On contraction of nonlinear difference equations with time-varying delays

General nonlinear difference equations with time-varying delays are considered. Explicit criteria for contraction of such equations are presented. Then some simple sufficient conditions for global exponential stability of equilibria and for stability of invariant sets are derived. Furthermore, explicit criteria for existence, uniqueness and global exponential stability of periodic solutions are derived. Finally, the obtained results are applied to time-varying discrete-time neural networks with delay.