Lyapunov functionals that lead to exponential stability and instability in finite delay Volterra difference equations (Q255675)

The article deals with some sufficient conditions of exponential stability/instability for the following finite delay Volterra difference equation \[ x(t + 1) = a(t)x(t) + \sum_{s=t-r}^{t-1} b(t,s)x(s), \quad t \geq 0. \] The main results are obtained in terms of Lyapunov functionals \(V(t,x)\) such that \[ V(t) = V(t,x(t)) = \bigg[x(t) + \sum_{s=t-r-1}^{t-1} A(t,s)x(s)\bigg]^2 + \delta \sum_{s=-r}^{-1} \sum_{z=t+s}^{t-1} A^2(t,z)x^2(z) \] and \[ V(t) = V(t,x(t)) = \bigg[x(t) + \sum_{s=t-r-1}^{t-1} A(t,s)x(s)\bigg]^2 - H \sum_{s=t-r-1}^{t-1} A^2(t,z)x^2(z) \] \[ \bigg(A(t,s) = \sum_{u=t-s}^r b(u + s,s)\bigg). \] Some numerical examples are presented.