On Stability of Volterra Difference Equations of Convolution Type

TL;DR

This paper investigates the stability of Volterra difference equations of convolution type, especially focusing on the challenging case where the power series has a radius of convergence equal to one, and extends existing stability characterizations.

Contribution

It analyzes the stability criteria for the case when the power series radius equals one and introduces new stability results using finite approximation methods.

Findings

Some parts of Elaydi's stability characterization hold when R=1.

The paper presents new stability results for R<1.

Finite approximation techniques are used to derive stability conditions.

Abstract

S. Elaydi obtained a characterization of the stability of the null solution of the Volterra difference equation $$ x_n=\sum_{i=0}^{n-1} a_{n-i} x_i\textrm{,}\quad n\geq 1\textrm{,} $$ by localizing the roots of its characteristic equation $$ 1-\sum_{n=1}^{\infty}a_nz^n=0\textrm{.} $$ The assumption that $(a_n)\in\ell^1$ was the single hypothesis considered for the validity of that characterization, which is an insufficient condition if the ratio $R$ of convergence of the power series of the previous equation equals one. In fact, when $R=1$, this characterization conflicts with a result obtained by Erd\"os, Feller and Pollard. Here, we analyze the $R=1$ case and show that some parts of that characterization still hold. Furthermore, studies on stability for the $R<1$ case are presented. Finally, we state some new results related to stability via finite approximation.