Solvable three-dimensional system of higher-order nonlinear difference equations

Abstract

In this work, we indicate three-dimensional system of difference equations
\small
\begin{equation*}
x_{n}=ay_{n-k}+\frac{dy_{n-k}x_{n-k-l}}{\widehat bx_{n-k-l}+\widehat cz_{n-l}},\ y_{n}=\alpha z_{n-k}+\frac{\delta
z_{n-k}y_{n-k-l }}{\widehat \beta y_{n-k-l}+\widehat \gamma
x_{n-l}},\
z_{n}=ex_{n-k}+\frac{hx_{n-k}z_{n-k-l}}{\widehat fz_{n-k-l}+\widehat gy_{n-l}}, \ n\in \mathbb{N}_{0},
\end{equation*}
\normalsize
where $k$ and $l$ are positive integers, the parameters $a$, $\widehat b$, $\widehat c$, $d$, $\alpha $, $\widehat \beta $, $\widehat \gamma $, $\delta $, $e$, $\widehat f$, $\widehat g$, $h$ and the
initial values $x_{-j}$, $y_{-j}$, $z_{-j}$ $j=\overline{1,k+l}$, are non-zero real numbers,
can be solved in the closed form. In addition, we obtained explicit formulas for the well-defined solutions of the aforementioned
system for the case $l=1$. Also, the forbidden set of solutions of the system is found. Finally, an application about a three-dimensional system of difference equations is given.