Abstract and Applied Analysis
Volume 2012 (2012), Article ID 763728, 14 pages
http://dx.doi.org/10.1155/2012/763728
Research Article

On the Stability Problem in Fuzzy Banach Space

1Faculty of Mathematical Sciences, University of Tabriz, Tabriz, Iran
2Department of Mathematics, University Politehnica of Timisoara, Piata Victoriei 2, 300006 Timisoara, Romania
3Department of Mathematics, Kangnam University, Suwon, Kyunggi 449-702, Republic of Korea

Received 6 February 2012; Accepted 17 May 2012

Copyright © 2012 G. Zamani Eskandani et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

We investigate the generalized Ulam-Hyers stability of the Cauchy functional equation and pose two open problems in fuzzy Banach space.

1. Introduction and Preliminaries

In 1940, Ulam [1] asked the first question on the stability problem. In 1941, Hyers [2] solved the problem of Ulam. This result was generalized by Aoki [3] for additive mappings and by Th. M. Rassias [4] for linear mappings by considering an unbounded Cauchy difference.

Theorem 1.1 (Th. M. Rassias). Let be a mapping from a normed vector space into a Banach space subject to the inequality: for all , where and are constants with and . Then, the limit exists for all and is the unique additive mapping which satisfies for all . Also, if for each the function is continuous in , then is linear.

In 1990, Th. M. Rassias [5] during the 27th International Symposium on Functional Equations asked the question whether such a theorem can also be proved for . In 1991, Gajda [6] gave an affirmative solution to this question for . It was shown by Gajda [6], as well as by Th. M. Rassias and Šemrl [7], that one cannot prove a Th. M. Rassias type theorem when . Găvruţa [8] proved that the function , if and satisfies (1.1) with but for any additive function . J. M. Rassias [9] replaced the factor by for with (see also [10, 11]) and has obtained the following theorem.

Theorem 1.2. Let be a real normed linear space and a real complete normed linear space. Assume that is an approximately additive mapping for which there exist constants and such that satisfies the inequality: for all . Then, there exists a unique additive mapping satisfying for all . If, in addition, is a mapping such that the transformation is continuous in for each fixed , then is an -linear mapping.

In the case , we do not have stability [12]. In 1994, a further generalization of Th. M. Rassias’ Theorem was obtained by Găvruţa [13], in which he replaced the bound by a general control function . Isac and Th. M. Rassias [14] replaced the factor by in Theorem 1.1 and solved stability problem when or , also they asked the question whether such a theorem can be proved for . Găvruţa [8] gave a negative answer to this question. Isac and Th. M. Rassias [15] applied the Ulam-Hyers-Rassias stability theory to prove fixed point theorems and study some new applications in nonlinear analysis. During the last two decades, a number of papers and research monographs have been published on various generalizations and applications of the generalized Ulam-Hyers stability to a number of functional equations and mappings (see [1640]). We also refer the readers to the books of Czerwik [41] and Hyers et al. [42].

Th. M. Rassias [43] has obtained the following theorem and posed a problem.

Theorem 1.3. Let and be two Banach spaces, and let be a mapping such that is continuous in for each fixed . Assume that there exist and such that for all . Let be a positive integer . Then, there exists a unique linear mapping such that for all , where

Th. M. Rassias Problem
What is the best possible value of in Theorem 1.3?
Găvruţa et al. have given a generalization of [13] and have answered to Th. M. Rassias problem [44].
In [45], J. M. Rassias et al. have investigated the generalized Ulam-Hyers “product-sum” stability of functional equations and have obtained the following theorem.

Theorem 1.4 (see [45]). Let be a mapping which satisfies the inequality
for all with , where and p are constants with and either ,  or , with , , and . Then, the limit exists for all and is the unique orthogonally Euler-Lagrange quadratic mapping such that for all .

Note that the mixed “product-sum” function was introduced by J. M. Rassias in 2008-2009 [4648].

We recall some basic facts concerning fuzzy normed space.

Let be a real linear space. A function (so-called fuzzy subset) is said to be a fuzzy norm on if for all and all ,

for ;

if and only if for all ;

if ;

;

is a nondecreasing function of and

The pair is called a fuzzy normed linear space. The properties of fuzzy normed vector spaces and examples of fuzzy norms are given in [4951].

Let be a fuzzy normed space and let be a sequence in . Then, is said to be convergent if there exists such that for all . In that case, is called the limit of the sequence and we denote it by .

A sequence in a fuzzy normed space is called Cauchy if, for each and , one can find some such that

for all .

It is known that every convergent sequence in a fuzzy normed space is Cauchy. If, in a fuzzy-normed space, 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.

Stability of Cauchy, Jensen, quadratic, and cubic function equation in fuzzy normed spaces have first been investigated in [5053].

In this paper, we give a generalization of the results from [13] and pose two open problems in fuzzy Banach space. For convenience, we use the following abbreviation for a given mapping :

2. Stability of the Cauchy Functional Equation

Hereafter, unless otherwise stated, we will assume that is real vector space, is a complete fuzzy norm space and is a fixed integer greater than 1.

Theorem 2.1. Let be a fuzzy normed space and be a mapping such that, for some with . Suppose that be mapping such that for all and all positive real number . Then, there is a unique additive mapping such that and where .

Proof. By induction on , we show that for all and all positive real number . Letting in (2.1), we get So we get (2.3) for .
Assume that (2.3) holds for with . Letting in (2.1), we get for all . By using (2.3) and (2.5), we get (2.3) for and this completes the induction argument. Replacing by in (2.3), we get Thus for all and all positive real number . Hence, Let and be given. Since , there is some such that . Since , there is some such that for all . It follows that
for all and all nonnegative integers and with . Therefore, the sequence is a Cauchy sequence in for all . Since is complete, the sequence converges in for all . So one can define the mapping by for all . Now, we show that is an additive mapping. It follows from (2.1) and (2.10) that
for all and all positive real number . Therefore, the mapping is additive. Moreover, if we put in (2.8), we observe that Therefore, It follows from (2.13), for large enough , that Now, we show that is unique. Let be another additive mapping from into , which satisfies the required inequality. Then, for each and , we have So, Hence, the right-hand side of the above inequality tends to as . It follows that for all .

Theorem 2.2. Let be a fuzzy normed space and, be a mapping such that for some with . Suppose that be mapping such that for all and all positive real number . Then, there is a unique additive mapping such that and where .

Proof. Similarly to the proof of Theorem 2.1, we have for all and all positive real number . Replacing by in (2.19), we get Thus, for all and all positive real number . Hence,
Let and be given. Since , there is some such that . Since , there is some such that for all . It follows from (2.22) that for all and all nonnegative integers and with . Therefore, the sequence is a Cauchy sequence in for all . Since is complete, the sequence converges in for all . So one can define the mapping by for all . The rest of the proof is similar to the proof of Theorem 2.1

Theorem 2.3. Let be a normed space, let be a fuzzy normed space, and let be a function such that
,
for all .
Suppose that a mapping satisfies the inequality: for all and all positive real number , where is a fixed vector of . Then, there exists a unique additive mapping satisfying and for all , where . Moreover, for all .

Proof. Let for all . So, where . By using Theorem 2.1, we can get (2.26). Now, we show that . It follows from (1) that . Replacing by in (2.26), we get for all . So we have Using (2) and passing the limit in (2.30), we get .

Theorem 2.4. Let be a normed space, let be a fuzzy normed space, and let be a function such that
,
for all .
Suppose that a mapping satisfies the inequality: for all and all positive real number , where is a fixed vector of . Then, there exists a unique additive mapping satisfying and for all , where
Moreover,    for all .

Proof. Let for all . So, we have where . It follows from (1) that . By using Theorem 2.2, we can get (2.32). Now, we show that . Replacing by in (2.32), we get for all . So we have Using (2) and passing the limit in (2.37), we get .

Theorem 2.5. Let be a normed space, let be a nonnegative real number such that , and let be a homogeneous function of degree . Suppose that be a fuzzy normed space and let be mapping such that for all and all positive real number , where is a fixed vector of . Then, there exists a unique additive mapping such that where .

Proof. The proof follows from Theorems 2.1 and 2.2.

For the particular cases , and , we have the following corollaries.

Corollary 2.6. Let be a normed space, let be a nonnegative real number such that . Suppose that be a fuzzy normed space and be mapping such that for all and all positive real number , where is a fixed vector of . Then, there exists a unique additive mapping such that

Corollary 2.7. Let be a normed space, be non-negative real numbers such that . Suppose that be a fuzzy normed space and be mapping such that for all and all positive real number , where is a fixed vector of . Then there exists a unique additive mapping such that

Corollary 2.8. Let be a normed space, and let be nonnegative real numbers such that . Suppose that be a fuzzy normed space and let be mapping such that for all and all positive real number , where is a fixed vector of . Then, there exists a unique additive mapping such that

Corollary 2.9. Let be a normed space, let be a nonnegative real number such that . Suppose that be a fuzzy normed space and let be mapping such that for all and all positive real number , where is a fixed vector of . Then, there exists a unique additive mapping such that

Problem 1. Whether Theorem 2.5 and/or such Corollaries can be proved for ?

Problem 2. What is the best possible value of in Corollaries 2.6 and 2.7?

Acknowledgment

G. H. Kim was partially supported by Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education, Science and Technology (Grant no. 2011-0005197).

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