Existence Of Positive Solutions Of Three-Point Boundary Value Problems For Singular Differential Equations

In this papar, we discuss the existence of positive solutions ofthree-point boundary value problems for singular differential equa-tions (1) and (2) as followswhere 0＜α＜1, 0＜η＜1, f may change sign and may be singularat x=0 and x′=0,where 0＜α＜1,η∈(0,+∞), f may change sign and may besingular at x=0 and x′=0.Three-point boundary value problems for differential equationsare that boundary value conditions of differential equations containthe value at interval endpoint and inside the interval. They arose in different fields of applicable mathematics and physics. Multi-pointboundary value problems have wide background.In [2], Barr D., Sherman T. firstly studied the multi-point bound-ary value problems and then Oupta.C.P did lots of work on second-order three-point boundary value problems. Ruyun Ma pre-sented the key conditions for these problems in 1999. After thenthere were lots of research in this aspect. But few papers studyexistence of positive solutions for three-point boundary value prob-lems when nonlinearity may change sign and may be singular. Com-pared to this results, this paper obtains the existence of positive so-lutions for three-point boundary value problems when nonlinearitymay change sign and may be singular at x=0, x′=0. So it is theimprovement of theory for existence of positive solutions for three-point boundary value problems. It has a great role in theory andpractice.When we study above problems, we mainly refer to[5-6], [9-17].The method what we used is firstly constructing proper integral op-erators and abciseing the part in which integral operators change signbecause of f sign changing. Then, using Arzela-Ascoli theorem, weconsider the set of the approximate solutions and obtain a convergentsubsequence. The limit is a positive solution for equation (1) or (2).This paper is divided into two chapters.In chapter 1, we first present the existence of minimal positive solutions of (1) when nonlinearity may be singular at x=0, x'=0 and f＞0. Nonlinearity af is less than three nonnegative functions which are bounded and integrabel. Then, the existence and compactness of positive solutions of (1) when nonlinearity may be singular at x=0, x'=0 and may change sign are obtained in two methods. The common of the two methods is that in order to assure the integral operators don't change sign, f is greater than a positive function when x' approaches to zero and the absolute value of f is less than three nonnegative functions. The difference between the two methods is that the nonnegative functions which are used to control f are integrabel and bounded in the first method, but the nonnegative functions in the second method are monotone. The second method is used more widely than the first one.In chapter 2, we mainly study the existence of positive solutions of (2) in half line. When f＞0, we discuss the existence of positive solutions of (2). Then, when f changes sign, we discuss the existence of positive solutions in finite interval which is compact. When it approaches to infinite interval, we obtain the existence of positive solution of (2).