Stability of a Mixed Type Functional Equation on Multi-Banach Spaces: A Fixed Point Approach
© LiguangWang et al. 2010
Received: 11 December 2009
Accepted: 29 March 2010
Published: 6 April 2010
Using fixed point methods, we prove the Hyers-Ulam-Rassias stability of a mixed type functional equation on multi-Banach spaces.
1. Introduction and Preliminaries
The stability problem of functional equations originated from a question of Ulam  concerning the stability of group homomorphisms. Hyers  gave a first affirmative partial answer to the question of Ulam for Banach spaces. Hyers's theorem was generalized by Aoki  for additive mappings and by Rassias  for linear mappings by considering an unbounded Cauchy difference. The paper of Rassias has provided a lot of influence in the development of what we call generalized Hyers-Ulam-Rassias stability of functional equations. In 1990, Rassias  asked whether such a theorem can also be proved for . In 1991, Gajda  gave an affirmative solution to this question when , but it was proved by Gajda  and Rassias and Šemrl  that one cannot prove an analogous theorem when . In 1994, a generalization was obtained by Gavruta , who replaced the bound by a general control function . Beginning around 1980, the stability problems of several functional equations and approximate homomorphisms have been extensively investigated by a number of authors, and there are many interesting results concerning this problem. Some of the open problems in this field were solved in the papers mentioned [9–15].
The notion of multi-normed space was introduced by Dales and Polyakov (see in [16–19]). 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 . Let be a complex linear space, and let , we denote by the linear space consisting of -tuples , where . The linear operations on are defined coordinate-wise. When we write for an element in , we understand that appears in the th coordinate. The zero elements of either E or are both denoted by when there is no confusion. We denote by the set and by the group of permutations on .
such that is a norm on for each , such that for each , and such that for each , the following axioms are satisfied:
In this case, we say that is a multi-normed space.
Suppose that is a multi-normed space and . It is easy to show that
It follows from (b) that if is a Banach space, then is a Banach space for each ; in this case is said to be a multi-Banach space.
In the following, we first recall some fundamental result in fixed-point theory.
Let be a set. A function is called a generalized metric on if satisfies
(1) if and only if ;
(2) for all ;
(3) for all .
We recall the following theorem of Diaz and Margolis .
Theorem 1.2 (see ).
for all nonnegative integers n or there exists a nonnegative integer such that
(1) for all ;
(2)the sequence converges to a fixed point of ;
(3) is the unique fixed point of J in the set ;
(4) for all .
Baker  was the first author who applied the fixed-point method in the study of Hyers-Ulam stability (see also ). In 2003, Cadariu and Radu applied the fixed-point method to the investigation of the Jensen functional equation (see [23, 24]). By using fixed point methods, the stability problems of several functional equations have been extensively investigated by a number of authors (see [25–27]).
In this paper, we will show the Hyers-Ulam-Rassias stability of a mixed type functional equation on multi-Banach spaces using fixed-point methods.
2. A Mixed Type Functional Equation
First we give some lemma needed later.
Lemma 2.1 (see  Lemma 6.1).
If an even function satisfies(2.1), then is quartic-quadratic function.
Lemma 2.2 (see  Lemma 6.2).
If an odd function satisfies (2.1), then is cubic-additive function.
for all .
and satisfies (2.1). Since is also even and , we have that is quadratic by Lemma 2.1. Then is quadratic.
for all .
The proof is similar to that of Theorem 2.3.
for all .
Let and for all . Then we have (2.17). The uniqueness of and is easy to show.
for all .
The proof is similar to that of Theorems 2.3 and 2.4.
for all .
for all . Let and . The rest is similar to that of the proof of Theorem 2.5.
for all .
By (2.26) and (2.28), we have (2.24).This completes the proof.
This work was supported in part by the Scientific Research Project of the Department of Education of Shandong Province (no. J08LI15). The authors are grateful to the referees for their valuable suggestions.
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