A new criterion on the non-Lyapunov method of stability analysis is developed for a large class of discrete time nonlinear systems. The amenable class is broad and includes systems mainly encountered in practice, for example, all plants described by functions of Lipschitz class, linear, bilinear, and polynomial functions, etc. Unlike other publications on the stability of discrete nonlinear systems, we consider the output feedback with time delays, and the control law from a broader class of functions. The derived sufficient criterion is simple and easily applicable.