On Stability of Volterra Difference Equations of Convolution Type

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.