Stability of additive-quadratic ρ-functional equations in Banach spaces: a fixed point approach

Abstract

Let

M(1)f (x, y) : = 3/4 f (x + y) -1/4 f (-x -y) + 1/4 f (x - y) + 1/4 f(y - x) -f (x) -f (y),

M(2)f(x, y) : = 2f( x + y/2) + f ( x - y/2 ) + f ( y - x/2 ) -f (x) -f (y).

We solve the additive-quadratic rho-functional equations

M(1)f (x, y) = rho M(2)f(x, y), (1)

and

M(2)f(x, y) = rho M(1)f (x, y), (2)

where rho is a fixed nonzero number with rho not equal 1.

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