Asymptotically polynomial solutions of difference equations of neutral type

TL;DR

This paper investigates the asymptotic behavior of solutions to a class of difference equations of neutral type, establishing conditions for solutions to be asymptotically polynomial using a novel approximation control technique.

Contribution

It introduces a new method to determine when solutions of neutral difference equations are asymptotically polynomial, expanding understanding of their long-term behavior.

Findings

All solutions are asymptotically polynomial under certain conditions

Solutions with polynomial growth are characterized

A new technique controls the degree of approximation

Abstract

Asymptotic properties of solutions of difference equation of the form \[ \Delta^m(x_n+u_nx_{n+k})=a_nf(n,x_{\sigma(n)})+b_n \] are studied. We give sufficient conditions under which all solutions, or all solutions with polynomial growth, or all nonoscillatory solutions are asymptotically polynomial. We use a new technique which allows us to control the degree of approximation.