Generalized Incremental Small Gain

This paper is directed on a using of Banach fixed-point theorem in the problem of stabilization of nonlinear systems. The theory is developed in Banach spaces with using general operator theory and may be applied both for continuous and for discrete systems. All theorems are proved and may be separated into two classes, global and local contraction operator theorems. It is studied possibility of parameters changes and continuity changes. Moreover there are derived a precision of convergence and a rate of convergence. It is interesting, that although it is almost nothing presupposed about the structure of controlled system and controller, the fact, that the operator of closed loop is k-contractive, allows to prove relatively strong assertions.