Stability of Quadratic Functional Equation in Two Variables

In this paper, we establish the general solution of a 2-variable quadratic functional equation and prove the generalized Hyers-Ulam stability of this functional equation.


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 F 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 1 G , a metric group for all 1 , x y G ∈ , then there exists a homomorphism 1 2 : such that ( ( ); ( )) d f x T x ε < for all 1 x G ∈ ? If the answer is affirmative, we would say that the equation of homomorphism ( ) ( ) ( ) T xy T x T y = 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] for all x X ∈ . 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.
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).

The Relation between (1.1) and (1.2) Theorem 2.1. Let :
f X X Y × → be a mapping satisfying (1.1) and let : g X Y → be the mapping given by , Proof. By (1.1) and (2.1), we can show that 4 g x y f x y x y f x y x y f x y x y f x x f y y g x y g x y g x g y for all , , x y X ∈ then f satisfies (1.1). Proof. By (1.2) and (2.2), we can show that 4 g x y z w g x y z w b cg z w ag x g x z g x z b cg z ag y g x z g x z cg w

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 : f X X Y × → satisfies (1.1) if and only if there exist two symmetric bi-additive mappings 1 2 , : S S X X Y × → and a bi-additive mapping  Conversely, we assume that f is a solution of (1.1). Define 1 2 , : One can easily verify that 1 2 , f f are quadratic. By [16], there exist two symmetric bi-additive mappings 1 2 , : for all , .
x y X ∈ Then, it is easy to investigate that B is bi-additive. This completes the proof.
In the following theorem, let X be a vector space and Y be a Banach space. Given a function : , for all , , , . x y z w X ∈ Then there exists a unique 2-variable quadratic mapping : for all , .
x y X ∈ The mapping A is given by ( ) ( )