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Abstract and Applied Analysis
Volume 2011, Article ID 536520, 11 pages
http://dx.doi.org/10.1155/2011/536520
Research Article

Stability of an Additive-Cubic-Quartic Functional Equation in Multi-Banach Spaces

1School of Science, Hubei University of Technology, Wuhan, Hubei 430068, China
2Department of Mathematics, Zhanjiang Normal University, Zhanjiang, Guangdong 524048, China
3Department of Mathematics, National Technical University of Athens, Zografou Campus, 15780 Athens, Greece

Received 5 August 2011; Accepted 30 September 2011

Academic Editor: Narcisa C. Apreutesei

Copyright © 2011 Zhihua Wang 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 prove the Hyers-Ulam stability of the additive-cubic-quartic functional equation in multi-Banach spaces by using the fixed point alternative method. The first results on the stability in the multi-Banach spaces were presented in (Dales and Moslehian 2007).

1. Introduction

Stability is investigated when one is asking whether a small error of parameters in one problem causes a large deviation of its solution. Given an approximate homomorphism, is it possible to approximate it by a true homomorphism? In other words, we are looking for situations when the homomorphisms are stable, that is, if a mapping is almost a homomorphism, then there exists a true homomorphism near it with small error as much as possible. This problem was posed by Ulam in 1940 (cf. [1]) and is called the stability of functional equations. For Banach spaces, the problem was solved by Hyers [2] in the case of approximately additive mappings. Later, Hyers' result was generalized by Aoki [3] for additive mappings and by Rassias [4] for linear mappings by allowing the Cauchy difference to be unbounded. During the last decade, stability of functional equations was studied by several mathematicians for mappings in various spaces including random normed spaces and fuzzy Banach spaces (cf. [5, 6]). For various other results on the stability of functional equations, one is referred to [726].

Most of the proofs of stability theorems in the Hyers-Ulam context have applied Hyers’ direct method. The exact solution of the functional equation is explicitly constructed as the limit of a sequence, which is originating from the given approximate solution. In 2003, Radu [27] proposed the fixed point alternative method for obtaining the existence of exact solutions and error estimations and noticed that a fixed point alternative method is essential for the solution of Ulam problem for approximate homomorphisms. Subsequently, some authors [28, 29] applied the fixed alternative method to investigate the stability problems of several functional equations.

The notion of multi-normed space was introduced by Dales and Polyakov [30] (or see [31, 32]). This concept is somewhat similar to operator sequence space and has some connections with operator spaces and Banach lattices. Motivations for the study of multi-normed spaces and many examples were given in [30, 31]. In 2007, stability of mappings on multi-normed spaces was first given in [31], and asymptotic aspect of the quadratic functional equation in multi-normed spaces was investigated in [33].

In this paper, we consider the following functional equation derived from additive, cubic, and quartic mappings: It is easy to see that the function satisfies the functional equation (1.1). Eshaghi Gordji et al. [34] established the general solution and proved the generalized Hyers-Ulam stability for (1.1). The main purpose of this paper is to establish the Hyers-Ulam stability of (1.1) in multi-Banach spaces by using the fixed point alternative method.

2. Preliminaries

In this section, some useful results are pointed out. We begin with the alternative of a fixed point of Diaz and Margolis, which we will refer to as follows.

Lemma 2.1 (cf. [27, 35]). Let be a complete generalized metric space and be a strictly contractive mapping, that is, for some . Then,  for each fixed element , either or for some natural number . Moreover, if the second alternative holds, thenthe sequence is convergent to a fixed point of ; is the unique fixed point of in the set and .

Following [30, 31], we recall some basic facts concerning multi-normed spaces and some preliminary results.

Let be a complex normed space, and let . We denote by the linear space consisting of -tuples , where . The linear operations are defined coordinatewise. The zero element of either or is denoted by 0. We denote by the set and by the group of permutations on symbols.

Definition 2.2 (cf. [30, 31]). A multi-norm on is a sequence such that is a norm on for each , for each , and the following axioms are satisfied for each with :
(N1), for , ;, for ; , for ;(N4), for .In this case, we say that is a multi-normed space.

Suppose that is a multi-normed space, and take . We need the following two properties of multi-norms. They can be found in [30], for ,, for .

It follows from that if is a Banach space, then is a Banach space for each ; in this case, is a multi-Banach space.

Lemma 2.3 (cf. [30, 31]). Suppose that and . For each , let be a sequence in such that . Then holds for all .

Definition 2.4 (cf. [30, 31]). Let be a multi-normed space. A sequence in is a multi-null sequence if for each , there exists such that

Let , we say that the sequence is multi-convergent to in and write if is a multi-null sequence.

3. Main Results

Throughout this section, let , be a linear space, and let be a multi-Banach space. For convenience, we use the following abbreviation for a given mapping : Before proceeding to the proof of the main results in this section, we shall need the following two lemmas.

Lemma 3.1 (cf. [34]). If an even mapping satisfies (1.1), then is quartic.

Lemma 3.2 (cf. [34]). If an odd mapping satisfies (1.1), then is cubic-additive.

Theorem 3.3. Suppose that an even mapping satisfies and for all . Then there exists a unique quartic mapping satisfying (1.1) and for all .

Proof. Letting in (3.2), we get for all . Replacing with in (3.2), we obtain for all .
It follows from (3.4) and (3.5) that for all .
Let and introduce the generalized metric defined on by Then, it is easy to show that is a complete generalized metric on (see the proof in [36] or [5]). We now define a function by We assert that is a strictly contractive mapping. Given let be an arbitrary constant with . From the definition of , it follows that for all . Therefore, for all . Hence, it holds that , that is, for all .
By using (3.6), we have . According to Lemma 2.1, we deduce the existence of a fixed point of , that is, the existence of a mapping such that for all . Moreover, we have , which implies that Also, implies the inequality
Set in (3.2), and divide both sides by . Then, using property , we obtain for all . Hence, by Lemma 3.1, is quartic.
The uniqueness of follows from the fact that is the unique fixed point of with the property that there exists such that for all . This completes the proof of the theorem.

Theorem 3.4. Suppose that an odd mapping satisfies for all . Then there exists a unique additive mapping and a unique cubic mapping satisfying (1.1) and for all .

Proof. Put in (3.15). Then, by the oddness of , we have for all . Replacing with in (3.15), we obtain for all . By (3.17) and (3.18), we have for all . Putting for all , we get for all .
Let the same definitions for and as in the proof of Theorem 3.3 such that becomes a complete generalized metric space. We now define a function by Applying a similar technique as in the proof of Theorem 3.3, we obtain , that is, for all .
By (3.20), we have . According to Lemma 2.1, we deduce the existence of a fixed point of , that is, the existence of a mapping such that for all . Moreover, we have , which implies that Also, implies the inequality
Hence, it follows that for all . This means that satisfies (1.1). Then, by Lemma 3.2, is additive. Thus, by , we conclude that is additive.
Putting for all , we get for all . We now define a function by Applying a similar technique as in the proof of Theorem 3.3, we obtain , that is, for all .
By (3.25), we have . According to Lemma 2.1, we deduce the existence of a fixed point of , that is, the existence of a mapping such that for all . Moreover, we have , which implies that Also, implies the inequality
Then we have for all . Hence, the mapping satisfies (1.1). Therefore, by Lemma 3.2, is cubic. Thus, implies that the mapping is cubic.
The uniqueness of and can be proved in the same reasoning as in the proof of Theorem 3.3. This completes the proof of the theorem.

Theorem 3.5. Suppose that an odd mapping satisfies for all . Then there exists a unique additive mapping and a unique cubic mapping such that for all .

Proof. By Theorem 3.4, there exist an additive mapping and a cubic mapping such that for all . Combining the above two equations in (3.32) yields that for all . So we obtain (3.31) by letting and .
To prove the uniqueness of and , let be other additive and cubic mappings satisfying (3.31). Let and . Then, using property , we obtain for all , then (3.34) implies that for all . Therefore, for all . By (3.34), we have for all . This completes the proof of the theorem.

Theorem 3.6. Suppose that a mapping satisfies and for all . Then there exists a unique additive mapping , a unique cubic mapping , and a unique quartic mapping such that for all .

Proof. Let for all , then and for all . From Theorem 3.5, it follows that there exists a unique additive mapping and a unique cubic mapping satisfying (3.31).
Let for all , then and for all . By Theorem 3.3, there exists a unique quartic mapping satisfying (3.3). Now it is obvious that (3.37) holds for all . This completes the proof of the theorem.

Acknowledgments

The authors are very grateful to the referees for their helpful comments and suggestions. This research is supported by Guangdong Provincial Natural Science Foundation (07301595), China.

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