Existence Of Multiple Periodic Solutions And Homoclinic Orbits For A Class Of Functional Differential Equations

The study of functional differential equations plays an important role in many systems in the world.Among them,the subject of periodic solutions and homoclinic orbits has also attracted widespread attention from scholars all over the world.In this article,we use the generalized Poincare-Birkhoff fixed point theorem and Mawhin's continuation theorem to study the existence of periodic solutions and homoclinic orbits of a class of functional differential equations.There is four chapters here.In the first chapter,we make an introduction about the research background and current research status of periodic solutions and homoclinic orbits of functional differential equations,and briefly describes the main content of this paper.In the second chapter,we use the generalized Poincare-Birkhoff fixed point theorem to study a class of delayed Duffing equations about the existence of multiple periodic solutions.x"(t)-f(t,x(t-?))=p(t)=p(t+2?)First,we construct the anti-periodic function space and define a Poincare mapping.By proving that the mapping is "twisted",it is concluded that the mapping has at least two fixed points in the region.Then we further prove the delayed Duffing equations have infinitely many periodic solutions in the whole domain.In the third chapter,we use the Mawhin continuation theorem to study a class of 2n+1 order functional differential equations about the existence of multiple periodic solutions.#12First,to estimate the bounds of the periodic solutions of the equations and their derivatives,we construct the relevant Green's functions.In the estimation process,we use the properties of inequalities and the constant variation method,and then use the Mawhin continuation theorem to prove this class 2n+1 order functional differential equations have periodic solutions.In the fourth chapter,we use the Mawhin continuation theorem to study a class of odd-order neutral functional differential equations about the existence of homoclinic orbits.#12First,we intercept the original equation in different intervals to make the original equation a periodic function.Then we use the Writinger's inequality technique to estimate the boundary of the periodic solution and derivative of the equation and use the Mawhin continuation theorem to prove the existence of periodic solution for the equation.Finally,through a series of subharmonic solutions to approximate the idea,it is proved that the equation has a homoclinic orbit.