# 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

Using the fixed point method, we prove the generalized Hyers-Ulam stability of the generalized quadratic functional equation in Banach modules, where are nonzero rational numbers with .

## 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*

for all

*Assume that*

*satisfies*

*for some*

*and all*.

*Then there exists a unique additive mapping*

*such that*

*for all*
.

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

(i) if and only if ,

(ii) for all ,

(iii) for all

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

(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 .

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.

for all .

## 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.

Case 1 ( ).

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

Case 2 ( ).

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.

for all .

Proof.

for all .

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.

for all .

Proof.

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

Remark 2.5.

for all .

for all .

Corollary 2.6.

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

Theorem 2.7.

for all .

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.

for all .

Proof.

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

Remark 2.9.

for all .

for all .

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.

for all .

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.

for all .

Proof.

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

Remark 2.13.

for all .

for all .

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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