Nonlinear Ordinary Differential Equations Positive Solution

Nonlinear functional analysis is an important branch of modern analysis mathematics, because it can explain all kinds of natural phenomena, more and more mathematicians are devoting their time to it. Among them, the nonlinear boundary value problem comes from a lot of branches of applied mathematics and physics. It is at present one of the most active fields that is studied in analysis mathematics.The aim of this article is, on basis of the partial order theory, to study the existence of positive solution boundary value problem of differential equations in Banach space by using nonlinear functional analysis method. We are concert with the study of semipositone and half-line problems, including some singular problems. By deep study, we obtained many new results. Most results of this paper are published or to be appeared in important journals of China, for example,《Applied Mathematics A Journal of Chinese Universities》,《Acta Analysis Functionalis Applicata》etc.The paper is divided into three chapters according to the contents:Chapter 1 is the introduction of this paper, which introduces the main contents of this paper. In chapter 2, we carry out study for the singular semipositoneproblem with profound background and give a new method to solve this type problems. In the first section of the second chapter,we study a class of nonresonant singular semipositone forth order boundary value problem .A new result on the existence of C~2[0,1]∩C~4(0, 1) positive solution for the question in the condition of second order derivative termβ<π~2 is gained.Essentially,we extend and improve those in the related reference. In the second section of the second chapter,in case of not requiring f(t, u) to be nonnegative, we consider the existence of positive solutions for the following singular boundary value problemsby transforming the boundary value problem into the integral equation system and using fixed point theory in cones. The existence of at least one positive solution to the boundary value problem is guaranteed. In chapter 3, we consider the existence of the positive solution for singular boundary value problem in half-line. In the first section of the third chapter,we study a class of singular second order Sturm - liouville boundary value problems by using fixed point theorem . A new result on the existence of C_p~1 [0, +∞) positive solution for the question is obtained. In the second section of the third chapter, the paper study a class of singular second order Sturm - liouville boundary value problemsby using fixed point theorem . A new result on the existence of C_p~1 [0, +∞) positive solution for the question is obtained.