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

Let M1f(x, y) := 3/4 f(x + y) − 1/4 f(−x − y) + 1/4 f(x − y) + 4 f(y − x) − f(x) − f(y), M2f(x, y) := 2f (x + y / 2)+ f (x − y /2) + f ( y − x /2) − f(x) − f(y). We solve the additive-quadratic ρ-functional equations M1f(x, y) = ρM2f(x, y), (1) and where ρ is a fixed nonzero number with ρ ≠ 1. M2f(x, y) = ρM1f(x, y), (2) Using the fixed point method, we prove the Hyers-Ulam stability of the additive-quadratic ρ-functional equations (1) and (2) in Banach spaces.