On a Solvable Two-Dimensional System of Difference Equations

Abstract

In this paper, we investigate the following

system of difference equations
\begin{equation*}
x_{n}=\frac{x_{n-3}y_{n-4}x_{n-5}}{y_{n-1}x_{n-2}(\alpha _{n}+\beta_nx_{n-3}y_{n-4}x_{n-5})},
y_{n}=\frac{y_{n-3}x_{n-4}y_{n-5}}{x_{n-1}y_{n-2}(\gamma _{n}+\delta_ny_{n-3}x_{n-4}y_{n-5})}, \ n\in
\mathbb{N}_{0},
\end{equation*}
where
$\left(\alpha_{n}\right)_{n\in \mathbb{N}_{0}}$, $\left(\beta_{n}\right)_{n\in \mathbb{N}_{0}}$, $\left(\gamma_{n}\right)_{n\in \mathbb{N}_{0}}$ and $\left(\delta_{n}\right)_{n\in \mathbb{N}_{0}}$ are real sequences and
initial values $x_{k}$,\ $y_{k}$ for $
k=\overline{-5,-1} $ are real numbers. Firstly, we obtain the general solutions of mentioned system of difference equations. Later, the solutions of the above system of difference equations are acquired when the sequences are constant. Additionally, the solutions are expressed when the parameters $\alpha$ and $\gamma$ are equal to 1 or not equal to 1. Further, we research the asymptotic behavior of the well- defined solutions of above system of difference equations. Lastly, the forbidden set of the initial conditions is acquired by using obtained formulas.