Open Access

Stability of a Mixed Type Functional Equation on Multi-Banach Spaces: A Fixed Point Approach

Fixed Point Theory and Applications20102010:283827

https://doi.org/10.1155/2010/283827

Received: 11 December 2009

Accepted: 29 March 2010

Published: 6 April 2010

Abstract

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 [1] concerning the stability of group homomorphisms. Hyers [2] gave a first affirmative partial answer to the question of Ulam for Banach spaces. Hyers's theorem was generalized by Aoki [3] for additive mappings and by Rassias [4] 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 [5] asked whether such a theorem can also be proved for . In 1991, Gajda [6] gave an affirmative solution to this question when , but it was proved by Gajda [6] and Rassias and Šemrl [7] that one cannot prove an analogous theorem when . In 1994, a generalization was obtained by Gavruta [8], 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 [915].

The notion of multi-normed space was introduced by Dales and Polyakov (see in [1619]). 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 [16]. 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 .

Definition 1.1.

A multi-norm on is a sequence
(1.1)

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

(a)

(b) .

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 [20].

Theorem 1.2 (see [20]).

let be a complete generalized metric space and let be a strictly contractive mapping with Lipschitz constant . Then for each given element , either
(1.2)

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 [21] was the first author who applied the fixed-point method in the study of Hyers-Ulam stability (see also [22]). 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 [2527]).

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

In this section, we investigate the stability of the following functional equation in multi-Banach spaces:
(2.1)
Let
(2.2)

First we give some lemma needed later.

Lemma 2.1 (see [28] Lemma 6.1).

If an even function satisfies(2.1), then is quartic-quadratic function.

Lemma 2.2 (see [28] Lemma 6.2).

If an odd function satisfies (2.1), then is cubic-additive function.

Theorem 2.3.

Let be a linear space and let be a multi-Banach space. Let and let be an even mapping with for which there exists a positive real number such that
(2.3)
for all . Then there exists a unique quadratic mapping satisfying (2.1) and
(2.4)

for all .

Proof.

Putting in (2.3), we have
(2.5)
Replacing with in (2.3), we get
(2.6)
By (2.5) and (2.6), we have
(2.7)
Let for all . Then we have
(2.8)
Set and define a metric on by
(2.9)
Define a map by . Let and let be an arbitrary constant with . From the definition of , we have
(2.10)
for . Then
(2.11)
for . So
(2.12)
Then is a strictly contractive mapping. It follows from (2.8) that
(2.13)
for . Then . According to Theorem 1.2, the sequence converges to a unique fixed point of in , that is,
(2.14)
Also we have for all , that is, for all . Also we have
(2.15)

and satisfies (2.1). Since is also even and , we have that is quadratic by Lemma 2.1. Then is quadratic.

Theorem 2.4.

Let be a linear space and let be a multi-Banach space. Let and let be an even mapping with for which there exists a positive real number such that (2.3) holds for all . Then there exists a unique quartic mapping satisfying (2.1) and
(2.16)

for all .

Proof.

The proof is similar to that of Theorem 2.3.

Theorem 2.5.

Let be a linear space and let be a multi-Banach space. Let and let be an even mapping with for which there exists a positive real number such that (2.3) holds for all . Then there exist a unique quadratic mapping and a unique quadratic mapping such that
(2.17)

for all .

Proof.

By Theorems 2.3 and 2.4, there exist a quadratic mapping and a unique quartic mapping such that
(2.18)
for all . By (2.18), we have
(2.19)

Let and for all . Then we have (2.17). The uniqueness of and is easy to show.

Theorem 2.6.

Let be a linear space and let be a multi-Banach space. Let and let be an odd mapping for which there exists a positive real number such that (2.3) holds for all . Then there exists a unique additive mapping and a unique cubic mapping satisfying (2.1) and
(2.20)

for all .

Proof.

The proof is similar to that of Theorems 2.3 and 2.4.

Theorem 2.7.

Let be a linear space and let be a multi-Banach space. Let and let be an odd mapping for which there exists a positive real number such that (2.3) holds for all . Then there exists a unique additive mapping and a unique cubic mapping satisfying (2.1) and
(2.21)

for all .

Proof.

By Theorem 2.6, there is an additive mapping and a cubic mapping such that
(2.22)
Thus
(2.23)

for all . Let and . The rest is similar to that of the proof of Theorem 2.5.

Theorem 2.8.

Let E be a linear space and let be a multi-Banach space. Let and let be an odd mapping satisfying and there exists a positive real number such that (2.3) holds for all . Then there exist a unique additive mapping , a unique cubic mapping a unique quadratic mapping and a unique quadratic mapping such that
(2.24)

for all .

Proof.

Let for all . Then and and
(2.25)
for all . By Theorem 2.5, there are a unique quadratic mapping and a unique quartic mapping satisfying
(2.26)
Let for all . Then is an odd mapping satisfying
(2.27)
for all . By Theorem 2.7, there are a unique additive mapping and a unique quartic mapping satisfying
(2.28)

By (2.26) and (2.28), we have (2.24).This completes the proof.

Declarations

Acknowledgments

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.

Authors’ Affiliations

(1)
School of Mathematical Sciences, Qufu Normal University

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© LiguangWang et al. 2010

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