Additive-Quadratic ρ-Functional Equations in β-Homogeneous Normed Spaces

Abstract

Let $M_1f(x,y) : = \frac {3}{4} f(x+y) - \frac {1}{4}f(-x-y) + \frac {1}{4} f(x-y) + \frac {1}{4} f(y-x) - f(x) - f(y)$ and $M_2 f(x,y): = 2 f\left ( \frac {x+y}{2} \right ) + f\left ( \frac {x-y}{2}\right ) + f\left ( \frac {y-x}{2}\right ) - f(x) - f(y).$ We solve the additive-quadratic ρ-functional inequalities

$$\displaystyle \begin{array}{@{}rcl@{}} {} {}\| M_1 f(x,y)\| \le \|\rho M_2f(x,y)\|, \end{array} $$

(1)

where ρ is a fixed complex number with $|\rho |<\frac {1}{2}$, and

$$\displaystyle \begin{array}{@{}rcl@{}} {} {}\| M_2 f(x,y)\| \le \|\rho M_1 f(x,y)\| , \end{array} $$

(2)

where ρ is a fixed complex number with |ρ| < 1. Using the direct method, we prove the Hyers–Ulam stability of the additive-quadratic ρ-functional inequalities (1) and (2) in β-homogeneous complex Banach spaces.