Asymptotic behavior and periodic nature of two difference equations

Abstract

We discuss the global asymptotic stability of the solutions of the difference equations $$x_{n+1} = \frac{x_{n−2}}{±1 + x_nx_{n−1}x_{n−2}}, \quad n = 0,1,…,$$ where the initial conditions $x_{−2}, x_{−1}, x_0$ are real numbers.