Abstract and Applied Analysis

Volume 2012 (2012), Article ID 205160, 10 pages

http://dx.doi.org/10.1155/2012/205160

## A Fixed Point Approach to the Stability of a Cauchy-Jensen Functional Equation

^{1}Graduate School of Education, Kyung Hee University, Yongin 446-701, Republic of Korea^{2}Department of Mathematics Education, College of Education, Mokwon University, Daejeon 302-729, Republic of Korea

Received 16 February 2012; Revised 6 April 2012; Accepted 20 April 2012

Academic Editor: Krzysztof Cieplinski

Copyright © 2012 Jae-Hyeong Bae and Won-Gil Park. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

We find out the general solution of a generalized Cauchy-Jensen functional equation and prove its stability. In fact, we investigate the existence of a Cauchy-Jensen mapping related to the generalized Cauchy-Jensen functional equation and prove its uniqueness. In the last section of this paper, we treat a fixed point approach to the stability of the Cauchy-Jensen functional equation.

#### 1. Introduction

In 1940, Ulam [1] gave a wide-range talk before the Mathematics Club of the University of Wisconsin in which he discussed a number of important unsolved problems. Among those was the question concerning the stability of homomorphisms.

Let

The case of approximately additive mappings was solved by Hyers [2] under the assumption that

Let

Note that the only substantial difference of the generalized metric from the metric is that the range of generalized metric includes the infinity.

In this paper, let

*Definition 1.1. * A mapping * Cauchy-Jensen mapping* if

When

For a mappings

In this paper, we find out the general solution and we prove the generalized Hyers-Ulam stability of the functional equation (1.2).

#### 2. General Solution of (1.2)

The following lemma ia a well-known fact (see, e.g., [6]).

Lemma 2.1. * A mapping *

*for all* if and only if it satisfies the generalized Jensen's functional equation:

*for all* .

Theorem 2.2. * A mapping *

*Proof. *If

Conversely, assume that

#### 3. Stability of (1.3) Using the Alternative of Fixed Point

In this section, let

Theorem 3.1 (The alternative of fixed point [9]). * Suppose that one is given a complete generalized metric space *

*Or there exists a positive integer* such that(i)

*(ii)* for all ;

*the sequence* is convergent to a fixed point of ;(iii)

*y* is the unique fixed point of* in the set ;(iv)

for all .

*From now on, let* be the set of all mappings satisfying .

Lemma 3.2. * Let *

*where* for all . Then, is complete.

* Proof. * Let

Using an idea of Cădariu and Radu (see [10] and also [4] where applications of different fixed point theorems to the theory of the Hyers-Ulam stability can be found), we will prove the generalized Hyers-Ulam stability of (1.3).

Theorem 3.3. *Let *

*for all* . Suppose that a mapping fulfils and the functional inequality:

*for all* . Then, there exists a unique mapping satisfying (1.3) such that

*where* is a function given by

*for all* .

*Proof. * By a similar method to the proof of Theorem 2.3 in [11], we have the inequality:

Theorem 3.4. *
for all *

*where* is a function given by

*for all* .

* Proof. *By a similar method to the proof of Theorem 2.3 in [11], we have the inequality

#### Acknowledgment

This research was supported by Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education, Science and Technology (Grant no. 2012003499).

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