### 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.

### Refbacks

- There are currently no refbacks.