Stability results for finite-dimensional discrete-time dynamical systems involving non-monotonic Lyapunov functions

In and in a more recent paper we established results for the uniform stability and the uniform asymptotic stability in the large involving non-monotonic Lyapunov functions for continuous-time dynamical systems. In the present paper we continue this work by addressing finite-dimensional discrete-time dynamical systems. Similarly as in and, we prove that in general, the results presented herein are less conservative than the corresponding standard Lyapunov stability results (henceforth called classical Lyapunov stability results) for finite-dimensional discrete-time dynamical systems. We present two specific examples to demonstrate the applicability of our results.