- Research Article
- Open Access

# Stability of a 2-Dimensional Functional Equation in a Class of Vector Variable Functions

- Won-Gil Park
^{1}and - Jae-Hyeong Bae
^{2}Email author

**2010**:167042

https://doi.org/10.1155/2010/167042

© Won-Gil Park and Jae-Hyeong Bae. 2010

**Received:**16 March 2010**Accepted:**24 June 2010**Published:**12 July 2010

## Abstract

We prove the Hyers-Ulam stability of a 2-dimensional quadratic functional equation in a class of vector variable functions in Banach modules over a unital -algebra.

## Keywords

- Continuous Function
- Real Number
- Normed Space
- Quadratic Mapping
- Stability Problem

## 1. Introduction

In 1940, Ulam proposed the stability problem (see [1]):

*Let*
*be a group and let*
*be a metric group with the metric*
*. Given*
*, does there exist a*
*such that if a mapping*
*satisfies the inequality*
*for all*
*then there is a homomorphism*
*with*
*for all*
*?*

In 1941, this problem was solved by Hyers [2] in the case of Banach space. Thereafter, we call that type the Hyers-Ulam stability. The authors investigated various functional equations and their Hyers-Ulam stability [3–8]. This Hyers-Ulam stability is a classical type of stability, but there is another kind of stability introduced by Risteski [9] for functional equations spanned over an -dimensional complex vector space too.

The quadratic form given by is a solution of (1.2). In 2008, the authors of [8] acquired the general solution and proved the stability of the -dimensional quadratic functional equation (1.2) for the case that and are real vector spaces as follows.

The results of [8, Theor ms 3 and 4] also hold for complex vector spaces and . In this paper, we investigate the stability of (1.2) with two module actions in Banach modules over a unital -algebra.

## 2. Preliminaries

Let be a unital -algebra with a norm , and let and be left Banach -modules with norms and , respectively. Put , , , , and .

Definition 2.1.

A
-dimensional vector variable quadratic mapping
satisfying (1.2) is called
*-quadratic* if
for all
and all
.

Definition 2.2.

A unital
-algebra
is said to have *real rank*
(see [11]) if the invertible self-adjoint elements are dense in
.

For any element , , where and are self-adjoint elements; furthermore, , where and are positive elements (see [12, Lemma ]).

## 3. Results

Theorem 3.1.

for all .

Proof.

as desired.

Lemma 3.2.

for all .

Proof.

for all .

for all . Hence the mapping is the unique 2-dimensional vector variable quadratic mapping, as desired.

for all . Hence the mapping is the unique 2-dimensional vector variable quadratic mapping, as desired.

Theorem 3.3.

for all and all . If is continuous in for each fixed , then there exists a unique -dimensional vector variable -quadratic mapping satisfying (1.2) and (3.10) for all .

Proof.

Suppose . By Lemma 3.2, it follows from the inequality of the statement for that there exists a unique -dimensional vector variable quadratic mapping satisfying (1.2) and inequality (3.10) for all .

for all nonzero and all . By (3.35), we get for all . Therefore the mapping is the unique -dimensional vector variable -quadratic mapping satisfying (1.2) and (3.10).

The proof of the case is similar to that of the case .

Theorem 3.4.

for all and all . For each fixed , let the sequence converge uniformly on . If is continuous in for each fixed , then there exists a unique -dimensional vector variable mapping satisfying (1.2) and (3.10) such that for all and all .

Proof.

for all and all .

for all . By equality (3.47) and the above convergence, we obtain for all and all .

The proof of the case is similar to that of the case .

## Authors’ Affiliations

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