On the System of Difference Equations $x_{n}=\frac{x_{n-2}y_{n-3}}{y_{n-1}\left(a_{n}+b_{n}x_{n-2}y_{n-3} \right) }$, $y_{n}=\frac{y_{n-2}x_{n-3}}{x_{n-1}\left(\alpha_{n}+\beta_{n}y_{n-2}x_{n-3} \right)

Abstract

In this paper, we show that the system of difference equations \begin{equation*}x_{n}=\frac{x_{n-2}y_{n-3}}{y_{n-1}\left(a_{n}+b_{n}x_{n-2}y_{n-3} \right) },\y_{n}=\frac{y_{n-2}x_{n-3}}{x_{n-1}\left(\alpha_{n}+\beta_{n}y_{n-2}x_{n-3} \right) }, \ n\in\mathbb{N}_{0},\end{equation*}%where the sequences $\forall n\in\mathbb{N}_{0}$, $\left( a_{n}\right) ,\left( b_{n}\right) , \left( \alpha_{n}\right) , \left( \beta_{n}\right) $ and the initial values $x_{-j}, y_{-j}, j\in\{1,2,3\}$ are non-zero real numbers, can be solvedin the closed form. For the case when all the sequences $\left( a_{n}\right) ,\left( b_{n}\right) , \left( \alpha_{n}\right) , \left( \beta_{n}\right) $ are constant we describe the asymptotic behavior and periodicity of solutions of above system is also investigated.