Some boundedness results for systems of two rational difference equations

TL;DR

This paper investigates boundedness in systems of two rational difference equations, extending known results to systems with non-negative parameters and initial conditions, using multiple analytical approaches.

Contribution

It introduces methods to extend boundedness results from single equations to systems of rational difference equations under specific conditions.

Findings

Extended boundedness results to systems of rational difference equations.

Developed multiple analytical approaches for boundedness analysis.

Applicable to systems with non-negative parameters and initial conditions.

Abstract

We study kth order systems of two rational difference equations $$x_n=\frac{\alpha+\sum^{k}_{i=1}\beta_{i}x_{n-i} + \sum^{k}_{i=1}\gamma_{i}y_{n-i}}{A+\sum^{k}_{j=1}B_{j}x_{n-j} + \sum^{k}_{j=1}C_{j}y_{n-j}},\quad n\in\mathbb{N},$$ $$y_n=\frac{p+\sum^{k}_{i=1}\delta_{i}x_{n-i} + \sum^{k}_{i=1}\epsilon_{i}y_{n-i}}{q+\sum^{k}_{j=1}D_{j}x_{n-j} + \sum^{k}_{j=1}E_{j}y_{n-j}},\quad n\in\mathbb{N}.$$ In particular we assume non-negative parameters and non-negative initial conditions. We develop several approaches which allow us to extend well known boundedness results on the kth order rational difference equation to the setting of systems in certain cases.