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

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 [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 [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 [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

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.