Stability and asymptoticity of Volterra difference equations: a progress report
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.
scientific article; zbMATH DE number 5561628