- Research Article
- Open Access

# Approximate Behavior of Bi-Quadratic Mappings in Quasinormed Spaces

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

**2010**:472721

https://doi.org/10.1155/2010/472721

© W.-G. Park and J.-H. Bae. 2010

**Received:**27 April 2010**Accepted:**18 June 2010**Published:**5 July 2010

## Abstract

We obtain the generalized Hyers-Ulam stability of the bi-quadratic functional equation in quasinormed spaces.

## Keywords

- Banach Space
- Functional Equation
- Stability Problem
- Additive Mapping
- Product Space

## 1. Introduction

In 1940, Ulam [1] gave a talk before the Mathematics Club of the University of Wisconsin in which he discussed a number of unsolved problems, containing the stability problem of homomorphisms as follows

*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*
*?*

is called the quadratic functional equation whose solution is said to be a quadratic mapping. A generalized stability problem for the quadratic functional equation was proved by Skof [5] for mappings , where is a normed space and is a Banach space. Cholewa [6] noticed that the theorem of Skof is still true if the relevant domain is replaced by an Abelian group. Czerwik [7] proved the generalized stability of the quadratic functional equation, and Park [8] proved the generalized stability of the quadratic functional equation in Banach modules over a -algebra.

Throughout this paper, let and be vector spaces.

Definition 1.1.

When , the function given by is a solution of (1.4).

Let
be a real linear space. A *quasinorm* is real-valued function on
satisfying the following

(i) for all and if and only if .

(ii) for all and all .

(iii)There is a constant such that for all .

for all and all .

*quasinormed space*if is a quasinorm on . The smallest possible is called the

*modulus of concavity*of . A

*quasi-Banach space*is a complete quasinormed space. A quasinorm is called a

*-norm*( ) if

for all
. In this case, a quasi-Banach space is called a
*-Banach space*.

Given a -norm, the formula gives us a translation invariant metric on . By the Aoki-Rolewicz theorem [10] (see also [9]), each quasinorm is equivalent to some -norm. Since it is much easier to work with -norms, henceforth we restrict our attention mainly to -norms. In [11], Tabor has investigated a version of Hyers-Rassias-Gajda theorem (see also [3, 12]) in quasi-Banach spaces. Since then, the stability problems have been investigated by many authors (see [13–18]).

The authors [19] solved the solutions of (1.4) and (1.5) as follows.

Theorem A.

A mapping satisfies (1.4) if and only if there exist a multi-additive mapping such that and for all .

Theorem B.

A mapping satisfies (1.4) if and only if it satisfies (1.5).

In this paper, we investigate the generalized Hyers-Ulam stability of (1.4) and (1.5) in quasi-Banach spaces.

## 2. Stability of (1.4) and (1.5) in Quasi-normed Spaces

Throughout this section, assume that is a quasinormed space with quasinorm and that is a -Banach space with -norm . Let be the modulus of concavity of .

for all .

for all .

Theorem 2.1.

for all .

Proof.

for all and all . Letting in the above two inequalities and using (2.1), is bi-quadratic.

for all . By the same method as above, define by for all . By the same argument as above, is a bi-quadratic mapping satisfying (2.7).

Corollary 2.2.

for all .

Proof.

In Theorem 2.1, putting and for all , we get the desired result.

for all .

We will use the following lemma in order to prove Theorem 2.4.

Lemma 2.3 (see [20]).

Theorem 2.4.

for all .

Proof.

for all .

for all . Hence the mapping is the unique bi-quadratic mapping, as desired.

Corollary 2.5.

for all .

Proof.

In Theorem 2.4, putting for all , we get the desired result.

## Authors’ Affiliations

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