- Research Article
- Open Access

# Fixed Points and Random Stability of a Generalized Apollonius Type Quadratic Functional Equation

- MinJune Kim
^{1}, - SeungWon Schin
^{1}, - Dohyeong Ki
^{1}, - Jaewon Chang
^{1}and - Ji-Hye Kim
^{2}Email author

**2011**:671514

https://doi.org/10.1155/2011/671514

© Min June Kim et al. 2011

**Received:**23 December 2010**Accepted:**28 February 2011**Published:**14 March 2011

## Abstract

Using the fixed-point method, we prove the generalized Hyers-Ulam stability of a generalized Apollonius type quadratic functional equation in random Banach spaces.

## Keywords

- Banach Space
- Functional Equation
- Normed Space
- Stability Problem
- Unique Fixed Point

## 1. Introduction

The stability problem of functional equations was 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 Th. M. Rassias [4] for linear mappings by considering an unbounded Cauchy difference. The paper of Th. M. Rassias [4] has provided a lot of influence in the development of what we call *generalized Hyers-Ulam stability* of functional equations. A generalization of the Th. M. Rassias theorem was obtained by Găvruţa [5] by replacing the unbounded Cauchy difference by a general control function in the spirit of the Th. M. Rassias' approach.

On the other hand, in 1982–1998, J. M. Rassias generalized the Hyers' stability result by presenting a weaker condition controlled by a product of different powers of norms.

for all .

The control function was introduced by Ravi et al. [13] and was used in several papers (see [14–19]).

The functional equation
is called a *quadratic functional equation*. In particular, every solution of the quadratic functional equation is said to be a *quadratic mapping*. The generalized Hyers-Ulam stability of the quadratic functional equation was proved by Skof [20] for mappings
:
, where
is a normed space and
is a Banach space. Cholewa [21] noticed that the theorem of Skof is still true if the relevant domain
is replaced by an Abelian group. Czerwik [22] proved the generalized Hyers-Ulam stability of the quadratic functional 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 [23–44]).

in Banach spaces.

## 2. Preliminaries

We define the notion of a random normed space, which goes back to Šerstnev et al. (see, e.g., [46, 47]).

Definition 2.1 (see [48]).

A function : is a continuous triangular norm (briefly, a -norm) if satisfies the following conditions:

(TN_{1})
is commutative and associative;

(TN_{2})
is continuous;

(TN_{3})
for all
;

(TN_{4})
whenever
and
for all
.

is defined as .

Definition 2.2 (see [47]).

A *random normed space* (briefly, RN-space) is a triple
, where
is a vector space,
is a continuous
-norm, and
is a mapping from
into
, such that the following conditions hold:

(RN_{1})
for all
if and only if
;

(RN_{2})
for all
,
;

(RN_{3})
for all
and all
.

Every normed space defines a random normed space , where for all , and is the minimum -norm. This space is called the induced random normed space.

Definition 2.3.

Let be an RN-space.

(1) A sequence
in
is said to be *convergent* to
in
if, for every
and
, there exists a positive integer
, such that
whenever
.

(2) A sequence
in
is called *Cauchy* if, for every
and
, there exists a positive integer
, such that
whenever
.

(3) An RN-space
is said to be *complete* if every Cauchy sequence in
is convergent to a point in
. A complete RN-space is said to be a *random Banach space*.

Theorem 2.4 (see [48]).

If is an RN-space and is a sequence, such that , then almost everywhere.

Starting with the paper [50], the stability of some functional equations in the framework of fuzzy normed spaces or random normed spaces has been investigated in [51–57].

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

(1) if and only if ;

(2) for all ;

(3) for all .

Let be a generalized metric space. An operator : satisfies a Lipschitz condition with Lipschitz constant if there exists a constant such that for all . If the Lipschitz constant is less than 1, then the operator is called a strictly contractive operator. Note that the distinction between the generalized metric and the usual metric is that the range of the former is permitted to include the infinity. We recall the following theorem by Diaz and Margolis.

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 Th. M Rassias [60] were the first to provide applications of stability theory of functional equations for the proof of new fixed-point theorems with applications. By using fixed point methods, the stability problems of several functional equations have been extensively investigated by a number of authors (see [61–67]).

In this paper, we prove the generalized Hyers-Ulam stability of the generalized Apollonius type quadratic functional equation (1.3) in random Banach space by using the fixed point method.

Throughout this paper, assume that is a vector spaces and is a complete RN-space.

## 3. Generalized Hyers-Ulam Stability of the Quadratic Functional Equation (1.3) in RN-Spaces

for all .

Using the fixed-point method, we prove the generalized Hyers-Ulam stability of the quadratic functional equation in complete RN-spaces.

Theorem 3.1.

for all and all .

Proof.

for all and all .

where, as usual, . It is easy to show that is a generalized complete metric space (see [68, Lemma 2.1]).

for all , that is, is a strictly contractive self-mapping on with the Lipschitz constant .

for all and all , which means that .

By Theorem 2.5, there exists a unique mapping : , such that is a fixed point of , that is, for all .

for all .

for all and all . Letting in (3.13), we find that for all , which implies . By [45, Lemma 2.1], the mapping, : is quadratic.

for all and all . This completes the proof.

Theorem 3.2.

for all and all .

Proof.

Let be the generalized metric space defined in the proof of Theorem 3.1.

for all , that is, is a strictly contractive self-mapping on with the Lipschitz constant .

By Theorem 2.5, there exists a unique mapping : , such that is a fixed point of , that is, for all .

for all .

for all and all , which implies that .

for all and all .

The rest of the proof is similar to the proof of Theorem 3.1.

## Declarations

### Acknowledgments

The fifth author 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) and by Hi Seoul Science Fellowship from Seoul Scholarship Foundation. This paper was dedictated to Professor Sehie Park.

## Authors’ Affiliations

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