Existence Of Periodic Solutions And Homoclinic Orbits For A Class Of Third Order Neutral Functional Differential Equations

Functional differential equations exist in various fields in the real world.The existence of periodic solutions and homoclinic orbits has become a central topic of concern for many mathematicians.In this paper,Kranoselskii fixed point theorem and Mawhin's continuous theorem are used to study the existence of periodic solutions and homoclinic orbits of a class of third-order neutral functional differential equations.The full text is divided into four chapters.The first chapter introduces the research background and development of the periodic solutions of the functional differential equations and the existence of homoclinic orbits,and introduces the main work of this paper.In the second chapter,the existence of periodic solutions for a class of third-order neutral functional differential equationsis discussed by using two different methods—Kranoselskii fixed point theorem and Mawhin's continuous theorem to obtaining new existence criteria.By constructing the relevant Green's function and using the inequality analysis technique,the bounds of the periodic solution and its derivatives are estimated,so that the equation has a T-periodic solution when some constraints are satisfied.In Chapter 3,we discuss the existence of periodic solutions and homoclinic orbits for a class of third-order neutral functional differential equations x(?)(t)+cx(?)(t-?)+a2(t)x(?)(t)+a1(t)x(t)+a0(t)x(t)=g(t,x(t-?1),x(t-?2),…,x(t-?n))+f(t)by means of Mawhin continuity theorem,and obtain some new existence criteria.Firstly,the existence of periodic solutions of equations is proved by estimating the bounds of periodic solutions and their derivatives.On the premise that the equation has periodic solutions,a nontrivial homoclinic orbit is obtained by approximating a series of subharmonic solutions.