# Fixed Points, Inner Product Spaces, and Functional Equations

- Choonkil Park
^{1}Email author

**2010**:713675

**DOI: **10.1155/2010/713675

© Choonkil Park. 2010

**Received: **1 February 2010

**Accepted: **5 July 2010

**Published: **20 July 2010

## Abstract

Rassias introduced the following equality , , for a fixed integer . Let be real vector spaces. It is shown that, if a mapping satisfies the following functional equation for all with , which is defined by the above equality, then the mapping is realized as the sum of an additive mapping and a quadratic mapping. Using the fixed point method, we prove the generalized Hyers-Ulam stability of the above functional equation in real Banach spaces.

## 1. Introduction and Preliminaries

The stability problem of functional equations originated from a question of Ulam [1] concerning the stability of group homomorphisms. Hyers [2] gave a first affirmative partial answer to the question of Ulam for Banach spaces. Hyers' theorem was generalized by Aoki [3] for additive mappings and by Rassias [4] for linear mappings by considering an unbounded Cauchy difference. The paper of Rassias [4] has provided a lot of influence on the development of what we call the *generalized Hyers-Ulam stability* or *Hyers-Ulam-Rassias stability* of functional equations. A generalization of the Rassias theorem was obtained by G
vruţa [5] by replacing the unbounded Cauchy difference by a general control function in the spirit of Rassias' approach.

is called a quadratic functional equation. A generalized Hyers-Ulam stability problem for the quadratic functional equation was proved by Skof [6] for mappings , where is a normed space and is a Banach space. Cholewa [7] noticed that the theorem of Skof is still true if the relevant domain is replaced by an Abelian group. Czerwik [8] proved the generalized Hyers-Ulam stability of the quadratic functional equation. The generalized Hyers-Ulam stability of the above quadratic functional equation and of two functional equations of quadratic type was obtained by C dariu and Radu [9].

for all with , where is a real vector space.

for all with (see [10]).

which is called a *functional equation of quadratic type*. In fact,
in
satisfies the functional equation of quadratic type. In particular, every solution of the functional equation of quadratic type is said to be a *quadratic-type mapping*. One can easily show that if
satisfies the quadratic functional equation then
satisfies the functional equation of quadratic type. 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 [11–24]).

Let
be a set. Then, a function
is called a *generalized metric* on
if
satisfies the following:

(1) if and only if ,

(2) for all ,

(3) for all .

We recall a fundamental result in fixed point theory.

for all nonnegative integers or there exists a positive 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 1996, Isac and Rassias [27] were the first to provide applications of stability theory of functional equations for the proof of new fixed point theorems with applications. By using the fixed point method, the stability problems of several functional equations have been extensively investigated by a number of authors (see [17, 28–31]).

Throughout this paper, assume that is a fixed integer greater than . Let be a real normed vector space with norm , and let be a real Banach space with norm .

In this paper, we investigate the functional equation (1.4). Using the fixed point method, we prove the generalized Hyers-Ulam stability of the functional equation (1.4) in real Banach spaces.

## 2. Fixed Points and Functional Equations Associated with Inner Product Spaces

We investigate the functional equation (1.4).

Lemma 2.1.

for all with , then the mapping is realized as the sum of an additive mapping and a quadratic-type mapping.

Proof.

Let and for all . Then, is an odd mapping and is an even mapping satisfying and (2.1).

for all . So, is an additive mapping.

for all . So, is a quadratic-type mapping.

for all .

Let and for all . Then, is an odd mapping and is an even mapping satisfying . If , then and .

Using the fixed point method, we prove the generalized Hyers-Ulam stability of the functional equation in real Banach spaces.

Theorem 2.2.

for all .

Proof.

*generalized metric*on :

By the same method given in [17, 28, 32], one can easily show that is complete.

for all .

for all .

for all . Hence, .

- (1)is a fixed point of ; that is,(2.14)

- (2)One has as . This implies the equality(2.17)

for all . Since is an even mapping, is an even mapping.

This implies that inequality (2.7) holds.

for all with . So, for all with . By Lemma 2.1, the mapping is a quadratic-type mapping.

Therefore, there exists a unique quadratic-type mapping satisfying (2.7).

Corollary 2.3.

for all .

Proof.

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

Remark 2.4.

for all .

for all .

Theorem 2.5.

for all .

Proof.

*generalized metric*on :

By the same method given in [17, 28, 32], one can easily show that is complete.

for all .

for all .

for all . Hence, .

- (1)is a fixed point of ; that is,(2.35)

- (2)One has as . This implies the equality(2.38)

- (3)Moreover , which implies the inequality(2.39)

This implies that inequality (2.28) holds.

for all with . So, for all with . By Lemma 2.1, the mapping is an additive mapping.

Therefore, there exists a unique additive mapping satisfying (2.28), as desired.

Corollary 2.6.

for all .

Proof.

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

Remark 2.7.

for all .

for all .

Combining Theorems 2.2 and 2.5, we obtain the following result.

Theorem 2.8.

for all .

Corollary 2.9.

for all .

Proof.

Define , and apply Theorem 2.8 to get the desired result.

Combining Remarks 2.4 and 2.7, we obtain the following result.

Remark 2.10.

for all .

for all .

## Declarations

### Acknowledgment

This paper was supported by Basic Science Research Program through the National Research Foundation of Korea funded by the Ministry of Education, Science and Technology (NRF-2009-0070788).

## Authors’ Affiliations

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