Global behavior of positive solutions of a third order difference equations system

In this paper, we investigate the global behavior of positive solutions of the system of difference equations \begin{equation*} x_{n+۱}=\alpha+ \dfrac{y^p_{n}}{y^p_{n-۲}},\y_{n+۱}=\alpha+ \dfrac{x^q_{n} }{x^q_{n-۲}},\ n=۰, ۱, ۲, ...\end{equation*}where parameters \alpha, p, q \in (۰, \infty) and the initial values x_{i}, y_{i} are arbitrary positive numbers for i= -۲,-۱, ۰. Moreover, the rate of convergence of positive solutions is established and some numerical examples are given to demonstrate our theoretical results.

Keywords:

difference equation , semi-cycle , equilibrium , boundedness , Global asymptotic stability , rate of convergence