Generalized quadratic mappings in several variables*1

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)(2d−2Cl−1−d−2Cl−d−2Cl−2)f∑j=1dxj+∑ι(j)=0,1∑j=1dι(j)=lf∑j=1d(−1)ι(j)xj=(d−1Cl+d−1Cl−1+2d−2Cl−1−d−2Cl−d−2Cl−2)∑j=1df(xj)for all x1,…,xd∈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.