- About this Journal ·
- Abstracting and Indexing ·
- Aims and Scope ·
- Annual Issues ·
- Article Processing Charges ·
- Articles in Press ·
- Author Guidelines ·
- Bibliographic Information ·
- Citations to this Journal ·
- Contact Information ·
- Editorial Board ·
- Editorial Workflow ·
- Free eTOC Alerts ·
- Publication Ethics ·
- Reviewers Acknowledgment ·
- Submit a Manuscript ·
- Subscription Information ·
- Table of Contents

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).

#### References

- S. M. Ulam,
*Problems in Modern Mathematics*, John Wiley & Sons, New York, NY, USA, 1964. View at MathSciNet - D. H. Hyers, “On the stability of the linear functional equation,”
*Proceedings of the National Academy of Sciences of the United States of America*, vol. 27, pp. 222–224, 1941. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - T. M. Rassias, “On the stability of the linear mapping in Banach spaces,”
*Proceedings of the American Mathematical Society*, vol. 72, no. 2, pp. 297–300, 1978. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - T. Aoki, “On the stability of the linear transformation in Banach spaces,”
*Journal of the Mathematical Society of Japan*, vol. 2, pp. 64–66, 1950. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - S. Czerwik,
*Functional Equations and Inequalities in Several Variables*, World Scientific, Singapore, 2002. View at Publisher · View at Google Scholar · View at MathSciNet - M. E. Gordji and M. Ramezani, “Erdös problem and quadratic equation,”
*Annals of Functional Analysis*, vol. 1, no. 2, pp. 64–67, 2010. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - D. H. Hyers, G. Isac, and T. M. Rassias,
*Stability of Functional Equations in Several Variables*, Progress in Nonlinear Differential Equations and their Applications, 34, Birkhäuser, Basel, Switzerland, 1998. View at Publisher · View at Google Scholar · View at MathSciNet - S.-M. Jung, “Hyers-Ulam-Rassias stability of Jensen's equation and its application,”
*Proceedings of the American Mathematical Society*, vol. 126, no. 11, pp. 3137–3143, 1998. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - S.-M. Jung,
*Hyers-Ulam-Rassias Stability of Functional Equations in Nonlinear Analysis*, vol. 48 of*Springer Optimization and Its Applications*, Springer, New York, NY, USA, 2011. View at Publisher · View at Google Scholar · View at MathSciNet - S. S. Zhang, R. Saadati, and G. Sadeghi, “Solution and stability of mixed type functional equation in non-Archimedean random normed spaces,”
*Applied Mathematics and Mechanics (English Edition)*, vol. 32, no. 5, pp. 663–676, 2011. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - J. A. Baker, “The stability of certain functional equations,”
*Proceedings of the American Mathematical Society*, vol. 112, no. 3, pp. 729–732, 1991. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - K. Ciepliński, “Applications of fixed point theorems to the Hyers-Ulam stability of functional equations—a survey,”
*Annals of Functional Analysis*, vol. 3, no. 1, pp. 151–164, 2012. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - M. Mirzavaziri and M. S. Moslehian, “A fixed point approach to stability of a quadratic equation,”
*Bulletin of the Brazilian Mathematical Society*, vol. 37, no. 3, pp. 361–376, 2006. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - S.-M. Jung, “On the Hyers-Ulam stability of the functional equations that have the quadratic property,”
*Journal of Mathematical Analysis and Applications*, vol. 222, no. 1, pp. 126–137, 1998. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet