On The Global Asymptotic Behavior of a System of Two Nonlinear Difference Equations

Abstract

In this paper, a sufficient condition is obtained for the global asymptotic stability of the following system of difference equations
\[x_{n+1} = \frac{x_ny_{n-1}}{x_ny_{n-1}+1} ,y_{n+1}=\frac{y_n x_{n-1}}{y_nx_{n-1} + 1} , \quad n = 0, 1, 2, \ldots,\]
where the initial values \((x_k, y_k) \in (0, \infty) (\text{for} k=-1,0)\).