Fixed points and quadratic rho-functional equations

Abstract

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

f (x + y) + f (x - y) - 2f (x) - 2f (y) = rho(2f (x + y/2) + 2f (x - y/2) - f (x) - f (y)), (1)

where rho is a fixed non-Archimedean number or a fixed real or complex number with rho not equal 1, 2, and

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

where rho is a fixed non-Archimedean number or a fixed real or complex number with rho not equal -1, 1/2.

Using the fixed point method, we prove the Hyers-Ulam stability of the quadratic rho-functional equations (1) and (2) in non-Archimedean Banach spaces and in Banach spaces.