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