Solvability Of Boundary Value Problem For Several Classes Of Nonlinear Ordinary Differential Equations

Nonlinear analysis and the solution of ordinary differential equation are two important research subjects of optimization problem. In this paper we consider the existence of multiple positive solutions and the necessary and sufficient condition of a class of singular nonlinear boundary value problem . This paper is composed of four chapters.Chapter 1 is the introduction of this paper, which introduces the basic theories of nonlinear analysis, and the main results obtained in this paper.In chapter 2, we consider the existence of multiple positive solutions of a class of singular nonlinear two-point boundary value problem with two parametersThis section establishes the existence of multiple positive solutions of singular nonlinear two-point boundary value problems (2.1.1) by means of the fixed point index theorem on cones, theorem 1.1.1 improves and generalizesthe existing conditions of [1]:lim/w=0, mmBt->o t t>o tlimZW=0,t tt->oc t i->oo twhen a — f3 = 0 ,lots of important results have been obtained, for example, [1 — 2] .The results presented in this paper essentially improve and generalize the known results of [2],so that the results can be applied to more kinds of functions.In chapter 3, we study the necessary and sufficient condition of existing positive solutions of the following nonlinear two-point boundary value problem:x"4\t) — hi(t)f(x,y), t G (0,1) f\t) = h2{t)g{x,y), i€(0,l) x{0) = x(l) = x"{0) = x"(l) = 0 ( y(0) = y(l) = y"(0) = y"(l) = 0 The results presented in this paper essentially extends the conclusion in[6]-In chapter 4,we consider the existence and multiplicity of positive solutionsof the following nonlinear three-point boundary value problem:(,)4.1.1 x(0) = x(l) = 0, ax"{0) - bx"'{0) = 0, cx"{I) + dx"'{\) = 0The existence and multiplicity of positive solutions is obtained by making use of the fixed point index theorem on cones and the properties of Green function, our results generalize and improve the results in [8].