Abstract

In this paper, we establish the general solution of a 2-variable quadratic functional equation f(2x+y,2z+w)=f(x+y,z+w)-f(x-y,z-w)+4f(x,z)+f(y,w) and prove the generalized Hyers-Ulam stability of this functional equation.

1. Introduction

One of the interesting questions in the theory of functional equations is the following (see ²):

When is it true that a function which approximately satisfies a functional equation F must be close to an exact solution of F?

If there exists an affirmative answer we say that the equation is stable. The stability problems of functional equations were raised by S. M. Ulam during his talk before a Mathematical Colloquium at the University of Wisconsin in 1940 ¹⁵:

Given a group, a metric group and a positive number , does there exist a number such that if a function satisfies the inequality for all, then there exists a homomorphism such that for all?

If the answer is affirmative, we would say that the equation of homomorphism is stable. The concept of stability for a functional equation arises when we replace the functional equation by an inequality which acts as a perturbation of the equation. Thus the stability question of functional equations is that how do the solutions of the inequality differ from those of the given functional equation?

Hyers ¹² gave a first affirmative partial answer to the question of Ulam for Banach spaces. Subsequently, his result was extended and generalized in several ways (see e.g. ¹³). Th.M. Rassias ¹⁵ extended Hyers’ theorem in the following form where Cauchy difference is allowed to be unbounded:

Let and be real normed spaces with complete, be a mapping such that for each fixed the mapping is continuous on , and Assume that there exist constants and such that

for all. In 1994, a generalization of Rassias' theorem was obtained by Gavruta P. Gavruta ¹⁰ in the spirit of Th. M. Rassias’ approach.

The stability problems of several functional equations have been extensively investigated by a number of authors and there are many interesting results concerning this problem. A large list of references can be found, the reader is referred to ^{5, 13, 14} and references therein for further information on stability.

Let X and Y be vector spaces. For a mapping consider the 2-variable quadratic functional equation:

(1.1)

When we see the quadratic form given by is a solution of (1.1). In fact, we can check that

For a mapping Now, we consider the quadratic functional equation:

(1.2)

In a one paper, by using the fixed point theorem method, C. Park ³ proved the generalized Hyers-Ulam stability of the quadratic functional equation (1.2).

In this paper, we investigate the relation between (1.1) and (1.2). And we find out the general solution and the generalized Hyers-Ulam stability of (1.1).

2. The Relation between (1.1) and (1.2)

Theorem 2.1. Let be a mapping satisfying (1.1) and let be the mapping given by

(2.1)

for all then satisfies (1.2).

Proof. By (1.1) and (2.1), we can show that

for all

Theorem 2.2. Let and be a mapping satisfying (1.2). If is the mapping given by

(2.2)

for all then satisfies (1.1).

Proof. By (1.2) and (2.2), we can show that

for all This completes the proof.

3. Solution and Stability Results

In the following theorem, we find out the general solution of the main functional equation (1.1).

Theorem 3.1. A mapping satisfies (1.1) if and only if there exist two symmetric bi-additive mappings and a bi-additive mapping such that

for all

Proof. We first assume that there exist two symmetric bi-additive mappings

and a bi-additive mapping

such that

for all Then we have

for all

Conversely, we assume that is a solution of (1.1). Define by and for all One can easily verify that are quadratic. By ¹⁶, there exist two symmetric bi-additive mappings

such that and for all Define by

for all Then, it is easy to investigate that B is bi-additive. This completes the proof.

In the following theorem, let be a vector space and be a Banach space. Given a function we set

for all

Theorem 3.2. Let be a mapping for which there exists a function such that

(3.1)

(3.2)

for all Then there exists a unique 2-variable quadratic mapping such that

(3.3)

for all The mapping A is given by

for all

Proof. Letting and in , we get

for all Thus we obtain

for all and all j. Replacing by in the above inequality, we see that

for all and all For given integers we get

(3.4)

for all and all It follows from (3.1) and (3.4) that the sequence is Cauchy. Due to the completeness of this sequence is convergent. So we can define the mapping by

for all By (3.2) and (3.1), we have

for all So Moreover, letting and passing the limit in (3.4), we get (3.3).

Now let be another 2-variable quadratic mapping satisfying (3.3). Then we have

which tends to zero as for all So we can conclude that for all This proves the uniqueness of A. This completes the proof.

Remark 3.3. Let be a mapping for which there exists a function satisfying (3.2) such that

for all By a similar method to the proof of Theorem 3.2, one can show that there exists a unique 2-variable quadratic mapping

such that

for all The mapping A is given by

for all

Acknowledgements

The authors wish to thank the editor and referees for their helpful comments and suggestions.

References