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 rf d j=1 xj r + ι(j)=0,1 d j=1 ι(j)=l rf d j=1(−1)ι(j)xj r = ( d−1Cl − d−1Cl−1 + 1) d j=1 f (xj ) for all x1, . . . , xd ∈ 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]. © 2007 Published by Elsevier Inc