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

Let X and Y   be Banach spaces and f:X→Yf:X→Y an odd mapping. We solve the following generalized additive functional equationrf(∑j=1dxjr)+∑ι(j)=0,1∑j=1dι(j)=lrf(∑j=1d(−1)ι(j)xjr)=(Cld−1−Cl−1d−1+1)∑j=1df(xj) for all x1,…,xd∈Xx1,…,xd∈X. Moreover we deal with the above functional equation in Banach modules over a C∗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].