QUADRATIC ρ-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 (t)), (0.1)

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(f(x + y) f (x - y) - 2f (x) - 2f (y)), (0.2)

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

Using the direct method, we prove the Hyers-Ulam stability of the quadratic rho-functional equations (0.1) and (0.2) in non-Archimedean Banach spaces and in Banach spaces.