The fixed point alternative to the stability of an additive (alpha,beta)-functional equation

Abstract

In this paper, we solve the additive (alpha, beta)-functional equationf(x) + f(y) + 2f(z) = alpha f(beta(x + y + 2z)), (0.1)where alpha,beta are fixed real or complex numbers with alpha not equal 4 and alpha beta = 1.Using the fixed point method and the direct method, we prove the Hyers-Ulam stability of the additive (alpha,beta)-functional equation (0.1) in Banach spaces.