Stability and asymptoticity of Volterra difference equations: A progress report

We survey some of the fundamental results on the stability and asymptoticity of linear Volterra difference equations. The method of Z -transform is heavily utilized in equations of convolution type. An example is given to show that uniform asymptotic stability does not necessarily imply exponential...

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Bibliographic Details
Published in:Journal of computational and applied mathematics 2009-06, Vol.228 (2), p.504-513
Main Author: Elaydi, Saber
Format: Article
Language:English
Subjects:
[formula omitted]-transform
Asymptotic equivalence
Convolution
Dichotomy
Difference and functional equations, recurrence relations
Exact sciences and technology
Finite differences and functional equations
Mathematical analysis
Mathematics
Numerical analysis
Numerical analysis. Scientific computation
Operator theory
Ordinary differential equations
Resolvent matrix
Sciences and techniques of general use
Stability
Volterra difference equations
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Summary:We survey some of the fundamental results on the stability and asymptoticity of linear Volterra difference equations. The method of Z -transform is heavily utilized in equations of convolution type. An example is given to show that uniform asymptotic stability does not necessarily imply exponential stabilty. It is shown that the two notions are equivalent if the kernel decays exponentially. For equations of nonconvolution type, Liapunov functions are used to find explicit criteria for stability. Moreover, the resolvent matrix is defined to produce a variation of constants formula. The study of asymptotic equivalence for difference equations with infinite delay is carried out in Section 6. Finally, we state some problems.
ISSN:0377-0427
1879-1778
DOI:10.1016/j.cam.2008.03.023