The stability of an additive (ρ1, ρ2)-functional inequality in Banach spaces

Abstract

In this paper, we introduce and solve the following additive (rho(1), rho(2))-functional inequality parallel to f(x + y) - f(x)-f(y)parallel to <= parallel to rho(1)(f(x+y)+f(x-y)-2f(x))parallel to (1) +parallel to rho(2)(2f(x+ y/2) - f(x) - f(y)parallel to, where rho(1) and rho(2) are fixed nonzero complex numbers with root 2 vertical bar rho(1)vertical bar+vertical bar rho 2 vertical bar < 1. Using the fixed point method and the direct method, we prove the Hyers-Ulam stability of the additive (rho(1), rho(2))-functional inequality (1) in complex Banach spaces.