Second Order Three-point Singular Boundary Value Problem

In this article, we consider the second-order three point singular boundary-value problem and the second-order three point singular boundary-value problem We show the existence of positive solution for BVP (1) when nonlinearities f are sign-changing and the singularity may appear at t=0, x=0, and investigate the existence of positive solution for BVP (2) when the sign-changing nonlinearities f are singular at t=0, x=0, but not x'=0, and also investigate the nonexistense and existence of positive solution for BVP (2) when nonlinearities f are sign-changing and the singularity may appear at t=0, x=0, x'=0.The existence of solutions of boundary value problem for nonlinear ordinary dif-ferential equation, especially the existence of positive solutions, is an interesting and critical question in application and theory, playing an important role in the research field of ordinary differential equations. Particularly, the boundary value problem for second order ordinary differential equations is always an important subject in the research field of differential equations. It has extensive applied backgrounds and significant theory value in many research field, such as physics, astronomy, bi-ology and sociology, etc. In the past decades, with nonlinear functional analysis arising, mathematicians using the upper and lower solution method, topological degree method, fixed point theorem on cone to solve the boundary value problem could always get good results and have made some great developments and success. More and more mathematicians get many significant achievements, also including some results about multi-point boundary value problems for nonlinear ordinary dif-ferential equations. And multi-point boundary value problem has received many researches. Multi-point boundary value problems arise from different areas of appli-cable mathematics and physics quite naturally. The study of multi-point boundary value problems for linear second order differential equations was initiated by Il'in and Moiseev [7] [8] and the study of three-point boundary value problems for non-linear ordinary differential equations was initiated by Gupta [5]. Since then, more general nonlinear multi-point boundary value problems have been studied by several authors, and for details, the readers are referred to [1,3-4,17-18] and the references therein. For example, when f(t, x(t))=g(t)f(x(t)) in (1) and g(t) is a sign-changing function in [0,1] and f is nondecreasing without any singular points, using the fixed point theorem of strict-set-contractions, Bing Liu [9] established the existence of at least two positive solutions for (1). When f(t,x,y) is nonnegative and may be su-perlinear in x at+∞, B.Yan [20] got the existence of at least two positive solutions for BVP (2) by using the fixed point index. When f is a nonnegative continuous function without any singular point, Y.Guo and W.Ge [4] established the existence of at least one positive solution for BVP (2) by the fixed point theorem in a cone.This paper is divided into two chapters. In chapter one, when the sign-changing nonlinearities f is singular at t=0, x=0, constructing a completely continuous operator and using the fixed theorem in a suitable Banach space, we establish so- lutions for a consequense of operators, and investigate a solution for BVP (1) by Ascoli-Arzela Theorem. Finally, we get the uniqueness of the solution for BVP (1) by the property of the concave functions.In chapter two, we establish the nonexistence and existence of positive solu-tion for second order singular three-point boundary value problem with derivative dependence and sign-changing nonlinearities. In section two, using the property of the concave function, we get a condition for the nonexistence of BVP (2). In section three, when the sign-changing nonlinearities f is singular at t=0, x=0 but not x'=0, constructing a completely continuous operator and using the fixed theorem in a suitable Banach space, we establish solutions for a consequense of operators, and investigate a solution for BVP (2) by Ascoli-Arzela Theorem. In section four, when the sign-changing nonlinearities f is singular at t=0, x=0, x'=0, constructing a completely continuous operator and using the fixed theorem in a Banach space and the property of the concave functions, we establish solutions for a consequense of operators, and investigate a solution for BVP (2) by Ascoli-Arzela Theorem.