### On the Stability of Volterra Difference Equations of Convolution Type

#### Abstract

In \cite{Elaydi-10}, S.\ Elaydi obtained a characterization of the stability of

the null solution of the Volterra difference equation

\beqae

x_n=\sum_{i=0}^{n-1} a_{n-i} x_i\textrm{,}\quad n\geq 1\textrm{,}

\eeqae

by localizing the roots of its characteristic equation

\beqae

1-\sum_{n=1}^{\infty}a_nz^n=0\textrm{.}

\eeqae

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 et al in \cite{Erdos}. 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.

#### Keywords

#### Full Text:

PDFDOI: https://doi.org/10.5540/tema.2017.018.03.337

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**TEMA - Trends in Applied and Computational Mathematics**

A publication of the Brazilian Society of Applied and Computational Mathematics (SBMAC)

ISSN: 1677-1966 (print version), 2179-8451 (online version)

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