Stability of additive-quadratic rho-functional inequalities in Banach spaces

Abstract

LetM(1)f(x,y): = 3/4f(x+y) - 1/4f(-x - y)+1/4f(x - y) + 1/4f(y - x) - f(x) - f(y),M(2)f(x,y) := 2f(x + y/2) + f(x - y/2) + f(y - x/2) - f(x) - f(y).We solve the additive-quadratic rho-functional inequalitiesparallel to M(1)f(x,y)parallel to <= parallel to rho M(2)f(x, y)parallel to, (0.1)where rho is a fixed complex number with vertical bar rho vertical bar < 1/2 andparallel to M(2)f(x,y)parallel to <= parallel to rho M(1)f(x, y)parallel to, (0.2)where rho is a fixed complex number with vertical bar rho vertical bar < 1.Using the fixed point method, we prove the Hyers-Ulam stability of the additive-quadratic rho-functional inequalities (0.1) and (0.2) in complex Banach spaces.