Nonlinear Ordinary Differential Equations Existence Of Positive Solutions,

In this paper, we use the fixed point index theory as well as the fixed point theorem of generalized cone expansion and compression, to study the positive solution of the several kinds of boundary value problems for nonlinear differential equation.The thesis is divided into four chapters according to contents.The first chapter chiefly narrates the background of the nonlinear analysis and innovation of this paper.In chapter 2, we use the fixed point theorem of generalized cone expansion and compression to investigate the following singular fourth-order boundary value problem with a sign-changing nonlinear term whereλ>0,f∈C((0,1)×[0,+∞),(-∞,+∞)), and f is allowed to be singular at t=0, t=1. moreover,ω∈C((0,1),(-∞,+∞)),αi(s)(i=1,2) is nonegative and integrable on [0,1], and∫01αi(s)ds∈(0,1). we obtain the existence of positive solution for the singular semipositone boundary value problem.In chapter 3, By applying the fixed-point index theory, we investigate the existence of symmetric positive solutions of the following singular boundary value problem where a,α,β,γ>0,α>β/2,β>γ/2,γ>α/2,η∈(0,1) and ,h(t)∈C((0,1),[0,+∞)), is allowed to be singular at t=0, t=1. moreover, f(t,u)∈C((0,1)×(0,+∞), [0,+∞)) is allowed to be singular at u= 0. we obtain the existence of symmetric positive solution for the singular boundary value problem.In chapter 4, we considered the existence of positive solutions for a class of n order m point boundary value problems with a sign-changing nonlinear term where f∈C([0,1]×[0,+∞), (-∞,+∞)),λ> 0,αi∈[0,+∞), i= 1,2,...,m-2, we obtain the existence of at least two positive solutions for the boundary value problem.