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Abstract and Applied Analysis
Volume 2012 (2012), Article ID 205160, 10 pages
http://dx.doi.org/10.1155/2012/205160
Research Article

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

1Graduate School of Education, Kyung Hee University, Yongin 446-701, Republic of Korea
2Department 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 be a group and let be a metric group with the metric . Given , does there exist a such that if a function satisfies the inequality for all , then there is a homomorphism with for all ?

The case of approximately additive mappings was solved by Hyers [2] under the assumption that and are Banach spaces. In 1978, Rassias [3] gave a generalization of Hyers's result. Many authors investigated solutions or stability of various functional equations (see [47]).

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

   if and only if ;

   for all ;

   for all .

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 and be two real vector spaces.

Definition 1.1. A mapping is called a Cauchy-Jensen mapping if satisfies the system of equations:
When , the function given by is a solution of (1.1).
For a mappings , consider the functional equation: where is a fixed integer greater than . In 2006, the authors [8] solved the functional equation: which is a special case of (1.2) for .

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

Theorem 2.2. A mapping satisfies (1.1) if and only if it satisfies (1.2).

Proof. If satisfies (1.1), then we get for all . Hence, we obtain that satisfies (1.2) by Lemma 2.1.
Conversely, assume that satisfies (1.2). Letting and in (1.2), we get for all . Putting , and in (1.2), we have for all . Setting and in (1.2), we obtain that for all . By Lemma 2.1, we see that for all .

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

In this section, let be a real Banach space. We investigate the stability of functional equation (1.3) using the alternative of fixed point. Before proceeding the proof, we will state the theorem which is the alternative of fixed point.

Theorem 3.1 (The alternative of fixed point [9]). Suppose that one is given a complete generalized metric space and a strictly contractive mapping with Lipschitz constant . Then, for each given , either Or there exists a positive integer such that(i) for all ;(ii)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 be a function. Consider the generalized metric on given by where for all . Then, is complete.

Proof. Let be a Cauchy sequence in . Then, given , there exists such that if . Let . Since , there exists such that for all . So, for each , is a Cauchy sequence in . Since is complete, for each , there exists such that as . So . Thus, we have . Taking the limit as in (3.3), we obtain that Hence, as .

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 and satisfy 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: for all . By (3.5), we get for all . Consider the generalized metric on given by for all . Then, we obtain By Lemma 3.2, the generalized metric space is complete. Now, we define a mapping by for all and all . Observe that, for all , Let , and . Then, there is a such that . By the above observation, we gain . So, we get for all . Thus, we have for all . By (3.5), we obtain that for all . Hence, . Therefore, we obtain that for all , that is, is a strictly contractive mapping of with Lipschitz constant . Applying the alternative of fixed point, we see that there exists a fixed point of in such that for all . Replacing by in (3.6), respectively, and dividing by , we have for all . By (3.5), the mapping satisfies (1.3). By (3.5) and (3.10), we obtain that for all and all , that is, for all . By the fixed point alternative, there exists a natural number such that the mapping is the unique fixed point of in the set . So, we have . Since we get . Thus, we have . Hence, we obtain for all and a . Again, using the fixed point alternative, we have By (3.12), we may conclude that which implies inequality (3.7).

Theorem 3.4. and satisfy for all . Suppose that a mapping fulfils and the functional inequality (3.6). 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 for all . So, we get for all . Consider the generalized metric on given by for all . Then, we obtain By Lemma 3.2, the generalized metric space is complete. Now, we define a mapping by for all and all . By the same argument as in the proof of Theorem 2.3 in [11], is a strictly contractive mapping of with Lipschitz constant . Applying the alternative of fixed point, we see that there exists a fixed point of in such that for all . Replacing by in (3.6), respectively, and multiplying by , we have for all . By (3.25), the mapping satisfies (1.3). By (3.25), we obtain that for all and all , that is, for all . By the same reasoning as in the proof of Theorem 2.3 in [11], we have By (3.31), we may conclude that which implies inequality (3.26).

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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