On linear Volterra difference equations with infinite delay

Ch G Philos,I K Purnaras

doi:10.1155/ade/2006/78470

Abstract

Linear neutral, and especially non-neutral, Volterra difference equations with infinite delay are considered and some new results on the behavior of solutions are established. The results are obtained by the use of appropriate positive roots of the corresponding characteristic equation.

Highlights

Introduction and notationsThroughout the paper, N stands for the set of all nonnegative integers and Z stands for the set of all integers
Let us consider the special case of the Volterra difference equation (2.3) and let λ0 be a positive root of the characteristic equation (2.9) with the property (2.11)
Our first main result is Theorem 3.1 below, which establishes a useful inequality for solutions of the neutral Volterra difference equation (2.2)

Summary

Preliminary notes

Motivated by the old but significant papers by Driver [3] and Driver et al [5], a number of relevant papers has recently appeared in the literature. In [33], Philos and Purnaras continued the study in [19, 21] and established some further results on the behavior of solutions of linear neutral integrodifferential equations with unbounded delay, and, especially, of linear (non-neutral) integrodifferential equations with unbounded delay. We choose to refer here to the papers by Jarosand Stavroulakis [13], Kiventidis [15], Kordonis and Philos [18], Ladas et al [22], and Philos [27] for some results concerning the existence and/or the nonexistence of positive solutions of certain linear Volterra difference equations.

Introduction and notations

Statement of the main results

Proofs of the main results