Novel development of the Lyapunov stability theory for discrete-time systems. I: Concepts and definitions (Q1582118)

The authors develop a theory of Lyapunov stability for discrete time nonlinear systems \[ x(t_{k+1})= f(t_k, x(t_k)). \] They are mainly concerned with the dependence of several stability concepts on the initial time (uniform and nonuniform stability). The analysis is carried out by means of certain invariant sets, called domains of stability. In this paper, they give a negative result about the existence of continuous Lyapunov functions if the stability domains are asymptotically contractive.