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 n i=1 aixi + c = n i=1 Aig(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.  2005 Elsevier Inc. All rights reserved