Periodic Solutions To Nonlinear Ordinary Differential Equations

This thesis is mainly composed of two chapters in which we consider the periodic boundary value problems for nonlinear ordinary differential equations.In Chapter I, we study the existence of positive periodic solutions to nonlinear second order differential equationFirstly, using the fixed point index theory, we discuss the existence of positive solutions to the following nonlinear periodic boundary value problem:Consequently, by extending the solution with ω-period, the existence of positive periodic solutions to equation (1.1.1) is obtained. Assume that is an ω-periodic continuous function and a(t)≠0; is continuous and is also an ω-periodic function for each To be convenient, we give some notations:Let M — maxt∈[0,w] a(t). The main result is the following.Theorem 1.3.1. Suppose that (H1) and (H2) hold. If M ∈ (0,(π/ω)2], then equation (1.1.1) has at least one positive periodic solution in each of the following cases:where λ1 is the first positive eigenvalue of the linear equation corresponding to equation(1.1.1).In [17], Li Yongxiang had discussed the existence of ω-periodic solutions to the differential equation (1.1.1) and gave an existence result of positive ω-periodic solution by using Krasnoselskii's cone operator fixed'point theorem. We improve the conditions in (H1), Lemma 1 and Lemma 2 of [17] and get a more essential result by the first positive eigenvalue of the corresponding linear problem.