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Lyapunov Functionals that Lead to Exponential Stability and Instability in Finite Delay Volterra Difference Equations
Catherine Kublik
,
Youssef Raffoul
Abstract
We use Lyapunov functionals to obtain sufficient conditions that guarantee exponential stability of the zero solution of the finite delay Volterra difference equation $$ x(t+1) = a(t)x(t)+\sum^{t-1}_{s=t-r}b(t,s)x(s). $$ 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 the condition $|a(t)| <1.$ Moreover we provide examples in which we show that our theorems provide an improvement of some recent results.