Generalized quadratic mappings of r-type 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 (∗) ( 2 C l − 1 d − 2 − C l d − 2 − C l − 2 d − 2 ) r 2 f ( ∑ j = 1 d x j r ) + ∑ ι ( j ) = 0 , 1 ∑ j = 1 d ι ( j ) = l r 2 f ( ∑ j = 1 d ( − 1 ) ι ( j ) x j r ) = ( C l d − 1 + C l − 1 d − 1 + 2 d − 2 C l − 1 − C l d − 2 − C l − 2 d − 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 (∗) in Banach spaces.