A qualitative investigation of a system of third-order difference equations with multiplicative reciprocal terms

Abstract

In this paper, we study the system of third-order difference equations \begin{equation*} x_{n+1}=a+\frac{a_{1}}{y_{n}}+\frac{a_{2}}{y_{n-1}}+\frac{a_{3}}{y_{n-2}}% ,\quad y_{n+1}=b+\frac{b_{1}}{x_{n}}+\frac{b_{2}}{x_{n-1}}+\frac{b_{3}}{% x_{n-2}},\quad n\in \mathbb{N}_{0}, \end{equation*}% where the parameters $a$, $a_{i}$, $b$, $b_{i}$, $i=1,2,3$, and the initial values $x_{-j}$, $y_{-j}$, $j=0,1,2$, are positive real numbers. We first prove a general convergence theorem. By applying this convergence theorem to the system, we show that positive equilibrium is a global attractor. We also study the local asymptotic stability of the equilibrium and show that it is globally asymptotically stable. Finally, we study the invariant set of solutions.