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

Abstract

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

Keywords

• Symmetric system of difference equations
• Solvable system
• Solution in closed form