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

Let X and Y be Banach spaces and f : X -> Y an odd mapping. We solve the following generalized additive functional equation [GRAPHICS] for all x(1,)..., x(d) E 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]. (c) 2007 Published by Elsevier Inc.