A simplification and generalization of Elsayed's two-dimensional system of third order difference equations

TL;DR

This paper performs a Lie analysis of a third-order difference equation system, deriving explicit solutions and conditions for periodicity, thereby generalizing and simplifying previous work by Elsayed and Ibrahim.

Contribution

It introduces a simplified and more general approach to analyzing third-order difference equations, including explicit solutions and periodicity conditions.

Findings

Explicit solutions in terms of initial values

Conditions for 2-periodic and 4-periodic solutions

Generalization of previous results by Elsayed and Ibrahim

Abstract

A full Lie analysis of a system of third-order difference equations is performed. Explicit solutions, expressed in terms of the initial values, are derived. Furthermore, we give sufficient conditions for existence of 2-periodic and 4-periodic solutions in certain cases. Our results generalize and simplify the work by Elsayed and Ibrahim [E. M. Elsayed and T.F. Ibrahim, Periodicity and solutions for some systems of nonlinear rational difference equations, Hacettepe Journal of Mathematics and Statistics 44:6 (2015), 1361-1390].