protected] Choonkil Park Research Institute for Natural Sciences, Hanyang University, Seoul, 133-791, Korea E-mail address: [email protected]

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CHOONKIL PARK ET AL 899-903

J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 18, NO.5, 2015, COPYRIGHT 2015 EUDOXUS PRESS, LLC

An iterative algorithm based on the hybrid steepest descent method for strictly pseudocontractive mappings Jong Soo Jung Department of Mathematics, Dong-A University, Busan 604-714, Korea

Abstract In this paper, we consider a general iterative method based on the hybrid steepest descent method for finding fixed points of a strictly pseudocontractive mapping in a Hilbert space. Utilizing weaker control conditions than previous ones, we establish the strong convergence of the sequence generated by the proposed iterative method to a fixed point of the mapping, which is the unique solution of a certain variational inequality. MSC: 47H09, 47H05, 47H10, 47J25, 49M05, 47J05..

Key words:

Iterative algorithm; Strictly pseudocontractive mapping; Fixed points; Weakly

asymptotically regular; ρ-Lipschitzian and η-strongly monotone operator; Variational inequality

1

Introduction

Let H be a real Hilbert space with inner product h·, ·i and induced norm k · k. Let C be a nonempty closed convex subset of H and S : C → C be a self-mapping on C. We denote by F (S) the set of fixed points of S. The class of pseudocontractive mappings is one of the most important classes of mappings among nonlinear mappings. We recall that a mapping T : C → H is said to be k-strictly pseudocontractive if there exists a constant k ∈ [0, 1)such that kT x − T yk2 ≤ kx − yk2 + kk(I − T )x − (I − T )yk2 ,

∀x, y ∈ C.

Note that the class of k-strictly pseudocontractive mappings includes the class of nonexpansive mappings as a subclass. That is, T is nonexpansive (i.e., kT x − T yk ≤ kx − yk, ∀x, y ∈ C) if and only if T is 0-strictly pseudocontractive. The mapping T is also said to be pseudocontractive if k = 1 and T is said to be strongly pseudocontractive if there exists a constant ν ∈ (0, 1) such that T − νI is pseudocontractive. Clearly, the class of k-strictly Email address: [email protected],

[email protected] (Jong Soo Jung).

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J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 18, NO.5, 2015, COPYRIGHT 2015 EUDOXUS PRESS, LLC

pseudocontractive mappings falls into the one between classes of nonexpansive mappings and pseudocontractive mappings. Also we remark that the class of strongly pseudocontractive mappings is independent of the class of k-strictly pseudocontractive mappings (see [1]). Recently, many authors have been devoting the studies on the problems of finding fixed points for pseudocontractive mappings, see, for example, [2–7] and the references therein. In 2010, by combining Yamada’s method [8] and Marino and Xu’s method [9], Tian [10] considered the following explicit iterative scheme for the nonexpansive mapping S: xn+1 = αn γV xn + (I − αn µF )Sxn , ∀n ≥ 0,

(1.1)

where F : H → H is a ρ-Lipschitzian and η-strongly monotone operator with constants ρ > 0 and η > 0 (i.e., kF x − F yk ≤ ρkx − yk and hF x − F y, x − yi ≥ ηkx − yk2 , x, y ∈ H, respectively), V : H → H is an l-Lipschitzian mapping with a constant l ≥ 0, 0 < µ < 2η ρ2 2

µρ and 0 ≤ γl < τ = µ(η using control conditions (i) {αn } ⊂ (0, 1), 2 ). In particular, byP P− αn+1 ∞ limn→∞ αn = 0, (ii) ∞ α = ∞, (iii) either n n=0 n=0 |αn+1 −αn | < ∞ or limn→∞ αn = 1 on {αn }, he proved that the sequence {xn } generated by (1.1) converges strongly to a fixed point x e of S, which is the unique solution of the following variational inequality related to the operator F : hµF x e − γV x e, x e − pi ≤ 0, ∀p ∈ F (S). (1.2)

His results improved the results of Tian [11] from the case of the contractive mapping f a constant α ∈ (0, 1) to the case of a Lipschitzian mapping V with a constant l ≥ 0. In 2011, Ceng et al. [12] also considered the following explicit iterative schemes for the nonexpansive mapping S: xn+1 = PC [αn γV xn + (I − αn µF )Sxn ], ∀n ≥ 0,

(1.3)

where PC is the metric projection of H ont C; F : C → H is a ρ-Lipschitzian and η-strongly monotone operator with constants ρ > 0 and η > 0; V : C → Hpis an l-Lipschitzian and 0 ≤ γl < τ = 1 − 1 − µ(2η − µρ2 ). In mapping with a constant l ≥ 0; 0 < µ < 2η ρ2 particular, by using the same control conditions on {αn } as in Tian [11,10], they proved that the sequence {xn } generated by (1.3) converges strongly to a fixed point x e of S, which is the unique solution of the variational inequality (1.2). Their results also improved the results of Tian [11] from the case of the contractive mapping f with a constant α ∈ (0, 1) to the case of a Lipschitzian mapping V with a constant l ≥ 0, and extended the range 2 0 < γα < τ = µ(η − µρ2 ) in [10, Theorem 3.1 and Theorem 3.2] to the case of range p 0 < γl < τ = 1 − 1 − µ(2η − µρ2 ). In this paper, motivated by the above-mentioned results, we consider the following explicit iterative scheme for a k-strictly pseudocontractive mapping T for some 0 ≤ k < 1: xn+1 = αn γV xn + (I − αn µF )Tn xn ,

∀n ≥ 0,

(1.4)

where Tn : H → H is a mapping defined by Tn x = λn x + (1 − λn )T x, ∀x ∈ H, with 0 ≤ k ≤ λn ≤ λ < 1 and limn→∞ λn = λ. By using weaker control conditions than previous ones, we establish the strong convergence of the sequence generated by the proposed scheme (1.4) to a fixed point of T , which is a solution of the variational inequality (1.2), where the constraint set is F (T ). The results in this paper improve and develop the corresponding results given in [3,4,6,9–14] and references therein.

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2

Preliminaries and Lemmas

Throughout this paper, when {xn } is a sequence in E, xn → x (resp., xn * x) will denote strong (resp., weak) convergence of the sequence {xn } to x. For every point x ∈ H, there exists a unique nearest point in C, denoted by PC (x), such that kx − PC (x)k ≤ kx − yk, ∀y ∈ C. PC is called the metric projection of H onto C. It is well known that PC is nonexpansive and that for x ∈ H, z = PC x ⇐⇒ hx − z, y − zi ≤ 0,

∀y ∈ C.

(2.1)

It is also well known that H satisfies the Opial condition, that is, for any sequence {xn } with xn * x, the inequality lim inf kxn − xk < lim inf kxn − yk n→∞

n→∞

holds for every y ∈ H with y 6= x. Lemma 2.1. In a real Hilbert space H, the following inequality holds: kx + yk2 ≤ kxk2 + 2hy, x + yi,

∀x, y ∈ H.

Let LIM be a Banach limit. According to time and circumstances, we use LIMn (an ) instead of LIM (a) for every a = {an } ∈ `∞ . The following properties are well-known: (i) for all n ≥ 1, an ≤ cn implies LIMn (an ) ≤ LIMn (cn ), (ii) LIMn (an+N ) = LIMn (an ) for any fixed positive integer N , (iii) lim inf n→∞ an ≤ LIMn (an ) ≤ lim supn→∞ an for all {an } ∈ l∞ . The following lemma was given in [15]. Lemma 2.2. Let a ∈ R be a real number and a sequence {an } ∈ l∞ satisfy the condition LIMn (an ) ≤ a for all Banach limit LIM . If lim supn→∞ (an+1 − an ) ≤ 0, then lim supn→∞ an ≤ a. We also need the following lemmas for the proof of our main results. Lemma 2.3 ([16]). Let {sn } be a sequence of non-negative real numbers satisfying sn+1 ≤ (1 − βn )sn + βn δn + rn ,

∀n ≥ 0,

where {βn }, {δn } and {rn } satisfy the following conditions:

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J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 18, NO.5, 2015, COPYRIGHT 2015 EUDOXUS PRESS, LLC

(i) {βn } ⊂ [0, 1] and

P∞

n=0 βn

= ∞,

P (ii) lim supn→∞ δn ≤ 0 or ∞ n=0 βn |δn | < ∞, P∞ (iii) rn ≥ 0 (n ≥ 0), n=0 rn < ∞. Then limn→∞ sn = 0. Lemma 2.4 ([17]). Let H be a Hilbert space and let C be a closed convex subset of H. Let T : C → H be a k-strictly pseudocontractive mapping on C. Then the following hold: (i) The fixed point set F (T ) is closed convex, so that the projection PF (T ) is well defined. (ii) F (PC T ) = F (T ). (iii) If we define a mapping S : C → H by Sx = λx + (1 − λ)T x for all x ∈ C. then, as λ ∈ [k, 1), S is a nonexpansive mapping such that F (T ) = F (S). The following lemmas can be easily proven, and therefore, we omit the proofs (see [5,8]). Lemma 2.5. Let H be a real Hilbert space. Let V : H → H be an l-Lipschitzian mapping with a constant l ≥ 0, and F : H → H be a ρ-Lipschitzian and η-strongly monotone operator with constants ρ > 0 and η > 0. Then for 0 ≤ γl < µη, h(µF − γV )x − (µF − γV )y, x − yi ≥ (µη − γl)kx − yk2 , ∀x, y ∈ C. That is, µF − γV is strongly monotone with a constant µη − γl. Lemma 2.6. Let H be a real Hilbert space H. Let F : H → H be a ρ-Lipschitzian and and η-strongly monotone operator with constants ρ > 0 and η > 0. Let 0 < µ < 2η ρ2 0 < t < ς ≤ 1. Then S := ςI − tµF : H → H is a contractive mapping with a constant p ς − tτ , where τ = 1 − 1 − µ(2η − µρ2 ). Finally, we recall that the sequence {xn } in H is said to be weakly asymptotically regular if w − lim (xn+1 − xn ) = 0, that is, xn+1 − xn * 0 n→∞

and asymptotically regular if lim kxn+1 − xn k = 0,

n→∞

respectively.

3

Main results

Throughout the rest of this paper, we always assume as follows: Let H be a real Hilbert space. Let T : H → H be a k-strictly pseudocontractive mapping with F (T ) 6= ∅ for some 0 ≤ k < 1, let F : H → H be a ρ-Lipschitzain and η-strongly monotone operator

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with constants ρ > 0 and η > 0, and let V : H → H be an l-Lipschitzian p mapping with and 1 − µ(2η − µρ2 ). a constant l ≥ 0. Let 0 < µ < 2η < τ , where τ = 1 − 0 < γl ρ2 Let Tn : H → H be a mapping defined by Tn x = λn x + (1 − λn )T x, ∀x ∈ H, where 0 ≤ k ≤ λn ≤ λ < 1 and limn→∞ λn = λ. By Lemma 2.4, Tn is nonexpansive. In this section, we consider the following explicit scheme which generates a sequence in an explicit way: xn+1 = αn γV xn + (I − αn µF )Tn xn , ∀n ≥ 0, (3.1) where {αn } ⊂ (0, 1) and x0 ∈ H is an arbitrary initial guess, and establish strong convergence of this sequence to a fixed point x e of T , which is the unique solution of the variational inequality: h(µF − γV )e x, x e − pi ≤ 0,

∀p ∈ F (T ).

(3.2)

First, we consider the following scheme that generates a net {xt }t∈(0,1) in an implicit way: xt = tγV xt + (I − tµF )Tt xt ,

(3.3)

where Tt x = λt x + (1 − λt )T x, ∀x ∈ H, with 0 ≤ k ≤ λt ≤ λ < 1 and limt→0 λt = λ. Indeed, for t ∈ (0, 1), consider a mapping Qt : H → H defined by Qt x = tγV x + (I − tµF )Tt x,

∀x ∈ H.

It is easy to see that Qt is a contractive mapping with constant 1 − t(τ − γl). Indeed, by Lemma 2.6, we have kQt x − Qt yk ≤ tγkV x − V yk + k(I − tµF )Tt x − (I − tµF )Tt yk ≤ tγlkx − yk + (1 − tτ )kx − yk = (1 − t(τ − γl))kx − yk. Hence Qt has a unique fixed point, denoted xt , which uniquely solves the fixed point equation (3.3). By utilizing the same method as in [10,12], we obtain the following theorem for strong convergence of the net {xt } as t → 0, which guarantees the existence of solutions of the variational inequality (3.2). Theorem 3.1. The net {xt } defined via (3.3) converges strongly to a fixed point x e of T as t → 0, which solves the variational inequality (3.2), equivalently, we have PF (T ) (I − µF + γV )e x) = x e. Proof. We first show the uniqueness of a solution of the variational inequality (3.2), which is indeed a consequence of the strong monotonicity of µF − γV . In fact, noting that 0 ≤ γl < τ and µη ≥ τ ⇐⇒ ρ ≥ η, it follows from Lemma 2.5 that h(µF − γV )x − (µF − γV )y, x − yi ≥ (µη − γl)kx − yk2 .

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That is, µF − γV is strongly monotone for 0 ≤ γl < τ ≤ µη. Suppose that x e ∈ F (T ) and x b ∈ F (T ) both are solutions to (3.2). Then we have h(µF − γV )e x, x e−x bi ≤ 0

(3.4)

h(µF − γV )b x, x b−x ei ≤ 0.

(3.5)

and Adding up (3.4) and (3.5) yields h(µF − γV )e x − (µF − γV )b x, x e−x bi ≤ 0. The strong monotonicity of µF − γV implies that x e=x b and the uniqueness is proved. Next, we prove that xt → x e as t → 0. Observing F (T ) = F (Tt ) by Lemma 2.4, from (3.3), we write, for given p ∈ F (T ), xt − p = t(γV xt − µF p) + (I − tµF )Tt xt − (I − tµF )p to derive that kxt − pk2 = thγV xt − µF p, xt − pi + h(I − tµF )Tt xt − (I − tµF )Tt p, xt − pi ≤ (1 − tτ )kxt − pk2 + thγV xt − µF p, xt − pi. It follows that

1 hγV xt − µF p, xt − pi τ 1 ≤ [γlkxt − pk2 + hγV p − µF p, xt − pi]. τ

kxt − pk2 ≤

Therefore kxt − pk2 ≤

1 hγV p − µF p, xt − pi. τ − γl

(3.6)

From (3.6), it follows that kxt − pk ≤

1 kγV p − µF pk, τ − γl

and so {xt }, {V xt }, {T xt }, {Tt xt }, {F xt } and {F Tt xt } are bounded. As a consequence, it follows that lim k(I − Tt )xt k = lim tkγV xt − µF Tt xt k = 0. (3.7) t→0

t→0

Since {xt } is bounded as t → 0, we show that if {tn } is a subsequence in (0,1) such that tn → 0 and xtn * x∗ , then x∗ ∈ F (T ). To this end, define S : H → H by Sx = λx + (1 − λ)T x, ∀x ∈ H. Then S is nonexpansive with F (S) = F (T ) by Lema 2.4. Notice that kSxtn − xtn k ≤ kSxtn − Ttn xtn k + kTtn xtn − xtn k ≤ |λ − λtn |kxtn − T xtn k + kxtn − Ttn xtn k. By (3.7) and λtn → λ, we have lim kSxtn − xtn k = 0.

n→∞

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Hence, if x∗ 6= Sx∗ , then, by Opial condition, we have lim inf kxtn − x∗ k < lim inf kxtn − Sx∗ k n→∞

n→∞

≤ lim inf (kxtn − Sxtn k + kSxtn − Sx∗ k) n→∞

≤ lim inf kxtn − x∗ k, n→∞

which is a contradiction. So get xtn → x∗ .

x∗

∈ F (S) = F (T ). Thus, by replacing p with x∗ in (3.6), we

Finally, we prove that x∗ is a solution of the variational inequality (3.2). Since xt = tγV xt + (I − tµF )Tt xt , we have

1 (µF − γV )xt = − (I − Tt )xt + µ(F xt − F Tt xt ). t From Tt p = p for p ∈ F (T ), it follows that 1 h(µF − γV )xt , xt − pi = − h(I − Tt )xt − (I − Tt )p, xt − pi t + µhF xt − F Tt xt , xt − pi ≤ µhF xt − F Tt xt , xt − pi

(3.8)

since I − Tt is monotone (i.e., hx − y, (I − Tt )x − (I − Tt )yi ≥ 0, x, y ∈ H, which is due to the nonexpansivity of Tt ). Now replacing t in (3.8) with tn and noticing that F xtn − F Ttn xtn → F x∗ − F x∗ = 0 as n → ∞ for x∗ ∈ F (T ), we obtain h(µF − γV )x∗ , x∗ − pi ≤ 0. That is, x∗ ∈ F (T ) is a solution of the variational inequality (3.2); hence x∗ = x e by uniqueness. In a summary, we have shown that each cluster point of {xt } (at t → 0) equals x e. Therefore, xt → x e as t → 0. The variational inequality (3.2) can be rewritten as h(I − µF + γV )e x−x e, x e − pi ≥ 0, ∀p ∈ F (T ). By reminding Lemma 2.4 and (2.1), this is equivalent to the fixed point equation PF (T ) (I − µF + γV )e x) = x e.

¤

Remark 3.1. 1) Theorem 3.1 improves the case of the nonexpansive mapping S in Theorem 3.1 of Tian [10] (and Ceng et al. [12]) to the case of the k-strictly pseudocontractive mapping T . 2) Theorem 3.1 includes the corresponding results of Tian [11], Marino and Xu [9], Moudafi [13] and Xu [14] as some special cases. First of all, we give the following result in order to establish strong convergence of the sequence generated by the explicit scheme (3.1). Theorem 3.2. Let {xn } be the sequence generated iteratively by the scheme (3.1) and let LIM be a Banach limit. If {αn } satisfies the following condition:

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(C1) {αn } ⊂ (0, 1) and limn→∞ αn = 0, then LIMn (hµF x e − γV x e, x e − xn i) ≤ 0, where x e = limt→0+ xt with xt being defined by xt = tγV xt + (I − tµF )Sxt ,

(3.9)

and Sx = λx + (1 − λ)T x, ∀x ∈ H, with 0 ≤ k ≤ λ < 1. Proof. First, note that from the condition (C1), without loss of generality, we assume that αn τ < 1 for all n ≥ 0. Let {xt } be the net generated by (3.9). By Theorem 3.1 with λt = λ for t ∈ (0, 1) and Lemma 2.4, there exists limt→0 xt ∈ F (T ). Denote it by x e. Moreover x e is the unique solution of the variational inequality (3.2). By (3.9), we have kxt − xn+1 k = ktγV xt + (I − tµF )Sxt − xn+1 k = k(I − tµF )Sxt − (I − tµF )xn+1 + t(γV xt − µF xn+1 )k. Applying Lemma 2.1 and Lemma 2.6, we have kxt − xn+1 k2 ≤ (1 − tτ )2 kSxt − xn+1 k2 + 2thγV xt − µF xn+1 , xt − xn+1 i.

(3.10)

From the proof of Theorem 3.1, we know that {xt }, {V xt }, {T xt }, {Sxt }, {F Sxt } and {F xt } are bounded. pk } for all n ≥ 0 and all p ∈ F (T ). Now we show that kxn − pk ≤ max{kx0 − pk, kµFτp−γV −γl Indeed, let p ∈ F (T ). Noticing p = Tn p, we have

kxn+1 − pk = kαn (γV xn − µF p) + (I − αn µF )Tn xn − (I − αn µF )Tn pk ≤ (1 − αn τ )kxn − pk + αn kγV xn − µF pk ≤ (1 − αn τ )kxn − pk + αn (kγV xn − γV pk + kγV p − µF pk) kγV p − µF pk ≤ [1 − (τ − γl)αn ]kxn − pk + (τ − γl)αn τ − γl ½ ¾ kγV p − µF pk ≤ max kxn − pk, . τ − γl pk }. Hence {xn } is bounded, Using an induction, we have kxn −pk ≤ max{kx0 −pk, kγVτp−µF −γl and so are {V xn }, {T xn } {Tn xn }, {F Tn xn }, and {F xn }. As a consequence of condition (C1), we get

kxn+1 − Tn xn k = αn kγV xn − µF Tn xn k → 0

(n → ∞).

From definitions of S and Tn with limn→∞ λn = λ, we deduce kSxt − xn+1 k ≤ kSxt − Sxn k + kSxn − Tn xn k + kTn xn − xn+1 k ≤ kxt − xn k + |λ − λn |kxn − T xn k + kTn xn − xn+1 k , ≤ kxt − xn k + |λ − λn |K1 + kTn xn − xn+1 k = kxt − xn k + en ,

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where K1 = sup{kxn − T xn k : n ≥ 0} and en = |λ − λn |K1 + kxn+1 − Tn xn k → 0 as n → ∞. Also observing that F is η-strongly monotone, we have hµF xt − µF xn , xt − xn i ≥ µηkxt − xn k2 ≥ τ kxt − xn k2 .

(3.11)

So, by combining (3.10) and (3.11), we obtain kxt − xn+1 k2 ≤ (1 − tτ )2 (kxt − xn k + en )2 + 2thγV xt − µF xt , xt − xn+1 i + 2thµF xt − µF xn+1 , xt − xn+1 i ≤ (t2 τ − 2t)τ kxt − xn k2 + kxt − xn k2 + (1 − tτ )2 en (2kxt − xn k + en ) + 2thγV xt − µF xt , xt − xn+1 i + 2thµF xt − µF xn+1 , xt − xn+1 i ≤ (t2 τ − 2t)hµF xt − µF xn , xt − xn i + kxt − xn k2 + en (K2 + en ) + 2thγV xt − µF xt , xt − xn+1 i + 2thµF xt − µF xn+1 , xt − xn+1 i

(3.12)

= t2 τ hµF xt − µF xn , xt − xn i + kxt − xn k2 + en (K2 + en ) + 2thγV xt − µF xt , xt − xn+1 i + 2t(hµF xt − µF xn+1 , xt − xn+1 i − hµF xt − µF xn , xt − xn i), where K2 = sup{2kxt − xn k : t, n ≥ 0}. Applying the Banach limit LIM to (3.12) together with limn→∞ en = 0, we have LIMn (kxt − xn+1 k2 ) ≤ t2 τ LIMn (hµF xt − µF xn , xt − xn i) + LIMn (kxt − xn k2 ) + 2tLIMn (hγV xt − µF xt , xt − xn+1 i) (3.13) + 2t[LIMn (hµF xt − µF xn+1 , xt − xn+1 i) − LIMn (hµF xt − µF xn , xt − xn i)]. Using the property LIMn (an ) = LIMn (an+1 ) of Banach limit in (3.12), we obtain LIMn (hµF xt − γV xt , xt − xn i) = LIMn (hµF xt − γV xt , xt − xn+1 i) tτ ≤ LIMn (hµF xt − µF xn , xt − xn i) 2 1 + [LIMn (kxt − xn k2 ) − LIMn (kxt − xn k2 )] 2t + [LIMn (hµF xt − µF xn , xt − xn i) − LIMn (hµF xt − µF xn , xt − xn i)] tτ = LIMn (hµF xt − µF xn , xt − xn i). 2

(3.14)

Since thµF xt − µF xn , xt − xn i ≤ tµρkxt − xn k2 ≤ tµρ(kxt − pk + kp − xn k)2 (3.15) ¶2 µ kγV p − µF pk + kx0 − pk → 0 (as t → 0), ≤ tµρ τ − γl we conclude from (3.14) and (3.15) that LIMn (hµF x e − γV x e, x e − xn )i) ≤ lim sup LIMn (hµF xt − γV xt , xt − xn i) t→0

≤ lim sup t→0

tτ LIMn (hµF xt − µF xn , xt − xn i) ≤ 0. 2

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This completes the proof.

¤

Now, using Theorem 3.2, we establish strong convergence of the sequence generated by the explicit scheme (3.1) to a fixed point x e of T , which is the unique solution of the variational inequality (3.2). Theorem 3.3. Let {xn } be the sequence generated iteratively by the scheme (3.1), where {αn } satisfies the following conditions: (C1) {αn } ⊂ (0, 1) and limn→∞ αn = 0. P∞ (C2) n=0 αn = ∞. If {xn } is weakly asymptotically regular, then {xn } converges strongly to x e ∈ F (T ), where is the unique solution of the variational inequality (3.2). Proof. First, note that from the condition (C1), without loss of generality, we assume n (τ −γl) < 1 for all n ≥ 0. that αn τ < 1 and 2α1−α n γl Let xt be defined by (3.9), that is, xt = tγV xt + (I − tµF )Sxt for 0 < t < 1, where Sx = λx + (1 − λ)T x, ∀x ∈ H, with 0 ≤ k ≤ λ < 1, and let limt→0 xt := x e ∈ F (S) = F (T ) (by using Theorem 3.1 and Lemma 2.4). Then x e is the unique solution of the variational inequality (3.2). We divides the proof several steps: ¾ ½ kγV p−µF pk for all n ≥ 0 and all p ∈ F (T ) Step 1. We see that kxn − pk ≤ max kx0 − pk, τ −γl as in the proof of Theorem 3.2. Hence {xn } is bounded and so are {Tn xn }, {F Tn xn } and {V xn }. Step 2. We show that lim supn→∞ hµF x e − γV x e, x e − xn i ≤ 0. To this end, put an := hµF x e − γV x e, x e − xn )i, ∀n ≥ 0. Then Theorem 3.2 implies that LIMn (an ) ≤ 0 for any Banach limit LIM . Since {xn } is bounded, there exists a subsequence {xnj } of {xn } such that lim sup(an+1 − an ) = lim (anj +1 − anj ) j→∞

n→∞

and xnj * v ∈ H. This implies that xnj +1 * v since {xn } is weakly asymptotically regular. Therefore, we have w − lim (e x − xnj +1 ) = w − lim (e x − xnj ) = (e x − v), j→∞

j→∞

and so lim sup(an+1 − an ) = lim hµF x e − γV x e, (e x − xnj +1 ) − (e x − xnj )i = 0. n→∞

j→∞

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Then Lemma 2.2 implies that lim supn→∞ an ≤ 0, that is, lim suphµF x e − γV x e, x e − xn )i ≤ 0. n→∞

Step 3. We show that limn→∞ kxn − x ek = 0. By using (3.1), we have xn+1 − x e = αn (γV xn − µF x e) + (I − αn µF )Tn xn − (I − αn µF )e x. Applying Lemma 2.1 and Lemma 2.6, we obtain kxn+1 − x ek2 = k(I − αn µF )Tn xn − (I − αn µF )e x + αn (γV xn − µF x e)k2 ≤ k(I − αn µF )Tn xn − (I − αn µF )Tn x ek2 + 2αn hγV xn − µF x e, xn+1 − x ei ≤ (1 − αn τ )2 kxn − x ek2 + 2αn hγV xn − γV x e, xn+1 − x ei + 2αn hγV x e − µF x e, xn+1 − x ei

(3.15)

≤ (1 − αn τ )2 kxn − x ek2 + αn γl(kxn − x ek2 + kxn+1 − x ek2 ) + 2αn hγV x e − µF x e, xn+1 − x ei. It then follows from (3.15) that 2αn (1 − αn τ )2 + αn γl kxn − x ek2 + hγV x e − µF x e, xn+1 − x ei 1 − αn γl 1 − αn γl µ ¶ 2αn (τ − γl) ≤ 1− kxn − x ek2 (3.16) 1 − αn γl µ ¶ αn τ 2 2αn (τ − γl) 1 ei + + hγV x e − µF x e, xn+1 − x K3 , 1 − αn γl τ − γl 2(τ − γl)

kxn+1 − x ek2 ≤

ek2 : n ≥ 0}. Put where K3 = sup{kxn − x 2αn (τ − γl) αn τ 2 1 and δn = hµF x e − γV x e, x e − xn+1 i + K3 . 1 − αn γl τ − γl 2(τ − γl) P From (C1), (C2) and Step 2, it follows that βn → 0, ∞ n=0 βn = ∞ and lim supn→∞ δn ≤ 0. Since (3.16) reduces to βn =

kxn+1 − x ek2 ≤ (1 − βn )kxn − x ek2 + βn δn , from Lemma 2.3 with rn = 0, we conclude that limn→∞ kxn − x ek = 0. This completes the proof. ¤ Corollary 3.1. Let {xn } be the sequence generated iteratively by the scheme (3.1), where {αn } satisfies the following conditions: (C1) {αn } ⊂ (0, 1) and limn→∞ αn = 0. P∞ (C2) n=0 αn = ∞.

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If {xn } is asymptotically regular, then {xn } converges strongly to x e ∈ F (T ), where is the unique solution of the variational inequality (3.2). Remark 3.2. If {αn } and {λn } in Corollary 3.1 satisfy conditions (C1), (C2), (C3)

P∞

n=0 |αn+1

− αn | < ∞; or

n+1 = 1 or, equivalently, limn→∞ αnα−α = 0; or, n+1 P∞ (C5) |αn+1 − αn | ≤ o(αn+1 ) + σn , n=0 σn < ∞ (the perturbed control condition); and P∞ (C6) n=0 |λn+1 − λn | < ∞,

(C4) limn→∞

αn αn+1

then the sequence {xn } generated by (3.1) is asymptotically regular. Now we give only the proof in case when {αn } satisfies the conditions (C1), (C2), (C5) and (C6). By Step 1 in the proof of Theorem 3.3, there exists a constant K4 > 0 such that for all n ≥ 0, µkF Tn xn k + γkV xn k ≤ K4 . Next, we notice that kTn xn − Tn−1 xn−1 k ≤ kTn xn − Tn xn−1 k + kTn xn−1 − Tn−1 xn−1 k ≤ kxn − xn−1 k + |λn − λn−1 |kxn−1 − T xn−1 k ≤ kxn − xn−1 k + |λn − λn−1 |K5 , where K5 = sup{kxn − T xn k : n ≥ 0}. So, we obtain, for all n ≥ 0, = ≤ ≤ ≤

kxn+1 − xn k k(I − αn µF )Tn xn − (I − αn µF )Tn−1 xn−1 + µ(αn − αn−1 )F Tn−1 xn−1 + γ[αn (V xn − V xn−1 ) + V xn−1 (αn − αn−1 )]k (1 − αn τ )kTn xn − Tn−1 xn−1 k + µ|αn − αn−1 |kF Tn−1 xn−1 k + γ[αn [lkxn − xn−1 k + kV xn−1 k|αn − αn−1 ] (1 − αn (τ − γl))kxn − xn−1 k + |λn − λn−1 |K5 + |αn − αn−1 |K4 (1 − αn (τ − γl))kxn − xn−1 k + |λn − λn−1 |K5 + (o(αn ) + σn−1 )K4 .

(3.17)

By taking sn+1 = kxn+1 − xn k, βn = αn (τ − γl), βn δn = o(αn )K4 and rn = σn−1 K4 + |λn − λn−1 |K5 , from (3.17) we have sn+1 ≤ (1 − βn )sn + βn δn + rn . Hence, by (C1), (C2), (C5), (C6) and Lemma 2.3, we obtain lim kxn+1 − xn k = 0.

n→∞

In view of this observation, we have the following: Corollary 3.2. Let {xn } be the sequence generated iteratively by the scheme (3.1), where {αn } and {λn } satisfy the conditions (C1), (C2), (C5) and (C6) (or the conditions (C1), (C2), (C3) and (C6), or the conditions (C1), (C2), (C4) and (C6)). Then {xn } converges strongly to x e ∈ F (T ), where is the unique solution of the variational inequality (3.2). Remark 3.3. 1) Theorem 3.3 extends Theorem 3.2 of Tian [10] and Ceng et al. [12] in the following ways:

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(a) The nonexpansive mapping S in [10, 12, Theorem 3.2] is extended to the case of a k-strictly pseudocontractive mapping T . P (b) The condition ∞ n=0 |αn+1 − αn | < ∞ in [10, 12, Theorem 3.2] is relaxed to the weak asymptotic regularity on {xn }. 2) Theorem 3.3 also generalizes the corresponding results of Cho et al. [3], Jung [6] and Marino and Xu [9] in following aspects: (a) A strongly positive bounded linear operator A in [3,6,9] is extended to the case of a ρ-Lipschitzian and η-strongly monotone operator F . (In fact, from the definitions, it follows that a strongly positive bounded linear operator A (i.e., there exists a constant γ > 0 with the property: hAx, xi ≥ γkxk2 , x ∈ H) is a kAk-Lipschitzian and γ-strongly monotone operator). (b) The contractive mapping f with a constant α ∈ (0, 1) in [3,6,9] is extended to the case of a Lipschizian mapping V with a constant l ≥ 0. (c) The nonexpansive mapping S in [3,9] is extended to the case of a k-strictly pseudocontractive mapping T . P (d) The condition ∞ n=0 |αn+1 − αn | < ∞ in [3,9] is weakened to the weak asymptotic regularity on {xn }.

Acknowledgments 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 (2013021600).

References [1] F. E. Browder and W. V. Petryshn, Construction of fixed points of nonlinear mappings Hilbert space, J. Math. Anal. Appl. 20 (1967) 197–228. [2] G. L. Acedo and H. K. Xu, Iterative methods for strictly pseudo-contractions in Hilbert space, Nonlinear Anal. 67 (2007) 2258–2271. [3] Y. J. Cho, S. M. Kang and X. Qin, Some results on k-strictly pseudo-contractive mappings in Hilbert spaces, Nonlinear Anal. 70 (2009) 1956–1964. [4] J. S. Jung, Strong convergence of iterative methods for k-strictly pseudo-contractive mappings in Hilbert spaces, Applied Math. Comput. 215 (2010) 3746-3753.

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[5] J. S. Jung, Some results on a general iterative method for k-strictly pseudo-contractive mappings, Fixed Point Theory Appl. 2011 (2011) 24 doi:10.1186/1687-1812-2011-24. [6] J. S. Jung, A general iterative method with some control conditions for k-strictly pseudo-contractive mappings, J. Computat. Anal. Appl. 14 (2012) no. 6, 1165–1177. [7] C. H. Morales and J. S. Jung, Convergence of paths for pseudo-contractive mappings in Banach spaces, Proc. Amer. math. Soc. 128 (2000) 3411–3419. [8] I. Yamada, The hybrid steepest descent method for the variational inequality problems over the intersection of fixed points sets of nonexpansive mappings, in: D. Butnariu, Y. Censor, S. Reich (Eds), Inherently Parallel Algorithms in Feasibility and Optimization anf Their Applications, Elservier, New York, 2001, pp. 473–504. [9] G. Marino and H. X. Xu, A general iterative method for nonexpansive mappings in Hilbert spaces, J. Math. Anal. Appl. 318 (2006) 43–52. [10] M. Tian, A general iterative method based on the hybrid steepest descent scheme for nonexpansive mappings in Hilbert spaces, in Proceedings of the International Conefrence on Computational Intelligence and Soft ware Engineering (CiSE 2010), art. no. 5677064, 2010. [11] M. Tian, A general iterative algorithm for nonexpansive mappings in Hilbert spaces, Nonlinear Anal. 73 (2010) 689–694. [12] L.-C, Ceng, Q. H. Ansari, J.-C. Yao, Some iterative methods for finding fixed points and for solving constrained convex minimization problems, Nonlinear Anal. 74 (2011) 5286–5302. [13] A. Moudafi, Viscosity approximation methods for fixed-points problems, J. Math. Anal. Appl. 241 (2000) 46–55. [14] H. K. Xu, Viscosity approximation methods for nonexpansive mappings, J. Math. Anal. Appl. 298 (2004) 279–291. [15] N. Shioji and W. Takahashi, Strong convergence of approximated sequences for nonexpansive mappings in Banach spaces, Proc. Amer. Math. Soc. 125 (1997), no. 12, 3641–3645. [16] H. K. Xu, Iterative algorithms for nonlinear operators, J. London Math. Soc. 66 (2002) 240–256. [17] H. Zhou, Convergence theorems of fixed points for k-strict pseudo-contractions in Hilbert spaces, Nonlinear Anal. 69 (2008) 456–462.

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BE-ALGEBRAS WITH ORDER REVERSING INVOLUTION SUN SHIN AHN AND YOUNG HEE KIM∗ AND JUNG HEE PARK

Abstract. The notions of a filter’s radical and extended filter are introduced in BE-algebras. Then some properties of filter’s radical and extended filter are obtained. Using a special set x−1 ∗ F , we give an equivalent condition for a filter to be prime.

1. Introduction In [6], H. S. Kim and Y. H. Kim introduced the notion of a BE-algebra. S. S. Ahn and K. S. So [4, 5] introduced the notion of ideals in BE-algebras. S. S. Ahn et al. [2] fuzzified the concept of BE-algebras and investigated some of their properties. Y. B. Jun and S. S. Ahn ([7]) provided several degrees in defining a fuzzy filter and a fuzzy implicative filter. It was a generalization of a fuzzy filter. In this paper, we introduce the notions of a filter’s radical and an extended filter in BEalgebras. Some properties of a filter’s radical and an extended filter are obtained. Using a special set x−1 ∗ F , we obtain an equivalent condition for a filter to be a prime filter. 2. Preliminaries An algebra (X; ∗, 1) of type (2, 0) is called a BE-algebra ([6]) 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 “ ∗ ”. Note 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. A mapping f : X → Y of BEalgebras is called a homomorphism if f (x ∗ y) = f (x) ∗ f (y) for any x, y ∈ X. A homomorphism f of BE-algebras is called an epimorphism if f is onto. Note that if f is a homomorphism of BE-algebras, then f (1) = 1. ∗

Corresponding author 2010 Mathematics Subject Classification. 06F35. Key words and phrases. BE-algebra, filter’s radical, (prime, primary) filter. Supported by Chungbuk National University Fund, 2013. 1

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Proposition 2.1([6]). Let (X; ∗, 1) be a self distributive BE-algebra. Then the 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). A BE-algebra (X; ∗, 1) is said to be transitive if it satisfies Proposition 2.1(iii). If a BE-algebra X is transitive, then Proposition 2.1(i) holds ([7]). Definition 2.2. Let X be a BE-algebra and let ∅ ̸= F ⊆ X of a BE-algebra X. F is called a filter ([6]) of X if (F1) 1 ∈ F ; (F2) if x ∗ y, x ∈ F , then y ∈ F . F is an implicative filter ([7]) of X if (F1) and (F3) if x ∗ (y ∗ z) ∈ F and x ∗ y ∈ F , then x ∗ z ∈ F . Note that every implicative filter is a filter in a BE-algebra. Proposition 2.3. Let X be a BE-algebra and let F be a filter of X. If x ≤ y and x ∈ F for any y ∈ F , then y ∈ F . Definition 2.4. Let X be a BE-algebra X and ∅ ̸= A ⊆ X. If B is the least filter containing A in X, then B is called the filter generated by A and denoted by (A]. It is trivial to verify that (A] = ∩{B|A ⊆ B ⊆ X, B is a filter}. In what follows, ({a}] is denoted by (a] and [a1 , a2 , · · · , an , x] := a1 ∗ (a2 ∗ (· · · (an ∗ x) · · · )). Specially, [a, x]0 := x, [a, x]1 := a ∗ x, and [a, x]n := a ∗ (a ∗ (· · · ∗ (a ∗x) · · · )) (n ≥ 2). | {z } n

Proposition 2.5([8]). Let X be a transitive BE-algebra and ∅ ̸= A ⊆ X. Then (A] = {x ∈ X|∃a1 , · · · an ∈ A, n ∈ N such that [a1 , a2 , · · · , an , x] = 1}. Definition 2.6([1]). Let X be a BE-algebra. X is said to be commutative if the following identity holds (C) (x ∗ y) ∗ y = (y ∗ x) ∗ x, i.e., x ∨ y = y ∨ x where x ∨ y = (y ∗ x) ∗ x, for all x, y ∈ X. Theorem 2.7([1]). If (X; ∗, 1) is a commutative BE-algebra X, then it is a semilattice with respect to ∨. 3. BE-algebra with order reversing involution Definition 3.1. A BE-algebra (X; ∗, 1) is said to have an order reversing involution “ ′ ” if

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(i) if x ≤ y, then y ′ ≤ x′ and (x′ )′ = x; (ii) 0 is the smallest element of X; (iii) x ∗ y = y ′ ∗ x′ for all x, y ∈ X, where 0′ := 1. In what follows, let X denote a BE-algebra with order reversing involution ′ unless otherwise specified. ′

Proposition 3.2. Suppose that X is a transitive BE-algebra with order reversing involution . Then the following hold: (i) x ∗ 0 = x′ ; (ii) 0 ∗ x = 1; (iii) x ≤ y ⇔ y ∗ z ≤ x ∗ z.

for any x, y, z ∈ X. Proof. (i) By Definition 3.1(iii), we have x ∗ 0 = 0′ ∗ x′ = 1 ∗ x′ = x′ . (ii) Let x ∈ X. Using Definition 3.1(iii) and (BE2), we get 0 ∗ x = x′ ∗ 0′ = x′ ∗ 1 = 1. (iii) By Proposition 2.1(i), x ≤ y imply y ∗ z ≤ x ∗ z. Conversely, suppose that y ∗ z ≤ x ∗ z for all x, y, z ∈ X. By Definition 3.1(i), we have x ∗ y = y ′ ∗ x′ = (y ∗ 0) ∗ (x ∗ 0) = 1, proving that x ≤ y.  Theorem 3.3. Let X be a transitive BE-algebra and let a, b, x ∈ X. (i) If a ≥ b, then [a, x]n ≤ [b, x]n for any n ∈ N; (ii) If n, m ∈ N with n ≥ m, then [a, x]n ≥ [a, x]m ; (iii) [a, x]n ≥ x for any n ∈ N. Proof. These conditions are trivial when n = 0 or m = 0. (i) We use induction on n to show [a, x]n ≤ [b, x]n . If n = 1, then a ≥ b imply [a, x]1 = a ∗ x ≤ [b, x]1 = b ∗ x. For n > 1, assume that [a, x]m ≤ [b, x]m for any m < n. Then [a, x]n = a ∗ [a, x]n−1 ≤ a ∗ [b, x]n−1 ≤ b ∗ [b, x]n−1 = [b, x]n . (ii) Suppose that n = m + p. Then p ≥ 0. We use induction on p to show [a, x]m+p ≥ [a, x]m . If p = 0, then [a, x]m+p ≥ [a, x]m holds. For p > 1, assume that [a, x]m+q ≥ [a, x]m for any q < p. It follows that [a, x]m+p = a ∗ [a, x]m+(p−1) ≥ a ∗ [a, x]m ≥ [a, x]m . 

(iii) The proof is similar to (i). Theorem 3.4. Let X be a transitive BE-algebra and let a, b ∈ X. Then (i) (a ∨ b] ⊆ (a] ∪ (b]; (ii) if a ≤ b, then (b] ⊆ (a].

Proof. (i) For any x ∈ (a ∨ b], there exists n ∈ N+ such that [a ∨ b, x]n = 1. Since a ≤ a ∨ b and b ≤ a ∨ b, by Theorem 3.3(i), [a ∨ b, x]n ≤ [a, x]n and [a ∨ b, x]n ≤ [b, x]n . Hence [a, x]n = 1 and [b, x]n = 1, i.e., x ∈ (a] ∩ (b]. Therefore x ∈ (a] ∩ (b]. Thus (a ∨ b] ⊆ (a] ∩ (b].

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(ii) If x ∈ (b], then there exists n ∈ N+ such that [b, x]n = 1. Since a ≤ b imply [b, x]n ≤ [a, x]n , we have [a, x]n = 1. Hence x ∈ (a].  Proposition 3.5. Let X1 , X2 be BE-algebras with order reversing involution ′ and f : X1 → X2 be a homomorphism of BE-algebras. If f (0) = 0, then f (x′ ) = (f (x))′ for any x ∈ X. Proof. For any x ∈ X1 , we have f (x′ ) = f (x ∗ 0) = f (x) ∗ f (0) = f (x) ∗ 0 = (f (x))′ .



Let X be a BE-algebra with an order reversing involution ′ . For any x, y ∈ X, we define a binary operation “ ⊕ ” as follows: x ⊕ y := x′ ∗ y. For any a ∈ X and n ∈ N, we denote (n + 1)a := a ⊕ (na). Proposition 3.6. Let X be a BE-algebra with order reserving ′ . Then for any a, b ∈ X and m, n ∈ N+ , we have (i) a, b ≤ a ⊕ b; (ii) if m ≤ n, then ma ≤ na. Proof. (i) By (BE4) and (BE2), we have b ∗ (a ⊕ b) = b ∗ (a′ ∗ b) = a′ ∗ (b ∗ b) = a′ ∗ 1 = 1 and so b ≤ a ⊕ b. Since a ∗ (a ⊕ b) = a ∗ (a′ ∗ b) = a ∗ (b′ ∗ (a′ )′ ) = a ∗ (b′ ∗ a) = b′ ∗ (a ∗ a) = b′ ∗ 1 = 1, we obtain a ≤ a ⊕ b. (ii) Using (i), we obtain na = ma ⊕ (n − m)a ≥ ma. This completes the proof.  Proposition 3.7. Let X be a transitive BE-algebra with order reserving involution ′ . Then, for any a, b, c ∈ X, a ≤ b implies a ⊕ c ≤ b ⊕ c. Proof. Let a ≤ b for any a, b ∈ X. By Definition 3.1(iii), we have b′ ≤ a′ . Since X is transitive, we obtain a′ ∗ c ≤ b′ ∗ c for any c ∈ X, proving that a ⊕ c ≤ b ⊕ c.  Let f : X1 → X2 be a homomorphism of BE-algebras with order reserving involution ′ . We define the dual kernel of f denoted by DKer f , as follows: DKer f := {x ∈ X1 |f (x) = 12 }.

4. The filter’s radical in BE-algebras Definition 4.1. Let J be a filter of X. A subset √ A := {x ∈ X|∃n ∈ N such that nx ∈ J} is called a filter’s radical of J and we denote it by J.

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Example 4.2. Let X := {0, a, b, c, d, 1} be a set with 0′ = 1, a′ = c, b′ = d, c′ = a, d′ = b, 1′ = 0. Define a binary operation ∗ as follows: ∗ 0 a b c d 1

0 1 c d a b 0

a 1 1 a a 1 a

b 1 b 1 1 1 b

c 1 c b 1 b c

d 1 b a a 1 d

1 1 1 1 1 1 1

′ Then (X; ∗,′ , 1, 0) is a BE-algebra with order reversing involution . Since {1} is a filter of X √ and 2b = 1, na = a, nc = c, nd = a for all n ∈ N, we obtain {1} = {b, 1}. √ Theorem 4.3. Let J be a filter of a BE-algebra X. Then J ⊆ J. √ J. Thus 1x = x ∈ J, i.e., ∃1 ∈ N such that 1x ∈ J. Hence x ∈ Proof. Let x ∈ J. Then √ J ⊆ J.  √ Note that 1 ∈ J whenever J is a filter of a BE-algebra X.

Theorem 4.4. Let J1 and J2 be filters of a BE-algebra X with an order reversing involution . Then the following hold: √ √ (i) if J1 ⊆ J2 , then J1 ⊆ J2 ; √ √ √ (ii) J1 ∩ J2 = √J1 ∩ J2 ; √ √ (iii) J1 ∪ √J2 ⊆ (J1 ∪ J2 ]; √ √ (iv) J1 ⊆ ( J1 ]. √ √ Proof. (i) Let x ∈ J1 . Then there exists n ∈ N such that nx ∈ J1 ⊆ J2 . Hence x ∈ J2 . √ √ Therefore J1 ⊆ J2 . √ √ (ii) Let x ∈ J1 ∩ J2 . Then there exist m, n ∈ N such that mx ∈ J1 and nx ∈ J2 and so m, n ≤ m + n. By Proposition 3.6(ii), we have mx ≤ (m + n)x and n ≤ (m + n)x. Since mx ∈ J1 and J1 is a filter of X, we get (m + n)x ∈ J1 . Since nx ∈ J2 and J2 is a filter of X, we obtain √ √ √ √ (m + n)x ∈ J2 . Hence x ∈ J1 ∩ J2 . Therefore J1 ∩ J2 ⊆ J1 ∩ J2 . Since J1 , J2 are filters of X, J1 ∩ J2 is a filter of X. Since J1 ∩ J2 ⊆ J1 and J1 ∩ J2 ⊆ J2 , by √ √ √ √ √ √ √ (i) we get J1 ∩ J2 ⊆ J1 and J1 ∩ J2 ⊆ J2 . Hence J1 ∩ J2 ⊆ J1 ∩ J2 . Thus we have √ √ √ J1 ∩ J2 = J1 ∩ J2 . √ √ √ √ (iii) Let x ∈ J1 ∪ J2 . Then x ∈ J1 or x ∈ J2 . Hence there exists n ∈ N such that nx ∈ J1 or there exists m ∈ N such 2 . In any case, √ we have n ∈ N such that √ that mx ∈ J√ √ nx ∈ J1 ∪ J2 ⊆ (J1 ∪ J2 ]. Thus x ∈ (J1 ∪ J2 ], i.e., J1 ∪ J2 ⊆ (J1 ∪ J2 ]. (iv) It follows immediately from Theorem 4.3 and Theorem 4.4(i).  √ √ Corollary 4.5. Let J1 , J2 , · · · , Jn be an implicative filter of a BE-algebra X. If J1 = J2 = √ √ √ · · · = Jn , then J1 = J1 ∩ · · · Jn . √ √ √ √ Proof. Using Theorem 4.4(ii), we have J1 = J1 ∩ · · · ∩ Jn = J1 ∩ · · · ∩ Jn .  ′

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The relations between the generated filters and their radicals are discussed as follows. Proposition 4.6. Let X be a transitive BE-algebra and let a, b ∈ X. Then the following hold: √ √ (b] ⊆ (a]; (i) if a ≤ b, then √ √ (ii) (a ∨ b] ⊆ (a] ∩ (b]. √Proof.√(i) If a ≤ b, then using Theorem 3.4(ii) we have (b] ⊆ (a]. By Theorem 4.4(i), we have (b] ⊆ (a]. √ √ (ii) Since (a ∨ b] ⊆ (a] ∩ (b], by Theorem 4.4(i), we obtain (a ∨ b] ⊆ (a] ∩ (b].  Theorem 4.7. Let J be an implicative filter of a transitive BE-algebra X with order reversing involution ′ . Then, for any n ∈ N and for any a ∈ X, na ∈ J implies a ∈ J. Proof. Use an induction on n. If n = 1, then it is trivial. Suppose the conclusion holds for p < n + 1 and (n + 1)a ∈ J. Then a′ ∗ (a′ ∗ (n − 1)a) = a′ ∗ na = (n + 1)a ∈ J. Since a′ ∗ a′ = 1 ∈ J and J is an implicative filter of X, we have na = a′ ∗ (n − 1)a ∈ J. By assumption, a ∈ J.  Theorem 4.8. Let J√be an implicative filter of a transitive BE-algebra X with order reversing involution ′ . Then J = J. √ √ Proof. By Theorem 4.3, J ⊆ J. Let x√∈ J. Then there√exists n ∈ N such that nx ∈ J. Using Theorem 4.7, we have x ∈ J. Hence J ⊆ J. Thus J = J.  Theorem 4.9. Let X1 , X2 be BE-algebras with order reversing involution ′ . Let f : X1 → X2 be a homomorphism of BE-algebras. If f (0) = 0, then for any n ∈ N, f (nx) = nf (x) for any x ∈ X1 . Proof. We use the induction on n to prove the conclusion. If n = 1, the conclusion is trivial. Now suppose that n > 1 and the conclusion holds for n. Then, by Proposition 3.5, we obtain f ((n + 1)x) =f (x ⊕ nx) = f (x′ ∗ nx) =f (x′ ) ∗ f (nx) = (f (x))′ ∗ nf (x) =f (x) ⊕ nf (x) = (n + 1)f (x), 

i.e., the conclusion holds for n + 1.

Theorem 4.10. Let (X1 , ∗1 , ′1 , 01 , 11 ) and (X2 , ∗2 , ′2 , 02 , 12 ) be BE-algebras with order reversing involutions ′i . Let f : X1 → X2 be an epimorphism of BE-algebras. If J is a filter of X1 such that DKer f ⊆ J, then f (J) is a filter of X2 . Proof. Suppose that J is a filter of X1 such that DKer f ⊆ J. Then 11 ∈ J and so 12 = f (x) ∗2 f (x) = f (x ∗1 x) = f (11 ) ∈ f (J). Assume that y∗2 z ∈ f (J) and y ∈ f (J) for any y, z ∈ X2 . Then there exist x, w ∈ J and v ∈ X1 such that f (x) = y ∗2 z, f (w) = y and f (v) = z. Hence f (x ∗1 (w ∗1 v)) = f (x) ∗2 (f (w ∗1 v)) = (y ∗2 z) ∗2 (f (w) ∗2 f (v)) = (y ∗2 z) ∗2 (y ∗2 z) = 12 and so x ∗1 (w ∗1 v) ∈ DKer f . Using DKer f ⊆ J, we have x ∗1 (w ∗1 v) ∈ J. Since x ∈ J and J is a filter of X, we get w ∗1 v ∈ J. Using w ∈ J, we obtain v ∈ J. Therefore z = f (v) ∈ f (J). Thus f (J) is a filter of X2 . 

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Proposition 4.11. Let (X1 , ∗1 , ′1 , 01 , 11 ) and (X2 , ∗2 , ′2 , 02 , 12 ) be BE-algebras with order reversing involutions ′i . Let f : X1 → X2 be an epimorphism √ of BE-algebras with f (0) = 0. If J √ is a filter of X1 such that DKer f ⊆ J, then f ( J) = f (J). Proof. √ Let J be a filter of X1 such √ that DKer f ⊆ J. By Theorem 4.10, √ f (J) is a filter of X2 . If y ∈ f ( J), then there exists x ∈ J such that f (x) = y. Since x ∈ J, there exists√n ∈ N such that nx√∈ J. Using √ Theorem 4.9, we have f (nx) = nf (x) = ny ∈ f (J). Hence y ∈ f (J). Therefore f ( J) ⊆ f√ (J). Conversely, let y ∈ f (J). Then there exists n ∈ N such that ny ∈ f (J). Since f is an epimorphism, there exists x ∈ X1 such that f (x) = y. Hence f (nx) = nf (x) = ny ∈ f (J) and so there exists z ∈ J such that f (z) = f (nx). Therefore f (z∗1 nx) = f (z)∗2 f (nx) = f (z)∗2 f (z) = 12 and √ so z ∗ nx ∈ DKer f ⊆ J. √ J is a filter √ Since z ∈ J and √ of X1 , we obtain nx ∈ J. Hence x ∈ J and so y = f (x) ∈ f ( J). Therefore f (J) ⊆ f ( J). This completes the proof.  Theorem 4.12. Let (X1 , ∗1 , ′1 , 01 , 11 ) and (X2 , ∗2 , ′2 , 02 , 12 ) be BE-algebras with order reversing involutions ′i . Let f : X1 → X2 be a homomorphism of BE-algebras. If J is a filter of X2 , √ √ then f −1 (J) is a filter of X1 . Furthermore, if f (01 ) = 02 , then f −1 ( J) = f −1 (J). Proof. Let J be a filter of X2 . Let x ∗1 y, x ∈ f −1 (J). Then f (x ∗1 y) = f (x) ∗2 f (y) ∈ J and f (x) ∈ J. Since J is a filter of X2 , we have f (y) ∈ J and so y ∈ f −1 (J). Since f (11 ) = 12 ∈ J, we obtain 11 ∈ f −1 (J). Therefore f −1 (J) is a √ filter of X1 . √ We assume that f (01 ) = 02 . Let x ∈ f −1 ( J). Then f (x) ∈ J√and so there exists √ n∈N −1 −1 −1 such √ that nf (x) = f (nx) ∈ J. Hence nx ∈ f (J). Therefore x ∈ f (J). Thus f ( J) ⊆ f −1 (J). √ Conversely, let x ∈ f −1 (J). Then there exists n ∈ N such√ that nx ∈ f −1 (J). Hence f (nx) = √ √ √ √ nf (x) ∈ J and so f (x) ∈ J. Therefore x ∈ f −1 ( J). Thus f −1 ( J) ⊆ f −1 ( J).  5. The extended filter in BE-algebras For any non-empty subset F of X and x ∈ X, we define x−1 ∗ F := {y ∈ X|x ∨ y ∈ F }. Note that if F is a filter of X, then 1 ∈ x−1 ∗ F . Proposition 5.1. Let X be a transitive commutative BE-algebra. If F is a filter of X, then x−1 ∗ F is a filter of X containing F . Proof. Let y ∈ x−1 ∗ F and y ∗ z ∈ x−1 ∗ F . Then x ∨ y ∈ F and x ∨ (y ∗ z) ∈ F . Now (x∨y)∗(x∨z) = ((y ∗x)∗x))∗((z ∗x)∗x) ≥ (z ∗x)∗(y ∗x) ≥ y ∗z and (x∨y)∗(x∨z) ≥ x∨z ≥ x. It follows form Theorem 2.6 that x ∨ (y ∗ z) ≤ (x ∨ y) ∗ (x ∨ z) so that (x ∨ y) ∗ (x ∨ z) ∈ F . Using the fact F is a filter of X and x ∨ y ∈ F , we get x ∨ z ∈ F , i.e., z ∈ x−1 ∗ F . Thus x−1 ∗ F is a filter of X. Let y ∈ F . Since y ≤ x ∨ y, it follows that x ∨ y ∈ F , i.e., y ∈ x−1 ∗ F . Hence F ⊆ x−1 ∗ F . This completes the proof. 

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8

Proposition 5.2. Let F and G be filters of a BE-algebra X. Then the following hold: for any x, y ∈ X, (i) x−1 ∗ F = X if and only if x ∈ F ; (ii) F ⊆ G imply x−1 ∗ F ⊆ y −1 ∗ F ; (iii) x−1 ∗ (F ∩ G) = (x−1 ∗ F ) ∩ (x−1 ∗ G) and x−1 ∗ (F ∪ G) = (x−1 ∗ F ) ∪ (x−1 ∗ G). Proof. (i) Let x ∈ F . Since x ≤ (y ∗ x) ∗ x = x ∨ y, we have x ∨ y ∈ F for all y ∈ X, i.e., y ∈ x−1 ∗ F . Thus x−1 ∗ F = X. Conversely, assume that x−1 ∗ F = X. Then x ∨ y ∈ F for all y ∈ X. In particular x = x ∨ x ∈ F . (ii) Assume that F ⊆ G. If z ∈ x−1 ∗ F , then x ∨ z ∈ F ⊆ G, i.e., x ∨ z ∈ G. Hence z ∈ x−1 ∗ G. Thus x−1 ∗ F ⊆ x−1 ∗ G. (iii) For any x, z ∈ X we have z ∈ x−1 ∗ (F ∩ G) ⇔ x ∨ z ∈ F ∩ G ⇔ x ∨ z ∈ F and x ∨ z ∈ G ⇔ z ∈ x−1 ∗ F and z ∈ x−1 ∗ G ⇔ z ∈ (x−1 ∗ F ) ∩ (x−1 ∗ G) and z ∈ x−1 ∗ (F ∪ G) ⇔ x ∨ z ∈ F ∪ G ⇔ x ∨ z ∈ F or x ∨ z ∈ G ⇔ z ∈ x−1 ∗ F or z ∈ x−1 ∗ G ⇔ z ∈ (x−1 ∗ F ) ∪ (x−1 ∗ G). 

This completes the proof.

Definition 5.3. A proper filter of a BE-algebra X is said to be prime if for any x, y ∈ X, x ∨ y ∈ P implies x ∈ P or y ∈ P . Proposition 5.4. Let P and F be filters of X such that F ⊆ P . If P is prime, then x−1 ∗F ⊆ P for all x ∈ X \ P . Proof. Let z ∈ x−1 ∗ F for all x ∈ X \ P . Then x ∨ z ∈ F ⊆ P . Since P is prime and x ∈ / P, −1 we have z ∈ P , i.e., x ∗ F ⊆ P .  Proposition 5.5. If P is a prime filter of X, then X \ P is ∨-closed, i.e., x ∨ y ∈ X \ P whenever x ∈ X \ P and y ∈ X \ P . 

Proof. Straightforward.

Theorem 5.6. Let X be a transitive commutative BE-algebra. A filter P of X is prime if and only if x−1 ∗ P = P for all X \ P . Proof. Suppose that P is a prime filter of X. Let x ∈ X \ P . The conclusion P ⊆ x−1 ∗ P follows from Proposition 5.1. If y ∈ x−1 ∗ P , then x ∨ y ∈ P . Since P is a prime filter of X, we have y ∈ P . Hence x−1 ∗ P ⊆ P . Thus x−1 ∗ P = P .

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Conversely, assume that x−1 ∗ P = P for all x ∈ X \ P . Let y ∨ z ∈ P and z ∈ / P . By the −1 −1 hypothesis, z ∗ P = P . Hence y ∈ z ∗ P = P . Thus P is prime.  Definition 5.7. Let P be a filter of a BE-algebra X. P is called a primary filter of X if x ∨ y ∈ P and x ∈ / P ⇒ ∃n ∈ N such that ny ∈ P, for any x, y ∈ X. Theorem 5.8. Let X be a transitive commutative BE-algebra and let J be a filter of X. If x ∗ J = J, ∀x ∈ / J, then J is a primary filter of X. −1

Proof. By Definition 5.7, a prime filter is a primary filter. Suppose that x−1 ∗ J = J, ∀x ∈ / J. Using Theorem 5.6, J is a prime filter of X. Hence J is a primary filter of X.  Theorem 5.9. Let X be a transitive commutative BE-algebra. If J is a filter of X, then J = ∩x∈X x−1 ∗ J. Proof. By Proposition 5.1, J ⊆ x−1 ∗ J for all x ∈ X. Hence J ⊆ ∩x∈X x−1 ∗ J. Assume that y ∈ ∩x∈X x−1 ∗ J. Then y ∈ y −1 ∗ J and so y ∈ J. Hence ∩x∈X x−1 ∗ J ⊆ J. This completes the proof.  References [1] S. S. Ahn, Y. H. Kim and J. M. Ko, Filters in commutative BE-algebras, Commun. Korean Math. Soc. 27 (2012), 233-242. [2] S. S. Ahn, Y. H. Kim and K. S. So, Fuzzy BE-algebras, J. Appl. Math. and Informatics 29 (2011), 1049-1057. [3] S. S. Ahn and J. M. Ko, On vague filters in BE-algebras, Commun. Korean Math. Soc. 26 (2011), 417-425. [4] S. S. Ahn and K. K. So, On ideals and upper sets in BE-algebras, Sci. Math. Japon. 68 (2008), 279-285. [5] S. S. Ahn and K. K. So, On generalized upper sets in BE-algebras, Bull. Korean Math. Soc. 46 (2009), 281-287. [6] H. S. Kim and Y. H. Kim, On BE-algebras, Sci. Math. Japon. 66 (2007), 113-116. [7] Y. B. Jun and S. S. Ahn, Fuzzy implicative filters of BE-algebras with degrees in the interval (0, 1], J. Computational Analsis and Applications 15(2013), 1456-1466. [8] B. L. Meng, On filters in BE-algebras, Sci. Math. Jpn. Online (e-2010), 105-111.

Sun Shin Ahn, Department of Mathematics Education, Dongguk University, Seoul, 100-715, Korea E-mail address: [email protected] Young Hee Kim, Department of Mathematics, Chungbuk National University, Chongju, 361-763, Korea E-mail address: [email protected] Jung Hee Park, Department of Mathematics, Chungbuk National University, Chongju, 361-763, Korea E-mail address: [email protected]

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SYMMETRY p-ADIC INVARIANT INTEGRAL ON Zp FOR q-EULER POLYNOMIALS DAE SAN KIM, TAEKYUN KIM, SANG-HUN LEE AND JONG-JIN SEO

Abstract. In this paper, we investigate several further interesting properties of symmetry for the p-adic fermionic integral on Zp . By using this symmetry of fermionic p-adic integral on Zp , we give some relations of symmetry between the power sum q-polynomials and q-Euler polynomials.

1. Introduction 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 completion of algebraic closure of Qp . Let vp be the normalized exponential valuation of Cp with |p|p = p−vp (p) = p1 . When one talks of q-extension, q is variously considered as an indeterminate, a complex number q ∈ C or p-adic number q ∈ Cp . If q ∈ C, one normally assumes |q| < 1; if q ∈ Cp , one normally assumes |1 − q|p < 1. We use the notation for x q-number as [x]q = 1−q 1−q . Note that lim [x]q = x. As is well known, the Euler q→1

polynomials are defined by the generating function to be ∞ ∑ 2 tn xt E(x)t e = e = , E (x) n et + 1 n! n=0

(1)

with the usual convention about replacing E n (x) by En (x) (see [1-13]). When x = 0, En = En (0) are called the Euler numbers. From (1), we note that n

(E + 1) + En = 2δ0,n, and

(n ≥ 0) ,

n ( ) ∑ n n−l En (x) = x El , l l=0

(see [1, 4, 7, 9]). Recently, Kim considered a q-extension of Euler polynomials (called q-Euler polynomials) as follows : (2)

2

∞ ∑

m

(−1) e[m+x]q t =

m=0

∞ ∑ n=0

En,q (x)

tn , n!

(see [6]). 1

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2

From (2), we can derive (3)

En,q (x) =

n ( ) l ∑ 2 n (−1) lx q n l 1 + ql (1 − q) l=0

=2

∞ ∑

m

n

(−1) [m + x]q .

m=0

When x = 0, En,q = En,q (0) are called q-Euler numbers. By (2), we easily get n

(qEq + 1) + En,q = 2δn,0 , with the usual convention about replacing Eqn by En,q . Let C (Zp ) be the space of continuous functions on Zp . For f ∈ C (Zp ), the fermionic p-adic integral on Zp is defined by Kim as follows : N ˆ p∑ −1 x (4) I−1 (f ) = f (x) dµ−1 (x) = lim f (x) (−1) , N →∞

Zp

(see [4, 5]). From (4), we note that ˆ ˆ n−1 (5) f (x + n) dµ−1 (x) + (−1) Zp

Zp

whre n ∈ N (see [1, 3, 4, 5]). By (5), we get ˆ (6) e(x+y)t dµ−1 (y) = Zp

From (6), we have ˆ (7) Zp

x=0

f (x) dµ−1 (x) = 2

n−1 ∑

(−1)

n−1−l

f (l) ,

l=0

∞ ∑ 2 tn xt e = En (x) . t e +1 n! n=0

n

(x + y) dµ−1 (y) = En (x) ,

(n ≥ 0) .

In [4], some relations of symmetry between the power sum polynomials and Euler polynomials were given by (7) and finding a q-extension of symmetry p-adic invariant integral on Zp for q-Euler polynomials was remained as an open question. In this paper, we investigate several further interesting properties of symmetry for the p-adic fermionic p-adic integral on Zp , and give some relations of symmetry between the power sum q-polynomials and q-Euler polynomials. Recently, several authors have studied the identities of symmetry and q-extensions of Euler polynomials (see [1-13]). 2. Identities of symmetry of the q-Euler polynomials From (4) and (5), we can derive the following equation : ˆ ∞ ∑ m (8) e[x+y]q t dµ−1 (y) = 2 (−1) e[m+x]q t . Zp

m=0

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3

By (2) and (8), we get ∞ ˆ ∑

(9)

n=0

n

Zp

[x + y]q dµ−1 (y)

∞ ∑ tn tn = En,q (x) . n! n=0 n!

By comparing coefficients on the both sides of (9), we get ˆ n

(10)

[x + y]q dµ−1 (y) = En,q (x)

Zp

=2

∞ ∑

m

n

(−1) [m + x]q

m=0

n ( ) ∑ 2 n 1 l = (−1) q lx . n l 1 + ql (1 − q) l=0

Let w1 , w2 ∈ N with w1 ≡ 1 (mod 2) and w2 ≡ 1 (mod 2). Then we observe that [ ] ˆ w [w1 ]q w2 x+ w2 j+y w t 1 q 1 dµ (11) e −1 (y) Zp

ˆ =

Zp

e[w1 w2 x+w2 j+w1 y]q t dµ−1 (y) N p∑ −1

= lim

N →∞

= lim

e[w1 w2 x+w2 j+w1 y]q t (−1)

y=0 N −1 w∑ 2 −1 p∑

N →∞

y

e[w1 w2 x+w2 j+w1 (i+w2 y)]q t (−1)

i+w2 y

.

y=0

i=0

From (11), we note that (12)

w∑ 1 −1

ˆ j

(−1)

e

[ ] w [w1 ]q w2 x+ w2 j+y 1

q w1

t

Zp

j=0

N −1 w∑ 1 −1 w 2 −1 p∑ ∑

= lim

N →∞

j=0

(−1)

i+j+y

dµ−1 (y)

e[w1 w2 (x+y)+w2 j+w1 i]q t .

y=0

i=0

By the same method as (12), we get (13)

w∑ 2 −1

ˆ j

(−1)

N →∞

2

Zp

j=0

= lim

e

] [ w [w2 ]q w1 x+ w1 j+y

N −1 w∑ 2 −1 w 1 −1 p∑ ∑

j=0

i=0

(−1)

i+j+y

q w2

t

dµ−1 (y)

e[w1 w2 (x+y)+w1 j+w2 i]q t .

y=0

Therefore, by (12) and (13), we obtain the following theorem.

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Theorem 1. For w1 , w2 ∈ N with w1 ≡ 1 (mod 2) and w2 ≡ 1 (mod 2), we have [ ] ˆ w∑ 1 −1 w [w1 ] w2 x+ w2 j+y w t j 1 q 1 dµ (−1) e q −1 (y) Zp

j=0

=

w∑ 2 −1

ˆ (−1)

j

e

2

Zp

j=0

Corollary 2. For n ≥ 0, we have ˆ w∑ 1 −1 n j [w1 ]q (−1) j=0 n = [w2 ]q

] [ w [w2 ]q w1 x+ w1 j+y

w∑ 2 −1

ˆ j

(−1)

j=0

q w2

t

dµ−1 (y) .

[ ]n w2 j+y dµ−1 (y) w2 x + w1 Zp q w1 [ ]n w1 w1 x + j+y dµ−1 (y) . w2 Zp q w2

By (9) and Corollary 2, we obtain the following theorem. Theorem 3. For n ≥ 0 and w1 , w2 ∈ N with w1 ≡ 1 (mod 2) and w2 ≡ 1 (mod 2), we have ( ) w∑ 1 −1 w2 n j w [w1 ]q (−1) En,q 1 w2 x + j w1 j=0 n = [w2 ]q

w∑ 2 −1

j

(−1) E

n,q w2

j=0

( ) w1 w1 x + j . w2

From (10), we note that ]n ˆ [ w2 (14) w2 x + j+y dµ−1 (y) w1 Zp q w1 ( )i ˆ n ( ) ∑ [w2 ]q n i n−i = [j]qw2 q w2 (n−i)j [w2 x + y]qw1 dµ−1 (y) i [w ] 1 Z q p i=0 )i ( n ( ) ∑ [w2 ]q n i = [j]qw2 q w2 (n−i)j En−i,qw1 (w2 x) . i [w ] 1 q i=0 Thus, by (14), we get n [w1 ]q

(15)

w∑ 1 −1

(−1)

Zp

j=0

=

n ( ) ∑ n i=0

=

i

n ( ) ∑ n i=0

i

where Sn,i,q (w1 ) =

[

ˆ j

n−i

[w1 ]q

i

[w2 ]q

w2 j+y w2 x + w1 w∑ 1 −1

j

]n dµ−1 (y) q w1

i

(−1) [j]qw2 q w2 (n−i)j En−i,qw1 (w2 x)

j=0 n−i

[w1 ]q w∑ 1 −1

i

[w2 ]q Sn,i,qw2 (w1 ) En−i,qw1 (w2 x) , j

i

(−1) q (n−i)j [j]q .

j=0

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By the same method as (15), we get ]n ˆ [ w∑ 2 −1 w1 n j (16) j+y dµ−1 (y) [w2 ]q (−1) w1 x + w2 Zp q w2 j=0 n ( ) ∑ n i n−i = [w1 ]q [w2 ]q Sn,i,qw1 (w2 ) En−i,qw2 (w1 x) . i i=0 Therefore, by Corollary 2, (15) and (16), we obtain the following theorem. Theorem 4. For n ≥ 0 and w1 , w2 ∈ N with w1 ≡ 1 (mod 2) and w2 ≡ 1 (mod 2), we have n ( ) ∑ n i=0

=

i

n ( ) ∑ n i=0

where Sn,i,q (w) =

w−1 ∑

i

n−i

[w1 ]q i

i

[w2 ]q En−i,qw1 (w2 x) Sn,i,qw2 (w1 ) n−i

[w1 ]q [w2 ]q j

En−i,qw2 (w1 x) Sn,i,qw1 (w2 ) ,

i

(−1) q (n−i)j [j]q .

j=0

ACKNOWLEDGEMENTS. The present Research has been conducted by the Research Grant of Kwangwoon University in 2014 and the first author was supported by the National Research Foundation of Korea(NRF) grant funded by the Korea government(MOE) (No.2012R1A1A2003786 ). References 1. S. Araci, M. Acikgoz, and J. J. Seo, Explicit formulas involving q-Euler numbers and polynomials, Abstr. Appl. Anal. (2012), Art. ID 298531, 11. 2. I. N. Cangul, V. Kurt, H. Ozden, and Y. Simsek, On the higher-order w-q-Genocchi numbers, Adv. Stud. Contemp. Math. (Kyungshang) 19 (2009), no. 1, 39–57. 3. D.S. Kim, N. Lee, J. Na, and K. H. Park, Identities of symmetry for higher-order Euler polynomials in three variables (I), Adv. Stud. Contemp. Math. (Kyungshang) 22 (2012), no. 1, 51–74. 4. T. Kim, Symmetry p-adic invariant integral on Zp for Bernoulli and Euler polynomials, J. Difference Equ. Appl. 14 (2008), no. 12, 1267–1277. 5. , Symmetry of power sum polynomials and multivariate fermionic p-adic invariant integral on Zp , Russ. J. Math. Phys. 16 (2009), no. 1, 93–96. 6. , Barnes-type multiple q-zeta functions and q-Euler polynomials, J. Phys. A 43 (2010), no. 25, 255201, 11. 7. , An identity of the symmetry for the Frobenius-Euler polynomials associated with the fermionic p-adic invariant q-integrals on Zp , Rocky Mountain J. Math. 41 (2011), no. 1, 239–247. 8. Y.-H. Kim and K.-W. Hwang, Symmetry of power sum and twisted Bernoulli polynomials, Adv. Stud. Contemp. Math. (Kyungshang) 18 (2009), no. 2, 127–133. 9. H. Ozden, Y. Simsek, S.-H. Rim, and I. N. Cangul, A note on p-adic q-Euler measure, Adv. Stud. Contemp. Math. (Kyungshang) 14 (2007), no. 2, 233–239. 10. S.-H. Rim and J. Jeong, On the modified q-Euler numbers of higher order with weight, Adv. Stud. Contemp. Math. (Kyungshang) 22 (2012), no. 1, 93–98. 11. Y. Simsek, Identities associated with generalized Stirling type numbers and Eulerian type polynomials, Math. Comput. Appl. 18 (2013), no. 3, 251–263.

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, Interpolation functions of the Eulerian type polynomials and numbers, Adv. Stud. Contemp. Math. (Kyungshang) 23 (2013), no. 2, 301–307. 13. H. J. H. Tuenter, A symmetry of power sum polynomials and Bernoulli numbers, Amer. Math. Monthly 108 (2001), no. 3, 258–261. 12.

Department of Mathematics, Sogang University, Seoul 121-742, Republic of Korea E-mail address : [email protected] Department of Mathematics, Kwangwoon University, Seoul 139-701, Republic of Korea E-mail address : [email protected] Division of General Education, Kwangwoon University, Seoul 139701, Republic of Korea E-mail address : [email protected] Department of Applied Mathematics, Pukyong National University, Pusan, Republic of Korea E-mail address : [email protected]

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Barnes’ multiple Bernoulli and poly-Bernoulli mixed-type polynomials Dmitry V. Dolgy Institute of Mathematics and Computer Sciences, Far Eastern Federal University Vladivostok, 27 Oktyabrskaya Str., Vladivostok, 690060, Russia d− [email protected]

Dae San Kim

∗

Department of Mathematics, Sogang University Seoul 121-741, Republic of Korea [email protected]

Taekyun Kim

†

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

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

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

Abstract ∗

The second author was supported by the National Research Foundation of Korea (NRF) grant funded by the Korean government (MOE) (No.2012R1A1A2003786). † The third author was supported by Kwangwoon University in 2014.

1

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In this paper, we consider Barnes’ multiple Bernoulli and poly-Bernoulli mixedtype polynomials. From the properties of Sheffer sequences of these polynomials arising from umbral calculus, we derive new and interesting identities.

1

Introduction (r,k)

In this paper, we consider the polynomials Sn is given by

(x|a1 , . . . , ar ) whose generating function

tr Lik (1 − e−t ) xt ∑ (r,k) tn ∏r e = S (x|a , , . . . , a ) 1 r n aj t − 1) 1 − e−t n! j=1 (e n=0 ∞

(1)

where r ∈ Z>0 , k ∈ Z, a1 , . . . , ar ̸= 0, and Lik (x) =

∞ ∑ xm mk m=1

(r,k)

is the kth polylogarithm function. Sn (x|a1 , . . . , ar ) will be called Barnes’ multiple (r,k) Bernoulli and poly-Bernoulli mixed-type polynomials. When Sn (a1 , . . . , ar ) (r,k) = Sn (0|a1 , . . . , ar ) will be called Barnes’ multiple Bernoulli and poly-Bernoulli mixedtype numbers. (k) Recall that, for every integer k, the poly-Bernoulli polynomials Bn (x) are defined by the generating function as Lik (1 − e−t ) xt ∑ (k) tn e = Bn (x) 1 − e−t n! n=0 ∞

(2)

([14], Cf.[4]). Also, recall that the Barnes’ multiple Bernoulli polynomials Bn (x|a1 , . . . , ar ) are defined by the generating function as ∑ tr tn xt ∏r e = B (x|a , . . . , , a ) n 1 r aj t − 1) n! j=1 (e n=0 ∞

(3)

where a1 , . . . , ar ̸= 0 ( see [1-14])). In this paper, we consider Barnes’ multiple Bernoulli and poly-Bernoulli mixed-type polynomials. From the properties of Sheffer sequences of these polynomials arising from umbral calculus, we derive new and interesting identities.

2

Umbral calculus

Let C be the complex number field and let F be the set of all formal power series in the variable t: } { ∞ ∑ ak k (4) F = f (t) = t ak ∈ C . k! k=0

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Let P = C[x] and let P∗ be the vector space of all linear functionals on P. ⟨L|p(x)⟩ is the action of the linear functional L on the polynomial p(x), and we recall that the vector space operations on P∗ are defined by ⟨L + M |p(x)⟩ = ⟨L|p(x)⟩ + ⟨M |p(x)⟩, ⟨cL|p(x)⟩ = c ⟨L|p(x)⟩, where c is a complex constant in C. For f (t) ∈ F , let us define the linear functional on P by setting ⟨f (t)|xn ⟩ = an , (n ≥ 0). (5) In particular,

⟨ k n⟩ t |x = n!δn,k

(n, k ≥ 0),

(6)

where δn,k is the Kronecker’s symbol. ∑ ⟨L|xk ⟩ k t , we have ⟨fL (t)|xn ⟩ = ⟨L|xn ⟩. That is, L = fL (t). The map For fL (t) = ∞ k=0 k! L 7→ fL (t) is a vector space isomorphism from P∗ onto F. Henceforth, F denotes both the algebra of formal power series in t and the vector space of all linear functionals on P, and so an element f (t) of F will be thought of as both a formal power series and a linear functional. We call F the algebra and the umbral calculus is the study of ( umbral ) umbral algebra. The order O f (t) of a power series ( f (t)(̸ ) = 0) is the smallest integer k k for which the coefficient of t does not vanish. If O f (t) = 1, then f (t) is called a delta ( ) series; f (t) is called an invertible series. For f (t), g(t) ∈ F with ( )if O f (t) =( 0, then ) O f (t) = 0, there exists a unique sequence sn (x) (deg sn (x) = n) such ⟨ = 1 and O g(t) ⟩ that g(t)f (t)( k |sn (x) )= n!δn,k , for n, k ≥ 0. Such a sequence s)n (x) is called the Sheffer ( sequence for g(t), f (t) which is denoted by sn (x) ∼ g(t), f (t) . For f (t), g(t) ∈ F and p(x) ∈ P, we have ⟨f (t)g(t)|p(x)⟩ = ⟨f (t)|g(t)p(x)⟩ = ⟨g(t)|f (t)p(x)⟩ and f (t) =

∞ ∑ ⟨ k=0

⟩ tk f (t)|x , k! k

(7)

∞ ∑ ⟨k ⟩ xk t |p(x) p(x) = k! k=0

(8)

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

(9)

([12, Theorem 2.2.5]). Thus, by (8), we get tk p(x) = p(k) (x) =

dk p(x) dxk

Sheffer sequences are characterized in the generating function ([12, Theorem 2.3.4]). ( ) Lemma 1 The sequence sn (x) is Sheffer for g(t), f (t) if and only if ∑ sk (y) 1 ¯ ) eyf (t) = tk ¯ k! g f (t) k=0 (

∞

(y ∈ C) ,

where f¯(t) is the compositional inverse of f (t).

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( ) For sn (x) ∼ g(t), f (t) , we have the following equations ([12, Theorem 2.3.7, Theorem 2.3.5, Theorem 2.3.9]): f (t)sn (x) = nsn−1 (x) (n ≥ 0), n ∑ 1 ⟨ ( ¯ )−1 ¯ j n ⟩ j g f (t) f (t) |x x , sn (x) = j! j=0 n ( ) ∑ n sn (x + y) = sj (x)pn−j (y) , j j=0

(10) (11) (12)

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

n ∑

Cn,m rm (x) (n ≥ 0) .

m=0

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

3

1 = m!

⟩ ⟨ ( ) h f¯(t) ( ¯ )m n ) l f (t) x ( . g f¯(t)

(13)

Main results (r,k)

(k)

We now note that Bn (x), Bn (x|a1 , . . . , ar ) and Sn (x|a1 , . . . , ar ) are the Appell sequences for ∏r ∏r aj t aj t − 1) − 1) 1 − e−t 1 − e−t j=1 (e j=1 (e gk (t) = , g (t) = , g (t) = . r r,k Lik (1 − e−t ) tr tr Lik (1 − e−t ) So, (

) 1 − e−t ,t , ∼ Lik (1 − e−t ) ) ( ∏r aj t (e − 1) j=1 Bn (x|a1 , . . . , ar ) ∼ ,t , tr ( ∏r ) aj t −t (e − 1) 1 − e j=1 Sn(r,k) (x|a1 , . . . , ar ) ∼ ,t . tr Lik (1 − e−t ) Bn(k) (x)

(14) (15) (16)

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In particular, we have d (k) (k) B (x) = nBn−1 (x) , dx n d Bn (x|a1 , . . . , ar ) tBn (x|a1 , . . . , ar ) = dx = nBn−1 (x|a1 , . . . , ar ) , d (r,k) S (x|a1 , . . . , ar ) tSn(r,k) (x|a1 , . . . , ar ) = dx n (r,k) = nSn−1 (x|a1 , . . . , ar ) . tBn(k) (x) =

Notice that

3.1

(17)

(18)

(19)

d 1 Lik (x) = Lik−1 (x) . dx x

Explicit expressions (r,k)

(r,k)

Write Bn (a1 , . . . , ar ) := Bn (0|a1 , . . . , ar ) and Sn (a1 , . . . , ar ) := Sn (n)j = n(n − 1) · · · (n − j + 1) (j ≥ 1) with (n)0 = 1.

(0|a1 , . . . , ar ). Let

Theorem 1 Sn(r,k) (x|a1 , . . . , ar )

n ( ) ∑ n (k) Bn−l (a1 , . . . , ar )Bl (x) , = l l=0 ( ) n ∑ n (k) = Bn−l Bl (x|a1 , . . . , ar ) , l l=0 ( )( ) m n n ∑∑∑ n 1 j m (−1) = B (a , . . . , ar )(x − j)l , k n−l 1 j l (m + 1) l=0 m=l j=0 ( n n−j ( )( ) n ∑∑ ∑ j n−m−j n (−1) = j l l=0 j=l m=0 ) m! × S2 (n − j, m)Bj−l (a1 , . . . , ar ) xl , (m + 1)k n ( ) ∑ n (r,k) = Sn−j (a1 , . . . , ar )xj . j j=0

(20) (21) (22)

(23) (24)

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Proof. By (1), (2) and (3), we have ⟩ ⟨∞ ∑ (r,k) ti n (r,k) Sn (y|a1 , . . . , ar ) = Si (y|a1 , . . . , ar ) x i! i=0 ⟨ ⟩ tr Lik (1 − e−t ) yt n = ∏r e x aj t − 1) 1 − e−t j=1 (e ⟨ ⟩ Li (1 − e−t ) tr k yt n = ∏r e x aj t − 1) 1 − e−t j=1 (e ⟨ ⟩ ∞ ∑ tr tl n (k) = ∏r Bl (y) x aj t − 1) l! j=1 (e l=0 ⟨ ⟩ ( ) n ∑ n tr (k) n−l = ∏r Bl (y)x aj t − 1) l j=1 (e l=0 ⟩ ⟨ n ( ) ∑ n tr (k) n−l x = Bl (y) ∏r aj t − 1) l j=1 (e l=0 ⟨∞ ⟩ n ( ) ∑ ∑ n ti n−l (k) Bl (y) Bi (a1 , . . . , ar ) x = l i! i=0 l=0 ( ) n ∑ n (k) = Bl (y)Bn−l (a1 , . . . , ar ) . l l=0 So, we get (20).

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We also have Sn(r,k) (y|a1 , . . . , ar ) =

⟨∞ ∑

t (r,k) Si (y|a1 , . . . , ar )

i

n x

⟩

i! ⟨ i=0 ⟩ Lik (1 − e−t ) tr = eyt xn ∏r aj t − 1) 1 − e−t (e j=1 ⟨ ⟩ ∞ Lik (1 − e−t ) ∑ tl n = Bl (y|a1 , . . . , ar ) x 1 − e−t l! l=0 ⟨ ⟩ ( ) n Lik (1 − e−t ) ∑ n n−l Bl (y|a1 , . . . , ar ) = x 1 − e−t l l=0 ⟩ ⟨ n ( ) ∑ n Lik (1 − e−t ) n−l = Bl (y|a1 , . . . , ar ) x −t l 1 − e l=0 ⟩ ⟨∞ n ( ) i ∑ ∑ n (k) t n−l = Bl (y|a1 , . . . , ar ) Bi x l i! i=0 l=0 n ( ) ∑ n (k) Bl (y|a1 , . . . , ar )Bn−l . = l l=0

Thus, we get (21). In [7] we obtained that ( ) n m ∑ Lik (1 − e−t ) n ∑ 1 j m x = (−1) (x − j)n . −t k 1−e (m + 1) j=0 j m=0 So, Lik (1 − e−t ) n tr x aj t − 1) 1 − e−t j=1 (e ( ) n m ∑ ∑ tr 1 j m ∏ = (−1) (x − j)n r aj t − 1) k j (m + 1) (e j=1 m=0 j=0 ( ) n m n ∑ ∑ ∑ (n) 1 j m = (−1) Bn−l (a1 , . . . , ar )(x − j)l k (m + 1) j l m=0 j=0 l=0 ( )( ) n ∑ n ∑ m ∑ m n 1 = B (a , . . . , ar )(x − j)l , (−1)j k n−l 1 (m + 1) j l l=0 m=l j=0

Sn(r,k) (x|a1 , . . . , ar ) = ∏r

which is the identity (22). In [7] we obtained that Lik (1 − e−t ) n ∑ x = 1 − e−t j=0 n

( n−j ) ∑ (−1)n−m−j (n) m!S2 (n − j, m) xj , k (m + 1) j m=0 7

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where S2 (l, m) are the Stirling numbers of the second kind, defined by (et − 1)m = m!

∞ ∑

tl S2 (l, m) . l! l=m

Thus, ( n−j ) ∑ (−1)n−m−j (n) tr ∏ Sn(r,k) (x|a1 , . . . , ar ) = xj m!S (n − j, m) 2 r ai t − 1) k j (e (m + 1) i=1 m=0 j=0 ( ) ( ) n−j n ∑ ∑ (−1)n−m−j n = m!S2 (n − j, m) Bj (x|a1 , . . . , ar ) k j (m + 1) m=0 j=0 ( n−j ) j ( ) n ∑ ∑ (−1)n−m−j (n) ∑ j m!S (n = − j, m) Bj−l (a1 , . . . , ar )xl 2 k j (m + 1) l m=0 j=0 l=0 ( ) ( )( ) n−j n n ∑ ∑∑ j m! n−m−j n S2 (n − j, m)Bj−l (a1 , . . . , ar ) xl , = (−1) k j l (m + 1) l=0 j=l m=0 n ∑

which is the identity (23). By (11) with (16), we have ⟨

⟩ tr Lik (1 − e−t ) j n ∏r t x aj t − 1) 1 − e−t j=1 (e ⟨ ⟩ Lik (1 − e−t ) n−j tr = (n)j ∏r x aj t − 1) 1 − e−t j=1 (e ⟩ ⟨∞ ∑ (r,k) ti n−j = (n)j Si (a1 , . . . , ar ) x i! i=0

⟩ ⟨ ( )−1 g f¯(t) f¯(t)j |xn =

(r,k)

= (n)j Sn−j (a1 , . . . , ar ) . Thus, we get (24).

3.2

Sheffer identity

Theorem 2 Sn(r,k) (x

n ( ) ∑ n (r,k) + y|a1 , . . . , ar ) = Sj (x|a1 , . . . , ar )y n−j . j j=0

(25)

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Proof. By (16) with

∏r

pn (x) =

j=1 (e

aj t

− 1)

tr = xn ∼ (1, t) ,

1 − e−t Sn(r,k) (x|a1 , . . . , ar ) −t Lik (1 − e )

using (12), we have (25).

3.3

Recurrence

Theorem 3 (r,k)

Sn+1 (x|a1 , . . . , ar ) = xSn(r,k) (x|a1 , . . . , ar ) ) r n ( 1 ∑∑ n + 1 (r,k) − (−aj )n+1−l Bn+1−l Sl (x|a1 , . . . , ar ) n + 1 j=1 l=0 l ) 1 ( (r+1,k) (r+1,k−1) − Sn+1 (x|a1 , . . . , ar , 1) − Sn+1 (x|a1 , . . . , ar , 1) , n+1 where Bn is the nth ordinary Bernoulli number.

(26)

Proof. By applying

( ) g ′ (t) 1 sn+1 (x) = x − sn (x) g(t) f ′ (t) ([12, Corollary 3.7.2]) with (16), we get ( ) ′ gr,k (t) (r,k) Sn+1 (x|a1 , . . . , ar ) = x − Sn(r,k) (x|a1 , . . . , ar ) . gr,k (t) Now, ′ gr,k (t) = (ln gr,k (t))′ gr,k (t) ( r )′ ∑ = ln(eaj t − 1) − r ln t + ln(1 − e−t ) − ln Lik (1 − e−t ) j=1 r ∑

( ) aj eaj t r e−t Lik−1 (1 − e−t ) = − + 1− eaj t − 1 t 1 − e−t Lik (1 − e−t ) j=1 ∑r ∏ ai t − 1)(aj teaj t − eaj t + 1) t Lik (1 − e−t ) − Lik−1 (1 − e−t ) j=1 i̸=j (e ∏ = + . r t j=1 (eaj t − 1) et − 1 tLik (1 − e−t ) Since ∑r j=1

∏

− 1)(aj teaj t − eaj t + 1) ∏r = aj t − 1) j=1 (e

i̸=j (e

ai t

1 2

1 = 2

( ∑r j=1

( r ∑

) a1 · · · aj−1 a2j aj+1 · · · ar tr+1 + · · · (a1 · · · ar )tr + · · · )

aj

t + ···

j=1

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is a series with order≥ 1 and Lik (1 − e−t ) − Lik−1 (1 − e−t ) = 1 − e−t

(

1 1 − 2k 2k−1

) t + ···

is a delta series, we have ′ gr,k (t) (r,k) = − S (x|a1 , . . . , ar ) gr,k (t) n ′ gr,k (t) Lik (1 − e−t ) n tr ∏r x = xSn(r,k) (x|a1 , . . . , ar ) − gr,k (t) j=1 (eaj t − 1) 1 − e−t

(r,k) Sn+1 (x|a1 , . . . , ar )

xSn(r,k) (x|a1 , . . . , ar )

= xSn(r,k) (x|a1 , . . . , ar ) Lik (1 − e−t ) tr − ∏r aj t − 1) 1 − e−t j=1 (e

∑r j=1

∏

ai t − 1)(aj teaj t − eaj t + 1) n ∏r x t j=1 (eaj t − 1)

i̸=j (e

tr t Lik (1 − e−t ) − Lik−1 (1 − e−t ) n x . aj t − 1) et − 1 t(1 − e−t ) j=1 (e

− ∏r Now,

∑r

∏

∑r =

− 1)(aj teaj t − eaj t + 1) n ∏ x t rj=1 (eaj t − 1)

i̸=j (e

j=1

∏

ai t

− 1)(aj teaj t − eaj t + 1) xn+1 ∏r aj t − 1) n+1 j=1 (e

i̸=j (e

j=1

ai t

1 ∑ aj teaj t − eaj t + 1 n+1 x n + 1 j=1 e aj t − 1 ) r ( 1 ∑ aj teaj t − 1 xn+1 n + 1 j=1 eaj t − 1 (∞ ) r 1 ∑ ∑ (−1)l Bl alj l t − 1 xn+1 n + 1 j=1 l=0 l! ( n+1 ( ) ) r 1 ∑ ∑ n+1 (−aj )l Bl xn+1−l − xn+1 n + 1 j=1 l=0 l ) r n+1 ( 1 ∑∑ n + 1 (−aj )l Bl xn+1−l n + 1 j=1 l=1 l ) r n ( 1 ∑∑ n + 1 (−aj )n+1−l Bn+1−l xl . n + 1 j=1 l=0 l r

= =

=

=

= = Also,

Lik (1 − e−t ) − Lik−1 (1 − e−t ) n 1 Lik (1 − e−t ) − Lik−1 (1 − e−t ) n+1 x . x = t(1 − e−t ) n+1 1 − e−t 10

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Thus, we get the identity (26).

3.4

One more relation

Theorem 4 (r,k)

Sn(r,k) (x|a1 , . . . , ar ) = xSn−1 (x|a1 , . . . , ar ) ( n−1 ) n r ∑ (−1)m−1 m−1 Bm ∑ (r,k) + am j Sn−m (x|a1 , . . . , ar ) m m=1 j=1 +

1 (r+1,k−1) 1 (x|a1 , . . . , ar , 1) − Sn(r+1,k) (x|a1 , . . . , ar , 1) . Sn n n

(27)

Proof. For n ≥ 1 we have Sn(r,k) (y|a1 , . . . , ar ) =

⟨∞ ∑

t (r,k) Sl (y|a1 , . . . , ar )

l

l!

⟨ l=0

n x

⟩

⟩ tr Lik (1 − e−t ) yt n = ∏r e x aj t − 1) 1 − e−t j=1 (e ⟨ ( ) ⟩ Lik (1 − e−t ) yt n−1 tr = ∂t ∏ r e x aj t − 1) 1 − e−t j=1 (e ) ⟩ ⟨( Lik (1 − e−t ) yt n−1 tr e x = ∂t ∏r aj t − 1) 1 − e−t j=1 (e ⟨ ⟩ ( ) tr Lik (1 − e−t ) yt n−1 + ∏r ∂t e x aj t − 1) 1 − e−t j=1 (e ⟩ ⟨ tr Lik (1 − e−t ) + ∏r (∂t eyt ) xn−1 aj t − 1) −t (e 1 − e j=1 (r,k)

= ySn−1 (y|a1 , . . . , ar ) ⟨( ) ⟩ tr Lik (1 − e−t ) yt n−1 + ∂t ∏r e x aj t − 1) 1 − e−t j=1 (e ⟨ ⟩ ( ) tr Lik (1 − e−t ) yt n−1 + ∏r ∂t e x . aj t − 1) 1 − e−t j=1 (e

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Observe that ( ∂t

tr ∏r aj t − 1) j=1 (e

) = =

=

=

=

=

∑ a eaj t rtr−1 − tr rj=1 eajj t −1 ∏r aj t − 1) j=1 (e ( ) r ∑ tr−1 aj teaj t ∏r r− aj t − 1) eaj t − 1 j=1 (e j=1 ( ) r ∑ tr−1 −aj t ∏r r− aj t − 1) e−aj t − 1 j=1 (e j=1 ( ) r ∑ ∞ r−1 m m ∑ t (−aj ) Bm t ∏r r− a t j − 1) m! j=1 (e j=1 m=0 ( ( ) ) ∞ r m ∑ ∑ B t tr−1 m ∏r r− (−aj )m aj t − 1) (e m! j=1 m=0 j=1 ( ) ∞ r ∑ ∑ tr (−1)m−1 Bm m−1 m ∏r a t . j aj t − 1) m! j=1 (e m=1 j=1

Thus, ⟨(

⟩ ) tr Lik (1 − e−t ) yt n−1 ∂t ∏r e x aj t − 1) 1 − e−t j=1 (e ⟨ ( r ) ⟩ n Lik (1 − e−t ) yt ∑ ∑ m (−1)m−1 Bm m−1 n−1 tr = ∏r e aj t x aj t − 1) 1 − e−t m! j=1 (e m=1 j=1 ⟨ ⟩ ( n−1 ) r n r −t ∑ (−1)m−1 m−1 Bm ∑ t Li (1 − e ) k ∏r am eyt xn−m = j aj t − 1) −t m 1 − e (e j=1 m=1 j=1 ( ) n r n−1 ∑ ∑ (−1)m−1 m−1 Bm (r,k) = Sn−m (y|a1 , . . . , ar ) am j . m m=1 j=1

Since

Lik−1 (1 − e−t ) − Lik (1 − e−t ) = 1 − e−t

(

1 2k−1

1 − k 2

) t + ···

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is a delta series, we have ⟩ ⟨ ( ) Lik (1 − e−t ) yt n−1 tr ∏r ∂t e x aj t − 1) 1 − e−t j=1 (e ⟨ ⟩ ( ) e−t Lik−1 (1 − e−t ) − Lik (1 − e−t ) yt n−1 tr = ∏r e x aj t − 1) (1 − e−t )2 j=1 (e ⟨ ⟩ t Lik−1 (1 − e−t ) − Lik (1 − e−t ) yt n−1 tr = ∏r e x aj t − 1) et − 1 t(1 − e−t ) j=1 (e ⟨ ⟩ tr+1 Lik−1 (1 − e−t ) − Lik (1 − e−t ) yt xn = ∏r e aj t − 1)(et − 1) 1 − e−t n j=1 (e ⟩ ⟨ Lik−1 (1 − e−t ) yt n 1 tr+1 ∏r e x = aj t − 1)(et − 1) n 1 − e−t j=1 (e ⟩ ⟨ 1 Lik (1 − e−t ) yt n tr+1 ∏r − e x aj t − 1)(et − 1) n 1 − e−t j=1 (e =

1 (r+1,k−1) 1 Sn (y|a1 , . . . , ar , 1) − Sn(r+1,k) (y|a1 , . . . , ar , 1) . n n

Therefore, we obtain the desired result. Remark. After simple modification, Theorem 4 becomes (r,k)

Sn+1 (x|a1 , . . . , ar ) = xSn(r,k) (x|a1 , . . . , ar ) (n) n+1 r ∑ (−1)l−1 l−1 Bl ∑ (r,k) + alj Sn+1−l (x|a1 , . . . , ar ) l j=1 l=1 +

1 1 (r+1,k−1) (r+1,k) (x|a1 , . . . , ar , 1) − Sn+1 S (x|a1 , . . . , ar , 1) . n+1 n + 1 n+1

which is the same as the above recurrence formula (26) upon replacing n by n − 1.

3.5

Relations with poly-Bernoulli numbers and Barnes’ multiple Bernoulli numbers

Theorem 5 ) n ( ∑ n+1 (r,k) (a1 , . . . , ar ) (−1)n−m Sm m m=0 =

n ∑ l ∑

l−m

(−1)

l=0 m=0

( )( ) l n+1 (k−1) Bn−l (a1 , . . . , ar ) . (28) Bm m l+1

13

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Proof. We shall compute

⟨

tr −t n+1 ∏r Li (1 − e ) x aj t − 1) k j=1 (e

⟩

in two different ways. On the one hand, ⟩ ⟨ ⟩ ⟨ r −t t Li (1 − e ) tr k ∏r Li (1 − e−t ) xn+1 = ∏r (1 − e−t )xn+1 aj t − 1) k aj t − 1) −t (e (e 1 − e j=1 j=1 ⟨ ⟩ tr Lik (1 − e−t ) n+1 = ∏r − (x − 1)n+1 x aj t − 1) −t (e 1 − e j=1 ⟨ ⟩ ( n r −t ∑ n + 1) t Li (1 − e ) k = (−1)n−m ∏r xm aj t − 1) −t m (e 1 − e j=1 m=0 ( ) n ∑ n+1 (r,k) = (−1)n−m Sm (a1 , . . . , ar ) . m m=0 On the other hand, ⟨ ⟩ ⟨ ⟩ tr tr −t n+1 −t n+1 ∏r Li (1 − e ) x = Lik (1 − e ) ∏r x aj t − 1) k aj t − 1) (e j=1 j=1 (e ⟨ ⟩ −t = Lik (1 − e ) Bn+1 (x|a1 , . . . , ar ) ⟨∫ t ⟩ ( ) −s ′ = Lik (1 − e ) ds Bn+1 (x|a1 , . . . , ar ) 0 ⟩ ⟨∫ t −s −s Lik−1 (1 − e ) ds Bn+1 (x|a1 , . . . , ar ) = e 1 − e−s 0 ⟨∫ ( ∞ ⟩ )( ∞ ) (k−1) t ∑ ∑ Bm (−s)j m = s ds Bn+1 (x|a1 , . . . , ar ) j! m! 0 m=0 j=0 ⟨∞ ( l ) ∫ ⟩ ( ) t ∑ ∑ l 1 (k−1) Bm = (−1)l−m sl ds Bn+1 (x|a1 , . . . , ar ) m l! 0 m=0 l=0 ( ) n ∑ l (k−1) ⟨ ⟩ ∑ l Bm (−1)l−m = tl+1 Bn+1 (x|a1 , . . . , ar ) m (l + 1)! l=0 m=0 ( ) (k−1) n ∑ l ∑ l Bm l−m = (−1) (n + 1)l+1 Bn−l (a1 , . . . , ar ) m (l + 1)! l=0 m=0 ( )( ) n ∑ l ∑ l n+1 l−m (k−1) = Bn−l (a1 , . . . , ar ) . (−1) Bm m l + 1 l=0 m=0 Here, Bn−l (a1 , . . . , ar ) = Bn−l (0|a1 , . . . , ar ). Thus, we get (28). 14

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3.6

Relations with the Stirling numbers of the second kind and the falling factorials

Theorem 6 Sn(r,k) (x|a1 , . . . , ar ) =

n ∑

( n ∑

m=0

) ( ) n (r,k) S2 (l, m) Sn−l (a1 , . . . , ar ) (x)m . l l=m

(29)

∑ (r,k) Proof. For (16) and (x)n ∼ (1, et − 1), assume that Sn (x|a1 , . . . , ar ) = nm=0 Cn,m (x)m . By (13), we have ⟩ ⟨ 1 1 t m n Cn,m = (e − 1) x ∏r aj t −1) 1−e−t m! j=1 (e tr Lik (1−e−t ) ⟩ ⟨ 1 Lik (1 − e−t ) t tr ∏r = (e − 1)m xn aj t − 1) −t (e m! 1 − e j=1 ⟨ ⟩ n 1 Lik (1 − e−t ) ∑ tl n tr ∏r = S2 (l, m) x m! aj t − 1) m! 1 − e−t l! j=1 (e l=m ⟨ ⟩ ( ) n ∑ n Lik (1 − e−t ) n−l tr ∏r S2 (l, m) = x aj t − 1) l 1 − e−t j=1 (e l=m ( ) n ∑ n (r,k) S2 (l, m) S = (a1 , . . . , ar ) . l n−l l=m Thus, we get the identity (29).

3.7

Relations with the Stirling numbers of the second kind and the rising factorials

Theorem 7 Sn(r,k) (x|a1 , . . . , ar ) =

n ∑ m=0

( n ∑

) ( ) n (r,k) S2 (l, m) Sn−l (−m|a1 , . . . , ar ) (x)(m) . l l=m

Proof. For (16) and (x)(n) = x(x+1) · · · (x+n−1) ∼ (1, 1−e−t ), assume that Sn

(r,k)

(30)

(x|a1 , . . . , ar ) =

15

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∑n m=0

Cn,m (x)(m) . By (13), we have ⟩ ⟨ 1 1 Cn,m = (1 − e−t )m xn ) ∏r aj t −1) 1−e−t m! j=1 (e tr Lik (1−e−t ) ⟩ ⟨ 1 Lik (1 − e−t ) −mt t tr ∏r = e (e − 1)m xn ) aj t − 1) −t (e m! 1 − e j=1 ⟩ ( )⟨ n r −t ∑ n t Li (1 − e ) k = S2 (l, m) e−mt ∏r xn−l aj t − 1) −t l (e 1 − e j=1 l=m ( ) n ⟨ ⟩ ∑ n −mt (r,k) = S2 (l, m) e Sn−l (x|a1 , . . . , ar ) l l=m ( ) n ∑ n (r,k) = S2 (l, m) Sn−l (−m|a1 , . . . , ar ) . l l=m

Thus, we get the identity (30).

3.8

Relations with higher-order Frobenius-Euler polynomials (r)

For λ ∈ C with λ ̸= 1, the Frobenius-Euler polynomials of order r, Hn (x|λ) are defined by the generating function (

1−λ et − λ

)r xt

e =

∞ ∑

t Hn(r) (x|λ)

n

n!

n=0

(see e.g. [6]). Theorem 8 Sn(r,k) (x|a1 , . . . , ar ) =

n ∑ m=0

(

) s ( ) ∑ s (r,k) (s) m (−λ)s−j Sn−m (j|a1 , . . . , ar ) Hm (x|λ) . (1 − λ)s j=0 j (n)

(31)

Proof. For (16) and

(( Hn(s) (x|λ)

∼

et − λ 1−λ

)s

) ,t

,

16

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∑ (s) (x|a1 , . . . , ar ) = nm=0 Cn,m Hm (x|λ). By (13), we have ⟩ ⟨( )s Lik (1 − e−t ) m n 1 et − λ tr ∏r t x aj t − 1) m! 1−λ 1 − e−t j=1 (e ⟩ ⟨ r −t 1 t Li (1 − e ) k (et − λ)s ∏r tm x n aj t − 1) −t m!(1 − λ)s (e 1 − e j=1 ⟨ ⟩ (n) ( ) s r −t ∑ s t Li (1 − e ) k m (−λ)s−j ejt ∏r xn−m aj t − 1) −t (e (1 − λ)s j=0 j 1 − e j=1 (n) ( ) s ∑ s (r,k) m (−λ)s−j Sn−m (j|a1 , . . . , ar ) . s (1 − λ) j=0 j

(r,k)

assume that Sn Cn,m = = = =

Thus, we get the identity (31).

3.9

Relations with higher-order Bernoulli polynomials (r)

Bernoulli polynomials Bn (x) of order r are defined by (

t t e −1

)r xt

e =

∞ (r) ∑ Bn (x) n=0

n!

tn

(see e.g. [12, Section 2.2]). Theorem 9

(( ) n−m ( ) ) n−m ∑ n (r,k) l (l+s ) S2 (l + s, s)Sn−m−l Sn(r,k) (x|a1 , . . . , ar ) = (a1 , . . . , ar ) B(s) m (x) . m l=0 l m=0 (32) n ∑

Proof. For (16) and

(( B(s) n (x)

∼

et − 1 t

)s

) ,t

,

17

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∑ (r,k) (s) assume that Sn (x|a1 , . . . , ar ) = nm=0 Cn,m Bm (x). By (13), we have ⟩ ⟨( )s 1 Lik (1 − e−t ) m n et − 1 tr ∏r Cn,m = t x aj t − 1) m! t 1 − e−t j=1 (e ⟩ ( )⟨ ( )s n tr Lik (1 − e−t ) et − 1 ∏r = xn−m aj t − 1) −t m (e 1 − e t j=1 ⟩ ( )⟨ n−m n tr Lik (1 − e−t ) ∑ s! ∏r = S2 (l + s, s)tl xn−m aj t − 1) −t m (e 1 − e (l + s)! j=1 l=0 ⟩ ⟨ ( ) n−m ∑ tr n s! Lik (1 − e−t ) n−m−l = S2 (l + s, s)(n − m)l ∏r x aj t − 1) m l=0 (l + s)! 1 − e−t j=1 (e ( ) n−m n ∑ s! (r,k) = S2 (l + s, s)(n − m)l Sn−m−l (a1 , . . . , ar ) m l=0 (l + s)! (n−m) ( ) n−m n ∑ (r,k) ( l ) S2 (l + s, s)Sn−m−l = (a1 , . . . , ar ) . m l=0 l+s l Thus, we get the identity (32).

References [1] A. A. Aygunes, Y. Simsek, Unification of multiple Lerch-zeta type functions , Adv. Stud. Contemp. Math. 21 (2011), 367-373. [2] A. Bayad, T. Kim, Results on Values of Barnes polynomials, Rocky Mountain J. Math. 43 (2013), no. 6, 1-10. [3] A. Bayad, T. Kim, W. J. Kim and S. H. Lee, Arithmetic properties of q-Barnes polynomials, J. Comput. Anal. Appl. 15 (2013), 111–117. [4] M. -A. Coppo and B. Candelpergher, The Arakawa-Kaneko zeta functions, Ramanujan J. 22 (2010), 153–162. [5] L. Jang, T. Kim, Y. -H. Kim, K. -W. Hwang, Note on the q q-extension of Barnes’ type multiple Euler polynomials, J. Inequal. Appl. 2009, Art. ID 136532, 8 pp. [6] D. S. Kim and T. Kim, Some identities of Frobenius-Euler polynomials arising from umbral calculus, Adv. Difference Equ. 2012 (2012), #196. [7] D. S. Kim, T. Kim and S. -H. Lee, Poly-Bernoulli polynomials arising from umbral calculus, available at http://arxiv.org/pdf/1306.6697.pdf [8] T. Kim, On Euler-Barnes multiple zeta functions, Russ. J. Math. Phys. 10 (2003), 261–267. 18

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[9] T. Kim, p-adic q-integrals associated with the Changhee-Barnes’ q-Bernoulli polynomials, Integral Transforms Spec. Funct. 15 (2004), 415–420. [10] T. Kim, Barnes-type multiple q-zeta functions and q-Euler polynomials, J. Phys. A 43 (2010), 255201, 11pp. [11] T. Kim and S. -H. Rim, On Changhee-Barnes’ q-Euler numbers and polynomials, Adv. Stud. Contemp. Math. (Kyungshang) 9 (2004), 81–86. [12] S. Roman, The umbral Calculus, Dover, New York, 2005. [13] Y. Simsek, T. Kim and I. -S. Pyung, Barnes’ type multiple Changhee q-zeta functions, Adv. Stud. Contemp. Math. (Kyungshang) 10 (2005), 121–129. [14] Y. Simsek, Generating functions of the twisted Bernoulli numbers and polynomials associated with their interpolation functions, Adv. Stud. Contemp. Math. 16 (2008), 251-278.

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TABLE OF CONTENTS, JOURNAL OF COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 18, NO. 5, 2015

An Umbral Calculus Approach to Poly-Cauchy Polynomials with a q Parameter, Dae San Kim, Taekyun Kim, Takao Komatsu, and Jong-Jin Seo,.....................................................................762 Tripled Fixed Point Theorems for Mixed Monotone Chatterjea Type Contractive Operators, Marin Borcut, Mădălina Păcurar, and Vasile Berinde,………………………………………..793 Soft Boolean Algebra and Its Properties, Rıdvan Şahin, and Ahmet Küçük,………………...803 Generating Functions for the Generalized Bivariate Fibonacci and Lucas Polynomials, Esra Erkuş-Duman, and Naim Tuglu,………………………………………………………………815 Integral norms of 𝑄𝐾,𝜔 (𝑝, 𝑞; 𝑛) Spaces and Weighted Bloch Spaces, A. El-Sayed Ahmed, and Aydah Ahmadi,………………………………………………………………………………...822 On two Dimensional q-Bernoulli and q-Genocchi Polynomials: Properties and location of zeros, N. I. Mahmudov, A. Akkeleș, and A. Öneren,…………………………………………………834 Existence Results of Sequential Derivatives of Nonlinear Quantum Difference Equations with a New Class of Three-Point Boundary Value Problems Conditions, Nichaphat Patanarapeelert, Thanin Sitthiwirattham,………………………………………………………………………...844 An Iterative Method for Solving Fourth-Order Boundary Value Problems of Mixed Type Integro-Differential Equations, Omar Abu Arqub,…………………………………………….857 An AQCQ-Functional Equation in Normed 2-Banach Spaces, Choonkil Park, Sun Young Jang, Reza Saadati, and Dong Yun Shin,……………………………………………………………875 Refined General Quadratic Equation with Four Variables and Its Stability Results, Hark-Mahn Kim, and Soon Lee,…………………………………………………………………………….885 Hyers-Ulam Stability of a Class of Differential Equations of Second Order, Mohammad Reza Abdollahpour, and Choonkil Park,…………………………………………………………….899 An Iterative Algorithm Based On the Hybrid Steepest Descent Method for Strictly Pseudocontractive Mappings, Jong Soo Jung,…………………………………………………904 BE-Algebras with Order Reversing Involution, Sun Shin Ahn, Young Hee Kim, and Jung Hee Park,……………………………………………………………………………………………918

TABLE OF CONTENTS, JOURNAL OF COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 18, NO. 5, 2015 (continued) Symmetry p-Adic Invariant Integral on ℤ𝑝 for q-Euler Polynomials, Dae San Kim, Taekyun Kim, Sang-Hun Lee, and Jong-Jin Seo,……………………………………………………….927 Barnes' Multiple Bernoulli and Poly-Bernoulli Mixed-Type Polynomials, Dmitry V. Dolgy, Dae San Kim, Taekyun Kim, Takao Komatsu, and Sang-Hun Lee,……………………………….933

Volume 18, Number 6 ISSN:1521-1398 PRINT,1572-9206 ONLINE

Journal of Computational Analysis and Applications EUDOXUS PRESS,LLC

June 2015

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

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J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 18, NO.6, 2015, COPYRIGHT 2015 EUDOXUS PRESS, LLC

FUNCTIONAL INEQUALITIES ASSOCIATED WITH INNER PRODUCT PRESERVING MAPPINGS GANG LU, GEORGE A. ANASTASSIOU, CHOONKIL PARK∗ , AND YUANFENG JIN Abstract. In this paper, we prove the Hyers-Ulam stability of inner product preserving mappings in Hilbert spaces for the following additive functional equation f (ax + by) = af (x) + bf (y).

1. Introduction and preliminaries The stability problem of functional equations originated from a question of Ulam [28] concerning the stability of group homomorphisms: Let (G1 , ∗) be a group and let (G2 , , d) be a metric group with the metric d(·, ·). Given  > 0, does there exist a δ() > 0 such that if a mapping h : G1 → G2 satisfies the inequality d(h(x ∗ y), h(x)  h(y)) < δ for all x, y ∈ G1 , then there is a homomorphism H : G1 → G2 with d(h(x), H(x)) <  for all x ∈ G1 ? If the answer is affirmative, we would say that the question of homomorphism H(x ∗ y) = H(x)  H(y) is stable. The concept of stability for a functional equation arises when we replace the functional equation by an inequality which acts as a perturbation of the equation. Thus the stability question of functional equation is that how do the solutions of the inequality differ from those of the given functional equation? Hyers [12] gave a first affirmative answer to the question of Ulam for Banach spaces. Let X and Y be Banach spaces. Assume that f : X → Y satisfies kf (x + y) − f (x) − f (y)k ≤  for all x, y ∈ X and some  ≥ 0. Then there exists a unique additive mapping T : X → Y such that kf (x) − T (x)k ≤  for all x ∈ X. Let X and Y be Banach spaces with norms k · k and k · k, respectively. Consider f : X → Y to be a mapping such that f (tx) is continuous in t ∈ R for each fixed x ∈ X. 2010 Mathematics Subject Classification. Primary 39B62, 39B52, 46B25. Key words and phrases. additive functional equation; inner product preserving mapping Hyers-Ulam stability; Hilbert space. ∗ Corresponding author.

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GANG LU ET AL 964-972

J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 18, NO.6, 2015, COPYRIGHT 2015 EUDOXUS PRESS, LLC

G. LU, G. A. ANASTASSIOU, C. PARK, AND Y. JIN

Rassias [22] introduced the following inequality: Assume that there exist constants λ ≥ 0 and p ∈ [0, 1) such that kf (x + y) − f (x) − f (y)k ≤ λ(kxkp + kykp ) for all x, y ∈ X. Rassias [22] showed that there exists a unique R-linear mapping T : X → Y such that 2λ kf (x) − T (x)k ≤ kxkp 2 − 2p for all x ∈ X. Beginning around the year 1980 the topic of approximate homomorphisms, or the stability of the equation of homomorphism, was studied by a number of mathematicians. Gˇ avruta [11] generalized the Rassias’ result. A square norm on an inner product space satisfies the important parallelogram equality kx + yk2 + kx − yk2 = 2kxk2 + 2kyk2 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 [27] for mappings f : X → Y , where X is a normed space and Y is a Banach space. Cholewa [4] noticed that the theorem of Skof is still true if the relevant domain X is replace by an Abelian group. In [5], Czerwik proved the Hyers-Ulam stability of the quadratic functional equation. Borelli and Forti [10] generalized the stability result as follows: Let G be an abelian group, E a Banach space. Assume that a mapping f : G → E satisfies the functional inequality kf (x + y) + f (x − y) − 2f (x) − 2f (y)k ≤ ϕ(x, y) for all x, y ∈ G, where ϕ : G × G → [0, ∞) is a function such that ∞ X 1 ϕ(2i x, 2i y) < ∞ Φ(x, y) := i+1 4 i=0

for all x ∈ G. The stability problems of several functional equations have been extensively investigated by a number of authors and there are many interesting results concerning this problem. A large list of references can be found in ([6, 7, 8, 9, 13, 14, 15, 16, 17, 18, 19, 20, 21, 23, 24, 25, 26]). Let X and Y be complex Hilbert spaces. An additive mapping f : X → Y is called an inner product preserving mapping if f satisfies the orthogonality equation hf (x), f (y)i = hx, yi for all x, y ∈ X. The inner product preserving mapping problem has been investigated in several papers (see [1, 2, 3]).

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GANG LU ET AL 964-972

J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 18, NO.6, 2015, COPYRIGHT 2015 EUDOXUS PRESS, LLC

INNER PRODUCT PRESERVING MAPPINGS

In this paper, we prove the Hyers-Ulam stability of inner product preserving mappings in Hilbert spaces for the following additive function equation. f (ax + by) = af (x) + bf (y),

a, b ∈ R \ {0}.

(1)

Throughout this paper, assume that X and Y are complex Hilbert spaces, and that a, b ∈ R \ {0} with |a| < 1 or |b| > 1 or a = b = 1. 2. Hyers-Ulam stability of (1) in Hilbert spaces We prove the Hyers-Ulam stability of inner product preserving mappings in Hilbert spaces for the additive function equation (1) when |a| < 1 or |b| > 1. Theorem 2.1. Let |a| < 1 and let f : X → Y be a mapping with f (0) = 0 for which there exists a function φ : X × X → [0, ∞) such that ∞ x y X j e φ(x, y) := |a |φ j , j < ∞, (2.1) a a j=0 kf (ax + by) − af (x) − bf (y)k ≤ φ(x, y),

(2.2)

|hf (x), f (y)i − hx, yi| ≤ φ(x, y)

(2.3)

for all x, y ∈ X. Then there exists a unique inner product preserving mapping I : X → Y such that x  (2.4) kf (x) − I(x)k ≤ φe , 0 a for all x ∈ X. Proof. Letting y = 0 in (2.2), we obtain kf (ax) − af (x)k ≤ φ(x, 0). Then

 x  x 

,0 .

≤φ

f (x) − af a a It follows from (2.5) that

x  x  m−1 x  x 

l

X j

m

a f j − aj+1 f j+1

a f l − a f m ≤ a a a a j=l ≤

m−1 X j=l

(2.5)

 x  |a |φ j+1 , 0 a j

for all nonnegative
integers m and l with m > l and all x ∈ X. It means that the sequence {an f axn } is a Cauchy sequence for all x ∈ X. Since Y is complete, the sequence {an f ( axn )} converges. We define the mapping I : X → Y by I(x) = limn→∞ {an f ( axn )} for all x ∈ X. Moreover, letting l = 0 and passing the limit m → ∞, we get (2.4).

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GANG LU ET AL 964-972

J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 18, NO.6, 2015, COPYRIGHT 2015 EUDOXUS PRESS, LLC

G. LU, G. A. ANASTASSIOU, C. PARK, AND Y. JIN

We show that I(x) is an additive mapping.

  x  y 

x+y n

kI(x + y) − I(x) − I(y)k = lim |a | f − f − f n→∞ an an an     x   y 

x + y

≤ lim |an | − af − bf f

n→∞ an aan ban

 x   x   y   y  o



+ af n − f n + bf − f

n n a an  a ba a     x x y y o n + φ n , 0 + φ 0, n ≤ lim |a | φ , n→∞ aan ban a a = 0. It follows from (2.1) and (2.3) that D D  x   y E D x y E x  y E n n − hx, yi = a2n · f n , f n − n, n a f n ,a f n a a a  a a  x ay  x y 2n n ≤ |a |φ n , n ≤ |a |φ n , n , a a a a which tends to zero as n → ∞ for all x, y ∈ X.  y E D x = hx, yi hI(x), I(y)i = lim an f n , an f n n→∞ a a for all x, y ∈ X. It only remains to show that the mapping I : X → Y is unique. Let g be another additive mapping satisfying (2.4). Then

x  y 

kg(x) − I(x)k = |an | g n − I n a  a        y   x x x

n ≤ |a | g n − f n + f n − I n a a a a ∞   x X ≤2 |ai+n |φe i+n , 0 a i=1 for all x ∈ X and n ∈ N. Thus from n → ∞, one establishes g(x) − I(x) = 0 for all x ∈ X. This completes the proof of uniqueness.



Corollary 2.2. Let |a| < 1 and let f : X → Y be a mapping for which there exist constants θ ≥ 0 and r ∈ [0, 1) such that kf (ax + by) − af (x) − bf (y)k ≤ θ(kxkr + kykr ),

(2.6)

|hf (x), f (y)i − hx, yi| ≤ θ(kxkr + kykr )

(2.7)

for all x, y ∈ X. Then there exists a unique inner product preserving mapping I : X → Y such that θ kf (x) − I(x)k ≤ r kxkr |a| − |a|

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J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 18, NO.6, 2015, COPYRIGHT 2015 EUDOXUS PRESS, LLC

INNER PRODUCT PRESERVING MAPPINGS

for all x ∈ X. Proof. Defining φ(x, y) = θ(kxk + kyk) and applying Theorem 2.1, we get the desired result.  Theorem 2.3. Let |b| > 1 and let f : X → Y be a mapping with f (0) = 0, (2.2) and (2.3) for which there exists a function φ : X × X → [0, ∞) such that ∞ x y X 2j |b |φ j , j < ∞ (2.8) b b j=0 for all x, y ∈ X. Then there exists a unique inner product preserving mapping I : X → Y such that  x kf (x) − I(x)k ≤ φe 0, (2.9) b for all x ∈ X, where ∞ x y X j e φ(x, y) := |b |φ j , j b b j=0 for all x, y ∈ X. Proof. Letting x = 0 and replacing y by x in (2.2), we obtain kf (bx) − bf (x)k ≤ φ(0, x), an so

x  x  m−1 x  x 

X j

l m

b f j − bj+1 f j+1

b f l − b f m ≤ b b b b j=l ≤

m−1 X

j



|b |φ 0,

j=l

x  bj+1

for all nonnegative
integers m and l with m > l and all x ∈ X. It means that the x n sequence {b f bn } is a Cauchy sequence for all x ∈ X. Since Y is complete, the sequence {bn f ( bxn )} converges. We define the mapping I : X → Y by I(x) = limn→∞ {bn f ( bxn )} for all x ∈ X. Moreover, letting l = 0 and passing the limit m → ∞, we get (2.9). It follows from (2.3) and (2.8) that D D  x   y E D x y E x  y E n − hx, yi = |b2n | · f n , f n − n, n b f n , bn f n b b b b b  x by  2n ≤ |b |φ n , n , b b which tends to zero as n → ∞ for all x, y ∈ X. D x  y E n n hI(x), I(y)i = lim b f n , b f n = hx, yi n→∞ b b for all x, y ∈ X. The rest of the proof is similar to the proof of Theorem 2.1. 

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GANG LU ET AL 964-972

J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 18, NO.6, 2015, COPYRIGHT 2015 EUDOXUS PRESS, LLC

G. LU, G. A. ANASTASSIOU, C. PARK, AND Y. JIN

Corollary 2.4. Let |b| > 1 and let f : X → Y be a mapping for which there exist constants θ ≥ 0 and r > 2 satisfying (2.6) and (2.7). Then there exists a unique inner product preserving mapping I : X → Y such that θ kf (x) − I(x)k ≤ r kxkr |b| − |b| for any x ∈ X. Proof. Defining φ(x, y) = θ(kxk + kyk) and applying Theorem 2.3, we get the desired result.  Now we prove the Hyers-Ulam stability of inner product preserving mappings in Hilbert spaces for the additive function equation (1) when a = b = 1. Theorem 2.5. Let f : X → Y be a mapping for which there exists a function φ : X ×X → [0, ∞) satisfying (2.3) and ∞ x y X e y) := φ(x, 4j φ j , j < ∞, (2.10) 2 2 j=1 kf (x + y) − f (x) − f (y)k ≤ φ(x, y)

(2.11)

for all x, y ∈ X. Then there exists a unique inner product preserving mapping I : X → Y such that 1 kf (x) − I(x)k ≤ φe (x, x) (2.12) 2 for all x ∈ X, where ∞ x y X e φ(x, y) := 2j φ j , j 2 2 j=1 for all x, y ∈ X. Proof. Letting y = x in (2.11), we obtain kf (2x) − 2f (x)k ≤ φ(x, x). Then

 x  x x

, .

f (x) − 2f

≤φ 2 2 2 It follows from (2.13) that

x  x  m−1 x  x 

l

X j

m

2 f l − 2 f m ≤

2 f j − 2j+1 f j+1 2 2 2 2 j=l ≤

m−1 X j=l

(2.13)

 x x  2 φ j+1 , j+1 2 2 j

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INNER PRODUCT PRESERVING MAPPINGS

for all nonnegative
integers m and l with m > l and all x ∈ X. It means that the x n sequence {2 f 2n } is a Cauchy sequence for all x ∈ X. Since Y is complete, the sequence {2n f ( 2xn )} converges. We define the mapping I : X → Y by I(x) = limn→∞ {2n f ( 2xn )} for all x ∈ X. Moreover, letting l = 0 and passing the limit m → ∞, we get (2.12). It follows from (2.3) and (2.10) that D  x   y E D x y E D x  y E n n 2n − hx, yi = |2 | · f n , f n − n, n 2 f n ,2 f n 2 2 2 2 2  x 2y  2n ≤ |2 |φ n , n , 2 2 which tends to zero as n → ∞ for all x, y ∈ X. D x  y E n n hI(x), I(y)i = lim 2 f n , 2 f n = hx, yi n→∞ 2 2 for all x, y ∈ X. The rest of the proof is similar to the proof of Theorem 2.1.  Corollary 2.6. Let f : X → Y be a mapping for which there exist constants θ ≥ 0 and r > 2 satisfying (2.7) and kf (x + y) − f (x) − f (y)k ≤ θ(kxkr + kykr )

(2.14)

for all x, y ∈ X. Then there exists a unique inner product preserving mapping I : X → Y such that 2θ kf (x) − I(x)k ≤ r kxkr 2 −2 for all x ∈ X. Proof. Defining φ(x, y) = θ(kxk + kyk) and applying Theorem 2.5, we get the desired result.  Theorem 2.7. Let f : X → Y be a mapping for which there exists a function φ : X ×X → [0, ∞) satisfying (2.3), (2.11) and ∞ X
1 j j e y) := φ(x, φ 2 x, 2 y 2j j=0 for all x, y ∈ X. Then there exists a unique inner product preserving mapping I : X → Y such that 1 kf (x) − I(x)k ≤ φe (x, x) 2 for all x ∈ X. Proof. It follows from (2.13) that



1

f (x) − f (2x) ≤ 1 φ (x, x) .

2 2 The rest of the proof is similar to the proofs of Theorems 2.1 and 2.5.

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GANG LU ET AL 964-972

J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 18, NO.6, 2015, COPYRIGHT 2015 EUDOXUS PRESS, LLC

G. LU, G. A. ANASTASSIOU, C. PARK, AND Y. JIN

Corollary 2.8. Let f : X → Y be a mapping for which there exist constants θ ≥ 0 and r ∈ (0, 1) satisfying (2.7) and (2.14). Then there exists a unique inner product preserving mapping I : X → Y such that 2θ kf (x) − I(x)k ≤ kxkr r 2−2 for all x ∈ X. Proof. Defining φ(x, y) = θ(kxk + kyk) and applying Theorem 2.7, we get the desired result.  Acknowledgments G. Lu was supported by supported by Doctoral Science Foundation of Liaoning Province, China, by Hall of Liaoning Province Science and Technology (No. 2012-1055) and C. Park was supported by Basic Science Research Program through the National Research Foundation of Korea funded by the Ministry of Education, Science and Technology (NRF2012R1A1A2004299). References [1] J. Chmieli` nski, On a singular case in the Hyers-Ulam-Rassias stability of the Wigner equation, J. Math. Anal. Appl. 289 (2004), 571–583. [2] J. Chmieli` nski, Linear mappings approximately preserving orthogonality, J. Math. Anal. Appl. 304 (2005), 158–169. [3] J. Chmieli` nski and S. Jung, The stability of the Wigner equation on a restricted domain, J. Math. Anal. Appl. 254 (2001), 309–320. [4] P. W. Cholewa, Remarks on the stability of functional equations, Aequationes Math. 27 (1984), 76–86. [5] S. Czerwik, On the stability of the quatradic mapping in normed spaces, Abh. Math. Sem. Univ. Hamburg 62 (1992), 59–64. [6] S. Czerwik, The stability of the quadratic functional equation, in Stability of Mappings of Hyers-Ulam Type (edited by Th. M. Rassias and J. Tabor), Hadronic Press, Palm Harbor, Florida, 1994, pp. 81–91. [7] S. Czerwik, Stability of Functional equations of Ulam-Hyers-Rassias Type, Hardronic Press, Palm Harbor, Florida, 2003. [8] M. Eshaghi Gordji, G. Kim, J. Lee and C. Park, Generalized ternary bi-derivations on ternary Banach algebras: a fixed point approach, J. Comput. Anal. Appl. 15 (2013), 45–54. [9] M. Eshaghi Gordji, G. Kim, J. Lee and C. Park, Nearly generalized derivations on non-Archimdean Banach algebras: a fixed point approach, J. Comput. Anal. Appl. 15 (2013), 308–315. [10] G. L. Forti, Comments on the core of the direct method for provinh Hyers-Ulam stability of functional equations, J. Math. Anal. Appl. 295 (2004), 127–133. [11] P. G˘ avruta, A generalization of the Hyers-Ulam-Rassias stability of approximately additive mappings, J. Math. Anal. Appl. 184 (1994), 431–436. [12] D. H. Hyers, On the stability of the linear functional equation, Proc. Natl. Acad. Sci. U.S.A. 27 (1941), 222–224. [13] D. H. Hyers, G.Isac and Th. M. Rassias, Stability of Functional Equations in Several Variables, Birkh¨ auser, Basel, 1998. [14] M. Kim, Y. Kim, G. A. Anastassiou and C. Park, An additive functional inequality in matrix normed modules over a C ∗ -algebra, J. Comput. Anal. Appl. 17 (2014), 329–335.

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[15] M. Kim, S. Lee, G. A. Anastassiou and C. Park, Functional equations in matrix normed modules, J. Comput. Anal. Appl. 17 (2014), 336–342. [16] J. Lee, S. Lee and C. Park, Fixed points and stability of the Cauchy-Jensen functional equation in fuzzy Banach algebras, J. Comput. Anal. Appl. 15 (2013), 692–698. [17] J. Lee, C. Park, Y. Cho and D. Shin, Orthogonal stability of a cubic-quartic functional equation in non-Archimedean spaces, J. Comput. Anal. Appl. 15 (2013), 572–583. [18] L. Li, G. Lu, C. Park and D. Shin, Additive functional inequalities in generalized quasi-Banach spaces, J. Comput. Anal. Appl. 15 (2013), 1165–1175. [19] G. Lu, Y. Jiang and C. Park, Additive functional equation in Fr´echet spaces, J. Comput. Anal. Appl. 15 (2013), 369–373. [20] C. Park, K. Ghasemi, S. G. Ghaleh, S. Jang, Approximate n-Jordan ∗-homomorphisms in C ∗ algebras, J. Comput. Anal. Appl. 15 (2013), 365–368. [21] 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. [22] Th. M. Rassias, On the stability of the linear mapping in Banach spaces, Proc. Amer. Math. Soc. 72 (1978), 297–300. [23] S. Shagholi, M. Eshaghi Gordji and M. Bavand Savadkouhi, Stability of ternary quadratic derivations on ternary Banach algebras, J. Comput. Anal. Appl. 13 (2011), 1097–1105. [24] S. Shagholi, M. Eshaghi Gordji and M. Bavand Savadkouhi, Nearly ternary cubic homomorphisms in ternary Fr´echet algebras, J. Comput. Anal. Appl. 13 (2011), 1106–1114. [25] D. Shin, C. Park and Sh. Farhadabadi, On the superstability of ternary Jordan C ∗ -homomorphisms, J. Comput. Anal. Appl. 16 (2014), 964–973. [26] 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. [27] F. Skof, Proprit` a locali e approssimazion di operatori, Rend. Sem. Mat. Fis. Milano 53 (1983), 113–129. [28] S. M. Ulam, problems in Modern mathematics, Wiley, New York, 1960. Gang Lu Department of Mathematics, School of Science, ShenYang University of Technology, Shenyang 110178, P.R. China E-mail address: [email protected] George A. Anastassiou Department of Mathematical Sciences, University of Memphis, Memphis, TN 38152, USA E-mail address: [email protected] Choonkil Park Department of Mathematics, Research Institute for Natural Sciences, Hanyang University, Seoul 133-791, Korea E-mail address: [email protected] Yuanfeng Jin Department of Mathematics, Yanbian University, Yanji 133001, P.R. China E-mail address: [email protected]

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. STABILITY AND SUPERSTABILITY OF (fr , fs )-DOUBLE DERIVATIONS IN QUASI-BANACH ALGEBRAS SUN YOUNG JANG, CHOONKIL PARK, PEGAH EFTEGHAR, AND SHAHROKH FARHADABADI∗ Abstract. In this paper, the following functional equation n h X k X k=2

k+1 X

n X

···

i1 =2 i2 =i1 +1

+f

f



in−k+1 =in−k +1 n X

n X i=1, i6=i1 ,··· ,in−k+1

ai xi −

n−k+1 X

air xir

i

r=1

 ai xi = a1 · 2n−1 f (x1 )

(0.1)

i=1

is considered (n ≥ 2), and the Hyers-Ulam stability and superstability of (fr , fs )-double derivations on quasi-Banach algebras associated with the functional equation (0.1) are proved.

1. Introduction and preliminaries The stability of functional equations theory discusses and studies about solutions of functional equations and analyzes the relationships between approximate and exact solutions of the functional equations. Actually, we say a functional equation is stable, if one can find an exact solution for any approximate solution of the functional equation. Subsequently, the concept of superstability has a near nature to the stability sense. In other words, it happens when any approximate solution is also an exact solution that in such situation the functional equation is called superstable. In 1940, the most preliminary kind of stability problems was proposed by Ulam [37]. He gave a talk and asked the following: “when and under what conditions does an exact solution of a functional equation near an approximately solution of that exist?” In 1941, Hyers [14] formulated and proved the Ulam’s problem for the Cauchy’s functional equation on Banach spaces. The result of Hyers was generalized by Aoki [3] for additive mappings and by Rassias [32] for linear mappings by considering an unbounded Cauchy difference. In 1994, G˘avrut¸a [13] provided a further generalization of Rassias’ theorem in which he replaced the unbounded Cauchy difference by a general control function for the existence of a unique linear mapping. It seems that considering stability problems concerning derivations returns to 1994 ˘ by Semrl [34] who had worked on derivations between operator algebras and afterwards Jun and Park [16]. They had investigated approximate derivations on Banach algebras 0

2010 Mathematics Subject Classification: 47B48, 47B47, 39B52, 39B82, 32A65, 17A36, 46Hxx. Key words and phrases. Functional equation; Hyers-Ulam stability; (fr , fs )-double derivation; Superstability; Quasi-Banach algebra. ∗ Corresponding author: Email address: shahrokh [email protected] (Sh. Farhadabadi).

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C n [0, 1]. More stability results in many kind of derivations can be found in (cf. [2, 4, 7, 8, 10, 24, 25, 26, 28, 36]). At present, the theory of stability is quickly being deployed by numerous mathematicians. They pose and investigate various stability problems including different functional equations, derivations and homomorphisms in various spaces and structures. For more epochal information and various aspects about the stability theory, the readers can refer to monographs (cf. [5, 6, 9, 11, 12, 15, 18, 19, 20, 21, 22, 23, 27, 29, 30, 31, 33, 35, 38]). Now we give briefly some useful definitions, preliminary and fundamental results of quasi-Banach spaces. Definition 1.1. ([17]) Let X be a real linear space. A function k · k : X → R is a quasi-norm (valuation) if it satisfies the following conditions: (N1 ) kxk ≥ 0 for all x ∈ X and kxk = 0 if only if x = 0; (N2 ) kλ · xk = |λ| · kxk for all λ ∈ R and all x ∈ X ; (N3 ) There is a constant K ≥ 1 such that kx + yk ≤ K(kxk + kyk) for all x, y ∈ X . In this case, the pair (X , k · k) is called a quasi-normed space and the smallest possible K is called the modulus of concavity of k · k. A complete quasi-normed space is a quasi-Banach space. Definition 1.2. ([17]) Let 0 < p ≤ 1 be a real number. A quasi-normed space (X , k · k) is called a p-normed space if kx + ykp ≤ kxkp + kykp for all x, y ∈ X . Definition 1.3. ([1]) Let X be an algebra and (X , k · k) be a quasi-normed space. The quasi-normed space (X , k · k) is called a quasi-normed algebra if there exist a constant C > 0 such that kxyk ≤ Ckxkkyk for all x, y ∈ X . In addition, if the quasi-norm k · k is a p-norm, then the quasi-normed algebra (X , k · k) is called a p-normed algebra. Definition 1.4. ([24]) Let X be an algebra and f2 , f3 : X → X be linear mappings. A linear mapping f1 : X → X is called an (f2 , f3 )-double derivation if f1 (xy) = f1 (x)y + xf1 (y) + f2 (x)f3 (y) + f3 (x)f2 (y) for all x, y ∈ X . By an f2 -double derivation we mean an (f2 , f2 )-double derivation. It is clear that f1 is an (f2 , f3 )-double derivation if and only if f1 is an (f3 , f2 )-double derivation. Now, consider the functional equation (0.1). This equation which is called the general n-dimensional additive functional equation, was introduced by Khodaei and Rassias [17]. In order to investigate (0.1), throughout this paper a1 , · · · , an (with n ≥ 2, a1 > 1), are fixed positive integers and X will be also a p-Banach algebra with p-norm k · k, as well as the integers 1 ≤ j, r, s ≤ 3 are assumed with j 6= r 6= s. 2. Hyers-Ulam stability of (fr , fs )-double derivations on quasi-Banach algebras In this section, we prove the Hyers-Ulam stability of (fr , fs )-double derivations on quasi-Banach algebras associated with the functional equation (0.1).

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SUPERSTABILITY OF (fr , fs )-DOUBLE DERIVATIONS IN QUASI-BANACH ALGEBRAS

First of all, for convenience, for given mappings fj : X → X , we define the difference operators: n hX k k+1 X X
Dλ fj x1 , · · · , xn := ··· k=2

fj



n X

i1 =2 i2 =i1 +1

ai λxi −

i=1, i6=i1 ,··· ,in−k+1

n−k+1 X

n X in−k+1 =in−k +1

air λxir

r=1

i

+ fj

n X

 ai λxi − a1 · 2n−1 λfj (x1 ),

i=1

Dfj , fr , fs (x, y) := fj (xy) − fj (x)y − xfj (y) − fr (x)fs (y) − fs (x)fr (y) for all x, y, x1 , · · · , xn ∈ X and all λ ∈ R. Lemma 2.1. ([17]) Let X and Y be real vector spaces. A mapping f : X → Y satisfies the functional equation (0.1) if and only if f : X → Y is additive. From now on, 0 < p ≤ 1 is a real number. Theorem 2.2. Let φ : X n → [0, ∞) be a function such that  ∞  X
p
1 i i φ a1 x1 , · · · , a1 xn < +∞ φ˜ x1 , · · · , xn := i a 1 i=0

(2.1)

for all x1 , · · · , xn ∈ X . Suppose that fj : X → X , j = 1, 2, 3, are mappings satisfying the inequalities n
o
max Dλ fj x1 , x2 , · · · , xn ≤ φ x1 , x2 , · · · , xn , (2.2) j n

o
max Dfj ,fr ,fs (x, y) ≤ φ x + y, · · · , x + y (2.3) j,r,s

for all x, y, x1 , · · · , xn ∈ X and all λ ∈ R. Then unique (Fr , Fs )-double derivations Fj : X → X defined by the limits 1 fj (am 1 x) m→∞ am 1

Fj (x) = lim exist and satisfy the inequalities



Fj (x) − fj (x) ≤

h
i p1 1 ˜ φ x, 0, · · · , 0 a1 · 2n−1

(2.4)

for all x ∈ X . Proof. Letting x1 = x, x2 = · · · = xn = 0 and λ = 1 in (2.2), we get





D1 f1 x, 0, · · · , 0 ≤ φ x, 0, · · · , 0 for all x ∈ X . From this inequality and the fact that   n−1  n−1  X X n−1 n−1 n−1 2 = =1+ , i i i=0 i=1

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we have




1 1

f1 (x) − f1 (a1 x) ≤ φ x, 0, · · · , 0

n−1 a1 a1 · 2 for all x ∈ X . Replacing x by ai1 x and then dividing both sides by ai1 , we get

1


1

f1 (ai1 x) − 1 f1 (ai+1

≤ x) φ ai1 x, 0, · · · , 0 1 i+1

ai

i+1 n−1 a1 · 2 a1 1

(2.5)

for all x ∈ X and all nonnegative integers i. Assume that m, t are positive integers with m > t. Since X is a p-Banach space, it follows from (2.5) that

p

p

m−1 X 1

1 1 1 m t i i+1

f1 (a1 x) − f1 (a1 x) ≤ i+1 f1 (a1 x) − i f1 (a1 x) t

am a1 a a 1 1 1 i=t  m−1 X 1
p 1 i ≤ φ a1 x, 0, · · · , 0 (2.6) (a1 · 2n−1 )p i=t a1 i for all x ∈ X . Now by the condition (2.1), we deduce nthat the right-hand side tends to o is Cauchy. Since X zero as t, m → ∞, and this implies that the sequence a1m f1 (am 1 x) 1 n o is complete, the sequence a1m f1 (am converges in X , and therefore we can define for 1 x) 1 all x ∈ X the mapping F1 : X → X by 1 f1 (am 1 x). m→∞ am 1

F1 (x) = lim

Now we claim that F1 : X → X is R-linear. In order to verify that, we first show that F1 is additive. It follows from (2.2) and (2.1) that




p
1

p m m m

D1 F1 x1 , x2 , · · · , xn = lim m D1 f1 a1 x1 , a1 x2 , · · · , a1 xn

m→∞ a1  
p 1 m m m φ a1 x1 , a1 x2 , · · · , a1 xn =0 ≤ lim m→∞ am 1 for all x1 , · · · , xn ∈ X . So D1 F1 (x1 , x2 , · · · , xn ) = 0 for all x1 , x2 , · · · , xn ∈ X , which means F1 : X → X satisfies the functional equation (0.1). Therefore, Lemma 2.1 clarifies that the mapping F1 is additive. So 1 f1 (am+1 x), 1 m→∞ am 1

a1 F1 (x) = F1 (a1 x) = lim F1 (x) =

1 f1 (am+1 x) 1 m+1 m→∞ a 1 lim

for all x ∈ X . For x1 = am 1 x and x2 = · · · = xn = 0, it follows from (2.2) that  

p


p 1 1

−(m+1) m+1 −m m m f1 (a1 λx) − a1 λf1 (a1 x) ≤ p (n−1)p m φ a1 x, 0, · · · , 0

a1 a1 · 2 a1

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for all x ∈ X and all λ ∈ R. By (2.1), the right-hand side tends to zero as m → ∞. Hence

p

F1 (λx) − λF1 (x) = 0 for all x ∈ X and all λ ∈ R. So F1 (λx) = λF1 (x) for all x ∈ X and all λ ∈ R, and thus F1 is R-linear. Putting t = 0 and passing the limit m → ∞ in (2.6), we get the inequality (2.4) for j = 1. Let F10 : X → X be another R-linear mapping satisfying (2.4). Then we have

p

0 m



1 

m p 0 m m p

+ F (a x) − f (a x) F (x) − F (x) ≤ F (a x) − f (a x)

1

1 1 1 1 1 1 1 1 1 amp 1
2 1 ˜ am x, 0, · · · , 0 ≤ mp · φ 1 a1 (a1 .2n−1 )p  ∞  X
p 2 1 i = · φ a1 x, 0, · · · , 0 (a1 .2n−1 )p i=m ai1 for all x ∈ X . By (2.1), the right-hand side tends to zero as m → ∞, which signifies the uniqueness of F1 . By a similar method, one can easily show that the unique and R-linear mappings F2 : X → X and F3 : X → X defined by 1 f2 (am 1 x), m→∞ am 1

F2 (x) = lim

1 f3 (am 1 x) m→∞ am 1

F3 (x) = lim

exist and satisfy (2.4) for all x ∈ X . To end the proof, it is just necessary to show that F1 is an (F2 , F3 )-double derivation. It follows from (2.3) that








Df1 ,f2 ,f3 (x, y) = Df1 ,f3 ,f2 (x, y) ≤ φ x + y, · · · , x + y
2 for all x, y ∈ X . We know a1 > 1, so a1 mp > 1 and a1 mp > a1 mp . Therefore, the last inequality and the condition (2.1) imply that

p



F1 (xy) − F1 (x)y − xF1 (y) − F2 (x)F3 (y) − F3 (x)F2 (y)
1

2m m m m f a xy − f1 (am = lim

1 1 1 x) a1 y − a1 xf1 (a1 y) m→∞ a1 2mp

p

m m m m −f2 (a1 x) f3 (a1 y) − f3 (a1 x) f2 (a1 y)

p 1

m m = lim

Df1 ,f3 ,f2 (a1 x, a1 y) 2mp m→∞ a1   p 1  m m < lim φ a1 (x + y), · · · , a1 (x + y) =0 m→∞ a1 m for all x, y ∈ X . Hence F1 (xy) = F1 (x)y + xF1 (y) + F2 (x)F3 (y) + F3 (x)F2 (y)

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for all x, y ∈ X . Similarly we can show that F2 and F3 are respectively (F1 , F3 )-double derivation and (F1 , F2 )-double derivation and so the proof is complete.  Theorem 2.3. Let φ : X n → [0, ∞) be a function such that p ∞  x X xn  1 i ˜ φ x1 , · · · , xn ) := a1 φ i , · · · , i < +∞, a a 1 1 i=0  p  x x  2m lim a1 φ m , · · · , m =0 m→∞ a1 a1

(2.7)

for all x, x1 , · · · , xn ∈ X . Suppose that fj : X → X , j = 1, 2, 3, are mappings satisfying (2.2) and (2.3). Then unique (Fr , Fs )-double derivations Fj : X → X defined by the limits  x  Fj (x) = lim am f 1 j m→∞ am 1 exist and satisfy the inequalities



Fj (x) − fj (x) ≤

1 2n−1

  1/p x ˜ φ , 0, · · · , 0 a1

(2.8)

for all x ∈ X . x Proof. Replacing x by a2i+1 in (2.5) and then multiplying by a2i+1 , we obtain 1 1

  x 

i+1  x 

ai1  x i

a1 f1

− a1 f1 i ≤ n−1 φ i+1 , 0, · · · , 0

a1 2 ai+1 a1 1

for all x ∈ X . Now by the same n method
owhich was done in the proof of Theorem 2.2, we x can assert that the sequence am is Cauchy and convergent in X and the unique 1 f1 am 1
x and R-linear mappings Fj (x) = limm→∞ am exist and satisfy (2.8). 1 fj am 1 The inequality (2.3) and the condition (2.7) imply that

p  x y 

p

2mp

DF1 ,F3 ,F2 (x, y) = lim a1

Df1 ,f3 ,f2 am , am m→∞ 1 1 p  x + y x + y 2m ≤ lim a1 φ ,··· , m =0 m→∞ am a1 1 for all x, y ∈ X . This shows that F1 is an (F2 , F3 )-double derivation. The arguments for j = 2, 3 are also the same as for j = 1 and so we will omit them.  Corollary 2.4. Let δ be a nonnegative real number and q be a positive real number such that q < 1 or q > 2. Suppose that fj : X → X , j = 1, 2, 3, are mappings satisfying the inequalities n
o

max Dλ fj x1 , x2 , · · · , xn ≤ δ (kx1 kq + · · · + kxn kq ) , j n

o max Dfj ,fr ,fs (x, y) ≤ nδkx + ykq j,r,s

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SUPERSTABILITY OF (fr , fs )-DOUBLE DERIVATIONS IN QUASI-BANACH ALGEBRAS

for all x, y, x1 , · · · , xn ∈ X , and all λ ∈ R. Then unique (Fr , Fs )-double derivations Fj : X → X exist and satisfy the inequalities

δkxkq

F (x) − f (x) ≤

j

j 1 p 2n−1 · ap1 − apq 1 for all x ∈ X . Proof. Defining φ(x1 , · · · , xn ) := δ (kx1 kq + · · · + kxn kq ) and applying Theorem 2.2 for the case q < 1, and Theorem 2.3 for the case q > 2, we get the result.  3. Superstability of (fr , fs )-double derivations on quasi-Banach algebras In this section, we prove the superstability of (fr , fs )-double derivations associated with the functional equation (0.1). Initially, we improve Lemma 2.1 to a stronger statement and afterwards we use it for the proof of superstability theorem of this section. Lemma 3.1. Let n ≥ 3 be a fixed integer and f : X → X be a mapping such that n hX k k+1

X X

n−1 ···

2 a1 λf (x1 ) − k=2

f



i1 =2 i2 =i1 +1 n X

n X in−k+1 =in−k +1

ai λxi −

n−k+1 X

n

X  i

ai λxi air λxir ≤ f

r=1

i=1, i6=i1 ,··· ,in−k+1

(3.1)

i=1

for all x1 , · · · , xn ∈ X and all λ ∈ R. Then f is R-linear. Proof. First, assume that n ≥ 4. Putting x4 = · · · = xn = 0 in (3.1), we get

 


n−1

2 λa1 f (x1 ) − 2n−3 − 1 f λ[a1 x1 + a2 x2 + a3 x3 ] − 2n−3 
f λ[a1 x1 − a2 x2 + a3 x3 ) − 2n−3 f λ[a1 x1 + a2 x2 − a3 x3 ]


n−3 −2 f λ[a1 x1 − a2 x2 − a3 x3 ]




≤ f λ[a1 x1 + a2 x2 + a3 x3 ]

(3.2)

for all x1 , x2 , x3 ∈ X and all λ ∈ R. Letting x1 = x2 = x3 = 0 and λ = 1 in (3.2), we obtain









(a1 − 1)2n−1 + 1 f (0) ≤ f (0) . So f (0) = 0, (since (a1 − 1)2n−1 > 0). Putting λ = 1 and substituting x1 , x2 , x3 by x/a1 , −x/a2 , 0 and then by x/a1 , −x/2a2 , −x/2a3 in (3.2), respectively, we get x n−1 a1 · 2 f = 2n−2 f (2x), a  x1  = 2n−2 f (x) + 2n−3 f (2x) a1 · 2n−1 f a1

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for all x ∈ X . Thus f (2x) = 2f (x) and f (x) = a1 f ax1 for all x ∈ X . Letting λ = 1, x1 = x + y/a1 , x2 = −x/a2 and x3 = −y/a3 in (3.2), we have x + y  a1 · 2n−1 f − 2n−3 f (2x) − 2n−3 f (2y) − 2n−3 f (2x + 2y) = 0 a1 for all x, y ∈ X , which implies that f (x + y) = f (x) + f (y) for all x, y ∈ X . Finally, letting x1 = x/a1 , x2 = −x/a2 and x3 = 0 in (3.2), we obtain x a1 λ · 2n−1 f − 2n−2 f (2λx) = 0 a1 for all x ∈ X and all λ ∈ R. So f (λx) = λf (x) for all x ∈ X and all λ ∈ R, and therefore f is R-linear. Now assume that n = 3 in (3.1). Then




4a1 λf (x1 ) − f λ[a1 x1 − a2 x2 − a3 x3 ]



−f λ[a1 x1 + a2 x2 − a3 x3 ] − f λ[a1 x1 − a2 x2 + a3 x3 ]




≤ f λ[a1 x1 + a2 x2 + a3 x3 ] for all x1 , x2 , x3 ∈ X and all λ ∈ R. As it is obvious that we can get an inequality similar to the normed inequality (3.2) for the case n ≥ 4, one can easily get the desired result.  Theorem 3.2. Let φ : X n → [0, ∞), n ≥ 3, be a function such that lim t−2l φ(tl x, · · · , tl x) = 0

l→∞

for all x ∈ X , where t 6= 1 is a real number. Suppose that fj : X → X , j = 1, 2, 3, are mappings satisfying (3.1) and (2.3). Then the mappings fj : X → X are (fr , fs )-double derivations. Proof. Since (3.1) holds for the mappings fj : X → X , Lemma 3.1 asserts that the mappings fj are R-linear. So it follows from (2.3) and the assumption on φ that

p

f (xy) − f (x)y − xf (y) − f (x)f (y) − f (x)f (y)

j

j j r s s r




1 = lim 2lp fj t2l xy − fj tl x tl y − tl xfj tl y l→∞ t





p l l l l −fr t x fs t y − fs t x fr t y
1

p l l = lim 2lp Dfj ,fk ,fi t x, t y l→∞ t  p 1 l l ≤ lim 2l φ t (x + y), · · · , t (x + y) = 0p l→∞ t for all x, y ∈ X , which implies that the mappings fj : X → X are (fr , fs )-double derivations, as desired. 

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Corollary 3.3. Let δ be a nonnegative real number and q1 , · · · , qn be positive real numbers such that q1 , · · · , qn > 2 or q1 , · · · , qn < 2. Suppose that fj : X → X , j = 1, 2, 3, are mappings satisfying (3.1) and the inequalities  n 

o max Dfj ,fr ,fs (x, y) ≤ δ kx + ykq1 + · · · + kx + ykqn j,r,s

for all x, y ∈ X . Then the mappings fj : X → X are (fr , fs )-double derivations. Proof. The proof follows from Theorem 3.2 by taking φ(x1 , · · · , xn ) := δ (kx1 kq1 + · · · + kxn kqn ) with t > 1 for the case q1 , · · · , qn < 2 and with t < 1 for the case q1 , · · · , qn > 2.  Corollary 3.4. Let δ be a nonnegative real number and q1 , · · · , qn be positive real numbers such that q1 + · · · + qn 6= 2. Suppose that fj : X → X , j = 1, 2, 3, are mappings satisfying (3.1) and the inequality n

o max Dfj ,fr ,fs (x, y) ≤ δkx + ykq1 +···+qn j,r,s

for all x, y ∈ X . Then the mappings fj : X → X are (fr , fs )-double derivations. Proof. The proof follows from Theorem 3.2 by taking φ(x1 , · · · , xn ) := δ (kx1 kq1 · · · kxp kqn ) with t > 1 for the case q1 + · · · + qn < 2 and with t < 1 for the case q1 + · · · + qn > 2.  The obtained results in this section can be simpler. Indeed, one can set q1 = · · · = qn = q in two last corollaries and get the better statements. Acknowledgments S. Y. Jang was supported by Basic Science Research Program through the National Research Foundation of Korea funded by the Ministry of Education, Science and Technology (NRF-2013007226) and has written during visiting the Research Institute of Mathematics, Seoul Natinal Univerity. C. Park was supported by Basic Science Research Program through the National Research Foundation of Korea funded by the Ministry of Education, Science and Technology (NRF-2012R1A1A2004299). References [1] J. M. Almira and U. Luther, Inverse closedness of opproximation algebras, J. Math. Anal. Appl. 314 (2006), 30–44. [2] M. Amyari, C. Park and M.S. Moslehian, Nearly ternary derivations, Taiwanese J. Math. 11 (2007), 1417–1424. [3] T. Aoki, On the stability of the linear transformation in Banach spaces, J. Math. Soc. Japan 2 (1950), 64–66. [4] H. Cao, J. Lv and J. M. Rassias, Superstability for generalized module left derivations and generalized module derivations on Banach module (II), J. Inequal. Pure Appl. Math. 10 (2009), 17 pages. [5] C.Y. Chou and J.-H. Tzeng, On approximate isomorphisms between Banach ∗-algebras or C ∗ algebras, Taiwanese J. Math. 10 (2006), 219–231. [6] P. Czerwik, Functional Equations and Inequalities in Several Variables, Word Scientific Publishing Company, New Jersey, Hong Kong, Singapore and London, 2002.

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[7] A. Ebadian, N. Ghobadipour and H. Baghban, Stability of bi-θ-derivations on JB ∗ -triples, Int. J. Geom. Methods Mod. Phys. 9 (2012), No. 7, Art. ID 1250051, 12 pages. [8] A. Ebadian, I. Nikoufar and M. Eshaghi Gordji, Nearly (θ1 , θ2 , θ3 , φ)-derivations on C ∗ -modules, Int. J. Geom. Methods Mod. Phys. 9 (2012), No. 3, Art. ID 1250019, 12 pages. [9] M. Eshaghi Gordji, A. Fazeli and C. Park, 3-Lie multipliers on Banach 3-Lie algebras, Int. J. Geom. Methods Mod. Phys. 9 (2012), No. 7, Art. ID 1250052, 15 pages. [10] M. Eshaghi Gordji and N. Ghobadipour, Stability of (α, β, γ)-derivations on Lie C ∗ -algebras, Int. J. Geom. Methods Mod. Phys. 7 (2010), 1097–1102. [11] M. Eshaghi Gordji, G. Kim, J. Lee and C. Park, Generalized ternary bi-derivations on ternary Banach algebras: a fixed point approach, J. Comput. Anal. Appl. 15 (2013), 45–54. [12] M. Eshaghi Gordji, G. Kim, J. Lee and C. Park, Nearly generalized derivations on non-Archimdean Banach algebras: a fixed point approach, J. Comput. Anal. Appl. 15 (2013), 308–315. ˘ [13] P. Gavrut ¸a, A generalization of the Hyers-Ulam-Rassias stability of approximately additive mappings, J. Math. Anal. Appl. 184 (1994), 431–436. [14] D. H. Hyers, On the stability of the linear functional equation, Proc. Natl Acad. Sci. U.S.A. 27 (1941), 222–224. [15] D. H. Hyers, G. Isac and Th. M. Rassias, Stability of Functional Equations in Several Variables, Birkh¨ auser, Basel, 1998. [16] K. Jun and D. Park, Almost derivations on the Banach algebra C n [0, 1], Bull. Korean Math. Soc. 33 (1996), 359–366. [17] H. Khodaei and Th. M. Rassias, Appriximately generalized additive functions in several variables, Int. J. Nonlinear Anal. Appl. 1 (2010), 22–41. [18] M. Kim, Y. Kim, G. A. Anastassiou and C. Park, An additive functional inequality in matrix normed modules over a C ∗ -algebra, J. Comput. Anal. Appl. 17 (2014), 329–335. [19] M. Kim, S. Lee, G. A. Anastassiou and C. Park, Functional equations in matrix normed modules, J. Comput. Anal. Appl. 17 (2014), 336–342. [20] J. Lee, S. Lee and C. Park, Fixed points and stability of the Cauchy-Jensen functional equation in fuzzy Banach algebras, J. Comput. Anal. Appl. 15 (2013), 692–698. [21] J. Lee, C. Park, Y. Cho and D. Shin, Orthogonal stability of a cubic-quartic functional equation in non-Archimedean spaces, J. Comput. Anal. Appl. 15 (2013), 572–583. [22] L. Li, G. Lu, C. Park and D. Shin, Additive functional inequalities in generalized quasi-Banach spaces, J. Comput. Anal. Appl. 15 (2013), 1165–1175. [23] G. Lu, Y. Jiang and C. Park, Additive functional equation in Fr´echet spaces, J. Comput. Anal. Appl. 15 (2013), 369–373. [24] M. Mirzavaziri and E. Omidvar Tehrani, δ-double derivations on C ∗ -algebras, Bull. Iranian Math. Soc. 35 (2009), 147–154. [25] T. Miura, G. Hirasawa and S.-E. Takahasi, A perturbation of ring derivations on Banach algebras, J. Math. Anal. Appl. 319 (2006), 522–530. [26] M. S. Moslehian, Ternary derivations, stability and physical aspects, Acta Appl. Math. 100 (2008), 187–199. [27] A. Najati and C. Park, On the stability of an n-dimensional functional equation originating from quadratic forms, Taiwanese J. Math. 12 (2008), 1609–1624.

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SUPERSTABILITY OF (fr , fs )-DOUBLE DERIVATIONS IN QUASI-BANACH ALGEBRAS

[28] A. Najati, C. Park and J. Lee, Homomorphisms and derivations in C ∗ -ternary algebras, Abs. Appl. Anal. 2009, Art. ID 612392, 16 pages (2009). [29] C. Park, Sh. Ghaffary Ghaleh, K. Ghasemi, N -Jordan ∗-homomorphisms in C ∗ -algebras, Taiwanese J. Math.16 (2012), 1803–1814. [30] C. Park, K. Ghasemi, S. G. Ghaleh, S. Jang, Approximate n-Jordan ∗-homomorphisms in C ∗ algebras, J. Comput. Anal. Appl. 15 (2013), 365–368. [31] 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. [32] Th. M. Rassias, On the stability of the linear mapping in Banach spaces, Proc. Amer. Math. Soc. 72 (1978), 297–300. [33] Th. M. Rassias, On the stability of functional equations and a problem of Ulam, Acta Appl. Math. 62 (2000), 23–130. ˘ [34] P. Semrl, The functional equation of multiplicative derivation is superstable on standard operator algebras, Integral Equations and Operator Theory 18 (1994), 118–122. [35] D. Shin, C. Park and Sh. Farhadabadi, On the superstability of ternary Jordan C ∗ -homomorphisms, J. Comput. Anal. Appl. 16 (2014), 964–973. [36] 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. [37] S. M. Ulam, Problems in Modern Mathematics, science ed, Wiley, New York, 1964, Chapter VI. [38] T. Xu and Z. Yang, Direct and fixed point approaches to the stability of an AQ-functional equation in non-Archimedean normed spaces, J. Comput. Anal. Appl. 17 (2014), 697–706. Sun Young Jang Department of Mathematics, University of Ulsan, Ulsan 680-749, Korea E-mail address: [email protected] Choonkil Park Department of Mathematics, Research Institute for Natural Sciences, Hanyang University, Seoul 133-791, Korea E-mail address: [email protected] Pegah Efteghar Department of Mathematics, Urmia University, Urmia, Iran E-mail address: P [email protected] Sharokh Farhadabadi Department of Mathematics, Urmia University, Urmia, Iran E-mail address: Shahrokh [email protected]

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THE FIXED POINT METHOD FOR PERTURBATION OF BIHOMOMORPHISMS AND BIDERIVATIONS IN NORMED 3-LIE SYSTEMS: REVISITED CHOONKIL PARK, JUNG RYE LEE, EON WHA SHIM, AND DONG YUN SHIN∗ Abstract. Shokri et al. [11] proved the Hyers-Ulam stability of bihomomorphisms and biderivations on normed 3-Lie systems by using the fixed point method. Under the conditions in the main theorems of [11, Section 2], we can show that the related mappings must be zero. In this paper, we correct the statements of the results in [11, Section 2], and prove the corrected theorems.

1. Introduction and preliminaries The stability problems of functional equations and inequalities has been studied in many mathmaticians (see [3, 4, 5, 6, 7, 9, 10, 12, 13, 14, 15]). Shokri et al. [11] defined bihomomorphisms and biderivations. Definition 1.1. ([11]) Let A and B be normed Lie triple systems. A C-bilinear mapping H : A × A → B is called a bihomomorphism if it satisfies H([x, y, z], w) = [H(x, w), H(y, w), H(z, w)], H(x, [y, z, w]) = [H(x, y), H(x, z), H(x, w)] for all x, y, z, w ∈ A. Note that if we replace w by 2w in the first equality of the definition of bihomomorphism then 2H([x, y, z], w) = 8[H(x, w), H(y, w), H(z, w)] and so H([x, y, z], w) = 0 for all x, y, z, w ∈ A. Similarly, one can show that H(x, [y, z, w]) = 0 for all x, y, z, w ∈ A. Thus we correct the definition of bihomomorphism as follows. Definition 1.2. Let A and B be normed Lie triple systems. A C-bilinear mapping H : A × A → B is called a bihomomorphism if it satisfies H([x, y, z], w3 ) = [H(x, w), H(y, w∗ ), H(z, w)], H(x3 , [y, z, w]) = [H(x, y), H(x∗ , z), H(x, w)] for all x, y, z, w ∈ A. Definition 1.3. ([11]) Let A and B be normed Lie triple systems. A C-bilinear mapping δ : A × A → A is called a biderivation if it satisfies δ([x, y, z], w) = [δ(x, w), y, z] + [x, δ(y, w), z] + [x, y, δ(z, w)], δ(x, [y, z, w], w) = [δ(x, y), z, w] + [y, δ(x, z), w] + [y, z, δ(x, w)] 2010 Mathematics Subject Classification. Primary 17A40, 39B52, 47H10, 39B82, 16W25. Key words and phrases. Hyers-Ulam stability; bi-additive mapping; fixed point; Lie triple system; bihomomorphism; biderivation. ∗ Corresponding author.

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J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 18, NO.6, 2015, COPYRIGHT 2015 EUDOXUS PRESS, LLC

C. PARK, J. LEE, E.W. SHIM, AND D. SHIN

for all x, y, z, w ∈ A. The w-variable of the left side in the first equality is C-linear and the x-variable of the left side in the second equality is C-linear. But the w-variable of the right side in the first equality is not C-linear and the x-variable of the right side in the second equality is not C-linear. Thus we correct the definition of biderivation as follows. Definition 1.4. Let A and B be normed Lie triple systems. A C-bilinear mapping δ : A × A → A is called a biderivation if it satisfies δ([x, y, z], w) = [δ(x, w), y, z] + [x, δ(y, w∗ ), z] + [x, y, δ(z, w)], δ(x, [y, z, w], w) = [δ(x, y), z, w] + [y, δ(x∗ , z), w] + [y, z, δ(x, w)] for all x, y, z, w ∈ A. All the mappings T and δ, given in [11, Section 2], satisfy the bi-additive functional equation (1.1) in [11]. Letting x = z = 0 in (1.1), we get f (y, −w) = f (y, w) for all y, w. Thus f is not bi-additive. So the results of [11, Section 2] are meaningless. In this paper, we will replace the equation (1.1), given in [11], by f (x + y, z + w) + f (x + y, z − w) = 2f (x, z) + 2f (y, z).

(1)

Moreover, we correct the statements of the results in [11, Section 2], and prove the corrected theorems. Let X be a set. A function d : X × X → [0, ∞] is called a generalized metric on X if d satisfies (1) d(x, y) = 0 if and only if x = y; (2) d(x, y) = d(y, x) for all x, y ∈ X; (3) d(x, z) ≤ d(x, y) + d(y, z) for all x, y, z ∈ X. We recall a fundamental result in fixed point theory. Theorem 1.5. [2] Let (X, d) be a complete generalized metric space and let J : X → X be a strictly contractive mapping with Lipschitz constant 0 < L < 1. Then for each given element x ∈ X, either d(J n x, J n+1 x) = ∞ for all nonnegative integers n or there exists a positive integer n0 such that (1) d(J n x, J n+1 x) < ∞, ∀n ≥ n0 ; n (2) the sequence {J x} converges to a fixed point y ∗ of J; (3) y ∗ is the unique fixed point of J in the set Y = {y ∈ X | d(J n0 x, y) < ∞}; 1 (4) d(y, y ∗ ) ≤ 1−L d(y, Jy) for all y ∈ Y . Throughout this paper, assume that A is a normed Lie triple system and B is a Banach Lie triple system. 2. Hyers-Ulam stability of bihomomorphisms in Banach Lie triple systems For a given mapping f : A × A → B, we define Dλ,µ f (x, y, z, w) = f (λx + λy, µz + µw) + f (λx + λy, µz − µw) − 2λµf (x, z) − 2λµf (y, z) for all x, y, z, w ∈ A and all λ, µ ∈ T1 := {ν ∈ C : |ν| = 1}. From now on, assume that f (0, z) = f (x, 0) = 0 for all x, z ∈ A. We need the following lemmas to obtain the main results.

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J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 18, NO.6, 2015, COPYRIGHT 2015 EUDOXUS PRESS, LLC

BIHOMOMORPHISMS AND BIDERIVATIONS IN NORMED 3-LIE SYSTEMS

Lemma 2.1. ([1]) Let f : A × A → B be a bi-additive mapping such that f (λx, µy) = λµf (x, y) for all x, y ∈ A and all λ, µ ∈T1 . Then the mapping f : A × A → B is C-bilinear. Lemma 2.2. Let f : A × A → B be a mapping satisfying Dλ,µ f (x, y, z, w) = 0 for all x, y, z, w ∈ A and all λ, µ ∈T1 . Then the mapping f : A × A → B is C-bilinear. Proof. Letting λ = µ = 1 in Dλ,µ f (x, y, z, w) = 0, we get (1). Letting y = 0 in (1), get f (x, z + w) + f (x, z − w) = 2f (x, z) for all x, z, w ∈ A. Letting w = 0 in (1), get 2f (x + y, z) = 2f (x, z) + 2f (y, z) for all x, y, z ∈ A. So f is bi-additive. Letting y = w = 0 in Dλ,µ f (x, y, z, w) = 0, we get 2f (λx, µz) = 2λµf (x, z) for x, z ∈ A and all λ, µ ∈ T1 . By Lemma 2.1, the mapping f : A×A → B is C-bilinear.

we we all 

Theorem 2.3. Let f : A × A → B be a mapping for which there exists a function ϕ : A4 → [0, ∞) such that kDλ,µ f (x, y, z, w)k ≤ ϕ(x, y, z, w),

(2)

kf ([x, y, z], w3 ) − [f (x, w), f (y, w∗ ), f (z, w)]k +kf (x3 , [y, z, w]) − [f (x, y), f (x∗ , z), f (x, w)]k ≤ ϕ(x, y, z, w)

(3)

for all λ, µ ∈ T
1 and all x, y, z, w ∈ A. If there exists an L < 1 such that ϕ(x, y, z, w) ≤ 4Lϕ x2 , y2 , z2 , w2 for all x, y, z, w ∈ A, then there exists a unique bihomomorphism H : A × A → B such that 1 ϕ(x, x, z, z) (4) kf (x, z) − H(x, z)k ≤ 4 − 4L for all x, z ∈ A. Proof. Letting λ = µ = 1, y = x and w = z in (2), we get kf (2x, 2z) − 4f (x, z)k ≤ ϕ(x, x, z, z)

(5)



1

f (x, z) − f (2x, 2z) ≤ 1 ϕ(x, x, z, z)

4 4

(6)

and so

for all x, z ∈ A. Consider the set S := {h : A × A → B} and introduce the generalized metric on S: d(g, h) = inf {µ ∈ R+ : kg(x, z) − h(x, z)k ≤ µϕ (x, x, z, z) , ∀x, z ∈ A} , where, as usual, inf φ = +∞. It is easy to show that (S, d) is complete (see [8]). Now we consider the linear mapping J : S → S such that 1 Jg(x, z) := g (2x, 2z) 4 for all x, z ∈ A. Let g, h ∈ S be given such that d(g, h) = ε. Then kg(x, z) − h(x, z)k ≤ εϕ (x, x, z, z)

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C. PARK, J. LEE, E.W. SHIM, AND D. SHIN

for all x ∈ A. Hence

1

1

kJg(x, z) − Jh(x, z)k = g (2x, 2z) − h (2x, 2z)

≤ εLϕ (x, x, z, z) 4 4 for all x, z ∈ A. So d(g, h) = ε implies that d(Jg, Jh) ≤ εL. This means that d(Jg, Jh) ≤ Ld(g, h) for all g, h ∈ S. It follows from (6) that d(f, Jf ) ≤ 41 . By Theorem 1.5, there exists a mapping H : A → B satisfying the following: (1) H is a fixed point of J, i.e., H (2x, 2z) = 4H(x, z) for all x, z ∈ A. The mapping H is a unique fixed point of J in the set M = {g ∈ S : d(h, g) < ∞}.

(7)

This implies that H is a unique mapping satisfying (7) such that there exists a µ ∈ (0, ∞) satisfying kf (x, z) − H(x, z)k ≤ µϕ (x, x, z, z) for all x, z ∈ A; (2) d(J n f, H) → 0 as n → ∞. This implies the equality 1 f (2n x, 2n z) = H(x, z) n→∞ 4n lim

for all x, z ∈ A; (3) d(f, H) ≤

1 d(f, Jf ), 1−α

(8)

which implies the inequality

1 . 4 − 4L This implies that the inequality (4) holds true. It follows from (2) and (8) that

1

n n n n

kDλ,µ H(x, y, z, w)k =

4n Dλ,µ f (2 x, 2 y, 2 z, 2 w) d(f, H) ≤

1 ϕ(2n x, 2n y, 2n z, 2n w), 4n which tends to zero as n → ∞ for all λ, µ ∈ T1 and all x, y, z, w ∈ A. By Lemma 2.2, the mapping H : A × A → B is C-bilinear. Now let T : A × A → B be another bi-additive mapping satisfying (4). Then 1 kH(x, z) − T (x, z)k = n kH(2n x, 2n z) − T (2n , 2n z)k 4 1 n n ≤ n (kH(2 x, 2 z) − f (2n , 2n z)k + kf (2n x, 2n z) − T (2n , 2n z)k) 4 2 ≤ n ϕ(2n x, 2n x, 2n z, 2n z), 4 which tends to zero as n → ∞. This proves the uniqueness of H. ≤

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J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 18, NO.6, 2015, COPYRIGHT 2015 EUDOXUS PRESS, LLC

BIHOMOMORPHISMS AND BIDERIVATIONS IN NORMED 3-LIE SYSTEMS

It follows from (3) that kH([x, y, z], w3 ) − [H(x, w), H(y, w∗ ), H(z, w)]k + kH(x3 , [y, z, w]) − [H(x, y), H(x∗ , z), H(x, w)]k 1 = lim n kf ([2n x, 2n y, 2n z], 8n w3 ) n→∞ 64 − [f (2n x, 2n w), f (2n y, 2n w∗ ), f (2n z, 2n w)]k + kf (8n x3 , [2n y, 2n z, 2n w])
− [f (2n x, 2n y), f (2n x∗ , 2n z), f (2n x, 2n w)]k 1 1 ≤ lim n ϕ(2n , 2n y, 2n z, 2n w) ≤ lim n ϕ(2n , 2n y, 2n z, 2n w) = 0 n→∞ 64 n→∞ 4 for all x, y, z, w ∈ A. So H([x, y, z], w3 ) = [H(x, w), H(y, w∗ ), H(z, w)] and

H(x3 , [y, z, w]) = [H(x, y), H(x∗ , z), H(x, w)]

for all x, y, z, w ∈ A. Therefore, the mapping H is a unique bi-homomorphism satisfying (4).



Corollary 2.4. Let p and θ be positive real numbers with p < 2, and let f : A×A → B be a mapping such that kDλ,µ f (x, y, z, w)k ≤ θ(kxkp + kykp + kzkp + kwkp ), (9) kf ([x, y, z], w3 ) − [f (x, w), f (y, w∗ ), f (z, w)]k +kf (x3 , [y, z, w]) − [f (x, y), f (x∗ , z), f (x, w)]k ≤ θ(kxkp + kykp + kzkp + kwkp )

(10)

for all λ, µ ∈ T1 and all x, y, z, w ∈ A. Then there exists a unique bihomomorphism H : A × A → B such that 2θ (kxkp + kykp ) kf (x, y) − H(x, y)k ≤ 4 − 2p for all x, y ∈ A. Proof. Defining ϕ(x, y, z, w) := θ(kxkp + kykp + kzkp + kwkp ) and letting L = 2p−2 in Theorem 2.3, we obtain the desired result.  Similarly, one can obtain the following. Theorem 2.5. Let f : A × A → B be a mapping for which there exists a function ϕ : A4 → [0, ∞) satisfying (2) and (3). If there exists an L < 1 such that ϕ(x, y, z, w) ≤ L ϕ(2x, 2y, 2z, 2w) for all x, y, z, w ∈ A, then there exists a unique bihomomorphism 64 H : A × A → B such that L kf (x, z) − H(x, z)k ≤ ϕ(x, x, z, z) (11) 64 − 64L for all x, z ∈ A.

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C. PARK, J. LEE, E.W. SHIM, AND D. SHIN

Note that

L ϕ(2x, 2y, 2z, 2w) 64

≤ L4 ϕ(2x, 2y, 2z, 2w) for all x, y, zw ∈ A.

Proof. Let (S, d) be the generalized metric space defined in the proof of Theorem 2.3. Now we consider the linear mapping J : S → S such that x z  Jg(x, z) := 4g , 2 2 for all x, z ∈ A. It follows from (5) that

 x z  x x z z  L L

, ≤ϕ , , , ≤ ϕ (x, x, z, z) ≤ ϕ (x, x, z, z)

f (x, z) − 4f 2 2 2 2 2 2 64 4 L for all x, z ∈ A. Thus d(f, Jf ) ≤ 4 . So L . 4 − 4L The rest of the proof is similar to the proof of Theorem 2.3. d(f, H) ≤



Corollary 2.6. Let p and θ be positive real numbers with p > 6, and let f : A×A → B be a mapping satisfying (9) and (10). Then there exists a unique bihomomorphism H : A × A → B such that 2θ kf (x, y) − H(x, y)k ≤ p (kxkp + kykp ) 2 −4 for all x, y ∈ A. Proof. Defining ϕ(x, y, z, w) := θ(kxkp + kykp + kzkp + kwkp ) and letting L = 22−p in Theorem 2.5, we obtain the desired result.  3. Hyers-Ulam stability of biderivations on Banach Lie triple systems In this section, we prove the Hyers-Ulam stability of biderivations on Banach Lie triple systems. Theorem 3.1. Let f : A × A → A be a mapping such that kDλ,µ f (x, y, z, w)k ≤ ϕ(x, y, z, w),

(12)

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

(13)

for all λ, µ ∈ T1 and all x, y,
z, w ∈ A. Assume that there exists an L < 1 such that ϕ(x, y, z, w) ≤ 4Lϕ x2 , y2 , z2 , w2 for all x, y, z, w ∈ A. If the mapping f : A × A → A satisfies 1 1 1 lim n f (2n x, 2n z) = lim n f (8n x, 2n z) = lim n f (2n x, 8n z) (14) n→∞ 4 n→∞ 16 n→∞ 16 for all x, z ∈ A, then there exists a unique biderivation δ : A × A → A such that 1 kf (x, z) − δ(x, z)k ≤ ϕ(x, x, z, z) (15) 4 − 4L for all x, z ∈ A.

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J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 18, NO.6, 2015, COPYRIGHT 2015 EUDOXUS PRESS, LLC

BIHOMOMORPHISMS AND BIDERIVATIONS IN NORMED 3-LIE SYSTEMS

Proof. By the same reasoning as in the proof of Theorem 2.3, we get a unique C-bilinear mapping δ : A × A → A given by δ(x, z) := limn→∞ 41n f (2n x, 2n z) satisfying (15). It follows from (13) and (14) that kδ([x, y, z], w) − [δ(x, w), y, z] − [x, δ(y, w∗ ), z] − [x, y, δ(z, w)]k + kδ(x, [y, z, w]) − [δ(x, y), z, w] − [y, δ(x∗ , z), w] − [y, z, δ(x, w)]k 1 = lim n (kf (8n [x, y, z], 2n w) − [f (2n x, 2n w), 2n y, 2n z] n→∞ 16 − [2n x, f (2n y, 2n w∗ ), 2n z] − [2n x, 2n y, f (2n z, 2n w)]k + kf (2n x, 8n [y, z, w]) − [f (2n x, 2n y), 2n z, 2n w] − [2n y, f (2n x∗ , 2n z), 2n w] − [2n y, 2n z, f (2n x, 2n w)]k) 1 1 ≤ lim n ϕ(2n x, 2n y, 2n z, 2n w) ≤ lim n ϕ(2n x, 2n y, 2n z, 2n w) = 0 n→∞ 16 n→∞ 4 for all x, y, z, w ∈ A. So δ([x, y, z], w) = [δ(x, w), y, z] + [x, δ(y, w∗ ), z] + [x, y, δ(z, w)] and δ(x, [y, z, w]) = [δ(x, y), z, w] + [y, δ(x∗ , z), w] + [y, z, δ(x, w)] for all x, y, z, w ∈ A. Therefore, the mapping δ : A×A → A is a biderivation satisfying (15), as desired.  Corollary 3.2. Let p and θ be positive real numbers with p < 2, and let f : A×A → A be a mapping satisfying (14) and kDλ,µ f (x, y, z, w)k ≤ θ(kxkp + kykp + kzkp + kwkp ), kf ([x, y, z], w) − [f (x, w), y, z] − [x, f (y, w∗ ), z] − [x, y, f (z, w)]k +kf (x, [y, z, w]) − [f (x, y), z, w] − [y, f (x∗ , z), w] − [y, z, f (x, w)]k ≤ θ(kxkp + kykp + kzkp + kwkp )

(16) (17)

for all λ, µ ∈ T1 and all x, y, z, w ∈ A. Then there exists a unique biderivation δ : A × A → A such that 2θ kf (x, y) − δ(x, y)k ≤ (kxkp + kykp ) p 4−2 for all x, y ∈ A. Proof. Defining ϕ(x, y, z, w) := θ(kxkp + kykp + kzkp + kwkp ) and letting L = 2p−2 in Theorem 3.1, we obtain the desired result.  Theorem 3.3. Let f : A × A → A be a mapping satisfying (12) and (13). Assume that L there exists an L < 1 such that ϕ(x, y, z, w) ≤ 16 ϕ(2x, 2y, 2z, 2w) for all x, y, z, w ∈ A. If the mapping f : A × A → A satisfies x z x z x z lim 4n f n , n = lim 16n f n , n = lim 16n f n , n (18) n→∞ n→∞ n→∞ 2 2 8 2 2 8 for all x, z ∈ A, then there exists a unique biderivation δ : A × A → A such that L kf (x, z) − δ(x, z)k ≤ ϕ(x, x, z, z) (19) 4 − 4L

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C. PARK, J. LEE, E.W. SHIM, AND D. SHIN

for all x, z ∈ A. Proof. By the same reasoning as in the proof of Theorem 2.3, we
get a unique C-bilinear mapping δ : A × A → A given by δ(x, z) := limn→∞ 4n f 2xn , 2zn satisfying (19). It follows from (13) and (18) that kδ([x, y, z], w) − [δ(x, w), y, z] − [x, δ(y, w∗ ), z] − [x, y, δ(z, w)]k + kδ(x, [y, z, w]) − [δ(x, y), z, w] − [y, δ(x∗ , z), w] − [y, z, δ(x, w)]k    h 

[x, y, z] w x w y z i n = lim 16 f − f , , , , n→∞ 8n 2n 2n 2n 2n 2n    h   z w i

x z x y y w∗

− n,f , − , , , f , 2 2n 2n 2n 2n 2n 2n 2n

  h 

x y  z wi x [y, z, w]

− f , , − + f , , 2n 8n 2n 2n 2n 2n   ∗    h  x w i

y x z w y z ,f , , n − n, n,f n, n

2n 2n 2n 2 2 2 2 2 x y z w ≤ lim 16n ϕ n , n , n , n = 0 n→∞ 2 2 2 2 for all x, y, z, w ∈ A. So δ([x, y, z], w) = [δ(x, w), y, z] + [x, δ(y, w∗ ), z] + [x, y, δ(z, w)] and δ(x, [y, z, w]) = [δ(x, y), z, w] + [y, δ(x∗ , z), w] + [y, z, δ(x, w)] for all x, y, z, w ∈ A. Therefore, the mapping δ : A×A → A is a biderivation satisfying (19), as desired.  Corollary 3.4. Let θ and p be positive real numbers with p > 4 and let f : A × A → A be a mapping satisfying (16), (17) and (18). Then there exists a unique biderivation δ : A × A → A such that kf (x, z) − δ(x, z)k ≤

2θ (kxkp + kzkp ) −4

2p

(20)

for all x, z ∈ A. Proof. Defining ϕ(x, y, z, w) := θ(kxkp + kykp + kzkp + kwkp ) and letting L = 22−p in Theorem 3.3, we obtain the desired result.  Acknowledgments C. Park was supported by Basic Science Research Program through the National Research Foundation of Korea funded by the Ministry of Education, Science and Technology (NRF-2012R1A1A2004299), and D. Y. Shin was supported by Basic Science Research Program through the National Research Foundation of Korea funded by the Ministry of Education, Science and Technology (NRF-2010-0021792)

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J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 18, NO.6, 2015, COPYRIGHT 2015 EUDOXUS PRESS, LLC

BIHOMOMORPHISMS AND BIDERIVATIONS IN NORMED 3-LIE SYSTEMS

References [1] J. Bae and W. Park, Approximate bi-homomorphisms and bi-derivations in C ∗ -ternary algebras, Bull. Korean Math. Soc. 47 (2010) 195–209. [2] 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. [3] M. Eshaghi Gordji and A. Bodaghi, On the stability of quadratic double centralizers on Banach algebras, J. Comput. Anal. Appl. 13 (2011), 724–729. [4] M. Eshaghi Gordji, R. Farokhzad Rostami and S.A.R. Hosseinioun, Nearly higher derivations in unital C ∗ -algebras, J. Comput. Anal. Appl. 13 (2011), 734–742. [5] M. Eshaghi Gordji, G. Kim, J. Lee and C. Park, Generalized ternary bi-derivations on ternary Banach algebras: a fixed point approach, J. Comput. Anal. Appl. 15 (2013), 45–54. [6] L. Li, G. Lu, C. Park and D. Shin, Additive functional inequalities in generalized quasi-Banach spaces, J. Comput. Anal. Appl. 15 (2013), 1165–1175. [7] G. Lu, Y. Jiang and C. Park, Additive functional equation in Fr´echet spaces, J. Comput. Anal. Appl. 15 (2013), 369–373. [8] 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. [9] C. Park, K. Ghasemi, S. G. Ghaleh, 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] J. Shokri, A. Ebadian and R. Aghalari, The fixed point method for perturbation of bihomomorphisms and biderivations in normed 3-Lie algebras, Int. J. Geom. Methods Mod. Phys. 10 (2013), No. 6, Art. ID 1350020, 13 pages. [12] S. Shagholi, M. Eshaghi Gordji and M. Bavand Savadkouhi, Stability of ternary quadratic derivations on ternary Banach algebras, J. Comput. Anal. Appl. 13 (2011), 1097–1105. [13] S. Shagholi, M. Eshaghi Gordji and M. Bavand Savadkouhi, Nearly ternary cubic homomorphisms in ternary Fr´echet algebras, J. Comput. Anal. Appl. 13 (2011), 1106–1114. [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. Choonkil Park Department of Mathematics, Research Institute for Natural Sciences, Hanyang University, Seoul 133-791, Korea E-mail address: [email protected] Jung Rye Lee Department of Mathematics, Daejin University, Kyeonggi 487-711, Korea E-mail address: [email protected] Eon Wha Shim Department of Mathematics, Hanyang University, Seoul 133-791, Korea E-mail address: [email protected] Dong Yun Shin Department of Mathematics, University of Seoul, Seoul 130-743, Korea E-mail address: [email protected]

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Dynamics of some Rational Difference Equations H. El-Metwally1,3 , E.M. Elsayed2,3 and H. El-Morshedy4 1 Department of Mathematics, Rabigh College of Science and Art, King Abdulaziz University, P.O. Box 344, Rabigh 21911, Saudi Arabia. 2 King Abdulaziz University, Faculty of Science, Mathematics Department, P. O. Box 80203, Jeddah 21589, Saudi Arabia. 3 Department of Mathematics, Faculty of Science, Mansoura University, Mansoura 35516, Egypt. 4 Department of Mathematics, Damietta Faculty of Science, Mansoura University, New damietta 34517, Egypt. 1

E-mail: [email protected] & [email protected], [email protected]. Abstract The main goal of this paper is to investigate the qualitative behavior of the solutions for the following rational difference equation: α+

k S

a2i xn−2i

i=0

xn+1 = β+

k S

,

n = 0, 1, 2, ...

b2i+1 xn−2i−1

i=0

where α, β, ai , bi ∈ (0, ∞), i = 0, 1, ..., k; with the initial conditions x0 , x−1 , ..., x−2k , x−2k−1 ∈ (0,∞). We determine the equilibrium points of the considered equation and then study their local stability. Also we study the boundedness and the permanence of the solutions. Finally we investigate the global asymptotically stable of the equilibrium points.

Keywords: permanence, global stability, difference equations. Mathematics Subject Classification: 39A10 ––––––––––––––––––––––

1

Introduction

Rational difference equations is an important class of difference equations where they have many applications, for example, the difference equation xn+1 =

a+bxn c+xn ,

n ≥ 0,

1

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has applications in Optics and Mathematical Biology and is known in the literature as the Riccati difference equation. The equation xn+1 =

1+xn xn−1 ,

n ≥ 0,

was discovered by Lyness [12] while he was working on a problem in Number Theory. Also this equation has many applications in geometry (see Leech [10]) and in frieze patterns (see Conway and Coxeter [4]). Also, we believe that the results about rational difference equations are of paramount importance in their on right and offer prototypes towards the development of the basic theory of the global behavior of solutions of nonlinear difference equations of order greater than one, so, there has been a great interest in studying the qualitative properties of rational difference equations by several authors such as Kulenovic and Ladas [9] presented some known results and derived several new ones on the boundedness, the global stability, and the periodicity of solutions of all rational difference equations of the form xn+1 =

α+βxn +γxn−1 A+Bxn +Cxn−1 .

Also, Camouzis and Ladas [2] presented the global character of solutions of the third-order rational difference equation. They presented a summary of the recent work and a large number of open problems and conjectures on the third order rational recursive sequence of the form xn+1 =

α+βxn +γxn−1 +δxn−2 A+Bxn +Cxn−1 +Dxn−2 .

Li and Sun [11] investigated the periodic character, invariant intervals and global stability of all positive solutions of the recursive sequence xn+1 =

pxn +xn−k q+xn−k .

Kocic et al. [8] examined the periodicity and oscillating properties of the positive solutions as well as the global attractivity of the nonnegative equilibrium of the difference equation a+bxn . xn+1 = d+x n−k In [6], the author studied the boundedness, the existence of prime period to solutions and the global attracitvity of solutions of the following recursive sequences l

xn+1

=

0

l

=

1

s

s

s

0

i

l

l

s

1

j

s

s

j i 0 1 0 1 cxn−k xn−k ...xn−k +dxn−r xn−r ...xn−r 0

yn+1

l

l

j i 0 1 0 1 axn−k xn−k ...xn−k +bxn−r xn−r ...xn−r 1

0

i

α0 yn +α1 yn−1 +...+αt yn−t β 0 yn +β 1 yn−1 +...+β t yn−t ,

1

,

j

n ≥ 0.

n ≥ 0,

(1) (2)

Some related results to rational difference equations can be found in [1,3,5,13] and the references therein.

2

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Let I be an interval of real numbers and let F : I k+1 → I be a continuously differentiable function. Consider the difference equation yn+1 = F (yn , yn−1 , ..., yn−k ) , n = 0, 1, ...,

(3)

with y−k , ..., y0 ∈ I. Recall that the point y ∈ I is called an equilibrium point of Eq.(3) if F (y, y, ..., y) = y. That is, yn = y for n ≥ 0, is a solution of Eq.(3), or equivalently, y is a fixed point of F . Let y be an equilibrium point of Eq.(3). Then the linearized equation of Eq.(3) about y is given by wn+1 =

k P

pi wn−i , n = 0, 1, ...,

(4)

i=0

where pi = Eq.(4) is

∂f ∂yn−i (y, ..., y), i

= 0, 1, 2..., k and the characteristic equation of

λ(k+1) − p1 λk − p2 λ(k−1) − ... − pk λ − p(k+1) = 0. Theorem A [8]: Assume that p, q ∈ R and k ∈ {0, 1, 2, ...}. Then |p| + |q| < 1 is a sufficient condition for the asymptotic stability of the difference equation un+1 + pun + qun−k = 0, n = 0, 1, ... . Remark: Theorem A can be easily extended to a general linear equations of the form (5) un+k + p1 un+k−1 + ... + pk un = 0, n = 0, 1, ... where p1 , p2 , ..., pk and k ∈ {1, 2, ...}. Then Eq.(5) is asymptotically stable proPk vided that i=1 |pi | < 1.

Theorem B [7]: Let {yn }∞ n=−k be a solution of Eq.(3), and suppose that there exist constants A ∈ I and B ∈ I such that A ≤ yn ≤ B for all n ≥ −k. Let ∞ 0 be a limit point of the sequence {yn }n=−k . Then the following statements are true: (i) There exists a solution {Ln }∞ n=−∞ of Eq.(3), called a full limiting sequence , such that L = , of {yn }∞ 0 0 and such that for every N ∈ {...,−1,0,1,...} LN n=−k is a limit point of {yn }∞ . n=−k ∞ (ii) For every i0 ≤ −k, there exists a subsequence {yri }∞ i=0 of {yn }n=−k such that LN = lim yri +N for every N ≥ i0 . i→∞

Theorem C [9]: Let [p, q] be an interval of real numbers and assume that g : [p, q]3 → [p, q], 3

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is a continuous function satisfying the following properties : (a) g(x, y, z) is non-decreasing in y and z in [p, q] for each x ∈ [p, q], and is non-increasing in x ∈ [p, q] for each y and z in [p, q]; (b) If (m, M ) ∈ [p, q] × [p, q] is a solution of the system M = g(m, M, M )

and

m = g(M, m, m),

then m = M. Then Eq.(5) has a unique equilibrium x ∈ [p, q] and every solution of Eq.(5) converges to x. In this paper we study the boundedness character and investigate the global stability for the solutions of the following difference equation: [ k+1 2 ]

α+

P

a2i xn−2i

i=0

xn+1 = β+

[ k+1 2 ] P

,

n = 0, 1, 2, ..,

(6)

b2i+1 xn−2i−1

i=0

where α, β ∈ [0, ∞), ai , bi ∈ (0, ∞) for i = 0, 1, ..., k; and k ∈ {10, 1, 2, ...} with the initial conditions x0 , x−1 , ..., x−2k , x−2k−1 ∈ (0,∞).

2

Local Stability of Eq.(6)

In this section we discuss the local stability of the equilibrium points of Eq.(6). [ k+1 [ k+1 2 ] 2 ] P P a2i and B = b2i+1 , then the following statements are true: Let A = i=0

i=0

(i) At α = 0 and β = 0, Eq.(6) has the equilibrium point x = 0 and the unique A . positive equilibrium point x = B (ii) At α = 0 and β < A, Eq.(6) has the equilibrium point x = 0 and the positive equilibrium point x = A−β B . (iii) At α = 0 and β ≥ A, Eq.(6) has the unique equilibrium point x = 0. (iv) At √α 6= 0 and β = 0, Eq.(6) has the unique positive equilibrium point 2 x = A+ A2B+4αB . (v) At α √ 6= 0 and β 6= 0, Eq.(6) has the unique positive equilibrium point A−B+

(A−B)2 +4αB

x= . 2B The following theorem deals with the local stability of the positive equilibrium point of Eq.(6). √ A−B+ (A−B)2 +4αB of Eq.(6) is locally Theorem 1 The equilibrium point x = 2B stable if A < β. Proof. √ A−B+

The linearized equation of Eq.(6) about the equilibrium point x =

(A−B)2 +4αB 2B

zn+1 =

k P

i=0

is given by

∂f (x,x,...,x) zn−i ∂xn−i

=

k P

i=0

a2i β+Bx zn−2i

−

k P

i=0

b2i+1 (α+Ax) (β+Bx)2 zn−2i−1

4

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for n = 0, 1, ... and the associated characteristic equation about x is k P

F (λ) = λ2k+2 −

2(k−i)+1 a2i β+Bx λ

i=0

+

k P

i=0

b2i+1 (α+Ax) 2(k−i) (β+Bx)2 λ

= 0.

Then it follows by Theorem A that x is locally stable if A β+Bx

+

B(α+Ax) (β+Bx)2

A β+Bx

< 1 ⇐⇒

+

Bx β+Bx

< 1 ⇐⇒ A < β.

This ends the proof of the theorem.

3

Boundedness of Solutions

Here we study the permanence of Eq.(6). Theorem 2 Assume that A < β. Then every solution of Eq.(6) is bounded and persists. Proof. Let {xn }∞ n=−2k−1 be a solution of Eq.(6). Then α+

[ k+1 2 ] P

a2i xn−2i

i=0

xn+1 =

≤

[ k+1 2 ]

β+

P

α β

+

1 β

b2i+1 xn−2i−1

i=0

Then lim sup xn ≤ n−→∞

α def β−A =

[ k+1 2 ] P

a2i xn−2i .

i=0

M. Thus xn ≤ M for all n ≥ 1.

Now we wish to find a constant m > 0 such that xn ≥ m for all n ≥ 1. The change of variables xn = y1n , gives Eq.(6) in the form 1

=

yn+1

a0 yn

+

b1 yn−1

+

α+ β+

a2 yn−2 b3 yn−3

+ ... + + ... +

a2k yn−2k b2k+1 yn−2k−1

,

or in the equivalent form yn+1

=

β+ α

b1 yn−1 + yan0

β+M ≤

+ +

k P

b3 yn−3 a2 yn−2

+ ... + + ... +

b2i+1

i=0

α

≤

b2k+1 yn−2k−1 a2k yn−2k

β + MB α

≤

β+

b1 yn−1

+

b3 yn−3

+ ... +

b2k+1 yn−2k−1

α

for all n ≥ 1.

Thus we obtain xn =

1 α ≥ yn β + BM

def

= m for all n ≥ 1.

Therefor we see that m ≤ xn ≤ M

for all n ≥ 1.

Then the proof is so complete. 5

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Theorem 3 Assume that A < β. Then every solution of Eq.(6) is bounded and persists. Proof. Let {xn }∞ n=−2k−1 be a solution of Eq.(6). α }, then it follows from (6) that Set H = max{x−2k−1 , x−2k , ..., x0 , β−A α+

[ k+1 2 ] P i=0

x1 =

a2i x−2i

[ k+1 2 ]

β+

P

i=0

α+H

[ k+1 2 ] P

a2i

i=0

≤

β

≤

α + AH ≤ H. β

b2i+1 x−2i−1

It follows by induction that xn ≤ H for all n ≥ 0. Now we wish to find a constant h > 0 such that xn ≥ h for all n ≥ 1. Again it follows from Eq.(7) that [ k+1 ] 2

α+

x1 =

P

i=0 [ k+1 ] 2

β+

P

i=0

a2i x−2i

α

≥

b2i+1 x−2i−1

≥

[ k+1 ] 2

β+H

P

def α β+BH =

h.

b2i+1

i=0

Then it follows by induction that xn ≥ h for all n ≥ 1.

4

Global Stability of the Equilibrium Points

Theorem 4 Let α = 0 and assume that A < β. Then every nonnegative solution of Eq.(6) converges to the unique equilibrium point of Eq.(6) x = 0. Proof. It follows by Theorem B that there exist solutions {In }∞ n=−∞ and of Eq.(6) with {Sn }∞ n=−∞ I = I0 = lim inf xn ≤ lim sup xn = S0 = S, n→∞

n→∞

where In , Sn ∈ [I,S] , n = 0, − 1,... .

Since x = 0 is the unique nonnegative equilibrium point of Eq.(6), then it suffices to show that S = 0. Suppose for the sake of contradiction that S > 0.Then it follows from Eq.(6) that S=

AS a0 S−1 + a2 S−3 + ... + a2k S−2k−1 , ≤ β + b1 S−2 + b3 S−4 + ... + b2k+1 S−2k−1 β + BI

and so 0 ≤ BSI ≤ (A − β)S < 0,

which is a contradiction. Hence

lim xn = x = 0.

n→∞

which is true by the hypothesis of the theorem and this completes the proof. 6

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Theorem 5 Assume that A < β, then the equilibrium point p A − B + (A − B)2 + 4αB x= 2B of Eq.(6) is global attractor of the solutions of Eq.(6). Proof. It was shown in Theorem 2 that m ≤ xn ≤ M for all n ≥ 1.Then it ∞ follows again by Theorem B that there exist solutions {In }∞ n=−∞ and {Sn }n=−∞ of Eq.(6) with I = I0 = lim inf xn ≤ lim sup xn = S0 = S, n→∞

n→∞

where In , Sn ∈ [I,S] , n = 0, − 1,... . It suffices to show that I ≥ S. Now it follows from Eq.(6) that I=

α + AI α + a0 I−1 + a2 I−3 + ... + a2k I−2k−1 ≥ β + b1 I−2 + b1 I−2 a2 I−3 + ... + a2k I−2k−1 β + BS

and so (β − A)I + BSI ≥ α.

(7)

Similarly, we see from Eq.(6) that S=

α+a0 S−1 +a2 S−3 +...+a2k S−2k−1 β+b1 S−2 +b3 S−4 +...+b2k+1 S−2k−1

≤

α+AS β+BI ,

and so (β − A)S + BSI ≤ α.

(8)

Then we obtain from relations (7) and (8) that (β − A)S + BSI ≤ α ≤ (β − A)I + BSI, thus (β − A)(I − S) ≥ 0 and since (β − A) > 0, then we should have I ≥ S.This completes the proof. We give the following two theorems which is a minor modification of Theorem A.0.2 in [9]. Theorem 6 Let [a, b] be an interval of real numbers and assume that f : [a, b]k+1 → [a, b], is a continuous function satisfying the following properties: © ª (i) f (x0 , x1 , ..., xk ) is non-decreasing in its arguments x2t with t ∈ 0, 1, 2, ..., [ k+1 2 ] for each xr (r 6= 2t) ª in [a, b] and non-increasing in its arguments x2r+1 with © ] for all xt (t 6= r) in [a, b]. r ∈ 0, 1, 2, ..., [ k+1 2 7

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(ii) If (M, m) ∈ [a, b] × [a, b] is a solution of the system M = f (M, m, M, m, ..., M, m, ..., M, m), m = f (m, M, m, M, ..., m, M..., m, M ), implies m = M. Then the difference equation xn+1 = f (xn , xn−1 , ..., xn−k ) has a unique equilibrium x ∈ [a, b] and every solution of the equation converges to x. Proof. Set m0 = a

and

M0 = b,

and for each i = 1, 2, ... set mi = f (mi−1 , Mi−1 , mi−1 , Mi−1 , ..., mi−1 , Mi−1 , mi−1 , ..., mi−1 , Mi−1 ), and Mi = f (Mi−1 , mi , Mi−1 , mi , ..., Mi−1 , mi−1 , Mi−1 , ..., Mi−1 , mi−1 ). Now observe that for each i ≥ 0, a = m0 ≤ m1 ≤ ... ≤ mi ≤ ... ≤ Mi ≤ ... ≤ M1 ≤ M0 = b, and mi ≤ xp ≤ Mi f or p ≥ (k + 1)i + 1. Set m = lim mi i→∞

and M = lim Mi . i→∞

Then M ≥ lim sup xi ≥ lim inf xi ≥ m i→∞

i→∞

and by the continuity of f , M = f (M, m, M, m, ..., M, m, .., M, m) and m = f (m, M, m, M, ..., m, M, ..., m, M ). In view of (ii), m = M = x, from which the result follows. Theorem 7 Let [a, b] be an interval of real numbers and assume that f : [a, b]k+1 → [a, b], is a continuous function satisfying the following properties: © ª (i) f (x0 , x1 , ..., xk ) is non-increasing in its arguments x2t with t ∈ 0, 1, 2, ..., [ k+1 2 ] for each xr (r 6= 2t)ª in [a, b] and non-decreasing in its arguments x2r+1 with © r ∈ 0, 1, 2, ..., [ k+1 2 ] for all xt (t 6= r) in [a, b]. (ii) If (M, m) ∈ [a, b] × [a, b] is a solution of the system M = f (m, M, m, M, ..., m, M, ..., m, M ), m = f (M, m, M, m, ..., M, m..., M, m), 8

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implies m = M. Then the difference equation xn+1 = f (xn , xn−1 , ..., xn−k ) has a unique equilibrium x ∈ [a, b] and every solution of the equation converges to x. Proof. As the proof of Theorem 6 and will be omitted. Theorem 8 Assume that A 6= β. Then every nonnegative solution of Eq.(6) √ A−B+ (A−B)2 +4αB . converges to the unique equilibrium point of Eq.(6) x = 2B Proof. Rewrite Eq. (6) in the following form α+

[ k+1 2 ] P

a2i xn−2i

i=0

xn+1 = f (xn , xn−1 , ..., xn−k ) = β+

[ k+1 2 ] P

,

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

b2i+1 xn−2i−1

i=0

Then the function f satisfies the hypotheses (i) of Theorem 6. Now consider the system M = f (M, m, M, m, ..., M, m) =

α+AM β+Bm ,

m = f (m, M, m, M, ..., m, M ) =

α+Am β+BM .

Thus it is easy to see that m = M. Therefore the function f satisfies the hypotheses (ii) of Theorem √ 6 too. Then it follows by Theorem 6 that the equilibrium

point x = Eq.(6).

A−B+

(A−B)2 +4αB 2B

of Eq.(6) is a global attractor of the solutions of

Theorem 9 Let α = 0 and A < β. Then every nonnegative solution of Eq.(6) converges to the unique equilibrium point of Eq.(6) x = 0. Proof. The proof is similar to the proof of Theorem 8 and will be omitted. Theorem 10 Let α = 0 and β = 0. Then every positive solution of Eq.(6) A . converges to the unique positive equilibrium point of Eq.(6) x = B Proof. The proof is similar to the proofs of Theorem 5 and Theorem 8 and will be omitted. Theorem 11 Let α = 0 and β < A. Then every nonnegative solution of Eq.(6) A−β . converges to the unique positive equilibrium point of Eq.(6) x = B Proof. The proof is similar to the proofs of Theorem 5 and Theorem 8 and will be omitted. Theorem 12 Let α 6= 0 and β = 0. Then every positive solution of√Eq.(6) conA + A2 + 4αB . verges to the unique positive equilibrium point of Eq.(6) x = 2B 9

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Proof. The proof is similar to the proofs of Theorem 5 and Theorem 8 and will be omitted. Remark: It is easy to obtain -by using Theorem B and Theorem 7- similar results for the following difference equation δ+ yn+1 =

[ k+1 2 ] P

p2i+1 yn−2i−1

i=0

γ+

[ k+1 2 ] P

. q2i yn−2i

i=0

Acknowledgements This article was funded by the Deanship of Scientific Research (DSR), King Abdulaziz University, Jeddah. The authors, therefore, acknowledge with thanks DSR technical and financial support.

References [1] M. Aloqeili, Dynamics of a rational difference equation, Appl. Math. Comp., 176 (2) (2006), 768-774. [2] E. Camouzis and G. Ladas, Dynamics of Third-Order Rational Difference Equations with Open Problems and Conjectures, vol. 5 of Advances in Discrete Mathematics and Applications, Chapman & Hall/CRC, Boca Raton, Fla, USA, 2008. [3] C. Cinar, R. Karatas and I. Yalcinkaya, On solutions of the difference xn−3 , Mathematica Bohemica, 132 (3) equation xn+1 = −1 + xn xn−1 xn−2 xn−3 (2007), 257-261. [4] J. H.Conway and H. S. M.Coxeter, Triangulated polygons and frieze patterns, math. Gaz. 57 (400) (1973), 87-94. [5] E. M. Elabbasy and E. M. Elsayed, Dynamics of a Rational Difference Equation, Chinese Annals of Math., Series B, 30 B (2), ( March — 2009), 187—198. [6] H. El-Metwally, Qualitative Proprieties of some Higher Order Difference Equations, Comput. Math. Appl., 58(4) (2009), 686-692. [7] H. El-Metwally, Qualitative Study of Nonlinear Difference Equations: Differential and Difference Equations, LAP Lambert Academic Publishing, 2010. 10

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[8] V. L. Kocic and G. Ladas, Global Behavior of Nonlinear Difference Equations of Higher Order with Applications, Kluwer Academic Publishers, Dordrecht, 1993. [9] M. R. S. Kulenovic and G. Ladas, Dynamics of Second Order Rational Difference Equations with Open Problems and Conjectures, Chapman & Hall / CRC Press, 2001. [10] J. Leech, The rational cuboid revisited, Amer. Math. Monthly, 84 (1977), 518-533. [11] W. Li and H. R. Sun, Dynamics of a rational difference equation, Appl. Math. Comp., 163 (2005), 577—591. [12] R. C. Lyness, Note 1581, Math. Gaz., 26:62 (1942). [13] I. Yalcinkaya, On the global attractivity of positive solutions of a rational difference equation, Selçuk J. Appl. Math., 9 (2) (2008), 3-8.

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Generalized integration operators from Hardy spaces to Zygmund-type spaces ∗†

Li u

Huiying Qu, Yongmin Liu ‡ and Shulei Cheng School of Mathematics and Statistics Jiangsu Normal University Xuzhou 221116, P.R. China Abstract Let H(D) denote the space of all holomorphic functions on the unit disk D of C. Let ϕ be a holomorphic self-map of D, n be a positive integer and g ∈ H(D). In this paper, we investigate the boundedness and compactness of a generalized integration operator Z z (n) Ig,ϕ = f (n) (ϕ(ζ))g(ζ)d(ζ) 0

1

in

from Hardy spaces to the Zygmund-type spaces Zµ .

Introduction

ng m

Let D denote the open unit disk of the complex plane C and H(D) the space of all analytic functions in D. For 0 < r < 1, f ∈ H(D), we set Mp (r, f ) =



1 2π

Z

2π

0

p 1/p iθ , f (re ) dθ

0 < p < ∞,

Yo

M∞ (r, f ) = max f (reiθ ) .

For 0 < p ≤ ∞, the Hardy space

0≤θ≤2π

Hp

consists of those functions f ∈ H(D), for which

kf kp = sup Mp (r, f ) < ∞. 0≤r 1, there exists a constant C(p) such that Z

0

2π

C(p) dθ ≤ , f or every z ∈ D. p |1 − z| (1 − |z|2 )p−1

Lemma 2.2 ([4, 5, 16]) Suppose that 0 < p < ∞, f ∈ H p , then k f kp (1 − |z|2 )1/p+n

,

Li u

(n) f (z) ≤ C

for every z ∈ D and all nonnegative n = 0, 1, 2, · · · .

The following criterion for the compactness is a useful tool and it follows from standard arguments, for example, [3, Proposition 3.11].

in

Lemma 2.3 Assume that n be a nonnegative integer and ϕ be a holomorphic self-map of (n) (n) D, 0 < p < ∞, µ be a weight. Then Ig,ϕ : H p → Zµ is compact if and only if Ig,ϕ : H p → Zµ is bounded and for any bounded sequence {fk } in H p which converges to zero uniformly on (n) compact subsets of D as k → ∞, we have kIg,ϕ fk kZµ → 0 as k → ∞.

ng m

The following lemma was proved in [7] similar to the corresponding lemma in [9]. Lemma 2.4 A closed set K in Zµ,0 is compact if and only if K is bounded and satisfies lim lim µ(z)|f ′′ (z)| = 0.

|z|→1 f ∈K

Yo

Lemma 2.5 Assume that 0 < p < ∞, then for a, b > 0

3

(a + b)p ≤ C (ap + bp ) .

(n)

Boundedness and compactness of Ig,ϕ from H p (0 < p < ∞) spaces to Zygmund-type spaces (n)

In this section, we study the boundedness and compactness of Ig,ϕ : H p → Zµ . Theorem 3.1. Let g ∈ H(D), n be a nonnegative integer and ϕ be a holomorphic self-map (n) of D, 0 < p < ∞, µ be a weight. Then Ig,ϕ : H p → Zµ is bounded if and only if the following conditions are satisfied µ(z)|g′ (z)| sup  (3.1) 1/p+n < ∞, 2 z∈D 1 − |ϕ(z)| 1006

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J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 18, NO.6, 2015, COPYRIGHT 2015 EUDOXUS PRESS, LLC

4

Products of generalized integration operators µ(z)|g(z)ϕ′ (z)| sup  1/p+n+1 < ∞. 2 z∈D 1 − |ϕ(z)|

(3.2)

Proof. Assume that (3.1) and (3.2) hold. Then for every z ∈ D and f ∈ H p , by Lemma 2.2 we have (n) ′′ µ(z) (Ig,ϕ f ) (z) = µ(z) f (n+1) (ϕ(z))ϕ′ (z)g(z) + f (n) (ϕ(z))g′ (z) ≤ µ(z)|g(z)ϕ′ (z)||f (n+1) (ϕ(z))| + µ(z)|g′ (z)||f (n) (ϕ(z))| µ(z)|g′ (z)| µ(z)|g(z)ϕ′ (z)| + Ckf k < ∞, ≤ Ckf kp p (1 − |ϕ(z)|2 )1/p+n+1 (1 − |ϕ(z)|2 )1/p+n

Li u

(3.3)

On the other hand , we have

(n) |(Ig,ϕ f )(0)| = 0

and

(n) ′ |(Ig,ϕ f ) (0)| = |f (n) (ϕ(0))g(0)| ≤ C

(3.4)

|g(0)| kf kp . (1 − |ϕ(0)|2 )1/p+n

(3.5)

(n)

zn n!

∈ H p , we have that

ng m

for all f ∈ H p . For f (z) =

in

Applying conditions (3.3), (3.4) and (3.5), we deduce that the operator Ig,ϕ : H p → Zµ is bounded. (n) Conversely we suppose that Ig,ϕ : H p → Zµ is bounded, that is there exists a constant C such that (n) kIg,ϕ f kZµ ≤ Ckf kp

sup µ(z)|g′ (z)| < ∞.

(3.6)

z∈D

Let f (z) =

z n+1 (n+1)!

∈ H p , we have that

Yo

sup µ(z)|ϕ′ (z)g(z) + ϕ(z)g′ (z)| < ∞.

(3.7)

z∈D

By (3.6), (3.7) and the boundedness of the function ϕ(z), we get sup µ(z)|g(z)ϕ′ (z)| < ∞.

(3.8)

z∈D

For a fixed ω ∈ D, set fω (z) = (1/p + n + 2)

1 − |ϕ(ω)|2 (1 − zϕ(ω))1/p+1

1007

− (1/p + 1)

(1 − |ϕ(ω)|2 )2 (1 − zϕ(ω))1/p+2

.

(3.9)

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5

Products of generalized integration operators

Li u

From Lemma 2.1 and Lemma 2.5 we have 1/p  Z 2π 1 iθ p |fω (re )| kfω kp = sup 0≤r − 12 , z ∈ U,

ζ ∈ U , and h(0, ζ) = 1. If λ, δ ≥ 0, m ∈ N, z ∈ U, ζ ∈ U , f ∈ A∗ζ and satisfies the strong differential subordination " ¡ ¢0 # 0 DRλm+1 f (z, ζ) z (DRλm f (z, ζ))z z 2 DRλm f (z, ζ) δ + 1 DRλm f (z, ζ) −2 z ≺≺ h(z, ζ), (2.5) ¡ ¡ ¢2 + ¢ δ δ DRm+1 f (z, ζ) 2 DRλm f (z, ζ) DRλm+1 f (z, ζ) DRm+1 f (z, ζ) λ

z ∈ U, ζ ∈ U , then z

λ

(

m DRλ f (z,ζ) 2 m+1 DRλ f (z,ζ)

is the best dominant.

)

≺≺ q(z, ζ), z ∈ U, ζ ∈ U , where q(z, ζ) =

m DRλ f (z,ζ)

δ zδ

Rz 0

h(t, ζ)tδ−1 dt is convex and it

, z ∈ U, ζ ∈ U , p ∈ H∗ [1, 1, ζ]. Differentiating with respect to z, we obtain ∙ ¸ 0 0 m f (z,ζ)) (DRm+1 (DRm DRλ DRm f (z,ζ) λ z 0 δ+1 z2 λ f (z,ζ))z z λ f (z,ζ) p (z, ζ) + δ pz (z, ζ) = z δ − 2 DRm+1 f (z,ζ) , z ∈ U, ζ ∈ U , and m f (z,ζ) 2+ δ 2 DRλ f (z,ζ)) (DRm+1 (DRλm+1 f (z,ζ)) λ λ (2.5) becomes p(z, ζ) + zδ p0z (z, ζ) ≺≺ h(z, ζ), z ∈ U, ζ ∈ U . Proof. Let p(z, ζ) = z

(DRλm+1 f (z,ζ))

2

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Using Lemma 1.2, we have p(z, ζ) ≺≺ q(z, ζ), z ∈ U, ζ ∈ U , i.e. z

DRm λ f (z,ζ)

(

≺≺ q(z, ζ) =

2 m+1 DRλ f (z,ζ)

)

δ zδ

z ∈ U, ζ ∈ U , and q is the best dominant.

Rz 0

h(t, ζ)tδ−1 dt,

Theorem 2.6 Let g be a convex function such that g(0, ζ) = 1 and let h be the function h(z, ζ) = g(z, ζ)+ zδ gz0 (z, ζ), z ∈ U, ζ ∈ U . If λ, δ ≥ 0, m ∈ N, f ∈ A∗ζ and the strong differential subordination z2

δ+2 δ

0 (DRλm f (z, ζ))z DRλm f (z, ζ)

z ∈ U, ζ ∈ U , holds, then z 2

3

+

m (DRλ f (z,ζ))0z m f (z,ζ) DRλ

z δ

⎡

00 (DRλm f (z, ζ))z 2 ⎣ DRλm f (z, ζ)

−

Ã

0 (DRλm f (z, ζ))z DRλm f (z, ζ)

!2 ⎤ ⎦ ≺≺ h(z, ζ),

(2.6)

≺≺ g(z, ζ), z ∈ U, ζ ∈ U . This result is sharp.

m (DRλ f (z,ζ))0z m f (z,ζ) . DRλ

We deduce that p ∈ H∗ [0, 1, ζ]. Differentiating with respect to z, we ∙ ¸ ³ m 00 m 0 ´2 f (z,ζ))0z (DRm (DRλ z 3 (DRλ f (z,ζ))z2 λ f (z,ζ))z , z ∈ U, ζ ∈ U . + − obtain p (z, ζ) + zδ p0z (z, ζ) = z 2 δ+2 m m m δ DR f (z,ζ) δ DR f (z,ζ) DR f (z,ζ) Proof. Let p(z, ζ) = z 2

λ

λ

λ

Using the notation in (2.6), the strong differential subordination becomes p(z, ζ) + 1δ zp0z (z, ζ) ≺≺ h(z, ζ) = g(z, ζ) + zδ gz0 (z, ζ), z ∈ U, ζ ∈ U . By using Lemma 1.1, we have p(z, ζ) ≺≺ g(z, ζ), z ∈ U, ζ ∈ U , i.e. z 2 and this result is sharp.

m f (z,ζ))0z (DRλ m f (z,ζ) DRλ

≺≺ g(z, ζ), z ∈ U, ζ ∈ U ,

³ 1+

´

> − 12 , z ∈ U,

⎡ Ã !2 ⎤ 0 00 0 m m m 3 (DR (DR f (z, ζ)) f (z, ζ)) f (z, ζ)) (DR z δ + 2 2 λ λ λ z z z ⎣ ⎦ ≺≺ h(z, ζ), + − z2 δ DRλm f (z, ζ) δ DRλm f (z, ζ) DRλm f (z, ζ)

(2.7)

Theorem 2.7 Let h be a holomorphic function which satisfies the inequality Re

zh00 (z,ζ) z2 h0z (z,ζ)

ζ ∈ U and h(0, ζ) = 1. If λ, δ ≥ 0, m ∈ N, f ∈ A∗ζ and satisfies the strong differential subordination

z ∈ U, ζ ∈ U , then z 2 the best dominant.

0 (DRm λ f (z,ζ))z m f (z,ζ) DRλ

≺≺ q(z, ζ), z ∈ U, ζ ∈ U , where q(z, ζ) =

δ zδ

Rz 0

h(t, ζ)tδ−1 dt is convex and it is

0 (DRm λ f (z,ζ))z m f (z,ζ) , DRλ

z ∈ U, ζ ∈ U , p ∈ H∗ [0, 1, ζ]. Differentiating with respect to z, we obtain ∙ ³ ´2 ¸ m 00 m 0 f (z,ζ))0z (DRm (DRλ z 3 (DRλ f (z,ζ))z2 λ f (z,ζ))z , z ∈ U, ζ ∈ U , and (2.7) becomes + − p (z, ζ) + zδ p0z (z, ζ) = z 2 δ+2 m m m δ DR f (z,ζ) δ DR f (z,ζ) DR f (z,ζ) Proof. Let p(z, ζ) = z 2

λ

λ

λ

p(z) + 1δ zp0z (z, ζ) ≺≺ h(z, ζ), z ∈ U, ζ ∈ U .

Using Lemma 1.2, we have p(z, ζ) ≺≺ q(z, ζ), z ∈ U, ζ ∈ U , i.e. z 2

m (DRλ f (z,ζ))0z DRm λ f (z,ζ)

z ∈ U, ζ ∈ U , and q is the best dominant.

≺≺ q(z, ζ) =

δ zδ

Rz 0

h(t, ζ)tδ−1 dt,

Theorem 2.8 Let g be a convex function such that g(0, ζ) = 1 and let h be the function h(z, ζ) = g(z, ζ)+zgz0 (z, ζ), z ∈ U, ζ ∈ U . If λ ≥ 0, m ∈ N, f ∈ A∗ζ and the strong differential subordination 1− holds, then

m DRλ f (z,ζ) m f (z,ζ) 0 z (DRλ )

z

z ∈ U, ζ ∈ U

(2.8)

≺≺ g(z, ζ), z ∈ U, ζ ∈ U . This result is sharp.

m DRλ f (z,ζ) . m f (z,ζ) 0 z (DRλ )z m m 00 DRλ f (z,ζ)·(DRλ f (z,ζ))z2

Proof. Let p(z, ζ) =

00

DRλm f (z, ζ) · (DRλm f (z, ζ))z2 ≺≺ h(z, ζ), £ ¤2 (DRλm f (z, ζ))0z

We deduce that p ∈ H∗ [1, 1, ζ]. Differentiating with respect to z, we

= p (z, ζ) + zp0z (z, ζ) , z ∈ U, ζ ∈ U . 2 [(DRλm f (z,ζ))0z ] Using the notation in (2.8), the strong differential subordination becomes p(z, ζ) + zp0z (z, ζ) ≺≺ h(z, ζ) = g(z, ζ) + zgz0 (z, ζ), z ∈ U , ζ ∈ U . m DRλ f (z,ζ) By using Lemma 1.1, we have p(z, ζ) ≺≺ g(z, ζ), z ∈ U, ζ ∈ U i.e. ≺≺ g(z, ζ), z ∈ U, ζ ∈ U , m f (z,ζ) 0 z (DRλ )z and this result is sharp.

obtain 1 −

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Theorem 2.9 Let h be a holomorphic function which satisfies the inequality Re

³ 1+

zh00 (z,ζ) z2 h0z (z,ζ)

´

ζ ∈ U , and h(0, ζ) = 1. If λ ≥ 0, m ∈ N, f ∈ A∗ζ and satisfies the strong differential subordination 1− then

m DRλ f (z,ζ) m f (z,ζ) 0 z (DRλ )

z

00

DRλm f (z, ζ) · (DRλm f (z, ζ))z2 ≺≺ h(z, ζ), £ 0 ¤2 (DRλm f (z, ζ))z

≺≺ q(z, ζ), z ∈ U, ζ ∈ U , where q(z, ζ) =

Proof. Let p(z) =

DRm λ f (z,ζ) m f (z,ζ) 0 z (DRλ )

z

m 00 DRm λ f (z,ζ)·(DRλ f (z,ζ))z2

1 z

Rz 0

> − 12 , z ∈ U,

z ∈ U, ζ ∈ U ,

(2.9)

h(t, ζ)dt is convex and it is the best dominant.

, z ∈ U, ζ ∈ U , p ∈ H∗ [0, 1, ζ]. Differentiating with respect to z, we obtain

= p (z, ζ) + zp0z (z, ζ) , z ∈ U, ζ ∈ U , and (2.9) becomes p(z, ζ) + zp0z (z, ζ) ≺≺ h(z, ζ), 0 2 [(DRλm f (z,ζ))z ] z ∈ U, ζ ∈ U . R DRm 1 z λ f (z,ζ) Using Lemma 1.2, we have p(z, ζ) ≺≺ q(z, ζ), z ∈ U, ζ ∈ U i.e. h(t, ζ)dt, 0 ≺≺ q(z, ζ) = z 0 z (DRm f (z,ζ) )z λ z ∈ U, ζ ∈ U , and q is the best dominant. 1−

Corollary 2.10 Let h(z, ζ) = ζ+(2β−ζ)z be a convex function in U × U , where 0 ≤ β < 1. If λ ≥ 0, m ∈ N, 1+z f ∈ A∗ζ and satisfies the strong differential subordination 1− then

DRm λ f (z,ζ) 0 z (DRm λ f (z,ζ))

z

DRλm f (z, ζ) · (DRλm f (z, ζ))00z2 ≺≺ h(z, ζ), £ 0 ¤2 (DRλm f (z, ζ))z

z ∈ U, ζ ∈ U ,

(2.10)

≺≺ q(z, ζ), z ∈ U, ζ ∈ U , where q is given by q(z, ζ) = (2β − ζ) + 2 (ζ − β) ln(1+z) , z ∈ U, z

ζ ∈ U . The function q is convex and it is the best dominant. Proof. Following the same steps as in the proof of Theorem 2.9 and considering p(z, ζ) =

m DRλ f (z,ζ) m f (z,ζ) 0 z (DRλ )

, the

z

strong differential subordination (2.10) becomes p(z, ζ) + zp0z (z, ζ) ≺≺ h(z, ζ) =

ζ+(2β−ζ)z , 1+z

m DRλ f (z,ζ) m f (z,ζ) 0 DRλ z

z ∈ U, ζ ∈ U . Rz ≺≺ q(z, ζ) = z1 0 h(t, ζ)dt =

By using Lemma 1.2 for γ = 1, we have p(z, ζ) ≺≺ q(z, ζ), i.e. z( ) i R R h 2(ζ−β) ln(1+z) 1 z ζ+(2β−ζ)t 1 z dt = , z ∈ U, ζ ∈ U . (2β − ζ) + dt = (2β − ζ) + 2 (ζ − β) z 0 1+t z 0 1+t z

Theorem 2.11 Let g be a convex function such that g(0, ζ) = 0 and let h be the function h(z, ζ) = g(z, ζ) + zgz0 (z, ζ), z ∈ U, ζ ∈ U . If λ ≥ 0, m ∈ N, f ∈ A∗ζ and the strong differential subordination

holds, then

£ ¤2 (DRλm f (z, ζ))0z + DRλm f (z, ζ) · (DRλm f (z, ζ))00z2 ≺≺ h(z, ζ),

m m DRλ f (z,ζ)·(DRλ f (z,ζ))0z z

z ∈ U, ζ ∈ U

(2.11)

≺≺ g(z, ζ), z ∈ U, ζ ∈ U . This result is sharp.

DRm f (z,ζ)·(DRm f (z,ζ))0

λ λ z . We deduce that p ∈ H∗ [0, 1, ζ]. Differentiating with respect to z, Proof. Let p(z, ζ) = z ¤ £ 2 0 we obtain (DRλm f (z, ζ))z + DRλm f (z, ζ) · (DRλm f (z, ζ))00z2 = p (z, ζ) + zp0z (z, ζ) , z ∈ U, ζ ∈ U . Using the notation in (2.11), the strong differential subordination becomes p(z, ζ) + zp0z (z, ζ) ≺≺ h(z, ζ) = g(z, ζ) + zgz0 (z, ζ), z ∈ U, ζ ∈ U . m 0 DRm λ f (z,ζ)·(DRλ f (z,ζ))z By using Lemma 1.1, we have p(z, ζ) ≺≺ g(z, ζ), z ∈ U, ζ ∈ U i.e. ≺≺ g(z, ζ), z ∈ U, z ζ ∈ U , and this result is sharp. ³ ´ zh00 (z,ζ) Theorem 2.12 Let h be a holomorphic function which satisfies the inequality Re 1 + h0z2(z,ζ) > − 12 , z ∈ U, z

ζ ∈ U and h(0, ζ) = 0. If λ ≥ 0, m ∈ N, f ∈ A∗ζ and satisfies the strong differential subordination £ 0 ¤2 00 (DRλm f (z, ζ))z + DRλm f (z, ζ) · (DRλm f (z, ζ))z2 ≺≺ h(z, ζ),

0 DRm f (z,ζ)·(DRm λ f (z,ζ))z z

λ then dominant.

≺≺ q(z, ζ), z ∈ U, ζ ∈ U , where q(z, ζ) = 1046

5

1 z

Rz 0

z ∈ U, ζ ∈ U ,

(2.12)

h(t, ζ)dt is convex and it is the best Andrei Loriana 1042-1048

J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 18, NO.6, 2015, COPYRIGHT 2015 EUDOXUS PRESS, LLC

DRm f (z,ζ)·(DRm f (z,ζ))0

λ λ z Proof. Let p(z, ζ) = , z ∈ U, ζ ∈ U , p ∈ H∗ [0, 1, ζ]. Differentiating with respect to z £ 0 ¤2 00 m m z, we obtain (DRλ f (z, ζ))z + DRλ f (z, ζ) · (DRλm f (z, ζ))z2 = p (z, ζ) + zp0z (z, ζ) , z ∈ U, ζ ∈ U , and (2.12) becomes p(z, ζ) + zp0z (z, ζ) ≺≺ h(z, ζ), z ∈ U, ζ ∈ U . m 0 DRλ f (z,ζ)·(DRm λ f (z,ζ))z Using Lemma 1.2, we have p(z, ζ) ≺≺ q(z, ζ), z ∈ U, ζ ∈ U , i.e. ≺≺ q(z, ζ) = z R 1 z z 0 h(t, ζ)dt, z ∈ U, ζ ∈ U , and q is the best dominant.

Corollary 2.13 Let h(z) = ζ+(2β−ζ)z be a convex function in U × U , where 0 ≤ β < 1. If λ ≥ 0, m ∈ N, f ∈ A∗ζ 1+z and satisfies the strong differential subordination £ ¤2 (DRλm f (z, ζ))0z + DRλm f (z, ζ) · (DRλm f (z, ζ))00z2 ≺≺ h(z, ζ), z ∈ U, ζ ∈ U, (2.13) DRm f (z,ζ)·(DRm f (z,ζ))0

λ λ z then ≺≺ q(z, ζ), z ∈ U, ζ ∈ U , where q is given by q(z, ζ) = (2β − ζ) + 2 (ζ − β) ln(1+z) , z z z ∈ U, ζ ∈ U . The function q is convex and it is the best dominant.

DRm f (z,ζ)·(DRm f (z,ζ))0z

λ λ Proof. Following the same steps as in the proof of Theorem 2.12 and considering p(z, ζ) = z ζ+(2β−ζ)z 0 the strong differential subordination (2.13) becomes p(z, ζ) + zpz (z, ζ) ≺≺ h(z, ζ) = , z ∈ U, ζ ∈ U . 1+z

DRm f (z,ζ)·(DRm f (z,ζ))0

λ λ z ≺≺ q(z, ζ) = By using Lemma 1.2 for γ = 1, weh have p(z, ζ) ≺≺ i q(z, ζ), i.e. z R R R 2(ζ−β) ln(1+z) 1 z 1 z ζ+(2β−ζ)t 1 z dt = z 0 (2β − ζ) + 1+t dt = (2β − ζ) + 2 (ζ − β) z , z ∈ U, ζ ∈ U . z 0 h(t, ζ)dt = z 0 1+t

Theorem 2.14 Let g be a convex function such that g(0, ζ) = 0 and let h be the function h(z, ζ) = g(z, ζ) + z 0 ∗ 1−δ gz (z, ζ), z ∈ U, ζ ∈ U . If λ ≥ 0, δ ∈ (0, 1), m ∈ N, f ∈ Aζ and the strong differential subordination á ! ¢0 ¶δ µ DRλm+1 f (z, ζ) z (DRλm f (z, ζ))0z DRλm+1 f (z, ζ) z −δ ≺≺ h(z, ζ), (2.14) DRλm f (z, ζ) 1−δ DRλm f (z, ζ) DRλm+1 f (z, ζ) z ∈ U, ζ ∈ U , holds, then

m+1 DRλ f (z,ζ) z

·

³

z

m f (z,ζ) DRλ

´δ

≺≺ g(z, ζ), z ∈ U, ζ ∈ U . This result is sharp.

³ ´δ m+1 DRλ f (z,ζ) Proof. Let p(z, ζ) = · DRmzf (z,ζ) . We deduce that p ∈ H∗ [1, 1, ζ]. Differentiating with respect z λ µ ¶ ´δ ³ m (DRλm+1 f (z,ζ))0z (DRλ f (z,ζ))0z f (z,ζ) DRm+1 z 1 λ = p (z, ζ) + 1−δ − δ DRm f (z,ζ) zp0z (z, ζ) , z ∈ U, to z, we obtain DRm f (z,ζ) 1−δ DRm+1 f (z,ζ) λ

λ

λ

ζ ∈ U. 1 zp0z (z, ζ) ≺≺ h(z, ζ) = Using the notation in (2.14), the strong differential subordination becomes p(z, ζ) + 1−δ z 0 g(z, ζ) + 1−δ gz (z, ζ), z ∈ U , ζ ∈ U . ³ ´δ DRm+1 f (z,ζ) z λ By using Lemma 1.1, we have p(z, ζ) ≺≺ g(z, ζ), z ∈ U, ζ ∈ U , i.e. · ≺≺ g(z, ζ), m z DR f (z,ζ) λ

z ∈ U, ζ ∈ U ,and this result is sharp.

Theorem 2.15 Let h be a holomorphic function which satisfies the inequality Re

³ 1+

zh00 (z,ζ) z2 h0z (z,ζ)

´

> − 12 , z ∈ U,

ζ ∈ U and h(0, ζ) = 1. If λ ≥ 0, δ ∈ (0, 1) , m ∈ N, f ∈ A∗ζ and satisfies the strong differential subordination á ! ¢0 ¶δ µ 0 DRλm+1 f (z, ζ) z (DRλm f (z, ζ))z DRλm+1 f (z, ζ) z −δ ≺≺ h(z, ζ), (2.15) DRλm f (z, ζ) 1−δ DRλm f (z, ζ) DRλm+1 f (z, ζ)

³ ´δ m+1 Rz DRλ f (z,ζ) · DRmzf (z,ζ) ≺≺ q(z, ζ), z ∈ U, ζ ∈ U , where q(z, ζ) = z1−δ h(t, ζ)t−δ dt is z ∈ U, ζ ∈ U , then 1−δ z 0 λ convex and it is the best dominant. ³ ´δ m+1 DRλ f (z,ζ) · DRmzf (z,ζ) , z ∈ U, ζ ∈ U , p ∈ H∗ [0, 1, ζ]. Differentiating with respect to Proof. Let p(z, ζ) = z µ λ m+1 ¶ 0 ´δ ³ m+1 m (DRλ f (z,ζ))z (DRλ f (z,ζ))0z f (z,ζ) DRλ z 1 = p (z, ζ) + 1−δ − δ zp0z (z, ζ) , z ∈ U, ζ ∈ U , z, we obtain DRm f (z,ζ) m m+1 1−δ DR f (z,ζ) DR f (z,ζ) λ

and (2.15) becomes p(z, ζ) +

λ

λ

1 0 1−δ zpz (z, ζ)

≺≺ h(z, ζ), z ∈ U, ζ ∈ U .

Using Lemma 1.2, we have p(z, ζ) ≺≺ q(z, ζ), z ∈ U, ζ ∈ U , i.e. Rz 1−δ h(t, ζ)t−δ dt, z ∈ U, ζ ∈ U , and q is the best dominant. z 1−δ 0 1047

6

m+1 DRλ f (z,ζ) z

·

³

z

m f (z,ζ) DRλ

´δ

≺≺ q(z, ζ) =

Andrei Loriana 1042-1048

,

J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 18, NO.6, 2015, COPYRIGHT 2015 EUDOXUS PRESS, LLC

References [1] A. Alb Lupa¸s, G.I. Oros, Gh. Oros, On special strong differential subordinations using Sa˘la˘gean and Ruscheweyh operators, Journal of Computational Analysis and Applications, Vol. 14, No. 2, 2012, 266-270. [2] A. Alb Lupa¸s, On special strong differential subordinations using a generalized Sa˘la˘gean operator and Ruscheweyh derivative, Journal of Concrete and Applicable Mathematics, Vol. 10, No.’s 1-2, 2012, 17-23. [3] A. Alb Lupa¸s, A note on strong differential subordinations using a generalized Sa˘la˘gean operator and Ruscheweyh operator, Acta Universitatis Apulensis No. 34/2013, 105-114. [4] A. Alb Lupa¸s, Certain strong differential superordinations using a generalized Sa˘la˘gean operator and Ruscheweyh operator, Journal of Applied Functional Analysis, Vol. 7, No.’s 1-2, 2012, 62-68. [5] A. Alb Lupa¸s, Certain strong differential subordinations using Sa˘la˘gean and Ruscheweyh operators, Advances in Applied Mathematical Analysis, Volume 6, Number 1 (2011), 27—34. [6] A. Alb Lupa¸s, A note on strong differential subordinations using Sa˘la˘gean and Ruscheweyh operators, Libertas Mathematica, tomus XXXI (2011), 15-21. [7] A. Alb Lupa¸s, Certain strong differential superordinations using Sa˘la˘gean and Ruscheweyh operators, Acta Universitatis Apulensis No. 30/2012, 325-336. [8] A. Alb Lupa¸s, A note on strong differential superordinations using Sa˘la˘gean and Ruscheweyh operators, Journal of Applied Functional Analysis, Vol. 7, No.’s 1-2, 2012, 54-61. [9] D.A. Alb Lupa¸s, Subordinations and Superordinations, Lap Lambert Academic Publishing, 2011. [10] L. Andrei, Differential subordination results using a generalized Sa˘la˘gean operator and Ruscheweyh operator, Acta Universitatis Apulensis, (to appear). [11] L. Andrei, Some differential subordination results using a generalized Sa˘la˘gean operator and Ruscheweyh operator, submitted. [12] J.A. Antonino, S. Romaguera, Strong differential subordination to Briot-Bouquet differential equations, Journal of Differential Equations, 114 (1994), 101-105. [13] G.I. Oros, Gh. Oros, Strong differential subordination, Turkish Journal of Mathematics, 33 (2009), 249-257.

1048

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J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 18, NO.6, 2015, COPYRIGHT 2015 EUDOXUS PRESS, LLC

On some differential sandwich theorems using a generalized Sa˘la˘gean operator and Ruscheweyh operator Andrei Loriana Department of Mathematics and Computer Science University of Oradea 1 Universitatii street, 410087 Oradea, Romania [email protected] Abstract In this work we define a new operator using the generalized S˘ al˘ agean operator and Ruscheweyh operator. Denote by DRλm,n the Hadamard product of the generalized S˘ al˘ agean operator Dλm and Ruscheweyh operator Rn , given by DRλm,n : A → A, DRλm,n f (z) = (Dλm ∗ Rn ) f (z) and An = {f ∈ H (U) : f (z) = z+an+1 z n+1 +..., z ∈ U} is the class of normalized analytic functions with A1 = A. The purpose of this paper is to introduce sufficient conditions for subordination and superordination involving the operator DRλm,n and also to obtain sandwich-type results.

Keywords: analytic functions, differential operator, differential subordination, differential superordination. 2010 Mathematical Subject Classification: 30C45.

1

Introduction

Let H (U ) be the class of analytic function in the open unit disc of the complex plane U = {z ∈ C : |z| < 1}. Let H (a, n) be the subclass of H (U ) consisting of functions of the form f (z) = an+ an z n + an+1 z n+1 + . . . . Let Ano= 00 (z) {f ∈ H(U ) : f (z) = z+an+1 z n+1 +. . . , z ∈ U } and A = A1 . Denote by K = f ∈ A : Re zff 0 (z) + 1 > 0, z ∈ U , the class of normalized convex functions in U . Let the functions f and g be analytic in U . We say that the function f is subordinate to g, written f ≺ g, if there exists a Schwarz function w, analytic in U , with w(0) = 0 and |w(z)| < 1, for all z ∈ U, such that f (z) = g(w(z)), for all z ∈ U . In particular, if the function g is univalent in U , the above subordination is equivalent to f (0) = g(0) and f (U ) ⊂ g(U ). Let ψ : C3 × U → C and h be an univalent function in U . If p is analytic in U and satisfies the second order differential subordination (1.1) ψ(p(z), zp0 (z), z 2 p00 (z); z) ≺ h(z), for z ∈ U, then p is called a solution of the differential subordination. The univalent function q is called a dominant of the solutions of the differential subordination, or more simply a dominant, if p ≺ q for all p satisfying (1.1). A dominant qe that satisfies qe ≺ q for all dominants q of (1.1) is said to be the best dominant of (1.1). The best dominant is unique up to a rotation of U . ¡ ¢ Let ψ : C2 × U → C and h analytic in U . If p and ψ p (z) , zp0 (z) , z 2 p00 (z) ; z are univalent and if p satisfies the second order differential superordination h(z) ≺ ψ(p(z), zp0 (z), z 2 p00 (z) ; z),

z ∈ U,

(1.2)

then p is a solution of the differential superordination (1.2) (if f is subordinate to F , then F is called to be superordinate to f ). An analytic function q is called a subordinant if q ≺ p for all p satisfying (1.2). An univalent subordinant qe that satisfies q ≺ qe for all subordinants q of (1.2) is said to be the best subordinant. Miller and Mocanu [15] obtained conditions h, q and ψ for which the following implication holds h(z) ≺ ψ(p(z), zp0 (z), z 2 p00 (z) ; z) ⇒ q (z) ≺Pp (z) . P∞ ∞ For two functions f (z) = z + j=2 aj z j and g(z) = z + j=2 bj z j analytic in the open unit disc U , the Hadamard product product) of f (z) and g (z), written as (f ∗ g) (z), is defined by f (z) ∗ g (z) = P (or convolution j (f ∗ g) (z) = z + ∞ j=2 aj bj z . 1049

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Andrei Loriana 1049-1056

J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 18, NO.6, 2015, COPYRIGHT 2015 EUDOXUS PRESS, LLC

Definition 1.1 (Al Oboudi [7]) For f ∈ A, λ ≥ 0 and n ∈ N, the operator Dλm is defined by Dλm : A → A, Dλ0 f (z) = f (z) Dλ1 f (z) = (1 − λ) f (z) + λzf 0 (z) = Dλ f (z) , ... ¡ ¢ Dλm f (z) = (1 − λ) Dλm−1 f (z) + λz (Dλm f (z))0 = Dλ Dλm−1 f (z) , z ∈ U. P∞ P∞ m Remark 1.1 If f ∈ A and f (z) = z + j=2 aj z j , then Dλm f (z) = z + j=2 [1 + (j − 1) λ] aj z j , for z ∈ U . Remark 1.2 For λ = 1 in the above definition we obtain the Sa˘la˘gean differential operator [18].

Definition 1.2 (Ruscheweyh [17]) For f ∈ A and n ∈ N, the operator Rn is defined by Rn : A → A, R0 f (z) R1 f (z) (n + 1) Rn+1 f (z) P j Remark 1.3 If f ∈ A, f (z) = z + ∞ j=2 aj z ,

= f (z) = zf 0 (z) , ... = z (Rn f (z))0 + nRn f (z) , z ∈ U. P (n+j−1)! j then Rn f (z) = z + ∞ j=2 n!(j−1)! aj z for z ∈ U .

The purpose of this paper is to derive the several subordination and superordination results involving a differential operator. Furthermore, we studied the results of M. Darus, K. Al-Shaqs [14], Shanmugam, Ramachandran, Darus and Sivasubramanian [19]. In order to prove our subordination and superordination results, we make use of the following known results.

Definition 1.3 [16] Denote by Q the set of all functions f that are analytic and injective on U \E (f ), where E (f ) = {ζ ∈ ∂U : lim f (z) = ∞}, and are such that f 0 (ζ) 6= 0 for ζ ∈ ∂U \E (f ). z→ζ

Lemma 1.1 [16] Let the function q be univalent in the unit disc U and θ and φ be analytic in a domain D containing q (U ) with φ (w) 6= 0 when w ∈³ q (U ).´ Set Q (z) = zq 0 (z) φ (q (z)) and h (z) = θ (q (z)) + Q (z). Suppose 0 (z) that Q is starlike univalent in U and Re zh > 0 for z ∈ U . If p is analytic with p (0) = q (0), p (U ) ⊆ D and Q(z) 0 0 θ (p (z)) + zp (z) φ (p (z)) ≺ θ (q (z)) + zq (z) φ (q (z)) , then p (z) ≺ q (z) and q is the best dominant. Lemma 1.2 [13] Let the function q be convex³univalent ´ in the open unit disc U and ν and φ be analytic in a 0 (q(z)) domain D containing q (U ). Suppose that Re νφ(q(z)) > 0 for z ∈ U and ψ (z) = zq 0 (z) φ (q (z)) is starlike univalent in U . If p (z) ∈ H [q (0) , 1] ∩ Q, with p (U ) ⊆ D and ν (p (z)) + zp0 (z) φ (p (z)) is univalent in U and ν (q (z)) + zq 0 (z) φ (q (z)) ≺ ν (p (z)) + zp0 (z) φ (p (z)) , then q (z) ≺ p (z) and q is the best subordinant.

2

Main results

Definition 2.1 Let λ ≥ 0 and n, m ∈ N. Denote by DRλm,n : A → A the operator given by the Hadamard product of the generalized Sa˘la˘gean operator Dλm and the Ruscheweyh operator Rn , DRλm,n f (z) = (Dλm ∗ Rn ) f (z) , for any z ∈ U and each nonnegative integers m, n. P∞ P m (n+j−1)! 2 j m,n j Remark 2.1 If f ∈ A and f (z) = z + ∞ j=2 aj z , then DRλ f (z) = z + j=2 [1 + (j − 1) λ] n!(j−1)! aj z , for z ∈ U. This operator was studied in [12]. Remark 2.2 For λ = 1, m = n, we obtain the Hadamard product SRn [1] of the Sa˘la˘gean operator S n and Ruscheweyh derivative Rn , which was studied in [2], [3]. Remark 2.3 For m = n we obtain the Hadamard product DRλn [4] of the generalized Sa˘la˘gean operator Dλn and Ruscheweyh derivative Rn , which was studied in [5], [6], [8], [9], [10], [11]. Using simple computation one obtains the next result. Proposition 2.1 For m, n ∈ N and λ ≥ 0 we have

DRλm+1,n f (z) = (1 − λ) DRλm,n f (z) + λz (DRλm,n f (z))

and

0

(2.1)

0

z (DRλm,n f (z)) = (n + 1) DRλm,n+1 f (z) − nDRλm,n f (z) . 1050

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J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 18, NO.6, 2015, COPYRIGHT 2015 EUDOXUS PRESS, LLC

P∞ m+1 (n+j−1)! 2 j Proof. We have DRλm+1,n f (z) = z + j=2 [1 + (j − 1) λ] n!(j−1)! aj z = P∞ m (n+j−1)! 2 j z + j=2 [(1 − λ) + λj] [1 + (j − 1) λ] n!(j−1)! aj z = P∞ P∞ m m (n+j−1)! 2 j 2 j z + (1 − λ) j=2 [1 + (j − 1) λ] (n+j−1)! j=2 [1 + (j − 1) λ] n!(j−1)! aj z + λ n!(j−1)! jaj z = 0

(1 − λ) DRλm,n f (z) + λz (DRλm,n f (z)) , and (n + 1) DRλm,n+1 f (z) − nDRλm,n f (z) = P∞ P∞ m m (n+j)! 2 j (n + 1) z + (n + 1) j=2 [1 + (j − 1) λ] (n+1)!(j−1)! a2j z j − nz − n j=2 [1 + (j − 1) λ] (n+j−1)! n!(j−1)! aj z = P P∞ ∞ m (n+j−1)! 2 j n+j (n+j−1)! 2 j z + (n + 1) j=2 [1 + (j − 1) λ]m n+1 j=2 [1 + (j − 1) λ] n!(j−1)! aj z − n n!(j−1)! aj z = P∞ 0 m (n+j−1)! m,n 2 j z + j=2 [1 + (j − 1) λ] n!(j−1)! jaj z = z (DRλ f (z)) . We begin with the following DRm+1,n f (z)

Theorem 2.2 Let DRλm,n f (z) ∈ H (U ) , z ∈ U , f ∈ A, m, n ∈ N, λ ≥ 0 and let the function q (z) be convex and λ univalent in U such that q (0) = 1. Assume that ¶ µ zq 00 (z) α 2β q (z) + 0 > 0, z ∈ U, (2.3) Re 1 + + µ µ q (z) for α, β, µ, ∈ C,µ 6= 0, z ∈ U, and ψ m,n λ

(α, β, µ; z) :=

µ

1 − λ(n + 1) µ+α λ

¶

DRλm+1,n f (z) + DRλm,n f (z)

(2.4)

DRλm,n+1 f (z) DRλm,n+2 f (z) µ +µ(n + 1) [1 − λ(n + 2)] + λµ(n + 1)(n + 2) + (β − ) DRλm,n f (z) DRλm,n f (z) λ

Ã

DRλm+1,n f (z) DRλm,n f (z)

!2

.

If q satisfies the following subordination (α, β, µ; z) ≺ αq (z) + β (q (z))2 + µzq 0 (z) , ψ m,n λ for, α, β, µ ∈ C, µ 6= 0 then

DRλm+1,n f (z) ≺ q (z) , DRλm,n f (z)

(2.5)

z ∈ U,

(2.6)

and q is the best dominant. Proof. Let the function p be defined by p (z) :=

m+1,n DRλ f (z) , m,n DRλ f (z)

z ∈ U , z 6= 0, f ∈ A. The function p is 0 m+1,n z (DRλ f (z)) − analytic in U and p (0) = 1. Differentiating this function, with respect to z,we get zp0 (z) = DRm,n f (z) λ 0 m,n m+1,n DRλ f (z) z (DRλ f (z)) . By using the identity (2.1) and (2.2), we obtain DRm,n f (z) DRm,n f (z) λ

λ

zp0 (z) =

DRλm,n+1 f (z) 1 − λ(n + 1) DRλm+1,n f (z) + (n + 1) [1 − λ(n + 2)] + λ DRλm,n f (z) DRλm,n f (z)

DRλm,n+2 f (z) 1 λ(n+1)(n+2) − DRλm,n f (z) λ

Ã

DRλm+1,n f (z) DRλm,n f (z)

!2

DRλm,n+2 f (z) 1 − +λ(n+1)(n+2) DRλm,n f (z) λ

Ã

DRλm+1,n f (z) DRλm,n f (z)

!2

(2.7)

By setting θ (w) := αw + βw2 and φ (w) := µ, α, β, µ ∈ C, µ 6= 0 it can be easily verified that θ is analytic in C, φ is analytic in C\{0} and that φ (w) 6= 0, w ∈ C\{0}. Also, by letting Q (z) = zq 0 (z) φ (q (z)) = µzq 0 (z) ,we find that Q (z) is starlike univalent in U. Let h (z) = θ (q (z)) + Q (z) = αq +´β (q (z))³2 + µzq 0 (z), z ∈ U. If we ´ derive ³ (z) 0 zh (z) zq 00 (z) 2β α the function Q, with respect to z, perform calculations, we have Re Q(z) = Re 1 + µ + µ q (z) + q0 (z) > 0. ³ ´ m+1,n DRλ f (z) µ + α + µ(n+1) [1 − λ(n + 2)] · By using (2.7), we obtain αp (z)+β (p (z))2 +µzp0 (z) = 1−λ(n+1) m,n λ DRλ f (z) ³ m+1,n ´2 m,n+1 m,n+2 DRλ f (z) DR f (z) DR f (z) 2 + λµ(n + 1)(n + 2) DRλm,n f (z) + (β − µλ ) DRλm,n f (z) . By using (2.5), we have αp (z) + β (p (z)) + DRm,n f (z) λ

2

λ

λ

µzp0 (z) ≺ αq (z) + β (q (z)) + µzq 0 (z) . Therefore, the conditions of Lemma 1.1 are met, so we have p (z) ≺ q (z),

z ∈ U, i.e.

m+1,n DRλ f (z) m,n DRλ f (z)

≺ q (z), z ∈ U, and q is the best dominant. 1051

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Corollary 2.3 Let q (z) =

1+Az 1+Bz ,

1+Az and ψ m,n (α, β, µ; z) ≺ α 1+Bz λ

defined in (2.4), then

−1 ≤ B < A ≤ 1, m, n ∈ N, λ ≥ 0, z ∈ U. Assume that (2.3) holds. If f ∈ A ³ ´2 (A−B)z 1+Az + β 1+Bz + µ (1+Bz) 6 0, −1 ≤ B < A ≤ 1, where ψ m,n is 2 , for α, β, µ ∈ C, µ = λ

m+1,n DRλ f (z) DRm,n f (z) λ

Proof. For q (z) =

1+Az 1+Bz ,

≺

1+Az 1+Bz

and

1+Az 1+Bz

is the best dominant.

−1 ≤ B < A ≤ 1, in Theorem 2.2 we get the corollary.

³ ´γ 1+z Corollary 2.4 Let q (z) = 1−z , m, n ∈ N, λ ≥ 0, z ∈ U. Assume that (2.3) holds. If f ∈ A and ψ m,n (α, β, µ; z) λ ³ ³ ´γ ´2γ ³ ´γ−1 2γz 1+z 1+z 1+z + β 1−z + µ 1−z , for α, µ, β ∈ C, 0 < γ ≤ 1, µ 6= 0, where ψ m,n is defined in (2.4), ≺ α 1−z 2 λ 1−z ³ ³ ´ ´ γ γ m+1,n DR f (z) 1+z 1+z then DRλm,n f (z) ≺ 1−z , and 1−z is the best dominant. λ

Proof. Corollary follows by using Theorem 2.2 for q (z) =

³

1+z 1−z

´γ

, 0 < γ ≤ 1.

Theorem 2.5 Let q be convex and univalent in U, such that q (0) = 1, m, n ∈ N, λ ≥ 0. Assume that µ 0 ¶ q (z) Re (α + 2βq (z)) > 0, for α, µ, β ∈ C, µ 6= 0, z ∈ U. µ

(2.8)

DRm+1,n f (z)

(α, β, µ; z) is univalent in U , where ψ m,n (α, β, µ; z) is as defined If f ∈ A, DRλm,n f (z) ∈ H [q (0) , 1] ∩ Q and ψ m,n λ λ λ in (2.4), then αq (z) + β (q (z))2 + µzq 0 (z) ≺ ψ m,n (α, β, µ; z) , z ∈ U, (2.9) λ implies q (z) ≺

DRλm+1,n f (z) , DRλm,n f (z)

z ∈ U,

(2.10)

and q is the best subordinant. Proof. Let the function p be defined by p (z) := 2

m+1,n DRλ f (z) , DRm,n f (z) λ

z ∈ U , z 6= 0, f ∈ A. By setting ν (w) :=

αw + βw and φ (w) := µ it can be easily verified that ν is analytic in³ C, φ is ´analytic³in C\{0} and that ´φ (w) 6= 0, 0 0 0 0 (q(z)) (q(z)) w ∈ C\{0}. Since νφ(q(z)) = q µ(z) (α + 2βq (z)), it follows that Re νφ(q(z)) = Re q µ(z) (α + 2βq (z)) > 0, for µ, ξ, β ∈ C, µ 6= 0. 2 2 By using (2.9) we obtain αq (z) + β (q (z)) + µzq 0 (z) ≺ αq (z) + β (q (z)) + µzq 0 (z) . Using Lemma 1.2, we have q (z) ≺ p (z) =

m+1,n DRλ f (z) , m,n DRλ f (z)

Corollary 2.6 Let q (z) = A,

m+1,n DRλ f (z) DRm,n f (z) λ

z ∈ U, and q is the best subordinant.

1+Az 1+Bz ,

−1 ≤ B < A ≤ 1, m, n ∈ N, λ ≥ 0. Assume that (2.8) holds. If f ∈ ³ ´2 (A−B)z m,n 1+Az 1+Az ∈ H [q (0) , 1] ∩ Q and α 1+Bz + β 1+Bz + µ (1+Bz) (α, β, µ; z) , for α, µ, β ∈ C, µ 6= 0, 2 ≺ ψλ

−1 ≤ B < A ≤ 1, where ψ m,n is defined in (2.4), then λ Proof. For q (z) =

1+Az 1+Bz ,

1+Az 1+Bz

≺

m+1,n DRλ f (z) DRm,n f (z) λ

and

1+Az 1+Bz

is the best subordinant.

−1 ≤ B < A ≤ 1 in Theorem 2.5 we get the corollary.

³ ´γ DRm+1,n f (z) 1+z Corollary 2.7 Let q (z) = 1−z , m, n ∈ N, λ ≥ 0. Assume that (2.8) holds. If f ∈ A, DRλm,n f (z) ∈ λ ³ ³ ´γ ´2γ ³ ´γ−1 m,n 2γz 1+z 1+z 1+z + µ 1−z2 1−z ≺ ψ λ (α, β, µ; z) , for α, µ, β ∈ C, 0 < γ ≤ 1, µ H [q (0) , 1] ∩ Q and α 1−z + β 1−z ³ ³ ´γ ´γ m+1,n DR f (z) m,n 1+z λ 6= 0, where ψ λ is defined in (2.4), then 1+z ≺ and is the best subordinant. m,n 1−z 1−z DR f (z) λ

³ ´γ Proof. Corollary follows by using Theorem 2.5 for q (z) = 1+z , 0 < γ ≤ 1. 1−z Combining Theorem 2.2 and Theorem 2.5, we state the following sandwich theorem.

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Theorem 2.8 Let q1 and q2 be analytic and univalent in U such that q1 (z) 6= 0 and q2 (z) 6= 0, for all z ∈ U , with zq10 (z) and zq20 (z) being starlike univalent. Suppose that q1 satisfies (2.3) and q2 satisfies (2.8). If m+1,n DRλ f (z) ∈ H [q (0) , 1] ∩ Q and ψ m,n (α, β, µ; z) is as defined in (2.4) univalent in U , then αq1 (z) + m,n λ DRλ f (z) 2 m,n 0 β (q1 (z)) + µzq1 (z) ≺ ψ λ (α, β, µ; z) ≺ αq2 (z) + β (q2 (z))2 + µzq20 (z) , for α, µ, β ∈ C, µ 6= 0, implies q1 (z) ≺ m+1,n DRλ f (z) ≺ q2 (z), δ ∈ C, δ 6= 0, and q1 and q2 are respectively the best subordinant and the best dominant. DRm,n f (z)

f ∈ A,

λ

For q1 (z) =

1+A1 z 1+B1 z ,

q2 (z) =

1+A2 z 1+B2 z ,

where −1 ≤ B2 < B1 < A1 < A2 ≤ 1, we have the following corollary.

1+A1 z 1+A2 z Corollary 2.9 Let m, n ∈ N, λ ≥ 0. Assume that (2.3) and (2.8) hold for q1 (z) = 1+B and q2 (z) = 1+B , 1z 2z ³ ´2 m+1,n DRλ f (z) (A1 −B1 )z m,n 1+A1 z 1+A1 z respectively. If f ∈ A, DRm,n f (z) ∈ H [q (0) , 1] ∩ Q and α 1+B1 z + β 1+B1 z + µ (1+B z)2 ≺ ψ λ (α, β, µ; z) 1 ³ ´2 λ (A2 −B2 )z 1+A2 z 1+A2 z is ≺ α 1+B2 z + β 1+B2 z + µ (1+B z)2 , for α, µ, β ∈ C, µ 6= 0, −1 ≤ B2 ≤ B1 < A1 ≤ A2 ≤ 1, where ψ m,n λ 2

1+A1 z 1+B1 z

defined in (2.4), then dominant, respectively.

≺

m+1,n DRλ f (z) m,n DRλ f (z)

1+A2 z 1+B2 z ,

≺

hence

1+A1 z 1+B1 z

and

1+A2 z 1+B2 z

are the best subordinant and the best

´δ ³ m+1,n DRλ f (z) ∈ H (U ) , f ∈ A, z ∈ U , δ ∈ C, δ 6= 0, m, n ∈ N, λ ≥ 0 and let the function Theorem 2.10 Let m,n DRλ f (z) q (z) be convex and univalent in U such that q (0) = 1, z ∈ U . Assume that µ ¶ α+β zq 00 (z) Re + 0 > 0, (2.11) β q (z) for α, β ∈ C, β 6= 0, z ∈ U, and ψ m,n λ

(α, β; z) :=

Ã

DRλm+1,n f (z) DRλm,n f (z)

!δ ∙

α + δβ

DRλm,n+1 f (z)

1 − λ(n + 1) + λ

DRλm,n+2 f (z)

δβ DRλm+1,n f (z) δβ(n + 1) [1 − λ(n + 2)] + δβλ(n + 1)(n + 2) − λ DRλm,n f (z) DRλm+1,n f (z) DRλm+1,n f (z)

#

(2.12)

If q satisfies the following subordination (α, β; z) ≺ αq (z) + βzq 0 (z) , ψ m,n λ for α, β ∈ C, β 6= 0, z ∈ U, then

Ã

DRλm+1,n f (z) DRλm,n f (z)

!δ

≺ q (z) ,

(2.13)

z ∈ U, δ ∈ C, δ 6= 0,

(2.14)

and q is the best dominant. Proof. Let the function p be defined by p (z) :=

³

m+1,n DRλ f (z) DRm,n f (z) λ

³

´δ

, z ∈ U , z 6= 0, f ∈ A. The func´δ ³ m+1,n ´0 m,n DRλ f (z) DRλ f (z) = m,n m+1,n DR f (z) DRλ f (z) λ ¶ ) . ´δ h³ ´ m,n m+1,n

m+1,n DRλ f (z) m,n DRλ f (z) 0 m,n DRλ f (z) DRm,n f (z) λ

tion p is analytic in U and p (0) = 1. We have zp0 (z) = δz µ ´δ ³ m+1,n 0 m+1,n m,n z (DRλ f (z)) DR f (z) f (z) DRm+1,n f (z) z ( DRλ − DRλm,n f (z) δ DRλm,n f (z) m,n m+1,n DR f (z) DR f (z) λ λ λ λ ³ m+1,n DR f (z) f (z) DRλ f (z) 1−λ(n+1) DRλ By using the identity (2.1) and (2.2), we obtain zp0 (z) = δ DRλm,n f (z) m+1,n λ DRm,n f (z) DRλ f (z) λ λ ³ m+1,n ´2 ¸ DRm,n+1 f (z) DRm,n+2 f (z) DR f (z) so, we obtain +(n + 1) [1 − λ(n + 2)] DRλm,n f (z) + λ(n + 1)(n + 2) DRλm,n f (z) − λ1 DRλm,n f (z) λ

λ

0

zp (z) = δ

Ã

DRλm+1,n f (z) DRλm,n f (z) 1053

5

λ

!δ ∙

1 − λ(n + 1) + λ

Andrei Loriana 1049-1056

J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 18, NO.6, 2015, COPYRIGHT 2015 EUDOXUS PRESS, LLC

DRλm,n+1 f (z)

DRλm,n+2 f (z)

1 DRλm+1,n f (z) (n + 1) [1 − λ(n + 2)] + λ(n + 1)(n + 2) − m,n DRλm+1,n f (z) DRλm+1,n f (z) λ DRλ f (z)

#

(2.15)

By setting θ (w) := αw and φ (w) := β, it can be easily verified that θ is analytic in C, φ is analytic in C\{0} 0 is and that φ (w) 6= 0, w ∈ C\{0}. Also, by letting Q (z) = zq 0 (z) φ (q (z)) = βzq ³ 0 (z)´, we find ³ that Q (z) ´ starlike zh (z) zq 00 (z) α+β 0 univalent in U. Let h (z) = θ (q (z)) + Q (z) = αq (z) + βzq (z).We have Re Q(z) = Re β + q0 (z) > 0. ³ m+1,n ´δ h DR f (z) + δβ(n + 1) [1 − λ(n + 2)] · α + δβ 1−λ(n+1) By using (2.15), we obtain αp (z) + βzp0 (z) = DRλm,n f (z) λ λ i m,n+1 m,n+2 m+1,n DRλ f (z) DRλ f (z) f (z) δβ DRλ + δβλ(n + 1)(n + 2) DRm+1,n f (z) − λ DRm,n f (z) . DRm+1,n f (z) λ

λ

λ

By using (2.13), we have αp (z) + βzp0 (z) ≺ αq (z) + βzq 0 (z) . From Lemma 1.1, we have p (z) ≺ q (z), z ∈ U, ³ m+1,n ´δ DRλ f (z) i.e. ≺ q (z), z ∈ U, δ ∈ C, δ 6= 0 and q is the best dominant. m,n DR f (z) λ

1+Az 1+Bz , z ∈ U, −1 ≤ B < A ≤ 1, m, n (A−B)z 1+Az α 1+Bz + β (1+Bz) 2 , for α, β ∈ C, β 6= 0, −1

Corollary 2.11 Let q (z) =

and then

ψ m,n λ ³

(α, β; z) ≺ ´δ m+1,n

DRλ f (z) m,n DRλ f (z)

≺

Proof. For q (z) =

1+Az 1+Bz ,

δ ∈ C, δ 6= 0, and

1+Az 1+Bz

∈ N, λ ≥ 0. Assume that (2.11) holds. If f ∈ A

≤ B < A ≤ 1, where ψ m,n is defined in (2.12), λ

is the best dominant.

1+Az 1+Bz ,

−1 ≤ B < A ≤ 1, in Theorem 2.10 we get the corollary. ³ ´γ 1+z Corollary 2.12 Let q (z) = 1−z , m, n ∈ N, λ ≥ 0. Assume that (2.11) holds. If f ∈ A and ψ m,n (α, β, µ; z) ≺ λ ³ m+1,n ´γ ³ ´γ−1 ´δ ³ DRλ f (z) m,n 2γz 1+z 1+z + β 1−z , for α, β ∈ C, 0 < γ ≤ 1, β = 6 0, where ψ is defined in (2.12), then α 1−z m,n 2 λ 1−z DRλ f (z) ³ ³ ´γ ´γ 1+z 1+z ≺ 1−z , δ ∈ C, δ 6= 0, and 1−z is the best dominant. Proof. Corollary follows by using Theorem 2.10 for q (z) =

³

1+z 1−z

´γ

, 0 < γ ≤ 1.

Theorem 2.13 Let q be convex and univalent in U such that q (0) = 1. Assume that ¶ µ α 0 q (z) > 0, for α, β ∈ C, β 6= 0. (2.16) Re β ´δ ³ m+1,n DR f (z) ∈ H [q (0) , 1] ∩ Q and ψ m,n (α, β; z) is univalent in U , where ψ m,n (α, β; z) is as defined If f ∈ A, DRλm,n f (z) λ λ λ in (2.12), then (α, β; z) (2.17) αq (z) + βzq 0 (z) ≺ ψ m,n λ implies

q (z) ≺

Ã

DRλm+1,n f (z) DRλm,n f (z)

!δ

,

δ ∈ C, δ 6= 0, z ∈ U,

(2.18)

and q is the best subordinant. ³ m+1,n ´δ DR f (z) Proof. Let the function p be defined by p (z) := DRλm,n f (z) , z ∈ U , z 6= 0, δ ∈ C, δ 6= 0, f ∈ A. The λ function p is analytic in U and p (0) = 1. By setting ν (w) := αw and φ (w) := β it can be easily verified that ν 0 (q(z)) 0 is analytic in C, φ is analytic in C\{0} and that φ (w) 6= 0, w ∈ C\{0}. Since νφ(q(z)) = α β q (z), it follows that ³ 0 ´ ´ ³ (q(z)) 0 Re νφ(q(z)) = Re α β q (z) > 0, for α, β ∈ C, β 6= 0. Now, by using (2.17) we obtain αq (z) + βzq 0 (z) ≺ αq (z) + βzq 0 (z) , z ∈ U. From Lemma 1.2, we have ³ m+1,n ´δ DR f (z) q (z) ≺ p (z) = DRλm,n f (z) , z ∈ U, δ ∈ C, δ 6= 0, and q is the best subordinant. λ

1+Az Corollary 2.14 Let q (z) = 1+Bz , −1 ≤ B < A ≤ 1, z ∈ U, m, n ∈ N, λ ≥ 0. Assume that (2.16) holds. If f ∈ A, ³ m+1,n ´δ DRλ f (z) (A−B)z m,n 1+Az ∈ H [q (0) , 1] ∩ Q, δ ∈ C, δ 6= 0 and α 1+Bz + β (1+Bz) (α, β; z) , for α, β ∈ C, β 6= 0, m,n 2 ≺ ψλ DRλ f (z) ³ m+1,n ´δ DR f (z) 1+Az 1+Az is defined in (2.12), then 1+Bz ≺ DRλm,n f (z) , δ ∈ C, δ 6= 0, and 1+Bz is the −1 ≤ B < A ≤ 1, where ψ m,n λ λ best subordinant.

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J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 18, NO.6, 2015, COPYRIGHT 2015 EUDOXUS PRESS, LLC

Proof. For q (z) =

1+Az 1+Bz ,

−1 ≤ B < A ≤ 1, in Theorem 2.13 we get the corollary.

³ ³ m+1,n ´γ ´δ DR f (z) 1+z Corollary 2.15 Let q (z) = 1−z , m, n ∈ N, λ ≥ 0. Assume that (2.16) holds. If f ∈ A, DRλm,n f (z) λ ³ ´γ ³ ´γ−1 m,n 2γz 1+z 1+z ∈ H [q (0) , 1] ∩ Q and α 1−z + β 1−z2 1−z ≺ ψ λ (α, β, µ; z) , for α, β ∈ C, 0 < γ ≤ 1, β 6= 0, where ³ ³ ´γ ³ m+1,n ´δ ´γ DRλ f (z) 1+z 1+z is defined in (2.12), then ≺ , δ ∈ C, δ = 6 0, and is the best subordinant. ψ m,n m,n λ 1−z 1−z DR f (z) λ

³ ´γ Proof. Corollary follows by using Theorem 2.13 for q (z) = 1+z , 0 < γ ≤ 1. 1−z Combining Theorem 2.10 and Theorem 2.13, we state the following sandwich theorem.

Theorem 2.16 Let q1 and q2 be convex and univalent in U such that q1 (z) 6= 0 and q2 (z) 6= 0, for all z ∈ U . ³ m+1,n ´δ DRλ f (z) Suppose that q1 satisfies (2.11) and q2 satisfies (2.16). If f ∈ A, ∈ H [q (0) , 1] ∩ Q , δ ∈ m,n DR f (z) λ

(α, β; z) is as defined in (2.12) univalent in U , then αq1 (z) + βzq10 (z) ≺ ψ m,n (α, β; z) ≺ C, δ 6= 0 and ψ m,n λ λ ³ m+1,n ´δ DR f (z) αq2 (z) + βzq20 (z) , for α, β ∈ C, β 6= 0, implies q1 (z) ≺ DRλm,n f (z) ≺ q2 (z), z ∈ U, δ ∈ C, δ 6= 0, and q1 and λ q2 are respectively the best subordinant and the best dominant. For q1 (z) =

1+A1 z 1+B1 z ,

q2 (z) =

1+A2 z 1+B2 z ,

where −1 ≤ B2 < B1 < A1 < A2 ≤ 1, we have the following corollary.

1z Corollary 2.17 Let m, n ∈ N, λ ≥ 0. Assume that (2.11) and (2.16) hold for q1 (z) = 1+A 1+B1 z and q2 (z) = ³ m+1,n ´δ DRλ f (z) 1+A2 z 1+A1 z 1 −B1 )z ∈ H [q (0) , 1] ∩ Q and α 1+B + β (A ≺ ψ m,n (α, β, µ; z) λ 1+B2 z , respectively. If f ∈ A, DRm,n f (z) (1+B z)2 1z 1

λ

(A2 −B2 )z m,n 2z ≺ α 1+A is defined in (2.4), 1+B2 z + β (1+B2 z)2 , z ∈ U, for α, β ∈ C, β 6= 0, −1 ≤ B2 ≤ B1 < A1 ≤ A2 ≤ 1, where ψ λ ³ m+1,n ´δ DR f (z) 1+A1 z 1+A2 z 1+A2 z 1z ≺ DRλm,n f (z) ≺ 1+B , z ∈ U, δ ∈ C, δ 6= 0, hence 1+A then 1+B 1+B1 z and 1+B2 z are the best subordinant and 1z 2z λ the best dominant, respectively.

References [1] A. Alb Lupas, Certain differential subordinations using Sa˘la˘gean and Ruscheweyh operators, Acta Universitatis Apulensis, No. 29/2012, 125-129. [2] A. Alb Lupas, A note on differential subordinations using Sa˘la˘gean and Ruscheweyh operators, Romai Journal, vol. 6, nr. 1(2010), 1—4. [3] A. Alb Lupas, Certain differential superordinations using Sa˘la˘gean and Ruscheweyh operators, Analele Universit˘ a¸tii din Oradea, Fascicola Matematica, Tom XVII, Issue no. 2, 2010, 209-216. [4] A. Alb Lupas, Certain differential subordinations using a generalized Sa˘la˘gean operator and Ruscheweyh operator I, Journal of Mathematics and Applications, No. 33 (2010), 67-72. [5] A. Alb Lupas, Certain differential subordinations using a generalized Sa˘la˘gean operator and Ruscheweyh operator II, Fractional Calculus and Applied Analysis, Vol 13, No. 4 (2010), 355-360. [6] A. Alb Lupas, Certain differential superordinations using a generalized Sa˘la˘gean and Ruscheweyh operators, Acta Universitatis Apulensis nr. 25/2011, 31-40. [7] F.M. Al-Oboudi, On univalent functions defined by a generalized Sa˘la˘gean operator, Ind. J. Math. Math. Sci., 27 (2004), 1429-1436. [8] L. Andrei, Differential subordination results using a generalized Sa˘la˘gean operator and Ruscheweyh operator, Acta Universitatis Apulensis (to appear). [9] L. Andrei, Some differential subordination results using a generalized Sa˘la˘gean operator and Ruscheweyh operator, submitted. [10] L. Andrei, Differential superordination results using a generalized Sa˘la˘gean operator and Ruscheweyh operator, submitted. 1055

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[11] L. Andrei, Some differential superordination results using a generalized Sa˘la˘gean operator and Ruscheweyh operator, submitted. [12] L. Andrei, Differential Sandwich Theorems using a generalized Sa˘la˘gean operator and Ruscheweyh operator, submitted. [13] T. Bulboac˘ a, Classes of first order differential superordinations, Demonstratio Math., Vol. 35, No. 2, 287-292. [14] M. Darus, K. Al-Shaqsi, Differential sandwich theorems with generalised derivative operator, Advanced Technologies, October, Kankesu Jayanthakumaran (Ed), ISBN:978-953-307-009-4 2009 [15] S.S. Miller, P.T. Mocanu, Subordinants of Differential Superordinations, Complex Variables, vol. 48, no. 10, 815-826, October, 2003. [16] S.S. Miller, P.T. Mocanu, Differential Subordinations: Theory and Applications, Marcel Dekker Inc., New York, 2000. [17] St. Ruscheweyh, New criteria for univalent functions, Proc. Amet. Math. Soc., 49(1975), 109-115. [18] G. St. S˘ al˘ agean, Subclasses of univalent functions, Lecture Notes in Math., Springer Verlag, Berlin, 1013(1983), 362-372. [19] T.N. Shanmugan, C. Ramachandran, M. Darus, S. Sivasubramanian, Differential sandwich theorems for some subclasses of analytic functions involving a linear operator, Acta Math. Univ. Comenianae, 16 (2007), no. 2, 287-294.

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Journal of Computational Analysis and Applications 00(0000), 000-000 http://JoCAAA/91-2015-Muhiuddin-Alroqi-JoCAAA-6-02-2014

Subalgebras of BCK/BCI-algebras based on (α, β)-type fuzzy sets G. Muhiuddina,∗ and Abdullah M. Al-roqib a b

Department of Mathematics, University of Tabuk, Tabuk 71491, Saudi Arabia

Department of Mathematics, King Abdulaziz University, Jeddah 21589, Saudi Arabia

Abstract. The notion of (ε, δ)-characteristic fuzzy sets is introduced. Given a subalgebra F of a BCK/BCI-algebra X, conditions for the (ε, δ)-characteristic fuzzy set in X to be an (∈, ∈ ∨ q)-fuzzy subalgebra, an (∈, q)-fuzzy subalgebra, an (∈, ∈ ∧ q)-fuzzy subalgebra, a (q, q)-fuzzy subalgebra, a (q, ∈)fuzzy subalgebra, a (q, ∈ ∨ q)-fuzzy subalgebra and a (q, ∈ ∧ q)-fuzzy subalgebra are provided. Using the (ε,δ)

notions of (α, β)-fuzzy subalgebra µF

, conditions for the F to be a subalgebra of X are investigated

where (α, β) is one of (∈, ∈ ∨ q), (∈, ∈ ∧ q), (∈, q), (q, ∈ ∨ q), (q, ∈ ∧ q), (q, ∈) and (q, q).

1. Introduction The idea of quasi-coincidence of a fuzzy point with a fuzzy set is given in [7] which played a vital role to generate some different types of fuzzy subgroups, called (α, β)-fuzzy subgroups, introduced by Bhakat and Das [1]. In particular, (∈, ∈ ∨ q )-fuzzy subgroup is an important and useful generalization of Rosenfeld’s fuzzy subgroup. Several authors [3, 4, 5, 8] have studied the concept of (α, β)-fuzzy subalgebras in BCK/BCI-algebras, which is an important and useful generalization of the well-known concepts, called fuzzy subalgebras. In this paper, we introduce the notion of (ε, δ)-characteristic fuzzy sets in BCK/BCI-algebras. Given a subalgebra F of a BCK/BCI-algebra X, we provide conditions for the (ε, δ)-characteristic fuzzy set in X to be an (∈, ∈ ∨ q)-fuzzy subalgebra, an (∈, q)-fuzzy subalgebra, an (∈, ∈ ∧ q)-fuzzy subalgebra, a (q, q)-fuzzy subalgebra, a (q, ∈)-fuzzy subalgebra, a (q, ∈ ∨ q)-fuzzy subalgebra and a (q, ∈ ∧ q)-fuzzy (ε,δ)

subalgebra. Using the notions of (α, β)-fuzzy subalgebra µF

, we investigate conditions for the F to be

a subalgebra of X where (α, β) is one of (∈, ∈ ∨ q), (∈, ∈ ∧ q), (∈, q), (q, ∈ ∨ q), (q, ∈ ∧ q), (q, ∈) and (q, q). 2010 Mathematics Subject Classification: 06F35; 03G25; 06D72. Keywords: (ε, δ)-characteristic fuzzy set, (Fuzzy) subalgebra, (α, β)-fuzzy subalgebra. *Corresponding author. E-mail: [email protected] (G. Muhiuddin), [email protected] (Abdullah M. Al-roqi)

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2. Preliminaries By a BCI-algebra we mean an algebra (X, ∗, 0) of type (2, 0) satisfying the axioms: (a1) ((x ∗ y) ∗ (x ∗ z)) ∗ (z ∗ y) = 0, (a2) (x ∗ (x ∗ y)) ∗ y = 0, (a3) x ∗ x = 0, (a4) x ∗ y = y ∗ x = 0 ⇒ x = y, for all x, y, z ∈ X. We can define a partial ordering ≤ by x ≤ y if and only if x ∗ y = 0. If a BCI-algebra X satisfies the axiom (a5) 0 ∗ x = 0 for all x ∈ X, then we say that X is a BCK-algebra. A nonempty subset S of a BCK/BCI-algebra X is called a subalgebra of X if x ∗ y ∈ S for all x, y ∈ S. We refer the reader to the books [2] and [6] for further information regarding BCK/BCI-algebras. A fuzzy set µ in a set X of the form ( µ(y) :=

t ∈ (0, 1]

if y = x,

0

if y 6= x,

is said to be a fuzzy point with support x and value t and is denoted by xt . For a fuzzy point xt and a fuzzy set µ in a set X, Pu and Liu [7] introduced the symbol xt αµ, where α ∈ {∈, q , ∈ ∨ q , ∈ ∧ q }. To say that xt ∈ µ (resp. xt q µ), we mean µ(x) ≥ t (resp. µ(x) + t > 1), and in this case, xt is said to belong to (resp. be quasi-coincident with) a fuzzy set µ. To say that xt ∈ ∨ q µ (resp. xt ∈ ∧ q µ), we mean xt ∈ µ or xt q µ (resp. xt ∈ µ and xt q µ). To say that xt α µ, we mean xt αµ does not hold, where α ∈ {∈, q, ∈ ∨ q , ∈ ∧ q }. A fuzzy set µ in a BCK/BCI-algebra X is called a fuzzy subalgebra of X if it satisfies: µ(x ∗ y) ≥ min{µ(x), µ(y)}

(2.1) for all x, y ∈ X.

Proposition 2.1 ([4]). Let X be a BCK/BCI-algebra. A fuzzy set µ in X is a fuzzy subalgebra of X if and only if the following assertion is valid. xt ∈ µ, ys ∈ µ =⇒ (x ∗ y)min{t,s} ∈ µ

(2.2) for all x, y ∈ X and t, s ∈ (0, 1].

3. Subalgebras of BCK/BCI-algebras based on (α, β)-type fuzzy sets In what follows, let X denote a BCK/BCI-algebra unless otherwise specified. (ε,δ)

Let F be a non-empty subset of X and ε, δ ∈ [0, 1] such that ε > δ. Define a fuzzy set µF

in X as

follows: ( (ε,δ) µF (x)

:=

ε

if x ∈ F,

δ

otherwise.

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We say that µF fuzzy set

(1,0) µF

3

is an (ε, δ)-characteristic fuzzy set in X over F. In particular, the (1, 0)-characteristic

in X over F is the characteristic function χF of F.

Theorem 3.1. For any non-empty subset F of X, the following are equivalent: (1) F is a subalgebra of X. (ε,δ)

(2) The fuzzy set µF

is a fuzzy subalgebra of X for all ε, δ ∈ [0, 1] with ε > δ.

Proof. Assume that F is a subalgebra of X and let ε, δ ∈ [0, 1] be such that ε > δ. Let x, y ∈ X. If x, y ∈ F , then x ∗ y ∈ F and so (ε,δ)

µF (ε,δ)

If x ∈ / F or y ∈ / F, then µF

(ε,δ)

Therefore µF

o n (ε,δ) (ε,δ) (x ∗ y) = ε = min µF (x), µF (y) . (ε,δ)

(x) = δ or µF

(y) = δ. Hence n o (ε,δ) (ε,δ) (ε,δ) µF (x ∗ y) ≥ δ = min µF (x), µF (y) .

is a fuzzy subalgebra of X for all ε, δ ∈ [0, 1] with ε > δ. (ε,δ)

Conversely, suppose n that (2) is valid. o Let x, y ∈ F. Then µF

(ε,δ) µF (x

∗ y) ≥ min

(ε,δ) (ε,δ) µF (x), µF (y)

(ε,δ)

(x) = ε and µF

(y) = ε. It follows that

= ε. Thus x ∗ y ∈ F, and therefore F is a subalgebra of X.



Definition 3.2 ([4]). A fuzzy set µ in X is said to be an (α, β)-fuzzy subalgebra of X, where α, β ∈ {∈ , q , ∈ ∨ q , ∈ ∧ q } and α 6= ∈ ∧ q , if it satisfies the following condition: xt1 α µ, yt2 α µ ⇒ (x ∗ y)min{t1 ,t2 } β µ.

(3.1) for all x, y ∈ X and t1 , t2 ∈ (0, 1].

Lemma 3.3 ([4]). A fuzzy set µ in X is an (∈, ∈ ∨ q )-fuzzy subalgebra of X if and only if it satisfies: (∀x, y ∈ X) (µ(x ∗ y) ≥ min{µ(x), µ(y), 0.5}) .

(3.2)

(ε,δ)

Theorem 3.4. If F is a subalgebra of X, then the fuzzy set µF

is an (∈, ∈ ∨ q )-fuzzy subalgebra of X

for all ε, δ ∈ [0, 1] with ε > δ. Proof. Assume that F is a subalgebra of X and let ε, δ ∈ [0, 1] such that ε > δ. For any x, y ∈ X, if x, y ∈ F , then x ∗ y ∈ F and so (ε,δ)

µF (ε,δ)

If x ∈ / F or y ∈ / F, then µF

n o (ε,δ) (ε,δ) (x ∗ y) = ε ≥ min µF (x), µF (y), 0.5 . (ε,δ)

(x) = δ or µF

(y) = δ. Hence n o (ε,δ) (ε,δ) (ε,δ) µF (x ∗ y) ≥ δ ≥ min µF (x), µF (y), 0.5 . (ε,δ)

It follows from Lemma 3.3 that µF

is an (∈, ∈ ∨ q )-fuzzy subalgebra of X for all ε, δ ∈ [0, 1] with

ε > δ.



Corollary 3.5. A non-empty subset F of X is a subalgebra of X if and only if the characteristic function χF of F is an (∈, ∈ ∨ q )-fuzzy subalgebra of X.

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Proof. The necessity is by taking ε = 1 and δ = 0 in Theorem 3.4. Conversely, suppose that the characteristic function χF of F is an (∈, ∈ ∨ q )-fuzzy subalgebra of X. Let x, y ∈ F. Then χF (x) = 1 = χF (y), which implies from (3.2) that χF (x ∗ y) ≥ min{χF (x), χF (y), 0.5} = min{1, 0.5} = 0.5. Hence x ∗ y ∈ F , and therefore F is a subalgebra of X.



We consider the converse of Theorem 3.4. (ε,δ)

Theorem 3.6. For any ε, δ ∈ [0, 1] such that δ < ε ≤ 0.5, if the fuzzy set µF

is an (∈, ∈ ∨ q )-fuzzy

subalgebra of X then F is a subalgebra of X. (ε,δ)

(ε,δ)

Proof. Let x, y ∈ F. Then µF

(x) = ε = µF (y). Using Lemma 3.3, we have n o (ε,δ) (ε,δ) (ε,δ) µF (x ∗ y) ≥ min µF (x), µF (y), 0.5 = min{ε, 0.5} = ε,

and so x ∗ y ∈ F. Therefore F is a subalgebra of X.

 (ε,δ)

Theorem 3.7. Let ε, δ ∈ [0, 1] such that ε > δ. If F is a subalgebra of X, then the fuzzy set µF (∈, q )-fuzzy subalgebra of X whenever if any element t in (0, 1] satisfies xt ∈

(ε,δ) µF

is an

for x ∈ X then δ < t

and 1 − t < ε. (ε,δ)

Proof. Let x, y ∈ X and t1 , t2 ∈ (0, 1] be such that xt1 ∈ µF

(ε,δ)

(ε,δ)

(x) ≥ t1 > δ (ε,δ) and ≥ t2 > δ. It follows that = ε = µF (y), and so x, y ∈ F. Since F is a subalgebra (ε,δ) (ε,δ) of X, we have x ∗ y ∈ F. Hence µF (x ∗ y) = ε, and thus µF (x ∗ y) + min{t1 , t2 } = ε + min{t1 , t2 } > 1 (ε,δ) (ε,δ) which shows that (x ∗ y)min{t1 ,t2 } q µF . Therefore µF is an (∈, q )-fuzzy subalgebra of X.  (ε,δ) µF (y)

and yt2 ∈ µF

. Then µF

(ε,δ) µF (x)

(ε,δ)

Theorem 3.8. Let ε, δ ∈ [0, 1] such that ε > δ. If ε + δ ≤ 1 and the fuzzy set µF

is an (∈, q)-fuzzy

subalgebra of X, then F is a subalgebra of X. (ε,δ)

Proof. Assume that ε + δ ≤ 1 and the fuzzy set µF Then

(ε,δ) µF (x)

which implies

(ε,δ) = ε = µF (y), and so (ε,δ) that µF (x ∗ y) + ε >

xε ∈

(ε,δ) µF

is an (∈, q)-fuzzy subalgebra of X. Let x, y ∈ F. (ε,δ)

and yε ∈ µF

1. Therefore

(ε,δ) µF (x

(ε,δ)

. Hence (x ∗ y)ε = (x ∗ y)min{ε,ε} q µF

∗ y) > 1 − ε ≥ δ, and thus

(ε,δ) µF (x

,

∗ y) = ε,

that is, x ∗ y ∈ F. Consequently, F is a subalgebra of X.



If we take ε = 1 and δ = 0 in Theorems 3.7 and 3.8, then we have the following corollary. Corollary 3.9. A non-empty subset F of X is a subalgebra of X if and only if the characteristic function χF of F is an (∈, q)-fuzzy subalgebra of X. (ε,δ)

Theorem 3.10. Let ε, δ ∈ [0, 1] such that ε > δ. If F is a subalgebra of X, then the fuzzy set µF (ε,δ)

is a (q, q)-fuzzy subalgebra of X whenever if any element t in (0, 1] satisfies xt ∈ µF

for x ∈ X then

δ ≤ 1 − t < ε.

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Proof. Let x, y ∈ X and t1 , t2 ∈ (0, 1] be such that xt1 q µF (ε,δ) and µF (y) + t2 > 1, which (ε,δ) (ε,δ) µF (x) = ε = µF (y) and (ε,δ) µF (x ∗ y) = ε. Thus

imply that

(ε,δ) µF (x)

(ε,δ)

and yt2 q µF

> 1 − t1 ≥ δ and

(ε,δ) µF (y)

5 (ε,δ)

. Then µF

(x) + t1 > 1

> 1 − t2 ≥ δ. It follows that

so that x, y ∈ F. Since F is a subalgebra of X, we have x ∗ y ∈ F and so (ε,δ)

µF (ε,δ)

that is, (x ∗ y)min{t1 ,t2 } q µF

(x ∗ y) + min{t1 , t2 } = ε + min{t1 , t2 } > 1, (ε,δ)

. This shows that µF

is a (q, q)-fuzzy subalgebra of X.

 (ε,δ)

Theorem 3.11. Let ε, δ ∈ [0, 1] such that ε > max{δ, 0.5} and ε + δ ≤ 1. If the fuzzy set µF

is a

(q, q)-fuzzy subalgebra of X, then F is a subalgebra of X. (ε,δ)

Proof. Let x, y ∈ F. Then µF

(ε,δ)

µF (ε,δ)

that is, xε q µF (x ∗

(ε,δ)

(x) = ε = µF

(ε,δ)

(x) + ε = ε + ε > 1 and µF

(ε,δ)

and yε q µF

(ε,δ) y)min{ε,ε} q µF .

Hence

(y), which implies that

(ε,δ)

. Since µF

(ε,δ) µF (x

(y) + ε = ε + ε > 1,

is a (q, q)-fuzzy subalgebra of X, it follows that (x ∗ y)ε = (ε,δ)

∗ y) > 1 − ε ≥ δ, and therefore µF

(x ∗ y) = ε. This proves that

x ∗ y ∈ F , and F is a subalgebra of X.



If we take ε = 1 and δ = 0 in Theorems 3.10 and 3.11, then we have the following corollary. Corollary 3.12. A non-empty subset F of X is a subalgebra of X if and only if the characteristic function χF of F is a (q, q)-fuzzy subalgebra of X. (ε,δ)

Theorem 3.13. Let ε, δ ∈ [0, 1] such that ε > δ. If F is a subalgebra of X, then the fuzzy set µF a (q, ∈)-fuzzy subalgebra of X whenever if any element t in (0, 1] satisfies xt ∈

(ε,δ) µF

is

for x ∈ X then

δ ≤ 1 − t and t < ε. (ε,δ)

Proof. Let x, y ∈ X and t1 , t2 ∈ (0, 1] be such that xt1 q µF (ε,δ) and µF (y) + t2 > 1, (ε,δ) (ε,δ) µF (x) = ε = µF (y),

which imply that

(ε,δ) µF (x)

(ε,δ)

and yt2 q µF

> 1 − t1 ≥ δ and

(ε,δ)

. Then µF

(ε,δ) µF (y)

(x) + t1 > 1

> 1 − t2 ≥ δ. Hence

and so x, y ∈ F. Since F is a subalgebra of X, we have x ∗ y ∈ F and thus (ε,δ)

µF (ε,δ)

that is, (x ∗ y)min{t1 ,t2 } ∈ µF

(x ∗ y) = ε ≥ min{t1 , t2 }, (ε,δ)

. This shows that µF

is a (q, ∈)-fuzzy subalgebra of X. (ε,δ)

Theorem 3.14. Let ε, δ ∈ [0, 1] such that ε > max{δ, 0.5}. If the fuzzy set µF



is a (q, ∈)-fuzzy

subalgebra of X, then F is a subalgebra of X. (ε,δ)

Proof. Let x, y ∈ F. Then µF

(ε,δ)

µF (ε,δ)

that is, xε q µF

(x ∗ y)min{ε,ε} ∈

(ε,δ) µF

(y), which implies that (ε,δ)

(x) + ε = ε + ε > 1 and µF

(ε,δ)

and yε q µF

(ε,δ)

(x) = ε = µF

(ε,δ)

. Since µF

and so that

(ε,δ) µF (x

(y) + ε = ε + ε > 1,

is a (q, ∈)-fuzzy subalgebra of X, it follows that (x ∗ y)ε =

∗ y) = ε, that is, x ∗ y ∈ F. Therefore F is a subalgebra of

X.

 If we take ε = 1 and δ = 0 in Theorems 3.13 and 3.14, then we have the following corollary.

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Corollary 3.15. A non-empty subset F of X is a subalgebra of X if and only if the characteristic function χF of F is a (q, ∈)-fuzzy subalgebra of X. (ε,δ)

Theorem 3.16. Let ε, δ ∈ [0, 1] such that ε > δ. If F is a subalgebra of X, then the fuzzy set µF (ε,δ) µF

(∈, ∈ ∧ q )-fuzzy subalgebra of X whenever if any element t in (0, 1] satisfies xt ∈

is an

for x ∈ X then

δ < t and 1 − t < ε. (ε,δ)

Proof. Let x, y ∈ X and t1 , t2 ∈ (0, 1] be such that xt1 ∈ µF and

(ε,δ) µF (y)

(ε,δ)

and yt2 ∈ µF

(ε,δ)

. Then µF

(x) ≥ t1 > δ

≥ t2 > δ, which imply that x, y ∈ F and ε ≥ min{t1 , t2 }. Since F is a subalgebra of X, we (ε,δ)

have x ∗ y ∈ F. Hence µF

(ε,δ)

(x ∗ y) = ε ≥ min{t1 , t2 }, i.e., (x ∗ y)min{t1 ,t2 } ∈ µF

min{t1 , t2 } = ε + min{t1 , t2 } > 1 and so (x ∗ and consequently

(ε,δ) µF

(ε,δ) y)min{t1 ,t2 } q µF .

(ε,δ)

. Now, µF

(x ∗ y) + (ε,δ)

Therefore (x ∗ y)min{t1 ,t2 } ∈ ∧ q µF

is an (∈, ∈ ∧ q )-fuzzy subalgebra of X.

,

 (ε,δ)

Theorem 3.17. Let ε, δ ∈ [0, 1] such that ε > δ. If ε + δ ≤ 1 and the fuzzy set µF

is an (∈, ∈ ∧ q )-fuzzy

subalgebra of X, then F is a subalgebra of X. (ε,δ)

Proof. Assume that ε + δ ≤ 1 and the fuzzy set µF Then

(ε,δ) µF (x)

=ε=

(ε,δ) µF (y),

and so xε ∈

that is, (x ∗ y)ε = (x ∗ y)min{ε,ε} ∈ (ε,δ) µF (x ∗ y) + ε > 1. (ε,δ) then µF (x ∗ y) >

If

(ε,δ) µF

(ε,δ) µF (x ∗ y)

(ε,δ) µF

(ε,δ)

and yε ∈ µF

and (x ∗ y)ε =

≥ ε, then

1 − ε ≥ δ and so

is an (∈, ∈ ∧ q )-fuzzy subalgebra of X. Let x, y ∈ F. (ε,δ) (x ∗ y)min{ε,ε} q µF .

(ε,δ) µF (x ∗ y)

(ε,δ) µF (x

(ε,δ)

. Hence (x∗y)ε = (x∗y)min{ε,ε} ∈ ∧ q µF Hence

= ε and thus x ∗ y ∈ F. If

,

(ε,δ) µF (x ∗ y) ≥ ε and (ε,δ) µF (x ∗ y) + ε > 1,

∗ y) = ε, which shows that x ∗ y ∈ F. Therefore F is a

subalgebra of X.



If we take ε = 1 and δ = 0 in Theorems 3.16 and 3.17, then we have the following corollary. Corollary 3.18. A non-empty subset F of X is a subalgebra of X if and only if the characteristic function χF of F is an (∈, ∈ ∧ q )-fuzzy subalgebra of X. (ε,δ)

is a

(ε,δ) µF

for

Theorem 3.19. Let ε, δ ∈ [0, 1] such that ε > δ. If F is a subalgebra of X, then the fuzzy set µF (q, ∈ ∧ q )-fuzzy subalgebra of X under the condition that if any element t in (0, 1] satisfies xt ∈ x ∈ X then δ ≤ 1 − t and t < ε. (ε,δ)

Proof. Let x, y ∈ X and t1 , t2 ∈ (0, 1] be such that xt1 q µF (ε,δ) and µF (y) + t2 > 1, (ε,δ) (ε,δ) µF (x) = ε = µF (y)

which imply that

(ε,δ) µF (x)

(ε,δ)

and yt2 q µF

> 1 − t1 ≥ δ and

(ε,δ)

. Then µF

(ε,δ) µF (y)

(x) + t1 > 1

> 1 − t2 ≥ δ. Hence

and ε > max{1 − t1 , 1 − t2 }, and so x, y ∈ F. Since F is a subalgebra of X, we

have x ∗ y ∈ F and thus (ε,δ)

µF that is, (x ∗ y)min{t1 ,t2 } ∈ (ε,δ) y)min{t1 ,t2 } q µF .

Hence

(ε,δ) µF .

Now,

(x ∗ y) = ε ≥ min{t1 , t2 },

(ε,δ) µF (x

∗ y) + min{t1 , t2 } = ε + min{t1 , t2 } > 1, and so (x ∗

(ε,δ) (x ∗ y)min{t1 ,t2 } ∈ ∧ q µF ,

(ε,δ)

and µF

is a (q, ∈ ∧ q )-fuzzy subalgebra of X.  (ε,δ)

Theorem 3.20. Let ε, δ ∈ [0, 1] such that ε > max{δ, 0.5}. If the fuzzy set µF

is a (q, ∈ ∧ q )-fuzzy

subalgebra of X, then F is a subalgebra of X. (ε,δ)

Proof. Let x, y ∈ F. Then µF

(ε,δ)

(x) = ε = µF

(y), which implies that

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µF (ε,δ)

that is, xε q µF (x ∗

(ε,δ)

(x) + ε = ε + ε > 1 and µF

(ε,δ)

and yε q µF

(ε,δ) y)min{ε,ε} ∈ ∧ q µF

(ε,δ)

. Since µF

and so that

7

(y) + ε = ε + ε > 1,

is a (q, ∈ ∧ q )-fuzzy subalgebra of X, it follows that (x∗y)ε =

(ε,δ) µF (x

∗ y) ≥ ε. Hence x ∗ y ∈ F and F is a subalgebra of X.



If we take ε = 1 and δ = 0 in Theorems 3.19 and 3.20, then we have the following corollary. Corollary 3.21. A non-empty subset F of X is a subalgebra of X if and only if the characteristic function χF of F is a (q, ∈ ∧ q )-fuzzy subalgebra of X. (ε,δ)

Theorem 3.22. Let ε, δ ∈ [0, 1] such that ε > δ. Assume that if any element t in (0, 1] satisfies xt ∈ µF (ε,δ)

for x ∈ X then δ ≤ 1−t. If F is a subalgebra of X, then the fuzzy set µF

is a (q, ∈ ∨ q )-fuzzy subalgebra

of X. (ε,δ)

Proof. Let x, y ∈ X and t1 , t2 ∈ (0, 1] be such that xt1 q µF

(ε,δ)

and yt2 q µF

(ε,δ)

. Then µF

(x) + t1 > 1 (ε,δ) (ε,δ) (ε,δ) and µF (y) + t2 > 1, which imply that µF (x) > 1 − t1 ≥ δ and µF (y) > 1 − t2 ≥ δ. Hence (ε,δ) (ε,δ) µF (x) = ε = µF (y), and so ε > max{1 − t1 , 1 − t2 } and x, y ∈ F. Since F is a subalgebra of X, we (ε,δ) (ε,δ) have x∗y ∈ F and thus µF (x∗y) = ε which implies that µF (x∗y)+min{t1 , t2 } = ε+min{t1 , t2 } > 1, (ε,δ) (ε,δ) (ε,δ) i.e., (x ∗ y)min{t1 ,t2 } q µF . It follows that (x ∗ y)min{t1 ,t2 } ∈ ∨ q µF . Therefore µF is a (q, ∈ ∨ q )-fuzzy subalgebra of X.

 (ε,δ)

Theorem 3.23. Let ε, δ ∈ [0, 1] such that ε > max{δ, 0.5} and ε + δ ≤ 1. If the fuzzy set µF

is a

(q, ∈ ∨ q )-fuzzy subalgebra of X, then F is a subalgebra of X. (ε,δ)

Proof. Let x, y ∈ F. Then µF

(ε,δ)

µF (ε,δ)

that is, xε q µF

(y), which implies that (ε,δ)

(x) + ε = ε + ε > 1 and µF

(ε,δ)

and yε q µF

(ε,δ)

(x) = ε = µF

(ε,δ)

. Since µF

(y) + ε = ε + ε > 1,

is a (q, ∈ ∨ q )-fuzzy subalgebra of X, it follows that (x∗y)ε =

(ε,δ) (ε,δ) (x∗y)min{ε,ε} ∈ ∨ q µF , that is, µF (x∗y) ≥ (ε,δ) (ε,δ) If µF (x ∗ y) + ε > 1, then µF (x ∗ y) > 1 − ε

(ε,δ)

ε or µF

(ε,δ)

(x∗y)+ε > 1. If µF

≥ δ and so

(ε,δ) µF (x ∗ y)

(x∗y) ≥ ε, then x∗y ∈ F.

= ε. Thus x ∗ y ∈ F, and therefore

F is a subalgebra of X.



If we take ε = 1 and δ = 0 in Theorems 3.22 and 3.23, then we have the following corollary. Corollary 3.24. A non-empty subset F of X is a subalgebra of X if and only if the characteristic function χF of F is a (q, ∈ ∨ q )-fuzzy subalgebra of X. Conclusions We have introduced the notion of (ε, δ)-characteristic fuzzy sets in BCK/BCI-algebras. Given a subalgebra F of a BCK/BCI-algebra X, we have provided conditions for the (ε, δ)-characteristic fuzzy set in X to be an (∈, ∈ ∨ q)-fuzzy subalgebra, an (∈, q)-fuzzy subalgebra, an (∈, ∈ ∧ q)-fuzzy subalgebra, a (q, q)-fuzzy subalgebra, a (q, ∈)-fuzzy subalgebra, a (q, ∈ ∨ q)-fuzzy subalgebra and a (q, ∈ ∧ q)-fuzzy (ε,δ)

subalgebra. Using the notions of (α, β)-fuzzy subalgebra µF

1063

, we have investigated conditions for the

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G. Muhiuddin and Abdullah M. Al-roqi

8

F to be a subalgebra of X where (α, β) is one of (∈, ∈ ∨ q), (∈, ∈ ∧ q), (∈, q), (q, ∈ ∨ q), (q, ∈ ∧ q), (q, ∈) and (q, q). In the consecutive research, we will discuss the following items: (1) Given a subalgebra F of a BCK/BCI-algebra X, we will provide conditions for the (ε, δ)characteristic fuzzy set in X to be an (∈ ∨ q, ∈ ∨ q)-fuzzy subalgebra, an (∈ ∨ q, ∈)-fuzzy subalgebra, an (∈ ∨ q, ∈ ∧ q)-fuzzy subalgebra, and an (∈ ∨ q, q)-fuzzy subalgebra. (ε,δ)

(2) Using the notions of (α, β)-fuzzy subalgebra µF

where (α, β) is one of (∈ ∨ q, ∈ ∨ q), (∈ ∨ q, ∈),

(∈ ∨ q, ∈ ∧ q) and (∈ ∨ q, q), we investigate conditions for the F to be a subalgebra of X. 4. Acknowledgements This research was partially supported by the Deanship of Scientific Research (DSR), King Abdulaziz University, Jeddah 21589, Ministry of Higher Education, Saudi Arabia.The authors would like to express their sincere thanks to the anonymous referees. References [1] S. K. Bhakat and P. Das, (∈, ∈ ∨ q)-fuzzy subgroup, Fuzzy Sets and Systems 80 (1996), 359–368. [2] Y. S. Huang, BCI-algebra, Science Press, Beijing, 2006. [3] Y. B. Jun, On (α, β)-fuzzy ideals of BCK/BCI-algebras, Sci. Math. Jpn. 60(3) (2004), 613–617. [4] Y. B. Jun, On (α, β)-fuzzy subalgebras of BCK/BCI-algebras, Bull. Korean Math. Soc. 42(4) (2005), 703–711. [5] Y. B. Jun, Fuzzy subalgebras of type (α, β) in BCK/BCI-algebras, Kyungpook Math. J. 47 (2007), 403–410. [6] J. Meng and Y. B. Jun, BCK-algebra, Kyungmoon Sa Co. Seoul, 1994. [7] P. M. Pu and Y. M. Liu, Fuzzy topology I, Neighborhood structure of a fuzzy point and Moore-Smith convergence, J. Math. Anal. Appl. 76 (1980), 571–599. [8] J. Zhan, Y. B. Jun and B. Davvaz, On (∈, ∈ ∨ q )-fuzzy ideals of BCI-algebras, Iran. J. Fuzzy Syst. 6(1) (2009), 81–94.

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Existence results for nonlinear fractional integrodifferential equations with antiperiodic type integral boundary conditions Xiaohong Zuo and Wengui Yang∗ Ministry of Public Education, Sanmenxia Polytechnic, Sanmenxia, Henan 472000, China

Abstract: This paper investigates the existence of solutions for a class of nonlinear boundary value problems involving fractional integrodifferential equations of fractional order α ∈ (2, 3] with antiperiodic type integral boundary conditions. Our results are based on contraction mapping principle and Krasnoselskii fixed point theorem. As an application, an interesting example is presented to illustrate the main results. Keywords: Fractional integrodifferential equations; antiperiodic boundary conditions; integral boundary conditions; fixed point theorem 2010 Mathematics Subject Classification: 34A08, 34B18.

1

Introduction

In the last few decades, the topic of fractional differential equations has gained a considerable attention and it has emerged as a popular field of research due to its extensive development and applications in several disciplines such as physics, mechanics, chemistry, engineering, etc. Therefore, there have been many papers and books dealing with the theoretical development of fractional calculus and the solutions or positive solutions of boundary value problems for nonlinear fractional differential equations; for examples and details, one can see [1, 2, 3, 4, 5, 6, 7] and references along these lines. For instance, Ahmad and Sivasundaram [8] proved the existence and uniqueness of solutions for a four-point nonlocal boundary value problem of nonlinear integro-differential equations of fractional order q ∈ (1, 2] by applying some standard fixed point theorems. Ahmad and Ntouyas [9, 10] studied the existence and uniqueness results for a class of nonlocal boundary value problems of nonlinear differential equations and inclusions of fractional order with strip conditions by using a variety of fixed point theorems. Recently, antiperiodic and/or integral boundary value problems of fractional differential equations occur in the mathematic modeling of a variety of physical processes and have been studied by a number of authors. For examples and details of antiperiodic and/or integral fractional boundary conditions, see [11, 12, 13, 14, 15, 16] and the references therein. For example, Ahmad and Nieto [17] obtained the existence and uniqueness results for antiperiodic boundary value problem for nonlinear fractional differential equation of order q ∈ (1, 2] by applying some standard fixed point principles. By using Schauder’s fixed point theorem and the contraction mapping principle, Wang and Liu [18] considered the existence and uniqueness results for antiperiodic fractional boundary value problem with fractional derivative. In [19], the authors investigated the following boundary value problem for a nonlinear fractional integrodifferential equation with integral boundary conditions c

Dq x(t) = f (t, x(t), (χx)(t)), t ∈ [0, 1], q ∈ (1, 2], Z 1 Z 1 q1 (x(s))ds, αx(1) + βx0 (1) = q2 (x(s))ds, αx(0) + β 0 (0) = 0

where f : [0, 1] × X × X → X, (χx)(t) =

0

Rt 0

γ(t, s)x(s)ds for γ : [0, T ] × [0, T ] → [0, +∞), q1 , q2 : X → X

∗ Corresponding

author. Email:[email protected] (X. Zuo) and [email protected] (W. Yang)

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and α > 0 and β ≥ 0 are real numbers. The authors established sufficient conditions for the existence of solutions for the above boundary value problems. In [20], Wang et al. studied the following fractional boundary value problem with antiperiodic boundary conditions c

Dα x(t) = f (t, x(t)), c

x(0) = −x(T ),

p

t ∈ [0, T ],

c

p

D x(0) = − D x(T ),

α ∈ (2, 3],

c

Dq x(0) = −c Dq x(T ),

where 0 < p < 1 < q < 2, and f : [0, T ] × R → R is a given continuous function. Some existence and uniqueness results are obtained contraction mapping principle and Leray-Schauder’s fixed point theorem. In [21], Chai concerned with the following antiperiodic boundary value problems of fractional differential equations c

x(0) = −x(T ),

Dα x(t) = f (t, x(t),c Dα1 (t),c Dα2 (t)),

β1 −1c

t

β1 −1c

β1

D x(t)|t→0 = −t

β1

D x(t)|t=1 ,

t ∈ (0, 1), β2 −1c

t

α ∈ (2, 3],

D x(t)|t→0 = −tβ2 −1c Dβ2 x(t)|t=1 , β2

where 2 < α ≤ 3, 0 < α1 ≤ 1 < α2 ≤ 2, 0 < β1 < 1 < β2 < 2, and f is a given continuous function. The author obtained some existence results by applying the Banach contraction mapping principle and the Leray-Schauder degree theory. In [22], Alsaedi discussed existence of solutions for the following integrodifferential equations of fractional order with antiperiodic boundary conditions c

Dq x(t) = f (t, x(t), (χx)(t)),

t ∈ [0, T ],

0

x(0) = −x(T ),

q ∈ (1, 2],

0

x (0) = −x (T ),

Rt

where f : [0, 1] × X × X → X, (χx)(t) = 0 γ(t, s)x(s)ds for γ : [0, 1] × [0, 1] → [0, +∞). In [23], Ahmad et al. considered the existence and uniqueness of the solutions for a new class of boundary value problems of nonlinear fractional differential equations with non-separated type integral boundary conditions c

Dq x(t) = f (t, x(t)), Z T x(0) − λ1 x(T ) = µ1 g(s, x(s))ds,

t ∈ [0, T ],

q ∈ (1, 2], Z 0 0 x (0) − λ2 x (T ) = µ2

0

T

h(s, x(s))ds,

0

where f, g, g : [0, T ] × R → R are given continuous functions and λj , µj ∈ R (λj 6= 0), j = 1, 2. Ahmad and Ntouyas [24] and Ahmad et al. [25] study a boundary value problem of fractional differential eqations and inclusions with anti-periodic type integral boundary conditions given by c

Dα x(t) = f (t, x(t)) (∈ F (t, x(t))), t ∈ [0, T ], α ∈ (2, 3], Z T x(j) (0) − λj x(j) (T ) = µj gj (s, x(s))ds, j = 0, 1, 2, 0

(j)

(0)

where x (·) denotes jth derivative of x(·) with x = x(·), gj : [0, T ] × R → R are given continuous functions and λj , µj ∈ R (λj 6= 0), respectively. In this paper, motivated greatly by the above mentioned works, we consider the following boundary value problem for a nonlinear fractional integrodifferential equation of fractional order α ∈ (2, 3] with antiperiodic type integral boundrary conditions Dα u(t) = f (t, u(t), (φu)(t), (ψu)(t)), t ∈ J = [0, T ] (T > 0), Z T Z T u(0) = µ0 u(T ) + ν0 g0 (s, u(s))ds, wi (0) = µi wi (T ) + νi gi (s, u(s))ds, c

0

(1.1) i = 1, 2,

0

where c Dα is the standard Caputo fractional derivative of fractional order α, 0 < α1 < 1 < α2 < 2, wi (t) = tαi −ic Dαi u(t), wi (0) = limt→0+ wi (t), wi (T ) = [wi (t)]t=T , µ0 , ν0 , µi , νi ∈ R (µ0 , µi 6= 1), i = 1, 2, the 2 1066

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nonlinear function f : J × R × R × R → R is continuous, gi : J × R → R are given continuous functions for i = 0, 1, 2, and for γ, δ : [0, T ] × [0, T ] → [0, ∞), Z (φu)(t) =

t

Z γ(t, s, u(s))ds,

T

(ψu)(t) =

0

δ(t, s, u(s))ds. 0

Let C = C(J , R) be Banach space of all continuous functions from J → R endowed with a topology of uniform convergence with the norm denoted by kϕk = sup{|ϕ(t)| : s ∈ J }.

2

Preliminaries

For the convenience of the reader, we give some background materials from fractional calculus theory to facilitate analysis of the problem (1.1). These materials can be found in the recent literature, see [27, 29, 30]. Definition 2.1. For at least n-times continuously differentiable function f : [0, ∞) → R, the Caputo derivative of fractional order α is defined as Z t 1 c α D f (t) = (t − s)n−α−1 f (n) (s)ds, n = [α] + 1, Γ(n − α) 0 where [α] denotes the integer part of the real number α. Definition 2.2. The Riemann-Liouville fractional integral of order α for a function f is defined as Z t 1 α (t − s)α−1 f (s)ds, α > 0, I f (t) = Γ(α) 0 provided that such integral exists. Definition 2.3. The Riemann-Liouville fractional derivative of order α for a function f is defined by  n Z t 1 d α D f (t) = (t − s)n−α−1 f (s)ds, n = [α] + 1, Γ(n − α) dt 0 provided that the right-hand side of the previous equation is pointwise defined on (0, ∞). Lemma 2.4 ([26, 28]). Let α > 0, then the fractional differential equation c

Dα u(t) = 0

has a unique solution given by the expression u(t) =

[α] X u(j) (0) j=0

j!

tj .

Lemma 2.5 ([26, 29]). Let α > 0, then I αc Dα u(t) = u(t) −

[α] X u(j) (0) j=0

j!

tj .

In view of Lemma 2.5, it follows that I αc Dα u(t) = u(t) + c0 + c1 t + c2 t2 + · · · + cn−1 tn−1 ,

(2.1)

for some ci ∈ R, i = 0, 1, 2, . . . , n − 1 (n = [α] + 1). 3 1067

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Lemma 2.6. For any y ∈ C[0, T ], the unique solution of the linear fractional boundary value problem c

Dα u(t) = y(t), t ∈ [0, T ], α ∈ (2, 3], Z T Z u(0) = µ0 u(T ) + ν0 g0 (s, u(s))ds, wi (0) = µi wi (T ) + νi 0

T

(2.2) gi (s, u(s))ds,

0

is given by Z u(t) =

T

Z G(t, s)y(s)ds + ν0 ξ0

0

T

Z g0 (s, u(s))ds + ν1 η1 (t)

0

where G(t, s)         G(t, s) =       

T

Z g1 (s, u(s))ds + ν2 η2 (t)

0

T

g2 (s, u(s))ds, 0

is Green’s function given by (t − s)α−1 + µ0 ξ0 (T − s)α−1 Γ(α) µ1 η1 (t)T α1 −1 (T − s)α−α1 −1 µ2 η2 (t)T α2 −2 (T − s)α−α2 −1 + + , Γ(α − α1 ) Γ(α − α2 ) α1 −1 α−α1 −1 α2 −2 α−1 µ1 η1 (t)T (T − s) µ2 η2 (t)T (T − s)α−α2 −1 µ0 ξ0 (T − s) + + , Γ(α) Γ(α − α1 ) Γ(α − α2 )

s ≤ t,

(2.3)

t ≤ s,

η1 (t) = ξ1 [µ0 T + (1 − µ0 )t], η2 (t) = ξ2 [µ0 [2µ1 + (1 − µ1 )(2 − α1 )]T 2 + 2µ1 T (1 − µ0 )t + (1 − µ0 )(1 − µ1 )(2 − α1 )t2 ], ξ0 =

1 , 1 − µ0

ξ1 =

Γ(2 − α1 ) , (1 − µ0 )(1 − µ1 )

ξ2 =

Γ(3 − α2 ) . 2(1 − µ0 )(1 − µ1 )(1 − µ2 )(2 − α1 )

Proof. Using (2.1), for some constants b0 , b1 , b2 ∈ R, we have u(t) = I α y(t) + b0 + b1 t + b2 t2 . Using the facts that c Dα1 b = 0 (b is constant and 0 < α1 < 1), c Dα1 t = c α1 α D I y(t) = I α−α1 y(t), we obtain c

Dα1 u(t) = I α−α1 y(t) +

In view of c Dα2 t = 0 (1 < α2 < 2), c Dα2 t2 = c

(2.4) t1−α1 c α1 2 Γ(2−α1 ) , D t

b1 2b2 t1−α1 + t2−α1 . Γ(2 − α1 ) Γ(3 − α1 )

2t2−α2 Γ(3−α2 ) ,

=

2t2−α1 Γ(3−α1 ) ,

and

(2.5)

and c Dα2 I α y(t) = I α−α2 y(t), we get

Dα2 u(t) = I α−α2 y(t) +

2b2 t2−α2 . Γ(3 − α2 )

(2.6)

From (2.5) and (2.6), we have tα1 −1c Dα1 u(t) = tα1 −1 I α−α1 y(t) +

b1 2b2 + t, Γ(2 − α1 ) Γ(3 − α1 )

(2.7)

2b2 . Γ(3 − α2 )

(2.8)

tα2 −2c Dα2 u(t) = tα2 −2 I α−α2 y(t) + From [?], we can know that lim tα1 −1 I α−α1 y(t) = 0,

t→0+

lim tα2 −2 I α−α2 y(t) = 0.

t→0+

(2.9)

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Applying the boundary conditions for problem (2.2) and (2.9) in (2.4), (2.7) and (2.8), we find that µ0 µ0 µ1 Γ(2 − α1 )T α1 α−α1 y(T ) I α y(T ) + I 1 − µ0 (1 − µ0 )(1 − µ1 )

b0

=

b1

Z T µ0 µ2 Γ(3 − α2 )[2µ1 + (1 − µ1 )(2 − α1 )]T α2 α−α2 ν0 g0 (s, u(s))ds I y(T ) + 2(1 − µ0 )(1 − µ1 )(1 − µ2 )(2 − α1 ) 1 − µ0 0 Z T Z µ0 ν1 Γ(2 − α1 )T µ0 ν2 Γ(3 − α2 )[2µ1 + (1 − µ1 )(2 − α1 )]T 2 T + g1 (s, u(s))ds + g2 (s, u(s))ds, (1 − µ0 )(1 − µ1 ) 0 2(1 − µ0 )(1 − µ1 )(1 − µ2 )(2 − α1 ) 0 µ1 Γ(2 − α1 )T α1 −1 α−α1 µ1 µ2 Γ(3 − α2 )T α2 −1 = I y(T ) + I α−α2 y(T ) 1 − µ1 (1 − µ1 )(1 − µ2 )(2 − α1 ) Z Z T µ1 ν2 Γ(3 − α2 )T ν1 Γ(2 − α1 ) T g1 (s, u(s))ds + g2 (s, u(s))ds, + 1 − µ1 (1 − µ1 )(1 − µ2 )(2 − α1 ) 0 0 Z µ2 Γ(3 − α2 )T α2 −2 α−α2 ν2 Γ(3 − α2 ) T g2 (s, u(s))ds. = I y(T ) + 2(1 − µ2 ) 2(1 − µ2 ) 0 +

b2

Thus, the unique solution of (2.2) is u(t)

= I α y(t) + µ0 ξ0 I α y(T ) + µ1 η1 (t)T α1 −1 I α−α1 y(T ) + µ2 η2 (t)T α2 −2 I α−α2 y(T ) Z T Z T Z T +ν0 ξ0 g0 (s, u(s))ds + ν1 η1 (t) g1 (s, u(s))ds + ν2 η2 (t) g2 (s, u(s))ds 0

Z =

0

T

Z G(t, s)y(s)ds + ν0 ξ0

0

0

T

Z g0 (s, u(s))ds + ν1 η1 (t)

0

T

Z

T

g1 (s, u(s))ds + ν2 η2 (t) 0

g2 (s, u(s))ds, 0

where G(t, s) is given by (2.3). The proof is completed.

3

Main results

For the sake of simplicity, we always consider the boundary value problem (1.1) together with the following assumptions. (H1 ) There exist positive constants Li and Mi (i = 0, 1, 2) such that kgi (t, u) − gi (t, v)k ≤ Li kx − yk, kgi (t, u)k ≤ Mi , ∀t ∈ J , u, v ∈ R. (H2 ) There exist continuous function Li : [0, T ] → R+ = [0, ∞) (i = 1, 2, 3) such that kf (t, u, φu, ψu)−f (t, v, φv, ψv)k ≤ L1 (t)ku−vk+L2 (t)kφu−φvk+L3 (t)kψu−ψvk, ∀t ∈ J , u, v ∈ R. (H3 ) There exist continuous functions p, q : [0, T ] → R+ such that

Z t



(γ(t, s, u(s)) − γ(t, s, v(s)))ds ≤ p(t)ku − vk, ∀t ∈ J , u, v ∈ R,

0

Z

T

(δ(t, s, u(s)) − δ(t, s, v(s)))ds ≤ q(t)ku − vk, ∀t ∈ J , u, v ∈ R.

0

(H4 ) ρ = (L0 |ν0 ξ0 | + L1 λ1 |ν1 | + L2 λ2 |ν2 |)T + L3 λ3 < 1, where λ1 = sup{|η1 (t)| : t ∈ J },

λ2 = sup{|η2 (t)| : t ∈ J },

L3 = sup{M (t) = L1 (t) + L2 (t)p(t) + L3 (t)q(t) : t ∈ J }. λ3 =

(1 + |µ0 ξ0 |)T α λ1 |µ1 |T α−1 λ2 |µ2 |T α−2 + + , Γ(α + 1) Γ(α − α1 + 1) Γ(α − α2 + 1) 5 1069

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(H5 ) kf (t, u(t), (φu)(t), (ψu)(t))k ≤ ω(t), for all (t, u, φu, ψu) ∈ [0, T ] × R × R × R, where ω ∈ L1 ([, T ], R+ ). Theorem 3.1. Assume that (H1 )-(H4 ) hold. Then the boundary value problem (1.1) has a unique solution on J . Proof. Define F : C → C by Z t (t − s)α−1 (F u)(t) = f (s, u(s), (φu)(s), (ψu)(s))ds Γ(α) 0 Z T (T − s)α−1 f (s, u(s), (φu)(s), (ψu)(s))ds +µ0 ξ0 Γ(α) 0 Z T (T − s)α−α1 −1 +µ1 η1 (t)T α1 −1 f (s, u(s), (φu)(s), (ψu)(s))ds Γ(α − α1 ) 0 Z T (T − s)α−α2 −1 α2 −2 f (s, u(s), (φu)(s), (ψu)(s))ds +µ2 η2 (t)T Γ(α − α2 ) 0 Z T Z T Z T +ν0 ξ0 g0 (s, u(s))ds + ν1 η1 (t) g1 (s, u(s))ds + ν2 η2 (t) g2 (s, u(s))ds, 0

0

t ∈ J.

0

Let us set ¯2, sup{L2 (t) : t ∈ J } = L

¯3, sup{L3 (t) : t ∈ J } = L

  Z t

: t ∈ J = M3 , γ(t, τ, 0)dτ sup

0

 Z

sup

0

 t

: t ∈ J = M4 , δ(t, τ, 0)dτ

sup{kf (t, 0, 0, 0)k : t ∈ J } = M5 .

According to the assumptions (H2 ) and (H3 ), we have

Z s

Z s





γ(s, τ, 0)dτ k(φu)(s)k ≤ (γ(s, τ, u(τ )) − γ(s, τ, 0))dτ +

0 0

Z s

γ(s, τ, 0)dτ ≤ p(s)kuk +

≤ p(s)kuk + M3 ,

0

Z

Z

T

T

δ(s, τ, 0)dτ k(ψu)(s)k ≤ (δ(s, τ, u(τ )) − δ(s, τ, 0))dτ +

0

0

Z

T

≤ q(s)kuk + δ(s, τ, 0)dτ ≤ q(s)kuk + M4 .

0

From the two above inequalities, we get kf (s, u(s), (φu)(s), (ψu)(s))k

≤

kf (s, u(s), (φu)(s), (ψu)(s)) − f (s, 0, 0, 0)k + kf (s, 0, 0, 0)k

≤ L1 (s)ku(s)k + L2 (s)k(φu)(s)k + L3 (s)k(ψu)(s))k + kf (s, 0, 0, 0)k ≤ (L1 (s) + L2 (s)p(s) + L3 (s)q(s))kuk + (L2 (s)M3 + L3 (s)M4 ) + M5 ¯ 2 M3 + L ¯ 3 M4 ) + M5 = M (s)kuk + M ∗ , ≤ M (s)kuk + (L ¯ 2 M3 + L ¯ 3 M4 ) + M5 . And consider Br = {u ∈ C : kuk ≤ r}, where r ≥ ρ1 /(1 − ρ), with where M ∗ = (L ρ1 = (M0 |ν0 ξ0 | + M1 λ1 |ν1 | + M2 λ2 |ν2 |)T + λ3 M ∗ ,

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and ρ is given by the assumption (H4 ). Now we show that F Br ⊂ Br . For u ∈ Br , we have ≤

t

(t − s)α−1 kf (s, u(s), (φu)(s), (ψu)(s))kds Γ(α) 0 Z T (T − s)α−1 +|µ0 ξ0 | kf (s, u(s), (φu)(s), (ψu)(s))kds Γ(α) 0 Z T (T − s)α−α1 −1 α1 −1 +|µ1 η1 (t)|T kf (s, u(s), (φu)(s), (ψu)(s))kds Γ(α − α1 ) 0 Z T (T − s)α−α2 −1 kf (s, u(s), (φu)(s), (ψu)(s))kds +|µ2 η2 (t)|T α2 −2 Γ(α − α2 ) 0 Z T Z T Z +|ν0 ξ0 | kg0 (s, u(s))kds + |ν1 η1 (t)| kg1 (s, u(s))kds + |ν2 η2 (t)| Z

k(F u)(t)k

0

kg2 (s, u(s))kds

0

t

Z T (t − s)α−1 (T − s)α−1 (M (s)kuk + M ∗ )ds + |µ0 ξ0 | (M (s)kuk + M ∗ )ds Γ(α) Γ(α) 0 0 Z T α−α1 −1 (T − s) (M (s)kuk + M ∗ )ds +|µ1 η1 (t)|T α1 −1 Γ(α − α1 ) 0 Z T (T − s)α−α2 −1 (M (s)kuk + M ∗ )ds +|µ2 η2 (t)|T α2 −2 Γ(α − α2 ) 0 Z T +|ν0 ξ0 | [kg0 (s, u(s)) − g0 (s, 0)k + kg0 (s, 0)k]ds Z

≤

0

T

0

Z

T

[kg1 (s, u(s)) − g1 (s, 0)k + kg1 (s, 0)k]ds

+|ν1 η1 (t)| 0

Z

T

[kg2 (s, u(s)) − g2 (s, 0)k + kg2 (s, 0)k]ds

+|ν2 η2 (t)| 0

Z T (T − s)α−1 (t − s)α−1 ds + |µ0 ξ0 | ds Γ(α) Γ(α) 0 0 ! Z T Z T α−α2 −1 α−α1 −1 (T − s) (T − s) ds + |µ2 η2 (t)|T α2 −2 ds +|µ1 η1 (t)|T α1 −1 Γ(α − α1 ) Γ(α − α2 ) 0 0

≤ (L3 r + M ∗ )

Z

t

+|ν0 ξ0 |(L0 r + M0 )T + |ν1 η1 (t)|(L1 r + M1 )T + |ν2 η2 (t)|(L2 r + M2 )T   (1 + |µ0 ξ0 |)T α λ1 |µ1 |T α−1 λ2 |µ2 |T α−2 ≤ (L3 r + M ∗ ) + + Γ(α + 1) Γ(α − α1 + 1) Γ(α − α2 + 1) +|ν0 ξ0 |(L0 r + M0 )T + λ1 |ν1 |(L1 r + M1 )T + λ2 |ν2 |(L2 r + M2 )T =

[(L0 |ν0 ξ0 | + L1 λ1 |ν1 | + L2 λ2 |ν2 |)T + L3 λ3 ]r +(M0 |ν0 ξ0 | + M1 λ1 |ν1 | + M2 λ2 |ν2 |)T + λ3 M ∗ ≤ ρr + ρ1 ≤ r.

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Now, for u, v and for each t ∈ [0, T ], we obtain k(F u)(t) − (F v)(t)k Z t (t − s)α−1 ≤ kf (s, u(s), (φu)(s), (ψu)(s)) − f (s, v(s), (φv)(s), (ψv)(s))kds Γ(α) 0 Z T (T − s)α−1 kf (s, u(s), (φu)(s), (ψu)(s)) − f (s, v(s), (φv)(s), (ψv)(s))kds +|µ0 ξ0 | Γ(α) 0 Z T (T − s)α−α1 −1 α1 −1 +|µ1 η1 (t)|T kf (s, u(s), (φu)(s), (ψu)(s)) − f (s, v(s), (φv)(s), (ψv)(s))kds Γ(α − α1 ) 0 Z T (T − s)α−α2 −1 +|µ2 η2 (t)|T α2 −2 kf (s, u(s), (φu)(s), (ψu)(s)) − f (s, v(s), (φv)(s), (ψv)(s))kds Γ(α − α2 ) 0 Z T Z T kg1 (s, u(s)) − g1 (s, v(s))kds kg0 (s, u(s)) − g0 (s, v(s))kds + |ν1 η1 (t)| +|ν0 ξ0 | 0

0

Z

T

kg2 (s, u(s)) − g2 (s, v(s))kds

+|ν2 η2 (t)| Z ≤ 0

t

0 α−1

(t − s) Γ(α)

Z M (s)ku − vkds + |µ0 ξ0 |

+|µ1 η1 (t)|T α1 −1

0

Z

T

0

T

(T − s)α−1 M (s)ku − vkds Γ(α)

(T − s)α−α1 −1 M (s)ku − vkds Γ(α − α1 )

T

(T − s)α−α2 −1 M (s)ku − vkds Γ(α − α2 ) 0 Z T Z T Z T +|ν0 ξ0 | L0 ku − vkds + |ν1 η1 (t)| L1 ku − vkds + |ν2 η2 (t) L2 |ku − vkds 0 0 0   λ1 |µ1 |T α−1 λ2 |µ2 |T α−2 (1 + |µ0 ξ0 |)T α + + ku − vk ≤ L3 Γ(α + 1) Γ(α − α1 + 1) Γ(α − α2 + 1) +(M0 |ν0 ξ0 | + M1 λ1 |ν1 | + M2 λ2 |ν2 |)T ku − vk +|µ2 η2 (t)|T α2 −2

=

Z

[(M0 |ν0 ξ0 | + M1 λ1 |ν1 | + M2 λ2 |ν2 |)T + L3 λ3 ]ku − vk = ρku − vk.

Observe that ρ depends only on the parameters involved in the problem. As ρ < 1 (H4 ), therefore F is a contraction. Thus, by the contraction mapping principle (Banach fixed point theorem), it follows that problem (1.1) has a unique solution on [0, T ]. Our next existence results is based on Krasnoselskii fixed point theorem [31]. Theorem 3.2. (Krasnoselskii). Let M be a closed, bounded, convex, and nonempty subset of a Banach space X. Let A, B be two operators such that (i) Ax + By ∈ M whenever x, y ∈ M ; (ii) A is compact and continuous; (iii) B is a contraction mapping. Then there exisz ∈ M such that z = Az + Bz. Theorem 3.3. Assume that (H1 )-(H3 ) and (H5 ) hold. Then the boundary value problem (1.1) has at least one solution on J provided   λ1 |µ1 |T α−1 λ2 |µ2 |T α−2 |µ0 ξ0 |T α + + + (L0 |ν0 ξ0 | + L1 λ1 |ν1 | + L2 λ2 |ν2 |)T < 1. L3 Γ(α + 1) Γ(α − α1 + 1) Γ(α − α2 + 1) Proof. Let us fix r ≥ (M0 |ν0 ξ0 | + M1 λ1 |ν1 | + M2 λ2 |ν2 |)T + λ3 kωkL1 .

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and consider Br = {u ∈ S : kuk ≤ r}. We define the operators Φ and Ψ on Br as Z t (t − s)α−1 (Φu)(t) = f (s, u(s), (φu)(s), (ψu)(s))ds, Γ(α) 0 Z T (T − s)α−1 (Ψu)(t) = µ0 ξ0 f (s, u(s), (φu)(s), (ψu)(s))ds Γ(α) 0 Z T (T − s)α−α1 −1 α1 −1 f (s, u(s), (φu)(s), (ψu)(s))ds +µ1 η1 (t)T Γ(α − α1 ) 0 Z T (T − s)α−α2 −1 +µ2 η2 (t)T α2 −2 f (s, u(s), (φu)(s), (ψu)(s))ds Γ(α − α2 ) 0 Z T Z T Z T +ν0 ξ0 g0 (s, u(s))ds + ν1 η1 (t) g1 (s, u(s))ds + ν2 η2 (t) g2 (s, u(s))ds. 0

0

0

Let us observe that if u, v ∈ Br , we find that

Z t

(t − s)α−1 k(Φu)(t) + (Ψv)(t)k = f (s, u(s), (φu)(s), (ψu)(s))ds

Γ(α) 0 Z T (T − s)α−1 f (s, v(s), (φv)(s), (ψv)(s))ds +µ0 ξ0 Γ(α) 0 Z T (T − s)α−α1 −1 f (s, v(s), (φv)(s), (ψv)(s))ds +µ1 η1 (t)T α1 −1 Γ(α − α1 ) 0 Z T (T − s)α−α2 −1 +µ2 η2 (t)T α2 −2 f (s, v(s), (φv)(s), (ψv)(s))ds Γ(α − α2 ) 0

Z T Z T Z T

+ν0 ξ0 g0 (s, v(s))ds + ν1 η1 (t) g1 (s, v(s))ds + ν2 η2 (t) g2 (s, v(s))ds

0 0 0   α−1 α−2 α λ1 |µ1 |T λ2 |µ2 |T (1 + |µ0 ξ0 |)T + + ≤ kωkL1 Γ(α + 1) Γ(α − α1 + 1) Γ(α − α2 + 1) +(M0 |ν0 ξ0 | + M1 λ1 |ν1 | + M2 λ2 |ν2 |)T =

(M0 |ν0 ξ0 | + M1 λ1 |ν1 | + M2 λ2 |ν2 |)T + λ3 kωkL1 ≤ r.

Thus, Φu + Ψv ∈ Br . It follows from the assumption (H1 )-(H3 ) that Ψ is a contraction mapping for   |µ0 ξ0 |T α λ1 |µ1 |T α−1 λ2 |µ2 |T α−2 L3 + + + (L0 |ν0 ξ0 | + L1 λ1 |ν1 | + L2 λ2 |ν2 |)T < 1. Γ(α + 1) Γ(α − α1 + 1) Γ(α − α2 + 1) Continuity of f implies that the operator Φ is continuous. Also, Φ is uniformly bounded on Br as kΦuk ≤ kωkL1 /Γ(α + 1). Now we prove the compactness of the operator Φ. In view of (H2 ), we define sup

kf (t, u, φu, ψu)k = fmax ,

Ω = J × Br × Br × Br ,

(t,u,φu,ψu)∈Ω

and consequently, for t1 , t2 ∈ J with t1 < t2 , we have

Z t1

(t2 − s)α−1 − (t1 − s)α−1

f (s, u(s), (φu)(s), (ψu)(s))ds k(Φu)(t2 ) − (Φu)(t1 )k ≤

Γ(α) 0

Z t2

(t2 − s)α−1 + f (s, u(s), (φu)(s), (ψu)(s))ds

Γ(α) t1 fmax α ≤ |2(t2 − t1 )α + tα 1 − t2 |, Γ(α + 1) which is independent of x. So Φ is relatively compact on Br . Hence, By Arzela Ascoli Theorem, Φ is compact on Br . Thus all the assumptions of Theorem 3.2 are satisfied and the conclusion of Theorem 3.2 implies that the boundary value problem (1.1) has at least one solution on J . 9 1073

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4

An example

Consider the following nonlinear fractional integrodifferential equations boundary value problem Z t Z 1 1 |u(t)| 1 |u(s)| |u(s)|e−t c 5/2 D u(t) = + + ds + ds, t ∈ J = [0, 1], 2 t 2 10Z (5 + t) 1 + |u(t)| 0 25e + s 0 25 + |u(s)| + s Z 1 1 1 |u(s)| 1 1 u(0) = u(1) + ds, lim [t−1/2c D1/2 u(t)] = D1/2 u(1) + ds, + 2 10 + |u(s)| + s 3 4 + |u(s)| +s t→0 0 0 Z 1 |u(s)| 1 lim [t−1/2c D3/2 u(t)] = D3/2 u(1) + ds. + 4 t→0 0 17 + |u(s)| + s

(4.1)

Here T = 1, α = 5/2, α1 = 1/2, α2 = 3/2, µ0 = 1/2, µ1 = 1/3, µ2 = 1/4, ν0 = ν1 = ν2 = 1, and f (t, x, y, z) =

x 1 1 + + y + z, 10 (5 + t)2 10 + x

g0 (s, u) =

|u| , 10 + |u| + s

g1 (s, u) =

As kf (t, x1 , y1 , z1 ) − f (t, x2 , y2 , z2 )k ≤

|u| , 25et + s

1 , 4 + |u| + s

δ(t, s, u) =

g2 (s, u) =

|u|e−t , 25 + |u|2 + s

|u| . 17 + |u| + s

1 kx1 − x2 k + ky1 − y2 k + kz1 − z2 k, (5 + t)2

1 1 kg1 (s, u) − g1 (s, v)k ≤ ku − vk, kg0 (s, u)k ≤ , 16 4

Z t  

1 |u(s)| 1 |v(s)|

kg2 (s, u) − g2 (s, v)k ≤ ku − vk, kg0 (s, u)k ≤ 1, − ds

≤ 25et ku − vk, t+s t+s 17 25e 25e 0

Z 1   −t −t

|u(s)|e |v(s)|e

≤ 1 ku − vk, − ds

25et 2+s 2+s 25 + |u(s)| 25 + |v(s)| 0

kg0 (s, u) − g0 (s, v)k ≤

1 ku − vk, 10

γ(t, s, u) =

kg0 (s, u)k ≤ 1,

therefore, (H1 )-(H3 ) are satisfied with L0 = 1/10, L1 = 1/16, L2 = 1/17, L1 (t) = 1/(5+t)2 , L2 (t) = L3 (t) = √ √ √ 1, p(t) = q(t) = 1/(25et ). Further, ξ0 = 2, ξ1 = 3 π/2, ξ2 = 4 π/3, λ1 = sup{|η1 (t)| = (3 π/4)|1 + t| : t ∈ √ √ √ √ √ J } = 3 π/2, λ2 = sup{|η2 (t)| = (2 π/9)|3t2 + 2t + 5| : t ∈ J } = 20 π/9, λ3 = 16/(15 π) + 29 π/36, L3 = sup{M (t) = L1 (t) + L2 (t)p(t) + L3 (t)q(t) = 1/(5 + t)2 + 2/(25et ) : t ∈ J } = 3/25, and √ 16 1 39307 π √ + ≈ 0.841414 < 1. ρ = (L0 |ν0 ξ0 | + L1 λ1 |ν1 | + L2 λ2 |ν2 |)T + L3 λ3 = + 5 125 π 122400 Thus, by Theorem 3.1, the boundary value problem (4.1) has a unique solution on [0, 1].

References [1] V. Lakshmikantham, Theory of fractional functional differential equations, Nonlinear Anal., 69, 33373343 (2008). [2] R.P. Agarwal, D. O’Regan, Svatoslav Stanˇek, Positive solutions for Dirichlet problems of singular nonlinear fractional differential equations, J. Math. Anal. Appl., 371, 57-68 (2010). [3] C. Li, X. Luo, Y. Zhou, Existence of positive solutions of the boundary value problem for nonlinear fractional differential equations, Comput. Math. Appl., 59, 1363-1375 (2010). [4] R.P. Agarwal, V. Lakshmikantham, J.J. Nieto, On the concept of solution for fractional differential equations with uncertainty, Nonlinear Anal., 72, 2859-2862 (2010). [5] B. Ahmad, J.J. Nieto, J. Pimentel, Some boundary value problems of fractional differential equations and inclusions, Comput. Math. Appl., 62, 1238-1250 (2011). 10 1074

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[6] 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 pages (2011). [7] B. Ahmad A. Alsaedi, Nonlinear fractional differential equations with nonlocal fractional integrodifferential boundary conditions, Bound. Value Probl., 2012, 124, 10 pages (2012). [8] B. Ahmad S. Sivasundaram On four-point nonlocal boundary value problems of nonlinear integrodifferential equations of fractional order, Appl. Math. Comput., 217, 480-487 (2010). [9] B. Ahmad, S.K. Ntouyas, Existence results for nonlocal boundary value problems of fractional differential equations and inclusions with strip conditions, Bound. Value Probl., 2012, 55, 21 pages (2012). [10] B. Ahmad, S.K. Ntouyas, Nonlinear fractional differential equations and inclusions of arbitrary order and multi-strip boundary conditions, Electron. J. Differ. Equ., 2012, No. 98, pp. 1-22, (2012). [11] R.P. Agarwal, B. Ahmad, Existence theory for anti-periodic boundary value problems of fractional differential equations and inclusions, Comput. Math. Appl., 62, 1200-1214 (2011). [12] A. Alsaedi,B. Ahmad, A. Assolami, On antiperiodic boundary value problems for higher-order fractional differential equations, Abstr. Appl. Anal., 2012, Art. ID 325984 (2012). [13] H.A.H. Salem, Fractional order boundary value problem with integral boundary conditions involving Pettis integral, Acta Math. Sci., 31B(2), 661-672 (2011). [14] X. Liu, M. Jia, B. Wu, Existence and uniqueness of solution for fractional differential equations with integral boundary conditions, Electron. J. Qual. Theory Differ. Equ., 2009, no. 69, pp. 1-10 (2009). [15] S. Hamani, M. Benchohra, J.R. Graef, Existence results for boundary value problems with nonlinear fractional inclusions and integral conditions, Electron. J. Differ. Equ., 2010, no. 20, pp. 1-16 (2010). [16] A. Guezane-Lakoud, R. Khaldi, Solvability of a fractional boundary value problem with fractional integral condition,Nonlinear Anal. 75, 2692-2700 (2012). [17] B. Ahmad, J.J. Nieto, Anti-periodic fractional boundary value problems. Comput. Math. Appl., 62, 1150-1156 (2011). [18] F. Wang, Z. Liu, Anti-periodic fractional boundary value problems for nonlinear differential equations of fractional order, Adv. Differ. Equ., 2012, 116 (2012). [19] B. Ahmad, J.J. Nieto, Existence results for nonlinear boundary value problems of fractional integrodifferential equations with integral boundary conditions, Bound. Value Probl., 2009, Art. ID 708576, 11 pages (2009). [20] X. Wang, X. Guo, G. Tang, Anti-periodic fractional boundary value problems for nonlinear differential equations of fractional order, J Appl. Math. Comput., 41(1-2), 367-375 (2013). [21] G. Chai, Existence results for anti-periodic boundary value problems of fractional differential equations, Adv. Differ. Equ, 2013, 53, 15 pages (2013). [22] A. Alsaedi, Existence of solutions for integrodifferential equations of fractional order with antiperiodic boundary conditions, Int. J. Differ. Equ., 2009, Art. ID 417606, 9 pages (2009). [23] B. Ahmad, J.J. Nieto, A. Alsaedi, Existence and uniqueness of solutions for nonlinear fractional differential equations with non-separated type integral boundary conditions, Acta Math. Sci., 31B(6), 2122-2130 (2011).

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[24] B. Ahmad, S.K. Ntouyas, A boundary value problem of fractional differential equations with anti-periodic type integral boundary conditions, J. Comput. Anal. Appl., 15(8), 1372-1380 (2013). [25] B. Ahmad, S.K. Ntouyas, A. Alsaedi, On fractional differential inclusions with anti-periodic type integral boundary conditions, Bound. Value Probl., 2013, 82 (2013). [26] A. Cabada, G. Wang, Positive solutions of nonlinear fractional differential equations with integral boundary value conditions, J. Math. Anal. Appl., 389, 403-411 (2012). [27] G.A. Anastassiou, Fractional Differentiation Inequalities, Springer, Dordrecht (2009). [28] M. Jia, X. Liu, Three nonnegative solutions for fractional differential equations with integral boundary conditions, Comput. Math. Appl., 62, 1405-1412 (2011). [29] A.A. Kilbas, H.M. Srivastava, J.J. Trujillo, Theory and Applications of Fractional Differential Equations, North-Holland Math. Stud., vol. 204, Elsevier Science B.V., Amsterdam, 2006. [30] I. Podlubny, Fractional Differential Equations, Math. Sci. Eng., Academic Press, New York, 1999. [31] D.R. Smart, Fixed Point Theorems, Cambridge University Press, Cambridge, 1980.

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IDENTITIES OF SYMMETRY FOR HIGHER-ORDER q-BERNOULLI POLYNOMIALS DAE SAN KIM AND TAEKYUN KIM

Abstract. Recently, the higher-order Carlitz’s q-Bernoulli polynomials are represented as q-Volkenborn integral on Zp by Kim. A question was asked in [13] as to finding the extended formulae of symmetries for Bernoulli polynomials which are related to Carlitz q-Bernoulli polynomials. In this paper, we give some new identities of symmetry for the higher-order Carlitz’s q-Bernoulli polynomials which are derived from multivariate q-Volkenborn integrals on Zp . We note that they are a partial answer to that question.

1. Introduction Let p be a fixed prime number. Throughout this paper, Zp , Qp and Cp will, respectively, denote the ring of p-adic rational integers, the field of p-adic rational numbers and the completion of algebraic closure of Qp . The p-adic absolute value in Cp is normalized so that |p|p = p−1 . Let q be an indeterminate in Cp with 1

|1 − q|p < p− p−1 . We say that f is uniformly differentiable function at a point a ∈ Zp , if the difference quotient, f (x) − f (y) , x−y have a limit l = f 0 (a) as (x, y) → (a, a). If f is uniformly differentiable on Zp , we denote this property by f ∈ U D (Zp ). For f ∈ U D (Zp ) , the q-Volkenborn integral is defined by Kim to be Z pN −1 1 X (1) Iq (f ) = f (x) dµq (x) = lim f (x) q x , N] N →∞ [p Zp q x=0 Ff : Zp × Zp → Zp

by Ff (x, y) =

x

where [x]q = 1−q 1−q , (see [12, 13, 14]). From (1), we note that (2)

qIq (f1 ) = Iq (f ) + (q − 1) f (0) +

q−1 0 f (0) log q

where f1 (x) = f (x + 1). As is well known, the Bernoulli polynomials are defined by the generating function to be ∞ X t tn xt B(x)t (3) e = e = B (x) . n et − 1 n! n=0 2000 Mathematics Subject Classification. 11B68; 11S80. Key words and phrases. Identities of symmetry; Higher-order Carlitz’s q-Bernoulli polynomial; Multivariate q-Volkenborn integral. 1

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2

DAE SAN KIM AND TAEKYUN KIM

When x = 0, Bn = Bn (0) are called the Bernoulli numbers. By (3), we get ( 1 if n = 1 n and B0 = 1. (4) (B + 1) − Bn = 0 if n > 1, In [3], Carlitz defined q-Bernoulli numbers as follows : ( 1, if n = 1 n (5) β0,q = 1, q (qβq + 1) − βn,q = 0, if n > 1, with the usual convention about replacing βqi by βi,q . From (4) and (5), we note that lim βn,q = Bn . q→1

The q-Bernoulli polynomials are given by n   X n lx n−l βn,q (x) = q βl,q [x]q (6) l l=0  n = q x βq + [x]q , (n ≥ 0) ,

(see [2, 3, 4, 14]).

In [12], Kim proved that Carlitz q-Bernoulli polynomials can be written by qVolkenborn integral on Zp as follows : Z n (7) [x + y]q dµq (y) βn,q (x) = Zp

=

n   X n

l

l=0

n−l

[x]q

q lx

Z

l

[y]q dµq (x) . Zp

Thus, by (7), we get Z (8)

n

[x]q dµq (x) ,

βn,q =

(n ≥ 0) .

Zp

From (2), we note that   q − 1 n n q [x + 1]q dµq (x) − [x]q dµq (x) = 1  Zp Zp  0 Z

(9)

Z

if n = 0 if n = 1 if n > 1.

By (7), (8) and (9), we see that (10)

β0,q = 1,

n

q (qβq + 1) − βn,q

( 1 if n = 1 = 0 if n > 1.

Let Z (11)

I1 (f ) = lim Iq (f ) = q→1

f (x) dµ1 (x) Zp N

p −1 1 X = lim N f (x) , N →∞ p x=0

(see [12, 23, 24]) .

Then, by (2), we get (12)

I1 (f1 ) − I1 (f ) = f 0 (0) .

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IDENTITIES OF SYMMETRY FOR HIGHER-ORDER q-BERNOULLI POLYNOMIALS

3

Let us take f (x) = etx . Then we have Z ∞ X t tn (13) ext dµ1 (x) = t = Bn , e − 1 n=0 n! Zp and Z (14)

e

(x+y)t

 dµ1 (y) =

Zp

t et − 1



ext =

∞ X

Bn (x)

n=0

tn . n!

For r ∈ N, the higher-order Bernoulli polynomials are defined by the generating function to be   r    ∞ X t tn t t xt xt e = e = Bn(r) (x) . (15) × · · · × t t t e −1 e −1 e −1 n! n=0 {z } | r−times By (14), we get Z Z (16) ···

e(x+y1 +···+yr ) dµ1 (y1 ) · · · dµ1 (yr ) =



Zp

Zp

=

t et − 1

∞ X n=0

r

ext

Bn(r) (x)

tn . n!

In [3, 4], Carlitz introduced the q-extension of higher-order Bernoulli polynomials as follows : !r n   X 1 n l+1 l lx (r) (17) βn,q (x) = (−1) q , n l [l + 1]q (1 − q) l=0

where n ≥ 0 and r ∈ N. (r) (r) Note that lim βn,q (x) = Bn (x). q→1

From (16), we note that Z Z n (x + y1 + · · · + yr ) dµ1 (y1 ) · · · dµ1 (yr ) = Bn(r) (x) , ··· (18) Zp

Zp

where n ≥ 0 and r ∈ N. In this paper, we consider q-extensions of (17) which are related to higher-order Carlitz’s q-Bernoulli polynomials. The purpose of this paper is to give some new and interesting identities of symmetry for the higher-order Carlitz’s q-Bernoulli polynomials which are derived from multivariate q-Volkenborn integral on Zp . 2. Identities of symmetry for higher-order q-Bernoulli polynomials In the sense of q-extension of (18), we observe the following equation (19) Z Z n ··· [x + y1 + · · · + yr ]q dµq (y1 ) · · · dµq (yn ) (19) Zp

Zp

n   X 1 n l = (−1) q lx n l (1 − q) l=0

1079

l+1 [l + 1]q

!r .

DAE SAN KIM ET AL 1077-1088

J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 18, NO.6, 2015, COPYRIGHT 2015 EUDOXUS PRESS, LLC

4

DAE SAN KIM AND TAEKYUN KIM

Thus, by (17) and (19), we get Z

(r) βn,q (x) =

(20)

Z

n

···

[x + y1 + · · · + yr ]q dµq (y1 ) · · · dµq (yr ) ,

Zp

Zp

where n ≥ 0 and r ∈ N. (r) Let us consider the generating function of βn,q (x) as follows : ∞ X

(21)

(r) βn,q (x)

n=0

tn = n!

Z

Z

e[x+y1 +···+yr ]q t dµq (y1 ) · · · dµq (yr ) .

··· Zp

Zp

For w1 , w2 ∈ N, we have (22) Z

1 r [w1 ]q

e

Zp

= lim

N →∞

= lim

N →∞

×

Z ···

[w1 w2 x+w2

Pr

l=1

jl +w1

Pr

l=1

yl ] t q

dµqw1 (y1 ) · · · dµqw1 (yr )

Zp

1 [w1 ]q [pN ]qw1

!r

e

[w1 w2 x+w2

e

[w1 w2 x+w2

Pr

l=1

jl +w1

Pr

l=1

yl ] t w1 (y1 +···+yr ) q

q

y1 ,··· , yr =0 !r

1 [w1 ]q [w2 pN ]qw1

w2X pN −1

N pX −1

Pr

l=1

jl +w1

Pr

l=1

yl ] t w1 (y1 +···+yr ) q

q

y1 ,··· ,yr =0

= lim

N →∞

×

!r

1 [w1 w2 pN ]q N pX −1

wX 2 −1

e

Pr

[w1 w2 x+

]

l=1 (w2 jl +w1 il +w1 w2 yl ) q t

q w1

Pr

l=1 (il +w2 yl )

.

i1 ,··· , ir =0 y1 ,··· ,yr =0

Thus, by (22), we get

(23)

wX 1 −1 Pr 1 q w2 l=1 jl r [w1 ]q j ,··· ,j =0 1 r Z Z P [w w x+ rl=1 (w2 jl +w1 yl )]q t dµqw1 (y1 ) · · · dµqw1 (yr ) × ··· e 1 2 Zp

= lim

N →∞

×

Zp

1 [w1 w2 pN ]q

N pX −1

e

!r

wX 1 −1

wX 2 −1

j1 ,··· , jr =0 i1 ,··· , ir =0 Pr

[w1 w2 x+

]

l=1 (w2 jl +w1 il +w1 w2 yl ) q t

q

Pr

l=1 (w1 il +w2 jl +w1 w2 yl )

.

y1 ,··· , yr =0

1080

DAE SAN KIM ET AL 1077-1088

J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 18, NO.6, 2015, COPYRIGHT 2015 EUDOXUS PRESS, LLC

IDENTITIES OF SYMMETRY FOR HIGHER-ORDER q-BERNOULLI POLYNOMIALS

5

By the same method as (23), we get wX 2 −1 Pr 1 q w1 l=1 jl r [w2 ]q j ,··· ,j =0 1 r Z Z P [w w x+ rl=1 (w1 jl +w2 yl )]q t × ··· e 1 2 dµqw2 (y1 ) · · · dµqw2 (yl )

(24)

Zp

= lim

N →∞

Zp

1 [w1 w2 pN ]q

N pX −1

×

e

!r

wX 2 −1

wX 1 −1

j1 ,··· ,jr =0 i1 ,··· ,ir =0

[w1 w2 x+

Pr

]

l=1 (w1 jl +w2 il +w1 w2 yl ) q t

q

Pr

l=1 (w2 il +w1 jl +w1 w2 yl )

.

y1 ,··· ,yr =0

Therefore, by (23), we obtain the following theorem. Theorem 1. For w1 , w2 ∈ N, we have wX 1 −1 Pr 1 q w2 l=1 jl r [w1 ]q j ,··· ,j =0 1 r Z Z P [w w x+ rl=1 (w2 jl +w1 yl )]q t × ··· e 1 2 dµqw1 (y1 ) · · · dµqw1 (yr ) Zp

=

Zp wX 2 −1

Pr 1 q w1 l=1 jl r [w2 ]q j ,··· ,j =0 1 r Z Z P [w w x+ rl=1 (w1 jl +w2 yl )]q t × ··· e 1 2 dµqw2 (y1 ) · · · dµqw2 (yr ) . Zp

Zp

It is easy to show that [w1 w2 x + w2 (j1 + · · · + jr ) + w1 (y1 + · · · + yr )]q   w2 = [w1 ]q w2 x + (j1 + · · · + jr ) + (y1 + · · · + yr ) . w1 q w1

(25)

Therefore, by (20), Theorem 1 and (25), we obtain the following corollary, and theorem. Corollary 2. For n ≥ 0 and w1 , w2 ∈ N, we have wX 1 −1

n−r

[w1 ]q

q w2

Pr

l=1

jl

j1 ,··· ,jr =0

 n w2 × ··· w2 x + (j1 + · · · + jr ) + y1 + · · · + yr dµqw1 (y1 ) · · · dµqw1 (yr ) w1 Zp Zp q w1 Z

Z

wX 2 −1

n−r

= [w2 ]q

q w1

Pr

l=1

jl

j1 ,··· ,jr =0

Z × Zp

 n w1 w1 x + (j1 + · · · + jr ) + y1 + · · · + yr dµqw2 (y1 ) · · · dµqw2 (yr ) . w2 Zp q w2

Z ···

1081

DAE SAN KIM ET AL 1077-1088

J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 18, NO.6, 2015, COPYRIGHT 2015 EUDOXUS PRESS, LLC

6

DAE SAN KIM AND TAEKYUN KIM

Theorem 3. For n ≥ 0 and w1 , w2 ∈ N, we have

wX 1 −1

n−r [w1 ]q

(r) q w2 (j1 +···+jr ) βn,qw1



 w2 w2 x + (j1 + · · · + jr ) w1



 w1 w1 x + (j1 + · · · + jr ) . w2

j1 ,··· ,jr =0 wX 2 −1

n−r = [w2 ]q

(r) q w1 (j1 +···+jr ) βn,qw2

j1 ,··· ,jr =0

Remark. Let w2 = 1. Then we have

(r) βn,q

(w1 x) =

n−r [w1 ]q

wX 1 −1

q

j1 +···+jr

(r) βn,qw1

j1 ,··· ,jr =0



j1 + · · · + jr x+ w1

 .

By (20), we see that

(26) Z

 n w2 (j1 + · · · + jr ) + y1 + · · · + yr w2 x + dµqw1 (y1 ) · · · dµqw1 (yr ) w1 Zp Zp q w1 !i n   X Pr [w2 ]q n i [j1 + · · · + jr ]qw2 q w2 (n−i) l=1 jl = [w1 ]q i i=0 #n−i Z Z " r X dµqw1 (y1 ) · · · dµqw1 (yr ) × ··· w2 x + yl Z

···

Zp n   X n = i i=0

Zp

[w2 ]q [w1 ]q

l=1

q w1

!i i

[j1 + · · · + jr ]qw2 q w2 (n−i)

1082

Pr

l=1

jl

(r)

βn−i,qw1 (w2 x) .

DAE SAN KIM ET AL 1077-1088

J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 18, NO.6, 2015, COPYRIGHT 2015 EUDOXUS PRESS, LLC

IDENTITIES OF SYMMETRY FOR HIGHER-ORDER q-BERNOULLI POLYNOMIALS

7

From Corollary 2 and (26), we have wX 1 −1

n−r

[w1 ]q

(27)

q w2

Pr

jl

l=1

j1 ,··· ,jr =0

Z ×

=

"

Z

r r X w2 X w2 x + jl + yl w1

··· Zp

Zp

wX 1 −1

w2

q

l=1

Pr

l=1

jl

j1 ,··· , jr =0

i

q w1

n−i−r

[w2 ]q [w1 ]q

i

i=0

dµqw1 (y1 ) · · · dµqw1 (yr )

l=1

n   X n

Pr

i jr ]qw2

#n

(r)

× [j1 + · · · + q w2 (n−i) l=1 jl βn−i,qw1 (w2 x)   n X n i n−i−r (r) = [w2 ]q [w1 ]q βn−i,qw1 (w2 x) i i=0 wX 1 −1

×

i

[j1 + · · · + jr ]qw2 q w2 (n−i+1)

Pr

l=1

jl

j1 ,··· ,jr =0

=

n   X n

i

i=0

n−i

[w2 ]q

wX 1 −1

×

i−r

[w1 ]q

(r)

βi,qw1 (w2 x) n−i

[j1 + · · · + jr ]qw2 q w2 (i+1)

Pr

l=1

jl

j1 ,··· , jr =0

=

n   X n

i

i=0

n−i

[w2 ]q

i−r

[w1 ]q

(r)

(r)

βi,qw1 (w2 x) Tn,i (w1 |q w2 ) ,

where w−1 X

(r)

(28)

Tn,i (w|q) =

n−i (i+1)

[j1 + · · · + jr ]q

q

Pr

l=1

jl

.

j1 ,··· ,jr =0

By the same method as (28), we get wX 2 −1

n−r

(29)

[w2 ]q

q w1

Pr

l=1

jl

j1 ,··· ,jr =0

Z ×

Z ···

Zp

=

Zp

n   X n i=0

"

i

r r X w1 X jl + yl w1 x + w2 l=1

n−i

[w1 ]q

i−r

[w2 ]q

l=1

#n dµqw2 (y1 ) · · · dµqw2 (yl ) q w2

(r)

(r)

βi,qw2 (w1 x) Tn,i (w2 |q w1 ) .

Therefore, by Corollary 2, (27) and (29), we obtain the following theorem.

1083

DAE SAN KIM ET AL 1077-1088

J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 18, NO.6, 2015, COPYRIGHT 2015 EUDOXUS PRESS, LLC

8

DAE SAN KIM AND TAEKYUN KIM

Theorem 4. For n ≥ 0 and r, w1 , w2 ∈ N, we have n   X n (r) n−i i−r (r) [w1 ]q [w2 ]q βi,qw2 (w1 x) Tn,i (w2 |q w1 ) i i=0 n   X n (r) n−i i−r (r) = [w2 ]q [w1 ]q βi,qw1 (w2 x) Tn,i (w1 |q w2 ) , i i=0 where w−1 X

(r)

Tn,i (w|q) =

n−i (i+1)

[j1 + · · · + jr ]q

q

Pr

l=1

jl

.

j1 ,··· ,jr =0

For h ∈ Z and r ∈ N, we have Z Z Pr n ··· [x + y1 + · · · + yr ]q q l=1 (h−l)yl dµq (y1 ) · · · dµq (yr ) Zp

Zp N

pX −1 Pr Pr 1 n q xj j q j l=1 yl q l=1 (h−l+1)yl lim = (−1) n r N j (1 − q) N →∞ [p ]q y ,··· ,y =0 j=0 1 r   n xj X n q (j + h) (j + h − 1) · · · (j + h − r + 1) j = (−1) n j (1 − q) [j + h]q [j + h − 1]q · · · [j + h − r + 1]q j=0
 n j+h X n 1 r! j xj r (−1) q j+h
, = n [r] (1 − q) j=0 j q! r n  X



q

  x [x] [x−1]q ···[x−r+1]q [x] [x−1] ···[x−r+1] = q where = [r]q [r−1]q ···[2] [1] q . [r]q ! q q q q r q From (18), we can also define q-extensions of higher-order Bernoulli polynomials as follows :

(30)

(h,r) βn,q

Z

Z ···

(x) = Zp

n

[x + y1 + · · · + yr ]q q

Pr

l=1 (h−l)yl

dµq (y1 ) · · · dµq (yr ) ,

Zp

where n ≥ 0 and h ∈ Z, r ∈ N. Let w1 , w2 ∈ N. Then we see that (31) wX 1 −1 Pr 1 q w2 l=1 (h−l+1)jl r [w1 ]q j ,··· ,j =0 1 r Z Z P Pr [w w x+ rl=1 (w2 jl +w1 yl )]q t × ··· q w1 l=1 (h−l)yl e 1 2 dµqw1 (y1 ) · · · dµqw1 (yr ) Zp

=

Zp wX 2 −1

Pr 1 q w1 l=1 (h−l+1)jl r [w2 ]q j ,··· ,j =0 1 r Z Z P Pr [w w x+ rl=1 (w1 jl +w2 yl )]q t × ··· q w2 l=1 (h−l)yl e 1 2 dµqw2 (y1 ) · · · dµqw2 (yr ) . Zp

Zp

1084

DAE SAN KIM ET AL 1077-1088

J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 18, NO.6, 2015, COPYRIGHT 2015 EUDOXUS PRESS, LLC

IDENTITIES OF SYMMETRY FOR HIGHER-ORDER q-BERNOULLI POLYNOMIALS

9

From (31), we have

(32) wX 1 −1

n−r

[w1 ]q

q w2

Pr

l=1 (h−l+1)jl

j1 ,··· ,jr =0

Z

"

Z

×

···

q

Zp

w1

Pr

l=1 (h−l)yl

Zp

l=1

wX 2 −1

n−r

= [w2 ]q

r r X w2 X jl + w2 x + yl w1

q w1

#n dµqw1 (y1 ) · · · dµqw1 (yr )

l=1

q w1

Pr

l=1 (h−l+1)jl

j1 ,··· ,jr =0

Z

"

Z

×

···

q

Zp

w2

Pr

l=1 (h−l)yl

Zp

r r X w1 X jl + yl w1 x + w2 l=1

#n

l=1

dµqw2 (y1 ) · · · dµqw2 (yr ) , q w2

where n ≥ 0 and r ∈ N, h ∈ Z. Therefore, by (30) and (32), we obtain the following theorem. Theorem 5. For n ≥ 0, h ∈ Z and w1 , w2 ∈ N, we have wX 1 −1

n−r

[w1 ]q

q w2

Pr

q w1

Pr

l=1 (h−l+1)jl

(h,r)

βn,qw1

j1 ,··· ,jr =0 wX 2 −1

n−r

= [w2 ]q

l=1 (h−l+1)jl

(h,r)

βn,qw2

j1 ,··· ,jr =0

  w2 (j1 + · · · + jr ) w2 x + w1   w1 w1 x + (j1 + · · · + jr ) . w2

From (30), we can derive the following equation :

(33) Z

"

Z ···

Zp

=

q

l=1 (h−l)yl

[w2 ]q

×

i

[j1 + · · · + jr ]qw2 q w2 (n−i) "

Z ···

Zp

q

w1

Pr

l=1 (h−r)yl

w2 x +

Zp

n   X n

i

l=1

[w2 ]q [w1 ]q

#n dµqw1 (y1 ) · · · dµqw1 (yr ) q w1

!i

[w1 ]q

i

Z

r r X w2 X yl w2 x + jl + w1 l=1

n   X n

i=0

Pr

Zp

i=0

=

w1

r X l=1

Pr

l=1

jl

#n−i dµqw1 (y1 ) · · · dµqw1 (yr )

yl q w1

!i i

[j1 + · · · + jr ]qw2 q w2 (n−i)

1085

Pr

l=1

jl

(h,r)

βn−i,qw1 (w2 x) .

DAE SAN KIM ET AL 1077-1088

J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 18, NO.6, 2015, COPYRIGHT 2015 EUDOXUS PRESS, LLC

10

DAE SAN KIM AND TAEKYUN KIM

By (33), we get (34) wX 1 −1

n−r

[w1 ]q

q w2

Pr

l=1 (h−l+1)jl

j1 ,··· ,jr =0

Z ×

···

q

Zp

Pr

w1

l=1 (h−l)yl

Zp

wX 1 −1

=

"

Z

q w2

r r X w2 X jl + yl w2 x + w1 l=1

Pr

l=1 (h−l+1)jl

j1 ,··· ,jr =0

n   X n

i

i=0

×q

w2 (n−i)

×q

w2

Pr

l=1 jl

i

l=1

n−i−r

[w2 ]q [w1 ]q

#n dµqw1 (y1 ) · · · dµqw1 (yr ) q w1 i

[j1 + · · · + jr ]qw2

(h,r) βn−i,qw1

(w2 x)   wX n 1 −1 X n i n−i−r (h,r) i = [w2 ]q [w1 ]q βn−i,qw1 (w2 x) [j1 + · · · + jr ]qw2 i i=0 j ,··· ,j =0 1

=

l=1 (n+h−l−i+1)jl

n   X n

i

i=0

r

Pr

n−i

[w2 ]q

i−r

[w1 ]q

(h,r)

(h,r)

βi,qw1 (w2 x) Tn,i

(w1 |q w2 ) ,

where (h,r)

(35)

Tn,i

w−1 X

(w|q) =

n−i

[j1 + · · · + jr ]q

q

Pr

l=1 (i+h−l+1)jl

.

j1 ,··· ,jr =0

By the same method as (34), we see that (36) wX 2 −1

n−r

[w2 ]q

q w1

Pr

l=1 (h−l+1)jl

j1 ,··· ,jr =0

···

×

n   X n i=0

q

w2

Pr

l=1 (h−l)yl

Zp

Zp

=

"

Z

Z

i

r r X w1 X jl + yl w1 x + w2 l=1

n−i

[w1 ]q

i−r

[w2 ]q

(h,r)

(h,r)

βi,qw2 (w1 x) Tn,i

l=1

#n dµqw2 (y1 ) · · · dµqw2 (yr ) q w1

(w2 |q w1 ) .

Therefore, by (34) and (36), we obtain the following theorem. Theorem 6. For n ≥ 0, h ∈ Z and r, w1 , w2 ∈ N, we have n   X n (h,r) n−i i−r (h,r) [w2 ]q [w1 ]q βi,qw1 (w2 x) Tn,i (w1 |q w2 ) i i=0 n   X n (h,r) n−i i−r (h,r) = [w1 ]q [w2 ]q βi,qw2 (w1 x) Tn,i (w2 |q w1 ) , i i=0 where (h,r)

Tn,i

(w|q) =

w−1 X

n−i

[j1 + · · · + jr ]q

q

Pr

l=1 (h+i−l+1)jl

.

j1 ,··· ,jr =0

1086

DAE SAN KIM ET AL 1077-1088

J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 18, NO.6, 2015, COPYRIGHT 2015 EUDOXUS PRESS, LLC

IDENTITIES OF SYMMETRY FOR HIGHER-ORDER q-BERNOULLI POLYNOMIALS

11

Remark. A p-adic approach to identities of symmetry for Carlitz’s q-Bernoulli polynomials has been studied in [10]. Acknowledgement This work was supported by the National Research Foundation of Korea (NRF) grant funded by the Korea government (MOE) (No.2012R1A1A2003786). References 1. M. A¸cikg¨ oz, D. Erdal, and S. Araci, A new approach to q-Bernoulli numbers and q-Bernoulli polynomials related to q-Bernstein polynomials, Adv. Difference Equ. (2010), Art. ID 951764, 9. 2. W. A. Al-Salam, q-Bernoulli numbers and polynomials, Math. Nachr. 17 (1959), 239–260 (1959). 3. L. Carlitz, q-Bernoulli numbers and polynomials, Duke Math. J. 15 (1948), 987–1000. 4. , Expansions of q-Bernoulli numbers, Duke Math. J. 25 (1958), 355–364. 5. M. Cenkci and V. Kurt, Congruences for generalized q-Bernoulli polynomials, J. Inequal. Appl. (2008), Art. ID 270713, 19. 6. J. Choi, T. Kim, and Y. H. Kim, A note on the extended q-Bernoulli numbers and polynomials, Adv. Stud. Contemp. Math. (Kyungshang) 21 (2011), no. 4, 351–354. 7. D. Ding and J. Yang, Some identities related to the Apostol-Euler and Apostol-Bernoulli polynomials, Adv. Stud. Contemp. Math. (Kyungshang) 20 (2010), no. 1, 7–21. 8. D. V. Dolgy and T. Kim, A note on the weighted q-Bernoulli numbers and the weighted q-Bernstein polynomials, Honam Math. J. 33 (2011), no. 4, 519–527. 9. K.-W. Hwang, D. V. Dolgy, D. S. Kim, T. Kim, and S. H. Lee, Some theorems on Bernoulli and Euler numbers, Ars Combin. 109 (2013), 285–297. 10. D. S. Kim, T. Kim, and S.-H. Lee, A p-adic approach to identities of symmetry for Carlitz’s q-Bernoulli polynomials (communicated), 2014. 11. D. S. Kim, N. Lee, J. Na, and K. H. Park, Abundant symmetry for higher-order Bernoulli polynomials (I), Adv. Stud. Contemp. Math. (Kyungshang) 23 (2013), no. 3, 461–482. 12. T. Kim, q-Volkenborn integration, Russ. J. Math. Phys. 9 (2002), no. 3, 288–299. 13. , On the symmetries of the q-Bernoulli polynomials, Abstr. Appl. Anal. (2008), Art. ID 914367, 7. , q-Bernoulli numbers and polynomials associated with Gaussian binomial coefficients, 14. Russ. J. Math. Phys. 15 (2008), no. 1, 51–57. 15. T. Kim and S.-H. Rim, Generalized Carlitz’s q-Bernoulli numbers in the p-adic number field, Adv. Stud. Contemp. Math. (Pusan) 2 (2000), 9–19. 16. T. Mansour, M. Shattuck, and C. Song, A q-analog of a general rational sum identity, Afr. Mat. 24 (2013), no. 3, 297–303. 17. H. Ozden, p-adic distribution of the unification of the Bernoulli, Euler and Genocchi polynomials, Appl. Math. Comput. 218 (2011), no. 3, 970–973. 18. H. Ozden, I. N. Cangul, and Y. Simsek, Remarks on q-Bernoulli numbers associated with Daehee numbers, Adv. Stud. Contemp. Math. (Kyungshang) 18 (2009), no. 1, 41–48. 19. H.-K. Pak and S.-H. Rim, q-Bernoulli numbers and polynomials via an invariant p-adic qintegral on Zp , Notes Number Theory Discrete Math. 7 (2001), no. 4, 105–110. 20. J.-W. Park, D. V. Dolgy, T. Kim, S.-H. Lee, and S.-H. Rim, A note on the modified Carlitz’s q-Bernoulli numbers and polynomials, J. Comput. Anal. Appl. 15 (2013), no. 4, 647–654. 21. S.-H. Rim, T. Kim, and B.-J. Lee, Some identities on the extended Carlitz’s q-Bernoulli numbers and polynomials, J. Comput. Anal. Appl. 14 (2012), 536–543. 22. S.-H. Rim, E.-J. Moon, S.-J. Lee, and J.-H. Jin, Multivariate twisted p-adic q-integral on Zp associated with twisted q-Bernoulli polynomials and numbers, J. Inequal. Appl. (2010), Art. ID 579509, 6. 23. Y Simsek, Twisted (h, q)-Bernoulli numbers and polynomials related to twisted (h, q)-zeta function and L-function, J. Math. Anal. Appl. 324 (2006), no. 2, 790–804. 24. H. M. Srivastava, T. Kim, and Y. Simsek, q-Bernoulli numbers and polynomials associated with multiple q-zeta functions and basic L-series, Russ. J. Math. Phys. 12 (2005), no. 2, 241–268.

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J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 18, NO.6, 2015, COPYRIGHT 2015 EUDOXUS PRESS, LLC

12

DAE SAN KIM AND TAEKYUN KIM

Department of Mathematics, Sogang University, Seoul 121-742, Republic of Korea E-mail address : [email protected] Department of Mathematics, Kwangwoon University, Seoul 139-701, Republic of Korea E-mail address : [email protected]

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DAE SAN KIM ET AL 1077-1088

J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 18, NO.6, 2015, COPYRIGHT 2015 EUDOXUS PRESS, LLC

FUZZY STABILITY OF FUNCTIONAL EQUATIONS IN MATRIX FUZZY NORMED SPACES CHOONKIL PARK, DONG YUN SHIN∗ , AND JUNG RYE LEE Abstract. Using the fixed point method, we prove the Hyers-Ulam stability of the Cauchy additive functional equation and the quadratic functional equation in matrix fuzzy normed spaces.

1. Introduction and preliminaries The abstract characterization given for linear spaces of bounded Hilbert space operators in terms of matricially normed spaces [65] implies that quotients, mapping spaces and various tensor products of operator spaces may again be regarded as operator spaces. Owing in part to this result, the theory of operator spaces is having an increasingly significant effect on operator algebra theory (see [19]). The proof given in [65] appealed to the theory of ordered operator spaces [12]. Effros and Ruan [20] showed that one can give a purely metric proof of this important theorem by using a technique of Pisier [55] and Haagerup [27] (as modified in [18]). The stability problem of functional equations originated from a question of Ulam [71] concerning the stability of group homomorphisms. The functional equation f (x + y) = f (x) + f (y) is called the Cauchy additive functional equation. In particular, every solution of the Cauchy additive functional equation is said to be an additive mapping. Hyers [28] 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 Th.M. Rassias [59] for linear mappings by considering an unbounded Cauchy difference. A generalization of the Th.M. Rassias theorem was obtained by G˘avruta [26] by replacing the unbounded Cauchy difference by a general control function in the spirit of Th.M. Rassias’ approach. In 1990, Th.M. Rassias [60] during the 27th International Symposium on Functional Equations asked the question whether such a theorem can also be proved for p ≥ 1. In 1991, Gajda [25] following the same approach as in Th.M. Rassias [59], gave an affirmative solution to this question for p > 1. It was shown by Gajda [25], as well as by Th.M. Rassias ˇ and Semrl [64] that one cannot prove a Th.M. Rassias’ type theorem when p = 1 (cf. the books of Czerwik [16], Hyers, Isac and Th.M. Rassias [29]). The functional equation f (x + y) + f (x − y) = 2f (x) + 2f (y) 2010 Mathematics Subject Classification. Primary 47L25, 47H10, 46S40, 39B82, 46L07, 39B52, 26E50. Key words and phrases. operator space; fixed point; Hyers-Ulam stability; matrix fuzzy normed space; Cauchy additive functional equation; quadratic functional equation. ∗ Corresponding author.

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CHOONKIL PARK ET AL 1089-1101

J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 18, NO.6, 2015, COPYRIGHT 2015 EUDOXUS PRESS, LLC

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

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 [70] for mappings f : X → Y , where X is a normed space and Y is a Banach space. Cholewa [13] noticed that the theorem of Skof is still true if the relevant domain X is replaced by an Abelian group. Czerwik [14] 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, 21, 30, 32, 33, 38, 39, 40, 41, 42, 43, 49, 53, 58, 61, 62, 63, 68, 69]). The theory of fuzzy space has much progressed as developing the theory of randomness. Some mathematicians have defined fuzzy norms on a vector space from various points of view [3, 24, 35, 37, 46, 72]. Following Cheng and Mordeson [8], Bag and Samanta [3] gave an idea of fuzzy norm in such a manner that the corresponding fuzzy metric is of Kramosil and Michalek type [36] and investigated some properties of fuzzy normed spaces [4]. We use the definition of fuzzy normed spaces given in [3, 46, 47] to investigate a fuzzy version of the Hyers-Ulam stability for the Cauchy-Jensen functional equation in the fuzzy normed algebra setting. Definition 1.1. [3, 46, 47, 48] 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; (N3 ) N (cx, t) = N (x, |c|t ) if c 6= 0; (N4 ) N (x + y, s + t) ≥ min{N (x, s), N (y, t)}; (N5 ) N (x, ·) is a non-decreasing function of R and limt→∞ N (x, t) = 1; (N6 ) for x 6= 0, N (x, ·) is continuous on R. The pair (X, N ) is called a fuzzy normed space. Definition 1.2. [3, 46, 47, 48] (1) Let (X, N ) be a fuzzy normed 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. (2) Let (X, N ) be a fuzzy normed 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 space is Cauchy. If each Cauchy sequence is convergent, then the fuzzy norm is said to be complete and the fuzzy normed space is called a fuzzy Banach space. We say that a mapping f : X → Y between fuzzy normed 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]). We introduce the concept of matrix fuzzy normed space. Definition 1.3. Let (X, N ) be a fuzzy normed space. (1) (X, {Nn }) is called a matrix fuzzy normed space if for integer n, (Mn (X), Nn ) is a fuzzy normed space  each positive  t and Nk (AxB, t) ≥ Nn x, kAk·kBk for all t > 0, A ∈ Mk,n (R), x = [xij ] ∈ Mn (X) and B ∈ Mn,k (R) with kAk · kBk 6= 0.

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CHOONKIL PARK ET AL 1089-1101

J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 18, NO.6, 2015, COPYRIGHT 2015 EUDOXUS PRESS, LLC

FUNCTIONAL EQUATIONS IN MATRIX FUZZY NORMED SPACES

(2) (X, {Nn }) is called a matrix fuzzy Banach space if (X, N ) is a fuzzy Banach space and (X, {Nn }) is a matrix fuzzy normed space. Example 1.4. Let (X, {k · kn }) be a matrix normed space. Let Nn (x, t) := t > 0 and x = [xij ] ∈ Mn (X). Then Nk (AxB, t) =

t t ≥ = t + kAxBkk t + kAk · kxkn · kBk

t t+kxkn

for all

t kAk·kBk t kAk·kBk

+ kxkn

for all t > 0, A ∈ Mk,n (R), x = [xij ] ∈ Mn (X) and B ∈ Mn,k (R) with kAk · kBk 6= 0. So (X, {Nn }) is a matrix fuzzy normed space. Let E, F be vector spaces. For a given mapping h : E → F and a given positive integer n, define hn : Mn (E) → Mn (F ) by hn ([xij ]) = [h(xij )] for all [xij ] ∈ Mn (E). Let X be a set. A function d : X × X → [0, ∞] is called a generalized metric on X if d satisfies (1) d(x, y) = 0 if and only if x = y; (2) d(x, y) = d(y, x) for all x, y ∈ X; (3) d(x, z) ≤ d(x, y) + d(y, z) for all x, y, z ∈ X. We recall a fundamental result in fixed point theory. Theorem 1.5. [5, 17] Let (X, d) be a complete generalized metric space and let J : X → X be a strictly contractive mapping with Lipschitz constant α < 1. Then for each given element x ∈ X, either d(J n x, J n+1 x) = ∞ for all nonnegative integers n or there exists a positive integer n0 such that (1) d(J n x, J n+1 x) < ∞, ∀n ≥ n0 ; n (2) the sequence {J x} converges to a fixed point y ∗ of J; (3) y ∗ is the unique fixed point of J in the set Y = {y ∈ X | d(J n0 x, y) < ∞}; 1 d(y, Jy) for all y ∈ Y . (4) d(y, y ∗ ) ≤ 1−α In 1996, G. Isac and Th.M. Rassias [31] were the first to provide applications of stability theory of functional equations for the proof of new fixed point theorems with applications. By using fixed point methods, the stability problems of several functional equations have been extensively investigated by a number of authors (see [6, 7, 22, 23, 34, 45, 50, 51, 54, 56]). Throughout this paper, let (X, {Nn }) be a matrix fuzzy normed space and (Y, {Nn }) a matrix fuzzy Banach space. In Section 2, we prove the Hyers-Ulam stability of the Cauchy additive functional equation in matrix fuzzy normed spaces by using fixed point method. In Section 3, we prove the Hyers-Ulam stability of the quadratic functional equation in matrix fuzzy normed spaces by using fixed point method. 2. Hyers-Ulam stability of the Cauchy additive functional equation in matrix fuzzy normed spaces Using the fixed point method, we prove the Hyers-Ulam stability of the Cauchy additive functional equation in matrix fuzzy normed spaces.

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CHOONKIL PARK ET AL 1089-1101

J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 18, NO.6, 2015, COPYRIGHT 2015 EUDOXUS PRESS, LLC

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

We will use the following notations: Mn (X) is the set of all n × n-matrices in X; ej ∈ M1,n (R) is that j-th component is 1 and the other components are zero; Eij ∈ Mn (R) is that (i, j)-component is 1 and the other components are zero; Eij ⊗ x ∈ Mn (X) is that (i, j)-component is x and the other components are zero. Lemma 2.1. Let (X, {Nn }) be a matrix fuzzy normed space. (1) Nn (Ekl ⊗ x, t) = N (x, t) for allPt > 0 and x ∈ X. (2) For all [xij ] ∈ Mn (X) and t = ni,j=1 tij , N (xkl , t) ≥ Nn ([xij ], t) ≥ min{N (xij , tij ) : i, j = 1, 2, · · · , n},     t N (xkl , t) ≥ Nn ([xij ], t) ≥ min N xij , 2 : i, j = 1, 2, · · · , n . n (3) limn→∞ xn = x if and only if limn→∞ xijn = xij for xn = [xijn ], x = [xij ] ∈ Mk (X). Proof. (1) Since Ekl ⊗ x = e∗k xel and ke∗k k = kel k = 1, Nn (Ekl ⊗ x, t) ≥ N (x, t). Since ek (Ekl ⊗ x)e∗l = x, Nn (Ekl ⊗ x, t) ≤ N(x, t). So N (E  kl ⊗ x, t) = N (x, t). t ∗ (2) N (xkl , t) = N (ek [xij ]el , t) ≥ Nn [xij ], kek k·kel k = Nn ([xij ], t). 

Nn ([xij ], t) = Nn 

n X



Eij ⊗ xij , t ≥ min{Nn (Eij ⊗ xij , tij ) : i, j = 1, 2, · · · , n}

i,j=1

= min{N (xij , tij ) : i, j = 1, 2, · · · , n}, where t =

Pn

i,j=1 tij .

n





o

So Nn ([xij ], t) ≥ min N xij , nt2 : i, j = 1, 2, · · · , n . n





o

(3) By N (xkl , t) ≥ Nn ([xij ], t) ≥ min N xij , nt2 : i, j = 1, 2, · · · , n , we obtain the result.  For a mapping f : X → Y , define Df : X 2 → Y and Dfn : Mn (X 2 ) → Mn (Y ) by Df (a, b) = f (a + b) − f (a) − f (b), Dfn ([xij ], [yij ]) := fn ([xij + yij ]) − fn ([xij ]) − fn ([yij ]) for all a, b ∈ X and all x = [xij ], y = [yij ] ∈ Mn (X). Theorem 2.2. Let ϕ : X 2 → [0, ∞) be a function such that there exists an α < 1 with α ϕ(a, b) ≤ ϕ (2a, 2b) (2.1) 2 for all a, b ∈ X. Let f : X → Y be a mapping satisfying t Nn (Dfn ([xij ], [yij ]), t) ≥ (2.2) Pn t + i,j=1 ϕ (xij , yij ) for all t > 0 and x = [xij ], y = [yij ] ∈ Mn (X). Then A(a) := N -liml→∞ 2l f each a ∈ X and defines an additive mapping A : X → Y such that N (fn ([xij ]) − An ([xij ]), t) ≥

2(1 − α)t P 2(1 − α)t + n2 α ni,j=1 ϕ(xij , xij )

  a 2l

exists for

(2.3)

for all t > 0 and x = [xij ] ∈ Mn (X).

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CHOONKIL PARK ET AL 1089-1101

J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 18, NO.6, 2015, COPYRIGHT 2015 EUDOXUS PRESS, LLC

FUNCTIONAL EQUATIONS IN MATRIX FUZZY NORMED SPACES

Proof. Let n = 1. Then (2.2) is equivalent to N (f (a + b) − f (a) − f (b), t) ≥

t t + ϕ (a, b)

(2.4)

for all t > 0 and a, b ∈ X. Letting b = a in (2.4), we get N (f (2a) − 2f (a), t) ≥

t t + ϕ (a, a)

(2.5)

and so t a t   ≥ ,t ≥ N f (a) − 2f α 2 t + 2 ϕ (a, a) t + ϕ a2 , a2 

 



(2.6)

for all t > 0 and a ∈ X. Consider the set S := {g : X → Y } and introduce the generalized metric on S: d(g, h) = inf{µ ∈ R+ : N (g(a) − h(a), µt) ≥

t , ∀a ∈ X, ∀t > 0}, t + ϕ (a, a)

where, as usual, inf φ = +∞. It is easy to show that (S, d) is complete (see the proof of [44, Lemma 2.1]). Now we consider the linear mapping J : S → S such that   a Jg(a) := 2g 2 for all a ∈ X. Let g, h ∈ S be given such that d(g, h) = ε. Then N (g(a) − h(a), εt) ≥

t t + ϕ (a, a)

for all a ∈ X and t > 0. Hence a a a α a − 2h , αεt = N g −h , εt N (Jg(a) − Jh(a), αεt) = N 2g 2 2 2 2 2 αt αt t 2 2  ≥ αt ≥ αt = α a a t + ϕ (a, a) + ϕ (a, a) +ϕ , 2 2 

 

2

 



  

 



2 2

for all a ∈ X and t > 0. So d(g, h) = ε implies that d(Jg, Jh) ≤ αε. This means that d(Jg, Jh) ≤ αd(g, h) for all g, h ∈ S. It follows from (2.4) that d(f, Jf ) ≤ α2 . By Theorem 1.5, there exists a mapping A : X → Y satisfying the following: (1) A is a fixed point of J, i.e., a 1 = A(a) 2 2

 

A

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J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 18, NO.6, 2015, COPYRIGHT 2015 EUDOXUS PRESS, LLC

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

for all a ∈ X. The mapping A is a unique fixed point of J in the set M = {g ∈ S : d(f, g) < ∞}. (2) d(J l f, A) → 0 as l → ∞. This implies the equality a N - lim 2 f l l→∞ 2 l

for all a ∈ X. (3) d(f, A) ≤

1 d(f, Jf ), 1−α





= A(a)

which implies the inequality α . d(f, A) ≤ 2 − 2α

(2.7)

By (2.2), !

l

N 2f

a+b a − 2l f l − 2l f l 2 2 



!

!

b t   , 2l t ≥ l 2 t + ϕ 2al , 2bl

for all a, b ∈ X and t > 0. So !

l

N 2f

a+b a − 2l f l − 2l f l 2 2 



for all a, b ∈ X and t > 0. Since liml→∞

!

t 2l l t + αl ϕ(a,b) 2l 2

t 2l

!

b ,t ≥ 2l

t 2l

+

αl ϕ (a, b) 2l

= 1 for all a, b ∈ X and t > 0,

N (A (a + b) − A(a) − A(b), t) = 1 for all a, b ∈ X and t > 0. Thus A (a + b) − A(a) − A(b) = 0. So the mapping A : X → Y is additive. By Lemma 2.1 and (2.7),     t Nn (fn ([xij ]) − An ([xij ]), t) ≥ min N f (xij ) − A(xij ), 2 : i, j = 1, 2, · · · , n n ) ( 2(1 − α)t : i, j = 1, 2, · · · , n ≥ min 2(1 − α)t + n2 αϕ(xij , xij ) 2(1 − α)t ≥ P 2(1 − α)t + n2 α ni,j=1 ϕ(xij , xij ) for all x = [xij ] ∈ Mn (X). Thus A : X → Y is a unique additive mapping satisfying (2.3), as desired.  Corollary 2.3. Let r, θ be positive real numbers with r < 1. Let f : X → Y be a mapping satisfying t (2.8) Nn (Dfn ([xij ], [yij ]), t) ≥ Pn t + i,j=1 θ(kxij kr + kyij kr ) for all t > 0 and x = [xij ], y = [yij ] ∈ Mn (X). Then A(a) := N -liml→∞ 2l f each a ∈ X and defines an additive mapping A : X → Y such that (2 − 2r )t N (fn ([xij ]) − An ([xij ]), t) ≥ P (2 − 2r )t + n2 · 2r ni,j=1 θkxij kr

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  a 2l

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CHOONKIL PARK ET AL 1089-1101

J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 18, NO.6, 2015, COPYRIGHT 2015 EUDOXUS PRESS, LLC

FUNCTIONAL EQUATIONS IN MATRIX FUZZY NORMED SPACES

for all t > 0 and x = [xij ] ∈ Mn (X). Proof. The proof follows from Theorem 2.2 by taking ϕ(a, b) = θ(kakr + kbkr ) for all a, b ∈ X. Then we can choose α = 2r−1 and we get the desired result.  Theorem 2.4. Let f : X → Y be a mapping satisfying (2.2) for which there exists a function ϕ : X 2 → [0, ∞) such that there exists an α < 1 with a b ϕ(a, b) ≤ 2αϕ , 2 2 for all a, b ∈ X. Then A(a) := N -liml→∞ additive mapping A : X → Y such that N (fn ([xij ]) − An ([xij ]), t) ≥

1 f 2l



!



2l a exists for each a ∈ X and defines an

2(1 − α)t P 2(1 − α)t + n2 ni,j=1 ϕ(xij , xij )

for all t > 0 and x = [xij ] ∈ Mn (X). Proof. Let (S, d) be the generalized metric space defined in the proof of Theorem 2.2. Now we consider the linear mapping J : S → S such that   a Jg(a) := 2g 2 for all a ∈ X. It follows from (2.5) that d(f, Jf ) ≤ 12 . So 1 . 2 − 2α The rest of the proof is similar to the proof of Theorem 2.2. d(f, A) ≤



Corollary 2.5. Let r, θ be positive real numbers  with  r > 1. Let f : X → Y be a mapping 1 l satisfying (2.8). Then A(a) := N -liml→∞ 2l f 2 a exists for each a ∈ X and defines an additive mapping A : X → Y such that (2r − 2)t N (fn ([xij ]) − An ([xij ]), t) ≥ r P (2 − 2)t + n2 · 2r ni,j=1 θkxij kr for all t > 0 and x = [xij ] ∈ Mn (X). Proof. The proof follows from Theorem 2.4 by taking ϕ(a, b) = θ(kakr + kbkr ) for all a, b ∈ X. Then we can choose α = 21−r and we get the desired result.  3. Hyers-Ulam stability of the quadratic functional equation in matrix fuzzy normed spaces Using the fixed point method, we prove the Hyers-Ulam stability of the quadratic functional equation in matrix fuzzy normed spaces. For a mapping f : X → Y , define Df : X 2 → Y and Dfn : Mn (X 2 ) → Mn (Y ) by Df (a, b) = f (a + b) + f (a − b) − 2f (a) − 2f (b), Dfn ([xij ], [yij ]) := fn ([xij + yij ]) + fn ([xij − yij ]) − 2fn ([xij ]) − 2fn ([yij ]) for all a, b ∈ X and all x = [xij ], y = [yij ] ∈ Mn (X).

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CHOONKIL PARK ET AL 1089-1101

J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 18, NO.6, 2015, COPYRIGHT 2015 EUDOXUS PRESS, LLC

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

Theorem 3.1. Let ϕ : X 2 → [0, ∞) be a function such that there exists an α < 1 with α ϕ(a, b) ≤ ϕ (2a, 2b) (3.1) 4 for all a, b ∈ X. Let f : X → Y be a mapping satisfying f (0) = 0 and t Nn (Dfn ([xij ], [yij ]), t) ≥ (3.2) Pn t + i,j=1 ϕ (xij , yij ) for all t > 0 and x = [xij ], y = [yij ] ∈ Mn (X). Then A(a) := N -liml→∞ 4l f each a ∈ X and defines a quadratic mapping Q : X → Y such that 4(1 − α)t N (fn ([xij ]) − Qn ([xij ]), t) ≥ P 4(1 − α)t + n2 α ni,j=1 ϕ(xij , xij )

  a 2l

exists for

(3.3)

for all t > 0 and x = [xij ] ∈ Mn (X). Proof. Let n = 1. Then (3.2) is equivalent to N (f (a + b) + f (a − b) − 2f (a) − 2f (b), t) ≥

t t + ϕ (a, b)

(3.4)

for all t > 0 and a, b ∈ X. Letting b = a in (3.4), we get N (f (2a) − 4f (a), t) ≥

t t + ϕ (a, a)

(3.5)

and so 

N f (a) − 4f

t t a   ≥ ,t ≥ α 2 t + 4 ϕ (a, a) t + ϕ a2 , a2

 



(3.6)

for all t > 0 and a ∈ X. Let (S, d) be the generalized metric space defined in the proof of Theorem 2.2. Now we consider the linear mapping J : S → S such that   a Jg(a) := 4g 2 for all a ∈ X. Let g, h ∈ S be given such that d(g, h) = ε. Then t N (g(a) − h(a), εt) ≥ t + ϕ (a, a) for all a ∈ X and t > 0. Hence             a a a a α N (Jg(a) − Jh(a), αεt) = N 4g − 4h , αεt = N g −h , εt 2 2 4 4 4 αt αt t 4 4  ≥ αt ≥ αt = α + 4 ϕ (a, a) t + ϕ (a, a) + ϕ a, a 4 4

2 2

for all a ∈ X and t > 0. So d(g, h) = ε implies that d(Jg, Jh) ≤ αε. This means that d(Jg, Jh) ≤ αd(g, h) for all g, h ∈ S.

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It follows from (3.4) that d(f, Jf ) ≤ α4 . By Theorem 1.5, there exists a mapping Q : X → Y satisfying the following: (1) Q is a fixed point of J, i.e.,   a 1 Q = Q(a) 2 4 for all a ∈ X. The mapping Q is a unique fixed point of J in the set M = {g ∈ S : d(f, g) < ∞}. (2) d(J l f, Q) → 0 as l → ∞. This implies the equality a N - lim 4 f l l→∞ 2 l

for all a ∈ X. (3) d(f, Q) ≤

1 d(f, Jf ), 1−α





= Q(a)

which implies the inequality α d(f, A) ≤ . 4 − 4α

(3.7)

By (3.2), !

l

N 4f

a+b + 4l f 2l

!



!



a−b a − 2 · 4l f l − 2 · 4l f l 2 2 

!

!

b t   , 4l t ≥ l 2 t + ϕ 2al , 2bl

for all a, b ∈ X and t > 0. So !

l

N 4f

a+b + 4l f 2l

a−b a − 2 · 4l f l − 2 · 4l f l 2 2

for all a, b ∈ X and t > 0. Since liml→∞



t 4l l t + αl ϕ(a,b) 4l 4

!

t 4l

!

b ,t ≥ 2l

t 4l

+

αl ϕ (a, b) 4l

= 1 for all a, b ∈ X and t > 0,

N (Q (a + b) + Q(a − b) − 2Q(a) − 2Q(b), t) = 1 for all a, b ∈ X and t > 0. Thus Q (a + b) + Q(a − b) − 2Q(a) − 2Q(b) = 0. So the mapping Q : X → Y is quadratic. By Lemma 2.1 and (3.7),     t Nn (fn ([xij ]) − Qn ([xij ]), t) ≥ min N f (xij ) − Q(xij ), 2 : i, j = 1, 2, · · · , n n ( ) 4(1 − α)t ≥ min : i, j = 1, 2, · · · , n 4(1 − α)t + n2 αϕ(xij , xij ) 4(1 − α)t ≥ P 4(1 − α)t + n2 α ni,j=1 ϕ(xij , xij ) for all x = [xij ] ∈ Mn (X). Thus Q : X → Y is a unique quadratic mapping satisfying (3.3), as desired.  Corollary 3.2. Let r, θ be positive real numbers with r < 2. Let f : X → Y be a mapping satisfying t Nn (Dfn ([xij ], [yij ]), t) ≥ (3.8) Pn t + i,j=1 θ(kxij kr + kyij kr )

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for all t > 0 and x = [xij ], y = [yij ] ∈ Mn (X). Then A(a) := N -liml→∞ 4l f each a ∈ X and defines a quadratic mapping Q : X → Y such that N (fn ([xij ]) − Qn ([xij ]), t) ≥

  a 2l

exists for

2(4 − 2r )t P 2(4 − 2r )t + n2 · 2r ni,j=1 θkxij kr

for all t > 0 and x = [xij ] ∈ Mn (X). Proof. The proof follows from Theorem 3.1 by taking ϕ(a, b) = θ(kakr + kbkr ) for all a, b ∈ X. Then we can choose α = 2r−2 and we get the desired result.  Theorem 3.3. Let f : X → Y be a mapping satisfying f (0) = 0 and (3.2) for which there exists a function ϕ : X 2 → [0, ∞) such that there exists an α < 1 with a b ϕ(a, b) ≤ 4αϕ , 2 2 for all a, b ∈ X. Then Q(a) := N -liml→∞ quadratic mapping Q : X → Y such that N (fn ([xij ]) − Qn ([xij ]), t) ≥

1 f 4l



!



2l a exists for each a ∈ X and defines a

4(1 − α)t P 4(1 − α)t + n2 ni,j=1 ϕ(xij , xij )

for all t > 0 and x = [xij ] ∈ Mn (X). Proof. Let (S, d) be the generalized metric space defined in the proof of Theorem 2.2. Now we consider the linear mapping J : S → S such that a 2

 

Jg(a) := 4g for all a ∈ X. It follows from (3.5) that d(f, Jf ) ≤ 14 . So d(f, Q) ≤

1 . 4 − 4α

The rest of the proof is similar to the proof of Theorem 3.1.



Corollary 3.4. Let r, θ be positive real numbers  with  r > 2. Let f : X → Y be a mapping 1 l satisfying (3.8). Then Q(a) := N -liml→∞ 4l f 2 a exists for each a ∈ X and defines a quadratic mapping Q : X → Y such that 2(2r − 4)t N (fn ([xij ]) − Qn ([xij ]), t) ≥ P 2(2r − 4)t + n2 · 2r ni,j=1 θkxij kr for all t > 0 and x = [xij ] ∈ Mn (X). Proof. The proof follows from Theorem 3.3 by taking ϕ(a, b) = θ(kakr + kbkr ) for all a, b ∈ X. Then we can choose α = 22−r and we get the desired result. 

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Acknowledgments C. Park was supported by Basic Science Research Program through the National Research Foundation of Korea funded by the Ministry of Education, Science and Technology (NRF-2012R1A1A2004299), and D. Y. Shin was supported by Basic Science Research Program through the National Research Foundation of Korea funded by the Ministry of Education, Science and Technology (NRF-2010-0021792). References [1] 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. Bag and S.K. Samanta, Finite dimensional fuzzy normed linear spaces, J. Fuzzy Math. 11 (2003), 687–705. [4] T. Bag and S.K. Samanta, Fuzzy bounded linear operators, Fuzzy Sets and Systems 151 (2005), 513–547. [5] L. C˘adariu and V. Radu, Fixed points and the stability of Jensen’s functional equation, J. Inequal. Pure Appl. Math. 4, no. 1, Art. ID 4 (2003). [6] L. C˘adariu and V. Radu, On the stability of the Cauchy functional equation: a fixed point approach, Grazer Math. Ber. 346 (2004), 43–52. [7] L. C˘adariu and V. Radu, Fixed point methods for the generalized stability of functional equations in a single variable, Fixed Point Theory and Applications 2008, Art. ID 749392 (2008). [8] S.C. Cheng and J.M. Mordeson, Fuzzy linear operators and fuzzy normed linear spaces, Bull. Calcutta Math. Soc. 86 (1994), 429–436. [9] Y. Cho, J. Kang and R. Saadati, Fixed points and stability of additive functional equations on the Banach algebras, J. Comput. Anal. Appl. 14(2012), 1103–1111. [10] Y. Cho, C. Park, Th.M. Rassias and R. Saadati, Inner product spaces and functional equations, J. Comput. Anal. Appl. 13(2011), 296–304. [11] Y. Cho, C. Park and R. Saadati, Functional inequalities in non-Archimedean Banach spaces, Appl. Math. Letters 23 (2010), 1238–1242. [12] M.-D. Choi and E. Effros, Injectivity and operator spaces, J. Funct. Anal. 24 (1977), 156–209. [13] P.W. Cholewa, Remarks on the stability of functional equations, Aequationes Math. 27 (1984), 76–86. [14] S. Czerwik, On the stability of the quadratic mapping in normed spaces, Abh. Math. Sem. Univ. Hamburg 62 (1992), 59–64. [15] S. Czerwik, The stability of the quadratic functional equation. in: Stability of mappings of HyersUlam type, (ed. Th.M. Rassias and J.Tabor), Hadronic Press, Palm Harbor, Florida, 1994, 81-91. [16] P. Czerwik, Functional Equations and Inequalities in Several Variables, World Scientific Publishing Company, New Jersey, Hong Kong, Singapore and London, 2002. [17] 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. [18] E. Effros, On multilinear completely bounded module maps, Contemp. Math. 62, Amer. Math. Soc.. Providence, RI, 1987, pp. 479–501. [19] E. Effros and Z.-J. Ruan, On approximation properties for operator spaces, Internat. J. Math. 1 (1990), 163–187. [20] E. Effros and Z.-J. Ruan, On the abstract characterization of operator spaces, Proc. Amer. Math. Soc. 119 (1993), 579–584. [21] 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. [22] M. Eshaghi Gordji, G. Kim, J. Lee and C. Park, Generalized ternary bi-derivations on ternary Banach algebras: a fixed point approach, J. Comput. Anal. Appl. 15 (2013), 45–54.

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[51] C. Park, Generalized Hyers-Ulam-Rassias stability of quadratic functional equations: a fixed point approach, Fixed Point Theory and Applications 2008, Art. ID 493751 (2008). [52] C. Park, Y. Cho and H.A. Kenary, Orthogonal stability of a generalized quadratic functional equation in non-Archimedean spaces, J. Comput. Anal. Appl. 14(2012), 526–535. [53] C. Park, K. Ghasemi, S. G. Ghaleh, S. Jang, Approximate n-Jordan ∗-homomorphisms in C ∗ algebras, J. Comput. Anal. Appl. 15 (2013), 365–368. [54] 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. [55] G. Pisier, Grothendieck’s Theorem for non-commutative C ∗ -algebras with an appendix on Grothendieck’s constants, J. Funct. Anal. 29 (1978), 397–415. [56] V. Radu, The fixed point alternative and the stability of functional equations, Fixed Point Theory 4 (2003), 91–96. [57] J.M. Rassias, On approximation of approximately linear mappings by linear mappings, J. Funct. Anal. 46 (1982) 126–130. [58] J.M. Rassias, Solution of a problem of Ulam, J. Approx. Theory 57 (1989), 268–273. [59] Th.M. Rassias, On the stability of the linear mapping in Banach spaces, Proc. Amer. Math. Soc. 72 (1978), 297–300. [60] Th.M. Rassias, Problem 16; 2, Report of the 27th International Symp. on Functional Equations, Aequationes Math. 39 (1990), 292–293; 309. [61] Th.M. Rassias (ed.), Functional Equations and Inequalities, Kluwer Academic, Dordrecht, 2000. [62] Th.M. Rassias, On the stability of functional equations in Banach spaces, J. Math. Anal. Appl. 251 (2000), 264–284. [63] Th.M. Rassias, On the stability of functional equations and a problem of Ulam, Acta Math. Appl. 62 (2000), 23–130. ˇ [64] Th.M. Rassias and P. Semrl, On the behaviour of mappings which do not satisfy Hyers-Ulam stability, Proc. Amer. Math. Soc. 114 (1992), 989–993. [65] Z.-J. Ruan, Subspaces of C ∗ -algebras, J. Funct. Anal. 76 (1988), 217–230. [66] R. Saadati and C. Park, Non-Archimedean L-fuzzy normed spaces and stability of functional equations, Computers Math. Appl. 60 (2010), 2488–2496. [67] D. Shin, S. Lee, C. Byun and S. Kim, On matrix normed spaces, Bull. Korean Math. Soc. 27 (1983), 103–112. [68] D. Shin, C. Park and Sh. Farhadabadi, On the superstability of ternary Jordan C ∗ -homomorphisms, J. Comput. Anal. Appl. 16 (2014), 964–973. [69] 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. [70] F. Skof, Propriet` a locali e approssimazione di operatori, Rend. Sem. Mat. Fis. Milano 53 (1983), 113–129. [71] S. M. Ulam, A Collection of the Mathematical Problems, Interscience Publ. New York, 1960. [72] J.Z. Xiao and X.H. Zhu, Fuzzy normed spaces of operators and its completeness, Fuzzy Sets and Systems 133 (2003), 389–399. Choonkil Park Department of Mathematics, Research Institute for Natural Sciences, Hanyang University, Seoul 133-791, Korea E-mail address: [email protected] Dong Yun Shin Department of Mathematics, University of Seoul, Seoul 130-743, Korea E-mail address: [email protected] Jung Rye Lee Department of Mathematics, Daejin University, Kyeonggi 487-711, Korea E-mail address: [email protected]

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J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 18, NO.6, 2015, COPYRIGHT 2015 EUDOXUS PRESS, LLC

On the stability of multi-additive mappings in non-Archimedean normed spaces

Tian Zhou Xu*

Chun Wang

School of Mathematics and Statistics, Beijing Institute of Technology, Beijing 100081, P. R. China E-mail: [email protected], [email protected]

Themistocles M. Rassias Department of Mathematics, National Technical University of Athens, Zografou Campus, Athens 15780, Greece E-mail: [email protected] Abstract. We establish some new stability results concerning multi-additive functional equation in non-Archimedean normed spaces. The results improve some recent results. Some applications of our result will be illustrated. In particular, we will see that some results about stability of multi-additive mappings in real normed spaces are not valid in nonArchimedean normed spaces. Keywords: Stability; Multi-additive mapping; Non-Archimedean normed space; Fixed point. MR(2000) Subject Classification. 39B22, 39B82, 46S10, 47S10

1. Introduction In 1897, Hensel discovered the p-adic numbers as a number theoretical analogue of power series in complex analysis. The most important examples of non-Archimedean spaces are p-adic numbers. During the last three decades, the theory of non-Archimedean spaces has gained the interest of physicists for their research, in particular, in problems deriving from quantum physics, p-adic strings and superstrings [10, 18]. Although many results in the classical normed space theory have a non-Archimedean counterpart, their proofs are essentially different and require an entirely new kind of approach. One may note that |n| ≤ 1 in each valuation field, every triangle is isosceles and there may be a no unit vector in a non-Archimedean normed space [10]. These facts show that the non-Archimedean framework is of special interest. A basic question in the theory of functional equations is the following: when is it true that a function, which approximately satisfies a functional equation must be close to an exact solution of the equation? If the problem accepts a unique solution, we say the equation is stable (see [12]). The first stability problem concerning group homomorphisms was raised by Ulam [17] in 1940 and affirmatively solved by Hyers [7]. The result of Hyers was generalized by Th.M. Rassias [14] for approximate linear mappings by allowing the Cauchy difference operator CDf (x, y) = f (x+y)−[f (x)+f (y)] to be controlled by ϵ(∥x∥p +∥y∥p ). In 1994, a generalization of Rassias’ theorem was obtained by G˘avrut¸a [6], who replaced ϵ(∥x∥p +∥y∥p ) by a general control function φ(x, y) by following Th.M. Rassias’ approach. Furthermore for an extensive account of methods and results concerning Hyers-Ulam stability of additive, multi-additive, multi-Jensen mappings and functional equations in a single variable as well as in several variables we refer the reader to ([1–4, 8, 9, 11, 15, 16, 19–24]) and references therein. In this paper, we determine some results concerning the stability of the multi-additive mappings in the nonArchimedean normed spaces. The presented results correspond to some outcomes from [1] and sometimes are their slight generalizations. 2. Preliminaries We recall some basic facts concerning non-Archimedean space and some basic results. 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. *Corresponding author. The first author was supported by the National Natural Science Foundation of China(Grant No. 11171022)

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Definition 2.1. Let K be a field. A non-Archimedean absolute value on K is a function | · | : K → R such that for any r, s ∈ K we have (i) |r| ≥ 0 and equality holds if and only if r = 0; (ii) |rs| = |r||s|; (iii) |r + s| ≤ max{|r|, |s|}. The condition (iii) is called the strong triangle inequality. Clearly, |1| = | − 1| = 1 and |n| ≤ 1 for all n ∈ N. We always assume in addition that | · | is non trivial, i.e., that (iv) there is an r0 ∈ K such that |r0 | ̸= 0, 1. The most important examples of non-Archimedean spaces are p-adic numbers. Example 2.2. Let p be a prime number. For any nonzero rational number x, there exists a unique integer nx such that x = ab pnx , where a and b are integers not divisible by p. Then |x|p := p−nx defines a non-Archimedean norm on Q. The completion of Q with respect to the metric d(x, y) = |x − y|p is denoted by Qp which is called ∑∞ k the p-adic number field. In fact, Qp is the set of all formal series x = k≥nx ak p , where |ak | ≤ p − 1 are integers. The addition and multiplication between any two elements of Qp are defined naturally. The norm ∑∞ | k≥nx ak pk |p = p−nx is a non-Archimedean norm on Qp and it makes Qp a locally compact field (see [18]). Note that if p > 2, then |2n |p = 1 for each integer n but |2|2 < 1. Throughout this paper, we assume that the base field is a non-Archimedean field and hence we can call it simply a field. Moreover, N stands for the set of all positive integers. Definition 2.3. Let X be a linear space over a field K with a non-Archimedean valuation | · |. A function ∥ · ∥ : X → [0, ∞) is a non-Archimedean norm if it satisfies the following conditions: (i) ∥x∥ = 0 if and only if x = 0; (ii) ∥rx∥ = |r|∥x∥ for all r ∈ K and x ∈ X ; (iii) the strong triangle inequality ∥x + y∥ ≤ max{∥x∥, ∥y∥},

∀x, y ∈ X .

Then (X , ∥ · ∥) is called a non-Archimedean normed space. Definition 2.4. Let X be a non-Archimedean normed space. Let {xn } be a sequence in X . Then {xn } is said to be convergent if there exists x ∈ X such that lim ∥xn − x∥ = 0. In that case, x is called the limit of n→∞

the sequence {xn } and we denote it by lim xn = x. A sequence {xn } in X is said to be a Cauchy sequence if n→∞

lim ∥xn+p − xn ∥ = 0 for all p = 1, 2, . . .. Due to the fact that

n→∞

∥xn − xm ∥ ≤ max{∥xj+1 − xj ∥ : m ≤ j ≤ n − 1} (n > m) a sequence {xn } is Cauchy if and only if {xn+1 − xn } converges to zero in a non-Archimedean normed space. It is known that every convergent sequence in a non-Archimedean normed space is a Cauchy sequence. If every Cauchy sequence in X converges, then the non-Archimedean normed space X is called a non-Archimedean Banach space. Definition 2.5. Let X denotes a linear space and Y represents a complete non-Archimedean normed space and n ≥ 1 is an integer. A function f : X n → Y is called a multi-additive mapping, if f is additive in each variable: f (x1 , . . . , xi−1 , xi + x′i , xi+1 , . . . , xn ) = f (x1 , . . . , xi−1 , xi , xi+1 , . . . , xn ) + f (x1 , . . . , xi−1 , x′i , xi+1 , . . . , xn ) for all i = 1, 2, . . . , n and all x1 , . . . , xi−1 , xi , x′i , xi+1 , . . . , xn ∈ X . Some basic facts on such mappings can be found for instance in [1, 2, 13], where their application to the representation of polynomial functions is also presented. Let Ω be a set. A function d : Ω × Ω → [0, ∞] is called a generalized metric on Ω if d satisfies the following conditions: (a) d(x, y) = 0 if and only if x = y; (b) d(x, y) = d(y, x) for all x, y ∈ Ω; (c) d(x, z) ≤ d(x, y) + d(y, z) for all x, y, z ∈ Ω. It will later on the following fixed point alternative theorem (cf. [5]) will be useful.

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On the stability of multi-additive mappings in non-Archimedean normed spaces

Theorem 2.6. Let (Ω, d) be a complete generalized metric space and J : Ω → Ω be a strictly contractive mapping with Lipschitz constant 0 < L < 1, that is d(Jx, Jy) ≤ Ld(x, y)

∀ x, y ∈ Ω.

Then, for each given x ∈ Ω, either d(J n x, J n+1 x) = ∞

∀ n ≥ 0,

or d(J n x, J n+1 x) < ∞,

∀ n ≥ n0 ,

for some natural number n0 . Actually if the second alternative holds, then the sequence {J n x} is convergent to a fixed point x∗ of J and (1) x∗ is the unique fixed point of J in the set Ω∗ = {y ∈ Ω : d(J n0 x, y) < ∞}; 1 (2) d(y, x∗ ) ≤ 1−L d(y, Jy) for all y ∈ Ω∗ . 3. Non-Archimedean stability of the multi-additive mapping: a direct method Let X be a linear space over a non-Archimedean field K with a valuation | · | and Y be a complete nonArchimedean normed space over K. For the given mapping f : X n → Y and every i ∈ {1, 2, . . . , n}, we define the difference operator Di f (x1 , . . . , xi , x′i , xi+1 , . . . , xn ) := f (x1 , . . . , xi−1 , xi + x′i , xi+1 , . . . , xn ) − f (x1 , . . . , xn ) − f (x1 , . . . , xi−1 , x′i , xi+1 , . . . , xn ) for all x1 , . . . , xi , x′i , xi+1 , . . . , xn ∈ X . Theorem 3.1. Let X be a linear space over a non-Archimedean field K with a valuation |·| and Y be a complete non-Archimedean normed space over K. Let n ∈ N and for every i ∈ {1, 2, . . . , n}, φi : X n+1 → [0, ∞) be a function. Let for some natural number k ∈ K,   lim |k|m φi (x1 /k m , . . . , xi , x′i , xi+1 , . . . , xn ) = 0,   m→∞   ..    .     m  lim |k| φ (x , . . . , xi−2 , xi−1 /k m , xi , x′i , xi+1 , . . . , xn ) = 0,  i 1   m→∞ lim |k|m φi (x1 , . . . , xi−1 , xi /k m , x′i /k m , xi+1 , . . . , xn ) = 0, (3.1) m→∞    m ′ m  lim |k| φi (x1 , . . . , xi , xi , xi+1 /k , xi+2 , . . . , xn ) = 0,   m→∞    ..   .     m  lim |k| φi (x1 , . . . , xi , x′ , xi+1 , . . . , xn−1 , xn /k m ) = 0 i

m→∞

for all x1 , . . . , xi , x′i , xi+1 , . . . , xn ∈ X and let for each (x1 , . . . , xn ) ∈ X n the limit

lim max{|k|s+1 max{φi (x1 , . . . , xi−1 , xi /k s+1 , jxi /k s+1 , xi+1 , . . . , xn ) : 1 ≤ j ≤ k − 1} : 0 ≤ s < m},

m→∞

denoted by φ˜i (x1 , . . . , xn ), exists. Suppose that f : X n → Y be a mapping satisfying ∥Di f (x1 , . . . , xi , x′i , xi+1 , . . . , xn )∥ ≤ φi (x1 , . . . , xi , x′i , xi+1 , . . . , xn )

(3.2)

for all x1 , . . . , xi , x′i , xi+1 , . . . , xn ∈ X and i ∈ {1, . . . , n}. Then for every i ∈ {1, . . . , n} there exists a multiadditive mapping Fi : X n → Y such that ∥f (x1 , . . . , xn ) − Fi (x1 , . . . , xn )∥ ≤

1 φ˜i (x1 , . . . , xn ) |k|

(3.3)

for all x1 , . . . , xn ∈ X . For every i ∈ {1, . . . , n} the function Fi is given by Fi (x1 , . . . , xn ) := lim k j f (x1 , . . . , xi−1 , xi /k j , xi+1 , . . . , xn ) j→∞

for all x1 , . . . , xn ∈ X , and if, in addition, lim |k|p φ˜i (x1 , . . . , xi−1 , xi /k p , xi+1 , . . . , xn ) = 0,

p→∞

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(3.4)

J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 18, NO.6, 2015, COPYRIGHT 2015 EUDOXUS PRESS, LLC

T. Z. Xu, C. Wang and Th. M. Rassias

then Fi is the unique multi-additive mapping satisfying (3.3). Proof. Fix j ∈ N and i ∈ {1, . . . , n}. Letting x′i = xi in (3.2), we get ∥f (x1 , . . . , xi−1 , 2xi , xi+1 , . . . , xn ) − 2f (x1 , . . . , xn )∥ ≤ φi (x1 , . . . , xi , xi , xi+1 , . . . , xn )

(3.5)

for all x1 , . . . , xn ∈ X . Letting x′i = 2xi in (3.2), we get ∥f (x1 , . . . , xi−1 , 3xi , xi+1 , . . . , xn ) − f (x1 , . . . , xn ) − f (x1 , . . . , xi−1 , 2xi , xi+1 , . . . , xn )∥ ≤ φi (x1 , . . . , xi , 2xi , xi+1 , . . . , xn )

(3.6)

for all x1 , . . . , xn ∈ X . Similarly, putting x′i = jxi ( 3 ≤ j ≤ k − 1) in (3.2), we get ∥f (x1 , . . . , xi−1 , (j + 1)xi , xi+1 , . . . , xn ) − f (x1 , . . . , xn ) − f (x1 , . . . , xi−1 , jxi , xi+1 , . . . , xn )∥ ≤ φi (x1 , . . . , xi , jxi , xi+1 , . . . , xn )

(3.7)

for all x1 , . . . , xn ∈ X . By (3.5)–(3.7), we see that ∥f (x1 , . . . , xi−1 , kxi , xi+1 , . . . , xn ) − kf (x1 , . . . , xn )∥ ≤ max{φi (x1 , . . . , xi , jxi , xi+1 , . . . , xn ) : 1 ≤ j ≤ k − 1}

(3.8)

for all x1 , . . . , xn ∈ X , and so ∥k m−1 f (x1 , . . . , xi−1 , xi /k m−1 , xi+1 , . . . , xn ) − k m f (x1 , . . . , xi−1 , xi /k m , xi+1 , . . . , xn )∥ ≤ |k|m−1 max{φi (x1 , . . . , xi−1 , xi /k m , jxi /k m , xi+1 , . . . , xn ) : 1 ≤ j ≤ k − 1}.

(3.9)

for all x1 , . . . , xn ∈ X and m = 0, 1, . . .. By (3.1), it follows that {k m f (x1 , . . . , xi−1 , xi /k m , xi+1 , . . . , xn )} is a Cauchy sequence in the complete non-Archimedean space Y . This sequence is convergent and we define Fi : X n → Y by Fi (x1 , . . . , xn ) := lim k m f (x1 , . . . , xi−1 , xi /k m , xi+1 , . . . , xn ) m→∞

(3.10)

for all x1 , . . . , xn ∈ X . Using (3.9), one can show that ∥f (x 1 , . . . , xn ) − k m f (x1 , . . . , xi−1 , xi /k m , xi+1 , . . . , xn )∥

m

∑

s−1 s−1 s s

= [k f (x1 , . . . , xi−1 , xi /k , xi+1 , . . . , xn ) − k f (x1 , . . . , xi−1 , xi /k , xi+1 , . . . , xn )]

s=1 1 s+1 |k| max{|k|

max{φi (x1 , . . . , xi−1 , xi /k s+1 , jxi /k s+1 , xi+1 , . . . , xn ) : 1 ≤ j ≤ k − 1} : 0 ≤ s < m}, (3.11) by taking limit as m → ∞ of both sides of (3.11), one can obtain the inequality (3.3). Now, we will show that for every i ∈ {1, . . . , n} the mapping Fi is multi-additive. By (3.2), we have   ∥k j Ds f (x1 , . . . , xs−1 , xs , x′s , xs+1 , . . . , xi−1 , xi /k j , xi+1 , . . . , xn )∥      for s < i, ≤ |k|j φs (x1 , . . . , xs−1 , xs , x′s , xs+1 , . . . , xi−1 , xi /k j , xi+1 , . . . , xn ),     ∥k j D f (x , . . . , x , x /k j , x′ /k j , x , . . . , x )∥ i 1 i−1 i i+1 n i (3.12) j j ′ j  ≤ |k| φ (x , . . . , x , x /k , x /k , x , . . . , xn ), for s = i,  i 1 i−1 i i+1 i     ∥k j Ds f (x1 , . . . , xi−1 , xi /k j , xi+1 , . . . , xs−1 , xs , x′s , xs+1 , . . . , xn )∥     ≤ |k|j φs (x1 , . . . , xi−1 , xi /k j , xi+1 , . . . , xs−1 , xs , x′s , xs+1 , . . . , xn ), for s > i ≤

for all x1 , . . . , xi , x′i , xi+1 , . . . , xn ∈ X , x′s ∈ X (s ∈ {1, 2, . . . , n}\{i}) and all j ∈ N. Letting j → ∞ in the above inequalities and using (3.1), we see that the mapping Fi is multi-additive. To prove the uniqueness of the mapping Fi , assume that Fi′ : X n → Y is another multi-additive mapping satisfying (3.3). Then we have ∥Fi (x1 , . . . , xn ) − Fi′ (x1 , . . . , xn )∥ = |k|p ∥Fi (x1 , . . . , xi−1 , xi /k p , xi+1 , . . . , xn ) − Fi′ (x1 , . . . , xi−1 , xi /k p , xi+1 , . . . , xn )∥ ≤ max{|k|p ∥f (x1 , . . . , xi−1 , xi /k p , xi+1 , . . . , xn ) − Fi (x1 , . . . , xi−1 , xi /k p , xi+1 , . . . , xn )∥, |k|p ∥f (x1 , . . . , xi−1 , xi /k p , xi+1 , . . . , xn ) − Fi′ (x1 , . . . , xi−1 , xi /k p , xi+1 , . . . , xn )∥} ≤ 2|k|p−1 φ˜i (x1 , . . . , xi−1 , xi /k p , xi+1 , . . . , xn )

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On the stability of multi-additive mappings in non-Archimedean normed spaces

for all p ∈ N and x1 , . . . , xn ∈ X . If lim |k|p φ˜i (x1 , . . . , xi−1 , xi /k p , xi+1 , . . . , xn ) = 0,

p→∞

we may conclude that Fi (x1 , . . . , xn ) = Fi′ (x1 , . . . , xn ) for all x1 , . . . , xn ∈ X , and the proof is complete.  The proof of the following result is similar to that in Theorem 3.1, hence it is omitted. Theorem 3.2. Let X be a linear space over a non-Archimedean field K with a valuation |·| and Y be a complete non-Archimedean normed space over K. Let n ∈ N and for every i ∈ {1, 2, . . . , n}, φi : X n+1 → [0, ∞) be a function. Assume for some natural number k ∈ K,   lim 1m φi (k m x1 , . . . , xi , x′i , xi+1 , . . . , xn ) = 0,   m→∞ |k|   ..    .     1 m  lim xi−1 , xi , x′i , xi+1 , . . . , xn ) = 0,  m φi (x1 , . . . , xi−2 , k   m→∞ |k| lim 1m φi (x1 , . . . , xi−1 , k m xi , k m x′i , xi+1 , . . . , xn ) = 0, m→∞ |k|     lim 1m φi (x1 , . . . , xi , x′i , k m xi+1 , xi+2 , . . . , xn ) = 0,   m→∞ |k|    ..   .     1 ′ m  lim xn ) = 0 m φi (x1 , . . . , xi , x , xi+1 , . . . , xn−1 , k m→∞ |k|

i

for all x1 , . . . , xi , x′i , xi+1 , . . . , xn ∈ X and let for each (x1 , . . . , xn ) ∈ X n the limit { } 1 s s lim max max{φi (x1 , . . . , xi−1 , k xi , jk xi , xi+1 , . . . , xn ) : 1 ≤ j ≤ k − 1} : 0 ≤ s < m , m→∞ |k|s

denoted by φ˜i (x1 , . . . , xn ), exists. Suppose that f : X n → Y is a mapping satisfying

∥Di f (x1 , . . . , xi , x′i , xi+1 , . . . , xn )∥ ≤ φi (x1 , . . . , xi , x′i , xi+1 , . . . , xn ) for all x1 , . . . , xi , x′i , xi+1 , . . . , xn ∈ X and i ∈ {1, . . . , n}. Then for every i ∈ {1, . . . , n} there exists a multiadditive mapping Fi : X n → Y such that ∥f (x1 , . . . , xn ) − Fi (x1 , . . . , xn )∥ ≤

1 φ˜i (x1 , . . . , xn ) |k|

for all x1 , . . . , xn ∈ X . For every i ∈ {1, . . . , n} the function Fi is given by Fi (x1 , . . . , xn ) := lim

j→∞

1 f (x1 , . . . , xi−1 , k j xi , xi+1 , . . . , xn ) kj

for all x1 , . . . , xn ∈ X , and if, in addition, lim

p→∞

1 φ˜i (x1 , . . . , xi−1 , k p xi , xi+1 , . . . , xn ) = 0, |k|p

then Fi is the unique multi-additive mapping satisfying (3.3). Corollary 3.3. Let K be a non-Archimedean field, (X , ∥ · ∥X ) be a non-Archimedean normed space over K, (Y , ∥ · ∥Y ) be a complete non-Archimedean normed space over K. Let k ∈ N with |k| < 1, ε, δ ≥ 0, 0 < ri < 1(1 ≤ i ≤ n), and f : X n → Y be a mapping such that ∥Di f (x1 , . . . , xi , x′i , xi+1 , . . . , xn )∥Y ≤ ε + δ[∥x1 ∥rX1 · · · ∥xi−1 ∥Xi−1 (∥xi ∥rXi + ∥x′i ∥rXi )∥xi+1 ∥Xi+1 · · · · ∥xn ∥rXn ] r

r

for all x1 , . . . , xi , x′i , xi+1 , . . . , xn ∈ X and every i ∈ {1, 2, . . . , n}. Then there exists a unique multi-additive mapping Fi : X n → Y such that ∥f (x1 , x2 , . . . , xn ) − Fi (x1 , x2 , . . . , xn )∥Y ≤ ε +

2δ (∥x1 ∥rX1 · ∥x2 ∥rX2 · · · · · ∥xn ∥rXn ) |k|ri

for all x1 , x2 , . . . , xn ∈ X . Corollary 3.4. Let K be a non-Archimedean field, (X , ∥ · ∥X ) be a non-Archimedean normed space over K, (Y , ∥ · ∥Y ) be a complete non-Archimedean normed space over K. Let k ∈ N with |k| < 1, ε, δ ≥ 0, ri , si > 0(1 ≤

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i ≤ n) with λi := ri + si < 1, and f : X n → Y be a mapping such that ∥Di f (x1 , . . . , xi , x′i , xi+1 , . . . , xn )∥Y ri +si ≤ ε + δ{∥x1 ∥λX1 · · · ∥xi−1 ∥Xi−1 [∥xi ∥rXi ∥x′i ∥sXi + (∥xi ∥rXi +si + ∥x′i ∥X )]∥xi+1 ∥Xi+1 · · · · ∥xn ∥λXn } λ

λ

for all x1 , . . . , xi , x′i , xi+1 , . . . , xn ∈ X and every i ∈ {1, 2, . . . , n}. Then there exists a unique multi-additive mapping Fi : X n → Y such that ∥f (x1 , x2 , . . . , xn ) − Fi (x1 , x2 , . . . , xn )∥Y ≤ ε +

3δ (∥x1 ∥λX1 · ∥x2 ∥λX2 · · · · · ∥xn ∥λXn ) |k|λi

for all x1 , x2 , . . . , xn ∈ X . Remark. Theorems 3.1 and 3.2 are generalized versions of Theorem 2.1 in [11]. In [2], Ciepli´ nski proved the following: Theorem 3.5. Let X be a commutative semigroup and Y be a Banach space. Assume also that n ∈ N and for every i ∈ {1, 2, . . . , n}, φi : X n+1 → [0, ∞) is a mapping such that for any (x1 , x2 , . . . , xn ) ∈ X n+1 we have φ˜i (x1 , . . . , xn+1 ) :=

∞ ∑

1 j j 2j [φi (2 x1 , x2 , . . . , xn+1 ) + · · · + φi (x1 , . . . , xi−2 , 2 xi−1 , xi , . . . , xn+1 ) j=0 + 12 φi (x1 , . . . , xi−1 , 2j xi , 2j xi+1 , xi+2 , . . . , xn+1 ) + φi (x1 , . . . , xi+1 , 2j xi+2 , xi+3 , . . . , xn+1 ) + · · · + φi (x1 , . . . , xn , 2j xn+1 ) < ∞.

If f : X n → Y is a function satisfying ∥f (x1 , . . . , xi−1 , xi + x′i , xi+1 , . . . , xn ) − f (x1 , . . . , xn ) − f (x1 , . . . , xi−1 , x′i , xi+1 , . . . , xn )∥ ≤ φi (x1 , . . . , xi , x′i , xi+1 , . . . , xn ), (x1 , . . . , xi , x′i , xi+1 , . . . , xn ) ∈ X n+1 , i ∈ {1, . . . , n}, then for every i ∈ {1, . . . , n} there exists a multi-additive mapping Fi : X n → Y such that for any (x1 , . . . , xn ) ∈ X n we have ∥f (x1 , . . . , xn ) − Fi (x1 , . . . , xn )∥ ≤ φ˜i (x1 , . . . , xi , xi , xi+1 , . . . , xn ). For every i ∈ {1, . . . , n} the function Fi is given by 1 f (x1 , . . . , xi−1 , 2j xi , xi+1 , . . . , xn ), j→∞ 2j

Fi (x1 , . . . , xn ) := lim

(x1 , . . . , xn ) ∈ X n .

The following example shows that the same result of Theorem 3.5 is not true in non-Archimedean normed spaces and the assumption |k| < 1 cannot be omitted in Corollaries 3.3 and 3.4. This example is a modification of the example of [22]. Example 3.6. Let p > 2 be a prime number and f : Q2p → Qp be defined by f (x1 , x2 ) = 2. Since, |2j |p = 1 for all j ∈ Z, then for φ1 (x1 , x2 , x3 ) = φ2 (x1 , x2 , x3 ) = 1 (the case when k = 2, ε = 1 and δ = 0 is considered), |D1 f (x1 , x′1 , x2 )|p = |D2 f (x1 , x2 , x′2 )|p = |2|p = 1 ≤ φ1 (x1 , x2 , x3 ) = φ2 (x1 , x2 , x3 ),

x1 , x′1 , x2 , x′2 ∈ Qp .

However |2j f (x1 /2j , x2 ) − 2j+1 f (x1 /2j+1 , x2 )|p = |2j+1 |p = 1 and |2j f (x1 , x2 /2j ) − 2j+1 f (x1 , x2 /2j+1 )|p = |2j+1 |p = 1 for all x1 , x2 ∈ Qp and j ∈ N. Hence neither {2j f (x1 /2j , x2 )} nor {2j f (x1 , x2 /2j )} is a Cauchy sequence. Hence these sequences are not convergent in Qp . 4. A fixed point approach to the stability In [2], Ciepli´ nski reduce the system of n Cauchy equations to a single functional equation (see [2, Theorem 2]).

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On the stability of multi-additive mappings in non-Archimedean normed spaces

Theorem 4.1. Assume that n ∈ N and let X be a commutative semigroup with the identity element 0 and Y be a linear space. A mapping f : X n → Y is multi-additive if and only if ∑ f (x11 + x12 , . . . , xn1 + xn2 ) = f (x1i1 , . . . , xnin ) (4.1) i1 ,...,in ∈{1,2}

for all x11 , x12 , . . . , xn1 , xn2 ∈ X .

Next, we will show the Hyers-Ulam stability of Eq.(4.1) by using the fixed point method. For the given mapping f : X n → Y , we define the difference operator ∑ Df (x11 , x12 , . . . , xn1 , xn2 ) := f (x11 + x12 , . . . , xn1 + xn2 ) − f (x1i1 , . . . , xnin ) i1 ,...,in ∈{1,2}

for all x11 , x12 , . . . , xn1 , xn2 ∈ X .

Theorem 4.2. Let X be a linear space over a non-Archimedean field K with a valuation |·| and Y be a complete non-Archimedean normed space over K. Let 0 ≤ L < 1 and φ : X 2n → [0, ∞) be a mapping such that φ(x11 , x12 , . . . , xn1 , xn2 ) ≤

L φ(2x11 , 2x12 , . . . , 2xn1 , 2xn2 ) |2|n

(4.2)

for all x11 , x12 , . . . , xn1 , xn2 ∈ X . If f : X n → Y is a function satisfying ∥Df (x11 , x12 , . . . , xn1 , xn2 )∥ ≤ φ(x11 , x12 , . . . , xn1 , xn2 )

(4.3)

for all x11 , x12 , . . . , xn1 , xn2 ∈ X , then there exists a unique multi-additive mapping F : X n → Y such that ∥f (x11 , x21 , . . . , xn1 ) − F (x11 , x21 , . . . , xn1 ∥ ≤

L φ(x11 , x11 , . . . , xn1 , xn1 ) |2|n (1 − L)

(4.4)

for all x11 , x21 , . . . , xn1 ∈ X . The function F is given by F (x11 , x21 , . . . , xn1 ) := lim 2mn f (x11 /2m , x21 /2m , . . . , xn1 /2m ) m→∞

(4.5)

for all x11 , x21 , . . . , xn1 ∈ X . Proof. It follows from (4.2) that lim |2|nj φ(x11 /2j , x12 /2j , . . . , xn1 /2j , xn2 /2j ) = 0

j→∞

(4.6)

for all x11 , x12 , . . . , xn1 , xn2 ∈ X . Consider the set Ω := {g | g : X n → Y } and introduce the generalized metric on Ω defined by d(g, h) := inf{c > 0|

∥g(x11 , x21 , . . . , xn1 ) − h(x11 , x21 , . . . , xn1 )∥ ≤ cφ(x11 , x11 , . . . , xn1 , xn1 ), ∀ x11 , x21 , . . . , xn1 ∈ X }.

It is easy to show that (Ω, d) is a complete generalized metric space. Now we consider the mapping J : Ω → Ω such that Jg(x11 , x21 , . . . , xn1 ) = 2n g(x11 /2, x21 /2, . . . , xn1 /2) for all x11 , x21 , . . . , xn1 ∈ X n . Let g, h ∈ Ω be given such that d(g, h) < β, by the definition, ∥g(x11 , x21 , . . . , xn1 ) − h(x11 , x21 , . . . , xn1 )∥ ≤ βφ(x11 , x11 , . . . , xn1 , xn1 ) for all x11 , x21 , . . . , xn1 ∈ X . Hence ∥Jg(x11 , x21 , . . . , xn1 ) − Jh(x11 , x21 , . . . , xn1 )∥ = ∥2n g(x11 /2, x21 /2, . . . , xn1 /2) − 2n h(x11 /2, x21 /2, . . . , xn1 /2)∥ = |2|n ∥g(x11 /2, x21 /2, . . . , xn1 /2) − h(x11 /2, x21 /2, . . . , xn1 /2)∥ ≤ |2|n βφ((x11 /2, x11 /2, . . . , xn1 /2, xn1 /2) ≤ βLφ(x11 , x11 , . . . , xn1 , xn1 )

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T. Z. Xu, C. Wang and Th. M. Rassias

for all x11 , x21 , . . . , xn1 ∈ X . By the definition it follows that d(Jg, Jh) ≤ βL. Therefore, d(Jg, Jh) ≤ Ld(g, h), for all g, h ∈ Ω. This means that J is a strictly contractive self-mapping of Ω with Lipschitz constant L. Putting xi2 = xi1 for i ∈ {1, 2, . . . , n} in (4.3), we get ∥f (2x11 , 2x21 , . . . , 2xn1 ) − 2n f (x11 , x21 , . . . , xn1 )∥ ≤ φ(x11 , x11 , . . . , xn1 , xn1 ) for all x11 , x21 , . . . , xn1 ∈ X . Hence ∥2n f (x11 /2, x21 /2, . . . , xn1 /2) − f (x11 , x21 , . . . , xn1 )∥

≤

φ(x11 /2, x11 /2, . . . , xn1 /2, xn1 /2)

≤

L |2|n φ(x11 , x12 , . . . , xn1 , xn2 )

for all x11 , x21 , . . . , xn1 ∈ X . Therefore d(Jf, f ) ≤ L/|2|n . By Theorem 2.6, there exists a mapping F : X n → Y such that (1) F is a fixed point of J, that is, F (2x11 , 2x21 , . . . , 2xn1 ) = 2n F (x11 , x21 , . . . , xn1 )

(4.7)

for all x11 , x21 , . . . , xn1 ∈ X . The mapping F is a unique fixed point of J in the set ∆ = {g ∈ Ω| d(f, g) < ∞}. This implies that F is a unique mapping satisfying (4.7) such that there exists c ∈ (0, ∞) satisfying ∥F (x11 , x21 , . . . , xn1 ) − f (x11 , x21 , . . . , xn1 )∥ ≤ c · φ(x11 , x11 , . . . , xn1 , xn1 ) for all x11 , x21 , . . . , xn1 ∈ X . (2) d(J m f, F ) → 0 as m → ∞. This implies the equality F (x11 , x21 , . . . , xn1 ) := lim 2mn f (x11 /2m , x21 /2m , . . . , xn1 /2m ) m→∞

(4.8)

for all x11 , x21 , . . . , xn1 ∈ X . (3) d(f, F ) ≤ d(f, Jf )/(1 − L), which implies the inequality d(f, F ) ≤ L/[|2|n (1 − L)]. Thus inequality (4.4) holds. It follows from (4.3), (4.6), and (4.8) that ∥F (x11 + x12 , . . . , xn1 + xn2 ) −

∑

i1 ,...,in ∈{1,2}

F (x1i1 , . . . , xnin )∥

= lim |2|nj ∥f ((x11 + x12 )/2j , . . . , (xn1 + xn2 )/2j ) − j→∞

≤ lim |2|nj φ(x11 /2j , x12 /2j , . . . , xn1 /2j , xn2 /2j ) = 0

∑

i1 ,...,in ∈{1,2}

f (x1i1 /2j , . . . , xnin /2j )∥

j→∞

for all x11 , x12 , . . . , xn1 , xn2 ∈ X . Therefore Theorem 4.1 now shows that F is multi-additive. This completes the proof.  Corollary 4.3. Let K be a non-Archimedean field, (X , ∥ · ∥X ) be a non-Archimedean normed space over K, (Y , ∥ · ∥Y ) be a complete non-Archimedean normed space over K. Let ε, δ ≥ 0, 0 < r < 1, |2| < 1, and f : X n → Y be a mapping such that nr nr nr ∥Df (x11 , x12 , . . . , xn1 , xn2 )∥Y ≤ ε + δ(∥x11 ∥nr X + ∥x12 ∥X + · · · + ∥xn1 ∥X + ∥xn2 ∥X )

for all x11 , x12 , . . . , xn1 , xn2 ∈ X . Then there exists a unique multi-additive mapping F : X n → Y such that ∥f (x11 , x21 , . . . , xn1 ) − F (x11 , x21 , . . . , xn1 )∥Y ≤

|2|nr

1 nr nr [ε + 2δ(∥x11 ∥nr X + ∥x21 ∥X + · · · + ∥xn1 ∥X )] − |2|n

for all x11 , x21 , . . . , xn1 ∈ X . The function F is given by F (x11 , x12 , . . . , xn1 ) := lim 2mn f (x11 /2m , . . . , xn1 /2m ) m→∞

for all x11 , x21 , . . . , xn1 ∈ X . Corollary 4.4. Let K be a non-Archimedean field, (X , ∥ · ∥X ) be a non-Archimedean normed space over K, (Y , ∥ · ∥Y ) be a complete non-Archimedean normed space over K, ε, δ ≥ 0 and r, s > 0 with λ := r + s < 1.

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On the stability of multi-additive mappings in non-Archimedean normed spaces

Assume also that f : X n → Y be a mapping satisfying ∥Df (x11 , x12 , . . . , xn1 , xn2 )∥Y ≤ ε + δ

n ∑

ns nλ nλ [∥xk1 ∥nr X · ∥xk2 ∥X + (∥xk1 ∥X + ∥xk2 ∥X )]

k=1

for all x11 , x12 , . . . , xn1 , xn2 ∈ X . Then there exists a unique multi-additive mapping F : X n → Y such that ∥f (x11 , x21 , . . . , xn1 ) − F (x11 , x21 , . . . , xn1 )∥ ≤

|2|nλ

1 nλ nλ [ε + 3δ(∥x11 ∥nλ X + ∥x21 ∥X + · · · + ∥xn1 ∥X )] − |2|n

for all x11 , x21 , . . . , xn1 ∈ X n . The function F is given by F (x11 , x12 , . . . , xn1 ) := lim 2mn f (x11 /2m , . . . , xn1 /2m ) m→∞

for all x11 , x21 , . . . , xn1 ∈ X . n

The following example shows that the assumption |2| < 1 cannot be omitted in Corollaries 4.3 and 4.4. Example 4.5. Let p > 2 be a prime number and f : Q2p → Qp be defined by f (x1 , x2 ) = 2. Since, |2j |p = 1 for all j ∈ Z, then for ε = 1 and δ = 0, |Df (x11 , x12 , x21 , x22 )|p = |6|p = |3|p ≤ 1 = ε,

x11 , x12 , x21 , x22 ∈ Qp .

However, |4j f (x1 /2j , x2 /2j ) − 4j+1 f (x1 /2j+1 , x2 )/2j+1 |p = |7 · 22j |p = |7|p for all x1 , x2 ∈ Qp and j ∈ N. Hence {4j f (x1 /2j , x2 /2j )} is not convergent in Qp . References [1] K. Ciepli´ nski, Stability of multi-additive mappings in non-Archimedean normed spaces, J. Math. Anal. Appl., 373(2011), 376–383. [2] K. Ciepli´ nski, Generalized stability of multi-additive mappings, Applied Mathematics Letters, 23(2010), 1291–1294. [3] K. Cieplinski, On the generalized Hyers-Ulam stability of multi-quadratic mappings, Comput. Math. Appl., 62(2011), 3418–3426. [4] K. Cieplinski, T.Z. Xu, Approximate multi-Jensen and multi-quadratic mappings in 2-Banach spaces, Carpathian Journal of Mathematics, 29(2)(2013), 159–166. [5] J.B. Diaz, B. Margolis, A fixed point theorem of the alternative for the contractions on generalized complete metric space, Bull. Amer. Math. Soc., 74(1968), 305–309. [6] P. G˘ avrut¸a, A generalization of the Hyers-Ulam-Rassias stability of approximately additive mappings, Journal of Mathematical Analysis and Applications, 184(1994), 431–436. [7] D.H. Hyers, On the stability of the linear functional equation, Proc. Nat. Acad. Sci. USA, 27(1941), 222–224. [8] D.H. Hyers, G. Isac and Th.M. Rassias, Stability of Functional Equations in Several Variables, Birkhauser, Boston, Basel, Berlin, 1998. [9] S.-M. Jung, Hyers-Ulam-Rassias Stability of Functional Equations in Nonlinear Analysis, Springer, New York, 2011. [10] A. Khrennikov, Non-Archimedean Analysis: Quantum Paradoxes, Dynamical Systems and Biological Models. Kluwer Academic Publishers, Dordrecht, 1997. [11] A.K. Mirmostafaee, M.S. Moslehian, Stability of additive mappings in non-Archimedean fuzzy normed spaces, Fuzzy Sets and Systems 160(2009) 1643-1652. [12] Z. Moszner, On the stability of functional equations, Aequationes Math., 77(2009), 33–88. [13] W. Prager, J. Schwaiger, Stability of the multi-Jensen equation, Bull. Korean Math. Soc., 45(2008), 133–142. [14] Th.M. Rassias, On the stability of the linear mapping in Banach spaces, Proc. Amer. Math. Soc., 72(1978), 297–300. [15] Th.M. Rassias, J. Brzd¸ek, Functional Equations in Mathematical Analysis, Springer, New York, 2011. [16] R. Saadati, Ch. Park, Non-Archimedian L -fuzzy normed spaces and stability of functional equations, Comput. Math. Appl., 60(2010), 2488–2496. [17] S.M. Ulam, A Collection of the Mathematical Problems, Interscience, New York, 1960. [18] V.S. Vladimirov, I.V. Volovich, E.I. Zelenov, p-adic Analysis and Mathematical Physics. World Scientific, 1994. [19] T.Z. Xu, On the stability of multi-Jensen mappings in β-normed spaces, Applied Mathematics Letters, 25(2012), 1866–1870. [20] T.Z. Xu, Approximate multi-Jensen, multi-Euler-Lagrange additive and quadratic mappings in n-Banach spaces, Abstract and Applied Analysis, 2013(2013), Article ID 648709, 1–12. [21] T.Z. Xu, J.M. Rassias, Approximate septic and octic mappings in quasi-β-normed spaces, Journal of Computational Analysis and Applications, 15(6)(2013), 1110–1119. [22] T.Z. Xu, J.M. Rassias, W.X. Xu, Stability of a general mixed additive-cubic functional equation in non-Archimedean fuzzy normed spaces, J. Math. Phys., 51(2010), Article ID 093508 1–19. [23] T.Z. Xu, J.M. Rassias, W.X. Xu, Stability of a general mixed additive-cubic equation in F -spaces, Journal of Computational Analysis and Applications, 14(6)(2012), 1026–1037. [24] T.Z. Xu, Z. Yang, J.M. Rassias, Direct and fixed point approaches to the stability of an AQ-functional equation in nonArchimedean normed spaces, Journal of Computational Analysis and Applications, 17(4)(2014), 697–706.

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Notes on Harmonic Functions for which the second Dilatation is α - spiral Melike Aydo˜gan 2014 Abstract In this study, we consider, f = h + g harmonic functions in the unit disc D. By applying S. S. Miller and P. M. Mocanu result, we ∗ obtain a new subclass of harmonic functions , such as SHP ST (α, β) We introduce this new class as defined in the following form, n ∗ SHP (α, β) = f = h(z) + g(z)|f ∈ SH , h(z) ∈ S ∗ , ST   o 0 π iα g (z) Re e 0 > β, |α| < , 0 ≤ β < Re(beiα ) h (z) 2 (0.1) We also use subordination principle , study on distortion theorems, some numerical examples and coefficient inequalities of this class.

1

Introduction

A planar harmonic mapping in the unit disc D = {z ∈ C||z| < 1} is a complexvalued harmonic function f which maps D onto some planar domain f (D). Since D is simply connected, the mapping f has a canonical decomposition f = h + g, where h and g are analytic in D, as usual, we call h the analytic part of f and g the co-analytic part of f . An elegant and complete account of the theory of planar harmonic mapping is given in Duren’s monograph [3]. 2000 Mathematics Subject Classification: 30C45, 30C55 Key words and phrases: Harmonic functions, growth theorem, distortion theorem, coefficient inequality

1

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Melike Aydogan 1111-1121

J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 18, NO.6, 2015, COPYRIGHT 2015 EUDOXUS PRESS, LLC

Lewy [5] proved in 1936 that the harmonic function f is locally univalent in a simply connected domain D1 if and only if its Jacobien 2

2

Jf (z) = |h0 (z)| − |g 0 (z)| > 0 is different from zero in D1 . In view of this result, locally univalent harmonic mappings in the unit disc are either sense-reversing if |g 0 (z)| > |h0 (z)| in D1 or sense-preserving if |g 0 (z)| < |h0 (z)| in D1 . Throughout this paper we will restrict ourselves to the study of sensepreserving harmonic mappings. However, since f is sense-preserving if and only if f is sense-reserving, all the results obtained in this article regarding sense-preserving harmonic mappings can be adapted to sense-reversing ones. Note that f = h + g is sense-preserving in D if and only if h0 (z) does not 0 vanish in the unit disc and the second-complex dilatation w(z) = hg 0(z) has (z) 2 the property |w(z)| < 1 in D, therefore we can take h(z) = z + a2 z + · · · , g(z) = b1 z + b2 z 2 + · · · . Thus the class of all harmonic mappings being sense-preserving in the unit disc can be defined by n SH = f = h(z) + g(z) |h(z) = z + a2 z 2 + · · · , g(z) = b1 z + b2 z 2 + · · · , o f sense-preserving Let Ω be the family of functions φ(z) which are regular in D and satisfying the conditions φ(0) = 0, |φ(z)| < 1 for all z ∈ D. Denote by P , the family of functions p(z) = 1 + p1 z + p2 z 2 + · · · which are regular in D such that p(z) =

1 + φ(z) 1 − φ(z)

(1.1)

for some function φ(z) ∈ Ω for all z ∈ D. Next, let S ∗ denote the family of functions s(z) = z + c2 z 2 + c3 z 3 + · · · which are regular in D such that s0 (z) z = p(z) (1.2) s(z) 2

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for some p(z) ∈ P for all z ∈ D. Let s1 (z) = z + α2 z 2 + α3 z3 + · · · and s2 (z) = z + β2 z 2 + β3 z 3 + · · · be analytic functions in D. If there exists φ(z) ∈ Ω such that s1 (z) = s2 (φ(z)) for all z ∈ D. Then we say that s1 (z) is subordinate to s2 (z) and we write s1 (z) ≺ s2 (z), then s1 (D) ⊂ s2 (D). Now, we consider the following class of harmonic mappings in the plane n ∗ (α, β) = f = h(z) + g(z)|f ∈ SH , h(z) ∈ S ∗ , SHP ST   o 0 π iα g (z) > β, |α| < , 0 ≤ β < Re(beiα ) Re e 0 h (z) 2 (1.3) ∗ In the present paper we will investigate the class SHP ST (α, β). We will need the following lemma and theorem in the sequel:

Theorem 1.1. ([4]) Let h(z) be an element of S ∗ , then r r ≤ |h(z)| ≤ , 2 (1 + r) (1 − r)2 for all |z| = r < 1. 1+r 1−r ≤ |h0 (z)| ≤ 3 (1 + r) (1 − r)3 These inequalities are sharp because the extremal function is z h(z) = (1−z) 2. Lemma 1.2. ([6]) Let M (z) and N (z) be regular in D with M (0) = N (0) = 0, and let γ be real. If N (z) maps D onto a (possibly many-sheeted) domain which is starlike with respect to the origin, then Re(

2

M (z) M 0 (z) ) > γ ⇒ Re( )>γ 0 N (z) N (z)

Main Results

Theorem 2.1. Let g(z) and h(z) be analytic in D with g(0) = h(0) = 0, and let 0 ≤ β < Re(eiα b1 ) and |α| < π2 . If h(z) maps D onto a (possibly many sheeted) domain which is starlike with respect to the origin, then Re(eiα

g 0 (z) g(z) ) > β(z ∈ D) ⇒ Re(eiα ) > β(z ∈ D) 0 h (z) h(z) 3

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Proof. If we take M (z) = g(z) and N (z) = e−iα h(z), then M (z) and N (z) are analytic in D and M (0) = N (0) = 0. Also, zh0 (z) zN 0 (z) ) = Re( ) > 0, (z ∈ D) Re( N (Z) h(z) Thus, appliying Lemma 1.2, we prove the theorem. Example 2.2. Let us consider the function g(z) such that eiα b1 > 0, β = 0 z and h(z) = (1−z) 2. Now , we consider g 0 (z) 1+z eiα 0 = eiα b1 h (z) 1−z Then, we easily see that, Re(eiα

g 0 (z) ) > 0, (z ∈ D). h0 (z)

For such g(z), we have that g 0 (z) =

b1 (1 + z)2 . (1 − z)4

Thus, we obtain that Z g(z) = b1 0

z

b1 z(3 + z 2 ) (1 + t)2 dt = (1 − t)4 3(1 − z)3

Using the above g(z) and h(z) = Re(eiα

z , (1−z)2

we see that

g(z) eiα b1 (3 + z 2 ) ) = Re( ) > 0, (z ∈ D) h(z) 3(1 − z)

∗ Theorem 2.3. If f (z) = h(z) + g(z) is in the class SHP ST (α, β), then g(z) |b1 | ei(φ−α) (1 + r2 e−i2φ ) − 2r2 β.eiα 2r(|b1 | cos φ − β) ≤ h(z) − 1 − r2 1 − r2

for |z| = r < 1, where φ = α + arg(b1 ).

4

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∗ Proof. For f (z) ∈ SHP ST (α, β), Theorem 1 gives us that

Re(eiα

g(z) ) > β(z ∈ D) h(z)

Let us define p(z) = eiα and φ(z) =

g(z) h(z)

p(z) − β − Imp(0) Rep(0) − β

Then, we see that p(0) = b1 eiα = |b1 | eiα (φ = α + arg(b1 )) and that φ(z) is analytic in D, φ(0) = 1, and Reφ(z) > 0, (z ∈ D). Therefore, φ(z) is Caratheodory function. It follows from the above that, φ(z) ≺

1+z , 1−z

that is, φ(z) =

1 + w(z) 1 − w(z)

where w(z) is analytic in D, w(0) = 0 and |w(z)| < 1, (z ∈ D). Therefore, using Schwarz lemma, we have that φ(z) − 1 ≤ r(|z| = r < 1) |w(z)| = φ(z) + 1 Note that

2 φ(z) − 1 + r ≤ 2r 1 − r2 1 − r2

and φ(z) =

g(z) eiα h(z) − β − i |b1 | φ

|b1 | cos φ − β

Therefore, we have that iα g(z) |b1 | eiφ (1 + r2 e−i2φ ) − 2r2 β 2r(|b1 | cos φ − β) e ≤ h(z) − 1 − r2 1 − r2 5

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Since φ = α + arg(b1 ), we obtain that g(z) |b1 | ei(φ−α) (1 + r2 e−i2φ ) − 2r2 β.e−iα 2r(|b1 | cos φ − β) ≤ . h(z) − 1 − r2 1 − r2

∗ Corollary 2.4. Let f (z) = h(z) + g(z) is in the class SHP ST (α, β); if arg(b1 ) = −α, then φ = 0. Therefore, we have that b1 (1 − r) − 2rβ g(z) b1 (1 + r) − 2rβ ≤ (2.1) h(z) ≤ 1+r 1−r

Proof. This is a simple consequence of Theorem 2.3. ∗ Corollary 2.5. Let f (z) = h(z) + g(z) is in the class SHP ST (α, β); if β = 0, then we have |b1 | (1 − r) g(z) |b1 | (1 + r) ≤ ≤ (2.2) 1+r h(z) 1−r

Proof. This is a simple consequence of Theorem 2.3. ∗ Corollary 2.6. Let f (z) = h(z) + g(z) is in the class SHP ST (α, β), then

r[|b1 | (1 − r) − 2rβ] r[|b1 | (1 + r) − 2rβ] ≤ |g(z)| ≤ 3 (1 + r ) (1 − r3 ) (1 − r)[|b1 | (1 − r) − 2rβ] (1 + r)[|b1 | (1 + r) − 2rβ] 0 ≤ |g (z)| ≤ (1 + r)4 (1 − r)4

(2.3)

(2.4)

for all |z| = r < 1. ∗ Proof. Let f (z) = h(z) + g(z) is in the class SHP ST (α, β), then by using Theorem 1.1, we can write

|h(z)|

|b1 | (1 − r)2rβ |b1 | (1 + r)2rβ ≤ |g(z)| ≤ |h(z)| (1 + r) (1 − r)

|h0 (z)|

|b1 | (1 − r)2rβ |b1 | (1 + r)2rβ ≤ |g 0 (z)| ≤ |h0 (z)| (1 + r) (1 − r)

Therefore we can take the result easliy. 6

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∗ Corollary 2.7. Let f (z) = h(z) + g(z) is in the class SHP ST (α, β), if β = 0, then we have r |b1 | (1 − r) r |b1 | (1 + r) ≤ |g(z)| ≤ 3 (1 + r) (1 − r)3

|b1 | (1 + r)2 |b1 | (1 − r)2 0 ≤ |g (z)| ≤ (1 + r)4 (1 − r)4 Proof. This is a simple consequence of Corollary 2.6. ∗ Corollary 2.8. Let f (z) = h(z) + g(z) is in the class SHP ST (α, β), then

F (|b1 | , −r, β) ≤ Jf (z) ≤ F (|b1 | , r, β) where F (|b1 | , r, β) =

[1 − |b1 | − r(1 + |b1 | + 2β)][1 + |b1 | + r(−1 − |b1 | + 2β)] (1 + r)6

Proof. Since 2

2

2

Jf (z) = |h0 (z)| − |g 0 (z)| = |h0 (z)| (1 − |w(z)|2 ) Using Corollary 2.6 and Theorem 2.3 we get the result easily. ∗ Theorem 2.9. If f (z) = h(z) + g(z) is in the class SHP ST (α, β), then

|bn − b1 an | ≤

(n − 1)(2n − 1) (|b1 | cos(α + arg(b1 )) − β) 3

(2.5)

(n = 2, 3, 4, ...) Proof. Since Re(eiα

g 0 (z) ) > β, (z ∈ D) h0 (z)

∗ for f (z) ∈ SHP ST (α, β), we see that

φ(z) =

p(z) − β − iImp(0) Rep(0) − β

is the Caratheodory function, where p(z) = eiα

g 0 (z) h0 (z)

7

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and p(0) = b1 eiα = |b1 | ei(α+arg(b1 )) If we write φ(z) = 1 + c1 z + c2 z 2 + ..., we have that |cn |}leq2, (1, 2, 3, ...). Note that p(z) = (Rep(0) − β)φ(z) + β + iImp(0), that is, that eiα g 0 (z) = h0 (z)[(Rep(0) − β)φ(z) + β + iImp(0)] It follows that iα

e b1 +

∞ X

nbn z n−1

n=2

= (1 +

∞ X

nan z

n−1

)[(Rep(0) − β)(1 +

n=2

Considering the coefficient for z

∞ X

cn z n ) + β + iImp(0)]

n=1 n−1

, we have that

meiα bm = (Rep(b1 eiα )−β)(cn−1 +2a2 cn−2 +3a3 cn−3 +...+(n−1)an−1 c1 )+b1 eiα nan This shows us that, 1 (Rep(b1 eiα )−β)(cn−1 +2a2 cn−2 +3a3 cn−3 +...+(n−1)an−1 c1 ). n (2.6) ∗ Since |an | ≤ n, (n = 2, 3, 4, ...) for h(z) ∈ S and |cn | ≤ 2, (n = 1, 2, 3, ...), we obtain that 2 |bn − b1 an | ≤ (|b1 | cos(α+arg(b1 ))−β)(1+2 |a2 |+3 |a3 |+...+(n−1) |an−1 |) n 2 ≤ (|b1 | cos(α + arg(b1 )) − β)(12 + 22 + 32 + ... + (n − 1)2 ) n (n − 1)(2n − 1) = (|b1 | cos(α + arg(b1 )) − β) 3 This completes the proof of the theorem. (bn −b1 an )eiα =

∗ Corollary 2.10. If f (z) = h(z) + g(z) is in the class SHP ST (α, β) with h(z) ∈ K, then

|bn − b1 an | ≤ (n − 1)(|b1 | cos(α + arg(b1 )) − β), (n = 2, 3, 4, ...) Proof. This is a simple consequence of Theorem 2.9. 8

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P P∞ n n Theorem 2.11. Let h(z) = z + ∞ n=2 an z and g(z) = b1 z + n=2 bn z be analytic in D. Also, let h(z) is starlike with respect to the origin in D, and f (z) = h(z) + g(z). If h(z) and g(z) satisfy, ∞ X

n[ (1 − β)an + eiα bn + (1 + β)an − eiα bn ]

n=2

≤ 1 − β + eiα b1 − 1 + β − eiα b1 for some real α (|α| < ∗ f (z) ∈ SHP ST (α, β).

π ) 2

and for some real β, (0 ≤ β ≤ Re(b1 eiα ), then

0

Then, if p(z) satisfies, Proof. Let p(z) = eiα hg 0(z) (z) 1 − (p(z) − β) 1 + (p(z) − β) < 1 ∗ then Rep(z) > β, so that, f (z) ∈ SHP ST (α, β). It follows that

|1 + (p(z) − β)| − |1 − (p(z) − β)| 1 0 iα 0 − (1 + β)h0 (z) − eiα g 0 (z) ] [ (1 − β)h (z) + e g (z) |h0 (z)| ∞ X 1 n((1 − β)an + eiα bn )z n−1 − (1 + β − eiα b1 ) [ (1 − β + eiα b1 ) + = 0 |h (z)| =

n=2

+

∞ X

n((1 + β)an − eiα bn )z n−1 ]

n=2

≥

∞ X 1 iα [ 1 − β + e b − n (1 − β)an + eiα bn |z|n−1 1 0 |h (z)| n=2 ∞ X iα (1 + β)an − eiα bn |z|n−1 ] − 1 + β − e b1 − n=2

>

1 |h0 (z)|

[ 1 − β + eiα b1 − 1 + β − eiα b1 ] 9

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Melike Aydogan 1111-1121

J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 18, NO.6, 2015, COPYRIGHT 2015 EUDOXUS PRESS, LLC

−

∞ X

n[ (1 − β)an + eiα bn + (1 + β)an − eiα bn ]

n=2

for z ∈ D. Therefore, if f (z) satisfies, ∞ X

n[ (1 − β)an + eiα bn + (1 + β)an − eiα bn ]

n=2

≤ 1 − β + eiα b1 − 1 + β − eiα b1 , ∗ then f (z) ∈ SHP ST (α, β).

References [1] S. D. Bernardi, Convex and Starlike Univalent Functions, Trans. Amer. Math. Soc. 1969, (135), 429-446. [2] J. Clunie, On Meromorphic Schlicht Functions, J.London. Math. Soc. 34, (1959), 215-216. [3] P. Duren, Harmonic Mapping in the Plane, Cambridge press 2004, Cambrdige. [4] A. W. Goodman, Univalent Functions, Volume I and Volume II, Mariner publishing Company INC, 1983. [5] H. Lewy, On the non-vanishing of the Jacobian in certain in one-to-one mappings, Bull. Amer. Math. Soc. 42 (1936), 689-692. [6] S. S. Miller and P. Mocanu, Differential Subordinations, Theory and Applications, Marcel Dekker, 2000, pp. 63.

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Melike Aydogan 1111-1121

J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 18, NO.6, 2015, COPYRIGHT 2015 EUDOXUS PRESS, LLC

˜ an Melike Aydog Department of Mathematics, ˙ I¸sık University, Me¸srutiyet Koyu, S ¸ ile Istanbul, Turkey e-mail: [email protected]

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Melike Aydogan 1111-1121

 

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TABLE OF CONTENTS, JOURNAL OF COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 18, NO. 6, 2015

Functional Inequalities Associated With Inner Product Preserving Mappings, Gang Lu, George A. Anastassiou, Choonkil Park, and Yuanfeng Jin,…………………………………………964 Stability and Superstability of (𝑓𝑟 , 𝑓𝑠 )-Double Derivations in Quasi-Banach Algebras, Sun Young Jang, Choonkil Park, Pegah Efteghar, and Shahrokh Farhadabadi,…………………………973 The Fixed Point Method for Perturbation of Bihomomorphisms and Biderivations in Normed 3Lie Systems: Revisited, Choonkil Park, Jung Rye Lee, Eon Wha Shim, and Dong Yun Shin,984 Dynamics of some Rational Difference Equations, H. El-Metwally, E.M. Elsayed, and H. El-Morshedy,………………………………………………………………………………993 Generalized Integration Operators from Hardy Spaces to Zygmund-Type Spaces, Huiying Qu, Yongmin Liu, and Shulei Cheng,……………………………………………………………1004 Approximation Properties of the Modification of Durrmeyer Type q-Baskakov Operators Which Preserve 𝑥 2 , Qing-Bo Cai,……………………………………………………………………1017 Qualitative Behavior of Two Systems of Second-Order Rational Difference Equations, A. Q. Khan, M. N. Qureshi, and Q. Din,……………………………………………………..1027

Strong Differential Subordination Results Using a Generalized Sălăgean Operator and Ruscheweyh Operator, Andrei Loriana, ……………………………………………………..1042 On Some Differential Sandwich Theorems Using a Generalized Sălăgean Operator and Ruscheweyh Operator, Andrei Loriana,………………………………………………………1049 Subalgebras of BCK/BCI-Algebras Based on (𝛼, 𝛽)-type Fuzzy Sets, G. Muhiuddin, and Abdullah M. Al-roqi,………………………………………………………………………….1057 Existence Results for Nonlinear Fractional Integrodifferential Equations with Antiperiodic Type Integral Boundary Conditions, Xiaohong Zuo, and Wengui Yang,………………………….1065 Identities of Symmetry for Higher-Order q-Bernoulli Polynomials, Dae San Kim,Taekyun Kim,…………………………………………………………………………………………..1077 Fuzzy Stability of Functional Equations in Matrix Fuzzy Normed Spaces, Choonkil Park, Dong Yun Shin, and Jung Rye Lee,…………………………………………………………………1089

TABLE OF CONTENTS, JOURNAL OF COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 18, NO. 6, 2015 (continued) On the Stability of Multi-Additive Mappings in Non-Archimedean Normed Spaces, Tian Zhou Xu, Chun Wang, and Themistocles M. Rassias,……………………………………………1102 Notes on Harmonic Functions for Which the Second Dilatation is 𝛼-Spiral, Melike Aydog̃an,……………………………………………………………………………………1111