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Abstract and Applied Analysis

Volume 2013 (2013), Article ID 472531, 5 pages

http://dx.doi.org/10.1155/2013/472531

## Generalized Hyers-Ulam Stability of a Mixed Type Functional Equation

^{1}Department of Mathematics Education, Gongju National University of Education, Gongju 314-711, Republic of Korea^{2}Mathematics Section, College of Science and Technology, Hongik University, Sejong 339-701, Republic of Korea

Received 20 April 2013; Accepted 28 May 2013

Academic Editor: Bing Xu

Copyright © 2013 Yang-Hi Lee and Soon-Mo Jung. 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 investigate the stability of a functional equation by applying the direct method in the sense of Hyers and Ulam.

#### 1. Introduction

In 1940, Ulam [1] gave a wide ranging 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 group 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 exists a homomorphism with for all ?

The Ulam’s problem for the Cauchy additive functional equation was solved by Hyers under the assumption that and are Banach spaces. Indeed, Hyers [2] proved that every solution of the inequality (for all and ) can be approximated by an additive function. In this case, the Cauchy additive functional equation, , is said to satisfy the Hyers-Ulam stability.

Thereafter, Rassias [3] attempted to weaken the condition for the bound of norm of the Cauchy difference as follows: and he proved that Hyers’ theorem is also true for this case. Indeed, Rassias proved the generalized Hyers-Ulam stability (or the Hyers-Ulam-Rassias stability) of the Cauchy additive functional equation between Banach spaces. We here remark that a paper of Aoki [4] was published concerning the generalized Hyers-Ulam stability of the Cauchy functional equation earlier than Rassias’ paper.

The stability concept that was introduced by Rassias’ theorem provided a large influence to a number of mathematicians to develop the notion of what is known today with the term generalized Hyers-Ulam stability of functional equations. Since then, the stability problems of several functional equations have been extensively investigated by several mathematicians (e.g*.*, see [5–10] and the references therein).

Almost all subsequent proofs in this very active area have used the Hyers’ method presented in [2]. Namely, starting from the given mapping that approximately satisfies a given functional equation, a solution of the functional equation is explicitly constructed by using the formula: which approximates the mapping . This method of Hyers is called the direct method.

We remark that another method for proving the Hyers-Ulam stability of various functional equations was introduced by Baker [11], which is called the fixed-point method. This method is very powerful technique of proving the stability of functional equations (see [12, 13]).

Now we consider the following functional equation: which is called the mixed type functional equation. The mapping is a solution of this functional equation, where are real constants. Every solution of (3) will be called a quadratic-additive mapping.

In 1998, Jung [14] proved the stability of (3) by decomposing into the odd and even parts. In his proof, using the direct method, an additive mapping and a quadratic mapping are separately constructed from the odd and even parts of , and then and are combined to provide a quadratic-additive mapping which is close to .

In this paper, we will prove the generalized Hyers-Ulam stability of (3) by making use of the direct method. In particular, we will approximate the given mapping by a solution of (3) without decomposing into its odd and even parts, while in the Jung’s paper [14] the mapping was decomposed into the odd and even parts, and each of them was separately approximated by the corresponding part of a solution of (3).

#### 2. Main Results

Throughout this paper, let be a (real or complex) normed space and a Banach space. For an arbitrary , we define and .

For a given mapping , we use the following abbreviations: for all .

As we stated in the previous section, is called a quadratic-additive mapping provided that satisfies the functional equation for all .

Proposition 1. *A mapping is a solution of (3) if and only if is a quadratic mapping and is an additive mapping. *

*Proof. *Assume that is a solution of (3). Then we have
for all , that is, is a quadratic mapping and is an additive mapping.

Conversely, assume that is a quadratic mapping and is an additive mapping. Then we get
for all ; that is, is a solution of (3).

We first prove the following lemma.

Lemma 2. *If a mapping satisfies for all and , then is a quadratic-additive mapping. *

*Proof. *Using the hypothesis, we have
for all . Furthermore, by the last equality, we get
for all . Since is invariant with respect to the permutation of , it holds that for all . It is also easy to show that , , , and for all as we desired.

In the following theorem, we can prove the generalized Hyers-Ulam stability of the functional equation (3) by making use of the direct method.

Theorem 3. *If a mapping satisfies and
**
for all with a real constant , then there exists a unique quadratic-additive mapping such that
**
for all . Moreover, if , then itself is a quadratic-additive mapping. *

*Proof. *Let us define , where . From the definitions of and , we have
for all and . It follows from (9) and (11) that
for all . So, it is easy to show that the sequence is a Cauchy sequence for all .

Since is complete and , the sequence converges for all . Hence, we can define a mapping by
for all . Moreover, if we put and let in (12), we obtain the inequality (10).

From the definition of , we get
for all . By Lemma 2, is a quadratic-additive mapping.

Now, we will show that is uniquely determined. Let be another quadratic-additive mapping satisfying (10). It is easy to show that for all quadratic-additive mapping . It follows from (11) that
for all and . Since and are quadratic-additive, if we replace with in (10), then we have
for all and . Taking the limit in the above inequality as , we can conclude that for all , which proves the uniqueness of .

Since
for all and , if , then we conclude that for all by letting in the previous inequality. From the fact that , is a quadratic-additive mapping.

Theorem 4. *If a mapping satisfies
**
for all and for a nonnegative real constant , then there exists a unique quadratic-additive mapping such that
**
for all . *

*Proof. *Since
we get for and for . From the definitions of and , we have
for all and , where is defined by and .

It follows from (18) and (21) that
for all . So, it is easy to show that the sequence is a Cauchy sequence for all .

Since is complete, the sequence converges for all . Hence, we can define a mapping by
for all . Moreover, putting and letting in (22), we get the inequality (19).

#### 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 (no. 2012R1A1A4A01002971).

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