Asymptotic Property And Periodic Solutions Of Functional Differential Equations

In much research of scientific fields, such as: dynamics, physics, biomathematics, economic mathematics, automatic control, and so on. ordinary differential equations already can't describe the objective phenomena precisely. Many phenomena utilize functional differential equations acting as their mathematical models. Therefore, the studies for functional differential equations attract many researchers' attention. However, the asymptotic property and periodic solutions of functional differential equations are two main research fields in functional differential equations. Recently, many scholars abroad and home devote themselves to the study of asymptotic property and periodic solutions of linear, nonlinear, neutral retarded functional differential equations in[26,27]. This paper discusses the asymptotic property and periodic solutions of a few classes of functional differential equations.The paper consists of three parts. In the first part, the asymptotic behavior of the solutions of functional differential equationx(t) = a(t)g(x(t)) + b(t)f(xt), t > 0, (1.1)are considered. Where a(t),b(t) : R+ - R+,f : C -> R,g : R - R. We establish the sufficient conditions that non-oscillatory solutions and oscillatory solutions tend to zero, respectively.At present, the periodic solutions of retarded differential equations with single delay have been studied by many authors. However, there has little research on the periodic solutions of equations with multiple and mixed delays. In the second part, we discuss the periodic solutions of equationWhere a(t), Ti(t), hi(t), σi(t)(i = 1,2.... ,n) are positive and continuous T-periodic functions. and is a parameter. In the end of this section, we give a typical example. When a(t) = S, g(y) = 1, gi(y) = 1, hi(t)σi(t) = pi, Ti(t) = n, fi(y) = ye~aiy. The equation (2.1) is simplified as the following equationWe discuss the existence of its positive periodic solutions.