The Research Of Qualitative Theory Of Some Classes Of Functional Differential Equations And Their Application

The solutions of functional differential equations reveal the behavior of dynamical systems, so it has widely applied in many fields, such as ecology, economics and physics, so the various dynamical properties of their solutions have attracted widely attentions of many authors, in particular, on the research of oscillation.First, we introduced the development, research content and significance of oscillation theory of functional differential equations, as well as the development of history of fractional functional differential equations in the first chapter. Next, four parts are consisted of the mainly study in this paper. From the second charpter to the forth chapter, we have discussesed respectively oscillation criteria of a class of second order nonlinear differential equation with damping term, a class of second order quasilinear neutral differential equations and a class of third order nonlinear neutral differential equations. Using some new methods and reduction to absurdity, some sufficient conditions for the oscillation of all solutions are given, new results improve and extend some similar oscillation criteria, and cannot be applied to a number of previous cases. Examples are inserted to illustrate the results. Using some fixed point theorems, we studied the existence and uniqueness of solutions for a class of initial value problems of fractional order nonlinear differential equation in the fifth chapter, new results improve and extend some similar theorems.