# A Convergence Criterion for the Solutions of Nonlinear Difference Equations and Dynamical Systems

## TL;DR

This paper establishes a general convergence criterion for solutions of nonlinear difference equations and dynamical systems, emphasizing the roles of delay patterns and system relations, with applications to population models.

## Contribution

It introduces a new sufficient condition for convergence of solutions in nonlinear difference equations and systems, considering delay structures and inter-equation relationships.

## Key findings

- Subsequences of solutions converge under specific delay conditions.
- System relations influence convergence criteria.
- Applications include models of population dynamics.

## Abstract

A general sufficient condition for the convergence of subsequences of solutions of non-autonomous, nonlinear difference equations and systems is obtained. For higher order equations the delay sizes and patterns play essential roles in determining which subsequences of solutions converge. For systems the specific manner in which the equations are related is important and lead to different criteria. Applications to discrete dynamical systems, including some that model populations of certain species are discussed.