An additive (alpha,beta)-functional equation and linear4 mappings in Banach spaces

Abstract

In this paper, we investigate the additive ( α,β )-functional equation f(x+y)+α¯f(αz)=β−1f(β(x+y+z)) for all complex numbers α with |α|=1 and for a fixed nonzero complex number β . Using the fixed point method and the direct method, we prove the Hyers–Ulam stability of this additive ( α,β )-functional equation in complex Banach spaces.