ADDITIVE (α,β)-FUNCTIONAL EQUATIONS AND LINEAR MAPPINGS

Abstract

In this paper, we investigate the additive (alpha, beta)-functional equation

f(x) + (alpha) over barf(alpha y) + f(z) = beta(-1)f(beta(x + y + z))

for all complex numbers alpha with vertical bar alpha vertical bar = 1 and for a fixed nonzero complex number beta.

Using the fixed point method and the direct method, we prove the Hyers-Ulam stability of the additive (alpha, beta)-functional equation (alpha, beta) in complex Banach spaces.