On the Cauchy–Rassias stability of a generalized additive functional equation

Abstract

Let X and Y be Banach spaces and f : X → Y an odd mapping. We solve the following generalized additive functional equation r f ( ∑ j = 1 d x j r ) + ∑ ι ( j ) = 0 , 1 ∑ j = 1 d ι ( j ) = l r f ( ∑ j = 1 d ( − 1 ) ι ( j ) x j r ) = ( C l d − 1 − C l − 1 d − 1 + 1 ) ∑ j = 1 d f ( x j ) for all x 1 , … , x d ∈ X . Moreover we deal with the above functional equation in Banach modules over a C ∗ -algebra and obtain generalizations of the Cauchy–Rassias stability. The concept of Cauchy–Rassias stability for the linear mapping was originated from Th.M. Rassias's stability theorem that appeared in his paper: [Th.M. Rassias, On the stability of the linear mapping in Banach spaces, Proc. Amer. Math. Soc. 72 (1978) 297–300].