# 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

## Keywords

## 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.

Proof.

as desired.

Lemma 3.2.

Proof.

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

## Authors’ Affiliations

## References

- Ulam SM:
*A Collection of Mathematical Problems, Interscience Tracts in Pure and Applied Mathematics, no. 8*. Interscience Publishers, New York, NY, USA; 1960:xiii+150.Google Scholar - Hyers DH:
**On the stability of the linear functional equation.***Proceedings of the National Academy of Sciences of the United States of America*1941,**27:**222–224. 10.1073/pnas.27.4.222MathSciNetView ArticleMATHGoogle Scholar - Bae J-H, Park W-G:
**On stability of a functional equation with variables.***Nonlinear Analysis. Theory, Methods & Applications*2006,**64**(4):856–868. 10.1016/j.na.2005.06.028MathSciNetView ArticleMATHGoogle Scholar - Bae J-H, Park W-G:
**On a cubic equation and a Jensen-quadratic equation.***Abstract and Applied Analysis*2007,**2007:**-10.Google Scholar - Park W-G, Bae J-H:
**On a Cauchy-Jensen functional equation and its stability.***Journal of Mathematical Analysis and Applications*2006,**323**(1):634–643. 10.1016/j.jmaa.2005.09.028MathSciNetView ArticleMATHGoogle Scholar - Park W-G, Bae J-H:
**A multidimensional functional equation having quadratic forms as solutions.***Journal of Inequalities and Applications*2007,**2007:**-8.Google Scholar - Park W-G, Bae J-H:
**A functional equation originating from elliptic curves.***Abstract and Applied Analysis*2008,**2008:**-10.Google Scholar - Park W-G, Bae J-H:
**A functional equation related to quadratic forms without the cross product terms.***Honam Mathematical Journal*2008,**30**(2):219–225.MathSciNetView ArticleMATHGoogle Scholar - Risteski IB:
**A new class of quasicyclic complex vector functional equations.***Mathematical Journal of Okayama University*2008,**50:**1–61.MathSciNetMATHGoogle Scholar - Aczél J, Dhombres J:
*Functional Equations in Several Variables, Encyclopedia of Mathematics and Its Applications*.*Volume 31*. Cambridge University Press, Cambridge, UK; 1989:xiv+462.View ArticleMATHGoogle Scholar - Davidson KR:
*C⋆-Algebras by Example, Fields Institute Monographs*.*Volume 6*. American Mathematical Society, Providence, RI, USA; 1996:xiv+309.MATHGoogle Scholar - Bonsall F, Duncan J:
*Complete Normed Algebras*. Springer, New York, NY, USA; 1973:x+301.View ArticleMATHGoogle Scholar - Kurepa S:
**On the quadratic functional.***Publications de l'Institut Mathématique*1961,**13:**57–72.MathSciNetMATHGoogle Scholar - Bogachev VI:
*Measure Theory. Vol. II*. Springer, Berlin, Germany; 2007.View ArticleMATHGoogle Scholar

## Copyright

This article is published under license to BioMed Central Ltd. 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.