Generalized quadratic mappings of r-type in several variables

Let X,Y be vector spaces. It is shown that if an even mapping f:X→Y satisfies f(0)=0, and (∗)(2Cl−1d−2−Cld−2−Cl−2d−2)r2f(∑j=1dxjr)+∑ι(j)=0,1∑j=1dι(j)=lr2f(∑j=1d(−1)ι(j)xjr)=(Cld−1+Cl−1d−1+2d−2Cl−1−Cld−2−Cl−2d−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 (∗) in Banach spaces.