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The Stability of the Quartic Functional Equation in Various Spaces View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by Elsevier - Publisher Connector Computers and Mathematics with Applications 60 (2010) 1994–2002 Contents lists available at ScienceDirect Computers and Mathematics with Applications journal homepage: www.elsevier.com/locate/camwa The stability of the quartic functional equation in various spaces Reza Saadati a,b, Yeol J. Cho c,∗, Javad Vahidi b a Department of Mathematics and Computer Science, Amirkabir University of Technology, 424 Hafez Avenue, Tehran 15914, Iran b Faculty of Sciences, Shomal University, Amol, Iran c Department of Mathematics Education and the RINS, Gyeongsang National University, Chinju 660-701, Republic of Korea article info a b s t r a c t Article history: The purpose of this paper is first to introduce the notation of intuitionistic random normed Received 20 January 2010 spaces, and then by virtue of this notation to study the stability of a quartic functional Received in revised form 21 July 2010 equation in the setting of these spaces under arbitrary triangle norms. Then we prove the Accepted 22 July 2010 stability of above quartic functional equation in non-Archimedean random normed spaces. Furthermore, the interdisciplinary relation among the theory of random spaces, the theory Keywords: of non-Archimedean spaces, the theory of intuitionistic spaces and the theory of functional Stability equations are also presented in the paper. Quartic functional equation Intuitionistic random normed space ' 2010 Elsevier Ltd. All rights reserved. Non-Archimedean random normed spaces 1. Introduction The study of stability problems for functional equations is related to a question of Ulam [1] concerning the stability of group homomorphisms, which was affirmatively answered for Banach spaces by Hyers [2]. Subsequently, the result of Hyers was generalized by Aoki [3] for additive mappings and by Rassias [4] for linear mappings by considering an unbounded Cauchy difference. The paper by Rassias has provided a lot of influence in the development of what we now call Hyers–Ulam–Rassias stability of functional equations. For more information on such problems, we refer interested readers to [5–9] and [10,11]. The functional equation f .2x C y/ C f .2x − y/ D 4f .x C y/ C 4f .x − y/ C 24f .x/ − 6f .y/ (1.1) is said to be a quartic functional equation since the function f .x/ D cx4 is a solution. Every solution of a quartic functional equation is said to be a quartic mapping. The stability problem for a quartic functional equation first was considered by Rassias [12] for mappings f V X −! Y , where X is a real normed space and Y is a Banach space. In addition, Alsina [13], Miheţ and Radu [14], Miheţ, Saadati and Vaezpour [15,16] Mirmostafaee, Mirzavaziri and Moslehian [17–19], Saadati et al., [20,21] and Shakeri [22] investigated the stability in the settings of probabilistic, random and fuzzy normed spaces. The usual uncertainty principle of Werner Heisenberg leads to a generalized uncertainty principle, which has been motivated by string theory and non-commutative geometry. In a strong quantum gravity regime, space–time points are determined in a random manner. Thus the impossibility of determining the position of articles gives space–time a fuzzy and random structure [23–27]. Because of this random structure, a position space representation of quantum mechanics breaks down and therefore a generalized normed space of a quasi-position eigenfunction is required [26]. Thus, one needs to discuss a new family of random norms. Also, the concept of topology introduced by a random norm has important applications in quantum particle physics, particularly in connection with both string and E-infinity theory; see [24,28]. One of the most ∗ Corresponding author. E-mail addresses: [email protected], [email protected], [email protected] (R. Saadati), [email protected] (Y.J. Cho). 0898-1221/$ – see front matter ' 2010 Elsevier Ltd. All rights reserved. doi:10.1016/j.camwa.2010.07.034 R. Saadati et al. / Computers and Mathematics with Applications 60 (2010) 1994–2002 1995 important problems in topology introduced by a random norm is to obtain an appropriate concept of intuitionistic random normed space. These problems have been investigated by Chang–Rassias–Saadati [29]. 2. Preliminaries In this section, we recall some definitions and results which will be used later on in the paper. A triangular norm (briefly, a t-norm) is a binary operation on the unit interval T0; 1U, i.e., a function T VT0; 1U × T0; 1U! T0; 1U such that for all a; b; c 2 T0; 1U the following four axioms are satisfied. (i) T .a; b/ D T .b; a/ (commutativity); (ii) T .a;.T .b; c/// D T .T .a; b/; c/ (associativity); (iii) T .a; 1/ D a (boundary condition); (iv) T .a; b/ ≤ T .a; c/ whenever b ≤ c (monotonicity). Basic examples are the Łukasiewicz t-norm TL; TL.a; b/ D max.a C b − 1; 0/ 8a; b 2 T0; 1U and the t-norms TP ; TM ; TD, where TP .a; b/ VD ab, TM .a; b/ VD minfa; bg, min.a; b/; if max.a; b/ D 1I T .a; b/ VD D 0; otherwise. .n/ 2 T U 2 [ f g D .n−1/ ≥ If T is a t-norm then xT is defined for every x 0; 1 and n N 0 by 1 if n 0 and by T .xT ; x/ if n 1. A t-norm 2 .n/ D T is said to be of Hadºi¢ type (we denote this by T H) if the family .xT /n2N is equicontinuous at x 1 (see [30]). Other important triangular norms are as follows (see [31]). f SW g SW D SW D The Sugeno–Weber family Tλ λ∈[−1;1U is defined by T−1 TD; T1 TP and x C y − 1 C λxy T SW .x; y/ D max 0; λ 1 C λ if λ 2 .−1; 1/. D The Domby family fTλ gλ2T0;1U is defined by TD, if λ D 0; TM , if λ D 1 and 1 T D.x; y/ D λ C 1−x λ C 1−y λ 1/λ 1 .. x / . y / / if λ 2 .0; 1/. AA The Aczel–Alsina family fTλ gλ2T0;1U is defined by TD, if λ D 0; TM , if λ D 1 and AA −.j log xjλCj log yjλ/1/λ Tλ .x; y/ D e if λ 2 .0; 1/. n A t-norm T can be extended (by associativity) in a unique way to an n-array operation. For any .x1; :::; xn/ 2 T0; 1U , the value T .x1; :::; xn/ is defined by 0 D n D n−1 D TiD1xi 1; TiD1xi T .TiD1 xi; xn/ T .x1; :::; xn/: T can also be extended to a countable operation. For any sequence .xn/n2N in T0; 1U, the value is defined by 1 D n TiD1xi lim TiD1xi: (2.1) n!1 f n g The limit on the right-hand side of (2.1) exists since the sequence TiD1xi n2N is non-increasing and bounded from below. Proposition 2.1 ([31]). .i/ For T ≥ TL, the following implication holds: 1 1 D () X − 1 lim TiD1xnCi 1 .1 xn/ < : n!1 nD1 .ii/ If T is of Hadºi¢ type, then 1 D lim TiD1xnCi 1 n!1 for every sequence fxngn2N in T0; 1U such that limn!1 xn D 1. AA D .iii/ If T 2 fTλ gλ2.0;1/ [ fTλ gλ2.0;1/, then 1 1 D () X − α 1 lim TiD1xnCi 1 .1 xn/ < : n!1 nD1 SW .iv/ If T 2 fTλ gλ∈[−1;1/, then 1 1 D () X − 1 lim TiD1xnCi 1 .1 xn/ < : n!1 nD1 1996 R. Saadati et al. / Computers and Mathematics with Applications 60 (2010) 1994–2002 3. Non-Archimedean random normed spaces In 1897, Hensel [32] introduced a field with a valuation norm which does not have the Archimedean property. Definition 3.1. Let K be a field. A non-Archimedean absolute value on K is a function j · j V K ! R such that, for any a; b 2 K, we have (i) jaj ≥ 0 and equality holds if and only if a D 0, (ii) jabj D jajjbj, (iii) ja C bj ≤ maxfjaj; jbjg (the strict triangle inequality). Note that jnj ≤ 1 for each integer n. We always assume in addition that j · j is non-trivial, i.e., that there is an a0 2 K such that ja0j 6D 0; 1. Definition 3.2. A non-Archimedean random normed space (briefly, a non-Archimedean RN-space) is a triple .X; µ, T /, where X is a linear space over a non-Archimedean field K, T is a continuous t-norm, and µ is a mapping from X into DC, such that the following conditions hold: (NA-RN1) µx.t/ D "0.t/ for all t > 0 if and only if x D 0; D t 2 6D (NA-RN2) µαx.t/ µx. jαj / for all x X, t > 0; α 0; (NA-RN3) µxCy.maxft; sg/ ≥ T (µx.t/; µy.s// for all x; y; z 2 X and t; s ≥ 0. It is easy to see that if (NA-RN3) holds then we have (RN3) µxCy.t C s/ ≥ T (µx.t/; µy.s//. As a classical example, if .X; k:k/ is a non-Archimedean normed linear space, then the triple .X; N; min/, where 0 if t ≤ kxk; µ .t/ D x 1 if t > kxk; is a non-Archimedean RN-space. Definition 3.3.
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