On a system of second-order difference equations

Abstract

We obtain explicit formulas for the solutions of the system of second-order difference equations of the form $x_{n+1}=\frac{x_n{y}_{n-1}}{y_n(a_n+b_n{x}_n{y}_{n-1})},y_{n+1}=\frac{x_{n-1}{y}_n}{x_n(c_n+d_n{x}_{n-1}{y}_n)}$, where $(a_n{)}_{n\in {\mathbb{N}}_0},(b_n{)}_{n\in {\mathbb{N}}_0},(c_n{)}_{n\in {\mathbb{N}}_0}$ and $(d_n{)}_{n\in {\mathbb{N}}_0}$ are real sequences. We use Lie symmetry analysis to derive non-trivial symmetries and thereafter, exact solutions are obtained.

11 pages