Global behavior of two-dimensional difference equations system with two period coefficients

Abstract

In this paper, we investigate the following system of difference equations \begin{equation*} x_{n+1}=\frac{\alpha _{n}}{1+y_{n}x_{n-1}},\ y_{n+1}=\frac{\beta _{n}} {1+x_{n}y_{n-1}}, n\in \mathbb{N}_{0}, \end{equation*} where the sequences $\left( \alpha _{n}\right) _{n\in \mathbb{N}_{0}}$, $\left( \beta_{n}\right) _{n\in \mathbb{N}_{0}}$ are positive, real and periodic with period two and the initial values $x_{-1}$, $x_{0}$, $y_{-1}$, $y_{0}$ are non-negative real numbers. We show that every positive solution of the system is bounded and examine their global behaviors. In addition, we give closed forms of the general solutions of the system by using the change of variables. Finally, we present a numerical example to support our results.