General solution to subclasses of a two-dimensional class of systems of difference equations

Abstract

We show practical solvability of the following two-dimensional systems of difference equations $$x_{n+1}=\frac{u_{n-2}v_{n-3}+a}{u_{n-2}+v_{n-3}},\quad y_{n+1}=\frac{w_{n-2}s_{n-3}+a}{w_{n-2}+s_{n-3}},\quad n\in\mathbb{N}_0,$$ where $u_n$, $v_n,$ $w_n$ and $s_n$ are $x_n$ or $y_n$, by presenting closed-form formulas for their solutions in terms of parameter $a$, initial values, and some sequences for which there are closed-form formulas in terms of index $n.$ This shows that a recently introduced class of systems of difference equations, contains a subclass such that one of the delays in the systems is equal to four, and that they all are practically solvable, which is a bit unexpected fact.

Keywords

• system of difference equations
• solvable systems
• practical solvability