| Functional differential equation is a mathematical model describing the phenomenon with time delays. The functional differential equations with periodic delays represent a natural framework for mathematical modeling of many real world phenomena such as biology, econnmy, ecology and so on. Therefore, the researches on the existence of periodic solutions and homoclinic solutions for functional differential equations with periodic delays have practical significance.By using Mawhin’s continuation theorem, the problem of existence of periodic solutions are studied for several functional differential equations with delays,and some sufficient conditions for the existence of periodic solutions are given. On the basis of periodic solutions,the existence of homoclinic solutions are proved. The full-text content is divided into three parts, the first chapter introduces the method of functional differential equations with delays, in chapter two, we discuss the existence of periodic solution about the neutral functional differential equation with nonlinear D-operator dDx1/dt=f(t,x1); and then discuss the the properties of nonlinear D-operator in the case of multiple delays, the third chapter studies the equation x"(t)=f(t,x(t))+g(t,x(t-τ(t)))+p(t), by using Mawhin’s continuation theorem, we get a new previous bound estimation method of periodic solution, in chapter four, we investigate the existence of homoclinic solutions of the equation such as u"(t)+f(u(t))u+(t)+g(u(t-γ(t)))=e(t), get a new result of homoclinic solutions, at the same time, This chapter explores the relationship between the delay τ (t) and homoclinic solutions. |
