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Abstract and Applied Analysis
Volume 2009, Article ID 907167, 11 pages
http://dx.doi.org/10.1155/2009/907167
Research Article

The Stability of a Quadratic Functional Equation with the Fixed Point Alternative

1Department of Mathematics, Research Institute for Natural Sciences, Hanyang University, Seoul 133-791, South Korea
2Department of Mathematics, Hanyang University, Seoul 133-791, South Korea

Received 15 September 2009; Accepted 1 December 2009

Academic Editor: W. A. Kirk

Copyright © 2009 Choonkil Park and Ji-Hye Kim. 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

Lee, An and Park introduced the quadratic functional equation and proved the stability of the quadratic functional equation in the spirit of Hyers, Ulam and Th. M. Rassias. Using the fixed point method, we prove the generalized Hyers-Ulam stability of the quadratic functional equation in Banach spaces.

1. Introduction

The stability problem of functional equations 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.

Theorem 1.1 (Th. M. Rassias). Let be a mapping from a normed vector space into a Banach space subject to the inequality for all , where and are constants with and . Then the limit exists for all , and is the unique additive mapping which satisfies for all . Also, if for each the function is continuous in , then is -linear.

The above inequality (1.1) has provided a lot of influence in the development of what is now known as a generalized Hyers-Ulam stability of functional equations. Beginning around the year 1980, the topic of approximate homomorphisms, or the stability of the equation of homomorphism, was studied by a number of mathematicians. Găvruta [5] generalized the Rassias’ result.

Theorem 1.2 (see [68]). Let be a real normed linear space and a real complete normed linear space. Assume that is an approximately additive mapping for which there exist constants and such that satisfies inequality for all . Then there exists a unique additive mapping satisfying for all . If, in addition, is a mapping such that the transformation is continuous in for each fixed , then is an -linear mapping.

The functional equation

is called a quadratic functional equation. In particular, every solution of the quadratic functional equation is said to be a quadratic function. A generalized Hyers-Ulam stability problem for the quadratic functional equation was proved by Skof [9] for mappings , where is a normed space and is a Banach space. Cholewa [10] noticed that the theorem of Skof is still true if the relevant domain is replaced by an Abelian group. Czerwik [11] proved the generalized Hyers-Ulam stability of the quadratic functional equation. Several functional equations have been investigated in [1225].

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 .

We recall a fundamental result in fixed point theory.

Theorem 1.3 (see [2628]). Let be a complete generalized metric space and let be a strictly contractive mapping with Lipschitz constant . Then for each given element , either 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 .

Lee et al. [29] proved that a mapping satisfies

for all if and only if the mapping satisfies

for all .

Using the fixed point method, Park [14] proved the generalized Hyers-Ulam stability of the quadratic functional equation

in Banach spaces.

In this paper, using the fixed point method, we prove the generalized Hyers-Ulam stability of the quadratic functional equation (1.8) in Banach spaces.

Throughout this paper, assume that is a normed vector space with norm and that is a Banach space with norm .

2. Fixed Points and Generalized Hyers-Ulam Stability of a Quadratic Functional Equation

For a given mapping , we define

for all .

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

Theorem 2.1. Let be a mapping for which there exists a function with such that for all . If there exists an such that for all , then there exists a unique quadratic mapping satisfying (1.8) and for all .

Proof. Consider the set and introduce the generalized metric on : It is easy to show that is complete.
Now we consider the linear mapping such that for all .
By [30, Theorem ], for all .
Letting in (2.2), we get for all . So for all . Hence .
By Theorem 1.3, there exists a mapping such that
(1) is a fixed point of , that is, for all . The mapping is a unique fixed point of in the set This implies that is a unique mapping satisfying (2.10) such that there exists satisfying for all .
(2) as . This implies the equality for all .
(3) , which implies the inequality This implies that the inequality (2.3) holds.
It follows from (2.2) and (2.13) that for all . So for all .
By [29, Proposition ], the mapping is quadratic, as desired.

Corollary 2.2. Let and be positive real numbers, and let be a mapping such that for all . Then there exists a unique quadratic mapping satisfying (1.8) and for all .

Proof. The proof follows from Theorem 2.1 by taking for all . Then , and we get the desired result.

Theorem 2.3. Let be a mapping for which there exists a function satisfying (2.2) and . If there exists an such that for all , then there exists a unique quadratic mapping satisfying (1.8) and for all .

Proof. We consider the linear mapping such that for all .
It follows from (2.8) that for all . Hence .
By Theorem 1.3, there exists a mapping such that
(1) is a fixed point of , that is, for all . The mapping is a unique fixed point of in the set This implies that is a unique mapping satisfying (2.22) such that there exists satisfying for all .
(2) as . This implies the equality for all .
(3) , which implies the inequality which implies that the inequality (2.19) holds.
The rest of the proof is similar to the proof of Theorem 2.1.

Corollary 2.4. Let and be positive real numbers, and let be a mapping satisfying (2.16). Then there exists a unique quadratic mapping satisfying (1.8) and for all .

Proof. The proof follows from Theorem 2.3 by taking for all . Then and, we get the desired result.

Theorem 2.5. Let be a mapping for which there exists a function satisfying (2.2). If there exists an such that for all , then there exists a unique quadratic mapping satisfying (1.8) and for all .

Proof. Consider the set and introduce the generalized metric on : It is easy to show that is complete.
Now we consider the linear mapping such that for all .
By [30, Theorem ], for all .
Letting in (2.2), we get for all . So for all . Hence .
By Theorem 1.3, there exists a mapping such that
(1) is a fixed point of , that is, for all . The mapping is a unique fixed point of in the set This implies that is a unique mapping satisfying (2.36) such that there exists satisfying for all .
(2) as . This implies the equality for all .
(3) , which implies the inequality This implies that the inequality (2.29) holds.
It follows from (2.2) and (2.39) that for all . So for all .
By [29, Proposition ], the mapping is quadratic, as desired.

Corollary 2.6. Let and be positive real numbers, and let be a mapping satisfying (2.16). Then there exists a unique quadratic mapping satisfying (1.8) and for all .

Proof. The proof follows from Theorem 2.5 by taking for all . Then and, we get the desired result.

Corollary 2.7. Let and be positive real numbers, and let be a mapping such that for all . Then there exists a unique quadratic mapping satisfying (1.8) and for all .

Proof. The proof follows from Theorem 2.5 by taking for all . Then and, we get the desired result.

Theorem 2.8. Let be a mapping for which there exists a function satisfying (2.2). If there exists an such that for all , then there exists a unique quadratic mapping satisfying (1.8) and for all .

Proof. We consider the linear mapping such that for all .
The rest of the proof is similar to the proof of Theorem 2.1.

Corollary 2.9. Let and be positive real numbers, and let be a mapping satisfying (2.16). Then there exists a unique quadratic mapping satisfying (1.8) and for all .

Proof. The proof follows from Theorem 2.8 by taking for all . Then , and we get the desired result.

Corollary 2.10. Let and be positive real numbers, and let be a mapping satisfying (2.44). Then there exists a unique quadratic mapping satisfying (1.8) and for all .

Proof. The proof follows from Theorem 2.8 by taking for all . Then , and we get the desired result.

Acknowledgment

The first author was supported by Hanyang University in 2009.

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