Additive ρ-functional inequalities in non-archimedean 2-normed spaces

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, parallel to 2f (x + y/2) - f(x) - f(y)parallel to <= parallel to rho(f(x + y) - f(x) - f(y))parallel to,

where rho is a fixed non-Archimedean number with vertical bar rho vertical bar < 1. More precisely, we investigate the solutions of these inequalities in non-Archimedean 2-normed spaces, and prove the Hyers-Ulam stability of these inequalities in non-Archimedean 2-normed spaces. Furthermore, we also prove the Hyers-Ulam stability of additive rho-functional equations associated with these inequalities in non-Archimedean 2-normed spaces.