# Fixed Points and Stability of a Generalized Quadratic Functional Equation

- Abbas Najati
^{1}and - Choonkil Park
^{2}Email author

**2009**:193035

**DOI: **10.1155/2009/193035

© A. Najati and C. Park. 2009

**Received: **26 November 2008

**Accepted: **11 February 2009

**Published: **24 February 2009

## Abstract

## 1. Introduction

*let*

*be a group and let*

*be a metric group with the metric*.

*Given*,

*does there exist*

*such that if a mapping*

*satisfies the inequality*

*Assume that*

*satisfies*

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

Theorem 1.1 (Th. M. Rassias [4]).

for all . If , then the inequality (1.5) holds for and (1.7) for . Also, if for each the mapping is continuous in , then is - linear.

Theorem 1.2 (J. M. Rassias [5–7]).

for all . If, in addition, is a mapping such that the transformation is continuous in for each fixed then is linear.

In 1994, a generalization of Theorems 1.1 and 1.2 was obtained by Găvruţa [8], who replaced the bounds and by a general control function .

*quadratic functional equation.*Quadratic functional equations were used to characterize inner product spaces [9–11]. In particular, every solution of the quadratic equation (1.10) is said to be a

*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 [9, 12]). The biadditive mapping is given by

In addition, J. M. Rassias [19] generalized the Euler-Lagrange quadratic mapping (1.12) and investigated its stability problem. The Euler-Lagrange quadratic mapping (1.12) has provided a lot of influence in the development of general Euler-Lagrange quadratic equations (mappings) which is now known as Euler-Lagrange-Rassias quadratic functional equations (mappings).

Jun and Lee [20] proved the generalized Hyers-Ulam stability of a pexiderized quadratic equation. The stability problems of several functional equations have been extensively investigated by a number of authors and there are many interesting results concerning this problem (see [8, 20–47]). We also refer the readers to the books [48–51].

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

We recall the following theorem by Margolis and Diaz.

Theorem 1.3 (see [52]).

for all nonnegative integers or there exists a nonnegative integer such that

(2)the sequence converges to a fixed point of ,

(3) is the unique fixed point of in the set ,

Throughout this paper, we assume that
are nonzero rational numbers with
and that
is a unital Banach algebra with unit
, norm
, and
. Assume that
is a normed left
-module and
is a (unit linked) Banach left
-module. A quadratic mapping
is called
-*quadratic* if
for all
and all
.

and using the fixed point method (see [24, 25, 38, 53–55]), we prove the generalized Hyers-Ulam stability of -quadratic mappings in Banach -modules associated with the functional equation (1.14). In 1996, Isac and Th. M. Rassias [56] were the first to provide applications of stability theory of functional equations for the proof of new fixed point theorems with applications.

## 2. Fixed Points and Stability of the Generalized Quadratic Functional Equation (1.14)

Proposition 2.1.

for all if and only if is quadratic.

Proof.

for all . Replacing by and by in (2.12), we infer that is additive. To complete the proof we have two cases.

Since is additive and satisfies (2.1), letting and replacing by in (2.1), we get for all . Since , we get

Since is additive and satisfies (2.2), we have for all . Since , we get

Hence and this proves that is quadratic.

for all . Hence satisfies (2.1).

Corollary 2.2.

for all and all If for each the mapping is continuous in , then is -quadratic.

Proof.

for all and all So the -quadratic mapping is also -quadratic. This completes the proof.

Now we prove the generalized Hyers-Ulam stability of -quadratic mappings in Banach -modules.

Theorem 2.3.

Proof.

It is easy to show that is a generalized complete metric space [24].

for all Hence for all and all By Corollary 2.2, the mapping is -quadratic.

Corollary 2.4.

Proof.

for all . Then we can choose and we get the desired result.

Remark 2.5.

Corollary 2.6.

for all and all . If for each the mapping is continuous in , then is -quadratic.

Theorem 2.7.

Proof.

Similar to the proof of Theorem 2.3, we deduce that the sequence converges to a fixed point of which is -quadratic. Also is the unique fixed point of in the set and satisfies (2.45).

Corollary 2.8.

Proof.

for all . Then we can choose and we get the desired result.

Remark 2.9.

Corollary 2.10.

Let and let be nonnegative real numbers such that and let be an even mapping satisfying the inequality (2.42) for all and all . If for each the mapping is continuous in , then is -quadratic.

We may omit the evenness of the mapping in Theorem 2.7.

Theorem 2.11.

Proof.

Also is odd since is odd. Therefore, since is quadratic too. Now (2.61) follows from (2.63) and (2.71).

Corollary 2.12.

Proof.

for all . Then we can choose and we get the desired result.

Remark 2.13.

For the case , we have the following counterexample which is a modification of the example of Czerwik [16].

Example 2.14.

which contradicts (2.88).

Corollary 2.15.

Let and let be nonnegative real numbers such that and let be a mapping satisfying the inequality (2.42) for all and all . If for each the mapping is continuous in , then is -quadratic.

## Declarations

### Acknowledgments

The authors would like to thank the referees for bringing some useful references to their attention. The second author was supported by Korea Research Foundation Grant KRF-2008-313-C00041.

## Authors’ Affiliations

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