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Lyapunov Functionals that Lead to Exponential Stability and Instability in Finite Delay Volterra Difference Equations

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Abstract

We use Lyapunov functionals to obtain sufficient conditions that guarantee exponential stability of the zero solution of the finite delay Volterra difference equation

$$x(t+1) = a(t)x(t)+\sum\limits^{t-1}_{s=t-r}b(t,s)x(s). $$

Also, by displaying a slightly different Lyapunov functional, we obtain conditions that guarantee the instability of the zero solution. The highlight of the paper is the relaxing of the condition |a(t)| < 1. Moreover, we provide examples in which we show that our theorems provide an improvement of some recent results.

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Acknowledgments

The authors would like to thank the referee for his/her valuable comments and careful reading of the manuscript which have greatly improved our paper.

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Authors and Affiliations

  1. Department of Mathematics, University of Dayton, Dayton, OH, 45469-2316, USA

    Catherine Kublik & Youssef Raffoul

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  1. Catherine Kublik
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  2. Youssef Raffoul
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Correspondence to Youssef Raffoul.

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Kublik, C., Raffoul, Y. Lyapunov Functionals that Lead to Exponential Stability and Instability in Finite Delay Volterra Difference Equations. Acta Math Vietnam 41, 77–89 (2016). https://doi.org/10.1007/s40306-014-0098-4

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  • DOI: https://doi.org/10.1007/s40306-014-0098-4

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