On the stability of quadratic functional equations

Abstract

Let X, Y be vector spaces and k a fixed positive integer. It is shown that a mapping f (kx + y) + f (kx - y) = 2k(2)f(x) + 2f(y) for all x, y is an element of X if and only if the mapping f : X -> Y satisfies f(x + y) + f(x - y) = 2f(x) + 2f(y) for all x, y. X. Furthermore, the Hyers-Ulam-Rassias stability of the above functional equation in Banach spaces is proven