Stability and asymptoticity of Volterra difference equations: A progress report |
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Authors: | Saber Elaydi |
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Institution: | Department of Mathematics, Trinity University, San Antonio, TX 78212, USA |
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Abstract: | 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. |
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Keywords: | Volterra difference equations Stability Resolvent matrix Z-transform" target="_blank">gif" overflow="scroll">Z-transform Asymptotic equivalence Convolution Dichotomy |
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