A new quadratic functional equation version and its stability and superstability

Abstract

Let X and y be vector spaces. It is shown that a mapping f : x -> y satisfies the functional equation f(x+y+z/2)+f(x-y-z/2)+f(y-x-z/2)+f(z-x-Y/2) = f (x) + f(Y)+ f (z) (0.1) if and only if f : x -> y is a quadratic mapping. Furthermore, we prove the superstability and the Hyers-Ulam stability for the quadratic functional equation (0.1) by using a direct method.