Uniform asymptotic stability of impulsive discrete systems with time delay (Q549929)

The authors consider the following impulsive discrete system with time delays \[ x(n+1) = f(n,\bar{x}_n) ,\quad n_0\in {\mathbb N} ,\;n\geq n_0 \] \[ \bar{x}(n) = \begin{cases} x(n), &n\neq N_k \\ x(N_k) + I(x(N_k)) , &n=N_k,\;k\in {\mathbb N}\\ x_{n_0} = \phi \end{cases} \] where \(x\in{\mathbb R}^m\), \(f:{\mathbb N}\times{\mathcal C}_\rho\mapsto{\mathbb R}^m\), \(I_k:{\mathcal C}_\rho\mapsto{\mathbb R}^m\), \(\phi\in{\mathcal C}\) and \[ \bar{x}_n(s) = \bar{x}(n+s) ,\quad s\in{\mathbb N}_{-r}=\{-r,-r+1,\dots,-1,0\} , \] \[ {\mathcal C}_\rho = \{\phi\in{\mathcal C},\;\|\phi\|<\rho\},\quad \|\phi\| = \max|\phi(s)|_{s\in{\mathbb N}_{-r}}. \] They prove uniform stability and uniform asymptotic stability of the zero solution of the above system using the approach of a Razumikhin type function.