The Multiplicity Results On The Periodic Solutions To A Generalized Kaplan-Yorke Type Delay Differential Equation

The study to the periodic solutions of nonlinear delay differential equations has been one of the most important subjects for delay differential equations all the time. In recent years, the results obtained in this filed have had a great effect on cybernetics, biomathematics, chemical reaction, etc.This paper mainly concern with the existence and multiplicity of periodic solutions for a generalized kaplan-yorke type delay differential equation by using the pseudo index theory of the critical point theory.This thesis is organized as follow. In chapter 1, we give an introduction to the background of the field we concerned.In chapter 2, we study the existence and multiplicity of periodic solutions of the equation, when f isn't resonance at infinity. Instead of studying the periodic solutions of the equation, we study the periodic solutions of Hamilton system which coupled with the equation. Also, the existence of periodic solutions of Hamiltonian system can be reduced to finding the critical points of a functional over a Hilbert space. By using the pseudo index theory of the critical point theory, we obtain some sufficient conditions for the existence and multiplicity of periodic solutions of the equation.In chapter 3, we study the existence and multiplicity of periodic solutions of the equation, when f is resonance at infinity. Using the similar method, we also obtain some sufficient conditions for the existence and multiplicity of periodic solutions of the equation.