In this paper, we set a new framework for the analysis of nonlinear discrete systems, to carry out the stabilization theorem, and to design stable controllers of the systems within that framework. This framework includes many practically important systems-described, for example, by Lipschitz functions, functions with arbitrary order bounded derivatives, bilinear functions and all polynomial functions. An important existence theorem for stabilizing has been proven for this framework. The theorem basically states that under definite, rather broad conditions, the zero solution of the plants can be made stable by a nonlinear output feedback. In addition, this nonlinear feedback is from a larger class of functions than these classes given by current literature. Following these stabilization theorems, a new design procedure for the output feedback control law is presented. All results presented are easy to use and convenient to apply. The application is illustrated by several examples, including the design of stabilizing control for a bilinear control system.