- Research Article
- Open Access

# A Fixed Point Approach to the Stability of a Quadratic Functional Equation in -Algebras

- Mohammad B. Moghimi
^{1}, - Abbas Najati
^{1}Email author and - Choonkil Park
^{2}

**2009**:256165

https://doi.org/10.1155/2009/256165

© Mohammad B. Moghimi et al. 2009

**Received:**18 May 2009**Accepted:**31 July 2009**Published:**19 August 2009

## Abstract

We use a fixed point method to investigate the stability problem of the quadratic functional equation in -algebras.

## Keywords

- Banach Space
- Functional Equation
- Positive Element
- Group Homomorphism
- Real Vector Space

## 1. Introduction and Preliminaries

*Under what conditions does there exist a group homomorphism near an approximately group homomorphism?*In 1941, Hyers [2] considered the case of approximately additive functions , where and are Banach spaces and satisfies

*Hyers inequality*

for all . Aoki [3] and Th. M. Rassias [4] provided a generalization of the Hyers' theorem for additive mappings and for linear mappings, respectively, by allowing the Cauchy difference to be unbounded (see also [5]).

Theorem 1.1 (Th. M. Rassias).

for all . If then inequality (1.2) holds for and (1.4) for . Also, if for each the mapping is continuous in , then is -linear.

*quadratic functional equation*is a functional equation of the following form:

*quadratic mapping.*It is well known that a mapping between real vector spaces is quadratic if and only if there exists a unique symmetric biadditive mapping such that for all (see [16, 21, 26, 27]. The biadditive mapping is given by

The Hyers-Ulam stability problem for the quadratic functional equation (1.5) was studied by Skof [28] for mappings where is a normed space and is a Banach space. Cholewa [8] noticed that the theorem of Skof is still true if we replace by an Abelian group. Czerwik [9] proved the generalized Hyers-Ulam stability of the quadratic functional equation (1.5). Grabiec [11] has generalized these results mentioned above. Jun and Lee [14] proved the generalized Hyers-Ulam stability of a Pexiderized quadratic functional equation.

Let
be a set. A function
is called a *generalized metric* on
if
satisfies

- (i)
if and only if ;

- (ii)
for all ;

- (iii)
for all

We recall the following theorem by Margolis and Diaz.

Theorem 1.2 (see [29]).

for all nonnegative integers or there exists a non-negative integer such that

- (1)
for all ;

- (2)
the sequence converges to a fixed point of ;

- (3)
is the unique fixed point of in the set ;

- (4)
for all .

in -algebras. A systematic study of fixed point theorems in nonlinear analysis is due to Hyers et al. [30] and Isac and Rassias [13].

## 2. Solutions of (1.8)

Theorem 2.1.

Let be a linear space. If a mapping satisfies and the functional equation (1.8), then is quadratic.

Proof.

for all Hence is quadratic.

Remark 2.2.

A quadratic mapping does not satisfy (1.8) in general. Let be the mapping defined by for all It is clear that is quadratic and that does not satisfy (1.8).

Corollary 2.3.

Let be a linear space. If a mapping satisfies the functional equation (1.8), then there exists a symmetric biadditive mapping such that for all

## 3. Generalized Hyers-Ulam Stability of (1.8) in -Algebras

In this section, we use a fixed point method (see [7, 15, 17]) to investigate the stability problem of the functional equation (1.8) in -algebras.

for all where is a linear space.

Theorem 3.1.

Moreover, if is continuous in for each fixed , then is -quadratic, that is, for all and all

Proof.

is a generalized complete metric space [7].

Let be the mapping defined by

for all By Theorem 2.1, the function is quadratic.

Moreover, if is continuous in for each fixed , then by the same reasoning as in the proof of [4] is -quadratic.

Corollary 3.2.

for all . Moreover, if is continuous in for each fixed , then is -quadratic.

The following theorem is an alternative result of Theorem 3.1 and we will omit the proof.

Theorem 3.3.

for all , where is defined as in Theorem 3.1. Moreover, if is continuous in for each fixed , then is -quadratic.

Corollary 3.4.

for all . Moreover, if is continuous in for each fixed , then is -quadratic.

For the case we use the Gajda's example [31] to give the following counterexample (see also [9]).

Example 3.5.

which contradicts (3.33).

## Declarations

### Acknowledgment

The third author was supported by Korea Research Foundation Grant funded by the Korean Government (KRF-2008-313-C00041).

## Authors’ Affiliations

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