Fixed points and additive ρ-functional equations

Abstract

In this paper, we solve the additive rho-functional equations

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

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

where rho is a fixed non-Archimedean number or a fixed real or complex number with rho not equal 1. Using the fixed point method, we prove the Hyers-Ulam stability of the above additive rho-functional equations in non-Archimedean Banach spaces and in Banach spaces.