Nenya O. Global stability of difference equations and functional-differential equations with impulsive action.

This dissertation research deals with the investigation of global stability of difference equations and functional-differential equations with impulsive action. We give sharp conditions that are sufficient for global stability of the zero solution of the difference equation with the nonlinear functions satisfy the negative feedback condition and have sublinear growth. We give simple sufficient conditions for global stability of the zero solution for the difference equation, where the nonlinear functions satisfy the Yorke condition. Interval (0,1] is presented as the union of [(2k+2)/3] disjoint subintervals and for q from each subinterval the global stability condition is given explicitely. Our conditions are sharp for the class of equations satisfying the Yorke condition. We give conditions that are sufficient for global stability of the zero solution of the functional-differential equation with impulsive action and nonlinear function satisfying the Yorke condition. By applying Krasnosel'skii fixed point theorem to a mapping on a cone, we find conditions for existence of positive piecewise smooth periodic solutions of functional-differential equations with impulsive effects. For Mackey-Glass equation with periodic coefficients, nonconstant delay and impulsive action conditions of boundedness, persistence, extinction and periodicity of positive solutions are presented.