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

TL;DR

This paper analyzes the global behavior, boundedness, and stability of positive solutions for a third-order difference equations system with parameters, providing conditions for stability and convergence, supported by numerical examples.

Contribution

It introduces a comprehensive semi-cycle analysis and stability criteria for the system, including convergence rates, which are novel for this class of equations.

Findings

Solutions are bounded under certain parameter conditions.

The equilibrium point is globally asymptotically stable when  and 0<p,q.

Convergence rate of solutions is explicitly established.

Abstract

\begin{abstract} In this paper, we consider the following system of difference equations \begin{equation*} x_{n+1}=\alpha+\dfrac{y_{n}^p}{y_{n-2}^p},\ y_{n+1}=\alpha+ \dfrac{x_{n}^q}{x_{n-2}^q}, \ n=0, 1, 2, ... \end{equation*} where parameters $\alpha, p, q \in (0, \infty)$ and the initial values $x_{-i}$, $y_{-i}$ are arbitrary positive numbers for $ i=-2,-1, 0$. Our main aim is to investigate semi-cycle analysis of solutions of above system. Also, we study the boundedness of the positive solutions and the global asymptotic stability of the equilibrium point in case $\alpha>1$, $ 0<p,\ q\leq 1$. Moreover, the rate of convergence of the solutions is established. Finally, some numerical examples are given to illustrate our theoretical results.