On solvability of a two-dimensional symmetric nonlinear system of difference equations

Abstract

We show that the system of difference equations $$ x_{n+k}=\frac{x_{n+l}y_{n}-ef}{x_{n+l}+y_{n}-e-f},\quad y_{n+k}= \frac{y_{n+l}x_{n}-ef}{y_{n+l}+x_{n}-e-f},\quad n\in {\mathbb{N}}_{0}, $$ where $k\in {\mathbb{N}}$ , $l\in {\mathbb{N}}_{0}$ , $l< k$ , $e, f\in {\mathbb{C}}$ , and $x_{j}, y_{j}\in {\mathbb{C}}$ , $j=\overline{0,k-1}$ , is theoretically solvable and present some cases of the system when the general solutions can be found in a closed form.