Positive Solutions To Boundary Value Problems For Several Kinds Of Nonlinear Differential Equations

In last few years, more and more nonlinear problems have resulted from mathematics,physics, chemistry, biology, medicine, economics, engineering, sybernctics and so on. In solving these problems, many important methods and theories such as partial ordering method, topological degree method, the theory of cone and the variational method have been developed gradually. They become very effective theoretical tool to solve many nonlinear problems in the fields of the science and technology.This paper deeply discusses the existence of one. solution and(or) multiple solutions to boundary value problems (BVP for short) for second order, third order and fourth order nonlinear differential equations mainly by making use of fixed point theorem and the theorem of cone. Because of the importance of singular, semipositone and parameters, we also study the effect of singular, semipositone and parameters on the solutions of nonlinear differential equations, and we obtain some useful results.There are four chapters in the dissertation.In the first chapter, by using Leggett-Williams fixed point theorem, we deal with singular second-order three-point BVPwhereα≥0,β≥0,0<η 0 is a parameter, M > 0, f(t,u), g(t,u) may be singular at t = 0,t = 1, g is allowed to change sign. In this chapter, by using the Krasnoselskii's fixed point theory in cones, we derive an interval ofλsuch that for anyλlying in this interval, the semipositone boundary value problem has at least one positive solution if f is superlinear or sublinear, and an example is worked out to indicate that our conditions are reasonable.In the third chapter, we discuss the following singular third-order three-point boundaryvalue problems with two parameterswhereη∈(0,1),α∈[0, (?)) are constants,λ₁,λ∈(0,+∞) are two parameters, f(t,u) may be singular at t = 0, t = 1 and u = 0. By using fixed point theory in cones, an explicit interval forλ₁ is derived such that for anyλ₁ lying in this interval, the existence of at least one positive solution or double positive solutions to the above boundary value problem is guaranteed whenλis small enough, and examples are worked out to indicate that our conditions are reasonable.In the last chapter, we deal with the existence of multiple solutions for the following fourth-order singular BVPwhere a≥0, b≥0, c≥0, d≥0, ac+bc + ad > 0, p(t), w(t) may be singular at t = 0, t = 1. In this chapter, by using the Krasnoselskii's fixed point theory in cones and localization method, the existence of n positive solutions is considered for a class of fourth-order singular boundary value problem in more general conditions.