Stability problem of Ulam for generalized forms of Cauchy functional equation

Let G1 be a vector space and G2 a Banach space. In this paper, we solve the generalized Hyers-Ulam-Rassias stability problem for a generalized form g(∑i=1naixi+c)=∑i=1nAig(xi)+C,∀xi∈G1, of Cauchy functional equation f(x+y)=f(x)+f(y) for a mapping f:G1→G2, where ai,Ai are scalars for all i=1,…,n and c∈G1, C∈G2 are vectors.