Generalized quadratic mappings in several variables

Abstract

Let X, Y be vector spaces. It is shown that if an even mapping f : X→Y satisfies f(0)=0, and (0.1) (2 d−2C l−1− d−2C l− d−2C l−2)f ∑ j=1 d x j + ∑ ι(j)=0,1 ∑ j=1 d ι(j)=l f ∑ j=1 d (−1) ι(j)x j = ( d−1C l+ d−1C l−1+2 d−2C l−1− d−2C l− d−2C l−2) ∑ j=1 d f(x j) for all x 1,…, x d ∈ X, then the even mapping f : X→Y is quadratic. Furthermore, we prove the Cauchy–Rassias stability of the functional equation (0.1) in Banach spaces.