Problems Of Periodic Solutions And Boundary Valuable Problem For Nonlinear Differential Equations

By applying Mawhin's continuation theorem of coincidence degree and fixed-point theorem in cones, this dissertation is mainly concerned with the problems of periodic solutions and boundary valuable problem for nonlinear differential equations.The whole thesis contains four chapters. The first chapter introduces concisely background of the subjects relevant to this dissertation, emphatically introduce the main work done in this thesis. Chapter 2 is the foundation of this dissertation. In this chapter, we analyses some properties of periodic functions and establish two important intergration inequalities. Furthermore, we give the a proof of ImL = {y|y ∈ Y, ∫₀^2Ï€(t)dt = 0}, and give six important theorems. All the above work is important for us to investigate the existence of positive solutions for boundary valuable problem of delay differential equation and the relation between the existence of periodic solutions and the delays in the following chapter. Chapter 3 devotes to the problems of periodic solutions for third-order functional differential equation with deviating argument. This Chapter contains three sections. The first section narrate the problems of periodic for third-order functional differential equation with deviating argument: x'''(t) + f(x'(t)) + g(x(t - T(t))) = p(t),The approaches estimate a priori dounds of periodic solutions are different from those used in previous literatures and the result obtained for the equation generalizes and extends the corresponding results of known. In the second section,we study the problem of periodic for third-order functional differential equation with deviating argument of the form x'"(t) + ax"(t) + bx'(t) + cx(t) + g(x(t - Τ(t))) = p(t).By means of several important lemmas and theorem of Fourier series we extimate a priori bounds of periodic solutions of Mawhin's continuation theorem of coincidence de-gree,and obtain sufficient conditions. The third section narrate the problem of periodic for third-order functional differential equation with deviating argument in the followingform:Wecan easily find they are the special cases of equation which we study. So the equation discussed by us is more general, and the result we give is relevant to the delay, which generalizes and extends the avaliable work.Chapter 4 mainly consider the existence of positive solutions for periodic boundary valuable problem of second-order delay differential equation.By using fixed-point theorem in cones,some new results are obtained,which generalize and extends the corresponding work in the past.