Ulam-stability of a generalized linear functional equation, a fixed point approach

We introduce the general functional equation and study its Ulam-Hyers-stability and hyperstability, using a fixed point approach, where m and n are positive integers, f is a mapping from a vector space X into a Banach space (Y, parallel to parallel to), and, for every i is an element of{1, 2,..., m}, phi(i) is a linear mapping from X-n into X, A(i) is a continuous endomorphism of Y and b. Y. Our result covers most of the former ones in the literature concerning the stability and hyperstability of linear functional equations, as well as new situations.