QUADRATIC ρ-FUNCTIONAL EQUATIONS IN β-HOMOGENEOUS NORMED SPACES

Abstract

In this paper, we solve the quadratic ρ-functional equations:

f(x+y)+f(x−y)−2f(x)−2f(y)

= ρ(4f (􏰀 x + y / 2) 􏰁 + f (x − y) − 2f (x) − 2f (y)􏰁 )----(1)

and

4f 􏰀(x + y / 2) 􏰁 + f (x − y) − 2f (x) − 2f (y)

= ρ(f(x+y)+f(x−y)−2f(x)−2f(y)),------(2)

where ρ is a fixed complex number with ρ≠1.

Using the direct method, we prove the Hyers-Ulam stability of the quadratic ρ-functional equations (1) and (2) in β-homogeneous complex Banach spaces.