Well-defined solutions of a system of difference equations

This note deals with the solution form of the system of difference equations $$\begin{aligned} x_{n+1}=\frac{a x_{n}y_{n-1}}{y_{n}-\alpha }+\beta ,\quad y_{n+1}=\frac{b x_{n-1}y_{n}}{x_{n}-\beta }+\alpha ,\quad n \in {\mathbb {N}}_{0}, \end{aligned}$$ where the parameters \(a,\,b,\,\alpha ,\,\beta \) and initial values \(x_{-i},\,y_{-i},\,i=0,1\), are non-zero real numbers. The special case \(a=b\) is treated separately, and the qualitative behavior of its solutions is examined. Also, conditions are determined so that the system admits periodic solutions. Finally, numerical examples are provided to support the theoretical results exhibited in the paper.