Karpenko O. The relation between the qualitative properties of solutions of differential equations and the corresponding difference equations.

The difference equations arise as independent mathematical models, whose evolution is discrete in nature, and arise in various difference schemes for numerical solution of differential equations. If the step of difference equation , than this equation is formally goes into the differential equation. There is a problem of preservation during the transition of properties for differential equations provided that these properties has a difference equations. In the thesis, thus, the questions of preservation of certain properties of the solutions at transition from differential equations to the corresponding difference equations and vice versa, and weakening of the previously known conditions are obtained. The study of this problem is especially actual in today's market economy conditions. It due to the fact that in various models (biological, financial, economic) real phenomena we observe no continuously but in particular discrete moments of time. As this study should be carried out at fairly large intervals time, then naturally arises problem of research of qualitative properties solutions of difference equations. The dissertation deals with the issue of bounded solution of differential and the corresponding difference equation. It is received the result that the existence of bounded solution on the whole axis of the differential equation implies the existence of a global bounded solution of the corresponding difference equation. Also, the inverse problem, that the existence of bounded solution of difference equation implies the existence of bounded solution of the corresponding difference equation, is considered. The behavior of bounded solutions of difference equations with is investigated. It is established the conditions that the existence of periodic solution of difference equation implies the existence of periodic solution of the corresponding differential equation. The smallness of the step of difference equation is received.