Stability and asymptoticity of Volterra difference equations: a progress report
type: Item
instance of: scholarly article
title: Stability and asymptoticity of Volterra difference equations: a progress report
zbMATH Open document ID: 1186.39023
published in: Journal of Computational and Applied Mathematics
publication date: 2009-06-04T00:00:00Z
review text: This paper can be viewed as a survey of the work done by the author and his collaborators on the stability of discrete Volterra equations. In the case that the Volterra difference equation under consideration is of convolution type, the \(Z\)-transform is used to study the stability of the equilibrium. It is shown by an example that uniform asymptotic stability does not necessarily imply exponential stability. However, the two notions are equivalent provided the kernel decays exponentially. For equations of nonconvolution type, Lyapunov functions are used to find explicit criteria for stability. Moreover, the resolvent matrix is defined to produce a variation of constant formula. The study of asymptotic equivalence for difference equations with infinite delay is carried out. Finally, some interesting open problems are posed.
Mathematics Subject Classification ID: 39A30
Mathematics Subject Classification ID: 39A06
Mathematics Subject Classification ID: 39A12
Mathematics Subject Classification ID: 45D05
zbMATH DE Number: 5561628
zbMATH Keywords: Volterra difference equation
zbMATH Keywords: stability
zbMATH Keywords: resolvent matrix
zbMATH Keywords: \(Z\)-transform
zbMATH Keywords: asymptotic equivalence
zbMATH Keywords: convolution
zbMATH Keywords: dichotomy
zbMATH Keywords: convolution type
zbMATH Keywords: nonconvolution type
zbMATH Keywords: Lyapunov functions
author: Saber Elaydi
MaRDI profile type: Publication
full work available at URL: j.cam.2008.03.023
OpenAlex ID: W2093718645
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DOI: 10.1016/J.CAM.2008.03.023
description: scientific article; zbMATH DE number 5561628