On the Asymptotic Behavior of Volterra Difference Equations

TL;DR

This paper investigates the long-term behavior of solutions to Volterra difference equations in Banach spaces, linking spectral properties of operators and sequences to stability and periodicity outcomes.

Contribution

It introduces a spectral approach based on the Z-transform to analyze asymptotic stability and almost periodicity of solutions, extending previous results.

Findings

Established a relation between Z-transform and spectrum of unilateral sequences.

Derived conditions for asymptotic stability in terms of spectral properties.

Extended previous results on the asymptotic behavior of Volterra difference equations.

Abstract

We consider the asymptotic behavior of solutions of the difference equations of the form $x(n+1)=Ax(n) + \sum_{k=0}^n B(n-k)x(k) + y(n)$ in a Banach space $\X$, where $n=0,1,2,...$; $A,B(n)$ are linear bounded operator in $\X$. Our method of study is based on the concept of spectrum of a unilateral sequence. The obtained results on asymptotic stability and almost periodicity are stated in terms of spectral properties of the equation and its solutions. To this end, a relation between the Z-transform and spectrum of a unilateral sequence is established. The main results extend previous ones.