Boundedness of Discrete Volterra Equations

Volterra discrete equations: summability of the fundamental matrix

“…In recent years the general problem of proving conditions which assure (1.8), i.e., the problem of boundedness of the solution of a VDE, has been investigated by numerical analysts in many papers (see e.g [4,7,10,11,13,19]). Their increasing interest in this topic is due to the fact that difference equations of the type (1.1) arise from the application of numerical methods to Volterra integral equations (VIEs) and integro-differential equations (VIDEs).…”

Stability Properties of Linear Volterra Discrete Systems with Nonlinear Perturbation

“…Their work heavily depended on showing or assuming the summability of the resolvent matrix. For more results on stability of the zero solution of Volterra discrete system we refer the reader to Crisci, Komanovskii and Vecchio [2], Elaydi [3] and Agarwal and Pang [1]. This research is a continuation of the research initiated by the authors in [6] and related to the work in [5].…”

Boundedness and Asymptotic Stability for the Solution of Homogeneous Volterra Discrete Equations

“…Most of the known results for the nonconvolution case are based on the hypothesis of double summability of the coefficients (∑ +∞ =0 ∑ =0 | , | < +∞); see [1,4,[13][14][15][16][17][18][19]. Another interesting approach, resembling the study of continuous VIDE (see, e.g., [20,21]), basically requires that the coefficient +1 of (5), assumed to be negative, in some sense "prevails" on the summation of the remaining coefficients . Here we would like to add another piece to the framework regarding the analysis of VDE behaviour, by considering hypotheses based on the sign of the coefficients and of their first and second differences.…”