On the Stability of Volterra Difference Equations of Convolution Type

In \cite{Elaydi-10}, S.\ Elaydi obtained a characterization of the stability ofthe null solution of the Volterra difference equation\beqaex_n=\sum_{i=0}^{n-1} a_{n-i} x_i\textrm{,}\quad n\geq 1\textrm{,}\eeqaeby localizing the roots of its characteristic equation\beqae1-\sum_{n=1}^{\infty}a_nz^n=0\textrm{.}\eeqaeThe assumption that $(a_n)\in\ell^1$ was the single hypothesis considered for the validity of that characterization, which is an insufficient condition if theratio $R$ of convergence of the power series of the previous equation equals one. In fact, when $R=1$, this characterization conflicts with a result obtainedby Erd\"os et al in \cite{Erdos}. Here, we analyze the $R=1$ case and show thatsome 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.

Weighted Asymptotically Periodic Solutions of Linear Volterra Difference Equations

A linear Volterra difference equation of the formx(n+1)=a(n)+b(n)x(n)+∑i=0nK(n,i)x(i),wherex:N0→R,a:N0→R,K:N0×N0→Randb:N0→R∖{0}isω-periodic, is considered. Sufficient conditions for the existence of weighted asymptotically periodic solutions of this equation are obtained. Unlike previous investigations, no restriction on∏j=0ω-1b(j)is assumed. The results generalize some of the recent results.

Existence of asymptotically periodic solutions of scalar Volterra difference equations

There is used a version of Schauder’s fixed point theorem to prove the existence of asymptotically periodic solutions of a scalar Volterra difference equation. Along with the existence of asymptotically periodic solutions, sufficient conditions for the nonexistence of such solutions are derived. Results are illustrated on examples.

Accurate solution estimates for vector difference equations

Accurate estimates for the norms of the solutions of a vector difference equation are derived. They give us stability conditions and bounds for the region of attraction of the stationary solution. Our approach is based on estimates for the powers of a constant matrix. We also discuss applications of our main results to partial reaction-diffusion difference equations and to a Volterra difference equation.