Nonlinear Ordinary Differential Equations The Solution Of The Problem

Along with science's and technology's development, various non-linear prob-lem has aroused people's widespread interest day by day, and so the nonlinear analysis has become one important research directions in modern mathematics. The nonlinear functional analysis is an important branch in nonlinear analysis, because it can explain well various the natural phenomenon. The boundary value problem of nonlinear differential equation stems from the applied mathematics, the physics, the cybernetics and each kind of application discipline. It is one of most active domains of functional analysis studies at present.second order equation origins the applied mathematics,the physics,the cy-bernetics and other disciplines. Ordinary differential equations have aroused people's widespread interest. In this paper, we use the upper and lower solu-tions method, fixed point index theory, to study several kinds of boundary value problems for nonlinear singular differential equation.The thesis is divided into three chapters according to contents.In chapter 1, we consider the exitence of positive solution for the singular second order boundary value problem, by means of the degree of fixed point theorem and upper and lower solutions. whereγ,η∈(0,1), b(t)∈C((0,1), [0,∞)), f(t, u)∈C([0,1]×[0,∞), [0,∞)), b(t) may be singular at t= 0, t=1.In chapter 2,we consider the exitence of positive solutions for the second order boundary value problem, by means of the degree of fixed point theorem and the property of Green function. where f(t, u(t)) may be singular at t= 0, t= 1 and u= 0.We introduce two height functions by fixed point theory.In weaker conditions,we obtain the existence of positive solution of (2.1.1). In chapter 3, we consider the exitence of positive solution for the singular second order boundary value problem, by means of the fixed point theorem. whereκ∈(0,π/2) is a constant, g(t, x) is monotone locally with respect to x and f(x),g(t,x) may be singular at t= 0, t=1 and x=0.We use the property of the Green's functions and fixed point theory to prove the existence of posi-tive solutions for the two point boundary value problem, g(t,x) just satifis local monotonous.