Additive rho-functional inequalities

Abstract

In this paper, we solve the additive rho-functional inequalities parallel to f(x + y) - f(x) - f(y)parallel to <= parallel to rho(2f(x+ y/2) - f(x) - f(y))parallel to, (1) parallel to 2f(x + y/2) - f(x) - f(y)parallel to <= parallel to rho(f(x + y) - f(x) - f(y))parallel to, (2) where rho is a fixed non-Archimedean number with vertical bar rho vertical bar < 1 or rho is a fixed complex number with vertical bar rho vertical bar < 1. Using the direct method, we prove the Hyers-Ulam stability of the additive rho-functional inequalities (I) and (2) in non-Archimedean Banach spaces and in complex Banach spaces, and prove the Hyers-Ulam stability of additive rho-functional equations associated with the additive rho-functional inequalities (1) and (2) in non-Archimedean Banach spaces and in complex Banach spaces.