Jensen type quadratic-quadratic mapping in Banach spaces

Abstract

Let X, Y be vector spaces. It is shown that if an even mapping f : X → Y satisfies f(0) = 0 and f(x + y/2 + z) + f(x + y/2 - z) + f(x - y/2 + z) (0.1) + f(x - y/2 - z) = f(x) + f(y) + 4f(z) for all x, y, z ∈ X, then the mapping f : X → Y is quadratic. Furthermore, we prove the Cauchy-Rassias stability of the functional equation (0.1) in Banach spaces.