On a solvable three-dimensional system of difference equations

Abstract

In this paper, we show that the following three-dimensional system of
difference equations
\begin{equation*}
x_{n}=\frac{z_{n-2}x_{n-3}}{ax_{n-3}+by_{n-1}}, \ y_{n}=\frac{x_{n-2}y_{n-3}}{cy_{n-3}+dz_{n-1}}, \ z_{n}=\frac{y_{n-2}z_{n-3}}{ez_{n-3}+fx_{n-1}}, \ n\in \mathbb{N}_{0},
\end{equation*}%
where the parameters $a, b, c, d, e, f$\ and the
initial values $x_{-i},y_{-i},z_{-i}$, $i \in \{1,2,3\}$, are real numbers, can be solved, extending further some results in literature. Also, we determine the asymptotic behavior of solutions and the forbidden set of the initial values by using the obtained formulae.