Quadratic rho-functional inequalities in Banach spaces

Abstract

In this paper, we solve the following quadratic $\rho$-functional inequalities ∥ ∥ ∥ f(x+y+z2 )+f(x−y−z2 )+f(y−x−z2 )+f(z−x−y2 )−f(x)−f(y)−f(z)∥ ∥ ∥ ≤∥ρ(f(x+y+z)+f(x−y−z)+f(y−x−z) +f(z−x−y)−4f(x)−4f(y)−4f(z))∥, ‖f(x+y+z2)+f(x−y−z2)+f(y−x−z2)+f(z−x−y2)−f(x)−f(y)−f(z)‖≤‖ρ(f(x+y+z)+f(x−y−z)+f(y−x−z) +f(z−x−y)−4f(x)−4f(y)−4f(z))‖, where $\rho$ is a fixed complex number with $|\rho|<\frac{1}{8}$, and ∥f(x+y+z)+f(x−y−z)+f(y−x−z)+f(z−x−y)−4f(x)−4f(y)−4f(z)∥≤∥ ∥ ∥ ρ(f(x+y+z2 )+f(x−y−z2 )+f(y−x−z2 )+f(z−x−y2 )−f(x)−f(y)−f(z))∥ ∥ ∥ , ‖f(x+y+z)+f(x−y−z)+f(y−x−z)+f(z−x−y)−4f(x)−4f(y)−4f(z)‖≤‖ρ(f(x+y+z2)+f(x−y−z2)+f(y−x−z2)+f(z−x−y2)−f(x)−f(y)−f(z))‖, where $\rho$ is a fixed complex number with $|\rho|<4$. Using the fixed point method, we prove the Hyers-Ulam stability of the quadratic $\rho$-functional inequalities (0.1) and (0.2) in complex Banach spaces.