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Abstract and Applied Analysis
Volume 2009, Article ID 923476, 11 pages
http://dx.doi.org/10.1155/2009/923476
Research Article

On the Generalized Hyers-Ulam-Rassias Stability of Quadratic Functional Equations

Department of Mathematics, Semnan University, P.O. Box 35195-363, Semnan, Iran

Received 17 December 2008; Revised 19 February 2009; Accepted 10 March 2009

Academic Editor: John Rassias

Copyright © 2009 M. Eshaghi Gordji and H. Khodaei. 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 achieve the general solution and the generalized Hyers-Ulam-Rassias and Ulam-Gavruta-Rassias stabilities for quadratic functional equations where are nonzero fixed integers with , and for fixed integers with and .

1. Introduction

In 1940, Ulam [1] proposed the stability problem for functional equations in the following question regarding to the stability of group homomorphism.

Let be a group and let be a metric group with the metric Given , does there exist a , such that if a mapping satisfies the inequality for all , then there exists a homomorphism with for all In other words, under what conditions does a homomorphism exist near an approximately homomorphism? Generally, the concept of stability for a functional equation comes up when we the functional equation is replaced by an inequality which acts as a perturbation of that equation. Hyers [2] answered to the question affirmatively in 1941 so if such that for all and for some where , are Banach spaces; then there exists a unique additive mapping such that for all However, if is a continuous mapping at for each fixed then is linear. In 1950, Hyers's theorem was generalized by Aoki [3] for additive mappings and independently, in 1978, by Rassias [4] for linear mappings considering the Cauchy difference controlled by sum of powers of norms. This stability phenomenon is called the Hyers-Ulam-Rassias stability.

On the other hand, Rassias [510] considered the Cauchy difference controlled by a product of different powers of norm. However, there was a singular case; for this singularity a counterexample was given by Găvruţa [11]. This stability phenomenon is called the Ulam-Găvruţa-Rassias stability (see also [12, 13]). In addition, J. M. Rassias considered the mixed product-sum of powers of norms control function [14]. This stability is called JMRassias mixed product-sum stability (see also [1522]).

The functional equation is related to symmetric biadditive function and is called a quadratic functional equation naturally, and every solution of the quadratic equation (1.3) is said to be a quadratic function. It is well known that a function between two real vector spaces is quadratic if and only if there exists a unique symmetric biadditive function such that for all where (see [23, 24]). Skof proved Hyers-Ulam-Rassias stability problem for quadratic functional equation (1.3) for a class of functions , where is normed space and is a Banach space, (see [25]). Cholewa [26] noticed that Skof's theorem is still true if relevant domain alters to an abelian group. In 1992, Czerwik proved the Hyers-Ulam-Rassias stability of (1.3) (see [27]) and four years later, Grabiec [28] generalized the result mentioned above.

Throughout this paper, assume that , are fixed integers with , we introduce the following functional equations, which are different from (1.3): where , and where

In this paper, we establish the general solution and the generalized Hyers-Ulam-Rassias and Ulam-Găvruţa-Rassias stabilities problem for (1.5), (1.6) which are equivalent to (1.3).

2. Solution of (1.5), (1.6)

Let and be real vector spaces. We here present the general solution of (1.5), (1.6).

Theorem 2.1. A function satisfies the functional equation (1.3) if and only if satisfies the functional equation (1.5). Therefore, every solution of functional equation (1.5) is also a quadratic function.

Proof. Let satisfy the functional equation (1.3). Putting in (1.3), we get . Set in (1.3) to get . Letting and in (1.3), respectively, we obtain that and for all . By induction, we lead to for all positive integers . Replacing and by and in (1.3), respectively, gives for all Using (1.3) and (2.1), we lead to for all Suppose that is a fixed integer by using (1.3), we get for all Using (2.2) and (2.3), we obtain for all Replacing and by and in (1.3), respectively, then using (1.3) and (2.3), we have for all By using the above method, by induction, we infer that for all and each positive integer . For a negative integer , replacing by one can easily prove the validity of (2.6). Therefore (1.3) implies (2.6) for any integer . First, it is noted that (2.6) also implies the following equation for all integers . Setting in (2.7) gives . Substituting with into (2.7), one gets for all Replacing by in (2.6), we observe that for all Hence, according to (2.8) and (2.9), we get for all In particular, if we substitute in (2.10) and dividing it by , we conclude that satisfies (1.5).
Let satisfy the functional equation (1.5), for nonzero fixed integers , with . Putting in (1.5), we get so but since and therefore . Setting in (1.5) gives for all . Letting in (1.5), we get for all If we compare (1.5) with (2.13), then since and we conclude that for all . Letting in (1.5) and using the evenness of give for all . Therefore for all , we get . Replacing and by and in (1.5), respectively, we have for all On the other hand, if we interchange with in (1.5), we obtain for all But since is even, it follows from (2.15) that for all Hence, according to (2.14) and (2.16), we obtain that for all So from (2.17), we have for all But since and we conclude that for all Therefore, satisfies (1.3).

Theorem 2.2. A function satisfies the functional equation (1.3) if and only if satisfies the functional equation (1.6). Therefore, every solution of functional equation (1.6) is also a quadratic function.

Proof. If satisfies the functional equation (1.3), then satisfies the functional equation (1.5). Now combining (1.3) with (1.5), we have for all So from (2.20), we conclude that satisfies (1.6).Let satisfy the functional equation (1.6) for fixed integers , with , and . Putting in (1.6), we get , and since , , therefore . Setting in (1.6) gives for all . Letting in (1.6), we have for all If we compare (1.6) with (2.21), then since and , we obtain that for all . Letting in (1.6) and using the evenness of gives for all . Therefore for all , we get . Replacing and by and in (1.6), respectively, we have for all Now, by using and , it follows from (2.22) that for all Which completes the proof of the theorem.

Corollary 2.3 ([29, Proposition 2.1]). A function satisfies the following functional equation: for all if and only if satisfies the functional equation (1.3) for all .

Proof. Assume that in functional equation (1.6) and apply Theorem 2.2.

3. Stability

We now investigate the generalized Hyers-Ulam-Rassias and Ulam-Gavruta-Rassias stabilities problem for functional equations (1.5), (1.6). From this point on, let be a real vector space and let be a Banach space. Before taking up the main subject, we define the difference operator by for all and fixed integers such that and where is a given function.

Theorem 3.1. Let be fixed, and let be a function such that for all . Suppose that be a function satisfies for all Furthermore, assume that in (3.4) for the case Then there exists a unique quadratic function such that for all .

Proof. For , putting in (3.4), we have for all So for all Replacing by in (3.7) and dividing by and summing the resulting inequality with (3.7), we get for all Hence for all nonnegative integers and with and for all It follows from (3.2) and (3.9) that the sequence is a Cauchy sequence for all Since is complete, the sequence converges. So one can define the function by for all By (3.3) for and (3.4), for all . So . By Theorem 2.1, the function is quadratic. Moreover, letting and passing the limit in (3.9), we get the inequality (3.5) for .
Now, let be another quadratic function satisfying (1.5) and (3.5). Then we have which tends to zero as for all So we can conclude that for all This proves the uniqueness of .
Also, for , it follows from (3.6) that for all Hence for all nonnegative integers and with and for all It follows from (3.14) that the sequence is a Cauchy sequence for all Since is complete, the sequence converges. So one can define the function by for all By (3.3) for and (3.4), for all . So . By Theorem 2.1, the function is quadratic. Moreover, letting and passing the limit in (3.14), we get the inequality (3.5) for . The rest of the proof is similar to the proof of previous section.

From Theorem 3.1, we obtain the following corollaries concerning the JMRassias mixed product-sum stability of the functional equation (1.5).

Corollary 3.2. Let and be real numbers such that and . Suppose that a function satisfies for all Then there exists a unique quadratic function such that for all

Proof. In Theorem 3.1, put and

Corollary 3.3. Let and be real numbers such that and . Suppose that a function with satisfies (3.17) for all Then there exists a unique quadratic function such that for all

Proof. In Theorem 3.1, put and

Theorem 3.4. Let be fixed, and let be a function such that for all . Suppose that be a function satisfies for all Furthermore, assume that in (3.21) for the case Then there exists a unique quadratic function such that for all .

Proof. The proof is similar to the proof of Theorem 3.1.

Acknowledgments

The authors would like to thank the referees for their valuable suggestions. Also, the second author would like to thank the office of gifted students at Semnan University for its financial support.

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