GENERALIZED QUADRATIC MAPPINGS IN 2d VARIABLES

Let X, Y be vector spaces. It is shown that if an even mapping f : X -> Y satisfies f(0) = 0, and 2(C-2d-2(d-1) -(2d-2) C-d)f (Sigma C-2d(j=1)d)f (Sigma(2d)(j=1) x(j)) + Sigma(tau(j)=0,1,Sigma j=12d tau(j) = d) f(Sigma(2d)(j=1) (-1)(tau(j))x(j)) = 2(C-2d-1(d) + C-2d-2(d-1) -C-2d-2(d)) Sigma(2d)(j=1) f(x(j)) for all x(1) , ... , x(2d) is an element of X, then the even mapping f : X -> Y is quadratic. Furthermore, we prove the Hyers-Ulam stability of the above functional equation in Banach spaces.