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 (2d−2Cl−1 −d−2Cl −d−2Cl−2)r2f d j=1 xj r + ι(j)=0,1 dj =1 ι(j)=l r2f d j=1(−1)ι(j)xj r = (d−1Cl + d−1Cl−1 + 2d−2Cl−1 − d−2Cl −d−2Cl−2) d j=1 f (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.  2005 Elsevier Inc. All rights reserved.