Reduction of Order, Periodicity and Boundedness in Nonlinear, Higher Order Difference Equations

TL;DR

This paper extends methods for analyzing nonlinear higher order difference equations, providing new results on boundedness and periodic solutions, especially for equations with complex roots in their characteristic polynomials.

Contribution

It introduces novel results on boundedness and periodicity for nonlinear difference equations with complex roots, expanding previous work to higher orders.

Findings

New boundedness criteria for third-order and higher equations.

Existence of periodic solutions under broader conditions.

Extension of semiconjugate factorization techniques.

Abstract

We consider the semiconjugate factorization and reduction of order for non-autonomous, nonlinear, higher order difference equations containing linear arguments. These equations have appeared in several mathematical models in biology and economics. By extending some recent results to cases where characteristic polynomials of the linear expressions have complex roots, we obtain new results on boundedness and the existence of periodic solutions for equations of order 3 or greater.