Positive Solutions For Several Kinds Of Multi-Point Boundary Value Problems Of Diferential Equations

Diferential equations have a profound and vivid actual background. They arise from the productionand science technology and are a powerful tool to analyze and solve problems in modern science andtechnology. The multi-point boundary value problems for diferential equations arise in a variety ofdiferent areas of applied mathematics and physics. For example, the vibrations of catenary wire of auniform cross-section which is composed of N parts of diferent densities can be converted into a multi-point boundary value problem; many problems in the theory of elastic stability can be handled by themethod of multi-point problems. The study of multi-point boundary value problems for linear second-order diferential equations was initiated by Ii’in and Moiseev in 1980s. In 1990s,Gupta began discussingthree-point boundary value problems of second-order nonlinear diferential equations.Since then,manyauthors have studied more general nonlinear multi-point boundary value problems and obtained manyachievements.It’s very important to investigate the existence of positive solutions of boundary value problems fordiferential equations,however, positive solutions are always paid more attention due to their practicalmeaning. The study of existence of positive solutions to diferential equations is often transformed intoinvestigating the existence of fixed points for integral operators on a cone. This work mainly appliessome fixed point theorems and fixed point index theory to study the existence of positive solutions forseveral kinds of multi-point boundary value problems of diferential equations.This paper is divided into three chapters.Chapter 1 discusses the existence of positive solutions fora type of m-point boundary value problem of diferential equations as follows whereξi∈(0,1),andξ1<ξ2<···<ξm 2<1.In[1], by using Krasnoselskii fixed point theorem, theauthors investigated the existence of the above boundary value problem,and obtained existence results ofone positive solution. In this section,by using the fixed point index theory, the problem has been studiedand the existence of two positive solutions has been obtained.Chapter 2 discusses the existence of positive solutions for the following semipositone second-orderm-point boundary value problem.whereλ∈(0,1).[2]discussed the existence of a positive solution for a second-order three-point boundaryvalue problem.[3]-[5] discussed the existence of positive solutions for a semipositone second-order three-point boundary value problem. In spite of these results, less research has been carried out for the existenceof positive solutions for a second-order m-point boundary value problem.In this section, the existence ofpositive solutions for the above semipositone second-order m-point boundary value problem was studied.Chapter 3 discusses the existence of multiple positive solutions for a kind of second-order m-pointboundary value problem aswhereλ∈(0,1).In[6], the authors have investigated the existence of one positive solution for the follow-ing m-point boundary value problem:Under the condition that f is monotonous and by using Krasnoselskii fixed point theorem,they obtainedthe existence results of multiple positive solutions.