On a solvable difference equations system

Abstract

In this paper, we study the following three dimensional system of difference equations
\begin{equation*}
x_n=\frac{y_{n-4}z_{n-5}x_{n-6}}{y_{n-1}z_{n-2}\left(\alpha +\beta x_{n-3}y_{n-4}z_{n-5}x_{n-6}\right)},
y_n=\frac{z_{n-4}x_{n-5}y_{n-6}}{z_{n-1}x_{n-2}\left(\gamma +\theta y_{n-3}z_{n-4}x_{n-5}y_{n-6}\right)},
z_n=\frac{x_{n-4}y_{n-5}z_{n-6}}{x_{n-1}y_{n-2}\left(\eta +\zeta z_{n-3}x_{n-4}y_{n-5}z_{n-6}\right)},
\end{equation*}
for $n\in \mathbb{N}_0$, the initial values $x_{-p}$, $y_{-p}$, $z_{-p}$ for $p=\overline{1,6}$ and the parameters $\alpha$, $\beta$, $\gamma$, $\theta$, $\eta$, $\zeta$ are real numbers. Firstly, we examine the solutions of the mentioned system depending on whether the parameters are equal to zero or non-zero. In addition, the solutions of the aforementioned system are obtained in closed form. Finally, we also describe the forbidden set of the solutions of the system of difference equations.