Motivated by the importance and TABLE application of discrete dynamical systems, this paper presents a new Lyapunov characterization which is an extension of conventional Lyapunov characterization for multistable discrete-time nonlinear systems.Based on a new type stability notion of W-stability introduced by D.Efimov, the estimates of solution and the Lyapunov stability theorem Stock Pots and converse theorem are proposed for multi-stable discrete-time nonlinear systems.