Periodic Solutions For Delay Differential Equations And Boundary Value Problems For Differential Equations

This Ph.D.Thesis is composed of five chapters, which mainly investigated the posi- tive periodic solutions for a nonautonomous delay differential equations, the existence of solutions for boundary value problems for differential equations with deviating ar- guments and p-Laplacian, solvability of multi-point boundary value problems at reso- nance, and the existence of solutions for rn-point boundary value problems of second order differential systems with impulses. In the first chapter, we introduce the historical background of problems which will be investigated and the main results of this paper. In Chapter 2, by using the Krasnoselskii axed point theorem, we study the existence of positive periodic solutions of a nonautonomous delay differential equation, and some sufficient conditions will be given. Two known dynemic mathematical models will be studied too. Chapter 3 mainly considers the boundary value problems for differential equations with deviating arguments. An existence result, obtained by the help of Leray-Schauder degree theory, have no restriction on the damping forces. Our result improves and generalizes the some known results. The purpose of Chapter 4 is to study the solvability of multi-point boundary value problems at resonance, and establishs some existence theorems under nonlinear growth restriction. Our method is based upon the coincidence degree theory. In Section 1, we consider the same sign case, and in Section 2, discuss the non-same sign case. In the last chapter, we consider the existence of solutions for rn-point boundary value problems of second order differential systems with impulses. We first obtain a general existence result, and give some sufficient conditions. The key tool in our approach is based on the coincidence degree theory and the concept of autonomouse curvature bound set. Our result is new even for the case without impulsive point.